microsoft word documento1 microsoft word documento1 microsoft word documento1 microsoft word r.m.5 cap.2.doc microsoft word documento1 microsoft word cap2.doc microsoft word cap5.doc microsoft word capitolo_7.doc microsoft word cap8.doc \documentstyle[12pt]{article} 61 chemical examples in hypergroups b. davvaz1 and a. dehghan-nezhad1 abstract hypergroups first were introduced by marty in 1934. up to now many researchers have been working on this field of modern algebra and developed it. it is purpose of this paper to provide examples of hypergroups associated with chemistry. the examples presented are connected to construction from chain reactions. ams subject classification: 20n20 keywords and phrases: hyperstructure, hypergroups, chain relation. 1 introduction the theory of algebraic hyperstructures which is a generalization of the concept of algebraic structures first was introduced by marty in 1934 [4], and had been studied in the following decades and nowadays by many mathematicians, and many papers concerning various hyperstructures have appeared in the literature, for example see [2,3,6,8]. the basic definitions of the object can be found in [1,7]. definition. a hyperstructure is a non-empty set s together with a function · : s × s → p*(s) called hyperoperation, where p*(s} denotes the set of all non-empty subsets of s. if a, b ⊆ s, x ∈ s then we define a · b = u bb,aa ∈∈ a · b, x · b = {x} · b, and a · x = a·{x}. the hyperoperation · is called associative in s if (x · y) · z = x · (y · z) for all x,y,z in s. definition. a hyperstructure (s , · ) is called a hypergroup [1] if i) ( · ) is associative. ii) a · s = s · a = s for all a ∈ s. definition. a non-empty subset k of the hypergroup s is called a subhypergroup of s if a · k = k · a = k for all a ∈ k. 1 department of mathematics, university of yazd, yazd, iran e-mail: davvaz@yazduni.ac.ir 62 in this paper, we will give some examples of hypergroups associated with chemistry. the examples presented are connected to construction from chian reactions. 2 preliminaries a) chain reactions an atom of group of atoms possessing an odd (unpaired) electron is called a free radical, such as cl, ch3, c2h5 the chlorination of methane is an example of a chain reaction, a reaction that involves a series of steps, each of which generates a reactive substance that brings about the next step. while chain reactions may vary widely in their details, they all have certain fundamental characteristics in common. 1) cl2 → 2clº (1) is called chain-initiating step. 2) clº + ch4 → hcl + ch3º 3) ch3º + cl2 → ch3cl + clº then (2), (3), (2), (3), etc, until finally: (2) and (3) are called chain-propagating steps. 4) clº + clº → cl2 or 5) ch3º + ch3º → ch3ch3 or 6) ch3º+clº → ch3cl. (4),(5) and (6) are called chain-terminating steps. first in the chain of reactions is a chain-initiating step, in which energy is absorbed and a reactive particle generated; in the present reaction it is the cleavage of chlorine into atoms (step 1). there are one or more chain-propagating steps, each of which consumes a reactive particle and generates another; there they are the reaction of chlorine atoms with methane (step 2), and of methyl radicals with chlorine (step 3). finally, there are chain-terminating steps, in which reactive particles are consumed but not generated; in the chlorination of methane these would involve the union of two of the reactive particles, or the capture of one of them by the walls of the reaction vessel. b) the halogens f, cl, br, and i the halogens are all typical non-metals. although their physical forms differfluorine and chlorine are gases, bromine is a liquid and iodine is a solid at room temperature, each consists of diatomic molecules; f2, cl2, br2 and i2. the halogens 63 all react with hydrogen to form gaseous compounds, with the formulas hf, hcl, hbr, and hi all of which are very soluble in water. the halogens all react with metals to give halides. .. .. .. .. .. .. .. .. : f f : , : cl cl : , : br br : , : i i : .. .. .. .. .. .. .. .. the reader will find in [5] a deep discussion of chain reactions and halogens. 3 chemical hypergroups in during chain reaction a2+ b2  →← lightorheat 2ab there exist all molecules a2, b2, ab and whose fragment parts aº, bº in experiment. elements of this collection can by combine with each other. all combinational probability for the set s = { aº, bº, a2, b2, ab} to do without energy can be displayed as follows: + aº bº a2 b2 ab aº aº,a2 aº,bº,ab aº,a2 aº,b2,bº,ab aº,ab,a2,bº bº aº, bº,ab bº, b2 aº,bº,ab, a2 bº,b2 aº, bº,ab, b2 a2 aº, a2 aº, bº,ab,a2 aº, a2 aº, bº,a2,b2,ab aº, bº, a2,ab b2 aº,bº,b2,a b bº,b2 aº, bº,a2, b2,ab bº, b2 aº, bº, b2,ab ab aº,ab,a2,bº aº,bº,ab,b2 aº, bº,a2,ab aº, bº, b2,ab aº, bº, a2, b2,ab theorem. (s , +) is a hypergroup. proof. clearly reproduction axiom and associativity are valid. as a sample of how to calculate the associativity, we illustrate some cases: (ab+a2)+b2 = { ab, a2, aº, bº }+b2 = {b2, ab, a2, aº, bº },    ab+(a2+b2)=ab+{a2, b2, aº, bº, ab } = { a2, b2, ab, aº, bº}, (ab+aº)+aº={ ab, aº, a2, bº }+aº = {a2, aº, ab, bº },    ab+(aº+aº)=ab+ {a2, aº } = { a2, ab, aº, bº}, (a2+bº)+b2 = { ab, aº, a2, bº }+b2 = {b2, ab, bº, aº, a2 },    a2+(bº+b2)=a2+{b2, bº } = { a2, aº, ab, bº, b2}. 64 corollary. s1={aº,a2 } and s2={bº,b2 } are only subhypergroups of (s , +). if we consider a=h and b ∈ { f, cl, br, i } (for example b = i), the complete reaction table becomes: + hº iº h2 i2 hi hº hº,h2 hº,iº,hi hº,h2 hº,i2,iº,hi hº,hi,h2,iº iº hº, iº,hi iº, i2 hº,iº,hi, h2 iº,i2 hº, iº,hi, i2 h2 hº, h2 hº, iº,hi,i2 hº, h2 hº, iº,h2,i2,hi hº, iº, h2,hi i2 hº,iº,i2,hi hº,i2 hº, iº,h2, i2,hi hº, i2 hº, iº, i2,hi hi hº,hi,h2,iº hº,iº,hi,i2 hº, iº,h2,hi hº, iº, h2,hi hº, iº, h2, i2,hi acknowledgment we appreciate the assistance and suggestions of dr. a. gorgi at the department of chemistry. references [1] p. corsini, prolegomena of hypergroup theory, second edition, aviani editor, (1993). [2] m.r. darafsheh and b. davvaz, hv-ring of fractions, italian j. pure appl. math., 5 (1999) 25-34. [3] b. davvaz, weak polygroups, proc. 28th annual iranian math. conf, (1977), 139-145. [4] f. marty, sur une generalization de la notion de groupe, 8iem congres math. scandinaves, stockhlom, (1934), 45-49. [5] morrison and boyd, organic chemistry, sixth eddition, prentice-hall, inc, 1992. [6] t. vougiouklis, a new class of hyperstructures, j. combin. inf. system sci, 20, (1995), 229-235. [7] t. vougiouklis, hyperstructures and their repesentations, hadronic press inc, (1994). [8] t. vougiouklis, convolution on wass hyperstructures, discrete math., (1997), 347-355. ratio mathematica volume 38, 2020, pp. 377-383 a note on α−irresolute topological rings madhu ram∗ abstract in [4], we introduced the notion of α−irresolute topological rings in mathematics. this notion is independent of topological rings. in this note, we point out that under certain conditions an α−irresolute topological ring is topological ring and vice versa. we prove that the minkowski sum a + b of an α−compact subset a ⊆ r and an α−closed subset b ⊆r of an α−irresolute topological ring (r, =) is actually a closed subset of r. in the twilight of this note, we pose several natural questions which are noteworthy. keywords: α−open sets, α−closed sets, α−irresolute topological rings. 2010 ams subject classifications: 16w80, 16w99. 1 ∗department of mathematics, university of jammu, jammu-180006, india; madhuram0502@gmail.com. 1received on april 27th, 2020. accepted on june 19th, 2020. published on june 30th, 2020. doi: 10.23755/rm.v38i0.523. issn: 1592-7415. eissn: 2282-8214. c©madhu ram. this paper is published under the cc-by licence agreement. 377 m. ram 1 introduction let’s first introduce some notations. we denote a topological space by (x, =) (or simply x) where = is a non-trivial topology on x. for every topological space (x, =), there is a finer topology on x which is called the α−topology on x and is denoted by =α. in fact, =α is the family of all α−open sets in x (with respect to =). njastad [3] showed that =α is a topology on x. in 2019 ([4]), we introduced a category of α−regular spaces called α−irresolute topological rings. an α−irresolute topological ring, denoted by (r, =), is a ring r that is endowed with a topology = such that the following mappings (1) (r, =α) × (r, =α) 7−→ (r, =α) (ς, ξ) −→ ς − ξ, for all ς, ξ ∈r and (2) (r, =α) × (r, =α) 7−→ (r, =α) (ς, ξ) −→ ς.ξ for all ς, ξ ∈r are continuous. the notion of α−irresolute topological rings has structural nuances and conceptual niceties with the notion of topological rings. structurally, it sounds that there may be a strong relationship between these concepts. in this note, we point out some conditions with which an α−irresolute topological ring is topological ring and vice versa. for subsets a, b of r, we can define the so-called minkowski addition of two sets as a + b = {a + b : a ∈ a, b ∈ b} we describe that the minkowski sum a+b of a ⊆r, an α−compact subset, and a ⊆r, an α−closed subset, is closed subset of r. definition 1.1. a subset u of a topological space (x, =) is said to be (1) α−open [3] if u ⊆ int(cl(int(u))). (2) semi-open [2] if u ⊆ cl(int(u)). the complement of an α−open (resp. semi-open) set is called α−closed (resp. semi-closed). 378 a note on α−irresolute topological rings 2 main results for an α−irresolute topological ring (r, =), let γ denotes the collection of all α−open subsets of r containing the additive identity 0 of r. in the sequel, we use the following lemma: lemma 2.1. ([4, corollary 3.9.1]) let (r, =) be an α−irresolute topological ring. then for every v ∈ γ, there exists a symmetric σ ∈ γ such that σ + σ ⊆ v. definition 2.1. a subset u of a topological space (x, =) is said to be α−compact [1] if every cover of u by α−open subsets of x has a finite subcover. theorem 2.1. let (r, =) be an α−irresolute topological ring. let a and b be any subsets of r such that a is α−compact and b is α−closed satisfying a∩b = ∅. then there exists a set σ ∈ γ with the property (a+σ)∩(b+σ) = ∅. proof. let ς be an element of a. by lemma 2.1, there exists a symmetric σς ∈ γ such that ς + σς + σς + σς ∩b = ∅ or that ς + σς + σς ∩b + σς = ∅. (∗) continuing in a similar vein, we obtain a family of α−open subsets, π = {ς + σς : σς ∈ γ, ς ∈ a}. since a is α−compact, we have a finite subset j ⊆ a such that a ⊆ ⋃ {ς + σς : σς ∈ γ, ς ∈ j}. consider the set σ = ⋂ {σς : ς ∈ j}. then σ is α−open subset of r. by virtue of (∗), (a + σ) ∩ (b + σ) = ∅. theorem 2.2. let (r, =) be an α−irresolute topological ring, a ⊆ r an α−compact, and b ⊆ r an α−closed. then a + b is α−closed subset of r. proof. we prove this theorem by contrapositive. let ς /∈ a + b. then, by theorem 2.1, for each ν ∈ a, there exists a set uν ∈ γ such that 379 m. ram (ς + uν)∩(ν + b + uν) = ∅. (*) consider the collection of sets obtained form this fixed ς, π = {ν + uν : ν ∈ a} by theorem 3.1 in [4], π is a cover of a by α−open subsets of r. therefore, it has a finite subcover π′ = {νi+uνi : νi ∈ a, i = 1, 2, ....., n}. (**) to obtain the sets, p = ⋂ {uνi : i = 1, 2, ......., n}, q = ⋃ {uνi : i = 1, 2, ......, n}. (@) by construction, p and q are α−open subsets of r. because of (*) and (**), we have (ς + uνi ) ∩ (νi + b + uνi ) = ∅, ∀ i = 1, 2, ....... n. (@@) then (@) and (@@) together complete the proof. theorem 2.3. let (r, =) be an α−irresolute topological ring, a ⊆ r an α−compact, and b ⊆r an α−closed. then a + b is closed subset of r. to prove theorem 2.3, we need the following fact: theorem 2.4. let (r, =) be an α−irresolute topological ring, σ ⊆ r an α−compact, and ∇ ⊆ r an α−closed satisfying σ ∩∇ = ∅. then there exist disjoint open subsets p and q of r such that σ ⊆ p and ∇⊆ q. proof. in light of theorem 2.1, there exist α−open subsets u and v of r such that σ ⊆ u, ∇⊆ v and u∩v = ∅. take p = int(cl(int(u))) and q = int(cl(int(v))). then p, q ∈= with p∩q = ∅. it finishes the proof. proof of theorem 2.3. let ς /∈ a + b. by theorem 2.2, a + b is α−closed subset of r. in view of theorem 2.4, there exists an open subset p ⊆r such that ς ∈ p and p∩ (a + b) = ∅. this proves that a + b is closed subset of r. proof is over. we now give two theorems, theorem 2.5 and theorem 2.6, having conceptual niceties or that indicate the relationship between the topological rings and the α−irresolute topological rings. 380 a note on α−irresolute topological rings theorem 2.5. let (r, =) be an α−irresolute topological ring such that every nowhere dense subset of r is α−closed. then (r, =) is a topological ring. proof. let b be any α−open subset of r. then b = o−∇ where o is an open subset of r and ∇ is nowhere dense subset of r. now, let ς ∈ b be any element. then ς /∈∇. by hypothesis, ∇ is α−closed subset of r. by dint of theorem 2.4, there exist open subsets u and v of r such that ς ∈ u, ∇⊆ v and u∩v = ∅. whence we easily conclude that b is open subset of r. hence (r, =) is a topological ring. theorem 2.6. let (r, =) be a topological ring such that every nowhere dense subset of r is closed. then (r, =) is α−irresolute topological ring. proof. let b be any α−open subset of r. then b = p −∇ for some open subset p ⊆ r and nowhere subset ∇ ⊆ r. by given condition, ∇ is a closed subset of r. thus, for any ς ∈ b, there exist u, v ∈= with the property that ς ∈ u, ∇⊆ v and u∩v = ∅. let o = p∩u. then evidently, o is open subset of r containing ς such that o ⊆ b. thereby it follows that (r, =) is an α−irresolute topological ring. theorem 2.7. let (r, =) be an α−irresolute topological ring. then, for every o ∈ γ, there exists σ ∈ γ such that αcl(σ) ⊆ o. proof. let o ∈ γ. by theorem 2.1, there exists a set σ ∈ γ such that σ ∩ (oc + σ) = ∅. this implies that σ ⊆ (oc + σ)c. consequently, αcl(σ) ⊆ o. definition 2.2. a topological space (x, =) is said to be (1) α−t0 if for distinct points x and y in x, there exists an α−open set u in x such that either x ∈ u, y /∈ u or y ∈ u, x /∈ u. (2) α −t1 if for distinct points x and y in x, there exist α−open sets u and v in x such that x ∈ u, y /∈ u and y ∈ v, x /∈ v . (3) α−t2 if for distinct points x and y in x, there exist disjoint α−open sets u and v in x such that x ∈ u, y /∈ u and y ∈ v, x /∈ v . 381 m. ram theorem 2.8. let (r, =) be an α−irresolute topological ring. then the following are equivalent: (1) (r, =) is α−t0 space. (2) (r, =) is α−t1 space. (3) (r, =) is α−t2 space. proof. we prove only (1) implies (3). other implications are obvious. (1) implies (3): let ς 6= 0. without loss of generality, we may assume that there is σ ∈ γ such that ς /∈ σ. then, by theorem 2.7, there exists o ∈ γ such that ς /∈ αcl(o). by theorem 2.1, there exists u ∈ γ such that (ς + u) ∩ (αcl(o) + u) = ∅. thence we infer that ς + u and u are disjoint α−open subsets of r containing ς and 0 respectively. hence (r, =) is α−hausdorff space. we end this note with some open questions. imbibing theorems, theorem 2.3, theorem 2.5, and theorem 2.6, we point out several interesting and pertinent questions which are worthy to the healthy discussion about the interconnection between the α−irresolute topological rings and the topological rings. question 1 . does there exist an α−irresolute topological ring (r, =) which is also a topological ring but = 6= =α? question 2 . can we have a finite α−irresolute topological ring which is not a topological ring? question 3 . is there any α−irresolute topological ring which is not a topological ring showing the essentiality of each condition in theorem 2.3? question 4 . does there exist a topological ring which is not an α−irresolute topological ring satisfying theorem 2.3? question 5 . can we replace ‘α−closedness’ in the statement of theorem 2.5 by a weaker form of it like semi-closedness, pre-closedness, etc.? question 6 . can we replace ‘closedness’ in the hypothesis of theorem 2.6 by a weaker form of it like α−closedness, semi-closedness, etc.? question 7 . does there exist an α−regular topological ring which is not an α−irresolute topological ring? 382 a note on α−irresolute topological rings 3 conclusions in the literature it is a well known result that the algebraic sum of a compact set and a closed set in a topological ring is closed set. in this note, we showed that the algebraic sum a + b of an α−compact subset a ⊆ r, and an α−closed subset b ⊆r of an α−irresolute topological ring r is a closed subset of r. in addition, we have shown the equivalence of α−ti spaces in α−irresolute topological rings for i = 0, 1, 2. references d. jangkovic, i.j. reilly and m. k. vamanamurthy, on strongly compact topological spaces, question and answer in general topology, 6(1) (1988), 29-40. n. levine, semi-open sets and semi-continuity in topological spaces, amer. math. monthly, 70 (1963), 36-41. o. njastad, on some classes of nearly open sets, pacific j. math. 15 (1965), 961-970. m. ram, s. sharma, s. billawria and t. landol, on α−irresolute topological rings, international journal of mathematics trends and technology, 65 (2) (2019), 1-5. 383 ratio mathematica volume 47, 2023 on the structure of composite odd integers and prime numbers giuseppe buffoni* abstract with the expression “structure of an odd integer” we mean the set of properties of the integer n which specifies the odd integer 2n + 1 and brings about its behaviour. these properties of n, for both composite and prime numbers, are expounded in detail, together with their geometrical implications. in this context, a set, in a two dimensional space, where all the composite odd integers in [2a + 1, 2n + 1] are localized, is illustrated. keywords: couples of divisors; no divisors. 2020 ams subject classifications: 11a00, 11y00.1 *cnr-imati “e. magenes”, milano, italy; giuseppe.buffoni9av2@gmail.com 1received on march 31, 2023. accepted on june 1, 2023. published on june 30, 2023. doi: 10.23755/rm.v41i0.967. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 437 g. buffoni 1 introduction let 2n + 1 be any composite odd integer, and 2i + 1, 2j + 1 a couple of its divisors. in [1] it is pointed out that 2n + 1 = (2i + 1)(2j + 1) ⇐⇒ n = kij, (1) where kij is the following symmetric form in i and j kij = i + j + 2ij, (i, j) ∈ n. (2) let k = {kij|(i, j) ∈ n} ⊂ n. since any odd integer is either a composite or a prime number, it follows that n ∈ k ⇐⇒ 2n + 1 is a composite number, n ∈ n \ k ⇐⇒ 2n + 1 is a prime number. some properties of the matrix {kij} of finite order are reported in the appendix 1 of [1], where the relation n = kij has been used to identify composite odd integers 2ν + 1, with 4 = k11 ≤ ν ≤ n, and having the same number of couples of divisors. the distributions of odd integers ≤ 2n + 1 vs. the number of their couple of divisors have been computed up to n ≃ 5 · 107. the motivation for this work is just the reference [1], unique reference for this paper, and very useful for the analysis of the problem under study. in section 2 we enlighten the basic properties of n for both composite odd numbers and primes. in section 3 their main geometrical implications are illustrated. in particular, the relation n = kij for composite odd integers suggested to investigate the localization of the pairs (i, j) producing kij in a given interval. thus, a set ω(a, n) of points (x, y) ∈ r2 containing all the points with integer coordinates (i, j) such that a ≤ kij ≤ n is defined. 2 the basic properties of n specifying the behaviour of 2n + 1 for a pair (i, j), identifying any possible couple of divisors of an odd integer 2n + 1, we should have n = kij, (3) where kij is defined in (2). furthermore, any couple (2i + 1, 2j + 1) of divisors of 2n + 1, with i ≤ j, must satisfy the following inequality 438 on the structure of composite odd integers and prime numbers 2i + 1 ≤ √ 2n + 1 ≤ 2j + 1 (equality holds iff i = j). it follows that 1 ≤ i ≤ in = 1 2 (−1 + √ 2n + 1) ≤ j. (4) let in = {1, 2, ..., in}, with in = [in] =integer part of in. the implications (1) state that the special form (2) of n is a necessary and sufficient condition in order for any odd integer 2n + 1 to be a composite number. moreover, if 2n + 1 is a prime, then n cannot be put in the form (2). in the following theorem, given a generic n, necessary and sufficient conditions are given for either n ∈ k or n ∈ n \ k. theorem 2.1. let n ∈ n. then n ∈ k iff ∃i ∈ in : (2i + 1) | (n − i), (5) n ∈ n \ k iff ∀i ∈ in : (2i + 1) ∤ (n − i). (6) proof. a non implicit relation between i and j is obtained by making explicit the variable j in (3). in doing so, (3) is written in the form of a real homographic function of the integer variable i y = ϕn(i) = n − i 2i + 1 , i ∈ in, y ∈ r. (7) if for some i we have y ∈ n, then the pair (i, j) with j = y = [y] identifies a couple of divisors of 2n + 1. thus, in this way we can identify all the couples of divisors of 2n + 1, if they exist. the theses follow directly from (7). 2 the in conditions assumed to have n ∈ n \ k are not superabundant. consequences of theorem 1 are the following two propositions. if n ∈ k, then n has a number of representations n = kij just equal to the number of couples of divisors of 2n + 1. if n ∈ n \ k, then n can be expressed as n = i + (2i + 1)qi + ri = kiqi + ri, 439 g. buffoni with qi, ri ∈ n and 1 ≤ ri < 2i + 1, ∀i = 1, 2, ..., in. remark 2.1. (r1) ϕn(i) is decreasing with i from ϕn(1) = (n − 1)/3 to ϕn(in). (r2) equation y = ϕn(i) is a simple algorithm to compute the couples of divisors of an integer 2n + 1, if they exist. it can also be used as a primality test. the order of the number of operations is √ n/2. (r3) by direct calculation it follows that 2n + 1 = 2kij + 1 = (i + j + 1) 2 − (j − i)2. this identity shows the well known fact that any composite odd integer may be written as a difference of two squares. possibly in more than two ways, while for a prime only holds the decomposition 2n + 1 = (n + 1)2 − n2. (r4) let mq be the sequence of integers defined recursively by mq = 2mq−1 + 1, q = 1, 2, ..., m0 = 0, which implies that mq = 2q − 1. when q is a prime, mq is a mersenne number. let q = 2r. since m2r = 22r − 1 = (2r + 1)(2r − 1) is composite, we have that m2r−1 ∈ k, r = 2, 3, .... 3 geometrical implcations of the properties of n the relation n = kij can be rewritten as n = i(j + 1) + (i + 1)j, (8) which can be interpreted as the determinant of the matrix a = ( i −j i+1 j+1 ) . since det(a) = n, n is the area of the parallelogram with the column vectors u = (i, j + 1)t , v = (−j, j + 1)t , as two of its sides. this property holds ∀(i, j) such that kij = n. in addition to the equality of determinants, no other relationships have been recognized between the matrices a related to the pairs (i, j) with kij = n. 440 on the structure of composite odd integers and prime numbers moreover (8) shows that n is also the area of the union of the two rectangles of sides i, (j + 1) and (i + 1), j. imagine that you are drawing, on a squared sheet, rectangles composed by an odd integer number of squares. the side of the squares is assumed as linear unit of measure, so that the rectangles have integer sides and area. obviously, only integers 2n + 1 with n satisfying (8) can represent rectangles. the number 2n + 1 is the area of the rectangle of sides (2i + 1), (2j + 1). moreover, with the help of (8), we can show that this rectangle is the union of five rectangles: two with sides i, j + 1, two with sides i + 1, j, plus one unit square (fig. 1). figure 1: decomposition of the rectangle of sides 2i + 1, 2j + 1. i = 3, j = 5, n = 3 · (5 + 1) + (3 + 1) · 5 = 38, 2n + 1 = 2 · 38 + 1 = 77, (2i + 1) · (2j + 1) = 7 · 11 = 77 thus we have shown that, when n ∈ k, then n represents also the area of the union of two rectangles, and 2n + 1 both the area of one rectangle and the union of five rectangles. when n /∈ k, then n can be represented as n = kiqi + ri, with ri ≥ 1, which leads to the following representation for 2n + 1 2n + 1 = 2kiqi + 1 + 2ri = (2i + 1)(2qi + 1) + 2ri, 1 ≤ ri < 2i + 1. since n /∈ k, this expression can not be reduced to the product of two odd integers. let us now consider the problem of the localization of composite odd integers 2ν + 1, with 4 ≤ a ≤ ν ≤ n. the discrete relation n = kij is viewed in the continuous form as n = κ(x, y) = x + y + 2xy, (x, y) ∈ r2, and in explicit form it is written as y = ϕn(x) = n − x 2x + 1 , 1 ≤ x ≤ in. (9) 441 g. buffoni ϕn(x) is decreasing with x from ϕn(1) = (n − 1)/3 to ϕn(in) = in. let the set ω(a, n) be defined as ω(a, n) = {(x, y) ∈ r2 | 1 ≤ x ≤ in, (x ≤ y) ∨ (a ≤ κ(x, y) ≤ n)}. (10) figure 2: set ω(a, n): n = 100, a = 50. continuous lines: y = ϕn(x) and y = ϕa(x); dotted lines: x = 1 and y = x; asterisk: points (ia, 0), (ia, ia); circle: points (in, 0), (in, in). different scales for x and y. it is a bounded, closed and convex set in a plane. from its implicit definition (10), it follows that ω(a, n) (fig. 2) can be defined explicity by the following inequalities as the union of two sets 1 ≤ x ≤ ia, ϕa(x) ≤ y ≤ ϕn(x), (11) ia < x ≤ in, x ≤ y ≤ ϕn(x). (12) both ϕa(x) and ϕn(x) are decreasing with x, and also their difference ∆n,a(x) = ϕn(x) − ϕa(x) = n − a 2x + 1 , 1 ≤ x ≤ ia. (13) thus, when n − a is small enough, we have that ∆n,a(x) < 1, and ω(a, n) may contain at the most one integer point. other details on this argument can be found in appendix. 4 concluding remarks it is worthy to draw a concluding remark. in other words, theorem 2.1 states that any prime has just one parent: if a generic n satisfies (6) in theorem 2.1, then it is the parent of 2n + 1. otherwise it is a composite number. 442 on the structure of composite odd integers and prime numbers references [1] g. buffoni. on odd integers and their couples of divisors. ratio mathematica, 40: 87-111, 2021. appendix cosider the limit case a = 4 (fig. 3, left), so that ia = 1, ϕ4(1) = 1. thus, the first set (11) consists of only one point x = ia = 1. since ϕa(ia) = ia, (11) and (12) reduce to 1 ≤ x ≤ in, x ≤ y ≤ ϕn(x). consider the case a large, i.e. n−a small (fig. 3, right). assume n−a << a. the difference in − ia may be written as in − ia = 0.5 √ 2n + 1 (1 − √ 2a + 1 2n + 1 ) = 0.5 √ 2n + 1 (1 − √ 1 − 2(n − a) 2n + 1 ). from the assumption n − a << a we have 2(n − a)/(2n + 1) << 1, so that in − ia ≃ n − a 2 √ 2n + 1 < 1. (14) thus, the second set (12) reduces to a very small interval. figure 3: set ω(a, n): n = 100. left: limit case a = 4; right: case a = 90, so that in − ia < 1. notes as in caption of fig. 2 consider now the set of points with integer coordinates. the set ω∗(a, n) of the pairs (i, j), such that 1 ≤ i ≤ j, a ≤ kij ≤ n, defined by 443 g. buffoni ω∗(a, n) = {(i, j) ∈ n | i = 1, 2, ..., in, (i ≤ j) ∨ (a ≤ kij ≤ n)}, (15) is the set of the points (x, y) ∈ ω(a, n) with x and y integer coordinates. afterwards, it is straightforward to formulate the explicit definition of the pairs (i, j) ∈ ω∗(a, n). then, ω∗(a, n) is again defined as the union of two sets i = 1, 2, ..., [ia], j = j(i), j(i) + 1, ..., [ϕn(i)], (16) where either j(i) = [ϕa(i)] + 1 when ϕa(i) /∈ n or j(i) = ϕa(i) when ϕa(i) ∈ n, and i = [ia] + 1, ..., [in], j = i, i + 1, ..., [ϕn(i)]. (17) when b−a is small enough, it follows that in−ia < 1, so that either [in] = [ia] or [in] = [ia] + 1. the second set (17) is missing when [in] = [ia], and consists of only one point when [in] = [ia] + 1; thus (16) and (17) reduce to i = 1, 2, ..., [in], j = j(i), j(i) + 1, ..., [ϕn(i)]. (18) if ν ∈ {a, a + 1, ..., n} \ {kij | (i, j) ∈ ω∗(a, n)}, then 2ν + 1 is a prime. 444 ratio mathematica vol. 32, 2017, pp. 37–44 issn: 1592-7415 eissn: 2282-8214 quasi-order hypergroups determinated by t -hypergroups šárka hošková-mayerová university of defence, faculty of military technology, department of mathematics and physics, kounicova 65, 612 00 brno, czech republic sarka.mayerova@unob.cz received on: 15-03-2017. accepted on: 02-05-2017. published on: 30-06-2017 doi: 10.23755/rm.v32i0.333 c©šárka hošková-mayerová abstract quasi-order hypergroups were introduced by jan chvalina in 90s of the twentieth century. he proved that they form a subclass of the class of all hypergroups, i.e. structures with one associative hyperoperation fulfilling the reproduction axiom. in this paper a theorem which allows an easy description of all quasi-order hypergroups is presented. moreover, some results concerning the relation of quasi-order and upper quasi-order hypergroups are given. furthermore, the transformation hypergroups acting on tolerance spaces are defined and an example of them is mentioned. keywords. quasi-order hypergroup, order hypergroup, tolerance relation, transformation semihypergroup, transformation hypergroup. 2010 ams subject classifications: 20f60, 20n20. 37 šárka hošková-mayerová the applications of mathematics in other disciplines, for example, in informatics, play a key role and they represent, in the last decades, one of the purpose, of the study of the experts of hyperstructures theory all over the world. hyperstructure theory was introduced in 1934 by the french mathematician marty [16], at the 8th congress of scandinavian mathematicians, where he defined hypergroups based on the notion of hyperoperation, began to analyze their properties, and applied them to groups. in the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics by many mathematicians. in a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. several books have been written on hyperstructure theory, see [6, 10, 17]. a recent book on hyperstructures [9] points out on their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs and hypergraphs. another book [10] is devoted especially to the study of hyperring theory. several kinds of hyperrings are introduced and analyzed. the volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: hyperstructures and transposition hypergroups. hypergroups in the sense of marty [16] form the largest class of multivalued systems that satisfies group-like axioms. it should be noted that various problems in non-commutative algebra lead to the introduction of algebraic systems in which the operations are not single-valued. the motivation for generalization of the notion of group resulted naturally from various problems in non-commutative algebra, another motivation for such an investigation came from geometry. hypergroups have been used in algebra, geometry, convexity, automata theory, combinatorial problems of coloring, lattice theory, boolean algebras, logic etc., over the years. over the following decades, new and interesting results again appeared, but it is above all that a more luxuriant flourishing of hyperstructures has been seen in the last 20 years. it is not surprising that hypergroups as well as hypergroupoids, quasi-hypergroups, semihypergroups, hyperfields, hyper vector spaces, hyperlattices etc. have been studied. the most complete bibliography up to 2002 can be found in the monograph of pierguilio corsini and violeta leoreanu: applications of hyperstructure theory [9]. another comprehensive list of literature is in monograph [17] and updated information is included in web site: http://aha.eled.duth.gr. in the paper [2] special types of hypergroups, so called quasi-order hypergroups (qohg) and order hypergroups (ohg), were introduced (cf. also [6, 9, 14, 5]). first of all recall some basic terms and definitions. a hyperoperation “◦” on a nonempty set h is a mapping from h ×h to p∗(h) (all nonempty subsets of 38 quasi-order hypergroups determinated by t -hypergroups h). the hypergroupoid is a pair (h,◦). the quasi-hypergroup is a hypergroupoid if the reproduction axiom (a◦h = h = h ◦a for any a ∈ h) is fulfilled. the quasi-hypergroup (h,◦) is called a hypergroup if moreover the hyperoperation “◦” is associative ( (a◦b)◦c = a◦(b◦c) for any a,b,c ∈ h ) . here for nonempty a,b ⊆ h we put a ◦ b = ⋃ a∈a,b∈b a ◦ b. we denote a ◦ b instead of {a}◦ b, a ∈ h. see, e.g. [5, 6, 9, 7, 11]. let (h,∗) and (h′,?) be hypergroupoids. then a mapping f : h −→ h′ is called inclusion homomorphism if it satisfies the condition: f(x∗y) ⊆ f(x) ? f(y) for all pairs x,y ∈ h. let x be a set and τ be a tolerance relation (i.e., reflexive and symmetric binary relation)—see [1]. then the pair (x,τ) is a tolerance space. definition 1. the hypergroup (h,◦) is called a quasi-order hypergroup—cf. [2, 4, 9]—if (i) a ∈ a3 = a2 for any a ∈ h, (1) (ii) a◦ b = a2 ∪ b2 for any a,b ∈ h. (2) the hypergroup (h,◦) is called an order hypergroup if moreover (iii) a2 = b2 implies a = b for any a,b ∈ h. (3) using the methods occurring in [2, 4] the following theorem characterizing all quasi-order hypergroups can be proved. for the prove see [13]. (by a2 we mean a◦a.) theorem 1. let (h,◦) is a quasi-order hypergroup. denote k(a) = a2 for any a ∈ h. then the system of sets k(a) fulfills the following conditions: (i) a ∈ k(a) for any a ∈ h, (4) (ii) if b ∈ k(a) then k(b) ⊆ k(a). (5) conversely, if any system of subsets k(a) of the set h, a ∈ h, fulfills (4) and (5), then there exists the only hyperoperation “◦” on h such that a ◦ a = k(a) and (h,◦) is a quasi-order hypergroup. with respect to (3) the following corollary evidently holds: corollary 1. under the assumptions of theorem 1 the quasi-order hypergroup (h,◦) is an order hypergroup if and only if for a 6= b there is k(a) 6= k(b). it is easy to show that if r is a quasi-ordering on a set h, then the pair (h,◦), where a ◦ b = r(a) ∪ r(b), a,b ∈ h, is a quasi-order hypergroup. (r(x) is 39 šárka hošková-mayerová an upper end of an element x ∈ h, i.e. the set {a ∈ h;arx for each element a ∈ h}). see e.g. [3, 11]. in [3] j. chvalina introduced the concept of an upper quasi-order and upper order hypergroup. definition 2. a hypergroup (h,◦) is said an upper quasi-order (upper order) hypergroup if there exists a quasi-ordering (ordering) r such that a◦ b = r(a)∪ r(b) for a,b ∈ h. it can be shown that the classes of all quasi-order hypergroups and upper quasi-order hypergroups coincide. the same is true for the classes of all order hypergroups and upper order hypergroups. see [2, theorem 1] or [9, proposition 2 on p. 96]. these results can be easily proved using theorem 1. as we will need the above mentioned result of prof. jan chvalina several times in this text we recall its formulation: proposition 1. [9, proposition 2 on p. 96] a hypergroupoid (h, ·) is a (quasi)order hypergroup if and only if there exists a (quasi)-ordeg ρ on the set h, such that ∀(a,b) ∈ h ×h, a · b = ρ(a)∪ρ(b), where ρ(a) = {x ∈ h,aρx}. theorem 2. every quasi-order (order) hypergroup is an upper quasi-order (upper order) hypergroup. proof. let (h,◦) be a quasi-order hypergroup. let us define a relation r on h as follows: arb iff b ∈ a2 for each a,b ∈ h. evidently arb iff b2 ⊆ a2. then (4) and (5) imply that r is a quasi-ordering. moreover, r(a) = a2. thus a◦ b = a2 ∪ b2 = r(a)∪r(b). if (h,◦) is even an order hypergroup, then by corollary 1 there is r(a) 6= r(b) for a 6= b, a,b ∈ h. thus r is an ordering. in [12] a more general concept of subquasi-order hypergroup is introduced. it is an open question whether a similar representation result as in theorem 1 can be found for this generalization. now let us recall the definition of a transformation hypergroup. it was introduced in [15]. recall first that tolerance relation is a reflective and symmetric relation on a set. this relation yields the concept of singularity in abstract mathematical expressions. this relation namely in connection with other structures moves corresponding mathematical theories to useful applications. many publications are devoted to systematic investigation to tolerances on algebraic structures compatible with 40 quasi-order hypergroups determinated by t -hypergroups all operations of corresponding algebras. a certain survey of important results including valuable investment can be found in [1]. tolerance space is a set endowed with a tolerance relation. definition 3. let x be a set, (g,•) be a hypergroup and π : x × g → x a mapping such that π(π(x,t),s) ∈ π(x,t•s), where π(x,t•s) = {π(x,u);u ∈ t•s)} for each x ∈ x, s,t ∈ g. then the triple t = (x,g,π) is called a discrete transformation hypergroup or an action of the hypergroup g on the phase set x. the mapping π is also usually said to be simply an action. more generally, it is possible to consider the situation, where the phase space x is endowed with some additional structure. the interesting case is given in the following definition. definition 4. let (x,τ) be a tolerance space (so called phase tolerance space), (g,•) be a semihypergroup (so called phase semihypergroup) and π : x×g → x a mapping such that (i) π(π(x,t),s) ∈ π(x,t•s), where π(x,t•s) = {π(x,u);u ∈ t•s)} for each x ∈ x, s,t ∈ g; (ii) if x,y ∈ x are such that xτ y, then π(x,g)τ π(y,g) holds for any g ∈ g. then t = (x,g,π) is a transformation semihypergroup with phase tolerance space. if, moreover, the pair (g,•) is a hypergroup (so called phase hypergroup), then the triple t = (x,g,π) is a transformation hypergroup with phase tolerance space. in case the tolerance τ is trivial, i.e., xτ y if and only if x = y, the preceding definition coincides in fact with definition 3. let us consider a discrete transformation hypergroup t = (x,g,π). it is possible to assign to each transformation hypergroup a commutative, extensive hypergroup with the support x (i.e., phase set of t ) as follows: let us define for arbitrary pair of elements x,y ∈ x a binary hyperoperation �: x ×x → p∗(x) in this way: x�y = π(x,g)∪π(y,g)∪{x,y}, where π(x,g) = {π(x,u),u ∈ g} and similarly for π(y,g). in the following we will need the next lemma. the proof can be found in [4]. 41 šárka hošková-mayerová lemma 1. a hypergroupoid (h, ·) such that a ∈ a3 ⊂ a2, a · b = a2 ∪ b2 for any a,b ∈ h is a quasi-order hypergroup. proposition 2. the pair (x,�) is an extensive, commutative hypergroup. the extensivity and commutativity of the hyperoperation is evident, so the pair (x,�) is an extensive, commutative hypergroupoid. the conditions of lemma 1 are satisfied too, so (x,�) is an extensive, commutative hypergroup. remark 1. even in case when t is a transformation semihypergroup we can assign a commutative, extensive hypergroup to this semihypergroup by the above described way. the considered mapping is a functorial assignment which is described in the following way: the above defined assignment determines a functor f from the category dth of all discrete transformation hypergroups into the category ah of all commutative (abelian) hypergroups. the functor f = (fo,fm) (o-as objects, m-as morphisms) is defined as follows: fo(t ) = (x,�); fm(hx,hg) = hx. consider ti = (xi,gi,πi) ∈ dth where (xi,�i) are hypergroups, i = 1,2 and the morhpisms hx : x1 → x2. then hx(x�1 y) = h ( π(x,g)∪π(y,g)∪{x,y} ) = { π(hx(x),hg(g),g ∈ g1) } ∪ { π(hx(y),hg(g),g ∈ g1) } ∪ { hx(x),hx(y) } ⊆ π(hx(x),g2)∪π(hx(y),g2)∪{hx(x),hx(y)} = hx(x)�2 hx(y) holds for all x,y ∈ x1. theorem 3. the pair (x,�) is a quasi-order hypergroup determined by t , shortly quasi-order t -hypergroup. proof. let us define on (x,�) a binary relation “ρ” as follows: xρy ⇔∃u ∈ g such that, π(x,u) = y or x = y. this relation is evidently reflexive. we will show, that it is transitive as well. let 1) x = y and y = z, then x = z and xρz, 2) x = y and π(y,v) = z, then π(x,v = z) so xρz, 3) π(x,u) = y and y = z, then π(x,u = z) so xρz, 42 quasi-order hypergroups determinated by t -hypergroups 4) π(x,u) = y and π(y,v) = z, then z = π(y,v) = π(π(x,u),v). from definition 4 we have π(π(x,u),v) ∈ π(x,u�v), thus there exists w ∈ u�v such that z = π(x,w). hence we have xρz. so ρ is a quasi-order. it is well known that ρ2 = ρ ⊃ diag(x), where diag(x) = {(x,x);x ∈ x}. now for any pair of elements x,y ∈ x we get x � y = ρ(x) ∪ ρ(y). so according proposition 1 the pair (x,�) is a quasi-order hypergroup determined by t . shortly it is a quasi-order t -hypergroup. references [1] i. chajda, algebraic theory of tolerance relations, the palacky university olomouc, czech republic (1991). [2] j. chvalina, commutative hypergroups in the sence of marthy, proceeding of the summer school (1994), 19–30, hornı́ lipová, czech republic. [3] j. chvalina, functional graphs, quasi-ordered sets and commutative hypergroups, mu brno, (1995), (in czech). [4] j. chvalina, l. chvalinová, state hypergroup of automata, acta mat. et. inf. univ. ostraviensis 4 (1996), 105–120. [5] j. chvalina, š. hošková, abelization of proximal hν-rings using graphs of good homomorphisms and diagonals of direct squares of hyperstructures, internat. congress on aha 8 (samothraki, greece 2002), 147–158. [6] p. corsini, prolegomena of hypergroup theory, aviani editore, 2nd edition, tricesimo, (1993). [7] p. corsini, hypergraphs and hypergroups, algebra universalis, 35, (1996), 548–555. [8] p. corsini, v. leoreanu, hypergroups and binary relations, algebra universalis, 43, (2000), 321–330. [9] p. corsini, v. leoreanu, applications of hyperstructure theory, kluwer academic publishers, dordrecht, hardbound, (2003). [10] b. davvaz, v. leoreanu-fotea, hyperring theory and applications, international academic publishers, palm harbor, usa, 2007. [11] š. hošková, abelization of weakly associative hyperstructures and their proximal modifications, ph.d. thesis, masaryk university brno (2003), 73p. 43 šárka hošková-mayerová [12] š. hošková, upper order hypergroups as a reflective subcategory of subquasiorder hypergroups, italian journal of pure and applied math., no. 20, 215– 222, (2006). [13] š. hošková, representation of quasi-order hypergroups, global journal of pure and applied mathematics (gjpam), vol.1 no. 2, 173–176, india, (2005). [14] š. hošková, j. chvalina, the unique square root condition for quasi-order hypergroups and the corresponding reflector for the category of all orderhypergroups, proc. of 3th international conference aplimat 2004, bratislava, 471–476, slovakia. [15] š. hošková, j. chvalina, transformation tolerance hypergroup, thai mathematical journal, 4(1), (2006), 63–72. [16] f. marty, : sur une generalisation de la notion de groupe, huitième congr. math. scan. (1934), stockholm, 45–49. [17] t. vougiouklis, : hyperstructures and their representations, hadronic press monographs in mathematics, palm harbor florida 1994. 44 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 37, 2019, pp. 61-68 61 on cyclic multigroup family johnson aderemi awolola* abstract in this paper, the concept of cyclic muligroup is studied from the preliminary knowledge of cyclic group which is a well-known concept in crisp environment. by using cyclic multigroups, we then delineate a cyclic multigroup family and investigate its structural properties. it is observed that the union of class of cyclic multigroups generated by 𝓐 is a cyclic multigroup. however, the union is an identity cyclic multigroup. in particular, we obtain a series of class of cyclic multigroups generated by 𝓐. keywords: multiset, multigroup, cyclic multigroup, cyclic multigroup family. 2010 ams subject classification: 08𝐴72, 03𝐸72, 94𝐷05.† * department of mathematics/statisics and computer science, college of science (university of agriculure, makurdi, nigeria), remsonjay@yahoo.com, awolola.johnson@uam.edu.ng. † received on august 23rd, 2019. accepted on december 25th, 2019. published on december 31th, 2019. doi: 10.23755/rm.v37i0.476. issn: 1592-7415. eissn: 2282-8214. ©johnson aderemi awolola. this paper is published under the cc-by licence agreement. mailto:remsonjay@yahoo.com j. a. awolola 62 1 introduction in set theory, repetition of objects are not allowed in a collection. this perspective rendered set almost irrelevant because many real life problems admit repetition. to remedy the inadequacy in the idea of sets, the concept of multisets was introduced in [6] as a generalization of sets by relaxing the restriction of distinctness on the nature of the objects forming a set. multiset is very promising in mathematics, computer science, website design, etc. see [4, 5] for details. generalization of algebraic structures is playing a prominent role in the sphere of mathematics. one of such generalization of algebraic structures is the notion of multigroups. multigroups are actually a generalization of groups and have come into the centre of interest. in [1], the multigroup proposed is analogous to fuzzy group [2] in that the underlying structure is a multiset. although multigroup concept was earlier used in [9, 12] as an extension of group theory, however the recent definition of multigroup in [1] is adopted in this paper because it shows a strong analogy in the behaviour of group and makes it possible to extend some of the major notions and results of groups to that of multigroups. some of the related works can be found in [3], [7], [8], [10], [11] etc. the aim of this paper is to promote research and the development of multiset knowledge by studying cyclic multigroup family based on the sufficient condition for a multiset to be a cyclic multigroup. 2 preliminaries in this section, we give the preliminary definitions and results that will be required in this paper from [1, 8]. definition 2.1 let ℧ be a non-empty set. a multiset 𝐴 drawn from ℧ is characterized by a count function 𝐶𝐴 defined as 𝐶𝐴 : ℧ ⟶ 𝓓 , where 𝒟 represents the set of non-negative integers. for each 𝑥 ∈ ℧, 𝐶𝐴(𝑥) is the characteristics value of 𝑥 in 𝐴 and indicates the number of occurrences of the element 𝑥 in 𝐴. an expedient notation of 𝐴 drawn from ℧ = {𝑥1, 𝑥2 , … , 𝑥𝑛 } is [𝑥1, 𝑥2 , … , 𝑥𝑛 ]𝐶𝐴(𝑥1), 𝐶𝐴(𝑥2) ,…, 𝐶𝐴(𝑥𝑛) such that 𝐶𝐴(𝑥𝑖 ) is the number of times 𝑥𝑖 occurs in 𝐴, (𝑖 = 1, 2, … , 𝑛). the class of all multisets over ℧ is denoted by 𝑀𝑆(℧). on cyclic multigroup family 63 definition 2.2 let 𝐴, 𝐵 ∈ ℧. then 𝐴 is a submultiset of 𝐵 written as 𝐴 ⊆ 𝐵 or 𝐵 ⊇ 𝐴 if 𝐶𝐴(𝑥) ≤ 𝐶𝐵 (𝑥), ∀ 𝑥 ∈ ℧. also, if 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵, then 𝐴 is called a proper submultiset of 𝐵 and denoted as 𝐴 ⊂ 𝐵. definition 2.3 let 𝐴, 𝐵 ∈ 𝑀𝑆(℧). then the union and intersection denoted by 𝐴 ⋃ 𝐵 and 𝐴 ⋂ 𝐵 are respectively defined as follows: 𝐶𝐴 ⋃ 𝐵 (𝑥) = 𝐶𝐴(𝑥) ⋁ 𝐶𝐵 (𝑥) = 𝑚𝑎𝑥{𝐶𝐴(𝑥), 𝐶𝐵 (𝑥)} and 𝐶𝐴 ⋂ 𝐵 (𝑥) = 𝐶𝐴(𝑥) ⋀ 𝐶𝐵 (𝑥) = 𝑚𝑖𝑛{𝐶𝐴(𝑥), 𝐶𝐵 (𝑥)}, ∀ 𝑥 ∈ ℧. definition 2.4 let {𝐴𝑖 }𝑖∈λ be an arbitrary family of multisets over ℧. then for each 𝑖 ∈ λ, ⋃𝑖∈λ𝐴𝑖 = ⋁𝑖∈λ𝐶𝐴𝑖 (𝑥) and ⋂𝑖∈λ𝐴𝑖 = ⋀𝑖∈λ𝐶𝐴𝑖 (𝑥). definition 2.5 the direct product of multisets 𝐴 and 𝐵 is defined as 𝐴 × 𝐵 = {[𝑥, 𝑦]𝐶𝐴×𝐵 [(𝑥, 𝑦)] ∣ 𝐶𝐴×𝐵 [(𝑥, 𝑦)] = 𝐶𝐴(𝑥)𝐶𝐴(𝑦)}. definition 2.6 let ℧ be a non-empty set. the sets of the form 𝐴𝑛 = {𝑥 ∈ ℧ ∣ 𝐶𝐴(𝑥) ≥ 𝑛, 𝑛 ∈ ℤ +} are called the 𝑛 – level sets of 𝐴. definition 2.7 let ℧ and 𝜉 be two non-empty sets and 𝑓 ∶ ℧ ⟶ 𝜉 be a mapping. then the image 𝑓(𝐴) of a multiset 𝐴 ∈ 𝑀𝑆(℧) is defined as 𝐶𝑓(𝐴)(𝑦) = { ⋁ 𝐶𝐴(𝑥), 𝑓 −1(𝑦) ≠ ∅𝑓(𝑥)=𝑦 0, 𝑓 −1(𝑦) = ∅ definition 2.8 let 𝒳 be a group. by a multigroup over 𝒳 we mean a count function 𝐶𝐴 ∶ 𝒳 ⟶ 𝒟 such that 𝐶𝐴(𝑥𝑦) ≥ 𝐶𝐴(𝑥) ⋀ 𝐶𝐴(𝑦), ∀ 𝑥, 𝑦 ∈ 𝒳 and 𝐶𝐴(𝑥 −1) ≥ 𝐶𝐴(𝑥), ∀ 𝑥 ∈ 𝒳. moreover, an abelian multigroup over 𝒳 is defined as a multigroup satisfying the condition 𝐶𝐴(𝑥𝑦) ≥ 𝐶𝐴(𝑦𝑥), ∀ 𝑥, 𝑦 ∈ 𝒳. let 𝑒 be the identity element of 𝒳. it can be easily verified that if 𝐴 is a multigroup over a group 𝒳, then 𝐶𝐴(𝑒) ≥ 𝐶𝐴(𝑥) and 𝐶𝐴(𝑥 −1) ≥ 𝐶𝐴(𝑥), ∀ 𝑥 ∈ 𝒳. we denote the class of all multigroups over 𝒳 by 𝑀𝐺(𝒳). proposition 2.1 let 𝐴 ∈ 𝑀𝑆(℧). then 𝐴 ∈ 𝑀𝐺(𝒳) if and only if 𝐶𝐴(𝑥𝑦 −1) ≥ 𝐶𝐴(𝑥) ⋀ 𝐶𝐴(𝑦), ∀ 𝑥, 𝑦 ∈ 𝒳. j. a. awolola 64 proposition 2.2 let 𝐴 ∈ 𝑀𝐺(𝒳). then 𝐴𝑛 , 𝑛 ∈ ℤ + are subgroups of 𝒳. proposition 2.3 let 𝒳, 𝒴 be groups and 𝑓 ∶ 𝒳 ⟶ 𝒴 be a homomorphism. if 𝐴 ∈ 𝑀𝐺(𝒳), then 𝑓(𝐴) ∈ 𝑀𝐺(𝒴). 3 cyclic multigroup family definition 3.1 let 𝒳 = 〈𝑎〉 be a cyclic group. if 𝒜 = {[𝑎𝑛]𝐶𝒜 (𝑎𝑛) ∣ 𝑛 ∈ ℤ} is a multigroup, then 𝒜 is called a cyclic multigroup generated by [𝑎]𝐶𝒜 (𝑎) and denoted by 〈[𝑎]𝐶𝒜 (𝑎)〉. proposition 3.1 if 𝒜 is a cyclic multigroup and 𝑚 ∈ ℤ+, then 𝒜𝑚 = {([𝑎𝑛]𝐶𝒜 (𝑎𝑛)) 𝑚 ∣ 𝑛 ∈ ℤ} is also a cyclic multigroup. proof. let us show that 𝒜𝑚 satisfies the two conditions in definition 2.8. we can consider only its count function because the 𝑚 − 𝑡ℎ power of 𝒜 effects just only the count function of 𝒜𝑚. since 𝒜 is a multigroup and 𝐶𝒜 (𝑎) ∈ 𝒟, we have (𝐶𝒜 (𝑎 𝑛1 𝑎𝑛2 ))𝑚 ≥ (𝐶𝒜 (𝑎 𝑛1 ) ⋀ 𝐶𝒜 (𝑎 𝑛2 )) 𝑚 = (𝐶𝒜 (𝑎 𝑛1 ) )𝑚 ⋀ (𝐶𝒜 (𝑎 𝑛2 ) )𝑚 and consequently, (𝐶𝒜 (𝑎 −𝑛)) 𝑚 ≥ (𝐶𝒜 (𝑎 𝑛)) 𝑚 . this completes the proof of the proposition. example 3.1 let 𝒳 = 〈𝑎〉 be a cyclic group of order 12 such that 𝐶𝒜 (𝑎 0) = 𝑡0, 𝐶𝒜 (𝑎 4) = 𝐶𝒜 (𝑎 8) = 𝑡1, 𝐶𝒜 (𝑎 2) = 𝐶𝒜 (𝑎 6) = 𝐶𝒜 (𝑎 10) = 𝑡2, 𝐶𝒜 (𝑥) = 𝑡3 for other elements 𝑥 ∈ 𝒳, where 𝑡𝑖 ∈ 𝒟, 0 ≤ 𝑖 ≤ 3 with 𝑡1 > 𝑡1 > 𝑡2 > 𝑡3. it is clear that 𝒜 is a multigroup over 𝒳. thus, 𝒜 = {[𝑎𝑛]𝐶𝒜 (𝑎𝑛) ∣ 𝑛 ∈ ℤ} is a cyclic multigroup generated by [𝑎]𝐶𝒜 (𝑎). definition 3.2 let 𝑒 be the identity element of the group 𝒳. we define the identity cyclic multigroup ℰ by ℰ = {[𝑒]𝐶𝒜 (𝑒) ∣ 𝐶𝒜 (𝑒) ≥ 𝐶𝒜 (𝑎 𝑛), 𝑛 ∈ ℤ}. proposition 3.2 if 𝑚 ≤ 𝑛, then the multigroup 𝒜𝑛 is a submultigroup of 𝒜𝑚. proof. clearly 𝒜𝑛 and 𝒜𝑚 are multigroups by definition 2.8. for every 𝑎 ∈ 𝒟, 𝑎𝑚 ≤ 𝑎𝑛 implies 𝒜𝑚 ⊆ 𝒜𝑛 (since 𝐶𝒜𝑚 (𝑎) ≤ 𝐶𝒜𝑛 (𝑎) ∀ 𝑎 ∈ 𝒳). on cyclic multigroup family 65 proposition 3.3 if 𝒜𝑖 and 𝒜𝑗 are cyclic multigroups, and 𝑖 < 𝑗, then 𝒜𝑖 ⋃ 𝒜𝑗 is also a cyclic multigroup for any 𝑖, 𝑗 ∈ ℤ+. proof. it is sufficient to consider only count functions. without loss of generality, let 𝑖 ≤ 𝑗. since 𝒜𝑖 ⊆ 𝒜𝑗 , we have 𝐶𝒜𝑖 ⋃ 𝒜𝑗 (𝑎 𝑛𝑎𝑚) = 𝐶𝒜𝑖 (𝑎 𝑛𝑎𝑚 ) ⋁ 𝐶𝒜𝑗 (𝑎 𝑛𝑎𝑚) = 𝐶𝒜𝑗 (𝑎 𝑛𝑎𝑚) ≥ 𝐶𝒜𝑗 (𝑎 𝑛) ⋀ 𝐶𝒜𝑗 (𝑎 𝑚) = 𝐶𝒜𝑖 ⋃ 𝒜𝑗 (𝑎 𝑛) ⋀ 𝐶𝒜𝑖 ⋃ 𝒜𝑗 (𝑎 𝑚) and 𝐶𝒜𝑖 ⋃𝒜𝑗 (𝑎 −𝑛) = 𝐶𝒜𝑖 (𝑎 −𝑛) ⋁ 𝐶𝒜𝑗 (𝑎 −𝑛) = 𝐶𝒜𝑖 (𝑎 𝑛) ⋁ 𝐶𝒜𝑗 (𝑎 𝑛) = 𝐶𝒜𝑖 ⋃ 𝒜𝑗 (𝑎 𝑛) hence, 𝒜𝑖 ⋃ 𝒜𝑗 is a cyclic multigroup. proposition 3.4 if 𝒜𝑖 and 𝒜𝑗 are cyclic multigroups, then 𝒜𝑖 ⋂ 𝒜𝑗 is also a cyclic multigroup. proof. similar to proposition 3.3. remark 3.1 since a cyclic group is an abelian group, it is obvious by definition 2.8 that the cyclic multigroups 𝒜𝑚, 𝒜𝑖 ⋃ 𝒜𝑗 and 𝒜𝑖 ⋂ 𝒜𝑗 are also abelian multigroups. definition 3.3 let 𝒜 be a cyclic multigroup, then the following class of cyclic multigroups {𝒜, 𝒜2, 𝒜3, … , 𝒜𝑚, … , ℰ} is called the cyclic multigroup family generated by 𝒜 and denoted by 〈𝒜〉. proposition 3.5 let 〈𝒜〉 = {𝒜, 𝒜2, 𝒜3, … , 𝒜𝑚, … , ℰ}. then ⋃ 𝒜𝑛 = 𝒜∞𝑛=1 and ⋂ 𝒜𝑛∞𝑛=1 = ℰ. proof. the proof is immediate from propositions 3.3 and 3.4. proposition 3.6 let 𝒜 be a cyclic multigroup. then 𝒜 ⊆ 𝒜2 ⊆ 𝒜3 ⊆ ⋯ ⊆ 𝒜𝑛 ⊆ ⋯ ⊆ ℰ. proof. it is known that 𝐶𝒜 (𝑎) ∈ 𝒟. hence, 𝐶𝒜 (𝑎) ≤ (𝐶𝒜2 (𝑎)) 2 , 𝐶𝒜 (𝑎 2) ≤ (𝐶𝒜2 (𝑎 2)) 2 , … , 𝐶𝒜 (𝑎 𝑛) ≤ (𝐶𝒜2 (𝑎 𝑛)) 2 . j. a. awolola 66 by definition 2.2, we have 𝒜 ⊆ 𝒜2 . by generalizing it for any 𝑖, 𝑗 ∈ ℤ+ with 𝑖 ≤ 𝑗, we then obtain (𝐶𝒜𝑖 (𝑎)) 𝑖 ≤ (𝐶𝒜𝑗 (𝑎)) 𝑗 , (𝐶𝒜𝑖 (𝑎 2)) 𝑖 ≤ (𝐶𝒜𝑗 (𝑎 2)) 𝑗 , … , (𝐶𝒜𝑖 (𝑎 𝑛)) 𝑖 ≤ (𝐶𝒜𝑗 (𝑎 𝑛 )) 𝑗 . so 𝒜𝑖 ⊆ 𝒜𝑗 for any 𝑖, 𝑗 ∈ ℤ+ with 𝑖 ≤ 𝑗, which means that 𝒜 ⊆ 𝒜2 ⊆ 𝒜3 ⊆ ⋯ ⊆ 𝒜𝑛 ⊆ ⋯ . finally, we have ℰ = ⋂ 𝒜𝑛∞𝑛=1 , which is immediate from proposition 3.5 since lim 𝑛⟶∞ 𝐶𝒜 (𝑎 𝑛) = { 𝑡0, 𝑖𝑓 𝑎 = 𝑒, 0, 𝑖𝑓 𝑎 ≠ 𝑒. this completes the proof for the required relations. corollary 3.1 let 〈𝒜〉 = {𝒜, 𝒜2, 𝒜3, … , 𝒜𝑚, … , ℰ}. then 𝒜 < 𝒜2 < 𝒜3 < ⋯ < 𝒜𝑚 < ⋯ < ℰ. proof. the proof is similar to proposition 3.6. proposition 3.7 let 𝜑 be a group homomorphism of a cyclic multigroup 𝒜. then the image of 𝒜 under 𝜑 is a cyclic multigroup. proof. it is well known that in the theory of classical cyclic groups, the image of any cyclic group is a cyclic group and the homomorphic image of a multigroup is a multigroup (from proposition 2.3). from these two results and definition 2.8, it is clearly seen that the image of 𝒜 under 𝜑 is a cyclic multigroup. proposition 3.8 let 𝒳𝑛 be the 𝑛 − level set of the cyclic group 𝒳. if 𝑖, 𝑗 ∈ ℤ + such that 𝑖 < 𝑗, then 𝒜𝑛 𝑖 is a subgroup of 𝒜𝑛 𝑗 . proof. it is obvious that sets 𝒳𝑛 and 𝒳𝑛 𝑚 are cyclic subgroups of 𝒳𝑛 in crisp sense. since 𝑖 < 𝑗, then 𝒜𝑛 𝑗 (𝑎) ≥ 𝒜𝑛 𝑖 (𝑎) ≥ 𝑛, ∀ 𝑎 ∈ 𝒳𝑛 𝑗 . thus, 𝒳𝑛 𝑖 ⊆ 𝒳𝑛 𝑗 . therefore, 𝒳𝑛 𝑖 is a subgroup of 𝒳𝑛 𝑗 . remark 3.2 from propositions 3.6 and 3.8, we have that a normal series of 𝒳 is a finite sequence 𝒳𝑛 𝑚, 𝒳𝑛 𝑚−1, … , 𝒳𝑛 of normal level subgroups of 𝒳 such that 𝒳𝑛 𝑚 > 𝒳𝑛 𝑚−1 > ⋯ > 𝒳𝑛 since 𝒳 is a cyclic group. on cyclic multigroup family 67 proposition 3.9 let {𝒜𝑚, 𝒜𝑚−1, … , 𝒜} be a finite cyclic multigroup family. then 𝒜𝑚 × 𝒜𝑚−1 × … × 𝒜 = 𝒜𝑚. proof. it is easily verified using the definition of product of multigroups and proposition 3.6. 4 conclusion the paper introduced the concept of cyclic multigroup family and investigated its related structure properties. for future studies, one can extend this idea to other non-classical algebraic structures such as soft group, rough group, neutrosophic group and smooth group. references [1] s. k. nazmul, p. majumdar and s. k. samanta. on multisets and multigroups. ann. of fuzzy math. and inform., 6, 643-656, 2013. 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[9] m. dresher and o. ore. theory of multigroups. american journal of mathematics, 60, 705-733, 1938. j. a. awolola 68 [10] p. a. ejegwa and a. m. ibrahim. some homomorphic properties of multigroups. buletinul academiei de stiinte a republicii moldova, 81, 67-76, 2017. [11] a. m. ibrahim and p. a. ejegwa. multigroup action on multiset. ann. fuzzy math. inform., 14(5), 515-526, 2017. [12] b. m. schein. multigroup. journal of algebra, 111, 114-132, 1987. ratio mathematica issue n. 30 (2016) pp. 59-66 issn (print): 1592-7415 issn (online): 2282-8214 the sum of the series of reciprocals of the quadratic polynomial with different negative integer roots radovan potůček department of mathematics and physics, faculty of military technology, university of defence, brno, czech republic radovan.potucek@unob.cz abstract this contribution, which is a follow-up to author’s paper [1] and [2] dealing with the sums of the series of reciprocals of some quadratic polynomials, deals with the series of reciprocals of the quadratic polynomials with different negative integer roots. we derive the formula for the sum of this series and verify it by some examples evaluated using the basic programming language of the cas maple 16. keywords: sequence of partial sums, telescoping series, harmonic number, computer algebra system maple. 2010 ams subject classifications: 40a05, 65b10 doi: 10.23755/rm.v30i1.12 1 introduction and basic notions let us recall the basic terms, concepts and notions. for any sequence {ak} of numbers the associated series is defined as the sum ∞∑ k=1 ak = a1 + a2 + a3 + · · · . 59 radovan potůček the sequence of partial sums {sn} associated to a series ∞∑ k=1 ak is defined for each n as the sum of the sequence {ak} from a1 to an, i.e. sn = n∑ k=1 ak = a1 + a2 + · · ·+ an . the series ∞∑ k=1 ak converges to a limit s if and only if the sequence of partial sums {sn} converges to s, i.e. lim n→∞ sn = s. we say that the series ∞∑ k=1 ak has a sum s and write ∞∑ k=1 ak = s. the telescoping series is any series where nearly every term cancels with a preceding or following term, so its partial sums eventually only have a fixed number of terms after cancellation. telescoping series are not very common in mathematics but are interesting to study. the method of changing series whose terms are rational functions into telescoping series is that of transforming the rational functions by the method of partial fractions. for example, the series ∞∑ k=1 1 k2 + k has the general nth term an = 1 n(n + 1) = a n + b n + 1 . after removing the fractions we get the equation 1 = a(n + 1) + bn. solving it for a and b we obtain an = 1/n−1/(n + 1). after that we arrange the terms of the nth partial sum sn = a1 + a2 + · · · + an in a form where can be seen what is cancelling. then we find the limit of the sequence of the partial sums sn in order to find the sum s of the infinite telescoping series as s = lim n→∞ sn. in our case we get sn = ( 1 1 − 1 2 ) + ( 1 2 − 1 3 ) +· · ·+ ( 1 n−1 − 1 n ) + ( 1 n − 1 n + 1 ) = 1− 1 n + 1 . so we have s = lim n→∞ ( 1− 1 n + 1 ) = 1. the nth harmonic number is the sum of the reciprocals of the first n natural numbers: hn = 1 + 1 2 + 1 3 + · · ·+ 1 n = n∑ k=1 1 k . 60 the sum of the series of reciprocals of the quadratic polynomial the values of the sequence {hn− lnn} decrease monotonically towards the limit γ . = 0.57721566, which is so-called the euler-mascheroni constant. basic information about harmonic numbers can be found e.g. in the web-sites [3] or [4], interesting information are included e.g. in the paper [5]. first ten values of the harmonic numbers are presented in this table: n 1 2 3 4 5 6 7 8 9 10 hn 1 3 2 11 6 25 12 137 60 49 20 363 140 761 280 7129 2520 7381 2520 2 the sum of the series of reciprocals of the quadratic polynomial with different negative integer roots now, we deal with the series formed by reciprocals of the normalized quadratic polynomial (k − a)(k − b), where a < b < 0 are integers. let us consider the series ∞∑ k=1 1 (k −a)(k − b) , and determine its sum s(a,b). we express the nth term an of this series as the sum of two partial fractions an = 1 (n−a)(n− b) = a n−a + b n− b . from the equality of two linear polynomials 1 = a(n− b) + b(n−a) for n = a we get a = 1/(a−b) and for n = b we get b = 1/(b−a) = −1/(a−b). so we have an = 1 a− b ( 1 n−a − 1 n− b ) = 1 b−a ( 1 n− b − 1 n−a ) . (1) for the nth partial sum of the given series so we get sn = 1 b−a [( 1 1− b − 1 1−a ) + ( 1 2− b − 1 2−a ) + · · · · · ·+ ( 1 n−1− b − 1 n−1−a ) + ( 1 n− b − 1 n−a )] . the first terms that cancel each other will be obviously the terms for which for the suitable index ` it holds 1/(1−a) = 1/(`−b). therefore the last term from the beginning of the expression of the nth partial sum sn, which will not cancel, will 61 radovan potůček be the term 1/(−a), so that the first terms from the beginning of the expression the sum sn, which will not cancel, will be the terms generating the sum 1 1− b + 1 2− b + · · ·+ 1 −a . analogously, the last terms that cancel each other will be the terms for which for the suitable index m it holds 1/(n − b) = 1/(m − a). therefore the first term from the ending of the expression of the nth partial sum sn, which will not cancel, will be the term 1/(n + 1 − b), so that the last terms from the ending in the expression of the sum sn, which will not cancel, will be the terms generating the sum − 1 n + 1− b − 1 n + 2− b −···− 1 n−a . after cancelling all the inside terms with the opposite signs we get the nth partial sum sn = 1 b−a ( 1 1− b + 1 2− b +· · ·+ 1 −a − 1 n + 1− b − 1 n + 2− b −···− 1 n−a ) . because for integer c it holds lim n→∞ 1 n + c = 0, then the searched sum, where a < b < 0, is s(a,b) = lim n→∞ sn = 1 b−a ( 1 1− b + 1 2− b + · · ·+ 1 −a ) = = 1 b−a [ 1 1 + 1 2 + · · ·+ 1 −a − ( 1 1 + 1 2 + · · ·+ 1 −b )] , so we get theorem 2.1. the series ∞∑ k=1 1 (k −a)(k − b) , where a < b < 0 are integers, has the sum s(a,b) = 1 b−a ( h−a −h−b ) , (2) where hn is the nth harmonic number. corolary 2.1. for the sum s(a,b) above it obviously hold: 1. s(a,b) = s(b,a), 2. s(a,a + 1) = h−a −h−a−1 = 1 −a , 62 the sum of the series of reciprocals of the quadratic polynomial 3. s(a,a+i) = 1 i ( h−a−h−a−i ) = 1 i ( 1 −a− i + 1 + 1 −a− i + 2 +· · ·+ 1 −a ) , i ∈ n. remark 2.1. let us note, that the formula (2) holds also in the case a < b = 0. because h0 is defined as 0, it has the form s(a,0) = 1 0−a ( h−a −h0 ) = h−a −a . (3) example 2.1. the series ∞∑ k=1 1( k − (−5) )( k − (−2) ) = ∞∑ k=1 1 (k + 2)(k + 5) , where a = −5, b = −2, has the nth partial sum sn = 1 3 ( 1 3 + 1 4 + 1 5 − 1 n + 3 − 1 n + 4 − 1 n + 5 ) . by the relation s(−5,−2) = lim n→∞ sn, since lim n→∞ 1 n + c = 0, or by theorem 2.1 we get its sum s(−5,−2) = 1 3 ( 1 3 + 1 4 + 1 5 ) = 1 3 ( h5 −h2 ) = 1 3 ( 137 60 − 3 2 ) = 47 180 = 0.261. example 2.2. the series ∞∑ k=1 1( k − (−5) ) k = ∞∑ k=1 1 k(k + 5) , where a = −5, b = 0, has the nth partial sum sn = 1 5 ( 1 1 + 1 2 + 1 3 + 1 4 + 1 5 − 1 n + 1 − 1 n + 2 − 1 n + 3 − 1 n + 4 − 1 n + 5 ) . by the relation s(−5,0) = lim n→∞ sn, since lim n→∞ 1 n + c = 0, or by theorem 2.1, or by the remark 2.1 we get its sum s(−5,0) = 1 5 ( 1 1 + 1 2 + 1 3 + 1 4 + 1 5 ) = h5 5 = 137/60 5 = 137 300 = 0.456 . 63 radovan potůček 3 numerical verification we solve the problem to determine the values of the sum s(a,b) = ∞∑ k=1 1 (k −a)(k − b) for a = −1,−2, . . . ,−9 and b = a + 1,a + 2, . . . ,−8. we use on the one hand an approximative direct evaluation of the sum s(a,b,t) = t∑ k=1 1 (k −a)(k − b) , where t = 108, using the basic programming language of the computer algebra system maple 16, and on the other hand the formula (2) for evaluation the sum s(a,b). we compare 45 = 9 + 8 + · · · + 1 pairs of these ways obtained sums s(a,b,108) and s(a,b) to verify the formula (2). we use following simple procedures hnum, rp2abneg and two for statements: hnum:=proc(h) local i,s; s:=0; if h=0 then s:=0 else for i from 1 to h do s:=s+1/i; end do; end if; end proc: rp2abneg:=proc(a,b,n) local k,sab,sumab; sumab:=0; sab:=(hnum(-a)-hnum(-b))/(b-a); print("n=",n,"s(",a,b,")=",evalf[20](sab)); for k from 1 to n do sumab:=sumab+1/((k-a)*(k-b)); end do; print("sum(",a,b,")=",evalf[20](sumab), "diff=",evalf[20](abs(sumab-sab))); end proc: for i from -1 by -1 to -9 do for j from i+1 by -1 to -8 do rp2abneg(i,j,100000000); end do; end do; 64 the sum of the series of reciprocals of the quadratic polynomial the approximative values of the sums s(a,b) rounded to 3 decimals obtained by these procedures are written into the following table: s(a, b) a =−1 a=−2 a=−3 a=−4 a=−5 a=−6 a=−7 a=−8 a=−9 b = 0 1.000 0.750 0.611 0.521 0.457 0.408 0.370 0.340 0.314 b=−1 × 0.500 0.417 0.361 0.321 0.290 0.266 0.245 0.229 b=−2 × × 0.333 0.292 0.261 0.238 0.219 0.203 0.190 b=−3 × × × 0.250 0.225 0.206 0.190 0.177 0.166 b=−4 × × × × 0.2000 0.183 0.170 0.159 0.149 b=−5 × × × × × 0.167 0.155 0.145 0.136 b=−6 × × × × × × 0.143 0.134 0.126 b=−7 × × × × × × × 0.125 0.118 b=−8 × × × × × × × × 0.111 computation of 45 couples of the sums s(a,b,108) and s(a,b) took over 18 minutes. the absolute errors, i.e. the differences ∣∣s(a,b)−s(a,b,108)∣∣, have here place value about 10−8. 4 conclusion we dealt with the sum of the series of reciprocals of the quadratic polynomials with different negative integer roots a and b, i.e. with the series ∞∑ k=1 1 (k −a)(k − b) , where a < b < 0 are integers. we derived that the sum s(a,b) of this series is given by the formula s(a,b) = 1 b−a ( h−a −h−b ) , where hn is the nth harmonic number. we verified this result by computing 45 various sums by using the cas maple 16. we also stated that this formula holds also for a < b = 0, when it has the form s(a,0) = 1 0−a ( h−a −h0 ) = h−a −a . the series of reciprocals of the quadratic polynomials with different negative integer roots so belong to special types of infinite series, such as geometric and telescoping series, which sums are given analytically by means of a simple formula. 65 radovan potůček references [1] r. potůček, the sums of the series of reciprocals of some quadratic polynomials. in: proceedings of afases 2010, 12th international conference ”scientific research and education in the air force” (cd-rom). brasov, romania, 2010, p. 1206-1209. isbn 978-973-8415-76-8. [2] r. potůček, the sum of the series of reciprocals of the quadratic polynomials with double non-positive integer root. in: proceedings of the 15th conference on applied mathematics aplimat 2016. faculty of mechanical engineering, slovak university of technology in bratislava, 2016, p. 919-925. isbn 978-80-227-4531-4. [3] wikipedia contributors: harmonic number. wikipedia, the free encyclopedia, [online], [cit. 2016-09-01]. available from: https://en.wikipedia.org/wiki/harmonic number. [4] e. w. weisstein, harmonic number. from mathworld – a wolfram web resource, [online], [cit. 2016-09-01]. available from: http://mathworld.wolfram.com/harmonicnumber.html [5] a. t. benjamin, g. o. preston, and j. j. quinn, a stirling encounter with harmonic numbers. mathematics magazine 75 (2), 2002, p. 95 –103, [online], [cit. 2016-09-01]. available from: https://www.math.hmc.edu/∼benjamin/papers/harmonic.pdf 66 ratio mathematica volume 38, 2020, pp. 329-339 on the planarity of line mycielskian graph of a graph keerthi g. mirajkar* anuradha v. deshpande† abstract the line mycielskian graph of a graph g, denoted by lµ(g) is defined as the graph obtained from l(g) by adding q + 1 new vertices e ′ = e ′ i : 1 ≤ i ≤ q and e, then for 1 ≤ i ≤ q, joining e ′ i to the neighbours of ei and to e. the vertex e is called the root of lµ(g). this research paper deals with the characterization of planarity of line mycielskian graph lµ(g) of a graph. further, we also obtain the characterization on outerplanar, maximal planar, maximal outerplanar, minimally nonouterplanar and 1-planar of lµ(g). keywords: planar graph, outerplanar, maximal planar, maximal outerplanar, minimally nonouterplanar and 1-planar. 2010 ams subject classifications: 05c07, 05c10, 05c38, 05c60, 05c76. 1 *department of mathematics, karnatak university’s karnatak arts college, dharwad 580001, karnataka, india; keerthi.mirajkar@gmail.com †department of mathematics, karnatak university’s karnatak arts college, dharwad 580001, karnataka, india; anudesh08@gmail.com. 1received on april 30th, 2020. accepted on june 19th, 2020. published on june 30th, 2020. doi: 10.23755/rm.v38i0.506. issn: 1592-7415. eissn: 2282-8214. ©keerthi g. mirajkar et al. this paper is published under the cc-by licence agreement. 329 keerthi g. mirajkar, anuradha v. deshpande 1 introduction all graphs considered in this paper are finite, undirected and without loops. the phraseologies are referred from [harary, 1969]. the graph g is said to be embedded in a surface s when its vertices are represented by points in s and each edge by a curve joining corresponding points in s, in such a way that no curve intersects itself and two curves intersect each other only at a common vertex. a graph g is said to be planar if it can be embedded in the plane. a plane representation of a planar graph divides the plane into number of plane areas called regions or faces. the regions enclosed by the planar graph are called interior faces of the graph. the region surrounding the planar graph is called the exterior face of the graph. a planar graph g is called maximal planar if the addition of any edge to g creates a nonplanar graph. a maximal planar graph is a planar graph in which every face (including the exterior face) is bounded by a triangle [dillencourt, 1991]. a planar graph is called outerplanar if it can be embedded in the plane so that all its points lie on the same face. an outerplanar graph is called maximal outerplanar if no line can be added without losing outerplanarity. the inner vertex number i(g) of a planar graph g is the minimum number of vertices not belonging to the boundary of the exterior region in any embedding of g in the plane. a graph g is said to be k minimally nonouterplanar if i(g) = k,k ≥ 1. an 1-minimally nonouterplnar graph is called minimally nonouterplanar [kulli and basavanagoud, 2004]. two graphs are said to be homeomorphic if one graph can be obtained from the other by insertion of vertex of degree two into its edges or by the merger of adjacent edges, where the incident vertex is of degree two. crossing number cr(g) of a graph g is the minimum number of crossings (of its edges) among the drawings of g in the plane. [ringel, 1965] introduced the concept of 1-planarity. a graph g is called 1-planar if it can be drawn in the plane so that all or any edge is crossed by at most one other edge. the line mycielskian graph of a graph g, denoted by lµ(g) is defined as the graph obtained from l(g) by adding q + 1 new vertices e ′ = e ′ i : 1 ≤ i ≤ q and e, then for 1 ≤ i ≤ q, joining e′i to the neighbours of ei and to e. the vertex e is called the root of lµ(g) [mirajkar and mathad, 2019]. 330 on the planarity of line mycielskian graph of a graph figure 1. c3, k1,3 and their line mycielskian graphs lµ(c1,3) and lµ(k1,3) motivated by the the research work [mirajkar and mathad, 2019], the present problem is initiated. further it is extended with the objective of obtaining the characterization results on planarity, outerplanar, maximal planar, maximal outerplanar, minimal nonouterplanar and 1-planar of lµ(g). 2 prelimnaries the following important theorems and remark are used for proving further results. theorem 2.1. [kuratowski, 1930] a graph is planar if and only if it has no subgraph homeomorphic to k5 or k3,3. theorem 2.2. [harary, 1969] a graph is outerplanar if and only if it has no subgraph homeomorphic to k4 or k2,3 except k4 − x. theorem 2.3. [czap and hudák, 2012] the complete graph kα1 , α1 ≤ 6 is 1planar. theorem 2.4. [mirajkar et al., 2019] the line mycielskian graph lµ(g) of a graph g is disconnected iff g = k2. remark 2.1. [mirajkar and mathad, 2019] l(g) is subgraph of lµ(g). 331 keerthi g. mirajkar, anuradha v. deshpande 3 results theorem 3.1. the line mycielskian graph lµ(g) is planar if and only if the graph g is cn, n = 3 or c3. proof. suppose lµ(g) is planar and g = cn. we consider the following cases. case 1. suppose n = 5, then g = c5. l(c5) is c5 and by remark 2.1, l(g) is subgraph of lµ(g). the construction of lµ(g) with five newly introduced vertices e ′ 1,e ′ 2,e ′ 3, e ′ 4 and e ′ 5 corresponding to vertices of l(c5) and root vertex e produces five mutually adjacent vertices with degree four which is a subgraph homeomorphic to k5. by theorm 2.1, a contradiction. case 2. suppose n = 4, then g = c4. l(c4) is c4 and by remark 2.1, l(g) is subgraph of lµ(g). here lµ(g) is constructed by introducing new vertices e ′ 1,e ′ 2,e ′ 3, and e ′ 4 corresponding to the vertices e1,e2,e3, and e4 of l(c4) and root vertex e. newly introduced vertices are connected with the vertices e1,e2,e3 and e4 of c4 in such a way that the, e ′ 1 and e ′ 3 are adjacent to the two opposite vertices e2 and e4 and e ′ 2 and e ′ 4 are adjacent to e1 and e3. root vertex e is connected to e ′ 1,e ′ 2,e ′ 3, e ′ 4. this construction produces the five mutually adjacent vertices with degree four and thus contains a subgraph homeomorphic to k5, a contradiction (from theorem 2.1). case 3. suppose n = 3, then g = c3. l(c3) is c3 by remark 2.1, l(g) is subgraph of lµ(g). to construct line mycielskian graph lµ(g), three new vertices e ′ 1,e ′ 2 and e ′ 3 corresponding to the edges of g and root vertex e are introduced. the edges between the vertices are drawn in such a way that the newly introduced vertices e ′ 1,e ′ 2 and e ′ 3 are adjacent to the corresponding adjacent edges of e1,e2 and e3 of g respectively. the root vertex e joins the vertices e ′ 1,e ′ 2 and e ′ 3 such that no crossing of edges occur as shown in figure 1. this implies that line graph lµ(g) is planar for g = c3. from the above cases, it is observed that lµ(g), for all g = cn, n ≥ 4, contains a subgraph homeomorphic to k5, a contradiction. thus lµ(g) is planar only if g is cn, n = 3. conversely, suppose g = c3, then the construction of lµ(g) as discussed in above case 3 which results into planar graph. theorem 3.2. the line mycielskian graph lµ(g) is planar if and only if the graph g is a path graph pn, n ≥ 3. proof. suppose lµ(g) is planar and g = pn. 332 on the planarity of line mycielskian graph of a graph we consider the following cases. case 1. suppose n = 2, then g = p2, then from theorem 2.4, lµ(g) is disconnected graph, a contradiction. case 2. suppose g = pn,n ≥ 3. by remark 2.1, l(g) is subgraph of lµ(g). l(pn) is pn−1. figure 2. p5 and its line mycielskian graph lµ(p5) for the construction of lµ(g), new vertices e ′ 1,e ′ 2,.., e ′ n−1 corresponding to edges of g and root vertex e are introduced. in lµ(g), the edges between the vertices are connected in such a way that, the newly introduced vertices e ′ 1 and e ′ n−1 are adjacent to e2 and en−2 of l(g) respectively and the other vertices e ′ 2,e ′ 3,.. , e ′ n−2 are adjacent to two vertices ei−1 and ei+1, i = 2, 3, .., (n − 2). the root vertex e is adjacent to the vertices e ′ 1,e ′ 2,.., e ′ n−1. all the faces are polygons without any crossings as shown in figure 2. i.e., lµ(g) is planar. it is clear from the above two cases that lµ(g) is planar only for g = pn,n ≥ 3. conversely, suppose g = pn, n ≥ 3. then the construction of lµ(g) results into planar graph (as discussed above). theorem 3.3. the line mycielskian graph lµ(g) of a graph g is planar if and only if one of the follwing conditions hold (i) ∆(g) = 2, except for cn, n ≥ 4 (ii) g = k1,3 proof. suppose lµ(g) of g is planar. we discuss the following cases based on the maximum degree of g. 333 keerthi g. mirajkar, anuradha v. deshpande case 1. suppose g is a graph with ∆(g) = 5. by remark 2.1, l(g) is subgraph of lµ(g). then the line mycielskian graph lµ(g) contains k5 as its subgraph. by theorem 2.1, lµ(g) is nonplanar, a contradiction. case 2. next suppose g is a graph with ∆(g) = 4. by remark 2.1, l(g) is subgraph of lµ(g) and contains k4 as its subgraph. lµ(g) is obtained by introducing new vertices e ′ 1,e ′ 2,e ′ 3 and e ′ 4 corresponding to the edges of subgraph k4 of l(g) and root vertex e by the definition. vertices of k4 and newly introduced vertices are connected. the edge from root vertex e is drawn in such a way that it is adjacent to the vertices e ′ 1,e ′ 2,e ′ 3 and e ′ 4. this construction produces subgraph k4 with each vertex degree ≥ 4. one of the newly introduced vertices of is lµ(g) is adjacent to the three vertices of k4 with path length 1. it is also adjacent to the remaining one vertex of k4 with path length 2. this produces the subgraph homeomorphic to k5, a contradiction. case 3. suppose ∆(g) = 3, g contains k1,3 as its subgraph. by remark 2.1, l(g) is subgraph of lµ(g). since l(k1,3) is c3, the construction of lµ(k1,3) is same as lµ(c3) as shown in figure 1. from theorem 3.1, lµ(c3) is planar. thus lµ(k1,3) is also planar. further in the construction of lµ(k1,3), presence of additional edge in k1,3 to any vertex leads to increase in the number of vertices and edges (2 vertices and 3 edges) in lµ(g) and thus contains k4. on constructing lµ(g), newly introduced vertices, root vertex and vertices of k4 produces five mutually adjacent vertices with degree 4 which is a subgraph homeomorphic to k5. by theorem 2.1, lµ(g) is nonplanar, a contradiction. case 4. suppose ∆(g) = 2, obiviously g is either path graph pn or cycle cn. by theorem 3.1 and theorem 3.2, lµ(cn), n = 3 and lµ(pn), n ≥ 3 are planar. from all the above cases, it is noted that, lµ(g) contains subgraph homeomorphic to k5 for ∆(g) ≥ 3, a contradiction and is planar only for ∆(g) = 2. converse is obivious. theorem 3.4. the line mycielskian graph lµ(g) is outerplanar if and only if g is p3. proof. suppose lµ(g) is outerplanar. then lµ(g) is planar. by theorems 3.1, 3.2 and 3.3, lµ(g) is planar only for the graphs c3,k1,3 and pn, n ≥ 3. suppose g=c3 or k1,3. from theorem 3.1 and figure 1, lµ(g) contains a subgraph homeomorphic to k2,3. from theorem 2.2, lµ(g) is not outerplanar, a contradiction. assume g = p4. line graph of p4 is p3. by remark 2.1, l(g) is subgraph of lµ(g). on constructing lµ(g) with newly introduced vertices and root vertex (as explained in case 2 of theroem 3.2) forms a subgraph homeomorphic to k2,3. by theorem 2.2 lµ(p4) is not outerplanar. which is contradiction. i.e., g cannot be p4. 334 on the planarity of line mycielskian graph of a graph similarly, for higher values of n i.e., for g = p4, n ≥ 5, the same construction (case 2 of theorem 3.2) repeates and generates a subgraph with every four vertices of lµ(g) which is homeomorphic to k2,3, a contradiction. next assume g = p3. line graph of p3 is p2 and by remark 2.1, l(g) is subgraph of lµ(g). lµ(g) is constructed by introducing root vertex e and new vertices e ′ 1 and e ′ 2 corresponding to the edges of e1 and e2 of g respectively. the edges from new vertices are drawn in such a way that e ′ 1 is adjacent to e2 and e ′ 2 to e1 which forms c5, as shown in figure 3, which is outerplanar. therefore, g = p3. by observing the above cases , it can be stated that lµ(g) for g = pn, n ≥ 4 contains subgraph homeomorphic to k2,3, a contradiction and hence is outerplanar only for g = pn, n = 3. figure 3. p3 and its line mycielskian graph lµ(p3) conversely, suppose g = p3. from theorem 3.2, lµ(g) is planar for g=pn, n ≥ 3. i.e., lµ(p3) produces c5 and is planar (as discussed above ). clearly which is outerplanar. theorem 3.5. the line mycielskian graph lµ(g) is maximal planar if and only if g is c3 or k1,3 or pn, n ≥ 4. proof. suppose lµ(g) is maximal planar, then it is planar. from theorem 3.1, theorem 3.2 and theorem 3.3, lµ(g) is planar if g is c3 or k1,3 or pn, n ≥ 4 . suppose g is c3 or k1,3. on the construction of lµ(c3) or lµ(k1,3) as shown in figure 1, it can be observed that, three of its interior faces are triangles and by the definition of homeomorphic graph, the other two faces can be triangularised by merging adjacent edges if the incident vertex is of degree two. i.e., lµ(g) has triangulation plane. lµ(g) is maximal planar. 335 keerthi g. mirajkar, anuradha v. deshpande suppose g is pn, n ≥ 3. from theorem 3.2, lµ(g) is planar. we consider the following cases. suppose g = p3. from theorem 3.4, the construction of lµ(p3) forms c5 and does not contain any triangle faces as shown in figure 3. thus lµ(p3) is not maximal planar, a contradiction. next, suppose g = pn, n ≥ 4. the construction of lµ(g) contains some of its interior faces as c4’s and some as c5’s. by definition of homeomorphic graph, all the faces can be triangularised by merging the adjacent edges. thus lµ(g) has triangulation plane and is maximal planar for g = pn, n ≥ 4. conversely, suppose g = c3 or k1,3 or pn, n ≥ 4. from theorem 3.1, theorem 3.3 and theorem 3.2, lµ(g) is planar. let us first consider g = c3 or k1,3. three interior faces of lµ(g) are triangles and two faces are c4’s and by the definition of homeomorphic graph, they can be triangularized by merging adjacent edges. i.e., lµ(g) has triangulation plane. hence, lµ(g) is maximal planar. next suppose g = pn, n ≥ 4. from figure 2, all the faces of lµ(g) can be triangularized by merging adjacent edges and thus lµ(g) has triangulation plane and it is maximal planar. 2 theorem 3.6. for any graph g, the line mycielskian graph lµ(g) of a graph g is not maximal outerplanar. proof. suppose lµ(g) is maximal outerplanar, then it is outerplanar. from theorem 3.4, lµ(g) is outerplanar only for g = p3. from figure 3, it is obvious that addition of an edge between any two non-adjacent vertices does not violate the property of planarity, a contradiction. 2 theorem 3.7. for any graph g, the line mycielskian graph lµ(g) of a graph g is not minimally nonouterplanar. proof. suppose lµ(g) is minimally nonouterplanar, then lµ(g) is planar. from theorem 3.1, theorem 3.2 and theorem 3.3, lµ(g) is planar only if g is cn, n = 3 or pn, n ≥ 3 or k1,3. we consider the following three cases. case 1. suppose g = cn, n = 3 or g = k1,3. from figure 1, lµ(g) contains at least three points belong to interior region, a contradiction. case 2. suppose g = pn, n ≥ 3. subcase 2.1 suppose n = 3 then g = p3. from figure 3, in lµ(g) all the points belong to exterior region. no point belong to interior region, a contradiction. 336 on the planarity of line mycielskian graph of a graph subcase 2.2. suppose g = pn, n ≥ 4, then lµ(g) is constructed with newly introduced vertices e ′ 1,e ′ 2,.., e ′ n−1 corresponding to edges of g and root vertex e (construction explained in case 2 of theorem 3.2).this construction generates subgraph with every four vertices of lµ(g) which is homeomorphic to k2,3. this process continues for the successive value of n in g and thus lµ(g) contains more than one subgraph homeomorphic to k2,3, a contradiction. in either cases, lµ(g) is not minimally nonouterplanar. 2 theorem 3.8. if the graph g is a cycle cn, n ≥ 4, then line mycielskian graph lµ(g) is 1-planar with n crossings . proof. let g be cn, n ≥ 4. by remark 2.1, l(g) is subgraph of lµ(g). from theorem 3.1, lµ(cn), n ≥ 4 is nonplanar. here lµ(g) is constructed by introducing new vertices e ′ 1,e ′ 2,e ′ 3, e ′ 4,.., e ′ n and root vertex e. root vertex, newly introduced vertices and vertices of l(cn) are connected as explained in the theorem 3.1. this construction produces the five mutually adjacent vertices with degree ≥ 4 and thus contains a subgraph homeomorphic to k5. from theorem 2.3, lµ(g) is 1-planar. figure 4. c4 and its line mycielskian graph lµ(c4) in lµ(g), every crossing arises from the edges drawn between the vertices l(cn) and newly introduced vertices. each newly introduced vertex is adjacent with two vertices of l(cn) and each crosing occurs from the two adjacent vertices of l(cn). as there are n vertices in l(cn), the number of crossings are n. 2 337 keerthi g. mirajkar, anuradha v. deshpande theorem 3.9. if g is k1,3•k2, then the line mycielskian graph lµ(g) is 1-planar with crossing number 1. proof . let g be k1,3 • k2. by remark 2.1, l(g) is subgraph of lµ(g). from theorem 3.3, lµ(k1,3) is planar. but inclusion of k2 to k1,3 increases number of vertices and edges in lµ(g) for which the construction is explained in theorem 3.3 case 3. this construction produces five mutually adjacent vertices with degree 4. thus lµ(g) contains a subgraph homeomorphic to k5. from theorem 2.3, lµ(g) is 1-planar. figure 5. k1,3 • k2 and its line mycielskian graph lµ((k1,3 • k2) in graph g = k1,3 • k2) shown in figure 5, the number of edges are four. since only one vertex e3 of g is adjacent to three edges, in lµ(g) the degree of newly introduced vertex corresponding to this edge is four as shown in figure 5, where as the degree of other newly introduced vertices e ′ 1,e ′ 2, and e ′ 4 is ≤ 3. thus, this results the graph lµ(g) to have only one crossing from the edges drawn between the root vertex e and newly introduced vertices e ′ 2,e ′ 3 and e ′ 4. 338 on the planarity of line mycielskian graph of a graph 4 conclusion in this research paper we obtained the characterization results on planarity, outerplanar, maximal planar, maximal outerplanar, minimal nonouterplanar and 1-planar of line mycielskian graph lµ(g). the results revealed that lµ(g) is planar only for the graphs c3, pn,n ≥ 3, k1,3 and maximal planar if g is c3, pn,n ≥ 4 and k1,3. it is outerplanar only if g= p3.there is no existance of any graph whose lµ(g) is maximal outerplanar and not minimally nonouterplanar. further we also obtained lµ(g) for 1-planar when g= cn, n ≥ 4 and (k1,3 •k2) with n crossings and 1 crossing respectively. future line of this research can be extended to study some more properties of planarity such as crossing numbers, genus, thickness and coarseness of line mycielskian graph of a graph. references j. czap and d. hudák. 1-planarity of complete multipartite graphs. discrete applied mathematics, 160(4-5):505–512, 2012. m. b dillencourt. an upper bound on the shortness exponent of 1-tough, maximal planar graphs. discrete mathematics, 90(1):93–97, 1991. f harary. graph theory, addison-wesley, reading, mass. 1969. v kulli and b basavanagoud. characterizations of planar plick graphs. discussiones mathematicae graph theory, 24(1):41–45, 2004. c. kuratowski. sur le probleme des courbes gauches en topologie. fundamenta mathematicae, 15(1):271–283, 1930. k. g. mirajkar and veena mathad. the line mycielskian graph of a graph. international journal of research and analytical reviews(ijrar), 6(1):277–280, 2019. k. g. mirajkar, veena mathad, and pooja b. miscellaneous properties of line mycielskian graph of a graph. international journal of research and analytical reviews(ijrar), 14(29):4552–4556, 2019. g. ringel. ein sechsfarbenproblem auf der kugel. in abhandlungen aus dem mathematischen seminar der universität hamburg, volume 29, pages 107– 117. springer, 1965. 339 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 39, 2020, pp. 147-163 147 frattini submultigroups of multigroups joseph achile otuwe* musa adeku ibrahim† abstract in this paper, we introduce and study maximal submultigroups and present some of its algebraic properties. frattini submultigroups as an extension of frattini subgroups is considered. a few submultigroups results on the new concepts in connection to normal, characteristic, commutator, abelian and center of a multigroup are established and the ideas of generating sets, fully and non-fully frattini multigroups are presented with some significant results. keywords: maximal, cyclic multigroup, commutator and generating set. 2010 ams subject classification: 55u10.‡ * department of mathematics, college of science (ahmadu bello university, zaria, nigeria), talk2josephotuwe2016@gmail.com. † department of mathematics, college of science (ahmadu bello university, zaria, nigeria), amibrahim@abu.edu.ng. ‡ received on october 13th, 2020. accepted on december 19th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.534. issn: 1592-7415. eissn: 22828214. ©otuwe et al. et al. this paper is published under the cc-by licence agreement. j. a otuwe and m.a ibrahim. 148 1 introduction the term multigroup was first mentioned in [7] as an algebraic structure that satisfied all the axioms of group except that the binary operation is multivalued. this concept was later redefined in [19] via count function of multisets and some of its properties were vividly discussed. the idea of submultigroup and its classes were established in [14]. concept of maximal subgroups is established in [3], [5] and [17] and some of its properties were investigated. also, normal and characteristic submultigroup were introduced in [8] and [13] respectively, and some of its properties were presented. frattini in [11], introduced a special subgroup named frattini subgroup and some results were obtained. other related work on frattini subgroup can be found in [1], [2], [6], [12], [15], [16], [18], [20] and [21]. furthermore, in [10] frattini subgroup was represented but in fuzzy environment called frattini fuzzy subgroup. in this paper, we focus on multiset setting to obtain frattini submultigroups and finally establish some related results. in general, the union of submultigroups of a multigroup may not be a multigroup, we therefore establish some conditions under which the union of all maximal submultigroups is a multigroup. when this occur, the frattini submultigroup obtained from such maximal submultigroups is called “fully frattini” otherwise it is called “non-fully frattini”. furthermore, other relevant concepts such as; cyclic multigroup, minimal generating set of a multigroup, generator and non-generator of a multigroup are introduced with reference to frattini submultigroups. finally, we study some properties of center of a multigroup, normal, commutator, minimal and characteristic submultigroups. 2 preliminaries definition 2.1 (|23|). let be a set. a multiset over is just a pair , where is a set and is a function. any ordinary set is actually a multiset , where is its characteristic function. the set is called the ground or generic set of the class of all multisets containing objects from . definition 2.2 (|22|). let and be two multisets over , is called a submultiset of written as if for all . also, if and , then is called a proper submultiset of and denoted as . frattini submultigroups of multigroups 149 definition 2.3 (|22|). let and be two multisets over , then and are equal if and only if for all . two multisets and are comparable to each other if or . definition 2.4 (|23|). suppose that , such that and . i. their intersection denoted by is the multiset , where , . ii. their union denoted by is the multiset , where , . iii. their sum denoted by is the multiset , where , . definition 2.5 (|19|). let be a group and . is said to be a multigroup of 𝑋 if the count function of or satisfies the following two conditions: i. (𝑥𝑦) ≥ (𝑥) (𝑦)], ∀𝑥, 𝑦 ∈𝑋. ii. ( ) ≥ (𝑥), ∀ 𝑥 ∈𝑋, where is a function that takes to a natural number, and denotes minimum operation. the set of all multigroups defined over 𝑋 is denoted by (𝑋). definition 2.6 (|19|). let . then is defined by . thus, . definition 2.7 (|19|). let . then is said to be abelian or commutative if . definition 2.8 (|19|). let . then the sets and are defined as and , where is the identity element of . definition 2.9 (|19|). let , be an arbitrary family of multigroups of a group then j. a otuwe and m.a ibrahim. 150 definition 2.10 (|14|). let . then the center of is defined as . definition 2.11 (|9|) commutator of two submultigroup: let and be submultigroups of . then the commutator of and is the multiset of defined as follows: that is, . since the supremum of an empty set is zero. if is not a commutator. definition 2.11 (|4|). let . then the order of denoted by is defined as . i.e., the total numbers of all multiplicities of its element. definition 2.12 (|14|). let . a submultiset of is called a submultigroup of denoted by if is a multigroup. a submultigroup of is a proper submultigroup denoted by , if and definition 2.13 (|14|). let . then a submultigroup of is said to be complete if , incomplete if , regular complete if is complete and and regular incomplete if is incomplete and . definition 2.14 (|8|). let such that . then is called a normal submultigroup of if . definition 2.15 (|10|). let and be two groups and let be a homomorphism. suppose and are multigroups of and respectively, then induces a homomorphism from to which satisfies i. . frattini submultigroups of multigroups 151 ii. where i. the image of under denoted by , is a multiset of defined by for each ii. the inverse image of under denoted by , is a multiset of defined by . definition 2.16 (|10|). let and be groups and let and respectively. then a homomorphism from to is called an automorphism of onto if is both injective and surjective, that is, bijective. definition 2.17 (|13|). let such that . then is called a characteristic (fully invariant) submultigroup of if for every automorphism, of . that is, for every . 3 frattini submultigroups and their properties in this section we propose the concept of minimal, maximal, frattini, commutator submultigroups, cyclic, fully and non-fully frattini multigroup and generating set of a multigroup with some illustrative examples. definition 3.1 a. minimal submultigroup: let be a group and . then a non trivial proper submultigroup denoted by of is said to be minimal if there exists no other non-trivial submultigroup of such that . remark 3.1. every minimal complete submultigroup of a multigroup is unique. b. maximal submultigroup: let be a group and . then a proper normal submultigroup denoted by of is said to be maximal if there exists no other proper submultigroup of such that . j. a otuwe and m.a ibrahim. 152 c. frattini submultigroup: let be a group. suppose is a multigroup of and, , , , (or simply for ) are maximal submultigroups of . then the frattini submultigroup of denoted by is the intersection of defined by or simply by . remark 3.2. i. let be a non abelian group and . if is a normal submultigroup of with an incomplete maximal submultigroups and for each are the maximal subgroups of , then the maximal submultigroups of are submultigroups of . ii. let . if is a submultigroup of and is the frattini submultigroup of then, . d. commutator submultigroup of a multigroup: let such that the commutator subgroup of is given as . then the commutator submultigroup of denoted by is defined as e. let . then the sets and are defined as and , where is the identity element of . remark 3.3. i. the commutator submultigroup of every abelian multigroup is ii. let and be the commutator submultigroup of . then . remark 3.4. let such that and be the commutator submultigroup of and . then . f. cyclic multigroup: let be a group generated by . then a multigroup over is said to be a cyclic multigroup if such that . the element is then called the generator of otherwise, a non generator of . frattini submultigroups of multigroups 153 g. generating set of a multigroup: let be a cyclic group and . a subset of is said to be a generating set for if all elements of and its inverses can be expressed as a finite product of elements in with and for some . h. minimal generating set of a multigroup: let be a cyclic group and . a subset of is termed minimal generating set of if such that and there is no proper subset of with , . i. fully frattini multigroup: let . then is called fully frattini if the union of the maximal submultigroups equals . otherwise, it is called non-fully frattini. in addition, every multigroup without incomplete maximal submultigroup is called trivial fully frattini. 4. some results on frattini submultigroups in this section, we present some results on frattini submultigroup of multigroups. theorem 4.1. let with complete maximal submultigroups. then every minimal submultigroup of is a submultigroup of . proof. suppose is the frattini submultigroup of then has maximal submultigroup say such that . since is multigroup over , has a minimal submultigroup say . if is not a submultigroup of then there exist at least an element such that which contradicts the fact that is a minimal submultigroup of . hence is a submultigroup of theorem 4.2 if for all , then is characteristic in . proof. since is an automorphism, the inverse is also an automorphism of . hence we have . applying , we have . then we obtain . by this j. a otuwe and m.a ibrahim. 154 fact, equality holds and so . hence the frattini submultigroup is characteristic in . theorem 4.3 every frattini submultigroup of a multigroup is characteristic. proof. by theorem 4.2, it suffices to proof that for every automorphism . let . then there exists such that . to show that , we consider an arbitrary element . then since is an automorphism, we have . thus there exists in such that . we have (since is a homomorphism) (since ) (since is a homomorphism) since this is true for all it follows that , and thus . hence the result. theorem 4.4 every frattini submultigroup of a multigroup is abelian. proof. let and be the frattini submultigroup of . it follows that is a normal submultigroup of by definition 2.14 consequently, . thus, . hence, the result follows by definition 2.7 theorem 4.5 every is a normal submultigroup of . proof. let and be the frattini submultigroup of . then frattini submultigroups of multigroups 155 , since . now, let , then since is a multigroup over by definition 2.14, we get now we proof that is a normal submultigroup of . let , then it follows that . hence, the result by definition 2.14 theorem 4.6 let be a multigroup over a non-abelian group , then . proof. , since at least . let then for all , and . consequently, where since . thus . now, let then . hence, . thus, therefore, is a subgroup of . to show that is a normal subgroup of . let and then . j. a otuwe and m.a ibrahim. 156 thus, and . hence, remark 4.1 if is a multigroup over an abelian group , is the root set of frattini submultigroup of and is the center of then is a normal subgroup of . theorem 4.7. if is a multigroup over a non-abelian group and is a normal submultigroup of , then . proof. clearly, and are submultigroups of . let be the maximal submultigroups of and be the maximal submultigroups of for each and . then by remark 3.2 ( ) we have for each and . | therefore, . to show that , suppose then the result holds trivially. but if then for any element , for each and . therefore, . theorem 4.8 if is a regular multigroup over a group . then . proof. since is a multigroup over , then . let and then . thus . therefore is a submultigroup of . now by theorem 4.5, and clearly . so let and , then implies . hence . frattini submultigroups of multigroups 157 theorem 4.9 if , is the commutator submultigroup of and is the frattini submultigroup of . then . proof. since is a multigroup over , then . let and , then . thus . therefore is a submultigroup of . now, by theorem 4.5, . let and , then implies . hence, . theorem 4.10. every frattini submultigroup of a cyclic multigroup is abelian. proof. let be the frattini submultigroup of a cyclic multigroup over a cyclic group , then there exists such that we have and for . it now follows that . theorem 4.11. if is a regular multigroup with an incomplete maximal submultigroups over a cyclic group . then is contained in the set of all non-generators of . in particular, coincide with the set of all nongenerators if has only one maximal submultigroup. proof. let be a cyclic group and and denotes the frattini submultigroup of . let be the set of all generators of and be the incomplete maximal submultigroups of , then for all , in fact, all is a non-generator. further, and . since , we have that j. a otuwe and m.a ibrahim. 158 for all , . this implies that . but is the largest set containing all non generators. hence is contained in the set of all non-generators. suppose has only one nontrivial maximal submultigroup say then, and . since , therefore for all non-generators . hence, is indeed the set of all non-generators. theorem 4.12. if a regular multigroup over a cyclic group has two incomplete maximal submultigroups, then the union of its generators coincide with the non-generating set of . proof. let and be the maximal submultigroups of and be the collection of all generators of . clearly, and (since and does not contain any generator). now, can be expressed as if is odd and if is even with for any maximal submultigroup of . also, . that is, for odd values of and for any but since for even we have for any . hence the result. theorem 4.13. if a regular multigroup over a cyclic group has two maximal submultigroups, then the union of the non-generators coincide with the generating set of . proof. let and be the maximal submultigroups of and be the collection of all nongenerators of . clearly, and so generates . , where . frattini submultigroups of multigroups 159 taking every , (for some ) if (i.e., generates distinct elements in ). since we have that for some . more explicitly, where . this yields for some . theorem 4.14. if a regular multigroup over a cyclic group has two incomplete maximal submultigroups and is the set of generators of , then form one of the root set of the maximal submultigroup of for some . proof. let be a multigroup over a cyclic group, be the maximal submultigroups of and be the generators of . then, . since , for all , , and but, . therefore, . in particular, and for any . remark 4.2 a. a generator of any multigroup over a cyclic group is not contained in any of its maximal submultigroups. b. the set of non-generators of any multigroup may not be a submultigroup. theorem if is a minimal generating set of a multigroup over a cyclic group , then . j. a otuwe and m.a ibrahim. 160 proof. suppose , then , where is a maximal submultigroup of . now, since contains at least one generator of , then every contains at least one generator of which is a contradiction. hence, . remarks 4.3 i. if is a multigroup over a cyclic group . then the union of all the minimal generating sets of is equal to . ii. every minimal generating set contains a non-generator. iii. given a multigroup over a cyclic group with order , if is a minimal generating set of then gives elements of . theorem 4.16 every irregular multigroup with complete maximal submultigroups over a group is fully frattini. proof. for multigroup to be irregular implies , . now let for each be the complete maximal submultigroups of . for to be complete in implies . since is complete in , then there exists such that . therefore for each . theorem 4.17 every irregular multigroup with an incomplete maximal submultigroups over a non-cyclic group is fully frattini. proof. is a non-cyclic group, implies it has no generator and . now let for each be the incomplete maximal submultigroups of . then for each , is contained in at least one of the with theorem every cyclic multigroup with incomplete maximal submultigroups is not fully frattini. proof. suppose is a cyclic group and is a multigroup with incomplete maximal submultigroups over . where has set of generators . now frattini submultigroups of multigroups 161 let for some finite then , where are the root sets of all the maximal submultigroups of for each and .by remark 4.2 , hence, is not fully frattini. theorem 4.19 every regular multigroup over a group is non-fully frattini. proof. let be a group and be a multigroup over . for to be a regular multigroup implies , . let for each be the maximal submultigroup of . since is regular and by definition , there exists at least an element such that for each and so remark 4.4. let be a nontrivial fully frattini multigroup over a group then has at least three maximalsubmultigroups if where is the identity element of . 5 conclusions most results in frattini subgroup are extended to multigroup. a number of new results 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[23] a. syropoulos, mathematics of multisets, springer-verlac berlin heidelberg. 2001. ratio mathematica volume 47, 2023 relation-theoretic contraction principle in metric spaces using multiplicative contraction radha yadav* balbir singh† abstract alam and imdad have presented a novel application of the banach contraction principle on a complete metric spaces with a binary relation. we have extended the concept of binary relation with the multiplicative contraction in a complete metric spaces. we have also included corollary to demonstrate our results. keywords: fixed point, metric spaces, binary relation, multiplicative contraction 2020 ams subject classifications: 47h10, 54h25.1 1 introduction in many scientific domains, particularly in fixed point theory, the concept of a metric space is extremely useful. this notion has been generalised in numerous directions in recent years, and many notions of a metric-type space have been introduced (b-metric, dislocated space, generalised metric space, quasi-metric space, symmetric space, etc.).the banach contraction principle’s [3]contraction condition has been generalised to numerous forms in the last fifty years. furthermore, the metric space in the banach contraction principle has been generalised to a variety of generalised metric spaces. many authors researched other sorts of fixed point theorems in metric spaces later on, as seen by the and references therein. *research scholar (starex university, gurugram, india); radha97yadav@gmail.com. †professor (starex university, gurugram, india); balbir.vashist007@gmail.com. 1received on march 28, 2022. accepted on december 28, 2022. published online on february 6, 2023. doi:10.23755/rm.v39i0.730. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 30 radha yadav and balbir singh 2 preliminaries we give the required background material needed to prove our results in this part to make our exposition self-contained. in what follows, n, n0, q and r denote the sets of positive integers, non-negative integers, rational numbers and real numbers, respectively. aftab and alam [1] proved new relation-theoretic fixed point theorems on metric spaces in 2015, and then inferred comparable findings in metric spaces. metric spaces are sets in which there is a defined a notion of ’distance between pair of points’. the concept of metric spaces was formulated in 1906 by m.frechet [7] ,though the definition presently in use given by the german mathematician, felix hausdorff. definition 2.1. let m be a non empty arbitrary set and d be a real function from m × m into r+ such that for all u, v, w ∈ m we have 1. d(u, v) ≥ 0, 2. d(u, v) = 0 ⇐⇒ u = v, 3. d(u, v) = d(v, u) and 4. d(u, w) ≤ d(u, v) + d(v, w), here (m, d) is called a metric in r and (r, d) is a metric space. example 2.1. 1. d(u, v) = |u − v| is a metric space in r. 2. if d(u, v) defined by d(u, v) = {1 if u ̸= v 0 if u = v} definition 2.2. [10] let m be a nonempty set. a subset r of m2 is called a binary relation on m. notice that for each pair u, v ∈ m, one of the following conditions holds: 1. (u, v) ∈ r; which amounts to saying that “u is r-related to v” or “u relates to v under r”. sometimes, we write urv instead of (u, v) ∈ r ; 2. (u, v) /∈ r; which means that “u is not r-related to v” or “u does not relate to v under r”. 31 relation-theoretic contraction principle in metric spaces using multiplicative contraction definition 2.3. [10] let r be a binary relation defined on a nonempty set m and u, v ∈ m. we say that u and v are r -comparative if either (u, v) ∈ r or (v, u) ∈ r. we denote it by [u, v] ∈ r. definition 2.4. [6, 10, 11, 12, 15] a binary relation r defined on a nonempty set m is called 1. reflexive if (u, u) ∈ r for all u ∈ m, 2. irreflexive if (u, u) /∈ r for all u ∈ m, 3. symmetric if (u, v) ∈ r implies (v, u) ∈ r, 4. antisymmetric if (u, v) ∈ r implies (v, u) /∈ r, 5. transitive if (u, w) ∈ r and (w, v) ∈ r implies (u, v) ∈ r, 6. complete, connected or dichotomous if [l, n] ∈ r for all l, m ∈ m, 7. weakly complete, weakly connected or trichotomous if [u, v] ∈ r or u = v for all u, v ∈ m. 8. strict order or sharp order if r is irreflexive and transitive, 9. near-order if r is antisymmetric and transitive, 10. pseudo-order if r is reflexive and antisymmetric, 11. quasi-order or preorder if r is reflexive and transitive, 12. partial order if r is reflexive, antisymmetric and transitive, 13. simple order if r if weakly complete strict order, 14. weak order if r is complete preorder, 15. total order, linear order or chain if r is complete partial order, 16. tolerance if r is reflexive and symmetric, 17. equivalence if r is reflexive, symmetric and transitive. definition 2.5. [4] let m be a nonempty set and r a binary relation on m. a sequence {un} ⊂ m is called r-preserving if (un, un+1) ∈ r ∀n ∈ n0 32 radha yadav and balbir singh the notion of dself closeness of a partial order ⪯ defined by turinici [16] is extended to an arbitrary binary relation in the following lines. now, we state and prove our main result, which is as follows: theorem 2.1. let (m, d) be a complete metric space, r a binary relation on m and t a self-mapping on m. suppose that the following conditions hold: a) m(f; r) is nonempty, b) r is f-closed, c) either f is continuous or r is p-self-closed, d) there exists λ ∈ [0, 1)d(f(u), f(v)) ≤ d(u, v)λ forallu, v ∈ m with(u, v) ∈ r then f has a fixed point. moreover, if e) υ(u, v, rs) is nonempty, for each u, v ∈ m, then f has a unique fixed point. proof. consider a point u0 ∈ m. now we define a sequence {un} of picard iterates, i.e., un = fun−1 for n = 1, 2, ... from the multiplicative contraction property [13] of f for all n ∈ n0. as (u0, fu0) ∈ r, using condition (b), we get (fu0, f 2u0), (f 2u0, f 3u0), ..., (f nu0, f n+1u0), ... ∈ r so that (un, un+1) ∈ r n ∈ n0. (1) thus the sequence {un} is r-preserving. applying the contractivity condition (d) to equation (1), we deduce, for all n ∈ n0, that d(un+1, un) ≤ d(un, un−1)λ ≤ d(un−1, un−2)λ 2 ≤ ... ≤ d(u1, u0)λ n . which by induction yields that d(un+1, un+2) ≤ d(u0, fu0)λ n+1 n ∈ n0. (2) using equation (2) and triangular inequality, for all n ∈ n0, p ∈ n, p ≥ 2, we have d(un+1, un+p) ≤ d(un+1, un+2) + d(un+2, un+3) + ... + d(un+p−1, un+p) ≤ d(u1, u0)λ n+1+...+λp ≤ d(u1, u0) λp 1−λ this implies that d(un, up) → 1 as (n, p → ∞). hence the sequence (xn) = (fnu0) is multiplicative cauchy. by the completeness of m, there is z ∈ m such that un → z as n → ∞. moreover, 33 relation-theoretic contraction principle in metric spaces using multiplicative contraction d(fz, z) ≤ d(fun, fz).d(fun, z) ≤ d(un, z)λ.d(fun, z) → 1 as n → ∞, which implies d(fz, z) = 0. therefore this says that z is a fixed point of f; that is fz = z. now, if there is another point y such that fy = y, then d(z, y) = d(fz, fy) ≤ d(z, y)λ. therefore d(z, y) = 0 and y = z. this implies that z is the unique fixed point of f. alternatively, let us assume that r is d-self-closed. as un is an r-preserving sequence and un →d u, there exists a subsequence {unk} of {un} with [unk, u] ∈ r k ∈ n0 using (d), [unk, u] ∈ r and un → d u, we obtain d(unk+1, fu) = d(funk, fu) ≤ d(unk, u) λ → 1 as k → ∞ so that unk+1 → d f(u). again, owing to the uniqueness of limit, we get f(u) = u so that u is a fixed point of f. to prove uniqueness, take u, v ∈ f(f), i.e., f(u) = u and f(v) = v. (3) by assumption (d), there exists a path (say {z0, z1, z2, ..., zk}) of some finite length k in rs from u to v so that z0 = u, zk = v, [zi, zi+1] ∈ r foreach i(0 ≤ i ≤ k − 1). (4) as r is f-closed, we have [fnzi, f nzi+1] ∈ r foreach i(0 ≤ i ≤ k − 1) and for each n ∈ n0. (5) making use of equations (3),(4),(5),, triangular inequality and assumption (d), we obtain d(u, v) = d(fnz0, f nzk) ≤ ∑i=0 k−1 d(f nzi, f nzi+1) ≤ ∑i=0 k−1 d(f n−1zi, f n−1zi+1) λ ≤ ∑i=0 k−1 d(f n−2zi, f n−2zi+1) λ2 34 radha yadav and balbir singh ≤ ... ≤ ∑i=0 k−1 d(zi, zi+1) λn → 0 as n → ∞ so that u = v. hence f has a unique fixed point.2 corolary 2.1. let (m, d) be a complete metric space. for ϵ with ϵ > 1 and u0 ∈ m, consider the multiplicative closed ball, bϵ(u0). suppose the mapping f : m → m satisfies the contraction condition d(f(u), f(v)) ≤ d(u, v)λ forallu, v ∈ bϵ(u0) where λ ∈ [0, 1) is a constant r is a relation such that d(fu0, u0) ≤ ϵ1−λ. then f has a unique fixed point in bϵ(u0). corolary 2.2. let (m, d) be a complete metric space. if a mapping f : m → m satisfies for some positive integer n, d(fnu, fnv) ≤ d(u, v)λ for all u, v ∈ m, where λ ∈ [0, 1) is a constant, then f has a unique fixed point in m. theorem 2.2. let (m, d) be a complete metric space, r a binary relation on m and t a self-mapping on m. suppose that the following conditions hold: a) m(f; r) is nonempty, b) r is f-closed, c) either f is continuous or r is p-self-closed, d) there exists λ ∈ [0, 1 2 ) d(fu, fv) ≤ (d(f(u, v).d(fv, u))λforallu, v ∈ m with(u, v) ∈ r then f has a fixed point. moreover, if e) υ(u, v, rs) is nonempty, for each u, v ∈ m, then f has a unique fixed point. proof. consider a point u0 ∈ m. now we define a sequence {un} of picard iterates, i.e., un = fun−1 for n = 1, 2, ... from the multiplicative contraction property of f for all n ∈ n0. as (u0, fu0) ∈ r, using condition 2, we get (fu0, f 2u0), (f 2u0, f 3u0), ..., (f nu0, f n+1u0), ... ∈ r we have d(un+1, un) = d(fun, fun−1) ≤ (d(fun, un).d(fun−1, un−1))λ = (d(un+1, un).d(un, un−1)) λ 35 relation-theoretic contraction principle in metric spaces using multiplicative contraction thus we have d(un+1, un) ≤ (d(un, un−1)) λ 1−λ = d(un, un−1) h, where h = λ 1−λ . for n > m, using triangular inequality, for all n ∈ n0, m ∈ n, m ≥ m, we have d(un, um) ≤ d(un, un−1).d(un−1, un−2)...d(um+1, um) ≤ d(u1, u0)h n−1+hn−2+...+hm ≤ d(u1, u0) hm 1−h this implies d(un, um) → 1 as (n, m → ∞). hence (un) is a cauchy sequence. by the completeness of m, there is z ∈ m such that un → z as n → ∞. since d(fz, z) ≤ d(fun, fz).d(fun, z) ≤ (d(fun, un).d(fz, z))λ.d(un+1, z), we have d(un+1, un) ≤ (d(un, un−1)) λ 1−λ = d(un, un−1) h, where h = λ 1−λ . for n > m, using triangular inequality, for all n ∈ n0, m ∈ n, m ≥ m, we have d(un, um) ≤ d(un, un−1).d(un−1, un−2)...d(um+1, um) ≤ d(u1, u0)h n−1+hn−2+...+hm ≤ d(u1, u0) hm 1−h this implies d(un, um) → 1 as (n, m → ∞). hence (un) is a cauchy sequence. by the completeness of m, there is z ∈ m such that un → z as n → ∞. since d(fz, z) ≤ d(fun, fz).d(fun, z) ≤ (d(fun, un).d(fz, z))λ.d(un+1, z), we have d(fz, z) ≤ (d(fun, un)λ.d(un+1, z)) 1 1−λ → 1 as n → ∞. hence d(fz, z) = 0. this implies fz = z. finally, it is easy to prove that the fixed point of f is unique. alternatively, let us assume that r is d-self-closed. as {un} is an r-preserving sequence and un →d u, 36 radha yadav and balbir singh there exists a subsequence {unk} of {un} with [unk, u] ∈ r k ∈ n0 using (d), [unk, u] ∈ r and unk → d u, we obtain d(unk+1, fu) = d(funk, fu) ≤ (d(unk+1, u).d(unk, u)) λ → 1 as k → ∞ so that unk+1 → d f(u). again, owing to the uniqueness of limit, we obtain f(u) = u, so that u is a fixed point of f. to prove uniqueness, take u, v ∈ f(f), i.e., f(u) = u and f(v) = v. (6) by assumption (e), there exists a path (say{z0, z1, z2, ..., zk}) of some finite length k in rs from u to v so that z0 = u, zk = v, [zi, zi+1] ∈ r for each i(0 ≤ i ≤ k − 1). (7) as r is f-closed, we have [fnzi, f nzi+1] ∈ r for each i(0 ≤ i ≤ k−1) and for each n ∈ n0. (8) making use of equations (6),(7),(8),, triangular inequality and assumption (d) we obtain d(u, v) = d(fnz0, f nzk) ≤ ∑i=0 k−1 d(f nzi, f nzi+1) ≤ ∑i=0 k−1(d(f n−1zi, f n−1zi+1).d(f n−1zi+1, f n−1zi)) λ ≤ ∑i=0 k−1(d(f n−2zi, f n−2zi+1).d(f n−2zi+1, f n−2zi)) λ2 ≤ ... ≤ ∑i=0 k−1(d(zi, zi+1).d(zi+1, zi)) λn → 0 as n → ∞ so that u = v. hence f has a unique fixed point.2 theorem 2.3. let (m, d) be a complete metric space, r a binary relation on m and t a self-mapping on m. suppose that the following conditions hold: a) m(f; r) is nonempty, b) r is f-closed, c) either f is continuous or r is p-self-closed, d) there exists λ ∈ [0, 1 2 ) d(fu, fv) ≤ (d(f(u, u).d(fv, v))λforallu, v ∈ m with(u, v) ∈ r then f has a fixed point. moreover, if e) υ(u, v, rs) is nonempty, for each u, v ∈ m, then f has a unique fixed point. 37 relation-theoretic contraction principle in metric spaces using multiplicative contraction proof. consider a point u0 ∈ m. now we define a sequence {un} of picard iterates, i.e., un = fun−1 for n = 1, 2, ... from the multiplicative contraction property of f for all n ∈ n0. as (u0, fu0) ∈ r, using condition 2, we get (fu0, f 2u0), (f 2u0, f 3u0), ..., (f nu0, f n+1u0), ... ∈ r we have d(un+1, un) = d(fun, fun−1) ≤ (d(fun, un).d(fun−1, un−1))λ = (d(un+1, un).d(un, un−1)) λ thus we have d(un+1, un) ≤ d(un, un−1)) λ 1−λ = d(un, un−1) h, where h = λ 1−λ . for n > m, using triangular inequality, for all n ∈ n0, m ∈ n, m ≥ m, we have d(un, um) ≤ d(un, un−1).d(un−1, un−2)...d(um+1, um) ≤ d(u1, u0)h n−1+hn−2+...+hm ≤ d(u1, u0) hm 1−h this implies d(un, um) → 1 as (n, m → ∞). hence (un) is a cauchy sequence. by the completeness of m, there is z ∈ m such that un → z as n → ∞. since d(fz, z) ≤ d(fun, fz).d(fun, z) ≤ (d(fun, un).d(fz, z))λ.d(un+1, z), we have d(fz, z) ≤ (d(fun, un)λ.d(un+1, z)) 1 1−λ → 1 as n → ∞. hence d(fz, z) = 0. this implies fz = z. finally, it is easy to prove that the fixed point of f is unique. alternatively, let us assume that r is d-self-closed. as {un} is an r-preserving sequence and un →d u, there exists a subsequence {unk} of {un} with [unk, u] ∈ r k ∈ n0 using (d), [unk, u] ∈ r and unk → d u, we obtain 38 radha yadav and balbir singh d(unk+1, fu) = d(funk, fu) ≤ (d(unk+1, u).d(unk, u)) λ → 1 as k → ∞ so that unk+1 → d f(u). again, owing to the uniqueness of limit, we obtain f(u) = u, so that u is a fixed point of f. to prove uniqueness, take u, v ∈ f(f), i.e., f(u) = u and f(v) = v. (9) by assumption (e), there exists a path (say{z0, z1, z2, ..., zk}) of some finite length k in rs from u to v so that z0 = u, zk = v, [zi, zi+1] ∈ r foreach i(0 ≤ i ≤ k − 1). (10) as r is f-closed, we have [fnzi, f nzi+1] ∈ r foreach i(0 ≤ i ≤ k − 1) and for each n ∈ n0. (11) making use of equations (9),(10),(11), triangular inequality and assumption (d), we obtain d(u, v) = d(fnz0, f nzk) ≤ ∑i=0 k−1 d(f nzi, f nzi+1) ≤ ∑i=0 k−1(d(f n−1zi, f n−1zi+1).d(f n−1zi+1, f n−1zi)) λ ≤ ∑i=0 k−1(d(f n−2zi, f n−2zi+1).d(f n−2zi+1, f n−2zi)) λ2 ≤ ... ≤ ∑i=0 k−1(d(zi, zi+1).d(zi+1, zi)) λn → 0 as n → ∞ so that u = v. hence f has a unique fixed point.2 acknowledgement the authors would like to express their gratitude to the journal’s editor and the entire team for this submission. 39 relation-theoretic contraction principle in metric spaces using multiplicative contraction references 1. a.alam and m.imdad. relation-theoretic contraction principle. j. fixed point theory appl., 17(4):693-702, 2015. 2. md.ahmadullah, m.imdad and r.gurban. relation-theoretic metrical fixed point theorems under nonlinear contractions. fixed point theory, 20(1):318, 2019. 3. s.banach. sur les operations dans les ensembles abstraits et leur applications aux equations integrales. fundamental mathematicae, 3(7):133-181, 1922. 4. l. e. j.brouwer. uber abbildung von mannigfaltigkeiten. math. ann.. 71(1):97-115, 1911. 5. l.b.ćirić. a new fixed point theorem for contractive mappings. inst. mathematique publ., nouvelle serie., 30(44):25-27, 1981. 6. v.flaska, j.jezek, t.kekpa and j.kortelainen. transitive closures of binary relations i. acta univ. carolin. math. phys., 44:55-59, 2007. 7. m.m.frechet. sur quelques points du calcul functionnel. rendiconti del circolo mathematico di palermo, 22(1):1-72, 1906. 8. r.kannan. some results on fixed points. bull. cal. math. soc., 60:71-76, 1968. 9. r. kannan. some results on fixed points ii. am. math. mon., 76:405-408, 1969. 10. s.lipschutz. schaum’s outlines of theory and problems of set theory and problems of set theory and related topics. mcgraw-hill, new york, 1964. 11. h.l.skala. trellis theory. algebra universalis, 1:218-233, 1971. 12. r.d.maddux and rodrigue-lopez. relation algebras. stud. logic found. math. 150, elsevier b. v., amesterdam, 2006. 13. m.ozavsar and a.c.cevikel. fixed point of multiplicative contraction mappings on multiplicative metric space. arxiv:1205.5131v [math. gm], 2012. 14. j.slapal. relations and topologies. czechoslovak mathematical journal, 43(1):141-150, 1993. 40 radha yadav and balbir singh 15. a.stouti and a. maaden. fixed points and common fixed points theorems in pseudo-ordered sets. proycecciones, 32:409-418, 2019. 16. m.turinici. product fixed point results in ordered metric spaces. arxiv:1110.3079v1, 2011. 41 ratio mathematica 24 (2013), 31–40 issn: 1592-7415 ordered polygroups mahmood bakhshi, radjab ali borzooei department of mathematics, bojnord university, bojnord, iran department of mathematics, shahid beheshti university, tehran, iran bakhshi@ub.ac.ir; borzooei@sbu.ac.ir abstract in this paper, those polygroups which are partially ordered are introduced and some properties and related results are given. key words: hypergroup, polygroup, ordered polygroup. msc2010: 20n20, 06f15, 06f05. 1 introduction and preliminaries the notion of a hyperstructure and hypergroup, as a generalization of group, was introduced by f. marty [5] in 1934 at the 8th congress of scandinavian mathematicians. in this definition for nonempty set h, a function · : h×h −→ p∗(h), where p∗(h) is the set of all nonempty subsets of h, is called a hyperoperation on h, and the system (h, ·) is called a hypergroupoid. if the hypergroupoid h satisfies a ·h = h ·a = h, for all a ∈ h, it is called a hypergroup. in a hypergroupoid h, for a,b ⊆ h and x ∈ h, a · b and a ·x are defined as a ·b = ⋃ a∈a,b∈b a · b,a ·x = a · {x}. an element e of hypergorupoid h is called an identity if for all a ∈ h, a ∈ a◦e∩e◦a. an element a′ ∈ h is called an inverse for a ∈ h if there is an identity e ∈ h such that e ∈ a◦a′ ∩a′ ◦a. by a subhypergroupoid of hypergroupoid h we mean a subset k of h that is closed with respect to the hyperoperation on h, and contains the unique identity of h and the inverses of its elements, provided there exist. m. bakhshi, r. a. borzooei hyperstructures have many applications to several sectors of both pure and applied sciences. a short review of the theory of hyperstructures appear in [2]. in [3] a wealth of applications can be found, too. there are applications to the following subjects: geometry, hypergraphs, binary relations, lattices, fuzzy set and rough sets, automata, cryptography, combinatorics, codes, artificial intelligence and probabilities. polygroups are certain subclasses of hypergroups which studied in 1981 by ioulidis in [4] and are used to study colour algebra. a polygroup is a system < g, ·,−1 ,e > where e ∈ g, ‘−1’ is a unary operation on g and ‘·’ is a binary hyperoperation on h satisfying the following: (1) (x ·y) ·z = x · (y ·z), (2) e ·x = x ·e = {x}, (3) x ∈ y ·z ⇔ y ∈ x ·z−1 ⇔ z ∈ y−1 ·x. in any polygroup the following hold: e ∈ x ·x−1 ∩x−1 ·x, e−1 = e, (x−1)−1 = x, (x ·y)−1 = y−1 ·x−1 where a−1 = {x−1 : x ∈ a}. some other concepts in polygroups is as follows. a nonempty subset k of polygroup g is said to be a subpolygroup if and only if e ∈ k and < k, ·,−1 ,e > is itself a polygroup. subpolygroup k of polygroup g is said to be normal if and only if a−1ka ⊆ k, for all a ∈ g. from now on, in this paper, g =< g, ·,−1 ,e > will denote a polygroup. 2 ordered hyperstructures: definition and properties this section is devoted to introduce the concept of a compatible order on a polygroup. it is first introduced the concept of an ordered hypergroupoid and some basic notions. then, the concept of ordered polygroups is introduced and some related results are given. for more details on compatible orders, specially ordered algebraic structures we refer to [1]. definition 2.1. let (h, ·) be a hypergroupoid. by a compatible order on h we mean an order “≤” with respect to which all translations x 7→ x ·y and x 7→ y ·x are isotone, that is x ≤ y implies b ·x ·a ≤ b ·y ·a, for all a,b ∈ h (2.1) 32 ordered polygroups where for a,b ⊆ h, a ≤ b means that for all a ∈ a there exists b ∈ b and for all b ∈ b there exists a ∈ a such that a ≤ b. definition 2.2. by an ordered hypergroupoid we mean a hypergroupoid on which is defined a compatible order. when “·” is commutative or associative, h is said to be an ordered commutative hypergroupoid or an ordered semihypergroup, respectively. example 2.3. (1) consider r1 = [1,∞), the set of all real numbers greater than 1, as a poset with the natural ordering, and define x ·y to be the set of all upper bounds of {x,y}. thus (r1, ·,≤) is an ordered commutative semihypergroup with 1 as the unique identity. (2) consider z, the additive group of all integers which is a chain with the natural ordering. for m,n ∈ z, let m·n be the subgroup of z generated by {m,n}. then (z, ·,≤) is an ordered commutative semihypergroup in which 0 is an identity. (3) let (g, ·,e,≤) be an ordered group, and let x◦y = 〈{x,y}〉, the subgroup of g generated by {x,y}. then, (g,◦,≤) is an ordered commutative hypergroup with an identity e. (4) let (l;∨,∧, 0) be a lattice with the least element 0. for a,b ∈ l, let a◦b = f(a∧b), where f(x) is the principal filter generated by x ∈ l. then, (l;◦) is an ordered hypergroup. also, 0 is an identity, and if x ∈ l be such that x∧y = 0, for some y ∈ l, then y is an inverse of x. definition 2.4. let h be an ordered hypergroupoid. (1) for every x,y ∈ h with x ≤ y, the set [x,y] = {z ∈ h : x ≤ z ≤ y} is said to be an interval in h. (2) a subset a of h is said to be convex if for all a,b ∈ a, where a ≤ b, we have [a,b] ⊆ a. definition 2.5. let (e;≤) be an ordered set. a subset d of e is said to be a down-set if y ≤ x and x ∈ d imply y ∈ d. down-set d is said to be principal if there exists x ∈ d such that d = {y ∈ e : y ≤ x} denoted by x↓. definition 2.6. let (g;◦g,≤g) and (h;◦h,≤h ) be ordered hypergroupoids and f : g −→ h be an isotone map, that is f(x) ≤h f(y) whenever x ≤g y. then, 33 m. bakhshi, r. a. borzooei (1) f is said to be an order homomorphism if f is a homomorphism of hypergroupoids (g; ,◦g) and (h;◦h ), (2) f is an order isomorphism if f is an isomorphism of hypergroupiods, and f−1 is isotone, (3) the kernel of f is defined by kerf = {(x,y) ∈ g×g : f(x) = f(y)}. 3 ordered polygroups in this section, we assume that g =< g, ·,−1 ,e > is a polygroup unless otherwise mentioned. hereafter, in this paper, we use xy for x ·y, and a for {a}. definition 3.1. by an ordered polygroup we mean a polygroup which is also a poset under the binary relation ≤ and in which (2.1) holds. definition 3.2. let h be an ordered hypergroupoid with a unique identity e. an element x ∈ h is called positive if e ≤ x. the set of all positive elements of h is called the positive cone of h and is denoted by h+. x ∈ h is called negative if x ≤ e. the set of all negative elements of h is called the negative cone of h and is denoted by h−. by an elementary consequence of translations we have proposition 3.3. in any ordered polygroup g, for each x,y ∈ g, we have x ≤ y ⇔ x−1y ∩g+ 6= ∅ ⇔ yx−1 ∩g+ 6= ∅ ⇔ xy−1 ∩g− 6= ∅ ⇔ y−1x∩g− 6= ∅⇔ y− ≤ x−. theorem 3.4. a subset p of a polygroup g is the positive cone with respect to some compatible order if and only if (1) p ∩p−1 = {e}, (2) p 2 = p , (3) for all x ∈ g, xpx−1 = p . moreover, if this order is total, p ∪p−1 = g. proof. (⇒) let ≤ be a compatible order on g and p = g+, the associated positive cone. (1) if x ∈ p∩p−1, on the one hand e ≤ x, and on the other hand x = y−1, for some y ∈ p . since, e ≤ y, then x = y−1 ≤ e proves that x = e. 34 ordered polygroups (2) since e ∈ p , p = pe ⊆ pp = p 2. now, let x,y ∈ p . then e ≤ x and e ≤ y and so e ≤ xy which implies that xy ⊆ p . hence, p 2 ⊆ p . (3) let y ∈ p , and x ∈ g. then, e ≤ y implies that e ∈ xex−1 ≤ xyx−1 proves that xyx−1 ⊆ p . since, this follows for all x ∈ g, replacing x by x−1, we have x−1px ⊆ p and so p ⊆ xpx−1, complete the proof. (⇐) let p be a subset of g that satisfies properties (1)-(3), and define the relation ≤ on g by x ≤ y ⇔ yx−1 ∩p 6= ∅. since, e ∈ p , by (3), xx−1 = xex−1 ⊆ xpx−1 = p implies that x ≤ x and so ≤ is reflexive. suppose that x ≤ y and y ≤ x, for x,y ∈ g. then yx−1∩p 6= ∅ and xy−1 ∩p 6= ∅ whence xy−1 ∩p−1 ∩p 6= ∅, implies that e ∈ xy−1, i.e., x = y proving ≤ is antisymmetric. now, assume that x ≤ y and y ≤ z, for x,y,z ∈ g. then yx−1 ∩ p 6= ∅ and zy−1 ∩ p 6= ∅. let u ∈ yx−1 ∩ p and v ∈ zy−1 ∩p . then uv ⊆ p 2 = p . on the other hand, ∈ zy−1 and v ∈ yx−1 imply y−1 ∈ z−1u and y ∈ vx whence e ∈ y−1y ⊆ z−1(uv)x. then, there is t ∈ uv and s ∈ tx such that e ∈ z−1s. this implies that z = s ∈ tx. hence, t ∈ zx−1, i.e., uv ∩ zx−1 6= ∅ whence zx−1 ∩ p 6= ∅ proving ≤ is transitive. thus, ≤ is an order. for compatibility, we first prove that px = xp , for all x ∈ g. let z ∈ g. then z ∈ px ⇒ z ∈ yx for some y ∈ p ⇒ x−1z ⊆ x−1yx = x−1y(x−1)−1 ⊆ p ⇒ z ∈ xp, i.e., px ⊆ xp . by a similar way, we can prove that xp ⊆ px. hence, xp = px, for all x ∈ g. now, assume that x ≤ y and a,b ∈ g. since, ≤ is reflexive, by (3) ayb(axb)−1 = aybb−1x−1a−1 ⊆ aypx−1a−1 ⊆ apyx−1a−1 ⊆ ap 2a−1 = apa−1 = p which shows that axb ≤ ayb. by the definition of ≤ we get x ∈ p if and only if e ≤ x and so p = g+. if g is totally ordered, then x ≤ e or e ≤ x, for all x ∈ g. so, e ∈ xx−1 ≤ ex−1 = x−1 and so x ∈ p or x ∈ p−1, observe that x = (x−1)−1. thus, g = p ∪p−1. 2 proposition 3.5. if g is an ordered polygroup with |g| > 1, then g can not have a top element or a bottom element. proof. let g = {e,a}. if e < a or a < e, then a = a−1 < e or e < a−1 = a, respectively, which is a contradiction. now, assume that 35 m. bakhshi, r. a. borzooei |g| > 2, t be the top element of g and e 6= a ∈ g. then a ≤ t and so ta ≤ t whence t ∈ te ⊆ taa−1 ≤ ta−1. hence, t ∈ ta−1. likewise, we conclude that t ∈ a−1t. by the uniqueness, we get a = e which is a contradiction. the proof of the other case is concluded as well. 2 definition 3.6. an element x of g is said to be of order n, n ∈ n, if e ∈ xn where xn = (· · ·(( n times︷ ︸︸ ︷ x◦x) ◦x) ◦ · · ·) ◦x). if such a natural number does not exist, we say that x is of infinite order. theorem 3.7. suppose that g is an ordered polygroup in which g+ 6= {e}. then every element of g+ \{e} is of infinite order. proof. suppose that x ∈ g+ \{e}. we first observe that if x = x−1, x can not belong to g+. then, e < x implies that e < x = ex < x2. moreover, this implies that e 6∈ x2. similarly, we conclude that e < x3 and e 6∈ x3. continuing this process we get e < xn and e 6∈ xn, for all n ∈ n, proving x is not of finite order. 2 corollary 3.8. any ordered polygroup in which every nontrivial element is of finite order is an antichain. proof. let g be an ordered polygroup satisfying the hypothesis. by theorem 3.7, we know that g+ = {e}. now, if a,b ∈ g be such that a ≤ b, then e ∈ a−1a ≤ a−1b and so e ≤ u, for some u ∈ a−1b. this implies that u ∈ g+ and so u = e. thus, e ∈ a−1b whence a = b. this means that g is an antichain. 2 corollary 3.9. every finite ordered polygroup is an antichain. example 3.10. let g = {e,a}. then g is a polygroup where the hyperoperation is given by the following table: ◦ e a e e a a a {e,a} in which a−1 = a i.e., a is an idempotent. now, if a is a positive element, so g+ = {e,a} and hence (g+)−1 ∩ g+ 6= {e}. this contradicts theorem 3.4. this example shows that the converse of theorem 3.7 does not hold in general. definition 3.11. if g is an ordered polygroup, by a convex subgroup of g we shall mean a subgroup which is also a convex subset, under the order of g. 36 ordered polygroups definition 3.12. a nonempty subset h of g is said to be s-reflexive if xy ∩h 6= ∅ implies that xy ⊆ h, for all x,y ∈ g. theorem 3.13. if h is a subpolygroup of an ordered polygroup g then h+ = h ∩ g+. moreover, if h+ is s-reflexive, the following statements are equivalent: (1) h is convex; (2) h+ is a down-set of g+. proof. since, eh = eg, it is clear that h + = h ∩g+. (1) ⇒ (2) suppose that eh ≤ y ≤ x where eh,x ∈ h+ ⊆ h. then (1) gives y ∈ h ∩g+ = h+ and so h+ is a down-set of g+. (2) ⇒ (1) suppose now that x ≤ y ≤ z where x,z ∈ h. then x−1x ≤ x−1y ≤ x−1z. thus, x−1z ⊆ h+ and so there is a ∈ h+ such that a ∈ x−1z. hence, there is b ∈ x−1y such that b ≤ a ∈ h+, and since h+ is a down-set of g+, b ∈ h+, i.e., x−1y∩h+ 6= ∅. since, h+ is s-reflexive, so x−1y ⊆ h+ ⊆ h whence y ∈ xh = h, proving h is convex. 2 if g is an ordered polygroup and h is a normal subpolygroup of g, then a natural candidate for a positive cone of g/h is \h (g +), where \h : g −→ g/h is the canonical projection. precisely when this occurs is the substance of the following result. theorem 3.14. let g be an ordered polygroup and let h be a normal subpolygroup of g. then \h (g +) = {ph : p ∈ g+} is the positive cone of a compatible order on the quotient polygroup g/h if and only if h is convex. proof. suppose that q = {ph : p ∈ g+} is the positive cone of a compatible order on g/h. to show that h is convex, suppose that c ≤ b ≤ a with c,a ∈ h. then (bh)−1 = (bh)−1 · ah = b−1ah. on the other hand, b ≤ a implies that b−1a∩g+ 6= ∅. hence (bh)−1∩q 6= ∅ and so bh∩q−1 6= ∅. similarly, we have bh = bh · c−1h = bc−1h and since bc−1 ∩ g+ 6= ∅, bh ∩q 6= ∅. thus, bh ∩ (q∩q−1) 6= ∅ whence bh = h, i.e., b ∈ h. conversely, suppose that h is convex and let q = {ph : p ∈ g+}. it is clear that q2 = q. suppose now that xh ∈ q ∩ q−1. then xh = ph = q−1h where p,q ∈ g+. these equalities also give pq ∩ h 6= ∅. now, since p ≤ pq, then eh ≤ p ≤ u, where u ∈ pq∩h whence the convexity of h gives p ∈ h. it follows that xh = ph = h and hence q∩q−1 = {h}. finally, since g+ is a normal subsemihypergroup of g it is clear that q = \h (g +) is a normal subsemihypergroup of g/h. it now follows by theorem 3.4 that q is the positive cone of a compatible order on g/h. 2 37 m. bakhshi, r. a. borzooei if h is a convex normal subpolygroup of an ordered polygroup g then the order ≤h on g/h that corresponds to the positive cone {ph : p ∈ g+} can be described as in the proof of theorem 3.4. we have xh ≤h yh ⇒ yx−1h ⊆ q ⇒ (∀a ∈ yx−1)(∃p ∈ g+)ah = ph ⇒ (∀a ∈ yx−1)(∃p ∈ g+)(∃h ∈ h)a ∈ ph ≥ h ⇒ (∀a ∈ yx−1)(∃h ∈ h) a ≥ h ⇒ yx−1 ≥ h. from the last inequality and that y ∈ ye ⊆ yx−1x it follows that y ≥ u, for some u ∈ hx. conversely, assume that there exists h ∈ h and u ∈ hx such that y ≥ u, and let a ∈ yx−1. from yx−1 ≥ yx−1 it follows that a ≥ t, for some t ∈ ux−1 and hence at−1 ≥ tt−1. this implies that v ≥ e, for some v ∈ at−1 and so vh ∈ at−1h ∩q. (3.1) now, t ∈ ux−1 implies that t−1 ∈ xu−1 ⊆ xx−1h−1 ⊆ xx−1h and so at−1 ⊆ axx−1h = axhx−1 = ah. thus, at−1h ⊆ ah. combining (3.1), we get {ah}∩q 6= ∅, i.e., ah ∈ q and so ah = ph, for some p ∈ g+. this implies yx−1h ⊆ q and hence xh ≤h yh, completes the proof. thus we see that ≤h can be described by xh ≤h yh ⇔ (∃h ∈ h)(∃u ∈ hx) y ≥ u. in referring to the ordered quotient polygroup g/h we shall implicitly infer that the order is ≤h as described above. here we give a characterization of polygroup homomorphisms that are isotone. theorem 3.15. let g and h be ordered polygroups. if f : g −→ h is a polygroup homomorphism, f is isotone if and only if f(g+) ⊆ h+. proof. assume that f is isotone. if x ∈ g+, i.e., x ≥ e then f(x) ≥ f(eg) = eh means that f(x) ∈ h+. conversely, assume that x ≤ y in g. then yx−1 ⊆ g+ and so f(y)f(x)−1 = f(yx−1) ⊆ f(g+) ⊆ h+. this implies that f(y) ≥ f(x) proving f is isotone. 2 corollary 3.16. if g is an ordered polygroup and h is a convex normal subpolygroup of g, then the natural homomorphism \h : g −→ g/h is isotone. 38 ordered polygroups proof. by theorem 3.15, it is enough to prove that \(g+) ⊆ (g/h)+. for this, let yh ∈ \(g+). then yh = gh, for some g ∈ g+ whence y ∈ gh ≥ h for some h ∈ h. this implies that eh ≤h yh and so yh ∈ (g/h)+. 2 definition 3.17. let g and h are ordered polygroups. a mapping f : g −→ h is said to be exact if f(g+) = h+. definition 3.18. two ordered polygroups g and h are said to be isomorphic if there is a polygroup isomorphism f : g −→ h that is also an order isomorphism. if two ordered polygroups g and h are isomorphic we write g '̇ h. theorem 3.19. for ordered polygroups g and h, the following are equivalent: (1) g'̇h, (2) there is an exact polygroup isomorphism f : g −→ h. proof. (1) ⇒ (2) if g and h are isomorphic, there is a polygroup isomorphism f : g −→ h which is also an order isomorphism. by theorem 3.15, f(g+) ⊆ h+. let g = f−1. obviously, g satisfies the conditions of theorem 3.15. hence, g(h+) ⊆ g+ whence h+ = f(g(h+)) ⊆ f(g+). thus h+ = f(g+) and so (2) holds. (2) ⇒ (1) it is obvious. 2 theorem 3.20. let g and h be ordered polygroups and f : g −→ h be an exact polygroup homomorphism. then imf '̇ g/kerf. proof. we first observe that kerf is a convex normal subpolygroup of g and so g/kerf is an ordered polygroup. by first isomorphism theorem of polygroups there is an isomorphism φ : g/kerf ' imf which φ(xk) = f(x) where k = kerf. it remains that we prove φ is exact. let xk ∈ (g/k)+. then egk ≤k xk whence k ≤ x, for some k ∈ k, and so eh = f(k) ≤ f(x) whence φ(xk) = f(x) ∈ (imf)+. conversely, if f(x) ∈ (imf)+ ⊆ h+, since f is exact, there exists g ∈ g+ such that f(x) = f(g). consequently, xk = gk and so x ∈ gk ≥ k, for some k ∈ k. thus, xk ∈ (g/k)+ ⇔ φ(xk) = f(x) ∈ (imf)+ proving φ is exact. it now follows by theorem 3.19 that g/kerf '̇ imf. 2 39 m. bakhshi, r. a. borzooei references [1] t. s. blyth, lattices and ordered algebraic structures, springer-verlag, london, 2005. [2] p. corsini, prolegomena of hypergroup theory, 2nd edition, aviani editor, 1993. [3] p. corsini and v. leoreanu, applications of hyperstructure theory, kluwer academic publishers, dordrecht, 2003. [4] s. ioulidis, polygroups et certain de leurs properties, bull. greek math. soc. 22 (1981), 95-104. [5] f. marty, sur une generalization de la notion de group, 8th congress math. scandenaves, stockholm, (1934), 45-49. 40 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 37, 2019, pp. 39-48 39 solving some specific tasks by euler's and fermat's little theorem viliam ďuriš* abstract euler's and fermat's little theorems have a great use in number theory. euler's theorem is currently widely used in computer science and cryptography, as one of the current encryption methods is an exponential cipher based on the knowledge of number theory, including the use of euler's theorem. therefore, knowing the theorem well and using it in specific mathematical applications is important. the aim of our paper is to show the validity of euler's theorem by means of linear congruences and to present several specific tasks which are suitable to be solved using euler's or fermat's little theorems and on which the principle of these theorems can be learned. some tasks combine various knowledge from the field of number theory, and are specific by the fact that the inclusion of euler's or fermat's little theorems to solve the task is not immediately apparent from their assignment. keywords: euler's theorem, coding, fermat's little theorem, linearcongruences, cryptology, primality testing, matlab 2010 ams subject classification: 11a07, 14g50† * department of mathematics, faculty of natural sciences constantine the philosopher university in nitra, tr. a. hlinku 1, 949 74 nitra, slovakia; vduris@ukf.sk. †received on december 1st, 2019. accepted on december 10th, 2019. published on december 31th, 2019. doi: 10.23755/rm.v37i0.485. issn: 1592-7415. eissn: 2282-8214. ©viliam ďuriš. this paper is published under the cc-by licence agreement. viliam ďuriš 40 1 introduction at present, mathematics provides apparatus for virtually all modern coding systems. the first coding system, with only two symbols a dot and a comma, was morse code which was used to send the first coding message by american inventor samuel f. b. morse in 1844 using an electric telegraph. binary code encoding has become a better code for message encryption at a later time, in which each coded word consists of blocks of ones and zeroes, and this encoding is still used today [1]. significant developments in coding occurred in the 20th century when euler's theorem was used for coding and the coded text could be broadcasted publicly with the message kept secret. the principle of this coding is that the sender assigns a number to a coded word (e.g. 74) and encodes that word using two additional numbers (e.g. 247 and 5), which may be public in such a way that 745(mod 247) = 120 is calculated. this will give you a message “120” that will be sent to the recipient. since numbers 247 and 5 are public keys, anyone can encode the message "74" to "120", but only the actual recipient can decode it correctly. the essence of the key to the cipher lies in the fact that only the recipient knows that number 247 was compiled as the product of primes 𝑝 = 13 and 𝑞 = 19 and using euler's theorem searches for the value x for which the congruence is 5𝑥 ≡ 1(mod [(𝑝 − 1)(𝑞 − 1)]). the recipient can easily get the result 𝑥 = 173. using this figure, the remainder by dividing 120173 by number 247 is found, thereby obtaining the original coded word 74 which can already be assigned to the message [2]. in practice, with this type of coding, the product of two very large primes is used, where the decomposition of the thus obtained number is very difficult, virtually impossible for someone who does not know the product of which two primes have been executed. despite the fact that the principle of this coding was discovered and started to be used practically in the 20th century, it is actually derived from euler's knowledge from the 18th century. most of the results in mathematics in the 18th century stemmed from efforts to solve various separate problems discovered in the 17th century. in this period, the theory of numbers remained more or less in the background, and the only mathematician who dealt with the issues of number theory after 1730 to a greater extent was euler. in 1736, he proved fermat's little theorem which claims that for any natural number a and prime p, 𝑎𝑝−1 ≡ 1(mod 𝑝). later in 1760, after the introduction of euler's totient function 𝜑(𝑛) he demonstrated the validity of congruence 𝑎𝜑(𝑚) ≡ 1(mod 𝑚) which is a generalization of fermat's little theorem. euler also dealt with many other fermat's claims. he also achieved several accomplishments related to the decomposition of certain expressions with the powers of natural numbers and to perfect and friendly numbers. he was also interested in the problem of integer roots of pell's equation, about which he published several articles, and presented his own solving some specific tasks by euler's and fermat's little theorem 41 method of solution. euler has introduced a number of concepts into number theory, such as the quadratic residue and the quadratic nonresidue in the law of quadratic reciprocity and his work and accomplishments, despite the lack of exact evidence in several areas, were generally accepted by respected mathematicians of the 18th and 19th centuries (e.g. gauss or legendre) [3]. we would like to mention there's also another principle of coding using fibonacci numbers and can be seen in [4]. 2 euler's theorem, fermat's little theorem let us consider two natural numbers a, m where (𝑎,𝑚) = 1. euler's theorem [5] then states that 𝑚|𝑎𝜑(𝑚) − 1, or that congruence 𝑎𝜑(𝑚) ≡ 1(mod 𝑚) applies. the symbol 𝜑(𝑛) denotes the number of natural numbers smaller than n and relatively prime to n and is called euler's totient function [6]. to show the validity of euler's theorem, we will use the basic properties of congruences and residue classes. let's write all relatively prime numbers to 𝑚 less than 𝑚. these are 𝑥1,𝑥2,⋯,𝑥𝜑(𝑚). let us further consider the sequence 𝑎𝑥1,𝑎𝑥2,⋯,𝑎𝑥𝜑(𝑚) and indirectly show that all its members are relatively prime to 𝑚. if ∃𝑖:(𝑎𝑥𝑖,𝑚) = 𝑑 > 1, then 𝑑|𝑎𝑥𝑖 ∧ 𝑑|𝑚. then (𝑑,𝑎) = 1, because (𝑎,𝑚) = 1 ∧ 𝑑|𝑚. in that 𝑑|𝑥𝑖 and numbers 𝑚,𝑥𝑖 are commensurable which is a controversy. furthermore, let us indirectly show that numbers 𝑎𝑥1,𝑎𝑥2,⋯,𝑎𝑥𝜑(𝑚) are noncongruent modulo 𝑚. ∃𝑖,𝑗:𝑎𝑥𝑖 ≡ 𝑎𝑥𝑗(mod 𝑚). then 𝑚|𝑎𝑥𝑖 − 𝑎𝑥𝑗 = 𝑎(𝑥𝑖 − 𝑥𝑗) ∧ (𝑎,𝑚) = 1, of which 𝑚|𝑥𝑖 − 𝑥𝑗 and then 𝑥𝑖 ≡ 𝑥𝑗(mod 𝑚), which is a controversy, because 𝑥𝑖 are differently lower from each other than 𝑚, and therefore cannot give the same remainder after division by 𝑚. before completing the evidence, we recall, that based on the basic properties of congruences, [7] we know that integers 𝑎 and 𝑏 belong to the same class 𝑅𝑖 modulo 𝑚 just when 𝑎 ≡ 𝑏 (mod 𝑚). if we first express the numbers a, b ∈ 𝑅𝑖 in the form 𝑎 = 𝑚 ∙ 𝑞 + 𝑖,𝑏 = 𝑚 ⋅ 𝑝 + 𝑖, then 𝑎 − 𝑏 = 𝑚(𝑞 − 𝑝), which means 𝑚|𝑎 − 𝑏, and thus 𝑎 ≡ 𝑏(mod 𝑚). on the other hand, let us assume that 𝑎 ≡ 𝑏 (mod 𝑚) and 𝑎 = 𝑚𝑞 + 𝑖,𝑏 = 𝑚𝑝 + 𝑗 (0 ≤ 𝑖, 𝑗 < 𝑚). for example, it is supposed that 𝑖 > 𝑗. since 𝑎 ≡ 𝑏 (mod 𝑚), 𝑚|𝑎 − 𝑏. but then 𝑚 (𝑎 − 𝑏) = [𝑚(𝑞 − 𝑝) + (𝑖 − 𝑗)], of which 𝑚 (𝑖 − 𝑗). this would be a controversy though, because 0 < 𝑖 − 𝑗 < 𝑚. similarly, a controversy arises even with the assumption 𝑖 < 𝑗. therefore 𝑖 = 𝑗 must hold, hence the numbers 𝑎 and 𝑏 belong to the same residual class modulo 𝑚 with 𝑎 ≡ 𝑏 (mod 𝑚). as the class representative does not matter, we can write 𝑎𝑥1 ∙ 𝑎𝑥2 ∙ ⋯∙ 𝑎𝑥𝜑(𝑚) ≡ 𝑥1 ∙ 𝑥2 ∙ ⋯∙ 𝑥𝜑(𝑚)(mod 𝑚). then 𝑚|𝑎𝑥1 ∙ 𝑎𝑥2 ∙ ⋯∙ 𝑎𝑥𝜑(𝑚) − 𝑥1 ∙ viliam ďuriš 42 𝑥2 ∙ ⋯∙ 𝑥𝜑(𝑚) = (𝑎 𝜑(𝑚) − 1)𝑥1 ∙ 𝑥2 ∙ ⋯∙ 𝑥𝜑(𝑚). since (𝑚,𝑥1 ∙ 𝑥2 ∙ ⋯∙ 𝑥𝜑(𝑚)) = 1, then 𝑚|𝑎 𝜑(𝑚) − 1. if 𝑚 is a prime number and 𝑝 ∤ 𝑎, then 𝜑(𝑚) = 𝑚 − 1 and we get fermat's little theorem 𝑎𝑝−1 ≡ 1(mod 𝑝) directly from euler's theorem. a variation of fermat's little theorem can be used to test primality [8]. if there exists 𝑎 ∈ {2,⋯,𝑛 − 1}, 𝑛 > 3, where 𝑎𝑛−1 ≢ 1(mod 𝑛), then n is a composite number and we call it fermat's witness for the compositeness of number 𝑛 [9]. fermat's primality test can be suitably algorithmically presented in a selected computational environment (e.g. matlab). the algorithm consists of two steps: a) we randomly select number a for which 1 < 𝑎 < 𝑛 b) it is tested whether congruence 𝑎𝑛−1 ≡ 1(mod 𝑛) is satisfied if congruence 𝑎𝑛−1 ≡ 1(mod 𝑛) is satisfied, the number 𝑛 may or may not be a prime number. if congruence is not satisfied, the number 𝑛 is not a prime and number 𝑎 is the fermat's witness for the compositeness of 𝑛. fermat's primality test works well for numbers that are not products of prime numbers different from each other. it can be demonstrated that if we test the number 𝑛, which is not the product of different prime numbers, hence there is such a prime 𝑝 where 𝑝2|𝑛, then with a probability of at least 75% we can choose between numbers 2,⋯,𝑛 − 1 such a number which will be the fermat's witness for the compositeness of 𝑛 [9]. first, in matlab, we create a function that helps us test congruence 𝑎𝑛−1 ≡ 1(mod 𝑛) generally for two given numbers 𝑎 and 𝑛. the function will calculate the value 𝑎𝑛−1 mod 𝑛 which we will compare with 1 within the residue classes. function res = test_congruence(a, n) expn = n 1; res = 1; while expn ~= 0 if rem(expn, 2) == 1 res = rem(res * a, n); end expn = floor(expn / 2); a = rem(a^2, n); end the second function randomly generates 𝑎 ∈ {2,⋯,𝑛 − 1} and we look for the fermat's witness for the compositeness of 𝑛. solving some specific tasks by euler's and fermat's little theorem 43 function test_fermat(n, cnt) fo = false; ii = 1; while (ii <= cnt) && (~fo) a = 1 + unique(ceil((n 2) * rand(1, 1))); tc = test_congruence(a, n); if(tc ~= 1) fermat_witness = a; fo = true; else ii = ii + 1; end end if fo disp(['number ' num2str(n) ' is a composite number.']); disp(['number ' num2str(fermat_witness) ' is a witness for the compositeness of ' num2str(n) '.']); else disp(['number ' num2str(n) ' can be a prime or a composite number.']); end the created test function is activated through the command line for any number 𝑛. >> test_fermat(223, 1) number 223 can be a prime or a composite number. >> test_fermat(273, 1) number 273 is a composite number. number 220 is a witness for the compositeness of 273. 3 euler's, fermat's little theorem applications in this section, we have selected and compiled a number of specific tasks [10], [11] that guide on how to solve certain types of tasks using euler's or fermat's viliam ďuriš 44 little theorem. we remark that for a natural number 𝑛 greater than 1 in canonical decomposition 𝑛 = 𝑝1 𝛼1 …𝑝𝑘 𝛼𝑘 it holds that 𝜑(𝑛) = 𝑛(1 − 1 𝑝1 )(1 − 1 𝑝2 )…(1 − 1 𝑝𝑘 ) [6] example 3.1. first, we demonstrate that if we divide number 1724 by number 39, the remainder 1 is obtained. solution. it is determined that 𝑎 = 17, 𝑚 = 39. (39,17) = 1 and euler's theorem can be applied. let us calculate 𝜑(𝑚) = 𝜑(39) = 39(1 − 1 3 )(1 − 1 13 ) = 24. then according to euler's theorem 39|1724 − 1, thus ∃𝑘 ∈ ℤ:1724 − 1 = 39𝑘. then we can write 1724 = 39𝑘 + 1, and 1 is obtained as a remainder. example 3.2. it is demonstrated that 𝑝 and 8𝑝2 + 1 are simultaneously prime just when 𝑝 = 3. solution. 1. first, 𝑝 = 3. then 8𝑝2 + 1 = 8 ∙ 9 + 1 = 73, which is a prime. 2. now let 𝑝 and 8𝑝2 + 1 be prime numbers simultaneously. 8𝑝2 + 1 is adjusted as 8𝑝2 + 1 = 8𝑝2 − 8 + 9 = 8(𝑝2 − 1) + 9. let 𝑝 be a prime number other than 3. then (𝑝,3) = 1 a 3|𝑝𝜑(3) − 1 = 𝑝2 − 1 . since 3|𝑝2 − 1, then 8(𝑝2 − 1) ∧ 3|9, then 3|8(𝑝2 − 1) + 9 = 8𝑝2 + 1 and 8𝑝2 + 1 would not be a prime number, which is a controversy, thus 𝑝 = 3. example 3.3. we show if 𝑎 is not divisible by 5, then only one number from 𝑎2–1, 𝑎2 + 1 is divisible by 5. solution. if 𝑎 is a multiple of 5, according to euler's theorem 𝑎4 − 1 is a multiple of 5. then only one of numbers 𝑎2 − 1 and 𝑎2 + 1 is a multiple of 5. they both concurrently cannot be, otherwise their difference would also be divisible by number 5, which is not, since (𝑎2 + 1) − (𝑎2 − 1) = 2. example 3.4. we find all primes 𝑝 for which 5𝑝 2 + 1 ≡ 0(𝑚𝑜𝑑 𝑝2). solution. the prime number 𝑝 = 5 does not satisfy the task and at the same time (𝑝,5) = 1. then according to euler's theorem 5𝑝−1 ≡ 1 (mod 𝑝). by exponentiation to 𝑝 + 1 we get 5𝑝 2−1 ≡ 1 (mod 𝑝), of which 5𝑝 2 ≡ 5 (mod 𝑝). next, the task assignment states that 5𝑝 2 + 1 ≡ 0 (mod 𝑝2), that implies 5𝑝 2 ≡ −1 (mod 𝑝2) and also 5𝑝 2 ≡ −1 (mod 𝑝). then congruences 5𝑝 2 ≡ solving some specific tasks by euler's and fermat's little theorem 45 5 (mod 𝑝) and 5𝑝 2 ≡ −1 (mod 𝑝) hold that 5 ≡ −1 (mod 𝑝). then 𝑝|6. in that 𝑝 = 2 or 𝑝 = 3. for 𝑝 = 2 it holds that 54 + 1 ≡ 14 + 1 = 2 ≢ 0 (mod 4). for 𝑝 = 3 it holds that 59 + 1 = 56 ∙ 53 + 1 ≡ 53 + 1 = 126 ≡ 0 (mod 9). then, the only prime number satisfying the task is 𝑝 = 3. example 3.5. for the odd number 𝑚 > 1 we find the remainder after division of 2𝜑(𝑚)−1 by number 𝑚. solution. euler's theorem implies that 2𝜑(𝑚) ≡ 1 ≡ 1 + 𝑚 = 2 ∙ 1+𝑚 2 = 2𝑟(mod 𝑚) where 𝑟 is a natural number 0 ≤ 𝑟 < 𝑚. the basic properties of congruences [7] determine that if 𝑎 ≡ 𝑏(mod 𝑚) and 𝑑 is an integer with properties 𝑑|𝑎, 𝑑|𝑏, (𝑑,𝑚) = 1, then 𝑎 𝑑 ≡ 𝑏 𝑑 (mod 𝑚). indeed 𝑎 = 𝑎1𝑑, 𝑏 = 𝑏1𝑑 and according to assumption 𝑚|(𝑎 − 𝑏), it holds that 𝑚|𝑑(𝑎1 − 𝑏1). since (𝑑,𝑚) = 1, it holds that 𝑚|(𝑎1 − 𝑏1). then 𝑎1 ≡ 𝑏1(mod 𝑚), thus 𝑎 𝑑 ≡ 𝑏 𝑑 (mod 𝑚). then, however, we can divide both sides of the congruence 2𝜑(𝑚) ≡ 2𝑟(mod 𝑚) by their common divisor, number 2, which is relatively prime to the modulo. then 2𝜑(𝑚)−1 ≡ 𝑟(mod 𝑚), and thus the remainder sought is 𝑟 = 1+𝑚 2 . example 3.6. we find the last two digits of number 13742. solution. the task leads to the search for the remainder when dividing number 13742 by number 100. since (137,100) = 1, according to euler's theorem it holds that 137𝜑(100) − 1 is a multiply of 100 (100|137𝜑(100) − 1). next 𝜑(100) = 100(1 − 1 2 )(1 − 1 5 ) = 40. then 13740 − 1 is a multiply of 100. therefore 13742 = 137213740 − 1372 + 1372 = 1372(13740 − 1) + +1372 = 1372(13740 − 1) + (100 + 37)2 = 100𝑘 + (100 + 37)2. next, we use the formula (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏2. then 13742 = 100𝑘 + 100𝑙 + 372 = 100𝑛 + 1369 = 100𝑛 + 1300 + 69 = 100𝑚 + 69. thus, the remainder sought is 69. example 3.7. we find the last 2 digits of number 𝑎 = 13747. solution. the last 2 digits of number a are again obtained as the remainder after dividing the number 𝑎 by 100. (100,137) = 1 and euler's theorem can be applied. then 100|137𝜑(100) − 1, thus 100|13740 − 1. then 13740 − 1 is a multiply of 100 and 13740 = 100𝑘 + 1. viliam ďuriš 46 let us calculate 13747 = 13740 ∙ 1377 = (100𝑘 + 1) ∙ 1377 = 100𝑘 ∙ 1377 + 1377 number 100𝑘 ∙ 1377 cannot specify last 2 digits (ending with 2 zeroes), and so just number 1377 has the last 2 digits of the given number 𝑎. next, the binomial theorem is applied. 1377 = (130 + 7)7 = ( 7 0 )1307 + ( 7 1 )1306 ∙ 7 + ⋯+ ( 7 6 )130 ∙ 76 + ( 7 7 )77. in this summation only the members ( 7 6 )130 ∙ 76 a ( 7 7 )77 decide the last two digits (other contribute zeroes in last two digits). their summation is calculated as ( 7 6 )130 ∙ 76 + ( 7 7 )77 = 130 ∙ 77 + 77 = 131 ∙ 77 = 107884133. overall, we get the last 2 digits of the number 𝑎 = 13747 which are 33. example 3.8. we find the remainder when dividing (8570 + 1932)16 by number 21. solution. according to the binomial theorem 8570 = (84 + 1)70 = ( 70 0 )8470 + +( 70 1 )8469 ∙ 1 + ⋯+ ( 70 69 )84 ∙ 169 + ( 70 70 )170. we see that number 21 can be removed from every member except the last one. then 8570 = (84 + 1)70 = 21𝑛 + 1. because 𝜑(21) = 12, 1912 − 1 is a multiply of 21 (applying euler's theorem), then 1932 = 198(1924 − 1) + 198 = 21𝑚 + 198. therefore (8570 + 1932)16 = (21𝑛 + 1 + 21𝑚 + 198)16 = (21𝑘 + 1 + (21 − 2)8)16 = (21𝑞 + 1 + 28)16 = (21𝑟 + 5)16 = 21𝑡 + 516 = 21𝑡 + 54(512 − 1) + 54 = 21𝑡 + 21𝑟 + 625 = 21𝑢 + 16. the remainder sought is 16. example 3.9. we demonstrate if 𝑥𝑝 + 𝑦𝑝 = 𝑧𝑝 where 𝑝 is a prime number, then 𝑥 + 𝑦 – 𝑧 is a multiply of 𝑝. solution. according to fermat's little theorem, if 𝑝 is a prime and 𝑝 ∤ 𝑥, then 𝑥𝑝−1 ≡ 1(mod 𝑝), that means 𝑝|𝑥𝑝−1 − 1 and thus 𝑝|𝑥(𝑥𝑝−1 − 1) = 𝑥𝑝 − 𝑥. similarly, 𝑝|𝑦𝑝 − 𝑦, 𝑝|𝑧𝑝 − 𝑧. therefore we can write 𝑥𝑝 = 𝑝𝑡1 + 𝑥, 𝑦 𝑝 = 𝑝𝑡2 + 𝑦 a 𝑧 𝑝 = 𝑝𝑡3 + 𝑧. if we substitute in the equation 𝑥 𝑝 + 𝑦𝑝 = 𝑧𝑝, we get 𝑝(𝑡3 − 𝑡1 − 𝑡2) = 𝑥 + 𝑦 − 𝑧 after adjustment, thus 𝑥 + 𝑦 – 𝑧 is a multiply of 𝑝. these examples are the basis for understanding the principle of working with large numbers using congruences through euler's and fermat's little theorem. congruences are a modern and irreplaceable security tool for protecting data by a public key. it is important to realize that the public key uses such large numbers for which there is no effective method of decomposing to primes even in today's modern computer age. that is why euler's theorem plays its role in encryption even today, when encryption uses keys of up to 256 bits in length solving some specific tasks by euler's and fermat's little theorem 47 and deciphering the word while trying out all the options would probably take more years than the age of the universe is. 4 conclusion the paper points out some specific applications suitable for presenting and understanding the basic principle of euler's and fermat's little theorems which are currently used in cryptography. leonhard paul euler was such a great mathematician that many of the principles he had known almost 300 years ago were actually used by contemporary society. euler, nicknamed as a "magician" in his time, had a great influence not only on number theory, but also on mathematical analysis or graph theory. he introduced many mathematical symbols such as the letter sigma σ to denote the sum, or introduced numbers such as 𝑒 and 𝑖, whereas 𝑒 is probably the most important number of the whole mathematics [12] and occurs in various areas. when mathematical intelligencer in 2004 asked readers to vote for "the most beautiful theorem of mathematics", euler's identity 𝑒𝑖𝜋 + 1 = 0 won by a large margin [13]. it is a formula that connects the five most important symbols of mathematics. several mathematicians have marked this equation as so mystical that it can only be reproduced and its consequences continually explored. in addition to euler's theorem itself and its evidence by means of linear congruences, we also wanted to highlight the work and the "size" of leonhard euler and his key contribution to number theory. viliam ďuriš 48 references [1] bose r. 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(1994). prime numbers and computer methods for factorisation. 2nd ed., progress in mathematics 126, birkhauser. [9] pommersheim j. e., marks t. k., flapan e. l. (2010). number theory. usa: wiley, 753 p., isbn 978-0-470-42413-1. [10] davydov u. s., znám š. (1972). teória čísel – základné pojmy a zbierka úloh. bratislava, slovak republic: spn. [11] apfelbeck a. (1968). kongruence. prague, czech republic: mladá fronta. [12] clifford a. p. (2011). the math book: from pythagoras to the 57th dimension, 250 milestones in the history of mathematics. new york, ny: sterling publishing, isbn: 9781402757969. [13] jackson t. (2017). mathematics: an illustrated history of numbers (ponderables: 100 breakthroughs that changed history) revised and updated edition. new york, ny: shelter harbor press, isbn: 9781627950954. approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 37, 2019, pp. 49-60 49 mahgoub transform on boehmians yogesh khandelwal* priti chaudhary† abstract boehmian’s space is established utilizing an algebraic way that approximates identities or delta sequences and appropriate convolution. the space of distributions can be related to the proper subspace. in this paper, firstly we establish the appropriate boehmian space, on which the mahgoub transformation can be described& function space k can be embedded. we add to more in this, our definitions enhance mahgoub transform to progressively wide spaces. we additionally explain the functional axioms of mahgoub transform on boehmians. lastly toward the finishing of topic, we analyze with specify axioms and properties for continuity and the enlarged mahgoub transform, also its inverse regards to ∆-convergence and δ. keywords: mahgoub transform; the space 𝔹(𝔛); the space𝔹(𝔛𝔐); boehmian spaces. 2010 ams subject classification: 44a99; 44a40;46f99; 20c20.‡ * department of mathematics, jaipur national university, jaipur, rajasthan (india); yogeshmaths81@gmail.com. † department of mathematics, jaipur national university, jaipur, rajasthan (india). ‡ received on july 3rd, 2019. accepted on august 9th, 2019. published on december 31th, 2019. doi: 10.23755/rm.v37i0.468. issn: 1592-7415. eissn: 2282-8214. ©khandelwal and chaudhary. this paper is published under the cc-by licence agreement. yogesh khandelwal and priti chaudhary 50 1. introduction the mahgoub transform [8] which is denoted by the operator 𝔐(. ) and mahgoub transform of 𝔗(𝓉∗) is defined by: 𝔐( 𝔗(𝓉∗)) = 𝔼(𝜗) = 𝜗 ∫ 𝔗(𝓉∗) ∞ 0 𝑒−𝜗𝓉 ∗ 𝑑𝓉 ∗, 𝓉 ∗ ≥ 0, (1.1) and 𝜌1 ≤ 𝜗 ≤ 𝜌2. in a set ; 𝔸 = {𝔗(𝓉∗): ∃𝕄, 𝜌1, 𝜌2 > 0 . |𝔗(𝓉 ∗)| < 𝕄𝑒 |𝓉 ∗| 𝜌𝑗 } , (1.2) where 𝜌1 and 𝜌2(may be finite or infinite), the constant 𝕄 must be finite. an existence’s mahgoub transform of 𝔗(𝓉∗) is essential for 𝓉∗ ≥ 0, a piece wise continuous and of exponential order is required, else it does not exist. convolution theorem for mahgoub transform [9-11]: if 𝔐( 𝔗(𝓉 ∗)) = 𝔼(𝜗) and 𝔐( 𝔓(𝓉∗)) = 𝕎(𝜗) then 𝔐(𝔗(𝓉∗) ⋆ 𝔓(𝓉 ∗)) = 1 ϑ 𝔐(𝔗(𝓉 ∗))𝔐(𝔓(𝓉∗)) = 1 ϑ 𝔼(ϑ)𝕎(ϑ) (1.3) linearity property of mahgoub transform: if 𝔐( 𝔗(𝓉∗)) = 𝔼(𝜗), 𝔐( 𝔓(𝓉∗)) = 𝕎(𝜗) then 𝔐{𝔞𝔗(𝓉∗) + 𝔟 𝔓(𝓉∗)} = 𝔞 𝔐(𝔗(𝓉 ∗)) + 𝔟 𝔐(𝔓(𝓉∗)) (1.4) 2. boehmian space boehmians was first developed as a generalization’s standard mikusinski operators [2]. the formation necessary for boehmians satisfying the following axioms. i. a non empty set 𝔄; ii. a semi group(𝜑,⊛) which is commutative; iii. ⊗: 𝔄 × 𝜑 → 𝔄 s.t.∀ ξ ∈ 𝔄 and 𝜂1, 𝜂2 ∈ 𝜑 , ξ ⊗ (𝜂1 ⊛ 𝜂2) = (ξ ⊗ 𝜂1) ⊗ 𝜂2; iv. a collection ∆ ⊂ 𝜑𝑁such that a) ifξ1, ξ2 ∈ 𝔄 , (𝜂𝑛) ∈ ∆ , ξ1 ⊗ 𝜂𝑛 = ξ2 ⊗ 𝜂𝑛 ∀ 𝑛then ξ1 = ξ2; b) if (𝜂𝑛), (𝜏𝑛) ∈ ∆, then (𝜂𝑛 ⊗ 𝜏𝑛) ∈ ∆. where elements of ∆ are known as delta sequences. consider ℋ = {(ξ𝑛, 𝜂𝑛 ): ξ𝑛 ∈ 𝔄 , (𝜂𝑛 ) ∈ ∆, ξ𝑛 ⊗ 𝜂𝑚 = ξ𝑚 ⊗ 𝜂𝑛 ∀ 𝑚, 𝑛 ∈ 𝑁}, now if (ξ𝑛, 𝜂𝑛 ), (∅𝑛, 𝜏𝑛 ) ∈ ℋ then ξ𝑛 ⊗ 𝜏𝑚 = ∅𝑚 ⊗ 𝜂𝑛 , ∀ 𝑚, 𝑛 ∈ 𝑁. an reckon of mahgoub transform for boehmians 51 we say that (ξ𝑛 , 𝜂𝑛 )~(∅𝑛, 𝜏𝑛). where ~ is an equivalence relation inℋ.the set of equivalence classes in ℋ is denoted asℌ. elements of ℌ are said to be boehmians. we assume that there is a canonical embedding between ℌ and𝔄, expressed as ξ → ξ𝑛⊗𝜂𝑛 𝜂𝑛 ,where⊗ can also be extended in ℌ × 𝔄 by ξ𝑛 𝜂𝑛 ⊗ 𝜏 = ξ𝑛⊗𝜏 𝜂𝑛 . in ℌ, there are two types of convergence is given by i. if ℶ𝑛 → ℶ as 𝑛 → ∞which belongs to𝔄,𝓀 ∈ 𝜑 is any fixed element, then ℶ𝑛 ⊗ 𝓀 → ℶ ⊗ 𝓀 as 𝑛 → ∞in 𝔄. ii. if ℶ𝑛 → ℶas 𝑛 → ∞ in 𝔄 and 𝜆𝑛 ∈ ∆ then ℶ𝑛 ⊗ 𝜆𝑛 → ℶas 𝑛 → ∞in 𝔄. an operation ⊗can be extended inℌ × 𝜑 as per condition: if [ ℶ𝑛 𝜂𝑛 ] ∈ ℌand 𝓀 ∈ 𝜑 then [ ℶ𝑛 𝜂𝑛 ] ⊗ 𝓀 = [ ℶ𝑛⊗𝓀 𝜂𝑛 ]. now convergence in ℌas following: 1. a sequence (𝜍𝑛)in ℌis called𝛿– convergent to 𝜍 in ℌ, i.e. 𝜍𝑛 𝛿 → 𝜍 if ∃ (𝜂𝑛) ∈ δ such that(𝜍𝑛 ⊗ 𝜂𝑛 ), (𝜍 ⊗ 𝜂𝑛 ) ∈ 𝔄, ∀ 𝑛 ∈ 𝑁 and (𝜍𝑛 ⊗ 𝜂𝓀 ) → (𝜍 ⊗ 𝜂𝓀) as 𝑛 → ∞ in 𝔄, ∀ 𝓀, 𝑛 ∈ 𝑁. 2. a sequence (𝜍𝑛) in ℌ is said to be ∆ convergent to 𝜍 in ℌ i.e. 𝜍𝑛 ∆ → 𝜍 , if ∃(𝜂𝑛) ∈ δ such that(𝜍𝑛 − 𝜍) → 0 as 𝑛 → ∞ which belongs to 𝔄 . for more details, see [3-6]. 3. the boehmian space 𝔹(𝖃): denoted by 𝔖+(ℝ) and 𝒞0+ ∞ (ℝ)are the space’s smooth function over ℝ and the schwarz space’s test function’s compact support over ℝ+ where ℝ+ = (0, ∞) respectively. we have found vital results for the structure of boehmian space 𝔹(𝔛)where 𝔛 = (𝔖+, 𝒞0+ ∞ , ∆+). lemma 3.1: 1) if 𝔡1, 𝔡2 ∈ 𝒞0+ ∞ (ℝ)then 𝔡1 ⋆ 𝔡2 ∈ 𝒞0+ ∞ (ℝ)(closure). 2) if 𝔉1, 𝔉2 ∈ 𝔖+(ℝ)and 𝔡1 ∈ 𝒞0+ ∞ (ℝ)then (𝔉1 + 𝔉2) ⋆ 𝔡1 = 𝔉1 ⋆ 𝔡1 + 𝔉2 ⋆ 𝔡1 (distributive). 3) 𝔡1 ⋆ 𝔡2 = 𝔡2 ⋆ 𝔡1∀𝔡1, 𝔡2 ∈ 𝒞0+ ∞ (ℝ) (commutative). yogesh khandelwal and priti chaudhary 52 4) if 𝔉 ∈ 𝔖+(ℝ), 𝔡1, 𝔡2 ∈ 𝒞0+ ∞ (ℝ)then (𝔉 ⋆ 𝔡1) ⋆ 𝔡2 = 𝔉 ⋆ (𝔡1 ⋆ 𝔡2)(associative). definition3.2: a sequence (𝜂𝑛) of function from 𝒞0+ ∞ (ℝ)is said to be in∆+. if ∆+ 1 : ∫ 𝜂𝑛 (𝜉)𝑑𝜉 = 1. ℝ+ ∆+ 2 : ∫ |𝜂𝑛(𝜉)|𝑑𝜉 ≤ 𝑚,ℝ+ where 𝑚 is a positive integer; ∆+ 3 ∶ 𝑆𝑢𝑝𝑝 𝜂𝑛 (𝜉) ⊂ (0, 𝜖𝑛), 𝜖𝑛 → 0 as 𝑛 → ∞. i.e.(𝜂𝑛) shrink to zero as 𝑛 → ∞.every member of ∆+ is known as an approximation identity or a delta sequences. in all manners delta sequences arise in numerous parts of mathematics, however likely the very important application are those in the presupposition’s generalized functions. the fundamental application of delta sequence is the regularization’s established functions and ahead we can be utilized to characterize the convolution product and its established functions. lemma 3.3: if(𝜂𝑛),(𝜏𝑛) ∈ ∆+, then 𝑠𝑢𝑝𝑝(𝜂𝑛 ⋆ 𝜏𝑛) ⊂ 𝑠𝑢𝑝𝑝𝜂𝑛 + 𝑠𝑢𝑝𝑝𝜏𝑛 . lemma 3.4: if 𝔡1, 𝔡2 ∈ 𝒞0+ ∞ (ℝ) then so is 𝔡1 ⋆ 𝔡2 and ∫ |𝔡1 ⋆ 𝔡2| ≤ℝ+ ∫ |𝔡1|ℝ+ ∫ |𝔡2|ℝ+ . theorem 3.5: let𝔉1, 𝔉2 ∈ 𝔖+(ℝ) and (𝜂𝑛 ) ∈ ∆+ such that 𝔉1 ⋆ 𝜂𝑛 = 𝔉2 ⋆ 𝜂𝑛 . where 𝑛 = 1,2,3, …, then 𝔉1 = 𝔉2 in 𝔖+(ℝ). proof: to prove that 𝔉1 ⋆ 𝜂𝑛 = 𝔉1. let k be a compact support accommodating the 𝑠𝑢𝑝𝑝𝜂𝑛 for each𝑛 ∈ 𝑁. by using ∆+ 1 , we write |𝜉 𝑘 𝐷𝑚(𝔉1 ⋆ 𝜂𝑛 − 𝔉1)(𝜉)| ≤ ∫ |𝜂𝑛(𝜏)| |𝜉 𝑘 𝐷𝑚(𝔉1(𝜉 − 𝜏) − 𝔉1(𝜉))| 𝑑𝜏 𝐾 (3.1) the mapping 𝜏 → 𝔉1 𝜏 where𝔉1 𝜏 = 𝔉1(𝜉 − 𝜏), is uniformly continuous from ℝ+ → ℝ+. by using axiom∆+ 3 that 𝑠𝑢𝑝𝑝𝜂𝑛 → 0 as 𝑛 → ∞, now we choose 𝑟 > 0;𝑠𝑢𝑝𝑝𝜂𝑛 ⊆ [0, 𝑟] for large 𝑛 and 𝜏 < 𝑟, that is |𝔉1(𝜉 − 𝜏) − 𝔉1(𝜉)| = |𝔉1 𝜏 − 𝔉1| < 𝜖𝑛 𝑀 (3.2) hence using ∆+ 2 and eq’s. (3.2), (3.1) we get an reckon of mahgoub transform for boehmians 53 |𝜉 𝑘 𝐷𝑚(𝔉1 ⋆ 𝜂𝑛 − 𝔉1)(𝜉)| < 𝜖𝑛 → 0 as 𝑛 → ∞. thus 𝔉1 ⋆ 𝜂𝑛 → 𝔉1 in𝔖+(ℝ). similarly, we prove that 𝔉2 ⋆ 𝜂𝑛 → 𝔉2in 𝔖+(ℝ) ⌂ theorem 3.6:if 𝔉𝑛 → 𝔉in 𝔖+(ℝ)as𝑛 → ∞ and 𝔡 ∈ 𝒞0+ ∞ (ℝ) then lim 𝑛→∞ 𝔉𝑛 ⋆ 𝔡 = 𝔉 ⋆ 𝔡. proof: using theorem we get |𝜉 𝑘 𝐷𝑚 ((𝔉𝑛 ⋆ 𝔡) − (𝔉 ⋆ 𝔡))(𝜉)| = |𝜉 𝑘 (𝐷𝑚(𝔉𝑛 − 𝔉) ⋆ 𝔡)(𝜉)| (3.3) the equation follows from [3] 𝐷𝑚𝔉 ⋆ 𝔡 = 𝐷𝑚𝔉 ⋆ 𝔡 = 𝔉 ⋆ 𝐷𝑚𝔡 for all 𝔡 ∈ 𝒞0+ ∞ (ℝ), we have |𝜉 𝑘 𝐷𝑚 ((𝔉𝑛 ⋆ 𝔡) − (𝔉 ⋆ 𝔡))(𝜉)| ≤ ∫ 𝜉 𝑘 |𝐷𝑚(𝔉𝑛 − 𝔉)(𝜉 − 𝜏)||𝔡(𝜏)|𝑑𝜏 𝐾 ≤ 𝑀𝛾𝑘 (𝔉𝑛 − 𝔉)for some constant m→ 0 as 𝑛 → ∞. ⌂ theorem 3.7:in 𝔖+(ℝ), let lim 𝑛→∞ 𝔉𝑛 = 𝔉 and (𝜂𝑛 ) ∈ ∆+⇒ lim 𝑛→∞ 𝔉𝑛 ⋆ 𝜂𝑛 = 𝔉. proof: by the hypothesis of the theorem 3.5, we get lim 𝑛→∞ 𝔉𝑛 ⋆ 𝜂𝑛 = 𝔉𝑛 → 𝔉as𝑛 → ∞. hence , we arrive, lim 𝑛→∞ 𝔉𝑛 ⋆ 𝜂𝑛 = 𝔉 as 𝑛 → ∞. ⌂ the canonical embedding between𝔹(𝔛)and𝔖+(ℝ), defined as𝜉 → [ 𝜉⋆𝜂 𝑛 𝜂 𝑛 ]. the extension of ⋆ to 𝔹(𝔛) × 𝔖+(ℝ) is given by [ ξ𝑛 𝜂𝑛 ] ⋆ 𝜏 = [ ξ𝑛⋆𝜏 𝜂𝑛 ]. convergence in 𝔹(𝔛)is followed: 𝜹– convergence: a sequence(𝜍𝑛)in 𝔹(𝔛)is called𝛿– convergent to 𝜍 in 𝔹(𝔛)denoted by 𝜍𝑛 𝛿 → 𝜍 if ∃ (𝜂𝑛) ∈ δ such that(𝜍𝑛 ⋆ 𝜂𝑛 ), (𝜍 ⋆ 𝜂𝑛) ∈ 𝔖+(ℝ), ∀ 𝑛 ∈ 𝑁 and (𝜍𝑛 ⋆ 𝜂𝓀 ) → (𝜍 ⋆ 𝜂𝓀 )as𝑛 → ∞ in 𝔖+(ℝ), ∀ 𝓀, 𝑛 ∈ 𝑁. 𝚫+ − convergence:a sequence (𝜍𝑛)in 𝔹(𝔛)is said to be δ+ − convergent to 𝜍 in 𝔹(𝔛)i.e.𝜍𝑛 ∆ → 𝜍 , if ∃(𝜂𝑛) ∈ δ+such that(𝜍𝑛 − 𝜍) ⊗ 𝜂𝑛 ∈ 𝔖+(ℝ)∀ 𝑛 ∈ 𝑁 and (𝜍𝑛 − 𝜍) ⊗ 𝜂𝑛 → 0 as 𝑛 → ∞in 𝔖+(ℝ). yogesh khandelwal and priti chaudhary 54 theorem 3.8:define 𝔉 → [ 𝔉⋆𝜂𝑛 𝜂𝑛 ] is continuous mapping which is embedding from 𝔖+(ℝ) into 𝔹(𝔛). proof: to show: the mapping is one one. we have [ 𝔉1⋆𝜂𝑛 𝜂𝑛 ] = [ 𝔉2⋆𝜏𝑛 𝜏𝑛 ], then (𝔉1 ⋆ 𝜂𝑛 ) ⋆ 𝜏𝑚 = (𝔉2 ⋆ 𝜏𝑚) ⋆ 𝜂𝑛 , 𝑚, 𝑛 ∈ 𝑁. ∵ (𝜏𝑛), (𝜂𝑛) ∈ δ+, 𝔉1 ⋆ (𝜂𝑚 ⋆ 𝜏𝑛) = 𝔉2 ⋆ (𝜏𝑛 ⋆ 𝜂𝑚) = 𝔉2 ⋆ (𝜂𝑚 ⋆ 𝜏𝑛). using theorem 3.5, we get 𝔉1 = 𝔉2. to prove: the mapping is continuous. let 𝔉𝑛 → 0 in𝔖+(ℝ)as 𝑛 → ∞. then we have[ 𝔉𝑛⋆𝜂𝑚 𝜂𝑚 ] 𝛿 → 0as 𝑛 → ∞. from the theorem 3.5, [ 𝔉𝑛⋆𝜂𝑚 𝜂𝑚 ] ⋆ 𝜂𝑚 = 𝔉𝑛 ⋆ 𝜂𝑚 → 0as 𝑛 → ∞. ⌂ theorem3.9: let 𝔡 ∈ 𝒞0+ ∞ (ℝ) and 𝔉 ∈ 𝔖+(ℝ)⇒𝔐(𝔉 ⋆ 𝔡)(𝜉) = 1 𝜉 𝔉𝔐(𝜉)𝔡𝔐(𝜉). 4. theboehmian space 𝔹(𝔛𝔐) we delineate boehmian space as ensues. let 𝔖+(ℝ) be the space’s immediately decreasing function [3]. we have 𝒞0+ ∞𝔐(ℝ) = {𝔡𝔐: ∀𝔡 ∈ 𝒞0+ ∞ (ℝ)} (4.1) here 𝔡𝔐 express the mahgoub transform of 𝔡 and also characterize 𝔉∎𝔡𝔐 by (𝔉∎𝔡𝔐)(𝜉) = 1 𝜉 𝔉(𝜉)𝔡𝔐(𝜉) (4.2) lemma 4.1 let 𝔉 ∈ 𝔖+(ℝ),𝔡 𝔐 ∈ 𝒞0+ ∞𝔐(ℝ)⇒𝔉∎𝔡𝔐 ∈ 𝔖+(ℝ). proof. let 𝔉 ∈ 𝔖+(ℝ), 𝔡 𝔐 ∈ 𝒞0+ ∞𝔐(ℝ), by leibnitz’ theorem and applying the definition of 𝔖+(ℝ), we found |𝜉 𝑘 𝐷𝜉 𝑚(𝔉∎𝔡𝔐)(𝜉)| ≤ |𝜉 𝑘 ∑ 𝐷𝑚−𝑗 ( 1 𝜉 𝔉(𝜉)) 𝐷𝑗 𝔡𝔐(𝜉) 𝑚 𝑗=1 | ≤ ∑ |𝜉 𝑘 𝐷𝑚−𝑗 ( 1 𝜉 𝔉(𝜉))| 𝑚 𝑗=1 |𝐷𝑗 𝔡𝔐(𝜉)| = ∑ |𝜉 𝑘 𝐷𝑚−𝑗 𝔉1(𝜉)| 𝑚 𝑗=1 |𝜗 ∫ 𝔡(𝜏)𝑒 − 𝜏 𝜗𝑑𝜏 𝐾 | an reckon of mahgoub transform for boehmians 55 where 𝔉1(𝜉) = 1 𝜉 𝔉(𝜉) ∈ 𝔖+(ℝ) and k is a compact support accommodating the 𝑠𝑢𝑝𝑝𝑢(𝜏). |𝜉 𝑘 𝐷𝜉 𝑚(𝔉∎𝔡𝔐)(𝜉)| ≤ 𝑀𝛾𝑘,𝑚−𝑗 (𝔉1) < ∞, for some positive constant m. ⌂ lemma 4.2 a mapping 𝔖+ × 𝒞0+ ∞𝔐 → 𝔖+ is defined by (𝔉, 𝔡𝔐) → 𝔉∎𝔡𝔐 satisfying the following axioms: (1) if 𝔡1 𝔐, 𝔡2 𝔐 ∈ 𝒞0+ ∞𝔐(ℝ), then 𝔡1 𝔐∎𝔡2 𝔐 ∈ 𝒞0+ ∞𝔐(ℝ). (2) if 𝔉1, 𝔉2 ∈ 𝔖+(ℝ), 𝔡 𝔐 ∈ 𝒞0+ ∞𝔐(ℝ), then (𝔉1+ 𝔉2)∎𝔡 𝔐 = 𝔉1∎𝔡 𝔐 + 𝔉2∎𝔡 𝔐. (3) for 𝔡1 𝔐, 𝔡2 𝔐 ∈ 𝒞0+ ∞𝔐(ℝ), 𝔡1 𝔐∎𝔡2 𝔐 = 𝔡2 𝔐∎𝔡1 𝔐. (4) for 𝔉 ∈ 𝔖+(ℝ),𝔡1 𝔐, 𝔡2 𝔐 ∈ 𝒞0+ ∞𝔐 (ℝ)then(𝔉∎𝔡1 𝔐)∎𝔡2 𝔐 = 𝔉∎(𝔡1 𝔐∎𝔡2 𝔐). proof .axioms of above lemma as follows: (1) let 𝔡1, 𝔡2 ∈ 𝒞0+ ∞ (ℝ) then 𝔡1∎𝔡2 ∈ 𝒞0+ ∞ (ℝ). ⇒ (𝔡1∎𝔡2) 𝔐 ∈ 𝒞0+ ∞𝔐(ℝ) by using theorem (3.9) implies 𝔡1 𝔐∎𝔡2 𝔐 ∈ 𝒞0+ ∞𝔐 (ℝ). (2) proof is straightforward. (3) we have (𝔡1 𝔐∎𝔡2 𝔐)(𝜉) = 1 𝜉 𝔡1 𝔐(𝜉)𝔡2 𝔐 (𝜉) = 1 𝜉 𝔡2 𝔐(𝜉)𝔡1 𝔐(𝜉) = (𝔡2 𝔐∎𝔡1 𝔐)(𝜉). (𝔡1 𝔐∎𝔡2 𝔐) = (𝔡2 𝔐∎𝔡1 𝔐) (4)let 𝔉 ∈ 𝔖+(ℝ)and 𝔡1 𝔐, 𝔡2 𝔐 ∈ 𝒞0+ ∞𝔐(ℝ), then ((𝔉∎𝔡1 𝔐)∎𝔡2 𝔐) (𝜉) = 1 𝜉 (𝔉∎𝔡1 𝔐)(𝜉)𝔡2 𝔐 (𝜉) = 1 𝜉 { 1 𝜉 𝔉(𝜉)𝔡1 𝔐 (𝜉) } 𝔡2 𝔐(𝜉) = 1 𝜉 𝔉(𝜉) 1 𝜉 𝔡1 𝔐 (𝜉) 𝔡2 𝔐(𝜉) yogesh khandelwal and priti chaudhary 56 = 1 𝜉 𝔉(𝜉)(𝔡1 𝔐∎𝔡2 𝔐)(𝜉) = (𝔉∎(𝔡1 𝔐∎𝔡2 𝔐)) (𝜉), (𝔉∎𝔡1 𝔐)∎𝔡2 𝔐 = 𝔉∎(𝔡1 𝔐∎𝔡2 𝔐) ⌂ denote by∆+ 𝔐 where∆+ 𝔐 is the collection of all mahgoub transform’s delta sequence in∆+. i.e., ∆+ 𝔐 = {(𝜂𝑛 𝔐): (𝜂𝑛) ∈ ∆+, ∀ 𝑛 ∈ 𝑁}. (4.3) lemma 4.3let 𝔉1, 𝔉2 ∈ 𝔖+(ℝ), (𝜂𝑛 𝔐) ∈ ∆+ 𝔐 such that 𝔉1∎𝜂𝑛 𝔐 = 𝔉2∎𝜂𝑛 𝔐 , ∀𝑛, then 𝔉1 = 𝔉2 in 𝔖+(ℝ). proof. let𝔉1, 𝔉2 ∈ 𝔖+(ℝ), (𝜂𝑛 𝔐) ∈ ∆+ 𝔐. since 𝔉1∎𝜂𝑛 𝔐 = 𝔉2∎𝜂𝑛 𝔐, using eq.(4.2) ⇒ 1 𝜉 𝔉1(𝜉)𝜂𝑛 𝔐(𝜉) = 1 𝜉 𝔉2(𝜉)𝜂𝑛 𝔐(𝜉) hence 𝔉1(𝜉) = 𝔉2(𝜉) for all𝜉. ⌂ lemma 4.4 for all (𝜏𝑛), (𝜂𝑛 ) ∈ δ+, (𝜂𝑛 𝔐∎𝜏𝑛 𝔐 ) ∈ ∆+ 𝔐. proof. since(𝜏𝑛), (𝜂𝑛) ∈ δ+,𝜂𝑛 ⋆ 𝜏𝑛 ∈ δ+, ∀𝑛 hence from theorem 3.9, we get 𝔐(𝜂𝑛 ⋆ 𝜏𝑛)(𝜉) = 1 𝜉 𝜂𝑛 𝔐(𝜉)𝜏𝑛 𝔐(𝜉) = 𝜂𝑛 𝔐∎𝜏𝑛 𝔐 ∈ ∆+ 𝔐, for each 𝑛. ⌂ lemma 4.5 let lim 𝑛→∞ 𝔉𝑛 = 𝔉in 𝔖+(ℝ), 𝔡 𝔐 ∈ 𝒞0+ ∞𝔐(ℝ) then 𝔉𝑛∎𝔡 𝔐 → 𝔉∎𝔡𝔐in 𝔖+(ℝ). proof.we know that 𝔡𝔐 is bounded in 𝒞0+ ∞𝔐(ℝ) we have (𝔉𝑛∎𝔡 𝔐)(𝜉) → 1 𝜉 𝔉(𝜉)𝔡𝔐(𝜉) → (𝔉∎𝔡𝔐)(𝜉). hence 𝔉𝑛∎𝔡 𝔐 → 𝔉∎𝔡𝔐. ⌂ lemma 4.6 let lim 𝑛→∞ 𝔉𝑛 = 𝔉 in 𝔖+(ℝ), (𝜂𝑛 𝔐) ∈ ∆+ 𝔐 then 𝔉𝑛 ∎𝜂𝑛 𝔐 → 𝔉 in 𝔖+(ℝ). proof.let (𝜂𝑛) ∈ δ+, 𝜂𝑛 𝔐(𝜉) → 𝜉 is uniformly on compact subsets of ℝ+. hence |𝜉 𝑘 𝐷𝜉 𝑚(𝔉𝑛∎𝜂𝑛 𝔐 − 𝔉)(𝜉)| = |𝜉 𝑘 𝐷𝜉 𝑚 ( 1 𝜉 𝔉𝑛 (𝜉)𝜂𝑛 𝔐(𝜉)) − 𝔉(𝜉)| → |𝜉 𝑘 𝐷𝜉 𝑚(𝔉𝑛 − 𝔉)(𝜉)| as 𝑛 → ∞ thus |𝜉 𝑘 𝐷𝜉 𝑚(𝔉𝑛 ∎𝜂𝑛 𝔐 − 𝔉)(𝜉)| → 0 as 𝑛 → ∞. an reckon of mahgoub transform for boehmians 57 this yield 𝔉𝑛 ∎𝜂𝑛 𝔐 → 𝔉 in𝔖+(ℝ). ⌂ lemma 4.7 define𝔉 → [ 𝔉∎𝜂𝑛 𝔐 𝜂𝑛 𝔐 ] is a continuous mapping which is embedding from 𝔖+(ℝ) into 𝔹(𝔛 𝔐). (4.4) proof. let 𝔉∎𝜂𝑛 𝔐 𝜂𝑛 𝔐 is a quotient of sequences where𝔉 ∈ 𝔖+(ℝ), 𝜂𝑛 𝔐 ∈ ∆+ 𝔐. we have (𝔉∎𝜂𝑛 𝔐)∎𝜂𝑚 𝔐 = 𝔉∎(𝜂𝑚 𝔐∎𝜂𝑛 𝔐).we show that the map (4.3) is one to one. let[ 𝔉1∎𝜂𝑛 𝔐 𝜂𝑛 𝔐 ] = [ 𝔉2∎𝜏𝑛 𝔐 𝜏𝑛 𝔐 ], then(𝔉1∎𝜂𝑛 𝔐)∎𝜏𝑚 𝔐 = (𝔉2∎𝜏𝑚 𝔐)∎𝜂𝑛 𝔐 , 𝑚, 𝑛 ∈ 𝑁. now using of lemma 4.2& 4.3, we get 𝔉1 = 𝔉2. to establish the continuity of eq.(4.4), let 𝔉𝑛 → 0as 𝑛 → ∞in 𝔖+(ℝ). then 𝔉𝑛 ∎𝜂𝑛 𝔐 → 0 as 𝑛 → ∞by lemma 4.6, and hence [ 𝔉∎𝜂𝑛 𝔐 𝜂𝑛 𝔐 ] → 0,as 𝑛 → ∞ in 𝔹(𝔛 𝔐). ⌂ 5. the mahgoub transform of bohemians let ℌ = [ 𝔉𝑛 𝜂𝑛 ] ∈ 𝔹(𝔛), then we delineate the mahgoub transform of ℌ by the relation ℌ1 𝔐 = [ 𝔉𝑛 𝔐 𝜂𝑛 𝔐 ]in 𝔹(𝔛 𝔐). (5.1) theorem 5.1 ℌ1 𝔐: 𝔹(𝔛) → 𝔹(𝔛𝔐) is well defined. proof. let ℌ1 = ℌ2 ∈ 𝔹(𝔛), where ℌ1 = [ 𝔉𝑛 𝜂𝑛 ] , ℌ2 = [ 𝑔𝑛 𝜏𝑛 ]then the concept of quotients yields 𝔉𝑛 ⋆ 𝜏𝑚 = 𝑔𝑚 ⋆ 𝜂𝑛 . applying theorem 3.9, we get 1 𝜉 𝔉𝑛 𝔐(𝜉)𝜏𝑚 𝔐(𝜉) = 1 𝜉 𝑔𝑚 𝔐(𝜉)𝜂𝑛 𝔐(𝜉), 𝑖. 𝑒. 𝔉𝑛 𝔐∎𝜏𝑚 𝔐 = 𝑔𝑚 𝔐∎𝜂𝑛 𝔐 ⇒ 𝑓𝑛 𝔐 𝜂𝑛 𝔐 ~ 𝑔𝑛 𝔐 𝜏𝑛 𝔐 . thus ℌ1 𝔐 = ℌ2 𝔐. ⌂ theorem 5.2 ℌ𝔐: 𝔹(𝔛) → 𝔹(𝔛𝔐) is continuous regards to 𝛿-convergence. proof. let ℌ𝑛 → 0 in 𝔹(𝔛)as 𝑛 → ∞. using [4] we get,ℌ𝑛 = [ 𝔉𝑛,𝓀 𝜂𝓀 ] and 𝔉𝑛,𝓀 → 0 in 𝔖+(ℝ) as 𝑛 → ∞ in 𝔖+(ℝ). now we apply the mahgoub transform to both sides revenue 𝔉𝑛,𝑘 𝔐 → 0 as 𝑛 → ∞. hence ℌ𝑛 𝔐 = [ 𝔉𝑛,𝓀 𝔐 𝜂𝓀 𝔐 ] → 0 as 𝑛 → ∞ in 𝔹(𝔛 𝔐). ⌂ yogesh khandelwal and priti chaudhary 58 theorem 5.3 ℌ𝔐 ∶ 𝔹(𝔛) → 𝔹(𝔛𝔐)is one-to-one mapping. proof. let ℌ1 𝔐 = [ 𝔉𝑛 𝔐 𝜂𝑛 𝔐] = [ 𝑔𝑛 𝔐 𝜏𝑛 𝔐 ] = ℌ2 𝔐,then 𝔉𝑛 𝔐∎𝜏𝑚 𝔐 = 𝑔𝑚 𝔐∎𝜂𝑛 𝔐 . hence (𝔉𝑛 ⋆ 𝜏𝑚) 𝔐 = (𝑔𝑚 ⋆ 𝜂𝑛 ) 𝔐. since the mahgoub transform is one to one, we get 𝔉𝑛 ⋆ 𝜏𝑚 = 𝑔𝑚 ⋆ 𝜂𝑛 .thus 𝔉𝑛 𝜂𝑛 ~ 𝑔𝑛 𝜏𝑛 . hence [ 𝔉𝑛 𝜂𝑛 ] = ℌ1 = [ 𝑔𝑛 𝜏𝑛 ] = ℌ2. ⌂ theorem 5.4 let ℌ1, ℌ2 ∈ 𝔹(𝔛), then (1) (ℌ1 + ℌ2) 𝔐 = ℌ1 𝔐 + ℌ2 𝔐; (2) (𝓀ℌ)𝔐 = 𝓀 ℌ𝔐 , 𝓀 ∈ ℂ . theorem5.5 ℌ𝔐 ∶ 𝔹(𝔛) → 𝔹(𝔛𝔐) is continuous regards to ∆+ convergence. proof. let ℌ𝑛 ∆ → ℌ in 𝔹(𝔛) as 𝑛 → ∞ then ∃𝔉𝑛 → 0 𝔖+(ℝ) and (𝜂𝑛) ∈ ∆+ such that (ℌ𝑛 − ℌ) ∗ 𝜂𝑛 = [ 𝔉𝑛∗𝜂𝓀 𝜂𝓀 ] and 𝔉𝑛 → 0 as 𝑛 → ∞.applying in eq.(5.1) , we get 𝔐((ℌ𝑛 − ℌ) ∗ 𝜂𝑛 ) = [ 𝔐(𝔉𝑛 ∗ 𝜂𝓀) 𝜂𝓀 𝔐 ]. hence we have 𝔐((ℌ𝑛 − ℌ) ∗ 𝜂𝑛 ) = [ 𝔉𝑛 𝔐𝜂𝓀 𝔐 𝜉𝜂𝓀 𝔐 ] → 0 as 𝑛 → ∞ in 𝔹(𝔛𝔐). therefore 𝔐((ℌ𝑛 − ℌ) ∗ 𝜂𝑛 ) = 1 𝜉 (ℌ𝑛 𝔐 − ℌ𝔐)𝜂𝑛 𝔐 → 0 as 𝑛 → ∞. ⌂ theorem5.6 let ℌ𝔐 ∶ 𝔹(𝔛) → 𝔹(𝔛𝔐) is onto. proof. let [ 𝔉𝑛 𝔐 𝜂𝑛 𝔐] ∈ 𝔹(𝔛 𝔐) be arbitrary then 𝔉𝑛 𝔐 ∎𝜂𝑚 𝔐 = 𝔉𝑚 𝔐 ∎𝜂𝑛 𝔐 for each 𝑚, 𝑛 ∈ 𝑁.then 𝔉𝑛 ⋆ 𝜂𝑚 = 𝔉𝑚 ⋆ 𝜂𝑛.that is, 𝔉𝑛 𝜂𝑛 is the corresponding quotient of sequences of 𝔉𝑛 𝔐 𝜂𝑛 𝔐. thus 𝔉𝑛 𝜂𝑛 ∈ 𝔹(𝔛) is such that 𝔐 [ 𝔉𝑛 𝜂𝑛 ] = [ 𝔉𝑛 𝔐 𝜂𝑛 𝔐 ] in 𝔹(𝔛 𝔐). let ℌ𝔐 = [ 𝔉𝑛 𝔐 𝜂𝑛 𝔐] ∈ 𝔹(𝔛 𝔐), then we express the inverse of mahgoub transform of ℌ𝔐 given by an reckon of mahgoub transform for boehmians 59 ℌ𝔐 −1 = [ 𝔉𝑛 𝜂𝑛 ] in the space 𝔹(𝔛). ⌂ theorem5.7 let [ 𝔉𝑛 𝔐 𝜂𝑛 𝔐] ∈ 𝔹(𝔛 𝔐) and 𝔡𝔐 ∈ 𝒞0+ ∞𝔐 (ℝ), 𝔡 ∈ 𝒞0+ ∞ (ℝ). ℌ ([ 𝔉𝑛 𝜂𝑛 ] ⋆ 𝔡) = [ 𝔉𝑛 𝔐 𝜂𝑛 𝔐] ∎𝔡 𝔐andℌ𝔐 −1 ([ 𝔉𝑛 𝔐 𝜂𝑛 𝔐] ∎𝔡 𝔐) = [ 𝔉𝑛 𝜂𝑛 ] ⋆ 𝔡. we can easily proof from the definitions. conflict of interests the authors declare that there is no conflict of interests regarding the publication of this paper. references [1] al-omari, s., kılıçman, a. an estimate of sumudu transforms for boehmians. advances in difference equations , 77(1),1-10.2013. [2] boehme, t.k: the support of mikusinski operators. trans. am. math. soc., 1973. [3] zemanian, a.h. generalized integral transformation. dover, new york, 1987. [4] mikusinski, p. fourier transform for integrable boehmians. rocky mt. j. math.,17(3),1987. [5] mikusinski, p. tempered boehmians and ultra distributions. proc. am. math. soc.,123(3),813-817.1995. [6] roopkumar, r. mellin transform for boehmians. bull. inst. math. acad. sin., 4(1),75-96.2009. 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[12] chaudhary,p.,chanchal,p.,khandelwal,y. duality of some famous integral transforms from the polynomial integral transform. international journal of mathematics trends and technology., 55(5), 345-349.2018. ratio mathematica 25 (2013), 3–14 issn:1592-7415 class of semihyperrings from partitions of a set a. asokkumar department of mathematics, aditanar college of arts and science, tiruchendur628216, tamilnadu, india. ashok a58@yahoo.co.in abstract in this paper we show that a partition {pα : α ∈ λ} of a nonempty set s, where λ is an ordered set with the least element α0 and pα0 is a singleton set, induces a hyperaddition + such that (s, +) is a commutative hypermonoid. also by using a collection of subsets of s, induced by the partition of the set s, we define hypermultiplication on s so that (s, +, ·) is a semihyperring. key words: hypermonoid, semihyperring, ∗-collection. msc 2010: 20n20. 1 introduction the theory of hyperstructures has been introduced by the french mathematician marty [11] in 1934 at the age of 23 during the 8thcongress of scandinavian mathematicians held in stockholm. since then many researchers have worked on this new area and developed it. the theory of hyperstructure has been subsequently developed by corsini [4, 5, 6], mittas [13], stratigopoulos [16] and various authors. basic definitions and results about the hyperstructures are found in [5, 6]. some researchers, namely, davvaz [7], massouros [12], vougiouklis [18] and others developed the theory of algebraic hyperstructures. there are different notions of hyperrings (r, +, ·). if the addition + is a hyperoperation and the multiplication · is a binary operation then we say the hyperring is an krasner (additive) hyperring [10]. rota [15] introduced 3 a. asokkumar a multiplicative hyperring, where + is a binary operation and · is a hyperoperation. de salvo [8] introduced a hyperring in which addition and multiplication are hyperoperations. these hyperrings are studied by rahnamani barghi [14] and by asokkumar and velrajan [1, 2, 17]. chvalina [3] and hoskova [3, 9], studied hν-groups, hν-rings. in this paper, by using different partitions of a set, we construct different semihyperrings (s, +, ·) where both + and · are hyperoperations. 2 preliminaries this section explains some basic definitions that have been used in the sequel. a hyperoperation ◦ on a non-empty set h is a mapping of h × h into the family of non-empty subsets of h (i.e., x◦y ⊆ h, for every x,y ∈ h). a hypergroupoid is a non-empty set h equipped with a hyperoperation ◦. for any two subsets a, b of a hypergroupoid h, the set a◦b means ⋃ a∈a b∈b (a◦b). a hypergroupoid (h,◦) is called a semihypergroup if x◦(y◦z) = (x◦y)◦z for all x,y,z ∈ h(the associative axiom). a semihypergroup h is said to be regular (in the sense of von neumann) if a ∈ a◦h ◦a for every a ∈ h. a hypergroupoid (h,◦) is called a quasihypergroup if x ◦ h = h ◦ x = h for every x ∈ h(the reproductive axiom). a reproductive semihypergroup is called a hypergroup(in the sense of marty).a comprehensive review of the theory of hypergroups appears in [5]. definition 2.1. a semihyperring is a non-empty set r with two hyperoperations + and · satisfying the following axioms: (1) (r, +) is a commutative hypermonoid, that is, (a) (x + y) + z = x + (y + z) for all x,y,z ∈ r, (b) there exists 0 ∈ r, such that x + 0 = 0 + x = {x} for all x ∈ r, (c) x + y = y + x for all x,y ∈ r. (2) (r, ·) is a semihypergroup, that is, x·(y·z) = (x·y)·z for all x,y,z ∈ r. (3) the hyperoperation · is distributive with respect to hyperoperation ’+’, that is, x · (y + z) = x · y + x · z and (x + y) · z = x · z + y · z for all x,y,z ∈ r. (4) there exists element 0 ∈ r, such that x · 0 = 0 ·x = 0 for all x ∈ r. definition 2.2. let s be a semihyperring, an element a ∈ s is said to be regular if there exists an element y ∈ s such that x ∈ xyx. a semihyperring s is said to be regular if each element of s is regular. 4 class of semihyperrings from partitions of a set 3 semihyperring constructed from a ∗-collection. in this section, for a given commutative hypermonoid (s, +), we define hyperoperation · on s suitably so that (s, +, ·) is a regular semihyperring. definition 3.1. let s be a commutative hypermonoid. a collection of nonempty subsets {sa : a ∈ s} of s satisfying the following conditions is called a ∗-collection if (i) sa = {0} if and only if a = 0, (ii) if a 6= 0 then {0,a}⊆ sa, (iii) ⋃ x∈sa sx = sa for every a ∈ s, (iv) sa + sa = sa for every a ∈ s and (v) ⋃ x∈a+b sx = sa + sb for every a,b ∈ s. example 3.2. consider the set s = {0,a,b}. if we define a hyperoperation + on s as in the following table, then (s,+) is a commutative hypermonoid. + 0 a b 0 0 a b a a {a,b} {a,b} b b {a,b} {a,b} now it is easy to see that s0 = {0}; sa = s; sb = s is a ∗-collection. example 3.3. consider the set s={0,a,b}. if we define a hyperoperation + on s as in the following table, then (s,+) is a commutative hypermonoid. + 0 a b 0 0 a b a a {a} {a,b} b b {a,b} {b} now it is easy to see that s0 = {0}; sa = s; sb = s is a ∗-collection. now we show that s0 = {0}; sa = {a, 0}; sb = {b, 0} is another ∗-collection. for each a ∈ s, ⋃ x∈sa sx = ⋃ x∈{a,0} sx = sa ⋃ s0 = {a, 0} ⋃ {0} = {a, 0} = sa. also s0 + s0 = {0} + {0} = {0} = s0; sa + sa = {0,a} + {0,a} = {0,a} = sa; sb + sb = {0,b} + {0,b} = {0,b} = sb. further, for a,b ∈ s, we get ⋃ x∈a+b sx = ⋃ x∈{a,b} sx = sa ⋃ sb = {0,a,b} = sa + sb. example 3.4. consider the set s={0,a,b}. if we define a hyperoperation + on s as in the following table, then (s,+) is a commutative hypermonoid. + 0 a b 0 0 a b a a {0,a} s b b s {0,b} 5 a. asokkumar it is easy to show that s0 = {0} ; sa = s for every a 6= 0 ∈ s, is a ∗-collection and s0 = {0} ; sa = {a, 0} for every a 6= 0 ∈ s is another ∗-collection example 3.5. consider the set s={0,a,b,c}. if we define a hyperoperation + on s as in the following table, then (s,+) is a commutative hypermonoid. + 0 a b c 0 {0} {a} {b} {c} a {a} {a} {a, b} {a, c} b {b} {a, b} {b} {b, c} c {c} {a, c} {b, c} {c} in this commutative hypermonoid, each one of the following is a ∗-collection. s0 = {0} ; sa = {a, 0} for every a 6= 0 ∈ s, s0 = {0} ; sa = s for every a 6= 0 ∈ s, s0 = {0}; sa = {0,a}; sb = {0,b,a}; sc = {0,c,a}, s0 = {0}; sa = {0,a,b}; sb = {0,b}; sc = {0,c,b}, s0 = {0}; sa = {0,a,c}; sb = {0,b,c}; sc = {0,c}. theorem 3.6. let s be a commutative hypermonoid with the additive identity 0 with the condition that x + y = {0} for x,y ∈ s implies either x = 0 or y = 0. let {sa : a ∈ s} be a ∗-collection on s. for a,b ∈ s, if we define a hypermultiplication on s as a · b = { sa if a 6= 0,b 6= 0, 0 otherwise then (s, +, .) is a regular semihyperring. proof. from the definition of the hypermultiplication, a ·0 = 0 ·a = 0 for all a ∈ s. let a,b,c ∈ s. if any one of a,b,c is 0, then a ·(b ·c) = {0} = (a ·b) ·c. if a 6= 0,b 6= 0 and c 6= 0, then a·(b·c) = a·sb = sa. also, (a·b)·c = sa ·c =⋃ x∈sa(x · c) = ⋃ x∈sa sx = sa. thus (a · b) · c = a · (b · c). therefore, (s, ·) is a semihypergroup. let a,b,c ∈ s. if a = 0 or b = 0 or c = 0, then it is obvious that a · (b + c) = a · b + a · c. suppose a 6= 0,b 6= 0 and c 6= 0. if 0 ∈ b + c, then a · (b + c) = s0 ∪ sa = sa = sa + sa = a · b + a · c. if 0 /∈ b + c, then a · (b + c) = sa = sa + sa = a · b + a · c. thus a · (b + c) = a · b + a · c. now we prove (a + b) · c = a · c + b · c. for, (a + b) · c = ⋃ x∈a+b x.c =⋃ x∈a+b sx = sa + sb = a · c + b · c. therefore, (a + b) · c = a · c + b · c. thus (s, +, ·) is a semihyperring. let x 6= 0 ∈ s. now, for any y 6= 0 ∈ s, we have x ∈ sx = x ·y ⊆ x ·sy = x · (y ·x). hence the semihyperring is regular. 6 class of semihyperrings from partitions of a set example 3.7. the semihyperring obtained by using the theorem 3.1 in the example 3.1 is as follows. + 0 a b 0 0 a b a a {a,b} {a,b} b b {a,b} {a,b} . 0 a b 0 0 0 0 a 0 s s b 0 s s example 3.8. the semihyperrings obtained by using the theorem 3.1 in the example 3.2 are as follows. + 0 a b 0 0 a b a a {a} {a,b} b b {a,b} {b} . 0 a b 0 0 0 0 a 0 s s b 0 s s . 0 a b 0 0 0 0 a 0 {0, a} {0,a} b 0 {0,b} {0,b} example 3.9. the semihyperrings obtained by using the theorem 3.1 in the example 3.3 are as follows. + 0 a b 0 0 a b a a {0,a} s b b s {0,b} . 0 a b 0 0 0 0 a 0 s s b 0 s s . 0 a b 0 0 0 0 a 0 {0, a} {0,a} b 0 {0,b} {0,b} theorem 3.10. let s be a commutative hypermonoid with the additive identity 0 with the condition that x + y = 0 for x,y ∈ s implies either x = 0 or y = 0. let {sa : a ∈ s} be a ∗-collection on s. for a,b ∈ s, if we define a hypermultiplication on s as a · b = { sb if a 6= 0,b 6= 0, 0 otherwise then (s, +, .) is a regular semihyperring. proof. the proof follows by the same lines as in the theorem 3.1. let x 6= 0 ∈ s. now, for any y 6= 0 ∈ s, we have x ∈ sx = y ·x ⊆ sy ·x = (x ·y) ·x. hence the semihyperring is regular. 7 a. asokkumar theorem 3.11. let s be a commutative hypermonoid with the additive identity 0 such that x + y = 0 for x,y ∈ s implies either x = 0 or y = 0. let {sa : a ∈ s} be a ∗-collection on s such that sa ∩ sb = x for all a 6= 0,b 6= 0 ∈ s where x is a subset of s such that x + x = x. for a,b ∈ s, if we define a hypermultiplication on s as a · b = { sa ∩sb = x if a 6= 0,b 6= 0, 0 otherwise then (s, +, .) is a regular semihyperring. proof. since 0 ∈ sa and 0 ∈ sb, we get 0 ∈ sa∩sb. this implies that the set x is non-empty. from the definition of hypermultiplication, a ·0 = 0 ·a = 0 for all a ∈ s. let a,b,c ∈ s. if any one of a,b,c is 0, then a·(b·c) = {0} = (a·b)·c. if a 6= 0,b 6= 0 and c 6= 0, then a·(b·c) = x = (a·b)·c. thus (a·b)·c = a·(b·c). therefore, (s, ·) is a semihypergroup. if a = 0 or b = 0 or c = 0, then it is obvious that a · (b + c) = a ·b + a ·c. suppose a 6= 0,b 6= 0 and c 6= 0 then, a · (b + c) = x = x + x = a ·b + a ·c. similarly we have (a+b)·c = x = a·c+b·c. thus (s, +, ·) is a semihyperring. let x 6= 0 ∈ s. since x ∈ sx, we have x ∈ sx = x ·x ⊆ x ·sx = x · (x ·x). hence the semihyperring is regular. example 3.12. using the theorem 3.3 in the commutative hypermonoid given in the example 3.4 and by using the following each ∗-collection s0 = {0}; sa = {0,a}; sb = {0,b,a}; sc = {0,c,a} with x = {0,a}, s0 = {0}; sa = {0,a,b}; sb = {0,b}; sc = {0,c,b} with x = {0,b}, s0 = {0}; sa = {0,a,c}; sb = {0,b,c}; sc = {0,c} with x = {0,c}, we get three hypermultiplications so that we get three semihyperrings. 4 semihyperrings induced by a partition. in this section we show that a partition of a non-empty set s induces a hyperaddition + such that, (s, +) is a commutative hypermonoid and also the partition induces a ∗-collection. using this ∗-collection,we define hypermultiplication · on the set s, so that (s, +, .) a regular semihyperring. theorem 4.1. let s be any non-empty set and {pα}α∈λ be a partition of s, where λ is an ordered set with the least element α0 ∈ λ and pα0 be a singleton set, say, {0}. define a hyperaddition ”+” on s as follows: for all a ∈ s, 0 + a = a + 0 = {a}. for a 6= 0,b 6= 0 ∈ s, suppose a ∈ pα and b ∈ pβ and γ = max {α,β}, a + b = { pγ if α 6= β, pα = pβ if α = β 8 class of semihyperrings from partitions of a set then (i) (s, +) is a commutative monoid and (ii) the partition {pα}α∈λ induces a ∗-collection. proof. it is clear that a + b = b + a for all a,b ∈ s. let a,b,c ∈ s. suppose that a ∈ pα, b ∈ pβ and c ∈ pγ, where α,β,γ ∈ λ. case 1 : suppose α < β < γ. then a + (b + c) = a + pγ = pγ. also, (a + b) + c = pβ + c = pγ. therefore, a + (b + c) = (a + b) + c. case 2 : suppose β < α < γ. then a + (b + c) = a + pγ = pγ. also, (a + b) + c = pα + c = pγ. therefore, a + (b + c) = (a + b) + c. case 3 : suppose α < γ < β. then a + (b + c) = a + pβ = pβ. also, (a + b) + c = pβ + c = pβ. therefore, a + (b + c) = (a + b) + c. case 4 : suppose γ < α < β. then a + (b + c) = a + pβ = pβ. also, (a + b) + c = pβ + c = pβ. therefore, a + (b + c) = (a + b) + c. case 5 : suppose γ < β < α. then a + (b + c) = a + pβ = pα. also, (a + b) + c = pα + c = pα. therefore, a + (b + c) = (a + b) + c. case 6 : suppose β < γ < α. then a + (b + c) = a + pγ = pα. also, (a + b) + c = pα + c = pα. therefore, a + (b + c) = (a + b) + c. case 7 : suppose α = β < γ. then a + (b + c) = a + pγ = pγ. also, (a + b) + c = pβ + c = pγ. therefore, a + (b + c) = (a + b) + c. case 8 : suppose γ < α = β. then a + (b + c) = a + pα = pα. also, (a + b) + c = pα + c = pα. therefore, a + (b + c) = (a + b) + c. case 9 : suppose α = γ < β. then a + (b + c) = a + pβ = pβ. also, (a + b) + c = pβ + c = pβ. therefore, a + (b + c) = (a + b) + c. case 10 : suppose β < α = γ. then a + (b + c) = a + pα = pα also, (a + b) + c = pα + c = pα. therefore, a + (b + c) = (a + b) + c. case 11 : suppose β = γ < α. then a + (b + c) = a + pγ = pα. also, (a + b) + c = pα + c = pα. therefore, a + (b + c) = (a + b) + c. case 12 : suppose α < β = γ. then a + (b + c) = a + pγ = pγ. also, (a + b) + c = pβ + c = pγ. therefore, a + (b + c) = (a + b) + c. 9 a. asokkumar case 13 : suppose α = β = γ. then a+(b+c) = pα = pβ = pγ = (a+b)+c. therefore, a+(b+c) = (a+b)+c. thus the hyperoperation + is associative. so, (s, +) is a commutative hypermonoid. let s0 = pα0 = {0}. for a 6= 0 ∈ s, then sa = ⋃ α0≤t≤α pt where a ∈ pα. it is clear that sa = ⋃ x∈sa sx. for a 6= 0 ∈ s, and a ∈ pα, then sa + sa = ⋃ α0≤t≤α pt + ⋃ α0≤t≤α pt = ⋃ α0≤t≤α pt = sa. also s0 + s0 = {0} + {0} = {0} = s0. if either a = 0 or b = 0, then ⋃ x∈a+b sx = sa + sb. let a 6= 0,b 6= 0 ∈ s. then a ∈ pα and b ∈ pβ for some α,β ∈ λ. case 1 : suppose α 6= β, say α < β, then a + b = pβ. now x ∈ a + b implies x ∈ pβ. therefore, sx = ⋃ α0≤t≤β pt. hence⋃ x∈a+b sx = ⋃ x∈a+b ( ⋃ α0≤t≤β pt) = ⋃ α0≤t≤α pt = ⋃ α0≤t≤α pt + ⋃ α0≤t≤β pt = sa + sb. case 2 : suppose α = β then a + b = pα. therefore, ⋃ x∈a+b sx =⋃ x∈pα sx = sa + sb. therefore, ⋃ x∈a+b sx = sa + sb. thus {sa : a ∈ s} is a ∗-collection. remark 4.2. let s be any non-empty set and x0 ∈ s. let p0 = {x0} and {p1,p2,p3, · · ·,pn, · · ·} be a partition of s \ {x0}. then the partition {p0,p1,p2, ···,pn, ···} of s induces a hyperoperation + on s so that (s, +) is a commutative hypermonoid and {p0,p1,p2, ···,pn, ···} induces a ∗-collection. theorem 4.3. let s be any non-empty set and {pα}α∈λ be a partition of s, where λ is an ordered set with the least element α0 and pα0 is a singleton set.then the partition induces a semihyperring. proof. by the theorem 4.1, the partition induces a hyperaddition + such that (s, +) is a commutative hypermonoid and it also induces a ∗-collection. hence by the theorem 3.1, we get a regular semihyperring. example 4.4. we illustrate the construction of semihyperrings from the following examples. let s = {0,a,b}. consider a partition p1 = {0},p2 = {a},p3 = {b} of s. here, the indexing set is λ = {1, 2, 3} which is an ordered set. the commutative hypermonoid induced by this partition is given by the following caley table. + 0 a b 0 0 a b a a {a} {b} b b {b} {b} 10 class of semihyperrings from partitions of a set the ∗-collection induced by this partition is s0 = {0},sa = {0,a},sb = {0,a,b} and the hypermultiplication induced by the ∗-collection is given in the caley table. . 0 a b 0 0 0 0 a 0 {0,a} {0,a} b 0 {0,a,b} {0,a,b} example 4.5. let s = {0,a,b}. consider a partition p1 = {0},p2 = {b},p3 = {a} of s. here, the indexing set is λ = {1, 2, 3} which is an ordered set. the commutative hypermonoid induced by this partition is given by the following caley table. + 0 a b 0 0 a b a a {a} {a} b b {a} {b} the ∗-collection induced by this partition is s0 = {0},sa = {0,a,b},sb = {0,b} and the hypermultiplication induced by the ∗-collection is given in the caley table. . 0 a b 0 0 0 0 a 0 {0,a,b} {0,a,b} b 0 {0,b} {0,b} example 4.6. let s = {0,a,b}. consider a partition p1 = {0},p2 = {a,b} of s. here, the indexing set is λ = {1, 2} which is an ordered set. the commutative hypermonoid induced by this partition is given by the following caley table. + 0 a b 0 0 a b a a {a,b} {a,b} b b {a,b} {a,b} the ∗-collection induced by this partition is s0 = {0},sa = {0,a,b},sb = {0,a,b} and the hypermultiplication induced by the ∗-collection is given in the caley table. . 0 a b 0 0 0 0 a 0 {0,a,b} {0,a,b} b 0 {0,a,b} {0,a,b} thus we have a regular semihyperring. 11 a. asokkumar conclusion : in the section 3 of this paper, for the given commutative hypermonoid, given ∗-collection, we construct three semihyperrings. in the section 4, by the theorem 4.1, a partition of a set s induces a hyperaddition + such that (s, +) is a commutative hypermonoid and it also induces a ∗collection. hence by the theorem 3.1, we get a semihyperring. thus we get semihyperrings depending on the partitions of the set satisfies the conditions of the theorem 4.2. all the semihyperrings so constructed are regular. references [1] a. asokkumar, hyperlattice formed by the idempotents of a hyperring, tamkang j. math., 38(3) (2007), 209–215. 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[9] s. hoskova, abelization of quasi-hypergroups, hv-rings and transposition hv-groups as a categorial reflection, global journal of pure and applied mathematics, vol. 3(2), (2007), 105–112. 12 class of semihyperrings from partitions of a set [10] m. krasner, a class of hyperrings and hyperfields, int. j. math and math. sci., 2 (1983), 307–312. [11] f. marty, sur une generalization de la notion de groupe, 8th congress math. scandenaves, stockholm, (1934), 45–49. [12] g. g.massouros, the hyperringoid, multi.val.logic., vol 3 (1998), 217– 234. [13] j. mittas, hypergroupes canoniques, mathematica balkanica t. 2 (1972), 165–179. [14] a. rahnamai barghi, a class of hyperrings, journal of discrete mathematical sciences & cryptography, 6(2003), 227–233. [15] r. rota, strongly distributive multiplicative hyperrings, j. geom. 39 (1990), 130–138. [16] d. stratigopoulos, certaines classes d’hypercorpes et hyperanneaux, atti convegno su ipergruppi, altri strutture multivoche e applicazioni, editor corsini, udine, 1985. [17] m. velrajan, a.asokkumar, construction of inclusive distributive hyperrings, algebra and its applications, editors: afzal beg, mohd. ashraf, narosa publishing house, new delhi, india (2011), 167–176. [18] t. vougiouklis, hyperstructures and their representations, hadronic press, inc., 115, palm harber, usa (1994). 13 14 ratio mathematica vol. 33, 2017, pp. 115-126 issn: 1592-7415 eissn: 2282-8214 finite hv-fields with strong-inverses theodora kaplani∗, thomas vougiouklis† ‡doi:110.23755/rm.v33i0.373 abstract the largest class of hyperstructures is the class of hv-structures. this is the class of hyperstructures where the equality is replaced by the non-empty intersection. this extremely large class can used to define several objects that they are not possible to be defined in the classical hypergroup theory. it is convenient, in applications, to use more axioms and conditions to restrict the research in smaller classes. in this direction, in the present paper we continue our study on hv-structures which have strong-inverse elements. more precisely we study the small finite cases. keywords: hyperstructure; hv-structure; hope; strong-inverse elements. 2010 ams subject classifications: 20n20, 16y99. ∗fotada, 42100 trikala, greece; dorakikaplani@gmail.com †democritus university of thrace, school of education, 68100 alexandroupolis, greece; tvougiou@eled.duth.gr ‡ c©theodora kaplani and thomas vougiouklis. received: 31-10-2017. accepted: 26-122017. published: 31-12-2017. 115 theodora kaplani and thomas vougiouklis 1 introduction first we present some basic definitions on hyperstructures, mainly on the weak hyperstructures introduced in 1990 [7]. definition 1.1. hyperstructures are called the algebraic structures equipped with, at least, one hyperoperation. abbreviate: hyperoperation=hope. the weak hyperstructures are called hv-structures and they are defined as follows: in a set h equipped with a hope · : h ×h → ℘(h)−{∅}, we abbreviate by wass the weak associativity: (xy)z ∩x(yz) 6= ∅,∀x,y,z ∈ h and by cow the weak commutativity: xy ∩yx 6= ∅,∀x,y ∈ h. the hyperstructure (h, ·) is called an hv-semigroup if it is wass, is called hv-group if it is reproductive hv-semigroup: xh = hx = h,∀x ∈ h. (r,+, ·) is called hv-ring if the hopes (+) and (·) are wass, the reproduction axiom is valid for (+), and (·) is weak distributive with respect to (+): x(y + z)∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅, ∀x,y,z ∈ r. an hv-group is called cyclic [6], [8], if there is an element, called generator, which the powers have union the underline set. the minimal power with this property is called the period of the generator. if there is an element and a special power, the minimum one, is the underline set, then the hv-group is called singlepower cyclic. for more definitions, results and applications on hyperstructures and mainly on the hv-structures, see books as [1], [2], [8] and papers as [6], [9], [8], [10], [11], [12], to mention but a few of them. an extreme class of hyperstructures is the following: an hv-structure is called very thin if and only if, all hopes are operations except one, with all hyperproducts to be singletons except only one, which is a subset with cardinality more than one. the fundamental relations β* and γ* are defined, in hv-groups and hvrings, respectively, as the smallest equivalences so that the quotient would be group and ring, respectively. normally to find the fundamental classes is very hard job. the basic theorems on the fundamental classes are analogous to the following: theorem 1.1. [8] let (h, ·) be an hv-group and let us denote by u the set of all finite products of elements of h. we define the relation β in h as follows: xβy iff {x,y} ⊂ u where u ∈ u. then the fundamental relation β* is the transitive closure of the relation β. 116 finite hv-fields with strong-inverses proof. see [7], [8]. an element of a hyperstructure is called single if its fundamental class is a singleton. motivation for hv-structures: the quotient of a group with respect to an invariant subgroup is a group. f. marty states that, the quotient of a group by any subgroup is a hypergroup. now, the quotient of a group with respect to any partition is an hv-group. we remark that in hv-groups (or even in hypergroups in the sense of f. marty) we do not have necessarily any ’unit’ element, consequently neither ’inverses’. however, we may have more than one unit elements and for each element of an hv-group we may have one inverse element or more than one inverse element. 2 definition 1.2. let (h, ·) be an hv-semigroup. an element e, is called left unit if ex 3 x,∀x ∈ h, it is called right unit if xe 3 x,∀x ∈ h and it is called unit element if it is both left and right unit element. for given unit e, an element x ∈ h, has a left inverse with respect to e, any element xle if xle ·x 3 e, it has a right inverse element xre if x ·xre 3 e, and it has an inverse xe with respect to e, if e ∈ xe ·x∩x ·xe. denote by el the set of all left unit elements, by er the set of all right unit elements, and by e the set of unit elements. definition 1.3. [16], [5] let (h, ·) be an hv-semigroup. an element is called strong-inverse if it is an inverse to x with respect to all unit elements. remark 1.1. we remark that an element xs is a strong-inverse to x, if e ⊂ xs · x ∩ x · xs. therefore the strong-inverse property it is not exists in the classical structures. definition 1.4. let (h, ·),(h,⊗) be hv-semigroups defined on the same h. (·) is smaller than (⊗), and (⊗) greater than (·), if and only if, there exists an automorphism f ∈ aut(h,⊗) such that xy ⊂ f(x ⊗ y), ∀x,y ∈ h. then (h,⊗) contains (h, ·) and write · ≤⊗. if (h, ·) is a structure, then it is basic and (h,⊗) is an hb-structure. the little theorem. in a set, greater hopes of the ones which are wass or cow, are also wass or cow, respectively. the fundamental relations are used for general definitions, thus, for example, in order to define the general hv-field one uses the fundamental relation γ*: definition 1.5. [7], [8], [9] the hv-ring (r,+, ·) is an hv-field if the quotient r/γ* is a field. the definition of the hv-field introduced a new class of hyperstructures [12]: the hv-semigroup (h, ·) is h/v-group if the quotient h/β* is a group. more complicated hyperstructures can be defined as well: 117 theodora kaplani and thomas vougiouklis definition 1.6. let [8] (r,+, ·) be an hv-ring, (m,+) be a cow hv-group and there exists an external hope · : r×m→ ℘(m) : (a,x) → ax such that ∀a,b ∈ r and ∀x,y ∈m we have a(x + y)∩ (ax + ay) 6= ∅, (a + b)x∩ (ax + bx) 6= ∅, (ab)x∩a(bx) 6= ∅, then m is called an hv -module over f. in the case of an hv-field f instead of an hv-ring r, then the hv-vector space is defined. in the above cases the fundamental relation �* is defined to be the smallest equivalence relation such that the quotient m/�* is a module (resp. vector space) over the fundamental ring r/γ* (resp. fundamental field f/γ*). the general definition of an hv-lie algebra was given in [14] as follows: definition 1.7. let (l,+) be an hv-vector space over the hv-field (f,+, ·), φ : f → f/γ* the canonical map and ωf = {x ∈ f : φ(x) = 0}, where 0 is the zero of the fundamental field f/γ*. similarly, let ωl be the core of the canonical map φ′ : l → l/�* and denote by the same symbol 0 the zero of l/�*. consider the bracket (commutator) hope: [, ] : l×l → ℘(l) : (x,y) → [x,y] then l is an hv-lie algebra over f if the following axioms are satisfied: (l1) the bracket hope is bilinear, i.e. [λ1x1 + λ2x2,y]∩ (λ1[x1,y] + λ2[x2,y]) 6= ∅ [x,λ1y1 + λ2y2]∩ (λ1[x,y1] + λ2[x,y2]) 6= ∅, ∀x,x1,x2,y,y1,y2 ∈ l,λ1,λ2 ∈ f (l2) [x,x]∩ωl 6= ∅, ∀x ∈ l (l3) ([x, [y,z]] + [y, [z,x]] + [z, [x,y]])∩ωl 6= ∅, ∀x,y ∈ l definition 1.8. [10] let (h, ·) be a hypergroupoid. we say that we remove the element h ∈ h, if we simply consider the restriction of (·) on h −{h}. we say that the element h ∈ h absorbs the element h ∈ h if we replace h, whenever it appears, by h. we say that the element h ∈ h merges with the element h ∈ h, if we take as product of x ∈ h by h, the union of the results of x with both h and h, and consider h and h as one class, with representative the element h. 118 finite hv-fields with strong-inverses 2 large classes of hv-structures and applications the large class of, so called, p-hyperstructures was appeared in 80’s to represent hopes of constant length [6]. since then several classes of p-hopes were introduced and studied [8], [4], [11]. definition 2.1. let (g, ·) be a groupoid, then for all p such that ∅ 6= p ⊂ g, we define the following hopes called p-hopes: ∀x,y ∈ g p : xpy = (xp)y ∪x(py), pr : xpry = (xy)p ∪x(yp), p l : xp ly = (px)y ∪p(xy). the (g,p),(g,pr), (g,p l) are called p-hyperstructures. the most usual case is when (g, ·) is semigroup, then we have xpy = (xp)y ∪x(py) = xpy and (g,p) is a semihypergroup. it is immediate the following: let (g, ·) be a group, then for all subsets p such that ∅ 6= p ⊂ g, the hyperstructure (g,p), where the p-hope is xpy = xpy, becomes a hypergoup in the sense of marty, i.e. the strong associativity is valid. the p-hope is of constant length, i.e. we have |xpy| = |p|. we call the hyperstructure (g,p), p-hypergroup. in [4], [15] a modified p-hope was introduced which is appropriate for the e-hyperstructures: construction 2.1. let (g, ·) be abelian group and p any subset of g with more than one elements. we define the hyperoperation ×p as follows: x×p y = { x ·p ·y = {x ·h ·y|h ∈ p} if x 6= e and c 6= e x ·y if x = e or y = e we call this hope pe-hope. the hyperstructure (g,×p) is an abelian hv-group. another large class is the one on which a new hope (∂) in a groupoid is defined. definition 2.2. [13]. let (g, ·) be groupoid (resp. hypergroupoid) and f : g → g be a map. we define a hope (∂), called theta-hope or simply ∂-hope, on g as follows x∂y = {f(x)·y,x·f(y)}, ∀x,y ∈ g. (resp. x∂y = (f(x)·y)∪(x·f(y)), ∀x,y ∈ g) 119 theodora kaplani and thomas vougiouklis if (·) is commutative then (∂) is commutative. if (·) is cow, then (∂) is cow. let (g, ·) be groupoid (resp. hypergroupoid) and f : g → p(g) −{∅} be multivalued map. we define the hope (∂), on g as follows x∂y = (f(x) ·y)∪ (x ·f(y)), ∀x,y ∈ g properties. if (g, ·) is a semigroup then: (a) for every f, the hope (∂) is wass. (b) if f is homomorphism and projection, or idempotent: f2 = f, then (∂) is associative. let (g, ·) be a groupoid and fi : g → g,i ∈ i, be a set of maps on g. we consider the map f∪ : g → p(g) such that f∪(x) = {fi(x)|i ∈ i,} called the the union of the fi(x). we define the union ∂-hope, on g if we consider as map the f∪(x). a special case for a given map f, is to take the union of this with the identity map. we consider the map f ≡ f ∪ (id), so f(x) = {x,f(x)},∀x ∈ g, which we call b−∂−hope. then we have x∂y = {xy,f(x) ·y,x ·f(y)},∀x,y ∈ g motivation for the definition of the ∂-hope is the map derivative where only the multiplication of functions can be used. therefore, in these terms, for given functions s(x), t(x), we have s∂t = {s′t,st′} where (′) denotes the derivative. proposition 2.1. let (g, ·) be group and f(x) = a, a constant map. then (g,∂)/β* is a singleton. proof. for all x in g we can take the hyperproduct of the elements, a−1 and a−1x a−1∂(a−1 ·x) = {f(a−1) ·a−1 ·x,a−1 ·f(a−1 ·x)} = {x,a}. thus xβa,∀x ∈ g, so β∗(x) = β∗(a) and (g,∂)/β* is singleton. 2 special case if (g, ·) be a group and f(x) = e, then x∂y = {x,y}, is the incidence hope. taking the application on the derivative, consider all polynomials of the first degree gi(x) = aix + bi. we have g1∂g2 = {a1a2x + a1b2,a1a2x + b1a2}, so it is a hope on the set of first degree polynomials. moreover all polynomials x + c, where c be a constant, are units. the lie-santilli isotopies born to solve hadronic mechanics problems. santilli [4], [15], proposed a ’lifting’ of the trivial unit matrix of a normal theory into a nowhere singular, symmetric, real-valued, new matrix. the original theory is 120 finite hv-fields with strong-inverses reconstructed such as to admit the new matrix as left and right unit. the isofields needed correspond to hv-structures called e-hyperfields which are used in physics or biology. therefore, in this theory the units and the inverses are playing very important role. the construction 2.1, is used, last years, in this theory. example 2.1. consider the ’small’ ring (z4,+, ·), suppose that we want to construct non-degenerate hv-field where 0 and 1 are scalars with respect for both addition and multiplication, and moreover every element of z4 has a unique opposite and every non-zero element has a unique inverse. then on the multiplication tables o these operations the lines and columns of the elements 0 and 1 remain the same. the sum 2+2=0 and the product 3 · 3 = 1, remain the same. in the results of the sums 2+3=1, 3+2=1 and 3+3=2 one can put respectively, the elements 3, 3 and 0. in the results of the products 2 ·2 = 0, 2 ·3 = 2 and 3 ·2 = 2 one can put respectively, the elements 2, 0 and 0. then in all those enlargements, even only one enlargement is used, we obtain (z4,+, ·)/β∗ ∼= z2 therefore this construction gives 49 hv-fields. 3 strong-inverse elements. we present now some hyperstructures, results and examples of hyperstructures with strong-inverse elements. properties 3.1. let (g, ·) be a group, take p such that ∅ 6= p ⊂ g and the p-hypergroup (g,p), where xpy = xpy. we have the following units: in order an element u to be right unit of the p-hypergroup (g,p), we must have xpu = xpu 3 x,∀x ∈ g. in fact the set pu must contain the unit element e of the group (g, ·). thus, all the elements of the set p−1, are right units. the same is valid for the left units, therefore, the set of all units is the p−1. inverses: let u be a unit in (g,p), then, for given x in order to have an inverse element x′ with respect to u, we must have xpx′ = xpx′ 3 u, so taking xpx′ = u, we obtain that all the elements of the form x′ = p−1x−1u are inverses to x with respect to the unit u. theorem 3.1. [16] let (g, ·) be a group, then for all normal subgroups p of g, the hyperstructure (g,p), where xpy = xpy,∀x,y ∈ g, is a hypergoup with strong inverses. moreover, for any inverse x′ of x ∈ g, with respect to any unit, we have xpx′ = p. 121 theodora kaplani and thomas vougiouklis proof. let x ∈ g, take an inverse x′ = p−1x−1u with respect to the unit u = p−1k , for any p. then we have xpx ′ = xpx′. but, since p is normal subgroup, we have xpx′ = xp−1x−1p−1k p = xp −1x−1p = xp−1px−1 = xpx−1 = p remark that in this case, p−1 = p , is the set of all units, thus all inverses are strong. 2 properties 3.2. let (g, ·) be groupoid and f : g → g be a map and (g,∂) the corresponding ∂-structure, then we have the following: units: in order an element u to be right unit, we must have x∂u = {f(x) ·u,x ·f(u)}3 x. but, the unit must not depend on the f(x), so f(u) = e, where e be unit in (g, ·) which must be a monoid. the same it is obtained for the left units. so the elements of kernf = {u : f(u) = e}, are the units of (g,∂). inverses: let u be a unit in (g,∂) , then (g, ·) is a monoid with unit e and f(u) = e. for given x in order to have an inverse element x′ with respect to u, we must have x∂x′ = {f(x) ·x′,x ·f(x′)}3 u and x′∂x = {f(x′) ·x,x′ ·f(x)}3 u. so the only cases, which do not depend on the image f(x′), are x′ = (f(x))−1u and x′ = u(f(x))−1 the right and left inverses, respectively. we have two-sided inverses iff f(x)u = uf(x). remark [16]: since the inverses are depending on the units, therefore they are not strong. the following constructions, originated from the properties the strong-inverse elements have, gives a minimal hyperstructure which have strong-inverse elements. this is a necessary enlargement in order all the elements to be stronginverses. construction 3.1. let (g, ·) be a group with unit e. consider a finite set e = {ei|i ∈ i}. on the set g = (g−{e})∪e we define a hope (×) as follows:  ei ×ej = {ei,ej}, ∀ei,ej ∈ e ei ×x = x×ej = x, ∀ei ∈ e,x ∈ g−{e} x×y = x ·y if x ·y ∈ g−{e} and x×y = e if x ·y = e then the hyperstructure (g,×) is a hypergroup. the set of unit elements is e and all the elements are strong-inverse. moreover we have (g,×)/β∗ ∼= (g, ·). 122 finite hv-fields with strong-inverses proof. for the associativity we have the cases (ei ×ej)×ek = ei × (ej ×ek) = {ei,ej,ek},∀ei,ej,ek ∈ e (x×y)×z = x×(y×z) = x ·y ·z or e, ∀x,y,z ∈ g and not all of them belong to e. in the second case there is no matter if the product of two inverse elements appears. the only difference is that the result is singleton and in some cases the result is equal to the set e. therefore the strong associativity is valid. moreover the reproductivity is valid and the set e is the set of units in (g,×). two elements of g are β* equivalent if they belong to any finite ×-product of elements of g. thus all fundamental classes are singletons except the set of units e. that means that we have (g,×)/β∗ ∼= (g, ·). 2 construction 3.2. let (g, ·) be an hv-group with only one unit element e and every element has a unique inverse. consider a finite set e = {ei|i ∈ i}. on the set g = (g−{e})∪e we define a hope (×) as follows:  ei ×ej = {ei,ej}, ∀ei,ej ∈ e ei ×x = x×ej = x, ∀ei ∈ e,x ∈ g−{e} x×y = x ·y if x ·y ∈ g−{e} and x×y = e if x ·y = e then the hyperstructure (g,×) is an hv-group. the set of unit elements is e and all the elements are strong-inverse. moreover we have (g,×)/β∗ ∼= (g, ·)/β∗. proof. for the associativity we have the cases (ei ×ej)×ek = ei × (ej ×ek) = {ei,ej,ek},∀ei,ej,ek ∈ e (x×y)×z = x×(y×z) = x ·y ·z or e, ∀x,y,z ∈ g and not all of them belong to e. therefore the wass is valid. moreover the reproductivity is valid and the set e is the set of units in (g,×). two elements of g are β* equivalent if they belong to any finite ×-product of elements of g. so, all fundamental classes correspond to the fundamental classes of (g, ·), with an enlargement of the class of e into e. thus, we have (g,×)/β∗ ∼= (g, ·)/β∗.2 we remark that the above constructions give a great number of hyperstructures with strong-inverses because we can enlarge then in any result except if the result is e. now we present a result on strong-inverses on a general finite case. 123 theodora kaplani and thomas vougiouklis theorem 3.2. the minimum non-degenerate, i.e. have non-degenerate fundamental field, h/v-fields with strong-inverses with respect to both sum-hope and product-hope, obtained by enlarging the ring (z2p,+, ·), where p > 2 is prime number, and which has fundamental field isomorphic to (zp,+, ·), is defined as follows: the sum-hope (⊕) is enlarged from (+) by setting (1). p(⊕)κ = κ(⊕)p = κ + e,∀κ ∈ z2p, where e = {0,p} be the set of zeros (2). whenever the result is 0 and p we enlarge it by setting p and 0, respectively. the product-hope (⊗) is enlarged from (·) by setting (3). (p + 1) ⊗κ = κ⊗ (p + 1) = κu,∀κ ∈ z2p, where u = {1,p + 1} be the set of units (4). whenever the result is 1 and p+1 we enlarge it by setting p+1 and 1, respectively. the fundamental classes are of the form κ = {κ,κ + p},∀κ ∈ z2p proof. in order to have non degenerate case, since we have 2p elements, in both, sum-hope and product-hope, is to take the zero-set e = {0,p} and unit-set u = {1,p + 1}. in order to have strong-opposites, we have to enlarge, according to remark 1.1, as in (2). moreover, in order to have strong-inverses, we have to enlarge, again according to remark 1.1, as in (4). from the above definition of the sum-hope it is to see that the fundamental classes are of the form κ = {κ,κ + p},∀κ ∈ z2p. for the above classes for the product-hope mod(2p), we have ∀κ,λ ∈ z2p, κ⊗λ = {κ,κ + p} ·{λ,λ + p} = {κλ,κ(λ + p),(κ + p)λ,(κ + p)(λ + p)} = = {κλ,κλ + κp,κλ + pλ,κλ + κp + pλ + pp} = {κλ,κλ + p} because, if κ or λ are odd numbers then κλ + κp or κλ + pλ, respectively, are equal mod2p to κλ+p. moreover, if both κ and λ are even numbers then we have, κλ + κp + pλ + pp = κλ + p. from the above we remark that the fundamental classes κ = {κ,κ + p},∀κ ∈ z2p, are formed from sum-hope and they are remain the same in the product-hope. finally the fundamental field is isomorphic to (zp,+, ·). 2 as example of the above theorem we present the case for p=5. example 3.1. in the case of the h/v-field (z10,⊕,⊗), i.e. p=5, we have the following multiplicative tables: 124 finite hv-fields with strong-inverses ⊕ 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5,0 6 7 8 9 1 1 2 3 4 5,0 6,1 7 8 9 0,5 2 2 3 4 5,0 6 7,2 8 9 0,5 1 3 3 4 5,0 6 7 8,3 9 0,5 1 2 4 4 5,0 6 7 8 9,4 0,5 1 2 3 5 5,0 6,1 7,2 8,3 9,4 0,5 1,6 2,7 3,8 4,9 6 6 7 8 9 0,5 1,6 2 3 4 5,0 7 7 8 9 0,5 1 2,7 3 4 5,0 6 8 8 9 0,5 1 2 3,8 4 5,0 6 7 9 9 0,5 1 2 3 4,9 5,0 6 7 8 and ⊗ 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1,6 2 3 4 5 6,1 7 8 9 2 0 2 4 6,1 8 0 2 4 6,1 8 3 0 3 6,1 9 2 5 8,3 1,6 4 7 4 0 4 8 2 6,1 0 4 8 2 6,1 5 0 5 0 5 0 5 0,5 5 0 5 6 0 6,1 2 8,3 4 0,5 6,1 2,7 8 4,9 7 0 7 4 1,6 8 5 2,7 9 6,1 3 8 0 8 6,1 4 2 0 8 6,1 4 2 9 0 9 8 7 6,1 5 4,9 3 2 1,6 moreover, it is easy to see that the fundamental field is isomorphic to (z5,+, ·). references [1] p. corsini, prolegomena of hypergroup theory, aviani editore, 1993. [2] p. corsini and v. leoreanu, application of hyperstructure theory, klower academic publishers, 2003. [3] b. davvaz and v. leoreanu, hyperring theory and applications int. acad. press, 2007 [4] b. davvaz, r.m santilli, and t. vougiouklis, studies of multi-valued hyperstructures for the characterization of matter-antimatter systems and their extension, algebras, groups and geometries 28(1), (2011), 105-116. 125 theodora kaplani and thomas vougiouklis [5] r. mahjoob, t. kaplani and t. vougiouklis, hv-groups with strong-inverses, submitted. [6] t. vougiouklis, generalization of p-hypergroups, rend. circolo mat. palermo, ser. ii, 36, (198), (1987), 114-121. [7] t. vougiouklis, (1991). the fundamental relation in hyperrings. the general hyperfield, 4th aha, xanthi 1990, world scientific, (1991), 203-211. [8] t. vougiouklis, hyperstructures and their representations, monographs in math., hadronic, 1994. [9] t. vougiouklis, some remarks on hyperstructures, contemporary math., amer. math. society, 184, (1995), 427-431. [10] t. vougiouklis, enlarging hv-structures, algebras and combinatorics, icac’97, hong kong, springer-verlag, (1999), 455-463. [11] t. vougiouklis, on hv-rings and hv-representations, discrete math., elsevier, 208/209, (1999), 615-620. [12] t. vougiouklis, the h/v-structures, j. discrete math. sciences and cryptography, v.6, n.2-3, (2003), 235-243. [13] t. vougiouklis, ∂-operations and hv-fields, acta math. sinica, (engl. ser.), v.24, n.7, (2008), 1067-1078. [14] t. vougiouklis, the lie-hyperalgebras and their fundamental relations, southeast asian bull. math., v. 37(4), (2013), 601-614. [15] t. vougiouklis, hypermathematics, hv-structures, hypernumbers, hypermatrices and lie-santilli admissibility, american j. modern physics, 4(5), (2015), 34-46. [16] t. vougiouklis, and t. kaplani, special elements on p-hopes and ∂-hopes, southeast asian bulletin mathematics, vol. 40(3), (2016), 451-460. 126 ratio mathematica vol. 33, 2017, pp. 167-179 issn: 1592-7415 eissn: 2282-8214 helix-hopes on s-helix matrices souzana vougioukli∗, thomas vougiouklis† ‡doi:10.23755/rm.v33i0.385 abstract a hyperproduct on non-square ordinary matrices can be defined by using the so called helix-hyperoperations. the main characteristic of the helixhyperoperation is that all entries of the matrices are used. such operations cannot be defined in the classical theory. several classes of non-square matrices have results of the helix-product with small cardinality. we study the helix-hyperstructures on the representations and we extend our study up to hv-lie theory by using ordinary fields. we introduce and study the class of s-helix matrices. keywords: hyperstructures; hv-structures; h/v-structures; hope; helix-hopes. 2010 ams subject classifications: 20n20, 16y99. ∗17 oikonomou str, exarheia, 10683 athens, greece; elsouvou@gmail.com †democritus university of thrace, school of education, 68100 alexandroupolis, greece; tvougiou@eled.duth.gr ‡ c©souzana vougioukli and thomas vougiouklis. received: 31-10-2017. accepted: 26-122017. published: 31-12-2017. 167 souzana vougioukli and thomas vougiouklis 1 introduction our object is the largest class of hyperstructures, the hv-structures, introduced in 1990 [10], satisfying the weak axioms where the non-empty intersection replaces the equality. definition 1.1. in a set h equipped with a hyperoperation (abbreviate by hope) · : h ×h → p(h)−{∅} : (x,y) → x ·y ⊂ h we abbreviate by wass the weak associativity: (xy)z ∩x(yz) 6= ∅, ∀x,y,z ∈ h and by cow the weak commutativity: xy ∩yx 6= ∅, ∀x,y ∈ h. the hyperstructure (h, ·) is called hv-semigroup if it is wass and is called hv−group if it is reproductive hv-semigroup: xh = hx = h, ∀x ∈ h. (r,+, ·) is called hv−ring if (+) and (·) are wass, the reproduction axiom is valid for (+) and (·) is weak distributive with respect to (+): x(y + z)∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅, ∀x,y,z ∈ r. for more definitions, results and applications on hv-structures, see [1], [2], [11], [12], [13], [17]. an interesting class is the following [8]: an hv-structure is very thin, if and only if, all hopes are operations except one, with all hyperproducts singletons except only one, which is a subset of cardinality more than one. therefore, in a very thin hv-structure in a set h there exists a hope (·) and a pair (a,b) ∈ h2 for which ab = a, with carda > 1, and all the other products, with respect to any other hopes, are singletons. the fundamental relations β* and γ* are defined, in hv-groups and hv-rings, respectively, as the smallest equivalences so that the quotient would be group and ring, respectively [8], [9], [11], [12], [13], [17]. the main theorem is the following: theorem 1.1. let (h, ·) be an hv-group and let us denote by u the set of all finite products of elements of h. we define the relation β in h as follows: xβy iff {x,y} ⊂ u where u ∈ u. then the fundamental relation β* is the transitive closure of the relation β. an element is called single if its fundamental class is a singleton. motivation: the quotient of a group with respect to any partition is an hvgroup. definition 1.2. let (h, ·),(h,⊗) be hv-semigroups defined on the same h. (·) is smaller than (⊗), and (⊗) greater than (·), iff there exists automorphism f ∈ aut(h,⊗) such that xy ⊂ f(x⊗y), ∀x,y ∈ h. 168 helix-hopes on s-helix matrices then (h,⊗) contains (h, ·) and write · ≤⊗. if (h, ·) is structure, then it is basic and (h,⊗) is an hb-structure. the little theorem [11]. greater hopes of the ones which are wass or cow, are also wass and cow, respectively. fundamental relations are used for general definitions of hyperstructures. thus, to define the general hv-field one uses the fundamental relation γ*: definition 1.3. [10] the hv-ring (r,+, ·) is called hv-field if the quotient r/γ* is a field. this definition introduces a new class of which is the following [15]: definition 1.4. the hv-semigroup (h, ·) is called h/v-group if h/β* is a group. the class of h/v-groups is more general than the hv-groups since in h/v-groups the reproductivity is not valid. the h/v-fields and the other related hyperstructures are defined in a similar way. an hv-group is called cyclic [8], if there is an element, called generator, which the powers have union the underline set, the minimal power with this property is the period of the generator. definition 1.5. [11], [14], [18]. let (r,+, ·) be an hv-ring, (m,+) be cow hv-group and there exists an external hope · : r ×m → p(m) : (a,x) → ax, such that, ∀a,b ∈ r and ∀x,y ∈ m we have a(x + y)∩ (ax + ay) 6= ∅, (a + b)x∩ (ax + bx) 6= ∅, (ab)x∩a(bx) 6= ∅, then m is called an hv-module over r. in the case of an hv-field f instead of an hv-ring r, then the hv-vector space is defined. definition 1.6. [16] let (l,+) be hv-vector space on (f,+, ·), φ : f → f/γ*, the canonical map and ωf = {x ∈ f : φ(x) = 0}, where 0 is the zero of the fundamental field f/γ*. similarly, let ωl be the core of the canonical map φ′ : l → l/�* and denote again 0 the zero of l/�*. consider the bracket (commutator) hope: [, ] : l×l → p(l) : (x,y) → [x,y] then l is an hv-lie algebra over f if the following axioms are satisfied: (l1) the bracket hope is bilinear, i.e. [λ1x1 + λ2x2,y]∩ (λ1[x1,y] + λ2[x2,y]) 6= ∅ [x,λ1y1 + λ2y2]∩ (λ1[x,y1] + λ2[x,y2]) 6= ∅, ∀x,x1,x2,y,y1,y2 ∈ l,λ1,λ2 ∈ f 169 souzana vougioukli and thomas vougiouklis (l2) [x,x]∩ωl 6= ∅, ∀x ∈ l (l3) ([x, [y,z]] + [y, [z,x]] + [z, [x,y]])∩ωl 6= ∅, ∀x,y,z ∈ l a well known and large class of hopes is given as follows [8], [9], [11]: definition 1.7. let (g, ·) be a groupoid, then for every subset p ⊂ g,p 6= ∅, we define the following hopes, called p-hopes: ∀x,y ∈ g p : xpy = (xp)y ∪x(py), pr : xpry = (xy)p ∪x(yp), p l : xp ly = (px)y ∪p(xy). the (g,p),(g,pr), (g,p l) are called p-hyperstructures. the usual case is for semigroup (g, ·), then xpy = (xp)y ∪x(py) = xpy, and (g,p) is a semihypergroup. a new important application of hv-structures in nuclear physics is in the santilli’s isotheory. in this theory a generalization of p-hopes is used, [4], [5], [22], which is defined as follows: let (g,) be an abelian group and p a subset of g with more than one elements. we define the hyperoperation ×p as follows: x×p y = { x ·p ·y = {x ·h ·y|h ∈ p} if x 6= e and c 6= e x ·y if x = e or y = e we call this hope pe-hope. the hyperstructure (g,×p) is an abelian hv-group. 2 small hypernumbers and hv-matrix representations several constructions of hv-fields are uses in representation theory and applications in applied sciences. we present some of them in the finite small case [18]. construction 2.1. on the ring (z4,+, ·) we will define all the multiplicative h/vfields which have non-degenerate fundamental field and, moreover they are, (a) very thin minimal, (b) cow (non-commutative), (c) they have the elements 0 and 1, scalars. 170 helix-hopes on s-helix matrices then, we have only the following isomorphic cases 2 ⊗ 3 = {0,2} or 3 ⊗ 2 = {0,2}. fundamental classes: [0] = {0,2}, [1] = {1,3} and we have (z4,+,⊗)/γ∗ ∼= (z2,+, ·). thus it is isomorphic to (z2 × z2,+). in this hv-group there is only one unit and every element has a unique double inverse. construction 2.2. on the ring (z6,+, ·) we define, up to isomorphism, all multiplicative h/v-fields which have non-degenerate fundamental field and, moreover they are: (a) very thin minimal, i.e. only one product has exactly two elements (b) cow (non-commutative) (c) they have the elements 0 and 1, scalars then we have the following cases, by giving the only one hyperproduct, (i) 2⊗3 = {0,3} or 2⊗4 = {2,5} or 2⊗5 = {1,4} 3⊗4 = {0,3} or 3⊗5 = {0,3} or 4⊗5 = {2,5} in all 6 cases the fundamental classes are [0] = {0,3}, [1] = {1,4}, [2] = {2,5} and we have (z6,+,⊗)/γ∗ ∼= (z3,+, ·). (ii) 2⊗3 = {0,2} or 2⊗3 = {0,4} or 2⊗4 = {0,2} or 2⊗4 = {2,4} or 2⊗5 = {0,4} or 2⊗5 = {2,4} or 3⊗4 = {0,2} or 3⊗4 = {0,4} or 3⊗5 = {1,3} or 3⊗5 = {3,5} or 4⊗5 = {0,2} or 4⊗5 = {2,4} in all 12 cases the fundamental classes are [0] = {0,2,4}, [1] = {1,3,5} and we have (z6,+,⊗)/γ∗ ∼= (z2,+, ·). hv-structures are used in representation theory of hv-groups which can be achieved by generalized permutations or by hv-matrices [11], [14], [18]. definition 2.1. hv-matrix is a matrix with entries of an hv-ring or hv-field. the hyperproduct of two hv-matrices (aij) and (bij), of type m × n and n × r respectively, is defined in the usual manner and it is a set of m× r hv-matrices. the sum of products of elements of the hv-ring is considered to be the n-ary circle hope on the hypersum. the hyperproduct of hv-matrices is not necessarily wass. the problem of the hv-matrix representations is the following: definition 2.2. let (h, ·) be hv-group (or h/v-group). find an hv-ring (r,+, ·), a set mr = {(aij)|aij ∈ r} and a map t : h → mr : h 7→ t(h) such that t(h1h2)∩t(h1)t(h2) 6= ∅,∀h1,h2 ∈ h. 171 souzana vougioukli and thomas vougiouklis t is hv -matrix (or h/v-matrix) representation. if t(h1h2) ⊂ t(h1)(h2) is called inclusion. if t(h1h2) = t(h1)(h2) = {t(h)|h ∈ h1h2}, ∀h1,h2 ∈ h, then t is good and then an induced representation t* for the hypergroup algebra is obtained. if t is one to one and good then it is faithful. the main theorem on representations is [11]: theorem 2.1. a necessary condition to have an inclusion representation t of an hv-group (h, ·) by n×n, hv-matrices over the hv-ring (r,+, ·) is the following: for all classes β∗(x),x ∈ h must exist elements aij ∈ h,i,j ∈ {1, . . . ,n} such that t(β∗(a)) ⊂{a = (a′ij)|aij ∈ γ ∗(aij), i,j ∈{1, . . . ,n}} inclusion t : h → mr : a 7→ t(a) = (aij) induces homomorphic representation t * of h/β* on r/γ* by setting t∗(β∗(a)) = [γ∗(aij)], ∀β∗(a) ∈ h/β∗, where γ∗(aij) ∈ r/γ∗ is the ij entry of the matrix t∗(β∗(a)). t * is called fundamental induced of t. in representations, several new classes are used: definition 2.3. let m = mm×n be the module of m × n matrices over r and p = {pi : i ∈ i}⊆ m. we define a p-hope p on m as follows p : m ×m → p(m) : (a,b) → apb = {ap ti b : i ∈ i}⊆ m where p t denotes the transpose of p. the hope p is bilinear map, is strong associative and the inclusion distributive: ap(b + c) ⊆ apb + apc,∀a,b,c ∈ m definition 2.4. let m = mm×n the m×n matrices over r and let us take sets s = {sk : k ∈ k}⊆ r, q = {qj : j ∈ j}⊆ m, p = {pi : i ∈ i}⊆ m. define three hopes as follows s : r×m → p(m) : (r,a) → rsa = {(rsk)a : k ∈ k}⊆ m q + : m ×m → p(m) : (a,b) → aq + b = {a + qj + b : j ∈ j}⊆ m p : m ×m → p(m) : (a,b) → apb = {ap ti b : i ∈ i}⊆ m then (m,s,q + ,p) is hyperalgebra on r called general matrix p-hyperalgebra. 172 helix-hopes on s-helix matrices 3 helix-hopes recall some definitions from [3], [4], [6], [7], [19], [20], [21]: definition 3.1. let a = (aij) ∈ mm×n be m×n matrix and s,t ∈ n be naturals such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. we define the map cst from mm×n to ms×t by corresponding to the matrix a, the matrix acst = (aij) where 1 ≤ i ≤ s, 1 ≤ j ≤ t. we call this map cut-projection of type st. thus acst = (aij) is matrix obtained from a by cutting the lines, with index greater than s, and columns, with index greater than t. we use cut-projections on all types of matrices to define sums and products. definition 3.2. let a = (aij) ∈ mm×n be an m × n matrix and s,t ∈ n, such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. we define the mod-like map st from mm×n to ms×t by corresponding to a the matrix ast = (aij) which has as entries the sets aij = {ai+κs,j+λt|1 ≤ i ≤ s,1 ≤ j ≤ t and κ,λ ∈ n,i + κs ≤ m,j + λt ≤ n}. thus, we have the map st : mm×n → ms×t : a → ast = (aij). we call this multivalued map helix-projection of type st. ast is a set of s × tmatrices x = (xij) such that xij ∈ aij,∀i,j. obviously amn = a. let a = (aij) ∈ mm×n be a matrix and s,t ∈ n such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. then it is clear that we can apply the helix-projection first on the rows and then on the columns, the result is the same if we apply the helix-projection on both, rows and columns. therefore we have (asn)st = (amt)st = ast. let a = (aij) ∈ mm×n be matrix and s,t ∈ n such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. then if ast is not a set but one single matrix then we call a cut-helix matrix of type s × t. in other words the matrix a is a helix matrix of type s × t, if acst= ast. definition 3.3. a. let a = (aij) ∈ mm×n,b = (bij) ∈ mu×v, be matrices and s=min(m,u), t=min(n,u). we define a hope, called helix-addition or helix-sum, as follows: ⊕ : mm×n ×mu×v → p(ms×t) : (a,b) → a⊕b = ast + bst = (aij) + (bij) ⊂ ms×t, where (aij) + (bij) = {(cij = (aij + bij)|aij ∈ aij and bij ∈ bij} 173 souzana vougioukli and thomas vougiouklis b. let a = (aij) ∈ mm×n and b = (bij) ∈ mu×v, be matrices and s=min(m,u). we define a hope, called helix-multiplication or helix-product, as follows: ⊗ : mm×n ×mu×v → p(mm×v) : (a,b) → a⊗b = ams ·bsv = (aij) · (bij) ⊂ mm×v, where (aij) · (bij) = {(cij = ( ∑ aitbtj)|aij ∈ aij and bij ∈ bij} the helix-sum is an external hope and the commutativity is valid. for the helix-product we remark that we have a ⊗ b = ams · bsv so we have either ams = a or bsv = b, that means that the helix-projection was applied only in one matrix and only in the rows or in the columns. if the appropriate matrices in the helix-sum and in the helix-product are cut-helix, then the result is singleton. remark. in mm×n the addition is ordinary operation, thus we are interested only in the ’product’. from the fact that the helix-product on non square matrices is defined, the definition of the lie-bracket is immediate, therefore the helix-lie algebra is defined [22], as well. this algebra is an hv-lie algebra where the fundamental relation �∗ gives, by a quotient, a lie algebra, from which a classification is obtained. in the following we restrict ourselves on the matrices mm×n where m < n. we have analogous results if m > n and for m = n we have the classical theory. notation. for given κ ∈ n −{0}, we denote by κ the remainder resulting from its division by m if the remainder is non zero, and κ = m if the remainder is zero. thus a matrix a = (aκλ) ∈ mm×n,m < n is a cut-helix matrix if we have aκλ = aκλ,∀κλ ∈ n−{0}. moreover let us denote by ic = (cκλ) the cut-helix unit matrix which the cut matrix is the unit matrix im. therefore, since im = (δκλ), where δκλ is the kronecker’s delta, we obtain that, ∀κ,λ, we have cκλ = δκλ. proposition 3.1. for m < n in (mm×n,⊗) the cut-helix unit matrix ic = (cκλ), where cκλ = δκλ, is a left scalar unit and a right unit. it is the only one left scalar unit. proof. let a,b ∈ mm×n then in the helix-multiplication, since m < n, we take helix projection of the matrix a, therefore, the result a⊗b is singleton if the matrix a is a cut-helix matrix of type m × m. moreover, in order to have a⊗b = amm·b = b, the matrix amm must be the unit matrix. consequently, ic = (cκλ), where cκλ = δκλ,∀κ,λ ∈ n−{0}, is necessarily the left scalar unit. let a = (auv) ∈ mm×n and consider the hyperproduct a ⊗ ic. in the entry κλ of this hyperproduct there are sets, for all 1 ≤ κ ≤ m, 1 ≤ λ ≤ n , of the form∑ aκscsλ = ∑ aκsδsλ = aκλ 3 aκλ. 174 helix-hopes on s-helix matrices therefore a⊗ ic 3 a,∀a ∈ mm×n. 2 4 the s-helix matrices definition 4.1. let a = (aij) ∈ mm×n be matrix and s,t ∈ n such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. then if ast is a set of upper triangular matrices with the same diagonal, then we call a an s-helix matrix of type s× t. therefore, in an s-helix matrix a of type s× t, the ast has on the diagonal entries which are not sets but elements. in the following, we restrict our study on the case of a = (aij) ∈ mm×n with m < n. remark. according to the cut-helix notation, we have, aκλ = aκλ = 0, for all κ > λ and aκλ = aκλ, for κ = λ. proposition 4.1. the set of s-helix matrices a = (aij) ∈ mm×n with m < n, is closed under the helix product. moreover, it has a unit the cut-helix unit matrix ic, which is left scalar. proof. it is clear that the helix product of two s-helix matrices, x = (xij),y = (aij) ∈ mm×n,x ⊗ y , contain matrices z = (zij), which are upper diagonals. moreover, for every zii, the entry ii is singleton since it is product of only z(i+km),(i+km) = zii, entries. the unit is, from proposition 3.1, the matrix ic = im×n, where we have im×n = imm = im. 2 an example of hyper-matrix representation, seven dimensional, with helixhope is the following: example 4.1. consider the special case of the matrices of the type 3 × 5 on the field of real or complex. then we have x =  x11 x12 x13 x11 x150 x22 x23 0 x22 0 0 x33 0 0   and y =  y11 y12 y13 y11 y150 y22 y23 0 y22 0 0 y33 0 0   x ⊗y =  x11 {x12, x15} x130 x22 x23 0 0 x33   ·  y11 y12 y13 y11 y150 y22 y23 0 y22 0 0 y33 0 0   = ( x11y11 x11y12 + {x12,x15}y22 x11y13 + {x12,x15}y23 + x13y33 x11y11 x11y15 + {x12,x15}y22 0 x22y22 x22y23 + x23y33 0 x22y22 0 0 x33y33 0 0 ) therefore the helix product is a set with cardinality up to 8. the unit of this type is ic =  1 0 0 1 00 1 0 0 1 0 0 1 0 0   175 souzana vougioukli and thomas vougiouklis definition 4.2. we call a matrix a = (aij) ∈ mm×n an s0-helix matrix if it is an s-helix matrix where the condition a11a22 . . .amm 6= 0, is valid. therefore, an s0-helix matrix has no zero elements on the diagonal and the set s0 is a subset of the set s of all s-helix matrices. we notice that this set is closed under the helix product not in addition. therefore it is interesting only when the product is used not the addition. proposition 4.2. the set of s0-helix matrices a = (aij) ∈ mm×n with m < n, is closed under the helix product, it has a unit the cut-helix unit matrix ic, which is left scalar and s0-helix matrices x have inverses x−1, i.e. ic ∈ x ⊗ x−1 ∩ x−1 ⊗x. proof. first it is clear that on the helix product of two s0-helix matrices, the diagonal has not any zero since there is no zero on each of them. therefore, the helix product is closed. the entries in the diagonal are inverses in the hv-field. in the rest entries we have to collect equations from those which correspond to each element of the entry-set. 2 example 4.2. consider the special case of the above example 4.1, of the matrices of the type 3 × 5. suppose we want to find the inverse matrix y = x−1, of the matrix x. then we have ic ∈ x ⊗y ∩y ⊗x. therefore, we obtain x11y11 = x22y22 = x33y33 = 1 x11y12 +{x12,x15}y22 3 0,x11y13 +{x12,x15}y23 + x13y33 3 0, x11y15 +{x12,x15}y22 3 0,x23y22 + x33y23 3 0, therefore a solution is y11 = 1 x11 ,y22 = 1 x22 ,y33 = 1 x33 y23 = −x23 x22x33 ,y12 = −x12 x11x22 ,y15 = −x15 x11x22 , and y13 = −x13 x11x33 + x23x12 x11x22x33 or y13 = −x13 x11x33 + x23x14 x11x22x33 thus, a left and right inverse matrix of x is x−1 =   1x11 −x12x11x22 −x13x11x33 + x23x12x11x22x33 1x11 −x15x110 1 x22 −x23 x22x33 0 1 x22 0 0 1 x33 0 0   an interesting research field is the finite case on small finite hv-fields. important cases appear taking the generating sets by any s0-helix matrix. 176 helix-hopes on s-helix matrices example 4.3. on the type 3 × 5 of matrices using the construction 2.1, on (z4,+, ·) we take the small hv-field (z4,+,⊗), where only 2 ⊗ 3 = {0,2} and fundamental classes {0,2},{1,3}. we consider the set of all s0-helix matrices and we take the s0-helix matrix: x =  1 2 2 1 00 3 1 0 3 0 0 1 0 0   then the powers of x are: x2 =  1 {0,2} {0,2} 1 {0,2}0 1 0 0 1 0 0 1 0 0   x3 =  1 {0,2} {0,2} 1 {0,2}0 3 1 0 3 0 0 1 0 0   , and so on we obtain that the generating set is the following 1 {0,2} {0,2} 1 {0,2}0 {1,3} {0,1} 0 {1,3} 0 0 1 0 0   where in the 22 and 25 entries appears simultaneously 1 or 3. 177 souzana vougioukli and thomas vougiouklis references [1] p. corsini and v. leoreanu, application of hyperstructure theory, klower academic publishers, 2003. 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[9] t. vougiouklis, groups in hypergroups, annals discrete math. 37, (1988), 459-468. [10] t. vougiouklis, (1991). the fundamental relation in hyperrings. the general hyperfield, 4th aha, xanthi 1990, world scientific, (1991), 203-211. [11] t. vougiouklis, hyperstructures and their representations, monographs in math., hadronic, 1994. [12] t. vougiouklis, some remarks on hyperstructures, contemporary math., amer. math. society, 184, (1995), 427-431. [13] t. vougiouklis, enlarging hv-structures, algebras and combinatorics, icac’97, hong kong, springer-verlag, (1999), 455-463. [14] t. vougiouklis, on hv-rings and hv-representations, discrete math., elsevier, 208/209, (1999), 615-620. 178 helix-hopes on s-helix matrices [15] t. vougiouklis, the h/v-structures, j. discrete math. sciences and cryptography, v.6, n.2-3, (2003), 235-243. [16] t. vougiouklis, the lie-hyperalgebras and their fundamental relations, southeast asian bull. math., v. 37(4), (2013), 601-614. [17] t. vougiouklis, from hv-rings to hv-fields, int. j. algebraic hyperstr. and appl. vol.1, no.1, (2014), 1-13. [18] t. vougiouklis, hypermatrix representations of finite hv-groups, european j. combinatorics, v.44 b, (2015), 307-315. [19] t. vougiouklis and s. vougioukli, the helix hyperoperations, italian j. pure appl. math., 18, (2005), 197-206. [20] t. vougiouklis and s. vougioukli, hyper-representations by non square matrices. helix-hopes, american j. modern physics, 4(5), (2015), 52-58. [21] t. vougiouklis and s. vougioukli, helix-hopes on finite hyperfields, ratio mathematica, v.31, (2016), 65-78. [22] t. vougiouklis and s. vougioukli, hyper lie-santilli admissibility, algebras groups and geometries (agg), v.33, n.4, (2016), 427-442. 179 ratio mathematica vol. 33, 2017, pp. 77-88 issn: 1592-7415 eissn: 2282-8214 vougiouklis contributions in the field of algebraic hyperstructures bijan davvaz∗ †doi:10.23755/rm.v33i0.380 abstract thomas vougiouklis was born in 1948, greece. he has many contributions to algebraic hyperstructures. hv-structures are some of his main contributions. in this article, we study some of vougiouklis ideas in the field of algebraic hyperstructures as follows: (1) semi-direct hyperproduct of two hypergroups; (2) representation of hypergroups; (3) fundamental relation in hyperrings; (4) commutative rings obtained from hyperrings; (5) hvstructures; (6) the uniting elements method; (7) the e-hyperstructures; (8) helix-hyperoperations. keywords: hyperoperation, hypergroup, hyperring, hv-group, hv-ring, fundamental relation. 2010 ams subject classifications: 20n20, 16y99. ∗department of mathematics, yazd university, yazd, iran; davvaz@yazd.ac.ir † c©bijan davvaz. received: 31-10-2017. accepted: 26-12-2017. published: 31-12-2017. 77 bijan davvaz 1 semi-direct hyperproduct of two hypergroups for all natural numbers n > 1, define the relation βn on a semihypergroup h, as follows: aβnb if and only if there exist x1, . . . ,xn ∈ h such that {a,b} ⊆∏n i=1 xi, and take β = ⋃ n≥1 βn, where β1 = {(x,x) | x ∈ h} is the diagonal relation on h. denote by β∗ the transitive closure of β. the relation β∗ is a strongly regular relation. this relation was introduced by koskas [10] and studied mainly by freni [9], proving the following basic result: if h is hypergroup, then β = β∗. note that, in general, for a semihypergroup may be β 6= β∗. moreover, the relation β∗ is the smallest equivalence relation on a hypergroup h, such that the quotient h/β∗ is a group. the heart ωh of a hypergroup h is defined like the set of all elements x of h, for which the equivalence class β∗(x) is the identity of the quotient group h/β∗. vougiouklis in [13] studied the fundamental relation introduced by koskas. he used the quotient set in order to define a semi-direct hyperproduct of two hypergroups. he obtained an extension of hypergroups by hypergroups. let a,b be two hypergroups and consider the group auta. let̂ : b/β∗ → auta be an arbitrary homomorphism, where we denote β̂∗(b) by b̂. then, in a × b a hyperproduct can be defined as follows: (a,b) � (c,d) = {(x,y) | x ∈ ab̂(c), y ∈ bd} then, a × b becomes a hypergroup called semidirect hyperproduct of a and b corresponding to ̂ and it is denoted by a×̂b. vougiouklis proved that a×̂b/β∗ a×̂b ∼= a/β∗a×̂b/β ∗ b [13]. 2 representation of hypergroups vougiouklis in a sequence of papers studied the representations of hypergroups. for instance, in [15], a class of hypermatrices to represent hypergroups is introduced and application on class of p -hypergroups is given. hypermatrices are matrices with entries of a semihyperring. the product of two hypermatrices (aij) and (bij) is the hyperoperation given in the usual manner (aij) · (bij) = {(cij) |cij ∈ ∑n k=1 aikbkj}. vougiouklis problem is the following one: for a given hypergroup h, find a semihyperring r such that to have a representation of h by hypermatrices with entries from r. recall that if mr = {(aij) | aij ∈ r}, then a map t : h → mr is called a representation if t(x) · t(y) = {t(z) | z ∈ xy} = t(xy), for all x,y ∈ h. he obtained an induced representation t∗ for the hypergroup algebra of h, see [14]. 78 vougiouklis contributions in the field of algebraic hyperstructures 3 fundamental relation in hyperrings vougiouklis introduced the notion of fundamental relation in the context of general hyperrings [16, 17]. a multivalued system (r,+, ·) is a ( general) hyperring if (1) (r,+) is a hypergroup; (2) (r, ·) is a semihypergroup; (3) (·) is (strong) distributive with respect to (+), i.e., for all x,y,z in r we have x · (y + z) = x·y+x·z and (x+y)·z = x·z+y ·z. in this paragraph, we use the term of a hyperring, instead of the term of a general hyperring, intending the above definition. a hyperring may be commutative with respect to (+) or (·). if r is commutative with respect to both (+) and (·), then it is a commutative hyperring. the above definition contains the class of multiplicative hyperrings and additive hyperrings as well. in the above hyperstructures, vougiouklis introduced the equivalence relation γ∗, which is similar to the relation β∗, defined in every hypergroup. let (r,+, ·) be a hyperring. he defined the relation γ as follows: aγb if and only if {a,b}⊆ u, where u is a finite sum of finite products of elements of r. denote the transitive closure of γ by γ∗. the equivalence relation γ∗ is called the fundamental equivalence relation in r. according to the distributive law, every set which is the value of a polynomial in elements of r is a subset of a sum of products in r. let u be the set of all finite sums of products of elements of r. we can rewrite the definition of γ∗ on r as follows: aγ∗b if and only if there exist z1, ...,zn+1 ∈ r with z1=a,zn+1=b and u1, ...,un ∈u such that {zi,zi+1}⊆ ui for i ∈{1, ...,n}. let (r,+, ·) be a hyperring. then the relation γ∗ is the smallest equivalence relation in r such that the quotient r/γ∗ is a ring [16]. the both ⊕ and � on r/γ∗ are defined as follows: γ∗(a) ⊕ γ∗(b) = γ∗(c), for all c ∈ γ∗(a) + γ∗(b) and γ∗(a) � γ∗(b) = γ∗(d), for all d ∈ γ∗(a) · γ∗(b). if u = ∑ j∈j( ∏ i∈ij xi) ∈ u, then for all z ∈ u, we have γ∗(u) = ⊕ ∑ j∈j(� ∏ i∈ij γ ∗(xi)) = γ ∗(z), where ⊕ ∑ and � ∏ denote the sum and the product of classes. 4 commutative rings obtained from hyperrings the commutativity in addition in rings can be related with the existence of the unit in multiplication. if e is the unit in a ring then for all elements a,b we have (a + b)(e + e) = (a + b)e + (a + b)e = a + b + a + b and (a + b)(e + e) = a(e + e) + b(e + e) = a + a + b + b. so a + b + a + b = a + a + b + b gives b + a = a + b. therefore, when we say (r,+, ·) is a hyperring, (+) is not commutative and there is not unit in the multiplication. so the commutativity, as well as the existence of the unit, it is not assumed in the fundamental ring. of course, we know there exist many rings (+ is commutative) while don’t have unit. davvaz and vougiouklis were interested in the fundamental ring to be commutative with respect to both sum and product, that is, the fundamental 79 bijan davvaz ring be an ordinary commutative ring. therefore they introduced the following definition. let r be a hyperring. define the relation α as follows: xαy if and only if ∃n ∈ n, ∃(k1, . . . ,kn) ∈ nn, ∃σ ∈ sn and [∃(xi1, . . . ,xiki) ∈ rki , ∃σi ∈ ski, (i = 1, . . . ,n)] such that x ∈ ∑n i=1( ∏ki j=1 xij) and y ∈ ∑n i=1 aσ(i), where ai = ∏ki j=1 xiσi(j). the relation α is reflexive and symmetric. let α ∗ be the transitive closure of α, then α∗ is a strongly regular relation both on (r,+) and (r, ·) [4]. the quotient r/α∗ is a commutative ring [4]. notice that they used the greek letter α for the relation because of the ‘a’belian. the relation α∗ is the smallest equivalence relation such that the quotient r/α∗ is a commutative ring [4]. 5 hv-structures during the 4th congress of algebraic hyperstructures and applications (xanthi, 1990), vougiouklis introduced the concept of the weak hyperstructures which are now named hv-structures. over the last 27 years this class of hyperstructure, which is the largest, has been studied from several aspects as well as in connection with many other topics of mathematics. the hyperstructure (h, ·) is called an hvgroup if (1) x·(y·z)∩(x·y)·z 6= ∅, for all x,y,z ∈ h; (2) a·h = h ·a = h, for all a ∈ h. a motivation to obtain the above structures is the following. let (g, ·) be a group and r an equivalence relation on g. in g/r consider the hyperoperation � such that x�y = {z|z ∈ x·y}, where x denotes the class of the element x. then (g,�) is an hv-group which is not always a hypergroup [20]. let (h1, ·), (h2,?) be two hv-groups. a map f : h1 → h2 is called an hv-homomorphism or weak homomorphism if f(x · y) ∩ f(x) ? f(y) 6= ∅, for all x,y ∈ h1. the map f is called an inclusion homomorphism if f(x · y) ⊆ f(x) ? f(y), for all x,y ∈ h1. finally, f is called a strong homomorphism if f(x ·y) = f(x)∗f(y), for all x,y ∈ h1. if f is onto, one to one and strong homomorphism, then it is called isomorphism, if moreover f is defined on the same hv-group then it is called automorphism. it is an easy verification that the set of all automorphisms in h, written auth, is a group. on a set h several hv-structures can be defined. a partial order on those hyperstructures is introduced as follows. let (h, ·), (h,?) be two hv-groups defined on the same set h. we call · less than or equal to ?, and write · ≤ ?, if there is f ∈ aut(h,∗) such that x · y ⊆ f(x ? y), for all x,y ∈ h [20]. a quasi-hypergroup is called a hypergroupoid (h, ·) if the reproduction axiom is valid. in [20], it is proved that all the quasi-hypergroups with two elements are hv-groups. it is also proved that up to the isomorphism there are exactly 18 different hv-groups. if a hyperoperation is weak associative then every greater hyperoperation, defined on the same set is also weak associative. in [21], using this property, the set of all hv-groups with a scalar unit defined 80 vougiouklis contributions in the field of algebraic hyperstructures on a set with three elements is determined, also, see [22]. let (h, ·) be an hvgroup. the relation β∗ is the smallest equivalence relation on h such that the quotient h/β∗, the set of all equivalence classes, is a group. β∗ is called the fundamental equivalence relation on h. according to [19] if u denotes the set of all the finite products of elements of h, then a relation β can be defined on h whose transitive closure is the fundamental relation β∗. the relation β is as follows: for x and y in h we write xβy if and only if {x,y} ⊆ u for some u ∈ u. we can rewrite the definition of β∗ on h as follows: aβ∗b if and only if there exist z1, . . . ,zn+1 ∈ h with z1 = a, zn+1 = b and u1, . . . ,un ∈ u such that {zi,zi+1} ⊆ ui (i = 1, . . . ,n). the product � on h/β∗ is defined as follows: β∗(a) � β∗(b) = {β∗(c)| c ∈ β∗(a) · β∗(b)}, for all a,b ∈ h. it is proved in [19] that β∗(a) �β∗(b) is the singleton {β∗(c)} for all c ∈ β∗(a) ·β∗(b). in this way h/β∗ becomes a hypergroup. if we put β∗(a) � β∗(b) = β∗(c), then h/β∗ becomes a group. a multi-valued system (r,+, ·) is an hv-ring if (1) (r,+) is an hv-group; (2) (r, ·) is an hv-semigroup; (3) (·) is weak distributive with respect to (+), i.e., for all x,y,z in r we have (x · (y + z)) ∩ (x · y + x · z) 6= ∅ and ((x + y) · z) ∩ (x · z + y · z) 6= ∅. let (r,+, ·) be an hv-ring. define γ∗ as the smallest equivalence relation such that the quotient r/γ∗ is a ring. let us denote the set of all finite polynomials of elements of r over n by u. define the relation γ as follows: xγy if and only if {x,y} ⊆ u, where u ∈ u. the fundamental equivalence relation γ∗ is the transitive closure of the relation γ [12]. vougiouklis also introduced hv-vector spaces in [18]. 6 the uniting elements method in 1989, corsini and vougiouklis [1], introduced a method, the uniting elements method, to obtain stricter algebraic structures, from given ones, through hyperstructure theory. this method was introduced before the introduction of the hv-structures, but in fact the hv-structures appeared in the procedure. this method is the following. let g be a structure and d be a property, which is not valid, and suppose that d is described by a set of equations. consider the partition in g for which it is put together, in the same class, every pair of elements that causes the non-validity of d. the quotient g/d is an hv-structure. then quotient of g/d by the fundamental relation β∗, is a stricter structure (g/d)/β∗ for which d is valid. an application of the uniting elements is when more than one properties are desired. the reason for this is some of the properties lead straighter to the classes than others. the commutativity and the reproductivity are easily applicable. one can do this because the following statement is valid. let (g, ·) be a groupoid, and f = {f1, ...,fm,fm+1, ...,fm+n} a system of equations on g consisting of two subsystems fm = {f1, ...,fm} and fn = fm+1, ...,fm+n}. let 81 bijan davvaz σ and σm be the equivalence relations defined by the uniting elements using the f and fm respectively, and let σn be the equivalence relation defined using the induced equations of fn on the groupoid gm = (gm/σn)/β∗. then, we have (g/σ)/β∗ ∼= (gm/σn)/β∗[19]. 7 the e-hyperstructures in 1996, santilli and vougiouklis point out that in physics the most interesting hyperstructures are the one called e-hyperstructures. the e-hyperstructures are a special kind of hyperstructures and, they can be interpreted as a generalization of two important concepts for physics: isotopies and genotopies. on the other hand, biological systems such as cells or organisms at large are open and irreversible because they grow. the representation of more complex systems, such as neural networks, requires more advances methods, such as hyperstructures. in this manner, e-hyperstructures can play a significant role for the representation of complex systems in physics and biology, such as nuclear fusion, the reproduction of cells or neural systems. they are the most important tools in lie-santilli theory too [2, 11]. a hypergroupoid (h, ·) is called an e-hypergroupoid if h contains a scalar identity (also called unit) e, which means that for all x ∈ h, x·e = e·x = x. in an e-hypergroupoid, an element x′ is called inverse of a given element x ∈ h if e ∈ x · x′ ∩ x′ · x. clearly, if a hypergroupoid contains a scalar unit, then it is unique, while the inverses are not necessarily unique. in what follows, we use some examples which are obtained as follows: take a set where an operation “·” is defined, then we “enlarge” the operation putting more elements in the products of some pairs. thus a hyperoperation “◦” can be obtained, for which we have x · y ∈ x ◦ y, ∀x,y ∈ h. recall that the hyperstructures obtained in this way are hb-structures. consider the usual multiplication on the subset {1, −1, i, −i} of complex numbers. then, we can consider the hyperoperation ◦ defined in the following table: ◦ 1 −1 i −i 1 1 −1 i −i −1 −1 1 −i i,−i i i −i −1 1 −i −i i 1, i −1, i we enlarged the products (−1)·(−i), (−i)·i and (−i)·(−i) by setting (−1)◦(−i)={i,−i}, (−i)◦i={1, i} and (−i)◦(−i)={−1, i}. we obtain an e-hypergroupoid, with the scalar unit 1. the inverses of the elements −1, i,−i are −1,−i, i respectively. moreover, the above structure is an 82 vougiouklis contributions in the field of algebraic hyperstructures hv-abelian group, which means that the hyperoperation ◦ is weak associative, weak commutative and the reproductive axiom holds. the weak associativity is valid for all hb-structures with associative basic operations [19]. we are interested now in another kind of an e-hyperstructure, which is the e-hyperfield. a set f , endowed with an operation “+”, which we call addition and a hyperoperation, called multiplication “·”, is said to be an e-hyperfield if the following axioms are valid: (1) (f,+) is an abelian group where 0 is the additive unit; (2) the multiplication · is weak associative; (3) the multiplication · is weak distributive with respect to +,i.e., for all x,y,z ∈ f , x(y+z)∩(xy+xz) 6= ∅, (x+y)z∩(xz+yz) 6= ∅; (4) 0 is an absorbing element, i.e., for all x ∈ f, 0·x = x·0 = 0; (5) there exists a multiplicative scalar unit 1, i.e., for all x ∈ f, 1·x = x·1 = x; (6) for every element x ∈ f there exists an inverse x−1, such that 1 ∈ x·x−1∩x−1·x. the elements of an e-hyperfield (f,+, ·) are called e-hypernumbers. we can define the product of two e-matrices in an usual manner: the elements of product of two e-matrices (aij),(bij) are cij = ∑ aik◦bkj, where the sum of products is the usual sum of sets. let (f,+, ·) be an e-hyperfield. an ordered set a = (a1,a2, . . . ,an) of n e-hypernumbers of f is called an e-hypervector and the e-hypernumbers ai, i ∈ {1,2, ·,n} are called components of the e-hypervector a. two e-hypervectors are equals if they have equal corresponding components. the hypersums of two e-hypervectors a,b is defined as follows: a + b = {(c1,c2, . . . ,cn) | ci ∈ ai + bi, i ∈ {1,2, ·,n}}. the scalar hypermultiplication of an e-hypervector a by an e-hypernumber λ is defined in a usual manner: λ◦a = {(c1,c2, . . . ,cn) | ci ∈ λ ·ai, i ∈ {1,2, . . . ,n}}. the set fn of all e-hypervectors with elements of f , endowed with the hypersum and the scalar hypermultiplication is called n-dimensional e-hypervector space. the set of m×n hypermatrices is an mn-dimensional e-hypervector space. we refer the readers to [5, 6, 7, 8] for more details. 8 helix-hyperoperations algebraic hyperstructures are a generalization of the classical algebraic structures which, among others, are appropriate in two directions: (a) to represent a lot of application in an algebraic model, (b) to overcome restrictions ordinary structures usually have. concerning the second direction the restrictions of the ordinary matrix algebra can be overcome by the helix-operations. more precisely, the helix addition and the helix-multiplication can be defined on every type of matrices [3, 23, 24]. let a = (aij) ∈ mm×n be a matrix and s,t ∈ n be two natural numbers such that 1 ≤ s ≤ m and 1 ≤ t ≤ n. then we define the characteristic-like map cst from mm×n to ms×t by corresponding to a the matrix acst = (aij), where 1 ≤ i ≤ s and 1 ≤ j ≤ t. we call this map cut-projection of type st. in other words, acst is a matrix obtained 83 bijan davvaz from a by cutting the lines and columns greater than s and t respectively. let a = (aij) ∈ mm×n be a matrix and s,t ∈ n be two natural numbers such that 1 ≤ s ≤ m and 1 ≤ t ≤ n. then we define the mod-like map st from mm×n to ms×t by corresponding to a the matrix ast = (aij) which has as entries the sets aij = {ai+ks,j+λt | k,λ ∈ n, i + ks ≤ m, j + λt ≤ n}, for 1 ≤ i ≤ s and 1 ≤ j ≤ t. we call this multivalued map helix-projection of type st. therefore, ast is a set of s×t-matrices x = (xij) such that xij ∈ aij for all i,j. obviously, amn = a. let us consider the following matrix: a =   2 1 3 4 2 3 2 0 1 2 2 4 5 1 −1 1 −1 0 0 8   . suppose that s = 3 and t = 2. then ac32 =   2 13 2 2 4   and a32 = (aij), where a11 = {a11,a13,a15,a41,a43,a45} = {2,3,2,1,0,8}, a12 = {a12,a14,a42,a44} = {1,4,−1,0}, a21 = {a21,a23,a25} = {3,0,2}, a22 = {a22,a24} = {2,1}, a31 = {a31,a33,a35} = {2,5,−1}, a32 = {a32,a34} = {4,1}. therefore, a32 = (aij) =   {2,3,1,0,8} {1,4,−1,0}{3,0,2} {2,1} {2,5,−1} {4,1}   = {(xij) | x11 ∈{0,1,2,3,8},x12 ∈{−1,0,1,4},x21 ∈{0,2,3}, x22 ∈{1,2},x31 ∈{−1,2,5},x32 ∈{1,4}}. therefore |a32| = 720. let a = (aij) ∈ mm×n and b = (aij) ∈ mu×v be two matrices and s = min(m,u), t = min(n,u). we define an addition, which we call cut-addition, as follows: ⊕c : mm×n ×mu×v −→ ms×t (a,b) 7→ a⊕c b = acst + bcst. 84 vougiouklis contributions in the field of algebraic hyperstructures let a = (aij) ∈ mm×n and b = (aij) ∈ mu×v be two matrices and s = min(n,u). then we define a multiplication, which we call cut-multiplication, as follows: ⊗c : mm×n ×mu×v −→ mm×v (a,b) 7→ a⊗c b = acms ·bcsv. the cut-addition is associative and commutative. let a = (aij) ∈ mm×n and b = (aij) ∈ mu×v be two matrices and s = min(m,u), t = min(n,v). we define a hyper-addition, which we call helixaddition or helix-sum, as follows: ⊕ : mm×n ×mu×v −→p(ms×t) (a,b) 7→ a⊕b = ast +h bst, where ast +h bst = {(cij) = (aij + bij) | aij ∈ aij,bij ∈ bij} . for illustration, suppose that a =   2 10 1 2 3   and b = [ 1 4 0 2 0 1 ] . then a22 = [ a11 a12 a21 a22 ] and b22 = [ b11 b12 b21 b22 ] , where a11 = {a11,a31} = {2}, b11 = {b11,b13} = {1,0}, a12 = {a12,a32} = {1,3}, b12 = {b12} = {4}, a21 = {a21} = {0}, b21 = {b21,b23} = {2,1}, a22 = {a22} = {1}, b22 = {b22} = {0}. so a22 = {[ 2 1 0 1 ] , [ 2 3 0 1 ]} , and b22 = {[ 1 4 2 0 ] , [ 1 4 1 0 ] , [ 0 4 2 0 ] , [ 0 4 1 0 ]} . therefore, we have a22 +h b22 = {[ 3 5 2 1 ] , [ 3 5 1 1 ] , [ 2 5 2 1 ] , [ 2 5 1 1 ] , [ 3 7 2 1 ] ,[ 3 7 1 1 ] , [ 2 7 2 1 ] , [ 2 7 1 1 ]} . the helix-addition is commutative. let a = (aij) ∈ mm×n and b = (aij) ∈ mu×v be two matrices and s = min(n,u). then we define a hyper-multiplication, 85 bijan davvaz which we call helix hyperoperation, as follows: ⊗ : mm×n ×mu×v −→p(mm×v) (a,b) 7→ a⊗b = ams ·h bsv, where ams ·h bsv = {(cij) = ( ∑ aitbtj) |aij ∈ aij,bij ∈ bij}. we consider the matrices a and b as follows: a = [ 1 0 2 0 3 1 3 2 ] and b = [ −1 1 0 2 ] . then a22 = [ {1,2} 0 3 {1,2} ] . therefore, a⊗b = [ {−1,−2} {1,2} −3 {5,7} ] the cut-multiplication ⊗c is associative, and the helix-multiplication ⊗ is weak associative [23]. note that the 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(new series), 1(2) (2009), 109-120. 88 ratio mathematica volume 42, 2022 d4-magic graphs anusha chappokkil* anil kumar vasu† abstract consider the set x = {1, 2, 3, 4} with 4 elements. a permutation of x is a function from x to itself that is both one one and on to. the permutations of x with the composition of functions as a binary operation is a nonabelian group, called the symmetric group s4. now consider the collection of all permutations corresponding to the ways that two copies of a square with vertices 1, 2, 3 and 4 can be placed one covering the other with vertices on the top of vertices. this collection form a nonabelian subgroup of s4, called the dihedral group d4. in this paper, we introduce a-magic labelings of graphs, where a is a finite nonabelian group and investigate graphs that are d4-magic. this did not attract much attention in the literature. keywords: a-magic labeling; dihedral group d4 ; d4-magic. 2020 ams subject classifications: 05c25, 05c78 .1 *department of mathematics, university of calicut, malappuram, kerala, india 673 635.; canusha235@gmail.com. †department of mathematics, university of calicut, malappuram, kerala, india 673 635.; anil@uoc.ac.in 1received on march 22th, 2022. accepted on june 12th, 2022. published on june 30th 2022. doi: 10.23755/rm.v41i0.738. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 167 c. anusha and v. anil kumar 1 introduction a graph g is an ordered pair (v (g), e(g)), where v (g) is a finite nonempty set whose elements are called vertices and e(g) is a binary irreflexive and symmetric relation on v (g) whose elements are called edges. for any abelian group a, written additively, any mapping ℓ : e(g) → a \ {0} is called a labeling. given a labeling on the edge set e(g), one can introduce a vertex set labeling ℓ+ : v (g) → a as follows: ℓ+(v) = ∑ uv∈e(g) l(uv) a graph g is said to be a-magic if there is a labeling ℓ : e(g) → a\{0} such that for each vertex v, the sum of the labels of the edges incident with v are all equal to the same constant, that is, ℓ+(v) = a for some fixed a ∈ a. the original concept of a-magic graph was introduced by sedláček[1]. according to him, a graph g is a-magic if there exists an edge labeling on g such that (i) distinct edges have distinct non-negative labels; and (ii) the sum of the labels of the edges incident to a particular vertex is same for all vertices. when a = z, the z-magic graphs are considered in stanley[7]. doob [5, 4] also considered a-magic graphs where a is an abelian group. also he determined which wheels are z-magic. observe that several authors studied v4-magic graphs[8, 6]. it is natural to ask does there exist graphs which admits a-magic labeling, when a is nonabelian? in this paper, we address this question and investigate graphs that are d4-magic. 2 main results let g = (v (g), e(g)) be a finite (p, q) graph and let (a, ∗) be a finite nonabelain group with identity element 1. let f : e(g) → nq = {1, 2, . . . , q} and let g : e(g) → a\{1} be two edge labelings of g such that f is bijective. define an edge labeling ℓ : e(g) → nq × a \ {1} by l(e) := (f(e), g(e)), e ∈ e(g). define a relation ≤ on the range of ℓ by: (f(e), g(e)) ≤ (f(e ′ ), g(e′)) if and only if f(e) ≤ f(e′). then obviously the relation ≤ is a partial order on the range of ℓ. let {(f(e1), g(e1)), (f(e2), g(e2)), . . . , (f(ek), g(ek))} be a chain in the range of ℓ. we define the product of elements of this chain as follows: k∏ i=1 (f(ei), g(ei)) := ((((g(e1) ∗ g(e2)) ∗ g(e3)) ∗ g(e4)) ∗ . . .) ∗ g(ek). 168 d4 magic graphs let u ∈ v and let n∗(u) be the set of all edges incident with u. note that the range of ℓ|n∗(u) is a chain, say (f(e1), g(e1)) ≤ (f(e2), g(e2)) ≤ · · · ≤ (f(en), g(en)). we define, ℓ∗(u) = n∏ i=1 (f(ei), g(ei)). (1) if ℓ∗(u) is a constant, say a for all u ∈ v (g), we say that the graph g is amagic. the map ℓ∗ is called an a-magic labeling of g and the corresponding constant a is called the magic constant. for example, consider the cycle graph c4 = (uv, vw, wx, xu) and the permutation group d4. note that the group d4 is a non abelian group of order 8 and its elements are given by ρ0 = ( 1 2 3 4 1 2 3 4 ) , µ1 = ( 1 2 3 4 4 3 2 1 ) , ρ1 = ( 1 2 3 4 2 3 4 1 ) , µ2 = ( 1 2 3 4 2 1 4 3 ) , ρ2 = ( 1 2 3 4 3 4 1 2 ) , δ1 = ( 1 2 3 4 1 4 3 2 ) , ρ3 = ( 1 2 3 4 4 1 2 3 ) , δ2 = ( 1 2 3 4 3 2 1 4 ) . define f : e(g) → n4 = {1, 2, 3, 4} as f(uv) = 1, f(wx) = 2, f(vw) = 3, f(xu) = 4 and g : e(g) → d4 \ {ρ0} as g(uv) = g(wx) = ρ1, g(vw) = g(xu) = δ1. thus ℓ∗(u) = (1, ρ1)(4, δ1) = ρ1δ1 = µ2, ℓ∗(v) = (1, ρ1)(3, δ1) = ρ1δ1 = µ2. similarly, ℓ∗(w) = µ2 and ℓ∗(x) = µ2. thus c4 is d4-magic with magic constant µ2. figure 1: d4-magic labeling of c4. in this paper, we will consider the symmetric group d4 and investigate graphs that are d4-magic. 169 c. anusha and v. anil kumar theorem 2.1. let a be a non abelian group having an element of order 2 and let g be a graph. if either the degree of the vertices of g are all even or odd. then g is a-magic. proof. let g be a (p, q) graph and a be a nonabelian group having an element of order 2. let a ∈ g is of order 2. let g : e(g) → a \ {1} be the constant map g(e) = a, ∀e ∈ e(g) and let f be any bijection from e(g) → nq. first assume that all the vertices of g are of even degree then l∗(u) = 1, ∀u ∈ v (g). similarly, if all the vertices of g are of odd degree then l∗(u) = a, ∀u ∈ v (g). hence the proof. corolary 2.1. all eulerian graphs are d4-magic. theorem 2.2. any regular graph is d4-magic. proof. let g = (v (g), e(g)) be a regular graph with |e(g)| = q. let f : e(g) → nq be any bijection and g be any constant map from e(g) → d4\{ρ0}. obviously, f and g will determine a d4-magic labeling of g. this completes the proof of the theorem. corolary 2.2. for any n ≥ 3, the cycle graph cn is d4-magic. corolary 2.3. for any n ≥ 2, the complete graph kn is d4-magic. corolary 2.4. the peterson graph is d4-magic. theorem 2.3. the star graph k1,n, n ≥ 2 is d4-magic iff n is odd. proof. let g = k1,n. suppose that n is odd. let f : e(g) → nn+1 be a bijection. define g : e(g) → d4 \ {ρ0} by g(e) = µ1. then clearly it is d4magic with magic constant µ1. conversely, suppose k1,n is d4-magic with magic constant, say ‘a’. so every pendent edge of k1,n should be mapped to a under g. let u be the vertex of k1,n with degree n. then ℓ∗(u) = aa · · · a︸ ︷︷ ︸ n times = a. this implies that an−1 = ρ0. if n is odd, the equation an−1 = ρ0 has five non trivial solutions in d4 viz. µ1, µ2, δ1, δ2 and ρ2. on the other hand, if n is even there are no element in d4 such that an−1 = ρ0. this completes the proof. a bistar graph bn is the graph obtained by connecting the apex vertices of two copies of star k1,n by a bridge. theorem 2.4. the bistar graph bn, n > 1 is d4-magic when n ̸≡ 1(mod 4). 170 d4 magic graphs proof. first, observe that there are 2n pendant edges and one bridge in bn. here we consider the following cases: case (i): n is even (n ≡ 2(mod 4) or n ≡ 0(mod 4)). if n is even, define g : e(bn) → d4 \ {ρ0} by g(e) = µ1, ∀e ∈ e(bn). let f be any bijective map from e(bn) → n2n+1. then obviously, bn is d4-magic with magic constant µ1. case (ii): n ≡ 3(mod 4). in this case we define g : e(bn) → d4 \ {ρ0} by g(e) = { ρ1, if e is a pendant edge, ρ2, if e is the bridge. let f be any bijective map from e(bn) to n2n+1. then obviously bn is d4-magic with the magic constant ρ1. case (iii): n ≡ 1(mod 4). suppose that n ≡ 1(mod 4). let k1 and k2 be the apex vertices of the bistar graph. assume that bn is d4-magic with magic constant µ1. therefore, g(e) = µ1 for all pendant edges e. assume that g(k1k2) = a, where a ∈ d4 \ {ρ0}. without loss of generality assume that f(k1k2) > f(b), ∀b ∈ e(g), where b denotes the pendant edge with one end point k1. then ℓ∗(k1) = µ1µ1 . . . µ1︸ ︷︷ ︸ (n times) a = µ1. the above equation tells us that a = ρ0, which is a contradiction. this contradiction shows that bn is not d4-magic with magic constant µ1. in a similar manner, we can prove that bn is not d4-magic with magic constants µ2, ρ1, ρ2, ρ3, δ1 or δ2. thus the bistar graph bn is not d4-magic when n ≡ 1(mod 4). this completes the proof of the theorem. theorem 2.5. the complete bipartite graph km,n is d4-magic, m, n > 1. proof. let g = km,n. suppose u = {u1, u2, . . . , un}, and v = {v1, v2, . . . , vm}. be the two partite sets of km,n. if m and n are both even or odd then the theorem is obvious by taking any constant map g : e(g) → {ρ2, µ1, µ2, δ1, δ2}. case (i): n ≡ 0(mod 2) and m ≡ 1(mod 4). let u = {u1, u2, . . . , u2l} and v = {v1, v2, . . . , v4r+1} where n = 2l, m = 4r + 1, and l, r ∈ n. for 1 ≤ i ≤ n and 1 ≤ j ≤ m define g(uiv5k+1) = µ1, where k < m, k = 0, 1, 2, 3, . . . g(uiv5k+2) = µ2, k < m, k = 0, 1, 2, 3, . . . g(uivj) = ρ2, j ̸= 5k + 1, 5k + 2 where k = 0, 1, 2, . . . f(uivj) = (i − 1)m + j, 1 ≤ i ≤ n, 1 ≤ j ≤ m. 171 c. anusha and v. anil kumar the maps f and g will determine a d4-magic labeling for km,n with magic constant ρ0. case (ii): n ≡ 0(mod 2) and m ≡ 3(mod 4). define g as follows: g(uivj) = { ρ1, if i is odd 1 ≤ j < m, ρ3, if i is even 1 ≤ j < m, and g(uivm) = ρ2, ∀i, 1 ≤ i ≤ n, and let f be any bijection from e(g) to {1, 2, . . . , mn}. then clearly f and g will determine a d4-magic labeling of km,n with magic constant ρ0. this completes the proof of the theorem. 3 cycle generated graphs in this section, we consider certain graphs which are constructed from cycles. a wheel wn of order n + 1, sometimes simply called an n wheel is a graph that contains a cycle of order n and for which every graph vertex in the cycle is connected to one other graph vertex (which is known as the hub). the edges of a wheel which include the hub are called spokes. the wheel wn can be defined as the graph join k1 + cn, where k1 is the singleton graph and cn is the cycle graph. theorem 3.1. if n ≥ 3, the wheel wn is d4-magic. proof. let the vertices of cn be u1, u2, . . . , un such that uiui+1 ∈ e(cn), i = 1, 2, . . . , n and un+1 = u1. denote the vertex of k1 by k. now we consider the following cases: case (i): n is odd. if n is odd then every vertex of wn is of odd degree. thus we can take g : e(wn) → d4\{ρ0} as any constant map from e(wn) to {ρ2, µ1, µ2, δ1, δ2}. since g is constant we can take f as any bijection from e(wn) to n2n. clearly this f and g will constitute a d4-magic labeling for wn. case (ii): n is even. suppose n is even define f : e(wn) → n2n as f(kui) = i, i = 1, 2, . . . , n, f(uiui+1) = n + i, 1 ≤ i ≤ n − 1, f(u1un) = 2n. 172 d4 magic graphs now we can define g : e(wn) → d4 \ {ρ0} by labeling each spokes by µ1 and all the outer edges by µ2 and ρ2 alternatively. then wn becomes d4-magic with magic constant ρ0. this completes the proof of the theorem. the helm hn is a graph obtained from a wheel wn by attaching a pendant edge at each vertex of the n cycle. theorem 3.2. the helm graph hn is d4-magic. proof. let {k, ui, vi : i = 1, 2, . . . , n} be the vertex set of hn, where k be the central vertex, u1, u2, . . . , un are the vertices of the cycle, v1, v2, . . . , vn are the pendant vertices adjacent to u1, u2, . . . , un. the edge set of hn is e(hn) = {uiui+1, kui, uivi : i = 1, 2, . . . , n, un+1 = u1}. now consider the following two cases: case(i): n is odd. suppose that n is odd. define f and g as follows: let g : e(g) → d4\{ρ0} be defined as g(kui) = ρ2, 1 ≤ i ≤ n, g(ujuj+1) = ρ1, 1 ≤ j ≤ n − 1, g(u1un) = ρ1, g(ukvk) = ρ2, 1 ≤ k ≤ n. now let f : e(g) → n2n+1 be any bijection. then clearly f and g will give a d4-magic labeling of hn, where n is odd. case(ii): n is even. let f be defined as above and define g : e(g) → d4 \ {ρ0} by g(uivi) = ρ2, 1 ≤ i ≤ n, g(v1vn) = ρ1 g(kuj) = { ρ2, if 1 ≤ j ≤ n − 2, ρ1, if j = n − 1, n. , g(ukuk+1) = { ρ1, if 1 ≤ k ≤ n − 2, ρ2, if k = n − 1. it follows that l∗(u) = ρ2, ∀u ∈ v (g). hence hn is d4-magic when n is even. this completes the proof of the theorem. the web graph w(2, n) is a graph obtained joining the pendant points of a helm to form a cycle and adding a single pendant edge to each vertex of this outer graph. theorem 3.3. the web graph w(2, n), n ≥ 3 is d4-magic. 173 c. anusha and v. anil kumar proof. let {k, ui, vi, wi : i = 1, 2, 3, . . . , n} be the vertex set of w(2, n), where k be the central vertex, u1, u2, u3, . . . , un are the vertices of inner cycle, v1, v2, v3, . . . , vn are the vertices of outer cycle and w1, w2, w3, . . . , wn are the pendant vertices adjacent to v1, v2, v3, . . . , vn of w(2, n). let e(w(2, n)) = {uiui+1, vivi+1, uivi, viwi : i = 1, 2, . . . , n and un+1 = u1, vn+1 = v1}. we define a d4-magic labeling for w(2, n) with magic constant ρ2 as follows: case (i): n is odd. let f : e(g) → n3n+1 be any bijection. define g : e(g) → d4 \ {ρ0} as g(kui) = ρ2 = g(uivi) = g(viwi), 1 ≤ i ≤ n, g(uiui+1) = ρ1 = g(vivi+1), 1 ≤ i ≤ n − 1, g(u1un) = ρ1 = g(v1vn). case (ii): n is even. let f : e(g) → n3n+1 be any bijection. define g : e(g) → d4 \ {ρ0} as g(kui) = ρ2, 1 ≤ i < n − 1, g(kun) = g(kun−1) = ρ1, g(vivi+1) = ρ1 = g(uiui+1) = ρ1, 1 ≤ i ≤ n − 1, g(viwi) = ρ2 = g(uivi), 1 ≤ i ≤ n, g(v1vn) = ρ1, g(u1un) = ρ2. this completes the proof of the theorem. a shell graph sn,n−3 of width n is a graph obtained by taking n−3 concurrent chords in a cycle cn of n vertices. the vertex at which all chords are concurrent is called is called the apex. the two vertices adjacent to the apex have degree 2, apex has degree n − 1 and all other vertices have degree 3. theorem 3.4. shell graphs sn,n−3 are d4-magic. proof. let us denote the vertices of the shell graph sn,n−3 by u1, u2, . . . , un such that ui is adjacent to ui+1, where i = 1, 2, . . . , n and un+1 = u1. without loss of generality let the apex be u1. now consider the following cases: case (i): n is even. we will define the map f : e(sn,n−3) → n2n−3 as f(uiui+1) = i, 1 ≤ i ≤ n − 1, f(unu1) = n, f(u1uj) = n + (j − 2), 3 ≤ j ≤ n − 1 174 d4 magic graphs and we define g : e(sn,n−3) → d4 \ {ρ0} as g(u1u2) = g(unu1) = ρ2, g(u1ui) = µ1, 3 ≤ i ≤ n − 1, g(uiui+1) = µ2, 2 ≤ i ≤ n − 1. clearly f and g define a d4-magic labeling with magic constant µ1. case (ii): n is odd. define f as f(uiui+1) = i, 1 ≤ i ≤ n − 1, f(u1un) = n, f(u1uj) = n + (j − 2), 3 ≤ j ≤ n − 1 and define g as g(u1u2) = g(u1un) = ρ2, g(u1uj) = µ1, 3 ≤ j ≤ n − 1, g(uiui+1) = { ρ2, if i is even, µ2, if i is odd, 1 < i ≤ n − 1. obviously the functions f and g define a d4-magic labeling of sn,n−3 with magic constant ρ0. this completes the proof of the theorem. when k copies of cn share a common edge it will form the n-gon book of k pages and is denoted by b(n, k). theorem 3.5. the graph n-gon book of k pages b(n, k) is d4-magic. proof. let g be the graph b(n, k). denote the vertices of common edge by k1 and kn and the edges of ith page other than k1 and kn by ui2, ui3, . . . , uin−1 such that ui2 is adjacent to k1 and uin−1 adjacent to kn and uij adjacent to uij+1 for all 2 ≤ j < n − 1. consider the following cases: case (i): k is even. define g : e(g) → d4 \ {ρ0} as g(k1kn) = ρ2, g(u1ju1j+1) = µ1, 2 ≤ j ≤ n − 2, g(u1n−1kn) = µ1 = g(k1u12), g(uijuij+1) = µ2, 2 ≤ i ≤ k, 2 ≤ j ≤ n − 1, g(k1ul2) = g(uln−1) = µ2, 2 ≤ l ≤ k. 175 c. anusha and v. anil kumar now define f as f(k1kn) = 1, f(k1u12) = 2, f(u1n−1kn) = n, f(u1ju1j+1) = j + 1, ∀ 2 ≤ j ≤ n − 2, f(k1ui2) = n + (i − 2)(n − 1) + 1, i ≥ 2, f(uijuij+1) = n + (i − 2)(n − 1) + j, 2 ≤ j ≤ n − 2, 2 ≤ i ≤ k, f(ui(n−1)kn) = n + (i − 2)(n − 1) + (n − 1), 2 ≤ i ≤ k. the functions f and g determine a d4-magic labeling with magic constant ρ0. case (ii): k is odd. here define g as g(e) = ρ2, ∀e ∈ e(g) then g together with any bijection f : e(g) → nkn−1 will define a d4-magic labeling of b(n, k) with magic constant ρ0. this completes the proof of the theorem. note that, for any n ≥ 3 the path graph of order n is not d4-magic. 4 path generated graphs in this section we will consider some graphs which are constructed from paths. we start with the splitting graph of path. a splitting graph s(g) of a graph g is the graph obtained from g by adding to g a new vertex z′ for each vertex z of g and joining z′ to the neighbors of z in g. theorem 4.1. splitting graph of the path graph pn, n ≥ 3 is d4-magic. proof. let pn be a path graph of order n, where n ≥ 3. let u1, u2, . . . , un be the vertices of pn, where uiui+1 ∈ e(pn), i = 1, 2, . . . , n − 1. there are 2n vertices and 3n − 3 edges in s(pn). let un+i be the vertex corresponding to the ith vertex in s(pn). observe that there are two pendant edges in s(pn), one with end points u2 and un+1 and the other with end points un−1 and u2n. case (i): n = 3. in this case, define f : e(s(p3)) → n6 as f(u1u2) = 1, f(u2u3) = 3, f(u3u5) = 2, f(u1u5) = 4, f(u2u4) = 5, f(u2u6) = 6. now define g : e(g) → d4 \ {ρ0} as g(u1u2) = g(u3u5) = ρ1, g(u2u3) = g(u1u5) = δ2, g(u2u4) = g(u2u6) = µ1. 176 d4 magic graphs case (ii): n > 3. in this case, define f and g as follows: f(uiui+1) = i, 1 ≤ i ≤ n − 1, f(uiun+(i−1)) = n + (i − 2), 2 ≤ i ≤ n, f(uiun+(i+1)) = (2n − 2) + i, 1 ≤ i ≤ n − 1 and g(u1u2) = ρ2, g(un−1un) = µ2, g(u2un+1) = g(un−1u2n) = µ1, g(uiui+1) = µ1, 2 ≤ i < n − 1, g(uiun+(i−1)) = ρ2, 3 ≤ i ≤ n, g(uiun+(i+1)) = µ2, 1 ≤ i ≤ n − 2. in all the above cases, we can prove that the functions f and g defines a d4-magic labeling of s(pn) with magic constant µ1. this completes the proof of the theorem. the middle graph of a connected graph g denoted by m(g) is the graph whose vertex set is v (g) ∪ e(g) where two vertices are adjacent if (i) they are adjacent edges of g or (ii) one is a vertex of g and the other is an edge incident with it. theorem 4.2. middle graph of the path graph pn is d4-magic for n ≥ 3. proof. let m(pn) be the middle graph of the path pn. denote the vertices of pn by u1, u2, . . . , un and edges by e1, e2, . . . , en−1 where ei incident with ui and ui+1. there are 2n − 1 vertices and 3n − 4 edges in m(pn). consider the following cases: case(i): n = 3. define f : e(m(p3)) → n3n−4 as f(e1u1) = 1, f(e1u2) = 2, f(e1e2) = 3, f(e2u2) = 4 and f(e2u3) = 5 and define g : e(m(p3)) → d4 \ {ρ0} as g(e1u1) = ρ2 = g(e2u3), g(e1u2) = ρ1 = g(e2u2), g(e1e2) = ρ3. then clearly the middle graph of the path p3 is d4-magic with magic constant ρ2. case(ii): n > 3. define f : e(m(pn)) → n3n−4 as follows: for 1 ≤ i ≤ n − 2, 1 ≤ j ≤ n, f(eiuj) = 2(i − 1) + j and f(eiei+1) = 3i. 177 c. anusha and v. anil kumar now define g : e(m(pn)) → d4 \ {ρ0 by g(e1u1) = ρ2 = g(en−1un), g(e1u2) = µ2, g(e2u2) = µ1, g(e2e3) = ρ3 g(e2u3) = ρ1 = g(e3u3), g(eiei+1) = µ2, where i ̸= 2 and 1 ≤ i ≤ n − 1, g(eiuj) = { µ1, if j = i + 1 and 3 ≤ i < n − 1, µ2, if i = j and 4 ≤ i ≤ n − 1. the above functions f and g will define a d4-magic labeling of m(pn) with magic constant ρ2. this completes the proof of the theorem. a triangular snake tn is obtained from the path pn by replacing each edge of the path by a triangle c3. theorem 4.3. the triangular snake tn is d4-magic. proof. note that every vertex of tn has even degree . so the proof is indisputable from theorem 2.1. the alternate triangular snake a(tn) is obtained from the path u1, u2, . . . , un by joining uiui+1(alternatively) to a new vertex vi. theorem 4.4. the alternate triangular graph a(tn) is d4-magic. proof. let us denote the vertices of the path pn be u1, u2, . . . , un and the vertex which join ui and ui+1 be denoted by vi. now consider the following cases: case (i): n is even and triangle starts from u1. suppose that n is even and the triangle starts from the first vertex ui, then there are n + n 2 vertices and 2n − 1 edges. suppose n = 2 then a(tn) is c3 itself. so there is nothing to prove. suppose n = 4 then take f be any bijection from e(a(tn)) to n7 and define g : e(a(tn)) → d4 \ {ρ0} by g(u1v1) = g(u4v3) = ρ1, g(u2u3) = ρ2, g(u2v1) = g(u3v3) = g(u1u2) = g(u3u4) = ρ3. then a(t4) becomes d4-magic with magic constant ρ0. suppose n > 4, then let f : e(a(tn)) → n2n−1 be any bijection and define 178 d4 magic graphs g : e(a(tn)) → d4 \ {ρ0} as g(u1u2) = g(un−1un) = g(u2v1) = g(un−1vn−1) = ρ3, g(u1v1) = g(un−1vn−1) = ρ1. for 2 ≤ i < n − 1, define g(uiui+1) = { ρ2, if i is even, µ2, if i is odd. g(u2k+1v2k+1) = g(u2k+2v2k+1) = µ1, k = 1, 2, . . . , n − 4 2 obviously the functions f and g will constitute a d4-magic labeling for a(tn) with l∗(u) = ρ0, ∀u ∈ v (a(tn)). case (ii): n is even and the triangle starts from the second vertex u2. we can define a magic labeling for a(tn), where n is even and cycle starts from u2 as follows: let f be any bijection as above and define g as g(uiui+1) = { ρ2, if i is odd, ρ3, if i is even, , 1 ≤ i ≤ n − 1, g(u2kv2k) = g(u2k+1v2k) = ρ1, k = 1, 2, 3, . . . , n − 2 2 . clearly l∗ is a constant map, i.e., l∗(u) = ρ2, ∀u ∈ v (a(tn)). case (iii): n is odd and the triangle starts from the first vertex. suppose n = 3 and the triangle starts from the first vertex u1. let f : e(a(tn)) → n4 be any bijection. now define g : e(a(tn)) → d4 \ {ρ0} by g(u1u2) = g(u2v1) = δ2, g(u1v1) = δ1, g(u2u3) = ρ2. using these maps we can show that the graph is d4-magic with magic constant ρ2. when n is odd and n > 3, there are n+ (n−1) 2 vertices and 2(n−1) edges in a(tn). suppose that n > 3, n is odd and the triangle of a(tn) starts from the first vertex u1. here we take f as any bijection and g : e(a(tn)) → d4 \ {ρ0} be defined as follows: g(u1u2) = g(u2v1) = δ2, g(u1v1) = δ1, g(uiui+1) = { ρ2, if i is even, ρ3, if i is odd, , 1 < i < n. g(uivi) = g(ui+1vi) = ρ1, 1 < i < n and i is odd. then clearly l∗(u) = ρ2, ∀v ∈ v (a(tn)) 179 c. anusha and v. anil kumar case (iv): n is odd and triangle starts from the second vertex. a(tn) with n odd and the triangle starts from the first vertex is just the mirror image of the a(tn) in case (iii). so we can define f and g similarly as in case(iii) and obtain a d4-magic labeling for a(tn) with magic constant ρ2. this completes the proof of the theorem. a double triangular snake d(tn) consists of two triangular snakes that have a common path pn. theorem 4.5. the double triangular graph is d4-magic. proof. let u1, u2, . . . , un be the vertices of the path pn and let v1, v2, . . . , vn−1, w1, w2, . . . , wn−1 be the remaining vertices of d(tn) such that the vertex vi is adjacent to ui and ui+1, where 1 ≤ i < n. similarly the vertex wi is adjacent to ui and ui+1. without loss of generality let v1, v2, . . . , vn−1 and w1, w2, . . . , wn−1 be the vertices of upper triangles and lower triangles respectively. now we define a d4-magic labeling for d(tn) as follows: let f : e(d(tn)) → n5(n−1) be any bijection and let g : e(d(tn)) → d4 \ {ρ0} be defined by g(uiui+1) = ρ2, g(uivi) = g(uiwi) = ρ1, and g(ui+1vi) = g(ui+1wi) = ρ3 where 1 ≤ i < n. thus we can see that l∗(u) = ρ0, ∀u ∈ v (d(tn)). this completes the proof. 5 conclusions in this paper, we introduced the concept of a-magic labeling of graphs, where a is a nonabelian group. furthermore, we characterised graphs which are d4 magic. 6 acknowledgements the first author acknowledges the financial support from university grants commission, government of india. references [1] sedláček, j., 1976. on magic graphs. mathematica slovaca, 26(4), pp.329– 335. 180 d4 magic graphs [2] fraleigh, j.b., 2003. a first course in abstract algebra. pearson education india. [3] parthasarathy, k.r., basic graph theory, 1994. tata mc-grawhill publishing company limited. [4] doob, m., 1978. characterizations of regular magic graphs. journal of combinatorial theory, series b, 25(1), pp.94–104. [5] doob, m., 1974. generalizations of magic graphs. journal of combinatorial theory, series b, 17(3), pp.205–217. [6] p. t. vandana and v. anil kumar, v4 magic labelings of wheel related graphs, british journal of mathematics and computer science, vol.8, issue 3,(2015). [7] richard, p., 1973. stanley, linear homogeneous diophantine equations and magic labelings of graphs, duke math. j., 40, pp.607–632. [8] lee, s.m., saba, f.a.r.r.o.k.h., salehi, e. and sun, h., 2002. on the v4-magic graphs. congressus numerantium, pp.59–68. 181 microsoft word documento1 microsoft word cap4.doc microsoft word documento1 microsoft word cap6.doc microsoft word cap1.doc microsoft word r.m.7 cap.10.doc microsoft word documento1 microsoft word documento1 ratio mathematica vol. 33, 2017, pp. 21-38 issn: 1592-7415 eissn: 2282-8214 an overview of topological and fuzzy topological hypergroupoids šarka hošková-mayerová∗ †doi:10.23755/rm.v33i0.389 it is my honour to dedicate this paper to professor thomas vougiouklis lifetime work. abstract on a hypergroup, one can define a topology such that the hyperoperation is pseudocontinuous or continuous. this concepts can be extend to the fuzzy case and a connection between the classical and fuzzy (pseudo)continuous hyperoperations can be given. this paper, that is his an overview of results received by s. hoskova-mayerova with coauthors i. cristea, m. tahere and b. davaz, gives examples of topological hypergroupoids and show that there is no relation (in general) between pseudotopological and strongly pseudotopological hypergroupoids. in particular, it shows a topological hypergroupoid that does not depend on the pseudocontinuity nor on strongly pseudocontinuity of the hyperoperation. keywords: hyperoperation, hypergroupoid, continuous, pseudocontinuous and strongly pseudocontinuous hyperoperation, topology, topological hypergroupoid, (fuzzy) pseudocontinuous hyperoperation, (fuzzy) continuous hyperoperation, fuzzy topological space. 2010 ams mathematics subject classification: 20n20, 22a22. ∗department of mathematics and physics, university of defence brno, czech republic; sarka.mayerova@unob.cz † c©šarka hošková-mayerová. received: 31-10-2017. accepted: 26-12-2017. published: 31-12-2017. 21 šarka hošková-mayerová 1 introduction as was mentioned e.g. in [32], in various branches of mathematics we encounter important examples of topologico-algebraical structures like topological groupoids, groups, rings, fields etc. therefore, there was a natural interest to generalize the concept of topological groupoid to topological hypergroupoid. first results of this type can be found e.g. in [6, 32]. hypergroups are generalizations of groups. group is a set with a binary operation on it satisfying a number of conditions. if this binary operation is taken to be multivalued, then we arrive at a hypergroup. the motivation for generalization of the notion of group resulted naturally from various problems in non-commutative algebra, another motivation for such an investigation came from geometry. hypergroups theory, born in 1934 with marty‘s paper [39] presented in the 8th congress of scandinavian mathematicians where he had given this renowned definition. ”marty, managed to do the greatest generalisation anybody would ever do, acting as a pure and clever researcher. he left space for future generalisations ”between” his axioms and other hypergroups, as the regular hypergroups, join spaces etc. the reproduction axiom in the theory of groups is also presented as solutions of two equations, consequently, marty got round that hitch, too. ” [53]. he was followed in 1938 by dresher with ore [23] as well as by griffiths [27] and in 1940 by eaton [24] is now studied from the theoretical point of view and for its applications to many subjects of pure and applied mathematics (see [9, 15, 16, 57]) like algebra, geometry, convexity, topology, cryptography and code theory, graphs and hypergraphs, lattice theory, boolean algebras, logic, probability theory, binary relations, theory of fuzzy and rough sets [12, 20], automata theory, economy, etc. [10, 11, 15]. hypergroupoids, [17] quasi-hypergroups, semihypergroups [41, 42], hypergroups [1, 2], hyperrings [40, 52], hyperfields, [60] hyper vector spaces, hyperlattices, up to all kinds of fuzzy hyperstructures [49], have been studied theoretically as well as from the perspective of particular applications, see e.g. [5, 18, 21, 30, 33, 56]. in 1990, th. vougiouklis introduced the class of hv-structures which satisfy the weak axioms where the non-empty intersection replaces the equality [55]. moreover, topological and algebraic structures in fuzzy sets are strategically located at the juncture of fuzzy sets, topology, [26] algebra [7], lattices, etc. they has these unique features: major studies in uniformities and convergence structures, fundamental examples in lattice-valued topology, modifications and extensions of sobriety, categorical aspects of lattice-valued subsets, logic and foundations of mathematics, t-norms and associated algebraic and ordered structures. in the last decade a number of interesting applications to social sciences appear, e.g. [3, 43, 44, 59, 61, 62]. 22 an overview of topological and fuzzy topological hypergroupoids in [6], ameri presented the concept of topological (transposition) hypergroups. he introduced the concept of a (pseudo, strong pseudo) topological hypergroup and gave some related basic results. r. ameri studied the relationships between pseudo, pseudo topological polygroups and topological polygroups. in [28], heidari et al. studied the notion of topological hypergroups as a generalization of topological groups. they showed by considering the quotient topology induced by the fundamental relation on a hypergroup that if every open subset of a topological hypergroup is complete part, then it’s fundamental group is a topological group. moreover, in [29], heidari et al. defined the notion of topological polygroups and they investigated the topological isomorphism theorems it. later on, salehi shadkami et al. [47, 48] established various relations between its complete parts and open sets and they used these facts to obtain some new results in topological polygroups. for example, they investigated some properties of cpresolvable topological polygroups. in [32], the author of this note introduced the concept of topological hypergroupoid and found necessary and sufficient conditions for having a τu -topological hypergroupoid, a τl-topological hypergroupoid and a τℵ-topological hypergroupoid by using the concepts of pseudocontinuity, strong pseudocontinuity and both respectively. when in 1965 zadeh [63] introduced the fuzzy sets, than the reconsideration of the concept of classical mathematics began. since then the connections between fuzzy sets and hyperstructures was studied. using the structure of a fuzzy topological space and that of a fuzzy group (introduced by rosenfeld [46]), foster [26] defined the concept of fuzzy topological group. later, ma and yu [38] changed foster’s definition in order to make sure that an ordinary topological group is a special case of a fuzzy topological group. an interesting book concerning fuzzy topology was published in 1997 by liu [36]. inspired by the definition of the topological groupoid i. cristea and s. hoskovamayerova in [19] extended these notions on a fuzzy topological space. this paper is an overview of results received by s. hoskova-mayerova in [32] as well as the results with coauthors i. cristea [19], m. tahere, b. davaz [50]. paper is organized as follows: firstly, we review some basic definitions and results on hypergroups and topology and fuzzy topological spaces. section 3 recall the results concerning topological hypergroupoids. in section 4 we recall some basic results on the fuzzy topological spaces that we use in the following section 6. in section 5 we recall the definition of fuzzy (pseudo)continuous hyperoperations, we explain relations between fuzzy continuous and continuous hyperoperations, between fuzzy continuous and fuzzy pseudocontinuous hyperoperations, respectively. moreover, we give the condition when a product hypergroupoid is a fuzzy pseudotopological hypergroupoid. finally, in section 6 we recall some results published in [19] concerning fuzzy topological hypergroupoids. 23 šarka hošková-mayerová 2 basic definitions in this section, we present some definitions related to hyperstructures and topology that are used throughout the paper. they can be found in e.g. [4, 19, 31, 22]. definition 2.1. let h be a non-empty set. then, a mapping ◦ : h×h →p∗(h) is called a binary hyperoperation on h, where p∗(h) is the family of all nonempty subsets of h. the couple (h,◦) is called a hypergroupoid. if a and b are two non-empty subsets of h and x ∈ h, then we define: a◦b = ⋃ a∈a b∈b a◦ b, x◦a = {x}◦a and a◦x = a◦{x}. a hypergroupoid (h,◦) is called a: • semihypergroup if for every x,y,z ∈ h, we have x◦ (y ◦z) = (x◦y) ◦z; • quasihypergroup if for every x ∈ h, x◦ = h = h ◦ x (this condition is called the reproduction axiom); • hypergroup if it is a semihypergroup and a quasihypergroup. definition 2.2. let (x,τ) be a topological space. then 1. (x,τ) is a t0-space if for all x 6= y ∈ x, there exists u ∈ τ such that x ∈ u and y is not in u or y ∈ u and x is not in u. 2. (x,τ) is a t1-space if for all x 6= y ∈ x, there exist u,v ∈ τ such that x ∈ u and y is not in u and y ∈ v and x is not in v . 3. (x,τ) is a t2-space if for all x 6= y ∈ x, there exist u,v ∈ τ such that x ∈ u, y ∈ v and u ∩v = ∅. so, every t2-topological space is a t1-topological space and every t1-topological space is a t0-topological space. definition 2.3. let (h1,◦1), (h2,◦2) be two hypergroupoids and define the topologies τ,τ ′ on h1,h2 respectively. a mapping f from h1 to h2 is said to be good topological homomorphism if for all x,y ∈ h1, 1. f(a◦1 b) = f(a) ◦2 f(b); 2. f is continuous; 24 an overview of topological and fuzzy topological hypergroupoids 3. f is open. a good topological homomorphism is a topological isomorphism if f is one to one and onto and we say that h1 is topologically isomorphic to h2. let (h,◦) be a hypergroupoid and a,b be non empty subsets of h. by a ≈ b we mean that a∩b 6= ∅. 3 topological hypergroupoids š. hošková-mayerová in [32] introduced some new definitions inspired by the definition of topological groupoid. her results are summarized in this section. definition 3.1. [32] let (h, ·) be a hypergroupoid and (h,τ) be a topological space. the hyperoperation “ · ” is called: 1. pseudocontinuous (p-continuous) if for every o ∈ τ, the set o? = {(x,y) ∈ h2 : x ·y ⊆ o} is open in h ×h. 2. strongly pseudocontinuous (sp-continuous) if for every o ∈ τ, the set o? = {(x,y) ∈ h2 : x ·y ≈ o} is open in h ×h. a simple way to prove that a hyperoperation “ · ” is pcontinuous (spcontinuous) is to take any open set o in τ and (x,y) ∈ h2 such that x · y ⊆ o (x ·y ≈ o) and prove that there exist u,v ∈ τ such that u.v ⊆ o (u.v ≈ o) for all (u,v) ∈ u ×v . definition 3.2. [32] let (h, ·) be a hypergroupoid, (h,τ) be a topological space and τ? be a topology on p∗(h). the triple (h, ·,τ) is called a pseudotopological or strongly pseudotopological hypergroupoid if the hyperoperation “ · ” is p-continuous or sp-continuous respectively. the quadruple (h, ·,τ,τ?) is called τ?-topological hypergroupoid if the hyperoperation “ · ” is τ?-continuous. let (h,τ) be a topological space, v,u1, . . . ,uk ∈ τ . we define sv , iv and ℵ(u1, . . . ,uk) as follows: • sv = {u ∈p∗(h) : u ⊆ v} = p∗(v ). • iv = {u ∈p∗(h) : u ≈ v}. • ℵ(u1, . . . ,uk) = {b ∈ p∗(h) : b ⊆ ⋃k i=1 ui and b ≈ ui for i = 1, . . . ,k}. 25 šarka hošková-mayerová s∅ = i∅ = ∅. for all v 6= ∅, we have sv = p∗(v ) and iv ⊇{h,p∗(v )}. lemma 3.1. let (h,τ) be a topological space. then {sv}v∈τ forms a base for a topology (τu ) on p∗(h). moreover, τu is called the upper topology. then {iv}v∈τ forms a subbase for a topology (τl) on p∗(h). moreover, τl is called the lower topology. let (h,τ) be a topological space. then {ℵ(u1, . . . ,uk)}ui∈τ forms a base for a topology (τℵ) on p∗(h). moreover, τℵ is called the vietoris topology [51]. following results was proved by s. hoskova-mayerova in [32]. theorem 3.1. let (h, ·) be a hypergroupoid and (h,τ) be a topological space. then the triple (h, ·,τ) is a pseudotopological hypergroupoid if and only if the quadruple (h, ·,τ,τu ) is a τu -topological hypergroupoid. then the triple (h, ·,τ) is a strongly pseudotopological hypergroupoid if and only if the quadruple (h, ·,τ,τl) is a τl-topological hypergroupoid. then the triple (h, ·,τ) is a pseudotopological hypergroupoid and strongly pseudotopological hypergroupoid if and only if the quadruple (h, ·,τ,τℵ) is a τℵ-topological hypergroupoid. 4 fuzzy topological spaces in this section we recall some basic results on the fuzzy topological spaces that we use in the following. let x be a nonempty set. a fuzzy set a in x is characterized by a membership function µa : x −→ [0, 1]. we denote by fs(x) the set of all fuzzy sets on x. in this paper we use the definition of a fuzzy topological space given by chang [8]. definition 4.1. [8] a fuzzy topology on a set x is a collection t of fuzzy sets in x satisfying (i) 0 ∈ t and 1 ∈ t (where 0, 1 : x −→ [0, 1], 0(x) = 0, 1(x) = 1, for any x ∈ x). (ii) if a1,a2 ∈t , then a1 ∩a2 ∈t . (iii) if ai ∈t for any i ∈ i, then ⋃ i∈i ai ∈t , 26 an overview of topological and fuzzy topological hypergroupoids where µa1∩a2 (x) = µa1 (x) ∧µa2 (x) and µ⋃i∈i ai (x) = ∨i∈i µai (x). the pair (x,t ) is called a fuzzy topological space. in the definition of a fuzzy topology of lowen [37], the condition (i) is substituted by (i’) for all c ∈ [0, 1], kc ∈t , where µkc (x) = c, for any x ∈ x. example 4.1. now we present some examples of fuzzy topologies on a set x. for more details see [19]. (i) the family t = {0, 1} is called the indiscrete fuzzy topology on x. (ii) the family of all fuzzy sets in x is called the discrete fuzzy topology on x. (iii) if τ is a topology on x, then the collection t = {ao | o ∈ τ} of fuzzy sets x, where µao is the characteristic function of the open set o, is a fuzzy topology on x. (iv) the collection of all constant fuzzy sets in x is a fuzzy topology on x, where a constant fuzzy set a in x has the membership function µa defined as follows : µa : −→ [0, 1], µa(x) = k, with k a fix constant in [0, 1]. definition 4.2. [8] given two topological spaces (x,t ) and (y,u), a function f : x −→ y is fuzzy continuous if, for any fuzzy set a ∈ u, the inverse image f−1[a] belongs to t , where µf−1[a](x) = µa(f(x)), for any x ∈ x. proposition 4.1. [8] a composition of fuzzy continuous functions is fuzzy continuous function. definition 4.3. [36] a base for a fuzzy topological space (x,t ) is a subcollection b of t such that each member a of t can be written as the union of members of b. a natural question is: ‘how to judge whether some fuzzy subsets just form a base of some fuzzy topological space?’ we have the following rule: proposition 4.2. [36] a family b of fuzzy sets in x is a base for a fuzzy topology t on x if and only if it satisfies the following conditions: (i) for any a1,a2 ∈b, we have a1 ∩a2 ∈b. (ii) ⋃ a∈b a = 1. 27 šarka hošková-mayerová if (x1,t1) and (x2,t2) are fuzzy topological spaces, we can speak about the product fuzzy topological space (x1 × x2,t1 × t2), where the product fuzzy topology is given by a base like in the following result, which can be generalized to a family of fuzzy topological spaces. proposition 4.3. [36] let (x1,t1) and (x2,t2) be fuzzy topological spaces. the product fuzzy topology t on the product space x = x1 × x2 has as a base the set of product fuzzy sets of the form a1 × a2, where ai ∈ ti, i = 1, 2, and µa1×a2 (x1,x2) = µa1 (x1) ∧µa2 (x2). proposition 4.4. [26] let {(xi,ti)}i∈i,{(yi,ui)}i∈i be two families of fuzzy topological spaces and (x,t ), (y,u) the respective product fuzzy topological spaces. for each i ∈ i, let fi : (xi,ti) −→ (yi,ui). then the product mapping f = ×fi : (xi) −→ (fi(xi)) of (x,t ) into (y,u) is fuzzy continuous if fi is fuzzy continuous, for each i ∈ i. 5 some results on relation between topological spaces on a set and topological spaces on its powerset in this section, we use the results presented in [32] to study topological hypergroupoids. first, we show that there is no relation (in general) between pseudotopological and strongly pseudotopological hypergroupoids. proposition 5.1. [50] let h = {a,b}, τ = {∅,{a},h} and define a hyperoperation “ ◦1 ” on h as follows: ◦1 a b a b h b h h then (h,◦1,τ) is a pseudotopological hypergroupoid. thus, the quadruple (h,◦1,τ,τu ) is a τu topological hypergroupoid. moreover, (h,◦1,τ) is not strongly pseudotopological hypergroupoid. proposition 5.2. [50] let h = {a,b}, τ = {∅,{a},h} and define a hyperoperation “ ◦2 ” on h as follows: ◦2 a b a h a b a b 28 an overview of topological and fuzzy topological hypergroupoids then (h,◦2,τ) is a strongly pseudotopological hypergroupoid. now we have: the quadruple (h,◦2,τ,τl) is a τltopological hypergroupoid. (h,◦2,τ) is not pseudotopological hypergroupoid. not every strongly pseudotopological hypergroupoid is a pseudotopological hypergroupoid. let (h,◦) be a hypergroupoid and τ a topology on h. then (h,◦,τ) may be neither a pseudotopological hypergroupoid nor a strongly pseudotopological hypergroupoid. we illustrate this fact by the following example. example 5.1. [50] let h = {a,b}, τ = {∅,{a},h} and define a hyperoperation “ ◦3 ” on h as follows: ◦3 a b a b a b a b it is easy to check, by taking o = {a} and a ◦3 b ∈ o, that (h,◦3,τ) is neither a pseudotopological hypergroupoid nor a strongly pseudotopological hypergroupoid. proposition 5.3. let (h,◦,τ,τu ) be a topological hypergroupoid. then (h,τ) is the trivial topology if and only if (p∗(h),τu ) is the trivial topology. for the proof see [50]. corolary 5.1. let (h,◦,τ,τℵ) be a topological hypergroupoid. then (h,τ) is the trivial topology if and only if (p∗(h),τℵ) is the trivial topology. proposition 5.4. let (h,◦,τ,τl) be a topological hypergroupoid. then (h,τ) is the trivial topology if and only if (p∗(h),τl) is the trivial topology. the proof is similar to that of proposition 5.3. proposition 5.5. let (h,◦,τ,τu ) be a topological hypergroupoid, |h| ≥ 2 and (h,τ) be the powerset topology. then (p∗(h),τu ) is not the powerset topology on p∗(h). then (p∗(h),τl) is not the powerset topology on p∗(h). proposition 5.6. let (h1,◦1,τ) and (h2,◦2,τ ′) be two topologically isomorphic hypergroupoids. if (h1,◦1,τ,τu ) is a τu topological hypergroupoid then (h2,◦2,τ ′,τ ′u ) is a τu topological hypergroupoid. if (h1,◦1,τ,τl) is a τltopological hypergroupoid then (h2,◦2,τ ′,τ ′l) is a τltopological hypergroupoid. 29 šarka hošková-mayerová corolary 5.2. let (h1,◦1,τ) and (h2,◦2,τ ′) be two topologically isomorphic hypergroupoids. if (h1,◦1,τ,τℵ) is a τℵtopological hypergroupoid then (h2,◦2,τ ′,τ ′ℵ) is a τℵtopological hypergroupoid. we present now some τℵtopological hypergroupoids. proposition 5.7. let (h,◦) be the total hypergroup (i.e., x◦y = h for all (x,y) ∈ h2) and τ be any topology on h. then (h,◦,τ) is both: pseudotopological hypergroupoid and strongly pseudotopological hypergroupoid. corolary 5.3. let (h,◦) be the total hypergroup and τ be any topology on h. then the quadruple (h,◦,τ,τℵ) is a τℵtopological hypergroupoid. proposition 5.8. let h = r, (h,◦) be the hypergroupoid defined by: x◦y = { {a ∈ r : x ≤ a ≤ y}, if x ≤ y; {a ∈ r : y ≤ a ≤ x}, if y ≤ x. and τ be the topology on h defined by: τ = {] −∞,a[: a ∈ r∪{±∞}}. then (h,◦,τ,τℵ) is a τℵtopological hypergroupoid. proposition 5.9. let h = r, (h,◦) be the hypergroupoid defined by: x◦y = { {a ∈ r : x ≤ a ≤ y}, if x ≤ y; {a ∈ r : y ≤ a ≤ x}, if y ≤ x. and τ be the topology on h defined by: τ = {]a,∞[: a ∈ r∪{±∞}}. then (h,◦,τ ′,τ ′ℵ) is a τ ′ ℵtopological hypergroupoid. proposition 5.10. let (h,?) be the hypergroupoid defined by x ? y = {x,y} and τ be any topology on h. then (h,?,τ,τℵ) is a τℵtopological hypergroupoid. example 5.2. [50] let h = {a,b}, τ = {∅,{a},h} and define a hyperoperation “ ◦4 ” on h as follows: ◦4 a b a a h b h b then, by proposition 5.10, (h,◦4,τ,τℵ) is a τℵtopological hypergroupoid. moreover, τℵ = {∅,{{a}},p∗(h)}. 30 an overview of topological and fuzzy topological hypergroupoids proposition 5.11. let (h, ·) be any hypergroupoid and τ be the power set topology on h. then (h, ·,τ,τℵ) is a τℵtopological hypergroupoid. proposition 5.12. let (h, ·) be any hypergroupoid and τ be the trivial topology on h. then (h, ·,τ,τℵ) is a τℵtopological hypergroupoid. next, we present a topological hypergroupoid that does not depend on the pseudocontinuity nor on strongly pseudocontinuity of the hyperoperation. proposition 5.13. let (h, ·) be any hypergroupoid, τ be any topology on h and τ? be the trivial topology on p∗(h). then (h, ·,τ,τ?) is a topological hypergroupoid. next, we present some results on t0,t1,t2topological spaces. proposition 5.14. let (h, ·,τ,τu ) be a τu -topological hypergroupoid. if (p∗(h),τu ) is a t0topological space then (h,τ) is a t0topological space. the converse of proposition 5.14 is not always true. an illustrating example is presented in [50]. proposition 5.15. let |h| ≥ 2 and (h, ·,τ,τu ) be a τu -topological hypergroupoid. then (p∗(h),τu ) is neither a t1topological space nor a t2topological space. proposition 5.16. let |h| ≥ 2 and (h, ·,τ,τl) be a τl-topological hypergroupoid. then (p∗(h),τl) is neither a t1topological space nor a t2topological space. proposition 5.17. let (h, ·,τ,τl) be a τl-topological hypergroupoid. if (p∗(h),τl) is a t0topological space then (h,τ) is a t0topological space. proposition 5.18. let (h, ·,τ,τl) be a τl-topological hypergroupoid. if (h,τ) is a t0topological space then (p∗(h),τl) may not be a t0topological space. it can be proved that: let (h, ·,τ,τℵ) be a τℵ-topological hypergroupoid. if (p∗(h),τℵ) is a t0topological space then (h,τ) is a t0topological space. let (h, ·,τ,τℵ) be a τℵ-topological hypergroupoid. then (p∗(h),τℵ) is neither a t2topological space nor a t1topological space. 6 fuzzy topological hypergroupoids in this section we recall some results published in [19] concerning fuzzy topological hypergroupoids. 31 šarka hošková-mayerová definition 6.1. let (h,◦) be a hypergroupoid, t and u be fuzzy topologies on h and p∗(h), respectively. the hyperoperation ”◦” is called u-fuzzy continuous if the map ◦ : h × h −→ p∗(h) is fuzzy continuous with respect to the fuzzy topologies t ×t and u. for any topology τ on a set x, we denote by tc the fuzzy topology formed with the characteristic functions of the open sets of τ. in the following result we give a relation between the continuity and fuzzy continuity of a hyperoperation. proposition 6.1. let (h,◦) be a hypergroupoid, τ and τ∗ be topologies on h and p∗(h), respectively. let tc and uc be the fuzzy topologies on h and p∗(h), respectively, generated by τ and τ∗, respectively. the hyperoperation ”◦” is τ∗continuous if and only if it is uc-fuzzy continuous. let (h,t ) be a fuzzy topological space. then the family b = {ã ∈ fs (p∗(h)) | a ∈t}, where µã(x) = ∧ x∈x µa(x), is a base for a fuzzy topology t ∗ on p∗(h). definition 6.2. let (h,◦) be a hypergroupoid endowed with a fuzzy topology t . the hyperoperation ”◦” is called fuzzy pseudocontinuous (or briefly fuzzy pcontinuous) if, for any a ∈t , the fuzzy set a∗ in h×h belongs to t ×t , where µa∗(x,y) = ∧ u∈x◦y µa(u). the triple (h,◦,t ) is called a fuzzy pseudotopological hypergroupoid if the hyperoperation ”◦” is fuzzy p-continuous. now we can characterize a fuzzy pseudotopological hypergroupoid (h,◦,t ) using the t ∗-fuzzy continuity of the hyperoperation ”◦”, where the fuzzy topology t ∗ is that one given in proposition 6.1. let (h,◦) be a hypergroupoid and t be a fuzzy topology on h. then the triple (h,◦,t ) is a fuzzy pseudotopological hypergroupoid if and only if the hyperoperation ”◦” is t ∗-fuzzy continuous. proposition 6.2. let (h1,t1) and (h2,t2) be two fuzzy topological spaces. we denote h = h1 ×h2 and t = t1 ×t2. then the mapping α : (h,t ) × (h,t ) −→ (h1 ×h1,t1 ×t1) × (h2 ×h2,t2 ×t2), defined by α((x1,x2), (y1,y2)) = ((x1,y1), (x2,y2)) is fuzzy continuous. let (h1,◦1) and (h2,◦2) be two hypergroupoids. the product hypergroupoid (h1 × h2,⊗) has the hyperoperation defined by (x1,x2) ⊗ (y1,y2) = (x1 ◦1 y1,x2 ◦2 y2), for any (x1,x2), (y1,y2) ∈ h1 ×h2. 32 an overview of topological and fuzzy topological hypergroupoids so, we get: if (h1,◦1,t1) and (h2,◦2,t2) are fuzzy pseudotopological hypergroupoids, then the product hypergroupoid (h1×h2,⊗,t1×t2) is a fuzzy pseudotopological hypergroupoid. 7 conclusions on a hypergroup, a topology such that the hyperoperation is pseudocontinuous can be defined. this paper highlighted the topological hypergroupoids by proving some of theirs properties. it illustrated the results achieved on topological hypergroupoids in [32] by examples and remarks. moreover, it was shown that there is no relation (in general) between pseudotopological and strongly pseudotopological hypergroupoids. in particular, we presented a topological hypergroupoid that does not depend on the pseudocontinuity nor on strongly pseudocontinuity of the hyperoperation. for future work, the existence of topological hypergroupoids on p∗(h) that are neither τu nor τl nor τℵ can be investigated or the existence of n-ary topological hypergroupoids can be studied. this work could be also continued in order to introduce the notion of fuzzy topological hypergroup as a generalization of a fuzzy topological group in the sense of foster [26] or in the sense of ma and yu [38]. the author would like to express a wish for this beautiful discipline of mathematics to be continue and to be developed. since there are already numbers of excellent mathematicians around the world who are concerned with this issue, lets believe their interest will not go away, and that the school of professor p. corsini and professor t. vougiouklis will 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[63] l. a. zadeh, fuzzy sets, inform and control, 8 (1965), 338–353. 38 ratio mathematica vol. 34, 2018, pp. 15-33 issn: 1592-7415 eissn: 2282-8214 on rough sets and hyperlattices ali akbar estaji∗, fereshteh bayati† received: 08-10-2017. accepted: 28-01-2018. published: 30-06-2018 doi:10.23755/rm.v34i0.350 c©ali akbar estaji et al. abstract in this paper, we introduce the concepts of upper and lower rough hyper fuzzy ideals (filters) in a hyperlattice and their basic properties are discussed. let θ be a hyper congruence relation on l. we show that if µ is a fuzzy subset of l, then θ(< µ >) = θ(< θ(µ) >) and θ(µ∗) = θ((θ(µ))∗), where < µ > is the least hyper fuzzy ideal of l containing µ and µ∗(x) = sup{α ∈ [0,1] : x ∈ i(µα)} for all x ∈ l. next, we prove that if µ is a hyper fuzzy ideal of l, then µ is an upper rough fuzzy ideal. also, if θ is a ∧−complete on l and µ is a hyper fuzzy prime ideal of l such that θ(µ) is a proper fuzzy subset of l, then µ is an upper rough fuzzy prime ideal. furthermore, let θ be a ∨-complete congruence relation on l. if µ is a hyper fuzzy ideal, then µ is a lower rough fuzzy ideal and if µ is a hyper fuzzy prime ideal such that θ(µ) is a proper fuzzy subset of l, then µ is a lower rough fuzzy prime ideal. keywords: rough set, upper and lower approximations, hyperlattice, hyper fuzzy prime ideal, hyper fuzzy prime filter. 2010 ams subject classifications: 03g10, 03e72, 46h10, 06d50, 08a72. ∗faculty of mathematics and computer sciences, hakim sabzevari university, sabzevar, iran. aaestaji@hsu.ac.ir †faculty of mathematics and computer sciences, hakim sabzevari university, sabzevar, iran.fereshte.bayati@yahoo.com 15 a. a. estaji and f. bayati 1 introduction in this paper, we are using three basic notions: hyperlattice, rough set and fuzzy subset. hyperstructure theory was born in 1934 when marty [14] defined hypergroups as a generalization of groups. extending lattices (also called hyperlattices) have been recently studied by a number of authors, in particular, koguep, nkuimi and lele [12], feng and zou [8], guo and xin [9], rahnamai-barghi [19], etc. rough set theory was introduced by pawlak in 1982 [17]. many authors have studied the general properties of generalized rough sets [1, 4, 5]. the concept of fuzzy subsets was first introduced by zadeh [22] in 1965 and then the fuzzy subsets have been used in the reconsideration of classical mathematics. the relationships between fuzzy subsets and algebraic hyperstructures had been already considered by many researchers (for example [2, 21]). also, there have been many papers studying the connections and differences of fuzzy subset theory and rough set theory [3, 15, 18]. in recent years, many efforts have been made to compare and combine the three theories [6, 7]. this paper is structured as follows. after the introduction, in section 2, we recall some basic notions and results on hyperlattices, rough sets and fuzzy subsets. in section 3, the notions of hyper congruence relation on a hyperlattice are introduced. next, some important properties of θ-upper approximations of a fuzzy subset will be studied. also by an example, we show that theorem 2.15 in [12] is incorrect (see example 3.20) and a corrected version is considered, proposition 3.21. finally, in section 4, θ-lower approximations of a fuzzy subset on a hyperlattice will be studied. 2 preliminaries of hyperlattices, rough sets and fuzzy subsets in the remainder of the paper we use some notation and results from the theory of hyperlattices, rough sets and fuzzy subsets. we present a few basic definitions here. let l be a set partially ordered by the binary relation ≤. the poset (l,≤) is a meet-semilattice if for all elements x and y of l, the greatest lower bound or the meet of the set {x,y}, denoted by x∧y, exists. for x and y in a meet-semilattice l, x ≤ y ⇔ x = x∧y. replacing greatest lower bound with least upper bound results in the dual concept of a join-semilattice. the least upper bound of {x,y} is called the join of x and y and is denoted by x ∨ y. a poset l is a lattice if and only if it is both a meetand a join-semilattice. 16 on rough sets and hyperlattices in this paper, we use the following notion of a hyperlattice. let l be a non-empty meet-semilattice and ∨ : l×l → p(l)∗ be a hyperoperation, where p(l) is the power set of l and p(l)∗ = p(l) \ {∅}. then (l,∧,∨) is a hyperlattice [19], if for all a,b,c ∈ l: 1. a ∈ a∨a and a = a∧a. 2. a∨ b = b∨a and a∧ b = b∧a. 3. (a∨ b) ∨ c = a∨ (b∨ c) and (a∧ b) ∧ c = a∧ (b∧ c). 4. a ∈ [a∧ (a∨ b)] ∩ [a∨ (a∧ b)]. 5. a ∈ a∨ b ⇔ a∧ b = b. where for all non-empty subsets a and b of l, a∧b = {a∧b|a ∈ a,b ∈ b}, a∨b = ⋃ {a∨ b|a ∈ a,b ∈ b}. throughout this paper, l is a hyperlattice with the least element 0 and the greatest element 1. for x ⊆ l and x ∈ l we write: 1. ↓ x = {y ∈ l : y ≤ x for some x ∈ x}. 2. ↑ x = {y ∈ l : y ≥ x for some x ∈ x}. 3. ↓ x =↓ {x}. 4. ↑ x =↑ {x}. a pair (l,θ), where θ is an equivalence relation on l, is called an approximation space [17] and for a ∈ l, the equivalence class (or coset) of a modulo θ is the set [a]θ = {x ∈ l|(a,x) ∈ θ} and also for a ⊆ l, we put [a]θ = ⋃ a∈a[a]θ. for an approximation space (l,θ), by an upper rough approximation in (l,θ) we mean a mapping apr : p(l) → p(l) which is defined for every x ∈ p(l) by apr(x) = {a ∈ l : [a]θ ∩x 6= ∅}. also, by a lower rough approximation in (l,θ) we mean a mapping apr : p(l) → p(l) defined for every x ∈p(l) by apr(x) = {a ∈ l : [a]θ ⊆ x}. then apr(x) = (apr(x),apr(x)) is called a rough subset in (l,θ) if apr(x) 6= apr(x). the following proposition is well known and easily seen. 17 a. a. estaji and f. bayati proposition 2.1. let (l,θ) be an approximation space. for every subsets x,y ⊆ l, we have 1. apr(x) ⊆ x ⊆ apr(x). 2. if x ⊆ y , then apr(x) ⊆ apr(y ) and apr(x) ⊆ apr(y ). 3. apr(x∪y ) = apr(x)∪apr(y ) and apr(x∩y ) ⊆ apr(x)∩apr(y ). 4. apr(x∩y ) = apr(x)∩apr(y ) and apr(x∪y ) ⊇ apr(x)∪apr(y ). 5. apr ( apr(x) ) = apr(x) and apr ( apr(x) ) = apr(x). proof. see [13]. proposition 2.2. [12] let (l,∨,∧) be a hyperlattice.then, for each pair (a,b) ∈ l×l there exist a1,b1 ∈ a∨ b, such that a ≤ a1 and b ≤ b1. definition 2.3. [19] a nonempty subset j of l is called an ideal of l if for all x,y ∈ l 1. x,y ∈ j implies x∨y ⊆ j. 2. if x ∈ j, then ↓ x ⊆ j. definition 2.4. [19] a nonempty subset f of l is called a filter of l if for all x,y ∈ l 1. x,y ∈ f implies x∧y ∈ f . 2. if x ∈ f and x ≤ y, then y ∈ f . given a hyperlattice l and a set x ⊆ l, let i(x) denote the least ideal containing x, called the ideal generated by x. a fuzzy subset of x is any function from x into [0, 1]. let f(l) be the set of all fuzzy subsets of l. for µ,λ ∈ f(x), we say µ ⊆ λ if and only if µ(x) ≤ λ(x) for all x ∈ x. definition 2.5. [12] let µ be a fuzzy subset of l. then 1. µ is a hyper fuzzy ideal of l if, for all x,y ∈ l, (a) ∧ a∈x∨y µ(a) ≥ µ(x) ∧µ(y). (b) x ≤ y ⇒ µ(x) ≥ µ(y). 18 on rough sets and hyperlattices 2. µ is a hyper fuzzy filter of l if, for all x,y ∈ l, (a) µ(x∧y) ≥ µ(x) ∧µ(y). (b) x ≤ y ⇒ µ(x) ≤ µ(y). definition 2.6. [12] let µ be a proper hyper fuzzy ideal of l. 1. µ is called a hyper fuzzy prime ideal, if µ(x ∧ y) ≤ µ(x) ∨ µ(y), for all x,y ∈ l. 2. µ is called a hyper fuzzy prime filter, if ∧ a∈x∨y µ(a) ≤ µ(x) ∨µ(y), for all x,y ∈ l. definition 2.7. [6] let θ be an equivalence relation on l and µ a fuzzy subset of l. then we define the fuzzy subsets θ(µ) and θ(µ) as follows: θ(µ)(x) = ∨ a∈[x]θ µ(a) and θ(µ)(x) = ∧ a∈[x]θ µ(a). the fuzzy subsets θ(µ) and θ(µ) are, respectively, called the θ-upper and θ-lower approximation of the fuzzy subset µ. then θ(µ) = (θ(µ),θ(µ)) is called a rough fuzzy subset with respect to µ if θ(µ) 6= θ(µ). proposition 2.8. [6] let θ be an equivalence relation on l and µ,λ ∈ f(l). then 1. θ(µ) ≤ µ ≤ θ(µ). 2. if µ ⊆ λ, then θ(µ) ≤ θ(λ) and θ(µ) ≤ θ(λ). 3. θ θ(µ) = θ(µ) and θ θ(µ) = θ(µ). 4. θ(µ)(x) = θ(µ)(a) and θ(µ)(x) = θ(µ)(a), for all x ∈ l and a ∈ [x]θ. 5. θθ(µ) = θ(µ) and θθ(µ) = θ(µ). the proofs of the following propositions are straightforward. proposition 2.9. let θ be an equivalence relation on set a. then the following statements hold: 1. for each x ∈p(a), apr(x) = ⋂{ y ∈p(a) : x ⊆ apr(y ) } = min { y ∈p(a) : x ⊆ apr(y ) } . 19 a. a. estaji and f. bayati 2. for each x ∈p(a), apr(x) = ⋃{ y ∈p(l) : apr(y ) ⊆ x } = max { y ∈p(l) : apr(y ) ⊆ x } . proposition 2.10. let θ be an equivalence relation on set a. then the following statements hold: 1. for each µ ∈f(a), θ(µ) = ∧ {λ ∈f(a) : θ(λ) ≥ µ} = min{λ ∈f(a) : θ(λ) ≥ µ} . 2. for each µ ∈f(a), θ(µ) = ∨{ λ ∈f(l) : θ(λ) ≤ µ } = max { λ ∈f(l) : θ(λ) ≤ µ } . 3 upper approximations of a fuzzy subset in this section we give some important properties of θ with many examples, starting with the following definition. definition 3.1. [16, 20] let θ be an equivalence relation on a hyperlattice l. then θ is called a hyper congruence relation if (a,b) ∈ θ implies that (a∨x)×(b∨x) ⊆ θ and (a∧x,b∧x) ∈ θ for all x ∈ l. it is clear that if l is a hyperlattice l and θ = l × l, then θ is a hyper congruence relation. also, in example 3.9, we’ll provide a non-trivial example. lemma 3.2. let θ be a hyper congruence relation on l. then, for every a,b,c,d ∈ l, 1. if (a,b) ∈ θ and (c,d) ∈ θ, then (a∧c,b∧d) ∈ θ and (a∨c)×(b∨d) ⊆ θ. 2. [a]θ ∨ [b]θ ⊆ [a∨ b]θ. 3. [a]θ ∧ [b]θ ⊆ [a∧ b]θ. proof. evident. proposition 3.3. let θ be an equivalence relation on l and x ⊆ l. if µ ∈f(l) is a hyper fuzzy ideal of l, then 1. µ(↓ x) ⊆↑ µ(x). 20 on rough sets and hyperlattices 2. µ(↑ x) ⊆↓ µ(x). 3. µ(apr(↓ x)) ⊆↑ µ(apr(x)). 4. µ(apr(↑ x)) ⊆↓ µ(apr(x)). 5. θ(µ)(apr(↓ x)) ⊆↑ θ(µ)(x). proof. (1) let a ∈↓ x. then there exists x ∈ x such that a ≤ x. since µ is a hyper fuzzy ideal, we conclude that µ(x) ≤ µ(a) which implies that µ(a) ∈↑ µ(x) and the proof is now complete. (2) the proof is similar to the proof of (1). (3) for each x ⊆ l, since ↓ apr(x) = apr(↓ x) =↓ apr(↓ x), we can then conclude from (1) that µ(apr(↓ x)) ⊆↑ µ(apr(x)). (4) for each x ⊆ l, we have ↑ apr(x) = apr(↑ x) =↑ apr(↑ x). by (2), µ(apr(↑ x)) ⊆↓ µ(apr(x)). (5) since θ(µ)(apr(↓ x)) = θ(µ)(↓ x), we can then conclude from (1) that θ(µ)(apr(↓ x)) ⊆↑ θ(µ)(x). proposition 3.4. let θ be a hyper congruence relation on l and x,y ∈ l. if µ ∈f(l) is a hyper fuzzy ideal of l, then∨ a∈[x]θ,b∈[y]θ ∨ µ(a∨ b) ≤ ∨ θ(µ)(x∨y). proof. by lemma 3.2,∨ a∈[x]θ,b∈[y]θ ∨ µ(a∨ b) ≤ ∨ z∈[x∨y]θ µ(z) = ∨ z∈ ⋃ a∈x∨y[a]θ µ(z) = ∨ a∈x∨y ∨ z∈[a]θ µ(z) = ∨ a∈x∨y θ(µ)(a) = ∨ θ(µ)(x∨y). lemma 3.5. let θ be a hyper congruence relation on l and x,y ∈ l. if µ ∈f(l) is a hyper fuzzy filter of l, then θ(µ)(x∧y) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b). proof. by lemma 3.2, ∨ a∈[x]θ,b∈[y]θ µ(a ∧ b) ≤ ∨ z∈[x∧y]θ µ(z) = θ(µ)(x ∧ y). now, assume that z ∈ [x∧y]θ. by lemma 3.2, (x∨ z) ×{x} ⊆ (x∨ z) × (x∨ (x∧y)) ⊆ θ and (y∨z)×{y}⊆ (y∨z)×(y∨(x∧y)) ⊆ θ. also, by proposition 2.2, there exist zx ∈ x ∨ z and zy ∈ y ∨ z such that z ≤ zx and z ≤ zy. since z ≤ zx∧zy and µ is a hyper fuzzy filter of l, we conclude that µ(z) ≤ µ(zx∧zy). therefore, θ(µ)(x∧y) = ∨ z∈[x∧y]θ µ(z) ≤ ∨ a∈[x]θ,b∈[y]θ µ(a∧b) and the proof is now complete. 21 a. a. estaji and f. bayati definition 3.6. let θ be an equivalence relation on l and µ ∈ f(l). then, µ is called an upper rough fuzzy (prime) filter if θ(µ) is a hyper fuzzy (prime) filter of l. proposition 3.7. let θ be a hyper congruence relation on l. if µ ∈ f(l) is a hyper fuzzy filter, then µ is an upper rough fuzzy filter. proof. let x,y ∈ l and x ≤ y. if z ∈ [x]θ, then (z,x) ∈ θ and (z∨y)×(x∨y) ⊆ θ. since y ∈ x ∨ y, we conclude that (z ∨ y) ×{y} ⊆ θ. also, by proposition 2.2, there exists z1 ∈ z ∨y such that z ≤ z1, and so µ(z) ≤ µ(z1) ≤ ∨ µ(z ∨y). therefore, θ(µ)(x) = ∨ z∈[x]θ µ(z) ≤ ∨ z∈[x]θ ∨ µ(z ∨y) ≤ ∨ t∈[y]θ µ(t) = θ(µ)(y). now, if x,y ∈ l, then θ(µ)(x∧y) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b) by lemma 3.5 ≥ ∨ a∈[x]θ,b∈[y]θ µ(a) ∧µ(b) µ is a hyper fuzzy filter = ∨ a∈[x]θ µ(a) ∧ ∨ b∈[y]θ µ(b) = θ(µ)(x) ∧θ(µ)(y). hence θ(µ) is a hyper fuzzy filter. example 3.8. let l = {0,a,b,c,d, 1} and define ∧ and ∨ by the following cayley tables: ∧ 0 a b c d 1 0 0 0 0 0 0 0 a 0 a a a a a b 0 a b a b b c 0 a a c c c d 0 a b c d d 1 0 a b c d 1 ∨ 0 a b c d 1 0 {0} {a} {b} {c} {d} {1} a {a} {a} {b} {c} {d} {1} b {b} {b} {b} {d} {d} {1} c {c} {c} {d} {c} {d} {1} d {d} {d} {d} {d} {d} {1} 1 {1} {1} {1} {1} {1} {d, 1} it is easy to see that the operations ∧ and ∨ on l are well-defined and l is a hyperlattice. let θ be an equivalence relation on the lattice l with the following equivalence classes: [0]θ = {0,a,b}; [d]θ = {d}; [c]θ = {c}; [1]θ = {1}. it is clear that θ is not a hyper congruence relation on the lattice l. if µ = ( 0 a b c d 1 0.1 0.2 0.7 0.2 0.7 0.9 ) , 22 on rough sets and hyperlattices then µ is a hyper fuzzy filter and θ(µ) = ( 0 a b c d 1 0.7 0.7 0.7 0.2 0.7 0.9 ) . since 0 ≤ c and θ(µ)(0) = 0.7 6≤ 0.2 = θ(µ)(c), we conclude that θ(µ) is not a hyper fuzzy filter. example 3.9. let the hyperlattice l be as in example 3.8. let θ be an equivalence relation on the lattice l with the following equivalence classes: [0]θ = {0} and [1]θ = {a,b,c,d, 1}. it is clear that θ is a hyper congruence relation on the lattice l. lemma 3.10. let θ be a hyper congruence relation on l and x,y ∈ l.then 1. if a ∈ [x]θ and b ∈ [y]θ, then [α]θ = [β]θ for every α ∈ [x ∨ y]θ and β ∈ [a∨ b]θ. 2. if µ is a fuzzy subset of l, then ∨ a∈[x]θ,b∈[y]θ ∧ z∈a∨b µ(z) ≤ ∧ z∈x∨y θ(µ)(z). proof. (1) let α ∈ [x ∨ y]θ and β ∈ [a ∨ b]θ. then, by lemma 3.2, (α,β) ∈ (a∨ b) × (x∨y) ⊆ θ, it follows that [α]θ = [β]θ. (2) by statement (1), we have µ(z) ≤ ∨ d∈[z]θ µ(d) = ∨ d∈[z′]θ µ(d) for every z ∈ a ∨ b and z′ ∈ x ∨ y. hence µ(z) ≤ ∧ z ′∈x∨y ∨ d∈[z′]θ µ(d) for every z ∈ a ∨ b. therefore ∧ z∈a∨b µ(z) ≤ ∧ z ′∈x∨y ∨ d∈[z′]θ µ(d), which follows that∨ a∈[x]θ,b∈[y]θ ∧ z∈a∨b µ(z) ≤ ∧ z∈x∨y θ(µ)(z). lemma 3.11. let θ be a hyper congruence relation on l and x,y ∈ l. if µ is a hyper fuzzy ideal of l and x ≤ y, then θ(µ)(x) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b). proof. it is clear that {a∧ b : a ∈ [x]θ,b ∈ [y]θ}⊆ [x]θ. therefore,∨ a∈[x]θ,b∈[y]θ µ(a∧ b) ≤ ∨ a∈[x]θ µ(a). now, we suppose that a ∈ [x]θ. then a ∧ y ∈ [x]θ and since µ is a hyper fuzzy ideal of l, we conclude that µ(a) ≤ µ(a∧y). therefore θ(µ)(x) = ∨ a∈[x]θ µ(a) ≤ ∨ a∈[x]θ µ(a∧y) ≤ ∨ a∈[x]θ,b∈[y]θ µ(a∧ b). hence θ(µ)(x) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b). 23 a. a. estaji and f. bayati definition 3.12. let θ be an equivalence relation on l and µ ∈f(l). then, µ is called an upper rough fuzzy (prime) ideal if θ(µ) is a hyper fuzzy (prime) ideal of l. proposition 3.13. let θ be a hyper congruence relation on l. if µ is a hyper fuzzy ideal of l, then µ is an upper rough fuzzy ideal. proof. let x,y ∈ l. then θ(µ)(x) ∧θ(µ)(y) = ∨ a∈[x]θ µ(a) ∧ ∨ b∈[y]θ µ(b) = ∨ a∈[x]θ,b∈[y]θ µ(a) ∧µ(b) ≤ ∨ a∈[x]θ,b∈[y]θ ∧ z∈a∨b µ(z) µ is a hyper fuzzy ideal of l ≤ ∧ z∈x∨y θ(µ)(z) by lemma 3.10. now, we suppose that x,y ∈ l and x ≤ y. hence θ(µ)(x) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b) by lemma 3.11 ≥ ∨ a∈[x]θ,b∈[y]θ µ(b) µ is a hyper fuzzy ideal of l = θ(µ)(y). example 3.14. let the hyperlattice l and the equivalence relation θ on l be as in example 3.8. if µ = ( 0 a b c d 1 0.3 0.3 0.2 0.3 0.2 0.1 ) , then µ is a hyper fuzzy ideal and θ(µ) = ( 0 a b c d 1 0.3 0.3 0.3 0.3 0.2 0.1 ) . since ∧ x∈b∨c θ(µ)(x) = 0.2 6≥ 0.3 = θ(µ)(b) ∧θ(µ)(c), we conclude that θ(µ) is not a hyper fuzzy ideal. definition 3.15. let θ be a hyper congruence relation on l. then θ is called ∨complete if [a∨b]θ = [a]θ∨[b]θ for all a,b ∈ l. likewise, θ is called ∧−complete if [a∧ b]θ = [a]θ ∧ [b]θ for all a,b ∈ l. a hyper congruence relation on l which is both ∨-complete and ∧−complete is called complete. proposition 3.16. let θ be a ∧−complete on l. if µ ∈ f(l) is a hyper fuzzy prime ideal of l such that θ(µ) is a proper fuzzy subset of l, then µ is an upper rough fuzzy prime ideal. 24 on rough sets and hyperlattices proof. if x,y ∈ l, then θ(µ)(x∧y) = ∨ a∈[x∧y]θ µ(a) = ∨ a∈[x]θ,b∈[y]θ µ(a∧ b) θ is ∧−complete ≤ ∨ a∈[x]θ,b∈[y]θ µ(a) ∨µ(b) µ is a hyper fuzzy prime ideal = ∨ a∈[x]θ µ(a) ∨ ∨ b∈[y]θ µ(b) = θ(µ)(x) ∨θ(µ)(y). now, by proposition 3.13, the proof is complete. example 3.17. let the lattice l be as in example 3.8. let θ be a hyper congruence relation on the lattice l with the following equivalence classes: [0]θ = {0,a}; [b]θ = {b}; [c]θ = {c}; [1]θ = {1,d}. since [b∧ c]θ = [a]θ = {0,a} 6= {a} = [b]θ ∧ [c]θ we conclude that θ is not ∧− complete. if µ = ( 0 a b c d 1 0.9 0.8 0.8 0.7 0.7 0.2 ) , then µ is a hyper fuzzy prime ideal and θ(µ) = ( 0 a b c d 1 0.9 0.9 0.8 0.7 0.7 0.7 ) . also, the ideal θ(µ) is not hyper fuzzy prime, because θ(µ)(b∧ c) = 0.9 � 0.8 = θ(µ)(b) ∨θ(µ)(c). definition 3.18. let µ be a fuzzy subset of l. the least hyper fuzzy ideal of l containing µ is called a hyper fuzzy ideal of l induced by µ and is denoted by < µ >. by remark 2.6 in [12], if µ is a fuzzy subset of l, then there exits < µ >. definition 3.19. for every µ ∈f(l), we define µ∗(x) = sup{α ∈ [0, 1] : x ∈ i(µα)} for all x ∈ l. with the following example, we prove that theorem 2.15 in [12] is incorrect. 25 a. a. estaji and f. bayati example 3.20. let the hyperlattice l be as in example 3.8. if µ = ( 0 a b c d 1 0.5 0.8 0.4 0.5 0.7 0.6 ) , then µ ∈f(l) and µ∗ = ( 0 a b c d 1 1 0.8 0.7 0.7 0.7 0.6 ) . if ν = ( 0 a b c d 1 0.9 0.8 0.7 0.7 0.7 0.6 ) , then ν is the hyper fuzzy ideal of l, µ ≤ ν and µ∗ 6≤ ν. therefore, µ∗ is not the hyper fuzzy ideal induced by µ. hence, theorem 2.15 in [12] is incorrect. also, if λn = 0.7 − 1n, for every n ∈ n, then b ∈ ⋂ n∈n i(µλn) =↓ d, but b 6∈ µλn for every n ≥ 10. hence the last paragraph of the proof of theorem 2.15 in [12] is incorrect. now we give the correct version of theorem 2.15 in [12]. proposition 3.21. let µ be a fuzzy subset of l. then the fuzzy subset µ∗ of l is the hyper fuzzy ideal of l and 1. µ ≤ µ∗. 2. µ∗ = ∧ {λ ∈fi(l)|µ ≤ λ and λ(0) = 1}. proof. for λ ∈ im(µ∗), let λn = λ − 1n, for n ∈ n, and let x ∈ µ ∗ λ. then µ∗(x) ≥ λ, which implies that µ∗(x) > λn. hence there exists β ∈ {α ∈ [0, 1]|x ∈ i(µα)} such that β > λn. thus µβ ⊆ µλn and so x ∈ i(µβ) ⊆ i(µλn) for all n ∈ n. therefore, x ∈ ⋂ n∈n i(µλn). conversely, if x ∈ ⋂ n∈n i(µλn), λn ∈ {α ∈ [0, 1] : x ∈ i(µα)}, for n ∈ n. therefore, λn = λ − 1n ≤ ∨ {α ∈ [0, 1] : x ∈ i(µα)} = µ∗(x). hence µ∗(x) ≥ λ, so that x ∈ µ∗λ. then we have µ∗λ = ⋂ n∈n i(µλn) which is an ideal of l. for x ∈ l, let β ∈ {α ∈ [0, 1] : x ∈ µα}. then x ∈ µβ, and so x ∈ i(µβ). thus β ∈{α ∈ [0, 1] : x ∈ i(µα)}, which implies that µ(x) = ∨ {α ∈ [0, 1] : x ∈ µα}≤ ∨ {α ∈ [0, 1] : x ∈ i(µα)} = µ∗(x). therefore, µ ≤ µ∗ (see [11, 12]). now, let ν be a hyper fuzzy ideal of l containing µ such that ν(0) = 1. then for every α ∈ [0, 1], since να 6= ∅, we conclude that i(µα) ≤ i(να) = να. hence µ∗(x) = ∨ {α ∈ [0, 1] : x ∈ i(µα)}≤ ∨ {α ∈ [0, 1] : x ∈ να} = ν(x) 26 on rough sets and hyperlattices for every x ∈ l. also, for every α ∈ [0, 1], 0 ∈ i(µα) and we infer that µ∗(0) =∨ {α ∈ [0, 1] : 0 ∈ i(µα)} = 1. therefore, µ∗ ∈{λ ∈fi(l)|µ ≤ λ and λ(0) = 1}. finally, we have µ∗ = ∧ {λ ∈fi(l)|µ ≤ λ and λ(0) = 1}. proposition 3.22. let θ be a hyper congruence relation on l and µ ∈ f(l). then θ(< µ >) = θ(< θ(µ) >) and θ(µ∗) = θ((θ(µ))∗). proof. since µ ≤< µ >≤ µ∗, we conclude from proposition 2.8 that θ(µ) ≤ θ(< µ >) ≤ θ(µ∗). it is clear that θ(µ∗)(0) = 1 and by propositions 3.13 and 3.21, we have < θ(µ) >≤ θ(< µ >) and (θ(µ))∗ ≤ θ(µ∗). again, by proposition 2.8, θ(< θ(µ) >) ≤ θ(< µ >) and θ((θ(µ))∗) ≤ θ(µ∗). since µ ≤ θ(µ), we conclude that < µ >≤< θ(µ) > and µ∗ ≤ (θ(µ))∗ and by proposition 2.8, θ(< µ >) ≤ θ(< θ(µ) >) and θ(µ∗) ≤ θ((θ(µ))∗). finally, we have θ(< µ >) = θ(< θ(µ) >) and θ(µ∗) = θ((θ(µ))∗). by the following example, we prove that the condition for an equivalence relation on l does not imply θ((θ(µ))∗) = θ(µ∗). example 3.23. let the hyperlattice l and the equivalence relation θ on l be as in example 3.8. if µ = ( 0 a b c d 1 0.5 0.8 0.4 0.7 0.5 0.6 ) , then θ(µ∗) = ( 0 a b c d 1 1 1 1 0.7 0.6 0.6 ) , 27 a. a. estaji and f. bayati and θ((θ(µ))∗) = ( 0 a b c d 1 1 1 1 0.7 0.7 0.6 ) . hence θ(µ∗) 6= θ((θ(µ))∗). therefore, in general θ(µ∗) = θ((θ(µ))∗) doesn’t hold. 4 lower approximations of a fuzzy subset in this section we give some important properties of θ with many examples. lemma 4.1. let θ be a hyper congruence relation on l and x,y ∈ l. if µ is a hyper fuzzy filter of l and x ≤ y, then θ(µ)(x) = ∧ a∈[x]θ,b∈[y]θ µ(a∧ b). proof. since x ≤ y, we can then conclude from lemma 3.2 that {µ(a∧ b) : a ∈ [x]θ,b ∈ [y]θ}⊆{µ(z) : z ∈ [x]θ}. hence θ(µ)(x) ≤ ∧ a∈[x]θ,b∈[y]θ µ(a∧b). also, since µ is a hyper fuzzy filter of l, we conclude that µ(a ∧ b) ≤ µ(a) for every a ∈ [x]θ and b ∈ [y]θ, which follows that θ(µ)(x) ≥ ∧ a∈[x]θ,b∈[y]θ µ(a∧b) and the proof is now complete. definition 4.2. let θ be an equivalence relation on l and µ ∈ f(l). then, µ is called a lower rough fuzzy (prime) ideal if θ(µ) is a hyper fuzzy (prime) ideal of l. proposition 4.3. let θ be a ∨-complete congruence relation on l. if µ ∈ f(l) is a hyper fuzzy ideal, then µ is a lower rough fuzzy ideal. proof. let x,y ∈ l. since ∧ t∈x∨y µ(t) ∈ { ∧ t∈a∨b µ(t) : a ∈ [x]θ,b ∈ [y]θ}, we conclude that θ(µ)(x) ∧θ(µ)(y) = ∧ a∈[x]θ µ(a) ∧ ∧ b∈[y]θ µ(b) = ∧ a∈[x]θ,b∈[y]θ µ(a) ∧µ(b) ≤ ∧ a∈[x]θ,b∈[y]θ ∧ t∈a∨b µ(t) µ is a hyper fuzzy ideal = ∧ t∈[x]θ∨[y]θ µ(t) = ∧ t∈[x∨y]θ µ(t) θ is ∨-complete = ∧ z∈x∨y ∧ t∈[z]θ µ(t) = ∧ z∈x∨y θ(µ)(z). let x,y ∈ l and x ≤ y. hence θ(µ)(x) = ∧ a∈[x]θ,b∈[y]θ µ(a∧ b) by lemma 4.1 ≥ ∧ b∈[y]θ µ(b) µ is a hyper fuzzy ideal = θ(µ)(y). 28 on rough sets and hyperlattices example 4.4. let the hyperlattice l and the hyper congruence relation θ on l be as in example 3.17. let µ = ( 0 a b c d 1 1 0.8 0.6 0.4 0.4 0 ) . it is clear that µ is a hyper fuzzy ideal on l and θ(µ) = ( 0 a b c d 1 0.8 0.8 0.6 0.4 0 0 ) . it is easy to see that θ is not ∨ complete, because [b∨ c]θ = {1.d} 6= {d} = [b]θ ∨ [c]θ. also, since θ(µ)(b) ∧θ(µ)(c) = 0.4 6≤ 0 = ∧ d∈b∨c θ(µ)(d) we conclude that θ(µ) is not a hyper fuzzy ideal. definition 4.5. let θ be an equivalence relation on l and µ ∈ f(l). then, µ is called a lower rough fuzzy (prime) filter if θ(µ) is a hyper fuzzy (prime) filter of l. proposition 4.6. let θ be a ∧−complete congruence relation on l. if µ ∈f(l) is a hyper fuzzy filter, then µ is a lower rough fuzzy filter. proof. let x,y ∈ l. θ(µ)(x∧y) = ∧ a∈[x∧y]θ µ(a) = ∧ a∈[x]θ,b∈[y]θ µ(a∧ b) θ is ∧−complete ≥ ∧ a∈[x]θ,b∈[y]θ µ(a) ∧µ(b) µ is a hyper fuzzy filter = ∧ a∈[x]θ µ(a) ∧ ∧ b∈[y]θ µ(b) = θ(µ)(x) ∧θ(µ)(y). let x,y ∈ l and x ≤ y. hence θ(µ)(x) = ∧ a∈[x]θ,b∈[y]θ µ(a∧ b) by lemma 4.1 ≤ ∧ b∈[y]θ µ(b) µ is a hyper fuzzy filter = θ(µ)(y). 29 a. a. estaji and f. bayati example 4.7. let the hyperlattice l and the hyper congruence relation θ on l be as in example 3.17. if µ = ( 0 a b c d 1 0.1 0.6 0.6 0.7 0.7 0.9 ) , then µ is a hyper fuzzy filter and θ(µ) = ( 0 a b c d 1 0.1 0.1 0.6 0.7 0.7 0.7 ) . since [b∧ c]θ = {0,a} 6= {a} = [b]θ ∧ [c]θ, we conclude that θ is not ∧− complete. also θ(ν) is not a hyper fuzzy filter, because θ(ν)(b∧ c) = θ(ν)(a) = 0.1 6≥ 0.6 = θ(ν)(b) ∧θ(ν)(c). proposition 4.8. let θ be a ∨-complete on l. if µ ∈f(l) is a hyper fuzzy prime ideal such that θ(µ) is a proper fuzzy subset of l, then µ is a lower rough fuzzy prime ideal. proof. let x,y ∈ l. θ(µ)(x∧y) = ∧ z∈[x∧y]θ µ(z) ≤ ∧ a∈[x]θ,b∈[y]θ µ(a∧ b) by lemma 3.2 ≤ ∧ a∈[x]θ,b∈[y]θ µ(a) ∨µ(b) µ is a hyper fuzzy prime ideal = ∧ a∈[x]θ µ(a) ∨ ∧ b∈[y]θ µ(b) = θ(µ)(x) ∨θ(µ)(y). by proposition 4.3, the proof is now complete. example 4.9. let the lattice l and the hyper congruence relation θ on l be as in example 3.17. if µ = ( 0 a b c d 1 0.9 0.8 0.8 0.7 0.7 0.2 ) , then µ is a hyper fuzzy prime ideal and θ(µ) = ( 0 a b c d 1 0.8 0.8 0.8 0.7 0.2 0.2 ) . since [b∨ c]θ = {1.d} 6= {d} = [b]θ ∨ [c]θ, we conclude that θ is not ∨ complete. also θ(µ) is not hyper fuzzy ideal, because θ(µ)(b) ∧θ(µ)(c) = 0.7 6≤ 0.2 = ∧ d∈(b∨c) θ(µ)(d). 30 on rough sets and hyperlattices proposition 4.10. let θ be a complete congruence relation on l. if µ ∈ f(l) is a hyper fuzzy prime filter such that θ(µ) is a proper fuzzy subset of l, then µ is a lower rough fuzzy prime filter. proof. let x,y ∈ l. θ(µ)(x) ∨θ(µ)(y) = ∧ a∈[x]θ µ(a) ∨ ∧ b∈[y]θ µ(b) = ∧ a∈[x]θ,b∈[y]θ µ(a) ∨µ(b) ≥ ∧ a∈[x]θ,b∈[y]θ ∧ t∈a∨b µ(t) µ is a hyper fuzzy prime filter = ∧ z∈x∨y ∧ t∈[z]θ µ(t) θ is ∨-complete = ∧ z∈x∨y θ(µ)(z). by proposition 4.6, the proof is now complete. example 4.11. let the hyperlattice l and the hyper congruence relation θ on l be as in example 3.17. it is clear that µ = ( 0 a b c d 1 0.1 0.3 0.7 0.3 0.7 0.9 ) is a hyper fuzzy prime filter and θ(µ) = ( 0 a b c d 1 0.1 0.1 0.7 0.3 0.7 0.7 ) . since θ(µ)(b) ∧θ(µ)(c) = 0.3 6≤ 0.1 = θ(µ)(b∧ c), we conclude that θ(µ) is not a hyper fuzzy ideal. also, as we have seen in example 4.4, θ is not ∨ complete. 5 conclusion rough set, fuzzy set and hyperlattice are different aspects of set theory. combining the three theories, one gets the rough concept fuzzy hyperlattice of a given context. we introduced the concepts of upper and lower rough hyper fuzzy ideals (filters) in a hyperlattice and its basic properties have been discussed. also, we discussed the relations between hyper fuzzy (prime) ideal and hyper fuzzy (prime) filter with their upper and lower approximations, respectively. in addition, by an example we show that theorem 2.15 in [12] is incorrect (see example 3.20) and a corrected version is considered, proposition 3.21. 31 a. a. estaji and f. bayati acknowledgement we express our gratitude to professor m. mehdi ebrahimi. references [1] s. m. 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[22] l. a. zadeh, fuzzy sets, inform. and control, 8 (1965) 338-353. 33 ratio mathematica volume 38, 2020, pp. 237-259 some characterizations of fuzzy comultisets and quotient fuzzy multigroups paul augustine ejegwa∗ abstract the idea of fuzzy multisets has been applied to some group theoretic notions. nonetheless, the notions of cosets and quotient groups have not been substantiated in fuzzy multigroup environment. the aim of this paper is to present the concepts of cosets and quotient groups in fuzzy multigroup context with some related results. to start with, the connection between fuzzy comultisets of fuzzy multigroups and the cosets of groups is established. some characterizations of fuzzy comultisets are outlined with proofs. in addition, quotient fuzzy multigroup is proposed and some of its properties are explored. it is proven that a normal fuzzy submultigroup, h̃ of a fuzzy multigroup, g̃ is commutative if and only if the quotient fuzzy multigroup, g̃ h̃ of g̃ by h̃ is commutative. finally, group theoretic isomorphism theorems are established in fuzzy multigroup setting. keywords: fuzzy multiset; fuzzy multigroup; fuzzy comultiset; quotient/factor fuzzy multigroup. 2010 ams subject classifications: 03e72, 08a72, 20n25. 1 ∗department of mathematics/statistics/computer science, university of agriculture, p.m.b. 2373, makurdi, nigeria; ocholohi@gmail.com; ejegwa.augustine@uam.edu.ng 1received on february 11th, 2020. accepted on april 26th, 2020. published on june 30th, 2020. doi: 10.23755/rm.v38i0.501. issn: 1592-7415. eissn: 2282-8214. c©p. a. ejegwa this paper is published under the cc-by licence agreement. 237 paul augustine ejegwa 1 introduction fuzzy set theory proposed in 1965 by zadeh (1965), although with vehement opposition as at then, has been extensively researched with applicative expressions ranging from engineering and computer science to medical diagnosis and social behavior, etc. in a way of extending the application of fuzzy sets to group theory, rosenfeld (1971) proposed the notion of fuzzy groups as an extension of group theory and some number of results were obtained. several studies have been carried out on some group theoretic notions in fuzzy group setting (see ajmal and prajapati, 1992; bhattacharya and mukherjee, 1987; ejegwa and otuwe, 2019; mukherjee and bhattacharya, 1986, 1984; onasanya and ilori, 2013). with the interest derived from fuzzy sets and multisets (see blizard, 1989), the idea of fuzzy multisets or fuzzy bags was proposed in (yager, 1986) as a generalization of fuzzy sets in multiset framework. myriad of works have been carried out on the fundamentals and properties of fuzzy multisets (see biswas, 1999; ejegwa, 2014, 2019a; miyamoto, 1996; miyamoto and mizutani, 2004; onasanya and sholabomi, 2019). in recent times, the concept of fuzzy multigroups was introduced as an application of fuzzy multisets to group theory (shinoj et al., 2015). the ideas of abelian fuzzy multigroups and order of fuzzy multigroups have been studied with some results (baby et al., 2015; ejegwa, 2018b), and the notions of center and centralizer in fuzzy multigroup context were established (ejegwa, 2018b). in the same vein, the concept of fuzzy multigroupoids was introduced and the idea of fuzzy submultigroups was explored with a number of results (ejegwa, 2018d). the concepts of normal subgroups, characteristic subgroups and frattini subgroups have been established in fuzzy multigroup setting with some results (ejegwa, 2018a; ejegwa et al., 2020; ejegwa, 2020c). in (ejegwa, 2018c), the idea of homomorphism in the environment of fuzzy multigroups was defined and some homomorphic properties of fuzzy multigroups were elaborated. subsequently, the idea of direct product of fuzzy multigroups was proposed and a number of results were established (ejegwa, 2019b). the idea of alpha-cuts of fuzzy multigroups and its homomorphic properties have been studied (ejegwa, 2020b,a). the concept of fuzzy multigroups was redefined by rasuli (2020) as an extension of the work in (anthony and sherwood, 1979). the present paper is a further study of fuzzy multigroups in group theoretic analogs. motivated by the researches done in fuzzy multigroups so far, it is expedient to investigate the notions of cosets and quotient groups in the light of fuzzy multigroups to strengthen the theory of fuzzy multigroups. establishing the ideas of fuzzy comultisets and quotient/factor multigroups shall enhance the plausibility of studying nilpotency and solvability in fuzzy multigroup setting. actually, this work is an application of fuzzy multisets to cosets and factor groups. in so doing, this paper assay to introduce fuzzy comultisets and quotient fuzzy multi238 fuzzy comultisets and quotient fuzzy multigroups groups with some analog results. the relationship between fuzzy comultisets of fuzzy multigroups and that of cosets of groups is examined, and the isomorphism theorems are duly established. by organization, the paper is thus presented: section 2 provides some preliminaries on fuzzy multisets and fuzzy multigroups. in section 3, the idea of fuzzy comultisets is proposed and some of its properties are discussed. section 4 discusses the concept of quotient or factor fuzzy multigroups with some results. finally, section 5 concludes the paper and provides direction for future studies. 2 preliminaries in this section, we present some existing definitions and results to be used in the sequel. 2.1 fuzzy multisets definition 2.1. (yager, 1986) assume x is a set of elements. then, a fuzzy bag/multiset, g̃ drawn from x can be characterized by a count membership function, cmg̃ such that cmg̃ : x → q, where q is the set of all crisp bags or multisets from the unit interval, i = [0,1]. a fuzzy multiset, g̃ can be characterized by a function cmg̃ : x → n i or cmg̃ : x → [0,1] → n, where i = [0,1] and n = n∪{0}. by miyamoto and mizutani (2004), it implies that cmg̃(x) for x ∈ x is given as cmg̃(x) = {µ 1 g̃ (x),µ2 g̃ (x), ...,µn g̃ (x), ...}, where µ1 g̃ (x),µ2 g̃ (x), ...,µn g̃ (x), ... ∈ [0,1] such that µ1 g̃ (x) ≥ µ2 g̃ (x) ≥ ... ≥ µn g̃ (x) ≥ ..., whereas in a finite case, we write cmg̃(x) = {µ 1 g̃ (x),µ2 g̃ (x), ...,µn g̃ (x)}, for µ1 g̃ (x) ≥ µ2 g̃ (x) ≥ ... ≥ µn g̃ (x). a fuzzy multiset, g̃ can be represented in the form g̃ = { 〈cmg̃(x)〉 x | x ∈ x}. we denote the set of all fuzzy multisets by fms(x). 239 paul augustine ejegwa definition 2.2. (yager, 1986) let g̃,h̃ ∈ fms(x). then, h̃ is called a fuzzy submultiset of g̃ written as h̃ ⊆ g̃ if cmh̃(x) ≤ cmg̃(x), ∀x ∈ x. also, if h̃ ⊆ g̃ and h̃ 6= g̃, then h̃ is called a proper fuzzy submultiset of g̃ and denoted as h̃ ⊂ g̃. definition 2.3. (yager, 1986) let g̃,h̃ ∈ fms(x). then, g̃ and h̃ are comparable to each other if and only if h̃ ⊆ g̃ or g̃ ⊆ h̃, and g̃ = h̃ if and only if cmg̃(x) = cmh̃(x), ∀x ∈ x. definition 2.4. (miyamoto, 1996) let g̃,h̃ ∈ fms(x). then, the intersection and union of g̃ and h̃, denoted by g̃∩h̃ and g̃∪h̃ are defined by the rules that for any object x ∈ x, (i) cmg̃∩h̃(x) = cmg̃(x)∧cmh̃(x), (ii) cmg̃∪h̃(x) = cmg̃(x)∨cmh̃(x), where ∧ and ∨ denote minimum and maximum operations, respectively. before finding the intersection and union of g̃ and h̃, the membership sequences of g̃ and h̃ should be equal. if not, it could be completed by affixing zero(s). 2.2 fuzzy multigroups we denote group by x and assume that all fuzzy multigroups are drawn from fmg(x), which is the set of all fuzzy multigroups of x. definition 2.5. (shinoj et al., 2015) a fuzzy multiset g̃ of x is said to be a fuzzy multigroup of x if it satisfies the following two conditions: (i) cmg̃(xy) ≥ cmg̃(x)∧cmg̃(y), ∀x,y ∈ x, (ii) cmg̃(x −1) = cmg̃(x), ∀x ∈ x. it can be easily verified that if g̃ is a fuzzy multigroup of x, then cmg̃(e) = ∨ x∈x cmg̃(x), that is, cmg̃(e) is the tip of g̃. remark 2.1. (shinoj et al., 2015) we notice that a fuzzy multiset, g̃ of a group x is a fuzzy multigroup if ∀x,y ∈ x, cmg̃(xy −1) ≥ cmg̃(x)∧cmg̃(y) holds. 240 fuzzy comultisets and quotient fuzzy multigroups definition 2.6. (shinoj et al., 2015) let g̃ be a fuzzy multigroup of a group x. then g̃−1 is defined by cmg̃−1(x) = cmg̃(x −1), ∀x ∈ x. by definition 2.5, we get cmg̃−1(x) = cmg̃(x −1) = cmg̃(x). that is, g̃−1 = g̃. thus, g̃ ∈ fmg(x) ⇔ g̃−1 ∈ fmg(x). proposition 2.1. (shinoj et al., 2015) let g̃,h̃ ∈ fmg(x). then, g̃ ∩ h̃ ∈ fmg(x). definition 2.7. (ejegwa, 2018d) let {g̃i}i∈i,i = 1, ...,n be an arbitrary family of fuzzy multigroups of x. then, cm⋂ i∈i g̃i (x) = ∧ i∈i cmg̃i(x), ∀x ∈ x and cm⋃ i∈i g̃i (x) = ∨ i∈i cmg̃i(x), ∀x ∈ x. the family of fuzzy multigroups {g̃i}i∈i of x is said to have inf or sup assuming chain if either g̃1 ⊆ g̃2 ⊆ ... ⊆ g̃n or g̃1 ⊇ g̃2 ⊇ ... ⊇ g̃n, respectively. definition 2.8. (baby et al., 2015) let g̃ ∈ fmg(x). then, g̃ is said to be commutative if for all x,y ∈ x, cmg̃(xy) = cmg̃(yx). definition 2.9. (ejegwa, 2018a) let g̃,h̃ ∈ fmg(x). then, the product, g̃◦h̃ of g̃ and h̃ is defined to be a fuzzy multiset of x as follows: cmg̃◦h̃(x) = { ∨ x=yz[cmg̃(y)∧cmh̃(z)], if ∃y,z ∈ x such that x = yz 0, otherwise. definition 2.10. (ejegwa, 2018d) let g̃ ∈ fmg(x). a fuzzy submultiset, h̃ of g̃ is called a fuzzy submultigroup of g̃ denoted by h̃ ⊆ g̃ if h̃ is a fuzzy multigroup. a fuzzy submultigroup, h̃ of g̃ is a proper fuzzy submultigroup denoted by h̃ ⊂ g̃, if h̃ ⊆ g̃ and h̃ 6= g̃. definition 2.11. (ejegwa, 2018a) let h̃,g̃ ∈ fmg(x) such that h̃ ⊆ g̃. then, h̃ is called a normal fuzzy submultigroup of g̃ if cmh̃(xyx −1) = cmh̃(y), ∀x,y ∈ x. definition 2.12. (ejegwa, 2018a) let h̃ be a fuzzy submultiset of g̃ ∈ fmg(x). then, the normalizer of h̃ in g̃ is the set given by n(h̃) = {g ∈ x | cmh̃(gy) = cmh̃(yg), ∀y ∈ x}. 241 paul augustine ejegwa theorem 2.1. (ejegwa, 2018a) let x be a finite group and h̃ be a fuzzy submultigroup of g̃ ∈ fmg(x). define h = {g ∈ x | cmh̃(g) = cmh̃(e)}, k = {x ∈ x | cmh̃x(y) = cmh̃e(y)}, where e denotes the identity element of x. then h and k are subgroups of x. again, h = k. definition 2.13. (ejegwa, 2018d; shinoj et al., 2015) let g̃ ∈ fmg(x). then, the set g̃∗ defined by g̃∗ = {x ∈ x | cmg̃(x) > 0} is the level set or support of g̃. it follows that g̃∗ is a subgroup of x. also, the set g̃∗ defined by g̃∗ = {x ∈ x | cmg̃(x) = cmg̃(e)} is a subgroup of x. definition 2.14. (ejegwa, 2018c) let x and y be groups and let f : x → y be a homomorphism. suppose g̃ and h̃ are fuzzy multigroups of x and y , respectively. then, f induces a homomorphism from g̃ to h̃ which satisfies (i) cmg̃(f −1(y1y2)) ≥ cmg̃(f −1(y1))∧cmg̃(f −1(y2)), ∀y1,y2 ∈ y , (ii) cmh̃(f(x1x2)) ≥ cmh̃(f(x1))∧cmh̃(f(x2)), ∀x1,x2 ∈ x, where (i) the image of g̃ under f, denoted by f(g̃), is a fuzzy multiset over y defined by cmf(g̃)(y) = { ∨ x∈f−1(y) cmg̃(x), f −1(y) 6= ∅ 0, otherwise for each y ∈ y . (ii) the inverse image of h̃ under f, denoted by f−1(h̃), is a fuzzy multiset over x defined by cmf−1(h̃)(x) = cmh̃(f(x)), ∀x ∈ x. proposition 2.2. (ejegwa, 2018c) let f : x → y be a homomorphism and g̃ ∈ fmg(x). if f is injective, then f−1(f(g̃)) = g̃. theorem 2.2. (ejegwa, 2018c) let x and y be groups and f : x → y be an isomorphism. then the following statements hold. (i) g̃ ∈ fmg(x) if and only if f(g̃) ∈ fmg(y ). (ii) h̃ ∈ fmg(y ) if and only if f−1(h̃) ∈ fmg(x). 242 fuzzy comultisets and quotient fuzzy multigroups 3 fuzzy comultiset and some of its properties in this section, we define fuzzy comultiset and characterize some of its properties. definition 3.1. suppose h̃ is a fuzzy submultigroup of a fuzzy multigroup g̃ of x. then, the fuzzy submultiset, yh̃ of g̃ for y ∈ x defined by cmyh̃(x) = cmh̃(y −1x), ∀x ∈ x is called the left fuzzy comultiset of h̃. similarly, the fuzzy submultiset, h̃y of g̃ for y ∈ x defined by cmh̃y(x) = cmh̃(xy −1), ∀x ∈ x is called the right fuzzy comultiset of h̃. the following result proves that the right and left fuzzy comultisets of a fuzzy submultigroup in a fuzzy multigroup are equal. proposition 3.1. if h̃ is a fuzzy submultigroup of g̃ ∈ fmg(x), then the right and left fuzzy comultisets of h̃ in g̃ are identical. proof. let x,y,∈ x. assume h̃ is a fuzzy submultigroup of g̃. then, we have cmyh̃(x) = cmh̃(y −1x) ≥ cmh̃(y)∧cmh̃(x) = cmh̃(x)∧cmh̃(y) = cmh̃(x)∧cmh̃(y −1). suppose by hypothesis, cmh̃(x)∧cmh̃(y −1) = cmh̃(xy −1). then, we have cmyh̃(x) ≥ cmh̃y(x). again, cmh̃y(x) = cmh̃(xy −1) ≥ cmh̃(x)∧cmh̃(y) = cmh̃(y)∧cmh̃(x) = cmh̃(y −1)∧cmh̃(x). by the same hypothesis, we get cmh̃y(x) ≥ cmyh̃(x). hence, cmyh̃(x) = cmh̃y(x) ⇒ yh̃ = h̃y. 243 paul augustine ejegwa remark 3.1. let h̃ be a fuzzy submultigroup of g̃ ∈ fmg(x). we notice that (i) the right and left fuzzy comultisets of h̃ in g̃ are fuzzy submultigroups of g̃. (ii) xh̃ = yh̃ = zh̃ = h̃, ∀x,y,z ∈ x. this is not applicable in the conventional case. (iii) there is a one-to-one correspondence between the set of right fuzzy comultisets and the set of left fuzzy comultisets of h̃ in g̃. (iv) the number of fuzzy comultisets of h̃ in g̃ equals the cardinality of h̃∗. (v) xh̃ ∩yh̃ ∩zh̃ = h̃ = xh̃ ∪yh̃ ∪zh̃, ∀x,y,z ∈ x. theorem 3.1. let h̃ be a fuzzy submultigroup of g̃ ∈ fmg(x). then, gh̃ = hh̃ for g,h ∈ x if and only if cmh̃(g −1h) = cmh̃(h −1g) = cmh̃(e). proof. let gh̃ = hh̃. then, cmgh̃(g) = cmhh̃(g) and cmgh̃(h) = cmhh̃(h) ∀g,h ∈ x. hence, cmh̃(g −1h) = cmh̃(h −1g) = cmh̃(e). conversely, let cmh̃(g −1h) = cmh̃(h −1g) ∀g,h ∈ x. for every x ∈ x, we have cmgh̃(x) = cmh̃(g −1x) = cmh̃(g −1hh−1x) ≥ cmh̃(g −1h)∧cmh̃(h −1x) = cmh̃(h −1x) = cmhh̃(x). similarly, cmhh̃(x) = cmh̃(h −1x) = cmh̃(h −1gg−1x) ≥ cmh̃(h −1g)∧cmh̃(g −1x) = cmh̃(g −1x) = cmgh̃(x). hence, cmgh̃(x) = cmhh̃(x) ⇒ gh̃ = hh̃. 244 fuzzy comultisets and quotient fuzzy multigroups corolary 3.1. let h̃ be a fuzzy submultigroup of g̃ ∈ fmg(x). then h̃g = h̃h for g,h ∈ x if and only if cmh̃(gh −1) = cmh̃(hg −1) = cmh̃(e). proof. straightforward from theorem 3.1. proposition 3.2. let h̃,g̃ ∈ fmg(x) such that h̃ ⊆ g̃. if gh̃ = hh̃, then cmh̃(g) = cmh̃(h), ∀g,h ∈ x. proof. let g,h ∈ x. suppose gh̃ = hh̃, then we have cmgh̃(g) = cmhh̃(g) ⇒ cmh̃(g −1g) = cmh̃(h −1g) ⇒ cmh̃(e) = cmh̃(h −1g) ∀g,h ∈ x. the fact that, cmh̃(e) = cmh̃(h −1g) ⇒ cmh̃(h) = cmh̃(g), the result follows. alternatively, suppose z ∈ x, we get cmgh̃(z) = cmhh̃(z) ⇒ cmh̃(g −1z) = cmh̃(h −1z) ⇒ cmh̃z−1(g) = cmh̃z−1(h) ⇒ cmh̃(g) = cmh̃(h), because h̃z−1 = h̃. theorem 3.2. let g̃ ∈ fmg(x). any fuzzy submultigroup h̃ of g̃ and for any z ∈ x, the fuzzy submultiset, zh̃z−1, where cmzh̃z−1(x) = cmh̃(z −1xz) for each x ∈ x is a fuzzy submultigroup of g̃. proof. let x,y ∈ x and h̃ ⊆ g̃. we prove that zh̃z−1 is a fuzzy submultigroup of g̃. now cmzh̃z−1(xy −1) = cmh̃(z −1xy−1z) = cmh̃(z −1xzz−1y−1z) ≥ cmh̃(z −1xz)∧cmh̃(z −1y−1z) = cmzh̃z−1(x)∧cmzh̃z−1(y −1) = cmzh̃z−1(x)∧cmzh̃z−1(y), ∀z ∈ x. hence, zh̃z−1 is a fuzzy submultigroup of g̃. 245 paul augustine ejegwa corolary 3.2. let {h̃i}i∈i ∈ fmg(x), then (i) ⋂ i∈i zh̃iz −1 ∈ fmg(x), ∀z ∈ x, (ii) ⋃ i∈i zh̃iz −1 ∈ fmg(x), ∀z ∈ x provided {h̃i}i∈i have sup/inf assuming chain. proof. the results follow from theorem 3.2. the following results are the application of product of fuzzy multigroups to the idea of fuzzy comultisets. proposition 3.3. suppose h̃ is a fuzzy submultigroup of h̃ ∈ fmg(x). then; (i) h̃g ◦ h̃g = h̃g. (ii) h̃g ◦ h̃h = h̃h◦ h̃g. (iii) (h̃g ◦ h̃h)−1 = (h̃h)−1 ◦ (h̃g)−1. (iv) (h̃g ◦ h̃h)−1 = h̃g ◦ h̃h. proof. using definition 2.9, the results follow. remark 3.2. proposition 3.3 also holds for left fuzzy comultisets. proposition 3.4. if h̃ is a fuzzy submultigroup of a commutative fuzzy multigroup g̃ of x, then (i) h̃y ◦ h̃z = h̃yz, ∀y,z ∈ x, (ii) yh̃ ◦zh̃ = yzh̃, ∀y,z ∈ x. proof. let h̃ ∈ fmg(x) and x,y,z ∈ x, then we have (i) cmh̃y◦h̃z(x) = ∨ x=zy [cmh̃y(z)∧cmh̃z(y)], ∀y,z ∈ x = ∨ x=zy [cmh̃(zy −1)∧cmh̃(yz −1)], ∀y,z ∈ x = ∨ x=zy [cmh̃∩h̃((zy −1)(yz−1))], ∀y,z ∈ x = cmh̃(xz −1y−1) = cmh̃yz(x). hence, h̃y ◦ h̃z = h̃yz. 246 fuzzy comultisets and quotient fuzzy multigroups (ii) similar to (i). corolary 3.3. suppose h̃ is a fuzzy submultigroup of a commutative fuzzy multigroup g̃ of x. then, the following statements are equivalent. (i) (h̃y ◦ h̃z)−1 = h̃y ◦ h̃z. (ii) h̃y ◦ h̃z = h̃yz. proof. the result is easy to see by combining definition 2.9 and proposition 3.4. remark 3.3. if (h̃y ◦ h̃y)−1 = h̃y ◦ h̃z and h̃y ◦ h̃z = h̃yz, then (h̃y ◦ h̃z)−1 = h̃yz. theorem 3.3. if h̃ is a fuzzy submultigroup of g̃ ∈ fmg(x) such that g̃ is commutative, then h̃g ◦ h̃h = h̃gh if and only if gh̃ ◦hh̃ = ghh̃, ∀ g,h ∈ x. consequently, h̃gh = ghh̃. proof. suppose h̃g ◦ h̃h = h̃gh. by definition 2.9, we get cmh̃gh(x) = cmh̃g◦h̃h(x) = ∨ y∈x [cmh̃g(y)∧cmh̃h(y −1x)] = ∨ y∈x [cmh̃(yg −1)∧cmh̃(y −1xh−1)] = ∨ y∈x [cmh̃(g −1y)∧cmh̃(h −1y−1x)] = ∨ y∈x [cmgh̃(y)∧cmhh̃(y −1x)] = cmgh̃◦hh̃(x) = cmghh̃(x) ⇒ gh̃ ◦hh̃ = ghh̃. 247 paul augustine ejegwa conversely, assuming gh̃ ◦hh̃ = ghh̃. then cmghh̃(x) = cmgh̃◦hh̃(x) = ∨ y∈x [cmgh̃(y)∧cmhh̃(y −1x)] = ∨ y∈x [cmh̃(g −1y)∧cmh̃(h −1y−1x)] = ∨ y∈x [cmh̃(yg −1)∧cmh̃(y −1xh−1)] = ∨ y∈x [cmh̃g(y)∧cmh̃h(y −1x)] = cmh̃g◦h̃h(x) = cmh̃gh(x) ⇒ h̃g ◦ h̃h = h̃gh. hence, the result follow. theorem 3.4. suppose g̃ ∈ fmg(x) and h̃ a fuzzy submultigroup of g̃. define h = {g ∈ x | cmh̃(g) = cmh̃(e)}. then hx = hy ⇔ h̃x = h̃y, ∀x,y ∈ x. similarly, xh = yh ⇔ xh̃ = yh̃. proof. this result gives a relationship between fuzzy comultisets of a fuzzy submultigroup of a fuzzy multigroup and the cosets of a subgroup of a given group. by theorem 2.1, we know that h is a subgroup of x and h = {x ∈ x | cmh̃x(z) = cmh̃e(z)}. now, suppose that hx = hy. then xy−1 ∈ h. thus cmh̃xy−1(z) = cmh̃e(z)∀z ∈ x and so cmh̃(zyx −1) = cmh̃(z). put z = zy−1, we get cmh̃(zy −1yx−1) = cmh̃(zy −1) ⇒ cmh̃(zx −1) = cmh̃(zy −1) ⇒ cmh̃x(z) = cmh̃y(z) and so, h̃x = h̃y. conversely, suppose that h̃x = h̃y, that is cmh̃x(z) = cmh̃y(z), ∀z ∈ x. this implies that cmh̃(zx −1) = cmh̃(zy −1). put z = y, we get cmh̃(yx −1) = cmh̃(e). so, yx−1 ∈ h. thus, hx = hy. the proof of xh = yh ⇔ xh̃ = yh̃ is similar. 248 fuzzy comultisets and quotient fuzzy multigroups 4 quotient fuzzy multigroups in this section, we present the notion of quotient groups in fuzzy multigroup setting and establish the isomorphism theorems. definition 4.1. suppose g̃ is a fuzzy multigroup of x and h̃ a normal fuzzy submultigroup of g̃. then, the union of the set of left/right fuzzy comultisets of h̃ such that the fuzzy comultisets satisfy xh̃ ◦yh̃ = xyh̃, ∀x,y ∈ x is called quotient or factor fuzzy multigroup of g̃ by h̃, denoted by g̃ h̃ . remark 4.1. suppose g̃ h̃ is a factor fuzzy multigroup of g̃ by h̃, it implies that h̃ is a normal fuzzy submultigroup of g̃ and g̃ h̃ = eh̃ = h̃. this property is not applicable in classical case. remark 4.2. suppose g̃ is a fuzzy multigroup of x, and h̃ a normal fuzzy submultigroup of g̃. then (i) if ĩ is a fuzzy submultigroup of g̃ such that h̃ ⊆ ĩ ⊆ g̃, then ĩ h̃ is a fuzzy submultigroup of g̃ h̃ . (ii) every fuzzy submultigroup of g̃ h̃ is of the form ĩ h̃ , for some fuzzy submultigroup ĩ of g̃ such that h̃ ⊆ ĩ ⊆ g̃. theorem 4.1. if h̃ is a normal fuzzy submultigroup of g̃ ∈ fmg(x). then h̃ is commutative if and only if g̃ h̃ is commutative. proof. let x,y ∈ x. suppose h̃ is commutative, then cmh̃(xyx −1y−1) = cmh̃(e), and hence, cmh̃(xy) = cmh̃(yx). consequently, h̃ is a normal fuzzy submultigroup of g̃ by definition 2.11. thus, since cmh̃(xy(yx) −1) = cmh̃(xyx −1y−1) = cmh̃(e), we have cmh̃(xy(yx) −1) = cmh̃(e) ⇒ cmh̃(xy(yx) −1) = cmh̃(xy(xy) −1) ⇒ cmh̃yx(xy) = cmh̃xy(xy). 249 paul augustine ejegwa thus, h̃xy = h̃yx. it follows that, h̃x◦h̃y = h̃y◦h̃x since h̃x◦h̃y = h̃xy and h̃y ◦ h̃x = h̃yx by proposition 3.4. hence, g̃ h̃ is commutative. conversely, assume g̃ h̃ is commutative, then h̃x◦ h̃y = h̃y ◦ h̃x ⇒ h̃xy = h̃yx. thus, cmh̃(xy(yx) −1) = cmh̃(e) ⇒ cmh̃(xy) = cmh̃(yx), completes the proof. theorem 4.2. suppose g̃ ∈ fmg(x) and h̃, ĩ are normal fuzzy submultigroups of g̃ and h̃ ⊆ ĩ, then ĩ h̃ is a normal fuzzy submultigroup of g̃ h̃ . proof. let x ∈ x. then cm ĩ h̃ (x) ≤ cm g̃ h̃ (x) since h̃ ⊆ ĩ and, h̃ and ĩ are normal fuzzy submultigroups of g̃. so, ĩ h̃ is a fuzzy submultigroup of g̃ h̃ . subsequently, cm ĩ h̃ (yxy−1) = cm ĩ h̃ (x)∀x,y ∈ x. hence, ĩ h̃ is a normal fuzzy submultigroup of g̃ h̃ by definition 2.11. remark 4.3. let g̃ be a fuzzy multigroup of x, and ĩ a normal fuzzy submultigroup of g̃. then, every normal fuzzy submultigroup of g̃ h̃ is of the form ĩ h̃ , for some normal fuzzy submultigroup h̃ of g̃ such that h̃ ⊆ ĩ ⊆ g̃. theorem 4.3. suppose g̃,h̃ ∈ fmg(x) and h̃ a normal fuzzy submultigroup of g̃. then h̃∩g̃ h̃∗ is a normal fuzzy submultigroup of h̃. proof. by definition 2.13, h̃∗ is a subgroup of x and h̃ ∩ g̃ ∈ fmg(x) by proposition 2.1. so, h̃∩g̃ h̃∗ is a fuzzy multigroup of x. since h̃ is a normal fuzzy submultigroup of g̃, then h̃∩g̃ is a fuzzy submultigroup of g̃ and h̃∩g̃ h̃∗ is a fuzzy submultigroup of h̃. we show that h̃∩g̃ h̃∗ is a normal fuzzy submultigroup of h̃. let x,y ∈ h̃∗. then xyx−1 ∈ h̃∗ since cmh̃(xyx −1) = cmh̃(y) > 0 by definition of h̃∗. this proves that h̃∗ is a normal subgroup of x. it is easy to see that h̃ ∩ g̃ is normal since cmh̃∩g̃(xyx −1) = cmh̃(xyx −1)∧cmg̃(xyx −1) = cmh̃(y)∧cmg̃(y) = cmh̃∩g̃(y). 250 fuzzy comultisets and quotient fuzzy multigroups in fact, h̃ ∩ g̃ is a normal fuzzy submultigroup since h̃ ∩ g̃ = h̃, and h̃ is a normal fuzzy submultigroup of g̃. hence, h̃∩g̃ h̃∗ is a normal fuzzy submultigroup of h̃. theorem 4.4. suppose g̃ is a fuzzy multigroup of x, h̃ ⊆ g̃ and n(h̃) is a normalizer. then n(h̃) is a subgroup of x and h̃ n(h̃) is a normal fuzzy submultigroup of g̃. proof. clearly, e ∈ n(h̃). let x,y ∈ n(h̃). then for any z ∈ x, we have cmh̃((xy −1)z) = cmh̃(x(y −1z)) = cmh̃((y −1z)x) = cmh̃(y −1(zx)) = cmh̃(y(zx) −1) = cmh̃(y(x −1z−1)) = cmh̃(z(xy −1)). hence, xy−1 ∈ n(h̃). therefore, n(h̃) is a subgroup of x. by definition 4.1, it follows that h̃ n(h̃) ∈ fmg(n(h̃)) and clearly, h̃ n(h̃) is a fuzzy submultigroup of h̃. since, cm h̃ n(h̃) (xyx−1) = cm h̃ n(h̃) (y), ∀x,y ∈ x, it implies that h̃ n(h̃) is a normal fuzzy submultigroup of h̃. theorem 4.5. suppose g̃ is a commutative fuzzy multigroup of x and h̃ a normal fuzzy submultigroup of g̃. then, there exists a natural homomorphism f : g̃ → g̃ h̃ defined by cmf(g̃)(y) = cmh̃(x −1y), ∀x,y ∈ x. proof. let f : g̃ → g̃ h̃ be a mapping defined by cmf(g̃)(y) = cmh̃(x −1y), ∀x,y ∈ x. that is, cmf(g̃)(y) = cmxh̃(y) ⇒ f(g̃) = xh̃ (consequently, f(g̃∗) = xh̃∗). since f : g̃ → g̃ h̃ is derived from f : g̃∗ → g̃∗h̃∗ such that h̃∗ is a normal subgroup of g̃∗, then to prove that f is a homomorphism, we show that cmxyh̃(z) = cmxh̃◦yh̃(z), ∀z ∈ x ⇒ f(xy) = f(x)f(y). since h̃ is commutative, then cmh̃(xz) = cmh̃(zx) ⇒ cmh̃(z −1xz) = cmh̃(x), ∀z ∈ x. 251 paul augustine ejegwa it is certain that, cmxh̃(z) = cmh̃(x −1z) and cmyh̃(z) = cmh̃(y −1z). then cmxyh̃(z) = cmh̃((xy) −1z). now, cmxh̃◦yh̃(z) = ∨ z=rs [cmxh̃(r)∧cmyh̃(s)] = ∨ z=rs [cmh̃(x −1r)∧cmh̃(y −1s)]. similarly, cmxyh̃(z) = cmh̃((xy) −1z) = cmh̃(y −1x−1z) ≥ ∨ z=rs [cmh̃(x −1r)∧cmh̃(y −1s)]. suppose by hypothesis, cmh̃(y −1x−1z) = ∨ z=rs [cmh̃(x −1r)∧cmh̃(y −1s)], then it follows that cmxyh̃(z) = cmxh̃◦yh̃(z), ∀z ∈ x. consequently, we have f(xy) = f(x)f(y), ∀x,y ∈ x. hence, f is a homomorphism. corolary 4.1. let g̃,h̃ ∈ fmg(x) such that cmg̃(x) = cmg̃(y), ∀x,y ∈ x and cmg̃(e) ≥ cmh̃(x), ∀x ∈ x. if f : g̃ → g̃ h̃ is a natural homomorphism defined by cmf(g̃)(y) = cmh̃(x −1y), ∀x,y ∈ x, then f−1(f(h̃)) = g̃◦ h̃. proof. let x ∈ x. to proof the result, we assume that f(x) = f(y), ∀x,y ∈ x. thus, cmf−1(f(h̃))(x) = ∨ x∈x [cmf(h̃)(f(x))] = ∨ x∈x [cmh̃(f −1(f(y))] = cmh̃(y). 252 fuzzy comultisets and quotient fuzzy multigroups again, cmg̃◦h̃(x) = ∨ x=zy [cmg̃(z)∧cmh̃(y)] = ∨ x∈x [cmg̃(xy −1)∧cmh̃(y)] = ∨ x∈x [cmg̃(e)∧cmh̃(y)] = cmh̃(y). hence, the proof follows. remark 4.4. assuming there is a bijective correspondence between every (normal) fuzzy submultigroup of g̃ that contains h̃ and the (normal) fuzzy submultigroups of g̃ h̃ . that is, if ĩ is a (normal) fuzzy submultigroup of g̃ containing h̃, then the corresponding (normal) fuzzy submultigroup of g̃ h̃ is f(ĩ). theorem 4.6. let x and y be groups, f : x → y be an isomorphism and h̃ a normal fuzzy submultigroup of g̃ ∈ fmg(x) such that cmg̃(x) = cmg̃(y)∀x,y ∈ x with kerf = {e}. then g̃ h̃ ∼= f(g̃) f(h̃) . proof. by theorem 2.2 and definition 4.1, g̃ h̃ and f(g̃) f(h̃) are fuzzy multigroups, respectively. let h : g̃ h̃ → f(g̃) f(h̃) be defined by h(h̃x) = f(h̃)(f(x)), ∀x ∈ x. if h̃x = h̃y, then cmh̃(xy −1) = cmh̃(e). since kerf = {e} meaning kerf ⊆ a ∗, then f−1(f(h̃)) = h̃ by proposition 2.2. thus, cmf−1(f(h̃))(xy −1) = cmf−1(f(h̃))(e) ⇒ cmf(h̃)(f(xy −1)) = cmf(h̃)(f(e)) ⇒ cmf(h̃)(f(x)(f(y)) −1) = cmf(h̃)(f(e)) ⇒ cmf(h̃)(f(x)) = cmf(h̃)(f(y)e ′) (where f(e) = e′). hence, cmf(h̃)(f(x)) = cmf(h̃)(f(y)) ⇒ f(h̃)(f(x)) = f(h̃)(f(y)). 253 paul augustine ejegwa hence, h is well-defined. it is also a homomorphism because h(h̃xh̃y) = h(h̃xy) = f(h̃)(f(xy)) = f(h̃)(f(x)f(y)) = f(h̃)(f(x))f(h̃)(f(y)) = h(h̃x)h(h̃y). suppose f is an epimorphism, then ∃x ∈ x such that f(x) = y. thus, h(h̃x) = f(h̃)(f(x)) = f(h̃)(y). moreover, f(h̃)(f(x)) = f(h̃)(f(y)) ⇒ cmf(h̃)(f(x)(f(y)) −1) = cmf(h̃)(e ′) ⇒ cmf(h̃)(f(xy −1)) = cmf(h̃)(f(e)) ⇒ cmf−1(f(h̃))(xy −1) = cmf−1(f(h̃))(e) implies cmh̃(xy −1) = cmh̃(e) ⇒ h̃x = h̃y, which proves that h is an isomorphism. hence, the result follows. corolary 4.2. let f : x → y be an isomorphism and h̃ a normal fuzzy submultigroup of g̃ ∈ fmg(y ) such that cmg̃(x) = cmg̃(y), ∀x,y ∈ y . then f(g̃) f(h̃) ∼= g̃ h̃ . proof. by theorem 2.2, f(g̃),f(h̃) ∈ fmg(x) and f(g̃) f(h̃) and f(g̃) f(h̃) are fuzzy multigroups by definition 4.1. again, since h̃ ∈ fmg(y ), then f(f−1(h̃)) = h̃. if x ∈ kerf, then f(x) = e′ = f(e), and so cmh̃(f(x)) = cmh̃(f(e)) ⇒ cmf−1(h̃)(x) = cmf−1(h̃)(e). hence, kerf ⊆ f−1(h̃∗). the proof is completed following the same logic as in theorem 4.6. theorem 4.7. let h̃,g̃ ∈ fmg(x) and h̃ a normal fuzzy submultigroup of g̃. then g̃ g̃∗ ≈ g̃ h̃ . proof. let f be a natural homomorphism from g̃∗ onto g̃∗ h̃∗ defined by f(xh̃∗) = xh̃∗ ∀x ∈ g̃∗. then, we have cm f( g̃ g̃∗ ) (xh̃∗) = ∨[cm g̃ g̃∗ (z)], ∀z ∈ g̃∗,f(z) = xh̃∗. 254 fuzzy comultisets and quotient fuzzy multigroups since g̃ g̃∗ and g̃ are bijective correspondence to each other (by remark 4.4) and z = f−1(xh̃∗) = xh̃∗, it follows that cm f( g̃ g̃∗ ) (xh̃∗) = ∨[cm g̃ g̃∗ (z)], ∀z ∈ g̃∗,f(z) = xh̃∗ = ∨[cmg̃(y)], ∀y ∈ xh̃∗ = cm g̃ h̃ (xh̃∗), ∀x ∈ g̃∗, because g̃ h̃ and g̃ are bijective correspondence to each other. hence, it follows that g̃ g̃∗ ≈ g̃ h̃ . lemma 4.1. suppose f : x → y and g̃ ∈ fmg(x), then (f(g̃))∗ = f(g̃∗). proof. straightforward. theorem 4.8. let g̃ ∈ fmg(x). suppose y is a group and ĩ ∈ fmg(y ) such that g̃ ≈ ĩ. then, there exists a normal fuzzy submultigroup h̃ of g̃ such that g̃ h̃ ∼= ĩ ĩ∗ . proof. since g̃ ≈ ĩ,∃ an epimorphism f of x onto y such that f(g̃) = ĩ. define h̃ ∈ fmg(x) as follows: ∀x ∈ x, cmh̃(x) = { cmg̃(x) if x ∈ kerf 0, otherwise clearly, h̃ ⊆ g̃. if x ∈ kerf, then yxy−1 ∈ kerf, ∀y ∈ x, and so cmh̃(yxy −1) = cmg̃(yxy −1) = cmg̃(x) = cmh̃(x), ∀y ∈ x. if x /∈ kerf, then cmh̃(x) = 0 and so cmh̃(yxy −1) = cmh̃(x) = 0, ∀y ∈ x. hence, h̃ is a normal fuzzy submultigroup of g̃. also, g̃ ≈ ĩ ⇒ f(g̃) = ĩ which further implies (f(g̃))∗ = ĩ∗ and f(g̃∗) = ĩ∗ by lemma 4.1. let f = g. then g is a homomorphism of g̃∗ onto ĩ∗ and ker g = h̃∗. thus, there exists an isomorphism h of g̃∗ h̃∗ onto ĩ∗ such that h(xh̃∗) = g(x) = f(x), ∀x ∈ g̃∗. for such an h, we have cm h( g̃ h̃ ) (z) = ∨[cm g̃ h̃ (xh̃∗)], ∀x ∈ g̃∗,h(xh̃∗) = z = ∨(∨[cmg̃(y)], ∀y ∈ xh̃∗),∀x ∈ g̃∗,g(x) = z = ∨[cmg̃(y)], ∀y ∈ g̃∗,g(y) = z = ∨[cmg̃(y)], ∀y ∈ x,f(y) = z = cmg̃(f −1(z)) = cmf(g̃)(z) = cmĩ(z), ∀z ∈ ĩ∗. therefore, g̃ h̃ ∼= ĩ ĩ∗ . 255 paul augustine ejegwa theorem 4.9. suppose g̃ is a fuzzy multigroup of x and h̃ a normal fuzzy submultigroup of g̃. then g̃ (h̃∩g̃) ' (h̃◦g̃) h̃ . proof. from definition 2.13, it is easy to infer that h̃∗ is also a normal subgroup of x. by the second isomorphism theorem for groups, we deduce g̃∗ h̃∗ ∩ g̃∗ ∼= h̃∗g̃∗ h̃∗ . assume that (h̃ ∩ g̃)∗ = h̃∗ ∩ g̃∗ and (h̃ ◦ g̃)∗ = h̃∗g̃∗. consequently, we have g̃∗ (h̃ ∩ g̃)∗ ∼= (h̃ ◦ g̃)∗ h̃∗ , where f is given by f(x(h̃ ∩ g̃)∗) = xh̃∗, ∀x ∈ g̃∗. thus, cm f( g̃ h̃∩g̃ ) (yh̃∗) = c g̃ h̃∩g̃ (y(h̃ ∩ g̃)∗) = ∨[cmg̃(z)], ∀z ∈ y(h̃ ∩ g̃)∗ ≤ ∨[cmh̃◦g̃(z)], ∀z ∈ y(h̃∗ ∩ g̃∗) ≤ ∨[cmh̃◦g̃(z)], ∀z ∈ yh̃∗ = cmh̃◦g̃ h̃ (yh̃∗), ∀y ∈ g̃∗. hence, f( g̃ (h̃∩g̃) ) ⊆ h̃◦g̃ h̃ . therefore, g̃ (h̃∩g̃) ' (h̃◦g̃) h̃ . theorem 4.10. let h̃, ĩ, g̃ ∈ fmg(x) such that h̃ ⊆ ĩ, and h̃ and ĩ are normal fuzzy submultigroups of g̃. then ( g̃ h̃ )/( ĩ h̃ ) ∼= g̃ ĩ . proof. if h̃, ĩ ∈ fmg(x) and h̃ is a normal fuzzy submultigroup of ĩ, then h̃∗ is a normal subgroup of ĩ∗ and both h̃∗ and ĩ∗ are normal subgroups of g̃∗. from the principle of third isomorphism theorem for groups, it follows that ( g̃∗ h̃∗ )/( ĩ∗ h̃∗ ) ∼= ( g̃∗ ĩ∗ ), where f is given by f(xh̃∗( ĩ∗ h̃∗ )) = xĩ∗, ∀x ∈ g̃∗. 256 fuzzy comultisets and quotient fuzzy multigroups then cm f(( g̃ h̃ )/( ĩ h̃ )) (xĩ∗) = cm( g̃ h̃ )/( ĩ h̃ ) (xh̃∗( ĩ∗ h̃∗ )) = ∨[cm g̃ h̃ (yh̃∗)], ∀y ∈ g̃∗,yh̃∗ ∈ xh̃∗( ĩ∗ h̃∗ ) = ∨[∨(cmg̃(z)),∀z ∈ yh̃∗], ∀y ∈ g̃∗, yh̃∗ ∈ xh̃∗( ĩ∗ h̃∗ ) = ∨[cmg̃(z)], ∀z ∈ g̃∗,zh̃∗ ∈ xh̃∗( ĩ∗ h̃∗ ) = ∨[cmg̃(z)], ∀z ∈ xh̃∗( ĩ∗ h̃∗ ) = ∨[cmg̃(z)], ∀z ∈ g̃∗,f(z) ∈ xĩ∗ = ∨[cmg̃(z)], ∀z ∈ g̃∗,f(z) = z = cmg̃ ĩ (xĩ∗), ∀x ∈ g̃∗, where the equalities hold since f is one-to-one. hence, the result follows. 5 conclusions in this paper, the ideas of fuzzy comultisets and quotient fuzzy multigroups have been proposed in an attempt to further strengthen the theory of fuzzy multigroups. a number of some related results were duly discussed in details. the connection between fuzzy comultisets of fuzzy multigroups and the cosets of groups has been proven. some characterizations of fuzzy comultisets 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r. r. yager. on the theory of bags. international journal of general systems, 13: 23–37, 1986. l. a. zadeh. fuzzy sets. information and control, 8:338–353, 1965. 259 microsoft word capitolo intero n 7.doc term functions and fundamental relation of fuzzy hyperalgebras r. ameri, t. nozari † school of mathematics, statistics and computer science college of sciences, university of tehran p.o. box 14155-6455, teheran, iran, e-mail:@umz.ac.ir ‡ department of mathematics, faculty of basic science, university of mazandaran, babolsar, iran abstract we introduce and study term functions over fuzzy hyperalgebras. we start from this idea that the set of nonzero fuzzy subsets of a fuzzy hyperalgebra can be organized naturally as a universal algebra, and constructing the term functions over this algebra. we present the form of generated subfuzzy hyperalgebra of a given fuzzy hyperalgebra as a generalization of universal algebras and multialgebras. finally, we characterize the form of the fundamental relation of a fuzzy hyperalgebra. keywords: hyperalgebra, fuzzy hyperalgebra, equivalence relation, term function, fundamental relation, quotient set. ratio mathematica, 20, 2010 43 1 introduction hyperstructure theory was born in 1934 when marty defined hypergroups, began to analysis their properties and applied them to groups, relational algebraic functions (see [15]). now they are widely studied from theoretical point of view and for their applications to many subjects of pure and applied properties ([7]). as it is well known, in 1965 zadeh ([28]) introduced the notion of a set µ on a nonempty set x as a function from x to the unite real interval i = [0, 1] as a fuzzy set. in 1971, rosenfeld ([25]) introduced fuzzy sets in the context of group theory and formulated the concept of a fuzzy subgroup of a group. since then, many researchers are engaged in extending the concepts of abstract algebra to the framework of the fuzzy setting ( for instance see [23]). the study of fuzzy hyperstructure is an interesting research topic of fuzzy sets and applied to the theory of algebraic hyperstructure. as it is known a hyperoperation assigns to every pair of elements of h a nonempty subset of h, while a fuzzy hyperoperation assigns to every pair of elements of h a nonzero fuzzy set on h. recently, sen, ameri and chowdhury introduced and analyzed fuzzy semihypergroups in [21]. this idea was followed by other researchers and extended to other branches of algebraic hyperstructures, for instance leoreanu and davvaz introduced and studied fuzzy hyperring notion in [13], chowdhury in [5] studied fuzzy transposition hypergroups and leoreanu studied fuzzy hypermodules in [15]. in this paper we follow the idea in [20] and introduced fuzzy hyperalgebras, as the largest class of fuzzy algebraic system. we introduce and study term functions over algebra of all nonzero fuzzy subsets of a fuzzy hyperalgebra, as an important tool to introduce fundamental relation on fuzzy hyperalgebra. finally, we construct fundamental relation of fuzzy algebras and investigate its basic properties. this paper is organized in four sections. in section 2 we gather the definitions and ratio mathematica, 20, 2010 44 basic properties of hyperalgebras and fuzzy sets that we need to develop our paper. in section 3 we introduce term functions over the algebra of nonzero fuzzy subsets of a fuzzy hyperalgebra and we obtained some basic results on fuzzy hyperalgebras, in section 4 we will present the form of the fundamental relation of a fuzzy hyperalgebra. 2 preliminaries in this section we present some definitions and simple properties of hyperalgebras from [2] and [3], which will be used in the next sections. in the sequel h is a fixed nonvoid set, p ∗(h) is the family of all nonvoid subsets of h, and for a positive integer n we denote for hn the set of n-tuples over h (for more see [6] and [7]). for a positive integer n a n-ary hyperoperation β on h is a function β : hn → p ∗(h). we say that n is the arity of β. a subset s of h is closed under the n-ary hyperoperation β if (x1, . . . , xn) ∈ sn implies that β(x1, . . . , xn) ⊆ s. a nullary hyperoperation on h is just an element of p ∗(h); i.e. a nonvoid subset of h. a hyperalgebraic system or a hyperalgebra 〈h, (βi : i ∈ i)〉 is the set h with together a collection (βi | i ∈ i) of hyperoperations on h. a subset s of a hyperalgebra h=〈h, (βi : i ∈ i)〉 is a subhyperalgebra of h if s is closed under each hyperoperation βi, for all i ∈ i, that is βi(a1, ..., ani ) ⊆ s, whenever (a1, ..., ani ) ∈ sni . the type of h is the map from i into the set n∗ of nonnegative integers assigning to each i ∈ i the arity of βi. in this paper we will assume that for every i ∈ i , the arity of βi is ni. for n > 0 we extend an n-ary hyperoperation β on h to an n-ary operation β on p ∗(h) by setting for all a1, ..., an ∈ p ∗(h) β(a1, ..., an) = ⋃ {β(a1, ..., an)|ai ∈ ai(i = 1, ..., n)} ratio mathematica, 20, 2010 45 it is easy to see that 〈p ∗(h), (βi : i ∈ i)〉 is an algebra of the same type of h. definition 2.1. let h=〈h, (βi : i ∈ i)〉 and h=〈h, (βi : i ∈ i)〉 be two similar hyperalgebras. a map h from h into h is called a (i) a homomorphism if for every i ∈ i and all (a1, ..., ani ) ∈ hni we have that h(βi((a1, ..., ani )) ⊆ βi(h(a1), ..., h(ani )); (ii) a good homomorphism if for every i ∈ i and all (a1, ..., ani ) ∈ hni we have that h(βi((a1, ..., ani )) = βi(h(a1), ..., h(ani )), for more details about homomorphism of hyperalgebras see [12]. let ρ be an equivalence relation on h. we can extend ρ on p ∗(h) in the following ways: (i) let {a, b} ⊆ p ∗(h). we write aρb iff ∀a ∈ a, ∃b ∈ b, such that aρb and ∀b ∈ b, ∃a ∈ a, such that aρb. (ii) we write aρb iff ∀a ∈ a, ∀b ∈ b we have aρb. definition 2.2. if h=〈h, (βi : i ∈ i)〉 be a hyperalgebra and ρ be an equivalence relation on h. then ρ is called regular (resp. strongly regular) if for every i ∈ i, and for all a1, ..., ani , b1, ..., bni ∈ h the following implication holds: a1ρb1, ..., ani ρbni ⇒ βi(a1, ..., ani )ρβi(b1, ..., bni ) (resp. a1ρb1, ..., ani ρbni ⇒ βi(a1, ..., ani )ρβi(b1, ..., bni )). definition 2.3. recall that for a nonempty set h, a fuzzy subset µ of h is a function µ : h → [0, 1]. if µi is a collection of fuzzy subsets of h, then we define the fuzzy subset ⋂ i∈i µi by: ( ⋂ i∈i µi)(x) = ∧ i∈i {µi(x)}, ∀x ∈ h. definition 2.4. let ρ be an equivalence relation on a hyperalgebra 〈h, (βi : i ∈ i)〉 and µ and υ be two fuzzy subsets on h. we say that µρυ if the following two conditions hold: (i) µ(a) > 0 ⇒ ∃b ∈ h : υ(b) > 0 , and aρb (ii) υ(x) > 0 ⇒ ∃y ∈ h : µ(y) > 0, and xρy. ratio mathematica, 20, 2010 46 3 fuzzy hyperalgebra and term functions definition 3.1. a fuzzy n-ary hyperoperation f n on s is a map f n : s×...×s −→ f ∗(s), which associated a nonzero fuzzy subset f n(a1, ..., an) with any n-tuple (a1, ..., an) of elements of s. the couple 〈s, f n〉 is called a fuzzy n-ary hypergroupoid. a fuzzy nullary hyperoperation on s is just an element of f ∗(s); i.e. a nonzero fuzzy subset of s. definition 3.2. let h be a nonempty set and for every i ∈ i, βi be a fuzzy ni-ary hyperoperation on h. then h=〈h, (βi : i ∈ i)〉 is called fuzzy hyperalgebra, where (ni : i ∈ i) is the type of this fuzzy hyperalgebra. definition 3.3. if µ1, ..., µni be ni nonzero fuzzy subsets of a fuzzy hyperalgebra h=〈h, (βi : i ∈ i)〉 , we define for all t ∈ h βi(µ1, ..., µni )(t) = ∨ (x1,...,xni )∈h ni (µ1(x1) ∧ ... ∧ µni (xni ) ∧ βi(x1, ..., xni )(t)) finally, if a1, ..., ank are nonempty subsets of h, for all t ∈ h βk(a1, ..., ank )(t) = ∨ (a1,...,ank )∈h nk (βk(a1, ..., ank )(t)). if a is a nonempty subset of h, then we denote the characteristic function of a by χa. note that, if f : h1 −→ h2 is a map and a ∈ h1, then f (χa) = χf (a). example 3.4. (i) a fuzzy hypergroupoid is a fuzzy hyperalgebra of type (2), that is a set h together with a fuzzy hyperoperation ◦. a fuzzy hypergroupoid 〈h, ◦〉, which is associative, that is x ◦ (y ◦ z) = (x ◦ y) ◦ z, for all x, y, z ∈ h is called fuzzy hypersemigroup[22]. in this ratio mathematica, 20, 2010 47 case for any µ ∈ f ∗(h), we define (a ◦ µ)(r) = ∨ t∈h ((a ◦ t)(r) ∧ µ(t)) and (µ ◦ a)(r) =∨ t∈h (µ(t) ∧ (t ◦ a)(r)) for all r ∈ h. (ii) a fuzzy hypergroup is a fuzzy hypersemigroup such that for all x ∈ h we have x ◦ h = h ◦ x = χh (fuzzy reproduction axiom)(for more details see [22]). (iii) a fuzzy hyperring r=〈r, ⊕, �〉 ([13]) is a fuzzy hyperalgebra of type (2, 2), which in that the following axioms hold: 1) a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c for all a, b, c ∈ r; 2) x ⊕ r = r ⊕ x = χr for all x ∈ r; 3) a ⊕ b = b ⊕ a for all a, b ∈ r; 4) a � (b � c) = (a � b) � c for all a, b, c ∈ r; 5) a � (b ⊕ c) = (a � b) ⊕ (a � c) and (a ⊕ b) � c = (a � c) ⊕ (b � c) for all a, b, c ∈ r. example 3.5. let h=〈h, (βi : i ∈ i)〉 be a hyperalgebra and µ be a nonzero fuzzy subset of h. define the following fuzzy n-ary hyperoperations on h, for every i ∈ i and for all (a1, ..., ani ) ∈ hni ; β�i (a1, ..., ani )(t) =   µ(a1) ∧ ... ∧ µ(ani ) t ∈ β(a1, ..., ani ) 0 otherwise and letting β◦i (a1, ..., ani ) = χ{a1,...,ani }. evidently h�=〈h, (β�i : i ∈ i)〉, h◦=〈h, (β◦i : i ∈ i)〉 are fuzzy hyperalgebras. theorem 3.6. let h=〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra, then for every i ∈ i and every a1, ..., ani ∈ h we have βi(χa1 , ..., χani ) = βi(a1, ..., ani ). definition 3.7. let h=〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra . a nonempty subset s of h is called a subfuzzy hyperalgebra if for ∀i ∈ i, ∀a1, ..., ani ∈ s, the following condition ratio mathematica, 20, 2010 48 hold: βi(a1, ..., ani )(x) > 0 then x ∈ s. we denote by s(u) the set of the subfuzzy hyperalgebras of h. definition 3.8. consider the fuzzy hyperalgebra h=〈h, (βi : i ∈ i)〉 and φ 6= x ⊆ h be nonempty. clearly, 〈x〉 = ⋂ {b : b ∈ s(h)| x ⊆ b} with the fuzzy hyperoperations of h form a subfuzzy hyperalgebra of h called the subfuzzy hyperalgebra of h generated by the subset x . evidently if x is a subfuzzy hyperalgebra for h then 〈x〉 = x. theorem 3.9. let h=〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra and φ 6= x ⊆ h. we consider x0 = x and for any k ∈ n, xk+1 = xk ∪ {a ∈ h | ∃i ∈ i, ni ∈ n, x1, ..., xni ∈ xk; βi(x1, ..., xni )(a) > 0}. then 〈x〉 = ⋃ k∈n xk. proof. let m = ⋃ k∈n xk, and ∀i ∈ i, consider t1, ..., tni ∈ m and βi(t1, ..., tni )(x) > 0. from x0 ⊆ x1 ⊆ ... ⊆ xk ⊆ ... it follows the existence of m ∈ n such that t1, ..., tni ∈ xm, which implies, according to the definition of xm+1 that x ∈ xm+1. thus x ∈ m and m = ⋃ k∈n xk is a subfuzzy hyperalgebra. from x = x0 ⊆ m , by definition of the generated subfuzzy hyperalgebra, it results 〈x〉 ⊆ 〈m〉 = m. to prove the inverse inclusion we will show by induction on k ∈ n that xk ⊆ 〈x〉 for any k ∈ n, and we have x0 = x ⊆ 〈x〉. we suppose that xk ⊆ 〈x〉. from 〈x〉 ∈ s(h) and the definition xk+1 we can deduce that xk+1 ⊆ 〈x〉. hence m ⊆ 〈x〉. the two inclusion lead us to m = 〈x〉.� let h=〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra then, the set of the nonzero fuzzy subsets of h denoted by f ∗(h), can be organized as a universal algebra with the operations; βi(µ1, ..., µni )(t) = ∨ (x1,...,xni )∈h ni (µ1(x1) ∧ ... ∧ µni (xni ) ∧ βi(x1, ..., xni )(t)) ratio mathematica, 20, 2010 49 for every i ∈ i, µ1, ..., µni ∈ f ∗(h) and t ∈ h. we denote this algebra by f∗(h). in [13] gratzer presents the algebra of the term functions of a universal algebra. if we consider an algebra b=〈b, (βi : i ∈ i)〉 we call n−ary term functions on b (n ∈ n) those and only those functions from bn into b, which can be obtained by applying (i) and (ii) from bellow for finitely many times: (i) the functions eni : b n → b, eni (x1, ..., xn) = xi, i = 1, ..., n are n−ary term functions on b; (ii) if p1, ..., pni are n−ary term functions on b, then βi(p1, ..., pni ) : bn → b, βi(p1, ..., pni )(x1, ..., xn) = βi(p1(x1, ..., xn), ..., pni (x1, ..., xn)) is also a n−ary term function on b. we can observe that (ii) organize the set of n−ary term functions over b (p (n)(b)) as a universal algebra, denoted by b(n)(b). if h is a fuzzy hyperalgebra then for any n ∈ n, we can construct the algebra of n−ary term functions on f∗(h), denoted by b(n)(f∗(h)) = 〈p (n)(f∗(h)), (βi : i ∈ i)〉. theorem 3.10. a necessary and sufficient condition for f∗(b) to be a subalgebra of f∗(u) is that b is to be a subfuzzy hyperalgebra for u. proof. obvious.� the next result immediately follows from theorem 3.10. corollary 3.11. (i) let h=〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra and b a subfuzzy hyperalgebra of h, and p ∈ p (n)(f∗(h)),(n ∈ n). if µ1, ..., µn ∈ f ∗(b) , then p(µ1, ..., µn) ∈ f ∗(b). (ii) let h= 〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra and b a subfuzzy hyperalgebra of h, and p ∈ p (n)(f∗(h)),(n ∈ n). if x1, ..., xn ∈ b, then p(χx1 , ..., χxn ) ∈ f ∗(b).� theorem 3.12. let h=〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra and φ 6= x ⊆ h. then a ∈ 〈x〉 if and only if ∃n ∈ n, ∃p ∈ p (n)(f∗(h)), and ∃x1, ..., xn ∈ x, such that ratio mathematica, 20, 2010 50 p(χx1 , ..., χxn )(a) > 0. proof. we denote m = {a ∈ h | ∃n ∈ n, ∃p ∈ p (n)(f∗(h)), ∃x1, ..., xn ∈ x : p(χx1 , ...χxn )(a) > 0}. for any x ∈ x we have e11(χx)(x) = χx(x) = 1, thus x ∈ x and hence x ⊆ m . also from corollary 3.11 (ii), it follows that p(χx1 , ..., χxn ) ∈ f∗(〈x〉), therefore m ⊆ 〈x〉. we will prove now that m is subfuzzy hyperalgebra of h. for any i ∈ i, if c1, ..., cni ∈ m and βi(c1, ..., cni )(x) > 0, we must show that x ∈ m. for c1, ..., cni ∈ m , it means that there exist mk ∈ n, pk ∈ p mk (f∗(h)), xk1, ..., xkmk ∈ x, k ∈ {1, ..., ni}, such that pk(χxk1 , ..., χxkmk )(ck) > 0, ∀k ∈ {1, ..., ni}. according to the corollary 8.2 from [12], for any n−ary term function p over f∗(h) and for m ≥ n there exists an m−ary term function q over f∗(h), such that p(µ1, ..., µn) = q(µ1, ..., µm), for all µ1, ..., µm ∈ f ∗(h); this allows us to consider instead of p1, ..., pni the term functions q1, ..., qni all with the same arity m = m1 + ... + mni and the elements y1, ..., ym ∈ x (which are the elements x11, ..., x 1 m1 , ..., xni1 , ..., x ni mni ), such that qk(χy1 , ..., χym )(ck) > 0,∀k ∈ {1, ..., ni}. but we have βi(q1(χy1 , ..., χym ), ..., qni (χy1 , ..., χym ))(x) =∨ (a1,...,ani )∈h ni (q1(χy1 , ..., χym )(a1) ∧ ... ∧ qni (χy1 , ..., χym )(ani ) ∧ βi(a1, ..., ani )(x)), and for (a1, ..., ani ) = (c1, ..., cni ) we have (βi(q1, ..., qni )(χy1 , ..., χym ))(x) > 0 . on the other hands we have βi(q1, ..., qni ) ∈ p (m)(f∗(h)), (m ∈ n) , y1, ..., ym ∈ x which implies that x ∈ m. therefore, m = 〈x〉 and this complete the proof.� remark 3.13. if h has a fuzzy nullary hyperoperation then < φ >= {a ∈ h | ∃µ ∈ p 0(f∗(h)), such that µ(a) > 0}. recall that if h=〈h, (βi : i ∈ i)〉 and b=〈b, (βi : i ∈ i)〉 are fuzzy hyperalgebras with the same type, then a map h : h → b is called a good homomorphism if for any i ∈ i we ratio mathematica, 20, 2010 51 have ; h(βi(a1, ..., ani )) = βi(h(a1), ..., h(ani )), ∀a1, ..., ani ∈ h. an equivalence relation on h ϕ is said to be an ideal if for any i ∈ i we have: βi(x1, ..., xni )(a) > 0 and xkϕyk(k ∈ {1, ..., ni}) ⇒ ∃b ∈ h : βi(y1, ..., yni )(b) > 0 and aϕb. for example the fuzzy regular relations on a fuzzy hypersemigroup are ideal equivalence. (for more details see [13, 21]) definition 3.14. let h=〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra and ϕ an equivalence relation on h. then h/ϕ can be described as a fuzzy hyperalgebra h/ϕ with the fuzzy hyperoperations: βi(ϕ(x1), ..., ϕ(xni ))(ϕ(xni+1)) = ∨ xkϕyk βi(y1, ..., yni )(yni+1). theorem 3.15. let h : h → b be a good homomorphism of fuzzy hyperalgebras h and b. then the relation ϕ = {(x, y) ∈ h|h(x) = h(y)} is an ideal relation on h. conversely, if ϕ is an ideal relation on h, then p = pϕ : h → h/ϕ is homomorphism (which is not strong). proof. straightforward.� remark 3.16. let h and b be fuzzy hyperalgebras of the same type and h be a homomorphism from h into b. we will construct the algebras f∗(h) and f∗(b). the homomorphism h induces a mapping h′ : f∗(h) → f∗(b) by h′(µ) = h(µ), for any µ ∈ f ∗(h). let us consider h a set and f ∗(h) the set of its nonzero fuzzy subsets. let ϕ be an equivalence on h and let us consider the relation ϕ on f ∗(h) as follows: ratio mathematica, 20, 2010 52 µϕν ⇔ ∀a ∈ h : µ(a) > 0 ⇒ ∃b ∈ h : ν(b) > 0 and aϕb and ∀b ∈ h : ν(b) > 0 ⇒ ∃a ∈ h : µ(a) > 0 and aϕb. it is immediate that ϕ is an equivalence relation on f ∗(h). the next result immediately follows. theorem 3.17. an equivalence relation ϕ on a fuzzy hyperalgebra h is ideal if and only if ϕ is a congruence relation on f∗(h). proof. let us suppose that ϕ is an ideal equivalence on h and let us consider i ∈ i and µk, νk ∈ f ∗(h) nonzero and µkϕνk, k ∈ {1, ..., ni} . then for any a ∈ h such that βi(µ1, ..., µni )(a) > 0, we have βi(µ1, ..., µni )(a) = ∨ (x1,...,xni )∈h ni µ1(x1) ∧ ... ∧ µni (xni ) ∧ βi(x1, ..., xni )(a). thus there exists (x1, ..., xni ) ∈ hni , such that µk(xk) > 0 for k ∈ {1, ..., ni} and βi(x1, ..., xni )(a) > 0. from the definition ϕ and hence there exists (y1, ..., yni ) ∈ hni , such that νk(yk) > 0 for k ∈ {1, ..., ni} and xkϕyk, and sice ϕ is an ideal and βi(x1, ..., xni )(a) > 0, there exists b ∈ h, such that βi(y1, ..., yni )(b) > 0 and aϕb. analogously, it can be proved that for all b ∈ h, such that βi(y1, ..., yni )(b) > 0, there exists a ∈ h, such that βi(x1, ..., xni )(a) > 0 and aϕb. hence, it is proved that ϕ is a congruence on f∗(h). conversely, let us take i ∈ i and a, xk, yk ∈ h, k ∈ {1, ..., ni} such that xkϕyk and βi(x1, ..., xni )(a) > 0. obviously, χxk ϕχyk , ∀k ∈ {1, ..., ni}, and because ϕ is a congruence on f∗(h) we can write βi(χx1 , ..., χxni )ϕβi(χy1 , ..., χyni ), hence βi(x1, ..., xni )ϕβi(y1, ..., yni ), which leads us to the existence b ∈ h, such that βi(y1, ..., yni )(b) > 0 and aϕb. this complete the proof.� corollary 3.18. (i) if h=〈h, (βi : i ∈ i)〉 is a fuzzy hyperalgebra, ϕ is an ideal equivalence relation on h and p ∈ p (n)(f∗(h)) if for any nonzero, µk, νk, such that µkϕνk ratio mathematica, 20, 2010 53 k ∈ {1, ..., n}, then p(µ1, ..., µn)ϕp(ν1, ..., νn). (ii) let h=〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra, ϕ an ideal equivalence relation on h. if xkϕyk, k ∈ {1, ..., n}, p ∈ p (n)(f∗(h)) , xk, yk ∈ h. then have p(χx1 , ..., χxn )ϕp(χy1 , ..., χyn ). let h be a homomorphism from h into b and take ϕ = {(x, y) ∈ h2 | h(x) = h(y)}. then we have ϕ = {(µ, ν) ∈ (f ∗(h))2 | h′(µ) = h′(ν)}. obviously, ϕ is an ideal of h if and only if ϕ is congruence on f∗(h). theorem 3.19. the map h is a homomorphism ofthe universal algebras f∗(h) and f∗(b) if and only if h is a good homomorphism between h and b. proof. straightforward.� the next result immediately follows from theorem 3.12. corollary 3.20. (i) let h=〈h, (βi : i ∈ i)〉 and b=〈b, (βi : i ∈ i)〉 be fuzzy hyperalgebras of the same type, h : h → b a homomorphism and p ∈ p (n)(f∗(h)). then for all µ1, ..., µn ∈ f ∗(h) we have h′(p(µ1, ..., µn)) = p(h′(µ1), ..., h′(µn)). (ii) let h=〈h, (βi : i ∈ i)〉 and b=〈b, (βi : i ∈ i)〉 be fuzzy hyperalgebras of the same type, h : h → b a homomorphism and p ∈ p (n)(f∗(h)). then for all a1, ..., an ∈ h, we have h′(p(χa1 , ..., χan )) = p(h ′(χa1 ), ..., h ′(χan )).� 4 fundamental relation of fuzzy hyperalgebra as it is known that if r is an strongly regular equivalence relation on a given hypergroup (resp. hypergroupoid, semihypergroup) h, then we can define a binary operation ⊗ on the quotient set h/r, the set of all equivalence classes of h with respect to r, such that (h/r, ⊗) consists a group (resp. groupoid, semigroup). in fact the relation β∗ is the ratio mathematica, 20, 2010 54 smallest equivalences relation such that the quotient h/β∗ is a group (resp. groupoid, semigroup) and it is called fundamental relation of h. the equivalence relation β∗ was studied on hypergroups by many authors( for more details see [6]). as the fundamental relation plays an important role in the theory of algebraic hyperstructure it extended to other classes of algebraic hyperstructure, such as hyperrings, hypermodules, hypervectorspaces( for more details see [25], [26] and [27]). in [20] pelea introduced and studied the fundamental relation of a multialgebra based on term functions. in the sequel we extend fundamental relation on fuzzy hyperalgebras and investigate its basic properties. let b=〈b, (βi : i ∈ i)〉 be an universal algebra. if we add to the set of the operations of b the nullary operations corresponding to the elements of b, the n−ary term functions of this new algebra are called the n−ary polynomial functions of b. the n−ary polynomial functions p n(b) of b form a universal algebra with the operations (βi : i ∈ i), denoted by p(n)(b), p(n)(b)=〈p n(b), (βi : i ∈ i)〉. let h=〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra. for any n ∈ n, we can construct the algebra p(n)(f∗(h)) of n−ary polynomial functions on f∗(h), ( p(n)(f∗(h)) = 〈p n(f∗(h)), (βi : i ∈ i)〉) . consider the subalgebra p (n) h (f ∗(h)) of p(n)(f∗(h)) obtained by adding to the operations (βi : i ∈ i) of f∗(h) only the nullary operations associated to the characteristic functions of the elements of h. thus the elements of p(n)h (f ∗(h)) (n ∈ n) are those and only those functions from (f ∗(h))n into f ∗(h) which can obtained by applying (i), (ii), (iii) from bellow for finitely many times: (i) the functions cnχa : (f ∗(h))n → f ∗(h), defined by setting cnχa (µ1, ..., µn) = χa, for all µ1, ..., µn ∈ f ∗(h) are elements of p (n) h (f ∗(h)), for every a ∈ h; (ii) the functions eni : (f ∗(h))n → f ∗(h), eni (µ1, ..., µn) = µi, for all µ1, ..., µn ∈ f ∗(h), i = 1, ..., n are elements of p(n)h (f ∗(h)); (iii) if p1, ..., pni are elements of p (n) h (f ∗(h)), and i ∈ i then βi(p1, ..., pni ) : (f ∗(h))n → ratio mathematica, 20, 2010 55 f ∗(h), defined by setting for all µ1, ..., µn ∈ f ∗(h), (βi(p1, ..., pni ))(µ1, ..., µn) = βi(p1(µ1, ..., µn), ..., pni (µ1, ..., µn)) is also an element of p (n) h (f ∗(h)). in the continue, we will use only polynomial functions from p(n)h (f ∗(h)). thus we will drop the subscript with no danger of confusion. definition 4.1. let α be the relation defined on h for x, y ∈ h set xαy follows: xαy ⇐⇒ p(χa1 , ..., χan )(x) > 0and p(χa1 , ..., χan )(y) > 0, for some p ∈ p n(f∗(h)), a1, ..., an ∈ h. it is clear that α is symmetric. because for any a ∈ h, e11(χa)(a) > 0, the relation α is also reflexive. we take α∗ to be the transitive closure of α. then α∗ is an equivalence relation on h. lemma 4.2. if f ∈ p 1(f∗(h)) and a, b ∈ h satisfy aα∗b then f (χa)α∗f (χb). proof. by the definition of α∗ : a = y1αy2α...αym = b for some m ∈ n and y2, ..., ym−1 ∈ h. let f (χyi )(ui) > 0, i = 1, ..., m. consider 1 ≤ j < m. clearly yjαyj+1 means that pj(χa1 , ..., χan )(yj) > 0 and pj(χa1 , ..., χan )(yj+1) > 0, for some nj ∈ n, pj ∈ p nj (f∗(h)), a1, ..., an ∈ h. define the nj−ary hyperoperation qj on f ∗(h) by setting qj(χx1 , ..., χxnj ) = ∨ {f (χt) : pj(χx1 , ..., χxnj )(t) > 0} for all x1, ..., xnj ∈ h. clearly qj ∈ p nj (f∗(h)) and for x ∈ h; qj(χa1 , ..., χan )(x) = ∨ pj (χa1 ,...,χan )(z)>0 f (χz)(x). from pj(χa1 , ..., χan )(yj) > 0 and pj(χa1 , ..., χan )(yj+1) > 0 we get 0 < f (χyj )(uj) ≤ qj(χa1 , ..., χan )(uj) and 0 < f (χyj+1 )(uj+1) ≤ qj(χa1 , ..., χan )(uj+1) proving ujαuj+1. thus u1α ∗um. since f (χa)(u1) = f (χy1 )(u1) > 0 and f (χb)(um) = f (χym )(um) > 0 were arbitrary, we obtain f (χa)α ∗f (χb).� remark 4.3. for a given fuzzy hyperalgebra h and equivalence relation ρ on h, the set h/ρ can be considered as a hyperalgebra with the hyperoperations ratio mathematica, 20, 2010 56 βi(ρ(a1), ..., ρ(ani )) = {ρ(z) | βi(b1, ..., bni )(z) > 0, bk ∈ ρ(ak), ∀k ∈ {1, ..., ni}} (1) for all i ∈ i. lemma 4.4. let ρ be an equivalence relation on h such that h/ρ be an universal algebra . then for any n ∈ n, p ∈ p n(f∗(h)) and a1, ..., an ∈ h the following gold: p(χa1 , ..., χan )(x) > 0 and p(χa1 , ..., χan )(y) > 0 =⇒ xρy. proof. we will prove this statement by induction over the steps of construction of an n−ary polynomial function( for n ∈ n arbitrary). if p = cnχa , from c n χa (χa1 , ..., χan )(x) > 0 and c n χa (χa1 , ..., χan )(y) > 0 we deduce that x = y = a, thus xρy. if p = eni with i ∈ {1, ..., n}, from eni (χa1 , ..., χan )(x) > 0 and eni (χa1 , ..., χan )(y) > 0 we deduce that x = y = ai, , and hence xρy. we suppose that the statement holds for the n−ary polynomial functions p1, ..., pnk and we will prove it for the n−ary polynomial function βk(p1, ..., pnk ). if 0 < βk(p1, ..., pnk )(χa1 , ..., χan )(x) = βk(p1(χa1 , ..., χan ), ..., pnk (χa1 , ..., χan ))(x) =∨ (x1,...,xnk )∈h nk (p1(χa1 , ..., χan )(x1) ∧ ... ∧ pnk (χa1 , ..., χan )(xnk ) ∧ βk(x1, ..., xnk )(x)) and if we set y instead of x, above statement is true. thus there exist x1, ..., xnk , y1, ..., ynk ∈ h, such that pi(χa1 , ..., χan )(xi) > 0 and pi(χa1 , ..., χan )(yi) > 0, for i ∈ {1, ..., nk} and βk(x1, ..., xnk )(x) > 0 and βk(y1, ..., ynk )(y) > 0. obviously, xiρyi for all i ∈ {1, ..., nk} and according to (1) and by the hypothesis we obtain that ρ(x) = ρ(y), i.e., xρy, as desired.� the next result immediately follows from previous two lemmas. theorem 4.5. the relation α∗ is the smallest equivalence relation on fuzzy hyperalgebra h such that h/ρ is an universal algebra. ratio mathematica, 20, 2010 57 we call h/ρ, fundamental universal algebra of fuzzy hyperalgebra h such that h/ρ. proof. at the first, we show that h/ρ is a universal algebra. for this we take any x, y ∈ h, such that α∗(x), α∗(y) ∈ βk(α∗(a1), ..., α∗(ank )) for k ∈ i and a1, ..., ank ∈ h. this means that there exist x1, ..., xnk , y1, ..., ynk ∈ h, such that βk(x1, ..., xnk )(x) > 0 and βk(y1, ..., ynk )(y) > 0 and xiα ∗aiα ∗yi for all i ∈ {1, ..., nk}. applying lemma 4.2 to the unary polynomial functions βi(z, c n χx2 , ..., cnχxnk ), βi(c n χy1 , z, cnχx3 , ..., cnχxnk ), ..., , βi(c n χy1 , ..., cnχynk−1 , z), we obtain the following relations: βi(χx1 , ..., χxnk )α ∗β(χy1 , χx2 , ..., χxnk ) βi(χy1 , χx2 , ..., χxnk )α ∗βi(χy1 , χ22 , χx3 ..., χxnk ) ... βi(χy1 , χy2 , ..., χxnk−1 )α ∗βi(χy1 , χy2 , ..., χynk ), which leads us to xα∗y (from definition α∗), i.e. α∗(x) = α∗(y). clearly, βi in (1) is an operation on h/α∗, for any i ∈ i, and h/α∗ is a universal algebra. now we prove that α∗ is smallest. if ρ is an arbitrary equivalence relation on h such that h/ρ is a universal algebra, we can show that α∗ ⊆ ρ. if xαy then there exist n ∈ n, p ∈ p n(f∗(h)) and a1, ..., an ∈ h for which p(χa1 , ..., χan )(x) > 0 and p(χa1 , ..., χan )(y) > 0, and hence by lemma 4.4 we have xρy, hence α ⊆ ρ, which implies α∗ ⊆ ρ.� remark 4.6. for a given fuzzy hyperalgebra h and equivalence relation α∗ on h. let us define the operations of the universal algebra h/α∗ as follows : βi(α ∗(a1), ..., α ∗(ani )) = {α∗(b) | βi(a1, ..., ani )(b) > 0}. moreover, we can write βi(α ∗(a1), ..., α ∗(ani )) = α ∗(b) βi(a1, ..., ani )(b) > 0. example 4.7. let h=〈h, ◦〉 be a fuzzy hypersemigroup, i.e. a fuzzy hyperalgebra with one binary fuzzy hyperoperation ◦, which is associative, that is x ◦ (y ◦ z) = (x ◦ y) ◦ z, ratio mathematica, 20, 2010 58 for all x, y, z ∈ h ( for more details see [21]). let f∗(h)=〈f ∗(h), ◦〉 be the universal algebra with one binary operation defined as follows: (µ ◦ ν)(r) = ∨ x,y∈h µ(x) ∧ ν(y) ∧ (x ◦ y)(r) ∀ µ, ν ∈ f ∗(h),r ∈ h. by distributivity of the lattice ([0, 1], ∨, ∧) and associativity of ◦ in h, we will prove that the operation ◦ in f∗(h) is associative. so for every µ, ν, η ∈ f ∗(h) and r ∈ h we have ((µ ◦ ν) ◦ η)(r) = ∨ x,y∈h [(µ ◦ ν)(x) ∧ η(y) ∧ (x ◦ y)(r)] =∨ x,y∈h [( ∨ p,q∈h µ(p) ∧ ν(q) ∧ (p ◦ q)(x)) ∧ η(y) ∧ (x ◦ y)(r)] =∨ p,q,y∈h [µ(p) ∧ ν(q) ∧ η(y) ∧ ( ∨ x∈h (p ◦ q)(x) ∧ (x ◦ y)(r))] =∨ p,q,y∈h [µ(p) ∧ ν(q) ∧ η(y) ∧ ( ∨ x∈h (p ◦ x)(r) ∧ (q ◦ y)(x))] = ∨ p,x∈h [µ(p) ∧ (p ◦ x)(r) ∧ ( ∨ q,y∈h ν(q) ∧ η(y) ∧ (q ◦ y)(x))] =∨ p,x∈h [µ(p) ∧ (p ◦ x)(r) ∧ (ν ◦ η)(x)] = (µ ◦ (ν ◦ η))(r). consider now the universal algebra of polynomial functions of 〈f ∗(h), ◦〉. the images of the elements of this algebra are the sums of nonzero fuzzy subsets of h. thus we can define α on h by: aαb ⇐⇒ ∃x1, ..., xn ∈ h(n ∈ n): (χx1 ◦ ... ◦ χxn )(a) > 0 and (χx1 ◦ ... ◦ χxn )(b) > 0. consider the quotient set h/α∗ with the hyperoperation α∗(a) ◦ α∗(b) = {α∗(c) | (a′ ◦ b′)(c) > 0, a′α∗a, b′α∗b}. really ◦ is an operation, because α∗ is the fundamental relation on h. also α∗(x) ◦ α∗(y) ◦ α∗(z)) = α∗(x) ◦ α∗(k) = α∗(l), where (y ◦ z)(k) > 0 and (x ◦ k)(l) > 0. therefore, 0 < (x ◦ (y ◦ z))(l) = ((x ◦ y) ◦ z)(l) = ∨ p∈h [(x ◦ y)(p) ∧ (p ◦ z)(l)]. thus ratio mathematica, 20, 2010 59 there exists p ∈ h, such that α∗(l) = α∗(p) ◦ α∗(z) = (α∗(x) ◦ α∗(y)) ◦ α∗(z), that the operation ◦ in h/α∗ is associative. moreover, if h=〈h, ◦〉 be a fuzzy hypergroup, that is x ◦ h = h ◦ x = χh , for every x ∈ h, since for every α∗(a), α∗(b) ∈ h/α∗, there exist α∗(t), α∗(s) ∈ h/α∗, such that, α∗(a) ◦ α∗(t) = α∗(b) and α∗(s) ◦ α∗(a) = α∗(b), it is concluded that h/α∗=〈h/α∗, ◦〉 is a group. example 4.8. let r=〈r, ⊕, �〉 be a fuzzy hyperring. this means that 〈r, ⊕〉 is a commutative fuzzy hypergroup, 〈r, �〉 is a fuzzy hypersemigroup and for all x, y, z ∈ r satisfies: x�(y⊕z) = (x�y)⊕(x�z) and (x⊕y)�z = (x�z)⊕(y�z) ( for more details see [13]). let f∗(r)=〈f ∗(r), ⊕, �〉 be the universal algebra with two binary operations defined as follows: (µ ⊕ ν)(r) = ∨ x,y∈h [µ(x) ∧ ν(y) ∧ (x ⊕ y)(r)], (µ � ν)(r) = ∨ x,y∈h [µ(x) ∧ ν(y) ∧ (x � y)(r)], for all µ, ν ∈ f ∗(r), r ∈ r. obviously, the operation ⊕ in f ∗(r) is commutative, and ⊕ and � in f ∗(r) are associative. by distributivity of the lattice [0, 1] and distributivity � with respect to ⊕ in r, we will prove that the operation � in f ∗(r) is distributive with respect to the operation ⊕, too. for every µ, ν, eta ∈ f ∗(r) and r ∈ r we have: (µ � (ν ⊕ η))(r) = ∨ x,y∈r [µ(x) ∧ (ν ⊕ η)(y) ∧ (x � y)(r)] =∨ x,y∈r [µ(x) ∧ ( ∨ s,t∈r ν(s) ∧ η(t) ∧ (s ⊕ t)(y)) ∧ (x � y)(r)] =∨ x,y∈r [ ∨ s,t∈r (µ(x) ∧ ν(s) ∧ η(t) ∧ (s ⊕ t)(y) ∧ (x � y)(r))] =∨ x,s,t∈r [µ(x) ∧ ν(s) ∧ η(t) ∧ ( ∨ y∈r (x � y)(r) ∧ (s ⊕ t)(y))] =∨ x,s,t∈r [µ(x) ∧ ν(s) ∧ η(t) ∧ ( ∨ p,q∈r (x � s)(p) ∧ (x � t)(q) ∧ (p ⊕ q)(r))] = ratio mathematica, 20, 2010 60 ∨ x,s,t∈r [ ∨ p,q∈r (µ(x) ∧ η(t) ∧ (x � t)(q) ∧ µ(x) ∧ ν(s) ∧ (x � s)(p) ∧ (p ⊕ q)(r))] =∨ p,q∈r [( ∨ x,t∈r µ(x) ∧ η(t) ∧ (x � t)(q)) ∧ ( ∨ x,s∈r µ(x) ∧ ν(s) ∧ (x � s)(p)) ∧ (p ⊕ q)(r)] =∨ p,q∈r [(µ � η)(q) ∧ (µ � ν)(p) ∧ (p ⊕ q)(r)] = ((µ � ν) ⊕ (µ � η))(r). and analogously, (µ ⊕ ν) � η = (µ � η) ⊕ (ν � η). now we can construct the universal algebra (with two binary operations) of the polynomial functions of f∗(r) for any n ∈ n. the images of the elements of this algebra are the sums of products of nonzero fuzzy subsets of r. thus we can define α on r by; aαb ⇐⇒ ∃xij ∈ r, i ∈ {1, ..., kj}, j ∈ {1, ..., l}, kj, l ∈ n: (⊕lj=1(� kj i=1χxij ))(a) > 0 and (⊕ l j=1(� kj i=1χxij ))(b) > 0. consider the quotient set r/α∗ withe two following hyperoperations : α∗(a) ⊕ α∗(b) = {α∗(c) | (a′ ⊕ b′)(c) > 0, a′α∗a, b′α∗b}, and α∗(a) � α∗(b) = {α∗(c) | (a′ � b′)(c) > 0, a′α∗a, b′α∗b} actually ⊕ and � are operations, because α∗ is the fundamental relation on r. by considering the previous example, obviously 〈r/α∗, ⊕〉 is a commutative group. we verify the distributivity of � with respect to ⊕ for the universal algebra r/α∗=〈r/α∗, ⊕, �〉. we have α∗(a) � (α∗(b) ⊕ α∗(c)) = α∗(a) � α∗(d) = α∗(e), where (b ⊕ c)(d) > 0 and (a � d)(e) > 0 0 < (a � (b ⊕ c))(e) = ∨ p∈r (a � p)(e) ∧ (b ⊕ c)(p). thus 0 < ((a � b) ⊕ (a � c))(e) = ∨ x,y∈r (a � b)(x) ∧ (a � c)(y) ∧ (x ⊕ y)(e). therefore, there exist x, y ∈ r such that α∗(e) = α∗(x) + α∗(y) = (α∗(a) + α∗(b))⊕(α∗(a)�α∗(c)), and hence it was proved that α∗(a) � (α∗(b) ⊕ α∗(c)) = (α∗(a) + α∗(b)) ⊕ (α∗(a) � α∗(c)). analogously, we can prove that (α∗(b) ⊕ α∗(c)) � α∗(a)) = (α∗(b) � α∗(a)) ⊕ (α∗(c) � α∗(a)). thus it concluded that r/α∗=〈r/α∗, ⊕, �〉 is a ring, as desired.� conclusion ratio mathematica, 20, 2010 61 we introduced and studied term functions over fuzzy hyperalgebras, as the largest class of fuzzy algebraic systems. we use the idea that the set of nonzero fuzzy subsets of a fuzzy hyperalgebra can be organized naturally as a universal algebra, and constructed the term functions over this algebra. we gave the form of generated subfuzzy hyperalgebra of a given fuzzy hyperalgebra as a generalization of universal algebras and multialgebras. finally, we characterized the form of the fundamental relation of a fuzzy hyperalgebra, to construct the fundamental universal algebra corresponding to a given fuzzy hyperalgebra, and this result guarantee that that fundamental relation on any fuzzy algebraic hyperstructures, such as fuzzy hypergroups, fuzzy hyperrings, fuzzy hypermodules,... exists. acknowledgement this research is partially supported by the “fuzzy systems and its applications center of excellence, shahid bahonar university of kerman, iran” and “research center in algebraic hyperstructures and fuzzy mathematics, university of mazandaran, babolsar, iran”. references [1] r. ameri, on categories of hypergroups and hypermodules , italian journal of pure and applid mathematics, vol. 6 (2003) 121-132. [2] r. ameri and i. g. rosenberg, congruences of multialgebras, multivalued logic and soft computing (to appaear). [3] r. ameri and m.m. zahedi, hyperalgebraic system, italian journal of pure and applid mathematics, vol. 6 (1999) 21-32. ratio mathematica, 20, 2010 62 [4] r. ameri and m.m. zahedi, fuzzy subhypermodules over fuzzy hyperrings, sixth international on aha, democritus university, 1996, 1-14,(1997). [5] s. burris, h. p. sankappanavar, a course in universal algebra, springer verlage 1981. [6] p. corsini, prolegomena of hypergroup theory, supplement to riv. mat.pura appl., aviani editor, 1993. [7] p. corsini, v. leoreanu, applications of hyperstructure theory, kluwer, dordrecht 2003. [8] p. corsini, i. tofan, on fuzzy hypergroups, pu.m.a., 8 (1997) 29-37. [9] b. davvaz, fuzzy hv-groups, fuzzy sets and systems, 101 (1999) 191-195. [10] b. davvaz, fuzzy hv-submodules, fuzzy sets and systems, 117 (2001) 477-484. [11] b. davvaz, p. corsini, generalized fuzzy sub-hyperquasigroups of hyperquasigroups, soft computing, 10 (11) (2006), 1109-1114. [12] m. mehdi ebrahimi, a. karimi and m. mahmoudion quotient and isomorphism theorems of universal hyperalgebras, italian journal of pure and applied mathematics, 18 (2005), 9-22. [13] g. gratzer, universal algebra, 2nd edition, springer verlage, 1970. [14] v. leoreanu-fotea, b. davvaz, fuzzy hyperrings, fuzzy sets and systems, 2008, doi 10.1016/j.fss.2008.11.007. [15] v. leoreanu-fotea, fuzzy hypermodules, computes and mathematics with applications, vol. 57 (2009) 466-475. ratio mathematica, 20, 2010 63 [16] f. marty, sur une generalization de la nation de groupe, 8th congress des mathematiciens scandinaves, stockholm (1934) 45-49. [17] j.n. mordeson, m.s. malik, fuzzy commutative algebra, word publ., 1998. [18] c. pelea, on the direct product of multialgebras, studia uni. babes-bolya, mathematica, vol. xlviii (2003) 93-98. 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[28] l. a. zadeh, fuzzy sets, inform. and control, vol. 8 (1965) 338-353. ratio mathematica, 20, 2010 65 79 an example of a join space associated with a relation laurenţiu leoreanu1 abstract in this paper a join spaces associated with a binary relation is presented. keywords join spaces, relations first of all, let us recall what a join space is. let h be a nonemptyset and o : h x h →p* (h), where p* (h) is the set of nonempty subsets of h. if a ⊂ h, b ⊂ h, then we set a o b = ∪a∈a ∪b∈b a o b. we denote a≈b if a∩b ≠ ø. if the hyperoperation �o� is associative and ∀ a∈h, we have a o h = h = h o a, then (h,o) is a hypergroup. denote a / b={x∈h a∈b o x}, for any (a,b)∈h2. a hypergroup (h,o) is called a join space if �o� is commutative and a / b ≈ c / d ⇒ a o d ≈ b o c. join spaces have been introduced by w. prenowitz and used by himself and j. jantosciak in order to rebuild some branches of non-euclidian geometries. afterwards, join spaces have also been used in the study of other topics (graphs and hypergraphs, lattices, binary relations and so on). here, a connection between join spaces and reflexive and symmetric relations is presented. first, we give an example: let f : h → u be an onto map. we define on h the following hyperoperation: ∀(x,y) ∈ h2, x o x = f-1(f(x)), x o y = x o x ∪ yoy (where ∀ y⊂u, f-1(y) = {x∈hf(x) ∈y}). proposition (h,o) is a join space. proof. for any (x,y,z) ∈ h3, we have (x o y)o z = x o (y o z) = f-1(f(x)) ∪ f-1(f(y)) ∪ f-1(f(z)) and x o h = ∪a∈h x o a = ∪a∈h f -1(f(x)) ∪ f-1(f(a)) = h, since f is onto. so, (h, o) is a commutative hypergroup. 1 grupul şcolar �miron costin � roman, românia 80 moreover, any x ∈ h is an identity of h (since ∀ y ∈ h, y ∈ x o y) and for any (x,y)∈h2, x is an inverse of y. let us check now that a / b ≈ c / d ⇒ a o d ≈ b o c. let x ∈ a / b ∩ c / d that is a ∈ f-1 (f(x)) ∪ f-1 (f(y)) and c ∈ f-1 (f(x)) ∪ f-1 (f(d)). it follows f(a) ∈ {f(x), f(b)} and f(c) ∈{f(x), f(d)}. we must prove that there is y ∈h, such that y∈ a o d ∩ b o c, that is f(y) ∈ {f(a), f(d)} ∩ {f(b), f(c)}. we have the following situations: 1) if f(a) = f(x) = f(c) then we can choose y = a ; 2) if f(a) = f(x) and f(c) = f(d), then we choose y = d ; 3) if f(a) = f(b), then we choose y = a. therefore, (h, o) is a join space. now, let us consider r a reflexive and symmetric relation on h. let us consider the following hyperoperation on h ∀(x,y) ∈ h2 , x or x = {z(z,x) ∈r}, x or y = x or x ∪ y or y. theorem (h, or) is a join space. proof. the associativity is immediate and ∀ x ∈ h, we have x orh = x orx ∪ ∪a∈h a ora = h, since ∪a∈h a ora = ∪a∈h {z(z,a) ∈ r} = h (r is reflexive). so, (h, or) is a commutative hypergroup. notice that a∈ a ora ⇔ (a,a)∈r. let us check now that a / b ≈ c / d ⇒ a ∈ a ord ≈ b orc. let x∈h, such that a ∈ x or b and c ∈ x or d. we have a ∈ {t(t,x) ∈ r or (t,b) ∈ r}, whence (a,x)∈r or (a,b)∈r. similarly, (c,x)∈r or (c,d)∈r. we must prove that there is y∈h, such that y∈ a ord and y∈ b or c, that is [(y,a) ∈r or (y,d) ∈r] and [(y,b) ∈r or (y,c) ∈r], or equivalently, [(y,a) ∈r and (y,b) ∈r] or [(y,a) ∈r and (y,c) ∈r] or [(y,d)∈r and (y,b)∈r] or [(y,d) ∈r and (y,c)∈r]. we have the following situations: 1) (a,x) ∈r and (c,x) ∈r. since r is symmetric, it follows (x,a) ∈r and (x,c) ∈r. in this case, we can choose y=x. 2) (a.x) ∈r and (c,d) ∈r. we take y=c and so, (y,d)=(c,d) ∈r and (y,c)=(c,c) ∈r. 3) (a,b) ∈r and [(c,x) ∈r or (c,d) ∈r]. 4) we take y=a, so (y,b)=(a,b) ∈r and (y,a)=(a,a) ∈r. therefore, (h, or) is a join space. 81 remark. if r is the relation defined as follows: (x,y) ∈r ⇔ f(x)=f(y) where f : h →u, then (h, or ) is the join space presented at the beginning. references [1] chvalina, j., relational product of join spaces determined by quasi-orders, sixth.int.congress on aha(1996), 15-23, democritus university of thrace press. [2] corsini, p., prolegomena of hypergroup theory, aviani editore, 1993. [3] corsini, p., on the hypergroups associated with binary relations, j. multiple-valued logic, 2000, vol.5, p.407-419. [4] corsini, p., binary relations and hypergroupoids, italian j. of pure and appl. math., n. 7, 2002. [5] corsini, p., leoreanu, v., hypergroups and binary relations, algebra universalis, 43, 2000, p. 321-330. [6] corsini, p., leoreanu, v., applications of hyperstructure theory, kluwer academic publishers, 2002. [7] leoreanu, v., leoreanu, l., hypergroups associated with hypergraphs, italian j. of pure and applied mathematics, 4, 1998, p. 119-126. [8] rosenberg, i.g., hypergroups and join spaces determined by relations, italian journal of pure and applied math., nr. 4, 1998, p. 93-101. microsoft word documento1 microsoft word documento1 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 36, 2019, pp. 5-26 5 how to define and test explanations in populations peter j. veazie* abstract solving applied social, economic, psychological, health care and public health problems can require an understanding of facts or phenomena related to populations of interest. therefore, it can be useful to test whether an explanation of a phenomenon holds in a population. however, different definitions for the phrase “explain in a population” lead to different interpretations and methods of testing. in this paper, i present two definitions: the first is based on the number of members in the population that conform to the explanation’s implications; the second is based on the total magnitude of explanation-consistent effects in the population. i show that claims based on either definition can be tested using random coefficient models, but claims based on the second definition can also be tested using the more common, and simpler, population-level regression models. consequently, this paper provides an understanding of the type of explanatory claims these common methods can test. keywords: explanation, statistical testing, population regression models, random coefficient models, mixture models 2010 ams subject classification: 62a01; 62f03.† * university of rochester, rochester ny, usa; peter_veazie@urmc.rochester.edu. † received on february 12th, 2019. accepted on may 3rd, 2019. published on june 30th, 2019. doi: 10.23755/rm.v36i1.463. issn: 1592-7415. eissn: 2282-8214. ©peter veazie this paper is published under the cc-by licence agreement. peter veazie 6 1. introduction science provides explanations for facts, phenomena, and other explanations. in applied research that draws on theories from disciplines such as economics, psychology, sociology, and organizational science, among others, this can require testing whether a proposed explanation explains a given fact, phenomenon, and other explanation in a specified population. for example, one might wish to test whether a proposed explanation based on psychology’s regulatory focus theory [1, 2] explains physician risk tolerance in treatment choice (the phenomenon) among primary care physicians in the united states (the population). however, what is meant by the phrase explains in a population? is it that the proposed explanation accounts for the behavior of every member of the population? this is a high bar: one member of the population for whom the explanation does not hold falsifies the claim. is it that the proposed explanation accounts for the behavior of at least one member? this is equally extreme: only one member of a population for whom the explanation holds warrants the claim. the claim is ambiguous. specific definitions are required if such claims are to be understood and tested. this paper provides definitions and identifies methods for testing corresponding explanatory claims. these definitions and the identification of corresponding methods are new contributions that provide conceptual and methodological guidance for researchers who seek to test explanations in populations. the methods themselves, however, are in common use: random coefficient models and population-level regression models. therefore, whereas a goal of this paper is to show which methods can be used to test specific explanatory claims, i do not present the implementation of the methods: there are many textbooks and articles that provide this information [e.g. 3, 4]. for simplicity of presentation, i only reference phenomena as the target of explanation rather than also facts and other explanations; however, any of these are applicable throughout. 2. defining explain before providing the required definitions, i will clarify what i mean by to explain and by an explanation. for this paper, to explain something is to provide a way of understanding it through a conceptual structure that accounts, at least in part, for that which is being explained [5, ch. 9]. the conceptual structure is the explanation. one might imagine there is a single explanation for any given phenomenon. however, for macro-level phenomena, such as organization and human behaviors, there may be multiple ways of understanding them. for example, a human behavioral phenomenon may have sociological explanations, psychological explanations, physiological explanations, and more. any one of the explanations could be referred to as how to define and test explanations in populations 7 an explanation, and no one of them referred to as exclusively the explanation. moreover, an explanation need not be complete. there may be many causal factors or mechanisms that contribute to the phenomenon; however, an explanation might focus only on a subset. an explanation can be intended to provide an understanding of a phenomenon as it is [6, ch. 4], a de re explanation; or, it can be intended to provide an understanding that, nonetheless, contains explicitly presumed falsehoods [7, 8], a de ficta explanation. all terms of a de re explanation refer to presumed real objects, qualities, characteristics, and relationships. designation as a de re explanation does not guarantee truth, nor does it imply the researcher believes it is true; indeed, if the researcher believed the explanation was in fact true, there is no need for further inquiry [9]. moreover, it is common to expect even a well-established theory-based explanation to be incorrect in some unknown way. it is the ontological commitments (the presumption that explanatory terms intend to have real referents) of the explanation’s terms that qualify it as a de re explanation. however, a de ficta explanation contains at least one identified term that is presumed to be false. these are often explanations that contain idealizations (e.g. the discrete energy levels in the bohr model of the atom [10-12], and the rationality of the rational choice model in classic microeconomics [13, 14]) or analogies (e.g. the computer analogy or corporate analogy of information processing in cognitive science [15]). given there need only be a single presumed false term to warrant designation as a de ficta explanation, the remaining terms have substantive ontological commitments. such de ficta explanations are presumed to be partially true [7]. although these definitions do not restrict explanations to those that are amenable to empirical investigation, this paper is written to provide guidance for empirical researchers. consequently, the focus of the discussion herein is on scientific explanations that have empirical implications. in the applied sciences, the goal of both de re and de ficta explanations is to guide interventions, actions, or policy. the pursuit and use of a de re explanation are based on the belief that understanding the world as it is provides assurance that consequent interventions, actions, and policies are more likely to work and generalize, and the causes for their failure are more likely to be identified. the de ficta explanation does not carry as great an assurance in these regards as it includes identified false claims. however, the de ficta explanation can be simpler, easier to develop and understand, and easier to apply. both types of explanation are usefully employed. explanations are often assessed in terms of explanatory power. explanatory power characterizes explanations in terms of explanatory virtues such as generality, coherence, accuracy, and predictive ability, among others [8, 16]. it has been qualitatively defined in terms of the scope of questions it peter veazie 8 can address [16], and it has been the basis for formal probability-based measures [17-20]. however, for the purposes of applied science another aspect of power can be useful: effective power. applied researchers often focus on the ability to influence specific outcomes and therefore seek explanations to inform actions that can produce specific effects. for example, researchers may seek to reduce systolic blood pressure, decrease expected expenditures, or expand social networks rather than seek to account for variation. to achieve such goals, it can be important to assess a phenomenon’s responsiveness to an explanation, its effective power. effective power is different from accuracy and predictive power (the abilities to account for and predict phenomena and behavior). consider an explanation of the relationship between behavior y and explanatory factor x for two individuals w and v. suppose the effect of the explanation on y can be modeled as a simple linear function of x with a positive coefficient, in which variable x completely determines y for individual w and only partially determines y for individual v: yw = βw⋅xw and yv = βv⋅xv + εv. the predictive power for w is greater than that for v; indeed, the predictive power for w is perfect, whereas it is only partial for v, due to the additional term ev. however, if βw = βv, then variable x has the same relationship with behavior y for both and thereby having the same effective power: a difference in x corresponds to the same difference in y for both w and v. if βv > βw, then the explanation has greater effective power for v, even though it has greater predictive power for w. effective power represents the responsiveness to the explanation whereas accuracy and predictive power represents the extent of y accounted for by the explanation. as an analogy, consider a regression analysis, in the above example effective power is analogous to β and predictive power is analogous to the coefficient of determination (commonly termed r-square) or an out-of-sample prediction metric. like schupbach and sprenger’s [18] definition of explanatory power, effective power can be negative for a proposed explanation, if the response is counter to that implied by the explanation: for example, the case in which the β’s in the preceding example were in fact negative, contrary to the explanatory implication of positive β’s. we can understand a population-level de re or de ficta explanatory claim as a reductive explanation: an explanation that applies to a population in virtue of an aggregation of the explanation’s application to its members. this is kin to what strevens terms an aggregative explanation [8]. for example, where i how to define and test explanations in populations 9 may seek to explain physician risk tolerance in treatment choice among primary care physicians in the united states, the proposed explanation is regarding its members’ relevant behaviors (the behaviors of individual physicians). so, regardless of the number of members in the population, which can be as few as one, our definition of the phrase a potential explanation explains a given phenomenon in a population represents an aggregation of an individual-level explanation across the members of the population. as stated in the introduction, definitions that require explanation of either every member or only one member of a population are extreme. appropriate definitions are likely somewhere in between. this paper focuses on two: definition 1. an explanation explains a phenomenon in a population if, and only if, it has positive effective power for most members of the population. definition 2. an explanation explains a phenomenon in a population if, and only if, its cumulative magnitudes of effective power among the members of the population for whom the explanation holds exceeds its cumulative magnitudes of effective power among the members of the population for whom the explanation does not hold. these definitions are based on minimal criteria. in the first case, it would be difficult to support an explanatory claim regarding scope if the possible explanation only applied to a minority of population members. in the second case, it would be difficult to support an explanatory claim regarding cumulative power if the possible explanation was associated with less cumulative power than the counter-explanation in a population. however, this is arbitrary, and we need not take the minimal stance. we can generalize the definitions to vary with a definitional parameter q: general definition 1. an explanation explains a phenomenon in a population if, and only if, it has effective power for at least q percent of the members of the population. general definition 2. an explanation explains a phenomenon in a population if, and only if, its cumulative magnitudes of effective power among the members of the population for whom the explanation holds exceeds q times its cumulative magnitudes of effective power among the members of the population for whom the explanation does not hold. the remaining sections focus on the minimal definitions, however the general testing method in section 4.1 can be used to test these general definitions as well. peter veazie 10 3. defining testable implications to test claims based on the preceding definitions, we required corresponding operational definitions in terms of testable implications: operational definition 1. if an explanation explains a phenomenon in a population, then the implications of the explanation hold for most of the members of the population. and, under reasonable presumption (i.e. credible alternative explanations are accounted for), if the implications of the explanation hold for most of the members of the population, then an explanation explains a phenomenon in a population. operational definition 2. if an explanation explains a phenomenon in a population, then the cumulative strength of the explanation’s implications among the members of the population for whom the explanation holds exceeds the cumulative strength of the counter-implications among the members of the population for whom the explanation does not hold. and, under reasonable presumption (i.e. credible alternative explanations are accounted for), if the cumulative strength of the explanation’s implications among the members of the population for whom the explanation holds exceeds the cumulative strength of the counter-implications among the members of the population for whom the explanation does not hold, then an explanation explains a phenomenon in a population. the first conditional in each operational definition allows evidence against each consequent (the testable implications) to provide evidence against the explanatory claim. the second conditional allows evidence for each antecedent (the testable implications) to provide evidence for the explanatory claim. the first conditionals are typically derived from the explanation. the second conditionals draw more upon the weaker condition of presumptionbased reasoning [21], which is grounded in current background knowledge and is thereby defeasible: future changes in scientific understanding can negate the conditional. a strong reasonable presumption for the second conditionals is achieved if there are no credible alternative explanations for the testable implications. regarding operational definition 1, we might say, for example, that a regulatory-focus-theory-based explanation explains physician risk tolerance in treatment choice among primary care physicians in the united states if a higher promotion focus (a term in regulatory focus theory [1, 22]) leads physicians to have higher risk tolerance (the explanation’s implication) for more than half of the physicians, accounting for alternative explanations. regarding operational definition 2, we might say that a regulatory-focustheory-based explanation explains physician risk tolerance in treatment choice among primary care physicians in the united states if the cumulative how to define and test explanations in populations 11 magnitudes of effect of promotion focus on risk tolerance among physicians for whom a higher promotion focus leads the physician to have higher risk tolerance exceeds the cumulative magnitudes of effect of promotion focus on risk tolerance among physicians for whom a higher promotion focus leads the physician to have lower risk tolerance (or no relationship). we can generalize the operational definitions, as we did with the original definitions, to vary with a definitional parameter q: general operational definition 1. if an explanation explains a phenomenon in a population, then the implications of the explanation hold for q percent of the members of the population. and, under reasonable presumption (i.e. credible alternative explanations are accounted for), if the implications of the explanation hold for q percent of the members of the population, then an explanation explains a phenomenon in a population general operational definition 2. if an explanation explains a phenomenon in a population, then the cumulative strength of the explanation’s implications among the members of the population for whom the explanation holds exceeds q times the cumulative strength of the counter-implications among the members of the population for whom the explanation does not hold. and, under reasonable presumption (i.e. credible alternative explanations are accounted for), if the cumulative strength of the explanation’s implications among the members of the population for whom the explanation holds exceeds q times the cumulative strength of the counter-implications among the members of the population for whom the explanation does not hold, then an explanation explains a phenomenon in a population. to test claims based on the preceding definitions, we start by identifying the proposed explanation’s implications. specifically, we presume an explanation-implied relationships g between variables y and x (as defined in the context of the phenomenon and explanation), with parameter θ : ( ; )y g x θ= , such that ( ; ) e g x x θ∂ ∈ ∂  , xx∀ ∈ . (1) this is to say that we have a proposed explanation e of a phenomenon that implies variables x and y are related by some, perhaps unknown, function g such that for all values x in range x the derivative of g with respect to x (or the difference quotient if x is a discrete set) is in the set e . note that the implications can be more general: the ( ; )g x x θ∂ ∂ term can be a vector of derivatives across multiple x variables. and, the implications for any given derivative can be multi-part, having different ranges for the derivative across peter veazie 12 different x-values. however, for ease of presentation this paper focuses on single-part implications. a simple example is g specified as a linear relationship, y = α + β⋅x, such that the proposed explanation e implies dy/dx > 0, i.e. e = (0,∞), for all positive values of x, i.e. x = (0,∞). applying this equation to all members of ω, we can say that if β is positive for most members of a population ω, then e explains by definition 1. if the sum of the magnitude of β’s across all members of ω for whom β>0 exceeds the sum of the magnitudes of β’s across all members for whom β≤0, then e explains by definition 2. to formalize the concept of explain, consider the following variable ∆ defined for w ∈ ω and x ∈ x : ( ; ( )) ( , ) g x w w x h x ∂ θ  ∆ =  ∂  . (2) the function h provides the relevant interpretation for explain. the two functions considered in this paper for h provide interpretations for explain as the scope of the explanation (definition 1 above) and as the power of the explanation (definition 2 above). these are detailed below. we can use two functions to separate the ∆’s into groups. the first picks out ∆ for the explanation-implied range of values for ∂g/∂x, and the second picks out ∆ for the range of values outside of the explanation-implied range— the counter-explanation range: ( ; ( )) ( , ) if ( , ) 0 otherwise e g x w w x w x x+ ∂ θ ∆ ∈ ∆ = ∂   (3) and ( ; ( )) ( , ) if ( , ) 0 otherwise e g x w w x w x x− ∂ θ ∆ ∉ ∆ = ∂   . (4) the sum of the magnitudes of ∆+ across population ω at value x reflects the extent of the proposed explanation’s implications in the population at x (the interpretation depending on h). the sum of the magnitudes of ∆across population ω at value x reflects the extent of counter-explanation implications in the population at x. for both specifications of h discussed below, a useful formalization of explain is to say that the proposed explanation explains a phenomenon in a population if the accumulated magnitudes of ∆ is larger in the explanationhow to define and test explanations in populations 13 implied region than in the counter-explanation region for all points in a specified set b of x-values. for arbitrary value x in b, this implies for both definitions 1 and 2 that ( ) ( ) { : ( ) } { : ( ) } ( , ) ( , ) w w x w x w w x w x w x w x+ − ∈ = ∈ = ∆ > ∆∑ ∑ . (5) for the generalized definitions this is ( ) ( ) { : ( ) } { : ( ) } ( , ) ( , ) w w x w x w w x w x w x q w x+ − ∈ = ∈ = ∆ > ⋅ ∆∑ ∑ , (6) where q° = q/(100 − q) for generalized definition 1, and q°= q for generalized definition 2. denoting the statement e explains p in ω on set b as ( , , , )e e p bω , the corresponding claims are ( , , , )e e p b trueω = and ( , , , )e e p b falseω = . the claim that the proposed explanation holds (i.e. ( , , , )e e p b trueω = ) is asserted if for all points x in the set b the proposed explanation’s implication exceeds that for the counter-explanation implication. the claim that the proposed explanation does not hold (i.e. ( , , , )e e p b falseω = ) is asserted if there exists at least one point in b for which the counter-explanation implication exceeds the proposed explanation’s implication. it is useful to take b to be one of two sets: either a singleton {x} or the phenomenologically-relevant range x . claims ( , , , )xe e p ω  are what we may consider when testing whether a proposed explanation explains, whereas point-wise claims ( , , ,{ })e e p xω are useful in understanding where in the range of x-values the claims ( , , , )xe e p ω  fail, if indeed they fail, or at which points of x is the underlying proposed explanation is either least or most powerful. there are occasions, however, when ( , , , )xe e p ω  is too strict: do we really want to say a proposed explanation does not explain in a population because it doesn’t hold at a single point x? for example, suppose economic demand follows the predicted relationship with price at all prices except at $1, do we say the price-demand theory does not hold in the population because of this singular exception? perhaps we should account for how important it is that the explanation hold at $1, or account for how many people face a price of $1 for the good being considered. we can address these concerns by taking a weighted average of x-specific effects across the range of x-values in x using a probability distribution for x conditional on x ∈ x . denoting this general explanatory claim as ( , , )e e p ω , it requires the weighted sum across all x-values being considered and thereby can balance non-explanatory points of x with other strongly explanatory points. its interpretation depends on peter veazie 14 the definition of the probability for x [23]. for example, it can be helpful to consider claims regarding ( , , )e e p ω in terms of random variables defined on population ω, with equal probabilities assigned to each member of ω. using ω as its domain, the variable x provides the value x that each member is facing. the probability distribution of x therefore represents the actual normalized frequency of x in the population, and consequently ( , , )e e p ω is based on the corresponding weighted average across this distribution. figure 1 presents an example in which the explanation implies negative derivatives of g with respect to x, i.e. ( , 0)e = −∞ for all values of x in x , but for which the actual g is as shown. it is clear, regarding the point-wise explanations, that the claim ( , , ,{ })e e p x trueω = holds true only for x less than x*, but ( , , ,{ })e e p x falseω = for all x greater than x*. consequently, due to the existing values of x for which the explanatory implications do not hold (i.e. for x > x*), the overall claim is therefore ( , , , )xe e p falseω = . on the other hand, for f(x) denoting the density of x based on ( | )xp x x ∈ , the general claim weighted by this probability is ( , , )e e p trueω = as there is little probability associated with x-values in the contra-explanatory range of derivatives. how to define and test explanations in populations 15 as mentioned above, two specifications for h are considered here. the first, for definition 1, specifies h as a constant function with value 1: ( , ) 1w x∆ = , for all w and x. (7) this leads to ( ; ( )) 1 if ( , ) 0 otherwise e g x w w x x+ ∂ θ ∈ ∆ = ∂   (8) and ( ; ( )) 1 if ( , ) 0 otherwise e g x w w x x− ∂ θ ∉ ∆ = ∂   . (9) by this definition, the sum of the absolute values of ∆+ is the number of people whose x and y relationship follows the proposed explanation’s prediction at specified x-values. the sum of absolute value of ∆is the number of people whose x and y relationship do not follow the proposed explanation’s prediction. a proposed explanation explains at x, by equation 5, if more people in the population follow the prediction than do not when x = x. the second specification, which is used for definition 2, is to define h as the identity function, and therefore ∆ is ( ; ( ))( , ) g x w w x x ∂ θ ∆ = ∂ . (10) this leads to ( ; ( )) ( ; ( )) if ( , ) 0 otherwise e g x w g x w d w x x x+ ∂ θ ∂ θ ∈ ∆ = ∂ ∂  (11) and ( ; ( )) ( ; ( )) if ( , ) 0 otherwise e g x w g x w d w x x x− ∂ θ ∂ θ ∉ ∆ = ∂ ∂  . (12) the corresponding definition for explain compares the accumulated magnitudes of ∆ between the explanation-implied region and the counterpeter veazie 16 explanation region, which reflects the cumulative effective power of the explanation in the population. the difference between these two corresponding specifications for h is that the first claim, e1, focuses on the scope (the number or proportion of the population consistent with the explanation), whereas the second claim, e2, focuses on the cumulative power of the explanation. it is possible for an explanation to apply to a minority of people in the population, but it does so with greater strength in the magnitude of ∆ among this minority than is the magnitude of ∆ for the majority, who are not in the implied region. in this case the explanation would be considered as explaining in terms of e2, which uses the identity function for h, but not in terms of e1, which uses the constant function for h. on the other hand, in the case where a majority has only a tiny magnitude of ∆ in the implied region but a minority has a large magnitude of ∆ in the non-implied region, the explanation would be considered as explaining in terms of e1 but not in terms of e2. this is analogous to considering the importance of whether a treatment has a larger total positive effect among those that benefit relative to the total negative affect among those who do not benefit (e2), or whether the treatment simply positively affects a greater proportion of people regardless of how small the effect (e1). which definition is appropriate depends on the research goal. these definitions are population-specific. consequently, it is possible for a proposed explanation to explain in one population but not another. moreover, it is possible to not explain in a population but to explain in one of its subpopulations, and vice versa. consider a population ω made up of two subpopulations ω1 and ω2: it is possible for ( , , , )xe e p falseω = , and yet ( , , , )x1e e p trueω = . this is often the advantage of doing subgroup analysis, to determine if a proposed explanation holds better in one group than another. indeed, the primary scientific aim of a study may be to identify for which population the proposed explanation holds. 4. testing explanations 4.1 general tests using random coefficient models how do we empirically test a hypothesis of the form ( , , , )xe e p trueω = or ( , , , )xe e p falseω = ? a general approach is conceptually straightforward, albeit empirically challenging. this approach is based on the idea that if we can estimate the distribution of ∆, we can estimate the conditions for ( , , , )xe e p trueω = and ( , , , )xe e p falseω = . to estimate how to define and test explanations in populations 17 the distribution of ∆, assuming our data generating process can support it, we can use a random coefficient model [3]. suppose we define random variables (or random vectors) y, x, θ, and  on the population ω, representing a population model such that ( ) ( ( ); ( )) ( )y w g x w w w= θ + , for w ∈ ω. (13) if we have a data generating process with n observations, i ∈ {1, …n}, we can consider the mixture model for the regression of y on x: ( | ) ( | , ) ( | )i i i i i i ie y x e y x df xθ θ= ⋅∫ . (14) substituting equation 13 for yi on the right-hand side of equation 14, yields ( | ) ( , ) ( | ) ( | , ) ( | )i i i i i i i i i i ie y x g x df x e x df xθ θ θ θ= ⋅ + ⋅∫ ∫  , (15) which is the expected value of g plus the expected value of  , each conditioned on x = x: ( | ) ( , ) ( | ) ( | )i i i i i i i ie y x g x df x e xθ θ= ⋅ +∫  . (16) under the assumption that the expected value of the error terms is 0 for all values of x, the regression is ( | ) ( , ) ( | )i i i i i ie y x g x df xθ θ= ⋅∫ . (17) the derivative of g and the estimated distribution for f can be used to obtain a distribution for ∆ and thereby estimate the conditions for the explanation to hold. notice, however, from equation 17 the function g must be the expected value of y conditional on values of x and θ, i.e. equation 14. consequently, if a statistically adequate model [24] for ( | , )i i ie y x θ can be empirically determined, an explicit a priori specification for g is not required, only hypotheses regarding implications (e.g. derivatives or difference quotients) are required a priori. estimation can be achieved using a mixture model, or random parameters model, if the study design and context allow for estimation of such a model. it is best to use a non-parametric estimator for f(θ | x) since results in this case are likely to be very sensitive to the distribution (we are integrating under different regions of the distribution, rather than merely estimating parameters of the distribution). for example, we may consider using fox et al’s nonparametric estimator for the distribution of random effects [25, 26]. suppose we can assume the error term is independent of x and that we have a relationship such that peter veazie 18 ( , ) i ixi ig x e θθ ⋅= , (18) which has the derivative ( , ) i ixi i i dg x e dx θθ θ ⋅= ⋅ . (19) the expected value of y conditional on x is ( )( | ) ( | )i ixi i i ie y x e df x θ θ⋅= ⋅∫ . (20) with an estimator for f, denoted as f̂ , we can estimate, using numeric integration, the population proportion of those whose derivative falls in the explanation-implied range for any x, ˆˆ ( ) ( 0) ( | )xp x i e df xθθ θ⋅= ⋅ > ⋅∫ , (21) in which ( )i ⋅ is an indicator function returning 1 if its argument is true, 0 otherwise. equation 21 can be used to test e1. for the general e1, based on the population distribution for x and representative sampling, we would average estimates from equation 21 for each observation in the data to obtain 1 1ˆ ˆ ( ) n i i p p x n = = ∑ . (22) in this case, because xe θ⋅ is always positive, the sign of the derivative is determined by the sign of θ. therefore, we can estimate p̂ based solely on an indicator of θ > 0: ˆˆ ( ) ( 0) ( | )p x i df xθ θ= > ⋅∫ . (23) if we can assume the distribution f is independent of x, i.e. ( | ) ( )f x fθ θ= for all x, then p̂ is not a function of x, and ˆ ( )p x is the same for all x; therefore ˆ ˆˆ ( 0) ( ) 1 (0)p i df fθ θ= > ⋅ = −∫ . (24) in this case we can base our test on ˆ1 (0)f− . using a bootstrap distribution for p̂ (for either equation 23 or equation 24), if a legitimate bootstrap method applies [27], we can test whether e1 is the case using the pvalue ˆ( | 0.5)p p p p≥ = if p̂ ≥ 0.5, and p-value ˆ( | 0.5)p p p p≤ = if p̂ ≤ 0.5 [28]. for testing e2 at specific x-values we calculate how to define and test explanations in populations 19 ( ) ( ) ˆ( ) ( 0) ( 0) ( )x xc x i e i e dfθ θθ θ θ θ θ⋅ ⋅ = > ⋅ ⋅ − ≤ ⋅ ⋅ ⋅ ∫ . (25) for testing the general e2 we average c(x) across the data. again, we can use the bootstrap distribution for f to obtain p-values ˆ( | 0)p c c c≥ = or ˆ( | 0)p c c c≤ = . 4.2 testing e2 using population-level regression models the preceding method, which uses random coefficient models and numeric integration, is complicated—particularly for e2, which represents definition 2. we can greatly simplify our method for testing e2, if the explanation’s implications are regarding positive vs non-positive (or negative vs nonnegative) derivatives. in this case, with an additional statistical assumption, we can use population-level regression models to test the explanation. the argument is as follows: as above, we say that e explains phenomenon p at x if inequality 5 holds. under the definition for e2, in the case of e being either positive, negative, non-positive or non-negative, the absolute values can be moved outside of the summations, ( ) ( ) ( : ( ) } ( : ( ) } ( , ) ( , ) w w x w x w w x w x w x w x+ − ∈ = ∈ = ∆ > ∆∑ ∑ . (26) consider e = (0,∞), i.e. the explanation implies positive derivatives. in this case, for the left-hand side of inequality 26 the summation of the ∆+ across the population with x = x is the same as the summation of the product of each ∆-value and its frequency for ∆-values greater than 0: ( ) ( : ( ) } ( , ) ( | ) w w x w x 0 w x freq x+ ∈ = ∆> ∆ = ∆⋅ ∆∑ ∑ . (27) similarly, regarding ∆−, ( ) ( : ( ) } ( , ) ( | ) w w x w x 0 w x freq x− ∈ = ∆≤ ∆ = ∆⋅ ∆∑ ∑ . (28) therefore, to determine e2 we can consider whether ( | ) ( | ) 0 0 freq x freq x ∆> ∆≤ ∆⋅ ∆ > ∆⋅ ∆∑ ∑ . (29) however, the inequality remains true if both sides are multiplied by the same positive constant. so, if we multiply by 1/nx, denoting the inverse of the population size with value x = x, then peter veazie 20 ( | ) ( | ) 0 0x x freq x freq x n n∆> ∆≤ ∆ ∆ ∆⋅ > ∆⋅∑ ∑ , (30) which is ( | ) ( | ) 0 0 f x f x ∆> ∆≤ ∆⋅ ∆ > ∆⋅ ∆∑ ∑ (31) for f denoting a probability mass function (however, the above logic and derivation also applies to ∆ as a continuous variable in which f is a density, and the summation is replaced with an integral). multiplying the left side of inequality 31 by 1 written as ( 0 | ) ( 0 | ) p x p x ∆ > ∆ > , and multiplying the right side by 1 written as ( 0 | ) ( 0 | ) p x p x ∆ ≤ ∆ ≤ , yields ( | ) ( | ) ( | ) ( | ) ( | ) ( | )0 0 p 0 x p 0 x f x f x p 0 x p 0 x∆> ∆≤ ∆ > ∆ ≤ ∆⋅ ∆ ⋅ > ∆⋅ ∆ ⋅ ∆ > ∆ ≤ ∑ ∑ . (32) because on the left side of this inequality ( | ) ( | , ) ( | ) f x f 0 x p 0 x ∆ = ∆ ∆ > ∆ > , (33) and on the right side of the inequality ( | ) ( | , ) ( | ) f x f 0 x p 0 x ∆ = ∆ ∆ ≤ ∆ ≤ , (34) the inequality can be rewritten as ( | , ) ( | ) ( | , ) ( | ) 0 0 f 0 x p 0 x f 0 x p 0 x ∆> ∆≤ ∆⋅ ∆ ∆ > ⋅ ∆ > > ∆⋅ ∆ ∆ ≤ ⋅ ∆ ≤∑ ∑ . (35) note that on the left side of inequality 35 ( | , ) ( | , ) 0 f 0 x e 0 x ∆> ∆⋅ ∆ ∆ > = ∆ ∆ >∑ , (36) and on the right side of the inequality how to define and test explanations in populations 21 ( | , ) ( | , ) 0 f 0 x e 0 x ∆≤ ∆⋅ ∆ ∆ ≤ = ∆ ∆ ≤∑ . (37) by substitution into equation 35, this yields ( | , ) ( | ) ( | , ) ( | )e 0 x p 0 x e 0 x p 0 x∆ ∆ > ⋅ ∆ > > ∆ ∆ ≤ ⋅ ∆ ≤ . (38) subtracting the right side of inequality 38 from both sides yields part a part b ( | , ) ( | ) ( | , ) ( | )e 0 x p 0 x e 0 x p 0 x 0∆ ∆ > ⋅ ∆ > − ∆ ∆ ≤ ⋅ ∆ ≤ > . (39) since part a of inequality 39 is the absolute value of a positive number (note we are conditioning on ∆ > 0), the absolute value function can be dropped. similarly, since part b is the absolute value of a non-positive number (note we are conditioning on ∆ ≤ 0), its subtraction from a is just the addition of the non-positive number. the absolute value operation can be dropped as well, if we add the components rather than subtract them. this yields ( | , ) ( | ) ( | , ) ( | )e 0 x p 0 x e 0 x p 0 x 0∆ ∆ > ⋅ ∆ > + ∆ ∆ ≤ ⋅ ∆ ≤ > . (40) however, the left-hand side of this inequality is the expected value of ∆ conditional on x. therefore, explanation e2 implies that ( | ) xe x 0 x∆ > ∀ ∈ . (41) since /g x∆ = ∂ ∂ and derivatives are linear operators (and assuming we can interchange the derivative and integral operations), we have ( ) ( ( ) | ) ( | ) g x de g x x e x e x x dx  ∂  ∆ = = ∂  , (42) and therefore, the implication of the explanation we seek to test is the direction of the derivative of the expected value of g: ( ( ) | ) x de g x x 0 x dx > ∀ ∈ . (43) unfortunately, whereas we are likely able to empirically evaluate e(y | x) in a regression analysis, we are not likely able to directly evaluate e(g | x). this is okay, if we can we use e(y | x) to evaluate e(g | x). when can we do this? the requirements are identified by taking the derivative of equation 16 with respect to x: part a part b ( | ) ( ; ) ( | ) ( | ) ( | ) ( ; ) de y x g x f x e x f x d g x d dx x x x θ θ θ θ θ θ ∂ ∂ ∂ = ⋅ ⋅ + ⋅ ⋅ + ∂ ∂ ∂∫ ∫  .(44) peter veazie 22 if the distribution of parameter θ is independent of x (which, in econometrics, is often considered as there is no selection on the gains [29]), then df/dx = 0 and consequently part a of equation 44 is zero. if the error is mean independent of x, then part b is zero (which in econometrics, is often considered as there is no selection on the outcome [29]). under these conditions we have ( | ) ( ; ) ( ) de y x g x f d dx x θ θ θ ∂ = ⋅ ⋅ ∂∫ . (45) but, the right-hand side of equation 45 is the e(∆ | x), which is what we seek to evaluate for our test. consequently, our empirical claim regarding ( , , , )xe e p trueω = for e2 is ( | ) , e x de y x x dx ∈ ∀ ∈ . (46) given the independence assumptions required for parts a and b to equal 0 in equation 44, we can test our proposed explanation e2 by evaluating the derivative of a population-level regression function (the left-hand side of equation 45). if an empirically identified statistically adequate regression function can be used, an explicit functional form for g need not be specified a priori. 5. conclusion knowing how to test a proposed explanation in a population requires having a definition for what is meant by explaining in a population. in this paper i gave definitions in terms of the scope of an explanation and in terms of the power of an explanation. i provided a general method for testing proposed explanations using random parameters models, and i showed when populationlevel regression models can be used to test proposed explanations in terms of effective power. although the tests were presented in terms of the minimal definitions, the tests can be extended to generalized definitions as described above. using the random parameters method, we can define our explanations in terms of the explanation-implied region being a multiple of that for the non-implied region. for example, the proposed explanation explains if it applies to at least 90 percent of the population (rather than at least 50 percent as used in the minimal definitions). i focused on defining and testing proposed explanations; however, in practice the requirements for such a test to provide evidence must be kept in mind. specifically, a proposed explanation’s testable empirical implications need to be specified such that alternative potential explanations for empirical how to define and test explanations in populations 23 implications are accounted for or ruled out, typically by statistical or experimental control. the extent of evidence provided by the test depends on the confidence we have that alternative explanations for empirical findings are indeed ruled out: the less confident we are, the less evidence is provided by the test. this concern is addressed by calibrating our interpretation accordingly. this paper addressed defining and testing explanations in populations. however, it should be noted that the general definition can be the basis for addressing estimation goals as well as testing goals. using the random coefficients method the proportion of a population that conforms to the explanation’s implications or the effective power can be estimated along with corresponding bootstrapped confidence intervals. peter veazie 24 references [1] e.t. higgins, promotion and prevention: regulatory focus as a motivational principle, in: m.p. zanna (ed.) adv exp soc psychol, academic press, new york, 1998, pp. 1-46. [2] p.j. veazie, s. mcintosh, b. chapman, j.g. dolan, regulatory focus affects physician risk tolerance, health psychology research, 2 (2014) 85-88. [3] e. demidenko, mixed models : theory and applications, wileyinterscience, hoboken, n.j., 2004. [4] r.b. darlington, a.f. hayes, regression analysis and linear models : concepts, applications, and implementation, guilford press, new york, 2017. [5] m. bunge, philosophy of science: from explanation to justification, rev. ed., transaction publishers, new brunswick, n.j., 1998. [6] t. sider, writing the book of the world, clarenson press ; oxford university press, oxford, new york, 2011. [7] n.c.a. da costa, s. french, science and partial truth : a unitary approach to models and scientific reasoning, oxford university press, oxford ; new york, 2003. [8] m. strevens, depth : an account of scientific explanation, harvard university press, cambridge, mass., 2008. 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[17] m.p. cohen, on three measures of explanatory power with axiomatic representations, brit j philos sci, 67 (2016) 1077-1089. [18] j.n. schupbach, j. sprenger, the logic of explanatory power, philosophy of science, 78 (2011) 105-127. [19] j.n. schupbach, comparing probabilistic measures of explanatory power, philosophy of science, 78 (2011) 813-829. [20] v. crupi, k. tentori, a second look at the logic of explanatory power (with two novel representation theorems), philosophy of science, 79 (2012) 365-385. [21] j.b. freeman, acceptable premises : an epistemic approach to an informal logic problem, cambridge university press, cambridge, uk ; new york, 2005. [22] e.t. higgins, beyond pleasure and pain, am. psychol., 52 (1997) 12801300. [23] p. veazie, what makes variables random : probability for the applied researcher, crc press, taylor & francis group, boca raton, 2017. [24] a. spanos, revisiting haavelmo's structural econometrics: bridging the gap between theory and data, journal of economic methodology, 22 (2015) 171-196. peter veazie 26 [25] j.t. fox, k.i. kim, c.y. yang, a simple nonparametric approach to estimating the distribution of random coefficients in structural models, journal of econometrics, 195 (2016) 236-254. [26] j.t. fox, k.i. kim, s.p. ryan, p. bajari, a simple estimator for the distribution of random coefficients, quant econ, 2 (2011) 381-418. [27] g. efron, r.j. tibshirani, an introduction to the bootstrap, chapman & hall, new york, 1993. [28] p.j. veazie, understanding statistical testing, sage open, 5 (2015). [29] j.j. heckman, e. vytlacil, econometric evaluation of social programs, part 1: causal models, structural models and econometric policy evaluation, in: j. heckman, e. leamer (eds.) handbook of econometrics, elsevier, amsterdam, 2007, pp. 4779-4874. abstract solving applied social, economic, psychological, health care and public health problems can require an understanding of facts or phenomena related to populations of interest. therefore, it can be useful to test whether an explanation of a phenomenon ... 4.2 testing e2 using population-level regression models microsoft word cap7.doc microsoft word cap 1 libro 10.doc microsoft word documento1 moderate-density burst error correcting linear codes moderate-density burst error correcting linear codes bal kishan dass and gangmei sobha* department of mathematics, university of delhi, delhi-110007, india abstract lower and upper bounds for the existence of linear codes which correct burst of length b (fixed) and whose weight lies between certain limits have been presented. keywords : error detecting codes, error correcting codes, burst errors, moderatedensity burst, lower and upper bounds. 1. introduction it is well known that during the process of transmission errors occur predominantly in the form of a burst. however, it does not generally happen that all the digits inside any burst length get corrupted. also when burst length is large then the actual number of errors inside the burst length is also not very less. keeping this in view, we study codes which detect/correct moderate-density burst errors. in the literature, various kinds of burst errors have been studied, viz. open loop bursts (c.f. peterson and weldon, jr. (1972), p.109), closed-loop bursts [campopiono, 1962], c.t. bursts [chien and tang, (1965)], low-density bursts [dass, 1975], etc. one important kind of bursts errors which has not drawn much attention is burst of specified length (fixed) [dass, 1982]. in this paper, we derive lower and upper bounds for linear codes that detect/correct moderate-density bursts of length b (fixed) for some positive integer b. in what follows we shall consider a linear code to be a subspace of n-tuples over gf(q). the weight of a vector shall be considered in the hamming's sense [hamming, 1950] and we shall mean by a burst of length b (fixed), is an n-tuple whose only nonzero components are confined to b consecutive positions, the first of which is nonzero and the number of its starting positions is the first (n-b+1) components. * on study leave from department of mathematics and statistics, college of agriculture, central agricultural university, imphal-795001, india. 47 2. bounds for codes correcting moderate-density bursts in this section, we consider codes correcting moderate-density burst errors. we first obtain a lower bound on the number of check digits which is a necessary conditions for the existence of codes capable of correcting bursts of length b (fixed) with weight lying between w1 and w2 (0 ≤ w1 ≤ w2 ≤ b). before this we prove the following lemma. lemma 1. if j (n, b, w1, w2) denotes the number of n-tuples over gf(q) which form bursts of length b (fixed) with weight lying w1 and w2 (0 ≤ w1 ≤ w2 ≤ b) then (1) ( ) ( ) ( 1i 1w 0w 1wi 21 1qi 1b 1bnw,w,b,nj 2 1 1 + − ≠ −= −      − +−= ∑ ) ) proof. the lemma follows immediately from the fact that the number of bursts of length b (fixed) with weight i is . ( ) ( .1bn1q 1 i 1b i +−−      − − theorem 1. the number of parity check symbols in an (n, k) linear code that corrects all bursts of length b (fixed) of weight lying between w1 and w2 (0 ≤ w1 ≤ w2 ≤ b) is at least. logq [1 + j (n, b, w1, w2)]. (2) proof. since the code has qn-k cosets in all, and all the error patterns are to be in different cosets of the standard array, therefore, in view of lemma 1, we must have qn-k ≥ 1 + j (n, b, w1, w2). (3) the result now follows by taking logarithm on both sides. remarks. if we put w1 = 0 and w2 = b in the above result then weight constraints imposed on the burst becomes redundant and we get qn-k ≥ 1 + [(n–b+1] (q–1)] qb-1, which gives the number of parity check digits in an (n, k) linear code over gf(q) that corrects all bursts of length b (fixed), a result due to dass [1980]. now, if we take, w1 = 0 and w2 = w in (3) we get qn-k ≥ 1 + (n-b+1) (q-1) [1+(q-1)](b-1, w-1), 48 which gives the number of check digits required for linear codes correcting bursts of length b (fixed) with weight w or less (w ≤ b) a result which is again due to dass [1983]. moreover, when w1 = w and w2 = b we obtain qn-k ≥ 1 + (n–b+1) , ( ) 1i 1b 0w 1wi 1q i 1b +− ≠ −= −      − ∑ which gives the number of check digits required in an (n, k) linear code that corrects all bursts of length b (fixed) with weight w or more (w ≤ b) which coincides with the result due to the authors [2000]. now, we first obtain a sufficient condition giving an upper bound for the existence of a code capable of detecting moderate-density burst errors, and then in the theorem following this result we shall obtain an upper bound for codes correcting such errors. theorem 2. given non-negative integers, w1, w2 and b such that 0 ≤ w1 ≤ w2 ≤ b, a sufficient condition that there exists an (n, k) linear code that has no burst of length b (fixed) whose weight lies between w1 and w2, as a code word is qn-k > 1 + . (4) ( i 1w 0w 1wi 1q i 1b2 1 1 −      − ∑ − ≠ −= ) proof. the existence of such a code will be proved by constructing a suitable (n–k) x n parity check matrix h for the desired code. for this we first construct an (n–k) x n matrix h′ and then h will be obtained by reversing altogether the columns of h′. we select any non-zero (n–k)-tuple as the first column of h′. subsequent columns are added to h′ in such a way that after having selected j–1 columns h1, h2,…,hj-1 suitably a nonzero (n–k)-tuple is chosen as the j-th column such that it is not a linear combination of any p columns (w1–1 ≤ p ≤ w2–1) from the immediately preceding b–1 columns hj-b+1, hj-b+2, … , hj-1. such a condition will ensure that a burst of length b (fixed) with weight lying between w1, w2 cannot be a code word in the code whose parity-check matrix is h to be obtained from h′ as prescribed earlier. in other words, hj ≠ a1hj-b+1 + a2hj-b+2 + … + ab-1hj-1, (5) where number of nonzero ai's lies between w1–1 and w2–2. since ai∈gf(q), the possible number of linear combinations on the r.h.s. of (5) including the case when all the αi's are zero is . ( )i 1w 0w 1wi 1q i 1b 1 2 1 1 −      − + ∑ − ≠ −= 49 therefore, a column hj can be added to h′ provided that this number is less than the total number of (n–k)-tuples. at worst, all these linear combinations might yield a distinct sum, therefore, hj can always be added to h′ provided that qn-k > 1 + . (6) ( i 1w 0w 1wi 1q i 1b2 1 1 −      − ∑ − ≠ −= ) it is important note that this relation is independent of j, therefore we can go on adding the columns as long as we wish but for the code of length j we shall stop after choosing j columns. so for j = n we shall added upto n columns. by reversing the order of columns of the matrix h′ = [h1,h2, …, hn], we get the required parity check matrix h = [h1,h2,…, hn] where hi = hn-i+1 (i.e. hn = h1, hn-1 = h2, …, h1 = hn]. thus we obtain the inequality as stated in (4). examples 1. consider the following 5x7 matrix of a (7, 2) code over gf (2).                 = 0100001 1100010 1100100 1101000 1010000 h this matrix has been constructed by the synthesis procedure outlined in the proof of theorem 2 by taking b = 4, w1 = 2 and w2 = 3. the code words of this code are 0000000, 0111101, 1000111, 1111010 which are not bursts of length 4 with weight lying between w1 = 2, w2 = 3. next we derive sufficient condition for codes correcting moderate-density bursts of length b (fixed). theorem 3. a sufficient condition for the existence of an (n, k) linear code over gf(q) which corrects all burst of length b (fixed) with weight lying between w1 and w2 (0 ≤ w1 ≤ w2 ≤ b) is 50 ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∑ ∑ ∑∑ ∑∑∑ − = +++ −≤≤−− −≤++ = −≥+ + − ≠ −= + = + − ≠ −= − ≠ −= −                 −      −−       −       − +           −      −−       − −      − +         −      − +           −      − +−           −      − +> 1b 1k 1rrr 32 2wr1kw 2w2rrr :r,r,r 1 1-b 1k 2wrr :r,r rr 32 1w 0w 1wr 1r 1 i p 0i 1i 1w 0w 1wi i 1w 0w 1wi kn 321 111 2321 322 132 32 32 2 1 11 1 2 1 1 2 1 1 1q r 1kb r 1k r kb 1q r 1kb r 1k 1q r kb 1q i 1b 1q i 1b 1b2n1q i 1b 1q (7) where p = 2w2–1, when b ≥ 2w1, q=2 = 2b – 2w1–1 when b < 2w1, q = 2 and w1–k ≤ r1 ≤ w2–1, w1–k–1 ≤ w2–1, 0 ≤ r2 ≤ 2w2–3, r1 + r2 + r3 ≤ 2w2 –2. proof. the existence of such a code shall be proved as in the previous theorem. a nonzero (n-k)-tuple is chosen as the first column of h′. subsequent columns are added such that after having selected j–1 columns, h1, h2, …, hj–1 suitably a column hj is added provided that it is not a linear combination of any number of columns lying between w1–1 and w2–1 among the immediately preceding b–1 columns hj–b+1, hj–b+2, …, hj–1 together with any number of columns lying between w1 and w2 among any b consecutive columns out of all the j–1 columns selected so far. in other words, hj can be added provided that hj ≠ (α1hj-b+1 + α2hj-b+2 + … + αb-1hj-1) + (βihi +βi+1hi+1 + … + βi+bhi+b-1) (8) where hj's are any b consecutive columns from all the j–1 previously chosen columns and the number of nonzero βi's lies between w1 and w2 whereas the number of nonzero αi's lies between w1–1 and w2–1 along with the case when all the αi's are zero. to compute the number of all possible linear combinations corresponding to r.h.s. of (8) for all possible choices of αj and βi we analyse three different cases as follows. case 1. when hj's are completely confined to the first j–b columns. the number of ways that the coefficients αj's can be selected is . (9) ( i 1w 0w 1wi 1q i 1b2 1 1 −      − ∑ − ≠ −= ) 51 further the number of ways that the coefficients of βi's which form a burst of length b (fixed) with weight lying between w1 and w2 in a vector of length j-b can be selected is (refer lemma 1). j(j–b, b, w1, w2) = (j–2b+1) . (10) ( ) 1i 1w 0w 1wi 1q i 1b2 1 1 + − ≠ −= −      − ∑ therefore, the total number of choices of coefficients in this case is . (11) ( ) ( ) ( )           −      − +−           −      − +− ≠ −= − ≠ −= ∑∑ 1i 1w 0w 1wi i 1w 0w 1wi 1q i 1b 1b2j1q i 1b 2 1 1 2 1 1 case ii. when hj's are completely confined to the immediately preceding b–1 columns. in this case the number of linear combinations corresponding to r.h.s. of (8) is , (12) ( i p 0i 1q i 1b −      − ∑ = ) where p = 2w2–1, when b ≥ 2w1, q=2 = 2b–2w1–1 when b < 2w1, q=2 case iii. when hj's are neither completely confined to the first (j–b) columns nor to the last b–1 columns. j–2b+2 j–2b+1+k j–b j–b+1 j–b+k–1 j–b+k j–b+k+1 j–1 b-1 b r3 r2 r1 let the burst starts from (j–2b+1+k)-th position which can continue upto (j–b+k)-th position, (1 ≤ k ≤ b –1). we select at least w1–1 and at the most w2–1 nonzero components corresponding to j–2b+1+k, j–2b+2+k, …, j–b+k–1 columns together with nonzero components lying between w1–1 and w2–1 corresponding to j–b+1, j–b+2, …, j–1 columns. let r1, r2 and r3 be the number of nonzero components corresponding to columns lying between (j–2b+1+k)-th to (j–b)-th, (j–b+1)-th to (j–b+k–1)-th and (j–b+k+1)-th to (j–1)-th column respectively, such that 52 w1–k ≤ r1 ≤ w2–1, w1–k–1 ≤ r3 ≤w2–1, 0≤r2 ≤ 2w2–3, r1 + r2 + r3 ≤ 2w2–2 (13) therefore total possible number of distinct choices of coefficients is ( ) ( ) ( )∑ ∑ ∑ ∑∑ − = +++ −≤≤−− −≤++ − = − −≥+ ++ − ≠ −=                 −      −−       −       − +           −      −−       − −      − + 1b 1k 1rrr 32 2wr1kw 2w2rrr :r,r,r 1 1b 1k 1w 2wrr :r,r rr 32 1r 1w 0w 1wr 1 321 111 2321 322 2 132 32 32i 2 1 11 1q r 1kb r 1k r kb 1q r 1kb r 1k 1q r kb (14) thus total possible number of distinct linear combinations corresponding to (8), which cannot be equal to hj including zero vector is ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∑ ∑ ∑∑ ∑∑∑ − = +++ −≤≤−− −≤++ − = + −≥+ + − ≠ −= = + − ≠ −= − ≠ −=                 −      −−       −       − +           −      −−       − −      − +         −      − +           −      − +−           −      − + 1b 1k 1rrr 32 2wr1kw 2w2rrr :r,r,r 1 1b 1k rr 3 2wrr :r,r 2 1r 1w 0w 1wr 1 i p 0i 1i 1w 0w 1wi i 1w 0w 1wi 321 111 2321 321 32 132 32 1 2 1 11 2 1 1 2 1 1 1q r 1kb r 1k r kb 1q r 1kb r 1k 1q r kb 1q i 1b 1q i 1b 1b2n1q i 1b 1 (15) therefore, the j-th column can be added to h′ provided that qn-k > m, (16) where m denotes expression (15). for the existence of an (n, k) desired code relation (16) should hold for j = n so that it is possible to add upto nth column to form an (n–k) x n matrix. thus we have constructed the matrix h′ = [hi], (hi denotes the i-th column from which we obtain the required parity check matrix h = [hi], (hi denotes the i-th column) by reversing its column altogether, i.e. hi → hn-i+1. this proves the result. 53 remarks 1. if we take w1 = 0, w2 = b in (16) the weight constraints becomes redundant. hence the bound gives qn-k > qb-1 [qb-1 (n–2b+1) (q–1)+1] which is a result due to dass [1980]. 2. if we put w1 = 0, w2 = w, in (16) we get the bound obtained by dass [1982], which is a sufficient condition for the existence of low-density burst correcting code that corrects all bursts of length b (fixed). example 2. consider the following matrix 6x9 of a (9,3) code over gf(2) which can correct all bursts of length 4 (fixed) with weight 2 or 3.                     = 011000001 101000010 001000100 101001000 001010000 001100000 h this matrix is constructed by the synthesis procedure outline in the proof of theorem 3. it can be seen from the table 1 that syndromes of all the correctable error patterns are distinct and therefore the null space of this matrix gives the desired code. 54 table 1 error pattern syndrome error pattern syndrome 110000000 000011 111000000 000111 011000000 000110 011100000 001110 001100000 001100 001110000 011100 000110000 011000 000111000 111000 000011000 110000 000011100 001111 000001100 011111 000001110 011110 101000000 000101 110100000 001101 010100000 001010 011010000 011010 001010000 010100 001101000 110100 000101000 101000 000110100 100111 000010100 101111 000011010 110001 000001010 100001 000001101 010101 100100000 001001 101100000 001011 010010000 010010 010110000 010110 001001000 100100 001011000 101100 000100100 110111 000101100 010111 000010010 010001 000010110 101110 000001001 101010 000001011 101011 references 1. chien, r.t. and d.t. tang (1965) : on definition of a burst, ibm j. research & develop., 9(4), 292-293. 2. dass, b.k. (1975) : a sufficient bound for codes correcting bursts with weight constraints, journal of the association for computing machinery (j. acm), vol.22, no.4, 501-503. 3. dass, b.k. (1980) : on a burst-error correcting linear codes, j. infor. & opt. sciences, vol.1, no.3, 291-295. 55 4. dass, b.k. (1983) : low density burst-error correcting linear codes, adv. in mngt. studies, vol.2, pp.375-385. 5. dass, b.k. and g. sobha (2000) : high-density burst error correcting linear codes, unpublished. 6. hamming, r.w. (1950) : error detecting and error correcting codes, bell system tec. j., 29, 147-160. 7. peterson, w.w. and e.j. weldon, jr. (1972) : error-correcting codes, cambridge, mass : the m.i.t. press. 8. sharma, b.d. and s.n. gupta (1975) : on the existence of moderatedensity burst codes, journal of mathematical sciences, vol.10, 8-12. 56 blockwise repeated burst error correcting linear codes b.k. dass department of mathematics university of delhi delhi 110 007, india dassbk@rediffmail.com surbhi madan ∗ department of mathematics shivaji college (university of delhi) new delhi 110 027, india surbhimadan@gmail.com abstract this paper presents a lower and an upper bound on the number of parity check digits required for a linear code that corrects a single subblock containing errors which are in the form of 2-repeated bursts of length b or less. an illustration of such kind of codes has been provided. further, the codes that correct m-repeated bursts of length b or less have also been studied. keywords: error locating codes, error correction, burst errors, repeated burst errors ams subject classification: : 94b20, 94b65, 94b25. ∗corresponding author ratio mathematica 20, 2010 97 i introduction error detecting codes and error correcting codes have been the traditional areas of study in the field of coding techniques on error control in digital data transmission. wolf and elspas [12] introduced a coding technique, error-locating codes (el codes), lying midway between error detection and error correction. in an error locating code, each block of received digits is regarded as being subdivided into mutually exclusive sub-blocks, and codes have been devised that permit the detection of errors occurring within a single sub-block, the sub-block containing errors being identified. in ordinary decision feedback systems using error detection the receiver tests each block of received digits for the presence of errors. if errors are detected, the receiver requests the retransmission of the corrupted block of digits alone and this process is repeated for each incoming block. one drawback of the conventional system is that long block lengths (which are desirable for increased coding efficiency) can result in a low data rate when the reception of large amount of data is called for. however, the use of el codes can soften this conflict between short and long block lengths by providing an additional design parameter. the overall constraint block length can be long to provide efficient coding while the length of the sub-blocks can be relatively short in order to keep the data rate up. codes developed at the early stages were meant mainly to detect and correct random errors. however, it was observed later that in many channels the likelihood of the occurrence of errors is more in adjacent positions rather than their occurrence in a random manner. in this spirit, abramson[1] developed codes correcting single and double adjacent errors. the concept of clustered errors, commonly called burst errors, was generalized further in the work due to fire [7]. a burst, also known as an open loop burst, of length b may be defined as follows: definition 1. a burst of length b is a vector whose all non-zero components are among some b consecutive components, the first and the last of which is non-zero. it was observed that in very busy communication channels, errors repeat themselves. similar is a situation when errors occur in the form of a burst. the development of codes for such kind of repeated burst errors is useful for ratio mathematica 20, 2010 98 improving upon the efficiency of some communication channels. not only do repeated bursts emerge as a natural generalization of bursts, but considering a recent study by srinivas, jain, saurav and sikdar [11], where the changes in the neuronal network properties during epileptiform activity in vitro in planar twodimensional neuronal networks cultured on a multielectrode array using the in vitro model of stroke-induced epilepsy have been explored, we observe that the study of these codes is significant. the study of codes that detect repeated open-loop bursts was initiated by berardi, dass and verma [2] and for correction of such errors by dass and verma [6] . an m-repeated burst (open-loop) of length b is defined as follows: definition 2. an m-repeated burst of length b is a vector of length n whose only non-zero components are confined to m distinct sets of b consecutive components, the first and the last component of each set being non-zero. for example, (001032000020310000313200) is a 3-repeated burst of length 4 over gf(4). in particular, a 2-repeated burst (open-loop) of length b is defined as: definition 3. a 2-repeated burst of length b is a vector of length n whose only non-zero components are confined to two distinct sets of b consecutive components, the first and the last component of each set being non-zero. wolf and elspas [12] obtained results in the form of bounds over the number of parity-check digits required for binary codes capable of detecting and locating a single sub-block containing random errors. a study of such error locating codes in which errors occur in the form of bursts was made by dass [3]. further, these results were extended to the codes correcting burst errors occurring within a subblock (refer dass and tyagi [5]). in our earlier paper [4] the authors obtained bounds over the number of parity-check digits required for codes detecting 2repeated and m-repeated bursts of length b or less occurring within a single subblock, the sub-block containing errors being identified. in this paper we extend our study to the correction of repeated bursts occurring within a sub-block. the development of codes correcting repeated burst errors within a sub-block improves the efficiency of the communication channel as it reduces the number of parity ratio mathematica 20, 2010 99 check digits required. the results that follow have been described in terms of the following parameters: the block of n digits, consisting of r check digits, and k = n−r information digits, is subdivided into s mutually exclusive sub-blocks, each sub-block containing t = n/s digits. ii bounds for codes correcting 2-repeated bursts in this section, we obtain bounds on the number of parity check digits of a code capable of correcting 2-repeated bursts of length b or less occurring within a single sub-block. we note that an (n,k) linear el code over gf(q) capable of detecting and locating a single sub-block containing 2-repeated burst of length b or less must satisfy the following two conditions: (i) the syndrome resulting from the occurrence of any 2-repeated burst of length b or less within any one sub-block must be non-zero. (ii) the syndrome resulting from the occurrence of any 2-repeated burst of length b or less within a single sub-block must be distinct from the syndrome resulting likewise from any 2-repeated burst of length b or less within any other sub-block. further, an (n,k) linear code over gf(q) capable of correcting an error requires the syndromes of any two vectors to be distinct irrespective of whether they belong to the same sub-block or different sub-blocks. so, in order to correct 2-repeated bursts of length b or less lying within a sub-block the following conditions need to be satisfied: (iii) the syndrome resulting from the occurrence of any 2-repeated burst of length b or less within a single sub-block must be distinct from the syndrome resulting from any other 2-repeated burst of length b or less within the same sub-block. (iv) the syndrome resulting from the occurrence of any 2-repeated burst of length b or less within a single sub-block must be distinct from the syndrome resulting likewise from any 2-repeated burst of length b or less within any other sub-block. remark 1. we observe that condition (ii) is the same as condition (iv). also, for computational purposes condition (i) is taken care of by condition (iii). from this we infer that correction of errors requires more strict conditions than location of ratio mathematica 20, 2010 100 errors. so we need to consider conditions (iii) and (iv) or equivalently conditions (ii) and (iii) for correction of the said type of errors. we first obtain a lower bound over the number of parity check digits required for such a code. theorem 1. the number of check digits r required for an (n,k) linear code over gf(q), subdivided into s sub-blocks of length t each, that corrects 2-repeated bursts of length b or less lying within a single corrupted sub-block is atleast logq { 1+s [ q2b−2 { q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )} −1 ]} . (1) proof. let v be an (n,k) linear code over gf(q) that corrects 2-repeated burst of length b or less within a single corrupted sub-block. the maximum number of distinct syndromes available using r check digits is qr. the proof proceeds by first counting the number of syndromes that are required to be distinct by the two conditions and then setting this number less than or equal to qr. since the code is capable of correcting all errors which are 2-repeated bursts of length b or less within any single sub-block, any syndrome produced by a 2repeated burst of length b or less in a given sub-block must be distinct from any such syndrome likewise resulting from another 2-repeated burst of length b or less in the same sub-block(refer to condition (iii)). moreover, syndromes produced by 2-repeated bursts of length b or less in different sub-blocks must also be distinct by condition (iv). thus, the syndromes of vectors which are 2-repeated bursts, whether in the same sub-block or in different sub-blocks, must be distinct. since there are q2b−2 { q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )} − 1 2-repeated bursts of length b or less within one sub-block of length t, excluding the vector of all zeros( refer dass and verma (2008)) and there are s sub-blocks ratio mathematica 20, 2010 101 in all, we must have at least 1 + s [ q2b−2 { q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )} − 1 ] distinct syndromes, including the all zeros syndrome. therefore, we must have qr ≥ 1 + s [ q2b−2 { q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )} − 1 ] i.e. r ≥logq { 1 + s [ q2b−2 { q + (q − 1)2 ( t − 2b + 2 2 ) + (q − 1) ( t − 2b + 1 1 )} − 1 ]} . remark 2. by taking s = 1 the bound obtained in (1) reduces to logq ( q2b−2 [ q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )]) which coincides with the result for correction of 2-repeated bursts obtained by dass and verma(2008). in the following result, we derive another bound on the number of check digits required for the existence of such a code. the proof is based on the technique used to establish varshamov-gilbert-sacks bound by constructing a parity check matrix for such a code ( refer sacks (1958) also theorem 4.7, peterson and weldon (1972)). this technique not only ensures the existence of such a code but also gives a method for the construction of the code. theorem 2. an (n,k) linear code over gf(q) capable of correcting 2-repeated burst of length b or less occurring within a single sub-block of length t (4b < t) can always be constructed using r check digits, where r is the smallest integer ratio mathematica 20, 2010 102 satisfying the inequality qr > q2(b−1) { q2(b−1) { (q − 1)3 ( t − 4b + 3 3 ) + (q − 1)2 ( t − 4b + 2 2 ) + q(q − 1) ( t − 4b + 1 1 ) + q2 } + { (s − 1) [ (t − 2b + 1)(q − 1) + 1 ] × [ q2(b−1) { q + (q − 1)2 ( t − 2b + 2 2 ) + (q − 1) ( t − 2b + 1 1 )} − 1 ]}} . (2) proof. we shall prove the result by constructing an appropriate (n − k) × n parity check matrix h for the desired code. suppose that the columns of the first s− 1 sub-blocks of h and the first j − 1 columns h1,h2, · · · ,hj−1 of the sth sub-block have been appropriately added. we now lay down conditions to add the jth column hj to the s th sub-block as follows: since the code is to correct 2-repeated bursts of length b or less within a single sub-block, therefore, by condition (iii), the syndrome of any 2-repeated burst in any sub-block must be different from the syndrome resulting from any other such burst within the same sub-block. therefore the jth column hj can be added provided that hj is not a linear combination of the immediately preceding b − 1 or fewer columns hj−b+1, · · · ,hj−1 of the sth sub-block together with any three distinct sets of b or fewer consecutive columns each from amongst the first j − b columns h1,h2, · · · ,hj−b. in other words, hj 6= (α1hj−b+1 + α2hj−b+2 + · · · + αb−1hj−1)+ 3∑ l=1 (βl1hl1 + βl2hl2 + · · · + βlbhlb ), (3) where αi,βli ∈ gf(q) and lb ≤ j − b. the number of ways in which the coefficients αi can be selected is clearly qb−1. to enumerate the coefficients βi is equivalent to enumerate the number of 3-repeated bursts of length b or less in a vector of length j − b which is (refer dass and verma(2008)) q3(b−1) { (q − 1)3 ( j − 4b + 3 3 ) + (q − 1)2 ( j − 4b + 2 2 ) + q(q − 1) ( j − 4b + 1 1 ) + q2 } . ratio mathematica 20, 2010 103 therefore, the total number of possible choices for αi and βi on the r.h.s of (3) is q4(b−1) { (q − 1)3 ( j − 4b + 3 3 ) + (q − 1)2 ( j − 4b + 2 2 ) + q(q − 1) ( j − 4b + 1 1 ) + q2 } . (4) further, by condition (iv), hj can be added to the s th sub-block provided hj is not a linear combination of the immediately preceding b − 1 or fewer columns together with one set of b or fewer columns from amongst the first j−b columns together with linear combination of any two sets of b or less consecutive columns within any other sub-block. i.e. hj 6= (α1hj−b+1 + α2hj−b+2 + · · · + αb−1hj−1)+ (β1hi + β2hi+1 + · · · + βbhi+b−1)+ (γ1hi1 + γ2hi1+1 + · · · + γbhi1+b−1)+ (δ1hi2 + δ2hi2+1 + · · · + δbhi2+b−1) (5) where αp,βp,γp,δp ∈ gf(q), i + b − 1 ≤ j − b and not all γp and δp are zero. (the last two terms in the above sum correspond to any two sets of b or less consecutive columns within any one of the other sub-block.) the number of ways in which the coefficients αp can be selected is clearly qb−1. to enumerate the coefficients βp is equivalent to enumerate the number of bursts of length b or less in a vector of length j−b which is qb−1[(j−2b+1)(q−1)+1] (refer fire [7]). therefore, the total number of possible choices for αp and βp on the r.h.s of (5) is q2(b−1)[(j − 2b + 1)(q − 1) + 1]. (6) also, the number of linear combinations corresponding to the last two terms on the r.h.s. of (5) is the same as the number of 2-repeated bursts of length b or less within a sub-block of length t, excluding the vector of all zeros; which is ( refer dass and verma (2008)) q2b−2 { q + (q − 1)2 ( t− 2b + 2 2 ) + (q − 1) ( t− 2b + 1 1 )} − 1. since there are s − 1 previously chosen sub-blocks, the number of such linear combinations becomes ratio mathematica 20, 2010 104 (s − 1) [ q2b−2 { q + (q − 1)2 ( t − 2b + 2 2 ) + (q − 1) ( t − 2b + 1 1 )} − 1 ] . (7) thus, the number of linear combinations to which hj can not be equal to is the product computed in expr. (6) and expr. (7). i.e. expr.(6) ×expr.(7). (8) thus, the total number of linear combinations that hj can not be equal to is the sum of linear combinations in (4) and (8). at worst, all these combinations might yield a distinct sum. therefore, hj can be added to the sth subblock of h provided that qr > q2(b−1) { q2(b−1) { (q − 1)3 ( j − 4b + 3 3 ) + (q − 1)2 ( j − 4b + 2 2 ) + q(q − 1) ( j − 4b + 1 1 ) + q2 } + { (s − 1) [ (j − 2b + 1)(q − 1) + 1 ] × [ q2(b−1) { q + (q − 1)2 ( t − 2b + 2 2 ) + (q − 1) ( t − 2b + 1 1 )} − 1 ]}} . for completing the sth sub-block of length t, replacing j by t gives the result as stated in (2). remark 3. by taking s = 1 in (2) the bound reduces to qr > q4(b−1) { (q − 1)3 ( t − 4b + 3 3 ) + (q − 1)2 ( t − 4b + 2 2 ) + q(q − 1) ( t − 4b + 1 1 ) + q2 } which coincides with the condition for existence of a code correcting 2-repeated bursts of length b or less( refer dass and verma(2008)). we conclude this section with an example. example 1 consider a (26, 10) binary code with a 16 × 26 parity-check matrix ratio mathematica 20, 2010 105 h given by h =   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0   this matrix has been constructed by the synthesis procedure outlined in the proof of theorem 2 by taking b = 3, s = 2, t = 13 over gf(2) ( ms excel program was used for the construction of the matrix). it can be seen from the table 1 that the syndromes of all distinct 2-repeated bursts of length 3 or less whether in the same sub-block or in different sub-blocks are different, showing thereby that the code that is the null space of this matrix corrects all 2-repeated bursts of length 3 or less occurring within a sub-block. ratio mathematica 20, 2010 106 table 1 error patterns syndrome vectors sub-block 1 s.no. error vector syndrome s. no. error vector syndrome 1 1111110000000 0000000000000 1111110000000000 44 0111101000000 0000000000000 0111101000000000 2 1110111000000 0000000000000 1110111000000000 45 0111010100000 0000000000000 0111010100000000 3 1110011100000 0000000000000 1110011100000000 46 0111001010000 0000000000000 0111001010000000 4 1110001110000 0000000000000 1110001110000000 47 0111000101000 0000000000000 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0011100010100000 36 1110000000000 0000000000000 1110000000000000 79 0011100001010 0000000000000 0011100001010000 37 0111111000000 0000000000000 0111111000000000 80 0011100000101 0000000000000 0011100000101000 38 0111011100000 0000000000000 0111011100000000 81 0011111000000 0000000000000 0011111000000000 39 0111001110000 0000000000000 0111001110000000 82 0011101100000 0000000000000 0011101100000000 40 0111000111000 0000000000000 0111000111000000 83 0011100110000 0000000000000 0011100110000000 41 0111000011100 0000000000000 0111000011100000 84 0011100011000 0000000000000 0011100011000000 42 0111000001110 0000000000000 0111000001110000 85 0011100001100 0000000000000 0011100001100000 43 0111000000111 0000000000000 0111000000111000 86 0011100000110 0000000000000 0011100000110000 ratio mathematica 20, 2010 107 sub-block 1 s.no. error vector syndrome s. no. error vector syndrome 87 0011100000011 0000000000000 0011100000011000 134 0000111100000 0000000000000 0000111100000000 88 0011110000000 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0010101000000000 226 0101000000111 0000000000000 0101000000111000 273 0010100100000 0000000000000 0010100100000000 227 0101101000000 0000000000000 0101101000000000 274 0010100010000 0000000000000 0010100010000000 ratio mathematica 20, 2010 109 sub-block 1 s.no. error vector syndrome s. no. error vector syndrome 275 0010100001000 0000000000000 0010100001000000 322 0000101000001 0000000000000 0000101000001000 276 0010100000100 0000000000000 0010100000100000 323 0000101000000 0000000000000 0000101000000000 277 0010100000010 0000000000000 0010100000010000 324 0000010111100 0000000000000 0000010111100000 278 0010100000001 0000000000000 0010100000001000 325 0000010101110 0000000000000 0000010101110000 279 0010100000000 0000000000000 0010100000000000 326 0000010100111 0000000000000 0000010100111000 280 0001011110000 0000000000000 0001011110000000 327 0000010110100 0000000000000 0000010110100000 281 0001010111000 0000000000000 0001010111000000 328 0000010101010 0000000000000 0000010101010000 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sub-block 2 s.no. error vector syndrome s. no. error vector syndrome 1104 0000000000000 1100011100000 0000111111110111 1151 0000000000000 0110000000101 0001101001000101 1105 0000000000000 1100001110000 0000111110110101 1152 0000000000000 0110110000000 0000110110110001 1106 0000000000000 1100000111000 0001110110110001 1153 0000000000000 0110011000000 0000101101111110 1107 0000000000000 1100000011100 0001001000100001 1154 0000000000000 0110001100000 0000011011010000 1108 0000000000000 1100000001110 0011101111001011 1155 0000000000000 0110000110000 0000110111101111 1109 0000000000000 1100000000111 0011101011001010 1156 0000000000000 0110000011000 0001100100111000 1110 0000000000000 1101010000000 0000101101101100 1157 0000000000000 0110000001100 0001101101000100 1111 0000000000000 1100101000000 0000011011001001 1158 0000000000000 0110000000110 0010101110010101 1112 0000000000000 1100010100000 0000110110101000 1159 0000000000000 0110000000011 0011000111010011 1113 0000000000000 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0000100001000 0001010011001011 1464 0000000000000 0000000010010 0010100111101010 1422 0000000000000 0000100000100 0000111110001100 1465 0000000000000 0000000010000 0000100101100000 1423 0000000000000 0000100000010 0010010000011010 1466 0000000000000 0000000001001 0000000100000001 1424 0000000000000 0000100000001 0001010111001010 1467 0000000000000 0000000001000 0001000001011011 1425 0000000000000 0000100000000 0000010010010000 1468 0000000000000 0000000000100 0000101100011100 1426 0000000000000 0000010011100 0001101100000101 1469 0000000000000 0000000000010 0010000010001010 1427 0000000000000 0000010001110 0011001011101111 1470 0000000000000 0000000000001 0001000101011010 1428 0000000000000 0000010000111 0011001111101110 ratio mathematica 20, 2010 122 iii bounds for codes correcting m-repeated bursts in this section, we extend the results of previous section to the case of m-repeated bursts of length b or less occurring within a single sub-block. similar to the case of correction of 2-repeated burst occurring within a sub-block, an (n,k) linear code over gf(q) capable of correcting any sub-block containing m-repeated burst of length b or less must satisfy the following two conditions: (v) the syndrome resulting from the occurrence of any m-repeated burst of length b or less within a single sub-block must be distinct from the syndrome resulting from any other m-repeated burst within the same sub-block. (vi) the syndrome resulting from the occurrence of any m-repeated burst of length b or less within a single sub-block must be distinct from the syndrome resulting likewise from any m-repeated burst of length b or less within any other sub-block. we now present a lower bound on the number of parity check digits required for such a code. theorem 3. the number of check digits r required for an (n,k) linear code over gf(q), subdivided into s sub-blocks of length t each, that corrects m-repeated bursts of length b or less lying within a single corrupted sub-block is atleast logq { 1 + s [ qm(b−1) (( t−mb + m m ) (q − 1)m+ m−1∑ l=0 ( t−mb + l l ) (q − 1)lqm−1−l ) − 1 ]} . (9) proof. the proof of this result is on the similar lines as that of proof of theorem 1 so we omit the proof. remark 4. by taking s = 1 the bound obtained in (9) reduces to logq { qm(b−1) (( t−mb + m m ) (q − 1)m + m−1∑ l=0 ( t−mb + l l ) (q − 1)lqm−1−l )} . ratio mathematica 20, 2010 123 which coincides with the result for correction of m-repeated burst obtained by dass and verma(2008). remark 5. for m = 2, the bound obtained in (9) coincides with the bound obtained in (1) for the case of 2-repeated bursts. in particular, for m = 1, the bound in (9) reduces to 1 + s ( qb−1 ( (t− b + 1)(q − 1) + 1 ) − 1 ) which reduces to the result for correction of burst of length b or less within a sub-block. in the following result, we present another bound on the number of check digits required for the existence of the code considered in theorem 3. theorem 4. an (n,k) linear code over gf(q) capable of correcting m-repeated burst of length b or less occurring within a single sub-block of length t (2mb < t) can always be constructed using r check digits where r is the smallest integer satisfying the inequality qr >qm(b−1) { qm(b−1) ( (q − 1)2m−1 ( t − 2mb + (2m − 1) 2m − 1 ) + 2m−2∑ l=0 (q − 1)lq2m−2−l ( t − 2mb + l l )) + ( (s − 1) × [ (q − 1)m−1 ( t − mb + (m − 1) m − 1 ) + m−2∑ l=0 (q − 1)lqm−2−l ( t − mb + l l )] × [ qm(b−1) (( t − mb + m m ) (q − 1)m+ m−1∑ l=0 ( t − mb + l l ) (q − 1)lqm−1−l ) − 1 ])} . (10) proof. as in theorem 3, we omit the proof of this result since it can be derived on lines similar to that of theorem 2. ratio mathematica 20, 2010 124 remark 6. by taking s = 1 in (10) the bound reduces to qr > q2m(b−1) ( (q − 1)2m−1 ( t − 2mb + (2m − 1) 2m − 1 ) + 2m−2∑ l=0 (q − 1)lq2m−2−l ( t − 2mb + l l )) which coincides with the sufficient condition for existence of a code correcting m-repeated bursts( refer dass and verma(2008)). remark 7. for m = 2, the result obtained in theorem 4 coincides with the result in theorem 2, for the case of 2-repeated burst of length b or less. for m = 1, the bound in (10) reduces to qb−1 ( qb−1 [ (q − 1)(t− 2b + 1) + 1 ] + (s− 1) [ qb−1 ( (t− b + 1)(q − 1) + 1 ) − 1 ]) which is the condition for existence of a code correcting bursts of length b or less within a sub-block. references [1] abramson, n.m., a class of systematic codes for non-independent errors, ire trans. on information theory it 5(4) (1959) 150-157. [2] berardi, l., dass, b.k. and verma, rashmi, on 2-repeated burst error detecting codes, journal of statistical theory and practice 3(2) (2009) 381-391. [3] dass, b.k., burst error locating codes, j. inf. and optimization sciences 3(1) (1982) 77-80. [4] dass, b.k., madan, surbhi, repeated burst error locating linear codes, communicated. [5] dass, b.k., verma, rashmi, repeated burst error correcting linear codes, asian-european journal of mathematics 1(3) (2008) 303-335. ratio mathematica 20, 2010 125 [6] fire, p., a class of multiple-error-correcting binary codes for nonindependent errors, sylvania report rsl-e-2, sylvania reconnaissance systems laboratory, mountain view, calif (1959). [7] hamming, r.w., error-detecting and error-correcting codes. bell system technical journal 29 (1950) 147160. [8] peterson, w.w., weldon, e.j., jr., error-correcting codes, 2nd ed., the mit press, mass (1972). [9] sacks, g.e., multiple error correction by means of parity-checks, ire trans. inform. theory it 4 (1958) 145-147. [10] srinivas, k.v., jain, r., saurav, s. and sikdar, s.k., small-world network topology of hippocampal neuronal network is lost, in an in vitro glutamate injury model of epilepsy, european journal of neuroscience, 25 (2007) 32763286. [11] wolf, j., elspas b., error-locating codes–a new concept in error control, ieee transactions on information theory 9(2) (1963) 113-117. ratio mathematica 20, 2010 126 ratio mathematica 25(2013), 59–66 issn:1592-7415 the lv-hyperstructures n. lygeros*, t. vougiouklis** *lgpc, university of lyon, lyon, france, **democritus university of thrace, school of education w@lygeros.org, tvougiou@eled.duth.gr abstract the largest class of hyperstructures is the one which satisfy the weak properties and they are called hv-structures introduced in 1990. the hv-structures have a partial order (poset) on which gradations can be defined. we introduce the lv-construction based on the levels variable. key words: hyperstructures, hv-structures, hopes, weak hopes. msc2010: 20n20. 1 fundamental definitions in a set h is called hyperoperation (abbreviation hyperoperation=hope) in a set h, is called any map · : h ×h →p(h) −{∅}. definition 1.1 (marty 1934). a hyperstructure (h, ·) is a hypergroup if (·) is an associative hyperoperation for which the reproduction axiom: hh = hh = h,∀x ∈ h, is valid. definition 1.2 (vougiouklis 1990). in a set h with a hope we abbreviate by wass the weak associativity : (xy)z ∩x(yz) 6= ∅,∀x,y,z ∈ h and by cow the weak commutativity : xy∩yx 6= ∅,∀x,y ∈ h. the hyperstructure (h, ·) is called hv-semigroup if it is wass, it is called hv-group if it is reproductive hv-semigroup, i.e. xh = hx = h,∀x ∈ h. the hyperstructure (r, +, ·) is called hv-ring if both (+) and (·) are wass, the reproduction axiom is valid for (+) and (·) is weak distributive with respect to (+) : x(y + z) ∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅,∀x,y,z ∈ r . n. lygeros, t. vougiouklis definition 1.3 (santilly-vougiouklis). a hyperstructure (h, ·) which contain a unique scalar unit e, is called e-hyperstructure. a hyperstructure (f, +, ·), where (+) is an operation and (·) is a hyperoperation, is called e-hyperfield if the following axioms are valid: 1. (f, +) is an abelian group with the additive unit 0, 2. (·) is wass, 3. (·) is weak distributive with respect to (+), 4. 0 is absorbing element: 0 ·x = x · 0 = 0,∀x ∈ f , 5. there exists a multiplicative scalar unit 1, i.e. 1 ·x = x ·1 = x,∀x ∈ f , 6. for every x ∈ f there exists a unique inverse x−1, such that 1 ∈ x ·x−1 ∩x−1 ·x. the elements of an e-hyperfield are called e-hypernumbers. in the case that the relation: 1 = x ·x−1 = x−1 ·x, is valid, then we say that we have a strong e-hyperfield. construction 1.4. the main e-construction. given a group (g, ·), where e is the unit, then we define in g, a large number of hyperoperations (⊗) as follows: x⊗y = {xy,g1,g2, . . .},∀x,y ∈ g−{e}, and g1,g2, . . . ∈ g−{e} g1,g2, . . . are not necessarily the same for each pair (x,y). then (g,⊗) becomes an hv-group, in fact is hb-group which contains the (g, ·). the hv-group (g,⊗) is an e-hypergroup. moreover, if for each x,y such that xy = e, so we have x⊗y = xy, then (g,⊗) becomes a strong e-hypergroup. for more definitions and applications on hv-structures, see the books and papers [1-20]. the main tool to study hyperstructures are the fundamental relations β∗, γ∗ and ε∗, which are defined, in hv-groups, hv-rings and hv-vector spaces, resp., as the smallest equivalences so that the quotient would be group, ring and vector space, resp. fundamental relations are used for general definitions. thus, an hv-ring (r, +, ·) is called hv-field if r/γ∗ is a field. definition 1.5. let (h, ·), (h,∗) be hv-semigroups defined on the same set h. then (·) is called smaller than (∗), and (∗) greater than (·), iff there exists an f ∈ aut(h,∗) such that xy ⊂ f(x∗ y),∀x,y ∈ h. then we write · ≤ ∗ and we say that (h,∗) contains (h, ·). if (h, ·) is a structure then it is called basic structure and (h,∗) is called hb-structure. 60 the lv-hyperstructures theorem 1.6 (the little theorem). greater hopes than the ones which are wass or cow, are also wass or cow, respectively. this theorem leads to a partial order on hv-structures, thus we have posets. the determination of all hv-groups and hv-rings is very interesting. to compare classes we can see the small sets. the problem of enumeration of classes of hv-structures was started very early but recently we have results by using computers. the partial order in hv-structures restricts the problem in finding the minimals. 2 enumeration theorems theorem 2.1 (chung-choi). there exists up to isomorphism, 13 minimal hv-groups of order 3 with scalar unit, i.e. minimal e-hyperstructures of order 3. theorem 2.2 (bayon-lygeros). • there exist, up to isomorphism, 20 hv-groups of order 2. • there exist, up to isomorphism, 292 hv-groups of order 3 with scalar unit, i.e. e-hyperstructures of order 3. • there exist, up to isomorphism, 6494 minimal hv-groups of order 3. • there exist, up to isomorphism, 1026462 hv-groups of order 3. theorem 2.3 (bayon-lygeros). • there exist, up to isomorphism, 631609 hv-groups of order 4 with scalar unit, i.e. e-hyperstructures of order 4. • there exist, up to isomorphism, 8.028.299.905 abelian hv-groups of order 4. theorem 2.4 (bayon-lygeros). • the number of abelian hv-groups of order 4 with scalar unit (i.e. abelian e-hyperstructures) in respect with their automorphism group are the following 61 n. lygeros, t. vougiouklis |aut (hv)| 1 2 3 4 6 8 12 24 — — — 32 — 46 5510 626021 • there are 63 isomorphism classes of hyperrings of order 2. • there are 875 isomorphism classes of hv-rings of order 2. • there are 33277642 isomorphism classes of hyperrings of order 3. in all the above results we construct the poset of hyperstructures of order 2 and 3 in the sense of inclusion for hyperproducts. we compute the betti numbers of the poset of hv-groups of order 2 and we have the following results: (1, 5), (2, 4), (3, 6), (4, 4), (5, 1). we also compute the betti numbers of the poset of hypergroups of order 3 and we have the following results: (1, 59), (2, 168), (3, 294), (4, 438), (5, 568), (6, 585), (7, 536), (8, 480), (9, 358), (10, 245), (11, 160), (12, 66), (13, 29), (14, 10), (15, 2), (16, 1). we explicitly compute the cayley subtables of the minimal e-hyperstructures with h = {e,a,b} and we have for the products (aa, ab, ba, bb) the following results: (b; e; e; a), (eb; a; a; e), (e; ab; ab; e), (a; eb; eb; a), (ab; ea; ea; e), (h; eb; a; ea), (h; a; eb; ea), (a; h; h; e), (b; h; h; e), (a; h; h; b), (h; b; a; h), (h; a; b; h), (h; e; ab; h). 3 construction theorems there are several ways to organize such posets using hyperstructure theory. we present now a new construction on posets and we name this lvconstruction since it is based on gradations where the levels are used as variable. thus lv means level variable. theorem 3.1. the lv-construction i consider the set pn of all hv-groups defined on a set of n elements. take the following gradation on pn based on posets: level 0 (or grade 0), denoted by g0, is the set of all minimals of pn. level (grade) 1, denoted by g1, is the set of all hv-groups obtained from minimals by adding one only element to anyone of the results of the products of two elements on the minimals of pn, i.e. of g0. level 2 (or grade 2), denoted by g2, is the set of all hv-groups obtained from minimals by adding only two elements to anyone of the results of the products of two elements of the minimals g0. then inductively the level k is defined, denoted by gk. in the 62 the lv-hyperstructures case that an hv-group is obtained by adding k1 elements of one minimal and by adding k2 elements of another minimal then we consider that it belongs to the level min(k1,k2). denote by r the cardinality of the minimals, |g0| = r, and by s the number of levels. take any hv-group with r elements corresponding to the r elements of g0, so we have an hv-group (g0,∗). then we define a hope on pn = g0 ∪ g1∪, . . . ,∪gs−1, as follows x⊗y = { x∗y, ∀x,y ∈ g0 gκ+λ, ∀x ∈ gκ,y ∈ gλ, where (κ,λ) 6= (0, 0) then the hyperstructure (pn,⊗) is an hv-group where its fundamental group is isomorphic to zs, thus we have pn/β ∗ ≈ zs. proof. let us correspond, numbered, the levels with the elements of zs : gi → i, i = 0, . . . ,s− 1. from the definition of (⊗) any hyperproduct of elements from several levels, apart of g0, equals to only one special set of hv-groups that constitute one level. moreover we have x⊗y = g0,∀x ∈ gκ,y ∈ g−κ, for any κ 6= 0. that means that the elements of g0are β*-equivalent. therefore all elements of each level are β∗-equivalent and there are no β∗-equivalent elements from different levels. that proves that pn/β ∗ ≈ zs. the above is a construction similar to the one from the book [15, p.27] a generalization of the above construction is the following: theorem 3.2. the lv-construction ii consider a graded finite poset with n elements: pn = g0∪g1∪, . . . ,∪gs−1, with s levels (grades) g0, g1, . . . , gs−1, such that s−1∑ i=0 |gi| = n. 63 n. lygeros, t. vougiouklis denoting |g0| = r, we consider two hv-groups (e, ·) and (s,∗) such that |e| = r, |s| = s and moreover s has a unit single element e. then we take 1:1 maps from e onto g0 and from s onto {g0, g1, . . . , gs−1}, so we obtain two hv-groups: (g0, ·) and ( g = {g0, g1, . . . , gs−1},∗ ) where e = g0 corresponds to the single element e. we define a hope on pn as follows: x⊗y = { x ·y, ∀x,y ∈ g0 gκ ∗ gλ, ∀gκ, gλ ∈ g, where (κ,λ) 6= (0, 0) then the hyperstructure (pn,⊗) is an hv-group where its fundamental group is isomorphic to the fundamental group of (s,∗), therefore we have (pn,⊗)/β ∗ ≈ (s,∗)/β∗. proof. from the reproductivity of (g,∗), for each gκ,κ 6= 0, there exists a gλ such that g0 ∈ gκ ∗ gλ. but g0 is a single element of (s,∗), therefore we have g0 = gκ ∗ gλ. then, by the definition, for any x ∈ gκ, y ∈ gλ we have, x⊗y = g0. therefore, all the elements of g0 are β∗-equivalent. on the other side, from the definition, all elements of each level are β*-equivalent and they are β∗-equivalent elements with different levels if and only if they are β∗-equivalent in (g,∗). in other wards they follow exactly the β∗-equivalence of (g,∗). that proves that (pn,⊗)/β ∗ ≈ (s,∗)/β∗. with this lv-construction we can define the poset for hv-groups of order 2. so we get a non-connected poset with betti numbers for the two subposets (1,4), (2,4), (3,1) and (1,1), (2, 4), (3,6). references [1] r. bayon, n. lygeros, les hypergroupes abéliens d’ordre 4. structure elements of hyper-structures, xanthi, (2005), 35–39. [2] r. bayon, n. lygeros, les hypergroupes d’ordre 3. italian j. pure and applied math., 20 (2006), 223–236. [3] r. bayon, n. lygeros, advanced results in enumeration of hyperstructures, j. algebra, 320 (2008), 821–835. [4] s-c. chung, b-m. choi. hv-groups on the set {e, a, b}. italian j. pure and applied math., 10 (2001), 133–140. 64 the lv-hyperstructures [5] p. corsini, v. leoreanu, applications of hypergroup theory, kluwer academic publ., 2003. [6] b. davvaz, v. leoreanu, hyperring theory and applications, int. academic press, 2007. [7] b. davvaz, r.m. santilli, t. vougiouklis, studies of multi-valued hyperstructures for characterization of matter and antimatter systems, j. computational methods in sciences and engineering 13, (2013), 37–50. [8] b. davvaz, s. vougioukli, t. vougiouklis, on the multiplicative-rings derived from helix hyperoperations, util. math., 84, (2011), 53–63. [9] s. hoskova, hv-structures are fifteen, proc. of 4th international mathematical workshop fast vut brno, czech republic, (2005), 55–57, http://math.fce.vutbr.cz/∼pribyl/workshop 2005/ prispevky/hoskova.pdf [10] p. kambaki-vougioukli, a. karakos, n. lygeros, t. vougiouklis, fuzzy instead of discrete, ann. fuzzy math. informatics, v.2, n.1 (2011), 81– 89. [11] p. kambaki-vougioukli, t. vougiouklis, bar instead of scale, ratio sociologica, 3, (2008), 49–56. [12] f. marty, sur un généralisation de la notion de groupe, 8ème congrès math. scandinaves, stockholm, (1934), 45–49. [13] t. vougiouklis, generalization of p-hypergroups, rend. circ. mat. palermo, s.ii,36 (1987), 114-121. [14] t. vougiouklis, the fundamental relation in hyperrings. the general hyperfield, 4thaha congress, world scientific (1991), 203–211. [15] t. vougiouklis, hyperstructures and their representations, monographs in math., hadronic press, 1994. [16] t. vougiouklis, some remarks on hyperstructures, contemp. math., 184, (1995), 427–431. [17] t. vougiouklis, on h v-rings and h v-representations, discrete math., 208/209 (1999), 615–620. [18] t. vougiouklis, a hyperoperation defined on a groupoid equipped with a map, ratio mathematica on line, n.1 (2005), 25–36. 65 n. lygeros, t. vougiouklis [19] t. vougiouklis, ∂-operations and h v-fields, acta math. sinica, (engl. ser.), v.24, n.7 (2008), 1067-1078. [20] t. vougiouklis, bar and theta hyperoperations, ratio mathematica, 21, (2011), 27–42. 66 ratio mathematica vol. 32, 2017, pp. 63-75 issn: 1592-7415 eissn: 2282-8214 teaching least squares in matrix notation guglielmo monaco1, aniello fedullo2∗ 1department of chemistry and biology ”a. zambelli”, university of salerno, italy gmonaco@unisa.it 2department of physics ”e. r. caianiello”, university of salerno, italy afedullo@unisa.it received on: 30-05-2017. accepted on: 27-06-2017. published on: 30-06-2017 doi:10.23755/rm.v32i0.335 c©g. monaco and a. fedullo abstract material for teaching least squares at the undergraduate level in matrix notation is reported. the weighted least squares equations are first derived in matrix form; equivalence with the standard results obtained by standard algebra are then given for the weighted average and the simplest linear regression. indicators of goodness of fit are introduced and interpreted. eventually a basic equation for resampling is derived. keywords: coefficient of determination, weighted sample mean, resampling, undergraduate education. 2010 ams subject classifications: 62j05. 63 g. monaco and a. fedullo 1 introduction statistics is a never missing topic in first degree courses of scientific programs. very soon, often at the second year undergraduate, the basic knowledge of random variables and distributions, is complemented by the simple linear regression, as a necessary tool for the interpretation of experimental data gathered in the laboratories. indeed, the critical practice of linear regressions often forms students’ basic awareness of data analysis. the advent of powerful and handy softwares on the one hand has reduced the effort required to the students for accomplishing the needed calculations, on the other hand has given them the possibility to easily perform more advanced statistical analyses [1, 2], which they cannot really understand on the grounds of the course. one of simplest of such more advanced analyses is the consideration of more regressors, the starting point of multivariate data analysis [3]. although a specific course at the last undergraduate or first graduate year can be much profitable, we experienced that, provided the students have a basic knowledge in linear algebra, the generalized least squares can be thought at the second year undergraduate with reasonable appreciation from the class. reference textbooks on the matter, seemingly more diffused in the community of econometrics [5] than in that of experimental sciences [6], are not missing. however, we needed to compact some fundamental concepts and equations, and still convince the students that the more general matrix form of the least squares allows to easily retrieve the results obtainable with standard algebra. thus, we prepared the following material, and we presented it effectively in a 12 hours module together with numerical exercises. although our lessons obviously have a significant overlap with reference textbooks, the revised simple linear regression and the introduction of the (adjusted) weighted coefficient of determination are not easily retrieved from any of the textbooks known to us. 2 matrix form of the weighted least squares we consider n measures {y1,y2, ...,yn} and for each of them, say the i-th one, the regressors {xi1,xi2, ...,xip}, here assumed constant, which are generally coming from different associated measures. we will assume that for each measure the first regressor equals one, xi1 = 1, in order to take into account the so called intercept. the linear regression model connects the above quantities by yi = p∑ j=1 xijβj + εi i = 1, 2, ...,n (1) 64 teaching least squares in matrix notation where β1,β2, ...,βp are the parameters to be estimated and ε1,ε2, ...,εn are random errors, assumed independent and possibly normally distributed, with mean 0 and standard deviations σ1,σ2, ...,σn. ordinary least squares (ols) and weighted least squares (wls), also called homoskedastic and heteroskedastic regressions, are the names used to distinguish the special case of equal values for all standard deviations from the case of different values. the equations for wls of course also apply to the special ols case. dividing eq. 1 by σi, i.e. given zi := yi σi , qij := xij σi , ςi := εi σi , and using the matrix notation, the model is written as z = qβ + ς, (2) or, equivalently, w 1 2 y = w 1 2 xβ + w 1 2 �, where w is a diagonal matrix whose elements wii := wi = σ −2 i are known as statistical weights, z and β are column matrices of n and p elements, respectively, q is a matrix of dimension n×p. it should be noticed that qβ is the expectation value of z, i.e. qβ =< z >. under these hypotheses the least squares method gives an estimate of the model parameters by the minimization with respect to β of the functional ss := ςtς = (z −qβ)t (z −qβ) (3) = (z −qβ)t (z −qβ) = ztz − 2βtqtz + βtqtqβ, (4) where it has been considered that βtqtz = ztqβ. the estimates of the parameters by the least squares method are the solutions of the equations ∂ss ∂βi = 0, for i = 1, 2, ...,p, one for each model parameter. the computation of the derivative with respect to the vector of the parameters gives: −qtz + qtqβ = 0, (5) whose solution β̂ = v qtz (6) is, by definition, the least squares estimator of β, where v := c−1, and c := qtq, which we will assume always invertible. 65 g. monaco and a. fedullo we note that β̂ is an unbiased estimator of β, indeed form eqs. 2 and 6 we have < β̂ >= v qt < z >= v qtqβ = v cβ = β. (7) an unbiased behavior also characterizes the weighted sample mean. indeed, eq. 5 for β = β̂ gives qtz = qt ẑ which, rewritten in the original variables, is xtwy = xtwŷ. from this and from the initial hypothesis xi1 = 1, for any i, one gets ∑ i wiyi = ∑ i wiŷi, which divided by ∑ i wi shows that the weighted sample mean of the fitted values equals the weighted sample mean of the measures: ȳw = ŷw. (8) given δ:= β̂ −β from eqs. 6 and 7 one gets δ = v qtς, (9) which allows to easily compute the covariance matrix of the parameters, showing that it coincides with v < δδt >= v qt < ςςt > qv = v qtiqv = v, where i denotes the identity matrix. the standard deviations of the estimators of the parameters are given by the square roots of the diagonal elements of v . using the fitted values, one can write z = ẑ + (z − ẑ) = qβ̂ + e, where e is known as the vector of residuals, whose analysis is object of much concern in literature. the fitted values are often written as ẑ = qβ̂ = qv qtz =: hz, (10) where we have introduced the symmetric matrix h, which is known as hat matrix as it ’puts the hat on z’. this matrix is readily verified to be idempotent, h2 = qv qtqv qt = h, a feature which readily allows to demonstrate the useful property of orthogonality of residuals and fitted values: (z − ẑ)t ẑ = zt (i −h)hz = 0. given 66 teaching least squares in matrix notation sse := minβss = e te, the expansion 4, with ς in place of z and δ in place of β, can be rewritten as: sse = (ς −qδ)t (ς −qδ) = ςtς−2δtqtς + δtcδ = ςtς − δtcδ, where we have considered that qtς = cδ thanks to eq. 9. given ssr := δtcδ, which as ss and sse is non-negative, the preceding equation becomes sse = ss −ssr whose interpretation is that the error in the estimation of the parameters, yielding a nonzero ssr, reduces the sum of squares ss which could have been computed with the expectation value 〈z〉 = qβ. the average of sse can be easily computed considering that δtcδ = tr [ δδtc ] , and then 〈sse〉 = 〈 ςtς 〉 −tr [〈 δδt 〉 c ] = n−tr (v c) = n−p, known as the number of degrees of freedom, denoted by ν. notation. in the following sxx,w, sxx,w e sxy,w indicate respectively the sample variance, sum of squares and weighted covariance, defined from the weighted sample mean ȳw := ∑ i wiyi∑ i wi in analogous manner to the corresponding unweighted means. we recall that their expressions are sxx,w = x2w− x̄2w, sxx,w = sxx,w ∑ i wi e sxy,w = xyw − x̄wȳw, where xy := (x1y1, ...,xnyn). 3 indicators for the goodness of fit besides reporting the best-fit parameters and the resulting fitted values, it is customary to give compact indicators of the goodness of fit. a method which is widely used in the analysis of experimental data consists in the chi-squared test: the hypothesis that the model is correct is not rejected, at the appropriate level of significance, if sse assumes values close to 〈sse〉, i.e., for any number of parameters, if χ2r = sse ν is close to 1. values of χ2r larger or smaller than 1 are then considered as indicators of a poor fit or, respectively, overfitting. a different approach considers weighted sample means. defining the weighted coefficient of determination r2w as the square of the weighted sample correlation 67 g. monaco and a. fedullo coefficient syŷ,w√ syy,wsŷŷ,w between data y and fitted values ŷ = xβ̂ and thus limited by 0 ≤ r2w ≤ 1, one has that 1 −r2w = sse syy,w = see syy,w , (11) showing that r2w = 1 iff sse = 0, i.e. iff all residuals are zero. therefore the greater the value of r2w the better the agreement. eq. 11 can be proven thanks to the orthogonality relation discussed above. the vector yww 1 2 , where w 1 2 is a column vector of elements w 1 2 i , is orthogonal to the vector of residuals z − ẑ, by virtue of eq. 8. therefore the orthogonality of residuals and fitted values, eq. 2, still holds if the fitted values are translated by yww 1 2 . the vector relationship z −yww 1 2 = (ẑ −yww 1 2 ) + (z − ẑ), (12) graphically sketched in figure 1, allows to assess that syy,w = sŷŷ,w + sse, (13) whose interpretation is that sŷŷ,w/syy,w is the fraction of variability of the data explained by the knowledge of q, i.e. by the regression, and sse/syy,w is the unexplained one, i.e. that coming from errors. still from eq. 12 one gets sŷy,w = (ẑ − ŷww 1 2 )t (z −yww 1 2 ) = (ẑ − ŷww 1 2 )t (ẑ − ŷww 1 2 ) = sŷŷ,w (14) and then r2w = s2ŷy,w sŷŷ,wsyy,w = sŷŷ,w syy,w . (15) insertion of eq. 15 in eq. 13 readily gives eq. 11. in order to discourage the introduction of models too complicated for the data examined, it has been introduced the adjusted determination coefficient r2a = 1 − (1 −r 2 w) n− 1 n−p , obtained substituting the unbiased variances in the rhs of eq. 11. it often happens that standard deviations of experimental data are only approximately known. a common assumption is that the standard deviations σi are known but for a factor k: σi = kσ̃i, with the σ̃i known a priori. if the adjustment 68 teaching least squares in matrix notation w1/2yw ẑ ẑ − w1/2yw z − w1/2yw z w1/2yw z − ẑ z − ẑ figure 1: the residuals z− ẑ are orthogonal to both the estimates ẑ and the vector yww 1 2 . 69 g. monaco and a. fedullo of k leads to a good fitting for the model, χ2r should be close to ν. using this value, one gets ν = n∑ i=1 (yi − ŷi)2 k2σ̃2i , and a trial value for k is obtained as k = √ 1 ν ∑ (yi − ŷi)2 σ̃2i . 4 basic applications 4.1 (weighted) mean the model y = β1 + ε has an n× 1 matrix of relative regressors, whose i-th element is qi1 = w 1 2 i application of eq. 7 soon gives as the best fit parameter the weighted mean β̂ = v qz = ∑ i wiyi∑ i wi = ȳw and its variance is the sum of the weights: σ2β = v11 = ∑ i wi. 4.2 wls for a straight line the standard linear regression considers the model y = a + bx. in the above notation a = β1 and b = β2 and the regressor matrix is x = [ 1 1 ... 1 x1 x2 ... xn ]t the matrix of relative regressors will be then q = [ √ w1 √ w2 ... √ wn√ w1x1 √ w2x2 ... √ wnxn ]t , the vector of relative data 70 teaching least squares in matrix notation z = [ √ w1y1 √ w2y2 ... √ wnyn ]t , and c = ∑ i wi [ 1 x̄w x̄w x2w ] , whose inverse gives the covariance matrix of the parameters v = 1 sxx,w [ x2w −x̄w −x̄w 1 ] . the standard deviations of the estimators of the parameters will be then [ σâ σb̂ ] = [ √ v11√ v22 ] = 1√ sxx,w [ √ x2w 1 ] , and the estimated parameters will be [ â b̂ ] = v qtz = 1 sxx,w [ x2w −x̄w −x̄w 1 ][ yw xyw ] = [ ȳw − sxy,w sxx,w x̄w sxy,w sxx,w ] , which in case of all equal weights (homoskedastic regression) have the simpler expression [ ȳ − sxy sxx x̄ sxy sxx ]t . 4.3 revised simple linear regression we now give a simplified approach for the bivariate weighted linear regression: given 1 := [1 1 ... 1]t , we subtract yw1 from the data and from the fitted data and, considering that yw = ŷw = a + bx̄w, we obtain y − yw1 = b(x− x̄w1) + ε, (16) which, with z := w 1 2 (y −yw1), q := w 1 2 (x− x̄w1) e ς := w 1 2 ε, can be written as in eq. 2, z = bq + ς, but here there is the single parameter b to be determined, as in the example of the weighted mean. this means that matrix c is the scalar sxx,w readily invertible, and then v = c−1 = 1 sxx,w . on the other hand, as 71 g. monaco and a. fedullo qtz = (x− x̄w1)tw(y − yw1) = sxy,w + yw ∑ i wi(xi − x̄w) = sxy,w, from eq. 6, one gets again b̂ = sxy,w sxx,w . writing now the model as y − bx = a1 + ε, the example in 4.1 gives for the intercept ȳw − bx̄w, from where, replacing b with its estimator1, one finally gets â = ȳw − b̂x̄w, as in 4.2. it is to be considered, however, that this simplification leads to loose information on the covariance of the a and b parameters, which should then be recover ex post (appendix). 4.4 resampling and the best-fit parameters a remarkable representation of the p best-fit parameters can be obtained if one tries to determine them from the ( n p ) p-elements subsets of the original set of n measures [4]. let s(s) be a p×n matrix obtained from the n×n identity matrix, upon selecting the p rows whose indices form subset s, with s = 1, . . . ( n p ) . let also m [k|v] be the matrix obtained from matrix m upon replacing its k-th column with vector v. for any p-elements subset s, the data needed for the wls are stored in vector z(s) = s(s)z and the square matrix q(s) = s(s)q; the best-fit parameters are β̂(s) = q −1 (s) z(s) = x −1 (s) w −1/2 (s) w 1/2 (s) y(s) = x −1 (s) y(s), (17) which shows that, for p measures, wls and ols give the same results. use of cramer’s rule on eqs. 5 and 17 gives β̂k = det qtq[k|z] det qtq , (18) and β̂(s)k = det q [k|z] (s) det q(s) = det x [k|y] (s) det x(s) , (19) use of the cauchy-binet theorem to expand the determinants of the equation 18 leads to β̂k = ∑ s det q(s) det q [k|z] (s)∑ s det q(s) det q(s) = ∑ s wsβ̂(s)k∑ s ws , (20) which is the equation for a weighted average of the ols results β̂(s)k with weights ws = (det q(s)) 2. (21) 1implicit use is made of the functional invariance of the estimator b̂. 72 teaching least squares in matrix notation the above representation of the best-fit parameters is the starting point for robust modifications of wls, where the basic idea is to exclude from the mean the more extreme values of β(s)k [7]. 5 conclusions the least squares method, a fundamental piece of knowledge for students of all scientific tracks, is often introduced considering the simple linear regression with only two parameters to be determined. however, the availability of ever more large data sets prompts even undergraduate students to a sounder and wider knowledge of linear regression. here, we have used the linear algebra formalism to compact the main results of the least squares method, encompassing ordinary and weighted least squares, goodness of fit indicators, and eventually a basic equation of re-sampling, which could be used to stimulate interested students in an even broader knowledge of data analysis. the compactness of the equations reported above allow their introduction at the undergraduate level, provided that basic linear algebra has been previously introduced. acknowledgements financial support from the miur (farb2015) is gratefully acknowledged. 73 g. monaco and a. fedullo references [1] r.j. carroll and d. ruppert. transformation and weighting in regression. chapmand and hall, 1988. [2] g. casella and r.l. berger. statistical inference. cengage learning, 2001. [3] a. j. dobson and a. g. barnett. an introduction to generalised linear models. chapman and hall, 2008. [4] r. w. farebrother. relations among subset estimators: a bibliographical note. technometrics, 27(1):85–86, 1985. [5] w.h. greene. econometric analysis. prentice hall, 2012. [6] j. r. taylor. an introduction to error analysis: the study of uncertainties in physical measurements. university science books, 1997. [7] c. f. j. wu. jackknife, bootstrap and other resampling methods in regression analysis. the annals of statistics, 14(4):1261–1295, 1986. 74 teaching least squares in matrix notation appendix moments of â e b̂ averages. < b̂ >= <sxy,w> sxx,w = (x−x̄w1)t w<y−ȳw1> sxx,w = (x−x̄w1)t w<y> sxx,w = (x−x̄w1)t w<a1+bx> sxx,w = b (x−x̄w1)t w(x−x̄w1) sxx,w = b; < â >=< ȳw > −x̄w < b̂ >= a + bx̄w − x̄wb = a the estimators are then unbiased. variances. we shall use the following auxiliary results: i) cov(yi,yj) = δijw −1 i ii) v ar(ȳw) = tr(w)−1 iii) cov(ȳw, b̂) = 0 given d := w(x − x̄w1), we have that dt1 = ∑ i di = 0 and then sxy,w = dt (y − ȳw1) = dty; then v ar(sxy,w) = ∑ ij didjcov(yi,yj) = ∑ i d 2 iw −1 i = ∑ i wi(xi − x̄w) 2 = sxx,w from which v ar(b̂) = v ar(sxy,w) s2xx,w = 1 sxx,w . v ar(â) = v ar(ȳw)−x̄wcov(ȳw, b̂)+x̄2wv ar(b̂) = tr(w)−1 + x̄2w sxx,w = x 2 w sxx,w . . cov(â, b̂) = cov(ȳw − x̄wb̂, b̂) = cov(ȳw, b̂) − x̄wv ar(b̂) = − x̄wsxx,w . proof of the auxiliary results i) cov(yi,yj) = cov(a + bxi + εi,a + bxj + εj) = cov(εi,εj) = δijσ2i = δijw −1 i . ii) v ar(ȳw) = tr(w)−2 ∑ ij wiwjcov(yi,yj) = tr(w) −2 ∑ i wi = tr(w) −1 iii) tr(w)cov(ȳw,sxy,w) = ∑ ij widjcov(yi,yj) = ∑ i di = 0 and then cov(ȳw, b̂) = 0. 2 75 ratio mathematica vol. 33, 2017, pp. 103-114 issn: 1592-7415 eissn: 2282-8214 on p-hv-structures in a two-dimensional real vector space ioanna iliou∗, achilles dramalidis† ‡doi:10.23755/rm.v33i0.381 abstract in this paper we study p-hv-structures in connection with hv-structures, arising from a specific p-hope in a two-dimensional real vector space. the visualization of these p-hv-structures is our priority, since visual thinking could be an alternative and powerful resource for people doing mathematics. using position vectors into the plane, abstract algebraic properties of these p-hv-structures are gradually transformed into geometrical shapes, which operate, not only as a translation of the algebraic concept, but also, as a teaching process. keywords: hyperstructures; hv-structures; hopes; p-hyperstructures. 2010 ams subject classifications: 20n20. ∗democritus university of thrace, school of education, 68100 alexandroupolis, greece; iiliou@eled.duth.gr †democritus university of thrace, school of education 68100 alexandroupolis, greece; adramali@psed.duth.gr ‡ c©ioanna iliou and achilles dramalidis. received: 31-10-2017. accepted: 26-12-2017. published: 31-12-2017. 103 ioanna iliou and achilles dramalidis 1 introduction in a set h 6= ∅, a hyperoperation (abbr. hyperoperation=hope) (·) is defined: · : h ×h → p(h)−{∅} : (x,y) 7→ x ·y ⊂ h and the (h, ·) is called hyperstructure. it is abbreviated by wass the weak associativity: (xy)z∩x(yz) 6= ∅,∀x,y,z ∈ h and by cow the weak commutativity: xy ∩yx 6= ∅,∀x,y ∈ h. the largest class of hyperstructures is the one which satisfy the weak properties. these are called hv-structures introduced by t. vougiouklis in 1990 [13], [14] and they proved to have a lot of applications on several applied sciences such as linguistics, biology, chemistry, physics, and so on. the hv-structures satisfy the weak axioms where the non-empty intersection replaces the equality. the hv-structures can be used in models as an organized devise. the hyperstructure (h, ·) is called hv -group if it is wass and the reproduction axiom is valid, i.e., xh = hx = h, ∀x ∈ h. it is called commutative hv-group if the commutativity is valid and it is called hvcommutative group if it is cow. the motivation for the hv-structures [13] is that the quotient of a group with respect to any partition (or equivalently to any equivalence relation), is an hvgroup. the fundamental relation β* is defined in hv-groups as the smallest equivalence so that the quotient is a group [14]. in a similar way more complicated hyperstructures are defined [14]. one can see basic definitions, results, applications and generalizations on both hyperstructure and hv-structure theory in the books and papers [1], [2], [3], [10], [12], [14], [18]. the element e ∈ h, is called left unit element if x ∈ ex,∀x ∈ h, right unit element if x ∈ xe,∀x ∈ h and unit element if x ∈ xe∩x ∈ ex,∀x ∈ h. an element x′ ∈ h is called left inverse of x ∈ h if there exists a unit e ∈ h, such that e ∈ x′x, right inverse of x ∈ h if e ∈ xx′ and inverse of x ∈ h if e ∈ x′x∩xx′. by el∗ is denoted the set of the left unit elements, by e r ∗ the set of the right unit elements and by e∗ the set of the unit elements with respect to hope (*) [7]. by il∗(x,e) is denoted the set of the left inverses, by i r ∗(x,e) the set of the right inverses and by i∗(x,e) the set of the inverses of the element x ∈ h associated with the unit e ∈ h with respect to hope (*) [7]. the class of p-hyperstructures was appeared in 80’s to represent hopes of constant length [16], [18]. then many applications appeared [1], [2], [4], [5], [6], [8], [9], [15]. vougiouklis introduced the following definition: 104 on p-hv-structures in a two-dimensional real vector space definition 1.1. let (g, ·) be a semigroup and p ⊂ gp 6= ∅. then the following hyperoperations can be defined and they are called p-hyperoperations: ∀x,y ∈ g p∗ : xp∗y = xpy, p∗r : xp ∗ r y = (xy)p p∗l : xp ∗ l y = p(xy). the (g,p∗),(g,p∗r ), (g,p ∗ l ) are called p-hyperstructures. one, combining the above definitions gets that the most usual case is if (g, ·) is semigroup, then xpy = xp∗y = xpy and (g,p) is a semihypergroup, but we do not know about (g,pr) and (g,p l). in some cases, depending on the choice of p, (g,pr) and (g,p l) can be associative or wass. (g,p), (g,pr) and (g,p l) can be associative or wass. in this paper we define in the ir2 a hope which is originated from geometry. this geometrically motivated hope in ir2 constructs hv-structures and p-hvstructures in which the existence of units and inverses are studied. one using the above hv-structures and p-hv-structures into the plane can easily combine abstract algebraic properties with geometrical figures [11]. 2 p-hv-structures on ir2 let us introduce a coordinate system into the ir2. we place a given vector −→p so that its initial point p determines an ordered pair (a1,a2). conversely, a point p with coordinates (a1,a2) determines the vector −→p = −→ op , where o the origin of the coordinate system. we shall refer to the elements x,y,z, . . . of the set ir2, as vectors whose initial point is the origin. these vectors are very well known as position vectors. in [7] dramalidis introduced and studied a number of hyperoperations originated from geometry. among them he introduced in ir2 the hyperoperation (⊕) as follows: definition 2.1. for every x,y ∈ ir2 ⊕ : ir2 × ir2 → p(ir2)−{∅} : (x,y) 7→ x⊕y = = [0,x + y] = {µ(x + y)/µ ∈ [0,1]}⊂ ir2 from geometrical point of view and for x,y linearly independent position vectors, the set x⊕y is the main diagonal of the parallelogram having vertices 0,x,x+y,y. 105 ioanna iliou and achilles dramalidis proposition 2.1. the hyperstructure (ir2,⊕) is a commutative hv-group. now, let p be the set p = [0,p] = {λp/λ ∈ [0,1]} ⊂ ir2, where p is a fixed point of the plane. geometrically, p is a line segment. consider the p-hyperoperation (p∗r(⊕)): definition 2.2. for every x,y ∈ ir2 p∗r(⊕) : ir 2 × ir2 → p(ir2)−{∅} : (x,y) 7→ xp∗r(⊕)y = (x⊕y)⊕p ⊂ ir 2 obviously, (p∗r(⊕)) is commutative and geometrically, for x,y linearly independent position vectors, the set xp∗r(⊕)y is the closed region of the parallelogram with vertices 0,x + y,x + y + p,p. proposition 2.2. the hyperstructure (r2,p∗r(⊕)) is a commutative p-hv-group. 106 on p-hv-structures in a two-dimensional real vector space proof. obviously, xp∗r(⊕)r 2 = r2p∗r(⊕)x = r 2,∀x ∈ r2. for x,y,z ∈ r2 (xp∗r(⊕)y)p ∗ r(⊕)z = {[(x⊕y)p ]⊕z}⊕p = [0,z,x + y + z,x + y + z + 2p,p] xp∗r(⊕)(yp ∗ r(⊕)z) = {x⊕ [(y ⊕z)⊕p ]}⊕p = = [0,x + y + z,x + y + z + 2p,x + y + 2p,x + 2p,p] so, (xp∗r(⊕)y)p ∗ r(⊕)z ∩xp ∗ r(⊕)(yp ∗ r(⊕)z) 6= ∅,∀x,y,z ∈ r 2.2 proposition 2.3. ep∗ r(⊕) = [−p,0] = {−λp/λ ∈ [0,1]} proof. let e ∈ elp∗ r(⊕) ⇔ xep∗r(⊕)x,∀x ∈ r 2 ⇔ x{µλe+µλx+µνp/µ,ν,λ[0,1]}. that means that, µλ = 1 and µλe + µνp = 0 ⇔ e + µνp = 0 ⇔ e = −µνp,−1 ≤ −µν ≤ 0, then e ∈ [−p,0]. so, elp∗ r(⊕) = [−p,0] and according to commutativity erp∗ r(⊕) = [−p,0] = ep∗ r(⊕) = [−p,0]. 107 ioanna iliou and achilles dramalidis proposition 2.4. i(p∗r(⊕))(x,e) = { 1 µλ e−x− ν λ p/µ,λ ∈ (0,1],ν ∈ [0,1]}, where e ∈ ep∗ r(⊕) . proof. let e ∈ ep∗ r(⊕) and x′ ∈ ilp∗ r(⊕) (x,e) ⇔ e ∈ x′p∗r(⊕)x ⇔ e{µλx ′ + µλx + µνp/λ,µ,ν[0,1]}. that means there exist λ1,µ1,ν1[0,1] : e = µ1λ1x ′ + µ1λ1x + µ1ν1p ⇒ x′ = 1 µ1λ1 e−x− ν1 λ1 p,µ1,λ1 6= 0. so, x′ ∈{ 1 µλ e−x− ν λ p/µ,λ ∈ (0,1],ν ∈ [0,1]}. since (p∗r(⊕)) is commutative, we get i(p ∗ r(⊕))(x,e) = { 1 µλ e − x − ν λ p/µ,λ ∈ (0,1],ν ∈ [0,1]}. the p-hyperoperation p∗l(⊕) = p ⊕ (x⊕y) is identical to (p ∗ r(⊕)). but the phyperoperation p∗(⊕) = x⊕p⊕y is different and even more p ∗l (⊕) = (x⊕p)⊕y 6= x⊕ (p ⊕y) = xp∗r(⊕)y, since (⊕) is not associative.2 108 on p-hv-structures in a two-dimensional real vector space definition 2.3. for every x,y ∈ ir2 p∗l(⊕) : r 2 ×r2 → (r2) : (x,y) 7→ xp∗l(⊕)y = (x⊕p)⊕y more specifically, xp∗l(⊕)y = {λκx + λy + λκµp/λ,κ,µ ∈ [0,1]},∀x,y ∈ r 2. geometrically, for x,y linearly independent position vectors, the set xp∗l(⊕) y is the closed region of the quadrilateral with vertices 0,x + y,x + y + p,y. on the other hand the set yp∗l(⊕)x is the closed region of the quadrilateral with vertices 0,x,x + y,x + y + p. so, (xp∗l(⊕)y)∩ (yp ∗l (⊕)x) = [0,x + y,x + y + p] 6= ∅,∀x,y ∈ r 2. proposition 2.5. the hyperstructure (r2,p∗l(⊕)) is a p-hvcommutative group. proof. obviously, xp∗l(⊕)r 2 = r2p∗l(⊕)x = r 2,∀x ∈ r2. for x,y,zr2 (xp∗l(⊕)y)p ∗l (⊕)z = {[(x⊕p)⊕y]p}⊕z ≡ [o,z,x+y+z,x+2p+y+z,y+p+z] xp∗l(⊕)(yp ∗l (⊕)z) = (x⊕p)⊕[(y⊕p)⊕z] ≡ [o,x,x+y+z,x+2p+y+z,y+p+z] 109 ioanna iliou and achilles dramalidis so, (xp∗l(⊕)y)p ∗l (⊕)z ∩xp ∗l (⊕)(yp ∗l (⊕)z) 6= ∅,x,y,z ∈ r 2. 2 proposition 2.6. i) el p∗l (⊕) = r2 ii) er p∗l (⊕) = [0,−p] = {−νp/ν ∈ [0,1]} = ep∗l (⊕) proof. i) notice that x ∈ ep∗l(⊕)x = [0,e+x,e+x+p,x],∀x,e ∈ r 2. so, el p∗l (⊕) = r2. ii) let e ∈ er p∗l (⊕) ⇔ x ∈ xp∗l(⊕)e,∀x ∈ r 2 ⇔ x ∈{λκx+λe+λκµp/λ,κ,µ ∈ [0,1]}. then, there exist µ1,λ1,κ1 ∈ [0,1] : x = λ1κ1x + λ1e + λ11µ1p ⇔ e = 1 / λ1[(1 − λ11)x − λ1κ1µ1p],λ1 6= 0. the last one is valid ∀x ∈ r2, so by setting x = 0 we get e = −κ1µ1p. since µ1,κ1 ∈ [0,1] there exists ν1 ∈ [0,1] : ν1 = κ1µ1 ⇒ e = −ν1p ⇒ e ∈{−νp/ν ∈ [0,1]} = [0,−p]. since er p∗l (⊕) ⊂ r2 = el p∗l (⊕) we get el p∗l (⊕) ∩ er p∗l (⊕) = {−νp/ν[0,1]} = ep∗l (⊕) .2 proposition 2.7. α) ir p∗l (⊕) (x,e) = {−κx − (ν λ + κµ)p/κ,µ,ν ∈ [0,1],λ ∈ (0,1]},e ∈ er p∗l (⊕) . β) ir p∗l (⊕) (x,e) = {e λ −κx−κµp/κ,µ ∈ [0,1],λ ∈ (0,1]},e ∈ el p∗l (⊕) γ) il p∗l (⊕) (x,e) = {−x κ − ( ν λκ + µ)p/κ,λ ∈ (0,1],µ ∈ (0,1]},e ∈ er p∗l (⊕) . 110 on p-hv-structures in a two-dimensional real vector space δ) il p∗l (⊕) (x,e) = {1 κ (e λ −x)−µp/κ,λ ∈ (0,1],µ ∈ [0,1]},e ∈ el p∗l (⊕) proof. α) let e ∈ er p∗l (⊕) = [0,−p] and x′ ∈ ir p∗l (⊕) (x,e), then e ∈ xp∗l(⊕)x ′ ⇒ e ∈{λκx+λx′ +λκµp/κ,λ,µ ∈ [0,1]}. that means there exist κ1,λ1,µ1 ∈ [0,1] : e = λ1κ1x + λ1x ′ + λ1κ1µ1p ⇒ x′ = eλ1 −κ1x−κ1µ1p,λ1 6= 0. but, e ∈{−νp/ν[0,1]}⇒3 ν1 ∈ [0,1] : e = −ν1p. so, x′ = −ν1 λ1 p−κ1x−κ1µ1p,λ1 6= 0 ⇒ x′ = −κ1x(ν1λ1 + κ1µ1)p,λ1 6= 0. then we get x′ ∈{−κx− (ν λ + κµ)p/κ,µ,ν ∈ [0,1],λ ∈ (0,1]}. β) similarly as above. γ) similarly as above. δ) similarly as above. 2 definition 2.4. for every x,y ∈ ir2 ∗r (⊕) : r 2 ×r2 → (r2) : (x,y) 7→ x∗r(⊕))y = x⊕ (p ⊕y) more specifically, x∗r(⊕)y = {λx + λκy + λκµp/λ,κ,µ ∈ [0,1]},∀x,y ∈ r 2 geometrically, for x,y linearly independent position vectors, the set xp∗r(⊕)y is the closed region of the quadrilateral with vertices 0,x,x + y,x + y + p. on the other hand the set yp∗r(⊕)x is the closed region of the quadrilateral with vertices 0,x + y,x + y + p,y. so, (xp∗r(⊕)y)∩ (yp ∗r (⊕)x) = [0,x + y,x + y + p] 6= ∅,∀x,y ∈ r 2. 111 ioanna iliou and achilles dramalidis proposition 2.8. the hyperstructure (r2,p∗r(⊕)) is a p-hvcommutative group. proof. obviously, xp∗r(⊕)r 2 = r2p∗r(⊕)x = r 2,∀x ∈ r2. for x,y,z ∈ r2 (xp∗r(⊕)y)p ∗r (⊕)z = [(x⊕(p⊕y)]⊕(p⊕z) ≡ [o,x,x+z,x+y+z,x+y+z+2p] xp∗r(⊕)(yp ∗r (⊕)z) = x⊕{p⊕[y⊕(p⊕z)]}≡ [o,x,x+y+z,x+y+z+2p,y+p+x] so, [(xp∗r(⊕)y)p ∗r (⊕)z]∩ [xp ∗r (⊕)(yp ∗r (⊕)z)] 6= ∅,∀x,y,z ∈ r 2.2 the following, are respective propositions of the propositions 2.6. and 2.7. : proposition 2.9. i) erp∗r (⊕) = r2 ii) elp∗r (⊕) = [0,−p] = {−νp/ν ∈ [0,1]} = ep∗r (⊕) . proposition 2.10. α) irp∗r (⊕) (x,e) = {1 κ (e λ −x)−µp/κ,λ ∈ (0,1],µ ∈ [0,1]},e ∈ erp∗r (⊕) β) irp∗r (⊕) (x,e) = {−x κ − ( ν λκ + µ)p/κ,λ ∈ (0,1],µ ∈ (0,1]},e ∈ elp∗r (⊕) . γ) ilp∗r (⊕) (x,e) = {e λ −κx−κµp/κ,µ ∈ [0,1],λ ∈ (0,1]},e ∈ erp∗r (⊕) δ) ilp∗r (⊕) (x,e) = {−κx− (ν λ + κµ)p/κ,µ,ν ∈ [0,1],λ ∈ (0,1]},e ∈ elp∗r (⊕) . remark 2.1. notice that, α)x∗l(⊕)y = y ∗r (⊕)x,∀x,y ∈ r 2 β)x∗r(⊕)y = y ∗l (⊕)x,∀x,y ∈ r 2 112 on p-hv-structures in a two-dimensional real vector space references [1] p. corsini, prolegomena of hypergroup theory, aviani editore, 1993. 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[18] t. vougiouklis and a. dramalidis, hv-modulus with external phyperoperations, proc. of the 5th aha, iasi, romania, (1993), 191-197. 114 ratio mathematica volume 31, 2019, pp. 79-88 the distinguishing number and the distinguishing index of co-normal product of two graphs saeid alikhani∗ samaneh soltani † abstract the distinguishing number (index) d(g) (d′(g)) of a graph g is the least integer d such that g has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. the co-normal product g ? h of two graphs g and h is the graph with vertex set v (g) × v (h) and edge set {{(x1,x2),(y1,y2)}|x1y1 ∈ e(g) or x2y2 ∈ e(h)}. in this paper we study the distinguishing number and the distinguishing index of the co-normal product of two graphs. we prove that for every k ≥ 3, the k-th co-normal power of a connected graph g with no false twin vertex and no dominating vertex, has the distinguishing number and the distinguishing index equal two. keywords: distinguishing number; distinguishing index; co-normal product. 2010 ams subject classifications: 05c15, 05c60. 1 ∗department of mathematics, yazd university, yazd, iran; alikhani@yazd.ac.ir †department of mathematics, yazd university, yazd, iran; s.soltani1979@gmail.com 1received on february 12th, 2019. accepted on may 3rd, 2019. published on june 30th, 2019. doi: 10.23755/rm.v36i1.452. issn: 1592-7415. eissn: 2282-8214. c©alikhani and soltani. this paper is published under the cc-by licence agreement. 79 saeid alikhani, samaneh soltani 1 introduction and definitions let g = (v,e) be a simple graph of order n ≥ 2. we use the the following notations: the set of vertices adjacent in g to a vertex of a vertex subset w ⊆ v is the open neighborhood n(w) of w . also n(w) ∪ w is called a closed neighborhood of w and denoted by n[w ]. a subgraph of a graph g is a graph h such that v (h) ⊆ v (g) and e(h) ⊆ e(g). if v (h) = v (g), we call h a spanning subgraph of g. any spanning subgraph of g can be obtained by deleting some of the edges from g. two distinct vertices u and v are called true twins if n[v] = n[u] and false twins if n(v) = n(u). two vertices are called twins if they are true or false twins. the number |n(v)| is called the degree of v in g, denoted as degg(v) or deg(v). a vertex having degree |v (g)|−1 is called a dominating vertex of g. also, aut(g) denotes the automorphism group of g, and graphs with |aut(g)| = 1 are called rigid graphs. a labeling of g, φ : v → {1,2, . . . ,r}, is said to be r-distinguishing, if no non-trivial automorphism of g preserves all of the vertex labels. the point of the labels on the vertices is to destroy the symmetries of the graph, that is, to make the automorphism group of the labeled graph trivial. formally, φ is r-distinguishing if for every non-trivial σ ∈ aut(g), there exists x in v such that φ(x) 6= φ(σ(x)). the distinguishing number of a graph g is defined by d(g) = min{r| g has a labeling that is r-distinguishing}. this number has defined in [1]. similar to this definition, the distinguishing index d′(g) of g has defined in [8] which is the least integer d such that g has an edge colouring with d colours that is preserved only by a trivial automorphism. if a graph has no nontrivial automorphisms, its distinguishing number is 1. in other words, d(g) = 1 for the asymmetric graphs. the other extreme, d(g) = |v (g)|, occurs if and only if g is a complete graph. the distinguishing index of some examples of graphs was exhibited in [8]. for instance, d(pn) = d′(pn) = 2 for every n ≥ 3, and d(cn) = d′(cn) = 3 for n = 3,4,5, d(cn) = d′(cn) = 2 for n ≥ 6, where pn denotes a path graph on n vertices and cn denotes a cycle graph on n vertices. a graph and its complement, always have the same automorphism group while their graph structure usually differs, hence d(g) = d(g) for every simple graph g. product graph of two graphs g and h is a new graph having the vertex set v (g) × v (h) and the adjacency of vertices is defined under some rule using the adjacency and the nonadjacency relations of g and h. the distinguishing number and the distinguishing index of some graph products has been studied in literature (see [2, 6, 7]). the cartesian product of graphs g and h is a graph, denoted by g2h, whose vertex set is v (g) × v (h). two vertices (g,h) and (g′,h′) are adjacent if either g = g′ and hh′ ∈ e(h), or gg′ ∈ e(g) and h = h′. 80 the distinguishing number and the distinguishing index of co-normal product of two graphs in 1962, ore [10] introduced a product graph, with the name cartesian sum of graphs. hammack et al. [4], named it co-normal product graph. the co-normal product of g and h is the graph denoted by g ? h, and is defined as follows: v (g ? h) = {(g,h)|g ∈ v (g) and h ∈ v (h)}, e(g ? h) = {{(x1,x2),(y1,y2)}|x1y1 ∈ e(g) or x2y2 ∈ e(h)}. we need knowledge of the structure of the automorphism group of the cartesian product, which was determined by imrich [5], and independently by miller [9]. theorem 1.1. [5, 9] suppose ψ is an automorphism of a connected graph g with prime factor decomposition g = g12g22 . . .2gr. then there is a permutation π of the set {1,2, . . . ,r} and there are isomorphisms ψi : gπ(i) → gi, i = 1, . . . ,r, such that ψ(x1,x2, . . . ,xr) = (ψ1(xπ(1)),ψ2(xπ(2)), . . . ,ψr(xπ(r))). imrich and klavžar in [7], and gorzkowska et.al. in [3] showed that the distinguishing number and the distinguishing index of the square and higher powers of a connected graph g 6= k2,k3 with respect to the cartesian product is 2. the relationship between the automorphism group of co-normal product of two non isomorphic, non rigid connected graphs with no false twin and no dominating vertex is the same as that in the case of the cartesian product. theorem 1.2. [12] for any two non isomorphic, non rigid graphs g and h, aut(g?h) = aut(g)×aut(h) if and only if both g and h have no false twins and dominating vertices. theorem 1.3. [12] for any two rigid isomorphic graphs g and h, aut(g?h) ∼= s2. theorem 1.4. [12]the graph g?h is rigid if and only if g � h and both g and h are rigid graphs. in the next section, we study the distinguishing number of the co-normal product of two graphs. in section 3, we show that the distinguishing index of the conormal product of two simple connected non isomorphic, non rigid graphs with no false twin and no dominating vertex cannot be more than the distinguishing index of their cartesian product. as a consequence, we prove that all powers of a connected graph g with no false twin and no dominating vertex distinguished by exactly two edge labels with respect to the co-normal product. 81 saeid alikhani, samaneh soltani 2 distinguishing number of co-normal product of two graphs we begin this section with a general upper bound for the co-normal product of two simple connected graphs. we need the following theorem. theorem 2.1. [12] let g and h be two graphs and λ : v (g ? h) → v (g ? h) be a mapping. (i) if λ = (α,β) defined as λ(g,h) = (α(g),β(h)), where α ∈ aut(g) and β ∈ aut(h), then λ is an automorphism on g ? h. (ii) if g is isomorphic to h and λ = (α,β) defined as λ(g,h) = (β(h),α(g)), where α is an isomorphism on g to h and β is an isomorphism on h to g, then λ is an automorphism on g ? h. theorem 2.2. if g and h are two simple connected graphs, then max { d(g2h),d(g),d(h) } ≤ d(g?h) ≤ min { d(g)|v (h)|, |v (g)|d(h) } . proof. we first show that max{d(g),d(h)}≤ d(g?h). by contradiction, we assume that d(g ? h) < max{d(g),d(h)}. without loss of generality we suppose that max{d(g),d(h)} = d(g). let c be a (d(g?h))-distinguishing labeling of g ? h. then the set of vertices {(g,h∗) : g ∈ v (g)}, where h∗ ∈ v (h) have been labeled with less than d(g) labels. hence we can define the labeling c′ with c′(g) := c(g,h∗) for all g ∈ v (g). since d(g ? h) < d(g), so c′ is not a distinguishing labeling of g, and so there exists a nonidentity automorphism α of g preserving the labeling c′. thus there exists a nonidentity automorphism λ of g?h with λ(g,h) := (α(g),h) for g ∈ v (g) and h ∈ v (h), such that λ preserves the distinguishing labeling c, which is a contradiction. now we show that d(g2h) ≤ d(g ? h), and so we prove the left inequality. by theorems 1.1 and 2.1, we can obtain that aut(g2h) ⊆ aut(g ? h), and since v (g2h) = v (g ? h), we have d(g2h) ≤ d(g ? h). now we show that d(g ? h) ≤ min{d(g)|v (h)|, |v (g)|d(h)}. for this purpose, we define two distinguishing labelings of g ? h with d(g)|v (h)| and |v (g)|d(h) labels, respectively. let c be a d(g)-distinguishing labeling of g and c′ be a d(h)-distinguishing labeling of h. we suppose that v (g) = {g1, . . . ,gn} and v (h) = {h1, . . . ,hm}, and define the two following distinguishing labelings l1 and l2 of g?h with d(g)|v (h)| and |v (g)|d(h) labels. l1(gj,hi) := (i−1)d(g) + c(gj), l2(gj,hi) := (j −1)d(h) + c′(hi). 82 the distinguishing number and the distinguishing index of co-normal product of two graphs we only prove that the labeling l1 is a distinguishing labeling, and by a similar argument, it can be concluded that l2 is a distinguishing labeling of g ? h. if f is an automorphism of g ? h preserving the labeling l1, then f maps the set hi := {(gj,hi) : gj ∈ v (g)} to itself, setwise, for all i = 1, . . . ,m. since the restriction of f to hi can be considered as an automorphism of g preserving the distinguishing labeling c, so for every 1 ≤ i ≤ m, the restriction of f to hi is the identity automorphism. hence f is the identity automorphism of g ? h. 2 the bounds of theorem 2.2 are sharp. for the right inequality it is sufficient to consider the complete graphs as the graphs g and h. in fact, if g = kn and h = km, then g ? h = knm. for the left inequality we consider the non isomorphic rigid graphs as the graphs g and h. then by theorem 1.4, we conclude that g ? h and g2h are a rigid graph and hence max { d(g2h),d(g),d(h) } = d(g ? h). with respect to theorems 1.1 and 1.2, we have that the automorphism group of a co-normal product of connected non isomorphic, non rigid graphs with no false twin and no dominating vertex, is the same as automorphism group of the cartesian product of them, so the following theorem follows immediately: theorem 2.3. if g and h are two simple connected, non isomorphic, non rigid graphs with no false twin and no dominating vertex, then d(g?h) = d(g2h). since the path graph pn (n ≥ 4), and the cycle graph cm (m ≥ 5) are connected, graphs with no false twin and no dominating vertex, then by theorem 2.3 we have d(pn ? pq) = d(pn ? cm) = d(cm ? cp) = 2 for any q,n ≥ 3, where q 6= n and m,p ≥ 5, where m 6= p. (see [7] for the distinguishing number of cartesian product of these graphs). to prove the next result, we need the following lemmas. lemma 2.1. [13] for any two distinct vertices (vi,uj) and (vr,us) in g ? h, n((vi,uj)) = n((vr,us)) if and only if (i) vi = vr in g and n(uj) = n(us) in h, or (ii) uj = us in h and n(vi) = n(vr) in g, or (iii) n(vi) = n(vr) in g and n(uj) = n(us). lemma 2.2. [13] a vertex (vi,uj) is a dominating vertex in g ? h if and only if vi and uj are dominating vertices in g and h, respectively. theorem 2.4. [12] for a rigid graph g and a non rigid graph h, |aut(g?h)| = |aut(h)| if and only if g has no dominating vertex and h has no false twin. 83 saeid alikhani, samaneh soltani now we are ready to state and prove the main result of this section. theorem 2.5. let g be a connected graph with no false twin and no dominating vertex, and ?gk the k-th power of g with respect to the co-normal product. then d(?gk) = 2 for k ≥ 3. in particular, if g is a rigid graph, then for k ≥ 2, d(?gk) = 2. proof. by lemmas 2.1 and 2.2, we can conclude that g ? g has no false twin and no dominating vertex. we consider the two following cases: case 1) let g be a non rigid graph. if h := g ? g, then d(?g3) = 2 by theorem 2.3. now by induction on k, we have the result. case 2) let g be a rigid graph. in this case, |aut(g ? g)| = 2, by theorem 1.3, and so d(g ? g) = 2. if h := g ? g, then |aut(g ? h)| = |aut(h)|, by theorem 2.4. hence |aut(?g3)| = 2. by induction on k and using theorem 2.4, we obtain d(?gk) = 2 for k ≥ 2, where g is a rigid graph. 2 3 distinguishing index of co-normal product of two graphs in this section we investigate the distinguishing index of co-normal product of graphs. pilśniak in [11] showed that the distinguishing index of traceable graphs, graphs with a hamiltonian path, of order equal or greater than seven is at most two. theorem 3.1. [11] if g is a traceable graph of order n ≥ 7, then d′(g) ≤ 2. we say that a graph g is almost spanned by a subgraph h if g−v, the graph obtained from g by removal of a vertex v and all edges incident to v, is spanned by h for some v ∈ v (g). the following two observations will play a crucial role in this section. lemma 3.1. [11] if a graph g is spanned or almost spanned by a subgraph h, then d′(g) ≤ d′(h) + 1. lemma 3.2. let g be a graph and h be a spanning subgraph of g. if aut(g) is a subgroup of aut(h), then d′(g) ≤ d′(h). proof. let to call the edges of g which are the edges of h, h-edges, and the others non-h-edges, then since aut(g) ⊆ aut(h), we can conclude that each automorphism of g maps h-edges to h-edges and non-h-edges to non-h-edges. so assigning each distinguishing edge labeling of h to g and assigning non-hedges a repeated label we make a distinguishing edge labeling of g. 84 the distinguishing number and the distinguishing index of co-normal product of two graphs since for two distinct simple non isomorphic, non rigid connected graphs, with no false twin and no dominating vertex we have aut(g?h) = aut(g2h), so a direct consequence of lemmas 3.1 and 3.2 is as follows: theorem 3.2. (i) if g and h are two simple connected graphs, then d′(g ? h) ≤ d′(g2h) + 1. (ii) if g and h are two simple connected non isomorphic, non rigid graphs with no false twin and no dominating vertex, then d′(g ? h) ≤ d′(g2h). theorem 3.3. let g be a connected graph with no false twin and no dominating vertex, and ?gk the k-th power of g with respect to the co-normal product. then for k ≥ 3, d′(?gk) = 2. in particular, if g is a rigid graph, then for k ≥ 2, d′(?gk) = 2. proof. by lemmas 2.1 and 2.2, we can conclude that g ? g has no false twin and no dominating vertex. we consider the two following cases: case 1) let g be a non rigid graph. if h = g ? g, then d(?g3) = 2 by theorem 3.2(ii). now by an induction on k, we have the result. case 2) let g be a rigid graph. in this case, |aut(g ? g)| = 2, by theorem 1.3, and so d(g ? g) = 2. if h := g ? g, then |aut(g ? h)| = |aut(h)|, by theorem 2.4. hence |aut(?g3)| = 2. by an induction on k and using theorem 2.4, we obtain d(?gk) = 2 for k ≥ 2, where g is a rigid graph. theorem 3.4. let g be a connected graph of order n ≥ 2. then d′(g?km) = 2 for every m ≥ 2, except d′(k2 ? k2) = 3. proof. since |aut(g ? km)| ≥ 2, so d′(g ≥ km) = 2. with respect to the degree of vertices g ? km we conclude that g ? km is a traceable graph. we consider the two following cases: case 1) suppose that n ≥ 2. if m ≥ 3, or m = 2, and n ≥ 4, then the order of g?km is at least 7, and so the result follows from theorem 3.1. if m = 2, n = 3, then g = p3 or k3. in each case, it is easy to see that d′(g ? km) = 2. case 2) suppose that n = 2. then g = k2, and so g ? km = k2m. thus d′(g ? km) = 2 for m ≥ 3, and d′(k2 ? k2) = d′(k4) = 3. 2 by the value of the distinguishing index of cartesian product of paths and cycles graphs in [3] and theorem 3.2, we can obtain this value for the co-normal product of them as the two following corollaries. corolary 3.1. (i) the co-normal product pm?pn of two paths of orders m ≥ 2 and n ≥ 2 has the distinguishing index equal to two, except d′(p2?p2) = 3. (ii) the co-normal product cm ? cn of two cycles of orders m ≥ 3 and n ≥ 3 has the distinguishing index equal to two. 85 saeid alikhani, samaneh soltani (iii) the co-normal product pm ? cn of orders m ≥ 2 and n ≥ 3 has the distinguishing index equal to two. proof. (i) if n,m ≥ 4, then the result follows from theorem 3.2 (ii). if n = 2 or m = 2, then we have the result by theorem 3.4. for the remaining cases, with respect to the degree of vertices in pm ? pn, we obtain easily the distinguishing index. (ii) if n,m ≥ 5, then the result follows from theorem 3.2 (ii). if n = 3 or m = 3, then we have the result by theorem 3.4. for the remaining cases we use of hamiltonicity of cm ? cn and theorem 3.1. (iii) if n ≥ 5 and m ≥ 4, then the result follows from theorem 3.2 (ii). if n = 3 or m = 2, then we have the result by theorem 3.4. the remaining cases are cn ? p3 and c4 ? pm. in the first case and with respect to the degree of vertices in cn ? p3, we obtain easily the distinguishing index. in the latter case, we use of hamiltonicity of c4 ? pm and theorem 3.1. 2 4 acknowledgements the authors would like to express their gratitude to the referee for her/his careful reading and helpful comments. references [1] m.o. albertson and k.l. collins, symmetry breaking in graphs, electron. j. combin. 3 (1996), #r18. [2] s. alikhani and s. soltani, the distinguishing number and distinguishing index of the lexicographic product of two graphs, discuss. math. graph theory 38 (2018) 853-865. [3] a. gorzkowska, r. kalinowski, and m. pilśniak, the distinguishing index of the cartesian product of finite graphs, ars math. contem. 12 (2017), 77-87. [4] r. hammack, w. imrich and s. klavžar, handbook of product graphs (second edition), taylor & francis group 2011. [5] w. imrich, automorphismen und das kartesische produkt von graphen, oesterreich. akad. wiss. math.-natur. kl. s.-b. ii 177 (1969), 203-214. 86 the distinguishing number and the distinguishing index of co-normal product of two graphs [6] w. imrich, j. jerebic and s. klavžar, the distinguishing number of cartesian products of complete graphs, european j. combin. 29 (4) (2008), 922-929. [7] w. imrich and s. klavžar, distinguishing cartesian powers of graphs, j. graph theory, 53.3 (2006), 250-260. [8] r. kalinowski and m. pilśniak, distinguishing graphs by edge colourings, european j. combin. 45 (2015), 124-131. [9] d.j. miller, the automorphism group of a product of graphs, proc. amer. math. soc. 25 (1970), 24-28. [10] o. ore, theory of graphs, amer. math. society 1962. [11] m. pilśniak, improving upper bounds for the distinguishing index, ars math. contemp. 13 (2017), 259-274. [12] s. rehman and i. javaid, fixing number of co-noraml product of graphs, arxiv:1703.00709 (2017). [13] s. rehman and i. javaid, resolving, dominating and locating dominating sets in co-normal product of graphs, submitted. 87 microsoft word capitolo intero n 8.doc approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 42, 2022 arboricity and span in m-fuzzy chromatic index s. yahya mohamed* s. suganthi† abstract a fuzzy matching is a set of edges in which no two edges incident on a same membership value. in this paper, arboricity and span of a fuzzy labeling graph are defined with suitable examples. also, the relation between arboricity and m-fuzzy chromatic index, span and m-fuzzy chromatic index are discussed. some results based on these concepts are stated and proved. keywords: m-fuzzy coloring; m-fuzzy chromatic index; arboricity; span annuity 2010 ams subject classification: 03e72, 05c72, 05c78‡ *dr. s. yahya mohamed, asst. professor in mathematics, government arts college, trichy-22, affiliated to bharathidasan university, trichy. email: yahya_md@yahoo.com. †department of mathematics, bharathiyar arts & science college for women, deviyakurichi, salem – 636112; sivasujithsuganthi@gmail.com. ‡ received on january 29th, 2022. accepted on june 22nd, 2022. published on june 30th, 2022. doi: 10.23755/rm.v39i0.710. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 115 s.yahya mohamed and s.suganthi 1. introduction in 1736, euler first invited the concept of graph theory. the theory of graphs is extremely useful for solving combinatorial problems in different areas such as geometry, algebra, number theory, topology, operations research, optimization and computer science. graph theory is the main tool to constitute many real-world problems. nowadays, graphs do not represent all systems properly due to the uncertainty or nebulousness of the framework of systems. this and many other circumstances stimulated to define fuzzy graphs. rosenfeld [8] first established the notion of fuzzy graphs in 1975. crisp graph and fuzzy graph are both systematically the same, but when there is an uncertainty on points and or lines then fuzzy graph has a segregated significance. since the world is brimming with uncertainty, so the fuzzy graph occurs in many real-life locations. fuzzy graph theory is elaborated with a large number of branches. the fuzzy graph model is used to constitute the traffic network. also it is used for job allocation, group structure and decision-making analysis etc. seethalakshmi et al. [10] defined perfect fuzzy matching with some examples, and they derived that if 𝐺 is a strong regular fuzzy graph with each vertex and is of degree at least two, then 𝐸 is not a perfect fuzzy matching in 𝐺. ananthanarayanan et al. [1] discussed the coloring of fuzzy graphs using α-cut with appropriate examples. anjaly kishore et al.[2] defined the chromatic number of a fuzzy graph and they discussed some results based on these chromatic numbers. arindam deyet al.[3] explained the concept of complement fuzzy graph and their edge coloring. r. jahir hussain et al.[5] found the fuzzy chromatic number of some special fuzzy graphs s.yahya mohamed et al.[11-14] introduced the concept of fuzzy matching in fuzzy labeling graphs through some illustrations. in this paper, we introduced the new concept of arboricity and span of a fuzzy labeling graph 𝐺. also, we discussed some properties based on arboricity and span of 𝐺. 2. preliminaries definition 2.1 let 𝐺 = (𝛼, 𝛽) be a fuzzy graph. the set of all subsets of 𝛽 in which they received the same color is called color class of 𝐺. it is denoted by 𝐶𝐶(𝐺). example 2.2 consider the fuzzy graph 𝐺 given in figure 2.1 in fig 2.1, the edge set is 𝛽 = {𝑒1(0.2), 𝑒2(0.5), 𝑒3(0.35), 𝑒4(0.14)} and all edges receive different colors. so all edges belong to the color class of 𝐺. 116 arboricity and span in m-fuzzy chromatic index figure 2.1: color class of a fuzzy graph 𝐺 definition 2.3 let 𝐺 = (𝛼, 𝛽) be a fuzzy labeling graph. the number of elements in the color class is called as coloring number of 𝐺. it is denoted by 𝐶𝑁(𝐺). definition 2.4 a family 𝜆 = {𝑀1, 𝑀2, 𝑀3 … … … . 𝑀𝑘 } of fuzzy matchings on a set 𝛽 is called a 𝑀 − fuzzy coloring of 𝐺 = (𝛼, 𝛽) if (i) ∨ 𝜆 = 𝛽. it means single edge does not belong to two distinct color classes. (ii) 𝜆𝑖 𝛬 𝜆𝑗 = 0. (iii) for every edge (𝑥, 𝑦) 𝑜𝑓 𝐺 𝑚𝑖𝑛 {𝜆𝑖 (𝑥), 𝜆𝑗 (𝑦)} = 0 (1 ≤ 𝑖 ≤ 𝑘). (this means any one of the edges does not receive different color). example 2.5 consider the fuzzy graph 𝐺 given in figure 2.2, figure 2.2: m-fuzzy coloring of 𝐺 in fig 2.2, the fuzzy matchings are 𝑀1 = { 𝑒1(0.4), 𝑒3(0.53 } and 𝑀2 = { 𝑒4(0.24), 𝑒2(0.62)}. here 𝜆 = {𝑀1, 𝑀2} is 𝑀 −fuzzy coloring of 𝐺. v1(0.4) v2(0.7) v3(0.9) v4(0.3) e3(0.3 5) e1(0.2) e2(0.5) e4(0.14) fig 2.1 v5(0.7) v3(0.8) v4(0.6) v1(0.5) v2(0.7) e2(0.62) e3(0.53 ) e4(0.24) e1(0.4) e5(0.18) fig 2.2 117 s.yahya mohamed and s.suganthi definition 2.6 let 𝐺 = (𝛼, 𝛽) be a fuzzy labeling graph .the minimum number 𝑘 for which there exists a 𝑀 −fuzzy coloring is called fuzzy matching chromatic index of 𝐺. it is denoted by 𝜒fmci (𝐺). example 2.7 consider the fuzzy graph given in figure 2.3 figure 2.3: fuzzy matching chromatic index of 𝐺 in fig 2.3, the fuzzy matching of a given graph 𝐺 are as follows 𝑀1 = { 𝑒1(0.04), 𝑒3(0.26), 𝑒5(0.48)}, 𝑀2 = { 𝑒2(0.15), 𝑒4(0.37), 𝑒6(0.09)}. then the fuzzy matching chromatic index 𝜒fmci (𝐺) = 2 because 𝜆 = {𝑀1, 𝑀2} is 𝑀 −fuzzy coloring of 𝐺. 3. main results definition 3.1 the minimum number of paths to cover all edges of a fuzzy graph g is called arboricity of .g it is denoted by ( ).a g example 3.2 consider the following fuzzy graph 𝐺 given in figure 3.1 figure 3.1: arboricity of 𝐺 v4(0.4) v5(0.5) v6(0.6) e5(0.48) e4(0.37) e3(0.26) v3(0.3) v2(0.2) e2(0.15) e1(0.04) v1(0.1) e6(0.09) fig 2.3 118 arboricity and span in m-fuzzy chromatic index in fig 3.1 the path cover all edges of a given graph are 1 1 2 2 3 v e v e v and 3 3 4 4 1 .v e v e v here each colour represents one path and two distinct paths cover all edges of .g hence, we obtain ( ) 2.a g = theorem 3.3 for any fuzzy labeling graph g with n vertices has arboricity . 2 n      proof consider a fuzzy labeling graph ( ),g  = with n vertices. now we construct a path to cover all edges of g. we obtain many paths to cover all edges of g. but we need minimum number of path to cover all edges. then we construct the path with maximum number of edges in .g this path occurs with maximum number of edges ( 1)n − . hence, we can find all paths with ( 1)n − edges and which is equal to 2 n      . therefore, g has arboricity 2 n      . example 3.4 consider the fuzzy graph given in figure 3.2, figure 3.2: fuzzy labeling graph 𝐺 in fig 3.2 the path cover all edges of a given graph are 3 5 1 4 4 6 2 v e v e v e v and 1 1 2 2 3 3 4 .v e v e v e v here, each color represents one path and there are two paths exist to cover all edges of g. hence, we get ( ) 4 2 . 2 a g   = =     v3(0.9) v4(0.8) v1(0.3) e1(0.2) v2(0.6) e2(0.4) e3(0.6) e4(0.1) fig 3.2 e6(0.5) e5(0.24) 119 s.yahya mohamed and s.suganthi theorem 3.5 for any fuzzy labeling graph ( ) ( ), .fmcig a g g proof consider a labeling fuzzy graph ( ),g  = and 1 2 3, , ... km m m m be the distinct fuzzy matching in .g here 1 2 3 , , ... k m m m m contains the set of independent elements and each matching receive different colors. now we find the path to cover all edges of .g here we need more than one path to cover all edges of .g already we have for any fuzzy graph with n vertices has arboricity . 2 n      here we require two matching or more than two matching to cover all edges of .g then the number of path to cover all edges is less than the number of matching in any color class of .g hence, we obtain ( ) ( ).fmcia g g example 3.6 consider the fuzzy graph given in figure 3.3, figure 3.3: fuzzy graph 𝐺 in fig 3.3 the fuzzy matching of a given graph g is as follows ( ) ( )  ( ) ( ) 1 4 6 2 1 30.53 , 0.29 , 0.15 , 0.29m e e m e e= = and v1(0.5) v5(0.9) v4(0.6) v3(0.3) v2(0.2) e4(0.41) e3(0.29) e4(0.53) e2(0.12) e6(0.29) e1(0.15) fig 3.3 120 arboricity and span in m-fuzzy chromatic index ( ) ( ) 3 2 40.12 , 0.41m e e= . also the color class ( )  1 2 3, , .cc g m m m= here ( ) 3fmci g = and the paths to cover all edges of g are 5 4 1 1 2 2 3v e v e v e v and 1 6 3 3 4 4 5 .v e v e v e v then ( ) 2.a g = hence, we obtain ( ) ( ).fmcia g g definition 3.7 the minimum number of interchanging path to cover all vertices of a fuzzy labeling graph 𝐺 is called spanning arboricity of g. example 3.8 consider the following fuzzy graph 𝐺 given in figure 3.4 figure 3.4: spanning arboricity of 𝐺 in fig 3.4 one of the fuzzy matching of a given graph g is ( ) ( ) 1 2 50.3 , 0.05 .m e e= then the interchanging path 2 2 3 1 1 5 4 4 5v e v e v e v e v is to cover all points of .g here spanning arboricity of g is one. theorem 3.9 a fuzzy labeling graph ( ),g  = with n vertices has spanning arboricity as unity. proof let us consider a fuzzy labeling graph ( ),g  = with n vertices. then we identify the fuzzy matching of g and an interchanging path with maximum lines to cover all vertices of g . now we can find an interchanging path to cover all vertices of .g an interchanging path is a path in which the edges are alternatively in m and ( ).m − but we can able to construct a fuzzy matching with maximum number of edges in the boundary of g . then an interchanging path in the boundary path of any graph is only enough to cover all vertices of g v2(0.8) v5(0.5) v4(0.6) v3(0.7) e3(0.2) e4(0.1) v1(0.9) e5(0.05) e1(0.4) e2(0.3) fig 3.4 121 s.yahya mohamed and s.suganthi example 3.10 consider the following fuzzy graph given in figure 3.5 figure 3.5: spanning arboricity of 𝐺 in fig 3.5 the fuzzy matching of a given graph g is ( ) ( ) 1 2 40.04 , 0.08m e e= and 1 1 2 2 3 3 4 4 5 v e v e v e v e v is an interchanging path to cover all vertices of .g hence, the spanning arboricity of g is one. definition 3.11 the number of edges in an interchanging path to cover all vertices of a fuzzy graph g is called span of g . it is denoted by ( ).gs example 3.12 consider the following fuzzy graph given in figure 3.6, figure 3.6 : span of 𝐺 v4(0.45) v5(0.56) e4(0.08) e3(0.06) v3(0.34) e2(0.04) v1(0.12) e5(0.03) e1(0.02) e6(0.07) v2(0.23) fig 3.5 v4(0.4) v3(0.3) e3(0.24) e4(0.07) v1(0.1) e1(0.05) v2(0.2) e2(0.11) e5(0.09) fig 3.6 122 arboricity and span in m-fuzzy chromatic index in fig 3.6 the fuzzy matching of a given graph g is ( ) ( ) 1 1 30.05 , 0.24m e e= and the number of edges in an interchanging path 1 1 2 2 3 3 4 v e v e v e v is 3. hence, we have ( ) 3.g =s theorem 3.13 for any fuzzy labeling graph ( ),g  = with n vertices, we have ( ) ( ).fmcig gs proof let us consider a fuzzy labeling graph ( ),g  = with n vertices. now we can form fuzzy matching of g and we find an interchanging path with maximum edges to cover all vertices of .g then the interchanging path in the boundary of any graph is only enough to cover all vertices of .g to cover all vertices of .g but this path occurs with maximum number of edges ( )1n − because each fuzzy matching receives distinct colors. therefore, span of g is ( )1 .n − here the fuzzy matching chromatic index is less than ( )1 .n − and hence, we obtain ( ) ( ).fmcis g g example 3.14 consider the following fuzzy graph given in figure 3.7, figure 3.7: span of 𝐺 in fig 3.7 the fuzzy matching is ( ) ( ) ( ) 1 2 5 80.05 , 0.11 , 0.17m e e e= and 1 1 2 2 3 4 4 5 5 6 6 v e v e v e v e v e v is an interchanging path to cover all vertices of g . then we have ( ) ( )5 2.fmcis g g=  = v4(0.45) v2(0.23) v5(0.56) v3(0.34) v1(0.19) v6(0.67) e6(0.13) e5(0.11) e4(0.09) e2(0.05) e1(0.03) e3(0.07) e7(0.15) e8(0.17) fig 3.7 123 s.yahya mohamed and s.suganthi 4. conclusion fuzzy set theory proposes to be a useful tool for handling vagueness and degrees of certainty, and for giving a consistent representation of linguistically formulated knowledge which allows the use of precise properties and algorithms. in this paper, we introduced the new concept of arboricity and span of fuzzy labeling graph. also we compared these values with m-fuzzy chromatic index with related examples. in future, we plan to extend our research work to operations like union, intersection and cartesian product of two fuzzy labeling graphs. references 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[2] anjaly kishore, m.s. sunitha. chromatic number of fuzzy graph. annals of fuzzy mathematics and informatics, 7(4), 543-551, 2014. [3] arindam dey, dhrubajyoti ghos, anita pal. edge coloring of complement fuzzy graph. international journal of modern engineering research, 2(4), 1929-1933, 2012. [4] frank harary, graph theory, indian student edition, narosa/addison wesley, 1988. [5] r. jahir hussain, k.s. kanzul fatima. fuzzy chromatic number of middle subdivision and total fuzzy graph. international journal of mathematical archive, 6(12), 90-94, 2015. [6] v. nivethana, a. parvathi. fuzzy total coloring and chromatic number of a complete fuzzy graph. international journal of emerging trends in engineering and development. 6(3), 377-384, 2013. [7] k. ranganathan, r. balakrishnan, a text book of graph theory, springer. [8] a. rosenfeld. fuzzy graphs. fuzzy sets and their applications to cognitive and decision process. new york. 75-95, 1975. 124 arboricity and span in m-fuzzy chromatic index [9] s. samanta, t. pramanik, m. pal, fuzzy coloring of fuzzy graphs. afrika matematika. 27(1-2), 37-50, 2016. [10] r. seethalakshmi and r.b. gnanajothi, a note on perfect fuzzy matching. international journal on pure and applied mathematics, 94(2), 2014, 155-161. [11] s.yahya mohamad, s. suganthi, properties of fuzzy matching in set theory, journal of emerging technologies and innovative research, 5(2), 1043-1047, 2018. [12] s.yahya mohamad, s. suganthi, energy of complete fuzzy labeling graph through fuzzy complete matching, international journal of mathematics trends and technology, 58(3), 178-181, 2018. [13] s.yahya mohamad, s. suganthi. matching and complete matching domination in fuzzy labelling graph. journal of applied science and computations. 5(10), 1052-1061, 2018. [14] s.yahya mohamad, s. suganthi. some parametres of fuzzy labeling tree using matching and perfect matching. malaya journal of matematik. s(1), 511-514, 2020. [15] l. a. zadeh. fuzzy sets. information and control. 8, 338-353, 1965. 125 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica 29 (2015) 41-52 issn: 1592-7415 41 application of mathematical software in solving the problems of electricity erika fechová 1 1 technical university of košice, faculty of manufacturing technologies with a seat in prešov, bayerova 1, 080 01 prešov, slovakia erika.fechova@tuke.sk abstract at the present time great emphasis is put on making accessible new knowledge to students through information and communication technologies in effort to facilitate and introduce objects, phenomena and reality. information and communication technologies complement and develop traditional methods such as direct observation, manipulation with objects, experiment. it is justified mainly at teaching natural sciences. the possibilities of solving physical problem with the use of software tools are presented in the paper. keywords: information and communication technologies, electrical circuit, kirchhoffov´s lows, ms excel, matlab. doi: 10.23755/rm.v29i1.21 1 introduction information and communication technologies currently present a set of modern means that are used for preparation, processing and distribution of data and information, but also process control with the aim of achieving more effective results and searching for optimal problem solutions at various fields and areas of human activities [1], [2]. information and communication technologies significantly influence even university education. information and communication technologies provide incomparably bigger information basis as http://dx.doi.org/10.23755/rm.v29i1.21 erika fechová 42 it was several years ago. this gradually changes the style of teaching and makes teachers implement new technologies not only in direct pedagogical activity, but also at its preparation. implementation of information and communication technologies into education enables new forms of university studies. we can stimulate the interest of students in studies of natural science subjects as mathematics, physics, chemistry, create conditions for educational individualization and improve conditions to raise the quality of education by a suitable combination of traditional and modern teaching methods [3]. in teaching physics there exist possibilities for effective and suitable integration of information and communication technologies into schooling system. one of them is physical problem solution with the support of computer. this paper concretely presents the solution of physical problem from the part physics – power and magnetism by the use of mathematical software ms excel a matlab. 2 physical analysis of the problem problem: figure out the currents in individual circuit branches in fig. 1, if source voltage and resistance are: u01=10 v, u02=20 v, u03=15 v, u04=10 v, r1=10 , r2=15 , r3=30 , r4=20 , r5=10 , r6=15 , r7=10 . solution: kirchhoff´s rules are used to figure out the currents in the circuit [4]. the first of kirchhoff´s rules describes the law of electric charge preservation: the sum of all the currents flowing into the junction point must equal the sum of all the currents leaving the point, i.e.    n k ki 1 0 . the second of kirchhoff´s rules forms the law of electric energy preservation for electric circuits: algebraic sum of electromotive voltages in any closed part of the electrical network is equal to the sum of ohmic voltages at individual branches of this closed part, i.e.    n k kk m i ei iru 11 . application of mathematical software in solving the problems of electricity 43 fig. 1. circuit based on the first and the second of kirchhoff´s rules (fig. 1) for the currents and electromotive voltages, it is valid 0 0 0 7653 5432 3211    iii:b iii:b iii:b 0 40 37766 0 3665544 0 2443322 0 20 12211 uuirir:iv uiririr:iii uiririr:ii uuirir:i     the set of 7 equations on 7 unknown quantities i1, i2, ..., i7 was obtained. numeric values are inducted for the known quantities and we have 51015 15151020 20203015 101510 0 0 0 76 654 432 21 765 543 321        ii iii iii ii iii iii iii 3 analytic solution of the problem based on analysis of the problem and use of electrical laws the system of 7 equations in 7 variables was obtained, where analytic solution is not simple. in general it is possible to solve the system of n equations in n variables in three ways: 1. solving the system of linear equations by means of cramer´s rule, 2. solving the system of linear equations by means of inversion matrix, 3. solving the system of linear equations by gauss elimination method. gauss elimination method appears to be a suitable method of solving the system of n equations in n variables, if 3>n [5]. by means of equivalent line adjustment the matrix of the system of equations, which is augmented by the second column (so called augmented matrix of the system) to a triangle shape, is modified. we write to such an augmented matrix an appropriate system, which is equivalent with the original system, i.e. it has the same family of solutions. frobenius norm and its consequences can be used to solve such a modified system. erika fechová 44 the system of heterogeneous equations can be solved only if the rank of a matrix is equal to the rank of an augmented matrix of the system. consequence 1: if nhh  )a((a) (n is the number of unknowns), then the system has only one solution. consequence 2: if nhh <)a((a)  (n is the number of unknowns), then the system has infinite number of solutions and hn  unknowns can be arbitrarily selected. consequence 3: if )a((a)  hh , then the system has no solution. we get the values of unknowns by gradual substitution into previous equations. the system of equations is written into the form of an augmented matrix and we get by means of equivalent line adjustment application of mathematical software in solving the problems of electricity 45                                                                                                                                          110 530 0 15 0 20 0 2020000000 1060159000000 1110000 0151020000 0011100 0002030150 0000111 420 530 0 15 0 20 0 960159000000 1060159000000 1110000 0151020000 0011100 0002030150 0000111 1010 5 0 15 0 20 0 000001510 101500000 1110000 0151020000 0011100 0002030150 0000111 5 15 20 10 0 0 0 101500000 0151020000 0002030150 000001510 1110000 0011100 0000111 6 1 r r  the rank of a matrix is equal to the rank of an augmented matrix, i.e. nhh  )a((a) (n is the number of unknowns), then the system has one solution that is determined from an appropriate system: 054460 202 11 1102020 77 ,ii  297030 101 30 1590 202 11 1060530 1590 1060530 53010601590 6 7 676 ,i i iii      erika fechová 46 242570 202 49 202 11 101 30 0 5 765765 ,i iiiiii   405940 101 41 20 101 30 15 202 49 1015 20 151015 15151020 4 65 4654 ,i ii iiii                    163370 202 33 202 49 101 41 0 3 543543 ,i iiiiii         465340 101 47 15 101 41 20 202 33 3020 15 203020 20203015 2 43 2432 ,i ii iiii                    301980 202 61 202 33 101 47 0 1 321321 ,i iiiiii         we get the following current values in the circuit: a054460 a297030 a242570 a405940 a163370 a465340 a301980 7 6 5 4 3 2 1 ,i ,i ,i ,i ,i ,i ,i        it results from the negative current values that currents in the circuit are in the opposite direction as we selected. analytic solution of the system of 7 equations in 7 variables by gauss elimination method requires not only knowledge of linear algebra (matrix algebra), but also good mathematical skills and time. numerical solution of the system of equations by means of various mathematical software tools such as ms excel, mathematica or matlab is much more easier. 4 using software tools at the problem solution 4.1 the problem solution by means of ms excel application of mathematical software in solving the problems of electricity 47 in current computing technique it is possible to use standard programs for the matrix inversion up to relatively big number of equations (hundreds of variables). one of the possibilities is the solution in ms excel [6]. we write the system of equations in matrix form ba                                             x. b b b x x x aaa aaa aaa nnn nnn n n      2 1 2 1 21 22 22 1 12 11 1 where a is the matrix of coefficients, x is the vector of unknowns and b is the vector of the second members. we get by multiplying a -1 from the left baaa -1-1  xx . if we calculate the inversion matrix, xk unknowns can be obtained by multiplication of the matrix and vector, which is procedure that is optimized very well and is the part of standard libraries of subprograms. to calculate the inversion matrix minverse functions from the offer of ms excel more functions is used. to calculate the roots of the system of equations (a -1 b) mmult function is used. the result of the solution can be found in fig. 2. fig. 2. numerical solution of the set of equations in ms excel another possibility to solve the system of equations is to use the ms excel solutionist [7], [8]. from the task and solution of the problem in the solutionist we get (fig. 3) erika fechová 48 fig. 3. numerical solution of the system of problems in the ms excel solutionist from both solutions in ms excel we get the following current values in the circuit a054460a297030a242570 a405940a163370a465350a301980 765 4321 ,i,,i,,i ,,i,,i,,i,,i   it results from the negative values of the current that currents have reverse directions as it was selected. 4.2 problem solution by means of matlab matlab presents highly functional language for technical calculations. it integrates the calculations, visualization and programming into simply usable environment where the problems and solutions are expressed in natural form [9], application of mathematical software in solving the problems of electricity 49 [10]. the field is the basic data type of this interactive system. this property together with number of built-in functions enables relatively easy solution of many technical problems, mainly those that lead to the vector or matrix formulations, in much shorter time as solution in classic program languages. to calculate the currents i1, i2, ..., i7 the method of node voltage is used. this method comes from the fact that )1( u equations is written by means of kirchhoff´s first law applied to suitably selected nodes. in these equations the equations of kirchhoff´s second law written for appropriate loops are implicitly included. that is why voltages on tree´s branches are selected as unknowns at the method of node voltage. to determine node voltages it is necessary to solve (u-1) equations. after calculation of node voltages the currents of the circuit are determined. we write for b1, b2 and b3 nodes according to kirchhoff´s first law 0 0 0 7653 5432 3211    iii:b iii:b iii:b it is possible to express above mentioned currents by means of known node voltages with regard to the selected reference node 0 0 0 7 0 43 6 0 33 5 32 3 5 32 4 2 3 21 2 3 21 2 0 21 1 10 1 1                 r uu r uu r uu :b r uu r u r uu :b r uu r uu r uu :b bbbb bbbbb bbbb we write the equations in the matrix form                                      70 460 3 20 210 1 3 2 1 7655 55433 3321 // 0 // /1/1/1/10 /1)/1/1/1(/1 0/1/1/1/1 ruru ruru u u u rrrr rrrrr rrrr b b b to solve such written equations the matrix solution in matlab is used. we form the m-file prudy.m (fig. 4) erika fechová 50 fig. 4. m-file prudy.m for calculation of matrices and currents after solving the system of equations we get the values of node voltages, which are converted to the currents in branches of the circuit. the result of solution is launching the script of prudy.m and print of results. another possibility of the problem solution in matlab is use of symbolic math toolbox, which provides functions for solution and graphic description of mathematical functions. tool panel provides libraries of functions in common mathematical areas such as mathematical analysis, linear algebra, algebraic and common differential equations and so on. symbolic math toolbox uses mupad language as a part of its calculus core. the language has a extensive set of functions, which are optimized to create and operate symbolic arithmetical muphlp://quickref.muphlp/glossary.xml#glossary application of mathematical software in solving the problems of electricity 51 expressions. to solve the system of equations linsolve ([eqs], [vars]) function was used, where eqs is a list or a set of linear equations or arithmetical expressions, vars is a list or a set of unknowns to solve for: typically identifiers or indexed identifiers. the solution of the system can be found in fig. 5, where x = i1, y = i2, z = i3, k = i4, l = i5, m = i6, n = i7: fig. 5. numeric solution of the system of equations in mupad the same values are obtained from the problem solution in matlab as in the case of the problem solution in ms excel. 5 conclusions it accrues from the solution results that solution of the system of equations of the physical problem in an analytic way as well as by using mathematical software tools leads to certain numeric values. analytic solution of the system of n equations in n variables requires certain mathematical knowledge and skills to solve matrices. use of modern software tools to solve the system of equations facilitates the problem solution. on the other side it requires certain computing skills. the physical problem being solved points out importance and necessity of using modern information and communication technology means and their utilization in educational process that makes “learning” for pupils and students more interesting and attractive. muphlp://datatypes.muphlp/dom_list.xml#dom_list muphlp://datatypes.muphlp/dom_set.xml#dom_set muphlp://quickref.muphlp/glossary.xml#glossary muphlp://quickref.muphlp/glossary.xml#glossary muphlp://datatypes.muphlp/dom_list.xml#dom_list muphlp://datatypes.muphlp/dom_set.xml#dom_set muphlp://datatypes.muphlp/dom_ident.xml#dom_ident muphlp://stdlib.muphlp/_index.xml#_index erika fechová 52 bibliography [1] kalaš i. (2001) čo ponúkajú informačné a komunikačné technológie iným predmetom, špú bratislava [2] turek i. (1997) zvyšovanie efektívnosti vyučovania, mc bratislava [3] brestenská b. et al. (2010) premena školy s využitím informačných a komunikačných technológií, elfa, s r. o košice, isbn 978-80-8086-1438 [4] bouche f. (1988) principle of physics. university of dayton, isbn 0-07303579-3 [5] kluvánek i., mišík l., švec m. (1971) matematika i, alfa bratislava [6] brož m. (2004) microsoft excel 2003, computer press brno, isbn 80251-0406-0 [7] hrehová s., mižáková j. (2008) využitie ms excel vo výpočtoch, informatech, košice, isbn 978-80-88941-32-3 [8] vagaská a. (2007) matlab and ms excel in education of numerical mathematics at technical universities, in: infotech 2007, votobia olomouc [9] dušek f. (2002) matlab a simulink, úvod do používaní, univerzita pardubice, isbn 80-7194-273-1 [10] karban p. (2007) výpočty a simulace v programoch matlab a simulink, computer press brno, isbn 80-2511-448-1 ratio mathematica volume 42, 2022 on the roots and stability of vertex connectivity polynomial priya krishnan* anil kumar vasu† abstract the introduction of the concept of vertex connectivity polynomial of graphs in [priya and anil kumar, 2021b] triggered the need to study the nature of roots as well as the stability properties of the same for various graph classes. this paper mainly deals with results about the nature of roots, stability and schur stability of the vertex connectivity polynomial. keywords: vertex connectivity polynomial, vertex-connected vertices, schur stability. 2020 ams subject classifications:05c31, 05c40.1 *department of mathematics, university of calicut, thenhipalam, kerala-673635, india; priyakrishna27.clt@gmail.com. †department of mathematics, university of calicut, thenhipalam, kerala-673635, india; anil@uoc.ac.in. 1received on march 17th. accepted on june 25th. published on june 30th. doi: 10.23755/rm.v41i0.731 . issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 145 k. priya, v. anil kumar 1 introduction the adjacency property of vertices in a graph is not sufficient to characterize a graph in terms of its connectedness as the deletion of the corresponding edge may or may not disconnect the graph. more generally, the connectivity property of vertices in a graph may not be preserved after the deletion of edges of the geodesics linking them. this motivated the authors to introduce the concept of “vertex-connected vertices”, which shares geodesics preserving the connectivity of a graph. the study of vertex-connected vertices in [priya and anil kumar, 2021a] triggered the need to define vertex connectivity polynomial discussed in [priya and anil kumar, 2021b] for simple finite connected graphs to explicitly reveal the number of vertex pairs that disconnect a graph. a polynomial p(z) is said to be stable, or a hurwitz polynomial, if all its zeros lie in the open left half-plane [fuhrmann, 2011] and is schur stable if all its zeros belong to the open unit disk. hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems [gajender and sharma, 2014]. thus the study of a graph polynomial is worthy only if it succeeds in predicting the behavior of some stable physical systems. this motivated the authors to study about the nature of roots and stability of the vertex connectivity polynomial of a graph. throughout this paper, g denotes a finite simple connected graph with vertex set and edge set denoted by v (g) and e(g) respectively. all the graph theoretic terminology and notations used in the paper are as in [harary, 1969]. 2 main results definition 2.1. let g be a graph of order n. then, the vertex connectivity polynomial of g, denoted by v [g; x], is defined in [priya and anil kumar, 2021b] as: v [g; x] = diam(g)+1∑ i=1 |di|xi, where di is the set of all vertex pairs which disconnects g into i components. definition 2.2. let g = (v, e) be a graph and let u, v ∈ v . then, the vertices u and v are said to be vertex-connected if the deletion of the edges of none of the geodesics connecting u and v disconnects g into multiple components. the graph g is vertex-connected if each of its vertex pairs are vertex-connected. [priya and anil kumar, 2021a] 146 on the roots and stability of vertex connectivity polynomial definition 2.3. let g = (v, e) be a graph. then, a pair of vertices (u, v) ∈ v ×v at a distance d > 0 are said to be d-metrically connective if it disconnects the graph g into d components. the graph g is metrically connective if every pair of its vertices are d-metrically connective for some d > 0.[priya and anil kumar, 2021a] theorem 2.4. (routh-hurwitz criteria [fuhrmann, 2011]) given a polynomial p(x) = a0x n + a1x n−1 + a2x n−2 + · · · + an−1x + an, where the coefficients ai’s are real constants, the n hurwitz matrices using the coefficients ai of the above polynomial are defined as h1 = [ a1 ] , h2 = [ a1 a0 a3 a2 ] , h3 =  a1 a0 0a3 a2 a1 a5 a4 a3   , · · · , hn =   a1 a0 0 0 · · · 0 a3 a2 a1 a0 · · · 0 a5 a4 a3 a2 · · · 0 ... ... ... ... · · · ... 0 0 0 0 · · · an   , where aj = 0 if j > n. all the roots of the polynomial p(x) are negative or have negative real part if and only if the determinants of all hurwitz matrices are positive: det(hj) > 0, j = 1, 2, . . . , n. definition 2.5. let g be a graph. then, the roots of the vertex connectivity polynomial of g are called the vertex connectivity roots of g. the following are some of the observations about the vertex connectivity roots of graphs. (i) (0,∞) is a zero-free interval of the vertex connectivity polynomial v [g; x] for any graph g. (ii) if g is metrically connective, then it has at most diam(g) vertex connectivity roots. (iii) if g is a tree, then it has diam(g) + 1 vertex connectivity roots counting multiplicities. theorem 2.6. let g be a graph. then, zero is a vertex connectivity root of g of multiplicity greater than one iff g is a tree. 147 k. priya, v. anil kumar proof. assume that zero is a vertex connectivity root of g of multiplicity k > 1. then, v [g; x] = xkg(x), a polynomial of degree greater than or equal to 2. this means that every vertex pair of g disconnects the graph into at least two components so that none of its edges belongs to a cycle. hence it follows that g is a tree. conversely, if g is a tree, all its edges being cut edges, none of the vertex pairs are vertex-connected. that is, v [g; x] is a polynomial free of linear coefficient. this completes the proof. theorem 2.7. let g be a graph of order n. then, zero is the only vertex connectivity root of g iff either g is vertex-connected or every vertex pair of g disconnects the graph into exactly two components. proof. assume that zero is the only vertex connectivity root of g. then, v [g; x] = rxk, where r, k > 0. but, v [g; 1] = ∑diam(g)+1 i=1 |di| = ( n 2 ) so that r = ( n 2 ) . that is,v [g; x] = ( n 2 ) xk. case (i): if k = 1, then v [g; x] = ( n 2 ) x so that g is vertex-connected. case (ii): if k > 1, then all the ( n 2 ) vertex pairs disconnects the graph into k components. but if a vertex pair disconnects g into k components, then corresponding to that there exists a vertex pair disconnecting the graph into two components. thus it follows that v [g; x] = ( n 2 ) x2. conversely, if either g is vertex-connected or every vertex pair of g disconnects the graph into exactly two components, then we get v [g; x] as ( n 2 ) x and ( n 2 ) x2 respectively. in both the cases, it follows that zero is the only vertex connectivity root of g . this completes the proof. corollary 2.8. let g be a non vertex-connected graph of order n. then, zero is the only vertex connectivity root of g iff g = p2. proof. let zero be the only vertex connectivity root of g. since g is not vertexconnected, from theorem 2.7 it follows that g is a tree and v [g; x] = ( n 2 ) x2. this means that all the ( n 2 ) vertex pairs disconnects g into two components. now if possible assume n > 2. then, since every edge of g is a cut edge, we can find a vertex pair at 2 units distance which disconnects g into three components. therefore n = 2 and g = p2. this completes the proof. 148 on the roots and stability of vertex connectivity polynomial theorem 2.9. let g be a graph and n1, n2 are respectively be the number of vertex-connected vertex pairs and the number of vertex pairs which disconnects g, where n1 > n2 . if n1 is prime, then g lacks non-zero real vertex connectivity roots. proof. let v [g; x] = xg(x), where g(x) = anxn + an−1xn−1 + . . . a1x + n1,∑n i=1 ai = n2, for some n. clearly all the zeros of g(x) are non-zero. let β be a zero of g(x) such that |β| ≤ 1. then, g(β) = anβ n + an−1β n−1 + . . . a1β + n1 = 0. that is, |n1| = |anβn + an−1βn−1 + . . . a1β| ≤ |an| + |an−1| + . . . |a1| = n2, a contradiction. therefore, |β| > 1. now, if possible, assume that β is real. then, g(x) is reducible over the set of all real numbers. that is, g(x) = (x − β)h(x), where h(x) is a nonconstant integer polynomial. now, n1 = g(0) = −βh(0). since n1 is prime and since |β| > 1, |h(0)| must be 1. let β1, . . . , βn−1 be the zeros of h(x) and a be its leading coefficient. then, |β1, . . . βn−1| = 1 |a| ≤ 1. but β1, . . . , βn−1 are the zeros of g(x) also so that |βi| > 1 ∀i = 1, . . . , n − 1, a contradiction. therefore g(x) has no real zeros and hence g lacks non-zero real vertex connectivity roots. this completes the proof. corollary 2.10. let g be a graph and n1, n2 are respectively be the number of vertex-connected vertex pairs and the number of vertex pairs which disconnects g, where n1 > n2. then, all the non-zero vertex connectivity roots of g lie strictly outside the unit circle. theorem 2.11. let g be a graph with |di| = m ∀i ≤ n, where 2 ≤ n ≤ diam(g) + 1. then, all the non-zero vertex connectivity roots of g are distinct and are of unit modulus. in addition, if g is a tree of diameter p, where p is an odd prime, then g lacks non-zero real vertex connectivity roots. 149 k. priya, v. anil kumar proof. without loss of generality, let g be a graph which is not a tree. since |di| = m, ∀ i = 1, 2, . . . , n, v [g, x] = m n∑ i=1 xi = mx[1 + x + . . . + xn−1]. the zeros of the polynomial 1 + x + . . . + xn−1 are the nth roots of unity except 1 and hence it follows that the vertex connectivity roots of g are distinct. now, if g is a tree, then v [g; x] is a polynomial of degree diam(g) + 1. that is, v [g, x] = m diam(g)+1∑ i=2 xi = mx2[1 + x + . . . + xdiam(g)−1]. since diam(g) = p, the polynomial 1 + x + . . . + xp−1 is irreducible over the set of all rationals and hence it has no rational roots. but the only possible real p − 1th root of unity is −1, which is rational. therefore, it can be concluded that g is free of non-zero real vertex connectivity roots. this completes the proof. corollary 2.12. if n is odd, then all the non-zero vertex connectivity roots of cn lies on the unit circle. in addition, if n−1 2 is even, then cn has exactly two real vertex connectivity roots. proof. this follows from the fact that v [cn; x] = ∑n−1 2 i=1 nx i for odd n and −1 is a kth root of unity if k is even. theorem 2.13. (i) let g1 and g2 be two disjoint graphs. if a is a vertex connectivity root of both g1 and g2, then a is also a vertex connectivity root of g1 + g2. (ii) let g be a vertex-connected graph of order n and let g ′ be the graph obtained by adjoining one pendent vertex to m distinct vertices of g, m ≤ n. then, the vertex connectivity roots of g ′ are the roots of the polynomial v [g; x] + nmx2 + ( m 2 ) x3. (iii) let g be metrically connective and let d = diam(g). then, g has exactly d vertex connectivity roots counting multiplicities. proof. (i) this follows from the fact that v [g1+g2; x] = v [g1; x]+v [g2; x]. (ii) since g is vertex-connected, the newly adjoined vertices along with every vertex of g disconnects g ′ into exactly two components and mutually disconnects g ′ into three components. therefore, v [g ′ ; x] = v [g; x] + nmx2 + ( m 2 ) x3. 150 on the roots and stability of vertex connectivity polynomial (iii) this follows from the fact that the degree of v [g; x] is d. this completes the proof. theorem 2.14. let g be a graph of order n and diameter d. then, any vertex connectivity root β of g either has nonpositive real part or satisfies |β| < 1 + √ 1 + 2n(n − 1) 2 . proof. let β be a non-zero vertex connectivity root of g. if |β| ≤ 1, then the result holds trivially since 1+ √ 1+2n(n−1) 2 > 1. case(i) : let g be a tree. then, the degree of v [g; x] is d + 1. we have, v [g; x] = x2g(x), where g(x) = ad−1xd−1 + . . . + a1x + a0. since ∑d−1 i=0 ai = ( n 2 ) , |ai| ≤ ( n 2 ) ∀i = 0, 1, . . . , d − 1. now, if |β| > 1 and re(β) > 0, then |g(β)| |βd−1| = |ad−1 + ad−2 β + . . . + a1 βd−2 + a0 βd−1 | ≥ |ad−1 + ad−2 β | − n(n − 1) 2 [ 1 |β2| + . . . + 1 |βd−1| ] > re(ad−1) + ad−2re( 1 β ) − n(n − 1) 2 [ 1 |β2| − |β| ] ≥ 1 − n(n − 1) 2 [ 1 |β2| − |β| ], since ad−1, ad−2 ≥ 1 and re(β) > 0. = |β2| − |β| − n(n−1) 2 |β2| − |β| ≥ 0, whenever |β2| − |β| − n(n−1) 2 ≥ 0. that is, if |β| ≥ 1+ √ 1+4 n(n−1) 2 2 = 1+ √ 1+2n(n−1) 2 and re(β) > 0, then |g(β)| |βd−1| > 0 so that β cannot be a zero of g(x). case(ii) let g be a graph which is not a tree. then at least one edge of g is a part of some cycle so that the linear coefficient of v [g; x] is non-zero. let v [g; x] = xh(x), where h(x) = am−1xm−1 + . . . + a1x + a0. here also∑m−1 i=0 ai = ( n 2 ) , |ai| ≤ ( n 2 ) ∀i = 0, 1, . . . , m − 1, so that similar calculation as in case(i) yields that if |β| ≥ 1+ √ 1+2n(n−1) 2 and re(β) > 0, then β cannot be a zero of g(x). . 151 k. priya, v. anil kumar therefore, from the above two cases we get that every vertex connectivity root of the graph g either has nonpositive real part or has modulus strictly less than 1+ √ 1+2n(n−1) 2 . this completes the proof. theorem 2.15. let g be a tree with diameter d ≤ 2. then, all the non-zero vertex connectivity roots of g lie in the left half plane. in addition, if the leading coefficient of v [g; x] dominates, then the polynomial is schur stable. proof. case(i): if d = 1, then g = p2 so that v [g; x] = x2. thus, the result holds trivially. case(ii): if d = 2, then v [g; x] = x2(ax + b), a cubic polynomial where a, b > 0. thus the only non-zero zero x = −b a of v [g; x] lies in the left half plane. now if the leading coefficient of v [g; x] dominates, then | b a | < 1 and thus all the vertex connectivity roots of g lies inside the unit circle. hence v [g; x] is schur stable. corollary 2.16. v [k1,n; x] is schur stable iff n > 3. theorem 2.17. every vertex connectivity root of the path graph pn lie in the closed left half plane iff n ≤ 5 proof. we have, v [pn; x] = ∑n−1 i=1 (n − i)x i+1. thus the result holds trivially for n = 2 and n = 3. now for n ≥ 4, let v [pn; x] = x2g(x). when n = 4, g(x) = x2 + 2x + 3 so that h1 = [ 2 ] , h2 = [ 2 1 0 3 ] are having positive determinants. if n = 5, then g(x) = x3 + 2x2 + 3x + 4 so that all the hurwitz matrices h1 = [ 2 ] , h2 = [ 2 1 4 3 ] , h3 =  2 1 04 3 2 0 5 4   have positive determinant. for n = 6, g(x) = x4 + 2x3 + 3x2 + 4x + 5. h3 =  2 1 04 3 2 0 0 4   152 on the roots and stability of vertex connectivity polynomial so that det(h3) < 0. for n ≥ 7, h3 =  2 1 04 3 2 6 5 4   so that det(h3) = 0. thus for n ≥ 6, g(x) is not stable so that all the vertex connectivity roots of pn does not belong to the closed left half plane. this completes the proof. a bistar graph bm,m is the union of two identical star graphs k1,m whose centres joined together to form a new edge. theorem 2.18. the vertex connectivity roots of the bistar graph bm,m always lie in the closed left half plane and are all real iff m ≥ 7. also, for all m ≥ 2, zero is the only vertex connectivity root of multiplicity > 1. proof. we have, v [bm,m; x] = x2[m2x2 + m(m + 1)x + (2m + 1)].[priya and anil kumar, 2021b] the zeros of the polynomial g(x) = m2x2 + m(m + 1)x + (2m + 1) are x = −m(m + 1) ± √ m2(m + 1)2 − 4m2(2m + 1) 2m2 = −(m + 1) ± √ m2 − 6m − 3 2m . since √ m2 − 6m − 3 < √ m2 < m + 1, all the zeros of g(x) lie in the left half plane. the zeros of g(x) are real iff m2 − 6m − 3 ≥ 0 iff m ≥ 3 + √ 12 or m ≤ 3 − √ 12. since m is a positive integer and since 6 < 3 + √ 12 < 7, we get that the vertex connectivity roots of bm,m are all real iff m ≥ 7. now, since m2 − 6m − 3 is never zero for any integer m, all the zeros of g(x) are distinct. this completes the proof. corollary 2.19. for m ≥ 3, v [bm,m; x] is schur stable. proof. in the proof of theorem 2.18, the zeros of g(x) are −(m+1)± √ m2−6m−3 2m . it can be observed that |−(m+1)± √ m2−6m−3 2m | < 1 for m ≥ 3 so that all the vertex connectivity roots of bm,m lie inside the unit circle. this completes the proof. 153 k. priya, v. anil kumar theorem 2.20. let p(z) = anzn + an−1zn−1 + . . . + a1z + a0 be a complex polynomial such that |ak| > |a0| + |a1| + . . . + |ak−1| + |ak+1| + . . . + |an| for some 0 ≤ k ≤ n. then exactly k zeros of p lie strictly inside the unit circle, and the other n − k zeros of p lie strictly outside the unit circle.[zhao, 1947] a helm hn is obtained by adding pendent edges to every vertex on the graph obtained as the join of the cycle cn−1 and k1. a webgraph wbn is obtained by joining the pendent vertices of hn to form a cycle and then adding a single pendent edge to each vertex of the outer cycle. a butterfly graph is a double shell with same apex along with exactly two pendent edges at the apex. theorem 2.21. (i) for n > 3, the helm hn is not schur stable. (ii) for n > 3, the web graph wbn is not schur stable. but all the vertex connectivity roots of wbn are real and belong to the closed left half plane. (iii) for n > 4, the butterfly graph bfn is not schur stable. but all the vertex connectivity roots of bfn are real and belong to the closed left half plane. proof. (i) from [priya and anil kumar, 2021b], we get the vertex connectivity polynomial of hn as v [hn; x] = ( n−1 2 ) x3 + n(n − 1)x2 + ( n 2 ) x . here, the quadratic coefficient n(n−1) > ( n−1 2 ) + ( n 2 ) so that from theorem 2.20 it follows that only two zeros of v [hn; x] lie inside the unit circle and one zero outside the unit circle. hence hn is not schur stable. (ii) we have, v [wbn; x] = ( n−1 2 ) x3 + (2n − 1)(n − 1)x2 + ( 2n−1 2 ) x. that is, v [wbn; x] = ( n−1 2 ) x[x2 + 2(2n−1) n−2 x + 2(2n−1) n−2 ]. since the discriminant of the quadratic polynomial p(x) = x2 + 2(2n−1) n−2 x + 2(2n−1) n−2 is 12(2n−1) (n−2)2 > 0, it follows that all the vertex connectivity roots of wbn are real. the remaining results follows easily from the fact that the zeros of p(x) are exactly −2(2n−1)± √ 12(2n−1) 2(n−2) . (iii) from [priya and anil kumar, 2021b], it follows that the vertex connectivity polynomial of bfn is v [bfn; x] = x3 + (2n − 4)x2 + ( n−2 2 ) x. for n = 5, 6, 7, the non-zero vertex connectivity roots of bfn are exactly −3 ± √ 6, −4 ± √ 10 and −5 ± √ 15 respectively so that in all those cases the graph is not schur stable. now for n > 7, the linear coefficient( n−2 2 ) > 1+2n−4 and hence by theorem 2.20, only one zero of v [bfn; x] lie inside the unit circle and the other two zeros lie outside the unit circle. hence for n > 4, bfn lacks schur stability. now, since 2n − 4 is greater than the magnitude of the square of the discriminant of the quadratic factor in v [bfn; x] and since the discriminant is positive, it follows that all the 154 on the roots and stability of vertex connectivity polynomial non-zero vertex connectivity roots of bfn belong to the left half plane. this completes the proof. 3 conclusions in this paper, the nature of roots and stability properties of some well known graphs were discussed in detail and inferences were made about its stability. more general properties regarding the graph stability can be inferred only after studying the same for graph operations. 4 acknowledgements the first author would like to thank the university grants commission of india for funding this research. references paul a fuhrmann. a polynomial approach to linear algebra. springer science & business media, 2011. gaurav gajender and himanshu sharma. hurwitz polynomial. international journal of innovative research in technology, 1(7):340–342, 2014. f harary. graph theory. addison-wesley, 1969. k priya and v anil kumar. on the vertex connectivity index of graphs. chinese journal of mathematical sciences, 1(1):49–59, 2021a. k priya and v anil kumar. on the vertex connectivity polynomial of graphs. advances and applications in discrete mathematics, 26(2):133–147, 2021b. yufei zhao. integer polynomials. mop 2007 black group, https://yufeizhao.com, 69(1):17–20, 1947. 155 capitolo 13 ratio mathematica volume 37, 2019, pp. 25-38 25 some results for volterra integrodifferential equations depending on derivative in unbounded domains giuseppe anichini1 giuseppe conti2 alberto trotta3 abstract in this paper we study the existence of continuous solutions of an integrodifferential equation in unbounded interval depending on derivative. this paper extends some results obtained by the authors using the technique developed in their previous paper. this technique consists in introducing, in the given problems, a function q, belonging to a suitable space, instead of the state variable x. the fixed points of this function are the solutions of the original problem. in this investigation we use a fixed point theorem in fréchet spaces. keywords: fréchet spaces, semi-norms, acyclic sets, ascoli-arzelà theorem. 2010 ams subject classification: 45g10, 47h09, 47h30* 1 università degli studi di firenze, department of mathematics dimai, viale morgagni 67/a, 50134 firenze. giuseppe.anichini@unifi.it 2 università degli studi di firenze, department of mathematics dimai, viale morgagni 67/a, 50134 firenze. gconti@unifi.it 3 iiss santa caterina-amendola, via l. lazzarelli 12, 84132 salerno. albertotrotta@virgilio.it * received on november 15th, 2019. accepted on december 30rd, 2019. published on december 31st, 2019. doi:10.23755/rm.v36i1.471. issn: 1592-7415. eissn: 2282-8214. doi: 10.23755/rm.v37i0.486. ©anichini et al. anichini, conti, and trotta 26 1 introduction in this paper we study, in abstract setting, the solvability of a nonlinear integrodifferential equation of volterra type with implicit derivative, defined in unbounded interval, like (1) 𝑥′(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑓(𝑠, 𝑥(𝑠), 𝑥′(𝑠))𝑑𝑠 𝑥(0) = 0, 𝑡 ∈ 𝐽 = [0, +∞) 𝑡 0 we will look for solutions of this equation in the fréchet space of all real c1 functions defined in the real unbounded interval 𝐽 = [0, +∞). equation (1) is a special case of integro-differential equations. these equations have been seen as an important tool in the study of many boundary problems that we can encounter in various applications, like, for exemple, heat flow in material, kinetic theory, electrical ingeneering, vehicular traffic theory, biology, population dynamics, control theory, mechanics, mathematical economics. the integro-differential equations have been studied in various papers with the help of several tools of functional analysis, topology and fixed point theory. for istance we can refer to [1], [2], [3], [4], [5], [11], [12] and the references therein. in [8] an hammerstein equation, similar to (1), is consideren in the multivalued setting and bounded intervals. our paper extend some results obtained by the authors anichini and conti, using the techniques developed in previous paper (see to [1], [2], [3], [4], [5]). the crucial key of our approach, in order to find solutions of equation (1), consists in the use of a very useful fixed point theorem for multivalued, compact, uppersemicontinuous maps with acyclic values in a fréchet space. 2 preliminaries and notations let c1(j, ℝ) be the fréchet the of all real c1 functions defined in the real unbounded interval 𝐽 = [0, +∞)  ℝ, equipped with the following family of semi-norms ‖𝑥‖1,𝑛 = max{‖𝑥‖𝑛 , ‖𝑥′‖𝑛} where ‖𝑥‖𝑛 = sup{|𝑥(𝑡)|, 𝑡 ∈ [0, 𝑛]} and ‖𝑥′‖𝑛 = sup{|𝑥′(𝑡)|, 𝑡 ∈ [0, 𝑛]}. we recall that the topology of c1(j, ℝ) coincides with the topology of a complete metric space {𝐹, 𝑑} where some results for volterra integro-differential equations 27 𝑑(𝑥, 𝑦) = ∑ 2−𝑛‖𝑥 − 𝑦‖1,𝑛 1 + ‖𝑥 − 𝑦‖1,𝑛 +∞ 𝑛=1 a subset a  c1(j, ℝ) is said to be bounded if, for every natural number n, there exists mn > 0 such that ‖𝑥‖1,𝑛 ≤ 𝑀𝑛 ∀ 𝑥 ∈ 𝐶 1(𝐽, ℝ). a subset a  c1(j, ℝ) is relatively compact set if and only if the functions of the set a are equicontinuous and uniformly bounded (with their derivatives) in any interval [0, n]. we will denote by c(f) the family of all nonempty and compact subset of a fréchet space f. let m be a subset of a fréchet space f; a multivalued map 𝑆: 𝑀 → 𝐶(𝐹) is said to be uppersemicontinuous (u.s.c.) if the graph is closed in 𝑀 × 𝐹, i.e. for any sequence {𝑥𝑛 }  𝑀, 𝑥𝑛→ 𝑥0 and 𝑦𝑛 ∈ 𝑆(𝑥𝑛), 𝑦𝑛→ 𝑦0, we have 𝑦0 ∈ 𝑆(𝑥0). a multivalued map 𝑆: 𝑀 → 𝐶(𝐹) is said to be compact if it sends bounded sets into relatively compact sets. we apply the same definition for singlevalued maps. a subset a of a metric space e is said to be an r set if a is the intersection of a countable decreasing sequence of absolute retracts contained in e (see [10]). it is known that an r set is an acyclic set, i.e. it is acyclic with respect to any cohomology theory (see [7]). let m be a subset of the fréchet space c1(j, ℝ) and consider an operator 𝑇: 𝑀 → 𝐶1(𝐽, ℝ) . let {𝜖𝑛} be an infinitesimal sequence of real numbers. a sequence {𝑇𝑛} of maps 𝑇𝑛 : 𝑀 → 𝐶 1(𝐽, ℝ) is said to be an 𝜖𝑛-approximation of t on m if ‖𝑇𝑛(𝑥) − 𝑇(𝑥)‖1,𝑛 ≤ 𝜖𝑛 for every 𝑥 ∈ 𝑀 and for any natural number n. define 𝑈𝑛 = {𝑥 ∈ 𝐹 ∶ ‖𝑥‖1,𝑛 < 1}. let t be a compact map 𝑇: 𝑀 → c1(j, ℝ), where m is a closed set of the fréchet space c1(j, ℝ), and let {𝑇𝑛} be a 𝜖𝑛-approximation of t on m, where 𝑇𝑛 : 𝑀 → 𝐶1(𝐽, ℝ) are compact maps; then the set of fixed point of t is a compact r set if the equation 𝑥 − 𝑇𝑛 (𝑥) = 𝑦 has at most a solution for every 𝑦 ∈ 𝜀𝑛𝑈𝑛 for any natural number n (see [5]). in the sequel we will use the following result (see [9]). proposition 1 (kirszebraun’s theorem) let f : m → ℝ be a lipschitz map defined on arbitrary subset m of ℝn. then f admits a lipschitz extension  : ℝn → ℝ with the same lipschtiz constant. the well known gronwall’s lemma, from the standard theory of ordinary differential equations, will be used. anichini, conti, and trotta 28 proposition 2 (gronwall’s lemma) let g, h : 𝐽 → 𝐽 be continuous functions such that the following inequality: 𝑔(𝑡) ≤ 𝑢(𝑡) + ∫ ℎ(𝑠)𝑔(𝑠)𝑑𝑠 𝑡 ∈ 𝐽 𝑡 0 , holds, where u : 𝐽 → 𝐽 is a continuous nondecreasing function. then we have: 𝑔(𝑡) ≤ 𝑢(𝑡)exp (∫ ℎ(𝑠)𝑑𝑠) 𝑡 ∈ 𝐽 𝑡 0 . in the sequel we will use the following proposition that can be deduced from theorem 1 of [6]. proposition 3 (a fixed poin theorem) let f be a fréchet space and m  𝑋 be a bounded, closed and convex subset; let 𝑆: 𝐹 → 𝑀 be a multivalued, uppersemicontinuous map with acyclic values. if s(f) is (relatively) compact, then s has a fixed point. 3 main result the following result holds. theorem consider integral equation (1). assume that i) k : 𝐽 × 𝐽 → ℝ is a c1 function; moreover we assume that there exists a continuous function h : 𝐽 → 𝐽 with |𝑘(𝑡, 𝑠)| ≤ ℎ(𝑠) and | 𝜕𝑘(𝑡,𝑠) 𝜕𝑡 | ≤ ℎ(𝑠). ii) f : 𝐽 × ℝ × ℝ → ℝ is a c1 function; moreover we assume that there exist continuous functions a, b : 𝐽 → 𝐽, with ∫ 𝑎(𝑠)𝑑𝑠 = 𝐴 < +∞ +∞ 0 and ∫ 𝑏(𝑠)𝑑𝑠 = 𝐵 < +∞ +∞ 0 , such that: |𝑓(𝑠, 𝑥, 𝑦)| ≤ 𝑎(𝑠) + 𝑏(𝑠)|𝑦|. iii) assume that ∫ ℎ(𝑠)𝑏(𝑠)𝑑𝑠 =  < 1 +∞ 0 . then equation (1) has at least one solution in the space c1(j, ℝ). some results for volterra integro-differential equations 29 proof let q be a function belonging to c1(j, ℝ) and consider the following integral equation: (2) 𝑦(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 𝑡 0 𝑡 ∈ 𝐽 = [0, +∞) let s : c1(j, ℝ) → c1(j, ℝ) be the multivalued map which associates to every q ∈ c1(j, ℝ) the set of solutions of equation (2). clearly, putting 𝑥(𝑡) = ∫ 𝑦(𝑠)𝑑𝑠 𝑡 0 (hence x’(t) = y(t) and x(0) = 0), we have that the fixed points of the map s are the solution of equation (1). in order to find the fixed points of multivalued map s, the following steps in the proof have to be established (proposition 3): a) there exists a bounded, closed and convex set m  c1(j, ℝ) such that s(c1(j, ℝ))  m. b) the set s(c1(j, ℝ)) is relatively compact. c) the map s is uppersemicontinuous. d) the set s(q) is an acyclic set for every q ∈ c1(j, ℝ). a) let q ∈ c1(j, ℝ) and consider equation (2); assume that t ∈ [0, n], from hypotheses we have, : |𝑦(𝑡)| = |∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 𝑡 0 | ≤ ≤ |∫ ℎ(𝑠)(𝑎(𝑠) + 𝑡 0 𝑏(𝑠)|𝑦(𝑠)|)𝑑𝑠| ≤ ≤ |∫ ℎ(𝑠)𝑎(𝑠)𝑑𝑠 𝑡 0 | + |∫ ℎ(𝑠)𝑏(𝑠)|𝑦(𝑠)|𝑑𝑠 𝑡 0 | ≤ ‖ℎ‖𝑛𝐴 +  ‖𝑦‖𝑛. so that, since  < 1, we have ‖𝑦‖𝑛 ≤ ‖ℎ‖𝑛𝐴 1− . moreover, we have for t ∈ [0, n]: anichini, conti, and trotta 30 𝑦′(𝑡) = ∫ 𝜕𝑘(𝑡, 𝑠) 𝜕𝑡 𝑡 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 + 𝑘(𝑡, 𝑡)𝑓 (𝑡, ∫ 𝑞(𝑠)𝑑𝑠 𝑡 0 , 𝑦(𝑡)) and we obtain: |𝑦′ (𝑡)| ≤ |∫ 𝜕𝑘(𝑡, 𝑠) 𝜕𝑡 𝑡 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠| + |𝑘(𝑡, 𝑡)𝑓 (𝑡, ∫ 𝑞(𝑠)𝑑𝑠 𝑡 0 , 𝑦(𝑡)) | ≤ ≤ ∫ ℎ(𝑠)𝑎(𝑠)𝑑𝑠 𝑡 0 + ∫ ℎ(𝑠)𝑏(𝑠)|𝑦(𝑠)|𝑑𝑠 𝑡 0 + ℎ(𝑡)(𝑎(𝑡) + 𝑏(𝑡)|𝑦(𝑡)|) ≤ ≤ ‖ℎ‖𝑛 𝐴 +  ‖𝑦‖𝑛 + ‖ℎ𝑎‖𝑛 + ‖ℎ𝑏‖𝑛‖𝑦‖𝑛 ≤ ≤ ‖ℎ‖𝑛 𝐴 + ‖ℎ𝑎‖𝑛 + ‖𝑦‖𝑛( + ‖ℎ𝑏‖𝑛) ≤ ‖ℎ‖𝑛𝐴 + ‖ℎ𝑎‖𝑛 + ‖ℎ‖𝑛𝐴 1− ( + ‖ℎ𝑏‖𝑛 ). so that there exists mn > 0 such that ‖𝑦‖1,𝑛 ≤ 𝑀𝑛 . then we have s(c1(j, ℝ))  m, where 𝑀 = {𝑦 ∈ 𝐶1(𝐽, ), ‖𝑦‖1,𝑛 ≤ 𝑀𝑛 }. b) now, we want to prove that the set s(c1(j, ℝ)) is relatively compact. let y ∈ s(c1(j, ℝ)) and fix 𝜀 > 0. for any u, w ∈ [0, n] we have: 𝑦′(𝑤) − 𝑦′(𝑢) = ∫ 𝜕𝑘(𝑤, 𝑠) 𝜕𝑡 𝑤 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 + 𝑘(𝑤, 𝑤)𝑓 (𝑤, ∫ 𝑞(𝑠)𝑑𝑠 𝑤 0 , 𝑦(𝑤)) − ∫ 𝜕𝑘(𝑢, 𝑠) 𝜕𝑡 𝑢 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 − 𝑘(𝑢, 𝑢)𝑓 (𝑢, ∫ 𝑞(𝑠)𝑑𝑠 𝑢 0 , 𝑦(𝑢)) some results for volterra integro-differential equations 31 = ∫ 𝜕𝑘(𝑤, 𝑠) 𝜕𝑡 𝑢 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 + 𝑘(𝑤, 𝑤)𝑓 (𝑤, ∫ 𝑞(𝑠)𝑑𝑠 𝑤 0 , 𝑦(𝑤)) − ∫ 𝜕𝑘(𝑢, 𝑠) 𝜕𝑡 𝑢 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 −𝑘(𝑢, 𝑢)𝑓 (𝑢, ∫ 𝑞(𝑠)𝑑𝑠 𝑢 0 , 𝑦(𝑢)) + ∫ 𝜕𝑘(𝑤, 𝑠) 𝜕𝑡 𝑤 𝑢 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 it follows that |𝑦′(𝑤) − 𝑦′(𝑢)| ≤ ≤ ∫ | 𝜕𝑘(𝑤, 𝑠) 𝜕𝑡 − 𝜕𝑘(𝑢, 𝑠) 𝜕𝑡 | 𝑢 0 (𝑎(𝑠) + 𝑏(𝑠)|𝑦(𝑠)|)𝑑𝑠 + ‖ℎ‖𝑛 |𝑓 (𝑤, ∫ 𝑞()𝑑 𝑤 0 , 𝑦(𝑤)) − 𝑓 (𝑢, ∫ 𝑞()𝑑 𝑢 0 , 𝑦(𝑢))| + ‖ℎ‖𝑛 |∫ 𝜕𝑘(𝑤, 𝑠) 𝜕𝑡 𝑤 𝑢 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠| by continuity of the functions q, h, f and 𝜕𝑘 𝜕𝑡 it follows that there exists  > 0 such that for |𝑤 − 𝑢| < , u, w ∈ [0, n], we have |𝑦′(𝑤) − 𝑦′(𝑢)| < 𝜀 since |𝑦(𝑤) − 𝑦(𝑢)| ≤ 𝑀𝑛 |𝑤 − 𝑢|, we can conclude that the set s(c 1(j, ℝ)) is relatively compact. c) let us now show that the map s is uppersemicontinuous. let {𝑞𝑚} be a sequence, 𝑞𝑚 ∈ c 1(j, ℝ), with ‖𝑞𝑚 − 𝑞0‖1,𝑛 → 0 , 𝑦𝑚 ∈ 𝑆(𝑞𝑚), i.e. 𝑦𝑚(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞𝑚()𝑑 𝑠 0 , 𝑦𝑚(𝑠)) 𝑑𝑠 𝑡 ∈ [0, 𝑛] 𝑡 0 assume that ‖𝑦𝑚 − 𝑦0‖1,𝑛 → 0 . we need to show that 𝑦0 ∈ 𝑆(𝑞0). from the dominated lebesgue convergence theorem it follows: anichini, conti, and trotta 32 𝑙𝑖𝑚𝑚→+∞ 𝑓 (𝑠, ∫ 𝑞𝑚()𝑑 𝑠 0 , 𝑦𝑚(𝑠)) = 𝑓 (𝑠, ∫ 𝑞0()𝑑 𝑠 0 , 𝑦0(𝑠)) and 𝑙𝑖𝑚𝑚→+∞ 𝑦𝑚(𝑡) = = 𝑙𝑖𝑚𝑚→+∞ ∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞𝑚()𝑑 𝑠 0 , 𝑦𝑚(𝑠)) 𝑑𝑠 = 𝑡 0 = ∫ 𝑙𝑖𝑚𝑚→+∞ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞𝑚()𝑑 𝑠 0 , 𝑦𝑚(𝑠)) 𝑑𝑠 = 𝑡 0 = ∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞0()𝑑 𝑠 0 , 𝑦0(𝑠)) 𝑑𝑠 . 𝑡 0 hence, we obtain 𝑦0(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞0()𝑑 𝑠 0 , 𝑦0(𝑠)) 𝑑𝑠 𝑡 0 i. e. 𝑦0 ∈ 𝑆(𝑞0) d) now we want to show that, for every fixed q ∈ c1(j, ℝ), the set s(q) is acyclic. consider equation (2) (with q fixed). put 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) = 𝑙(𝑠, 𝑦). then equation (2) can be written in the following way: 𝑦(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑙(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 ∈ [0, +∞) 𝑡 0 we have: |𝑦(𝑡)| ≤ |∫ 𝑘(𝑡, 𝑠)𝑙(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 | ≤ ∫ ℎ(𝑠)𝑎(𝑠)𝑑𝑠 + ∫ ℎ(𝑠)𝑏(𝑠)|𝑦(𝑠)|𝑑𝑠. 𝑡 0 𝑡 0 some results for volterra integro-differential equations 33 from gronwall’s lemma it follows that: |𝑦(𝑡)| ≤ ∫ ℎ(𝑠)𝑎(𝑠)𝑑𝑠 exp(∫ ℎ(𝑠)𝑏(𝑠)𝑑𝑠) = 𝑚(𝑠) 𝑡 0 𝑡 0 where m is a continuous function. let 𝑈: ℝ → [0, 1] the uryshon (continuous) function defined by 𝑈(𝑧) = 1 if |𝑧| ≤ 1 and 𝑈(𝑧) = 0 if |𝑧| ≥ 2. now we define the function 𝑔(𝑠, 𝑦) = 𝑈 ( 𝑦 𝑚(𝑠) + 1 ) 𝑙(𝑠, 𝑦). clearly 𝑔(𝑠, 𝑦) = 𝑙(𝑠, 𝑦) when |𝑦| ≤ 𝑚(𝑠). hence the set of solutions of the following equation 𝑦(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 ∈ [0, +∞) 𝑡 0 coincides with the set of solutions of equation (2) with q fixed. consider now the integral operator 𝐻: 𝐶1(𝐽, ℝ)→ 𝐶1(𝐽, ℝ): (𝐻(𝑦))(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 𝑡 ∈ [0, +∞) if z = h(y), we have 𝑧(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑈 ( 𝑦(𝑠) 𝑚(𝑠) + 1 ) 𝑙(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 . notice that 𝑈 ( 𝑦(𝑠) 𝑚(𝑠)+1 ) 𝑙(𝑠, 𝑦(𝑠)) = 𝑙(𝑠, 𝑦(𝑠)) if 𝑦(𝑠) ≤ 𝑚(𝑠) + 1 anichini, conti, and trotta 34 and 𝑈 ( 𝑦(𝑠) 𝑚(𝑠)+1 ) 𝑙(𝑠, 𝑦(𝑠)) = 0 if 𝑦(𝑠) ≥ 2𝑚(𝑠) + 2. so that: ‖𝑧‖𝑛 ≤ ‖ℎ‖𝑛𝐴 + 2(‖𝑚‖𝑛 + 1). moreover we obtain: |𝑧′(𝑡)| ≤ ∫ ℎ(𝑠)𝑈 ( 𝑦(𝑠) 𝑚(𝑠) + 1 ) |𝑙(𝑠, 𝑦(𝑠))|𝑑𝑠 𝑡 0 + ℎ(𝑡)𝑈 ( 𝑦(𝑡) 𝑚(𝑡) + 1 ) |𝑙(𝑡, 𝑦(𝑡))| hence ‖𝑧′‖𝑛 ≤ ‖ℎ‖𝑛𝐴 + 2(‖𝑚‖𝑛 + 1) + ‖ℎ𝑎‖𝑛 + 2‖ℎ𝑏‖𝑛(‖𝑚‖𝑛 + 1) = 𝐴𝑛 it follows that ‖𝑧‖1,𝑛 ≤ 𝐴𝑛, where z = h(y). so that the set of solutions of equation 𝑦(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 ∈ [0, +∞) 𝑡 0 coincides with the set of fixed points of operator h in the set 𝐴 = {𝑧 ∈ 𝐶1(𝐽, ℝ), ‖𝑧‖1,𝑛 ≤ 𝐴𝑛}. it is easy to see (again as consequence of the ascoliarzelà theorem) that the set h(a) is relatively compact set. moreover h is a continuous operator; to show the last assertion, let us take 𝑦0 , 𝑦𝑚 ∈ 𝐴, ‖𝑦𝑚 − 𝑦0‖1,𝑛 → 0, 𝑧𝑚 ∈ 𝐻(𝑦𝑚), ‖𝑧𝑚 − 𝑧0‖1,𝑛 → 0; we are going to prove that 𝑧0 ∈ 𝐻(𝑦0). for every 𝑡 ∈ [0, 𝑛] we have: some results for volterra integro-differential equations 35 𝑙𝑖𝑚𝑚→+∞ |∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦𝑚(𝑠))𝑑𝑠 − ∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦0(𝑠))𝑑𝑠 𝑡 0 𝑡 0 | ≤ (from the dominated lebesgue convergence theorem and the continuity of function g) ≤ ∫ 𝑙𝑖𝑚𝑚→+∞ ℎ (𝑠)|𝑔(𝑠, 𝑦𝑚(𝑠)) − 𝑔(𝑠, 𝑦0(𝑠))|𝑑𝑠 . 𝑡 0 hence 𝑧0(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦0(𝑠))𝑑𝑠 = (𝐻(𝑦0))(𝑡). 𝑡 0 fix now a natural number n. we know (proposition 1) that there exists a lipschitz function 𝑔𝑛: [0, 𝑛] × [−𝐴𝑛, 𝐴𝑛 ] → ℝ such that, for every (𝑠, 𝑦) ∈ [0, 𝑛] × [−𝐴𝑛 , 𝐴𝑛 ] , we have: |𝑔𝑛(𝑠, 𝑦) − 𝑔(𝑠, 𝑦)| ≤ 1 (𝑛 + 1)2‖ℎ‖𝑛 and |𝑔𝑛(𝑠, 𝑦) − 𝑔𝑛(𝑠, 𝑦1)| ≤ 𝐿𝑛|𝑦 − 𝑦1| for every (𝑠, 𝑦), (𝑠, 𝑦1) ∈ [0, 𝑛] × [−𝐴𝑛, 𝐴𝑛 ] , let 𝐺𝑛 : 𝐽 × ℝ → ℝ be the lipschitz extension of the function 𝑔𝑛; hence 𝐺𝑛(𝑠, 𝑦) = 𝑔𝑛(𝑠, 𝑦) for every (𝑠, 𝑦) ∈ [0, 𝑛] × [−𝐴𝑛 , 𝐴𝑛] and |𝐺𝑛 (𝑠, 𝑦) − 𝐺𝑛(𝑠, 𝑦1)| ≤ 𝐿𝑛|𝑦 − 𝑦1| for every (𝑠, 𝑦), (𝑠, 𝑦1) ∈ 𝐽 × ℝ. let 𝐻𝑛 ∶ 𝐴 → 𝐶 1(𝐽, ℝ)) be the operator defined as follows: (𝐻𝑛(𝑦))(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝐺𝑛(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 𝑡 ∈ [0, +∞) anichini, conti, and trotta 36 clearly this operator is compact for every natural number n. moreover, for every t ∈ [0, n] and y ∈ a, we have: |(𝐻𝑛(𝑦))(𝑡) − (𝐻(𝑦))(𝑡)| ≤ ∫ 𝑘(𝑡, 𝑠)|𝐺𝑛(𝑠, 𝑦(𝑠)) − 𝑔(𝑠, 𝑦(𝑠))|𝑑𝑠 ≤ 𝑡 0 ≤ 𝑛‖ℎ‖𝑛 1 (𝑛+1)2‖ℎ‖𝑛 < 1 𝑛 . so that ‖𝐻𝑛(𝑦) − 𝐻(𝑦)‖𝑛 < 1 𝑛 . moreover we have for every y ∈ a: |(𝐻′𝑛(𝑦))(𝑡) − (𝐻 ′(𝑦))(𝑡)| ≤ ≤ |∫ 𝜕𝑘(𝑡, 𝑠) 𝜕𝑡 𝐺𝑛(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 − 𝑘(𝑡, 𝑡)𝐺𝑛(𝑡, 𝑦(𝑡)) + ∫ 𝜕𝑘(𝑡, 𝑠) 𝜕𝑡 𝑔(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 − 𝑘(𝑡, 𝑡)𝑔(𝑡, 𝑦(𝑡))| ≤ ∫ ℎ(𝑠)|𝐺𝑛(𝑠, 𝑦(𝑠)) − 𝑔(𝑠, 𝑦(𝑠))|𝑑𝑠 𝑡 0 + ℎ(𝑡)|𝐺𝑛(𝑡, 𝑦(𝑡)) − 𝑔(𝑡, 𝑦(𝑡))| ≤ ≤ 𝑛‖ℎ‖𝑛 1 (𝑛+1)2‖ℎ‖𝑛 + ‖ℎ‖𝑛 1 (𝑛+1)2‖ℎ‖𝑛 = 𝑛+1 (𝑛+1)2 < 1 𝑛 . hence ‖𝐻′𝑛(𝑦) − 𝐻′(𝑦)‖𝑛 < 1 𝑛 . let now b ∈ a. we consider the equation 𝑦 − 𝐻𝑛(𝑦) = 𝑏. we want to prove that it has at most one solution. consider the equation 𝑧 − 𝐻𝑛(𝑧) = 𝑏; then, for every t ∈ j and by gronwall’s lemma we have: |𝑦(𝑡) − 𝑧(𝑡)| ≤ ∫ ℎ(𝑠)|𝐺𝑛(𝑠, 𝑦(𝑠)) − 𝐺𝑛 (𝑠, 𝑧(𝑠))|𝑑𝑠 ≤ 𝑡 0 ≤ ∫ ℎ(𝑠)𝐿𝑛|𝑦(𝑠) − 𝑧(𝑠)|𝑑𝑠 ≤ 0 𝑡 0 . some results for volterra integro-differential equations 37 so that we can say that y(t) = z(t) for every t ∈ j. finally, we are able to conclude that, for every q ∈ c1(j, ℝ), the set s(q) is acyclic and the theorem is proved. 4 an example consider the following integro-differential equation: (3) 𝑥′(𝑡) = ∫ 3𝑡 𝑒−𝑠+2 1 + 𝑡3 ( 3𝑠2 𝑒−2𝑠 1 + (sin(𝑥(𝑠))) 2 + 𝑠 𝑒−𝑠−2𝑥′(𝑠)) 𝑑𝑠 𝑡 0 𝑥(0) = 0, 𝑡 ∈ 𝐽 = [0, +∞). we have 𝑘(𝑡, 𝑠) = 3𝑡 𝑒−𝑠+2 1 + 𝑡3 , 𝑓(𝑠, 𝑥(𝑠), 𝑥′(𝑠)) = 3𝑠2 𝑒−2𝑠 1 + (sin(𝑥(𝑠))) 2 + 𝑠 𝑒−𝑠−2𝑥′(𝑠) ℎ(𝑠) = 3 𝑒−𝑠+2, 𝑎(𝑠) = 3𝑠2 𝑒−2𝑠, 𝑏(𝑠) = 𝑠 𝑒−𝑠−2. hence, we obtain: ∫ 𝑎(𝑠)𝑑𝑠 = 3 4 +∞ 0 ∫ 𝑏(𝑠)𝑑𝑠 = 𝑒−2 +∞ 0 ∫ ℎ(𝑠)𝑏(𝑠)𝑑𝑠 = ∫ 3𝑠𝑒−2𝑠𝑑𝑠 = 3 4 < 1 +∞ 0 +∞ 0 so that the assumptions of our theorem are satisfied and integro-differential equation (3) has solutions. anichini, conti, and trotta 38 references [1] anichini g. and conti g. (2014). “on the existence of solutions for quadratic integral equations on unbounded intervals for quasibounded maps”. rend. sem. mat. univ. politec. torino, vol. 72, 3–4, 151-160. [2] anichini g. and conti g. (2016). “existence of solutions for volterra integral equations depending on derivative”. pioneer journal of mathematics and mathematical sciences, 18 (1), 45-60. [3] anichini g. and conti g. (2018). “existence of solutions for volterra integro-differential equations with implicit derivative”. international journal of scientific and innovative mathematical research, 6 (4), 10-15. [4] anichini g. and conti g. (2018). “existence of solutions for volterra integro-differential equations in unbounded domains”. pioneer journal of mathematics and mathematical sciences, 22 (1), 53-68. [5] czarnowski k. and pruszko t. (1991). “on the structure of fixed point sets of compact maps in b0 spaces with applications to integral and differential equations in unbounded domain”. journal of mathematical analysis and applications, vol. 154, no 1, 151-163. [6] fitzpratrick p. m. and petryshyn w. v. (1974). “fixed point theorems for multivalued noncompact acyclic mappings”. pacific journal of mathematics, vol. 54, no. 2, 17-23. [7] gabor g. (1999). “on the acyclicity of fixed point sets of multivalued maps”. topological methods in nonlinear analysis, 14, 327-343. [8] hoai n. t. and loi n. v. (2007). “existence of solutions for some hammerstein type integro-differential inclusions”. electronic journal of differential equations, 178, 1-8. [9] kirszbraun m. d. (1934). “über die zusammenziehende und lipschitzche transformationem”. fund. math., 22, 77-108. [10] lasry j. m. and robert r. (1976). analyse non lineaire mutivoque. publ. n. 7611, centre rech. math. décision (ceremade), université de paris ix (dauphine). [11] pachpatte b. g. (2010). “implicit type volterra integro-differential equation”. tamkang journal of mathematics, 4 (1), 97-107. [12] sikorska a. (2001). “existence theory for nonlinear volterra integral and differential equations”. journal inequalities and applications, 6, 325-338. ratio mathematica 27 (2014) 81-90 issn:1592-7415 some applications of linear difference equations in finance with wolfram|alpha and maple dana ř́ıhová, lenka viskotová department of statistics and operation analysis, faculty of business and economics, mendel university in brno, zemědělská 1, 613 00 brno dana.rihova@mendelu.cz, lenka.viskotova@mendelu.cz abstract the principle objective of this paper is to show how linear difference equations can be applied to solve some issues of financial mathematics. we focus on the area of compound interest and annuities. in both cases we determine appropriate recursive rules, which constitute the first order linear difference equations with constant coefficients, and derive formulas required for calculating examples. finally, we present possibilities of application of two selected computer algebra systems wolfram|alpha and maple in this mathematical area. key words: linear difference equation, compound interest, future value of an annuity, periodic payment, computer algebra systems. 2000 ams: 39a06, 39-04. 1 introduction the values of most economic variables are given as a sequence of values observed at discrete time intervals or periods. these sequences are often specified by recursion with some initial elements. but it is preferable to know a rule in the form of an equation for the n-th element to calculate the values of sequence elements. the recursive rule of a sequence represents a difference equation and the functional notation for the n-th element can be obtained by solving this difference equation (see [7]). 81 d. ř́ıhová, l. viskotová many formulas used in financial mathematics can be derived from the recursive rules between two consecutive elements which constitute difference equations of the first order. this includes for example simple and compound interest calculation, the present and future value of an annuity and loan amortization. 2 compound interest 2.1 derivation of formula a sum of money deposited in a bank earns interest which is added to the principal at regular intervals and the new amount is used for calculating the interest for the next conversion period. we shall develop a formula for the total amount of money that is accumulated by a given principal after a certain number of conversion periods, see also [3], [4]. let r stand for the annual interest rate and k denote the number of conversion periods in a year. let n be equal to the number of conversion periods in the term of the deposit. let yn represent the amount on deposit at the end of n conversion periods and p the initial sum deposited (i.e. principal). we obtain the following recursive rule yn+1 = yn + r k yn = ( 1 + r k ) yn, n = 0, 1, 2, . . . with y0 = p , where the fraction r k stands for the interest rate per conversion period and r k yn is the interest generated during (n+1)st period. the previous formula represents the first order homogeneous linear difference equation with constant coefficients yn+1 − ( 1 + r k ) yn = 0 (1) with initial condition y0 = p. (2) the above problem (1), (2) can be solved by using the properties of a geometric sequence, see [1] and [8]. but our approach will be different due to the use of a difference equation. the characteristic equation of (1) takes form z − ( 1 + r k ) = 0 with real root z = 1 + r k . 82 some applications of linear difference equations in finance according to [5], the general solution is a geometric sequence yn = c ( 1 + r k )n , c ∈ r. a constant c can be specified from the initial condition (2) for period n = 0, hence c = p. thus the general solution is given by yn = p ( 1 + r k )n (3) and represents the compound interest formula. this formula gives the amount yn into which principal p grows when it earns compound interest for n conversion periods at an interest rate of r k per conversion period. 2.2 illustrative examples example 2.1. an amount of eur 1, 000 is deposited into a savings account at an annual interest rate of 2.5%, compounded yearly. what will the value of the account be worth after 20 years? to find the amount we use formula (3). we have principal p = 1, 000, annual interest rate r = 0.025, number of conversion periods per year k = 1 and total number of conversion periods n = 20. after plugging those figures into the formula, we get y20 = 1000 ( 1 + 0.025 1 )20 . = 1638.62 example 2.2. find the number of years required for a given sum of money to double itself if the interest rate is 3%, compounded quarterly. substituting yn = 2p in the compound interest formula (3), we have 2p = p ( 1 + r k )n which implies 2 = ( 1 + r k )n . (4) taking natural logarithms on both sides and using properties of logarithms gives n = ln 2 ln ( 1 + r k ). 83 d. ř́ıhová, l. viskotová to calculate the number of years n we have to divide the total number of conversion periods n by their number in a year k n = 1 k ln 2 ln ( 1 + r k ). (5) setting r = 0.03, k = 4 we get the required number of years n = 1 4 ln 2 ln ( 1 + 0.03 4 ) .= 23.19 3 future value of an annuity 3.1 derivation of formula an annuity is essentially a sequence of periodic payments, usually equal in amount, payable at equal intervals of time over the course of a fixed time period. the future value of an annuity is the total value of its periodic payments enhanced at interest rate for given number of conversion periods. it is defined as the sum of the amounts of all payments and the total compound interest earned on these payments to the time of the last payment. see for example [1], [4]. suppose the constant sum r is deposited at the end of each conversion period in a bank which credits interest at the annual rate r. the deposits are made k times each year over n conversion periods. let yn denote the total amount in the account at the end of n conversion periods. we shall find the total worth of an annuity after n deposits. the recursive rule for the future value of an annuity can be written as yn+1 = yn + r k yn + r = ( 1 + r k ) yn + r, n = 0, 1, 2, . . . with y0 = 0, where r k is the interest rate per conversion period. this equation constitutes the first order nonhomogeneous linear difference equation with constant coefficients yn+1 − ( 1 + r k ) yn = r (6) with initial condition y0 = 0. (7) in financial mathematics, the above problem (6), (7) is solved by using the properties of a geometric sequence. but we will proceed by means of 84 some applications of linear difference equations in finance difference equations like in the case of compound interest. to solve nonhomogeneous difference equation (6) we consider the corresponding homogeneous difference equation yn+1 − ( 1 + r k ) yn = 0 (8) which is the same as (1) in the case of compound interest. hence the general solution ȳn of this homogeneous difference equation is given by ȳn = c ( 1 + r k )n , c ∈ r. (9) the right-hand side of the nonhomogeneous difference equation (6) is a constant r which is a polynomial of degree zero. thus a particular solution yn can be estimated by yn = b, b ∈ r. for more details see [5]. using the method of undetermined coefficients (see [2]) we substitute the above estimate into (6). we get b− ( 1 + r k ) b = r and solving for b we obtain b = −r k r . therefore the particular solution of (6) takes the form yn = −r k r . (10) using (9), (10) according to the superposition principle (see [6]), the general solution of the nonhomogeneous linear difference equation (6) is the sum yn = yn + ȳn = −r k r + c ( 1 + r k )n , c ∈ r. a constant c can be specified from the initial condition (7) for period n = 0. hence we obtain c = r k r . consequently, the general solution of (6) takes the form yn = −r k r + r k r ( 1 + r k )n 85 d. ř́ıhová, l. viskotová which can be written as yn = r ( 1 + r k )n − 1 r k . (11) the above relation represents the future value of an annuity formula which gives the amount of an annuity of n payments of r at the compound rate r k per conversion period under the assumption that the payment interval equals the conversion period. the future value of an annuity formula is used to calculate what value at a future date would be for a series of periodic payments. in financial mathematics, it is common to use the following form of the formula (11) setting i = r k where i represents the interest rate per compounding interval (see [8] and [10]) yn = r (1 + i) n − 1 i . (12) 3.2 illustrative examples example 3.1. suppose eur 500 is deposited at the end of every six-month period in a bank, whose annual rate is 3.4%, compounded semiannually. how much will this account be worth after 7 years? we get the solution using (11), where r = 500, r = 0.034, k = 2, n = 14. then we obtain y14 = 500 ( 1 + 0.034 2 )14 − 1 0.034 2 . = 7828.64 example 3.2. find the payment amount that you should deposit at the end of each month in a bank so that eur 35, 000 will be available after 10 years if the interest rate is 1.6%, compounded monthly after each deposit. solving for r from the future value of an ordinary annuity formula (11) we get r = yn r k( 1 + r k )n − 1. in our case we have r = 0.016, k = 12, n = 120, y120 = 35, 000. hence the monthly payment is calculated as follows r = 35, 000 0.016 12( 1 + 0.016 12 )120 − 1 .= 269.149 86 some applications of linear difference equations in finance 4 solving with computer algebra systems in mathematics of finance, excel is commonly used for calculations. in this paper the quoted calculations of compound interest and annuitity are completed by computational tool wolfram|alpha and mathematical software maple, respectively. 4.1 wolfram|alpha we will demonstrate the computation of compound interest (3) and example 2.2 through the free online service wolfram|alpha, which is available via any web browser at http://wolframalpha.com. this tool provides mathematical computations based on software mathematica and accepts completely free-form input, commands are specified by the name of operation in english. to solve the difference equation (1) with the initial condition (2) we type both equations together separated by comma into an input field writing indexes in parentheses. the provided general solution (3) is shown in figure 1. fig. 1. compound interest formula further, in figure 2 you can see calculation of number of years from example 2.2 using derived formula (5). we get the same result by solving equation (4) and using the command solve and the reserved word for as shown in figure 3. 87 d. ř́ıhová, l. viskotová fig. 2. calculation of number of years by using derived formula fig. 3. calculation of number of years by solving equation 4.2 maple now we show the computation of the future value of an annuity (11) and illustrative example 3.2. we assign the recurrence relation (6) to the name req. > req:=y(n+1)-y(n)*(1+r/k)=r; maple returns the output: req := y(n + 1) −y(n) ( 1 + r k ) = r then we make the assignment of the initial condition (7) to the name ic. > ic:=y(0)=0: to solve the given difference equation we execute the command rsolve which solves among others the first order linear difference equations. a single recurrence relation and a boundary condition are the first argument, the second argument indicates the function that is solved for. indexes are written in 88 some applications of linear difference equations in finance parentheses. > rsolve({req,ic},y(n)); rk ( k+r k )n r − kr r the above obtained expression corresponds to the future value of an annuity (11). for determining the payment r we type the following command, where fv equals to the total amount yn in the account upon the last deposit (i.e the future value of an annuity). > isolate(%=fv,r):simplify(%); r = fv r k (( k+r k )n − 1) to make the calculation of example 3.2 we use command subs. > subs(k=12,r=0.016,fv=35000,n=120,%); r = 269.1493510 4.3 comparison of used systems the professional maple is very powerful tool which enables to make new procedures and modules, save and read them or together with other data store in a library. on the other hand, it requires certain programming skills. in comparison with maple, wolfram|alpha does not allow to save and reload the results of computations and make own procedures, also its performance is rather slow. but its significant advantage is that it is free online and very simple to use. moreover, wolfram|alpha provides a variety of computations from other fields, for example from money and finance. 5 conclusion this paper has discussed linear difference equations and their applications in economics (see also [9]). these equations are frequently used especially in financial mathematics and some of their typical applications have been presented here. our main aim was to show relationship between some formulas of financial topics and mathematical knowledge which is required for their deriving. we have focused on derivation of the compound interest and the future value 89 d. ř́ıhová, l. viskotová of an annuity formula by means of solution of difference equations. the simultaneous application of mathematical software has been demonstrated, the supplementary computations have been performed through maple and wolfram|alpha. finally, the paper emphasizes the need for mathematics in economic subjects. the presented approach can be used in teaching of mathematics at economic universities and helps to provide students with the opportunities to apply their mathematics in relevant economics contexts. references [1] j. j. costello, o. s. gowdy and m. a. rash, mathematics for the management, life, and social sciences, new york: harcourt brace jovanovich, inc., 1982. [2] s. elaydi, an introduction to difference equations, new york: springer science+business media, inc., 2005. [3] g. fulford, p. forrester and a. jones, modelling with differential and difference equations, cambridge: cambridge university press, 1997. [4] s. goldberg, introduction to difference equations: with illustrative examples from economics, psychology and sociology, new york: dover publications, inc., 2010. [5] j. moučka and p. rádl, matematika pro studenty ekonomie, praha: grada publishing, inc., 2010. [6] k. neusser, difference equations for economists, bern: university of bern, 2012, [online], [cited 2015-07-15]. available from: http://www.neusser.ch/downloads/differenceequations.pdf. [7] p. pražák, diferenčńı rovnice s aplikacemi v ekonomii, hradec králové: gaudeamus, 2013. [8] j. radová, p. dvořák and j. málek, finančńı matematika pro každého, praha: grada publishing, inc., 2005. [9] d. ř́ıhová and l. viskotová, compound interest and annuities with linear difference equations and cas, mitav 2015 (proceedings of abstracts), brno: university of defence, 2015. [10] o. šoba, m. šir̊uček and r. ptáček, finančńı matematika v praxi, praha: grada publishing, inc., 2013. 90 ratio mathematica volume 38, 2020, pp. 349-365 uniqueness of an entire function sharing fixed points with its derivatives md majibur rahaman * imrul kaish† abstract the uniqueness problems of an entire functions that share a nonzero finite value have been studied and many results on this topic have been obtained. in this paper we prove a uniqueness theorem for an entire function, which share a linear polynomial, in particular fixed points, with its higher order derivatives. keywords: uniqueness; entire functions; fixed points; sharing; derivatives 2010 ams subject classifications: 30d35. 1 *department of mathematics and statistics, aliah university, kolkata, west bengal 700160, india; e-mail :majiburjrf107@gmail.com †department of mathematics and statistics, aliah university, kolkata, west bengal 700160, india; e-mail:imrulksh3@gmail.com 1received on april 19th, 2020. accepted on june 19th, 2020. published on june 30th, 2020. doi: 10.23755/rm.v38i0.520. issn: 1592-7415. eissn: 2282-8214. ©md majibur rahaman. this paper is published under the cc-by licence agreement. 349 md majibur rahaman and imrul kaish 1 introduction, definitions and results let f be a non-constant meromorphic function in the open complex plane c. a meromorphic function a = a(z) is called a small function of f if t(r,a) = s(r,f), where t(r,f) is the nevanlinna characteristic function of f and s(r,f) = ◦{t(r,f)}, as r →∞, possibly outside a set of finite linear measure. let f and g be two non-constant meromorphic functions and a = a(z) be a polynomial. we say that f and g share a cm if f − a and g − a have the same zeros with same multiplicities. on the other hand, we say that f and g share a im if f − a and g − a have the same zeros ignoring multiplicities. we express the cm sharing and im sharing respectively by the notations f = a g = a and f = a ⇔ g = a. let zk(k = 1, 2, . . .) be zeros of f − a and tk be the multiplicity of the zero zk. if zk(k = 1, 2, . . .) are also zeros of g−a and the multiplicity of the zero zk is at least tk then we use the notation f = a → g = a. for standared definitions and notations of the distribution theory we refer the reader to hayman [1964]. the problem of uniqueness of meromorphic functions sharing values with their derivatives is a special case of the uniqueness theory of meromorphic function. there are some results related to value sharing. in the begining, jank, mues and volkmann jank et al. [1986] considered the situation that an entire function shares a nonzero value with its derivatives and they prove the following result. theorem a. jank et al. [1986]. let f be a non-constant entire function and a be a non-zero finite value. if f, f(1) and f(2) share a cm, then f ≡ f(1). following example shows that in theorem a the second derivative cannot be replaced by any higher order derivatives. example 1.1. let k(≥ 3) be an integer and ω(6= 1) is a (k−1)th root of unity. we put f = eωz + ω − 1. then f, f(1) and f(k) share the value ω cm, but f 6≡ f(1). on the basis of this example, zhong improved theorem a by considering higher order derivetives in the following way. theorem b. let f be a non-constant entire function and a be a non-zero finite number. also let n(≥ 1) be a positive integer. if f and f(1) share the value a cm, and if f(n)(z) = f(n+1)(z) = a whenever f(z) = a, then f ≡ f(n). in 2002, chang and fang [2002] extendeed theorem a by considering shared fixed points. 350 uniqueness of an entire function sharing fixed points with its derivatives theorem c. chang and fang [2002]. let f be a non-constant entire function. if f, f(1) and f(2) share z cm, then f ≡ f(1). later in 2003, wang and yi [2003] improved theorem a and generalize theorem b by considering higher order derivatives in the following way. theorem d. wang and yi [2003]. let f be a non-constant entire function and a be a non-zero finite constant. also let m and n be positive integers satisfying m > n. if f and f(1) share the value a cm, and if f(m)(z) = f(n)(z) = a whenever f(z) = a, then f(z) = aeλz + a− a λ , where a(6= 0) and λ are constants satisfying λn−1 = 1 and λm−1 = 1. in this paper we improve theorem d by considering the situation when a nonconstant entire function f shares a linear plynomial a(z) = αz + β, α( 6= 0) and β are constants, with higher order derivatives. the main result of the paper is the following theorem. theorem 1.1. let f be a non-constant entire function and a(z) = αz + β be a polynomial, where α(6= 0) and β are constants. also let m amd n be two positive integers satisfying m > n > 1. if f(z) = a(z) f(1)(z) = a(z) and f(z) = a(z) → f(m)(z) = f(n)(z) = a(z), then f(z) = cez or f(z) = ceλz + a(z) − a(z) λ + α(1 −λ) λ2 , where c and λ are non-zero constants. 351 md majibur rahaman and imrul kaish 2 lemmas in this section we state some necessary lemmas. lemma 2.1. ngoan and ostrovskii [1965]. let f be an entire function of order at most 1 and k be a positive integer, then m ( r, f(k) f ) = o(log r), as r →∞. the above lemma motivates us to prove the following: lemma 2.2. let f be an entire function of finite order and k be a positive integer. then for any small function a(z) with respect to f(z), m ( r, f(k)(z) −a(k)(z) f(z) −a(z) ) = o(log r), as r →∞. proof. let g(z) = f(z) −a(z). then g(k)(z) = f(k)(z) −a(k)(z). now by lemma 2.1 and using above equality, we have m ( r, g(k)(z) g(z) ) = o(log r), as r →∞. this implies m ( r, f(k)(z) −a(k)(z) f(z) −a(z) ) = o(log r), as r →∞. this proves the lemma. lemma 2.3. clunie [1962]. let f be a transcendental meromorphic solution of the equation fnp(f) = q(f), where p(f) and q(f) are polynomials in f and its derivatives with meromorphic coefficients aj (say). if the total degree of q(f) is at most n, then m(r,p (f)) ≤ ∑ j m(r,aj) + s(r,f). 352 uniqueness of an entire function sharing fixed points with its derivatives lemma 2.4. chen and li [2014]. let a(z) be an entire function of finite order and q(z) be a non-constant polynomial. if f is an entire solution of the equation f(k) −eq(z)f = a(z) such that ρ(f) > ρ(a), then ρ(f) = ∞. we use this lemma to prove the following one. lemma 2.5. let f be a non-constant entire function of finite order and a(z) = αz +β be a polynomial, where α(6= 0) and β are constant. also let k be a positive integer. if f(z) and f(k)(z) share a(z) cm, then f(k)(z) −a(z) f(z) −a(z) ≡ c, (2.1) for some nonzero constant c. proof. since f has finite order and since f(z) and f(k)(z) share a(z) cm, it follows from the hadamard factorization theorem that f(k)(z) −a(z) f(z) −a(z) ≡ eq(z), (2.2) where q(z) is a polynomial. suppose that f(z) = f(z) −a(z). then f(k)(z) = f(k)(z). from (2.2) and above equality, we have f(k)(z) −eq(z)f(z) = a(z). if q(z) is non-constant, then from above equality and by lemma 2.4, we get f has infinite order. since f has finite order, this is impossible. hence q(z) is a constant. therefore from (2.2), we obtain (2.1) for a non-zero constant c. this proves the lemma. lemma 2.6. let f be a transcendental entire function of finite order and a(z) = αz + β be a polynomial, where α(6= 0) and β are constants. also let m be a positive integer. if (i) m ( r, 1 f(z)−a(z) ) = s(r,f), (ii) f(z) = a(z) f(1)(z) = a(z) and (iii) f(z) = a(z) → f(m)(z) = a(z), then f(z) = cez, where c is a non-zero constant. 353 md majibur rahaman and imrul kaish proof. let h(z) = f(1)(z) −a(z) f(z) −a(z) . (2.3) since f(z) and f(1)(z) share a(z) cm, we see that h(z) is an entire function. now by lemma 2.1, lemma 2.2 and from the hypothesis of lemma 2.6, we deduce that t(r,h(z)) = m(r,h(z)) = m ( r, f(1)(z) −a(z) f(z) −a(z) ) ≤ m ( r, f(1)(z) −a(1)(z) f(z) −a(z) ) + m ( r, a(1)(z) −a(z) f(z) −a(z) ) + log 2 = s(r,f). (2.4) we rewrite (2.3), as f(1)(z) = h(z)f(z) + a(z)(1 −h(z)) = ξ1(z)f(z) + η1(z), (2.5) where ξ1(z) and η1(z) are defined by ξ1(z) = h(z), η1(z) = a(z)(1 −h(z)). by (2.5), we have f(2)(z) = ξ1(z)f (1)(z) + ξ (1) 1 (z)f(z) + η (1) 1 (z) = ξ1(z)[ξ1(z)f(z) + η1(z)] + ξ (1) 1 (z)f(z) + η (1) 1 (z) = [ξ (1) 1 (z) + ξ1(z)ξ1(z)]f(z) + η (1) 1 (z) + η1(z)ξ1(z) = ξ2(z)f(z) + η2(z), where ξ2(z) = ξ (1) 1 (z) + ξ1(z)ξ1(z) and η2(z) = η (1) 1 (z) + η1(z)ξ1(z). now from above equality and using (2.5), we get f(3)(z) = ξ2(z)f (1)(z) + ξ (1) 2 (z)f(z) + η (1) 2 (z) = [ξ (1) 2 (z) + ξ1(z)ξ2(z)]f(z) + η (1) 2 (z) + η1(z)ξ2(z) = ξ3(z)f(z) + η3(z), 354 uniqueness of an entire function sharing fixed points with its derivatives where ξ3(z) = ξ (1) 2 (z) + ξ1(z)ξ2(z) and η3(z) = η (1) 2 (z) + η1(z)ξ2(z). similarly, f(k)(z) = ξk(z)f(z) + ηk(z), (2.6) where ξk+1(z) = ξ (1) k (z) + ξ1(z)ξk(z) (2.7) and ηk+1(z) = η (1) k (z) + η1(z)ξk(z). (2.8) puting k = 1 in (2.7), we have ξ2(z) = ξ (1) 1 (z) + ξ1(z)ξ1(z) = h2(z) + h(1)(z). again puting k = 2 in (2.7), we get ξ3(z) = ξ (1) 2 (z) + ξ1(z)ξ2(z) = [ h2(z) + h(1)(z) ](1) + h(z)[h2(z) + h(1)(z)] = h3(z) + h(2)(z) + 3h(z)h(1)(z). similarly, ξ4(z) = h 4(z) + h(3)(z) + 4h(z)h(2)(z) + 3 [ 2h2(z) + h(1)(z) ] h(1)(z). hence using mathematical induction, one can easily check ξk(z) = h k(z) + pk−1(z,h(z)), (2.9) where pk−1(z,h(z)) is a polynomial such that total degree degpk−1(z,h(z)) ≤ k−1 in h(z) and its derivatives, and all coefficients in pk−1(z,h(z)) are constants. now putting k = 1 in (2.8), we have η2(z) = η (1) 1 (z) + η1(z)ξ1(z) = [a(z)(1 −h(z))](1) + a(z)(1 −h(z))h(z) = −a(z)h2(z) −a(z)h(1)(z) + (a(z) −α)h(z) + α. 355 md majibur rahaman and imrul kaish again putting k = 2 in (2.8), we get η3(z) = η (1) 2 (z) + η1(z)ξ2(z) = [ −a(z)h2(z) −a(z)h(1)(z) + (a(z) −α)h(z) + α ](1) +a(z)(1 −h(z))(h2(z) + h(1)(z)) = −a(z)h3(z) −a(z)h(2)(z) + [2a(z) − 3a(z)h(z) − 2α] h(1)(z) +(a(z) −α)h2(z) + αh(z). similarly, = −a(z)h4(z) −a(z)h(3)(z) + [3a(z) − 4a(z)h(z) − 3α] h(2)(z) + [ 5a(z)h(z) − 5αh(z) − 6a(z)h2(z) − 3a(z)h(1)(z) + 3α ] h(1)(z) +(a(z) −α)h3(z) + αh2(z). like the previous one, it can be easily verified that ηk(z) = −a(z)hk(z) + qk−1(z,h(z)), (2.10) where qk−1(z,h(z)) is a polynomial such that total degree degqk−1(z,h(z)) ≤ k − 1 in h(z) and its derivatives, and all coefficients in qk−1(z,h(z)) are either constants or polynomial a(z). from (2.4) and (2.9), for k = 1, 2, · · · , we have t(r,ξk(z)) = t(r,h k(z) + pk−1(z,h(z))) ≤ t(r,hk(z)) + t(r,pk−1(z,h(z))) + log 2 = s(r,f). similarly, t(r,ηk(z)) = s(r,f). from hypothesis of lemma 2.6, we have n ( r, 1 f(z) −a(z) ) = t(r,f(z)) −m ( r, 1 f(z) −a(z) ) + o(1) = t(r,f(z)) + s(r,f), (2.11) which implies that f(z) −a(z) must have zeros. let zj be a zero of f(z) −a(z) with multiplicity δ(j). since f(z) = a(z) → f(m)(z) = a(z), we see that zj is also a zero of f(m)(z) −a(z) with multiplicity at least δ(j). hence f(zj) = a(zj) and f(m)(zj) = a(zj). 356 uniqueness of an entire function sharing fixed points with its derivatives it follows from (2.6) that, for k = m, f(m)(z) = ξm(z)f(z) + ηm(z) (2.12) and then a(zj) = a(zj)ξm(zj) + ηm(zj). now we shall prove that, a(z) ≡ a(z)ξm(z) + ηm(z). (2.13) otherwise, a(z)ξm(z) + ηm(z) −a(z) 6≡ 0. from (2.12), we have a(z)ξm(z) + ηm(z) −a(z) = (f(m)(z) −a(z)) − ξm(z)(f(z) −a(z)). by the reasoning as mentioned above, we deduce that zj is a zero of (f(m)(z)− a(z)) − ξm(z)(f(z) − a(z)), that is, a zero of a(z)ξm(z) + ηm(z) − a(z) with multiplicity at least δ(j). it follows from this and the fact that ξm(z) and ηm(z) are small functions of f(z), n ( r, 1 f(z) −a(z) ) ≤ n ( r, 1 a(z)ξm(z) + ηm(z) −a(z) ) ≤ t ( r, 1 a(z)ξm(z) + ηm(z) −a(z) ) = s(r,f), which contradicts (2.11). thus a(z) ≡ a(z)ξm(z) + ηm(z), which is (2.13). now by induction we prove that ηk+1(z) + a(z)ξk+1(z) = (a(z) −α)hk(z) + rk−1(z,h(z)), (2.14) where rk−1(z,h(z)) is a polynomial such that degrk−1(z,h(z)) ≤ k− 1 in h(z) and its derivatives, and all the coefficients in rk−1(z,h(z)) are constants or polynomial a(z). 357 md majibur rahaman and imrul kaish firstly, from (2.7), (2.8) and for k = 1, we have η2(z) + a(z)ξ2(z) = η (1) 1 (z) + η1(z)ξ1(z) + a(z) [ ξ (1) 1 (z) + ξ1(z)ξ1(z) ] = [a(z)(1 −h(z))](1) + a(z)(1 −h(z))h(z) + a(z)h(1)(z) +a(z)h2(z) = a(z)(−h(1)(z)) + α(1 −h(z)) + a(z)h(z) −a(z)h2(z) +a(z)h(1)(z) + a(z)h2(z) = (a(z) −α)h(z) + α. secondly, we suppose that the following equation holds ηk(z) + a(z)ξk(z) = (a(z) −α)hk−1(z) + rk−2(z,h(z)). now, by (2.7)–(2.10), we deduce that ηk+1(z) + a(z)ξk+1(z) = η (1) k (z) + η1(z)ξk(z) + a(z)(ξ (1) k (z) + ξ1(z)ξk(z)) = [ −a(z)hk(z) + qk−1(z,h(z)) ](1) + a(z)(1 −h(z))ξk(z) +a(z) [ hk(z) + pk−1(z,h(z)) ](1) + a(z)h(z)ξk(z) = −ka(z)hk−1(z) −αhk(z) + [qk−1(z,h(z))] (1) + a(z)ξk(z) −a(z)h(z)ξk(z) + ka(z)hk−1(z) + a(z) [pk−1(z,h(z))] (1) +a(z)h(z)ξk(z) = a(z) [ hk(z) + (pk−1(z,h(z))) ] −αhk(z) + [qk−1(z,h(z))] (1) +a(z) [pk−1(z,h(z))] (1) = (a(z) −α)hk(z) + rk−1(z,h(z)), which proves (2.14). from (2.13) and (2.14), we obtain (a(z) −α)hm−1(z) + rk−2(z,h(z)) ≡ a(z). (2.15) clearly, rk−2(z,h(z)) 6≡ a(z). othewise, from (2.3), (2.15) and the hypothesis of lemma 2.6, we have a contradiction. hence by lemma 2.3 and from (2.15), we can deduce that h(z) must be constant. from (2.7) and ξ1(z) = h(z), we have ξ2(z) = h 2(z), ξ3(z) = h 3(z), ξ4(z) = h 4(z). similarly, ξk(z) = h k(z), for k = 1, 2, . . . . (2.16) 358 uniqueness of an entire function sharing fixed points with its derivatives now, from (2.8) and η1(z) = a(z)(1 −h(z)), we get η2(z) = (1 −h(z))(α + a(z)h(z)), η3(z) = (1 −h(z))(α + a(z)h(z))h(z), η4(z) = (1 −h(z))(α + a(z)h(z))h2(z). similarly, ηk(z) = (1 −h(z))(α + a(z)h(z))hk−2(z), for k = 2, 3, . . . . (2.17) from (2.13), (2.16) and (2.17), we have a(z) ≡ a(z)hm(z) + (1 −h(z))(α + a(z)h(z))hm−2(z) ≡ hm−2(z) [ a(z)h2(z) + α(1 −h(z)) + a(z)h(z) −a(z)h2(z) ] ≡ hm−2(z) [a(z)h(z) + α(1 −h(z))] , which implies that h(z) = 1. hence from (2.3) and h(z) = 1, we can obtain f(1)(z) = f(z). this implies f(z) = cez. where c(6= 0) ia a constant. this proof the lemma 2.6. 3 proof of the theorem 1.1 first we verify that f(z) cannot be a polynomial. let f(z) be a polynomial of degree 1. suppose that f(z) = a1z + b1, where a1(6= 0) and b1 are constants. then f(1)(z) = a1, f(m)(z) ≡ 0 ≡ f(n)(z). now β−b1a1−α is the only zero of f(z) − a(z), a1−β α is the only zero of f(1)(z) − a(z) and −β α is the only zero of f(m)(z)−a(z). since f(z) and f(1)(z) share polynomial a(z) cm and the zeros of f(z) −a(z) are the zeros of f(m)(z) −a(z), we have a1−β α = −β α and so a1 = 0, a contradiction. now let f(z) be a polynomial of degree greater than 1. suppose that deg(f(z)) = p. then deg(f(z)−a(z)) = p and deg(f(1)(z)−a(z)) = p−1, it contradicts the fact that f(z) and f(1)(z) share polynomial a(z) cm. hence f(z) is a transcendental entire function. thus t(r,a(z)) = s(r,f). 359 md majibur rahaman and imrul kaish to prove the theorem let us consider two functions defined as follows. φ(z) = (a(z) −a(1)(z))f(m)(z) −a(z)(f(1)(z) −a(1)(z)) f(z) −a(z) (3.1) and ψ(z) = (a(z) −a(1)(z))f(n)(z) −a(z)(f(1)(z) −a(1)(z)) f(z) −a(z) . (3.2) then φ(z) 6≡ ψ(z). we know from the hypothesis of theorem 1.1 that φ(z) and ψ(z) are entire functions. then, by lemma 2.1 and lemma 2.2, we have t(r, φ(z)) = m(r, φ(z)) = m ( r, (a(z) −a(1)(z))f(m)(z) −a(z)(f(1)(z) −a(1)(z)) f(z) −a(z) ) ≤ m ( r, (a(z) −a(1)(z)) f(m)(z) f(z) −a(z) ) + m ( r,a(z) (f(1)(z) −a(1)(z)) f(z) −a(z) ) + log 2 = s(r,f). similarly, t(r, ψ(z)) = s(r,f). we shall the following three cases. case 1. first we suppose that φ(z) 6≡ 0. then by (3.1), we have f(z) = a(z) + 1 φ(z) {(a(z) −a(1)(z))f(m)(z) −a(z)(f(1)(z) −a(1)(z))}. (3.3) from (3.1) and (3.2), we get f(1)(z) = (a(z) −a(1)(z)) a(z)(φ(z) − ψ(z)) (φ(z)f(n)(z) − ψ(z)f(m)(z)) + a(1)(z). therefore f(1)(z) −a(z) = (a(z) −a(1)(z)) a(z)(φ(z) − ψ(z)) (φ(z)f(n)(z) − ψ(z)f(m)(z)) +a(1)(z) −a(z). (3.4) 360 uniqueness of an entire function sharing fixed points with its derivatives first we suppose that m > n > 2. then from (3.4), we get 1 f(1)(z) −a(z) = 1 a(z)(φ(z) − ψ(z)) (φ(z)f(n)(z) − ψ(z)f(m)(z)) f(1)(z) −a(z) + 1 a(1)(z) −a(z) . (3.5) using lemma 2.1 and from (3.5), we have m ( r, 1 f(1)(z) −a(z) ) = s(r,f). (3.6) next we suppose m > n = 2. then from (3.4), we get (f(1)(z) −a(z))(φ(z) − ψ(z))a(z) = γ(z) + (a(z) −a(1)(z))φ(z)(f(2)(z) −a(1)(z)) −(a(z) −a(1)(z))ψ(z)f(m)(z), (3.7) where γ(z) = (a(z) −a(1)(z))(φ(z)a(1)(z) − (φ(z) − ψ(z))a(z)). clearly γ(z) 6≡ 0. if γ(z) ≡ 0, then φ(z) ≡ a(z) a(z) −a(1)(z) ψ(z), which is a contradiction because φ(z) and ψ(z) are entire functions and ψ(z) 6= 0 when a(z) −a(1)(z) = 0. now from (3.7) we get 1 f(1)(z) −a(z) = (φ(z) − ψ(z))a(z) γ(z) − a(z) −a(1)(z) γ(z) φ(z) f(2)(z) −a(1)(z) f(1)(z) −a(z) + a(z) −a(1)(z) γ(z) ψ(z) f(m)(z) f(1)(z) −a(z) . (3.8) again using lemma 2.1 and from (3.8), we get m ( r, 1 f(1)(z) −a(z) ) = s(r,f), which is (3.6). 361 md majibur rahaman and imrul kaish since φ(z) 6≡ 0, it follows from (3.3) and lemma 2.1 that t(r,f(z)) = m(r,f(z)) = m ( r,a(z) + 1 φ(z) {(a(z) −a(1)(z))f(m)(z) −a(z)(f(1)(z) −a(1)(z))} ) = m ( r,a(z) + (a(z) −α)f(m)(z) −a(z)f(1)(z) + a(z)α φ(z) ) ≤ m(r,a(z)) + m ( r, (a(z) −α)f(m)(z) −a(z)f(1)(z) φ(z) ) + m ( r, αa(z) φ(z) ) + log 3 = m  r,a(z)f(1)(z) (a(z)−α)a(z) f(m)(z)f(1)(z) − 1 φ(z)   + s(r,f) ≤ m  r, (a(z)−α)a(z) f(m)(z)f(1)(z) − 1 φ(z)   + m(r,f(1)(z)) + s(r,f) ≤ m ( r, f(m)(z) f(1)(z) − 1 ) + m(r,f(1)(z)) + s(r,f) = t(r,f(1)(z)) + s(r,f). (3.9) applying lemma 2.1, we can easily see that t(r,f(1)(z)) = m(r,f(1)(z)) = m ( r, f(1)(z) f(z) ·f(z) ) ≤ m ( r, f(1)(z) f(z) ) + m(r,f(z)) = m(r,f(z)) + s(r,f) ≤ t(r,f(z)) + s(r,f). (3.10) combining (3.9) and (3.10), we have t(r,f(1)(z)) = t(r,f(z)) + s(r,f). (3.11) since f(z) and f(1)(z) share a(z) cm, by using (3.6) and (3.11) together with 362 uniqueness of an entire function sharing fixed points with its derivatives the first fundamental theorem, we obtain m ( r, 1 f(z) −a(z) ) = t(r,f(z)) −n ( r, 1 f(z) −a(z) ) + o(1) = t(r,f(1)(z)) −n ( r, 1 f(1)(z) −a(z) ) + s(r,f) = m ( r, 1 f(1)(z) −a(z) ) + n ( r, 1 f(1)(z) −a(z) ) −n ( r, 1 f(1)(z) −a(z) ) + s(r,f) = m ( r, 1 f(1)(z) −a(z) ) + s(r,f) = s(r,f). hence by lemma 2.6, we have f(z) = cez, where c(6= 0) is a constant. case 2. now we suppose that ψ(z) 6≡ 0. then following the similar arguments of case-1 and using lemma 2.6, we have f(z) = cez. where c(6= 0) is a constant. case 3. finally we suppose that φ(z) ≡ 0 and ψ(z) ≡ 0. then from (3.1) and (3.2), we get (a(z) −a(1)(z))f(m)(z) −a(z)(f(1)(z) −a(1)(z)) ≡ 0 (3.12) and (a(z) −a(1)(z))f(n)(z) −a(z)(f(1)(z) −a(1)(z)) ≡ 0. (3.13) now subtracting (3.13) from (3.12), we have (a(z) −a(1)(z))(f(m)(z) −f(n)(z)) ≡ 0. since a(z) 6≡ a(1)(z), we get f(m)(z) ≡ f(n)(z). 363 md majibur rahaman and imrul kaish solving this we have f(z) = p0 + p1e t1z + p2e t2z + · · · + pm−netm−nz, where t1, t2, · · · , tm−n are distinct (m − n)th roots of unity and p0, p1, p2, · · · pm−n are constants. since f(z) and f(1)(z) share a(z) cm, applying lemma 2.5, we get f(1)(z) −a(z) f(z) −a(z) = λ, for some nonzero constant λ. solving above equality, we obtain f(z) = ceλz + a(z) − a(z) λ + α(1 −λ) λ2 , where c(6= 0) is a constant. this completes the proof of theorem 1.1. 4 conclusions after the above discussion we arrive at the conclusion that if an entire function and its first derivative share a linear polynomial with counting multiplicity and it partially shares the linear polynomial with its two higher order derivatives then the funtion is either one of the following two forms. (i) f(z) = cez , (ii) f(z) = ceλz + a(z) − a(z) λ + α(1−λ) λ2 , where c and λ are non-zero constants. 5 acknowledgements authors are thankful to the referee for valuable suggestions and observations towards the improvement of the paper. 364 uniqueness of an entire function sharing fixed points with its derivatives references j chang and m fang. uniquness of entire functions and fixed points. kodai math. j., 25:309–320, 2002. b chen and s li. some results on the entire function sharing problem. math. solovaca., 64:1217–1226, 2014. j clunie. on integral and meromorphic functions. j. london math. soc., 37: 17–27, 1962. hayman. meromorphic functions. the clarendon press, oxford, oxford, 1964. g jank, e mues, and l volkmann. meromorphe funktionen, die mit ihrer ersten und zweiten ableitung einen endlichen wert teilen. complex var. theory appl., 6:51–71, 1986. v ngoan and i. v ostrovskii. the logarithmic derivative of meromorphic function. akad. nauk. armjan. ssr. dokl., 41:272–277, 1965. j. p wang and h. x yi. entire functions that share one value cm with their derivatives. j. math. anal. appl., 277:155–163, 2003. 365 ratio mathematica volume 48, 2023 fuzzy soft set connected mappings alpa singh rajput* mahima thakur† samajh singh thakur‡ abstract in this paper, the concepts of fuzzy soft connectedness between fuzzy soft sets and fuzzy soft set connected mappings in fuzzy soft topological spaces has been introduced. it is shown that a fuzzy soft topological space is fuzzy soft connected if and only if it is fuzzy soft connected between every pair of its nonempty fuzzy soft sets. every fuzzy soft continuous mapping is fuzzy soft set-connected a counter example is given to show the converse may not be true. several properties of fuzzy soft set-connected mappings in fuzzy soft topological spaces have been studied. keywords: fuzzy soft sets; fuzzy soft connectedness; fuzzy soft connectedness between fuzzy soft sets and fuzzy soft set-connected mappings. 2020 ams subject classifications:54a40, 54d05, 54c08. 1 *department of science and humanities, vignan foundation for science, technology and research, guntur, india ; asr sh@vignan.ac.in. †department of applied mathematics, jabalpur engineering college jabalpur, india ; mthakur@jecjabalpur.ac.in. ‡former professor department of applied mathematics, jabalpur engineering college jabalpur, india; ssthakur@jecjabalpur.ac.in. 1received on april 17, 2023. accepted on july 1, 2023. published on august 1,2023. doi:10.23755/rm.v41i0.1180. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 1 1 introduction the notion of a fuzzy set was introduced by zadeh [1965] as a generalization of classical set in the year 1965. chang [1968] gave the definition of fuzzy topology and extended some topological concepts to fuzzy sets. in 1999, molodtsov [1999] introduced the concept of soft sets to deal with uncertainties while modelling the problems with incomplete information. in 2011, shabir and naz [2011] initiated the study of soft topological spaces as a generalization of topological spaces. the hybrid structure of fuzzy sets and soft set called fuzzy soft set was created by maji et al. [2001]. to continue the investigation on fuzzy soft sets, ahmad and kharal [2009] presented some more properties of fuzzy soft sets and introduced the notion of a mapping on fuzzy soft sets. tanay and kandemir [2011] introduced the fuzzy soft topology as an extension of fuzzy topology and soft topology. fuzzy topological spaces are further investigated by varol and aygun [2012] and shown that a fuzzy soft topological space gives a parametrized family of fuzzy topological spaces. roy and samanta [2011] and tridiv et al. [2012] are also investigated various topological concepts in fuzzy soft topological spaces. the study of connectedness in fuzzy soft topology was initiated by karatas et al. [2015] and further studied by kandil et al. [2017]. connectedness between sets and set connected mappings is one of the important topic of research in topology. in 2018, thakur and rajput [2018] extended and studied these concepts to soft topology. till the date these concepts are not studied in fuzzy soft topology. therefore to fill up this gap, the present paper introduces the concept of connectedness between fuzzy soft sets and studied some of its properties in fuzzy soft topological spaces. further the concepts of fuzzy soft set-connected mappings are defined and established some theorems related to its characterizations and properties. the results established in this paper generalized many results which are already available in the literature. 2 preliminaries throughout this paper, x refers to an initial universe, e is the set of all parameters for x, i = [0,1] and ix is the set of all fuzzy sets on x. the reader should refer molodtsov [1999] and zadeh [1965] for the basic concepts on fuzzy sets and soft sets. definition 2.1. (maji et al. [2001]) let a ⊂ e. a pair (f,a) is called a fuzzy soft set (in short fss) over x, where f : a → ix defined by , (f,a)(e) = µe(f,a) ,where µe(f,a) = 0̃ if e /∈ a and µ e (f,a) ̸= 0̃ if e ∈ a . the family of all fsss over (x,e) will be denoted by fs(x, e). for the fuzzy soft set connected mappings notations, basic operations and properties of fsss the reader should refer ahmad and kharal [2009], maji et al. [2001]. definition 2.2. (tanay and kandemir [2011]) a subfamily τ of fs(x, e) is called a fuzzy soft topology on x if: (1) ϕ̃, x̃ belong to τ. (2) the union of any number of soft sets in τ belongs to τ. (3) the intersection of any two soft sets in τ belongs to τ. the triplet (x, τ, e) is called a fuzzy soft topological space (in short fsts). the members of τ are called fuzzy soft open sets in x and their complements called fuzzy soft closed sets in x. the basic fuzzy soft topological concepts can be seen in kandil et al. [2017], karatas et al. [2015], roy and samanta [2011], tridiv et al. [2012] and varol and aygun [2012]). lemma 2.1. (roy and samanta [2011]) let (f,a) and (g,b) be two fsss. then, (f, a) ⊆ (g, b)c ⇔ (f, a)q̃(g, b). where (f, a)q̃(g, b) means (f,a) is not quasicoincident with (g,b). lemma 2.2. (tridiv et al. [2012]) let (y, τy , e) be a fuzzy soft subspace of a fsts (x, τ, e) and (f, e) be a fuzzy soft open set in y. if ỹ ∈ τ then (f, e) ∈ τ. lemma 2.3. (tridiv et al. [2012]) let (x,τ,e) be a fsts and (y, τy ,e) be a fuzzy soft subspace of (x, τ, e), then a fuzzy soft closed set (fy , e) of y is fuzzy soft closed in x if and only if ỹ is fuzzy soft closed in x. 3 connectedness between fuzzy soft sets throughout this paper fuzzy soft clopen means fuzzy soft closed open. definition 3.1. (kandil et al. [2017], karatas et al. [2015]) a fsts (x,τ,e) is fuzzy soft connected if and only if there is no nonempty fss of (x,τ,e) which is both fuzzy soft open and fuzzy soft closed in (x,τ,e). definition 3.2. a fsts (x ,τ, e) is said to be fuzzy soft connected between fsss (f1,e) and (f2,e) if and only if there is no fuzzy soft clopen set (f,e) over x such that (f1,e) ⊂ (f,e) and (f,e) q̃ (f2,e). theorem 3.1. a fsts (x,τ,e) is fuzzy soft connected between fsss (f1,e) and (f2,e) if and only if there is no fuzzy soft clopen set (f,e) over x such that (f1,e) ⊂ (f,e) ⊂ (f2, e)c. proof. follows from definition 3.2. and lemma 2.1.2 theorem 3.2. if a fsts (x,τ,e) is fuzzy soft connected between fsss (f1,e) and (f2,e) then (f1,e) ̸= ϕ ̸= (f2,e). proof. if any fss (f1,e) = ϕ, then (f1,e) is a fuzzy soft clopen set over x such that (f1,e) ⊂ (f2,e) and (f1,e) q̃ (f2,e) and hence (x,τ,e) can not be fuzzy soft connected between fsss (f1,e) and (f2,e), which is contradiction.2 theorem 3.3. if a fsts (x,τ,e) is fuzzy soft connected between fsss (f1,e) and (f2,e) and if (f1,e) ⊂ (f3,e) and (f2,e) ⊂ (f4,e) then (x,τ,e) is fuzzy soft connected between fsss (f3,e) and (f4,e). proof. suppose fsts (x,τ,e) is not fuzzy soft connected between fsss (f3,e) and (f4,e) then there is a fuzzy soft clopen set (f,e) over x such that (f3,e) ⊂ (f,e) and (f,e) q̃ (f4,e) .clearly (f1,e) ⊂ (f,e). now we claim that (f,e) q̃ (f2,e) .if (f,e) q (f2,e) then there exists a point x ∈ x such that µe(f,e)(x ) + µ e (f2,e) (x) ≻ 1. therefore µe(f,e)(x ) + µ e (f4,e) (x) ≻ µe(f,e)(x ) + µ e (f2,e) (x) ≻ 1 and (f,e) q (f4,e), a contradiction. consequently, (x,τ,e) is not fuzzy soft connected between fsss (f1,e) and (f2,e).2 theorem 3.4. a fsts (x,τ,e) is fuzzy soft connected between fsss (f1,e) and (f2,e) if and only if (x,τ,e) is fuzzy soft connected between fsss cl(f1,e) and cl(f2,e). proof. necessity : follows from theorem 3.3. sufficiency : suppose fsts (x,τ,e) is not fuzzy soft connected between fsss (f1,e) and (f2,e), then there exists fuzzy soft clopen set (f,e) over x such that (f1,e) ⊂ (f,e) and (f,e) q̃ (f2,e). since (f,e) is fuzzy soft closed, cl(f1,e) ⊂ cl(f,e) = (f,e). clearly, by lemma 2.1, (f,e) q̃ (f2,e) ⇔ (f,e) ⊂ (f2, e)c. therefore (f,e) = int(f,e) ⊂ int((f2, e)c) = (cl(f2, e))c. hence, (f,e) q̃ cl(f2,e) and (x,τ,e) is not fuzzy soft connected between fsss cl(f1,e) and cl(f2,e).2 theorem 3.5. if (f1,e) and (f2,e) are two fsss in fsts (x,τ,e) and (f1,e) q (f2,e), then (x,τ,e) is fuzzy soft connected between (f1,e) and (f2,e). proof. if (f,e) is any fuzzy soft clopen set over x such that (f1,e) ⊂ (f,e), then (f1,e) q (f2,e) ⇒ (f,e) q (f2,e). this proves the theorem.2 remark 3.1. the converse of theorem 3.5 need not be true . example 3.1. let x = {a,b} be universe set and e = {e1 ,e2 } be the set of fuzzy soft set connected mappings parameters. the fsss let (f,e), (f1,e) and (f2,e) over x are defined as follows: f1(e1) = {(a, 0.3), (b, 0.4)} f1(e2) = {(a, 0.2), (b, 0.3)} f2(e1) = {(a, 0.5), (b, 0.4)} f2(e2) = {(a, 0.6), (b, 0.5)} f3(e1) = {(a, 0.3), (b, 0.5)} f3(e2) = {(a, 0.3), (b, 0.2)}. let τ = {0̃e , 1̃e, (f1, e) } be a fuzzy soft topology on x, then fsts (x,τ ,e) is fuzzy soft connected between the fsss (f2,e) and (f3,e), but (f2,e) q̃ (f3,e). theorem 3.6. if a fsts (x,τ,e) is neither fuzzy soft connected between (f,e) and (g0,e) nor fuzzy soft connected between (f,e) and (g1,e) then it is not fuzzy soft connected between (f,e) and (g0,e) ∪ (g1,e). proof. since a fsts (x,τ,e) is not fuzzy soft connected between (f ,e) and (g0,e), there is a fuzzy soft clopen set (h0,e) over x such that (f,e) ⊂ (h0,e) and (h0,e) q̃ (g0,e). also since (x,τ,e) is not fuzzy soft connected between (f ,e) and (g1,e) there exists a fuzzy soft clopen set (h1,e) over x such that (f,e) ⊂ (h1,e) and (g1,e) q̃ (h1,e). put (h,e) = (h0,e) ∩ (h1,e). since any intersection of fuzzy soft closed sets is fuzzy soft closed, (h,e) is fuzzy soft closed. again intersection of finite family of fuzzy soft open sets is fuzzy soft open, (h,e) is fuzzy soft open. therefore (h,e) is fuzzy soft clopen set over x such that (f,e) ⊂ (h,e) and (h,e) q̃((g0,e) ∪ (g1, e)). if (h,e) q ((g0,e) ∪ (g1, e)) there exists x ∈ x such that µe(h,e) (x) + (µe(g0,e) ∪ µ e (g1,e) )(x) ≻ 1. this implies that (h,e) q (g0,e) or (h,e) q (g1,e) a contradiction. hence, (x,τ,e) is not fuzzy soft connected between (f,e) and (g0, e) ∪ (g1, e). 2 theorem 3.7. a fsts (x,τ,e) is fuzzy soft connected if and only if it is fuzzy soft connected between every pair of its nonempty fsss. proof. let (f,e) and (g,e) be a pair of nonempty fsss over x. suppose (x,τ,e) is not fuzzy soft connected between (f,e) and (g,e). then there is a fuzzy soft clopen set (h,e) over x such that (f,e) ⊂ (h,e) and (g,e) q̃ (h,e). since (f,e) and (g,e) are nonempty it follows that (h,e) is a nonempty fuzzy soft proper clopen set over x. hence, (x,τ,e) is not fuzzy soft connected. conversely, suppose that (x,τ,e) is not fuzzy soft connected. then there exists a nonempty proper fss (h,e) over x which is both fuzzy soft open and fuzzy soft closed. consequently, (x,τ,e) is not fuzzy soft connected between (h,e) and (h, e)c. thus, (x,τ,e) is not fuzzy soft connected between arbitrary pair of its nonempty fsss.2 remark 3.2. if a fsts (x,τ,e) is fuzzy soft connected between a pair of its fsss, then it is not necessarily that it is fuzzy soft connected between each pair of its fsss and so it is not necessarily fuzzy soft connected. example 3.2. let x = {a,b} be an universe set, e = {e1, e2} be the set of parameter and the soft sets (f1, e) , (f2, e) , (f3,e), (f4,e), (f5,e) and (f6,e) over x are defined as follows: f1(e1) = {(a, 0.3), (b, 0.4)} f1(e2) = {(a, 0.2), (b, 0.3)} f2(e1) = {(a, 0.7), (b, 0.6)} f2(e2) = {(a, 0.8), (b, 0.7)} f3(e1) = {(a, 0.3), (b, 0.1)} f3(e2) = {(a, 0.4), (b, 0.2)} f4(e1) = {(a, 0.8), (b, 0.7)} f4(e2) = {(a, 0.9), (b, 0.8)} f5(e1) = {(a, 0.2), (b, 0.1)} f5(e2) = {(a, 0.1), (b, 0.2)} f6(e1) = {(a, 0.4), (b, 0.3)} f6(e2) = {(a, 0.5), (b, 0.4)}. let τ = {0̃e, 1̃e, ( f1, e), (f2, e)} be a fuzzy soft topology over x. then the fsts (x, τ, e) is fuzzy soft connected between the fsss (f3, e) and (f4, e) but it is not fuzzy soft connected between (f5,e) and (f6,e). also the fsts (x, τ, e) is not fuzzy soft connected. theorem 3.8. let (y,τy ,e) be a fuzzy soft subspace of a fsts (x,τ,e). if (y,τy ,e) is fuzzy soft connected between the fsss (f ,e) and (g,e) over y, then fsts (x,τ,e) is fuzzy soft connected between (f ,e) and (g,e). proof. suppose fsts (x,τ,e) is not fuzzy soft connected between fsss (f,e) and (g,e),then there is fuzzy soft clopen set (h,e) over x such that (f,e) ⊂ (h,e) and (h,e) q̃ (g,e). then ỹ ∩ (h ,e) is fuzzy soft clopen over y such that (f,e) ⊂ (h,e) ∩ ỹ and {(h,e) ∩ ỹ } q̃ (g ,e). consequently, (y,τy ,e) is not fuzzy soft connected between (f,e) and (g,e), a contradiction.2 theorem 3.9. let (y,τy ,e) be a fuzzy soft clopen subspace of a fsts (x,τ,e) and (f ,e), (g,e) ⊂ ỹ . if (x,τ,e) is fuzzy soft connected between (f,e) and (g,e) then (y,τy ,e) is fuzzy soft connected between(f,e) and (g,e). proof. suppose (y,τy ,e) is not fuzzy soft connected between (f,e) and (g,e). then there is fuzzy soft clopen set (h,e) of (y,τy ,e) such that (f,e) ⊂ (h,e) and (h,e) q̃ (g,e). since, (y,τy ,e) is fuzzy soft clopen in (x,τ,e), by lemma 2.2 and lemma 2.3 (h,e) is fuzzy soft clopen set of (x,τ,e) such that (f,e) ⊂ (h,e) and (h,e) q̃ (g,e). consequently, (x,τ,e) is not fuzzy soft connected between (f,e) and (g,e), a contradiction.2 4 fuzzy soft set-connected mappings definition 4.1. a fuzzy soft mapping ϱpu : (x, τ, e) → (y, ϑ, k) is said to be fuzzy soft set-connected provided, if fsts(x, τ, e) is fuzzy soft connected between fsss (f,e) and (g,e) then fuzzy soft subspace (ϱpu(x), ϑϱpu(x), k) is fuzzy fuzzy soft set connected mappings soft connected between ϱpu(f, e) and ϱpu(g, e) with respect to fuzzy soft relative topology. theorem 4.1. a fuzzy soft mapping ϱpu : (x, τ, e) → (y, ϑ, k) is fuzzy soft setconnected mapping if and only if ϱ−1pu (f, k) is a fuzzy soft clopen set over x for any fuzzy soft clopen set (h,k) of (ϱpu(x) ,ϑϱpu(x), k). proof. necessity : let ϱpu be fuzzy soft set-connected mapping and (h,k) be fuzzy soft clopen set in (ϱpu(x), ϑϱpu(x), k). suppose ϱ −1 pu (h, k) is not fuzzy soft clopen in (x ,τ,e). then (x ,τ,e) is fuzzy soft-connected between ϱ−1pu (h, k) and (ϱ−1pu (h, k)) c. therefore, (ϱpu(x), ϑϱpu(x), k) is fuzzy soft-connected between ϱpu(ϱ −1 pu (h, k)) and ϱpu((ϱ −1 pu (h, k)) c) because ϱpu is fuzzy soft set-connected. but, ϱpu(ϱ−1pu (h, k)) = (h, k)∩(ϱpu(x), ϑϱpu(x), k) = (h, k) and ϱpu((ϱ−1pu (h, k))c = (h, k)c imply that (h,k) is not fuzzy soft clopen in (ϱpu(x), ϑϱpu(x), k), a contradiction. hence, ϱ−1pu (h, k) is fuzzy soft clopen in (x ,τ,e). sufficiency : let (x ,τ,e) be fuzzy soft-connected between (f,e) and (g,e). if (ϱpu(x), ϑϱpu(x), k) is not fuzzy soft-connected between ϱpu(f, e) and ϱpu(g, e) then there exists a fuzzy soft clopen set (h,k) in (ϱpu(x), ϑϱpu(x), k) such that ϱpu(f, e) ⊂ (h, k) ⊂ (ϱpu(g, e))c. by hypothesis, ϱ−1pu (h, k) is fuzzy soft clopen set over x and (f, e) ⊂ ϱ−1pu (h, k) ⊂ (g, e)c. therefore, (x ,τ,e) is not fuzzy softconnected between (f,e) and (g,e). this is a contradiction. hence, ϱpu is fuzzy soft set-connected.2 theorem 4.2. every fuzzy soft continuous mapping ϱpu : (x ,τ,e) → (y,ϑ,k) is a fuzzy soft set-connected mapping. proof. it is obvious.2 remark 4.1. the converse of theorem 4.2 need not be true. example 4.1. let x = {x1, x2} , e = {e1, e2} and y = {y1, y2} , k = {k1, k2}. the soft sets (f,e) and (g,k) defined as follows: f(e1) = {x1 = 0.3, x2 = 0}, f(e2) = {x1 = 0, x2 = 0.4} g(k1) = {y1 = 0.6, y2 = 0}, g(k2) = {y1 = 0, y2 = 0.5} let τ = {0̃e,1̃e,(f,e) } and υ = {0̃k , 1̃k , (g ,k) } are fuzzy soft topologies on x and y respectively. then the fuzzy soft mapping ϱpu : (x, τ, e) → (y, υ, k) defined by u(x1) = y1 , u(x2 )= y2 and p(e1 )= k1, p(e2 ) = k2 is fuzzy soft set-connected but it is not fuzzy soft continuous, because fuzzy soft set (g,k) is fuzzy soft open set in y not fuzzy soft open in x. theorem 4.3. every fuzzy soft mapping ϱpu : (x, τ, e) → (y, ϑ, k) such that (ϱpu(x), ϑϱpu(x), k) is a fuzzy soft connected set is a fuzzy soft set-connected mapping. proof. let (ϱpu(x), ϑϱpu(x), k) be fuzzy soft connected. then by lemma 3.1, no nonempty proper fss of (ϱpu(x), ϑϱpu(x), k) which is fuzzy soft clopen. hence , ϱpu is fuzzy soft set-connected.2 theorem 4.4. let ϱpu : (x, τ, e) → (y, ϑ, k) be a fuzzy soft set-connected mapping. if (x ,τ,e) is fuzzy soft connected set, then (ϱpu(x), ϑϱpu(x), k) is a fuzzy soft connected set of (y,ϑ,k). proof. suppose (ϱpu(x), ϑϱpu(x), k) is not fuzzy soft connected in (y,ϑ,k), then by lemma 3.1, there is a nonempty proper fuzzy soft clopen set (h,k) of (ϱpu(x), ϑϱpu(x), k). since ϱpu is fuzzy soft set-connected, ϱ −1 pu (h, k) is a nonempty proper fuzzy soft clopen set over x. consequently,(x ,τ,e) is not fuzzy soft connected.2 theorem 4.5. let ϱpu : (x, τ, e) → (y, ϑ, k) be a fuzzy soft set-connected mapping and (f,e) be a fuzzy soft set over x such that ϱpu(f, e) is fuzzy soft clopen set of (ϱpu(x), ϑϱpu(x), k). then ϱpu/(f, e) : (f, e) → (y, ϑ, k) is fuzzy soft set-connected mapping. proof: let (f,e) be fuzzy soft connected between (g,e) and (h,e). then by theorem 3.8, (x, τ, e) is fuzzy soft connected between (g,e) and (h,e). since ϱpu is fuzzy soft set-connected, (ϱpu(x), ϑϱpu(x), k) is fuzzy soft connected between ϱpu(g, e) and ϱpu(h, e). now, since ϱpu(f, e) is fuzzy soft clopen set of (ϱpu(x), ϑϱpu(x), k), it follows by theorem 3.9 that ϱpu(f, e) is fuzzy soft connected between ϱpu(g, e) and ϱpu(h, e). this proves the theorem.2 theorem 4.6. let ϱpu : (x, τ, e) → (y, ϑ, k) be a fuzzy soft set-connected surjection. then for any fuzzy soft clopen set (h,k) of (y, ϑ, k) is fuzzy soft connected if ϱ−1pu (h, k) is fuzzy soft connected in (x ,τ,e). in particular, if (x ,τ,e) is fuzzy soft connected then (y,ϑ,k) is fuzzy soft connected. proof. by theorem 4.5 ϱpu/ϱ−1pu (h, k) : ϱ −1 pu (h, k) → (y, ϑ, k) is fuzzy soft set-connected. and, since ϱ−1pu (h, k) is fuzzy soft connected by theorem 4.4, ϱpu/ϱ −1 pu (h, k)[ϱ −1 pu (h, k)] = (h, k) is fuzzy soft connected.2 theorem 4.7. let ϱpu : (x, τ, e) → (y, ϑ, k) be a fuzzy soft set connected fuzzy soft open surjection and ϱ−1pu ((y α k )k) is fuzzy soft connected for each soft point (yαk )k of y. then for every fuzzy soft clopen set (h,k) of y is fuzzy soft connected if and only if ϱ−1pu ((h, k)) is fuzzy soft connected. proof. necessity: let (h,k) be a fuzzy soft clopen fuzzy soft connected set of y. suppose ϱ−1pu (h, k) is not fuzzy soft connected in x. then there are fuzzy soft open sets (f,e) and (g,e ) of x such that ϱ−1pu (h, k) ∩ ((f, e) ∩ (g, e)) = ϕ, ϱ−1pu (h, k) = ((f, e) ∪ (g, e)) and ϱ−1pu (h, k) ∩ (f, e) ̸= ϕ ̸= ϱ−1pu (h, k) ∩ (g, e). since, ϱ−1pu ((y α k )k) is fuzzy soft connected either ϱ −1 pu ((y α k )k) ⊂ (f, e) or ϱ−1pu ((y α k )k) ⊂ (g, e) for every fuzzy soft point (y α k )k ∈ (h, k). therefore (h, k)∩ϱpu(f, e)∩ϱpu(g, e) = ϕ. (h, k) ⊂ ϱpu(f, e)∪ϱpu(g, e) and (h, k)∩ fuzzy soft set connected mappings ϱpu(f, e) ̸= ϕ ̸= (h, k) ∩ ϱpu(g, e) . since, ϱpu is fuzzy soft open mapping ϱpu(f, e) and ϱpu(g, e) are fuzzy soft open set over y. hence, (h,k) is not fuzzy soft connected. sufficiency : follows from theorem 4.6.2 theorem 4.8. let ϱp1u1 : (x ,τ,e) → (y,ϑ,k) be a surjective fuzzy soft set-connected and σp2u2 :(y,ϑ,k) → (z,η,t) a fuzzy soft set-connected mapping. then (σp2u2oϱp1u1) : (x ,τ,e) → (z,η,t) is fuzzy soft set-connected. proof. let (h,t) be a fuzzy soft clopen set in σp2u2(y ). then σ −1 p2u2 (h,t) is fuzzy soft clopen over y = ϱp1u1 (x) and so ϱ −1 p1u1 (σ−1p2u2 (h,t)) is fuzzy soft clopen in (x ,τ,e). now (σp2u2oϱp1u1)(x ) = σp2u2 (y) and (σp2u2oϱp1u1) −1 (h,t) = ϱ−1p1u1 (σ−1p2u2 (h,t)) is fuzzy soft clopen in (x ,τ,e). hence, (σp2u2oϱp1u1) is fuzzy soft set connected.2 definition 4.2. a fuzzy soft mapping ϱpu : (x, τ, e) → (y, ϑ, k) is said to be fuzzy soft weakly continuous if for each fuzzy soft point (xαe )e ∈ x and each fuzzy soft open set (g,k) over y containing ϱpu((xαe )e), there exists a fuzzy soft open set (f,e) over x containing (xαe )e such that ϱpu(f,e) ⊂ cl(g,k). theorem 4.9. a soft mapping ϱpu : (x, τ, e) → (y, ϑ, k) is fuzzy soft weakly continuous if and only if for each fuzzy soft open set (h,k) over y, ϱ−1pu (h, k) ⊂ int(ϱ−1pu (cl(h, k))). proof. necessity : let (h,k) be a fuzzy soft open set over y and let (xαe )e ∈ ϱ−1pu (h, k) then ϱpu((x α e )e) ∈ (h, k). then, there exists a fuzzy soft open set (f,e) over x such that (xαe )e ∈ (f, e) and ϱpu(f, e) ⊂ cl(h, k). hence, (xαe )e ∈ (f, e) ⊂ ϱ−1pu (cl(h, k)) and (xαe )e ∈ int(ϱ−1pu (cl(h, k))) since (f,e) is fuzzy soft open. sufficiency : let (xαe )e ∈ x and ϱpu((xe)e) ∈ (h, k). then (xe)e ∈ ϱ−1pu (h, k) ⊂ int(ϱ−1pu (cl(h, k))). let (f, e) = int(ϱ−1pu (cl(h, k))) then (f,e) is fuzzy soft open set containing (xαe )e and ϱpu(f, e) = ϱpu(int(ϱ −1 pu (cl(h, k)))) ⊂ ϱpu(ϱ −1 pu (cl(h, k))) ⊂ cl(h, k). hence, ϱpu is fuzzy soft weakly continuous.2 theorem 4.10. if a fsts space (x ,τ,e) is fuzzy soft connected and ϱpu : (x, τ, e) → (y, ϑ, k) is a fuzzy soft weakly continuous surjection, then (y,ϑ,k) is fuzzy soft connected. proof . suppose (y, ϑ, k) is not fuzzy soft connected. then, there exist nonempty fuzzy soft open sets (h1, k) and (h2, k) in y such that (h1, k)∩(h2, k) = ϕ and (h1, k) ∪ (h2, k) = x̃. hence we have ϱ−1pu (h1, k) ∩ ϱ−1pu (h2, k) = ϕ and ϱ−1pu (h1,k) ∪ ϱ−1pu (v2,k) = x̃. since ϱpu is surjective, ϱ−1pu (vj,k) ̸= ϕ for j = 1, 2. by theorem 4.9, we have ϱ−1pu (hj,k) ⊂int (ϱ−1pu (cl(hj,k))) because ϱpu is fuzzy soft weakly continuous. since (hj,k) is fuzzy soft open and also fuzzy soft closed, we have ϱ−1pu (hj,k) ⊂ int( ϱ−1pu (hj,k. hence, ϱ−1pu (hj,k) is fuzzy soft open in x for j = 1,2. this implies that x is not fuzzy soft connected. this is contrary to the hypothesis that x is fuzzy soft connected. hence, (y,ϑ,k) is fuzzy soft connected.2 theorem 4.11. a fuzzy soft mapping ϱpu : (x, τ, e) → (y, ϑ, k) is fuzzy soft weakly continuous, then cl(ϱ−1pu (h, k)) ⊂ (ϱ−1pu (cl(h, k)) for each fuzzy soft open set (h,k) over y. proof. suppose there exists a fuzzy soft point (xαe )e ∈ cl(ϱ−1pu (h, k)) − ϱ−1pu (cl(h, k)). then ϱpu((xe)e) /∈ cl(h, k). hence there exists a fuzzy soft open set (g ,k) containing ϱpu((xe)e) such that (g, k) ∩ (v, k) = ϕ. since (h,k) is fuzzy soft open set over y, we have (h, k) ∩ cl(g, k) = ϕ. since ϱpu is fuzzy soft weakly continuous ,there exists a fuzzy soft open set (f,e) over x containing (xαe )e such that ϱpu(f, e) ⊂ cl(g, k). thus, we obtain ϱpu(f, e) ∩ (h, k) = ϕ. on the other hand, since (xαe )e ∈ cl(ϱ−1pu (h, k)), we have (f, e) ∩ ϱ−1pu (h, k) ̸= ϕ and hence, ϱpu(f, e) ∩ (h, k) ̸= ϕ. thus we have a contradiction. hence cl(ϱ−1pu (h, k)) ⊂ (ϱ−1pu (cl(h, k)).2 theorem 4.12. if a fuzzy soft surjection ϱpu : (x ,τ,e) → (y,ϑ,k) is fuzzy soft weakly continuous ,then ϱpu is fuzzy soft set-connected. proof. let (h,k) be any fuzzy soft clopen set over y. since (h,k) is fuzzy soft closed, we have cl(h,k) = (h,k). thus, by theorem 4.9, we obtain ϱ−1pu (h, k) ⊂ int(ϱ−1pu (h, k)). this shows that ϱ −1 pu (h, k) is fuzzy soft open set over x. moreover , by theorem 4.11, we obtain cl(ϱ−1pu (h, k)) ⊂ ϱ−1pu (h, k). this shows that ϱ−1pu (h, k) is a fuzzy soft closed set over x. since ϱpu is fuzzy soft surjection, by theorem 4.1, we observe that ϱpu is a fuzzy soft set-connected mapping.2 remark 4.2. the converse of theorem 4.12 is not true. example 4.2. letx = {x1, x2}, e = {e1, e2} and y = {y1, y2} , k = {k1, k2}. the fuzzy soft sets (f,e)and (g,k) are defined as follows : f(e1) = {x1 = 0.3, x2 = 0}, f(e2) = {x1 = 0, x2 = 0.4} g(k1) = {y1 = 0.4, y2 = 0}, g(k2) = {y1 = 0, y2 = 0.5} let τ = {0̃e,1̃e,(h,e) } and υ = {0̃k , 1̃k , (g ,k) } are fuzzy soft topologies on x and y respectively. then fuzzy soft mapping ϱpu : (x, τ, e) → (y, υ, k) defined by u(x1) = u(x2)=y1 and p(e1 )= k1 , p(e2 ) = k2 is fuzzy soft set-connected but it is not fuzzy soft weakly continuous. theorem 4.13. let (y,ϑ,k) be an fuzzy soft extremally disconnected space .if a fuzzy soft mapping ϱpu : (x, τ, e) → (y, ϑ, k) is fuzzy soft set-connected, then ϱpu is fuzzy soft weakly continuous. proof. let (xαe )e be a fuzzy soft point of x and (g,k) any fuzzy soft open set over y containing ϱpu((xαe )e). since (y,ϑ,k) is fuzzy soft extremally disconnected, fuzzy soft set connected mappings cl(g,k) is fuzzy soft clopen set over y. thus cl(g, k)∩ϱpu(x̃) is fuzzy soft clopen set in the fuzzy soft subspace (ϱpu(x), ϑϱpu(x), k). put ϱ −1(cl(g, k)∩ϱpu(x̃)) = (f, e). then, since ϱpu is fuzzy soft set-connected, it follows from theorem 4.1 that (f,e) is fuzzy soft clopen set over x. therefore, (f,e) is a fuzzy soft open set containing (xαe )e over x such that ϱpu(f, e) ⊂ cl(g, k). this implies that ϱpu is fuzzy soft weakly continuous.2 theorem 4.14. let (y, ϑ, k) be a fuzzy soft extremally disconnected space. a fuzzy soft surjection mapping ϱpu : (x, τ, e) → (y, ϑ, k) is fuzzy soft setconnected if and only if ϱpu is fuzzy soft weakly continuous. proof. it follows from theorem 4.12 and theorem 4.13.2 5 conclusions connectedness is an important and major area of topology and it can give many relationships between other scientific areas and mathematical models. the notion of connectedness captures the idea of hanging-togetherness of image elements in an object by assigning a strength of connectedness to every possible path between every possible pair of image elements. this paper, introduces the notion of fuzzy soft connectedness between fuzzy soft sets in fuzzy soft topological spaces. it is shown that a fuzzy soft topological space is fuzzy soft connected if and only if it is fuzzy soft connected between every pair of its nonempty fuzzy soft sets. further two new classes of fuzzy soft mappings called fuzzy soft set connected and soft weakly continuous have been introduced. it is shown that the class of fuzzy soft set connected (respt. fuzzy soft weakly continuous) mappings properly contains the class of all fuzzy soft continuous mappings. several properties and characterizations of fuzzy soft set connected and fuzzy soft weakly continuous mappings have been studied. hope that the concepts and results established in this paper will help researcher to enhance and promote the further study on fuzzy soft topology to carry out a general framework for the development of information systems. references b. ahmad and a. kharal. on fuzzy soft sets. adv. fuzzy syst., 9:1–6, 2009. c. l. chang. fuzzy topological spaces. j. math. anal. appl., 24(1):182–189, 1968. a. kandil, o. a. el-tantawy, s. a. el-shiekh, and s. s. s. el-sayed. fuzzy soft connected sets in fuzzy soft topological spaces. j. egyptian math. soc., 25(2): 171–177, 2017. s. karatas, b. kilcc, and m. tellioglu. on fuzzy soft connected topological spaces. j. linear topol. algebra, 4(3):229–240, 2015. p. k. maji, r. biswas, and a. r. roy. fuzzy soft sets. j. fuzzy math., 9(3): 589–602, 2001. d. molodtsov. soft set theory first results. comput. math.appl., 37:9–31, 1999. s. roy and t. k. samanta. a note on fuzzy soft topological spaces. ann. fuzzy math. inform., 3(2):305–311, 2011. m. shabir and m. naz. on soft topological spaces. comput. math. appl., 61: 1786–1799, 2011. b. tanay and m. b. kandemir. topological structures of fuzzy soft sets. comput. math. appl., 61:412–418, 2011. s. s. thakur and a. s. rajput. connectedness between soft sets. new mathematics and natural computation, 14(1):53–71, 2018. j. n. tridiv, d. k. sut, and g. c. hazarika. fuzzy soft topological spaces. int. j. latest trend math., 2(1):407–419, 2012. b. p. varol and h. aygun. fuzzy soft topology. hacet. j. math. stat., 41(3): 407–419, 2012. l. a. zadeh. fuzzy sets. inf. control, 8:338–353, 1965. ratio mathematica volume 37, 2019, pp. 5-23 studies on the classical determinism predicted by a. einstein, b. podolsky and n. rosen ruggero maria santilli∗ abstract in this paper, we continue the study initiated in preceding works of the argument by a. einstein, b. podolsky and n. rosen according to which quantum mechanics could be “completed” into a broader theory recovering classical determinism. by using the previously achieved isotopic lifting of applied mathematics into isomathematics and that of of quantum mechanics into the isotopic branch of hadronic mechanics, we show that extended particles appear to progressively approach classical determinism in the interior of hadrons, nuclei and stars, and appear to recover classical determinism at the limit conditions in the interior of gravitational collapse. keywords: epr argument, isomathematics, isomechanics. 2010 ams subject classifications: 05c15, 05c60. 1 ∗institute for basic research, palm harbor, fl 34683 u.s.a., research@i-b-r.org 1received on september 21st, 2019. accepted on december 20rd, 2019. published on december 31st, 2019. issn: 1592-7415. eissn: 2282-8214. doi: 10.23755/rm.v37i0.477 c©ruggero maria santilli 5 ruggero maria santilli 1. introduction 1.1. the epr argument as it is well known, albert einstein did not consider quantum mechanical uncertainties to be final, for which reason he made his famous quote “god does not play dice with the universe.” more particularly, einstein accepted quantum mechanics for atomic structures and other systems, but believed that quantum mechanics is an “incomplete theory,” in the sense that it could be broadened into such a form to recover classical determinism at least under limit conditions. einstein communicated his views to b. podolsky and n. rosen and they jointly published in 1935 the historical paper [1] that became known as the epr argument. soon after the appearance of paper [1], n. n. bohr published paper [2] expressing a negative judgment on the possibility of “completing” quantum mechanics along the lines of the epr argument. bohr’s paper was followed by a variety of papers essentially supporting bohr’s rejection of the epr argument, among which we recall bell’s inequality [3] establishing that the su(2) spin algebra does not admit limit values with an identical classical counterpart. we should also recall von neumann theorem [4] achieving a rejection of the epr argument via the uniqueness of the eigenvalues of quantum mechanical hermitean operators under unitary transforms. the field became known as local realism and was centered on the rejection of the epr argument via additional claims that hidden variables [5] are not admitted by quantum axioms (see the review [6]). 1.2. the 1998 apparent proof of the epr argument in 1998, the author published paper [7] presenting an apparent proof of the epr argument based on the following main steps that we here outline to render this paper minimally self-sufficient: step 1: the proof that bell’s inequality, von neumann’s theorem and other similar objections against the epr argument [6] are indeed correct, but under the generally tacit assumptions of point-like particles moving in vacuum under sole potential/hamiltonian interactions (exterior dynamical systems) when the systems are treated via quantum mechanics and its underlying 20th century mathematics, including lie’s theory and the 6 apparent proof of the epr argument newton-leibnitz differential calculus; step 2: the proof that the above treatments are not applicable for extended, therefore deformable and hyperdense particles under conditions of mutual penetration or entanglement occurring in the structure of hadrons, nuclei, stars, and gravitational collapse such as for black holes, with novel non-linear, non-local, and non-potential/non-hamiltonian interactions (interior dynamical systems); step 3: the treatment of interior systems via the axiom-preserving lifting of 20th century applied mathematics known as isomathematics, whose study was initiated by the author in the late 1970’s when he was at harvard university under doe support, refs. [8] to [12] and then continued by various mathematicians. isomathematics is based on: 3-a) the axiom-preserving isotopy of the conventional associative product between generic quantities a,b (numbers, functions, operators, etc.) first introduced in eq. (5), p. 71 of ref. [11] ab → a ? b = at̂b, (1) where t̂ is a positive-definite quantity called the isotopic element providing a representation of the dimension, deformability and density of particles and physical media in which they are immersed via realizations of the type t̂ = diag.( 1 n21 , 1 n22 , 1 n23 , 1 n24 )e−γ, (2) where: n24 represents the density; n 2 k, k = 1, 2, 3 represents the deformable share of particles; n2µ,µ = 1, 2, 3, 4, and γ are solely restricted to be positivedefinite but otherwise admit a functional dependence on any needed local variables, such as time t, coordinates r, momenta p, energy e, density d, temperature τ, pressure π, wavefunctions ψ, their derivatives ∂ψ, etc. nµ = nµ(t,r,p,e,d,τ,π,ψ,∂ψ,....) > 0, µ = 1, 2, 3, 4, (3) γ = γ(t,r,p,e,d,τ,π,ψ,∂ψ,....) � 0. (4) e− γ(t,r,p,e,d,τ,π,ψ,∂ψ,....) � 1. (5) 3-b) the formulation of isoassociative algebras on an isofield f̂(n̂,?, î) first introduced in ref. [13] (see also independent work [14]), with isounit î = 1t̂, (6) 7 ruggero maria santilli and isoreal, isocomplex and isoquaternionic isonumbers n̂ = nî under isoproduct (1), with ensuing isooperations such as the isosquare n̂2̂ = n̂ ? n̂. (7) isofields also imply the lifting of functions into isofunctions [11] [20] f̂(r̂) = [f(rî)]î, (8) among which we quote the isoexponentiation êx = (ext̂ )î = î (et̂x), (9) where x is a hermitean operator. 3-c) the ensuing axiom-preserving lifting of lie’s theory into a nonlinear, non-local and non-hamiltonian form first introduced in ref. [11] (see also the recent paper [15] and independent work [16]), which theory is today known as the lie-santilli isotheory, with isobrackets at the foundation of ref. [7] [x,̂y ] = x ? y −y ? x = xt̂y −y t̂x. (10) 3-d) the isotopic lifting of the newton-leibnitz differential calculus, from its historical definition at isolated points, into a form defined on volumes, first introduced in ref. [17] (see refs. [18] for vast independent works) with isodifferential d̂r̂ = t̂(r, ...)dr̂ = = t̂(r, ...)d[rî(r, ...)] = dr + rt̂dî(r, ...), (11) and corresponding isoderivatives ∂̂f̂(r̂) ∂̂r̂ = î ∂f̂(r̂) ∂r̂ . (12) step 4: the axiom-preserving lifting of quantum mechanics into the isotopic branch of hadronic mechanics, or isomechanics for short, whose study was initiated in refs. [8] to [12] (see the 1995 monographs [19] [20] [21] with 2008 upgrade [22] and independent studies [23][24]). isomechanics is formulated on a hilbert-myung-santilli (hms) isospace [25] ĥ over the isofield of isocomplex isonumbers ĉ, and it is based on the 8 apparent proof of the epr argument iso-heisenberg isoequations for the time evolution of a hermitean operator q̂ in the infinitesimal form î ? d̂q̂ d̂t̂ = [q̂,̂ĥ] = q̂ ? ĥ − ĥ ? q̂ = = q̂t̂ĥ − ĥt̂q̂, (13) and the finite form q̂(t̂) = û(t̂)† ? q̂(0) ? û(t̂) = = êĥ?t̂?̂i ? q̂(0) ? ê−î?t̂?ĥ = = eĥt̂tiq(0)e−itt̂ĥ, (14) with the following rules for the basic isounitary isotransforms û(t̂)† ? û(t̂) = û(t̂) ? û(t̂)† = î, (15) where t̂ = tît is the isotime which is assumed hereon to coincide with conventional time, ît = 1. dynamical equations (13) to (15) were first presented in eq. (4.16.49), page 752 of ref. [9] over conventional fields and reformulated via the full use of isomathematics in ref. [17]). isomechanics is also based on the iso-schrödinger isorepresentation characterized by the fundamental representation of the isomomentum permitted by the isodifferential isocalculus, eq. (12), p̂̂|ψ(t̂, r̂) >= −î ? ∂̂t̂,r̂|̂ψ(t̂, r̂) >= = −iî∂r̂|̂ψ(t̂, r̂) >, (16) from which one can derive the iso-schrödinger isoequation, [12] [17] [20] î ? ∂̂t̂|ψ̂(t̂, r̂) >= ĥ ? |ψ̂(t̂, r̂) >= = ĥ(r,p)t̂(t,r,p,e,d,τ,π,ψ,∂ψ,....)|ψ̂(t̂, r̂ >) = = ê ? |ψ̂(t̂, r̂) >= e|ψ̂(t̂, r̂) > (17) and the isocanonical isocommutation rules, [r̂î,p̂j]|ψ̂ >= î ? δ̂i.j ? |ψ̂ >= iδij|ψ̂ > (18) 9 ruggero maria santilli [r̂î,r̂j]|ψ̂ >= [p̂î,p̂j]|ψ̂ >= 0. (19) note that the characterization of extended particles at mutual distances smaller than their size requires the knowledge of two quantities, the conventional hamiltonian h for the representation of potential interactions, and the isotopic element t̂ for the representation of dimension, shape, density as well as of non-linear, non-local and non-potential interactions. step 5: the proof in ref. [7] that the isotopic ŝu(2)-spin symmetry for extended particles immersed within a dense hadronic medium admits an explicit and concrete realization of hidden variables [5], e.g., of the type t̂ = diag.(λ, 1/λ), dett̂ = 1. (20) in particular, the isotopic ŝu(2)-spin isosymmetry admits limit conditions with identical classical counterpart, eq. (5.4) page 189 ref. [7]. one aspect of isomathematics and isomechanics which is crucial for this paper is that in all applications to date, the isotopic element t̂ has values much smaller than 1, eqs. (4) (5), as it has been the case for: the synthesis of the neutron from the hydrogen in the core of stars; the representation of nuclear magnetic moments and spin; new clean energies; and other applications [21]. it should be also noted that thanks to the new interactions represented by t̂ , isomathematics and isomechanics have permitted the first known identification of the attractive force between identical valence electron pairs in molecular structures [26]. a significant confirmation of values |t̂| � 1 is provided by the fact that exact representations of binding energies for the hydrogen and water molecules have been achieved with isoseries based on isoproduct (1) that are at least one thousand times faster than conventional quantum chemical series [27] [28]. we should finally indicate that the numerical invariance of the isotopic element t̂ and therefore, of the isounit î = 1/t̂ , under isounitary time evolutions (14) (15) was proved in ref. [29]. detailed reviews and upgrades of isomathematics, isomechanics, and their applications to interior problems which are specifically written for the epr argument should soon be available in refs. [30] [31]. 1.3. aim of the paper in this work, we shall attempt to complete the proof of the epr argument of ref. [7] by showing that extended particles in interior dynamical 10 apparent proof of the epr argument conditions appear to progressively recover classical determinism in interior dynamical conditions with the increase of the density and other characteristics, as indicated at the end of ref. [7]. it should be stressed that a technical understanding of this work requires technical knowledge of hadronic mechanics, e.g., from refs. [19] [20] [21] or from the forthcoming reviews and upgrades [30] [31]. we should indicate that the words “completion of quantum mechanics” is used in einstein’s sense for the intent of honoring his memory. for instance, the conventional associative product ab of eq. (1), which is at the foundation of quantum mechanics, admits a “completion” into the equally associative, yet more general isoproduct at̂b. under no conditions einstein’s word “completed theory” should be confused with a ’final theory,’ that is a theory admitting no additional einstein’s “completions.” in fact, the time-reversal invariant, lie-isotopic isomathematics and isomechanics studied in this work admit the “completion” into the covering, irreversible lie-admissible genomathematics and genomechanics (in which t̂ is no longer hermitean) which, in turn, admit a covering via the most general mathematics and mechanics conceived by the human mind, the multi-valued hypermathematics and hypermechanics [32] [33], with additional “coverings” remaining possible in due time [19] [20] [21] . the reader should be finally aware that the isotopic element t̂ and isounit î = 1/t̂ are inverted in some of the early quoted literature not dealing with determinism without affecting their consistency. an important aim of this paper has been that of achieving the final selection of isotopic element and isounit which is compatible with studies on determinism. 2. recovering of determinism in interior conditions? 2.1. heisenberg uncertainty principle consider an electron in empty space represented with the 3-dimensional euclidean space e(r,δ,i), where r represents coordinates, δ = diag.(1, 1, 1) represents the euclidean metric and i = dian(1, 1, 1, ) is the space unit. let the operator representation of said electron be done in a hilbert space h over the field of complex numbers c with states ψ(r) and familiar normalization < ψ(r)| |ψ(r) >= ∫ +∞ −∞ ψ(r)†ψ(r)dr = 1. (21) as it is well known, the primary objections against the epr argument 11 ruggero maria santilli [2] [3] [4] were based on the uncertainty principle formulated by werner heisenberg in 1927, according to which the position r and the momentum p of said electron cannot both be measured exactly at the same time. by introducing the standard deviations ∆r and ∆p, the uncertainty principle is generally written in the form (see, e.g., [5]) ∆r∆p ≥ 1 2 ~, (22) easily derivable via the vacuum expectation value of the canonical commutation rule ∆r∆p ≥ | 1 2i < ψ| [r,p] |ψ > | = 1 2 ~. (23) the standard deviations have the known form [34] (with ~ = 1) ∆r = √ < ψ(r)|[ r − (< ψ(r)| r |ψ(r) >)]2|ψ(r) >, ∆p = √ < ψ(p)| [p− (< ψ(p)| p |ψ(p) >)]2|ψ(p) >, (24) where ψ(r) and ψ(p) are the wavefunctions in coordinate and momentum spaces, respectively. 2.2. particle in interior conditions we consider now the electron, this time, in the core of a star classically represented with the iso-euclidean isospace ê(r̂, δ̂, î) [17] with basic isounit î = 1/t̂ > 0, isocoordinates r̂ = rî, isometric δ̂ = t̂δ, (25) and isotopic element of type (2) under conditions (3) to (5). besides being immersed in the core of a star, the electron has no hamiltonian interactions. consequently, we can represent the electron in the hms isospace ĥ [25] over the isofield of isocomplex isonumbers ĉ [13], and introduce the time independent isoplanewave [20] ψ̂(r̂) = ψ̂(r̂)î = = n̂ ? (êî?k̂?r̂)î = n(eikt̂r̂)î, (26) where n̂ = nî is an isonormalization isoscalar, k̂ = kî is the isowavenumber, and the isoexponentiation is given by eq. (9). 12 apparent proof of the epr argument the corresponding representation in isomomentum isospace is given by ψ̂(p̂) = m̂ ? êî?n̂?p̂, (27) where m̂ = mî is an isonormalization isoscalar and n̂ = nî is the isowavenumber in isomomentum isospace. 2.3. isodeterministic isoprinciple the isopropability isofunction is given by [20] p̂ = <̂|? |>̂ =< ψ̂(r̂)| t |ψ̂(r̂) > i = = [ ∫ +∞ −∞ ψ̂(r̂) † ? ψ̂(r̂) ? d̂r̂]î = = [ ∫ +∞ −∞ ψ̂(r̂) †ψ̂(r̂)d̂r̂]î, (28) where one should keep in mind that the isodifferential d̂r̂ is now given by eqs. (11). the isoexpectation isovalues of a hermitean operator q̂ are then given by [20] <̂|? q̂ ? |>̂ =< ψ̂(r̂)|? q̂ ? |ψ̂(r̂) > î = = [ ∫ +∞ −∞ ψ̂(r̂) † ? q̂ ? ψ̂(r̂)d̂r̂]î = = [ ∫ +∞ −∞ ψ̂(r̂) †q̂ψ̂(r̂)d̂r̂]î, (29) with corresponding expressions for the isoexpectation isovalues in isomomentum isospace. we now introduce, apparently for the first time in this paper, the isotopic operator t̂ = t̂ î = i, (30) that, despite its seemingly irrelevant value, is indeed the correct operator formulation of the isotopic element for the transition of the isoproduct from its scalar form (1) into the isoscalar form n̂2̂ = n̂ ? n̂ = n̂ ? t̂ ? n̂ = n2î. (31) since the identity i can be inserted anywhere in the expectation values of quantum mechanics without altering the results, realization (33) illustrates the central feature of the isotopies, namely, the property that the abstract axioms of quantum mechanics admit a “hidden” realization broader 13 ruggero maria santilli than that of the copenhagen school whose degrees of freedom have been used in ref.[7] for the proof of the epr argument [1]. we now introduce the isoexpectation isovalue of the isotopic operator <̂|? t̂ ? |>̂ =< ψ̂(r̂)|? t̂ ? |ψ̂(r̂) > î = = [ ∫ +∞ −∞ ψ̂(r̂) †t̂ψ̂(r̂)d̂r̂]î, (32) and assume the isonormalization <̂|? t̂ ? |>̂ = = ∫ +∞ −∞ ψ̂(r̂) †t̂ψ̂(r̂)d̂r̂ = t̂. (33) we then introduce, in this paper apparently for the first time, the isostandard isodeviation for isocoordinates ∆r̂ = ∆rî and isomomenta ∆p̂ = ∆pî, where ∆r and ∆p are the standard deviations in our space. by using isocanonical isocommutation rules (18), we obtain the expression ∆r̂ ? ∆p̂ = ∆r∆pî ≈ 1 2 | < ψ̂(r̂)|? [r̂̂,p̂] ? ψ̂(r̂) > | = = 1 2 | < ψ̂(r̂)|t̂ [r̂̂,p̂]t̂|ψ̂(r̂) > . (34) by eliminating the common isounit î, we then have the desired isodeterministic isoprinciple here proposed apparently for the first time ∆r∆p ≈ 1 2 | < ψ̂(r̂)|? [r̂̂,p̂] ? |ψ̂(r̂) >= = 1 2 | < ψ̂(r̂)|t̂[r̂̂,p̂]t̂|ψ̂(r̂) >= ∫ +∞ −∞ ψ̂(r̂) †t̂ψ̂(r̂)d̂r̂ = t � 1 (35) where the property ∆r∆p � 1 follows from the fact that the isotopic element t̂ has always a value smaller than 1 (section 1.2). it is now necessary to verify isoprinciple (35) by proving that the isostandard isodeviations tend to null values when t̂ → 0. for this purpose, we introduce the following simple isotopy of eqs. (24) (where we ignore the common multiplication by the isounit) ∆r = √ < ψ̂(r̂)|[ r̂− < ψ̂(r̂)|? r̂ ? |ψ̂(r̂) >]2̂̂|ψ(r̂) >, ∆p = √ < ψ̂(p̂)| [p̂− < ψ̂(p̂)|? p̂ ? |ψ̂(p̂) >]2̂|ψ̂(p̂) >, (36) 14 apparent proof of the epr argument where the differentiation between the isotopic elements for isocoordinates and isomomenta is ignored for simplicity. it is then easy to see that the isosquare (7) implies the covering forms of the isostandard isodeviations ∆r = √ t̂ < ψ̂(r̂)|[ r̂− < ψ̂(r̂)|? r̂ ? |ψ̂(r̂) >]2|ψ̂(r̂) >, ∆p = √ t̂ < ψ̂(p̂)| [p̂− < ψ̂(p̂)|? p̂ ? |ψ̂(p̂) >]2|ψ̂(p̂) >, (37) that indeed approach null value under the limit conditions limt̂=0∆r = 0, limt̂=0∆p = 0, (38) thus confirming isodeterministic isoprinciple (35). 2.4. particles under pressure to illustrate the above expressions, we consider an electron in the center of a star, thus being under extreme pressures π from the surrounding hadronic medium in all radial directions, while ignoring particle reactions in first approximation or under a sufficiently short period of time. these conditions are here rudimentarily represented by assuming that the γ > 0 function of the the isotopic element (2) is a constant linearly dependent on the pressure π, resulting in a realization of the isotopic element of the type t̂ = e−wπ � 1, î = e+wπ � 1, (39) where w is a positive constant. the isodeterministic isoprinciple for the considered particle is then given by ∆r∆p ≈ 1 2 e−wπ � 1, (40) and tends to null values for diverging pressures. the above example illustrates the consistency of isorenormalization (33) because, a constant isotopic element implies the consistent expression <̂ψ̂(r̂)|t̂|ψ̂(r̂) > î = t < ψ̂(r̂)| |ψ̂(r̂) > î = < ψ̂(r̂)| |ψ̂(r̂) >, (41) 15 ruggero maria santilli while, by contrast, the following alternative isonormalization <̂ψ̂(r̂)|t̂|ψ̂(r̂) > î = î, (42) would imply the expression < ψ̂(r̂)||ψ̂(r̂) > î = î, (43) which is manifestly inconsistent since < ψ̂(r̂)||ψ̂(r̂) > is an ordinary number while î is a matrix with integro-differential elements. note that we have considered a free particle immersed in a hadronic medium, rather than a bound state of extended particles in condition of mutual penetration. consequently, in our view, isotopic element (2) represents a subsidiary constraint caused by the pressure of the hadronic medium encompassing the particle considered, by therefore restricting the values of the isostandard isodeviations for isocoordinates and isomomenta. illustrations of the isodeterministic isoprinciple in specific structure models of hadrons and related aspects have been studied in ref. [21] and their interpretation in terms of the isodeterministic isoprinciple will be studied in future works. 2.5. gravitational example to provide a gravitational illustration, recall that isotopic element (2) contains as particular cases all possible symmetric metrics in (3+1)-dimensions, thus including the riemannian metric [20]. we then consider the 3-dimensional sub-case of isotopic element (2) and factorize the space component of the schwartzchild metric gs(r) according to isotopic rule introduced in refs. [35] [36] gs(r) = t̂(r)δ, (44) where δ is the euclidean metric. we reach in this way the following realization of the isotopic element t̂ = 1 1 − 2m r = r r − 2m , (45) where m is the gravitational mass of the body considered, with ensuing isodeterministic isoprinciple ∆r̂∆p̂ ≈ t̂ = r r − 2m ⇒r→0= 0, (46) 16 apparent proof of the epr argument which confirms the statement in page 190 of ref. [7], on the possible recovering of full classical determinism in the interior of gravitational collapse (see ref. [37], chapter 6 in particular, for a penetrating critical analysis of black holes). it should perhaps be indicated that refs. [35] [36] introduced the factorization of a full riemannian metric g(x), x = (r,t) in (3 + 1)-dimensions g(x) = t̂gr(x)η, (47) where t̂gr is the gravitational isotopic element, and η is the minkowski metric η = diag.(1, 1, 1,−1). refs. [35] [36] then reformulated the riemannian geometry via the transition from a formulation over the field of real numbers r to that over the isofield of isoreal isonumbers r̂ where the gravitational isounit is evidently given by îgr(x) = 1/t̂gr(x). (48) the above reformulation turns the riemannian geometry into a new geometry called iso-minkowskian isogeometry, which is locally isomorphic to the minkowskian geometry, while maintaining the mathematical machinery of the riemannian geometry (covariant derivative, connection, geodesics, etc.) us fully maintained, although reformulated in terms of the isodifferential isocalculus [38]. the apparent advantages of the identical iso-minkowskian reformulation of riemannian metrics and einstein’s field equations (see, e.g., eqs. (2.9), page 390 of ref. [38]) are: 1) the achievement of a consistent operator form gravity in terms of relativistic hadronic mechanics [39] whose axioms are those of quantum mechanics, only subjected to a broader realization; 2) the achievement of a universal symmetry of all non-singular riemannian metrics, which symmetry is locally isomorphic to the lorentzpoincaré symmetry, today known as the lorentz-poincaré-santilli (lps) isosymmetry [40], and it is notoriously impossible on a conventional riemannian space over the reals; 3) the achievement of clear compatibility of einstein’s field equation with 20th century sciences, such as a clear compatibility of general relativity with special relativity via the simple limit îgr = i implying the transition from the universal lps isosymmetry to the poincaré symmetry of special relativity with ensuing recovering of conservation and other special relativity laws [41] [42]; the achievement of axiomatic compatibility of gravitation with electroweak interactions thanks to the replacement of curvature into the new notion of isoflatness with the ensuing, currently 17 ruggero maria santilli impossible, foundations for a grand unification [43]; and other intriguing advances. 3. concluding remarks t in this paper, we have continued the study of the epr argument [1] conducted in ref. [7] and preceding works, with particular reference to the study of the uncertainties for extended particles immersed within hyperdense medias with ensuing linear and non-linear, local and non-local and hamiltonian as well as non-hamiltonian interactions. this study has been conducted via the use of isomathematics and isomechanics characterized by the isotopic element t̂ of eq. (1) which represents the non-linear, non-local and non-hamiltonian interactions of the particles with the medium [19] [20] [21]. the main result of this paper is that the standard deviations of coordinates and momenta for particles within hyperdense media are characterized by the isotopic element that, being always very small, t̂ � 1, reduces the uncertainties in a way inversely proportional to a non-linear increase of the density, pressure, temperature, and other characteristics of the medium, while admitting the value t̂ = 0 under extreme/limit conditions with ensuing recovering of full determinism as predicted by a. einstein, b. podolsky and n. rosen [1]. we can, therefore, tentatively summarize the content of this paper with the following: isodeterministic isoprinciple: the product of isostandard isodeviations for isocoordinates ∆r̂ and isomomenta ∆p̂, as well as the individual isodeviations, progressively approach classical determinism for extended particles in the interior of hadrons, nuclei, and stars, and achieve classical determinism at the extreme densities in the interior of gravitational collapse. acknowledgments sincere thanks are due to thomas vougiouklis, svetlin georgiev and jeremy dunning davies for penetrating critical comments. additional thanks are also due to mrs. sherri stone for an accurate proofreading of the manuscript. 18 apparent proof of the epr argument references [1] a. einstein, b. podolsky , and n. rosen, “can quantum-mechanical description of physical reality be considered complete?,” phys. rev., vol. 47 , p. 777 (1935), http://www.galileoprincipia.org/docs/epr-argument.pdf [2] n. bohr, “can quantum mechanical description of physical reality be considered complete?” phys. rev. vol. 48, p. 696 (1935), www.informationphilosopher.com/solutions/scientists/bohr/eprbohr.pdf [3] j.s. bell: “on the einstein podolsky rosen paradox” physics vol. 1, 195 (1964), www.santilli-foundation.org/docs/bell.pdf [4] j. von neumann, mathematische grundlagen der quantenmechanik, springer, berlin (1951). 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[19] r. m. santilli, elements of hadronic mechanics, ukraine academy of sciences, kiev, volume i (1995), mathematical foundations, href=”http://www.santilli-foundation.org/docs/santilli-300.pdf 20 apparent proof of the epr argument [20] r. m. santilli, elements of hadronic mechanics, ukraine academy of sciences, kiev, volume ii (1995), theoretical foundations, http://www.santilli-foundation.org/docs/santilli-301.pdf [21] r. m. santilli, elements of hadronic mechanics, ukraine academy of sciences, kiev, volume iii (2016), experimental verifications, http://www.santilli-foundation.org/docs/elements-hadronicmechanics-iii.compressed.pdf [22] r. m. santilli, hadronic mathematics, mechanics and chemistry, volumes i to v, international academic press, (2008), http://www.i-b-r.org/hadronic-mechanics.htm [23] raul m. falcon ganfornina and juan nunez valdes, fundamentos de la isdotopia de santilli, international academic press (2001), http://www.i-b-r.org/docs/spanish.pdf english translations algebras, groups and geometries vol. 32, pages 135-308 (2015), http://www.i-b-r.org/docs/aversa-translation.pdf [24] i. gandzha and j. kadeisvili, new sciences for a new era: mathematical, physical and chemical discoveries of ruggero maria santilli, kathmandu university, sankata printing press, nepal (2011), http://www.santilli-foundation.org/docs/rms.pdf [25] h. c. myung and r. m. santilli, “modular-isotopic hilbert space formulation of the exterior strong problem,” hadronic journal vol. 5, p. 1277-1366 (1982), http://www.santilli-foundation.org/docs/santilli-201.pdf [26] r. m. santilli, foundations of hadronic chemistry, with applications to new clean energies and fuels, kluwer academic publishers (2001), http://www.santilli-foundation.org/docs/santilli-113.pdf russian translation by a. k. aringazin http://i-b-r.org/docs/santilli-hadronic-chemistry.pdf [27] r. m. santilli and d. d. shillady,, “a new isochemical model of the hydrogen molecule,” intern. j. hydrogen energy vol. 24, pages 943956 (1999), http://www.santilli-foundation.org/docs/santilli-135.pdf [28] r. m. santilli and d. d. shillady, “a new isochemical model of the water molecule,” intern. j. hydrogen energy vol. 25, 173-183 (2000), http://www.santilli-foundation.org/docs/santilli-39.pdf 21 ruggero maria santilli [29] r. m. santilli, “invariant lie-isotopic and lie-admissible formulation of quantum deformations,” found. phys. vol. 27, p. 11591177 (1997) , http://www.santilli-foundation.org/docs/santilli-06.pdf [30] r. m. santilli, “studies on einstein-podolsky-rosen argument that “quantum mechanics is not a complete theory,” i: basic formalism,” ibr preprint rms-7-19 (2019), to appear. [31] r. m. santilli,“studies on einstein-podolsky-rosen argument that “quantum mechanics is not a complete theory,” ii: apparent proof of the epr argument.” ibr preprint rms-9-19 (2019), to appear. [32] thomas vougiouklis hypermathematics, “hv-structures, hypernumbers, hypermatrices and lie-santilli addmissibility,” american journal of modern physics, vol. 4, no. 5, 2015, pp. 3846.11 special issue i;foundations of hadronic mathematics dedicated to the 80th birthday of prof. r. m. santilli, http://www.santillifoundation.org/docs/10.11648.j.ajmp.s.2015040501.15.pdf [33] bijan davvaz and thomas vougiouklis, a walk through weak hyperstructures, hv-structures, world scientific (2018) [34] löve,m. probability theory, in graduate texts in mathematics, volume 45, 4th edition, springer-verlag (1977). [35] r. m. santilli, “isotopic quantization of gravity and its universal isopoincaré symmetry” in the proceedings of the seventh marcel grossmann meeting in gravitation, slac 1992, r. t. jantzen, g. m. keiser and r. ruffini, editors, world scientific publishers pages 500-505(1994), http://www.santilli-foundation.org/docs/santilli-120.pdf [36] r. m. santilli, “unification of gravitation and electroweak interactions” in the proceedings of the eight marcel grossmann meeting in gravitation, israel 1997, t. piran and r. ruffini, editors, world scientific, pages 473-475 (1999), http://www.santilli-foundation.org/docs/santilli-137.pdf [37] jeremy dunning-davies, exploding a myth, ”conventional wisdom” or scientific truth? horwood publishing (2007). [38] r. m. santilli, “isominkowskian geometry for the gravitational treatment of matter and its isodual for antimatter,” intern. j. modern phys. d vol. 7, 351 (1998), http://www.santilli-foundation.org/docs/santilli-35.pdf 22 apparent proof of the epr argument [39] r. m. santilli, “relativistic hadronic mechanics: nonunitary, axiompreserving completion of relativistic quantum mechanics,” found. phys. vol. 27, 625-729 (1997), http://www.santilli-foundation.org/docs/santilli-15.pdf [40] r. m. santilli, “nonlinear, nonlocal and noncanonical isotopies of the poincaré symmetry,” moscow phys. soc. vol. 3, 255 (1993), http://www.santilli-foundation.org/docs/santilli-40.pdf [41] r. m. santilli, “rudiments of isogravitation for matter and its isodual for antimatter,” american journal of modern physics vol. 4, no. 5, 2015, pp. 59, http://www.santillifoundation.org/docs/10.11648.j.ajmp.s.2015040501.18.pdf [42] r. m. santilli, “isominkowskian reformulation of einstein’s gravitation and its compatibility with 20th century sciences,” ibr preprint 19-gr-07 (2019), to appear. [43] r.m. santilli, . isodual theory of antimatter with applications to antigravity, grand unification and cosmology, springer (2006). http://www.santilli-foundation.org/docs/santilli-79.pdf 23 ratio mathematica volume 38, 2020, pp. 367-375 on commutativity of prime and semiprime rings with generalized derivations md hamidur rahaman* abstract let r be a prime ring, extended centroid c and m,n,k ≥ 1 are fixed integers. if r admits a generalized derivation f associated with a derivation d such that (f(x) ◦ y)m + (x ◦ d(y))n = 0 or (f(x) ◦m y)k +x◦nd(y)=0 for all x,y ∈ i, where i is a nonzero ideal of r, then either r is commutative or there exist b ∈ u, utumi ring of quotient of r such that f(x) = bx for all x ∈ r. moreover, we also examine the case r is a semiprime ring. keywords: prime rings, semiprime rings, extended centroid, utumi quotient rings, generalized derivations. 2010 ams subject classifications: 16w25, 16n60, 16u80. 1 1 introduction r is always an associative ring with centre z(r), extended centroid c and utumi quotient ring u. for further information, definitions and properties on these concepts refer to beidar and martindale [1996]. for any x,y ∈ r, the symbols [x,y] and x◦y stand for the lie commutator xy−yx and jordan commutator xy + yx respectively. given x,y ∈ r, we set x◦0 y = x, x◦1 y = x◦y = xy + yx and inductively x◦m y = (x◦m−1 y)◦y for m > 2. recall that a ring r is prime if for any a,b ∈ r, arb = {0} implies that either a = 0 or b = 0 and is semiprime if for any a ∈ r, ara = {0} implies that a = 0. every prime ring is a semiprime ring but semiprime ring need not be prime ring. the socle of a ring r denoted by soc(r) is the sum of the minimal left (right) ideals of r, if r has minimal left *department of mathematics, aligarh muslim university, aligarh, india; rahamanhamidmath@gmail.com. 1received on february 15th, 2020. accepted on june 1st, 2020. published on june 30th, 2020. doi: 10.23755/rm.v38i0.502. issn: 1592-7415. eissn: 2282-8214. ©md hamidur rahaman this paper is published under the cc-by licence agreement. 367 md hamidur rahaman (right) ideals, otherwise soc(r) = (0). the goal of this paper is to establish that there is a relationship between the structure of the ring r and the behaviour of suitable additive mappings defined on r that satisfy certain special identities. in particular we study the case when the map is a generalized derivation of r. we recall that an additive map d : r → r is a derivation of r if d(xy) = d(x)y +xd(y) for all x,y ∈ r. in particular, d is an inner derivation induced by an element a ∈ r, if ia(x) = [a,x] for all x ∈ r and d is outer derivation if it is not inner derivation. an additive map f : r → r is said to be a generalized derivation if there is a derivation d of r such that for all x,y ∈ r, f(xy) = f(x)y + xd(y). all derivations are generalized derivations but generalized derivation need to be derivation. during the past few decades, there has been an ongoing interest concerning the relationship between the commutativity of a ring and the existence of certain specific types of derivations (see ashraf and rehman [2002] , where further references can be found). in argaç and inceboz [2009], argaç and inceboz proved that: if r is a prime ring, i a nonzero ideal of r, k a fixed positive integer and r admits a nonzero derivation d with the property (d(x◦y))k = x◦y for all x,y ∈ i, then r is commutative. in [quadri et al., 2003, theorem 2.3], quadri et al. discussed the commutativity of prime rings with generalized derivations. more precisely, quadri et al. proved that if r is a prime ring, i a nonzero ideal of r and f a generalized derivation associated with a nonzero derivation d such that f(x◦y) = x◦y for all x,y ∈ i, then r is commutative. in ashraf and rehman [2002], ashraf and rehman proved that if r is a 2-torsion free prime ring, i a nonzero ideal of r and d a nonzero derivation of r such that d(x) ◦d(y) = x◦y for all x,y ∈ i, then r is commutative. the present paper is motivated by the previous results and we here generalized the result obtained in ashraf and rehman [2002] . moreover, we continue this line of investigation by examining what happens if a ring r satisfies the following identities: (i)(f(x) ◦y)m + (x◦d(y))n = 0, (ii)(f(x) ◦m y)k + x◦n d(y) = 0 for all x,y ∈ i, a nonzero ideal of r. we obtain some analogous results for semiprime ring in the case i = r. more precisely, we shall prove the following theorems: theorem 1.1. let r be a prime ring, i a nonzero ideal of r and m,n are fixed positive integers. if r admits a generalized derivation f associated with a nonzero derivation d such that (f(x) ◦y)m + (x◦d(y))n = 0 for all x,y ∈ i, then either r is commutative or there exist b ∈ u, utumi ring of quotient of r such that f(x) = bx for all x ∈ r. 368 on commutativity of prime and semiprime rings with generalized derivations theorem 1.2. let r be a prime ring, i a nonzero ideal of r and m,n,k are fixed positive integers. if r admits a generalized derivation f associated with a nonzero derivation d such that (f(x) ◦m y)k + x◦n d(y) = 0 for all x,y ∈ i, then either r is commutative or there exist b ∈ u, utumi ring of quotient of r such that f(x) = bx for all x ∈ r. theorem 1.3. let r be a semiprime ring, u the left utumi quotient ring of r and m,n are fixed positive integers. if r admits a generalized derivation f associated with a nonzero derivation d such that (f(x)◦y)m + (x◦d(y))n = 0 for all x,y ∈ r, then r is commutative. theorem 1.4. let r be a semiprime ring, u the left utumi quotient ring of r and m,n,k are fixed positive integers. if r admits a generalized derivation f associated with a nonzero derivation d such that (f(x) ◦m y)k + x ◦n d(y) = 0 for all x,y ∈ r, then r is commutative. 2 preliminary results before starting our results, we state some well known facts which are very crucial for developing the proof of our main result. in particular, we will make frequent use of the following facts. fact 2.1. [lee, 1992, theorem 2] if i is a two-sided ideal of r, then r,i and u satisfy the same differential identities. fact 2.2. [lee, 1999, theorem 4] let r be a semiprime ring. then every generalized derivation f on a dense right ideal of r is uniquely extended to u and assumes the form f(x) = ax + d(x) for some a ∈ u and a derivation d on u. moreover, a and d are uniquely determined by the generalized derivation f . fact 2.3. [chuang, 1988, theorem 2] if i is a two-sided ideal of r, then r, i and u satisfy the same generalized polynomial identities with coefficients in u. fact 2.4. let r be a prime ring and d a nonzero derivation on r and i be a nonzero ideal of r. by kharchenko’s theorem [kharchenko, 1978, theorem 2], if i satisfies the differential polynomial identity p(x1,x2, ....,xn,d(x1),d(x2), ....,d(xn)) = 0, then either d is an inner derivation or d is outer derivation and i satisfies the generalized polynomial identity p(x1,x2, ....,xn,y1,y2, ....,yn) = 0. fact 2.5. [chuang, 1994, page no. 38] if r is semiprime then so is its left utumi quotient ring. the extended centroid c of a semiprime ring coincides with the center of its left utumi quotient ring. 369 md hamidur rahaman fact 2.6. [lee, 1992, lemma 2] any derivation of a semiprime ring r can be uniquely extended to a derivation of its left utumi quotient ring u, and so any derivation of r can be defined on the whole u. fact 2.7. [chuang, 1994, page no. 42] let b be the set of all the idempotents in c, the extended centroid of r. assume r is a b −algebra orthogonal complete. for any maximal ideal p of b, pr forms a minimal prime ideal of r, which is invariant under any nonzero derivation of r. 3 main results theorem 1.1 let r be a prime ring, i a nonzero ideal of r and m,n are fixed positive integers. if r admits a generalized derivation f associated with a derivation d such that (f(x) ◦ y)m + (x ◦ d(y))n = 0 for all x,y ∈ i, then either r is commutative or there exist b ∈ u, utumi ring of quotient of r such that f(x) = bx for all x ∈ r. proof. by hypothesis (f(x) ◦y)m + (x◦d(y))n = 0 for all x,y ∈ i. (1) by the fact 2.1 i, r and u satisfy the same generalized polynomial identity (gpi), we have (f(x) ◦y)m + (x◦d(y))n = 0 for all x,y ∈ u. (2) since r is a prime ring and f a generalized derivation of r, by fact 2.2, f(x) = ax + d(x) for some a ∈ u and a derivation d on u. then u satisfies ((ax + d(x)) ◦y)m + (x◦d(y))n = 0 for all x,y ∈ u. (3) that is (ax◦y + d(x) ◦y)m + (x◦d(y))n = 0 for all x,y ∈ u. (4) in the light of kharchenko’s theorem [kharchenko, 1978, theorem 2], we divide the proof into two cases:case i let d be an inner derivation of u, that is, d(x) = [q,x] for all x ∈ u and for some q ∈ u. then u satisfies f(x,y) = (ax◦y + [q,x] ◦y)m + (x◦ [q,y])n = 0 for all x,y ∈ u. (5) in case c is infinite, we have f(x,y) = 0 for all x,y ∈ u ⊗ c c̄, where c̄ is the algebraic closure of c. since both u and u ⊗ c c̄ are prime and centrally 370 on commutativity of prime and semiprime rings with generalized derivations closed erickson et al. [1975], we may replace r by u or u ⊗ c c̄ according to c is finite or infinite. thus we may assume that r is centrally closed over c which is either finite or algebraically closed and f(x,y) = 0 for all x,y ∈ r. by martindale’s theorem [martindale, 1969, theorem 3], r is then a primitive ring having nonzero socle, soc(r) with d as associative division ring. hence by jacobson’s theorem [jacobson, 1956, p.75], r is isomorphic to dense ring of linear transformations of vector space v over c. then the density of r on v implies that r ∼= mk(d), where k = dimdv . assume that dimdv > 2, otherwise we are done. suppose that there exists v ∈ v such that v and qv are linearly dindependent. since dimdv > 2, then there exists w ∈ v such that {v,qv,w} is linearly independent over d. by density of r there exist x,y ∈ r such that xv = 0,xqv = w,yw = v,xw = 0,yv = 0,yqv = v. (6) multiplying equation (5) by v from right and using conditions in equation (6), we get (−1)mv = 0, a contradiction. now we want to show that qv = vβ for some β ∈ d. let v,w be linearly independent. then by the precedant argument, there exist βv, βw, βv+w ∈ d, such that qv = vβv, qw = wβw, q(v+w) = (v+w)βv+w. moreover, vβv +wβw = (v+w)βv+w and hence v(βv−βv+w)+w(βw−βv+w) = 0. since v,w are linearly independent, we have βv = βw = βv+w that is β does not depend on the choice of v. let now for r ∈ r, v ∈ v , by precedant calculation, qv = vα, r(qv) = r(vα) and also q(rv) = (rv)α. thus 0 = [q,r]v for any v ∈ v , that is, [q,r]v = 0 . since v is a left faithful irreducible r -module, [q,r] = 0 for all r ∈ r i.e., q ∈ z(r) and d = 0. case 2 let d be an outer derivation of r. then by fact 2.4, i satisfies the generalized polynomial identity (ax◦y + t◦y)m + (x◦ t)n = 0 for all x,y,t ∈ i. (7) in particular, for y = 0, i satisfies (xt + tx)n = 0. by chuang [chuang, 1988, theorem 2], this polynomial identity is also satisfied by q and hence r as well. by lemma 1 lanski [1993], there exists a field f such that r ⊆ mk(f), the ring of k × k matrices over a field f , where k ≥ 1. moreover, r and mk(f) satisfy the same polynomial identity, that is (xt + tx)n = 0 for all t,x ∈ mk(f). let eij be the usual matrix unit with 1 in the (i,j) entry and zero elsewhere. by choosing t = e12,x = e21, we see that (xt + tx)n = (e11 + e22)n 6= 0, a contradiction. theorem 1.2 let r be a prime ring, i a nonzero ideal of r and m,n,k are fixed positive integers. if r admits a generalized derivation f associated with a nonzero derivation d such that (f(x) ◦m y)k + x◦n d(y) = 0 for all x,y ∈ i, then either 371 md hamidur rahaman r is commutative or there exist b ∈ u, utumi ring of quotient of r such that f(x) = bx for all x ∈ r. proof. by hypothesis (f(x) ◦m y)k + x◦n d(y) = 0 for all x,y ∈ i. (8) by the fact 2.1 i, r and u satisfy the same generalized polynomial identity (gpi), we have (f(x) ◦m y)k + x◦n d(y) = 0 for all x,y ∈ u. (9) by fact 2.2, f(x) = ax + d(x) for some a ∈ u and derivation d on u. then u satisfies ((ax + d(x)) ◦m y)k + x◦n d(y) = 0 for all x,y ∈ u. (10) that is (ax◦m y + d(x) ◦m y)k + x◦n d(y) = 0 for all x,y ∈ u. (11) in the light of kharchenko’s theorem [kharchenko, 1978, theorem 2], we divide the proof into two cases:case i let d be an inner derivation of u that is d(x) = [q,x] for all x ∈ u and for some q ∈ u. then u satisfies (ax◦m y + [q,x] ◦m y)k + x◦n [q,y] = 0 for all x,y ∈ u. (12) as in the proof of theorem 1.1, we have (ax◦m y + [q,x] ◦m y)k + x◦n [q,y] = 0 for all x,y ∈ r. (13) where r is a primitive ring with d as the associated division ring. if v is finite dimensional over d, then the density of r implies that r ∼= mk(d), where k = dimdv . assume that dimdv > 2, otherwise we are done. suppose that there exists v ∈ v such that v and qv are linearly d-independent. since dimdv > 2, then there exists w ∈ v such that {v,qv,w} is linearly independent over d. by density of r there exist x,y ∈ r such that xv = 0,xqv = w,yw = v,xw = 0,yv = 0,yqv = v. multiplying equation (13) by v from right, we get (−1)mkv = 0 which is a contradiction to the linearly independent of the set {v,qv}. therefore, {v,qv} is linearly 372 on commutativity of prime and semiprime rings with generalized derivations dependent and so q ∈ z(r), i.e, d = 0. case 2 let d be an outer derivation. then (ax◦m y + t◦m y)k + x◦n s = 0 for all x,y,s,t ∈ i. (14) in particular, choosing y = 0, we have s ◦n x = 0. by chuang [chuang, 1988, theorem 2], this polynomial identity is also satisfied by q and hence r as well. by lemma 1 lanski [1993], there exists a field f such that r ⊆ mk(f), the ring of k × k matrices over a field f , where k ≥ 1. moreover, r and mk(f) satisfy the same polynomial identity, that is, s ◦n x = 0 for all s,x ∈ mk(f). denote eij the usual matrix unit with 1 in (i,j)-entry and zero elsewhere. by choosing s = e12,x = e11, we see that s◦n x = e12 6= 0, a contradiction. the following examples demonstrate that r to be prime can not be omitted in the hypothesis of theorem 1.1 and theorem 1.2. example 3.1. for any ring s, let r = {( x y 0 0 ) | x,y ∈ s } and i ={( 0 y 0 0 ) | y ∈ s } . then r is a ring under usual addition and multiplication of matrices and i is a nonzero ideal of r. define maps f,d : r → r by f( ( x y 0 0 ) ) = ( x 2y 0 0 ) and d( ( x y 0 0 ) ) = ( 0 y 0 0 ) . then f is a generalized derivation on r associated with the nonzero derivation d satisfying (f(x)◦y)m + (x◦d(y))n = 0 for all x,y ∈ i. however r is not commutative as well as f can not be written as f(x) = bx for all x ∈ r as d is nonzero. hence theorem 1.1 is not true for arbitrary rings. example 3.2. let r = {( x y o z ) | x,y,z ∈ z2 } and i = {( 0 y 0 0 ) | y ∈ z2 } . then r is a ring under usual addition and multiplication of matrices and i is a nonzero ideal of r. define maps f,d : r → r by f( ( x y 0 z ) ) =( x 0 0 0 ) and d( ( x y 0 z ) ) = ( 0 y 0 0 ) . then f is a generalized derivation on r associated with the nonzero derivation d satisfying (f(x)◦m y)k + x◦n d(y) = 0 for all x,y ∈ i. however r is not commutative as well as f can not be written as f(x) = bx for all x ∈ r as d is nonzero . hence theorem 1.2 is not true for arbitrary rings. theorem 1.3 let r be a semiprime ring, u the left utumi quotient ring of r and m,n are fixed positive integers. if r admits a generalized derivation f associated 373 md hamidur rahaman with a nonzero derivation d such that (f(x)◦y)m + (x◦d(y))n = 0 for all x,y ∈ r, then r is commutative. proof. by fact 2.6, any derivation of a semiprime ring r can be uniquely extended to a derivation of its left utumi quotient ring u and so any derivation of r can be defined on the whole of u. moreover, by fact 2.3 i, r and u satisfy the same gpis and by fact 2.1 i, r, u satisfy same differential identities. also by fact 2.2, we have f(x) = ax + d(x) for some a ∈ u and derivation d of u. then ((ax + d(x)) ◦y)m + (x◦d(y))n = 0 for all x,y ∈ u. (15) by fact 2.5, we have z(u) = c. let m(c) be the set of all maximal ideals of c and p ∈ m(c). by fact 2.7, we have pu is a prime ideal of u invariant under all derivations of u. moreover, ⋂ {pu | p ∈ m(c) } = 0. set u = u/pu. then derivation d canonically induce derivation d on u defined by d(x) = d(x) for all x ∈ ū. therefore, ((āx̄ + d(x)) ◦ ȳ)m + (x◦d(y))n = 0 for all x,y ∈ u. it is obvious that u is prime. therefore, by theorem 1.1, we have for each p ∈ m(c) either [u,u] ⊆ pu or d(u) ⊆ pu. in any case d(u)[u,u] ⊆ pu for all p ∈ m(c). thus d(u)[u,u] ⊆ ⋂ {pu | p ∈ m(c) } = 0, we obtain d(u)[u,u] = 0. therefore, [u,u] = 0 since ⋂ {pu | p ∈ m(c) } = 0 and d 6= 0. since r is subring of u, so in particular [r,r] = 0. hence r is commutative. this completes the proof of the theorem. using the similar arguments as used in the proof of the above theorem, we can prove the following theorem. theorem 1.4 let r be a semiprime ring, u the left utumi quotient ring of r and k,m,n are fixed positive integers. if r admits a generalized derivation f associated with a nonzero derivation d such that (f(x) ◦m y)k + x ◦n d(y) = 0 for all x,y ∈ r, then r is commutative. references n. argaç and h.g inceboz. derivation of prime and semiprime rings. j. korean math. soc., 46:997–1005, 2009. m. ashraf and n. rehman. on commutativity of rings with derivations. results math., 42:3–8, 2002. 374 on commutativity of prime and semiprime rings with generalized derivations k.i. beidar and w.s. martindale. rings with generalized identities. dekker, new york, 1996. c.l. chuang. gpi’s having coefficients in utumi quotient rings. proc. amer. math. soc., 103:723–728, 1988. c.l. chuang. hypercentral derivations. j. algebra, 166:34–71, 1994. t. erickson, w.s. martindale, and j.m. osborn. prime nonassociative algebras. pacific j. math., 60:49–63, 1975. n. jacobson. structure of rings. colloquium publications, amer. math. soc. vii, provindence, ri, 1956. v.k. kharchenko. differential identities of prime rings. algebra and logic, 17: 155–168, 1978. c. lanski. an engel condition with derivation. proc. amer. math. soc., 118: 731–734, 1993. t.k. lee. semiprime rings with differential identities. bull. inst. math. acad. sin., 8:27–38, 1992. t.k. lee. generalized derivations of left faithful rings. communications in algebra, 27:4057–4073, 1999. w.s. martindale. prime rings satisfying a generalized polynomial identity. j. algebra, 12:576–584, 1969. m.a. quadri, s. khan, and n. rehman. generalized derivations and commutativity of prime rings. indian j. pure appl. math., 34:1393–1396, 2003. 375 ratio mathematica 25 (2013), 15–28 issn:1592-7415 classification of hyper m v -algebras of order 3 r. a. borzooei∗, a. radfar∗∗ ∗department of mathematics, shahid beheshti university, g. c.,tehran, iran ∗∗department of mathematics, payame noor university, tehran, iran borzooei@sbu.ac.ir, ateferadfar@yahoo.com abstract in this paper, we investigated the number of hyper m v -algebras of order 3. in fact, we prove that there are 33 hyper m v -algebras of order 3, up to isomorphism. key words: hyper m v -algebra msc 2010: 97u99. 1 introduction the concept of m v -algebras was introduced by chang in [1] in order to show lukasiewicz logic to be standard complete, i.e. complete with respect to evaluations of propositional variables in the real unit interval [0, 1]. in [6], mundici showed that any m v -algebra is an interval of an abelian lattice ordered group with a strong unit. also, he introduced the concept of state on m v -algebra. georgescu and iorgulescu [2] introduced a new noncommutative algebraic structures, which were called pseudo m v -algebras. it can be obtained by dropping commutative axioms in m v -algebras, which are a generalization of m v -algebras. the hyper structure theory was introduced by f. marty [5] at the 8th congress of scandinavian mathematicians in 1934. since then many researches have worked in these areas. recently in [4], sh. ghorbani, a. hasankhni and e. eslami applied the hyper structure to m v -algebras and introduced the concept of a hyper m v -algebra which is a generalization of an m v -algebra and investigated some related results. now, in this paper we find all hyper m v -algebras of order 3. 15 r. a. borzooei, a. radfar 2 preliminary definition 2.1. [1] an m v -algebra (x,⊕,∗ , 0) is a set x equipped with a binary operation ⊕, a unary operation ∗ and a constant 0 satisfying the following equations: (m v1) x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z, (m v2) x ⊕ y = y ⊕ x, (m v3) x ⊕ 0 = x, (m v4) (x ∗)∗ = x, (m v5) x ⊕ 0∗ = 0∗, (m v6) (x ∗ ⊕ y)∗ ⊕ y = (y∗ ⊕ x)∗ ⊕ x, for all x, y, z ∈ x. definition 2.2. [3] a hyperalgebra (m,⊕,∗ , 0) with a hyperoperation ⊕ : m × m −→ p∗(m ), a unary operation ∗ : m −→ m and a constant 0, is said to be a hyper m v -algebra if and only if satisfies the following axioms, for all x, y, z ∈ m : (hm v1) x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z, (hm v2) x ⊕ y = y ⊕ x, (hm v3) (x ∗)∗ = x, (hm v4) 0 ∗ ∈ x ⊕ 0∗, (hm v5) (x ∗ ⊕ y)∗ ⊕ y = (y∗ ⊕ x)∗ ⊕ x, (hm v6) 0 ∗ ∈ x ⊕ x∗, (hm v7) if x 6 y and y 6 x, then x = y, where x 6 y is defined by 0∗ ∈ x∗ ⊕ y. for every x, y ⊆ m , x 6 y if there exist x ∈ x and y ∈ y such that x 6 y. we define 1 = 0∗ theorem 2.3. [3] let (m,⊕,∗ , 0) be a hyper-m v algebra. then for all x, y, z ∈ m and for all non-empty subsets a, b and c of m the following hold: (i) (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c), (ii) 0 6 x 6 1, x 6 x and a 6 a, (iii) if x 6 y then y∗ 6 x∗ and a 6 b implies b∗ 6 a∗, (iv) if x 6 0 or 1 6 x, then x = 0 or x = 1, respectively, (v) 0 ⊕ 0 = {0}, (vi) x ∈ x ⊕ 0, (vii) if x ⊕ 0 = y ⊕ 0, then x = y. 16 classification of hyper m v -algebras of order 3 3 classification of hyper m v -algebras of order 3 in this section we try to find all hyper m v -algebras of order 3, up to isomorphism. theorem 3.1. let m be a hyper m v -algebra and x be an element of m such that 0 ⊕ x = {x} and x∗ = x. then the following statements hold: (i) (1 ⊕ x)∗ ⊕ x = {x}, (ii) (1 ⊕ x)∗ ⊕ 1 = x ⊕ x, (iii) x 6∈ 1 ⊕ x and 0 6∈ 1 ⊕ x. proof. since 0∗ = 1, then by hypothesis and (hm v 5); (1⊕x)∗⊕x = (0∗⊕x)∗⊕x = (x∗⊕0)∗⊕0 = (x⊕0)∗⊕0 = x∗⊕0 = x⊕0 = {x} (1 ⊕ x)∗ ⊕ 1 = (x ⊕ 1)∗ ⊕ 1 = ((x∗)∗ ⊕ 1)∗ ⊕ 1 = = (1∗ ⊕ x∗)∗ ⊕ x∗ = (0 ⊕ x)∗ ⊕ x∗ = x∗ ⊕ x∗ = x ⊕ x and so (i) and (ii) hold. (iii) if x ∈ 1⊕x, then x = x∗ ∈ (1⊕x)∗ and so x⊕x = x∗⊕x ⊆ (1⊕x)∗⊕x. by (i), x ⊕ x ⊆{x}. hence x ⊕ x = {x}. now, since by (hm v6), 1 = 0∗ ∈ x ⊕ x∗ = x ⊕ x = {x}, then x = 1 and so 0 = 1∗ = x∗ = x = 1, which is a contradiction. hence x /∈ 1 ⊕ x. now, let 0 ∈ 1 ⊕ x. then 1 = 0∗ ∈ (1 ⊕ x)∗ and so 1 ⊕ x ⊆ (1 ⊕ x)∗ ⊕ x. by (i), 1 ⊕ x ⊆{x}. thus 1 ⊕ x = {x}, which is a contradiction. hence 0 /∈ 1 ⊕ x. note. from now one in this paper, we let m = {0, a, 1} be a hyper m v -algebra of order 3. theorem 3.2. (i) 1 ≤ 1, 0 ≤ 0, a ≤ a, 0 ≤ 1 and 0 ≤ a, (ii) a 6≤ 0, (iii) a∗ = a, (iv) 1 ∈ 1 ⊕ a. proof. (i). by theorem 2.3(ii), the proof is clear. (ii). by theorem 2.3(iv), the proof is clear. (iii). by definition 2.2, 0∗ = 1 and by (hm v3), 0 = (0 ∗)∗ = 1∗. now, if a∗ = 1, then 0 = 1∗ = (a∗)∗ = a, which is a contradiction. by similar way, if a∗ = 0, then 1 = 0∗ = (a∗)∗ = a, which is a contradiction. hence, a∗ = a. (iv). by (hm v4), 1 = 0 ∗ ∈ 0∗ ⊕ a = 1 ⊕ a. theorem 3.3. if 0 ⊕ a = {a} or 1 ⊕ a = {1}, then m is an m v -algebra. 17 r. a. borzooei, a. radfar proof. let 0 ⊕ a = {a}. since a∗ = a, then by theorem 3.1(iii), a 6∈ 1 ⊕ a and 0 6∈ 1 ⊕ a and so 1 ⊕ a = {1}. moreover, by theorem 3.1(iii) and (i), 0 6∈ 1⊕0 and (1⊕0)∗ ⊕0 = {0}. since 0 6∈ {a} = 0⊕a and 0 6∈ 1⊕0, then (1⊕0)∗ = {0} and so 1⊕0 = {1}. by theorem 3.1(i) and (ii), 0 ⊕ 1 = {1} = (1 ⊕ a)∗ ⊕ 1 = a ⊕ a. hence a ⊕ a = {1}. now, by (hm v1), 1 ⊕ 1 = (a ⊕ a) ⊕ 1 = a ⊕ (1 ⊕ a) = a ⊕ 1 = {1}. therefore, x⊕y is singleton for all x, y ∈ m and so m is an m v -algebra. now, if 1⊕a = {1}, then {0} = {1∗} = (1⊕a)∗ and so 0⊕a = (1⊕a)∗⊕a. by (hm v5), 0 ⊕ a = (1 ⊕ a)∗ ⊕ a = 0 ⊕ (0 ⊕ a)∗. by theorem 3.2, a 6< 0, 1 6∈ 0 ⊕ a. if 0 ∈ 0 ⊕ a, then 0 ⊕ a = {0, a} and {0, a} = 0 ⊕ a = 0 ⊕ (0 ⊕ a)∗ = 0 ⊕{0, a}∗ = = 0 ⊕{1, a} = (0 ⊕ 1) ∪ (0 ⊕ a) = (0 ⊕ 1) ∪{0, a}. hence 0 ⊕ 1 ⊆ {0, a}. by (hm v 4), 1 ∈ 0 ⊕ 1. thus 1 ∈ {0, a}, which is a contradiction. thus 0 6∈ 0 ⊕ a and so 0 ⊕ a = {a}. therefore, m is a same m v -algebra, which is as follows: ⊕1 0 a 1 0 {0} {a} {1} a {a} {1} {1} 1 {1} {1} {1} definition 3.4. we call a hyper m v -algebra is proper, if it is not an m v algebra. lemma 3.5. let m = {0, a, 1} be a proper hyper m v -algebra of order 3. then (i) 0 ⊕ a = {0, a}, (ii) 0 ⊕ 1 = {1}, {0, 1} or m , (iii) a ⊕ a = {1}, {0, 1}, {1, a} or m , (iv) 1 ⊕ a = {0, 1}, {1, a} or m , (v) 1 ⊕ 1 = {1}, {0, 1} {1, a} or m , (vi) if a ⊕ a = {1}, then 0 ⊕ 1 = m . 18 classification of hyper m v -algebras of order 3 proof. (i). since a 6< 0, then 1 6∈ 0⊕a. by theorem 2.3 (vi), a ∈ 0⊕a. thus 0 ⊕ a = {a} or {0, a}. if 0 ⊕ a = {a}, then by theorem 3.3, m is not proper. thus 0 ⊕ a = {0, a} (ii). since 0 ≤ 0, then 1 = 0∗ ∈ 0∗ ⊕ 0 = 1 ⊕ 0 = 0 ⊕ 1. hence it is sufficient to show that 0 ⊕ 1 6= {1, a}. let 0 ⊕ 1 = {1, a}, by the contrary. then by (hm v1), {1, a} = 0 ⊕ 1 = (0 ⊕ 0) ⊕ 1 = (0 ⊕ 1) ⊕ 0 = {1, a}⊕ 0 = {0, a, 1}, which is impossible. therefore, 0 ⊕ 1 6= {1, a} and so 0 ⊕ 1 = {1}, {0, 1} or m . (iii), (v). since a ≤ a and 0 ≤ 1, then 1 ∈ a⊕a and 1 ∈ 1⊕1 and so (v) and (iii) are hold. (iv). since 0 ≤ a, then 1 ∈ 1 ⊕ a. by theorem 3.3, if a ⊕ 1 = {1}, then m is an m v algebra which is impossible. hence 1 ⊕ a = {0, 1}, {1, a} or m . (vi). let a ⊕ a = {1}. then by (hm v1), 0 ⊕ 1 = 0 ⊕ (a ⊕ a) = (0 ⊕ a) ⊕ a = {0, a}⊕ a = (0 ⊕ a) ∪ (a ⊕ a) = m. by lemma 3.5 (ii), we know that 0 ⊕ 1 = {1}, {0, 1} or m . so, for the classification of all hyper m v -algebras of order 3, we consider the following three cases. case 1: 0 ⊕ 1 = {1} lemma 3.6. let m = {0, a, 1} be a proper hyper m v -algebra of order 3 and 0 ⊕ 1 = {1}. then (i) a ⊕ a = {1, a} or m , (ii) 1 ⊕ 1 = {1}, (iii) 1 ⊕ a = m . proof. (i). by lemma 3.5 (i) and (iii), 0⊕a = {0, a} and 1 ∈ a⊕a. hence (0 ⊕ a) ⊕ a = {0, a}⊕ a = (0 ⊕ a) ∪ (a ⊕ a) = {0, a}∪ (a ⊕ a) = m. since by (hm v1), (0 ⊕ a) ⊕ a = 0 ⊕ (a ⊕ a), then 0 ⊕ (a ⊕ a) = m . by lemma 3.5(iii), a ⊕ a = {1}, {0, 1}, {1, a} or m . if a ⊕ a = {1}, then 0 ⊕ (a ⊕ a) = 0 ⊕ 1 = {1}, which is a contradiction. if a ⊕ a = {0, 1}, then by theorem 2.3(v), 0 ⊕ (a ⊕ a) = 0 ⊕{0, 1} = (0 ⊕ 0) ∪ (0 ⊕ 1) = {0, 1}, which is a contradiction. hence, a ⊕ a = {1, a} or m . 19 r. a. borzooei, a. radfar (ii). by (hm v5), and theorem 2.3(v), (1 ⊕ 1)∗ ⊕ 1 = (0∗ ⊕ 1)∗ ⊕ 1 = (1∗ ⊕ 0)∗ ⊕ 0 = (0 ⊕ 0)∗ ⊕ 0 = 1 ⊕ 0 = {1}. if 0 ∈ 1 ⊕ 1, then 1 ⊕ 1 ⊆ (1 ⊕ 1)∗ ⊕ 1 = {1} and so 0 /∈ 1 ⊕ 1, which is a contradiction. if a ∈ 1⊕1, then a⊕1 ⊆ (1⊕1)∗⊕1 = {1}. thus a⊕1 = {1} and so by theorem 3.3, m is an m v -algebra, which is a contradiction. hence, 1 ⊕ 1 = {1}. (iii). by lemma 3.5, 1 ⊕ a = {0, 1}, {1, a} or m . if 1 ⊕ a = {0, 1}, since by (hm v1), 1 ⊕ (1 ⊕a) = (1 ⊕ 1) ⊕a = 1 ⊕a, then 1 ⊕ (1 ⊕a) = {1}, which is a contradiction. if 1 ⊕ a = {1, a}, since by (hm v1), 0 ⊕ (1 ⊕ a) = (0 ⊕ 1) ⊕ a = 1 ⊕ a, then 0 ⊕ (1 ⊕ a) = (0 ⊕ 1) ∪ (0 ⊕ a) = m , which is a contradiction. hence, 1 ⊕ a = m . theorem 3.7. there are two non-isomorphic proper hyper m v -algebras of order 3 such that 0 ⊕ 1 = {1}. proof. according theorem 3.6, if m is a proper hyper m v -algebra of order 3 and 0 ⊕ 1 = {1}, then we must investigate two following tables, which both of them are non-isomorphic hyper m v -algebras. ⊕2 0 a 1 0 {0} {0, a} {1} a {0, a} {1, a} {0, a, 1} 1 {1} {0, a, 1} {1} ⊕3 0 a 1 0 {0} {0, a} {1} a {0, a} {0, a, 1} {0, a, 1} 1 {1} {0, a, 1} {1} case 2: 0 ⊕ 1 = {0, 1} lemma 3.8. let m = {0, a, 1} be a proper hyper m v -algebra of order 3 and 0 ⊕ 1 = {0, 1}. then (i) (a ⊕ a) ∪ (1 ⊕ a) = m , (ii) a ⊕ 1 = {a, 1} or m , (iii) a ⊕ a = {a, 1} or m , (iv) 1 ⊕ 1 = {0, 1} or {1}. proof. (i). let 0⊕1 = {0, 1}. by theorem 3.5(iv), since 1 ∈ 1⊕a, by (hm v1), (0⊕a)⊕1 = (0⊕1)⊕a = {0, 1}⊕a = (0⊕a)∪(1⊕a) = {0, a}∪(1⊕a) = m. on the other hands (0 ⊕ a) ⊕ 1 = {0, a}⊕ 1 = (0 ⊕ 1) ∪ (a ⊕ 1) = {0, 1}∪ (a ⊕ 1). 20 classification of hyper m v -algebras of order 3 thus {0, 1}∪ (a ⊕ 1) = m and so a ∈ a ⊕ 1. new, we consider two cases 0 ∈ a ⊕ 1 or 0 6= a ⊕ 1. if 0 ∈ a ⊕ 1, since by theorem 3.5, 1 ∈ a ⊕ 1, then a⊕1 = m and so (a⊕a)∪(1⊕a) = m . now, if 0 6= a⊕1, then by theorem 3.5, a ∈ a ⊕ 1. hence by theorem 3.2(iv), {1, a}⊆ a ⊕ 1. thus m = (0 ⊕ 1) ∪ (a ⊕ 1) = {0, a}⊕ 1 = {1, a}∗ ⊕ 1 ⊆ (a ⊕ 1)∗ ⊕ 1 ⊆ m and so (a ⊕ 1)∗ ⊕ 1 = m . on the other hands, by (hm v5), (a ⊕ 1)∗ ⊕ 1 = (0 ⊕ a)∗ ⊕ a. hence (0 ⊕ a)∗ ⊕ a = m . since 0 ⊕ a = {0, a}, then m = (0 ⊕ a)∗ ⊕ a = {1, a}⊕ a = (1 ⊕ a) ∪ (a ⊕ a). (ii). by lemma 3.5(iv), it is enough to show that 1 ⊕ a = {0, 1}. let 0 ∈ a ⊕ 1, by the contrary. since by lemma 3.5(iv) and (i), 0 ⊕ a = {0, a} and 1 ∈ 1 ⊕ a, then (0 ⊕ 1) ⊕ a = {0, 1}⊕ a = (0 ⊕ a) ∪ (1 ⊕ a) = m. thus by (hm v1), m = (0 ⊕ 1) ⊕ a = (0 ⊕ a) ⊕ 1 = {0, 1}∪ (1 ⊕ a). and so a ∈ 1⊕a. hence a⊕1 6= {0, 1} and so by lemma 3.5(iv), a⊕1 = {a, 1} or m . (iii). by lemma 3.5(i), 0 ⊕ a = {0, a}. now, since 1 ∈ a ⊕ a, then (0 ⊕ a) ⊕ a = {0, a}⊕ a = (0 ⊕ a) ∪ (a ⊕ a) = m. hence, by (hm v1), 0 ⊕ (a ⊕ a) = (0 ⊕ a) ⊕ a = m . since a 6∈ 0 ⊕ 0 and a 6∈ 0 ⊕ 1, then a ∈ a ⊕ a. hence a ⊕ a = {a, 1} or m . (iv). let a ∈ 1 ⊕ 1. by (hm v5), a ⊕ 1 = a∗ ⊕ 1 ⊆ (1 ⊕ 1)∗ ⊕ 1 = (0 ⊕ 0)∗ ⊕ 0 = {0, 1}. which is a contradiction by (i). hence a 6∈ 1 ⊕ 1 and so by lemma 3.5(v), 1 ⊕ 1 = {0, 1} or {1}. theorem 3.9. there are 6 non-isomorphic proper hyper m v -algebras of order 3 such that 0 ⊕ 1 = {0, 1}. proof. by lemma 3.8 (iii), a⊕a = {a, 1} or m . if a⊕a = {a, 1}, then by lemma 3.8 (ii), a⊕1 = {a, 1} or m . by lemma 3.8 (i), if a⊕a = {a, 1}, 21 r. a. borzooei, a. radfar then a⊕1 6= {a, 1}. hence we must investigate 2 following tables which both of them are hyper m v -algebras. ⊕4 0 a 1 0 {0} {0, a} {0, 1} a {0, a} {a, 1} {0, a, 1} 1 {0, 1} {0, a, 1} {1} ⊕5 0 a 1 0 {0} {0, a} {0, 1} a {0, a} {a, 1} {0, a, 1} 1 {1} {0, a, 1} {0, 1} now, if a⊕a = m , then by lemma 3.8 (ii) and (iv), a⊕1 = {a, 1} or m and 1⊕1 = {0, 1} or {1}. thus we must investigate 4 following tables, which all of them are hyper m v -algebras. ⊕6 0 a 1 0 {0} {0, a} {0, 1} a {0, a} {0, a, 1} {a, 1} 1 {0, 1} {a, 1} {1} ⊕7 0 a 1 0 {0} {0, a} {0, 1} a {0, a} {0, a, 1} {a, 1} 1 {1} {a, 1} {0, 1} ⊕8 0 a 1 0 {0} {0, a} {0, 1} a {0, a} {0, a, 1} {0, a, 1} 1 {0, 1} {0, a, 1} {1} ⊕9 0 a 1 0 {0} {0, a} {0, 1} a {0, a} {0, a, 1} {0, a, 1} 1 {0, 1} {0, a, 1} {0, 1} case 3: 0 ⊕ 1 = m lemma 3.10. let m = {0, a, 1} be a proper hyper m v -algebra of order 3 such that 0 ⊕ 1 = m . then (i) (a ⊕ a) ∪ (1 ⊕ a) = m , (ii) if a ⊕ a = {1}, then a ⊕ 1 = 1 ⊕ 1 = m , (iii) if a ⊕ a = {0, 1}, then a ⊕ 1 = {a, 1} or m and if a ⊕ 1 = {a, 1}, then 1 ⊕ 1 = {1},{0, 1} or m , (iv) if a ⊕ a = {a, 1}, then a ⊕ 1 = {0, 1} or m and if a ⊕ 1 = {0, 1}, then 1 ⊕ 1 = {a, 1} or m , (v) if a ⊕ a = m and a ⊕ 1 = {1, a}, then 1 ⊕ 1 = {1},{0, 1} or m , (vi) if a ⊕ a = m and a ⊕ 1 = {0, 1}, then 1 ⊕ 1 = {0, 1},{a, 1} or m . proof. (i). since by lemma 3.5(iv), 1 ∈ 1 ⊕ a, then m = 0 ⊕ 1 = 1∗ ⊕ 1 ⊆ (a ⊕ 1)∗ ⊕ 1 and so (a ⊕ 1)∗ ⊕ 1 = m . hence by (hm v5), (0 ⊕ a)∗ ⊕ a = (a ⊕ 1)∗ ⊕ 1 = m and so by lemma 3.5(i), m = (0 ⊕ a)∗ ⊕ a = {0, a}∗ ⊕ a = {1, a}⊕ a = (1 ⊕ a) ∪ a ⊕ a. 22 classification of hyper m v -algebras of order 3 (ii). let a ⊕ a = {1}. since 1 ∈ 1 ⊕ a, then by (hm v5) and lemma 3.5(i), 1 ⊕ a = (1 ⊕ a) ∪ (a ⊕ a) = {1, a}⊕ a = {0, a}∗ ⊕ a = (0 ⊕ a)∗ ⊕ a = (a ⊕ 0)∗ ⊕ 0 = {1, a}⊕ 0 = (1 ⊕ 0) ∪ (a ⊕ 0) = m now, since a ⊕ a = {1} and 1 ⊕ a = m , then by (hm v1), 1 ⊕ 1 = (a ⊕ a) ⊕ (a ⊕ a) = a ⊕ (a ⊕ (a ⊕ a)) = a ⊕ (a ⊕ 1) = a ⊕ m = (a ⊕ 1) ∪ (a ⊕ a) ∪ (a ⊕ 0) = m. (iii). if a ⊕ a = {0, 1}, then by (i) and lemma 3.5(iv), a ⊕ 1 = {a, 1} or m . let a ⊕ 1 = {a, 1}. if 1 ⊕ 1 = {a, 1}, then by (hm v1) and (i), m = (a ⊕ a) ∪ (1 ⊕ a) = {a, 1}⊕ a = (1 ⊕ 1) ⊕ a = 1 ⊕ (1 ⊕ a) = 1 ⊕{1, a} = (1 ⊕ 1) ∪ (1 ⊕ a) = (1 ⊕ 1) ∪ {1, a} hence 0 ∈ 1 ⊕ 1 = {a, 1}, which is a contradiction. thus 1 ⊕ 1 6= {a, 1} and so by lemma 3.5(v), 1 ⊕ 1 = {1},{0, 1} or m . (iv). by (i), if a ⊕ a = {a, 1}, then a ⊕ 1 = {0, 1} or m . if a ⊕ 1 = {0, 1}, then by (hm v1), m = {0, a}∪ (1 ⊕ a) = {0, 1}⊕ a = (1 ⊕ a) ⊕ a = 1 ⊕ (a ⊕ a) = 1 ⊕{a, 1} = (1 ⊕ a) ∪ (1 ⊕ 1) = {0, 1}∪ (1 ⊕ 1) hence a ∈ 1 ⊕ 1. by lemma 3.5(v), 1 ⊕ 1 = {1, a} or m . (v). let a⊕a = m and 1⊕a = {1, a}. if 1⊕1 = {a, 1}, then by (hm v1), m = (a ⊕ a) ∪ (1 ⊕ a) = {1, a}⊕ a = (1 ⊕ 1) ⊕ a = 1 ⊕ (1 ⊕ a) = 1 ⊕{1, a} = (1 ⊕ 1) ∪ (1 ⊕ a) = (1 ⊕ 1) ∪{1, a} hence 0 ∈ 1 ⊕ 1 = {a, 1}, which is impossible. thus 1 ⊕ 1 6= {1, a} and so by lemma 3.5(v), 1 ⊕ 1 = {1}, {0, 1} or m . (vi). let a ⊕ a = m and 1 ⊕ a = {0, 1}. then by (hm v1), (1 ⊕ 1) ⊕ a = 1 ⊕ (1 ⊕ a) = 1 ⊕{0, 1} = (0 ⊕ 1) ∪ (1 ⊕ 1) = m. now, if 1⊕1 = {1}, then 1⊕a = (1⊕1)⊕a = m , which is a contradiction. hence 1 ⊕ 1 6= {1} and so by theorem 3.5(v), 1 ⊕ 1 = {0, 1},{a, 1} or m 23 r. a. borzooei, a. radfar theorem 3.11. there are 24 non-isomorphic proper hyper m v -algebras of order 3 such that 0 ⊕ 1 = m . proof. by lemma 3.5 (iii), a ⊕ a = {1}, {0, 1}, {1, a} or m . if a ⊕ a = {1} , then by lemma 3.10 (ii), a ⊕ 1 = 1 ⊕ 1 = m and so we must investigate the following table, which is a hyper m v -algebra. ⊕10 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {0, a, 1} if a ⊕ a = {0, 1}, then by lemma 3.10 (iii), a ⊕ 1 = {a, 1} or m and if a ⊕ 1 = {a, 1}, then 1 ⊕ 1 = {1}, {0, 1} or m . thus we must investigate the following 3 cases which all of them are hyper m v -algebras. ⊕11 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, 1} {a, 1} 1 {0, a, 1} {a, 1} {1} ⊕12 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, 1} {a, 1} 1 {0, a, 1} {a, 1} {0, 1} ⊕13 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, 1} {a, 1} 1 {0, a, 1} {a, 1} {0, a, 1} if a ⊕ 1 = m , then by lemma 3.5 (v), 1 ⊕ 1 = {1}, {0, 1}, {1, a} or m . hence we must investigate the following 4 cases which all of them are hyper m v -algebras. ⊕14 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {1} ⊕15 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {0, 1} ⊕16 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {a, 1} ⊕17 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {0, a, 1} 24 classification of hyper m v -algebras of order 3 now, if a ⊕ a = {a, 1}, then by lemma 3.10 (iv), a ⊕ 1 = {0, 1} or m and if a ⊕ 1 = {0, 1}, then 1 ⊕ 1 = {a, 1} or m . hence we must investigate the following 2 cases which both of them are hyper m v -algebras. ⊕18 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {a, 1} {0, 1} 1 {0, a, 1} {0, 1} {a, 1} ⊕19 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {a, 1} {0, 1} 1 {0, a, 1} {0, 1} {0, a, 1} if a ⊕ 1 = m , then by lemma 3.5 (v), 1 ⊕ 1 = {1}, {0, 1}, {a, 1} or m and so we must investigate the following 4 cases which all of them are hyper m v -algebras. ⊕20 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {a, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {1} ⊕21 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {a, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {0, 1} ⊕22 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {a, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {a, 1} ⊕23 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {a, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {0, a, 1} now, let a ⊕ a = m . then by lemma 3.10 (v), a ⊕ 1 = {1, a}, {0, 1} or m . if a ⊕ 1 = {1, a}, then 1 ⊕ 1 = {1},{0, 1} or m . thus we must investigate the following 3 cases which all of them are hyper m v -algebras. ⊕24 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, a, 1} {a, 1} 1 {0, a, 1} {a, 1} {1} ⊕25 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, a, 1} {a, 1} 1 {0, a, 1} {a, 1} {0, 1} ⊕26 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, a, 1} {a, 1} 1 {0, a, 1} {a, 1} {0, a, 1} also by lemma 3.10 (v), if a ⊕ 1 = {0, 1}, then 1 ⊕ 1 = {0, 1},{a, 1} or m . hence we must investigate the following 3 cases which all of them are hyper 25 r. a. borzooei, a. radfar m v -algebras. ⊕27 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, a, 1} {0, 1} 1 {0, a, 1} {0, 1} {0, 1} ⊕28 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, a, 1} {0, 1} 1 {0, a, 1} {0, 1} {a, 1} ⊕29 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, a, 1} {0, 1} 1 {0, a, 1} {0, 1} {0, a, 1} finally, if a⊕1 = m , then by lemma 3.5 (v), 1⊕1 = {1},{0, 1},{a, 1} or m . hence we must investigate the following 4 cases which all of them are hyper m v -algebras. ⊕30 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, a, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {1} ⊕31 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, a, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {0, 1} ⊕32 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, a, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {a, 1} ⊕33 0 a 1 0 {0} {0, a} {0, a, 1} a {0, a} {0, a, 1} {0, a, 1} 1 {0, a, 1} {0, a, 1} {0, a, 1} corolary 3.12. there are 33 non-isomorphic hyper m v -algebras of order 3. proof. by theorems 3.3, 3.7, 3.9 and 3.11, we have 33 non-isomorphic hyper m v -algebras of order 3. references [1] c. c. chang, algebraic analysis of many valued logics, trans. amer. math. soc, 88 (1958), 467–490. [2] g. georgescu, a. iorgulescu, pseudo-m v algebras, multi valued logic, 6, (2001), 95-135. 26 classification of hyper m v -algebras of order 3 [3] s. ghorbani, e. eslami and a. hasankhani, quotient hyper mvalgebras, scientiae mathematicae japonicae, 3 (2007) 371–386. [4] sh. ghorbani, a. hasankhani, and e. eslami, hyper mv-algebras, setvalued math. appl, 1 (2008), 205–222. [5] f. marty, sur une generalization de la notion de groupe, 8th congress math. scandin aves, stockholm (1934), 45–49. [6] d. mundici, interpretation of af c∗-algebras in lukasiewicz sentential calculus, j. funct. anal, 65, (1986), 15–63. 27 28 microsoft word r.m.6 cap.8.doc microsoft word capitolo intero n 4.doc microsoft word documento1 microsoft word capitolo intero n 3.doc microsoft word documento1 microsoft word documento1 ratio mathematica volume 39, 2020, pp. 237-252 on a class of sets between a-open sets and gδ-open sets jagadeesh b.toranagatti∗ abstract in this paper, a new class of sets called da-open sets are introduced and investigated with the help of gδ-open and δ-closed sets. relationships between this new class and other related classes of sets are established and as an application da-continuous and almost dacontinuous functions have been defined to study its properties in terms of da-open sets. finally, some properties of da-closed graph and (d,a)-closed graph are investigated. keywords:a-open set,δ-open set,gδ-open,da-open set,da-closed set. 2010 ams subject classifications: 54a05, 54a10.1 ∗department of mathematics, karnatak university’s karnatak college, dharwad 580001, karnataka, india; jagadeeshbt2000@gmail.com 1received on august 30th, 2020. accepted on december 17th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.535. issn: 1592-7415. eissn: 2282-8214. c©toranagatti. this paper is published under the cc-by licence agreement. 237 jagadeesh b.toranagatti 1 introduction the concept of generalized open sets introduced by levine[levine, 1970] plays a significant role in general topology. the study of generalized open sets and its properties found to be useful in computer science and digital topology[khalimsky et al., 1990, kovalevsky, 1994, smyth, 1995]. since professor elnaschie has recently shown in [el naschie, 1998, 2000, 2005] that the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and �∞ theory.so,the fuzzy topological version of the notions and results introduced in this paper are very important. recently, ekici [ekici, 2008] introduced the notion of a-open sets as a continuation of research done by velicko [velicko, 1968] on the notion of δ-open sets.dontchev et al., introduced gδ-closed sets and gδ-continuity.in this paper,new generalizations of a-open sets by using gδ-open and δ-closed sets called da-open sets are presented. also da-continuous functions,almost da-continuous functions,da-closed graphs and (d,a)-closed graphs have been defined to study its properties in terms of da-open sets. 2 prerequisites, definitions and theorems in what follows,spaces always mean topological spaces on which no separation axioms are assumed unless explicitly stated and f:(x,τ) → (y,η) or simply f:x →y denotes a function f of a space (x,τ) into a space (y,η). the δ-closure of a subset a of x is the intersection of all δ-closed sets containing a and is denoted by clδ(a). definition 2.1. in (x,τ),let n ⊂ x.then n is called: (i)regular closed[stone, 1937] (resp.,a-closed[ekici, 2008], δ-preclosed[raychaudhuri and mukherjee, 1993], e∗-closed[ekici, 2009], δ-semiclosed[park et al., 1997], β-closed[abd el-monsef, 1983], semiclosed[levine, 1963], preclosed[mashhour, 1982]) if n = cl(int(n)) (resp., cl(int(clδ(n))) ⊂ n, cl(intδ(n)) ⊂ n, int(cl(intδ(n)) ⊂ n, int(clδ(n)) ⊂ n, int(cl(int(n)) ⊂ n, int(cl(n)) ⊂ n, cl(int(n)) ⊂ n). (ii) δ-closed [velicko, 1968] if n = clδ(n) where clδ(n) = {p∈x:int(cl(o))∩n6=φ,o∈τ and p∈o}. (iii)generalized δ-closed (briefly,gδ-closed)[dontchev et al., 2000] if cl(n)) ⊂ g whenever n ⊂ g and g is δ-open in x. (iv)generalized closed (briefly,g-closed)[levine, 1970] if cl(n)) ⊂ g whenever n ⊂ g and g is open in x. the complements of the above mentioned closed sets are their respective open sets. 238 on a class of sets between a-open sets and gδ-open sets the set of all regular open (resp.,δ-open, β-open, δ-preopen, preopen, semiopen, δ-semiopen,e∗-open,gδ-open and a-open) sets of (x,τ) is denoted by ro(x) (resp.δo(x), βo(x), δpo(x), po(x), so(x), δso(x), e∗o(x), gδo(x) and ao(x)). the a-closure[ekici, 2008](resp, gδ-closure,δ-closure) of a set n is the intersection of all a-closed(resp, gδ-closed,δ-closed) sets containing n and is denoted by a-cl(n) (resp., clgδ(n),clδ(n)). the a-interior[ekici, 2008](resp,gδ-interior,δinterior) of a set n is the union of all a-open(resp, gδ-open,δ-open) sets contained in m and is denoted by a-int(m)(resp, intgδ(m),intδ(m)) definition 2.2. [ekici, 2005] a topological space (x,τ) is said to be: (1) r-t1 if for each pair of distinct points x and y of x, there exist regular open sets u and v such that x ∈ u, y /∈u and x /∈ v, y ∈ v. (2) r-t2 if for each pair of distinct points x and y of x, there exist regular open sets u and v such that x ∈ u, y ∈ v and u∩v =φ . theorem 2.1. let c and d be subsets of a topological space (x,τ).then (i)if c is gδ-closed,then clgδ(c) = c. (ii) if c⊂d,then clgδ(c)⊂ clgδ(d). (iv) x ∈clgδ(c) if and only if for each gδ-open set o containing x,o ∩ c6=φ, (v)clgδ(c)∪clgδ(d)⊂ clgδ(a∪d). (vi)clgδ(c∩d)⊆clgδ(c)∩clgδ(d). 3 da-open sets. definition 3.1. a subset m of a topological space (x,τ) is said to be: (1) da-open if m ⊂ intgδ(clδ(intgδ(m)), (2) da-closed if clgδ(intδ(clgδ(m))⊂m. the collection of all da-open(resp,da-closed) sets in (x,τ) is denoted by dao(x) (resp,dac(x)). theorem 3.1. let (x,τ) be a space.then for any n⊂x, (i) n∈δo(x) implies n∈ao(x)[ekici, 2008]. (ii) n∈δo(x) implies n ∈gδo(x)[dontchev et al., 2000]. (iii)n∈go(x) implies n ∈gδo(x)[dontchev et al., 2000]. (iv) n ∈ao(x) implies n∈dao(x). (v) n∈gδo(x) implies n∈dao(x). proof: (iv) since δo(x)⊂gδo(x), intδ(n) ⊂ intgδ(n). now,let n∈ao(x), then n ⊂ int(cl(intδ(n)). therefore, n ⊂ int(cl(intδ(n))=intδ(cl(intδ(n))⊂intgδ(clδ(intgδ(n)). hence n ∈dao(x). (v) suppose n is gδ-open. then intgδ(n)=n. 239 jagadeesh b.toranagatti therefore, intgδ(n)⊂ clδ(intgδ(n).then n=intgδ(n)=intgδ(intgδ(n)) ⊂ intgδ(clδ(intgδ(n)). hence n ∈dao(x). remark 3.1. the following diagram holds for any subset of a space (x,τ). open set ←− δ-open set −→ a-open set ↙ ↓ ↓ da-open set ↗ g-open set −→ gδ-open set none of these implications is reversible example 3.1. let x={p,q,r,s} and τ={x,φ,{p},{q},{p,q},{p,r}.{p,q,r}},then ao(x)={x,φ,{q},{p,r},{p,q,r}} gδo(x)={x, φ,{p},{q},{r},{p,q},{p,r}{q,r},{p,q,r}}. dao(x)={x, φ,{p},{q},{r},{p,q},{p,r},{q,r}{p,q,r}{p,q,s},{q,r,s}}. therefore, {q,r,s}∈dao(x) but {q,r,s}/∈ao(x) and {q,r,s}/∈ gδo(x). lemma 3.1. if there exists a m ∈ gδo(x) such that m ⊂ n ⊂intgδ(clδ(m)),then n is da-open. proof: since m is gδ-open, intδg(m)=m. therefore, intgδ(clδ(intgδ(n)) ⊃intgδ(clδ(intgδ(m)) = intgδ(clδ(m)) ⊃ n. hence n is da-open. converse of the lemma 3.1 is not true as shown in example 3.1. example 3.2. in example 3.1, {p,q,r}∈dao(x) and {p,r}∈gδo(x) but {p,r}⊆ {p,q,r} 6⊆ intgδ(clδ({p,r}))={p,r} . lemma 3.2. for a family { bλ:λ∈∧} of subsets of a space (x,τ),the following hold: (1) clgδ( ⋂ {bλ:λ∈∧}) ⊂ ⋂ {clgδ(bλ):λ∈∧}. (2) clgδ( ⋃ {vλ:λ∈∧}) ⊃ ⋃ {clgδ(bλ):λ∈∧}. (3) clδ( ⋂ {bλ:λ∈∧}) ⊂ ⋂ {clδ(bλ):λ∈∧}. (4) clδ( ⋃ {bλ:λ∈∧}) ⊃ ⋃ {clδ(bλ):λ∈∧} theorem 3.2. if {gα:λ∈∧} is a collection of da-open sets in a space (x,τ),then⋃ α∈∧ gα is a da-open set in (x,τ) : proof: since each gαis da-open, gα ⊂ intgδ(clδ(intgδ(gα)) for each α∈∧ and hence ⋃ α∈∧ gα ⊂ ⋃ α∈∧ intgδ(clδ(intgδ(gα))⊂intgδ(clδ(intgδ( ⋃ α∈∧ gα)). thus ⋃ α∈∧ gα is da-open. 240 on a class of sets between a-open sets and gδ-open sets corolary 3.1. if {fα:α∈∧} is a collection of da-closed sets in a space (x,τ),then⋂ α∈∧ fα is a da-closed set in (x,τ) remark 3.2. m and n ∈ dao(x) 6⇒ m ∩ n ∈ dao(x) as seen from example 3.1, where both m = {q,r,s} and n = {p,q,s}∈ dao(x) but m ∩ n = {q,s} /∈ dao(x). corolary 3.2. if m∈ dao(x) and b∈ao(x),then m∪b∈ dao(x). proof:follows from theorem 3.1(iv) and theorem 3.2 corolary 3.3. if m∈ dao(x) and b∈gδo(x),then m∪b∈ dao(x). proof:follows from theorem 3.1(v) and theorem 3.2 definition 3.2. in (x,τ),let m ⊂ x. (1)the da-interior of m, denoted by intda (m) is defined as intda (m)= ⋃ {g:g⊆m and m∈dao(x)}; (2)the da-closure of m, denoted by clda (m) is defined as clda (a)= ⋂ {f:m⊆f and f∈dac(x)}. theorem 3.3. in (x,τ),let m, n,f ⊂ x.then: (1)m ⊂ clda (m)⊂acl(m), clda (m)⊂clgδ(m). (2) clda (m) is a da-closed set. (3) if f is a da-closed set, and f ⊃ m,then f ⊃ clda (m). i.e.,clda (m) is the smallest da-closed set containing m. (4)m is da-closed set if and only if clda (m)=m. (5) clda (cl d a (m)) = cl d a (m). (6)m ⊆ n implies clda (m) ⊆ clda (n). (7)p ∈clda (m) if and only if for each da-open set v containing p,v ∩ m 6=φ. (8) clda (m) ∪ clda (n) ⊂ clda (m ∪ n). (9) clda (m ∩ n) ⊂ clda (m) ∩ clda (n). proof: (1)it follows from theorem 3.1(iv) and (v) (2)it follows from definition 3.2 and corollary 3.1 (3)let f be a da-closed set,containing m.clda (m) is the intersection of da-closed sets containing m, and f is one among these;hence f ⊃ clda (m). (4) let m be da-closed,then by definition 3.2(2),clda (m)=m. conversely,let clda (m)=m. then by (2) above,m is da-closed. (5)it follows from (2) and (4). (6) obvious. (7) p /∈clda (m) ⇔ (∃ g∈dac(x))(m⊂g)(p /∈g) ⇔ (∃ g∈dac(x))(m⊂g)(p ∈gc) ⇔ (∃ gc∈dao(x))(m∩gc=φ)(p ∈gc) ⇔ (∃ gc∈dao(x,p))(m∩gc=φ) 241 jagadeesh b.toranagatti i.e.,(∃ u(=gc)∈ dao(x,p))(m∩u=φ) (8) and (9) follows from (6). remark 3.3. (1) clda (m) ∪ clda (n) 6= clda (m ∪ n), in general, as seen from example 3.1 where m = {p}, n = {r} and m ∪ n = {p,r}.then clda (m)={p}, clda (n)={r},clda (m)∪clda (n)={p,r} and clda (m∪n)={p,r,s}; (2) clda (m ∩ n)6= clda (m) ∩ clda (n), in general,as seen from example 3.1 where,m = {p,q,r}, n = {s} and m∩n = φ.then clda (m) = x, clda (n) = {s}, clda (m)∩clda (n) = {s} and clda (m∩n)=φ lemma 3.3. in (x,τ),let m ⊂ x.then (1) clda (x\m) = x\intda (m), (2) intda (x\m) = x\clda (m). theorem 3.4. in (x,τ),let m,n,g ⊂ x, (1)aint(m) ⊆ intda (m)⊆m, intgδ(m)⊆intda (m). (2) intda (m) is a da-open set. (3) if g is a da-open set, and g ⊂ m,then g ⊂ intda (m). i.e.,intda (m) is the largest da-open set contained in m. (4)m is da-open set if and only if intda (m)=m. (5) intda (int d a (m)) = int d a (m). (6)m ⊆ n implies intda (m) ⊆ intda (n). (7) p ∈ intda (m) if and only if there exists da-open set n containing p such that n ⊆ m. (8) intda (m ∩ n)⊆ intda (m) ∩ intda (n). (9) intda (m) ∪ intda (n) ⊆intda (m ∪ n). proof:similar to the proof of theorem 3.3 remark 3.4. (8)intda (m ∩ n)6= intda (m) ∩ intda (n), in general, as seen from example 3.1,where m = {p,q,s}, n = {q,r,s} and m ∩ n = {q,s}.then intda (m) = {p,q,s}, intda (n) = {q,r,s}, intda (m) ∩ intda (n) = {q,s} and intda (m∩n) = {q}. (9) intda (m) ∪ intda (n) 6= intda (m ∪ n),in general, as seen from example 3.1, where m = {p,q,r}, n = {s} and m ∪ n = x.then intda (m) = {p,q,r}, intda (n) = φ, intda (m) ∪ intda (n) = {p,q,r} and intda (m ∪ n) = x. lemma 3.4. in (x,τ),let m ⊂ x. then (1)m is da-open if and only if m = m ∩ intgδ(clδ(intgδ(m)). (2)m is da-closed if and only if m = m∪ clgδ(intδ(clgδ(m)). proof:(1) let m be an da-open. then, m⊆intgδ(clδ(intgδ(m)) implies m∩ intgδ(clδ(intgδ(m))=m. conversely,let m = m∩ intgδ(clδ(intgδ(m)) implies m ⊂ intgδ(clδ(intgδ(m)). (2)it follows from (1) 242 on a class of sets between a-open sets and gδ-open sets lemma 3.5. in (x,τ),let m ⊂ x. then (i)m ∩ intgδ(clδ(intgδ(m)) is da-open (ii)m∪ clgδ(intδ(clgδ(m)) is da-closed. proof: (i) intgδ(clδ(intgδ(m ∩ intgδ(clδ(intgδ(m)))))) = intgδ(clδ(intgδ(a)∩ intgδ(clδ(intgδ(m))))) = intgδ(clδ(intgδ(m))). this implies that m ∩ intgδ(clδ(intgδ(m))) = m ∩ intgδ(clδ(intgδ(m ∩ intgδ(clδ(intgδ(m)))))) ⊆ intgδ(clδ(intgδ(m ∩ intgδ(clδ(intgδ(m)))))) . therefore m ∩ intgδ(clδ(intgδ(m))) is da-open. (ii) from (i) we have x\(m∪clgδ(intδ(clgδ(m))) = (x\m) ∩ clgδ(intδ(clgδ(x\m))) is da-open so that m ∪clgδ(intδ(clgδ(m))) is da-closed. lemma 3.6. in (x,τ),let m ⊂ x. then (i)intda (m)=m ∩ intgδ(clδ(intgδ(m)). (ii)clda (m)=m∪ clgδ(intδ(clgδ(m)). proof:(i)let n=intda (m),then n⊂m.since n is da-open,n⊂intgδ(clδ(intgδ(n)) ⊂intgδ(clδ(intgδ(m)).then n⊂m∩intgδ(clδ(intgδ(m))⊂m.therefore,by lemma 3.5, it follows that m∩intgδ(clδ(intgδ(m)) is a da-open set contained in m. but intda (m) is the largest da-open set contained in m it follows that m∩intgδ(clδ(intgδ(m))⊂ intda (m)=n.then n=m∩intgδ(clδ(intgδ(m)). therefore,intda (m)=m ∩ intgδ(clδ(intgδ(m)). (ii)it follows from (i) 4 da-continuous functions. definition 4.1. a function f:(x,τ) → (y,η) is said be a da-continuous if for each p∈x and each n∈o(y,f(p)), there exists m ∈ dao(x,p) such that f(m)⊂ n. theorem 4.1. for a function f:(x,τ) → (y,η),the following are equivalent (1)f is da-continuous; (2)for each n∈o(y),f−1(v)∈dao(x). proof:(1)−→(2)let n∈o(y) and p∈f−1(n). since f(p) ∈ n,then by(1),there exists mp ∈ dao(x,p) such that f(mp) ⊂ n.it follows that f−1(n)=∪{mp: p∈f−1(n)}∈dao(x), by theorem 3.2 . (2)−→(1) let p ∈ x and n ∈o(y,f(p)).then,by (2),f−1(n)∈dao(x,p). take m = f−1(n), then f(m) ⊂ n. corolary 4.1. a function f:(x,τ) → (y,η) is da-continuous if and only if f−1(f)∈dac(x) for each f∈c(y). 243 jagadeesh b.toranagatti remark 4.1. the following implications hold for a function f:(x,τ) → (y,η): continuity ←− δ-continuity −→ a-continuity ↙ ↓ ↓ da-continuity ↗ g-continuity −→ gδ-continuity example 4.1. consider (x,τ) as in example 3.1 and η={x,φ,{p},{q},{p,q},{p,q,r}}. define f:(x,σ)→(x,η) by f(p)=s,f(q)=p,f(r)=q and f(s)=r.then f is da-continuous but neither a-continuous nor gδ-continuous since {p,q,r} is open in (x,η), f−1({p,q,r}) = {q,r,s}∈dao(x) but {q,r,s}/∈ao(x) and {q,r,s}/∈ gδo(x). the other examples are shown in[3,5,21] theorem 4.2. the following conditions are equivalent for a function f:(x,τ) → (y,η): (1) f is da-continuous; (2) for each subset n of y, clgδ(intδ(clgδ(f−1(n))) ⊂ f−1(cl(n); (3)for each subset n of y, f−1(int(n)) ⊂ intgδ(clδ(intgδ(f−1(n)); (4)for each subset n of y,clda (f −1(n)) ⊂ f−1(cl(n)); (5)for each subset m of x,f(clda (m)) ⊂ cl(f(m)); (6)for each subset n of y, f−1(int(n)) ⊂ intda (f−1(n)). proof: (1)→(2) let n ⊂ y.then by (1),f−1(cl(n)) ∈ dac(x) implies f−1(cl(n)⊃clgδ(intδ(clgδ(f−1(cl(n)))⊃ clgδ(intδ(clgδ(f−1(n))). (2)→(3).replace n by y\n in (2), we have clgδ(intδ(clgδ(f−1(y\n)))⊂f−1(cl(y\n), and therefore f−1(int(n)) ⊂ intgδ(clδ(intgδ(f−1(n)) for each subset n of y. (3)→(1). clear (1)→(4). let n ⊂ y .then by (1), f−1(cl(n))∈dac(x). thus clda (f −1(n)) ⊂ clda (f−1(cl(n))=f−1(cl(n) by theorem 3.3(4). (4)→(1). let n ∈c(y).then by (4), clda (f −1(n)) ⊂ f−1(cl(n)=f−1(n) implies clda (f−1(n))=f−1(n). then by theorem 3.3(4), f−1(n) ∈ dac(x). (4)→(5).let m ⊂ x.then f(m) ⊂ y.by (4), we have f−1(cl(f(m))) ⊃ clda (f−1(f(m))) ⊃ clda (m). therefore, f(clda (m)) ⊂ f(f−1(cl(f(m))) ⊂ cl(f(m). (5)→(4).let n ⊂ y and m=f−1(n) ⊂ x.then by (5), f(clda (f −1(n))) ⊂ cl(f(f−1(n)) ⊂ cl(n) implies clda (f−1(n)) ⊂ f−1(cl(n)). (4)→(6).replace n by y \n in (4), we get clda (f −1(y\n)) ⊂ f−1(cl(y\n)) implies clda (x\f−1(n)) ⊂ f−1(y\int(n)) therefore,f−1(int(n)) ⊂ intda (f−1(n)) for each subset n of y. 244 on a class of sets between a-open sets and gδ-open sets (6)→(1).let g⊂y be open.then f−1(g)=f−1(int(g)) ⊂ intda (f−1(g) implies intda (f −1(g)=f−1(g).so by theorem 3.4(4),f−1(g)∈dao(x). definition 4.2. two non-empty subsets a and b of a topological space (x,τ) are said to be da-separated if there exist two da-open sets g and h,such that a⊂g,b⊂h, a∩h=φ and b∩g=φ. definition 4.3. two non-empty subsets a and b of a topological space (x,τ) are said to be strongly da-separated if there exist two da-open sets u and v,such that a⊂u,b⊂v and u∩v=φ. definition 4.4. a topological space (x,τ) is said to be (1) da-t2 if any two distinct points are strongly da-separated in (x,τ) (2) da-t1 if every pair of distinct points is da-separated in (x,τ). remark 4.2. the following implications are hold for a topological space (x,τ) a-t2 −→da-t2←− t2 ↓ ↓ ↓ a-t1 −→da-t1←− t1 theorem 4.3. if an injective function f:(x,τ) → (y,η) is da-continuous and (y,η) is t1, then (x,τ) is da-t1. proof: let (y,σ) be t1 and p,q∈x with p 6=q. then there exist open subsets g, h in y such that f(p) ∈ g, f(q) /∈ g, f(p) /∈ h and f(q) ∈ h. since f is da-continuous, f−1(g) and f−1(h) ∈ dao(x) such that p ∈f−1(g), q /∈f−1(g), p /∈f−1(h) and q ∈ f−1(h). hence,(x,σ) is da-t1 . theorem 4.4. if an injective function f: (x,τ) → (y,η) is da-continuous and (y,η) is t2, then (x,τ) is da-t2. proof: similar to the proof of theorem 4.3 recall that for a function f:(x,τ) → (y,η), the subset gf ={(x,f(x)):x ∈x}⊂ x×y is said to be graph of f. definition 4.5. a graph gf of a function f:(x,τ) → (y,η) is said to be da-closed if for each (p,q) /∈ gf , there exist u∈dao(x,p) and v∈o(y,q) such that (u×v)∩ gf = φ. as a consequence of definition 4.5 and the fact that for any subsets c ⊂ x and d ⊂ y, (c×d)∩ gf =φ if and only if f(c)∩d = φ,we have the following result. lemma 4.1. for a graph gf of a function f:(x,τ) → (y,η), the following properties are equivalent: (1)gf is da-closed in x×y; (2)for each (p,q) /∈gf , there exist u∈dao(x,p) and v∈o(y,q) such that f(u)∩v = φ. 245 jagadeesh b.toranagatti theorem 4.5. if f:(x,τ) → (y,η) is da-continuous and (y,η) is t2 , then gf is da-closed in x×y. proof: let (p,q) /∈gf , f(p) 6=q. since y is t2, there exist v,w ∈o(y) such that f(p)∈ v, q∈w and v∩w=φ. since f is da-continuous, f−1(v)∈dac(x,p).set u =f−1(v), we have f(u)⊂ v. therefore, f(u)∩w=φ and gf is da-closed in x×y theorem 4.6. let f:(x,τ) → (y,η) have a da-closed graph gf . if f is injective, then (x,τ) is da-t1. proof:let x1,x2∈x with x1 6=x2.then f(x1)6=f(x2) as f is injective so that (x1,f(x2)) /∈gf .thus there exist u∈dao(x,x1) and v∈o(y,f(x2)) such that f(u)∩v = φ.then f(x2)/∈f(u) implies x2 /∈u and it follows that x is da-t1. theorem 4.7. let f:(x,τ) → (y,η) have a da-closed graph gf . if f is surjective, then (y,η) is t1. proof:let y1,y2∈y with y1 6=y2.since f is surjective,f(x)=y2 for some x∈x and (x,y2)/∈gf .by lemma 4.1,there exist u∈dao(x,x) and v∈o(y,y1) such that f(u)∩v = φ.it follows that y2 /∈v.hence y is t1. theorem 4.8. let f:(x,τ) → (y,η) have a da-closed graph gf . if f is surjective, then (y,η) is da-t1. proof:similar to the proof of theorem 4.7 corolary 4.2. let f:(x,τ) → (y,η) have a da-closed graph gf . if f is bijective, then both (x,τ) and (y,η) are da-t1 proof:follows from theorems 4.6 and 4.8 definition 4.6. a graph gf of a function f:(x,τ) → (y,η) is said to be (d,a)closed if for each (p,q) /∈ gf , there exist u∈dao(x,p) and v∈ao(y,q) such that (u×acl(v))∩ gf = φ. lemma 4.2. for a graph gf of a function f:(x,τ) → (y,η), the following properties are equivalent: (1)gf is da-closed in x×y; (2)for each (p,q) /∈gf , there exist u∈dao(x,p) and v∈ao(y,q) such that f(u)∩acl(v)) = φ. theorem 4.9. let m ⊂ x.then x∈ a-cl(m) if and only if g ∩ m 6= φ, for every a-open set g containing x. proof:similar to the proof of theorem 3.3(7) theorem 4.10. let f:(x,τ) → (y,η) have a (d,a)-closed graph gf . if f is surjective, then (y,η) is a-t2(resp,a-t1). proof:let y1,y2∈y with y1 6=y2.since f surjective, f(x1)=y1 x1∈x and hence (x1,y2)/∈gf . by lemma 4.2,there exist e∈dao(x,x1) and f∈ao(y,y2) such that f(e)∩ acl(f) = φ. now, x1∈e implies f(x1)=y1∈f(e) so that y1 /∈acl(f).by theorem 4.9,there exists d∈ao(y,y1) such that d∩f=φ.hence y is a-t2. 246 on a class of sets between a-open sets and gδ-open sets theorem 4.11. let f:(x,τ) → (y,η) have a (d,a)-closed graph gf . if f is surjective, then (y,η) is da-t2(resp,da-t1). proof:similar to the proof of theorem 4.10 theorem 4.12. let f:(x,τ) → (y,η) have a (d,a)-closed graph gf . if f is injective, then (x,τ) is da-t1. proof:similar to the proof of theorem 4.6 corolary 4.3. let f:(x,τ) → (y,η) have a (d,a)-closed graph gf . if f is bijective, then both (x,τ) and (y,η) are da-t1 proof:follows from theorems 4.11 and 4.12 5 almost da-continuous functions. definition 5.1. a function f:(x,τ) → (y,η) is said to be almost da-continuous if for each point p ∈ x and each open subset v of y containing f(p), there exists u ∈ dao(x,p) such that f(u) ⊂ int(cl(v)). theorem 5.1. if f:(x,τ) → (y,η) is da-continuous function , then f is an almost da-continuous,but not conversely. proof:obvious example 5.1. consider (x,τ) and (x,η) as in 4.1. define f:(x,τ) → (x,η) by f(p)=p,f(q)=s,f(r)=q and f(s)=r then f is almost da-continuous but not da-continuous since {p,q,r} is open in (x,η), f−1({p,q,r})={p,r,s}/∈dao(x,τ) definition 5.2. [noiri and popa, 1998] a space x is said to be semi-regular if for any open set u of x and each point x ∈ u there exists a regular open set v of x such that x ∈ v ⊂ u. theorem 5.2. if f:(x,τ) → (y,η) is an almost da-continuous function and y is semi-regular, then f is da-continuous. proof: let p ∈ x and let v ∈ o(y,f(p)). by the semi-regularity of y , there exists g∈ro(y,f(p)) such that g ⊂ v . since f is almost da-continuous, there exists u ∈ dao(x, x) such that f(u) ⊂ int(cl(g)) = g ⊂ v and hence f is da-continuous. lemma 5.1. let (x,τ) be a space and let a be a subset of x. the following statements are true: (1) a ∈ po(x) if and only if scl(a) = int(cl(a)) [janković, 1985]. (2) a ∈ βo(x) if and only if cl(a) is regular closed [abd el-monsef, 1983]. 247 jagadeesh b.toranagatti theorem 5.3. let f:(x,τ) → (y,η) be a function. then the following conditions are equivalent: (1) f is almost da-continuous; (2) for every n∈ro(y), f−1(n)∈dao(x); (3) for every m∈rc(y), f−1(m)∈dac(x); (4) for each subset c of x, f(clda (c)) ⊂ clδ(f(c)); (5)for each subset d of y, clda (f −1(d)) ⊂ f−1(clδ(d)); (6)for every g∈δc(y), f−1(g)∈dac(x); (7)for every h∈δo(y), f−1(h)∈dao(x); (8) for every n∈o(y), f−1(int(cl(n)∈dao(x); (9) for every m∈c(y), f−1(cl(int(m)∈dac(x); (10) for every n∈βo(y), clda (f−1(n)) ⊂ f−1(cl(n)); (11) for every m∈βc(y), f−1(int(m)) ⊂ intda (f−1(m)); (12) for every m∈sc(y), f−1(int(m)) ⊂ intda (f−1(m)); (13) for every n∈so(y), clda (f−1(n)) ⊂ f−1(cl(n)); (14) for every m∈po(y), f−1(m) ⊂ intda (f−1(int(cl(m)); (15) for each p∈ x and each n∈o(y,f(p)), there exists m ∈ dao(x,p) such that f(m) ⊂ scl(n); (16) for each p∈ x and each n∈ro(y,f(p)), there exists m ∈ dao(x,p) such that f(m) ⊂ n; (17) for each p∈ x and each n∈δo(y,f(p)), there exists m ∈ dao(x,p) such that f(m) ⊂ n. proof: (1)−→(2) similar to the proof of (1)−→(2) of theorem 4.1. (2)−→(3) it follows from the fact that f−1(y\f) = x \f−1(f). (3)−→(4) suppose that d∈ δc(y) such that f(c)⊂ d. observe that d = clδ(d) = ⋂ {f:d⊂f and f∈rc(y)} and so f−1(d) = ⋂ {f−1(f):d⊂f and f∈rc(y)}. by (3) and corollary 3.1,we have f−1(d)∈dac(x) and c⊂f−1(d). hence clda (c) ⊂f−1(d), and it follows that f(clda (c) ) ⊂ d. since this is true for any δ-closed set d containing f(c), we have f(clda (c))⊂ clδ(f(c)). (4)−→(5) let d ⊂ y, then f−1(d) ⊂ x. by (4), f(clda (f −1(d)))⊂ clδ(f(f−1(d)))⊂clδ(d). so that clda (f −1(d)) ⊂ f−1(clδ(d)). (5)−→(6) let g∈δc(y) then by (5), clda (f−1(g)) ⊂ f−1(clδ(g))=f−1(g). in consequence, clda (f −1(g))=f−1(g) and hence by theorem 3.3(4), f−1(g)∈dac(x). (6)−→(7):clear. (7)−→(1): let p∈ x and let o∈o(y,f(p)). set d = int(cl(o)) and c =f−1(d). since d∈ δo(y), then by (7), c = f−1(d) ∈ dao(x). now, f(p) ∈ o= int(o)⊂ int(cl(o)) = d it follows that p∈f−1(d)=c and f(c)=f(f−1(d)⊂d=int(cl(o). (2)←→(8): let n ∈o(y). since int(cl(n))∈ro(y),by (2), f−1(int(cl(n))∈dao(x). the converse is similar. (3)←→(9)it is similar to (8)←→(2). 248 on a class of sets between a-open sets and gδ-open sets (3)−→ (10): let n∈βo(y).then by lemma 5.1(2),cl(n) ∈ rc(y).so by(3),f−1(cl(n)) ∈dac(x) .since f−1(n) ⊂f−1(cl(n)) and by theorem 3.3(4),clda (f−1(n))⊂f−1(cl(n)). (10)−→ (11): and (12)−→ (13):follows from lemma 3.3 (11)−→ (12):it follows from the fact that sc(y)⊂βc(y) (13)−→ (3):it follows from the fact that rc(y)⊂so(y). (2)←→ (14): let n ∈po(y). since int(cl(n)) ∈ ro(y),then by (2), f−1(int(cl(n))) ∈ dao(x) and hence f−1(n) ⊂ f−1(int(cl(n))) = intda (f−1(int(cl(n)))). conversely,let n∈ro(y). since n ∈ po(y), f−1(n) ⊂ intda (f−1(int(cl(n)))) =intda (f−1(n)). in consequence, intda (f −1(n))=f−1(n) and by theorem 3.4, f−1(n) ∈ dao(x). (1)−→ (15): let p∈x and n∈o(y,f(p)). by (1), there exists m∈ dao(x,p) such that f(m) ⊂ int(cl(n)).since n∈po(y),by lemma 5.1, f(m) ⊂ scl(n). (15)−→ (16): let p∈ x and n∈ro(y,f(p)). since n∈o(y,f(p)) and by (15), there exists m∈dao(x,p) such that f(m)⊂ scl(n). since n ∈po(y), then by lemma 5.1, f(m) ⊂int(cl(n)) = n. (16)−→ (17):let p∈ x and v∈δo(y,f(p)). then, there exists g∈o(y.f(p))such that g ⊂ int(cl(g)) ⊂ n. since int(cl(g))∈ro(y,f(p)), by (16), there exists m∈ dao(x,p) such that f(m) ⊂ int(cl(g))⊂ n. (17)−→(1). let p∈ x and n∈o(y,f(p)). then int(cl(n))∈ δo(y,f(p)). by (17), there exists m∈ dao(x,p) such that f(m) ⊂ int(cl(n)). therefore,f is almost continuous theorem 5.4. if f:(x,τ) → (y,η) is an almost da-continuous injective function and (y,η) is r-t1 , then (x,σ) is da-t1 . proof: it is similar to the proof of theorem 4.3 theorem 5.5. if f:(x,τ) → (y,σ) is an almost da-continuous injective function and (y,σ) is r-t2 , then (x,τ) is da-t2 . proof: it is similar to the proof of theorem 4.4 lemma 5.2. [ayhan and ozkoç, 2016] let (x,τ) be a space and let a be a subset of x. then: a ∈e∗o(x) if and only if clδ(a) is regular closed. theorem 5.6. for a function f:(x,τ) → (y,η),the following are equivalent: (a) f is almost da-continuous; (b) for every e∗-open set n in y,f−1(clδ(n)) is da-closed in x; (c) for every δ-semiopen subset n of y,f−1(clδ(n)) is da-closed set in x; (d) for every δ-preopen subset n of y,f−1(int(clδ(n))) is da-open set in x; (e) for every open subset n of y,f−1(int(clδ(n))) is da-open set in x; (f) for every closed subset n of y,f−1(cl(intδ(a))) is da-closed set in x . proof: (a)→(b):let n∈e∗o(y) then by lemma 5.2,clδ(n)∈rc(y). 249 jagadeesh b.toranagatti by (a),f−1(clδ(n))∈dac(x). (b)→(c):obvious since δso(y)⊂ e∗o(y). (c)→(d):let n ∈δpo(y),then intδ(y\n)∈δ-so(y).by (c), f−1(clδ(intδ(y\n))∈dac(x) which implies f−1(int(clδ(n))∈dao(x). (d)→(e):obvious since o(y)⊂ δpo(y). (e)→(f):clear (f)→(a):let n∈ro(y).then n=int(clδ(n)) and hence y\n∈c(x). by (f), f−1(y\n)=x\f−1(int(clδ(n)))=f−1(cl(intδ(y\n))∈dac(x). thus f−1(n)∈dao(x). lemma 5.3. [ayhan and ozkoç, 2016] let (x,τ) be a space and let a ⊂ x. the following statements are true: (a) for each a∈e∗o(x), a-cl(a)=clδ(a) (b)for each a∈δso(x), δ-pcl(a)=clδ(a). (c)for each a∈δpo(x),δ-scl(a)=int(clδ(a)). as a consequence of theorem 5.6 and lemma 5.3, we have the following result: theorem 5.7. the following are equivalent for a function f:(x,τ) → (y,η): (a) f is almost da-continuous; (b) for every e∗-open subset g of y,f−1(a-cl(g)) is da-closed set in x; (c) for every δ-semiopen subset g of y,f−1(δ-pcl(g)) is da-closed set in x; (d) for every δ-preopen subset g of y,f−1(δ-scl(g))) is da-open set in x; references me abd el-monsef. β-open sets and β-continuous mappings. bull. fac. sci. assiut univ., 12:77–90, 1983. burcu sünbül ayhan and murad ozkoç. almost e-continuous functions and their characterizations. journal of nonlinear sciences & applications (jnsa), 9(12), 2016. julian dontchev, i arokiarani, and krishnan balachandran. on generalized deltaclosed sets and almost weakly hausdorff spaces. questions and answers in general topology, 18(1):17–30, 2000. erdal ekici. generalization of perfectly continuous, regular set-connected and clopen 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topology. theoretical computer science, 151(1):257–276, 1995. 251 jagadeesh b.toranagatti marshall harvey stone. applications of the theory of boolean rings to general topology. transactions of the american mathematical society, 41(3):375–481, 1937. nv velicko. h-closed topological spaces. amer. math. soc. transl., 78:103–118, 1968. 252 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 39, 2020, pp. 7-32 7 a simple goodness-of-fit test for continuous conditional distributions peter veazie* zhiqiu ye† abstract this paper presents a pragmatic specification test for conditional continuous distributions with uncensored data. we employ monte carlo (mc) experiments and the 2011 medical expenditure panel survey data to examine coverage and the power to discern deviations from the correct model specification in distribution and parameterization. we carry out mc experiments using 2000 runs for sample sizes 500 and 1000. the experiments show that the test has accurate coverage under correct specification, and that the test can discern deviations from the correct specification in both the distributional family and parameterization. the power increases as sample size increases. the empirical example shows the test’s ability to identify specific distributions from other candidates using real cost data. although the test can be used as a goodness-of-fit test for marginal distributions, it is particularly useful as an easyto-use test for conditional continuous distributions, even those with one observation per pattern of explanatory variables. keywords: goodness-of-fit test; model specification test; conditional continuous distributions. 2010 ams subject classification: 62f03‡ * university of rochester, rochester new york, usa; peter_veazie@urmc.rochester.edu. † university of rochester, rochester new york, usa; sophieye999@gmail.com. ‡ received on june 10th, 2020. accepted on december 17th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.524. issn: 1592-7415. eissn: 2282-8214. ©peter veazie et al. this paper is published under the cc-by licence agreement. p. veazie, z. ye 8 1 introduction to determine whether a probability model is statistically adequate for representing a data generating process (dgp), it is common to test whether the model fits with a data set produced by that process. the investigation into the model specification of a conditional distribution is fundamental for methods such as maximum likelihood estimation (mle), which is consistent and asymptotically efficient only if the distribution is correctly specified (amemiya, 1985). however, there are two key challenges for a general test of continuous conditional distribution models, if it is to be broadly adopted in applied sciences such as social and health sciences: first, is the sparse empirical information regarding the conditional distribution when patterns of the explanatory variables have few corresponding observations. second, is the ease of use: many researchers do not have the background, time, or inclination to engage in complicated programming in order to implement a statistical test—to be useful to such researchers a test must be easily implemented. regarding sparse information, consider the data shown in figure 1: although some data points appear close to each other, for most of the data there is no more than one observation at each value of x. consequently, the empirical distribution of random variable y conditioned on such a value for variable x is based on a trivial point mass. how then can we test a model of the conditional distribution of y for such sparse data? regarding ease of use, existing tests for conditional distributions require more mathematical and computational skill than many applied researchers may have to make their implementation generally accepted. some of these tests require the use of kernel or local polynomial functions with arbitrary smoothing parameters (zheng, 2000, fan et al., 2006). others, such as the conditional kolmogorov test, compare model and distribution functions additionally incorporating the empirical distribution functions of the conditioning set of variables (andrews, 1997). transformations to the unit interval have been applied to construct tests for goodness of fit such as the rincon-gallardo et al. test for multivariate normality (rincon-gallardo et al., 1979). however, their method is also technically difficult and computationally intense in general applications due to procedures involved in the transformation (o'reilly and quesenberry, 1973). additionally, some are dependent on the order of the data being transformed (o'reilly and stephens, 1982); therefore, researchers may obtain variant test results if the same data were ordered differently. what is needed for the applied researcher who does not have the mathematical or programming skills to meaningfully implement complex algorithms is a simple pragmatic test. this paper presents a pragmatic a simple goodness-of-fit test 9 general goodness-of-fit statistic for continuous conditional models using uncensored data. in the next section, we introduce the goodness-of-fit test and the rationale behind it. we then evaluate the performance of the goodness-of-fit statistic in section 3 using two groups of monte carlo experiments. the first group of experiments focuses on discerning deviations from correct specification in the distributional family; the second group focuses on discerning deviations in parameterization. we choose these investigations because they represent the two misspecification issues in estimating conditional probability models. in section 4, we apply the goodness-of-fit test to the 2011 medical expenditures panel survey (meps) dataset, modeling three health care expenditure outcomes as functions of patient characteristics. finally, in section 5, we conclude our paper with a summary of the findings and discussions about the applications of the goodness-of-fit test. the appendix provides the expected value of the statistic and the procedure for the calculation of the degrees of freedom for the test statistics, the data generating process for each monte carlo experiment, and the analyses modelling cost data from meps. figure 1. a typical conditional gumbel distribution with sparse observations for each conditional value of observed x. -1 0 -5 0 5 1 0 y -5 0 5 10 x f( y | x ) a gumbel distribution p. veazie, z. ye 10 2 a proposed goodness-of-fit test the pearson chi-square goodness-of-fit statistic is based on comparing the number of actual observations within each set of a partition of the random variable’s range to the number of observations that would be expected to show up in those sets if the model correctly represents the dgp (schervish, 1995). if the model is correct, then the expected number of observations is the expected number for the dgp; consequently, the observed and predicted number in each set should be different merely by random variation. the chi-square goodness-of-fit statistic for continuous distributions is created by partitioning the range of a continuous random variable y into k regions. denote each region k{1, 2, …k} as the number of observations with values of y in region rk as nk, and the total sample size as n. the probability of an observation with y in region rk is then in which fy(y) is the probability density for y associated with a cumulative distribution function (cdf) fy(y). the chi-square statistic is defined as the corresponding sample statistic is in which is a consistent estimator of . if fy(y;) accurately represents the data generating process, converges to c with increasing n and the corresponding asymptotic distribution of cn is a chi-square with degrees of freedom equal to the number of groups in the partition minus the number of estimated parameters plus one (schervish, 1995). if we are interested in a model of the conditional distribution f(y|x;), the preceding statistic is not generally applicable because nk can contain insufficient observations to inform the conditional distribution. indeed, with x a simple goodness-of-fit test 11 containing precisely measured continuous variables, there may be only one value y for some observed x values (see figure 1 as an example). however, we can take advantage of the probability integral transform and consequent fact that the cdf of a continuous random variable is itself a random variable with a uniform distribution on the unit interval. because the uniform distribution is the same regardless of underlying cdf, a set of random variables from independent observations with different conditional distributions can all be converted by their cdfs to the same uniform distribution. we can use this fact to construct a test of the conditional distribution; even if each observation has a different conditioning value (i.e. the data in figure 1 will pose no problem for this test). because the cdf for each random variable has a uniform distribution, the cdf values of sample results from a correctly specified model for each random variable will produce a single realization from a uniform distribution. therefore, the full sample results should together provide a histogram that deviates only by chance from a uniform distribution. we can use a pearson chi-square type statistic applied to the uniform distribution to test of the specification for the conditional distributions. the process is quite simple. for each observation i we have a model specification for the distribution f(yi|xi) and therefore can obtain from the estimated model the sample quantity ui = f(yi|xi) for which (xi, yi) are the observed values for observation i. the random variable underlying ui has a uniform distribution on the unit interval if f is correctly specified. we can construct a goodness-of-fit test by partitioning the unit interval into k subintervals defined by equally spaced boundary points, which for k = 10 is s.t. k {1, 2, … 10}. the statistic is then for which n is the total sample size, and nk is the observed number of u values in the rk interval. this statistic can alternatively be written as p. veazie, z. ye 12 in which is the observed proportion in interval rk. because the statistic is based on a partition of the uniform into k equal sized intervals, pk = 1 k ; therefore, 21ˆ( ) k k u k n p=   − . as shown in the appendix, the expected value of u, which is the degrees of freedom for its approximating chi-square distribution, is equal to the degrees of freedom for the usual pearson chi-square test (i.e. k − 1) minus a factor due to the estimation of model parameters. since pk is known, which in the case of k = 10 intervals is 0.1, we can simply state the statistic for k = 10 as the selection of k = 10 is arbitrary, as it is with the hosmer-lemeshow test for logistic regression (hosmer and lemeshow, 1980). for other values of k, the degrees of freedom can be directly estimated as shown in the appendix or determined by monte carlo simulation (see box 2). the u statistic has a distribution proportional to the sum of gamma random variables with different parameters. specifically, denoting 1ˆ k k p − as zk, as shown in the appendix zk is asymptotically normally distributed with mean 0 and variance k 2. consequently, the ratio of zk squared to k 2 has an asymptotic chi-square distribution with degrees of freedom 1, which is a gamma distribution with parameters 0.5 and 2 (i.e. (0.5, 2)). therefore, zk 2 has a distribution k 2(0.5, 2), which is (0.5, 2k 2). u is therefore proportional to the sum of k differently scaled gamma random variables. moschopoulos shows that sum of such variates can be express as a gamma series in which the series coefficients can be recursively determined (moschopoulos, 1985). the use of this recursive coefficient determination and gamma series is overly complex for the practical application of this statistic among many applied researchers. however, ease of use is the purpose of this goodness-of-fit statistic. fortunately, the monte carlo experiments presented below indicate that for a correct specification the statistic is approximately chi-square in distribution with degrees of freedom 7.5 when k = 10 and calculated as shown in the appendix or as shown in box 2 if k is not 10. a simple goodness-of-fit test 13 3 simulation experiments 3.1 methods we investigated finite sample performance of the proposed statistic using monte carlo experiments of conditional normal, gumbel, gamma, and weibull models, each applied to data generating processes based on the same set of distributions. the first set of experiments comprised a total of sixteen model/dgp comparisons. we evaluated each model/dgp pair for sample sizes 500 and 1000, each using 2000 monte carlo samples from the dgp (see appendix table a1 for parameter specifications). we inspected rejection rates for significance levels spanning between 0 and 0.2 for each comparison. for each correct model/dgp pair (i.e. normal/normal, gumbel/gumbel, gamma/gamma, and weibull/weibull), the plot of the empirical cumulative distribution function (ecdf) of the calculated p values, across the 2000 mc samples, should approximately match the significance level (i.e. this plot should be approximately a straight line). for example, the use of a significance level of 0.01 should reject the model for approximately 1 per cent of the 2000 samples; using a significant level of 0.05 should reject approximately 5 per cent of the samples; and a 0.1 significance level should result in approximately 10 per cent rejections. for mismatched pairs (e.g. weibull/gumbel), if the fit test is useful it should produce rejection rates that are higher than the significance levels and increase with sample size; consequently, the ecdf of the test’s p-value should be above the significance level. the second set of experiments compared models in which parameters are specified as linear in conditioning variables to the dgp having the same distributional family but with parameters quadratic in the conditioning variables (see appendix table a2 for parameter specifications). for the normal distribution, we estimated models with homoscedasticity and heteroscedasticity. in the case of heteroscedasticity both the mean and variance were generated as quadratic in x in the dgp, but they were modeled as linear in the misspecified model. similarly, we carried out experiments for sample sizes 500 and 1000. these experiments provide evidence regarding whether the test can identify deviations in parameterization as well as distributional family. in the monte carlo experiments reported below, we applied the steps presented in box 1 to obtain p-values for each of 2000 data sets generated for each model/dgp being considered. we calculated both a p-value using degrees of freedom equal to 7.5 and also using the mean of the 2000 calculated u values for each dgp considered when using the correct model (remember that the degrees of freedom are associated with the distribution of u given the model is correct). p. veazie, z. ye 14 3.2 results because we tested continuous conditional distributions, it is difficult to see the differences between the model and dgp for all patterns of explanatory variables. however, table 1 shows the probability density functions for the true dgp (in the solid line) and the estimation model (the dotted line, using the average parameter values across the 2000 estimated models) evaluated at the mean of x. this gives some sense of the differences between the distributions being tested in the first set of experiments; however, the deviation of the model from the underlying distribution that drives larger values of u may be from other regions of the conditioning set than at the mean of x. tables 2 and 3 present the ecdfs of the statistic’s p-values for each indicated model applied to the indicated dgp plotted for significance levels up to 0.2. table 2 presents results for sample sizes of 500; table 3 presents results for sample sizes of 1000. the thin straight lines show the points where the ecdfs would be if it corresponded to the significance level. the thick dark lines (or curves) are the ecdfs associated with p values based on degrees of freedom set to 7.5. the thick light lines (or curves) are the ecdfs associated with the monte carlo based empirical degrees of freedom. we determined the 7.5 degrees of freedom approximation by the average of the four empirical degrees of freedom across the dgps using sample sizes of 1000. we also ran monte carlo experiments for correct model specifications using 10 correlated explanatory variables (results not presented); these box 1. how to calculate u and its p-value using k = 10 step 1. for a candidate model f(yi | xi ; ), estimate the parameters, obtaining̂ . step 2. calculate the cdf value of ui = f(yi | xi ; ̂ ) for each observation (yi , xi) in the data. step 3. calculate the proportion ( ˆkp ) of ui in each of the ten intervals rk for k  {1, 2, … 10}. step 4. calculate the statistic u using the equation . step 5. calculate the p-value as the upper tail area of a chi-square distribution with degrees of freedom set to 7.5 or set to the estimated value determined by the equations presented in the appendix or the modelspecific monte carlo determined empirical degrees of freedom (see box 2 for the algorithm). a simple goodness-of-fit test 15 experiments showed that the empirical degrees of freedom remained around 7.5 in multivariable models. specifically, the means of the u statistics, and therefore the degrees of freedom, in these experiments for the normal, gamma, weibull, and gumbel were 7.52, 7.32, 7.54, and 7.27 respectively. the figures on the diagonals of tables 2 and 3 show the coverage of the test when the model is correctly specified. the results fell along the line representing accurate coverage: the ecdf corresponds to the significance level. not surprisingly, the empirical degrees of freedom (the thick lighter line) were more accurate than using the approximate degrees of freedom of 7.5; however, the differences were slight, particularly up to the 0.1 significance level. the off-diagonal figures in tables 2 and 3 show the rejection rate for the test of misspecified models across significance levels. the test was sufficiently powerful for some of the model/dgp combinations to reject the model for all 2000 samples at all significance levels greater than 0.001. results for these combinations are simply indicated by the phrase ‘all data sets rejected at significance level 0.001’. not surprisingly, comparing table 2 to table 3, the curve has a greater departure from the straight line in table 3; it is evident that the power of the test increases with sample size. it is also clear that using the approximate 7.5 table 1. probability density functions of the true data generating process (solid curve) and the estimated model (dashed curve) evaluated at x=0 for the monte carlo simulations. p. veazie, z. ye 16 table 2. monte carlo simulation: empirical cdfs of experiments on distribution specifications (n=500). degrees of freedom provide similar results to that of using the monte carlo determined empirical degrees of freedom. table 4 presents results for the second set of experiments, which tested deviations from correct specification in the parameterization. the upper two rows show results for sample sizes of 500; the lower two rows show results for sample sizes of 1000. similar to the first set of experiments, results showed accurate coverage for the test when the model was correctly specified and the ability to discern deviations from correct specification in parameterization. as the sample size went up, the power of the test to discern such deviations increased. the approximate 7.5 degrees of freedom yields results that were similar to the monte carlo calculated empirical degrees of freedom. 4 example to present an example with real data, we used a random sample of 2000 individuals from the household component of the 2011 medical expenditure panel survey data file (meps). as one of the largest national health survey, a simple goodness-of-fit test 17 meps has been widely used to study the patterns of health care access, utilization and expenditures in the united states (cohen et al., 2009). we modeled each of the three outcomes – annual total health care expenditure, total office-based visits expenditure, and total dental care expenditure – as a function of individual demographics, socioeconomic status, self-rated health status, common chronic conditions, presence of usual source of care provider, and health insurance coverage. these covariates were selected in accordance with prior studies focusing on modeling health care costs using meps survey data (fenton et al., 2012, fleishman and cohen, 2010). for each model, we included all individuals who reported an expense on the outcome of interest and took the log of the expenditure as the dependent variable. there were 1527 and 1215 individuals reporting expenses on health care and office-based services, which represented 76.4% and 60.8% of the total sample, respectively. much fewer individuals reported any expenses on dental care (n= 724, 36.2%). appendix table a3 presents the descriptive statistics and distribution of the outcome variables and the covariates that we employed in the model. we used pregibon’s link test (pregibon, 1980) to identify a statistically table 3. monte carlo simulation: empirical cdfs of experiments on distribution specifications (n=1000). p. veazie, z. ye 18 adequate specification of the explanatory variables for each model. we then computed u to test the hypothesis that the specified distribution was correct. this allows us to use the test to focus on testing deviations in the distributional family. we calculated the p-value based on the approximate degrees of freedom of 7.5 and the empirical degrees of freedom calculated from the parameter estimates of the specified model, based on 500 monte carlo samples. the algorithm for computing the empirical degrees of freedom is shown in box 2. table 5 presents the results from the empirical example for the three health care expenditure outcomes. the test clearly discerns the goodness-of-fit performance of different distributions. results for the model of the logarithm of total health care expenditure strongly rejected the hypothesis that the conditional distribution follows a gamma, weibull or gumbel distribution (u ranges from 21.985 to 116.578, p-value < 0.001 for all), and unequivocally failed to reject the hypothesis for normal (u = 5.304, p-value = 0.676 with approximate degrees of freedom of 7.5). for the model of officebased visits expenditure, we strongly rejected the hypotheses for the gumbel and weibull distribution (u equals to 59.626 and 33.776, respectively, p-value table 4. monte carlo simulation: empirical cdfs of experiments on parameter specifications. a simple goodness-of-fit test 19 < 0.001 for both) and fail to reject the normal (u=12.467, p-value = 0.107) or gamma (u = 8.897, p-value = 0.305). for the model of dental care expenditure, we rejected all distributions except for the gumbel (u = 14.333, p-value = 0.058 with degrees of freedom of 7.5). figures a1-a3, in the appendix, show the histograms of the residuals obtained from these models, standardized by the estimated standard deviations. figure a1 shows the symmetry expected of a normal distribution, which was not rejected by the test that unambiguously rejected the other distributions. figure a2 shows a right-skewedness characteristic of a gamma distribution (model 1), but it is insufficiently skewed to reject the normal at a significance level of 0.05. however, u is smaller in the gamma indicating a better fit to the data. under certain circumstances (i.e. shape parameter sufficiently large, >15), the gamma distribution is approximately a normal distribution (rothschild and logothetis, 1986). in this real-data example, the estimated shape parameter equaled to 35 in the model assuming gamma distribution. it is therefore not surprising the test did not reject either the gamma or the normal distributions. figure a3 demonstrates the clear right-skewedness of the residual from the model of dental care expenditure, which is expected of a gumbel distribution. the calculated monte carlo empirical degrees of freedoms were approximately 7.5 for all three outcomes and therefore yielded similar results. as there were 21 variables in the empirical model, these results again show that the degrees of freedom for the statistic distribution based on 10 categories is approximately 7.5 in multivariable models. table 5. empirical example: goodness-of-fit tests on conditional probability models for log-transformed health expenditures from meps. 5 conclusion in this paper, we presented a simple specification test for conditional continuous distributions using uncensored data (see box 1). we showed, using simulation experiments, that the test has accurate coverage under correct specification, and that the test can discern deviations from correct specification in both the distributional family as well as parameterization. the empirical example shows its ability to distinguish specific distributions from other candidates using real data. p. veazie, z. ye 20 the results of our analysis indicate that u is approximately distributed chisquare with degrees of freedom 7.5. we also provide a monte carlo method for an empirical determination of degrees of freedom in box 2 and a direct estimator in the appendix should the researcher not wish to use the approximating 7.5, for example when the p-value using the approximating 7.5 degrees of freedom is close to the test’s designated significance level. however, comparing the empirical degrees of freedom to 7.5 across all monte carlo experiments and real-data analyses of our study, the differences were slight and not likely to impact inferences. if a researcher does not wish to approximate the distribution using a chi-square, a p-value based on the monte carlo distribution of statistic values generated in the process of box 2 can be used as a parametric bootstrap test (davison et al., 2003). because the test discerns deviations in parameterization as well as the distributional family, an extra step is required to investigate the distributional family alone. specifically, the researcher should engage in standard tests to identify the best parameter specification within each proposed model (e.g. we used pregibon’s link test in the preceding example). using the best withinfamily model specification, the test will then primarily be identifying deviations in the distributional family. it is important to note that our results using multiple explanatory variables in the models indicate the degrees of freedom for the statistic’s distribution is not a function of the number of estimated parameters. this is different from box 2. how to calculate the empirical degrees of freedom step 1. obtain the parameter estimates predicted from the estimated model ( ). step 2. generate outcome values as random draws from the distribution defined by the estimated parameters for all xi in the data. step 3. re-estimate the model using the generated outcomes. step 4. obtain the predicted parameter estimates ( ) from using the ‘correctly’ specified model in step 3. step 5. calculate the value of for each observation. step 6. calculate u. step 7. repeat the steps 2 through 6 multiple times (e.g. we repeated 500 times in the empirical example), saving the statistic values. step 8. set the degrees of freedom to the mean of the calculated u values. a simple goodness-of-fit test 21 the direct application of the pearson chi-square test to distributions with multiple parameters in which the degrees of freedom depend on the number of parameters m. this is an advantage since the degrees of freedom in the latter case is typically k−m−1, which implies m must be less than k −1 for those applications (schervish, 1995): our test does not have this constraint. although our test can be used as a goodness-of-fit test for marginal distributions, it is particularly useful as an easy-to-use model fit test of continuous conditional distributions for uncensored data, particularly in the case of few observations, indeed even one observation per pattern of explanatory variables, such as a time-series. p. veazie, z. ye 22 references amemiya, t. 1985. advanced econometrics, cambridge, ma, harvard university press. andrews, d. w. k. 1997. a conditional kolmogorov test. econometrica, 65, 1097-1128. cohen, j. w., cohen, s. b. & banthin, j. s. 2009. the medical expenditure panel survey: a national information resource to support healthcare cost research and inform policy and practice. med. care., 47, s44-50. davison, a. c., hinkley, d. v. & young, g. a. 2003. recent developments in bootstrap methodology. statistical science, 18, 141-157. fan, y. q., li, q. & min, i. 2006. a nonparametric bootstrap test of conditional distributions. economet. theor., 22, 587-613. fenton, j. j., jerant, a. f., bertakis, k. d. & franks, p. 2012. the cost of satisfaction: a national study of patient satisfaction, health care utilization, expenditures, and mortality. arch. intern. med., 172, 405-11. fleishman, j. a. & cohen, j. w. 2010. using information on clinical conditions to predict high-cost patients. health. serv. res., 45, 532-552. hosmer, d. w. & lemeshow, s. 1980. a goodness-of-fit test for the multiple logistic regression model. communications in statistics part atheory and methods, 10, 1043-1069. moschopoulos, p. g. 1985. the distribution of the sum of independent gamma random variables. annals of the institute of statistical mathematics, 37, 541-544. o'reilly, f. j. & quesenberry, c. p. 1973. the conditional probability integral transformation and applications to obtain composite chi-square goodness-of-fit tests. ann. statist., 1, 74-83. o'reilly, f. j. & stephens, m. a. 1982. characterizations and goodness of fit tests. j. roy. statist. soc. ser. b, 44, 353-360. pregibon, d. 1980. goodness of link tests for generalized linear models. j. roy. statist. soc. ser. c, 29, 15-24. rincon-gallardo, s., quesenberry, c. p. & o'reilly, f. j. 1979. conditional probability integral transformations and goodness-of-fit tests for multivariate normal distributions. ann. statist., 7, 1052-1057. rothschild, v. & logothetis, n. 1986. probability distributions, wiley. schervish, m. j. 1995. theory of statistics, new york, springer-verlag. zheng, j. x. 2000. a consistent test of conditional parametric distributions economet. theor., 16, 667-691. a simple goodness-of-fit test 23 appendix a1 the expected value of the u-statistic the expected value of u is the expected value associated with the distribution of the standard pearson chi-square goodness-of-fit statistic minus a factor due to estimating the parameters of the model. in this appendix we provide the determination of the expected value, and we provide an estimator for the adjustment factor and thereby an estimator of the expected value for the proposed statistic. the expected value of u is proportional to the sum of expected values across the k equal-length regions of the partition of the unit interval being considered: 2 2 ˆ( ) [ ( ) ] ˆ[( ) ] k k k k k k e u e k n p p k n e p p =   − =   −   the expected values under the summation sign on the right-hand side of this equation are variances. this is seen by denoting an indicator of whether observation i falls in region k as , 1 (( 1) 0.1, 0.1) 0 i k i u k k i otherwise  −   =   and noting that the expected value of the estimated proportion in category k is 1 , 1 1 , 1 1 , 1 , ˆ ˆˆ ˆ[ ] [ | ] ( ) ˆ ˆ[ | ] ( ) ˆ ˆ[ | ] ( ) ˆ ˆ( ) ( ) ˆ ˆ( ) ( ), for for all ˆ( ( )) k k n k in i n k in i n k in i k k i k k e p e p df e i df e i df p df p df p p i e p            = = = = = =  =  = = =      to determine ˆ( ( )) k e p  , consider a first order taylor series approximation around the true value  p. veazie, z. ye 24 ˆ ˆ( ) ( ) ( )k k k p p p      = +  −  , which yields ˆ ˆ( ( ) ( )) ( )k k k p n p p n       − =   −  . for an estimator, such as the maximum likelihood estimator, for which ˆ( )n   − converges to a normal distribution n(0,) by a central limit theorem, the left-hand side converges in distribution to a normal as well: ˆ( ( ) ( )) (0, ) d k k k k p p n p p n       − ⎯⎯→    . therefore, ˆ( ) k p  has an asymptotic distribution with expected value of ˆ[ ( )] ( ) k k e p p = and variance of 1ˆ[ ( )] k k k p p v p n       =       . consequently, since ˆ[ ( )] ( ) k k e p p = , 2ˆ ˆ[( ) ] [ ] k k k e p p v p− = . the expected value of the u is then proportional to the sum of variances: 2ˆ ˆ[ ( ) ] [ ] k k k k k e k n p p k n v p  − =    . the variance terms under the summation sign on the right-hand side are 2 1 , 1 1 , 1 1 21 21 1 ˆ ˆˆ ˆ[ ] [ | ] ( ) ˆ ˆ[ | ] ( ) ˆ ˆ[ | ] ( ), for independent observations ˆ ˆ ˆ( )(1 ( )) ( ) ˆ ˆ ˆ( ( ) ( ) ) ( ) ˆ ˆ ˆ[ ( ( )) ( ( )) ( ( )) [ k k n k in i n k in i k kn k kn k k kn n v p v p df v i df v i df p p df p p df e p e p v p                = = = =  =  =  − =  − =  − − =       2 ˆ( ) ( ) ( ( )] k k k p p v p  − − therefore, a simple goodness-of-fit test 25 2 21 1 2 ˆˆ[ ( ) ] [( ) ( ( ))] 1 1 ˆ[( ) ( ( ))] ˆ( 1) ( ( )) k k k k kn k k kn k k k e k n p p k n p p v p k n v p k k k k v p      − =    − − =    − − = − −      the expected value of u is the degrees of freedom for a common pearson chi-square test statistic (i.e. k − 1) minus a factor due to estimation of the distribution parameters. for k = 10, the expected value of u is then ˆ9 10 ( ( )) k k v p −  . a2 estimation of the shrinkage factor the variance terms in the shrinkage factor can be estimated by using consistent estimators for the derivatives k p    and the covariance matrix . the derivative of kp is determined by noting that * * 1 [ ( ( ) | ; ) ( ( ) | ; )] ( ) k k k x p f y x x f y x x df x  − = − , for which * k y are the critical values * 1 ( ) ( | ) k k y x f x k − = . therefore, assuming we can interchange the order of integration and differentiation, * * 1 ( ( ) | ; ) ( ) ( ( ) | ; ) ( )k k x k x p f y x x df x f y x x df x     −    = −      . estimating the integrals on the right-hand side of the equation by sample means yields the estimator * * 1 1 1 1 1ˆ ˆ( ( ) | ; ) ( ( ) | ; ) n n k k i i k i i i i p f y x x f y x x n n      − = =    =  −       . the estimator for the variances in the shrinkage factor is therefore 1ˆˆ ˆ[ ( )] k k k p p v p n       =        . p. veazie, z. ye 26 for the maximum likelihood estimator, note that the scaled deviation of the estimator converges in distribution to a normal: 11ˆ( ) (0,[ ( ( ))] ) d n n n e h   −  − ⎯⎯→ − , for h denoting the matrix of second derivatives of the log-likelihood with respect to the parameters. therefore, 11 1 [ ( ( ))] [ ( ( ))] n e h n e h   − −  = − =  − using the sample mean for the expectation of the hessian, evaluated at the estimated parameter values, yields the estimator 2 1ˆˆ [ ( )]n h  −  =  − . the estimated variance of ˆ( ) k p  is then 1ˆ ˆˆ[ ( )] [ ( )]k k k p p v p n h    −    =   −       . for example, consider the weibull distribution specified in table a1. the weibull cdf is ( ) ( | ) b b x0 1a a x e0 1e y f y x 1 e +  +  −  = − . the derivatives with respect to the parameters are ( ) ln( ) ( ) ln( ) 0 1 0 1 0 1 b b x 0 b b x 1 f d a f d x a f d e y b f d e y x b +  +   =   =    =     =     where, b b x0 1b b x a a xe0 1 0 1 0 1a a xe y ed y e e +  +  +  + −  =   . a simple goodness-of-fit test 27 evaluating each of these derivatives and each observation in the sample i  {1, … n} at the estimated parameter values, data values ix , and the corresponding critical values * 0 ( ) i y x , and * ( ) k i y x for each k  {1,…10} creates variables for which the sample means can be used to determine k p    . these estimated derivatives combined with the estimated parameter covariance matrix ̂ provide the information to calculate the shrinkage factor as shown above. table a0 presents the means of the estimated expected value of u using the above equations and means of the calculated u values across 100,000 data sets of sample sizes 100, 1000, and 10,000. the mean estimated e(u) was very similar to the mean of u values, rounding to 7.37 for each. an alternative for estimating the expected value of u (i.e. degrees of freedom for an approximating chi square distribution) is the monte carlo method shown in box 2 of the main text. table a0. mean estimated e(u) and mean u across 100,000 samples. a3 additional tables and figures p. veazie, z. ye 28 table a1. simulation process: conditional distribution of the data for the test of incorrect distributional family. true data generating process normal/normal (homoscedasticity) normal/normal (heteroscedasticity) gumbel/gumbel location scale gamma/gamma shape scale weibull/weibull shape scale table a2. simulation process: conditional distribution of the data for the test of incorrect parameterization a simple goodness-of-fit test 29 table a3. distribution of the cost-related outcome variables and patient characteristics. p. veazie, z. ye 30 0 .1 .2 .3 .4 d e n s it y -4 -2 0 2 4 standardized residual figure a1. histogram of the standardized residual from the model for annual total health care expenditure. model: mle assuming normal distribution with heteroskedasticity. a simple goodness-of-fit test 31 figure a2. histogram of the standardized residual from the model for annual total expenditures on officebased visits. p. veazie, z. ye 32 figure a3. histogram of the standardized residual from the model for annual total expenditures on dental care. hyperstructures as models in social sciences ratio mathematica, 21, 2011, pp. 27-42 27 bar and theta hyperoperations thomas vougiouklis democritus university of thrace, school of education 681 00 alexandroupolis, greece tvougiou@eled.duth.gr abstract in questionnaires the replacement of the scale of likert by a bar was suggested in 2008 by vougiouklis & vougiouklis. the use of the bar was rapidly accepted in social sciences. the bar is closely related with fuzzy theory and has several advantages during both the filling-in questionnaires and mainly in the research processing. in this paper we relate hyperstructure theory with questionnaires and we study the obtained hyperstructures which are used as an organising device of the problem. key words: hyperstructures, hv-structures, hopes, -hopes. subject classification: 20n20, 16y99. 1. introduction hyperstructures are called the algebraic structures equipped with at least one hyperoperation i.e. a multivalued operation. we have abbreviated the ‘hyperoperation’ by ‘hope’ [24]. therefore, if in a set h at least one hope :hhp(h)-{} is defined, then (h,) is called a hypergroupoid. the hvstructures introduced in 1990 [15], is the largest class of hyperstructures. the hvstructures satisfy the weak axioms where the non-empty intersection replaces the equality. in (h,) we abbreviate by wass the weak associativity: (xy)zx(yz)  , x,y,zh and by cow the weak commutativity: xyyx  , x,yh. the hyperstructure (h,) is called hv-semigroup if it is wass, and it is called hv-group if it is reproductive hv-semigroup, i.e. xh=hx=h, xh. the hyperstructure (r,+,) is called hv-ring if both hopes (+) and () are wass, the reproduction axiom is valid for (+) and () is weak distributive with respect to (+): x(y+z)(xy+xz)  , (x+y)z(xz+yz)  , x,y,zr. mailto:tvougiou@eled.duth.gr 28 the main tool to study all hyperstructures are the fundamental relations β*, γ* and ε*, which are defined, in hv-groups, hv-rings and hv-vector spaces, resp., as the smallest equivalences so that the quotient would be group, ring and vector space, resp., [17]. an element is called single if its fundamental class is singleton. a way to find the fundamental classes is given by analogous theorems to the following [17],[18],[19],[20],[21],[5]: theorem. let (h,) be an hv-group and denote u the set of all finite products of elements of h. we define the relation β in h by setting xβy iff {x,y}u where uu. then β* is the transitive closure of β. analogous theorems for the relations γ* in hv-rings and ε* in hv-modules and hv-vector spaces, are also proved. these relations were introduced and first studied by t.vougiouklis, see [17]. one can see generalizations of the classical hyperstructure theory in several papers and books as [3],[4],[6],[17]. fundamental relations are used for general definitions [17],[20]. thus, in the general definition of the hv-field, the γ* is used: an hv-ring (r,+,) is called hv-field if r/γ* is a field. this definition includes all the well known definitions of hyperfields [15], [17], as special cases. motivations. the motivation for hv-structures is the following: we know that the quotient of a group with respect to an invariant subgroup is a group. f. marty from 1934, states that, the quotient of a group by any subgroup is a hypergroup. now, the quotient of a group by any partition (or equivalently to any equivalence relation) is an hv-group. this is the motivation to introduce the hv-structures [15]. specifying this motivation we remark: let (g,) be a group and r be an equivalence relation (or a partition) in g, then (g/r,) is an hv-group, therefore we have the quotient (g/r,)/β* which is a group, the fundamental one. remark that the classes of the fundamental group (g/r,)/β* are a union of some of the rclasses. otherwise, the (g/r,)/β* has elements classes of g where they form a partition which classes are larger than the classes of the original partition r. let (h,), (h,*) be hv-semigroups on the same set. () is called smaller than (*), and (*) greater than (), iff there exists faut(h,*) such that xyf(x*y), x,yh. then, we write * and say that (h,*) contains (h,). if (h,) is a structure then it is called basic structure and (h,*) is called hb-structure. theorem (the little theorem), [17],[18]. in all hv-structures and for all hopes, which are defined on them, greater hopes than the ones which are wass or cow, are also wass or cow, respectively. 29 this theorem leads to a partial order on hv-structures so, to a correspondence between hyperstructures and posets. the determination of all hvgroups and hv-rings is hard. in this direction there are many results by r. bayon and n. lygeros [1]. definitions 1 [19],[20]. let (h,) be hypergroupoid. we remove hh, if we consider the restriction of () in the set h-{h}. hh absorbs hh if we replace h by h and h does not appear in the structure. hh merges with hh, if we take as product of any xh by h, the union of the results of x with both h, h, and consider h and h as one class with representative h, therefore, h does not appear in the hyperstructure. most of hv-structures are used in representation (abbreviate by rep) theory. reps of hv-groups can be considered either by generalized permutations or by hv-matrices [16],[17]. reps by generalized permutations can be achieved by using translations. in the rep theory the singles are playing a crucial role. the rep problem by hv-matrices is the following: hv-matrix is called a matrix if has entries from an hv-ring. the hyperproduct of hv-matrices a=(aij) and b=(bij), of type mn and nr, respectively, is a set of mr hv-matrices, defined in a usual manner: ab = (aij)(bij) = { c = (cij)  cij  σaikbkj }, where () denotes the n-ary circle hope on the hyperaddition [17]. definition 2. let (h,) be hv-group, (r,+,) hv-ring, mr={(aij)aijr}, then any map t:hmr: ht(h) with t(h1h2)t(h1)t(h2)  , h1,h2h, is called hv-matrix rep. if t(h1h2)t(h1)t(h2), then t is an inclusion rep, if t(h1h2)=t(h1)t(h2), then t is a good rep. hyperoperations on any type of matrices can be defined: definition 3 [13],[8]. let a=(aij)mmn be matrix and s,tn, with 1sm, 1tn. then helix-projection is a map st: mmnmst: aast = (aij), where ast has entries aij = { ai+s,j+t 1is, 1jt and ,n, i+sm, j+tn } let a=(aij)mmn, b=(bij)muv be matrices and s=min(m,u), t=min(n,v). we define a hyper-addition, called helix-addition, by 30 :mmnmuvp(mst):(a,b)ab=ast+bst=(aij)+(bij)mst where (aij)+(bij)= {(cij)=(aij+bij) aijaij and bijbij)}. let a=(aij)mmn, b=(bij)muv and s=min(n,u). we define the helixmultiplication, by :mmnmuv  p(mmv): (a,b)  ab = amsbsv = (aij)(bij)  mmv where (aij)(bij)= {(cij)=(aitbtj) aijaij and bijbij)}. the helix-addition is commutative, wass but not associative. the helixmultiplication is wass, not associative and it is not distributive, not even weak, to the helix-addition. for all matrices of the same type, the inclusion distributivity, is valid. 2. basic definitions one can see basic definitions, results, applications and generalizations on hyperstructure theory, not only for hv-structures, in the books [3],[4],[6],[17] and the survey papers [2],[5],[7],[14],[20],[21]. here we present some definitions related to our problem. in a hypergroupoid (h,) the powers of hh are h 1 ={h},…, h n = hh…h, where () denotes the n-ary circle hope, i.e. take the union of hyperproducts with all possible patterns of parentheses put on them. an hv-semigroup (h,) is called cyclic of period s, if there exists a g (generator) and a number s, the minimum, such that h= h 1  …  h s . the cyclicity for the infinite period is defined in [14]. if there is an h and a number s, the minimum, such that h=h s , then (h,) is called single-power cyclic of period s. in 1989 corsini & vougiouklis introduced a method to obtain stricter algebraic structures from given ones through hyperstructure theory. this method was introduced before of the hv-structures, but in fact the hv-structures appeared in the procedure. definition. the uniting elements method is the following: let g be a structure and d be a property, which is not valid, and it is described by a set of equations. consider the partition in g for which it is put together, in the same class, every pair of elements that causes the non-validity of d. the quotient g/d is an hvstructure. then quotient of g/d by the fundamental relation β*, is a stricter structure (g/d)β* for which d is valid. 31 an application of the uniting elements is if more than one property desired. the reason for this is some of the properties lead straighter to the classes: commutativity and the reproductivity are easily applicable. one can do this because the following is valid: theorem [17],[21]. let (g,) be groupoid, and f={f1,…,fm,fm+1,…,fm+n} a system of equations on g consisting of two subsystems fm={f1,…,fm}, fn={fm+1,…, fm+n}. let σ and σm be the equivalence relations defined by the uniting elements using the f and fm respectively, and let σn be the equivalence relation defined using the induced equations of fn on the grupoid gm = (g/σm)/β*. then we have (g/σ)/β*  (gm/σn)/β*. definition 4 [17],[21]. let (f,+,) be an hv-field, (v,+) be a cow hv-group and there exists an external hope  : fv  p(v ): (a,x)  ax such that, for all a,b in f and x,y in v we have a(x+y)  (ax+ay)  , (a+b)x  (ax+bx)  , (ab)x  a(bx)  , then v is called an hv-vector space over f. in the case of an hv-ring instead of hv-field then the hv-modulo is defined. in the above cases the fundamental relation ε* is the smallest equivalence relation such that the quotient v/ε* is a vector space over the fundamental field f/γ*. the general definition of an hv-lie algebra over a field f was given in as follows: definition 5. let (l,+) be an hv-vector space over the field (f,+,), φ: ff/γ* be the canonical map and ωf={xf:φ(x)=0}, where 0 is the zero of the fundamental field f/γ*. similarly, let ωl be the core of the canonical map φ: ll/ε* and denote by the same symbol 0 the zero of l/ε*. consider the bracket (commutator) hope: [ , ] : ll  p(l): (x,y)  [x,y] then l is an hv-lie algebra over f if the following axioms are satisfied: (l1) the bracket hope is bilinear, i.e. [λ1x1+ λ2x2,y]  ( λ1[x1,y] + λ2[x2,y])   [x, λ1y1+ λ2y]  ( λ1[x,y1] + λ2[x,y2])  ,  x,x1,x2,y,y1,y2 l, λ1,λ2  f 32 (l2) [x,x]  ωl   , xl (l3) ([x,[y,z]]+[y,[z,x]]+[z,[x,y]])  ωl  , x,yl. we remark that this is a very general definition therefore one can use special cases in order to face several problems in applied sciences [12],. moreover, from this definition we can see how the weak properties can be defined as the above weak linearity (l1), anti-commutativity (l2) and the jacobi identity (l3). 3. -hopes in [22] an extremely large class of hopes introduced called theta: definition 6. let h be a set equipped with n operations (or hopes) 1,…,n and a map (or multivalued map) f:hh (or f:hp(h)-{}, resp.), then n hopes 1,…,n on h are defined, called theta-hopes, (-hopes), by putting xiy = {f(x)iy, xif(y) }, x,yh and i{1,2,…,n} or, in the case where i is hope or f is multivalued map, we have xiy = ( f(x)iy)(xif(y) ), x,yh and i{1,2,…,n} if i is associative, then i is wass. remark that one can use several maps f instead of only one, in a similar way. in a groupoid (g,), or a hypergroupoid, with a -hope, one can study several properties like the following ones: reproductivity. for the reproductivity we must have xg= gg{f(x)g, xf(g)}= g and gx= gg{f(g)x, gf(x)}= g. if () is reproductive, then () is reproductive: gg{f(x)g} = g. commutativity. if () is commutative then () is commutative. if f is into the centre, then () is commutative. if () is cow then () is cow. unit elements. u is right unit if xu={f(x)u,xf(u)}x. so f(u)=e, if e is a unit in (g,). the elements of the kernel of f, are the units of (g,). inverse elements. let (g,) be a monoid with unit e and u be a unit in (g,), then f(u)=e. for given x, the x is an inverse with respect to u, if xx={f(x)x,xf(x)}u and xx={f(x)x,xf(x)}u. so, x=(f(x)) -1 u and 33 x=u(f(x)) -1 , are the right and left inverses, respectively. we have two-sided inverses iff f(x)u = uf(x). proposition 7. let (g,) be a group then, for all maps f: gg, the (g,) is an hvgroup. one can define -hopes on rings and more complicated structures (or hyperstructures or hv-structures), where more than one -hopes can be defined. motivation for the definition of the theta-hope is the map derivative where only the multiplication of functions can be used. therefore, in these terms, for two functions s(x), t(x), we have st = {st, st} where () denotes the derivative. example. let (g,) be a group and f(x)=a the constant map on g, then xy={ay,xa}, x,yg. the (g,)/* is singleton, indeed, we have a -1 (a 1 x)={x,e} xg, so xe, xg, thus *(x)=*(e). for f(x)=e we obtain xy={x,y}, the smallest incidence hope. propositions 8. let gg is a generator of the group (g,). then, (a) for every f, g is a generator in (g,), with period at most n. (b) suppose that there exists an element w such that f(w)=g, then the element w is a generator in (g,), with period at most n. definitions 9. let (g,) be a groupoid and fi:gg, ii, be a set of maps. take the map f:gp(g) such that f(x)={fi(x)ii }, i.e. the union of fi(x). we call union -hopes, if we consider the map f(x). special case: the union of f with the identity, i.e. f= f(id), so f(x)={x,f(x)}, xg, which is called b-hope. we denote the b-hope by (), so xy = { xy, f(x)y, xf(y) }, x,yg. remark that  contains the operation () so it is a b-hope. if f: gp(g){}, then the b-hope is defined by using the map f(x)={x}f(x), xg. a construction between  and  is the one which obtained by using special maps. definition 10. let (g,) be a groupoid and f:gg be a map, we call basic set of the map f the set b = bf = { xg: f(x)=x }. then, if b, we have xy = xy, x,yb, xy = { xy, xf(y) }, xb, yg-b, xy = { f(x)y, xy }, xg-b, yb, 34 xy = { f(x)y, xf(y) }, x,yg-b. for (g,) groups, we obtain the following: if b is a subgroup of (g,), then (b,) is a sub-hv-group of (g,). if eb, then e is a unit of (b,) because it belongs into the kernel of f. inverses: if u is a unit of (b,), then xg, has an inverse in (g,) if f(x)u = uf(x). therefore an element xb has an inverse iff xu=ux. if eb then the element x -1 is an inverse of x in (g,) as well. proposition 11. let gg is a generator of the group (g,). if gb then g is a generator in (g,), f, with period at most n. there is connection of -hopes with other hyperstructures: example. merging and . if (h,) is a groupoid and hh merges with the hh, then h does not appeared and we have for the merge (h,◦), h◦x = {hx, hx}, x◦h = {xh, xh}, h◦h = {hh, hh, hh, hh} and in rest cases (◦) coincides with (), so we have merge (h-{h},◦). similarly, if (h,) is a hypergroupoid then we have h◦x= (hx)(hx), x◦h= (xh)(xh), h◦h= (hh)(hh)(hh)(hh) in order to see a connection of merge with the -hope, consider the map f such that f(h)=h and f(x)=x in the rest cases. then in (h-{h},) we have, x,yh-{h} hx = {hx, hx}, xh = {xh, xh} and hh = {hh, hh} and in the rest cases () coincides with (). therefore,  ◦, or hh = {hh, hh}  {hh, hh, hh, hh}= h◦h and in the remaining cases we have ◦. example. p-hopes [14]. let (g,) be a commutative semigroup and pg. consider the multivalued map f such that f(x) = px, xg. then we have xy = xyp, x,yg. so the -hope coincides with the well known class of p-hopes [20]. one can define theta-hopes on rings and other more complicate structures, where more than one -hopes can be defined. moreover, one can replace structures by hyper ones or by hv-structures, as well [23],[24]. 35 definition 12. let (r,+,) be a ring and f:rr, g:rr be two maps. we define two hyperoperations (+) and (), called both theta-operations, on r as follows x+y = {f(x)+y, x+f(y) } and xy = {g(x)y, xg(y) }, x,yg. a hyperstructure (r,+,), where (+), () be hyperoperations which satisfy all hv-ring axioms, except the weak distributivity, will be called hv-near-ring. proposition 13. let (r,+,) ring and f:rr, g:rr maps. the hyperstructure (r,+,), called theta, is an hv-near-ring. moreover (+) is commutative. proof. first, one can see that all properties of an hv-ring, except the distributivity, are valid. for the distributivity we have, x,y,zr, x(y+z)  (xy)+(xz) = . ■ in order more properties to be valid, the  can be replaced by . proposition 14. let (r,+,) ring and f:rr, g:rr maps, then (r,+,), is an hv-ring. proof. the only one axiom we have to see is the distributivity. so, x,y,zr, x(y+z) = {g(x)(y+z), g(x)(f(y)+z), g(x)(y+f(z)), xg(y+z), xg(f(y)+z), xg(y+f(z))} and (xy)+(xz) = {g(x)(y+z), f(g(x)y)+g(x)z, g(x)y+f(g(x)z), g(x)y+xg(z), f(g(x)y)+xg(z), g(x)y+f(xg(z)), xg(y)+g(x)z, f(xg(y))+g(x)z, xg(y)+f(g(x)z), x(g(y)+g(z)), f(xg(y))+xg(z), xg(y)+f(xg(z))}. so x(y+z)  (xy)+(xz) ={ g(x)(y+z)}  . therefore, (r,+,) is an hv-ring. ■ remark. if (r,+,) ring and f:rr, g:rr maps, then (r,+,) is still an hvnear-ring. theorems 15. (a) consider the group of integers (z,+) and let n0 be a natural number. take the map f such that f(0)=n and f(x)=x, xz-{0}. then (z,)/*  (zn,+). (b) consider the ring of integers (z,+,) and let n0 be a natural. consider the map f such that f(0)=n and f(x)=x, xz-{0}. then (z,+,) is an hv-near-ring, with (z,+,)/γ*  zn. 36 special case of the above is for n=p, prime, then (z,+,) is an hv-field. proposition 16. let (v,+,) be an algebra over the field (f,+,) and f:vv be a map. consider the -hope defined only on the multiplication of the vectors (), then (v,+,) is an hv-algebra over f, where the related properties are weak. if, moreover f is linear then we have more strong properties. definition 17. let l be a lie algebra, defined on an algebra (v,+,) over the field (f,+,) with lie bracket [x,y]=xy-yx. consider a map f:ll, then the -hope is defined by xy = {f(x)y-f(y)x, f(x)y-yf(x), xf(y)-f(y)x, xf(y)-yf(x)} proposition 18. let (v,+,) be an algebra over the field (f,+,) and f:vv be a linear map. consider the -hope defined only on the multiplication of the vectors (), then (v,+,) is an hv-algebra over f, with respect to lie bracket, where the weak anti-commutatinity and the inclusion linearity is valid. if (g,) is a semigroup then, for every f, the b-operation () is wass. 4. hyprestructures in questionnaires during last decades hyperstructures seem to have a variety of applications not only in other branches of mathematics but also in many other sciences including the social ones. these applications range from biomathematics and hadronic physics to automata theory, to mention but a few. this theory is closely related to fuzzy theory; consequently, hyperstructures can now be widely applicable in industry and production, too. in several papers, such as [2],[4],[11],[12] one can find numerous applications; similarly, in the books [4], [6] a wide variety of applications is also presented. an important new application, which combines hyperstructure theory and fuzzy theory, is to replace in questionnaires the scale of likert by the bar of vougiouklis & vougiouklis. the suggestion is the following [10]: definition 19. “in every question substitute the likert scale with ‘the bar’ whose poles are defined with ‘0’ on the left end, and ‘1’ on the right end: 0 1 the subjects/participants are asked instead of deciding and checking a specific grade on the scale, to cut the bar at any point s/he feels expresses her/his answer to the specific question”. 37 the use of the bar of vougiouklis & vougiouklis instead of a scale of likert has several advantages during both the filling-in and the research processing. the final suggested length of the bar, according to the golden ratio, is 6.2cm, see [9], [25]. now we state our main problem for this paper by using this bar and we can describe in mathematical model using theta-hopes. problem 20. in the research processing suppose that we want to use likert scale dividing the continuum [01] both by, first, into equal steps (segments) and, second, into equal-area spaces according to gauss distribution [9], [25]. if we consider both types of divisions into n segments, then the continuum [01] is divided into 2n-1 segments, if n is odd number and into 2(n-1) segments, if n is even number. we can number the segments and we can consider as an organized devise the group (zk,) where k=2n-1 or 2(n-1). then we can obtain several hyperstructures using -hopes as the following way: we can have two partitions of the final segments, into n classes either using the division into equal steps or the gauss distribution by putting in the same class all segments that belong (a) to the equal step or (b) to equal-area spaces according to gauss distribution. then we can consider two kinds of maps (i) a multi-map where every element corresponds to the hole class or (ii) a map where every element corresponds to one special fixed element of the same class. using these maps we define the -hopes and we obtain the corresponding hv-structure. an example for the case (i) is the following: example 21. suppose that we take the case of the likert scale with 5 equal steps: [0-1.24-2.48-3.72-4.96-6.2] and the gauss 5 equal areas: [0-2.4-2.9-3.3-3.8-6.2] we have 9 segments as follows [0 –1.24 – 2.4 – 2.48 – 2.9 – 3.3 – 3.72 – 3.8 – 4.96 – 6.2] therefore, if we consider the set z9 and if we name the segments by 1, 2,…, 8, 0, then if we consider the equal steps partition: {1}, {2,3}, {4,5,6}, {7,8}, {0} we take, according to the above construction the multi-map f such that f(1)={1}, f(2)={2,3}, f(3)={2,3}, f(4)={4,5,6}, f(5)={4,5,6}, f(6)={4,5,6}, f(7)={7,8}, f(8)={7,8}, f(0)={0}, then we obtain the following table: 38  1 2 3 4 5 6 7 8 0 1 2 3,4 3,4 5,6,7 5,6,7 5,6,7 0,8 0,8 1 2 3,4 4,5 4,5,6 6,7,8 6,7,8 6,7,8,9 0,1 0,1,2 2,3 3 3,4 4,5,6 5,6 0,6,7,8 0,6,7,8 0,6,7,8 0,1,2 0,1,2 2,3 4 5,6,7 6,7,8 0,6,7,8 0,1,8 0,1,2,8 0,1,2,3,8 2,3,4 2,3,4,5 4,5,6 5 5,6,7 6,7,8 0,6,7,8 0,1,2,8 0,1,2 0,1,2,3 2,3,4 3,4,5 4,5,6 6 5,6,7 6,7,8,9 0,6,7,8 0,1,2,3,8 0,1,2,3 1,2,3 2,3,4,5 3,4,5 4,5,6 7 0,8 0,1 0,1,2 2,3,4 2,3,4 2,3,4,5 5,6 5,6,7 7,8 8 0,8 0,1,2 0,1,2 2,3,4,5 3,4,5 3,4,5 5,6,7 6,7 7,8 0 1 2,3 2,3 4,5,6 4,5,6 4,5,6 7,8 7,8 0 5. hyperstructures in several scales obtained from the bar now we represent a mathematic model on obtained from the problem 20: construction 22. consider a group (g,) and suppose take a partition gi,ii of the g. select and fix an element gi of each partition class gi, and consider the map f: gg such that f(x)= gi , xgi, then (g,) is an hv-group. moreover the fundamental group (g/r,)/β* is (up to isomorphism) a subgroup of the corresponding fundamental group (g,)/β*. proof. first, we remark that the -hv-group (g,) is an hv-group because this is true for all given maps. now, let us call r the given partition. for all xgi and ygj we have xy={giy,xgj}, thus we remark that the elements giy and xgj belong to the same r class. therefore, the β*-classes with respect to , are subsets of the β*-classes with respect to the r-classes. the fundamental group (g/r,)/β* is (up to isomorphism) a subgroup of the corresponding fundamental group (g,)/β*. ■ theorem 23. in the above construction, if one of the selected elements is the unit element e of the group (g,), otherwise, if there exist an element zg such that f(z)=e, then we have (g/r,)/β* = (g,)/β*. proof. since there exist zg such that f(z)=e, then for all xgi, we have f(x)=gi, consequently, f(e)=e. moreover, for all xgi, we have xe = {gie, xe}= {gi, x}, 39 thus, x belongs to the fundamental class to gi with respect to -hope. so gi β*(gi) and from the above theorem we obtain that (g/r,)/β* = (g,)/β*. ■ in hypergroups does not necessarily exist any unit element and if there exists a unit this is not necessarily unique. moreover the -hopes do not have always the unit element of the group as unit for the corresponding -hope. this is so because ee = {f(e)e, ef(e)} = {f(e)}. however for the above hyperstructure we have the following: proposition 24. suppose (g,) be a group and gi,ii be a partition of g. for any class we fix a gigi, and take the map f: gg: f(x)=gi, xgi. if for the unit element e, in (g,), we have f(e)=e, i.e. e is any fixed element, then e is also a unit element of the hv-group (g,). moreover (f(x)) -1 is an inverse element in the hv-group (g,), of x. proof. for all xg we have xe = {f(x)e, xe} = {f(x), x} x. thus, e is a unit element in (g,). moreover, xg, denoting (f(x)) -1 the inverse of f(x) in (g,), we have x(f(x)) -1 = {f(x)(f(x)) -1 , xf((f(x)) -1 )} = {e, xf((f(x)) -1 )} e. therefore the element (f(x)) -1 is an inverse of x with respect to . this theorem states that the inverse gi -1 in (g,), of every fixed element gi, is also an inverse in (g,) of all elements which belong to their partition class gi. finally, remark that some of the elements of g may have more than one inverse in (g,). ■ now we conclude with an example of the above construction 22 on our main problem 20: example 11. in the case of the likert scale with 6 equal steps: [0-1-2.1-3.1-4.15.1-6.2] and the gauss 6 equal areas: [0-2.23-2.73-3.1-3.47-3.97-6.2] we have 10 segments as follows [0 – 1 – 2.1 – 2.23 – 2.73 – 3.1 – 3.47 – 3.97 – 4.1 – 5.1 – 6.2] therefore, if we consider the set z10 and if we name the segments by 1, 2,…, 9, 0, then if we consider the gauss partition: {1,2,3}, {4}, {5}, {6}, {7}, 40 {8,9,0} we take, according to the above theorem, the map f such that f(1)={1}, f(2)={1}, f(3)={1}, f(4)={4}, f(5)={5}, f(6)={6}, f(7)={7}, f(8)={0}, f(9)={0}, f(0)={0}, then we obtain the following table: references [1] r.bayon, n.lygeros, advanced results in enumeration of hyperstructures, to appear in journal of algebra. [2] j.chvalina, s.hoskova, modelling of join spaces with proximities by firstorder linear partial differential operators, italian journal of pure and applied math., n.21 (2007), 177-190. [3] p.corsini, prolegomena of hypergroup theory, aviani editore, 1993. [4] p.corsini, v.leoreanu, applications of hypergroup theory, kluwer academic publishers, 2003.  1 2 3 4 5 6 7 8 9 0 1 2 2,3 2,4 5 6 7 8 1,9 0,1 1 2 2,3 3 3,4 5,6 6,7 7,8 8,9 2,9 0,2 1,9 3 2,4 3,4 4 5,7 6,8 7,9 0,8 3,9 0,3 1,3 4 5 5,6 5,7 8 9 0 1 2,4 3,4 4 5 6 6,7 6,8 9 0 1 2 3,5 4,5 5 6 7 7,8 7,9 0 1 2 3 4,6 5,6 6 7 8 8,9 0,8 1 2 3 4 5,7 6,7 7 8 1,9 2,9 3,9 2,4 3,5 4,6 5,7 8 8,9 0,8 9 0,1 0,2 0,3 3,4 4,5 5,6 6,7 8,9 9 0,9 0 1 1,9 1,3 4 5 6 7 0,8 0,9 0 41 [5] b.davvaz, a brief survey of the theory of hv-structures, 8 th aha congress, spanidis press (2003), 39-70. [6] b.davvaz, v.leoreanu, hyperring theory and applications, international academic press, 2007. [7] b.davvaz, t.vougiouklis, n-ary hypergroups, iranian journal of science & technology, transaction a, v.30, n.a2, (2006), 165-174. [8] b.davvaz, s.vougioukli, t.vougiouklis, on the multiplicative-rings derived from helix hyperoperations, utilitas mathematica, 84, 2011, 53-63. [9] p.kambaki-vougioukli, a.karakos, n.lygeros, t.vougiouklis, fuzzy instead of discrete, annals of fuzzy mathematics and informatics, v.2, n.1 (2011), 81-89. [10] p. kambaki-vougioukli, t.vougiouklis, bar instead of scale, ratio sociologica, 3, (2008), 49-56. [11] a.maturo, e.sciarra, i.tofan, a formalization of some aspects of social organization by means of the fuzzy set theory,, ratio sociologica, 1, (2008), 5-20. [12] r.m.santilli, t.vougiouklis, isotopies, genotopies, hyperstructures and their applications, proc. new frontiers in hyperstructures and related algebras, hadronic press (1996), 177-188. [13] s.vougioukli, hv-vector spaces from helix hyperoperations, int. journal of mathematics and analysis (new series), v.1, no 2 (2009), 109-120. [14] t.vougiouklis, generalization of p-hypergroups, rendiconti circolo matematico di palermo, s.ii,36 (1987), 114-121. [15] t.vougiouklis, the fundamental relation in hyperrings. the general hyperfield, 4 th aha congress, world scientific (1991), 203-211. [16] t.vougiouklis, representations of hypergroups by generalized permutations, algebra universalis, 29 (1992), 172-183. [17] t.vougiouklis, hyperstructures and their representations, monographs in mathematics, hadronic press, 1994. [18] t.vougiouklis, some remarks on hyperstructures, contemporary mathematics, american mathematical society, 184, (1995), 427-431. [19] t.vougiouklis, enlarging hv-structures, algebras and combinatorics, icac’97, hong kong, springer verlag (1999), 455-463. [20] t.vougiouklis, on hv-rings and hv-representations, discrete mathematics, elsevier, 208/209 (1999), 615-620. [21] t.vougiouklis, finite hv-structures and their representations, rendiconti seminario matematico di messina s.ii, v.9 (2003), 245-265. [22] t.vougiouklis, a hyperoperation defined on a groupoid equipped with a map, ratio mathematica, n.1 (2005), 25-36. 42 [23] t.vougiouklis, -operations and hv-fields, acta mathematica sinica, english series, v.24, n.7 (2008), 1067-1078. [24] t.vougiouklis, the relation of the theta-hyperoperation () with the other classes of hyperstructures, journal of basic science 4, n.1 (2008), 135-145. [25] t.vougiouklis, p.kambaki-vougioukli, on the use of the bar, china-usa business review, vol.10, no. 6 (2011), 484-489. ratio mathematica vol. 33, 2017, pp. 151-165 issn: 1592-7415 eissn: 2282-8214 hyperstructures in lie-santilli admissibility and iso-theories ruggero maria santilli∗, thomas vougiouklis† ‡doi:10.23755/rm.v33i0.374 abstract in the quiver of hyperstructures professor r. m. santilli, in early 90’es, tried to find algebraic structures in order to express his pioneer lie-santilli’s theory. santilli’s theory on ’isotopies’ and ’genotopies’, born in 1960’s, desperately needs ’units e’ on left or right, which are nowhere singular, symmetric, real-valued, positive-defined for n-dimensional matrices based on the so called isofields.these elements can be found in hyperstructure theory, especially in hv-structure theory introduced in 1990. this connection appeared first in 1996 and actually several hv-fields, the e-hyperfields, can be used as isofields or genofields so as, in such way they should cover additional properties and satisfy more restrictions. several large classes of hyperstructures as the p-hyperfields, can be used in lie-santilli’s theory when multivalued problems appeared, either in finite or in infinite case. we review some of these topics and we present the lie-santilli admissibility in hyperstructures. keywords: lie-santilli iso-theory, hyperstructures, hope, hv-structures. 2010 ams mathematics subject classification: 20n20; 16y99. ∗the institute for basic research, 35246 us 19 north, p.o.box 34684, cpalm harbor, florida 34684, usa; research@i-b-r.org †democritus university of thrace, school of education, 68100 alexandroupolis, greece; tvougio@eled.duth.gr ‡ c©ruggero maria santilli and thomas vougiouklis. received: 31-10-2017. accepted: 2612-2017. published: 31-12-2017. 151 ruggero maria santilli and thomas vougiouklis 1 introduction in t. vougiouklis, ”the santilli’s theory ’invasion’ in hyperstructures” [24], there is a first description on how santilli’s theories effect in hyperstructures and how new theories in mathematics appeared by santilli’s pioneer research. we continue with new topics in this direction. last years hyperstructures have applications in mathematics and in other sciences as well. the applications range from biomathematics -conchology, inheritanceand hadronic physics or on leptons, in the santilli’s iso-theory, to mention but a few. the hyperstructure theory is closely related to fuzzy theory; consequently, can be widely applicable in linguistic, in sociology, in industry and production, too. for all the above applications the largest class of the hyperstructures, the hv-structures, is used, they satisfy the weak axioms where the non-empty intersection replaces the equality. the main tools of this theory are the fundamental relations which connect, by quotients, the hv-structures with the corresponding classical ones. these relations are used to define hyperstructures as hv-fields, hvvector spaces and so on. hypernumbers or hv-numbers are called the elements of hv-fields and they are important for the representation theory. the hyperstructures were introduced by f. marty in 1934 who defined the hypergoup as a set equipped with an associative and reproductive hyperoperation. m. koskas in 1970 was introduced the fundamental relation β∗, which it turns to be the main tool in the study of hyperstructures. t. vougiouklis in 1990 was introduced the hv-structures, by defining the weak axioms. the class of hvstructures is the largest class of hyperstructures. motivation for hv-structures: the quotient of a group with respect to an invariant subgroup is a group. the quotient of a group with respect to any subgroup is a hypergroup. the quotient of a group with respect to any partition is an hv-group. the lie-santilli theory on isotopies was born in 1970’s to solve hadronic mechanics problems. santilli proposed a ’lifting’ of the n-dimensional trivial unit matrix of a normal theory into a nowhere singular, symmetric, real-valued, positive-defined, n-dimensional new matrix. the original theory is reconstructed such as to admit the new matrix as left and right unit. according to santilli’s iso-theory [14], [8] on a field f = (f,+, ·), a general isofield f̂ = f̂(â,+̂,×̂) is defined to be a field with elements â = a × 1̂, called isonumbers, where a ∈ f , and 1̂ is a positive-defined element generally outside f, equipped with two operations +̂ and ×̂ where +̂ is the sum with the conventional additive unit 0, and ×̂ is a new product â×̂b̂ := â× t̂ × b̂, with 1̂ = t̂−1,∀â, b̂ ∈ f̂. 152 hyperstructures in lie-santilli admissibility and iso-theories called iso-multiplication, for which 1̂ is the left and right unit of f̂, 1̂×̂â = â× 1̂ = â,∀â ∈ f̂ called iso-unit. the rest properties of a field are reformulated analogously. the isofields needed in this theory correspond into the hyperstructures were introduced by santilli & vougiouklis in 1996 [15], and called e-hyperfields. they point out that in physics the most interesting hyperstructures are the one called e-hyperstructures which contain a unique left ant right scalar unit. 2 basic definitions on hyperstructures in what follows we present the related hyperstructure theory, enriched with some new results. however one can see the books and related papers for more definitions and results on hyperstructures and related topics: [2], [4], [17], [18], [19], [20], [23], [31], [33]. in a set h is called hyperoperation (abbreviated: hope) or multivalued operation, any map from h ×h to the power set of h. therefore, in a hope · : h ×h → ℘(h) : (x,y) → x ·y ⊂ h the result is subset of h, instead of element as we have in usually operations. in a set h equipped with a hope · : h ×h → ℘(h)−{∅}, we abbreviate by wass the weak associativity: (xy)z ∩x(yz) 6= ∅,∀x,y,z ∈ h and by cow the weak commutativity: xy ∩yx 6= ∅,∀x,y ∈ h. the hyperstructure (h, ·) is called hv-semigroup if it is wass and it is called hv-group if it is reproductive hv-semigroup, i.e. xh = hx = h,∀x ∈ h. the hyperstructure (r,+, ·) is called hv-ring if (+) and (·) are wass, the reproduction axiom is valid for (+), and (·) is weak distributive to (+): x(y + z)∩ (xy + xz) 6= ∅,(x + y)z ∩ (xz + yz) 6= ∅,∀x,y,z ∈ r. an hv-structure is very thin iff all hopes are operations except one, with all hyperproducts singletons except one, which is set of cardinality more than one. the main tool to study all hyperstructures are the fundamental relations β*, γ* and �*, which are defined, in hv-groups, hv-rings and hv-vector spaces, respectively, as the smallest equivalences so that the quotient would be group, ring and vector space, respectively [17], [18]. a way to find fundamental classes is given by analogous to the following: theorem 2.1. let (h, ·) be hv-group and u all finite products of elements of h. define the relation β by setting xβy iff {x,y} ⊂ u,u ∈ u. then β* is the transitive closure of β. 153 ruggero maria santilli and thomas vougiouklis let (r,+, ·) be hv-ring, u all finite polynomials of r. define γ in r as follows: xγy iff {x,y} ⊂ u where u ∈ u. then γ* is the transitive closure of γ. an element is called single if its fundamental class is singleton. the fundamental relations are used for general definitions. thus, to define the hv-field the γ* is used [17], [18]: a hv-ring (r,+, ·) is called hv-field if r/γ* is a field. in the sequence the hv-vector space is defined. let (f,+, ·) be hv-field, (v,+) a cow hv-group and there exists an external hope · : f ×v → ℘(v) : (a,x) → ax such that, ∀a,b ∈ f and ∀x,y ∈ v, we have a(x + y)∩ (ax + ay) 6= ∅,(a + b)x∩ (ax + bx) 6= ∅,(ab)x∩a(bx) 6= ∅, then v is called an hv-vector space over f. in the case of an hv-ring instead of hv-field then the hv-modulo is defined. in the above cases the fundamental relation �* is the smallest equivalence such that the quotient v/�* is a vector space over the fundamental field f/γ*. let (h, ·), (h,∗) be hv-semigroups defined on the same set h. (·) is called smaller than (∗), and (∗) greater than (·), iff there exists an f ∈ aut(h,∗) such that xy ⊂ f(x∗y),∀x,y ∈ h then we write · ≤ ∗ and we say that (h,∗) contains (h, ·). if (h, ·) is a structure then it is called basic structure and (h,∗) is called hb-structure. the little theorem. greater hopes than the ones which are wass or cow, are also wass or cow, respectively. the definition of hv-field introduced a new class of hyperstructures: the hv-semigroup (h, ·) is called h/v-group if the quotient h/β* is a group. in [20] the ’enlarged’ hyperstructures were examined if an element, outside the underlying set, appears in one result. in enlargement or reduction, most useful in representations are hv-structures with the same fundamental structure. the attach construction. let ((h, ·) be an hv-semigroup and v /∈ h. we extend (·) into h = h∪{v} as follows: x·v = v ·x = v,∀x ∈ h, and v ·v = h. then (h, ·) is an h/v-group where (h, ·)/β∗ ∼= z2 and v is single element. we call the hyperstructure (h, ·) attach h/v-group of (h, ·) . definition 2.1. let (h, ·) be a hypergroupoid. we say that remove h ∈ h, if simply consider the restriction of (·) on h −{h}. we say that h ∈ h absorbs h ∈ h if we replace h, whenever it appears, by h. we say that h ∈ h merges with h ∈ h, if we take as product of x ∈ h by h, the union of the results of x with both h and h, and consider h and h as one class, with representative h. 154 hyperstructures in lie-santilli admissibility and iso-theories the uniting elements method was introduced by corsini & vougiouklis [3]. with this method one puts in the same class more elements. this leads, through hyperstructures, to structures satisfying additional properties. the uniting elements method is the following: let g be algebraic structure and d be a property, which is not valid and it is described by a set of equations; then, consider the partition in g for which it is put in the same partition class, all pairs that causes the non-validity of d. the quotient g/d is an hv-structure. then, quotient out the hv-structure g/d by the fundamental relation β*, a stricter structure (g/d)/β* for which the property d is valid, is obtained. an application is when more than one properties are desired then: theorem 2.2. [18] let (g, ·) be a groupoid, and f = {f1, . . . ,fm,fm+1, . . . ,fm+n} be a system of equations on g consisting of two subsystems fm = {f1, . . . ,fm} and fn = {fm+1, . . . ,fm+n}. let σ, σm be the equivalence relations defined by the uniting elements procedure using the systems f and fm resp., and let σn be the equivalence relation defined using the induced equations of fn on the groupoid gm = (g/σm)/β∗. then (g/σ)/β∗ ∼= (gm/σn)/β∗. in a groupoid with a map on it, a hope is introduced [22]: definition 2.2. let (g, ·) be groupoid (resp., hypergroupoid) and f : g → g be map. we define a hope (∂), called theta and we write ∂-hope, on g as follows x∂y = {f(x) ·y,x ·f(y)},∀x,y ∈ g. (resp.x∂y = (f(x) ·y)∪ (x ·f(y),∀x,y ∈ g) if (·) is commutative then (∂) is commutative. if (·) is cow, then (∂) is cow. motivation for a ∂-hope is the map derivative where only the product of functions is used. thus for two functions s(x), t(x), we have s∂t = {s′t,st′} where (′) is the derivative. a large class of hyperstructures based on classical ones are defined by [18]: definition 2.3. let (g, ·) be groupoid, then for every p ⊂ g, p 6= ∅, we define the following hopes called p-hopes: ∀x,y ∈ g p : xpy = (xp)y ∪x(py), pr : xpry = (xy)p ∪x(yp), p l : xp ly = (px)y ∪p(xy). the (g,p),(g,pr) and (g,p l) are called p-hyperstructures. the usual case is for (g, ·) semigroup, then xpy = (xp)y ∪x(py) = xpy and (g,p) is a semihypergroup. 155 ruggero maria santilli and thomas vougiouklis 3 representations. hv-lie algebras. representations of hv-groups, can be faced either by hv-matrices or by generalized permutations [18], [20], [31]. hv-matrix (or h/v-matrix) is called a matrix with entries elements of an hvring or hv-field (or h/v-field). the hyperproduct of hv-matrices a = (aij) and b = (bij), of type m×n and n× r, respectively, is a set of m× r hv-matrices, defined in a usual manner: a ·b = (aij) · (bij) = {c = (cij)|cij ∈⊕ ∑ aik · bkj}, where (⊕) is the n-ary circle hope on the hypersum: the sum of products of elements is considered to be the union of the sets obtained with all possible parentheses. in the case of 2 × 2 hv-matrices the 2-ary circle hope which coincides with the hypersum in the hv-ring. notice that the hyperproduct of hv-matrices does not nessesarily satisfy wass. the representation problem by hv-matrices is the following: definition 3.1. let (h, ·) be hv-group, (r,+, ·) be hv-ring and mr = {(aij)|aij ∈ r}, then any t : h → mr : h → t(h) with t(h1h2)∩t(h1)t(h2) 6= ∅,∀h1,h2 ∈ h, is called hv-matrix representation if t(h1h2) ⊂ t(h1)t(h2), then t is an inclusion representation, if t(h1h2) = t(h1)t(h2), then t is a good representation. if t is one to one and good then it is a faithful representation. the main theorem of representations of hv-structures is the following: theorem 3.1. a necessary condition in order to have an inclusion representation t of an hv-group (h, ·) by n × n hv-matrices over the hv-ring (r,+, ·) is the following: for all β∗(x),x ∈ h there must exist elements aij ∈ h,i,j ∈ {1, . . . ,n} such that t(β∗(a)) ⊂{a = (a′ij)|a ′ ij ∈ γ ∗(aij), i,j ∈{1, . . . ,n}} therefore, every inclusion representation t : h → mr : a 7→ t(a) = (aij) induces an homomorphic representation t * of h/β* over r/γ* by setting t∗(β∗(a)) = [γ∗(aij)],∀β∗(a) ∈ h/β∗, where the element γ∗(aij) ∈ r/γ∗ is the ij entry of the matrix t∗(β∗(a)). then t * is called fundamental induced representation of t . the helix hopes can be defined on any type of ordinary matrices [33], [34]: 156 hyperstructures in lie-santilli admissibility and iso-theories definition 3.2. let a = (aij) ∈ mm×n be matrix and s,t ∈ n, with 1 ≤ s ≤ m, 1 ≤ t ≤ n. the helix-projection is a map st : mm×n → ms×t : a → ast = (aij), where ast has entries aij = {ai+κs,j+λt|1 ≤ i ≤ s,1 ≤ j ≤ t and κ,λ ∈ n,i + κs ≤ m,j + λt ≤ n} let a = (aij) ∈ mm×n,b = (bij) ∈ mu×v be matrices and s = min(m,u), t = min(n,v). we define a hyper-addition, called helix-sum, by ⊕ : mm×n ×mu×v → ℘(ms×t) : (a,b) → a⊕b = = ast + bst = (aij) + (bij) ⊂ ms×t where (aij) + (bij) = {(cij) = (aij + bij)|aij ∈ aij and bij ∈ bij)}. let a = (aij) ∈ mm×n,b = (bij) ∈ mu×v and s = min(n,u). define the helix-product, by ⊗ : mm×n ×mu×v → ℘(mm×v) : (a,b) → a⊗b = = ams ·bsv = (aij) · (bij) ⊂ mm×v where (aij) · (bij) = {(cij) = ( ∑ aitbtj)|aij ∈ aij and bij ∈ bij)}. the helix-sum is commutative, wass, not associative. the helix-product is wass, not associative and not distributive to the helix-addition. using several classes of hv-structures one can face several representations. some of those classes are as follows [18], [19], [7]: definition 3.3. let m = mm×n, the set of m × n matrices on r and p = {pi : i ∈ i}⊆ m. we define, a kind of, a p-hope p on m as follows p : m ×m → ℘(m) : (a,b)apb = {ap ti b : i ∈ i}⊆ m where p t is the transpose of p. p is bilinear rees’ like operation where instead of one sandwich matrix a set is used. p is strong associative and inclusion distributive to sum: ap(b + c) ⊆ apb + apc,∀a,b,c ∈ m. so (m,+,p) defines a multiplicative hyperring on non-square matrices. definition 3.4. let m = mm×n be module of m×n matrices on r and take the sets s = {sk : k ∈ k}⊆ r,q = {qj : j ∈ j}⊆ m,p = {pi : i ∈ i}⊆ m. 157 ruggero maria santilli and thomas vougiouklis define three hopes as follows s : r×m → ℘(m) : (r,a) → rsa = {(rsk)a : k ∈ k}⊆ m q + : m×m → ℘(m) : (a,b) → aq + b = {a + qj + b : j ∈ j}⊆ m p : m×m → ℘(m) : (a,b) → apb = {ap ti b : i ∈ i}⊆ m then (m,s,q , p) is a hyperalgebra on r called general matrix p-hyperalgebra. the general definition of an hv-lie algebra is the following [26], [31], [16]: definition 3.5. let (l,+) be hv-vector space on (f,+, ·), φ : f → f/γ*, canonical map and ωf = {x ∈ f : φ(x) = 0}, where 0 is the zero of the fundamental field f/γ∗. similarly, let ωl be the core of the canonical map φ′ : l → l/�* and denote by the same symbol 0 the zero of l/�*. consider the bracket hope (commutator): [, ] : l×l → ℘(l) : (x,y) → [x,y] then l is an hv-lie algebra over f if the following axioms are satisfied: (l1) the bracket hope is bilinear, i.e.∀x,x1,x2,y,y1,y2 ∈ l, and λ1,λ2 ∈ f [λ1x1 + λ2x2,y]∩ (λ1[x1,y] + λ2[x2,y]) 6= ∅ [x,λ1y1 + λ2y2]∩ (λ1[x,y1] + λ2[x,y2]) 6= ∅, (l2) [x,x]∩ωl 6= ∅, ∀x ∈ l (l3) ([x, [y,z]] + [y, [z,x]] + [z, [x,y]])∩ωl 6= ∅,∀x,y,z ∈ l 4 the santilli’s: e-hyperstructures, iso-hyper theory. the e-hyperstructures where introduced in [15], [25] and where investigates in several aspects depending from applications [5], [6], [16], [31]. definition 4.1. a hyperstructure (h, ·) which contains a unique scalar unit e, is called e-hyperstructure. in an e-hyperstructure, we assume that for every element x, there exists an inverse x−1, i.e. e ∈ x ·x−1 ∩x−1 ·x. 158 hyperstructures in lie-santilli admissibility and iso-theories definition 4.2. a hyperstructure (f,+, ·), where (+) is an operation and (·) a hope, is called e-hyperfield if the following axioms are valid: (f,+) is an abelian group with the additive unit 0, (·) is wass, (·) is weak distributive with respect to (+), 0 is absorbing element: 0·x = x·0 = 0,∀x ∈ f , there exists a multiplicative scalar unit 1, i.e. 1 · x = x · 1 = x,∀x ∈ f , and ∀x ∈ f there exists a unique inverse x−1, such that 1 ∈ x ·x−1 ∩x−1 ·x. the elements of an e-hyperfield are called e-hypernumbers. in the case that the relation: 1 = x · x−1 = x−1 · x, is valid, then we say that we have a strong e-hyperfield. definition 4.3. main e-construction. given a group (g, ·), where e is the unit, we define in g, an extremely large number of hopes (⊗) as follows: x⊗y = {xy,g1,g2, ...},∀x,y ∈ g−{e}, and g1,g2, ... ∈ g−{e} g1,g2,... are not necessarily the same for each pair (x,y). (g,⊗) is an hv-group, it is an hb-group which contains the (g, ·). (g,⊗) is an e-hypergroup. moreover, if for each x,y such that xy = e, so we have x⊗ y = xy, then (g,⊗) becomes a strong e-hypergroup the proof is immediate since for both cases we enlarge the results of the group by putting elements from the set g and applying the little theorem. moreover it is easy to see that the unit e is unique scalar element and for each x in g, there exists a unique inverse x−1, such that 1 ∈ x · x−1 ∩ x−1 · x. finally if the last condition is valid then we have 1 = x · x−1 = x−1 · x, so the hyperstructure (g,⊗) is a strong e-hypergroup. example 4.1. consider the quaternion group q = {1,−1, i,−i,j,−j,k,−k} with defining relations i2 = j2 = −1, ij = −ji = k. denoting i = {i,−i},j = {j,−j},k = {k,−k} we may define a very large number (∗) hopes by enlarging only few products. for example, (−1)∗k = k,k∗i = j and i∗j = k. then the hyperstructure (q,∗) is a strong e-hypergroup. construction 4.1. [31], [32]. on the ring (z4,+, ·) we will define all the multiplicative h/v-fields which have non-degenerate fundamental field and, moreover they are, (a) very thin minimal, (b) cow (non-commutative), (c) they have 0 and 1, scalars. 159 ruggero maria santilli and thomas vougiouklis we have the isomorphic cases: 2⊗3 = {0,2} or 3⊗2 = {0,2}. the fundamental classes are [0] = {0,2}, [1] = {1,3} and we have (z4,+,⊗)/γ∗ ∼= (z2,+, ·). thus it is isomorphic to (z2 × z2,+). in this hv-group there is only one unit and every element has a unique double inverse. we can also define the analogous cases for the rings (z6,+, ·),(z9,+, ·), and (z10,+, ·). in order to transfer santilli’s iso-theory theory into the hyperstructure case we generalize only the new product ×̂ by replacing it by a hope including the old one [15], [27], [29], [32] and [1], [5], [6], [13], [14], [21], [24]. we introduce two general constructions on this direction as follows: construction 4.2. general enlargement. on a field f = (f,+, ·) and on the isofield f̂ = f̂(â,+̂,×̂) we replace in the results of the iso-product â×̂b̂ = â× t̂ × b̂, with 1̂ = t̂−1 of the element t̂ by a set of elements ĥab = {t̂, x̂1, x̂2, . . .} where x̂1, x̂2, . . . ∈ f̂, containing t̂ , for all â×̂b̂ for which â, b̂ /∈ {0̂, 1̂} and x̂1, x̂2, . . . ∈ f̂ −{0̂, 1̂}. if one of â, b̂, or both, is equal to 0̂ or 1̂, then ĥab = {t̂}. therefore the new iso-hope is â×̂b̂ = â× ĥab × b̂ = â×{t̂, x̂1, x̂2, . . .}× b̂,∀â, b̂ ∈ f̂ f̂ = f̂(â,+̂,×̂) becomes isohv-field. the elements of f are called isohvnumbers or isonumbers. more important hopes, of the above construction, are the ones where only for few ordered pairs (â, b̂) the result is enlarged, even more, the extra elements x̂i, are only few, preferable one. thus, this special case is if there exists only one pair (â, b̂) for which â×̂b̂ = â×{t̂, x̂}× b̂,∀â, b̂ ∈ f̂ and the rest are ordinary results, then we have a very thin isohv-field. the assumption ĥab = {t̂}, â or b̂, is equal to 0̂ or 1̂, with that x̂i, are not 0̂ or 1̂, give that the isohv-field has one scalar absorbing 0̂, one scalar 1̂, and ∀â ∈ f̂ one inverse. a generalization of p-hopes, used in santilli’s isotheory, is the following [5], [28], [31]: let (g, ·) be abelian group and p a subset of g with #p > 1. we define the hope (×p) as follows: x×p y = { x ·p ·y = {x ·h ·y|h ∈ p} if x 6= e and y 6= e x ·y if x = e or y = e we call this hope pe-hope. the hyperstructure (g,×p) is abelian hv-group. 160 hyperstructures in lie-santilli admissibility and iso-theories construction 4.3. the p-hope. consider an isofield f̂ = f̂(â,+̂,×̂) with â = a× 1̂, the isonumbers, where a ∈ f , and 1̂ is positive-defined outside f, with two operations +̂ and ×̂, where +̂ is the sum with the conventional unit 0, and ×̂ is the iso-product â×̂b̂ = â× t̂ × b̂, with 1̂ = t̂−1,∀â, b̂ ∈ f̂ take a set p̂ = {t̂, p̂1, ..., p̂s}, with p̂1, . . . , p̂s ∈ f̂ −{0̂, 1̂}, we define the isophv-field, f̂ = f̂(â,+̂,×̂p) where the hope ×̂p as follows: â×̂p b̂ := { â× ˆ̂p × b̂ = {â× ˆ̂h× b̂|ˆ̂h ∈ ˆ̂p} if â 6= 1̂ and b̂ 6= 1̂ â× ˆ̂t × b̂ if â = 1̂ or b̂ = 1̂ the elements of f̂ are called isop-hv-numbers. remark. if p̂ = {t̂, p̂}, that is that p̂ contains only one p̂ except t̂ . the inverses in isop-hv-fields, are not necessarily unique. example 4.2. non degenerate example on the above constructions: in order to define a generalized p-hope on ẑ7 = ẑ7(â,+̂,×̂), where we take p̂ = {1̂, 6̂}, the weak associative multiplicative hope is described by the table: ×̂ 0̂ 1̂ 2̂ 3̂ 4̂ 5̂ 6̂ 0̂ 0̂ 0̂ 0̂ 0̂ 0̂ 0̂ 0̂ 1̂ 0̂ 1̂ 2̂ 3̂ 4̂ 5̂ 6̂ 2̂ 0̂ 2̂ 4̂,3̂ 6̂,1̂ 1̂,6̂ 3̂,4̂ 5̂, 2̂ 3̂ 0̂ 3̂ 6̂,1̂ 2̂,5̂ 5̂,2̂ 1̂,6̂ 4̂,3̂ 4̂ 0̂ 4̂ 1̂,6̂ 5̂,2̂ 2̂,5̂ 6̂,1̂ 3̂,4̂ 5̂ 0̂ 5̂ 3̂,4̂ 1̂,6̂ 6̂,1̂ 4̂,3̂ 2̂,5̂ 6̂ 0̂ 6̂ 5̂,2̂ 4̂,3̂ 3̂,4̂ 2̂,5̂ 1̂,6̂ the hyperstructure ẑ7 = ẑ7(â,+̂,×̂) is commutative and associative on the product hope. moreover the β* classes on the iso-product ×̂ are {1̂, 6̂},{5̂, 2̂},{3̂, 4̂}. 5 the lie-santilli’s admissibility. another very important new field in hypermathematics comes straightforward from santilli’s admissibility. we can transfer santilli’s theory in admissibility for representations in two ways: using either, the ordinary matrices and a hope on them, or using hypermatrices and ordinary operations on them [10], [11], [12], [14], [16] and [7], [9], [30], [31], [34]. 161 ruggero maria santilli and thomas vougiouklis definition 5.1. let l be hv-vector space over the hv-field (f,+, ·), φ : f → f/γ∗, the canonical map and ωf = {x ∈ f : φ(x) = 0}, where 0 is the zero of the fundamental field f/γ∗. let ωl be the core of the canonical map φ′ : l → l/�* and denote by the same symbol 0 the zero of l/�*. take two subsets r,s ⊆ l then a lie-santilli admissible hyperalgebra is obtained by taking the lie bracket, which is a hope: [, ]rs : l×l → ℘(l) : [x,y]rs = xry−ysx = {xry−ysx|r ∈ r and s ∈ s} special cases, but not degenerate, are the ’small’ and ’strict’ ones: (a) when only s is considered, then [x,y]s = xy −ysx = {xy −ysx|s ∈ s} (b) when only r is considered, then [x,y]r = xry−yx = {xry−yx|r ∈ r} (c) when r = {r1,r2} and s = {s1,s2} then [x,y]rs = xry−ysx = {xr1y−ys1x,xr1y−ys2x,xr2y−ys1x,xr2y−ys2x}. (d) we have one case which is ’like’ p-hope for any subset s ⊆ l: [x,y]s = {xsy −ysx|s ∈ s} on non square matrices we can define admissibility, as well: construction 5.1. let l = (mm×n,+) be hv-vector space of m × n hypermatrices on the hv-field (f,+, ·),φ : f → f/γ∗, canonical map and ωf = {x ∈ f : φ(x) = 0}, where 0 is the zero of the field f/γ*. similarly, let ωl be the core of φ′ : l → l/�∗ and denote by the same symbol 0 the zero of l/�*. take any two subsets r,s ⊆ l then a santilli’s lie-admissible hyperalgebra is obtained by taking the lie bracket, which is a hope: [, ]rs : l×l → ℘(l) : [x,y]rs = xrty −ystx. notice that [x,y]rs = xrty −ystx = {xrty −ystx|r ∈ r and s ∈ s} special cases, but not degenerate, is the ’small’: r = {r1,r2} and s = {s1,s2} then [x,y]rs = xr ty −ystx = = {xrt1y −ys t 1x,xr t 1y −ys t 2x,xr t 2y −ys t 1x,xr t 2y −ys t 2x} 162 hyperstructures in lie-santilli admissibility and iso-theories references [1] r. anderson, a. a. bhalekar, b. davvaz, p. s. muktibodh, t. vougiouklis, an introduction to santilli’s isodual theory of antimatter and the open problem of detecting antimatter asteroids, numta b., 6 (2012-13), 1–33. 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[34] t. vougiouklis, s. vougiouklis, hyper lie-santilli admisibility, agg, 33, n.4, (2016), 427–442. 165 microsoft word capitolo intero n 9.doc microsoft word documento1 ratio mathematica vol. 33, 2017, pp. 181-201 issn: 1592-7415 eissn: 2282-8214 hv-fields, h/v-fields thomas vougiouklis∗ †doi:10.23755/rm.v33i0.386 abstract in the last decades, the hyperstructures have had a lot of applications in mathematics and in other sciences. these applications range from biomathematics and hadronic physics to linguistic and sociology. for applications the largest class of the hyperstructures, the hv-structures, is used, they satisfy the weak axioms where the non-empty intersection replaces the equality. the main tools in the theory of hyperstructures are the fundamental relations which connect, by quotients, the hv-structures with the corresponding classical ones. these relations are used to define hyperstructures as hv-fields, hv-vector spaces and so on, as well. the extension of the reproduction axiom, from elements to fundamental classes, introduces the extension of hv-structures to the class of h/v-structures. we focus our study mainly in the relation of these classes and we present some constructions on them. keywords: hope; hv-structure; h/v-structure; hv-field; h/v-field. 2010 ams subject classifications: 20n20, 16y99. ∗democritus university of thrace, school of education, 68100 alexandroupolis, greece; tvougiou@eled.duth.gr † c©thomas vougiouklis. received: 31-10-2017. accepted: 26-12-2017. published: 31-122017. 181 thomas vougiouklis 1 introduction the main object in this paper is the largest class of hyperstructures called hvstructures introduced in 1990 [35], which satisfy the weak axioms where the nonempty intersection replaces the equality. abbreviation: hyperoperation=hope. definition 1.1. an algebraic hyperstructure is called a set h equipped with at least one hope · : h × h → p(h) −{∅}. we abbreviate by wass the weak associativity: (xy)z ∩ x(yz) 6= ∅,∀x,y,z ∈ h and by cow the weak commutativity: xy ∩ yx 6= ∅,∀x,y ∈ h. the hyperstructure (h, ·) is called an hv semigroup if it is wass, it is called hv-group if it is reproductive hv-semigroup, i.e., xh = hx = h,∀x ∈ h. motivation. the quotient of a group by an invariant subgroup, is a group. f. marty (1934), ’sur une generalization de la notion de groupe’. 8eme congres math. scandinaves, stockholm, pp.45-49, states: the quotient of a group by a subgroup is a hypergroup. the quotient of a group by a partition (or equivalently to any equivalence) is an hv-group. in an hv-semigroup the powers are defined by: h1 = {h},h2 = h ·h,...,hn = h◦h◦...◦h, where (◦) is the n-ary circle hope, i.e. take the union of hyperproducts, n times, with all possible patterns of parentheses put on them. an hv-semigroup (h, ·) is cyclic of period s, if there is an element h, called generator, and a natural number s, the minimum : h = h1 ∪ h2... ∪ hs. analogously the cyclicity for the infinite period is defined [30], [33], [39]. if there is an h and s, the minimum: h = hs, then (h, ·), is called single-power cyclic of period s. definition 1.2. an (r, +, ·) is called hv−ring if (+) and (·) are wass, the reproduction axiom is valid for (+) and (·) is weak distributive with respect to (+): x(y + z) ∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅, ∀x,y,z ∈ r. let (r, +, ·) be an hv-ring, (m, +) be a cow hv-group and there exists an external hope · : r×m → p(m) : (a,x) → ax such that ∀a,b ∈ r and ∀x,y ∈ m we have a(x + y) ∩ (ax + ay) 6= ∅, (a + b)x∩ (ax + bx) 6= ∅, (ab)x∩a(bx) 6= ∅, then m is called an hv-module over f. in the case of an hv-field f, which is defined later, instead of an hv-ring r, then the hv−vector space is defined. for more definitions and applications on hyperstructures one can see books [4], [5], [9], [10], [11], [39] and papers as [3], [7], [8], [15], [16], [20], [21], [27], [38], [40], [41], [43], [48], [55], [68]. 182 hv-fields, h/v-fields definition 1.3. let (h, ·), (h,∗) be hv-semigroups on the same set h, the hope (·) is called smaller than the (∗), and (∗) greater than (·), iff there exists an f ∈ aut(h,∗) such that xy ⊂ f(x∗y), ∀x,y ∈ h. then we write · ≤ ∗ and we say that (h,∗) contains (h, ·). if (h, ·) is a structure then it is called basic structure and (h,∗) is called hb − structure. the little theorem. greater hopes than ones which are wass or cow, are also wass or cow, respectively. this theorem leads to a partial order on hv-structures and to posets [39], [42], [43], [21]. let (h, ·) be hypergroupoid. we remove h ∈ h, if we take the restriction of (·) in the set h −{h}. h ∈ h absorbs h ∈ h if we replace h by h and h does not appear. h ∈ h merges with h ∈ h, if we take as product of any x ∈ h by h, the union of the results of x with both h, h, and consider h and h as one class with representative h. the main tool in hyperstructures is the fundamental relation. m. koscas 1970, [20], defined in hypergroups the relation β and its transitive closure β*. this relation is defined in hv-groups, as well, and connect hyperstructures with the classical structures. t. vougiouklis [34], [35], [39], [40], [41], [53], [54], [60], introduced the γ* and �* relations, which are defined, in hv-rings and hv-vector spaces, respectively. he also named all these relations, fundamental. (see also [4], [5], [1], [8], [10], [11]). definition 1.4. the fundamental relations β*, γ* and �*, are defined, in hvgroups, hv-rings and hv-vector spaces, respectively, as the smallest equivalences so that the quotient would be group, ring and vector spaces, respectively. specifying the above motivation we remark that: let (g, ·) be a group and r be an equivalence relation (or a partition) in g, then (g/r, ·) is an hv-group, therefore we have the quotient (g/r, ·)/β∗ which is a group, the fundamental one. the main theorem to find the fundamental classes is the following: theorem 1.1. let (h, ·) be an hv-group and denote by u the set of all finite products of elements of h. we define the relation β in h by setting xβy iff {x,y}⊂ u where u ∈ u. then β* is the transitive closure of β. notation. we denote by [x] the fundamental class of the element x ∈ h. therefore β∗(x) = [x]. analogous theorems are for hv-rings, hv-vector spaces and so on. for proof, see [34], [39]. an element is called single [39] if its fundamental class is singleton so, [x] = {x}. 183 thomas vougiouklis more general structures can be defined by using the fundamental structures. an application in this direction is the general hyperfield. there was no general definition of a hyperfield, but from 1990 [35] there is the following [38], [39]: definition 1.5. an hv-ring (r, +, ·) is called hv-field if r/γ* is a field. since the algebras are defined on vector spaces, the analogous to theorem 1.1, on hv-vector spaces is the following: let (v, +) be an hv-vector space over the hv-field f. denote by u the set of all expressions consisting of finite hopes either on f and v or the external hope applied on finite sets of elements of f and v. we define the relation �, in v as follows: x�y iff {x,y} ∈ u where u ∈ u. then the relation �* is the transitive closure of the relation �. definition 1.6. [53], [54], [57]. let (l, +) be an hv-vector space over the hvfield (f, +, ·), φ : f → f/γ* the canonical map and ωf = {x ∈ f : φ(x) = 0}, where 0 is the zero of the fundamental field f/γ*. let ωl be the core of the canonical map φ′ : l → l/�* and denote by the same symbol 0 the zero of l/�*. consider the bracket (commutator) hope: [, ] : l×l → p(l) : (x,y) → [x,y] then l is an hv-lie algebra over f if the following axioms are satisfied: (l1) the bracket hope is bilinear, i.e. [λ1x1 + λ2x2,y] ∩ (λ1[x1,y] + λ2[x2,y]) 6= ∅ [x,λ1y1 + λ2y2] ∩ (λ1[x,y1] + λ2[x,y2]) 6= ∅, ∀x,x1,x2,y,y1,y2 ∈ l,λ1,λ2 ∈ f (l2) [x,x] ∩ωl 6= ∅, ∀x ∈ l (l3) ([x, [y,z]] + [y, [z,x]] + [z, [x,y]]) ∩ωl 6= ∅, ∀x,y ∈ l in the definition 1.5, was introduced a new class of which is the following [45] (for a preliminary report see: t. vougiouklis. a generalized hypergroup, abstracts ams, vol. 19.3, issue 113, 1998, p.489): definition 1.7. the hv-semigroup (h, ·) is called h/v-group if h/β∗ is a group. an important and well known class of hyperstructures defined on classical structures are defined as follows [30], [33], [36], [57], [60]: definition 1.8. let (g, ·) be groupoid, then for every p ⊂ g,p 6= ∅, we define the following hopes called p-hopes: ∀x,y ∈ g p : xpy = (xp)y ∪x(py), 184 hv-fields, h/v-fields pr : xpry = (xy)p ∪x(yp), p l : xp ly = (px)y ∪p(xy). the (g,p),(g,pr), (g,p l) are called p-hyperstructures. the most usual case is if (g, ·) is semigroup, then xpy = (xp)y ∪ x(py) = xpy and (g,p) is a semihypergroup. a generalization of p-hopes, used in santilli’s isotheory, is the following [12], [13], [14]: let (g, ·) be abelian group and p a subset of g with #p > 1. we define the hope (×p ) as follows: x×p y = { x ·p ·y = {x ·h ·y|h ∈ p} if x 6= e and c 6= e x ·y if x = e or y = e we call this hope pe-hope. the hyperstructure (g,×p ) is abelian hv-group. definition 1.9. [36]. an hv-structure is called very thin if all hopes are operations except one, which has all hyperproducts singletons except one, which is a subset of cardinality more than one. therefore, in a very thin hv-structure in h there exists a hope (·) and a pair (a,b) ∈ h2 for which ab = a, with carda > 1, and all the other products, are singletons. from the properties of the very thin hopes the attach construction is obtained [43], [54]: let (h, ·) be an hv-semigroup and v /∈ h. we extend the (·) into h = h ∪{v} by: x ·v = v ·x = v,∀x ∈ h, and v ·v = h. the (h, ·) is an hv-group, where (h, ·)/β∗ ∼= z2 and v is a single. a class of hv-structures is the following [47], [49], [57], [60]: definition 1.10. let (g, ·) be groupoid (resp. hypergroupoid) and f : g → g be a map. we define a hope (∂), called theta-hope, we write ∂-hope, on g as follows x∂y = {f(x)·y,x·f(y)}, ∀x,y ∈ g. (resp. x∂y = (f(x)·y)∪(x·f(y)), ∀x,y ∈ g) if (·) is commutative then ∂ is commutative. if (·) is cow, then ∂ is cow. if (g, ·) is a groupoid (or hypergroupoid) and f : g → p(g) −{∅} be any multivalued map. we define the ∂-hope on g as follows: x∂y = (f(x) ·y) ∪ (x ·f(y)), ∀x,y ∈ g. the ∂-hopes can be defined in hv-vector spaces and hv-lie algebras: 185 thomas vougiouklis let (a, +, ·) be an algebra over the field f. take any map f : a → a, then the ∂-hope on the lie bracket [x,y] = xy −yx, is defined as follows x∂y = {f(x)y −f(y)x,f(x)y −yf(x),xf(y) −f(y)x,xf(y) −yf(x)}. then (a, +,∂) is an hv-algebra over f, with respect to the ∂-hopes on lie bracket, where the weak anti-commutativity and the inclusion linearity is valid. motivation for the theta-hope is the map derivative where only the multiplication of functions can be used. basic property: if (g, ·) is semigroup then ∀f, the ∂-hope is wass. example. (a) in integers (z, +, ·) fix n 6= 0, a natural number. consider the map f such that f(0) = n and f(x) = x, ∀x ∈ z − {0}. then (z,∂+,∂·), where ∂+ and ∂· are the ∂-hopes refereed to the addition and the multiplication respectively, is an hv-near-ring, with (z,∂+,∂·)/γ* ∼= zn. (b) in (z, +, ·) with n 6= 0, take f such that f(n) = 0 and f(x) = x, ∀x ∈ z−{n}. then (z,∂+,∂·) is an hv-ring, moreover, (z,∂+,∂·)/γ* ∼= zn. special case of the above is for n = p, prime, then (z,∂+,∂·) is an hv-field. the uniting elements method was introduced by corsini-vougiouklis [6] in 1989. with this method one puts in the same class, two or more elements. this leads, through hyperstructures, to structures satisfying additional properties. the uniting elements method is the following: let g be algebraic structure and d, a property which is not valid. suppose that d is described by a set of equations; then, take the partition in g for which it is put together, in the same class, every pair of elements that causes the non-validity of the property d. the quotient by this partition g/d is an hv-structure. then, quotient out the hv-structure g/d by the fundamental relation β*, a stricter structure (g/d)/β* for which the property d is valid, is obtained. it is very important if more properties are desired, then we have the following [39]: theorem 1.2. let (r, +, ·) be a ring, and f = {f1, ...,fm,fm+1, ...,fm+n} be a system of equations on r consisting of two subsystems fm = {f1, ...,fm} and fn = {fm+1, ...,fm+n}. let σ, σm be the equivalence relations defined by the uniting elements procedure using the systems f and fm respectively, and let σn be the equivalence relation defined using the induced equations of fn on the ring rm = (r/σm)/γ*. then, (r/σ)/γ∗ ∼= (rm/σn)γ∗. 186 hv-fields, h/v-fields combining the uniting elements procedure with the enlarging theory or the ∂-theory, we can obtain analogous results [39], [51], [54], [60], [22]. theorem 1.3. in the ring (zn, +, ·), with n=ms we enlarge the multiplication only in the product of the special elements 0 · m by setting 0 ⊗ m = {0,m} and the rest results remain the same. then (zn, +,⊗)/γ∗ ∼= (zm, +, ·). remark that we can enlarge other products as well, for example 2·m by setting 2⊗m = {2,m + 2}, then the result remains the same. in this case 0 and 1 remain scalars. corollary. in the ring (zn, +, ·), with n=ps where p is prime, we enlarge only the product 0 ·p by 0⊗p = {0,p} and the rest remain the same. then (zn, +,⊗) is very thin hv-field. 2 constructions of hv-fields and h/v-fields the class of h/v-groups is more general than the hv-groups since in h/v-groups the reproductivity is not valid. the reproductivity of classes is valid, i.e. if h is partitioned into equivalence classes, then x[y] = [xy] = [x]y,∀x,y ∈ h, because the quotient is reproductive. in a similar way the h/v-rings, h/v-fields, h/v-modulus, h/v-vector spaces etc are defined. remark 2.1. from definition of the hv-field, we remark that the reproduction axiom in the product, is not assumed, the same is also valid for the definition of the h/v-field. therefore, an hv-field is an h/v-field where the reproduction axiom for the sum is also valid. we know that the reproductivity in the classical group theory is equivalent to the axioms of the existence of the unit element and the existence of an inverse element for any given element. from the definition of the h/v-group, since a generalization of the reproductivity is valid, we have to extend the above two axioms on the equivalent classes. definition 2.1. let (h, ·) be an hv-semigroup, and denote [x] the fundamental, or equivalent class, of the element x ∈ h. we call unit class the class [e] if we have ([e] · [x]) ∩ [x] 6= ∅ and ([x] · [e]) ∩ [x] 6= ∅,∀x ∈ h, and for each element x ∈ h, we call inverse class of [x], the class [x′], if we have ([x] · [x′]) ∩ [e] 6= ∅ and ([x′] · [x]) ∩ [e] 6= ∅. 187 thomas vougiouklis the ’enlarged’ hyperstructures were examined in the sense that a new element appears in one result. in enlargement or reduction, most useful are those hvstructures or h/v-structures with the same fundamental structure [43], [53]. construction 2.1. (a) let (h, ·) be an hv-semigroup and v /∈ h. we extend the (·) into h = h ∪{v} as follows: x ·v = v ·x = v,∀x ∈ h, and v ·v = h. the (h, ·) is an h/v-group, called attach, where (h, ·)/β∗ ∼= z2 and v is a single element. we have core (h, ·) = h. the scalars and units of (h, ·) are scalars and units (resp.) in (h, ·). if (h, ·) is cow (resp. commutative) then (h, ·) is also cow (resp. commutative). (b) let (h, ·) be an hv-semigroup and {v1, . . . ,vn}∩ h = ∅, is an ordered set, where if vi < vj, when i < j. extend (·) in hn = h ∪{v1, . . . ,vn} as follows: x ·vi = vi ·x = vi,vi ·vj = vj ·vi = vj,∀i < j and vi ·vi = h ∪{v1, . . . ,vi−1},∀x ∈ h,i ∈{1, . . . ,n}. then (hn, ·) is h/v-group, called attach elements, where (hn, ·)/β∗ ∼= z2 and vn is single. (c) let (h, ·) be an hv-semigroup, v /∈ h, and (h, ·) be its attached h/v-group. take an element 0 /∈ h and define in ho = h ∪{v, 0} two hopes: hypersum (+): 0 + 0 = x + v = v + x = 0, 0 + v = v + 0 = x + y = v, 0 + x = x + 0 = v + v = h, ∀x,y ∈ h hyperproduct (·): remains the same as in h moreover 0·0 = v ·x = x·0 = 0,∀x ∈ h then (ho, +, ·) is h/v-field with (ho, +, ·)/γ∗ ∼= z3. (+) is associative, (·) is wass and weak distributive with respect to (+). 0 is zero absorbing and single but not scalar in (+). (ho, +, ·) is called the attached h/v-field of the hv-semigroup (h, ·). let us denote by u the set of all finite products of elements of a hypergroupoid (h, ·). consider the relation defined as follows: xly iff there exists u ∈ u such that ux∩uy 6= ∅. then the transitive closure l∗ of l is called left fundamental reproductivity relation. similarly, the right fundamental reproductivity relation r∗ is defined. 188 hv-fields, h/v-fields theorem 2.1. if (h, ·) is a commutative semihypergroup, i.e. the strong commutativity and the strong associativity is valid, then the strong expression of the above l relation: ux = uy, has the property: l∗ = l . proof. suppose that two elements x and y of h are l* equivalent. therefore, there are u1, . . . ,un+1 elements of u, and z1, . . . ,zn elements of h, such that u1x = u1z1,u2z1 = u2z2, . . . ,unzn−1 = unzn,un+1zn = un+1y. from these relations, using the strong commutativity, we obtain un+1 . . .u2u1x = un+1 . . .u2u1z1 = un+1 . . .u1u2z1 = = un+1 . . .u2u1z2 = · · · = un+1 . . .u2u1y therefore, setting u = un+1 . . .u2u1 ∈ u, we have ux = uy. 2 corollary. let (s, ·) be commutative semigroup which has an element w ∈ s such that the set ws is finite. consider the transitive closure l* of the relation l defined by: xly iff there exists z ∈ s such that zx = zy. then < s/l∗,◦ /β∗ is a finite commutative group, where (◦) is the induced operation on classes of s/l*. open problem: prove that l*, is the smallest equivalence: h/l*, is reproductive. we present now the small non-degenerate hv-fields on (zn, +, ·) which satisfy the following conditions, appropriate in santilli’s iso-theory: 1. multiplicative very thin minimal, 2. cow (non-commutative), 3. they have the elements 0 and 1, scalars, 4. when an element has inverse element, then this is unique. remark that last condition means than we cannot enlarge the result if it is 1 and we cannot put 1 in enlargement. moreover we study only the upper triangular cases, in the multiplicative table, since the corresponding under, are isomorphic since the commutativity is valid for the underline rings. from the fact that the reproduction axiom in addition is valid, we have always hv-fields. theorem 2.2. all multiplicative hv-fields defined on (z4, +, ·), which have nondegenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: the only product which is set is 2 ⊗ 3 = {0, 2} or 3 ⊗ 2 = {0, 2}. the fundamental classes are [0] = {0, 2}, [1] = {1, 3} and we have (z4, +,⊗)/γ∗ ∼= (z2, +, ·). 189 thomas vougiouklis example. let us denote by eij the matrix with 1 in the ij-entry and zero in the rest entries. then take the following 2×2 upper triangular hv-matrices on the above hv-field (z4, +, ·) of the case that only 2 ⊗ 3 = {0, 2} is a hyperproduct: i = e11 +e22,a = e11 +e12 +e22,b = e11 + 2e12 +e22,c = e11 + 3e12 +e22, d = e11+3e22,e = e11+e12+3e22,f = e11+2e12+3e22,g = e11+3e12+3e22, then, we obtain for x = {i,a,b,c,d,e,f,g}, that (x,⊗) is non-cow hv-group and the fundamental classes are a = {a,c},d = {d,f},e = {e,g} and the fundamental group is isomorphic to (z2 ×z2, +). in this hv-group there is only one unit and every element has a unique double inverse. theorem 2.3. all multiplicative hv-fields defined on (z6, +, ·), which have nondegenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: we have the only one hyperproduct, (i) 2 ⊗ 3 = {0, 3} or 2 ⊗ 4 = {2, 5} or 3 ⊗ 4 = {0, 3} or 3 ⊗ 5 = {0, 3} or 4 ⊗ 5 = {2, 5} fundamental classes: [0] = {0, 3}, [1] = {1, 4}, [2] = {2, 5}, and (z6, +, ·)/γ∗ ∼= (z3, +, ·). (ii) 2 ⊗ 3 = {0, 2} or 2 ⊗ 3 = {0, 4} or 2 ⊗ 4 = {0, 2} or 2 ⊗ 4 = {2, 4} or 2 ⊗ 5 = {0, 4} or 2 ⊗ 5 = {2, 4} or 3 ⊗ 4 = {0, 2} or 3 ⊗ 4 = {0, 4} or 3 ⊗ 5 = {3, 5} or 4 ⊗ 5 = {0, 2} or 4 ⊗ 5 = {2, 4} fundamental classes: [0] = {0, 2, 4}, [1] = {1, 3, 5}, and (z6, +,⊗)/γ∗ ∼= (z2, +, ·). theorem 2.4. all multiplicative hv-fields defined on (z9, +, ·), which have nondegenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: we have the only one hyperproduct, 2 ⊗ 3 = {0, 6} or {3, 6}, 2 ⊗ 4 = {2, 8} or {5, 8}, 2 ⊗ 6 = {0, 3} or {3, 6}, 2 ⊗ 7 = {2, 5} or {5, 8}, 2 ⊗ 8 = {1, 7} or {4, 7}, 3 ⊗ 4 = {0, 3} or {3, 6}, 3 ⊗ 5 = {0, 6} or {3, 6}, 3 ⊗ 6 = {0, 3} or {0, 6}, 3 ⊗ 7 = {0, 3} or {3, 6}, 3 ⊗ 8 = {0, 6} or {3, 6}, 4 ⊗ 5 = {2, 5} or {2, 8}, 4 ⊗ 6 = {0, 6} or {3, 6}, 4 ⊗ 8 = {2, 5} or {5, 8}, 5 ⊗ 6 = {0, 3} or {3, 6}, 5 ⊗ 7 = {2, 8} or {5, 8}, 5 ⊗ 8 = {1, 4} or {4, 7}, 6 ⊗ 7 = {0, 6} or {3, 6}, 6 ⊗ 8 = {0, 3} or {3, 6}, 7 ⊗ 8 = {2, 5} or {2, 8}, fundamental classes: [0] = {0, 3, 6}, [1] = {1, 4, 7}, [2] = {2, 5, 8}, and (z9, +,⊗)/γ∗ ∼= (z3, +, ·). 190 hv-fields, h/v-fields theorem 2.5. all hv-fields defined on (z10, +, ·), which have non-degenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: (i) we have the only one hyperproduct, 2⊗4 = {3, 8}, 2⊗5 = {2, 5}, 2⊗6 = {2, 7}, 2⊗7 = {4, 9}, 2⊗9 = {3, 8}, 3⊗4 = {2, 7}, 3⊗5 = {0, 5}, 3⊗6 = {3, 8}, 3⊗8 = {4, 9}, 3⊗9 = {2, 7}, 4⊗5 = {0, 5}, 4⊗6 = {4, 9}, 4⊗7 = {3, 8}, 4⊗8 = {2, 7}, 5⊗6 = {0, 5}, 5⊗7 = {0, 5}, 5⊗8 = {0, 5}, 5⊗9 = {0, 5}, 6⊗7 = {2, 7}, 6⊗8 = {3, 8}, 6 ⊗ 9 = {4, 9}, 7 ⊗ 9 = {3, 8}, 8 ⊗ 9 = {2, 7}. fundamental classes: [0] = {0, 5}, [1] = {1, 6}, [2] = {2, 7}, [3] = {3, 8}, [4] = {4, 9} and (z10, +,⊗)/γ∗ ∼= (z5, +, ·). (ii) the cases where we have two classes [0] = {0, 2, 4, 6, 8} and [1] = {1, 3, 5, 7, 9}, thus we have fundamental field (z10, +,⊗)/γ∗ ∼= (z2, +, ·), can be described as follows: taking in the multiplicative table only the results above the diagonal, we enlarge each of the products by putting one element of the same class of the results. we do not enlarge setting the element 1, and we cannot enlarge only the product 3 ⊗ 7 = 1. the number of those hv-fields is 103. example 2.1. in order to see how hard is to realize the reproductivity of classes and the unit class and inverse class, we consider the above hv-field (z10, +,⊗) where we have 2 ⊗ 4 = {3, 8}. then the multiplicative table of the hyperproduct is the following: ⊗ 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 3,8 0 2 4 6 8 3 0 3 6 9 2 5 8 1 4 7 4 0 4 8 2 6 0 4 8 2 6 5 0 5 0 5 0 5 0 5 0 5 6 0 6 2 8 4 0 6 2 8 4 7 0 7 4 1 8 5 2 9 6 3 8 0 8 6 4 2 0 8 6 4 2 9 0 9 8 7 6 5 4 3 2 1 on this table it is easy to see that the reproductivity is not valid but it is very hard to see that the reproductivity of classes is valid. we can see the reproductivity of classes easier if we reformulate the multiplicative table according to the fundamental classes, [0] = {0, 5}, [1] = {1, 6}, [2] = {2, 7}, [3] = {3, 8}, [4] = {4, 9}. then we obtain: 191 thomas vougiouklis ⊗ 0 5 1 6 2 7 3 8 4 9 0 0 0 0 0 0 0 0 0 0 0 5 0 5 5 0 0 5 5 0 0 5 1 0 5 1 6 2 7 3 8 4 9 6 0 0 6 6 2 2 8 8 4 4 2 0 0 2 2 4 4 6 6 3,8 8 7 0 5 7 2 4 9 1 6 8 3 3 0 5 3 8 6 1 9 4 2 7 8 0 0 8 8 6 6 4 4 2 2 4 0 0 4 4 8 8 2 2 6 6 9 0 5 9 4 8 3 7 2 6 1 from this it is easy to see the unit class and the inverse class of each class. 3 the h/v-representations and applications hv-structures are used in representation theory of hv-groups which can be achieved either by generalized permutations or by hv-matrices [31], [32], [38], [39], [44], [46], [57], [58]. the representations by generalized permutations can be faced by translations [37]. moreover in hyperstructure theory we can define hyperproduct on non-square ordinary matrices by using the so called helix hopes where we use all entries of them [65], [28], [29] and [13], [14], [66], [67]. thus, we face the representations of the hyperstructures by non-square matrices as well. hv-matrix (or h/v-matrix) is a matrix with entries of an hv-ring or hv-field (or h/v-ring or h/v-field). the hyperproduct of two hv-matrices (aij) and = (bij), of type m × n and n × r respectively, is defined in the usual manner and it is a set of m × r hv-matrices. the sum of products of elements of the hv-ring is considered to be the n-ary circle hope on the hyperaddition. the hyperproduct of hv-matrices is not necessarily wass. the problem of the hv-matrix (or h/v-group) representations is the following: definition 3.1. let (h, ·) be an hv-group (or h/v-group). find an hv-ring (or h/vring) (r, +, ·), a set mr={(aij)|aij∈r} and a map t : h → mr : h 7→ t(h) such that t(h1h2) ∩t(h1)t(h2) 6= ∅,∀h1,h2 ∈ h. t is an hv-matrix (or h/v matrix) representation. if t(h1h2) ⊂ t(h1)t(h2),∀h1,h2 ∈ h, then t is an inclusion representation. if t(h1h2) = t(h1)t(h2),∀h1,h2 ∈ h, then t is a good representation and an induced representation t * of the hypergroup algebra is obtained. if t is one to one and the good condition is valid then it is called faithful representation. 192 hv-fields, h/v-fields the main theorem of the theory of representations is the following [31], [32], [38]: theorem 3.1. a necessary condition in order to have an inclusion representation t of an h/v-group (h, ·) by n×n, h/v-matrices over the h/v-ring (r, +, ·) is the following: for all classes β*(x), x ∈ h there must exist elements aij ∈ h,i,j ∈ {1, ...,n} such that t(β*(a)) ⊂{a = (a′ij)|a ′ ij ∈ γ*(aij), i,j ∈{1, ...,n}} thus, inclusion representation t : h → mr : a 7→ t(a) = (aij) induces an homomorphic t * of h/β* over r/γ* by setting t *(β*(a)) = [γ*(aij)],∀β*(a) ∈ h/β*, where γ*(aij)r/γ* is the ij entry of t *(β*(a)). t * is called fundamental induced representation of t . let t a representation of an h/v-group h by h/v-matrices and trφ(t(x)) = γ∗(txii) be the fundamental trace, then is called fundamental character, the mapping xt : h → r/γ* : x 7→ xt (x) = trφ(t(x)) = trt∗(x) in representations of hv-groups there are two difficulties: first to find an hv-ring or an hv-field and second, an appropriate set of hv-matrices. notice that the more interesting cases are for the small hv-fields, where the results have one or few elements. example 3.1. in the case of the hv-field (z6, +,⊗) where the only one hyperproduct is 2 ⊗ 4 = {2, 5} we consider the 2 × 2 h/v-matrices of type i = e11 + ie12 + 4e22, where i = 0, 1, 2, 3, 4, 5, then an h/v-group is obtained and the multiplicative table of the hyperproduct of those hv-matrices is given by ⊗ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 4 5 0 1 2 3 2 2 0,3 1,4 2,5 0,3 1,4 3 0 1 2 3 4 5 4 4 5 0 1 2 3 5 2 3 4 5 0 1 where the fundamental classes are (0) = {0, 3}, (1) = {1, 4}, (2) = {2, 5} and the fundamental group is isomorphic to (z3, +). remark that (z6,⊗) is an h/vgroup which is cyclic where the elements 2 and 4 are generators of period 4. notice that the hope (⊗) is a hyperproduct of h/v-matrices although (0) stands for the unit matrix, this is so because the symbolism follows the entry 12. 193 thomas vougiouklis example 3.2. let us denote by eij the matrix with 1 in the ij-entry and zero in the rest entries. then take the following 2 × 2 upper triangular h/v-matrices on the above h/v-field (z4, +,⊗) of the case that only 2 ⊗ 3 = {0, 2} is a hyperproduct: i = e11 +e22,a = e11 +e12 +e22,b = e11 + 2e12 +e22,c = e11 + 3e12 +e22, d = e11+3e22,e = e11+e12+3e22,f = e11+2e12+3e22,g = e11+3e12+3e22, then, we obtain the following multiplicative table for the set x={i,a,b,c,d,e,f,g} ⊗ i a b c d e f g i i a b c d e f g a a b c i g d e f b b c i a d,f e,g d,f e,g c c i a b e f g d d d e f g i a b c e e f g d c i a b f f g d e i,b a,c i,b a,c g g d e f a b c i the (x,⊗) is non-cow, hv-group and we can see that the fundamental classes are the a = {a,c}, d = {d,f}, e = {e,g} and the fundamental group is isomorphic to (z2 × z2, +). in this hv-group there is only one unit and every element has a unique double inverse. only f has one more right inverse element, the d, since f ⊗d = {i,b}. remark that if we need h/v-fields where the elements have at most one inverse element, then we must exclude the case of 2 ⊗ 5 = {1, 4} from (i), and the case 3 ⊗ 5 = {1, 3} from (ii). last decades hv-structures have applications in other branches of mathematics and in other sciences. these applications range from biomathematics -conchology, inheritanceand hadronic physics or on leptons to mention but a few. the hyperstructure theory is related to fuzzy theory; consequently, hyperstructures can be widely applicable in industry and production, too [2], [5], [11], [12], [23], [25], [43], [47], [59]. the lie-santilli theory on isotopies was born in 1970’s to solve hadronic mechanics problems. santilli proposed a ’lifting’of the n-dimensional trivial unit matrix of a normal theory into a nowhere singular, symmetric, real-valued, positivedefined, n-dimensional new matrix. the original theory is reconstructed such as to admit the new matrix as left and right unit. the isofields needed, correspond into the hyperstructures were introduced by santilli & vougiouklis in 1996 [25] and they are called e-hyperfields, [12], [24], [52], [56], [61]. 194 hv-fields, h/v-fields definition 3.2. a hyperstructure (h, ·) which contains a unique scalar unit e, is called e-hyperstructure. in an e-hyperstructure, we assume that for every element x, there exists an inverse x−1, i.e. e ∈ x ·x−1 ∩x−1 ·x. definition 3.3. a hyperstructure (f, +, ·), where (+) is an operation and (·) is a hope, is called e-hyperfield if the following axioms are valid: (f, +) is an abelian group with the additive unit 0, (·) is wass, (·) is weak distributive with respect to (+), 0 is absorbing element: 0·x = x·0 = 0,∀x ∈ f , there exists a multiplicative scalar unit 1, i.e. 1 · x = x · 1 = x,∀x ∈ f , and ∀x ∈ f there exists a unique inverse x−1, such that 1 ∈ x ·x−1 ∩x−1 ·x. the elements of an e-hyperfield are called e-hypernumbers. in the case that the relation: 1 = x · x−1 = x−1 · x, is valid, then we say that we have a strong e-hyperfield. definition 3.4. main e-construction. given a group (g, ·), where e is the unit, then we define in g, a large number of hopes (⊗) as follows: x⊗y = {xy,g1,g2, ...},∀x,y ∈ g−{e}, and g1,g2, ... ∈ g−{e} g1,g2,... are not necessarily the same for each pair (x,y). (g,⊗) is an hv-group, in fact it is an hb-group which contains the (g, ·). (g,⊗) is an e-hypergroup. moreover, if for each x,y such that xy = e, then (g,⊗) becomes a strong ehypergroup the main e-construction gives an extremely large number of e-hopes. example. consider the quaternions q = {1,−1, i,−i,j,−j,k,−k}, with i2 = j2 = −1, ij = −ji = k, and denote i = {i,−i},j = {j,−j},k = {k,−k}. we define a lot of hopes (∗) by enlarging few products. for example, (−1) ∗k = k,k∗i = j and i∗j = k. then the hyperstructure (q,∗) is a strong e-hypergroup. the lie-santilli admissibility on non-quare matrices [12], [14], [24], [26], [57], [61]: construction 3.1. let l = (mm×n, +) be an hv-vector space of m × n hypermatrices over the hv-field (f, +, ·),φ : f → f/γ∗, the canonical map and ωf = {x ∈ f : φ(x) = 0}, where 0 is the zero of the fundamental field f/γ*. similarly, let ωl be the core of the canonical map φ′ : l → l/�∗ and denote by the same symbol 0 the zero of l/�*. take any two subsets r,s ⊆ l then a santilli’s lieadmissible hyperalgebra is obtained by taking the lie bracket, which is a hope: [, ]rs : l×l → p(l) : [x,y]rs = xrty −ystx. notice that [x,y]rs = xrty −ystx = {xrty −ystx|r ∈ r and s ∈ s} 195 thomas vougiouklis an application, which combines the ∂-structures and fuzzy theory, is to replace in questionnaires the scale of likert by the bar of vougiouklis & vougiouklis [19]: definition 3.5. in every question substitute the likert scale with ’the bar’ whose poles are defined with ’0’ on the left end, and ’1’ on the right end: 0 1 the subjects/participants are asked instead of deciding and checking a specific grade on the scale, to cut the bar at any point s/he feels expresses her/his answer to the specific question the use of the vougiouklis & vougiouklis bar instead of a likert scale has several advantages during both the filling-in and the research processing. the final suggested length of the bar, according to the golden ratio, is 6.2cm, [17], [18], [50], [51], [62], [63], [64]. 196 hv-fields, h/v-fields references [1] n. antampoufis, s. spartalis and t. vougiouklis, fundamental relations in special extensions, 8th aha, samothraki, greece 2002, spanidis, (2003), 81-89. 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[68] j. zhan, v. leoreanu-fotea and t. vougiouklis,fuzzy soft γ-hypermodules, u.p.b. sci. bull., series a, vol. 73, iss. 3, (2011), 13-28. 201 ratio mathematica volume 42, 2022 to some structural properties of ∞-languages ivan mezník1 abstract properties of catenation of sequences of finite (words) and infinite (𝜔-words) lengths are largely studied in formal language theory. these operations are derived from the mechanism how they are accepted or generated by the corresponding devices. finite automata accept structures containing only words, 𝜔-automata accept only 𝜔-words. structures containing both words and 𝜔-words (∞-words) are mostly generated by various types of ∞-automata (∞-machines). the aim of the paper is to investigate algebraic properties of operations on ∞-words generated by igk-automata, where k is to model the depth of memory. it has importance in many applications (shift registers, discrete systems with memory…). it is shown that resulting algebraic structures are of „pure“ groupoid or partial groupoid type. keywords: ∞-words; ∞-language; ρn,p,r-catenation; closure of an ∞language; ρ-operation. mathematics subject classification: 68q70, 08a552 1 institute of informatics, faculty of business and management, brno university of technology, kolejní 2906/4, 612 00 brno, czech republic; meznik@vutbr.cz. 2 received on january 23th, 2022. accepted on june 4th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v41i0.721. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 127 ivan mezník 1. introduction the notion of an ∞-language was introduced by nivat ([10]) as a free-monoid structure containing words finite („clasical“ languages) and infinite (ω-languages) lengths. the theory of ω-languages has been intensively developed so far, mostly as a generalization of acceptance conditions of various types of automata ([1], [8], [9], [11], [12], [13], [14], [15] among others). devices capable to accept (or generate) simultaneously words or finite or infinite (ω-words) lengths were described and investigated in [2], [3]. such devices (k-machines, igk-machines) also provide to implement the depth of memory and possess various applications (shift registers, modelling of phenomena working in a discrete time scale). they also make a lot of properties. in [3] lattice structures were described. the way how they generate words and ω-words motivates to study various types of catenation of words, which is the aim of the paper. the paper is organized as follows. in section 2 we introduce basic concepts. in sections 3 and 4 we examine properties of 𝜌-closure and 𝜌-operation, respectively. 2. preliminaries an alphabet is any finite set (including the empty set) and is denoted by 𝛴; its elements are called letters. let ω be the least infinite ordinal and n, 0 ≤ 𝑛 ≤ 𝜔 be an ordinal. for 𝛴 ≠ ∅ the set of all finite sequences of elements of σ including the empty sequence λ is denoted by 𝛴∗, the set of all infinite sequences of elements of σ by 𝛴𝜔 and the set 𝛴∗ ∪ 𝛴𝜔 by 𝛴∞. for 𝛴 = ∅, by definition, 𝛴∗ = 𝛴𝜔 = 𝛴∞ = ∅. the elements of 𝛴∗ are words, the elements of 𝛴𝜔 are ω-words, the elements of 𝛴∞ are ∞-words. instead of (𝑎0, 𝑎1, … , 𝑎𝑛−1) ∈ 𝛴 ∗, 𝑛 ≥ 1 and (𝑎0, 𝑎1, … ) ∈ 𝛴 𝜔 we write simply 𝑎0𝑎1 … 𝑎𝑛−1 and 𝑎0𝑎1 …. for 𝑤 ∈ 𝛴 ∞, the length of w, denoted by |𝑤| is defined as follows: if 𝑤 = 𝑎0𝑎1 … 𝑎𝑛−1 ∈ 𝛴 ∗ then |𝑤| = 𝑛, if 𝑤 = 𝑎0𝑎1 … ∈ 𝛴 𝜔 then |𝑤| = 𝜔 and if 𝑤 = 𝜆 then |𝑤| = 0. a subset of 𝛴∗ and 𝛴𝜔 and 𝛴∞ is referred to as a language and an ω-language and an ∞-language (over σ) respectively. for 𝐿 ⊆ 𝛴∞ we define 𝑚(𝑙) = 𝑖𝑛𝑓{|𝑤|; 𝑤 ∈ 𝐿}. let 𝑤 ∈ 𝛴∞ − {𝜆}, 1 ≤ 𝑚 ≤ 𝑛 < 1 + |𝑤|; by 𝑤(𝑛) we denote the n-th letter of w, by w[𝑚, 𝑛] the word 𝑤(𝑚) … 𝑤(𝑛) and if |𝑤| = 𝜔 by 𝑤[𝑚, 𝜔] the word 𝑤(𝑚)𝑤(𝑚 + 1) …. instead of 𝑤[1, 𝑛] we write only 𝑤[𝑛]. in case 𝑚 > 𝑛 we formally put 𝑤[𝑚, 𝑛] = 𝜆. the usual operation of catenation on 𝛴∗ may be extended to 𝛴∞ as a partial operation as follows. let 𝑤 ∈ 𝛴∗, |𝑤| = 𝑛, 𝑤 ′ ∈ 𝛴∞; if 𝑤 ′ ∈ 𝛴∗, then 𝑤𝑤 ′ is defined by catenation on 𝛴∗ and if 𝑤 ′ ∈ 𝛴𝜔, then 𝑤𝑤 ′ = 𝑤(1)𝑤(2) … 𝑤(𝑛)𝑤 ′(1)𝑤 ′(2) …. for 𝑤 ∈ 𝛴∗, 𝑘 ≥ 1, the symbol 𝑤 𝑘 denotes the result of k catenations of w and the symbol 𝑤 𝜔 denotes the result of infinite number of catenations of w; by definition, 𝑤 0 = 𝜆. 3. closure of an ∞-language in this section we define the notion of a 𝜌-catenation. subsequently the concept of a 𝜌-closure is introduced and its closure characterization is derived. 128 to some structural properties of ∞-languages 2.1 definition. let 𝑛, 𝑝, 𝑟 be positive integers and 𝑢 ∈ 𝛴∞, 𝑣 ∈ 𝛴∞. for u,v satisfying the property 𝑢[𝑝, 𝑝 + 𝑛 − 1] = 𝑣[𝑟, 𝑟 + 𝑛 − 1] we define the operation 𝜌𝑛,𝑝,𝑟 by (2.1) 𝜌𝑛,𝑝,𝑟 (𝑢, 𝑣) = 𝑢[𝑝 + 𝑛 − 1]𝑣[𝑟 + 𝑛, |𝑣|] called a 𝜌𝑛,𝑝,𝑟-catenation or only ρ-catenation if n,p,r are given by the context. apparently, each 𝜌𝑛,𝑝,𝑟-catenation is a partial operation in 𝛴 ∞. in other words, each 𝜌𝑛,𝑝,𝑟-catenation defines a partial groupoid in 𝛴 ∞. in this manner (regarding the given n,p,r) the set of partial operations (groupoids) in 𝛴∞ is given. instead of 𝜌𝑛,𝑝,𝑟 (𝑢, 𝑣) we write as customary 𝑢𝜌𝑛,𝑝,𝑟 𝑣 or only 𝑢𝜌𝑣, if 𝑛, 𝑝, 𝑟 are clear from the context. to simplify the text, by stating 𝑢𝜌𝑛,𝑝,𝑟 𝑣 it is supposed that (𝑢, 𝑣) ∈ 𝐷𝑜𝑚(𝜌𝑛,𝑝,𝑟 ). 2.2 lemma. suppose 𝑢 ∈ 𝛴∞ and 𝑢𝜌𝑛,𝑝,𝑟 𝑢 for some 𝑛, 𝑝, 𝑟 . then it holds 𝑢𝜌𝑛,𝑝,𝑟 𝑢 = 𝑢. proof. it is an immediate consequence of definition 2.1 remarks. 1∘ suppose 𝑢 ∈ 𝛴∞ and 𝑛, 𝑝, 𝑟 ≤ |𝑢|. then there is obviously an infinite number of words 𝑥 ∈ 𝛴∞ such that 𝑢𝜌𝑛,𝑝,𝑟 𝑥 = 𝑢 playing the role of the „identity“ element of 𝜌𝑛,𝑝,𝑟-catenation. 2∘ operation 𝜌𝑛,𝑝,𝑟 is in general not commutative. for example consider words u,v over 𝛴 = {𝑎, 𝑏}, 𝑢 = (𝑎𝑏)3, 𝑣 = 𝑎3. applying the previous definition we get 𝑢𝜌1,3,1𝑣 = 𝑎𝑏𝑎3, 𝑣𝜌1,3,1𝑢 = 𝑎 2(𝑎𝑏)3, 𝑢𝜌1,3,1𝑣 ≠ 𝑣𝜌1,3,1𝑢. 3∘ operation 𝜌𝑛,𝑝,𝑟 is in general not associative. for example consider 𝜌1,3,2 − 𝑐𝑎𝑡𝑒𝑛𝑎𝑡𝑖𝑜𝑛 in {𝑎, 𝑏}∞ and let 𝑢 = (𝑎𝑏)4, 𝑣 = 𝑎5, 𝑤 = 𝑎7. construct (𝑢𝜌1,3,2𝑣)𝜌1,3,2𝑤, 𝑢𝜌1,3,2(𝑣𝜌1,3,2𝑤). due to definition 2.1 we get (𝑢𝜌1,3,2𝑣)𝜌1,3,2𝑤 = 𝑎𝑏𝑎 5, whereas 𝑢𝜌1,3,2(𝑣𝜌1,3,2𝑤) = 𝑎𝑏𝑎 7. of course, some of catenations in the given expressions need not be defined. 2.3 theorem let 𝑢 ∈ 𝛴∞, 𝑣 ∈ 𝛴∞ and suppose 𝑢𝜌𝑛,𝑝,𝑟 𝑣, 𝑣𝜌𝑛,𝑟,𝑠𝑤 for some 𝑛, 𝑝, 𝑟, 𝑠 ≥ 1. then 𝑢𝜌𝑛,𝑝,𝑠𝑤 and there holds (2.2) 𝑢𝜌𝑛,𝑝,𝑠𝑤 = 𝑢𝜌𝑛,𝑝,𝑟 (𝑣𝜌𝑛,𝑟,𝑠𝑤). proof. suppose that 𝑢𝜌𝑛,𝑝,𝑟 𝑣, 𝑣𝜌𝑛,𝑟,𝑠 𝑤 hold for given 𝑛, 𝑝, 𝑟, 𝑠 ≥ 1. from definition 2.1 it follows that 𝑢[𝑝, 𝑝 + 𝑛 − 1] = 𝑣[𝑟, 𝑟 + 𝑛 − 1] and 𝑣[𝑟, 𝑟 + 𝑛 − 1] = 𝑤[𝑠, 𝑠 + 𝑛 − 1]. then evidently 𝑢[𝑝, 𝑝 + 𝑛 − 1] = 𝑤[𝑠, 𝑠 + 𝑛 − 1], applying definition 2.1 we get 𝑢𝜌𝑛,𝑝,𝑠𝑤 = 𝑢[𝑝 + 𝑛 − 1]𝑤[𝑠 + 𝑛, |𝑤|] and the first part of the statement is verified. rewriting this expression we obtain (2.3) 𝑢𝜌𝑛,𝑝,𝑠𝑤 = 𝑢[1] … 𝑢[𝑝 + 𝑛 − 1]𝑤[𝑠 + 𝑛]𝑤[𝑠 + 𝑛 + 1] … 𝑤[|𝑤|]. now we costruct the right part of (2.2). by definition 2.1 we have 129 ivan mezník 𝑣𝜌𝑛,𝑟,𝑠𝑤 = 𝑣[1] … 𝑣[𝑟] … 𝑣[𝑟 + 𝑛 − 1]𝑤[𝑠 + 𝑛] … 𝑤[|𝑤|], where 𝑣[𝑟, 𝑟 + 𝑛 − 1] = 𝑣[𝑟] … 𝑣[𝑟 + 𝑛 − 1] = 𝑤[𝑠] … 𝑤[𝑠 + 𝑛 − 1] = 𝑤[𝑠, 𝑠 + 𝑛 − 1] and (2.4) 𝑢𝜌𝑛,𝑝,𝑟 (𝑣𝜌𝑛,𝑟,𝑠𝑤) = 𝑢[1] … 𝑢[𝑝] … 𝑢[𝑝 + 𝑛 − 1]𝑤[𝑠 + 𝑛] … 𝑤[|𝑤|]. from (2.3) and (2.4) the statement (2.2) holds and the proof is completed. 2.4 theorem let 𝑢, 𝑣 ∈ 𝛴∞ and suppose 𝑢𝜌𝑛,𝑝,𝑟 𝑣 for fixed 𝑛 > 1, 𝑝 ≥ 1, 𝑟 ≥ 1. then 𝑢𝜌𝑚,𝑝,𝑟 𝑣 for any 𝑚 < 𝑛 and it holds (2.5) 𝑢𝜌𝑛,𝑝,𝑟 𝑣 = 𝑢𝜌𝑚,𝑝,𝑟 𝑣 . proof. let 𝑢𝜌𝑛,𝑝,𝑟 𝑣 for the given 𝑛, 𝑝, 𝑟. from definition 2.1 it follows that 𝑢[𝑝, 𝑝 + 𝑛 − 1] = 𝑣[𝑟, 𝑟 + 𝑛 − 1]. since 𝑚 < 𝑛 , then apparently 𝑢[𝑝, 𝑝 + 𝑚 − 1] = 𝑣[𝑟, 𝑟 + 𝑚 − 1] holds for any 𝑚 < 𝑛 as well and thus 𝑢𝜌𝑚,𝑝,𝑟 𝑣. due to (2.1) 𝑢𝜌𝑛,𝑝,𝑟 𝑣 = 𝑢[𝑝 + 𝑛 − 1]𝑣[𝑟 + 𝑛, |𝑣|]. in a detailed version we have (2.6) 𝑢𝜌𝑛,𝑝,𝑟 𝑣 = 𝑢[1] … 𝑢[𝑝 + 𝑛 − 1]𝑣[𝑟 + 𝑛] … 𝑣[|𝑣|]. with a view to 𝑚 < 𝑛, (2.6) may be rewritten as (2.7) 𝑢𝜌𝑛,𝑝,𝑟 𝑣 = 𝑢[1] … 𝑢[𝑝 + 𝑚 − 1]𝑢[𝑝 + 𝑚] … 𝑢[𝑝 + 𝑛 − 1]𝑣[𝑟 + 𝑛]𝑣[𝑟 + 𝑛 + 1] … 𝑣[|𝑣|]. now, we construct 𝑢𝜌𝑚,𝑝,𝑟 𝑣 for 𝑚 < 𝑛. it holds 𝑢[𝑝, 𝑝 + 𝑚 − 1] = 𝑣[𝑟, 𝑟 + 𝑚 − 1] and by (2.1) (2.8) 𝑢𝜌𝑚,𝑝,𝑟 𝑣 = 𝑢[𝑝 + 𝑚 − 1]𝑣[𝑟 + 𝑚, |𝑣|]. in detail (2.9) 𝑢𝜌𝑚,𝑝,𝑟 𝑣 = 𝑢[1] … 𝑢[𝑝 + 𝑚 − 1]𝑣[𝑟 + 𝑚] … 𝑣[|𝑣|]. with a view to 𝑚 < 𝑛, (2.9) may be rewritten as (2.10) 𝑢𝜌𝑚,𝑝,𝑟 𝑣 = 𝑢[1] … 𝑢[𝑝] … 𝑢[𝑝 + 𝑚 − 1]𝑣[𝑟 + 𝑚] … 𝑣[𝑟 + 𝑛 − 1]𝑣[𝑟 + 𝑛] … 𝑣[|𝑣|]. due to (2.7) and (2.10) 𝑢[1] … 𝑢[𝑝 + 𝑚 − 1] and 𝑣[𝑟 + 𝑛] … 𝑣[|𝑣|] are common parts. it remains to verify that 𝑣[𝑟 + 𝑚] … 𝑣[𝑟 + 𝑛 − 1] = 𝑢[𝑝 + 𝑚] … 𝑢[𝑝 + 𝑛 − 1]. using assumptions of definition 2.1 we have 𝑢[𝑝, 𝑝 + 𝑛 − 1] = 𝑣[𝑟, 𝑟 + 𝑛 − 1] and thus also 𝑣[𝑟 + 𝑚] … 𝑣[𝑟 + 𝑛 − 1] = 𝑢[𝑝 + 𝑚] … 𝑢[𝑝 + 𝑛 − 1]. hence (2.7) and (2.10) are identical words and the proof is completed. 2.5 definition. let a 𝜌𝑛,𝑝,𝑟 − catenation be given. define a relation 𝑅𝑛,𝑝,𝑟 on 𝛴 ∞ by (2.11) 𝑅𝑛,𝑝,𝑟 = {𝑢, 𝑣 ∈ 𝛴 ∞; (𝑢, 𝑣) ∈ 𝐷𝑜𝑚(𝜌𝑛,𝑝,𝑟 )} ⊆ 𝛴 ∞ × 𝛴∞. 2.6 lemma. the relation 𝑅𝑛,𝑝,𝑟 is (i) reflexive, (ii) not symmetric, (iii) not antisymmetric, 130 to some structural properties of ∞-languages (iv) not transitive. proof. (i) reflexivity of 𝑅𝑛,𝑝,𝑟 follows immediately from lemma 2.2. (ii) consider 𝑢 = 𝑎𝑏𝑏𝑏, 𝑣 = 𝑎𝑎𝑏𝑏𝑏. by definition 2.1 it holds 𝑢 = 𝑎𝑏𝑏𝑏𝜌2,3,3𝑎𝑎𝑏𝑏𝑏 = 𝑣, whereas 𝑣 = 𝑎𝑎𝑏𝑏𝑏𝜌2,3,3𝑎𝑏𝑏𝑏 = 𝑢 does not hold, so the relation 𝑅𝑛,𝑝,𝑟 is not transitive. (iii) put 𝑢 = 𝑎𝑏𝑎𝑏𝑎𝑏𝑎𝑏 = (𝑎𝑏)4, 𝑣 = 𝑏𝑎𝑏𝑎𝑏𝑎𝑏𝑎𝑏𝑎 = (𝑏𝑎)5. by definition 2.1 we have 𝑢𝜌1,2,5𝑣, 𝑣𝜌1,2,5𝑢, but 𝑢 ≠ 𝑣 and hence the relation 𝑅𝑛,𝑝,𝑟 is not antisymmetric. (iv) let 𝑢 = 𝑎𝑏𝑏𝑏, 𝑣 = 𝑏𝑎𝑏𝑎𝑏𝑎, 𝑤 = 𝑏𝑏𝑏𝑏. from definition it follows 𝑢𝜌1,1,2𝑣, 𝑣𝜌1,1,2𝑤, but 𝑢𝜌1,1,2𝑤 does not hold and the relation 𝑅𝑛,𝑝,𝑟 is not transitive. 2.5 definition. let 𝐿 ⊆ 𝛴∞ be an ∞-language, 𝑛 ≥ 1 integer and 𝑢, 𝑣 ∈ 𝛴∞. put 𝐶𝑛 𝜌 (𝑢, 𝑣) = ∪ 𝑝, 𝑟 𝑢𝜌𝑛,𝑝,𝑟 𝑣, 𝐶𝑛 𝜌 (𝐿) = ∪ 𝑢, 𝑣 𝐶𝑛 𝜌 (𝑢, 𝑣), 𝐶𝜌(𝐿) = ∪ 𝑛 𝐶𝑛 𝜌 (𝐿). the set 𝐶𝑛 𝜌 (𝐿) is called the n-th ρ-closure of l and the set 𝐶𝜌(𝐿) = ∪ 𝑛 𝐶𝑛 𝜌 (𝐿) the ρclosure of l respectively. 2.6 lemma. let 𝐿 ⊆ 𝛴∞ − {𝜆} be an ∞-language. then 𝐿 ⊆ 𝐶𝜌(𝐿) holds true. proof. suppose 𝑤 ∈ 𝐿. trivially 𝑤(1) = 𝑤(1) and by definition 2.1 it holds 𝑤𝜌1,1,1𝑤 = 𝑤[1]𝑤[2, |𝑤|] = 𝑤 and hence by definition 2.5 𝑤 ∈ 𝐶1 𝜌 (𝐿) and also 𝑤 ∈ 𝐶𝜌(𝐿) and the statement holds true. 2.7 theorem let 𝐿 ⊆ (𝛴∞ − {𝜆}) be an ∞-language. then for every 𝑖 ≥ 1 there holds 𝐶𝑖+1 𝜌 (𝐿) ⊆ 𝐶𝑖 𝜌 (𝐿). proof. let 𝑤 ∈ 𝐶𝑖+1 𝜌 (𝐿). according to definition 2.1 there exist 𝑢, 𝑣 ∈ 𝛴∞ and 𝑖, 𝑝, 𝑟 ≥ 1 with the property 𝑢[𝑝, 𝑝 + 𝑖] = 𝑣[𝑟, 𝑟 + 𝑖] for which 𝑢𝜌𝑖+1,𝑝.𝑟 𝑣 = 𝑢[𝑝 + 𝑖]𝑣[𝑟 + 𝑖 + 1, |𝑣|] = 𝑤 holds. obviously if 𝑢[𝑝, 𝑝 + 𝑖] = 𝑣[𝑟, 𝑟 + 𝑖] then also 𝑢[𝑝, 𝑝 + 𝑖 − 1] = 𝑣[𝑟, 𝑟 + 𝑖 − 1] holds. by definition 2.5 we get 𝑢𝜌𝑖,𝑝,𝑟 𝑣 = 𝑢[𝑝 + 𝑖 − 1]𝑣[𝑟 + 𝑖, |𝑣|] = 𝑤 ′ ∈ 𝐶𝑖 𝜌 (𝐿). but apparently 𝑤, 𝑤 ′ are identical words. hence 𝑤 ∈ 𝐶𝑖 𝜌 (𝐿) and the statement is valid. as a consequence of definition 2.5 and theorem 2.7 the following corollary 2.8 holds: 2.8 corollary let 𝐿 ⊆ (𝛴∞ − {𝜆}) be an ∞-language. then 𝐶𝜌(𝐿) = 𝐶1 𝜌 (𝐿) holds true. 2.9 example let 𝐿 = {𝑎𝑏, 𝑏𝑎𝑘 , 𝑎𝜔 ; 𝑘 ≥ 1} ⊆ {𝑎, 𝑏}∞ be an ∞ language. to find 𝐶𝑛 𝜌 (𝐿) and 𝐶𝜌(𝐿) applying definition 2.5 we get the results as follows. (i) 𝐶1 𝜌 (𝐿): 𝐶1 𝜌 (𝑎𝑏, 𝑎𝑏) = {𝑎𝑏}, 𝐶1 𝜌 (𝑎𝑏, 𝑏𝑎𝑘 ) = {𝑎𝑘 , 𝑎𝑏𝑎𝑘 ; 𝑘 ≥ 1}, 𝐶1 𝜌 (𝑏𝑎𝑘 , 𝑎𝑏) = {𝑏, 𝑏𝑎𝑘 𝑏; 𝑘 ≥ 1}, 𝐶1 𝜌 (𝑏𝑎𝑘 , 𝑏𝑎𝑘 ) = {𝑏𝑎𝑘 ; 𝑘 ≥ 1}, 𝐶1 𝜌 (𝑎𝑏, 𝑎𝜔 ) = {𝑎𝜔 }, 𝐶1 𝜌 (𝑎𝜔 , 𝑎𝑏) = {𝑎𝑘 𝑏; 𝑘 ≥ 1}, 𝐶1 𝜌 (𝑎𝜔 , 𝑎𝜔 ) = {𝑎𝜔 }, 𝐶1 𝜌 (𝑏𝑎𝑘 , 𝑎𝜔 ) = {𝑏𝑎𝜔 }, 𝐶1 𝜌 (𝑎𝜔 , 𝑏𝑎𝑘 ) = {𝑎𝑘 ; 𝑘 ≥ 1}; therefore 𝐶1 𝜌 (𝐿) = {𝑎𝑏, 𝑎𝑘 , 𝑎𝑏𝑎𝑘 , 𝑏𝑎𝑘 𝑏, 𝑏𝑎𝑘 , 𝑎𝜔 , 𝑎𝑘 𝑏, 𝑏𝑎𝜔 ; 𝑘 ≥ 1}. 131 ivan mezník (ii) 𝐶2 𝜌 (𝐿): 𝐶2 𝜌 (𝑎𝑏, 𝑎𝑏) = {𝑎𝑏}, 𝐶2 𝜌 (𝑎𝑏, 𝑏𝑎𝑘 ) = ∅, 𝐶2 𝜌 (𝑏𝑎𝑘 , 𝑎𝑏) = ∅, 𝐶2 𝜌 (𝑎𝜔 , 𝑎𝜔 ) = {𝑎𝜔 }, 𝐶2 𝜌 (𝑏𝑎𝑘 , 𝑎𝜔 ) = {𝑏𝑎𝜔 } for 𝑘 ≥ 2, 𝐶2 𝜌 (𝑎𝜔, 𝑏𝑎𝑘 ) = {𝑎𝑘 ; 𝑘 ≥ 2}, 𝐶2 𝜌 (𝑏𝑎𝑘 , 𝑏𝑎𝑘 ) = {𝑏𝑎𝑘 ; 𝑘 ≥ 1}, 𝐶2 𝜌 (𝑎𝑏, 𝑎𝜔 ) = ∅, 𝐶2 𝜌 (𝑎𝜔 , 𝑎𝑏) = ∅; therefore 𝐶2 𝜌 (𝐿) = {𝑎𝑏, 𝑏𝑎𝑘 , 𝑎𝜔 , 𝑏𝑎𝜔 , 𝑎𝑘+1; 𝑘 ≥ 1}. (iii) 𝐶3 𝜌 (𝐿): 𝐶3 𝜌 (𝑎𝑏, 𝑎𝑏)=𝐶3 𝜌 (𝑎𝑏, 𝑏𝑎𝑘 ) = 𝐶3 𝜌 (𝑏𝑎𝑘 , 𝑎𝑏) = 𝐶3 𝜌 (𝑎𝑏, 𝑎𝜔 ) = 𝐶3 𝜌 (𝑎𝜔 , 𝑎𝑏) = ∅, 𝐶3 𝜌 (𝑏𝑎𝑘 , 𝑏𝑎𝑘 ) = {𝑏𝑎𝑘 ; 𝑘 ≥ 2}, 𝐶3 𝜌 (𝑎𝜔 , 𝑎𝜔 ) = {𝑎𝜔 }, 𝐶3 𝜌 (𝑏𝑎𝑘 , 𝑎𝜔 ) = {𝑏𝑎𝜔 } for 𝑘 ≥ 3, 𝐶3 𝜌 (𝑎𝜔 , 𝑏𝑎𝑘 ) = {𝑎𝑘 ; 𝑘 ≥ 3}; therefore 𝐶3 𝜌 (𝐿) = {𝑏𝑎𝑘 , 𝑎𝜔 , 𝑏𝑎𝜔 , 𝑎𝑘+1; 𝑘 ≥ 2}. (iv) 𝐶𝑛 𝜌 (𝐿) for 𝑛 ≥ 4: 𝐶𝑛 𝜌 (𝑎𝑏, 𝑎𝑏)=𝐶𝑛 𝜌 (𝑎𝑏, 𝑏𝑎𝑘 ) = 𝐶𝑛 𝜌 (𝑏𝑎𝑘 , 𝑎𝑏) = 𝐶𝑛 𝜌 (𝑎𝑏, 𝑎𝜔 ) = 𝐶𝑛 𝜌 (𝑎𝜔 , 𝑎𝑏) = ∅, 𝐶𝑛 𝜌 (𝑏𝑎𝑘 , 𝑏𝑎𝑘 ) = {𝑏𝑎𝑘 ; 𝑘 ≥ 𝑛 − 1}, 𝐶𝑛 𝜌 (𝑎𝜔 , 𝑎𝜔 ) = {𝑎𝜔 }, 𝐶𝑛 𝜌 (𝑏𝑎𝑘 , 𝑎𝜔 ) = {𝑏𝑎𝜔 } for 𝑘 ≥ 𝑛 − 1, 𝐶𝑛 𝜌 (𝑎𝜔, 𝑏𝑎𝑘 ) = {𝑎𝑘 ; 𝑘 ≥ 𝑛 − 1}; therefore 𝐶𝑛 𝜌 (𝐿) = {𝑏𝑎𝑘 , 𝑎𝜔 , 𝑏𝑎𝜔 , 𝑎𝑘+1; 𝑘 ≥ 𝑛 − 1}. conclusion: 𝐶𝜌(𝐿) = {𝑎𝑏, 𝑎𝑘 , 𝑎𝑏𝑎𝑘 , 𝑏𝑎𝑘 𝑏, 𝑏𝑎𝑘 , 𝑎𝜔 , 𝑎𝑘 𝑏, 𝑏𝑎𝜔 ; 𝑘 ≥ 1} = 𝐶1 𝜌 (𝐿). 2.10 theorem. the set of ρ-closures is not closed under set union. proof. we state an counterexample. consider 𝐿1 = {𝑎𝑏}, 𝐿2 = {𝑎 𝜔} over {𝑎, 𝑏}∞and put 𝐿 = 𝐿1 ∪ 𝐿2 = {𝑎𝑏, 𝑎 𝜔 }. applying definition 2.1 and corollary 2.8 we get 𝐶𝜌(𝐿1) = 𝐶1 𝜌 (𝐿1) = 𝐶 𝜌({𝑎𝑏}) = {𝑎𝑏}, 𝐶𝜌(𝐿2) = 𝐶1 𝜌 (𝐿2) = 𝐶 𝜌({𝑎𝜔 }) = {𝑎𝜔 }. further, 𝐶𝜌(𝐿) = 𝐶𝜌(𝐿1 ∪ 𝐿2) = 𝐶 𝜌({𝑎𝑏, 𝑎𝜔 }) = {𝑎𝜔 , 𝑎𝑘 𝑏; 𝑘 ≥ 1}. obviously 𝐶𝜌(𝐿1 ∪ 𝐿2) ≠ 𝐶 𝜌(𝐿1) ∪ 𝐶𝜌𝐿2) and the statement is verified. 2.11 theorem. let 𝐿1, 𝐿2 ⊆ 𝛴 ∞ . then 𝐶𝜌(𝐿1) ∪ 𝐶 𝜌𝐿2) ⊆ 𝐶 𝜌(𝐿1 ∪ 𝐿2). proof. with a view to corollary 2.6 we may consider 𝐶1 𝜌 instead of 𝐶𝜌. let 𝑤 ∈ 𝐶1 𝜌 (𝐿1) ∪ 𝐶1 𝜌 (𝐿2). according to definition 2.3 then (a) there exist 𝑢 ∈ 𝐿1, 𝑣 ∈ 𝐿1 and positive integers 𝑝, 𝑟 such that 𝑢𝜌1,𝑝,𝑟 𝑣 = 𝑤 ∈ 𝐶1 𝜌 (𝐿1) or (b) there exist 𝑢 ∈ 𝐿2, 𝑣 ∈ 𝐿2 and positive integers 𝑝, 𝑟 such that 𝑢𝜌1,𝑝,𝑟 𝑣 = 𝑤 ∈ 𝐶1 𝜌 (𝐿2). assuming (a), the statement there exist 𝑢 ∈ 𝐿1 ∪ 𝐿2, 𝑣 ∈ 𝐿1 ∪ 𝐿2 and positive integers 𝑝, 𝑟 such that 𝑢𝜌1,𝑝,𝑟 𝑣 = 𝑤 ∈ 𝐶1 𝜌 (𝐿1 ∪ 𝐿2) is obviously also valid for an arbitrary set 𝐿2. assuming (b), the statement there exist 𝑢 ∈ 𝐿2 ∪ 𝐿1, 𝑣 ∈ 𝐿2 ∪ 𝐿1 and positive integers 𝑝, 𝑟 such that 𝑢𝜌1,𝑝,𝑟 𝑣 = 𝑤 ∈ 𝐶1 𝜌 (𝐿2 ∪ 𝐿1) is valid as well for ab arbitrary set 𝐿1. thus 𝑤 ∈ 𝐶1 𝜌 (𝐿1 ∪ 𝐿2) and the proof is completed. 2.12 theorem. the set of ρ-closures is not closed under set intersection. proof. we state an counterexample. consider 𝐿1 = {𝑎 𝜔, 𝑎3, 𝑏}, 𝐿2 = {𝑎 3, 𝑎𝑏} over {𝑎, 𝑏}∞and put 𝐿 = 𝐿1 ∩ 𝐿2 = {𝑎 3}. applying definition 2.1 and corollary 2.8 we get 𝐶𝜌(𝐿1) = 𝐶1 𝜌 (𝐿1) = 𝐶1 𝜌 ({𝑎𝜔, 𝑎3, 𝑏}) = {𝑎𝜔 , 𝑏, 𝑎𝑘 ; 𝑘 ≥ 1}, 𝐶𝜌(𝐿2) = 𝐶1 𝜌 (𝐿2) = 𝐶1 𝜌 ({𝑎3, 𝑎𝑏}) = {𝑎, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑏, 𝑎𝑏, 𝑎2𝑏, 𝑎3𝑏}, 𝐶𝜌(𝐿1) ∩ 𝐶 𝜌(𝐿2) = {𝑎, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑏}. further,𝐶𝜌(𝐿) = 𝐶𝜌(𝐿1 ∩ 𝐿2) = 𝐶 𝜌({𝑎3}) = {𝑎𝑘 ; 1 ≤ 𝑘 ≤ 5}. obviously 𝐶𝜌(𝐿1 ∩ 𝐿2) ≠ 𝐶 𝜌(𝐿1) ∩ 𝐶 𝜌(𝐿2) and the statement is verified. 2.13 theorem. let 𝐿1, 𝐿2 ⊆ 𝛴 ∞. then 𝐶𝜌(𝐿1 ∩ 𝐿2) ⊆ 𝐶 𝜌(𝐿1) ∩ 𝐶 𝜌(𝐿2). 132 to some structural properties of ∞-languages proof. with a view to corollary 2.6 we may work with 𝐶1 𝜌 instead of 𝐶𝜌. let 𝑤 ∈ 𝐶1 𝜌 (𝐿1 ∩ 𝐿2). according to definition 2.3 there exist 𝑢 ∈ (𝐿1 ∩ 𝐿2), 𝑣 ∈ (𝐿1 ∩ 𝐿2) and positive integers 𝑝, 𝑟 such that 𝑢𝜌1,𝑝,𝑟 𝑣 = 𝑤 ∈ 𝐶1 𝜌 (𝐿1 ∩ 𝐿2). since 𝑢 ∈ (𝐿1 ∩ 𝐿2), 𝑣 ∈ (𝐿1 ∩ 𝐿2), then 𝑤 ∈ 𝐶1 𝜌 (𝐿1) and also 𝑤 ∈ 𝐶1 𝜌 (𝐿2). thus 𝑤 ∈ 𝐶1 𝜌 (𝐿1) ∩ 𝐶1 𝜌 (𝐿2) and the statement holds. 2.14 example. using the setting of the counterexample from the proof of theorem 2.12, we have 𝐶𝜌(𝐿1 ∩ 𝐿2) = 𝐶 𝜌({𝑎3}) = {𝑎𝑘 ; 1 ≤ 𝑘 ≤ 5} ⊆ 𝐶𝜌(𝐿1) ∩ 𝐶 𝜌(𝐿2) = {𝑎, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑏} to illustrate theorem 2.13. further, we have 𝐶𝜌(𝐿1) ∪ 𝐶 𝜌𝐿2) = {𝑎𝜔 , 𝑏, 𝑎𝑏, 𝑎2𝑏, 𝑎3𝑏, 𝑎𝑘 ; 𝑘 ≥ 1} ⊆ 𝐶𝜌(𝐿1 ∪ 𝐿2) = {𝑎 𝜔 , 𝑎𝑘 , 𝑎𝑘 𝑏, 𝑏; 𝑘 ≥ 1} to illustrate theorem 2.11. 3. operation 𝝆𝒏 3.1 definition. given 𝐿1, 𝐿2 ⊆ 𝛴 ∞ and 𝑛 ≥ 1, an operation 𝜌𝑛 is defined as follows: 𝜌𝑛(𝐿1, 𝐿2) = {𝑥𝑢𝑦; 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑢 ∈ 𝛴 𝑛 𝑤𝑖𝑡ℎ 𝑥𝑢 ∈ 𝐿1𝑎𝑛𝑑 𝑢𝑦 ∈ 𝐿2}. clearly, for each n, 𝜌𝑛 is the operation on 2 𝛴∞ . in this manner the set of operations on 2𝛴 ∞ is given. instead of 𝜌𝑛 (𝐿1, 𝐿2) we also write 𝐿1𝜌𝑛 𝐿2. 3.2 theorem. (i) let 𝐿1, 𝐿2 ⊆ 𝛴 𝜔 . then for all 𝑛 ≥ 1 there holds 𝐿1𝜌𝑛 𝐿2 = ∅. (ii) given 𝐿1, 𝐿2 ⊆ 𝛴 ∗ and let 𝐿1 ∪ 𝐿2 be a finite set. then for all 𝑛 > max 𝑤∈𝐿1∪𝐿2 |𝑤| here holds 𝐿1𝜌𝑛 𝐿2 = ∅. proof. both statements (i), (ii) follow immediately from definition 3.1. 3.3 example. (a) let 𝐿1, 𝐿2 ⊆ {𝑎, 𝑏} ∗, 𝐿1 = {(𝑎𝑏) 𝑘 ; 𝑘 ≥ 1},𝐿2 = {𝑎 𝑘 , 𝑏𝑘 ; 𝑘 ≥ 1}. applying definition 3.1 we have 𝐿1𝜌1𝐿2 = {𝑎𝑏 𝑘 , (𝑎𝑏)𝑘 , (𝑎𝑏)𝑘 𝑏𝑚; 𝑘, 𝑚 ≥ 1}. similarly, and with accordance to theorem 3.2(ii) we get 𝐿1𝜌𝑛𝐿2 = ∅ and 𝐿2𝜌𝑛 𝐿1 = ∅ for any 𝑛 ≥ 2. (b) let 𝐿1, 𝐿2, 𝐿3 ⊆ {𝑎, 𝑏} ∞, 𝐿1 = {𝑎 𝑘 , 𝑏; 𝑘 ≥ 1}, 𝐿2 = {𝑎 3, 𝑏2}, 𝐿3 = {𝑎 𝜔 , 𝑎𝑏}. applying definition 3.1 we have 𝐿1𝜌1𝐿2 = {𝑎 𝑘 , 𝑏2; 𝑘 ≥ 3}, 𝐿2𝜌1𝐿3 = {𝑎𝜔 , 𝑎3𝑏}, ( 𝐿1𝜌1𝐿2)𝜌1𝐿3 = {𝑎 𝜔 , 𝑎𝑘 𝑏; 𝑘 ≥ 1}, 𝐿1𝜌1(𝐿2𝜌1𝐿3) = {𝑎 𝜔 , 𝑎3𝑏}. 3.4 theorem. the operation 𝜌𝑛 is generally (i) not commutative, (ii) not associative. proof. it follows immediately from the results of example 3.3. 3.4 remark. theorem 3.4 justifies the conclusion that the set 2𝛴 ∞ with the operation 𝜌𝑛 forms a „pure“ groupoid. also nonexistence of an identity element may be simply verified. 4. conclusion in this paper we examined algebraic properties of operations on ∞-words having direct relation to ∞-languages generated by ∞automata. it may motivate to consider further types of operations, particularly modeling the depth of memory of such devices. 133 ivan mezník as a generalization a variant structure of ∞-automata may be considered and the corresponding structures of their ∞-languages studied. references [1] j. chvalina, š. hošková-mayerová, general 𝜔-hyperstructures and certain applications of those, ratio mathematica 23, 3–20(2012) [2] z. grodzki, the k-machines, bull. acad. polon. sci. sér. sci. math. astronom. phys., vol. xviii, 7, 541–544(1970) [3] m. juráš and i. mezník, on ig-languages, mathematics university of oulu(1992) [4] w. kwasowiec, generable sets, information and control 17, 257–264(1970) [5] m. linna, on ω-sets associated with context-free languages, inform. and control 31, 273-293(1976) [6] r. mcnaughton, testing and generating infinite sequences by a finite automaton. inform. and control 9, 521–530 (1966) [7] i. mezník, on a subclass of ∞-regular languages, theoretical computer science 61, 25–32(1988) [8] i. mezník, on some structural properties of a subclass of ∞-regular languages, discrete applied mathematics 18, 315–319(1987) [9] d.e. muller, infinite sequences and finite machines, in: ieee proc. fourth ann. symp. on switching theory and logical design, 3–16(1963) [10] m. nivat, infinite words, infinite trees, infinite computations, in: j. w. de bakker and j. van leeuwen, eds., foundations of computer science iii (mathematisch centrum, amsterdam), 5–52(1979) [11] m. novotný, sets constructed by acceptors, inform. and control 26, 116–133(1974) [12] z. pawlak, stored program computers. algorytmy 10, 7–22(1969) [13] d. perrin, an introduction to finite automata on infinite words, lecture notes in computer science 192, 2–17(1984) [14] a. skowron, languages determined by machine systems, bull. acad. polon. sci. sér. sci. math. astronom. phys., vol.xix, 4, 327–329(1971) [15] l. staiger, finite-state ω-languages, j. of comp. and syst. sci. 27, 434–448(1983) 134 ratio mathematica 27 (2014) 61-68 issn:1592-7415 on weights of 2-repeated bursts barkha rohtagia, bhu dev sharmab adepartment of applied sciences, kiet, nh-58, p.box-02, ghaziabad-201206, india barkha.rohtagi@gmail.com bdepartment of mathematics, jiit (deemed university), a-10, sector-62, noida-201301, india bhudev\_sharma@yahoo.com abstract it is generally seen that the behavior of the bursts depend upon the nature of the channel. in a very busy communication channel bursts repeat themselves. in this communication we are exploring the idea of weight consideration of 2-repeated bursts of length b(fixed). some results on weights of 2-repeated bursts of length b(fixed) are derived and some combinatorial results with weight constraint for 2-repeated bursts of length b(fixed) are also given. key words: repeated burst errors, weight of bursts, burst error correcting codes. 2000 ams: 94b20, 94b25, 94b75. 1 introduction in most of the communication channels disturbances due to lightning, break downs and loose connections affect successive digits for some length of the word, causing errors in bursts. abramson (1959) initiated the idea of such errors and developed a class of error correcting codes which correct all double adjacent errors. later, a systematic study in this direction was made by fire (1959), regier (1960) and elspas (1960). these studies were based on the assumption that if errors occur in the form of bursts then all digits within a burst may not be corrupted. easy implementation and efficient functioning 61 b. rohtagi and b. d. sharma are the added advantages with burst error correcting codes. stone (1961) and bridwell and wolf (1970) considered multiple bursts. it was noted by chien and tang (1965) hat in several channels errors occur in the form of a burst but not in the end digit of the burst. channels due to alexander, gryb and nast (1960) belong to this category. in the view of this chien and tang modified the definition of a burst which in literature is known as ct burst. although, this definition was further modified by dass (1980). in general communication the messages are long and the strings of bursts may be short repeating in a vector itself. the notion of repeated burst was introduced by berardi, dass and verma (2009). they defined 2-repeated bursts and obtained results for correction and detection of such type of errors. dass, garg and zannetti (2008) introduced a different type of repeated burst, termed as repeated burst of length b(fixed). later on dass and garg (2009) defined 2-repeated burst of length b(fixed) and gave codes for correcting and detecting such type of errors. sharma and dass (1976) were first to study bursts in terms of weight. the area of 2-repeated burst of length b(fixed) with weight w was explored by dass and garg (2011). in this paper, we obtain results regarding the weight of all vectors having 2-repeated bursts of length b(fixed). the paper has been organized as follows: in section 2 basic definitions are stated with some examples. in section 3 some results on weights of 2-repeated bursts of length b(fixed) are derived. in this correspondence, we shall consider the space of all n-tuples whose nonzero components are taken from the field of q code characters with elements 0, 1, 2, . . . , q − 1. the weight of a vector is considered in hamming sense as the number of non-zero entries. 2 preliminaries we give definition of a burst, defined by fire (1959): definition 2.1. a burst of length b is a vector all of whose nonzero components are confined to some b consecutive components, the first and the last of which is nonzero. a vector may have not just one cluster of errors, but more than one. lumping them into one burst, amounts to neglecting the nature of communication and unnecessarily considering longer burst which may have a part, which is not of cluster in-between. for example in a very busy communication channel, sometimes, bursts repeat themselves. berardi, dass and verma (2009) introduced the idea of repeated bursts. in particular they defined ‘2-repeated burst ’. 62 on weights of 2-repeated bursts a 2-repeated burst of length b may be defined as follows: definition 2.2. a 2-repeated burst of length b is a vector of length n whose only nonzero components are confined to two distinct sets of b consecutive components, the first and the last component of each set being nonzero. example 2.1. (0001204100300) is a 2-repeated burst of length 4 over gf(5). chien and tang (1965) defined a burst of length b which is called as ct burst of length b and may be defined as follows: definition 2.3. a ct burst of length b is a vector whose only non-zero components are confined to some b consecutive positions, the first of which is non-zero. this definition was further modified by dass (1980) as follows: definition 2.4. a burst of length b(fixed ) is an n-tuple whose only non-zero components are confined to b consecutive positions, the first of which is nonzero and the number of its starting positions in an n-tuple is among the first n − b + 1 components. following is the definition of a 2-repeated burst of length b(fixed) as given by dass and garg (2009): definition 2.5. a 2-repeated burst of length b(fixed ) is an n-tuple whose only non-zero components are confined to two distinct sets of b consecutive digits, the first component of each set is non-zero and the number of its starting positions is amongst the first n − 2b + 1 components. for example, (10000010000) is a 2-repeated burst of length up to 5(fixed) whereas (0000100100) is a 2-repeated burst of length at most 3 (fixed). dass and garg (2011) defined a 2-repeated burst of length b(fixed) with weight w as follows: definition 2.6. a 2-repeated burst of length b(fixed ) with weight w or less is an n-tuple whose only non-zero components are confined to two distinct sets of b consecutive components the first component of each set is non-zero where each set can have at most w non-zero components (w ≤ b), and the number of its starting positions is among the first n − 2b + 1 components. for example, (001111000000100000) is a 2-repeated burst of length up to 6(fixed) with weight 4 or less. weight structure being of quite some interest, in the next section, we present some results on weights of 2-repeated bursts of length b(fixed). 63 b. rohtagi and b. d. sharma 3 results on weights of 2-repeated bursts let w2b denotes the total weight of all vectors having 2-repeated bursts of length b in the space of all n-tuples. before obtaining w2b in terms of n and b we give two results in the lemmas below, on counting the 2-repeated bursts. lemma 3.1. the total number of 2-repeated bursts length b > 1(fixed ), in the space of all n-tuple over gf(q), is (n − 2b + 1)(n − 2b + 2) 2 (q − 1)2q2(b−1) . (1) proof. total number of 2-repeated bursts of length b(fixed) in the space of all n-tuples over gf (q) is, refer theorem 1 of dass, garg and zannetti (2008), 1+ ( b 1 ) (q−1)qb−1 + n−2b+1∑ i=1 (q−1)qb−1 [ 1 + ( n−2b−i+2 1 ) (q−1)qb−1 ] . (2) eqn. (2) includes the cases when all vectors are zero and when in the last 2b − 1 position there remains only a single burst of length b(fixed). as we are counting the number of 2-repeated bursts of length b(fixed) only, eqn. (2) reduces to the following form n−2b+1∑ i=1 (q − 1)qb−1 [ 1 + ( n − 2b − i + 2 1 ) (q − 1)qb−1 ] or (n − 2b + 1)(n − 2b + 2) 2 (q − 1)2q2(b−1). this proves the result. next we impose weight restriction on 2-repeated bursts and count their numbers. the results are given in the lemma below. lemma 3.2. the total number of vectors having 2-repeated bursts of length b(fixed ) with weight w (1 ≤ w ≤ b) in the space of all n-tuples is: (n − 2b + 1)(n − 2b + 2) 2 [lb−1w,q ] 2, (3) where [ w∑ s=1 ( b − 1 s − 1 ) (q − 1)s ] = lb−1w,q (4) is the incomplete binomial expansion of [1 + (q − 1)]b−1 up to the (q − 1)w in the ascending powers of (q − 1), w ≤ b. 64 on weights of 2-repeated bursts proof. let us consider a vector having 2-repeated bursts of length b(fixed) with weight w. its only nonzero components are confined to two distinct sets of consecutive components, the first component of each set is nonzero, where each set can have at most w non-zero components (w ≤ b), and the number of its starting positions is among the first n − 2b + 1 components. now, each of these, the first component of each set may be any of the q − 1 nonzero field elements. as we are considering only 2-repeated bursts of length b(fixed) with weight w, in a vector of length n, this will have non-zero positions as follows: (i) first position of first burst. (ii) first position of second burst. (iii) some r − 1 amongst the b − 1 in-between positions of first burst (1 ≤ r ≤ w) and then some s − 1 in the in-between b − 1 positions of the second burst (1 ≤ s ≤ w). (iv) other positions have the value 0. thus analyzing in combinatorial ways, in the earlier counting factor [(q − 1)qb−1]2 replacing one factor qb−1 by ( b−1 r−1 ) (q − 1)r−1 and the other by ( b−1 s−1 ) (q − 1)s−1 each 2-repeated burst will give its number by: (q − 1)(q − 1) w∑ r=1 ( b − 1 r − 1 ) (q − 1)r−1 w∑ s=1 ( b − 1 s − 1 ) (q − 1)s−1 = w∑ r=1 ( b − 1 r − 1 ) (q − 1)r [ w∑ s=1 ( b − 1 s − 1 ) (q − 1)s ] . then from eqn. (4) the number of each 2-repeated burst of length b(fixed) with weight w is given by, [lb−1w,q ] 2. therefore, the total number of 2-repeated bursts of length b(fixed) and weight w, with sum of their starting position (n − 2b + 1)(n − 2b + 2) 2 is (n − 2b + 1)(n − 2b + 2) 2 [lb−1w,q ] 2 . this proves the lemma. now we return to finding an expression for w2b, the total weight of all vectors having 2-repeated bursts of length b(fixed) in the space of all n-tuples. 65 b. rohtagi and b. d. sharma theorem 3.1. for n ≥ b w2 = n(n − 1) 2 (q − 1)2 (5) and w2b = (n − 2b + 1)(n − 2b + 2) 2 w2[lb−1w,q ] 2 . (6) proof. the value of w2 follows simply by considering all vectors having any two non-zero entries out of n. their number clearly is given by( n 2 ) (q − 1)2 = n(n − 1) 2 (q − 1)2 . this gives the value of w2 as stated. next, for b > 1, using the lemma 3.2, the total weight of all vectors having 2-repeated bursts of length b(fixed) each with weight of each burst at most w, is given by w∑ i=1 w∑ j=1 (n − 2b + 1)(n − 2b + 2) 2 i[lb−1w,q ] · j[l b−1 w,q ] = (n − 2b + 1)(n − 2b + 2) 2 w2[lb−1w,q ] 2 . this completes the proof of the theorem. further, in coding theory, an important criterion is to look for minimum weight in a group of vectors. our following theorem is a result in that direction. theorem 3.2. the minimum weight of a vector having 2-repeated burst of length b > 1(fixed ) in the space of all n-tuples is at most[ wlb−1w,q (q − 1)qb−1 ]2 . (7) proof. from lemma 3.1, it is clear that the number of 2-repeated bursts of length b(fixed) in the space of all n-tuples with symbols taken from the field of q elements is [q(b−1)(q − 1)]2 (n − 2b + 1)(n − 2b + 2) 2 . 66 on weights of 2-repeated bursts also from theorem 3.1, their total weight is (n − 2b + 1)(n − 2b + 2) 2 w2[lb−1w,q ] 2 . since the minimum weight element can at most be equal to the average weight, an upper bound on minimum weight of a 2-repeated burst of length b(fixed) is given by (n − 2b + 1)(n − 2b + 2) 2 w2[lb−1w,q ] 2 · 2 (n − 2b + 1)(n − 2b + 2)(q − 1)2q2(b−1) = [ wlb−1w,q (q − 1) q(b−1) ]2 . this proves the result. 4 concluding remarks here we have considered vectors having two bursts of equal lengths b(fixed), with or without weight constraints. studies generalizing these considerations have also attracted our attention that will be reported separately. with these bursts as error patterns in block-wise manner will be a part of later study as codes capable of correcting such type of error patterns will improve the communication rate. references [1] n. m. abramson, a class of systematic codes for non-independent errors, ire trans. on information theory it-5 (4) (1959), 150-157. [2] a. a. alexander, r. m. gryb and d. w. nast, capabilities of the telephone network for data transmission, bell system tech. j. 39 (3) (1960), 431-476. [3] l. berardi, b. k. dass and r. verma, on 2-repeated burst error detecting codes, journal of statistical theory and practice 3 (2009), 381-391. [4] j. d. bridwell and j. k. wolf, burst distance and multiple-burst correction, bell system tech. j. 49 (1970), 889-909. [5] r. t. chien and d. t. tang, on definitions of a burst, ibm journal of research and development 9 (4) (1965), 229-293. 67 b. rohtagi and b. d. sharma [6] b. k. dass, on a burst-error correcting code, journal of information and optimization sciences 1 (3) (1980), 291-295. [7] b. k. dass, p. garg and m. zanneti, on repeated burst error detecting and correcting codes, in special volume of east-west j. of mathematics: contributed in general algebra ii (eds. nguyen van sanh and nittiya pabhapote) (2008), 79-98. [8] b. k. dass and p. garg, on repeated low-density burst error detecting linear codes, mathematical communications 16 (2011), 37-47. [9] b. k. dass and p. garg, on 2-repeatted burst codes, ratio mathematica 19 (2009), 11-24. [10] b. elspas, a note on p-nary adjacent error correcting codes, ire trans. it-6 (1960), 13-15. [11] p. fire, a class of multiple-error-correcting binary codes for nonindependent errors, sylvania report rsl-e-2, sylvania reconnaissance system laboratory, mountain view, calif., (1959). [12] w. w. peterson and e. j. weldon, jr., error-ccorrecting codes, 2nd edition, the mit press, mass. (1972). [13] s. h. reiger, codes for the correction of clustered errors, ire trans. inform. theory, it-6 (1960), 16-21. [14] b. d. sharma and b. k. dass (1972), on weight of bursts, presented at 38th annul. conf. of ims, bhopal, india. [15] j. j. stone, multiple burst error correction, information and control 4 (1961), 324-331. 68 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 36, 2019, pp. 43-52 43 the inclusion and exclusion principle in view of number theory viliam ďuriš* tomáš lengyelfalusy † abstract the inclusion and exclusion (connection and disconnection) principle is mainly known from combinatorics in solving the combinatorial problem of calculating all permutations of a finite set or other combinatorial problems. finite sets and venn diagrams are the standard methods of teaching this principle. the paper presents an alternative approach to teaching the inclusion and exclusion principle from the number theory point of view, while presenting several selected application tasks and possible principle implementation into the matlab computing environment. keywords: inclusion, exclusion, number theory, combinatorics, matlab 2010 ams subject classification: 11b75.‡ * department of mathematics, faculty of natural sciences constantine the philosopher university in nitra, tr. a. hlinku 1, 949 74 nitra, slovakia; vduris@ukf.sk. † department of didactics, technology and educational technologies, dti university sládkovičova 533/20, 018 41 dubnica nad váhom, slovakia; lengyelfalusy@dti.sk. ‡received on may 2nd, 2019. accepted on june 3rd, 2019. published on june 30th, 2019. doi: 10.23755/rm.v36i1.465. issn: 1592-7415. eissn: 2282-8214. ©ďuriš, lengyelfalusy. this paper is published under the cc-by licence agreement. v. ďuriš, t. lengyelfalusy 44 1 introduction in traditional secondary school mathematics (in combinatorics, number theory or even in probability theory), the notion of factorial and combinatorial numbers is introduced [1]. if n and k are two natural numbers with 𝑛 ≥ 𝑘, then we call a combinatorial number the following notation ( 𝑛 𝑘 ) = 𝑛! (𝑛 − 𝑘)! 𝑘! = 𝑛(𝑛 − 1) … (𝑛 − 𝑘 + 1) 1 ∙ 2 ∙ … ∙ 𝑘 while (factorial of the number n) 𝑛! = 1 ∙ 2 ∙ ⋯ ∙ 𝑛, where 𝑛 > 1, 0! = 1, 1! = 1. for combinatorial numbers, the basic properties apply: ( 𝑛 1 ) = 𝑛 ( 𝑛 0 ) = 1 ( 0 0 ) = 1 ( 𝑛 𝑘 ) = ( 𝑛 𝑛 − 𝑘 ) ( 𝑛 𝑘 ) + ( 𝑛 𝑘 + 1 ) = ( 𝑛 + 1 𝑘 + 1 ) the relation ( 𝑛 𝑘 ) + ( 𝑛 𝑘 + 1 ) = ( 𝑛 + 1 𝑘 + 1 ) is the basis for placing combinatorial numbers in the plane in the shape of a triangle (a so-called pascal’s triangle) [2], in which combinatorial numbers can be gradually calculated using the fact that ( 𝑛 0 ) = ( 𝑛 𝑛 ) = 1 for each n. ( 0 0 ) ( 1 0 ) ( 1 1 ) ( 2 0 ) ( 2 1 ) ( 2 2 ) ( 3 0 ) ( 3 1 ) ( 3 2 ) ( 3 3 ) ⋯ if n is a natural number, and if a, b are arbitrary complex numbers, then the binomial theorem can be applied by using the form: (𝑎 + 𝑏)𝑛 = ( 𝑛 0 ) 𝑎𝑛 + ( 𝑛 1 ) 𝑎𝑛−1𝑏 + ⋯ + ( 𝑛 𝑛 − 1 ) 𝑎𝑏𝑛−1 + ( 𝑛 𝑛 ) 𝑏𝑛 the special cases of the binomial theorem are as follows: the inclusion and exclusion principle in view of number theory 45 a) if 𝑎 = 1, 𝑏 = −1: 1 − ( 𝑛 1 ) + ⋯ + (−1)𝑛−1 ( 𝑛 𝑛 − 1 ) + (−1)𝑛 = 0 b) if 𝑎 = 1, 𝑏 = 1: (1 + 1)𝑛 = ( 𝑛 0 ) + ( 𝑛 1 ) + ⋯ + ( 𝑛 𝑛 − 1 ) + ( 𝑛 𝑛 ) = 2𝑛 let us consider now n given objects and k properties 𝑎1, … , 𝑎𝐾. let us denote 𝑁(0) as the number of objects that do not have either of these properties, 𝑁(𝑎𝑖 ) as the number of those that have the property 𝑎𝑖, 𝑁(𝑎𝑖 𝑎𝑗 ) as the number of those that have the property 𝑎𝑖 as well as 𝑎𝑗 etc. then 𝑁(0) = 𝑁 − ∑ 𝑁(𝑎𝑖 ) + ∑ 𝑁(𝑎𝑖 𝑎𝑗 ) − ∑ 𝑁(𝑎𝑖 𝑎𝑗 𝑎𝑠) + ⋯ + (−1)𝐾 𝑁(𝑎1𝑎2 … 𝑎𝐾 ), where, in the first addition, we sum up using numbers 𝑖 = 1, 2, … , 𝐾, in the second addition, using all pairs of these numbers, in the third addition, using all threesomes of these numbers, etc. we call this relationship the inclusion and exclusion principle [3]. the validity of the inclusion and exclusion principle can be shown from the number theory point of view the way that if an object has no property from the properties 𝑎𝑖 , 𝑖 = 1, ⋯ , 𝐾, so it contributes by the unit value to the left equality, though contributing at the same time to the right side, that is, to the number n (in the following additions it does not reappear). let an object now have t properties (𝑡 ≥ 1). then, it does not contribute to the left side as there is a number of objects on the left side that do not have any of the properties. let us calculate the contribution of this object to the right side. in the first addition, it appears t-times. in the second addition, it appears ( 𝑡 2 )–times because from t properties it is possible to choose pairs of the properties in ( 𝑡 2 ) ways. in the third addition, it appears ( 𝑡 3 )–times, etc., so the total contribution to the right side is as follows: 1 − 𝑡 + ( 𝑡 2 )-( 𝑡 3 )+...+(−1)𝑡−1( 𝑡 𝑡−1 ) + (−1)𝑡 = 0, which is a special case of the binomial theorem. thus, the total contribution of such an object to both sides is zero and the right side is actually equal to the number of objects that do not have any of the given properties. v. ďuriš, t. lengyelfalusy 46 2 selected examples of the inclusion and exclusion principle the first example requires some mathematical concepts to be recalled. by the cartesian product of sets a, b we mean set 𝐴 × 𝐵 = {[𝑥, 𝑦]: 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵}, with the symbol |𝐴| we denote the number of elements (so-called cardinality) of the finite set a. if |𝐴| = 𝑎, |𝐵| = 𝑏, the cartesian product then contains 𝑎 ∙ 𝑏 of ordered pairs. since the cartesian product contains ordered pairs, 𝐴 × 𝐵 is not the same set as 𝐵 × 𝐴. [4] the relation f of set a to set b is called a function of set a to set b if ∀𝑥 ∈ 𝐴∃𝑦 ∈ 𝐵: [𝑥, 𝑦] ∈ 𝑓 and simultaneously if [𝑥, 𝑦] ∈ 𝑓 ∧ [𝑥, 𝑧] ∈ 𝑓, so 𝑦 = 𝑧. the symbol 𝐵 𝐴 denotes a set of all functions 𝐴 → 𝐵. if f is a function of set a into set b and ∀𝑥1, 𝑥2 ∈ 𝐴: 𝑥1 ≠ 𝑥2 ⇒ 𝑓(𝑥1) ≠ 𝑓(𝑥2), the function f is called an injective function of set a into set b (or simply an injection; we also say that the function f is ordinary). let us now consider two finite sets a, b, where |𝐴| = 𝑛 and |𝐵| = 𝑚. then the number of all injective functions from a into b is 𝑚 ∙ (𝑚 − 1) ∙ ⋯ ∙ (𝑚 − 𝑛 + 1) = ∏ (𝑚 − 𝑖)𝑛−1𝑖=0 . injections from set 𝐴 = {1,2, ⋯ , 𝑛} into set b, where |𝐵| = 𝑚, are called variations without repetition (or simply variations) of the n-th class from m elements (of the set b). for these functions, the term 𝑉𝑛(𝑚) is used in practice. it is easier to write the expression 𝑚 ∙ (𝑚 − 1) ∙ ⋯ ∙ (𝑚 − 𝑛 + 1) with the following factorial notation 𝑉𝑛(𝑚) = 𝑚! (𝑚−𝑛)! . variations of the n-th class from n elements of the set b are bijective functions 𝐴 → 𝐵 and their number is 𝑛 ∙ (𝑛 − 1) ∙ ⋯ ∙ 2 ∙ 1 = 𝑛!. they are called permutations (of set b) and denote 𝑃(𝑛) = 𝑛!. let us now consider basic set a with the cardinality |𝐴| = 𝑛. combinations (without repetition) of the k-th class (or k-combinations) from n elements are kelement subsets of set a. we denote them as 𝐶𝑘 (𝑛). if a is a finite set, with |𝐴| = 𝑛, then, the number of k-combinations of elements of set a is 𝐶𝑘 (𝑛) = ( 𝑛 𝑘 ) = 𝑛! (𝑛−𝑘)!𝑘! = 𝑛(𝑛−1)⋯(𝑛−𝑘+1) 𝑘(𝑘−1)⋯1 . [5] example 2.1. a group of n men is to take part in a chess tournament. before entering the room, they place their coats in the locker room. however, when they are about to leave, they are unable to recognize their coats. what is the probability that none of them will take their own coat? solution. let us denote the coats 1,2, ⋯ , 𝑁. then the distribution of the coats on the chess players can be made 𝑁!, since these are the permutations of the set {1,2, ⋯ , 𝑁}. first, we determine the number 𝑁(0) of permutations, for which there is no coat on the right player. the number of permutations that do not leave the inclusion and exclusion principle in view of number theory 47 in place the k-element set of coats is (𝑁 − 𝑘)! the number of k-sets can be chosen in ( 𝑁 𝑘 ) ways. then, based on the inclusion and exclusion principle, there applies 𝑁(0) = 𝑁 − ( 𝑁 1 ) (𝑁 − 1)! + ( 𝑁 2 ) (𝑁 − 2)! − ⋯ + (−1)𝑁 ( 𝑁 𝑁 ) (𝑁 − 𝑁)! 𝑁(0) = ∑(−1)𝑘 ( 𝑁 𝑘 ) (𝑁 − 𝑘)! 𝑁 𝑘=0 next, we get 𝑁(0) = ∑(−1)𝑘 𝑁! 𝑘! (𝑁 − 𝑘)! (𝑁 − 𝑘)! = 𝑁 𝑘=0 𝑁! ∑ (−1)𝑘 𝑘! 𝑁 𝑘=0 all permutations of n elements is n!, hence the likelihood that no chess player is wearing his coat when leaving the tournament is 𝑁! ∑ (−1)𝑘 𝑘! 𝑁 𝑘=0 𝑁! = ∑ (−1)𝑘 𝑘! 𝑁 𝑘=0 example 2.2. a tennis centre has a certain number of players and 4 groups a, b, c, d. each player trains in at least one group, while some players train in multiple groups at once according to the table. a.............26 ac...........18 abc...........5 b.............17 ad...........3 abd...........0 c.............58 bc...........9 acd...........2 d.............19 bd...........0 bcd...........0 ab...........7 cd...........5 abcd........0 we will show how many players have a tennis centre. solution. let us denote 𝑀1 as the set of all players in group a, 𝑀2 as the set of all players in group b, 𝑀3 as the set of all players in group c and 𝑀4 as the set of all players in group d. then, set 𝑁 = 𝑀1 ∪ 𝑀2 ∪ 𝑀3 ∪ 𝑀4 is a set of all players in the centre. v. ďuriš, t. lengyelfalusy 48 based on the inclusion and exclusion principle, there applies: 0 = |𝑀1 ∪ 𝑀2 ∪ 𝑀3 ∪ 𝑀4| − (26 + 17 + 59 + 19) + (7 + 18 + 3 + 9 + 5) − (5 + 2) + 0 from which |𝑀1 ∪ 𝑀2 ∪ 𝑀3 ∪ 𝑀4| = 26 + 17 + 59 + 19 − 7 − 18 − 3 − 9 − 5 + 5 + 2 = 85. as a result, the tennis centre has 85 players. example 2.3. let 𝑛 > 1 be a natural number. in number theory, the symbol 𝜑(𝑛) denotes the number of natural numbers smaller than n and relatively prime s n, where 𝜑(𝑛) is called euler’s function [3]. let 𝑛 = 𝑝1 𝛼1 … 𝑝𝑘 𝛼𝑘 be a canonical decomposition of the number n. we will show that the following relation applies: 𝜑(𝑛) = 𝑛 (1 − 1 𝑝1 ) (1 − 1 𝑝2 ) … (1 − 1 𝑝𝑘 ) solution. once more, we will use the inclusion and exclusion principle. let 𝑛 = 𝑝1 𝛼1 𝑝2 𝛼2 … 𝑝𝑘 𝛼𝑘 is a canonical decomposition of the number n. the natural numbers that are relatively prime with the number n are those that are not divisible by either of the prime numbers 𝑝1, 𝑝2, … , 𝑝𝑘. so, let 𝑎𝑖 mean the property that “the number m is divisible by the prime number 𝑝𝑖 , 𝑖 = 1, … , 𝑘“. the number of numbers that are smaller or equal to the number n and are divisible by the number 𝑝𝑖 is 𝑁(𝑎𝑖 ) = 𝑛 𝑝𝑖 . it is an integer since 𝑝𝑖 ⃓𝑛. next, we get 𝑁(𝑎𝑖 𝑎𝑗 ) = 𝑛 𝑝𝑖𝑝𝑗 and other members of the notation. then: 𝜑(𝑛) = 𝑛 − ∑ 𝑛 𝑝𝑖 + ∑ 𝑛 𝑝𝑖 𝑝𝑗 − ∑ 𝑛 𝑝𝑖 𝑝𝑗 𝑝𝑠 + ⋯ + (−1)𝑘 𝑛 𝑝1𝑝2 … 𝑝𝑘 this expression can be simplified to the form: 𝜑(𝑛) = 𝑛 (1 − 1 𝑝1 ) (1 − 1 𝑝2 ) … (1 − 1 𝑝𝑘 ) several other interesting tasks and applications of the inclusion and exclusion principle can be found e.g. in the resources [6], [7]. the inclusion and exclusion principle in view of number theory 49 3 implementation of the inclusion and exclusion principle in the matlab computing environment when solving various practical tasks with pupils, it is possible and appropriate to use some computing environment, e.g. matlab. we will now solve a simple task of divisibility. example 3.1. we will show how many numbers there are up to 1000 that are not divisible by three, five, or seven. solution. before proceeding to the solution of the task, we will use divisibility relations to determine the number of all natural numbers smaller than 1000, each of which can be divided simultaneously by three, five, and seven. first, we will generally show that if 3|𝑎, 5|𝑎, then 3 ∙ 5 = 15|𝑎, being valid if 3|𝑎, so 𝑎 = 3𝑏, if 5|𝑎, so 𝑎 = 5𝑐. the left sides are equal, so the right sides must be equal, too. then 3𝑏 = 5𝑐 since (3,5) = 1 ⇒ 3|c ⇒ 𝑐 = 3𝑑. then 𝑎 = 5𝑐 = 15𝑑 ⇒ 15|𝑎. now, we will show that if 15|𝑎, 7|𝑎, then 15 ∙ 7 = 105|𝑎 is valid if 15|𝑎, so 𝑎 = 15𝑒, if 7|𝑎, so 𝑎 = 7𝑓. since 𝑎 = 𝑎, it holds true that 15𝑒 = 7𝑓 from the relation (15,7) = 1 ⇒ 15|f ⇒ 𝑓 = 15𝑔. then 𝑎 = 7𝑓 = 105𝑔 ⇒ 105|𝑎. we will do the division 1000 105 = 9 + 55 105 and we see that there exist 9 numbers with the required property. let us get back to our basic task. there, we have 𝑁 = 1000. let 𝑎1 be the property that “the number n is divisible by three“, property 𝑎2 stand for “the number n is divisible by five“, property 𝑎3 stand for “the number n is divisible by seven“. at the same time, 𝑁(0) is the number of searched numbers not divisible by any of the numbers 3, 5, 7. every third natural number is divisible by three since 1000 = 3 ∙ 333 + 1. we have the number 𝑁(𝑎1) = 333, that is 333 numbers up to 1000 are divisible by three. by similar consideration, we determine 𝑁(𝑎2) = 200, 𝑁(𝑎3) = 142. v. ďuriš, t. lengyelfalusy 50 based on the previous considerations, we determine the number 𝑁(𝑎1𝑎2). it holds true that if a number is divisible by three and five, it is also divisible by its product, i.e. by the number 15 (inasmuch as the numbers 3 and 5 are relatively prime). hence, 𝑁(𝑎1𝑎2) equals the number of numbers up to 1000 divisible by 15 and 𝑁(𝑎1𝑎2) = 66. similarly, we determine 𝑁(𝑎2𝑎3) = 28 and 𝑁(𝑎1𝑎3) = 47. for the number 𝑁(𝑎1𝑎2𝑎3) it is valid that it will be equal to the number of numbers up to 1000 that are divisible by the product 3 ∙ 5 ∙ 7 = 105, hence 𝑁(𝑎1𝑎2𝑎3) = 9. then, based on the inclusion and exclusion principle, we have in total 𝑁(0) = 1000 − (333 + 200 + 142) + (66 + 28 + 47) − 9 = 457 now we implement the given task into the matlab computing environment to verify the result. first we create the function “count_the_divisors”, which is the application of the inclusion and exclusion principle: function cnt = count_the_divisors(n, a, b, c) cnt_3 = floor(n / a); %counts of numbers divisible by a cnt_5 = floor(n / b); %counts of numbers divisible by b cnt_7 = floor(n / c); %counts of numbers divisible by c cnt_3_5 = floor(n / (a * b)); %counts of numbers divisible by a and b cnt_5_7 = floor(n / (b * c)); %counts of numbers divisible by b and c cnt_3_7 = floor(n / (a * c)); %counts of numbers divisible by a and c cnt_3_5_7 = floor(n / (a * b * c)); %counts of numbers divisible by a, b and c %and now inclusion-exclusion principle applied cnt = n (cnt_3 + cnt_5 + cnt_7) + (cnt_3_5 + cnt_5_7 + cnt_3_7) cnt_3_5_7; we will call the function from the command line: >> n = 1000; >> count_the_divisors(n, 3, 5, 7) the inclusion and exclusion principle in view of number theory 51 ans = 457 when creating functions or scripts solving various problems based on the inclusion and exclusion principle, it is possible to use various set operations (functions) built directly in matlab without the need to create one’s own structures. [8] 4 conclusion the principle of inclusion and exclusion is a “set problem“ that falls within the field of discrete mathematics with different applications in combinatorics. however, this principle also plays a significant role in number theory when defining the so-called euler’s function or fermat’s theorem, or in clarifying and exploring the fundamental problems of number theory, such as expressing the distribution of prime numbers among natural numbers on the numerical axis and many other questions still open today. the paper offered something different than just a set view of the inclusion and exclusion principle and its definition using number theory knowledge and the properties of combinatorial numbers. our work is a guideline for solving selected practical tasks in which the involvement of the principle might not be expected at first sight. we also showed the possible application of ict and the matlab computing environment in solving computational problems in the field of number theory, which can be concurrently involved in mathematics teaching. in conclusion, the inclusion and exclusion principle has much more application than we allege in our short contribution and can be used to solve more difficult tasks, e.g. in algebra to solve specific systems of equations or to solve various problems in combination with the dirichlet principle. some research shows that the ability to solve problems also depends on the substitution thinking, which makes possible to use mathematical knowledge effectively in various areas of number theory [9]. v. ďuriš, t. lengyelfalusy 52 references [1] j. sedláček. faktoriály a kombinační čísla. praha, mladá fronta, 1964. [2] a. vrba. kombinatorika. praha: mladá fronta, 1980. [3] š. znám. teória čísel. bratislava, spn, 1975. [4] m.t. keller, w.t. trotter. applied combinatorics. american institute of mathematics, 2017. [5] m. škoviera. úvod do diskrétnej matematiky. bratislava, katedra informatiky fmfi uk, 2007. [6] k. h. rosen. discrete mathematics and its applications. 4th ed., wcb/mcgraw hill, boston, 1999. [7] m.j. erickson. introduction to combinatorics. john wiley & sons, new york, isbn: 0-471-15408-3, 1996. [8] mathwork. online documentation. 2019. available at: https://www.mathworks.com/help/matlab/set-operations.html, accessed 15th of april 2019. [9] d. gonda: the elements of substitution thinking and its impact on the level of mathematical thinking. in: iejme — mathematics education, vol. 11, no. 7, p. 2402-2417, look academic publishers, 2016. ratio mathematica 26 (2014), 21-38 issn:1592-7415 on multiplication γ-modules a. a. estaji1, a. as. estaji2, a. s. khorasani3, s. baghdari4 1 2 3 4 department of mathematics and computer sciences, hakim sabzevari university, po box 397, sabzevar, iran. 1 aaestaji@hsu.ac.ir, 2 a$ -$aestaji@yahoo.com 3 saghafiali21@yahoo.com, 4 m.baghdari@yahoo.com abstract in this article, we study some properties of multiplication mγmodules and their prime mγ-submodules. we verify the conditions of acc and dcc on prime mγ-submodules of multiplication mγmodule. key words: γ-ring, multiplication mγ-module, prime mγ-submodule, prime ideal. msc 2010: 13a15, 16d25, 16n60. 1 introduction the notion of a γ-ring was first introduced by nobusawa [17]. barnes [5] weakened slightly the conditions in the definition of γ-ring in the sense of nobusawa. after the γ-ring was defined by barnes and nobusawa, a lot of researchers studied on the γ-ring. barnes [5], kyuno [15] and luh [16] studied the structure of γ-rings and obtained various generalizations analogous of corresponding parts in ring theory. recently, dumitru, ersoy, hoque, öztürk, paul, selvaraj, have studied on several aspects in gammarings (see [10, 8, 12, 14, 18, 19, 20]). mccasland and smith [14] showed that any noetherian module m contains only finitely many minimal prime submodules. d. d. anderson [2] generalized the well-known counterpart of this result for commutative rings, i.e., he abandoned the noetherianness and showed that if every prime ideal minimal over an ideal i is finitely generated, then r contains only finitely many prime ideals minimal over i. behboodi and koohy [7] showed that this 21 a. a. estaji, a. as. estaji, a. s. khorasani, s. baghdari result of anderson was true for any associative ring (not necessarily commutative) and also, they extended it to multiplication modules, i.e., if m is a multiplication module such that every prime submodule minimal over a submodule k is finitely generated, then m contains only finitely many prime submodules minimal over k. in this paper, we study some properties of multiplication left mγ-modules and their prime mγ-submodules. this paper is organized as follows: in section 2, we review some basic notions and properties of γ-rings. in section 3, the concept of a moltiplication mγ-module is introduced and its basic properties are discussed. also, we show that if l is a left operator ring of the γ-ring m and a is a multiplication unitary left mγ-module, then a is a multiplication left l-module. in section 4, we proved that in fact this result was true for γ-rings and mγ-modules. 2 preliminaries in this section we recall certain definitions needed for our purpose. recall that for additive abelian groups m and γ we say that m is a γ-ring if there exists a mapping · : m × γ ×m −→ m (m,γ,m′) −→ mγm′ such that for every a,b,c ∈ m and α,β ∈ γ, the following hold: 1. (a+b)αc = aαc+bαc, a(α+β)c = aαc+aβc and aα(b+c) = aαb+aαc; 2. (aαb)βc = aα(bβc). note that any ring r, can be regarded as an r-ring. a γ-ring m is called commutative, if for any x,y ∈ m and γ ∈ γ, we have xγy = yγx. m is called a γ-ring with unit, if there exists elements 1 ∈ m and γ0 ∈ γ such that for any m ∈ m, 1γ0m = m = mγ01. if a and b are subsets of a γ-ring m and θ ⊆ γ, we denote aθb, the subset of m consisting of all finite sums of the form ∑ aiγibi, where (ai,γi,bi) ∈ a × θ × b. for singleton subsets we abbreviate this notation for example, {a}θb = aθb. a subset i of a γ-ring m is said to be a right ideal of r if i is an additive subgroup of m and iγm ⊆ i. a left ideal of m is defined in a similar way. if i is both a right and left ideal, we say that a is an ideal of m. for each subset s of a γ-ring m, the smallest right ideal containing s is called the right ideal generated by s and is denoted by |s〉. similarly 22 on multiplication γ-modules we define 〈s| and 〈s〉, the left and two-sided (respectively) ideals generated by s. for each a of a γ-ring m, the smallest right ideal containing a is called the principal right ideal generated by a and is denoted by |a〉. we similarly define 〈a| and 〈a〉, the principal left and two-sided (respectively) ideals generated by a. we have |a〉 = za + aγm, 〈a| = za + mγa, and 〈a〉 = za + aγm + mγa + mγaγm, where za = {na : n is an integer}. let i be an ideal of γ-ring m. if for each a + i, b + i in the factor group m/i, and each γ ∈ γ, we define (a + i)γ(b + i) = aγb + i, then m/i is a γ-ring which we shall call the difference γ-ring of m with respect to i. let m be a γ-ring and f the free abelian group generated by γ × m. then a = { ∑ i ni(γi,xi) ∈ f : a ∈ m ⇒ ∑ l niaγixi = 0} is a subgroup of f. let r = f/a, the factor group, and denote the coset (γ,x) + a by [γ,x]. it can be verified easily that [α,x] + [β,x] = [α + β,x] and [α,x] + [α,y] = [α,x + y] for all α, β ∈ γ and x, y ∈ m. we define a multiplication in r by ∑ i[αi,xi] ∑ j[βj,yj] = ∑ ij [αi,xiβjyj]. then r forms a ring. if we define a composition on m×r into m by a ∑ l[αi,xi] = ∑ i aαixi for a ∈ m,∑ i[αi,xi] ∈ r, then m is a right r-module, and we call r the right operator ring of the γ -ring m. similarly, we may construct a left operator ring l of m so that m is a left l-module. clearly i is a right (left) ideal of m if and only if i is a right r-module (left lmodule) of m. also if a is a right (left) ideal of r(l), then ma(am) is an ideal of m. for subsets n ⊆ m, φ ⊆ γ, we denote by [φ,n] the set of all finite sums ∑ i[γi,xi] in r, where γi ∈ φ, xi ∈ n, and we denote by [(φ,n)] the set of all elements [ϕ,x] in r, where ϕ ∈ φ, x ∈ n. thus, in particular, r = [γ,m]. an ideal p of m is prime if, for any ideals u and v of m, uγu ⊆ p implies u ⊆ p or v ⊆ p . a subset s of m is an m-system in m if s = ∅ or if a,b ∈ s implies < a > γ < b > ∩s 6= ∅. the prime radical p(a) is the set of x in m such that every m-system containing x meets a. the prime radical of the zero ideal in a γ-ring m is called the prime radical of the γ-ring m which we denote by p(m). an ideal q of m is semi-prime if, for any ideals u of m, uγu ⊆ q implies u ⊆ q. proposition 2.1. [15] if q is an ideal in a commutative γ-ring with unit m, then p(q) is the smallest semi-prime ideal in m which contains q, i.e. p(q) = ⋂ p where p runs over all the semi-prime ideals of m such that q ⊆ p . let p be a proper ideal in a commutative γ-ring with unit m. it is clear that the following conditions are equivallent. 23 a. a. estaji, a. as. estaji, a. s. khorasani, s. baghdari 1. p is semi-prime. 2. for any a ∈ m, if aγ0a ∈ p , then a ∈ p . 3. for any a ∈ m and n ∈ n, if (aγ0)na ∈ p , then a ∈ p . proposition 2.2. [13] let q be an ideal in a commutative γ-ring with unit m and a be the set of all x ∈ m such that (xγ0)nx ∈ q for some n ∈ n∪{0}, where (xγ0) 0x = x. then a = p(q). 3 mγ-module let m be a γ-ring. a left mγ-module is an additive abelian group a together with a mapping · : m × γ × a −→ a ( the image of (m,γ,a) being denoted by mγa), such that for all a,a1,a2 ∈ a, γ,γ1,γ2 ∈ γ, and m,m1,m2 ∈ m the following hold: 1. mγ(a1 + a2) = mγa1 + mγa2; 2. (m1 + m2)γa = m1γm + m2γa; 3. m1γ1(m2γ2a) = (m1γ1m2)γ2a. a right mγ-module is defined in analogous manner. if i is a left ideal of a γ-ring m, then i is a left mγ-module with rγa (r ∈ m,γ ∈ γ,a ∈ i) being the ordinary product in m. in particular, {0} and m are mγ-modules. let a be a left mγ-module and b a nonempty subset of a. b is a mγsubmodule of a, which we denote by b ≤ a, provided that b is an additive subgroup of a and mγb ∈ b, for all (m,γ,b) ∈ m × γ ×b. definition 3.1. let a be a left mγ-module and x a subset of a. let {aλ}λ∈λ be the family of all mγ-submodule of a which contain x. then ⋂ λ∈λ aλ is called the mγ-submodule of a generated by the set x and denoted 〈x|. if b ⊆ a, n ⊆ m and θ ⊆ γ, we denote nθb, the subset of a consisting of all finite sums of the form ∑ niγibi where (ni,γi,bi) ∈ n × θ × b. for singleton subsets we abbreviate this notation for example, {n}θb = nθb. if x = {a1, . . . ,an}, we write 〈a1, . . . ,an| in place of 〈x|. if a = 〈a1, . . . ,an|, (ai ∈ a), a is said to be finitely generated. if a ∈ a, the mγ-submodule 〈a| of a is called the cyclic mγ-submodule generated by a. we have 〈x| = zx + mγx, where zs = { ∑k i=1 nixi : ni ∈ z,xi ∈ s and k is an integer}. finally, if {bλ}λ∈λ is a family of mγ-submodules of a, then the mγsubmodule generated by x = ⋃ λ∈λ bλ is called the sum of the mγ-modules 24 on multiplication γ-modules bλ and usually denoted 〈x| = ∑ λ∈λ bλ. if the index set λ is finite, the sum of b1, . . . , bk is denoted b1 + b2 + ... + bk. it is clear that if {bλ}λ∈λ is a family of mγ-submodules of a, then ∑ λ∈λ bλ consists of all finite sums bλ1 + . . . + bλk with bλj ∈ bλl. proposition 3.1. let m be a γ-ring and {iλ}λ∈λ be a family of left ideals of m. if a is a left mγ-module, then ( ∑ λ∈λ iλ)γa = ∑ λ∈λ (iλγa). proof. let x ∈ ( ∑ λ∈λ iλ)γa. then there exists a1, . . . ,ak ∈ a and γ1, . . . ,γk ∈ γ and x1, . . . ,xk ∈ ∑ λ∈λ iλ such that x = ∑k t=1 xtγtat, it follows that for 1 ≤ t ≤ k, xt = ∑kt j=1 iλjt with iλjt ∈ iλjt. hence x = ∑k t=1 ∑kt j=1 iλjtγtat ∈∑ λ∈λ(iλγa). therefore ( ∑ λ∈λ iλ)γa ⊆ ∑ λ∈λ(iλγa). also, since for every λ ∈ λ, iλγa ⊆ ( ∑ λ∈λ iλ)γa, we conclude that ∑ λ∈λ(iλγa) ⊆ ( ∑ λ∈λ iλ)γa. hence ( ∑ λ∈λ iλ)γa = ∑ λ∈λ(iλγa). definition 3.2. if a is a left mγ-module and s is the set of all mγ-submodules b of a such that b 6= a, then s is partially ordered by set-theoretic inclusion. b is a maximal mγ-submodule if and only if b is a maximal element in the partially ordered set s. proposition 3.2. if a is a non-zero finitely generated left mγ-module, then the following statements are hold. 1. if k is a proper mγ-submodule of a, then there exists a maximal mγsubmodule of a such that contain k. 2. a has a maximal mγ-submodule. proof. (1) let a = 〈a1, . . . ,an| and s = {l : k ⊆ l and l is a proper mγ-submodule of a}. s is partially ordered by inclusion and note that s 6= ∅, since k ∈ s. if {lλ}λ∈λ is a chain in s, then l = ⋃ λ∈λ lλ is a mγ-submodule of a. we show that l 6= a. if l = a, then for every 1 ≤ i ≤ n, there exists λi ∈ λ such that ai ∈ lλi. since {lλ}λ∈λ is a chain in s, we conclude that there exists 1 ≤ j ≤ n such that a1, . . . ,an ∈ lλj . therefore a = lλj ∈ s which contradicts the fact that a 6∈ s. it follows easily that l is an upper bound {lλ}λ∈λ in s. by zorn’s lemma there exists a proper mγ-submodule b of a that is maximal in s. it is a clear that b a maximal mγ-submodule of a such that contain k. (2) by part (1), it suffices we put k = 〈0|. 25 a. a. estaji, a. as. estaji, a. s. khorasani, s. baghdari definition 3.3. a left mγ-module a is unitary if there exists an element, say 1 in m and an element γ0 ∈ γ, such that, 1γ0a = a and 1γ0m = m = mγ01 for every (a,m) ∈ a×m. corolary 3.1. if m is a unitary left (right) mγ-module, then m has a left (right) maximal ideal. proof. it is evident by proposition 3.2. let a be a left mγ-module. let x ⊆ a and let b ≤ a. then the set (b : x) := {m ∈ m : mγx ⊆ b} is a left ideal of m. in particular, if a ∈ a, then (0 : a) := ((0) : {a}) is called the left annihilator of a and (0 : a) := ((0) : a) is an ideal of m called the annihilating ideal of a. furthermore a is said to be faithful if and only if (0 : a) = (0). definition 3.4. a left mγ-module a is called a multiplication left mγ-module if each mγ-submodule of a is of the form iγa, where i is an ideal of m. proposition 3.3. let b be a mγ-submodule of multiplication left mγ-module a. then b = (b : a)γa. proof. it is a clear that (b : a)γa ⊆ b. since a is a multiplication left mγmodule, we conclude that there exists ideal i of γ-ring m such that b = iγa, it follows that b = iγa ⊆ (b : a)γa ⊆ b. therefore b = (b : a)γa. proposition 3.4. let a be a left mγ-module. a is multiplication if and only if for every a ∈ a, there exists ideal i in m such that 〈a| = iγa. proof. in view of definition 3.4, it is enough to show that if for every a ∈ a, there exists ideal i in m such that 〈a| = iγa, then a is multiplication. let b be an mγ-submodule of a. then for every b ∈ b, there exists ideal ib in m such that 〈b| = ibγa. by proposition 3.1, ( ∑ b∈b ib)γa = ∑ b∈b(ibγa) =∑ b∈b〈b| = b, it follows that a is multiplication. proposition 3.5. let m be a γ-ring which has a unique maximal ideal q and a be a unitary multiplication left mγ-module. if every ideal i in m is contained in q, then for every a ∈ a\qγa, 〈a| = a. proof. suppose that a ∈ a\qγa. since a is multiplication left mγ-module, we conclude that there exists ideal i in m such that 〈a| = iγa. clearly i 6⊆ q and hence i = m, which implies 〈a| = mγa = a. corolary 3.2. let γ-ring m be a unitary left mγ-module which has a unique maximal ideal q and a be a unitary multiplication left mγ-module. then for every a ∈ a\qγa, 〈a| = a. 26 on multiplication γ-modules proof. by propositions 3.2 and 3.5, it is evident. proposition 3.6. let l be a left operator ring of the γ-ring m and let a be a unitary left mγ-module. if we define a composition on l × a into a by ( ∑ l[xi,αi])a = ∑ i xiαia for a ∈ a, ∑ i[xi,αi] ∈ l, then a is a left lmodule. also, for every b ⊆ a, b is a mγ-submodule of a if and only if b is a l-submodule of a. proof. suppose that 1 ∈ m and γ0 ∈ γ such that for every (a,m) ∈ a×m, 1γ0a = a and 1γ0m = m = mγ01. let ∑t i=1[xi,αi] = ∑s j=1[yj,βj] ∈ l and a = b ∈ a. by definition of left operator ring of the γ-ring m, we conclude that ∑t i=1 xiαi1 = ∑s j=1 yjβj1, it follows that ( ∑t i=1[xi,αi])a = ∑t i=1 xiαia = ∑t i=1(xiαi(1γ0a)) = ∑t i=1(xiαi1)γ0a = ( ∑t i=1 xiαi1)γ0a = ( ∑s j=1 yjβj1)γ0b = ∑s j=1 yjβjb = ( ∑s j=1[yj,βj])b hence composition on l × a into a is a well-defined. let r = ∑t i=1[xi,αi] and s = ∑s j=1[yj,βj]. then for every a ∈ a, (rs)a = ( ∑ i,j[xiαiyj,βj])a = ∑ i,j(xiαiyj)βja = ∑ i,j xiαi(yjβja) = ∑t i=1 xiαi( ∑s j=1 yjβja) = ( ∑t i=1[xi,αi])( ∑s j=1 yjβja) = r(( ∑s j=1[yj,βj])a) = r(sa) the remainder of the proof is now easy. proposition 3.7. let l be a left operator ring of the γ-ring m. if a is a multiplication unitary left mγ-module, then a is a multiplication left lmodule. proof. let b be a l-submodule of a. by proposition 3.6, b is a mγsubmodule of a and there exists ideal i of γ-ring m such that b = iγa. it well known that [γ,i] is an ideal of l. we show that b = [i, γ]a. suppose that a1, . . . ,at ∈ a, and for every 1 ≤ i ≤ t, ∑ki j=1[xij,αij ] ∈ [i, γ]. then we 27 a. a. estaji, a. as. estaji, a. s. khorasani, s. baghdari have ∑t i=1( ∑ki j=1[xij,αij ])ai = ∑t i=1 ∑ki j=1 xijαijai) ∈ b and it follows that [i, γ]a ⊆ b. also, if b ∈ b, then there exists x1, . . . ,xt ∈ i,γ1, . . . ,γt ∈ γ, and a1, . . . ,at ∈ a such that b = ∑t i=1 xiγiai = ∑t i=1[xi,γi]ai ∈ [i, γ]a and we conclude that b = [i, γ]a. proposition 3.8. let a be a unitary cyclic left mγ-module. if l is a left operator ring of the γ-ring m and for every l, l′ ∈ l, there exists l′′ ∈ l such that ll′ = l′′l, then a is a multiplication left l-module. proof. let b be a l-submodule of a and i = {l ∈ l : la ⊆ b}, then ia ⊆ b. since a is a unitary cyclic left mγ-module, we conclude that there exists a ∈ a such that a = mγa. let b ∈ b. hence there exists m1, . . . ,mt ∈ m and γ1, . . . ,γt ∈ γ such that b = ∑t i=1 miγia. in view of operations of addition and multiplication in left l-module a, we have b = ∑t i=1[mi,γi]a = ( ∑t i=1[mi,γi])a. we put l = ∑t i=1[mi,γi] and it follows that b = la. if a′ ∈ a, then a similar argument shows that there exists l′ ∈ l such that a′ = l′a. by hypothesis, there exists l′′ ∈ l such that ll′ = l′′l. therefore la′ = ll′a = l′′la = l′′b ∈ b and it follows that l ∈ i, this is b = la ∈ ia. hence b = ia and the proof is now complete. definition 3.5. let a be a unitary left mγ-module and b be a mγ-submodule in a and p ∈ max(m). a is called p -cyclic if there exist p ∈ p and b ∈ b such that (1−p)γ0b ⊆ mγb and also, it is clear that (1−p)γ0b = (1−p)γb. define tpb as the set of all b ∈ b such that (1−p)γ0b = 0, for some p ∈ p . lemma 3.1. let a be a unitary left mγ-module and b be a mγ-submodule in a and p ∈ max(m). if m is a commutative γ-ring, then tpb is a mγ-submodule in a. proof. suppose b1,b2 ∈ tpb. so there exist p1,p2 ∈ p such that b1 = p1γ0b1 and b2 = p2γ0b2. let p0 = p1 +p2−p1γ0p2. it is clear that (1−p0)γ0(b1−b2) = 0. hence b1 − b2 ∈ tpb. let x ∈ mγ(tpb). so x = ∑n i=1 miγibi, where n ∈ n, bi ∈ tpb, γi ∈ γ and mi ∈ m (1 ≤ i ≤ n). suppose i ∈ {1, · · · ,n}. since bi ∈ tpn, there exists pi ∈ p such that (1 − pi)γ0miγibi = 0. hence miγibi ∈ tpn. thus x ∈ tpb. hence mγtpb = tpb. proposition 3.9. let m be a commutative γ-ring and let a be a unitary left mγ-module. a is multiplication mγ-module if and only if for any ideal p ∈ max(m), either a = tpa or a is p -cyclic. proof. let a be a multiplication ideal and p ∈ max(m). first suppose that a = pγa. since a is multiplication ideal, we conclude that for every a ∈ a, there exists an ideal i in m such that < a >= iγa. hence < a >= pγ < 28 on multiplication γ-modules a >. so there exists p ∈ p such that (1−p)γ0a = 0, it follows that a ∈ tpb and then a = tpa. now suppose that a 6= pγa and x ∈ a\pγa. then there exists an ideal i in m such that < x >= iγa and p + i = m. obviously, if we assume that p ∈ p , then (1 −p)γ0a ⊆ mγx. therefore a is p-cyclic. conversely, suppose that b is a mγ-submodule in a. define i as the set of all m ∈ m, where mγ0a ∈ b for any a ∈ a. clearly i is an ideal in m and iγa ⊆ b. let b ∈ b. define k as the set of all m ∈ m, where mγ0b ∈ iγa. we claim k = m. assume that k ⊂ m. hence by zorns lemma there exists q ∈ max(m) such that k ⊆ q ⊂ m. by hypothesis a = tqa or a is q-cyclic. if a = tqa, then there exists s ∈ q such that (1−s)γ0b = 0. hence (1−s) ∈ k ⊆ q, it follows that 1 ∈ q, a contradiction. if a is q-cyclic, then there exist t ∈ q and c ∈ a such that (1−t)γ0a ⊆ mγc =< c >. define l as the set of all m ∈ m such that mγ0c ∈ (1−t)γ0b. clearly l is an ideal in m and lγ0c ⊆ (1 − t)γ0b ⊆< c >. hence (1 − t)γ0b ⊆ lγ0c. so (1 − t)γ0b = lγ0c, it follows that (1 − t)γ0lγ0a ⊆ (1 − t)γ0b ⊆ b and (1 − t)γ0l ⊆ i. therefore (1 − t)γ0(1 − t)γ0b ⊆ iγa. hence (1 − t)γ0(1 − t) ∈ k ⊆ q. thus 1 − t ∈ q, it follows that 1 ∈ q, a contradiction. hence k = m and b ∈ iγa. thus a is a multiplication ideal. let a be a left mγ-module. a is said to have the intersection property provided that for every non-empty collection of ideals {iλ}λ∈λ of m,⋂ λ∈λ iλγa = ( ⋂ λ∈λ iλ)γa. if left mγ-module of a has intersection property, then for every non-empty collection of ideals {iλ}λ∈λ of m,⋂ λ∈λ iλγa = ( ⋂ λ∈λ (iλ + ann(a)))γa. proposition 3.10. let m be a commutative γ-ring and let a be a unitary left mγ-module. 1. if a has intersection property and for any mγ-submodule n in a any ideal i in m which n ⊂ iγa, there exists ideal j in m such that j ⊂ i and n ⊆ jγa, then a is multiplication left mγ-module. 2. if a is faithful left multiplication mγ-module, then a has intersection property and for any mγ-submodule n in a any ideal i in m which n ⊂ iγa, there exists ideal j in m such that j ⊂ i and n ⊆ jγa. 29 a. a. estaji, a. as. estaji, a. s. khorasani, s. baghdari proof. (1) let n be a mγ-submodule in a and s = {i : i is an ideal of m and n ⊆ iγa}. clearly m ∈ s. since a has intersection property, we conclude from zorns lemma that s has a minimal member i (say). since n ⊆ iγa and i is minimal element of s, we can conclude that n = iγa. it follows that a is a multiplication ideal. (2) let {iλ}λ∈λ be a nonempty collection of ideal in m and i = ⋂ λ∈λ iλ. clearly iγa ⊆ ⋂ λ∈λ(iλγa). let x ∈ ⋂ λ∈λ(iλγa) and we put l = {m ∈ m : mγ0x ∈ iγa}. we claim l = m. assume that l ⊂ m. by proposition 3.2, there exists p ∈ max(m) such that l ⊆ p . it is clear that x 6∈ tpa. hence tpa 6= a and by proposition 3.9, a is p-cyclic. hence there exist a ∈ a and p ∈ p such that (1−p)γ0a ⊆ mγa =< a >. thus (1−p)γ0x ∈ ⋂ λ∈λ(iλγ0a) and so for any λ ∈ λ, (1 −p)γ0x ∈ iλγ0a. it is clear that (1 −p)γ0(1 −p) ∈ l ⊆ p , in view of the fact that a is faithful. hence 1 ∈ p , a contradiction. therefore l = m, it follows that x = 1γ0x ∈ iγa and a has intersection property. now suppose n be a mγ-submodule in a and i be an ideal in m which n ⊂ iγa. since a is multiplication mγ-module, there exists an ideal j in m such that n = jγa. let k = i ∩ j. clearly, k ⊂ i and since a has intersection property, we conclude that n ⊆ kγa. the proof is now complete. proposition 3.11. let a be a faithful multiplication mγ-module and i,j be two ideals in m. iγa ⊆ jγa if and only if either i ⊆ j or a = [j : i]γa. proof. let i 6⊆ j. note that [j : i] = ⋂ i∈x[j :< i >] where x is the set of all elements i ∈ i with i 6∈ j. by proposition 3.10, [j : i]γa = ⋂ i∈x ([j :< i >]γa) if for every i ∈ x, a = [j :< i >]γa, then a = [j : i]γa, which finishes the proof. let i ∈ x and q = [j :< i >]. it is clear that q 6= m. let ω denote the collection of all semi-prime ideals p in m containing q. suppose that there exists p ∈ ω such that a 6= pγa and x ∈ a\pγa. since a is a multiplication mγ-module, we conclude that there exists ideal d in m such that < x >= dγa and d 6⊆ p . thus cγa ⊆< x > for some c ∈ d \ p . now we have cγaγa ⊆ jγ < x >. it is easily to show that for any γ ∈ γ, there exists γ1 ∈ γ and b ∈ j such that (cγa − 1γ1b)γ0x = 0, it follows that (cγa − 1γ1b)γcγa = 0. hence cγc ∈ [j :< i >] = q. since p is a semi-prime ideal containing q, we conclude that c ∈ p , a contradiction. therefore for every p ∈ ω, a = pγa and by propositions 2.1 and 3.10, 30 on multiplication γ-modules a = p(q)γa. let j ∈ a. it is easily to show that < j >= p(q)γ < j >. then there exists s ∈ p(q) such that for every n ∈ n, j = (sγ0)nj. by proposition 2.2, there exists t ∈ n ∪{0} such that (sγ0)ts ∈ q, it follows that j = (sγ0) tsγ0j ∈ qγa, i.e., a ⊆ qγa. hence qγa = a. the converse is evident. 4 prime mγ-submodule through this section m and a will denote a commutative γ-ring with unit and an unitary left mγ-module, respectively. definition 4.1. a prime ideal p in m is called a minimal prime ideal of the ideal i if i ⊆ p and there is no prime ideal p ′ such that i ⊆ p ′ ⊂ p . let min(i) denote the set of minimal prime ideals of i in γ-ring m, and every element of min((0)) is called minimal prime ideal. proposition 4.1. if an ideal i of γ-ring m is contained in a prime ideal p of m, then p contains a minimal prime ideal of i. proof. let a = {q : q is prime ideal of m and i ⊆ q ⊆ p}. by zorn’s lemma, there is a prime ideal q of r which is minimal member with respect to inclusion in a. therefore q ∈ min(i) and i ⊆ q ⊆ p . lemma 4.1. let γ be a finitely generated group. if i and j are finitely generated ideals of m, then iγj is finitely generated ideal of m. proof. let i = 〈a1, . . . ,an〉, j = 〈b1, . . . ,bm〉, and γ = 〈γ1, . . . ,γk〉. it is clear that iγj = 〈aiγtbj : 1 ≤ i ≤ n, 1 ≤ t ≤ k, 1 ≤ j ≤ m〉. proposition 4.2. let γ be a finitely generated group. if i is a proper ideal of m and each minimal prime ideal of i is finitely generated, then min(i) is finite set. proof. consider the set s = {p1γp2...pn; n ∈ n and pi ∈ min(i), for each 1 ≤ i ≤ n} and set ∆ = {k; k is an ideal of m and q 6⊆ k, for each q ∈s} 31 a. a. estaji, a. as. estaji, a. s. khorasani, s. baghdari which is the non-empty set, since i ∈ ∆. (∆,⊆) is the partial ordered set. suppose {kλ}λ∈λ is the chain of ∆ in which λ 6= ∅ and set k = ⋃ λ∈λ kλ. it is clear that k is an ideal of m. also, if there exits q ∈s such that q ⊆ k, then by lemma 4.1, q = p1γp2...pn is finitely generated ideal of m, i.e., q = 〈x1, . . . ,xn〉. but q ⊆ k implies that x1,x2, ...,xn ∈ k. thus there exists λ ∈ λ such that x1,x2, ...,xn ∈ kλ and so q ⊆ kλ, contradiction. hence, for each q ∈s, q 6⊆ k and k ∈ ∆ is the upper band of this chain. by zorhn’s lemma ∆ has maximal element such as q. now if a 6∈ q and b 6∈ q for a,b ∈ m, then q ⊆〈q∪{a}〉 and q ⊆〈q∪{b}〉. maximality of q implies that 〈q∪{a}〉, 〈q∪{b}〉 6∈ ∆. so there exists q1 and q2 in s such that q1 ⊆〈q∪{a}〉 and q2 ⊆〈q∪{b}〉. it is clear that q1γq2 ⊆ q which is contradiction, since q1γq2 ∈s. therefore 〈a〉γ〈b〉 6⊆ q and q is a prime ideal of m contained i. by proposition 4.1, there exists a minimal prime ideal p ⊆ q. but p ∈ s, contradictory with q ∈ ∆. above contradicts show that there exists q′ = p1γp2...pm ∈s such that q′ ⊆ i. now for each p ∈ min(i) we have q′ ⊆ i ⊆ p and p1γp2...pm ⊆ p . it is clear that pj ⊆ p for some 1 ≤ j ≤ m. thus pj = p , since p is minimal. hence min(i) = {p1,p2, ...,pm} is finite. proposition 4.3. for proper mγ-submodule b of a, the following statements equivalent: 1. for every mγ-submodule c of a, if b ⊂ c, then (b : a) = (b : c). 2. for every (m,a) ∈ m ×a, if mγa ⊆ b, then a ∈ b or m ∈ (b : a). proof. (1) ⇒ (2) let (m,a) ∈ m × a such that mγa ⊆ b and a 6∈ b. it is clear that b ⊂ b + mγa. since mγ(b + mγa) ⊆ mγb + mγ(mγa) = mγb + mγ(mγa) ⊆ b, we conclude from statement (1) that m ∈ (b : b + mγa) = (b : a) and the proof is now complete. (2) ⇒ (1) let c be a mγ-submodule of a such that b ⊂ c. it is clear that (b : a) ⊆ (b : c). now, suppose that m ∈ (b : c). since b ⊂ c, we infer that there exists a ∈ c \b such that mγa ⊆ b. by statement (2), m ∈ (b : a) and the proof is now complete. definition 4.2. a proper mγ-submodule b of a is said to be prime if mγa ⊆ b for (m,a) ∈ m ×a implies that either a ∈ b or m ∈ (b : a). proposition 4.4. if b is a prime mγ-submodule of a, then (b : a) is a prime ideal of γ-ring m. 32 on multiplication γ-modules proof. let x,y ∈ m such that 〈x〉γ〈y〉 ⊆ (b : a) and x 6∈ (b : a). then there exists γ ∈ γ and a ∈ a such that xγa 6∈ b, and also, yγ(xγa) = (yγx)γa = (xγy)γa ⊆ b. since b is a prime mγ-submodule of a and xγa 6∈ b, we conclude that yγa ⊆ b, i. e., y ∈ (b : a). the proof is now complete. proposition 4.5. let a be a multiplication left mγ-module, and b, b1, . . . , bk be mγ-submodules of a. if b is a prime mγ-submodule of a, then the following statements are equivalent. 1. bj ⊆ b for some 1 ≤ j ≤ k. 2. ⋂k i=1 bi ⊆ b. proof. (1) ⇒ (2) it is clear. (2) ⇒ (1) we have bi = iiγa for some ideals ii, (1 ≤ i ≤ k) of γ-ring m. then ( ⋂k i=1 ii)γa ⊆ ⋂k i=1(iiγa) = ⋂k i=1 bi ⊆ b and so ⋂k i=1 ii ⊆ (b : a). since m is a commutative γ-ring, we infer that for every permutations θ of {1, 2, . . . ,k}, i1γi2 · · ·ik = iθ(1)γiθ(2) · · ·iθ(k), it follows that i1γi2 · · ·ik ⊆⋂k i=1 ii ⊆ (b : a). since by proposition 4.4, (b : a) is prime ideal of γ-ring m, we conclude that ij ⊆ (b : a) for some 1 ≤ j ≤ k. therefore, by proposition 3.3, bj = ijγa ⊆ b for some 1 ≤ j ≤ k. proposition 4.6. if a is a multiplication left mγ-module, then for mγsubmodule b of a, the following statements are equivalent. 1. b is prime mγ-submodule of a. 2. (b : a) is prime ideal of γ-ring m. 3. there exists prime ideal p of γ-ring m such that b = pγa and for every ideal i of m, iγa ⊆ b implies that i ⊆ p . proof. (1) ⇒ (2) by proposition 4.4, it is evident. (2) ⇒ (3) we put m = {p : b = pγa and p is an ideal of γ-ring m } since a is multiplication left mγ-module, we conclude that (m,⊆) is a nonempty partial order set. let {pλ}λ∈λ ⊆m be a chain. by proposition 3.10,⋂ λ∈λ pλ ∈ m is an upper bound of {pλ}λ∈λ. by zorn’s lemma m has a maximal element. thus, we can pick a p to be maximal element of m. let x,y ∈ m and 〈x〉γ〈y〉 ⊆ p . hence (〈x〉γ〈y〉)γa ⊆ pγa = b and we infer that 〈x〉γ〈y〉⊆ (b : a). now, by statement (2), x ∈ (b : a) or y ∈ (b : a). since a is multiplication left mγ-module, we conclude from the proposition 33 a. a. estaji, a. as. estaji, a. s. khorasani, s. baghdari 3.3 that b = (b : a)γa, it follows that (b : a) ∈ m. on the other hand, clearly p ⊆ (b : a) and so p = (b : a), i.e., x ∈ p or y ∈ p , thus p is prime ideal of γ-ring m. (3) ⇒ (1) let prime ideal p of γ-ring m such that b = pγa and for every ideal i of γ-ring m, iγa ⊆ b implies that i ⊆ p . it is clear that p = (b : a). let m ∈ m and a ∈ a such that mγa ⊆ b. since a is a multiplication s-act, we conclude that there exists an ideal i of γ-ring m such that 〈a〉 = iγa, it follows that (mγi)γa = mγ(iγa) = mγ(mγa) = (mγm)γa = (mγm)γa = mγ(mγa) ⊆ b. therefore mγi ⊆ (b : a) = p and it is easy to see directly that 〈m〉γi ⊆ (b : a). then mγa ⊆ b or a ∈ iγa ⊆ b and the proof is now complete. lemma 4.2. let a be a finitely generated left mγ-module. if i is an ideal of m such that a = iγa, then there exists i ∈ i such that (1 − i)γ0a = 0. proof. if a =< a1, . . . ,an >, then for every 1 ≤ i ≤ n, there exists yi1, . . . ,yin ∈ i such that ai = ∑n j=1 yijγ0aj, it follows that −yi1γ0a1−···−yi(i−1)γ0ai−1 + (1−yii)γ0ai−yi(i+1)γ0ai+1−···−yinγ0an = 0. if b =  1 −y11 −y12 · · · −y1n... ... ... ... −yn1 −yn2 · · · 1 −ynn   , then there exists y ∈ i such that detγ(b) = (1 + y), where detγ(b) = ∑ sign(σ)b 1,σ(1) γ0b2,σ(2)γ0 · · ·γ0bn,σ(n) and σ runs over all the permutation on {1, 2, . . . ,n} (see [13]). since for every 1 ≤ i ≤ n, detγ(b)γ0ai = 0, we conclude that (1 + y)γ0a = 0 and by setting i = −y the proof will be completed. proposition 4.7. let a be a finitely generated faitfull multiplication left mγmodule. for proper ideal of p in m, the following statements are equivalent. 1. p is a prime ideal of m. 2. pγa is a prime mγ-submodule of a. proof. (1) ⇒ (2) let i be an ideal of m such that iγa ⊆ pγa. then by proposition 3.11, either i ⊆ p or a = [p : i]γa. if a = [p : i]γa, then by lemma 4.2, there exists i ∈ [p : i] such that (1 − i)γ0a = 0. since a is a 34 on multiplication γ-modules faitfull mγ-module, we conclude that i = 1 and i ⊆ p . hence by proposition 4.6, pγa is a prime mγ-submodule of a. (2) ⇒ (1) since a is a faitfull mγ-module and [pγa : a]γa ⊆ pγa, we conclude from the proposition 3.11 and lemma 4.2 that [pγa : a] ⊆ p . hence [pγa : a] = p and by proposition 4.6, p is a prime ideal of m. proposition 4.8. let a be a multiplication left mγ-module. then 1. if m satisfies acc (dcc) on prime ideals, then a satisfies acc (dcc) on prime mγ-submodules. 2. if a is faitfull mγ-module and (b : a) is a minimal prime ideal in m, then b is a minimal prime mγ-submodule of a. proof. (1) assume that b1 ⊆ b2 ⊆ . . . is a chain of prime mγ-submodule of a. by proposition 4.4, (b1 : a) ⊆ (b2 : a) ⊆ . . . is a chain of prime ideal of γ-ring m. by hypothesis there exists k ∈ n such that for every i ≥ k, (bi : a) = (bk : a). it follows from proposition 3.3 that bi = (bi : a)γa = (bk : a)γa = bk. thus a satisfies acc on prime mγ-submodules. (2) assume that b′ is a prime mγ-submodule of a such that b ′ ⊆ b. by proposition 4.6, (b′ : a) ⊆ (b : a) is a chain of prime ideal of γ-ring m. by hypothesis (b′ : a) = (b : a), it follows from proposition 3.3 that b′ = (b′ : a)γa = (b : a)γa = b. thus b is a minimal prime mγ-submodule of a. proposition 4.9. let a be a finitely generated faitfull multiplication left mγ-module. then 1. if a satisfies acc (dcc) on prime mγ-submodules, then γ-ring m satisfies acc (dcc) on prime ideals. 2. if b is a minimal prime mγ-submodule of a, then (b : a) is a minimal prime ideal of γ-ring m. proof. (1) assume that p1 ⊆ p2 ⊆ . . . is a chain of prime ideals of γ-ring m. by proposition 4.7, p1γa ⊆ p2γa ⊆ . . . is a chain of prime mγ-submodule of a. by hypothesis there exists k ∈ n such that for every i ≥ k, pkγa = piγa. since a is a finitely generated faitfull multiplication mγ-module, we conclude from the proposition 3.11 and lemma 4.2 that pk = pi. (2) by proposition 4.6, (b : a) is a prime ideal of γ-ring m. assume that p is a prime ideal of γ-ring m such that p ⊆ (b : a). hence by proposition 3.3, pγa ⊆ (b : a)γa = b. since by proposition 4.7, pγa is a prime mγsubmodule of a, we conclude from our hypothesis that pγa = (b : a)γa. 35 a. a. estaji, a. as. estaji, a. s. khorasani, s. baghdari since a is a finitely generated faitfull multiplication mγ-module, we conclude from the proposition 3.11 and lemma 4.2 that p = (b : a). the proof is now complete. proposition 4.10. let γ be a finitely generated group. let a be a finitely generated faitfull multiplication left mγ-module. 1. if every prime ideal of γ-ring m is finitely generated, then a contains only a finitely many minimal prime mγ-submodule. 2. if every minimal prime mγ-submodule of a is finitely generated, then γ-ring m contains only a finite number of minimal prime ideal. proof. (1) assume that {bλ}λ∈λ is the family of minimal prime mγ-submodules of a. set iλ = (bλ : a) for λ ∈ λ. by proposition 4.9, each iλ is a minimal prime ideal of γ-ring m. on the other hand, by proposition 4.2, m contains only a finite number of minimal prime ideal as {i1,i2, . . .in}. now suppose that λ ∈ λ. so iλ = ii, for some 1 ≤ i ≤ n and by proposition 3.3, bλ = iλγa = iiγa. thus {i1γa,i2γa,. . . ,inγa} is the finite family of minimal prime mγ-submodule of a. (2) suppose that i and j are two distinct minimal prime ideal of γ-ring m. by proposition 3.11 and lemma 4.2, a 6= iγa 6= jγa and also, by proposition 4.7, iγa and jγa are prime mγ-submodules of a. assume that b1 and b2 are two prime mγ-submodules of a such that b1 ⊆ iγa and b2 ⊆ jγa. by proposition 3.3, b1 = (b1 : a)γa and b2 = (b2 : a)γa. by proposition 3.11 and lemma 4.2, (b1 : a) ⊆ i and (b2 : a) ⊆ j. since i and j are two distinct minimal prime ideal of γ-ring m, we conclude from the proposition 4.4 that (b1 : a) = i and (b2 : a) = j. this says that iγa and jγa are two distinct minimal prime mγ-submodules of a. now if γ-ring m contains infinite many minimal prime ideals, then a must have infinitely many minimal prime mγ-submodules which is contradiction. references [1] j. aliro and p. penea, a note on prime module, divulgaciones matematicas 8 (2000), 31-42. [2] d. d. anderson, a note on minimal prime ideals, proc. amer. math. soc., 122 (1994), 13-14. 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[12] b. a. ersoy, fuzzy semiprime ideals in gamma-rings, international journal of physical sciences vol. 5(4) (2010), 308-312. [13] a. a. estaji, a. saghafi khorasani and s. baghdari, multiplication ideals in γ-rings, journal of hyperstructures 2 (1) (2013), 30-39. [14] m. f. hoque and a. c. paul, on centralizers of semiprime gamma rings, international mathematical forum, vol. 6 (2011), no. 13, 627 638. [15] s. kyuno, on prime gamma ring, pacific j. math. 75 (1978), 185-190. [16] l. luh, on the theory of simple γ-rings, michigan math. j. 16 (1969), 65-75. [17] n. nobusawa, on a generalization of the ring theory, osaka j. math. 1 (1964), 81-89. [18] m. a. öztürk and h. yazarl, modules over the generalized centroid of semi-prime gamma rings, bull. korean math. soc. 44 (2007), no. 2, 203-213. 37 a. a. estaji, a. as. estaji, a. s. khorasani, s. baghdari [19] a.c. paul and md. sabur uddin, lie structure in simple gamma rings, int. j. pure appl. sci. technol., 4(2) (2010), 63-70. [20] c. selvaraj and s. petchimuthu, strongly prime gamma rings and morita equivalence of rings, southeast asian bulletin of mathematics 32 (2008), 1137-1147. 38 ratio mathematica vol. 33, 2017, pp. 89-102 issn: 1592-7415 eissn: 2282-8214 special classes of hb-matrices achilles dramalidis∗ †doi:10.23755/rm.v33i0.382 abstract in the present paper we deal with constructions of 2 × 2 diagonal or uppertriangular or lower-triangular hb-matrices with entries either of an hb-field on z2 or on z3. we study the kind of the hyperstructures that arise, their unit and inverse elements. also, we focus our study on the cyclicity of these hyperstructures, their generators and the respective periods. keywords: hope; hv-structure; hb-structure; hv-matrix 2010 ams subject classifications: 20n20. ∗democritus university of thrace, school of education, 68100 alexandroupolis, greece; adramali@psed.duth.gr † c©achilles dramalidis. received: 31-10-2017. accepted: 26-12-2017. published: 31-122017. 89 achilles dramalidis 1 introduction f. marty, in 1934 [13], introduced the hypergroup as a set h equipped with a hyperoperation · : h × h → p(h) −{∅} which satisfies the associative law: (xy)z = x(yz), for all x,y,z ∈ h and the reproduction axiom: xh = hx = h, for all x ∈ h. in that case, the reproduction axiom is not valid, the (h, ·) is called semihypergroup. in 1990, t. vougiouklis [19] in the fourth aha congress, introduced the hvstructures, a larger class than the known hyperstructures, which satisfy the weak axioms where the non-empty intersection replaces the equality. definition 1.1. [21], the (·) in h is called weak associative, we write wass, if (xy)z ∩x(yz) 6= ∅,∀x,y,z ∈ h. the (·) is called weak commutative, we write cow, if xy ∩yx 6= ∅,∀x,y ∈ h. the hyperstructure (h, ·) is called hv-semigroup if (·) is wass. it is called hvgroup if it is hv-semigroup and the reproduction axiom is valid. further more, it is called hv-commutative group if it is an hv-group and a cow. if the commutativity is valid, then h is called commutative hv-group. analogous definitions for other hv-structures, as hv-rings, hv-module, hv-vector spaces and so on can be given. for more definitions and applications on hyperstructures one can see books [3], [4], [5], [6], [21] and papers as [2], [7], [9], [10], [12], [14], [20], [22], [23], [24], [26], [27]. an element e ∈ h is called left unit if x ∈ ex,∀x ∈ h and it is called right unit if x ∈ xe,∀x ∈ h. it is called unit if x ∈ ex∩xe,∀x ∈ h. the set of left units is denoted by e` [8]. the set of right units is denoted by er and by e = e` ∩er the set of units [8]. the element a′ ∈ h is called left inverse of the element a ∈ h if e ∈ a′a, where e unit element (left or right) and it is called right inverse if e ∈ aa′. if e ∈ a′a∩aa′ then it is called inverse element of a ∈ h. the set of the left inverses is denoted by i`(a,e) and the set of the right inverses is denoted by ir(a,e)[8]. by i(a,e) = i`(a,e)∩ ir(a,e), the set of inverses of the element a ∈ h, is denoted. in an hv-semigroup the powers are defined by: h1 = {h},h2 = h·h, · · · ,hn = h◦ h◦· · ·◦h, where (◦) is the n-ary circle hope, i.e. take the union of hyperproducts, n times, with all possible patterns of parentheses put on them. an hv-semigroup (h, ·) is cyclic of period s, if there is an h, called generator and a natural s, the minimum: h = h1∪h2∪·· ·∪hs. analogously the cyclicity for the infinite period 90 special classes of hb-matrices is defined [17],[21]. if there is an h and s, the minimum: h = hs, then (h, ·), is called single-power cyclic of period s. definition 1.2. the fundamental relations β∗,γ∗ and �∗, are defined, in hv-groups, hv-rings and hv-vector spaces, respectively, as the smallest equivalences so that the quotient would be group, ring and vector spaces, respectively [18],[19],[21],[22], (see also [1],[3],[4]). more general structures can be defined by using the fundamental structures. an application in this direction is the general hyperfield. there was no general definition of a hyperfield, but from 1990 [19] there is the following [20], [21]: definition 1.3. an hv-ring (r,+, ·) is called hv-field if r/γ∗ is a field. hv-matrix is a matrix with entries of an hv-ring or hv-field. the hyperproduct of two hv-matrices (aij) and (bij), of type m × n and n × r respectively, is defined in the usual manner and it is a set of m × r hv-matrices. the sum of products of elements of the hv-ring is considered to be the n-ary circle hope on the hyperaddition. the hyperproduct of hv-matrices is not necessarily wass. hv-matrices is a very useful tool in representation theory of hv-groups [15],[16], [25],[28] (see also [11], [29]). 2 constructions of 2×2 hb-matrices with entries of an hv-field on z2 consider the field (z2,+, ·). on the set z2 also consider the hyperoperation (�) defined by setting: 1�1 = {0,1} and x�y = x ·y for all (x,y) ∈ z2 ×z2 −{(0,1)}. then (z2,+,�) becomes an hb-field. all the 2×2 hb-matrices with entries of the hb-field (z2,+,�), are 24 = 16. let us denote them by: 0 = ( 0 0 0 0 ) ,a1 = ( 1 0 0 0 ) ,a2 = ( 0 1 0 0 ) ,a3 = ( 0 0 1 0 ) , a4 = ( 0 0 0 1 ) ,a5 = ( 1 1 0 0 ) ,a6 = ( 1 0 1 0 ) ,a7 = ( 1 0 0 1 ) , a8 = ( 0 1 1 0 ) ,a9 = ( 0 1 0 1 ) ,a10 = ( 0 0 1 1 ) ,a11 = ( 1 1 1 0 ) , 91 achilles dramalidis a12 = ( 1 1 0 1 ) ,a13 = ( 1 0 1 1 ) ,a14 = ( 0 1 1 1 ) ,a15 = ( 1 1 1 1 ) . by taking a2i , i = 1, · · · ,15 there exist 15 closed sets, let us say hi , i = 1, · · · ,15. two of them are singletons, h2 = h3 = {0}. also, h7 = h8 and h11 = h14 = h15. so, we shall study, according to the hyperproduct (·) of two hb-matrices, the following sets: h1 = {0,a1},h4 = {0,a4},h5 = {0,a1,a2,a5},h6 = {0,a1,a3,a6}, h7 = {0,a1,a4,a7},h9 = {0,a2,a4,a9},h10 = {0,a3,a4,a10}, h12 = {0,a1,a2,a4,a5,a7,a9,a12},h13 = {0,a1,a3,a4,a6,a7,a10,a13}, h15 = {0,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10a11,a12,a13,a14,a15}. 2.1 the case of diagonal 2×2 hb-matrices every set of h1,h4,h7 consists of diagonal 2 × 2 hb-matrices. then, the multiplicative tables of the hyperproduct, are the following: · 0 a1 0 0 0 a1 0 h1 , · 0 a4 0 0 0 a4 0 h4 · 0 a1 a4 a7 0 0 0 0 0 a1 0 0,a1 0 0,a1 a4 0 0 0,a4 0,a4 a7 0 0,a1 0,a4 h7 in all cases: x ·y = y ·x, ∀x,y ∈ hi, i = 1,4,7 (x ·y) ·z = x · (y ·z), ∀x,y,z ∈ hi, i = 1,4,7 so, we get the next propositions: proposition 2.1. every set h, consisting of diagonal 2 × 2 hb-matrices with entries of the hb-field (z2,+,�), equipped with the usual hyperproduct (·) of matrices, is a commutative semihypergroup. 92 special classes of hb-matrices notice that h1,h4 ⊂ h7 and since h1 ·h1 ⊆ h1 , h4 ·h4 ⊆ h4 then h1,h4 are sub-semihypergroups of (h7, ·). proposition 2.2. for all commutative semihypergroups (h, ·), consisting of diagonal 2×2 hb-matrices with entries of the hb-field (z2,+,�): e = {ai}, i(ai,ai) = {ai}, where a2i = h. remark 2.1. according to the above construction, the commutative semihypergroups (h1, ·),(h4, ·) and (h7, ·), are single-power cyclic commutative semihypergroups with generators the elements a1,a4 and a7, respectively, with singlepower period 2. 2.2 the case of upperand lowertriangular 2×2 hb-matrices every set of h5,h9,h12 consists of upper-triangular 2 × 2 hb-matrices and every set of h6,h10,h13 consists of lower-triangular 2 × 2 hb-matrices. then, the multiplicative tables of the hyperproduct, are the following: · 0 a1 a2 a5 0 0 0 0 0 a1 0 0,a1 0,a2 h5 a2 0 0 0 0 a5 0 0,a1 0,a2 h5 , · 0 a2 a4 a9 0 0 0 0 0 a2 0 0 0,a2 0,a2 a4 0 0 0,a4 0,a4 a9 0 0 h9 h9 · 0 a1 a2 a4 a5 a7 a9 a12 0 0 0 0 0 0 0 0 0 a1 0 0,a1 0,a2 0 0,a1, 0,a1 0,a2 0,a1, a2,a5 a2,a5 a2 0 0 0 0,a2 0 0,a2 0,a2 0,a2 a4 0 0 0 0,a4 0 0,a4 0,a4 0,a4 a5 0 0,a1 0,a2 0,a2 0,a1, 0,a1, 0,a2 0,a1, a2,a5 a2,a5 a2,a5 a7 0 0,a1 0,a2 0,a4 0,a1, 0,a1, 0,a2, h12 a2,a5 a4,a7 a4,a9 a9 0 0 0 0,a2, 0 0,a2, 0,a2, 0,a2, a4,a9 a4,a9 a4,a9 a4,a9 a12 0 0,a1 0,a2 0,a2, 0,a1, h12 0,a2, h12 a4,a9 a2,a5 a4,a9 93 achilles dramalidis · 0 a1 a3 a6 0 0 0 0 0 a1 0 0,a1 0 0,a1 a3 0 0,a3 0 0,a3 a6 0 h6 0 h6 , · 0 a3 a4 a10 0 0 0 0 0 a3 0 0 0 0 a4 0 0,a3 0,a4 h10 a10 0 0,a3 0,a4 h10 · 0 a1 a3 a4 a6 a7 a10 a13 0 0 0 0 0 0 0 0 0 a1 0 0,a1 0 0 0,a1 0,a1 0 0,a1 a3 0 0,a3 0 0 0,a3 0,a3 0 0,a3 a4 0 0 0,a3 0,a4 0,a3 0,a4 0,a3, 0,a3, a4,a10 a4,a10 a6 0 0,a1, 0 0 0,a1, 0,a1, 0 0,a1, a3,a6 a3,a6 a3,a6 a3,a6 a7 0 0,a1 0,a3 0,a4 0,a1, 0,a1, 0,a3, h13 a3,a6 a4,a7 a4,a10 a10 0 0,a3 0,a3 0,a4 0,a3 0,a3, 0,a3, 0,a3, a4,a10 a4,a10 a4,a10 a13 0 0,a1, 0,a3 0,a4 0,a1, h13 0,a3, h13 a3,a6 a3,a6 a4,a10 in all cases: (x ·y)∩ (y ·x) 6= ∅, ∀x,y ∈ hi, i = 5,6,9,10,12,13 (x ·y) ·z = x · (y ·z), ∀x,y,z ∈ hi, i = 5,6,9,10,12,13 so, we get the next proposition: proposition 2.3. every set h, consisting either of upper-triangular or lowertriangular 2 × 2 hb-matrices with entries of the hb-field (z2,+,�), equipped with the usual hyperproduct (·) of matrices, is a weak commutative semihypergroup. notice that h5,h9 ⊂ h12 and h6,h10 ⊂ h13. since h5·h5 ⊆ h5 , h9·h9 ⊆ h9 , h6 ·h6 ⊆ h6 , h10 ·h10 ⊆ h10 , then h5,h9 are sub-semihypergroups of (h12, ·) and h6,h10 are sub-semihypergroups of (h13, ·). proposition 2.4. for all weak commutative semihypergroups (h, ·), consisting either of upper-triangular or lower-triangular 2 × 2 hb-matrices with entries of the hb-field (z2,+,�), the following assertions hold i) if ai,aj ∈ h : ai ·aj = h, ai ∈ a2i , a2j = h, ai ∈ aj ·ai , then a)e` = {ai,aj}, b)i(ai,ai) = i(aj,ai) = {ai,aj} 94 special classes of hb-matrices c)i(aj,aj) = i r(ai,aj) = {aj}, d)i`(ai,aj) = ∅ ii) if ai,aj ∈ h : aj ·ai = h, ai ∈ a2i , a2j = h, ai ∈ ai ·aj, then a)er = {ai,aj}, b)i(ai,ai) = i(aj,ai) = {ai,aj} c)i(aj,aj) = i `(ai,aj) = {aj}, d)ir(ai,aj) = ∅ iii) if ai,aj ∈ h : ai ·aj = aj ·ai = h, ai ∈ a2i , a2j = h, then a)e = {ai,aj}, b)i(ai,ai) = i(aj,ai) = i(aj,aj) = {ai,aj}, c)i(ai,aj) = {aj} remark 2.2. according to the above construction, the weak commutative semihypergroups (hi, ·), i=5,6,9,10,12,13 are single-power cyclic weak commutative semihypergroups with generators the elements a5,a6,a9,a10,a12,a13 respectively, with single-power period 2. 3 constructions of 2×2 hb-matrices with entries of an hb-field on z3 consider the field (z3,+, ·). on the set z3, we consider four cases for the hyperoperation (�i), i = 1,2,3,4 defined, each time, by setting: 1) 1�1 2 = {1,2} and x�1 y = x ·y for all (x,y) ∈ z3 ×z3 −{(1,2)}. 2) 2�2 1 = {1,2} and x�2 y = x ·y for all (x,y) ∈ z3 ×z3 −{(1,2)}. 3) 1�3 1 = {1,2} and x�3 y = x ·y for all (x,y) ∈ z3 ×z3 −{(1,2)}. 4) 2�4 2 = {1,2} and x�4 y = x ·y for all (x,y) ∈ z3 ×z3 −{(1,2)}. then, each time, (z3,+,�i), i = 1,2,3,4 becomes an hb-field. now, consider the set h of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hbmatrices, with entries of the hb-field (z3,+,�i). let us denote them by: a11 = ( 1 0 0 1 ) ,a12 = ( 1 0 0 2 ) ,a21 = ( 2 0 0 1 ) ,a22 = ( 2 0 0 2 ) . so, h = {a11,a12,a21,a22}. 95 achilles dramalidis 3.1 the case of 1�1 2 = {1,2} the multiplicative table of the hyperproduct, is the following: · a11 a12 a21 a22 a11 a11 a11,a12 a11,a21 h a12 a12 a11 a12,a22 a11,a21 a21 a21 a21,a22 a11 a11,a12 a22 a22 a21 a12 a11 notice that in the above multiplicative table: i) x ·h = h ·x = h,∀x ∈ h ii) (x ·y)∩ (y ·x) 6= ∅,∀x,y ∈ h iii) (x ·y) ·z ∩x · (y ·z) 6= ∅,∀x,y,z ∈ h so, we get the next proposition: proposition 3.1. the set h, consisting of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�1), equipped with the usual hyperproduct (·) of matrices, is an hv-commutative group. proposition 3.2. for the hv-commutative group (h, ·), consisting of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�1) : i) e = {a11} ii) ir(x,a11) = {a22},∀x ∈ h iii) i`(x,a11) = {a11},∀x ∈ h proposition 3.3. the hv-commutative group (h, ·), consisting of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�1), is a single-power cyclic hv-commutative group with generator the element a22, with single-power period 3. 3.2 the case of 2�2 1 = {1,2} the multiplicative table of the hyperproduct, is the following: · a11 a12 a21 a22 a11 a11 a12 a21 a22 a12 a11,a12 a11 a21,a22 a21 a21 a11,a21 a12,a22 a11 a12 a22 h a11,a21 a11,a12 a11 as in the paragraph 3.1: proposition 3.4. the set h, consisting of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�2), equipped with the usual hyperproduct (·) of matrices, is an hv-commutative group. 96 special classes of hb-matrices now, take a map f onto and 1:1, f : h → h , such that f(a11) = a22, f(a12) = a21, f(a21) = a12, f(a22) = a11 then, the successive transformations of the above multiplicative table are: · a22 a21 a12 a11 a22 a11 a12 a21 a22 a21 a11,a12 a11 a21,a22 a21 a12 a11,a21 a12,a22 a11 a12 a11 h a11,a21 a11,a12 a11 · a22 a21 a12 a11 a11 h a11,a21 a11,a12 a11 a12 a11,a21 a12,a22 a11 a12 a21 a11,a12 a11 a21,a22 a21 a22 a11 a12 a21 a22 · a11 a12 a21 a22 a11 a11 a11,a12 a11,a21 h a12 a12 a11 a12,a22 a11,a21 a21 a21 a21,a22 a11 a11,a12 a22 a22 a21 a12 a11 then, the last multiplicative table is the table of the paragraph 3.1. so, we get: proposition 3.5. the hv-commutative group (h, ·) consisting of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�2), is isomorphic to hv-commutative group (h, ·) consisting of the diag(b11,b22) , b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�1). 3.3 the case of 1�3 1 = {1,2} the multiplicative table of the hyperproduct, is the following: · a11 a12 a21 a22 a11 h a12,a22 a21,a22 a22 a12 a12,a22 a11,a21 a22 a21 a21 a21,a22 a22 a11,a12 a12 a22 a22 a21 a12 a11 notice that in the above multiplicative table: i) x ·h = h ·x = h, ∀x ∈ h 97 achilles dramalidis ii) x ·y = y ·x, ∀x,y ∈ h iii) (x ·y) ·z ∩x · (y ·z) 6= ∅, ∀x,y,z ∈ h so, we get the next proposition: proposition 3.6. the set h, consisting of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�3), equipped with the usual hyperproduct (·) of matrices, is a commutative hvgroup. proposition 3.7. for the commutative hv-group (h, ·), consisting of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�3) : i) e = er = e` = {a11} ii) i(x,a11) = ir(x,a11) = i`(x,a11) = {x},∀x ∈ h proposition 3.8. the commutative hv-group (h, ·), consisting of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�3) : i) is a single-power cyclic commutative hv-group with generator the element a11, with single-power period 2. ii) is a single-power cyclic commutative hv-group with generator the element a22, with single-power period 4. iii) is a cyclic commutative hv-group of period 3 to each of the generators a12 and a21. 3.4 the case of 2�4 2 = {1,2} the multiplicative table of the hyperproduct, is the following: · a11 a12 a21 a22 a11 a11 a12 a21 a22 a12 a12 a11,a12 a22 a21,a22 a21 a21 a22 a11,a21 a12,a22 a22 a22 a21,a22 a12,a22 h notice that in the above multiplicative table: i) x ·h = h ·x = h, ∀x ∈ h ii) x ·y = y ·x, ∀x,y ∈ h iii) (x ·y) ·z = x · (y ·z), ∀x,y,z ∈ h so, we get the next proposition: proposition 3.9. the set h, consisting of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�4), equipped with the usual hyperproduct (·) of matrices, is a commutative hypergroup. 98 special classes of hb-matrices proposition 3.10. for the commutative hypergroup (h, ·), consisting of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�4) : i) e = {a11} ii) i(x,a11) = {x},∀x ∈ h proposition 3.11. the commutative hypergroup (h, ·), consisting of the diag(b11,b22), b11,b22 ∈ z3 with b11b22 6= 0 hb-matrices, with entries of the hb-field (z3,+,�4) is a single-power cyclic commutative hypergroup with generator the element a22, with single-power period 2. 4 construction of 2×2 upper-triangular hb-matrices with entries of an hb-field on z3 on the set z3, consider the hyperoperation (�1) defined, by setting: 1�1 2 = {1,2} and x�1 y = x ·y for all (x,y) ∈ z3 ×z3 −{(1,2)} now, consider the set h of the 2 × 2 upper-triangular hb-matrices with b11,b22 ∈ z3 and b11b22 6= 0, with entries of the hb-field (z3,+,�1). let us denote the elements of h by: a1 = ( 1 0 0 1 ) ,a2 = ( 1 0 0 2 ) ,a3 = ( 1 1 0 1 ) ,a4 = ( 1 1 0 2 ) , a5 = ( 1 2 0 1 ) ,a6 = ( 1 2 0 2 ) ,a7 = ( 2 0 0 1 ) ,a8 = ( 2 0 0 2 ) , a9 = ( 2 1 0 1 ) ,a10 = ( 2 1 0 2 ) ,a11 = ( 2 2 0 1 ) ,a12 = ( 2 2 0 2 ) so, h = {a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12}. since the multiplicative table is long enough, it is omitted. from this table we get: i) x ·h = h ·x = h, ∀x ∈ h ii) (·) is non-commutative iii) (x ·y) ·z ∩x · (y ·z) 6= ∅, ∀x,y,z ∈ h so, we get the next proposition: proposition 4.1. the set h, consisting of the 2×2 upper-triangular hb-matrices with b11,b22 ∈ z3 and b11b22 6= 0, with entries of the hb-field (z3,+,�1), equipped with the usual hyperproduct (·) of matrices, is a non-commtative hvgroup. 99 achilles dramalidis proposition 4.2. for the non-commtative hv-group (h, ·), consisting of the 2×2 upper-triangular hb-matrices with b11,b22 ∈ z3 and b11b22 6= 0, with entries of the hb-field (z3,+,�1) : e = e` = er = {a1},∀x ∈ h. proposition 4.3. the non-commtative hv-group (h, ·), consisting of the 2 × 2 upper-triangular hb-matrices with b11,b22 ∈ z3 and b11b22 6= 0, with entries of the hb-field (z3,+,�1) : i) is a single-power cyclic non-commutative hv-group with generator the element a12, with single-power period 4. ii) is a single-power cyclic non-commutative hv-group with generator the element a10, with single-power period 3. now, take any hb-field (zp,+,�1) , p = prime 6= 2 and then consider a set h consisting of the 2 × 2 upper-triangular hb-matrices with entries of this hb-field, with b11b22 6= 0 , b11,b22 ∈ zp. then, for any such a set zp, take for example the elements a3,a7 ∈ h, then: a7 ·a3 = a11 and a3 ·a7 = {a1,a7} so, we get the next general proposition: proposition 4.4. any set h, consisting of the 2×2 upper-triangular hb-matrices with b11b22 6= 0 , b11,b22 ∈ zp, p = prime 6= 2, with entries of the hbfield (zp,+,�1), equipped with the usual hyperproduct (·) of matrices, is a noncommutative hyperstructure. remark 4.1. the above proposition means that, the minimum non-commutative hvgroup, equipped with the usual hyperproduct (·) of matrices and consisting of the 2 × 2 upper-triangular hb-matrices with b11b22 6= 0, is that with entries of the hb-field 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[29] t. vougiouklis and s. vougiouklis, the helix hyperoperations, italian j. pure appl. math., 18, (2005), 197-206. 102 microsoft word r.m.7 cap.9.doc microsoft word r.m.5 cap.10.doc microsoft word capitolo intero 5.doc volume 36, 2019, pp. 27-42 ratio mathematica 27 solution of two-point fuzzy boundary value problems by fuzzy neural networks mazin hashim suhhiem* basim nasih abood+ mohammed hadi lafta++ abstract in this work, we have introduced a modified method for solving second-order fuzzy differential equations. this method based on the fully fuzzy neural network to find the numerical solution of the two-point fuzzy boundary value problems for the ordinary differential equations. the fuzzy trial solution of the two-point fuzzy boundary value problems is written based on the concepts of the fully fuzzy feed-forward neural networks which containing fuzzy adjustable parameters. in comparison with other numerical methods, the proposed method provides numerical solutions with high accuracy. keywords: two-point fuzzy boundary value problem; fully fuzzy neural network; fuzzy trial solution; minimized error function; hyperbolic tangent activation function. ـــ .ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ * department of statistics, university of sumer, alrifaee, iraq; mazin.suhhiem@yahoo.com + department of mathematics, university of wasit, alkut, iraq; basimabood@yahoo.com ++ department of statistics, university of sumer, alrifaee, iraq; mohamnedhadi@yahoo.com received on march 5th, 2019. accepted on april 29rd, 2019. published on june 30th, 2019. doi:10.23755/rm.v36i1.455. issn: 1592-7415. eissn: 2282-8214. ©mazin suhhiem et al. this paper is published under the cc-by licence agreement. mazin h. suhhiem, basim n. abood, mohammed h. lafta 28 1. introduction many methods have been developed so far for solving fuzzy differential equations (fdes) since it is utilized widely for the purpose of modelling problems in science and engineering. most of the practical problems require the solution of the fde which satisfies fuzzy initial conditions or fuzzy boundary conditions, therefore, the fde must be solved. many fde could not be solved exactly, thus considering their approximate solutions is becoming more important. the theory of fde was "first formulated by kaleva and seikkala..kaleva was formulated fde in terms of the "hukuhara derivative" (h-derivative). buckley and feuring have given a very general formulation of a first order" fuzzy "initial value problem. they first find the crisp solution, make it fuzzy and then check if it satisfies the fde. in 1990 researchers began using the artificial neural network (ann) for solving ordinary differential equation (ode) and partial differential equation (pde) such as: lee and kang in [1]; meade and fernandez in [2,3]; lagaris and likas in [4]; liu and jammes in [5]; tawfiq in [6]; malek and shekari in [7]; pattanaik and mishra in [8]; baymani and kerayechian in [9]; and other researchers. in 2010 researchers began using ann for solving a fuzzy differential equation such as: effati and pakdaman in [10]; mosleh and otadi in [11]; ezadi and parandin in [12]. in 2012 researchers began using partially (non-fully) fuzzy artificial neural network(fann) for solving a fuzzy differential equation such as mosleh and otadi in [13,14,15]. in (2016) suhhiem [16] developed and used partially fann for solving fuzzy and non-fuzzy differential equations. in this work, we have used fully feed forward fuzzy neural network to find the numerical solution of the two-point fuzzy boundary value problems for the ordinary differential equations. the fuzzy trial solution of the fuzzy boundary value problem is written as a sum of two parts. the first part satisfies the fuzzy boundary condition, it contains no fuzzy adjustable parameters. the second part involves fully fuzzy feed-forward neural networks which containing fuzzy adjustable parameters. solution of two-point fuzzy boundary value problems by fuzzy neural networks 29 2 basic definitions in this section, the basic notations which are used in fuzzy calculus are introduced definition(𝟏),[𝟏𝟔]: the r level ( or r cut ) set of a fuzzy set ã labeled by ar is the crisp set of all x in x (universal set) such that : µã(x) ≥ r ; i. e. ar = {x ∈ x ∶ µã(x) ≥ r , r ∈ [0,1] } . (1) definition(𝟐), 𝐅𝐮𝐳𝐳𝐲 𝐍𝐮𝐦𝐛𝐞𝐫[𝟏𝟔]: a fuzzy number ũ is completely determined by an ordered pair of functions (u (r) , u (r)), 0 ≤ r ≤ 1, which satisfy the following requirements: 𝟏) u (r) is a bounded left continuous and non-decreasing function on [0,1]. 𝟐) u (r) is a bounded left continuous and non-increasing function on [0,1]. 𝟑) u (r) ≤ u (r) , 0 ≤ r ≤ 1. (2) the crisp number (a) is simply represented by: u (r) = u (r) = a , 0 ≤ r ≤ 1 . the set of all the fuzzy numbers is denoted by e1. remark(𝟏),[𝟏𝟎]: for arbitrary ũ = (u , u) , ṽ = (v , v) and k ∈ r, the addition and multiplication by k for all r ∈ [0,1] can be defined as: 𝟏) (u + v) (r) = u (r) + v (r). 𝟐) (u + v) (r) = u (r) + v (r). 𝟑) (ku) (r) = k u (r), (ku) (r) = k u (r) , if k ≥ 0. 𝟒) (ku) (r) = k u (r), (ku) (r) = k u (r), if k < 0. (3) remark(𝟐),[𝟏𝟔]: the distance between two arbitrary fuzzy numbers ũ = (u , u) and ṽ = (v , v) is given as: d (ũ , ṽ) = [∫ ( u (r) v (r) 1 0 ) 2 dr + ∫ ( u (r) v (r) 1 0 ) 2 dr] 1 2 (4) remark(𝟑),[𝟏𝟔]: (e1,d) is a complete metric space. mazin h. suhhiem, basim n. abood, mohammed h. lafta 30 definition (𝟑) , 𝐅𝐮𝐳𝐳𝐲 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧 [𝟏𝟔] : the function f: r ⟶ e1 is called a fuzzy function. we call every function defined in set ã ⊆ e1 to b̃ ⊆ e1 a fuzzy function. definition(𝟒),[𝟏𝟎]: the fuzzy function f: r ⟶ e1 is said to be continuous if: for an arbitrary t1 ∈ r and ϵ > 0 there exists a δ > 0 such that: |t t1| < δ ⇒ d (f (t), f(t1)) < ϵ, where d is the distance between two fuzzy numbers. definition (5),[16]: let i be a real interval. the r-level set of the fuzzy function y: i → e1 can be denoted by: [y(x)]r = [y1 r(x), y2 r (x)], x ∈ i , r ∈ [0,1] (5) the seikkala derivative yˊ(x) of the fuzzy function y(x) is defined by: [yˊ(x)]r = [(y1 r)ˊ(x), (y2 r )ˊ(x)], x ∈ i, r ∈ [0,1] (6) definition (6),[𝟏𝟎]: let u and v ∈ e1. if there exist w ∈ e1 such that: u = v+w then w is called the h-difference (hukuhara-difference) of u and v and it is denoted by w = u ⊝ v. in this work, the ⊝ sign stands always for h-difference, and let us remark that u ⊝ v ≠ u + (-1) v . definition (7), 𝐅𝐮𝐳𝐳𝐲 𝐃𝐞𝐫𝐢𝐯𝐚𝐭𝐢𝐯𝐞[𝟏𝟐]: let f : (a,b) → e1 and t0 ∈ (a,b).we say that f is h-differential (hukuhara-differential) at x0, if there exists an element fˊ(x0) ∈ e 1 such that for all h> 0 (sufficiently small), ∃ f (x0 +h)⊝f(x0), f(x0) ⊝ f (x0 h) and the limits (in the metric d) lim h→0 f(x0 + h) ⊝f(x0) h = lim h→0 f(x0) ⊝ f(x0 − h) h = fˊ(x0) (7) then fˊ(x0) is called fuzzy derivative (h-derivative) of f at x0. where d is the distance between two fuzzy numbers. solution of two-point fuzzy boundary value problems by fuzzy neural networks 31 3 fully fuzzy neural network [6,16] artificial neural networks are learning machines that can learn any arbitrary functional mapping between input and output. they are fast machines and can be implemented in parallel, either in software or in hardware. in fact, the computational complexity of ann is polynomial in the number of neurons used in the network. parallelism also brings with it the advantages of robustness and fault tolerance. (i.e.) ann is a simplified mathematical model of the human brain. it can be implemented by both electric elements and computer software. it is a parallel distributed processor with large numbers of connections it is an information processing system that has certain performance characters in common with biological neural networks. a fuzzy neural network or neuro-fuzzy system is a learning machine that finds the parameters of a fuzzy system (i.e., fuzzy set, fuzzy rules) by exploiting approximation techniques from neural networks. combining fuzzy systems with neural networks. both neural networks and fuzzy systems have some things in common. they can be used for solving problems (e. g. fuzzy differential equations, fuzzy integral equations, etc ). if all the adjustable parameters (weights and biases) are fuzzy numbers, then the fuzzy neural network is called fully fuzzy neural network; otherwise it is called partially fuzzy neural network. 4 solution of fdes by fully fuzzy neural network to solve any fuzzy ordinary differential equation, we consider a threelayered fully fuzzy neural network with one unit entry x, one hidden layer consisting of m activation functions and one unit output n(x). the activation function for the hidden units of our fully fuzzy neural network is the hyperbolic tangent function (s(∝) = tanh (∝)). here the dimension of a fully fuzzy neural network is (1 × m × 1) (figure1). mazin h. suhhiem, basim n. abood, mohammed h. lafta 32 figure1: (1 × m × 1) fully fuzzy feed-forward neural network. for every entry x (where x ≥ 0) the mathematical operations in the fully fuzzy neural network can be described as: input unit: x = x, (8) hidden units : [zj]r = [[zj]r l , [zj]r u ] = [s ([netj]r l ) , s ([netj]r u )] (9) where [netj]r l = x [wj]r l + [bj]r l (10) [netj]r u = x [wj]r u + [bj]r u (11) solution of two-point fuzzy boundary value problems by fuzzy neural networks 33 output unit: [n(x)]r = [[n(x)]r l , [n(x)]r u] (12) where [n(x)]r l=∑ min {mj=1 [vj]r l [zj]r l , [vj]r l [zj]r u , [vj]r u [zj]r l , [vj]r u [zj]r u } (13) [n(x)]r u=∑ max {mj=1 [vj]r l [zj]r l , [vj]r l [zj]r u , [vj]r u [zj]r l , [vj]r u [zj]r u } (14) where [zj]r l = s (x [wj]r l + [bj]r l ) (15) [zj]r u = s (x [wj]r u + [bj]r u ) (16) 5 description of the proposed method for illustration the proposed method, we will consider the two points fuzzy boundary value problems: y´´(x) = f (x, y(x) , y´(x) ) , x ∈ [a , b] (17) with the fuzzy boundary conditions: y(a) = a and y(b) = b, where a and b are fuzzy numbers in e1 with r-level sets: [a]r = [a , a] and [b]r = [b , b] . the fuzzy trial solution for this problem is: [yt(x)]r = b − x b − a [a]r + x − a b − a [b]r +(x − a) (x − b) [n(x)]r (18) this fuzzy trial solution by intention satisfies the fuzzy boundary conditions in (17). the error function that must be minimized for problem (17) is in the form: e= ∑ (eir l + eir u) g i=1 (19) where mazin h. suhhiem, basim n. abood, mohammed h. lafta 34 eir l = [ [ d2yt (xi) dx2 ] r l − [f (xi , yt (xi), d yt (xi) dx )] r l ] 2 (20) eir u = [ [ d2yt (xi) dx2 ] r u − [f (xi , yt (xi), d yt (xi) dx )] r u ] 2 (21) where {xi}i=1 g are discrete points belonging to the interval [a , b] (training set) and in the cost function (19), er l and er u can be viewed as the squared errors for the lower limits and the upper limits of the r – level sets, respectively. now, to drive the minimized error function for problem (17): from (18) we can find: [yt(x)]r l = b − x b − a [a]r l + x − a b − a [b]r l +( x2 − (a + b)x + ab)[n(x)]r l (22) [yt(x)]r u = b − x b − a [a]r u + x − a b − a [b]r u +( x2 − (a + b)x + ab)[n(x)]r u (23) then we get: d[yt(x)]r l dx = −1 b − a [a]r l + 1 b − a [b]r l +( x2 − (a + b)x + ab) d[n(x)]r l dx +(2x−a − b)[n(x)]r l (24) d[yt(x)]r u dx = −1 b − a [a]r u + 1 b − a [b]r u +( x2 − (a + b)x + ab) d[n(x)]r u dx +(2x−a − b)[n(x)]r u (25) therefore, we have: [ d2yt (x) dx2 ] r l = ( x2 − (a + b)x + ab) d2[n(x)]r l dx2 +2(2x − a − b ) d[n(x)]r l dx + 2[n(x)]r l (26) [ d2yt (x) dx2 ] r u = ( x2 − (a + b)x + ab) d2[n(x)]r u dx2 +2(2x − a − b ) d[n(x)]r u dx + 2[n(x)]r u (27) then (20) and (21) can be rewritten as: solution of two-point fuzzy boundary value problems by fuzzy neural networks 35 eir l = [( xi 2 − (a + b)xi + ab) d2[n(xi)]r l dx2 + 2(2xi − a − b ) d[n(xi)]r l dx + 2[n(xi)]r l – f(xi , b − xi b − a [a]r l + xi − a b − a [b]r l + ( xi 2 − (a + b)xi + ab)[n(xi)]r l , −1 b − a [a]r l+ 1 b − a [b]r l+( xi 2 − (a + b)xi + ab) d[n(xi)]r l dx + (2xi − a − b)[n(xi)]r l) ]2 (28) eir u = [( xi 2 − (a + b)xi + ab) d2[n(xi)]r u dx2 + 2(2xi − a − b ) d[n(xi)]r u dx + 2[n(xi)]r u – f(xi , b − xi b − a [a]r u + xi − a b − a [b]r u + ( xi 2 − (a + b)xi + ab)[n(xi)]r u , −1 b − a [a]r u+ 1 b − a [b]r u+( xi 2 − (a + b)xi + ab) d[n(xi)]r u dx + (2xi − a − b)[n(xi)]r u) ]2 (29) where [n(xi)]r l = ∑ min {mj=1 [vj]r l s (xi [wj]r l + [bj]r l ) , [vj]r l s (xi [wj]r u + [bj]r u ) , [vj]r u s (xi [wj]r l + [bj]r l ) , [vj]r u s (xi [wj]r u + [bj]r u ) } (30) [n(xi)]r l = ∑ max {mj=1 [vj]r l s (xi [wj]r l + [bj]r l ) , [vj]r l s (xi [wj]r u + [bj]r u ) , [vj]r u s (xi [wj]r l + [bj]r l ) , [vj]r u s (xi [wj]r u + [bj]r u ) } (31) d[n(xi)]r l dx = ∑ min {mj=1 [vj]r l [wj]r l s´ (xi [wj]r l + [bj]r l ) , [vj]r l [wj]r u s´ (xi [wj]r u + [bj]r u ) , [vj]r u [wj]r l s´ (xi [wj]r l + [bj]r l ) , [vj]r u [wj]r u s´ (xi [wj]r u + [bj]r u ) } (32) d[n(xi)]r u dx = ∑ max {mj=1 [vj]r l [wj]r l s´ (xi [wj]r l + [bj]r l ) , [vj]r l [wj]r u s´ (xi [wj]r u + [bj]r u ) , [vj]r u [wj]r l s´ (xi [wj]r l + [bj]r l ) , [vj]r u [wj]r u s´ (xi [wj]r u + [bj]r u ) } (33) d2[n(xi)]r l dx2 = ∑ min{mj=1 [vj]r l ([wj]r l )2s ́ ́ (xi [wj]r l +[bj]r l ) , [vj]r l ([wj]r u )2 s ́ ́ (xi [wj]r u + [bj]r u ) , [vj]r u ([wj]r l )2s ́ ́ (xi [wj]r l +[bj]r l ) , [vj]r u ([wj]r u )2s ́ ́ (xi [wj]r u + [bj]r u ) } (34) mazin h. suhhiem, basim n. abood, mohammed h. lafta 36 d2[n(xi)]r u dx2 = ∑ max{mj=1 [vj]r l ([wj]r l )2s ́ ́ (xi [wj]r l +[bj]r l ) , [vj]r l ([wj]r u )2 s ́ ́ (xi [wj]r u + [bj]r u ) , [vj]r u ([wj]r l )2s ́ ́ (xi [wj]r l +[bj]r l ) , [vj]r u ([wj]r u )2s ́ ́ (xi [wj]r u + [bj]r u ) } (35) where s´ and s ́ ́ are the first and second derivative of the hyperbolic tangent function. then we substitute (28) and (29) in (19) to find the error function that must be minimized for problem (17). 6. numerical example in this section, we will solve two problems about two-point fuzzy boundary value problem. we have used (1 × 10 × 1) fully fuzzy feed-forward neural network. the activation function of each hidden unit is the hyperbolic tangent activation function. the analytical solutions [ya(x)]r l and [ya(x)]r u has been known in advance. therefore, we test the accuracy of the obtained solutions by computing the deviation: e (x , r) = |[ya(x)]r u − [yt (x)]r u| , e (x , r)= |[ya(x)]r l − [yt(x)]r l| to minimize the error function, we have used bfgs quasi-newton method (for more details, see [16]). the computer programs which we have used in this work are coded in matlab 2015. example (1): consider the linear fuzzy boundary value problem: y´´ (x) − y´(x) = 1 . with x ∈ [0, 0.5] y(0) = [ 2 + r , 4 − r ] , y(0.5)= [5 + r , 7 − r] . where r ∈ [0, 1]. the analytical solutions for this problem are: [ya(x)]r l = ( 2 + r − 3 e0.5−1 ) + ( 3 e0.5−1 )ex [ya(x)]r u= ( 4 − r − 3 e0.5−1 ) + ( 3 e0.5−1 )ex the trial solutions for this problem are: [yt(x)]r l= (1 − 2x) (2 + r) + 2x (4 − r)+(x2 − 0.5 x ) [n(x)]r l solution of two-point fuzzy boundary value problems by fuzzy neural networks 37 [yt(x)]r u= (1 − 2x) (5 + r) + 2x (7 − r) + (x2 − 0.5 x ) [n(x)]r u the fully fuzzy feed forward neural network has been trained by using a grid of ten equidistant points in [0, 0.5]. the error function that must be minimized for this problem will be: e= ∑ (eir l + eir u)11i=1 (36) where eir l =[( xi 2 − 0.5xi) d2[n(xi)]r l dx2 + (4xi − 1) d[n(xi)]r l dx + 2[n(xi)]r l − ( xi 2 − 0.5xi) d[n(xi)]r l dx − (2xi − 0.5)[n(xi)]r l + 4r − 5 ]2 (37) eir u =[( xi 2 − 0.5xi) d2[n(xi)]r u dx2 + (4xi − 1) d[n(xi)]r u dx + 2[n(xi)]r u − ( xi 2 − 0.5xi) d[n(xi)]r u dx − (2xi − 0.5)[n(xi)]r u + 4r − 5 ]2 (38) numerical solutions for this problem can be found in table (1). table (1): numerical result for example (1), x=1. r [yt(x)]r l e (x , r) [yt(x)]r u e (x , r) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 9.946164141 10.04616401 10.14616481 10.24616458 10.34616447 10.44616422 10.54616396 10.64616391 10.74616385 10.84616389 10.94616389 3.29137e-7 1.96846e-7 9.95565e-7 7.63284e-7 6.60993e-7 4.09513e-7 1.47232e-7 9.75941e-8 3.39072e-8 7.52383e-8 7.39070e-8 11.94616425 11.84616411 11.74616478 11.64616385 11.54616387 11.44616389 11.34616391 11.24616382 11.14616384 11.04616386 10.94616386 4.33916e-7 2.93475e-7 9.70548e-7 3.95104e-8 5.67802e-8 7.56011e-8 9.53493e-8 1.15291e-8 2.63433e-8 5.26859e-8 4.56782e-8 mazin h. suhhiem, basim n. abood, mohammed h. lafta 38 example (2): consider the non-linear fuzzy boundary value problem: y´´ (x) = (y´(x)) 2 . with x ∈ [0 , 2] y(0) = [ r , 2 − r ] , y(2) = [1 + r , 3 − r] and r ∈ [0 , 1]. the analytical solutions for this problem are: [ya(x)]r l = ln (x + 2 e−1 ) + r − ln 2 e−1 [ya(x)]r u= ln (x + 2 e−1 ) + 2 − r − ln 2 e−1 the trial solutions for this problem are: [yt(x)]r l = r 2−x 2 + (1 + r) x 2 + x (x − 2 ) [n(x)]r l [yt(x)]r u= (2 − r) 2−x 2 + (3 − r ) x 2 + x (x − 2 ) [n(x)]r u the fully fuzzy feed forward neural network has been trained by using a grid of ten equidistant points in [0, 2]. the error function that must be minimized for this problem will be: e= ∑ (eir l + eir u)11i=1 (39) where eir l =[( xi 2 − 2xi) d2[n(xi)]r l dx2 + (4xi − 4 ) d[n(xi)]r l dx + 2[n(xi)]r l + (( xi 2 − 2xi) d[n(xi)]r l dx + (2xi − 2 )[n(xi)]r l + 0.5)2 ]2 (40) eir l =[( xi 2 − 2xi) d2[n(xi)]r u dx2 + (4xi − 4 ) d[n(xi)]r u dx + 2[n(xi)]r u + (( xi 2 − 2xi) d[n(xi)]r u dx + (2xi − 2 )[n(xi)]r u + 0.5)2 ]2 (41) then we use (39) to update the weights and biases. numerical solution for this problem can be found in table (2). solution of two-point fuzzy boundary value problems by fuzzy neural networks 39 table (2): numerical result for example (2), x=1. r [yt(x)]r l e (x , r) [yt(x)]r u e (x , r) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.620114507 0.720114507 0.820114507 0.920114507 1.020114507 1.120114507 1.220114507 1.320114516 1.420114512 1.520114507 1.620114514 3.24734e-10 4.66221-10 2.03208e-10 3.80684e-10 4.09557e-10 3.50405e-10 4.59008e-10 9.46681e-9 5.06564e-9 8.21899e-10 7.88763e-9 2.620114507 2.520114507 2.420114507 2.320114513 2.220114514 2.120114508 2.020114507 1.920114507 1.820114507 1.720114514 1.620114508 8.46634e-10 9.79602e-10 6.85555e-10 6.62032e-9 7.59010e-9 1.74006e-9 9.00817e-10 9.21604e-10 4.99811e-10 7.15955e-9 1.02988e-9 for the above two problems we have [n(xi)]r l = ∑ min {10j=1 [vj]r l s (xi [wj]r l + [bj]r l ) , [vj]r l s (xi [wj]r u + [bj]r u ) , [vj]r u s (xi [wj]r l + [bj]r l ) , [vj]r u s (xi [wj]r u + [bj]r u ) } [n(xi)]r l = ∑ max {10j=1 [vj]r l s (xi [wj]r l + [bj]r l ) , [vj]r l s (xi [wj]r u + [bj]r u ) , [vj]r u s (xi [wj]r l + [bj]r l ) , [vj]r u s (xi [wj]r u + [bj]r u ) } d[n(xi)]r l dx = ∑ min {10j=1 [vj]r l [wj]r l s´ (xi [wj]r l + [bj]r l ) , [vj]r l [wj]r u s´ (xi [wj]r u + [bj]r u ) , [vj]r u [wj]r l s´ (xi [wj]r l + [bj]r l ) , [vj]r u [wj]r u s´ (xi [wj]r u + [bj]r u ) } mazin h. suhhiem, basim n. abood, mohammed h. lafta 40 d[n(xi)]r u dx = ∑ max {10j=1 [vj]r l [wj]r l s´ (xi [wj]r l + [bj]r l ) , [vj]r l [wj]r u s´ (xi [wj]r u + [bj]r u ) , [vj]r u [wj]r l s´ (xi [wj]r l + [bj]r l ) , [vj]r u [wj]r u s´ (xi [wj]r u + [bj]r u ) } d2[n(xi)]r l dx2 = ∑ min{10j=1 [vj]r l ([wj]r l )2s ́ ́ (xi [wj]r l +[bj]r l ) , [vj]r l ([wj]r u )2 s ́ ́ (xi [wj]r u + [bj]r u ) , [vj]r u ([wj]r l )2s ́ ́ (xi [wj]r l +[bj]r l ) , [vj]r u ([wj]r u )2s ́ ́ (xi [wj]r u + [bj]r u ) } d2[n(xi)]r u dx2 = ∑ max{10j=1 [vj]r l ([wj]r l )2s ́ ́ (xi [wj]r l +[bj]r l ) , [vj]r l ([wj]r u )2 s ́ ́ (xi [wj]r u + [bj]r u ) , [vj]r u ([wj]r l )2s ́ ́ (xi [wj]r l +[bj]r l ) , [vj]r u ([wj]r u )2s ́ ́ (xi [wj]r u + [bj]r u ) } 7 conclusion in this work, we have introduced a modified method to find the numerical solution of the two-point fuzzy boundary value problems for the ordinary differential equations. this method based on the fully fuzzy neural network to approximate the solution of the second-order fuzzy differential equations. for future studies, one can extend this method to find a numerical solution of the higher order fuzzy differential equations. also, one may use this method for solving a fuzzy partial differential equation. solution of two-point fuzzy boundary value problems by fuzzy neural networks 41 references [1] h. lee, i.s. kang. neural algorithms for solving differential equations. journal of computational physics, 91, 110-131. 1990. 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[16] suhhiem. fuzzy artificial neural network for solving fuzzy and nonfuzzy differential equations. ph.d. thesis, college of sciences, almustansiriyah university, iraq. 2016. ratio mathematica volume 42, 2022 some edge domination parameters in bipolar hesitancy fuzzy graph jahir hussain rasheed* mujeeburahman thacharakavil chemmala † dhamodharan durairaj‡ abstract in this article, we establish edge domination in bipolar hesitancy fuzzy graph(bhfg). various domination parameters such as inverse edge domination and total edge domination in bhfg are determined. some theorems related to edge domination and examples are also discussed. keywords: bipolar fuzzy graph; hesitant fuzzy graph; edge domination number; total edge domination; inverse edge domination; 2020 ams subject classifications: 05c72, 05c69, 94d05.1 *(pg and research department of mathematics, jamal mohamed college(autonomous), affiliated to bharathidasan university, tiruchirapplli-620020, india); hssn jhr@yahoo.com. †(pg and research department of mathematics, jamal mohamed college(autonomous), affiliated to bharathidasan university, tiruchirapplli-620020, india); mohdmujeebtc@gmail.com. ‡(pg and research department of mathematics, jamal mohamed college(autonomous), affiliated to bharathidasan university, tiruchirapplli-620020, india); dharan raj28@yahoo.co.in. 1received on march 26th, 2022. accepted on june 26th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v41i0.732. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 157 r. jahir hussain, t.c. mujeeburahman, d. dhamodharan 1 introduction the concept of fuzzy sets was first originated by l.a. zadeh [zadeh [1965]]. in 1973, kaufmann established fuzzy graph using zadeh’s fuzzy relation. the domination concept in fuzzy graph was first established by a. somasundaram and s. somasundaram [somasundaram and somasundaram [1998]]. the edge domination in fuzzy graphs was initiated by s. velammal and k.thiagarajan [vellamal and thiagarajan [2012]]. the notion of bipolar fuzzy graph(bfg) was established by m.akram [akram [2002]].the approach of domination in bipolar fuzzy graphs was proposed by m.g. karunambigai, palanivel and akram [karunambigai et al. [2013]].the book by akram, sarwar and dudek entitled ”graphs for the analysis of bipolar fuzzy information” [akram et al. [2021]] is a great tool for understanding the concepts of domination in bfgs. s. ramya and s. lavanya developed edge domination in bipolar fuzzy graphs [ramya and lavanya [2017]].the notion of hesitant fuzzy sets was first introduced by v.torra [torra [2010]] in the year 2010. hesitancy fuzzy graph, a new approach to fuzzy graph theory was first established by t. pathinathan,et.al [pathinathan et al. [2015]].the idea of domination in hesitancy fuzzy graph was investigated by r. sakthivel et.al,[sakthivel et al. [2019]]. in the year 2021, k. anantha kanaga jothi and k. balasangu [anantha kanaga jothi and balasangu [2021]]defined the idea of irregular and totally irregular bipolar hesitancy fuzzy graphs and some of its properties. 2 preliminaries definition 2.1 (akram [2002]). let x be a non empty set. a bipolar fuzzy set b in x is an object having the form b = {(x, µpb(x), µ n b (x))|x ∈ x} where, µpb : x → [0, 1] and µ n b : x → [−1, 0] are mapppings. definition 2.2 (akram [2002]). a bipolar fuzzy graph (bfg) is of the form g = (v, e) where 1. v = {v1, v2, ...vn} such that µp1 : v → [0, 1] and µn1 : v → [−1, 0] 2. e ⊂ v × v where µp2 : v × v → [0, 1] and µn2 : v × v → [−1, 0] such that µp2 (vi, vj) ≤ min(µ p 1 (vi), µ p 1 (vj)) and µn2 (vi, vj) ≥ max(µ n 1 (vi), µ n 1 (vj)) for all (vi, vj) ∈ e. 158 some edge domination parameters in bipolar hesitancy fuzzy graph definition 2.3 (akram [2002]). let g = (v, e) be a bfg is said to be strong then µp2 = min(µ p 1 (vi), µ p 1 (vj)) and µ n 2 = max(µ n 1 (vi), µ n 1 (vj)) ∀vi, vj ∈ v. definition 2.4 (akram [2002]). let g = (v, e) be a bfg is said to be complete then, µp2 (vi, vj) = min(µ p 1 (vi), µ p 1 (vj)) µn2 (vi, vj) = max(µ n 1 (vi), µ n 1 (vj)) for all vi, vj ∈ v. definition 2.5 (karunambigai et al. [2013]). an arc (a, b) is said to be strong edge in a bfg, if µp2 (a, b) ≥ (µ p 2 ) ∞(a, b) and µn2 (a, b) ≥ (µ n 2 ) ∞(a, b) whereas (µp2 ) ∞(a, b) = max{(µp2 )k(a, b)|k = 1, 2, ..., n} and (µn2 ) ∞(a, b) = min{(µn2 )k(a, b)|k = 1, 2, ..., n}. definition 2.6 (karunambigai et al. [2013]). let g = (v, e) be a bfg, then cardinality of g is defined as |g| = ∑ vi∈v (1 + µp1 (vi) + µ n 1 (vi)) 2 + ∑ (vi,vj)∈e (1 + µp2 (vi, vj) + µ n 2 (vi, vj)) 2 . definition 2.7 (karunambigai et al. [2013]). the cardinality of v, i.e., amount of nodes is termed as the order of g = (v, e) and is signified by |v|(or o(g)) and determined by o(g) = |v| = ∑ vi∈v (1 + µp1 (vi) + µ n 1 (vi)) 2 the no. of elements in a set of s, i.e., amount of edges is termed as size of g = (v, e) and signified as |s|(or s(g)) and determined by s(g) = |s| = ∑ (vi,vj)∈e (1 + µp2 (vi, vj) + µ n 2 (vi, vj)) 2 for all (vi, vj) ∈ e. 159 r. jahir hussain, t.c. mujeeburahman, d. dhamodharan 3 bipolar hesitancy fuzzy graph definition 3.1 (anantha kanaga jothi and balasangu [2021]). let x be a nonempty set. a bipolar hesitancy fuzzy set b = {x, µp1 (x), µ n 1 (x), γ p 1 (x), γ n 1 (x), β p 1 (x), β n 1 (x)/x ∈ x} where µp1 , γ p 1 , β p 1 : x → [0, 1] and µn1 , γn1 , βn1 : x → [−1, 0] are mappings such that, 0 ≤ µp1 (x) + γ p 1 (x) + β p 1 (x) ≤ 1 and −1 ≤ µn1 (x) + γ n 1 (x) + β n 1 (x) ≤ 0 . definition 3.2 (anantha kanaga jothi and balasangu [2021]). let x be a non empty set.then we call mappings µp2 , γ p 2 , β p 2 : x × x → [0, 1], µn2 , γn2 , βn2 : x × x → [−1, 0] are bipolar hesitancy fuzzy relation on x such that, µp2 (x, y) ≤ µp1 (x) ∧ µp1 (y); µn2 (x, y) ≥ µn1 (x) ∨ µn1 (y); γp2 (x, y) ≤ γp1 (x) ∧ γp1 (y); γ n 2 (x, y) ≥ γn1 (x) ∨ γn1 (y); βp2 (x, y) ≤ βp1 (x) ∧ βp1 (y); βn2 (x, y) ≥ βn1 (x) ∨ βn1 (y). definition 3.3 (anantha kanaga jothi and balasangu [2021]). a bipolar hesitancy fuzzy relation a on x is called symmetric relation if µp2 (x, y) = µp2 (x, y), µn2 (x, y) = µ n 2 (x, y) ,γ p 2 (x, y) = γ p 2 (x, y), γ n 2 (x, y) = γ n 2 (x, y), β p 2 (x, y) = βp2 (x, y), β n 2 (x, y) = β n 2 (x, y) for all (x, y) ∈ x definition 3.4 (pathinathan et al. [2015]). a hesitancy fuzzy graph is of the form g = (v, e) where, v = {v!, v2, . . . vn} such that µ1, γ1, β1 : v → [0, 1] denote the degree of membership, non-membership and hesitancy of the vertex vi ∈ v respectively and µ1(vi)+γ1(vi)+β1(vi) = 1 for every vi ∈ v where β1(vi) = 1−[µ1(vi)+γ1(vi)] and e ⊆ v × v where µ2, γ2, β2 : v × v → [0, 1] denote the degree of membership, non-membership and hesitancy of the edge (vi, vj) ∈ e respectively such that, µ2(vi, vj) ≤ µ1(vi) ∧ µ1(vj); γ2(vi, vj) ≤ γ1(vi) ∨ γ1(vj); β2(vi, vj) ≤ β1(vi) ∧ β1(vj) and 0 ≤ µ2(vi, vj) + γ2(vi, vj) + β2(vi, vj) ≤ 1 for every (vi, vj) ∈ e. definition 3.5 (anantha kanaga jothi and balasangu [2021]). a bipolar hesitancy fuzzy graph (bhfg) is of the form g = (v, e) where (i) v = {v1, v2, . . . , vn} such that µp1 , γp1 , βp1 : v → [0, 1] denote the degree of positive membership, positive non-membership and positive hesitancy of 160 some edge domination parameters in bipolar hesitancy fuzzy graph the vertex vi ∈ v respectively, µn1 , γn1 , βn1 : v → [−1, 0] denote the degree of negative membership,negative non-membership and negative hesitancy of the vertex vi ∈ v. for every vi ∈ v, µp1 (vi) + γ p 1 (vi) + β p 1 (vi) = 1 and µ n 1 (vi) + γ n 1 (vi) + β n 1 (vi) = −1 βp1 (vi) = 1 − [µp1 (vi) + γp1 (vi)] and βn1 (vi) = −1 − [µn1 (vi) + γn1 (vi)] (ii) e ⊆ v×v where, µp2 , γp2 , βp2 : v×v → [0, 1]; µn2 , γn2 , βn2 : v×v → [−1, 0] are mappings such that µp2 (vi, vj) ≤ µ p 1 (vi) ∧ µ p 1 (vj) µn2 (vi, vj) ≥ µ n 1 (vi) ∨ µ n 1 (vj) γp2 (vi, vj) ≤ γ p 1 (vi) ∨ γ p 1 (vj) γn2 (vi, vj) ≥ γ n 1 (vi) ∧ γ n 1 (vj) βp2 (vi, vj) ≤ β p 1 (vi) ∧ β p 1 (vj) βn2 (vi, vj) ≥ β n 1 (vi) ∨ β n 1 (vj) denote the degree of positive, negative membership, degree of positive, negative non membership and degree of positive, negative hesitancy of the edge (vi, vj) ∈ e respectively and 0 ≤ µp2 (vi, vj) + γ p 2 (vi, vj) + β p 2 (vi, vj) ≤ 1 , −1 ≤ µn2 (vi, vj) + γ n 2 (vi, vj) + β n 2 (vi, vj) ≤ 0 for every (vi, vj) ∈ e. figure 1: bipolar hesitancy fuzzy graph 161 r. jahir hussain, t.c. mujeeburahman, d. dhamodharan example 3.1. from fig 1, for vertex v1, µp1 (v1) + γ p 1 (v1) + β p 1 (v1) = 0.4 + 0.2 + 0.4 = 1 µn1 (v1) + γ n 1 (v1) + β n 1 (v1) = −0.6 − 0.2 − 0.2 = −1. for edge (v1, v2); µp2 (v1, v2) + γ p 2 (v1, v2) + β p 2 (v1, v2) = 0.8 ≤ 1 µn2 (v1, v2) + γ n 2 (v1, v2) + β n 2 (v1, v2) = −0.7 ≥ −1. definition 3.6. a bipolar hesitancy fuzzy graph g = (v, e) is said to be complete when, µp2 (vi, vj) = µ p 1 (vi) ∧ µp1 (vj),µn2 (vi, vj) = µn1 (vi) ∨ µn1 (vj), γp2 (vi, vj) = γp1 (vi) ∨ γp1 (vj), γn2 (vi, vj) = γn1 (vi) ∧ γn1 (vj) , βp2 (vi, vj) = βp1 (vi) ∧ βp1 (vj) , βn2 (vi, vj) = β n 1 (vi) ∨ βn1 (vj) for every vi, vj ∈ v. definition 3.7. a bipolar hesitancy fuzzy graph g = (v, e) is said to be strong when, µp2 (vi, vj) = µ p 1 (vi) ∧ µp1 (vj),µn2 (vi, vj) = µn1 (vi) ∨ µn1 (vj) γp2 (vi, vj) = γp1 (vi) ∨ γp1 (vj), γn2 (vi, vj) = γn1 (vi) ∧ γn1 (vj) , βp2 (vi, vj) = βp1 (vi) ∧ βp1 (vj) , βn2 (vi, vj) = β n 1 (vi) ∨ βn1 (vj) for every (vi, vj) ∈ e. definition 3.8. let g be a bipolar hesitancy fuzzy graph. the neighbourhood of a vertex x in g is defined by n(x) = (npµ (x), n n µ (x), n p γ (x), n n γ (x), n p β (x), n n β (x)) where npµ (x) = {y ∈ v/µp2 (x, y) ≤ µp1 (x) ∧ µp1 (x)}; nnµ (x) = {y ∈ v/µn2 (x, y) ≥ µn1 (x) ∨ µn1 (x)};npγ (x) = {y ∈ v/γp2 (x, y) ≤ γp1 (x) ∧ γp1 (x)}; nnγ (x) = {y ∈ v/γn2 (x, y) ≥ γn1 (x) ∨ γn1 (x)};npβ (x) = {y ∈ v/β p 2 (x, y) ≤ βp1 (x) ∧ βp1 (x)}; nnβ (x) = {y ∈ v/β n 2 (x, y) ≥ βn1 (x) ∨ βn1 (x)}. definition 3.9. let g be a bipolar hesitancy fuzzy graph. the neighborhood degree of a vertex x in g is defined by deg(x) = [deg µp (x), deg µn(x), deg γp (x), deg γn(x), deg βp (x), deg βn(x)] y ∈ v, where deg µp (x) = ∑ y∈n(x) µp1 (y), deg µ n(x) = ∑ y∈n(x) µn1 (y), deg γ p (x) = ∑ y∈n(x) γp1 (y) deg γn(x) = ∑ y∈n(x) γn1 (y), deg β p (x) = ∑ y∈n(x) βp1 (y), deg β n(x) = ∑ y∈n(x) βn1 (y) . definition 3.10. let g = (v, e) be a bhfg. the edge cardinality of g is given by, |e| = r = ∑ (u,v)∈e 3 + µp2 (u, v) + µ n 2 (u, v) + γ p 2 (u, v) + γ n 2 (u, v) + β p 2 (u, v) + β n 2 (u, v) 3 . 162 some edge domination parameters in bipolar hesitancy fuzzy graph definition 3.11. an arc (u,v) is said to be strong edge in bhfg. then, µp2 (u, v) ≥ (µp2 )∞(u, v),µn2 (u, v) ≥ (µn2 )∞(u, v),γp2 (u, v) ≥ (γp2 )∞(u, v), γn2 (u, v) ≥ (γn2 )∞(u, v),βp2 (u, v) ≥ (βp2 )∞(u, v), βn2 (u, v) ≥ (βn2 )∞(u, v) whereas (µp2 ) ∞(u, v) = max{(µp2 )k(u, v)|k = 1, 2, . . . , n}; (µn2 ) ∞(u, v) = min{(µn2 )k(u, v)|k = 1, 2, . . . , n}; (γp2 ) ∞(u, v) = max{(γp2 )k(u, v)|k = 1, 2, . . . , n}; (γn2 ) ∞(u, v) = min{(γn2 )k(u, v)|k = 1, 2, . . . , n}; (βp2 ) ∞(u, v) = max{(βp2 )k(u, v)|k = 1, 2, . . . , n}; (βn2 ) ∞(u, v) = min{(βp2 )k(u, v)|k = 1, 2, . . . , n}. 4 edge domination in bipolar hesitancy fuzzy graph definition 4.1. let g = (v, e) be a bipolar hesitancy fuzzy graph. a set s ⊆ e is said to be an edge dominating set of g if every edge not in s is incident to some edge in s. definition 4.2. an edge dominating set s ⊆ e is said to be minimal if no proper subset of s is an edge dominating set. definition 4.3. the minimum cardinality out of all minimal dominating sets of bhfg g is said to be lower domination number of g and denoted as dbh(g). definition 4.4. the maximum cardinality out of all minimal dominating sets of bhfg g is said to be upper domination number of g and denoted as dbh(g). figure 2: edge domination in bhfg 163 r. jahir hussain, t.c. mujeeburahman, d. dhamodharan example 4.1. in the above figure 2, {e1, e2, e4, e5},{e2, e3, e5},{e1, e3, e4} are edge dominating sets of g.{e1, e4} {e3, e2},{e5} are minimal edge dominating sets of g. among all the minimal dominating sets, {e5} has minimum cardinality and edge domination number γbh(g) = 1.06. theorem 4.1. let s be a minimal edge dominating set of a bhfg g = (v, e). if for any edge e ∈ s, one of the following condition hold a) n(e) ∩ s = ϕ b) ∃e′ ∈ e − s such that n(e) ∩ s = {e}. proof. given g = (v, e) is a bhfg and s is a minimal edge dominating set of g, then for every edge e ∈ s, s −{e} is not an edge dominating set and hence there exists an edge e′ ∈ e − s which is not adjacent to any element of s − {e}. thus if e′ = e we get (a) and if e′ ̸= e we get (b). 2 definition 4.5. an edge e of a bhfg g is called an an isolated edge if no effective edges is incident with the vertices of e and hence it doesn’t dominate any other vertex in g. theorem 4.2. if g = (v, e) is a bhfg without any isolated edges, then for every minimal edge dominating set s, prove that e − s is also an edge dominating set. proof. given g = (v, e) a bhfg without any isolated edges. let s be minimal edge dominating set of g, then there exists an edge e′ ∈ n((e). from theorem 5.4 we get e′ ∈ e − s which implies every edge in e − s is adjacent to an edge in s. hence e − s is also an edge dominating set. 2 corolary 4.1. for any graph g without isolated edges γbh(g) ≤ r 3 . definition 4.6. let g = (v, e) be a bhfg. let s be a minimum edge set of g. if e − s contains an edge dominating set s′ of g, then s′ is said to be inverse edge dominating set of g. the minimum cardinality out of all minmal inverse edge dominating sets is said to be inverse edge domination number and is denoted as γ−1bh (g). proposition 4.1. for any graph g without isolated edges and vertices γbh(g) ≤ γ−1bh (g) . proposition 4.2. if g is a graph without isolated edges and vertices and if number of vertices are greater than or equal to 3, then γbh(g) + γ −1 bh (g) ≤ r . 164 some edge domination parameters in bipolar hesitancy fuzzy graph definition 4.7. let g = (v, e) be bhfg without isolated edges. an edge dominating set s is called as total edge dominating set if < s > has no isolated edge.the minimum cardinality of all minimal total edge dominating sets is said to be total edge domination number of g and is denoted as γtbh. a set f ⊆ e is said to be a total edge dominating set of g if for every edge in e is adjacent to at least one edge in f. theorem 4.3. for any bipolar hesitancy fuzzy graph g, γbh(g) ≤ γtbh(g). 2 theorem 4.4. for any bipolar fuzzy graph g with r edges then prove that γtbh = r iff every edge of g has a unique neighbor. proof. given a bhfg g with r edges.let us consider every edge of g has a unique neighbor, then s is the only total edge dominating set of g which implies γtbh = r. conversely, suppose γtbh = r and if there exists an edge with neighbors s and t then s − {s} gives a total edge dominating set of g. thus γtbh < r which is a contradiction. 2 5 conclusions we have established edge domination in bipolar hesitancy fuzzy graph(bhfg). along with various domination parameters such as inverse and total edge domination were also discussed. we have also given various examples and theorems supporting the main result. our result can be extended to other domination parameters as well. 6 acknowledgements the authors thanks the management, ratio mathematica for their constant support towards the successful completion of this work. we wish to thank the anonymous reviewers for a careful reading of manuscript and for very useful comments and suggestions. references m. akram. bipolar fuzzy graphs. information science, 181:5548–5564, 2002. m. akram, m. sarwar, and w. a. dudek. graphs for the analysis of bipolar fuzzy information. springer, singapore, 2021. 165 r. jahir hussain, t.c. mujeeburahman, d. dhamodharan k. anantha kanaga jothi and k. balasangu. irregular and totally irregular bipolar hesitancy fuzzy graphs and some of its properties. advances and applications in mathematical sciences, 20:1685–1696, 2021. m. karunambigai, m. akram, and k. palanivel. domination in bipolar fuzzy graphs. international conference on fuzzy systems, 2013:7–10, 2013. t. pathinathan, jonadoss jon arockiaraj, and jesintha rosline. hesitancy fuzzy graphs. indian journal of science and technology, 8:1–5, 2015. s. ramya and s. lavanya. edge contraction in bipolar fuzzy graphs. international journal of trend in research and development, 4:475–477, 2017. r. sakthivel, r. vikramaprasad, and n. vinothkumar. domination in hesitancy fuzzy graphs. international journal of advanced science and technology, 28: 1142 – 1156, 2019. a. somasundaram and s. somasundaram. domination in fuzzy graphs-i. pattern recognition letters, 19:787–791, 1998. v. torra. hesitant fuzzy sets. international journal of intelligent systems, 25: 529–539, 2010. s. vellamal and k. thiagarajan. edge domination in fuzzy graphs. international journal of theoretical and applied physics, 2:33–40, 2012. l. zadeh. fuzzy sets. information and control, 8:195–204, 1965. 166 ratio mathematica volume 38, 2020, pp. 223-236 pairwise paracompactness pallavi s. mirajakar* p. g. patil† abstract the purpose of this paper is to introduce and study a new paracompactness in bitopological spaces using (τi,τj)-g∗ωα-closed sets. further, the properties of (τi,τj)-g∗ωα-closed sets, (τi,τj)-g∗ωαcontinuous functions and (τi,τj)-g∗ωα-irresolute maps and (τi,τj)g∗ωα-paracompact spaces are discussed in bitopological spaces. keywords: (τi,τj)-g∗ωα-closed sets, (τi,τj)-g∗ωα-open sets, (τi,τj)-g∗ωα-continuous and (τi,τj)-g∗ωα-irresolute maps, (τi,τj)-g∗ωα-paracompact spaces. 2010 ams subject classifications: 54e55. 1 *department of mathematics (p. c. jabin science college, hubbali,karnatak, india); psmirajakar@gmail.com. †department of mathematics (karnatak university, dharwad, karnatak, india); pgpatil@kud.ac.in 1received on january 10th, 2020. accepted on may 3rd, 2020. published on june 30th, 2020. doi: 10.23755/rm.v38i0.495. issn: 1592-7415. eissn: 2282-8214. ©pallavi s. mirajakar et al. this paper is published under the cc-by licence agreement. 223 p. s. mirajakar and p. g. patil 1 introduction the research in topology over last two decades has reached a high level in many directions. topological methods are widely used in many other branches of modern mathematics such as differential equation, functional analysis, classical mechanics, general theory of relativity, mathematical economics, quantum theory, biology etc. bitopological space is a triplet (x,τ1,τ2), where x is a non empty set and τ1 and τ2 are topologies on a space x. in 1963, j. c. kelly [8] initiated the study of bitopological spaces. in 1985, fututake [5] studied the concept of generalized closed (briefly g-closed) sets in bitopological spaces. after that, several authors turned their attention towards the generalizations of various concepts in topology by considering bitopological spaces. in this paper, (τi,τj)-g∗ωα-closed sets, (τi,τj)-g∗ωα-continuous functions and (τi,τj)-g∗ωα-irresolute maps are defined and studied in bitopological spaces. also, the concept of (τi,τj)-g∗ωα-paracompactness in bitopological spaces is introduced and studied. 2 preliminaries throughout this present paper, let x, y and z always represents non-empty bitopological spaces (x,τ1,τ2), (y,σ1,σ2) and (z,γ1,γ2) on which no separation axioms are assumed unless explicitly mentioned and the integers i,j,k ∈{1, 2}. definition 2.1. [13] a space x is said to be g∗ωα-paracompact if every open cover of x has a g∗ωα-locally finite g∗ωα-refinement. definition 2.2. let a ⊆ x. then a is said to be a (a) ωα-closed [2] if αcl(a) ⊆ u whenever a ⊆ u and u is ω-open in x. (b) g∗ωα-closed[12] if cl(a) ⊆ u whenever a ⊆ u and u is ωα-open in x. definition 2.3. a subset a of a bitopological space (x,τ1,τ2) is called a (a) (τi,τj)-g-closed [5] if τj-cl(a) ⊆ u whenever a ⊆ u and u is open in τi. (b) (τi,τj)-rg-closed [1] if τj-cl(a) ⊆ u whenever a ⊆ u and u is regular open in τi. (c) (τi,τj)-αg-closed [3] if τj-αcl(a) ⊆ u whenever a ⊆ u and u is open in τi. (d) (τi,τj)-gα-closed [3] if τj-αcl(a) ⊆ u whenever a ⊆ u and u is α-open in τi. (e) (τi,τj)-gpr-closed [7] if τj-pcl(a) ⊆ u whenever a ⊆ u and u is regular open in τi. (f) (τi,τj)-g∗-closed [14] if τj-cl(a) ⊆ u whenever a ⊆ u and u is g-open in τi. 224 pairwise paracompactness (g) (τi,τj)-ωα-closed [11] if τj-cl(a) ⊆ u whenever a ⊆ u and u is ω-open in τi. in all the above definitions i 6= j. definition 2.4. a map f : (x,τ1,τ2) → (y,µ1,µ2) is called a (a) τj-µk-continuous [10] if f−1(g) ∈ τj for every open set g in µk. (b) d(τi,τj)-µk-continuous [10] if the inverse image of every µk-closed set in (y,µ1,µ2) is (τi,τj)-g-closed in (x,τ1,τ2). (c) dr(τi,τj)-µk-continuous [1] if the inverse image of every µk-closed set in (y,µ1,µ2) is (τi,τj)-rg-closed in (x,τ1,τ2). (d) c(τi,τj)-µk-continuous [6] if the inverse image of every µk-closed set in (y,µ1,µ2) is (τi,τj)-ω-closed in (x,τ1,τ2). (e) d∗(τi,τj)-µk-continuous [14] if the inverse image of every µk-closed set in (y,µ1,µ2) is (τi,τj)-g∗-closed in (x,τ1,τ2). (f) (τi,τj)-αg-continuous [4] if the inverse image of every µk-closed set in (y,µ1,µ2) is (τi,τj)-αg-closed in (x,τ1,τ2). definition 2.5. [8] a bitopological space (x,τ1,τ2) is said to be pairwise hausdorff if for each pair of distinct points x and y of x, there exist u ∈ pi and v ∈ pj such that x ∈ u, y ∈ v and u ∩v = φ. 3 (τi,τj)-g∗ωα-closed sets this section deals with the concept of g∗ωα-closed sets in bitopological spaces and some of their properties. definition 3.1. let (i,j) ∈ {1, 2} where i 6= j. a subset a of a bitopological space (x,τ1,τ2) is said to be (τi,τj)-g∗ωα-closed if τj-cl(a) ⊆ u whenever a ⊆ u and u ∈ τi-ωα-open in x. example 3.1. let x = {m,n,p}, τ1 = {x,φ,{m},{n,p}} and τ2 = {x,φ,{m}}. consider a set in the space (x,τ1,τ2), a = {n,p} which is (τ1,τ2)-g∗ωα-closed. remark 3.1. if τ1 = τ2 = τ in definition 3.1, then (τi,τj)-g∗ωα-closed set in (x,τ1,τ2) is same as g∗ωα-closed [12] in (x,τ). the family of all (τi,τj)-g∗ωα-closed sets in (x,τ1,τ2) is denoted by p(τi,τj). theorem 3.1. every τj-closed (resp. (τi,τj)-regular closed) is (τi,τj)-g∗ωαclosed. however the converse need not be true in general as shown in the following example. 225 p. s. mirajakar and p. g. patil example 3.2. let x = {m,n,p}, τ1 = {x,φ,{m},{n,p}} and τ2 = {x,φ,{m}}. consider the set, a = {m,p} is (τ1,τ2)-g∗ωα-closed but not τ2-closed (resp. (τi,τj)-regular closed). we have the following implification: τj-closed → (τi,τj)-g∗ωα-closed → (τi,τj)-αg-closed remark 3.2. if a and b are (τi,τj)-g∗ωα-closed in (x,τ1,τ2) then a∪b is also (τi,τj)-g∗ωα-closed. theorem 3.2. if a subset a of (x,τ1,τ2) is (τi,τj)-g∗ωα-closed then τj-cl(a)−a does not contain any non empty ωα-closed set in τi. proof. let a ⊆ (τi,τj)-g∗ωα-closed and f ⊆ τi-ωα-closed set such that f ⊆ τj-cl(a) −a. now f ⊆ τj-cl(a) and f ⊆ x −a. then a ⊆ x −f and by hypothesis a is (τi,τj)-g∗ωα-closed and x − f is τi − ωα-open. thus from definition 3.1, τj-cl(a) ⊆ x − f , that is f ⊆ (x − τj-cl(a)). then f ⊆ (τjcl(a)) ∩ (x − τj-cl(a)) = φ and so f = φ which is a contradiction. hence τj-cl(a) −a does not contain any non empty ωα-closed set. 2 remark 3.3. a (τi,τj)-g∗ωα-closed set need not be τi-g∗ωα-closed or τj-g∗ωαclosed. example 3.3. let x = {m,n,p}, τ1 = {x,φ,{m}} and τ2 = {x,φ}. then the set a = {n,p} is (τi,τj)-g∗ωα-closed but not τ2-g∗ωα-closed. also, if x = {m,n,p}, τ1 = {x,φ,{p}} and τ2 = {x,φ,{m},{m,n}} be topology on x. then the set a = {m,p} is(τi,τj)-g∗ωα-closed but not τ1-g∗ωαclosed in (x,τ1,τ2). remark 3.4. in general p(τi,τj) 6= p(τj,τi). example 3.4. let x = {m,n,p}, τ1 = {x,φ,{p}} and τ2 = {x,φ,{m},{m,n}}. then g∗ωαc(τ1,τ2) = {x,φ,{m,n}} and g∗ωαc(τ2,τ1) = {x,φ,{n,p},{p}}. hence we can observe that g∗ωαc(τ1,τ2) 6= g∗ωαc(τ2,τ1). remark 3.5. if τ1 ⊆ τ2, then p(τ2,τ1) ⊆ p(τ1,τ2) but converse is not true. example 3.5. let x = {m,n,p}, τ1 = {x,φ,{p}} and τ2 = {x,φ,{m},{m,n}}. then p(τ2,τ1) = {x},φ,{m,n},{n,p}} and p(τ1,τ2) = {x,φ,{m,n},{n,p},{p}}. then p(τ2,τ1) ⊆ g(τ1,τ2) but τ1 * τ2. theorem 3.3. a τi-ωα-open and (τi,τj)-g∗ωα-closed set is τj-closed. proof. now a ⊆ a. then τj-cl(a) ⊆ a and a ⊆ τj-cl(a). therefore τj-cl(a) = a and hence a ∈ τj-closed. 2 226 pairwise paracompactness theorem 3.4. let a be τi-ωα-open and (τi,τj)-g∗ωα-closed. suppose f is τjclosed, then a∩f is (τi,τj)-g∗ωα-closed. proof. let a be τi-ωα-open and a be (τi,τj)-g∗ωα-closed and f be τj-closed. then from theorem 3.3, a is τj-closed. so a∩f is τj-closed and hence (τi,τj)g∗ωα-closed. 2 theorem 3.5. if a is (τi,τj)-g∗ωα-closed and a ⊆ b ⊆ τj-cl(a), then τj-cl(b)− b contains no non empty τi-closed set. proof. let a be (τi,τj)-g∗ωα-closed and a ⊆ b ⊆ τj-cl(a). then b is (τi,τj)-g∗ωα-closed follows from theorem 3.19 [12]. hence τj-cl(b) − b contains no non empty τi-closed set. 2 corolary 3.1. if a is (τi,τj)-g∗ωα-closed and a ⊆ b ⊆ τj-cl(a), then τj-cl(b)− b contains no non empty τi-ωα-closed set. theorem 3.6. arbitrary union of (τi,τj)-g∗ωα-closed sets {ai : i ∈ i} is (τi,τj)g∗ωα-closed if the family {ai : i ∈ i} is τj-locally finite. proof. let {ai : i ∈ i} is τj-locally finite and {ai : i ∈ i} is (τi,τj)g∗ωα-closed. let ∪ai ⊆ u where u ∈ τi-ωα-open. then ai ⊆ u and ωαopen in τi. since a is (τi,τj)-g∗ωα-closed the for each i ∈ i, τj-cl(ai) ⊆ u. consequently ∪τj-cl(ai) ⊆ u. since the family, {ai : i ∈ i} is τj-locally finite τj-cl(∪ai) = ∪(τj-cl(ai)) ⊆ u. therefore ∪ai is (τi,τj)-g∗ωα-closed. 2 theorem 3.7. for an element x in x, the set x −{x} is (τi,τj)-g∗ωα-closed or x −{x} is ωα-open in τi. proof. suppose x −{x} is not ωα-open in τi, then x is the only ωα-open set containing x −{x}, that is τj-cl(x −{x}) ⊆ τj-cl({x}) = x. hence τjcl(x −{x}) ⊆ x. thus x −{x} is τi,τj)-g∗ωα-closed. 2 definition 3.2. a subset a of a bitopological space (x,τ1,τ2) is (τi,τj)-g∗ωαopen if its complement is (τi,τj)-g∗ωα-closed. definition 3.3. for a subset a of a bitopological space (x,τ1,τ2), (τi,τj)-g∗ωαinterior of a is denoted by (τi,τj)-g∗ωα-int(a) and is defined as (τi,τj)-g∗ωα-int(a) = ∪{f : f ∈ (τi,τj)-g∗ωα-open and f ⊆ a}. theorem 3.8. let a be (τi,τj)-g∗ωα-open. then p = x whenever g is τi-ωαopen and τj-g∗ωα-int(a) ∪ac ⊆ g. theorem 3.9. a set a is (τi,τj)-g∗ωα-open if and only if f ⊆ τj-int(a) whenever f is τi-closed and f ⊆ a. 227 p. s. mirajakar and p. g. patil theorem 3.10. if a and b are separated (τi,τj)-g∗ωα-open sets then a ∪ b is also (τi,τj)-g∗ωα-open. proof.suppose a and b are (τi,τj)-g∗ωα-open sets. let f be an τi-closed set such that f ⊆ a∪b. since a and b are separated, τi-cl(a) ∩b = a∩ τi-cl(b) = φ and τj-cl(a) ∩b = a∩ τj-cl(b) = φ. then f ∩ τj-cl(a) ⊆ (a∪b) ∩ τjcl(a) = a. similarly, f ∩ τj-cl(b) ⊆ b. since f is τi-closed, we have f ∩ τicl(a),f ∩ τi-cl(b) are also τi-closed and from hypothesis a and b are (τi,τj)g∗ωα-open sets, f ∩ τj-cl(a) ⊆ τj-int(a) and f ∩ τj-cl(b) ⊆ τj-int(b). now f = f ∩ (a∪b) ⊆ (f ∩ τj-cl(a)) ∪ (f ∩ τj-cl(b)) ⊆ τj-int(a∪b). hence a∪b is (τi,τj)-g∗ωα-open. 2 definition 3.4. a bitopological space (x,τ1,τ2) is said to be a (τi,τj)-tg∗ωαspace if every (τi,τj)-g∗ωα-closed set is τj-closed. example 3.6. let x = {m,n,p}, τ1 = {x,φ,{m},{m,n}} and τ2 = {x,φ,{m}, {p},{m,p}}. then (x,τ1,τ2) is (τ1,τ2)-tg∗ωα-space. theorem 3.11. if a bitopological space (x,τ1,τ2) is (τi,τj)-tg∗ωα space, then for each x ∈ x, {x} is τi-ωα-closed or τj-open. proof. suppose {x} is not (τi,τj)-g∗ωα-open, then {x}c is (τi,τj)-g∗ωαclosed. as x is is (τi,τj)-tg∗ωα-space, {x}c is τj-closed and hence {x} is τj-open. 2 remark 3.6. every singleton subset of (x,τ1,τ2) is τj-closed or τi-ωα-closed but (x,τ1,τ2) is not (τi,τj)-tg∗ωα-space. example 3.7. let x = {m,n,p}, τ1 = {x,φ,{m},{m,n}} and τ2 = {x,φ,{m}, {p},{m,p}}. then every singleton set {x} of x is either τ2-open or τ1-ωα-closed. however, (x,τ1,τ2) is not (τ1,τ2)-tg∗ωα-space. remark 3.7. if (x,τ1) and (x,τ2) are both tg∗ωα-space, then it need not imply (τ1,τ2)-tg∗ωα-space. example 3.8. let x = {m,n,p}, τ1 = {x,φ,{n},{n,p}} and τ2 = {x,φ,{m}, {m,n}}. then (x,τ1) and (x,τ2) are tg∗ωα-space, but (x,τ1,τ2) is not (τ1,τ2)tg∗ωα-space. remark 3.8. the space (x,τ1) is not generally tg∗ωα-space if (x,τ1,τ2) is (τ1,τ2)tg∗ωα-space. example 3.9. let x = {m,n,p}, τ1 = {x,φ,{m},{n,p}} and τ2 = {x,φ,{m}}. then (x,τ1,τ2) is (τ1,τ2)-tg∗ωα-space, but (x,τ1) is not tg∗ωα-space. 228 pairwise paracompactness 4 (τi,τj)-g∗ωα-continuous and (τi,τj)-g∗ωα-irresolute maps definition 4.1. a map f : (x,τ1,τ2) → (y,µ1,µ2) is called p(τi,τj)-µk-continuous (pairwise g∗ωα-continuous) if the inverse image of every µk-closed set in (y,µ1,µ2) is (τi,τj)-g∗ωα-closed in (x,τ1,τ2). theorem 4.1. every is τj-µk-continuous function is p(τi,τj)-µk-continuous. proof. follows from theorem 3.1. 2 the converse need not be true as seen from the following example. example 4.1. let x = y = {m,n,p}, τ1 = {x,φ,{m},{n,p}}, τ2 = {x,φ, {m},{p},{m,p}}, µ1 = {y,φ,{n}} and µ2 = {y,φ,{m}}. let f : (x,τ1,τ2) → (y,µ1,µ2) be the identity map. then f is p(τ1,τ2)-µ1-continuous but not τ2-µ1continuous, since for the µ1-closed set a = {m,p} in y, f−1({m,p}) = {m,p} is not τ2-closed in x. remark 4.1. let f : (x,τ1,τ2) → (y,µ1,µ2) be p(τi,τj)-µk-continuous g : (y,µ1,µ2) → (z,γ1,γ2) be p(µ1,µ2)-γm-continuous but their composition need not be p(τi,τj)-γm-continuous. example 4.2. let x = y = {m,n,p}, τ1 = {x,φ,{m},{m,n}}, τ2 = {x,φ, {m},{p},{m,p}}, µ1 = {y,φ,{m},{n,p}}, µ2 = {y,φ,{m}}, γ1 = {z,φ, {m},{m,p}} and γ2 = {z,φ,{m},{m,n},{m,p}}. let f : (x,τ1,τ2) → (y,µ1,µ2) be identity map and define a map g : (y,µ1,µ2) → (z,γ1,γ2) by g(m) = n,g(n) = m,g(p) = p. then f and g are pairwise g∗ωα-continuous maps but their composition is not pairwise g∗ωα-continuous, since for the γ1-closed set {n,p} in (z,γ1,γ2), (gof)−1({n,p}) = f−1(g−1({n,p})) = f−1({m,p}) = {m,p} is not (τ1,τ2)-g∗ωα-closed in (x,τ1,τ2). definition 4.2. a map f : (x,τ1,τ2) → (y,µ1,µ2) is called pairwise g∗ωαirresolute if for every a ∈ p(µk,µe) in (y,µ1,µ2), f−1(a) ∈ p(τi,τj) in (x,τ1,τ2). theorem 4.2. if a map f : (x,τ1,τ2) → (y,µ1,µ2) pairwise g∗ωα-irresolute if f is p(τi,τj)-µe-continuous. proof. let f be µe-closed, then f is (µk,µe)-g∗ωα-closed in (y,µ1,µ2). from theorem 3.1, f ∈ p(µk,µe). since f is pairwise g∗ωα-irresolute, f−1(f) ∈ p(τi,τj). therefore f is p(τi,τj)-µe-continuous. 2 the converse of this theorem need not be true as seen from the following example. 229 p. s. mirajakar and p. g. patil example 4.3. let x = y = {m,n,p}, τ1 = {x,φ,{m},{m,n}}, τ2 = {x,φ, {m},{p},{m,p}}, µ1 = {y,φ,{n},{n,p}} and µ2 = {y,φ,{m},{m,p}}. let f : (x,τ1,τ2) → (y,µ1,µ2) be the identity map. then f is p(τ1,τ2)-µ1continuous map but not pairwise g∗ωα-irresolute map, since for the (µ1,µ2)g∗ωα-closed set {m,p} in (y,µ1,µ2), f−1({m,p}) = {m,p} is not (τi,τj)-g∗ωαclosed set in (x,τ1,τ2). theorem 4.3. let f : (x,τ1,τ2) → (y,µ1,µ2) be a map and (y,µ1,µ2) be (µk,µe)-tg∗ωα-space. then f is pairwise g∗ωα-irresolute if and only if f is p(τi,τj)µe-continuous. proof. suppose f is pairwise g∗ωα-irresolute. from theorem 4.2, f is p(τi,τj)µe-continuous. conversely, let f be p(τi,τj)-µe-continuous map. let f be (µk,µe)-g∗ωα-closed in (y,µ1,µ2). by hypothesis (y,µ1,µ2) is (µk,µe)-tg∗ωα-space, f is µe-closed set in (y,µ1,µ2). again, since f is p(τi,τj)-µe-continuous, f−1(f) is (τi,τj)-g∗ωαclosed set in (x,τ1,τ2). hence f is pairwise g∗ωα-irresolute. 2 5 (τi,τj)-g∗ωα-paracompact spaces we recall that, a collection ξ = {fλ : λ ∈ γ} of subsets of a space x is called a locally finite with respect to the topology τi, if for each x ∈ x there exists ux ∈ τi containing x and ux which intersects at most finitely many members of ξ. definition 5.1. a collection ξ = {fλ : λ ∈ γ} of subsets of a space x is called (τi,τj)-p-locally finite if for each x ∈ x there exist (τi,τj)-g∗ωα-open ux in x and ux intersects at most finitely many members of ξ. theorem 5.1. let ξ = {fλ : λ ∈ γ} be a collection of subsets of (x,τ1,τ2) then (a) ξ is (τi,τj)-g∗ωα-locally finite if and only if {(τi,τj)g∗ωα-cl(fλ) : λ ∈ γ} is (τi,τj)-p-locally finite. (b) if ξ is (τi,τj)-p-locally finite, then ∪(τi,τj)g∗ωα-cl(fλ) = (τi,τj)g∗ωα-cl(∪fλ). (c) ξ is locally finite with respect to the topology τi if and only if the collection {(τi,τj)g∗ωα-cl(fλ : λ ∈ γ)} is locally finite with respect to the topology τi. proof. (a) suppose ξ is (τi,τj)-p-locally finite. then for each x ∈ x, there exists (τi,τj)-g∗ωα-open set ux containing x, which meets only finitely many of the sets fλ, say fλ1,fλ2, ...,fλn . since fλk ⊆ (τi,τj)g ∗ωα-cl(fλk ) for each k = 1,2,...,n and ux meets (τi,τj)-g∗ωα-cl(fλ1 ), ..., (τi,τj)g ∗ωα-cl(fλn ). therefore g∗ωα-cl(fλ) where λ ∈ γ is (τi,τj)-p-locally finite. conversely, let x ∈ x. then there exists (τi,τj)-g∗ωα-open ux, which meets only finitely many of the sets (τi,τj)-g∗ωα-cl(fλn ), say (τi,τj)-g ∗ωα-cl(fλ1 ), ..., 230 pairwise paracompactness (τi,τj)g ∗ωα-cl(fλn ). then ux ∩ (τi,τj)-g∗ωα-cl(fλk ) 6= φ. let q ∈ ux and q ∈ (τi,τj)-g∗ωα-cl(fλk ), implies that for every (τi,τj)-g ∗ωα-open set vq, we have vq ∩fλk 6= φ. but, we have ux is (τi,τj)-g ∗ωα-open set containing q and so ux ∩fλk 6= φ for each k=1,2,...,n. thus ξ is (τi,τj)-p locally finite. (b) suppose ξ is (τi,τj)-p-locally finite, then ∪(τi,τj)-g∗ωα-cl(fλ) ⊆ (τi,τj)g∗ωαcl(∪fλ). on the other hand, let q ∈ (τi,τj)-g∗ωα-cl(∪fλ). then for every (τi,τj)g∗ωα-open set vq such that vq ∩(∪fλ) 6= φ. but from the hypothesis, there exists (τi,τj)-g∗ωα-open set uq such that uq meets only finitely many of the sets fλ, say fλ1,fλ2, ...,fλn . thus for each (τi,τj)-g ∗ωα-open set vq containing q, we have vq ∩ (∪fλk ) 6= φ where k=1,2,...,n. that is, for each q ∈ (τi,τj)-g ∗ωα-cl(∪fλk ), there exists h such that q ∈ (τi,τj)-g∗ωα-cl(fλh ). therefore q ∈ ∪(τi,τj)-g ∗ωαcl(fλ) and hence (τi,τj)-g∗ωα-cl(∪fλ) = ∪(τi,τj)-g∗ωα-cl(fλ). (c) suppose ξ is locally finite with respect to the topology τi, then for each x ∈ x there exists τi-open set ux which meets only finitely many set fλ, say fλ1,fλ2, ... ,fλn , but fλk ⊆ (τi,τj)-g∗ωα-cl(fλk ). then ux meets (τi,τj)-g ∗ωα-cl(fλ1 ), ..., (τi,τj)-g∗ωα-cl(fλn ). thus (τi,τj)-g ∗ωα-cl(fλ : λ ∈ γ) is locally finite with respect to the topology τi. conversely, let x ∈ x. then there exists τ1-open set ux which meets only finitely many of the sets (τi,τj)-g∗ωα-cl(fλ), that is (τi,τj)-g∗ωα-cl(fλ1 ), ..., (τi,τj)g∗ωα-cl(fλn ). let q ∈ ux and q ∈ (τi,τj)-g∗ωα-cl(fλk ) where k=1,2,...,n. then for each (τi,τj)-g∗ωα-open set vq containing q such that vq∩fλk 6= φ. but q ∈ ux and so ux meets only finitely many of the sets fλ. hence ξ is locally finite with respect to the topology τi. 2 lemma 5.1. let f : (x,τ1,τ2) → (y,σ1,σ2) be a function. then f is (τi,τj)g∗ωα-closed if and only if for every y ∈ y and u ∈ τ1o(x) which contains f−1(y) there exists v ∈ (τi,τj)-g∗ωα-open set in (y,σ1,σ2) such that y ∈ y and f−1(v ) ⊆ u. theorem 5.2. let f : (x,τ1,τ2) → (y,σ1,σ2) be (τi,τj)-g∗ωα-irresolute. if ξ = {fλ : λ ∈ γ} be a (τi,τj)-p-locally finite collection in y, then f−1(ξ) = {f−1(fλ) : λ ∈ γ} is (τi,τj)-p locally finite collection in x. theorem 5.3. let f : (x,τ1,τ2) → (y,σ1,σ2) be (τi,τj)-g∗ωα-continuous. if ξ = {fλ : λ ∈ γ} is (τi,τj)-p locally finite collection in y, then f−1(ξ) = {f−1(fλ) : λ ∈ γ} is locally finite collection with respect to the topology τi. definition 5.2. a non empty collection ξ = {ai, i ∈ i, an index set} is called a (τi,τj)-g∗ωα-open cover of a bitopological space (x,τ1,τ2) if x = ∪ai and ξ ⊆ τ1-g∗ωαo(x,τ1,τ2) ∪ τ2-g∗ωαo(x,τ1,τ2) and ξ contains at least one member of τ1-g∗ωαo(x,τ1,τ2) and one member of τ2-g∗ωαo(x,τ1,τ2). definition 5.3. a subset a of a bitopological space (x,τ1,τ2) is said to be (τi,τj)g∗ωα-compact if every cover of a by (τi,τj)-g∗ωα-open sets has a finite subcover. 231 p. s. mirajakar and p. g. patil example 5.1. let x = {m,n,p,q}, τ1 = {x,φ,{m},{m,n}} and τ2 = {x,φ, {m,n},{m,n,p},{m,p,q}}. let ξ = {{m},{m,n},{m,n,p},{m,p,q}} be a g∗ωα-open cover of (x,τ1,τ2). then (x,τ1,τ2) is (τi,τj)-g∗ωα-compact. definition 5.4. a set a of a bitopological space (x,τ1,τ2) is said to be (τi,τj)g∗ωα-compact relative to x if every (τi,τj)-g∗ωα-open cover of a has a finite subcover as a subspace. theorem 5.4. every (τi,τj)-g∗ωα-compact space is (τi,τj) compact. proof: let (x,τ1,τ2) be (τi,τj)-g∗ωα-compact. let ξ = {ai : i ∈ i} be (τi,τj) open cover of x. then x = ∪ai and ξ ⊆ τi ∪ τj, so ξ contains at least one member of τi and one member of τj. since, every τi-open set is τi-g∗ωα-open, we have x = ∪ai and ξ ⊆ τi-g∗ωαo(x) ∪ τj-g∗ωαo(x) and by definition ξ contains at least one member of τi-g∗ωαo(x,τ1,τ2) and one member of τjg∗ωαo(x,τ1,τ2). therefore ξ is (τi,τj)-g∗ωα-open cover of x. as x is (τi,τj)g∗ωα-compact then ξ has the finite subcover and hence x is (τi,τj) compact. theorem 5.5. if y is τi-g∗ωα closed subset of a (τi,τj)-g∗ωα-compact space (x,τ1,τ2) then y is τj-g∗ωα compact. proof: let (x,τ1,τ2) be (τi,τj)-g∗ωα-compact. let ξ = {ai : i ∈ i} be a τj-g∗ωα open cover of y. as y is τi-g∗ωα closed, y c is τi-g∗ωα open. also, ξ ∪ y c = y c ∪ {ai : i ∈ i} be a (τi,τj)-g∗ωα-open cover of x. since x is (τi,τj)-g∗ωα-compact, we have x = y c ∪a1 ∪ ...∪an, so y = a1 ∪ ....∪an. hence, y is τj-g∗ωα compact. theorem 5.6. let f : (x,τ1,τ2) → (y,σ1,σ2) be a (τi,τj) continuous, bijective and (τi,τj)-g∗ωα-irresolute. then the image of a (τi,τj)-g∗ωα-compact space under f is (τi,τj)-g∗ωα-compact. proof: let f : (x,τ1,τ2) → (y,σ1,σ2) be a (τi,τj) continuous surjective and (τi,τj)-g∗ωα-closed. let x be (τi,τj)-g∗ωα-compact. let ξ = {ai, i ∈ i} be a (τi,τj)-g∗ωα-open cover of y. then y = ∪ai and ξ ⊆ σ1-g∗ωαo(y ) ∪ σ2g∗ωαo(y ) and ξ contains at least one member of σ1-g∗ωαo(y ) and one member of σ2-g∗ωαo(y ). therefore x = f−1(∪(ai)) = ∪f−1(ai) and f−1(ξ) ⊆ τ1g∗ωαo(x) ∪ τ2-g∗ωαo(x) and f−1(ξ) contains at least one member of τ1g∗ωαo(x) and one member of τ2-g∗ωαo(x). therefore f−1(ξ) is the (τi,τj)g∗ωα-open cover of x. since x is (τi,τj)-g∗ωα-compact, we have x = ∪f−1(ai) for each i = 1,...,n, that is y = f(x) = ∪(ai), i=1,...,n. hence, ξ has the finite subcover. therefore y is (τi,τj)-g∗ωα-compact. definition 5.5. a bitopological space (x,τ1,τ2) is said to be (τi,τj)-g∗ωα-paracompact (pairwise g∗ωα paracompact) if every τi-open cover of x has a (τi,τj)-p-locally finite (τi,τj)-g∗ωα-open refinement. 232 pairwise paracompactness example 5.2. let x = {m,n,p}, τ1 = {x,φ,{m},{m,n}} and τ2 = {x,φ, {m},{p},{m,p}}. let ξ = {{m},{m,n}}. then the space (x,τ1,τ2) is (τi,τj) g∗ωα-paracompact. definition 5.6. let (x,τ1,τ2) be a bitopological space. then x is said to be: (a) (τi,τj)-g∗ωα-regular if for each τi-closed set f and x ∈ x there exist τig∗ωα-open set u and τj-g∗ωα-open set v such that x ∈ u and f ⊆ v . (b) (τi,τj)-g∗ωα-normal if there exist two disjoint τi-closed sets a and b , there exist disjoint (τi,τj)-g∗ωα-open sets u and v such that a ⊆ u and b ⊆ v . example 5.3. let x = {m,n,p}, τ1 = {x,φ,{m},{m,n}} and τ2 = {x,φ, {m},{p},{m,p}}. then (x,τ1,τ2) is (τi,τj)-g∗ωα-regular. example 5.4. let x = {m,n,p}, τ1 = {x,φ,{m},{n,p}} and τ2 = {x,φ, {m}}. then (x,τ1,τ2) is (τi,τj)-g∗ωα-normal. proposition 5.1. a space (x,τ1,τ2) is (τi,τj)-g∗ωα-regular if and only if for each τi-open set u and x ∈ u there exists v ∈ (τi,τj)-g∗ωαo(x) such that x ∈ v ∈ (τi,τj)g∗ωα-cl(v ) ⊆ u. theorem 5.7. every (τi,τj)-g∗ωα-paracompact pairwise hausdorff bitopological space is (τi,τj)-g∗ωα-regular. proof. let a be τi-closed set with x /∈ a. then for each y ∈ a, choose τi-open sets uy and hx such that y ∈ uy, x ∈ hx and uy ∩ hx = φ, that is x /∈ (τi,τj)g∗ωα-cl(uy). therefore the family u = {uy : y ∈ a}∪{x − a} is an τi-open cover of x and so it has a (τi,τj)-p-locally finite (τi,τj)-g∗ωα-open refinement say ρ. let v = {h ∈ ρ : h ∩a /∈ φ}, then v is a (τi,τj)-g∗ωα-open set containing a and (τi,τj)-g∗ωα-cl(v ) = ∪{(τi,τj)-g∗ωα-cl(h) : h ∈ ρ and h ∩a /∈ φ}. therefore u = x − (τi,τj)-g∗ωα-cl(v ) is a (τi,τj)-g∗ωα-open set containing x such that u and v are disjoint subsets of x. thus x is (τi,τj)-g∗ωαregular. 2 corolary 5.1. every (τi,τj)-g∗ωα-paracompact pairwise hausdorff bitopologcal space (τi,τj)-g∗ωα-normal. theorem 5.8. let (x,τ1) and (x,τ2) be two regular spaces. then (x,τ1,τ2) is (τi,τj)-g∗ωα-paracompact if and only if every τi-open cover ξ of x has a (τi,τj)-p locally finite (τi,τj)-g∗ωα-closed refinement say ρ. proof. necessity: let ξ be an τ1-open cover of x. then for each x ∈ x choose a member ux ∈ ξ. since (x,τ1) and (x,τ2) are τi-regular, there exists τi-open set vx containing x such that τi ⊆ cl(vx) ⊆ ux. therefore ψ = {vx : x ∈ x} is an τi-open cover of x and by hypothesis ψ has a (τi,τj)-p-locally 233 p. s. mirajakar and p. g. patil finite (τi,τj)-g∗ωα-refinement say ω = {wλ : λ ∈ γ}. consider the collection (τi,τj)g ∗ωα− ω = {(τi,τj)g∗ωα-cl(wλ) : λ ∈ γ} is a (τi,τj)-p-locally finite of (τi,τj)-g∗ωα-closed subsets of (x,τ1,τ2). since for every λ ∈ γ, (τi,τj)-g∗ωαcl(wλ) ⊆ (τi,τj)g∗ωα-cl(vx) ⊆ τ1-cl(vx) ⊆ ux for some ux ∈ ξ, therefore (τi,τj)-g∗ωα-cl(ω) is a refinement of ξ. sufficiency: let ξ be an τi-open cover of x and ψ be a (τi,τj)-p locally finite (τi,τj)-g∗ωα-closed refinement of ξ. then for each x ∈ x choose wx ∈ (τi,τj)g∗ωαo(x) such that x ∈ wx and wx intersects at most finitely many member of ψ. let σ be (τi,τj)-g∗ωα-closed (τi,τj)-p-locally finite refinement of ω = {wx : x ∈ x}. then for each v ∈ ψ, v 1 = x − h, where h ∈ σ and h ∩ v = φ. then {v 1 : v ∈ ψ} is a (τi,τj)-g∗ωα-open cover of x. now for each v ∈ ψ, let us choose uv ∈ ξ such that v ⊆ uv. hence the collection {uv ∩ v 1 : v ∈ ψ} is a (τi,τj)-p-locally finite (τi,τj)-g∗ωα-open refinement of ξ. thus (x,τ1,τ2) is (τi,τj)-g∗ωα-paracompact. 2 theorem 5.9. let a be a (i,j)regular closed subset of a bitopological space (x,τ1,τ2). then (a,τia,τja ) is (i,j)-g ∗ωα-paracompact. proof. let σ = {vλ : λ ∈ γ} is an τi-open cover of a in (a,τia,τja ). then for each λ ∈ γ, choose an uλ ∈ τi such that vλ = a ∩ uλ. then the collection ξ = {uλ : λ ∈ γ}∪{x −a} which is an τi-open cover of the (i,j)g∗ωα-paracompact space x and so it has a (i,j)-p-locally finite (i,j)-g∗ωα-open refinement say σ = {wδ : δ ∈ ∆}. but we have (i,j)−ro(x) ⊆ (i,j)−o(x), then the collection {a∩wδ : δ ∈ ∆} is a (i,j)-p-locally finite (i,j)-g∗ωα-open refinement of σ in (a,τia,τja ). 2 theorem 5.10. let f : (x,τ1,τ2) → (y,σ1,σ2) be (τ1,σ1) and (τ2,σ2) closed (τi,τj)-g∗ωα-irresolute surjective function such that f−1(y) is τi-compact in (x,τ1) for each y ∈ y . if (y,σ1,σ2) is (τi,τj)-g∗ωα-paracompact, then (x,τ1,τ2) is also (τi,τj)-g∗ωα-paracompact. proof. let ξ = {uλ : λ ∈ γ} be an τi-open cover of a bitopological space (x,τ1,τ2). then for each y ∈ y , ξ is an τi-open cover of the τi-compact subspace f−1(y). so there exist a finite subcover γy of γ such that f−1(y) ⊆ ∪uλ for each λ ∈ ξλ. let uλ = ∪uλ which is an τi-open in (x,τ1). as f is (τ1,σ1)closed, then for each y ∈ y there exists σ1-open set vy in y such that y ∈ vy and f−1(vy) ⊆ uλ. then the collection ψ = {vy : y ∈ y} is an σ1-open cover of the (τi,τj)-g∗ωα-paracompact space (y,σ1,σ2) and so it has a (τi,τj)p locally finite (τi,τj)-g∗ωα-open refinement say ω = {wγ : γ ∈ ∆}. as f is (τi,τj)-g∗ωα-irresolute, the collection f−1(ω) = {f−1(wγ) : γ ∈ ∆} which is an (τi,τj)-g∗ωα-open (τi,τj)-p-locally finite cover of (x,τ1,τ2) such that for each γ ∈ δ, f−1(wγ) ⊆ uy for some y ∈ y . then the collection {f−1(wγ)∩uγ : γ ∈ 234 pairwise paracompactness δ,λ ∈ γy} is an (τi,τj)-p-locally finite (τi,τj)-g∗ωα-open refinement of ξ. thus (x,τ1,τ2) is (τi,τj)-g∗ωα-paracompact. 2 conclusion the notions of sets and functions in topological spaces are extensively developed and used in many fields such as particle physics, computational topology, quantum physics. 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[14] m. sheik john and p. sundaram, g∗-closed sets in bitopological spaces, indian jl. pure appl. math., 35(1)(2004), 71-80. 236 kalman filters and arma models 41 kalman filters and arma models aniello fedullo1 abstract. the kalman filter is the celebrated algorithm giving a recursive solution of the prediction problem for time series. after a quite general formulation of the prediction problem, the contributions of its solution by the great mathematicians kolmogorov and wiener are shorthly recalled and it is showed as kalman filter furnishes the optimal predictor, in the sense of least squares, for processes which satisfy the linear models with a finite number of parameters, that are the arma models. 1. introduction: time series a time series, in our study, is considered like a finite part (a sample) of a single realization of a stochastic process. the fundamental problem of the anaysis of the time series is the following: given a time series, infer, al least in part, the characteristics of the process. remember that, in general, a stochastic process is characterized by the joint distributions of all the finite sub-families of its random variables. if there is no other information known on the process, or if no other hypothesis is made about it, the problem is unsolvable or perhaps hill-posed. conversely, if we limit ourselves to particular families of processes, the above-mentioned statistical inference is possible. in particular this happens for weak (second order) stationary, ergodic, invertible and with a gaussian residual processes (see papoulis 1965). in the following, unless otherwise indicated, we will consider only such processes. 2. the prediction problem the information on the process inferred from the time series allows to resolve problems particularly important for applications, such as those of prediction, filtration and control. in the present work we will concentrate our 1 dipartimento di matematica e informatica, università di salerno, italia. 42 attention on the first of the three. the problem of prediction is to estimate a future value x(t+m) (m>0) of a time series whose past values x(t), x(t-1) , …, x(t-n) are known. the estimation )(ˆ mtx + of x(t+m) will be a suitable function of these latter values which minimizes m(m):=e[( x̂ (t+m)-x(t+m))2]. it can be shown that in the absence of imposed constraints on the form of the aforementioned function, the preceding minimum problem has the solution given by the conditional expectation e[x(t+m)x(t),x(t-1),…,x(t-n)]. the calculation of the latter, in general, requires the knowledge of the joint distribution of the random variables involved, that is a very detailed knowledge of the process, one rather difficult to arrive at. observe that if the joint distributions are normal then the conditional expectation is linear, for which the so called predictor is written x̂ (t+m)=a0x(t) + a1x(t-1)+ … + anx(t-n) where a0, …, an are constants (dependent only on m and n) to determine. this is done in such a way to minimize m and in order to do so it suffices to know the autocovariance. if the process is gaussian then the linear predictor (2) is optimal, in the sense of least squares. otherwise the conditional expectation is not in general linear and, as said, we could not know how to calculate it easily. nevertheless, in such a case, we can restrict to the afore mentioned linear predictors and search among them for those which are optimal. the originators of the researches on linear predictors were kolmogorov and wiener in the early '40s. the publication of their works, however, was held up until 1949 because of military concerns (automatic poynting of anti-aircraft weapons and fire control). it became clear then that kolmogorov and wiener had resolved independently the same problem, using different techniques. they, as it was understood afterwards, related to different choises of the coordinates related up to the same geometric problem in hilbert space. furthemore, both the authors also studied the semi-infinite version of the problem (n → ∞), for the major simplicity of the mathematical treatment. that, nevertheless, can be used as a good approximation of the real case where n is large but finite. 43 3. recursive algorithms kolmogorov's and wiener's approaches to the problem of prediction has the advantage of furnishing an explicit expression of x(t+m), but a notable disadvantage is that it must be recalculated ex novo for different values, also consecutive, of m. this carries enormous complications in the computations, especially for real time applications. such problems were overcome by the development of recursive algorithms. these calculate the estimation at time n+1 by means of a simple correction of that at time n, with a notable saving of memory and of computation time. the two most famous recursive approaches are the one by box and jenkins and the one by kalman. both conducted to the optimal linear predictor in the sense of least squares, and both are applied only to the processes which satisfy the linear models with a finite number of parameters. 4. linear models with a finite numbers of parameters a) arma models let x(t) be a stochastic process which satisfies the properties indicated at the end of section 3 and which we assume, without loss of generality, has mean 0. let h(ω) be the spectral density of x(t); that is a non negative real function of ω which we will also assume to be a rational function of exp (i ω ). for the spectral factorization theorem there exist the polynomials a e c such that h(ω) = cost )(*)( )(*)( ωω ωω ii ii ecec eaea where a(z):= ;1:a ;:)(* , 0 00 == ∑∑ = − = p i ip i p i i i zazaza ai ∈ /r for every i 44 c(z):= ;1:c ;:)(* , 0 00 == ∑∑ = − = q i iq i q i i i zczczc ci ∈ /r for every i we observe that a(w) = 0 if and only if a(1/w)=0 , by which, in virtue of the fundamental theorem of algebra, we can choose a and c in such a way that their zeros are all in modulus greater than 1. then the process x(t) can be written as x(t)= )( )( β β c a e(t) where β is the shift operator defined as βx(t):= x(t-1) for every t and e(t) is a white noise, that is a noise made up of random variables with mean 0, variance σ2 and pairwise uncorrelated. in fact it is easy to verify that the right hand of (4) has the spectrum (3). multiplying both sides of (4) by c one has c(β)x(t)=a(β)e(t) called autoregressive moving average or arma model. b) models with state space they are described by equations of the type s(t+1)=fs(t)+gu(t)+w(t), x(t)=hs(t)+e(t) where s(t) is a n dimensional vector called state, and f, g, h are matrices of adequate dimensions, u(t) is the input (which is considered relevant in the problems of control; here we may assume to be 0), w(t) is the so-called noise of process, made up of random variables pairwise uncorrelated with covariance matrix e[w(t) w~ ( s) ]= δts r1, e(t) is the so-called noise of 45 mausure, also made up of random variables pairwise uncorrelated, with covariance matrix e[e(t) e~ (s)] = δts r2 ; between the two noises there is in general some correlation given by e[w(t) e~ (s)] = δts r12 . beyond this, one assumes that s(0) , the initial state, is a random vector independent of the future terms of noise, with mean s and covariance matrix ∏0 . observe that the model described above is general enough to include the multivariate case in which x(t) has a dimension greater than 1. even if it is possible to associate such a model to an arma model (eventually vectorial) and vice versa, the use of the state space is revealed more versatile and powerfull. 5. the kalman filter algorithm let return to the recursive algorithms of the preceeding section 3. box and jenkins approach can be regarded as a special case of the most general and most powerfull algorithm of the kalman filter(cfr. caines 1972). kalman's algorithm, based on the description of the linear model by means of the state space, lends itself to be extended to multivariate processes, with little additional strength, differently from box and jenkins approach. consequently in the following we only illustrate kalman's algorithm. we are considering for the model (6) the problem of estimating the state vector, given x(t) (and possibly u(t) ). let ŝ (t):=e[s(t)x(0),u(0), …,x(t-1), u(t-1)]. in the case the initial state and the involved noises are gaussian, the preceding estimation is obtained by the following recursive procedure (kalman filter): ŝ (0):=s(0) ŝ (t+1) = f ŝ (t)+gu(t)+k(t)[x(t)-h ŝ (t)]. the matrix k(t) is called kalman gain and is given by k(t):=[fp(t) h ~ + 12 ~ r ][ hp(t) h ~ +r2]-1 where the matrix p(t) is a solution of the riccati equation p(0):=π0 46 p(t+1)=fp(t) f ~ +r1-[fp(t) h ~ + 12 ~ r ][hp(t) h ~ +r2]-1[fp(t) h ~ + 12 ~ r ]. in general the algorithm, when the disturbances are not gaussian, does not furnish an estimation coinciding with the conditional expectation (7); but a minimum covariance estimation among those which are linear in x and u . finally we can note that kalman filter works also in the most general cases where all matrices are time-dependent; but a detailed study of this would be beyond the imposed limits of the present paper. bibliography box g.e.p. e g.m. jenkins (1970), time series analysis, forecasting and control, holden-day, san francisco. caines p.e. (1972), relationship between box-kenkins-astrom control and kalman linear regulator, proc. ieee, 119, 615--620. kalman r.e. (1960), a new approach to linear filtering and prediction problems, trans. asme j. basic engrg., ser. d, 82, 35--45. kalman r.e. e r.s. bucy (1961), new results in linear filtering and prediction problems, trans. asme j. basic engrg., ser. d, 83, 95--108. kolmogoroff a.n. (1941), interpolation und extrapolation von stationaren zufallingen folgen, bull. acad. sci. (nauk), u.s.s.r., ser. math., 5, 3--14. papoulis a. (1965), probability, random variables and stochastic processes, mc graw hill, new york. wiener n. (1949), the extrapolation, interpolation and smoothing of stationary time series with engineering applications, wiley, new york. ratio mathematica 29 (2015) 25-40 issn:1592-7415 fundamental hoop-algebras r. a. borzooei1, h. r. varasteh2, k. borna2 1 department of mathematics, shahid beheshti university, tehran, iran borzooei@sbu.ac.ir 2 faculty of mathematics and computer science, kharazmi university, tehran, iran varastehhamid@gmail.com, borna@khu.ac.ir abstract in this paper, we investigate some results on hoop algebras and hyper hoop-algebras. we construct a hoop and a hyper hoop on any countable set. then using the notion of the fundamental relation we define the fundamental hoop and we show that any hoop is a fundamental hoop and then we construct a fundamental hoop on any non-empty countable set. keywords: hoop algebras, hyper hoop algebras, (strong) regular relation,fundamental relations. 2000 ams subject classifications: 20n20, 14l17, 97h50, 03g25,06f35. doi:10.23755/rm.v29i1.20 1 introduction hoop-algebras are naturally ordered commutative residuated integral monoids were originally introduced by bosbach in [7] under the name of complementary semigroups. it was proved that a hoop is a meet-semilattice. hoop-algbras then investigated by büchi and owens in an unpublished manuscript [8] of 1975, and they have been studied by blok and ferreirim[2],[3], and aglianò et.al.[1]. the study of hoops is motivated by researchers both in universal algebra and algebraic logic.in recent years, hoop theory was enriched with deep structure theorems. many of these results have a strong impact with fuzzy logic. particularly, from the structure theorem of finite basic hoops one obtains an elegant short proof of 25 r. a. borzooei, h. r. varasteh, k. borna the completeness theorem for propositional basic logic(see theorem 3.8 of [1]) introduced by hájek in [13]. the algebraic structures corresponding to hájek’s propositional (fuzzy) basic logic, bl-algebras, are particular cases of hoops and mv-algebras, product algebras and gödel algebras are the most known classes of bl-algebras. recent investigations are concerned with non-commutative generalizations for these structures. hypersructure theory was introduced in 1934[15], by marty. some fields of applications of the mentioned structures are lattices, graphs, coding, ordered sets, median algebra, automata, and cryptography[9]. many researchers have worked on this area. the authors applied hyper structure theory on hyper hoop and introduced and studied hyper hoop algebras in [17]and[16]. in this paper, we investigate some new results on hoop-algebras and hyper hoop-algebras. we construct a hoop and a hyper hoop on any countable set. then using the notion of the fundamental relation we define the fundamental hoop. 2 preliminaries first, we recall following basic notions of the hypergroup theory from[10]: let a be a non-empty set. a hypergroupoid is a pair (a,�), where � : a × a −→ p(a) −{∅} is a binary hyperoperation on a. if associativity low holds, then (a,�) is called a semihypergroup, and it is said to be commutative if � is commutative. an element 1 ∈ a is called a unit, if a ∈ 1 � a ∩ a � 1, for all a ∈ a and is called a scaler unit, if 1�a = a�1 = {a}, for all a ∈ a. note that if b,c ⊆ a, then we consider b �c by b �c = ⋃ b∈b,c∈c (b� c). (see [10]) definition 2.1. [3] a hoop-algebra or briefly hoop is an algebra (a,�,→,1) of type (2,2,0) such that, (hp1): (a,�,1) is a commutative monoid and for all x,y,z ∈ a, (hp2): x → x = 1, (hp3): (x � y) → z = x → (y → z) and (hp4): (x → y) �x = (y → x) �y. on hoop a we define ”x ≤ y” if and only if x → y = 1. it is easy to see that ≤ is a partial order relation on a. definition 2.2. [17] a hyper hoop-algebra or briefly, a hyper hoop is a nonempty set a endowed with two binary hyperoperations �,→ and a constant 1 such that, for all x,y,z ∈ a satisfying the following conditions, (hha1) (a,�,1) is a commutative semihypergroup with 1 as the unit, (hha2) 1 ∈ x → x, (hha3) (x → y)�x = (y → x)�y, (hha4) x → (y → z) = (x�y) → z, (hha5) 1 ∈ x → 1, 26 fundamental hoop-algebras (hha6) if 1 ∈ x → y and 1 ∈ y → x then x = y, (hha7) if 1 ∈ x → y and 1 ∈ y → z then 1 ∈ x → z. in the sequel we will refer to the hyper hoop (a,�,→,1) by its universe a. on hyper hoop a, we define x ≤ y if and only if 1 ∈ x → y. if a is a hyper hoop, it is easy to see that ≤ is a partial order relation on a. moreover, for all b,c ⊆ a we define b � c iff there exist b ∈ b and c ∈ c such that b ≤ c and define b ≤ c iff for any b ∈ b there exists c ∈ c such that b ≤ c. a hyper hoop a is bounded if there is an element 0 ∈ a such that 0 ≤ x, for all x ∈ a. proposition 2.3. in any hyper hoop (a,�,→,1), if x � y and x → y are singletons, for any x,y ∈ a, then (a,�,→,1) is a hoop. then hyper hoops are a generalization of hoops and every hoop is a trivial hyper hoop. proposition 2.4. [17] let a be a hyper hoop. then for all x,y,z ∈ a and b,c,d ⊆ a, the following hold, (hha8) x�y � z ⇔ x ≤ y → z, (hha9) b �c � d ⇔ b � c → d, (hha10) z → y ≤ (y → x) → (z → x), (hha11) z → y � (x → z) → (x → y), (hha12) 1�1 = {1}. notations: let r be an equivalence relation on hyper hoop a and b,c ⊆ a. then brc, brc and brc denoted as follows, (i) brc if there exist b ∈ b and c ∈ c such that brc, (ii) brc if for all b ∈ b there exists c ∈ c such that brc and for all c ∈ c there exists b ∈ b such that brc, (iii) brc if for all b ∈ b and c ∈ c, we have brc. remark 2.5. it is clear that brc and crd imply that brd, for all b,c,d ⊆ a. definition 2.6. [17] let r be an equivalence relation on hyper hoop a. then r is called a regular relation on a if and only if for all x,y,z ∈ a, (i) if xry, then x�zry �z, (ii) if xry, then x → zry → z and z → xrz → y, (iii) if x → yr{1} and y → xr{1}, then xry. definition 2.7. [17] let r be an equivalence relation on hyper hoop a. then r is called a strong regular relation on a if and only if, for all x,y,z ∈ a, (i) if xry, then x�zry �z, (ii) if xry, then x → zry → z and z → xrz → y, 27 r. a. borzooei, h. r. varasteh, k. borna theorem 2.8. [17] let r be a regular relation on hyper hoop a and ar be the set of all equivalence classes respect to r, that is ar = {[x]|x ∈ a}. then ( a r ,⊗, ↪→, [1]) is a hyper hoop, which is called the quotient hyper hoop of a respect to r, where for all [x], [y] ∈ ar , [x]⊗ [y] = {[t]|t ∈ x�y} and [x] ↪→ [y] = {[z]|z ∈ x → y} theorem 2.9. [17] let r be a strong regular relation on hyper hoop a. then (ar ,⊗, ↪→, [1]) is a hoop which is called the quotient hoop of a respect to r. theorem 2.10. [4] let x and y be two sets such that |x| = |y |. if (y,≤,0) is a well-ordered set, then there exists a binary order relation on x and x0 ∈ x, such that (x,≤,x0) is a well-ordered set. lemma 2.11. [14] let x be an infinite set. then for any set {a,b}, we have |x ×{a,b}| = |x|. 3 constructing of hoops in this section, we show that we can construct a hoop on any non-empty countable set. lemma 3.1. let a and b be two sets such that |a| = |b|. if a is a hoop, then we can construct a hoop on b by using of a. proof. since |a| = |b|, there exists a bijection ϕ : a → b. for any b1,b2 ∈ b. we define the binary operations �b and →b on b by, b1 �b b2 = ϕ(a1 �a a2) and b1 →b b2 = ϕ(a1 →a a2) where b1 = ϕ(a1), b2 = ϕ(a2) and a1,a2 ∈ a. it is easy to show that �b and →b are well-defined. moreover, for any b ∈ b we define 1b as 1b = ϕ(1a). now, by some modification we can show that (b,�b,→b,1b) is a hoop.2 lemma 3.2. for any k ∈ n, we can construct a hoop on wk = {0,1,2,3, ...,k− 1}. proof. let k ∈ n. we define the operations ”�” and ”→”, on wk as follows, for all a,b ∈ wk, a� b= { 0 if a + b ≤ k −1, a + b−k + 1 otherwise a → b = { k −1 if a ≤ b, k −1−a + b otherwise 28 fundamental hoop-algebras now, we show that (wk,�,→,k −1) is a hoop, (hp1): since, + is commutative, hence � is commutative. now, we show that � is associative on wk. for all a,b,c ∈ wk, case 1: if a + b ≤ k −1 and b + c ≤ k −1, then (a� b)� c = (0)� c = 0 and a� (b� c) = a�0 = 0 and so (a� b)� c = a� (b� c). case 2: if a + b > k − 1 and b + c ≤ k − 1, since a + b + c ≤ 2(k − 1) and so a + b + c−k + 1 ≤ k−1, we get (a�b)�c = (a + b−k + 1)�c = 0. on the other hand, a� (b� c) = a�0 = 0 and then (a� b)� c = a� (b� c). case 3: if a+b > k−1 and b+c > k−1, then (a�b)�c = (a+b−k+1)�c and a�(b�c) = a�(b+c−k+1). if a+b+c ≤ 2k then (a�b)�c = a�(b�c) = 0 and if a + b + c > 2k then (a� b)� c = a� (b� c) = a + b + c−2k + 2. case 4: let a + b ≤ k −1 and b + c > k −1. this case is similar to the case 2. now, we have 0�k−1 = 0 and if 0 6= a ∈ wk, we have a+ (k−1) > k−1 and so a� (k −1) = a + k −1−k + 1 = a. then (k −1) is the identity of (wk,�) and so (wk,�,k −1) is a commutative monoid. (hp2): it is clear that, for all a ∈ wk, a → a = k −1. (hp3): let a,b,c ∈ wk. we show that (a� b) → c = a → (b → c). case 1: if a + b ≤ k −1 and a ≤ b ≤ c, then (a� b) → c = 0 → c = k −1 and a → (b → c) = a → (k −1) = k −1. hence, (a� b) → c = a → (b → c). case 2: if a + b ≤ k − 1 and a ≤ c < b, (a � b) → c = 0 → c = k − 1 and since k − 1 − b + c ≥ a, a → (b → c) = a → (k − 1 − b + c) = k − 1. hence, (a� b) → c = a → (b → c). case 3: if a + b ≤ k −1 and b ≤ a ≤ c, then (a� b) → c = 0 → c = k −1 and a → (b → c) = a → (k −1) = k −1. hence, (a� b) → c = a → (b → c). case 4: if a + b ≤ k −1 and b ≤ c < a, then (a� b) → c = 0 → c = k −1 and a → (b → c) = a → (k −1) = k −1. hence, (a� b) → c = a → (b → c). case 5: if a + b ≤ k −1 and c ≤ b ≤ a, then (a� b) → c = 0 → c = k −1. on the other hand since a+b ≤ k−1, we get a+b−c ≤ k−1, a ≤ (k−1−b+c) and a → (k−1−b+c) = k−1. then a → (b → c) = a → (k−1−b+c) = k−1. hence, (a� b) → c = a → (b → c). case 6: if a+b ≤ k−1 and c ≤ a < b, then (a�b) → c = 0 → c = k−1. on the other hand since a+b ≤ k−1, we get a+b−c ≤ k−1, a ≤ (k−1−b+c) and a → (k−1−b+c) = k−1. then a → (b → c) = a → (k−1−b+c) = k−1. hence, (a� b) → c = a → (b → c). case 7: let a + b > k − 1 and a ≤ b ≤ c. since a ≤ b ≤ c, we get a + b− c ≤ a ≤ k − 1 and so a + b − k + 1 ≤ c. then (a � b) → c = (a + b − k + 1) → c = k − 1. on the other hand, a → (b → c) = a → (k − 1) = k − 1. hence, (a� b) → c = a → (b → c). case 8: let a+b > k−1 and a ≤ c < b. since a ≤ c < b we get a+b−c ≤ b ≤ k−1 and so a+b−k +1 ≤ c. then (a�b) → c = (a+b−k +1) → c = k−1. on the other hand, since k − 1 − b + c ≥ c ≥ a, we get a → (b → c) = a → 29 r. a. borzooei, h. r. varasteh, k. borna (k −1− b + c) = k −1. hence, (a� b) → c = a → (b → c). case 9: let a+b > k−1 and b ≤ a ≤ c. since b ≤ a ≤ c, we get a+b−c ≤ a ≤ k−1 and so a+b−k +1 ≤ c. then (a�b) → c = (a+b−k +1) → c = k−1. on the other hand since k − 1 − b + c ≥ c ≥ a, we get a → (b → c) = a → (k −1− b + c) = k −1. hence, (a� b) → c = a → (b → c). case 10: let a + b > k − 1 and b ≤ c < a. since b ≤ c < a, we get a + b− c ≤ a ≤ k − 1 and so a + b − k + 1 ≤ c. then (a � b) → c = (a + b − k + 1) → c = k − 1. on the other hand a → (b → c) = a → (k − 1) = k − 1. hence, (a� b) → c = a → (b → c). case 11: if a+b > k−1 and c ≤ b ≤ a, then (a�b) → c = (a+b−k +1) → c and a → (b → c) = a → (k − 1 − b + c). hence, if a + b − c ≤ k − 1, then (a � b) → c = a → (b → c) = k − 1 and if a + b − c > k − 1, then (a� b) → c = a → (b → c) = 2k −2−a− b + c. case 12: if a+b > k−1 and c ≤ a < b, then (a�b) → c = (a+b−k + 1) → c and a → (b → c) = a → (k − 1 − b + c). hence, if a + b − c ≤ k − 1, then (a � b) → c = a → (b → c) = k − 1 and if a + b − c > k − 1, then (a� b) → c = a → (b → c) = 2k −2−a− b + c (hp4): now, we show that (a → b)�a = (b → a)� b, for all a,b ∈ wk. case 1: if a ≤ b, then (a → b) � a = (k − 1) � a = a and (b → a) � b = (k−1−b+a)�b = k−1−b+a+b−k+1 = a. hence, (a → b)�a = (b → a)�b. case 2: if a > b, then (a → b)�a = (k−1−a+b)�a = k−1−a+b+a−k+1 = b and (b → a)� b = (k −1)� b = b. hence, (a → b)�a = (b → a)� b. therefore, (wk,�,→,k −1) is a hoop.2 theorem 3.3. let a be a finite set. then there exist binary operations � and → and constant 1 on a, such that (a,�,→,1), is a hoop. proof. let a be a finite set. then, there exists k ∈ n such that |a| = |wk|. now, by lemma 3.2, (wk,�,→,1) is a hoop and so by lemma 3.1, there exist binary operations � and →, and constant 1 on a , such that (a,�,→,1) is a hoop.2 lemma 3.4. let 1 < n ∈ q. then there exist binary operations � and → on e = q∩ [1,n], such that (e,�,→,n) is a hoop. proof. for any 1 < n ∈ e, we define the binary operations � and → on e as follows, for all a,b ∈ e, a� b= { 1 if ab ≤ n, ab n otherwise a → b = { n if a ≤ b, nb a otherwise clearly, � and → are well-defined on e. now, we show that (e,�,→,n) is a hoop. 30 fundamental hoop-algebras (hp1): for all a ∈ e, if a 6= 1, since an > n we have a�n= n�a = an n = a and if a = 1, we have a�n = 1 �n = 1 = a. then n is the identity element of (e,�). now, we show that � is associative on e. let a,b,c ∈ e, case 1: if ab ≤ n and bc ≤ n, then (a� b) � c = 1 � c = 1. on the other hand a� (b� c) = a� (1) = 1. then (a� b)� c = a� (b� c). case 2: if ab ≤ n and bc > n, then (a� b) � c = 1 � c = 1. on the other hand b � c = bc n and then a � (b � c) = a � (bc n ). since abc n = ab n c ≤ c ≤ n, we get a� (b� c) = 1 and so (a� b)� c = a� (b� c). case3: if ab > n and bc > n, then (a � b) � c = (ab n ) � c. on the other hand a � (b � c) = a � (bc n ). if abc n ≤ n, then (a � b) � c = a � (b � c) = 1 and if abc n > n, then (a�b)�c = a� (b�c) = abc n2 . hence, (a�b)�c = a� (b�c). case 4: let ab > n and bc ≤ n. this case is similar to the case 2. it is clear that, for all a,b ∈ e, a�b = b�a. hence, (e,�,n) is a commutative monoid. (hp2): it is clear that, for all a ∈ e, we have a → a = n. (hp3): for all a,b,c ∈ e, we have the following cases, case 1: if b ≤ c and ab ≤ n, then a → (b → c) = a → n = n and (a � b) → c = 1 → c = n. then a → (b → c) = (a� b) → c. case 2: if b ≤ c and ab > n, then a → (b → c) = a → n = n and since a n < 1 , we get ab n < b ≤ c and so (a � b) → c = ab n → c = n. then a → (b → c) = (a� b) → c. case 3: if b > c and ab ≤ n, since ab ≤ n ≤ nc and so a ≤ nc b , then a → (b → c) = a → nc b = n. on the other hand, (a � b) → c = 1 → c = n. then a → (b → c) = (a� b) → c. case 4: if b > c and ab > n, then a → (b → c) = a → nc b and (a � b) → c = ab n → c. we have, a ≤ nc b if and only if ab n ≤ c, and so a → (b → c) = (a�b) → c. hp4: for all a,b ∈ e, we have the following cases, case 1: if a ≤ b, then a � (a → b) = a � n = an n = a and b � (b → a) = b� na b = bna bn = a and so a� (a → b) = b� (b → a). case 2: if a > b, then a � (a → b) = a � nb a = anb an = b and b � (b → a) = b�n = bn n = b and so a� (a → b) = b� (b → a). therefore, (e,�,→,n) is a hoop.2 theorem 3.5. let a be an infinite countable set. then there exist binary operations � and → and constant 1 on a, such that (a,�,→,1) is a hoop. proof. let a be an infinite countable set and e = q∩[1,n]. then by lemma 3.4, (e,�,→,1) is an infinite countable hoop and |a| = |e|. hence, by lemma 3.1, there exist binary operations � and → and constant 1, such that (a,�,→,1) is a hoop.2 corollary 3.6. for any non-empty countable set a, we can construct a hoop on a. 31 r. a. borzooei, h. r. varasteh, k. borna proof. let a be a non-empty countable set. then, a is a finite set, or an infinite countable set . then by the theorems 3.3 and 3.5, the proof is clear.2 4 constructing of some hyper hoops in this section first we show that the cartesian product of hoops is a hyper hoop and then we construct a hyper hoop by any non-empty countable set. theorem 4.1. let (a,�a,→a,1a) and (b,�b,→b,1b) be two hoops. then there exist hyperoperations �, → and constant 1 on a × b such that (a × b,�,→,1) is a hyper hoop. proof. for any (a1,b1),(a2,b2) ∈ a×b, we define the binary hyperoperations �, → on a×b by, (a1,b1)� (a2,b2) = {(a1 �a a2,b1),(a1 �a a2,b2)}, (a1,b1) → (a2,b2) = { {(a1 →a a2,b2),(a1 →a a2,1b)} if b1 = b2, {(a1 →a a2,b2)} otherwise and constant 1 = (1a,1b). it is easy to show that the hyperoperations are welldefined. now, we show that (a×b,�,→,1) is a hyper hoop. (hha1): since �a , is associative and commutative, we get � is associative and commutative. moreover, for all (a,b) ∈ a × b, we have (a,b) � (1a,1b) = {(a�a 1a,b),(a�a 1a,1b)} 3 (a,b). then (a×b,�,→,1) is a commutative semihypergroup with 1 as the unit, where 1 = (1a,1b). (hha2): for all (a,b) ∈ a×b, we have (a,b) → (a,b) = {(a →a a,b),(a →a a,1b)} = {(a →a a,b),(1a,1b)}3 (1a,1b) = 1 (hha3): for all (a1,b1),(a2,b2) ∈ a×b, we have the following cases, case 1: if b1 6= b2, then, ((a1,b1) → (a2,b2))� (a1,b1) = {(a1 → a2,b2)}� (a1,b1) = {((a1 → a2)�a a1,b1),((a1 → a2)�a a1, b2)} = {((a2 → a1)�a a2,b1),((a2 → a1)�a a2, b2)} = ((a2,b2) → (a1,b1))� (a2,b2) 32 fundamental hoop-algebras case 2: if b1 = b2, then, ((a1,b1) → (a2,b2))� (a1,b1) = {(a1 → a2,b2),(a1 → a2,1b)}� (a1,b1) = {((a1 → a2)�a a1,b1),((a1 → a2)�a a1, b2),((a1 → a2)�a a1,1b)} = {((a2 → a1)�a a2,b1),((a2 → a1)�a a2, b2),((a2 → a1)�a a2,1b)} = ((a2,b2) → (a1,b1))� (a2,b2) (hha4): for all (a1,b1),(a2,b2),(a3,b3) ∈ a×b, we have the following cases, case 1: if b1 = b2 = b3, (a1,b1) → ((a2,b2) → (a3,b3)) = (a1,b1) →{((a2 →a a3),b3),((a2 →a a3), 1b)} = {(a1 →a (a2 →a a3),1b),(a1 →a (a2 →a a3),b3)} = {((a1 �a a2) →a a3,1b),((a1 �a a2) →a a3),b3)} = ((a1,b1)� (a2,b2)) → (a3,b3) case 2: if b1 6= b2 = b3, (a1,b1) → ((a2,b2) → (a3,b3)) = (a1,b1) →{((a2 →a a3),b3),((a2 →a a3), 1b)} = {(a1 →a (a2 →a a3),1b),(a1 →a (a2 →a a3),b3)} = {(a1 �a a2) →a (a3,1b),((a1 �a a2) →a a3),b3)} = ((a1,b1)� (a2,b2)) → (a3,b3) case 3: if b1 = b2 6= b3, (a1,b1) → ((a2,b2) → (a3,b3)) = (a1,b1) →{((a2 →a a3),b3)} = {a1 →a (a2 →a a3),b3)} = {((a1 �a a2) →a a3,b3)} = ((a1,b1)� (a2,b2)) → (a3,b3) 33 r. a. borzooei, h. r. varasteh, k. borna case 4: if b1 6= b2 6= b3, (a1,b1) → ((a2,b2) → (a3,b3)) = (a1,b1) →{((a2 →a a3),b3)} = {(a1 →a (a2 →a a3),b3)} = {((a1 �a a2) →a a3,b3)} = ((a1,b1)� (a2,b2)) → (a3,b3) (hha5): for all (a,b) ∈ a×b, we have the following cases, case 1: if b = 1b, then (a,b) → (1a,1b) = {(a → 1a,1b),(a → 1a,b → 1b)} = {(1a,1b)}3 (1a,1b). case 2: if b 6= 1b, then (a,b) → (1a,1b) = {(a → 1a,1b)} = {(1a,1b)} 3 (1a,1b). (hha6): for all (a1,b1),(a2,b2) ∈ a×b, if (1a,1b) ∈ (a1,b1) → (a2,b2) and (1a,1b) ∈ (a2,b2) → (a1,b1), then we have the following cases, case 1: if b1 6= b2, then (1a,1b) ∈ {(a1 →a a2,b2)} and (1a,1b) ∈ {(a2 →a a1,b1)}. hence, 1a = a1 →a a2 and 1a = a2 → a1 and 1b = b1 = b2. since a is a hoop, we get a1 = a2 and so (a1,b1) = (a2,b2) case 2: if b1 = b2, then (1a,1b) ∈ {(a1 →a a2,b2),(a1 →a a2,1b)} and (1a,1b) ∈ {(a2 →a a1,b1),(a2 →a a1,1b)}. hence 1a = a1 →a a2 and 1a = a2 → a1. since a is a hoop, we get a1 = a2 and by assumption, we have b1 = b2. so (a1,b1) = (a2,b2). (hha7): for all (a1,b1),(a2,b2),(a3,b3) ∈ a × b, let (1a,1b) ∈ (a1,b1) → (a2,b2) and (1a,1b) ∈ (a2,b2) → (a3,b3). then we consider the following cases: case 1: if b1 = b2 = b3, then (1a,1b) ∈ {(a1 →a a2,1b),(a1 →a a2,b2)} and (1a,1b) ∈ {(a2 →a a3,1b),(a2 →a a3,b3)}. hence 1a = a1 →a a2 and 1a = a2 → a3. since a is a hoop, we get 1a = a1 →a a3. hence, (a1,b1) → (a3,b3) = {(a1 →a a3,b3),(a1 →a a3,1b)} = {(1a,b3),(1a,1b)}3 (1a,1b). case 2: if b1 6= b2 = b3, then (1a,1b) ∈ {(a1 →a a2,b2)} and (1a,1b) ∈ {(a2 →a a3,1b),(a2 →a a3,b3)}. hence 1a = a1 →a a2 and 1a = a2 → a3 and b2 = b3 = 1b. since a is a hoop, we get 1a = a1 →a a3. hence, (a1,b1) → (a3,b3) = {(a1 →a a3,b3)} = {(1a,1b)}3 (1a,1b). case 3: let b1 = b2 6= b3. then proof is similar to the case 2. case 4: if b1 6= b2 6= b3, then (1a,1b) ∈ {(a1 →a a2,b2)} and (1a,1b) ∈ {(a2 →a a3,b3)}. hence, 1a = a1 →a a2 and 1a = a2 → a3 and b2 = b3 = 1b. since a is a hoop, we get 1a = a1 →a a3. hence, (a1,b1) → (a3,b3) = {(a1 →a a3,b3)} = {(1a,1b)}3 (1a,1b). therefore,(a×b,�,→,1) is a hyper hoop, where 1 = (1a,1b).2 lemma 4.2. let a and b be two sets such that |a| = |b|. if (a,�a,→a,1a) is a hyper hoop, then there exist hyperoperations �b , →b and constant 1b on b, such that (b,�b,→b,1b) is a hyper hoop and (a,�a,→a,1a) ∼= (b,�b,→b,1b). 34 fundamental hoop-algebras proof. since |a| = |b|, then there exists a bijection ϕ : a → b . for any b1,b2 ∈ b, there exist a1,a2 ∈ a such that b1 = ϕ(a1) and b2 = ϕ(a2). then we define the hyperoperations �b,→b on b by, b1 �b b2 = {ϕ(a)|a ∈ a1 �a2}, and b1 →b b2 = {ϕ(a)|a ∈ a1 → a2}. it is easy to show that �b,→b are well-defined and (b,�b,→b,1b) is a hyper hoop, where 1b = ϕ(1a). now, we define the map θ : (a,�a,→a,1a) → (b,�b,→b,1b) by θ(x) = ϕ(x). since ϕ is a bijection then θ is a bijection and it is easy to see that θ is a homomorphism and so it is an isomorphism.2 corollary 4.3. for any non-empty countable set a and any hoop b, we can construct a hyper hoop on a×b. proof. by corollary 3.6, we can construct a hoop on a and by theorem 4.1, we can construct a hyper hoop on a×b.2 corollary 4.4. let a be an infinite countable set. we can construct a hyper hoop on a. proof. let a be an infinite countable set. then by corollary 3.6, we can construct a hoop on a. now, by theorem 3.3, for arbitrary elements x,y not belonging to a, we can define operations � and → on the set {x,y}, such that ({x,y},�,→) is a hoop. then by theorem 4.1, we can construct a hyper hoop on a×{x,y}. then by lemma 2.11 and 4.2, there exists a hyper hoop on a.2 5 fundametal hoops in this section we apply the β∗ relation to the hyper hoops and obtain some results. then we show that any hoop is a fundamental hoop. let (a,�,→,1) be a hyper hoop and u(a) denote the set of all finite combinations of elements of a with respect to � and →. then, for all a,b ∈ a, we define aβb if and only if {a,b}⊆ u, where u ∈ u(a), and aβ∗b if and only if there exist z1, ...,zm+1 ∈ a with z1 = a,zm+1 = b such that {zi,zi+1} ⊆ ui ⊆ u(a), for i = 1, ...,m (in fact β∗ is the transitive closure of the relation β). theorem 5.1. let a be a hyper hoop. then β∗ is a strong regular relation on a. proof. let aβ∗b, for a,b ∈ a. then there exist x1, ...,xn+1 ∈ a with x1 = a,xn+1 = b and ui ∈ u(a) such that {xi,xi+1} ⊆ ui, for 1 ≤ i ≤ n. let zi ∈ xi → c, for all 1 ≤ i ≤ n + 1,c ∈ a. then we have, {zi,zi+1}⊆ (xi → c)∪ (xi+1 → c) ⊆ ui → c ⊆ u(a), for all 1 ≤ i ≤ n. hence, z1β∗zn+1, where z1 ∈ a → c and zn+1 ∈ b → c and so a → cβ∗b → c. similarly, we can show that c → aβ∗c → b. now, by the same way we can prove 35 r. a. borzooei, h. r. varasteh, k. borna that aβ∗b implies a�cβ∗b�c, for all c ∈ a. hence, β∗ is a strong regular relation on a.2 corollary 5.2. let a be a hyper hoop. then ( a β∗ ,⊗, ↪→) is a hoop, where ⊗ and ↪→ are defined by theorem 2.8. proof. by theorem 2.9 the proof is clear.2 theorem 5.3. let a be a hyper hoop. then the relation β∗ is the smallest equivalence relations γ defined on a such that the quotient a γ is a hoop with operations γ(x)⊗γ(y) = γ(t) : t ∈ x�y and γ(x) ↪→ γ(y) = γ(z) : z ∈ x → y where γ(x) is equivalence class of x with respect to the relation γ. proof. by corollary 5.2, a β∗ is a hoop. now, let γ be an equivalence relation on a such that a γ is a hoop. let xβy, for x,y ∈ a and π : a → a γ be the natural projection such that π(x) = γ(x). it is clear that π is a homomorphism of hyper hoops. then there exists u ∈ u(a) such that {x,y} ⊆ u. since π is a homomorphism of hyper hoops, we get |π(u)| = |γ(u)| = 1. since {π(x),π(y)} ⊆ π(u) and |π(u)| = 1, we get π(x) = π(y) and so γ(x) = γ(y) i.e. xγy. hence, β ⊆ γ. now, let aβ∗b, for a,b ∈ a. then there exist x1, ...,xn+1 ∈ a, such that a = x1βx2, ...,βxn+1 = b. since β ⊆ γ, we get a = x1γx2, ...,γxn+1 = b. then since γ is a transitive relation on a, we get aγb and so β∗ ⊆ γ.2 corollary 5.4. the relation β∗ is the smallest strong regular relation on hyper hoop a. proof. the proof is straightforward.2 lemma 5.5. if a1 and a2 are two hyper hoops, then the cartesian product a1 × a2 is a hyper hoop with the unit (1a1,1a2) by the following hyperoperations, for (x1,y1),(x2,y2) ∈ a1 ×a2, (x1,y1)� (x2,y2) = {(a,b)|a ∈ x1 �x2,b ∈ y1 �y2}, (x1,y1) → (x2,y2) = {(a′,b′)|a′ ∈ x1 → x2,b′ ∈ y1 → y2} proof. the proof is straightforward.2 lemma 5.6. let a1 and a2 be two hyper hoops. then, for a,c ∈ a1 and b,d ∈ a2, we have (a,b)β∗a1×a2(c,d) if and only if aβ ∗ a1 c and bβ∗a2d. proof. we know that u ∈ u(a1 ×a2) if and only if there exist u1 ∈ u(a1) and u2 ∈ u(a2) such that u = u1 × u2. then (a,b)β∗a1×a2(c,d) if and only if there exist u1 ∈ u(a1) and u2 ∈ u(a2) such that {(a,b),(c,d)}⊆ u1×u2 if and only if {a,c}⊆ u1 and {b,d}⊆ u2 if and only if aβ∗a1c and bβ ∗ a2 d.2 36 fundamental hoop-algebras theorem 5.7. let a1 and a2 be two hyper hoops. then a1×a2β∗ a1×a2 ∼= a1 β∗ a1 × a2 β∗ a2 . proof. let ϕ : a1×a2 β∗ → a1 β∗ a1 × a2 β∗ a2 be defined by ϕ(β∗(x,y)) = (β∗a1(x),β ∗ a2 (y)), where β∗ = β∗a1×a2 by lemma 5.5, a1×a2 β∗ is well-define. it is clear that ϕ is onto. by lemma 5.6, we have β∗(x1,y1) = β∗(x2,y2) if and only if β∗a1(x1) = β ∗ a2 (x2) and β∗a2(y1) = β ∗ a2 (y2), for any (x1,y1),(x2,y2) ∈ a1 ×a2. so, ϕ is well defined and one to one. also, by considering the hyperoperations ⊗ and ↪→ defined in theorem 2.8, we have, ϕ(β∗(x1,y1) ↪→ β∗(x2,y2)) = ϕ({β∗(a,b)|a ∈ x1 → x2,b ∈ y1 → y2}) = {ϕ(β∗(a,b))|a ∈ x1 → x2,b ∈ y1 → y2} = {(β∗a1(a),β ∗ a2 (b))|a ∈ x1 → x2,b ∈ y1 → y2} = (β∗a1(x1) ↪→ β ∗ a1 (x2),β ∗ a2 (y1) ↪→ β∗a2(y2)) = (β∗a1(x1),β ∗ a2 (y1)) ↪→ (β∗a1(x2),β ∗ a2 (y2)) = ϕ(β∗(x1,y1)) ↪→ ϕ(β∗(x2,y2)) similarly, we can show that ϕ(β∗(x1,y1)⊗β∗(x2,y2)) = ϕ(β∗(x1,y1))⊗ϕ(β∗(x2, y2)). moreover, it is clear that ϕ(β∗(1a1,1a2)) = (β ∗(1a1),β ∗(1a2)). hence, ϕ is an isomorphism.2 corollary 5.8. let a1,a2, ....,an be hyper hoops. then, a1×a2×....×an β∗ a1×a2×....×an ∼= a1 β∗1 × a2 β∗2 × .......× an β∗n proof. the proof is straightforward.2 theorem 5.9. let a and b be two sets such that |a| = |b|. if (a,�a,→a,1a) is a hyper hoop, then there exist hyperoperations �b and →b and constant 1b on b such that (b,�b,→b,1b) is a hyper hoop and (a,�a,→a,1a) β∗ a ∼= (b,�b,→b,1b) β∗ b . proof. since |a| = |b|, then by lemma 4.2, there exist binary hyperoperations �b and →b, such that (b,�b,→b,1b) is a hyper hoop. moreover, there exists an isomorphism f : (a,�a,→a,1a) → (b,�b,→b,1b), such that f(1a) = 1b. now, we define ϕ : (a,�a,→a,1a) β∗ a → (b,�b,→b,1b) β∗ b by ϕ(β∗a(x)) = β ∗ b(f(x)). since f is an isomorphism, ϕ is onto. let y1 , y2 ∈ b. then there exist a1,a2 ∈ a such that b1 = f(a1) and b2 = f(a2). then β∗a(a1) = β ∗ a(a2) iff a1β ∗ aa2 iff there exists u ∈ u(a) such that {a1,a2} ⊆ u iff there existes f(u) ∈ u(b) : {f(a1),f(a2)} ⊆ f(u) iff β∗b(b1) = β ∗ b(b2) iff β∗b(f(a1)) = β ∗ b(f(a2)). then ϕ is well-defined and one to one. also, by consid37 r. a. borzooei, h. r. varasteh, k. borna ering the hyperoperations ⊗ and ↪→ defined in theorem 2.8, we have, ϕ(β∗a(a1)⊗β ∗ a(a2)) = ϕt∈a1�a2(β ∗ a(t)) = β ∗ t∈a1�a2(f(t)) = β∗t′∈f(a1�a2)(t ′) = β∗t′∈f(a1)�f(a2)(t ′) = β∗b(f(a1))⊗β ∗ b (f(a2)) = ϕ(β∗a(a1))⊗ϕ(β ∗ a(a2)) by the same way, we can show that ϕ(β∗a(a1) ↪→ β ∗ a(a2)) = ϕ(β ∗ a(a1)) ↪→ ϕ(β ∗ a(a2)) since f is an isomorphism, we get ϕ(β∗a(1a)) = β ∗ b(f(1a)) = β ∗ b(1b). hence, ϕ is an isomorphism.2 definition 5.10. let a be a hoop algebra. then a is called a fundamental hoop, if there exists a nontrivial hyper hoop b, such that b β∗ b ∼= a theorem 5.11. every hoop is a fundamental hoop. proof. let a be a hoop. then by theorem 4.1, for any hoop b, a × b is a hyper hoop. by considering the hyperoperations � and → defined in theorem 4.1, we get that any finite combination u ∈ u(a×b) is the form of u = {(a,xi)|a ∈ a,xi ∈ b}. hence, for any (a1,b1),(a2,b2) ∈ a×b, (a1,b1)β ∗(a2,b2) ⇔∃u ∈ u(a×b) such that {(a1,b1),(a2,b2)}⊆ u ⇔ a1 = a2 hence, for any (a,b) ∈ a×b, β∗(a,b) = {(a,x)|x ∈ b}. now, we define the map ψ : a×b β∗ → a by, ψ(β∗(a,b)) = a. it is clear that, β∗(a1,b1) = β ∗(a2,b2) ⇔ a1 = a2 ⇔ ψ(β∗(a1,b1)) = ψ(β∗(a2,b2)). then, ψ is well defined and one to one. in the following, we show that ψ is a homomorphism. for this we have, ψ(β∗(a1,b1)⊗β∗(a2,b2)) = ψ(β∗(u,v)) : (u,v) ∈ (a1,b1)� (a2,b2) = ψ(β∗(u,v)) : (u,v) ∈{((a1 �a2),b1),((a1 � a2),b2)} = {u|u ∈ a1 �a2} = a1 �a2 = ψ(β∗(a1,b1))�ψ(β∗(a2,b2)) 38 fundamental hoop-algebras and similarly, for the operation ↪→, we have the following cases, case 1: if b1 6= b2, then, ψ(β∗(a1,b1) ↪→ β∗(a2,b2)) = ψ(β∗(u,v)) : (u,v) ∈ (a1,b1) → (a2,b2) = ψ(β∗(u,v)) : (u,v) ∈{((a1 → a2),b2)} = {u|u ∈ a1 → a2} = a1 → a2 = ψ(β∗(a1,b1)) → ψ(β∗(a2,b2)) case 2:if b1 = b2, then, ψ(β∗(a1,b1) ↪→ β∗(a2,b2)) = ψ(β∗(u,v)) : (u,v) ∈ (a1,b1) → (a2,b2) = ψ(β∗(u,v)) : (u,v) ∈{((a1 → a2),b2),((a1 → a2),1b)} = {u|u ∈ a1 → a2} = a1 → a2 = ψ(β∗(a1,b1)) → ψ(β∗(a2,b2)) clearly, ψ(β∗(1a,1b) = 1a and ψ is onto. therefore, ψ is an isomorphism i.e. a×b β∗ ∼= a and so a is fundamental.2 corollary 5.12. for any non-empty countable set a, we can construct a fundamental hoop on a. proof. by corollary 3.6 and theorem 5.11 the proof is clear.2 references [1] p. aglianò, i. m. a. ferreirim, f. montagna, basic hoops: an algebraic study of continuous t-norms, studia logica, 87(1) (2007), 73-98. 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[9] p. corsini, v. leoreanu, applications of hyperstructure theory, advances in mathematics, kluwer academic publishers, dordrecht, (2003). [10] p. corsini,prolegomena of hypergroup theory. aviani editore tricesimo , (1993). [11] sh. ghorbani, a. hasankhani, e. eslami, quotient hyper mv-algebras, scientiae mathematicae japonicae online, (2007), 521-536. [12] sh. ghorbani, a. hasankhani, e. eslami, hyper mv-algebras, set-valued math, appl, 1, (2008), 205-222. [13] p. hájek, metamathematics of fuzzy logic, trends in logic-studia logica library, dordrecht/boston/london,(1998). [14] p. r. halmos, naive set theory, springer-verlag, new york, (1974). [15] f. marty, sur une generalization de la notion de groupe, 8th congress mathematiciens scandinaves, stockholm, (1934), 45-49. [16] h. r. varasteh, r. a. borzooei, fuzzy regular relation on hyper hoops, journal of intelligent and fuzzy systems, 30 (2016), 12751282.. [17] m. m. zahedi, r. a. borzooei, h. rezaei, some classifications of hyper kalgebras of order 3, scientiae mathematicae japonicae, 53(1) (2001), 133142. 40 ratio mathematica volume 42, 2022 abel fractional differential equations using variation of parameters method n. nithyadevi* p. prakash† abstract the variation of parameters method (vpm) is utilized throughout the research to identify a numerical model for a nonlinear fractional abel differential equation (fade). the approach given here is used to solve the initial problem of fractional abel differential equations. there is no conversion, quantization, disturbance, structural change, or precautionary concerns in the proposed method, although it is easy with numerical solutions. the measured values are graphed and tabulated to be compared with the numerical model. keywords: abel fractional differential equations; reimann-liouville fractional integral; reimann-liouville fractional derivative; variation of parameters method.1 *n.nithyadevi (research scholar, department of mathematics, periyar university, salem636011, india); nithyadevi84bu@gmail.com. †p. prakash (associate professor, department of mathematics, periyar university, salem636011, india); pprakashmaths@gmail.com. 1received on february 25th, 2022. accepted on june 20th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v41i0.723. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 135 n. nithyadevi and p. prakash 1 introduction abel differential equations is well established, perturbed of implementations in structure and function in dynamics, linear systems with stochastic, modeling technique, and linear algebra. numerous research has been done mostly on methods of the abel differential equations. we suggest several methods, including the iterative method of the abel differential equation achieved from the variation iteration method. we consider the following nonlinear fade: dβf(x) = p1f 3(x) + p2f 2(x) + p3f(x) + p4, 0 < β ≤ 1, 0 ≤ x ≤ r. (1) with the initial condition fk(0) = uk, k = 1, 2, 3, ....n − 1. (2) where p1, p2, p3 and p4 are arbitrary number, dβ is the fractional derivative for order β and f(x) is unknown function of the crisp variable x, k is an integer. however, assume ivps (0.2) and each x > 0 has a unique fuzzy solution. the vpm calculation is a novel numeric plan created to examine and decipher the arrangement of first and second request dubious ivps. computer simulation, biophysics, synthetic science, applied mathematics, geophysics, material science, harmonies, and other domains rely heavily on linear and non-linear fractional equations. this is seeing fractional derivatives must store the record of the parameter under deliberation. the proposed approach targets constructing an answer of a vpm development just as limiting remaining blunder capacities for processing the obscure coefficients of vpm by applying a specific differential administrator without linearly or constraint on the structure. again, we refer to see numerous qualities to show and reconsider some radical strategies for managing the various problems that occur in ordinary miracles. 2 preliminaries the definitions of significance and associated characteristics of the hypothesis are examined in this section. definition 2.1. the fractional component of riemann-liouville, f the valued function of the fuzzy number β is considered to include jβαf(x) = 1 γ(β) ∫ x 0 f(ξ) (x−ξ)1−β dξ , x > a where γ(β) is the famous gamma characteristic. definition 2.2. the riemann-liouville fractional order derivative β of the crisp function f almost everywhere on i exists and can be represented by rla d βf(x) = 1 γ(m−β) dm dxm ∫ x 0 f(ξ)(x − ξ)m−β−1dξ, where m − 1 ≤ β < m ∈ z+. 136 abel fractional differential equations using variation of parameters method definition 2.3. the modified riemann-liouville fractional order derivative β of the crisp function f almost everywhere on i exists and can be represented by rl a d βf(x) = 1 γ(m−β) dm dxm ∫ x 0 (f(ξ) − f(0))(x − ξ)m−β−1dξ, where m − 1 ≤ β < m, and m ≥ 1. remark 2.1. . (i) riemann-liouville derivative does not satisfy dαα(1) = 0 (d α α(1) = 0 for the caputo derivative), if α is not a natural number (ii) all fractionals do not satisfy the known formula of the derivative of product of the two functions: dαα(fg) = f(d α α(g)) + g(d α α(f)) (ii) all fractionals do not satisfy the known formula of the derivative of quotient of the two functions: dαα( f g ) = g(dαα(f))−f(dαα(g)) g2 3 variation of parameters method we consider the extensive expression to derive the main definition of the vpm l(u) + n(u) + r(u) = f(x), a ≤ x ≤ b (3) where l, n operators are linear and non-linear. r is a linear differential operator but l has the highest order than r, f(x) is a source term in the given domain [a, b]. we have the following equation solution by using the vpm u(x) = ∑k−1 l=0 pl+1x l l! + ∫ x 0 f(x, α)(−n(u)(α) − r(u)(α) + f(α))dα, (4) where k represent the order of given differential equation and cl where l = 1, 2, 3,are unknown. so u(x) = ∑k−1 l=0 pl+1x l l! (5) for homogeneous solution which is used by l(u) = 0. (6) 137 n. nithyadevi and p. prakash another component obtained from definition (0.2) from vpm is∫ x 0 f(x, α)(−n(u)(α) − r(u)(α) + f(α))dα. (7) where, f(x, α) is a lagrange multiplier which that eliminates the incremental use of integers in the inverse problem and is dependent on the order of equations. the explore description is based on calculating the function of the variable f(x, α) from either a set of numbers. f(x, α) = ∑k−1 l=0 (−1)l−1αl−1xk−1 (l−1)!(k−1)! = (x−α)k−1 (k−1)! , (8) the sequence of the specific differential equations differs with k. we have always had the following criteria to explore: k = 1, f(x, α) = 1, k = 2, f(x, α) = (x − α) k = 3, f(x, α) = x 2 2! + α 2 2! − αx (9) as a result, we utilize its investigation to improve, the system to solve equations un+1 = u0 + ∫ x 0 f(x, α)(−n(u)(α) − r(u)(α) + f(α))dα. (10) using initial conditions, we can obtain the initial guess u0(x). we improve our estimate by using a specific value for the input parameter in each iteration. we are using reimann-liouville to solve the fractional abel differential condition. when we combine vpm with a fractional integral for the arrangement process, the iterative plan for fractional equations is un+1 = u0 + 1 γ(β) ∫ x 0 f(x, α)x−α(−n(u)(α) − r(u)(α) + f(α))dα. (11) 4 numerical examples example 4.1. we consider the following fractional abel problem, dβf(x) − 3f3(x) + f(x) = 1, 0 < β ≤ 1, x > 0, (12) with initial condition f(0) = 1 3 can be found as follows: f(x) = 1√ 6e2x+3 . (13) for the above problem, we create the iterative scheme shown below fn+1 = f0 + 1 γ(β) ∫ x 0 (x − α)β−1(3f3(x) − f(x))dα (14) 138 abel fractional differential equations using variation of parameters method x exact value vpm absolute error 0.0 [0.3333] [0.2916] [0.0417] 0.1 [0.3111] [0.3062] [0.0049] 0.2 [0.2892] [0.2730] [0.0162] 0.3 [0.2679] [0.2531] [0.0148] 0.4 [0.2472] [0.2374] [0.0098] 0.5 [0.2275] [0.2164] [0.0111] 0.6 [0.2088] [0.1943] [0.0145] 0.7 [0.1912] [0.1804] [0.0108] 0.8 [0.1748] [0.1692] [0.0056] 0.9 [0.1595] [0.1439] [0.0156] 1.0 [0.1453] [0.1379] [0.0074] table 1: value of f(x), β = 1 x β = 0.7 β = 0.8 β = 0.9 0.0 0.3201 0.3197 0.2932 0.2 0.2817 0.2809 0.2782 0.4 0.2402 0.2397 0.2381 0.6 0.2007 0.1991 0.1962 0.8 0.1731 0.1706 0.1699 1.0 0.1497 0.1399 0.1388 table 2: different values of β taking initial condition f(0) = 1 3 , the following results for β = 1 are produced: f1(x) = 1 3 − 2 9 x f2(x) = 7 24 − 4 9 x + 4 81 x3 − 2 243 x4 f3(x) = 7 24 − 3049 4608 x + 5 96 x2 + 2 9 x3 − 133 1728 x4 − 881 3880 x5 + 13 729 x6 − 2 1701 x7 − 13 8748 x8 + 5 13122 x9 + 8 885735 x10 − 32 1948617 x11 + 4 1594323 x12 − 8 62178597 x13. table 1 show a approximate solution and exact solution for β = 1. table 2 shows different values of β. fig 1 represents the exact and approximate solution. 139 n. nithyadevi and p. prakash figure 1: value of f(x) example 4.2. we consider the following fractional abel problem, dβf(x) + f3(x) − f(x) = 1, 0 < β ≤ 1, x > 0, (15) with initial condition f(0) = 1 3 can be found as follows: f(x) = e t √ e2t+8 . (16) for the above problem, we create the iterative scheme shown below fn+1 = f0 + 1 γ(β) ∫ x 0 (x − α)β−1(f(x) − f3(x))dα (17) taking initial condition f(0) = 1 3 , the following results for β = 1 are produced: f1(x) = 1 3 + 8 27 x f2(x) = 31 96 + 16 27 x + 8 81 x2 − 64 2187 x3 − 128 19683 x4 f3(x) = 31 96 + 780193 884736 x+ 1045 3456 x2− 11201 93312 x3− 38591 419904 x4− 2657 157464 x5+ 784 177147 x6+ 3776 1594323 x7 − 2240 14348907 x8 − 100864 1162261467 x9 − 851968 52301766015 x10 − 131072 345191655699 x11 + 262144 847288609443 x12 + 2097152 99132767304831 x13. 140 abel fractional differential equations using variation of parameters method x exact value vpm absolute error 0.0 [0.3333] [0.3229] [0.0104] 0.1 [0.3639] [0.3508] [0.0131] 0.2 [0.3964] [0.3872] [0.0092] 0.3 [0.4307] [0.4139] [0.0168] 0.4 [0.4665] [0.4477] [0.0188] 0.5 [0.5035] [0.4972] [0.0063] 0.6 [0.5415] [0.5128] [0.0287] 0.7 [0.5799] [0.5524] [0.0275] 0.8 [0.6183] [0.6106] [0.0077] 0.9 [0.6561] [0.6392] [0.0169] 1.0 [0.6929] [0.6548] [0.0381] table 3: value of f(x), β = 1 x β = 0.7 β = 0.8 β = 0.9 0.0 0.3301 0.3299 0.3271 0.2 0.3956 0.3907 0.3882 0.4 0.4641 0.4596 0.4481 0.6 0.5364 0.5209 0.5197 0.8 0.6180 0.6159 0.6132 1.0 0.6877 0.6792 0.6674 table 4: different values of β table 3 shows a approximate solution and exact solution for β = 1. table 4 shows different values of β. fig 2 represents the exact and approximate solution. 141 n. nithyadevi and p. prakash figure 2: value of f(x) 5 conclusions in this study, we examined the formulation to fractional abel equation with the variation of parameters method. we demonstrate that vpm is a functional, efficient approach for the achievement of empirical and numerical testing for a wide variety of nonlinear fractional equations. we discovered that obtained estimate effects for different beta values interact simultaneously before the first-order derivative is exceeded. 6 references [1] a. khastan, j. j. nieto and r. rodriguez-lopez,” schauder fixed-point theorem in semilinear spaces and its application to fractional differential equations with uncertainty”, fixed point theory and applications, 21(2014). 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[14] j. a. nanware and d. b. dhaigude, existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions, j. nonlinear sci. appl. 7 (2014), 246–254. 143 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 47, 2023 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices roberta martino* viviana ventre† giacomo di tollo+ abstract due to the major crises of the past decade, the need to introduce consumer profiling operations to protect individuals from committing superficial business transactions has been realized in 2014/65/eu directive. the present paper investigates how consumer behavioral attitudes influence the decision-making process so that the choice results, from a normative point of view, non-rational in the context of intertemporal choice. in addition, the particular focus by european institutions on closing the gender gap in the economic and financial sector motivated this research to enrich the analysis with gender assessments. the study of the relationship between cognitive characteristics and consumer decision-making are deepened with a multidisciplinary approach involving mathematics, behavioral finance, temperament theory and multi-criteria analysis. the results of an experimental investigation confirm that there is not a better temperament or a more adept gender in economic and financial choices. keywords: analytic hierarchy process, behavioral anomalies, economic behavior, gender analysis, impatience, intertemporal choice, personalized strategies. 2010 ams subject classification: 91b08.‡ * dept. of mathematics and physics university of campania “l. vanvitelli” viale a. lincoln, 5 – 81100 caserta, italy; roberta.martino@unicampania.it . corresponding author. 0000-0002-2195-8715. † dept. of mathematics and physics university of campania “l. vanvitelli” viale a. lincoln, 5 – 81100 caserta, italy; viviana.ventre@unicampania.it 0000-0001-5314-5770. + dept. of law, economics, management and quantitative methods (demm),university of sannio, benevento, italy; giacomo.ditollo@unisannio.it. 0000-0001-7044-6014. ‡ received on february 18, 2023. accepted on may 1, 2023. published online on may 10, 2023. doi:10.23755/rm.v39i0.1137. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 408 mailto:roberta.martino@unicampania.it mailto:giacomo.ditollo@unisannio.it roberta martino, viviana ventre and giacomo di tollo 1. introduction theoretical and empirical investigation of the mechanisms that guide decisionmaking has become necessary to improve the functioning of economic and financial markets. the directive 2014/65/eu of the european parliament and of the council of 15 may 2014 on markets in financial instruments stresses, due to shortcomings in the functioning and transparency of financial markets, the need to protect investors and to regulate financial instruments. therefore, following the major crises of the last decade, the creation of investor profiles was necessary to protect savers and to offer products that fit the client profile. in addition to customer profiling, the eu aims to promote equality between men and women in all social, economic, and political activities. hebert simon (1982) was the first to point out that individuals have limited cognitive resources that unconsciously lead to a simplification of the decision-making problem. assuming that the decision-maker is rationally limited, research is turning to the personalization of decision-making strategies. in this regard, the study and analysis of the errors that characterize the dynamics of individual decisions must be conducted through personality theory. in addition, studies of personal financial skills have revealed an important gender gap (goldsmith and goldsmith, 2006) both in terms of approach with respect to investment strategies (barber and odean, 2001) and knowledge of financial topics (hira and mugenda, 2000). to improve the quality of women's choices and to reduce the gap, it is necessary to accept the existence of the many differences between the male and female genders and to investigate their reasons and consequences (fonseca et al., 2012). the present paper proposes a practical application of the multidisciplinary approach presented in martino and ventre (2022) addressed to two complementary purpose: the first purpose is to understand the extent to which emotional, cognitive and informational factors, which intervene during the evaluation and selection phase of an alternative, weigh on the economic decision-making process; the second aim is to investigate the effects and differences between the genders to promote the implementation of strategies aimed at closing the gender gap in all areas such as employment, social, political and economic. the disciplines involved in the present work are: mathematical finance for quantifying consumer cognitive and emotional biases; temperament theory for projecting characteristics of individuals into identifying classes; personalized finance for explaining relationships between bias and behavioral attitudes; and multicriteria analysis for quantitatively comparing class and gender characteristics. the multi-criteria decision support technique used is the analytical hierarchical process (ahp) (saaty, 1980) because it allows to compare elements difficult to compare directly, as shown in ventre et al. (2022, c) and ventre et al. (2023). in fact, its mathematical structure makes it possible to break down the analysis of choice into all the components that weigh on the decision. the originality of the present paper consists of an extension of ventre et al. (2022, c) with gender analysis and in give to it an applicable context of the multidisciplinary approach used. the analysis conducted by ahp proves that it is not possible to define an optimal strategy for everyone because of different behavioral characteristics. therefore, consultants and automated devices that provide digital advice must consider the importance of the decision-maker’s personality to whom service is provided. with respect to gender analysis, the ahp makes it clear that there is no privileged gender but there are only different attitudes, mostly complementary inclinations dictated by the known “distance between mars and venus” (del giudice et al., 2012). the paper is structured as follows: in section 2, a literature review about gender 409 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices analysis will be briefly exposed; in section 3 the main results about anomalies in intertemporal choice will be shown and discussed in terms of emotional impulses; in section 4, the structure of the ahp will be constructed with reference to keirsey's temperament theory (1998); in section 5 the experimental part aimed at constructing the weight of anomalies for each temperament, both in general and by gender, will be implemented; finally, there will be a discussion section and a conclusion section with ideas for future development. 2. gender differences many differences between the behavior of men and women were observed in various domains leading to the hypothesis that they were caused by different preferences between the genders (croson and gneezy, 2009). most theoretical and empirical work analyses gender difference with a comparison between the environment in which the individual develops (eagly and wood, 1999) and evolutionary origins (geary, 2010). in this regard, two main theories can be pointed out: the sociocultural theory (eagly and wood, 1999), which focuses the reasons for gender difference on a historical division of labour between men and women; the cognitive theory of social learning (bussey and bandura, 1999), on the other hand, which focuses gender differences with children's internalization of existing social norms. even in the workplace, gender discrimination is fueled by stereotypes and organizational factors. the study conducted by bobbit-zeher (2011) proves how the combination of cultural, structural and organizational factors, together with gender stereotypes, can have a strong impact on gender discrimination. for example, with respect to financial knowledge, chen and volpe's study (2002) proved that women are actually even less interested in learning about financial topics and that, also because of social stereotypes, female students are less oriented than their male peers towards financial studies (hawash et al., 2020). these differences could lead to women being more exposed to financial problems. however, the studies in the literature on social preferences remain incomplete if the interaction between gender and intra-gender variability is not considered (thoni and volk, 2021). for this reason, gender differences in preferences have been investigated to deepen social and economic outcomes (croson and gneexy, 2009) and they are a good starting point for understanding gender differences in many areas such as education, the labor market and financial decision-making (buser et al., 2014). the first differences are found in social preferences that express how individuals interact with others. the study conducted by kamas and preston (2015) not only proves that gender differences in economic behavior can be deepened and understood by social preferences, but also proves important findings. first, women tend to avoid inequality and this aversion to inequity could be linked to the fact that in trust games they send less. in addition, women are more inclined to charitable activities and equal pay, a preference that can be explained by inequity aversion and lower self-confidence. some studies also prove that women are more altruistic than males, less present-oriented and reliable (horn et al., 2022). a determining factor for social preferences is also age as proven by a study conducted on swedish and austrian children and adolescents (martinsson et al., 2011). risk preferences are another of the most studied factors for which it is possible to conclude that men are more risk-prone than women. this conclusion confirms previous results of an economic nature (eckel and grossman, 2008) and psychological nature (brody, 2022). in this regard, studies focusing on the portfolio selection problem have also provided direct evidence of gender differences. for example, graham et al. (2002) have proven 410 roberta martino, viviana ventre and giacomo di tollo that men invest in pension planning in a less conservative way than women do, and in the domain of financial risk, single women are less risk prone than single man (sunden et al., 1998; finucane et al., 2013). a study conducted by jianakoplos (2002) on gender differences in risk preferences emphasizes that the many differences must be observed and considered by advisors to make appropriate recommendations as risk appetite has a decisive impact on the construction of an asset allocation. precisely because risk tolerance is a key element in determining appropriate financial advice, some studies have focused on factors that have a strong impact on measurement. the study conducted by marinelli et al. (2017) on 2,374 investors proves a relationship between low literacy, income, and economic behavior with inconsistency among the most used risk tolerance metrics. therefore, the analysis of the gender difference in this respect must also incorporate the factors of the individual subjective structure. this gender gap is very evident, and many explanations have been proposed. a first explanation is referred to the concept of "risk as a feeling" developed by loewenstein (2001) referring to the reaction that instinctively develops in a risky decision-making context and from a psychological point of view, women feel emotions more intensely than men. moreover, in emotional terms, the difference between males and females is not only the intensity of the emotions they feel, but also the emotions felt in the same situation are different (grossman and wood, 1993). in this regard, lerner et al. (2003) have shown that different emotions lead to different attitudes, explaining the different approach to risk. the bias of overconfidence is the second reason that could explain the gender differences because even if overconfidence is shared by both, men are more overconfident than women (lundeberg, 1994). in particular, niederle and vesterlund (2007) have shown that men's more insistent overconfidence than women's leads to men being more convinced of their victories so that everything is seen as a competition in which the belief to win is greater and consequently so is the risk attitude. the way in which the risk situation is interpreted can also be another determining factor in investigating the underlying causes of these differences. while men perceive a risky situation as a challenge, women perceive the context as a threat (arch, 1993). this implies that the gender difference does not consist of different abilities with respect to the decision-making environment but rather of different motivations driving both genders. at the same time, factors that may be motivating for men may be harmful to women (block, 1983). confirming the hypothesis that the gender gap is not a lack of aptitude are numerous studies looking at a sample of professionals for whom the difference between men and women is largely small and almost insignificant (dwyer et al., 2002; master and maier, 1988). biological differences, such as different testosterone concentrations, also play an important role in this context (sapienza et al., 2009). the studies outlined so far emphasize that to promote gender equality and achieve the main goals proposed by the european union, it is not necessary to reduce diversity but to turn gender inequality into a strength for any sector. zimmer (1988) suggests that theories that neutralize the gender gap, such as tokenism used to create the illusion of inclusiveness (hogg and vaughan, 2008), have a limitation that may even hinder inclusion and gender equality. in fact, in roles where approach and behavior are essential, inattention to gender differences can generate employment bottlenecks. organizations can therefore reap many benefits from diversity. indeed, not only does diversity enable them to reach different markets and customers, but it also promotes innovation and better performance (nair and vohra, 2015) some empirical evidence shows that diversity has its advantages (herring, 2009). in addition, a recent study conducted by shrader et al. (2020) also proves a relationship between the percentage of women at the general management level and better financial performance. in conclusion, to promote gender equality, the classic factors used to argue 411 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices gender difference, such as biological, intellectual, emotional, and behavioral aspect, should not be regarded with a negative connotation as caused by prejudice against women (benschop and verloo, 2011). fletcher (2001), with an original and alternative point of view, conceptualizes the main characteristics of the female gender (empathy, vulnerability, connection, relational skills) as strengths and not weaknesses. from a strategic point of view, bridging the gap requires the use of continuous innovation processes (benschop and verloo, 2011), based on the combination of gender mainstreaming and intersectionality. intersectionality refers to the evaluation of each element or trait of an individual as the intrinsic union of all the elements that characterize her identity (palczewski and defrancisco, 2014). however, holvino (2010) noted that intersectionality is not enough acknowledged by organization studies and gender change projects in organizations must consider the implications of other factors as class, ethnicity, sexuality, and nation. 3. intertemporal choice and impatience most of the decision-making problems we face throughout our lives can be traced back to the typology of intertemporal choices, i.e., those whose consequences manifest only over time. this argument is particularly complex mainly for two reasons: the first one is that, deciding to get a result in the future necessarily implies renouncing to something in the present (noor, 2011); the second aspect concerns the difficulty of keeping the coherence constant in future objectives (sayman and onculer, 2009). intertemporal choices are studied in various fields such as economics and psychology skylark et al.(2021) and read et al. (2003) state that “the goal of research into intertemporal choice is to understand how these choices are made, and how they should be made”. the discounted utility model (samuelson, 1937,1952) assumes that the decision-making process of individuals is based on associating a level of utility to each alternative and then choosing the alternative with greater utility. the model introduced by samuelson has some weaknesses when it refers to the effective decision-making behavior. with the term anomalies, we refer to those empirical results that are difficult to rationalize. to achieve the objective of customer profiling, the concept of investor’s impatience will be used to measure the weight of anomalies. in the context of intertemporal choice, the impatience is defined as “the amount of money that the agent is willing to lose in exchange for anticipating the availability of a $1 reward” (cruz rambaud and muñoz torrecillas, 2016). this analysis is inspired by the experimental method proposed by ventre et al. (2022, b) in which, after demonstrating a correspondence between the anomaly and the alteration of the degree of impatience, the authors investigate the emotional drives of choice using the value of the hyperbolic factor (rohde, 2010). comparing the impatience of different discount functions, a relationship emerges between the speed with which the function varies and the degree of investor impatience: the faster the discount function decreases, the greater the associated impatience (cruz rambaud and muñoz torrecillas, 2016). prelec (2004) proved that a decreasing degree of impatience has a significant effect on the selection of suboptimal choices and that variation in impatience underlying the anomalies in the discounted utility model can be seen as a reflection of the decision-maker's bounded rationality. the originality of the present work lies in the fact that the analysis is not limited to the study of abnormal attitudes but deepens their properties by investigating behavioral characteristics in a context of gender to enrich the motivations presented about gender gap and financial anomalies. in practice, by quantifying the weight of emotional factors driving decision-making, it is possible to measure the different impact of cognitive and 412 roberta martino, viviana ventre and giacomo di tollo emotional biases between men and women. for these reasons, the analysis submitted by ventre et al. (2022, b) of anomalies will be enriched by behavioral classification and gender analysis. 3.1 economic model from an operational perspective, assuming that the decision-makers' preferences are continuous, monotonous, and impatient, the study of intertemporal choices is based on the model of discounted utility (samuelson 1937,1952). fixed x= 𝑅𝑛 the set of available results and t=𝑅+ the set of points of time : • the n-pla (𝑥0, 𝑡0; … ; 𝑥𝑛 , 𝑡𝑛) ∈ (𝑋 ⨯ 𝑇) 𝑛 indicates an intertemporal prospectus: if the prospectus is accepted, the decision-maker will receive the result 𝑥𝑖 at the time 𝑡𝑖 and a null result for each 𝑡𝑗 with j ≠ i; • the intertemporal utility defined on the prospectus is given by ∑ 𝑓(𝑡𝑖 )𝑈(𝑥𝑖 ) 𝑛 𝑖=0 where 𝑈(𝑥𝑖 ) is the cardinal utility function associated with the consumption of the outcome 𝑥𝑖 at the time 𝑡𝑖 and 𝑓(𝑡𝑖 ) is the individual discount function. the time lag between the decision and review of the result creates a risk due to uncertainty. the individual perception of this indeterminacy is enclosed in the discount rate 𝜌(𝑡) = − 𝑓′(𝑡) 𝑓(𝑡) that indicates “the proportional variation of f in a standard period” (read et al., 2003). empirical evidence demonstrates that people violate the discountedutility theory systematically (rao and li, 2011). in fact, the first formulation of the model hypothesized a constant discount rate over time but, due to the systematic deviation from the exponential model predictions, alternative functions were formulated to provide a more accurate empirical description. the main difference between the linear, exponential, and hyperbolic models in table 1 is that, while exponential and linear functions present a constant discount rate for all future periods, hyperbolic models manifest a discount rate varying over time. linear 𝑓(𝑡) = 1 − 𝑟𝑡 exponential 𝑓(𝑡) = ( 1 1 + 𝜌 ) 𝑡 hyperbolic generalized quasi-hyperbolic one parameter 𝑓(𝑡) = (1 + 𝛼𝑡) − 𝛽 𝛼 𝑓(𝑡) = 𝛽 ( 1 1 + 𝜌 ) 𝑡 𝑓(𝑡) = ( 1 1 + 𝜌𝑡 ) table 1. common discount functions in intertemporal choice reported by read et al. (2003). to understand the consequences of this aspect from a rational choice perspective, this study considers two outcomes: the first one is minor and anticipated (ss), and the second one is major and delayed (ll). a constant discount rate implies that, whatever the individual's choice, her preference remains stable over time in line with the profile of a 413 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices rational investor assumed by classical finance. empirically, however, since the imminent results appear disproportionately attractive, the decision-maker reverses her preferences, making the choice inconsistent as shown in figure 1. in fact, with the term temporal inconsistency, we refer to a situation in which a subject varies her preferences over time. figure 1. comparison between ss and ll. the images (a) and (b) describe different situations in which choices are temporally consistent, i.e., preferences remain constant over time. the image (c) shows instead as a variation of the course of the curve involves an inversion of the preferences and therefore temporally inconsistent choices. prelec (2004), analyzing the psychological characteristics underlying inconsistency, proved equivalence between the selection of dominated prospects, i.e. those outcomes that are not optimal from any temporal viewpoint , and a degree of impatience that decreases over time. in particular: definition 1. the preference ≥ shows a decreasing impatience (di) if ∀𝜎 > 0, 𝑦 > 𝑥 > 0, (𝑥, 𝑠)~ (𝑦, 𝑡) we have (𝑥, 𝑠 + 𝜎) ≤ (𝑦, 𝑡 + 𝜎) (strictly decreasing if (𝑥, 𝑠 + 𝜎) < (𝑦, 𝑡 + 𝜎). definition 2. an intertemporal prospect (x, s) dominates (y, t) with s<t, i.e. (x, s) >> (y, t), if (x, s) ≥ (y, s) and (x, s) > (x, t). prelec (2004), studying dominance violation on two-step decision problems, describes the following attitudes, shown in figure 2 and figure 3: definition 3. the preference ≥ allows a sophisticated two-step violation of dominance at the time (t, s, r, τ) if the relationship is di and ∃ x, y: today (0): (x, t+τ) >> (x, r) ≥ (y, s+τ) tomorrow (τ): (x, t) < (y, s) figure 2. sophisticated decision-maker (dm) predicts the selection of the lower outcome at time τ and commits in advance to the best available outcome at time 0 (prelec 2004). 414 roberta martino, viviana ventre and giacomo di tollo definition 4. the preference ≥ allows a naïve two-step violation of dominance at the time (t,s,r, τ) if the relationship is di and ∃ x, y: today (0): (y, s+τ) ≥ (x, r) >> (x, t+τ) tomorrow (τ): (y, s) < (x, t) figure 3. naïve decision-maker (dm) at time 0 the dm prefers the most considerable available outcome at time τ, but his preferences change, and the choice will fall on the lower outcome (prelec, 2004). such attitudes are not only related to a greater degree of decrease in impatience but also to a greater convexity of the discount function. applying the previous definitions to infinitesimal intervals, it is possible to prove that the greater the degree of di, the more significant the difference between the rates of impatience and temporal preference. the latter means that the decrease in impatience quantitatively represents the difference between preferring an event to occur and preferring an event to occur sooner. moreover, the expression that prelec (2004) proposes for calculating the degree of di is equivalent to the speed with which the discount rate varies over time, determining a link between the subjective perception of time and impatience. in particular, the more distorted this perception is, the more hyperbolic will be the trend of the discount rate. a direct relationship between subjective perception of time and temporal inconsistency has been proven in ventre and martino (2022) and ventre et al. (2022, a). the tool we exploit in the experimental phase to calculate the degree of di is the hyperbolic factor formalized by rohde (2010). definition 5. consider (𝑥, 𝑠)~(𝑦, 𝑡) (𝑥, 𝑠 + 𝜎)~(𝑦, 𝑡 + 𝜏) s<t, x<y, τ>0. the hyperbolic factor is: 𝐻(𝑠, 𝑡, 𝑦, 𝜏) = 𝜏 − 𝜎 𝑡𝜎 − 𝑠𝜏 theorem 1. a pair of indifference can be constructed as follows: step i. fix y≁0 and fix s, t, τ with s<t, and τ>0; step ii. find x such that (x, s) ∼ (y, t); step iii. find σ such that (x, s+σ) ∼ (y, t+τ). 3.2 anomalies and decreasing in impatience by virtue of the link between inconsistency and decreased impatience, it is possible to relate anomalies in the discounted utility model and the cognitive mechanisms responsible. the cognitive elements that influence our attitude are behavioral biases, and they can be of emotional or cognitive nature: the former is due to factors of the emotional sphere; the latter, instead, are “shortcuts” that the decision-maker creates in his mind during the interpretation and processing of information. jordan and rand (2018) 415 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices emphasized the importance of character in the decision-making process in a broader sense. given that behavioral anomalies are proper to the nature of decision-makers, even assuming a reasonable and rational investor, it is always possible to analyze anomalous situations from a mathematical viewpoint. in the literature, researchers have identified various types of anomalies; the latter can be related to investors’ emotional impulses through the concept of impatience by comparing the preferences associated with a hyperbolic discount and those associated with an exponential discount. the delay effect (ventre and ventre, 2012) consists in the inversion of the preferences due to the increase of the period between the decision and the reception of the result. for this anomaly, it is possible to prove that the speed with which the function of a hyperbolic discount is decreasing and, therefore, the impatience of the investor is intense according to the period of evaluation of the results. in particular, by ventre et al. (2022, b): theorem 2. if (𝑥, 𝑠) ∼ (𝑦, 𝑡) ⇒ (𝑥, 𝑠 + ℎ) < (𝑦, 𝑡 + ℎ) holds, then the rate of variation of the discount function 𝑓(𝑡) is decreasing. the interval effect (read et al., 2003; read and roelofsma, 2003) is the anomaly for which the applied discount rate decreases as the length of the interval increases. therefore, the phenomenon can be studied as a direct consequence of the subjective perception of time whose mathematical proof is in section 4.2 in ventre et al. (2022, b). the preferences of individuals are also conditioned by the size of the outcome. the anomaly in question, defined magnitude effect (benzion et al., 1989), consists in an inversion of preferences due to the size of the results and from a psychological vantage point can be explained considering that the more the figures are important, the more the patience of the investor increases. the investor's attitude is determined by the fact that smaller figures are associated with immediate consumption while larger figures are associated with a future investment idea. it can be proved that the reversal of the preferences in question is related to the difference of the figures considered and not to their ratio, i.e. (ventre et al., 2022, b): theorem 3. let 𝑥 and 𝑦, with 𝑥 < 𝑦, and 𝑋 and 𝑌, with 𝑥 < 𝑋 < 𝑌, such that 𝑈(𝑦) 𝑈(𝑥) = 𝑈(𝑌) 𝑈(𝑋) . if (𝑥, 𝑠) ∼ (𝑦, 𝑡), then (𝑋, 𝑠) ≺ (𝑌, 𝑡) if, and only if, 𝑈(𝑦) − 𝑈(𝑥) < 𝑈(𝑌) − 𝑈(𝑋). finally, the sign effect (loewenstein and thaler, 1989), also called gain-loss asymmetry, consists of the tendency of individuals to anticipate losses more than gains of the same magnitude. the first approach to studying the anomaly in question is the emotional loss aversion bias for which the disutility associated with a loss is greater than the utility associated with a gain of the same amount. the discount rate is, therefore, steeper in the context of a loss because the investor's haste toward payment is greater. the second approach is based on prospect theory (kahneman and tversky, 2013). in this regard, the anomaly can be justified by replacing the instantaneous cardinal utility function with the utility function proposed by the theory concerned as proven by ventre et al. (2022, b): theorem 4. if the utility of a result is calculated as 𝑉(𝑥), where 𝜀 > 1 (according to the prospectus theory) and 𝑈 is concave (𝑈′′ < 0), then, for every 0 < 𝑥 < 𝑦 and 𝑠 < 𝑡, (𝑥, 𝑠) ∼ (𝑦, 𝑡) implies (−𝑥, 𝑠) ≻ (−𝑦, 𝑡). 416 roberta martino, viviana ventre and giacomo di tollo 3. the ahp model applied to economic behavior to implement an ahp, it is first necessary to define a structure representing the problem. our aim is to define strategies that best fit the investor's profile. the first level representing the general objective, i.e. "selection of the best strategy" will be followed by behavioral criteria taken by keirsey's temperament theory. keirsey believed that individuals can be divided into four major temperaments, each with specific behavioral characteristics. pompian (2012) overlaid keirsey's four temperaments with four types of behavioral investors through behavioral finance studies with the goal of relating cognitive biases to an individual's attitude in the financial market. table 2 shows the relationship that exists between the four keirsey’s temperaments and the bit. keirsey’s temperaments artisan guardian idealist rational types of behavioral investors accumulators conservative followers independent table 2. relationship between keirsey’s temperaments and types of behavioral investors since the financial personality is related to temperament and each behavioral investor is characterized by specific behavioral biases, the third level is composed of the anomalies of the discounted utility model studied in ventre et al. (2022, b). the last level has as vertices the investment strategies assuming that a strategy can be viewed as combinations of the domains in which the anomalies develop. the final structure is shown in figure 4. figure 4. ahp structure based on four different levels. the present paper aims to determine the weights of level 3 compared with level 2. "different drummers" and "understand me" refer to the book "please understand me ii. temperament character intelligence" (keirsey, 1998) of which was used the keirsey fourtypes sorter test for the determination of temperament. the quiz, consisting 417 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices of sixteen questions with four alternatives to be ordered by decreasing preference, provides us with the intensity of the four temperaments in each individual. the obtained scores, observing that the main temperament corresponds to the minimum value, allow calculating the weights of level 2 with the following pairwise comparison matrix reported in table 3. go artisan guardian idealist rational artisan 1 g/a i/a r/a guardian a/g 1 i/g r/g idealist a/i g/i 1 r/i rational a/r g/r i/r 1 table 3. pairwise comparison matrix for the weights of go with respect to each temperament. the score of each temperament is given by the keirsey four types sorter test (keirsey, 1998, p.348). in addition to empirically testing the relationships in section 3, this study tries to estimate the relative importance of anomalies with respect to higher-level items. to accomplish this, assuming that the degree of decrease in impatience reflects the nonrationality underlying the preference reversal phenomenon, the questionnaire is created by constructing the indifference pairs given in theorem 1. the hyperbolic factor provides us with a measure of the altered degree of impatience for each anomaly that makes up level 3. for each trait that composes in level 2, the median of the hyperbolic factors is calculated based on the main temperament of each individual. in this way, for each main trait, the matrix of the pairwise comparison shown in the table 4 is obtained. in the table 4, 𝐻(𝑠, 𝑡, 𝑦, 𝜏) denotes the hyperbolic factor in a prospectus in which s, t, y and 𝜏 are fixed. the respondent is free to choose for each question the values of x and σ obtaining the indifference pairs (𝑥, 𝑠)~(𝑦, 𝑡) and (𝑥, 𝑠 + 𝜎)~(𝑦, 𝑡 + 𝜏) identified by the theorem 1. dominant trait delay effect magnitude effect sign effect delay effect 1 medianh(0,6,500,12) medianh(0,6,50,12) medianh(0,6,500,12) medianh(0,6, −500,12) magnitude effect medianh(0,6,50,12) medianh(0,6,500,12) 1 medianh(0,6,50,12) medianh(0,6, −500,12) sign effect medianh(0,6, −500,12) medianh(0,6,500,12) medianh(0,6, −500,12) medianh(0,6,50,12) 1 table 4. pairwise comparison matrix for the weights of the temperaments with respect to the individual anomalies. underlying this construction is the hypothesis that the degree of decrease in impatience represents the non-rationality underlying the inconsistency. 418 roberta martino, viviana ventre and giacomo di tollo 4. results to analyze the relationships among the four temperaments we consider the median of the hyperbolic factors of individuals according to the dominant trait. the median is recommended because the data exhibit high variability. in fact, it is possible to prove that the maximum reference value that can be obtained on the basis of the data set is 𝐻(𝑠, 𝑡, 𝑦, 𝜏)= 66.50 while the lower value is not defined. before proceeding with the presentation of the main results of the experimental phase, in table 5 we report the distribution of the sample. temperament rational idealist guardian artisan distribution 38.5% 26.9% 19.2% 15.4% table 5. distribution of the sample. the quiz administration sample consisted of 52 persons, ages ranging from 18 years old and 65 years old, of which 48.08% were women. table 6 shows the results for the hyperbolic factor h(0,6,500, 12). the maximum degree of decrease in impatience indicates strong impulsiveness. artisans, in general, are more focused on the short term and therefore prefer more immediate gratification. in fact, artisans manifest the major median while idealists, due to their general disinterestedness, stand out among all temperaments being the only ones not to realize in maximum value. main temperament minimum value and distribution maximum value and distribution median artisan 0.50 (12.50%) 66.50 (25.00%) 7.83 guardian 0 (10%) 66.50 (20%) 2.83 idealist 0 (7%) 28.40 (7.14%) 1.83 rational -0.02 (5%) 66.50% (25.00%) 1.83 table 6. h(0,6,500,12). distribution and median for each temperament. table 7 shows the results for the hyperbolic factor where lesser digits are involved in the indifference pairs of theorem 1. comparing the values in table 6 and table 7, the distribution of the maximum value is doubled in the case of less important digits. in line with predictions, the medians are all greater than the factor h(0,6,500,12) . as for artisans, their strong impulsiveness is even more evident with a distribution of the maximum value of 62.50%. the case of the guardians is particularly interesting. in fact, this category presents the maximum value for the ratio h(50) /h(500) which, from a cognitive perspective point of view, can be explained by the mental accounting bias for which money is categorized by subjective criteria. this is equivalent to saying that the prevalence of impulsiveness over self-control breeds inconsistency. also, the idealists realize h=66.50 unlike the case reported in table 6. the rationales present less inconsistency among the choices involving figures of different entity. 419 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices main temperament minimum value and distribution maximum value and distribution median artisan 2.69 (12.50%) 66.50 (62.50%) 28.4 guardian 0 (10%) 66.50 (30%) 18.12 idealist 0.06 (7.14%) 66.50 (35.71%) 4.83 rational -0.02 (5%) 66.50 (40.00%) 4.17 table 7. h(0,6,50,12). distribution and median for each temperament. table 8 shows the comparison between hyperbolic factors h(0,1,50,1) and h(0,6,50,12) with the purpose of understanding how much and how the applied discount varies with the length of the intervals involved. it is evident that the greater part of the variation of the degree of impatience happens in the first period confirming the hypothesis of a decreasing speed in the time. moreover, as the width of the interval increases, the applied discount also increases. this outcome involves all temperaments. h (0,1,50,1) h (0,6,50,6) minimum value and distribution 0 (7.69%) 0 (3.85%) m (25.00%) f (75.00%) m (50%) f (50%) maximum value and distribution 32.33 (48.08%) 66.50 (40.38%) m (64.00%) f (36.00%) m (61.90%) f (38.10%) median 13.29 18.12 table 8. comparison of h(0,1,50,1) and h(0,6,50,12). table 9 highlights how many individuals, just by varying the sign of the outcomes considered, exhibit different hyperbolic discounts. analyzing the median for temperaments, particular attitudes characterize each category. the artisans , for example, generally show a high value for the hyperbolic factor, underlining that the serenity with which they face the payment prevails over the displeasure of the payment itself. this is because artisans are the most flexible with respect to losses. on the other hand, the guardians have a different approach, as evidenced by the finding of the smallest maximum value of the entire experimental phase. this temperament suffers the payment because it is seen as a reduction of its assets. in fact, when investigating the timing with which individuals would prefer to act in a loss and gain situation, guardians are the ones that most postpone losses among the four temperaments as shown in figure 5. main temperament minimum value and distribution maximum value and distribution median h (-500) artisan 0.03 (25.00%) 66.50 (62.50%) 66.50 guardian 0 (20.00%) 7.83 (20%) 2.83 idealist 0 (7.14%) 66.50 (14.29%) 1.83 rational 0 (5%) 66.50% (50.00%) 31.17 table 9. h(0,6,-500,12). 420 roberta martino, viviana ventre and giacomo di tollo figure 5. distribution of individuals who defer payment. guardians take an attitude that sets them apart from other temperaments in such a situation. figure 6 shows that rationals despite being the least emotional are prone to particular distortions such as the overconfidence bias. in fact, rationals present themselves as more irrational than the other temperaments in this circumstance in that, although they manifested a moderate hyperbolic factor, they were the individuals who would pay higher amounts than they had stated they would receive. figure 6. distribution of individuals who would unconsciously increase the amount payable. for the bias of excess of confidence that characterizes them, the rationales prefer to pay more than they would like to receive. 421 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices the analysis ends by calculating pairwise comparison matrices for each temperament and determining the relative weight of the anomalies relative to the top level. the values obtained are shown in figure 7. figure 7. final structure. by coasting the pairwise comparison matrices for each temperament as shown in table 4, the local priorities of the affected level were estimated. these priorities did not vary across individuals. therefore, it could be a method to homogenize heterogeneous categories. 5.1 results by genders to investigate and deepen the differences between the genders, we repeat the analysis with the aim of creating two different ahp. in this way, it will be possible to quantify the strengths and weaknesses of both genders. the distribution of the sample with respect to gender (48.08% female, 51.92% male) is sufficient and this allows us to proceed with the analysis. the table 10 shows the medians of hyperbolic factors with respect to the genders. h(0,500,6,12) h(0,50,6,12) h(0,1,50,1) h(0,-500,6,12) male 1.83 6.50 32.33 1.83 female 7.83 28.41 3.00 6.50 table 10. medians of hyperbolic factors with respect to the genders. the h(0,500,6,12) column in table 10 shows that women are generally more impatient than men. in particular, the value corresponding to women is approximately 4.28 times greater than men. what has been said is equivalent to affirming that women are more prone to exhibit a dynamic inconsistency. this trait is still evident when we consider the magnitude effect. first, we observe that both genders undergo an important variation of the hyperbolic factor when smaller figures intervene, leading us to conclude that the magnitude effect has an equivalent impact on men and women. in fact, the ratio of the medians reported in column two of the table 10 is 4.37, similar to the previous one, still highlighting the greater impatience of women. the last column of the table 10, 422 roberta martino, viviana ventre and giacomo di tollo relating to the median of h(0,-500,6,12), again underlines a different approach in intertemporal choices between the genders, in this case with respect to losses. to investigate this result, the figure 8 and the figure 9 show who would pay more and who would postpone payment more according to gender. figure 8. distribution of individuals who would unconsciously increase the amount payable according to gender. figure 9. distribution of individuals who defer payment according to gender. 423 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices women, in accordance with a lower aptitude for the financial environment, account for a higher percentage in figure 8, while, with respect to figure 9, the percentages are almost similar, and no further differences are evident. finally, about the hyperbolic factor h(0,1,50,1) it is possible to observe the only case in which men show greater impatience than women. although the gender distribution is uniform, we cannot be satisfied with our analysis after observing in section 5 many important differences between the four temperaments. in fact, we believe that the distribution of temperaments with respect to gender has a nonnegligible impact on the medians shown in the table 10. table 11 shows the gender distribution in the four temperaments, whose inhomogeneity confirms our hypothesis. artisan guardian idealist rational female 25.00% 70.00% 64.29% 35.00% male 75.00% 30.00% 35.71% 65.00% table 11. distribution of gender with respect to each temperament. to understand to what extent, the gender-temperament combination affected the gender analysis, we propose again an analysis of the medians of the four temperaments with respect to gender. let’s beginning again with h(0,500,6,12) for each temperament. the medians are reported in table 12. it can be observed that the medians of the female gender are much higher than the respective medians of the male gender. for the male gender, in this case, we still find that the highest median is manifested by artisans and the lowest by guardians, as was the case with the general evaluation. for women, on the other hand, the highest median is manifested by guardians and the lowest median by idealists. this result underlines that gender difference has a great impact on the temperament of the guardians in the delay effect. artisan guardian idealist rational female 18.12 28.40 1.83 7.83 male 5.83 0.17 0.83 1.83 table 12. h(0,6,500,12)distribution and median for each temperament by genders. to see if the attitude of guardians is uniform with respect to anomalies, we proceed with the magnitude effect. from table 13, which shows the medians for each temperament according to the gender of h(0, 6, 500, 12), it is possible to say that one cannot generalize what happens for the delay effect. in fact, although the magnitude effect is confirmed by the fact that the values of the medians of h(0, 6, 50, 12) are for the most part greater than the medians of h(0, 6,500, 12), it is evident that the financial behavior of the genders is different. the lack of uniformity in the survey confirms that each anomaly is driven by different psychological principles. artisan guardian idealist rational female 28.40 28.40 7.83 1.83 male 66.50 0.17 1.83 6.50 table 13. h(0,6,50,12) distribution and median for each temperament by genders. 424 roberta martino, viviana ventre and giacomo di tollo the first thing to note is the relationship between the temperament of the guardians and the artisans. in fact, while the female gender of these two temperaments has the highest median, the male gender shows totally opposite values: it is highest for artisans and lowest for guardians. moreover, the rational female gender shows a median that is about four times lower than the respective male temperament. therefore, women are not always more impatient than men. to analyse the financial behaviour with respect to the length of the intervals involved, we compare h(0,6,50,12) with h(0,1,50,1). in table 14 we can observe that the temperament of the artisans is confirmed for both genders with a greater impulsiveness evident in the high degree of decreasing in impatience. artisan guardian idealist rational female 22.81 3.00 3.00 13.29 male 32.33 1.00 33.33 3.35 table 14. h(0,1,50,1) distribution and median for each temperament by genders particularly interesting are the idealists and the rationales: the former, of the male gender, show a decrease in impatience equal to that of the artisans; the latter, of the female gender, suffer a very strong impact from this type of anomaly as can be seen by the wide variation ranging from 1.83 to 13.29. however, it is clear to all that the change in the degree of impatience occurs in the first period of the prospectus. the behavior of the male guardians stands out: emphasizes the cautious nature of this temperament. from table 15 it can be seen how strong the impact of the sign effect is on guardians, the only temperament that shows a lower median of h(-500) than h(500). when analyzing the data with respect to gender, however, those who suffer most from the payment are women of this temperament, a result also shown in figure 10. in fact, women, much more than men, tend to postpone payment. artisan guardian idealist rational female 66.50 3.83 4.83 7.83 male 47.50 0 8.18 65.00 table 15. h(0,6,-500,12) distribution and median for each temperament by genders. figure 10. distribution of individuals who defer payment divided by gender. 425 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices to investigate how the overconfidence bias that characterized the rationales in the general analysis in section 5 is distributed over gender, figure 6 has been enriched with the gender in figure 11. it can be observed immediately that most of the rationales who would pay more than they want to receive are about 60% male. figure 11. distribution of individuals who would unconsciously increase the amount payable divided by gender. figure 12 and figure 13 show the final structures of the female and male genders respectively. figure 12. final structure for female. 426 roberta martino, viviana ventre and giacomo di tollo figure 13. final structure for male. 5. discussion this paper has applied ahp method in the context of intertemporal choices to decompose the economic behavior and to promote a better understanding of the dynamics of the decision-making. in particular, the applicability of the present research addresses to two necessary eu goals: costumer profiling profiles and gender equality. the directive 2014/65/eu (art. 24 section 11 and art. 25) emphasizes the need for information to understand whether a specific financial transaction meets the following objectives with respect to the client: matching the client's objectives and risk tolerance; a financial nature such that the client is able to bear the risks in relation to its investment objectives; a financial nature such that the client has the necessary knowledge to understand the transaction and the risks involved. to address these needs, in this paper proposed a method to classify the investor by interweaving keirsey's temperament theory and the concept of decreasing impatience in intertemporal choices. the anomalies of the discounted utility model studied in ventre et al. (2022, b) are deepened and investigated from a behavioral point of view according to the temperament and gender of the respondents. the short section 2 on gender difference underlines the need to break down the decision-making process to understand the differences between men and women. in fact, to implement a strategy that aims at gender equality in any field and customer management, it is necessary to consider the diversity that characterizes men and women as a starting point. in section 3, the theoretical concepts necessary to understand the experimental framework were outlined, emphasizing that the decrease in impatience is a measure of the non-rationality of the decision-maker due to the emotional drives that develop during the evaluation and selection phase of an intertemporal prospectus. in this context, the theoretical results proven in ventre et al. (2022, b) with respect to the interval effect, delay effect, magnitude effect and sign effect were recalled in section 3.2. after presenting the essential tools, the construction of the experiment is based on by the implementation of an analytical hierarchical process. the mathematical structure of the ahp makes it possible to 427 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices decompose the decision-making process into all its elements and to explain their relationships in quantitative terms. indeed, figure 7 shows important differences between the four temperaments, confirming the need to design strategies that can adhere to the profile of the individual investor without ever generalizing their effectiveness. in fact, it immediately emerges that: the greatest weight obtained by ahp is shown by the rationales with respect to the sign effect anomaly, with a value of 0.804; guardians, have similar weights to the rationales but with a greater influence of the magnitude effect; guardians, on the other hand, have lower weights than the sign effect anomaly but are the temperament with the highest weight with respect to the magnitude effect; the idealists, finally, is the temperament with the highest weight with respect to the delay effect. the reported results can be further deepened and analyzed by considering not only the first behavioral trait but also the remaining ones. in fact, from the obvious differences between the temperaments we expect that a rational with guardian as second trait would behave differently towards the magnitude effect than a rational with artisan as second trait. gender analysis made it possible to further investigate the relationship between temperaments and anomalies. it emerges that there is not a better temperament or a more capable gender. in some temperaments, the male and female genders behave in almost opposite ways. for example, as can be seen for the rationales with respect to the delay effect (0.45 for women and 0.03 for men), or for the guardians with respect to the sign effect (0.54 for women and 0.01 for men). however, these results do not work against women. it is enough to look at the behavior of artisans with respect to the magnitude effect (0.55 for men, 0.25 for women), or idealists with respect to the sign effect (0.75 for men, 0.33 for women). the analysis presented and the quantification of differences in intertemporal preference can foster gender mainstreaming in the financial world through the inclusion of specific measures aimed toward women. finally, we stress that the structure of the ahp can be expanded with other nonnegligible information, such as age and initial wealth. in this way it is possible to describe more and more defined classes of clients, describing the individual as the combination of multiple dimensions of identity and recalling the concept of intersections, allowing dominant and subordinate factors to be included at the same time (atewologun et al., 2016) with respect to the distribution of the data, the inhomogeneity of temperaments with respect to gender does not allow to draw definitive conclusions with respect to the combination of gender and temperament, but certainly highlights the importance of the subjective structures of the decision-maker during a choice. moreover, during data collection, women were more reluctant to submit to the test, except for the temperament of the guardians and idealists. therefore, in order not to distort the male-female distribution, it was difficult to obtain homogeneity in the temperaments. this attitude of refusal is certainly linked to a cultural gap, in which women feel less skilled and qualified in the field of finance than men. in this respect, various other factors such as age, social class and level of education play a role. our work is an invitation to consider these elements to understand why it is necessary to define consumer classes. in fact, the analysis of the data did no more than experimentally confirm the requirements of the decree 2014/65/eu. our work offers an original method that quantifies diversity and legitimizes anomaly. 428 roberta martino, viviana ventre and giacomo di tollo 6 conclusions the multidisciplinary approach used made it possible to conduct an original analysis on gender differences and consumers classification. such attitudes have defined over the years a break between the theoretical model and actual actions of individuals. the basis of this discrepancy is the assumption of a rational investor, capable of making decisions always favoring her interests. behavioral finance, which studies investor behavior in the financial market, describes "special cases of deviation from rational action" (rubaltelli, 2006) through regular and systematic patterns of behavior for which the subject is rationally limited. classical theories overlook the numerous distortions of individual decision-making arising from emotional drives and the complexity of the environment in which the choice is made. this is equivalent to state that non-rationality is part of human nature. decision-makers do not always process information correctly and impartially and, as a result, make choices that worsen their condition. from a pratical point of view, consumer profiling operations have been introduced in european directives as written in directive 2014/65/ue of the european parliament and of the council. the aim is to protect clients by designing financial products that correspond to individual profiles, client by client. to exploit the full potential of personalized advice, it is then necessary to know how to construct a strategic communication plan to improve client choices. the intersection between strategic customization and the understanding of gender differences is evident in different risk preferences and risk management. in fact, advisors consider the risk appetite index a key element for asset allocation modelling and the numerous empirical evidences presented in section 2 underline the need to take into account the different risk perceptions between men and women in order to improve personalised finance services. in addition, the analysis of gender differences using the intertemporal choice theory can also contribute to understanding important behavioral differences such as the propensity to depression, addictions and procrastination mechanisms. in fact, prelec (2004) also proved that irrational degeneration of definition 3 and definition 4 lead to attitudes of procrastination and promotion of consumption, described in figure 14 and figure 15. figure 14. procrastination: the procrastinating decision-maker prefers to do the work immediately rather than not do it at all but keep putting it off (prelec 2004, p.221). 429 analytic hierarchy process for classes of economic behavior in the context of intertemporal choices figure 15. pre-emptive consumption: the decision-maker anticipates the choice by deciding to consume today despite this being a less preferred option than abstaining (prelec 2004, p.221). the work presented can provide methodological support for studies that aim to understand gender differences in topics such as suicidal ideation, stress, alcohol use and depression (e.g., rich et al., 1992). understanding the differences that exist between individuals and accepting that each identifying element (such as gender, age, culture...) has a non-negligible impact on decision-making are necessary research processes to then to nudge the individual toward a better option using the same cognitive boundaries (thaler and sunstein, 2014). nudges are intended to influence social behavior without significantly limiting freedom of choice. various studies have shown that these interventions have positive results (pe’er et al., 2019). nudging has emerged as an efficient technique to improve the quality of individuals’ choices to address this problem. the nudge technique is based on using behavioral biases to the individual's advantage through a variation in the decision-making environment. in this way, the decision-maker is nudged toward the best option without denying the freedom to choose the other alternatives. such an intervention has been studied and implemented by rubaltelli and lotto (2021) to increase the pension funds of freelancers. to improve the nudging technique, two important observations must be considered. first, individuals’ response to pushing can vary from subject to subject based on age, gender, and character; second, change necessarily involves an alteration in one's state. therefore, a determining factor in optimizing interventions of this type is an analysis of the tendency to maintain one's status. in this regard, gal (2006) refers to a "psychological law of inertia" to indicate, in the context of loss aversion, the tendency of individuals not to alter their habits. so, to optimize choice architecture, we need to move towards customized interventions. describing anomalous attitudes in terms of inconsistency among decision-making processes involving choices defined over a short or long period allowed us to quantify the inconsistency of individuals' choices. the direct consequence is the possibility of being able to quantify how much behavioral bias weighs in individual decision-making. decision theory makes it possible to implement a personalized choice architecture. choice architecture plays the key role in implementing nudging. indeed, if decisionmakers are influential because they are rationally constrained, the decision-making environment can be organized to influence toward the best individual option. the methodology proposed in the present paper could be a framework to help the selection of the best choice architecture because behavioral differences show the necessity to diversify the strategies. the use of ahp allows for a construction of the decision context that fits the individual by modeling the structuring of options and the description of options based on the cognitive mechanisms involved. our approach can be expanded in three different ways. first, one can consider sixteen temperaments instead of four. keirsey divided the four temperaments into two categories, each with two variants by relating each group to 430 roberta martino, viviana ventre and giacomo di tollo the sixteen types of the mbti (myers, 1980). this makes it possible to study the relationship between anomalies and personality in more details. the second development of our approach requires the inclusion within the structure of other levels that contain increasingly personal information such as age. other anomalies in the discounted utility model such as delay-anticipation asymmetry (loewenstein, 1988). and preference for sequences of outcomes (loewenstein and sicherman, 1991; chapman, 2000) can also be included, too. anticipation-delay asymmetry consists of the phenomenon whereby the discount function is steeper if an 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permutes with every subgroup of b and b permutes with every subgroup of a. we say that the product is totally permutable if every subgroup of a permutes with every subgroup of b. in this paper we prove the following theorem. let g=ab be the mutually permutable product of the super soluble subgroups a and b. if coreg(a∩b)=1, then g is super soluble. keywords: quasinormal; permutable product; super soluble; etc. 2010 ams subject classification: 20f32^ * assistant professor of islamic azad university talesh branch. talesh, iran.; b_razzagh@yahoo.com. ^received on march 12th, 2020. accepted on june 3rd, 2020. published on june 30th, 2020. doi: 10.23755/rm.v38i0.503. issn: 1592-7415. eissn: 2282-8214. ©b. razzaghmaneshi. https://groupprops.subwiki.org/wiki/subgroup https://groupprops.subwiki.org/wiki/group https://groupprops.subwiki.org/wiki/product_of_subgroups https://groupprops.subwiki.org/wiki/permuting_subgroups behnam razzaghmaneshi 386 1 introduction all groups considered in this paper are finite. it is known that a group which is the product of two super soluble groups is not necessarily super soluble, even if the two factors are normal subgroups of the group. baer proved in [3] that if a group g is the product of two normal supersoluble groups and g′ is nilpotent, then g is super soluble. the search for generalisations of baer’s result has been a fruitful topic of investigation recently (see [5,7]). most of the generalisations centre around replacing normality of the factors by different permutability conditions. in [2], asaad and shaalan considered products satisfying one of the following conditions. we will follow carocca [6], and say that g=ab is the mutually permutable product of the subgroups a and b if a permutes with every subgroup of b and b permutes with every subgroup of a. we say that the product is totally permutable if every subgroup of a permutes with every subgroup of b. essentially, the results by asaad and shaalan are devoted to obtaining sufficient conditions for a mutually permutable product of two supersoluble subgroups to be supersoluble. they prove in [2, theorem 3.8] the following generalisation of baer’s theorem: let g be the mutually permutable product of the supersoluble subgroups a and b. if g′ is nilpotent, then g is supersoluble.they also show that the result remains true if “g′ nilpotent” is replaced by “bnilpotent”[2, theorem 3.2]. in addition, they prove [2, theorem 3.1]: if g is the totally permutable product of the supersoluble subgroups a and b, then g is supersoluble. it is well known that if g=ab is a mutually permutable product of two supersoluble subgroups a and b such that a∩b=1, then the product is in fact totally permutable [6,proposition 3.5], and therefore g is supersoluble. our main theorem is a generalisation of this last property. theorem 1. let g=ab be the mutually permutable product of the supersoluble subgroups a and b. if coreg(a∩b) =1, then g is supersoluble. the second aim of the present paper has been to obtain more complete information about the structure of mutually permutable products of two supersoluble groups. as a straightforward consequence of theorem 1, we have that, in the notation used above, g/coreg(a∩b) is always supersoluble. therefore, every mutually permutable product of two supersoluble subgroups is metasupersoluble. it is possible to obtain more precise information about its structure, as our second main theorem shows. theorem 2. let g=ab be the mutually permutable product of the supersoluble subgroups a and b. then g/f (g)is supersoluble and properties of quasinormal groups (pqg) 387 metabelian.this last theorem can not be improved easily, as the following example shows. example. let s3 be the symmetric group of degree 3, given by s3=〈α, β:α2=β3=1;βα=β2〉 and let t7 be the non-abelian group of order 73 and exponent 7. write t7=〈a,b〉with a7=b7=[a,b]7=1 and let c=[a,b]. we have that s3 acts on t7 in the following way: aα=b, bα=a, cα=c−1, aβ=a2, bβ=b4, cβ=c. thus, we can consider the semidirect product g=[t7] s3. take now the subgroups a= t7〈β〉and b=t7〈α〉of g. clearly both a and b are supersoluble, and it is easy to check that g=ab is the mutually permutable product of a and b. finally, we show that theorem 1 provides elementary proofs for the results of asaad and shaalan about mutually permutable products.2. main results: the following four lemmas are needed to prove theorem 1. lemma 1[4, theorem 2]. if g=ab is the mutually permutable product of the supersoluble subgroups a and b, then g is soluble. lemma 2. let g=ab be the mutually permutable product of the supersoluble subgroups a and b. then, either g is supersoluble or na < g and nb < g for every minimal normal subgroup n of g. proof. assume that g is not supersoluble. then both a and b are proper subgroups of g. let n be a minimal normal subgroup of g and for contradiction assume that na=g. then, as n is abelian, n∩a is a normal subgroup of 〈n,a〉= g. since n is a minimal normal subgroup of g and a<g, we have that n∩a=1 and consequently a is a maximal subgroup of g. clearly, we can also assume that b is not contained in a. it is not difficult to argue that we can choose an element b of b\a such that bq∈a for some prime q. since the product g=ab is mutually permutable, a〈b〉is a subgroup of g and the maximality of a implies that g=a 〈 b 〉 . we now take orders to reach our final contradiction: |a||n|=|g|=|a||〈b〉||a∩〈b〉|=q|a|. consequently, we have that |n|=q and then g is supersoluble, a contradiction. lemma 3. let g=ab be the mutually permutable product of the subgroups a and b and let n be any minimal normal subgroup of g. then either n∩a=n∩b=1 or n=(n∩a)(n∩b). proof. let n be a minimal normal subgroup of g. clearly a(n∩b) and (n∩a)b are both subgroups of g. note that a normalizes n∩(a(n∩b))=(n∩a)(n∩b) and b normalizes n∩((a∩n)b)= (n∩a)(n∩b). behnam razzaghmaneshi 388 therefore (n∩a)(n∩b) is a normal subgroup of g and the minimality of n yields the result. lemma 4. let g be a group, and n a minimal normal subgroup of g such that |n|=pn, where p is a prime and n>1. denote c=cg(n) and assume that g/cis supersoluble. then, if q/cis a subgroup of g/c containing op′(g/c), we have that q is normal in g and n=∏ti=1ni, where ni are non-cyclic minimal normal subgroups of nq for i=1,...,t. proof. since by [8, lemma a.13.6], we have that op(g/c)=1 and the commutator subgroup (g/c)′ of g/c is nilpotent because e g/c is supersoluble, it follows that (g/c)′is a p′-group. therefore (g/c)′is contained in op′(g/c) and thus op′(g/c)is a hall p′-subgroup of g/c. consequently, q/cis a normal subgroup of g/c and hence q is normal in g. consider now n as a g-module over gf (p)by conjugation. then, by clifford’s theorem [8, theorem b.7.3], n viewed as a q-module is a direct sum n=∏ti=1ni, where ni are irreducible qmodules for i=1,...,t. suppose that there exists i∈{1,...,t}such that |ni|=p. then clearly |nj|= p for all j. therefore q/cq(ni) is abelian of exponent dividing p−1, and the same is true for q/c. in particular, q/c=op′(g/c) is a hall p′-subgroup of g/c. since n is not cyclic, it follows that q = g and thus p divides |g/c|. hence p is the largest prime dividing |g/c|. from the supersolubility of g/c, we obtain that 1= op(g/c) is a sylow subgroup of g/c, a contradiction. consequently, ni is a non-cyclic minimal normal subgroup of nq for all i∈{1,t},as we wanted to prove. proof of theorem 1. let g=ab be the mutually permutable product of the supersoluble subgroups a and b, with coreg(a∩b)=1, and suppose that g has been chosen minimal such that its supersoluble residual gu is non-trivial. let n be a minimal normal subgroup of g contained in gu. note that n is an elementary abelian p-group for some prime p. applying lemma 2, we have that both na and nb are proper subgroups of g. moreover, using lemma 3, we have that either n=(n∩a)(n∩b ) or n∩a=n∩b=1. assume first that n=(n∩a)(n∩b). (i) if n∩a=1, then n is cyclic. assume that n∩a=1. it follows that n is contained in b. let n0 be a non-trivial cyclic subgroup of n. since an0 is a subgroup of g, we have that n0 =an0∩n is anormal subgroup of an0. hence every cyclic subgroup of n is normalised by a. now let n1 be a minimal normal subgroup of b contained in n. since b is supersoluble, it follows that n1 is cyclic and thus normalised by a. hence n1 is a normal subgroup of g. the minimality of n implies that n=n1 and consequently n is cyclic. (ii) n∩a=1 and n∩b=1.on the contrary, assume that n∩a=1. from (i), we know that n is cyclic. moreover, nis contained in b. hence an∩b= (a∩b)n. properties of quasinormal groups (pqg) 389 let l=coreg(a∩b)n). clearly, n is contained in land l=l∩((a∩b)n)=(l∩a∩b)n. it is clear that g/l=(al/l)(bl/l)is a mutually permutable product of al/l and bl/l suchthat coreg/l((al/l)∩(bl/l))=1. by the minimality of g, it follows that g/l is supersoluble. on the other hand, since n is cyclic, we have that g/cg(n) is abelian. hence g/cl(n) is supersoluble and gucl(n)=c. note that c=n×(c∩a∩b). therefore c∩a∩b contains a hall p′-subgroup of c. since coreg(a∩b)=1 and op′(c) is a normal subgroup of g contained in c∩a∩b, we have that op′(c)=1. moreover, c′=(c∩a∩b)′ is a normal subgroup of g contained in a∩b. consequently, c′=1 and c is an abelian p-group. in particular, gu is abelian and thus gu is complemented in g by a supersoluble normalizer d which is also a supersoluble projector of g, by [8, theorems v.4.2 and v.5.18]. since n is cyclic, we know that n is central with respect to the saturated formation of all supersoluble groups. by [8,theorem v.3.2.e], dcovers n and thus n is contained in d. it follows nd∩gu=1, a contradiction. (iii) either n=n∩a or n=n∩b. if we have n=n∩a=n∩b, then n is contained in a∩b, contradicting the factthat coreg(a∩b)=1. we may assume without loss of generality that n∩a=n. (iv) an and bn are both supersoluble. since n=(n∩a)(n∩b) and n=n∩a, it follows that n∩b is not contained in n∩a. let n be any element of n∩b such that n/∈n∩a, and write n0 =〈n〉. note that an0 is a subgroup of g, and an0∩n=(n∩a)n0. therefore n0(n∩a) is a normal subgroup of an0, and consequently a normalizes (a∩n)n0. this yields that a/ca(n/n∩a) acts as a power automorphism group on n/n∩a. this means that an is supersoluble. if n∩b=n, then bn=b is supersoluble. on the contrary, if n∩b=n, we can argue as above and we obtain that bn is supersoluble. consequently, acg(n)/cg(n) and bcg(n)/cg(n) are both abelian groups of exponent dividing p−1. but then g/cg(n)=(acg(n)/cg(n))(bcg(n)/cg(n)) is a π-group for some set of primes π such that if q∈π, then q divides p−1. (v) let b0 be a hall π-subgroup of b. then ab0∩n= a∩n. this follows just by observing that ab0∩nis contained in each hall π′subgroup of ab0 and every hall π′-subgroup of a is a hall π′-subgroup of ab0. note that |g/cg(n)| is a π-number and ab0 contains a hall π-subgroup of g. therefore g=(ab0)cg(n). but then a∩n is a normal subgroup of g. the minimality of g yields either a∩n= 1or a∩n= n. this contradicts our assumption 1=n∩a=n, and so we cannot have n=(a∩n)(b∩n). thus, by lemma 3, we may assume n∩a=n∩b=1. let m= coreg(an∩bn). then n∩m= n and g/m is supersoluble by the minimality of g. again, we reach a contradiction after several steps. (vi) m=n. suppose that m=n. since g/m is supersoluble, we know that n cannot be cyclic. let us write c=cg(n), and consider the quotient group g/c. it is clear that g/c is supersoluble. let q/c=op(g/c). since op(g/c)=1 and behnam razzaghmaneshi 390 (g/c)′is nilpotent, it follows that q/c is a normal hall p′-subgroup of g/c. let bp′ be a hall p′-subgroup of b. since |n| divides |b:a∩b|, we have that (a∩b)bp′ is a proper subgroup of b. let t be a maximal subgroup of b containing (a∩b)bp′. then at is a maximal subgroup of g and |g:at|= p= |b:t|. if n is not contained in at, we have g=(at )n and at∩n=1. then |n|=p, a contradiction. therefore, n is contained in at. in particular, the family s={x:x is a proper subgroup of b, (a∩b)bp′x and nax} is non-empty. let r be an element of s of minimal order. observe that ar has p-power index in g and thus arc/c contains op′(g/c). regarding n as a ar-module over gf (p), we know, by lemma 4, that n is a direct sum n=∏ti=1ni, where ni is an irreducible ar-module whose dimension is greater than 1, for all i∈{1,...,t}. assume that (a∩b)bp′=r. then ar=abp′ and thus n is contained in a, a contradiction. therefore abp′∩b=(a∩b)bp′ is a proper subgroup of r. let s be a maximal subgroup of r containing (a∩b)bp′. from the minimality of r, we know that n is not contained in as. consequently, there exists some i∈{1,...,t} such that ni is not contained in as, which is a maximal subgroup of ar. hence ar=(as)ni. since ni is a minimal normal subgroup of ar, it follows that as∩ni= 1and |ni|= |ar:as|= |r:s|= p, a contradiction. (vii) m is an elementary abelian p-group. note that m=n(m∩a)=n(m∩b) and |m∩a|=|m∩b|=|m|/|n|. moreover, a(m∩b)is a subgroup of g such that a(m∩b)∩m= (m∩a)(m∩b). hence (m∩a)(m∩b) is also a subgroup of g. if m∩a= m∩b, then m∩a is a normal subgroup of g contained in a∩b. this implies that m∩a=1 and consequently m=n, a contradiction. it yields that m∩a=m∩b. next we see that (m∩a)(m∩b) is a normal subgroup of g. since (m∩a)(m∩b)= m∩a(m∩b), we have that a normalizes (m∩a)(m∩b). similarly, b normalises (m∩a)(m∩b) since (m∩a)(m∩b)= m∩b(m∩a). this implies normality of (m∩a)(m∩b) in g. let x=(m∩a)(m∩b). since we cannot have m∩a= m∩b, m∩a must be strictly contained in x. thus x=x∩m=(x∩n)(m∩a) > m∩a gives us x∩n=1. but then x∩n=n, giving nx. suppose that q is a hall p′-subgroup of m∩b. then qa is a subgroup and so qa∩m=q(m∩a) is also a subgroup which contains q. hence, as |m:m∩a|=pk for some k, we have that qm∩a∩b. thus qb∩mm∩a∩b and similarly qa∩mm∩a∩b. consequently, qm is contained in m∩a∩b. since qm=op(m), it follows that op( m) is a normal subgroup of g contained in a∩b. hence op(m)=1, a contradiction, and consequently q=1andmis a p-group. hence n is contained in z(m) and m=n×(m∩a)=n×(m∩b). thus φ(m)=φ(m∩a)=φ(m∩b) is a normal subgroup of g contained in a∩b. this implies that φ(m)=1 and m is an elementary abelian p-group, as claimed. (viii) final contradiction. we have from the previous steps that m∩a is not contained in m∩b and that m∩b is not contained in m∩a because otherwise, since |m∩a|=|m∩b|, it follows that properties of quasinormal groups (pqg) 391 m∩a=m∩b is a normal subgroup of g contained in a∩b. this would imply m∩a=m∩b=1, and m=(m∩a)n=n. this fact contradicts step (vi). let x be an element of m∩b such that x/∈m∩a. then a〈x〉is a subgroup of g, and so is m0=a〈x〉∩m=(a∩m)〈x〉. therefore, m0 is an ainvariant subgroup of g. in particular, since m=(m∩a)(m∩b), we have that every subgroup of m/m∩a is a-invariant; that is, a/ca(m/m∩a) acts as a group of power automorphisms on m/m∩a. it is clear that m/m∩a is aisomorphic to n. consequently, a/ca(n) acts as a group of power automorphisms on n. this implies that a normalises each subgroup of n. a nalogously, b normalises each subgroup of n. it follows that n is a cyclic group. we argue as in step (ii) above to reach a final contradiction. we have that g/m is supersoluble and m is abelian. therefore gum and thus gu is abelian and complemented in g by a supersoluble normaliser, d say, by [8, theorem v.5.18]. since n is cyclic, we know that d covers n and thus ngu∩d=1, a contradiction. proof of theorem 2. let m=gu denote the supersoluble residual of g. theorem 1yields that g/coreg(a∩b) is supersoluble. therefore, m is contained in coreg(a∩b). in particular, m is supersoluble. let f(m) be the fitting subgroup of m. since a and bare supersoluble, we have that [m,a]f(a)∩mf(m) and [m,b]f(b)∩mf(m). consequently, [m,g] is contained in f(m). note now that the chief factors of g between f(m) and mare cyclic,and recall that g/m is supersoluble. therefore, we have that g/f (m) is supersoluble. this implies that m=f(m) and thus m is nilpotent. consequently, g/f (g) is supersoluble. we now show that g/f (g) is metabelian. we prove first that a′ and b′ both centralise every chief factor of g. let h/k be a chief factor of g. if h/k is cyclic, then as g′ centralizes h/k, so do a′ an db′. hence we may assume that h/k is a non-cyclic p-chief factor of g for some prime p. note that we may assume that h is contained in m because g/m is supersoluble and h/k is non-cyclic. to simplify notation, we can consider k=1. since f(g) centralizes h [8, theorem a.13.8.b], g/cg(h ) is supersoluble. let ap′ be a hall p′-subgroup of a. by maschke’s theorem [8, theorem a.11.5],h is a completely reducible ap′module and hap′ is supersoluble because h is contained in a. therefore ap′/cap′(h ) is abelian of exponent dividing p−1. this implies that the primes involved in |a/ca(h )| can only be p or divisors of p−1.the same is true for |b/cb(h )|. this implies that if p divides |g/cg(h )|, then p is the largest prime dividing |g/cg(h )|. but since op(g/cg(h ))=1 and g/cg(h ) is supersoluble, it follows that g/cg(h ) must be a p′-group. consider h as a-module over gf (p). since acg(h )/cg(h ) is a p′-group, we have that h is a completely reducible a-module and every irreducible a-submodule of h is cyclic. consequently a′ centralizes h, and the same is true for b′. let now u/v be a chief factor of g. then g/cg(u/v )is the product of the abelian subgroups acg(u/v )/cg(u/v ) and bcg(u/v )/cg(u/v ). by itô’s theorem [9], we behnam razzaghmaneshi 392 have that g/cg(u/v )is metabelian. since f(g)is the intersection of the centralisers of all chief factors (again by [8, theorem a.13.8.b]), we can conclude that g/f (g) is metabelian.3. final remarks finally, theorem 1 enables us to give succinct proofs of earlier results on mutually permutable products. corollary 1[2, theorem 3.2]. let g=ab be the mutually permutable product of the subgroups a and b. if a is supersoluble and b is nilpotent, then g is supersoluble. proof. assume that the assertion is false, and let g be a minimal counterexample. we have that g is a primitive group, and so g has a unique minimal normal subgroup, n say, with n=cg(n) a p-group for some prime p. since g is not supersoluble, applying theorem 1, we know that coreg(a∩b)=1. this yields that n is contained in a∩b. now, since n is contained in b, which is nilpotent, it follows that any p′-element of b must centralize n. since cg(n)=n, we have that b itself is a p-group. consequently, a must contain a hall p′-subgroup of g. now let t/n=op′(g/n). the previous argument yields that t/n is contained in a/n. note that if b=n, then g =an= a is supersoluble, a contradiction. thus, n is a proper subgroup of b. this implies that p must divide |g:t|. since g/n is supersoluble, p must divideq−1 for some prime q∈π(t/n). it is clear then that q can not divide p−1. therefore, there exists a sylow q-subgroup aq of a which centralizes n. using that cg(n)=n, it yields that aq=1 and thus q does not divide |g|,a contradiction. corollary 2[2, theorem 3.8]. let g=ab be the mutually permutable product of thes upersoluble subgroups a and b. if g′ is nilpotent, then g is supersoluble. proof. we assume the result to be false, and choose a minimal counterexample g. thus, g is a primitive group with unique minimal normal subgroup n. we also have that g=nm, where m is a maximal subgroup of g,n∩m=1 and n=f(g)=op(g) for some prime p. now g′ is nilpotent and thus g′=f(g)=n. therefore, m is an abelian group. since n is self-centralising, arguing as we did in the previous corollary, we have that n is contained in a∩b. note that m∼=g/n, and thus op(m)=1. since m is abelian, this yields that m is a p′-group. thus m is in fact a hall p′-subgroup of g. applying [1, theorem 1.3.2], wehave that there exist a hall p′-subgroup ap′ of a and a hall p′subgroup bp′ of b suchthat m=ap′bp′. since na∩b, it follows that both ap′ and bp′ must have exponent dividing p−1.regarding n as a m-module, it is easy to see that m must be a cyclic group. now, since m=ap′bp′ has exponent properties of quasinormal groups (pqg) 393 dividing p−1, it follows that n is a cyclic group as well. this implies that g is supersoluble, a contradiction. references [1] b. amberg, s. franciosi, f. de giovanni, products of groups, clarendon, oxford, 1992. [2] m. asaad, a. shaalan, on the supersolvability of finite groups, arch. math. 53 (1989) 318–326. [3] r. baer, classes of finite groups and their properties, illinois j. math. 1 (1957) 115–187. [4] a. ballester-bolinches, j. cossey, m.c. pedraza-aguilera, on products of finite supersoluble groups, comm. algebra 29 (7) (2001) 3145–3152. [5] a. ballester-bolinches, m.d. pérez ramos, m.c. pedraza-aguilera, totally and mutually permutable products of finite groups, in: groups st. andrews 1997 in bath i, in: london math. soc. lecture note ser., vol. 260, cambridge university press, cambridge, 1999, pp. 65–68. [6] a. carocca, p-supersolvability of factorized finite groups, hokkaido math. j. 21 (1992) 395–403. [7] a. carocca, r. maier, theorems of kegel–wielandt type, in: groups st. andrews 1997 in bath i, in: london math. soc. lecture note ser., vol. 260, cambridge university press, cambridge, 1999, pp. 195–201. [8] k. doerk, t.o. hawkes, finite soluble groups, in:de gruyter exp. math., vol. 4, de gruyter, berlin, 1992. [9] n. itô, über das produkt von zwei abelschen gruppen, math. z. 62 (1955) 400–401. ratio mathematica vol. 33, 2017, pp. 139-150 issn: 1592-7415 eissn: 2282-8214 multiple ways of processing in questionnaires pipina nikolaidou∗ †doi:10.23755/rm.v33i0.376 abstract in social sciences when questionnaires are used, there is a new tool, the bar instead of likert scale. the bar has been suggested by vougiouklis & vougiouklis in 2008, who have proposed the replacement of likert scales, usually used in questionnaires, with bar. this new tool, gives the opportunity to researchers to elaborate the questionnaires in different ways, depending on the filled questionnaires and of course on the problem. moreover, we improve the procedure of the filling the questionnaires, using the bar instead of likert scale, on computers where we write down automatically the results, so they are ready for research. this new kind of elaboration is being applied on data obtained by a survey, studying the new results. the hyperstructure theory is being related with questionnaires and we study the obtained hyperstructures, which are used as an organized device of the problem and we focus on special problems. keywords: hyperstructures; questionnaires; bar; 2010 ams subject classifications: 20n20, 16y99. ∗democritus university of thrace, school of education, 68100 alexandroupolis, greece; pnikolai@eled.duth.gr † c©pipina nikolaidou. received: 31-10-2017. accepted: 26-12-2017. published: 31-12-2017. 139 pipina nikolaidou 1 basic definitions the main object of this paper is the class of hyperstructures called hv-structures introduced in 1990 [17], which satisfy the weak axioms where the non-empty intersection replaces the equality. some basic definitions are the following: in a set h equipped with a hyperoperation (abbreviation hyperoperation = hope) · : h×h → p(h)−{∅}, we abbreviate by wass the weak associativity: (xy)z∩x(yz) 6= ∅,∀x,y,z ∈ h and by cow the weak commutativity: xy∩yx 6= ∅,∀x,y ∈ h. the hyperstructure (h, ·) is called an hv-semigroup if it is wass, it is called hv-group if it is reproductive hv-semigroup, i.e., xh = hx = h,∀x ∈ h. motivation. in the classical theory the quotient of a group with respect to an invariant subgroup is a group. f. marty from 1934, states that, the quotient of a group with respect to any subgroup is a hypergroup. finally, the quotient of a group with respect to any partition (or equivalently to any equivalence relation) is an hv-group. this is the motivation to introduce the hv-structures [17], [18]. (r,+, ·) is called an hv-ring if (+) and (·) are wass, the reproduction axiom is valid for (+) and (·) is weak distributive with respect to (+): x(y + z)∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅, ∀x,y,z ∈ r. let (r,+, ·) be an hv-ring, (m,+) be a cow hv-group and there exists an external hope · : r×m → p(m) : (a,x) → ax such that ∀a,b ∈ r and ∀x,y ∈ m we have a(x + y)∩ (ax + ay) 6= ∅, (a + b)x∩ (ax + bx) 6= ∅, (ab)x∩a(bx) 6= ∅, then m is called an hv-module over f. in the case of an hv-field f, which is defined later, instead of an hv-ring r, then the hv-vector space is defined. for more definitions and applications on hv-structures one can see [2], [3], [4], [5], [6], [10], [14], [16], [18]. the main tool to study hyperstructures is the fundamental relation. in 1970 m. koscas defined in hypergroups the relation β and its transitive closure β*. this relation connects the hyperstructures with the corresponding classical structures and is defined in hv-groups as well. t. vougiouklis introduced the γ* and �* relations, which are defined, in hv-rings and hv-vector spaces, respectively [17]. he also named all these relations β*, γ* and �*, fundamental relations because they play very important role to the study of hyperstructures especially in the representation theory of them. for similar relations see [18], [22], [4]. 140 multiple ways of processing in questionnaires definition 1.1. the fundamental relations β*, γ* and �*, are defined, in hvgroups, hv-rings and hv-vector space, respectively, as the smallest equivalences so that the quotient would be group, ring and vector space, respectively. specifying the above motivation we remark the following: let (g, ·) be a group and r be an equivalence relation (or a partition) in g, then (g/r, ·) is an hvgroup, therefore we have the quotient (g/r, ·)/β* which is a group, the fundamental one. remark that the classes of the fundamental group (g/r, ·)/β* are a union of some of the r-classes. otherwise, the (g/r, ·)/β* has elements classes of g where they form a partition which classes are larger than the classes of the original partition r. the way to find the fundamental classes is given by the following [17], [20], [21], [22]: theorem 1.1. let (h, ·) be an hv-group and denote by u the set of all finite products of elements of h. we define the relation β in h by setting xβy iff {x,y}⊂ u where u ∈ u. then β* is the transitive closure of β. a well known and large class of hopes is given as follows [15], [18], [12]: let (g, ·) be a groupoid then for every p ⊂ g, p 6= ∅, we define the following hopes called p-hopes: for all x,y ∈ g p : xpy = (xp)y ∪x(py), pr : xpry = (xy)p ∪x(yp),p l : xp ly = (px)y ∪p(xy). the (g,p),(g,pr) and (g,p l) are called p-hyperstructures. the most usual case is if (g, ·) is semigroup, then xpy = (xp)y∪x(py) = xpy and (g,p) is a semihypergroup. we do not know what hyperstructures are (g,pr) and (g,p l). in some cases, depending on the choice of p, the (g,pr) and (g,p l) can be associative or wass. if more operations are defined in g, then for each operation several p -hopes can be defined. 2 the bar in questionnaires last decades hyperstructures seem to have a variety of application not only in mathematics, but also in many other sciences [1], [2], [9], [13], [19], [25], including the social ones. an important application which can be used in social sciences is the combination of hyperstructure theory with fuzzy theory, by the replacement of the likert scale by the bar. the suggestion is the following [9]: 141 pipina nikolaidou definition 2.1. in every question substitute the likert scale with ’the bar’ whose poles are defined with ’0’ on the left end, and ’1’ on the right end: 0 1 the subjects/participants are asked instead of deciding and checking a specific grade on the scale, to cut the bar at any point s/he feels expresses her/his answer to the specific question. the use of the bar of vougiouklis & vougiouklis instead of a scale of likert has several advantages during both the filling-in and the research processing. the final suggested length of the bar, according to the golden ratio, is 6.2cm, see [7], [8], [23]. several advantages on the use of the bar instead of scale one can find in [9]. there are certain advantages concerning the use of the bar comparing to the likert-scale during all stages of developing, filling and processing. the most important maybe advantage of the bar though is the fact that it provides the potential for different types of processing. therefore, it gives the initiative to the researcher to explore if the given answers follow a special kind of distribution, as gauss or parabola for example. in this case the researcher has the opportunity to correct any kind of tendency appeared, for more accurate results. a possibility of choosing among a number of alternatives is offered, by using fuzzy logic in the same way as it has already been done combining mathematical models with multivalued operation. 3 evaluation the following survey is based on the described theory that has been established in the department of elementary education of democritus university of thrace, in the frame of course evaluation, and especially of algebra of first semester. the sample was 152 students, who were asked to answer questions related to the course, to the teacher and to the teaching of the course. the questionnaire used the bar, which was firstly divided into six equal-segments according to the first questionnaires which used a six-grade likert scale. the use of histograms helped in order to explore if the answers follow any kind of distribution or they present any kind of tendency. in this case, the bar is redivided into equal-area segments, for more accurate results. the filling questionnaire procedure has been accomplished using computers, and especially a software developed for this purpose. using this software the results can automatically be transferred for research elaboration. there are several advantages of the bar, the only disadvantage is to the data collection for further 142 multiple ways of processing in questionnaires elaboration. the implemented program has been developed to overcome the problems raised during the data collection, inputting of data from questionnaires to processing. it eliminates the time of data collection, transferring data directly for any kind of elaboration [10]. 3.1 question category: course the first question category is about the course and consists of 9 questions. gathering the answers on the bar, it is obvious that there is an upward trend, a fact that becomes even more obvious on the following histograms: figure 1: question category: course in the majority of the questions, on can notice a vast concentration in the last 2 or 3 grades and in some of the questions this is more obvious, as the concentration is the last grades is much higher. more specifically, question number 1,2,3,5,6 and 9 present the biggest concentration rate in the last 2 grades, while in question 4, there is a remarkable concentration in the center of the bar. based mainly on this histograms and some other parameters that have been obtained by the correspondence analysis, the answers of questions 2,4 and 6 will be redistributed on the bar, which will be now divided in equal-area segments: for question 2, the bar will be divided in 6 equal-ares segments according to the increasing-low parabola for question 4, the be will be divided in 6 equal-ares segments according to the 143 pipina nikolaidou gauss distribution and, for question 6, the bar will be divided in 6 equal-ares segments according to the increasing-upper parabola. the new obtained histograms are the following: figure 2: equal segments figure 3: equal-area segments one can see that the use of the upper-low parabola on question 2, reveals that in question 2, more than 50% was concentrated at the last two grades of the scale, but with the new distribution there exists a tendency to the first grades. for questions number 4 and 6 the new histograms give no more information. 3.2 question category:teaching the second question category consists of 6 questions relevant to the ’teaching of the lesson’ and to the extend that some factors contributed to its comprehension. the related histograms are the following: 144 multiple ways of processing in questionnaires figure 4: question category: teaching in this question category, there is also a general upward trend with the exception of question number 12. more specifically, in questions 10, 11 and 13 the biggest concentration rate appears in the last grades, in opposition to question 12, in which the biggest rate appears in the first 2 grades. question 14 present a vast rate in the last grade. so, for question 12 the bar will be divided in equal-area segments according to decreasing-low parabola and for question 14, according to increasing upper parabola. the new obtained histograms are the following: figure 5: equal segments figure 6: equal-area segments 145 pipina nikolaidou from the new distribution, the bar gives different results for question 12, as it reveals that the increasing-low trend not exists anymore. this fact is very important for the researcher as it gives him information he couldn’t have only through the first subdivision of the bar. the second question leads to the same results. 3.3 question category:teacher in the penultimate category there are 3 questions concerning the teacher. figure 7: question category: teacher once again, there is an obvious trend to the respondents according to the histograms, even more remarkable in the first question: there is a vast concentration rate in the last grade. because of that, the bar will be divided into equal area segments following the increasing-upper parabola: figure 8: equal segments figure 9: equal-area segments the new histogram is just confirming the first result. 146 multiple ways of processing in questionnaires 4 questionnares and hyperstructures in the research processing suppose that we want to use likert scale through the bar dividing the continuum [0,62] into equal segments and into equal area division of gauss distribution [9] or parabola distribution [24]. if we consider that the continuum [0,62] is divided into n segments, we can number the n segments starting with 0. we can define a hope on the segments as follows [11] : definition 4.1. for all i,j ∈{0,1, ...,n−1}, if en the nth segment ,then ei ⊕ej = {ek : x + y ∈ ek,∀x ∈ ei,y ∈ ej} therefore, we can consider as an organized device the group (zn,⊕) where n the number of segments, as we have a modulo like hyperoperation. the multiplication tables obtained by this hyperoperation , referred in mm, are the following: 6 equal segments 0:[0,10.33], 1:(10.33,20.66], 2:(20.66,30.99], 3:(30.99,41.32], 4:(41.32,51.65], 5:(51.65,62] ⊕ 0 1 2 3 4 5 0 0,1 1,2 2,3 3,4 4,5 0,5 1 1,2 2,3 3,4 4,5 0,5 0,1 2 2,3 3,4 4,5 0,5 0,1 1,2 3 3,4 4,5 0,5 0,1 1,2 2,3 4 4,5 0,5 0,1 1,2 2,3 3,4 5 0,5 0,1 1,2 2,3 3,4 4,5 6 equal-area segments (gauss distribution) 0:[0,22], 1:(22,27], 2:(27,31], 3:(31,35], 4:(35,40], 5:(40,62], ⊕ 0 1 2 3 4 5 0 0,1,2,3,4,5 1,2,3,4,5 2,3,4,5 3,4,5 4,5 0,5 1 1,2,3,4,5 5 5 5 0,5 0,1 2 2,3,4,5 5 5 0,5 0 0,1,2 3 3,4,5 5 0,5 0 0 0,1,2,3 4 4,5 0,5 0 0 0 0,1,2,3,4 5 0,5 0,1 0,1,2 0,1,2,3 0,1,2,3,4 0,1,2,3,4,5 147 pipina nikolaidou increasing low parabola x = y2 0:[0,34], 1:(34,43], 2:(43,49], 3:(49,54], 4:(54,58], 5:(58,62] ⊕ 0 1 2 3 4 5 0 0,1,2,3,4,5 0,1,2,3,4,5 0,2,3,4,5 0,3,4,5 0,4,5 0,5 1 0,1,2,3,4,5 0 0 0,1 0,1 0,1 2 0,2,3,4,5 0 0,1 0,1 1,2 1,2 3 0,3,4,5 0,1 0,1 1,2 1,2,3 2,3 4 0,4,5 0,1 1,2 1,2,3 2,3 3,4 5 0,5 0,1 1,2 2,3 3,4 4,5 increasing upper parabola 1−y = (1−x)2 0:[0,22], 1:(22,32], 2:(32,40], 3:(40,48], 4:(48,55], 5:(55,62] ⊕ 0 1 2 3 4 5 0 0,1,2,3, 1,2,3,4 2,3,4,5 0,3,4,5 4,5,0 0,5 1 1,2,3,4 0,3,4,5 0,4,5 0 0,1 0,1 2 2,3,4,5 0,4,5 0 0,1 0,1,2 1,2 3 0,3,4,5 0 0,1 0,1,2 1,2,3 2,3 4 0,4,5 0,1 0,1,2 1,2,3 2,3 3,4 5 0,5 0,1 1,2 2,3 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[25] t. vougiouklis and s. vougiouklis, the helix hyperoperations, italian journal of pure and applied mathematics, 18, (2005), 197. 150 ratio mathematica volume 38, 2020, pp. 261-285 geometrical foundations of the sampling design with fixed sample size pierpaolo angelini ∗ abstract we study the sampling design with fixed sample size from a geometric point of view. the first-order and second-order inclusion probabilities are chosen by the statistician. they are subjective probabilities. it is possible to study them inside of linear spaces provided with a quadratic and linear metric. we define particular random quantities whose logically possible values are all logically possible samples of a given size. in particular, we define random quantities which are complementary to the horvitz-thompson estimator. we identify a quadratic and linear metric with regard to two univariate random quantities representing deviations. we use the α-criterion of concordance introduced by gini in order to identify it. we innovatively apply to probability this statistical criterion. keywords: tensor product; linear map; bilinear map; quadratic and linear metric; α-product; α-norm 2010 ams subject classifications: 62d05. 1 ∗dipartimento di scienze statistiche, università la sapienza, roma, italia; pier.angelini@uniroma1.it 1received on may 12th, 2020. accepted on june 3rd, 2020. published on june 30th, 2020. doi: 10.23755/rm.v38i0.511. issn: 1592-7415. eissn: 2282-8214. c©p. angelini this paper is published under the cc-by licence agreement. 261 pierpaolo angelini 1 introduction given a finite population having n elements, we are only interested in considering samples containing units of this population where no element of the population under consideration can be selected more than once in the same sample (basu [1971]). we are not interested in considering ordered samples of a given size selected from a finite population (basu [1958]). on the other hand, when we consider not ordered samples where repetitions are not allowed we have no loss of information about a given parameter of the population under consideration (conti and marella [2015]). all logically possible samples of a given size belong to a given set. we suppose that we are always able to number them. it is known that if the number of all logically possible samples of a given set is very large then it could be a very hard or impossible work to give to them a number (godambe and joshi [1965]). a sampling design is characterized by a pair of elements (joshi [1971]). the first element of this pair represents the set of all logically possible samples selected from a finite population. the second element of this pair represents all probabilities assigned to the samples of the set of all logically possible samples of a given size. we consider a distribution of probability in this way (hartley and rao [1962]). each element of the set of all logically possible samples of a given size can be viewed as a logically possible event of a finite partition of incompatible and exhaustive elementary events. it is then possible to assign a subjective probability to each logically possible event of this partition (good [1962]). a probability subjectively assigned to each logically possible event of a finite partition of events must only be coherent. it is inadmissible when it is not coherent. a probability is subjectively assigned to each logically possible event of a finite partition of events even when it is an equal probability assigned to each of them. an equal probability assigned to each logically possible event of a finite partition of events is always a subjective judgment. we have to note a very important point: when we say that it is possible to assign a subjective and coherent probability to every logically possible event of a given set of events we mean that the choice of any value in the interval from 0 to 1 is allowed. such an interval includes both endpoints. it would therefore be possible to assign to every logically possible event of a given set of events a probability equal to 0. this choice is absolutely coherent. we will however introduce a restriction that is concerned with this point. we have to note another very important point: we methodologically distinguish what it is logically possible from what it is subjectively probable. what it is logically possible at a given instant it is not either certainly true or certainly false. one and only one element of the elements belonging to the set containing all logically possible elements at a given instant will be true a posteriori. a subjective probability is then assigned to each element of the set containing all logically possible elements before knowing this thing. 262 the α-criterion of concordance applied to probability 2 events as points in the space of random quantities we consider a finite set of vectors denoted by s into the field r of real numbers. we enumerate them. we consequently write s1, . . . ,sn, (1) where it turns out to be si ∈ s, i = 1, . . . ,n. we consider a linear space over r of all linear combinations of elements of s expressed in the form c1s1 + . . . + cnsn, (2) where every ci, i = 1, . . . ,n, is a real number. we observe that (2) is completely determined by the real numbers c1, . . . ,cn . each number ci is associated with the element si of the set s. it is known that an association is exactly a function. for each si ∈s and c ∈ r we then consider csi (3) to be the function that associates c with si and 0 with sj, where we have j 6= i. given a ∈ r, we obtain a(csi) = (ac)si. (4) given c′ ∈ r, we obtain (c + c′)si = csi + c ′si. (5) thus, it is possible to consider a linear space over r. it is the set of all functions of s into r. these functions can be written in the form given by (2). the functions 1s1, . . . , 1sn (6) are linearly independent so they represent a basis of the linear space under consideration. we have then to suppose that c1, . . . ,cn are elements of r such that it is possible to obtain the zero function given by c1s1 + . . . + cnsn = 0. (7) this means that we have ci = 0 for every ci, i = 1, . . . ,n. this thing consequently proves the linear independence under consideration. moreover, it is always possible to write si instead of 1si. a sample belonging to the set of all logically possible samples of a given size is then expressed by the vector δ(s′) =   δ(1; s′) δ(2; s′) ... δ(n; s′)   (8) 263 pierpaolo angelini having n components, where s′ is a sample of the set of all logically possible samples denoted by s′ (godambe [1955]). we will always consider vectors viewed as ordered lists of real numbers within this context. a sample can be expressed by the real numbers of a linear combination of n-dimensional vectors by means of which another n-dimensional vector is obtained. if a sample is identified with an n-dimensional vector then its components express the real numbers of a linear combination of the elements of a basis of the linear space under consideration. this linear space is denoted by rn . its basis is denoted by s = {ej}, j = 1, . . . ,n. we always consider orthonormal bases within this context. we therefore write δ(1; s′)e1 + δ(2; s ′)e2 + . . . + δ(n; s ′)en = y, (9) where we have y ∈ rn . we consider as many linear combinations of the elements of s = {ej}, j = 1, . . . ,n, as logically possible samples there are into the set of all logically possible samples of a given size denoted by s′. we note that the real numbers of every linear combination of the elements of s = {ej}, j = 1, . . . ,n, represent one of the logically possible samples of s′. we have evidently δ(i; s′) = { 1 if i ∈ s′ 0 if i /∈ s′ (10) for every i = 1, . . . ,n, where the elements of the population under consideration are overall n. we consider all logically possible samples of s′ having the same size denoted by n. since the population has got n elements we observe that the number of n-combinations is equal to the binomial coefficient denoted by ( n n ) . we observe that s′ whose elements are elementary events is a subset of rn . we say that s′ is embedded in rn . 3 finite partitions of logically possible elementary events given n, all logically possible samples whose size is equal to n belong to the set denoted by s′. we have n = n∑ i=1 δ(i; s′) (11) for every s′ ∈s′. every sample of the set of all logically possible samples corresponds to a vertex denoted by δ(s′) of an n-dimensional unit hypercube denoted by [0, 1]n . all logically possible samples of s′ can be viewed as possible events 264 the α-criterion of concordance applied to probability of a finite partition of incompatible and exhaustive elementary events (de finetti [1982b]). we are consequently able to define a univariate random quantity whose logically possible values are represented by all logically possible samples of s′. the logically possible values of it are not real numbers but they are n-dimensional vectors of an n-dimensional linear space over r. every logically possible sample belonging to s′ has a subjective probability of being selected (de finetti [1989]. it represents the degree of belief in the selection of a logically possible sample assigned by a given individual (the statistician) at a certain instant with a given set of information. an evaluation of probability known over a set of possible and elementary events coinciding with all logically possible samples of s′ is admissible when it is coherent. this means that it must be∑ s′∈s′ p(s′) = 1. (12) it is essential to note a very important point: we have to introduce an unusual restriction with regard to the coherence because we exclude of choosing a subjective probability equal to 0 with respect to any possible and elementary event. this implies that any logically possible sample of s′ has always a probability greater than zero of being selected. we have consequently 0 < p(s′) ≤ 1 (13) for every s′ ∈ s′ (coletti et al. [2015]). thus, conditions of coherence coincide with positivity of each probability of a random event and finite additivity of probabilities of incompatible and exhaustive events (gilio and sanfilippo [2014]). we will also consider bivariate random quantities whose components are two univariate random quantities (de finetti [2011]). if the logically possible values of these univariate random quantities are the same vectors of the same n-dimensional linear space over r then these random quantities have the same marginal distributions of probability. they represent the same finite partition of incompatible and exhaustive elementary events. putting them into a two-way table we observe that it is always a table having the same number of rows and columns. 4 first-order inclusion probabilities viewed as a coherent prevision of a univariate random quantity we consider a univariate random quantity denoted by s whose logically possible values are vectors of rn . given n and n, the number of the logically possible values of s coincides with the binomial coefficient expressed by( n n ) = k. (14) 265 pierpaolo angelini the set of the logically possible values of s is then given by i(s) = {s′1, . . . ,s′k}, with s′i ∈s′, i = 1, . . . ,k. a nonzero probability is assigned to each sample of the set of all logically possible samples. let p(s′1), . . . ,p(s ′ k) be these probabilities. it must therefore be k∑ i=1 p(s′i) = 1, (15) with 0 < p(s′i) ≤ 1 (16) for every i = 1, . . . ,k. it is possible to obtain an n-dimensional vector after assigning a nonzero probability to each sample of s′. we denote it with π. it represents the first-order inclusion probabilities of all units of the population under consideration. thus, we write π =   π1 π2 ... πn   = p(s′1)   δ(1; s′1) δ(2; s′1) ... δ(n; s′1)   + . . . + p(s′k)   δ(1; s′k) δ(2; s′k) ... δ(n; s′k)   , (17) where we have πi > 0 for every i = 1, . . . ,n. we have evidently written a convex combination of the vertices of the n-dimensional unit hypercube [0, 1]n corresponding to the samples of s′. each vertex is a sample having a nonzero weight representing a subjective probability. it is essential to note that π is a coherent prevision of s denoted by p(s). we therefore write π =   π1 π2 ... πn   = p(s) = k∑ i=1 δ(s′i)p(s ′ i). (18) we observe that the logically possible values of s are represented by vectors having n components so its coherent prevision must also be represented by a vector having n components. the logically possible values of s belong to the set denoted by i(s). each element of i(s) contains first-order inclusion “a posteriori” probabilities. this implies that π must contain first-order inclusion “a priori” probabilities based on the degree of belief in the selection of all logically possible samples attributed by the statistician at a certain instant with a given set of information. an “a posteriori” probability of a unit of the population of being included in a given sample is always predetermined. if a unit of the population is contained “a posteriori” in the sample that has been selected then its probability is equal to 1. if a unit of the population does not belong “a posteriori” to the sample that 266 the α-criterion of concordance applied to probability has been selected then its probability is equal to 0. a convex combination coinciding with p(s) has conveniently been taken under consideration because the logically possible values of s are incompatible and exhaustive elementary events of a finite partition of random events. in general, if we consider an event divided into two or more than two incompatible events then we obtain that its coherent probability is the sum of two or more than two coherent probabilities. this sum is a linear combination of probabilities (de finetti [1980]). we evidently consider a convex combination coinciding with p(s) within this context, where its weights or coefficients are “a priori” subjective probabilities connected with the samples of s′ (de finetti [1981]). this convex combination is characterized by k column vectors viewed as k matrices. each row of every n × 1 matrix is a first-order inclusion “a posteriori” probability. we therefore consider a linear combination of probabilities (de finetti [1982a]). 5 first-order inclusion probabilities obtained by means of linear maps we consider all logically possible samples belonging to the set s′. given n and n, let k be the number of all elements of s′. we are consequently able to determine an n × k matrix in r. we denote it by b. it is therefore possible to define a linear map expressed by lb : rk → rn. (19) it depends on b. moreover, it also depends on the choice of bases for rk and rn . we choose standard bases for rk and rn . we consider all probabilities assigned to the logically possible samples of s′ whose size is equal to n. they can be viewed as a column vector. we denote it by q. we have then q =   p(s′1) p(s′2) ... p(s′k)   . (20) it therefore turns out to be lb(q) = bq = π =   π1 π2 ... πn   . (21) 267 pierpaolo angelini we note that if k = n then we are able to define a linear map expressed by lb : rn → rn. (22) we observe that b is a square matrix. this linear map is an endomorphism. it is also an isomorphism. it is then an automorphism, so we write b−1π =   p(s′1) p(s′2) ... p(s′k)   . (23) given b, each row of q can subjectively vary because an evaluation of probability known over a set of logically possible events must only be coherent. this means that the sum of all probabilities of the samples of s′ must be equal to 1. we consequently observe that there are infinite ways of choosing all probabilities of the samples of s′. they are conveniently caught by lb. it is hence possible to obtain π as a multiplication of matrices according to a linear map depending on b and the standard bases of the linear spaces under consideration. also, we always obtain n∑ i=1 πi = n. (24) 6 first-order and second-order inclusion probabilities obtained by means of tensor products we consider a bivariate random quantity denoted by s12 whose components are two univariate random quantities denoted by 1s and 2s. we therefore write s12 = {1s, 2s}. given n and n, the logically possible values of each univariate random quantity coincide with k samples belonging to the set s′. they are all logically possible samples of s′ whose size is equal to n. each sample of s′ is a vector of rn . we have to note a very important point: we suppose that the logically possible values of 1s and 2s are the same n-dimensional vectors of the same n-dimensional linear space over r. these univariate random quantities have then the same marginal distributions of probability. putting them into a two-way table we observe that it is always a square table. we observe that all probabilities of the joint distribution of probability outside of the main diagonal of this table are always equal to 0. the nonzero probabilities of the joint distribution of probability coincide with p(s′1), . . . ,p(s ′ k). they are on the main diagonal of the table under consideration. a coherent prevision of s12 denoted by p(s12) 268 the α-criterion of concordance applied to probability is obtained by means of the sum of k square matrices. the number of rows and columns of every square matrix of this sum is equal to n. each square matrix of this sum derives from a tensor product belonging to the same linear space denoted by rn ⊗rn . it is an n2-dimensional linear space over r. we always consider as many tensor products as joint probabilities are associated with the samples of s′. we have then p(s′i)     δ(1; s′i) δ(2; s′i) ... δ(n; s′i)   ,   δ(1; s′i) δ(2; s′i) ... δ(n; s′i)     7→ p(s′i)     δ(1; s′i) δ(2; s′i) ... δ(n; s′i)  ⊗   δ(1; s′i) δ(2; s′i) ... δ(n; s′i)     (25) for every i = 1, . . . ,k. we note that it turns out to be p(s′i)     δ(1; s′i) δ(2; s′i) ... δ(n; s′i)  ⊗   δ(1; s′i) δ(2; s′i) ... δ(n; s′i)     = p(s′i)   δ(1; s′i) δ(2; s′i) ... δ(n; s′i)  [δ(1; s′i) δ(2; s′i) . . . δ(n; s′i)] . (26) if we consider a coherent prevision of s12 then we deal with a bilinear map expressed by rn × rn → mn, n (r), where the linear space over r of the n × n matrices in r is denoted by mn, n (r). this linear space is isomorphic to rn2 . the matrix product resulting from this bilinear map is factorized by means of the tensor product of vectors of rn . it is also factorized by means of a unique linear map whose domain coincides with rn ⊗ rn . this is because we are able to know a basis of rn ⊗ rn as well as the value of the linear map under consideration on basis elements. we suppose that a basis of rn ⊗ rn results from the standard basis of rn , where rn is evidently considered two times. it is therefore possible to say that there exists a unique linear map given by rn ⊗ rn → mn, n (r). it coincides with the product of a joint probability viewed as a scalar and a square matrix. we consider k products of a joint probability and a square matrix. we obtain k square matrices in this way. we consider the sum of these k square matrices in order to obtain a coherent prevision of s12. we observe that rn × rn → mn, n (r) and rn ⊗ rn → mn, n (r) have the same codomain. a factorization of rn × rn → mn, n (r) is then realized by means of a bilinear map given by rn × rn → rn ⊗ rn and a linear map given by rn ⊗rn →mn, n (r). these two maps are connected, so we obtain a composition of functions identified with rn × rn →mn, n (r). the following 269 pierpaolo angelini commutative diagram rn ×rn rn ⊗rn mn, n (r) permits of visualizing what we have said. a coherent prevision of s12 is then bilinear and homogeneous. it is given by p(s12) = π =   π1 π12 . . . π1n π21 π2 . . . π2n . . . . . . . . . . . . πn1 πn2 . . . πn   =   π1 π12 . . . π1n π12 π2 . . . π2n . . . . . . . . . . . . π1n π2n . . . πn   . (27) it coincides with the symmetric matrix of the first-order and second-order inclusion probabilities. the trace of this matrix is evidently equal to n (angelini [2020]). 7 the covariance of two univariate random quantities obtained by considering two bilinear maps given s12 = {1s, 2s}, the covariance of 1s and 2s is expressed by c(1s, 2s) = p(s12) −p(1s)p(2s), (28) where p(s12) represents the prevision or mathematical expectation or expected value of s12, while p(1s) and p(2s) represent the prevision or mathematical expectation or expected value of 1s and 2s. we note that a coherent prevision of s12 derives from a bilinear map because we have p(s12) =   π1 π12 . . . π1n π21 π2 . . . π2n . . . . . . . . . . . . πn1 πn2 . . . πn   . (29) moreover, since we have p(1s) =   π1 π2 ... πn   (30) 270 the α-criterion of concordance applied to probability as well as p(2s) =   π1 π2 ... πn   , (31) we note that the product of these two linear maps is evidently bilinear. such a product is expressed in the form  π1 π2 ... πn  [π1 π2 . . . πn] =   π1π1 π1π2 . . . π1πn π2π1 π2π2 . . . π2πn . . . . . . . . . . . . πnπ1 πnπ2 . . . πnπn   . (32) it is then evident that the covariance of 1s and 2s derives from two bilinear maps because we can write c(1s, 2s) =   π1 π12 . . . π1n π21 π2 . . . π2n . . . . . . . . . . . . πn1 πn2 . . . πn  −   π1π1 π1π2 . . . π1πn π2π1 π2π2 . . . π2πn . . . . . . . . . . . . πnπ1 πnπ2 . . . πnπn   . (33) by writing c(1s, 2s) =   (π1 −π1π1) (π12 −π1π2) . . . (π1n −π1πn ) (π21 −π2π1) (π2 −π2π2) . . . (π2n −π2πn ) . . . . . . . . . . . . (πn1 −πnπ1) (πn2 −πnπ2) . . . (πn −πnπn )   (34) we note that it is possible to consider as many random components as inclusion probabilities are studied. a unit of the population under consideration can be included, or not, in a given sample (bondesson [2010]). this thing is uncertain until a given sample is selected (hájek [1958]). two different units of the population under consideration can be included, or not, in the same sample (deville and tillé [1998]). this thing is uncertain until a given sample is selected. a component associated with one or two different units of the population under consideration is evidently random for this reason (connor [1966]). this means that each random component is characterized by a subjective probability. it is an “a priori” probability. it is also characterized by an “a posteriori” probability coinciding with one of the two logically possible values of a random event, 0 and 1. one and only one of these two logically possible values of a random event will be true “a posteriori”. on the other hand, it is known that the notion of probability basically 271 pierpaolo angelini deals with an aspect that is included between two extreme aspects. the first extreme aspect deals with situations of non-knowledge or ignorance or uncertainty determining the set of all logically possible samples of a given size viewed as elementary events. they are evidently all logically possible alternatives that can be considered. the second extreme aspect deals with definitive certainty expressed in the form of what it is certainly true or certainly false. thus, every logically possible sample of a given size definitively becomes true or false. probability is subjectively distributed by the statistician as a mass over the domain of all logically possible samples of a given size before knowing which is the true sample to be selected “a posteriori”. having said that, the variance of every random component as well as the covariance of two random components are dealt with by means of the first-order and second-order inclusion probabilities. the variance of each random component is represented by every element on the main diagonal of the symmetric matrix given by (34). the covariance of two random components is represented by every element outside of the main diagonal of the square matrix given by (34). 8 a univariate random quantity representing deviations we define a univariate random quantity representing deviations. we denote it by d. we firstly consider s whose values are all logically possible samples of a given size viewed as elementary events belonging to the set s′. given n and n, the number of the logically possible values of s is equal to ( n n ) = k. the set of the logically possible values of s is then given by i(s) = {s′1, . . . ,s′k}, with s′i ∈ s′, i = 1, . . . ,k. a nonzero probability denoted by p(s′i), i = 1, . . . ,k, is assigned to each sample of s′. we therefore obtain an n-dimensional vector denoted by π. it represents the first-order inclusion probabilities of all units of the population under consideration. they are all greater than zero. this vector is always independent of the origin of the coordinate system that we could consider. we note that the number of the logically possible values of d is equal to k. it is the same of the one of s. the set of the logically possible values of d is given by i(d) = {d′1, . . . ,d′k}, with d′i =     δ(1; s′i) δ(2; s′i) ... δ(n; s′i)  −   π1 π2 ... πn     , (35) 272 the α-criterion of concordance applied to probability where we have i = 1, . . . ,k. it follows that we have p(s′1)d ′ 1 + . . . + p(s ′ k)d ′ k =   0 0 ... 0   . (36) this means that p(s) is an n-dimensional vector such that all deviations from it that are multiplied by the corresponding probabilities represent n-dimensional vectors whose sum coincides with the zero vector of rn . we are now able to calculate the variance of s by using d. we refer to the α-criterion of concordance introduced by gini. it is a statistical criterion that we innovatively apply to probability viewed as a mass. an absolute maximum of concordance is then realized when each d′i, i = 1, . . . ,k, is multiplied by itself. if each d ′ i, i = 1, . . . ,k, is multiplied by itself then we obtain k square matrices. every multiplication that we consider is a tensor product of two vectors of rn . these two vectors represent two deviations which are the same. the components of these two vectors are then the same. hence, the variance of s coincides with the sum of k traces of k square matrices. each trace of the square matrix under consideration is an inner product viewed as an α-product. an α-product is a bilinear form. we consider each p(s′i), i = 1, . . . ,k, as a scalar. each p(s′i), i = 1, . . . ,k, is firstly a subjective probability. thus, it always characterizes a random quantity. it is nevertheless viewed as a scalar within this context. we can therefore multiply all components of d′i by p(s′i), i = 1, . . . ,k. we note that the components of each d ′ i, i = 1, . . . ,k, are always independent of the origin of the coordinate system that we could consider. we therefore write σ2s = tr ( d′1 t (p(s′1)d ′ 1) ) + . . . + tr ( d′k t (p(s′k)d ′ k) ) . (37) we have evidently introduced a quadratic and linear metric in this way. we therefore note that σ2s is the sum of the squares of k α-norms. it is possible to verify that every trace of a square matrix is an α-product which is an α-commutative product, an α-associative product, an α-distributive product and an α-orthogonal product. we have to note a very important point: s and d are two different quantities from a geometric point of view because they are represented by different sets of n-dimensional vectors. they are nevertheless the same quantity from a randomness point of view. they are characterized by the same probabilities. we therefore observe the same events because we consider only a change of origin. 273 pierpaolo angelini 9 intrinsic properties of a univariate random quantity representing deviations translations and rotations of vectors identifying a given univariate random quantity representing deviations are intrinsic properties of it. they do not depend on the choice of a basis of a given linear space. we say that all vectors of s′ are subjected to the same translation when we consider k sums of two vectors. we consider k sums of two vectors because the number of the elements of s′ is equal to k. the first vector of each sum of them is given by s′i, i = 1, . . . ,k. the second vector of each sum of them is given by an arbitrary n-dimensional vector which is always the same. we say that all vectors of s′ are then subjected to the same change of origin. it follows that σ2s is invariant with respect to a translation of all vectors of s′. we say that a quadratic and linear metric is invariant with respect to a translation of all vectors of s′. concerning a rotation, let a = (ai′j ) be an n×n orthogonal matrix. each element of this matrix is denoted by two indices. we use contravariant and covariant indices without loss of generality. the contravariant indices represent the rows of the matrix. we have i′ = 1, . . . ,n. the covariant indices represent the columns of the matrix. we have j = 1, . . . ,n. we observe that rotations of all vectors contained in i(d) = {d′1, . . . ,d′k} are characterized by a. we write ra(d′i) : d ′ i ⇒ ad ′ i = (d ′ i) ∗, (38) where we have i = 1, . . . ,k. we evidently denote by (d′i) ∗ the result of the rotation of the vector d′i obtained by means of the orthogonal matrix denoted by a. the vector (d′i) ∗ is an n-dimensional vector. its components are originated by n linear and homogeneous relationships. we have to note a very important point: p(s) is an n-dimensional vector such that all rotated deviations from it that are multiplied by the corresponding probabilities represent n-dimensional vectors whose sum coincides with the zero vector of rn . we have then p(s′1)(d ′ 1) ∗ + . . . + p(s′k)(d ′ k) ∗ =   0 0 ... 0   . (39) if we consider rotated deviations then we write σ2s∗ = tr ( (d′1) ∗t (p(s′1)(d ′ 1) ∗) ) + . . . + tr ( (d′k) ∗t (p(s′k)(d ′ k) ∗) ) , (40) where s∗ represents a univariate random quantity connected with rotated deviations. since it turns out to be σ2s = σ 2 s∗, (41) 274 the α-criterion of concordance applied to probability we say that the variance of s is invariant with respect to all rotated deviations obtained by means of the same orthogonal matrix denoted by a. we have therefore introduced a quadratic and linear metric which is invariant with respect to translations and rotations of vectors identifying a univariate random quantity representing deviations. 10 a univariate random quantity representing variations and its intrinsic properties we define a univariate random quantity representing variations. we denote it by v . given d, the set of the logically possible values of v is expressed by i(v ) = {v′1, . . . ,v′k}, with v′i = d ′ i 1√ σ2s , (42) where we have i = 1, . . . ,k. we therefore note that s, d and v are different quantities from a geometric point of view. they are conversely the same quantity from a randomness point of view. it is possible to verify that it turns out to be σ2v = 1. (43) this index is always equal to 1 independently of the components of d′i, i = 1, . . . ,k. it is evident that these components identify σ2s, so we say that σ 2 v = 1 is also independent of σ2s. we observe that rotations of all vectors belonging to i(v ) = {v′1, . . . ,v′k} are always characterized by an n × n orthogonal matrix. we write (v′i) ∗ = (d′i) ∗ 1√ σ2s , (44) where we have i = 1, . . . ,k. if we consider translations and rotations of vectors identifying a univariate random quantity representing variations then we observe intrinsic properties that we have already considered. we note that v can be subjected to an affine transformation. if v is subjected to an affine transformation then we write v ⇒ av + b, (45) where we have a 6= 0. we therefore observe that each vector of i(av + b) is equal to the corresponding vector of i(v ). this means that the components of each vector of i(av + b) are the same of the ones of the corresponding vector of i(v ). hence, we say that univariate random quantities representing variations are invariant with respect to an affine transformation. given s12 = {1s, 2s}, we note 275 pierpaolo angelini that we have 1v = 2v = v if and only if it turns out to be 1s = 2s = s. it is possible to verify that the covariance of 1v and 2v is an α-product. it is always equal to 1. on the other hand, it coincides with the bravais-pearson correlation coefficient in the case of a perfect direct linear relationship between two quantities. it is possible to verify that the bravais-pearson correlation coefficient is invariant with respect to rotations of vectors belonging to i(v ). it is therefore invariant with respect to an affine transformation of v . we have to note a very important point: intrinsic properties that we have considered can be related to the random quantities themselves or to specific metric indices based on these quantities. specific metric indices are evidently based on random quantities representing deviations or variations because we calculate them after taking such random quantities into account. we have to note another very important point: we are not interested in translating or rotating a geometric object in real terms but we are interested in studying its intrinsic properties because these properties are a fundamental consequence of its geometric representation. 11 metric aspects of an estimate of the population mean we want to wonder what happens from a metric point of view when we study one or more than one attribute with respect to each element of the population under consideration. we suppose of observing three different and independent characteristics of each element of the population under consideration. we admit this thing without loss of generality. we therefore consider three different and independent variables denoted by x, y and z. we note that x is the variable concerning the first attribute of each element of the population under consideration. the variable concerning the second attribute of each element of the population under consideration is denoted by y . the variable concerning the third attribute of each element of the population under consideration is denoted by z. if we study only one attribute of each element of the population under consideration then we estimate the population mean by using the univariate horvitz-thompson estimator. it is defined by t (x) ht = 1 n n∑ i=1 1 πi δ(i; s′)xi, (46) where we have s′ ∈ s′. it is linear and homogeneous (horvitz and thompson [1952]). we note that s′ is one of the logically possible samples of s′. also, the weight of the generic unit i of the population under consideration never depends on s′. it is obtained beginning from (17). we have conversely considered all 276 the α-criterion of concordance applied to probability logically possible samples of s′ when we have defined s, d and v . we did not consider only one of them. these random quantities are complementary to the univariate horvitz-thompson estimator for this reason. also, we have always taken p(s) = π into account when we have defined s, d and v . on the other hand, a coherent prevision of s is itself linear and homogeneous. the expected value of the univariate horvitz-thompson estimator is given by e[t (x) ht ] = µx. (47) it is equal to the population mean denoted by µx for any vector (x1 x2 . . . xn )t ∈ rn . we have µx = 1 n n∑ i=1 xi. (48) the variance of the univariate horvitz-thompson estimator is given by v(t (x) ht ) = 1 n2 n∑ i=1 n∑ j=1 xi πi xj πj ∆ij, (49) where we have ∆ij = πij −πiπj, with i,j = 1, . . . ,n. we note that ∆ij, i,j = 1, . . . ,n, is obtained by means of (34). since we consider all logically possible samples whose size is equal to n we can also write v(t (x) ht ) = − 1 2n2 n∑ i=1 n∑ j=1 ( xi πi − xj πj )2 ∆ij, (50) where we have again ∆ij = πij − πiπj, with i,j = 1, . . . ,n (yates and grundy [1953]). this variance is estimated by the univariate yates-grundy estimator given by v̂y g(t (x) ht ) = 1 2n2 ∑ i∈s′ ∑ j∈s′ ( xi πi − xj πj )2 πiπj −πij πij , (51) where we have πij > 0 because we assume that the sampling design is measurable and πij ≤ πiπj, with i,j = 1, . . . ,n. the same thing goes when we consider y and z. we have to note a very important point: the variance of s denoted by σ2s coincides with the variance of the univariate horvitz-thompson estimator given by (50) when the absolute values of each deviation of xi from xj, with i 6= j = 1, . . . ,n, are multiples of n. in addition to this thing, the variance of s coincides with the variance of the univariate horvitz-thompson estimator given by (50) when the entropy h of the sampling design with fixed sample size is maximum (tillé and wilhelm [2017]), where we have h = − ∑ s′∈s′ p(s′) log p(s′). (52) 277 pierpaolo angelini we note that h is maximum when we have p(s′1) = p(s ′ 2) = . . . = p(s ′ k), (53) with ∑k i=1 p(s ′ i) = 1. it does not turn out to be p(s ′) = 0 within this context. however, if we observe p(s′) = 0 with regard to (52) then it turns out to be [0 log 0] = 0 by convention. we therefore say that the weights of the univariate horvitz-thompson estimator are based on a coherent prevision of s. we have obtained a linear and quadratic metric by considering two univariate random quantities representing deviations. we have obtained the variance of s by using this metric. the same thing goes when we consider y and z. we have to note another very important point: by studying three different and independent attributes of each element of the population under consideration we do not jointly consider three variables but we jointly consider two variables at a time. this is because it is not appropriate to use a trilinear form when we deal with metric relationships. if we jointly study two attributes of each element of the population under consideration then we estimate the bivariate population mean by using the bivariate horvitz-thompson estimator. we write t (xy) ht = 1 n2 n∑ i=1 n∑ j=1 1 πi δ(i; s′)xi 1 πj δ(j; s′)yj (54) when we jointly consider x and y , where all first-order inclusion probabilities are greater than zero. they are obtained by means of (17). we write t (xz) ht = 1 n2 n∑ i=1 n∑ j=1 1 πi δ(i; s′)xi 1 πj δ(j; s′)zj (55) when we jointly consider x and z, where all first-order inclusion probabilities are greater than zero. they are obtained by means of (17). we write t (yz) ht = 1 n2 n∑ i=1 n∑ j=1 1 πi δ(i; s′)yi 1 πj δ(j; s′)zj (56) when we jointly consider y and z, where all first-order inclusion probabilities are greater than zero. they are obtained by means of (17). the bivariate horvitzthompson estimator is obtained by multiplying two linear and homogeneous expressions. this means that what we have said concerning the weights of the univariate horvitz-thompson estimator does not change. the expected value of the bivariate horvitz-thompson estimator concerning x and y is given by e[t (xy) ht ] = 1 n2 n∑ i=1 n∑ j=1 1 πi e[δ(i; s′)]xi 1 πj e[δ(j; s′)]yj. (57) 278 the α-criterion of concordance applied to probability we observe that it turns out to be e[δ(i; s′)] = πi as well as e[δ(j; s′)] = πj for every s′ ∈ s′, i,j = 1, . . . ,n. it is therefore evident that (57) is equal to the population mean denoted by µ(xy) for any vector (x1 x2 . . . xn )t ∈ rn and (y1 y2 . . . yn ) t ∈ rn , where we have µ(xy) = 1 n2 n∑ i=1 n∑ j=1 xi yj. (58) the same thing goes when we consider the expected value of the bivariate horvitzthompson estimator concerning x and z as well as the expected value of the bivariate horvitz-thompson estimator concerning y and z. we consider an auxiliary variable denoted by x′ related to x when the values of x given by xi, i = 1, . . . ,n, are unknown. we consider an auxiliary variable denoted by y ′ related to y when the values of y given by yi, i = 1, . . . ,n, are unknown. we consider an auxiliary variable denoted by z′ related to z when the values of z given by zi, i = 1, . . . ,n, are unknown. the known values of x′ are given by x′i, i = 1, . . . ,n. we write µx′ = 1 n n∑ i=1 x′i. (59) if x and x′ are approximately proportional then it turns out to be xi x′i ≈ constant, (60) where we have i = 1, . . . ,n. the first-order inclusion probabilities chosen by the statistician are then given by πi = nx′i nµx′ , (61) where we have i = 1, . . . ,n. we note that such probabilities are used into (23) in order to obtain p(s′i), i = 1, . . . ,k, when we have k = n. we observe that p(s′i), i = 1, . . . ,k, are used in order to obtain a coherent prevision of s. if we have k 6= n then we consider a system of n linear equations with k unknowns, where π1, . . . ,πn are constant terms. we evidently refer to (21). we therefore observe that π1, . . . ,πn represent a coherent prevision of s obtained beginning from p(s′i), i = 1, . . . ,k. we observe that α-products and α-norms use p(s ′ i), i = 1, . . . ,k, as scalars. also the second-order inclusion probabilities characterize our metric structure. they are obtained by means of tensor products having p(s′i), i = 1, . . . ,k, as scalars. they are chosen by the statistician because he subjectively chooses p(s′i), i = 1, . . . ,k. he is consequently able to observe πij > 0, i,j = 1, . . . ,n. we have established them in (27). the same thing goes when we consider y ′ and z′. 279 pierpaolo angelini 12 a metric homoscedasticity of different variables identifying different and independent attributes of the units of the population we have jointly to consider two variables at a time for a metric reason. when we jointly consider x and y we have firstly to disaggregate t(xy)ht . given t (x) ht = 1 n n∑ i=1 1 πi δ(i; s′)xi (62) and t (y) ht = 1 n n∑ j=1 1 πj δ(j; s′)yj, (63) the covariance of these two univariate horvitz-thompson estimators is therefore expressed by c(t (x) ht , t (y) ht ) = 1 n2 n∑ i=1 n∑ j=1 xi πi yj πj ∆ij, (64) where we have ∆ij = πij −πiπj, with i,j = 1, . . . ,n. we note that ∆ij, i,j = 1, . . . ,n, is obtained by means of (34). the same thing goes when we jointly consider x and z as well as y and z. we note that c(t (x) ht , t (x) ht ) = v(t (x) ht ) = 1 n2 n∑ i=1 n∑ j=1 xi πi xj πj ∆ij, (65) where we have ∆ij = πij − πiπj, i,j = 1, . . . ,n. we observe that ∆ij, i,j = 1, . . . ,n, is obtained by means of (34). we note that c(t (y) ht , t (y) ht ) = v(t (y) ht ) = 1 n2 n∑ i=1 n∑ j=1 yi πi yj πj ∆ij, (66) where we have ∆ij = πij − πiπj, with i,j = 1, . . . ,n. we observe that ∆ij, i,j = 1, . . . ,n, is obtained by means of (34). it is also possible to write c(t (z) ht , t (z) ht ) = v(t (z) ht ) = 1 n2 n∑ i=1 n∑ j=1 zi πi zj πj ∆ij, (67) where we have ∆ij = πij − πiπj, with i,j = 1, . . . ,n. we observe that ∆ij, i,j = 1, . . . ,n, is obtained by means of (34). we are interested in knowing 280 the α-criterion of concordance applied to probability what happens from a metric point of view when we study three different and independent attributes with respect to each element of the population under consideration. we have defined s, d and v . in particular, we consider a bivariate random quantity representing deviations. it is expressed by d12 = {1d, 2d}. its components are two univariate random quantities, 1d and 2d, identifying two sets of n-dimensional vectors. each vector of a set of n-dimensional vectors is equal to the corresponding vector of the other set of n-dimensional vectors. we have consequently i(1d) = i(2d) = {d′1, . . . ,d′k}. given p(s ′ i), i = 1, . . . ,k, we observe that 1d is equal to 2d, so the covariance of 1d and 2d is equal to the variance of s denoted by σ2s. we observe this thing regardless of any pair of variables that we consider. we could indifferently consider x and y or x and z or y and z. on the other hand, if we take 1v and 2v into account then we note that their covariance is equal to 1. since it turns out to be 1v = 2v = v we say that the variance of v is equal to 1. we observe this thing regardless of any pair of variables that we consider. we could indifferently consider x and y or x and z or y and z. we therefore say that x, y and z are homoscedastic from a metric point of view. we say this thing after considering all logically possible samples having a given size belonging to s′. we say this thing after defining s with respect to x, y , z. we say this thing because, given p(s′i), i = 1, . . . ,k, the variance of s is always the same. it is obtained by virtue of the metric structure that we have introduced. 13 what is all this for? all the first-order inclusion probabilities derive from a coherent prevision of s. a coherent prevision of s always depends on p(s′i), i = 1, . . . ,k, where these probabilities are coherently chosen by the statistician. all the second-order inclusion probabilities derive from a coherent prevision of s12. a coherent prevision of s12 always depends on p(s′i), i = 1, . . . ,k. a coherent prevision of s is linear and homogeneous. a coherent prevision of s12 is bilinear and homogeneous. the bivariate horvitz-thompson estimator is obtained by multiplying two linear and homogeneous expressions. this means that what we are going to say concerning the weights of the univariate horvitz-thompson estimator continues to be valid even when we make reference to the bivariate horvitz-thompson estimator. we therefore make reference to the first-order inclusion probabilities. if there exists a direct linear relationship between x′ and x then the statistician chooses high inclusion probabilities denoted by πi with respect to the units of the population under consideration having high attributes of x′ denoted by x′i, i = 1, . . . ,n. this is because they are likely associated with high attributes of x denoted by xi, i = 1, . . . ,n. the same thing goes when we consider a direct linear relationship 281 pierpaolo angelini between y ′ and y as well as between z′ and z. if x and x′ are approximately proportional then the first-order inclusion probabilities chosen by the statistician are given by πi = nx′i∑n j=1 x ′ j , (68) where we have i = 1, . . . ,n. if it turns out to be πi > 1 for some unit of the population under consideration then we have πi = 1 for all units of the population under consideration having i as a label and such that it turns out to be nx′i ≥ ∑n j=1 x ′ j because x ′ i is high. we consider n > 1 within this context. the statistician consequently chooses πi = (n−na) x′i∑n j=1 j /∈a x′j , (69) where we have i = 1, . . . ,n, i /∈ a, concerning the remaining units of the population under consideration. the set of the units of the population under consideration such that it turns out to be nx′i ≥ ∑n j=1 x ′ j is denoted by a, while their number is denoted by na. the same thing goes when we consider y ′ and y as well as z′ and z. having said that, we evidently establish a linear relationship between p(s′i), i = 1, . . . ,k, and πi, i = 1, . . . ,n. if the statistician chooses p(s ′ i), i = 1, . . . ,k, with ∑k i=1 p(s ′ i) = 1, then it is possible to get πi, i = 1, . . . ,n, with∑n i=1 πi = n. we write   π1 π2 ... πn   = k∑ i=1 δ(s′i)p(s ′ i). (70) he is consequently able to obtain πi > 0 for every i = 1, . . . ,n. conversely, if the statistician chooses πi, i = 1, . . . ,n, then it is possible to get p(s′i), i = 1, . . . ,k. we observe that α-products and α-norms use p(s′i), i = 1, . . . ,k, as scalars. we obtain different metric relationships by using α-norms whose scalars are p(s′i), i = 1, . . . ,k. we note that π1, . . . ,πn are used into b−1p(s) =   p(s′1) p(s′2) ... p(s′k)   (71) in order to obtain p(s′i), i = 1, . . . ,k, when we have k = n. we note that b is a square matrix, while b−1 is its inverse. if we have k 6= n then we consider 282 the α-criterion of concordance applied to probability a system of n linear equations with k unknowns, where π1, . . . ,πn are constant terms. we evidently refer to lb(q) = b   p(s′1) p(s′2) ... p(s′k)   =   π1 π2 ... πn   = p(s). (72) it is known that if the statistician chooses appropriate inclusion probabilities then he is able to obtain a more efficient estimator of the population mean. 14 conclusions we have considered random quantities whose logically possible values are all logically possible samples of a given size belonging to a given set. every logically possible sample belonging to a given set has a subjective probability of being selected. we have obtained the first-order inclusion probabilities by means of coherent previsions of univariate random quantities. we have defined bivariate random quantities whose components are two univariate random quantities having all logically possible samples of a given size as their logically possible values. all univariate random quantities which we have defined are complementary to the univariate horvitz-thompson estimator. it is linear and homogeneous like a coherent prevision of a univariate random quantity whose logically possible values are all logically 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anti-fuzzy graphs s. firthous fatima* k. janofer† abstract in this paper, the notion of connected end anti-fuzzy equitable dominating set of an antifuzzy graph is discussed. the connected end anti-fuzzy equitable domination number for some standard graphs are obtained. the relation between anti-fuzzy equitable domination number, end anti-fuzzy equitable domination number and connected end anti-fuzzy equitable domination number are established. theorems related to these parameters are stated and proved. keywords: dominating set; end anti-fuzzy equitable dominating set; connected end anti-fuzzy equitable dominating set 2010 ams subject classification: 05c62, 05e99, 05c07‡ * assistant professor, department of mathematics, sadakathullah appa college (autonomous), rahmath nagar, tirunelveli, india, 627011; kitherali@yahoo.co.in. † part-time research scholar, reg. no: 19131192092019, department of mathematics, sadakathullah appa college (autonomous), rahmath nagar, tirunelveli, india, 627011, affiliated to manonmaniam sundaranar university, abishekaptti, tirunelveli 627012, india; janofermath@gmail.com. ‡received on september 25, 2022. accepted on march 1, 2023. published on march 21, 2023. doi:10.23755/rm.v39i0.861. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 271 mailto:kitherali@yahoo.co.in mailto:janofermath@gmail.com s. firthous fatima and k. janofer 1. introduction a graph is an advantageous method for representing the data including connection between objects. the objects are represented by nodes and relations by arcs. whenever there is uncertainty or vagueness in the description of items or in its connections or in both, it is common that we have to plan an anti-fuzzy graph model. a fuzzy set, as a superset of a crisp set, owes its origin to the work of zadeh [14] in 1965 that has been introduced to deal with uncertainty. m. akram [1] defined the concept of anti-fuzzy graph structures in 2012. a. somasundaram and s.somasundaram [12] presented several types of domination parameters such as independent domination, total domination, connected domination and domination in cartesian product and composition of fuzzy graphs. r.muthuraj and a. sasireka [7] introduced domination in anti-fuzzy graphs. the concept of equitable domination in graphs was introduced by swaminathan and dharmalingam [13]. the end equitable domination number in graph has introduced by j.h.hattingh and m.h.henning [8]. further results were extended by murthy and puttaswamy [10]. some works in extension of fuzzy graphs can be found in [5, 6, 9]. s.firthous fatima and k.janofer [2, 3, 4] introduced the concept of anti-fuzzy equitable dominating set, connected anti-fuzzy equitable dominating set and end antifuzzy equitable dominating set of an anti-fuzzy graphs. in this paper, the connected end anti-fuzzy equitable domination set of an anti-fuzzy graph is introduced. the connected end anti-fuzzy equitable domination number of an anti-fuzzy graphs is also obtained. 2. preliminaries definition 2.1[1] a fuzzy graph is said to be an anti-fuzzy graph with a pair of functions 𝜎 ∶ 𝑉 → [0,1] and 𝜇 ∶ 𝑉 × 𝑉 → [0,1], where for all 𝑢, 𝑣 ∈ 𝑉, we have 𝜇(𝑢, 𝑣) ≥ 𝜎(𝑢) ∨ 𝜎(𝑣) and it is denoted by 𝐺𝐴𝐹 (𝜎, 𝜇) or 𝐺(𝜎, 𝜇). definition 2.2 [1] the order 𝑝 and size 𝑞 of an anti-fuzzy graph 𝐺𝐴𝐹 = (𝜎, 𝜇) are defined to be 𝑝 = ∑ 𝜎(𝑢)𝑢∈𝑉 and 𝑞 = ∑ 𝜇(𝑢, 𝑣)𝑢𝑣∈𝐸 . it is denoted by 𝑂(𝐺) and 𝑆(𝐺). note 2.1 in all the examples, 𝜎 is chosen suitably and the function 𝜇 considered as reflexive and symmetric and 𝐺 is an undirected anti-fuzzy graph. definition 2.3 an anti-fuzzy graph 𝐺𝐴𝐹 is said to be bipartite if the vertex set 𝑉 can be partitioned into two sets 𝜎1 on 𝑉1 and 𝜎2 on 𝑉2 such that 𝜇(𝑣1, 𝑣2) = 0 if (𝑣1, 𝑣2) ∈ 𝑉1 × 𝑉1 or (𝑣1, 𝑣2) ∈ 𝑉2 × 𝑉2. definition 2.4 a bipartite anti-fuzzy graph 𝐺𝐴𝐹 is said to be complete bipartite antifuzzy graph if 𝜇(𝑣1, 𝑣2) = 𝜎(𝑣1) ∨ 𝜎(𝑣2) for all 𝑣1 ∈ 𝑉1 and 𝑣2 ∈ 𝑉2 and is denoted by 𝐾𝜎1,𝜎2 . definition 2.5 a path in an anti-fuzzy graph 𝐺𝐴𝐹 is a sequence of distinct vertices 𝑢0, 𝑢1, 𝑢2, . . . , 𝑢𝑛 such that µ(𝑢𝑖−1 , 𝑢𝑖 ) = 𝜎(𝑢𝑖−1) ∨ 𝜎(𝑢𝑖 ), 1 ≤ 𝑖 ≤ 𝑛, 𝑛 > 0 is called the length of the path. the path in an anti-fuzzy graph is called an anti-fuzzy cycle if 𝑢0 = 𝑢𝑛, 𝑛 ≥ 3. 272 connected end anti-fuzzy equitable dominating set in anti-fuzzy graphs definition 2.6 an anti-fuzzy graph 𝐺𝐴𝐹 is said to be cyclic if it contains at least one anti-fuzzy cycle, otherwise it is called acyclic. definition 2.7 an anti-fuzzy graph 𝐺𝐴𝐹 is said to be connected if there exists at least one path between every pair of vertices. a connected acyclic anti-fuzzy graph is said to be an anti-fuzzy tree. definition 2.8 [4] let 𝐺𝐴𝐹 be an anti-fuzzy graph and let 𝑢, 𝑣 ∈ 𝑉. if 𝜇(𝑢, 𝑣) = 𝜎(𝑢) ∨ 𝜎(𝑣) then 𝑢 dominates 𝑣 (or 𝑣 dominates 𝑢) in 𝐺𝐴𝐹 . a set 𝐷 ⊆ 𝑉 is said to be a dominating set of an anti-fuzzy graph 𝐺𝐴𝐹 if for every vertex 𝑣 ∈ 𝑉 − 𝐷 there exists 𝑢 ∈ 𝐷 such that 𝑢 dominates 𝑣. definition 2.9 [4] a dominating set 𝐷 of an anti-fuzzy graph 𝐺𝐴𝐹 is called a minimal dominating set if there is no dominating set 𝐷′ such that 𝐷′ ⊂ 𝐷. definition 2.10 [5] the maximum scalar cardinality taken over all minimal dominating set is called anti-fuzzy domination number of an anti-fuzzy graph 𝐺𝐴𝐹 and is denoted by 𝛾𝐴𝐹𝐺 𝑑 (𝐺𝐴𝐹 ). definition 2.11 [2] let 𝐺𝐴𝐹 be an anti-fuzzy graph. let 𝑣1 and 𝑣2 be two vertices of 𝐺𝐴𝐹 . a subset 𝐷 of 𝑉 is called a anti-fuzzy equitable dominating set if every 𝑣2 ∈ 𝑉 − 𝐷 there exist a vertex 𝑣1 ∈ 𝐷 such that 𝑣1𝑣2 ∈ 𝐸 and |𝑑(𝑣1) − 𝑑(𝑣2)| ≤ 1 where 𝑑(𝑣1) denotes the degree of vertex 𝑣1 and 𝑑(𝑣2) denotes the degree of vertex 𝑣2 with 𝜇(𝑣1, 𝑣2) = 𝜎(𝑣1) ∨ 𝜎(𝑣2). definition 2.12 [2] an anti-fuzzy equitable dominating set 𝐷 of an anti-fuzzy graph 𝐺𝐴𝐹 is called a minimal anti-fuzzy equitable dominating set if there is no anti-fuzzy equitable dominating set 𝐷′ such that 𝐷′ ⊂ 𝐷. the maximum scalar cardinality taken over all minimal anti-fuzzy equitable dominating set is called anti-fuzzy equitable domination number and is denoted by 𝛾𝐴𝐹𝐺 𝑒𝑑 . definition 2.13 [4] an anti-fuzzy equitable dominating set 𝑆 of a connected anti-fuzzy graph 𝐺𝐴𝐹 is called the end anti-fuzzy equitable dominating set if 𝑆 contains all the terminal vertices. definition 2.14 [4] the maximum scalar cardinality taken over all minimal end antifuzzy equitable dominating set is called end anti-fuzzy equitable domination number of 𝐺𝐴𝐹 and it is denoted by 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 . definition 2.15 if for each 𝑥 ∈ 𝑉 − 𝑆 there exist a vertex 𝑦 ∈ 𝑆 such that 𝑥𝑦 ∈ 𝐸(𝐺𝐴𝐹) and either one of the vertex 𝑥 or 𝑦 is with degree 𝑘 and other vertex is with degree 𝑘 + 1, then 𝐺𝐴𝐹 is called a bi-regular anti-fuzzy graph. 3. connected end anti-fuzzy equitable dominating set definition 3.1 an end anti-fuzzy equitable dominating set 𝑆 of an anti-fuzzy graph 𝐺𝐴𝐹 is called the connected end anti-fuzzy equitable dominating set (ceafed-set) if induced anti-fuzzy subgraph < 𝑆 > is connected. 273 s. firthous fatima and k. janofer definition 3.2 the maximum scalar cardinality taken over all minimal connected end anti-fuzzy equitable dominating set is called connected end anti-fuzzy equitable domination number of 𝐺𝐴𝐹 and it is denoted by 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑. example 3.3 consider the following anti-fuzzy graph 𝐺𝐴𝐹 , in the anti-fuzzy graph 𝐺𝐴𝐹 , given in figure 3.1, the minimal connected end anti-fuzzy equitable dominating sets are 𝑆1 = {𝑣3, 𝑣4, 𝑣5, 𝑣6, 𝑣7} and 𝑆2 = {𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣6}. the scalar cardinality of 𝑆1 = |{𝑣3, 𝑣4, 𝑣5, 𝑣6, 𝑣7}| = 0.4 + 0.2 + 0.3 + 0.6 + 0.6 = 2.1 the scalar cardinality of 𝑆2 = |{𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣6}| = 0.5 + 0.4 + 0.2 + 0.3 + 0.6 = 2 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 = max{|𝑆1|, |𝑆2|} = max{2.1, 2} = 2.1 𝑣6(0.6) 𝑣5(0.3) 0.6 0.4 𝑣7(0.6) 0.6 0.4 𝑣4(0.2) 𝑣3(0.4) 0.6 0.6 0.5 𝑣1(0.3) 0.5 𝑣2(0.5) figure 3.1: anti-fuzzy graph 𝐺𝐴𝐹 therefore, connected end anti-fuzzy equitable domination number is 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 = 2.1 corresponding to the connected end anti-fuzzy equitable dominating set 𝑆1. theorem 3.4 let 𝐺𝐴𝐹 be any connected anti-fuzzy graph. then 𝛾𝐴𝐹𝐺 𝑑 (𝐺𝐴𝐹 ) ≤ 𝛾𝐴𝐹𝐺 𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹). proof: let 𝐺𝐴𝐹 be any connected anti-fuzzy graph. let 𝑆 ⊆ 𝑉(𝐺𝐴𝐹 ) be any 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 −set in 𝐺𝐴𝐹 . then obviously, 𝑆 is also an end anti-fuzzy equitable dominating set in 𝐺. therefore, 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹 ) = |𝑆| ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) (1) suppose let 𝑆′ be any 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 −set of 𝐺𝐴𝐹 . by definition of end anti-fuzzy equitable dominating set, 𝑆′ is also an anti-fuzzy equitable dominating set of 𝐺𝐴𝐹 . therefore, 𝛾𝐴𝐹𝐺 𝑒𝑑 (𝐺𝐴𝐹 ) = |𝑆′| ≤ 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹 ) (2) we know that, every anti-fuzzy equitable dominating set is an anti-fuzzy dominating set. therefore, 𝛾𝐴𝐹𝐺 𝑑 (𝐺𝐴𝐹 ) ≤ 𝛾𝐴𝐹𝐺 𝑒𝑑 (𝐺𝐴𝐹 ) (3) hence from (1), (2) and (3), we have 274 connected end anti-fuzzy equitable dominating set in anti-fuzzy graphs 𝛾𝐴𝐹𝐺 𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑒𝑑 (𝐺𝐴𝐹 ) ≤ 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹 ) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ). remarks 3.5 the equality of theorem 3.4 can be hold when the anti-fuzzy graph 𝐺 has no isolated vertices. for example, anti-fuzzy cycle and complete anti-fuzzy graphs can hold the equality condition. theorem 3.6 for any connected anti-fuzzy graph 𝐺𝐴𝐹 , 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹). proof: let 𝑆 ⊆ 𝑉 be the minimum connected anti-fuzzy equitable dominating set of a connected anti-fuzzy graph 𝐺𝐴𝐹 . then 𝑆 is an anti-fuzzy dominating set of 𝐺 and the induced anti-fuzzy subgraph < 𝑆 > is connected. therefore, 𝑆 is also a connected anti-fuzzy dominating set. clearly, any connected end anti-fuzzy equitable dominating set is also connected equitable dominating set. hence, 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹). theorem 3.7 for any 𝑘-regular anti-fuzzy graph for 𝑘 > 1 then 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ) proof: let us assume that 𝐺𝐴𝐹 be a k-regular anti-fuzzy graph. then each vertex of 𝐺𝐴𝐹 has a same degree 𝑘. let 𝑆 be the minimal connected anti-fuzzy equitable dominating set of 𝐺𝐴𝐹 then cardinality of 𝑆 = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹). if 𝑢 ∈ 𝑉 − 𝑆 then 𝑆 is connected anti-fuzzy equitable dominating set, then there exists 𝑣 ∈ 𝑆 and 𝑢𝑣 be the effective edge, also 𝑑(𝑢) = 𝑑(𝑣) = 𝑘. therefore |𝑑(𝑢) − 𝑑(𝑣)| = |𝑘 − 𝑘| = 0 ≤ 1. hence 𝑆 is a connected end anti-fuzzy equitable dominating set of 𝐺 such that 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹 ) ≥ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) (1) by theorem 3.6, we have 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹). (2) hence, from (1) and (2), 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ) corollary 3.8 let 𝐺𝐴𝐹 be (𝑘, 𝑘 + 1) bi-regular anti-fuzzy graph. then, 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺). proof: by theorem 3.6, 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺) ≤ 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑(𝐺). now, let 𝑆 be minimum connected end antifuzzy equitable set of (𝑘, 𝑘 + 1) bi-regular anti-fuzzy graph. by the definition, the connected end anti-fuzzy equitable dominating set 𝑆 is also an anti-fuzzy equitable 275 s. firthous fatima and k. janofer dominating set and 〈𝑆〉 is connected, since 𝐺 is (𝑘, 𝑘 + 1) bi-regular anti-fuzzy graph. therefore, 𝑆 is also a connected end anti-fuzzy equitable dominating set. theorem 3.9 let 𝐺𝐴𝐹 be an anti-fuzzy graph with 𝑛 vertices then 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺) = 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺) if and only if 𝐺𝐴𝐹 has no end vertex and there is atleast one vertex 𝑣 ∈ 𝑉 adjacent to (𝑛 − 1) vertices in 𝐺𝐴𝐹 . proof: let 𝐺𝐴𝐹 be an connected anti-fuzzy graph with n vertices and without end vertex. then, every vertex adjacent to atleast two vertices and there exists a one vertex say 𝑢 ∈ 𝑉(𝐺𝐴𝐹 ) adjacent to (𝑛 − 1) vertices, then the set 𝑆 = {𝑢} is connected end antifuzzy equitable dominating set 𝐺. by theorem 3.4, we get 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹 ) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹). conversely, suppose 𝐺𝐴𝐹 is connected anti-fuzzy graph and 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) = 𝛾𝐴𝐹𝐺 𝑒𝑒𝑑 (𝐺𝐴𝐹 ) then 𝐺 has no end vertex and there is 𝑆 = {𝑣} which is connected end anti-fuzzy equitable dominating set. therefore atleast any one vertex adjacent to (𝑛 − 1) vertices in 𝐺. corollary 3.10 let 𝐺𝐴𝐹 be an anti-fuzzy cycle with order 𝑝 then 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) = 𝑝 − max {𝑚𝑖𝑛𝑢𝑣∈𝐸 {𝜎(𝑢), 𝜎(𝑣)}}. proof: since 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹 ) = 𝑝 − max {𝑚𝑖𝑛𝑢𝑣∈𝐸 {𝜎(𝑢), 𝜎(𝑣)}} and 𝑆 = 𝑉 – {𝑢, 𝑣} is any subset of the vertices on the anti-fuzzy cycle 𝐺𝐴𝐹 such that 𝑢 and 𝑣 are adjacent vertices. clearly 𝑆 is connected end anti-fuzzy equitable set of 𝐺 it means 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝑝 − max {𝑚𝑖𝑛𝑢𝑣∈𝐸 {𝜎(𝑢), 𝜎(𝑣)}} and by the theorem 3.6, we have, 𝑝 − max {𝑚𝑖𝑛𝑢𝑣∈𝐸 {𝜎(𝑢), 𝜎(𝑣)}} = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ). hence 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺) = 𝑝 − max {𝑚𝑖𝑛𝑢𝑣∈𝐸 {𝜎(𝑢), 𝜎(𝑣)}}. theorem 3.11 for any complete bipartite anti-fuzzy graph then 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹 ) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ) = { max 𝑢∈𝑉1 {𝜎(𝑢)} + max 𝑣∈𝑉2 {𝜎(𝑣)}, |𝑑(𝑢) + 𝑑(𝑣)| ≤ 1 𝑝 + 𝑞 , |𝑑(𝑢) + 𝑑(𝑣)| > 1 proof: case (1) : if 𝐺 ≅ 𝐾𝑚,𝑛 and |𝑑(𝑢) + 𝑑(𝑣)| > 1 for all 𝑢 ∈ 𝑉1 and 𝑣 ∈ 𝑉2 then the antifuzzy graph 𝐺𝐴𝐹 is totally anti-fuzzy equitable disconnected. therefore, 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹 ) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) = 𝑝 + 𝑞. case (2): if 𝐺 ≅ 𝐾𝑚,𝑛 and |𝑑(𝑢) + 𝑑(𝑣)| ≤ 1 then if 𝑉1 and 𝑉2 be the partite sets of 𝐺𝐴𝐹 , be selecting one vertex 𝑢 ∈ 𝑉1 and 𝑣 ∈ 𝑉2 then 𝑆 = {𝑢, 𝑣} is connected end antifuzzy equitable dominating set. 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) ≤ 𝑚𝑎𝑥𝑢∈𝑉1 𝜎(𝑢) + 𝑚𝑎𝑥𝑣∈𝑉2 𝜎(𝑣) , but 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹 ) ≠ 𝑚𝑎𝑥𝑢∈𝑉 𝜎(𝑢). 276 connected end anti-fuzzy equitable dominating set in anti-fuzzy graphs 𝛾𝐴𝐹𝐺 𝑐𝑒𝑑 (𝐺𝐴𝐹) = 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) = max 𝑢∈𝑉1 {𝜎(𝑢)} + max 𝑣∈𝑉2 {𝜎(𝑣)}. theorem 3.12 let 𝐺𝐴𝐹 be connected anti-fuzzy graph with order 𝑚 and 𝑁 be the set of all end antifuzzy vertices of 𝐺𝐴𝐹 then 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) ≥ |𝑁| + max {𝜎(𝑥)}, where 𝑥 ∈ 𝑉 which is not an end anti-fuzzy vertex. proof: clearly if 𝐺𝐴𝐹 is connected anti-fuzzy graph with order 𝑚. let 𝑁 be the set of all end anti-fuzzy vertices of 𝐺𝐴𝐹 . let 𝑆 be any connected end antifuzzy equitable dominating set of 𝐺𝐴𝐹 then all the end vertices and also supporting vertices must along to end anti-fuzzy equitable dominating set. hence 𝛾𝐴𝐹𝐺 𝑐𝑒𝑒𝑑 (𝐺𝐴𝐹) ≥ |𝑁| + max {𝜎(𝑥)}. 4 conclusions a mathematical model helps to accomplish the problem in a complex situation. the possible solution is to convert the problem into a graph model. anti-fuzzy graph theory has been used to model many decision making problems in uncertain situations. it have numerous applications in modern science in technology, computer science, especially in the field of information theory, neural network, cluster analysis, diagnosis and control theory etc., in this paper, the connected end anti-fuzzy equitable dominating set of an anti-fuzzy graph is defined. the relation between anti-fuzzy equitable domination number, end anti-fuzzy equitable domination number and connected end anti-fuzzy equitable domination number are established. theorems related to these parameters are established. in future, we are going to establish these types of parameters in edge dominating sets of anti-fuzzy graphs. acknowledgements the authors express their gratitude to the editor-in-chief and also thank the anonymous reviewers for their valuable suggestions and comments. references [1] m.akram. anti fuzzy structures on graphs: middle east journal of scientific research, 11(12), 1641-1648. 2012. 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[14] l. a. zadeh. fuzzy sets: information sciences. 8, 338-353. 1965. 278 ratio mathematica vol. 33, 2017, pp. 39-46 issn: 1592-7415 eissn: 2282-8214 contributions in mathematics: hyperstructures of professor thomas vougiouklis violeta leoreanu fotea ∗ †doi:10.23755/rm.v33i0.391 abstract after presenting some basic notions of hyperstructures and their applications, i shall point out on the contribution of professor thomas vougiouklis to this field of research: algebraic hyperstructures. keywords: weak hyperstructure 2010 ams subject classifications: 20n20. ∗faculty of mathematics, al.i. cuza university of iaşi, bd. carol i, no. 11; foteavioleta@gmail.com † c©violeta leoreanu fotea. received: 31-10-2017. accepted: 26-12-2017. published: 3112-2017. 39 violeta leoreanu fotea 1 hyperstructures and applications theory of hyperstructures is a field of algebra, around 80 years old and very rich in applications, for instance in geometry, fuzzy and rough sets, automata, cryptography, codes, probabilities, graphs and hypergraphs (see [2], [3]). some basic definitions: a hyperoperation on a nonempty set h is a map ◦ : h ×h →p∗(h), where p∗(h) denotes the set of nonempty subsets of h. for subsets a,b of h, set a◦b = ⋃ a∈a;b∈b a◦b, and for h ∈ h write h◦a and a◦h for {h}◦a and a◦{h}. the pair (h,◦) is a hypergroup if for all a,b,c of h we have (a◦ b)◦ c = a◦ (b◦ c) and a◦h = h ◦a = h. if only the associativity is satisfied then (h,◦) is a semihypergroup. the condition a◦h = h ◦a = h for all a of h is called the reproductive law. a nonempty subset k of h is a subhypergroup if k ◦ k ⊆ k and for all a ∈ k, k ◦a = k = a◦k. a commutative hypergroup (h,◦) is a join space iff the following implication holds: for all a,b,c,d,x of h, a ∈ b◦x,c ∈ d◦x ⇒ a◦d∩ b◦ c 6= ∅. a semijoin space is a commutative semihypergroup satisfying the join condition. hypergroups have been introduced by marty [5] and join spaces by prenowitz [6]. join spaces are an important tool in the study of graphs and hypergraphs, binary relations, fuzzy and rough sets and in the reconstruction of several types of noneuclidean geometries, such as the descriptive, spherical and projective geometries [3], [6]. several interesting books have been writen on hyperstructures [2], [3], [4], [6], [8]. 2 emeritus professor thomas vougiouklis and his contribution to hyperstructures professor thomas vougiouklis is an author of more than 150 research papers and seven text books in mathematics. he have over 3000 references. he also wrote eight books on poetry, one cd music and lyrics. he participated in congresses (invited) about 60 congresses, over 20 countries. his monograph: 40 contributions in mathematics: hyperstructures of professor thomas vougiouklis hyperstructures and their representations, hadronic press monograph in mathematics, usa (1994) is an important book on the theory of algebraic hyperstructures. let us mention here some of his main contributions in hyperstructures, especially hv -structures, lie algebras of infinite dimension, ring theory, mathematical models. he first introduced and studied: • the term hope=hyperoperation (2008) • p-hypergroups, single-power cyclicity (1981). • fundamental relations in hyper-rings (γ∗-relation) and representations of hypergroups by generalized permutations and hypermatrices (1985). • very thin hyperstructures, s-construction (1988). • uniting elements procedure (1989), with p.corsini. • general hyperring, hyperfield (1990). • the weak properties and the hv-structures (1990). • general hypermodules, hypervector spaces(1990). • representation theory by hv-matrices (1990). • fundamental relations in hyper-modulus and hyper-vector spaces (ε∗ relation) (1994). • the e-hyperstructures, hv-lie algebras (1996). • the h/v-structures (1998). • ∂ operations (2005), • the helix hyperoperations, with s. vougiouklis, • n-ary hypergroups (2006), with b.davvaz., • bar instead of scale, (2008), with p. vougioukli, etc 41 violeta leoreanu fotea let us present here some of these notions. hv structures these notions were introduced in 1990 and they satisfy the weak axioms, where the non-empty intersection replaces the equality. wass means weak associativity: ∀x,y,z ∈ h, (xy)z ∩x(yz) 6= ∅. cow means weak commutativity: ∀x,y ∈ h, xy ∩yx 6= ∅. a hyperstructure (h, ·) is called hv -semigroup if it is wass and it is called hv group if it is a reproductive hv -semigroup, i.e. xh = hx = h, ∀x ∈ h. similarly, hv -vector spaces, hv -algebras and hv -lie algebras are defined and their applications are mentioned in the above books. fundamental relations the fundamental relations β∗, γ∗ and �∗ are defined in hv -groups, hv -rings and hv -vector spaces being the smallest equivalences, such that the quotient structures are a group, a ring or a vector space respectively. the following theorem holds: theorem. let (h, ·) be an hv -group and denote by u the set of all finite products of elements of h. we define the relation β in h as follows: xβy ⇔∃u ∈ u : {x,y}⊆ u then β∗ is the transitive closure of β. in a similar way, relation γ∗ is defined in an hv -ring and relation �∗ is defined in an hv -vector space. an hv -ring (r,+, ·) is called an hv -field if r/γ∗ is a field. 42 contributions in mathematics: hyperstructures of professor thomas vougiouklis if (h, ·), (h,∗) are hv -semigroups defined on the same set h, then the hyperoperation (·) is smaller than (∗) (and (∗) is greater than (·) if there exists an f ∈ aut(h,∗), such that x ·y ⊆ f(x∗y). theorem. greater hopes than the ones which are wass or cow are also wass or cow, respectively. this theorem leads to a partial order on hv -structures and mainly to a correspondence between hyperstructures and posets. the determination of all hv -groups and hv -rings is very interesting, but difficult. there are many results of r. bayon and n. lygeros in this direction. in paper [1] one can see how many hv -groups and hv -rings there exist, up to isomorphism, for several chasses of hyperstructures of two, three or four elements. ∂operations the hyperoperations, called theta-operations, are motivated from the usual property, which the derivative has on the derivation of a product of functions. if h is a set endowed with n operations (or hyperoperations) ◦1,◦2, ...,◦n and with one map or multivalued map f : h → h (or f : h → p(h) respectively), then n hyperoperations ∂1,∂2, ...,∂n on h can be defined as follows: ∀x,y ∈ h,∀i ∈{1,2, ...,n}, x∂iy = {f(x)◦i y, x◦i f(y)} or in the case ◦i is a hyperoperation or f is a multivalued map, we have ∀x,y ∈ h,∀i ∈{1,2, ...,n}, x∂iy = (f(x)◦i y)∪ (x◦i f(y)). if ◦i is wass, then ∂i is wass too. n-ary hypergroups a mapping f : h ×···×h︸ ︷︷ ︸ n −→ p∗(h) is called an n-ary hyperoperation, where p∗(h) is the set of all the nonempty subsets of h. an algebraic system (h,f), where f is an n-ary hyperoperation defined on h, is called an nary hypergroupoid. 43 violeta leoreanu fotea we shall use the following abbreviated notation: the sequence xi,xi+1, ...,xj will be denoted by x j i . for j < i, x j i is the empty symbol. when yi+1 = · · · = yj = y the last expression will be written in the form f(xi1,y (j−i),znj+1). for nonempty subsets a1, ...,an of h we define f(an1) = f(a1, ...,an) =⋃ {f(xn1) | xi ∈ ai, i = 1, ...,n}. an n-ary hyperoperation f is called associative if f(xi−11 ,f(x n+i−1 i ),x 2n−1 n+i ) = f(x j−1 1 ,f(x n+j−1 j ),x 2n−1 n+j ), hold for every 1 ≤ i < j ≤ n and all x1,x2, ...,x2n−1 ∈ h. an n-ary hypergroupoid with the associative n-ary hyperoperation is called an n-ary semihypergroup. an n-ary hypergroupoid (h,f) in which the equation b ∈ f(ai−11 ,xi,ani+1) has a solution xi ∈ h for every ai−11 ,ani+1,b ∈ h and 1 ≤ i ≤ n, is called an n-ary quasihypergroup. moreover, if (h,f) is an n-ary semihypergroup, (h,f) is called an n-ary hypergroup. an n-ary hypergroupoid (h,f) is commutative if for all σ ∈ sn and for every an1 ∈ h we have f(a1, ...,an) = f(aσ(1), ...,aσ(n)). let (h,f) be an n-ary hypergroup and b be a non-empty subset of h. b is called an n-ary subhypergroup of (h,f), if f(xn1) ⊆ b for xn1 ∈ b, and the equation b ∈ f(bi−11 ,xi,bni+1) has a solution xi ∈ b for every b i−1 1 ,b n i+1,b ∈ b and 1 ≤ i ≤ n. 44 contributions in mathematics: hyperstructures of professor thomas vougiouklis references [1] r. bayon and n. lygeros, advanced results in enumeration of hyperstructures, journal of algebra (computational algebra), 320/2 (2008),821-835 [2] p. corsini, prolegomena of hypergroup theory, aviani editore, italy, (1993) [3] p. corsini, p and v. leoreanu, applications of hyperstructure theory, kluwer aademic publishers, advances in mathematics, vol. 5, 2003. [4] b. davvaz and v. leoreanu fotea, hyperring theory and applications, hadronic press, 2008. [5] f. marty, sur une généralisation de la notion de groupe, actes du iv congrès des mathématiciens scandinaves, stockholm, (1934), 45-49. [6] w. prenowitz and j. jantosciak, join geometries, springer-verlag, utm, 1979. [7] t. vougiouklis, phd: cyclicity in hypergroups, in greek, xanthi, 1980 [8] t. vougiouklis, hyperstructures and their representations, hadronic press, inc., 1994. [9] t. vougiouklis, generalization of p-hypergroups, rendiconti del circolo matematico di palermo, serie ii, tomo xxxvi¡ (1987), 114-121 [10] t. vougiouklis, representation of hypergroups by generalized permutations, algebra universalis 29 (1992), 172-183. [11] t. vougiouklis, representations of hv-structures, proceedings of the international conference on group theory, timisoara, (1992), 159-184. [12] t. vougiouklis, hv-structures:birth and... childhood, j. of basic science, 4, no.1 (2008), 119-133. [13] t. vougiouklis, ∂ operations and hv -fields, acta math. sinica, english series, vol. 24 (2008), no. 7, 1067-1078. [14] t. vougiouklis, on the hyperstructures called ∂ hope, 10th int. congress on algebraic hyperstructures and appl., (2008), 97-112, [15] t. vougiouklis, hypermatrix representations of finite hv groups, european j. of combinatorics, 44 (2015), 307-3113. 45 violeta leoreanu fotea [16] t. vougiouklis, hypermathematics, hv structures, hypernumbers, hypermatrices and lie-santilli addmissibility, american j. of modern physics, (2015), 4(5), 38-46. [17] t. vougiouklis, special elements on p-hopes and ∂ hopes, southeast asian bull. of math. (2016), 40, 451-460. [18] t. vougiouklis, on the hyperstructure theory, southeast asian bull. of math. (2016), 40, 603-620. [19] t. vougiouklis, hypernumbers, finite hyper-fields , algebra, groups and geometries, vol. 33, no. 4 (2016), 471-490. [20] s .vougioukli and t. vougiouklis, helix-hopes on finite hv-fields , algebra, groups and geometries, vol. 33, no. 4 (2016), 491-506. [21] r. majoob and t. vougiouklis, applications of the uniting elements method, it. j. of pure and appl. math., 36 (2016), 23-34. 46 ratio mathematica 25 (2013), 67–76 issn:1592-7415 some properties of certain subhypergroups christos g. massouros 54, klious street, 15561 cholargos, athens, greece ch.massouros@gmail.com http://www.teihal.gr/gen/profesors/massouros/index.htm abstract the structure of the hypergroup is much more complicated than that of the group. thus there exist various kinds of subhypergroups. this paper deals with some of these subhypergroups and presents certain properties of the closed, invertible and ultra-closed subhypergroups. key words: hypergroup, subhypergroup. msc2010: 20n20. 1 introduction in 1934 f. marty, in order to study problems in non-commutative algebra, such as cosets determined by non-invariant subgroups, generalized the notion of the group, thus defining the hypergroup [11]. an operation or composition in a non-void set h is a function from h × h to h, while a hyperoperation or hypercomposition is a function from h × h to the powerset p (h ) of h. an algebraic structure that satisfies the axioms: i. a · (b · c) = (a · b) · c for every a, b, c ∈ h (associative axiom) and ii. a · h = h · a = h for every a ∈ h (reproductive axiom). is called group if · is a composition [16] and hypergroup if · is a hypercomposition [11]. when there is no likelihood of confusion · can be omitted. if a and b are subsets of h, then ab signifies the union ⋃ (a,b)∈a×b ab, in particular if a=∅ or b =∅ then ab =∅. ab and ab have the same meaning as a{b} and {a}b. in general, the singleton {a} is identified with its member a. 67 christos g. massouros proposition 1.1. if a non-void set h is endowed with a composition which satisfies the associative and the reproductive axioms, then h has a bilateral neutral element and any element in h has a bilateral symmetric. proof. let x ∈ h. per reproductive axiom x ∈ xh. therefore there exists e ∈ h such that xe = x. next, let y be an arbitrary element in h. per reproductive axiom there exists z ∈ h such that y = zx. consequently ye = (zx) e = z (xe) = zx = y. hence e is a right neutral element. in an analogous way there exists a left neutral element e′. then the equality e = e′e = e′ is valid. therefore e is the bilateral neutral element of h. now, per reproductive axiom e ∈ xh. thus there exists x′ ∈ h, such that e = xx′. hence any element in h has a right symmetric. similarly any element in h has a left symmetric and it is easy to prove that these two symmetric elements coincide. remark 1.2. an analogous proposition to proposition 1.1 is not valid when h is endowed with a hypercomposition. in hypergroups there exist different types of neutral elements [15] (e.g. scalar [4], strong [8,17] ect). there also exist special types of hypergroups which have a neutral element and each one of their elements has one or more symmetric. such hypergroups are for example the canonical hypergroups [21], the quasicanonical hypergroups [12], the fortified join hypergroups [17], the fortified transposition hypergroups [8], the transposition polysymmetrical hypergroups [19], the canonical polysymmetrical hypergroups [14], etc. proposition 1.3. if h is a hypergroup, then ab 6= ∅ is valid for all the elements a, b of h. proof. suppose that ab = ∅ for some a, b ∈ h. per reproductive axiom, ah = h and bh = h. hence, h = ah = a (bh) = (ab) h = ∅h = ∅ , which is absurd. in [11], f. marty also defined the two induced hypercompositions (right and left division) that result from the hypercomposition of the hypergroup, i.e. a | b = {x ∈ h|a ∈ xb} and a b | = {x ∈ h|a ∈ bx}. it is obvious that the two induced hypercompositions coincide, if the hypergroup is commutative. for the sake of notational simplicity, a/b or a : b is used for right division and b\a or a..b for left division [7, 13]. proposition 1.4. if h is a hypergroup, then a/b 6= ∅ and b\a 6= ∅ for all the elements a, b of h. 68 some properties of certain subhypergroups proof. per reproductive axiom, hb = h for all b ∈ h . hence, for every a ∈ hthere exists x ∈ h, such that a ∈ xb . thus, x ∈ a/b and, therefore, a/b 6= ∅ . dually, b\a 6= ∅ . in proposition 2.3 of [13] the following properties were proved for any hypergroup h (see also proposition 1 in [7]) proposition 1.5. i) (a/b) /c = a/(cb) and c\(b\a) = (bc)\a, for all a, b, c ∈ h. ii) b ∈ (a/b)\a and b ∈ a/ (b\a), for all a, b ∈ h. in [7] and then in [8] a principle of duality is established in the theory of hypergroups and in the theory of transposition hypergroups as follows: given a theorem, the dual statement which results from the interchanging of the order of the hypercomposition . (and necessarily interchanging of the left and the right division), is also a theorem. this principle is used throughout this paper. 2 closed, invertible and ultra-closed subhypergoups the structure of the hypergroup is much more complicated than that of the group. there are various kinds of subhypergroups. in particular a nonempty subset k of h is called semi-subhypergroup when it is stable under the hypercomposition, i.e. it has the property xy ⊆ k for all x, y ∈ k. k is a subhypergroup of h if it satisfies the reproductive axiom, i.e. if the equality xk = kx = k is valid for all x ∈ k(for the fuzzy case see e.g [3]). this means that when k is a subhypergroup and a, b ∈ k, the relations a ∈ bx and a ∈ yb always have solutions in k. although the non-void intersection of two subhypergroups is stable under the hypercomposition, it usually is not a subhypergroup since the reproductive axiom fails to be valid for it. this led, from the very early steps of hypergroup theory, to the consideration of more special types of subhypergroups. one of them is the closed subhypergroup (e.g. see [5], [9]). a subhypergroup k of h is called left closed with respect to h if for any two elements a and b in k, all the solutions of the relation a ∈ yb lie in k. this means that k is left closed if and only if a/b ⊆ k, for all a, b ∈ k (see [13]). similarly k is right closed when all the solutions of the relation a ∈ bx lie in k or equivalently if b\a ⊆ k, for all a, b ∈ k [13]. finally k is closed when it is both right and left closed. in the case of the closed subhypergroups, the non-void intersection of any family of closed 69 christos g. massouros subhypergroups is a closed subhypergroup. it must be mentioned though that a hypergroup may have subhypergroups, but no proper closed ones. for example if q is a quasi-order hypergroup [6], a2 is a subhypergroup of q, for each a ∈ q, but a/a = a\a = q for all a ∈ q. also fortified transposition hypergroups [8, 17] consisting only of attractive elements have no proper closed subhypergroups [18]. proposition 2.1. if k is a subset of a hypergroup h such that a/b ⊆ k and b\a ⊆ k, for all a, b ∈ k, then k is a subhypergroup of h. proof. let a be an element of k. it must be shown that ak = ka = k. suppose that x ∈ k. then a\x ⊆ k, therefore x ∈ ak, hence k ⊆ ak. for the reverse inclusion now suppose that y ∈ ak. then k/y ⊆ k/ak. so k ∩ (k/ak) y 6= ∅. thus, y ∈ (k/ak)\k. per proposition 1.4 (i) the equality k/ak = (k/k) /a is valid. thus (k/ak)\k = ((k/k) /a)\k ⊆ (k/a)\k ⊆ (k/k)\k ⊆ k\k ⊆ k. hence y ∈ k and so ak ⊆ k. therefore ak = k. the equality ka = k follows by duality. in [13] it is also proved that the equalities k = k/a = a/k = a\k = k\a are valid for every element a of a closed subhypergroup k. next some properties of these subhypergroups will be presented. proposition 2.2. if k is a subhypergroup of h, then h − k ⊆ (h − k) s and h − k ⊆ s (h − k), for all s ∈ k. proof. let r be an element in h − k which does not belong to (h − k) s. because of the reproductive axiom, r ∈ hs and since r /∈ (h − k) s, r must be a member of ks. thus, r ∈ ks ⊆ kk = k. this contradicts the assumption and so h − k ⊆ (h − k) s. the second inclusion follows by duality. proposition 2.3. (i) a subhypergroup k of h is left closed in h, if and only if (h − k) s = h − k for all s ∈ k. (ii) a subhypergroup k of h is right closed in h, if and only if s (h − k) = h − k for all s ∈ k. (iii) a subhypergroup k of h is closed in h, if and only if s (h − k) = (h − k) s = h − k for all s ∈ k. 70 some properties of certain subhypergroups proof. (i) let k be left closed in h. suppose that z lies in h−k and assume that zs ∩ k 6= ∅. then, there exists an element y in k such that y ∈ zs, or equivalently, z ∈ y/s. therefore z ∈ k, which is absurd. hence (h − k) s ⊆ h − k. next, because of proposition 1, h − k ⊆ (h − k) s and therefore h − k = (h − k) s. conversely now. suppose that (h − k) s = h − k for all s ∈ k. then (h − k) s ∩ k = ∅ for all s ∈ k. hence x /∈ rs and so r /∈ x/s for all x, s ∈ k and r ∈ h − k. therefore x/s ∩ (h − k) = ∅ which implies that x/s ⊆ k. thus k is closed in h. (ii) follows by duality and (iii) is an obvious consequence of (i) and (ii). corolary 2.4. (i) if k is a left closed subhypergroup in h, then xk∩k = ∅, for all x ∈ h − k. (ii) if k is a right closed subhypergroup in h, then kx ∩ k = ∅, for all x ∈ h − k. (iii) if k is a closed subhypergroup in h, then xk ∩k = ∅ and kx∩k = ∅, for all x ∈ h − k. proposition 2.5. if k is a subhypergroup of h, a ⊆ k and b ⊆ h, then (i) a (b ∩ k) ⊆ ab ∩ k and (ii) (b ∩ k) a ⊆ ba ∩ k. proof. let t ∈ a (b ∩ k). then t ∈ ax, with a ∈ a and x ∈ b ∩ k. since x lies in b ∩ k, it derives that x ∈ b and x ∈ k. hence ax ⊆ ab and ax ⊆ ak = k. thus ax ⊆ ab ∩k and therefore t ∈ ab ∩k. duality gives (ii) and so the proposition. proposition 2.6. (i) if k is a left closed subhypergroup in h, a ⊆ k and b ⊆ h, then (b ∩ k) a = ba ∩ k. (ii) if k is a right closed subhypergroup in h, a ⊆ k and b ⊆ h, then a (b ∩ k) = ab ∩ k. proof. (i) let t ∈ ba ∩ k. since k is right closed, for any element y in b − k, it is valid that ya ∩ k ⊆ yk ∩ k = ∅. hence t ∈ (b ∩ k) a ∩ k. but (b ∩ k) a ⊆ kk = k. thus t ∈ (b ∩ k) a. therefore ba ∩ k ⊆ (b ∩ k) a. next the inclusion becomes equality because of proposition 2.5. (ii) derives from the duality. proposition 2.7. (i) if k is a left closed subhypergroup in h, a ⊆ k and b ⊆ h, then (b ∩ k) /a = (b/a) ∩ k. (ii) if k is a right closed subhypergroup in h, a ⊆ k and b ⊆ h, then (b ∩ k)\a = b\a ∩ k. proof. (i) since b ∩ k ⊆ b, it derives that (b ∩ k) /a ⊆ b/a. moreover a ⊆ k and b ∩ k ⊆ k, thus (b ∩ k) /a ⊆ k. hence (b ∩ k) /a ⊆ 71 christos g. massouros (b/a)∩k. for the reverse inclusion now suppose that x ∈ (b/a)∩k. then, there exist a ∈ a, b ∈ b such that x ∈ b/a or equivalently b ∈ ax. since ax ⊆ k it derives that b ∈ k and so b ∈ b∩k. therefore b/a ⊆ (b ∩ k) /a. thus x ∈ (b ∩ k) /a. hence (b/a)∩k ⊆ (b ∩ k) /a, qed. duality gives (ii) and so the proposition. krasner generalized the notion of the closed subhypergroups, considering closed subhypergroups in other subhypergroups [9]. let us define the restriction of the right and left division in subset a of a hypergroup h as follows: a/ab = {x ∈ a|a ∈ xb} and b\aa = {x ∈ a|a ∈ bx} thus, if k is a subhypergroup of h and k ⊆ a, then k is right closed in a, if b\aa ⊆ k for all a, b ∈ k and k is left closed in a, if a/ab ⊆ k for all a, b ∈ k. proposition 2.8. let k, m be two subhypergroups of a hypergroup h, such that k ⊆ m . if k is left (or right) closed in m and m is left (or right) closed in h, then k is left (or right) closed in h. proof. since k is left closed in m , the inclusion a/m b ⊆ k is valid, for all a, b ∈ k. this means that if x is an element of m such that a ∈ xb, then x ∈ k. next if there exists y ∈ h − m such that a ∈ yb, then a/b will not be a subset of m . hence m will not be left closed in h. this contradicts the assumption, and so the proposition. corolary 2.9. let k, m be two subhypergroups of a hypergroup h, such that k ⊆ m . if k is closed in m and m is closed in h, then k is closed in h. proposition 2.10. let k, m be two subhypergroups of a hypergroup hand suppose that k is left (or right) closed in h. then k ∩ m is left (or right) closed in m . proof. let a, b ∈ k ∩ m . then a/b = {x ∈ h|a ∈ xb} ⊆ k. hence {x ∈ m|a ∈ xb} ⊆ k ∩ m . therefore a/m b ⊆ k ∩ m . thus k ∩ m is left closed in m . corolary 2.11. let k, m be two subhypergroups of a hypergroup hand suppose that k is closed in h. then k ∩ m is closed in m . proposition 2.12. if two subhypergroups k, m of a hypergroup hare left (or right) closed in hand their intersection is not void, then k ∩ m is left (or right) closed in m . 72 some properties of certain subhypergroups proof. let a, b ∈ k ∩ m . since k, m are left closed in h, a/b = {x ∈ h|a ∈ xb} is a subset of both k and m . hence a/b ⊆ k ∩ m and so the proposition. corolary 2.13. the non-void intersection of two closed subhypergroups is a closed subhypergroup. the next type of hypergroups was introduced by dresher and ore in [5] and immediately after that, m. krasner used them in [9]. in both [5] and [9] they are named reversible subhypergoups. in our days these subhypergroups are called invertible. the definition that follows was given by jantosciak in [7]. definition 2.14. a subhypergroup k of a hypergroup h is right invertible if a/b ∩ k 6= ∅, implies that b/a ∩ k 6= ∅, a, b ∈ h. k is left invertible if b\a ∩ k 6= ∅, implies that a\b ∩ k 6= ∅, a, b ∈ h. if k is both right and left invertible, then it is called invertible. theorem 4 in [1] gives an interesting example of an invertible subhypergroup in a join hypergroup of partial differential operators. moreover the closed subhypergroups of the quasicanonical or of the canonical hypergroups are invertible [21]. direct consequences of the above definition are the following propositions: proposition 2.15. (i) k is right invertible in h, if and only if the following implication is valid: b ∈ ka ⇒ a ∈ kb, a, b ∈ h. (ii) k is left invertible in h, if and only if the following implication is valid: b ∈ ak ⇒ a ∈ bk, a, b ∈ h. proposition 2.16. (i) k is right invertible in h, if and only if the following implication is valid: ka 6= kb ⇒ ka ∩ kb = ∅, a, b ∈ h. (ii) k is left invertible in h, if and only if the following implication is valid: ak 6= bk ⇒ ak ∩ bk = ∅, a, b ∈ h. proposition 2.17. if k is right (left) invertible in h, then k is right (left) closed in h. in [2] one can find examples of closed hypegroups that are not invertible. definition 2.18. a subhypergroup k of a hypergroup h is right ultraclosed if it is right closed and a/a ⊆ k for each a ∈ h. k is left ultraclosed if it is left closed and a\a ⊆ k for each a ∈ h. if k is both right and left ultra-closed, then it is called ultra-closed. 73 christos g. massouros proposition 2.19. (i) if k is right ultra-closed in h, then either a/b ⊆ k or a/b ∩ k = ∅, for all a, b ∈ h. moreover if a/b ⊆ k, then b/a ⊆ k. (ii) if k is left ultra-closed in h, then either b\a ⊆ k or b\a ∩ k = ∅, for all a, b ∈ h. moreover if b\a ⊆ k, then a\b ⊆ k. proof. suppose that a/b ∩ k 6= ∅, a, b ∈ h. then a ∈ kb, for some k ∈ k. next assume that b/a ∩ (h − k) 6= ∅. then b ∈ ra, r ∈ h − k. thus a ∈ k (ra) = (kr) a. since k is right closed, per proposition 2.3, kr ⊆ h−k. so a ∈ va, for some v ∈ h − k. therefore a/a ∩ (h − k) 6= ∅, which is absurd. hence b/a ⊆ k. now let there be x in k such that b ∈ xa. if a/b ∩ (h − k) 6= ∅, there exists y ∈ h − k such that a ∈ yb. therefore b ∈ x (yb) = (xy) b. since k is right closed, per proposition 2.3, xy ⊆ h−k. so b ∈ zb, for some z ∈ h−k. therefore b/b∩(h − k) 6= ∅, which is absurd. hence a/b ⊆ k. duality gives (ii). corolary 2.20. if k is right (left) ultra-closed in h, then k is right (left) invertible in h. ultra-closed subhypergroups were introduced by y. sureau [22] (see also [2, 20]). the following proposition proves that the above given definition is equivalent to the definition used by sureau: proposition 2.21. (i) k is right ultra-closed in h, if and only if ka ∩ (h − k) a = ∅ for all a ∈ h. (ii) k is left ultra-closed in h, if and only if ak ∩a (h − k) = ∅ for all a ∈ h. proof. suppose that k is right ultra-closed in h. then a/a ⊆ k for all a ∈ h. since k is right closed, (a/a) /k ⊆ k is valid, or equivalently a/ (ak) ⊆ k for all k ∈ k. proposition 2.19 yields (ak) /a ⊆ k for all k ∈ k. if ka ∩ (h − k) a 6= ∅ , then there exist k ∈ k and v ∈ h − k, such that ka ∩ va 6= ∅, which implies that v ∈ ak/a. but (ak) /a ⊆ k, hence v ∈ k which is absurd. conversely now: let ka ∩ (h − k) a = ∅ for all a ∈ h. if a ∈ k, then k ∩ (h − k) a = ∅. therefore k /∈ ra, for each k ∈ k and r ∈ h − k. equivalently k/a ∩ (h − k) = ∅, for all k ∈ k. hence k/a ⊆ k for all k ∈ k and a ∈ k. so k is right closed. next suppose that a/a ∩ (h − k) 6= ∅ for some a ∈ h. then a ∈ (h − k) a, or ka ⊆ k (h − k) a. since k is closed, per proposition 2.3, k (h − k) ⊆ h −k is valid. thus ka ⊆ (h − k) a, which contradicts the assumption. duality gives (ii). 74 some properties of certain subhypergroups references [1] j. chvalina, s. hoskova, modelling of join spaces with proximities by first-order linear partial differential operators, italian journal of pure and applied mathematics, no. 21 (2007) pp. 177–190. 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[16] ch. g. massouros, g. g. massouros, on certain fundamental properties of hypergroups and fuzzy hypergroups mimic fuzzy hypergroups , international journal of risk theory, vol. 2, no. 2 (2012), pp. 71–82. [17] g. g. massouros, ch. g. massouros and j. d. mittas, fortified join hypergroups. annales matematiques blaise pascal, 3, no. 2 (1996), pp. 155–169. [18] g. g. massouros, the subhypergroups of the fortified join hypergroup, italian j. pure & appl. maths, no 2 (1997), pp. 51–63. [19] g. g. massouros, f. zafiropoulos, ch. g. massouros, transposition polysymmetrical hypergroups, proc. of the 8th aha (2002), spanidis press (2003), pp. 191–202. [20] g. g. massouros, on the hypergroup theory, fuzzy systems & a.i. reports and letters, academia romana, vol. iv, no. 2/3 (1995) pp. 13–25, scripta scientiarum mathematicarum, tom. i, fasc. i, (1997) pp. 153-170. [21] j. mittas, hypergroupes canoniques, mathematica balkanica, 2 (1972), pp. 165–179. [22] y. sureau, sous-hypergroupe engendre par deux sous-hypergroupes et sous-hypergroupe ultra-clos d’un hypergroupe, c. r. acad. sc. paris, t. 284, serie a, (1977) pp. 983–984. 76 ratio mathematica 27 (2014) 69-79 issn:1592-7415 the sum of the series of reciprocals of the cubic polynomials with triple non-positive integer root radovan pot̊uček department of mathematics and physics, faculty of military technology, university of defence, brno, czech republic radovan.potucek@unob.cz abstract this contribution, which is a follow-up to author’s paper [1] dealing with the sums of the series of reciprocals of some quadratic polynomials, deals with the series of reciprocals of the cubic polynomials with triple non-positive integer root. three formulas for the sum of this kind of series expressed by means of harmonic numbers are derived and presented, together with one approximate formula, and verified by several examples evaluated using the basic programming language of the computer algebra system maple 16. this contribution can be an inspiration for teachers who are teaching the topic infinite series or as a subject matter for work with talented students. key words: telescoping series, harmonic numbers, cas maple, riemann zeta function. 2000 ams: 40a05, 65b10. 1 introduction let us recall some basic terms. for any sequence {ak} of numbers the associated series is defined as the sum ∞∑ k=1 ak = a1 + a2 + a3 + · · · . the sequence of partial sums {sn} associated to a series ∞∑ k=1 ak is defined for each n 69 radovan pot̊uček as the sum sn = n∑ k=1 ak = a1 + a2 + · · · + an. the series ∞∑ k=1 ak converges to a limit s if and only if the sequence of partial sums {sn} converges to s, i.e. lim n→∞ sn = s. we say that the series ∞∑ k=1 ak has a sum s and write ∞∑ k=1 ak = s. the nth harmonic number is the sum of the reciprocals of the first n natural numbers: hn = 1 + 1 2 + 1 3 + · · · + 1 n = n∑ k=1 1 k . the generalized harmonic numbers of order n in power r is the sum hn,r = n∑ k=1 1 kr , (1) where hn,1 = hn are harmonic numbers. every generalized harmonic number of order n in power m can be written as a function of generalized harmonic number of order n in power m− 1 using formula (see [2]): hn,m = n−1∑ k=1 hk,m−1 k(k + 1) + hn,m−1 n , (2) whence hn,2 = n−1∑ k=1 hk k(k + 1) + hn n , hn,3 = n−1∑ k=1 hk,2 k(k + 1) + hn,2 n . therefore hn,3 = n−1∑ k=1 1 k(k + 1) (k−1∑ i=1 hi i(i + 1) + hk k ) + 1 n n−1∑ k=1 hk k(k + 1) + hn n2 , thus hn,3 = n−1∑ k=1 1 k(k + 1) (k−1∑ i=1 hi i(i + 1) + hk k + hk n ) + hn n2 . (3) from formula (1), where r = 1, 2, 3 and n = 1, 2, . . . , 8, we get this table: n 1 2 3 4 5 6 7 8 hn 1 3 2 11 6 25 12 137 60 49 20 363 140 761 280 hn,2 1 5 4 49 36 205 144 5269 3600 5369 3600 266681 176400 1077749 705600 hn,3 1 9 8 251 216 2035 1728 256103 216000 28567 24000 9822481 8232000 78708473 65856000 70 the sum of the series of reciprocals of the cubic polynomials 2 the sum of the series of reciprocals of the cubic polynomials with triple non-positive integer root we deal with the problem to determine the sum s(a,a,a) of the series ∞∑ k=1 1 (k −a)3 for non-positive integers a, i.e. to determine the sum s(0, 0, 0) of the series ∞∑ k=1 1 k3 = 1 13 + 1 23 + 1 33 + 1 43 + · · · , (4) the sum s(−1,−1,−1) of the series ∞∑ k=1 1 (k + 1)3 = 1 23 + 1 33 + 1 43 + · · · = s(0, 0, 0) −s1(0, 0, 0) = s(0, 0, 0) − 1 , the sum s(−2,−2,−2) of the series ∞∑ k=1 1 (k + 2)3 = 1 33 + 1 43 + · · · = s(0, 0, 0) −s2(0, 0, 0) = s(0, 0, 0) − 9 8 etc. clearly, we get the formula ∞∑ k=1 1 (k −a)3 = s(0, 0, 0) −s−a(0, 0, 0) , (5) where s−a(0, 0, 0) is the (−a)th partial sum of the series (4). several values of the nth partial sums sn(0, 0, 0), briefly denoted by sn, are: s100 . = 1.2020074, s1000 . = 1.2020564, s10000 . = 1.2020569, s100000 . = 1.2020569. let us note that the series s(0, 0, 0) converges to the apéry’s constant 1.202056903159 . . . , which represents the value ζ(3) of the riemann zeta function ζ(3) = ∞∑ k=1 1 k3 = 1 13 + 1 23 + 1 33 + 1 43 + · · · . the partial sums sn(0, 0, 0) so present the generalized harmonic numbers hn,3. according to formula (5) is s(a,a,a) = ζ(3) −h−a,3 , (6) then using formula (3) we get 71 radovan pot̊uček theorem 2.1. the series ∞∑ k=1 1 (k −a)3 , where a is a negative integer, has the sum s(a,a,a) = ζ(3) − −a−1∑ k=1 1 k(k + 1) (k−1∑ i=1 hi i(i + 1) + hk k − hk a ) − h−a a2 . (7) now, we express formula (3) in another form. we have hn,3 = 1 2 ( h1 1 + h1 n ) + 1 6 ( h1 2 + h2 2 + h2 n ) + + 1 12 ( h1 2 + h2 6 + h3 3 + h3 n ) + 1 20 ( h1 2 + h2 6 + h3 12 + h4 4 + h4 n ) + + 1 30 ( h1 2 + h2 6 + h3 12 + h4 20 + h5 5 + h5 n ) + · · · · · · + 1 (n− 3)(n− 2) ( h1 2 + h2 6 + · · · + hn−4 (n− 4)(n− 3) + hn−3 n− 3 + hn−3 n ) + + 1 (n− 2)(n− 1) ( h1 2 + h2 6 + · · · + hn−3 (n− 3)(n− 2) + hn−2 n− 2 + hn−2 n ) + + 1 (n− 1)n ( h1 2 + h2 6 + · · · + hn−2 (n− 2)(n− 1) + hn−1 n− 1 + hn−1 n ) + hn n2 , i.e. hn,3 = h1 1 · 2 ( 1 1 + 1 2 · 3 + 1 3 · 4 + · · · + 1 (n− 2)(n− 1) + 1 (n− 1)n + 1 n ) + + h2 2 · 3 ( 1 2 + 1 3 · 4 + 1 4 · 5 + · · · + 1 (n− 2)(n− 1) + 1 (n− 1)n + 1 n ) + + h3 3 · 4 ( 1 3 + 1 4 · 5 + 1 5 · 6 + · · · + 1 (n− 2)(n− 1) + 1 (n− 1)n + 1 n ) + · · · · · · + hn−3 (n− 3)(n− 2) ( 1 n− 3 + 1 (n− 2)(n− 1) + 1 (n− 1)n + 1 n ) + + hn−2 (n− 2)(n− 1) ( 1 n− 2 + 1 (n− 1)n + 1 n ) + hn−1 (n− 1)n ( 1 n− 1 + 1 n ) + hn n2 . (8) because 1 k(k + 1) = 1 k − 1 k + 1 , then the nth partial sum tn of the telescoping 72 the sum of the series of reciprocals of the cubic polynomials series ∞∑ k=2 1 k(k + 1) = 1 2 · 3 + 1 3 · 4 + · · ·+ 1 n(n + 1) + 1 (n + 1)(n + 2) + · · · is tn = ( 1 2 − 1 3 ) + ( 1 3 − 1 4 ) +· · ·+ ( 1 n − 1 n + 1 ) + ( 1 n + 1 − 1 n + 2 ) = 1 2 − 1 n + 2 , for the expressions in the first three parentheses of formula (8) we get 1 1 + 1 2 · 3 + 1 3 · 4 + · · · + 1 (n− 1)n + 1 n = 1 + tn−2 + 1 n = = 1 + ( 1 2 − 1 n ) + 1 n = 3 1 · 2 , 1 2 + 1 3 · 4 + 1 4 · 5 + · · · + 1 (n− 1)n + 1 n = 1 2 + tn−2 − t1 + 1 n = = 1 2 + 1 2 − ( 1 2 − 1 3 ) = 5 2 · 3 , 1 3 + 1 4 · 5 + 1 5 · 6 + · · · + 1 (n− 1)n + 1 n = 1 3 + tn−2 − t2 + 1 n = = 1 3 + 1 2 − ( 1 2 − 1 4 ) = 7 3 · 4 and analogously for the expressions in the last three parentheses of formula (8) we get 1 n−3 + 1 (n−2)(n−1) + 1 (n−1)n + 1 n = 1 n−3 + tn−2 − tn−4 + 1 n = = 1 n−3 + 1 2 − ( 1 2 − 1 n−2 ) = 2n−5 (n−3)(n−2) , 1 n−2 + 1 (n−1)n + 1 n = 1 n−2 + tn−2 − tn−3 + 1 n = = 1 n−2 + 1 2 − 1 2 + 1 n−1 = 2n−3 (n−2)(n−1) , 1 n−1 + 1 n = 2n−1 (n−1)n . therefore hn,3 = h1 1 · 2 · 3 1 · 2 + h2 2 · 3 · 5 2 · 3 + h3 3 · 4 · 7 3 · 4 + · · · · · · + hn−2 (n−2)(n−1) · 2n−3 (n−2)(n−1) + hn−1 (n−1)n · 2n−1 (n−1)n + hn n2 , 73 radovan pot̊uček hence hn,3 = 3h1 (1 · 2)2 + 5h2 (2 · 3)2 + · · · + (2n− 3)hn−2 [(n− 2)(n− 1)]2 + (2n− 1)hn−1 [(n− 1)n]2 + hn n2 , thus hn,3 = n−1∑ k=1 (2k + 1)hk [k(k + 1)]2 + hn n2 . (9) from formulas (6) and (9) we obtain theorem 2.2. the series ∞∑ k=1 1 (k −a)3 , where a is a negative integer, has the sum s′(a,a,a) = ζ(3) − −a−1∑ k=1 (2k + 1)hk [k(k + 1)]2 − h−a a2 . (10) remark 2.1. in [4] it is derived that a good approximation for the partial sum sn(0, 0, 0) is the expression n∑ k=1 1 k3 ≈ ζ(3) − 1 4 ( 2 n2 − 2 n3 + 1 n4 ) . (11) it is stated that for small n, say, n = 5, the relative error in the above approximation is vanishingly small, i.e. about 0.03%, and that for larger n ∼ 1000, the error is swamped by machine precision. if we use formulas (11) and (5), where a is a negative integer, we get s(a,a,a) = s(0, 0, 0) −s−a(0, 0, 0) = = ζ(3) − −a∑ k=1 1 k3 ≈ ζ(3) − [ ζ(3) − 1 4 ( 2 (−a)2 − 2 (−a)3 + 1 (−a)4 )] , so we have an approximate formula s(a,a,a) ≈ 1 4 ( 2 a2 + 2 a3 + 1 a4 ) and an approximate sum s(a,a,a) = 2a2 + 2a + 1 4a4 . (12) example 2.1. evaluate the sum of the series ∞∑ k=1 1 (k + 5)3 74 the sum of the series of reciprocals of the cubic polynomials by means of formula: i) (7), ii) (10), iii) (5), iv) (12) and compare obtained results. solution: i) the series has by theorem 2.1, where a = −5, the sum s(−5,−5,−5) = ζ(3) − 4∑ k=1 1 k(k + 1) (k−1∑ i=1 hi i(i + 1) + hk k + hk 5 ) − h5 25 . the last summand h5 25 = 137/60 25 = 137 1500 . now, we evaluate the middle summand: 4∑ k=1 1 k(k + 1) (k−1∑ i=1 hi i(i + 1) + hk k + hk 5 ) = 1 1 · 2 ( h1 1 + h1 5 ) + + 1 2 · 3 ( h1 1 · 2 + h2 2 + h2 5 ) + 1 3 · 4 ( h1 1 · 2 + h2 2 · 3 + h3 3 + h3 5 ) + + 1 4 · 5 ( h1 1 · 2 + h2 2 · 3 + h3 3 · 4 + h4 4 + h4 5 ) . if we denote this summand as s and use the values of the first five harmonic numbers from the table above, we get s = 1 2 ( 1 1 + 1 5 ) + 1 6 ( 1 2 + 3/2 2 + 3/2 5 ) + 1 12 ( 1 2 + 3/2 6 + 11/6 3 + 11/6 5 ) + + 1 20 ( 1 2 + 3/2 6 + 11/6 12 + 25/12 4 + 25/12 5 ) + = 1 2 · 6 5 + 1 6 · 31 20 + 1 12 · 311 180 + 1 20 · 265 144 = 1891 1728 . altogether we have s(−5,−5,−5) = ζ(3) − 1891 1728 − 137 1500 = ζ(3) − 256103 216000 . = 0.016394866122 . ii) by theorem 2.2 we get an easy and effective way how to obtain the required sum: s′(−5,−5,−5) = ζ(3) − 4∑ k=1 (2k + 1)hk [k(k + 1)]2 − h5 25 = = ζ(3) − 3h1 (1 · 2)2 − 5h2 (2 · 3)2 − 7h3 (3 · 4)2 − 9h4 (4 · 5)2 − h5 25 . 75 radovan pot̊uček by means of the first five values of the harmonic numbers we have s′(−5,−5,−5) = ζ(3) − 3 4 · 1 − 5 36 · 3 2 − 7 144 · 11 6 − 9 400 · 25 12 − 1 25 · 137 60 = = ζ(3) − 256103 216000 . = 0.016394866122 . iii) the third and in this case much more easily way, how to determine the sum s(−5,−5,−5), is to use formula (5) and the value of s5(0, 0, 0) = h5,3 from the table above. so we immediately obtain the required result: s(−5,−5,−5) = s(0, 0, 0) −s5(0, 0, 0) = ζ(3) − 256103 216000 . = 0.016394866122 . iv) if we use formula (12), we get the approximate sum s(−5,−5,−5) = 2(−5)2 + 2(−5) + 1 4(−5)4 = 2 · 52 − 2 · 5 + 1 4 · 54 = 41 2500 = 0.0164 . formulas (7), (10), and (5) give identical result 0.016394866122 , while formula (12) gives approximate result 0.0164 . the relative error of the fourth approximate result is 3.13 · 10−4 ∼ 0.03 %. 3 numerical verification we solve the problem to determine the values of the sum s(a,a,a) of the series ∞∑ k=1 1 (k −a)3 for a = −1,−2, . . . ,−10,−99,−100,−500,−999,−1000. we use on the one hand an approximate evaluation of the sum s(a,a,a,t) = t∑ k=1 1 (k −a)3 , where t = 106, and formula (12) for approximate evaluation sum s(a,a,a), and on the other hand formulas (7) and (10) for evaluation the sum s(a,a,a) we compare 15 quadruplets of the sums s(a,a,a), s′(a,a,a), s(a,a,a, 106), and s(a,a,a) to verify formulas (7) and (10) and to determine the relative error of two approximate sums s(a,a,a, 106) and s(a,a,a). we use procedures hnum and rp3aaaneg written in the basic programming language of the cas maple 16 and one for statement: 76 the sum of the series of reciprocals of the cubic polynomials hnum:=proc(n) local m,h; h:=0; for m from 1 to n do h:=h+1/m; end do; end proc; rp3aaaneg:=proc(a,t) local i,k,a,a2,s,s1,s2,s3,saaa,s2aaa,sumaaa,sumaaaline,z3; a:=-a; a2:=a*a; s:=0; saaa:=0; s2aaa:=0; sumaaa:=0; z3:=1.20205690315959428540; for k from 1 to a-1 do s1:=0; s2:=0; s3:=0; if k-1=0 then s2:=0 else for i from 1 to k-1 do s2:=s2+hnum(i)/(i*(i+1)); end do; end if; s2:=s2+hnum(k)/k+hnum(k)/a; s1:=s1+s2/(k*(k+1)); s:=s+s1; s3:=s3+((2*k+1)*hnum(k))/(k*k*(k+1)*(k+1)); end do; saaa:=z3-s-hnum(a)/a2; s2aaa:=z3-s3-hnum(a)/a2; print("a=",a,":saaa=",evalf[20](saaa),s2aaa=",evalf[20](s2aaa)); for k from 1 to t do sumaaa:=sumaaa+1/((k-a)*(k-a)*(k-a)); end do; print("sumaaa(",t,")=",evalf[20](sumaaa)); sumaaaline:=(2*a2+2*a+1)/(4*a2*a2); print("sumaaaline=",evalf[20](sumaaaline)); print("rerrsumaaa=",evalf[20]((abs(sumaaa-saaa))/saaa)); print("rerrsumaaaline=",evalf[20]((abs(sumaaaline-saaa))/saaa)); end proc: a:=[-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,-99,-100,-500,-999,-1000]; for a in a do rp3aaaneg(a,1000000); end do; the approximate values of the sums s(a,a,a) and s(a,a,a, 106), denoted briefly s and s(106), and the sum s(a,a,a), denoted s, obtained by the procedures above and rounded to 9 decimals, are written into the following table (the sums s′(a,a,a) give identically values as the sums s(a,a,a)) : 77 radovan pot̊uček a −1 −2 −3 −4 −5 s 0.202056903 0.077056903 0.040019866 0.024394866 0.016394866 s(106) 0.202056903 0.077056903 0.040019866 0.024394866 0.016394866 s 0.250000000 0.078125000 0.040123457 0.024414063 0.016400000 a −6 −7 −8 −9 −10 s 0.011765236 0.008849784 0.006896659 0.005524917 0.004524917 s(106) 0.011765236 0.008849785 0.006896660 0.005524917 0.004524917 s 0.011766975 0.008850479 0.006896973 0.005525072 0.004525000 a −99 −100 −500 −999 −1000 s 0.000050502 0.000049502 0.000001996 0.000000501 0.000000500 s(106) 0.000050502 0.000049502 0.000001996 0.000000500 0.000000499 s 0.000050503 0.000049503 0.000001996 0.000000501 0.000000500 computation of 15 quadruplets of the sums s(a,a,a), s′(a,a,a), s(a,a,a, 106) and s(a,a,a) took about 21 hours and 30 minutes. the relative errors of the approximate sums s(a,a,a, 106), i.e. the ratios∣∣[s(a,a,a, 106) −s(a,a,a)]/s(a,a,a)∣∣, range from 10−10 (for a = −1) to 10−5 (for a = −1000), and the relative errors of the approximate sums s(a,a,a), i.e. the ratios∣∣[s(a,a,a) −s(a,a,a)]/s(a,a,a)∣∣, range from 10−1 (for a = −1) to 10−5 (for a = −1000). 4 conclusion we dealt with the sum of the series of reciprocals of the cubic polynomials with triple non-positive integer root a, i.e. with the series ∞∑ k=1 1 (k −a)3 . we stated that its sum clearly can be for great number of members t (we used t = 106) approximately computed by formula s(a,a,a,t) = t∑ k=1 1 (k −a)3 , we derived that the approximate value of its sum is for a negative a given by simple formula s(a,a,a) = 2a2 + 2a + 1 4a4 , 78 the sum of the series of reciprocals of the cubic polynomials and we derived that the precise value of the sum is for a negative a given by formula s(a,a,a) = ζ(3) −h−a,3 , i.e. by formula s(a,a,a) = ζ(3) − −a−1∑ k=1 1 k(k + 1) (k−1∑ i=1 hi i(i + 1) + hk k − hk a ) − h−a a2 , and also by easier formula s′(a,a,a) = ζ(3) − −a−1∑ k=1 (2k + 1)hk [k(k + 1)]2 − h−a a2 . we verified these results by computing 15 quadruplets of the four sums above for a = −1,−2, . . . ,−10, −99,−100,−500,−999,−1000 by using the cas maple 16 and compared their values. the series of reciprocals of the cubic polynomials with triple non-positive integer root so belong to special types of infinite series, such as geometric and telescoping series, which sums are given analytically by means of a formula which can be expressed in closed form. references [1] r. pot̊uček, the sums of the series of reciprocals of some quadratic polynomials . in: proceedings of afases 2010, 12th international conference ”scientific research and education in the air force”. (cd-rom). brasov, romania, 2010, 1206-1209. isbn 978-973-8415-76-8. [2] wikipedia contributors, harmonic number . wikipedia, the free encyclopedia, [online], [cit. 2015-06-25]. available from: https://en.wikipedia.org/wiki/harmonic number. [3] e. w. weisstein, harmonic number . from mathworld – a wolfram web resource, [online], [cit. 2015-06-15]. available from: http://mathworld.wolfram.com/harmonicnumber.html. [4] mathematics stack exchange – a question and answer website for people studying math. [online], [cit. 2015-06-15]. available from: http://math.stackexchange.com/questions/361386/is-therea-formula-for-sum-n-1k-frac1n3?rq=1. 79 e:\uziv\sarka\clanky\rm_26\final\rm_26_4.dvi ratio mathematica 26 (2014), 65–76 issn:1592-7415 recognizability in stochastic monoids a. kalampakasa, o. louscou-bozapalidoub, s. spartalisc a,cdepartment of production engineering and management, democritus university of thrace, 67100, xanthi, greece bsection of mathematics and informatics, technical institute of west macedonia, 50100, kozani, greece akalampakas@gmail.duth.gr sspar@pme.duth.gr abstract stochastic monoids and stochastic congruences are introduced and the syntactic stochastic monoid ml associated to a subset l of a stochastic monoid m is constructed. it is shown that ml is minimal among all stochastic epimorphisms h : m → m ′ whose kernel saturates l. the subset l is said to be stochastically recognizable whenever ml is finite. the so obtained class is closed under boolean operations and inverse morphisms. key words: recognizability, stochastic monoids, minimization. msc 2010: 68r01, 68q10, 20m32. 1 introduction a stochastic subset of a set m is a function f : m → [0, 1] with the additional property σm∈m f (m) = 1, i.e., f is a discrete probability distribution. the corresponding class is denoted by stoc(m). our subject of study, in the present paper, are stochastic monoids which were introduced in [4]. a stochastic monoid is a set m equipped with a stochastic multiplication m × m → stoc(m) which is associative and unitary. it can be viewed as a nondeterministic monoid (cf. [1, 2, 3]) with multiplication m ×m →p(m) such that for all m1, m2 ∈ m a discrete probability distribution is assigned on the set m1 · m2. 65 kalampakas, louscou-bozapalidou, spartalis a congruence on a stochastic monoid m is an equivalence ∼ on m such that m1 ∼ m ′ 1 and m2 ∼ m ′ 2 imply ∑ n∈c (m1 · m2)(n) = ∑ n∈c (m′1 · m ′ 2)(n) for all ∼-classes c. the quotient m/ ∼ admits a stochastic monoid structure rendering the canonical function m 7→ [m] an epimorphism of stochastic monoids. the classical isomorphism theorem of algebra still holds in the stochastic setup, namely for any epimorphism of stochastic monoids h : m → m ′ and every stochastic congruence ∼ on m ′ its inverse image h−1(∼) defined by m1h −1(∼)m2 iff h(m1) ∼ h(m2), is again a stochastic congruence and the quotient stochastic monoids m/h−1(∼) and m ′/ ∼ are isomorphic. in particular if ∼ is the equality, then h−1(=) is the kernel congruence of h (denoted by ∼h) m1 ∼h m2 iff h(m1) = h(m2), and the stochastic monoids m/ ∼h and m ′ are isomorphic. we show that stochastic congruences are closed under the join operation. this allows us to construct the greatest stochastic congruence included in an equivalence ∼. it is the join of all stochastic congruences on m included into ∼ and it is denoted by ∼stoc. the quotient stochastic monoid m/ ∼stoc is denoted by m stoc and has the following universal property: given an epimorphism of stochastic monoids h : m → m ′ whose kernel ∼h saturates the equivalence ∼ there exists a unique epimorphism of stochastic monoids h′ : m ′ → m stoc such that h′ ◦ h = hstoc, where hstoc : m → m stoc is the canonical epimorphism into the quotient. this result states that hstoc is minimal among all epimorphisms saturating ∼. let m be a stochastic monoid and l ⊆ m. denote by ∼l the greatest congruence of m included in the partition (equivalence) {l, m − l}, i.e., ∼l= {l, m − l} stoc. the quotient stochastic monoid ml = m/ ∼l will be called the syntactic stochastic monoid of l and it is characterized by the following universal property. for every stochastic monoid m and every epimorphism h : m → m ′ verifying h−1(h(l)) = l, there exists a unique epimorphism h′ : m ′ → ml such that h ′ ◦ h = hl where hl : m → ml is the canonical projection into the quotient. 66 recognizability in stochastic monoids a subset l of a stochastic monoid m is stochastically recognizable if there exist a finite stochastic monoid m ′ and a morphism h : m → m ′ such that h−1(h(l)) = l. by taking into account the previous result we get that l is recognizable if and only if its syntactic stochastic monoid is finite. moreover stochastically recognizable subsets are closed under boolean operations and inverse morphisms. 2 stochastic subsets some useful elementary facts are displayed. let (xi)i∈i , (xij )i∈i,j∈j , (yj)j∈j be families of nonnegative reals, then sup i∈i,j∈j xij = sup i∈i sup j∈j xij = sup j∈j sup i∈i xij , sup i∈i,j∈j xiyj = sup i∈i xi · sup j∈j yj, provided that the above suprema exist. if sup i′⊆f ini σi∈i′ xi exists, then we say that the sum σi∈i xi exists and we put σ i∈i xi = sup i′⊆f ini σ i∈i′ xi where the notation i′ ⊆f in i means that i ′ is a finite subset of i. it holds ∑ i∈i,j∈j xij = ∑ i∈i ∑ j∈j xij = ∑ j∈j ∑ i∈i xij , ∑ i∈i,j∈j xiyj = ∑ i∈i xi ∑ j∈j yj. let m be a non empty set and [0, 1] the unit interval, a stochastic subset of m is a function f : m → [0, 1] with the additional property that the sum of its values exists and is equal to 1 ∑ m∈m f (m) = 1. we denote by stoc(m) the set of all stochastic subsets of m. let fi : m → r+, i ∈ i, be a family of functions such that for every m ∈ m the sum ∑ i∈i fi(m) exists. then the assignment m 7→ ∑ i∈i fi(m) defines a function from m to r+ denoted by ∑ i∈i fi, i.e., ( ∑ i∈i fi ) (m) = ∑ i∈i fi(m), m ∈ m. 67 kalampakas, louscou-bozapalidou, spartalis now let (λi)i∈i be a family in [0, 1] such that ∑ i∈i λi = 1 and fi ∈ stoc(m), i ∈ i. for any finite subset i′ of i and any m ∈ m, we have ∑ i∈i λifi(m) = sup i′⊆f ini ∑ i∈i′ λifi(m) ≤ 1. thus ∑ i∈i λifi is defined and belongs to stoc(m) because ∑ m∈m ( ∑ i∈i λifi ) (m) = ∑ m∈m ∑ i∈i λifi(m) = ∑ i∈i ∑ m∈m λifi(m) = ( ∑ i∈i λi ) ( ∑ m∈m fi(m)) = 1 · 1 = 1. thus we can state: strong convexity lemma (scl). the set stoc(m) is a strongly convex set, i.e., for any stochastic family λi ∈ [0, 1], fi ∈ stoc(m), i ∈ i the function ∑ i∈i λifi is in stoc(m). for arbitrary sets m, m ′ any function h : m → stoc(m ′) can be extended into a function h̄ : stoc(m) → stoc(m ′) by setting h̄(f ) = ∑ m∈m f (m) · h(m). in particular, any function h : m → m ′ is extended into a function h̄ : stoc(m) → stoc(m ′) by the same as above formula. this formula is legitimate since by the strong convexity lemma ∑ m∈m f (m) = 1 and h(m) is a stochastic subset of m. hence, for any stochastic subset f : m → [0, 1] we have the expansion formula f = ∑ m∈m f (m)m̂ where m̂ : m → [0, 1] stands for the singleton function m̂(n) = { 1, if n = m; 0, if n 6= m. often m̂ is identified with m itself. 68 recognizability in stochastic monoids 3 stochastic congruences our main interest is focused on equivalences in the stochastic setup. any equivalence relation ∼ on the set m, can be extended into an equivalence relation ≈ on the set stoc(m) as follows: for f, f ′ ∈ stoc(m) we set f ≈ f ′ if and only if for each ∼-class c it holds ∑ m∈c f (m) = ∑ m∈c f ′(m), that is both f, f ′ behave stochastically on c in similar way. the above sums exist because f, f ′ are stochastic subsets of m: ∑ m∈c f (m) ≤ ∑ m∈m f (m) = 1. the equivalence ≈ has a fundamental property, it is compatible with strong convex combinations. proposition 3.1. assume that (λi)i∈i is a stochastic family of numbers in [0, 1] and fi, f ′ i ∈ stoc(m), for all i ∈ i. then fi ≈ f ′ i , for all i ∈ i, implies ∑ i∈i λifi ≈ ∑ i∈i λif ′ i . proof. by hypothesis we have ∑ m∈c fi(m) = ∑ m∈c f ′i (m) for any ∼-class c in m, and thus ∑ m∈c ( ∑ i∈i λifi ) (m) = ∑ m∈c ∑ i∈i λifi(m) = ∑ i∈i λi ∑ m∈c fi(m) = ∑ i∈i λi ∑ m∈c f ′i (m) = ∑ m∈c ∑ i∈i λif ′ i (m) = ∑ m∈c ( ∑ i∈i λif ′ i ) (m) that is ∑ i∈i λifi ≈ ∑ i∈i λif ′ i as wanted. 69 kalampakas, louscou-bozapalidou, spartalis 4 stochastic monoids a stochastic monoid is a set m equipped with a stochastic multiplication, i.e. a function m × m → stoc(m), (m1, m2) 7→ m1m2 which is associative ∑ n∈m (m1m2)(n)(nm3) = ∑ n∈m (m2m3)(n)(m1n) and unitary i.e. there is an element e ∈ m such that me = m = em, for all m ∈ m. for instance any ordinary monoid can be viewed as a stochastic monoid. in the present study it is important to have a congruence notion. more precisely, let m be a stochastic monoid and ∼ an equivalence relation on the set m, such that: m1 ∼ m ′ 1 and m2 ∼ m ′ 2 implies ∑ m∈c (m1m2)(m) = ∑ m∈c (m′1m ′ 2)(m) for all ∼-classes c, then ∼ is called a stochastic congruence on m. this condition can be reformulated as follows: m1 ∼ m ′ 1 and m2 ∼ m ′ 2 implies m1m2 ≈ m ′ 1m ′ 2. proposition 4.1. the quotient set m/ ∼ is structured into a stochastic monoid by defining the stochastic multiplication via the formula ([m1][m2])([n]) = ∑ m∈[n] (m1m2)(m). proof. first observe that the above multiplication is well defined. next for every ∼-class [b] we have (([m1][m2]) [m3]) ([b]) = ∑ [n]∈m/∼ ([m1][m2])([n])([n][m3])([b]) = ∑ [n]∈m/∼ ∑ n1∈[n] (m1m2)(n1) ∑ b′∈[b] (nm3)(b ′) 70 recognizability in stochastic monoids since n ∼ n1 we get = ∑ [n]∈m/∼ ∑ n1∈[n] (m1m2)(n1) ∑ b′∈[b] (n1m3)(b ′) = ∑ [n]∈m/∼ ∑ b′∈[b] ∑ n1∈[n] (m1m2)(n1)(n1m3)(b ′) = ∑ b′∈[b] ∑ n1∈m (m1m2)(n1)(n1m3)(b ′). by taking into account the associativity of m we obtain: = ∑ b′∈[b] ∑ n1∈m (m2m3)(n1)(m1n1)(b ′) = ([m1]([m2][m3]))([b]). congruences on an ordinary monoid m coincide with stochastic congruences when m is viewed as a stochastic monoid. the first question arising is whether stochastic congruence is a good algebraic notion. this is checked by the validity of the known isomorphism theorems in their stochastic variant. given stochastic monoids m and m ′, a strict morphism from m to m ′ is a function h : m → m ′ preserving stochastic multiplication and units, i.e., h̄(m1m2) = h(m1)h(m2), h(e) = e ′, for all m1, m2 ∈ m, where e, e ′ are the units of m, m ′ respectively, and h̄ : stoc(m) → stoc(m ′) the canonical extension of h defined in section 2. theorem 4.1. given an epimorphism of stochastic monoids h : m → m ′ and a stochastic congruence ∼ on m ′, its inverse image h−1(∼) defined by m1h −1(∼)m2 if h(m1) ∼ h(m2) is also a stochastic congruence and the stochastic quotient monoids m/h−1(∼ ) and m ′/ ∼ are isomorphic. proof. assume that m1h −1(∼)m′1 and m2h −1(∼)m′2 that is h(m1) ∼ h(m ′ 1) and h(m2) ∼ h(m ′ 2). 71 kalampakas, louscou-bozapalidou, spartalis then h̄(m1m2) = h(m1)h(m2) ≈ h(m ′ 1)h(m ′ 2) = h̄(m ′ 1m ′ 2), that is for all c ∈ m ′/ ∼, we have ∑ c∈c h̄(m1m2)(c) = ∑ c∈c h̄(m′1m ′ 2)(c), but ∑ c∈c h̄(m1m2)(c) = ∑ c∈c ∑ m∈m (m1m2)(m)h(m)(c) = ∑ m∈m (m1m2)(m) ∑ c∈c h(m)(c) = ∑ m∈h−1(c) (m1m2)(m). recall that all h−1(∼)-classes are of the form h−1(c), c ∈ m ′/ ∼. consequently, = ∑ m∈h−1(c) (m1m2)(m) = ∑ m∈h−1(c) (m′1m ′ 2)(m) which shows that h−1(∼) is indeed a congruence of the stochastic monoid m. the desired isomorphism ĥ : m/h−1(∼) → m ′/ ∼ is given by ĥ([m]h−1(∼)) = [h(m)]∼. corolary 4.1. let h : m → m ′ be an epimorphism of stochastic monoids. then the kernel equivalence m1 ∼h m2 if h(m1) = h(m2) is a congruence on m and the stochastic quotient monoid m/ ∼h is isomorphic to m ′. given stochastic monoids m1, . . . , mk the stochastic multiplication [(m1, . . . , mk) · (m ′ 1, . . . , m ′ k)](n1, · · · , nk) = (m1m ′ 1)(n1) · · ·(mkm ′ k)(nk) structures the set m1×·· ·×mk into a stochastic monoid so that the canonical projection πi : m1 ×·· ·× mk → mi, πi(m1, . . . , mk) = mi becomes a morphism of stochastic monoids. notice that the above multiplication is stochastic because ∑ ni∈mi 1≤i≤k (m1m ′ 1)(n1) · · ·(mkm ′ k)(nk) = ∑ n1∈m1 (m1m ′ 1)(n1) · · · ∑ nk∈mk (mkm ′ k)(nk) = 1 · · ·1 = 1. 72 recognizability in stochastic monoids theorem 4.2. let ∼i be a stochastic congruence on the stochastic monoid mi (1 ≤ i ≤ k). then ∼1 ×·· ·× ∼k is a stochastic congruence on the stochastic monoid m1×·· ·×mk and the stochastic monoids m1×·· ·×mk/ ∼1 ×·· ·×∼k and m1/ ∼1 ×·· ·× mk/ ∼k are isomorphic. 5 greatest stochastic congruence saturating an equivalence first observe that, due to the symmetric property which an equivalence relation satisfies, the sumability condition in the definition of a congruence can be replaced by the weaker condition: m1 ∼ m ′ 1 and m2 ∼ m ′ 2 implies ∑ m∈c (m1m2)(m) ≤ ∑ m∈c (m′1m ′ 2)(m) for all ∼-classes c. lemma 5.1. the equivalence ∼ on the stochastic monoid m is a congruence if and only if the following condition is fulfilled: m ∼ m′, implies ∑ b∈c (m · n)(b) ≤ ∑ b∈c (m′ · n)(b) and ∑ b∈c (n · m)(b) ≤ ∑ b∈c (n · m′)(b). proof. one direction is immediate whereas for the opposite direction we have: m1 ∼ m ′ 1 and m2 ∼ m ′ 2 imply ∑ b∈c (m1 · m2)(b) ≤ ∑ b∈c (m′1 · m2)(b) ≤ ∑ b∈c (m′1 · m ′ 2)(b). next we demonstrate that stochastic congruences are closed under the join operation. we recall that the join ∨ i∈i ∼i of a family of equivalences (∼i)i∈i on a set a is the reflexive and transitive closure of their union: ∨ i∈i ∼i= ( ⋃ i∈i ∼i )∗ . theorem 5.1. if (∼i)i∈i is a family of stochastic congruences on m, then their join ∨ i∈i ∼i is also a stochastic congruence. proof. let ∼1,∼2 be two congruences on m and ∼=∼1 ∨∼2. first we show that m ∼1 m ′ implies ∑ b∈c (m · n)(b) ≤ ∑ b∈c (m′ · n)(b), 73 kalampakas, louscou-bozapalidou, spartalis for all ∼-classes c. from the inclusion ∼1⊆∼ we get that c is the disjoint union c = m ⋃ j=1 c1j where c1j denote ∼1-classes. then ∑ b∈c (m · n)(b) = m ∑ j=1 ∑ b∈c1 j (m · n)(b) ≤ m ∑ j=1 ∑ b∈c1 j (m′ · n)(b) = ∑ b∈c (m′ · n)(b). by a similar argument we show that m ∼2 m ′ implies ∑ b∈c (m · n)(b) ≤ ∑ b∈c (m′ · n)(b), for all ∼-classes c. now, if m ∼ m′, without any loss we may assume that m ∼1 m1 ∼2 m2 ∼1 · · · ∼1 m2λ−1 ∼2 m ′ for some elements m1, . . . , m2λ−1 ∈ m. applying successively the previous facts, we obtain ∑ b∈c (m · n)(b) ≤ ∑ b∈c (m1 · n)(b) ≤ ·· ·≤ ∑ b∈c (m2λ−1 · n)(b) ≤ ∑ b∈c (m′ · n)(b). for an arbitrary set of congruences we proceed in a similar way. the previous result leads us to introduce the greatest stochastic congruence included into an equivalence ∼ of m. it is the join of all stochastic congruences on m included into ∼ and it is denoted by ∼stoc. the quotient stochastic monoid m/ ∼stoc is denoted by m stoc and has the following universal property theorem 5.2. given an epimorphism of stochastic monoids h : m → m ′ whose kernel ∼h saturates the equivalence ∼ there exists a unique epimorphism of stochastic monoids h′ : m ′ → m stoc rendering commutative the triangle m stocm ′ h′ m h hstoc 74 recognizability in stochastic monoids where hstoc : m → m stoc is the canonical projection m 7→ [m]stoc sending every element m ∈ m on its ∼stoc-class. proof. by virtue of the isomorphism theorem the stochastic monoid m ′ is isomorphic to the quotient m/ ∼h. since by assumption ∼h⊆∼ stoc, h′ is the following composition m ′ ∼ → m/ ∼h f → m/ ∼stoc= m stoc, with f ([m]h) = [m]stoc, [m]h being the ∼h-class of m. the previous result states that hstoc is minimal among all epimorphisms saturating ∼. 6 syntactic stochastic monoids let m be a stochastic monoid and l ⊆ m. denote by ∼l the greatest congruence of m included in the partition (equivalence) {l, m − l}, i.e., ∼l= {l, m − l} stoc. the quotient stochastic monoid ml = m/ ∼l will be called the syntactic stochastic monoid of l and it is characterized by the following universal property. theorem 6.1. for every stochastic monoid m and every epimorphism h : m → m ′ verifying h−1(h(l)) = l, there exists a unique epimorphism h′ : m ′ → ml rendering commutative the triangle mlm ′ h′ m h hl where hl is the canonical morphism sending every element m ∈ m to its ∼l-class. proof. the hypothesis h−1(h(l)) = l means that ∼h saturates l and so the statement follows immediately by theorem 5.2. given stochastic monoids m, m ′ we write m ′ < m if there is a stochastic monoid m̄ and a situation 75 kalampakas, louscou-bozapalidou, spartalis m ′ h ←− m̄ i −→ m where i (resp. h) is a monomorphism (resp. epimorphism). theorem 6.2. given subsets l1, l2, l of a stochastic monoid m it holds i) ml1∩l2 < ml1 × ml2 , ii) ml = ml̄, where l̄ designates the set theoretic complement of l, iii) ml1∪l2 < ml1 × ml2 , iv) if h : m → n is an epimorphism of nd-monoids and l ⊆ n, then mh−1(l) = ml. proof. the proof follows by applying theorem 6.1. a subset l of a stochastic monoid m is stochastically recognizable if there exist a finite stochastic monoid m ′ and a morphism h : m → m ′ such that h−1(h(l)) = l. the class of stochastically recognizable subsets of m is denoted by stocrec(m). by taking into account theorem 6.1 we get proposition 6.1. l ⊆ m is recognizable if and only if its syntactic stochastic monoid is finite, card(ml) < ∞. putting this result together with theorem 6.2 we yield proposition 6.2. the class stocrec(m) is closed under boolean operations and inverse morphisms. references [1] i.p. cabrera, p. cordero, m. ojeda-aciego, non-deterministic algebraic structures for soft computing, advances in computational intelligence, lecture notes in computer science 6692(2011) 437-444. [2] j. s. golan, semirings of formal series over hypermonoids: some interesting cases, kyungpook math. j. 36(1996) 107-111. [3] a. kalampakas and o. louskou-bozapalidou, syntactic nondeterministic monoids, submitted in pure mathematics and applications. [4] o. louskou-bozapalidou, stochastic monoids, applied mathematical sciences 3(2007) 443-446. 76 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 36, 2019, pp. 53-68 53 mathematics and radiotherapy of tumors luciano corso* abstract the present work takes inspiration from the scientific degree plan of the italian ministry of education and has a didactic and cultural character. it pursues three objectives: the first is to make young people understand the importance of mathematics in medicine; the second is to stimulate students to use mathematical tools to give rational answers in the therapeutic field, in particular in the treatment of some types of nodular tumors; the third is to inform people on the effectiveness of mathematical methods and their indispensability in the rigorous treatment of some human pathologies. using the experimental data about the development of a tumor, we move on to the analysis of the mathematical models able to allow a rational control of its behavior. the method we used in the development of this therapeutic process is essentially deterministic, even if some passages implicitly have a probabilistic nature. keywords: population, cells, tumor, carrying capacity, differential equation 2010 ams subject classification: 92b08, 92d08, 34k02, 39a06, 62p07, 97d06. † *founder and director of the journal matematicamente president of the verona section of mathesis – verona, italy; e-mail: lcorso@iol.it. †received on may 1st, 2019. accepted on june 20rd, 2019. published on june 30th, 2019. doi:10.23755/rm.v36i1.471. issn: 1592-7415. eissn: 2282-8214. ©luciano corso. this paper is published under the cc-by licence agreement. luciano corso 54 1. premise usually, when we talk about the therapeutic treatment of serious pathologies it is difficult to consider the contribution of mathematics and statistics to the success of the interventions. most often it is thought that positive results correspond to the abilities and knowledge of the luminaries of surgery and medicine. this article aims to provide additional information: to demonstrate that applied mathematics (in particular statistics) offers indispensable tools for a rational approach to these therapies. the method we used in the development of this therapeutic process is essentially deterministic, although some passages implicitly provide a probabilistic reference; in particular, when the least squares principle is applied for the research of the theoretical model of interpolation. the basic hypothesis is that the deviations of the experimental values from the theoretical values of the model have a normal distribution. 2. mathematics as a measure of the world the field in which mathematics moves has become vast. usually, it is divided into two major sectors: the pure and that applied mathematics. the first sector has a purely speculative nature and is concerned with a rigorous arrangement of the basic principles of the discipline; the second, instead, relates to the applications of mathematical methods to natural sciences, medicine, engineering and economics. it is in this second sector that interesting applications can be found that can help man solve several technical-scientific problems. it is necessary, however, to warn this is only an exemplifying division. actually, mathematics is a unitary whole and it is difficult to know where its theoretical part ends and its experimental soul begins and vice versa. often, problems arise in an application environment that requires in-depth theoretical analysis. so, it is necessary to refer to an experience, to a useful operational path. a wider approach, not only descriptive, to natural phenomena requires a considerable knowledge of the mathematics that allows: their measurement (analysis, probability calculus, statistics); the study of their possible forms (analysis, geometry, statistics); the coherent arrangement of the rules followed (logic, algebra). all scientific methodologies require compliance with these three points. 3. problem analysis biology is one of the sciences that is proving to be very ductile to use mathematical techniques for a rational response to problems. it enables, with mathematics and radiotherapy of tumors 55 genetics good practices and good procedures to improve the lives of human beings. the mathematical fields that can be applied to biology range from combinatorial calculus to probability calculus, to geometry, to statistics and they offer a vast set of procedures. the problem i am presenting is, certainly, of undoubted effect. it is an efficient and effective treatment to counteract, and eventually block, the progress of a particular type of tumor: the glioblastoma. it is a nodular tumor that lurks in the brain tissues and soon leads to the death of the host (the patient). we start from an experimental model of the tumor nodule, which, growing in the laboratory, gives us a lot of biological and kinetic measures of its growth (figure 1). in particular, we can determine the growth time, the number of the cells for each instant of time and the critical limit of their growth beyond which there is nothing left to do (for example, for the compression of the tissues or for metastasis). in the dynamics of the tumor, we also consider the necrosis of many of its cells for the lack of food and of oxygen. it is also necessary to know the clinical picture of the patient and his immune response. after that, we analyze the mathematical models able to guarantee a rigorous control of the behavior of this type of tumor. 4. the choice of mathematical models on the basis of what we previously analyzed, the process requires the selection of mathematical models, as the first approach, in order to quantitatively describe the natural growth of the tumor mass over time and to find a mathematical model that allows to give to the patient a therapy that increases his life expectancy compared to the natural one, starting from the observation of the neoplasm. the mathematical models able to control the growth of biological populations are studied by that part of mathematics that is known as population dynamics [8]. when dealing with a problem of growth of biological populations, we take on known and tested standard models. usually, any changes to be made to the models are arranged during the work, keeping the standard model used as fixed as possible. one of the most well-known growth models is that of verhulst [8]. in our case, however, the verhulst equation does not adapt well to describe the growth dynamics of the glioblastoma tumor cells. it has been observed, from previous studies, that the most suitable model to describe this growth is given by the differential equation of b. gompertz. luciano corso 56 figure 1. photomicrograph of an experimental tumor nodule (tumor spheroid). the reference bar is 400 µm long. the central area of the nodule, darker and denser, is mainly formed by dead cells because of the poor availability of oxygen and the accumulation of toxic substances produced by the cells themselves with their metabolism, due to problems related to the diffusion of these molecules in the tissue. this area is generally referred to as the necrotic heart. photo courtesy of dr roberto chignola, department of biotechnology, university of verona. 5. gompertz model and tumor growth this model can be expressed as a system of differential equations { 𝑑𝑋(𝑡) 𝑑𝑡 = 𝑘𝑝(𝑡) ∙ 𝑋(𝑡) 𝑑𝑘𝑝(𝑡) 𝑑𝑡 = −𝛽 ∙ 𝑘𝑝(𝑡) (1) or as a differential equation that includes both equations (2). 𝑑𝑋(𝑡) 𝑑𝑡 = 𝛽 ∙ 𝑋(𝑡) ∙ 𝐿𝑜𝑔[( 𝑋(𝑡) 𝐾 ) −1 ] (2) model (2) derives from (1), as can be demonstrated. we now present the parameters and variables of models (1) and (2). x (t) is the number of tumor cells at time t; k is the carrying capacity of the environment in which the tumor cells live and is equal to k = max (x (t)): it represents the critical limit beyond which a tumor mass cannot go (otherwise would kill the host); x (t) / k is the occupancy rate of the environment; kp (t) is the timedependent growth rate of the tumor cell population; 𝛽 is a parameter that dampens the genetic growth of the population of individuals considered. mathematics and radiotherapy of tumors 57 the differential equation (2) admits an integral curve in a closed form. it is given by: 𝑋(𝑡) = 𝐾 ∙ 𝑒−𝐶∙𝑒 −𝛽∙𝑡 . (3) as shown, (3) depends on the parameters k, 𝛽, c. the tumor has a mass whose volume is estimated on an experimental basis as follows: 𝑉𝑜𝑙(𝑡) = 4 3 ∙ 𝜋 ∙ 𝑟0 3 , 𝑟0 = 1 2 ∙ √𝑑𝑚𝑖𝑛 ∙ 𝑑𝑚𝑎𝑥 , (4) where vol (t) is the volume of the tumor mass at time t, 𝑟0 is the geometric mean of the two rays dmin / 2 and dmax / 2, where dmin and dmax are the minimum and the maximum of the diameters of the spheroid. once the volume is known, taking into account that a tumor cell has a known size (usually estimated in 10−9𝑐𝑚3), one can determine the number of cells in the nodule in the following way: 𝑋(𝑡) = 𝑉𝑜𝑙(𝑡) 𝑉𝑜𝑙𝑐𝑒𝑙𝑙𝑢𝑙𝑎⁄ . (4bis) x (t) of (4bis) is a very large value and therefore not very useful for calculations. since the volume of a cell is known and is constant, the size of the population of tumor cells is conveniently replaced by the volume of the tumor mass vol (t). starting from this substitution, x (t) becomes vol (t) and, considering the multiplicative constant (1 / volcellula), is also the population numerousness. it is now necessary to estimate the parameters of the model (3). 6. discretization and parameter estimation the inevitable step to estimate the parameters of the model (2) or (3) with the least squares method is the discretization of the model. in practice, it consists to replacing the derivative with the incremental ratio and with the application of the finite difference operator first. let ∆𝑋𝑡 = 𝑋𝑡+1 − 𝑋𝑡, from (2) we obtain: ∆𝑋𝑡 ∆𝑡 = 𝛽 ∙ 𝑋𝑡 ∙ 𝐿𝑜𝑔[( 𝑋𝑡 𝐾 ) −1 ] , (5) where ∆𝑡 = 1. with easy algebraic steps, we get to: 𝑋𝑡+1 𝑋𝑡 = 1 + 𝛽 ∙ 𝐿𝑜𝑔( 𝐾 𝑋𝑡 ) . (6) equation (6) can be set in the following way: �̂�𝑡 = 𝐴 + 𝐵 ∙ 𝐿𝑜𝑔(𝑋𝑡) , (7) where �̂�𝑡 = 𝑋𝑡+1 𝑋𝑡⁄ , 𝐴 = 1 + 𝛽 ∙ 𝐿𝑜𝑔(𝐾) , 𝐵 = −𝛽 . luciano corso 58 equation (7) is a linear model in the parameters. thus we can apply the least squares method to estimate parameters a and b based on the experimental data in our possession. we obtain: 𝑆(𝐴,𝐵) = ∑ (𝑌𝑗 − �̂�𝑗) 2 𝑛 𝑗=1 = ∑ (𝑌𝑗 − 𝐴 − 𝐵 ∙ 𝐿𝑜𝑔(𝑋𝑡)) 2 𝑛 𝑗=1 . (8) passing to the partial derivatives with respect to a and to b, setting them equal to zero and solving the system, we have: ( 𝑛 ∑ 𝐿𝑜𝑔(𝑋𝑗) 𝑛 𝑗=1 ∑ 𝐿𝑜𝑔(𝑋𝑗) 𝑛 𝑗=1 ∑ [𝐿𝑜𝑔(𝑋𝑗)] 2 𝑛 𝑗=1 ) ∙ ( 𝐴 𝐵 ) = ( ∑ 𝑌𝑗 𝑛 𝑗=1 ∑ 𝑌𝑗 ∙ 𝐿𝑜𝑔(𝑋𝑗) 𝑛 𝑗=1 ) . (9) in this case, it is not necessary to proceed to the calculation of the second derivatives since the hessian is a positive semidefinite matrix and therefore the solutions of the system (9) give precisely the minimum of s(a, b) [9]. once we have found the values for a and b, β and k are easily obtained. it is then calculated 𝑋𝑡. for the calculation of the constant c in (3), the initial condition is taken into account: at time 𝑡 = 0 we have 𝑋(0) = 𝐾 ∙ 𝑒−𝐶, and hence we get 𝐶 = 𝐿𝑜𝑔𝐾 − 𝐿𝑜𝑔𝑋(0). 7. processing to verify the validity of the method presented above, one uses the experimental measurements daily obtained with glioblastoma tumor nodules grown in laboratory (spheroids). the measures are relative to the variations of nodular size, taken for 77 days. we start, therefore, from the set w of the experimental data, where the first term of each pair represents the discrete time expressed in days of each observation and the second the volume of the tumor mass expressed in mm3: w= {{0, 3.57}, {1, 7.37}, {2, 10.9025},{3, 14.435},{4, 21.5},{5, 28.6}, {6, 37.14}, {7, 41.98}, {8, 52.89}, {9, 57.805},{10, 62.72},{11, 72.55}, {12, 88}, {13, 105.6}, {14, 96.5}, {15, 105.6},{16, 116.05},{17, 126.5}, {18, 147.4}, {19, 147.4}, {20, 185.2},{21, 172},{22, 199},{23, 199}, {24, 199}, {25, 199}, {26, 213.6}, {27, 199},{28, 199},{29, 199},{30, 199}, {31, 199}, {32, 199}, {33, 213.6}, {34, 199},{35, 213.6}, {36, 206.5}, {37, 199.4}, {38, 193}, {39, 185.2}, {40, 199}, {41, 199}, {42, 213.6}, mathematics and radiotherapy of tumors 59 {43, 213.6}, {44, 213.6}, {45, 213.6}, {46, 213.6}, {47, 185.2}, {48, 213.6}, {49, 199}, {50, 213.6}, {51, 209.95}, {52, 206.3}, {53, 199}, {54, 199}, {55, 213.6}, {56, 199}, {57, 199}, {58, 199}, {59, 199}, {60, 199}, {61, 213.6}, {62, 185.2}, {63, 185.2}, {64, 185.2}, {65, 185.2}, {66, 185.2}, {67, 213.6}, {68, 213.6}, {69, 199}, {70, 213.6}, {71, 203.2}, {72, 192.8}, {73, 172}, {74, 199}, {75, 185.2},{76, 199}, {77, 199}}. from (9) we get: a = 1.93967, 𝛽  0.18076, k  180.991 mm3, x0 ≅ 3.57 mm3, c ≅ 3.92588. it, therefore, turns out to be �̂�(𝑡) = 180.991 ∙ 𝑒−3.92588∙𝑒 −0.18076∙𝑡 . (10) it is not linear and therefore the goodness of fit is measured by the following fit index (which is a particular coefficient of variation): 𝐼2 = 1 𝑀(�̂�) ∙ √ ∑ (𝑋𝑗 − �̂�𝑗) 2𝑛 𝑗=1 𝑛 , (11) where xj are the second terms of the data pairs w, �̂�𝑗 are the theoretical results of the application of (10), m is the average of the theoretical values �̂�𝑗 and n is the sample size. in our case the value is 𝐼2 ≅ 0.147582. the value of i2 seems acceptable; moreover, given the difficulty of data collection, we can be satisfied with this approach even if, according to the international standard, a value lower than 0.1 should be recommended [10]. we now present the graph of the theoretical model and the distribution of experimental data around it (figure 2). figure 2. on the t-axis there is time in days, on the ordinates there is the volume of tumor. calculating the second derivative of (10) and placing it equal to zero, we obtain the inflection point [7]. it is equal to (7.56578 days, 66.5829 mm3). we have thus finished studying the gompertz model applied to our experimental luciano corso 60 data. let us now turn to the study of the optimal therapy to be applied to the nodule to control its growth. 8. the radiobiological treatment of tumor the goal of radiological treatment of the cancer is to reduce its mass by killing its cells, without simultaneously damaging healthy cells. radiotherapies aim to achieve this goal. this treatment, however, is rather dangerous since, in the irradiation of the tumor mass, healthy tissue cells are unfortunately also affected. in short, the following problem must be addressed: how much minimum radiant dose should be given to the patient to maximize the number of cancer cells killed with minimal damage to healthy cells? to answer this question, we need to address some preliminary aspects on the subject. we have shown that the gompertz model is valid in the interpretation of the dynamics of the tumor mass of an experimental nodule of glioblastoma. at this point we apply the model also to evaluate the dynamic behavior of the same tumor in a patient. before tackling the preliminaries, we consider that 𝑋0 = 𝐾 ∙ 𝑒 −𝑐 and we put it in (3), obtaining the following formula (algebraic steps are simple and are omitted): 𝑋(𝑡) = 𝑋0 ∙ 𝑒 𝛼0 𝛽 ∙(1−𝑒−𝛽∙𝑡) , (12) where 𝛼0 𝛽 = 𝐶⁄ , the parameter 𝛼0 assumes the meaning of instantaneous spheroid growth rate at time t = 0 and 𝛽 is a generic factor that deaden the tumour growth. from (12) it is confirmed that 𝑀𝑎𝑥[𝑋(𝑡)] = lim 𝑡→+∞ 𝑋0 ∙ 𝑒 𝛼0 𝛽 ∙(1−𝑒−𝛽∙𝑡) = 𝑋0 ∙ 𝑒 𝛼0 𝛽 = 𝐾. (13) equation (13) represents a constraint on the growth of the spheroid. on the basis of a consolidated case series, it is believed that the maximum volume of the tumour borne by a patient can reach 25 cm3, after which the effects are devastating and lead to the death of the guest in a short time. then from (13) we have: 𝐿𝑜𝑔(𝐾) = 𝐿𝑜𝑔(𝑋0) + 𝛼0 𝛽 , 𝛼0 𝛽 = 𝐿𝑜𝑔( 𝐾 𝑋0 ) ≅ 𝐿𝑜𝑔( 25 𝑐𝑚3 10−9𝑐𝑚3 ) ≅ 23.94, (14) where 𝑋0 in this case corresponds to the volume in cm 3 of a tumor cell at the beginning of the process; that is 𝑋0 = 𝑉𝑜𝑙𝑐𝑒𝑙𝑙𝑢𝑙𝑎. mathematics and radiotherapy of tumors 61 9. some notions of radiobiology often only possible therapy in the treatment of tumors is the radiotherapy, especially when the tumor involves important tissues of the human body or is located in places of difficult surgical access. from a clinical point of view, radiotherapy is an indispensable treatment even when it is considered necessary to intervene with more invasive therapies such as surgery and chemotherapy. currently, biomedical research is further progressing with promising studies on the interaction between tumor cells and subatomic particles obtained with appropriate accelerators. at the moment encouraging results have been achieved, but the journey is still long. the treatment of tumor masses with radiation has the purpose of inducing massive molecular damage to the diseased cells so as to lead them to death. the decisive problem is to avoid as far as possible damage to healthy cells when one intervenes on sick cells. the damage induced by radiotherapy treatment depends on the intensity of the radiant dose. there are international indications that establish the effects of any radiation therapy. the radiant dose is expressed in gray (gy), which corresponds to the energy of 1 joule absorbed by 1 kg of biological tissue. moreover, this basic unit must be multiplied by a suitable parameter that allows to take into account the effect on biological tissues of different nature of this radiant dose (rbe = relative biological effectiveness). finally, the product between gy and rbe gives the equivalent biological dose to be administered, which is measured in sievert (sv). it should be considered that for radiations of clinical interest, radiation 𝛾 [4], we consider rbe = 1 and gy = sv. table 1 highlights from a descriptive point of view the effects on human beings of exposure to radiant doses of different degrees of intensity [5]. dose (sv) effects (0.05 0.2] no symptoms, but risk of dna mutations (0.2 0.5] temporary drop in red blood cells (0.5 1] drop in immune system cells and risk of infection (1 2] immunodepression, nausea and vomiting. mortality of 10% at 30 days from exposure (2 3] severe immunodepression, nausea and vomiting 1-6 hours after exposure. latency phase of 7-14 days after which symptoms appear such as hair loss. mortality of 35% at 30 days from exposure (3 – 4] bleeding of the mouth and urinary tract. mortality of 50% at 30 days from exposure (4 – 6] mortality of 60% at 30 days from exposure. female infertility. the convalescence lasts from a few months to a year (6 – 10] complete injury of the bone marrow (the organ that produces red blood cells and all cells of the immune system). symptoms appear between 15 and 30 minutes after exposure and mortality is 100% at 14 days after exposure luciano corso 62 (10 – 50] immediate nausea, bleeding from the gastrointestinal tract and diarrhea, coma and death within 7 days. no medical intervention is possible (50 – 80] immediate coma. death occurs in a few hours due to the collapse of the nervous system > 80 exposure to these doses occurred in two circumstances. both subjects died within 49 hours of the accident table 1: effects of radiation on human beings 10. the modeling of therapy at this point, we must find a therapeutic process that allows us to stop the growth of the tumor or, even better, to reduce its mass to extinction. the model should take into account the disposition of the cells within the tumor mass, their microenvironment and the toxic effects induced on the healthy tissues of the surrounding cells and any other factor that may inform about the dynamics of the tumor. studies conducted so far in various research institutes around the world have led to confirm, as an acceptable model to be considered in the treatment of tumors with radiant dose, the following one: 𝑆�̂�(𝐷) = 𝑒−𝑎∙𝐷−𝑏∙𝐷 2 , (15) where 𝑆�̂� is the survival rate, a and b are two arbitrary parameters and d is the radiant dose. we must estimate the parameters a and b of the model as a function of the experimental data. even in this case we linearize the model and apply the least squares method. dose (gy) sf dose (gy) sf 0.0000 1.0000 5.5036 0.19609 0.53957 0.87780 6.0072 0.18372 1.0072 0.84048 6.5108 0.14785 1.5468 0.73778 7.0144 0.11642 2.0144 0.78746 7.5180 0.097850 2.5180 0.62009 7.9856 0.073780 3.0216 0.55627 8.4892 0.058100 3.5252 0.46753 8.9928 0.043800 3.9928 0.36816 9.4964 0.036020 4.5324 0.33752 10.000 0.033750 5.0360 0.26007 10.504 0.026010 table n. 2: numerical data relating to the graph in figure 3, further on mathematics and radiotherapy of tumors 63 11. assumptions for the radiotherapy when we face the problem of finding the relationship between a dynamic model of natural growth of a tumor and its radio-therapeutic treatment, collateral effects inevitably arise that create states other than those we would have liked to encounter. the complete modeling of a radiotherapy treatment requires the consideration of numerous variables that influence the interaction between tumor cells and radiant doses. for this reason, as a first approximation, we put some valid hypotheses to simplify the method. the choice of the hypotheses useful for the simplification of an effective model for the treatment of a tumor is in any case indispensable every time the control of the final results is desired. if we consider the analysis of the problem from a mathematical point of view, it is necessary to think about the implication of having to replace differential equations, defined in the continuous, with equivalent equations defined in the discrete. at this point we present the list of the necessary hypotheses to get on with the analysis of the process. assumption 1: the gompertz model is a good representation of the growth dynamic of a tumor mass, starting from a first degenerated cell up to asymptotically reaching a volume of 25 cm3. thus, it is possible to simulate tumor growth using the equations (1), (2) and (3). assumption 2: a solid tumor, in general, consists of proliferating cells p, quiescent cells q and dead cells u. the number of total cells n at time t is therefore given by n(t) = p(t) + q(t) + u(t). (16) table 3 and figure 5 refer only to proliferating cells since ionizing radiations are much less effective if directed against quiescent cells. assumption 3: in a solid tumor, on an experimental basis, it is possible to state that the number of quiescent and dead cells becomes significant with respect to the total of cells at the inflection point of the gompertz curve (3) and (12). assumption 4: radiation therapy has instantaneous effects, causing the immediate death of the cancer cells. these effects should at least be faster than the growth of tumor cells. this avoids a detailed kinetic analysis of the toxicity of radiation. assumption 5: after undergoing radiotherapy treatment, the tumor grows with the same dynamic modalities that preceded the treatment. it is a common convention in scientific treatises; however, there are also different points of view on this matter [6]. assumption 6: the maximum dose in a single treatment is 3 gy. you can also perform multiple treatments if and only if they are repeated at 24-hour intervals. it is not possible, however, to exceed 65 gy. this assumption is luciano corso 64 indicated by the radiotherapeutic protocols followed in the therapy of some tumors. the 3 gy dose allows healthy tissues affected by radiation to recover from damage. assumption 7: we assume the existence of two critical thresholds in the treatment phase: 1) if after treatment a tumor falls below 1 mm3, then we consider a therapy to be successful; 2) if, on the other hand, the volume increases beyond the dimension corresponding to the inflection point of the gompertz curve, the therapy must be considered as failed. in practice, nothing justifies this assumption from a clinical or biological point of view and yet we accept it as work hypothesis. based on these hypotheses, we can proceed with the estimation of the model parameters (15) and with the application of the programmed therapy. 12. procedure for a rational therapy the method we will use for the treatment of glioblastoma, meets the following two objectives: 1) check if there is a relationship between the effectiveness of the radiotherapy treatment used and the rate of tumor growth. 2) in case of an affirmative answer to the first objective, find a specific treatment protocol that allows to optimize the relationship between the benefits of the therapy and the costs due to the induction of toxic effects; in concrete terms, it is necessary to find the minimum amount of radiation to be used with the maximum destructive effect of cancer cells. we start with the estimation of the parameters of the model (15) using the well-known method of least squares and, also in this case, evaluating the goodness of fit with index i2 (11). the model (15) must be linearized: 𝐿𝑜𝑔[𝑆�̂�(𝐷)] = −𝑎 ∙ 𝐷 − 𝑏 ∙ 𝐷2 (17) and applying the least squares method we have: 𝑆(𝑎,𝑏) = ∑ (𝐿𝑜𝑔(𝑆𝐹(𝐷𝑗)) − 𝐿𝑜𝑔(𝑆�̂�(𝐷𝑗))) 2𝑛 𝑗=1 = = ∑ (𝐿𝑜𝑔(𝑆𝐹(𝐷𝑗)) + 𝑎 ∙ 𝐷 + 𝑏 ∙ 𝐷 2 ) 2𝑛 𝑗=1 . calculating the partial derivatives of s(a, b) with refer to a and to b, we obtain the system mathematics and radiotherapy of tumors 65 { (∑ 𝐷𝑗 2 𝑛 𝑗=1 ) ∙ 𝑎 + (∑ 𝐷𝑗 3 𝑛 𝑗=1 ) ∙ 𝑏 = −∑ 𝐷𝑗 𝑛 𝑗=1 ∙ 𝐿𝑜𝑔[𝑆𝐹(𝐷𝑗)] (∑ 𝐷𝑗 3 𝑛 𝑗=1 ) ∙ 𝑎 + (∑ 𝐷𝑗 4 𝑛 𝑗=1 ) ∙ 𝑏 = −∑ 𝐷𝑗 2 𝑛 𝑗=1 ∙ 𝐿𝑜𝑔[𝑆𝐹(𝐷𝑗)] (18) considering the data in table 2 and solving (18) with refer to a and b we obtain: 𝑎 ≅ 0.124275; 𝑏 ≅ 0.0264028. the model adapted to the data in table 2 is, therefore: 𝑆�̂�(𝐷) ≅ 𝑒−0.124275 𝐷−0.0264028 𝐷 2 (19) using the coefficient of variation: 𝐼2,𝑆𝐹 = 1 𝑀(𝑆�̂�(𝐷)) ∙ √ ∑ (𝑆𝐹(𝐷𝑗) − 𝑆�̂�(𝐷𝑗)) 2𝑛 𝑗=1 𝑛 (20) and taking into account both the data in table 2, and the theoretical values calculated with (19), we get the goodness of fit: 𝐼2,𝑆𝐹 ≅ 0.0998765. this value shows that our approach is good. figure 3 presents both the trend of experimental data and the interpolated model. figure 3: the graph shows the link between the radiant dose and the fraction of surviving individuals (19). the points represent, in cartesian coordinates, the data of table 2. we put on the d-axis the radiant dose, on the ordinate axis the survival rate sf(d). 13. research of the inflection point at this point, it is necessary to start the therapy taking into account what has resulted from these preliminary procedures. we consider again the model (10) and figure 3. furthermore, on the basis of the assumption 3, the most effective radiotherapy treatment is the one which begins at the inflection point of the gompertz curve. luciano corso 66 we consider the model (12): �̂�(𝑡) = 𝑋0 ∙ 𝑒 𝛼0 𝛽 ∙(1−𝑒−𝛽∙𝑡) , (21) and taking into account that a cancer cell has a volume of 10−9 cm3 and that 𝛽 = 0.016 we have: �̂�(𝑡) = 10−9 ∙ 𝑒23.9421 (1−𝑒 −0.016 𝑡) . (22) calculating the second derivative of x(t) and setting it equal to zero we get the inflection point (198.477 days, 9.19654 cm3) [7]. we note that this result is different from that obtained using the model (10). here, in fact, the parameters of the gompertz model are changed, which are now imposed not by the experimental data of the single experimental nodule (which in our case led to the model (10)), but by a different operating standard that requires both a start from a single tumor cell, whose volume is fixed at 10 –9 cm3, and from a critical maximum limit of tumor expansion equal to 𝐾 ≅ 25 cm3. figure 4 presents the function with the flex point. at this point the radiant doses should be applied at intervals that allow the patient's average life to be maximized. the first simulation (fig. 4) considers a single-dose therapy to hit the tumor mass with a single dose of radiation (from 1 to 3 gy with intervals of 0.4). starting at the time of the cancer diagnosis observation, when the tumor mass can vary from a minimum of 0.0050 cm3 to a maximum marked by the flex point, we have to measure the effect of the therapy on the cancer using the delay time of its growth. this time corresponds to the one that the tumor mass needs, after having been treated with radiotherapy, to return to the mass it had before the treatment was carried out. methods and procedures are reported in [2]. in the last two graphs we report two other simulations in which, with respect to the protocol for the search for an optimal result, two different outcomes are observed. in figure 5, the result is not satisfactory; instead, in figure 6 the protocol gives a favorable outcome and the mass of the glioblastoma is reduced below the desired minimum threshold. figure 4. gompertz curve (22) related to the investigated tumor. the inflection point is (198.477 days, 9.19654 cm3). on the t-axis there is the time in days and on the ordinate axis the tumor volume in cm3. mathematics and radiotherapy of tumors 67 figure 5. the graph describes the effect of 12 radiotherapy treatments on a glioblastoma. the doses, in gray, is (1, 2, 3, 1, 2, 0, 0, 1, 2, 3, 1, 2). note that the treatment did not give the desired result. the mass of the tumor has not been reduced below the critical threshold set by the protocol. figure 6. the graph describes the effect of 12 radiotherapy treatments on a glioblastoma. the doses, in gray, is (2, 2, 3, 3, 3, 0, 0, 2, 2, 3, 3, 3). in this case, it should be noted that the treatment has reached the desired result. the tumor mass was reduced under the critical threshold established by the protocol. each cusp corresponds to the flex point of the various curves that sequentially describe the progression of tumor growth after each treatment. luciano corso 68 references [1] chignola r, castelli f., corso l., pezzo g., zuccher s., la biomatematica in un problema di oncologia sperimentale, i quaderni del marconi, itis g. marconi di verona, anno 2006, isbn 88-902125-9-4. piano lauree scientifiche univr [2] burato a., chignola r., castelli f., corso l., pezzo g., zuccher s., matematica e radioterapia dei tumori, sviluppo e applicazioni di un modello predittivo semplificato, i quaderni del marconi, itis g. marconi di verona, anno 2007, isbn 978-88-95539-01-0. piano lauree scientifiche univr. [4] http://en.wikipedia.org/wiki/gamma_ray [5] http://en.wikipedia.org7wiki/radiation_poisoning [6] guirardo d. et al., dose dependence of growth rate of multicellular tumour spheroids after irradiation, the british journal of radiology, vol. 76, 2003, pp. 109-116. [7] mathematica 5. 1, wolfram research – www.wolfram.com. [8] smith j. m., l’ecologia e i suoi modelli, a. mondadori editore, milano, 1975, isbn 0303-2752. [9] apostol tom m., calcolo, vol. 1 e 3, boringhieri ed, torino, 1985, isbn 88-339-5033-6. [10] gambotto manzone a. m., susara longo c., probabilità e statistica 2, ed. tramontana, milano, 2010. isbn 978-88-23322-98-1. microsoft word capitolo_5.doc ratio mathematica vol. 33, 2017, pp. 127-138 issn: 1592-7415 eissn: 2282-8214 a fuzzy coding approach to data processing using the bar angelos markos∗ †doi:10.23755/rm.v33i0.387 abstract the bar is an alternative to likert-type scale as a response format option used in closed-form questionnaires. an important advantage of using the bar is that it provides a variety of data post-processing options (i.e., ways of partitioning the values of a continuous variable into discrete groups). in this context, continuous variables are usually divided into equal-length or equalarea intervals according to a user-specified distribution (e.g. the gaussian). however, this transition from continuous into discrete can lead to a significant loss of information. in this work, we present a fuzzy coding of the original variables which exploits linear and invertible triangular membership functions. the proposed coding scheme retains all of the information in the data and can be naturally combined with an exploratory data analysis technique, correspondence analysis, in order to visually investigate both linear and non-linear variable associations. the proposed approach is illustrated with a real-world application to a student course evaluation dataset. keywords: likert scale; bar; correspondence analysis; fuzzy coding; triangular membership functions 2010 ams subject classifications: 62p25. ∗department of primary education, alexandroupolis, greece; amarkos@eled.duth.gr † c©angelos markos. received: 31-10-2017. accepted: 26-12-2017. published: 31-12-2017. 127 angelos markos 1 introduction the closed-form questionnaire is the most commonly used data collection tool or instrument in quantitative studies. likert scales are commonly used to measure attitude, providing a range of responses to a given question or statement (see e.g., [5]). typically, likert scales have an odd number of response categories, 3, 5 or 7, representing the degree of agreement with the corresponding statements. for example, a five-point scale ranges from 1 = strongly disagree to 5 = strongly agree. although the response categories have a rank order, the invervals between values are frequently presumed equal. this assumption is often convenient in that it permits the calculation of descriptive and inferential statistics suitable for continuous variables. the processing of questionnaire data obtained via likert scales has certain advantages, but there are also major shortcomings [4, 5, 13, 27]. firstly, the desicion on the number of categories of a likert-type scale may affect the outcome of statistical analysis. too many or too few response categories, may cause respondent fatigue with a corresponding drop-off in response rate and reliability [3]. second, there is evidence that participants would give different ratings when using different versions of the same likert-type scale [12]. this indicates that the decision on the verbal labels that will be used to describe the numerical values of a likert scale is not a trivial one. such a problem also involves a number of social and psychologica factors. [27]. a third issue is related to the legitimacy of assuming a continuous or interval scale for likert-type categories, instead of an ordinal level of measurement. in fact, many authors advocate against this practice, given that the appropriate descriptive and inferential statistics differ for ordinal and interval variables [13]. therefore, if the wrong statistical technique is used, researchers increase the chance of coming to the wrong conclusions about their findings. finally, the fixed number of response categories limits the options for data processing and does not allow the direct comparison with the results of similar studies, where the same questions but with a different number of response categories were used [27]. kambaki-vougioukli & vougiouklis [14] introduced the “bar”, an alternative to the likert scale as a measurement instrument of a characteristic or attitude that is believed to range across a continuum of values. the bar is a straight horizontal line of fixed length, usually 62mm. the ends are defined as the extreme limits of the characteristic to be measured, orientated from the left (0) to the right (62). the study participants are asked to mark the bar at any point that expresses their answer to the specific question. although similar to other concepts in the field of psychology, such as the visual analogue scale [6], the idea of the bar originates from hyperstructure theory, a branch of mathematics that has recently found a wide range of applications in the social sciences (see e.g., [2, 7, 8, 21]). conse128 a fuzzy coding approach to data analysis using the bar quently, the bar marks a transition from discrete into continuous and from single valued into fuzzy or multivalued [27]. a series of studies, mostly in quantitative linguistics [15, 16, 17, 18, 19], have shown that the bar can be widely used to a broad range of populations and settings due to its simplicity and adaptability. a questionnaire developed using the bar instead of a likert scale takes less time to complete and no training or special skill of the participants is required other than to possess an understanding of distance on a ruler. moreover, minimal translation difficulties can easily lead to a crosscultural adaptation of a questionnaire. recently, [19] developed a software for using the bar in online questionnaires. the most important merit of using the bar, however, is the flexibility it offers to practitioners with regard to data analysis, without having to re-administer the questionnaire. after data collection, the analyst can decide how to split each variable at appropriate intervals. instead, in the case of likert-type scales, such a decision has to be taken before data collection and does not give any room for testing alternative ways of data processing. the number of groups per variable is chosen according to the distribution of the variable at hand. in this context, continuous variables are usually divided into equal-length or equal-area intervals according to a desirable distribution (e.g., the gaussian or the parabola). a detailed justification of such a discretization scheme is given in [27]. hereafter, we will refer to this procedure by crisp coding. crisp coding of a continuous value to a category obviously loses a substantial part of the original information and, subsequently, the advantage of continuity provided using the bar. this is because the original values are usually not uniformly distributed in the newly created intervals. to alleviate this problem, we discuss an alternative, fuzzy coding of the original data, which exploits linear and invertible triangular membership functions. a side advantage of the proposed fuzzy coding scheme is that the resulting data matrix can be given as input to correspondence analysis, a multivariate technique that can visualize both linear and non-linear variable associations. section 2 presents the rationale behind utilizing a fuzzy instead of a crisp coding scheme to data obtained from questionnaires using the bar. section 3 offers a brief introduction to correspondence analysis applied on fuzzy coded data. the proposed approach is illustrated with a real-world application in section 4. section 5 concludes the paper. 2 crisp versus fuzzy coding of continuous variables let a,b,c,. . . be a number of continuous variables whose values range from 0 to 62 and were collected for a number of survey participants or subjects using 129 angelos markos the bar. a common discretization scheme is to split each variable into five intervals of equal length, 1 to 5, as follows: 1 : [0−12.4],2 : [12.4−24.8),3 : [24.8−37.2),4 : [37.2−49.6),5 : (49.6−62]. then, for each subject a binary vector can be formed to summarize any value of each variable. for example, the value 35.7 for variable a lies in the third interval and can be coded into [0 0 1 0 0]. this type of binary coding is commonly referred to as crisp coding (e.g. see [1]) and a row-wise concatenation of all binary vectors forms a table, za, for variable a. the row margins of za are the same, equal to a column of ones. the so-called indicator matrix, denoted by z, is composed of a set of subtables, za, zb, zc, . . . stacked side by side, one for each variable. table 1 shows an example of crisp coding for some subjects on variable a with five categories (on the left), and their coding into a dummy variable (on the right). the matrix on the right is the subtable za and z = [za; zb; zc; . . .] denotes the full indicator matrix. this matrix can be subsequently analyzed with correspondence analysis, a well-established exploratory data analysis technique (see e.g., [9] and section 3). table 1: an example of crisp coding of a categorical variable with five categories into a dummy or indicator variable a 3 1 5 . . . a1 a2 a3 a4 a5 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 . . . . . . . . . . . . . . . in the case of crisp coding, it is assumed that the original continuous values are uniformly distributed within each interval. however, this is a strong assumption to make and the discrete assignment of continuous values to categories obviously loses a substantial part of the original information. this problem can be alleviated by using a fuzzy instead of crisp coding scheme. fuzzy coding (codage flou in french) has been successfully used in a variety of data analysis techniques and settings (see e.g., [1, 10, 11, 26]. the idea is to convert a continuous variable into a pseudo-categorical (i.e., fuzzy) variable using appropriate membership functions [11]. this is called “fuzzification” of the data. for example, 35.7 can be fuzzy coded into [0 0 0.75 0.25 0], instead of [0 0 1 0 0]. an important decision to make is the choice of membership 130 a fuzzy coding approach to data analysis using the bar functions that will be used for fuzzification. following [1], we adopt the system of the so-called “three-point triangular membership functions”, also known as piecewise linear functions, or second order b-splines [26]. triangular membership functions have two nice properties that will be further illustrated below: they are linear and invertible. a simple example of triangular membership functions is shown in figure 1, defining a fuzzy variable with five categories. on the horizontal axis is the scale of the original variable and five hinge points or knots, chosen as the minimum, 1st quartile, median, 3rd quartile and maximum values of the variable. this choice of hinge points is a simple one and corresponds to the quantiles of the distribution; it has been argued that such a choice ensures robustness [24]. the five functions shown in figure 1 are used for the recoding, and 35.7 is graphically shown to be recoded as 0 for category 1, 0 for category 2, 0.75 for category 3, 0.25 for category 4 and 0 for category 5. this coding scheme is linear and invertible, as shown below: 35.7 = 0.0×0 + 0.0×21 + 0.75×31 + 0.25×50 + 0.0×62. (1) given the fuzzy observation [0 0 0.75 0.25 0], the value of the original variable is unique and equals to 35.7. an algebraic description of the proposed scheme is given below. using triangular membership functions, the fuzzy values z1,z2, . . . ,z5 for a five-category fuzzy coding, where x is the original value on the continuous scale and the hinge points are m1,m2, . . . ,m5 are given by: z1(x) = { m2−x m2−m1 , for x ≤ m2 0, otherwise z2(x) =   x−m1 m2−m1 , for x ≤ m2 m3−x m3−m2 , for m2 ≤ x ≤ m3 0 otherwise z3(x) =   x−m2 m3−m2 , for m2 ≤ x ≤ m3 m4−x m4−m3 , for m3 ≤ x ≤ m4 0 otherwise z4(x) =   x−m3 m4−m3 , for m3 ≤ x ≤ m4 m5−x m5−m4 , for x > m4 0 otherwise z5(x) = { x−m4 m5−m4 , for x > m4 0 otherwise table 2 shows the corresponding subtable za in the case of fuzzy coding of some values of variable a. let z∗ denote the full fuzzy indicator matrix, which is 131 angelos markos figure 1: triangular membership functions to code a continuous variable (horizontal axis) into five fuzzy categorical variables. an example is shown of a value on the orginal scale (35.7) being fuzzy coded as [0 0 0.75 0.25 0]. composed of a set of subtables stacked side by side, one for each fuzzy indicator variable. as it is obvious from eq. 1, fuzzy coding transforms continuous variables into fuzzy categories with no loss of information, since a fuzzy-coded variable can be back-transformed to its original value. this is an improvement over crisp coding, where the information about the value of the variable within each interval is lost. alternatives to triangular membership functions can be, for example, trapezoidal, gaussian and generalized bell membership functions [1, 25]. a thorough investigation of their properties in the context of questionnaire data obtained using the bar is beyond the scope of this work. table 2: an example of fuzzy coding of a continuous variable into a fuzzy indicator variable with five categories a 35.7 43.1 25.0 . . . a1 a2 a3 a4 a5 0 0 0.75 0.25 0 0 0 0.36 0.64 0 0 0.60 0.40 0 0 . . . . . . . . . . . . . . . 132 a fuzzy coding approach to data analysis using the bar 3 correspondence analysis on fuzzy-coded data the fuzzy coding scheme described in section 2, can be combined with correspondence analysis (ca), a well-established method of geometric data analysis [23] for visualizing the rows and columns of a matrix of nonnegative data as points in a spatial representation. for a detailed treatment of ca we refer the reader to [9], for example. aşan and greenacre [1] showed that ca on the fuzzy indicator matrix z∗ (see table 2) can visualize nonlinear relationships between variables and that this property holds for all forms of membership functions. the core of the ca algorithm is the singular value decomposition (svd) of a suitably transformed matrix. next, we briefly present the algorithmic steps of ca on the fuzzy indicator matrix z∗ [20]. step 1. given a data table with continuous variables, apply the fuzzy coding scheme of section 2 to obtain the fuzzy indicator matrix, z∗. step 2. compute the matrix p as z∗ divided by its grand total, with row and column sums of p defined as r = p1, ct = 1tp, where 1 denotes a column vector of 1’s of appropriate order and t denotes vector and matrix transpose. the elements of r and c are called row and column masses in ca terminology. step 3. compute the matrix of standardized residuals s: s = d−1/2r (p − rc) td−1/2c where dr and dc denote diagonal matrices of the respective masses. step 4. compute the svd of s: s = udαv t where the singular vectors in u and v are normalized as utu = vtv = i, and dα is the diagonal matrix of the singular values, which are positive and in descending order, α1 ≥ α2 ≥ . . . > 0. step 5. compute the coordinates of the row and column points to obtain the socalled “symmetric” ca map: rows: f = d−1/2r udα, columns: γ = d −1/2 c vdα. 4 application to real data the real data set considered here consists of 159 pre-service teachers’ evaluation ratings of an introductory statistics university course. the focus of the analysis is on the following 5 statements, a to e, that were used to evaluate the quality of the teaching-learning process. 133 angelos markos how much has each of the following contributed to your understanding of the main ideas covered in this course? a: the tutor’s description of the aim, syllabus content and course objectives. b: the tutor’s encouragement of students to ask questions. c: the connection of the course material with everyday life examples. d: your own effort and engagement in the course. e: your own consistency in attending classes. the original five-point likert-type scale was substituted by the bar (0 to 62mm). after data collection, each one of the five statements was coded into five fuzzy categories using triangular membership functions, as described in section 2. data analysis was perfomed using the r package ca [22] and r code written by the author. the correspondence analysis symmetric map for these data (first and second dimension) is shown in figure 2. this map explains a total of 27.3% of the variance (or inertia) in the data. triangle points correspond to the fuzzy categories of each variable (a1 to a5, b1 to b5, etc). variable category points close to each other indicate similar response profiles to the corresponding statements. the origin of the map corresponds to the average response profile. the main interpretation of the ca map is carried out by evaluating the positions of the category points to each axis. on the left part of the first dimension (horizontal axis) lies a group of students who attribute their understanding of the course content to the tutor’s quality of teaching and practices (strong agreement with statements a, b and c) but not to their own efforts and consistency in attending classes (strong disagreement with statements d and e). on the right part of the first dimension, there is a group of students that contrasts the one on the left. these students express strong agreement with statements d and e but strong disagreement with statements a, b and c. the second dimension, when projected on the vertical axis, separates extreme values on top from moderate responses below the cross of the axes. the resulting parabolic shape or “horse shoe” is a typical structure in ca that has a unidimensional structure and confirms that the items are articulated around a hierarchical scale (for more details on the horseshoe effect, see [23]). to sum up, ca on the fuzzy-coded data obtained using the bar, reveals an interesting negative association between statements {a, b, c} and {d, e}. 5 conclusions the bar of kambaki-vougioukli and vougiouklis is a suitable and useful continuous scale, similar to a rule, that serves to collect survey data. after data collection the analyst can decide how to split each variable at appropriate intervals. 134 a fuzzy coding approach to data analysis using the bar −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 − 1 .0 − 0 .5 0 .0 0 .5 1 .0 1 .5 ca factor map dim 1 (15.36%) d im 2 ( 1 1 .9 4 % ) a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 c1 c2 c3 c4 c5 d1 d2 d3 d4 d5 e1 e2 e3 e4 e5 figure 2: correspondence analysis symmetric map (1st and 2nd dimension). this type of discrete or crisp coding, however, can lead to a significant loss of information and negate the important advantages of using the bar. fuzzy instead of crisp coding preserves the original information lying in the original data. the original values are mapped, via triangular membership functions, to a 5-category recoding, using the minimum, quartiles and maximum as the hinge points, with the first and last functions not being “shouldered”. the proposed scheme is linear and invertible and can be paired with a well-established exploratory data analysis method, correspondence analysis, for the visual investigation of both linear and non-linear relationships among variables. a side advantage of fuzzy coding is that it transforms continuous data to a form that is comparable to categorical data, and so enables analysis of mixed measurement scales. exploring this possibility is an important step for future work. 135 angelos markos references [1] aşan, z. and greenacre, m. 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(2015). questionnaires with the ‘bar’ in social sciences. science & philosophy, 3(2), pp.47–58. 138 e:\uziv\sarka\clanky\rm_25\chaudha\final\rm_25_3_.dvi ratio mathematica 25 (2013), 29–46 issn:1592-7415 optimal control policy of a production and inventory system for multi-product in segmented market kuldeep chaudhary, yogender singh, p. c. jha department of operational research, university of delhi, delhi-110007, india chaudharyiitr33@gmail.com, aeiou.yogi@gmail.com, jhapc@yahoo.com abstract in this paper, we use market segmentation approach in multiproduct inventory production system with deteriorating items. the objective is to make use of optimal control theory to solve the production inventory problem and develop an optimal production policy that maximize the total profit associated with inventory and production rate in segmented market. first, we consider a single production and inventory problem with multi-destination demand that vary from segment to segment. further, we described a single source production multi destination inventory and demand problem under the assumption that firm may choose independently the inventory directed to each segment. this problem has been discussed using numerical example. key words: market segmentation, production-inventory system, optimal control problem msc2010: 97u99. 1 introduction market segmentation is an essential element of marketing in industrialized countries. goods can no longer be produced and sold without considering customer needs and recognizing the heterogeneity of these needs [1]. earlier 29 k. chaudhary, y. singh, p. c. jha in this century, industrial development in various sectors of economy induced strategies of mass production and marketing. those strategies were manufacturing oriented, focusing on reduction of production costs rather than satisfaction of customers. but as production processes become more flexible, and customer’s affluence led to the diversification of demand, firms that identified the specific needs of groups of customers were able to develop the right offer for one or more submarkets and thus obtained a competitive advantage. segmentation has emerged as a key planning tool and the foundation for effective strategy formulation. nevertheless, market segmentation is not well known in mathematical inventory-production models. only a few papers on inventory-production models deal with market segmentation [2, 3]. optimal control theory, a modern extension of the calculus of variations, is a mathematical optimization tool for deriving control policies. it has been used in inventory-production [4, 6] to derive the theoretical structure of optimal policies. apart from inventory-production, it has been successfully applied to many areas of operational research such as finance [7, 8], economics [9, 10, 11], marketing [12, 13, 14, 15], maintenance [16] and the consumption of natural resources [17, 18, 19] etc. the application of optimal control theory in inventory-production control analysis is possible due to its dynamic behaviour. continuous optimal control models provide a powerful tool for understanding the behaviour of production-inventory system where dynamic aspect plays an important role. several papers have been written on the application of optimal control theory in production-inventory system with deteriorating items [20, 21, 22, 23]. in this paper, we assume that firm has defined its target market in a segmented consumer population and that it develop a production-inventory plan to attack each segment with the objective of maximizing profit. in addition, we shed some light on the problem in the control of a single firm with a finite production capacity (producing a multi-product at a time) that serves as a supplier of a multi product to multiple market segments. segmented customers place demand continuously over time with rates that vary from segment to segment. in response to segmented customer demand, the firm must decide on how much inventory to stock and when to replenish this stock by producing. we apply optimal control theory to solve the problem and find the optimal production and inventory policies. the rest of the paper is organized as follows. following this introduction, all the notations and assumptions needed in the sequel is stated in section 2. in section 3, we described the single source production-inventory problem with multidestination demand that vary from segment to segment and developed the optimal control theory problem and its solution. in section 4 of this paper we introduce optimal control formulation of a single source productionmulti 30 optimal control policy of a production and inventory system for ... destination demand and inventory problem and discussion of solution. numerical illustration is presented in the section 5 and finally conclusions are drawn in section 6 with some future research directions. 2 notations and assumptions here we begin the analysis by stating the model with as few notations as possible. let us consider a manufacturing firm producing m product in segmented market environment. we introduce the notation that is used in the development of the model: notations: t : length of planning period, pj (t) : production rate for j th product, ij (t) : inventory level for j th product, iij (t) : inventory level for j th product in ith segment, dij (t) : demand rate for j th product in ith segment, hj (ij (t)) : holding cost rate for j th product, (single source inventory) hij (iij (t)) : holding cost rate for j th product in ith segment, (multi destination) cj : the unit production cost rate for j th product, θj (t, ij (t)) : deterioration rate for j th product, (single source inventory) θij (t, iij (t)) : the deterioration rate for j th product in ith segment, (multi destination) kj (pj(t)) : cost rate corresponding to the production rate for j th product, rij : the revenue rate per unit sale for j th product in ith segment, ρ : constant non-negative discount rate. the model is based on the following assumptions: we assume that the time horizon is finite. the model is developed for multi-product in segmented market. the production, demand, and deterioration rates are function of time. the holding cost rate is function of inventory level & production cost rate depends on the production rate. the functions hij (iij(t)) (in case of single source hj (ij (t)) and θij (t, iij(t)) (in case of single source θj (t, ij(t))) are convex. all functions are assumed to be non negative, continuous and differentiable functions. this allows us to derive the most general and robust conclusions. further, we will consider more specific cases for which we obtain 31 k. chaudhary, y. singh, p. c. jha some important results. 3 single source production and inventorymulti-destination demand problem many manufacturing enterprises use a production-inventory system to manage fluctuations in consumers demand for the product. such a system consists of a manufacturing plant and a finished goods warehouse to store those products which manufactured but not immediately sold. here, we assume that once a product is made and put inventory into single warehouse, and demand for all products comes from each segment. let there be m products and n segments. (i.e., j = 1, . . . , m and i = 1, . . . , n). therefore, the inventory evolution in segmented market is described by the following differential equation: d dt ij (t) = pj(t) − n ∑ i=1 dij(t) − θj (t, ij (t)), ∀j = 1, . . . , m. (1) so far, firm want to maximize the total profit during planning period in segmented market. therefore, the objective functional for all segments is defined as max pj (t)≥ p m i=1 dij (t)+θj (t,ij (t)) j = ∫ t 0 e−ρt m ∑ j=1 [ n ∑ i=1 rijdij (t) − kj(pj (t)) − hj (ij (t)) ] dt + ∫ t 0 e−ρt m ∑ j=1 [ cj ( n ∑ i=1 dij (t) − pj(t) )] dt (2) subject to the equation (1).this is the optimal control problem with mcontrol variable (rate of production) with m-state variable (inventory states). since total demand occurs at rate ∑n i=1 dij (t) and production occurs at controllable rate pj(t) for j th, it follows that ij(t) evolves according to the above state equation (1). the constraints pj(t) ≥ ∑m i=1 dij(t) − θj (t, ij(t)) and ij (0) = ij0 ≥ 0 ensure that shortage are not allowed. using the maximum principle [10], the necessary conditions for (p ∗j , i ∗ j ) to be an optimal solution of above problem are that there should exist a piecewise continuously differentiable function λ and piecewise continuous function µ , 32 optimal control policy of a production and inventory system for ... called the adjoint and lagrange multiplier function, respectively such that h(t, i∗, p ∗, λ) ≥ h(t, i∗, p, λ), for all pj (t) ≥ n ∑ i=1 dij (t) − θj (t, ij(t) (3) d dt λj (t) = − ∂ ∂ij l(t, ij , pj, λj, µj) (4) ij (0) = ij0, λj(t ) = 0 (5) ∂ ∂pj l(t, ij , pj, λj, µj) = 0 (6) pj (t) − n ∑ i=1 dij(t) − θj (t, ij (t)) ≥ 0, µj(t) ≥ 0, µj(t) [ pj(t) − n ∑ i=1 dij(t) − θj (t, ij (t)) ] = 0 (7) where, h(t, i, p, λ) and l(t, i, p, λ, µ) are hamiltonian function and lagrangian function respectively. in the present problem hamiltonian function and lagrangian function are defined as h = m ∑ j=1 [ n ∑ i=1 rijdij (t) + cj ( n ∑ i=1 dij (t) − pj(t) ) − kj (pj(t)) − hj (ij(t)) ] + m ∑ j=1 [ λj(t) ( pj(t) − n ∑ i=1 dij (t) − θj (t, ij(t)) )] (8) l = m ∑ j=1 [ n ∑ i=1 rijdij (t) + cj ( n ∑ i=1 dij (t) − pj(t) ) − kj (pj(t)) − hj (ij(t)) ] + m ∑ j=1 [ (λj(t) + µj (t)) ( pj(t) − n ∑ i=1 dij(t) − θj (t, ij (t)) )] (9) a simple interpretation of the hamiltonian is that it represents the overall profit of the various policy decisions with both the immediate and the future effects taken into account and the value of λj(t) at time t describes the future effect on profits upon making a small change in ij(t) . let the hamiltonian h for all segments is strictly concave in pj(t) and according to the mangasarian sufficiency theorem [4, 10]; there exists a unique production rate. 33 k. chaudhary, y. singh, p. c. jha from equation (4) and (6), we have following equations respectively d dt λj(t) = ρλj (t) − { − ∂hj (ij(t)) ∂ij − (λj(t) + µj (t)) ∂θj (t, ij(t)) ∂ij } , (10) for all j = 1, · · · , m λj (t) + µj(t) = cj + d dpj kj (pj(t)). (11) now, consider equation (7). then for any t, we have either pj (t) − n ∑ i=1 dij (t) − θj (t, ij (t)) = 0 or pj (t) − n ∑ i=1 dij (t) − θj (t, ij (t)) > 0 ∀ j = 1, · · · , m. 3.1 case 1: let s is a subset of planning period [0, t ], when pj (t) − ∑n i=1 dij (t) − θj (t, ij(t)) = 0. then d dt ij (t) = 0 on s, in this case i ∗(t) is obviously constant on s and the optimal production rate is given by the following equation p ∗j (t) = n ∑ i=1 dij (t) − θj (t, i ∗ j (t)), for all t ∈ s (12) by equation (10) and (11), we have d dt λj(t) = ρλj(t) − { − ∂hj (ij(t)) ∂ij − ( cj + d dpj kj (pj(t)) ) ∂θj (t, ij (t)) ∂ij } (13) after solving the above equation, we get a explicit from of the adjoint function λj(t). from the equation (10)), we can obtain the value 0f lagrange multiplier µj(t). 3.2 case 2: pj(t) − ∑n i=1 dij(t) − θj (t, ij (t)) > 0, for t ∈ [0, t ]\s. then µj (t) = 0 on t ∈ [0, t ]\s. in this case the equation (10) and (11) becomes d dt λj(t) = ρλj (t) − { − ∂hj (ij(t)) ∂ij − λj(t) ∂θj (t, ij (t)) ∂ij } , ∀ j = 1, · · · , m (14) λj (t) = cj + d dpj kj(pj (t)) (15) 34 optimal control policy of a production and inventory system for ... cobining these equation with the state equation, we have the following second order differential equation: d dt pj(t) d2 dp 2j kj (pj) − [ ρ + ∂θj (t, ij (t)) ∂ij ] d dpj kj (pj) =cj ( ρ + ∂θj (t, ij(t)) ∂ij ) + ∂hj (t, ij (t)) ∂ij (16) and ij(0) = ij0, cj + d dpj kj (pj(t )) = 0. for illustration purpose, let us assume the following forms the exogenous functions kj(pj ) = kj p 2 j /2, hj(t, ij (t)) = hjij (t) and θj (t, ij (t)) = θj ij(t), where kj hj θj are positive constants for all j = 1, · · · , m. for these functions the necessary conditions for (p ∗j , i ∗ j ) to be optimal solution of problem (2) with equation (1) becomes d2 dt2 ij (t) − ρ d dt ij (t) − (ρ + θj )θ1j ij (t) = ηj (t) (17) with ij(0) = ij0, cj + d dpj kj(pj (t )) = 0. where, ηj(t) = − ∑n i=1 ( d dt dij (t) ) + (ρ + θ1j ) ( ∑n i=1 dij(t) ) + (cj (ρ+θ1j )+hj ) kj . this problem is a two point boundary value problem. proposition 3.1. the optimal solution of (p ∗j , i ∗ j ) to the problem is given by i∗j (t) = a1j e m1j t + a2j e m2j t + qj (t), (18) and the corresponding p ∗j is given by p ∗j (t) =a1j (m1j + θ1j )e m1j t + a2j (m2j + θj )e m2j t + d dt qj (t) + θ1j qj (t) + n ∑ i=1 dij. (19) the values of the constant a1j , a2j , m1j , m2j are given in the proof, and qj(t) is a particular solution of the equation (17). proof. the solution of the two point boundary value problem (17) is given by standard method. its characteristic equation m2j − ρmj − (ρ + θj )θ1j = 0, has two real roots of opposite sign, given by m1j = 1 2 ( ρ − √ ρ2 + 4(ρ + θ1j )θ1j ) < 0, m2j = 1 2 ( ρ + √ ρ2 + 4(ρ + θ1j )θ1j ) > 0, 35 k. chaudhary, y. singh, p. c. jha and therefore i∗j (t) is given by (18), where qj (t) is the particular solution. then initial and terminal condition used to determineed the values of constant a1j and a2j as follows a1j + a2j + qj (0) = ij0, a1j (m1j + θ1j )e m1j t + a2j (m1j + θ1j )e m2j t + ( cj kj + d dt qj (t ) + θ1j qj(t ) + n ∑ i=1 dij(t ) ) = 0. by putting b1j = ij0 − qj(0) and b2j = −( cj kj + d dt qj (t ) + θ1j qj (t ) + ∑n i=1 dij(t )), we obtain the following system of two linear equation with two unknowns a1j + a2j = b1j a1j (m1j + θ1j )e m1j t + a2j (m1j + θ1j )e m2j t = b2j (20) the value of p ∗j is deduced using the values of i ∗ j and the state equation. 4 single source productionmulti destination demand and inventory problem we assume the single source production and multi destination demandinventory system. hence, the inventory evolution in each segmented is described by the following differential equation: d dt iij(t) = γijpj(t) − dij (t) − θij (t, iij(t)), ∀ j = 1, · · · , m; i = 1, · · · , n. (21) here, γij > 0, ∑n i=1 γij = 1, ∀ j = 1, · · · m with the conditions iij (0) = i 0 ij and γijpj (t) ≥ dij (t) − θij (t, iij(t)). we called γij > 0 the segment production spectrum and γijpj(t) define the relative segment production rate of j th product towards ith segment. we develop a marketing-production model in which firm seeks to maximize its all profit by properly choosing production and market segmentation. therefore, we defined the profit maximization 36 optimal control policy of a production and inventory system for ... objective function as follows: max γij pj (t)≥dij (t)−θij (t,iij (t)) j = = ∫ t 0 e−ρt m ∑ j=1 [ n ∑ i=1 rij dij(t) + cj ( n ∑ i=1 (dij (t) − γijpj(t)) )] dt − ∫ t 0 e−ρt m ∑ j=1 [ n ∑ i=1 hij (iij(t)) − kj(pj (t)) ] dt (22) subject to the equation (21). this is the optimal control problem (production rate) with m control variable with mn state variable (stock of inventory). to solve the optimal control problem expressed in equation (21) and (22), the following hamiltonian and lagrangian are defined as h = m ∑ j=1 [ n ∑ i=1 rijdij (t) + cj ( n ∑ i=1 (dij(t) − γijpj (t)) )] − m ∑ j=1 [ n ∑ i=1 hij (iij(t)) + kj (pj(t)) ] + m ∑ j=1 n ∑ i=1 λij (t)[γipj(t) − dij (t) − θij (t, iij(t))] (23) l = m ∑ j=1 [ n ∑ i=1 rij dij(t) + cj ( n ∑ i=1 (dij (t) − γijpj(t)) )] − m ∑ j=1 [ n ∑ i=1 hij (iij (t)) + kj(pj (t)) ] + m ∑ j=1 n ∑ i=1 (λij + µij(t))[γipj(t) − dij (t) − θij (t, iij(t))] (24) equation (4), (6) and (21) yield d dt λij (t) = ρλij (t) − { − ∂hij (iij (t)) ∂iij − λij(t) ∂θij (t, iij (t)) ∂iij } , (25) for all i = 1, · · · , n, j = 1, · · · , m n ∑ i=1 (λij (t) + µij(t))γi = cj + d dpj kj (pj(t)) (26) in the next section of the paper, we consider only case when γijpj(t) − dij(t) − θij (t, iij (t)) > 0, ∀ i, j. 37 k. chaudhary, y. singh, p. c. jha 4.1 case 2: γijpj(t)−dij (t)−θij (t, iij(t)) > 0 ∀ i, j, for t ∈ [0, t ]\s. then µij(t) = 0 on t ∈ [0, t ]\s. in this case, the equation (25) and (26) becomes d dt λij (t) = ρλij (t) − { − ∂hij (iij (t)) ∂iij − λij(t) ∂θij (t, iij(t)) ∂iij } (27) n ∑ i=1 γijλij (t) = cj + d dpj kj (pj(t)) (28) cobining these equation with the state equation, we have the following second order differential equation: d dt pj(t) d2 dp 2j kj(pj ) − 1 n n ∑ i=1 ( ρ + ∂θi(t, iij (t)) ∂iij ) d dpj kj (pj) = n ∑ i=1 cjγi ( ρ + ∂θij (t, iij(t)) ∂iij ) + n ∑ i=1 γi ∂hij (t, iij(t)) ∂iij (29) with ij (0) = i 0 ij , ∑n i=1 γij λij(t ) = 0 → λij(t ) = 0 ∀ i and j, cj + d dpj kj (pj(t )) = 0. for illustration, let us assume the following forms the exogenous functions kj(pj) = kj p 2 j /2, hij (t, iij(t)) = hij iij (t) and θij (t, iij (t)) = θij iij(t), where kj hij θij are positive constants. for these functions the necessary conditions for (p ∗j , i ∗ ij) to be optimal solution of problem (19) with equation (18) becomes d2iij (t) dt2 + (θij − a) diij(t) dt − aθiiij (t) = ηij (t) (30) with iij(0) = i 0 ij , λij (t ) = 0 ∀ i, cj + d dpj kj(pj (t )) = 0. where, ηij (t) = −dij (t)a + γj kj [ ∑n i=1 γi(hij + cj (ρ + θij )) ] + ddij (t) dt , a = ∑n i=1 (ρ+θij ) n . this problem is a two point boundary value problem. the above system of two point boundary value problem (29) is solved by same method that we used in to solve (17). 5 numerical illustration in order to demonstrate the numerical results of the above problem, the discounted continuous optimal problem (2) is transferred into equivalent discrete problem [24] that is solved to present numerical solution. the discrete 38 optimal control policy of a production and inventory system for ... optimal control can be written as follows: j = t ∑ k=1 ( m ∑ j=1 [ n ∑ i=1 (rij(k − 1)dij(k − 1)) ])( 1 (1 + ρ)k−2 ) + t ∑ k=1 ( m ∑ j=1 cj ( n ∑ i=1 dij(k − 1) − pj (k − 1) ))( 1 (1 + ρ)k−2 ) − t ∑ k=1 ( m ∑ j=1 [kj (pj(k − 1) + hj(ij (k − 1)))] )( 1 (1 + ρ)k−2 ) such that ij(k) = ij(k − 1) + pj(k − 1) − n ∑ i=1 dij (k − 1) − θj (k − 1, ij(k − 1)) for all j = 1, · · · , m. similar discrete optimal control problem can be written for single source production multi destination and inventory control problem. these discrete optimal control problems are solved by using lingo11. we assume that the duration of all the time periods are equal and demand are equal from segment for each product. the number of market segmentsis 4 and the number of products is 3. the value of parameters are ri1 = 2.55, 2.53, 2.53, 2.54; table 1: the optimal production and inventory rate in segment market t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 p1 100 86 80 73 64 53 39 21 5 0 p2 110 81 76 70 62 52 38 21 5 0 p3 140 79 75 69 61 51 38 21 5 0 i1 20 98 154 199 232 254 262 255 231 193 i2 20 107 156 194 222 238 241 231 205 166 i3 20 137 179 211 233 244 244 231 203 161 ri2 = 2.52, 2.53, 2.54, 2.53; ri3 = 2.51, 2.54, 2.54, 2.52 for segments i = 1 to 4; cj = 1; kj = 2; θj = 0.10, 0.12, 0.13; hj = 1; for all the three products. the optimal production rate and inventory for every product for each segment is shown in table 1 and their corresponding total profit is $177402.70. 39 k. chaudhary, y. singh, p. c. jha the optimal trajectories of production and inventory rate for every product for each segment are shown in fig1, fig2 and fig3 respectively (appendix). in case of single source production-multi destination demand and inventory, the number of market segments m is 4 and the number of products is 3. the values of additional parameters are each segment is shown in table 2. table 2: the values of parameter of deteriorating rate and holding cost rate constant segment θi1 θi2 θi3 hi1 hi2 hi3 m1 0.10 0.11 0.11 1.0 1.1 1.0 m2 0.11 0.12 0.12 1.1 1.2 1.1 m3 0.13 0.11 0.11 1.2 1.1 1.2 m4 0.11 0.13 0.11 1.1 1.0 1.3 table 3: values of the parameter for single source production-multi destination demand and inventory problem in each segment t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 p1 100 85 79 73 65 54 41 23 5 0 p2 110 82 77 70 62 52 38 21 5 0 p3 140 83 77 71 63 52 38 21 5 0 i11 20 98 153 197 231 254 263 258 236 197 i12 20 97 152 195 227 247 255 248 225 185 i13 20 97 150 192 223 242 247 239 214 173 i14 20 97 149 190 218 236 240 230 204 162 i21 20 108 158 198 227 245 250 242 217 178 i22 20 107 157 195 222 238 242 232 206 167 i23 20 108 158 198 227 245 250 242 217 178 i24 20 107 156 192 218 232 235 223 196 156 i31 20 138 186 223 250 265 268 258 231 191 i32 20 138 184 220 244 258 260 248 219 178 i33 20 138 185 223 250 265 268 258 231 191 i34 20 138 186 223 250 265 268 258 231 191 the optimal production rate and inventory for every product for each segment is shown in table 3 with production spectrum γ11 = 0.10, γ12 = 0.10, γ13 = 0.77, γ14 = 0.03; γ21 = 0.12, γ22 = 0.12, γ23 = 0.75, γ24 = 0.01; γ31 = 0.14, γ32 = 0.14, γ33 = 0.72, γ34 = 0.04. the optimal value of 40 optimal control policy of a production and inventory system for ... total profit for all products is $185876.90. in case of single source productionmulti destination demand and inventory, the optimal trajectories of production and inventory rate for every product for each segment are shown in fig4, fig5, fig6 and fig7 respectively (appendix). 6 conclusion in this paper, we have introduced market segmentation concept in the production inventory system for multi product and its optimal control formulation. we have used maximum principle to determine the optimal production rate policy that maximizes the total profit associated with inventory and production rate. the resulting analytical solution yield good insight on how production planning task can be carried out in segmented market environment. in order to show the numerical results of the above problem, the discounted continuous optimal problem is transferred into equivalent discrete problem [24] that is solved using lingo 11 to present numerical solution. in the present paper, we have assumption that the segmented demand for each product is a function of time only. a natural extension to the analysis developed here is the consideration of segmented demand that is a general functional of time and amount of onhand stock (inventory). references [1] kotler, p., marketing management, 11th edition, prentice hall, englewood cliffs, new jersey, 2003. 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[24] rosen, j. b., numerical solution of optimal control problems. in g. b. dantzig & a. f. veinott(eds.), mathematics of decision science: part-2, american mathematical society, 1968, pp.37–45. appendix 0 20 40 60 80 100 120 140 160 1 2 3 4 5 6 7 8 9 10 p 1 � � fig-1 43 k. chaudhary, y. singh, p. c. jha 0 50 100 150 200 250 300 1 2 3 4 5 6 7 8 9 10 i 1 i 2 i 3 fig-2 � �� ��� ��� ��� ��� ��� � � � � � � � � �� p � p � p � � fig-3 � �� � �� �� ��� ��� � � ��� � � � � � � � � �� � � � � � � fig-4 44 optimal control policy of a production and inventory system for ... 0 50 100 150 200 250 300 1 2 3 4 5 6 7 8 9 10 i 11 i 12 fig-5 � �� ��� ��� ��� ��� ��� � � � � � � � � �� �� �� �� �� fig-6 � �� �� �� ��� ��� ��� � � � � � � � � � � �� � �� � �� � �� fig-7 45 46 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 36, 2019, pp. 97-108 97 creation of the concept of zero-point method in teaching mathematics tomáš lengyelfalusy* dalibor gonda† abstract pupils learn different calculating algorithms. the effective use of learned algorithms requires creativity in their application to solving diverse tasks. to achieve this goal, it is necessary to create a concept of the calculating algorithm for pupils. the present paper describes a method of creating a zero-point method. the teaching of this method is divided into two stages. in the first stage, the student masters the basic algorithm and becomes familiar with the main ideas of this method, while in the second stage a student learns how to apply this method with some modifications in other types of tasks. in our article, we present the application of a zero-point method in solving quadratic inequalities. keywords: concept creation, method of the zero-point, algorithm, pupils’ understanding. 2010 ams subject classification: 97d40. ‡ * department of didactics, technology and educational technologies, dti university, sládkovičova 533/20, 018 41, dubnica nad váhom. lengyelfalusy@dti.sk. † faculty of humanities, university of žilina, univerzitná 8215/1, 010 26 žilina. daliborgonda@gmail.com. ‡ received on may 12nd, 2018. accepted on june 24th, 2019. published on june 30th, 2019. doi: 10.23755/rm.v36i1.466. issn: 1592-7415. eissn: 2282-8214. ©lengyelfalusy et al. this paper is published under the cc-by licence agreement. tomáš lengyelfalusy and dalibor gonda 98 1 introduction recently the education at primary and secondary schools has undergone several reforms. one of the essential features of these reforms has been a reduction of the curriculum of individual subjects and reducing the number of lessons, especially science lessons. the main aim of reducing the curriculum and thus reducing the demands was monitoring the improvements in educational achievements of our students [1]. but pisa 2015 test results say otherwise. slovak students achieved in 2015, on average, significantly worse results than the oecd average. it is worthy of reflection that our students achieve the best results-the results almost on an average of the best students in the oecd. another feature of this educational reform is teaching a "playful" way. pupils should acquire new knowledge and skills not by memorizing and practising, but above all by the playful way. pisa testing in 2015 showed that, in terms of pupils' attitudes to learning, our students declare significantly lower endurance to solve complex problems, lower openness to solve tasks and less belief in their own abilities. it can also negatively be reflected on their results in mathematics. compared to 2003, many of slovak students' attitudes to learning significantly deteriorated. 2015 pisa test results are in substantial agreement with the results of the external part of the school leaving examination (maturita). all slovak students have to pass maturita from slovak language and literature and a foreign language. only those students have to pass maturita from mathematics, who choose math as a maturita subject. nevertheless, over the past three years, the average percentage of school maturita exam in mathematics is always worse than the average percentage of school maturita exam in compulsory subjects. we think that the ideas of school reforms are correct, but it turns out that it is not right to use the same methods to achieve the goals for all subjects. mathematics affects almost every area of human life. in the education of our youth, who should be, according to the reference of john paul ii., our hope for the future. math is challenging in its own way but at the same time can also be beautiful. we think it is necessary to seek such forms and methods of teaching mathematics [2], that we make the beauty of math available to students [3]. in the following lines, we will outline one possible way of teaching mathematics. 2 two stages of mathematical education mathematical education can be divided into two stages. the first, basic stage is the acquisition of basic calculating algorithms. these calculating algorithms are acquired by students, who practice them on the appropriate number of tasks. we can talk about math "drill", without which it is creation of the concept of zero-point method in teaching mathematics 99 impossible to be a successful solver of mathematical problems. the information-receptive didactic method with a combination of the reproductive method is mainly used in this stage. it is very important that the student acquires the necessary skill of how to use them by repeated use of basic calculating algorithms. the teacher, by the right choice of tasks, ensures that pupils acquire these calculating algorithms at least at the level of understanding, not only at the level of memorization. the second, application stage is the application of the acquired algorithms in different areas of mathematics and other disciplines or in practical everyday life. at this stage, the mathematical "drill" is replaced by mathematical thinking. based on the assignment a student considers what math knowledge and skills he can use to solve the task. unlike the first stage, he must learn that the first step of task solution is not to count but to think. based on a detailed consideration and possible task mathematization the student chooses a suitable calculating algorithm. at this stage, the teacher becomes a moderator of solution and uses a heuristic didactic method. at this stage, in terms of the taxonomy of educational objectives, the level of acquirement of calculating algorithms will be increased for the minimum to the application level. if the teaching is correct, we can say, that at this stage, the students do not learn new calculating algorithms. at this stage, students gain new, mainly theoretical knowledge of mathematics, and also learn how to apply already gained calculating algorithms in a new context. the above-described stages are illustrated on the example of the method of zero points. 3 method of zero points solving of the most mathematical problems includes solving of various equations and inequalities, or their systems. the tasks, where it is necessary to solve equations, inequalities and their systems belong to the declaratory mathematical tasks [4]. declaratory mathematical tasks are historically the oldest mathematical tasks. when solving these tasks the mathematical concepts and methods. those are the tasks that require finding, calculating, constructing etc. of all mathematical objects of a particular type, having the desired properties. in each declaratory task, we can define as the frame of considerations some nonempty set m of mathematical objects, which is a carrier of a particular structure. using the terms belonging to this structure, it is then possible to express the desired properties of those objects of the set m that we are looking for. to characterize the elements of the set m we use propositional form v (x) which verity domains then create subsets of the set m. in each determinative task there is a subset k of set m, which elements have the characteristics required by task assignment. the task and the objective of the investigator are to determine the set p by naming of its elements or to operate tomáš lengyelfalusy and dalibor gonda 100 with already known subsets of the set m. we can solve the mathematical declaratory task with the direct and indirect methods. the direct method of solving means a process by which we determine the set of solutions k so that we work exclusively with sets that belong to the chain of sets inclusions   …  𝑲  …  𝑴, where m is a non-empty set of mathematical objects, among which elements we are looking for the solving of the task. indirect methods consist in the fact that instead of solving the task that is defined we solve the other task or other tasks (using some direct method) and the results are used to obtain the results of the original task. one of the indirect methods is to switch to subtasks on the same set. we divide the set m to individual subsets and we investigate the specific location of each original task. we will obtain partial solutions to the original task on each of these subsets. the overall result for the task will be obtained by the unification of partial results. method of zero points can be included precisely into that category of indirect methods (in some literature this method is also called the method of intervals). the essential feature of the method of zero points is the attempt to divide tasks into several "sub-tasks", solving them on the corresponding subsets intervals. to deal with this method it is necessary to learn the algorithms of expression modifying, polynomial factorization to the product of the root factors and solving various types of equations [5]. 4 teaching the method of zero points the teaching of this method is recommended to be realized in three levels. level 1: acquisition of the method the students meet the method of zero points for the first time when they solve inequalities with an unknown in the denominator. its basic steps are learned through leading example. example 1: on the set r solve the inequality 2𝑥+3 𝑥−1 < 1 solution: most students have the following knowledge on solving the inequalities: inequalities are solved using the same equivalent adjustment as the equations. if the inequality is multiplied or divided by a negative number, the sign of inequality is changed to the opposite. on the basis of this knowledge the first step of solving is an attempt to remove a fraction of the assigned inequality, that is, they multiply the inequality by the expression (x1). already in the introduction of the model example, students learn another difference between solving the equations and inequalities. inequalities, unlike equations, cannot be multiplied by the expression of which i cannot clearly decide whether it is positive or negative. if we want students to use the creation of the concept of zero-point method in teaching mathematics 101 proposed adjustment, it is first necessary to determine for which values of the variable the expression (𝑥 − 1) is positive and for which negative. consequently, it is necessary to divide the solving of the inequalities to, in this case, two parts when the expression (𝑥 − 1) is positive and the sign of inequality does not change after multiplication, and when the expression (𝑥 − 1) is negative and the sign of inequality changes to opposite one after multiplication. basically, the assigned inequality should be tackled twice. we recommend concentrating on the issue of "multiplying inequalities" and pay sufficient attention, because it is needed to change students fixed “definition” of solving the inequalities. the method of zero points does not require multiple solving of the same inequalities and therefore it, is considered to be mora effective method. it can be divided into the following steps: 1. annulling the right side of the equation: 2𝑥 + 3 𝑥 − 1 − 1 < 0 2. simplifying the expression on the left side of the inequality: 𝑥 + 4 𝑥 − 1 < 0 (1) after these adjustments, we draw the students’ attention to the intermediate target of our solutions. we compare the fraction to zero. therefore, we only need to determine the sign of the expression 𝑥+4 𝑥−1 . our partial objective is to determine for what value of 𝑥 it is positive and for what value negative. 3. determining the zero points: zero points are the values of variable x for which numerator and denominator separately on the left side of the inequality takes the zero value. zero points can be determined based on solving the equation 𝑥 + 4 = 0; 𝑥 − 1 = 0. zero points are nb: -4; 1. 4. adjusted numerical axis: we come to the core of the method. first, we explain the function of zero points. zero points divided real numbers, in this case, into the three sets intervals. for each interval is true: the expression 𝑥+4 𝑥−1 is positive or negative in the whole interval, in other words, it does not change the resulting sign. the adjacent intervals the expression 𝑥+4 𝑥−1 has different resulting signs. based on the above it is sufficient, if we want to determine the final sign, to substitute any number belonging to this interval to the expression. if we know the final sign in one of the intervals, we automatically recognize the resulting sign in all intervals as signs alternate. using that knowledge, we can create a customized numerical axis (fig. 1): tomáš lengyelfalusy and dalibor gonda 102 figure 1. there are numbers under the axis to be substituted for variable x in the expression; above the axis are values of the expression after substitution. that is, if we substitute any number from the interval (-4, 1) the resulting sign of the expression 𝑥+4 𝑥−1 is negative, after the substituting 𝑥 = −4, the resulting value of the expression is zero. symbol ∅ means that for the value 𝑥 = 1 the expression is not defined. the adjusted numerical axis can be created as follows. first, on the numerical axis (from the bottom), we mark zero points. (students often automatically show zero even if it is not the zero point on the numerical axis. there should be only zero points on the numerical axis). we substitute any number different from zero points to the expression on the left side of the inequality. if the zero point is not zero, we substitute number zero to the variable. after substituting the number zero to the variable x, the expression 𝑥+4 𝑥−1 has the value of 4. then we write a minus sign above the numerical axis in the part corresponding to the interval, from which we substituted the number zero. the signs in the other intervals will be completed without calculations, whilst complying with the principle of alternating signs. we complete 0 above the zero point “of the numerator“ and the sign ∅ above the zero point "of the denominator". 5 determination of results those values of variable x for which the expression 𝑥+4 𝑥−1 acquire negative values will be the solution to the inequality (1). based on the adjusted numbering axis, the search solution to the assigned inequality is the interval from -4 to 1. finally, we determine the "brackets" of the final interval. zero point, above which is symbol ∅, cannot be the solution, therefore it will be at zero point "of the denominator" always round bracket. if there is the symbol 0 above the zero point, it means, that after its substituting, the resulting value of the expression is zero. however, we are looking for negative values of the expression and therefore the number -4 has a round bracket. the ultimate solution is 𝑥∈ (-4.1). after solving the model example we recommend to discuss with students how the solution would change if we solve the inequality 2𝑥+3 𝑥−1 > 1 and the inequality 2𝑥+3 𝑥−1 ≤ 1. students should be aware, that in both cases, the first four steps will be identical with the model example. in the fifth step, based on the same considerations, the solution of the inequality would be 2𝑥+3 𝑥−1 > 1 𝑥 ∈ creation of the concept of zero-point method in teaching mathematics 103 (−∞; −4) ∪ (1; ∞). the solution of inethe quality 2𝑥+3 𝑥−1 ≤ 1 is 𝑥 ∈ ⟨−4; 1) level 2: understanding of the method the main idea behind the method of zero points can be considered a comparison of the fraction with zero. a student knows that if a numerator and a denominator have the same final sign, so the fraction is positive if they have different sign fraction is negative. the correct application of this idea leads to an understanding of the method of zero points and also to a more efficient using of this method. the correct application of the main idea it is essential to understand the "functioning" of zero points. the zero point for this expression, in principle, divides the set of real numbers (na) into three subsets. on one of the subsets, it acquires only positive values, on another one just negative. the third subset is only composed of zero point and the expression of the set acquires a value of 0. for example, the expression 𝑥 − 5 has a zero point 5. then, the expression acquires negative values on the set 𝑀1 = (−∞; 5), on the set 𝑀2 = (5; ∞) it acquires positive values and on the set 𝑀 = {5} it takes the value 0. thus, we can simplistically say, that there is a different sign of the expression from the various sides of the zero point. if the expression is in productive form, the zero points of individual members of the product create the zero points of all expression. example 2: on the set r solve the inequality (𝑥−9)(𝑥+1)2 (𝑥−4)(𝑥+5) > 0 solution: zero points -5; -1; 4; 9. at first, we draw attention to the expression (𝑥 + 1)2. this expression acquires for all x ∈ r non-negative values. therefore, it has no influence to the final sign of the expression (𝑥−9)(𝑥+1)2 (𝑥−4)(𝑥+5) . the zero point of the expression (𝑥 + 1)2 can be described as "unnecessary" zero point and it will not be showed on the adjusted numbering axis. (if we showed it there, the theory of alternation marks would not apply.) to obtain the solution of the inequality we only need to know the final sign of the expression (𝑥−9)(𝑥+1)2 (𝑥−4)(𝑥+5) . therefore. after substituting, for example 𝑥 = 0, it is not necessary to know the numerical value. at the same time, we know that it is not necessary to substitute to the expression (𝑥 + 1)2. by applying the above mentioned ideas after substituting x = 0 we obtain "a signed" value of the expression: − −.+ . we set the adjusted numbering axis (fig. 2): tomáš lengyelfalusy and dalibor gonda 104 figure 2. the solution of the assigned inequality based on the adjusted numbering line and the sign of inequality is 𝑥 ∈ (−5; 4) ∪ (9; ∞). to obtain the final solution, we must once again pay attention to "needless" zero point. we know that for = −1, the expression acquires the resulting value zero on the left side. therefore, the number −1 does not belong to the solution of our set of ithe nequality. the ultimate solution of the inequality 𝑥 ∈ (−5; −1) ∪ (−1; 4) ∪ (9; ∞). level 3: application of the method after mastering the basic algorithm and understanding the method of zero points we recommend to focus on the teaching of its application in other types of examples, such as those in which students can penetrate into its mysteries. the closest type of tasks is inequalities in the productive form. the student already knows that there are the same rules for comparison zero to the product as for the comparison of the quotient to zero. therefore, in solving qualities in productive form, the method of zero points can be used identically as in solving the inequalities in productive form. quadratic inequality can be seen as inequality in the productive form. in example 3 we show a sample solution. example 3: on the set r solve the inequality 𝑥2 + 3𝑥 − 4 ≥ 0. solution: quadratic trinomial on the left side of the inequality must be adjusted to the product of the root factors, and therefore we obtain the inequality in the form of productive form (𝑥 − 1)(𝑥 + 4) ≥ 0 zero points are -4; 1. the quadratic trinomial, after substitution x = 0, acquires negative value. in fact, zero is not necessary to be substituted, because for x = 0 is the final "a signed" value of quadratic trinomial, identical to the sign in front of the absolute member. we set the adjusted numbering axis (fig. 3): figure 3. based on the sign of inequality, in assigned inequality, we search for which values of unknown x the expression 𝑥2 + 3𝑥 − 4 acquires positive or zero values. therefore, the solution is inequality is 𝑥 ∈ (−∞; −4⟩ ∪ ⟨1; ∞). if the quadratic equation corresponding to the assigned inequality has less than two real roots, the method of zero points is modified. at this creation of the concept of zero-point method in teaching mathematics 105 modification, we primarily rely on understanding the "functioning" of zero points. example 4: on the set r solve the inequality 𝑥2 − 4𝑥 + 4 ≥ 0. solution: the inequality should be adjusted to productive form (𝑥 − 2)(𝑥 − 2) ≥ 0. the left side of inequality will not be left in this form, because the students would incorrectly use the principle of alternation marks around the zero points. the expression on the left side of the inequality will be written in simplified form, and we receive the inequality (𝑥 − 2)2 ≥ 0 zero point is 2. since the expression (𝑥 − 2)2 is for all 𝑥 ∈ 𝑹 nonnegative, number 2 is “unnecessary" zero point. number 2 is the only zero point and so it is not needed to set the adjusted numbering axis. the solution of the inequality is 𝑥 ∈ 𝑹 and it was discovered when we were considering the zero point. after solving example 4 we suggest a discussion on solving inequalities: 𝑥2 − 4𝑥 + 4 > 0, 𝑥2 − 4𝑥 + 4 ≤ 0 , 𝑥2 − 4𝑥 + 4 < 0. note: a common mistake at solving the inequality (𝑥 − 2)2 ≥ 0 is the extract of the root of both sides of the inequality, after which students have the wrong inequality 𝑥 − 2 ≥ 0. the following consideration can bring them to the fact, that the inequality is incorrect. both sides of the inequality were non-negative before extracting the root and the left side can also takes negative values. if we want, even after extracting, both sides being nonnegative, we must put the left side of inequality to an absolute value(√𝑎2 = |𝑎|). after correct extracting, we get the inequality with absolute value which can also be solved by the method of zero points. example 5: on the set r solve the inequality 𝑥2 + 2𝑥 + 6 < 0. solution: on the set r it is not possible to modify the quadratic trinomial to the product, as the appropriate quadratic equation 𝑥2 + 2𝑥 + 6 = 0 have no real roots. based on the understanding of the function of zero points we know, that expression 𝑥2 + 2𝑥 + 6 has for all x ∈ r a signed value. it is identical with the sign in front of the absolute term. so the expression on the left side of the inequality is for all real numbers positive. the solution of the inequality is x = {}. even after solving this inequality we recommend the discussion about solutions for different variants of the sign of inequality. if we want to see if the students understand the method, they must be able to apply the basic ideas of the method to solving the task. in other words, tomáš lengyelfalusy and dalibor gonda 106 we understand the method of solving if it developed our mathematical thinking. the following example can be solved by applying the basic ideas of the method of zero points. example 6: for which parameter values 𝑎 ∈ 𝑅 is each x ∈ r the solution of inequality solution: the expression 𝑥2 − 8𝑥 + 20 has no zero points and according to the sign in front of the absolute member we know, that it acquires positive values for all 𝑥 ∈ 𝑅. if all real numbers should be the solution of the assigned inequality, the expression in the denominator of the inequality fraction must be negative for all 𝑥 ∈ 𝑅. using the basic ideas of the method of zero points, we consider the following. we need the expression 𝑎𝑥2 + 2(𝑎 + 1)𝑥 + 9𝑎 + 4 "still" negative, and that does not change the final sign, and therefore we cannot have the zero points. that is, the quadratic equation 𝑎𝑥2 + 2(𝑎 + 1)𝑥 + 9𝑎 + 4 = 0 has no solution. thus, discriminant has to be negative. this way we get the inequality 𝑥2 − 8𝑥 + 20 𝑎𝑥2 + 2(𝑎 + 1)𝑥 + 9𝑎 + 4 < 0 the solution to this inequality that we solve using the method of zero points is 𝑎 ∈ (−∞; − 1 2 ) ∪ (1; ∞). now, we secure the final sign will be negative. we know from the method of zero points, that by substituting zero to quadratic trinomial, the final sign is identical with a sign in front of the absolute term. the denominator in the assigned inequality is a quadratic trinomial with parameter. for 𝑎1 ∈ (−∞; − 1 2 ) ∪ (1; ∞) has the constant sign for all x ∈ r. if the absolute member is negative, the resulting sign of trinomial will be negative. therefore we solve the inequality 9𝑎 + 4 < 0. its solution is 𝑎2 ∈ (−∞; − 4 9 ). based on the previous considerations, the parameter 𝑎 must meet both conditions. the ultimate solution is 𝑎 ∈ 𝑎1 ∩ 𝑎2 = (−∞; − 1 2 ). conclusion the basis for the success of a student in solving mathematical tasks is acquiring the calculating algorithms [6], [7]. to achieve this goal it is necessary to solve, especially alone, the sufficient number of tasks, more or less, of the same type. we believe that the mastery of basic calculating algorithms is necessary but not sufficient condition for student success in creation of the concept of zero-point method in teaching mathematics 107 dealing with the tasks. it is not enough just to learn the calculating algorithm, it is necessary, after its acquisition, also think about its individual elements. this is the way when the basic ideas, used in the algorithm, occur. the discovering these main ideas of calculating algorithm lead to understanding, as well as acquiring the algorithm at a higher level. the understanding causes the method to be is a powerful tool in students dealing with tasks. it affects his mathematical thinking. the method of zero points is a method that should be understood and not only learned. if a student enters its secrets, it becomes flexible and he will be able to use it in different types of tasks and, as appropriate, be adapted. by understanding the method will become effective tool in the hands of the investigator. the students know that the method of zero points is mainly used to solve inequalities. if the students know the method, it heads their initial ideas, when solving inequality, to adjust the inequality to a productive or quotient form. this fact can be used in teaching solutions to quadratic inequalities. using the method of zero points the student does not learn new calculating algorithm, but he learns how to apply already acquired knowledge and skills. we think that one of the possible ways to increase the efficiency in mathematical learning is the emphasis on understanding the calculating algorithms and their subsequent application in various areas of mathematics. while we make sure that we choose those tasks, where the main ideas can be applied. this way helps us to create the thought linking of mathematics as a whole and mathematics with other disciplines, e.g. those involving computers into the pedagogical process [8], in the mind of the students. basically, there is no need to reduce the amount of subject matter, just to organize the mathematical knowledge better in the mind of the students. tomáš lengyelfalusy and dalibor gonda 108 references [1] bušek, i. řešené maturitní úlohy z matematiky. praha: spn. 1985. [2] šedivý, o., ďuriš, v., fulier, j. konštruktivizmus vo vyučovaní matematiky. in. veda-vzdelávanie-prax 2 diel: zborník z medzinárodnej vedeckej konferencie. nitra, ukf 2007, p. 319-323, isbn 978-80-8094-2038. [3] fulier, j., šedivý, o. motivácia a tvorivosť vo vyučovaní matematiky. nitra: fakulta prírodných vied ukf v nitre. 2001. [4] hejný, m. teória vyučovania matematiky 2. bratislava: spn. 1991. [5] odvárko, o. metody řešení matematických úloh. praha: spn. 1990. [6] turek, i. didaktika. bratislava: iura edition. 2008. [7] hošková-mayerová, š., rosická, z. programmed learning. in: procedia social and behavioral sciences, vol. 31, p. 782-787, 2012 doi: 10.1016/j.sbspro.2011.12.141. [8] ďuriš, v.: prečo začleňovať počítačové programy do vyučovania. in. technológia vzdelávania: vedecko-pedagogický časopis, issn 1335-003x, vol. 6, 2005 ratio mathematica 24 (2013), 3–10 issn:1592-7415 fuzzy hyperalgebras and direct product r. ameri, t. nozari school of mathematics, statistics and computer sciences, college of sciences, university of tehran, tehran, iran departement of mathematics, faculty of science, golestan university, gorgan, iran rameri@ut.ac.ir,t.nozari@gu.ac.ir abstract we introduce and study the direct product of a family of fuzzy hyperalgebras of the same type and present some properties of it. key words: fuzzy hyperalgebras, term function, direct product. msc2010: 97u99. 1 introduction in this section we present some definitions and simple properties of hyperalgebras which will be used in the next section. in the sequel h is a fixed nonvoid set, p∗(h) is the family of all nonvoid subsets of h, and for a positive integer n we denote for hn the set of n-tuples over h (for more see [1]). recall that for a positive integer n a n-ary hyperoperation β on h is a function β : hn → p∗(h). we say that n is the arity of β. a subset s of h is closed under the n-ary hyperoperation β if (x1, . . . ,xn) ∈ sn implies that β(x1, . . . ,xn) ⊆ s. a nullary hyperoperation on h is just an element of p∗(h); i.e. a nonvoid subset of h. a hyperalgebra h = 〈h, (βi, | i ∈ i)〉 (which is called hyperalgebraic system or a multialgebra ) is the set h with together a collection (βi, | i ∈ i) of hyperoperations on h. r. ameri, t. nozari a subset s of a hyperalgebra h=〈h, (βi, : i ∈ i)〉 is a subhyperalgebra of h if s is closed under each hyperoperation βi, for all i ∈ i, that is βi(a1, ...,ani ) ⊆ s, whenever (a1, ...,ani ) ∈ sni . the type of h is the map from i into the set n∗ of nonnegative integers assigning to each i ∈ i the arity of βi. two hyperalgebras of the same type are called similar hyperalgrbras. for n > 0 we extend an n-ary hyperoperation β on h to an n-ary operation β on p∗(h) by setting for all a1, ...,an ∈ p∗(h) β(a1, ...,an) = ⋃ {β(a1, ...,an)|ai ∈ ai(i = 1, ...,n)} it is easy to see that 〈p∗(h), (βi : i ∈ i)〉 is an algebra of the same type of h. definition 1.1 let h=〈h, (βi : i ∈ i)〉 and h=〈h, (βi : i ∈ i)〉 be two similar hyperalgebras. a map h from h into h is called a (i) a homomorphism if for every i ∈ i and all (a1, ...,ani ) ∈ hni we have that h(βi((a1, ...,ani )) ⊆ βi(h(a1), ...,h(ani )); (ii) a good homomorphism if for every i ∈ i and all (a1, ...,ani ) ∈ hni we have that h(βi((a1, ...,ani )) = βi(h(a1), ...,h(ani )). definition 1.2 let h be a nonempty set. a fuzzy subset µ of h is a function µ : h → [0, 1]. definition 1.3 a fuzzy n-ary hyperoperation fn on s is a map fn : s × ···× s −→ f∗(s), which associated a nonzero fuzzy subset fn(a1, . . . ,an) with any n-tuple (a1, . . . ,an) of elements of s. the couple (s,f n) is called a fuzzy n-ary hypergroupoid. a fuzzy nullary hyperoperation on s is just an element of f∗(s); i.e. a nonzero fuzzy subset of s. definition 1.4 let h be a nonempty set and for every i ∈ i, βi be a fuzzy ni-ary hyperoperation on h, then h=〈h, (βi : i ∈ i)〉 is called fuzzy hyperalgebra, where (ni : i ∈ i) is type of this fuzzy hyperalgebra. definition 1.5 if µ1, . . . ,µni be ni nonzero fuzzy subsets of a fuzzy huperalgebra h=〈h, (βi : i ∈ i)〉, we define for all t ∈ h βi(µ1, . . . ,µni )(t) = ∨ (x1,...,xni )∈h ni (µ1(x1) ∧ . . . ∧ µni (xni ) ∧ βi(x1, . . . ,xni )(t)) finally, for nonempty subsets a1, . . . ,ank of h, set a = a1 × . . . × ani . then for all t ∈ h βk(a1, . . . ,ank )(t) = ∨(a1,...,ank )∈a(βk(a1, . . . ,ank )(t)). 4 fuzzy hyperalgebras and direct product for nonempty subset a of h, χa denote the characteristic function of a . note that, if f : h1 −→ h2 is a map and a ∈ h1, then f(χa) = χf(a). definition 1.6 let h = 〈h, (βi : i ∈ i)〉 and h′ = 〈h′, (β′i : i ∈ i)〉 be two fuzzy hyperalgebras with the same type, and f : h −→ h′ be a map. we say that f is a homomorphism of fuzzy hyperalgebras if for every i ∈ i and every a1, . . . ,ani ∈ h we have f(βi(a1, . . . ,ani )) ≤ β′i(f(a1), . . . ,f(ani )). let h=〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra then, the set of the nonzero fuzzy subsets of h denoted by f∗(h), can be organized as a universal algebra with the operations; βi(µ1, . . . ,µni )(t) = ∨ (x1,...,xni )∈h ni (µ1(x1) ∧ . . . ∧ µni (xni ) ∧ βi(x1, . . . ,xni )(t)) for every i ∈ i, µ1, . . . ,µni ∈ f∗(h) and t ∈ h. we denote this algebra by f∗(h). in [3] gratzer presents the algebra of the term functions of a universal algebra. if we consider an algebra b=〈b, (βi : i ∈ i)〉 we call n−ary term functions on b (n ∈ n) those and only those functions from bn into b, which can be obtained by applying (i) and (ii) from bellow for finitely many times: (i) the functions eni : b n → b, eni (x1, . . . ,xn) = xi, i = 1, . . . ,n are n−ary term functions on b; (ii) if p1, . . . ,pni are n−ary term functions on b, then βi(p1, . . . ,pni ) : bn → b, βi(p1, . . . ,pni )(x1, . . . ,xn) = βi(p1(x1, . . . ,xn), . . . ,pni (x1, . . . ,xn)) is also a n−ary term function on b. we can observe that (ii) organize the set of n−ary term functions over b (p (n)(b)) as a universal algebra, denoted by b(n)(b). if h is a fuzzy hyperalgebra then for any n ∈ n, we can construct the algebra of n−ary term functions on f∗(h), denoted by b(n)(f∗(h)) = 〈p (n)(f∗(h)), (βi : i ∈ i)〉. 2 on the direct product of fuzzy hyperalgebras proposition 2.1 let h=〈h, (βi : i ∈ i)〉 and b=〈b, (βi : i ∈ i)〉 are fuzzy hyperalgebras of the same type, h : h → b a fuzzy homomorphism and p ∈ p (n)(f∗(h)). then for all a1, . . . ,an ∈ h we have h(p(χa1, . . . ,χan )) ⊆ p(h(χa1 ), . . . ,h(χan )). 5 r. ameri, t. nozari proof. the prove is by induction over the steps of construction of a term.2 remark 2.1 if h : h → b be fuzzy good homomorphism then h(p(χa1, . . . ,χan )) = p(h(χa1 ), . . . ,h(χan )). remark 2.2 we can easily construct the category of the fuzzy hyperalgebras of the same type, where the morphisms are considered to be the fuzzy homomorphisms and the composition of two morphisms is the usual mapping composition and we will denote it by fha definition 2.1 let q,p ∈ p (n)(f∗(h)). the n−ary (strong) identity p = q is said to be satisfied on a fuzzy hyperalgebra h if p(χa1, . . . ,χan ) = q(χa1, . . . ,χan ) for all a1, . . . ,an ∈ h. we can also consider that a weak identity p∩ q 6= ∅ is said to be satisfied on a fuzzy hyperalgebra h if p(χa1, . . . ,χan ) ∧ q(χa1, . . . ,χan ) > 0 for all a1, . . . ,an ∈ h. definition 2.2 let ((hk, (β k i : i ∈ i)),k ∈ k) be an indexed family of fuzzy hyperalgebras with the same type. the direct product ∏ k∈k hk is a fuzzy hyperalgebra with univers πk∈khk and for every i ∈ i and (a1k)k∈k, . . . , (a ni k )k∈k ∈ πk∈khk : β q i ((a 1 k)k∈k, . . . , (a ni k )k∈k)(tk)k∈k = ∧ k∈k βki (a 1 k, . . . ,a ni k )(tk) theorem 2.1 the fuzzy hyperalgebra ∏ k∈k hk constructed this way, together with the canonical projections, is the product of the fuzzy hyperalgebras (hk,k ∈ k) in the category fha. proof. for any fuzzy hyperalgebra (b, (βbi : i ∈ i)) and for any family of fuzzy hyperalgebra homomorphisms (αk : b → hk|k ∈ k) there is only one homomorphism α : b → πk∈khk such that αk = πkk ◦α for any k ∈ k. indeed, there exists only one mapping α such that the diagram is commutative. 6 b hk � αk πk∈khk πk k α 6 fuzzy hyperalgebras and direct product this mapping is defined by α(b) = (αk(b))k∈k. now we have to do is to verify that α is fuzzy hyperalgebra homomorphism. if we consider i ∈ i and b1, . . . ,bni ∈ b, (tk)k∈k ∈ πk∈khk then if r ∈ α−1((tk)k∈k) we have α(r) = (tk)k∈k and α(r) = (αk(r))k∈k, hence ∀k ∈ k; tk = αk(r), it means that ∀k ∈ k; r ∈ α−1k (tk), therefore ∀k ∈ k; α −1((tk)k∈k) ⊆ α−1k (tk). we have α(βbi (b1, . . . ,bni ))(tk)k∈k = ∨ r∈α−1((tk)k∈k ) (βbi (b1, . . . ,bni ))(r) ≤ ∨ s∈α−1 k (tk)) βbi (b1, . . . ,bni ))(s) = αk(β b i (b1, . . . ,bni ))(tk) then α(βbi (b1, . . . ,bni ))(tk)k∈k ≤ ∧ k∈k αk(β b i (b1, . . . ,bni ))(tk) ≤ ∧ k∈k βki (αk(b1), . . . ,αk(bni ))(tk) = β q i (α(b1), . . . ,α(bni ))(tk)k∈k. which finishes the proof.2 proposition 2.2 for every n ∈ n, p ∈ p (n)(f∗(h)) and (a1k)k∈k, . . . , (a n k)k∈k, we have p(χ(a1 k )k∈k , . . . ,χ(an k )k∈k )(tk)k∈k = ∧ k∈k p(χa1 k , . . . ,χan k )(tk) proof. we will use the steps of construction of a term. i. if p = ejn(j = 1, 2, . . . ,n) then p(χ(a1 k ) k∈k , . . . ,χ(an k ) k∈k )(tk)k∈k = e j n(χ(a1k)k∈k , . . . ,χ(an k ) k∈k )(tk)k∈k = χ (a j k ) k∈k (tk)k∈k = ∧ k∈k ejn(χa1k, . . . ,χa n k )(tk) = ∧ k∈k p(χa1 k , . . . ,χan k )(tk) ii. suppose that the statement has been proved for p1, . . . ,pni and that p = βi(p1, . . . ,pni ). then we have p(χ(a1 k )k∈k , . . . ,χ(an k )k∈k )(tk)k∈k = βi(p1, . . . ,pni )(χ(a1k)k∈k , . . . ,χ(an k )k∈k )(tk)k∈k = βi(p1(χ(a1 k )k∈k , . . . ,χ(an k )k∈k ), . . . ,pni (χ(a1k)k∈k , . . . ,χ(an k )k∈k ))(tk)k∈k = ∨ (s1 k )k∈k,...,(s ni k )k∈k [p1(χ(a1 k )k∈k , . . . ,χ(an k )k∈k )(s1k)k∈k∧. . .∧pni (χ(a1k)k∈k, . . . ,χ(ank )k∈k) (snik )k∈k ∧βi((s 1 k)k∈k, . . . , (s ni k )k∈k)(tk)k∈k] 7 r. ameri, t. nozari = ∨ (s1 k )k∈k,...,(s ni k )k∈k [ ∧ k∈k p1(χa1 k , . . . ,χan k )(s1k)∧ . . .∧ ∧ k∈k pni (χa1k, . . . ,χa n k )(snik )∧∧ k∈k βi(s 1 k, . . . ,s ni k )(tk)] = ∧ k∈k [ ∨ (s1 k )k∈k,...,(s ni k )k∈k p1(χa1 k , . . . ,χan k )(s1k)∧. . .∧pni (χa1k, . . . ,χank )(s ni k )∧βi(s 1 k, . . . ,s ni k )(tk)] = ∧ k∈k βi(p1(χa1 k , . . . ,χan k ), . . . ,pni (χa1k, . . . ,χa n k ))(tk) = ∧ k∈k βi(p1, . . . ,pni )(χa1k, . . . ,χa n k )(tk) = ∧ k∈k p(χa1 k , . . . ,χan k )(tk). which finishes the proof of the proposition.2 theorem 2.2 if ((hk, (β k i : i ∈ i)),k ∈ k) be an indexed family of fuzzy hyperalgebras with the same type i such that p∩q 6= ∅ is satisfied on each fuzzy hyperalgebra hk, then is also satisfied on the fuzzy hyperalgebra ∏ k∈k hk. proof. let p,q ∈ p (n)(f∗(h)) and suppose that p∩ q 6= ∅ is satisfied on each fuzzy hyperalgebra hk. this means that for all k ∈ k and for any a1k, . . . ,a n k ∈ hk we have p(χa1k, . . . ,χank )∧q(χa1k, . . . ,χank ) > 0. by proposition 3.7 , we conclude that p(χ(a1 k )k∈k , . . . ,χ(an k )k∈k ) ∧ r(χ(a1 k )k∈k , . . . ,χ(an k )k∈k ) = = ∧ k∈k p(χa1 k , . . . ,χan k ) ∧ ∧ k∈k q(χa1 k , . . . ,χan k ) = ∧ k∈k (p(χa1 k , . . . ,χan k ) ∧ q(χa1 k , . . . ,χan k )) > 0 and the proof is finished.2 theorem 2.3 if ((hk, (β k i : i ∈ i)),k ∈ k) be an indexed family of fuzzy hyperalgebras with the same type i such that p = q is satisfied on each fuzzy hyperalgebra hk, then p = q is also satisfied on the fuzzy hyperalgebra ∏ k∈k hk. proof. let p,q ∈ p (n)(f∗(h)) and suppose that p = q is satisfied on each fuzzy hyperalgebra hk. this means that for all k ∈ k and for any a1k, . . . ,a n k ∈ hk we have p(χa1k, . . . ,χank ) = q(χa1k, . . . ,χank ). by proposition 3.7 , we conclude that p(χ(a1 k )k∈k , . . . ,χ(an k )k∈k ) = ∧ k∈k p(χa1 k , . . . ,χan k ) = ∧ k∈k q(χa1 k , . . . ,χan k ) 8 fuzzy hyperalgebras and direct product = r(χ(a1 k )k∈k , . . . ,χ(an k )k∈k ) and the proof is finished.2 3 acknowledgement the first author partially has been supported by the ”research center in algebraic hyperstructures and fuzzy mathematics, university of mazandaran, babolsar, iran” and ”algebraic hyperstructure excellence, tarbiat modares university, tehran, iran”. references [1] p. corsini, prolegomena of hypergroup theory, second edition, aviani editor (1993). [2] p. corsini and i. tofan, on fuzzy hypergroups, pu.m.a, 8 (1997), 2937. [3] g. gratzer, universal algebra, 2nd edition, springer verlage, 1970. [4] j. n. mordeson,m.s. malik, fuzzy commutative algebra, word publ. 1998. [5] c. pelea, on the direct product of multialgebras, studia uni, babes-bolya, mathematica, vol. xlviii (2003), 93-98. [6] c. pelea, multialgebras and term functions over the algebra of their nonvoid subsets, mathematica (cluj), 43 (2001), 143-149. [7] m.k. sen, r.ameri, g. chowdhury, fuzzy hypersemigroups, soft computing, soft comput, 12 (2008), 891-900. [8] t. vougiuklis, hyperstructures and their representations, hardonic press (1994). 9 10 ratio mathematica 27 (2014) 91-110 issn:1592-7415 the intertemporal choice behavior: the role of emotions in a multiagent decision problem viviana ventre dipartimento di economia, management e metodi quantitativi università degli studi del sannio, italy ventre@unisannio.it abstract traditional discounted utility model assumes an exponential delay discount function, with a constant discount rate: this implies dynamic consistency and stationary intertemporal preferences. contrary to the normative setting, decision neuroscience stresses a lack of rationality, i.e., inconsistency, in some intertemporal choice behaviors. we deal with both models are dealt with in the framework of some relevant decision problems. key words: time preference, exponential discounting, hyperbolic discounting. 2000 ams: 91c99. 1 introduction the traditional discounted utility model (du model) (samuelson, 1937) [20] fails in being both normative and descriptive. indeed several studies, especially carried out in psychology and neuroeconomics, reveal the existence of relevant anomalies violating the axioms of the traditional model (section 3). bechara and colleagues [2] show that decision making processes are guided by emotional signaling, which allow people to choose advantageously before they realized the strategy that worked best. this fact justifies the presence of anomalies in intertemporal choice and the use of hyperbolic delay discounting 91 viviana ventre (declining as the length of the delay increases), so, people have the tendency to increasingly choose a smaller-sooner reward over a larger-later reward as the delay occurs sooner in time. this entails intertemporal inconsistency and preferences reversal. even so, an impatient behavior not necessarily can be considered incoherent (section 4). the results of some studies by shiv et al. [21] and naqvi et al. [17] have demonstrated that patients with lesions in specific components of a neural circuitry critical for the processing of emotions will make more advantageous decisions than normal subjects when faced with the types of positive-expected-value gambles that most people routinely shun (section 5). recent neuroeconomic and econophysical studies have explored neurobiological and psychological factors, e.g. impulsivity and inconsistency that determined individual differences in intertemporal choice. takahashi et al. [25] attempt to dissociate impulsivity and inconsistency in their econophysical studies proposing a quasi-exponential delay discount function. other behavioral economists propose multiple selves models attempting to measure the strength of the internal conflict within the decision maker, best known as quasi-hyperbolic discount model (laibson, 1997) [11] (section 6). to fight impulsivity strotz [23] proposed two strategies that might be adopted by a person who foresees how her preferences will change over time; thaler and shefrin [26] built a structure in which the individual is treated as if he contained two distinct psyches denoted as planner and doer (section 7). in a multiagent decision context the objective for a decision group is to choose a common decision, that is an alternative which is judged the best by the majority of the decision makers. so in most strategic decisions, it is important to be able to estimate the characteristics and behaviors of others. if the characteristics of other players are unknown, estimating them is a critical task (section 8). moreover, psychological evidence suggests people own beliefs, values, and habits tend to bias their perceptions of how widely they are shared (false consensus effect ). this effect demonstrates an inability of individuals to process information rationally (section 9). therefore when we use the aggregation of the agent preferences to assess consensus, we obtain a coefficient which includes the false consensus effect that depends on the subjectivity and also increases the degree of consensus. in order to eliminate the component of human judgment vagueness a procedure defined by ordered weighted averaging (owa) operators, introduced by yager [29], can be applied (section 10). an experiment performed by engelmann and strobel [8] demonstrates that a false consensus effect is present only if information about decision 92 thtemporal e interchoice behavior: the role of emotions in a multi-agent decision problem of other members of the group is implicit. so the consensus effect is not always false but only when people, forming expectations concerning decisions of others, weight their own decision more heavily than that of a randomly selected person from the same population (see [6], [7]), (section 11). the result is linked with the analysis of false consensus effect in cooperative and non-cooperative decision problem. indeed, in a cooperative decision problem, agents know choices of other members, while in a non-cooperative one they have to judge choices of others (section 12). 2 traditional discounting model and decision neuroscience the standard economic model of discounted utility assumes that economic agents make intertemporal choices over consumption profiles (ct, . . . ,ct ) and such preferences can be represented by an intertemporal utility function ut(ct, . . . ,ct ), which can be described by the following special functional form: ut(ct, . . . ,ct ) = t−t∑ k=0 d(k)u(ct+k) where d(k) = ( 1 1 + ρ )k so the du model assumes an exponential temporal discounting function and a constant discount rate (ρ), which represents the pure rate of time preference of the individual. an important implication of constant discount rate and exponential discounting function is that intertemporal preferences of the individual are timeconsistent: if at time t a person prefers c2 at t + 2 to c1 at t + 1, then at time t + 1 she must prefer c2 at t + 2 to c1 instantly. so, with the same temporal options and the same information, later preferences confirm earlier preferences. however, several empirical studies have documented various inadequacies of the du model as a descriptive model of behavior. behavioral economic theories on decision process have found that there are a number of behavior patterns that violate the rational choice theory [27]. decision neuroscience is an emerging area of research whose goal is to integrate research in neuroscience and behavioral decision making. it calls into question the theories of choice that assume decisions derive from an assessment of the future outcomes of various options and alternatives through 93 viviana ventre some type of cost-benefit analysis, which ignore influence of emotions on decision-making. this investigation explores the neural road map for the physiological processes intervening between knowledge and behavior, and the potential interruptions that lead to a disconnection between what one knows and what one decides to do. decision making studies in neurological patients, who can no longer process emotional information, normally suggest that people make judgments not only by evaluating the consequences and their probability of occurring, but also and even sometimes primarily at a gut or emotional level (see [1]). 3 behavioral finance: empirical anomalies violating du model some studies concerning the individual behavior from the psychological perspective, e.g. related with discounting real or hypothetical rewards, show the existence of violations of the du model. a first empirical remark is that discount rates are not constant over time, but appear to decline a pattern often referred to as hyperbolic discounting ([22], [23]). furthermore, even for a given delay, discount rates vary across different locations of intertemporal choices [28]. delay effect, magnitude effect, sign effect and sequence effect are among the relevant anomalies in intertemporal choice, we will deal with. the delay effect rests on the evidence that as waiting time increases, the discount rates tend to be higher in the short intervals than in the longer ones. prelec and loewenstein [18] define this anomaly as common difference effect and immediacy effect. we can set out delay effect as: (x,s) ∼ (y,t) but (x,s + h) < (y,t + h) for y > x,s < t and h > 0 if two capitals, (x,s) and (y,t), are indifferent, (x,s) ∼ (y,t), their projections onto a common instant p have to coincide: xa(s,p) = ya(t,p) if and only if x y = a(t,p) a(s,p) = v(s,t,p) being a(t,p) the discount function which represents the amount available at p instead of one euro available at t, and v(s,t,p) the corresponding financial 94 thtemporal e interchoice behavior: the role of emotions in a multi-agent decision problem factor. in the same way, if (x,s + h) ∼ (y,t + h), this implies that xa(s + h,p) = ya(t + h,p) if and only if x y = a(t + h,p) a(s + h,p) = v(s + h,t + h,p) then: v(s,t,p) < v(s + h,t + h,p) the magnitude effect can be described as follows. larger outcomes are discounted at a lower rate than smaller outcomes. let us suppose that the instantaneous discount rate is inversely proportional to the discounted amount: a(c,z) = ce− ∫ z 0 k c dx = ce− k c z prelec and loewenstein [18] formulate the magnitude effect as follows: (x,s) ∼ (y,t) implies (ax,s) < (ay,t) for y > x > 0, s < t and (−x,s) ∼ (−y,t) implies (−ax,s) > (−ay,t) the sign effect. gains are discounted at a higher rate than losses of the same magnitude. prelec and loewenstein [18] proposed the amplification loss property implying that, changing the sign of an amount from gains to losses, the weight of this amount increases: (x,s) ∼ (y,t) implies (−x,s) > (−y,t) for y > x > 0, s < t. increasing sequences of consumption are preferred over decreasing ones even if the total amount is the same. in general, when subjects choose among different sequences of two events people tend to save the better thing for last, contradicting the standard assumption of a positive interest rate. in the improving sequence effect, for all s and t, and s < t, there is a c0 such that, for all y > x > c0, the following preference holds {(x,s), (y,t)} >p {(y,s), (x,t)} in the instant p ([16], [26]). 95 viviana ventre 4 anticipation of future events and hyperbolic discounting in contrast with the historically dominant view of emotions as a negative influence in human behavior, recent research in neuroscience and psychology has highlighted the positive roles played by emotions in decision making (bechara et al. [2]; damasio [5]; loewenstein and lerner [12]). although strong negative emotions can lead destructive patterns of behavior, some authors (see [2]; [5]; [21]) have shown that individuals with emotional dysfunction tend to perform poorly compared with those who have intact emotional processes. an experiment exhibited in [2] leads to the conclusion that decision making is guided by emotional signaling generated in anticipation of future events. without the ability to generate these emotional signals, the patients fail to avoid choices that lead to losses, and instead continue to sample from the disadvantageous choices until they go broke in a manner that is akin to how they behave in real life. in normal individuals, unconscious biases guide behavior before conscious knowledge does. without the help of such biases, overt knowledge may be insufficient to ensure advantageous behavior. decision maker preferences are inconsistent and change over time, because normal people possess anticipatory indices of somatic states, that represent unconscious biases that are linked to prior experiences with reward and punishment. these biases alarm the normal subject about selecting a disadvantageous course of action, even before the subject becomes aware of the goodness or badness of the choice he is about to make [1]. indeed, when normal people won or lost money on an investment round, they adopted a conservative strategy and became more reluctant to invest on the subsequent round [21]. furthermore the preference for more immediate rewards per se is not always irrational, because there are opportunity costs and risk associated with non-gaining in delaying the rewards. as a consequence there is considerable agreement among psychologists and economists that the notion of exponential discounting should be replaced by some form of hyperbolic discounting, which can represent the tendency of the individuals to increasingly choose a smaller-sooner reward over a largerlater reward as the delay occurs sooner in time (delay effect ). many authors proposed different hyperbolic discount functions, in which temporal discount function δ increases with the delay to an outcome. loewen96 thtemporal e interchoice behavior: the role of emotions in a multi-agent decision problem stein and prelec [13] proposed the form: d(t) = ( 1 1 + αt )β α where β > 0 is the degree of discounting and α > 0 is the departure from exponential discounting. hyperbolic discounting has been applied to a wide range of phenomena, including consumption-saving behavior. consistent with hyperbolic discounting, people’s investment behavior exhibits patience in the long run and impatience in the short run [28]. a second type of empirical support for hyperbolic discounting comes from experiments on dynamic inconsistency. studies and empirical evidences show that delay effect can derive in preference reversal between two rewards as the time distance to these rewards diminishes. a hyperbolic discount model can demonstrate this; indeed, non-exponential time preference curves can cross [23] and consequently the preference for one future reward over another may change in time [28]. 5 the negative side of emotions: impulsivity the positive roles played by emotions when making decisions are in contrast with some contexts in which individuals deprived of normal emotional reactions might actually make better decisions than normal individuals. for instance, consider the case of a patient with ventromedial prefrontal damage (which involves severe impairments in judgment and emotion) who was driving under hazardous road conditions [5]. when other drivers reached an icy patch, they hit their brakes in panic, causing their vehicles to skid out of control, but the patient crossed the icy patch unperturbed, gently pulling away from a tailspin and driving ahead safely. the patient remembered the fact that not hitting the brakes was the appropriate behavior, and his lack of fear allowed him to perform optimally [21]. other evidences suggest that even relatively mild negative emotions that do not result in a loss of self-control can play a counterproductive role among normal individuals in some situations. when gambles that involve some possible loss are presented one at a time, most people display extreme levels of risk aversion toward the gambles, a condition known as myopic loss aversion [3]. if myopic loss aversion does indeed have an emotional basis, then any dysfunction in neural systems subserving emotion ought to result in reduced levels of risk aversion and, thus, lead to more advantageous decisions in cases 97 viviana ventre in which risk taking is rewarded. furthermore individuals deprived of normal emotional reactions might, in certain situations, make more advantageous decisions than those not deprived of such reactions; so the lack of emotional reactions may lead to more advantageous decisions [21]. indeed in many cases, indeed, temptations induce disadvantageous behavior, and when temptation becomes too great, what the person knows to be his best long run interests conflicts with his short run desires. sociologists and psychologists have persistently studied impulsivity relative to its resultant behaviors such as drug addiction, suicide, aggression and violence. these studies suggests that individuals who frequently engage in impulsive behavior may fail to appropriately evaluate the consequences of their behavior [28]. 6 neuroeconomics: impulsivity and inconsistency in intertemporal choice the greatest contradiction to rational theory, in intertemporal choice, is inconsistent preference, usually manifested as temporary preference for options that are extremely costly or harmful in the long run. this behavior can be typically seen in psychiatric disorders (alcoholism, drug abuse), but also in more ordinary phenomena (overeating, credit card debt) [28]. some investigations in neuroeconomics, a specialized field of decision neuroscience, have found that addicts are more myopic, i.e., they have large time-discount rates, in comparison with non-addict populations [4]. it results that hyperbolic discounting may explain various human problematic behaviors [11]: loss of self-control, failure in planned abstinence from addictive substances and relapse, a deadline rush due to procrastination, failure in saving enough before retirement and risky sexual behavior. addiction and financial mismanagement frequently co-occur, and elevated delay discounting may be a common mechanism contributing to both of these problematic behaviors. we have noted that the preference for more immediate rewards per se is not always irrational or inconsistent (section 4); therefore, impulsivity in intertemporal choice is rationalizable for several categories of persons. the behaviors of addicts are clinically problematic, but economically rational when their choices are time-consistent, if they have large discount rates with an exponential discount function. however, it is known that addicts also discount delayed outcomes hyperbolically, suggesting the intertemporal choices of addicts are time-inconsistent, resulting in a loss of self-control [4]: 98 thtemporal e interchoice behavior: the role of emotions in a multi-agent decision problem they act more impulsively at the moment of the choice, against their own previously-intended plan. moreover if large discount rates are due to habitual drug intake, it is expected that discount rates decreased after long term abstinence. however, recent studies on alcoholics and smokers report that abstinence does not dramatically reduce discount rates of former alcoholics and smokers [24]. behavioral neuroeconomic and econophysical studies have proposed two discount models, in order to clarify the neural and behavioral correlates of impulsivity and inconsistency in intertemporal choice, namely, a quasiexponential discount model and a quasi-hyperbolic discount model. quasi-exponential discount model. takahashi et al. [25] have proposed and examined the following function for subjective value v (d) of delayed reward: v (d) = a expq(kqd) = a [1 + (1 − q)kqd] 1 1−q where d denotes a delay until receipt of a reward, a the value of a reward at d = 0, and kq a parameter of impulsivity at delay d = 0 (q-exponential discount rate) and the q-exponential function is defined as: expq(x) = (1 + (1 − q)x) 1 1−q this function can distinctly parametrize impulsivity and inconsistency [28]. quasi-hyperbolic discount model. behavioral economists have proposed that the inconsistency in intertemporal choice may be attributed to an internal conflict between multiple selves within a decision maker. as a consequence, there are at least two exponential discounting selves (with two exponential discount rates) in a single individual; and when delayed rewards are at the distant future (> 1 year), the self with a smaller discount rate wins, while delayed rewards approach to the near future (within a year), the self with a larger discount rate wins, resulting in preference reversal over time. this intertemporal choice behavior can be parametrized in a quasihyperbolic discount model (also as a β − δ model). for discrete time τ (the unit assumed is one year) the quasi-hyperbolic discount factor is defined [11] as: f(τ) = βδt for τ = 1, 2, 3 . . . and f(0) = 1, 0 < β < δ < 1. a discount factor between the present and one-time period later β is smaller than that between two future time-periods δ. 99 viviana ventre in the continuous time, the proposed model is equivalent to the linearlyweighted two-exponential functions (generalized quasi-hyperbolic discounting): v (d) = a[w exp(−k1d) + (1 −w) exp(−k2d)] where 0 < w < 1, is a weighting parameter and k1 and k2 are two exponential discount rates (k1 < k2). note that the larger exponential discount rate of the two k2, corresponds to an impulsive self, while the smaller discount rate k1 corresponds to a patient self [28]. 7 self-control against impulsivity: strotz model and thaler and shefrin model a number of mechanisms of self-control are predicted by hyperbolic discounting. strotz proposed two strategies that might be adopted by a person who foresees how her preferences will change over time. 1. the strategy of precommitment. a person commits himself to perform a plan of action. for instance, consider a consumer with an initial endowment k0 of consumer goods which has to be allocated over the finite interval (0,t). at time t he wishes to maximize his utility function: j0 = ∫ t 0 λ(t− 0)u[c̄(t), t] dt subject to ∫ t 0 c(t) dt = k0, where [c̄(t), t] is the instantaneous rate of consumption at time period t, and λ(t − 0) is a discount factor, whose value depends on the elapsing time between a past or future date and present. this implies that the discounted marginal utility of consumption should be the same for all periods. but, at a later date, the consumer may reconsider his consumption plan. then the problem is to maximize j0 = ∫ t 0 λ(t− τ)u[c(t), t] dt subject to ∫ t τ c(t) dt = kτ = k0 − ∫ τ 0 c(t) dt. the optimal pattern of consumption will change with changes in τ and if the original plan is altered, the individual is said to display dynamic inconsistency. strotz showed that individuals will not alter the original plan only if λ(t,τ) is an exponential in |t− τ|. 100 thtemporal e interchoice behavior: the role of emotions in a multi-agent decision problem 2. the strategy of consistent planning. since precommitment is not always a feasible solution to the problem of intertemporal conflict, an individual may adopt a different strategy: take into account future changes in the utility function and reject any plan that he will not follow through. his problem is then to find the best plan among those he will actually follow. in the setting of multiple selves models, in order to control impulsivity, thaler and shefrin [26] proposed a planner-doer model which draws upon principal agent theory. they treat an individual as if he contained two distinct psyches: one planner, which pursue longer-run results, and multiple doers, which are concerned only with short-term satisfactions, so they care only about their own immediate gratification (and have no affinity for future or past doers). for instance, consider an individual with a fixed income stream y = [y1,y2, . . . ,yt ], where∑ t yt = y has to be allocated over the finite interval (0,t). the planner would choose a consumption plan to maximize his utility function v (z1,z2. . . . ,zt ) subject to ∑ t ct ≤ y , where zt is a utility function of level consumption in t (ct). on the other hand, the unrestrained doer 1 would borrow y −y1 on the capital market and therefore choose c1 = y ; the resulting consequence is naturally c2 = c3 = . . . = ct = 0. such an action would suggest a complete absence of psychic integration. then the model focuses on the strategies employed by the planner to control the behavior of the doers, and it proposes two tools at his disposal. (a) he can impose rules on the doers behavior, which operate by altering the constraints imposed on any given doer; or (b) he can use discretion accompanied by some method of altering the incentives or rewards to the doer without any self-imposed constraints [28]. 101 viviana ventre 8 multiagent decision problems: consensus and agreement in a multiagent decision problem an individual needs to take his intertemporal choice considering others’ preferences, in order to achieve a consensus over a common decision. group decision problems, indeed, consist in finding the best alternative(s) from a set of feasible alternatives a = {a1, . . . ,am} according to the preferences provided by a group of agents e = {e1, . . . ,en}. the objective is to obtain the maximum degree of agreement among the agents overall performance judgements on the alternatives (see [22]). specifically, every agent assesses each alternative in his preference system. furthermore the group of agents has to verify if there is a possibility to rank the alternative set in a way shared by (a majority in the group). if such an operation succeeds, the group has reached a consensus about the ranking of the alternative set. in real situations, humans rarely come to a unanimous agreement: this has led to evaluate not only crisp degrees of consensus, but also intermediate degrees between 0 and 1, corresponding to partial agreement among all agents. however, full consensus can be considered not necessarily as a result of unanimous agreement, but it can be obtained ever in the case of agreement among a fuzzy majority of agents (see [9], [10]). 9 false consensus it is well known, not only in the areas of social sciences, that people are egocentric. as pointed out in several experiments, in a multiagent decision problem each decision maker overestimates his own opinion. social psychology has founded that people with a certain preference tend to make higher judgements of the popularity of that preference in others, compared to the judgements of those with different preferences. this empirical result has been termed the false consensus effect (see [19], [16]). it states that individuals overestimate the number of the people who possess the same attributes as they do. people often believe that others are more like themselves than they really are. thus, their predictions about others’ beliefs or behaviors, based on casual observation, are very likely to err in the direction of their own beliefs or behavior. for example, college students who preferred brown bread estimated that over 50% of all other college students preferred brown bread, while white-bread eaters estimated that 37% showed brown bread preference. as the consequence, in multi-agent decision problem we often have to deal with different opinions, different importance of criteria and agents, who 102 thtemporal e interchoice behavior: the role of emotions in a multi-agent decision problem are not fully impartial objective. in this sense, the false consensus effect produces partial objectivity and incomplete impartiality, which perturbs the agreements over the evaluation. 10 assessing consensus and false consensus different methods to compute a degree of a consensus in fuzzy, or imprecise, environments have been defined, and some approaches have been proposed to measure consensus in the context of fuzzy preference relations (see [9], [10]). however, as we have seen, the false consensus effect can lead to an absence of objectivity in the evaluation process. then just a numerical indication seems not to be sufficient to synthesize the degree of consensus of agents which incorporate both the true knowledge generated in the agent opinion and the subjective component that produces false consensus outputs. the opinion of each agent is decomposed into two components: a vector, made of the ranking of the alternatives, built by means of a classical procedure, e.g., a hierarchical procedure [14], and a fuzzy component that represents the contribution of the false consensus effect, which we assume to be fuzzy in nature [15]. this allows us to consider aggregation operators, such as owa operators, useful when synthesis among fuzzy variables is to be built [22]. a formal model considers the set n of decision makers, the set a of the alternatives, and the set c of the criteria. let any decision maker i ∈ n be able to assess the relevance of each criterion. precisely, for every i, a function hi : c → [0, 1] with ∑ c∈c hi(c) = 1, denoting the evaluation or weight that the decision maker assigns to the criterion c, is defined. furthermore, the function gi : a×c → [0, 1] is defined, such that gi(a,c) is the value of the alternative a with respect to the criterion c, in the perspective of i. let n, p, and m denote the (positive integer) numbers of the elements of the sets n, c, and a, respectively. the value hi(c)c∈c denotes the evaluation of the p-tuple of the criteria by the decision maker i and the value gi(c,a)c∈c,a∈a, defines the matrix p×m whose elements are the evaluations, made by i, of the alternatives with respect to each criterion in c. the function: a → [0, 1], defined by( fi(a) ) a∈a = hi(c)c∈c ·gi(c,a)c∈c,a∈a 103 viviana ventre is the evaluation, made by i, of the alternative a ∈ a. a euclidean metric that acts between couples of decision makers i and j, i.e., between individual rankings of alternatives, is defined by d(fifj) = √ 1 |a| ∑ a∈a ( fi(a) −fj(a) )2 if the functions hi, gi range in [0, 1], then also 0 ≤ d(fifj) ≤ 1. if we set δ? = max{d(fi,fj)|i,j ∈ n}, then a degree of consensus δ? can be defined as the complement to one of the maximum distance between two positions of the agents: δ? = 1 − δ? = 1 − max{d(fi,fj)|i,j ∈ n} now to identify the portion of the false consensus effect internal to the consensus reaching process, we have to consider a vector that represents the components of the consensus p(a)p +q(a)q. this polynomial representation of the measure of the effect is composed by a numeric component p(a)p , that contains all quantitative information available derived from the consensus reaching process, and q(a)q that reflects the false consensus effect. then the measure of the effect is: q(a) = 1 n(d?)2 n∑ i=1 (fi −fj)2 with 0 ≤ q(a) ≤ 1, ∀i, j ∈ n. this component can be estimated by means of owa operators (a class of decision support tools for providing heuristic solution to situations where several trade-offs should be taken into consideration). in yager [29] is introduced an approach for multiple criteria aggregation, based on ordered weighted averaging (owa) operators. by ranking the alternatives, the operators provide an enhanced methodology for evaluating actions on a qualitative basis [22]. 11 study on false consensus effect under varying information conditions. engelmann and strobel experiment in section 9, false consensus has been defined as an egocentric bias that occurs when people estimate consensus for their own behaviors. the judgements of each agent, indeed, are frequently based, in part, on intuition or 104 thtemporal e interchoice behavior: the role of emotions in a multi-agent decision problem subjective beliefs, rather than detailed data on the preferences of the people being predicted. however such intuitive judgements become more pervasive judgements when people lack necessary data to base their judgements. therefore, according to dawes (see [6], [7]), classical definition of false consensus does not justify the attribute false. he argues that it is perfectly rational to use the information about one’s own decision in the same way as the information about any other randomly selected from a sample. the effect is only false if too much weight is assigned to one’s own decision compared to a randomly selected person from the same population. engelmann and strobel [8] refer to the effect as defined above as a consensus effect and affirms that people exhibit a false consensus effect if among those with the same total information (i.e. that includes the information about their own decision) the estimates are biased in the direction of their own decision. to demonstrate this and investigate whether a false consensus effect depends on the cognitive effort needed to retrieve information, engelmann and strobel compared two treatments in a simple one-shot experiment. results are in opposite direction to a false consensus effect when in a decision group the agents have explicit information about the choice of other members of their own group, while results are in line with a false consensus effect in all groups in which the information were implicit. this shows that most subjects are unwilling or unable to use information that is not handed to them on a silver platter. it appears to us that in the implicit information treatment it does not occur to many subjects that the other subjects’ choices are valuable information and that this information is rather easily available, while the prominent information in the explicit information treatments is recognized as valuable information by virtually all subjects (or leads them to unconsciously update their beliefs). in conclusion, engelmann and strobel affirm that there is no false consensus effect if representative information is highly prominent and retrievable without any effort. indeed, there is even a significant effect in the opposite direction, indicating that subjects consider others’ choices as more informative than their own. 12 false consensus effect and emotions in a multiagent decision problem multiagent decision problems are characterized by interplay between intertemporal considerations and strategic interactions: two or more agents could have to take a common decision for a future time and in this pro105 viviana ventre cess they are influenced by emotional signal, which arise with impulsivity and with false or true consensus effect. theory of games provide tools for describe strategic interaction. indeed, in non-cooperative interaction each agent makes decisions independently, without collaboration or communication with any of the others. this can be assimilated to situations in which information about decision of other members of decision group is implicit. in this kind of strategic decision the consensus effect is false. as in engelmann and strobel experiment, if members of group decision do not cooperate they do not possess information about the choices of others, so the influence of psychological aspects lead to judge others in the same way that they judge themselves. then two situations are possible: 1) each agent have the same preference and they will reach a common decision that is given by the unanimous choice, 2) the agents have different preferences and do not assign any weight to the other preferences, so it is not possible to aggregate them (see section 10). then the influence of emotions has no negative consequences if the choices of the agents are unanimous, and then the final decision will be also the best decision in the paretian sense. if this does not happen, it is impossible to achieve a common strategy without arresting impulsivity, and unanimity becomes increasingly difficult to obtain when the number of agents increases. on the contrary in a cooperative decision problem the influence of false consensus effect is present at period-one, while the loss of self-control of each agent is fought by the imposition of a rule [26]. the rationality of the equilibrium choice of the cooperative game is saved by the possibility of making an arrangement among agents, which represents a pure rule to maintain self-control at later time in thaler and shefrin model (section 7). moreover with an arrangement the agents have explicit information about the choices of other members, so the lack of false consensus effect is in line with the result of engelmann and strobel experiment. consider the classic example of coordination game: the battle-of-the sexes. in this game an engaged couple must choose what to do in the evening: the man prefers to watch a baseball game and the women prefers to attend an opera. in terms of utility the payoff for each strategy is: man opera (o) baseball (b) woman opera (o) 3, 1 0, 0 baseball (b) 0, 0 1, 3 106 thtemporal e interchoice behavior: the role of emotions in a multi-agent decision problem in the example there are multiple outcomes that are equilibriums: (b,b) and (o,o). however both players would rather do something together than go to separate events, so no single individual has an incentive to deviate if others are conforming to an outcome: the man would attend the opera if he thinks the woman will be there even though he prefers the other equilibrium outcome in which both attend the baseball game. in this context, a consensus decision making process can be considered as an instrument to choose the best strategy in a coordination game. the final decision is often not the first preference of each individual 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[18] d. prelec and g. loewenstein, decision making over time and under uncertainty: a common approach, management science 37 (1991), 770786. [19] l. ross, d. green and p. house, the false consensus effect: an egocentric bias in social perception and attribution processes, journal of experimental social psychology 13 (1977), 279-301. [20] p. a. samuelson, a note on the measurement of utility, the review of economic studies 4 (1937), 155-161. [21] b. shiv, g. loewenstein, a. bechara, h. damasio and a. r. damasio, investment behavior and the negative side of emotion, psychological science 16 (2005), 435-439. [22] m. squillante and v. ventre, assessing false consensus effect in a consensus enhancing procedure, international journal of intelligent systems 25 (2010), 274-285. [23] r. h. strotz, myopia and inconsistency in dynamic utililty maximization, review of economic studies 23(3) (1955-1956), 165-80. 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[28] a. g. s. ventre and v. ventre, the intertemporal choice behaviour: classical and alternative delay discounting models and control techniques, atti accademia peloritana dei pericolanti classe scienze fisiche matematiche naturali 90 (1) c3 (2012), 1-21. 109 viviana ventre [29] r. r. yager on ordered weighted averaging aggregation operators in multi criteria decision making, ieee trans systems man cybernetics 18(1) (1988), 183-190. 110 ratio mathematica vol. 33, 2017, pp. 61-76 issn: 1592-7415 eissn: 2282-8214 some remarks on hyperstructures their connections with fuzzy sets and extensions to weak structures piergiulio corsini∗ †doi:10.23755/rm.v33i0.384 abstract a brief excursus on the last results on hyperstructures and their connections with fuzzy sets. at the end a calculation of the fuzzy grade of hvstructures of order two. keywords: hyperstructure 2010 ams mathematics subject classification: 20n20 ∗udine university, italy; piergiuliocorsini@gmail.com † c©piergiulio corsini. received: 31-10-2017. accepted: 26-12-2017. published: 31-12-2017. 61 piergiulio corsini 1 introduction one knows that to every fuzzy set (h,µ0) one hypergroup can be associated (which i proved [9], is a join space) in the following way: ∀(x,y) ∈ h2, one sets x◦0 y = {z | min{µ0(x),µ0(y)}≤ µ0(z) ≤ max{µ0(x),µ0(y)}. i proved also [18] that to every hypergroupoid (h,◦) a fuzzy set corresponds, defined as you can see below: set ∀u ∈ h, q(z) = {(x,y) | u ∈ x◦y}, q(u) = |q(u)|. a(u) = ∑ (x,y)∈q(u) 1/|x◦y|, µ1(u) = a(u)/q(u). i proved that h0 = (h,◦) is a join space. so, to every hypergroupoid, a sequence of hypergroupoids and fuzzy sets is associated: (h,µ0), (h,µ1), ... if |h| < ℵ0, then the sequence is clearly finite. we call fuzzy grade [20] of (h,◦) the minimum natural number of k, such that two consecutive join spaces are isomorphic. for the hv-structures, notion introduced by t. vougiouklis, one can proceed in a similar way. so, one defines the fuzzy grade of a hv-hypergroupoid as min{k | hk ' hk+1}. thomas vougiouklis is author of many papers on hyperstructures. just at the beginnng of his activity he invented and studied a structure, defining the following hyperoperation: given a hypergroupoid (h;∗) and a non empty subset p of h, he set x ◦ y = x ∗ p ∗ y and found several interesting results on this hyperoperation. but the most important theory that he introduced is that one of the hv-hyperstructures. he replaced the notion of associativity with that one of ”weak associativity”. that is instead of supposing for every x,y,z ∈ h, (x∗y)∗z = x∗ (y ∗z), one supposes (x∗y)∗z ∩x∗ (y ∗z) 6= ∅. one has considered also weak rings. it is enough to set for every a,b,c in r, a◦ (b + c)∩ (a◦ b + a◦ c) 6= ∅. 62 some remarks on hyperstructures their connections with fuzzy sets and extensions to weak structures the idea by vougiouklis of considering weak hyperstructures opened a new branch of mathematics. many significant results have been obtained in this field and probably many others will be found in the future. a theme which deserves to be considered in this context is that one of hx structures. hx-groups were born in china, invented by li hongxing [81], and studied by him, wang and others, see [79], [80], [87], [117], [118], [119]. in italy, corsini extended this notion to hyperstructures. he and cristea in italy, fotea in romania, kellil and bouaziz in saudi arabia worked in this direction. given a group g and the set p∗(g) of all nonempty subsets of g, endowed with the operation ∀(a,b) ∈p∗(g)×p∗(g), a◦b = {xy | x ∈ a,y ∈ b} a subgroup of p∗(g) is called an hx-group. one has calculated the fuzzy grade for z/nz for n ≤ 16 and also for other structures, for instance for the multiplicative group z2,22 and the direct product of some z/nz, see [22], [23], [24]. it would very interesting to consider the same problems in the such general context of weak structures, that is to calculate the fuzzy grade of hx-hypergroup zn. given an hx-group f , one considers the set f ′ of all nonempty subsets of f . let us suppose that k is a subgroup of f. we define the following hyperoperation x⊗y = ⋃ x∈a, y∈b, {a,b}⊆k ab in the set ∪a∈ka. the structure (h,⊗) is called an hx-hypergroupoid. one can extend the notion of hx-hypergroup to weak hyperstructures. some open problems on weak structures: • find conditions for an hx-hypergroupoid to be a hypergroup; • the fuzzy grade of hxweak hypergroups already considered in the classic case. 63 piergiulio corsini 2 hv-hypergroupoids of order 2 the hv-hypergroupoids of order 2, which are not associative, are 10. the following [12], [13], [15] have fuzzy grade 1. the others [9], [10], [11], [14], [16], [17], [18] have fuzzy grade 2. • h12 a b a h h b b a we have q1(a) = 3, a1(a) = 2, so µ1(a) = 2/3. q1(b) = 3, a1(b) = 2, so µ1(b) = 2/3. then ∂h12 = 1. • h13 a b a h b b h a we have q1(a) = 3, a1(a) = 2, so µ1(a) = 2/3. q1(b) = 3, a1(b) = 2, so µ1(b) = 2/3. then ∂h13 = 1. • h15 a b a h a b b h indeed we find q1(a) = 3, a1(a) = 2, so µ1(a) = 2/3. q1(b) = 3, a1(b) = 2, so µ1(b) = 2/3. hence h1 = t , the total hypergroup. therefore ∂h15 = 1. 64 some remarks on hyperstructures their connections with fuzzy sets and extensions to weak structures • h9 a b a h b b b a we have q1(a) = 2, a1(a) = 3/2, so µ1(a) = 3/4 = 0.75. q1(b) = 3, a1(b) = 5/2, so µ1(b) = 5/6 = 0.8333. so we obtain h19 a b a a h b h b therefore µ2(a) = µ2(b), whence h29 is the total hypergroup, whence ∂h9 = 2. • h10 a b a a h b b a we find q1(a) = 2, a1(a) = 5/2, so µ1(a) = 0.833. q1(b) = 2, a1(b) = 3/2, so µ1(b) = 3/4 = 0.75. we obtain h110 a b a a h b h b so µ2(a) = µ2(b), whence h210 is the total hypergroup, whence ∂h10 = 2. • h11 a b a b h b a b we find q1(a) = 2, a1(a) = 3/2, so µ1(a) = 0.75. q1(b) = 3, a1(b) = 5/2, so µ1(b) = 0.833. 65 piergiulio corsini it follows h111 a b a a h b h b so µ2(a) = µ2(b), so ∂h11 = 2. • h14 a b a h a b a h we find q1(a) = 4, a1(a) = 3, so µ1(a) = 0.75. q1(b) = 2, a1(b) = 1, so µ1(b) = 0.50. whence we hve h114 a b a a h b h b by consequence h214 is the total hypergroup, whence ∂h14 = 2. • h16 a b a a h b h a we find q1(a) = 4, a1(a) = 3, so µ1(a) = 3/4 = 0.75. q1(b) = 2,a1(b) = 1,µ1(b) = 0.50. it follows h116 a b a a h b h b so µ2(a) = µ2(b), whence h216 is the total hypergroup, whence ∂h16 = 2. 66 some remarks on hyperstructures their connections with fuzzy sets and extensions to weak structures • h17 a b a h h b a h we find q1(a) = 4, a1(a) = 5/2, so µ1(a) = 0.625. q1(b) = 3, a1(b) = 3/2, so µ1(b) = 0.50. by cnsequence h117 a b a a h b h b whence h217 is the total hypergroup, therefore ∂h17 = 2. • h18 a b a h h b b h we find q1(a) = 3, a1(a) = 3/2, so µ1(a) = 0.50. q1(b) = 4, a1(b) = 5/2, so µ1(b) = 0.625. we obtain h118 a b a a h b h b by consequence, h218 is the total hypergroup, whence ∂h18 = 2. references [1] ameri r., nozari t., a connection between categories of (fuzzy) multialgebras and (fuzzy) algebras, italian journal of pure and applied mathematics, 27 (2010) [2] ameri r., shafiyan n., fuzzy prime and primary hyperideals in hyperrings. advances in fuzzy mathematics, n. 1-2. 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(1994) 76 ratio mathematica volume 43, 2023 application of homotopy to the ageing process of human bone parvathy chennimalai ramanathan * vidhya kuduva rengaraj † abstract this article presents the ageing process of human bone which can play a major role in the structure of the human body from the concept of homotopy in algebraic topology. this article is about applications. hence this title. in 2019, william obeng-denteh et al., wrote the literature on the application of homotopy within the framework of algebraic topology on the ageing of the human body. we applied this application, which was generally created for the human body to the ageing of the human bone. the result was good. so we have solved this problem in the literature topologically. also described by the cartesian function. as a result of the literature, we have explained the assumption that bone age increases as human age increases with the use of homotopy. the structure of the human bone, which is precisely connected is considered here to be topologically equivalent to a cylinder [9]. the process of continuous ageing bone is considered to be a family of homotopy based on its functions. the study discusses the algebraic topology of homotopes through the homotopy of stable functions of the human bone from infancy to old age. keywords: homomorphism, homotopy, homology, chain complex, topological space. 2020 ams subject classification: 51a10, 55p15, 55r40, 55u15, 18f60. * (psgr krishnammal college for women, coimbatore, tamilnadu india); parvathytopo@gmail.com † (psgr krishnammal college for women, coimbatore, tamilnadu india); krvidhyamani@gmail.com § received on october 14, 2023. accepted on february 3, 2023. published on xxx xx, xxxx. doi:10.23755/rm.v41i0.874. issn: xxxx-xxxx. eissn: xxxx-xxxx. ©the authors. this paper is published under the cc-by licence agreement. 300 mailto:krvidhyamani@gmail.com parvathy chennimalai ramanathan and vidhya kuduva rengaraj 1. introduction a lot of research articles have been published using the homotopy application. in particular, hpm also known as the homotopy perturbation method, plays in engineering, applied maths, biology, physics, quantum physics and real-world problems. it only began to expand in the 18th and 19th centuries. homotopy analysis method for solving the biological population sample in 2011 used by anas. a. m. arafa et al., [2]. in 2012 f. guerrero et al., as real-world applications of how smoking has evolved in spain [4]. in 2019 william obeng-denteh et al., published a research paper application of homotopy to the ageing process of the human body [9]. based on this, in this paper, we have described the ageing process of bone in the human body with homotopy application in algebraic topology. the growth of human body organs or body growth occurs as a result of genetic, hormonal, dietary and other environmental factors. bones play an important role in supporting the growing or developing organs. what is our condition if there are no bones? at birth, a baby has about 350 bones in its body and as the baby grows there is a large bone formed by the combination of tiny little bones found in many parts of the body. physiological research reports that this results in a change in the number of bones reduced to 206. the development of human body organs from infancy to old age is considered a developmental change that does not only aggravate the appearance of the body. the internal organs of the body also continue to mature. the elegant but complex skeletal shape of the human bone is described by the cartesian functions of and [9]. the bone shape of the human body at an early stage. that is the bone shape of the newborn is called homotopy. we will explore the continuous changes between this childhood bone and the bone seen in old age using the properties of homotopy in mathematics. especially in algebraic topology. for example, vector calculus is used to demonstrate the relationship between an integral line around a simple closure. homotopy is used to define the surface area of an unaltered ( ) human bone that can be calculated in algebraic topology where is attached and closed. thus, if the surface of a bone is attached and closed ( ) a few developed variations have been derived from algebraic topology. they reflect the connecting properties of spaces or objects. more complex spatial biological systems are described by topological collection techniques. spaces are objects connected in mathematics that have been considered for the introduction of topology. homotopy theory is used to describe the ageing process of unchanged human bone. it offers numerous applications when fredholm equations use homotopy analysis on integrated equations. the functions of the values of the given and parameters are given by selecting the appropriate value for the time parameters 301 appication of homotopy to the ageing process of human bone of the process functions of the ageing bones. in which . in this study the growth or age of human bone is compared with the age of the bone is the same as the age of the body. the homotopy ( ) from ( ) to ( ) such that ( ) ( ) and ( ) ( ) is currently the same age as bone. that is the age of humans is one. as the age of bone grows and it is functions change. this is what we call homotopy. this is because continuous transformations take place from one function ( ) to another function ( ). thus, we can take . where . if we consider the first age as a function, we can refer to it as ( ) ( ) and the old or mature age as ( ) ( ). the bone undergoes a series of changes from ( ) to ( ). in this shape, the bone develops as homotopy. that is ( ) ( ) to ( ) ( ) as the bone age expanse to . hence the interval the human bone is closed and connected surface at each change in the variable the age of the human bone growth from one level to another. for instance, if ( ) ( ) and ( ) ( ) then and are connected by a line that is a homotopy path ( ) given by a continuous map in . 2. homotopy definition 2.1. [7] let be topological spaces and be continuous maps and homotopy from to is continuous function ; satisfying, ( ) ( ) and ( ) ( ) for all . if such a homotopy exists then is homotopic to and it is denoted by . definition 2.2. [7] two paths and . mapping the interval into are said to be path homotopy. if they have the same primary point and the same endpoint and if there is a continuous map . such that ( ) ( ) and ( ) ( ). ( ) and ( ) for each . then is a path homotopy between and . if is pate homotopic to then notation . definition 2.3. [5] let be topological spaces and the mapping . a continuous real-valued function ( ), at any point in . if for each neighbourhood ( ). then there exists a neighbourhood ( ), such that ( ) definition 2.4. [11] let be topological spaces. let we say that is continuous at . if for every there exists such that | | then | ( ) ( )| where . a function is said to be continuous if for each open subset of the set ( ) is an open subset of space . 302 parvathy chennimalai ramanathan and vidhya kuduva rengaraj definition 2.5. [8] let be topological spaces. then the function and , are continuous functions from to then is homotopic to . if there is a continuous family of functions, : for . then the following conditions are satisfied; a. b. c. ( ) is continuous both as a function and theorem 2.6. [7] let and be two functions. suppose and are continuous at , such that ( ) then ( ) is continuous at . theorem 2.7. [1] a topological space is a path connected if any two points could be connected by a line. for instance, if ( ) and ( ) then are connected by a line given by a continuous map in a topological space . 3. homology the fundamental group ( ) is especially good for low-dimensional spaces. because it’s concerning loops. the definition of objects in the 2 dimensions expresses itself to the maximum. for example, when is a cw complex then ( ) depends only on the 2skeleton of . homotopy is the best way to differentiate and construct all dimensions and spheres. however, these high-dimensional homotopy groups have some drawbacks. these are a bit difficult to calculate. there is an alternative to this. it is homology groups. yes. homology is a commutative alternative to homotopy. the calculation or definition of homology groups is less explicit than the calculation or definition of homotopy groups. the chain complex is the algebraic structure of the abelian groups that form the image of each homomorphism and the sequence of homomorphisms between each group added to the next kernel homotopy is related to the chain complex. definition 3.1. [3] a chain complex is a sequence of additive abelian groups and homomorphism’s → . the elements of are called -chains and these maps are boundary maps. such that composition of 2 successive homomorphism’s zero. that is, for each . such a sequence is called a chain complex. definition 3.2. [3] the homology group of the chain complex is the quotient group . that is, ( ) ( ) ( ) 303 appication of homotopy to the ageing process of human bone here, is the homology group of . elements of are called cycles. elements of are called boundaries. elements of are cosets of called homology classes. 4. description of ageing human bone function the inherent properties of the human body can be explored topologically. that is various factors such as the functions, development and transformation of different body parts can be explored using the properties of algebraic topology. we can now compare the ageing functions of bone with the properties of homotopy in algebraic topology at intervals ranging from the age of human bone to the age of onset. the time interval is considered to be and the human bone . the importance of algebraic invariants such as homotopy that reflect the connectivity of the bone. the structure of human bone and let and define the growth of the bone and the age of the bone respectively. although we take the total duration of human bone to be from early age to final age. we cannot accurately determine the final bone age. that is, we must assume that the functions of the bone will last as long as man exists. so is the final age of the bone. the topological shape of the bone at the beginning ( ) undergoes various changes every year. that is, ( ) ( ) ( ) ( ). so, the full-scale development of this topological shape of the human bone can be referred to as ( ) ( ) and ( ) ( ) from this, the age of the human which is the duration of the bone is determined as . the total period time of a bone is divided into two functions and the total bone life of a human being is assumed to be . we know the definition of path homotopy. the paths specified in it are given the starting point ( ) and the endpoint ( ) which have the same starting point and ending point even after the paths have undergone various changes and transformations. and the relationship between the two is an equivalence relation. here it’s called path homotopy. similarly, the homotopic and equivalence relationship between ( ) bone in childhood and ( ) bone in old age is similar. we can extend the n-period in which the closed attached human bones divided into two functions and . defined by the equivalence functions of homotopy on the whole interval . 304 parvathy chennimalai ramanathan and vidhya kuduva rengaraj the interval indicates the initial age and height or growth of the bone. since it satisfies the initial condition ( ) ( ) and ( ) ( ) the intervals also change with increasing age. that is there is a continuous function ( ) = ( ) and ( ) = ( ) , similarly, there is a continuous function ( ) = ( ) and ( ) = ( ) there is continuous function ( ) ( ) and ( ) ( ) etc., these functions are satisfied for the homotopy ( ) from ( ) to ( ). the total lifespan of human bone is ( ) for every function we have ( ) ( ), ( ) ( ) , ( ) are continuous functions from to . and these are homotopy to each other. strictly speaking, ( ) is homotopic to ( ), ( ) is homotopic to ( ), ( ) is homotopic to ( ) etc. if there is homotopy ( ) where from ( ) to ( ). such that; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), …, ( ) ( ) these are all continuous but undergo many changes at different intervals here we see information about the age of the bone found in the body with homotopy. homotopic functions have been shown to change the shape and age of the bone each year. that is the function ( ). no matter how many changes occur in bone function or growth, they all occur within a period of n]. to be clear, in the first year, in the second year, etc., if the homotopy ( ) is denoted by ( ) and ( ). that is, the age of human bone increased from ( ) to ( ) followed by ( ) etc., in this case, if we consider this series of functions available to us in homotopy as a common element it will be available as ( ) where . that is, ( ) ( ) represents the initial age of the bone. ( ) ( ) ( ) ( ) ( ) ( ) etc.,where all are represents the age of the bone. all of these continuous activities show different stages of bone age and development. the increasing sequence of the function ( ) of the bone provides the chains of the bone. in which human bone is a 2-cell complex and each subsequent growth of each bone forms 2 chains which can be referred to as ( ). thus with each age of the bone, an additional 2 chain complexes are formed. that as the bone ages many 2 chains are formed. these 2 chains are all referred to as ( ) where is a positive integer. bone is a closed structure. hence the set of all closed bones is called the kernel ( ) at this point, the boundary of cell 2 is zero. that is ( ) because any 2 successive homomorphisms then the composition to each other we get the zero map. so the homotopy groups ( ) ( ) ( ) where ( ) characterizes the connected 2-cell at each human age level. the homological 305 appication of homotopy to the ageing process of human bone value of the topologically unchanged ( ) is the closed and attached bone in the body it changes with age. so the homology theory gives the sequence of abelian groups of bone and the sequence of homomorphisms ( ) ( ) for the image that follows it. all the maps we have are continuous and homomorphism, the composition of two homotopic functions is homotopic. therefore homotopic function is obtained by assembling two or more homotopic functions in this ageing process of human bone. 306 parvathy chennimalai ramanathan and vidhya kuduva rengaraj figure 1. the shape of the bone n-years human from ( ) to ( ) remark 4.1. let be the maps and be the space. proposition 4.2. if and are two continuous functions of the body. show that , then ( ) ( ). . theorem 4.3. prove that and be continuous and homotopic. then ( ) ( ) be continuous and homotopic. 5. conclusions in this article, we talked about the ageing process of human bone. this is because the application of homotopy, which is generally accepted for the human body is to know whether it corresponds to a particular body organ. this paper is designed to stabilize human bone growth in the event of physical and to state its function and structure. the age of man increases as the time variable increases. the age of the bone also increases with age. it also describes the continuous process of ( ) in the ageing process. the concepts of homotopy were very helpful in accessing the ageing function of the human bone. we add the basic definition of homotopy to the definition of homology in the literature to justify the publication of the paper. as a result of the literature, we have explained with the applications of homotopy that bone age increases as human age increases. since we have included the basic definition of homotopy, the interval is t. so here the ageing process is considered by choosing suitable values for the time parameters t of the functions ( ) hence ( ) is increases from ( ) ( ) to ( ) ( ), . the homology theory is slightly incorporated here because the human bone is denoted as a 2-cell complex and bone growths are also 2 chains. in the future, the development of homotopy will play an important role in the research of biologically interacting substances or elements that change over time. references [1] allen hatcher, algebraic topology, cambridge university press – 2002. [2] anas a. m. arafa, s. z. rida, hegagi mohamed ali. homotopy analysis method for solving biological population model. vol. 56, no. 5, november 15, 2011. [3] andrew h. wallace, algebraic topology: homology and cohomology, university of pennsylvania – 1970. 307 appication of homotopy to the ageing process of human bone [4] f. guerrero, f. j. santonja, r. j. villanueva. solving a model for the evolution of smoking habit in spain with homotopy analysis method. 14(2013) 549-558. [5] j. dugundji, topology, allen and bacon, inc. – 1966. [6] j. l. giavitto and o. michel. (2003) modeling the topological organization of cellular processes, bio-systems. 70(2), 149-163. [7] james r. munkres, a first course in topology, prencite–hall. inc. new jersey – 1967. [8] kinsey, l. (1993). topology of surfaces, springer-erlag, new york, inc. usa, 428. [9] laurence boxer, ismet karaca, ahmet oztel. (2011). topological invariants in digital images, journal of mathematical sciences: advances and application. 11(2), 109140. [10] lewis brew, william obeng-denteh, david delali zigli. application of homotopy to the ageing process of the human body within the framework of algebraic topology. journal of mathematics research – vol. 11, no. 4; august 2019. [11] simmons, g. f. (1963). introduction to topology and modern analysis,mcgraw-hill book company, inc. new york, usa, 362. [12] w. s. massey, (1991). a basic course in algebraic topology, springer-verlag, new york, inc. usa, 428. 308 ratio mathematica volume 43, 2022 analysis of finite population stochastic modeling with state-dependent arrival and service facilities k lakshmanan* s padmasekaran† m bhuvaneshwari‡ r asokan§ abstract this paper investigates a stock-dependent arrival process(sdap) and queuedependent service process(qdsp) in the stochastic queueing-inventory system(sqis). the arriving units in the system generated from the finite source population. the arrival process holds the properties of quasi-random process and its intensity rate is defined based on the two-component demand rate(tcdr). the customers departure time is exponentially distributed. the concepts of non-sdap and sdap, non-qdsp and qdsp are to be generalized. the inventory system may have the perishable quality of the products. it adopts the (s, q) reordering policy whenever the replenishment is required. further, the join probability distribution of a markov process is derived and necessary system performance measures are computed. the comparative discussion is presented to improve the quality of this model. keywords: stock-dependent arrival process; non-stock dependent arrival process; two component demand rate; finite population; queue-dependent service process 2020 ams subject classifications: 90c15, 60g07. 1 *professor of mathematics, kuwait american school of education, salmiya, pc 22063, kuwait; lakshmanlingam@kas.edu.kw †corresponding author; assistant professor of mathematics, periyar university, salem, tamilnadu, india; padmasekarans@periyaruniversity.ac.in. ‡p. g. assistant in mathematics, government higher secondary school, sathirakudi, ramanathapuram, tamilnadu, india; bhabhu1977@gmail.com. §professor of mathematics, madurai kamaraj university, madurai, tamilnadu, india; asokan.maths@mkuniversity.org. 1received on february 9, 2022. accepted on december 1, 2022. published on december 30, 2022. doi: 7 k. lakshmanan, s padmasekaran, m bhuvaneshwari, r asokan 1 introduction as the continuous increment in the population growth, the demand of the people such as foods, cloths and public service etc are also skyrocketed in this century. so the analysis of queueing-inventory management is inevitable in this current situation, especially in that food items related queueing-inventory systems. the results can be applied effectively in queue and inventory management systems and the optimum cost will increase the economy also. this model consists of a perishable inventory system with queue-dependent service rate and customers from a finite source. due to the decay of the food items after some certain period, its must to reorder it at some fixed inventory level to avoid economical loss. the arrival rate of customer is dependent on the number of items available in the shops. also service rate is dependent on the mood of server. the following illustration will give the exact model idea. in the shopping malls and theaters, some stores are used to sell snack items for the customers. the possibility of customer occurrence is dependent on inventory level and also service rate can be dependent on the queue length. here the population size is finite, since the only possible customers to purchase the snacks are from theater or the mall. though there are many shops are available, most of the customers are interested to purchase from the shop where many items are available comparing to the shops where less items are available. here the customer arrival depends on the inventory level. when a small queue is formed in the shop, server may serve slowly, speaking with someone or using mobile phone. also server would provide a quick service if the queue length is big, in order to decrease the customer loss. though the formed queue is bigger, customer won,t leave the queue as the service rate is high. this realistic situation motivates the author to develop the proposed model. in this model, stocks are replenished according to (s, q) policy. 2 literature review as many researchers show their interest of research on the stochastic queueing inventory modeling, this area has developed enormously in certain period. queuing inventory systems with retrial is in-fusible in all walk of life. reshmi and jose studied the queuing inventory model, considering perishable items and customer retrial on reshmi and jose [2019]. periyasamy considered a finite population perishable inventory system where server is looking for the customers from the orbit to provide service after completing service to each primary customer on periyasamy [2017]. berman and sapna berman and sapna [2002] discussed the rate of optimal service with perishable inventory in which instantaneous reordering policy was assumed. considering the negative 10.23755/rm.v42i0.715. issn: 1592-7415. eissn: 2282-8214. ©k lakshmanan et al.. this paper is published under the cc-by licence agreement. 8 analysis of finite population stochastic modeling exponential rate for the life time of stocks, kalpagam and arivagam kalpakam and arivarignan [1988] analyzed the (s, s) inventory system in which stock one is evicted from the inventory whenever the demand or failure of item occurs. sangeetha investigated the production optimal control of production time of perishable inventory system with finite source in order to get the minimal total cost on n. sangeetha and arivarignan [2015]. alfres introduced the concept of occurring demand rate depends upon the stock level in the inventory system on alfares [2007] and determined the total cost by variable holding cost assuming holding cost per unit item to be a monotonically increasing function of spending time in the storage. diana tom varghese and dhanya shajin varghese and shajin [2018] studied the state dependent demand on the continuous review m/m/1/s inventory model. k. venkata subbaiah et al. k. venkata subbaiah and satyanarayana [2004] developed the perishable inventory model with stock dependent demand rate. rathod and bhathawala rathod and bhathawala [2013] analyzed the inventory system with stock dependent demand having variable holding cost and shortages. the effect of demand rate depending on stock level was discussed through the proposed logistical growth model of tsoularis tsoularis [2014]. a shortage free inventory model with stock dependent demand was analyzed by datta and pal on datta and pal [1990]. sudhir kumar sahu et al. sudhir kumar sahu and sahoo [2008] developed an inventory system with stock dependent demand rate and constant deterioration with the possibilities of partial or complete backlog and without it. shib sankar sana sana and sankar [2010] proposed an eoq model for the perishable inventory item with discount rate and the demand depending on stock level. mandal mandal and s. [1989] derived an inventory system with consumption rate depending on stock level. for analyzing the local area management, falin and artalejo falin and artalejo [1998] proposed a retrial queue with finite source customer. shophia lawrence et al. a. shophia lawrence and arivarignan [2013] discussed the perishable queueing-inventory system with demands from finite homogeneous source. attahiru sule alfaa and sapna isotupa ? discussed an m/ph/k retrial queue with the finite source. k. jeganathan k. jeganathan and vigneshwaran [2015] analyzed the perishable inventory system with the possibility of server interruption and the multiple server vacation and customer is provided service only when customer level reaches to a particular n and no customer is left behind the system after service started. jeganathan jeganathan [2015] discussed finite source inventory system with an additional service for some customers which is called bonus service. artalejo and lopez-herrero investigated retrial queue involving finite population with an bsde approach on artalejo and lopez-herrero [2012]. sivakumar analyzed the perishable inventory system with retrial demand from finite source without service on sivakumar [2009]. shanthikumar and yao shanthikumar and yao [1988] studied the upper and lower bounds on a closed queuing network with the queue dependent service rate. menich ronald ronald [1987] derived the optimal of shortest queue routing to the queue depen9 k. lakshmanan, s padmasekaran, m bhuvaneshwari, r asokan dent service station considering a general markovian system. avhishek chatterjee et al. avhishek chatterjee and varshney lav [2017] studied the information-theoretic limit of reliable information processing using queue dependent service facility. jeganathan et al.[2021] proposed a finite inventory single server system and analyzed the queue dependent service rate. though the large number of researches have been done in this area, there is a research gap in analyzing stock dependent demand rate on the finite source queuing inventory system with the queue dependent service rate and retrial customer. as the demand rate depending upon stock level on the inventory and service rate depends on the queue length, this model simulates the realistic situation. 3 model developing 3.1 mathematical formulation of the model this model deals the state dependent arrival and queue dependent service processes in a single server markovian queueing-inventory system(ssmqis) in a finite source environment. the system holds maximum of s units of inventory product in its storage place. it allows the customers to buy the product from a finite source, n only. it admits the arriving customers into the finite waiting hall of size n. there is only two possible choice of a customer such that they must be either free or in the waiting hall at any time. the appearance of arrival process generates a output process called quasi-random process; that is, the probability that any particular customer generates a request for demand in any interval (t, t + dt) is θrdt + o( dt)(r ∈ bs0 ) as d t → 0 if the customer is free at time t and zero if the customer is in waiting hall at time t, independently of the behavior of any other customers. the arrival process of any individual customer is nonhomogeneous, since the generation of arrivals must dependent upon the current stock level of the system. this non-homogeneous arrival streams come under the category of state dependent arrival stream. next, the service pattern is processed following a first come and first serve(fcfs) service discipline. the service time of any customer at time t is non-homogeneous and exponentially distributed. that is, µs (s ∈ bn1 ) is the service rate of an individual at any time. this service process comes under the category of state dependent service processes. after each service completion, there will be one unit dropped in the storage place. the stored products in the system does not have any guarantee about its life time till it will be sold. it may have the deteriorating quality. so this deterioration process follows exponential distribution and have the intensity rate rα1 where r ∈ bs1 . the service and deterioration processes cause the depletion of an inventory product unit by unit. at one fine stage, the current number of product in the storage system will reach the predetermined value s. as and when the maximum inventory level reduced to s or 10 analysis of finite population stochastic modeling less than s, then the replenishment process will be triggered immediately. each time there are q = (s − s) items will be replaced whenever the reorder required. this policy is known as (s, q) reordering policy and this processing time is exponentially distributed with an intensity rate α. the defined arrival and service rates are ordered, θ0 ≤ θ1 ≤ θ2 ≤ · · · ≤ θs and µ1 ≤ µ2 ≤ · · · ≤ µn as an increasing manner. when the case θ1 = θ2 = · · · = θs = θ, the considered model comes under the category of two component demand rate. that is the arrival rate is homogeneous in the positive stock period and during the stock out period, it is θ0. when the case µ1 = µ2 = · · · = µn = µ means that the service rate become homogeneous. remark 3.1. • for a numerical computation θr can be defined by θr rβ1, 0 < β1 ≤ 1 and r ∈ bs1 . • for a numerical computation µs can be defined by µs sβ2, 0 ≤ β2 ≤ 1 and s ∈ bn1 . • the case β1 = 0 and β2 = 0 explores the result of non-stock dependent arrival process and nonqueue dependent service process of the proposed model. 4 analytical discussion of the model let { (r1(t), r2(t)) ; t ≥ 0 } be a stochastic process having state space {(r1, r2) : r1 ∈ bs0 and r2 ∈ bn0 } satisfies the markov process, where r1(t) denotes the level of inventory at time t and r2(t) denotes the number of customers in the orbit at time t. the transition from any state (r1, r2) to other state (r′1, r ′ 2) at any interval is denoted by p ((r1, r2) , (r ′ 1, r ′ 2)). any y items in the inventory perish alone at the rate of r1γ and the occurrence of primary demand is (n − r2) θr1 from any one of the sources (n − r2). hence, the probability of transition is p ((r1, r2) , (r1 − 1, r2)) = r1α1 r1 ∈ bs1 and r2 ∈ bk0 . since the service rate is queue dependent service rate p ((r1, r2) , (r1 − 1, r2 − 1)) = µr2 where r1 ∈ bs1 and r2 ∈ bk1 . if the arrival rate is dependent on inventory, arriving customers enter into the waiting hall. so the probability of the transition from the state (r1, r2) to the state (r1, r2 + 1) is p ((r1, r2) , (r1, r2 + 1)) = (k − r2) θr1 where r1 ∈ bs0 , r2 ∈ b k−1 0 . when q items are ordered, the probability of transition from the state (r1, r2) to state (r1 + q, r2) for all r2 and r1 ∈ bs0 is given by p ((r1, r2) , (r1 + q, z)) = α. the rate of other transitions is zero. the sum of each row of this matrix should be zero. hence, the diagonal entry is multiplied by a negative sign after summing all the entries from the row. all the possible transitions are given below. 11 k. lakshmanan, s padmasekaran, m bhuvaneshwari, r asokan r ((r1, r2) , (r ′ 1, r ′ 2)) =   r1α1, r ′ 1 = r1 − 1, r1 ∈ bs1 , r′2 = r2, r2 ∈ bk0 , µr2, r ′ 1 = r1 − 1, r1 ∈ bs1 , r′2 = r2 − 1, r2 ∈ bk1 , (k − r2)θr1, r′1 = r1, r1 ∈ bs0 , r′2 = r2 + 1, r1 ∈ b k−1 0 , α, r′1 = r1 + q, r1 ∈ bs0, r′2 = r2, r2 ∈ bk0 , −(δ̄k,r2(k − r2)θr1 + α), r′1 = r1, r1 ∈ b00, r′2 = r2, , r2 ∈ bk0 , −(δ̄k,r2(k − r2)θr1 + α + δ̄0,r2µr2 + r1α1), r′1 = r1, r1 ∈ bs1, r′2 = r2, , r2 ∈ bk0 , −(δ̄k,r2(k − r2)θr1 + δ̄0,r2µr2 + r1α1), r′1 = r1, r1 ∈ bss+1, r′2 = r2, , r2 ∈ bk0 , 0, otherwise. the block partitioned matrices of the proposed model is structured as follows: r =   ly, r ′ 1 = r1, r1 ∈ bs0 , my, r ′ 1 = r1 − 1, r1 ∈ bs1 , n, y′ = q + r1 r1 ∈ bs0, 0, otherwise. for r1 ∈ bs1 , mr1 =   r1α1, r ′ 2 = r2, r2 ∈ bk0 , , µr2 r ′ 2 = r2 − 1, r2 ∈ bk1 , 0, otherwise. for r1 ∈ bs0, n = { α, r′2 = r2, r2 ∈ bk0 , 0, otherwise. for r1 = 0, lr1 =   (k − r2)θ0, r′2 = r2 + 1, r2 ∈ b k−1 0 , −((k − r2)θ0 + α), r′2 = r2, r2 ∈ b k−1 0 , −α, r′2 = r2, r2 ∈ bkk , 0, otherwise. 12 analysis of finite population stochastic modeling for r1 ∈ bs1, lr1 =   (k − r2)θr1, r′2 = r2 + 1, r2 ∈ b k−1 0 , −(δ̄k,r2(k − r2)θr1 + α + δ̄0,r2µr2 + r1α1), r′2 = r2, r2 ∈ bk0 , 0, otherwise. for r1 ∈ bss+1, lr1 =   (k − r2)θr1, r′2 = r2 + 1, r2 ∈ b k−1 0 , −(δ̄k,r2(k − r2)θr1 + δ̄0,r2µr2 + r1α1), r′2 = r2, r2 ∈ bk0 , 0, otherwise. 4.1 steady state analysis the structure of the homogeneous markov process {(r1(t), r2(t); t ≥ 0} with finite state space indicates that it is irreducible. hence, the limiting distribution is ξ(r1,r2) = lim t→∞ pr {(r1(t) = r1, r2(t) = r2)|(r1(0), r2(0))} let ξ = (ξ(0), ξ(1), . . . , ξ(s)) where each ξ(r1) = (ξ(r1,0), ξ(r1,1), . . . , ξ(r1,k)) for r1 ∈ bs0 which satisfies ξp = 0 and ξe = 1 (1) from the above we get the following equation ξ(r1)lr1 + ξ (r1+1)mr1+1 = 0 r1 ∈ b q−1 0 , (2) ξ(r1)lr1 + ξ (r1+1)mr1+1 + ξ (r1−q)n = 0 r1 = q (3) ξ(r1)lr1 + ξ (r1+1)mr1+1 + ξ (r1−q)n = 0 r1 ∈ bs−1q+1, (4) ξ(r1)lr1 + ξ (r1−q)n = 0 r1 = s (5) except the r1 = q case, solving other equations recursively, we get, ξ(r1) = ξ(q)∆r1, r1 ∈ b s 0 , where ∆i =   (−1)(q−r1)(mqmq−1 . . . mr1+1)(l −1 q−1l −1 q−2 . . . l −1 r1 ), r1 ∈ b q−1 0 , i r1q, (−1)2q+1−r1 c−r1∑ j=0 (mqmq−1 . . . ms+1−j)(l −1 q−1l −1 q−2 . . . l −1 s−j)nl −1 s−j (ms−jms−j−1 . . . mr1+1)(l −1 s−j−1l −1 s−j−2 . . . l −1 r1 ) r1 ∈ bsq+1, 13 k. lakshmanan, s padmasekaran, m bhuvaneshwari, r asokan ξ(q) can be yield solving ξ(q) [ (−1)2q+1−r1 s−r1∑ j=0 [ (mql −1 q−1mq−1 . . . ms+1−jl −1 s−j)nl −1 s−j(ms−jl −1 s−j−1ms−j−1 . . . . . . mq+2l −1 q+1) ] mq+1 + lq + (−1)qmql−1q−1mq−1 . . . m1l −1 0 n ] = 0 ξ(q) [ s∑ r1=q+1 ( (−1)2q−r1+1 s−r1∑ j=0 [ (mql −1 q−1mq−1 . . . ms+1−jl −1 s−j)nl −1 s−j(ms−jl −1 s−j−1 ms−j−1 . . . mr1+1l −1 r1 ) ] mq+1 ) + q−1∑ r1=0 ( (−1)q−r1mql−1q−1 . . . mr1+1l −1 r1 ) +i ] e = 1. 5 system performance measures to make a detailed investigation of the proposed model, some significant system characteristics are to be computed as follows: 1. expected present stock level e[psl] = ∑s r1=1 ∑k r2=0 r1ξ (r1,r2). 2. expected reorder level e[reorder] = ∑k r2=1 µr2ξ (s+1,r2)+ ∑k r2=0 (s+1)α1ξ ((s+1),r2). 3. expected perishable rate e[perishable] = ∑s r1=1 ∑k r2=0 r1α1ξ (r1,r2). 4. expected number of customers in the waiting hall e[cwh] = ∑s r0=1 ∑k r2=1 r2ξ (r1,r2). 5. expected number of customers enter into the waiting hall e[cewh] = ∑s r1=0 ∑k−1 r2=0 (k− r2)θr1ξ (r1,r2). 6. expected waiting time of a customer in the waiting hall e[wt] = e[cwh] e[cewh] 7. probability that the server is busy p(busy) = ∑s r0=1 ∑k r2=1 ξ(r1,r2). 8. probability that the server is idle p(idle) = 1 − p(busy) 9. the total expected cost value of the proposed model is defined as tcv = cae[psl]+ cbe[reorder] + cce[perishable] + cde[wt] where ca− holding cost per unit, cb− setup cost per unit, cc− perishable cost per unit and cd− waiting cost per customer. 14 analysis of finite population stochastic modeling table 1: tcv for the case of sdap and qdsp s 7 8 9 10 11 12 13 s 58 2.805974 2.805623 2.806103 2.807504 2.809919 2.813445 2.818190 59 2.806534 2.805602 2.805445 2.806146 2.807788 2.810462 2.814264 60 2.807988 2.806522 2.805778 2.805835 2.806769 2.808661 2.811598 61 2.810287 2.808327 2.807044 2.806509 2.806792 2.807966 2.810111 62 2.813383 2.810970 2.809190 2.808110 2.807794 2.808309 2.809726 63 2.817234 2.814403 2.812165 2.810584 2.809717 2.809625 2.810374 64 2.821800 2.818583 2.815924 2.813880 2.812506 2.811856 2.811991 6 simulation analysis in this section, the optimum cost analysis, monotonic behavior of some system characteristics are to be discussed by the numerical illustrations. this will be helpful to deliver a effective decision making polices for every inventory business tycoons. for knowing such curious results of our proposed model, we need to fix the value of the parameters and the cost values such that θ = 5, θ0 = 2, µ = 9, α = 0.9, γ = 0.07, β1 = 0.5, β2 = 0.5, s = 61, s = 10, n = 10, ca = 0.05, cb = 0.9, cc = 0.1, and cd = 7. example 6.1. optimum cost analysis this example briefly investigate the minimum optimal tcv for the category of both arrival and service processes of homogeneous and non-homogeneous cases as shown in table (1)-(2). in table (1), s ∈ b6458 and s ∈ b137 are used to find the minimal optimum tcv under the case of discussion between sdap and qdsp. in this case, the tcv ∗ = 2.805445 and corresponding optimum s∗ = 59 and s∗ = 9 are obtained. next, the output values of the case non-sdap and non-qdsp are given in table (2). here, s ∈ b6357 and s ∈ b1610 are varied to get an optimum tcv. in this case, the tcv ∗ = 8.966159 and corresponding optimum s∗ = 60 and s∗ = 13 are obtained. as we expected due to the assumption of the proposed model, the case non-sdap and nonqdsp have a higher tcv ∗ than the sdap and qdsp case. that is, the minimal optimum tcv obtained in the case of sdap and qdsp. hence the arrival and service rates influence the cost value become a minimum one. k1 = 0, k2 = 0 k1 = 0, k2 = 0.6 example 6.2. the variation of tcv under the parameter variation in this example, we describe the path of tcv with each parameter considered in the model. in such a way, the major objective of this example is discussed with the 15 k. lakshmanan, s padmasekaran, m bhuvaneshwari, r asokan table 2: tcv for the case of non-sdap and non-qdsp s 10 11 12 13 14 15 16 s 57 8.986034 8.976046 8.970983 8.970903 8.975952 8.986375 9.002534 58 8.985814 8.975121 8.969166 8.967979 8.971672 8.980449 8.994615 59 8.986685 8.975367 8.968613 8.966432 8.968907 8.976202 8.988575 60 8.988575 8.976700 8.969232 8.966159 8.967537 8.973497 8.984257 61 8.991414 8.979048 8.970942 8.967068 8.967457 8.972214 8.981518 62 8.995143 8.982342 8.973667 8.969073 8.968572 8.972242 8.980232 63 8.999705 8.986521 8.977338 8.972097 8.970793 8.973483 8.980284 scaling factors β1 and β2, because they are deciding factors whether the arrival and service processes are non-sdap and non-qdsp or not respectively. in table (3), the scaling factor β1 increases the total cost if it is increasing. that is, β1 increases means, the arriving customers in the system is increased. subsequently, the sales of number of product in the inventory is raised. so the management is often ready to store or making reorder for their requirement. these jobs cause the increase of total cost. the same characteristics are holds the parameter θ. simultaneously, when we are focusing the another scaling factor β2, more interestingly it reduces the tcv. if β2 increases means that the service time of an individual become reduced. so the number of customers leaves the system after a successful service completion of them is increasing. this helps to reduce the mean service time of a customer. so this is the reason for tcv is reduced if β2 increases. if β2 and µ are directly proportional to each other µ holds the same behavior as β2. then the perishable parameter α1 affects the item life time. if α1 raises, the number of current stock level starts falling down. if it happens, the management has to store more number of products which cause the extra expenditure to maintain the system. so this expenditure cause the increase of total cost. finally, the reorder intensity rate α minimize the total cost when it is increasing. the successive mean reorder time reduced means the number of available product of the system become positive. therefore, the service completion will be done as soon as possible. hence, all the parameters involved in table (3) and table (4) are satisfies their own properties. example 6.3. graphical analysis • the scaling factors β1 and β2 shows the increasing/decreasing path due to its sdap and non-sdap, qdsp and non-qdsp. we observe that 0.2 ≤ β1 ≤ 1 the β2 curves deviation is high and 0.5 ≤ β2 ≤ 1.0 the β2 curves deviation is low for all β1 ∈ (0.2, 1). • the graph of expected waiting time is shown in figure (2) when β1 and β2 are varying together. here, the deviation of β2 curves coincides with the characteristics as we said in figure (1). 16 analysis of finite population stochastic modeling table 3: the variation of tcv under the parameter variation β2 θ α1 β1 0 0.5 1 µ 7 9 11 7 9 11 7 9 11 α 0 4.5 0.05 0.70 11.08 8.92 7.55 44.64 35.40 29.67 136.60 104.43 85.33 0.90 10.83 8.66 7.28 44.78 35.11 29.13 145.45 109.22 87.88 1.10 10.71 8.52 7.14 45.25 35.24 29.04 154.70 114.88 91.46 0.07 0.70 11.18 9.02 7.64 44.31 35.24 29.60 132.94 102.29 83.98 0.90 10.92 8.74 7.36 44.40 34.91 29.03 141.43 106.85 86.36 1.10 10.79 8.60 7.21 44.86 35.02 28.92 150.45 112.34 89.81 0.09 0.70 11.29 9.12 7.73 44.01 35.10 29.54 129.67 100.36 82.73 0.90 11.01 8.82 7.43 44.06 34.73 28.93 137.82 104.70 84.96 1.10 10.87 8.68 7.28 44.50 34.82 28.80 146.61 110.02 88.30 5 0.05 0.70 11.21 9.06 7.69 44.81 35.59 29.87 137.00 104.88 85.82 0.90 10.95 8.79 7.41 44.93 35.28 29.31 145.74 109.58 88.28 1.10 10.83 8.65 7.27 45.38 35.38 29.20 154.89 115.16 91.78 0.07 0.70 11.31 9.15 7.78 44.48 35.43 29.80 133.34 102.75 84.46 0.90 11.04 8.87 7.49 44.55 35.08 29.20 141.73 107.21 86.76 1.10 10.91 8.73 7.34 44.98 35.16 29.08 150.65 112.62 90.14 0.09 0.70 11.42 9.25 7.86 44.19 35.29 29.74 130.08 100.82 83.22 0.90 11.13 8.95 7.57 44.21 34.90 29.11 138.12 105.06 85.36 1.10 10.99 8.80 7.41 44.63 34.96 28.96 146.81 110.30 88.62 5.5 0.05 0.70 11.31 9.17 7.80 44.95 35.74 30.03 137.33 105.26 86.22 0.90 11.06 8.89 7.52 45.04 35.41 29.45 145.97 109.87 88.61 1.10 10.93 8.76 7.38 45.48 35.50 29.33 155.04 115.37 92.04 0.07 0.70 11.42 9.26 7.89 44.62 35.59 29.96 133.68 103.13 84.87 0.90 11.14 8.98 7.60 44.67 35.21 29.35 141.96 107.50 87.09 1.10 11.01 8.83 7.45 45.08 35.28 29.21 150.80 112.84 90.40 0.09 0.70 11.52 9.36 7.97 44.33 35.45 29.91 130.41 101.20 83.63 0.90 11.23 9.06 7.67 44.33 35.03 29.26 138.36 105.35 85.69 1.10 11.09 8.91 7.52 44.73 35.08 29.10 146.97 110.52 88.89 0.5 4.5 0.05 0.70 2.02 1.83 1.70 5.58 4.87 4.41 18.73 16.13 14.48 0.90 2.04 1.85 1.73 5.31 4.61 4.16 17.71 15.08 13.43 1.10 2.06 1.88 1.76 5.16 4.48 4.03 17.18 14.51 12.85 0.07 0.70 2.06 1.87 1.74 5.640 4.92 4.46 18.78 16.19 14.54 0.90 2.08 1.89 1.76 5.36 4.66 4.21 17.75 15.13 13.48 1.10 2.10 1.92 1.80 5.22 4.53 4.08 17.21 14.55 12.89 0.09 0.70 2.10 1.90 1.77 5.69 4.98 4.50 18.83 16.25 14.59 0.90 2.12 1.93 1.80 5.42 4.71 4.25 17.79 15.17 13.52 1.10 2.15 1.96 1.84 5.27 4.58 4.12 17.23 14.59 12.93 5 0.05 0.70 2.02 1.83 1.70 5.60 4.89 4.44 18.80 16.21 14.56 0.90 2.04 1.85 1.73 5.33 4.63 4.18 17.773 15.15 13.50 1.10 2.06 1.88 1.76 5.18 4.50 4.05 17.22 14.57 12.91 0.07 0.70 2.06 1.86 1.73 5.66 4.95 4.48 18.85 16.27 14.62 0.90 2.08 1.89 1.76 5.38 4.68 4.23 17.80 15.19 13.55 1.10 2.11 1.92 1.80 5.24 4.55 4.10 17.25 14.61 12.95 17 k. lakshmanan, s padmasekaran, m bhuvaneshwari, r asokan table 4: the variation of tcv under the parameter variation β2 θ α1 β1 0 0.5 1 µ 7 9 11 7 9 11 7 9 11 α 0.09 0.70 2.106 1.904 1.769 5.719 5.003 4.532 18.908 16.327 14.678 0.90 2.125 1.931 1.801 5.441 4.737 4.276 17.845 15.242 13.596 1.10 2.154 1.967 1.843 5.294 4.600 4.148 17.283 14.649 12.999 5.5 0.05 0.70 2.029 1.832 1.699 5.618 4.918 4.460 18.860 16.278 14.637 0.90 2.046 1.855 1.727 5.347 4.656 4.205 17.818 15.205 13.562 1.10 2.071 1.888 1.765 5.202 4.519 4.076 17.266 14.616 12.965 0.07 0.70 2.068 1.868 1.733 5.678 4.970 4.506 18.913 16.335 14.693 0.90 2.087 1.893 1.763 5.403 4.706 4.250 17.854 15.250 13.609 1.10 2.114 1.928 1.803 5.256 4.568 4.121 17.293 14.655 13.008 0.09 0.70 2.107 1.903 1.766 5.737 5.022 4.552 18.966 16.392 14.749 0.90 2.127 1.931 1.798 5.457 4.755 4.294 17.891 15.296 13.656 1.10 2.156 1.967 1.840 5.309 4.617 4.165 17.321 14.695 13.051 1.0 4.5 0.05 0.70 1.021 0.981 0.957 1.290 1.190 1.124 3.460 3.200 3.026 0.90 1.124 1.086 1.064 1.329 1.228 1.160 3.231 2.976 2.805 1.10 1.207 1.173 1.153 1.382 1.282 1.215 3.129 2.882 2.716 0.07 0.70 1.045 1.004 0.979 1.313 1.210 1.142 3.489 3.224 3.047 0.90 1.152 1.113 1.090 1.355 1.251 1.181 3.261 3.001 2.827 1.10 1.239 1.203 1.183 1.410 1.307 1.239 3.160 2.908 2.739 0.09 0.70 1.069 1.027 1.002 1.335 1.230 1.161 3.518 3.248 3.068 0.90 1.179 1.140 1.116 1.380 1.274 1.202 3.290 3.026 2.849 1.10 1.270 1.233 1.212 1.438 1.333 1.262 3.191 2.935 2.763 5 0.05 0.70 0.998 0.955 0.929 1.283 1.182 1.114 3.469 3.209 3.036 0.90 1.101 1.060 1.035 1.322 1.219 1.150 3.239 2.985 2.814 1.10 1.185 1.147 1.125 1.374 1.272 1.204 3.136 2.890 2.725 0.07 0.70 1.022 0.978 0.951 1.306 1.202 1.132 3.498 3.234 3.057 0.90 1.129 1.086 1.061 1.347 1.241 1.170 3.269 3.010 2.836 1.10 1.216 1.177 1.154 1.402 1.297 1.227 3.167 2.916 2.748 0.09 0.70 1.045 1.000 0.973 1.328 1.221 1.150 3.527 3.258 3.078 0.90 1.156 1.113 1.087 1.372 1.264 1.191 3.298 3.035 2.858 1.10 1.247 1.207 1.183 1.429 1.322 1.250 3.198 2.942 2.771 5.5 0.05 0.70 0.978 0.932 0.904 1.277 1.175 1.106 3.477 3.218 3.045 0.90 1.081 1.037 1.010 1.315 1.211 1.141 3.246 2.992 2.822 1.10 1.166 1.125 1.100 1.367 1.264 1.194 3.143 2.897 2.732 0.07 0.70 1.001 0.955 0.926 1.300 1.194 1.124 3.506 3.242 3.066 0.90 1.108 1.063 1.036 1.340 1.233 1.161 3.276 3.017 2.844 1.10 1.197 1.154 1.129 1.395 1.289 1.217 3.174 2.923 2.755 0.09 0.70 1.024 0.977 0.947 1.322 1.214 1.142 3.534 3.266 3.086 0.90 1.135 1.089 1.060 1.365 1.256 1.181 3.305 3.042 2.866 1.10 1.227 1.183 1.157 1.423 1.314 1.240 3.204 2.949 2.778 18 analysis of finite population stochastic modeling figure 1: impact of tcv on β1 vs β2 • figure (3) explores expected present stock level of the system on the combination of β1 vs β2. both β1 vs β2 reduce the e[psl] when they are increasing. here, the deviation of β2 curves is high when 0.2 ≤ β1 ≤ 1. • the parameters α and α1 are affects the e[wt] as shown in figure 4. in this graph, the beta curve has the high deviation with themselves and low deviation with α1. • the e[wt] is shown in figure 5 if θ and µ are increasing together. the θ curves are decreasing when µ is increasing and it means that the increased service rate cause less mean service time of an individual.therefore, the e[wt ] is decreased. if θ and µ are inversely proportional each other, θ reacts against µ. • the average waiting time of a customer is discussed for the case of θ vs k in figure 6 and µ vs k in figure 7. as we have enough discussion about θ and µ on the e[wt], we shall move to analyses the impact of k.when the number of finite source population is increases, for the e[wt], the θ curve will be straight line. example 6.4. impact of e[psl], e[reorder] and e[perishable] with the parameter variation this example describes the important system performance measures, e[psl], e[reorder] and e[perishable] are to be discussed with the parameter analysis of α, α1, θ, µ and β2 as shown in table 5-7. as per the scaling factor, β2 = 0, β2 = 0.5 and β2 = 1 are to be explored in table 5, 6 and 7 respectively. if we increase the reorder rate, the expected present stock level increases. for every replenishment, there are q items replaced as it reaches the system. so it makes the e[psl] is increasing when it is increase. when α 19 k. lakshmanan, s padmasekaran, m bhuvaneshwari, r asokan figure 2: impact of e[wt] on β1 vs β2 figure 3: impact of e[psl] onβ1 vs β2 20 analysis of finite population stochastic modeling figure 4: impact of e[wt] on α vs α1 figure 5: impact of e[wt] on µ vs θ 21 k. lakshmanan, s padmasekaran, m bhuvaneshwari, r asokan figure 6: impact of impact of e[wt] on k vs θ figure 7: impact of e[wt] on µ vsk 22 analysis of finite population stochastic modeling is increasing, for the value of µ = 7 e[reorder] behaves first increasing and then decreasing but at µ = 11 it is increasing only. further, for both µ, the e[perishable] will increase. here, the perishable quality of the products depends on the number of present stock level of the system. the parameter α1 affects the e[psl] to fall down if it is increasing. perishable products starts deterioration process depending the existing current stock level. so it is decreased. since the items in the inventory storage system are perished, the system requires more number of products to provides the sales service. hence the expected reorder level is increased. here the raise of a perishable rate obviously influence the increase of e[perishable] . then the parameter θ changes the e[psl] and e[reorder] by direct variation where as with e[perishable] it varies by indirect variation. for every arrival, there will be an unit in the system getting down when they leave the system. to fulfill such required number of items, there must be a reorder needed. since the inventory reduces by the more sales, there must be less number of items remaining in the inventory storage place. this cause the e[perishable] become less. more interestingly, as we predicted earlier, the intensity rate µ is inversely proportional to each e[psl], e[perishable]. if we increase µ, each of them starts falling down. if mean service time of individual customer too short, number of inventory falls down fast and less inventory requires more reorder and less number of perishable items. 23 k. lakshmanan, s padmasekaran, m bhuvaneshwari, r asokan table 5: impact of e[psl], e[reorder and e[perishable] with the parameter variation β2 θ α1 e[stock] e[reorder] e[perishable] µ 7 11 7 11 7 11 α 0 4.5 0.05 0.70 24.9645 22.6523 0.3136 0.4386 1.2482 1.1326 0.90 26.3157 24.3173 0.3138 0.4543 1.3158 1.2159 1.10 27.2531 25.5090 0.3089 0.4602 1.3627 1.2755 0.07 0.70 24.4147 22.2220 0.3277 0.4503 1.7090 1.5555 0.90 25.8047 23.9103 0.3293 0.4679 1.8063 1.6737 1.10 26.7722 25.1230 0.3252 0.4751 1.8741 1.7586 0.09 0.70 23.9075 21.8168 0.3408 0.4614 2.1517 1.9635 0.90 25.3337 23.5268 0.3439 0.4809 2.2800 2.1174 1.10 26.3299 24.7591 0.3408 0.4893 2.3697 2.2283 5 0.05 0.70 24.9645 22.6523 0.3188 0.4503 1.2482 1.1326 0.90 26.3157 24.3173 0.3191 0.4665 1.3158 1.2159 1.10 27.2531 25.5090 0.3142 0.4726 1.3627 1.2755 0.07 0.70 24.4147 22.2220 0.3330 0.4621 1.7090 1.5555 0.90 25.8047 23.9103 0.3347 0.4803 1.8063 1.6737 1.10 26.7722 25.1230 0.3307 0.4877 1.8741 1.7586 0.09 0.70 23.9075 21.8168 0.3463 0.4733 2.1517 1.9635 0.90 25.3337 23.5268 0.3495 0.4934 2.2800 2.1174 1.10 26.3299 24.7591 0.3464 0.5022 2.3697 2.2283 5.5 0.05 0.70 24.9645 22.6523 0.3211 0.4553 1.2482 1.1326 0.90 26.3157 24.3173 0.3214 0.4718 1.3158 1.2159 1.10 27.2531 25.5090 0.3165 0.4780 1.3627 1.2755 0.07 0.70 24.4147 22.2220 0.3353 0.4672 1.7090 1.5555 0.90 25.8047 23.9103 0.3371 0.4856 1.8063 1.6737 1.10 26.7722 25.1230 0.3330 0.4932 1.8741 1.7586 0.09 0.70 23.9075 21.8168 0.3486 0.4785 2.1517 1.9635 0.90 25.3337 23.5268 0.3519 0.4988 2.2800 2.1174 1.10 26.3299 24.7591 0.3488 0.5078 2.3697 2.2283 24 analysis of finite population stochastic modeling table 6: impact of e[psl], e[reorder and e[perishable] with the parameter variation β2 θ α1 e[stock] e[reorder] e[perishable] µ 7 11 7 11 7 11 α 0.5 4.5 0.05 0.70 18.5347 15.4582 0.2337 0.2938 0.9267 0.7729 0.90 20.5232 17.5264 0.2533 0.3280 1.0262 0.8763 1.10 22.0246 19.1546 0.2662 0.3532 1.1012 0.9577 0.07 0.70 18.2647 15.2755 0.2384 0.2973 1.2785 1.0693 0.90 20.2574 17.3398 0.2589 0.3323 1.4180 1.2138 1.10 21.7661 18.9683 0.2725 0.3582 1.5236 1.3278 0.09 0.70 18.0051 15.0982 0.2430 0.3007 1.6205 1.3588 0.90 20.0014 17.1583 0.2644 0.3366 1.8001 1.5442 1.10 21.5169 18.7869 0.2787 0.3631 1.9365 1.6908 5 0.05 0.70 18.2833 15.0507 0.2366 0.2977 0.9142 0.7525 0.90 20.2883 17.1217 0.2573 0.3339 1.0144 0.8561 1.10 21.8085 18.7631 0.2712 0.3609 1.0904 0.9382 0.07 0.70 18.0201 14.8771 0.2412 0.3009 1.2614 1.0414 0.90 20.0281 16.9432 0.2628 0.3380 1.4020 1.1860 1.10 21.5549 18.5840 0.2773 0.3657 1.5088 1.3009 0.09 0.70 17.7668 14.7084 0.2456 0.3041 1.5990 1.3238 0.90 19.7774 16.7695 0.2681 0.3420 1.7800 1.5093 1.10 21.3101 18.4095 0.2834 0.3705 1.9179 1.6569 5.5 0.05 0.70 18.1713 14.8689 0.2379 0.2993 0.9086 0.7434 0.90 20.1830 16.9399 0.2591 0.3364 1.0092 0.8470 1.10 21.7112 18.5861 0.2733 0.3642 1.0856 0.9293 0.07 0.70 17.9110 14.6992 0.2424 0.3025 1.2538 1.0289 0.90 19.9254 16.7649 0.2644 0.3404 1.3948 1.1735 1.10 21.4596 18.4102 0.2794 0.3689 1.5022 1.2887 0.09 0.70 17.6605 14.5342 0.2468 0.3056 1.5894 1.3081 0.90 19.6770 16.5946 0.2697 0.3443 1.7709 1.4935 1.10 21.2168 18.2387 0.2854 0.3736 1.9095 1.6415 25 k. lakshmanan, s padmasekaran, m bhuvaneshwari, r asokan table 7: impact of e[psl], e[reorder and e[perishable] with the parameter variation β2 θ α1 e[stock] e[reorder] e[perishable] µ 7 11 7 11 7 11 α 1 4.5 0.05 0.70 12.2895 10.2694 0.1562 0.1954 0.6145 0.5135 0.90 14.2804 12.1335 0.1792 0.2281 0.7140 0.6067 1.10 15.9157 13.7102 0.1973 0.2549 0.7958 0.6855 0.07 0.70 12.1776 10.1919 0.1586 0.1974 0.8524 0.7134 0.90 14.1615 12.0488 0.1822 0.2306 0.9913 0.8434 1.10 15.7933 13.6209 0.2007 0.2578 1.1055 0.9535 0.09 0.70 12.0681 10.1157 0.1610 0.1994 1.0861 0.9104 0.90 14.0450 11.9654 0.1851 0.2330 1.2640 1.0769 1.10 15.6733 13.5331 0.2041 0.2606 1.4106 1.2180 5 0.05 0.70 11.2674 9.0422 0.1473 0.1796 0.5634 0.4521 0.90 13.2040 10.7934 0.1709 0.2123 0.6602 0.5397 1.10 14.8221 12.3059 0.1899 0.2400 0.7411 0.6153 0.07 0.70 11.1733 8.9823 0.1494 0.1812 0.7821 0.6288 0.90 13.1024 10.7265 0.1734 0.2144 0.9172 0.7509 1.10 14.7161 12.2343 0.1928 0.2424 1.0301 0.8564 0.09 0.70 11.0811 8.9232 0.1514 0.1828 0.9973 0.8031 0.90 13.0026 10.6606 0.1759 0.2164 1.1702 0.9594 1.10 14.6120 12.1636 0.1958 0.2448 1.3151 1.0947 5.5 0.05 0.70 10.7714 8.4426 0.1426 0.1709 0.5386 0.4221 0.90 12.6732 10.1258 0.1662 0.2033 0.6337 0.5063 1.10 14.2751 11.5941 0.1856 0.2311 0.7138 0.5797 0.07 0.70 10.6852 8.3903 0.1445 0.1723 0.7480 0.5873 0.90 12.5794 10.0669 0.1686 0.2051 0.8806 0.7047 1.10 14.1766 11.5305 0.1883 0.2332 0.9924 0.8071 0.09 0.70 10.6007 8.3388 0.1464 0.1738 0.9541 0.7505 0.90 12.4872 10.0088 0.1709 0.2069 1.1239 0.9008 1.10 14.0798 11.4676 0.1910 0.2354 1.2672 1.0321 26 analysis of finite population stochastic modeling 7 conclusion the finite source population is considered to explore the non-sdap and non-qdsp in the sqis. the generalization of homogeneous and non-homogeneous arrival and service processes are given in the steady state of the model. also, the comparative discussion is made in the numerical investigations.the illustrations given in the examples enhance minimum optimal total cost for the qdsp category. the sdap will increase the number of units arriving to the inventory system. this increased units of arrival will produce the more sales of the inventory. when we are focusing the development of the inventory business, the first step has to be initialized to attract the customers towards the system. for such process, displayed stock level will assure the increase of customers in the inventory system. and maintaining the sufficient current stock level will play the crucial role for the development of an inventory system. simultaneously, the qdsr contribute the reduce of waiting time of a customer in the system. if a management provides some polices to reduce the waiting time of an individual in the servicing system, the customers are impressed and they will come to the same system often. in such a way that, the proposed model will applicable in a economically. acknowledgement the second author is supported by the fund for improvement of science and technology infrastructure (fist) of dst (sr/fst/msi-115/2016). the authors thank the editor and reviewers of the ratio mathematica for their constant support towards the successful completion of this work. references b. sivakumar a. shophia lawrence and g. arivarignan. a perishable inventory system with service facility and finite source. applied mathematical modelling, 37:4771– 4786, 2013. h. k. alfares. inventory model with stock-level dependent demand rate and variable holding cost. int j. prod. econ, 108:259–265, 2007. r. jesus artalejo and maria jesus lopez-herrero. the single server retrial queue with 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with the demand rate dependent on stock and constant deterioration. journal of indian academic mathematics, 30:221–228, 2008. a. tsoularis. deterministic and stochastic optimal inventory control with logistic stockdependent demand rate. international journal of mathematics in operational research, 6, 2014. diana tom varghese and dhanya shajin. state dependent admission of demands in a finite storage system. international journal of pure and applied mathematics, 118: 917–922, 2018. 29 ratio mathematica 26 (2014), 77–94 issn:1592-7415 dynamique du problème 3x + 1 sur la droite réelle nik lygeros,∗ olivier rozier∗∗ * lgpc (umr 5285), universitè de lyon, 69616, villeurbanne, france ** ipgp (umr 7154), 75238, paris, france nlygeros@gmail.com,rozier@ipgp.fr abstract the 3x + 1 problem is a difficult conjecture dealing with quite a simple algorithm on the positive integers. a possible approach is to go beyond the discrete nature of the problem, following m. chamberland who used an analytic extension to the half-line r+. we complete his results on the dynamic of the critical points and obtain a new formulation the 3x + 1 problem. we clarify the links with the question of the existence of wandering intervals. then, we extend the study of the dynamic to the half-line r−, in connection with the 3x − 1 problem. finally, we analyze the mean behaviour of real iterations near ±∞. it follows that the average growth rate of the iterates is close to (2 + √ 3)/4 under a condition of uniform distribution modulo 2. key words : 3x + 1 problem, one-dimensional dynamics, attracting cycles, asymptotic analysis. msc 2010 : 37e05. 1 introduction gènéralement attribué à lothar collatz, le problème 3x + 1 est aussi appelé conjecture de syracuse, en référence à l’université du màme nom. il se rapporte à la fonction t définie sur les entiers positifs par (1.1) t(n) := { (3n + 1)/2 si n est impair, n/2 sinon. 77 n. lygeros, o. rozier 12 6 3 5 8 40 20 10 4 13 42 21 32 16 128 64 2 1 fig. 1 – arbre inverse du problème 3x + 1 représentant l’ensemble des antécédents de 1 sur sept itérations. il s’agit de prouver que toute itération de t à partir d’un entier positif n arbitraire conduit nécessairement à la valeur 1. cette valeur est cyclique de période 2 : t(t(1)) = 1. conjecture 1.1. problème 3x + 1 pour tout entier n > 0, il existe un entier k ≥ 0 tel que tk(n) = 1.1 la figure 1 représente toutes les orbites qui aboutissent à 1 en un maximum de sept itérations. le problème 3x + 1 se ramène entièrement aux deux conjectures 1.2 et 1.3 sur la dynamique de la fonction t. conjecture 1.2. absence de trajectoires divergentes tout entier positif n a une orbite {t i(n)}∞i=0 bornée. conjecture 1.3. absence de cycles non-triviaux il n’existe pas d’entiers n > 2 et k > 0 tels que tk(n) = n. 1on note t k(n) le kàme itéré de t . 78 dynamique du problème 3x + 1 sur la droite réelle la conjecture 1.2 implique que tout entier positif a une orbite cyclique à partir d’un certain rang par itération de t. la conjecture 1.3 stipule que le seul cycle possible est le cycle (1, 2). généralement, on convient de stopper les itérations lorsque la valeur 1 est atteinte. ainsi on appelle temps de vol de n le plus petit entier k tel que tk(n) = 1. t. oliveira e silva a vérifié par des calculs sur ordinateur que tout entier positif n < 5 · 260 a un temps de vol fini [7]. les conjectures 1.2 et 1.3, bien qu’abondamment étudiées, ne sont toujours pas résolues. on pourra se référer aux ouvrages de j. lagarias [7] et g.j. wirsching [10] pour une synthèse détaillée des résultats partiels relatifs au problème 3x + 1 et diverses variantes. r. e. crandall [4] a avancé un argument heuristique basé sur l’idée de promenade aléatoire : si l’on considère uniquement la sous-suite des itérés impairs d’un entier n assez grand, on s’attend à ce que l’ensemble des rapports possibles entre deux termes successifs impairs, à savoir 3/2, 3/4, 3/8, . . ., aient pour probabilités respectives les valeurs 1/2, 1/4, 1/8, . . .. on obtient comme rapport moyen la valeur 3/4. ceci découle de l’égalité (1.2) ( 3 2 )1 2 · ( 3 4 )1 4 · ( 3 8 )1 8 · · · = 3 4 . cet argument plaide fortement en faveur de la conjecture 1.2. dans le cadre de notre étude, nous appellerons vitesse moyenne d’une séquence finie { n,t(n), · · · ,tk(n) } la quantité ( tk(n)/n )1/k . un raisonnement analogue [2] à celui de crandall suggère que la vitesse moyenne d’une séquence arbitraire non-cyclique a statistiquement une valeur proche de √ 3/2 ' 0.866 . . ., moyenne géométrique de 1/2 et 3/2. en effet, la croissance d’une séquence dépend principalement de la parité des itérés successifs. or, on s’attend à ce que les parités soient équiréparties sur un grand nombre d’itérations. ainsi le temps de vol k d’un entier n serait tel que (1/n)1/k ≈ √ 3/2 et l’on obtiendrait la valeur moyenne k ≈ 2 ln n ln ( 4 3 ) en l’absence de cycle [7, p. 7]. ces estimations sont confortées par les calculs numériques. il semble donc qu’un tel raisonnement permette de saisir l’essentiel de la dynamique asymptotique du problème 3x + 1. 79 n. lygeros, o. rozier 2 extension sur les réels positifs une approche possible du problème 3x + 1 est de sortir du cadre discret et d’étendre t par une fonction analytique sur l’ensemble des nombres réels [3] ou complexes [5, 8]. nous opterons pour l’extension réelle2 qui nous parait la plus naturelle, définie par l’équation (2.1) ci-après, et nous expliciterons les liens étroits qu’entretiennent la dynamique sur les réels et le problème 3x + 1. chamberland [3] a étudié la dynamique sur la demi-droite r+ de la fonction analytique (2.1) f(x) := x + 1 4 − ( x 2 + 1 4 ) cos(πx) qui vérifie f(n) = t(n) pour tout entier n, et f (r+) = r+ . il a ainsi obtenu plusieurs résultats significatifs : le point fixe 0 est attractif ainsi que les cycles a1 := {1, 2} et(2.2) a2 := {1.192 . . . , 2.138 . . .} de période 2. la dérivée schwartzienne de f est négative sur r+.(2.3) les intervalles [0,µ1] et [µ1,µ3] sont invariants par f, où(2.4) µ1 = 0.277 . . . et µ3 = 2.445 . . . sont des points fixes répulsifs. tout cycle d’entiers positifs est attractif.(2.5) il existe des orbites monotones non-bornées sur r+.(2.6) par ailleurs, il énonce la conjecture “stable set” [3] ci-dessous : conjecture 2.1. cycles attractifs sur r+ la fonction f n’admet aucun cycle attractif sur l’intervalle [µ3, +∞). une conséquence immédiate de (2.5) est que la conjecture 2.1 entrâıne la conjecture 1.3 du problème 3x + 1. puis, il définit l’ensemble des orbites non-bornées (2.7) u∞f := { x ∈ r+ : lim sup k→∞ fk(x) = ∞ } . 2le deuxième auteur (o. rozier) avait antérieurement suggéré l’étude de l’extension (2.1) dans le plan complexe et obtenu des représentations graphiques des bassins d’attraction [1]. 80 dynamique du problème 3x + 1 sur la droite réelle le résultat (2.6) prouve que u∞f est infini, et l’on démontre que u ∞ f contient un ensemble de cantor dans chaque intervalle [n,n+1] pour tout entier n ≥ 2 [8]. il suit que u∞f n’est pas dénombrable. conjecture 2.2. orbites non-bornées sur r+ l’ensemble u∞f est d’intérieur vide. la conjecture 2.2 est une formulation faible de la conjecture “unstable set” [3]. nous allons montrer qu’elle a des liens logiques avec le problème 3x + 1. lemme 2.1. soit {cn} ∞ n=0 l’ensemble des points critiques de f dans r +, ordonnés de telle sorte que 0 < c1 < c2 < .. .. alors on a n− 1 π2n < cn < n, si n est pair ; n < cn < n + 3 π2n , si n est impair. démonstration. (indications) soit n un entier positif. on a f ′(x) = 1 − 1 2 cos(πx) + π ( x 2 + 1 4 ) sin(πx) et on vérifie facilement que n− 1 2 < cn < n si n est pair, et n < cn < n + 1 2 si n est impair. de plus, on a toujours f ′(n) > 0 et on montre que f ′ ( n− 1 π2n ) < (20 − 6π2n) n + 1 24π2n3 < 0, si n est pair, f ′ ( n + 3 π2n ) < (18 − 6π2n) n + 9 8π2n3 < 0, si n est impair, en utilisant les encadrements 1 − t 2 2 < cos t < 1 et t − t 3 6 < sin t < t pour 0 < t < 1. lemme 2.2. on considère la famille d’intervalles jan := [ n,n + a π2n ] pour tout entier n > 0 et tout réel a tel que 27 8 < a < 6. alors on a f (jan) ⊂ jaf(n) pour tout entier n assez grand. si de plus a = 7 2 , alors l’inclusion est vraie pour tout n > 0. 81 n. lygeros, o. rozier démonstration. soit un entier n > 0 et un réel a tel que 27 8 < a < 6. 1er cas : n est pair, f(n) = n 2 et f est croissante sur jan. on vérifie alors que f ( n + a π2n ) ≤ f(n) + a π2f(n) + a ·b avec a = a 8π4n3 et b = π2n (2 (a− 6) n + a) + 2a2 en utilisant l’inégalité 1−cos t < t 2 2 pour 0 < t < 1. comme a−6 < 0, il est clair que a ·b < 0 pour n suffisamment grand. si de plus a = 7 2 , alors b ≤ 49 2 − 13π2 < 0 pour tout n. 2e cas : n est impair, f(n) = 3n+1 2 et f est croissante sur [n,cn] et décroissante sur [ cn,n + a π2n ] . on vérifie alors que f ( n + a π2n ) ≥ f(n) −a ·b a et b étant défini comme précédemment, donc a ·b < 0 pour n suffisamment grand. si de plus a = 7 2 , alors a ·b < 0 pour tout n ≥ 3, et dans le cas n = 1, on a f ( 1 + 7 2π2 ) = 2.013 . . . > f(1). d’après le lemme 2.1, on a cn = n + b π2n avec 0 < b < 3. il vient f(cn) −f(n) ≤ 3b 2π2n − n 2 ( 1 − cos ( b πn )) puis en utilisant l’inégalité 1 − cos t > t 2 2 − t 4 24 pour 0 < t < 1, f(cn) −f(n) < b(6 − b) 4π2n + b4 48π4n3 ≤ 9 4π2n + 27 16π4n3 . on obtient f(cn) < f(n) + a π2f(n) + c d avec c = 4π2n2 ((27 − 8a)n + 9) + 81n + 27 et d = 16π4n3(3n + 1). on voit que c < 0 pour n suffisamment grand. si de plus a = 7 2 et n ≥ 11, on a alors c = 4π2n2(9 −n) + 81n + 27 < 0 82 dynamique du problème 3x + 1 sur la droite réelle et dans les cas où n = 1, 3, 5, 7 ou 9, on vérifie numériquement que f(cn) −f(n) − 7 (3n + 1)π2 < 0 en utilisant les valeurs c1 = 1.180938 . . ., c3 = 3.084794..., c5 = 5.054721..., c7 = 7.040311... et c9 = 9.031889.... on déduit du lemme 2.2 un lien logique entre les conjectures 1.2 et 2.2 : théorème 2.1. la conjecture 2.2 implique la conjecture 1.2 (absence d’orbites non-bornées) du problème 3x + 1 . démonstration. supposons que la conjecture 2.2 soit vraie et que la conjecture 1.2 soit fausse. alors il existe un entier positif n0 tel que lim sup k→∞ fk(n0) = ∞. d’après le lemme 2.2, une simple récurrence donne fk ( j 7 2 n0 ) ⊂ j 7 2 fk(n0) pour tout entier k ≥ 0. donc l’ensemble u∞f contient l’intervalle j 7 2 n0 , ce qui est en contradication avec notre hypothèse que u∞f soit d’intérieur vide. 3 dynamique des points critiques les résultats (2.3) et (2.5) entrainent que le bassin d’attraction immédiat de tout cycle d’entiers strictement positifs contient au moins un point critique [3]. pour cette raison, chamberland a effectué des calculs numériques relatifs aux orbites des points critiques cn pour n ≤ 1000. il énonce la conjecture “critical points” ci-dessous : conjecture 3.1. points critiques tous les points critiques cn, n > 0, sont attirés par l’un des cycles a1 ou a2. nous complétons ici les résultats numériques de chamberland. une précision de 1500 chiffres décimaux en virgule flottante est requise pour le calcul de certaines orbites (c646 par exemple). nous avons vérifié nos résultats avec deux logiciels différents, mathematica et maple. d’après nos calculs, les cycles a1 et a2 attirent tous les points critiques cn pour n ≤ 2000. plus précisément, cn est attiré par a2 pour 3 n = 1, 3en gras les valeurs déjà obtenues par chamberland. 83 n. lygeros, o. rozier 3, 5, 382, 496, 502, 504, 508, 530, 550, 644, 646, 656, 666, 754, 830, 874, 1078, 1150, 1214, 1534, 1590, 1598, 1614, 1662, 1854, et par a1 pour toutes les autres valeurs de n ≤ 2000. nous avons observé que l’orbite de cn est toujours proche de l’orbite de n, sauf pour n ≡−2 (mod 64) et pour n=54, 334, 338, 366, 390, 442, 444, 470, 484, 486, 496, 500, . . .. les résultats numériques suggèrent la conjecture suivante4 : conjecture 3.2. points critiques d’ordre impair les points critiques cn sont attirés par le cycle a1 = {1, 2} pour tout entier n ≥ 7 impair. nous montrons à présent que la conjecture 3.2 suffit pour reformuler complètement le problème 3x + 1. théorème 3.1. soit un entier impair n ≥ 7 dont l’orbite contient 1. alors le point critique cn est attiré par le cycle a1. démonstration. considérons un entier impair n ≥ 7 dont l’orbite contient 1. la construction de l’arbre des orbites inverses de 1, représenté sur la figure 1, montre que l’orbite de n contient l’un des entiers 12, 13, 16 ou 40. on déduit de règles itératives modulo 3 sur les entiers que les antécédents de 12 sont des entiers pairs. il vient que fk(n) = 13, 16 ou 40 pour un entier k ≥ 0. les lemmes 2.1 et 2.2 entrâınent que cn appartient à j 7 2 n et f k(cn) se trouve dans j 7 2 13 ∪j 7 2 16 ∪j 7 2 40. 1er cas : fk(n) = 13, fk(cn) ∈ j 7 2 13. la séquence des itérés de f k(n) est 13 → 20 → 10 → 5 → 8 → 4 → 2 → 1. soit m un entier pris dans cette séquence. la fonction f est unimodale sur j 7 2 m avec un maximum en cm lorsque m est impair, et strictement croissante lorsque m est pair. ce comportement permet de déterminer les images successives de j 7 2 13 en fonction de c13 = 13.022478 . . .. f ( j 7 2 13 ) = [20,f(c13)] f3 ( j 7 2 13 ) = [ 5,f3(c13) ] avec f3(c13) = 5.0249 . . . < c5 = 5.0547 . . .. f7 ( j 7 2 13 ) = [ 1,f7(c13) ] 4dans [5], une conjecture analogue avec davantage d’hypothèses est formulée relativement à une autre extension de la fonction t sur les réels. 84 dynamique du problème 3x + 1 sur la droite réelle avec f7 (c13) = 1.0184 . . .. de plus la fonction f2 est strictement croissante sur l’intervalle (1,c1) avec une unique point fixe x1 = 1.023686 . . . qui est répulsif. il suit que l’intervalle [1,x1) fait partie du bassin d’attraction immédiat du cycle a1 et que cn est attiré par a1. 2e cas : fk(n) = 16, fk(cn) ∈ j 7 2 16. on a la séquence 16 → 8 → 4 → 2 → 1. comme précédemment, on obtient l’image f4 ( j 7 2 16 ) = [ 1,f4 ( 16 + 7 32π2 )] avec f4 ( 16 + 7 32π2 ) = 1.0227 . . . < x1. donc cn est attiré par a1. 3e cas : fk(n) = 40, fk(cn) ∈ j 7 2 40, et la séquence des itérés est 40 → 20 → 10 → 5 → 8 → 4 → 2 → 1. de la màme manière, on itère les images successives f3 ( j 7 2 40 ) = [ 5,f3 ( 40 + 7 80π2 )] avec f3 ( 40 + 7 80π2 ) = 5.0118 . . . < c5 = 5.0547 . . ., f7 ( j 7 2 40 ) = [ 1,f7 ( 40 + 7 80π2 )] avec f7 ( 40 + 7 80π2 ) = 1.0047 . . . < x1. ainsi cn est attiré par a1 dans tous les cas. remarque 3.1. dans cette démonstration, il n’est pas possible de fusionner les cas 1 et 3 en partant de l’entier 20 car f6 ( j 7 2 20 ) = [ 1,f6 ( 20 + 7 40π2 )] = [1, 1.023691 . . .] n’est pas inclus (de très peu) dans le bassin d’attraction de a1 délimité par x1 = 1.023686 . . .. corollaire 3.1. la conjecture 3.2 est logiquement équivalente au problème 3x + 1. démonstration. une conséquence immédiate du théorème 3.1 est que la conjecture 1.1 (problème 3x + 1) implique la conjecture 3.2 sur la dynamique des points critiques d’ordre impair. on démontre à présent la réciproque. considérons un entier n > 0. son orbite contient au moins un entier impair fk1 (n), k1 ≥ 0. si fk1 (n) ≤ 5, alors l’orbite de n contient le point 1 (cf. figure 1). on considère à présent le cas fk1 (n) ≥ 7. 85 n. lygeros, o. rozier supposons que la conjecture 3.2 soit vraie. alors il existe un entier positif k2 tel que fk2 ( cfk1 (n) ) < 2. de plus, le lemme 2.2 donne par récurrence l’inclusion fk2 ( cfk1 (n) ) ∈ j 7 2 fk1+k2 (n) . il découle l’égalité fk1+k2 (n) = 1. 4 intervalles errants l’existence d’intervalles errants [9] dans la dynamique de l’extension f est une question ouverte avec d’importantes implications pour le problème 3x + 1. conjecture 4.1. absence d’intervalles errants la fonction f n’admet pas d’intervalles errants dans r+. elle est au cœur du théorème ci-dessous. théorème 4.1. on a les relations suivantes entre conjectures : (a) la conjecture 2.2 entrâıne la conjecture 4.1, (b) la conjecture 4.1 entrâıne la conjecture 1.2. démonstration. par l’absurde. (a) supposons que la conjecture 2.2 soit vraie et que la conjecture 4.1 soit fausse. cela implique que la fonction f admette une famille d’intervalles errants sur une partie bornée de r+. or ce serait en contradiction avec la propriété (2.3) : la dérivée schwartzienne de f est négative sur r+. (b) supposons que la conjecture 1.2 soit fausse. alors il existe un entier positif n tel que limi→∞ f i(n) = +∞. d’après le lemme 2.2, les intervalles{ fi ( j 7/2 n )}∞ i=0 sont inclus dans les intervalles { j 7/2 fi(n) }∞ i=0 , deux à deux disjoints. il s’agit d’une famille d’intervalles errants. une synthèse des liens logiques entre conjectures est donnée en annexe. 86 dynamique du problème 3x + 1 sur la droite réelle 5 extension sur les réels négatifs l’ensemble r− des réels négatifs est également invariant par la fonction f définie par (2.1). la dynamique sur les entiers négatifs est alors identique, au signe près, à celle de la fonction “3x−1”, notée u et définie sur les entiers positifs par (5.1) u(n) := { (3n− 1)/2 si n est impair, n/2 sinon. en effet, on a la relation de conjugaison f(−n) = −u(n) pour tout entier n positif. la fonction u admet le point fixe 1 et a deux cycles connus : {5, 7, 10} de période 3 et {17, 25, 37, 55, 82, 41, 61, 91, 136, 68, 34} de période 11. cela conduit à formuler le “problème 3x− 1” : conjecture 5.1. problème 3x− 1 pour tout entier n > 0, il existe un entier k ≥ 0 tel que uk(n) = 1, 5 ou 17. les valeurs de f sur r+ et (−∞,−1] sont liées pas l’équation fonctionnelle (5.2) f(x) −f(−1 −x) = 2x + 1 de sorte que les points fixes de f sur (−∞,−1] sont exactement les points νi := −1 − µi, où {µi}∞i=0 désigne l’ensemble des points fixes de f sur r+, µ0 = 0 < µ1 < 1 < µ2 < 2 < .. .. néanmoins, la dynamique de f sur r− diffère partiellement de celle que l’on a pu décrire sur r+, comme le montrent les propriétés (5.3) à (5.7). les points fixes 0 et ν1 = −1.277 . . . sont attractifs, ainsi que les(5.3) cycles b1 := { x,f(x),f2(x) } où x = −5.046002 . . . , b2 := { x,f(x),f2(x) } où x = −4.998739 . . . , b3 := { x,f(x), . . . ,f10(x) } où x = −17.002728 . . . , b4 := { x,f(x), . . . ,f10(x) } où x = −16.999991 . . . .. la dérivée schwartzienne de f n’est pas partout négative sur r−.(5.4) les intervalles [−1, 0] et [ν1,−1] sont invariants par f.(5.5) tout cycle d’entiers négatifs est répulsif.(5.6) il existe des orbites monotones non-bornées sur r−.(5.7) 87 n. lygeros, o. rozier point ou cycle attractif période multiplicateur 0 1 0.5 ν1 1 0.385708 . . . b1 3 0.036389 . . . b2 3 0.866135 . . . b3 11 0.003773 . . . b4 11 0.926287 . . . tab. 1 – coefficients multiplicateurs des points et cycles attractifs sur les réels négatifs. démonstration. (indications) propriété (5.3) : les vitesses d’attraction sont données dans le tableau 1. propriété (5.4) : la dérivée schwartzienne est positive sur un intervalle contenant le point -0.2. on a en effet sf(−0.2) = 39.961 . . ., où sf(x) = f ′′′(x) f ′(x) − 3 2 ( f ′′(x) f ′(x) )2 . propriété (5.5) : la fonction f est strictement croissante sur l’intervalle [ν1, 0] contenant le point fixe répulsif -1. propriété (5.6) : voir les indications dans [3, p.16]. propriété (5.7) : la démonstration est similaire à celle de (2.6). remarque 5.1. les cycles b2 et b4 sont très faiblement attractifs car leur multiplicateur est proche de 1 (cf. tableau 1). on vérifie également que les cycles contenant les points -5 et -17 sont très faiblement répulsifs, avec pour multiplicateurs respectifs les rationnels 9/8 et 2187/2048. comme précédemment, on note cn les points critiques proches des entiers n < 0, et on peut montrer que les itérés successifs de cn pour n impair négatif restent proches des itérés de n, par valeurs inférieures. nous avons vérifié numériquement pour tout entier n, −1000 < n < 0, que 88 dynamique du problème 3x + 1 sur la droite réelle – si n est impair et fk(n) = −1 (resp. -5, -17) pour un entier k, alors l’orbite de cn converge vers ν1 (resp. b1, b3) ; – si n est pair et fk(n) = −1 (resp. -5, -17) pour un entier k, alors l’orbite de cn converge vers 0 (resp. b2, b4), sauf pour n=-34, -66, -98, -130, -132, -162, -174, -194, -202, -226, . . . où l’orbite de cn converge vers b3, b1, b3, ν1, b3, b1, ν1, b1, ν1, ν1, . . . respectivement. on note que les entiers n ≡−2 (mod 32) semblent toujours faire partie des exceptions. le plus souvent, lorsque n < 0 est pair, l’orbite de cn reste proche de l’orbite de n, par valeurs supérieures. pour n=-34, -98, -132, -162, -202, . . . les itérés de cn finissent pas àtre inférieurs aux itérés de n, sans s’en éloigner pour autant. pour n=-66, -130, -174, -194, -258, . . . les orbites de n et de cn sont décorrélées après un nombre fini d’itérations. dans ce dernier cas, on observe une répartition des orbites de cn dans chacun des six bassins d’attraction de r− : 0, ν1, b1, b2, b3 et b4. conjecture 5.2. points critiques d’ordre négatif impair les points critiques cn sont attirés soit par le point fixe ν1, soit par l’un des cycles b1 ou b3, pour tout entier n < 0 impair. 6 dynamique asymptotique dans cette partie, nous étudions le comportement moyen de séquences finies ou infinies d’itérations de f, afin de déterminer la vitesse moyenne asymptotique (i.e. au voisinage de ±∞). nous dirons ainsi qu’une séquence infinie s = {fi(x)}∞i=0 est uniformément distribuée modulo 2 (u. d. mod 2) si et seulement si la discrépance à l’origine de {fi(x) mod 2}n−1i=0 dans l’intervalle [0, 2], notée d ∗ n(s mod 2), vérifie 5 lim n→∞ d∗n(s mod 2) = 0. dans le cas d’une séquence finie s = {fi(x)}ni=0, nous dirons de manière informelle que s est u. d. mod 2 dès lors que d∗n(s mod 2) � 1. on rappelle que la notion de discrépance est une mesure de l’uniformité de la distribution d’une séquence de points x = {x1, . . . ,xn} ∈ [a,b]n et est définie par (6.1) d∗n(x) := sup a≤c≤b ∣∣∣∣|{x1, . . . ,xn}∩ [a,c)|n − c−ab−a ∣∣∣∣ elle intervient notamment dans l’inégalité de koksma [6] : 5on note x mod 2 la valeur modulo 2 de tout réel x, définie par x mod 2 := x − 2bx 2 c. 89 n. lygeros, o. rozier théorème 6.1. (koksma) soit f : [a,b] → r une fonction à variation (totale) v (f) bornée. alors pour toute séquence x = {x1, . . . ,xn} ∈ [a,b]n, on a ∣∣∣∣∣1n n∑ i=1 f(xi) − 1 b−a ∫ b a f(t) dt ∣∣∣∣∣ < v (f)d∗n(x) nous considérons dorénavant que la fonction f définie par (2.1) s’applique sur r tout entier. comme f ne s’annule qu’en 0, il suit que fn(x) est de màme signe que x pour tout réel x 6= 0 et tout entier n. notre approche consiste à approximer f(x)/x par son asymptote sinusöıdale (6.2) g(x) := 1 − cos(πx) 2 dont on détermine la moyenne géométrique. lemme 6.1. la moyenne géométrique τ de la fonction réelle g(x) = 1 − cos(πx)/2 sur [0, 2] est égale à α/4, où α = 2 + √ 3 est racine du polynôme x2 − 4x + 1. démonstration. on cherche à calculer τ := exp ( 1 2 ∫ 2 0 ln (g(t)) dt ) avec g(t) = 1 − cos(πt)/2 = ( α−eiπt )( α−e−iπt ) /(4α) = ∣∣α−eiπt∣∣2 /(4α). on obtient ln τ = ∫ 2 0 ln ∣∣α−eiπt∣∣ dt − ln (4α) . la formule de jensen relative aux fonctions analytiques sur le disque de centre α et de rayon 1 donne le résultat attendu ln τ = 2 ln α− ln(4α) = ln (α 4 ) . on montre à présent qu’au voisinage de ±∞ toute séquence d’itérations u. d. mod 2 de f décroit avec une vitesse moyenne proche de τ = (2+ √ 3)/4 ' 0.933 . . .. théorème 6.2. soit une séquence finie d’itérations s = {fi(x)}ni=0 telle que min{|fi(x)|}n−1i=0 ≥ m pour un réel m > 1 3 . alors on a∣∣∣∣1n ln ( fn(x) x ) − ln τ ∣∣∣∣ < 2 (ln 3) d∗n(s mod 2) − ln ( 1 − 1 3m ) . 90 dynamique du problème 3x + 1 sur la droite réelle démonstration. on considère la formulation f(t) = g(t) (t + h(t)) où h est la fonction périodique h(t) := 1 − cos(πt) 4g(t) = 1 − cos(πt) 4 − 2 cos(πt) . on a donc fn(x) x = n−1∏ i=0 fi+1(x) fi(x) = n−1∏ i=0 g ( fi(x) )( 1 + h (fi(x)) fi(x) ) il vient alors 1 n ln ( fn(x) x ) − ln τ = a + b avec a = 1 n n−1∑ i=0 ln ( g ( fi(x) )) − ln τ et b = 1 n n−1∑ i=0 ln ( 1 + h (fi(x)) fi(x) ) d’après le lemme 6.1, ln τ = 1 2 ∫ 2 0 ln (g(t)) dt on applique l’inégalité de koksma : |a| ≤ v (φ)d∗n(s mod 2) où v (φ) est la variation totale de la fonction φ(t) := ln (g(t)) sur [0, 2], soit v (φ) = 2φ(1) −φ(2) −φ(0) = 2 ln 3. pour majorer |b|, on vérifie que la fonction h(t) est à valeur dans [0, 1/3] avec un maximum en t = 1. on en déduit que |b| ≤ max ( − ln ( 1 − 1 3m ) , ln ( 1 + 1 3m )) = − ln ( 1 − 1 3m ) . le théorème 6.2 est inopérant pour les séquences d’entiers, dont la vitesse moyenne attendue est √ 3/2, strictement inférieure à τ. il permet toutefois d’établir un lien entre la vitesse moyenne et la distribution modulo 2 des itérations. 91 n. lygeros, o. rozier théorème 6.3. soit x un réel d’orbite {fi(x)}∞i=0 telle que lim inf i→∞ |fi(x)| > 1 3(1 − τ) ' 4.97 . . . alors l’orbite de x n’est pas uniformément distribuée modulo 2. démonstration. il existe un entier positif n et un réel a > 1 tels que |fi(x)| ≥ a 3(1 − τ) pour tout i ≥ n. on considère les séquences finies sn = {fi(x)}n+ni=n pour tout n entier positif, et on pose mn := min{|fi(x)|}n+ni=n . d’après le théorème 6.2, 1 n ln ( fn+n (x) fn (x) ) − ln τ < 2(ln 3)d∗n(sn mod 2) − ln ( 1 − 1 3mn ) . il vient 2(ln 3)d∗n(sn mod 2) > an + bn avec an = 1 n ln ( fn+n (x) fn (x) ) et bn = − ln τ + ln ( 1 − 1 3mn ) . d’une part, on vérifie aisément que lim infn→∞ an ≥ 0. d’autre part, on a bn ≥− ln τ + ln ( 1 − 1 − τ a ) = ln ( 1 + (a− 1)(1 − τ) aτ ) > 0. on obtient donc le résultat souhaité : lim inf n→∞ d∗n(sn mod 2) ≥ ln ( 1 + (a−1)(1−τ) aτ ) 2 ln 3 > 0. l’existence d’orbites tendant vers l’infini a été prouvée par chamberland pour la fonction f et le corollaire 6.1 donne une condition nécessaire sur l’ensemble des valeurs modulo 2 d’une telle orbite. 92 dynamique du problème 3x + 1 sur la droite réelle corollaire 6.1. soit x un réel d’orbite {fi(x)}∞i=0 divergente telle que lim i→∞ |fi(x)| = +∞. alors l’orbite de x n’est pas u. d. mod 2. ce résultat renforce la conjecture 2.2. en effet, on peut s’attendre à ce que la condition de distribution uniforme modulo 2 des itérations de f soit le plus souvent valide au voisinage de ±∞, compte tenu des propriétés suivantes : – le diamètre et la densité des zones contractantes tend vers 0, – l’amplitude des oscillations devient infiniment grande. références [1] a. aoufi, o. rozier, le problème de syracuse dans c, singularité no5 (1990) 26. [2] e. barone, una argumentazione euristica probabilistica sulla successione di collatz, ital. j. pure appl. math., 4 (1998) 151–153. [3] m. chamberland, a continuous extension of the 3x+1 problem to the real line, dynamics of continuous, discrete and impulsive systems, 2 (1996) 495–509. [4] r. e. crandall, on the ”3x+ 1” problem, math. comp., 32 (1978) 1281– 1292. [5] j. dumont, c. reiter, real dynamics of a 3-power extension of the 3x+1 function, dynamics of continuous, discrete and impulsive systems, 10 (2003) 875–893. [6] l. kuipers, h. niederreiter, uniform distribution of sequences, john wiley & sons, 1974. [7] j. lagarias, the ultimate challenge : the 3x+1 problem, american mathematical monthly, 2010. [8] s. letherman, d. schleicher, r. wood, the 3n + 1 problem and holomorphic dynamics, experiment. math., 8, (1999) 241–251. [9] w. de melo, s. van strien, one-dimensional dynamics, springer-verlag, 1993. [10] g. j. wirsching, the dynamical system generated by the 3n + 1 function, springer-verlag, 1998. 93 n. lygeros, o. rozier annexe la figure 2 ci-dessous résume quelques-uns des principaux résultats de cet article sous la forme de liens logiques entre diverses conjectures. conjecture 3.2 points critiques d'ordre impair conjecture 1.1 problème 3x+1 conjecture 2.1 cycles attractifs positifs conjecture 1.2 trajectoires divergentes conjecture 2.2 orbites non-bornées positives conjecture 4.1 absence d'intervalles errants conjecture 1.3 cycles non-triviaux fig. 2 – liens logiques entre conjectures. la partie gauche concerne le cadre continu r+ et la partie droite le cadre discret z+. 94 ratio mathematica volume 47, 2023 some topological properties of revised fuzzy cone metric spaces a. muraliraj * r. thangathamizh† abstract in this paper, we introduced revised fuzzy cone metric space with its topological properties. likewise a necessary and sufficient condition for a revised fuzzy cone metric space to be precompact is given. we additionally show that each distinct revised fuzzy cone metric space is second countable and that a subspace of a separable revised fuzzy cone metric space is separable. keywords: revised fuzzy metric space; revised fuzzy cone metric space; separable; second countable. 2020 ams subject classifications: 54a40, 54e35, 54e15, 54h25.1 *assistant professor, department of mathematics, urumu dhanalakshmi college trichy, india; karguzali@gmail.com †assistant professor, department of mathematics, k. ramakrishnan college of engineering, trichy, india; thamizh1418@gmail.com 1received on march 19, 2022. accepted on september 12, 2022. published online on january 13, 2023. doi: 10.23755/rm.v39i0.734. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 42 a. muraliraj and r. thangathamizh 1 introduction after zadeh [1965] introduced the idea of fuzzy sets, several authors have introduced and studied many notions of metric indistinctness [huang and zhang, 2007, kramosil and michalek, 1975, ahmad and mesiarová-zemánková, 2007, navara, 2007, muraliraj and thangathamizh, 2021a,b,d, oner and tanay, 2015, grigorenko et al., 2020] and metric cone indistinctness. by modifying the idea of metric indistinctness introduced by george and veeramani [1994], zadeh [1965] studied the notion of fuzzy cone metric areas. especially, they evidenced that every fuzzy cone topological space generates a hausdorf first-countable topology. here we tend to study additional topological properties of these areas whose fuzzy metric version are usually found in george and veeramani [1994], ghareeb and al-omeri [2018], grabiec [1988], gregori et al. [2011]. sostak [2018] additionally represented the idea of “george–veeramani fuzzy metrics revised”. presently olga grigorenko, juan jose minana, alexander sostak, muraliraj and thangathamizh [2021c] have introduced “on t-conorm primarily based fuzzy (pseudo) metrics”. recently muraliraj and thangathamizh [2021a,c,d] proved various fixed point theorems in revised fuzzy metric spaces. muraliraj and thangathamizh [2021b] introduce the concept of revised fuzzy modular metric space. moreover, we tend to prove that a revised fuzzy cone topological space is precompact if and providing each sequence in it’s a cauchy subsequence. further, we tend to show that x1 × x2 may be a complete revised fuzzy cone topological space if and providing x1 and x2 are complete revised fuzzy cone metric areas. finally it’s tried that each divisible revised fuzzy cone topological space is second calculable and a mathematical space of a separable revised fuzzy cone topological space is separable. 2 preliminaries definition 2.1 (gregori et al. [2011]). let e be a real banach space, θ the zero of e and p a subset of e. then p is called a cone if and only if (i) p is closed, nonempty, and p ̸= {θ}, (ii) if ab ∈ r, ab ≥ 0 and xy ∈ p , then ax + by ∈ p , (iii) if both x ∈ p and −x ∈ p , then x = θ. given a cone p , a partial ordering ≾ on e with respect to p is defined by x ≾ y if only if y − x ∈ p . the notation x ≺ y will stand for x ≾ y and x ̸= y, while x ≪ y will stand for y − x ∈ int(p). throughout this paper, we assume that all the cones have nonempty interiors. 43 some topological properties of revised fuzzy cone metric spaces there are two kinds of cones: normal and non-normal ones. a cone p is called normal if there exists a constant k ≥ 1 such that for all t, s ∈ e, θ ≾ t ≾ s implies ∥t∥ ≤ k ∥s∥, and the least positive number k having this property is called normal constant of p gregori et al. [2011]. it is clear that k ≥ 1. definition 2.2 (sostak [2018]). a binary operation ⊕ : [0, 1] × [0, 1] → [0, 1] is a t-conorm if it satisfies the following conditions: (i) ⊕ is associative andcommutative, (ii) ⊕ is continuous, (iii) a ⊕ 0 = a for all a ∈ [0, 1], (iv) a ⊕ b ≤ c ⊕ d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1]. 2.1 examples sostak and öner [2020] (i) lukasievicz t-conorm: a ⊕ b = max{a, b}; (ii) product t-conorm: a ⊕ b = a + b − ab; (iii) minimum t-conorm: a ⊕ b = min(a + b, 1). definition 2.3 (oner and tanay [2015]). a 3-tuple (u, m0, ∗) is called a fuzzy cone metric space (fcm space) if c is a cone of e, u is an arbitrary set, (∗) is a continuous t-norm, and m0 is a fuzzy set on u2 × int(p) satisfying the following conditions: (i) m0(λ1, λ2, t) > 0 and m0(λ1, λ2, t) = 1 ⇔ λ1 = λ2, (ii) m0 (λ1, λ2, t) = m0(λ2, λ1, t) (iii) m0 (λ1, λ2, t) ∗m0(λ2, λ3, s) ≤ m0(λ1, λ3, t + s), (iv) m0 (λ1, λ2, ) : int(p) → [0, 1] is continuous ∀λ1, λ2, λ3 ∈ u and t, s ∈ int(p). definition 2.4 (sostak [2018]). a revised fuzzy metric space is an ordered triple (x, µ, ⊕) such that x is a non empty set, ⊕ is a continuous t-conorm and µ is a revised fuzzy set on µ : x2 × r+ → [0, 1] satisfies the following conditions: (rfm1) µ (x, y, t) < 1; (rfm2) µ(x, y, t) = 0 if and only if x = y; (rfm3) µ(x, y, t) = µ(y, x, t); (rfm4) µ(x, z, t + s) ≤ µ(x, y, t) ⊕ µ(y, z, s); (rfm5) µ(x, y, −) : (0, ∞) → [0, 1) is continuous ∀x, y, z ∈ x and t, s ∈ r+. then µ is called a revised fuzzy metric on x. 44 a. muraliraj and r. thangathamizh 3 main results definition 3.1. a revised fuzzy cone metric space is an 3-triple (x, µ, ⊕) such that p is a cone of e, x is a non empty set, ⊕ is a continuous t-conorm and µ is a revised fuzzy set on x2 × int(p) satisfies the following conditions, ∀x, y, z ∈ x and s, t ∈ int(p) (that is s ≫ θ, t ≫ θ), (rfcm 1) µ (x, y, t) < 1 , (rfcm 2) µ(x, y, t) = 0 if and only if x = y, (rfcm 3) µ(x, y, t) = µ(y, x, t), (rfcm 4) µ(x, z, t + s) ≤ µ(x, y, t) ⊕ µ(y, z, s), (rfcm 5) µ(x, y, −) : int(p) → [0, 1] is continuous. then µ is called a revised fuzzy cone metric on x. if (x, µ, ⊕) is a revised fuzzy cone metric space, we will say that µ is a revised fuzzy cone metric on x. every revised fuzzy cone metric space (x, µ, ⊕) induces a hausdorff firstcountable topology τfc on x which has as a base the family of sets of the form {b(x, r, t) : x ∈ x; 0 < r < 1, t ≫ θ}, where {b(x, r, t) : y ∈ x; µ(x, y, t) < r} for every r with 0 < r < 1 and t ≫ θ. a revised fuzzy cone metric space (x, µ, ⊕) is called complete if every cauchy sequence in it is convergent, where a sequence {xn} is said to be a cauchy sequence if for any ε ∈ (0, 1) and any t ≫ θ there exists a natural number n0 such that µ (xn, xm, t) < ε for all n, m ≥ n0, and a sequence {xn} is said to converge to x if for any t ≫ θ and any r ∈ (0, 1) there exists a natural number n0 such that µ (xn, x, t) < r for all n ≥ n0. a sequence {xn} converges to x if and only if µ (xn, x, t) → 0 for each t ≫ θ. definition 3.2. let (x, µ, ⊕) be a revised fuzzy cone metric space. for t ≫ θ, the closed ball b[x, r, t] with center x and radius r ∈ (0, 1) is defined by b[x, r, t] = {y ∈ x; µ(x, y, t) < r} . lemma 3.3. every closed ball in a revised fuzzy cone metric space (x, µ, ⊕) is a closed set. proof. let y ∈ b[x, r, t]. since x is first countable, there exits a sequence {yn} in b[x, r, t] converging to y. therefore µ (yn, y, t) converges to 0 for all t ≫ θ. for a given ϵ ≫ 0, we have, µ(x, y, t + ϵ) ≤ µ (x, yn, t) ⊕ µ (yn, y, ϵ) hence, µ(x, y, t + ϵ) ≤ µ (x, yn, t) ⊕ µ (yn, y, ϵ) ≤ 0 ⊕ 0 = 0. 45 some topological properties of revised fuzzy cone metric spaces (if µ (x, yn, t) is bounded, then the sequence yn has a subsequence, which we again denote by yn, for which µ (x, yn, t) exists.) in particular for n ∈ n, take ϵ = t n . then, µ ( x, y, t + t n ) < r. hence, µ(xyt) ≤ µ ( x, y, t + t n ) < r. thus y ∈ b[x, r, t]. therefore b[x, r, t] is a closed set. definition 3.4. a revised fuzzy cone metric space (x, µ, ⊕) is called precompact if for each r, with 0 < r < 1, and each t ≫ θ, there is a finite subset a of x, such that x = ⋃ a∈a b(a, r, t). in this case, we say that µ is a precompact revised fuzzy cone metric on x. lemma 3.5. a revised fuzzy cone metric space is precompact if and only if every sequence has a cauchy subsequence. proof. suppose that (x, µ, ⊕) is a precompact revised fuzzy cone metric space. let xn be a sequence in x. for each m ∈ n there is a finite subset am of x such that x = ⋃ a∈a b ( a, a m , t0 m∥t0∥ ) where t0, t ≫ θ is a constant. hence, for m = 1, there exists an a1 ∈ a1 and a subsequence x1(n) of xn such that x1(n) ∈ b ( a1, 1, t0 m∥t0∥ ) for every n ∈ n. similarly, there exist an a2 ∈ a2 and a subsequence { x2(n) } of x1(n) such that x2(n) ∈ b ( a2, 1 2 , t0 m∥t0∥ ) for every n ∈ n. by continuing this process, we get that for m ∈ n, m > 1, there is an am ∈ am and a subsequence { xm(n) } of xm−1(n) such that xm(n) ∈ b ( am, 1 m , t0 m∥t0∥ ) for every n ∈ n. now, consider the subsequence xn(n) of xn. given r with 0 < r < 1 and t ≫ θ there is an n0 ∈ n such that 1n0 ⊕ 1 n0 < r and 2t0 n0∥t0∥ ≪ t. then, for every km ≥ n0, we have µ ( xk(k), xm(m), t ) ≤ µ ( xk(k), xm(m), t0 n0 ∥t0∥ ) ≤ µ ( xk(k), an0, t0 n0 ∥t0∥ ) ⊕ µ ( an0, xm(m), t0 n0 ∥t0∥ ) ≤ 1 n0 ⊕ 1 n0 < r hence ( xn(n) ) is a cauchy sequence in (x, µ, ⊕). conversely, suppose that (x, µ, ⊕) is a nonprecompact revised fuzzy cone metric space. then there exist an r with 0 < r < 1 and t ≫ θ such that for each finite subset a of x, we have x ̸= ⋃ a∈a b(a, r, t) fix x1 ∈ x. there is x2 ∈ x − b(x1rt). moreover, there is an x3 ∈ x − 2⋃ k=1 b(xk, r, t). by continuing this process, we construct a sequence xn of distinct points in x such that xn+1 /∈ x − n⋃ k=1 b(xk, r, t) for every n ∈ n. therefore xn has no cauchy subsequence. this completes the proof. 46 a. muraliraj and r. thangathamizh lemma 3.6. let (x, µ, ⊕) be a revised fuzzy cone metric space. if a cauchy sequence clusters around a point x ∈ x, then the sequence converges to x. proof. let xn be a cauchy sequence in (x, µ, ⊕) having a cluster point x ∈ x. then, there is a subsequence { xk(n) } of xn that converges to x with respect to τfc. thus, given r with 0 < r < 1 and t ≫ θ, there is an n0 ∈ n such that for each n ≥ n0, µ ( x, xk(n), t 2 ) < s where s > 0 satisfies s ⊕ s < r. on the other hand, there is n0 ≥ k(n0) such that for each nm ≥ n1, we have µ ( xn, xm, t 2 ) < s. therefore, for each n ≥ n1, we have µ (x, xn, t) ≤ µ ( x, xk(n), t 2 ) ⊕ µ ( xk(n), x, t 2 ) ≤ s ⊕ s < r. we conclude that the cauchy sequence xn converges to x. proposition 3.7. let (x1, µ1, ⊕) and (x2, µ2, ⊕) be revised fuzzy cone metric spaces. for (x1, x2) , (y1, y2) ∈ x1, x2, let µ ((x1, x2) , (y1, y2) , t) = µ1 (x1, y1, t)⊕ µ2 (x2, y2, t), then µ is a revised fuzzy cone metric on x1 × x2. proof. rfcm 1: since µ1 (x1, y1, t) < 1 and µ2 (x2, y2, t) < 1, this implies that µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t) < 1. therefore, µ ((x1, x2) , (y1, y2) , t) < 1. rfcm 2: suppose that for all t ≫ θ, (x1, y1, t) = (x2, y2, t). this implies that x1 = y1 and x2 = y2 for all t ≫ θ. hence, µ1 (x1, y1, t) = 0 and µ2 (x2, y2, t) = 0. it follows that, µ ((x1, x2) , (y1, y2) , t) = 0. conversely, suppose that µ ((x1, x2) , (y1, y2) , t) = 0. this implies that µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t) = 0. since, 0 < µ1 (x1, y1, t) < 1 and 0 < µ2 (x2, y2, t) < 1. it follows that, µ1 (x1, y1, t) = 0 and µ2 (x2, y2, t) = 0. thus x1 = y1 and x2 = y2. therefore (x1, x2) = (y1, y2). rfcm 3: to prove that µ ((x1, x2) , (y1, y2) , t) = µ ((y1, y2) , (x1, x2) , t) we observe that µ1 (x1, y1, t) = µ1 (y1, x1, t) and µ2 (x2, y2, t) = µ2 (y2, x2, t). it follows that for all (x1, x2) (y1, y2) ∈ x1 × x2 and t ≫ θ, µ ((x1, x2) , (y1, y2) , t) = µ ((y1, y2) , (x1, x2) , t) rfcm 4: since (x1, µ1, ⊕) and (x2, µ2, ⊕) are revised fuzzy cone metric spaces, we have that, µ1 (x1, z1, t + s) ≤ µ1 (x1, y1, t) ⊕ µ1 (y1, z1, s) and µ2 (x2, z2, t + s) ≤ µ2 (x2, y2, t) ⊕ µ2 (y2, z2, s) , for all (x1, x2) (y1, y2) (z1, z2) ∈ x1 × x2 and t, s ≫ θ. therefore, µ ((x1, x2) , (z1, z2) , t + s) = µ1 (x1, z1, t + s) ⊕ µ2 (x2, z2, t + s) ≤ µ1 (x1, y1, t) ⊕ µ1 (y1, z1, s) ⊕ µ2 (x2, y2, t) ⊕ µ2 (y2, z2, s) ≤ µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t) ⊕ µ1 (y1, z1, s) ⊕ µ2 (y2, z2, s) ≤ µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t) ⊕ µ1 (y1, z1, s) ⊕ µ2 (y2, z2, s) ≤ µ ((x1, x2) , (y1, y2) , t) ⊕ µ ((y1, y2) , (z1, z2) , t) 47 some topological properties of revised fuzzy cone metric spaces rfcm 5: note that µ1 (x1, y1, t) and µ2 (x2, y2, t) are continuous with respect to t and ⊕ is continuous too. it follows that, µ ((x1, x2) , (y1, y2) , t) = µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t) is also continuous. proposition 3.8. let (x1, µ1, ⊕) and (x2, µ2, ⊕) be revised fuzzy cone metric spaces. we define, µ ((x1, x2) , (y1, y2) , t) = µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t). then µ is a complete revised fuzzy cone metric on x1 × x2 if and only if (x1, µ1, ⊕) and (x2, µ2, ⊕) are complete. corollary 3.9. every separable revised fuzzy cone metric space is second countable. proof. let (x, µ, ⊕) be the given separable revised fuzzy cone metric space. let a = {an : n ∈ n} be a countable dense subset of x. consider b = {( aj, 1 k , t1 k ∥t1∥ ) : j, k ∈ n } where t1 ≫ θ is constant. then b is countable. we claim that b is a base for the family of all open sets in x. let g be an open set in x. let x ∈ g then there exists r with 0 < r < 1 and t ≫ θ such that b(x, rt) ⊂ g. since r ∈ (0, 1), we can find an s ∈ (0, 1) such that s⊕s < r. choose m ∈ n such that 1 m < s and t1 m∥t1∥ ≪ t 2 . since a is dense in x, there exists an aj ∈ a such that aj ∈ b ( x, 1 m , t1 m∥t1∥ ) . now if y ∈ b ( aj, 1 m , t1 m∥t1∥ ) , then µ(x, y, t) ≤ µ ( x, aj, t 2 ) ⊕ µ ( y, aj, t 2 ) ≤ µ ( x, aj, t1 k ∥t1∥ ) ⊕ µ ( x, aj, t1 k ∥t1∥ ) ≤ 1 m ⊕ 1 m ≤ s ⊕ s < r < r. thus y ∈ b(x, r, t) and hence b is a basis. proposition 3.10. a subspace of a separable revised fuzzy cone metric space is separable. proof. let x be a separable revised fuzzy cone metric space and y a subspace of x. let a = {xn : n ∈ n} be a countable dense subset of x. for arbitrary but fixed nk ∈ n, if there are points x ∈ x such that µ ( xn, x, t1 k∥t1∥ ) < 1 k where t1 ≫ θ is constant, choose one of them and denote it by xnk . 48 a. muraliraj and r. thangathamizh let b = {xnk : n, k ∈ n} then b is countable. now we claim that y ⊂ b̄. let x ∈ y . given r with 0 < r < 1 and t ≫ θ we can find k ∈ n such that 1 k ⊕ 1 k < r and t1 k∥t1∥ ≪ t 2 . since a is dense in x, there exists an m ∈ n such that µ ( xm, y, t1 k∥t1∥ ) < 1 k . but by definition of b, there exists an xmk such that µ ( xmk, xm, t1 k∥t1∥ ) < 1 k . now µ ( xmk, y, t1 k ∥t1∥ ) ≤ µ ( xmk, xm, t 2 ) ⊕ µ ( xm, y, t 2 ) ≤ µ ( xmk, xm, t1 k ∥t1∥ ) ⊕ µ ( xm, y, t1 k ∥t1∥ ) ≤ 1 k ⊕ 1 k < r. thus y ⊂ b̄ and hence y is separable. corollary 3.11. let (x, µ, ⊕) be a revised fuzzy cone metric space. then (x, τfc) is hausdorff. corollary 3.12. let (x, µ, ⊕) be a revised fuzzy cone metric space. define τfc= a ⊂ x : x ∈ a if and only if there exist r ∈ (0, 1), and t ≫ θ such that b(x, r, tt) ⊂ a, then τfc is a topology on x. corollary 3.13. in a revised fuzzy cone metric space, every compact set is closed and rfc-bounded. 4 conclusion in this paper we proved a necessary and sufficient condition for a revised fuzzy cone metric space to be precompact. we also show that every separable revised fuzzy cone metric space is second countable and that a subspace of a separable revised fuzzy cone metric space is separable. acknowledgements we are grateful to the professor fabrizio maturo, chief editor of ratio mathematica and the reviewers for their interesting comments on this paper. 49 some topological properties of revised fuzzy cone metric spaces references k. ahmad and a. mesiarová-zemánková. chosing t-norms and t-conorms for fuzzy controllers. in fourth international conference on fuzzy systems and knowledge discovery, haikou, hainan, china, august 24–27 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and t. öner. on metric-type spaces based on extended t-conorms. mathematics, 8, 2020. doi: 10. 3390/math8071097. l. a. zadeh. fuzzy sets. inform. control, 8:338–353, 1965. 51 ratio mathematica 24 (2013), 11–30 issn: 1592-7415 on fuzzy gamma hypermodules r. ameri, r. sadeghi school of mathematics, statistic and computer sciences, college of science, university of tehran, iran department of mathematics, faculty of mathematical sciences, university of mazandaran, babolsar, iran rameri@ut.ac.ir, razi-sadeghi@yahoo.com abstract let r be a γ-hyperring and m be an γ-hypermodule over r. we introduce and study fuzzy rγ-hypermodules. also, we associate a γhypermodule to every fuzzy γ-hypermodule and investigate its basic properties. key words: γ-hyperring, γ-hypermodule, fundamental relation, fuzzy γ-hypermodule. msc2010: 20n20. 1 introduction hyperstructure theory was born in 1934 when marty [13] defined hypergroups, began to analysis their properties and applied them to groups. algebraic hyperstructures are a suitable generalization of classical algebraic structures. zadeh [18] introduced the notion of a fuzzy subset of a non-empty set x, as a function from x to [0, 1]. rosenfeld [15] defined the concept of fuzzy group. since then many papers have been published in the field of fuzzy algebra. in [16], sen, ameri and chowdhury introduced the notions of fuzzy hypersemigroups and obtained a characterization of them. then in [10], leoreanu-fotea and davvaz introduced and analyzed the fuzzy hyperring notion and in [11], leoreanu-fotea introduced the fuzzy hypermodule notion and obtained a connection between hypermodules and fuzzy hypermodules (for more information about fuzzy hypersrtuctures see [1]-[6]). the notion r. ameri, r. sadeghi of a γ-ring was introduced by n. nobusawa in [14]. recently, w.e. barnes [7], j. luh [12], w.e. coppage studied the structure of γ-rings and obtained various generalization analogous of corresponding parts in ring theory. in [3] ameri, sadeghi introduced the notion of γ-module over a γ-ring. now in this paper we introduced and study fuzzy γ-hypermodules as generalization of γ-hypermodule as well as fuzzy modules. the paper has been prepared in 5 sections. in section 2, we introduce some definitions and results of γ-hypermodules and fuzzy sets which we need to developing our paper. in section 3, we introduced and study fuzzy γ-hypermodules and obtain its basic results. in section 4, we study fundamental relation of fuzzy γ-hypermodules. 2 preliminaries in this section, we present some definitions which need to developing our paper. as it is well known a hypergroupoid is a set together with a function ◦ : h × h −→ p?(h), which is called a hyperoperation, where p?(h) denotes the set of all nonempty subsets of h. a hypergroupoid (h,◦), which is associative, that is x ◦ (y ◦ z) = (x ◦ y) ◦ z for all x,y,z ∈ h is called a semihypergroup. a hypergroup is a semihypergroup such that for all x ∈ h we have x ◦ h = h = h ◦ x (called the reproduction axiom). we say that a hypergroup h is canonical hypergroup if it is commutative, it has a scalar identity, every element has a unique inverse and it is reversible (for more details of hypergroups see [9]). definition 2.1. the triple (r, +, .) is a hyperring (in the sense of krasner) if the following hold: (i) (r, +) is a commutative hypergroup; (ii) (r,.) is a semihypergroup; (iii) the hyperoperation ”.” is distributive over the hyperoperation ”+”, which means that for all r,s,t of r we have: r.(s + t) = r.s + r.t and (r + s).t = r.t + s.t ( for more about hyperrings see [9] and [11]). definition 2.2. let (r,],◦) be a hyperring. a nonempty set m, endowed with two hyperoperations ⊕,� is called a left hypermodule over (r,],◦) if the following conditions hold: (1) (m,⊕) is a commutative hypergroup; (2) � : r×m −→ p∗(m) is such that for all a,b ∈ m and r,s ∈ r we have (i) r � (a⊕ b) = (r �a) ⊕ (r � b); (ii) (r ]s) �a = (r �a) ⊕ (s�a); (iii) (r ◦s) �a = r � (s�a). for more details about hypermodules see [8], [9], [?] and [18]). 12 on fuzzy gamma hypermodules definition 2.3. ([7]) let r and γ be additive abelian groups. we say that r is a γ − ring if there exists a mapping · : r× γ ×r −→ r (r,γ,r′) 7−→ r.γ.r′ (= rγr′) such that for every a,b,c ∈ r and α,β ∈ γ, the following conditions hold: (i) (a + b)αc = aαc + bαc; a(α + β)c = aαc + aβc; aα(b + c) = aαb + aαc; (ii) (aαb)βc = aα(bβc). definition 2.4. let r be a γ-ring. a (left )gamma module over r is an additive abelian group m together with a mapping . : r× γ ×m −→ m ( the image of (r,γ,m) being denoted by rγm), such that for all m,m1,m2 ∈ m and γ,γ1,γ2 ∈ γ and r,r1,r2 ∈ r the following conditions are satisfied: (gm1) r.γ.(m1 + m2) = r.γ.m1 + r.γ.m2; (gm2) (r1 + r2).γ.m = r1.γ.m + r2.γ.m; (gm3) r.(γ1 + γ2).m = r.γ1.m + r.γ2.m; (gm4) r1.γ1.(r2.γ2.m) = (r1.γ1.r2).γ2.m. a right gamma module over r is defined in analogous manner. in this case we say that m is a left(or right) rγ-module (for more details about gamma modules see [2]). let (h,◦) be a hypergroupoid. if {a,b} ⊆ p∗(h) and ρ is an equivalence relation on h, then we denote aρ̄b if ∀a ∈ a, ∃b ∈ b : aρb, and, ∀b ∈ b, ∃a ∈ a : aρb. we denote a ¯̄ρ b if ∀a ∈ a, ∀b ∈ b we have aρb. an equivalence relation ρ on h is called regular (strongly regular ) if for all a,a′,b,b′ of h. the following implication holds: aρb,a′ρb′ =⇒ (a◦a′)ρ̄(b◦ b′) (aρb,a′ρb′ =⇒ (a◦a′) ¯̄ρ(b◦ b′)). theorem 2.1. ([17]) let (m, +, .) be a hypermodule over a hyperring r, let δ be an equivalence relation on m and let ρ be an strongly regular relation on r. the following statements hold: (1) if δ is strongly regular on m and ∀x,y ∈ m and ∀r ∈ r the hyperoperations: 13 r. ameri, r. sadeghi δ(x) ⊕ δ(y) = {δ(z) | z ∈ x + y} and ρ(r) � δ(x) = {δ(z) | z ∈ r.x}, is define a module structure on m/δ over r/ρ; (2) if (m/δ,⊕,�) is a module over r/ρ, then δ is strongly regular on m. the relation δ∗ is the smallest strongly regular relation on the hypermodule (m, +, .) such that (m/δ,⊕,�) the quotient structure (m/δ,⊕,�) is a module over the ring r/ρ, and it is called the fundamental relation over hypermodule m. hence, δ∗ is the smallest equivalence relation on m, such that m/δ∗ is a module over the ring r/ρ∗, where ρ∗ is fundamental relation on r. if we denote by u the set of all expressions consisting of finite hyperoperations either on r and m or the external hyperoperation applied on finite sets of elements of r and m, then we have xδy ⇐⇒∃u ∈u, such that {x,y}⊂ u. δ∗ is the transitive closure of δ. in the fundamental module (m/δ∗,⊕,�) over r/ρ∗, the hyperoperations ⊕ and � are defined as follows: ∀x,y ∈ m and ∀z ∈ δ∗(x) ⊕ δ∗(y), we have δ∗(x) � δ∗(y) = δ∗(z); ∀r ∈ r, ∀x ∈ m and ∀z ∈ δ∗(r).δ∗(x), we have ρ∗(r) � δ∗(x) = δ∗(z), (for more details about the fundamental relation on hyperstructures see [8] and [9]). definition 2.5. a multivalued system (r, +, .) is a γ-hyperring if the following hold: (i) (r, +) and γ are canonical hypergroups; (ii) (r,.) is semihypergroup. (iii) (.) is distributive with respect to (+), i.e., for all x,y,z in r we have x.(y + z) = (x.y) + (x.z) and (x + y).z = (x.z) + (y + z). definition 2.6. let (r,],◦) be a γ-hyperring and (γ,∗) be a canonical hypergroup. we say that (m, +, .) is a left γ − hypermodule over r, if (m, +) be a canonical hypergroup and there exists a mapping · : r× γ ×m −→ p?(m) (r,γ,m) 7−→ r ·γ ·m such that for every r,s ∈ r and α,β ∈ γ and a,b ∈ m, the following conditions are satisfied: (ghm1) (i) (r ]s).α.a = r.α.a + s.α.a; (ii) r.(α∗β).a = r.α.a + r.β.a; (iii) r.α.(a + b) = r.α.a + r.α.b; (ghm2) (r ◦α◦s).β.a = r.α.(s.β.a). 14 on fuzzy gamma hypermodules a right γ-hypermodule of r is defined in a similar way. in this case we say that m is a rγ-hypermodule. 3 fuzzy gamma subhypermodules in the sequel r is a γ-hyperring and all gamma hypermodules are considered over r. in [16] m.k. sen, r. ameri, g. chowdhury introduced the notion of fuzzy semihypergroups, in [10] v. leoreanu-fotea, b. davvaz study fuzzy hyperrings and v. leoreanu-fotea in [11] studied fuzzy hypermodules. now in this section we follows these and introduce and studied fuzzy gamma hypermodules. let s and γ be two nonempty sets. f∗(s) denotes the set h of all nonzero fuzzy subset of s. a fuzzy γ − hyperoperation on s is a map ◦ : s × γ × s −→ f∗(s), which associates a nonzero subset a ◦ γ ◦ b for all a,b ∈ s and γ ∈ γ. (s,◦) is called a fuzzy γ − hypergroupoid . a fuzzy γ-hypergroupoid (s,◦) is called a fuzzy γ-hypersemigroup if for all a,b,c ∈ s and α,β ∈ γ, we have a ◦ α ◦ (b ◦ β ◦ c) = (a ◦ α ◦ b) ◦ β ◦ c, where for any µ ∈ f∗(s), we have (a◦γ ◦µ)(r) = ∨ t∈s((a◦γ ◦ t)(r) ∧µ(t)) and (µ◦γ ◦a)(r) = ∨ t∈s(µ(t) ∧ (t◦γ ◦a)(r)) for all r ∈ s,γ ∈ γ. if a is a nonempty subset of s and x ∈ s, then for all r ∈ s,γ ∈ γ we have: (x◦γ ◦a)(r) = ∨ a∈a (x◦γ ◦a)(r), and (a◦γ ◦x)(r) = ∨ a∈a (a◦γ ◦x)(r). a fuzzy γ-hypersemigroup (s,◦) is called a fuzzy γ-hypergroup if for all a ∈ s and γ ∈ γ, we have a◦γ◦s = s◦γ◦a = χs. we say that an element e of (s,◦) is identity (resp. scalar identity) if for all s,r ∈ s,γ ∈ γ, we have (e◦γ ◦ r)(r) > 0, and (r ◦γ ◦e)(r) > 0, ((e◦γ ◦ r)(s) > 0, and (r ◦γ ◦e)(s) > 0 itfollowsr = s). let (s,◦) be a fuzzy hypergroup, endowed with at least an identity. an element a′ ∈ s is called an inverse of a ∈ s if there is an identity e ∈ s, such that 15 r. ameri, r. sadeghi (a◦a′)(e) > 0, and (a′ ◦a)(e) > 0. definition 3.1. a fuzzy hypergroup s is regular if it has at least one identity and each element has at least one inverse. a regular fuzzy hypergroup (s,◦) is called reversible if for any x,y,a ∈ s, it satisfies the following conditions: (1) if (a◦x)(y) > 0, then there exists an inverse a1 of a, such that (a1◦y)(x) > 0; (2) if (x◦a)(y) > 0, then there exists an inverse a2 of a, such that (y◦a2)(x) > 0. definition 3.2. we say that a fuzzy hypergroup s is a fuzzy canonical if (1) it is commutative; (2) it has an scalar identity; (3) every element has a unique inverse; (4) it is reversible. let µ and ν be two nonzero fuzzy subsets of a fuzzy γ-hypergroupoid (s,◦). we define (µ◦γ ◦ν)(t) = ∨ p,q∈s (µ(p) ∧ (p◦γ ◦ q)(t) ∧ν(q),∀t ∈ s,γ ∈ γ. in the following we introduce and study fuzzy gamma hyperrings . definition 3.3. let r, γ be two nonempty sets and �,� be two fuzzy hyperoperations on r and ⊗ be a fuzzy hyperoperation on γ. let (r,�) and (γ,⊗) be two canonical fuzzy hypergroups. r is called a fuzzy γ-hyperring if there exists the mapping: � : r× γ ×r −→ f∗(r) (r,γ,s) 7−→ r �γ �s, such that for all r,s,t ∈ r,α,β ∈ γ, the following conditions are satisfied: (i) r �α� (s� t) = (r �α�s) � (r �α� t); (ii) r � (α⊗β) �s = (r �α�s) � (r �β �s); (iii) (r �s) �α� t = (r �α� t) � (s�α� t); (iv) r �α� (s�β � t) = (r �α�s) �β � t. definition 3.4. let (γ,⊗) be a fuzzy canonical hypergroups. let (r,�,�) be a fuzzy γ-hyperring. a nonempty set m, endowed with two fuzzy γhyperoperation ⊕,� is called a left fuzzy γ-hypermodule over (r,�.�) if the following conditions hold: 16 on fuzzy gamma hypermodules (1) (m,⊕) is a canonical fuzzy γ-hypergroup; (2) � : r × γ × m −→ f∗(m) is such that for all a,b ∈ m,r,s ∈ r and α,β ∈ γ we have (i) r �α� (a⊕ b) = (r �α�a) ⊕ (r �α� b); (ii) (r �s) �α�a = (r �α�a) ⊕ (s�α�a); (iii) r � (α⊗β) �a = (r �α�a) ⊕ (r �β �a); (iv) r �α� (s�β �a) = (r ·α ·s) �β �a. if both (r,�), (γ,⊗) and (m,⊕) have scaler identities, denoted by 0r, 0γ and 0m , then the fuzzy γ-hypermodule (m,⊕,�) also satisfies the condition: 0r �γ �a = χ0m , r � 0γ �a = χ0γ, r �γ � 0m = χ0m , for all r ∈ r,γ ∈ γ,a ∈ a. moreover, if (r,�) has an identity, say 1, then the fuzzy γ-hypermodule (m,⊕,�) is called unitary if it satisfies the condition: for all a of m, we have 1 �γ �a = χa. clearly, any fuzzy γ-hyperring is a fuzzy γ-hypermodule over itself. proposition 3.5. let (m, +, .) be a module over a ring (r,],◦) and γ = r. we define the following fuzzy γ-hyperoperations: for a,b of m, a⊕ b = χ{a,b}, for all a of m and r ∈ r,γ ∈ γ, r �γ �a = χ{r.γ.a}, for all r,s of r, r �s = χ{r,s} and r �γ �s = χ{r◦γ◦s}. then (m,⊕,�) is a fuzzy γ-hypermodule over the fuzzy γ-hyperring (r,�,�). note that the last theorem is satisfied, when m is a γ-module over a γ-ring r, such that γ 6= r. proposition 3.6. let (r,◦) and (s,•) be two fuzzy γ-hyperrings. let (m,⊕,�) be a left fuzzy γ-hypermodule over r and a right fuzzy γ-hypermodule over s. then a = { ( r m 0 s ) | r ∈ r,s ∈ s,m ∈ m} is a fuzzy γ-hyperring and fuzzy γ-hypermodule over a, under the mappings ? : a× γ ×a −→ f∗(a) ( ( r m 0 s ) ,γ, ( r1 m1 0 s1 ) ) 7−→( r ◦γ ◦ r1 r �γ �m1 ⊕m�γ �s1 0 s•γ •s1 ) . 17 r. ameri, r. sadeghi such that( r ◦γ ◦ r1 r �γ �m1 ⊕m�γ �s1 0 s•γ •s1 )( r2 m2 0 s2 ) =( (r ◦γ ◦ r1)(r2) (r �γ �m1 ⊕m�γ �s1)(m2) 0 (s•γ •s1)(s2) ) ={ 1, r2,m2,s2 6= 0 0, otherwise. . proof. straightforward.2 example 3.7. let r be a γ-ring and (m, +, .) a γ-module. consider the mapping α : m −→ r. then m is an fuzzy γ-hypermodule over m, under the following operations: m⊕n = m+n. and ◦ : m×γ×m −→ f∗(m)(m,γ,n) 7−→ m◦γ◦n = χα(m).γ.n, for all m,n ∈ m,γ ∈ γ. proposition 3.8. let (m, +, .) be a γ-module over γ-ring r and ν be a nonzero fuzzy γ-semigroup on m. let µ and ρ be two nonzero fuzzy γ-semigroups on r. for r ∈ r, a,b ∈ m and γ ∈ γ, define a fuzzy γhyperoperation � on m by (r �γ �a)(t) = { µ(r) ∧ρ(γ) ∧ν(a), if t = r.γ.a 0 , otherwise. also, a ⊕ b = χ{a+b}. it is easy to verify that (m,⊕,�) is a fuzzy γhypermodule. let s, γ be nonempty sets, and s endowed with a fuzzy γ-hyperoperation ◦. for all a,b ∈ s,γ ∈ γ and p ∈ [0, 1] consider the p-cuts: (a◦γ ◦ b)p = {t ∈ s : (a◦γ ◦ b)(t) ≥ p} of a◦γ ◦ b, where p ∈ [0, 1]. for all p ∈ [0, 1], we define the following crisp γ-hyperoperation on s: a◦p γ ◦p b = (a◦γ ◦ b)p. example 3.9. let r = γ = z and m = zn for n ∈ n. define following fuzzy γ-hyperoperations for all a,b ∈ m,γ ∈ γ: a⊕ b = χ{a,b},∀a ∈ m,∀r ∈ r,γ ∈ γ, 18 on fuzzy gamma hypermodules r �γ �a = χ{rγa}, ∀r,s ∈ r,∀γ ∈ γ, r.γ.s = χ{rγs} and r + s = χ{r,s}, for all α,β ∈ γ, and α�β = χ{α,β}, such that x is denote a typical element in zn. then it is easy to verify that (m,⊕,�) is a fuzzy γ-hypermodule over fuzzy γ-hyperring r and canonical fuzzy hypergroup (γ,�). proposition 3.10. let (m,◦) be a fuzzy γ-hyperoperation. for all a,b,c,u ∈ m and α,β ∈ γ and for all p ∈ [0, 1] the following equivalence holds: (a◦α◦ (b◦β ◦ c)) ≥ p ⇐⇒ u ∈ a◦p α◦p (b◦p β ◦p c). ((a◦α◦ b) ◦β ◦ c) ≥ p ⇐⇒ u ∈ (a◦p α◦p b) ◦p β ◦p c.) proof. clearly, (a◦α◦ (b◦β ◦ c))(u) = ∨ t∈m (a◦α◦ t)(u) ∧ (b◦β ◦ c)(t) ≥ p, if and only if there exists t0 ∈ m, such that (a ◦ α ◦ t0)(u) ≥ p and (b◦β◦c)(t0) ≥ p, which means that u ∈ a◦pα◦pt0, t0 ∈ b◦pβ◦pc. therefore, u ∈ a◦p α◦p (b◦p β ◦p c).2 proposition 3.11. let (m,⊕,�) be a fuzzy γ-hypermodule over a fuzzy γ-hyperring (r,�,�). then for all a ∈ m,r ∈ r,γ ∈ γ, conditions are equivalence: (1) a⊕m = χm ⇐⇒∀p ∈ [0, 1], a⊕p m = m; (2) r �γ �m = χm ⇐⇒∀p ∈ [0, 1], r �p γ �p m = m. proof. we only proof (2). let r � γ � m = χm . then for all t ∈ m and p ∈ [0, 1], we have ∨ u∈m (r�γ �u)(t) = 1 ≥ p, whence there exists m ∈ m, such that (r � γ � m)(t) ≥ p, which means that t ∈ r �p γ �p m. hence, ∀p ∈ [0, 1], r�pγ�pm = m. conversely, for p = 1 we have r�1γ�1m = m, whence for all t ∈ m, there exists u ∈ m, such that t ∈ r �1 γ �1 u, which means that (r �γ �u)(t) = 1. in other words, r �γ �m = χm .2 proposition 3.12. the structure (m,⊕,�) is a fuzzy γ-hypermodule over a fuzzy γ-hyperring (r,�,�) if and only if ∀p ∈ [0, 1], (m,⊕p,�p) is a γ-hypermodule over the hyperring (r,�p,�p). proof. it is straightforward.2 19 r. ameri, r. sadeghi consider (m,⊕,�) as a fuzzy γ-hypermodule over a fuzzy γ-hyperring (r,�,�) and canonical fuzzy hypergroup (γ,⊗). now we follow [8], and define a new types of γ-hyperoperations on m,r, γ, as follows: ∀a,b ∈ m, a + b = {x ∈ m|(a⊕ b)(x) > 0}, ∀r,s ∈ r, r ]s = {t ∈ r | (r �s)(t) > 0},forallα,β ∈ γ, α∗β = {γ ∈ γ | (α∗β)(γ) > 0}, ∀a ∈ m, ∀r ∈ r,∀γ ∈ γ, r.γ.a = {b ∈ m | (r �γ �a)(b) > 0}, ∀r,s ∈ r, ∀γ ∈ γ, r ◦γ ◦s = {t ∈ r | (r �γ �s)(t) > 0}. proposition 3.13. if (m,⊕,�) is a fuzzy γ-hypermodule over a fuzzy γhyperring (r,�,�) and canonical fuzzy hypergroup (γ,⊗), then (m, +, .) is a γ-hypermodule over the γ-hyperring (r,],◦) and canonical hypergroup (γ,?). proof. by [10], it is obtained that (r,]), (γ,∗) and (m, +) are canonical hypergroups. it is sufficient to verify (m,.) is a γ-hypermodule. we consider the following cases: case: (i) (r ]s).γ.a = (r.γ.a) + (s.γ.a), for all r,s ∈ r,γ ∈ γ,a ∈ m. suppose that x ∈ (r ] s).γ.a = ⋃ y∈r]s y �γ �a. then (y �γ �a)(x) > 0 and (r � s)(y) > 0, for some y ∈ r ] s, and hence ∨p∈m ((r � s)(p) ∧ (p � γ � a)(x) > 0. thus ((r � s) � γ � a)(x) > 0, which implies that ((r � γ � a) ⊕ (s � γ � a))(x) > 0. thus there exist z,t ∈ m, such that (z⊕t)(x) > 0, (r�γ�a)(z) > 0 and (s�γ�a)(t) > 0 i.e., x ∈ z+t,z ∈ r.γ.a and t ∈ s.γ.a and hence x ∈ (r.γ.a) + (s.γ.a). therefore, (r ] s).γ.a ⊆ (r.γ.a) + (s.γ.a). similarly, we can show that (r.γ.a) + (s.γ.a)t ⊆ (r]s).γ.a. therefore, (r ] s).γ.a = (r.γ.a) + (s.γ.a). the other conditions are verified similarly and omitted. 2 20 on fuzzy gamma hypermodules on the other hands, if (m, +, .) is a γ-hypermodule over a γ-hyperring (r,],◦), then we define the following fuzzy γ-hyperoperations: a⊕ b = χ{a+b},∀a,b ∈ m,r �s = χ{r]s},∀r,s ∈ r,γ ∈ γ,r �γ �a = χ{r.γ.a},∀a ∈ m,r ∈ r,r �γ �s = χ{r◦γ◦s},∀r,s ∈ r,∀γ ∈ γ,β = χ{α∗β}∀α,β ∈ γ,α⊗β. the next result is immediately follows from above discussion and [14]. proposition 3.14. for every hypergroup (m, +), there is an associated fuzzy hypergroup. proposition 3.15. let (m, +, .) be a γ-hypermodule over a γ-hyperring. let (r,],◦) be a canonical hypergroup (γ,?). then (m,⊕,�) is a fuzzy γhypermodule over a fuzzy γ-hyperring (r,�,�) and canonical fuzzy hypergroup (γ,⊗), where the fuzzy hyperoperations ⊕,�,�,� and ⊗ are defined above. proof. by proposition 3.14, (m,⊕) is a commutative fuzzy γ-hypergroup. we show that (m,⊕) is canonical. since (m, +) is canonical γ-hypergroup, then there exists e ∈ m,∀a ∈ m, a = e + a = a + e =⇒ (e ⊕ a)(a) = χ{e+a}(a) > 0, (a ⊕ e)(a) = χ{e+a}(a) > 0 and because for all a ∈ m there exists b ∈ m, such that e ∈ a + b∩ b + a, b) is the inverse of a with respect to +). then (a⊕ b)(e) = χ{a+b}(e) = χ{b+a}(e) = (b⊕a)(e) > 0. let (a⊕x)(y) = χ{a+x}(y) > 0 =⇒ y ∈ a + x =⇒ ∃ b ( the inverse of a such that x ∈ b + y =⇒ (b⊕y)(x) = χ{b+y}(x) > 0. the other cases is can be proved in a similar way and omitted. then (m,⊕) is a canonical fuzzy γ-hypergroup. now, we show that (m,⊕,�) is a fuzzy γ-hypermodule. we investigate only the condition (iv) of definition 3.4. first , we show that that for all r,s ∈ r,α,β ∈ γ,a ∈ m, we have (r �α� (s�β �a)) = (r �α�s) �β �a, ∀t ∈ m. then (r �α� (s�β �a))(t) = ∨ p∈m , [(r �α�p)(t) ∧ (s�β �a)(p)] = ∨ p∈m [χr.α.p(t) ∧χs.βa(p)] = 21 r. ameri, r. sadeghi { 1, t ∈ r.α.(s.β.a) 0, otherwise = { 1, t ∈ (r.α.s).β.a 0, otherwise = ((r �α�s) �β �a)(t), for all t ∈ m. it is easy to verify that the other conditions of definition 3.4 can be obtained in a similar way.2 proposition 3.16. let m an rγ-module and µ be a fuzzy γ-module of m. then the set m will be a fuzzy γ-hypermodule. proof. let (γ,∗) be an abelian group and (m, +, .) be a γ-module over γ-ring (r,],◦). we define fuzzy γ-hyperoperations on m as follows: (a⊕ b)(t) = χ{a+b}, (r �γ �a)(t) = µ(r.γ.a− t), (α⊗β)(γ) = χ{α∗β}(γ) = χ{r]s}r �s)(z)(r �α�s)(z) = χ{r◦α◦s}(z), ∀a,b,t ∈ m,r,s,z ∈ r,α,β,γ ∈ γ. it is easy to verify that (m,⊕) is a canonical fuzzy hypergroup. now, we show (m,⊕,�) is a fuzzy γ-hypermodule with µ(0) = 1. (i) ((r �s) �γ �a)(t) = ∨p∈r(r �s)(p) ∧ (p�γ �a)(t) = ∨p∈rχr]s(p) ∧µ(p.γ.a− t) = µ((r ]s).γ.a− t) if p = r ]s. also, ((r �γ �a) ⊕ (s�γ �a))(t) = = ∨p,q∈m (r �γ �a)(p) ∧ (p⊕ q)(t) ∧ (s�γ �a)(q) = ∨p,q∈mµ(r.γ.a−p) ∧χ{p+q}(t) ∧µ(s.γ.a− q) = ∨p,q∈m,t=p+qµ(r.γ.a−p) ∧µ(s.γ.a− q) ≤ µ(r.γ.a−p + s.γ.a− q) = µ((r ]s).γ.a− (p + q)), on the other hands, if q = s.γ.a, p = t−s.γ.a., then ∨p,q∈m,t=p+qµ(r.γ.a−p) ∧µ(r.γ.a− q) ≥ ∨p∈mµ(r.γ.a−p) ≥ µ(r.γ.a− t + s.γ.a) = µ((r ]s).γ.q − t). (ii) (r � (α⊗β) �a)(t) = ∨γ∈γ[(r �γ �a)(t) ∧ (α⊗β)(γ)] = ∨µ(r.γ.a− t) ∧χ{α∗β}(γ) = µ(r.(α∗β).a− t). 22 on fuzzy gamma hypermodules also, ((r �α�a) ⊕ (r �β �a))(t) = = ∨p,q∈m [(r �α�a)(p) ∧ (p⊕ q)(t) ∧ (r �β �a)(q) = ∨p,q∈m [µ(r.α.a−p) ∧χ{p+q}(t) ∧µ(r.β.a− q)] = ∨t=p+q µ(r.α.a−p) ∧µ(r.βa− q) ≤ µ(r.αa−p + r.βa− q) = µ(r.(α∗β).a− (p + q)). on the other hands, suppose that q = r.β.a, then for p = t − r.β.a we have ∨t=p+qµ(r.α.a−p) ∧µ(r.βa− q) = ∨p∈mµ(r.αa−p) ≥ µ(r.αa− (t− rβa)) = µ(r.(α∗β).a− (p + q)), (iii) r �γ � (a⊕ b) = ∨p∈m (r �γ �p)(t) ∧ (a⊕ b)(p) = ∨p∈mµ(r.γ.p− t) ∧χ{a+b}(p) = µ(r.γ.(a + b) − t) and ((r �γ �a) ⊕ (r �γ � b))(t) = ∨p,q∈m (r �γ �a)(p) ∧ (p⊕ q)(t) ∧ (r �γ � b)(q) = ∨p,q∈mµ(r.γ.a−p) ∧χ{p+q}(t) ∧µ(r.γ.b− q) = ∨p,q∈m,t=p+qµ(r.γ.a−p) ∧µ(r.γ.b− q) ≤ µ(r.γ.a−p + r.γ.b− q) = µ(r.γ.(a + b) − t). on the other hands, for q = r.γ.b,p = t− r.γ.b. we have ∨p,q∈m,t=p+qµ(r.γ.a−p) ∧µ(r.γ.b− q) ≥ ∨p∈mµ(r.γ.a−p) ≥ µ(r.γ(a + b) − t). (iv) (r �α� (s�β �a))(t) = ∨p∈m (r �α�p)(t) ∧ (s�β �a)(p) = ∨p∈mµ((r.α.p) − t) ∧µ((s.β.a) −p) = µ(r.α.(s.β.a) − t), and ((r �α�s) �β �a)(t) = ∨p∈r(r �α�s)(p) ∧ (p�β �a)(t) = ∨p∈rχ{r◦α◦s}(p) ∧µ(p.β.a− t) = µ(r ◦α◦s · (β ·a) − t) if p = r ◦α◦s. 23 r. ameri, r. sadeghi 2 remark. let h = 〈h, (βi : i ∈ i)〉 be a fuzzy hyperalgebra. denote by f∗(h) the set of the nonzero fuzzy subsets of h. then h can be organized as a universal algebra under the following operations: βi(µ1, ...,µni )(t) = ∨ (x1,...,xni )∈h ni [(µ1(x1) ∧ ...µni (xni ) ∧ βi(x1, ...,xni )(t))], for every i ∈ i,µ1, ...,µni ∈ f ∗ (h) and t ∈ h. we denote this algebra by f∗(h). proposition 3.17. if (m,⊕,♦) is a fuzzy γ-hypermodule, then (f∗(m),∗,©) is a γ-module. proof. we define operations ∗,♦ on f∗(m) by µ∗ν = µ⊕ν, and r♦γ♦µ = r�γ�µ for all µ,ν ∈ f∗(m),r ∈ r,γ ∈ γ. it is easy to see that (f∗(m),∗) is a group. clearly, (f∗(m),⊕) is a semigroup. (i) identity: we must prove that there exists a ν ∈ f∗(m) such that ,µ∗ν = µ. we have (µ∗ν)(t) = (µ⊕ν)(t) = ∨p,q∈mµ(p) ∧ (p⊕ q)(t) ∧ν(q) = ∨p∈mµ(p) ∧ (p⊕e)(t) = µ(t) ⊕ if q = e; ν(q) = 1,p = t. thus it is enough we choose ν = χe. (ii) inverse: it must prove that for µ ∈ f∗(m), there exists a ν ∈ f∗(m), such that µ∗ν = χe. it is sufficient to consider ν = −µ, then we have (µ∗ν)(t) = (µ⊕ν)(t) = ∨p,q∈mµ(p) ∧ (p⊕ q)(t) ∧ (−µ)(q) = ∨p,q∈mµ(p) ∧ (p⊕ q)(t) ∧µ(−q) ≤ µ(p− (−q)) ∧ (p⊕ q)(t) ≤ (p⊕ q)(t) = χe(t) where, p is inverse of q. on the other hands, we have ∨p,q∈mµ(p) ∧ (p⊕ q)(t) ∧µ(−q) ≥ ∨p∈mµ(p) ∧ (p⊕−p)(t) ≥ (p⊕−p)(t) = χe(t). 24 on fuzzy gamma hypermodules other cases are easy and omitted. 2 definition 3.18. let (m,⊕,�) be a fuzzy γ-hypermodule over a fuzzy γ-hyperring (r,�,�). a nonempty subset n of m is called a subfuzzy γhypermodule if for all x,y ∈ n,r ∈ r and γ ∈ γ, the following conditions hold: (1) (x⊕y)(t) > 0 ⇒ t ∈ n; (2) x⊕n = χn ; (3) (r �γ �x)(t) > 0 ⇒ t ∈ n. proposition 3.19. (i) if (n,⊕,�) is a subfuzzy γ-hypermodule of (m,⊕,�) over a fuzzy γ-hyperring (r,�,�), then the associated γ-hypermodule (n, +, .) is a γ-hypersubmodule of (m, +, .) over (r,],◦); (ii) (n, +, .) is a γ-hypersubmodule of (m, +, .) over (r,],◦) if and only if (n,⊕,�) is a subfuzzy γ-hypermodule of (m,⊕,�) over (r,�,�). 4 fundamental relation of fuzzy γ-hypermodule in [14], fuzzy regular relations are introduced in the context of fuzzy hypersemigroups. in this section we extend this notion to fuzzy γ-hypermodules. let ρ be an equivalence relation on a fuzzy γ-hypersemigroup (m,◦) and µ,ν be two fuzzy subsets on m. we say that µρν if the following conditions hold: (1) if µ(a) > 0, then there exists b ∈ m, such that ν(b) > 0 and aρb and; (2) if ν(x) > 0, then there exists y ∈ m, such that µ(y) > 0 and xρy. an equivalence relation ρ on a fuzzy γ-hypersemigroup (m,◦) is called a fuzzy regular relation (or fuzzy hypercongruence) on (m,◦) if, for all a,b,c ∈ m,γ ∈ γ, the following implication holds: aρb =⇒ (a◦γ ◦ c) ρ (b◦γ ◦ c) and (c◦γ ◦a) ρ (c◦γ ◦ b). this condition is equivalent to aρa′,bρb′ ⇒ (a◦γ ◦ b)ρ(a′ ◦γ ◦ b′),∀a,b,a′,b′ ∈ m,γ ∈ γ. definition 4.1. an equivalence relation ρ on a fuzzy γ-hypermodule (m,⊕,�) over a fuzzy γ-hyperring (r,�,�) and a canonical fuzzy hypergroup (γ,⊗) is called a fuzzy regular relation on (m,⊕,�) if it is a fuzzy regular relation on (m,⊕) and for all x,y ∈ m,r ∈ r,γ ∈ γ, the following implication holds: xρy =⇒ (r �γ �x)ρ(r �γ �y). 25 r. ameri, r. sadeghi let (m,⊕,�) be a fuzzy γ-hypermodule over a fuzzy γ-hyperring (r,�,�) and a canonical fuzzy hypergroup (γ,⊗). suppose (m, +, .) is the associated γ-hypermodule over the γ-hyperring (r,],◦) and the canonical hypergroup (γ,∗). then we have the next result. theorem 4.2. an equivalence relation ρ is a fuzzy regular relation on (m,⊕,�) over a fuzzy γ-hyperring (r,�,�) and canonical fuzzy hypergroup (γ,⊗) if and only if ρ is a regular relation on (m, +, .) over the γ-hyperring (r,],◦) and canonical hypergroup (γ,∗). proof. letting xρy and x′ρy′, where x,x′,y,y′ ∈ m. we have (x⊕x′)ρ(y+y′) if and only if the following conditions hold: (x⊕x′)(u) > 0,⇒∃v ∈ m : (y ⊕y′)(v) > 0 and uρv, and (y ⊕y′)(t) > 0 ⇒ ∃w ∈ m : (x⊕x′)(w) > 0 and atρw. these are equivalent to: if u ∈ x + x′, then there exists v ∈ y + y′, such that uρv; if t ∈ y + y′, then there exists w ∈ x + x′, such that tρw; which mean that (x + x′)ρ̄(y + y′). hence ρ is fuzzy regular on (m,⊕) if and only if ρ is regular on (m, +). on the other hands, if xρy and r ∈ r,γ ∈ γ. we have (r�γ�x)ρ(r�γ�y) if and only if the next conditions hold: if (r �γ �x)(u) > 0, then there exists v ∈ m, such that (r �γ �y)(v) > 0 and uρv; if (r�γ �y)(t) > 0, then there exists w ∈ m, such that (r�γ �x)(w) > 0 and tρw. these are equivalent to: if u ∈ r.γ.x, then there exists v ∈ r.γ.y, such that uρv; if t ∈ r.γ.y, then there exists w ∈ r.γ.x, such that tρw; which means that (r.γ.x)ρ(r.γ.y).2 definition 4.3. an equivalence relation ρ on a fuzzy γ-hypersemigroup (m,◦) is called a fuzzy strongly regular relation on (m,◦) if, for all a,a′,b,b′ of m and for all γ ∈ γ, such that aρb and a′ρb′, the following condition holds: (a◦γ ◦ c)(x) > 0, (b◦γ ◦d)(y) > 0 ⇒ xρy, for all x,y ∈ m. note that if ρ is a fuzzy strongly relation on a fuzzy γhypersemigroup (m,◦), then it is a fuzzy regular on (m,◦). an equivalence relation ρ on a fuzzy γ-hyperring (r,�,�) is called a fuzzy strongly regular 26 on fuzzy gamma hypermodules relation on (r,�,�) if it is a fuzzy strongly regular relation both on (r,�) and on (r,�). definition 4.4. let ρ be a fuzzy strongly regular relation on a fuzzy γhyperring (r,�,�) and θ be a fuzzy strongly regular relation on a canonical fuzzy γ-hypergroup (γ,∗). an equivalence relation δ on a fuzzy γhypermodule (m,⊕,�) over a fuzzy γ-hyperring (r,�,�) and canonical fuzzy γ-hypergroup (γ,⊗) is called a fuzzy strongly regular relation on (m,⊕,�) if it is a fuzzy strongly regular relation on (m,⊕) and if xδy, rρs and αθβ, then the next condition holds: for all u ∈ m, such that (r � α � x)(u) > 0 and for all v ∈ m, such that (s�β �y)(v) > 0, we have uδv. theorem 4.5. an equivalence relation δ is a fuzzy strongly regular relation on (m,⊕,�) if and only if δ is a strongly regular relation on (m, +, .). proof. set xδy and x′δy′, where x,x′,y,y′ ∈ m and set rρs, where r,s ∈ r and αθβ, where α,β ∈ γ. the relation δ is strongly regular on (m,⊕,�) if and only if the following conditions are satisfied: ∀u ∈ m, such that (x⊕x′)(u) > 0 and ∀v ∈ m, such that (y ⊕ y′)(v) > 0, we have uδv; ∀t ∈ m, such that (r�α�x)(t) > 0 and ∀w ∈ m, such that (s�β�y)(w) > 0, we have tδw. these conditions are equivalent to the following ones: ∀u ∈ m, such that u ∈ x + x′ and ∀v ∈ m, such that v ∈ y + y′, we have uδv; ∀t ∈ m, such that t ∈ r.α.x and ∀w ∈ m, such that w ∈ s.β.y, we have tδw, which mean that (x + x′)¯̄δ(y + y′) and (r.α.x)¯̄δ(s.β.y). hence δ is strongly regular on (m,⊕,�) if and only if δ is strongly regular on (m, +, .). now, let δ be a fuzzy regular relation on a fuzzy γ-hypermodule (m,⊕,�) over a fuzzy γ-hyperring (r,�,�) and canonical fuzzy γ-hypergroup (γ,⊗) and ρ,θ be fuzzy strongly regular relations on the γ-hyperring (r,�,�) and canonical fuzzy γ-hypergroup. (γ,⊗). we consider the following γ-hyperoperations on the quotient set m/δ: x̄ ? ȳ = {z̄ | z ∈ x + y} = {z̄ | (x⊕y)(z) > 0}, r̄ } ᾱ} x̄ = {z̄ | z ∈ r.α.x} = {z̄ | (r �α�x)(z) > 0}. theorem 4.6. let (m,⊕,�) be a fuzzy γ-hypermodule over a fuzzy γhyperring (r,�,�) and canonical fuzzy hypergroup (γ,∗). let (m, +, .) be the associated γ-hypermodule over the corresponding γ-hypergroup (r,],◦) and canonical hypergroup (γ,∗). then we have: 27 r. ameri, r. sadeghi (i) the relation δ is a fuzzy regular relation on (m,⊕,�) if and only if (m/δ,?,}) is a γ-hypermodule over (r,],◦) and (γ,∗). (ii) the relation δ is a fuzzy strongly regular relation on (m,⊕,�) over (r,�,�) and (γ,⊗) if and only if (m/δ,?,}) is a γ-module over r/ρ and γ/θ. if we denote by u the set of all expressions consisting of finite fuzzy γhyperoperations either on r, γ and m or the external fuzzy γ-hyperoperations applied on finite sets of elements of r, γ and m, then we have x�y ⇐⇒∃u ∈ u : {x,y}⊂ u. now, we introduced fundamental relation on fuzzy γ-hypemodules. definition 4.7. an equivalence relation �∗ is called fundamental relation on a fuzzy γ-hypermodule (m,⊕,�) if �∗ is fundamental relation on the associated γ-hypermodule (m, +, .). hence, �∗ is fundamental relation on a fuzzy γ-hypermodule (m,⊕,�) if and only if �∗ is the smallest fuzzy strongly equivalence relation on (m,⊕,�). denote by uf the set of all expressions consisting of finite fuzzy γ-hyperoperations either on r, γ and m or the external fuzzy γ-hyperoperation applied on finite sets of elements of r, γ and m. we obtain x�y ⇐⇒∃ µf ∈ uf : {x,y}⊆ µfγ ⇐⇒ µfγ(x) > 0 and µfγ(y) > 0. the relation �∗ is the transitive closure of �. denote by ∑∗ ⊕ any finite fuzzy hypersum and by ∏∗ � any finite fuzzy γhyperproduct of the fuzzy γ-hypemodule (m,⊕,�). as above, we obtain that ( ∑∗ i⊕ ∏∗ j � aji)(p) > 0 if and only if p ∈ ∑∗ i⊕ ∏∗ j� aji. hence, {x,y} ⊂ ∑∗ i⊕ ∏∗ j � aji if and only if ( ∑∗ i⊕ ∏∗ j � aji)(x) > 0 and ( ∑∗ i⊕ ∏∗ j � aji)(y) > 0. therefore, we obtain x�y ⇐⇒ ∃µfγ ∈ uf such that µfγ(x) > 0 and µfγ(y) > 0. so, in order to obtain a fuzzy γ-module starting from a fuzzy γ-hypermodule, we consider first the relation �, then the transitive closure �∗ of � and finally the quotient structure (m/�∗,?,}) of the fuzzy γ-hypermodule (m,⊕,�). acknowledgements the first author partially has been supported by the ”research center in algebraic hyperstructures and fuzzy mathematics, university of mazandaran, babolsar, iran” and ”algebraic hyperstructure excellence, tarbiat modares university, tehran, iran”. 28 on fuzzy gamma hypermodules references [1] ameri, r., norouzi, m., prime and primary hyperideals in krasner (m,n)-ary hyperrings, european j. combin. 34 (2013) 379-390. 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[18] zadeh. l. a. fuzzy sets, inform control 8, 338-353(1965). 30 e:\uziv\sarka\clanky\rm_26\final\rm_26_6.dvi ratio mathematica 26 (2014), 95–112 issn:1592-7415 a multiobjective optimization model for optimal supplier selection in multiple sourcing environment m. k. mehlawat, s. kumar department of operational research, university of delhi, delhi, india mukesh0980@yahoo.com,santoshaor@gmail.com abstract supplier selection is an important concern of a firm’s competitiveness, more so in the context of the imperative of supply-chain management. in this paper, we use an approach to a multiobjective supplier selection problem in which the emphasis is on building supplier portfolios. the supplier evaluation and order allocation is based upon the criteria of expected unit price, expected score of quality and expected score of delivery. a fuzzy approach is proposed that relies on nonlinear s-shape membership functions to generate different efficient supplier portfolios. numerical experiments conducted on a data set of a multinational company are provided to demonstrate the applicability and efficiency of the proposed approach to real-world applications of supplier selection. key words: multiobjective optimization, fuzzy supplier selection, nonlinear optimization, membership functions. msc 2010: 90c30, 90c70. 1 introduction supplier selection or vendor selection is a multi-criteria decision making (mcdm) problem. one of the well known studies on supplier selection by dickson [10] discusses 23 important evaluation criteria for supplier selection. it has been pointed out that quality, delivery, and performance history are the three most important criteria. other important studies that highlights the 95 m. k. mehlawat, s. kumar importance of evaluation criteria for supplier selection includes the works of ghodsypour and o’brien [13], ho et al. [16], weber et al. [35]. many authors have discussed optimization models of supplier selection problem. parthiban et al. [26] developed an integrated model based on 10 criteria including quality, delivery, productivity, service, costs for the supplier selection problem. punniyamoorthy et al. [27] applied 10 criteria for supplier evaluation including quality, technical capability, financial position. karpak et al. [19] used a goal programming model to minimize costs and maximize delivery reliability and quality in supplier selection when assigning order quantities to each supplier. weber and current [36] used multi-objective linear programming for supplier selection to systematically analyze the trade-off between conflicting factors. recently, feng et al. [12] proposed a multiobjective model to select desired suppliers and also developed a multiobjective algorithm based on tabu search for solving it. reviews of supplier selection criteria and methods can be found in studies carried out by aissaoui et al. [1] and chai et al. [8]. in real-world, for supplier selection problem, decision makers do not have exact and complete information related to various input parameters. in such cases the fuzzy set theory (fst) [38] is considered one of the best tools to handle uncertainty. the supplier selection formulations have benefited greatly from the fst in terms of integrating quantitative and qualitative information, subjective preferences and knowledge of the decision maker. a review of literature on applications of fst in supplier selection shows that a variety of approaches are being used. kumar et al. [20] presented fuzzy goal programming models to capture uncertainty related to the supplier selection problem. amid et al. [2, 3] developed a weighted additive fuzzy model for supplier selection problem. bayrak et al. [6] presented a fuzzy multi-criteria group decision making approach to supplier selection based on fuzzy arithmetic operation. chen et al. [9] extended the concept of topsis method to develop a methodology for solving supplier selection problems in fuzzy environment. erol et al. [11] and li et al. [24] discussed the applications of fst in supplier selection. kwang et al. [21] introduced a combined scoring method with fuzzy expert systems approach for determination of best supplier. kahraman et al. [18] developed a fuzzy ahp model to select the best supplier firm providing the most satisfaction for the criteria determined. shaw et al. [30] proposed an integrated approach that combines fuzzy ahp and fuzzy multiobjective linear programming for selecting the appropriate supplier. toloo and nalchigar [32] proposed a new integrated data envelopment analysis model which is able to identify most appropriate supplier in presence of both cardinal and ordinal data. tsai and hung [33] proposed a fuzzy goal programming approach that integrates activity-based costing and performance evaluation in a value-chain structure for optimal green supply 96 a multiobjective optimization model for optimal supplier selection chain supplier selection and flow allocation. yücel and güneri [37] developed a weighted additive fuzzy programming approach for multi-criteria supplier selection. recently, amid et al. [4] developed a weighted maxmin fuzzy model to handle effectively the vagueness of input data and different weights of criteria in a supplier selection problem. arikan [5] proposed a fuzzy mathematical model and a novel solution approach to satisfy the decision maker’s aspirations for fuzzy goals. in all the studies mentioned thus far, supplier selection is driven by nonportfolio based approaches only. this type of framework is restrictive as it does not provide the decision maker with an opportunity to leverage the supplier diversity with reference to preferences in respect of cost, quality and delivery. recently, guu et al. [15] discussed supplier selection problem with interval coefficients using portfolio based approach. in this paper, we consider three supplier’s selection criteria, namely, expected unit price, expected score of quality and expected score of delivery. the proposed fuzzy optimization model simultaneously minimize the expected unit cost and maximize the expected score of quality and expected score of delivery. the model is constrained by several realistic constraints, namely, demand constraint, maximal and minimal fraction of the total order allocation to a single supplier, number of suppliers held in the portfolio. note that in comparison to the approach used in guu et al. [15], the proposed approach is capable of generating many efficient supplier portfolios using different shape parameters of the nonlinear s-shape membership functions from which the decision maker may choose the one according to his/her preferences. the paper is organized as follows. in section 2, we present multiobjective programming model of supplier selection based on portfolio theory. in section 3, we present fuzzy optimization models of supplier selection using nonlinear s-shape fuzzy membership functions. the proposed models are test-run in section 4. this section also includes a discussion of the results obtained. finally in section 5, we submit our concluding observations. 2 the supplier selection problem here, we assume that the decision maker allocate orders among n suppliers offering different price, quality and delivery. we use the following variables and parameters in the supplier selection model: xi: the proportion of total order allocated to i-th supplier , pi: the per unit net purchase price from i-th supplier , qi: the percentage of quality level of i-th supplier , 97 m. k. mehlawat, s. kumar di: the percentage of on-time-delivery level of i-th supplier , yi: the binary variable indicating whether the i-th supplier is contained in the supplier portfolio or not, i.e., yi = { 1, if i-th supplier is contained in the supplier portfolio, 0, otherwise, ui: the maximal fraction of the total order allocated to the i-th supplier , li: the minimal fraction of the total order allocated to the i-th supplier . 2.1 objectives • expected unit price the expected unit cost is the weighted average of the prices quoted by different suppliers, the fractions of the overall quantity ordered to them serving as the respective weights. here, we consider the overall demand as 1 which overcomes the dependence of supplier selection problem on the units of measurement of the commodities [15]. the expected unit price of the supplier portfolio is expressed as f1(x) = n ∑ i=1 pixi . • expected score of quality quality of the supplies is measured in terms of the extent of satisfaction (fraction) with quality. we use the expected score of quality which in effect is the average of the satisfaction of the established standards by different suppliers as an objective of supplier selection [15]. the expected score of quality of the supplier portfolio is expressed as f2(x) = n ∑ i=1 qixi . • expected score of delivery a supplier’s compliance (fraction of 1) with on-time-delivery schedule is regarded as his/her score of delivery. using the fraction of quantity allocated to different suppliers as weight [15], the expected score of delivery of the supplier portfolio is expressed as f3(x) = n ∑ i=1 dixi . 98 a multiobjective optimization model for optimal supplier selection 2.2 constraints • total order constraint on the suppliers: n ∑ i=1 xi = 1 . • maximal fraction of the total order that can be allocated to a single supplier: xi ≤ uiyi , i = 1, 2, . . . , n . • minimal fraction of the total order that can be allocated to a single supplier: xi ≥ liyi , i = 1, 2, . . . , n . the constraints corresponding to lower bounds li and upper bounds ui on the allocation to individual suppliers (0 ≤ li, ui ≤ 1, li ≤ ui , ∀i) are included to avoid a large number of very small allocations (lower bounds) and at the same time to ensure a sufficient diversification of the allocation (upper bounds) [15]. • number of suppliers held in a supplier portfolio: n ∑ i=1 yi = h where h is the number of suppliers that the decision maker chooses to include in the supplier portfolio [15]. of all the suppliers from a given set, the decision maker would pick up the ones that are likely to yield the desired satisfaction of his/her preferences. it is not necessary that all the suppliers from a given set may configure in the supplier portfolio as well. • no negative proportions of total orders: xi ≥ 0 , i = 1, 2, . . . , n . 99 m. k. mehlawat, s. kumar 2.3 the decision problem the mixed-integer model for purchasing a single item in multiple sourcing networks is presented as follows: (p1) min f1(x) = n ∑ i=1 pixi max f2(x) = n ∑ i=1 qixi max f3(x) = n ∑ i=1 dixi subject to n ∑ i=1 xi = 1 , (1) n ∑ i=1 yi = h , (2) xi ≤ uiyi , i = 1, 2, . . . , n , (3) xi ≥ liyi , i = 1, 2, . . . , n , (4) xi ≥ 0 , i = 1, 2, . . . , n , (5) yi ∈ {0, 1} , i = 1, 2, . . . , n . (6) it may be noted that the basic framework of the supplier selection model (p1) is similar to the one used in [15]; however, instead of using interval coefficients for an uncertain environment as in [15], we rely on fuzzy membership functions to generate supplier selection strategies that meets the preferences of the decision maker. 3 supplier portfolio selection models based on fuzzy set theory operationally, formulating an supplier portfolio requires estimation of distributions of price, quality and delivery for the various suppliers. distributed randomly as they are over the chosen time horizon, such estimates, at best, represent decision maker’s subjective interpretation of the information available at the time of decision making. note that the same information may be interpreted differently by different decision makers. under such circumstances, the issue of constructing a supplier portfolio becomes the one of a 100 a multiobjective optimization model for optimal supplier selection choice from a ‘fuzzy’ set of subjective interpretations, the term ‘fuzzy’ being suggestive of the diversity of both the decision maker’s objective functions as well as that of the constraints. here, we formulate fuzzy multiobjective supplier portfolio selection problem based on vague aspiration levels of decision makers to determine a satisfying supplier portfolio selection strategy. we assume that decision makers indicate aspiration levels on the basis of their prior experience and knowledge. as the aspiration levels are vague, we may refer to the fuzzy membership functions, for example, linear [39, 40], piecewise linear [17], exponential [23], tangent [22]. a linear membership function is most commonly used because it is simple and it is defined by fixing two points: the upper and lower levels of acceptability. however, there are some difficulties in using linear membership functions as pointed out by watada [34]. further, if the membership function is interpreted as fuzzy utility of the decision maker, describing the behavior of indifference, preference or aversion towards uncertainty, then a nonlinear membership function provides a better representation. it may also be noted that nonlinear membership functions are much more desirable for real-world decision making, as unlike linear membership functions, for nonlinear membership functions, the marginal rate of increase (or decrease) of membership values as a function of model parameters is not constant-a technique that reflects reality better than the linear case. in this paper, we use logistic function [34], i.e., a nonlinear s-shape membership function to express vague aspiration levels of decision makers. this function has several advantages over other nonlinear membership functions and is considered an appropriate choice in portfolio selection, see gupta et al. [14]. we now define the following nonlinear s-shape membership function of the goal of net price: • µp(x) = 1 1 + exp ( αp ( n ∑ i=1 pixi − pm )) , where pm is the mid-point (middle aspiration level for the net price) at which the membership function value is 0.5 and αp is provided by decision makers based on their degree of satisfaction of the goal (see fig. 1). 101 m. k. mehlawat, s. kumar 0�� 1 0 figure 1. membership function of the goal of net price the membership function of the goal of quality is given by • µq(x) = 1 1 + exp ( −αq ( n ∑ i=1 qixi − qm )) , where qm is the mid-point and αq is provided by decision makers based on their degree of satisfaction regarding the level of quality (see fig. 2). ��� � � figure 2. membership function of the goal of quality similarly, we define membership functions of the goal of delivery as follows: • µd(x) = 1 1 + exp ( −αd ( n ∑ i=1 dixi − dm )) , where dm is the respective mid-point and αd is provided by decision makers. note that the membership function of the goal of delivery as described above, have shape similar to that of the membership function defining the goal of quality. 102 a multiobjective optimization model for optimal supplier selection using bellman and zadeh’s maximization principle [7] with the above defined fuzzy membership functions, the fuzzy supplier portfolio selection problem for selecting suppliers is formulated as follows: (p2) max η subject to η ≤ µp(x) , η ≤ µq(x) , η ≤ µd(x) , 0 ≤ η ≤ 1 , and constraints (1) − (6) . the problem (p2) is a nonlinear programming problem. it can be transformed into a linear programming problem by letting θ = log η 1 − η , so that η = 1 1 + exp(−θ) . since, the logistic function is monotonically increasing, hence, maximizing η makes θ maximize. therefore, the problem (p2) can be transformed into the following equivalent linear programming problem: (p3) max θ subject to θ ≤ αp ( pm − n ∑ i=1 pixi ) , θ ≤ αq ( n ∑ i=1 qixi − qm ) , θ ≤ αd ( n ∑ i=1 dixi − dm ) , and constraints (1) − (6) . note that θ ∈] − ∞, +∞[. the fuzzy supplier portfolio selection problem (p2)/(p3) leads to a fuzzy decision that simultaneously satisfies all the fuzzy objectives. then, we determine the maximizing decision as the maximum degree of membership for the fuzzy decision. in this approach, the relationship between various objectives in a fuzzy environment is considered fully symmetric [40], i.e., all fuzzy objectives are treated equivalent. this approach is efficient in computation but it may provide ‘uniform’ membership degrees for all fuzzy objectives even when achievement of some objective(s) is more stringently required. therefore, we use the ‘weighted additive model’ proposed in [31] to incorporate relative importance of various fuzzy objectives 103 m. k. mehlawat, s. kumar in supplier portfolio selection. the weighted additive model of the fuzzy supplier portfolio selection problem is formulated as follows: (p4) max 3 ∑ r=1 ωrηr subject to η1 ≤ µp(x) , η2 ≤ µq(x) , η3 ≤ µd(x) , 0 ≤ ηr ≤ 1 , r = 1, 2, 3 and constraints (1) − (6) , where ωr is the relative weight of the r-th objective given by decision makers such that ωr > 0 and 3 ∑ r=1 ωr = 1. the max-min approach used in the formulation of the problems (p2)/(p3) and (p4) possesses good computational properties. however, the approach does not ensure fuzzy-efficient solution. to ensure efficiency of the solution, we take recourse to the two-phase approach proposed in [25]. as a result, it becomes possible to choose explicitly a minimum degree of satisfaction (taken to be equal to the solution of the max-min approach) for each fuzzy objective function and examine whether the same can be improved upon or not. hence, we solve the problems (p5) and (p6) corresponding to the problems (p3) and (p4) respectively in the second-phase. (p5) max 3 ∑ r=1 ωrθr subject to log µp(x ∗) 1 − µp(x∗) ≤ θ1 ≤ αp ( pm − n ∑ i=1 pixi ) , log µq(x ∗) 1 − µq(x∗) ≤ θ2 ≤ αq ( n ∑ i=1 qixi − qm ) , log µd(x ∗) 1 − µd(x∗) ≤ θ3 ≤ αs ( n ∑ i=1 dixi − dm ) , and constraints (1) − (6) , where x∗ is an optimal solution of (p3), ω1 = ω2 = ω3, ωr > 0, 3 ∑ r=1 ωr = 1 and θr ∈] − ∞, +∞[ r = 1, 2, 3. 104 a multiobjective optimization model for optimal supplier selection (p6) max 3 ∑ r=1 ωrηr subject to µp(x ∗∗) ≤ η1 ≤ µp(x) , µq(x ∗∗) ≤ η2 ≤ µq(x) , µd(x ∗∗) ≤ η3 ≤ µd(x) , 0 ≤ ηr ≤ 1 , r = 1, 2, 3 and constraints (1) − (6) , where x∗∗ is an optimal solution of (p4), ωr is the relative weight of the r-th objective given by decision makers such that ωr > 0 and 3 ∑ r=1 ωr = 1. the problems (p3) and (p5) are linear programming problems which can be solved using the lindo software [28]. the problems (p4) and (p6) are nonlinear programming problems. although, for medium or large-sized problems, one may suspect that solving these nonlinear programming problems could be computationally difficult, this is not the case, as many excellent softwares are available to solve them. we can use lingo [29] to solve (p4) and (p6). 4 numerical illustration in this section, we present an illustration of the developed supplier portfolio selection decision procedure for a multinational company. the purchasing manager of the company have identified 10 potential suppliers. the manager will select the most favorable suppliers(s) and allocate various proportion of total order among selected suppliers(s) such that to minimize the net price of purchasing and to maximize total quality and delivery level of purchased items. 4.1 supplier allocation the 10 suppliers form the population from which we attempt to construct a supplier portfolio comprising 5 suppliers. the suppliers profiles shown in table 1 represents the estimated values of their net price (pi), quality level (qi) and delivery level (di) along with the estimated values of lower and upper bounds. 105 m. k. mehlawat, s. kumar table 1 input data of suppliers price quality delivery lower bound upper bound (rs.) (%) (%) (li) (ui) supplier 1 13 0.82 0.80 0.03 0.22 supplier 2 12.5 0.78 0.75 0.06 0.33 supplier 3 11.5 0.70 0.80 0.03 0.20 supplier 4 14 0.88 0.90 0.027 0.22 supplier 5 15 0.84 0.92 0.2 1.17 supplier 6 16 0.95 0.88 0.06 0.27 supplier 7 14.5 0.80 0.78 0.05 0.4 supplier 8 15.5 0.92 0.84 0.017 0.17 supplier 9 13.5 0.85 0.85 0.03 0.25 supplier 10 12 0.75 0.78 0.06 0.30 we now present the computational results. corresponding to pm = 13.3, qm = 0.83 and dm = 0.82, we obtain supplier portfolio selection strategy by solving the problem (p3). to check efficiency of the solution obtained, we use the two-phase approach and solve the problem (p5). if the purchasing manager is not satisfied with the supplier portfolio obtained, more supplier portfolios can be generated by varying the values of the shape parameters in the problem (p3). the computational results summarized in table 2 are based on three different sets of values of the shape parameters. note that all the three solutions obtained are efficient, i.e., their criteria vector are nondominated. table 3 presents proportions of the total order allocated to suppliers in obtained supplier portfolios table 2 summary results of supplier portfolio selection shape parameters & variables net price quality level delivery level η θ αp αq αd 0.85900 1.80700 200 600 600 13.29095 0.83301 0.84703 0.58128 0.32803 100 100 100 13.29671 0.83328 0.84720 0.52087 0.08353 6 30 30 13.28609 0.83278 0.84688 106 a multiobjective optimization model for optimal supplier selection table 3 the proportions of the total order allocated to suppliers in obtained supplier portfolios shape parameters suppliers αp αq αd s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 200 600 600 0.22 0.27635 0 0.22 0 0 0 0.03365 0.25 0 100 100 100 0.22 0.27443 0 0.22 0 0 0 0.03557 0.25 0 6 30 30 0.22 0.27797 0 0.22 0 0 0 0.03203 0.25 0 next, we present computational results considering preferences of the purchasing manager for the three objectives. • case 1 we consider the following weights of the fuzzy goals of expected net price (ω1), expected quality level (ω2) and expected delivery level (ω3): ω1 = 0.6, ω2 = 0.25, ω3 = 0.15. corresponding to pm = 13.3, qm = 0.81 and dm = 0.88, we obtain supplier portfolio selection strategy by solving the problem (p4). the efficiency of the solution is verified by solving the problem (p6) in the second phase. the corresponding computational results are listed in tables 4-5. the achievement levels of the various membership functions are η1 = 0.95744, η2 = 0.41261, η3 = 0.31576. note that these achievement levels are consistent with the purchasing manager preferences, i.e., (η1 > η2 > η3) agrees with (ω1 > ω2 > ω3). • case 2 here, we consider the weights as ω1 = 0.15, ω2 = 0.6, ω3 = 0.25. by taking pm = 13.3, qm = 0.81 and dm = 0.88, we obtain supplier portfolio selection strategy by solving the problem (p4). the solution is verified for efficiency. the corresponding computational results are listed in tables 4-5. the achievement levels of the various membership functions are η1 = 0.00023, η2 = 0.90362, η3 = 0.70285 which are consistent with the purchasing manager preferences. • case 3 as performed above in case 1 and case 2, corresponding to the weights ω1 = 0.15, ω2 = 0.2, ω3 = 0.65 and pm = 13.3, qm = 0.81, dm = 0.88, we obtain portfolio selection strategy by solving the problem (p4). the solution is found to be efficient. the corresponding computational results are listed in tables 4-5. the achievement levels of the various membership functions are η1 = 0.00028, η2 = 0.77664, η3 = 0.78516 which are consistent with the purchasing manager preferences. 107 m. k. mehlawat, s. kumar table 4 summary results of supplier portfolio selection incorporating purchasing manager preferences case shape parameters price quality level delivery level αp αq αd case 1 6 30 30 12.78110 0.79823 0.85422 case 2 6 30 30 14.69752 0.88460 0.90870 case 3 6 30 30 14.66650 0.85154 0.92320 table 5 the proportions of the total order allocated to suppliers in obtained supplier portfolios incorporating purchasing manager preferences class suppliers s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 class 1 0.0661 0 0.2 0.22 0 0 0 0 0.25 0.2639 class 2 0 0 0 0.22 0.21496 0.27 0 0.04504 0.25 0 class 3 0 0 0 0.027 0.646 0.06 0 0.017 0.25 0 the foregoing analysis of the various decision situations from the stand point of decision makers preferences demonstrates that the supplier portfolio selection models developed in this paper discriminate among decision makers. thus, it is possible to construct efficient portfolios with reference to the diversity of decision maker preferences. 5 conclusions this paper proposed a flexible approach to multiobjective supplier selection problems. we used the criteria of expected unit price, expected score of quality and expected score of delivery for supplier evaluation and order allocation. further, the benefits of supplier diversification using trade-offs among the three chosen criteria have been achieved. the upper bounds and lower bounds are used for fractions of order that may be assigned to a particular supplier in order to ensure supplier diversification as well as to avoid the situations where very small fractions of the ordered quantity are obtained. recognizing that supplier selection involves mcdm in an environment that befits more fuzzy approximation than deterministic formulation, we have transformed the supplier portfolio selection model into a fuzzy model using nonlinear s-shape fuzzy membership functions. numerical illustrations based on 10-supplier universe have been presented to illustrate the effectiveness of the proposed models. the efficiency of the obtained solutions was 108 a multiobjective optimization model for optimal supplier selection verified using the two-phase approach. the main advantage of the proposed models is that if decision maker is not satisfied with any of the supplier portfolios, more portfolios can be generated by varying the values of the shape parameters. 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[40] h. -j. zimmermann, fuzzy programming and linear programming with several objective functions, fuzzy sets and systems 1 (1978), 45-55. 112 ratio mathematica 25 (2013), 77–94 issn:1592-7415 hypermatrix based on krasner hypervector spaces ∗maedeh motameni, reza ameri, razieh sadeghi ∗ department of mathematics, qaemshahr branch, islamic azad university,qaemshahr, iran, school of mathematics, statistics and computer sciences, university of tehran, tehran, iran, faculty of mathematics, university of mazandaran, babolsar, iran motameni.m@gmail.com,rameri@ut.ac.ir,razi$\ $sadeghi@yahoo.com abstract in this paper we extend a very specific class of hypervector spaces called krasner hypervector spaces in order to obtain a hypermatrix. for reaching to this goal, we will define dependent and independent vectors in this kind of hypervector space and define basis and dimension for it. also, by using multivalued linear transformations, we examine the possibility of existing a free object here. finally, we study the fundamental relation on krasner hypervector spaces and we define a functor. key words: hypermatrix, hypervector spaces, basis of a hypervector space, multivalued linear transformations. msc2010: 15a33. 1 introduction the notion of a hypergroup was introduced by f. marty in 1934 [5]. since then many researchers have worked on hyperalgebraic structures and developed this theory (for more details see [2],[3]). using hyperstructures ∗ corresponding author 77 m. motameni, r. ameri, r. sadeghi theory, mathematicians have defined and studied variety of algebraic structures. among them the notion of hypervector spaces has been studied mainly by vougiuklis [8, 9], tallini [6, 7] and krasner [3].(see also [1]). there are differences mainley about operation or a hyperoperation in these three type of hypervector spaces. vougiuklis has studied hv vector spaces which deals with a very weak condition regarding intersections. tallini defined a hypervector spaces considering a crisp sum and using a hyperexternal operation which assigns to the production every element of a field and every element of the abelian geroup (v, +), a non empty subset of v , while krasner in the definition of a hypervector space used a hypersum to make a canonical hypergroup and by using a singlevalued operation he defined the krasner hypervector space with some definitions. in this paper we have chosen the definition of krasner and we defined the generalized subset of it. also, to make a correct logical relation between definitions we had to define the notion of a multivalued linear transformation and by using this notion we could talk about basis and dimension of a krasner hypervector space. in the sequel, considering the multivalued functions, we have constructed a kind of matrix whith hyperarrays with coefficients taken from the hyperfield of krasner and elements of the basis. also, we studied the notion of singular and nonsingular transformations. finally, we studied the category of krasner hypervector spaces and defines the fundamental relation on it. in the last part we have defines a functor. 2 preliminaries in this section we present definitions and properties of hypervector spaces and subsets, that we need for developing our paper. a mapping ◦ : h × h −→ p∗(h) is called a hyperoperation (or a join operation), where p∗(h) is the set of all non-empty subsets of h. the join operation is extended to subsets of h in natural way, so that a◦b is given by a◦b = ⋃ {a◦ b : a ∈ a and b ∈ b} the notations a◦a and a◦a are used for {a}◦a and a◦{a}, respectively. generally, the singleton {a} is identified by its element a. a hypergroupoid (h,◦), which is associative, i. e, x◦(y◦z) = (x◦y)◦z , ∀x,y,z ∈ h is called a semihypergroup. a hypergroup is a semihypergroup such that for all x ∈ h, we have x ◦ h = h = h ◦ x, which is called reproduction axiom. 78 hypermatrix based on krasner hypervector spaces definition 2.1. [3] a semihypergroup (h, +) is called a canonical hypergroup if the following conditions are satisfied: (i)x + y = y + x,∀x,y ∈ r; (ii)∃0 ∈ r(unique) such that for every x ∈ r,x ∈ 0 + x = x; (iii) for every x ∈ r , there exists a unique element, say x́ such that 0 ∈ x + x́.(we denote x́ by -x); (iv) for every x,y,z ∈ r, z ∈ x + y ⇐⇒ x ∈ z −y ⇐⇒ y ∈ z −x. from the definition it can be easily verified that −(−x) = x and −(x + y) = −x−y. definition 2.2. [3] a krasner hyperring is a hyperstructure (r,⊕,?) where (i) (a,⊕) is a canonical hypergroup; (ii) (a,?) is a semigroup endowed with a two-sided absorbing element 0; (iii) the product distributes from both sides over the sum. a hyperfield is a krasner hyperring (k,⊕,?), such that (k −{0},?) is a group. definition 2.3. [3] let (k,⊕,?) be a hyperskewfield and (v,⊕) be a canonical hypergroup. we define a krasner hypervector space over k to be the quadrupled (v,⊕, ·,k), where ” · ” is a single-valued operation · : k ×v −→ v, such that for all a ∈ k and x ∈ v we have a · x ∈ v , and for all a,b ∈ k and x,y ∈ v the following conditions hold: (h1) a · (x⊕y) = a ·x⊕a ·y; (h2) (a⊕ b) ·x = a ·x⊕ b ·x; (h3) a · (b ·x) = (a ? b) ·x; (h4) 0 ·x = 0; (h5) 1 ·x = x. we say that (v,⊕, ·,k) is anti-left distributive if for all a,b ∈ k, x ∈ v, (a + b) · x ⊇ a · x + b · x, and strongly left distributive, if for all a,b ∈ k, x ∈ v, (a⊕ b) ·x = a ·x⊕ b ·x, in a similar way we define the anti-right distributive and strongly right distributive hypervector spaces, respectively. v is called strongly distributive if it is both strongly left and strongly right distributive. in the sequel by a hypervector space we mean a krasner hypervector space. 79 m. motameni, r. ameri, r. sadeghi 3 krasner subhypervector space here we study some basic results of krasner hypervector spaces and after defining the category of krasner hypervector spaces, we continue to find a free object in the category of krasner hypervector spaces. definition 3.1. a nonempty subset s of v is a subhyperspace if (s,⊕) is a canonical subhypergroup of v and for all a ∈ k, x ∈ s, we have a ·x ∈ s. here we present example of a krasner hypervector spaces. example 3.2. let f be a field , v be a vector space and f∗ be a multiplicative subgroup of f . for all x,y ∈ v we define the equivalence relation ∼ on v as follows: x ∼ y ⇐⇒ x = ty t ∈ f∗ now, let v̄ be the set of all classes of v modulo ∼. v̄ together with the hypersum ⊕, construct a canonical hypergroup: x̄⊕ ȳ = {v̄ ∈ v̄ | v̄ ⊆ x̄⊕ ȳ} here we consider the external composition · : f̄ × v̄ −→ v̄ ā · v̄ 7−→ āv now, (v̄ ,⊕, ·,f) is a hypervector space. lemma 3.3. let vi be a hypervector space, for all i ∈ i, then ⋂ vi is also a hypervector space. definition 3.4. let v be a hypervector spaces and s a nonempty subset of it, then the smallest subhypervector space of v containing s is called linear space generated by s and is denoted by < s >. moreover, < s >=⋂ s⊆w≤v w . theorem 3.5. let v be a hypervector space and s a nonempty subset of it, then < s > = {t ∈ v |t ∈ n∑ i=1 ai ·si,ai ∈ k,si ∈ s,n ∈ n} = = {t1 ⊕ ...⊕ tn|ti = ai ·si}. proof. let a = {t ∈ v |t ∈ ∑n i=1 ai · si,ai ∈ k,si ∈ s,n ∈ n}. we claim that (a,⊕, ·,k) is the smallest hypervector space generated by s. first we show that (a,⊕) is a canonical hypergroup. commutativity is obvious. for all x ∈ a, we have x ∈ ∑n i=1 ai ·si. suppose there exists a scalar identity 80 hypermatrix based on krasner hypervector spaces 0a ∈ a such that 0a ∈ ∑n i=1 bi · ri, for bi ∈ k and ri ∈ s, we should have x⊕ 0a = ∑n i=1 ai ·si ⊕ ∑n i=1 bi · ri = ∑n i=1 ai ·si 3 x. since for all si ∈ a, we have si ∈ s ⊆ v , and (v,⊕) is a canonical hypergroup, then there exists a scalar identity in v called 0v such that si⊕0v = si. hence in the above equation it is enough to choose bi = ai and ri = 0v , we obtain x⊕0a = ∑n i=1 ai·si⊕ ∑n i=1 ai·0v = ∑n i=1 ai·(si⊕0v ) = ∑n i=1 ai·si 3 x. now for all x ∈ a we define −x = ∑n i=1 ai · (−si), then we have 0s = ∑n i=1 ai·0v ∈ ∑n i=1 ai·si⊕ ∑n i=1 ai·(−s)i = ∑n i=1 ai·(si⊕(−si)) hence every element in (a,⊕) has a unique identity. moreover, every element in (a,⊕) is reversible, because suppose for all x,y,z ∈ a, we have x = ∑n i=1 ai ·si, y = ∑n i=1 ái · śi, z = ∑n i=1 ´́ai · ´́si. since for si, śi, ´́si ∈ s ⊆ v , if ´́si ∈ si⊕śi we have śi ∈ ´́si⊕(−si), then it is sufficient to choose ´́ai = ái = ai. therefore (a,⊕) is a canonical subhypergroup. now for all t ∈ a, k ∈ k, we have k · t ⊆ k · n∑ i=1 ai ·si = n∑ i=1 (k ? ai) ·si ⊆ a. then (a,⊕, ·,k) is a subhypervector space of v . let w be another subhypervector space of v containg s. let t ∈ a, then t ∈ ∑n i=1 ai ·si, for ai ∈ k,si ∈ s,n ∈ n. since w is a subhypervector space of v containing s, then ∑n i=1 ai ·si ⊆ w and a ⊆ w. so, a is the smallest subhypervector space of v . also, for all s ∈ s, we have s = 1 ·s, then s ∈ a, therefore s ⊆ a. definition 3.6. let (v,⊕, ·), (w,⊕, ·) be two hypervector spaces over a hyperskewfield k, then the mapping t : v −→ p∗(w) is called (i) multivalued linear transformation if t(x⊕y) ⊆ t(x) ⊕t(y) and t(a ·x) = a ·t(x). (ii) multivalued good linear transformation if t(x⊕y) = t(x) ⊕t(y) and t(a ·x) = a ·t(x). where, p∗(w) is the nonempty power set of w. from now on, by mvlinear transformation we mean a multivalued linear transformation. remark 3.7. we define t(0v ) = 0w . definition 3.8. let v,w be two hypervector spaces over a hyperskewfield k, and t : v −→ p(w) be a mv-linear transformation. then the kernel of t is denoted by kert and defined by kert = {x ∈ v | 0w ∈ t(x)} 81 m. motameni, r. ameri, r. sadeghi theorem 3.9. let v,w be two hypervector spaces on a hyperskewfield k and t : v −→ w be a linear transformation. then kert is a subhypervector space of v . proof. by remark 4.18, we have t(0v ) = 0w which means that 0v ∈ kert and kert 6= ∅, then we have x ∈ x⊕ 0v = x, for all x ∈ kert . the other properties of a canonical subhypervector space will inherit from v . theorem 3.10. let v,u be two hypervector spaces and t : v −→ p∗(u) be a mv-linear transformation : (i) if w is a subhypervector space of v , then t(w) is also a subhypervector space of u. (ii) if l is a subhypervector space of u, then t−1(l) is also a subhypervector space of v containing kert . proof. (i) let a ∈ k and x́, ý ∈ t(w), such that x́ = t(x), ý = t(y) for some x,y ∈ w. then x́ ⊕ ý = t(x) ⊕ t(y) = t(y) ⊕ t(x) = ý ⊕ x́, hence commutativity holds. for all x ∈ v we have x = x⊕ 0v , then we obtain t(x) = t(x⊕ 0v ) ⊆ t(x) ⊕t(0v ) = t(x) ⊕ 0u . also, for all x ∈ v , there exists x́ = −x ∈ v such that 0v ∈ x⊕ (−x). by remark 4.18 we have 0u = t(0v ) ∈ t(x⊕ (−x)) ⊆ t(x) ⊕t(−x) = x́⊕ x́. where x́ = t(−x) is the unique inverse of x́. now suppose for all x,y,z ∈ v we have x ∈ y ⊕z =⇒ y ∈ x⊕ (−z) this is equivalent to t(x) ∈ t(y⊕z) ⊆ t(y)⊕t(z) =⇒ t(y) ∈ t(x)⊕t(−z). so, (t(w),⊕) is a canonical hypergroup. now for a ∈ k and x́ ∈ t(w), we have a · x́ = a ·t(x) = t(a ·x) ⊆ t(w). hence, (t(w),⊕, ·), is a subhypervector space of v . (ii) let a ∈ k and x,y ∈ t−1(l). suppose x́ = t(x), ý = t(y), for x́, ý ∈ l. since (u,⊕) is a canonical hypergroup, then we have x⊕y = t−1(x́) ⊕t−1(ý) = t−1(ý) ⊕t−1(x́) = y ⊕x. also, we have x⊕ 0v = t−1(x́) ⊕t−1(0u ) ⊇ t−1(x́⊕ 0u ) ⊇ t−1(x́) = x. for all x́ ∈ v , there exists −́x such that 0u ∈ x́⊕(−́x), hence for x ∈ t−1(x́), there exists t−1(−x́) ∈ t−1(l) such that x⊕ (−x) = t−1(x́) ⊕t−1(−x́) = t−1(x⊕ (−x́)) = t−1(0u ) = 0v . now for all x́, ý, ź ∈ l, we have x́ ∈ ý ⊕ ź =⇒ ý ∈ x́⊕ (−ź) 82 hypermatrix based on krasner hypervector spaces suppose x,y,z ∈ t−1(l). the above relation is equivalent to y ⊕z = t−1(ý) ⊕t−1(ź) ⊇ t−1(ý ⊕ ź) ⊇ t−1(x́) = x =⇒ x⊕ (−z) = t(x́) ⊕t(−ź) ⊇ t(x́⊕ (−ź)) ⊇ t(ý) = y. which means that x ∈ y ⊕z =⇒ y ∈ x⊕ (−z). moreover, a ·x = a ·t−1(x́) = t−1(a · x́) ⊆ t−1(l). hence (t−1(l),⊕, ·) is a subhypervector space of v . now for x ∈ kert we have t(x) = 0u ∈ l, then we obtain x ∈ t−1(l), hence kert ⊆ t−1(l). theorem 3.11. let u, v be two hypervector spaces on a hyperskewfield k and t : v −→ p∗(u) be a good linear transformation. then there is a one to one correspondence between sunhypervector spaces of v containing kert and subhypervector spaces of u. proof. suppose a = {w|w ≤ v, w ⊇ kert} and b = {l|l ≤ u}. we show that the following map is one to one and onto: φ : a −→ b w −→ t(w) by theorem 3.10, t(w) belongs to b, for all w ∈ a. now let w1,w2 be two elements of a such that w1 6= w2, then there exists w1 ∈ w1 − w2 or w2 ∈ w2 −w1. let w1 ∈ w1 −w2, then t(w1) ∈ t(w1)−t(w2) and hence t(w1) 6= t(w2). if w2 ∈ w2 − w1, then t(w1) 6= t(w2), too. so, φ is well defined and one to one. now for l ∈ b, put w = t−1(l), then by theorem 3.9 we have w ∈ a and t(w) = l. therefore, φ is onto, hence the result. 4 construction of a hypermatrix now, we will talk about the basis of a hypervector space and verify that considering a multivalued linear transformation will imply some conditions to this definition. finally, with the elements of hyperfield and basis we will construct a hypermatrix. definition 4.1. a subset s of v is called linearly independent if for every vectors v1, ...,vn ∈ s, and c1, ...,cn ∈ k, if we have 0v ∈ c1 ·v1 ⊕ ...⊕ cn ·vn , implies that c1 = ... = cn = 0k. otherwise s is called linearly dependent. theorem 4.2. let v be a hypervector space and v1, ...,vn be independent in v . then every element in the linear space < v1, ...,vn > belongs to a unique sum of the form ∑n i=1 ai ·vi where ai ∈ k. 83 m. motameni, r. ameri, r. sadeghi proof. every element of < v1, ...,vn > belongs to a set of the form ∑n i=1 ai ·vi where ai ∈ k. we will show that this form is unique. let u ∈ v such that u ⊆ ∑n i=1 ai·vi and u ⊆ ∑n i=1 bi·vi, where ai,bi ∈ k. since v is a hypervector space we have : 0v ∈ u−u ⊆ ∑n i=1 ai·vi− ∑n i=1 bi·vi = ∑n i=1 ai·vi⊕ ∑n i=1(−b)i·vi. therefore, 0v ⊆ ∑n i=1(ai⊕(−bi)) ·vi. and since v1, ...vn are independent we have ai ⊕ (−bi) = 0, ∀i, then ai = −(−bi) = bi. theorem 4.3. let v be a hypervector space. then vectors v1, ...vn ∈ v are independent or vj for some 1 ≤ j ≤ r, belongs to the linear combination of the other vectors. proof. let v1, ...,vnbe dependent and let 0v ⊆ ∑n i=1 ai ·vi such that at least one of the scalars such as aj is not zero. then there exists ti, (i = 1, ...,n) such that 0v ∈ t1 ⊕ t2 ⊕ ...⊕ tn, where ti = ai ·vi, which means that tj ∈ 0 ⊕ (−(t1 ⊕ ...⊕ tj−1 ⊕ tj+1 ⊕ ...⊕ tn)) =⇒ tj ∈ 0 ⊕ ((−t1) ⊕ ...⊕ (−tj−1) ⊕ (−tj+1) ⊕ ...⊕ (−tn)) moreover, for at least one vj we have vj = (a −1 j ) · tj. which means vj ∈ (a−1j ) · (−t1 ⊕ ...⊕ (−tj−1) ⊕ (−tj+1) ⊕ ...⊕ (−tn)) ∈ ∈ ((a−1j ) · (−t1)) ⊕ ((a −1 j ) · (−tj−1)) ⊕ ((a −1 j ) · (−tj+1)) ⊕ ...⊕ ((a −1 j ) · (−tn)) ∈ ((a−1j ) · (−a1 ·v1)) ⊕ ...⊕ ((a −1 j ) · (−a1 ·vj−1)) ⊕ ((a −1 j ) · (−aj+1 ·vj+1)) ⊕ ...⊕ ((a−1j ) · (−an · tn)) ∈ ((a−1j ? (−a1)) ·v1) ⊕ ...⊕ ((a −1 j ? (−a1)) ·vj−1) ⊕ ((a −1 j ? (−aj+1)) ·vj+1) ⊕ ...⊕ ((a−1j ? (−an)) · tn) ∈ (c1 ·v1) ⊕ ...⊕ (cj ·vj−i) ⊕ (cj ·vj+1) ⊕ ...⊕ (cj ·vn) where cj = (a −1 j ? (−an)). therefore vj belongs to a linear combination of v1, ...,vj−1,vj+1, ...,vn as desired. definition 4.4. we call β a basis for v if it is a linearly independent subset of v and it spans v . we say that v has finite dimensional if it has a finite basis. the following results are the generalization of the same results for vector spaces, also the methods here are adopted from those in the ordinary vector spaces. 84 hypermatrix based on krasner hypervector spaces theorem 4.5. let v be a hypervector space. if w is a subhypervector space of v generated by β = {v1, ...,vn}, then w has a basis contained in β. corolary 4.6. if v is a hypervector space, then every generating subset of v , contains a basis of v , which means every independent subset of v is included in a finite basis. theorem 4.7. let v be a hypervector space. if v has a finite basis with n elements, then the number of elements of every independent subset of v is smaller or equal to n. corolary 4.8. let v be strongly left distributive and hypervector space. if v is finite dimensional then every two basis of v have the same elements. lemma 4.9. let v be a hypervector space. if v is finite dimensional, then every linearly independent subset of v is contained in a finite basis. now, we want to determine that what is a free object in the category of hypervector spaces. first, notice that if we denote the category of hypervector spaces by krh-vect, we define the category as follows: (i) the objects in this category are hypervector spaces over a hyperskew field k; (ii) for the objects v,w of krh-vect, the set of morphisms from v to p∗(w) is the multivalued linear transformations which we show by home(v , w ). (iii) combination of morphism is defined as usual; (iv) for all objects v in the category, the morphism 1v : v −→ v is the identity. according to the definition of a free object in the category of hypersets [2], and considering the category of hypervector spaces, if x is a basis for the hypervector space v , then we say that f is a free object in krh-vect then for every function f : x −→ v , there exists a homomorphism f̄ : f −→ v , such that f̄ ◦ i = f, where i is the inclusion function. now, we have (f̄◦i)(x) = f̄(i(x)) = f̄(x) (?) since the homomorphism f̄ is defined in h-vect, it is a multivalued transformation, then we define f̄(x) = {f(x)} we obtain f̄ ◦ i = f. let g : f −→ v be another homomorphism such that g(xi) = f(xi), then for t ∈ ∑n i=1 ai ·xi, let f̄ be defined by f̄(t) = ∑n i=1 ai ·f(xi), we have g(t) ⊆ g( n∑ i=1 ai ·xi) = n∑ i=1 ai ·g(xi) = f̄(t). 85 m. motameni, r. ameri, r. sadeghi hence f̄ defined above is the maximum homomorphism such that (?) is satisfied. suppose t ∈ ∑n i=1 ai · xi and t ∈ ∑n i=1 bi · xi, for ai,bi ∈ k, we have f̄(t) = ∑n i=1 ai ·f(xi), and also f̄(t) = ∑n i=1 bi ·f(xi), then ∑n i=1 ai ·f(xi) =∑n i=1 bi ·f(xi), we obtain 0 ∈ ∑n i=1 ai ·f(xi) − bi ·f(xi) = ∑n i=1(ai − bi) ·f(xi) so ai = bi. therefore, f̄ is a unique mv-transformation. hence we have the following corollary: corolary 4.10. every hypervector space with a basis is a free object in the category of hypervector spaces. theorem 4.11. let (v,⊕, ·), (w,⊕, ·) be two hypervector spaces on a hyperskewfiled k . if t : v −→ p∗(w) and u : v −→ p∗(w) be two mvtransformations. we define l(v,w) = {t|t : v −→ p∗(w)} and the hyperoperation ” � ” as follows: (t �u)(α) = t(α) �u(α) also, we define the external composition as (c�t)(α) = c�t(α) then (l(v,w),�,�)) as defined above is a hypervectorspace over a hyperskewfield k. proof. the external composition ” � ” is defined as follows: � : k ×l(v,w) −→ p∗(l(v,w)) (α,t) 7−→ α�t first we show that (l(v,w),�) is a canonical hypergroup. communativity and associativity is obvious. we consider the transformation 0 : v −→ 0 as a ”0” for the group and 1 : v −→ p∗(v ) as the identity. then there exists a unique inverse (−t) such that 0 ∈ (t � (−t))(α). now, let t,u,z be three linear transformations that belong to l(v,w) then if z ∈ t � u then we have z(α) ∈ (t � u)(α), which means z(α) ∈ t(α) � u(α). now since w is hypervector space then we obtain t(α) ∈ z(α) � (−u)(α), hence t ∈ z � (−u),∀α ∈ k. therefore, (l(v,w),�) is a canonical hypergroup. now, we check that l(v,w) is a hypervector space. let x,y ∈ k and t,u ∈ l(v,w) then we have (1) (x� (t �u))(α) = x� (t �u)(α) = (x�t(α)) � (x�u(α)) (2) ((x�y) �t)(α) = ⋃ z∈x�y z �t(α) = (x�t(α)) � (y �t(α)). 86 hypermatrix based on krasner hypervector spaces the other conditions will be obtained immediately. therefore, (l(v,w),�,�) is a hypervector space. theorem 4.12. let (v,⊕, ·), (w,⊕, ·) be two hypervector spaces on a hyperskewfield k, if a = {α1, ...,αn} be a basis for v and β1, ...,βn be any vectors in w , then there is a unique linear transformation t : v −→ p∗(w) such that t(αi) = βi, 1 ≤ i ≤ n. in other words, every linear transformation can be characterized by its operation on the basis of v . proof. since for every v ∈ v , there exists scalars c1, ...,cn ∈ k such that (∗) v ∈ n∑ i=1 ci ·αi then we define a map t : v −→ p∗(w) as follows: t(v) = ∑n i=1 ci ·t(αi) = ∑n i=1 ci ·βi since (∗) is unique then t is well-defined. now, we check that t is a linear transformation. let v,w ∈ v and scalars d1, ...,dn ∈ k then v ∈ ∑n i=1 ci ·αi and w ∈ ∑n i=1 di · αi, then we have t(v) = ∑n i=1 ci · t(αi) and t(w) =∑n i=1 di ·t(αi). now since v ⊕w ∈ ∑n i=1(ci ⊕di) ·αi, then we obtain t(v ⊕w) ⊆ t( ∑n i=1(ci ⊕di) ·αi) = ∑n i=1(ci ⊕di) ·t(αi) = ∑n i=1 ci·t(αi)⊕ ∑n i=1 di·t(αi) = t(v)⊕t(w). also, it is clear that (c◦t)(α) = c◦t(α). hence, t is a linear transformation. now, we shall check that t is unique. let s : v −→ p∗(w) be another linear transformation that satisfies s(αi) = βi. we will show that s = t. we have s(α) = n∑ i=1 ci ·s(αi) = n∑ i=1 ci ·βi = n∑ i=1 ci ·t(αi) = t(α) so, s = t as desired. remark 4.13. let t : v −→ p∗(w) be a linear transformation. we denote kert = {α ∈ v | 0 ∈ t(α)} by nt and by imt we mean rt = {t(α)|α ∈ v}. we call dimension of rt , rank of t and it is denoted by r(t). notice that nt is a subhypervector space of v and rt is a subhypervector space of w. theorem 4.14. let v,w be two hypervector spaces over a field k. let t : v −→ p∗(w) be a linear transformation and dimv = n < ∞. then dimrt + dimkert = dimv 87 m. motameni, r. ameri, r. sadeghi proof. let w = nt and let β1 = {α1, ...,αk} be a basis for w. we extend β1 to β2 = {α1, ...,αk,αk+1, ...,αn}. we will show that β = {t(αk+1), ...,t (αn)} is a basis for rt . let c1, ...,cn be scalars in k such that 0 ∈ n∑ i=k+1 ci ·t(αi) then there exists γ ∈ ∑n i=k+1(ci · αi) such that 0 ∈ t(γ), this implies that γ ∈ kert = nt , hence γ ∈ ∑k i=1(ci ·αi). therefore 0 = γ −γ ∈ k∑ i=1 (ci ·αi) ⊕ n∑ i=k+1 ((−ci) ·αi) =⇒ ci = 0 now, we claim that β generates rt because if for all α ∈ v we have t(α) = β, and since 0 ∈ ∑k i=1 ci ·t(αi), hence β = t(α) ⊆ t( ∑n i=1 ci ·αi) = ∑n i=1 ci ·t(αi) = ∑k i=1 ci ·t(αi) + ∑n i=k+1 ci · t(αi) = ∑n i=k+1 ci ·t(αi) therefore, dimrt + dimnt = (n−k) + k = n = dimv . for all 1 6 j 6 n and 1 6 p 6 m, we define cpj as the coordinator of t(αj) on the ordered basis b = {β1, ...,βp} which means t(αj) = m∑ p=1 cpj ·βp where for cpj = (cpj),βp = (βp1). now, if we notice the following matrix with a crisp product and hypersum, we will have a hypermatrix as the following:  c11 ... c1p... ... ... cj1 ... cjp   ︸ ︷︷ ︸   β11... βp1   ︸ ︷︷ ︸ =   c11 ·β11 ⊕ ...⊕ c1p ·βp1... cj1 ·β11 ⊕ ...⊕ cjp ·βp1  =   t(α1)... t(αj)   ︸ ︷︷ ︸ cpj βp theorem 4.15. let v,w be two hypervector spaces. if dimv = n and dimw = m, then diml(v,w) = mn. proof. let a = {α1, ...,αn} and b = {β1, ...,βm} be the basis of v,w respectively. for all (p,q), where p,q ∈ z, and 1 6 q 6 n, 1 6 p 6 m by theorem 4.12 we have a unique linear transformation tpq : v −→ p∗(w) which we define by tpq(αi) = βp, when i = q and otherwise it is defined 0. since we have mn linear transformation from v to p∗(w), it is sufficient to 88 hypermatrix based on krasner hypervector spaces show that β ′′ = {tpq|1 6 p 6 m, 1 6 q 6 n} is a basis for l(v,w). let t : v −→ p∗(w) be a linear transformation. for all 1 6 j 6 n, let c1j, ...,cmj be the coordinate of t(αj) in the ordered basis β́, i.e, t(αj) =∑m p=1 cpj ·βp. we will show that t = ∑m p=1 ∑n q=1 cpq ·tpq generates l(v,w). because if we suppose u = ∑m p=1 ∑n q=1 cpq · tpq, then if suppose i = q we obtain u(αj) = ∑m p=1 ∑n q=1 cpq·tpq(αj) = ∑m p=1 ∑n q=1 cpq·βp = ∑m p=1 apj·βp = t(αj). otherwise it will be 0. also, it is obvious that β ′′ is independent. hence the result. remark 4.16. let t : v −→ p∗(w) and s : w −→ p∗(z) be two linear transformations and α ∈ v , we define (s ◦t)(α) = s(t(α)) = ⋃ β∈t(α) s(β) then s ◦t is also a linear transformation. definition 4.17. let t : v −→ p∗(v ) be a linear transformation, we call t a linear operator (or shortly an operator) on v , and if we have t ◦t , we denote it by t 2. lemma 4.18. let v be a hypervector space on a field k. if u,t,s be three operators on v and k ∈ k, then the following results are immediate: (i) i ◦u = u ◦ i = u; (ii) (s ⊕t) ◦u = s ◦u ⊕t ◦u, u ◦ (s ⊕t) = u ◦s ⊕u ◦t ; (iii) k ⊕ (u ◦t) = (ku) ◦t = u ◦ (kt ).� example 4.19. let β = {α1, ...,αn} be an ordered basis for the hypervector space v . consider the operators t(p,q) regarding the proof of theorem 4.15. these n2 operators construct a basis for the space of operators of v . let s,u be two operators on v then we have s = ∑ p ∑ q cpq ·spq, u = ∑ r ∑ s brs ·srs. now by lemma 4.18, we have (s ◦u)(αi) = s(u(αi)) = ⋃ β∈u(αi) s(β) = ⋃ β∈σr σsbrs·trs(αi) s(β) = s( ∑ r ∑ s brs ·t(r,s)(αi)) = s( ∑ r ∑ s brs ·αr) when i = s we have∑ r ∑ s bri ·s(αr) = ∑ r ∑ s brs · ( ∑ p ∑ q cpq ·tpq(αr)) = ∑ r ∑ s ∑ p ∑ q(briapq) ◦αp and when r = q we have = ∑ r ∑ s ∑ p ∑ q(bricpr) ·αp = ∑ r ∑ s ∑ p ∑ q(cprbri) ·αp and since 1 6 i 6 p then we have ∑ r ∑ s ∑ p ∑ q(bc)n2 ·αi hence when we compose two operators s and u, the result is obtained by multiplying two matrices of them.� 89 m. motameni, r. ameri, r. sadeghi now, it is time to talk about the inverse of a transformation. as it is usual for defining an inverse we have: definition 4.20. let t : v −→ p∗(w) be one to one and onto. t is said to have an inverse when there exists u : w −→ p∗(v ) such that t ◦ u = iv and u ◦t = iw . also, the inverse of t is denoted by t−1 and obviously is not unique. we have (u ◦t)−1 = t−1 ◦u−1. we say that a linear transformation t is called nonsingular if 0 ∈ t(α) implies that α = {0}, which means that the null space of t is equal to {0}. lemma 4.21. let t : v −→ p∗(w) be a linear transformation then t is one to one if and only if t is nonsingular if and only if kert = 0 proof. let t be one to one and suppose 0 ∈ t(α), then since t(0) = 0, we have t(0) ∈ t(α) then t(0) ∈ t(α + 0) ⊆ t(α) + t(0) =⇒ t(α) ∈ t(0) + (−t(0)) = 0 hence α = 0. conversely, let t is nonsingular and suppose for x,y ∈ v , we have t(x) = t(y) then, 0 ∈ t(x) − t(y) = t(x − y) and since t is nonsingular we obtain x−y = 0, which means x = y. now let for all α ∈ kert we have 0 ∈ t(α), then since t is nonsingular we obtain α = 0 which means kert = 0. conversely, if kert = 0, then suppose 0 ∈ t(α) implies that α ∈ kert = 0, hence α = 0. theorem 4.22. let v,w be two hypervector spaces on a hyperfiled k and let t : v −→ p∗(w) be a linear transformation. if t is good reversible linear transformation, then the reverse of t is also a good linear transformation. proof. let w1,w2 ∈ w and k ∈ k, then there exists v1,v2 ∈ v such that t−1(w1) = v1,t −1(w2) = v2, where t(v1) = w1, and t(w2) = v2. we have t−1(w1 ⊕ w2) = t−1(t(v1) ⊕ t(v2)) ⊇ t−1(t(v1 ⊕ v2)) = v1 ⊕ v2 = t−1(w1) ⊕t−1(w2) and when t is a good linear transformation, t−1 is also a good linear transformation. theorem 4.23. let t : v −→ p∗(w) be a linear transformation. t is nonsingular if and only if t corresponds every linearly independent subset of v onto a linearly independent subset of w . proof. let t be nonsingular and s be a linearly independent subset of v . we show that t(s) is independent. let śi ∈ t(s) and for all i there exists si ∈ s such that t(si) = śi. we assume∑n i=1 ci · śi = 0 =⇒ ∑n i=1 ci ·t(si) = 0 =⇒ t( ∑n i=1 ci ·si) = 0 because t is nonsingular we have ∑n i=1 ci · si = 0, and since si, for all i are 90 hypermatrix based on krasner hypervector spaces linearly independent then ci = 0, hence t(s) is linearly independent. conversely, let 0 6= α ∈ v , then {0} is an independent set. hence by hypothesis t corresponds this independent set to a linearly in dependent set such as t(α) ∈ p∗(w), then we have t(α) 6= 0. therefore, t is nonsingular. theorem 4.24. let v,w be two hypervector spaces with finite dimension on a hyperskewfield k and dimv = dimw . if t : v −→ p∗(w) is a linear transformation, then the followings are equivalent: (i) t is reversible; (ii) t is nonsingular; (iii) t is onto. (iv) if {α1, ...,αn} is a basis for v , then {t(α1), ...,t(αn)} is a basis for w . (v) there exists a basis like {α1, ...,αn} for v such that {t(α1), ...,t(αn)} is a basis for w . lemma 4.25. let v be a hypervector space with finite dimension on a hyperfield k, then v ∼= kn. 5 fundamental relations let (v,⊕, ·) be a hypervector space, we define the relation ε∗ as the smallest equivalence relation on v such that the set of all equivalence classes,v/ε∗, is an ordinary vector space. ε∗ is called fundamental equivalence relation on v and v/ε∗ is the fundamental ring. let ε∗(v) is the equivalence class containing v ∈ v , then we define � and � on v/ε∗ as follows: ε∗(v) �ε∗(w) = ε∗(z), for all z ∈ ε∗(v) ⊕ε∗(w) a�ε∗(v) = ε∗(z), for all z ∈ a ·ε∗(v), a ∈ k let u be the set of all finite linear combinations of elements of v with coefficients in k, which means u = { ∑n i=1 ai ·vi; ai ∈ k, vi ∈ v, n ∈ n} we define the relation ε as follows: vεw ⇐⇒∃u ∈ u;{v,w}⊆ u koskas [4] introduced the relation β∗ on hypergroups as the smallest equivalence relation such that the quotient r/β∗+ is a group. we will denote β+ the relation in r as follows: vβ+w ⇐⇒∃(c1, ...,cn) ∈ v n such that {v,w}⊆ c1 ⊕ ...⊕ cn 91 m. motameni, r. ameri, r. sadeghi freni proved that for hyperrings we have β∗+ = β+. since in here (v,⊕) is a canonical hypergroup the we will have: theorem 5.1. in the hypervector space (v,⊕, ·), we have ε∗ = β∗+. vougiouklis [9] has proved that the sets {ε∗(z) : z ∈ ε∗(v) ⊕ ε∗(w)} and {ε∗(z) : z ∈ a · ε∗(v)} are singletons. with a similar method we can prove the following theorem: theorem 5.2. let (v,⊕, ·) be a hypervector space, then for all a ∈ k , v,w ∈ v , we have the followings: (i) ε∗(v) �ε∗(w) = ε∗(z), ∀z ∈ ε∗(v) ⊕ε∗(w) a�ε∗(v) = ε∗(z), ∀z ∈ a ·ε∗(w) (ii)ε∗(0v ) is the zero element of (v/� ∗,�). (iii) (v/ε∗,�,�) is a hypervector space and is called the fundamental hypervector space of v . proof. (i) the proof is the same as [9], and we omit it. (ii) since from (i) we obtain ε∗(v)�ε∗(w) = ε∗(v⊕w) and a�ε∗(v) = ε∗(a·v) we have ε∗(v) � �∗(0) = ε(v ⊕ 0) = ε∗(v) (iii) the conditions for the vector space (v/�∗,�,�) will be obtained from the hypervector space (v,⊕, ·). theorem 5.3. let (v,⊕, ·,k) be a hypervector space and (v/�∗,�,�) be the fundamental relation of it then dimv = dimv/�∗. proof. let b = {v1, ...,vn} be a basis for v . we show that the set b∗ = {ε∗(v1), ...,ε∗(vn)} is a basis for v/�∗. for this let ε∗(v) ∈ v/ε∗, then for every v ∈ v there exists a1, ...,an ∈ k such that x ∈ ∑n i=1 ai · vi, then v = t1 ⊕ ... ⊕ tn, where ti = ai · vi, i ∈ {1, ...,n}. now by theorem 5.2 we have ε∗(ti) = ai ·ε∗(vi) then ε∗(v) = ε∗(t1⊕....⊕tn) = ε∗(t1)�....�ε∗(tn) = (a1�ε∗(v1))�(an�ε∗(vn)). hence, v/ε∗ is spanned by b∗. now we show that b∗ is linearly independent. for this let (a1 �ε ∗(v1)) � ...� (an �ε ∗(vn)) = ε ∗(0) =⇒ ε∗(a1 ·v1) � ...�ε∗(an ·vn) = ε∗(0) =⇒ ε∗(a1 ·v1 ⊕ ...⊕an ·vn) = ε∗(0) =⇒ 0 ∈ a1 ·v1 ⊕ ...⊕an ·vn since b in linearly independent in v , then a1 = ... = an = 0. therefore, b ∗ is also linearly independent. 92 hypermatrix based on krasner hypervector spaces lemma 5.4. let v , w be two hypervector spaces and t : v −→ p∗(w) be a linear transformation, then (i) t(ε∗(v)) ⊆ ε∗(t(v)), for all v ∈ v ; (ii) the map t∗ : v/ε∗ −→ w/ε∗ defined as t∗(ε∗(v)) = ε∗(t(v)) is a linear transformation. proof. (i) straightforward. (ii) it is obvious that t∗ is well defined. now we show that t∗ is a linear transformation. let a ∈ k, x,y ∈ v , then by theorem 5.2 we have t∗(ε∗(x) � ε∗(y)) = t∗(ε∗(x ⊕ y)) = ε∗(t(x ⊕ y)) ⊆ ε∗(t(x) ⊕ t(y)) = ε∗(t(x)) �ε∗(t(y)) = t∗(ε∗(x)) �t∗(ε∗(y)) and t∗(a�ε∗(x)) = t∗(ε∗(a·x)) = ε∗(t(a·x)) = ε∗(a·t(x)) = a�ε∗(t(x)) = a�t∗(ε∗(x)) hence, t∗ is a linear transformation. theorem 5.5. the map f : hv −→ v defined by f(v ) = v/ε∗ and f(t) = t∗ is a functor, where hv and v denote the category of hypervector spaces and vector spaces respectively. moreover, f preserves the dimension. proof. by lemma 5.4 f is well-defined. lett : v −→ p∗(w) and u : w −→ p∗(z) be two linear transformations, then f(u ◦t) = (u ◦t)∗ such that for all v ∈ v we have (u ◦t)∗(ε∗(v)) = ε∗((u ◦t)(v)) = ε∗(u(t(v))) = u∗ε∗(t∗(x)) = u∗t∗(ε∗(x)) = f(u)f(t)(ε∗(v)) =⇒ f(u ◦t) = f(u)f(t) also, the identity is f(1∗v ) : v/ε ∗ −→ v/ε∗ such that 1∗v (ε ∗(v)) = ε∗(v). hence, f is a functor and by theorem 4.14 we have dim(f(v )) = dim(v/ε∗) = dim(v ). theorem 5.6. let t : v −→ p∗(w) be a liner transformation in hv. then the following diagram is commutative: v t−→ w ϕv ↓ ↓ ϕw v/ε∗ t ∗ −→ w/ε ∗ where βv ,βw are the canonical projections of v and w . proof. let v ∈ v then ϕw (t(v)) = ε∗(t(v)) = t∗(ε∗(v)) = t∗(ϕv (v)) = t∗ϕv (v). hence, the diagram is commutative. acknowledgement. the first author has been financially supported by ” office of vice chancellor of research and technology of islamic azad university-qaemshahr branch”. 93 m. motameni, r. ameri, r. sadeghi references [1] r. ameri and o. r. dehghan, on dimension of hypervector spaces, european j. pure appl. math., vol. 1, no. 2, 32–50, (2008). [2] p. corsini, prolegomena of hypergroup theory, second edition, aviani editor (1993). [3] p. corsini and v. leoreanu, applications of hyperstructure theory, kluwer academic publishers, (2003). [4] m. koskas, groupoids, demi-groupes et hypergroups, j. math. pures appl. 49 155–192 (1970). [5] f. marty, sur une generalization de la notion de groupe, 8th congress des mathematiciens scandinaves, stockholm, 45–49 (1934). [6] m. s. tallini, hypervector spaces, 4th int. cong. aha, 167–174 (1990). [7] m. s. tallini, weak hypervector spaces and norms in such spaces, algebraic hyperstructures and applications, hadronic press, 199–206 (1994). [8] t. vougiouklis, hyperstructures and their representations, hadronic, press, inc. (1994). [9] t. vougiouklis, hv vector spaces, 5 th int. cong. on aha, iasi, romania, 181–190, 1994. 94 ratio mathematica 25(2013), 47-58 issn:1592-7415 the divisors’ hyperoperations achilles dramalidis democritus university of thrace, school of education, 681 00 alexandroupolis, greece adramali@psed.duth.gr abstract in the set n of the natural numbers we define two hyperoperations based on the divisors of the addition and multiplication of two numbers. then, the properties of these two hyperoperations are studied together with the resulting hyperstructures. furthermore, from the coexistence of these two hyperoperations in n∗, an hv-ring is resulting which is dual. key words: hyperstructures, hv-structures msc2010: 20n20, 16y99. 1 introduction in 1934, f. marty introduced the definitions of the hyperoperation and of the hypergroup as a generalization of the operation and the group respectively. let h be a set and ◦ : h ×h → p′(h) be a hyperoperation, [2], [3], [5], [6], [8]: the hyperoperation (◦) in h is called associative, if (x◦y) ◦z = x◦ (y ◦z),∀x,y,z ∈ h. the hyperoperation (◦) in h is called commutative, if x◦y = y ◦x,∀x,y ∈ h. an algebraic hyperstructure (h,◦), i.e. a set h equipped with the hyperoperation (◦), is called hypergroupoid . if this hyperoperation is associative, then the hyperstructure is called semihypergroup. the semihypergroup (h,◦), is called hypergroup if it satisfies the reproduction axiom: achilles dramalidis x◦h = h ◦x,∀x ∈ h. one of the topics of great interest, in the last years, is the hv -structures, which was introduced by t. vougiouklis in 1990 [7]. the class of hv structures is the largest class of algebraic hyperstructures. these structures satisfy weak axioms, where the non-empty intersection replaces the equality, as bellow [8]: i) the (◦) in h is called weak associative, we write wass , if (x◦y) ◦z ∩x◦ (y ◦z) 6= ∅,∀x,y,z ∈ h. ii) the (◦) is called weak commutative, we write cow , if (x◦y) ∩ (y ◦x) 6= ∅,∀x,y ∈ h. iii) if h is equipped with two hyperoperations (◦) and (∗), then (∗) is called weak distributive with respect to (◦), if [x∗ (y ◦z)] ∩ [(x∗y) ◦ (x∗z)] 6= ∅,∀x,y,z ∈ h. the hyperstructure (h,◦) is called h v-semigroup if it is wass and it is called h v-group if it is a reproductive (i.e. x◦h = h ◦x = h,∀x ∈ h) hv-semigroup. it is called commutative h v-group if (◦) is commutative and it is called h vcommutative group if (◦) is weak commutative. the hyperstructure (h,◦,∗) is called h v-ring if both hyperstructures (◦) and (∗) are wass, the reproduction axiom is valid for (◦), and (∗) is weak distributive with respect to (◦). it is denoted [4] by e∗ the set of the unit elements with respect to (∗) and by i∗(x,e) the set of the inverse elements of x associated with the unit e, with respect to (∗). an hv-ring (r, +, ·) is called dual h v-ring, if (r, ·, +) is an hv-ring, too [4]. let (h, ·) be a hypergroupoid. an element e ∈ h is called right unit element if a ∈ a ·e,∀a ∈ h and is called left unit element if a ∈ e ·a,∀a ∈ h. the element e ∈ h is called unit element if a ∈ a ·e∩e ·a,∀a ∈ h. let (h, ·) be a hypergroupoid endowed with at least one unit element. an element a′ ∈ h is called an inverse element of the element a ∈h, if there exists a unit element e ∈ h such that e ∈ a′ ·a∩a ·a′. moreover, let us define here: if x ∈ x · y(resp.x ∈ y ·x)∀y ∈ h, then, x is called left absorbing-like element (resp. right absorbing-like element). an element a ∈ h is called idempotent element if a2 = a. the nth power of an element h, denoted hs, is defined to be the union of all expressions of n times 48 the divisors’ hyperoperations of h, in which the parentheses are put in all possible ways. an hv-group (h,·) is called cyclic with finite period with respect to h ∈ h, if there exists a positive integer s, such that h = h1 ∪ h2 . . . ∪ hs. the minimum such s is called period of the generator h. if all generators have the same period, then h is cyclic with period. if there exists h ∈ h and s positive integer, the minimum one, such that h = hs, then h is called single-power cyclic and h is a generator with single-power period s. the cyclicity in the infinite case is defined similarly. thus, for example, the hv-group (h, ·) is called single-power cyclic with infinite period with generator h if every element of h belongs to a power of h and there exists s0 ≥ 1, such that ∀s ≥ s0 we have: h1 ∪h2 ∪·· ·∪hs−1 ⊂ hs. 2 the divisors’ hyperoperation due to addition in n let n be the set of the natural numbers. let us define the hyperoperation (�) in n as follows: definition 2.1. for every x,y ∈ n �: n × n →p(n) −{∅} : (x,y) 7→ x�y ⊂ n such that x�y = {z ∈ n : x + y = z ·λ,λ ∈ n} where (+) and (·) are the usual operations of the addition and multiplication in n, respectively. we call the above hyperoperation, divisors’ hyperoperation due to addition. some properties of the divisors’ hyperoperation due to addition 1. x�y = y �x,∀x,y ∈ n 2. 0 � 0 = n 3. 0 � 1 = 1 � 0 = 1 4. {1,x + y}⊂ x�y,∀x,y ∈ n 5. if x + y = κ ·ν ⇒{1,κ,ν,κ ·ν}⊂ x�y,x,y,κ,ν ∈ n. 49 achilles dramalidis remark 2.2. if x + y = p, where p ∈ n is a prime number, then x � y = {1,p},x,y ∈ n. proposition 2.3. the number 0 is a unit element of the divisors’ hyperoperation due to addition. proof. indeed, for x ∈ n x� 0 = {z ∈ n : x + 0 = z ·λ,λ ∈ n} = {z ∈ n : x = z ·λ,λ ∈ n}3 x. then, x ∈ (x� 0) ∩ (0 �x),∀x ∈ n. remark 2.4. since, there is no x′ ∈ n such that 0 ∈ (x � x′) ∩ (x′ � x) when x 6= 0, the number 0 is the only one in n having an inverse element (and that is 0) associated with the unique unit element 0 of the divisors’ hyperoperation due to addition, i.e. 0 ∈ 0 � 0. proposition 2.5. the number 1 is an absorbing-like element of the divisors’ hyperoperation due to addition. proof. indeed, 1 ∈ x�y,∀x,y ∈ n ⇒ 1 ∈ 1�y,∀y ∈ n ⇒ 1 ∈ (1�y)∩(y�1),∀y ∈ n. proposition 2.6. if y = n ·x, x,n ∈ n then {1,x, 1 + n,x(1 + n)}⊂ x�y. proof. let y = n ·x, x,n ∈ n then x�y = {z ∈ n : x + y = z ·λ,λ ∈ n} = {z ∈ n : x + nx = z ·λ,λ ∈ n} = {z ∈ n : x(1 + n) = z ·λ,λ ∈ n}⊃{1,x, 1 + n,x(1 + n)}. proposition 2.7. if x ∈ n is a prime number then x2 = {1, 2,x, 2x}. proof. let x ∈ n, be a prime number then x2 = x�x = {z ∈ n : x + x = z ·λ,λ ∈ n} = {z ∈ n : 2x = z ·λ,λ ∈ n}. according to property 5, {1, 2,x, 2x} ⊂ x2, but since x is prime, x2 = {1, 2,x, 2x}. proposition 2.8. x� n∗ = n∗ �x = n∗,∀x ∈ n∗. proof. let x ∈ n∗, then x� n∗ ⊃ x� (nx) 3 n + 1, n ∈ n∗, according to proposition 2.6. 50 the divisors’ hyperoperations so, we proved that n + 1 ∈ x�n∗,∀x,n ∈ n∗ and since 1 ∈ x�n∗,∀x ∈ n∗, we get x� n∗ = n∗ �x = n∗,∀x ∈ n∗. remark 2.9. notice that, for x ∈ n∗ x� n = ⋃ n∈in (x�n) = ⋃ n∈in {z ∈ n : x + n = z ·λ,λ ∈ n∗}⊇ ⊇ ⋃ n∈in {z ∈ n : x + nx = z ·λ,λ ∈ n∗} = ⋃ n∈in (x�nx). but from proposition 2.6,⋃ n∈in (x�nx) ⊃ ⋃ n∈in {1,x,n + 1,x(n + 1)}⊃ ⋃ n∈in {n + 1} = n∗. so, x�n = n �x = n∗,∀x ∈ n∗. proposition 2.10. the divisors’ hyperoperation due to addition is a weak associative one in n∗. proof. for x,y,z ∈ n∗ (x�y) �z = {w ∈ n∗ : x + y = w ·λ,λ ∈ n∗}�z = ⋃ w∈in∗ (w �z) = = ⋃ w∈in∗ {w′ ∈ n∗ : w + z = w′ ·λ′,λ′ ∈ n∗} ⊃{w′ ∈ n∗ : x + y + z = w′ ·λ′,λ′ ∈ n∗} (i) on the other hand x� (y �z) = x�{v ∈ n∗ : y + z = v ·ρ,ρ ∈ n∗} = ⋃ v∈in∗ (x�v) = = ⋃ v∈in∗ {v′ ∈ n∗ : x + v = v′ ·ρ′,ρ′ ∈ n∗} ⊃{v′ ∈ n∗ : x + y + z = v′ ·ρ′,ρ′ ∈ n∗} (ii) from (i) and (ii) we get: (x�y)�z∩x�(y�z) = {n ∈ n∗ : x+y+z = n·µ,µ ∈ n∗} 6= ∅,∀x,y,z ∈ n∗. since the divisors’ hyperoperation due to addition is commutative, according to propositions 2.8 and 2.10, we get the following: 51 achilles dramalidis proposition 2.11. the hyperstructure (n∗,�) is a commutative hv-group. proposition 2.12. for (x,y,z) ∈ n∗×n∗×n∗, if x = z, then the divisors’ hyperoperation due to addition is strong associative. proof. let (x,y,z) ∈ n∗ × n∗ ×n∗ such that x = z, then due to commutativity we get: (x�y) �z = (x�y) �x = x� (x�y) = x� (y �x) = x� (y �z). proposition 2.13. the (n∗,�), is a single-power cyclic hv-group with infinite period where every x ∈ n∗ is a generator. proof. for x ∈ n∗, notice that x1 = {x} x2 = x�x = {z ∈ n∗ : 2x = z ·λ,λ ∈ n∗}⊃{1, 2} x3 = x2 �x = {z ∈ n∗ : 2x = z ·λ,λ ∈ n∗}�x = ⋃ z∈in∗ (z �x) = ⋃ z∈in∗ {w ∈ n∗ : z + x = w ·ρ,ρ ∈ n∗}⊃ ⊃{w′ ∈ n∗ : 2x + x = w′ ·ρ′,ρ′ ∈ n∗}∪ ∪{w′′ ∈ n∗ : x + x = w′′ ·ρ′′,ρ′′ ∈ n∗}⊃ ⊃{1, 3}∪{1, 2} = {1, 2, 3}. we shall prove that xn ⊃ {1, 2, 3, . . . ,n}, ∀x ∈ n∗, n ∈ n∗,n ≥ 2, by induction. suppose that for n = k,k ∈ n∗,k ≥ 2 : xk ⊃{1, 2, 3, . . .,k} we shall prove that the above is valid for n = k + 1, i.e. xk+1 ⊃{1, 2, 3, . . .,k,k + 1}. indeed, xk+1 = (xk �x) ∪ (xk−1 �x2) ∪ . . .∪ (x�xk). then xk+1 ⊃ (xk−1 �x2) ⊃{1, 2, 3, . . . ,k − 1}�{1, 2}⊃ ⊂{1, 2, 3, . . . ,k}∪{k + 1} = {1, 2, 3, . . . ,k,k + 1}. therefore every element of n∗ belongs to a special power of x, thus, is a generator of the single-power cyclic hv-group. 52 the divisors’ hyperoperations 3 the divisors’ hyperoperation due to multiplication in n now, let us define the hyperoperation (⊗) in n as follows: definition 3.1. for every x,y ∈ in ⊗ : n × n → p(n) −{∅} : (x,y) 7→ x⊗y ⊂ n such that x�y = {z ∈ n : x ·y = z ·λ,λ ∈ n} where (·) is the usual operation of the multiplication in n. we call the above hyperoperation, divisors’ hyperoperation due to multiplication. some properties of the divisors’ hyperoperation due to multiplication 1. x⊗y = y ⊗x,∀x,y ∈ n 2. 0 ⊗x = x⊗ 0 = n,∀x ∈ n 3. 1 ⊗ 1 = 1, i.e 1 is an idempotent element 4. {1,x,y,xy}⊂ x⊗y,∀x,y ∈ n remark 3.2. if x is a prime number, then 1 ⊗x = x⊗ 1 = {1,x}. proposition 3.3. e⊗ = n. proof. for x,e ∈ n,x⊗e = {z ∈ n : x ·e = z ·λ,λ ∈ n}3 x. so, according to property 1, we get x ∈ (x⊗e) ∩ (e⊗x),∀x,e ∈ n that means that the set of the unit elements with respect to (⊗) is the set n, i.e. e⊗ = n. proposition 3.4. i) i⊗(x, 0) = {0},y ∈ n ii) i⊗(x, 1) = n. proof. i) straightforward from property 2. ii) take the unit element 1, then from property 4, we get 1 ∈ (x⊗y) ∩ (y ⊗x),∀x,y ∈ n which means that i⊗(x, 1) = n. 53 achilles dramalidis proposition 3.5. if a unit element p is a prime number, then i⊗(x,p) = { n, x = np,n ∈ n pn, x 6= np,n ∈ n. proof. let p ∈ n be a unit element and p = prime number. then p has no other divisors than 1 and itself. so, let x = np, n ∈ n, then for x′ ∈ n x⊗x′ = {z ∈ n : x ·x′ = z ·λ,λ ∈ n} = = {z ∈ n : (np) ·x′ = z ·λ,λ ∈ n}3 p,∀x′ ∈ n. that means that i⊗(x,p) = n. let x 6= np,n ∈ n, then p ∈ {z ∈ n : x ·x′ = z ·λ,λ ∈ n}⇔ x′ = pn,n ∈ n ⇔ i⊗(x,p) = pn. seems to be particularly interesting, one to study cases where the unit element is not a prime number. the following two examples study the cases where the unit element is 6 and 9. example 3.6. let 6 be the unit element. assume that x = 6n, n ∈ n, then x⊗x′ = {z ∈ n : (6n) ·x′ = z ·λ,λ ∈ n}3 6,∀x′ ∈ n. then, i⊗(x, 6) = n. assume that x = 3m 6= 6n, n,m ∈ n, then 6 ∈ x⊗x′ = {z ∈ n : (3m) ·x′ = z ·λ,λ ∈ n}⇔ x′ = 2n,n ∈ n ⇔ i⊗(x, 6) = 2n. assume that x = 2m 6= 6n, n,m ∈ n, then 6 ∈ x⊗x′ = {z ∈ n : (2m) ·x′ = z ·λ,λ ∈ n}⇔ x′ = 3n,n ∈ n ⇔ i⊗(x, 6) = 3n. assume that x = 2m + 1 6= 3n, n,m ∈ n, then 6 ∈ x⊗x′ = {z ∈ n : (2m + 1) ·x′ = z ·λ,λ ∈ n} ⇔ x′ = 6n,n ∈ n ⇔ i⊗(x, 6) = 6n. example 3.7. let 9 be the unit element. assume that x = 9n, n ∈ n, then x⊗x′ = {z ∈ n : (9n) ·x′ = z ·λ,λ ∈ n}3 9,∀x′ ∈ n. then, i⊗(x, 9) = n. assume that x = 3m 6= 9n, n,m ∈ n, then 9 ∈ x⊗x′ = {z ∈ n : (3m) ·x′ = z ·λ,λ ∈ n} ⇔ x′ = 3n,n ∈ n ⇔ i⊗(x, 9) = 3n. assume that x 6= 3m, m ∈ n, then 9 ∈ x⊗x′ = {z ∈ n : x·x′ = z·λ,λ ∈ n}⇔ x′ = 9n,n ∈ n ⇔ i⊗(x, 9) = 9n. 54 the divisors’ hyperoperations proposition 3.8. every element x ∈ n is an absorbing-like element of the divisors’ hyperoperation due to multiplication. proof. according to property 4, x ∈ x ⊗ y, ∀x,y ∈ n, which means, that for every x∈ n, x ∈ x ⊗ y, ∀y ∈ n and due to property 1, ∀x ∈ n,x ∈ (x ⊗ y) ∩ (y ⊗ x),∀y ∈ n. then, every natural number is an absorbing-like element of the divisors’ hyperoperation due to multiplication. proposition 3.9. the divisors’ hyperoperation due to multiplication is a strong associative one in n. proof. for x,y,z ∈ n (x⊗y) ⊗z = {w ∈ n : x ·y = w ·λ,λ ∈ n}⊗z = ⋃ w∈in (w ⊗z) = = ⋃ w∈in {w′ ∈ n : w ·z = w′ ·λ′,λ′ ∈ n} = = ⋃ λ∈in {w′ ∈ n : [1 λ (xy) ] z = w′ ·λ′,λ′ ∈ n} = = ⋃ λ∈in {w′ ∈ n : x [1 λ (yz) ] = w′ ·λ′,λ′ ∈ n} = = ⋃ v∈in {w′ ∈ n : x ·v = w′ ·λ′,λ′ ∈ n} = = ⋃ v∈in (x⊗v) = x⊗{v ∈ n : y ·z = v ·λ,λ ∈ n} = x⊗ (y ⊗z). so, (x⊗y) ⊗z = x⊗ (y ⊗z),∀x,y,z ∈ n. proposition 3.10. the hyperstructure (n,⊗) is a commutative hypergroup. proof. indeed, for x ∈ n, x⊗ n = (x⊗ 0) ∪ [ ⋃ n∈in∗ (x⊗n)] = n ∪ [ ⋃ n∈in∗ (x⊗n) ] = n. so, x⊗ n = n ⊗x = n, ∀x ∈ n. also, according to property 1 and proposition 3.9 we get that (n,⊗) is a commutative hypergroup. remark 3.11. for x ∈ n∗, x⊗n∗ = ⋃ n∈in∗ (x⊗n) = ⋃ n∈in∗ {z ∈ n : x·n = z·λ,λ ∈ n∗}⊃ ⋃ n∈in∗ {n} = n∗. 55 achilles dramalidis so, x⊗ n∗ = n∗ ⊗x = n∗,∀x ∈ n∗. proposition 3.12. for every x ∈ n,xn−1 ⊆ xn,n ∈ n,n ≥ 2. proof. for x ∈ n and n ∈ n,n ≥ 2 xn = (xn−1 ⊗x) ∪ (xn−2 ⊗x2) ∪ . . .∪ (xn−p ⊗xp) where p = [ n 2 ] the integer part of n 2 , [1]. then, xn ⊇ xn−1 ⊗x ⊇ xn−1 ⊗ 1 ⊇ xn−1. 4 on a dual hv-ring in n∗ proposition 4.1. (x⊗y) � (x⊗z) ⊃ x⊗ (y �z),∀x,y,z ∈ n∗. proof. for x,y,z ∈ n∗, we get x⊗ (y �z) = x⊗{w ∈ n∗ : y + z = w ·λ,λ ∈ n∗} = ⋃ w∈in∗ (x⊗w) = = ⋃ w∈in∗ {w′ ∈ n∗ : x ·w = w′ ·λ′,λ′ ∈ n∗} = = ⋃ λ∈in∗ {w′ ∈ n∗ : x · y + z λ = w′ ·λ′,λ′ ∈ n∗}. on the other hand, (x⊗y) � (x⊗z) = = {v ∈ n∗ : x ·y = v ·ρ,ρ ∈ n∗}�{v′ ∈ n∗ : x ·z = v′ ·ρ′,ρ′ ∈ n∗} = ⋃ v,v′∈in∗ (v �v′) = ⋃ v,v′∈in∗ {k ∈ n∗ : v + v′ = k ·µ,µ ∈ n∗} = = ⋃ ρ,ρ′∈in∗ {κ ∈ n∗ : xy ρ + xz ρ′ = κ ·µ,µ ∈ n∗}⊃ ⊃ ⋃ µ′∈in∗ {κ′ ∈ n∗ : x · y + z µ′ = κ′ · τ ′,τ ′ ∈ n∗} = x⊗ (y �z). so, (x⊗y) � (x⊗z) ⊃ x⊗ (y �z) and then, x⊗ (y �z) ∩ (x⊗y) � (x⊗z) 6= ∅,∀x,y,z ∈ n∗. proposition 4.2. the divisors’ hyperoperation due to addition is weak distributive with respect to the divisors’ hyperoperation due to multiplication in n∗. 56 the divisors’ hyperoperations proof. for x,y,z ∈ n∗, we get x� (y ⊗z) = x�{w ∈ n∗ : y ·z = w ·λ,λ ∈ n∗} = ⋃ w∈in∗ (x�w) = = ⋃ w∈in∗ {w′ ∈ n∗ : x + w = w′ ·λ′,λ′ ∈ n∗}⊃ ⊃{w′′ ∈ n∗ : x + y = w′′ ·λ′′,λ′′ ∈ n∗}. on the other hand, (x�y) ⊗ (x�z) = = {v ∈ n∗ : x + y = v ·ρ,ρ ∈ n∗}⊗{v′ ∈ n∗ : x + z = v′ ·ρ′,ρ′ ∈ n∗} = = ⋃ v,v′∈in∗ (v ⊗v′) = ⋃ v,v′∈in∗ {v′′ ∈ n∗ : v ·v′ = v′′ ·ρ′′,ρ′′ ∈ n∗}⊃ ⊃{κ ∈ n∗ : (x + y) · (x + z) = κ ·µ,µ ∈ n∗}⊃ ⊃{κ′ ∈ n∗ : x + y = κ′ ·µ′,µ′ ∈ n∗}. so, x�(y⊗z)∩(x�y)⊗(x�z) ⊃{τ ∈ n∗ : x + y = τ ·σ,σ ∈ n∗} and then x� (y ⊗z) ∩ (x�y) ⊗ (x�z) 6= ∅,∀x,y,z ∈ n∗. proposition 4.3. the hyperstructure (n∗,�,⊗) is a commutative dual hvring. proof. indeed, according to propositions 2.11 and 3.10 the hyperstructures (n∗,�) and (n∗,⊗) are commutative hv-group and commutative hypergroup respectively. from propositions 4.1 and 4.2 we get that (⊗) is weak distributive with respect to (�) and (�) is weak distributive with respect to (⊗), respectively. references [1] n. antampoufis, widening complex number addition, proc. “structure elements of hyper-structures”, alexandroupolis, greece, spanidis press, (2005), 5–16. [2] p. corsini, v. leoreanu, applications of hypergroup theory, kluwer academic publishers, 2003. [3] b. davvaz, v. leoreanu-fotea, hyperring theory and applications, international academic press, 2007. 57 achilles dramalidis [4] a. dramalidis, dual h v-rings (mr1413019), rivista di matematica pura ed applicata, italy, v. 17, (1996), 55–62. [5] a. dramalidis, on geometrical hyperstructures of finite order, ratio mathematica, italy, v. 21(2011), 43–58. [6] a. dramalidis, t. vougiouklis,h v-semigroups an noise pollution models in urban areas, ratio mathematica, italy, v. 23(2012), 39–50. [7] t. vougiouklis, the fundamental relation in hyperrings. the general hyperfield, 4thaha congress, world scientific (1991), 203–211. [8] t. vougiouklis, hyperstructures and their representations, monographs in mathematics, hadronic press, 1994. 58 ratio mathematica volume 44, 2022 further diversification of nano binary open sets j. jasmine elizabeth1 g. hari siva annam2 abstract the purpose of this paper is to introduce and study the nano binary exterior, nano binary border and nano binary derived in nano binary topological spaces. also studied their characterizations. keywords: 𝑁𝐵derived, 𝑁𝐵exterior, 𝑁𝐵border. 2010 ams subject classification: 54a05, 54a993. 1assistant professor, kamaraj college, thoothukudi, (part time research scholar [reg no. 19122102092008], manonmaniam sundaranar university, tirunelveli-627012) tamil nadu, india. jasmineelizabeth89@gmail.com 2assistant professor, pg and research department of mathematics, kamaraj college, thoothukudi628003, tamil nadu, india. hsannam@yahoo.com. affiliated to manonmaniam sundaranar university, tirunelveli-627012, tamil nadu, india. 3received on june 4th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.884. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement 15 mailto:hsannam@yahoo.com j. jasmine elizabeth, g. hari siva annam 1. introduction m. lellis thivagar [1] introduced the concept of nano topological space with respect to a subset x of a universe u. s. nithyanantha jothi and p. thangavelu [2] introduced the concept of binary topological spaces. by combining these two concepts dr. g. hari siva annam and j. jasmine elizabeth [3] introduced nano binary topological spaces. in this paper we have introduced the nano binary border, nano binary derived and nano binary exterior in nano binary topological spaces. also studied their properties and characterizations with suitable examples. 2. preliminaries definition 2.1: [3] let (𝑈1, 𝑈2) be a non-empty finite set of objects called the universe and r be an equivalence relation on (𝑈1, 𝑈2)named as the indiscernibility relation. elements belonging to the same equivalence class are said to be indiscernible with one another. the pair (𝑈1, 𝑈2, 𝑅)is said to be the approximation space. let (𝑋1, 𝑋2) ⊆ (𝑈1, 𝑈2). 1. the lower approximation of (𝑋1, 𝑋2) with respect to r is the set of all objects, which can be for certain classified as (𝑋1, 𝑋2)with respect to r and it is denoted by 𝐿𝑅 (𝑋1, 𝑋2). that is,𝐿𝑅 (𝑋1, 𝑋2) = ⋃ {𝑅(𝑥1, 𝑥2)𝜖(𝑈1,𝑈2) (𝑥1, 𝑥2): 𝑅(𝑥1, 𝑥2) ⊆ (𝑋1, 𝑋2)} where 𝑅(𝑥1, 𝑥2)denotes the equivalence class determined by (𝑥1, 𝑥2). 2. the upper approximation of (𝑋1, 𝑋2) with respect to r is the set of all objects, which can be possibly classified as (𝑋1, 𝑋2)with respect to r and it is denoted by 𝑈𝑅 (𝑋1, 𝑋2). that is,𝑈𝑅 (𝑋1, 𝑋2) = ⋃ {𝑅(𝑥1,𝑥2)𝜖(𝑈1,𝑈2) (𝑥1, 𝑥2): 𝑅(𝑥1, 𝑥2)⋂(𝑋1, 𝑋2) ≠ ϕ}. 3. the boundary region of (𝑋1, 𝑋2) with respect to r is the set of all objects, which can be classified neither as (𝑋1, 𝑋2) nor as not −(𝑋1, 𝑋2) with respect to r and it is denoted by 𝐵𝑅 (𝑋1, 𝑋2). that is, 𝐵𝑅 (𝑋1, 𝑋2) = 𝑈𝑅 (𝑋1, 𝑋2) − 𝐿𝑅 (𝑋1, 𝑋2). definition 2.2: [3] let (𝑈1, 𝑈2) be the universe, r be an equivalence on (𝑈1, 𝑈2) and 𝜏𝑅 (𝑋1, 𝑋2) = {(𝑈1, 𝑈2), (𝜙, 𝜙), 𝐿𝑅 (𝑋1, 𝑋2), 𝑈𝑅 (𝑋1, 𝑋2), 𝐵𝑅 (𝑋1, 𝑋2)} where (𝑋1, 𝑋2) ⊆ (𝑈1, 𝑈2). then by the property r(𝑋1, 𝑋2)satisfies the following axioms 1. (𝑈1, 𝑈2)and (𝜙, 𝜙) 𝜖 𝜏𝑅 (𝑋1, 𝑋2). 2. the union of the elements of any sub collection of 𝜏𝑅 (𝑋1, 𝑋2) is in 𝜏𝑅 (𝑋1, 𝑋2). 3. the intersection of the elements of any finite sub collection of 𝜏𝑅 (𝑋1, 𝑋2) is in 𝜏𝑅 (𝑋1, 𝑋2). that is,𝜏𝑅 (𝑋1, 𝑋2) is a topology on (𝑈1, 𝑈2) called the nano binary topology on (𝑈1, 𝑈2) with respect to(𝑋1, 𝑋2). 16 further diversification of nano binary open sets we call (𝑈1, 𝑈2, 𝜏𝑅 (𝑋1, 𝑋2))as the nano binary topological spaces. the elements of 𝜏𝑅 (𝑋1, 𝑋2)are called as nano binary open sets and it is denoted by 𝑁𝐵 open sets. their complement is called 𝑁𝐵 closed sets. definition 2.3: [3] if (𝑈1, 𝑈2, 𝜏𝑅 (𝑋1, 𝑋2))is a nano binary topological spaces with respect to (𝑋1, 𝑋2) and if (𝐻1, 𝐻2) ⊆ (𝑈1, 𝑈2), then the nano binary interior of (𝐻1, 𝐻2) is defined as the union of all 𝑁𝐵open subsets of (𝐴1, 𝐴2) and it is defined by 𝑁𝐵 ∘(𝐻1, 𝐻2). that is, 𝑁𝐵 ∘(𝐻1, 𝐻2) is the largest 𝑁𝐵open subset of(𝐻1, 𝐻2). the nano binary closure of (𝐻1, 𝐻2) is defined as the intersection of all 𝑁𝐵closed sets containing (𝐻1, 𝐻2) and it is denoted by 𝑁𝐵 (𝐻1, 𝐻2). that is, 𝑁𝐵 (𝐻1, 𝐻2) is the smallest 𝑁𝐵closed set containing(𝐻1, 𝐻2). 3. nano binary derived definition 3.1: a point (𝑥1, 𝑥2) ∈ (𝑈1, 𝑈2) is said to be a 𝑁𝐵 limit point of (𝐴1, 𝐴2) if for each 𝑁𝐵-open set (𝐾1, 𝐾2) containing (𝑥1, 𝑥2) satisfies (𝐾1, 𝐾2) ∩ ((𝐴1, 𝐴2) − (𝑥1, 𝑥2)) ≠ (∅, ∅). definition 3.2: the set of all 𝑁𝐵 limit points of (𝐴1, 𝐴2) is said to be nano binary derived set and is denoted by 𝑁𝐵 _𝐷(𝐴1, 𝐴2). theorem 3.3: in (𝑈1, 𝑈2, 𝜏𝑅 (𝑋1, 𝑋2)), let (𝐴1, 𝐴2) and (𝐵1, 𝐵2) be two subsets of (𝑈1, 𝑈2). then the following holds: 1) 𝑁𝐵 _𝐷(∅, ∅) = (∅, ∅). 2) if (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐷(𝐴1, 𝐴2) then (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐷((𝐴1, 𝐴2) − (𝑥1, 𝑥2)). 3) if (𝐴1, 𝐴2) ⊆ (𝐵1, 𝐵2), then 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ⊆ 𝑁𝐵 _𝐷(𝐵1, 𝐵2). 4) 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐵1, 𝐵2) = 𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)). proof: 1) let (𝑥1, 𝑥2) ∈ (𝑈1, 𝑈2) and (𝐺1, 𝐺2) be a 𝑁𝐵-open set containing (𝑥1, 𝑥2). then ((𝐺1, 𝐺2) − (𝑥1, 𝑥2)) ∩ (∅, ∅) = (∅, ∅) ⇒ (𝑥1, 𝑥2) ∉ 𝑁𝐵 _𝐷(∅, ∅). therefore, for any (𝑥1, 𝑥2) ∈ (𝑈1, 𝑈2), (𝑥1, 𝑥2) is not a𝑁𝐵 limit point of (∅, ∅). hence 𝑁𝐵 _𝐷(∅, ∅) = (∅, ∅). 2)let (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐷(𝐴1, 𝐴2). then (𝐺1, 𝐺2) ∩ ((𝐴1, 𝐴2) − (𝑥1, 𝑥2)) ≠ (∅, ∅), for every 𝑁𝐵-open set (𝐺1, 𝐺2) containing (𝑥1, 𝑥2) implies every 𝑁𝐵-open set (𝐺1, 𝐺2) of (𝑥1, 𝑥2), contains at least one point other than (𝑥1, 𝑥2) of (𝐴1, 𝐴2). therefore (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐷((𝐴1, 𝐴2) − (𝑥1, 𝑥2)). 3)let (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐷(𝐴1, 𝐴2). then (𝐺1, 𝐺2) ∩ ((𝐴1, 𝐴2) − (𝑥1, 𝑥2)) ≠ (∅, ∅), for every 𝑁𝐵-open set (𝐺1, 𝐺2) containing (𝑥1, 𝑥2). since (𝐴1, 𝐴2) ⊆ (𝐵1, 𝐵2) implies (𝐺1, 𝐺2) ∩ ((𝐵1, 𝐵2) − (𝑥1, 𝑥2)) ≠ (∅, ∅) ⇒ (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐷(𝐵1, 𝐵2). thus (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ⇒ (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐷(𝐵1, 𝐵2). therefore 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ⊆ 𝑁𝐵 _𝐷(𝐵1, 𝐵2). 4)since (𝐴1, 𝐴2) ⊆ (𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2) and (𝐵1, 𝐵2) ⊆ (𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2). by (3), 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ⊆ 𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) and 𝑁𝐵 _𝐷(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∪ 17 j. jasmine elizabeth, g. hari siva annam (𝐵1, 𝐵2)). therefore, 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)).… (1). let (𝑥1, 𝑥2) ∉ 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐵1, 𝐵2). then (𝑥1, 𝑥2) ∉ 𝑁𝐵 𝐷(𝐴1,𝐴2) and (𝑥1, 𝑥2) ∉ 𝑁𝐵 _𝐷(𝐵1, 𝐵2). therefore, there exists 𝑁𝐵-open sets (𝐺1, 𝐺2) and (𝐻1, 𝐻2) containing (𝑥1, 𝑥2) such that (𝐺1, 𝐺2) ∩ ((𝐴1, 𝐴2) − (𝑥1, 𝑥2)) = (∅, ∅) and (𝐻1, 𝐻2) ∩ ((𝐵1, 𝐵2) − (𝑥1, 𝑥2)) = (∅, ∅). since (𝐺1, 𝐺2) ∩ (𝐻1, 𝐻2) ⊆ (𝐺1, 𝐺2) and (𝐻1, 𝐻2), ((𝐺1, 𝐺2) ∩ (𝐻1, 𝐻2)) ∩ ((𝐴1, 𝐴2) − (𝑥1, 𝑥2)) = (∅, ∅) and ((𝐺1, 𝐺2) ∩ (𝐻1, 𝐻2)) ∩ ((𝐵1, 𝐵2) − (𝑥1, 𝑥2)) = (∅, ∅). also (𝐺1, 𝐺2) ∩ (𝐻1, 𝐻2) is a 𝑁𝐵-open set containing(𝑥1, 𝑥2). therefore, ((𝐺1, 𝐺2) ∩ (𝐻1, 𝐻2)) ∩ (((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) − (𝑥1, 𝑥2)) = (∅, ∅). that is, (𝑥1, 𝑥2) is not a 𝑁𝐵 limit point of (𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2). hence (𝑥1, 𝑥2) ∉ 𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)). therefore, 𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐵1, 𝐵2)…. (2). from (1) and (2),𝑁𝐵 _𝐷(𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐵1, 𝐵2) = 𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)). theorem 3.4: let (𝐴1, 𝐴2)and (𝐵1, 𝐵2) be two subsets of 𝑁𝐵 topological space (𝑈1, 𝑈2, 𝜏𝑅 (𝑋1, 𝑋2)).then 𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐷(𝐵1, 𝐵2). proof: since (𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2) ⊆ (𝐴1, 𝐴2)and (𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2) ⊆ (𝐵1, 𝐵2). by the previous theorem,𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐷(𝐴1, 𝐴2)and 𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐷(𝐵1, 𝐵2). therefore,𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐷(𝐵1, 𝐵2). remark 3.5: the reverse inclusion may not true as shown in the following example. example 3.6: u1 = {𝑎, 𝑏, 𝑐}, u2 = {1, 2} with (u1, u2) r ⁄ = {({a, b}, {2}), ({c}, {1})}. let (x1, x2) =({b}, {2}). then τr(x1, x2) ={(φ, φ), (u1, u2), ({a, b}, {2})}. here (𝐴1, 𝐴2) = ({a, b}, {1})and (𝐵1, 𝐵2) = ({b, c}, {1,2}), (𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)=({b}, {1}) and hence 𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2))={({a}, {1}), ({a}, {2}), ({b}, {2}), ({c}, {1}), ({c}, {2})}. also 𝑁𝐵 _𝐷(𝐴1, 𝐴2)={({a}, {1}), ({a}, {2}), ({b}, {1}), ({b}, {2}), ({c}, {1}), ({c}, {2})} and 𝑁𝐵 _𝐷(𝐵1, 𝐵2)={({a}, {1}), ({a}, {2}), ({b}, {1}), ({c}, {1}), ({c}, {2})}. but 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐷(𝐵1, 𝐵2)={({a}, {1}), ({a}, {2}), ({b}, {1}), ({c}, {1}), ({c}, {2})}. thus, 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐷(𝐵1, 𝐵2)⊈𝑁𝐵 _𝐷((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)). theorem 3.7: 𝑁𝐵 (𝐴1, 𝐴2) = (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐴1, 𝐴2), where (𝐴1, 𝐴2) ⊆ (u1, u2). proof: if (𝑥1, 𝑥2) ∈ (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐴1, 𝐴2), then (𝑥1, 𝑥2) ∈ (𝐴1, 𝐴2) or (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐷(𝐴1, 𝐴2). let (𝑥1, 𝑥2) ∉ (𝐴1, 𝐴2). then (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐷(𝐴1, 𝐴2). therefore, for every 𝑁𝐵-open set (𝐺1, 𝐺2)containing (𝑥1, 𝑥2), (𝐺1, 𝐺2) ∩ ((𝐴1, 𝐴2) − (𝑥1, 𝑥2)) ≠ (∅, ∅). since (𝑥1, 𝑥2) ∉ (𝐴1, 𝐴2), (𝐺1, 𝐺2) ∩ (𝐴1, 𝐴2) ≠ (∅, ∅). therefore, (𝑥1, 𝑥2) ∈ 𝑁𝐵 (𝐴1, 𝐴2). therefore, (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ⊆ 𝑁𝐵 (𝐴1, 𝐴2)…. (1). let (𝑥1, 𝑥2) ∈ 𝑁𝐵 (𝐴1, 𝐴2)and (𝑥1, 𝑥2) ∈ (𝐴1, 𝐴2). then the result is obvious. if (𝑥1, 𝑥2) ∈ 𝑁𝐵 (𝐴1, 𝐴2)and (𝑥1, 𝑥2) ∉ (𝐴1, 𝐴2). therefore, (𝐺1, 𝐺2) ∩ (𝐴1, 𝐴2) ≠ (∅, ∅) for every 𝑁𝐵-open set (𝐺1, 𝐺2) containing (𝑥1, 𝑥2) and hence (𝐺1, 𝐺2) ∩ ((𝐴1, 𝐴2) − (𝑥1, 𝑥2)) ≠ 18 further diversification of nano binary open sets (∅, ∅). therefore, (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐷(𝐴1, 𝐴2) and hence (𝑥1, 𝑥2) ∈ (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐴1, 𝐴2). therefore, 𝑁𝐵 (𝐴1, 𝐴2) ⊆ (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐴1, 𝐴2)… (2). from (1) and (2), 𝑁𝐵 (𝐴1, 𝐴2) = (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐴1, 𝐴2). result 3.8: 𝑁𝐵 𝑜 (𝐴1, 𝐴2) = (𝐴1, 𝐴2) − 𝑁𝐵 _𝐷[(u1, u2) − (𝐴1, 𝐴2)], where (𝐴1, 𝐴2) ⊆ (u1, u2). proof: by the previous theorem, 𝑁𝐵 (𝐴1, 𝐴2) = (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐷(𝐴1, 𝐴2)⇒(u1, u2) − 𝑁𝐵 (𝐴1, 𝐴2) = ((u1, u2) − (𝐴1, 𝐴2)) ∩ ((u1, u2) − 𝑁𝐵 _𝐷(𝐴1, 𝐴2)) ⇒ (u1, u2) − 𝑁𝐵 (𝐴1, 𝐴2) = ((u1, u2) − (𝐴1, 𝐴2)) − 𝑁𝐵 _𝐷(𝐴1, 𝐴2) ⇒ 𝑁𝐵 𝑜 ((u1, u2) − (𝐴1, 𝐴2)) = ((u1, u2) − (𝐴1, 𝐴2)) − 𝑁𝐵 _𝐷(𝐴1, 𝐴2). by replacing (u1, u2) − (𝐴1, 𝐴2) by (𝐴1, 𝐴2) and (𝐴1, 𝐴2)by (u1, u2) − (𝐴1, 𝐴2), 𝑁𝐵 𝑜 (𝐴1, 𝐴2) = (𝐴1, 𝐴2) − 𝑁𝐵 _𝐷[(u1, u2) − (𝐴1, 𝐴2)]. 4. nano binary exterior definition 4.1: for a subset(𝐴1, 𝐴2) ⊆ (𝑈1, 𝑈2), the nano binary exterior of (𝐴1, 𝐴2) is defined as𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − (𝐴1, 𝐴2)). it is denoted by 𝑁𝐵 _𝐸(𝐴1, 𝐴2). definition 4.2: for a subset(𝐴1, 𝐴2) ⊆ (𝑈1, 𝑈2), the nano binary border of (𝐴1, 𝐴2) is defined as (𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2). it is denoted by 𝑁𝐵 _𝐵(𝐴1, 𝐴2). theorem 4.3: let (𝐴1, 𝐴2)and (𝐵1, 𝐵2) be two subsets of 𝑁𝐵 topological space (𝑈1, 𝑈2, 𝜏𝑅 (𝑋1, 𝑋2)). then the following holds: 1) if (𝐴1, 𝐴2) ⊆ (𝐵1, 𝐵2), then 𝑁𝐵 _𝐸(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐸(𝐴1, 𝐴2). 2) 𝑁𝐵 _𝐸((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐸(𝐵1, 𝐵2). 3) 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐸(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐸((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)). proof: 1) if (𝐴1, 𝐴2) ⊆ (𝐵1, 𝐵2)then (𝑈1, 𝑈2) − (𝐵1, 𝐵2) ⊆ (𝑈1, 𝑈2) − (𝐴1, 𝐴2) ⇒ 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − (𝐴1, 𝐴2)) ⇒ 𝑁𝐵 _𝐸(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐸(𝐴1, 𝐴2). 2) since (𝐴1, 𝐴2) ⊆ (𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)and (𝐵1, 𝐵2) ⊆ (𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2). by (1), 𝑁𝐵 _𝐸((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐸(𝐴1, 𝐴2)and 𝑁𝐵 _𝐸((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐸(𝐵1, 𝐵2). therefore, 𝑁𝐵 _𝐸((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐸(𝐵1, 𝐵2). 3) since (𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2) ⊆ (𝐴1, 𝐴2)and (𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2) ⊆ (𝐵1, 𝐵2). by (1) 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ⊆ 𝑁𝐵 _𝐸((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2))and 𝑁𝐵 _𝐸(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐸((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)). therefore, 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐸(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐸((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)). remark 4.4: the inclusion may be strict. we can see in the following example. example 4.5: let u1 = {𝑎, 𝑏, 𝑐}, u2 = {1, 2} with (u1, u2) r ⁄ {({a, b}, {2}), ({c}, {1})}. let (x1, x2) =({b}, {2}). then τr(x1, x2) ={(φ, φ), (u1, u2), ({a, b}, {2})}. 2) take (𝐴1, 𝐴2) = ({𝑎, 𝑏}, {2}) 𝑎𝑛𝑑 (𝐵1, 𝐵2) = ({c}, {1}). 𝑁𝐵 _𝐸({𝑎, 𝑏}, {2}) ∪ 𝑁𝐵 _𝐸({c}, {1}) = 𝑁𝐵 𝑜 ({c}, {1}) − 𝑁𝐵 𝑜 ({𝑎, 𝑏}, {2}) = (φ, φ) − ({𝑎, 𝑏}, {2}) = ({𝑎, 𝑏}, {2}). also, 𝑁𝐵 _𝐸(({𝑎, 𝑏}, {2}) ∪ ({c}, {1})) = 𝑁𝐵 _𝐸(u1, u2) = 𝑁𝐵 𝑜 (φ, φ) = 19 j. jasmine elizabeth, g. hari siva annam (φ, φ). therefore, ({a, b}, {2}) ⊈ (φ, φ) and hence 𝑁𝐵 _𝐸((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) ⊂ 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐸(𝐵1, 𝐵2). 3) take (𝐴1, 𝐴2) = ({𝑐}, {1,2}) 𝑎𝑛𝑑 (𝐵1, 𝐵2) = ({a, c}, {1}). 𝑁𝐵 _𝐸(({𝑐}, {1,2}) ∩ ({a, c}, {1})) = 𝑁𝐵 _𝐸({c}, {1}) = 𝑁𝐵 𝑜 ({𝑎, 𝑏}, {2}) = ({𝑎, 𝑏}, {2})and 𝑁𝐵 _𝐸({𝑐}, {1,2}) ∩ 𝑁𝐵 _𝐸({a, c}, {1}) = 𝑁𝐵 𝑜 ({a, b}, {∅}) ∩ 𝑁𝐵 𝑜 ({b}, {2}) = (φ, φ) ∩ (φ, φ) = (φ, φ). therefore, ({a, b}, {2}) ⊈ (φ, φ) and hence 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐸(𝐵1, 𝐵2) ⊂ 𝑁𝐵 _𝐸((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)). theorem 4.6: let (𝐴1, 𝐴2) and (𝐵1, 𝐵2) be two subsets of 𝑁𝐵 topological space (𝑈1, 𝑈2, 𝜏𝑅 (𝑋1, 𝑋2)). then the following holds: 1) 𝑁𝐵 _𝐸(𝐴1, 𝐴2) = (𝑈1, 𝑈2) − 𝑁𝐵 (𝐴1, 𝐴2) 2)𝑁𝐵 _𝐸(𝑁𝐵 _𝐸(𝐴1, 𝐴2)) = 𝑁𝐵 𝑜 (𝑁𝐵 (𝐴1, 𝐴2)) 3) 𝑁𝐵 _𝐸(𝑈1, 𝑈2) = (∅, ∅)and𝑁𝐵 _𝐸(∅, ∅) = (𝑈1, 𝑈2) 4) 𝑁𝐵 _𝐸(𝐴1, 𝐴2) = 𝑁𝐵 _𝐸[(𝑈1, 𝑈2) − 𝑁𝐵 _𝐸(𝐴1, 𝐴2)] 5) 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ⊆ 𝑁𝐵 _𝐸(𝑁𝐵 _𝐸(𝐴1, 𝐴2) 6) 𝑁𝐵 𝑜 (𝐴1, 𝐴2), 𝑁𝐵 _𝐸(𝐴1, 𝐴2), 𝑁𝐵 _𝐹(𝐴1, 𝐴2) are mutually disjoint and (𝑈1, 𝑈2) = 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐹(𝐴1, 𝐴2). 7) (𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐸(𝐴1, 𝐴2) = (∅, ∅) 8) 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ⊆ (𝑈1, 𝑈2) − (𝐴1, 𝐴2) proof: 1)𝑁𝐵 _𝐸(𝐴1, 𝐴2) = 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − (𝐴1, 𝐴2)) = (𝑈1, 𝑈2) − 𝑁𝐵 (𝐴1, 𝐴2). hence (1) is proved. 2)𝑁𝐵 _𝐸(𝑁𝐵 _𝐸(𝐴1, 𝐴2)) = 𝑁𝐵 _𝐸[𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − (𝐴1, 𝐴2))] = 𝑁𝐵 _𝐸[(𝑈1, 𝑈2) − 𝑁𝐵 (𝐴1, 𝐴2)] = 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − [(𝑈1, 𝑈2) − 𝑁𝐵 (𝐴1, 𝐴2)]) = 𝑁𝐵 𝑜 (𝑁𝐵 (𝐴1, 𝐴2)).therefore, 𝑁𝐵 _𝐸(𝑁𝐵 _𝐸(𝐴1, 𝐴2)) = 𝑁𝐵 𝑜 (𝑁𝐵 (𝐴1, 𝐴2)). 3)𝑁𝐵 _𝐸(𝑈1, 𝑈2) = 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − (𝑈1, 𝑈2)) = 𝑁𝐵 𝑜 (∅, ∅) = (∅, ∅)and 𝑁𝐵 _𝐸(∅, ∅) = 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − (∅, ∅)) = 𝑁𝐵 𝑜 (𝑈1, 𝑈2) = (𝑈1, 𝑈2). therefore, 𝑁𝐵 _𝐸(𝑈1, 𝑈2) = (∅, ∅)and 𝑁𝐵 _𝐸(∅, ∅) = (𝑈1, 𝑈2). 4)𝑁𝐵 _𝐸[(𝑈1, 𝑈2) − 𝑁𝐵 _𝐸(𝐴1, 𝐴2)] = 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − [(𝑈1, 𝑈2) − 𝑁𝐵 _𝐸(𝐴1, 𝐴2)]) = 𝑁𝐵 𝑜 (𝑁𝐵 _𝐸(𝐴1, 𝐴2)) = 𝑁𝐵 𝑜 [𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − (𝐴1, 𝐴2))] = 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − (𝐴1, 𝐴2)) = 𝑁𝐵 _𝐸(𝐴1, 𝐴2). therefore, 𝑁𝐵 _𝐸(𝐴1, 𝐴2) = 𝑁𝐵 _𝐸[(𝑈1, 𝑈2) − 𝑁𝐵 _𝐸(𝐴1, 𝐴2)]. 5)since (𝐴1, 𝐴2) ⊆ 𝑁𝐵 (𝐴1, 𝐴2) ⇒ 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ⊆ 𝑁𝐵 𝑜 (𝑁𝐵 (𝐴1, 𝐴2)) = 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − 𝑁𝐵 𝑜 [(𝑈1, 𝑈2) − (𝐴1, 𝐴2)]) = 𝑁𝐵 _𝐸(𝑁𝐵 𝑜 [(𝑈1, 𝑈2) − (𝐴1, 𝐴2)]) = 𝑁𝐵 _𝐸(𝑁𝐵 _𝐸(𝐴1, 𝐴2)). therefore, 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ⊆ 𝑁𝐵 _𝐸(𝑁𝐵 _𝐸(𝐴1, 𝐴2)). 6)assume that 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ∩ 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ≠ (∅, ∅). then there exists (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ∩ 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ⇒ (𝑥1, 𝑥2) ∈ 𝑁𝐵 _𝐸(𝐴1, 𝐴2) and (𝑥1, 𝑥2) ∈ 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ⇒ (𝑥1, 𝑥2) ∈ (𝑈1, 𝑈2) − (𝐴1, 𝐴2)and (𝑥1, 𝑥2) ∈ (𝐴1, 𝐴2), which is not possible. hence our 20 further diversification of nano binary open sets assumption is wrong. therefore, 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ∩ 𝑁𝐵 𝑜 (𝐴1, 𝐴2) = (∅, ∅). in the same way we can prove the others. 𝑁𝐵 _𝐸(𝐴1, 𝐴2) = (𝑈1, 𝑈2) − 𝑁𝐵 (𝐴1, 𝐴2) = (𝑈1, 𝑈2) − 𝑁𝐵 (𝐴1, 𝐴2) = (𝑈1, 𝑈2) − [𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐹(𝐴1, 𝐴2)]. therefore, (𝑈1, 𝑈2) = 𝑁𝐵 _𝐸(𝐴1, 𝐴2) ∪ 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐹(𝐴1, 𝐴2). 7)(𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐸(𝐴1, 𝐴2) = (𝐴1, 𝐴2) ∩ 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − (𝐴1, 𝐴2)) ⊆ (𝐴1, 𝐴2) ∩ ((𝑈1, 𝑈2) − (𝐴1, 𝐴2)) = (∅, ∅). therefore, (𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐸(𝐴1, 𝐴2) = (∅, ∅). 8)𝑁𝐵 _𝐸(𝐴1, 𝐴2) = 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − (𝐴1, 𝐴2)) ⊆ (𝑈1, 𝑈2) − (𝐴1, 𝐴2). note 4.7: if (𝐴1, 𝐴2) is 𝑁𝐵 closed, then equality holds in (5). theorem 4.8: in (𝑈1, 𝑈2, 𝜏𝑅 (𝑋1, 𝑋2)), (𝐴1, 𝐴2)and (𝐵1, 𝐵2) be two subsets of (𝑈1, 𝑈2). then the following holds: 1) 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐵(𝐴1, 𝐴2) = (∅, ∅) 2) (𝐴1, 𝐴2) is 𝑁𝐵-open if and only if 𝑁𝐵 _𝐵(𝐴1, 𝐴2) = (∅, ∅) 3) 𝑁𝐵 𝑜 (𝑁𝐵 _𝐵(𝐴1, 𝐴2)) = (∅, ∅) 4) 𝑁𝐵 _𝐵(𝑁𝐵 𝑜 (𝐴1, 𝐴2)) = (∅, ∅) 5) 𝑁𝐵 _𝐵(𝑁𝐵 _𝐵(𝐴1, 𝐴2)) = 𝑁𝐵 _𝐵(𝐴1, 𝐴2) 6) 𝑁𝐵 _𝐵(𝐴1, 𝐴2) = (𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2) = (𝐴1, 𝐴2) ∩ 𝑁𝐵 ((𝑈1, 𝑈2) − (𝐴1, 𝐴2)). 7) if (𝐴1, 𝐴2) ⊆ (𝐵1, 𝐵2), then 𝑁𝐵 _𝐵(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐵(𝐴1, 𝐴2). 8) 𝑁𝐵 _𝐵((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐵(𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐵(𝐵1, 𝐵2). 9) 𝑁𝐵 _𝐵(𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐵(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐵((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)). 10) 𝑁𝐵 _𝐵(𝐴1, 𝐴2) = 𝑁𝐵 _𝐷((𝑈1, 𝑈2) − (𝐴1, 𝐴2))and 𝑁𝐵 _𝐷(𝐴1, 𝐴2) = 𝑁𝐵 _𝐵((𝑈1, 𝑈2) − (𝐴1, 𝐴2)). 11) (𝐴1, 𝐴2) = 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐵(𝐴1, 𝐴2). proof: 1) 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐵(𝐴1, 𝐴2) = 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∩ [(𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2)] = 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∩ [(𝐴1, 𝐴2) ∩ ((𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2))] = 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∩ ((𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2)) ∩ (𝐴1, 𝐴2) = (∅, ∅) ∩ (𝐴1, 𝐴2) = (∅, ∅).therefore, 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐵(𝐴1, 𝐴2) = (∅, ∅) 2)any subset (𝐴1, 𝐴2) of 𝑁𝐵 topological space (𝑈1, 𝑈2, 𝜏𝑅 (𝑋1, 𝑋2)) is 𝑁𝐵-open ⇔(𝐴1, 𝐴2) = 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ⇔ (𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2) = (∅, ∅) ⇔ 𝑁𝐵 _𝐵(𝐴1, 𝐴2) = (∅, ∅). 3)𝑁𝐵 𝑜 (𝑁𝐵 _𝐵(𝐴1, 𝐴2)) = 𝑁𝐵 𝑜 ((𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2)) = 𝑁𝐵 𝑜 [(𝐴1, 𝐴2) ∩ ((𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2))] ⊆ 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∩ 𝑁𝐵 𝑜 ((𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2)) ⊆ 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∩ ((𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2)) = (∅, ∅). therefore, 𝑁𝐵 𝑜 (𝑁𝐵 _𝐵(𝐴1, 𝐴2)) = (∅, ∅). 4)𝑁𝐵 _𝐵(𝑁𝐵 𝑜 (𝐴1, 𝐴2)) = 𝑁𝐵 𝑜 (𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝑁𝐵 𝑜(𝐴1, 𝐴2)) = 𝑁𝐵 𝑜 (𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2) = (∅, ∅). therefore, 𝑁𝐵 _𝐵(𝑁𝐵 𝑜 (𝐴1, 𝐴2)) = (∅, ∅). 5)𝑁𝐵 _𝐵(𝑁𝐵 _𝐵(𝐴1, 𝐴2)) = 𝑁𝐵 _𝐵(𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝑁𝐵 _𝐵(𝐴1, 𝐴2)) = 𝑁𝐵 _𝐵(𝐴1, 𝐴2) − (∅, ∅) (by (3)) = 𝑁𝐵 _𝐵(𝐴1, 𝐴2). therefore, 𝑁𝐵 _𝐵(𝑁𝐵 _𝐵(𝐴1, 𝐴2)) = 𝑁𝐵 _𝐵(𝐴1, 𝐴2). 21 j. jasmine elizabeth, g. hari siva annam 6)𝑁𝐵 _𝐵(𝐴1, 𝐴2) = (𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2) = (𝐴1, 𝐴2) ∩ ((𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2)) = (𝐴1, 𝐴2) ∩ 𝑁𝐵 ((𝑈1, 𝑈2) − (𝐴1, 𝐴2)). therefore, 𝑁𝐵 _𝐵(𝐴1, 𝐴2) = (𝐴1, 𝐴2) ∩ 𝑁𝐵 ((𝑈1, 𝑈2) − (𝐴1, 𝐴2)). 7)if (𝐴1, 𝐴2) ⊆ (𝐵1, 𝐵2), then 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ⊆ 𝑁𝐵 𝑜 (𝐵1, 𝐵2) ⇒ (𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐵1, 𝐵2) ⊆ (𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ⇒ (𝐴1, 𝐴2) ∩ ((𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐵1, 𝐵2)) ⊆ (𝐴1, 𝐴2) ∩ ((𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2)) ⇒ (𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐵1, 𝐵2) ⊆ (𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ⇒ 𝑁𝐵 _𝐵(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐵(𝐴1, 𝐴2) (by (6)) 8)since (𝐴1, 𝐴2) ⊆ (𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)and (𝐵1, 𝐵2) ⊆ (𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2). by (4) 𝑁𝐵 _𝐵((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐵(𝐴1, 𝐴2)and 𝑁𝐵 _𝐵((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐵(𝐵1, 𝐵2). therefore, 𝑁𝐵 _𝐵((𝐴1, 𝐴2) ∪ (𝐵1, 𝐵2)) ⊆ 𝑁𝐵 _𝐵(𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐵(𝐵1, 𝐵2). 9)since (𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2) ⊆ (𝐴1, 𝐴2)and (𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2) ⊆ (𝐵1, 𝐵2). by (4), 𝑁𝐵 _𝐵(𝐴1, 𝐴2) ⊆ 𝑁𝐵 _𝐵((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2))and 𝑁𝐵 _𝐵(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐵((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)). therefore, 𝑁𝐵 _𝐵(𝐴1, 𝐴2) ∩ 𝑁𝐵 _𝐵(𝐵1, 𝐵2) ⊆ 𝑁𝐵 _𝐵((𝐴1, 𝐴2) ∩ (𝐵1, 𝐵2)). 10)𝑁𝐵 _𝐵(𝐴1, 𝐴2) = (𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2). by result 3.8, (𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2) = (𝐴1, 𝐴2) − [(𝐴1, 𝐴2) − 𝑁𝐵 _𝐷((𝑈1, 𝑈2) − (𝐴1, 𝐴2))] = 𝑁𝐵 _𝐷((𝑈1, 𝑈2) − (𝐴1, 𝐴2))by replacing (𝐴1, 𝐴2)by (𝑈1, 𝑈2) − (𝐴1, 𝐴2), 𝑁𝐵 _𝐷(𝐴1, 𝐴2) = 𝑁𝐵 _𝐵((𝑈1, 𝑈2) − (𝐴1, 𝐴2)). 11)𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐵(𝐴1, 𝐴2) = 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∪ ((𝐴1, 𝐴2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2)) = 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∪ ((𝐴1, 𝐴2) ∩ ((𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2))) = (𝑁𝐵 𝑜(𝐴1, 𝐴2) ∪ (𝐴1, 𝐴2)) ∩ (𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∪ ((𝑈1, 𝑈2) − 𝑁𝐵 𝑜 (𝐴1, 𝐴2))) = (𝐴1, 𝐴2) ∩ (𝑈1, 𝑈2) = (𝐴1, 𝐴2). therefore, (𝐴1, 𝐴2) = 𝑁𝐵 𝑜 (𝐴1, 𝐴2) ∪ 𝑁𝐵 _𝐵(𝐴1, 𝐴2). 5. conclusion nano binary derived, nano binary border and nano binary exterior in nano binary topological spaces were introduced and their properties were discussed. in future we will discuss generalized closed sets in nano binary topological spaces. references [1] lellis thivagar. m and carmel richard, on nano forms of weakly open sets, international journal of mathematics and statistics invention, 1(1) 2013, 31-37. [2] nithyanantha jothi. s and p. thangavelu, on binary topological spaces, pacific asian journal of mathematics. [3] hari siva annam. g and j. jasmine elizabeth, cognition of nano binary topological spaces, global journal of pure and applied mathematics, 6(2019), 10451054. 22 further diversification of nano binary open sets [4] jasmine elizabeth. j and g. hari siva annam, some notions on nano binary continuous, malaya journal of matematik, vol. 9, n0. 1,592-597, 2021. [5] jasmine elizabeth. j and g. hari siva annam, nano binary contra continuous functions in nano binary topological spaces, j. math. comput. sci. 11 (2021), no. 4, 4994-5011. [6] lellis thivagar. m and sutha devi. v, on multi-granular nano topology, accepted (2015), south east asian bulletin of mathematics, springer verlag. [7] c. janaki and a. jeyalakshmi, a new form of generalized closed sets via regular local function in ideal topological spaces, malaya journal of matematik, s (1) (2015), 1-9. [8] o. njastad, on some classes of nearly open sets, pacific j. math.,15 (1965), 961970. [9] e. f. lashin and t. medhat, topological reduction of information systems, chaos, solitions and fractals, 25 (2015), 277-286. [10] c. richard, studies on nano topological spaces, ph.d. thesis, madurai kamaraj university, india (2013). 23 microsoft word capitolo intero n 6.doc microsoft word cap6.doc ratio mathematica volume 48, 2023 products on maximal compact frames jayaprasad p n* abstract in topological spaces, many topological properties such as separation properties, paracompactness etc. are preserved under the act of taking product of topological spaces. the well known tychonoff theorem in topological spaces which states that product of compact spaces is compact. many of these results can be extended to the “generalized topological spaces”, known as locales(frames). according to tychonoff product theorem for locales, locale product(coproduct of frames) of compact locales(frames) is compact. in this paper, we examine whether the coproduct of maximal compact frames is maximal compact. we examine the case for a finite coproduct and for an arbitrary coproduct of maximal compact frames. every subframe of a compact frame is compact but a sublocale need not be. we provide a characterization for a sublocale of a maximal compact frame to be a maximal compact sublocale. keywords: frame, locale, spatial frame, maximal compact frame, subframe, sublocale. 2020 ams subject classifications: 06d22, 54e 1 *department of mathematics, rajiv gandhi institute of technology, velloor p. o., pampdy, kottayam, india, e-mail: jayaprasadpn@rit.ac.in 1received on february 18, 2023. accepted on july 10, 2023. published on august 1, 2023. doi: 10.23755/rm.v39i0.1140. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. jayaprasad p n 1 introduction garrett birkhoff, in 1936, pointed out the notion of the comparison of two different topologies on the same basic set. he had done this by ordering these topologies as a lattice under set inclusion. a topological space (x,t) with property r is said to be maximal r if t is a maximal element in the set r(x) of all topologies on the set x having property r with the partial ordering of set inclusions. the set of all topologies sharing a given property may not have a greatest element, but it may have maximal elements. in topological spaces, a closed subspace of a compact space is compact and a compact subspace of a hausdorff space is closed. thus in a compact hausdorff space, closed subspaces coincide with compact subspaces. ”a topological space is maximal compact if and only if its compact subsets are precisely the closed sets”, as proved in [12]. norman levine named those spaces in which closed subsets coincide with compact subsets as c-c spaces. a detailed analysis of its properties are discussed in [10]. a locale is a categorical extension of a topological space whereas a frame is an object in its opposite category. in 1972, through the paper [3], j.r.isbell pointed out the need for a seperate terminology for the dual category of frames. the objects of this dual category are named as “locales” by him and they are actually the “generalized spaces”. a pointfree analogue of the maximal compact space and its associated properties are discussed in [6]. a maximal compact frame is a compact frame which is not properly contained in any other compact frame. a characterization of the maximal compact frame is that a subframe a of any non-compact frame l is maximal compact if and only if the closed sublocales of a are exactly the compact sublocales of a. it is also proved that a compact hausdorff frame is maximal compact. we know that many of the topological properties such as separation properties, paracompactness etc. are preserved under the act of taking product of topological spaces. the well known tychonoff theorem in topological spaces which states that product of compact spaces is compact is equivalent to the axiom of choice by [9]. coproducts(products) in frames(locales) are the categorical counterpart of product of topological spaces. frame theory has the advantage that many results in topology requiring axiom of choice or some of its variants can be proved without its use. the tychonoff’s theorem for locales is constructively proved by [7] without using the axiom of choice. in this paper, we examine whether the coproduct of maximal compact frames is maximal compact. we examine the case for a finite coproduct and for an arbitrary coproduct of maximal compact frames. every subframe of a compact frame is compact but a sublocale need not be. we provide a characterization for a sublocale of a maximal compact frame to be a maximal compact sublocale. products on maximal compact frames 2 preliminaries the term frame was coined by c.h. dowker. a remarkable study on frames was done by them in [1]. a frame is a complete lattice l in which the infinite distributive law a∧ ∨ s = ∨ {a∧s : s ∈ s} holds for all a ∈ l,s ⊆ l. a map between frames that preserves arbitrary joins and finite meets is called a frame homomorphism. associated with a frame homomorphism h : m → l is its right adjoint h∗ : l → m given by h∗(b) = ∨ {x ∈ m : h(x) ≤ b}. we denote the top element and the bottom element of a frame by 1 and 0 respectively. the category of frames and frame homomorphisms is denoted by frm. the dual category frmop is referred to as the category of locales denoted by loc. the morphisms in loc, called localic maps, are given by the right adjoints of frame homomorphisms between two objects. a frame is said to be spatial, if it is isomorphic to the topology ωx of a topological space (x, ωx). a subset of a frame which is closed under arbitrary joins and finite meets in that frame is called a subframe. a sublocale m of a locale l can be represented in terms of an onto frame homomorphism h : l → m in the sense that the image of m under the right adjoint h∗ : m → l will represent that sublocale. for a locale l, denote ↑ a = {x ∈ l : x ≥ a} and ↓ b = {x ∈ l : x ≤ b}. then the sublocale given by the frame homomorphism j : l →↑ a defined by x → a ∨ x for any a ∈ l is called a closed sublocale of l. a cover in a frame l is a subset s of l with ∨ s = 1l. a frame l is said to be compact if each cover a of l has a finite subcover. the construction of coproducts in frames which was first presented in [1]. the construction is provided below through the following definitions. definition 2.1. let r ⊆ a×a be an arbitrary binary relation on a frame a. an element s ∈ a is r-saturated if arb ⇒ (a∧c ≤ s ⇔ b∧c ≤ s) foralla,b,c ∈ a let a/r denotes the set of all saturated elements of a. define ν : a → a/r by ν(a) = νr(a) = ∧ {s ∈ a : a ≤ s} where s is saturated. then ν is a surjective frame homomorphism. for a semilattice a, define d(a) = {u ⊆ a : φ 6= u =↓ u}. then (d(a),⊆) is a frame. define λa : a → da by λa(a) =↓ a which is a semilattice homomorphism between them. let ai, i ∈ i be frames. set∏′ i∈i ai = {(ai)i∈i ∈ ∏ i∈i ai : ai = 1 for all but finitely many i } ⋃ {(0)i∈i} define γj : aj → ∏′ i∈i ai by setting (γj(a))i = { a if i = j 1 otherwise consider the frame d(a) where a = ∏′ i∈i ai. r = { (λaγj( ∨ m∈m am), ∨ m∈m λaγj(am)) : j ∈ i,am ∈ aj } where m is any set is a relation. jayaprasad p n definition 2.2. the frame ⊕ i∈i ai = d( ∏′ i∈i ai)/r, containing all r-saturated elements of the frame d( ∏′ i∈i ai) is called the frame coproduct. definition 2.3. the mapping ν : d( ∏′ i∈i ai) → ⊕ i∈i ai be as defined above. the maps pj = ν ◦ λ ◦ γj : aj → ⊕ i∈i ai are frame homomorphisms, called coproduct injections. also ∧ i∈ipi(ai) = ⊕(ai)i∈i . thus the set of all elements of the form ⊕(ai)i∈i is a join basis of ⊕(ai)i∈i . the right adjoint of pj denoted by pj ∗ is called the projection of the locale ⊕ i∈i ai to the locale aj. a frame a is called a hausdorff frame if for any u ∈ a⊕a, the codiagonal ∇ : a⊕a → a defined by ∇(u) = ∨ {a∧ b : (a,b) ∈ u} is a closed sublocale. the frame l is said to be a regular frame if a = ∨ {x ∈ l/x ≺ a} for all a ∈ l. for a detailed reading regarding frames we refer to [11]. 3 maximal compact frames and frame coproducts an intense research has been done on maximal and minimal topologies so far. the properties of maximal topologies for compactness, countable compactness, and sequential compactness etc. are investigated in [2] and the relations between these spaces are investigated. the paper also presented discussions on some interesting product theorems. analogously maximal compactness in the pointfree counterpart is studied by [6]. a maximal compact frame is defined here as given below definition 3.1. [6] a frame a is said to be maximal compact if, 1. a is compact, 2. if a is a proper subframe of the frame l, then l is not compact. a characterization for maximal compact frames is obtained in [6]. theorem 3.1. [6] a frame is maximal compact if and only if its closed sublocales are exactly the compact sublocales. for a topological space x, the frame of open sets is denoted by ωx. a relation between a c-c space and the associated frame of open sets is established in the following result. products on maximal compact frames theorem 3.2. [6] the topological space (x, ωx) is a c-c space if and only if ωx is a maximal compact frame. a compact hausdorff topological space is regular and the same is true locales too. theorem 3.3. [11] a compact hausdorff locale is regular. the property of being a c-c spaces is productive if the component spaces are hausdorff. the product hausdorff c-c spaces are c-c and vice-versa by the following result. theorem 3.4. [10] let (x,τ) be a topological space and let (x ×x,τ∗) be the cartesian product of (x,τ) with itself. then (x × x,τ∗) is c-c if and only if (x,τ) is c-c and hausdorff. the following three results regarding productive properties of compact frame as well as regular frames are proved in [8]. they are as expected as one learned in topological spaces. theorem 3.5. a compact regular frame(locale) is spatial. theorem 3.6. the coproduct(product) of compact frames(locales) is compact. theorem 3.7. the coproduct(product) of regular frames(locales) is regular. the frame coproduct of component spaces is isomorphic to frame of open sets of the product topology of these spaces. theorem 3.8. [11] let xi, i ∈ i be family of spaces. then ⊕i(ωxi) is isomorphic to ω(⊕xi) if and only if it is a spatial frame. 3.1 coproducts on maximal compact frames most of the topological properties are preserved under the act of taking product. the productive properties of c-c spaces are discussed and a characterization is obtained in [10]. the paper also provides some results associated with the productive properties of c-c spaces. the previous section contains characterization for a maximal compact frame and some results associated with it. in this section, we are conducting a study on productive properties of maximal compact frames. the inverse image of a compact sublocale of a compact frame under the localic projection map in the coproduct a given nonempty family of nonempty frames is again compact by the following result. jayaprasad p n theorem 3.9. let {ai : i ∈ i} be a nonempty family of nonempty compact frames and let a be the frame coproduct. if cj is a compact sublocale of aj, then pj ∗−1(cj) is compact in a. proof. take aj = cj in ⊕ i∈i ai, then we have pj ∗( ⊕ i∈i ai) = cj. thus pj ∗−1(cj) = ⊕ i∈i ai where aj = cj. since cj is a compact sublocale and all the other a,is are compact, by tychonoff theorem for locales, ⊕ i∈i ai is compact. hence pj∗ −1(cj) is compact in a. theorem 3.10. a locale l is compact if and only if the product projection p∗ : l ⊕ m → m( the coproduct injection p : m → l ⊕ m in frame language) is closed for every locale m the above result is known as the kuratowski-mrowka theorem for locales[11] and is used for proving the next result.. theorem 3.11. let {ai : i ∈ i} be a non empty family of non empty compact frames and let a be the frame coproduct. if a is a maximal compact frame, then each ai is a maximal compact frame. proof. if cj is closed in aj, then it is compact as a closed sublocale of a compact frame is compact. now suppose that cj is a compact sublocale of aj. then by theorem 3.9, c = pj∗ −1(cj) is compact in a and hence it is closed in a as a is a cce frame. then pj∗(c) = cj and since a is compact by tychonoff theorem for locales cj is closed as the projections of a to ai being closed maps bytheorem 3.10. hence each ai is a maximal compact frame. the converse of the above result need not be true. we prove this through the next theorem which tells that the coproduct of a compact frame a with itself is a maximal compact frame if and only if a is a maximal compact frame that is hausdorff. hence, if the condition hausdorffness is dropped, then the coproduct may not be a maximal compact frame. theorem 3.12. let a be any compact frame and let a⊕a be the coproduct of a with itself. then a ⊕ a is a maximal compact frame if and only if a is maximal compact and hausdorff. proof. suppose that a ⊕ a is a maximal compact frame. then a is a maximal compact frame by theorem 3.11. let ∆(u) = ∨ {a∧ b : (a,b) ∈ u} where u ∈ a ⊕ a. then ∆ : a ⊕ a → a called the codiagonal is a surjective frame homomorphism and hence its right adjoint ∆∗ is a sublocale map by which a is a sublocale of a⊕a. since a is a maximal compact frame, it is compact. hence a is a compact sublocale of a⊕a. since a⊕a is a maximal compact frame, a is products on maximal compact frames closed in a⊕a, by theorem 3.1. thus a is hausdorff as the diagonal ∆ embeds a as a closed sublocale of a⊕a, by definition of a hausdorff frame. assume that a is a maximal compact frame and hausdorff. a compact hausdorff frame is regular by theorem 3.3. since a compact regular frame is spatial by theorem 3.5, a is spatial. then the topological space corresponding to a is a c-c space by corollary 3.2. also the space corresponding to a regular frame is regular and hence hausdorff. thus a is a c-c space that is hausdorff and hence by theorem 3.4, the product topological space is c-c. then by corollary 3.2, the product topology is a maximal compact frame. since a is compact and regular, by theorem 3.6 and theorem 3.7, a ⊕ a is compact and regular. then, by theorem 3.5, a ⊕ a is spatial. now, by theorem 3.8, a ⊕ a is isomorphic to the product topology. hence a⊕a is a maximal compact frame. we know that every subframe of a compact frame is compact. but every sublocale of a compact locale need not be compact. it happens when the sublocale becomes a closed sublocale. we know that a sublocale is different from a subframe. a sublocale is a quotient frame and hence it cannot be regarded as a subframe of a frame. hence a maximal compact frame can have a sublocale which in its own respect may become a maximal compact frame. in the next theorem, we prove that the above situation occurs when the sublocale is closed. theorem 3.13. let a be a maximal compact frame. a sublocale k of a is maximal compact if and only if k is closed in a. proof. suppose that the sublocale k of a is maximal compact. then it is a compact sublocale of a and hence closed, as a is a maximal compact frame. conversely, suppose that k is a closed sublocale of a. then k is a compact sublocale of a as it is maximal compact. thus any closed sublocale of k is compact. now assume that k1 is a compact sublocale of k. then it is a compact sublocale of a. since a is a maximal compact frame k1 is closed in a. therefore k1 = ↑aa where a ∈ a. since k1 is a sublocale of k, we have ↑aa = ↑ka. hence k1 is closed in k. thus k is a maximal compact locale. 4 conclusions as discussed above, the study of productive properties in topological spaces were extensively studied in [2]. but the extension of these results to the “generalized spaces” were not done to that extent. at first we characterized the maximal compact frames in [6] and consequently took the first step to examine the productive property of maximal compact frames. we conclude that if an arbitrary coproduct of compact frames is maximal compact then each component frame is jayaprasad p n maximal compact and the converse need not hold. we also examined the same result in finite case and observed that hausdorffness of the component frames is necessary for the two way existence of the result. a frame is said to be reversible[5], if every order preserving self bijection is a frame isomorphism. a characterization for reversible frames is given in [5]. it is also proved that a frame that is maximal or minimal with respect to some frame isomorphic property is reversible. hence the characterization for maximal compact frames can be used as a method to identify reversible frames . thereby one can get some information about the automorphism groups possible on finite frames that are maximal compact[4]. the possible automorphism groups on infinite frames is still an open problem. acknowledgements i would like to thank dr. t. p johnson, professor and head, department of applied sciences and humanities, school of engineering, cochin university of science and technology, cochin, kerala, india for giving me valuable suggestions for improving this paper. i am equally indebted to the valuable comments of the reviewers in shaping the paper to its final version. references [1] c. h. dowker and d. strauss. sums in the category of frames. houston j. math., 3:7–15, 1977. [2] douglas e. cameron. maximal and minimal topologies. trans. amer. math. soc., 160:229–248, 1971. [3] j.r. isbell. atomless parts of spaces. math. scand., 31:5–32, 1972. [4] p. n. jayaprasad and t.p. johnson. automorphism group of finite frames. international journal of algebra and statistics, 1(2):118–123, 2012. [5] p. n. jayaprasad and t.p. johnson. reversible frames. journal of advanced studies in topology, 3(2):7–13, 2012. [6] p. n. jayaprasad, n. m. madhavan namboothiri, p. k.santhosh, and varghese jacob. on maximal compact frames. korean j. math., 29(3):493–499, 2021. products on maximal compact frames [7] p. t. johnstone. tychonoff’s thoerem without the axiom of choice. fund. math., 113:21–35, 1981. [8] p. t. johnstone. stone spaces. camb. univ. press, united kingdom, 1982. [9] j. l. kelley. the tychonoff product theorem implies the axiom of choice. fund. math., 37:75–76, 1950. [10] n. levine. when are compact and closed equivalent? amer. math. month., 71(1):41–44, 1965. [11] j. picado and a. pultr. frames and locales-topology without points. birkhäuser, switzerland, 2012. [12] a. ramanathan. minimal-bicompact spaces. j.indian math. soc., 12:40–46, 1948. microsoft word cap4.doc microsoft word cap8.doc ratio mathematica vol.34, 2018, pp. 35-47 issn: 1592-7415 eissn: 2282-8214 on a geometric representation of probability laws and of a coherent prevision-function according to subjectivistic conception of probability pierpaolo angelini∗, angela de sanctis† received: 22-03-2018. accepted: 30-05-2018. published: 30-06-2018 doi:10.23755/rm.v34i0.401 c©pierpaolo angelini and angela de sanctis abstract we distinguish the two extreme aspects of the logic of certainty by identifying their corresponding structures into a linear space. we extend probability laws p formally admissible in terms of coherence to random quantities. we give a geometric representation of these laws p and of a coherent previsionfunction p which we previously defined. this work is the foundation of our next and extensive study concerning the formulation of a geometric, wellorganized and original theory of random quantities. keywords: metric; collinearity; vector subspace; convex set; linear dependence 2010 ams subject classifications: 51a05; 60a02; 60b02. ∗miur, roma, italia. pierpaolo.angelini@istruzione.it †dea, univerity “g. d’annunzio” of chieti-pescara, pescara, italia angela.desanctis@unich.it 35 pierpaolo angelini and angela de sanctis 1 introduction an event e is conceptually a mental separation between subjective sensations: it is actually a proposition or statement such that, by betting on it, we can establish in an unmistakable fashion whether it is true or false, that is to say, whether it has occurred or not and so whether the bet has been won or lost ([9], [15]). for any individual who does not certainly know the true value of a quantity x, which is random in a non-redundant usage for him, there are two or more than two possible values for x. the set of these values is denoted by i(x). in any case, only one is the true value of each random quantity and the meaning that you have to give to random is the one of unknown by the individual of whom you consider his state of uncertainty or ignorance. thus, random does not mean undetermined but it means established in an unequivocal fashion, so a supposed bet based upon it would unmistakably be decided at the appropriate time. when one wonders if infinite events of a set are all true or which is the true event among an infinite number of events, one can never verify if such statements are true or false. these statements are infinite in number, so they do not coincide with any mental separation between subjective sensations: they are conceptually meaningless. hence, we can understand the reason for which it is not a logical restriction to define a random quantity as a finite partition of incompatible and exhaustive events, so one and only one of the possible values for x belonging to the set i(x) = {x1, . . . ,xn} is necessarily true. a random quantity is dealt with by the logic of certainty as well as by the logic of probable ([8]). we recognize two different and extreme aspects concerning the logic of certainty. at first we distinguish a more or less extensive class of alternatives which appear objectively possible to us in the current state of our information: when a given individual outlines the domain of uncertainty he does not use his subjective opinions on what he does not know because the possible values of x depend only on what he objectively knows or not. afterwards we definitively observe which is the true alternative to be verified among the ones logically possible. the probability is an additional notion, so it comes into play after constituting the range of possibility and before knowing which is the true alternative to be verified: the logic of probable will fill in this range in a coherent way by considering a probabilistic mass distributed upon it. an individual correctly makes a prevision of a random quantity when he leaves the objective domain of the logically possible in order to distribute his subjective sensations of probability among all the possible alternatives and in the way which will appear most appropriate to him ([7], [12], [13]). given an evaluation of probability pi, i = 1, . . . ,n, a prevision of x turns out to be p(x) = x1p1 + . . . + xnpn, where we have 0 ≤ pi ≤ 1, i = 1, . . . ,n, and ∑n i=1 pi = 1: it is rendered as a function of the probabilities of the possible values for x and it is admissible in terms of coherence because it is a barycenter of these values. it is usually called the 36 on a geometric representation of probability laws and of a coherent prevision-function mathematical expectation of x or its mean value ([14]). it is certainly possible to extend this result by using more advanced mathematical tools such as stieltjes integrals. nevertheless, such an extension adds nothing from conceptual and operational point of view and for this reason we will not consider it. conversely, the possible values of any possible event are only two: 0 and 1. therefore, each event is a specific random quantity. the same symbol p denotes both prevision of a random quantity and probability of an event ([10]). 2 space of alternatives as a linear space when we consider one random quantity x, each possible value of it, for a given individual at a certain instant, is a real number in the space s of alternatives coinciding with a line on which an origin, a unit of length and an orientation are chosen. every point of this line is assumed to correspond to a real number and every real number to a point: the real line is a vector space over the field r of real numbers, that is to say, over itself, of dimension 1. when we consider two random quantities, x1 and x2, a cartesian coordinate plane is the space s of alternatives: possible pairs (x1,x2) are the cartesian coordinates of a possible point of this plane. every point of a cartesian coordinate plane is assumed to correspond to an ordered pair of real numbers and vice versa: r2 is a vector space over the field r of real numbers of dimension 2 and it is called the two-dimensional real space. when we consider three random quantities, x1, x2 and x3, the threedimensional real space r3 corresponds to the set s of alternatives and possible triples (x1,x2,x3) are the cartesian coordinates of a possible point of this linear space. there is a bijection between the points of the vector space r3 over the field r of real numbers and the ordered triples of real numbers. more generally, in the case of n random quantities, where n is an integer > 3, one can think of the cartesian coordinates of the n-dimensional real space rn. there is a bijection between the points of the vector space rn over the field r and the ordered ntuples of real numbers. it is always essential that different pairs of real numbers are made to correspond to distinct points or different triples of real numbers are made to correspond to different points or, more generally, distinct n-tuples of real numbers are made to correspond to dissimilar points ([11]). those alternatives which appear possible to us are elements of the set q and they are embedded in the space s of alternatives. such a space is conceptually a set of points whose subset q consists of those possible points non-themselves subdivisible for the purposes of the problem under consideration. sometimes, the set q coincides with s. there is a very meaningful point among the points of q: it represents the true alternative, that is to say, the one which will turn out to be verified “a posteriori”. it is a random point “a priori” and it expresses everything there is to 37 pierpaolo angelini and angela de sanctis be said. 3 two different aspects of the logic of certainty into a linear space we study the two aspects of the logic of certainty into a linear space coinciding with the n-dimensional real space rn where we consider n random quantities x1, . . . ,xn. therefore, we have n orthogonal axes to each other: a same cartesian coordinate system is chosen on every axis. thus, the real space rn has a euclidean structure and it is evidently our space s of alternatives. into the logic of certainty exist certain and impossible and possible as alternatives with respect to the temporary knowledge of each individual: each random quantity justifies itself “a priori” because every finite partition of incompatible and exhaustive events referring to it shows the possible ways in which a certain reality may be expressed. a multiplicity of possible values for every random quantity is only a formal construction that precedes the empirical observation by means of which a single value among the ones of the set q is realized. the set q of every random quantity is a subset of a vector subspace of dimension 1 into the n-dimensional real space rn. in general, given x, we have q = i(x) = {x1, . . . ,xn}. it is absolutely the same thing if every possible value of each random quantity is viewed as a particular n-tuple of real numbers or as a single real number. every possible value for a random quantity definitively becomes 0 or 1 when we make an empirical observation referring to it: into the logic of certainty also exist true and false as final answers ([2], [3]). logical operations are applicable to idempotent numbers 0 and 1. if a and b are events, the negation of a is ā = 1 −a and such an event is true if a is false, while if a is true it is false; the negation of b is similarly b̄ = 1 −b. the logical product of a and b is a∧b = ab and such an event is true if a is true and b is true, otherwise it is false; the logical sum of a and b is (a∨b) = ( ā∧ b̄ ) = 1 − (1 −a)(1 −b), from which it follows that such an event is true if at least one of events is true, where we have a∨b = a + b when a and b are incompatible events because it is impossible for them both to occur. concerning the logical product and the logical sum, we have evidently the same thing when we consider more than two events. an algebraic structure (l,∧,∨), where the logical product ∧ and the logical sum ∨ are two binary operations on the set l whose elements are 0 and 1, is a boolean algebra because commutative laws, associative laws, absorption laws, idempotent laws and distributive laws hold for 0 and 1 of l. it admits both an identity element with respect to the logical product and an identity element with respect to the logical sum, so we have (x∧ 1) = x, (x∨ 0) = x, 38 on a geometric representation of probability laws and of a coherent prevision-function for all x of l. it admits that every x of l has a unique complement x̄, so we have (x∧ x̄) = 0, (x∨ x̄) = 1. we can extend the logical operations into the field of real numbers when we make the following definitions: x ∧ y = min(x,y), x ∨ y = max(x,y), x̄ = 1 − x. therefore, it is not true that the logical operations are applicable only to idempotent numbers 0 and 1 because they are also applicable to all real numbers. on the other hand, it is not true that the arithmetic operations are applicable only to natural, rational, real, complex numbers or integers because they are also applicable to idempotent numbers 0 and 1 identifying events. for instance, the arithmetic sum of many events coincides with the random number of successes given by y = e1 + . . . + en. therefore, we observe that the set q of every random quantity considered into a linear space becomes a boolean algebra whose two idempotent numbers are on every axis of rn. these two numbers are elements of a subset of a vector subspace of dimension 1 into the n-dimensional real space rn over the field r of real numbers. that being so, it is evident that to postulate that the field over which the probability is defined be a σ-algebra is not natural. hence, what we will later say is conceptually and mathematically well-founded. 4 probability laws p formally admissible in terms of coherence the probability p of an event e, in opinion of a given individual, is operationally a price in terms of gain and a bet is the real or conceptual experiment to be made oneself in order to obtain its measurement ([4]). if p = p(e) is a coherent assessment expressed by this individual, then such a bet is fair because it is acceptable in both senses indifferently. therefore, he considers as fair an exchange, for any s positive or negative, between a certain sum ps and the right to a sum s dependent on the occurrence of e ([5]). from notion of fairness it follows that the two possible values of the random quantity g′ = (λ−p)s, where λ is a random quantity whose possible values are 0 and 1, do not have the same sign. given s, if these values of g′ are only positive or negative, then we have an incoherent assessment and the bet on the event under consideration is not fair. if e is a certain event, then we have p = 1 in a coherent fashion. if e is an impossible event, then we have p = 0 in a coherent fashion. if e is a possible event because it is not either certain or impossible, then we have 0 ≤ p ≤ 1 in a coherent fashion. even the probability p of the trievent e = e′′|e′ is a price in terms of gain. it is the price to be paid for a bet that can be won or lost or annulled if e′ does not occur. nevertheless, we will not consider the notion of conditional 39 pierpaolo angelini and angela de sanctis probability from now on, because it is not essential to this context. given n events e1, . . . ,en of the set e of events, a certain individual assigns to them, respectively, the probabilities p1 = p(e1), . . . ,pn = p(en) in a coherent way. thus, by betting on e1, . . . ,en, this individual considers as fair an exchange, for any s1, . . . ,sn positive or negative, between a certain sum p1s1 + . . . + pnsn and the right to a sum e1s1 + . . . + ensn dependent on the occurrence of e1, . . . ,en, where we have ei = 1 or ei = 0, i = 1, . . . ,n, whether ei occurs or not. evidently, if p1 = p(e1), . . . ,pn = p(en) are not coherent assessments, then the possible values of the random quantity g = (λ1 −p1)s1 + . . . + (λn −pn)sn are all positive or negative. probability laws p formally admissible in terms of coherence allow to extend in a logical or coherent way the probabilities of the events of e which are already evaluated in a subjective way. these laws allow to determine which is the most general set of events whose probabilities are uniquely determined, in accordance with theorems of probability calculus, because one knows the probability of each event of e. moreover, probability laws p allow to determine which is the most general set of events for which their probabilities lie between two numbers, which are not 0 and 1, after evaluating the probability of each event of e in a subjective way, while for the remaining events can be said nothing in addition to the banal observation that their probabilities are included between 0 and 1. if e is a finite set of incompatible and exhaustive events e1, . . . ,en, then p is a probability law formally admissible in terms of coherence with regard to events of e if and only if the theorem of total probability is valid, so we have p(e1) +. . .+p(en) = 1. probability laws p formally admissible are evidently ∞n and a given individual may subjectively choose one of these laws depending on the circumstances. given a, its probability p(a) is uniquely determined when a is a logical sum of two or more than two incompatible events of e: a is linearly dependent on these events. otherwise, we can only say that p(a) is greater than or equal to the sum of the probabilities of the events ei which imply a and less than or equal to the sum of the probabilities of the events ei which are compatible with a. if e is a finite set of events e1, . . . ,en whatsoever, then the 2n constituents c1, . . . ,cs form a finite set of incompatible and exhaustive events for which it is certain that one and only one of them occurs. these constituents are elementary or atomic events and they are obtained by the logical product e1 ∧ . . . ∧ en: each time we substitute in an orderly way one event ei, i = 1, . . . ,n, or more than one event with its negation ēi or their negations, we obtain one constituent of the set of constituents generated by e1, . . . ,en. it is possible that some constituent is impossible, so the number of possible constituents is s ≤ 2n. the most general probability law assigns to the possible constituents c1, . . . ,cs the probabilities q1, . . . ,qs which sum to 1, while the probability of an impossible constituent is always 0. conversely, every probability law which is valid for the events of e can be extended to the constituents c1, . . . ,cs, so the 40 on a geometric representation of probability laws and of a coherent prevision-function probabilities p1 = p(e1), . . . ,pn = p(en) are admissible if and only if the nonnegative numbers q1, . . . ,qs satisfy a system of n + 1 linear equations in the s variables q1, . . . ,qs expressed by  ∑(1) i qi = p1 ...∑(n) i qi = pn∑s i=1 qi = 1. the notation ∑(h) i qi represents the sum concerning those indices i for which ci is an event implying eh. if a is a logical sum of some constituent, then we have x = ∑(a) i qi and we can say that the probability of a is uniquely determined because x = ∑(a) i qi is linearly dependent on the n + 1 linear equations of the system under consideration. if a is not a logical sum of constituents, then a′ is the greatest logical sum of the ones which are contained in a and a′′ is the lowest logical sum of the ones which contain a, so we have x′ ≤ x ≤ x′′, where x′ is the lowest admissible probability of a′, while x′′ is the greatest admissible probability of a′′. if e is an infinite set of events, then p is a probability law formally admissible with regard to events of e if and only if p is a probability law formally admissible with regard to any finite subset of e. therefore, given a, its probability p(a) is uniquely determined or bounded from above and below or absolutely undetermined because we have 0 ≤ p(a) ≤ 1. now we extend probability laws p formally admissible in terms of coherence to random quantities we defined in the beginning. the set x can be a finite set of n random quantities x1, . . . ,xn or it can be an infinite set of random quantities. in general, given x, i(x) = {x1, . . . ,xn} is the set of its possible values. thus, after assigning to every possible value xi of x its subjective and corresponding probability pi, with ∑n i=1 pi = 1, we have infi(x) ≤ p(x) ≤ supi(x) in accordance with convexity property of p. given z = x1 + . . . + xn which is a linear combination of n random quantities x1, . . . ,xn of x , i(z) = {z1, . . . ,zn} is the set of its possible values. therefore, its coherent prevision must satisfy convexity property of p, so we have infi(z) ≤ p(z) ≤ supi(z), where it turns out to be p(z) = p(x1) + . . . + p(xn) in accordance with linearity property of p. linearity property can clearly be of interest to any linear combination of n random quantities. we may also consider less than n random quantities. the possibility of certain consequences whose unacceptability appears recognizable to everyone is excluded when convexity property of p and its linearity property are valid. they are the foundation of the whole theory of probability because they are necessary and sufficient conditions for coherence: decisions under conditions of uncertainty lead to a certain loss when linearity and convexity of p are broken ([1]). the 41 pierpaolo angelini and angela de sanctis probabilities of every possible value of a given random quantity belonging to a finite or infinite set of random quantities sum to 1 in a coherent way according to probability laws p formally admissible in terms of coherence with regard to these possible values. 5 a coherent prevision-function p from mathematical point of view, p is a function. we define it by taking into account its objective coherence. the domain of p is the arbitrary set x = {x1, . . . ,xn} consisting of a finite number of random quantities: for each of them, the set of possible values is i(xi) = {xi1, . . . ,xin}, with xi1 < .. . < xin, i = 1, . . . ,n. moreover, we suppose xi1 6= xj1 and xin 6= xjn, with i 6= j, i,j = 1, . . . ,n. the codomain of p is the set y consisting of as many intervals as random quantities are found into the set x of p, with infi(xi) ≤ p(xi) ≤ supi(xi) for every interval referring to the random quantity xi, i = 1, . . . ,n. therefore, both x and y are sets whose elements are themselves sets. the coherent function p is called prevision-function and it is a bijective function because each element of x , xi ∈x , is paired with exactly one element of y, for which it turns out to be infi(xi) ≤ p(xi) ≤ supi(xi), and each element of y is paired with exactly one element of x . there are no unpaired elements, with p(xi) which is a prevision of xi on the basis of the state of information of a certain individual at a given instant. given the set i(x) = {x1, . . . ,xn}, with x1 < .. . < xn, the image of x under p is p(x) = x1p1 +. . .+xnpn, with 0 ≤ pi ≤ 1, i = 1, . . . ,n, and ∑n i=1 pi = 1: such an image coincides with all weighted arithmetic means calculated in a coherent fashion when pi varies while xi is constant. all coherent previsions of x satisfy the inequality infi(x) ≤ p(x) ≤ supi(x). the image of the entire domain x of p is the image of p and it coincides with the entire codomain y. if x is an infinite set of random quantities, we can always consider a restriction of the prevision-function p which is a new function obtained by choosing a smaller and finite domain. therefore, the above observations remain unchanged because such a new function coincides with p whose domain is a finite set of random quantities. in the case in which the domain of p is the arbitrary set e = {e1, . . . ,en} consisting of a finite number of possible events, its codomain is the set y consisting of as many intervals as events are found into the set e of p, with infei ≤ p(ei) ≤ supei, i = 1, . . . ,n, for each of such intervals. nevertheless, since we have infei = 0 and supei = 1, i = 1, . . . ,n, it turns out to be 0 ≤ p(ei) ≤ 1 for every interval of y. the coherent function p is called probability-function and it is a bijective function because each element of e, ei ∈e, is paired with exactly one element of y, for which it turns out to be 0 ≤ p(ei) ≤ 1, and each element of y is paired with exactly one element of e. 42 on a geometric representation of probability laws and of a coherent prevision-function there are no unpaired elements, with p(ei) which is an evaluation of probability of ei. the image of ei under p is an interval. if e is an infinite set of events, we can always consider a restriction of the probability-function p as above. we admit that p can be evaluated by anybody for every event e or random quantity x. thus, it is not true that it would make sense to speak of probability only when all events under consideration are repeatable, as well as it is not true that it would make sense to speak of prevision only when all random quantities under consideration belong to a measurable set i. we cannot pretend that p is actually imagined as determined, by any individual, for all events or random quantities which could be considered in the abstract. we must recognize if p includes or not any incoherence. if so the individual, when made aware of such an incoherence, should eliminate it. thus, the subjective evaluation is objectively coherent and can be extended to any larger set of events or random quantities. it is necessary to interrogate a given individual in order to force him to reveal his evaluation of elements of the codomain y of p, p(xi) or p(ei), i = 1, . . . ,n: both prevision of a random quantity and probability of an event always express what an individual chooses in his given state of ignorance, so it is wrong to imagine a greater degree of ignorance which would justify the refusal to answer. if a prevision-function is not understood as an expression of the opinion of a certain individual, we can interrogate many individuals in order to study their common opinion which is denoted by p. therefore, p will exist in the ambit of those random quantities x for which all evaluations pi(x), i = 1, . . . ,n, coincide. such evaluations will define p(x) in this way. evidently, p will not exist elsewhere, for other random quantities x for which the subjective evaluations pi(x) do not coincide. the above observations remain valid when a given individual confines himself to evaluations which conform to more restrictive criteria coinciding with classical definition of probability and with the statistical one ([6], [16]). 6 geometric representation of p given the set x = {x1, . . . ,xn} or the set e = {e1, . . . ,en}, the possible values of each random quantity or random event can geometrically be represented on n lines for which a cartesian coordinate system has been chosen. such lines belong to the vector space rn over the field r of real numbers. rn has a euclidean structure characterized by a metric. hence, the standard basis of rn is given by {e1, . . . ,en}, where we have e1 = (1, . . . , 0), . . . , en = (0, . . . , 1), and it consists of orthogonal vectors to each other having a euclidean norm equal to 1. the point of rn where n lines meet is the origin of rn given by (0, . . . , 0). we have an oneto-one correspondence between the points of rn and the n-tuples of real numbers. we consider n coordinate subspaces of dimension 1 in the vector space rn. in fact, 43 pierpaolo angelini and angela de sanctis when we project every point of rn referring to (x1,x2, . . . ,xn) and expressed by (x1,x2, . . . ,xn) onto the coordinate axis x1, it becomes (x1, . . . , 0). when we project the same point onto the coordinate axis x2, it becomes (0,x2, . . . , 0) and so on. after projecting all the possible points of x1 onto the coordinate axis x1, . . . , all the possible points of xn onto the coordinate axis xn, every point onto the coordinate axis x1, coinciding with a particular n-tuple of real numbers of rn, can be viewed as a real number of r, . . . , every point onto the coordinate axis xn, coinciding with a particular n-tuple of real numbers of rn, can be viewed as a real number of r. it is finally clear that n projected points onto the coordinate axis x1 can be viewed as n real numbers of an one-dimensional vector space, . . . , n projected points onto the coordinate axis xn can be viewed as n real numbers of an one-dimensional vector space. the possible points of ei projected onto the coordinate axis xi, i = 1, . . . ,n, are evidently only two. for instance, if n = 3, we have the points (1, 0, 0) and (0, 0, 0) onto the coordinate axis x1 referring to e1 which can respectively be viewed as 1 and 0, . . . , the points (0, 0, 1) and (0, 0, 0) onto the coordinate axis x3 referring to e3 which can respectively be viewed as 1 and 0. nevertheless, we have always three real lines, so we do not get confused. in any case, it is conceptually the same thing if we make use only of particular n-tuples of real numbers of rn without seeing them as real numbers of r. the codomain of p is the set y consisting of n intervals which coincide with n line segments belonging to n different real lines. these line segments could become increasingly larger by virtue of linearity of p extended to any finite number of random quantities considered on a same line. indeed, we observe that all weights or probabilistic masses, which are non-negative and sum to 1, remain unchanged with respect to starting point characterized by only one random quantity. nevertheless, they are paired with real numbers whose absolute values are evidently greater. such numbers can be interpreted as the possible values of one random quantity considered on a same line. it is evident that the set of all coherent previsions of every random quantity xi as well as the set of all coherent probabilities of every random event ei, i = 1, . . . ,n, is a subset of a vector subspace of dimension 1. such a subset is however a convex set while the set of the possible values for every random quantity considered into rn is not a convex set. the same thing goes when we consider the set of the possible values for every random event represented into rn. we already saw that it is always possible to consider a finite number of events or random quantities in order to study probability laws p formally admissible in terms of coherence. as a first step we refer to events. given n events e1, . . . ,en of e, we represent them by means of n axes of rn. nevertheless, instead of concentrating our attention on n axes of rn as above, we consider only one of them which we choose in an arbitrary fashion. such an axis is an one-dimensional vector subspace of rn. it is generated by a vector of the standard basis of rn. every point of rn on a same line 44 on a geometric representation of probability laws and of a coherent prevision-function is obtained multiplying by a real number the vector of the standard basis of rn which we have arbitrarily chosen. therefore, we can always multiply by any real number a same n-tuple of real numbers in order to obtain points of rn which are said to be collinear. now the real number or coefficient of the linear combination under consideration, characterized by only one scalar, represents the probability of an event a into our geometric scheme of representation. given p(e1), . . . , p(en), we know that p(a) can be uniquely determined or bounded from above and below or absolutely undetermined depending on the circumstances. if it is uniquely determined, then we have a precise point of rn on the axis under consideration. if it is bounded from above and below, then we have two points of rn on this axis and an admissible probability is found between them. if it is absolutely undetermined, then we have a larger interval on this axis which is included between the lowest admissible probability of any event and the greatest admissible one. in particular, given p(e1) = p1, . . . , p(en) = pn, after choosing the vector en of the standard basis of rn, by means of the linear combination given by [(λ1−p1)s1 + . . . + (λn−pn)sn]en, with si 6= 0, i = 1, . . . ,n, we can obtain the possible values of the random quantity g = (λ1−p1)s1 +. . .+ (λn−pn)sn referring to n bets concerning n events as special n-tuples of rn. the same thing goes if we choose another vector of the standard basis of rn. thus, we even represent n bets concerning n events into a linear space. by examining n random quantities x1, . . . ,xn of x , we similarly represent them by means of n axes of rn. nevertheless, by considering another random quantity z which is again bounded from above and below, instead of concentrating our attention on n axes of rn, we consider only one of them which we choose in an arbitrary way. after individuating two points of rn on this axis which are respectively the lowest possible value of the random quantity under consideration and the greatest possible one, p(z) can be viewed as a point of rn coherently included between the two points of rn already individuated. probability laws p formally admissible in terms of coherence are those laws for which the probabilities of the possible values of the random quantity under consideration sum to 1. 7 conclusions we distinguished the two extreme aspects of the logic of certainty by identifying their corresponding structures into a linear space. we extended probability laws p formally admissible in terms of coherence to random quantities. we proposed a geometric representation of these laws and of a coherent previsionfunction p which we previously defined. we connected the convex set of all coherent previsions of a random quantity as well as the convex set of all coherent probabilities of an event with a specific algebraic structure: such a structure is an 45 pierpaolo angelini and angela de sanctis one-dimensional vector subspace over the field r of real numbers because events of any finite set of events can be viewed as special points of a vector space of dimension n over the field r of real numbers. it is exactly the linear space of random quantities having a euclidean structure characterized by a metric coinciding with the dot product in a natural way. overall, we pointed out that linearity is the most meaningful concept concerning probability calculus whose laws gain a more extensive rigour by means of the geometric scheme of representation we showed. on the other hand, it is possible to extend linearity concept in order to formulate a geometric, well-organized and original theory of random quantities: we will make this into our next works. references [1] g. coletti and r. scozzafava, probabilistic logic in a coherent setting, kluwer academic publishers, dordrecht/boston/london, 2002. [2] b. de finetti, teoria delle probabilità: sintesi introduttiva con appendice critica, voll. i e ii, einaudi, torino, 1970. [3] b. de finetti, probability, induction and statistics (the art of guessing), j. wiley & sons, london-new york-sydney-toronto, 1972. [4] b. de finetti, la probabilità: guardarsi dalle contraffazioni!, scientia, 111 (1976), 255–281. [5] b. de finetti, the role of “dutch books” and of “proper scoring rules”, the british journal of psychology of sciences, 32 (1981), 55–56. [6] b. de finetti, probability: the different views and terminologies in a critical analysis, logic, methodology and philosophy of science, vi (hannover, 1979) (1982), 391–394. [7] i. j. good, subjective probability as the measure of a non-measureable set, logic, methodology and philosophy of science, proc. 1960 internat. congr. (1962), 319–329. [8] h. jeffreys, theory of probability, 3rd edn., clarendon press, oxford, 1961. [9] b. o. koopman, the axioms and algebra of intuitive probability, annals of mathematics 41, (1940), 269–292. [10] h. e. kyburg jr. and h. e. smokler, studies in subjective probability, j. wiley & sons, new york, london, sydney, 1964. 46 on a geometric representation of probability laws and of a coherent prevision-function [11] g. pompilj, teoria affine delle variabili casuali, l’industria, 2 (1956), 143– 163. [12] f. p. ramsey, the foundations of mathematics and other logical essays. edited by r. b. braithwaite with a preface by g. e. moore, littlefield, adams & co, paterson, n. j., 1960. [13] l. j. savage, the foundations of statistics, j. wiley & sons, new york, 1954. [14] b. de finetti, la prévision: ses lois logiques, ses sources subjectives, ann. inst. h. poincaré 7, 1, (1937), 1–68. [15] b. de finetti, sulla proseguibilità di processi aleatori scambiabili, rend. ist. mat. univ. trieste 1, (1969), 53–67. [16] b. de finetti, probability and statistics in relation to induction, from various points of view, induction and statistics, cime summer schools, springer, heidelberg 18, (2011), 1–122. 47 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 38, 2020, pp. 341-348 341 approximation of functions by (c,2)(e,1) product summability method of fourier series jitendra kumar kushwaha* abstract various investigators such as leindler [10], chandra [1], mishra et al. [7], khan [11], kushwaha [6] have determined the degree of approximation of 2 -periodic functions belonging to classes lip , ),( rlip  , )),(( rtlip  of functions through trigonometric fourier approximation using different summability means. recently nigam [12] has determined that the fourier series is summable under the summability means (c,2)(e,1) but he did not find the degree of approximation of function belonging to various classes. in this paper a theorem concerning the degree of approximation of function f belonging to )),(( rtlip  class by (c,2)(e,1) product summability method of fourier series has been established which in turn generalizes the result of h. k. nigam [12]. keywords: degree of approximation; fourier series; pruduct summability methods. 2010 ams subject classification: 42b05; 42b08.† *deen dayal upadhyaya gorakhpur university, gorakhpur, india. jitendra.mathstat@ddugu.ac.in. † received on march 29th, 2020. accepted on june 19th, 2020. published on june 30th, 2020. doi: 10.23755/rm.v38i0.504. issn: 1592-7415. eissn: 2282-8214. ©jitendra kumar kushwaha. jitendra kumar kushwaha 342 1. introduction the study of the theory of trigonometric approximation is of great mathematical interest and of great practical importance. broadly speaking, signals are treated as function of single variable and images are represented by function of two variables. the study of these concepts is directly related to the emerging area of information technology. studies on trigonometric approximation of functions in pl -norm using different linear operators such hölder,nörlund, euler, riesz, borel etc. were made by several researchers like mohapatra & chandra [9], holland, mohapatra & sahney [8], chandra [2]. the degree o approximation of a function belonging to different class of functions by product summability methods were made by lal & singh [5], lal & kushwaha [6]. the aim of this paper is to study fourier series and conjugate series by product operators. the advantage of considering product operators over linear operators can be understood with the observation that the infinite series, which is neither summable by left linear operators nor by right linear operators individually, is summable to some number by the product operators obtained from the same linear operators placed in the same sequential order. moreover , in studies of error estimates )( fen through trigonometric fourier approximation, product operators give better approximation than individual linear operators. generalizing the result of nigam [12], the degree of approximation of function f belonging to )),(( rtlip  class by (c,2)(e,1) product summability method of fourier series has been established. therefore, in this paper, (c,2)(e,1) product summability method is introduced and a theorem on the approximation of functions belonging to ( )rtl ),( class has been established. let   =0n n u be given infinite series with ns for its th n partial sum. let  1ent denote the sequence of (e,1) mean of the sequence  ns . if the (e,1) transform of ns is defined as →→      =  = nassxfs k n xft k n k n e n );( 2 1 );( 0 1 (1.1) the series   =0n n u is said to summable to the number s by the (e,1) method (hardy [14]). approximation of functions by (c,2)(e,1) product summability method of fourier series 343 let  2c n t denote the sequence of (c, 2) mean of the sequence   n s . if the (c, 2) transform of ns is defined as ( ) →→+− ++ =  = nassxfskn nn xft k n k c n );(1 )2)(1( 2 );( 0 2 (1.2) the series   =0n n u is said to be summable to the number s by (c, 2) method (cesàro method). thus if ( ) →→      +− ++ =  == nassxfs v n kn nn xft v n v k n k ec n );( 2 1 1 )2)(1( 2 );( 00 . 12 ( 1.3) where 12 . ec n t denotes the sequence of (c,2)(e,1) product mean of the sequence ns . the series   =0n n u is said to summable to the number s by (c,2)(e,1) method. we observe that (c,2)(e,1) method is regular. let f be 2 -periodic and lebesgue integrable function. the fourier series associated with f at a point x is defined by ( ) )(sincos 2 ~)( 01 0 xanxbnxa a xf n n n nn   =  = ++ (1.4) with partial sum );( xfs n . throughout this paper, we use following notations: )()()(),()( xftxftxftxt −−++==   = =               +      +− ++ = n k k v kn t tv v kkn nn tm 0 0 )2/sin( )2/1sin( 2 1 )2)(1( 1 )(  . 2. main theorem we prove the following theorem theorem . if rrf →: is 2 -periodic, lebesgue integrable on ],[ − and belonging to ( )rtlip ),( class then the degree of approximation of f by the (c,2)(e,1) product means ( ) );( 2 1 1 )2)(1( 2 );( 00 . 12 xfs v n kn nn xft v n v k n k ec n  ==       +− ++ = of its fourier series (1.4) is given by jitendra kumar kushwaha 344             + =− 1 1 12 . n oft r ec n  . 3. lemmas 3.1 lemma 1 for )1/(10 + nt , )1()( += notk n . proof for )1/(10 + nt , tnnt sinsin    = =       +      +− ++  n k k kn t kkn nn tm 0 0 )2/sin( )2/1sin( 2 )1( )2)(1( 1 )(      = =       +      +− ++  n k k kn t tkkn nn tm 0 0 )2/sin( )2/sin()12( 2 )1( )2)(1( 1 )(      = =             + +− ++  n k k k k k kn nn 0 0 )12( 2 )1( )2)(1( 1   =  = ++− ++ n k kkn nn 0 )]12)(1[( )2)(1( 1   == + ++ −+ ++ + = n k n k kk nn k nn n 00 )]12([ )2)(1( 1 )12( )2)(1( 1        + ++ −+ + =  === n k n k n k kk nn k n 00 2 0 2 )2)(1( 1 )12( )2( 1        + + ++ ++ − + + = 2 )1( 3 )12)(1( )2)(1( 1 )2( )1( 2 nnnnn nnn n  ).1( += no 3.2 lemma 2 for + tn )1/(1 , )./1()( totk n = proof for + tn )1/(1 , applying jordan’s lemma, /)2/sin( tt  and 1sin nt .   = =             ++ +  n k k k t k nn n 0 0 )/( 1 2 1 )2)(1( )1(     = =       +      +− ++  n k k kn t kkn nn tm 0 0 )2/sin( )2/1sin( 2 )1( )2)(1( 1 )(    approximation of functions by (c,2)(e,1) product summability method of fourier series 345   = =             ++ n k k k t kk nn 0 0 )/( 1 2)2)(1( 1    == ++ − + = n k n k k nntnt 00 )2)(1( 1 1 )2( 1 )./1( to= 4. proof of the theorem following titchmarsh [13] and using riemann lebesgue theorem, );( xfs n of the series (1.4) is given by dt t tn txfxfs n )2/sin( )2/1sin( )( 2 1 )();( 0 + =−     using (1), the (e,1) transforms of );( xfs n is given by dt t tn k n txft n k n e n         +       =−  = + )2/sin( )2/1sin( )( 2 1 )( 00 1 )1,(    the (c,2) (e,1) transform of );( xfs n is given by   = =               +     +− ++ =− n k k k ec n dtt k t tkn nn xft 0 00 . )2/1sin( )2/sin( )( 2 )1( )2)(1( 1 )(12       dttkt n )()( 0 =   dttktdttkt n n n n )()()()( )1/(1 )1/(1 0  + + +=   21 ii += (4.1) now , dttkti n n )()( )1/(1 0 1  + =  =         + + dtnto n )1()( )1/(1 0  , by lemma (1)         +=  + )1/(1 0 )()1( n dttno  jitendra kumar kushwaha 346  +             + += )1/(1 1 1 )1( n dt n no   , where )1/(10 + n by first mean value theorem of calculus             + = 1 1 n o  . (4.2) lastly,         =  + dttktoi n n )()( )1/(1 2           + =  + dt tn t o n   )1/(1 )1( )( , by lemma ( 2)             + = 1 1 n o  . (4.3) combining (4.1)-(4.3), we get             + =− 1 1 12 . n oft r ec n  . this completes the proof of the theorem. 5. conclusions the result of main theorem is . 1 1 12 .             + =− n oft r ec n  from which the results of h.k. nigam [12] can be derived directly. acknowledgement author is highly thankful to professor shyam lal, department of mathematics, institute of science, banaras hindu university, varanasi, india for his encouragement and support to this work. approximation of functions by (c,2)(e,1) product summability method of fourier series 347 references [1] p. chandra, approximation by nörlund operators, mat. vestnik, vol. 8, 263-269, 1986. [2] p. chandra, functions of classes p l and ),( plip  and their riesz means, riv. mat. univ. parm, vol. 4, no. 12, 275-282, 1986. [3] ] p. chandra , on the degree of approximation of a class of function by means of fourier series , acta mathematica hungarica, vol. 52 , no. 3-4, 199205, 1988. [4] a.b.s. hollend, r.n. mohapatra and b.n. sahney , pl approximation of function by euler means, rendiconti di mathematica (rome)(2), vol. 3, 341355, 1983. [5] s. lal and p.n. singh, degree o approximation of conjugate of function by (c,1)(e,1) means of conjugate series of fourier series, tamkang journal of mathematics, vol. 33, no. 3, 269-274, 2002. [6] s. lal and j. k. kushwaha, degree of approximation of lipschitz function by (c,1)(e,q) product summability method, int. math. forum vol.4 (no. 43), 2009, pp. 2101-2107. 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[12] h.k. nigam, on (c,2)(e,1) product means of fourier series, electronic journal of mathematical analysis an application, vol.1(2) july 2013, pp. 334344. jitendra kumar kushwaha 348 [13] e.c. titchmarsh, the theory of functions, oxford university press, 402403, 1939. [14] g.h. hardy, divergent series, oxford university press, oxford, 1949. ratio mathematica volume 44, 2022 on the effect of doubling of intervals on the 0 distributive property of the lattice of weak congruences of chains gladys mano amirtha v1 d. premalatha2 abstract alan day’s doubling construction of intervals has been found to affect some properties of the lattice of weak congruences of chains. here, in this paper, we study how it affects the property of 0-distributivity of the lattice of weak congruences of chains. keywords: doubling construction in lattices, weak congruence lattices, 0-distributive lattices. 2010 ams subject classification: 06b10, 06d993. 1reg. no.: 20111172092013, ph.d. research scholar (full time), pg and research department of mathematics, (rani anna government college for women, tirunelveli-627008, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india.) gladyspeter3@gmail.com. 2associate professor, pg and research department of mathematics, (rani anna government college for women, tirunelveli-627008, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india.) lathaaedward@gmail.com. 3 received on june 7th, 2022. accepted on aug 10th, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.888. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 42 mailto:gladyspeter3@gmail.com mailto:lathaaedward@gmail.com gladys mano amirtha v & d. premalatha 1. introduction vojvodić g. and šešelja b. initiated the study of the concept of weak congruences of a lattice in the year 1988 [12]. j. c. varlet [9] was the first to introduce the concept of 0-distributive lattices. several other researchers made various contributions in different aspects of 0-distributivity. for example, one can refer to [1], [8], [11]. a. veeramani [10] in his thesis, has studied about the lattice of weak congruences of a finite chain, boolean lattices and the lattices 𝐶𝑛, 𝑀3, 𝑁5first by considering 0 and 1 as non-constants and then considering them as constants, again by considering the boolean lattice as an algebra and he studied about some weaker properties like 0-distributivity, 0-modularity, consistency, etc. g. gratzer constructed a new lattice lufrom a given lattice l by adding an element au called the double of a ≠ 0 or 1 in l where lu = l ∪ {au}with a new order denoted by ≤u [6]. following that construction, a day [3] introduced a similar construction l[i]by doubling an interval i of a given lattice l. after that it witnessed many developments, e.g., see [4], [5], [7]. alan day in [2] proved that a distributive lattice remains distributive when it is doubled by either a lower interval or an upper interval. in our present study, we analyse the effect of doubling of intervals on the property of 0distributivity in the lattice of weak congruences of chains. 2. preliminaries definition 2.1 [6] a lattice 𝐿 satisfying the following identities • 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧) • 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧) for all 𝑥, 𝑦, 𝑧 ∈ 𝐿 is called a distributive lattice. if not, it is a non-distributive lattice. definition 2.2 [4] a lattice 𝐿 is said to be 0 distributive, if for all 𝑥, 𝑦, 𝑧 ∈ 𝐿, whenever 𝑥 ∧ 𝑦 = 0 and 𝑥 ∧ 𝑧 = 0, then𝑥 ∧ (𝑦 ∨ 𝑧) = 0. lemma 2.3 [10] if 𝐿𝑛 is a chain of 𝑛 elements, then 𝐶𝑊(𝐿𝑛) is 0 distributive. definition 2.4 [6] an equivalence relation 𝜃 on a lattice 𝐿 is said to be a congruence relation on 𝐿, if it is compatible with both meet and join, that is, for all 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝐿, 𝑎 ≡ 𝑏 (𝜃) and 𝑐 ≡ 𝑑 (𝜃) imply that 𝑎 ∨ 𝑐 ≡ 𝑏 ∨ 𝑑 (𝜃) and 𝑎 ∧ 𝑐 ≡ 𝑏 ∧ 𝑑 (𝜃). definition 2.5 [12] a weak congruence relation on an algebra 𝐴 is a symmetric and transitive sub-universe of 𝐴2. note 2.6 the lattice of all weak congruence relations of 𝐿including 𝜙 with respect to the relation ⊆ is denoted by 𝐶𝑊(𝐿). we consider 0 and 1 of 𝐿 as non-constants in this paper. remark 2.7 [12] in 𝐶𝑊(𝐿), we have • [𝜙, 𝛥] ≅ 𝑆𝑢𝑏(𝐿), the lattice of all sublattices of 𝐿. 43 on the effect of doubling of intervals on the 0-distributive property of the lattice of weak congruences of chains • [𝛥, 𝜏 ] ≅ 𝐶𝑜𝑛(𝐿), the lattice of all congruences of 𝐿. definition 2.8 [6] let 𝐼 = [𝑎, 𝑏] be an interval of a lattice 𝐿. the set 𝐼 × 𝐶2 is formed using the two-element chain 𝐶2 = {0,1}. the set 𝐿[𝐼] = (𝐿 ∖ 𝐼) ∪ (𝐼 × 𝐶2) is the lattice given by the ordering: for 𝑥, 𝑦 ∈ 𝐿[𝐼] and 𝑖, 𝑗 ∈ 𝐶2; 𝑥 ≤ 𝑦 if 𝑥 ≤ 𝑦 in 𝐿; (𝑥, 𝑖) ≤ 𝑦 if 𝑥 ≤ 𝑦 in 𝐿; 𝑥 ≤ (𝑦, 𝑗) if 𝑥 ≤ 𝑦 in 𝐿; (𝑥, 𝑖) ≤ (𝑦, 𝑗) if 𝑥 ≤ 𝑦 in 𝐿 and 𝑖 ≤ 𝑗 in 𝐶2. 𝐿[𝐼] is the lattice got by doubling of the interval 𝐼in𝐿. this is day's definition of doubling of intervals. 2. results and discussions in this section, we examine whether [𝐶𝑊(𝐿𝑛)](𝐼) is 0 distributive or not. it turns out that [𝐶𝑊(𝐿𝑛)](𝐼) remains 0-distributive in case of lower and upper intervals in 𝐶𝑊(𝐿𝑛), whereas in the case of an intermediate interval, 0 – distributivity gets affected. theorem 3.1 if ln is a chain of n elements, then [cw(ln)](i) is 0 – distributive where i is a lower interval in cw(ln). proof. let 𝐿𝑛 = { 0 ≺ 𝑥1 ≺ 𝑥2 ≺ 𝐼 ≺ 𝑥𝑛−1 = 1 } be a chain of 𝑛 elements. let 𝐶𝑊(𝐿𝑛) be the lattice of all weak congruences of 𝐿𝑛. let 𝐼 = [𝜙, 𝜃] where 𝜃 is a proper congruence relation of 𝐿𝑛. let [𝐶𝑊(𝐿𝑛)](𝐼) be the doubling of 𝐶𝑊(𝐿𝑛) by the interval 𝐼. let 𝐴, 𝐵, 𝐶 ∈ [𝐶𝑊(𝐿𝑛)](𝐼), where 𝐼 is a lower interval of 𝐶𝑊(𝐿𝑛) such that 𝐴 ∧ 𝐵 = (𝜙, 0) and 𝐴 ∧ 𝐶 = (𝜙, 0). to prove that, [𝐶𝑊(𝐿𝑛)](𝐼) is 0 – distributive. that is, we have to prove that 𝐴 ∧ (𝐵 ∨ 𝐶) = (𝜙, 0) (3.1) suppose({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ (𝐵 ∨ 𝐶) ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴and ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 𝑎𝑛𝑑 ({(𝑥𝑖 , 𝑥𝑖 ) }, 0) ≤ 𝐵 or ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 or({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ both 𝐵 and 𝐶 or incomparable with both. now,𝐴, 𝐵, 𝐶 ∈ [𝐶𝑊(𝐿𝑛)](𝐼) ⇒ 𝐴 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 or 𝐴 ∈ 𝐼 × 𝐶2, 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 or 𝐵 ∈ 𝐼 × 𝐶2, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 or𝐶 ∈ 𝐼 × 𝐶2. the following cases arise: i.𝐴, 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 ii.𝐴, 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐶 ∈ 𝐼 × 𝐶2 iii.𝐴 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐵, 𝐶 ∈ 𝐼 × 𝐶2 iv.𝐴, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐵 ∈ 𝐼 × 𝐶2 v.𝐴, 𝐵, 𝐶 ∈ 𝐼 × 𝐶2 vi.𝐴, 𝐵 ∈ 𝐼 × 𝐶2 𝑎𝑛𝑑 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 vii.𝐴, 𝐶 ∈ 𝐼 × 𝐶2 𝑎𝑛𝑑 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 viii.𝐴 ∈ 𝐼 × 𝐶2 𝑎𝑛𝑑 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 case (i): let 𝐴, 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. 𝐴, 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 implies that 𝐴, 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛). therefore, (3.1) follows as 𝐶𝑊(𝐿𝑛) is 0distributive. 44 gladys mano amirtha v & d. premalatha case (ii): let 𝐴, 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 and 𝐶 ∈ 𝐼 × 𝐶2, that is, 𝐶 = (𝜃3, 𝑗) where 𝑗 = either 0 or 1. (i) let({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐵 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐵. this is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (ii) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 = (𝜃3, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃3. also, {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∧ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃3 , 𝑗) = 𝐴 ∧ 𝐶. this is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (iii) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 and 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 and(𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃3 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐵 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐵. this is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (iv) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. but, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = 𝐵 ∨ (𝜃3, 0) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 ∨ 𝜃3 ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐵 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃3. by symmetry and transitivity, (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 or(𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶.again we have 𝐴 ∧ 𝐵 ≠ (𝜙, 0)or 𝐴 ∧ 𝐶 ≠ (𝜙, 0).this contradiction proves (3.1). case (iii): let 𝐴 ∈ 𝐶𝑊 (𝐿𝑛) ∖ 𝐼 and 𝐵, 𝐶 ∈ 𝐼 × 𝐶2.let 𝐵 = (𝜃2, 𝑗) and 𝐶 = (𝜃3, 𝑗) where 𝑗 = either 0 or 1. (i) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 = (𝜃2, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2. also, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃2 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃2, 𝑗) = 𝐴 ∧ 𝐵. this is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (ii) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 = (𝜃3, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃3.also, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃3, 𝑗) = 𝐴 ∧ 𝐶. this is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (iii) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 and 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2and {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃3 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃2 and{(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃2, 0) = 𝐴 ∧ 𝐵 and ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃3, 𝑗) = 𝐴 ∧ 𝐶 this is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0) and 𝐴 ∧ 𝐶 = (𝜙, 0). (iv) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. but, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = (𝜃2 ∨ 𝜃3, 𝑗) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 ∨ 𝜃3 ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃3 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃3, by symmetry and transitivity in 𝜃2 and 𝜃3. ⇒ ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ (𝜃2, 0) 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ (𝜃3, 0). that is, ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ 𝐵 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ 𝐶. therefore, we have ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶. this is a contradiction to our assumption. therefore, (3.1) holds. case (iv): let 𝐴, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 and 𝐵 ∈ 𝐼 × 𝐶2, that is, 𝐵 = (𝜃2, 𝑗) where 𝑗 = either 0 or 1. (i) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 = (𝜃2, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2 . also, {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃2 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃2, 𝑗) = 𝐴 ∧ 𝐵. this is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (ii) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐶 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐶. this is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (iii) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 and 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2and (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐶 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐶. this is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). 45 on the effect of doubling of intervals on the 0-distributive property of the lattice of weak congruences of chains (iv) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. but, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = (𝜃2 ∨ 𝐶, 0) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 ∨ 𝐶 ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐶. by symmetry and transitivity, (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 𝑜𝑟 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶. this is a contradiction to our assumption. therefore, (3.1) holds. case (v): let 𝐴, 𝐵, 𝐶 ∈ 𝐼 × 𝐶2, that is, 𝐴 = (𝜃1, 𝑗), 𝐵 = (𝜃2, 𝑗), 𝐶 = (𝜃3, 𝑗) where 𝑗 = either 0 or 1. (i) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 = (𝜃2, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2. also, {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∧ 𝜃 2 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∧ 𝜃2, 𝑗) = 𝐴 ∧ 𝐵. this is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (ii) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 = (𝜃3, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃3. also,{(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃 3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃3, 𝑗) = 𝐴 ∧ 𝐶. this is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (iii) let({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 𝑎𝑛𝑑 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2 𝑎𝑛𝑑 {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃3 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃2and{(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃2, 0) = 𝐴 ∧ 𝐵 and ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∩ 𝜃3, 𝑗) = 𝐴 ∧ 𝐶. this is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0) 𝑎𝑛𝑑 𝐴 ∧ 𝐶 = (𝜙, 0). (iv) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. but, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = (𝜃2 ∨ 𝜃3, 0) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 ∨ 𝜃3 ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that(𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ (𝜃2, 0) 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ (𝜃3, 0). that is, ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ 𝐵 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑘 )}, 0) ≤ 𝐶. by symmetry and transitivity in 𝐵 and 𝐶, we have ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 𝑜𝑟 ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶. this is a contradiction to our assumption. therefore, (3.1) holds. case (vi): let 𝐴, 𝐵 ∈ 𝐼 × 𝐶2, that is, 𝐴 = (𝜃1, 𝑗), 𝐵 = (𝜃2, 𝑗) where j = either 0 or 1 and 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. (i) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 = (𝜃2, 𝑖) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝜃2. also,{(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∧ 𝜃2 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∧ 𝜃2, 𝑗) = 𝐴 ∧ 𝐵. this is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (ii) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐶 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐶. this is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (iii) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 𝑎𝑛𝑑 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 𝑎𝑛𝑑 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 ∩ 𝐶 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐶. this is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (iv) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. but, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = (𝜃2 ∨ 𝐶, 0) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 ∨ 𝐶 ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐶. by symmetry and transitivity, we have (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃2 𝑜𝑟 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶. this is a contradiction to our assumption.therefore, (3.1) holds. case (vii): let 𝐴, 𝐶 ∈ 𝐼 × 𝐶2, that is, 𝐴 = (𝜃1, 𝑗), 𝐶 = (𝜃3, 𝑗) where 𝑗 = either 0 or 1 and 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. (i) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐵 46 gladys mano amirtha v & d. premalatha ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐵. this is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (ii) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐶 = (𝜃3, 𝑗) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐶. also, {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ⊆ 𝐴 ∩ 𝜃3 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ (𝐴 ∧ 𝜃3, 𝑗) = 𝐴 ∧ 𝐶. this is a contradiction, since 𝐴 ∧ 𝐶 = (𝜙, 0). (iii) let ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 𝑎𝑛𝑑 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 𝑎𝑛𝑑 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝜃3 ⇒ (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐴 ∩ 𝐵 ⇒ ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐴 ∧ 𝐵.this is a contradiction, since 𝐴 ∧ 𝐵 = (𝜙, 0). (iv) ({(𝑥𝑖 , 𝑥𝑖 )}, 0) is incomparable with both 𝐵 and 𝐶. but, ({(𝑥𝑖 , 𝑥𝑖 )}, 0) ≤ 𝐵 ∨ 𝐶 = (𝐵 ∨ 𝜃3, 0) implies (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 ∨ 𝜃3. ⇒ there exists a 𝑥𝑘 ≠ 𝑥𝑖 such that (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐵 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝜃3. by symmetry and transitivity, we have (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵 𝑜𝑟 (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐶.this is a contradiction to our assumption. therefore, (3.1) holds. case (viii): let 𝐴 ∈ 𝐼 × 𝐶2, 𝐵 and 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. let 𝐴 = (𝜃1, 𝑗)where 𝑗 = either 0 or 1. this case follows, since 𝐵 𝑎𝑛𝑑 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 implies 𝐵 𝑎𝑛𝑑 𝐶 ∈ 𝐶𝑊(𝐿𝑛) which is 0-distributive. hence, [𝐶𝑊(𝐿𝑛)](𝐼) is 0-distributive whenever 𝐼 is a lower interval of 𝐶𝑊 (𝐿𝑛). example 3.2 consider the 0-distributive lattice 𝐶𝑊(𝐿4) where 𝐿4 is {0 ≺ 𝑎 ≺ 𝑏 ≺ 1}. figure 1. 𝐶𝑊(𝐿4). consider the interval 𝐼 = [𝜙, 𝑙32] in the above lattice figure 1. let 𝐶2 = {0,1} be the two-element chain. we can form the new lattice [𝐶𝑊(𝐿4)](𝐼) = {𝐶𝑊(𝐿4) ∖ 𝐼} ∪ (𝐼 × 𝐶2) given in figure 2. 47 on the effect of doubling of intervals on the 0-distributive property of the lattice of weak congruences of chains figure 2. [𝐶𝑊(𝐿4)](𝐼) where 𝐼 = [𝜙, 𝑙32]. theorem 3.3 [𝐶𝑊(𝐿𝑛)](𝐼) is 0 distributive, when 𝐼 is an upper interval of 𝐶𝑊(𝐿𝑛). proof. let 𝐼 = [{(1,1)}, 𝜏]. let 𝐴, 𝐵, 𝐶 ∈ [𝐶𝑊(𝐿𝑛)](𝐼) such that 𝐴 ∧ 𝐵 = 𝜙, 𝐴 ∧ 𝐶 = 𝜙. claim:𝐴 ∧ (𝐵 ∨ 𝐶) = 𝜙. (3.2) there can be two possibilities, that is, 𝐴 maybe in 𝐼 × 𝐶2 or 𝐴 maynot be in 𝐼 × 𝐶2. the following cases arise: i.𝐴 ∈ 𝐼 × 𝐶2 𝑎𝑛𝑑 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 ii.𝐴 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 iii.𝐴 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐵, 𝐶 ∈ 𝐼 × 𝐶2 iv.𝐴, 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 𝑎𝑛𝑑 𝐶 ∈ 𝐼 × 𝐶2 case (i): let 𝐴 ∈ 𝐼 × 𝐶2 and 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. we note that either 𝐴 ∧ (𝐵 ∨ 𝐶) ∈ 𝐼 × 𝐶2 or 𝐴 ∧ (𝐵 ∨ 𝐶) ∉ 𝐼 × 𝐶2. suppose 𝐴 ∧ (𝐵 ∨ 𝐶) ≠ 𝜙. therefore, there exists {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 ∧ (𝐵 ∨ 𝐶) ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 and{(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 ∨ 𝐶 ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 and {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 or {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐶 or{(𝑥𝑖 , 𝑥𝑖 )} ≤ both 𝐵 and 𝐶 or incomparable with both. ⇒ {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 𝑎𝑛𝑑 {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 which is a contradiction, since 𝐴 ∧ 𝐵 ≠ 𝜙. similarly, we get a contradiction, when {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 𝑎𝑛𝑑 {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐶 and when {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐴 and {(𝑥𝑖 , 𝑥𝑖 )} ≤ 𝐵 & 𝐶.so, 𝐴 ∧ (𝐵 ∨ 𝐶) = 𝜙. when {(𝑥𝑖 , 𝑥𝑖 )} ≰ 𝑏𝑜𝑡ℎ 𝐵 & 𝐶, then there exists 𝑥𝑘 such that {(𝑥𝑖 , 𝑥𝑘 )} ≤ 𝐵 or {(𝑥𝑖 , 𝑥𝑘 )} ≤ 𝐶 ⇒ (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐵 𝑜𝑟 (𝑥𝑖 , 𝑥𝑘 ) ∈ 𝐶. so, by symmetry and transitivity, we have (𝑥𝑖 , 𝑥𝑖 ) ∈ 𝐵.so, 𝐴 ∧ 𝐵 ≥ {(𝑥𝑖 , 𝑥𝑖 )}, a contradiction again. so, (3.2) holds. case (ii): let 𝐴 ∈ 𝐶𝑊 (𝐿𝑛) ∖ 𝐼 and 𝐵, 𝐶 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼. as 𝐶𝑊(𝐿𝑛) is 0 distributive, it follows that 𝐴 ∧ (𝐵 ∨ 𝐶) = 𝜙. case (iii): let 𝐴 ∈ 𝐶𝑊 (𝐿𝑛) ∖ 𝐼 and 𝐵, 𝐶 ∈ 𝐼 × 𝐶2. suppose 𝐴 ∧ (𝐵 ∨ 𝐶) ≠ 𝜙. 48 gladys mano amirtha v & d. premalatha therefore, there exists (𝑥𝑖 , 𝑥𝑖 ) ≤ 𝐴 ∧ (𝐵 ∨ 𝐶).as in case (i),𝐴 ∧ (𝐵 ∨ 𝐶) = 𝜙 follows. case (iv): let 𝐴, 𝐵 ∈ 𝐶𝑊(𝐿𝑛) ∖ 𝐼 and 𝐶 ∈ 𝐼 × 𝐶2. suppose 𝐴 ∧ (𝐵 ∨ 𝐶) ≠ 𝜙. therefore, there exists (𝑥𝑖 , 𝑥𝑖 ) ≤ 𝐴 ∧ (𝐵 ∨ 𝐶). as in case (i),𝐴 ∧ (𝐵 ∨ 𝐶) = 𝜙 follows. hence, for an upper interval 𝐼, [𝐶𝑊(𝐿𝑛)](𝐼) is 0-distributive. example 3.4 consider the interval [𝑙1, 𝜏] in figure 1. the lattice formed by the doubling of the interval is given in figure 3. figure 3. [𝐶𝑊(𝐿4)](𝐼) where 𝐼 = [𝑙1, 𝜏]. remark 3.5 the property of 0 distributivity doesn’t hold if we consider an intermediate interval i of [𝐶𝑊(𝐿𝑛)]. example 3.6 consider the intermediate interval 𝐼 = [𝑙4, 𝑙20] in figure 1. consider the elements (𝑙4, 1) and (𝑙10, 0) disjoint with 𝑙3, that is, 𝑙3 ∧ (𝑙4, 1) = 𝜙 and 𝑙3 ∧ (𝑙10, 0) = 𝜙. now, 𝑙3 ∧ [(𝑙4, 1) ∨ (𝑙10, 0)] = 𝑙3 ∧ (𝑙10, 1) = 𝑙3 ≠ 𝜙. figure 4. [𝐶𝑊(𝐿4)[𝐼] where 𝐼 = [𝑙4, 𝑙20]. 49 on the effect of doubling of intervals on the 0-distributive property of the lattice of weak congruences of chains references [1] balasubramani, p., stone topologies of the set of prime filters of a 0distributive lattice, indian journal of pure and applied math., 35(2) (2004), 149 158. [2] chajda.i., congruence distributivity in varieties with constant, archivum mathematicum., 22 (1983), 121 124. [3] a, day, a simple solution of the word problem for lattice, canad. math. bull. 13 (1970), 253-254 [4] a. day, herb gaskill and werner poguntke, distributive lattice with finite projective covers, pacific journal of math., 81 (1979). [5] a. day, doubling constructions in lattie theory, can. j. math., vol. 44(2) (1992), 252-269. [6] g. grätzer, lattice theory: foundation, birkh�̈�user (2011). [7] j b nation, alan day’s doubling construction, algebra universalis, 34 (1995), 24 35. [8] pawar, y. s., 0-1-distributive lattices, indian journal of pure and applied math., 24(3) (1993), 173 178. [9] varlet, j. c., a generalization of notion of pseudo complemented ness, bull. soc. roy. li�́�ge., 37 (1968), 149 158. [10] a. veeramani, a study on characterisations of some lattices, phd thesis, bharathidasan university (2012). [11] a. vethamanickam, topics in universal algebra, phd thesis, madurai kamaraj university, 1994. [12] vojvodić, g., šešelja, b., on the lattice of weak congruence relations, algebra universalis, 25 (1988), 121 130. 50 ratio mathematica volume 43, 2022 controlling measles transmission dynamics with optimal control analysis chinwendu e. madubueze* isaac o. onwubuya† iorwuese mzungwega‡ abstract in this paper, a deterministic model for the transmission dynamics of measles infection with two doses of vaccination and isolation is studied. the disease-free equilibrium state and basic reproduction number, 𝑅0, of the model are computed. the sensitivity analysis of the model parameters is carried out using the latin hypercube sampling (lhs) scheme in other to ascertain the parameters that contribute to the spread of measles in the population. the result of the sensitivity analysis shows that transmission rates, vaccination rates and isolation of the infected persons in the prodromal stage are significant parameters to be targeted for the eradication of measles infection. based on the result of sensitivity analysis, an optimal control model with nutritional support as a control is developed. the analysis of optimal control model is carried out using pontryagin’s maximum principle to identify the optimal control strategies to be adopted by public health practitioners and health policy makers in curtailing the spread of measles infection. the result of the optimal control analysis via numerical simulations revealed that combined timely implementation of correct administration of the two doses of vaccination, isolation of infected persons in the prodromal stage and mass distribution of nutritional support would curtail the measles disease outbreak in the population. however, in a situation where there is a limited facility to isolate the infected persons in * chinwendu e. madubueze (joseph sarwuan tarka univerisity, makurdi, nigeria); ce.madubueze@gmail.com. † isaac o. onwubuya (joseph sarwuan tarka univerisity, makurdi, nigeria); isaacobiajulu@gmail.com. ‡ iorwuese mzungwega (joseph sarwuan tarka univerisity, makurdi, nigeria). ‡ received on april 17th, 2022. accepted on august 12th, 2022. published on september 25th, 2022. doi:10.23755/rm.v41i0.742. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. mailto:ce.madubueze@gmail.com mailto:isaacobiajulu@gmail.com c. e. madubueze, i. o. onwubuya, and i. mzungwega the prodromal stage, the combined implementation of mass distribution of nutritional support and administration of the two doses of vaccination will still eradicate measles infection in the population. keywords: measles; nutritional support; vaccination; isolation; sensitivity analysis; pairwise comparison; optimal control analysis. 2010 ams subject classification: 49k15, 49k40, 90c31, 34d20, 34c60. 1. introduction measles is a viral infectious disease caused by a single-stranded rna virus that belongs to the group of morbilliviruses of the paramyxoviridae family. it is a seasonal disease that occurs mostly during the dry season in tropical zones where it is endemic and it peaks during late winter and early spring in temperate zones [24]. non-immune people are infected via direct contact with the nasal and oral secretions or inhaling the aerosol droplets of an infected person. ninety percentage of non-immune people are exposed to an infective and have the chance of being infected with measles disease [24, 27]. measles has an average of 10 12 days incubation period. the incubation period is the interval from exposure to the prodromal stage [28], which spans for seven days of the infection period, after which the infective recovers with lifelong immunity against the disease. the symptoms of measles are based on different stages of the disease. prodromal stage symptoms include high fever, runny nose (coryza), cough, and red eyes (conjunctivitis) that lasts 2 to 4 days with a range of 17 days, while the rash stage symptoms occur a few days after the initial symptoms. the rash stage can lead to fatal complications or death if not treated early [10,12]. annually, measles affects up to 20 million people worldwide and most cases are from africa and asia [24]. according to a report from world health organization and united state center for diseases control and prevention (cdc) [44, 45], 869,770 infection cases with 207,500 deaths of measles were recorded globally in 2019, making it the highest number since the 1996 outbreak and has 50% increment as of 2016. for sub-saharan africa, about 134,200 measles deaths were recorded in 2015, while nigeria recorded a significant increase of 28,400 cases in 2019 compared with 5,067 cases in 2018. despite the cases dropping to 9,316 in 2020, the confirmed cases of measles remain high, and the case fatality is yet to be eased anytime there is an outbreak. this implies that comprehensive efforts and intervention strategies to reduce the menace of measles is crucial. therefore, it is imperative to examine the optimal strategy that can be implemented to control measles disease in high-burden countries. according to who, the two major strategies to eradicate measles are vaccine and treatment [21, 23]. isolation of infected people is also important in preventing further spread of the disease. however, increasing population immunity through vaccination remains the most effective way to prevent outbreaks of measles in a community [22]. the vaccination is mainly based on mmr (measles, mumps, and rubella) and mmrv (measles, mumps, rubella, and varicella) vaccines. these vaccines are about 95% effective as they globally prevent 4.5 million deaths yearly [16]. there are two doses of controlling measles transmission dynamics with optimal control analysis mmr vaccine. the first dose produces 90% to 95% immunity to measles while the second dose produces a stronger immunity for those that do not respond to the first dose [17]. among the childhood vaccine-preventable diseases, measles causes the most deaths in children. measles outbreaks is prevented in a community if 90% to 95% of children are vaccinated. mathematical models of infectious diseases are useful in studying transmission dynamics of diseases, testing theories, planning, implementing, evaluating and comparing various control programs that will prevent the further spread of diseases and their epidemics. a notable number of mathematical models have been elaborated and applied to infectious diseases like measles [4, 7, 9, 18, 25]. some authors, like [8, 10, 19], developed an seir model of measles where testing and diagnosis therapy was incorporated as in [10] at the latent period. authors [15, 20] considered the effect of supplemental immunization activities as an optimal policy for measles using an agestratified compartmental model. stephen et al. [17] revealed that the spread of measles disease largely depends on the contact rates with infected people within a population and the disease dies out in the population if the proportion of the population that is immune exceeds the herd immunity level. vaccination is considered in the autonomous models [3, 4, 11, 13, 19, 29] as constant parameter or compartment for vaccinated people, while in [27-31], it is examined as a time-dependent control function to determine the optimal vaccination strategy that can be implemented to control measles in high-burden countries. although [30 – 35] considered optimal control of vaccination for measles, the effect of two doses of vaccination and nutritional support are not studied. however, the authors [3, 37, 40] examined the effect of two doses of vaccination and isolation on measles disease as constant parameters without nutritional support impact and optimal control analysis. as advised by who [46], it is important to consider the effect of two doses of vaccination and nutritional support on measles transmission dynamics, which forms the study’s motivation. this involves modification of the model by [3] to investigate the impact of the two doses of vaccination, nutritional support and isolation on measles dynamics using sensitivity analysis and optimal control analysis approaches. this will help to provide the mathematical analysis of the possible control strategy (vaccine and nutritional support) that will help the public health practitioners to achieve the best strategy for the prevention and control of the spread of measles in community. the rest of the paper is organized as follows: section 2 is the model formulation for measles with constant control measures. the model analysis is discussed in section 3 which includes sensitivity analysis. we obtained the optimal control of the formulated model in section 4. in section 5, we carried out numerical simulation to verify some analytic results and their discussion, while section 6 is the conclusion. 2. model formulation a deterministic model for measles disease is presented by modifying the model by aldila and asrianti [3]. the total population at any time (𝑡) denoted by 𝑁(𝑡), is subdivided into susceptible persons, 𝑆(𝑡), exposed persons, 𝐸(𝑡), infected persons in prodromal stage, 𝑃(𝑡), infected persons in rash stage, 𝐼(𝑡), isolated persons, 𝐽(𝑡), 1st c. e. madubueze, i. o. onwubuya, and i. mzungwega dose of vaccinated persons, 𝑉1(𝑡), 2 nd dose of vaccinated persons, 𝑉2(𝑡), and recovered persons, 𝑅(𝑡) such that 𝑁 = 𝑆 + 𝐸 + 𝑃 + 𝐼 + 𝐽 + 𝑉1 + 𝑉2 + 𝑅. (1) the susceptible persons,𝑆(𝑡), decreases when they come in contact with the infected persons at a force of infection, 𝜆1 and become exposed person or by vaccination with 1 st dose vaccine at a rate, 1. the persons vaccinated with 1 st dose vaccine may be infected at a force of infection, 𝜆2 since vaccine is not 100% efficacy. the force of infections, 𝜆1 and 𝜆2, are given by 𝜆1 = 𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽) 𝑁 𝜆2 = 𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽) 𝑁 } (2) where 𝛽1 and 𝛽2 are the transmission rate for the susceptible and 1 st dose vaccinated persons respectively, 𝑛1 and 𝑛2 are the parameters that reduce the infectivity of the infected persons in the rash stage and isolated persons respectively. the 1st dose of vaccinated persons, 𝑉1(𝑡) receive 2 nd dose vaccine at a rate, 2 and achieve immunity at the rate, 𝜎. the exposed persons, 𝐸(𝑡), becomes infected persons in the prodromal stage, 𝑃(𝑡) at a rate, 𝑘 after incubation period of measles disease. the infected persons in the prodromal stage, 𝑃(𝑡), then progress to rash stage at a rate, 𝛼 while some are isolated for further treatment at a rate, 𝜑1 or they recovered at a rate, 𝛿1. in a similar way, the infected persons at the rash stage, 𝐼(𝑡), are isolated at a rate, 𝜑2 or recovered from measles at a rate, 𝛿2. meanwhile, the isolated persons recovered at a rate, 𝛿3. it is assumed that all the subpopulations experience natural death at a rate, 𝜇 and the subpopulations, 𝐼(𝑡) and 𝐽(𝑡) may die of measles disease at 𝑑1 and 𝑑2 respectively. the model description and details of the model parameters are presented in figure 1 and table 1. respectively. with figure 1 and table 1, the transition within subpopulations are expressed by the following system of first order differential equations; controlling measles transmission dynamics with optimal control analysis 𝑑𝑆 𝑑𝑡 = λ − 𝜆1𝑆 − ( 1 + 𝜇)𝑆, 𝑑𝑉1 𝑑𝑡 = 1𝑆 − (𝜆2 + 2 + 𝜇)𝑉1, 𝑑𝑉2 𝑑𝑡 = 2𝑉1 − (𝜇 + 𝜎)𝑉2, 𝑑𝐸 𝑑𝑡 = 𝜆1𝑆 + 𝜆2𝑉1 − (𝜇 + 𝑘)𝐸, 𝑑𝑃 𝑑𝑡 = 𝑘𝐸 − (𝛼 + 𝜇 + 𝜑1 + 𝛿1)𝑃, 𝑑𝐼 𝑑𝑡 = 𝛼𝑃 − (𝜑2 + 𝛿2 + 𝜇 + 𝑑1)𝐼, 𝑑𝐽 𝑑𝑡 = 𝜑1𝑃 + 𝜑2𝐼 − (𝛿3 + 𝜇 + 𝑑2)𝐽, 𝑑𝑅 𝑑𝑡 = 𝜎𝑉2 + 𝛿1𝑃 + 𝛿2𝐼 + 𝛿3𝐽 − 𝜇𝑅} (3) where 𝜆1 = 𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽) 𝑁 , 𝜆2 = 𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽) 𝑁 and the initial conditions, 𝑆(0) > 0,𝑉1(0) ≥ 0,𝑉2(0) ≥ 0,𝐸(0) ≥ 0,𝑃(0) ≥ 0,𝐼(0) ≥ 0,𝐽(0) ≥ 0, 𝑅(0) ≥ 0. the model parameters are assumed to be nonnegative except recruitment rate, λ, that is strictly positive. figure 1. model flow diagram for transmission dynamics of measles disease. c. e. madubueze, i. o. onwubuya, and i. mzungwega table 1. parameters and their descriptions parameters parameters description (ranges)nominal values sources 𝚲 𝜷𝟏 𝜷𝟐 𝜺𝟏 𝜺𝟐 𝒏𝟏 𝒏𝟐 𝝁 𝒌 𝜶 𝜹𝟏 𝜹𝟐 𝜹𝟑 𝒅𝟏 𝒅𝟐 𝝎𝟏 𝝎𝟐 𝝈 recruitment rate transmission rate for 𝑆(𝑡) class transmission rate for 𝑉(𝑡) class vaccination rate of first dose vaccine vaccination rate of second dose vaccine infectivity reduction rate for 𝐼(𝑡) class infectivity reduction rate for 𝐽(𝑡) class natural death rate progression rate from 𝐸(𝑡) to 𝑃(𝑡) progression rate from 𝑃(𝑡) to 𝐼(𝑡) recovery rate for 𝑃(𝑡) class recovery rate for 𝐼(𝑡)class recovery rate for 𝐽(𝑡) class disease-related death rate for 𝐼(𝑡)class disease-related death rate for 𝐽(𝑡) class isolation rate for 𝑃(𝑡) class isolation rate for 𝐼(𝑡) class immunity rate due to 2nd dose of vaccine (−)2000 (0.0004 − 0.5)0.6 (0.0003 − 0.4)0.5 (0.01 − 0.95)0.6 (0.01 − 0.95)0.01 (−)0.1 (−)0.01 (−)1 65.365⁄ (−)0.09 (−)0.003 (−)0.2 (−)0.06 (−)0.3121 (−)0.125 (−)0.1 (0.0001 − 0.05)0.01 (0.001 − 0.5)0.001 (−)0.01 [1] assumed assumed [38] assumed [3] [3] [3] [1] assumed [3] assumed [36] [39] [1] [1] assumed [38] 3. model analysis here, the well-poseness of system (3) is established which implies that the model makes biological sense. this is done by proving the existence of nonnegative solutions and boundedness of the model (3) when given initial solutions of the model. theorem 1. with the initial solutions, 𝑆(0) > 0, 𝑉1(0) ≥ 0, 𝑉2(0) ≥ 0, 𝐸(0) ≥ 0, 𝑃(0) ≥ 0, 𝐼(0) ≥ 0, 𝐽(0) ≥ 0,𝑅(0) ≥ 0, the model equation (3) has non-negative solutions for all time, 𝑡 > 0. proof. let 𝑡1 = 𝑠𝑢𝑝{𝑡 > 0: 𝑆(0) > 0, 𝑉1(0) ≥ 0, 𝑉2(0) ≥ 0, 𝐸(0) ≥ 0, 𝑃(0) ≥ 0, 𝐼(0) ≥ 0, 𝐽(0) ≥ 0,𝑅(0) ≥ 0} ∈ [0,𝑡]. controlling measles transmission dynamics with optimal control analysis from the first equation of system (3), we have 𝑑𝑆 𝑑𝑡 = λ − 𝜆1𝑆 − 1𝑆 − 𝜇𝑆 ≥ −( 1 + 𝜇 + 𝜆1)𝑆. applying the method of integrating factor with initial condition, 𝑆(0), we have 𝑆(𝑡) ≥ 𝑆(0)exp {−∫ ( 1 + 𝜇 + 𝜆1) 𝑡 0 𝑡1} > 0 which is always positive for 𝑡 > 0. in similar way, 𝑉1(𝑡) > 0,𝑉2(𝑡) > 0,𝐸(𝑡) > 0, 𝑃(𝑡) > 0, 𝐼(𝑡) > 0, 𝐽(𝑡) > 0, 𝑅(𝑡) > 0 for 𝑡 > 0. this means that the solution set 𝑆(𝑡),𝑉1(𝑡),𝑉2(𝑡),𝐸(𝑡),𝑃(𝑡),𝐼(𝑡),𝐽(𝑡),𝑅(𝑡) of the system (3) is non-negative for all 𝑡 > 0. to show the boundedness of the solutions of the system (3), we state and prove feasible region of the system (3). theorem 2. the solutions of system (3) are contained in the feasible region, ω = {(𝑆, 𝑉1,𝑉2,𝐸,𝑃,𝐼, 𝐽,𝑅) ∈ ℜ+ 8 : 𝑁 ≤ λ 𝜇 } with the non-negative initial conditions. proof. to obtain the total population, 𝑁(𝑡), we sum up the equations of system (3) to yields 𝑑𝑁 𝑑𝑡 = λ − 𝜇𝑁 − 𝑑1𝐼 − 𝑑2𝐽 ≤ λ − 𝜇𝑁. (4) applying gronwall’s inequality with the initial condition, 𝑁(0) = 𝑁0 in equation (4) gives 𝑁(𝑡) ≤ λ 𝜇 + [𝑁0 − λ 𝜇 ]𝑒−𝜇𝑡. (5) if 𝑁0 > (<) λ 𝜇 , the total population, 𝑁, tends to λ 𝜇 as 𝑡 → ∞. thus, in either case, the total population, 𝑁(𝑡) → λ 𝜇 as 𝑡 → ∞ in (5). hence, the solution set of system (3) will enter the feasible region, ω that is positively invariant. 3.1 existence of disease-free equilibrium state and basic reproduction number disease-free equilibrium state occurs when there is no infection in the population, that is when the infected state variables are zero. solving simultaneously at equilibrium state, 𝑑𝑆 𝑑𝑡 = 0, 𝑑𝑉1 𝑑𝑡 = 0, 𝑑𝑉2 𝑑𝑡 = 0, 𝑑𝐸 𝑑𝑡 = 0, 𝑑𝑃 𝑑𝑡 = 0, 𝑑𝐼 𝑑𝑡 = 0, 𝑑𝐽 𝑑𝑡 = 0, 𝑑𝑅 𝑑𝑡 = 0 of the system (3) gives the disease-free equilibrium state, c. e. madubueze, i. o. onwubuya, and i. mzungwega 𝐸0 = (𝑆 0,𝑉1 0,𝑉2 0,𝐸0,𝑃0, 𝐼0, 𝐽0,𝑅0) = ( λ (𝜀1+𝜇) , 𝜀1λ 𝑓(𝜀1+𝜇) , 𝜀1𝜀2λ 𝑓(𝜎+𝜇)(𝜀1+𝜇) ,0,0,0,0, 𝜎𝜀1𝜀2λ 𝜇𝑓(𝜎+𝜇)(𝜀1+𝜇) ) (6) with 𝑓 = 2 + 𝜇. basic reproduction number, 𝑅0 the basic reproduction number is a threshold quantity that determines the persistence and eradication of the infectious disease in the population, making it the most important quantity in infectious disease epidemiology. it is defined as the mean number of persons infected when a single infective is introduced into a wholly susceptible population [6]. 𝑅0 is computed using the next-generation matrix approach [6]. following the approach in [6], the rate of new infection, ℱ𝑖, and the rate of transitional terms, 𝒱𝑖, in compartment 𝑖, of the system (3) are given as ℱ𝑖 = ( 𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽)𝑆 𝑁 + 𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽)𝑉1 𝑁 0 0 0 ) , 𝒱𝑖 = ( 𝑔𝐸 −𝑘£ + ℎ𝑃 −𝛼𝑃 + 𝑝𝐼 −𝜑1𝐼 − 𝜑2𝑃 + 𝑞𝐽 ), where 𝑖 = 1,…,4 is the number of infected compartments and 𝑔 = (𝜇 + 𝑘),ℎ = (𝛼 + 𝜑1 + 𝛿1 + 𝜇),𝑝 = (𝜑2 + 𝛿2 + 𝜇 + 𝑑1),𝑞 = (𝛿3 + 𝜇 + 𝑑2). (7) taking the partial derivative of ℱ𝑖 and 𝒱𝑖 with respect to 𝐸,𝑃,𝐼 𝑎𝑛𝑑 𝐽 at dfe, 𝐸0, we have respective jacobian matrices 𝐹 = ( 0 (𝛽1𝑆 0+𝛽2𝑉1 0) 𝑁0 𝑛1(𝛽1𝑆 0+𝛽2𝑉1 0) 𝑁0 𝑛2(𝛽1𝑆 0+𝛽2𝑉1 0) 𝑁0 0 0 0 0 0 0 0 0 0 0 0 0 ) , 𝑉 = ( 𝑔 0 0 0 −𝑘 ℎ 0 0 0 −𝛼 𝑝 0 0 −𝜑1 −𝜑2 𝑞 ), where 𝑁0 = 𝑆0 + 𝑉1 0 + 𝑉2 0 + 𝐸0 + 𝑃0 + 𝐼0 + 𝐽0 + 𝑅0 = λ 𝜇 . controlling measles transmission dynamics with optimal control analysis the inverse of 𝑉 is given as 𝑉−1 = ( 1 𝑔 0 0 0 𝑘 𝑔ℎ 1 ℎ 0 0 𝛼𝑘 𝑔ℎ𝑝 𝛼 ℎ𝑝 1 𝑝 0 𝑘(𝛼𝜑2+𝑝𝜑1) 𝑔ℎ𝑝𝑞 (𝛼𝜑2+𝑝𝜑1) ℎ𝑝𝑞 𝜑2 𝑝𝑞 1 𝑞) . with definition of basic reproduction number, 𝑅0, as the spectral radius of matrix, 𝐹𝑉−1, we have 𝑅0 = (𝛽1𝑆 0+𝛽2𝑉1 0) 𝑁0 [ 𝑘 𝑔ℎ + 𝛼𝑘𝑛1 𝑔ℎ𝑝 + 𝑘𝑛2(𝛼𝜑2+𝑝𝜑1) 𝑔ℎ𝑝𝑞 ]. upon substitution of 𝑆0 = λ 𝜀1+𝜇 , 𝑉1 0 = 𝜀1λ 𝑓(𝜀1+𝜇) and 𝑁0 = λ 𝜇 , we have 𝑅0 = 𝜇(𝛽1𝑓+𝛽2𝜀1) 𝑓(𝜀1+𝜇) [ 𝑘 𝑔ℎ + 𝛼𝑘𝑛1 𝑔ℎ𝑝 + 𝑘𝑛2(𝛼𝜑2+𝑝𝜑1) 𝑔ℎ𝑝𝑞 ]. by the virtue of next-generation matrix approach [6], the disease-free equilibrium, 𝐸0, of system (3) is locally asymptotically stable if 𝑅0 < 1 and unstable if 𝑅0 > 1. this means that the measles infection will die out in the population if 𝑅0 < 1 while it will persist in the population when 𝑅0 > 1. 3.2 sensitivity analysis sensitivity analysis plays an important role in examining the effect, influence and contribution of the parameters of a mathematical model to the model output. to know the type of intervention strategies to adopt in reducing the transmission and prevalence of any infectious disease, sensitivity analysis is carried-out to determine the biological significance of the model parameters in relation to the reproduction number, 𝑅0. we adopt the latin hypercube sampling (lhs) scheme used by [41,42,43] with the partial rank correlation coefficients (prccs) procedure to assess the biological implications of each input parameter to the output parameter, the disease threshold, 𝑅0. this type of sensitivity analysis approach provides numerical results that enable us to explore the entire parameter space simultaneously, thereby producing an unbiased selection of the parameter values. the signs (positive or negative) of the prccs indicate the precise strength of the relationship between the input variables (parameters of the model) and the output variable, 𝑅0 in this case. it also provides an insight to the degree of monotonicity between the parameters of the model and 𝑅0. thus, comparing the values of prccs enabled us to directly evaluate the impact of the model parameters on 𝑅0. c. e. madubueze, i. o. onwubuya, and i. mzungwega figure 2 shows the prccs for some important parameters of the model. the parameters 𝛽1 and 𝛽2 have positive prccs meaning increasing their values increase 𝑅0, which in return increase the spread of measles infection in the population. whereas, the parameters 1 , 2,𝜔1 and 𝜔2 with negative prccs reduce the value of 𝑅0 when they are increased. they have the capacity of ameliorate the spread of measles infection in the population, which leads to the eradication of the disease in the population. however, the parameter 𝜔2 has a small magnitude of prcc that is non-monotonically related to 𝑅0 but it can still produce a change in the transmission dynamics of measles infection. in other to identify the model parameters that are significant in curtailing or enhancing the spread of measles disease, the fisher transformation is applied to the prccs to compute the pvalues of each of the model parameters as used in [42]. this is shown in table 2. it is observed in table 2 that the parameters (𝛽1,𝛽2, 1 , 2,𝜔1) have p-values that are significant while the parameter, 𝜔2, has an insignificance p-value. this is further shown in figure 3 as scatterplots for 𝑅0 against some model parameters. from figure 3, it is observed that the parameters (𝛽1,𝛽2, 1 , 2,𝜔1) have a significant impact on 𝑅0 than 𝜔2. figure 2. tornados plot for some significant model parameters. controlling measles transmission dynamics with optimal control analysis figure 3. monte carlo simulations for some important parameters of the model generated using the parameter values in table 1. in each simulation run, 1000 randomly selected parameters are used. table 2. parameter prcc significance (unadjusted p-value) 0 0.5 1 -6 -4 -2 0 2  1 lo g (r 0 ) 0 0.5 1 -6 -4 -2 0 2  2 lo g (r 0 )0 0 0.5 1 -6 -4 -2 0 2  1 lo g (r 0 ) 0 0.5 1 -6 -4 -2 0 2  2 lo g (r 0 ) 0 0.05 0.1 -6 -4 -2 0 2  1 lo g (r 0 ) 0 0.005 0.01 -6 -4 -2 0 2  2 lo g (r 0 ) parameter prcc p-value keep 𝜷𝟏 0.65564141 0.0000 true 𝜷𝟐 0.57787305 0.0000 true 𝜺𝟏 −0.67496222 0.0000 true 𝜺𝟐 −0.65937154 0.0000 true 𝝎𝟏 −0.54171573 0.0000 true 𝝎𝟐 −0.01142558 0.7049 false c. e. madubueze, i. o. onwubuya, and i. mzungwega table 3. pairwise prcc comparisons (unadjusted p-values) 𝜷𝟏 𝜷𝟐 𝜺𝟏 𝜺𝟐 𝝎𝟏 𝜷𝟏 0.00506 0 0 0 𝜷𝟐 0 0 0 𝜺𝟏 0.5314 2.048 × 10 −6 𝜺𝟐 3.743 × 10 −5 𝝎𝟏 table 4. pairwise prcc comparisons (fdr adjusted p-values) 𝜷𝟏 𝜷𝟐 𝜺𝟏 𝜺𝟐 𝝎𝟏 𝜷𝟏 0.005622 0 0 0 𝜷𝟐 0 0 0 𝜺𝟏 0.5314 2.926 × 10 −6 𝜺𝟐 4.679 × 10 −5 𝝎𝟏 table 5. parameters different after fdr adjustment? 𝜷𝟏 𝜷𝟐 𝜺𝟏 𝜺𝟐 𝝎𝟏 𝜷𝟏 true true true true 𝜷𝟐 true true true 𝜺𝟏 fasle true 𝜺𝟐 true 𝝎𝟏 tables 3 and 4 show the pairwise comparison of the important parameters of the model, whose p-values are less than 0.05.𝑇ℎ𝑖𝑠 𝑖𝑠 to establish if there exist any difference between the processes describing the compared parameters. the results of the pairwise prcc comparison for the unadjusted p-values and the false discovery rate (fdr) adjusted p-values are presented in table 3 and table 4, respectively. with the fdr adjusted p-values in table 4, we present the parameters different in table 5. if the pvalues of the compared pair of significant parameters are less than 0.05, we say that they controlling measles transmission dynamics with optimal control analysis are different (true); otherwise not different (false). we also noted from table 5, that apart from 1 − 2 pair, all other pairs of parameters are significantly different. thus, the parameters 𝛽1,𝛽2, 1, 2, 𝜔1 play a vital role in the eradication of the measles disease. hence, the spread of measles infection will reduce drastically if the value of 𝑅0 is less than a unity (𝑅0 < 1), which implies reducing the values of 𝛽1 and 𝛽2 as well as increasing the values of 1, 2, 𝜔1 . this establishes that isolating the infected individuals in the prodromal stage and minimizing contact with infected persons (both at the prodromal and rash stage) will eradicate the spread of measles in the population. also, increasing the rate of correct administration of vaccines (first and second dose) will go a long way in reducing the number of infected individuals as many susceptible people will be protected by vaccination thereby minimize the spread of measles before and during the epidemic. 4. optimal control analysis optimal control has been extensively applied as a strategy in controlling many epidemic outbreaks. the main idea of applying the optimal control to disease epidemics is to choose among the available strategies, the most suitable and effective strategies that will reduce disease infection rate to a minimum level while optimizing the cost of deploying these strategies [26]. in terms of measles epidemics, such strategy can include therapies, vaccines, isolation and educational campaigns [5]. based on the result of the sensitivity analysis, the functions, 𝑢1(𝑡),𝑢2(𝑡),𝑢3(𝑡),𝑢4(𝑡), are considered as time-dependent control functions where 𝑢1(𝑡) is mass distribution of nutrition (supplement) support that reduces the transmission rates, 𝑢2(𝑡) is the first dose vaccination control, 𝑢3(𝑡) is the second dose vaccination control and 𝑢4(𝑡) is the isolation of infected people in the prodromal stage. the nutritional support is to boost the immune system of the body. thus, the optimal control model of the system (3) is given by 𝑑𝑆 𝑑𝑡 = λ − (1−𝑢1(𝑡))𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽)𝑆 𝑁 − 1𝑢2(𝑡)𝑆 − 𝜇𝑆, 𝑑𝑉1 𝑑𝑡 = 1𝑢2(𝑡)𝑆 − (1−𝑢1(𝑡))(1−𝑢2(𝑡))𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽)𝑉1 𝑁 − 2𝑢3(𝑡)𝑉1 − 𝜇𝑉1, 𝑑𝑉2 𝑑𝑡 = 2𝑢3(𝑡)𝑉1 − (𝜇 + 𝜎)𝑉2, 𝑑𝐸 𝑑𝑡 = (1−𝑢1(𝑡))𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽)𝑆 𝑁 + (1−𝑢1(𝑡))(1−𝑢2(𝑡))𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽)𝑉1 𝑁 − (𝜇 + 𝑘)𝐸, 𝑑𝑃 𝑑𝑡 = 𝑘𝐸 − (𝛼 + 𝜇 + 𝜑1 + 𝑢4(𝑡) + 𝛿1)𝑃, 𝑑𝐼 𝑑𝑡 = 𝛼𝑃 − (𝜑2 + 𝛿2 + 𝜇 + 𝑑1)𝐼, 𝑑𝐽 𝑑𝑡 = 𝜑1𝑃 + 𝑢4(𝑡)𝑃 + 𝜑2𝐼 − (𝛿3 + 𝜇 + 𝑑2)𝐽, 𝑑𝑅 𝑑𝑡 = 𝜎𝑉2 + 𝛿2𝐼 + 𝛿1𝑃 + 𝛿3𝐽 − 𝜇𝑅. } (8) c. e. madubueze, i. o. onwubuya, and i. mzungwega these control functions are bounded, lebesgue integrable functions that satisfy 0 ≤ 𝑢1 ≤ 1, 0 ≤ 𝑢2 ≤ 0.95, 0 ≤ 𝑢3 ≤ 0.95 and 0 ≤ 𝑢4 ≤ 1 with assumption that the highest vaccination coverage will be 95%. the goal is to reduce the number of infected people (𝑃(𝑡),𝐼(𝑡),𝐽(𝑡)) and increase the number of susceptible people 𝑆(𝑡) while minimizing the cost of implementing controls. therefore, the objective function is given as γ(𝑢1 ,𝑢2 ,𝑢3 ,𝑢4 ) = ∫ (𝑏𝑃 + 𝑐𝐼 + 𝑑𝐽 + 1 2 ∑ 𝑚𝑖𝑢𝑖 2(𝑡)4𝑖=1 ) 𝑡𝑓 0 𝑑𝑡 (9) and is subject to equation (8) with the initial conditions of the system (3). in equation (9), the constants, 𝑏,𝑐,𝑑,𝑚1,𝑚2,𝑚3,𝑚4, are positive weights to balance the size of the terms attached with them and 𝑡𝑓 is the final time to implement the controls, 𝑢1(𝑡),𝑢2(𝑡),𝑢3(𝑡),𝑢4(𝑡). the terms, 𝑏𝑃,𝑐𝐼,𝑑𝐽 are the cost related to reducing the number of infected people (𝑃,𝐼,𝐽) such as cost of the mass distribution of nutrition (supplement) support, isolation and first dose and second dose vaccination at the due time. we seek optimal controls 𝑢1 ∗,𝑢2 ∗,𝑢3 ∗,𝑢4 ∗ such that γ(𝑢1 ∗,𝑢2 ∗,𝑢3 ∗,𝑢4 ∗) = 𝑚𝑖𝑛{γ(𝑢1 ,𝑢2 ,𝑢3 ,𝑢4 )|𝑢1 ,𝑢2 ,𝑢3 ,𝑢4 ∈ 𝑈}. (10) with the application of pontryagin’s maximum principle [14], the equations (8) and (9) are converted into a problem of minimizing pointwise a hamiltonian, 𝐻 with respect to 𝑢1 ,𝑢2 ,𝑢3 ,𝑢4 . this is given by 𝐻 = 𝑏𝑃 + 𝑐𝐼 + 𝑒𝐽 + 𝑚1𝑢1 2(𝑡) 2 + 𝑚2𝑢2 2(𝑡) 2 + 𝑚3𝑢3 2(𝑡) 2 + 𝑚4𝑢4 2(𝑡) 2 + 1 (λ − (1 − 𝑢1(𝑡))𝛽1(𝑃 + 𝑛1𝐼 + 𝑛2𝐽)𝑆 𝑁 − 1𝑢2(𝑡)𝑆 − 𝜇𝑆) + 2 ( 1𝑢2(𝑡)𝑆 − (1 − 𝑢1(𝑡))(1 − 𝑢2(𝑡))𝛽2(𝑃 + 𝑛1𝐼 + 𝑛2𝐽)𝑉1 𝑁 − 2𝑢3(𝑡)𝑉1 − 𝜇𝑉1) + 3( 2𝑢3(𝑡)𝑉1 − (𝜇 + 𝜎)𝑉2) + 4 ( (1 − 𝑢1(𝑡))𝛽1(𝑃 + 𝑛1𝐼 + 𝑛2𝐽)𝑆 𝑁 + (1 − 𝑢1(𝑡))(1 − 𝑢2(𝑡))𝛽2(𝑃 + 𝑛1𝐼 + 𝑛2𝐽)𝑉1 𝑁 − (𝜇 + 𝑘)𝐸) + 5 (𝑘𝐸 − (𝛼 + 𝜇 + 𝜑1 + 𝑢4(𝑡) + 𝛿1)𝑃) + 6 (𝛼𝑃 − (𝜑2 + 𝛿2 + 𝜇 + 𝑑1)𝐼) + 7 (𝜑1𝑃 + 𝑢4(𝑡)𝑃 + 𝜑2𝐼 − (𝛿3 + 𝜇 + 𝑑2)𝐽) + 8( 𝜎𝑉2 + 𝛿2𝐼 + 𝛿1𝑃 + 𝛿3𝐽 − 𝜇𝑅) controlling measles transmission dynamics with optimal control analysis with 1, 2, 3, 4, 5, 6, 7, 8 as respective adjoint variables for the state variables, 𝑆,𝑉1,𝑉2,𝐸,𝑃,𝐼,𝐽,𝑅. the system of adjoint variables are derived by taking the partial derivative of 𝐻 with respect to each of their corresponding state variables. this is given by 𝑑𝜁1 𝑑𝑡 = − 𝜕𝐻 𝜕𝑆 = 𝑎 + 𝐴(1 − 𝑆 𝑁 )( 1 − 4) + 𝐵𝑉1( 4 − 2) + 1𝜇 + 1𝑢2(𝑡)( 1 − 2), 𝑑𝜁2 𝑑𝑡 = − 𝜕𝐻 𝜕𝑉1 = 𝐵𝑁(1 − 𝑆 𝑁 )( 2 − 4) + 𝐴𝑆 𝑁 ( 4 − 1)+ 2𝜇 + 2𝑢3(𝑡)( 2 − 3), 𝑑𝜁3 𝑑𝑡 = − 𝜕𝐻 𝜕𝑉2 = 𝐵𝑉1( 4 − 2)+ 𝐴𝑆 𝑁 ( 4 − 1) + 𝜎( 3 − 8) + 3𝜇, 𝑑𝜁4 𝑑𝑡 = − 𝜕𝐻 𝜕𝐸 = 𝐵𝑉1( 4 − 2) + 𝐴𝑆 𝑁 ( 4 − 1)+ ( 4 − 5)𝑘 + 4𝜇, 𝑑𝜁5 𝑑𝑡 = − 𝜕𝐻 𝜕𝑃 = −𝑏 + 𝐵𝑉1( 4 − 2)+ 𝐴𝑆 𝑁 ( 4 − 1) + ( 5 − 8)𝛿1 + (1−𝑢1(𝑡))(1−𝑢2(𝑡))𝛽2𝑉1 𝑁 ( 2 − 4) + ( 5 − 7)(𝜑1 + 𝑢4(𝑡)) + (1−𝑢1(𝑡))𝛽1𝑆 𝑁 ( 1 − 4)+ ( 5 − 6)𝛼 + 5𝜇, 𝑑𝜁6 𝑑𝑡 = − 𝜕𝐻 𝜕𝐼 = −𝑐 + 𝐵𝑉1( 4 − 2)+ 𝐴𝑆 𝑁 ( 4 − 1)+ 6(𝜇 + 𝑑1) + (1−𝑢1(𝑡))(1−𝑢2(𝑡))𝑛1𝛽2𝑉1 𝑁 ( 2 − 4)+ ( 6 − 7)𝜑2 + (1−𝑢1(𝑡))𝑛1𝛽1𝑆 𝑁 ( 1 − 4)+ ( 6 − 8)𝛿2, 𝑑𝜁7 𝑑𝑡 = − 𝜕𝐻 𝜕𝐽 = −𝑑 + 𝐵𝑉1( 4 − 2) + 𝐴𝑆 𝑁 ( 4 − 1)+ 7(𝜇 + 𝑑2 ) + (1−𝑢1(𝑡))(1−𝑢2(𝑡)) 𝑛2𝛽2𝑉1 𝑁 ( 2 − 4)+ 8𝜇 + ( 7 − 8)𝛿3 + (1−𝑢1(𝑡))𝑛2𝛽1𝑆 𝑁 ( 1 − 4), 𝑑𝜁8 𝑑𝑡 = − 𝜕𝐻 𝜕𝑅 = 𝐵𝑉1( 4 − 2) + 𝐴𝑆 𝑁 ( 4 − 1), } (11) where 𝐴 = (1−𝑢1(𝑡))𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽) 𝑁 and 𝐵 = (1−𝑢1(𝑡))(1−𝑢2(𝑡))𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽) 𝑁2 with tranversality conditions 1(𝑡𝑓) = 2(𝑡𝑓) = 3(𝑡𝑓) = 4(𝑡𝑓) = 5 (𝑡𝑓) = 6(𝑡𝑓) = 7(𝑡𝑓) = 8(𝑡𝑓) = 0. (12) furthermore, the respective controls, 𝑢1 ∗,𝑢2 ∗,𝑢3 ∗,𝑢4 ∗ are obtained by solving 𝜕𝐻 𝜕𝑢1 = 0, 𝜕𝐻 𝜕𝑢2 = 0, 𝜕𝐻 𝜕𝑢3 = 0, 𝜕𝐻 𝜕𝑢4 = 0 and these are given by c. e. madubueze, i. o. onwubuya, and i. mzungwega 𝑢1 ∗ = (𝑃+𝑛1𝐼+𝑛2𝐽)(𝛽1𝑆(𝜁4− 𝜁1)+(1−𝑢2(𝑡))𝛽2𝑉1(𝜁4− 𝜁2)) 𝑚1𝑁 , 𝑢2 ∗ = (𝑃+𝑛1𝐼+𝑛2𝐽)(1−𝑢1(𝑡))𝛽2𝑉1(𝜁4− 𝜁2) 𝑚1𝑁 + (𝜁1−𝜁2)𝜀1𝑆 𝑚2 , 𝑢3 ∗ = (𝜁2−𝜁3)𝜀2𝑉1 𝑚3 , 𝑢4 ∗ = (𝜁5−𝜁7)𝑃 𝑚4 . with the controls 𝑢1 ∗,𝑢2 ∗,𝑢3 ∗,𝑢4 ∗, the optimality condition is given by 𝑢1 𝑜𝑝𝑡 = 𝑚𝑎𝑥{0 ,𝑚𝑖𝑛(1 ,𝑢1 ∗)}, 𝑢2 𝑜𝑝𝑡 = 𝑚𝑎𝑥{0 ,𝑚𝑖𝑛(0.95 ,𝑢2 ∗)}, 𝑢3 𝑜𝑝𝑡 = 𝑚𝑎𝑥{0 ,𝑚𝑖𝑛(0.95 ,𝑢3 ∗)}, 𝑢4 𝑜𝑝𝑡 = 𝑚𝑎𝑥{0 ,𝑚𝑖𝑛(1 ,𝑢4 ∗)}. } (13) the optimality system consists of the state system (8), the adjoint system (11) with initial conditions of (3) and transversality condition (12) together with the characterization of the optimality condition (13). the restrictions of obtaining the uniqueness of the optimal control based on the length of time follow the approach in [2, 9, 14]. 5. numerical simulations in this section, the solutions of the optimality system are solved numerically using the forward and backward fourth-order runge-kutta method that is coded in matlab software. the parameter values in table 1 with the constants 𝑏 = 𝑐 = 𝑑 = 100,𝑚1 = 10000,𝑚2 = 2000, 𝑚3 = 2000,𝑚4 = 5000 and the initial conditions, 𝑆(0) = 200000, 𝑉1(0) = 2000, 𝑉2(0) = 1800, 𝐸(0) = 80, 𝑃(0) = 60, 𝐼(0) = 100, 𝐽(0) = 20, 𝑅(0) = 10000 are used for the numerical simulations purpose. 5.1 discussion figure 4 shows the population dynamics of infected persons in the prodromal stage, 𝑃(𝑡), infected persons in the rash stage, 𝐼(𝑡), isolated persons, 𝐽(𝑡), for with and without control and the control profile. according to figure 4𝑎, with control measures, measlesfree population is achieved for 𝑃(𝑡) population faster, and thus reducing the number of persons moving to the infected people in the rash stage, while figure 4b, the number of infected persons in the rash stage, 𝐼(𝑡), decrease to zero within 25 weeks with control measures in place. also, figure 4𝑐 reveals that with the implementation of the control measures, there is a sharp increase in the number of isolated persons before decreasing to zero by 25 weeks and achieves measles-free population. controlling measles transmission dynamics with optimal control analysis figure 4. the population dynamics of (a) infected persons in the prodromal stage, 𝑃(𝑡), (b) infected persons in the rash stage, 𝐼(𝑡), (c) isolated persons, 𝐽(𝑡), (d) controls, 𝑢2, 𝑢3, and (e) controls, 𝑢1, 𝑢4. here, w/c means “with control” while w/o/c means “without control”. 0 50 100 0 500 1000 1500 2000 time (weeks) p (t ) (a) w/c w/o/c 0 50 100 0 50 100 time (weeks) i( t) (b) w/c w/o/c 0 50 100 0 20 40 60 time (weeks) j (t ) (c) w/c w/o/c 0 50 100 0 2 4 6 8 x 10 -4 time (weeks) c o n tr o l p ro fi le (d) u 2 u 3 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (weeks) c o n tr o l p ro fi le (e) u 1 u 4 c. e. madubueze, i. o. onwubuya, and i. mzungwega figure 5. the population dynamics of (a) infected persons in the prodromal stage, 𝑃(𝑡), (b) infected persons in the rash stage, 𝐼(𝑡), (c) isolated persons, 𝐽(𝑡) when triple controls are implemented together. here, the numbers 1,2,3, 𝑎𝑛𝑑 4, are subscripts of the control functions, 𝑢1,𝑢2,𝑢3,𝑢4 while w/o/c means “without control”. figure 6. control profile for implementation of triple optimal controls. 0 50 100 0 500 1000 1500 2000 (a) 123 124 134 234 w/o/c 0 50 100 0 50 100 time (weeks) i( t) (b) 123 124 134 234 w/o/c 0 50 100 0 20 40 60 time (weeks) j (t ) (c) 123 124 134 234 w/o/c 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 x 10 -4 time (weeks) c o n tr o l p ro fi le u 4 =0 u 2 u 3 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 time (weeks) u 1 u 4 =0 u 1 controlling measles transmission dynamics with optimal control analysis the control profile for achieving the result in figures 4a-4c are displayed in figures 4d and 4e. they show that the upper bounds for 𝑢1,𝑢2,𝑢3 are 9.9, 6.1 × 10 −4 and 2 × 10−5, respectively, where the control, 𝑢3 maintains a bound of 6.9 for 50 weeks before it gradually increases to 0.9 as at 90 weeks. for without control measures, figures 4a – 4c show the endemicity of the measles infection in the population. in figure 5, the implementation of triple control measures for the dynamics of the infected compartments (𝑃(𝑡),𝐼(𝑡),𝐽(𝑡)) are evaluated. we observed in figures 5a -5b that simultaneous implementation of any triple control measures reduce the number of infected persons (𝑃(𝑡),𝐼(𝑡)) in the population as they achieve a measles-free population within a short time while for isolated persons, 𝐽(𝑡), the combine implementation of 𝑢1,𝑢2,𝑢3(123) yields a faster and better result in achieving a measles-free population compared with other combinations (see figure 5c). with this, it implies that combined implementation of control measures, 𝑢1,𝑢2,𝑢3(123), reduces the number of infected persons in the prodromal stage, 𝑃(𝑡), rash stage, 𝐼(𝑡) and the isolated persons, (𝐽(𝑡)) compare with any other combination of control measures. the control profile for triple control measures are display in figure 6. the control profile when 𝑢4 = 0 is the combined implementation of 𝑢1,𝑢2,𝑢3(123) that gives the best result in figure 5, which indicates that mass distribution of nutritional (supplement) support, administration of first and second dose vaccine control measures have much effect on controlling measles in the population. to achieved this, 𝑢1 maintains an upper bound that declines after 85 weeks, whereas 𝑢2 maintains a bound of 1.2 × 10 −4 that decreases gradually till 85 weeks where it declines to the final time. for 𝑢3, it starts with a bound of 3.2 × 10 −5 that slightly increases to 4.0 × 10−5 at 70 weeks before declining to the final time. the numerical simulations imply that combined implementation of mass distribution of nutritional support, complete vaccination with the first and second dose of vaccine and isolation of infected persons in the prodromal stage will help eradicate the spread of measles in the population. this is in agreement with the sensitivity analysis result and the results in [3]. this indicates that implementation of control measures will help prevent the spread of measles infection in the population. however, if there are limited facilities to isolate the infected persons in the prodromal stage, the triple control measures, mass distribution of nutritional support and complete vaccination with first and second doses of vaccine will reduce the spread of measles infection as fewer people will be infected and thus help the health practitioners achieve the best strategy in the control of the spread of measles in the community. 6. conclusion in this paper, an autonomous system for the transmission dynamics of measles disease involving isolated persons and two doses of vaccination is formulated. the model diseasefree equilibrium and basic reproduction number (𝑅0) are computed. the sensitivity analysis of the basic reproduction number, 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[44] cdc. morbidity and mortality weekly report (mmwr). progress toward regional measles elimination – world, 2000-2019. weekly / november 13, 2020 / 69(45): 17001705, 2020. access 27th february 2022 on https://www.cdc.gov/mmwr/volumes/69/wr/mm6945a6.htm?s_cid=mm6945a6_w [45] who. worldwide measles deaths climb 50% from 2016 to 2019 claiming over 207,500 lives in 2019. access 27th february 2022 on https://www.who.int/news/item/1211-2020-worldwide-measles-deaths-climb-50-from-2016-to-2019-claiming-over-207500-lives-in-2019 [46] who. measles, 2019. access 27th february 2022 on https://www.who.int/newsroom/fact-sheets/detail/measles https://www.ephi.gov.et/images/guidelines/guideline-on-measles-surveillance-and-outbreak-management2012.pdf https://www.ephi.gov.et/images/guidelines/guideline-on-measles-surveillance-and-outbreak-management2012.pdf https://www.cdc.gov/mmwr/volumes/69/wr/mm6945a6.htm?s_cid=mm6945a6_w https://www.who.int/news/item/12-11-2020-worldwide-measles-deaths-climb-50-from-2016-to-2019-claiming-over-207-500-lives-in-2019 https://www.who.int/news/item/12-11-2020-worldwide-measles-deaths-climb-50-from-2016-to-2019-claiming-over-207-500-lives-in-2019 https://www.who.int/news/item/12-11-2020-worldwide-measles-deaths-climb-50-from-2016-to-2019-claiming-over-207-500-lives-in-2019 https://www.who.int/news-room/fact-sheets/detail/measles https://www.who.int/news-room/fact-sheets/detail/measles approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 36, 2019, pp.89-95 89 the theorem of the complex exponentials alberto daunisi* abstract this paper describes a new theorem that relates the lengths of the legs of a right triangle with the ratio of three complex exponentials. the big novelty of the theorem consists in transforming two real measures of legs derived from euclidean geometry into a combination of imaginary elements obtained from the complex analysis. keywords: complex analysis, complex geometry 2010 subject classification: 37k20.† * researcher in mathematics, bologna (italy). alberto.daunisi@gmail.com † received on december 10th, 2018. accepted on april 24th, 2019. published on june 30th, 2019. doi: 10.23755/rm.v36i1.472. issn: 1592-7415. eissn: 2282-8214. ©alberto daunisi this paper is published under the cc-by licence agreement. alberto daunisi 90 1. introduction it seems difficult, apparently, to imagine that the difference between the lengths of the legs of a right triangle may have some connection with the ratio of complex numbers, or vice-versa, that a ratio of complex numbers may be obtained from the difference of the legs of a right triangle, but this theorem, that we will call, precisely, theorem of the complex exponentials, shows that it is possible. let’s start with some historical and mathematical considerations, from which our research is inspired. in ancient greece, the right triangles were basically solved by the first and second theorem of euclide (iv-iii century b.c.) and by the theorem of pythagoras (about 575-495 b.c.). this happened because the greek trigonometry, that was only applied to the study of astronomy, was based on the measurement of the ropes of a circle (subtended by a certain angle), rather than on that of sines and cosines. the functions sine and cosine, developed by the indians in the iv-v century a.c., have been imported in the arab world around the viii century a.c., and then, to the west world, a few centuries later. from this moment, the triangles started to be solved by the relations that bind the lengths of the sides of the triangle with the values of the trigonometric functions of its angles. in particular, two fundamental trigonometric theorems were introduced, through which it has been possible to solve any problem related to the elements of a triangle: the theorem of sines and the theorem of carnot. the first one states that in any triangle the ratio between one side and the sine of the opposite angle is always constant and equal to the diameter of the circle circumscribed to the given triangle; the second one states that in any triangle the square of one side is equal to the sum of the squares of the other two, plus their product to the cosine of the angle included. from the theorem of sines, applied to the right triangles, it descends the theorem according to which in a right triangle a leg is equal to the product of the hypotenuse for the sine of the angle opposite to the leg. finally, we arrive at the xviii century a.c., where, in another branch of mathematics completely different from the above one, is developed, in all its entirety, the theory of complex numbers of the form x+iy, with x and y real numbers and i = √-1 the imaginary unit. in particular, the studies of abraham de moivre (1667-1754) and leonhard euler (17071783) provided to the complex numbers a definitive and systematic structure from which descended the complex trigonometric functions and the complex exponential functions. de moivre left us the famous formula (1739) that calculates the power of a complex number expressed in the form trigonometric (cos α + sin α)ᵐ= cos(mα) + i sin(mα), while euler left us the equally famous formula (1748) that binds the trigonometric functions sine and cosine to the complex exponential function = cos w + i sin w. the theorem of the complex exponentials 91 our theorem is inspired by a very specific motivation: considering that the sides of a right triangle may be expressed by a trigonometric function, and this one by a complex variable, we wanted to discover if the same sides may have a relationship with the elements of the complex analysis and, in case of positive response, in which way and form. with this purpose, we discovered two important results: the first one is that the ratio between the difference of the legs of a right triangle and the difference of their projections on the hypotenuse, multiplied by the cosine of half-difference of two angles opposite to the legs, is always constant; the second one is that this constant is given by the ratio between complex exponentials, or their powers, where the most important constants of the all mathematics are appearing: the constant of napier e (or of euler), introduced by john napier on 1618 and used systematically by euler (1736) for its exponentials; the imaginary unit i, officially introduced by friedrich gauss (1777-1855) in an essay of 1832; the constant of archimedes (287-212 b.c.) π, calculated with approximation by the greek mathematician in the iii century b.c. and definitively calculated, with 35 decimal digits, by ludolf van ceulen on 1610. both the above important results are set forth and proved in the following theorem. 2. the theorem of the complex exponentials statement: in a right triangle cab (rectangle in a), where a is the hypotenuse, c and b the legs, m and n the projections of the respective legs c and b on the hypotenuse, γ and β the angles respectively opposed to c and b, it results: where e=2,71….is the napier’s constant, π=3,14…..is the pi and i= is the imaginary unit. proof. let us consider the right triangle cab of figure 1, rectangle in a (α=90°), having hypotenuse a, height h, minor leg b and major leg c, n and m the respective projections of b and c on the hypotenuse a, γ the angle opposed to c and β the angle opposed to b. alberto daunisi 92 figure 1 with reference to the right triangle of figure 1, we know that the first euclid’s theorem asserts: from which, subtracting member to member, it derives: namely: taking into account that it is: c=a sinγ and b=a sinβ, from (3) it derives: simplifying and applying the formulas prosthaphaeresis to the denominator of second member (4), from (4) it’s obtained: (1) (2) (4) (3) the theorem of the complex exponentials 93 but we know that γ + β = 90°, so in the denominator of (5) it’s 2 sin , therefore from (5) it derives: namely: we know, from complex number’s theory, that the trigonometric form of the complex number 1+i is: 1+i = for the formula of euler it’s: cos replacing (9) in (8), we obtain: 1+i= namely: (5) (6) (7) (8) (9) (10) alberto daunisi 94 let us remind now that it is: 1= and i= replacing (12) in (11), we obtain: finally, replacing the second member of (7) with the second member of (13), we obtain: and the theorem is thus proven. conclusions we have shown a theorem born from the motivation to investigate and solve a problem: to link a geometric result of iii century b.c., although it reworked by the trigonometric functions of xvi century, to the last theories of complex numbers of xviii century, apparently irreconcilables with the euclidean geometry. we think to have got two relevant teachings: on the one hand we have bound the elements of a right triangle (legs and angles) to a constant of complex analysis, given by the combination of three most important constants of mathematics; on the other hand we have notably pointed out a precise methodological procedure of the proof, based strictly on the deductive method, where, starting from a general axiom alleging geometric structure of the right triangles, we reached, through a series of rigorous logical concatenations, a particular result alleging new structure of complex analysis. we finally think that from this article we also can draw another useful teaching: to discover this theorem allowed us to investigate on three completely different (among their) branches of mathematics (euclidean geometry, trigonometric functions, complex analysis), born and developed in different (11) (12) (13) the theorem of the complex exponentials 95 ages, transmitted by several men separated by time and by different languages, cultures and religions, who, although not knowing themselves with each other, have always improved the ideas of their predecessors and transmitted it to the future generations. they have been united only by their love for mathematics, in addition to the desire to contribute to its development. we think that we all must pick up an example from this act of faith, that only mathematics, between all sciences, is able to provide. references [1] alberto daunisi (2014). l’ultimo teorema di fermat, storia matematica, booksprint edizioni, salerno-italy. [2] howard levi (1960). foundations of geometry and trigonometry, prentice-hall, inc. englewood cliffs, new jersey. [3] joseph bak and donald newman (1997). complex analysis, springerverlag new york inc. [4] morris kline (1998). calculus, dover publications, inc. mineola, new york. [5] barry mazur (2004). imaging numbers, picador (farrar, straus and giroux), new york. ratio mathematica volume 39, 2020, pp. 79-110 on some computational and applications of finite fields jean pierre muhirwa* abstract finite field is a wide topic in mathematics. consequently, none can talk about the whole contents of finite fields. that is why this research focuses on small content of finite fields such as polynomials computational, ring of integers modulo p where p is prime or a power of prime. most of the times, books which talk about finite fields are rarely to be found, therefore one can know how arithmetic computational on small finite fields works and be able to extend to the higher order. this means how integer and polynomial arithmetic operations are done for zp such as addition, subtraction, division and multiplication in zp followed by reduction of p (modulo p). only addition and multiplication arithmetic operations are considered for a small range of finite fields (z2 − z17). with polynomials, one can learn how arithmetic computational through polynomials over finite fields are performed as their coefficients are drawn from finite fields. the paper includes also construction of polynomials over finite fields as an extension of finite fields with polynomials i.e fq[x]/f(x), where f(x) is irreducible over fq. from the past decades, many researchers complained about the applications of some topics in pure mathematics and therefore the finite fields play more important role in coding theory, such as error-coding detection and error-correction as well as cyclic codes. hence, this paper shows these applications. keywords: finite fields; error-detection; error-correction; coding; decoding; codewords; cosets; syndromes.1 *university of rwanda, college of science and technology, school of science, department of mathematics, kigali, rwanda; muhijeapi@gmail.com. 1received on january 20th, 2020. accepted on june 19th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.521. issn: 1592-7415. eissn: 2282-8214. ©jean pierre muhirwa. this paper is published under the cc-by licence agreement. 79 jean pierre muhirwa 1 introduction the structure of this research paper includes the introductory part where some preliminary properties of set theory, group theory, ring theory and fields theory are discussed. in reality we can not know what is a field without defining a group and a ring since the field is a special case of the ring. apart from introductory, the second section consist of computational in the first seven finite fields. the third, the fourth and the fifth parts of this paper discuss and compare the usual polynomial arithmetic computational and the finite field polynomial computational. the sixth part of this paper explains some of the applications of finite fields with the typical examples in coding and decoding theories, the seventh section gives the conclusion of the research paper while the last part acknowledges the financial support received from the eastern africa universities mathematics programme-international science programme, university of rwanda node (eaump-isp, ur-node). 1.1 preliminaries definition 1.1. a set is a collection of distinct objects, considered as an object in it own rights. sets are the one of the most fundamental concepts of mathematics. example 1.1. the set r, denote the set of all real numbers, and this set includes rational numbers and irrational numbers (example π, √ 2, and e) z, denote the set of all integers for both sign (negative and positive). definition 1.2. group theory, a set r together with a binary operation is called a group if it satisfies the conditions such that closure, associative, admits identity element and inverse element under the operation within the elements of r. definition 1.3. abelian group, a set r is an abelian group if it is a group for which commutative law within an operation together with r to the elements of r is verified. definition 1.4. ring theory, a set r together with two binary operations (addition and multiplication) on the elements of r is called a ring if the following conditions are satisfied: 1. (r, +) is an abelian group. 2. associative law for multiplication and distributive law are also satisfied. definition 1.5. commutative ring, a commutative ring is a ring for which the multiplication is commutative. 80 on some computational and applications of finite fields definition 1.6. commutative ring with unity, a commutative ring with unity is a ring for which there exists a non-zero multiplicative identity element. example 1.2. the set of integers z is commutative ring with 1 as a multiplicative identity element. definition 1.7. field, a field is a commutative ring with unity and for which every non-zero element of that commutative ring is invertible. example 1.3. in the set of rational numbers, q, every non-zero element has its inverse i.e (every non-zero element is invertible). definition 1.8. finite field, a finite field is a field with a finite number of elements. example 1.4. consider the set of integers modulo p (zp), where p is prime integers). this set consists of p−1 elements and all non-zero elements of this set are invertible. definition 1.9. galois group, the galois group of an extension of fields f/k, is the set of all automorphisms obtained by fixing the elements of k. definition 1.10. codewords, codewords are string of digits that can be interpreted by any machine as words or characters. example 1.5. the string 100110 is a codeword of the vector space v (6,2) of the length 6 over the finite field f2. definition 1.11. prime number, a prime number is a natural number that can be divisible only by 1 and itself (i.e, a prime number has two divisors namely 1 and the number itself). example 1.6. the first ten prime numbers are 2,3,5,7,11,13,17,19,23,29. definition 1.12. algorithm, an algorithm is a scientific term for solving an instance or a set of instructions that can be followed for solving a problem. example 1.7. to find the greatest common divisor (gcd) of two numbers a and b, we can apply division algorithm, and the gcd is the last non-zero remainder. all steps that are followed to determine the gcd will make an algorithm. 1.2 mathematical definition of a group a set r together with a binary operation (∗) is said to be a group if it satisfies the following properties: for a,b and c ∈ r, 81 jean pierre muhirwa 1. a∗ b ∈ r (closure) 2. (a∗ b)∗ c = a∗ (b∗ c) (associativity) 3. there exists additive identity element e of r such that a ∗ e = e ∗ a = a , for all a ∈ r ( for (*) operation, identity is always e ( identity element) ) 4. there exists inverse element a−1 of r such that a ∗ a−1 = a−1 ∗ a = e ( inverse element) 5. furthermore if a∗b = b∗a, then r is said to be a commutative group or an abelian group. note: this operation is not always (∗) it can be also addition, and it may be another operation defined on a set r. however, in this research paper we are restricted on the usual addition and multiplication operators. 1.3 mathematical definition of a ring a set r together with two binary operations namely addition (+) and multiplication (∗) is said to be a ring if the following 3 conditions are satisfied: fora,b and c ∈ r, 1. ( r, +) must be an abelian group 2. a∗ (b∗ c) = (a∗ b)∗ c: associativity law for multiplication 3. a∗ (b + c) = a∗ b + a∗ c ( left distributive law) (a + b)∗ c = a∗ c + b∗ c (right distributive law) note: the above two operations (+) and (∗) are not necessarily the ordinary addition and multiplication operations, reason why the definition of these operations may be needed in mathematical expressions. but this paper considers them as ordinary addition and multiplication. if there exists multiplicative identity element of r for each every non-zero element of r, always denoted 1 such that a∗1 = 1∗a = a, then we can call the ring r to be the ring with unity. the inverse of an element a for the abelian group (r,+) is denoted (−a). in addition if a ∗ b = b ∗ a, then r is called a commutative ring with unity. if every nonzero element of a commutative ring r with unity is invertible, then r 82 on some computational and applications of finite fields becomes a field. 1.4 classification of fields fields can be classified by size or by the number of elements that a field possesses. if a field contains a finite number of elements then that field is called finite field, otherwise it is an infinite field. for the rest of the work we will proceed with the finite field only. for example consider the commutative ring, zp, where p is a prime number, is a commutative ring with unity which is the field hence finite field because it possesses finite number of element. this is the most popular example of finite field. then, definition of this topic as the name indicated above, a finite field is a field with a finite order (i.e number of elements is finite). it is also called galois field (so named in honor of evariste galois). the order of a finite field is always a prime number or a power of a prime number. a finite field of order pn is denoted gf(pn), often written as f(pn) in current usage. gf(pn) is called the prime field of order p, where the p elements are denoted 0,1,2,3, ...,p − 1. in the finite field gf(p) if two elements are written as a = b this is the same as a ≡ b(mod p). finite fields are therefore denoted by gf(pn) instead of gf(k) where k = pn, for clarity. the finite field gf(2) consists of elements 0,1 which satisfy the addition and multiplication modulo 2. let us first consider the addition and multiplication of elements in gf(2) as shown in following two tables below: + 0 1 0 0 1 1 1 0 table 1: the table shows the addition in gf(2) * 0 1 0 0 0 1 0 1 table 2: the table describes the multiplication in gf(2) 83 jean pierre muhirwa clearly gf(2) is finite field since it contains two elements 0 and 1 which is a finite number of elements and also by the rule that every non-zero element is invertible, in the table it is clear that 1 is the only non-zero element and it is invertible. the finite fields are classified by size, as follows: 1. the order or number of elements of finite fields is of the form pn, where p is a prime number called the characteristic of the field, and n is a positive integer. 2. for every prime number p and a positive integer n, there exists a finite field with pn elements. 3. any two finite fields with the same number of elements are isomorphic. for example z/(3) is isomorphic to f3. that is under some renaming of the elements of one of these two fields, its addition and multiplication tables become identical to the corresponding tables of the other one. this classification is justified by using a naming scheme for finite fields that specifies only the order of the field. note: finite fields are important and very useful in number theory, algebraic geometry, galois theory, cryptography, coding theory and quantum error correction. its applications may also be appearing in the electrical circuits. 2 computational over finite fields with first seven rings (zp, where p = 2,3,5,7,11,13,17) arithmetic in a finite field is different from standard integers arithmetic. there are a limited number of elements in the finite field; all operations performed in the finite field result in an element within that field. while each finite field is itself not infinite, there are infinitely many different finite fields; their number of elements (which is also called cardinality) is necessarily of the form pn, where p is a prime number and n is a positive integer, and two finite fields of the same size are isomorphic. consider z/(3) is isomorphic to z3. the prime p is called the characteristic of the finite field, and the positive integer n is called the dimension of the field over its prime field. the finite field with pn elements is denoted gf(pn) and is also called the galois field, in honor of the founder of finite field theory, evariste galois [cox, 2011]. gf(p), where p is a prime number, is simply the ring of integers modulo p. that 84 on some computational and applications of finite fields is, one can perform operations (addition, subtraction, division and multiplication) by using the usual operation on integers, followed by reduction modulo p. for instance, in gf(5), 4+3 = 7 is reduced to 2 modulo 5. division is multiplication by the inverse modulo p, which may be computed using the extended euclidean algorithm. a particular case is gf(2), as addition and multiplication have been shown above in table 1 and table 2 respectively, and the only invertible element is 1. now arithmetic operations in this paper are done on the first seven rings of integers modulo p (zp), where p is a prime number, and those are z2,z3,z5,z7,z11,z13 and z17. 2.1 arthmetic operation in the ring of integers (z3) the class of residues in z3 are 0,1,2 + 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1 table 3: this is a table that shows the addition in z3 * 0 1 2 0 0 0 0 1 0 1 2 2 0 2 1 table 4: this table describes the multiplication in z3 2.2 arthmetic operation in the ring of integers (z5) the class residues in z5 are 0,1,2,3,4 85 jean pierre muhirwa * 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1 table 6: this a multiplication table in z5 + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 table 5: this is a table that illustrates how an addition is done in z5 2.3 arthmetic operation in the ring of integers (z7) the class residues of z7 are 0,1,2,3,4,5,6 + 0 1 2 3 4 5 6 0 0 1 2 3 4 5 6 1 1 2 3 4 5 6 0 2 2 3 4 5 6 0 1 3 3 4 5 6 0 1 2 4 4 5 6 0 1 2 3 5 5 6 0 1 2 3 4 6 6 0 1 2 3 4 5 table 7: an addition table in z7 86 on some computational and applications of finite fields * 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 1 3 5 3 0 3 6 2 5 1 4 4 0 4 1 5 2 6 3 5 0 5 3 1 6 4 2 6 0 6 5 4 3 2 1 table 8: multiplication table in z7 2.4 arthmetic operation in the ring of integers (z11) the class residues of z11 are 0,1,2,3,4,5,6,7,8,9,10 + 0 1 2 3 4 5 6 7 8 9 10 0 0 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 0 2 2 3 4 5 6 7 8 9 10 0 1 3 3 4 5 6 7 8 9 10 0 1 2 4 4 5 6 7 8 9 10 0 1 2 3 5 5 6 7 8 9 10 0 1 2 3 4 6 6 7 8 9 10 0 1 2 3 4 5 7 7 8 9 10 0 1 2 3 4 5 6 8 8 9 10 0 1 2 3 4 5 6 7 9 9 10 0 1 2 3 4 5 6 7 8 10 10 0 1 2 3 4 5 6 7 8 9 table 9: this table demonstrates the addition in z11 87 jean pierre muhirwa * 0 1 2 3 4 5 6 7 8 9 10 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 2 0 2 4 6 8 10 1 3 5 7 9 3 0 3 6 9 1 4 7 10 2 5 8 4 0 4 8 1 5 9 2 6 10 3 7 5 0 5 10 4 9 3 8 2 7 1 6 6 0 6 1 7 2 8 3 9 4 10 5 7 0 7 3 10 6 2 9 5 1 8 4 8 0 8 5 2 10 7 4 1 9 6 3 9 0 9 7 5 3 1 10 8 6 4 2 10 0 10 9 8 7 6 5 4 3 2 1 table 10: multiplication table in z11 2.5 arthmetic operation in the ring of integers (z13) the class residues of z13 are 0,1,2,3,4,5,6,7,8,9,10,11,12 + 0 1 2 3 4 5 6 7 8 9 10 11 12 0 0 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 0 2 2 3 4 5 6 7 8 9 10 11 12 0 1 3 3 4 5 6 7 8 9 10 11 12 0 1 2 4 4 5 6 7 8 9 10 11 12 0 1 2 3 5 5 6 7 8 9 10 11 12 0 1 2 3 4 6 6 7 8 9 10 11 12 0 1 2 3 4 5 7 7 8 9 10 11 12 0 1 2 3 4 5 6 8 8 9 10 11 12 0 1 2 3 4 5 6 7 9 9 10 11 12 0 1 2 3 4 5 6 7 8 10 10 11 12 0 1 2 3 4 5 6 7 8 9 11 11 12 0 1 2 3 4 5 6 7 8 9 10 12 12 0 1 2 3 4 5 6 7 8 9 10 11 table 11: this is an addition table in z13 88 on some computational and applications of finite fields * 0 1 2 3 4 5 6 7 8 9 10 11 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 11 12 2 0 2 4 6 8 10 12 1 3 5 7 9 11 3 0 3 6 9 12 2 5 8 11 1 4 7 10 4 0 4 8 12 3 7 11 2 6 10 1 5 9 5 0 5 10 2 7 12 4 9 1 6 11 3 8 6 0 6 12 5 11 4 10 3 9 2 8 1 7 7 0 7 1 8 2 12 3 10 4 11 5 12 6 8 0 8 3 11 6 1 8 4 12 7 2 10 5 9 0 9 5 1 10 6 2 11 7 3 12 8 4 10 0 10 7 4 1 11 8 5 2 12 9 6 3 11 0 11 9 7 5 3 1 12 10 8 6 4 2 12 0 12 11 10 9 8 7 6 5 4 3 2 1 table 12: multiplication table in z13 from this table, each non-zero element has its multiplicative inverse, the multiplicative inverse of 8 for example is 5, the multiplicative inverse of 11 is 6, and so on. 2.6 arithmetic operation in the ring of integers (z17) the class residues of z17 are 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 89 jean pierre muhirwa + 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 3 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 4 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 5 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 7 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 8 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 9 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 10 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 11 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 12 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 13 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 14 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 15 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 table 13: this table points out how to perform an addition in z17 90 on some computational and applications of finite fields * 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2 0 2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15 3 0 3 6 9 12 15 1 4 7 10 13 16 2 5 8 11 14 4 0 4 8 12 16 3 7 11 15 2 6 10 14 1 5 9 13 5 0 5 10 15 3 8 13 1 6 11 16 4 9 14 2 7 12 6 0 6 12 1 7 13 2 8 14 3 9 15 4 10 16 5 11 7 0 7 14 4 11 1 8 15 5 12 2 9 16 6 13 3 10 8 0 8 16 7 15 6 14 5 13 4 12 3 11 2 10 1 9 9 0 9 1 10 2 11 3 12 4 13 5 14 6 15 7 16 8 10 0 10 3 13 6 16 9 2 12 5 15 8 1 11 4 14 7 11 0 11 5 16 10 4 15 9 3 14 8 2 13 7 1 12 6 12 0 12 7 2 14 9 4 16 11 6 1 13 8 3 15 10 5 13 0 13 9 5 1 14 10 6 2 15 11 7 3 16 12 8 4 14 0 14 11 8 5 2 16 13 10 7 4 1 15 12 9 6 3 15 0 15 13 11 9 7 5 3 1 16 14 12 10 8 6 4 2 16 0 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 table 14: this a multiplication table in z17 apart from the 14 tables represented above, one may proceed in the same way up to the finite fields of p − 1 class residues with p being a prime number or a power of a prime number. 3 arithmetic computational of polynomials over finite fields the theory of polynomials over finite fields is important for investigating the algebraic structure of finite fields as well as for many applications. above all, irreducible polynomials, the prime elements of polynomial rings over finite fields are indispensable for constructing finite fields and computing with the elements of finite fields [rónyai, 1992]. a polynomial is an expression of the form anxn + an−1xn−1 + ... + a1x + a0, for some non-negative integer n and where the coefficients a0,a1...,an are drawn from some designated set s, which is in particular finite field and called the coefficient set. polynomial arithmetic deals with the addition, subtraction, multiplication, and 91 jean pierre muhirwa division of polynomials. 3.1 what problems does polynomial arithmetic adress? given two polynomials whose coefficients are derived from a set s, what can we say about the coefficients of the polynomial that results from an arithmetic operation on the two polynomials? if we insist that the polynomial coefficient all come from a particular s, then which arithmetic operations are permitted and which prohibited? let us say that the coefficient set is a finite field f with its own rules for addition, subtraction, multiplication and division, and let us further say that when we carry out an arithmetic operation on two polynomials, we subject the operations on the coefficients to those that apply to the finite field f . now what can be said about the set of such polynomials? all these questions will have their answers as we move on in this paper. 3.2 ordinary addition and subtraction of polynomials let f(x) = a2x2 + a1x + a0 and g(x) = b1x + b0 then f(x) + g(x) = a2x2 + (b1 + a1)x + (a0 + b0) let f(x) = a2x2 + a1x + a0 and g(x) = b3x3 + b0, then f(x)−g(x) = −b3x3 + a2x 2 + a1x + (a0 − b0) 3.3 ordinary multiplication of polynomials let f(x) = a2x2 + a1x + a0 and g(x) = b1x + b0, then f(x) ∗g(x) = a2b1x3 + (a2b0 + a1b1)x 2 + (a1b0 + a0b1)x + a0b0. 3.4 ordinary division of polynomials 3.4.1 when is division of polynomials permited? polynomial division is obviously not allowed for polynomials that are not defined over certain fields. for example, for polynomials defined over the set of all integers, you cannot divide 4x2 + 5 by the polynomial 5x. if you tried, the first term of the quotient would be (4 5 )x where the coefficient of x is not an integer. you can always divide polynomials defined over a certain field. what that means is that the operation of division is legal when the coefficients are drawn from a finite field. note that, in general, when you divide such polynomial by another, you will end up with a remainder, and when, in general you divide one integer by another 92 on some computational and applications of finite fields integer it is possible in purely integer arithmetic. therefore, in general, for polynomials defined over a field, the division of a polynomial f(x) of a degree m by another polynomial g(x) of a degree n − m can be expressed by f(x)/g(x) = q(x)g(x) + r(x), where q(x) is the quotient and, r(x) the remainder, so we can write for any two polynomials defined over a field, f(x) = q(x) ∗ g(x) + r(x) assuming that the degree of f(x) is not less than that of g(x). when r(x) is zero, we say that g(x) divides f(x). this fact can also be expressed by saying that g(x) is a divisor of f(x) and by notation, g(x)|f(x). 3.4.2 division of a polynomial by another upon using long division let us divide the polynomial 8x2 + 3x + 2 by the polynomial 2x + 1: in this example, our dividend is 8x2 + 3x + 2 and the divisor is 2x + 1. we now need to find the quotient. long division for polynomials consists of the following steps: step 1: arrange both the dividend and the divisor in the descending powers of the variable. step 2: divide the first term of the dividend by the first term of the divisor and write the result as the first term of the quotient. in our example, the first term of the dividend is 8x2 and the first term of the divisor is 2x , so the first term of the quotient is 4x. step 3: multiply the divisor with the quotient term just obtained and arranges the result under the dividend so that the same powers of x match up. subtract the expression just laid out from the dividend. in our example, 4x times 2x + 1 is equal to 8x2 + 4x. subtracting this from the dividend yields −x + 2. consider the result of the above subtraction as the new dividend and go back to the first step. (the new dividend in our case is (−x + 2). in our example, dividing the polynomial 8x2 +3x+2 by the polynomial 2x+1, yield quotient of 4x−0.5 and a remainder of 2.5. 3.5 arithmetic operations on polynomials whose coefficients belong to a defined finite fields the arithmetic operations on polynomials whose coefficients are drawn from finite fields is not the same as the usual operations of polynomials. to see this, let us consider the set of all polynomials whose coefficients belong to the finite field z7 (which is the same as gf(7)). here is an example of adding two such polynomials: f(x) = 5x2 +4x+6, g(x) = 5x+6 we get f(x)+g(x) = 5x2 +9x+12 = 93 jean pierre muhirwa 5x2 + 2x + 5 if we perform the difference of both polynomials, f(x) = 5x2 + 4x + 6 and g(x) = 5x + 6 then f(x)−g(x) = 5x2 −x = 5x2 + 6x since the additive inverse of 5 in z7 is 2 and that of 6 is 1. so 4x− 5x is the same as 4x + 2x and 6 − 6 is the same as 6 + 1, with both additions modulo 7. the multiplication of polynomials f(x) = 5x2 + 4x + 6, and g(x) = 5x + 6 is given by f(x)∗g(x) = 4x3 + x2 + 5x + 1 lastly the divison of polynomials f(x) = 5x2 +4x+6, g(x) = 2x+1 is given by f(x)/g(x) = 6x + 6. if you multiply the divisor 2x + 1 with the quotient 6x + 6 , you get the dividend 5x2 + 4x + 6. let consider also the polynomials defined over gf(2). recall that the notation gf(2) means the same thing as z2. we are obviously talking about arithmetic modulo 2. first of all, gf(2) is a sweet basic finite field. recall that the number 2 is the first prime. (a prime has exactly two distinct divisors, 1 and itself). gf(2) consists of the set 0,1. the two elements of this set obey the following addition and multiplication rules: 0 + 0 = 0 0 x 0 = 0 0 + 1 = 1 0 x 1 = 0 1 + 0 = 1 1 x 0 = 0 1 + 1 = 0 1 x 1 = 1 so the addition over gf(2) is equivalent to the logical xor operation, and multiplication to the logical and operation. some examples of polynomials defined over gf(2): are x3 + x2 −1;−x5 + x4 −x2 + 1;x + 1, etc. 3.5.1 arithmetic computational of polynomials defined over gf(2) here is an example of adding two such polynomials: f(x) = x2 + x + 1,g(x) = x + 1, therefore f(x) + g(x) = x2 + 2x + 2 = x2 • here is an example of subtracting two such polynomials, f(x) = x2 + x + 1,g(x) = x + 1, then f(x)−g(x) = x2 • here is an example of multiplying two such polynomials, f(x) = x2+x+1, and g(x) = x + 1, then f(x)×g(x) = x3 + 1 94 on some computational and applications of finite fields • here is an example of dividing two such polynomials,f(x) = x2 + x + 1,g(x) = x + 1, then f(x)/g(x) = x. if you multiply the divisor, x + 1 with the quotient x, you get x2 + x. that when added to the remainder 1 gives us back the dividend x2 + x + 1 3.6 division of polynomials defined over finite fileds first, note that a polynomial is defined over a field if all its coefficients are drawn from that field. dividing polynomials defined over a finite field is a little bit more frustrating than performing other arithmetic operations on such polynomials. now your mental gymnastics must include both additive inverses and multiplicative inverses.consider again the polynomials defined over gf(7). let’s say we want to divide 5x2 + 4x + 6 by 2x + 1. in a long division, we must start by dividing 5x2 by 2x. this requires that we divide 5 by 2 in gf(7). dividing 5 by 2 is the same as multiplying 5 by the multiplicative inverse of 2. multiplicative inverse of 2 is 4 since 2 ≡ 4 mod 7 is 1. so we have 5 ≡ 2−1 = 5 ≡ 4 = 20 mod 7 = 6. therefore, the first term of the quotient is 6x. since the product of 6x and 2x + 1 is 5x2 + 6x, we need to subtract 5x2 + 6x from the dividend 5x2 + 4x + 6. the result is (4 − 6)x + 6, which (since the additive inverse of 6 is 1) is the same as (4 + 1)x + 6, and that is the same as 5x + 6. our new dividend for the next round of long division is therefore 5x + 6. to find the next quotient term, we need to divide 5x by the first term of the divisor, that is by 2x. reasoning as before, we see that the next quotient term is again 6. the final result is that when the coefficients are drawn from the set gf(7)), 5x2 + 4x + 6 divided by 2x + 1 yields a quotient of 6x + 6 with the remainder zero. so we can say that as a polynomial defined over the field, gf(7), 5x2 + 4x + 6 is a product of two factors, 2x+1 and 6x+6. we can therefore write 5x2 +4x+6 = (2x + 1) ≡ (6x + 6) 4 irreducible polynomials or prime polynomials definition 4.1. according to [rónyai, 1992], a polynomial f ∈ f [x] is said to be irreducible over f (or irreducible in f[x], or prime in f [x]) if f has positive degree and f = g ∗ h, with g,h ∈ f [x] implies that either g or f is a constant polynomial, otherwise it is reducible over f . the reducibility or irreducibility of a given polynomial depends heavily on the field under considerations. for instance, the polynomial x2 −2 ∈ q(x) is irreducible over the field q of rational numbers, but x2 − 2 = (x + √ 2)(x − √ 2) but reducible over the field of real numbers 95 jean pierre muhirwa (r). for polynomials over finite fields, the same argument hold except that the coefficients are reduced in mod p. example 4.1. f(x) = x2 + x + 1 is irreducible over f2 but g(x) = x2 + 1 is reducible over f2 to see this g(x) = x2 + 1 = (x + 1)(x + 1) = x2 + 2x + 1, since 2 ≡ 0 mod(2), and then 2x ≡ 0 mod(2). in few words we can say, when g(x) divides f(x) without leaving a remainder, we say g(x) is a factor of f(x). a polynomial f(x) over a field f is called irreducible, if f(x) cannot be expressed as a product of two polynomials, both over f and both of degree lower than that of f(x). an irreducible polynomial is also referred to as a prime polynomial. 5 some computational tables of quotient polynomials over finite fields to represent the elements of an extension fields over finite fields in a computational table, we must have the quotient? fq[x]/f(x), where f(x) is irreducible over fq[x]. this form of polynomials are looked like powers of prime [lidl and niederreiter, 1994]. example 5.1. let f(x) = x2 + 1 ∈ f3[x]. thus to find the computational tables of f3[x]/(f(x)), we need to find the residue class ring as pn where n is the degree of polynomial f(x), and then we have a set of residue class ring of 32 = 9 elements, as it looks like a representation of f(9), such as 0,1,2,x,1 +x,2 +x,1 + 2x,2x,2 + 2x, these are precisely the polynomials of degree less than 2 over f3 by equating x2 + 1 = 0 and this implies that x2 = −1 = 2, but remember that computational in finite fields are followed by mod p [gong et al., 2013] . + 0 1 2 x 1+x 2+x 1+2x 2x 2+2x 0 0 1 2 x 1+x 2+x 1+2x 2x 2+2x 1 1 2 0 1+x 2+x x 2+2x 1+2x 2x 2 2 0 1 2+x x 1+x 2x 2+2x 1+2x x x 1+x 2+x 2x 1+2x 2+2x 1 0 2 1+x 1+x 2+x x 1+2x 2+2x 2x 2 1 0 2+x 2+x x 1+x 2+2x 2x 1+2x 0 2 1 1+2x 1+2x 2+2x 2x 1 2 0 2+x 1+x x 2x 2x 2x+1 2+2x 0 1 2 1+x x 2+x 2+2x 2+2x 2x 1+2x 2 0 1 x 2+x 1+x table 15: addition table for f3[x]/(f(x)) 96 on some computational and applications of finite fields * 0 1 2 x 1+x 2+x 1+2x 2x 2+2x 0 0 0 0 0 0 0 0 0 0 1 0 1 2 x 1+x 2+x 1+2x 2x 2+2x 2 0 2 1 2x 2+2x 1+2x 2+x x 1+x x 0 x 2x 2 x+2 2x+2 x+1 1 2x+1 1+x 0 1+x 2+2x x+2 2x 1 2 2x+1 x 2+x 0 2+x 1+2x 2x+2 1 x 2x x+1 2 1+2x 0 1+2x 2+x x+1 2 2x x 2x+2 1 2x 0 2x x -2 2x+1 x+1 2x+2 x x+2 2+2x 0 2+2x 1+x 2x+1 x 2 1 x+2 2x table 16: multiplication table for f3[x]/(f(x)) 6 applications of finite fields 6.1 algebraic coding theory it is one of the major applications of finite field. this theory has its origin in famous theorem of shannon that guarantees the existence of codes that can transmit information at rates close to the capacity of a communication channel with an arbitrary small probability of error. one of the purposes of algebraic coding theory, the theory of error-correcting and error-detecting codes is to devise methods for construction of such codes [von zur gathen et al.]. during the last two decades more and more abstract algebraic tools such as the theory of finite fields and the theory of polynomials over finite fields have influenced coding. in particular, the description of redundant codes by polynomials over fq is a milestone in this development. the fact that one can use shift registers for coding and decoding establishes a connection with linear recurring sequences. in our discussion of algebraic coding theory we do not consider any of the problems of the implementation or technical realization of the codes. we restrict ourselves to the study of basic properties of block codes and the description of some interesting classes of block codes. 6.1.1 linear coding the problem of communicating the information, in particular the coding and decoding of information for the reliable transmission over a ”noisy” channel is of great importance today. typically, one has to transmit a message which consists of finite string of symbols that are elements of some finite alphabet. for instance, if this alphabet consists of simply 0 and 1, the message can be described as binary 97 jean pierre muhirwa number. generally the alphabet is assumed to be finite fields. now the transmission of finite string of elements of the alphabet over a communication channel need not to be perfect in the sense that each bit of information is transmitted unaltered over this channel. as there is no ideal channel without ”noise” the receiver of the transmitted message may obtain distorted information and may make errors in interpreting the transmitted signal. one of the main problems of coding theory is to make the errors, which occur for instance because of noisy channel, extremely improbable. the methods of improve the reliability of transmission depend on properties of finite fields. a basic idea in algebraic coding theory is to transmit redundant information together with the message one wants to communicate; that is, one extends the string of message symbols to a longer string in a systematic way. a simple model of communication system is shown in the figure bellow: we assume that the symbols of the message and the coded message are elements of the same finite field fq. coding means to encode a block of k message symbols a1,a2, ...,ak where ai ∈ fq into a code word c1,c2, ...,cn of n symbols, where cj ∈ fq, with n > k. we regard the code word as an n-dimensional row vector c ∈ fnq . thus f in the figure below is a function from fkq into fnq , called a coding scheme, and g : fnq → fkq is a decoding scheme. figure 1: communication figure that shows how a message is coded, transmitted and decoded a simple type of coding scheme arises when each block a1a2...ak of message symbols is encoded into a code word of the form a1a2...akck+1...cn, where the first k 98 on some computational and applications of finite fields symbols are the original message symbols and the additional n−k symbols in fq are control symbols. such coding schemes are often presented in the following way. let h be a given (n−k)×n matrix with entries in fq that is of the special form h = (a,in−k), where a is (n−k)×k matrix and in−k is the identity matrix of order n − k. the control symbols ck+1, ...,cn can then be calculated from the system of the equations hct = 0, for code word c. the equations of this system are called parity-check equations. the examples of this theory will be given later. 6.2 error-correcting codes (practice of linear code) since the theory of codes was developed in order to ensure reliability of transmitted information, as an example, consider the isbn (international standard book number) of published book. this number usually appears on the back of the book in the bottom right-hand corner. the isbn consists of a nine-digits 0,1, ...,9 or the symbol x (standing for 10). this final symbol may be calculated from the other nine as follows: from an integer n by adding together the first digit, twice the second digit, three times the third and so on. the check digit is the remainder when n is divided by 11. for example, a book with first 9 digits 019853453 will have n = 0 + 2 + 27 + 32 + 25 + 18 + 28 + 40 + 27 = 199, and so the check digit should be 1, giving isbn 01953453 1. the point about such a number is that if it is inaccurately copied, and an error is made in any of the digits in the first nine locations (such as the last ”5” being copied as a ”3”), then the resulting number will not have ”1” as its check digit. this is an example of errordetecting code: the isbn detects when a single error is made after transcribing the number. another example of finding check digit is that of 102463798, then n = 1×1 + 0×2 + 2×3 + 4×4 + 6×5 + 3×6 + 7×7 + 9×8 + 8×9 = 264, and divide this number by 11 to get the check digit which is 0, and hence giving isbn 102463798 0 in this part we shall explain methods which not only detect errors, but also enables us to correct it. definition 6.1. let p be prime integer. denote by v (n,p) the set of all sequences of length n of the elements from the set zp of congruence classes modulo p, so that v (n,p) has pn elements. we will usually omit the commas and brackets commonly used to denote elements of the vector spaces, so that (1,0,1), will be written as 101. thus v (3,2) consists of the eight sequences 000,001,010,011,100,101,110,111 while v (2,3) consists of the nine sequences 00,01,02,10,11,12,20,21,22. we add sequences by adding the corresponding terms, by just remembering that we are adding congruence classes. thus, for example in v (3,2), 110 + 011 = 101 99 jean pierre muhirwa while in v (2,3), 12 + 11 = 20. we can also multiply an element in v (n,p) by a congruence class by multiplying each term in the sequence by the representative for the congruence class and reducing modulo p. for example, in the space v (3,3) we see that 2(102) =201. in fact v (n,p) is a vector space of dimension n over the field zp definition 6.2. a linear (n,k)-code is any k-dimensional subspace c of the vector space v (n,p). thus c satisfies the following two conditions: the difference of any two elements of c is an element of c, and the product of any element of c with an element of zp is also an element of c. the elements of c are called codewords. note: a subspace of a vector space is necessary non-empty, so condition (1) ensures that the zero element of the vector space is in the subspace c. it then follows by the additive version that c is a group under addition. example 6.1. consider the four elements 000,001,010,011 of v (3,2). these are precisely the four sequences which start with 0. this subspace of v (3,2) satisfies condition one, that subtracting any two of these gives a sequence starting with 0. also condition (2) holds, since 0 and 1 are the only elements of z2 and then multiply each sequence by any of these two elements we get an element starting with 0. therefore the four elements form a linear (3,2)-code. definition 6.3. let v be any element of v (n,p). the weight of v is the number of non-zero terms in the sequence v. if v and w are two elements of v (n,p), the distance d(v,w) is the number of places at which v and w differ. example 6.2. in v (4,3) the weight of 1201 is three, since there are three nonzero entries. the distance from 1201 to 2211 is two, since these two vectors differ in two places. in v (5,5) the weight of 13402 is four and so on. proposition 6.1. let u,v and w be any elements of v (n,p). then 1. d(u,v) ≥ 0 with equality if and only if u = v; 2. d(u,v) = d(v,u); and 3. d(u,v) + d(v,w) ≥ d(u,w). 100 on some computational and applications of finite fields proof. 1. it follows directly from the definition that d(u,v) is positive except u and v do not differ anywhere. 2. this is always true for u and v. 3. in each location at which u and w differ, v cannot agree with both u and w. thus every contribution to the value of d(u,w) provides a contribution to either d(u,v) or to d(v,w). 2 definition 6.4. let c be subspace of v (n,p). the minimum distance d of c is the least distance between different codewords: d = minu,v{d(u,v)}. the next result shows that for a linear code, the minimum distance d can be calculated from the code words. proposition 6.2. let c be a linear (n−k)-code. then the minimum distance of c is equal to the smallest possible weight of any non-zero codeword. proof. let f be the smallest possible weight of any non-zero codeword, and let 0 denote the sequence consisting entirely of zeros. suppose that w is a codeword of weight f. then d(w,0) if and only if so f ≥ d. now let u and v be pair of codewords with d(u,v) = d. since c is a linear code, the word u−v is a codeword of weight d, so d ≥ f. it follows that d = f. the importance of the minimum distance lies in the detecting the errors and correction of those errors. to see this, consider the following proposition. 2 proposition 6.3. let c be linear code with minimum distance d. then c detects d−1 or fewer errors, and corrects e errors for any e with 2e + 1 ≤ d. proof. let v be a vector which has distance f from a codeword c, where f ≤ d− 1. we think of c as the transmitted word and v as the received word, so that there are f errors in transmission. since d is the minimum distance for c, the received v cannot be a codeword. we express this by saying that the code c detects d or fewer errors. suppose now that v has distance e from a codeword c and also that 2e + 1 ≤ d. then there can be no other codeword near to v: if c1 was in c and d(v,c1) ≤ e, 101 jean pierre muhirwa then by property of triangle inequality d(c,c1) ≤ d(c,v) + d(v,c1) ≤ e + e < d, which contradicts the definition. thus there is a unique nearest codeword to v, and we say that c corrects e errors in this case. 2 definition 6.5. let n and k be any positive integers with n > k. let p be a prime number. a (standard) generator matrix g over zp is a k ×n matrix with entries in zp, in which the first k columns form an identity k × k matrix. given such a matrix, we obtain a linear code by regarding the rows as sequences and taking all possible linear combinations of these. alternatively, we can consider the code as consisting of all sequences obtained from matrix multiplications of the form u.g as u varies over all sequences of length k over zp. example 6.3. consider the generator matrix over z2 g = ( 1 0 1 0 1 1 ) the corresponding code consists of the combinations of the rows and so has four elements: 000; 101; 011 and 110. the codewords can also be described as the vectors of the form ug, as u varies over the four vectors 00; 01; 10; 11. every non-zero codeword has weight 2, so the codeword detects one error, but does not correct errors. for example, 111 is not among codewords (so it is detected) but it is of equal distance from the two codewords 101 and 011 in g, so it cannot be corrected. example 6.4. another example of a binary code (code over z2) is provided by the matrix g =  1 0 0 1 1 00 1 0 1 0 1 0 0 1 0 1 1   there are 8 code words obtained from the rows of this matrix: 000000; 100110; 010101; 110011; 001011; 101101; 011110; 111000. there are four code words of weight 3, three code words of weight 4 and one of weight 0. the minimum distance (d) of this code is therefore 3, so the code detects d − 1 errors means two errors and corrects one error. for example, 100111 lies at distance one from a unique codeword, 100110 and so there is unique way to correct one error. the vector 100001, however has distance two from 000000 and 110011, so cannot be corrected. 102 on some computational and applications of finite fields example 6.5. consider the following generator matrix over z3 : g = ( 1 0 2 1 0 1 1 2 ) in this case, the codeword consists of the linear combinations of the rows of the matrix, including multiplication by 1 and 2 since p = 3. there are 9 code words: 0000; 1021; 2012; 0112; 1100; 2121; 0221; 1212 and 2200. since there is a codeword of weight 2, this code detects one error. note that the minimum distance is 2 despite the fact that each row of the generator matrix has weight 3. example 6.6. consider also the following important code over z3  1 0 0 0 0 0 0 1 2 2 1 0 1 0 0 0 0 1 0 1 2 2 0 0 1 0 0 0 2 1 0 1 2 0 0 0 1 0 0 2 2 1 0 1 0 0 0 0 1 0 1 2 2 1 0 0 0 0 0 0 1 1 1 1 1 1   by considering this matrix, the minimum distance of this code is at most 5 since there is a row of the generator matrix of weight 5. it can be shown that the minimum distance is exactly 5, so that code corrects two errors. this is the ternary golay code and is one of the most important code. more details and its descriptions are found in [cohen et al., 2013]. we now consider the problem of decoding a linear (n,k)-code c. this is done by listing the left cosets of the subgroup c of v (n,p) in a table known as the cosets decoding table. the table is organized by writing the codewords as its first row with the zero codeword first. each subsequent row is a left coset of c. the entries in the first column are the coset representatives, now called cosets leaders. the algorithm for choosing the rth-coset leader is to choose any word of minimum weight not already included in the first (r − 1) rows. then to decode a given vector, locate it in the table, and correct it to the codeword standing in the same column of the coset decoding table. example 6.7. consider example 6.6, above there are eight code words which form a subgroup c of the vector space v (6,2). since v has 26 = 64 elements, 103 jean pierre muhirwa this subgroup has index 64/8=8. to form a complete coset decoding table, we list the elements of c in a row. we then choose any element v2 which is of smallest weight among those not in the first row and write this at the left hand end of the second row. the second row is obtained by adding each element of c in turn to this. thus the second row is just the coset of c with respect to v2. continue this process by choosing v3 to be of the smallest weight among the elements not in the first two rows, and so on. this process is not unique, but depends upon the choice of coset representatives [pless, 1998]. one example of these choices is given in the following table 000000, 100110, 010101, 110011, 001011, 101101, 011110, 111000 100000, 000110, 110101, 010011, 101011, 001101, 111110, 011000 010000, 110110, 000101, 100011, 011011, 111101, 001110, 101000 001000, 101110, 011101, 111011, 000011, 100101, 010110, 110000 000100, 100010, 010001, 110111, 001111, 101001, 011010, 111100 000010, 100100, 010111, 110001, 001001, 101111, 011100, 111010 000001, 100111, 010100, 110010, 001010, 101100, 011111, 111001 100001, 000111, 110100, 010010, 101010, 001100, 111111, 011001 to decode any element v of v (6,2), we locate v in the table and then correct it to the element in the first row of the column containing v. thus to use the table to decode 011010, we need to locate it (it is in the fifth row and seventh column) and correct it to the element in the first row and the same column, giving 011110. note that the cosets representative for the last row is not easy to find. according to the algorithm, we need a word of weight 2 not in the first seven rows. the representative we choose, 100001, is not unique. this is actually a somewhat cumbersome way to arrange the decoding, since an exhaustive search is required. the calculation can be made more systematic for codes given by (standard) generator matrices using (standard) parity check matrices [sayed, 2011]. definition 6.6. let c be an (n,k)-linear code over zp defined using k ×n generator matrix g of the form g =   1 0 0 ... 0 0 1 0 ... 0 a ... 0 0 0 ... 1   where, a is k × (n−k) matrix. the parity check matrix associated with g is the (n−k)×n matrix 104 on some computational and applications of finite fields p =   1 0 0 ... 0 −at 0 1 0 ... 0 ... 0 0 0 ... 1   note: the generator matrix g above is often written, in a block matrix form as g = (ik|a). similarly, the parity check matrix is written as p = (−at |i(n−k)) [kar, 2012]. example 6.8. the parity check matrix of the generator matrix over z2. the parity check of the matrix of g =  1 0 0 1 1 00 1 0 1 0 1 0 0 1 0 1 1   , is the matrix p =  1 1 0 1 0 01 0 1 0 1 0 0 1 1 0 0 1   [kar, 2012]. this matrix is obtained by considering the matrix a =  1 1 01 0 1 0 1 1   , and then after computing −at get the above matrix p given by −at |i(n−k) [kar, 2012]. definition 6.7. let c be a linear (n,k)-code with generator matrix g and associated parity matrix p . for any v in v (n,p), let vt denote the transpose of v, the column vector obtained by writing the members of the sequence v vertically. then the syndrome of v is the element of v (n−k,p) given by pvt . thus in the above example, the syndrome of v = 100000 is 110 and the syndrome of v = 110011 is 000. note: if c is a code with standard parity check matrix p , then an element v in v (n,p) is a codeword if and only if the syndrome of v is the zero sequence. 105 jean pierre muhirwa example 6.9. we need not store or the complete coset decoding table, but merely a table of two columns, the coset representatives and their syndromes. in our previous example in which p was 1 1 0 1 0 01 0 1 0 1 0 0 1 1 0 0 1   this table would be as the following, coset representatives syndromes 000000 000 100000 110 010000 101 001000 011 000100 100 000010 010 000001 001 100001 111 table 17: this a table of syndromes and cosets representatives thus to decode a given vector such as 100111, calculate its syndrome to obtain 001. this is the syndrome for the seventh row, so this vector is not a codeword, but the word 100110 obtained by subtracting 000001 is a codeword. the advantage of listing coset representatives together with syndromes is that, it is much easier to find any missing coset representatives, since each sequence in v (n−k,p) occurs as syndrome. thus in this above example, the syndrome for the last row must be 111 because the other seven sequences of length 3 have already been used as syndromes. this enables us to find a representative relatively easily (compared with searching through the first seven rows), by seeing how to combine known coset leaders and their syndromes to obtain 111. 6.3 cyclic codes definition 6.8. in the paper of [peterson and brown, 1961], a linear code c is called a cyclic code if it has the following property: if (c0,c1,c2, ...,cn−1) ∈ c, then it is also reality that (c1,c2, ...,cn−1,c0) ∈ c. from this definition the automorphism group aut(c) of a code c is the set of permutations δ ∈ sn such that δ(c) ∈ c for all c ∈ c, where δ(c0,c1,c2, ...,cn−1) = 106 on some computational and applications of finite fields (cδ(0), ...,cδ(n−1)). in other words, the code, c is cyclic if and only if the permutation δ = (0,1,2, ...,n−1) is in aut (c)[roberts and vivaldi, 2005]. example 6.10. let c be a subspace of a vector space v (6,7) and consider the code words v = (345601) of c, then c is cyclic code if (4,5,6,0,1,3); (5,6,0,1,3,4); (6,0,1,3,4,5); (0,1,3,4,5,6); (1,3,4,5,6,0), all are elements of c. we can define an algebraic structure by looking at cyclic code if we let c to be a cyclic code over the field fq and we set rn + fq[x]/(xn − 1). we can take the elements of rn as polynomials of degree at most n− 1 over fq, where multiplication can be happen except that xn = 1,xn+1 = x , and so on. from this, we can deduce one to one correspondence between polynomials and the code words of cyclic code as can be seen in [sziklai, 2013]. example 6.11. let c be a subspace of a vector space v (5,7) over f7 = z/7z and let consider the code word (1,2,3,5,6). then we can find the polynomial of degree less than 5 correspond to this code word which is given by 1 + 2x + 3x2 + 5x3 + 6x4. to find the elements of rn + fq[x]/(xn − 1), we do it as found for the previous case of quotient finite fields, and these are precisely the polynomials of degree at most n − 1, hence the total number of the elements of rn + fq[x]/(x n −1), are qn elements. example 6.12. let find the elements of r3 + f2[x]/(x3 − 1), here our q = 2 and n = 3, therefore the total number of the elements of this polynomial field are qn = 23 = 8 polynomials of degree less than 3 whose coefficients are in f2. so the elements r3 + f2[x]/(x3−1) are 0,1,x,1 +x,x2,x2 + 1,1 +x+x2,x+x2. theorem 6.1. from this kind of cyclic codes we define also an ideal of rn given by ic + (c(x) + c0 + c1x + ... + cn−1xn−1) ∈ rnc + (c0,c1, ...,cn−1) ∈ c proof. let c,d ∈ ic, a ∈ rn, then we want to show that c − d ∈ ic and ac ∈ ic, therefore c(x) = c0 + c1x + ... + cn−1x(n−1),d(x) = d0 + d1x + ... + d(n−1)x(n−1) and a(x) = a0 +a1x+ ...+an−1x(n−1). so, c(x)−d(x) = c0−d0 +(c1−d1)x+ ... + (cn−1 −dn−1)x(n−1) ∈ ic ⇒ (c0 −d0,c1 −d1, ...,cn−1 −dn−1) ∈ c. c ∈ ic ⇔ (c0,c1, ...,cn−1) ∈ c d ∈ ic ⇔ (d0,d1, ...,dn−1) ∈ c. (c0−d0,c1−d1, ...,cn−1−dn−1) ∈ c, is a code word of cyclic code c (since c is a vector space of v (n,q) over fq. it remains to show that a(x)c(x) is an element of ic. then a(x)c(x) = (a0+a1x+...+an−1x(n−1))(c0+c1x+...+cn−1x(n−1)) = a0c0+a0c1+a0c2+...a1c0+a1c1+...+a2c0+..., is also a code word of length n−1. 107 jean pierre muhirwa let illustrate by using example, let a(x) ∈ r3 + f2[x]/(x3 − 1), and c(x) ∈ ic, we have a(x) = a0 + a1x + a2x2 and c(x) = c0 + c1x + c2x2, where ai ∈ fq for i = 0,1,2 and ci ∈ c for i = 1,2,3. then a(x)c(x) = (a0+a1x+a2x2)(c0+c1x+ c2x 2) = a0c0+(a0c1+a1c0)x+(a0c2+a1c1+a2c0)x 2+(a1c2+a2c1)x 3+(a2c2)x 4. but x3 = 1 and x4 = x, then we have a0c0 +a1c2 +a2c1 +(a0c1 +a1c0 +a2c2)x+ (a0c2 + a1c1 + a2c0)x 2 ∈ ic. ⇒ (a0c0 + a1c2 + a2c1;a0c1 + a1c0 + a2c2;a0c2 + a1c1 + a2c0) ∈ c. ⇒ (a0c0,a0c1,a0c2) + (a1c2,a1c0,a1c1) + (a2c1,a2c2,a2c0). ⇒ a0(c0,c1,c2) + a1(c2,c0,c1) + a2(c1,c2,c0). but (c0,c1,c2),(c2,c0,c1),(c1,c2,c0) ∈ c since c is cyclic code. therefore ic is an ideal of rn. 2 theorem 6.2. let ic be an ideal of r(n) and let g(x) ∈ c be monic polynomial of minimal degree l = deg(g(x)). then a. g(x) is the only monic polynomial of degree l in ic. b. g(x) generates ic as an ideal of rn. proof. let f be any other nonzero monic polynomial of minimal of i with degree less than l then f − g ∈ i, but f 6= g ⇒ f − g 6= 0, f(x) − g(x) = ckx k + ...+c1x+c0 and this polynomial is not monic, it becomes monic if we divide it by c−1k with ck 6= 0 , and then we get 1/ck(f(x)−g(x)) = x k+...+d1x+d0, where d = ci/ck for i = 0,1, ...,k. hence k < l which contradicts that l is the minimal degree. therefore, g(x) is unique monic polynomial of the minimal degree. g(x) generates i means that i =< g >= gh,h ∈ rn, this also means if f ∈ i, then f = gh for some h ∈ rn . let f ∈ i ⊂ rn = fq[x]/(xn − 1), write f(x) = g(x)q(x) + r(x) ∈ i with deg(r(x)) < deg(g(x)) = l. ⇔ f(x)−g(x)q(x) = r(x) ∈ i (since q(x),g(x) ∈ i ). ⇒ r(x) = 0 ⇒ f(x) = g(x)q(x) ⇒ f ∈< g > and i ∈< g > but g ∈ i, so < g >∈ i. hence i =< g >. 2 7 conclusion this paper has discussed about finite fields whereby some important definitions, propositions, theorems and their proofs have been given in order to capture 108 on some computational and applications of finite fields what finite fields are and how finite fields deal with operations in different ways from usual known operations that may be performed for a set of integers. the operations procedure required any arithmetic followed by reduction of p, and this is the reason why several tables from finite fields z2 to z17 are computed to highlight how one may compute in finite fields. it includes polynomials arithmetic operations over finite fields such as addition, subtraction, multiplication, and division. the arithmetic polynomials over finite fields are computed by using the reduction of p to its coefficients, because their coefficients are drawn from finite fields that are taken into consideration. besides polynomials computational over finite fields, this paper also explains what are cyclic codes and their applications. this research paper has further shown the applications of finite fields in the most important domain of communication regarding algebraic coding theory, code error-detection and error-correction, whereby coding and decoding schemes using cosets representative and syndromes table are discussed by using tangible examples. from this paper one may learn about finite fields and its applications and be able to extend up to p− 1 class residues with p being any prime number or any power of a prime number. 8 acknowledgement i would like to acknowledge the financial support from the eastern africa universities mathematics programme-international science programme, university of rwanda node (eaump-isp, ur-node). references parity check matrix recognition from noisy codewords. arxiv preprint arxiv:1205.4641, 2012. arjeh m cohen, hans cuypers, and hans sterk. some tapas of computer algebra, volume 4. springer science & business media, 2013. david a cox. galois theory., volume 61. john wiley & sons, 2011. guang gong, katalin gyarmati, fernando hernando, sophie huczynska, dieter jungnickel, gohar m kyureghyan, gary mcguire, harald niederreiter, alina ostafe, and igor e shparlinski. finite fields and their applications: character sums and polynomials, volume 11. walter de gruyter, 2013. 109 jean pierre muhirwa rudolf lidl and harald niederreiter. introduction to finite fields and their applications. cambridge university press, 2 edition, 1994. doi: 10.1017/cbo9781139172769. william wesley peterson and daniel t brown. cyclic codes for error detection. proceedings of the ire, 49(1):228–235, 1961. vera pless. introduction to the theory of error-correcting codes, volume 48. john wiley & sons, 1998. john ag roberts and franco vivaldi. signature of time-reversal symmetry in polynomial automorphisms over finite fields. nonlinearity, 18(5):2171, 2005. lajos rónyai. galois groups and factoring polynomials over finite fields. siam journal on discrete mathematics, 5(3):345–365, 1992. mohamed sayed. coset decomposition method for decoding linear codes. international journal of algebra, 5(28):1395–1404, 2011. péter sziklai. applications of polynomials over finite fields. phd thesis, elte ttk, 2013. joachim von zur gathen, igor e shparlinski, and henning stichtenoth. finite fields: theory and applications. 110 microsoft word capitolo intero n 1.doc microsoft word rm_35_4_lolja_final---corrected.docx ratio mathematica issn: 1592-7415 vol. 35, 2018, pp. 53-74 eissn: 2282-8214 53 the proof of the fermat’s conjecture in the correct domain saimir a. lolja1 received: 25-09-2018. accepted: 30-11-2018. published: 31-12-2018. doi: 10.23755/rm.v35i0.426 ©saimir a. lolja abstract the distinction between the domain of natural numbers and the domain of line gets highlighted. this division provides the new perception to the fermat’s conjecture, where to place it and how to prove it. the reasons why the fermat’s conjecture remained unproven for 382 years are examined. the fermat’s conjecture receives the proof in the domain of natural numbers only. the equation a n + b n = c n with positive integers a, b, c, n is not the fermat’s conjecture in the domain of line. keywords: fermat’s conjecture; fermat’s last theorem; domain of natural numbers; domain of line 2010 ams subject classification: 11d41 1 introduction there are two fundamental domains in mathematics: the domain of natural numbers (positive whole numbers or positive integers) and the domain of line. they appear in table 1. 1 faculty of natural sciences, university of tirana, blvd. zogu i, tirana 1001, albania. email: slolja@hotmail.com. saimir a. lolja 54 the domain of natural numbers the domain of line onedimensional filled space numbered unit squares (squarits) all kinds of line euclidian, hyperbolic, elliptic, dashed, etc. twodimensional filled space the squared circle from the centre, it is the same distance equal to a specific integer or number of squarits (unit squares). a rotation brings the position to the same beginning squarit. the lined circle from the centre, it is the same distance equal to a specific decimal or integer number. a rotation brings the position to the same beginning point. threedimensional filled space the cube from the centre, it is the same distance equal to a specific integer number of cubits (unit cubes). a rotation brings the position to the same beginning cubit. the sphere from the centre, it is the same distance equal to a specific decimal or integer number. a rotation brings the position to the same beginning point. zero it is the impassable wall at the bottom. zero refers to none, nothing, no-one. zero gets used for counting the natural numbers to mark the new set of 9. thus, the numbers that contain zeros can be viewed as multiples of nine plus one in all accepted combinations and expressions. the proof of the fermat’s conjecture in the correct domain 55 digit from 1 to 9, e.g. 10 = 9 + 1, 103 = 9×9 + 2×9 + 4. numbers the whole positive numbers and their ratios (rational numbers) only. the array of even numbers 2n starts at zero: 0, 2, 4, 6, 8, … the collection of odd numbers 2n-1 begins at one: 1, 3, 5, 7, 9... on the graph, the positive integers constitute a straight collection of dots, at the same pace, stretching at geometrically 45 o , and numbering ndots. the real and complex numbers, whole and rational numbers, positive and negative numbers, logarithmic and decimal numbers, irrational and transcendental numbers. on the graph, the function y = x is a continuous line stretching at geometrically 45 o and containing an uncounted number of dots. relations the relations with and for positive integers only. the algebraic relations and mathematical analyses. functions and equations of all possible lines, groups, rings, and fields. euclidian and non-euclidian geometries. diophantine and algebraic geometry. calculus and analytical geometry. table 1. the domain of natural numbers versus the domain of line in the domain of line, zero gets assigned to the origin or the beginning point. in one-dimensional space, the geometry determines the distance within two points or one point on the coordinative axis and zero (the origin) and what kinds of lines are passing through those points: parallel (euclidian), hyperbolic (lobachevskian) or elliptic. in two-dimensional space, the geometry determines the inner area between three points or one point on each of the two coordinative axes and zero (the origin). in threedimensional space, the geometry determines the inner volume between four points or one point on each of the two coordinative axes and zero (the origin). the domain of line makes use of the conclusions that come from the domain of natural numbers, but not the opposite. our existence starts at one. below zero, n < 0, the meaning of life and existence loses. things, living and self-thinking entities get numbered positively. we exist as numbers and get shaped by lines. in the domain of natural numbers, the oneor twoor three-dimensional entities are geometrically saimir a. lolja 56 unconnected objects. numbers connect them, because of numbers bond numbers. after squarits or cubits get packed in their respective spaces, there are no void spaces left in between. in one-dimensional space, for example, three connects one and two, because 1 + 2 = 3. in two-dimensional space, 5 2 connects 3 2 and 4 2 , because 9 + 16 = 25. this heavenly-existed set 3-4-5 is the first square set in the unique sequence commonly called the pythagorean triples. these can, for example, be generated by the fibonacci’s method (since the year 1225), by the michael stifel’s method (since the year 1544) and jacques ozanam’s technique (since the year 1694) of the progressions of whole and fractional numbers. the pythagorean triples get also produced using either the leonard eugene dickson’s method (since the year 1920) or euclid’s algebraic quadratic equations or the matrices and linear transformations, etc. the first set of positive integers 3-4-5 is followed by 6-8-10, 5-12-13, 9-12-15, 8-15-17, 12-1620, 15-20-25, 7-24-25, 10-24-26, 20-21-29, 18-24-30, and so on. [1] any relationship in the pythagorean triples can be proved using squared circles. only for the pythagorean triples, the three squared circles form between the geometrical shape of the right-angled triangle with sides taking integer numbers. otherwise, the rightangled triangles are geometrical lines and have the length of at least one of their sides taking a non-integer number. in three-dimensional space, 6 3 connects 3 3 , 4 3 and 5 3 , because 27 + 64 + 125 = 216. this essentially natural cubic set is the first in the unique cubic sequence 3-4-56, 6-8-01-9, 6-8-10-12, 12-16-02-18, 9-12-15-18, 12-16-20-24, 18-24-03-27, and so on. a lined circle cannot take positive integers and get converted to a lined (geometrical) square with positive integers. because, a lined square consists of four equal sides with either an odd or an even integer number of steps, which so produce either an odd or an even integer number of squarits. thus, a lined square fundamentally falls into the domain of natural numbers at a time when the lined circle divided into an irrational number π = 3.14159265358… of steps remains in the domain of line. the geometric irrational number π = 3.14159265358… mirrors the ratio 22/7 in the domain of natural numbers. a lined circle and a lined square bond only when they have an equal geometrical inner area or by inscribing the lined circle inside the lined square and vice versa. 2 the fermat’s last theorem around the year of 1636, pierre de fermat (1607-1665) wrote a comment on the margin of a page in a copy of 1621 edition of the book arithmetica, that translations since the third century a.d. had brought as written by diophantus of alexandria. the the proof of the fermat’s conjecture in the correct domain 57 first part of the comment stated that four positive integers or natural numbers a, b, c, n when n > 2 cannot be a solution to the following equation: �� + �� = �� 1) the second part of the comment stated that he, pierre de fermat, had the proof for eq. (1) but he could not write it because the page margin did not have enough space for it. likely, pierre de fermat had a flash that could prove eq. (1), because he did not write anytime later a general proof of eq. (1). what he communicated in detail was the use of an original logic known as “the infinite descent” to derive a contradiction to an invented counterexample from himself. [1-10] he stated that if the area of a right-angled triangle were equal to the square of an integer, e.g., r2, then there would exist two numbers p, q in the fourth power the difference of which equals r 2 . [3, 10, 11] and without his assertion what the numbers p and q were, the following was his equation: � − � = � 2) in the domain of line, if by wish r 2 is chosen equal to s, then eq. (2) appears as p 4 q 4 = s. if by wish r = t 2 , then p 4 q 4 = t 4 which is a form of eq. (1) for n = 4. if by wish t = u 2 then p 4 q 4 = u 8 , and so on. eq. (2) is inaccurately taken as the specific case of n = 4 for eq. (1), because by command it puts the condition of r = t 2 . also, the counterexample built by pierre de fermat or his eq. (2) falls in the domain of line, while the mathematical relationship bodied in eq. (1) is in the domain of natural numbers. as a sort of indirect proof, the technique of infinite descent is more a wording logic looking for a contradiction to its start than a mathematical method of proof. though it relies on geometry and numbers, the purpose of this technique is to decide by language. the contradiction emerges since the start is either non-existent or untrue or unproven. the infinite descent by pierre de fermat trailed the logic of reductio ad absurdum (reduction to absurdity) by ancient aristotle. though reductio ad absurdum has full power in philosophical perception, it is not enough in the mathematics of numbers. it is so because reasoning is subjective (coming or accepted from the thinking) and numbers are objective (existing independently of thought). he activated his proving approach using the formula of the pythagorean triples, where the sides of the right-angle triangles are sets of specific positive integers and belong to the domain of natural numbers. also, he guessed that the edges of such triangles were relatively prime numbers. then through further calculations and assumptions, e.g., any time the difference of two integers in fourth power was assumed a squared integer, a descending spiral of infinite smaller and smaller such right-angled triangles emerged. the only way to stop the descending loop or the saimir a. lolja 58 infinite descent was by the wording, as pierre de fermat wrote, “…this is impossible since there is not an infinitude of positive integers than a given one”. thus, in accord with pierre de fermat, the infinite descent was in contradiction to the original counterexample, and so it proved that a right triangle could not have an area equal to a squared integer. [3, 5, 11] the proof for a problem that stays within the domain of natural numbers is not enough or valid to become credible proof for the domain of line. reversibly, a general proof extracted in the domain of line is larger than the gate of the domain of natural numbers, and thus unacceptable there. the infinite descent or the descending spiral did not produce anything new, except the need to stop it verbally on purpose. the infinite descent generated rightangled triangles with decreasing size and headed to infinitely small such triangles. it is equivalent to the direction of the infinite ascent, which creates right-angled triangles with increasing size and leading to unbelievably big such triangles. geometrically, as pierre de fermat created his counterexample and the procedure for finding its contradiction, there are not any contradictions going down to infinitely small or up to infinitely big right-angled triangles. as such, both the infinite descent and infinite ascent cannot be stopped verbally except than on purpose. in the domain of line, the area of a right-angled triangle equal to a squared integer is possible and can be only when the lengths of the adjacent sides to the right angle relates in the ratio 2:1. in which case, the length of the side opposite the right angle equals the unit number multiplying √5. it means that such a right-angled triangle is not one of the pythagorean triples and precisely it appears within the domain of line. in the domain of natural numbers, the sequence of natural numbers begins at one and has zero its bottom limit. the chain of natural numbers has no top boundary and increases infinitely by an increment of one. the existence of the bottom base cannot constitute a contradiction in the process of the infinite descent for the invented counterexample because it is just an arrival at the lower limit. it is just a trial in the engineering optimization. after the death of pierre de fermat, his son clément-samuel examined his father’s papers, letters, and notes and published them as a book in 1670. [8] then, eq. (1) came into sight for other mathematicians who began a pursuit to prove it. equation (1) is known as the fermat’s last theorem or the fermat’s conjecture, because since then in century xvii it has not been proved in a general form. the proof of the fermat’s conjecture in the correct domain 59 3 the endeavours for proving the fermat’s last theorem the first effort for the specific case n = 4 to prove the relation embodied in eq. (1) appeared in 1676 and accelerated in century xix and early century xx. due to its outward ease, eq. (1) attracted all mathematicians and leaders in mathematics. [1, 2, 6, 8, 12-15] the diving efforts of brilliant minds into the ocean of mathematics for solving the fermat’s conjecture advanced the science of mathematics in new directions. [10, 16, 17] there have been many publications related to the efforts for proving the fermat’s conjecture. they cover a range of peer-reviewed top mathematical journals to the simplest personal trials and progress reports posted on the internet. such relevant publications keep coming into the scientific view. [7-9, 13, 14, 18-29] it is impossible to cite for reference all of them. however, it is possible to praise all researchers for the time spent for searching to prove the fermat’s last theorem. the shared characteristics of the efforts exerted to prove the fermat’s conjecture and the root reasons why not a final general proof has been reached unfold below. first – the proofs have been searched geometrically (e.g., using elliptic curves) or algebraically (e.g., using bernoulli or complex numbers) in the domain of line at a time that eq. (1) is inside the domain of natural numbers; please refer to table 1 and associated elucidations. likewise, the proof of eq. (1) has been examined on algebraic equations, abstract functions, and conditions noticeable other than eq. (1). [2, 4, 6, 8, 18, 19, 23-26, 29-41] second – the logic of conclusion has been the logic of contradiction to the one assumed either counterexample or new starting conditions; please refer to figure 1. figure 1. the two paths of the solution, where: po is the original point of conditions of the problem. ps is the solution point of the problem. pa is the point of the assumed to-be-original-conditions of the problem. pca is the contradicting point to pa. saimir a. lolja 60 the path or vector of solution pops is the path that preserves the original conditions of the problem. while the imaginary route papca starts with an assumed identity of conditions at point pa that is detached from the point po, the original status of conditions. and sometimes, one counterexample or invented supposition is planted at point pa. then, a solution is accepted if a contradiction to point pa comes across in the path papca. the rejection by contradiction at point pca proves only that the assumed to-be-original-conditions or the counterexample at point pa were not accurate or could not exist. that is, an encounter at a point pca will undoubtedly contradict its self-non-existence that rooted at point pa. the emerged contradiction relates to the false assumption made at point pa and ruins only the characteristics of position pa, which stays detached from the point po. thus, the emerged contradiction at point pca has no connection with path pops and conditions of the solution at the position ps. also, a counterexample is specific, and there is not any general counterexample. third – the proofs have progressed on steps that incorporated the assumption or supposition of specific conditions or properties for variables, equations, and functions. [1, 2, 4-8, 10, 12, 14, 16-18, 22, 23, 25, 27, 29, 31-35, 37-48, 50] that is, the conditions or properties or counterexamples have been created on purpose, taken for granted, personally accepted or assigned, thought or imagined to be that way. the examination of the natural eq. (1) in imaginary systems or the endeavours to reach its proof with tools of the imaginative mathematics beget misleading results. it reaffirms figure 1. fourth – the proofs of the fermat’s conjecture have been researched for isolated power numbers, for example, n = 3, 4, 5, 7, 6, 10, 14 or ideal numbers, and especially for prime numbers. [1, 2, 4, 6, 8, 12, 14, 18, 24-27, 30-39, 41, 42, 46-49] the trail of attempts to prove the fermat’s conjecture by selecting prime numbers for the exponent in eq. (1) started by sophie germain in 1823. sophie germain grouped in case one the prime numbers p that cannot divide a, b, c in eq. (1) and in case two those that do. moreover, she reformulated eq. (1) into the following equation, which both had different conditions from eq. (1) and it was not the fermat’s last theorem anymore [18, 24, 31-36, 47]: �� + �� + �� = 0 3) in 1847, gabriel lamé tried unsuccessfully to factorize the fermat’s last theorem in the cyclotomic field of complex numbers. based on that experience, ernst e. kummer developed the theory of ideal numbers in 1849. within that theory, and using the compound bernoulli numbers, ernst e. kummer defined the set of regular prime numbers. he used them to prove the first case of fermat’s last theorem. [1, 2,4, 6, 8, 18, 31, 33, 38, 39, 47-49] the proof of the fermat’s conjecture in the correct domain 61 the ideal numbers are algebraic integers, which means they are complex numbers. they are part of the ring theory studied in the abstract algebra. they represent the ideals (subsets) in the rings of integers of algebraic number fields, which have finite dimensions. as such, the bernoulli, complex and ideal numbers differ totally from natural numbers and do not reside in the domain of natural numbers. their incorporation in the form of regular prime numbers for proving eq. (1) cannot give the proof or at least a general solution. above all, the past and modern researchers that try to find a proof for sophie germain’s first case embodied in eq. (3) have tried to find a proof of a relationship which is not the fermat’s conjecture embodied in eq. (1). fifth – a wording instrument linked to integer numbers, known as modulus operandi, has been used in algebraic or number formulas. [2, 4, 7, 18, 24-26, 30, 31, 33-35,37, 38, 42-44, 47, 49] the modulo operation depicts the integer remaining after another integer number divides one integer. thus, for two integers x, y that give the same remainder r after divided by another shared integer z, it gets worded that both x and y are congruent modulo z and x – y is a multiple of z. it becomes mathematically visible with the following wordy phrase: x ≡ y (mod z) 4) arithmetically, the relations among the integers x, y, z are generalized as the following: � � = � + � 5) � � = � + � 6) (� − �) � = � − � 7) the wording phrase (4) is not a numeral operator, a numeralis operandi, and only describes the ratio (x – y)/z in eq. (7) by implying that it is equal to an integer number. as just a notation, the wording phrase (4) does not display the values of v – w and r. it is not a mathematical formula or a line equation or a numerical function. the wording phrase (4) is a verbum operandi and does not bring anything new mathematically. the eqs. (5-7) give the complete explicit information. in the domain of natural numbers, mathematics gets explicitly expressed through numeral operators of plus, minus, multiplication, division (ratio), power, equal and sum. the use of the verbum operandi (4) in numeralis operandi for proving the fermat’s conjecture does not fit. it does not offer any specific sets of natural numbers that can be examples for eq. (1). [19, 31, 34, 35] the arithmetic is an explicit and exact science, while modulo operation is both a wording phrase and an implying saimir a. lolja 62 operator. the modulo itself deals with cyclic numbers and all integers, while the natural numbers a, b, c, n in eq. (1) are only positive integers and not cyclic. a modulo solution used for proving eq. (1) must be congruent with a proof using arithmetic operators and mathematical formulas. it just complicates a mathematical expression, e.g., eq. (7), by making invisible and undetermined the integers v-w and r in eqs. (5-7). even when eq. (1) is arranged in the following rational-number form, ��� � � + ���� � = 1 8) there is not any condition in the fermat’s conjecture that the first term is congruent to the second term or a is congruent to b modulo c in eq. (8). anyway, a solution must keep or provide the variables a, b, c, n as positive integers. sixth – the effort to use the elliptic curves and imaginary galois representations to prove the fermat’s conjecture gets separately examined here. between 1955 and 1967, goro shimura, yutaka taniyama, and andré weil set forth the modularity theorem, also known as the taniyama-shimura-weil conjecture. it claimed that all elliptic curves in the field of rational numbers (at rational number coordinates) associated with the modular forms; that is, they were modular. [2, 4, 6, 12, 16, 42, 50] yves hellegouarch in 1974 and gerhard frey in 1982 claimed that the following algebraic equation of the geometrical semi-stable elliptic curves, where p is an odd prime number, is correlated with fermat’s last theorem or eq. (1). [2, 6, 12, 42, 51] � = �(� − ��)(� + �� ) 9) gerhard frey proposed that if a solution for a, b, c, p exists from eq. (1) then a, b of it would give a semi-stable elliptic curve from eq. (9), referred to as the freyhellegouarch curve, which would not be modular. thus, referring to point pa in figure 1, gerhard frey established a counterexample to fermat’s conjecture. in 1985, gerhard frey deepened the mathematical abstraction by articulating that the taniyama-shimura-weil conjecture implied fermat's last theorem. [2, 4, 6, 7, 12] so, referring to point pca in figure 1, gerhard frey laid down the imaginary path of solution papca. on it, someone could investigate for a proof of the taniyamashimura-weil conjecture that would contradict the counterexample flagged at point pa, thus proving the fermat’s last theorem. [46] in 1985, jean-pierre serre wrote that a frey-hellegouarch curve could not be modular and since he did not offer a solid proof for his proposition it turned to be known as the epsilon conjecture. in the summer of 1986, kenneth a. ribet proved the epsilon conjecture for a semi-stable elliptic curve, which meant that the taniyama-shimura-weil conjecture implied the fermat's last theorem. [2, 4, 6, 13, 22, 39, 41, 46] the proof of the fermat’s conjecture in the correct domain 63 a highlighted effort for proving eq. (1) emerged when andrew j. wiles published a final article 108-page-long in parallel with a supportive article co-authored with richard taylor 19-page-long in the annals of mathematics in 1995. [43, 44] using those two pieces, andrew j. wiles confirmed the modularity theorem for semistable elliptic curves to be adequate for contradicting the gerhard frey’s proposition and thus implying the truth of fermat's last theorem. very a few mathematicians seem to understand the depths of abstract mathematics contained in those two published papers and the connection to the proof of fermat’s last theorem. [2, 7, 13] the whole approach summarizes in the following figure 2: figure 2. the paths associated with the efforts to prove the fermat’s conjecture using geometric elliptic curves. as a preface, the proposed solution first guessed by gerhard frey and later laid out by andrew j. wiles did not provide a general proof because they treated prime numbers instead of the natural numbers for the exponent in eq. (1). also, the elliptic curves, modular forms or galois representations incorporated by them are tools for inside the domain of line while the fermat’s conjecture is inside the domain of natural numbers. the counterexample proposed by yves hellegouarch and gerhard frey was a false assumption because the solution to fermat’s conjecture never existed. something cannot both exist and not to be at the same time, place, and under the same conditions. ancient aristotle had summarized this in his principle of non-contradiction, as well. that is, a solution cannot be both known and unknown at the same time, place, and conditions. that is, it was and is impossible to find a set of four natural numbers a, b, c, n that can prove eq. (1). figure 2 confirms the figure 1 and both figures endorse the principle of explosion ex contradictione sequitur quodlibet (from a contradiction, anything follows). since saimir a. lolja 64 both the right and left paths started from a false point or non-existing key, their timeshifted final points had neither any connection with nor an authority on the precise spot of the fermat’s conjecture. even if both branches are opposite, their disagreement is dual and not general. both right and left routes did not comply with the gottfried w. leibniz’s principle of the truth of reasoning, in which an object is resolved into its simplest ideas and truths, into its primitives, to prove it. as brilliant mathematicians, yves hellegouarch, gerhard frey, jean-pierre serre and kenneth a. ribet on the right route and yutaka taniyama, goro shimura, andré weil, andrew j. wiles and richard taylor on the left path were correct in their conclusions about the modularity of geometrical semis-stable elliptic curves. they built their conjectures on detached assumptions, conditions, and tools, independently. therefore, they produced various products (conclusions). otherwise, they should have reached the same conclusions. their right and left approaches to exploration were not even contradicting. their findings in conceptual mathematics were only different in seeing the geometrical semi-stable elliptic curves from diverse viewpoints. their research brought highlighted advancements in theoretical mathematics. as a natural science, mathematics is an explicitly exact science that makes unfit the implying proposition that the modularity theorem can imply the fermat's last theorem. both routes do not end at the precise point of the fermat’s conjecture. the course for going to the correct spot of the fermat’s conjecture is explicitly apparent. eq. (1) was not born from eq. (9) or some modular forms, or vice versa. there is no genetic connection between eq. (1) and eq. (9), independently that the two pairs a n , b n and a p , b p seem of the same gender. whatever solution that the values a p , b p can for elliptic curves in the field of rational numbers, the pair a p , b p does not deliver the duo a n , b n . and this, at a time that cn is not known, and so even the sum ap + bp cannot be evaluated. along with eq. (9), a solution to any other elliptic or non-elliptic equation y = f(x) that combines a n , b n , c n is not a condition of eligibility for giving any hint how to prove eq. (1). also, a galois field is a theoretic finite-field enclosing a limited number of elements, while the array of natural numbers is a chain without end. therefore, any discovery on eq. (9) has no sway on eq. (1). the elliptic eq. (9) is a specific equation and the other elliptic curves are twodimensional geometric functions y2 = f(x3) that give continuous geometric lines, which contain an incalculable amount of numbers of all kinds. the properties that the elliptic curves might have at rational number coordinates have no link to eq. (1), which contains only four arrays of positive integers. while eq. (1) has as variables the natural number a, b, c, n, eq. (9) has geometrical variables x, y, a, b and prime number variable p. [4, 6, 22, 46] a solution for eq. (9) is an optimum solution that incorporates and belongs to the set of the geometrical variables x, y, a, b and prime number variable p. that is, even when a and b in eq. (9) are positive integers, they get the proof of the fermat’s conjecture in the correct domain 65 processed and so lose their originality and individuality as positive integers. therefore, such a solution has no authority over the solution of eq. (1). also, by definition, a modular form is a complex analytic function (a holomorphic function) on the upper half-plane, which itself is a set of complex numbers with the positive imaginary part. furthermore, a meromorphic function, expressed as a ratio between two holomorphic functions, is a complex-valued function and unlinked to the chain of natural numbers. a modular form is a function that has superior symmetries and complexity on a unit disk. [7, 22, 42, 46, 51] which means that a modular form is not an array of natural numbers. a function can be symmetric. on the other side, the collection of natural numbers has no symmetries because it is a chain of increasing positive integers. the modular forms are absolutely part of the domain of line and not part of the domain of natural numbers. in the article by andrew j. wiles, there is no conclusive formula where any substitution with concrete natural numbers a, b, c, n would confirm the fermat’s conjecture. except mentioning the fermat’s last theorem by name six times in the title and introduction, eq. (1) was not engaged in the article. it was so because andrew j. wiles theoretically proved using related galois representations only that the semi-stable elliptic curves were modular. [4, 12, 39, 43, 46, 51] christophe breuil, brian conrad, fred diamond, and richard taylor advanced the path laid down by andrew j. wiles and proved the modularity theorem for all elliptic curves in 2001 [45]. both right and left pathways in figure 2 constitute a non-constructive proving endeavour for the fermat’s conjecture because they provide no numeral examples for eq. (1). 4 the proof of the fermat’s last theorem 4.1 the initial cases for n < 3 to prove the fermat’s conjecture expressed in eq. (1), initially, means to assume (to bear the error) that eq. (1) will remain the same for all n > 2: two terms on the left and one term on the right. but the fact of the unique cubic sequence 3-4-5-6, 6-8-019, 6-8-10-12… is the example just at the beginning for n > 2 that eq. (1) does not exist with two terms on the left and one term on the right when a, b, c are positive integers. it means that the efforts for proving the fermat’s conjecture have conveyed the untruth that eq. (1) with the natural numbers a, b, c, n has only two terms on the left and one term on the right. with the knowledge of this error, summing up eq. (1) side by side for all n gives the following: saimir a. lolja 66 � �� � ��� + � �� � ��� = � �� � ��� 10) �� � − � � − 1 + �� � − � � − 1 = �� � − � � − 1 11) the naturalness and conditions of natural numbers a, b, c, n of eq. (1) are kept undisturbed in eq. (11). however, eq. (11) cannot be used for proving the fermat’s conjecture because it is untrue that eq. (1) will remain with only two terms on the left and one term on the right for all n. for n = 1, eq. (1) or eq. (11) becomes a + b = c that is true for unlimited cases in which the numbers a, b, and c form the bond a + b = c. this. this relation also tells that always c > {a, b}. for n = 1 and a = b eq. (1) becomes 2a = c, which is true for all cases when c = 2a. both cases for n = 1 refers to the situation of one-dimensional array of unit squares (squarits) in table 1. for n = 2, a ≠ b, a + b ≠ c and c > {a, b}, eq. (1) is true only for the pythagorean triples. these are generated when a, b, and c relate through, for example, euclid’s algebraic quadratic equations with a = k(p 2 – q 2 ), b = k(2pq), c = k(p 2 + q 2 ) and where p, q are coprime and not both odd, and k is an additional positive integer. it refers to the situation of a two-dimensional collection of squarits in table 1 that comply with the rule in figure 3. figure 3. the pythagorean rule in the domain of natural numbers, e.g., 3 2 + 4 2 = 5 2 . when n = 2 and a = b then eq. (1) becomes 2a 2 = c 2 , which cannot make c be a natural number (positive integer) because 2 1/2 cannot be a positive integer. since 2 1/2 is an irrational number, 21/2 cannot be constructed as a ratio of two integer numbers. as such, 2 1/ a cannot give an integer number of squarits for c, which would make the dimension of the squared field 2 1/ a an integer number of squarits. in other words, 2a 2 squarits cannot be arranged in a square field. the proof of the fermat’s conjecture in the correct domain 67 for n ≥ 3, the situation in the domain of natural numbers belong to a ndimensional space. for instance, for n = 3, the space is cubic and filled by cubits (unit cubes). the three dimensions of the cube are equal to an integer number of cubits. finding the value (a 3 ) 1/3 = a means finding the dimension an of the cube that contains a3 cubits. in general, the value (an)1/n = a means finding the dimension an of the ndimensional body that contains a n space units. therefore, for n ≥ 3 and a = b, eq. (1) becomes 2an = cn. as such, c cannot have an integer value because 2 1/n is an irrational number and cannot be either a positive integer or expressed as a ratio of two integers. that is, 2 1/n a cannot give an integer number of space units for c. which means that the dimension of the equally-shaped spatial body 2 1/n a cannot bear an integer number of space units. in other words, 2a n space units cannot be arranged in an equally-shaped spatial body. so, the fermat’s conjecture is proved for these initial scenarios. the remaining general set, which is the epical quest of mathematicians to prove the fermat’s conjecture, is the case for n ≥ 3, a ≠ b, and a < b < c. 4.2 eq. (1) arranged in a fractional form the fermat’s last theorem provides only one equation, the eq. (1), with four variables and no specific link between them. as such and since there are no fixed pair distances in the set {a, b, c, n}, the eq. (1) does not get measured. the use of modulus operandi does not help either because the bonds among a, b, c, n are undefined and unconditioned. staying in the domain of natural numbers and without disturbing the identity of natural numbers, the only equations that can be used to prove the fermat’s conjecture are eq. (1) or those like eq. (8). let’s arrange eq. (1) as follows: 1 + ���� � = ���� � summing up both side from n = 1 to n = n as follows: 12) � 1 � ��� + � ���� �� ��� = � ���� �� ��� it results to: 13) ! + � ��� � � − ���� − 1 = � ��� � � − ���� − 1 14) saimir a. lolja 68 after some arrangements, it becomes the following: ���� � = (� − �)(!� − �! + �)�(� − �) or, 15) � = � "(� − �)(!� − �! + �)�(� − �) # � �⁄ 16) the left side of eq. (15) is a positive rational number. since 0 < a < b < c and n ≥ 3, the right side of eq. (15) will be positive only if � � % ≤ 3 2% . in this event, it will be a rational number too. the nth root of a rational number with at least either its nominator or denominator being not at nth power gives an irrational number. which means that, the nth root of the expression inside the square bracket in eq. (16) is an irrational number. the multiplication of an irrational number with an integer produces an irrational number as well. thus, b is an irrational number in eq. (16), meaning not an integer number. it so proves the fermat’s conjecture. the other event is when � �% > 3 2% and thus the right side of eq. (15) will be a negative number. it also proves the fermat’s conjecture because the path paved by � �% > 3 2% meets a contradiction with �� �% � � > 0. besides, both these events border at the value 3 2% that is the perfect fifth interval or the tone g in the diatonic musical scale; or the note sol at the solfeggio system. after the fully-consonant octave interval 1 : 2, the next best harmony ratio is the perfect fifth 2 : 3. the just perfect fifth and octave intervals are the foundation of the pythagorean musical tuning. the border value 3 2% holds the number 2 that replicates n = 2 in the pythagorean triples and the number 3 that replicates n ≥ 3 in the endeavour to prove the fermat’s conjecture. 4.3 eq. (1) arranged in a squared form with a general setting of a < b < c and n ≥ 3, another technique to verify the fermat’s conjecture is to start with the following modified eq. (1): +�� ⁄ , + +�� ⁄ , = +�� ⁄ , 17) the proof of the fermat’s conjecture in the correct domain 69 only when the three squared terms are bonded in the domain of natural numbers in the form of the pythagorean triples through euclid’s algebraic quadratic equations, they can contain integer numbers. that is, they relate to the following equations: � = -.(� − � )/ �⁄ 18) � = -.(2��)/ �⁄ 19) � = -.(� + � )/ �⁄ 20) in eq. (18-19), p and q are coprime, not both odd and 0 < q < p, n ≥ 3, while k is an additional positive integer. it is enough for proving the fermat’s conjecture to look only at eq. (19). wherein, no matter what the value of (2kpq) is, there will be no integer value for b because of the power 2/n at -.(2��)/ �⁄ . furthermore, no matter what the value of -.��/ �⁄ will be, b will not be an integer number because 2 �⁄ is an irrational number. this is adequate to affirm that for n > 2, the values of a, b, c discovered with eqs. (17-20) will not simultaneously be all positive integers. therefore, the fermat’s conjecture holds true in the domain of natural numbers wherein the eq. (1) does not have a solution for positive integer values of a, b, c, n when n > 2. 4.4 incorporating a new integer in eq. (1) for a < b < c and n ≥ 3, another approach is to discover, e.g., whether b will be an integer when c = a + d and a, c, d are the known integers. then, eq. (1) becomes: �� + �� = (� + 0)� then 21) � = � "�1 + 0�� � − 1# � �⁄ and 22) � = -(� + 0)� − �� /� �⁄ = = 1!��2�0 + !(! − 1)2! ��2 0 + ⋯ + !�0�2� + 0�5 � �⁄ 23) saimir a. lolja 70 in the domain of natural numbers, b is the dimension of an equally-shaped spatial body with volume bn space units and unit subsection having an space units. the removal of a unit subsection from an equally-shaped spatial body with volume (a + d) n space units leaves a number of space units that cannot be finitely divided into an integer number of identical unit subsections needed for the new equally-shaped spatial body. the multiplication of an integer or rational number with an irrational number gives an irrational number. saying it differently from eq. (22), the dimension b cannot be an integer number because 6�1 + 78� � − 19� �⁄ is an irrational number; that is, not an integer number. therefore, the spatial units in the resulting spatial body cannot be arranged in a way that the spatial body will be equally-shaped, having a dimension b equal to an integer number, and containing an integer number bn of spatial units. in addition, whatever is the value of the sum inside the bracket in eq. (23), it cannot give an integer value for b because a n has been cancelled out and the power of the big bracket is 1/n. which means that b will be an irrational number, so not an integer. thus, eq. (1) cannot be true for simultaneous integer values for a, b, c and n ≥ 3 in the domain of natural numbers. this is proof of the fermat’s conjecture. 4.5 incorporating a multiple in eq. (1) having a < b < c and n ≥ 3, any such three numbers in the series of natural numbers may relate in pairs in the forms of c = ga and b = ha. the positive coefficients g, h are larger than one. they can be integers (e.g., a = 3, c = 9, then g = 9/3 = 3; and e.g., a = 3, b = 6, thus h = 6/3 = 2) or non-integers (e.g., a = 3, c = 8, then g = 8/3 > 1; and e.g., a = 3, b = 5, thus h = 5/3 > 1). the search for the proof means to discover, using eq. (1), whether the third term can be an integer when the two other terms are integers. let’s take the case of c = ga with g >1. it means that the positive integers a, c are known and the discovery will be whether b can be another natural number. now, eq. (1) appears in the following form: �� + �� = (:�)� 24) then � = �(:� − 1)� �⁄ 25) with g being either an integer or a non-integer, since g n = ggggggg… n-times and (g n – 1) < g n by one, then gn – 1 = egn-1 = gn(e/g), where 1 < e < g or ae < c. the quantity e is a non-integer because: the proof of the fermat’s conjecture in the correct domain 71 ; = :� − 1:�2� 26) then, the eq. (25) becomes: � = � �:� ;:� � �⁄ = �: �;:� � �⁄ = � �;:� � �⁄ = = (��2��;)� �⁄ = �� �⁄ �(�2�) �⁄ ; � �⁄ 27) while the multiplication of an integer with a non-integer can give either an integer or a non-integer number, the eq. (27) produces only a non-integer value for b. because of no matter whether (��2��;) will give an integer value or not, its power 1/n omit the option that b will have an integer value. the multiplication of an integer or rational number with an irrational number gives an irrational number. it explicitly means that b cannot be an integer because �<=� � �⁄ is an irrational number; so, confirming the fermat’s conjecture in the domain of natural numbers. the proof of the fermat’s conjecture that concluded by using eqs. (12 27) make evident that for a < b < c and n ≥ 3 it stays true in the domain of natural numbers only. whereas a general eq. (1) has its field of the degrees of freedom in the domain of line where a, b, c, n can be real or complex numbers. in the domain of line, eq. (1) can be analysed with all possible mathematical, geometrical, algebraic, analytical, complex and imaginary tools. in the domain of line, the eq. (1) is not the fermat’s conjecture anymore. 5 conclusion a mathematical conjecture or any formula and equation needs be first defined to which domain it belongs: to the domain of natural numbers or the domain of line. then, this will determine the point of view and tools directed to the analysed conjecture or equation. if a conjecture or equation is entirely on natural numbers (it is inside the domain of natural numbers), then the mathematical tools should be extracted from the domain of natural numbers. if a conjecture or equation gets defined for the domain of line, then the precise tools should be derived from the domain of line and the domain of natural numbers if they fit. the fermat’s last theorem preserves its original identity if it is proved within the domain of natural numbers and with mathematical tools from this domain. pierre de fermat was correct that eq. (1) having positive integers a, b, c, n cannot be possible for n > 2. however, he missed defining both in which domain he was conjuring the saimir a. lolja 72 eq. (1) and any relationship among numbers a, b, c, n. it took 382 years to outline and prove the fermat last theorem correctly. acknowledgment the author likes to thank all mathematicians engaged with the proof of the fermat last theorem and the reviewers of this paper. references [1] rideout, d. (june 2006). pythagorean triples, and fermat’s last theorem, lecture at canadian management center seminar, waterloo, canada. [2] frey, g. (1997). the way to the proof of fermat’s last theorem, lecture at the international symposium on information theory, ulm, germany. [3] dickson, l. e. (1999). history of the theory of numbers, volume 2, american mathematical society, pp. 615–626. [4] boston, n. (2003). the proof of fermat’s last theorem, university of wisconsinmadison. [5] delahaye, d., mayero, m. 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(2012). the generalized fermat equation: a progress report, lecture at the university of british columbia. [43] wiles, a.j. (1995). modular elliptic curves, and fermat’s last theorem, annals of mathematics, 141, pp. 443-551. [44] taylor, r., wiles, a.j. (1995). ring-theoretic properties of certain hecke algebras, annals of mathematics, 141, pp. 553-572. [45] breuil, ch., conrad, b., diamond, f., taylor, r. (2001). on the modularity of elliptic curves over q: wild 3-adic exercises, journal of the american mathematical society, 14(4), pp. 843-939. [46] rubin, k. (2007). the solving of fermat’s last theorem, physical sciences breakfast lecture, the university of california at irvine. [47] granville, a., monagan, m.b. (1988). the first case of fermat’s last theorem is true for all prime exponents up to 714’591’416’091’389, transactions of the american mathematical society, 306(1), pp. 329-359. [48] kummer, e.e. (1850). allgemeiner beweis des fermatschen satzes, dafs die gleichung x 2 + y 2 = z durch ganze zhalen unlösbar ist, für alle diejenigen potenzexponenten λ, welche ungerade primzahlen sind und in den zählern der ersten (λ3)/2 bernoullischen zahlen als factoren nicht vorkommen, journal für die reine und angewandte mathematik, pp. 138-146. [49] stewart, i., tall, d. (2002). algebraic number theory and fermat's last theorem, 3rd ed., a. k. peters. [50] http://mathworld.wolfram.com/taniyama-shimuraconjecture.html [51] faltings, g. (1995). the proof of fermat’s last theorem by r. taylor & a. wiles, notices of the american mathematical society, 42(7), pp. 743-746. ratio mathematica vol. 34, 2018, pp. 77–83 issn: 1592-7415 eissn: 2282-8214 notes on the solutions of the first order quasilinear differential equations alena vagaská∗, dušan mamrilla† received: 11-05-2018. accepted: 24-06-2018. published: 30-06-2018 doi:10.23755/rm.v34i0.414 c©dušan mamrilla et al. abstract the system of the quasilinear differential first order equations with the antisymetric matrix and the same element f (t, x (t)) on the main diagonal have the property that r′ (t) = f (t, x (t)) r (t), where r (t) ≥ 0 is the polar function of the system. in special cases, when values f (t, x (t)) and g (t, x (t)) are only dependent on r2 (t), t ∈ j0 we can find the general solution of the system (1) explicitly. keywords: nonlinear; quasilinear; differential equation; differential system; 2010 ams subject classifications: 34c10. ∗technical university of košice, faculty of manufacturing technologies, prešov, slovakia. alena.vagaska@tuke.sk †prešov, slovakia. dusan.mamrilla@gmail.com 77 a. vagaská and d. mamrilla 1 introduction norkin, s. b. and tchartorickij, j. a. [1] and kurzweill, j. [2] investigated the oscillatory properties of the 1,2-nontrivial solutions x(t) of systems of two first order linear differential equations applying polar coordinates. mamrilla, d. and norkin, s. b. [3] investigated the oscilatory properties of the 1,2,3-nontrivial solutions x(t) of systems of three first order linear differential equations applying spherical coordinates. applying polar (spherical) coordinates, the boundedness and oscillatority of the 1,2 (1,2,3)-nontrivial solutions x(t) of systems of two (three) first order quasilinear differential equations have been investigated by mamrilla, d. [4], [5], [6] and mamrilla, d. and seman, j. and vagaská, a. [7], while special attention was paid to the study of the properties of the x(t) solutions of the systems, the matrix of which has the same element f (t,x(t)) on the main diagonal. this paper gives some asymptotical and oscillatory properties of the solutions to the system of the nonlinear differential equations:  x1x2 x3   ′ =   f (t,x(t)) 0 g (t,x(t))0 f (t,x(t)) 0 −g (t,x(t)) 0 f (t,x(t))   ·   x1x2 x3   , (1) where t > 0, 0 6= f (t,x(t)), 0 6= g (t,x(t)) ∈ c0 (d ≡ j ×r3,r). we assume that each solution x(t) = (x1 (t) ,x2 (t) ,x3 (t)) , (2) x1 (t0) = x 0 1, x2 (t0) = x 0 2, x3 (t0) = x 0 3, t0 ∈ j exists on the interval j and we denote h > t0 > 0 the right endpoint of the interval j and j0 = [t0,h) . we shall denote g1 (t,x) = f (t,x)x1 + g (t,x)x3, g2 (t,x) = f (t,x)x2, (3) g3 (t,x) = −g (t,x)x1 + f (t,x)x3. it is known that if d0 ⊂ d is open nonempty set and derivatives (∂gi(t,x)/∂xj) are continuous functions on d0 for every i,j ∈ {1,2,3} then each point (t0,x 0 1,x 0 2,x 0 3) ∈ d0 is passed by one and only one integral curve x ∈ d of the system (1) [3]. 78 notes on the solutions of the first order quasilinear differential equations definition 1.1. the solution x(t) to the system (1) is called i − trivial, i ∈ {1,2,3} is fixed, if xi (t) = 0 on the interval j0. otherwise x(t) is i − nontrivial solution. if for at least one i ∈ {1,2,3} the solution to the system (1) is i−nontrivial, shortly so solution x(t) is said to be nontrivial. it is obvious that system (1) has 1,2,3 − trivial solution; 1,3 − trivial and 2 − nontrivial solution; 1,3 − nontrivial and 2 − trivial solution; 1,2,3 − nontrivial solution. definition 1.2. the solution x(t) to the system (1) is called i − positive (i − negative), i ∈ {1,2,3} is fixed, if xi (t) is positive (negative) function on the interval j0. definition 1.3. the solution x(t) to the system (1) is called i − nondecreasing (i−nonincreasing), i ∈{1,2,3} is fixed, if xi (t) is nondecreasing (nonincreasing) function on the interval j0. it is obvious that if f (t,x)x2 ≥ 0 (f (t,x)x2 ≤ 0) for any point (t,x) ∈ d then arbitrary solution x(t), t ∈ j0 to the system (1) is 2 − nondecreasing (2−nonincreasing). definition 1.4. the solution x(t) to the system (1) is called i − bounded, i ∈ {1,2,3} is fixed, if xi (t) is the bounded function on interval j0. at other cases x(t) is i − unbounded one which is called i − from above (i − from below) unbounded, i ∈ {1,2,3} is fixed, if xi (t) is from above (from below) unbounded function on interval j0. it is obvious that if for every continuous function y defined on interval j0 : a) sup y (∫ h t0 |f (t,y)y2|dt ) < ∞, then any solution x(t), t ∈ j0 to the system (1) is 2− bounded, b) sup y (∫ h t0 f (t,y)y2dt ) = −∞ ( inf y (∫ h t0 f (t,y)y2dt ) = ∞ ) then there exists a point t∗ ≥ t0 and 2−negative (2−positive) solution x(t), t ∈ [t∗,h) to the system (1) such that it is 2 − from below (2 − from above) unbounded. definition 1.5. the solution x(t) to the system (1) is called i − oscillatory, i ∈ {1,2,3} is fixed, if xi (t) is the oscillatory function, i. e. if there exists the increasing sequence {tn} ∞ n=1 such that tn ∈ j0, tn → h and xi (tn) .xi (tn+1) < 0 for each n ∈ n. the solution x(t) is called i − nonoscillatory if there exists h1 < h such that xi (t) is not changing its sign on the interval [h1,h), resp. if it has maximally finite number of zero point on the interval [t0,h). 79 a. vagaská and d. mamrilla 2 main results theorem 2.1. the general solution to the system (1) is generated by the trinity of the functions: x1 (t) = ( c2 cos (∫ t t0 g (s,x(s))ds ) −c3 sin (∫ t t0 g (s,x(s))ds )) ×exp   t∫ t0 f (s,x(s))ds   , x2 (t) = c1 exp (∫ t t0 f (s,x(s))ds ) , x3 (t) = ( −c2 sin (∫ t t0 g (s,x(s))ds ) −c3 cos (∫ t t0 g (s,x(s))ds )) ×exp (∫ t t0 f (s,x(s))ds ) , where ci (i = 1,2,3) ∈ r are arbitrary constants. proof. the characteristic quasipolynomial of the system (1) is det(a(t,x(t))−λ(t,x(t))e) = = (f (t,x(t))−λ(t,x(t)))3 + g2 (t,x(t)) (f (t,x(t))−λ(t,x(t))) = 0 the solutions of which are the functions λ1 (t,x(t)) = f (t,x(t)) and λ2,3 (t,x(t)) = f (t,x(t))± ig (t,x(t)) . the fundamental system of the solutions to the system (1) is generated by the vector functions x1 (t,x(t)), rexc2 (t,x(t)), imx c 2 (t,x(t)), where x1 (t,x(t)) =   01 0  exp(∫ t t0 f (s,x(s))ds ) xc2 (t,x(t)) =     10 0   + i   00 −1    exp(∫ t t0 (f (s,x(s))− ig (s,x(s)))ds ) , 80 notes on the solutions of the first order quasilinear differential equations e.g., x2 (t,x(t)) = =     10 0  cos(∫ t t0 g (s,x(s))ds ) +   00 −1  sin(∫ t t0 g (s,x(s))ds ) × exp (∫ t t0 f (s,x(s))ds ) , x3 (t,x(t)) = =     00 −1  cos(∫ t t0 g (s,x(s))ds ) −   10 0  sin(∫ t t0 g (s,x(s))ds ) × exp (∫ t t0 f (s,x(s))ds ) . this proves the theorem.2 corolary 2.1. if we put g (t,x(t)) = 1 in theorem (2.1), we obtain assertion of theorem (2.1) in [7]. theorem 2.2. let for all continuous functions y defined on the interval j0: a) sup y (∫ h t0 |f (s,y) |ds ) < ∞, then each solution x(t), t ∈ j0 to the system (1) is 1,2,3− bounded, b) sup y (∫ h t0 f (s,y)ds ) = −∞, then each solution x(t), t ∈ j0 to the system (1) is 1,2,3 − bounded and such that x1 (t) → 0, x2 (t) → 0, x3 (t) → 0 for t → h, c) inf y (∫ h t0 f (s,y)ds ) = ∞, then each solution x(t), t ∈ j0 to the system (1) is such that it is i−unbounded at least for one i ∈{1,2,3} . proof. theorem (2.1)implies that the general solution to the system (1) fulfils a condition x21 (t) +x 2 2 (t) +x 2 3 (t) = (c 2 1 + c 2 2 + c 2 3) exp ( 2 ∫ t t0 f (s,x(s))ds ) , and this implies the assertion of the theorem. 2 we assume that for each nontrivial solution x(t), t ∈ j0 to the system (1) there exists the trinity of the functions r (t) > 0, u(t), v (t) ∈ c1 (j0,r) such that the coordinates xi (t), t ∈ j0, i = 1,2,3 fulfil [7]: 81 a. vagaská and d. mamrilla x1 (t) = r (t) cosu(t) , x2 (t) = r (t) sinu(t) cosv (t) , x3 (t) = r (t) sinu(t) sinv (t) , (4) r ′ (t) = x ′ 1 (t) cosu(t) + x ′ 2 (t) sinu(t) cosv (t) + + x ′ 3 (t) sinu(t) sinv (t) , r (t)u′ (t) = −x ′ 1 (t) sinu(t) + x ′ 2 (t) cosu(t) cosv (t) + + x ′ 3 (t) cosu(t) sinv (t) , r (t) sinu(t)v′ (t) = −x ′ 2 (t) sinv (t) + x ′ 3 (t) cosv (t) . the function r (t) is called the polar, u(t) the first angle function and v (t) the second angle function. from this after equivalent arrangement for nontrivial solutions to the system (1) we get: r′ (t) = f (t,x(t))r (t) , u′ (t) = −g (t,x(t)) sinv (t) , (5) sinu(t)v′ (t) = −g (t,x(t)) cosu(t) cosv (t) . 3 conclusions the paper deals with qualitative and quantitative properties of the solutions of special differential equations and systems of differential equations. non-linear and quasi-linear equations are less researched in mathematical publications, so the goal of this paper was to investigate some asymptotical and oscillatory properties of non-trivial solutions of such differential equations and systems thus contributing to knowledge in this field of research. special attention was focused on the study of the asymptotic and oscillatory properties of the x(t) solutions of the systems, the matrix of which has the same element on the main diagonal. we have achieved new results due to the investigation of this subject by applying of polar or spherical coordinates. 4 acknowledgements the research work is supported by the project kega 026tuke-4/2016. title of the project: implementation of modern information and communication technologies in education of natural science and technical subjects at technical faculties. 82 notes on the solutions of the first order quasilinear differential equations references [1] s. b. norkin and j. a. tchartorickij, investigation of oscillatory properties of a system of two linear differential equations by means of angle function, differential equations and approximation theory, madi (1977), moscow, pp. 19-32 [2] j. kurzweill, ordinary differential equations, sntl, praha, 1978 [3] d. mamrilla, s. b. norkin, investigation of oscillatory properties of linear third order differential equations systems by means of angle functions, investigations of differential equations , madi 1986, moscow, pp. 97-106 [4] d. mamrilla on boundedness and oscillatoricity of certain differential equations systems, fasciculi mathematici 24 (1994), pp. 27-35, poznan [5] d. mamrilla the theory of angle functions and some properties of certain nonlinear differential systems, fasciculi mathematici 24 (1994), pp. 55-65, poznan [6] d. mamrilla on the systems of first order quasi linear differential equations, tribun eu brno (2008) [7] d. mamrilla and j. seman and a. vagaská on the solutions of the first order nonlinear differential equations, journal of the applied mathematics, statistics and informatics 1(jamsi), 3(2007), pp. 63 70, trnava 83 ratio mathematica volume 47, 2023 minimal reinhard zumkeller divisor cordial graphs a. ruby priscilla * s. firthous fatima † abstract in this paper, the notion of minimal reinhard zumkeller divisor cordial labeling has been introduced. let g = (v, e) be a simple graph and γ : v (g) → minimum {2i × 3, 2j+1 × 5, 2k+1 × 7, 2l × 3 × 5, 2m × 3 × 7 where i, j, k, l, m ≥ 1} be an injection such that the sum of the cardinality of exponent of γ(v (g)) should be equal to the order of the graph g. for each edge uv, assign the label 1 if γ(u)|γ(v) or γ(v)|γ(u) where γ(u) and γ(v) are zumkeller numbers and the label 0 if γ(u) ∤ γ(v) and also if |eγ|(0) − eγ|(1)| ≤ 1 then γ is called minimal reinhard zumkeller divisor cordial labeling. this paper elucidates how the zumkeller number, which is the generalization of the perfect number, goes along with the divisibility concept of the number theory and the cordial labeling technique. it also probes the existence of minimal reinhard zumkeller divisor cordial labeling of path, cycle, star k1,s, complete bipartite, complete graph kn for n < 17 , tadpole graph tn,k for all values of n and k. keywords: zumkeller graph, divisor cordial labeling, zumkeller divisor cordial graph. 2010 ams subject classifications: 05c78. 1 *research scholar, reg.no:18221192092003, department of mathematics, sadakathullah appa college (autonomous), affiliated to manonmaniam sundaranar university, tirunelveli, tamilnadu, india and assistant professor, department of mathematics, sarah tucker college (autonomous), tirunelveli-7. ruby@sarahtuckercollege.edu.in †assistant professor, department of mathematics, sadakathullah appa college (autonomous), rahmath nagar, tirunelveli-627011, kitherali@yahoo.co.in 1received on october 07, 2022. accepted on june 10, 2023. published on june 30, 2023. doi: 10.23755/rm.v39i0.866. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 279 a. ruby priscilla and s. firthous fatima 1 introduction graphs regarded here are finite, undirected and simple. the symbols v (g) and e(g) denote the vertex set and the edge set of a graph g. most of the graph labeling methods trace their origin to the one introduced by rosa [1967]. a graph labeling is an assignment of integer to the vertices or edges or both subject to certain conditions. labelled graph has many branch out applications such as coding theory, missile guidance, x-ray, crystallography analysis, communication network addressing systems, astronomy, radar, circuit design, database management etc., the concept of cordial labeling was introduced by cahit [1987]. varatharajan et al. [2011] introduced divisor cordial labeling. if the sum of all the proper positive divisors of a positive integer is equal to the number, then the number is called perfect number. generalizing the concept of perfect numbers r.h.zumkeller defined a new type of number as a zumkeller number. peng and rao [2013] established several results and conjectures on zumkeller numbers. the notion of zumkeller labeling of some cycle related graphs was investigated by balamurugan et al. [2014]. murali et al. [2017] proved results about zumkeller cordial labeling of cycle related graphs. shahbaz and mahmood [2020] proved that zumkeller number is either a super totient or a hyper totient number. graph labeling has a potent communication between the number theory and graph network. the idea behind this work fosters us to develop a graph labeling technique called minimal reinhard zumkeller divisor cordial labeling by pooling the divisor cordial graph labeling technique and characteristics of zumkeller number which is one of the engrossing parts of the number theory. because of the existence of diverse γ vertex labeling design for some graph structure, minimal condition is emphasized. as the work has been focused on minimization condition, zumkeller number chosen for the concept depicted herein is a sequence of least even zumkeller numbers. an added reason for not using sequence of odd zumkeller numbers is mainly due to point up the minimum sequence of zumkeller numbers. mahanta et al. [2020] stated that 945 is the smallest odd zumkeller number. in this paper we discuss the existence of minimal reinhard zumkeller divisor cordial labeling of path, cycle, star k1,s, complete bipartite, complete graph kn for n < 17, tadpole graph tn,k for all values of n and k. 2 preliminaries definition 2.1. varatharajan et al. [2011] let g = (v, e) be a simple graph and f : v → {1, 2, . . . |v |} be a bijection. for each edge uv, assign the label 1 if either f(u) | f (v) or f(v) | f(u) and the label 0 if f(u) ∤ f(v). f is called a divisor cordial labeling if |ef (0) − ef (1)| ≤ 1. a graph with a divisor cordial labeling is called a divisor cordial graph. 280 minimal reinhard zumkeller divisor cordial graphs 1. ef|(0)is the number of edges of the graph g having label 0 under f | 2. ef|(1) is the number of edges of the graph g having label 1 under f | definition 2.2. peng and rao [2013] a positive integer n is said to be a zumkeller number if the positive divisors of n can be partitioned into two disjoint subsets of equal sum. a zumkeller partition for a zumkeller number n is a partition {a, b} of the set of positive divisors of n so that each of a and b sums to the same value. proposition 2.3. peng and rao [2013] for any prime p ̸= 2 and positive integer k with p ≤ 2k+1 − 1, the number 2kp is a zumkeller number. fact 2.4. peng and rao [2013] let the prime factorization of an even zumkeller number n be 2kpk11 p k2 2 . . . p km m where k is a positive integer. then at least one of ki must be odd. definition 2.5. balamurugan et al. [2014] a simple graph g = (v, e), where v is vertex set and e is edge set of g is said to admit a zumkeller labeling if there exists an injective function f : v → n such that f∗ : e → n defined as f∗(xy) = f(x)f(y) is a zumkeller number for xy ∈ e; x, y ∈ v . the labelled graph g is called as a zumkeller graph. definition 2.6. murali et al. [2017] let g = (v, e) be a graph. an injective function f : v → n is said to be a zumkeller cordial labeling of the graph g if there exists an induced function f∗ : e → {0, 1} defined by f∗(xy) = f(x)f(y) satisfies the following conditions 1. for every xy∈e, f∗(xy) = { 1 , f (x) f (y) is a zumkeller number ; 0 , otherwise 2. |ef∗ (0) − ef∗ (1)| ≤1 definition 2.7. murali et al. [2017] a graph g = (v, e) which admits a zumkeller cordial labeling is called a zumkeller cordial graph. 3 main results the vertex labeling γ mention in the definition 3.1 is defined by using proposition 2.3 and fact 2.4. definition 3.1. let g = (v, e) be a simple graph and γ : v (g) → minimum{ 2i × 3, 2j+1 × 5, 2k+1 × 7, 2l × 3 × 5, 2m × 3 × 7 where i, j, k, l, m ≥ 1 } be an injection satisfying any one of the following conditions |i|+|j| = |v (g)| or |i| + |k| = |v (g)| or |i| + |l| = |v (g)| or |i| + |m| = |v (g)| or |i| + |j| + |k| = 281 a. ruby priscilla and s. firthous fatima |v (g)| or |i|+|j|+|l| = |v (g)| or |i|+|j|+|m| = |v (g)| or |i|+|j|+|k|+|l| = |v (g)| or |i|+|j|+|k|+|m| = |v (g)| or |i|+|j|+|k|+|l|+|m| = |v (g)| i.e., the sum of the cardinality of exponent of γ(v (g)) should be equal to the order of the graph g. for each edge uv, assign the label 1 if γ(u) | γ(v) or γ(v) | γ(u) where γ(u) and γ(v) are zumkeller numbers and the label 0 if γ(u) ∤ γ(v) and also if |eγ| (0) − eγ| (1)| ≤ 1 then γ is called minimal reinhard zumkeller divisor cordial labeling. a graph with a minimal reinhard zumkeller divisor cordial labeling is called a minimal reinhard zumkeller divisor cordial graph. theorem 3.2. the path pn is a minimal reinhard zumkeller divisor cordial when n ≡ 0, 1 (mod 2) proof. let v1, v2, . . . .vn be the vertices of the path pn. label those consecutive adjacent vertices in the order as 2i × 3 and 2l × 3 × 5 where 1 ≤ i ≤ n 2 and 1 ≤ l ≤ n 2 for the path having even number of vertices and for the path having odd number of vertices 1 ≤ i ≤ n+1 2 and 1 ≤ l ≤ n−1 2 and also |i| + |l| = |v (g)|. if 2i × 3 |2l × 3 × 5 then the consecutive adjacent vertices contribute 1 to each edge and if 2i × 3 ∤ 2l × 3 × 5 then the consecutive adjacent vertices contribute 0 to each edge. thus eγ|(1) = n 2 and eγ|(0) = n−2 2 if n is even and eγ|(1) = eγ|(0) = n−1 2 if n is odd. hence |eγ|(0) − eγ|(1)| ≤ 1.thus pn is a minimal reinhard zumkeller divisor cordial graph. theorem 3.3. the cycle cn is a minimal reinhard zumkeller divisor cordial when n ≡ 1 (mod 2) , n ≥ 3, n ∈ n proof. let v1, v2, . . . .vn be the vertices of the cycle cn. by making use of the similar pattern described as for path, the cycle of odd order is investigated as a minimal reinhard zumkeller divisor cordial graph. theorem 3.4. the cycle cn admits a minimal reinhard zumkeller divisor cordial when the vertex vn is labelled with 2 × 3 × 7 where n ≡ 0 (mod 2) , n ≥ 4, n ∈ n. proof. let v1, v2, . . . .vn be the vertices of the cycle cn. label the vertex vn with 2 × 3 × 7 and appertain with the similar pattern described as for path for the remaining vertices results in a minimal reinhard zumkeller divisor cordial labeling for the cycle graph. 282 minimal reinhard zumkeller divisor cordial graphs theorem 3.5. the wheel graph wn = k1 + cn is a minimal reinhard zumkeller divisor cordial proof. let vo be the center vertex of wn and label the center vertex as 22 × 5 case 1. n is even. label the vertices v1, . . . .vn of cn as 2i × 3 where 1≤i≤n such that gcd( (20,γ (v1)) , (20,γ (v2)) , . . . . (20,γ (vn)) = 1 and gcd (γ (vg) , γ (vg+1)) > 1 where 1≤g≤n−1 and also |i| + |j| =n+1 =n+1 = |v (wn)| . we observe that, eγ|(0) =eγ| (1) =n. (1) case 2. n is odd. label the center vertex vo as 22×5 and label v1, . . . vn of cn with the same labeling design mentioned in case 1 here also, eγ| (0) = eγ| (1) = n (2) hence, from (1) and (2) we get that |eγ|(0) − eγ|(1)| = { 0 if n is even 0 if n is odd theorem 3.6. the star graph k1,s is a minimal reinhard zumkeller divisor cordial labeling when s ≡ 0, 1(mod 2) proof. let g = k1,s be the star graph with vertex set v (g) = {v0 ∪ {vg : 1 ≤ g ≤ s}} where v0is a center vertex and vg’s are pendant vertices and an edge set e(g) = {eg = v0vg : 1 ≤ g ≤ s}. here we notice that the order of the graph |v (g)| = s + 1 and the size of the graph |e(g)| = s. case 1. s ≡ 1 (mod 2) assume γ(v0) = 2 × 3 which is a zumkeller number. the pendant vertices contribute 1 to its adjacent edges are labelled as follows. γ(vi) = 2 i+1 ×3 for 1≤i≤s−1 2 + 1 and the pendant vertices contributes 0 to its adjacent edges are labelled as follows γ(vj) = 2 j+1×5 for 1≤j≤s−1 2 . and also |i| + |j| = s + 1 case 2. s ≡ 0 (mod 2) assume γ(v0) = 2 × 3 which is a zumkeller number. the pendant vertices contribute 1 to its adjacent edges are labelled as follows. γ(vi) = 2 i+1 ×3 for 1≤i≤s 2 and the pendant vertices contributes 0 to its adjacent edges are labelled as follows γ(vj) = 2 j+1×5 for 1≤i≤s 2 . and also |i| + |j| =s+1 hence from cases 1 and 2, we get that eγ|(0) = s+1 2 and eγ|(1) = s−1 2 when m is odd and eγ|(0) = eγ|(1) = s 2 when s is even. 283 a. ruby priscilla and s. firthous fatima hence ∣∣eγ| (0) − eγ| (1)∣∣ = { 0 if s is even 1 if s is odd thus ∣∣eγ| (0) − eγ| (1)∣∣ ≤1 . hence, k1,s is a minimal reinhard zumkeller divisor cordial. theorem 3.7. the complete bipartite graph kx,z is a minimal reinhard zumkeller divisor cordial graph for all values of x, y ≥ 2 proof. let v = v1 ∪ v2 be the bipartition of kx,z such that v1 = {v1, v2, . . . vx} and v2 = {w1, w2, . . . wz} .the order of the complete bipartite graph kx,z is x + z = f . case 1. x = z where x and z are even obviously, there are f 2 vertices in v1 and f 2 vertices in v2. then label f 4 vertices out of f 2 vertices as 2i×3, where 1≤i≤f 2 and the remaining f 4 vertices get the label as 2j+1×5 , where 1≤j≤f 2 . label f 4 vertices in v2 as 2i×3 , where f2 +1≤i≤z and the remaining f 4 vertices as 2j+1×5 where f 2 +1≤i≤z . then the cordiality condition |eγ|(0) − eγ|(1)| = 0 . case 2. x = z, when x and z are odd. label f 2 vertices in v1 as follows: label f 2 +1 2 vertices out of f 2 in v1 as 2i×3, where 1≤i≤ f 2 +1 2 and label the remaining vertices f 2 −( f 2 +1) 2 vertices are labelled as 2i+1×5, where 1≤j≤f 2 − f 2 +1 2 . then label f 2 vertices in v2 as follows: label f 2 +1 2 vertices out of f 2 in v2 as 2i × 3, where (f2 +1) 2 + 1 ≤ i ≤ z + 1 and label the remaining vertices z − f 2 +1 2 in v2 as 2j+1 × 5, where f2 − (f2 +1) 2 + 1 ≤ i ≤ z − 1 then the cordiality condition |eγ|(0) − eγ|(1)| = 1. case 3. x < z and x + z where z = x + 1, x is odd and z is even there are f+1 2 − 1 vertices in v1 and f+12 vertices in v2 .label f+1 2 − 1 vertices in v1 as follows: label f+1 2 2 vertices out of f+1 2 − 1 in v1 as 2i × 3, where 1 ≤ i ≤ f+1 2 2 and the remaining vertices f+1 2 − 1 − f+1 2 2 are labelled as 2j+1 × 5, where 1 ≤ j ≤ f+1 2 − 1 − f+1 2 2 .label f+1 2 vertices in v2 as follows: label f+1 2 2 vertices as 2i × 3 , where f+1 2 2 + 1 ≤ i ≤ z and label the left over vertices f+1 2 − f+1 2 2 as 2j+1 × 5, where f+1 2 − f+1 2 2 ≤ j ≤ z − 1 . following the labeling pattern results in |eγ|(0) − eγ|(1)| = 1 case 4. x<z and z = x + 1, where x is even and z is odd. there are f+1 2 −1 vertices in v1 and f+12 vertices in v2. label f+1 2 − 1 vertices in v1 as follows: label f+1 2 −1 2 vertices as 2i × 3 , where 1 ≤ i ≤ f+1 2 −1 2 and label 284 minimal reinhard zumkeller divisor cordial graphs f+1 2 − 1 − f+1 2 −1 2 vertices as 2j+1 × 5, where 1 ≤ j ≤ f+1 2 −1 2 .now proceed to label f+1 2 vertices in v2 as follows: label f+1 2 −1 2 + 1 vertices as 2i × 3 , where f+1 2 −1 2 + 1 ≤ i ≤ z and label the left over vertices f+1 2 − f+1 2 −1 2 + 1 as 2j+1 × 5, where f+1 2 − f+1 2 −1 2 +1 ≤ j ≤ z − 1 . then the cordiality condition∣∣eγ| (0) − eγ| (1)∣∣ = 0. case 5. x > z and x = z + 2. obviously, there are f 2 +1 vertices in v1 and f 2 −1 vertices in v2. then label f 2 +1 2 vertices out of f 2 + 1 vertices as 2i × 3 , where 1 ≤ i ≤ f 2 +1 2 and the remaining f 2 +1 2 vertices get the label as 2j+1 × 5, where 1 ≤ j ≤ f 2 +1 2 . likewise label f 2 − 2 vertices out of f 2 − 1 in v2 as 2i × 3, where f 2 +1 2 + 1 ≤ i ≤ z + 1 and the remaining f 2 −1 − f 2 −2 vertices as 2j+1 × 5 where f 2 +1 2 + 1 ≤ j ≤ z + 1. then the cordiality condition ∣∣eγ| (0) − eγ| (1)∣∣ = 0. proceeding like this for all values of x and z, the cordiality condition is satisfied. hence the complete bipartite is a minimal reinhard zumkeller divisor cordial graph. theorem 3.8. the tadpole tn, k is a minimal reinhard zumkeller divisor cordial graph for all values of n and k proof. let v1, v2, . . . , vn be the vertices of cycle cn and w1, w2, . . . , wk be the vertices of the path pk. let tn,k be the repercussion graph obtained by recognizing a vertex of cycle cn to an end vertex of the path pk .then the order of tn,k graph is |v (tn,k)| = n+k and the size of tn,k graph is |e (tn,k)| = n+k. concatenate the pendant vertex of pk to one of the vertices of cn with an edge in such a way that   vn+3 2 =w1 for n ≡ 1 (mod 2) if n+3 2 is even vn+3 2 −1=w1 for n ≡ 1 (mod 2) if n+3 2 is odd vn+2 2 = w1 for n ≡ 0 (mod 2) if n+22 is even vn+2 2 − 1 = w1 for n ≡ 0 (mod 2) if n+22 is odd . we contemplate the following cases.let v1, v2, . . . , vn be the vertices of cycle cnbe labelled as follows: case 1. n ≡ 1 (mod 2) and k = 1. let v1, v2, . . . , vn be the vertices of cycle cnbe labelled as follows: γ(vg) = { 2i × 3 where g ≡ 1 (mod 2) and 1 ≤ i ≤ n+1 2 2l × 3 × 5 where g ≡ 0 (mod 2) and 1 ≤ l ≤ n+1 2 − 1 (3) let the vertices of the path ph w1, w2 be labelled as follows. γ ( vn+3 2 ) = γ(w1) (4) 285 a. ruby priscilla and s. firthous fatima γ(w2) = 2 n+1 2 +1 × 3 (5) in regards to the labeling designs (3),(4),(5), we get that e|γ(0) = n+k2 ; e | γ(1) = n+k 2 . hence ∣∣∣e|γ(0) − e|γ(1)∣∣∣ = 0. case 2. n ≡ 1 (mod 2) and k ≥ 2, where k is even γ(vg) = { 2i × 3 where g ≡ 1 (mod 2) and 1 ≤ i ≤ n+1 2 2l × 3 × 5 where g ≡ 0 (mod 2) and 1 ≤ l ≤ n+1 2 − 1 (6) γ(wh) = { 2i × 3 where h ≡ 1 (mod 2) , 3 ≤ h ≤ k + 1 andn+1 2 + 1 ≤ i ≤ n+1+k 2 2l × 3 × 5 where h ≡ 0 (mod 2) , 2 ≤ h ≤ k andn+1 2 ≤ l ≤ n+1+k 2 − 1 (7) hence from (6) and (7), we get that e|γ(1) = n+k+12 ; e | γ(0) = n+k−1 2 . hence ∣∣∣e|γ(0) − e|γ(1)∣∣∣ = 1. case 3. n ≡ 0 (mod 2) and k ≥ 1 where k is odd γ(vg) = { 2i × 3 where g ≡ 1 (mod 2) and 1 ≤ i ≤ n 2 2l × 3 × 5 where g ≡ 0 (mod 2) and 1 ≤ l ≤ n 2 (8) let the vertices of the path ph, w1, w2, . . . , wh be labelled as follows, γ(wh) = { 2i × 3 where h ≡ 0 (mod 2) , 2 ≤ h ≤ k + 1 and n 2 + 1 ≤ i ≤ n+k+1 2 2l × 3 × 5 where h ≡ 1 (mod 2) , 3 ≤ h ≤ k and n 2 + 1 ≤ l ≤ n+k+1 2 − 1 (9) in regards to the above labeling design (8) and (9), we get that e|γ(1) = n+k−12 ; e | γ(0) = n+k+1 2 hence ∣∣∣e|γ(0) − e|γ(1)∣∣∣ = 1. case 4. n ≡ 1 (mod 2) where k ≥ 1 where k is odd γ(vg) = { 2i × 3 where g ≡ 1 (mod 2) and 1 ≤ i ≤ n+1 2 2l × 3 × 5 where g ≡ 0 (mod 2) and 1 ≤ l ≤ n+1 2 − 1 (10) γ(wh) = { 2i × 3 where h ≡ 1 (mod 2) , 3 ≤ h ≤ k and n+1 2 + 1 ≤ i ≤ n+k 2 2l × 3 × 5 where h ≡ 0 (mod 2) , 2 ≤ h ≤ k − 1 and n+1 2 + 1 ≤ l ≤ n+k 2 − 1 (11) γ(wk+1) = 2 2 × 5 (12) in regards to the labeling design (10), (11) and (12), we get that e|γ (1) = n+k2 ; e | γ (0) = n+k 2 hence ∣∣∣e|γ (0) − e|γ (1)∣∣∣ = 0 286 minimal reinhard zumkeller divisor cordial graphs case 5. n ≡ 0 (mod 2) where k ≥ 2 and k is even γ(vg) = { 2i × 3 where g ≡ 1 (mod 2) and 1 ≤ i ≤ n 2 2l × 3 × 5 where g ≡ 0 (mod 2) and 1 ≤ l ≤ n 2 (13) γ(wh) = { 2i × 3 where h ≡ 0 (mod 2) , 2 ≤ h ≤ k and n 2 + 1 ≤ i ≤ n+k 2 2l × 3 × 5 where h ≡ 1 (mod 2) , 3 ≤ h ≤ k − 1 and n 2 + 1 ≤ l ≤ n+k 2 − 1 (14) γ(wk+1) = 2 2 × 5 (15) in regards to the labeling design (13), (14) and (15), we get that e|γ (1) = n+k2 ; e | γ (0) = n+k 2 hence ∣∣∣e|γ (0) − e|γ (1)∣∣∣ = 0 hence from all cases we get that the tadpole tn,k is a minimal reinhard zumkeller divisor cordial graph for all values of n and k. theorem 3.9. the complete graph kn is a minimal reinhard zumkeller divisor cordial if n ≤ 16 proof. obviously k1, k2, k3 are minimal reinhard zumkeller divisor cordial graph. the following table 1 brings forth a minimal reinhard zumkeller divisor cordial labeling of kn for 4 ≤ n < 17 order of vertex labels cordiality condition kn 4 6,12,20,24 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 5 6,12,28,30,60 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 6 6,12,24,28,30,60 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 7 6,12,24,28,30,48,60 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 8 6,12,24,28,30,48,60,96 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 9 6,12,24,28,30,48,60,96,168 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 10 6,12,24,28,30,48,60,96,168,192 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 11 6,12,24,28,30,48,60,84,96,168,192 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 12 6,12,24,28,30,48,60,84,96,120,168,192 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 13 6,12,24,28,30,48,60,84,96,120,168,192,384 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 14 6,12,24,28,30,48,60,84,96,120,168,192,240,384 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 15 6,12,24,28,30,48,60,84,96,120,168,192,240,384,768 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 16 6,12,24,28,30,48,60,84,96,120,168,192,240,336,384,768 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 table 1: minimal reinhard zumkeller divisor cordial labeling of knfor 4 ≤ n < 17 k17 is not reinhard zumkeller divisor cordial graph. since by following the labeling pattern of k16 the vertex labels to be selected for the vertex v17 must be 287 a. ruby priscilla and s. firthous fatima anyone of them: 480 or 1536 or 672 or by choosing 20 instead of 28 from the above labeling pattern then the cordiality condition is ∣∣eγ| (1) − eγ| (0)∣∣ = 69 − 67 = 2 and ∣∣eγ| (0) − eγ| (1)∣∣ = 69 − 67 = 2 respectively. since the labeling pattern for each complete graph kn follows the labeling pattern of its predecessor, for all higher order complete graphs the cordiality condition increases by 1 for each n ≥ 17. 4 discussion for the notion of minimal reinhard zumkeller divisor cordial labeling ,this effort has produced several fresh findings. in order to create a minimal reinhard zumkeller divisor cordial graph and introduce a new element to the labeling pattern of various graph structures, the traits of the zumkeller number are unified with the divisor cordial graph labeling technique. the results that are established in this paper are amalgamated and motivated us to get into the conclusion that for every connected minimal reinhard zumkeller divisor cordial graph g, γ(u) ≡ 0(mod6) for some vertex u ∈ v (g) . deriving similar results for other graph families is an open problem . 5 conclusions in the present investigation, minimal reinhard zumkeller divisor cordial labeling has been introduced and probed for the existence of reinhard zumkeller divisor cordial labeling of path, cycle, star k1,s, complete graph kn for n < 17, complete bipartite and tadpole graph tn,k for all values of n and k. in future research work, we will develop findings to construct dense minimal reinhard zumkeller divisor cordial graphs, book graphs with polygonal pages, generalized petersen graphs, wheel graphs and product related graphs. acknowledgement the authors are thankful to the anonymous referees for putting forth their valuable suggestions to refine this article. references b. balamurugan and r. m. meenakshi. zumkeller labeling of complete graphs. international journal of engineering and advanced technology (ijeat), issn, pages 2249–8958, 2019. b. balamurugan, k. thirusangu, and d. thomas. zumkeller labeling of some 288 minimal reinhard zumkeller divisor cordial graphs cycle related graphs. proceedings of international conference on mathematical sciences (icms—2014), elsevier, pages 549–553, 2014. j. bondy and u. murty. graph theory (graduate texts in mathematics) springer. new york, page 244, 2008. d. m. burton. elementary number theory, wm. c, 1980. i. cahit. cordial graphs-a weaker version of graceful and harmonious graphs. ars combinatoria, 23:201–207, 1987. j. a. gallian. a dynamic survey of graph labeling. electronic journal of combinatorics, 1(dynamicsurveys):ds6, 2018. p. j. mahanta, m. p. saikia, and d. yaqubi. some properties of zumkeller numbers and k-layered numbers. journal of number theory, 217:218–236, 2020. b. murali, k. thirusangu, and b. balamurugan. zumkeller cordial labeling of cycle related graphs. international journal of pure and applied mathematics, 116(3):617–627, 2017. h. patodia and h. k. saikia. generalization of zumkeller numbers. advances in mathematics: scientific journal, 9(12):10613–10619, 2020. y. peng and k. b. rao. on zumkeller numbers. journal of number theory, 133 (4):1135–1155, 2013. a. rosa. on certain valuations of the vertices of a graph, theory of graphs (internat. symposium, rome, july 1966), 1967. a. shahbaz and m. k. mahmood. new numbers on euler’s totient function with applications. 2020. r. varatharajan, s. navanaeethakrishnan, and k. nagarajan. divisor cordial graphs. international journal of mathematical combinatorics, 4:15, 2011. 289 microsoft word r.m.7 cap.11.doc ratio mathematica 26 (2014), 03–20 issn:1592-7415 rough set theory applied to hyper bck-algebra r. ameria, r. moradianb and r. a. borzooeic aschool of mathematics, statistics and computer science, college of sciences, university of tehran, p.o. box 14155-6455, teheran, iran ameri@ut.ac.ir bdepartment of mathematics, payam noor university, tehran, iran rmoradian58@yahoo.com cdepartment of mathematics, shahid beheshti university, tehran, iran borzooei@sbu.ac.ir abstract the aim of this paper is to introduce the notions of lower and upper approximation of a subset of a hyper bck-algebra with respect to a hyper bck-ideal. we give the notion of rough hyper subalgebra and rough hyper bck-ideal, too, and we investigate their properties. key words: rough set, rough (weak, strong) hyper bck-ideal, rough hyper subalgebra, regular congruence relation. msc 2010: 20n20, 20n25. 1 introduction in 1966, y. imai and k. iseki [2] introduced a new notion, called a bckalgebra. the hyper structure theory (called also multi algebras ) was introduced in 1934 by f. marty [6] at the 8th congress of scandinavian mathematicians. in [3], y. b. jun, m. m. zahedi, x. l. xin, r. a. borzooei applied the hyper structures to bck-algebras and they introduced the notion of hyper bck-algebra (resp. hyper k-algebra) which is a generalization of bck-algebra (resp. hyper bck-algebra). they also introduced the notion of hyper bck-ideal, weak hyper bck-ideal, hyper k-ideal and weak 3 ameri, moradian, borzooei hyper k-ideal and gave relations among them. in 1982, pawlak introduced the concept of rough set (see [7]). recently jun [5] applied rough set theory to bck-algebras. in this paper, we apply the rough set theory to hyper bck-algebras. 2 preliminaries let u be a universal set. for an equivalence relation θ on u, the set of elements of u that are related to x ∈ u, is called the equivalence class of x and is denoted by [x]θ. moreover, let u/θ denote the family of all equivalence classes induced on u by θ. for any x ⊆ u, we write xc to denote the complement of x in u, that is the set u\x. a pair (u, θ) where u 6= φ and θ is an equivalence relation on u is called an approximation space. the interpretation in rough set theory is that our knowledge of the objects in u extends only up to membership in the class of θ and our knowledge about a subset x of u is limited to the class of θ and their unions. this leads to the following definition. definition 2.1. [7] for an approximation space (u, θ), by a rough approximation in (u, θ) we mean a mapping apr : p(u) −→ p(u) ×p(u) defined for every x ∈ p(u) by apr(x) = (apr(x),apr(x)), where apr(x) = {x ∈ u|[x]θ ⊆ x}, apr(x) = {x ∈ u|[x]θ ∩x 6= φ}. apr(x) is called a lower rough approximation of x in (u, θ), whereas apr(x) is called an upper rough approximation of x in (u, θ). definition 2.2. [7] given an approximation space (u, θ), a pair (a,b) ∈ p(u) ×p(u) is called a rough set in (u, θ) if and only if (a,b) = apr(x) for some x ∈ p(u). definition 2.3. ([7]) let (u, θ) be an approximation space and x be a non-empty subset of u. (i) if apr(x) = apr(x), then x is called definable. (ii) if apr(x) = φ, then x is called empty interior. 4 rough set theory applied to hyper bck-algebra (iii) if apr(x) = u, then x is called empty exterior. let h be a non-empty set endowed with a hyper operation “◦”, that is ◦ is a function from h ×h to p∗(h) = p(h) −{φ}. for two subsets a and b of h, denote by a◦b the set ⋃ a∈a,b∈b a◦ b. we shall use x◦ y instead of x◦{y}, {x}◦y, or {x}◦{y}. definition 2.4. ([3]) by a hyper bck-algebra we mean a nonempty set h endowed with a hyper operation “◦”and a constant 0 satisfying the following axioms: (hk1) (x◦z) ◦ (y ◦z) � x◦y, (hk2) (x◦y) ◦z = (x◦z) ◦y, (hk3) x◦h �{x}, (hk4) x � y and y � x imply x = y, for all x,y,z ∈ h, where x � y is defined by 0 ∈ x◦y and for every a,b ⊆ h, a � b is defined by ∀a ∈ a,∃b ∈ b such that a � b. in such case, we call “�”the hyper order in h. theorem 2.5. ([3]) in any hyper bck-algebra h, the following hold: (a1) 0 ◦ 0 = {0}, (a2) 0 � x, (a3) x � x, (a4) a � a, (a5) a � 0 implies a = {0}, (a6) a ⊆ b implies a � b, (a7) 0 ◦x = {0}, (a8) x◦y � x, (a9) x◦ 0 = {x}, (a10) y � z implies x◦z � x◦y, (a11) x◦y = {0} implies (x◦z) ◦ (y ◦z) = {0} and x◦z � y ◦z, (a12) a◦{0} = {0} implies a = {0}, for all x,y,z ∈ h and for all non-empty subsets a and b of h. 5 ameri, moradian, borzooei definition 2.6. ([3]) let h be a hyper bck-algebra and let s be a subset of h containing 0. if s be a hyper bck-algebra with respect to the hyper operation “◦”on h, we say that s is a hyper subalgebra of h. theorem 2.7. ([3]) let s be a non-empty subset of hyper bck-algebra h. then s is a hyper subalgebra of h if and only if x◦y ⊆ s, for all x,y ∈ s. definition 2.8. ([3]) let i be a non-empty subset of hyper bck-algebra h and 0 ∈ i. (i) i is said to be a hyper bck-ideal of h if x◦ y � i and y ∈ i imply x ∈ i for all x,y ∈ h. (ii) i is said to be a weak hyper bck-ideal of h if x ◦ y ⊆ i and y ∈ i imply x ∈ i for all x,y ∈ h. (iii) i is called a strong hyper bck-ideal of h if (x◦y) ∩ i 6= φ and y ∈ i imply x ∈ i for all x,y ∈ h. theorem 2.9. ([3]) if h be a hyper bck-algebra, then (i) every hyper bck-ideal of h is a weak hyper bck-ideal of h. (ii) every strong hyper bck-ideal of h is a hyper bck-ideal of h. definition 2.10. ([4]) let h be a hyper bck-algebra. a hyper bckideal i of h is called reflexive if x◦x ⊆ i for all x ∈ h. definition 2.11. ([1]) let θ be an equivalence relation on hyper bckalgebra h and a,b ⊆ h. then, (i) aθb means that, there exist a ∈ a and b ∈ b such that aθb, (ii) aθ̄b means that, for all a ∈ a there exists b ∈ b such that aθb and for all b ∈ b there exists a ∈ a such that aθb, (iii) θ is called a congruence relation on h, if xθy and x′θy′ imply x ◦ x′θ̄y ◦y′ for all x,y,x′,y′ ∈ h. (iv) θ is called a regular relation on h, if x◦yθ{0} and y ◦xθ{0} imply xθy for all x,y ∈ h. 6 rough set theory applied to hyper bck-algebra example 2.12. let h1 = {0, 1, 2}, h2 = {0,a,b} and hyper operations “◦1”and “◦2”on h1 and h2 are defined respectively, as follow: ◦1 0 1 2 0 {0} {0} {0} 1 {1} {0} {1} 2 {2} {2} {0, 2} ◦2 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,b} then (h1,◦1) and (h2,◦2) are hyper bck-algebras. define the equivalence relation θ1 and θ2 on h1 and h2, respectively, as θ1 = {(0, 0), (1, 1), (2, 2), (0, 2), (2, 0)}, and θ2 = {(0, 0), (a,a), (b,b), (0,a), (a, 0)}. it is easily checked that θ1 is a congruence relation on h1. but θ2 is not a congruence relation on h2, since bθ2b and 0θ2a but b◦ 0θ̄2b◦a is not true. example 2.13. let (h1,◦1) be a hyper bck-algebra as example 2.12. let h2 = {0,a,b,c} and define the hyper operation “◦2”on h2 as follow: ◦2 0 a b c 0 {0} {0} {0} {0} a {a} {0,a} {0} {a} b {b} {b} {0,a} {b} c {c} {c} {c} {0,c} then (h2,◦2) is a hyper bck-algebra. define the congruence relation θ1 and θ2 on h1 and h2, respectively, by θ1 = {(0, 0), (1, 1), (2, 2), (0, 1), (1, 0)}, and θ2 = {(0, 0), (a,a), (b,b), (c,c), (0,b), (b, 0)}. it is easily checked that θ1 is a regular congruence relation on h1, but θ2 is not a regular relation on h2, since a◦bθ2{0} and b◦aθ2{0} but (a,b) 6∈ θ2. theorem 2.14. ([1]) let θ be a regular congruence relation on hyper bck-algebra h. then [0]θ is a hyper bck-ideal of h. 7 ameri, moradian, borzooei theorem 2.15. ([1]) let θ be a regular congruence relation on h,i = [0]θ and h i = {ix : x ∈ h}, where ix = [x]θ for all x ∈ h. then hi with hyper operation “◦”and hyper order “<”which is defined as follow, is a hyper bckalgebra which is called quotient hyper bck-algebra, ix ◦ iy = {iz : z ∈ x◦y}, and ix < iy ⇐⇒ i ∈ ix ◦ iy. theorem 2.16. ([1]) let i be a reflexive hyper bck-ideal of h and relation θ on h be defined as follow: xθy ⇐⇒ x◦y ⊆ i and y ◦x ⊆ i for all x,y ∈ h. then θ is a regular congruence relation on h and i = [0]θ. 3 rough hyper bck-ideals throughout this section h is a hyper bck-algebra. in this section first we define lower and upper approximation of the subset a of h with respect to hyper bck-ideal of h and prove some properties. then we give the definition of (weak, strong) rough hyper bck-ideals and investigate the relation between them and (weak, strong) hyper bck-ideals of h. definition 3.1. let θ be a regular congruence relation on hyper bckalgebra h, i = [0]θ, ix = [x]θ and a be a non-empty subset of h. then the sets apr i (a) = {x ∈ h|ix ⊆ a}, apri (a) = {x ∈ h|ix ∩a 6= φ}. are called lower and upper approximation of the set a with respect to the hyper bck-ideal i, respectively. proposition 3.2. for every approximation space (h, θ) and every subsets a,b ⊆ h, we have: (1) apr i (a) ⊆ a ⊆ apri (a), (2) apr i (φ) = φ = apri (φ), 8 rough set theory applied to hyper bck-algebra (3) apr i (h) = h = apri (h), (4) if a ⊆ b, then apr i (a) ⊆ apr i (b) and apri (a) ⊆ apri (b), (5) apr i (apr i (a)) = apr i (a), (6) apri (apri (a)) = apri (a), (7) apri (apri (a)) = apri (a), (8) apr i (apri (a)) = apri (a), (9) apr i (a) = (apri (a c))c, (10) apri (a) = (apri (a c))c, (11) apri (a∩b) ⊆ apri (a) ∩apri (b), (12) apr i (a∩b) = apr i (a) ∩apr i (b), (13) apri (a∪b) = apri (a) ∪apri (b), (14) apr i (a∪b) ⊇ apr i (a) ∪apr i (b), (15) apr i (ix) = h = apri (ix) for all x ∈ h. proof. (1), (2) and (3) are straightforward. (4) for any x ∈ apr i (a) we have ix ⊆ a ⊆ b and so x ∈ apri (b). now, suppose that x ∈ apri (a). then ix ∩a 6= φ and so ix ∩b 6= φ. hence x ∈ apri (b). (5) since apr i (a) ⊆ a, by (4) we have apr i (apr i (a)) ⊆ apr i (a). now, let x ∈ apr i (a). then ix ⊆ a. since for any y ∈ ix, we have ix = iy, then iy ⊆ a and so y ∈ apri (a). therefore, ix ⊆ apri (a) and then we obtain x ∈ apr i (apr i (a)). (6) by (1) and (4), apri (a) ⊆ apri (apri (a)). on the other hand, we assume that x ∈ apri (apri (a)). then we have ix ∩apri (a) 6= φ and so there exist a ∈ ix and a ∈ apri (a). hence ia = ix and ia ∩a 6= φ which imply ix ∩a 6= φ. therefore, x ∈ apri (a). 9 ameri, moradian, borzooei (7) by (1), we have apr i (a) ⊆ apri (apri (a)). now, let x ∈ apri (apri (a)). then ix ∩ apri (a) 6= φ and so there exist a ∈ ix and a ∈ apri (a). hence ia = ix and ia ⊆ a which imply ix ⊆ a. therefore, x ∈ apr i (a). (8) by (1), we have apr i (apri (a)) ⊆ apri (a). now, we assume that x ∈ apri (a). then ix ∩ a 6= φ. for every y ∈ ix, we have iy = ix and so iy ∩ a 6= φ. hence y ∈ apri (a) which implies ix ⊆ apri (a). therefore, x ∈ apr i (apri (a)). (9) for any subset a of h we have: (apri (a c))c = {x ∈ h : x 6∈ apri (a c)} = {x ∈ h : ix ∩ac = φ} = {x ∈ h : ix ⊆ a} = {x ∈ h : x ∈ apr i (a)} = apr i (a). (10) for any subset a of h we have: (apr i (ac))c = {x ∈ h : x 6∈ apr i (ac)} = {x ∈ h : ix 6⊂ ac} = {x ∈ h : ix ∩a 6= φ} = {x ∈ h : x ∈ apri (a)} = apri (a). (11) since a∩b ⊆ a and a∩b ⊆ b, then by (4), apri (a∩b) ⊆ apri (a) and apri (a∩b) ⊆ apri (b). hence apri (a∩b) ⊆ apri (a)∩apri (b). 10 rough set theory applied to hyper bck-algebra (12) for any subset a and b of h we have: x ∈ apr i (a∩b) ⇐⇒ ix ⊆ a∩b ⇐⇒ ix ⊆ a and ix ⊆ b ⇐⇒ x ∈ apr i (a) and x ∈ apr i (b) ⇐⇒ x ∈ apr i (a) ∩apr i (b). (13) for any subset a and b of h we have x ∈ apri (a∪b) ⇐⇒ ix ∩ (a∪b) 6= φ ⇐⇒ (ix ∩a) ∪ (ix ∩b) 6= φ ⇐⇒ ix ∩a 6= φ or ix ∩b 6= φ ⇐⇒ x ∈ apri (a) or x ∈ apri (b) ⇐⇒ x ∈ apri (a) ∪apri (b). (14) since a ⊆ a∪b and b ⊆ a∪b, then by (4), apr i (a) ⊆ apr i (a∪b) and apr i (b) ⊆ apr i (a∪b), which imply that apr i (a) ∪apr i (b) ⊆ apr i (a∪b). (15) the proof is straightforward. corollary 3.3. let (h, θ) be an approximation space. then (i) for every a ⊆ h, apr i (a) and apri (a) are definable sets, (ii) for every x ∈ h,ix is definable set. proof. (i) by proposition 3.2 (5) and (7), we have apr i (apr i (a)) = apr i (a) = apri (apri (a)). hence apri (a) is a definable set. on the other hand by proposition 3.2 (6) and (8), we have apri (apri (a)) = apri (a) = apr i (apri (a)). therefore apri (a) is a definable set. (ii) by proposition 3.2 (15) the proof is clear. 11 ameri, moradian, borzooei theorem 3.4. let θ be a regular congruence relation on h, i = [0]θ be a hyper bck-ideal of h and a,b are non-empty subsets of h. then (i) apri (a) ◦apri (b) = apri (a◦b), (ii) apr i (a) ◦apr i (b) ⊆ apr i (a◦b). proof. (i) let z ∈ apri (a) ◦apri (b). then there exist a ∈ apri (a) and b ∈ apri (b) such that z ∈ a◦b. hence ia ∩a 6= φ and ib ∩b 6= φ and so there exist c ∈ ia ∩a and d ∈ ib ∩b such that aθc and bθd. since θ is a congruence relation on h, then we have a◦bθ̄c◦d and because z ∈ a ◦ b, then there exist y ∈ c ◦ d such that zθy. hence y ∈ iz. on the other hand, y ∈ c◦d ⊆ a◦b which implies iz ∩ (a◦b) 6= φ and so z ∈ apri (a◦b). therefore apri (a)◦apri (b) ⊆ apri (a◦b). now, suppose that x ∈ apri (a ◦ b). then ix ∩ (a ◦ b) 6= φ. let z ∈ ix ∩ (a◦b), then there exist a ∈ a and b ∈ b such that z ∈ a◦ b and ix = iz. thus we have iz ∈ ia ◦ ib and so ix ∈ ia ◦ ib. hence x ∈ a ◦ b ⊆ a ◦ b ⊆ apri (a) ◦ apri (b). therefore, apri (a ◦ b) ⊆ apri (a) ◦apri (b). 2 (ii) let z ∈ apr i (a) ◦ apr i (b). then there exist a ∈ apr i (a) and b ∈ apr i (b) such that z ∈ a◦ b, ia ⊆ a and ib ⊆ b. for every y ∈ iz, we have iz = iy ∈ ia ◦ib and so y ∈ a◦b ⊆ a◦b. then y ∈ a◦b and so iz ⊆ a◦b. therefore z ∈ apri (a◦b). example 3.5. let h = {0, 1, 2} and define the hyper operation “◦”on h as follow: ◦ 0 1 2 0 {0} {0} {0} 1 {1} {0} {1} 2 {2} {2} {0, 2} then (h,◦) is a hyper bck-algebra. define the equivalence relation θ by θ = {(0, 0), (1, 1), (2, 2), (0, 1), (1, 0)}. then θ is a regular congruence relation on h and so we have: i = [0]θ = {0, 1},i1 = [1]θ = {0, 1},i2 = [2]θ = {2}. 12 rough set theory applied to hyper bck-algebra now, if we let a = {1, 2} and b = {0, 2}, then we have a◦b = {0, 1, 2} and so apr i (a) = {x ∈ h|ix ⊆ a} = {2}, apri (a) = {x ∈ h|ix ∩a 6= φ} = {0, 1, 2}, apr i (b) = {x ∈ h|ix ⊆ b} = {2}, apri (b) = {x ∈ h|ix ∩b 6= φ} = {0, 1, 2}, apr i (a◦b) = {x ∈ h|ix ⊆ a◦b} = {0, 1, 2}, apri (a◦b) = {x ∈ h|ix ∩ (a◦b) 6= φ} = {0, 1, 2}, apri (a) ◦apri (b) = {0, 1, 2}, apr i (a) ◦apr i (b) = {0, 2}. therefore, we see that apr i (a) ◦ apr i (b) 6= apr i (a ◦ b) but apri (a) ◦ apri (b) = apri (a◦b). definition 3.6. let θ be a regular congruence relation on h, i = [0]θ be a hyper bck-ideal of h and a be a non-empty subset of h. if apr i (a) and apri (a) are hyper subalgebra of h, then a is called a rough hyper subalgebra of h. theorem 3.7. if i be a hyper bck-ideal and j be a hyper subalgebra of h, then (i) apri (j) is a hyper subalgebra of h. (ii) if i ⊆ j, then apr i (j) is a hyper subalgebra of h. proof. (i) since 0 ∈ j ⊆ apri (j), then apri (j) 6= φ. now, we assume that x,y ∈ apri (j). we must prove that x ◦ y ⊆ apri (j). since ix ∩ j 6= φ and iy ∩ j 6= φ, we can let t ∈ ix ∩ j, s ∈ iy ∩ j and z ∈ x◦ y. hence iz ∈ ix ◦ iy = it ◦ is and so z ∈ t◦ s ⊆ j. thus we have z ∈ j and z ∈ iz and so iz ∩j 6= φ. therefore, z ∈ apri (j) and so x◦y ⊆ apri (j). (ii) since i = i0 ⊆ j, we have 0 ∈ apri (j) 6= φ. now, suppose that a,b ∈ apr i (j). then ia ⊆ j and ib ⊆ j. for every z ∈ a◦b and every y ∈ iz, we have iz = iy ∈ ia ◦ ib and so y ∈ a◦ b ⊆ j. hence iz ⊆ j, which implies that z ∈ apr i (j). therefore, a◦ b ⊆ apr i (j). 13 ameri, moradian, borzooei theorem 3.8. let θ and φ be two regular congruence relations on h and i = [0]θ, j = [0]φ be two hyper bck-ideals of h such that i ⊆ j. then for any nonempty subset a of h, we have: (i) apr j (a) ⊆ apr i (a), (ii) apri (a) ⊆ aprj (a). proof. (i) first we show that if i ⊆ j, then ix ⊆ jx. let y ∈ ix. then xθy. since θ is a congruence relation on h and xθx, then x◦xθ̄x◦y. since 0 ∈ x ◦ x, then there exist t ∈ x ◦ y such that 0θt and so t ∈ [0]θ = i ⊆ j = [0]φ. thus by hypothesis, t ∈ [0]φ and so x◦yφ{0}. by the similar way, we can show that y ◦xφ{0}. since φ is a regular congruence relation, we get xφy and so y ∈ [x]φ = jx. therefore, ix ⊆ jx. now, let x ∈ aprj (a). then jx ⊆ a and so ix ⊆ a which implies x ∈ apr i (a). (ii) assume that x ∈ apri (a). then ix ∩a 6= φ. since ix ⊆ jx, we have jx ∩a 6= φ. therefore, x ∈ aprj (a). corollary 3.9. let θ and φ are two regular congruence relations on h, i = [0]θ, j = [0]φ be two hyper bck-ideals of hyper bck-algebra h and a be a non-empty subset of h. then (i) apr i (a) ∩apr j (a) ⊆ apr i∩j (a), (ii) apri∩j (a) ⊆ apri (a) ∩aprj (a). proof. by theorem 3.8, the proof is clear. definition 3.10. let θ be a regular congruence relation on h, i = [0]θ be a hyper bck-ideal of h, a be a non-empty subset of h and apri (a) = (apr i (a),apri (a)) be a rough set in the approximation space (h, θ). if apr i (a) and apri (a) are hyper bck-ideals (resp, weak, strong) of h, then a is called a rough hyper bck-ideal (resp, weak, strong) of h. 14 rough set theory applied to hyper bck-algebra example 3.11. let h = {0, 1, 2, 3} and hyper operation “◦”on h is defined as follow: ◦ 0 1 2 3 0 {0} {0} {0} {0} 1 {1} {0, 1} {0} {1} 2 {2} {2} {0, 1} {2} 3 {3} {3} {3} {0, 3} then (h,◦, 0) is a hyper bck-algebra. we define the regular congruence relation on h as follow: θ = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 1), (1, 0)}. so we have: i = i0 = i1 = {0, 1},i2 = {2},i3 = {3}. now, let a = {0, 1, 3} be a subset of h, then apr i (a) = {x ∈ h|ix ⊆ a} = {0, 1, 3}, apri (a) = {x ∈ h|ix ∩a 6= φ} = {0, 1, 3}. easily we give that apr i (a) and apri (a) are hyper bck-ideals. therefore, a is a rough hyper bck-ideal of h. example 3.12. let h = {0,a,b,c}. by the following table (h,◦) is a hyper bck-algebra. ◦ 0 a b c 0 {0} {0} {0} {0} a {a} {0,a} {0} {a} b {b} {b} {0,a} {b} c {c} {c} {c} {0,c} now, let relation θ on h is defined as follow: θ = {(0, 0), (a,a), (b,b), (c,c), (0,b), (b, 0), (0,a), (a, 0), (a,b), (b,a)}. then, i0 = ia = ib = {0,a,b},ic = {c}. let j1 = {0,c}, j2 = {0,b} and j3 = {c}. then, apr i (j1) = {c},apri (j1) = {0,a,b,c}, apr i (j2) = {},apri (j2) = {0,a,b}, apr i (j3) = {c},apri (j3) = {c}. 15 ameri, moradian, borzooei hence we can see that j1 is a hyper bck-ideal of h but apri (j1) is not a hyper bck-ideal. moreover j2 is not a hyper bck-ideal but apri (j2) is a hyper bck-ideal of h. in follows, j3 is not a hyper bck-ideal and neither apr i (j3) nor apri (j3) is a hyper bck-ideal of h. theorem 3.13. let θ be a regular congruence relation on h and i = [0]θ be a hyper bck-ideal of h. then (i) if j be a weak hyper bck-ideal of h containing i, then apr i (j) is a weak hyper bck-ideal of h, (ii) if j be a hyper bck-ideal of h containing i, then apr i (j) is a hyper bck-ideal of h, (iii) if j be a strong hyper bck-ideal of h containing i, then apr i (j) is a strong hyper bck-ideal of h. proof. (i) since i = i0 ⊆ j, then 0 ∈ apri (j). now, let x,y ∈ h be such that x ◦ y ⊆ apr i (j) and y ∈ apr i (j). we must prove that ix ⊆ j. let a ∈ ix and b ∈ iy. then aθx and bθy. since θ is a congruence relation on h, we have a◦bθx◦y and so for every z ∈ a◦b, there exist t ∈ x◦ y such that zθt. since x◦ y ⊆ apr i (j), we have t ∈ apr i (j) and so it = iz ⊆ j which implies z ∈ j. thus a ◦ b ⊆ j. on the other hand, b ∈ iy ⊆ j. since j is a weak hyper bck-ideal, we have a ∈ j and so ix ⊆ j. hence x ∈ apri (j). therefore, apri (j) is a weak hyper bck-ideal of h. (ii) let x,y ∈ h be such that x ◦ y � apr i (j) and y ∈ apr i (j). we must prove that ix ⊆ j. let a ∈ ix and b ∈ iy. then aθx and bθy. since θ is a congruence relation on h, we have a◦ bθx◦ y and so for every z ∈ a ◦ b, there exist z′ ∈ x ◦ y such that zθz′. since z′ ∈ x ◦ y � apr i (j), then there exists t ∈ apr i (j) ⊆ j such that z′ � t and so from zθz′, we have i0 ∈ iz′ ◦it = iz ◦it. hence 0 ∈ z◦ t and then z � t. thus we have proved that for every z ∈ a ◦ b, there exist t ∈ j such that z � t which means that a◦b � j. on the other hand we have b ∈ iy ⊆ j. since j is a hyper bck-ideal of h, we 16 rough set theory applied to hyper bck-algebra have a ∈ j. thus ix ⊆ j which implies that x ∈ apri (j). therefore, apr i (j) is a hyper bck-ideal of h. (iii) suppose that x,y ∈ h be such that (x ◦ y) ∩ apr i (j) 6= φ and y ∈ apr i (j). let a ∈ ix and b ∈ iy. then aθx and bθy. since θ is a congruence relation on h, we have a◦bθx◦y. since (x◦y)∩apr i (j) 6= φ, then there exist t ∈ h such that t ∈ x ◦ y and t ∈ apr i (j). now, t ∈ x◦yθa◦ b implies that there exist z ∈ a◦ b such that zθt and so it = iz ⊆ j. hence z ∈ j and so (a◦b)∩j 6= φ. on the other hand, we have b ∈ iy ⊆ j. since j is a strong hyper bck-ideal of h, then we have a ∈ j which implies ix ⊆ j that means x ∈ apri (j). therefore, apr i (j) is a strong hyper bck-ideal of h. theorem 3.14. suppose that i be a hyper bck-ideal of h and θ be a regular congruence relation on h which is defined as follow: xθy ⇔ x◦y ⊆ i and y ◦x ⊆ i. (i) if j be a weak hyper bck-ideal of h containing i, then apri (j) is a weak hyper bck-ideal of h, (ii) if j be a hyper bck-ideal of h containing i, then apri (j) is a hyper bck-ideal of h, (iii) if j be a strong hyper bck-ideal of h containing i, then apri (j) is a strong hyper bck-ideal of h. proof. (i) since i ⊆ j ⊆ apri (j), then we have 0 ∈ apri (j). let x,y ∈ h be such that x◦y ⊆ apri (j) and y ∈ apri (j). then iy ∩j 6= φ and for every z ∈ x◦y, we have z ∈ apri (j) which means iz ∩j 6= φ. thus there exist a,b ∈ h such that a ∈ iy ∩ j and b ∈ iz ∩ j which imply that aθy, bθz and a,b ∈ j. thus y◦a ⊆ i ⊆ j and z◦b ⊆ i ⊆ j and so we get y,z ∈ j, since j is a weak hyper bck-ideal. thus we have proved that for any z ∈ x◦y, we have z ∈ j and so x◦y ⊆ j. since j is a weak hyper bck-ideal and y ∈ j, obviously we have x ∈ j. since x ∈ ix, then ix ∩ j 6= φ. therefore x ∈ apri (j) and so apri (j) is a weak hyper bck-ideal of h. 17 ameri, moradian, borzooei (ii) let x,y ∈ h be such that x ◦ y � apri (j) and y ∈ apri (j). then iy ∩j 6= φ and for every z ∈ x◦ y, there exist t ∈ apri (j) such that z � t and it ∩j 6= φ. thus, there exist c,d ∈ h such that c ∈ it ∩j and d ∈ iy ∩ j and so cθt, dθy and c,d ∈ j. hence t ◦ c ⊆ i ⊆ j and y ◦ d ⊆ i ⊆ j. since j is a hyper bck-ideal and c,d ∈ j, we have y,t ∈ j. thus, we have proved that for every z ∈ x ◦ y, there exist t ∈ j such that z � t which means that x◦ y � j and so from y ∈ j we get x ∈ j. consequently, ix ∩ j 6= φ and so x ∈ apri (j). therefore, apri (j) is a hyper bck-ideal. (iii) let x,y ∈ h be such that (x ◦ y) ∩ apri (j) 6= φ and y ∈ apri (j). then iy ∩ j 6= φ and so there exist z ∈ h such that z ∈ x ◦ y and z ∈ apri (j). hence iz ∩j 6= φ and so there exist c,d ∈ h such that c ∈ iz ∩j and d ∈ iy ∩j. hence cθz and dθy where c,d ∈ j. thus we have z◦c ⊆ i ⊆ j and y◦d ⊆ i ⊆ j. since j is a strong hyper bckideal and c,d ∈ j, we have z ∈ j and y ∈ j. thus we have proved that (x◦y) ∩j 6= φ and y ∈ j. since j is a strong hyper bck-ideal, we have x ∈ j and so ix ∩j 6= φ which means that apri (j) is a strong hyper bck-ideal of h. 4 conclusion this paper is intend to built up connection between rough sets and hyper bck-algebras. we have presented a definition of the lower and upper approximation of a subset of a hyper bck-algebra with respect to a hyper bck-ideal. this definition and main results are easily extended to other algebraic structures such as hyper k-algebra, hyper i-algebra, etc. acknowledgements this work was partially supported by ”center of excellence of algebraic hyperstructures and its applications of tarbiat modares university (ceaha)” and ”research center in algebraic hyperstructures and fuzzy mathematics, university of mazandaran, babolsar, iran”. 18 rough set theory applied to hyper bck-algebra references [1] r. a. borzooei and h. harizavi, regular congrucence relation on hyper bck-algebra, sci. math. jpn., 61(1)(2005), 83-98. [2] y. imai, k. iseki, on axiom system of propositional calculi xiv, proc. japan academy, 42(1966), 19-22. [3] y. b. jun, m. m. zahedi, x. l. xin, r. a. borzooei, on hyper bckalgebra, italian journal of pure and applied mathematics, no. 10(2000), 127-136. [4] y. b. jun, x. l. xin, e. h. roh and m. m. zahedi, strong hyper bckideals of hyper bck-algebra, mathematicae japonicae, vol. 51, no. 3(2000), 493-498. [5] y. b. jun, roughness of ideals in bck-algebras, scientiae mathematicae japonicae, 57, no. 1(2003), 165-169. [6] f. marty, surune generalization de la notion de groups, 8th congress math. scandinaves, stockhholm, (1934). 45-49. [7] z. pawlak, rough sets, internet. j. comput. inform. sci., 11(1982) 341356. kluwer academic publishing, dorderecht(1991). 19 ratio mathematica vol. 33, 2017, pp. 47-60 issn: 1592-7415 eissn: 2282-8214 a brief survey on the two different approaches of fundamental equivalence relations on hyperstructures nikolaos antampoufis∗, šarka hošková-mayerovᆠ‡doi:10.23755/rm.v33i0.388 this paper is dedicated to prof. thomas vougiouklis lifetime work. abstract fundamental structures are the main tools in the study of hyperstructures. fundamental equivalence relations link hyperstructure theory to the theory of corresponding classical structures. they also introduce new hyperstructure classes.the present paper is a brief reference to the two different approaches to the notion of the fundamental relation in hyperstructures. the first one belongs to koskas, who introduced the β∗ relation in hyperstructures and the second approach to vougiouklis, who gave the name fundamental to the resulting quotient sets. the two approaches, the necessary definitions and the theorems for the introduction of the fundamental equivalence relation in hyperstructures, are presented. keywords: fundamental equivalence relations, strongly regular relation, hyperstructures, hv structures. 2010 ams subject classifications: 20n20, 01a99. ∗lyceum of avdera, p. christidou 42, 67132 xanthi, greece; antanik@otenet.gr. †department of mathematics and physics, university of defence brno, czech republic; sarka.mayerova@unob.cz ‡ c©nikolaos antampoufis and šarka hošková-mayerová. received: 31-10-2017. accepted: 26-12-2017. published: 31-12-2017. 47 nikolaos antampoufis and sarka hoskova-mayerova 1 introduction dealing with classical algebraic structures often leads to the study of the behaviours of the elements of these sets with respect to the introduced operation(s). this study focuses, very often, on looking for elements with similar behaviour. therefore, the use of the quotient set is intertwined with the search for regularity and symmetry between elements of algebraic structures and ’similar’ algebraic structures too. it is well known that ”... the most powerful tool in order to obtain a stricter structure from a given one is the quotient out procedure. to use this method in ordinary algebraic domains, one needs special equivalence relations. if one suggests a method that can applied for every equivalence relation, has to use the hyperstructures” [6]. in the commutative algebra, many problems of algebraic structures are not always visible, resulting in a large number of questions and obstacles appearing in the non-commutative algebra. for example, in classical theory if g is a group and h ⊆ g is a subgroup, then g/h quotient is a group only when h is a normal subgroup. this obstacle [21] is overcome by the definition of fr. marty (1934) [17], since ”if g is a group and h ⊆ g is a subgroup of it, then the quotient g/h is a hypergroup.” the previous proposition is generalized by the definition [26] of the weak hyperstructures by th. vougiouklis (1990), as follows: ”if g is a group and s is any partition of g, then the quotient g/h is a hv-group”. in these cases, the quotient set functioned as a process that led to ’looser’ structures than classic algebraic ones, but increased complexity. the utility of outmost importance of the quotient set in hyperstructures is its use as a bridge between classical structures and hyperstructures. in 1970, this connection was achieved by m. koskas [16] using the β relation and its transitive closure. observing the similar behaviour of elements belonging to the same hyperproduct leads to the introduction of the β relation which, clearly, is reflective, symmetric but not always transitive. the next step is to use the transitive closure of β to obtain equivalence relation and partition in equivalence classes. using the usual definition of operations between classes, we return to classical algebraic structures. this relation studied mainly by corsini [5], vougiouklis [25], davvaz [8], leoreanou-fotea [7], freni [12], migliorato [19] and many others. the quotient set not only links the hyperstructures with the classical structures 48 a brief survey on the two different approaches of fundamental equivalence relations on hyperstructures as a bridge, but also enhances the view of hyperstructures as a generalization of the corresponding classical algebraic structures. in this way, as reported in [22], the algebraic structures are contained in the corresponding hyperstructures as subcases. it seems that fr. marty defined the hypergroup replacing the axiom of the existence of a unitary and inverse element with the axiom of reproduction because he had ”sensed” this connection and chose the widest possible generalization in order to create space for the introduction of new types of hypergroups. in 1988 at a congress in italy, th. vougiouklis presents a paper titled ”how a hypergroup hides a group” [22], [27] and finds out that a) vougiouklis, eighteen years after koskas worked on the same subject without having knowledge of his work. b) much of the study was completed with the work of koskas and especially p. corsini and his school. c) vougiouklis approach was different from that of koskas and the others. in the following we will present the two different approaches. 2 preliminaries in a set h 6= ∅ equipped with a hyperoperation (·) : h × h → ℘∗(h) we abbreviate by wass the weak associativity: x · (y ·z) ∩ (x ·y) ·z 6= ∅,∀(x,y,z) ∈ h3. cow the weak commutativity: x ·y ∩y ·x 6= ∅,∀(x,y) ∈ h2. definition 2.1. the hyperstructure (h, ·) is called hv-semigroup if it is wass and it is called hv-group if it is reproductive hv-semigroup, that is xh = hx = h,∀x ∈ h. definition 2.2. the hyperstructure (h, ·) is called semihypergroup if x · (y ·z) = (x ·y) ·z,∀(x,y,z) ∈ h3 and it is called hypergroup if it is reproductive semihypergroup. definition 2.3. a hv-group is called hb-group if its hyperoperation contains operation which define a group. we define analogously hb-ring, hb-vector space. definition 2.4. let (h, ·) be a hv structure. an element e ∈ h is called identity if x ∈ ex∩xe,∀x ∈ h. we define analogously the left (right) identity. definition 2.5. let φ : h → h/β∗ be the fundamental map of a hv-group then, the kernel of φ is called core and it is denoted by ωh . definition 2.6. a hv-semigroup or a semihypergroup h is called cyclic if there exists s ∈ h, called generator, such that: h = s1∪s2∪...∪sn∪...,n ∈ n,n > 0. for more definitions and applications on hv-structures, see also the papers [4], [9], [10], [14], [15], [18], [20], [28]. 49 nikolaos antampoufis and sarka hoskova-mayerova 3 the two approaches of fundamental relations searching for the quotient set, the definition of the relation between the elements of the hyperstructure plays an important role. the observation of a hyperoperation leads to the conclusion that elements belonging to the same hyperproduct act in a similar way with respect to the hyperoperation. this observation is the basis for defining the relation β. this definition is common to both approaches. another common finding of the two approaches is the fact that the relation β is reflective, symmetric but not always transitive. the need for an equivalence relation that produces a quotient set such that it is a classical algebraic structure, makes it necessary to consider the β∗ relation that is the transitive closure of the β relation and is, obviously, an equivalence relation. the last common point of the two approaches is the search for the smallest (with respect to the inclusion) equivalence relation having as quotient set the corresponding algebraic structure. 3.1 koskas approach koskas, in his approach, introduces the equivalence relation that obtains as quotient set the corresponding algebraic structure by using the strongly regular equivalence relation. it is then shown that the transitive closure of the β relation is the smallest strongly regular equivalence relation, i.e. β∗ is the targeted relation. the proof is completed by the obvious finding that β∗ is the desired equivalence relation such that the h/β∗ is the corresponding algebraic structure. one can say that koskas approach is a deductive way of defining fundamental relation on hyperstructures, since he starts considering a general definition and then specifying the β∗-relation as a subcase. taking into consideration the approach, the necessary definitions and theorems are mentioned, having as main sources the books [5], [7], [8]. definition 3.1. let (h,◦) be a hypergroupoid, a,b elements of h and ρ be an equivalence relation on h. then ρ is strongly regular on the left if the following implication holds: aρb ⇒∀u ∈ h,∀x ∈ u◦a,∀y ∈ u◦ b : xρy. similarly, the strong regularity on the right can be defined. we call ρ strongly regular if it is strongly regular on both the left and the right. definition 3.2. let (h, ·) be a semihypergroup and n > 1 be a natural number. we define the βn relation as follows: xβny if there exist a1,a2, ...,an elements of h, so subsets {x,y}⊆ n∏ i=1 ai. 50 a brief survey on the two different approaches of fundamental equivalence relations on hyperstructures and let β = ⋃ n≥1 βn, where β1 = {(x,x)/x ∈ h} is the diagonal relation on h. notice that relation β is reflexive and symmetric but, generally, not a transitive one. definition 3.3. we denote β∗ the transitive closure of relation β. theorem 3.1. β∗ is the smallest strongly regular equivalence relation on h with respect to the inclusion. theorem 3.2. let (h, ·) be a semihypergroup (hypergroup), then the transitive closure of relation β is the smallest equivalence relation such that the quotient h/β∗is a semigroup (group). definition 3.4. β∗ is called the fundamental relation on h and h/β∗ is called the fundamental semigroup (group). notice that [22] the term fundamental, given by vougiouklis, is subsequent of koskas definitions but totally used nowadays. 3.2 vougiouklis approach unlike the previous ones, vougiouklis approaches the issue in a straightforward way starting with the acquired question about the appropriate equivalence relations. he defines the relation that has as quotient set the corresponding algebraic structure. he then defines the relation β in a more general manner than previously defined. the approach has been completed by proving that the fundamental relation is no other than the transitive closure of the relation β. we can assume that vougiouklis approach is an inductive way of defining the fundamental relation in hyperstructures because it starts with the partial and ends in a more general result. it is important to note that vougiouklis definitions were given for hv-groups which are a wider class than the one of hypergroups. also, the proof of theorem about β∗ relation (see below) follows a remarkable strategy [3]. taking into consideration the approach, the necessary definitions and theorems are mentioned, having as main source the book [25] and the papers [24], [26]. definition 3.5. let (h, ·) be a hv-group. the relation β∗ on h is called fundamental equivalence relation if it is the smallest equivalence relation on h such that the quotient set h/β∗ is a group, called fundamental group of h. 51 nikolaos antampoufis and sarka hoskova-mayerova notice that the proof of the fundamental groups existence for any hv-group is an obvious result of the following theorem’s proof. let us denote by u the set of all finite hyperproducts of elements of h. definition 3.6. let (h, ·) be a hv-group. we define the relation β on h as follows: xβy iff {x,y}⊆ u,u ∈ u. theorem 3.3. the fundamental equivalence relation β∗ is the transitive closure of the relation β on h. definition 3.7. let (r, +, ·) be a hv-ring. the relation γ∗ on r is called fundamental equivalence relation on r if it is the smallest equivalence relation on r such that the quotient set r/γ∗ is a ring, called fundamental ring of r. let us denote by u the set of all finite polynomials of elements of r, over n. definition 3.8. let (r, +, ·) be a hv-ring. we define the relation γ on h as follows: xγy iff {x,y}⊆ u,u ∈ u. theorem 3.4. the fundamental equivalence relation γ∗ is the transitive closure of the relation γ on r. definition 3.9. [26] a hv-ring is called hv-field if its fundamental ring is a field. definition 3.10. let v be a hv-vector space over a hv-field r. the relation ε∗ on v is called fundamental equivalence relation if it is the smallest equivalence relation on v such that the quotient set v/ε∗ is a vector space over the field r/γ∗, called fundamental vector space of v over r. 4 fundamental classes searching for the classes of fundamental equivalence relations is a central question in studying the fundamental structures derived from hyperstructures. this quest is intertwined with the exploration of the conditions that must be accomplished so that the β relation is transitive, that is, β = β∗. it is clear that the two different approaches to the fundamental equivalence relation in hyperstructures settle on two different ways of searching or constructing the fundamental equivalence classes in a hyperstructure. we could also talk, in a similar way with 3.1 and 3.2, about the deductive and inductive way of finding equivalence classes. 52 a brief survey on the two different approaches of fundamental equivalence relations on hyperstructures 4.1 complete parts according to the deductive way that koskas used and corsini’s school continued, the equivalence classes, that occur when the equivalence relation is strongly regular, are used. the notion of complete part of a hyperstructure’s subset plays a key role in finding the β∗ class of each element. the complete closure c(a) of the part a is connected with an increasing chain of subsets of the hyperstructure which, in turn, are related to hyperproducts containing a. it then turns out that the introduction, in a natural way, of the equivalence relation k is essentially a consideration of the equivalence β∗. in this way the complete closure coincides to the fundamental equivalence class. in particular, the definition of the complete part is used in the case of the singleton {x}, for each element x of the hyperstructure, so that we find ourselves in the environment of the fundamental equivalence relation. the increasing chain of sets created by the set {x} constructs the fundamental class of the arbitrary element x. it is evident that, as in the introduction of the β∗ relation [16], a notion is used as a mediator, which comes in between the questions ”how is the class” and ”what is the class”. this notion is k relation. we now present the necessary propositions in order to describe the step by step approach of the fundamental classes notion. the main references we use are the books [5], [7], [8] and the papers [13], [19]. definition 4.1. let (h, ·) be a semihypergroup and a be a nonempty subset of h. we say that a is a complete part of h if for any nonzero natural number n and for all a1,a2, ,an elements of h, the following implication holds: a∩ n∏ i=1 ai 6= ∅ ⇒ n∏ i=1 ai ⊆ a. notice that complete part a absorbs every hyperproduct containing one, at least, element of a. according to theorem 3.1, β∗(x) is a complete part of h,∀x ∈ h. (step 1) definition 4.2. let (h, ·) be a semihypergroup and a be a nonempty subset of h. the intersection of the complete parts of h which contain a is called the complete closure of a in h; it will be denoted by c(a). denote k1(a) = a and for all n > 0 kn+1(a) ={ x ∈ h| ∃p ∈ n,∃(h1,h2, ...,hp) ∈ hp : x ∈ p∏ i=1 (hi), kn(a) ∩ p∏ i=1 (hi) 6= ∅ } . 53 nikolaos antampoufis and sarka hoskova-mayerova obviously, kn(a),n > 0 is an increasing chain of subsets of h as we mentioned. if x ∈ h, we denote kn({x}) = kn(x). this implies that kn(a) = ⋃ a∈a kn(a) in particular, if p is the set of all finite hyperproducts of elements of h, and x ∈ h we have: k1(x) = {x}, k2(x) = ⋃ u∈p u : x ∈ u, k3(x) = ⋃ u∈p u : u∩k2(x) 6= ∅, ...,kn+1(x) = ⋃ u∈p u : u∩kn(x) 6= ∅ and c(a) = ⋃ a∈a c(a),a ⊆ h. notice that k2(x) = {z ∈ h|zβx} = β(x). (step 2) theorem 4.1. let (h, ·) be a semihypergroup and k a binary relation defined as follows: xky ⇔ x ∈ c(y), (x,y) ∈ h2. then,k is an equivalence relation that coincides with β∗. (step 3) thus, the relation k and the chain of sets kn behave as a mediator, which comes in between c(x) and β∗, connecting the construction of the class with the class itself. now we present some propositions mainly about the transitivity of β-relation. theorem 4.2. [12] let (h, ·) be a hypergroup then, β = β∗. theorem 4.3. let (h, ·) be a hypergroupoid. then, β = β∗ ⇔ c(x) = k2(x),∀x ∈ h. theorem 4.4. [13] let (h, ·) be a hv-group having, at least, one identity. then, β = β∗. theorem 4.5. [13] let (h, ·) be a hb-group then, β = β∗. 4.2 constructing fundamental classes according to the inductive way that vougiouklis introduced, the direct approach to the fundamental class is used. vougiouklis, while studying the fundamental classes, follows the same philosophy and strategy as he does in his approach to the introduction of the fundamental relationship and the corresponding 54 a brief survey on the two different approaches of fundamental equivalence relations on hyperstructures structure. he does not attempt to create an environment of desirable definitions in which he will then place the fundamental class of each element, as koskas has chosen to do. on the contrary, he considers the fundamental class of each element as a set and tries to specify its elements through their common properties. he prefers, in short, a straightforward reference to the common behaviour of these elements in the generation of hyperproducts. therefore, he develops propositions that relate each other the elements which behave in a similar way, thus achieving the accumulation of all the equivalent, with respect to β∗, elements which belong to the same fundamental class. it is clear that this straightforward dealing with the class study is an inductive type of answer to the question of the nature and form of the fundamental classes. the fundamental class which is a singleton, whenever it exists, plays an essential role in the study of hv-groups and hvrings. the element of each such class is called single element. its value lies in the fact that each hyperproduct having a single element as a factor, is an entire fundamental class. thus, finding the classes is achieved by multiplying a single element with each element of the hyperstructure. in fact, it is not necessary to perform the hyperoperation with all the elements. moreover, the existence of one, at least, single element is a sufficient condition such that β = β∗ holds in hv-groups. finally, the above mentioned approach also includes the technique of ”translation” of hyperproducts which allows us to find the fundamental structure of an hv-group and its classes using isomorphism between the quotient sets. we now present the necessary propositions in order to describe the direct approach of fundamental classes. the main references we use are the book [25] and the papers [24], [26]. theorem 4.6. let (h, ·) be a hvgroup, then xβ∗y iff there exist a,a ′ ⊆ β∗(a), b,b ′ ⊆ β∗(b), (a,b) ∈ h2, such that xa∩b 6= ∅ and ya′ ∩b′ 6= ∅. theorem 4.7. let (h, ·) be a hv-group, then u ∈ ωh iff there exist a ⊆ β∗(a), for some a ∈ h, such that ua∩a 6= ∅. definition 4.3. let h be a hv-structure. an element s ∈ h is called single if β∗(s) = {s}. we denote by sh the set of singles elements of h theorem 4.8. let (h, ·) be a hv-group and s ∈ sh . let (a,v) ∈ h2 such that s ∈ av, then β∗(a) = {h ∈ h : hv = s} and the core of h is ωh = {z ∈ h : zs = s}. 55 nikolaos antampoufis and sarka hoskova-mayerova theorem 4.9. let (h, ·) be a hv-group and sh 6= ∅, then β∗ = β. definition 4.4. (translations) [25] let (h, ·) be a hv-group and x ∈ h, then h/β∗ ∼= (h/lx)/β∗, where lx is the translation equivalence relation. 4.3 using the fundamental equivalence relations the fundamental equivalence relations on hyperstructures, on the one hand, connect the theory of hyperstructures to that of the corresponding classical structures, and on the other hand, are a tool for the introduction of new hyperstructure classes. the two approaches to the concept of the fundamental equivalence relations were initially referred to semihypergroups and hypergroups. however, they form the driving lever to apply similar definitions to other hyperstructures as hyperrrings and hyperfields or to study specific behaviour of some hyperstructures using the quotient set. freni [11] and others (p. corsini, b. davazz) introduced and studied equivalence relations that refer to individual properties or characteristics of elements of a hyperstructure. as an example, we mention the equivalence relations ∆h∗ and α∗ which are related to the cyclicity and commutativity of a hyperstructure, respectively. our main reference is [8]. relation α was introduced by freni who used the letter γ instead of α. thus, there was a confusion on symbolism since vougiouklis had already used the letter γ for the fundamental equivalence relation on hyperrings. vougiouklis focused on the study of hyperstructures which have a desirable quotient.(h/v)-structures are a typical example of generalization using the fundamental structures. they are a larger class than that of hv-structures. also, the use of the fundamental classes of equivalence as hyperproducts, led to constructions of new hyperstructure classes. as additional examples of hyperstructures with desirable quotient we mention the s1,s2, ...,sn-hyperstructures [3], the complete (in the sense of corsini) hv-groups and the constructions s1 and s2 [25]. definition 4.5. [23] the hv-semigroup (h, ·) is called h/v-group if h/β∗ is a group. notice that the reproductivity is not necessarily valid. however, the reproductivity of classes is valid. definition 4.6. [1], [2]. we shall say that a hyperstructure is an sn-hyperstructure if all its fundamental classes are singleton except for n of them, n ∈ n,n > 0. 56 a brief survey on the two different approaches of fundamental equivalence relations on hyperstructures 5 conclusions findings the aim of this paper was to present the two different approaches to the concept of the fundamental equivalence relation through a comparison of the choices and methods applied by koskas and vougiouklis, but also through the results they have achieved. summarizing our conclusions and findings, we can note that: a) the approach of koskas is a deductive way based on the introduction of general notions willing to draw conclusions to more specific ones. on the contrary, vougiouklis makes the reverse. beginning with the specific notion leads to propositions generalizing his conclusions. b) the directness of vougiouklis approach allows for the ”transfer” of definitions and corresponding theorems from the hv-groups to hv-rings, hv-modules and the other weak hyperstructures. in essence, it is a widely applied model that imposed, among other things, on the terminology of the ”fundamental” relations. c) the nature of the step-by-step approach of koskas has led to frequent use of the mathematical induction method in proving many basic theorems. on the other hand, due to the direct reference of vougiouklis to the equivalence classes and their view as sets of elements, the most frequent proving method used is the proof by contradiction. d) the koskas approach raises the question: what is the quotient set of the hyperstructure that we are studying? that is why increasing chains of relations are the main study tool. vougiouklis approach reverses the question with the following one: which hyperstructure produces a particular desired fundamental quotient? in this inversion, there is a trigger for the introduction of weak and h/v-structures. finally, we consider that the mathematical value of fundamental equivalence relations in hyperstructures is important and their asynchronized approaches by koskas and vougiouklis is an interesting moment of the short history of algebraic hyperstructures from their beginning at 1934 until today. 57 nikolaos antampoufis and sarka hoskova-mayerova references [1] n. antampoufis, s2−hv(h/v)-structures and s2-hyperstructures, int. j. of alg. hyperstructures and appl. (ijaha), 2(1), (2016), 163–174. [2] n. antampoufis, s1-hv-groups, s1-hypergroups and the ∂ operation, proc. of 10th aha congress 2008, brno, czech republic, (2009), 99–112. [3] n. antampoufis, contribution to the study of hyperstructures with applications in compulsory education, doctoral thesis, ed. telia+pavla, xanthi, greece, 2008. [4] n. antampoufis, t. vougiouklis and a. dramalidis, geometrical and circle hyperoperations in urban applications, ratio sociologica 4(2) (2011), 53– 66. [5] p. corsini, prolegomena of hypergroup theory, aviani editore, 1993. [6] p. corsini and th. vougiouklis, from groupoids to groups through hypergroups, renticcoti mat. s. vii, 9 (1989), 173–181. [7] p. corsini and v. leoreanu, applications of hyperstructure theory, kluwer academic publishers, 2002. [8] j. chvalina and s. hoskova-mayerova, general ω-hyperstructures and certain applications of those, ratio mathematica, 23, (2012), 3–20. [9] b. davvaz, semihypergroup theory, elsevier, academic press, 2016. [10] a. dramalidis, geometical hv-strucures, stucture elements of hyperstructures, proceedings (eds. n. lygeros and th. vougiouklis), spanidis press, alexandoupolis, greece, (2005), 41–52. [11] d. freni, hypergroupoids and fundamental relations, proceedings of aha, ed. m. stefanescu, hadronic press, (1994), 81–92. 58 a brief survey on the two different approaches of fundamental equivalence relations on hyperstructures [12] d. freni, a note on the core of a hypergroup and the transitive closure β∗ of β, riv. mat. pura appl. 8, (1991), 153–156. [13] m. gutan, properties of hyperproducts and the relation β in quasihypergroups, ratio mathematica 12, (1997), 19–34. [14] s. hoskova-mayerova, quasi-order hypergroups determinated by t-hypergroups, ratio mathematica, 32, (2017), 37–44. [15] s. hoskova-mayerova and a. maturo, algebraic hyperstructures and social relations, ital. j. pure appl. math. 39, (2018), in print. [16] m. koskas, groupoids, demi-hypergroupes et hypergroupes, j. math. pures appl., 49(9), (1970), 155–192. [17] f. marty, sur une gènèralisation de la notion de groupe, huitieme congrès mathématiciens scandinaves, stockholm, (1934), 45–49. [18] a. maturo, probability, utility and hyperstructures, proceedings of the 8th international congress on aha , samothraki, greece, (2003), 203–214. [19] r. migliorato, fundamental relation on non-associative hypergroupoids, ital. j. pure appl. math., 6 (1999), 147–160. [20] m. novák, el-hyperstructures: an overview, ratio mathematica, 23 (2012), 65–80. [21] o. ore, structures and group theory i, duke math. j., 3, (1937), 149–174. [22] th. vougiouklis, an hv-interview, i.e. weak (interviewer n. lygeros), structure elements of hyperstructures, proceedings,(eds. n. lygeros and th. vougiouklis), spanidis press, (2005), alexandroupolis, greece, 5–15. [23] th. vougiouklis, the h/v structures, j. discrete math. sci. cryptogr., 6, (2003), 235–243. 59 nikolaos antampoufis and sarka hoskova-mayerova [24] th. vougiouklis, hv-groups defined on the same set, discrete mathematics 155, (1996), 259–265. [25] th. vougiouklis, hyperstructures and their representations, monographs, hadronic press, usa, 1994. [26] th. vougiouklis, the fundamental relation in hyperrings. the general hyperfield, 4th aha cong. xanthi, world scientific (1991), 209–217. [27] th. vougiouklis, groups in hypergroups, annals discrete math., 37, (1988), 459–468. [28] t. vougiouklis, s. vougiouklis, helix-hopes on finite hyperfields, ratio mathematica, 31 (2016), 65–78. 60 microsoft word capitolo intero n 7.doc error locating codes dealing with repeated low-density burst errors b. k. dass department of mathematics university of delhi delhi-110 007, india e-mail: dassbk@rediffmail.com ritu arora∗ department of mathematics jdm college (university of delhi) sir ganga ram hospital marg new delhi-110 060, india e-mail: rituaroraind@gmail.com abstract. this paper presents a study of linear codes which are capable to detect and locate errors which are repeated low-density bursts of length b(fixed) with weight w or less. an illustration for such a kind of code has also been provided. keywords: error locating codes, burst errors, burst errors of length of b(fixed), repeated low-density burst errors of length b(fixed). ams subject classification: 94b20, 94b65, 94b25. ∗corresponding author. ratio mathematica, 20, 2010 67 1 introduction burst errors are the type of errors that occur quite frequently in several communication channels. codes developed to detect and correct such errors have been studied extensively by many authors. abramson [1959] developed codes which dealt with the correction of single and double adjacent errors, which was extended by fire [1959] as a more general concept called ‘burst errors’. a burst of length b is defined as follows: definition 1. a burst of length b is a vector whose only non-zero components are among some b consecutive components, the first and the last of which is non-zero. the nature of burst errors differs from channel to channel depending upon the kind of channel. chien and tang [1965] proposed a modification in the definition of a burst and they defined a burst of length b, which shall be called as ct-burst of length b, as follows: definition 2. a ct-burst of length b is a vector whose only non-zero components are confined to some b consecutive positions, the first of which is non-zero. channels due to alexander, gryb and nast [1960] fall in this category. this definition was further modified by dass [1980] as follows: definition 3. a burst of length b(fixed) is an n-tuple whose only non-zero components are confined to b consecutive positions, the first of which is non-zero and the number of its starting positions is among the first n−b+1 components. ratio mathematica, 20, 2010 68 this definition is useful for channels not producing errors near the end of a code word. in very busy communication channels errors repeat themselves. so is a situation when errors occur in the form of bursts. dass, garg and zannetti [2008] studied this kind of repeated burst errors. they termed such errors as m-repeated burst errors of length b(fixed) which has been defined as follows: definition 4. an m-repeated bursts of length b(fixed) is an n-tuple whose only non-zero components are confined to m distinct sets of b consecutive digits, the first component of each set is non-zero and the number of its starting positions is among the first n − mb + 1 components. in particular a 2-repeated bursts of length b(fixed) has been defined by dass and garg [2009(a)] as follows: definition 5. a 2-repeated bursts of length b(fixed) is an n-tuple whose only non-zero components are confined to 2 distinct sets of b consecutive digits, the first component of each set is non-zero and the number of its starting positions is among the first n − 2b + 1 components. during the process of transmission some disturbances cause occurrence of burst errors in such a way that over a given length, some digits are received correctly while others get corrupted i.e. not all the digits inside a burst are in error. such bursts are termed as low-density bursts [wyner (1963)]. a low-density burst of length b(fixed) with weight w or less has been defined as follows: ratio mathematica, 20, 2010 69 definition 6. a low-density burst of length b(fixed) with weight w or less is an n-tuple whose only non-zero components are confined to some b consecutive positions, the first of which is non-zero with at most w (w ≤ b) non-zero components within such b consecutive digits and the number of starting positions of the burst is among the first n − b + 1 components. dass and garg [2009(b)] studied codes which are capable to detect and/or correct m-repeated low-density bursts of length b(fixed) with weight w or less. they defined such codes as follows: definition 7. an m-repeated low-density burst of length b(fixed) with weight w or less is an n-tuple whose only non-zero components are confined to m distinct sets of b consecutive positions, the first component of each set is non-zero where each set can have at most w non-zero components (w ≤ b), and the number of its starting positions in an n-tuple is among the first n − mb + 1 positions. in particular, a 2-repeated low-density burst of length b(fixed) with weight w or less has been defined as follows: definition 8. a 2-repeated low-density burst of length b(fixed) with weight w or less is an n-tuple whose only non-zero components are confined to two distinct sets of b consecutive positions, the first component of each set is non-zero where each set can have at most w non-zero components (w ≤ b), and the number of its starting positions in an n-tuple is among the first n − 2b + 1 positions. ratio mathematica, 20, 2010 70 as an illustration, (21010000102000) is a 2-repeated low-density burst of length up to 6(fixed) with weight 3 or less over gf(3) whereas (001000011110) is a 2-repeated low-density burst of length at most 5(fixed) with weight 4 or less over gf(2). in this paper we have presented a study of codes dealing with the location of such kind of errors occurring within a sub-block. the concept of error-locating codes, lying midway between error detection and error correction, was introduced by wolf and elspas [1963]. in this technique the block of received digits is to be regarded as subdivided into mutually exclusive sub-blocks and while decoding it is possible to detect the error and in addition the receiver is able to identify which particular sub-block contains error. such codes are referred to as error-locating codes (elcodes). wolf and elspas [1963] studied binary codes which are capable of detecting and locating a single sub-block containing random errors. a study of codes locating burst errors of length b(fixed) has been made by dass and kishanchand [1986]. dass and arora [2010] obtained bounds for codes capable of locating repeated burst errors of length b(fixed) occurring within a sub-block. in this paper we have obtained bounds on the number of check digits required to locate 2-repeated low-density bursts of length b(fixed), and m-repeated low-density bursts of length b(fixed). an illustration of such a code has also been given. development of such codes will economize in the number of parity-check digits required in comparison to the usual lowdensity burst error locating codes while considering such repeated bursts as single bursts. ratio mathematica, 20, 2010 71 the paper has been organized as follows. in section 2 the necessary condition for the detection and location of 2-repeated low-density burst of length b(fixed) with weight w or less has been derived. this is followed by a sufficient condition for the existence of such a code. an illustration of 2-repeated low-density burst of length b(fixed) with weight w or less over gf(2) has also been given. in section 3 a necessary condition for the detection and location of m-repeated low-density burst of length b(fixed) with weight w or less has been given followed by a sufficient condition for the existence of such a code. in what follows we shall consider a linear code to be a subspace of ntuples over gf(q). the block of n digits, consisting of r check digits and k = n − r information digits, is considered to be divided into s mutually exclusive sub-blocks. each sub-block contains t = n/s digits. 2 2-repeated low-density burst error locating codes in this section, we consider (n, k) linear codes over gf(q) that are capable of detecting and locating all 2-repeated low-density burst of length b(fixed) with weight w or less within a single sub-block. it may be noted that an el-code capable of detecting and locating a single sub-block containing an error which is in the form of a 2-repeated low-density bursts of length b(fixed) with weight w or less must satisfy the following conditions: (a) the syndrome resulting from the occurrence of a 2-repeated lowdensity burst of length b(fixed) with weight w or less within any one 6 ratio mathematica, 20, 2010 72 sub-block must be distinct from the all zero syndrome. (b) the syndrome resulting from the occurrence of any 2-repeated lowdensity burst of length b(fixed) with weight w or less within a single sub-block must be distinct from the syndrome resulting likewise from any 2-repeated low-density burst of length b(fixed) with weight w or less within any other sub-block. in this section we shall derive two results. the first result derives a lower bound on the number of check digits required for the existence of a linear code over gf(q) capable of detecting and locating a single sub-block containing errors that are 2-repeated low-density burst of length b(fixed) with weight w or less. in the second result, an upper bound on the number of check digits which ensures the existence of such a code has been derived. as the code is divided into several blocks of length t each and we wish to detect a 2-repeated low-density burst of length b(fixed) with weight w or less, we may come across with a situation when the difference between 2b and t (b + w and t) becomes narrow. we note that if t−b + 1 < b + w and if we consider any two 2-repeated low-density bursts x1 and x2 of length b(fixed) with weight w or less such that their non-zero components are confined to first t − b + 1 positions with w components confining to some fixed w positions out of first b consecutive positions then their difference x1 x2 is again a 2-repeated low-density burst of length b(fixed) with weight w or less. however if we do not restrict ourselves to first t − b + 1 positions then we may not get a 2-repeated burst of length b(fixed) with weight w or less. this may be better understood with the help of the following examples: ratio mathematica, 20, 2010 73 example 1. let t = 9, b = 4, w = 3 and q = 2. so that t − b + 1 = 6 < b + w(= 7). let x1 = (101101001) and x2 = (100101011). then x1 and x2 are 2-repeated low-density burst of length 4(fixed) with weight 3 or less whereas x1 − x2 = (001000010) is not a 2-repeated burst of length 4(fixed). example 2. let t = 11, b = 5, w = 3 and q = 2. let x1 = (10101010010) and x2 = (10101010001) then x1 and x2 are 2-repeated low-density burst of length 5(fixed) with weight 3 or less whereas x1 − x2 = (00000000011) which is not even a 2-repeated burst of length 4(fixed) what to talk of its weight. so, accordingly we discuss the following cases: case 1: when t − b + 1 ≥ 2b. let x be the collection of all those vectors in which all the nonzero components are confined to some fixed w positions out of two sets of b consecutive positions each i.e. l -th to (l + b)-th position and j -th to (j + b)-th position where j > l + b. we observe that the syndromes of all the elements of x should be different; else for any x1, x2 belonging to x having the same syndrome would imply that the syndrome of x1 − x2 which is also an element of x and hence a 2-repeated low density burst of length b(fixed) with weight w or less within the same sub-block becomes zero; in violation of condition (a). also, since the error locates a single sub-block containing errors that are 2-repeated low-density bursts of length b(fixed) of weight w or less, ratio mathematica, 20, 2010 74 the syndromes produced by similar vectors in different sub-blocks must be distinct by condition (b). thus the syndromes of vectors which are 2-repeated low-density burst of length b(fixed) with weight w or less in fixed positions, whether in the same sub-block or in different sub-blocks, must be distinct. (it may be noted that the choice of different fixed components in different sub-blocks will also yield the same result). as there are (q2w − 1) distinct non-zero syndromes corresponding to the vectors in any one sub-block and there are s sub-blocks in all, so we must have atleast (1 + s(q2w −1)) distinct syndromes counting the all zero syndrome. as maximum number of distinct syndromes available using r check bits is qr , so there are qr distinct syndromes in all, therefore we must have qr ≥ {1 + s(q2w − 1)} (1) where t − b + 1 ≥ 2b. case 2: when b + w ≤ t − b + 1 < 2b. let x be the collection of all those vectors in which all the nonzero components are confined to some w fixed positions out of first b components i.e first and b-th position and another set of w fixed positions out of (b + 1)-th to (t − b + 1)-th positions. as discussed in case 1 the syndromes of all the elements of x is different. in this case also, there are (q2w − 1) distinct non-zero syndromes corresponding to the vectors in any one sub-block and there are s subratio mathematica, 20, 2010 75 blocks in all, so we must have atleast (1 + s(q2w − 1)) distinct syndromes counting the all zero syndrome. so, in this case also, we must have qr ≥ {1 + s(q2w − 1)} (2) where b + w ≤ t − b + 1 < 2b. case 3: when t − b + 1 < b + w . in this case consider x to be collection of all those vectors in which all the non-zero components are confined to some w fixed positions out of first b positions and t − 2b + 1 components from (b + 1)-th to (t − b + 1)th positions. in this case there are (qw+(t−2b+1) − 1) distinct non-zero syndromes corresponding to the vectors in any one sub-block. as and there are s sub-blocks in all, so we must have atleast (1+s(qw+(t−2b+1)−1)) distinct syndromes counting the all zero syndrome. therefore in this case, we must have qr ≥ {1 + s(qw+(t−2b+1) − 1)} (3) where t − b + 1 < b + w . from (1), (2), and (3) we have r ≥    logq{1 + s(q2w − 1)} where t − b + 1 ≥ 2b and b + w ≤ t − b + 1 < 2b logq{1 + s(qw+(t−2b+1) − 1)} where t − b + 1 < b + w. thus we have proved: theorem 1. the number of parity check digits r in an (n, k) linear code subdivided into s sub-blocks of length t each, that locates a single corrupted ratio mathematica, 20, 2010 76 sub-block containing errors that are 2-repeated low density burst of length b(fixed) with weight w or less is at least    logq{1 + s(q2w − 1)} where t − b + 1 ≥ 2b and b + w ≤ t − b + 1 < 2b logq{1 + s(qw+(t−2b+1) − 1)} where t − b + 1 < b + w . remark 1. for w = b, the weight consideration over the burst becomes redundant and the result coincides with theorem 1[dass and arora [2010]], when the bursts considered are 2-repeated bursts of length b(fixed). in the following result we derive another bound on the number of check digits required for the existence of such a code. the proof is based on the technique used to establish varshomov-gilbert sacks bound by constructing a parity check matrix for such a code [refer sacks[1958], also theorem 4.7 peterson and weldon[1972]]. this technique not only ensures the existence of such a code but also gives a method for the construction of such a code. theorem 2. an (n, k) linear el-code over gf(q) capable of detecting a 2-repeated low density burst of length b(fixed) with weight w or less (w ≤ b) within a single sub-block and of locating that sub-block can always be constructed provided that qn−k > [1 + (q − 1)](b−1,w−1){1 + (q − 1)(t − 2b + 1)[1 + (q − 1)](b−1,w−1)} · { 1 + (s − 1) 2∑ i=1 ( t − ib + i i ) (q − 1)i{[1 + (q − 1)](b−1,w−1)}i } (4) where [1 + x](m,r) denotes the incomplete binomial expansion of (1 + x)m up to the term xr in ascending power of x, viz. [1 + x](m,r) = 1 + ( m 1 ) x + ( m 2 ) x2 + . . . + ( m r ) xr. ratio mathematica, 20, 2010 77 proof. the existence of such a code will be shown by constructing an appropriate (n − k × n) parity check matrix h by a synthesis procedure. for that we first construct a matrix h1 from which the requisite parity check matrix h shall be obtained by reversing the order of the columns of each sub-block. after adding (s−1)t columns appropriately corresponding to the first (s − 1) sub-blocks, suppose that we have added the first j − 1 columns h1, h2, . . . , hj−1 of the s-th sub-block also, out of which the first b − 1 columns h1, h2, . . . , hb−1 may be chosen arbitrarily (non-zero). we now lay down the condition to add the j -th column hj to h1 as follows: according to condition (a), for the detection of 2-repeated lowdensity burst of length b(fixed) with weight w or less in the s-th sub-block hj should not be a linear combination of any w−1 or fewer columns among the immediately preceding b − 1 columns hj−b+1, hj−b+2, . . . hj−1 together with any w or fewer columns from amongst some b consecutive columns from the first j − b columns of the s-th sub-block. i.e. hj 6= (αj1 hj1 +αj2 hj2 + . . . +αjw−1 hjw−1 ) +(βl1 hl1 +βl2 hl2 + . . . +βlw hlw ) (5) where hj1 , hj2 , . . . , hjw−1 are any w−1 columns among hj−b+1, hj−b+2, . . . hj−1 and hl ’s are any w columns from a set of b consecutive columns among the first j−b columns of the s-th sub-block such that either all the coefficients βli ’s are zero or if the p-th coefficient βlp is the last non-zero coefficients then b ≤ p ≤ j − b; αji ’s, βli ’s ∈ gf(q). ratio mathematica, 20, 2010 78 the number of ways in which the coefficients αi ’s can be selected is [1 + (q − 1)](b−1,w−1) . to enumerate the coefficients βi ’s is equivalent to enumerate the number of bursts of length b(fixed) with weight w or less in a vector of length j − b. this number including the vector of all zeros [refer theorem 1, dass [1983]] is 1 + (j − 2b + 1)(q − 1)[1 + (q − 1)](b−1,w−1) so, the number of linear combinations on the right hand side of (5) is [1 + (q − 1)](b−1,w−1)[1 + (j − 2b + 1)(q − 1)[1 + (q − 1)](b−1,w−1)] (6) according to condition (b), for the location of 2-repeated low-density bursts of length b(fixed) with weight w or less, hj should not be a linear combination of any w − 1 or fewer columns among the immediately preceding the b−1 columns and any w columns from a set of b consecutive columns from the remaining j − b columns of the s-th sub-block along with any w or less columns each from any of the two sets of b consecutive columns out of any one of the previously chosen s − 1 sub-blocks, the coefficient of the last column of either both or one of the sets being nonzero. the number of 2-repeated low-density bursts of length b(fixed) with weight w or less in a sub-block of length t [refer dass and garg [2009(b)]] is 2∑ i=1 ( t − ib + i i ) (q − 1)i{[1 + (q − 1)](b−1,w−1)}i (7) since there are (s−1) previous sub-blocks, therefore number of such linear ratio mathematica, 20, 2010 79 combinations becomes (s − 1) 2∑ i=1 ( t − ib + i i ) (q − 1)i{[1 + (q − 1)](b−1,w−1)}i (8) so, for the location of 2-repeated low-density burst of length b(fixed) with weight w or less the number of linear combinations to which hj can not be equal to is the product of expr.(6) and expr.(8) i.e. expr.(6) × expr.(8) (9) thus the total number of linear combinations to which hi can not be equal to is the sum of exp.(6) and exp.(9) at worst all these combinations might yield distinct sum. therefore hi can be added to the s-th sub-block provided that qn−k > [1 + (q − 1)](b−1,w−1){1 + (q − 1)(j − 2b + 1)[1 + (q − 1)](b−1,w−1)} · { 1 + (s − 1) 2∑ i=1 ( t − ib + i i ) (q − 1)i{[1 + (q − 1)](b−1,w−1)}i } to obtain the length of the block as t we replace j by t in the above expression. the required parity-check matrix h can be obtained from h1 by reversing the order of the columns in each sub-block. remark 2. for w = b, the weight consideration over the burst becomes redundant and the inequality in theorem 2 reduces to qn−k > qb−1{1 + (q − 1)(t − 2b + 1)qb−1} × { 1 + (s − 1) 2∑ i=1 ( t − ib + i i ) (q − 1)iqi(b−1) } ratio mathematica, 20, 2010 80 which coincides with the condition for the location of 2-repeated burst of length b(fixed) [refer theorem 2, dass and arora [2010]]. we conclude this section with the following example: example 3. for an (27,15) linear code over gf(2) consider the following 12 × 27 matrix h which has been constructed by the synthesis procedure given in the proof of theorem 2 by taking s = 3, t = 9, b = 3, w = 2. h =   0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 1 1   the null space of this matrix can be used as a code to locate a subblock of length t = 9 containing 2-repeated burst of length 3(fixed). from the error pattern syndrome table 1 we observe that: the syndromes of 2-repeated burst of length 3(fixed) within any subblock are all non-zero showing thereby that the code detects all 2-repeated low-density bursts of length 3(fixed) with weight 2 or less occurring within a sub-block. ratio mathematica, 20, 2010 81 it has been verified through ms-excel program that the syndromes of the 2-repeated bursts of length 3(fixed) with weight 2 or less in any sub-block is different from the syndrome of a 2-repeated burst of length 3(fixed) with weight 2 or less within any other sub-block. table 1 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 1 1 100100000 000000000 000000000 0000 0100 1000 2 100101000 000000000 000000000 0001 0100 1000 3 100110000 000000000 000000000 0000 1100 1000 4 101100000 000000000 000000000 0000 0110 1000 5 101101000 000000000 000000000 0001 0110 1000 6 101110000 000000000 000000000 0000 1110 1000 7 110100000 000000000 000000000 0000 0101 1000 8 110101000 000000000 000000000 0001 0101 1000 9 110110000 000000000 000000000 0000 1101 1000 10 100010000 000000000 000000000 0000 1000 1000 11 100010100 000000000 000000000 0010 1000 1000 12 100011000 000000000 000000000 0001 1000 1000 13 101010000 000000000 000000000 0000 1010 1000 14 101010100 000000000 000000000 0010 1010 1000 15 101011000 000000000 000000000 0001 1010 1000 16 110010000 000000000 000000000 0000 1001 1000 17 110010100 000000000 000000000 0010 1001 1000 18 110011000 000000000 000000000 0001 1001 1000 19 100001000 000000000 000000000 0001 0000 1000 20 100001010 000000000 000000000 0101 0000 1000 21 100001100 000000000 000000000 0011 0000 1000 22 101001000 000000000 000000000 0001 0010 1000 23 101001010 000000000 000000000 0101 0010 1000 24 101001100 000000000 000000000 0011 0010 1000 25 110001000 000000000 000000000 0001 0001 1000 26 110001010 000000000 000000000 0101 0001 1000 27 110001100 000000000 000000000 0011 0001 1000 28 100000100 000000000 000000000 0010 0000 1000 29 100000101 000000000 000000000 1010 0000 1000 ratio mathematica, 20, 2010 82 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 1 30 100000110 000000000 000000000 0110 0000 1000 31 101000100 000000000 000000000 0010 0010 1000 32 101000101 000000000 000000000 1010 0010 1000 33 101000110 000000000 000000000 0110 0010 1000 34 110000100 000000000 000000000 0010 0001 1000 35 110000101 000000000 000000000 1010 0001 1000 36 110000110 000000000 000000000 0110 0001 1000 37 010010000 000000000 000000000 0000 1001 0000 38 010010100 000000000 000000000 0010 1001 0000 39 010011000 000000000 000000000 0001 1001 0000 40 010110000 000000000 000000000 0000 1101 0000 41 010110100 000000000 000000000 0010 1101 0000 42 010111000 000000000 000000000 0001 1101 0000 43 011010000 000000000 000000000 0000 1011 0000 44 011010100 000000000 000000000 0010 1011 0000 45 011011000 000000000 000000000 0001 1011 0000 46 010001000 000000000 000000000 0001 0001 0000 47 010001010 000000000 000000000 0101 0001 0000 48 010001100 000000000 000000000 0011 0001 0000 49 010101000 000000000 000000000 0001 0101 0000 50 010101010 000000000 000000000 0101 0101 0000 51 010101100 000000000 000000000 0011 0101 0000 52 011001000 000000000 000000000 0001 0011 0000 53 011001010 000000000 000000000 0101 0011 0000 54 011001100 000000000 000000000 0011 0011 0000 55 010000100 000000000 000000000 0010 0001 0000 56 010000101 000000000 000000000 1010 0001 0000 57 010000110 000000000 000000000 0110 0001 0000 58 010100100 000000000 000000000 0010 0101 0000 59 010100101 000000000 000000000 1010 0101 0000 60 010100110 000000000 000000000 0110 0101 0000 61 011000100 000000000 000000000 0010 0011 0000 62 011000101 000000000 000000000 1010 0011 0000 63 011000110 000000000 000000000 0110 0011 0000 64 001001000 000000000 000000000 0001 0010 0000 65 001001010 000000000 000000000 0101 0010 0000 66 001001100 000000000 000000000 0011 0010 0000 ratio mathematica, 20, 2010 83 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 1 67 001011000 000000000 000000000 0001 1010 0000 68 001011010 000000000 000000000 0101 1010 0000 69 001011100 000000000 000000000 0011 1010 0000 70 001101000 000000000 000000000 0001 0110 0000 71 001101010 000000000 000000000 0101 0110 0000 72 001101100 000000000 000000000 0011 0110 0000 73 001000100 000000000 000000000 0010 0010 0000 74 001000101 000000000 000000000 1010 0010 0000 75 001000110 000000000 000000000 0110 0010 0000 76 001010100 000000000 000000000 0010 1010 0000 77 001010101 000000000 000000000 1010 1010 0000 78 001010110 000000000 000000000 0110 1010 0000 79 001100100 000000000 000000000 0010 0110 0000 80 001100101 000000000 000000000 1010 0110 0000 81 001100110 000000000 000000000 0110 0110 0000 82 000100100 000000000 000000000 0010 0100 0000 83 000100101 000000000 000000000 1010 0100 0000 84 000100110 000000000 000000000 0110 0100 0000 85 000101100 000000000 000000000 0011 0100 0000 86 000101101 000000000 000000000 1011 0100 0000 87 000101110 000000000 000000000 0111 0100 0000 88 000110100 000000000 000000000 0010 1100 0000 89 000110101 000000000 000000000 1010 1100 0000 90 000110110 000000000 000000000 0110 1100 0000 91 100000000 000000000 000000000 0000 0000 1000 92 101000000 000000000 000000000 0000 0010 1000 93 110000000 000000000 000000000 0000 0001 1000 94 010000000 000000000 000000000 0000 0001 0000 95 010100000 000000000 000000000 0000 0101 0000 96 011000000 000000000 000000000 0000 0011 0000 97 001000000 000000000 000000000 0000 0010 0000 98 001010000 000000000 000000000 0000 1010 0000 99 001100000 000000000 000000000 0000 0110 0000 100 000100000 000000000 000000000 0000 0100 0000 101 000101000 000000000 000000000 0001 0100 0000 102 000110000 000000000 000000000 0000 1100 0000 ratio mathematica, 20, 2010 84 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 1 103 000010000 000000000 000000000 0000 1000 0000 104 000010100 000000000 000000000 0010 1000 0000 105 000011000 000000000 000000000 0001 1000 0000 106 000001000 000000000 000000000 0001 0000 0000 107 000001010 000000000 000000000 0101 0000 0000 108 000001100 000000000 000000000 0011 0000 0000 109 000000100 000000000 000000000 0010 0000 0000 110 000000101 000000000 000000000 1010 0000 0000 111 000000110 000000000 000000000 0110 0000 0000 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 2 112 000000000 100100000 000000000 0011 0000 1100 113 000000000 100101000 000000000 1111 0000 1100 114 000000000 100110000 000000000 1111 0000 1111 115 000000000 101100000 000000000 1001 1010 0110 116 000000000 101101000 000000000 0101 1010 0110 117 000000000 101110000 000000000 0101 1010 0101 118 000000000 110100000 000000000 1101 1110 0010 119 000000000 110101000 000000000 0001 1110 0010 120 000000000 110110000 000000000 0001 1110 0001 121 000000000 100010000 000000000 0011 1100 0011 122 000000000 100010100 000000000 0011 1100 0010 123 000000000 100011000 000000000 1111 1100 0011 124 000000000 101010000 000000000 1001 0110 1001 125 000000000 101010100 000000000 1001 0110 1000 126 000000000 101011000 000000000 0101 0110 1001 127 000000000 110010000 000000000 1101 0010 1101 128 000000000 110010100 000000000 1101 0010 1100 129 000000000 110011000 000000000 0001 0010 1101 130 000000000 100001000 000000000 0011 1100 0000 131 000000000 100001010 000000000 0011 1100 0010 132 000000000 100001100 000000000 0011 1100 0001 133 000000000 101001000 000000000 1001 0110 1010 134 000000000 101001010 000000000 1001 0110 1000 135 000000000 101001100 000000000 1001 0110 1011 136 000000000 110001000 000000000 1101 0010 1110 ratio mathematica, 20, 2010 85 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 2 137 000000000 110001010 000000000 1101 0010 1100 138 000000000 110001100 000000000 1101 0010 1111 139 000000000 100000100 000000000 1111 1100 0001 140 000000000 100000101 000000000 1111 1100 0101 141 000000000 100000110 000000000 1111 1100 0011 142 000000000 101000100 000000000 0101 0110 1011 143 000000000 101000101 000000000 0101 0110 1111 144 000000000 101000110 000000000 0101 0110 1001 145 000000000 110000100 000000000 0001 0010 1111 146 000000000 110000101 000000000 0001 0010 1011 147 000000000 110000110 000000000 0001 0010 1101 148 000000000 010010000 000000000 0010 1110 1101 149 000000000 010010100 000000000 0010 1110 1100 150 000000000 010011000 000000000 1110 1110 1101 151 000000000 010110000 000000000 1110 0010 0001 152 000000000 010110100 000000000 1110 0010 0000 153 000000000 010111000 000000000 0010 0010 0001 154 000000000 011010000 000000000 1000 0100 0111 155 000000000 011010100 000000000 1000 0100 0110 156 000000000 011011000 000000000 0100 0100 0111 157 000000000 010001000 000000000 0010 1110 1110 158 000000000 010001010 000000000 0010 1110 1100 159 000000000 010001100 000000000 0010 1110 1111 160 000000000 010101000 000000000 1110 0010 0010 161 000000000 010101010 000000000 1110 0010 0000 162 000000000 010101100 000000000 1110 0010 0011 163 000000000 011001000 000000000 1000 0100 0100 164 000000000 011001010 000000000 1000 0100 0110 165 000000000 011001100 000000000 1000 0100 0101 166 000000000 010000100 000000000 1110 1110 1111 167 000000000 010000101 000000000 1110 1110 1011 168 000000000 010000110 000000000 1110 1110 1101 169 000000000 010100100 000000000 0010 0010 0011 170 000000000 010100101 000000000 0010 0010 0111 171 000000000 010100110 000000000 0010 0010 0001 172 000000000 011000100 000000000 0100 0100 0101 173 000000000 011000101 000000000 0100 0100 0001 ratio mathematica, 20, 2010 86 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 2 174 000000000 011000110 000000000 0100 0100 0111 175 000000000 001001000 000000000 0110 1010 1010 176 000000000 001001010 000000000 0110 1010 1000 177 000000000 001001100 000000000 0110 1010 1011 178 000000000 001011000 000000000 1010 1010 1001 179 000000000 001011010 000000000 1010 1010 1011 180 000000000 001011100 000000000 1010 1010 1000 181 000000000 001101000 000000000 1010 0110 0110 182 000000000 001101010 000000000 1010 0110 0100 183 000000000 001101100 000000000 1010 0110 0111 184 000000000 001000100 000000000 1010 1010 1011 185 000000000 001000101 000000000 1010 1010 1111 186 000000000 001000110 000000000 1010 1010 1001 187 000000000 001010100 000000000 0110 1010 1000 188 000000000 001010101 000000000 0110 1010 1100 189 000000000 001010110 000000000 0110 1010 1010 190 000000000 001100100 000000000 0110 0110 0111 191 000000000 001100101 000000000 0110 0110 0011 192 000000000 001100110 000000000 0110 0110 0101 193 000000000 000100100 000000000 1100 1100 1101 194 000000000 000100101 000000000 1100 1100 1001 195 000000000 000100110 000000000 1100 1100 1111 196 000000000 000101100 000000000 0000 1100 1101 197 000000000 000101101 000000000 0000 1100 1001 198 000000000 000101110 000000000 0000 1100 1111 199 000000000 000110100 000000000 0000 1100 1110 200 000000000 000110101 000000000 0000 1100 1010 201 000000000 000110110 000000000 0000 1100 1100 202 000000000 100000000 000000000 1111 1100 0000 203 000000000 101000000 000000000 0101 0110 1010 204 000000000 110000000 000000000 0001 0010 1110 205 000000000 010000000 000000000 1110 1110 1110 206 000000000 010100000 000000000 0010 0010 0010 207 000000000 011000000 000000000 0100 0100 0100 208 000000000 001000000 000000000 1010 1010 1010 209 000000000 001010000 000000000 0110 1010 1001 210 000000000 001100000 000000000 0110 0110 0110 211 000000000 000100000 000000000 1100 1100 1100 ratio mathematica, 20, 2010 87 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 2 212 000000000 000101000 000000000 0000 1100 1100 213 000000000 000110000 000000000 0000 1100 1111 214 000000000 000010000 000000000 1100 0000 0011 215 000000000 000010100 000000000 1100 0000 0010 216 000000000 000011000 000000000 0000 0000 0011 217 000000000 000001000 000000000 1100 0000 0000 218 000000000 000001010 000000000 1100 0000 0010 219 000000000 000001100 000000000 1100 0000 0001 220 000000000 000000100 000000000 0000 0000 0001 221 000000000 000000101 000000000 0000 0000 0101 222 000000000 000000110 000000000 0000 0000 0011 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 3 223 000000000 000000000 100100000 1111 1110 0111 224 000000000 000000000 100101000 1101 0111 0011 225 000000000 000000000 100110000 1000 1110 1001 226 000000000 000000000 101100000 0111 0111 1000 227 000000000 000000000 101101000 0101 1110 1100 228 000000000 000000000 101110000 0000 0111 0110 229 000000000 000000000 110100000 1111 1110 1101 230 000000000 000000000 110101000 1101 0111 1001 231 000000000 000000000 110110000 1000 1110 0011 232 000000000 000000000 100010000 1000 1111 0110 233 000000000 000000000 100010100 0110 0000 0001 234 000000000 000000000 100011000 1010 0110 0010 235 000000000 000000000 101010000 0000 0110 1001 236 000000000 000000000 101010100 1110 1001 1110 237 000000000 000000000 101011000 0010 1111 1101 238 000000000 000000000 110010000 1000 1111 1100 239 000000000 000000000 110010100 0110 0000 1011 240 000000000 000000000 110011000 1010 0110 1000 241 000000000 000000000 100001000 1101 0110 1100 242 000000000 000000000 100001010 0011 0110 1011 243 000000000 000000000 100001100 0011 1001 1011 244 000000000 000000000 101001000 0101 1111 0011 ratio mathematica, 20, 2010 88 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 3 245 000000000 000000000 101001010 1011 1111 0100 246 000000000 000000000 101001100 1011 0000 0100 247 000000000 000000000 110001000 1101 0110 0110 248 000000000 000000000 110001010 0011 0110 0001 249 000000000 000000000 110001100 0011 1001 0001 250 000000000 000000000 100000100 0001 0000 1111 251 000000000 000000000 100000101 1001 0000 1110 252 000000000 000000000 100000110 1111 0000 1000 253 000000000 000000000 101000100 1001 1001 0000 254 000000000 000000000 101000101 0001 1001 0001 255 000000000 000000000 101000110 0111 1001 0111 256 000000000 000000000 110000100 0001 0000 0101 257 000000000 000000000 110000101 1001 0000 0100 258 000000000 000000000 110000110 1111 0000 0010 259 000000000 000000000 010010000 0111 0000 0100 260 000000000 000000000 010010100 1001 1111 0011 261 000000000 000000000 010011000 0101 1001 0000 262 000000000 000000000 010110000 0111 0001 1011 263 000000000 000000000 010110100 1001 1110 1100 264 000000000 000000000 010111000 0101 1000 1111 265 000000000 000000000 011010000 1111 1001 1011 266 000000000 000000000 011010100 0001 0110 1100 267 000000000 000000000 011011000 1101 0000 1111 268 000000000 000000000 010001000 0010 1001 1110 269 000000000 000000000 010001010 1100 1001 1001 270 000000000 000000000 010001100 1100 0110 1001 271 000000000 000000000 010101000 0010 1000 0001 272 000000000 000000000 010101010 1100 1000 0110 273 000000000 000000000 010101100 1100 0111 0110 274 000000000 000000000 011001000 1010 0000 0001 275 000000000 000000000 011001010 0100 0000 0110 276 000000000 000000000 011001100 0100 1111 0110 277 000000000 000000000 010000100 1110 1111 1101 278 000000000 000000000 010000101 0110 1111 1100 279 000000000 000000000 010000110 0000 1111 1010 280 000000000 000000000 010100100 1110 1110 0010 281 000000000 000000000 010100101 0110 1110 0011 ratio mathematica, 20, 2010 89 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 3 282 000000000 000000000 010100110 0000 1110 0101 283 000000000 000000000 011000100 0110 0110 0010 284 000000000 000000000 011000101 1110 0110 0011 285 000000000 000000000 011000110 1000 0110 0101 286 000000000 000000000 001001000 1010 0000 1011 287 000000000 000000000 001001010 0100 0000 1100 288 000000000 000000000 001001100 0100 1111 1100 289 000000000 000000000 001011000 1101 0000 0101 290 000000000 000000000 001011010 0011 0000 0010 291 000000000 000000000 001011100 0011 1111 0010 292 000000000 000000000 001101000 1010 0001 0100 293 000000000 000000000 001101010 0100 0001 0011 294 000000000 000000000 001101100 0100 1110 0011 295 000000000 000000000 001000100 0110 0110 1000 296 000000000 000000000 001000101 1110 0110 1001 297 000000000 000000000 001000110 1000 0110 1111 298 000000000 000000000 001010100 0001 0110 0110 299 000000000 000000000 001010101 1001 0110 0111 300 000000000 000000000 001010110 1111 0110 0001 301 000000000 000000000 001100100 0110 0111 0111 302 000000000 000000000 001100101 1110 0111 0110 303 000000000 000000000 001100110 1000 0111 0000 304 000000000 000000000 000100100 1110 1110 1000 305 000000000 000000000 000100101 0110 1110 1001 306 000000000 000000000 000100110 0000 1110 1111 307 000000000 000000000 000101100 1100 0111 1100 308 000000000 000000000 000101101 0100 0111 1101 309 000000000 000000000 000101110 0010 0111 1011 310 000000000 000000000 000110100 1001 1110 0110 311 000000000 000000000 000110101 0001 1110 0111 312 000000000 000000000 000110110 0111 1110 0001 313 000000000 000000000 100000000 1111 1111 1000 314 000000000 000000000 101000000 0111 0110 0111 315 000000000 000000000 110000000 1111 1111 0010 316 000000000 000000000 010000000 0000 0000 1010 317 000000000 000000000 010100000 0000 0001 0101 318 000000000 000000000 011000000 1000 1001 0101 ratio mathematica, 20, 2010 90 low density 2-repeated bursts of length 3(fixed) syndromes sub-block 3 319 000000000 000000000 001000000 1000 1001 1111 320 000000000 000000000 001010000 1111 1001 0001 321 000000000 000000000 001100000 1000 1000 0000 322 000000000 000000000 000100000 0000 0001 1111 323 000000000 000000000 000101000 0010 1000 1011 324 000000000 000000000 000110000 0111 0001 0001 325 000000000 000000000 000010000 0111 0000 1110 326 000000000 000000000 000010100 1001 1111 1001 327 000000000 000000000 000011000 0101 1001 1010 328 000000000 000000000 000001000 0010 1001 0100 329 000000000 000000000 000001010 1100 1001 0011 330 000000000 000000000 000001100 1100 0110 0011 331 000000000 000000000 000000100 1110 1111 0111 332 000000000 000000000 000000101 0110 1111 0110 333 000000000 000000000 000000110 0000 1111 0000 remark 3. the space visible between vectors in the first column in table 1 has been given to distinguish between different sub-blocks whereas the space given in the syndrome vector is for convenience. observation. syndromes of some of the 2-repeated bursts of length 3(fixed) occurring within the second sub-block are same. for coding efficiency it is desired that the syndromes of the error patterns within any sub-block is identical whenever possible. 3 location of m-repeated low-density burst of length b(fixed) in this section a necessary and sufficient condition for the location of an m-repeated low-density burst of length b(fixed) with weight w or less has been given. ratio mathematica, 20, 2010 91 it may be noted that an el-code capable of detecting and locating a single sub-block containing an error which is in the form of an m-repeated low-density burst of length b(fixed) with weight w or less (w ≤ b) must satisfy the following conditions: (c) the syndrome resulting from the occurrence of an m-repeated lowdensity burst of length b(fixed) with weight w or less within any one sub-block must be distinct from the all zero syndrome. (d) the syndrome resulting from the occurrence of any m-repeated lowdensity burst of length b(fixed) with weight w or less within a single sub-block must be distinct from the syndrome resulting likewise from any m-repeated low-density burst of length b(fixed) with weight w or less within any other sub-block. in this section we shall derive two results. the first result gives a lower bound on the number of check digits required for the existence of a linear code over gf(q) capable of detecting and locating a single sub-block containing errors that are m-repeated low-density bursts of length b(fixed) with weight w or less. in the second result, we derive an upper bound on the number of check digits which ensures the existence of such a code. theorem 3. the number of parity check digits r in an (n, k) linear code subdivided into s sub-blocks of length t each, that locates a single corrupted sub-block containing errors that are 2-repeated low density bursts of length b(fixed) with weight w or less is at least    logq{1 + s(qmw − 1)} where t − b + 1 ≥ mb and (m−1)b+w≤t−b+1 < mb logq{1 + s(q(m−1)w+(t−mb+1) − 1)} where t − b + 1 < (m − 1)b + w. (10) ratio mathematica, 20, 2010 92 the proof of this result is on the similar lines as that of proof of theorem 1 so we omit the proof. remark 4. for m = 2 the result coincides with that of theorem 1 when 2-repeated low-density bursts of length b(fixed) with weight w or less are considered. remark 5. for m = 1, the result obtained in (10) reduces to    logq{1 + s(qw − 1)} where t − b + 1 ≥ b and w ≤ t − b + 1 < b logq{1 + s(q(t−b+1) − 1)} where t − b + 1 < w . which is a case of detecting and locating a sub-block containing errors which are usual low-density bursts of length b(fixed) with weight w or less. remark 6. for w = b, the result obtained in (10) reduces to r ≥ { logq{1 + s(qmb − 1)} where t − b + 1 ≥ mb logq{1 + s(q(t−b+1) − 1)} where t − b + 1 < mb which coincides with the result due to dass and arora [theorem 3, 2010]. in the following result we derive another bound on the number of check digits required for the existence of such a code. as earlier the proof is based on the technique used to establish varshomov-gilbert sacks bound by constructing a parity check matrix for such a code (refer sacks, theorem 4.7 peterson and weldon(1972)). theorem 4. an (n, k) linear el-code over gf(q) capable of detecting an m-repeated low density burst of length b(fixed) with weight w or less ratio mathematica, 20, 2010 93 (w ≤ b) within a single sub-block and of locating that sub-block can always be constructed provided that qn−k > [1 + (q − 1)](b−1,w−1) · { m−1∑ i=0 ( t − (i + 1)b + i i ) (q − 1)i[1 + (q − 1)](b−1,w−1) } · { 1 + (s − 1) m∑ i=1 ( t − ib + i i ) (q − 1)i{[1 + (q−1)](b−1,w−1)}i } (11) where [1 + x](m,r) denotes the incomplete binomial expansion of (1 + x)m up to the term xr in ascending power of x, viz. [1 + x](m,r) = 1 + ( m 1 ) x + ( m 2 ) x2 + . . . + ( m r ) xr. as in theorem 3 we omit the proof because proof of this result is on the similar lines as that of proof of theorem 2. remark 7. for m = 2 the result coincides with that of theorem 2 when 2-repeated low-density bursts of length b(fixed) with weight w or less are considered. remark 8. for m = 1, the result obtained in (11) reduces to qn−k>[1 + (q − 1)](b−1,w−1){1 + (s − 1)(t − b + 1)(q − 1)[1 + (q − 1)](b−1,w−1)} which is a necessary condition for detecting and locating a sub-block containing errors which are usual low-density bursts of length b(fixed) with weight w or less. ratio mathematica, 20, 2010 94 remark 9. for w = b, the result obtained in (11) reduces to qn−k > qb−1 { m−1∑ i=0 ( j − (i + 1)b + i i ) (q − 1)iqi(b−1) } · { 1 + (s − 1) m∑ i=1 ( j − (i + 1)b + i i ) (q − 1)iqi(b−1) } which coincides with the result due to dass and arora [theorem 4, 2010]. references [1] abramson, n.m. [1959] a class of systematic codes for nonindependent errors, ire trans. on information theory, it-5 (4), 150– 157. [2] alexander, a.a., gryb, r.m. and nast, d.w. [1960] capabilities of the telephone network for data transmission, bell system tech j., 39(3), 431-476. [3] chien, r.t. and tang, d.t. [1965] on definitions of a burst, ibm journal of research and development, 9(4), 292–293. [4] dass, b.k. [1980] on a bursterror correcting codes, j. inf. optimization sciences, 1(3), 291–295. [5] dass, b.k. [1983] low-density burst error correcting linear codes, advances in management studies, 2(4), 375–385. [6] dass, b.k. and arora, ritu [2010] error locating codes dealing with repeated burst errors, accepted for publication in italian journal of pure and applied mathematics, no. 30. ratio mathematica, 20, 2010 95 [7] dass, b.k. and chand, kishan [1986] linear codes locating/correcting burst errors, dei journal of science and engineering research, 4(2), 41–46. [8] dass, b.k., garg, poonam and zannetti, m. [2008] some combinatorial aspects of m-repeated burst error detecting codes, journal of statistical theory and practice, 2(4), 707–711. [9] dass, b.k. and garg, poonam [2009(a)] on 2-repeated burst codes, ratio mathematica journal of applied mathematics, 19, 11–24. [10] dass, b.k. and garg, poonam [2009(b)] bounds for codes correcting/detecting repeated low-density burst errors, communicated. [11] dass, b.k. and garg, poonam [2010], on repeated low-density burst error detecting linear codes, communicated. [12] fire, p. [1959] a class of multiple-error-correcting binary codes for non-independent errors, sylvania report rsl-e-2, sylvania reconnaissance systems laboratory, mountain view, calif. [13] hamming, r.w. [1950] error-detecting and error-correcting codes, bell system tech. j., 29, 147160. [14] peterson, w.w., weldon, e.j., jr. [1972] error-correcting codes, 2nd edition, the mit press, mass. [15] sacks, g.e. [1958] multiple error correction by means of parity-checks, ire trans. inform. theory it, 4(december), 145–147. [16] wyner, a.d. [1963] low-density-burst-correcting codes, ieee trans. informatiom theory, (april), 124. [17] wolf, j., elspas, b. [1963] error-locating codes a new concept in error control, ieee transactions on information theory, 9(2), 113–117. ratio mathematica, 20, 2010 96 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 36, 2019, pp. 69-78 69 en route for the calculus of variations jan coufal* jiří tobíšek† abstract optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. an optimal control is an extension of the calculus of variations. it is a mathematical optimization method for deriving control policies. the calculus of variations is concerned with the extrema of functionals. the different approaches tried out in its solution may be considered, in a more or less direct way, as the starting point for new theories. while the true “mathematical” demonstration involves what we now call the calculus of variations, a theory for which euler and then lagrange established the foundations, the solution which johann bernoulli originally produced, obtained with the help analogy with the law of refraction on optics, was empirical. a similar analogy between optics and mechanics reappears when hamilton applied the principle of least action in mechanics which maupertuis justified in the first instance, on the basis of the laws of optics. keywords: calculus of variations, johann bernoulli, jacob bernoulli, euler, lagrange, maupertuis, principle of least action. 2010 ams subject classification: 01a50, 49k50‡ * university of economics and management, nárožní 2600/9a, 158 00 praha 5, czech republic, e-mail: jan.n.coufal@seznam.cz. † university of economics and management, nárožní 2600/9a, 158 00 praha 5, czech republic, e-mail: jiri.tobisek@vsem.cz. ‡received on may 4th, 2019. accepted on may 27th, 2019. published on june 30th, 2019. doi: 10.23755/rm.v36i1.467. issn: 1592-7415. eissn: 2282-8214. ©coufal and tobíšek. this paper is published under the cc-by licence agreement. jan coufal and jiří tobíšek 70 1 introduction our intention here is to write the history of the brachistrone and its remarkable consequences. in the contemporary socio-cultural context, the question would essentially be formulated in the following text: what shape should we make slides in children’s playgrounds so that the time of descent should be minimized? the considerable importance of this question is well understood when we consider how children behave, and they want to obtain the best performance, but the question is also important in a more general way, and a great number of scholars have attempted to solve this problem. unfortunately the problem appears to be particularly tricky, and it depends upon a number of parameters, including the variable value of the friction between the clothes of the child and the surface of the slide. we shall not attempt to solve that particular problem here, but content ourselves with theory of the idealized problem, simplifying the situation sufficiently in order to be able to find a solution. in fact we shall replace the child by a perfectly smooth marble, and we assume that it rolls down a smooth surface, thus assuming that friction forces are negligible with respect to gravity. now, we are simply confronted with the problem of brachistrone as johann bernoulli expressed it in the acta eruditorum published in leipzig in june 1696 ([1], vol. 1, p. 161): datis in plano vertacali duobus punctis a & b, assignare mobili m viam amb, per quam gravitate sua descenden, & moveri incipiens a puncto a, brevissimo tempore perveniat ad alterum punctum b. the expression brevissimo tempore is the latin translation of the greek term brachistochrone (brachys is brief, brachisto is quickest, chronos is time and brachistochrone is the shortes time). in a modern style: given two points a and b in a vertical plane, what is the curve traced out by a point subject only to the force gravity, starting from rest at a, such that it arrives at b in the shortest time? common sense suggests that this curve is necessarily situated in the vertical plane containing the points a and b. common sense also leads us to think that the quickest route is the shortest, and is given by the line segment joining the points a and b. but this is not the case. we know, for example that a longer journey on a motorway be faster than going a shorter distance on an ordinary road. here, in order to try to solve the problem of brachistochrome, it is necessary to consider all the curves joining points a and b and compare all the corresponding times of travel. taking everything into account, even under these restrictions, the problem turns out to be a subtle one. the brachistochrone problem, a priori a simple game for mathematicians, turns out in the end to be a considerable problem. en route for the calculus of variations 71 2 falling bodies, reflection and refraction in 1638, well before the problem had been explicitly stated, galileo gave his solution to the brachistochrone problem in the course of the third day of his [5]. it is here that he studied uniform acceleration – galileo called it “natural acceleration” – comparing it with uniform motion, and showed that a body falling in space traverses a distance proportional to the square of the time of descent (theorem ii in [4]). with regard to bodies moving on inclined planes he deduced ([5]): theorem v. the times of descent along planes of different length, slope and height bear to one another a ratio which is equal to the product of the ratio of the lengths by square root of inverse ratio of their heights. we interpret the proportionality to be: a body travels a distance l and descends a height h in time t such that: 𝑡 = 𝑘 ∙ 𝐿 √𝐻 . galileo then proves the following neat result ([5]): theorem vi. if from the highest or lowest point in a vertical circle there be drawn any inclined planes meeting the circumference, the times of descent along these chords are each equal to the other. at the end of the third day, galileo shows that it is also possible to improve on this descent ([5]): theorem xxii. if from the lowest point of a vertical circle, a chord is drawn subtending an arc not greater than a quadrant, and if from the two ends of this chord two other chords be drawn to any point on the arc, the time of descent along the two later chords will be shorter than along the first, and shorter also, by the same amount, than along the lower of these two latter chords. this result is false, since arguing the case from two to three segments is based on a faulty intuition from arguing from one to two segments. the brachistrochrone problem is considerably more subtle than the one of the research into optimum inclination of planes, which is a simple problem of the extremum for a function of single variable. the demonstration by johann bernoulli [1] also derives from an intuitive approach. this approach, an analogy with the law of refraction, leads to the curve solution which one cannot find a priori, without an arsenal of sufficiently sophisticated techniques. let us begin by recalling the first laws of optics, which are in fact consequences of the principles of optimization. experience tells us that light travels in straight lines. this phenomenon is stated as a principle: light chooses the shortest path. this formulation led to a real theoretical advance since it allowed hero of alexandria in the first century ad to explain the law of reflection, namely, the equality of the angles of incidence and reflection. in the case of reflection, the speed remains constant. it jan coufal and jiří tobíšek 72 is not so for refraction, where the speed of light n c varies as a function of the index n of the medium traversed. however, the principle stated above could have been stated in the following form as the fermat’s principle: light chooses the fastest route, which in a homogenous medium where its speed is constant, is equivalent to the previous principle. so, to go from a to b, passing from a medium of index 1 n to medium of index 2n , the trajectory of the light will not be the line segment ab, but broken line aib such that the trajectory aib will have the shortest time of all trajectories from a to b. using the initial conditions we calculate that the angle of incidence i and the angle of refraction r are related to the respective speeds by the formula: sin 𝑖 𝑣𝑖 = sin 𝑟 𝑣𝑟 , (1) or using the indices in and rn we have the sine formula 𝑛𝑖 ∙ sin 𝑖 = 𝑛𝑟 ∙ sin 𝑟. this formula, discovered by the dutch scientist snell in 1621, received its correct interpretation with fermat. in a letter of the 1st of january 1662 to m de la chambre, fermat explains ([4], vol. ii, pp. 457-463): as i said in my previous letter, m. descartes has never demonstrated his principle; because not only do the comparisons hardly serve as a foundation for the demonstrations, but he uses them in the opposite sense and supposes that the passage of light is more easy in dense bodies than in rare bodies, which is clearly false. i will not say anything to you about the shortcomings of the demonstration itself … fermat puts his principle to work, and proves the sine formula using his method ‘de maximis et minimis’ ([4]). another example of a non-homogeneous medium where the shortest trajectory is not the quickest occurs in mechanics, where the effect of gravity is in the vertical direction. and this is the context for johann bernoulli brachistochrone problem. johann bernoulli in the acta eruditorum of may 1697 ([1], vol. 1, pp. 187-193). his method typically corresponds to what we now call a discretisation of the problem. he imagines space carved into small lamina, sufficiently fine so that within each one it is possible to imagine that the speed is constant. within each strip the trajectory becomes the shortest route, and necessarily a segment. the complete trajectory appears as a sequence of segments. but how we move from one strip to another? we must always optimize the time of travel. as in refraction of light, this is done by using fermat’s principle. thus, if iv is the speed in a given band and rv in the band immediately below, the angle i is the angle made with the vertical by segment of the trajectory in the first band, an the angle r in the neighboring band, then they are connected by the rule of sines (1). if we now imagine that the horizontal strips become progressively thinner, and their number increases indefinitely, the line of segments tends towards a curve. the tangents at each en route for the calculus of variations 73 point of this curve approach the sequence of segments. the angle u which the tangent makes with the vertical is then connected to the speed v by the relation: sin 𝑢 𝑣 = 𝑐𝑜𝑛𝑠𝑡. here, the speed v of a particle is known; it is result of the action of gravity and, as we know from galileo, it is a function of the distance fallen y, according to the formula 𝑣 = √2𝑔𝑦. and so the rule of sines leads to the equation: sin 𝑢 √𝑦 = 𝑐𝑜𝑛𝑠𝑡. in particular, for y = 0, the tangent is vertical. that is a characteristic equation of a well-known curve of the time, the cycloid. we have just seen that the solution to the curve is a cycloid. but how can we construct such a curve, starting from a point a, an arriving exactly at a point b? newton gave a simple solution in a letter to montague on the 30th of january 1697 (see [10], p. 223). in addition to newton’s contribution to the solution of the problem of the brachistochrone, we must also mention leibniz, and in a lesser role, the marquis de l’hospital, and most of all, jacob bernoulli, the older brother of johann ([1], vol. 1, p. 194-204): … my elder brother made up the fourth of these, that the three great nations, germany, england, france, have given us each one of their own to unite with myself in such a beautiful search, all finding the same truth. the method used by jacob bernoulli is laborious, but quite general. also, jacob, in wanting to show the singular character of johann’s method, extended the problem by posing new questions. indeed, johann’s method, founded on an analogy, does not work except in a particular case, and cannot be used for more general problems of this type. in particular, jacob bernoulli put the following question to his brother” given a vertical line which of all the cycloids having the same starting point and the same horizontal base, is the one which will allow a heavy body passing along it to arrive at the vertical line the soonest? such statement reminds us of calileo’s first version, which was about finding the inclined plane through a given point which gave the shortest time to reach a given vertical. johann bernoulli ([1], vol. 1, p. 206-213) replied and showed that the cycloid in question is the one which meets the given line horizontally. more generally, the cycloid which allows us to achieve the swiftest possible descent to a given oblique line is the one which meets the line at right angles. this cycloid which, as we have just said, is a brachistochrone curve, was also known to huygens fro 1659 as the tautochrone curve: bodies which fall in an inverted cycloid arrive at the bottom at the same time, no matter from what height they are released. this property was perhaps closer to that observed by galileo: the jan coufal and jiří tobíšek 74 equality of the times for the distance on the chords of the same circle. among the other problems posed by jacob bernoulli to johann are those which are called isoperimetric problems, which together with brachistochrone problem are prototypes of optimization problems. these scientific exchanges between the two brothers were carried out in the form of letters. here is a sample of johann’s response to same criticisms by jacob ([1], vol. 1, p. 194-204): so there it is, his imagination, stronger and more vivid than those claiming to be sorcerers who believe they have found themselves bodily present at a sabbath, has seduced him; he is carried along by a torrent of vain conjectures; in a word, he is longer ready to give reign to reason … the resolution of these problems is then the object – reason or excuse? – for a long dispute between the two brothers; a dispute which developed into a major row, but which gave birth to new area in mathematics, the calculus of variations. 3 the calculus of variations when we look for boundary values of a function f of a variable x, i.e. when we look for values of the variable x for which the value 𝑓(𝑥) is a maximum or minimum, we look for the points where the graph of f has a horizontal tangent, or we say we look for the values where 𝑓 ′(𝑥) = 0. in the case of a function f of two variables x and y, we have to consider the points where the tangent plane is horizontal to the surface which has the equation 𝑧 = 𝑓(𝑥, 𝑦). alternatively we could say we seek the number pairs [ x, y ] for which 𝜕𝑓 𝜕𝑥 (𝑥, 𝑦) = 𝜕𝑓 𝜕𝑦 (𝑥, 𝑦) = 0. or we can say we are looking for the points where the function f has a stationary value. in the case of a finite number of variables, the difficulties seem surmountable, and the approach to the problem may be effected with the aid of the differential calculus of newton and leibniz. here the object which changes is not a number or a point, but a curve, a function, and the corresponding quantity to maximize or minimize is a number depending on this curve or on this function. it is necessary to conceive an extension of the differential calculus. the new theory which was created is called the calculus of variations, the variations being those of the function. but, in 1696, this theory had not been formulated and our problem becomes a priori somewhat subtle. a problem in the calculus of variations can be presented generally in the following fashion: we try to find a curve, being the graphical representation of a function y of x, which minimizes or maximizes a certain quantity among all the curves en route for the calculus of variations 75 constrained by certain conditions§. the quantity whose extreme value has to be found** is expressed generally in the form of an integral: 𝐼(𝑦) = ∫ 𝐹(𝑥, 𝑦, 𝑦′) 𝑑𝑥 𝑏 𝑎 where y represented the unknown function, 𝑦′ its derivative, x variable and f a particular function. among the typical problems of the calculus of variations, besides the isoperimetric problems above are investigations of the geodesic lines on surface, i.e. the curves of minimum length joining two points of a surface. also, the investigation of the shapes of the surfaces of revolution which offer the least resistance to movement, a problem which newton tackled in 1687 in the principia. the statement of the brachistochrone problem in 1696 could be considered as the definitive origin of the calculus of variations, for it is the problem which generated general methods of investigation which were gradually developed in a competitive context. johann bernoulli himself posed the problem of geodetics to euler. euler reworked the ideas of jacob bernoulli, simplified them, and finally was the first to formulate the general methods which allowed them to be applied to the principal problems of the calculus variation. he developed these ideas systematically in 1744 in [3]. in a way like jacob bernoulli, euler tackles the problem as a problem of limits in an investigation of the ordinary extremum. euler derived the differential equation: 𝜕𝐹 𝜕𝑦 (𝑥, 𝑦, 𝑦′) − 𝑑 𝑑𝑥 ( 𝜕𝐹 𝜕𝑦′ (𝑥, 𝑦, 𝑦′)) = 0 (2) which satisfies each solution y. it is only a necessary condition and the method does not establish the existence of a solution. the equation (2), today called the euler-lagrange equation, is a second order differential equation in y: 𝜕𝐹 𝜕𝑦 (𝑥, 𝑦, 𝑦′) − 𝜕2𝐹 𝜕𝑦′𝜕𝑥 (𝑥, 𝑦, 𝑦′) − 𝜕2𝐹 𝜕𝑦′𝜕𝑦 (𝑥, 𝑦, 𝑦′) − 𝜕2𝐹 𝜕𝑦′2 (𝑥, 𝑦, 𝑦′) = 0. in 1760, lagrange greatly simplified matters by introducing the differential symbol δ, specifically for the calculus of variations, corresponding to a variation of the complete function. he makes the point of it in the introduction to [6]: for as little as we know the principles of the differential calculus, we know the method for determining the largest and smallest ordinates of curves; but there are questions of maxima and minima at a higher level which, although depending on the same method, are not able to be applied so easily. they are § for brachistochrone problem – the curve joining two points a and b. ** here – the time of the journey. jan coufal and jiří tobíšek 76 those where it is needed to find the curves themselves, in which a given integral expression becomes a maximum or minimum with respect to all the other curves. … now here is a method which only requires a straightforward use of the principles of the differential and integral calculus; but above all i must give warning that while this method requires that the same quantities vary in two different ways, in order not to mix up these variations, i have introduced into my calculations a new symbol δ. in this way, δz expressed a difference of z which is not the same as dz, but which, however, will be formed by the same rules; such that where we have for any equation dz=m dx, we can equally have δz=m δx, and likewise for other cases. a century later, mach was able to write in [7]: in this way, by analogy, johann bernoulli accidentally found a solution to the problem. jacob bernoulli developed a geometric method for the solution of analogous problems in one stroke, euler generalized the problem and the geometrical method, lagrange finally freed it completely from the consideration of diagrams, and provided an analytical method. 4 the principle of least action we shall make a digression, the purpose of which will soon become clear maupertuis stated his principle of least action in 1744 in [8]. he explains and justifies his principle from the law of refraction: in thinking deeply upon this matter, i reflected that light, as it passes from one medium to another, yet not taking the shortest path, which is a straight line, might just as well not take the shortest time. actually, why should there be a preference here for time over space? light cannot go at the same time by the shortest path and by the quickest route, so why does it go by one route rather than another? in fact, it does not take either of these; it takes a route that has the greater real advantage: the path taken is the one where the quantity of action is the least. now i must explain what i mean by the quantity of action. when a body is moved from one place to another, a certain action is needed: this action depends neither on the speed of the body and the distance travelled; but it depends on the speed nor the distance taken separately. the quantity of action is moreover greater when the speed of the body is greater and when the path travelled is greater; it is proportional to the sum of the distance multiplied respectively by the speed travelled over each space. … it is quantity of action which is the true expenditure of nature, and which she uses as sparingly as possible in the motion of light. let there be two different media, separated by a surface represented by the line cd, such that the speed of light in the medium above is m. and the speed in the medium below is n. let a ray of light, starting from point a, reach a point b: to find the point r where the ray changes course, we look for the point where if the ray bends the en route for the calculus of variations 77 quantity of action is the least: and i have 𝑚 ∙ 𝐴𝑅 = 𝑛 ∙ 𝑅𝐵 which must be a minimum. … that is to say, the sine of the angle of incidence to the sine of the angle of refraction is in inverse proportion to the speed with the light traverses each medium. all the phenomena of refraction now agree with the central principle that nature, in the production of its effects, always tends towards the most simple means. so this principle follows, that when light passes from one medium to another the sine of the angle of refraction to the sine of the angle of incidence is in inverse ratio to the speed with which the light traverses each medium. and so for maupertuis, light is propagated so as to minimize 𝐴𝑅 ∙ 𝑣1 = 𝑅𝐵 ∙ 𝑣2 and not the quantity 𝐴𝑅 𝑣1 = 𝑅𝐵 𝑣2 . for these conclusions to agree with the experimental results of the time, and so that his principle would lead to the sine law. it is true that at that time no one knew how to measure the speed of light and no one could find a way of deciding between the different theories. the experimental proof that light travels faster in air than in water was not established until 1850 foucault. in 1746, maupertuis extended his principle from optics to mechanics ([9]): when a body is carried from one place to another, the action is greater when the mass is heavier, when the speed is faster, when the distance over which it is carried is longer. … whenever a change in nature takes place, the quantity of action necessary for this change is the smallest possible. with this general principle, maupertuis established a kind of union between philosophy, physics and mathematics: nature works in such a way as to minimize its action; the idea of causality is abandoned in favor of the idea achieving an aim, characterized by a harmony between the physical world and rational thought. 5 conclusion it would be right to conclude by revisiting our initial problem of the slides in the playground. we are circumspect, and content ourselves with noticing that in the course of this wander through diverse disciplines, the theme of minimization or maximization briefly the problem of optimalization is ever present, and should not be underestimated during these unhappy times. acknowledgements this contribution is a follow-up to the project of the centre of economic studies of university of economics and management jan coufal and jiří tobíšek 78 references [1] bernoulli, j. opera omnia. lausanne and geneva, 1742. [2] chabert, j.-l. the brachistrone problem. history of mathematics, histories of problems, elipse, paris, 1997, pp. 183-202. [3] euler, l. methodus inveniendi lineas curvas maximi minimive proprietate gaudentes: sive solution problematic isoperimetrici lattissimo sensu accepti, lausanne and geneva, 1774 (œvres, vol. 24, berne, orel füssli, 1952). [4] fermat, p. œvres, ed. tannery, p. and henry, c., gauthier-villars, paris, 1894. [5] galileo, g. discorsi e dimostrazioni matematiche intorno a duo nuove scienze, leyden, 1638. [6] lagrange, j.-l. essai d’une nouvelle méthode pour déterminer les maxima et les minima des formules integrals defines. miscellanea taurinensia, vol ii, 1760-1761 (œvres, vol. i, paris, pp. 333-468). [7] mach, e. die mechanik in ihrer entwickelung : historisch-kritisch dargestellt, f. a. brockhaus, leipzig, 1883. [8] maupertius, p. l. m. de. accord de différentes lois de la nature qui avaient jusqu'ici paru incompatibles. mémoires de l’acadenie des sciences de paris, 1744, pp. 417-426. [9] maupertius, p. l. m. de. les lois du movement et du repos déduites d’un principe métaphysique. mémoires de l’acadenie des sciences de berlin, 1746, pp. 267-294. [10] newton, i. correspondence, ed. by scott, j. f., vol. iv (1694-1709), cambridge university press, cambridge, 1967. microsoft word cap5.doc microsoft word di bartolomeo.doc 1 existence and policy effectiveness in feedback nash lq-games* nicola acocella university of rome la sapienza and giovanni di bartolomeo university of teramo september 2007 abstract. this paper illustrates how the classical theory of economic policy can profitably be used to verify some properties of the linear nash feedback equilibrium in difference lq-games. in particular, we find that both a necessary condition for the equilibrium existence and a sufficient condition for policy ineffectiveness can be defined in the terms of the simple tinbergen counting rule. jel classification: c72, e52, e61. keywords: lq-policy games, policy ineffectiveness, controllability, linear feedback nash equilibrium existence. 1. introduction one fundamental question of many economic debates is that of effectiveness of public or private agents’ action. the most prominent examples are the long debates on the neutrality of monetary and fiscal policy. however, the issue is not exclusive of economic policy; on the contrary, ineffectiveness has to be interpreted in a broader sense. for instance, debates on the effectiveness of advertising strategies in marketing or firm’s price policies are related to a similar general problem. * we are grateful to j. capaldo, j. engwerda, a. hughes hallett, r. neck for their comments. nicola acocella acknowledges the financial support of the university of rome la sapienza. giovanni di bartolomeo acknowledges the financial support of the european union (marie curie programme). this research project has been supported by a marie curie transfer of knowledge fellowship of the european community's sixth framework programme under contract number mtkd-ct-014288. 2 in so far as the decision-maker’s action does not take account of the reactions of other agents (parametric approach), the debate is a purely empiric issue. however, especially after lucas’ (1976) criticism, a parametric approach can hardly be justified and a strategic one seems to be more coherent in almost all economic applications. in that case, the need for describing the mechanisms leading to ineffectiveness in general terms becomes a crucial theoretical modeling issue. some recent studies in that direction provide some general conditions for policy ineffectiveness and equilibrium existence in static lq-games (see acocella and di bartolomeo, 2004, 2006). this new approach shows how the classical theory of economic policy can profitably be used to define some general properties of policy games. this paper is one of the first attempts to extend this approach to a context of dynamic games. 1 more specifically, we consider feedback equilibrium in lq-difference games and illustrate how a necessary condition for the equilibrium existence and a sufficient condition for policy ineffectiveness can be defined in terms of a simple counting rule of targets and instruments. in a companion paper, acocella et al. (2007) have investigated the same problem in the case of sparse-matrix dynamic systems, when, written in its structural form, a large dynamic system governing the economy relates each endogenous variable to just a few other endogenous variables and a small number of lagged endogenous variables, control variables, or predetermined variables. in such a context, acocella et al. (2007) provide an example to illustrate the usefulness of this line of research by considering a model incorporating a taylor rule, a description of expectations formation and a relation that can be interpreted as either a dynamic open-economy phillips curve or a new-keynesian is curve with dynamics. small variations in the model specification can bring, or take away, policy effectiveness – allowing the policy makers the possibility to disagree on none, one or several of the target values in their (common) targets – and possibly make institutional and policy independence counterproductive. this paper focuses on a case opposite to that of a sparse matrix system, i.e. a case of full rank matrices, where all the economic variables are highly interdependent. 1 difference and differential games are extensively studied and used in many economic applications. for applications to macroeconomic theory, see among others levine and brociner (1994), neck and dockner (1995), aarle et al. (1997), başar et al. (1988), levine and smith (2000), engwerda et al. (2002), pappa (2004), di bartolomeo et al. (2006), plasmans et al. (2005). for a more complete overview considering also different areas, see dockner et al. (2000). see başar and olsder (1995), dockner et al. (2000), engwerda (2000a, 2000b) for some methodological aspects. 3 the outline of the paper is as follows. section 2 defines basic concepts and introduces a formal framework to describe lq-difference games. section 3 derives two theorems, stating a sufficient condition for policy ineffectiveness and a necessary condition for the equilibrium existence in the traditional tinbergen’s terms. the paper ends with some concluding remarks. 2. controllability and lq-policy games 2.1 some preliminary definitions in order to apply the traditional theory of economic policy to study the properties of nash feedback equilibrium, we first recall the traditional tinbergen’s golden rule. definition (golden rule): a policymaker satisfies the golden rule of economic policy if the number of its independent instruments equals the number of its independent targets. second, we need to redefine policy ineffectiveness, since its classical definition2 cannot be maintained in the realm of policy games as policy instruments are endogenous variables, whose values really depend on the preferences of the decision-makers. the following definition of ineffectiveness can be accepted instead.3 definition (ineffectiveness): a policy is ineffective if the equilibrium values of the targets are never affected by changes in the parameters of its criterion. we are now ready to introduce our policy game. 2.2 a formal framework we consider the problem where n players try each to minimize their respective quadratic performance criterion. the game is played for t periods, where t may be arbitrarily large. each player controls a different set of inputs to a single system, which is described by the following difference equation: (1) ( 1) ( ) ( )i i i n x t ax t b u t     2 the classical definition of policy ineffectiveness implies that autonomous changes in policymaker’s instruments have no influence on the targets 3 see gylfason and lindbeck (1994). 4 where n is the set of the players; mx � is the vector of the states of the system; ( )m iiu � is the (control) variable vector that player i can manipulate; m ma � and ( )m m iib � are full rank matrices describing the constant system parameters. the performance criterion player i n aims to minimize is (2)      1 2 0 , ,..., ( ) ( ) t i n i i i t j u u u x t x q x t x     where mix � is a vector of given target values and q is an appropriate constant symmetric positive definite matrix of weights. we assume that i jx x for all i n j n   . for reasons that we shall clarify we keep targets and instruments formally separate. however, in order to take account of the costs or limits in the use of some instruments, we could simply introduce an additional loss into equation (2) due to deviations of the instruments from the vector of their target values or equality (static) constraints concerning the instruments into equation (1). 3. controllability, ineffectiveness and equilibrium existence controllability, in the terms of the golden rule of economic policy, ineffectiveness and linear feedback nash equilibrium existence are related together by the following two theorems. theorem 1 (ineffectiveness): if one (and only one) player satisfies the golden rule, all the other players’ policies are ineffective. theorem 2 (non-existence): no linear feedback nash equilibrium4 of the policy game described exists if at least two players satisfy the golden rule (unless they share the same target values). the proofs of the theorems are simple and can be combined into a single one. proof. we start by guessing that the policymakers’ value functions are quadratic, 5      ( ) ( ) ,i i i iv x x t x p x t x   where ip are positive definite symmetric matrices (for the 4 we restrict ourselves to the case of the feedback nash equilibrium based on a linear rule as defined in engwerda (2006). in principle our results can be extended to the case of non-linear feedback rules by assuming an affine function as value function. however, the task is not without computational costs because the difficulties of computing non linear rules, which usually are infinite (see tsutsui and mino, 1990). see also dockner et al. (2000) and fujiwara and matsueda (2007), who also show how to derive the markov feedback nash equilibrium from a linear rule. in addition notice that the linear rule result to be optimal for the case of a finite period lq games (see başar et al., 1988; or başar and olsder, 1995, chapter 6). 5 sake of simplicity, time indexes will be omitted). by using the transition law to eliminate the next period state, the n bellman equations become: (3)        min i i i i i i i i i i i i i iu i n i n x x p x x x x q x x ax b u x p ax b u x                               a linear feedback nash equilibrium must satisfy the first-order conditions: (4)   / i i i i i i i i i j j j n i b p b u b p ax x b p b u         to which the following policy rule corresponds: (5)      1 1 / i i i i i i i i i i i i j j j n i u b p b b p ax x b p b b p b u            now, to demonstrate theorem 1, we focus on player 1 without loss of generality. if player 1 satisfies the golden rule, then (1)m m and 1 m mb � is a square matrix. equation (5) becomes: (6)  1 11 1 1 1 2 n j j j u b ax x b b u        which implies: (7) 1( 1)x t x  for  0,t   thus, if a linear feedback nash equilibrium exists, the value of the target variables only depends on the preferences of player 1, since, in this case, condition (6) must be verified for all  0, .t   this completes the proof of theorem 1. as to theorem 2, we only need to show that, if also another player (e.g. player 2) satisfies its golden rule, the equilibrium does not exist. as above, if also player 2 satisfies its golden rule, the following reaction function must hold in equilibrium: (8)  1 1 12 2 2 2 1 1 2 3 n j j j u b ax x b b u b b u          by plugging equation (8) into equation (6), it is clear that they cannot be mutually satisfied unless: (9) 1 2 0x x  5 recall that we are looking for the linear feedback nash equilibrium. see engwerda (2006: section 4) for more technical details. see also sargent (1987: 42-48) or ljungqvist and sargent (2004: chapter 5) for the single policymaker case. 6 which is not true in general, since 1 2x x .  theorem 1 gives a sufficient condition for policy ineffectiveness, but this does not assure the equilibrium existence, which may fail to occur. by contrast, theorem 2 gives a necessary condition for the equilibrium existence since it states a sufficient condition for non-existence. however, it may be not sufficient.6 note that if theorem 1 is satisfied, theorem 2 is not. this directly derives from the sentence in bracket in theorem 1. 3. the instrument cost issue it is useful to compare our results to a well-known theorem of existence of nash equilibrium, dasgupta and maskin’s (1986), which relates existence to the costs of the instruments since, in a similar manner, we have expressed the necessary condition for existence in terms of an instruments/targets counting rule. dasgupta and maskin (1986) shows that a sufficient condition for the nash equilibrium existence is that the space of strategies of each player is convex and compact. if the players’ controls are unbounded, the nash equilibrium may not exist. in static linear quadratic games, the introduction of quadratic instrument costs would make them bounded, thus assuring equilibrium existence. in our terms, the introduction of quadratic instrument costs would imply that the dimensions of matrices iq become ( )m m i . thus, the number of instruments would always be less than that of targets, the system would be not controllable by any player and equilibrium would exist. it is worth noticing, however that theorem 2 is more general than the mentioned theorem of existence, since that of instrument costs is a particular case. 4. concluding remarks this paper represents a first attempt to generalize some recent results developed in static policy games to a dynamic context. in the fashion of the classical theory of economic policy, we have shown that if one player satisfies the well-known simple tinbergen’s counting rule 6 existence is a rather complex matter in this context. moreover, being in a dynamic system also stability is required. see engwerda (2000a, 2000b). 7 (golden rule), either all the other players’ policies are ineffective or no linear nash feedback equilibrium exists, without exact agreement on all the (dynamic) target values, (i.e., unless all the players satisfying the golden rule have equal targets values). in doing that, since difference games imply many technical complications, we have introduced a number of simplifications. some of them are not crucial. others cannot be easily relaxed. for instance, discounting, a finite time, and additive uncertainty can be easily introduced. of course, in the case of stochastic games we should discuss our results in terms of expected values. we have also assumed that all the policy-makers share all the targets. however, our theorems are probably weak, in the sense that they are limit cases based on the strong concept of static controllability. it is well known, in fact, that in general fewer instruments than targets are needed to control a dynamic system. once the theorems are reformulated in terms of dynamic controllability, by following preston (1974) and aoki (1975), it may be possible to define more general and less stringent conditions. this seems to be a promising line for future research. references aarle, b. van, a.l. bovenberg, and m.g. raith (1997), “is there a tragedy for a common central bank? a dynamic analysis,” journal of economic dynamics and control, 21: 417-447. acocella, n. and g. di bartolomeo (2004), “non-neutrality of monetary policy in policy games,” european journal of political economy, 20: 695-707. acocella, n. and g. di bartolomeo (2006), “tinbergen and theil meet nash: controllability in policy games,” economics letters, 90: 213-18. acocella, n., g. di bartolomeo, and a. hughes hallett (2007), “dynamic controllability with overlapping targets: or why target independence may not be good for you,” macroeconomic dynamics, 11, 202-213. aoki, m. (1975), “on a generalization of tinbergen’s condition in the theory of policy to dynamic models,” review of economic studies, 42: 293-296. başar, t. and g.j. olsder (1995), dynamic noncooperative game theory, second edition, academic press limited, london. başar, t., v. d’orey, and s.j. turnovsky (1988), “dynamic strategic monetary policies and coordination in interdependent economies,” american economic review, 78: 341-61. dasgupta, p. and e. maskin, (1986), “the existence of the equilibrium in discontinuous economic games. i: theory,” review of economic studies, 53: 1-26. 8 di bartolomeo, g., j.c. engwerda, j. plasmans, b. van aarle, and t. michalak (2006), “macroeconomic stabilization policies in the emu: spillovers asymmetries and institutions,” scottish journal of political economy, 53: 461-483. dockner, e., s. jorgensen, n. van long, and g. sorger (2000), differential games in economics and management sciences, cambridge university press, cambridge. engwerda, j.c. (2000a), “feedback nash equilibria in the scalar infinite horizon lq-game,” automatica, 36: 135-139. engwerda, j.c. (2000b), “the solution set of the n-player scalar feedback nash algebraic riccati equations,” ieee transactions on automatic control, 45: 2363-2369. engwerda, j.c. (2006), “linear quadratic differential games: an overview,” university of tilburg, center discussion paper no. 110. engwerda, j.c., b. van aarle, and j. plasmans (2002), “cooperative and non-cooperative fiscal stabilisation policies in the emu,” journal of economic dynamics and control, 26: 451-481. fujiwara k. and n. matsueda (2007), “on a nonlinear feedback strategy equilibrium of a dynamic game,” economics bulletin, 3, no. 8. gylfason, t. and a. lindbeck (1994), “the interaction of monetary policy and wages,” public choice, 79: 33-46. ljungqvist l. and t. sargent (2004), recursive macroeconomic theory, second edition, the mit press, cambridge. levine, p. and a. brociner (1994), “fiscal policy coordination and emu,” journal of economic dynamics and control, 18: 699-729. levine, p. and r. smith (2000), “the arms trade game: from laissez faire to a common defence policy,” oxford economic papers, 52: 357-380. lucas, r.e. (1976), “econometric policy evaluation. a critique,” journal of monetary economics, supplement, carnegie-rochester conference series on public policy, 1: 19-46. neck, r. and e.j. dockner (1995), “commitment and coordination in a dynamic game model of international economic policy-making,” open economies review, 6: 5-28. pappa, e. (2004), “do the ecb and the fed really need to cooperate? optimal monetary policy in a two-country world,” journal of monetary economics, 51: 753-779. plasmans, j., j.c. engwerda, b. van aarle, g. di bartolomeo, and t. michalak (2005), dynamic modeling of monetary and fiscal cooperation among nations, springer, berlin. preston, a.j. (1974), “a dynamic generalization of tinbergen’s theory of policy,” review of economic studies, 41: 65-74. sargent, t.j. (1987), dynamic macroeconomic theory, harvard university press, cambridge. tsutsui, s. and k. mino (1990), “nonlinear strategies in dynamic duopolistic competition with sticky prices,” journal of economic theory, 52: 136-161. microsoft word cap9.doc microsoft word r.m.7 cap.13.doc microsoft word r.m.7 cap.16.doc microsoft word capitolo intero n 11.doc approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 39, 2020, pp. 33-54 33 modelling the shape of sunspot cycle using a modified maxwell-boltzmann probability distribution function amaranathan sabarinath * girija puthumana beena † ajimandiram krishnankuttynair anilkumar‡ abstract the 11-year sunspot number cycle has been a fascinating phenomenon for many scientists in the last three centuries. various mathematical functions have been used for modelling the 11-year sunspot number cycles. in this paper, we present a new model, which is derived from the well known maxwell-boltzmann probability distribution function. a modification has been carried out by introducing a new parameter, called area parameter to model sunspot number cycle using maxwellboltzmann probability distribution function. this parameter removes the normality condition possessed by probability density function, and fits an arbitrary sunspot cycle of any magnitude. the new model has been fitted in the actual monthly averaged sunspot cycles and it is found that, the hathaway, wilson and reichmann measure, the goodness of fit is high. the estimated parameters of the sunspot number cycles 1 to 24 have been presented in this paper. a monte carlo based simple random search is used for nonlinear parameter estimation. the prediction has been carried out for the next sunspot number cycle 25 through a model by averaging of recent cycle's model parameters. this prediction can be used for simulating a more realistic sunspot cycle profile. through extensive monte carlo simulations, a large number of sunspot cycle profiles could be generated and these can be used in the studies of the orbital dynamics. keywords: solar cycle; modelling; sunspot number 2010 ams subject classification: 70f15; 97m10.§ * vikram sarabhai space centre, thiruvananthapuram, kerala, india; a_sabarinath@yahoo.co.in. † assistant professor, st.gregorios college, kottarakkara, kerala, india; beenamabhi@gmail.com. ‡ director, dssam, isro head quarters, bangalore, karnataka, india; ak_anilkumar@isro.gov.in. sabarinath.a, beena.g.p, anilkumar.a.k 34 1 introduction to know in advance the multitude of atmospheric processes that cause concern to mankind, in particular phenomena occurring in the solar plasma receive great consideration from the scientific world. since the 18th century, scientists are conducting systematic research on a multitude of processes caused by solar activity. solar activity forecasting is crucial in scientific and technological fields such as spacecraft orbital life time prediction, airline communications and geophysical applications, mainly it is the energy source behind all phenomena driving space weather. the low earth orbiting satellites are also influenced by solar activity (seeds,m.a,backman,d,[2015]; hathaway,d.h., [2010]). however, predicting the solar cycle is challenging on the basis of time series of various proposed indicators, due to the high frequency contents, noise contamination, high dispersion level and high variability both in phase and amplitude. the prediction of solar activity is complicated by the lack of a quantitative theoretical model of the sun's magnetic cycle. the effect of solar activity is greater on space activities especially on the operations of low earth orbiting satellites which provide significant contribution in communication, national defence and earth mapping. such satellites also handle a large quantity of scientific data. during higher solar activity, the maximum ultraviolet rays are emitted from the sun that heat up earth's upper atmosphere, and expands the atmosphere. this affects the life time of operational space crafts in the low earth orbits (whitlock,d, [2006]). therefore better predictions of solar activity are essential to help spacecraft mission planning and design. 2 satellite life time estimation and re-entry prediction in spacecraft mission design, orbital life time estimation is a critical activity (whitlock, d, [2006]). many uncertain parameters need to be considered while doing orbital life time estimation. the upper atmospheric density variation is the primary factor which is so difficult to predict. many studies have been taken place to model the atmospheric density accurately. orbital life time estimation community has always been looking up for better models of atmospheric density. atmospheric models generally use parameters such as ap or kp, and f10.7. solar flux receives a lot of attention because it is an important parameter in determining atmospheric density. most predictions rely § received on october 24th, 2020. accepted on december 17th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.532. issn: 1592-7415. eissn: 2282-8214. ©sabarinath et al. this paper is published under the cc-by licence agreement. modelling the shape of sunspot cycle using a modified maxwellboltzmann probability distribution function 35 on the sunspot activity happening in the sun. this has been monitored since the 17th century regularly. an empirical relationship exists between the sunspot number, r, averaged over a month, and f10.7 (david a.vallado et.al ,[2014]). f10.7 = 63.7 + 0.728 r + 0.000 89 r2, (1) from the above equation, we can see that 10.7 cm radio flux has a base level of about 63.7 solar flux units. to understand and estimate the radio emissions effectively we can use the following equation (david a.vallado et.al, [2014]) f10.7 = 145 + 75 cos (0.001696 t + 0.35 sin (0.00001695)), (2) where t is the number of days from january 1, 1981. we can summarise it as, atmospheric density is directly related to the solar flux, which in turn can be related to the solar activity. studies done by different scientists and academicians shows that solar activity and solar flux have affirmed relation, a monthly estimate of f10.7 and sunspot number has been well established. predicting the solar flux accurately can generate more accurate atmospheric density models that will help in fine tuning the fuel budget for longer satellite life. the discussion went so far reminds that the accurate prediction of the life time requires a very good predicted solar flux profile. in turn, it is sufficient to have a predicted sunspot number cycle. since, via equation (1) one can transform sunspot numbers into solar flux. in this paper we try to predict sunspot number cycle in a simple and powerful technique. initially, we model the sunspot cycle using a skew-symmetric probability distribution. the maxwell-boltzmann distribution is considered for this purpose. then a preliminary level prediction is proposed as an average (mean) cycle of some recent cycles. then a varying error band is derived from the past cycles. within this error profiles, via monte carlo sampling, the predicted averaged cycle is transformed into many profiles. sample profiles are taken and plotted. before venturing into the details, a brief review of sunspot data and review some of the recent models are provided. 3 sunspot number cycles and sunspot number data in 1848 the swiss astronomer johann rudolph wolff introduced actual measurements of sunspot number. his method uses still today. total number sabarinath.a, beena.g.p, anilkumar.a.k 36 of spots visible on the face of the sun is 'n' and the number of groups into which they cluster is 'g' then the sunspot number rn is defined as rn = 10 g + n. (3) to compensate the observational limitations like earth's atmosphere variability above the observing site and sun's rotation, each daily sunspot number is computed as a weighted average of measurements made from a network of observatories. the 11-year cyclic variation in the sunspot numbers was first noted by schwabe, m., [1844]. in 1848 rudolf wolf at swiss federal observatory in zurich, switzerland devised his measure of sunspot numbers that continues to this day as the international sunspot number. wolf recognised that it is far easier to identify sunspot groups than to identify each individual sunspot. this relative sunspot number,rz with emphasis on sunspot groups is defined as, rz = k (10 g + n), (4) where k the correction factor for the observer, g is the number of identified sunspot groups, and n is the number of individual sunspots. these sunspot numbers are called the zurich or international sunspot numbers have been obtained daily since 1848. sunspot cycle time series is one of the longest time series which was studied by many experts for various reasons. first of all, this time series is non-stationary, cyclic and highly nonlinear in the time domain. in the present study, the prediction of sunspot cycles is carried out with the monthly averaged sunspot number values. the monthly averaged sunspot data were available from, http://www.sidc.be/silso/versionarchive at the royal observatory, belgium is being used for the present study. it may be noted that, the scientific community recently recalibrated the entire historical sunspot number record and that silso (sunspot index and long-term solar observations) maintains this new definitive record as well as the original version of sunspot numbers. figure 1: sunspot cycle evolution-monthly averaged sunspot numbers from the year 1749 to december 2016. 1749 1791 1832 1874 1916 1957 1999 0 50 100 150 200 250 300 m o n th ly a v e ra g e d s u n s p o t n u m b e rs time (year) modelling the shape of sunspot cycle using a modified maxwellboltzmann probability distribution function 37 4 existing models of sunspot number cycles several mathematical functions were introduced to model the shape of the sunspot number cycle. due to the exponential rise and decay, the exponential function was used by nordemann, [1992], nordemann, et.al.,[1992]. the bell shaped nature of the sunspot cycle was explored by hathaway et.al.,[1994]. few statistical probability distribution functions were also proposed for the shape modelling by various authors. de mayer, f.,[1981], proposed a model using periodic functions. in prediction, averaged models are used as an initial estimate of the future cycle. we have an exhaustive list of details and voluminous data literature available at hand pertaining to the attempts to predict the future behaviour of solar activity (hathaway, et.al., [1999]). it can be categorised under five heads, based on the nature of the prediction methods. they are: 1) curve fitting, 2) precursor, 3) spectral, 4) neural networks and 5) climatology (sello, s.,[2001]). mcnish-lincoln curve fitting was the first attempt on the methodology of curve fitting (de meyer,[1981], mcnish, a.g., lincoln, j.v.,[1949]). over the years, various techniques and models have been proposed by several authors working in the field for the prediction of the nonlinear behaviour of sunspot cycles. the first breakthrough in the field of modelling the shape of the sunspot cycles by fitting an exponential function over the sunspot number cycle time series was due to nordemann,[1992]. in this method, fitting the rise to maximum and the fall to minimum were fitted with a function of exponential function demanding six free parameters. later a modified version of f-distribution density function with five parameters was proposed by elling and schwentek[1992]. nordemann's[1992] method suggests exponential fitting and explain the solar behaviour. hathaway, wilson, and reichmann[1994] substantiated the superiority of a new model along with a measure for the goodness of fit. number of free parameters in this model is reduced to four. all these models introduce high amount of error in the prediction, due to the incompetence to fit the peak locations of the sunspot cycle. the continuous nature of the model at the high solar activity period contributes a large amount of uncertainty and hence in the applications such as the orbital re-entry predictions these models are not suitable. the next subsection surveys the literature pertaining to some models, especially on the shape of sunspot cycles. 4.1 stewart and panofsky model stewart and panofsky [1938] proposed a function for the shape of the cycle with the form sabarinath.a, beena.g.p, anilkumar.a.k 38 r(t) = a(t − t0) be−c(t−t0), (5) where a, b, c, and t0 are parameters that vary from cycle to cycle. the important thing to be noticed is that, this model gives a power law for the rising phase of a cycle and an exponential for the declining phase of a cycle. the model parameters for cycle 1to 16 were computed and there by the maximum amplitude, the epoch of the peak sunspot number, etc. was predicted. 4.2 nordemann model nordemann used the solution of the differential equation dn dt = kn, in analogy with the nuclear decay process. thus the declining phase of a sunspot cycle is represented by: n = n0e kt k < 0 (6) and the solution of dn dt = a + kn, is used to represent the ascent phase of a sunspot cycle. thus the model for the ascent phase is: n = a k (1 − ekt) k < 0 (7 ) where n represents sunspot numbers, k decay constant and a a production parameter. the estimated values of the parameters n0, k and a for all the 22 sunspot cycles were given in nordemann [1992]. 4.3 elling and schwentek model instead of using yearly means, quarterly averages of sunspot numbers were utilised by elling and schwentek[1992] for optimal fitting of each cycle. they used a modified f-distribution density function that required five free parameters. this approach is much more worth than the previous models. in this model fitting concluded only for modern era of sunspot cycles (10 to 21). by considering the maxima and minima of mean sunspot number as a function of time, affinity can be observed in each cycles. while considering different sunspot cycles the ascending phase take dwindle time than the descending phase, that means starting from a minimum, time taken for reaching the maximum is always shorter as compared to the time from maximum down to minimum . modelling the shape of sunspot cycle using a modified maxwellboltzmann probability distribution function 39 they explained very effectively, ascending and descending branches of the various cycle curves have curvatures which are rather similar to those of the fdistribution curves. for this reason, each sunspot cycles from cycle 10 to cycle 21 has been approximated by a modified f-distribution, f(t) which is defined by: f(t) = p4 γ [ p2 + p3 2 ] γ ( p2 2 ) γ ( p3 2 ) p 2 p2 2 p 3 p3 2 [p1(t + p5)] p2 2 −1 [p3 + p2p1(t + p5)] (p2+p3) 2 , (8) where t is the time and γ(x) is the gamma function. p1 is the length or duration of the sunspot cycle, that is, the time interval from one minimum to the next, p2 to the curvature of the ascending branch of f(t), p3 to the curvature of the descending branch of f(t), p4 to the amplitude of the maximum of f(t), p5 to the time shift of the f(t) curve. through least square fit all the five parameters are estimated. 4.4 hathaway, wilson, and reichmann model hathaway et.al [1994] suggested a model with free parameters fewer than the models which we had come across. they utilised a four-parameter quasiplanck function to fit the monthly mean sunspot numbers of a solar cycle, similar to that of stewart and panofsky[1938]. but the only difference we can see that a fixed power law for the initial rise of the sunspot cycle and the phase starting from maximum down to minimum can be well represented by a function that decreases as e−t 2 . by combining these, the model as a function of time can be written as: f(t) = a(t − t0) 3 e [ (t−t0) 2 b2 ] − c , (9) this model has four parameters. a represents the amplitude and is directly related to the rate of rise from minimum; b is related to the time in months from minimum to maximum; c gives the asymmetry of the cycle; and a starting time t0 . along with the early detection of parameters to predict the solar activity they examine the relationship between the parameters. it is similar to the plank function but contains four free parameters and has a more rapid decrease after maximum, but causes lack of accuracy. the estimation of these parameters was obtained through levenberg-marquardt methods (press, w, h., [1992]). sabarinath.a, beena.g.p, anilkumar.a.k 40 4.5 volobuev’s one-parameter fit in 2009, volobuev introduced a function of two-parameters and he refers to this as a one parameter fit. we can see that the parameters are correlated (r = 0.88) for all the 23 solar cycles. the correlation between the parameters provides the possibility of a one-parameter fit by neglecting the need to determine the best starting time. he showed that a one-parameter fit can also be derived from truncated dynamo models. due to the unavoidable uncertainty of starting time goodness of fit value is not better as compared to the empirical fit. we can see that this model is also similar to that of stewart and panofsky [1938] proposed pearson's type iii curves by putting b =2 and modifying the growth multiplier and decay multiplier properly by introducing the new parameters ts and td. the empirical model used is written as: r = ( t − t0 ts ) 2 e −( t−t0 td ) 2 , (10) 4.6 sabarinath and anilkumar model sabarinath and anilkumar[2008] proposed a model consist of a mixture of laplace distribution with six parameters (later reduced to two). this model fits the multiple sharp peaks in a solar cycle. the model for a generic cycle is: f = a1 33.2 exp ( −|t − 41.7| 16.6 ) + a2 46 exp ( −|t − 67.3| 23 ) , (11) where t is the time. 5 skew symmetrical distributions sunspot cycles are asymmetric with respect to their maxima (hathaway, d.h., [2010]). starting from minimum the time taken to reach maximum is 48 months and 84 months to fall back to minimum again. an average cycle can be constructed by stretching and contracting each cycle to the average length and normalising each to the average amplitude. in general, if we survey any model of the shape of the sunspot cycle, it is evident that, all functions are a product of a polynomial and a negative exponential function. then the goodness of fit solely depends on how the model parameters are chosen in the model. in this context, we propose a skew modelling the shape of sunspot cycle using a modified maxwellboltzmann probability distribution function 41 symmetrical function from the class of skew symmetrical probability functions. 6 maxwell-boltzmann probability distribution function in statistical physics, maxwell-boltzmann distribution is a probability distribution named after the famous scottish physicist james clerk maxwell and ludwig boltzmann. it is used in atomic physics for describing particle speeds in idealised gas. the maxwell-boltzmann distribution function is given as (balakrishnan, n., nevzorov, v.b., [2003]) f(v) = √( m 2πkt ) 3 4πv2e − mv2 2kt , (12) where m the particle mass and kt is the product of boltzmann's constant and thermodynamic temperature. from equation (12), if we put α = √ kt m , then the maxwell-boltzmann probability distribution function can be simplified as f(x; α) = 1 α3 √ 2 π x2e − x2 2α2 , (13) where the variable v is replaced with a generic random variable x with x ≥ 0 and it can be noted that the parameter α ≥ 0 is a real quantity. typical shape of maxwell-boltzmann distribution is given in figure-2, for a value of =30. one can clearly see from figure-2 that the ascend phase is of 47 units and the descent phase is 85 units. there by, a skew symmetrical process or phenomenal could be modelled by the maxwell-boltzmann distribution. our interest is in modelling the sunspot cycle. by observing all the cycles individually one can easily see that the rise time (starting minimum to maximum sunspot number) and fall time (maximum sunspot number to cycle end) are not equal or not symmetrical about the peak sunspot number occurring epoch during the 11 year sunspot cycle period. sabarinath.a, beena.g.p, anilkumar.a.k 42 figure-2. maxwell-boltzmann distribution for a value of =30.0 7 modified maxwell-boltzman probability distribution function (mmpdf) since equation (13) being a probability density function, we know that, mathematically the area under the probability density function is 1, that is, ∫ f(x)dx = 1, (14) ∞ −∞ so, if we want to fit this equation (13) into an arbitrary set s of n data points, s = {(xi, yi); xi ∈ r, yi ∈ r, i = 1,2, … , n}, where r is the set of real numbers, we need to de-normalise the property of f(x) given by equation (14). this is because; the area under the curve determined by the set of points in s need not be equal to one. that is, ∑ [(xi − xi−1) (yi + yi−1) 2 ] n i=2 = a, (15) where a need not be equal to 1. in this case we can modify equation (13) to fit into any arbitrary set as equation (16) by introducing a new parameter called area parameter a. f(x; α; a) = a α3 √ 2 π x2e − x2 2α2 , (16) modelling the shape of sunspot cycle using a modified maxwellboltzmann probability distribution function 43 now, it may be noted that, ∫ f(x)dx = a, (17) ∞ −∞ modified model for the sunspot cycles is f(x; α; a) = a α3 √ 2 π x2e − x2 2α2 , (18) where a is the area parameter. modified maxwell-boltzmann distribution with a value of =30 and a=6000 is given in figure-3. figure-3. modified maxwell-boltzmann distribution for a value of =30 and a=6000. 8 estimation of model parameters the function in which parameters to be estimated is, f(x; α; a) = a α3 √ 2 π x2e − x2 2α2 . (19) the maximum likelihood estimate of the parameters α and a are considered to be the best unbiased, consistent and sufficient estimate of the parameters sabarinath.a, beena.g.p, anilkumar.a.k 44 (sorenson, h.w., [1980]). practically, the least square estimate is considered to be the maximum likelihood estimate. the simple mathematical procedure to estimate the parameters is to minimise the sum of squared error function j, j = ∑ er 2 r , (20) where er is the error. the minimum of j can be found by differentiating j with respect to the parameters α and a. in the present study, if we consider without loss of generality, a sunspot cycle having a length of 132 months( 11 year), and if we assume {sn: n = 1,2, … ,132} as the realised sunspot number values, then the j function can be written as, j = ∑[sn − f(xn, α, a)] 2 132 n=1 , (21) where, xn = 1,2, … ,132, represents the months for each n = 1,2, … . ,132. then our objective is to compute and solve α and a from ∂j ∂α = 0, (22) ∂j ∂a = 0, (23) analytically solving the equations (22) and (23) for α and a is not possible due to the nonlinear terms involved in the equations. hence we go with numerical procedures for estimating the parameters. monte carlo based simple random search based procedure is considered here to estimate the parameters. this procedure is described below as an algorithm. step-1. start with a search region α and a. let sα and sa are the bounded search regions of α and a. our objective is to find an α0 ∈ sα and a0 ∈ sa, such that, jα0,a0 = ∑[sn − f(xn, α0, a0)] 2 132 n=1 , (24) is minimum or jα0,a0 ≤ jα,a (25) for any α ∈ sα and a ∈ sa. step-2. start with a random initial value of α in sα and a in sa. compute j and in each iteration keep the minimum value of j, α and a. after a very large number of iterations take the value of, α and a corresponds to the global minimum value of j. modelling the shape of sunspot cycle using a modified maxwellboltzmann probability distribution function 45 9 fitting of mmpdf on sunspot cycles using the method described in section 8, the model parameters are estimated for all the past 24 cycles. it is noticed that the fit is very much close to the actual sunspot numbers. this is evident in the goodness of fit computed for each of the 24 cycles, which is discussed in the next section in detail. figure 4 and 5 shows the model and actual data of sunspot cycles 20 and 22. figure-4. fitting of sunspot cycle 20 by the model sabarinath.a, beena.g.p, anilkumar.a.k 46 figure-5. fitting of sunspot cycle 22 by the model figure-6. the parameters α and a for all the 24 cycles. modelling the shape of sunspot cycle using a modified maxwellboltzmann probability distribution function 47 10 models of sunspot cycles 1 to 24 the estimated model for all the past 24 cycles is given in table-1. in figure-6, the variation trends of the parameters α and a for all the 24 cycles are plotted. it may be noted that, the average of the parameters are 36.25 units of α and 7095.76 of a. table-1. estimated parameters of cycles 1 to 24 cycle no α a 1 48.76 5883.33 2 33.40 6251.65 3 30.56 7309.46 4 35.80 8619.41 5 43.79 3525.58 6 48.75 3067.09 7 48.16 5322.72 8 32.65 7552.73 9 44.70 8234.25 10 40.63 6410.82 11 33.80 7381.50 12 37.13 4433.49 13 32.55 4933.80 14 38.57 4356.00 15 36.00 5390.27 16 35.73 4882.42 17 38.95 7341.83 18 36.35 9228.41 19 33.79 11420.62 20 40.02 7959.33 21 35.49 9907.72 22 31.48 9075.22 23 38.99 8006.39 24 38.65 5023.60 mean 1 to 24 38.11 6729.90 mean 11 to 24 36.25 7095.76 it may be noted that variation in α is less and variation in a is more. so a is a more sensitive parameter than α. variation in a is not much significant as its sensitivity is less. sabarinath.a, beena.g.p, anilkumar.a.k 48 10.1 goodness of fit goodness of fit by hathaway, wilson, and reichmann [1994] is measured by the following function χ = √ ( ∑ (ri − fi) 2n i=1 si 2⁄ ) n , (26) where, ri and si is the monthly averaged sunspot number and its standard deviation respectively , fi gives the functional fit value, n is the number of months in the cycle. using this equation, computed χ value for all the 23 cycles. for checking the goodness of fit of the proposed model we have to consider other popular methods available in the literature. the second column of table 2 gives the goodness of fit of the proposed modified maxwellboltzmann distribution function; the third and the fourth column gives the goodness of fit by three and two parameter fit of hathaway, wilson, and reichmann [1994], respectively; the fifth column gives the goodness of fit by the five parameter function of elling and schwentek[1992] who considered cycles 10 to 21 for their study. figure-7, shows the goodness of fit of 3 different models along with the modified maxwell-boltzmann distribution function model. it may be observed from the goodness of fit value, that the present model proposed in this study has a very good fitness compared with other models. especially the modern cycles (cycles 11 to 24) shows very good fitness for the modified maxwell-boltzmann distribution function model. figure-7. the goodness of fit of 3 different models and mmpdf model modelling the shape of sunspot cycle using a modified maxwellboltzmann probability distribution function 49 table 2: hathaway, wilson and reichmann χ -measure of the goodness of fit value computed for all the 22 sunspot cycles with different models. mmpdf shows good fit compared with other models. cycle number mmpdf model threeparameter fit by hathaway et.al. twoparameter fit by hathaway et.al. ellingschwentek f-distribution fit 1 0.69 0.71 0.75 2 1.38 1.42 1.50 3 1.64 1.70 1.56 4 0.93 0.89 0.95 5 2.87 2.34 2.50 6 1.72 1.90 2.14 7 1.80 0.94 1.01 8 1.16 0.96 0.99 9 0.86 0.99 0.97 10 0.72 0.74 0.76 0.70 11 0.75 0.88 0.83 1.35 12 2.06 2.08 2.12 2.17 13 0.70 0.90 0.91 0.90 14 0.97 1.11 1.09 1.12 15 0.80 0.88 0.89 1.16 16 0.76 0.89 0.97 0.89 17 0.98 0.86 0.87 1.10 18 1.21 1.05 1.04 1.27 19 0.90 0.91 0.89 1.61 20 0.79 0.87 0.95 0.66 21 0.94 0.89 0.89 1.11 22 0.82 1.05 1.06 23 0.79 11 prediction of sunspot cycle 25 as an attempt to predict the sunspot cycle 25, we consider the average of the model parameters by considering cycles-11 to 24. this computed average is given in table-1. thus, the parameter values of cycle 25 are: α = 36.25, and a = 7095.76. hence the model is, sabarinath.a, beena.g.p, anilkumar.a.k 50 f(x; α; a) = a α3 √ 2 π x2e − x2 2α2 , (27) where, α = 36.25, and a = 7095.76. that is, f(x; 36.74; 6608.04) = 0.119 x2e−0.00038x 2 , (28) is the model for cycle-25. figure-8 shows the shape of cycle 25 in an average sense. it may be observed that cycle 25 may peak up to 105 units and it is also fairly a slow cycle as cycle 24. figure-8. preliminary level prediction of sunspot cycle 25 . 12 prediction error and simulated sunspot cycles any prediction or forecast is partial, if it is not supplemented with a prediction error. here, for our study we propose a prediction error band based on the statistical variation of all the cycles. for this, consider all the monthly averaged cycles. we propose the error band each month data as ±s, where s is the standard deviation of the sunspot numbers for that month. figure-9 shows the mean along with the mean+s, the upper bound, and mean-s, the lower bound profile. modelling the shape of sunspot cycle using a modified maxwellboltzmann probability distribution function 51 figure-9. mean cycle from the actual monthly sunspot cycle along with the mean+s, the upper bound and mean-s, the lower bound profile once we are having a prediction error and a prediction model, we can generate any number of forecast profile based on simple monte carlo method. here we consider the envelop derived above as the envelope with 99.7% confidence or 3sigma confidence level, since all the realised cycles falls inside the proposed confidence interval band. hence in the monte carlo simulation a typical profile will be generated using equation (29). sn ′ (i) = mn(i) + rand(i) × ( env(i) 3 ) , (29) where sn ′ (i) , is the simulated n-th sunspot cycle, i = 1,2, … , cycle length, mn(i) is the model value, rand(i) is the random number and env(i) is the envelop value given in figure-10. sabarinath.a, beena.g.p, anilkumar.a.k 52 figure-10. simulated sunspot cycle 20 by the model the same methodology proposed in the study can be implemented to the f10.7 cm solar flux value and one can easiliy forecast an entire cycle and subsequently it can be applied in the life time computation of satellites. 13 conclusions the 11-year sunspot number cycles have been a fascinating phenomenon for many in the last three centuries. different mathematical models have been derived for modelling the shape of the 11-year sunspot number cycles. in the present study, we introduced a new model which is derived from the well known maxwell-boltzmann probability distribution function. the modification has been carried out by introducing a new parameter, called area parameter. the new model has been fitted in the original monthly averaged sunspot cycles data and it is found that a very high goodness of fit through the hathaway, wilson and reichmann measure. the models estimated for all the sunspot cycles from 1 to 24 have been presented. detailed discussion on the nonlinear parameter estimation carried out for fitting the function in the original data is also summarised. an attempt has been carried out for predicting the next sunspot cycles 25. the sunspot cycle 25 may peak up to 105 units and it is also fairly a slow cycle as the previous cycle 24. modelling the shape of sunspot cycle using a modified maxwellboltzmann probability distribution function 53 references [1].balakrishnan, n., nevzorov, v.b., a primer on statistical distributions, john wiley & sons, inc., hoboken, new jersey. 2003. 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[18].stewart, j.q., panofsky, h.a.a., the mathematical characteristics of sunspot variations, astrophys. j., vol. 88, 385–407. 1938. [19].volobuev, d.m., the shape of the sunspot cycle: a one parameter fit, solar phys., vol. 258, 319-330.2009. [20].whitlock, d, modeling the effect of high solar activity on the orbital debris environment, orbital debris quarterly news, 10(2):4.2006. ratio mathematica volume 44, 2022 fuzzy translation in fuzzy d-ideals and fuzzy d-subalgebra r. g. keerthana1 k. r. sobha2 abstract in this paper, we discuss about some properties such as fuzzy translation in fuzzy dideals and fuzzy d-subalgebra. keywords: d algebra, d subalgebra, d ideal, fuzzy d ideal, fuzzy – 𝛼 – translation, fuzzy d subalgebra. 2010 ams subject classification: 94d05, 08a723. 1research scholar, reg.no.20213182092001, sree ayyappa college for women, chunkankadai, nagercoil-62900. [affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627012, tamilnadu, india.]. keerthanarajagopal.rg@gmail.com 2assistant professor, department of mathematics, sree ayyappa college for women, chunkankadai, nagercoil. [affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627012, tamilnadu, india.] vijayakumar.sobha9@gmail.com. 3received on june 18th, 2022. accepted on aug 10th, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.885. issn: 1592 -7415. eissn: 2282 – 8214. ©the authors. this paper is published under the cc by licence agreement. 24 mailto:keerthanarajagopal.rg@gmail.com mailto:vijayakumar.sobha9@gmail.com r. g. keerthana, k. r. sobha 1. introduction fuzzy set theory was introduced by zadeh in 1965 [6]. the study of fuzzy subsets and its applications to various mathematical contexts has given rise to what is now commonly called fuzzy mathematics. it forms a branch of mathematics including fuzzy set theory and fuzzy logic. fuzzy set theory was guided by the assumption that the classical sets were not natural appropriate or useful notions in describing the real-life problems because every object encountered in the real physical world carries some degree of fuzziness. hence fuzzy set has become strong area of research in engineering, medical science, graph theory etc. algebraic structures play an important role in mathematics with wide range of application in many disciplines such as computer sciences, control engineering, theoretical physics, information systems and topological spaces. since these ideas have been applied to other algebraic structures such as group, semigroup, ring, modules, vector spaces and topologies. it gives enthusiasm to the researchers to view various concepts and results from the area of abstract algebra in the broader frame work of fuzzy setting. fuzzy algebra is an important branch of fuzzy mathematics. in 1996, j. negger and h.s. kim [3] introduced the class of d-algebra which is a generalization of bckalgebras and investigated relation between d-algebra and bck-algebra. m. akram and k.h. dar [1] introduced the concepts fuzzy d-algebra, fuzzy subalgebra and fuzzy dideals of d-algebra. 2. preliminaries definition:2.1 a d-algebra is a non-empty set 𝑋 with a constant 0 and a binary operation ∗ satisfies the following axioms: i.𝑥 ∗ 𝑥 = 0 ii.0 ∗ 𝑥 = 0 iii.𝑥 ∗ 𝑦 = 0 and 𝑦 ∗ 𝑥 = 0 ⇒ 𝑥 = 𝑦, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋. definition:2.2 a non-empty subset of a d-algebra 𝑋 is called d-subalgebra of 𝑋 if 𝑥 ∗ 𝑦 ∈ 𝑋, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋. definition: 2.3 let (𝑋,∗ ,0) be a d-algebra and 𝑎 ∈ 𝑋. define 𝑎 ∗ 𝑋 = {𝑎 ∗ 𝑥/𝑥 ∈ 𝑋}. then 𝑋 is said to be edge if 𝑎 ∗ 𝑋 = {0, 𝑎 } for all 𝑎 ∈ 𝑋. 25 fuzzy translation in fuzzy d-ideals and fuzzy d-subalgebra definition: 2.4 let 𝑋 be a d-algebra and 𝐼 be a subset of 𝑋, then 𝐼 is called d-ideal of 𝑋 if it satisfies the following conditions: i.0 ∈ 𝐼 ii.𝑥 ∗ 𝑦 ∈ 𝐼 𝑎𝑛𝑑 𝑦 ∈ 𝐼 ⇒ 𝑥 ∈ 𝐼 iii.x ∈ i and y ∈ x ⇒ x ∗ y ∈ i. definition: 2.5 a fuzzy subset 𝜇𝐴 of 𝑋 is called a fuzzy d-ideal of 𝑋 if it satisfies the following condition: i.𝜇𝐴(0) ≥ 𝜇𝐴(𝑥) ii.𝜇𝐴(𝑥) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦), 𝜇𝐴(𝑦)} iii.𝜇𝐴(𝑥 ∗ 𝑦) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋. 3. fuzzy translation in fuzzy d ideals definition: 3.1 let 𝜇𝐴 be a fuzzy subset of 𝑋 and 𝛼 ∈ [0, 𝑇]. a mapping (𝜇𝐴)𝛼 𝑇 : 𝑋 → [0,1] is said to be a fuzzy-𝛼-translation of 𝜇𝐴 if it satisfies (𝜇𝐴)𝛼 𝑇 (𝑥) = 𝜇𝐴(𝑥) + 𝛼, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋. theorem: 3.2 if 𝜇𝐴 is a fuzzy d-ideal of 𝑋, then the fuzzy-𝛼-translation (𝜇𝐴)𝛼 𝑇 of 𝜇𝐴 is a fuzzy d-ideal of x, for all 𝛼 ∈ [0,1]. proof: let 𝜇𝐴 be a fuzzy d-ideal of 𝑋 and let 𝛼 ∈ [0,1] then (𝜇𝐴)𝛼 𝑇 (0) = 𝜇𝐴(0) + 𝛼 ≥ 𝜇𝐴(𝑥) + 𝛼 = (𝜇𝐴)𝛼 𝑇 (𝑥) (𝜇𝐴)𝛼 𝑇 (0) ≥ (𝜇𝐴)𝛼 𝑇 (𝑥) (𝜇𝐴)𝛼 𝑇 (𝑥) = 𝜇𝐴(𝑥) + 𝛼 ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦), 𝜇𝐴(𝑦)} + 𝛼 = 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{(𝜇𝐴)𝛼 𝑇 (𝑥 ∗ 𝑦), (𝜇𝐴)𝛼 𝑇 (𝑦)} (𝜇𝐴)𝛼 𝑇 (𝑥) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼 𝑇 (𝑥 ∗ 𝑦), (𝜇𝐴)𝛼 𝑇 (𝑦)} (𝜇𝐴)𝛼 𝑇 (𝑥 ∗ 𝑦) = 𝜇𝐴(𝑥 ∗ 𝑦) + 𝛼 ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} + 𝛼 = 𝑚𝑖𝑛{𝜇𝐴(𝑥) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{(𝜇𝐴)𝛼 𝑇 (𝑥), (𝜇𝐴)𝛼 𝑇 (𝑦)} (𝜇𝐴)𝛼 𝑇 (𝑥 ∗ 𝑦) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼 𝑇 (𝑥), (𝜇𝐴)𝛼 𝑇 (𝑦)} hence(𝜇𝐴)𝛼 𝑇 of 𝜇𝐴 is a fuzzy d-ideal of x for all 𝛼 ∈ [0,1]. 26 r. g. keerthana, k. r. sobha theorem: 3.3 let 𝜇𝐴 be a fuzzy subset of x such that the fuzzy-𝛼-translation (𝜇𝐴)𝛼 𝑇 of 𝜇𝐴 is a fuzzy d-ideal of 𝑋for some 𝛼 ∈ [0,1], then 𝜇𝐴 is a fuzzy d-ideal of 𝑋. proof: let (𝜇𝐴)𝛼 𝑇 is a fuzzy d-ideal of 𝑋 for some 𝛼 ∈ [0,1] let 𝑥, 𝑦 ∈ 𝑋 𝜇𝐴(0) + 𝛼 = (𝜇𝐴)𝛼 𝑇 (0) ≥ (𝜇𝐴)𝛼 𝑇 (𝑥) = 𝜇𝐴(𝑥) + 𝛼 𝜇𝐴(0) ≥ 𝜇𝐴(𝑥) 𝜇𝐴(𝑥) + 𝛼 = (𝜇𝐴)𝛼 𝑇 (𝑥) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼 𝑇 (𝑥 ∗ 𝑦), (𝜇𝐴)𝛼 𝑇 (𝑦)} = 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦), 𝜇𝐴(𝑦)} + 𝛼 𝜇𝐴(𝑥) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦), 𝜇𝐴(𝑦)} 𝜇𝐴(𝑥 ∗ 𝑦) + 𝛼 = (𝜇𝐴)𝛼 𝑇 (𝑥 ∗ 𝑦) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼 𝑇 (𝑥), (𝜇𝐴)𝛼 𝑇 (𝑦)} = 𝑚𝑖𝑛{𝜇𝐴(𝑥) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} + 𝛼 𝜇𝐴(𝑥 ∗ 𝑦) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} hence 𝜇𝐴 is a fuzzy d-ideal of 𝑋. 4. fuzzy translation in fuzzy d subalgebra definition: 4.1 a fuzzy set 𝜇𝐴in d-algebra 𝑋 is called fuzzy d-subalgebra of 𝑋 if it satisfies, 𝜇𝐴(𝑥𝑦) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋. theorem: 4.2 if 𝜇𝐴 is a fuzzy d-subalgebra of 𝑋, then the fuzzy-𝛼-translation 𝜇𝛼 𝑇 of 𝜇𝐴 is also a fuzzy d-subalgebra of x for all 𝛼 ∈ [0,1]. proof: let 𝑥, 𝑦 ∈ 𝑋 and let 𝛼 ∈ [0,1] let 𝜇𝐴 be a fuzzy d-subalgebra of 𝑋 . then 𝜇𝐴(𝑥𝑦) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} (𝜇𝐴)𝛼 𝑇 (𝑥𝑦) = 𝜇𝐴(𝑥𝑦) + 𝛼 ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} + 𝛼 = 𝑚𝑖𝑛{𝜇𝐴(𝑥) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{(𝜇𝐴)𝛼 𝑇 (𝑥), (𝜇𝐴)𝛼 𝑇 (𝑦)} (𝜇𝐴)𝛼 𝑇 (𝑥𝑦) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼 𝑇 (𝑥), (𝜇𝐴)𝛼 𝑇 (𝑦)} hence (𝜇𝐴)𝛼 𝑇 of 𝜇𝐴 is a fuzzy d-subalgebra of x for all 𝛼 ∈ [0,1]. 27 fuzzy translation in fuzzy d-ideals and fuzzy d-subalgebra theorem: 4.3 let 𝜇𝐴 be a fuzzy subset of x such that the fuzzy-𝛼-translation (𝜇𝐴)𝛼 𝑇 of 𝜇 is a fuzzy d-subalgebra of 𝑋 for some 𝛼 ∈ [0,1], then 𝜇𝐴 is a fuzzy d-subalgebra of 𝑋. proof: let (𝜇𝐴)𝛼 𝑇 is a fuzzy d-subalgebra of 𝑋 for some 𝛼 ∈ [0,1]. let 𝑥, 𝑦 ∈ 𝑋. 𝜇𝐴(𝑥𝑦) + 𝛼 = (𝜇𝐴)𝛼 𝑇 (𝑥𝑦) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼 𝑇 (𝑥), (𝜇𝐴)𝛼 𝑇 (𝑦)} = 𝑚𝑖𝑛{𝜇𝐴(𝑥) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} + 𝛼 𝜇𝐴(𝑥𝑦) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} hence 𝜇𝐴 is a fuzzy d-subalgebra of 𝑋. 5. conclusions in this paper, we have given some ideas on fuzzy translation in fuzzy d-ideals and fuzzy d-subalgebras. our further research will be focus on dot product and level set. references [1] m. akram and k.h. dar, on fuzzy d-algebra, punjab university journal of math, 37 (2005), 61-76. [2] j. neggers; y.b. jun; h.s. kim, “on d – ideals in d – algebras”, mathematica slovaca.49 (1999), no.3, 243-251 [3] j. neggers and h.s. kim, “on d – algebra”, math. slovaca 49 (1999) no.1, 19 – 26. [4] t. priya and t. ramachandran, homomorphism and cartesian product of fuzzy psalgebra, applied mathematical sciences, 8, vol (67) (2014) 3321 – 3330. [5] t. priya and t. ramachandran, homomorphism and cartesian product on fuzzy translation and fuzzy multiplication of ps-algebras, annals of pure and mathematics, vol.8, no.1, 2014, 93 – 104. [6] l.a. zadeh,” fuzzy set”, inform and control.8 (1965), 338 – 353. 28 microsoft word documento1 ratio mathematica issue n. 30 (2016) pp. 35-43 issn (print): 1592-7415 (online): 2282-8214 a recursive variant of schwarz type domain decomposition methods frantǐsek bubeńık, petr mayer faculty of civil engineering, czech technical university in prague, czech republic frantisek.bubenik@cvut.cz, petr.mayer@cvut.cz abstract in this paper a slightly different approach to the use of the domain decomposition method of the schwarz type is proposed. instead of the standard coarse space construction we propose to use a recursive solution on each domain. thus we do not need to construct a coarse space but nevertheless we are still keeping o(1) convergence speed. for local problems we use the standard iterative solvers for which the amount of the work for one step is o(n), where n is the number of equations. due to the fact that the overlapping is under our control we can keep total work in o(n(1+γ)) operations with arbitrary positive γ. keywords: domain decomposition, finite element method, linear systems. 2000 ams subject classifications: 97u99 doi: 10.23755/rm.v30i1.4 1 introduction this paper deals with some aspects of the classic schwarz alternating method. there are analyzed ways how to arrange with the deceleration of algorithms if the stepsize of a mesh for the finite element method is decreasing. the standard two-level method is described and some alternative approaches to solve a number of local problems by the same method are proposed. 35 frantǐsek bubeńık and petr mayer 2 the schwartz alternating method as a model problem we will solve a poisson problem −4u(x) = f(x), x ∈ ω ⊆ rd. we use the finite element method to solve the problem. it leads to a linear system ax = b. (1) consider functions ϕ1, . . . ,ϕn as a basis and denote by vn = span(ϕ1, . . . , ϕn) the linear hull of the functions, that is the set of all linear combinations of the functions. let us remind that aij = a(ϕi,ϕj) and a(u,v) = ∫ ω uv dω. the matrix a for our problem is symmetric and positive definite. moreover, for many interesting choices of the basis of vn, the matrix a is sparse, but large. one of the possible strategies to solve the system (1) is to use a sparse version of lu-decomposition. in most cases, fill-in which takes place along with gaussian elimination makes such an approach unusable. a usual choice is then to use some iterative method. since our matrix is a symmetric positive definite, it seems to be more advantageous to employ the conjugate gradient method. but this choice is still problematic because the amount of the work is rapidly increasing with the size of the problem. therefore there is then more convenient to use a preconditioned conjugate gradient method with an appropriate preconditioner. as a preconditioner we can choose a slightly modified the schwarz alternatig method, see [2]. the original description is in [1]. we can see it as a kind of some block symmetrized gauss-seidel method. 2.1 formulation of the algorithm let us denote by nd the number of domains. for each i ∈ i = {1, 2, . . . ,nd} we define an index set ii = { i (i) 1 , i (i) 2 , . . . , i (i) ni } . these index sets realize a covering of i, i. e. i = nd⋃ i=1 ii. this covering is not required to be disjoint. we define subspaces of vn so that the subspace v (i) n is the linear hull of a corresponding part of the basis of vn, that is v (i) n = span j ∈ ii {ϕj} . finally we 36 a recursive variant of schwarz type domain decomposition methods define subdomains ω(i) = ⋃ j∈ii supp (ϕj) , (2) where supp (f) denotes the support of a function f. the sizes of individual domains are n1, . . . ,nnd. we can write a (i) = a(ii,ii) in terms of matlab-like notation. matrix interpretation of local problems is n(i) = {n(i)k,l} ∈ r n×ni, where n (i) k,l = { 1 if l = i (i) k , 0 otherwise. then a(i) = n(i) t an(i). (3) the following algorithm describes the transition from x(k) to x(k+1). algorithm 2.1. one step of the symmetrized schwarz method x (k+ 0 2nd ) := x(k) for i = 1, . . . ,nd r := b−ax(k+ i−1 2nd ) r̃ := n(i) t r a(i) := n(i) t an(i) (♣) solve a(i)c = r̃, i. e. c = a(i)−1r̃ x (k+ i 2nd ) := x (k+ i−1 2nd ) + n(i)c end for (4) for i = 1, . . . ,nd (5) r := b−ax(k+ 1 2 + i−1 2nd ) r̃ := n(nd+1−i) t r a(nd+1−i) := n(nd+1−i) t an(nd+1−i) (♣) solve a(nd+1−i)c = r̃ x (k+ 1 2 + i 2nd ) := x (k+ 1 2 + i−1 2nd ) + n(nd+1−i)c end for end algorithm the method used in the algorithm 2.1 can also be viewed as a variant of the block gauss-seidel method, but with the fact that the individual blocks can overlap. put p (i) = a1/2n(i)(n(i) t an(i))−1n(i) t a1/2. (6) 37 frantǐsek bubeńık and petr mayer then p (i) 2 = a1/2n(i)(n(i) t an(i))−1n(i) t a1/2a1/2n(i)(n(i) t an(i))−1n(i) t a1/2 = a1/2n(i)a(i) −1 a(i)a(i) −1 n(i) t a1/2 = a1/2n(i)a(i) −1 n(i) t a1/2 = p (i). it follows that p (i) is a projection, moreover a-orthogonal. further p (i) = p (i) t , then the projection is symmetric. let us denote ε(k) = x(k) −x∗, where x∗ denotes the solution of the problem. errors are analyzed in terms of the energy norm ||x||a = √ (x,x)a, where (x,y)a = x tay. let us note that a is a symmetric positive definite matrix and a(i) is a principal minor of a. then a(i) is also a symmetric positive definite matrix. we have ||ε(k)||2a = ε (k)taε(k) = ε(k) t a1/2 a1/2 ε(k) = ||a1/2ε(k)||2a. thus a1/2ε(k−1) = (i −p (1)) . . . (i −p (nd))(i −p (nd)) . . . (i −p (1))a1/2ε(k) = m a1/2ε(k). (7) since i−p (i) is a symmetric a−orthogonal projection then m is a symmetric matrix. and moreover, m is, according to the definition, a positive semidefinite matrix. it can be proved that m is even a positive definite matrix. 2.2 dependence on the dimension of vn for the following considerations we suppose that piecewise linear finite elements are used. in that case n = o(1/hd), where h is the stepsize of a mesh. if we try to keep domains with the same geometry, the amount of elements will increase as o((h/h)d), where h is the typical size of a domain. then, the amount of iterations is the same, but the amount of work for one step will increase. on the other hand, when we keep equal the number of elements inside a domain, then the size of the domain will decrease and then the number of domains will increase. this leads to increasing amount of iterations and slightly increasing work for one full step. usual solution is to use a coarse space. it typically means to replace each domain by a base function. we create the coarse space and the solution of the 38 a recursive variant of schwarz type domain decomposition methods problem for the corrections on the coarse level is inserted between steps (4) and (5) of the algorithm 2.1. a detailed analysis is introduced, for example, in [2]. when it is used in a right way, we get o(1) convergence speed. 2.3 basic convergence let us denote e(x) = xtax− 2xtb. (8) it is known that ax = b if and only if e(x) assumes its minimum at x. each step in algorithm 2.1 means the minimization of functional (8) on a corresponding subspace and the following inequality holds for the successive terms of the minimizing sequence e(x(k+1)) ≤ e(x(k)). (9) we prove the following equivalence: lemma 2.1. the equality in (9) occurs ⇐⇒ x(k) is the accurate solution of ax(k) = b. proof. it is clear that an accurate solution is equivalent to r(k) = 0, where r(k) = b−ax(k). we prove one direction of the equivalence: suppose that x(k) is an accurate solution and we prove the equality required. it is easy to see that if x(k) is an accurate solution then r(k) = 0 and then the equality in (9) occurs. now we prove the opposite direction of the equivalence. we apply the proof by contradiction: suppose that x(k) is not an accurate solution and suppose that the equality in (9) holds. if x(k) is not an accurate solution then r(k) 6= 0. then there exists the least index im such that n(im) t r(k) 6= 0. it causes a decrease at this step of algorithm 2.1 and then the strict inequality e(x(k+1)) < e(x(k)). this contradicts to the assumption of equality in (9) and the proof of the equivalence in lemma 2.1 is complete. 2 3 recursive approach another possibility comes from the idea that the local problems are conceptually identical as the original one. it opens a possibility to use the same schwarz algorithm for solving them. it means to retain domains in the same geometry, then ω(i) in (2) remain unchanged. on the other hand it means that the number of degrees of freedom for a domain increases. in this case we recommend to use the same algorithm for each domain separately. 39 frantǐsek bubeńık and petr mayer 3.1 two level variant we start from the algorithm 2.1, and we replace both steps denoted by (♣) in the algorithm by an iterative solution for c. it means that we replace a(i) −1 r̃ by an approximate solution of the problem a(i)c = r̃. as the method we use again the algorithm 2.1 with ` steps. then the error operator has the form m̃ = (i − p̃ (1)) . . . (i − p̃ (nd))(i − p̃ (nd)) . . . (i − p̃ (1)), where p̃ (i) = a1/2n(i)q̃(i)n(i) t a1/2. (10) expression (n(i) t an(i))−1 in (6) is for short denoted by q(i) and it is replaced in (10) by q̃(i) = a(i) −1/2 [ i − ( i −a(i) 1/2 m(i)a(i) 1/2 )`] a(i) −1/2 , (11) where a(i) is from (3) and m(i) denotes the error operator to the algorithm 2.1 applied on the i−th domain. this replacement comes from the following: let us solve a problem ax = b and let us use the following iterative method x(i+1) = x(i) + m(b−ax(i)), where m is a symmetric positive definite matrix. the initial approximation is x(0) = 0. (12) let x∗ = a−1b. then x∗ −x(i+1) = x∗ −x(i) −m(b−ax(i)) = (i −ma)(x∗ −x(i)) and then a1/2(x∗ −x(i+1)) = (i −a1/2ma1/2)a1/2(x∗ −x(i)). after `−iterations we get a1/2(x∗ −x(`)) = (i −a1/2ma1/2)`a1/2(x∗ −x(0)) and thus x∗ −x(`) = a−1/2(i −a1/2ma1/2)`a1/2(x∗ −x(0)). 40 a recursive variant of schwarz type domain decomposition methods since x(0) = 0 we get x∗ −x(`) = a−1/2(i −a1/2ma1/2)`a1/2x∗ = a−1/2(i −a1/2ma1/2)`a−1/2b. thus x(`) = x∗ −a−1/2(i −a1/2ma1/2)`a−1/2b = a−1/2 [ i − ( i −a1/2ma1/2 )`] a−1/2b and that is why the form of q̃(i) in (11) and p̃ (i) in (10). 3.2 recursive multilevel method when we use a two-level method we need to compute p̃ (i) in (10) and for it there is required to know q̃(i) from (11). for q̃(i) there is necessary to know m(i) and its realization comes from the solution of a local problem on subdomains of the i−th domain. in case when these local problems are still large then the process may be repeated again and a two-level method becomes a recursive-multilevel method. as to convergence of a recursive variant of the method the same facts as in section 2.3 can be used. then we can state that a recursive multilevel method converges as well and we have proved the following theorem. theorem 3.1. a recursive multilevel method is convergent. 4 cost analysis 4.1 two levels the work required for solving a problem of size n is w = k n(1+β) with k and β positive constants. let α be a relative overlapping in one dimension. the number of domains is nd. then the size of a local problem is nloc = n nd (1 + α)d and the work needed for its solving w = k ( n nd (1 + α)d )(1+β) . 41 frantǐsek bubeńık and petr mayer thus the work for one iteration is w = 2nd k ( n nd (1 + α)d )(1+β) = 2(1 + α)d(1+β) n β d k n(1+β) and for ` iterations on this level w = 2` (1 + α)d(1+β) n β d k n(1+β). 4.2 k levels in the k−th level we repeat the previous considerations. we have nkd subdomains and the size of one subdomain is nloc,k = n ( 1 + α nd )k . the problem is solved on each subdomain (2`)k times. total work is w = (2`)knkd k ( n ( 1 + α nd )k)(1+β) = k ( 2`n −β d (1 + α) (1+β) )k n(1+β). 4.3 full recursion we want to find such k that nloc,k = 1. we take the greatest possible k. this is k = ln n ln nd − ln(1 + α) . we obtain w = k ( 2`n −β d (1 + α) (1+β) ) ln n ln nd−ln(1+α) n(1+β) which is w = k exp (ln 2+ln `−β ln nd+(1+β) ln(1+α)) ln nln nd−ln(1+α) +(1+α) ln n . this expression can be improved and after some manipulations we get w = k n(1+γ), with γ = ln(1 + α) + ln 2 + ln ` ln nd − ln(1 + α) . (13) we can see from (13) that it is possible to achieve γ arbitrary small by an appropriate choice of α, `,nd. 42 a recursive variant of schwarz type domain decomposition methods 5 conclusion an alternative process to the classic two-level method with a coarse space is proposed in this paper. one of the significant advantages of the method presented here is the fact that we can extremely reduce the memory requirements if this is called for. references [1] h. a. schwarz, gessamelte mathematische abhandlungen, volume 2, pages 133 − 143, springer, berlin, (1890). first published in vierteljahrsschrift der naturforschenden gesellschaft in zürich, über einen grenzübergang durch alternierendes verfahren, volume 15, (1870), pp. 272 − 286. [2] a. toselli and o. widlund, domain decomposition methods algorithms and theory. springer-verlag, berlin heidelberg, (2005). [3] m. brezina and p. vaněk, a black-box iterative solver based on a twolevel schwarz method. computing 63(3), (1999), 233 − 263. [4] r. varga, matrix iterative analysis. prentice hall, first edition, (1962). 43 ratio mathematica volume 38, 2020, pp. 287-311 287 analytical and numerical solution of differential equations with generalized fuzzy derivative basim nasih abood* abstract the aim of this work is to present a novel approach based on the fuzzy neural network for finding the numerical solution of the first order fuzzy differential equations under generalized h-derivation.the differentiability concept that used in this paper is the generalized differentiability since a first order fuzzy differential equation under this differentiability can have two solutions.the fuzzy trial solution of the fuzzy initial value problem is written as a sum of two parts. the first part satisfies the fuzzy condition, it contains no fuzzy adjustable parameters. the second part involves fuzzy feed-forward neural networks containing fuzzy adjustable parameters. this method, in comparison with existing numerical methods and the analytical solutions, shows that the use of fuzzy neural networks provides solutions with good generalization and high accuracy. keywords: fuzzy differential equations; generalized h-derivation; fuzzy neural network; fuzzy trial solution; error function. ــ ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ * department of mathematics, university of wasit, alkut, iraq; basim.nasih@yahoo.com received on december 10th, 2019. accepted on march 3rd, 2020. published on june 30th, 2020. doi: 10.23755/rm.v38i0.519. issn: 1592-7415. eissn: 2282-8214. ©basim nasih abood. this paper is published under the cc-by licence agreement. basim n. abood 288 1 introduction nowadays, fuzzy differential equations (fdes) is a popular topic studied by many researchers since it is utilized widely for the purpose of modeling problems in science and engineering. most of the practical problems require the solution of a fde which satisfies fuzzy initial or fuzzy boundary conditions, therefore, a fuzzy initial or fuzzy boundary problem should be solved. however, many fuzzy initial or fuzzy boundary value problems could not be solved exactly, sometimes it is even impossible to find their analytical solutions. thus, considering their approximate solutions is becoming more important [1]. the theory of fde was first formulated by kaleva and seikkala. kaleva had formulated fde in terms of the hukuhara derivative (h-derivative). buckley and feuring have given a very general formulation of a first-order fuzzy initial value problem. they first find the crisp solution, make it fuzzy and then check if it satisfies the fuzzy differential equation [2]. in recent years artificial neural network (ann) for estimation of the ordinary differential equation (ode) and partial differential equation (pde) has been used. we briefly review some articles in the literature concerning the differential equations. lee and kang in [3] used parallel processor computers to solve a first order differential equation with hopfield neural network models. meade and fernandez in [4,5] solved linear and non-linear odes by using feed-forward neural networks (ffnn) architecture and b-splines of degree one. lagaris and likas in [6,7] used ann for solving odes and pdes with the initial / boundary value problems. liu and jammes in [8] developed some properties of the trial solution to solve the odes by using ann. ali and ucar in [9] solved the vibration control problems by using ann. tawfiq in [10] presented and developed supervised and unsupervised algorithms for solving ode and pde. malek and shekari in [11] presented numerical method based on ann and optimization techniques which the higher-order ode answers approximates by finding a package form analytical of specific functions. pattanaik and mishra in [12] applied and developed some properties of ann for solution of pde in rf engineering. baymani and kerayechian in [13] proposed ann approach for solving stokes problems. oraibi in [14] designed ffnn for solving ordinary initial value problem. ali in [15] analytical and numerical solution of differential equations with generalized fuzzy derivative 289 designed fast ffnn to solve two-point boundary value problems. hussein in [16] designed fast ffnn to solve singular boundary value problems. tawfiq and al-abrahemee in [17] designed ann to solve singular perturbation problems, and other researchers. numerical solution of fde by using ann is the subject of a very modern because it only goes back to 2010. effati and pakdaman in [18] used ann for solving fde, they used for the first time the ann to approximate fuzzy initial value problems. mosleh and otadi in [19] used ann for solving fuzzy fredholm integro-differential equations. ezadi and parandin in [20] used ann based on semi-taylor series to solve first order fde. numerical solution of fde by using fuzzy artificial neural network (fann) is more modern than the previous subject, where it goes back to 2012. mosleh and otadi in [21] used fann for solving first order fde, they used for the first time fann to approximate fuzzy initial value problems. mosleh in [22] used fann for solving a system of fde. mosleh and otadi in [23] used fann for solving second order fde. suhhiem in [24] developed and used fann for solving fuzzy and non-fuzzy differential equations. in 2008, the concept of the generalized hukuhara – differentiability is studied by chalco-cano and roman flores [25,26] to solve fde. in this work, for solving fde under generalized h – derivation, we present modified method which relies on the function approximation capabilities of fuzzy ffnn and results in the construction of a solution written in a differentiable, closed analytic form. this form employs fuzzy ffnn as the basic approximation element, whose fuzzy parameters (weights and biases) are adjusted to minimize an appropriate error function. to train the fann which we design, we employ optimization techniques, which in turn require the computation of the gradient of the error with respect to the network parameters. in this proposed approach the model function is expressed as the sum of the two terms: the first term satisfies the fuzzy initial / fuzzy boundary conditions and contains no fuzzy adjustable parameters. the second term can be found by using fuzzy ffnn, which is trained so as to satisfy the fde. basim n. abood 290 2 basic definitions 2.1 fuzzy concepts in this subsection, the basic notations which are used in fuzzy calculus are introduced. definition (𝟐.𝟏.𝟏), [𝟐] : the r level ( or r cut ) set of a fuzzy set ã labeled by ar, is the crisp set of all x in x (universal set) such that : µã(x) ≥ r ; i.e. ar = {x ∈ x ∶ µã(x) ≥ r, r ∈ [0,1] }. (1) definition(2.1.2),[𝟐]: extension principle let x be the cartesian product of universes x 1 , x 2, …, x m and ã1, ã2, …, ãm be m fuzzy subset in x 1, x 2, …, x m respectively, with cartesian product ã = ã1 × ã2 × … × ãm and f is a function from x to a universe y, ( y = f ( x1, x 2, …, x m ) ). then, the extension principle allows to define a fuzzy subset b̃ = f (ã) in y by b̃ = {( y,µ b̃ ( y )) : y = f ( x1, x 2, …, x m ), ( x1, x 2, …, x m ) ∈ x}, where µ b̃ (y)={ sup ( x1,…,xm )∈f −1(y) min{µ ã1 (x1),…, µãm (xm) } , if f −1(y)≠∅ 0, otherwise. (2) and f −1 is the inverse image of f. for m = 1, the extension principle will be : b̃ = f(ã) = {(y, µ b̃ (y)) ∶ y = f(x), x ∈ x }, where µ b̃ (y) = { sup x ∈ f −1(y) µ ã (x) , if f −1(y)≠∅ 0, otherwise. (3) analytical and numerical solution of differential equations with generalized fuzzy derivative 291 definition (𝟐.𝟏.𝟑), [𝟐𝟒]: fuzzy number a fuzzy number ũ is completely determined by an ordered pair of functions (u (r), u (r)), 0 ≤ r ≤ 1, which satisfy the following requirements: 𝟏) u (r) is a bounded left continuous and non-decreasing function on [0,1]. 𝟐) u (r) is a bounded left continuous and non-increasing function on [0,1]. 𝟑) u (r) ≤ u (r), 0 ≤ r ≤ 1. the crisp number a is simply represented by: u (r) = u (r) = a, 0 ≤ r ≤ 1. the set of all the fuzzy numbers is denoted by e1. remark (𝟏), [𝟐𝟒]: for arbitrary ũ = (u, u), ṽ = (v, v) and k ∈ r, the addition and multiplication by k can be defined as : 𝟏) (u + v) (r) = u (r) + v (r) (4) 𝟐) (u + v) (r) = u (r) + v (r) (5) 𝟑) (ku) (r) = k u (r), (ku) (r) = k u (r), if k ≥ 0 (6) 𝟒) (ku) (r) = k u (r), (ku) (r) = k u (r), if k < 0. (7) for all r ∈ [0,1] . remark (𝟐), [𝟏]: the distance between two arbitrary fuzzy numbers ũ = (u, u) and ṽ = (v, v) is given as: d (ũ, ṽ) = [∫ ( u (r) v (r) 1 0 ) 2 dr + ∫ ( u (r) v (r) 1 0 ) 2 dr] 1 2 (8) remark (𝟑), [𝟏]: (e1,d) is a complete metric space. basim n. abood 292 remark (𝟒), [𝟐]: the operations of fuzzy numbers (in parametric form) can be generalized from that of crisp intervals. let us have a look at the operations of intervals. ∀ a1, b1, a2,b2 ∈ r, a = [a1, b1] and b = [a2,b2 ]. assuming a and b numbers expressed as interval, main operations of intervals are : 𝟏) addition: a + b = [a1, b1] + [a2,b2 ] = [a1 + a2, b1 + b2 ]. 𝟐) subtraction: a b = [a1, b1] [a2,b2 ] = [a1 b2, b1 a2 ]. 𝟑) multiplication: a. b = [min{a1 a2, a1 b2, b1 a2, b1 b2},max{a1 a2, a1 b2, b1 a2, b1 b2}] 𝟒) division : a/b =[min{a1 / a2, a1 / b2, b1 / a2, b1 / b2},max{a1 / a2, a1 / b2, b1 / a2, b1 / b2}] excluding the case a2 = 0 or b2 = 0. 𝟓) inverse : a-1 = [a1,b1]-1 = [min{ 1 a1 , 1 b1 } ,max{ 1 a1 , 1 b1 }] excluding the case a1 = 0 or b1 = 0. in the case of 0 ≤ a2 ≤ b2, multiplication operation can be simplified as: a. b = [min{a1 a2, a1 b2 },max{ b1 a2, b1 b2}] when previous sets a and b is defined in the positive real number r+, the operations of multiplication, division and inverse are written as : 𝟑 ́) multiplication: a. b = [a1, b1]. [a2,b2 ] = [a1 a2, b1 b2] 𝟒 ́) division: a / b = [a1, b1] / [a2,b2 ] = [ a1 b2 , b1 a2 ]. 𝟓 ́) inverse: a-1 = [a1,b1] -1 = [ 1 b1 , 1 a1 ]. definition (𝟐.𝟏.𝟒), [𝟐𝟒] : triangular fuzzy number among the various shapes of fuzzy numbers, triangular fuzzy numbers is the most popular one. a triangular fuzzy number is a fuzzy number represented with three points as follows: ã = (a1,a2, a3), where a1 ≤ a2 ≤ a3. analytical and numerical solution of differential equations with generalized fuzzy derivative 293 this representation is interpreted as membership functions: μã(x) = { 0 , if x < a1 x a1 a2 a1 , if a1 ≤ x ≤ a2 a3-x a3-a2 , if a2 ≤ x ≤ a3 0 , if x > a3 (9) now if you get crisp interval by r cut operation, interval [a]r shall be obtained as follows ∀ r ∈ [0,1] from: a − a1 a2 − a1 = r, a3 − a a3 – a2 =r, we get: a= (a2 – a1)r + a1, a= (a2 – a3) r + a3. thus: [a]r = [a, a ] = [(a2 – a1)r + a1, (a2 – a3) r + a3] (10) which is the parametric form of triangular fuzzy number ã. definition (𝟐.𝟏.𝟓), [𝟏𝟖] : fuzzy function a classical function f : x ⟶ y maps from a fuzzy domain ã ⊆ x into a fuzzy range b̃ ⊆ y if and only if ∀ x ∈ x , μb̃(f (x)) ≥ μã(x) . remark (𝟓) , [𝟏𝟖] : (1) the function f : r ⟶ e1 is called a fuzzy function . (2) we call every function defined in set ã ⊆ e1 to b̃ ⊆ e1 a fuzzy function. definition (𝟐.𝟏.𝟔), [𝟏𝟖]: the fuzzy function f : r ⟶ e1 is said to be continuous if : for an arbitrary t1 ∈ r and ϵ > 0 there exists a δ > 0 such that: |t t1| < δ ⇒ d (f (t), f(t1)) < ϵ, where d is the distance between two fuzzy numbers. definition (2.1.7),[25]: let i be a real interval. the r-level set of the fuzzy function y ∶ i → e1 can be denoted by : [y(t)]r = [y1 r(t),y2 r(t)] t ∈ i (11) the seikkala derivative yˊ(t) of the fuzzy function y(t) is defined by: basim n. abood 294 [yˊ(t)]r = [(y1 r)ˊ(t),(y2 r)ˊ(t)] t ∈ i (12) 2.2 h – differentiability in this subsection, the basic definitions which are used in h – differentiability are introduced. definition (2.2.1), [𝟏𝟖]: let u, v ∈ e1 . if there exist w ∈ e1 such that u = v+w then w is called the h-difference (hukuhara-difference) of u, v and it is denoted by w= u ⊝ v. in this work the ⊝ sign stands always for h-difference, and let us remark that u ⊝ v ≠ u + (-1) v. definition (2.2.2), [𝟐𝟒] let f : (a,b) → e1 and t0 ∈ (a,b).we say that f is h-differential (hukuhara-differential) at t0, if there exists an element fˊ(t0) ∈ e 1 such that for all h> 0 (sufficiently small), ∃ f (t0 +h)⊝f(t0), f(t0) ⊝ f (t0 h) and the limits (in the metric d) lim h→0 f(t0 + h) ⊝f(t0) h = lim h→0 f(t0) ⊝ f(t0 − h) h = fˊ(t0) (13) then fˊ(t0) is called fuzzy derivative (h-derivative) of f at t0. where d is the distance between two fuzzy numbers. it is necessary to note that the definition (2.2.2) is the classical definition of the h-derivative (or differentiability in the sense of hukuhara ). definition (2.2.3), [𝟐𝟓,𝟐𝟔]: let f ∶ t → e1 and t0 ∈ t ⊂ r . f is differentiable at t0, if (1) there exist an element fˊ(t0) ∈ e 1, such that for all h > 0 sufficiently small, there are f(t0 + h) ⊝f(t0),f(t0)⊝f(t0 − h) and the limits (in the metric d ) lim h→0 f(t0 + h) ⊝f(t0) h = lim h→0 f(t0) ⊝ f(t0 − h) h = fˊ(t0) (14) analytical and numerical solution of differential equations with generalized fuzzy derivative 295 (in this case, f is called (1)-differentiable) or (2) there exist an element fˊ(t0) ∈ e 1, such that for all h > 0 sufficiently small, there are f(t0)⊝f(t0 + h), f(t0 − h) ⊝ f(t0) and the limits (in the metric d ) lim h→0 f(t0)⊝f(t0 + h) −h = lim h→0 f(t0 − h) ⊝ f(t0) −h = fˊ(t0) (15) (in this case, f is called (2)-differentiable) where the relation (1) is the classical definition of the h-derivative. theorem (1) : let f ∶ i → e1 be a function and denote [f(t)]r = [ fr(t),gr(t)], for each r ∈ [0,1]. then (i) if f is differentiable in the first form (1) of definition (2.2.3.), then fr and gr are differentiable functions and [fˊ(t)]r = [ fr ˊ(t),gr ˊ (t) ] (ii) if f is differentiable in the second form (2) of definition (2.2.3), then fr and gr are differentiable functions and [fˊ(t)]r = [gr ˊ (t), fr ˊ(t) ] proof: see [25] □ 3 fuzzy neural network [24] a fuzzy neural network (fnn) or neuro – fuzzy system is a learning machine that finds the parameters of a fuzzy system (i.e., fuzzy set, fuzzy rules) by exploiting approximation techniques from neural networks. combining fuzzy systems with neural networks. both neural networks and fuzzy systems have some things in common. they can be used for solving a problem (e.g. fuzzy differential equations, fuzzy integral equations, etc. ). basim n. abood 296 before 2005 fnn called fuzzy weight neural networks was developed by pabisek, jakubek and et al. membership functions of fwnn were formulated by the multi – layered perceptron (mlp) network training, separately for each learning pattern and then the interval arithmetic was applied to process crisp or fuzzy data. 3.1 input – output relations of each unit [21,22] let us consider a three-layer fuzzy ffnn with n input units, m hidden units and s output units. target vector, connection weights and biases are fuzzy numbers and input vector is real numbers. for convenience in this discussion, fnn with an input layer, a single hidden layer, and an output layer in fig. (1) is represented as a basic structural architecture. here, the dimension of fnn is denoted by the number of neurons in each layer, that is n × m × s, where n, m and s are the number of the neurons in the input layer, the hidden layer and the output layer, respectively. the architecture of the model shows how the fnn transforms the n inputs (x1,x2,…,xi,…,xn) into the s fuzzy outputs ([y1]r, [y2]r,…,[yk]r,…,[ys]r) throughout the m hidden fuzzy neurons ([z1]r, [z2]r,…[zj]r,… [zm]r), where the cycles represent the neurons in each layer. let [bj]r be the fuzzy bias for the fuzzy neuron [zj]r, [ck]r be the fuzzy bias for the fuzzy neuron [yk]r, [wji]r be the fuzzy weight connecting crisp neuron xi to fuzzy neuron [zj]r, and [wkj]r be the fuzzy weight connecting fuzzy neuron [zj]r to fuzzy neuron [yk]r. when an n – dimensional input vector (x1,x2,…,xi,…,xn) is presented to our fnn, its input – output relations can be written as follows, where h : rn ⟶ es : input units: oi = xi , i = 1,2,3, …n (16) hidden units: [zj]r = h ([netj]r) , j = 1,2,3, …,m, (17) where [netj]r = ∑ oi [wji]r + [bj]r n i=1 (18) analytical and numerical solution of differential equations with generalized fuzzy derivative 297 output units: [yk]r =h ([netk]r), k = 1,2,3, …, s, (19) where [netk]r=∑ [wkj]r [zj]r+ m j=1 [ck]r (20) fig. (1) three-layer feed forward fuzzy neural network. the architecture of our fuzzy neural network is shown in fig. (1), where connection weights, biases, and targets are fuzzy numbers and inputs are real numbers. from the operations of fuzzy numbers (which we have described in section two), the above relations are rewritten as follows when the inputs xi´s are non – negative, i.e., xi ≥ 0 : input units: oi = xi (21) basim n. abood 298 hidden units: [zj]r = h ([netj]r) = [[zj]r l , [zj]r u ] = [h([netj]r l ),h([netj]r u )] (22) where [netj]r l = ∑ oi [wji]r l + [bj]r ln i=1 (23) [netj]r u =∑ oi [wji]r u + [bj]r un i=1 (24) output units: [yk]r = h ([netk]r) = [[yk]r l, [yk]r u] = [h([netk]r l),h([netk]r u)] (25) where [netk]r l = ∑ [wkj]r l [zj]r l j∈a + ∑ [wkj]r l [zj]r u j∈b + [ck]r l (26) [netk]r u = ∑ [wkj]r u [zj]r u j∈c + ∑ [wkj]r u [zj]r l j∈d + [ck]r u (27) for [zj]r u ≥ [zj]r l ≥ 0 , where a = {j ∶ [wkj]r l ≥ 0}, b = {j ∶ [wkj]r l < 0} c = {j ∶ [wkj]r u ≥ 0}, d = {j ∶ [wkj]r u < 0}, a ∪ b = {1,2,3,…,m} and c ∪ d = {1,2,3,…,m}. 4 technique of the proposed method 4.1 first order fuzzy differential equation to solve any fuzzy ordinary differential equation we consider a three – layered fuzzy ffnn with one unit entry x, one hidden layer consisting of m activation functions and one unit output n(x,p). the activation function for the hidden units of our fnn is the hyperbolic tangent function. here, the dimension of fnnm is (1 × m × 1) . analytical and numerical solution of differential equations with generalized fuzzy derivative 299 for every entry x (where x ≥ 0 ) equations (21-27) will be : input unit: o = x, (28) hidden units : [zj]r = [[zj]r l , [zj]r u ] = [s([netj]r l ),s([netj]r u )] (29) where [netj]r l = o [wj]r l + [bj]r l (30) [netj]r u = o [wj]r u + [bj]r u (31) output unit: [n]r = [[n]r l, [n]r u] (32) where [n]r l=∑ [vj]r l [zj]r l + ∑ [vj]r l [zj]r u j∈b j∈a (33) [n]r u=∑ [vj]r u [zj]r u + ∑ [vj]r u [zj]r l j∈d j∈c (34) for illustration the solution steps of our proposed method, we will consider the first order fuzzy differential equation : dy(x) dx = f(x,y) , x ∈ [a,b] , y (a) = a (35) where a is a fuzzy number in e 1 with r – level sets: [a]r = [[a]r l, [a]r u] , r ∈ [0,1]. the fuzzy trial solution for this problem is: [yt(x,p)]r l = [a]r l + (x − a)[n(x,p)]r u [yt(x,p)]r u = [a]r u + (x − a)[n(x,p)]r l (36) this fuzzy solution by intention satisfies the fuzzy initial condition in (35). basim n. abood 300 in this work we use the error function : e = ∑ (eir l + eir u) g i=1 , where eir l and eir u can be viewed as the squared errors for the lower limits and the upper limits of the r-level sets, respectively. therefore, the error function that must be minimized for the problem (35) is in the form [23] : e= ∑ (eir l + eir u) g i=1 (37) where eir l = [ [ d yt (xi,p) dx ] r l − [f (xi,yt (xi,p))]r l ] 2 eir u = [ [ d yt (xi,p) dx ] r u − [f (xi,yt (xi,p))]r l ] 2 (38) where {xi}i=1 g are discrete points belonging to the interval [a,b] (training set) and in the cost function (37), er l and er u can be viewed as the squared errors for the lower limits and the upper limits of the r – level sets, respectively. it is easy to express the first derivative of [n(x,p)]r u and [n(x,p)]r l in terms of the derivative of the hyperbolic tangent activation function, i.e., ∂ [n]r l ∂x = ∑ [vj]r l ∂ [zj]r l ∂ [netj]r l ∂ [netj]r l ∂xa + ∑ [vj]r l ∂ [zj]r u ∂ [netj]r u ∂ [netj]r u ∂xb (39) ∂ [n]r u ∂x = ∑ [vj]r u ∂ [zj]r u ∂ [netj]r u ∂ [netj]r u ∂xc + ∑ [vj]r u ∂ [zj]r l ∂ [netj]r l ∂ [netj]r l ∂xd (40) from (29-31) we can get ∂[netj]r l ∂x = [wj]r l (41) ∂[zj]r l ∂ [netj]r l = 1 ([zj]r l ) 2 (42) ∂[netj]r u ∂x = [wj]r u (43) analytical and numerical solution of differential equations with generalized fuzzy derivative 301 ∂ [zj]r u ∂ [netj]r u = 1 ([zj]r u ) 2 (44) from (36) we can get ∂[yt (x,p)]r l ∂x = [n (x,p)]r u + (x − a) ∂ [n (x,p)]r u ∂x (45) ∂[yt (x,p)]r u ∂x = [n (x,p)]r l + (x − a) ∂ [n (x,p)]r l ∂x (46) then we have eir u = [ ∑ [vj]r l [zj]r l + ∑ [vj]r l [zj]r u + (xi − a)ba (∑ [vj]r l [wj]r l (1 −a ([zj]r l ) 2 ) + ∑ [vj]r l [wj]r u (1 − ([zj]r u ) 2 )b )− f(xi, [a]r u + (xi − a) (∑ [vj]r l [zj]r l + ∑ [vj]r l [zj]r u ba )) ] 2 (47) eir l = [ ∑ [vj]r u [zj]r u + ∑ [vj]r u [zj]r l + (xi − a)dc (∑ [vj]r u [wj]r u (1 −c ([zj]r u ) 2 ) + ∑ [vj]r u [wj]r l (1 − ([zj]r l ) 2 )d ) − f(xi, [a]r l + (xi − a) (∑ [vj]r u [zj]r u + ∑ [vj]r u [zj]r l dc )) ] 2 (48) then we substitute (47) and (48) in (37) to find the error function that must be minimized for problem (35). note : for reducing the complexity of the learning algorithm in (37), we have used partially fuzzy neural network (pfnn) architecture where connection weights to the output unit are fuzzy numbers while connection weights and biases to the hidden units are real numbers . the input – output relation of each unit of the pfnn can be rewritten for r – level sets as follows: input unit: o = x (49) hidden units: zj = s (netj) , j = 1,2,3,…m (50) where netj = o wj + bj (51) output unit: basim n. abood 302 [n]r=[[n]r l, [n]r u]=[∑ [vj]r l zj m j=1 ,∑ [vj]r u zj m j=1 ] (52) now, to find the minimized error function for problem (35) : ∂ [n]r l ∂x = ∑ [vj]r l ∂ zj ∂ netj ∂ netj ∂x m j=1 = ∑ [vj]r l mj=1 wj (1 − zj 2) (53) ∂ [n]r u ∂x = ∑ [vj]r u ∂ zj ∂ netj ∂ netj ∂x m j=1 = ∑ [vj]r u mj=1 wj (1 − zj 2) (54) then we obtain : eir u = [ ∑ zj m j=1 [vj]r l + (xi − a) ∑ wj m j=1 (1 − zj 2)[vj]r l − f(xi, [a]r u+ (xi − a) ∑ zj m j=1 [vj]r l ) ]2 (55) eir l = [ ∑ zj m j=1 [vj]r u + (xi − a) ∑ wj m j=1 (1 − zj 2)[vj]r u − f(xi, [a]r l+ (xi − a) ∑ zj m j=1 [vj]r u ) ]2 (56) then we substitute (55) and (56) in (37) to find the error function that must be minimized for problem (35) under pfnn. 4.2 reducing a fde to a system of odes [24,25] the solution of the fuzzy differential equation (35) is depend on the choice of the derivative (in the first form or in the second form of definition (10)). let us explain the proposed method, if we denote [y(x)]r = [y1 r(x),y2 r(x)], [y0] r = [y01 r , y02 r ] and [f(x,y(x))]r = [ f1 r(x,y1 r(x),y2 r(x)),f2 r(x,y1 r(x),y2 r(x)) ] (57) we have the following results: case i. if we consider yˊ(x) in the first form (1) of definition (2.2.3), then we have to solve the following system of odes analytical and numerical solution of differential equations with generalized fuzzy derivative 303 d dx (y1 r(x)) = f1 r(x,y1 r(x),y2 r(x)) y1 r(a) = y01 r d dx (y2 r(x)) = f2 r(x,y1 r(x),y2 r(x)) y2 r(a) = y02 r case ii. if we consider yˊ(t) in the second form (2) of definition (2.2.3) then we have to solve the following system of odes d dx (y1 r(x)) = f2 r(x,y1 r(x),y2 r(x)) y1 r(a) = y01 r d dx (y2 r(x)) = f1 r(x,y1 r(x),y2 r(x)) y2 r(a) = y02 r the existence and uniqueness of the two solutions (for problem (35)) which described above are given by the following theorem theorem (2) : let f ∶ i × e1 → e1 be a continuous fuzzy function such that there exists k > 0 such that d(f(x,w),f(x,z)) ≤ k d(w,z) for all t ∈ i and w,z ∈ e1 then the problem (35) has two solutions (one (1)-differentiable and the other one (2)differentiable) on i, where i = [a,b] . proof: see [26] □ to illustrate how we can find the two solutions for a fuzzy differential equation under generalized h-derivation, we present the following problems: problem (1): consider the fuzzy initial value problem y ́ = −y(x) , y(0) = [ 0.96 + 0.04r,1.01 − 0.01r ] (1) according to subsection (4.2), case i., after reducing the above problem , we have the following system of odes d dx (y1 r(x)) = −y1 r(x), y1 r(0) = 0.96 + 0.04r d dx (y2 r(x)) = −y2 r(x), y2 r(0) = 1.01 − 0.01r basim n. abood 304 which gives the following fuzzy analytical solution y(x,r) = [ (0.96 + 0.04r)e−x, (1.01 − 0.01r)e−x ] (2) according to subsection (4.2), case ii., after reducing the above problem , we have the following system of odes d dx (y1 r(x)) = −y2 r(x), y1 r(0) = 0.96 + 0.04r d dx (y2 r(x)) = −y1 r(x), y2 r(0) = 1.01 − 0.01r which gives the following fuzzy analytical solution y(x,r) = [ (0.985 + 0.015r)e−x − (1 − r)0.025ex, (0.985 + 0.015r)e−x + (1 − r)0.025ex ]. problem (2): consider the fuzzy initial value problem y ́ = −3y(x) + e2x , y(0) = [ 0.75 + 0.25r,1.25 − 0.25r ] (1) according to subsection (4.2), case i., after reducing the above problem , we have the following system of odes d dx (y1 r(x)) = −3y1 r(x) + e2x, y1 r(0) = 0.75 + 0.25r d dx (y2 r(x)) = −3y2 r(x) + e2x, y2 r(0) = 1.25 − 0.25r which gives the following fuzzy analytical solution y(x,r) = [ (0.55 + 0.25r)e−3x + 0.2e2x, (1.05 − 0.25r)e−3x + 0.2e2x ] (2) according to subsection (4.2), case ii., after reducing the above problem , we have the following system of odes d dx (y1 r(x)) = −3y2 r(x) + e2x, y1 r(0) = 0.75 + 0.25r d dx (y2 r(x)) = −3y1 r(x) + e2x, y2 r(0) = 1.25 − 0.25r analytical and numerical solution of differential equations with generalized fuzzy derivative 305 which gives the following fuzzy analytical solution y(x,r) = [ (0.75 + 0.25r)cosh3x − (1.25 − 0.25r)sinh3x + 0.2e2x − 0.2e−3x , (1.25 − 0.25r)cosh3x − (0.75 + 0.25r)sinh3x + 0.2e2x − 0.2e−3x ]. 5 numerical examples to show the behavior and properties of the proposed method, two problem will be solved in this section. we have used a multilayer perceptron having one hidden layer with ten hidden units and one output unit. the activation function of each hidden unit is hyperbolic tangent activation function. the analytical solution [ya(x)]r l and [ya(x)]r u has been known in advance. therefore, we test the accuracy of the obtained solutions by computing the deviation (absolute error): e (x, r) = |[ya(x)]r u − [yt (x)]r u|, e (x, r)= |[ya(x)]r l − [yt(x)]r l| in order to obtain better results, more hidden units or training points may be used. to minimize the error function we have used bfgs quasi-newton method (for more details, see [24]). the computer programs which we have used in this work are coded in matlab 2015. example 1: consider the following linear fuzzy initial value problem: y´ = y + x + 1, with x ∈ [0, 1] y(0) = [0.96 + 0.04r,1.01 – 0.01r], where r ∈ [0, 1]. the analytical solution (according to subsection (4.2), case ii. ) are : [ya(x)]r l = x + (0.985 + 0.015r)e−x − (1 − r)0.025ex [ya(x)]r u= x + (0.985 + 0.015r)e−x + (1 − r)0.025ex the trial solution (according to the proposed method in this work) are: [yt(x)]r l = (0.96 + 0.04r) + x [n(x,p)]r u [yt(x)]r u= (1.01 − 0.01r) + x [n(x,p)]r l basim n. abood 306 the ann trained using a grid of ten equidistant points in [0, 1]. the error function that must be minimized for this problem will be: e = ∑ ( [11i=1 xi ∑ [vj]r l wj s´ 10 j=1 (xi wj + bj)+ (1 + xi) ∑ [vj]r l s10j=1 (xi wj + bj ) – xi 0.01r + 0.01 ] 2 +[xi ∑ [vj]r u wj s´ 10 j=1 (xi wj + bj) + (1 + xi) ∑ [vj]r u s10 j=1 (xi wj + bj)– xi + 0.04 r 0.04 ] 2 ) (58) where s´ is first derivative of hyperbolic tangent activation function. then we use (58) to update the weights and biases. analytical and trial(numerical) solutions for this problem can be found in table (1) and table (2). example 2: consider the following non-linear fuzzy initial value problem: y´ = cos (x2 + y2), with x ∈ [0, 1] y(0) = [0.75 + 0.25r,1.25 – 0.25r], where r ∈ [0, 1]. the trial solutions (according to the proposed method in this work) for this problem are : [yt(x)]r l= (0.75 + 0.25r) + x [n(x,p)]r u [yt(x)]r u= (1.25 − 0.25r) + x [n(x,p)]r l the ann trained using a grid of ten equidistant points in [0, 1]. the error function that must be minimized for this problem will be e = ∑ ( [11i=1 xi ∑ [vj]r l wj s´ 10 j=1 (xi wj + bj)+ ∑ [vj]r l s10j=1 ( xi wj + bj) – cos ( xi 2+(1.25 − 0.25r + xi ∑ [vj]r l s10j=1 ( xi wj + bj)) 2) ]2+[ analytical and numerical solution of differential equations with generalized fuzzy derivative 307 xi ∑ [vj]r u wj s´ 10 j=1 (xi wj + bj)+ ∑ [vj]r u s10j=1 (xi wj + bj) –cos ( xi 2 + (0.75 + 0.25r + xi ∑ [vj]r u s10j=1 (xi wj + bj)) 2) ]2 ) (59) then we use (59) to update the weights and biases. trial (numerical) solutions for this problem can be found in table (3). table (1): trial solutions for example (1), x = 2. table (2): trial solutions for example (1), r = 0.5. r [yt(x)]r l e (x,r) [yt(x)]r u e (x,r) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.948579274 1.967254967 1.985930525 2.004605986 2.023282053 2.041957703 2.060632805 2.079308402 2.097984071 2.116660365 2.135335791 0.000000422 0.000000472 0.000000387 0.000000205 0.000000629 0.000000635 0.000000094 0.000000048 0.000000074 0.000000725 0.000000508 2.318032417 2.299762662 2.281492466 2.263223455 2.244953513 2.226683566 2.208413946 2.190144557 2.171875271 2.153604974 2.135336111 0.000000761 0.000000643 0.000000085 0.000000711 0.000000406 0.000000096 0.000000114 0.000000362 0.000000713 0.000000054 0.000000828 x [yt(x)]r l e (x,r) [yt(x)]r u e (x,r) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.98 0.984236574 0.997322850 1.018388968 1.046645251 1.081372866 1.121919133 1.167689342 1.218140638 1.272775429 1.331142019 0 0.000000074 0.000000112 0.000000119 0.000000414 0.000000202 0.000000069 0.000000337 0.000000903 0.000000081 0.000000196 1.005 1.011865864 1.027858634 1.052136129 1.083941124 1.122591654 1.167472118 1.218032891 1.273778484 1.334265671 1.399099381 0 0.000000090 0.000000827 0.000000810 0.000000670 0.000000958 0.000000084 0.000000068 0.000000226 0.000000245 0.000000513 basim n. abood 308 table (3): trial solutions for example (2), r = 0.25. 6 conclusion in this paper, we have presented numerical method based on fuzzy neural network for solving first order fuzzy initial value problem under generalized h-derivation. we have demonstrated the ability of the fuzzy neural network to approximate the solution of the fuzzy differential equations. therefore, we can conclude that the method which we proposed can handle effectively all types of the fuzzy differential equations and provide accurate approximate solution throughout the whole domain and not only at the training set. as well, one can use the interpolation techniques to find the approximate solution at points between the training points or at points outside the training set. 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[26] cano y. c., flores h. r.,et al., "fuzzy differential equations with generalized derivative",fuzzy sets and systems, 160, 1517-1527,2008. ratio mathematica 26 (2014), 39–64 issn:1592-7415 fuzzy bi-objective optimization model for multi-echelon distribution network kanika gandhi, p. c. jha bhavan’s usha & lakshmi mittal institute of management, new delhi, india department of operational research, university of delhi, delhi, india gandhi.kanika@gmail.com,jhapc@yahoo.com abstract it is important for modern businesses to search the ways for continuous improvement in performance of their supply chains. the effective coordination and integrated decision making across the supply chain enhances the performance among its various partners in a multi stage network. the partners considered in this paper are product suppliers, processing points (pp), distribution centres (dc) and retail outlets (ro). the network addresses an uncertain environment threatened by different sources in order to captivate the real world conditions. the uncertain demand of deteriorating products and its dependent costs develop uncertainties in the environment. on the other hand, suppliers and processing points have restricted capacities for the retail outlets’ order amount happened in each period. a bi-objective non-linear fuzzy mathematical model is developed in which the uncertainties are represented by the fuzzy set theory. the proposed model shows cost minimization and best supplier selection coordination under the conditions of capacity constraints, uncertain parameters and product’s deteriorating nature. the fish and fish products give good examples for the proposed model. to solve, the model is converted into crisp form and solved with the help of fuzzy goal programming. key words: multi stage, supplier selection, processing point, fuzzy goal programming, supply chain, bi-objective. 39 gandhi, jha 1 introduction with the growing importance of supply chain management (scm) in enterprise development and in the operation of socio-economic systems, cost management has become a strategic business issue in recent years. it involves not only the financial flows but also the associated material flows and information flows among supply chain partners. moreover, it plays an indispensable role in bringing profits and competitive advantage to firms, and consequently receives increasing attention from both supply chain managers and academics. activities in supply chain system consist of transforming natural resources, raw materials and components into finished product and their final delivery to the end customers. most of these economic activities form an integral part of the value chain. from this view point, cost management in supply chains is not limited to individual enterprises, but extends to all the purchasing, warehousing, production and distribution activities along the chain. its goal is to provide a management tool and method to design the integrated chain, to promote its development and to reduce the total cost of supply chain system. however, a lot more complexity is involved in effectively integrating all the supply chain activities in a cost efficient manner owing to shorter life cycle of products and increased competition among suppliers who are offering different opportunities to the retailer. the uncertain demand of deteriorating products and their dependent costs creates uncertainty in the environment and consequently results in an indecisive and unsure environment for the decision makers. choosing high level of procured quantity and inventory to avoid shortages will definitely lead to an immense increase in the cost of purchase and inventory holding. in this regard, operations management practices and mathematical models provide a sound framework for effective and integrative decision making across supply chain. for minimizing the cost and improving the overall performance, major functions considered are economic ordered quantity decisions, supplier selection decisions, inventory & capacity decisions and transportation policies in multi periods and for multi products. while economic ordered quantity decisions aim to minimize the cost of procurement, inventory and transportation, the intent of supplier selection and transportation policy selection decisions is to maximize inbound logistics performance by attaining a high degree of quality and delivery performance. due to the inherent interdependency among these decisions, a firm cannot optimize them separately. hence the main purpose of this paper is to develop a model addressing above issues i.e. to characterize the optimal decisions that each partner in supply chain should adopt to motivate the chain partners to coordinate so that everyone benefits from the improved performance of the system. 40 fuzzy bi-objective optimization model though procurement functions need to consider cost minimization objective, yet in doing so one cannot compromise on quality and delivery related criteria. nowadays, quality and delivery related objectives are being given higher priority than cost criterion during procurement decisions. suppliers’ performance on quality and delivery criteria has a significant influence on the ordered quantity and the total transportation costs. taking into account the above observations, in this study we develop a fuzzy bi-objective non-linear programming model for an integrated economic ordered quantity, supplier selection and transportation policy problem. we investigate a problem in which multi products are procured from multiple suppliers in multiple periods considering limitations on capacity at supplier point and processing point for deteriorating products. we also incorporate cost of inventory at distribution centres & retail outlets and transportation cost and policy concepts in one stage to another. imprecise demand and other uncertain known parameters make the environment of model uncertain and fuzzy. to summarize the above discussions, the present work shows (1) a fuzzy bi-objective multi stage non-linear optimization model that includes computation of cost of procurement, processing, holding and transportation as first objective and the other objective shows the process to choose best supplier on the basis of delivery and quality; (2) the coordination among multi stages, i.e. (i) procurement stage; (ii) processing stage constituted of (a) receiving & scanning, (b) sorting & packaging & (c) scanning & dispatching; (iii) distribution centres and (iv) retail outlets; (3) transportation policies and minimum cost per weight from processing stage to distribution centres and transportation cost per unit from distribution centre to retail outlet; (4) fuzzy set theory to coordinate uncertain parameters; (5) coordination in procurement, demand and inventory so the zero shortage is ensured. 2 literature review there are vast researches working on supplier selection problems with different approaches. one of the most important decisions related to procurement operations is supplier evaluation and selection. there are several factors involved such as price offered by the supplier, lead time, the quality of items, the capacity of supplier and the geographical location of supplier while making supplier evaluation and selection decisions (ho et al., 2010). ho et al. (2010), the three most important criteria considered while selecting suppliers are product quality, delivery lead time and price. hassini (2008) studies a lot sizing and supplier selection problem when supplier capacity reservation dependent on lead time. ravindran, bilsel, wadhwa, and yang (2010) study 41 gandhi, jha supplier selection and order allocation considering incremental price breaks. liao and rittscher (2007) propose a multi objective programming model for supplier selection, procurement lot sizing and carrier selection decisions. razmi and maghool (2010) propose a fuzzy bi-objective model for multiple items, multiple period, supplier selection and purchasing problem under capacity constraint and budget limitation. zhang and zhang (2011) formulate a mixed integer programming model for selecting suppliers and allocating the ordering quantity properly among the selected suppliers to minimize the selection, purchase and inventory costs. jolai, yazdian, shahanaghi, and khojasteh (2011) proposed a two-phase approach for supplier selection and order allocation problem under fuzzy environment for multiple products from multiple suppliers in multiple periods. pal, sana, and chaudhuri (2012) addressed a multi-echelon suppler chain with two suppliers in which the main supplier may face supply disruption and the secondary supplier is reliable but more expensive, and the manufacturer may produce defective items. kilic (2013) discussed an integrated approach including fuzzy technique for order preference by similarity to ideal solution (topsis) and a mixed integer linear programming model is developed to select the best supplier in a multiitem/multi-supplier environment. few of the studies have addressed problems having multi objectives and with fuzziness. madronero, peidro, and vassant (2010) used s-curve membership functions for fuzzy aspiration levels for objective functions, maximum capacity of the vendors as rhs, budget amount allocated to vendors as rhs with fuzzy programming by using modified werner’s fuzzy or operator. wu, zhang, wu and olson (2010) used trapezoidal membership functions for fuzzy model parameters as objective function coefficients and right hand side (rhs) constants with sequential quadratic programming. arikan (2011) used triangular and right triangular membership functions for fuzzy aspiration levels for objective functions and demand level as rhs with lai and hwang’s augmented max–min model. concerning with multichoice goals, decision-making behaviour and limit of resources, lee, kang, and chang (2009) develop a fuzzy multiple goal programming model to help downstream companies to select thin film transistor liquid display suppliers for cooperation. they used triangular membership functions for fuzzy aspiration levels for objective functions. further, a multi-objective model for supplier selection in multi-service outsourcing is developed by feng, fan, and li (2011). a multi objective mathematical model has been discussed by seifbarghy and esfandiari (2013), which includes minimizing the transaction costs of purchasing from suppliers as well as other objectives as minimizing the purchasing cost, rejected units, and late delivered units, and maximizing the evaluation scores of the selected suppliers. the problem is converted into 42 fuzzy bi-objective optimization model single objective using weighting method and solved using meta-heuristics. aghai, mollaverdi and saddagh (2014), outlined a fuzzy multi-objective programming model to propose supplier selection taking quantitative, qualitative, and risk factors into consideration. also quantity discount has been considered to determine the best suppliers and to place the optimal order quantities among them. from the literature, it is evident that most studies have not paid much attention to uncertainty in supplier’s information and many problematic criteria in the conditions of multi product, transportation modes and multiple sourcing. the main purpose of this paper has been outlined as (1) to propose a fuzzy bi-objective mathematical model to choose the supplier with best performance on the basis of quality & delivery percentages and to keep the cost optimum while procurement, processing of products and transportation, the ideal number of inventory items so that shortages does not take place, and optimum quantity from suppliers subject to the constraints pertaining to demand, suppliers capacity, processing capacity and inspection, (2) the objectives are conflicting in nature as minimization of cost and performance maximization of the supplier. because of uncertain parameters the environment of the problem becomes fuzzy, for which, fuzzy goal programming method has been used to solve the mathematical model of cost minimization and suppliers selection with maximum performance. 3 problem definition to manage different entities to minimize their cost and simultaneously measuring the suppliers’ performances in the environment of uncertainty, the current paper presents a fuzzy bi-objective mixed integer non-liner model. the first objective of the proposed model minimizes the cost of integration of procurement and distribution. this comprises of multi source (suppliers), two processing points, multi distribution centres & multi retail outlets and incorporating transportation costs and policies. the second objective focuses on performance and selection of suppliers on the bases of on-time delivery percentage and acceptance percentage of the ordered quantity. the first stage of first objective explains procurement cost as per optimum procured quantity from the active suppliers, processing cost per unit in three levels at processing point. at this point receiving, scanning, sorting and packing of goods takes time, hence holding cost is included in the processing cost. the second stage shows the fuzzy cost of holding at distribution centres and cost of transportation of goods from processing points to distribution centres which is completed through two modes of transportation as full truck 43 gandhi, jha load (tl) mode and truck load (tl) & less than truck load (ltl) mode. in truck load transportation mode, the cost is fixed of one truck up to a given capacity. in this mode, the company may use less than the capacity available but cost per truck will not be reduced. however, sometimes the weighted quantity may not be large enough to corroborate the cost associated with a tl mode. in such situation, a ltl mode may be used. ltl is defined as a shipment of weighted quantity which does not fill a truck. in such a case, transportation cost is taken on the bases of per unit weight. the third stage includes inspection, fuzzy holding cost at retail outlet and transportation cost per unit in the account from distribution centres to retail outlet. the second objective is to find best suppliers with the combination of fuzzy ontime delivery percentage and fuzzy acceptance percentage of the ordered quantity. the model integrates inventory, procurement and transportation mechanism to minimize all costs discussed above and also chooses the best supplier. in the model, all the co-ordinations among supply chain partners are being managed under one buyer who is taking care of processing points, distribution centres and retail outlets but not sources (suppliers) directly. the total cost of the model becomes fuzzy due to fuzzy holding cost and demand. on the other hand, performance level is also fuzzy as percentage of on-time delivery and acceptances are fuzzy. hence, the model discussed above is fuzzy biobjective mixed integer non-linear model. in the solution process, the fuzzy model is converted into crisp and further fuzzy goal programming approach is employed where each objective could be assigned a different weight. 4 proposed model formulation the model is based on following assumptions: • finite planning horizon • demand at retail outlet is uncertain and no shortages are allowed • initial inventory at the beginning of planning horizon is zero • inventory at retail outlet deteriorates at constant rate • inspection cost of received goods at retail out is fixed • no transportation cost is discussed as it is considered as part of purchasing cost • holding cost is part of processing cost at processing point 44 fuzzy bi-objective optimization model 4.1 sets set cardinality index product p i supplier j j processing point z z distribution centre m m retail outlet o o time period t t 4.2 parameters ∼ c : fuzzy total cost c0 & c ∗ 0 : aspiration & tolerance level of fuzzy total cost ∼ pr : fuzzy performance of supplier pr0 & pr ∗ 0 : aspiration & tolerance level of fuzzy performance of supplier ∼ hd imt & hdimt : fuzzy & defuzzified holding cost per unit of product i for tth period at mth distribution centre ϕijzt : unit purchase cost for i th product in tth period from supplier j for zth processing point a: cost per weight of transportation in ltl policy kzmt : fixed freight cost for each truck load in period tfrom processing point z to distribution centre m tcimot : transportation cost for unit in period tfrom distribution centre m to retail outlet o ∼ hr iot & hriot : fuzzy & defuzzified holding cost per unit of product i for tth period at retail outlet o λiot : inspection cost per unit of product i in period t at retail outlet o ∼ d iot & diot : fuzzy & defuzzified demand at retail outlet o for product i in period t in izt: initial inventory processing point z in beginning of planning horizon for product i η : deterioration percentage of ith product at retail outlet wi: per unit weight of product i ω : weight transported in each full truck 45 gandhi, jha ∼ dt ijzt & dt ijzt : fuzzy & defuzzified percentage of on-time delivery time for product i in period t for supplier j for processing point z ∼ ac ijzt & acijzt : fuzzy & defuzzified percentage of acceptance for product i in period t for supplier j for processing point z δijz: capacity at supplier j for product ifor z th processing point αizrt : capacity of receiving & scanning level (r) at z th processing point for product i in period t cizrt : cost of receiving & scanning level (r) at z th processing point for product i in period t βizst : capacity of sorting & packing level (s) at z th processing point for product i in period t cizst : cost of sorting & packing (s) at z th processing point for product i in period t γizdt : capacity of scanning & dispatching level (d) at z th processing point for product i in period t cizdt : cost of scanning & dispatching (d) at z th processing point for product i in period t 4.3 decision variable xijzt : optimum ordered quantity of product i ordered in period tfrom supplier j transported to processing point z vijt: if ordered quantity is procured by active supplier j for product i in period tthen the variable takes value 1 otherwise zero uzmt: usage of modes, either tl & ltl mode (value is 1) or only tl mode (value is 0) 4.4 operating variables yizt : procured quantity reached at receiving & scanning level of zth processing point from all the active suppliers aizt : goods moved to sorting & packaging from receiving & scanning level at zth processing point eimt : goods reaching at m th distribution centre from all processing points jzmt : total number of truck loads in period t from processing point z to distribution centre m qzmt : weighted quantity in excess of truckload capacity giot : total quantity reached at retail outlet o from all distribution centres iizt : inventory at processing point in period t for product i iimt : inventory at distribution centre in period t for product i 46 fuzzy bi-objective optimization model iiot : inventory at retail outlet in period t for product i bizmt : quantity of product i shipped from z th processing point to mth distribution centre in period t fimot : quantity of product i shipped from m th distribution centre to oth retail outlet in period t lzmt : weighted quantity transported from z th processing point to mth distribution centre in period t 4.5 fuzzy optimization model formulation fuzzy dependent environment with respect to uncertain independent variables cannot be quantified by crisp mathematical programming approaches. fuzzy optimization approach permits adequate solutions of real problems in the presence of vague information by defining the mechanisms to quantify uncertainties directly. therefore, we formulate fuzzy optimization model for vague aspiration levels on cost, demand, on-time delivery percentage and acceptance percentage the decision maker may decide the aspiration and tolerance levels on the basis of past experience and knowledge. 4.5.1 formulation of objectives initially a bi-objective fuzzy model is formulated which discusses about fuzzy total cost and performance of the suppliers. the first objective of the model minimizes the total cost, consisting of procurement cost of goods from supplier, processing cost, holding cost at distribution centres, transportation cost from processing point to distribution centres and further to retail outlets, holding cost at retail outlets and finally inspection cost of the reached quantity at retail outlets. minimize, ∼ c = t∑ t=1 z∑ z=1 j∑ j=1 p∑ i=1 ϕijztxijztvijzt + t∑ t=1 z∑ z=1 p∑ i=1 [ r∑ r=1 cizrtyizt + ( s∑ s=1 cizst + d∑ d=1 cizdt ) aizt ] + t∑ t=1 m∑ m=1 p∑ i=1 ∼ hd imt eimt 47 gandhi, jha + t∑ t=1 m∑ m=1 z∑ z=1 [(aqzmt + jzmtkzmt) uzmt + (jzmt + 1) kzmt (1 −uzmt)] + t∑ t=1 o∑ o=1 m∑ m=1 p∑ i=1 tcimotfimot + t∑ t=1 o∑ o=1 p∑ i=1 ∼ hr iot iiot + t∑ t=1 o∑ o=1 p∑ i=1 λiotgiot the second objective discusses the performance of suppliers and maximizes the performance percentage of supplier as per on-delivery time percentage and acceptance percentage of ordered quantity. maximize ∼ pr = t∑ t=1 z∑ z=1 j∑ j=1 p∑ i=1 ( ∼ dt ijzt + ∼ ac ijzt ) vijzt 4.5.2 constraint formulation all the suppliers must have enough capacity to fulfil the orders. the following equation ensures that the active supplier shall have enough capacity to complete the orders from processing point. xijzt ≤ δijzvijzt ∀i,j, z,t next equation ensures that only one supplier can be active for a particular product in a period. however, same supplier can be active again in next period. j∑ j=1 vijzt = 1 ∀ i, t,z goods are reaching at zth processing point from all the suppliers. yizt = j∑ j=1 xijzt ∀i, t,z at receiving & scanning level in processing point, 2% from each lot is rejected and removed. aizt = 0.98yizt ∀i, t,z quantity dispatched from zth processing point is being transported to all distribution centres. aizt = m∑ m=1 bizmt ∀i, z, t 48 fuzzy bi-objective optimization model goods reaching at mth distribution centre are transported from all the processing points. eimt = z∑ z=1 bizmt ∀i, m, t goods are transported from mth distribution centre to all the retail outlets. eimt = o∑ o=1 fimot ∀i, m, t goods reaching at oth retail outlets eiot are transported from all the distribution centres giot = m∑ m=1 fimot ∀i, o, t following three equations explain the capacities in processing point at all the levels respectively i.e. receiving and scanning level, sorting & packaging level and scanning and dispatching level. yizt ≤ αizrt ∀i, z,t,r aizt ≤ βizst ∀i, t,z,s aizt ≤ γizdt ∀i, t,z,d next three equations show balancing equations at processing point, which also takes care of no shortages assumption. first two equations of the set calculate inventory at end of the period with respect to quantity reached at receiving and scanning level from the supplier and quantity sent to sorting & packaging level. the third equitation takes care of the shortages by balancing the quantity between the two levels discussed above. iizt = iizt−1 + yizt −aizt ∀i, t > 1,z iizt = inizt + yizt −aizt ∀i, t = 1,z t∑ t=1 iizt + t∑ t=1 yizt ≥ t∑ t=1 aizt ∀i,z balancing at distribution centres have been discussed in next three equation, where assumption of no shortages has also been taken care of. iimt = iimt−1 + eimt − o∑ o=1 fimot ∀i, t > 1,m 49 gandhi, jha iim1 = 0 ∀i,m t∑ t=1 iimt + t∑ t=1 eimt ≥ t∑ t=1 o∑ o=1 fimot ∀i,m at retail outlets also, inventory has been balanced with respect to the received quantity and demand. iiot = iiot−1 + giot − ∼ d iot −ηiiot ∀i, t > 1,o iio1 = 0 ∀i,o (1 −η) t∑ t=1 iiot + t∑ t=1 giot ≥ ∼ t∑ t=1 ∼ d iot ∀i,o following equation is an integrator and calculates the weighted quantity which is to be transported from processing point to distribution centres. lzmt = p∑ i=1 ωibizmt ∀z,t,m the next equation finds out transportation policy as per the weighted quantity. here, the costs of tl policy and tl&ltl policy are compared as per the weight. lzmt ≤ (qzmt + jzmtw) uzmt + (jzmt + 1) w (1 −uzmt) ∀z,m,t the calculation of overhead quantity in tl&ltl policy is calculated by comparing total weighted quantity with total number of full truck loads as per weight is discussed in following equation. lzmt = qzmt + jzmtw ∀z,m, t lastly, describing the nature of decision variables and enforcing the binary and non-negative restrictions to them. xijzt, yizt, aizt, eimt, fimot, giot, lzmt ≥ 0; vijzt, uzmt ∈ [0, 1] ; iimt, iiot, iizt, qzmt, jzmt are integer. 50 fuzzy bi-objective optimization model 4.5.3 formulated model minimize ∼ c = t∑ t=1 z∑ z=1 j∑ j=1 p∑ i=1 ϕijztxijztvijzt + t∑ t=1 z∑ z=1 p∑ i=1 [ r∑ r=1 cizrtyizt + ( s∑ s=1 cizst + d∑ d=1 cizdt ) aizt ] + t∑ t=1 m∑ m=1 p∑ i=1 ∼ hd imt eimt + t∑ t=1 m∑ m=1 z∑ z=1 [(aqzmt + jzmtkzmt) uzmt + (jzmt + 1) kzmt (1 −uzmt)] + t∑ t=1 o∑ o=1 m∑ m=1 p∑ i=1 tcimotfimot + t∑ t=1 o∑ o=1 p∑ i=1 ∼ hr iot iiot + t∑ t=1 o∑ o=1 p∑ i=1 λiotgiot maximize ∼ pr = t∑ t=1 z∑ z=1 j∑ j=1 p∑ i=1 ( ∼ dt ijzt + ∼ ac ijzt ) vijzt. subject to xijzt ≤ δijzvijzt ∀i,j, z,t j∑ j=1 vijzt = 1 ∀ i, t,z yizt = j∑ j=1 xijzt ∀i, t,z aizt = 0.98yizt ∀i, t,z aizt = m∑ m=1 bizmt ∀i, z, t eimt = z∑ z=1 bizmt ∀i, m, t eimt = o∑ o=1 fimot ∀i, m, t giot = m∑ m=1 fimot ∀i, o, t yizt ≤ αizrt ∀i, z,t,r aizt ≤ βizst ∀i, t,z,s aizt ≤ γizdt ∀i, t,z,d iizt = iizt−1 + yizt −aizt ∀i, t > 1,z iizt = inizt + yizt −aizt ∀i, t = 1,z t∑ t=1 iizt + t∑ t=1 yizt ≥ t∑ t=1 aizt ∀i,z 51 gandhi, jha iimt = iimt−1 + eimt − o∑ o=1 fimot ∀i, t > 1,m iim1 = 0 ∀i,m t∑ t=1 iimt + t∑ t=1 eimt ≥ t∑ t=1 o∑ o=1 fimot ∀i,m iiot = iiot−1 + giot − ∼ d iot −ηiiot ∀i, t > 1,o iio1 = 0 ∀i,o (1 −η) t∑ t=1 iiot + t∑ t=1 giot ≥ ∼ t∑ t=1 ∼ d iot ∀i,o lzmt = p∑ i=1 ωibizmt ∀z,t,m lzmt ≤ (qzmt + jzmtw) uzmt + (jzmt + 1) w (1 −uzmt) ∀z,m,t lzmt = qzmt + jzmtw ∀z,m, t xijzt, yizt, aizt, eimt, fimot, giot, lzmt ≥ 0; vijzt, uzmt ∈ [0, 1] ; iimt, iiot, iizt, qzmt, jzmt are integer. 5 solution algorithm 5.1 fuzzy solution algorithm in following algorithm by zimmermann (1976) specifies the sequential steps to solve the fuzzy mathematical programming problems. step 1. compute the crisp equivalent of the fuzzy parameters using a defuzzification function. here, ranking technique is employed to defuzzify the parameters as f2(a) = (al + 2am + au)/4, where al,am,au are the triangular fuzzy numbers (tfn). let − d iot be the defuzzified value of ∼ d iot and (d1iot,d 2 iot,d 3 iot) for each i,o & t be triangular fuzzy numbers then,d iot = (d1iot + 2d 2 iot + d 3 iot) /4. similarly, − hd imt and − hr iot are defuzzified aspired holding cost at warehouse and destination. step 2. since industry is highly volatile and customer demand changes in every short span, a precise estimation of cost and performance aspirations is a major area of discussion. hence, a better way to come out of such situation is to incorporate tolerance and aspiration level with the main objectives. the model discussed in section 4.5.3 can thus be re-written as follows: find x, x ∈ s 52 fuzzy bi-objective optimization model (1 −η) t∑ t=1 iiot + t∑ t=1 giot ≥ ∼ t∑ t=1 − d iot ∀i,o c(x)≤ ∼ c0 pr≥ ∼ pr0 xijzt, yizt, aizt, eimt, fimot, giot, lzmt ≥ 0; vijzt, uzmt ∈ [0, 1] ; iimt, iiot, iizt, qzmt, jzmt are integer. step3. define appropriate membership functions for each fuzzy inequalities as well as constraint corresponding to the objective functions. µc (x) =   1 ; c(x) ≤ c0 c∗0−c(x) c∗0−c0 ; c0 ≤ c(x) < c∗0 0 ; c(x) > c∗0 , µp r(x) =   1 ; pr ≥ pr0 p r−p r∗0 p r0−p r∗0 ; pr∗0 ≤ pr < pr0 0 ; pr < pr∗0 µiiot (x) =   1 ; iiot(x) ≥ d0 iiot(x)−d ∗ 0 d0−d ∗ 0 ; d ∗ 0 ≤ iiot(x) < d0 0 ; iiot(x) > d ∗ 0 where d0 = t∑ t=1 o∑ o=1 diotis the aspiration and d ∗ 0is the tolerance level to inventory constraints. step4. employ extension principle to identify the fuzzy decision, which results in a crisp mathematical programming problem given by maximize α subject to µc(x) ≥ α, µp r(x) ≥ α, µiiot (x) ≥ α, x ∈ s where α represents the degree up to which the aspiration of the decisionmaker is met. the above problem can be solved by the standard crisp mathematical programming algorithms. step5. following bellman and zadeh (1970), while solving the problem following steps 1-4, the objective of the problem is also treated as a constraint. each constraint is considered to be an objective for the decision-maker and the problem can be looked as a fuzzy bi-objective mathematical programming problem. further, each objective can have a different level of importance and can be assigned weight to measure the relative importance. the resulting 53 gandhi, jha problem can be solved by the weighted min max approach. on substituting the values for µp r(x) and µc (x)the problem becomes maximize α subject to pr(x) ≥ pr0 − (1 −w1α)(pr0 −pr∗0) c(x) ≤ c0 + (1 −w2α)(c∗0 −c0) (p1) µiiot (x) ≥ α x ∈ s w1 ≥ 0, w2 ≥ 0, w1 + w2 = 1,α ∈ [0, 1] step6. if a feasible solution is not obtained for the problem in step 5, then we can use the fuzzy goal programming approach to obtain a compromised solution given by mohamed (1997). the method is discussed in detail in the next section. 5.2 fuzzy goal programming approach on solving the problem, we found that the problem (p1) is not feasible; hence the management goal cannot be achieved for a feasible value of α[0,1]. then, we use the fuzzy goal programming technique to obtain a compromised solution. the approach is based on the goal programming technique for solving the crisp goal programming problem given by mohamed (1997). the maximum value of any membership function can be 1; maximization of α[0,1] is equivalent to making it as close to 1 as best as possible. this can be achieved by minimizing the negative deviational variables of goal programming (i.e., η) from 1. the fuzzy goal programming formulation for the given problem (p1) introducing the negative and positive deviational variables ηj & ρj is given as minimize u subject to µp r(x) + η1 −ρ1 = 1 µc (x) + η2 −ρ2 = 1 u ≥ wj ∗ηj j = 1, 2 ηj ∗ρj = 0 j = 1, 2 w1 + w2 = 1 α = 1 −u ηj,ρj ≥ 0; x ∈ s; u ∈ [0, 1]; w1,w2 ≥ 0 6 case study fish is a highly perishable food which needs proper handling and preservation if it is to have a long shelf life and also retain a desirable quality and 54 fuzzy bi-objective optimization model its nutritional value. the central concern of fish processing is to prevent fish from deterioration. when fish are captured or harvested for commercial purposes, they need some pre-processing so they can be delivered to the next part of the supply chain in a fresh and undamaged condition. this means, for example, that fish caught by a fishing vessel need handling so they can be stored safely until the boat lands the fish on shore. some of the methods to preserve and process fish and fish products include control of temperature using ice, refrigeration or freezing, sorting and grading, chilling, storing the chilled fish. the model is validated for the case on fish and fish products. case is taken for two suppliers, two processing points, three distribution centres and three retail outlets for three time periods. each processing point has its own internal three stages i.e. receiving & scanning, sorting & packing and scanning & dispatching. at processing point, fish products are received and scanned, which have been pre-processed to reduce the deterioration percentage. afterwards, they are sorted as per quality checks and packed and further sent to the next stage for final scanning before dispatching to the distribution centres. the objectives include minimizing the cost of procurement, processing, transportation and inventory by obtaining the optimal ordered quantity, transportation weights & minimum inventory and maximizing the performance of procurement by choosing the best supplier on the basis of delivery and quality. the data on cost of procurement from suppliers, processing cost, transportation cost from one stage to another, cost of inspection and inventory carrying cost has been discussed. three types of fish have been discussed in the case are rohu, katle and pomfret which are ranging from rs.80 to rs.190 per kg. in the case, uncertain parameters are performance parameters, holding cost and demand. further, defuzzified holding costs at all distribution centres and retail outlets are rs.14, rs.8 and rs.8 for three fish types respectively in all the periods. the capacity at both the suppliers is 300 and 380 packets for fish type ‘rohu’, 370 and 390 packets for fish type ‘katle’ and 360 and 380 packets for fish type ‘pomfret’. in processing stage, the costs of receiving & scanning, sorting & packing and scanning & dispatching are rs.1, rs.2 and rs.2.5 respectively per packet. inspection cost per packet is rs.2 and deterioration percentage is constant with 3% deterioration cost. 55 gandhi, jha product type supplier rohu katle pomfret supplier 1 134 90 190 supplier 2 185 85 185 table 1: purchase cost in all periods and at all processing points product type processing point rohu katle pomf pp 1 320 310 300 pp 2 355 275 245 table 2: capacity at all stages in processing point for all periods supplier 1 to pp1 & pp2 product type period 1 period 2 period 3 ac dt ac dt ac dt rohu 0.93 0.98 0.93 0.98 0.93 0.98 katle 0.99 0.98 0.99 0.98 0.99 0.98 pomfret 0.95 0.98 0.95 0.98 0.95 0.98 supplier 2 to pp1 & pp2 product type period 1 period 2 period 3 ac dt ac dt ac dt rohu 0.95 0.99 0.95 0.99 0.95 0.99 katle 0.93 0.97 0.93 0.97 0.93 0.97 pomfret 0.95 0.97 0.95 0.97 0.95 0.97 table 3: de-fuzzified delivery time (dt) and acceptance (ac) probabilities distribution centre processing point dc 1 dc 2 dc 3 pp 1 2000 2500 2500 pp 2 2200 2900 2400 table 4: transportation cost per truck 56 fuzzy bi-objective optimization model retail outlet distribution centre ro 1 ro 2 ro 3 dc 1 2 2.2 1.9 dc 2 2.2 2.5 2.1 dc 3 1.9 1.8 2 table 5: transportation cost per packet from dc to ro product type retail outlet rohu katle pomfret ro 1 100 160 140 ro 2 110 150 135 ro 3 105 170 150 table 6: de-fuzzified demand in all time periods truckload per truck is 250kg. overhead quantity transportation cost is rs.9 per packet. 6.1 results and managerial implications the model helps company to provide minimum total cost incurred coordinating all the entities. rs. 1085767 is the total cost which consists of holding cost at distribution centres as rs.65758, procurement cost of rs.856600, processing cost of rs.33001, cost of transportation from processing point to distribution centres of rs.76588, holding cost at retail outlets of rs.28015.63, cost of transportation from distribution centres to retail outlets of rs.13848.80 and finally inspection cost of rs.11956. it is observed from the results that highest proportion is of the cost of procurement, which clearly validates the requirement of supplier selection. further, keeping a valid track of transportation polices is equally important as the second highest portion in the cost is due to the transportation cost only. next observation is towards the impact of the product’s nature as holding cost at distribution centre contributes towards the third highest portion in the cost. to prevent the over valuation of cost, the aspiration and tolerance level have been considered as rs.950000 and rs.1220000. as validated with the help of cost, the suppliers’ performance is second objective of the model which is a combination of 57 gandhi, jha on-time delivery and acceptance percentage of the suppliers. the higher the performance of the supplier, better the performance of the company. keeping the aspiration level of suppliers’ performance as 39 and tolerance as 30, the performance level of suppliers obtained is 35.04. the model tries to activate the high performers to procure ordered quantity so that uncertainty in the environment can be managed. nearby 78% of the aspiration level of cost and performance has been attained which makes the environment more certain and crisp for future decisions. processing point 1 per. 1 per. 2 per. 3 pr.t. s1 s2 s1 s2 s1 s2 rohu 0 350 0 350 0 350 katle 350 0 350 0 350 0 pomfret 350 0 350 0 350 0 processing point 2 per. 1 per. 2 per. 3 pr.t. s1 s2 s1 s2 s1 s2 rohu 0 350 0 350 0 150 katle 350 0 350 0 350 0 pomfret 350 0 350 0 350 0 table 7: optimum ordered quantity from supplier (s1-s2) in table 7, the positive ordered quantity indicates the active supplier to supply goods as he has the highest performance percentage between the two suppliers on the bases of on-time delivery, acceptance percentage and capacity. it can help in reducing the procurement cost and making the process smooth in further echelon. tables 8 and 9 shows ending inventory at processing points and retail outlets, which ensures no shortages in the case of unexpected demand. it is observed that at second retail outlet, storage capacity and infrastructure is better as well as the cost of holding is also low, hence inventory is higher at this outlet in comparison to others. inventory at distribution is not discussed as no inventory was leftover at any of the distribution centres. while transporting weighted quantity to distribution centres, the policy type, number of trucks and overhead weights are to be checked as each of them incurs cost. in the table 10 it is observed that while transporting 58 fuzzy bi-objective optimization model processing point period 1 period 2 period 3 product type pp1 pp2 pp1 pp2 pp1 pp2 rohu 7 7 14 14 21 21 katle 7 7 14 14 21 21 pomfret 3 7 10 14 17 21 table 8: inventory at processing points (in packets) retail outlet period 1 period 2 period 3 producttype ro1 ro2 ro3 ro1 ro2 ro3 ro1 ro2 ro3 rohu 0 0 0 112 171 78 11 698 1 katle 0 0 0 131 69 0 2 317 75 pomfret 0 0 0 144 58 51 5 487 8 table 9: inventory at retail outlets (in packets) from processing point 1 to distribution centre 1 in period 2, only truckload (t*) policy is used as 250kg can be transported by 1 truck. in this case, ltl policy will become expensive. on the other side, transporting from processing point 1 to distribution centre 1 in period 1, tl & ltl? policy is used as 49kg should be transported as per unit weight. in the case of tl&ltl policy, if overhead weighted quantity is transported through full truckload, the cost of transportation will become much higher than using ltl policy. where tl & ltl is indicated as tlt and only tl is indicated as t. some more operational variables who helped in smooth process of goods from one level to other are as follows: 7 conclusion in the emerging business scenario, the concepts of time, volume and capacity become even more essential to the managerial decision-making. customers are more sensitive to delivery times and service quality. the coordination among the members of the chain helps them to make a cost-effective procurement and distribution network as well as better response to the cus59 gandhi, jha distribution centre 1 period 1 period 2 period 3 pp1 pp2 pp1 pp2 pp1 pp2 tpt quantity 49 7 250 0 329 250 no. of trucks 0 0 1 0 1 0 tpt mode tlt? tlt t* t tlt t qty overhead 49 7 0 0 79 0 distribution centre 2 period 1 period 2 period 3 pp1 pp2 pp1 pp2 pp1 pp2 tpt quantity 749 761 752 1000 686 500 no. of trucks 2 3 3 4 2 2 tpt mode tlt tlt tlt t t t qty overhead 249 11 2 0 186 0 distribution centre 3 period 1 period 2 period 3 pp1 pp2 pp1 pp2 pp1 pp2 tpt quantity 35 261 27 29 14 279 no. of trucks 0 1 0 0 0 1 tpt mode tlt tlt tlt tlt tlt tlt qty overhead 35 11 27 29 14 29 table 10: transported quantity, no. of trucks, transportation mode, overhead quantity e imt period 1 period 2 period 3 dis.c. rohu katle pomf rohu katle pomf rohu katle pomf dc 1 18 16 22 94 144 12 1 0 578 dc 2 451 634 425 544 534 674 657 421 108 dc 3 217 36 43 48 8 0 28 265 0 table 11: 60 fuzzy bi-objective optimization model giot period 1 period 2 period 3 r.o. rohu katle pomf rohu katle pomf rohu katle pomf ro 1 0 35 0 215 295 288 0 31 1 ro 2 641 647 426 286 221 195 658 408 578 ro 3 45 4 64 185 170 203 28 247 107 table 12: tomers’ demand. the authors explain the coordination among many entities of supply chain. as mentioned in the objectives of this study, the main plan of this research is to find optimum quantity from the best suppliers under fuzzy environment to develop an optimum coordination among multi supplier, multi processing points, multi distribution centres and multiple number of retail outlets. to attain the objective, a fuzzy bi-objective mathematical model is formulated with objective functions of cost and combination of timely delivery & acceptance of lot, keeping the constraints as supplier capacity, processing capacity, deteriorating nature of the product and truck capacity. the parameters in study as holding cost, consumption, delivery time and acceptance percentage are fuzzy in nature. to handle the issues of uncertainty and fuzziness, the model is converted into crisp form with the help of membership functions of fuzzy modeling. the parameters are also converted into crisp form by using triangular fuzzy numbers. to obtain the solutions, a fuzzy goal programming is employed. hence, the current study is able to find a balance between minimum cost and best performed supplier. the proposed model was validated by applying to the real case study data. references [1] ho, w., xu, x., and dey, p.k. (2010). multi-criteria decision making approaches for supplier evaluation and selection: a literature review. european journal of operational research 202 (1), 16-24. [2] hassini, e. (2008). order lot sizing with multiple capacitated suppliers offering lead time-dependent capacity reservation and unit price discounts. production planning & control, 19(2), 142–149. [3] ravindran, a. r., bilsel, r. u., wadhwa, v., and yang, t. (2010). risk adjusted multi criteria supplier selection models with applications. international journal of production research, 48(2), 405–424. 61 gandhi, jha [4] liao, z., and rittscher, j. (2007). integration of supplier selection, procurement lot sizing and carrier selection under dynamic demand conditions. international journal of production economics, 107, 502–510. [5] razmi, j., and maghool, e. (2010). multi-item supplier selection and lot-sizing planning under multiple price discounts using augmented econstraint and tchebycheff method. international journal of advances in manufacturing technology, 49, 379–392. [6] zhang, j.-l., and zhang, m.-y. (2011). supplier selection and purchase problem with fixed cost and constrained order quantities under stochastic demander. international journal of production economics, 129, 1–7. [7] jolai, f., yazdian, s. a., shahanaghi, k., and khojasteh, m. a. (2011). integrating fuzzy topsis and multi-period goal programming for purchasing multiple products from multiple suppliers. journal of purchasing & supply management, 17, 42–53. [8] pal, b., sana, s. s., and chaudhuri, k. (2012). a multi-echelon supply chain model for reworkable items in multiple-markets with supply disruption. economic modelling, 29(5), 1891–1898. [9] kilic, h. s. (2013). an integrated approach for supplier selection in multi-item/ multi-supplier environment. applied mathematical modelling, 37, 7752–7763. [10] lee, a. h. i., kang, h. y., and chang, c. t. (2009). fuzzy multiple goal programming applied to tft-lcd supplier selection by downstream manufacturers. expert system with applications, 36, 6318–6325. [11] madronero, m. d., peidro, d., and vassant, p. (2010). vendor selection problem by using an interactive fuzzy multi-objective approach with modified s-curve membership functions. computers and mathematics with applications, 60,1038–1048. [12] wu, d. d., zhang, y., wu, d., and olson, d. l. (2010). fuzzy multiobjective programming for supplier selection and risk modeling: a possibility approach. european journal of operations research, 200, 774– 787. [13] arikan, f. (2011). an augmented max–min model for multiple objective supplier selection, 9th international congress on logistics and 62 fuzzy bi-objective optimization model supply chain management, 27–29 october, yas ar university, ces meizmir-turkey. the congress proceeding including full-texts is in the publication process. [14] feng, b., fan, z., and li, y. (2011). a decision method for supplier selection in multi-service outsourcing, international journal of production economics, 132, pp 240-250. [15] seifbarghy, m. and esfandiari, n. (2013). modeling and solving a multi-objective supplier quota allocation problem considering transaction costs, journal of intelligent manufacturing, 24(1), 201-209. [16] aghai, s., mollaverdi, n., and sabbagh, m. s. (2014), a fuzzy multiobjective programming model for supplier selection with volume discount and risk criteria, the international journal of advanced manufacturing technology, 71(5-8), 1483-1492. [17] zimmermann, h. j. 1976. “description and optimization of fuzzy systems.” international journal of general systems 2 (4): 209–215. [18] bellman, r. e., and zadeh, l. a. (1970). “decision-making in a fuzzy environment.” management science 17 (4): 141-164. [19] mohamed, r. h. 1997. “the relationship between goal programming and fuzzy programming.” fuzzy sets and systems 89 (2): 215-222. 63 gandhi, jha 64 microsoft word capitolo intero n 10.doc microsoft word cap3.doc microsoft word cap10.doc approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica issn: 1592-7415 vol. 33, 2017, pp. 5-20 eissn: 2282-8214 5 on some applications of the vougiouklis hyperstructures to probability theory antonio maturo1, fabrizio maturo2 3doi: 10.23755/rm.v33i0.372 abstract some important concepts about algebraic hyperstructures, especially from a geometric point of view, are recalled. many applications of the hv structures, introduced by vougiouklis in 1990, to the de finetti subjective probability theory are considered. we show how the wealth of probabilistic meanings of hv-structures confirms the importance of the theoretical results obtained by vougiouklis. such results can be very meaningful also in many application fields, such as decision theory, highly dependent on subjective probability. keywords: algebraic hyperstructures; subjective probability; hv structures, join spaces. 2010 ams subject classification: 20n20; 60a05; 52a10. 1 department of architecture, university of chieti-pescara, italy; antomato75@gmail.com 2 department of business administration, university of chieti-pescara, italy; f.maturo@unich.it 3 ©antonio maturo and fabrizio maturo. received: 15-10-2017. accepted: 26-12-2017. published: 31-12-2017. antonio maturo and fabrizio maturo 6 1 introduction the theory of the algebraic hyperstructures was born with the paper (marty, 1934) at the viii congress of scandinavian mathematicians and it was developed in the last 40 years. in the book "prolegomena of hypergroup theory" (corsini, 1993) all the fundamental results on the algebraic hyperstructures, until 1992, have been presented. a complete bibliography is given in the appendix. a review of the results until 2003 is in (corsini, leoreanu, 2003). perhaps the most important motivation for the study of algebraic hyperstructures comes from the basic text "join geometries" by prenowitz and jantosciak (1979), which in addition to giving an original and general approach to the study of geometry, introduces an interdisciplinary vision of geometry and algebra, showing how the euclidean spaces can be drawn as join spaces, i.e. commutative hypergroups that satisfy an axiom called "incidence property". moreover, various other geometries, such as the projective geometries (beutelspacher, rosembaum, 1998), are also join spaces. considering, for example, the affine geometries, it is seen that associative property is not satisfied in many important geometric spaces. this and other important geometric and algebraic issues have led to the study of weak associative hyperstructures. the theory of such hyperstructures, called hvstructures, was carried out by thomas vougiouklis, who introduced the concept of hv-structures in the work “the fundamental relation in hyperrings. the general hyperfield” (1991),. presented at the 4th aha conference, xanthi, greece, 1990. subsequently vougiouklis found many fundamental results on the hv-structures in numerous works (e.g. vougiouklis, 1991, 1992, 1994a, 1994b; spartalis, vougiouklis, 1994). a collection of all the results on the subject until 1994 is in the important book “hyperstructures and their representations” (vougiouklis, 1994c). subsequent insights into hv-structures were made by vougiouklis in many subsequent works (1996a, 1996b, 1996c, 1997, 1999a, 1999b, 2003a, 2003b, 2008, 2014), also in collaboration with other authors (dramalidis, vougiouklis, 2009, 2012; vougiouklis et al., 1997; nikolaidou, vougiouklis, 2012). from the hv-structures of vougiouklis, the idea in the chieti-pescara research group was conceived to interpret some important structures of subjective probability as algebraic structures. some paper on this topic are (doria, maturo, 1995, 1996; maturo, 1997a, 1997b, 1997c, 2000a, 2000b, 2001a, 2001b, 2003c, 2008, 2010). the study of applications of hyperstructures to the treatment of uncertainty and decision-making problems in architecture and social sciences begins with on some applications of the vougiouklis hyperstructures to probability theory 7 a series of lectures held at the faculty of architecture in pescara by giuseppe tallini in 1993, on hyperstructures seen from a geometric point of view, and was developed at various aha conferences (algebraic hyperstructures and applications) as well as various seminars and conferences with piergiulio corsini from 1994 to 2014. for example, in december 1994 and october 1995, two conferences on "hyperstructures and their applications in cryptography, geometry and uncertainty treatment" were organized by corsini, eugeni and maturo, respectively in chieti and pescara, with which it was initiated the systematic study of the applications of hyperstructures to the treatment of uncertainty and architecture. in (corsini, 1994), it is proved that the fuzzy sets are particular hypergroups. this fact leads us to examine properties of fuzzy partitions from a point of view of the theory of hypergroups. in particular, crisp and fuzzy partitions given by a clustering could be well represented by hypergroups. some results on this topic and applications in architecture are in the papers of ferri and maturo (1997, 1998, 1999a, 1999b, 2001a, 2001b). applications of hyperstructures in architecture are also in (antampoufis et al., 2011; maturo, tofan, 2001). moreover, the results on fuzzy regression by fabrizio maturo, sarka hoskova-mayerova (2016) can be translate as results on hyperstructures. a new research trend concerns the applications of hypergroupoid to social sciences. vougiouklis, in some of his papers (e.g. 2009, 2011), propose hyperstructures as models in social sciences; hoskova-mayerova and maturo analyze social relations and social group behaviors with fuzzy sets and hvstructures (2013, 2014), and introduce some generalization of the moreno indices. 2 fundamental definitions on hyperstructures let us recall some of the main definitions on the hyperstructures that will be applied in this paper to represent concepts of logic and subjective probability. for further details on hyperstructure theory, see, for example, (corsini, 1993; corsini, leoreanu, 2003; vougiouklis, 1994c). definition 2.1 let h be a non-empty set and let *(h) be the family of non-empty subsets of h. a hyperoperation on h is a function  hh  *(h), such that to every ordered pair (a, b) of elements of h associates a non-empty subset of h, noted ab. the pair (h, ) is called hypergroupoid with support h and hyperoperation . antonio maturo and fabrizio maturo 8 if a and b are non-empty subsets of h, we put ab = {ab: aa, bb}. moreover, a, bh, we put, ab = {a}b and ab = a{b}. definition 2.2 a hypergroupoid (h, ) is said to be: • a semihypergroup, if a, y, ch, a(bc) = (ab)c (associativity); • a quasihypergroup, if ah, ah = h = ha (riproducibility); • a hypergroup if it is both a semihypergroup and a quasihypergroup; • commutative, if a, bh, ab = ba; • idempotent, if ah, aa={a}. • weak associative, if a, b, ch, a(bc)  (ab)c  ; • weak commutative, if a, bh, ab  ba ≠. the weak associative hypergroupoid, called also hv-semigroup by vougiouklis (1991), appear to be particularly significant in the theory of subjective probability, and all results found by vougiouklis in later papers (e.g.1992, 1994a, 1994b), should have important logic and probabilistic meanings. vougiouklis (1991) introduced also the notation “hv-group” for the weak associative quasihypergroups. a hv-semigroup is said to be left directed if a, b, ch, a(bc)  (ab)c, and right directed if a(bc)  (ab)c. let (h, ) a hypergroupoid. using a geometric language, a singleton {a}, ah, is said to be a block of order 1 (briefly 1-block) generated by a. every hyperproduct ab, a, bh, is a block of order 2 (2-block), called block generated by (a, b). for every a1, a2, a3 h, the hyperproducts a1  (a2  a3) and (a1  a2)  a3 are the 3-blocks generated by (a1, a2, a3). for recurrence, for every a1, a2, …, anh, n > 2, the blocks generated by (a1, a2, …, an) are the hyperproducts ab, with a block of order s < n, generated by (a1, …, as), and b block of order n-s generated by (as+1, …, an). in general, for every n > 1, a block of order n (or n-block) is a hyperproduct ab, with a block of order s < n, b block of order n s. for every nn, let n be the set of all the blocks of order n, and let 0 = {n, nn}. then for every nn0, a geometric space (h, n) is associated to the hypergroupoid (h, ). a polygonal with length m of (h, n) is a n-tuple (a1, a2, …, am) of blocks of n such that ai  ai+1 ≠ . let n be the set of all the polygonals of (h, n). the relation n and n* are defined as: a, bh, a n b   an: {a, b} a, a, bh, a n* b   pn: {a, b} p. n is reflexive and symmetric, n* is the transitive closure of n. for n = 0 we have the classical relations  and * considered in many papers, e.g. (freni,1991; corsini, 1993; gutan, 1997; vougiouklis 1999b), on some applications of the vougiouklis hyperstructures to probability theory 9 a more restrictive condition than the weak associativity is the “strong weak associativity”, called also feeble associativity. definition 2.3 a hypergroupoid (h, ) is said to be feeble associative if, for every a1, a2, …, anh, the intersection of all the blocks generated by (a1, a2, …, an) is not empty. if (h, ) is a commutative quasihypergroup, the -division is defined in h, as the hyperoperation /  hh  *(h) that to every pair (a, b) hh associates the nonempty set {xh: abx}. definition 2.4 a commutative hypergroup (h, ) is said to be a join space if the following “incidence property” hold: a, b, c, d h, a / b  c / d ≠  ad  bc ≠ . (2.1) a join space (h, ) is: • open, if, a, b h, a ≠ b, a  b  {a, b} = ; • closed, if, a, b h, {a, b}  a  b; • -idempotent, if, ah, a  a = {a}; • / -idempotent, if, ah, a / a = {a}. a join space (h, ) is said to be a join geometry if it is -idempotent and / -idempotent. we have the following theorem. theorem 2.1 let h = rn and  the hyperoperation that to every (a, b)hh associates the open segment with extremes a and b if a ≠ b, and a  a = {a}. (h, ) is a join geometry, called euclidean join geometry. let (h, ) be a join geometry. we can note that it is open. using a notation like that of euclidean join geometry, in this paper the elements of h are called points and a block ab, with a ≠ b, is called (open) “-segment” with extremes a and b or simply “segment” if only the hyperoperation  is considered in the context. the concept of join space leads to a unified vision of algebra and geometry, that can be very useful from the point of view of advanced didactics (di gennaro, maturo, 2002). also, as some of our papers show, join geometries have important applications in subjective probability. moreover, we can introduce general uncertainty measures in join geometries such that in the euclidean join geometries reduce to the de finetti coherent probability (maturo, 2003a, 2003b, 2006, 2008; maturo et al., 2010). antonio maturo and fabrizio maturo 10 3 subjective probability and hyperstructures let us recall the concept of coherent probabilty and its geometric representation with the notation given in (maturo, 2006). the coherence of an assessment of probabilities p = (p1, p2, …, pn) on a n-tuple e = (e1, e2, …, en) of events is defined by an hypothetical bet with a n-tuple of wins s = (s1, s2, …, sn) (de finetti, 1970; coletti, scozzafava, 2002; maturo 2006). for every i {1, 2, …, n} an individual a, called the better, pays the stake pisi to an individual b, called the bank, and, if the event ei occurs, a receives from b the win si. if si < 0 the verse of the bet on ei is inverted, i. e. b pays the stake and a pays the win. the total random gain ga of a is given by the formula: ga, p, s = (|e1| – p1) s1 + (|e2| – p2) s2 + … + (|en| – pn) sn. (3.1) where |ei| = 1 if the event ei is verified and |ei| = 0 if the event ei is not verified. the atoms associated with the set of events e = {e1, e2, …, en} are the intersections a1a2…an, where ai{ei, -ei}, different from the impossible event . let at(e) be the set of the atoms. then ga(p, s) can be interpret as the function: ga, p, s: a = a1a2…an  at(e)  (|e1| – p1) s1 + (|e2| – p2) s2 + … + (|en| – pn) sn. (3.2) definition 3.1 the probability assessment p = (p1, p2, …, pn) on the ntuple e = (e1, e2, …, en) of events is said to be coherent if, for every s = (s1, s2, …, sn)  r n, there are a, bat(e) such that ga, p, s(a)  0 and ga, p, s(b)  0. we note that the previous definition implies a hyperoperation. let  be an algebra of events containing the set e. then  also contains at(e) and we can define the hyperoperation  on : : (a, b) at(a, b). (3.3) the above considerations show that it may be important, in a probabilistic context, to know the properties of the algebraic hyperstructure (, ), introduced in (doria, maturo, 2006), and called hypergroupoid of atoms. the coatoms associated with e are the nonimpossible complementary events of the atoms. let co(e) be the set of coatoms, and k be the number of atoms. for k = 1, at(e) = {}, where  is the certain event and co(e) is empty. for k = 2, at(e) = co(e) and for k >2 the sets at(e) and co(e) are disjoint and with the some number of elements. for every a, b, c  , we have (we write x y to denote x  y): (ab)c = ({x c, x (-c), xat(a, b)}  {y c, y (-c), yco(a, b)})-{}, on some applications of the vougiouklis hyperstructures to probability theory 11 a(bc) = ({a z, (-a) z, zat(b, c)}  {a t, (-a) t, tco(b, c)}-{}, at{a, b, c} = {x c, x (-c), xat(a, b)}-{} = {a z, (-a) z, zat(b, c)}-{}. then: at{a, b, c}  (ab)c  a(bc). therefore, the following theorem applies: theorem 3.1 let  be an algebra of events, and  the hyperoperation defined by (3.3). then (, ) is a commutative hv-semigroup. the algebra associated with the set of events e = {e1, e2, …, en}, denoted with alg(e) is the set containing the impossible event  and all the unions of the elements of at(e), i.e. xalg(e) iff y(at(e)) such that x is the union of the elements of y. if |at(e)|=s, then |alg(e)| = 2s. let  be an algebra of events. we can introduce the following hyperoperation on : : (a, b) alg(a, b) (3.4) the hyperoperation  is commutative, and, since {a, b}  alg(a, b), (, ) is a quasihypergroup. moreover at(a, b)  alg(a, b) and so  is an extension of the operation  and we have: at{a, b, c}  (a  b)  c  a  (b  c). theorem 3.2 let  be an algebra of events, and  the hyperoperation defined by (3.4). then (, ) is a commutative hv-group. suppose a, b, c are logically independent events, then |at(a, b| = 4, |alg(a, b| = 24 = 16, |at(a, b, c)| = 8, alg(a, b, c)| = 28 = 256. moreover alg(a, b) contains ,  and other 7 elements with their complements. if x is one of these elements, then x  c contains , , c, -c and other 12 elements. then (a  b)  c has 712+4= 88 elements and 168 elements are in alg(a, b, c) but not in (a  b)  c. so, in general, we can write: at{a, b, c}  co{a, b, c}  (a  b)  c, a  (b  c)  alg(a, b, c). let (h, ) be a join geometry. from the associative and commutative properties, for every a1, a2, …, anh there is only a block a1 a2 …an generated by (a1, a2, …, an) and this block depend only by on the set {a1, a2, …, an} and not on the order of the elements. by the idempotence we can reduce to the case in which a1, a2, …, an are distinct. definition 3.2 for every a  h, a ≠ , the convex hull of a, in (h, ), is the set [a] = {xh: nn,  a1, a2, …, ana : x  a1 a2 …an}. if a is finite then [a] is said to be the polytope generated by a. antonio maturo and fabrizio maturo 12 let e = (e1, e2, …, en) be a n-tuple of events set and let at(e) the set of atoms associated to e. for every a = a1a2…an at(e), let xi(a) =1 if ai = ei and xi(a) = 0 if ai = ei. the atom a is identified with the point (x1(a), x2(a), …, xn(a))r n. from definition 3.1, the following theorem applies (maturo, 2006, 2008, 2009). theorem 3.3 let (rn, ) the euclidean join geometry. the probability assessment p = (p1, p2, …, pn) on the n-tuple e = (e1, e2, …, en) of events is coherent iff p[at(e)]. the theorem 3.3 opens the way to introduce measures of uncertainty that are different from the probability and coherent with respect non-euclidean join geometries. we can introduce many possible join geometries. the following is an example. example 3.1 let h = rn and  the hyperoperation that to every (a = (a1, a2, …, an), b = (b1, b2, …, bn))hh associated the cartesian product of the open segments ir with extremes ar and br belonging to (r, ). we can prove that (h, ) is a join geometry, called the cartesian join geometry. some applications of the cartesian join geometry to problems of architecture and town-planning are in (ferri, maturo, 2001a, 2001b). in a general join geometry with support rn we can introduce the following definition: definition 3.3 let (rn, ) be a join geometry. the measure assessment m = (m1, m2, …, mn) on the n-tuple e = (e1, e2, …, en) of events is said to be coherent with respect to (rn, ) iff m[at(e)]. for example,  can be the hyperoperation that to every (a, b)hh associates a particular curve with extremes a and b, and the polytope [at(e)] is a deformation of the euclidean polytope, obtained by replacing the segments with curves. it can have important meanings in appropriate contexts of physics or social sciences. in a generic join geometry (rn, ) can happen that some of the most intuitive properties of the euclidean join geometry fall. to avoid this, you should restrict yourself to join geometries where some additional properties apply. important is the following: ordering condition. if a, b, c, are distinct elements of rn, at most one of the following formulas occurs: abc, bac, cab. a join geometry (rn, ) with the order condition is said to be an ordered join geometry. on some applications of the vougiouklis hyperstructures to probability theory 13 4 conditional events, conditional probability and hyperstructures the “axiomatic probability” by kolmogorov, usually considered as the “true probability” is based on the assessment of a universal set u, whose elements are called the atoms, a -algebra s of subsets of u, whose elements are called the events, and a finite measure p on s, called the probability, such that p(u) = 1. let s* = s-{}. in the kolmogorov approach to probability no consideration is given to the logical concept of conditional event e|h, with es and hs*, but only the conditional probability p(e|h) is defined, only in the case in which p(h) > 0, by the formula: p(e|h) = p(e h)/p(h). (4.1) on the contrary, the “subjective probability” (de finetti,1970; dubins, 1975; coletti, scozzafava 2002; maturo, 2003b, 2006, 2008b), don’t consider the events as subsets of a given universal set u, but they are logical propositions that can assume only one of the truth values: true and false. a sharp separation is given among the concepts concerning the three areas of the logic of the certain, the logic of the uncertain and the measure theory. the conditional event e|h is a concept belonging to the logic of the certain and it is a proposition that can assume three values: true if both e and h are verified, false if h is verified but e is not and empty (or undetermined) if h is not verified. the conditional event e|h reduces to the event e if h is the certain event . in the appendix of his fundamental book (1970) de finetti presents also some different interpretations of the logical concept of three valued proposition. by the point of view of reichenbach (1942) the value “empty” is replaced by the value “undetermined”. in the following we assume the notation of reichenbach and we write t for true, f for false and u for undetermined. the set v = {f, u, t} is also ordered by putting f < u < t. a numerical representation of the ordered set v is given by associating 0 to f, 1 to t and the number 1/2 to u. an alternative, in a fuzzy contest, we can associate to u is the fuzzy number u with support and core the interval [0, 1], then the relation 0 < u <1 is a consequence of the usual order relation among the trapezoidal fuzzy numbers. in the subjective probability, the conditional probability p(e|h) of the conditional event e|h is given by an expert and no condition is given about the belonging of the events e and h to a structured set, e.g. like an algebra. the only condition of h ≠ is required, because if h =  we have the totally undetermined conditional event. antonio maturo and fabrizio maturo 14 if c is a set of conditional events the assessment of a subjective conditional probability to the elements of c must satisfy some coherence conditions. the coherence of an assessment of probabilities p = (p1, p2, …, pn) on a ntuple k = (e1|h1, e2|h2, …, en|hn) of conditional events is defined by an hypothetical bet with a n-tuple of wins s = (s1, s2, …, sn) (de finetti, 1970; coletti, scozzafava, 2002; maturo, 2006). for every i {1, 2, …, n} an individual a, called the better, pays the stake pisi to an individual b, called the bank, and, • if the event eihi occurs, a receives from b the win si; • if the event -hi occurs, the amount paid pisi is refunded to a; • if the event (-ei) hi occurs, no payment is made to a. the total random gain ga of a is given by the formula: ga, p, s = |h1| (|e1| – p1) s1 +…+|hn|(|en| – pn) sn. (4.2) where |ei| = 1 if the event ei is verified and |ei| = 0 if the event ei is not verified, and similarly to h. the atoms associated with the set of conditional events k = {e1|h1, e2|h2, …, en|hn} are the intersections a1a2…an, where ai{ei hi, -ei hi, -hi}, different from the impossible event . the complement of h = {hi, i {1, 2, …, n}} is said to be the inactive atom. let atc(e) be the set of the atoms associated to k. then ga, p, s can be interpret as the function: ga, p, s: a = a1a2…an  atc(e)  (|a1| – p1) s1 + (|a2| – p2) s2 + … + (|an| – pn) sn (4.3) where |ai| = 1, 0, pi, if ai = ei hi, -ei hi, -hi, respectively. definition 4.1 the conditional probability assessment p = (p1, p2, …, pn) on the n-tuple k = (e1|h1, e2|h2, …, en|hn) of conditional events is said to be quasi-coherent if, for every s = (s1, s2, …, sn)  r n, there are a, batc(e) such that ga, p, s(a)  0 and ga, p, s(b)  0. moreover, p = (p1, p2, …, pn) is said to be coherent if, for any s  n and for any {i1, i2, …, is}  {1, 2, …, n}, the conditional probability assessment pi1, i2, …, is = (pi1, pi2, …, pis) on (ei1|hi1, ei2|hi2, …, eis|his) is quasi-coherent. let k = (e1|h1, e2|h2, …, en|hn) be a n-tuple of conditional events and let atc(k) the set of atoms associated to k = {e1|h1, e2|h2, …, en|hn}. for every a = a1a2…anatc(e), let xi(a) = |ai|. the atom a is identified with the point (x1(a), x2(a), …, xn(a))r n. from definition 4.1, the following theorems applies: theorem 4.1 let (rn, ) the euclidean join geometry. the probability assessment p = (p1, p2, …, pn) on the n-tuple k = (e1|h1, e2|h2, …, en|hn) of conditional events is quasi-coherent iff p[atc(k)]. on some applications of the vougiouklis hyperstructures to probability theory 15 theorem 4.2 the probability assessment p = (p1, p2, …, pn) on k = (e1|h1, e2|h2, …, en|hn) is coherent iff for any s  n and for any {i1, i2, …, is}  {1, 2, …, n}, the conditional probability assessment pi1, i2, …, is = (pi1, pi2, …, pis) on (ei1|hi1, ei2|hi2, …, eis|his) belongs to [atc(ei1|hi1, ei2|hi2, …, eis|his)]  rs. let  be an algebra of events. an axiomatic formalization of the coherence conditions in the case in which k = {e|h, e, h {}} is in dubins (1975). in terms of hyperstructures, conditional events can be defined by the following hyperstructure, introduced in (doria, maturo, 1996) and studied in (maturo, 1997c). definition 4.2 let  be an algebra of events. we define on  the hyperoperation: : (e, h)   {e h, h}. we have: e  h  h  e = {e h}; (e  h)  k = {e h k, h k, k}, e  (h  k) = {e h k, h k, e k, k}; e  e = {e}. then we have the following theorem. theorem 4.3 the hyperstructure (, ), let us call the hyperstructure of conditional events, is a weak commutative and idempotent hv-semigroup. moreover (, ) is right directed, i.e. (e  h)  k  e  (h  k). any singleton {h} is the conditional event h|h and any set {e, h} with e  h is the conditional event e|h, true if e is verified, false if h is verified but not e, and it is not undetermined if h is not verified. many other meanings, of the finite subsets of , are considered in (maturo, 1997c). the coherence conditions of definition 4.1 and theorems 4.1 and 4.2 lead us to associate the n-tuple k = (e1|h1, e2|h2, …, en|hn) of conditional events with the set all the conditional events a|b with aat{e1, e2, …, en} and b an union of elements of {h1, h2, …, hn}. then, if  is an algebra of events, and    is a set of nonempty events, closed with respect to the union, the following hyperoperation can be introduced: : (e|h, f|k)  ()()  {a|b: aat{e, f}, b{h, k, hk}}. we can prove the following thorem theorem 4.4 the hyperstructure (, ) is a commutative hvsemigroup, called hypergroupoid of conditional atoms and, for  = {}, is isomorphic to (, ). antonio maturo and fabrizio maturo 16 5 conclusions and perspectives of research we have shown that all logical operations related to subjective probability can reduce to vougiouklis hyperstructures. (, ) and (, ) are commutative hv-semigroups, and (, ) is a commutative hv-group. the hyperoperation  isweak commutative and idempotent and (, ) is a right directed hv-semigroup. to verify the coherence of a subjective probability assignment p = (p1, p2, …, pn) on the n-tuple e = (e1, e2, …, en) of events, we represent the atoms as points of the space rn, in which the i-th axis is associated with the event ei. the assessment p is coherent iff p belongs to the polytope of the join geometry (rn, ) generate from the atoms. more complex is the coherence check of a conditional probability assessment p = (p1, p2, …, pn) on the n-tuple k = (e1|h1, e2|h2, …, en|hn) of conditional events, as in this case we must consider polytopes in all the join geometries (rs, ), s  n associated to subsets of k = {e1|h1, e2|h2, …, en|hn}. a research perspective is to investigate the properties of the considered vougiouklis structures, highlighting their meanings from the point of view of logic and subjective probability. a further research perspective is studying the measures that can be obtained by applying the geometric coherence conditions in ordered join geometries other than the euclidean join geometry. on some applications of the vougiouklis hyperstructures to probability theory 17 references antampoufis n., vougiouklis t., dramalidis a., (2011), geometrical and circle hyperoperations in urban applications, ratio sociologica, vol 4, n.2, 2011,53-66. beutelspacher a., rosembaum u., (1998), projective geometry, cambridge university press coletti g., scozzafava r., (2002), probabilistic logic in a coherent setting, kluwer academic publishers, london corsini p., (1993), prolegomena of hypergroup theory, aviani ed. udine, (1993). corsini p., (1994) join spaces, power sets, fuzzy sets, proc. of the fifth international congress on algebraic hyperstructures and applications, jasi, romania, corsini p., leoreanu l., (2003), applications of the hyperstructure theory, kluver academic publishers london de finetti b. 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(2014), from hv-rings to hv-fields, int. j. algebraic hyperstructures appl. vol.1, no.1, 2014, 1-13. approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 37, 2019, pp. 69-84 69 a conceptual proposal on the undecidability of the distribution law of prime numbers and theoretical consequences gianfranco minati* in music and in mathematics there is the same perfume of eternity abstract† within the conceptual framework of number theory, we consider prime numbers and the classic still unsolved problem to find a complete law of their distribution. we ask ourselves if such persisting difficulties could be understood as due to theoretical incompatibilities. we consider the problem in the conceptual framework of computational theory. this article is a contribution to the philosophy of mathematics proposing different possible understandings of the supposed theoretical unavailability and indemonstrability of the existence of a law of distribution of prime numbers. tentatively, we conceptually consider demonstrability as computability, in our case the conceptual availability of an algorithm able to compute the general properties of the presumed primes’ distribution law without computing such distribution. the link between the conceptual availability of a distribution law of primes and decidability is given by considering how to decide if a number is prime without computing. the supposed distribution law should allow for any given prime knowing the next prime without factorial computing. factorial properties of numbers, such as their property of primality, require their factorisation (or equivalent, e.g., the sieves), i.e., effective computing. however, we have factorisation techniques available, but there are no (non-quantum) known algorithms which can effectively factor arbitrary large integers. then factorisation is undecidable. we consider the theoretical unavailability of a distribution law for factorial properties, as being prime, equivalent to its non-computability, undecidability. * italian systems society, milan, italy; gianfranco.minati.it. † received on november 15th, 2019. accepted on december 17th, 2019. published on december 30th, 2019. doi: 10.23755/rm.v37i0.480. issn: 1592-7415. eissn: 2282-8214. ©gianfranco minati. this paper is published under the cc-by licence agreement. g. minati 70 the availability and demonstrability of a hypothetical law of distribution of primes are inconsistent with its undecidability. the perspective is to transform this conjecture into a theorem. keywords: algorithm; computation; decidability; incompleteness; indemonstrability; law of distribution; prime numbers; symbolic; undecidability. 2010 ams subject classification: 11n05; 00a30; 11a51. 1 introduction number theory is an antique and fascinating discipline. number theory considers endless properties of numbers such as perfect numbers, golden ratios, and fibonacci numbers. an endless list of approaches, problems, properties, and results added one to the other over time deal with prime numbers and the possibility to find a suitable law of their distribution. with regards to prime numbers, mathematicians introduced several conjectures, and not definitive, proven partial results. to name a few, we recall properties and results relating to prime number generation such as the fundamental theorem of arithmetic (by gauss in the 1801), the goldbach's conjecture (approximately 1742), the classic sieve of eratosthenes (275–194 b.c.), the sieve of sundaram (approximately 1934), the sieve of atkin (approximately 2003), and the mersenne prime (1536) of the form mn = 2 n – 1 for pseudorandom number generators, all used for applications such as cryptography. throughout history, several important mathematicians have tentatively contributed to the identification of the asymptotic law of distribution of prime numbers and its proof. we just mention legendre (approximately 1808), dirichlet (approximately 1837), gauss (approximately in 1849 reported the connection between prime numbers and logarithms), riemann (in 1859) wrote his very famous article (riemann, 1859), euler's theorem (approximately 1763) as a generalisation of fermat's little theorem, chebyshev (approximately 1850), and yitang zhang’s contributions to the twin-prime conjecture (approximately 2013). however, since providing a complete review of the literature is beyond the scope of this article, we leave it to the reader to familiarize themselves with the literature on this subject. the contribution to the philosophy of mathematics of the present article is to propose different possible understandings of the unavailability and a conceptual proposal on the undecidability of the distribution law of prime numbers and theoretical consequences 71 indemonstrability of the existence of the law of distribution of prime numbers. further research is expected to allow suitable formalisations. in section 2 we consider generic indemonstrability as a fact of incompleteness and platonic consistency of knowledge. this is further explored in section 4, in a constructivist understanding, where we propose indemonstrability to prevent inevitably implicit inconsistencies because a paradigm shift is required instead. in section 3 we propose to consider demonstrability as having symbolic nature and as decidability. indemonstrability cannot be demonstrated and it can be intended as a fact of incompleteness, case of undecidability. the link between the conceptual availability of a distribution law of primes and decidability is given by considering how to decide if a number is prime without computing. the supposed distribution law should allow for any given prime knowing what the next prime with without computing such sequences. however, factorial properties of numbers, such as their property of being prime, require their factorisation (or equivalent, e.g., the sieves), i.e., effective computing. because of that it is not possible to know in advance the properties of the factorisation, in the same way as it is not possible to solve the alt of a turing machine (tm) -the halting problem consists on determining if an arbitrary computer program and its input will finish running or continue to run forever (such as being in loop). a general algorithm to solve the halting problem for all possible program-input couples cannot exist-, it is not possible to know the result of the processing of a neural network without performing the entire processing, and to know the patterns generated by a cellular automata without performing the entire processing. in section 4, regarding the research relating to a prime’s distribution law (pdl), we present, for the general reader, a short, partial overview of the situation as it currently consists mainly of a list of conjectures. such conjectures have been not falsified but, rather, computationally confirmed by considering numerically large cases. in section 5 we tentatively conceptually consider demonstrability as computability, i.e., in our case the conceptual existence of an algorithm able to compute the general properties of the presumed primes’ distribution law without computing such distribution. we tentatively consider generic indemonstrability, unavailability as undecidability of the law of distribution and the probabilistic nature of the prime number theorem (pnt) as an aspect of its undecidability. we consider then the usability of such undecidability, in the historical conceptual framework of the very effective usability of imaginary numbers. we ask ourselves if the non-demonstrability of existence of the pdl and its nondiscovery can be intended as a prototype of the non-distribution and of possible different non-equivalent non-distributions. besides, such non-demonstrability g. minati 72 of existence and the persisting non-availability of the pdl may be considered as a prototype of the generic non-demonstrability, of theoretical incompleteness, and theoretical incomprehensibility. we conclude by mentioning how from the issues considered above it is possible to use such incompleteness in order to introduce paradigm shifts and non-equivalent, mutually irreducible, incommensurable approaches. 2 indemonstrability as a fact of consistency we consider here a kind of platonic consistency of the knowledge, as theoretical incompleteness [1, 2] which manifests when dealing with incomplete problems or indemonstrability of incomplete or wrong theses. in a constructivist understanding it is a kind of experiment having no reaction as a result, stating that the experiment is inadmissible, inconsistent, wrong. as a classic example, consider the unsuccessful attempts to demonstrate the fifth postulate in euclidian geometry. the history of the attempts to demonstrate the fifth postulate reveals how the conclusion was obtained by appealing to a new proposition that was equivalent to the fifth postulate itself. the italian mathematician eugenio beltrami discovered the giovanni girolamo saccheri’s article euclides ab omni naevo vindicatus (euclid freed of every flaw), published in 1733 in which he tried to prove the euclid's postulate of parallel lines. by using a similar approach, beltrami, among others, inadvertently introduced a paradigm shift towards the non-euclidean geometries by reasoning per reductio ad absurdum, i.e., as a result of the impossibility of proving the absurdity of the negation of the fifth postulate [3, 4]. an example of a relationship between theoretical incompleteness and indemonstrability is given by the two celebrated gӧdel’s syntactic incompleteness theorems [5]. the meaning of the first theorem states that within any mathematical theory, having at least the power of arithmetic, there exists a formula that, neither the formula nor its negation is syntactically provable. in other words, it is possible to construct a formally correct proposition that, however, cannot be proven or disproved. this is logically equivalent to the construction of a logical formula that denies its provability. the meaning of the second theorem is that no coherent system is able to demonstrate its own syntactic coherence. the two theorems can be intended to prove the inexhaustibility in principle of pure mathematics [6-8]. “in other words, infinite-state logical theories when sufficiently complex are necessarily incomplete. whether this result implies a sort of incompleteness of other kinds of theories (for instance, those of physics) is still an open question [9, p. 7]. a conceptual proposal on the undecidability of the distribution law of prime numbers and theoretical consequences 73 as for incompleteness in physics, it is closely related to the uncertainty principles. it relates to the well-known uncertainty principle, first introduced by werner heisenberg [10]. furthermore, there is the principle of complementarity introduced by neils bohr [11] stating that the corpuscular and undulatory aspects of a physical phenomenon cannot be observed simultaneously. this is the case of the measurement of homologous components such as position and momentum. from now on we consider a tentative relationship among some generic concepts such as indemonstrability, incompleteness and undecidability: theoretical incompleteness and indemonstrability; indemonstrability as a fact of incompleteness; demonstrability of incompleteness; the other issue is that of indemonstrability and (as?) undecidability. 3 indemonstrability and undecidability a problem is considered as “undecidable” when there is no algorithm that produces the corresponding solution in a finite time for each instance of the input data. a typical example is the classic halting problem for the turing machine [12]. the set of decidable problems is incomplete. in this regard, turing himself introduced an issue of ‘completion’ by inserting the concept of oracle [13], representing another logic, possibly incommensurable, that, however, combines, interferes, superimposes, and acts on that in use. all this in the framework of a general theory of truth, e.g., tarskian semantics, see, for instance [1]. however, even in case of availability of effective computational algorithms, the finite precision or finite memory (in case for symbolic manipulation) implies theoretical incompleteness [14-16]. moreover, another example is given by the non-explicit, non-symbolic computation, for instance, of artificial neural networks (anns), see, for example [17, 18]. the computational processing is represented and performed in a nonanalytical, non-symbolic way through weighted connections and levels. if we look instant per instant at the calculation performed, it is incomprehensible and we have to wait for the final result. this also applies to other computational processes such as cellular automata. the computation acquires properties not formally prescribed like learning [19, 20]. particular classes of anns, such as those with non-turing computable weights, and recurrent-anns [21, 22] show a non-turing behaviour for which the principles of hypercomputation [23-25] and naturally-inspired computation [26] apply. g. minati 74 indemonstrability cannot be symbolically demonstrated and is intended to be a fact of incompleteness, case of undecidability. furthermore, it is possible to conceptually consider symbolic demonstrability as having logical equivalences with decidability. 4 prime numbers please download the latex template and see the .pdf file to see how to format editing definitions, theorems, corollaries. at this point we may ask ourselves how to interpret the non-comprehension, the non-availability of the pdl, which is used in areas such as cryptography [27]? as incompleteness of the theory of numbers, undecidability, and indemonstrability [28]? the problem has been frequented by mathematicians for centuries, with important, but not definitive results. at this point we may consider two questions: in a constructivist understanding, can we intend such barrier to prevent an inevitably, implicitly inconsistent demonstration because a paradigm shift is required instead? in a platonic understanding, can we intend such a barrier to protect from an inevitably wrong demonstration contrasting with the general consistency and requiring different entry points? 4.1 a brief summary of the current situation attention to prime numbers first focused on the question whether they were infinite or not, and then turned to the understanding how they are distributed within natural numbers. it dates back to the 3rd century bc and to the euclid’s first proof that infinitely many primes exist (see the elements, book ix, proposition 20), see the polignac’s conjecture below. in modern times euler gave an alternative proof of this result by using, for the first time, concepts coming from infinitesimal mathematical analysis. gauss understood the still fundamental key to the understanding of a crucial characteristic of the prime numbers: their density. riemann introduced his conjecture, listed below, which concerns the distribution of the zeros of a particular complex function, known as the zeta function, which has a very close connection with the distribution of primes. in particular, the distribution of the zeros of the zeta function is linked to the possibility of accurately counting the prime numbers. in what follows, we propose a very short overview on the very large world of attempts to deal with the still unsolved problem of finding a pdl. this world a conceptual proposal on the undecidability of the distribution law of prime numbers and theoretical consequences 75 includes mainly conjectures and few theorems. we give approximate reference dates for the convenience of the general reader. 4.2 an overview the overview [29-31] includes the following subjects. 1) goldbach’s conjecture 1742: every even integer greater than 2 can be expressed as the sum of two primes. 2) cramér’s conjecture, 1936: it gives an asymptotic estimate for the size of gaps between consecutive prime numbers where: pn denotes the nth prime; ln is the natural logarithm. this is based on a probabilistic model assuming that the probability that a natural number x is prime is 1/ ln x, from which it can be shown that the conjecture is true with probability 1. in other words, if the prime numbers follow a "random" distribution, it is very likely that the conjecture is true. in short, the cramér's conjecture states that the difference between two consecutive prime numbers always remains less than the square of the natural logarithm of the smaller of the first two. this conjecture implies the following: 3) opperman's conjecture, approximately 1882: the conjecture states that, for every integer x > 1, there is at least one prime number between x(x − 1) and x2. this conjecture in turn implies the next conjecture: 4) legendre’s (1752 – 1833) conjecture: it states that there exists at least one prime number between n2 and (n + 1)2 for all natural numbers. the previous conjectures are all more restrictive than the bertrand postulate (which has been proven and is now a theorem): 5) bertrand postulate, approximately 1845: in its less restrictive formulation it states that for every n>1 there is always at least one prime p such that n<p<2n. 6) polignac’s twin prime conjecture (approximately 1846 and previously considered by euclid): it states that there are infinitely many twin primes, or pairs of primes that differ by 2. as numbers get larger, primes become less frequent and twin primes become rarer as well. in this regard in 1919 brun’s theorem showed that the sum of the reciprocals of the twin primes converges to a sum, now known as brun’s constant. in 2010, the value of brun’s constant was approximately 1.902160583209 ± g. minati 76 0.000000000781 based on all twin primes less than 2 × 1016. conversely, the sum of the reciprocals of all primes diverges to infinity [32]. 7) riemann hypothesis, 1859: it deals with the distribution of the zeros of a particular complex function, now called "riemann zeta function", which has a very close connection with the distribution of primes. in particular, the distribution of its zeros is linked to the possibility of accurately counting the prime numbers. the riemann hypothesis can be described geometrically by saying that the zeros of the riemann zeta function are confined to two lines in the complex plane [33, 34]. 8) the prime number theorem (pnt) describes the asymptotic distribution of prime numbers: it states a general view of how primes are distributed among positive integers and also states that the primes become less common as they become larger. let π(x) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x. the pnt then states that x / log x is a good approximation to π(x), that is π(x) ∼ x log x. this notation means only that the quotient limit of the two functions π (x) and x / ln (x) for x which tends to infinity is 1, but not that the limit of the difference of the two functions, as x tends to infinity, is 0. this means that for large enough n, the probability that a random integer not greater than n is prime is very close to 1 / log(n). the pnt is based on several previous and subsequent, increasingly specifying contributions, such as legendre’s conjecture stating that π(a) is approximated by the function a / (a log a + b), where a and b are unspecified constants; gauss studied the problem; dirichlet introduced a logarithmic integral li(x) as approximating function; the connection between the prime number theorem and the riemann zeta function is very deep and allowed by the euler product. the plausibility of such conjectures and approaches is supported by a large number of computational simulations which did not lead to falsifying cases. 5 indemonstrability as undecidability of the distribution? the main conclusions of the study may be presented in a short conclusions section, which may stand alone or form a subsection of a discussion or results. we are tentatively proposing to consider here the incomplete, probabilistic or approximate nature of pnt not as much as a limit to be solved by more appropriated approaches, but as an unavoidable theoretical aspect, price to be paid for consistency within the theory of computation rather than within number a conceptual proposal on the undecidability of the distribution law of prime numbers and theoretical consequences 77 theory itself. we consider here that number theory and its properties and theorems may be not incompatible with the availability of regularities in the distribution of prime numbers, while such availability can be considered incompatible with other properties and theories, such as general forms of theoretical, structural incompleteness, such as the halting problem for the turing machine in theory of computation. we consider here, reasoning in proof by contradiction, that the computability of such distribution is possible. 5.1 demonstrability as computability the question relates to the conceptual availability of an algorithm able to compute general properties of the primes’ distribution. such properties are supposed to allow to know for any number the properties of the following sequence of prime numbers without computing each item of such sequences. we just mention that the case of the knowledge of properties of a function, e.g., continuity, differentiability, minimum and maximum points, asymptotes, is different. properties of a function are known from its formal definition and not from the knowledge of the properties of the distribution of all the values assumed in its domain of validity, i.e., law of distribution. we may know the analytical properties of an exponential function without computing its values in any points on the abscissa axis. the same holds for sequences of numbers such as the fibonacci sequence defined as fn=fn-1+fn-2, with f1=f2=1 (two successive fibonacci numbers are relatively prime). how do we decide if a number is prime without computing? when considering a number, we may take into account, for instance, some of its properties 1) will not require its factorisation –we consider here the case of factorization of an integer. we do not consider here the cases related to polynomial factorization and ringsor 2) will consider its factorisation. as stated by the fundamental theorem of arithmetic every integer > 1 either is prime itself or is the product of prime numbers. this product is unique regardless of the order of the factors. the first explicit proof of the theorem of arithmetic, namely that the set of integer numbers has a unique factorization, is due to carl friederich gauss, who inserted it in the disquisitiones arithmeticae, published in 1798, but already introduced by euclid. examples of properties of the first kind (not requiring factorisation) are generic properties such as considering if a number is greater or less than another, the number of its digits, and if it is even or odd. similarly, properties of values of a function are known from its formal definition and do not require the g. minati 78 effective computation of values. positions within the sequence of natural numbers correspond to properties. examples of properties of the second kind (requiring the factorisation of the number) relate the identification of the number as given by the exponentiation of a base and prime numbers. in the first case, it is possible to detect a property without computing and factorise. however, factorial properties of numbers, as their factorial breakdown, exponential factor values and their being prime, i.e., non-decomposability, require their factorisation (or equivalent, e.g., the sieves), i.e., effective computing. properties of a distribution law, e.g., the graph of a function, its continuity, regularity, domains, and values of its derivatives, allow to know subsequent values moving along the graph without computing each value corresponding to the punctual abscissas. in the case of factorisations, each of them must be computed since not made available by any property of a distribution law. in the second case, factorisation is then necessary. for instance, each value of the function f=xn is available on its graph. rather, each factorisation of an integer (factorisation is different from "combinatorial calculus" when factors are known) is in principle unknown and must be computed case by case, being not available from sequences or any graph. in the first case, we have available the complete computational procedure, i.e., an algorithm. in the second case, we have factorisation techniques available, but there are no known algorithms (can integer factorization be solved in polynomial time on a non-quantum computer [35]?) which can effectively factor arbitrary large integers, see, for instance, [36] and [37]. the adjective effectively refers to the definition of tm for which the algorithm should produce the solution in a finite time for each instance of the input data [12]. this also refers to tractable problems that can be solved by algorithms in polynomial time, i.e., for a problem of size n, the time or number of steps needed to find the solution is a polynomial function of n. conversely, algorithms for solving intractable problems require times that are exponential functions of the problem size n. then factorisation is undecidable. furthermore, we mention that sieves, such as the eratosthenes, legendre (it is an extension of eratosthenes' idea), brun, selberg, and turán sieves [38], have an exponential time complexity with regard to input size, making them pseudopolynomial algorithms. we consider the theoretical unavailability of a distribution law for factorial properties, as being prime, equivalent to its non-computability, undecidability. a conceptual proposal on the undecidability of the distribution law of prime numbers and theoretical consequences 79 the availability and demonstrability of a hypothetical pdl are inconsistent with its undecidability. in the second case, factorisation is then necessary. because of that it is not possible to know in advance the properties of the factorisations, in the same way as it is not possible to solve the alt of a tm (see the introduction), it is not possible to know the result of the processing of a neural network without performing the entire processing, and to know the patterns generated by a cellular automata without performing the entire processing. positions within the sequence of natural numbers do not correspond to the distributed property of being prime number. in light of that, we tentatively propose the speculative conjecture that the complete knowledge of the pdl, that allows the availability of a rule, is not possible since it would disprove the alt problem for a tm. we conclude that the pdl is undecidable. we may conclude the indemonstrability of the riemann hypothesis (millennium problem), the riemann hypothesis is undecidable in arithmetic. conceptual non-availability of an algorithm defines all undecidable problems as correspondent to the alt problem for a turing machine. the probabilistic nature of pnt should be considered an aspect of its undecidability. this will theoretically provide reassurance about the usage of prime numbers for a large variety of applications such as cryptography and pseudorandom number generation. 5.2 using the indemonstrability a theoretical incompletable list of non-equivalent models and approaches are necessary to deal with the endless acquisitions and modality of acquisition of properties in complexity and emergent phenomena. this is the case for uncertainty principles and theoretical incompleteness such as that of mathematics, of the turing machines, and of the so-called logical openness in the dynamic usage of models -dysam [39, pp. 64-88], based on established approaches in the literature, such as ensemble learning [40, 41] and evolutionary game theory [42, 43]. other cases relate to the undecidability and irreducibility of emergence [17, 44], the usage of the non-computable and unknowable imaginary numbers, however very effective and used, and the nonsymbolic computation of ann and ca. the non-demonstrability of the pdl primes’ distribution law is well used in cryptography in the same way as some pharmaceutical products are used for their side-effects. g. minati 80 this relates to the usage of the theoretically incomprehensible [45] which is suitable for introducing paradigm-shifts and non-equivalent, incommensurable, mutually irreducible approaches. can the non-demonstrability of the primes’ distribution law become the prototype of the non-distribution(s) having some possible different levels of equivalence; the prototype of the non-demonstrability, of theoretical incompleteness, and of theoretical incomprehensibility? conclusions we shortly considered the research about primes in mathematics and the theoretical, still elusive, results looking for a pdl. we considered as these endless difficulties may be interpreted as logical consistency, since the availability of such distribution law could be theoretically incompatible with other consolidated theories and properties. this is the case for the theoretical incompleteness of mathematics, the turing machines, and of the so-called logical openness in the use of dynamic usage of models (dysam). we considered the conceptual incompatibility of the availability of a pdl and the alt problem for a tm, i.e., implying that the pdl is undecidable. the link between the conceptual availability of a pdl and decidability is given by considering how to decide if a number is prime without its computation. the supposed pdl should allow to know the sequence of primes without their computation, but considering only their sequential positions which coincide, however, with the numbers in question. however, factorial properties of numbers, such as their primality, require their factorisation (or equivalent, e.g., the sieves), i.e., effective computing. because of that it is not possible to know in advance the properties of the factorisation, in the same way as it is not possible to solve the alt of a tm, it is not possible to know the result of the processing of a neural network without performing the entire processing, and to know the patterns generated by a cellular automata without performing the entire processing. positions within the sequence of natural numbers do not correspond to the distributed property of a prime number. we may conclude that the availability and demonstrability of a hypothetical pdl are inconsistent with its undecidability. the perspective is to transform this conjecture into a theorem. furthermore, we considered the unavailability of a pdl as corresponding, representing 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[45] minati, on theoretical incomprehensibility. philosophies, 4(3), https://www.mdpi.com/2409-9287/4/3/49 (accessed on 11 october 2019). https://www.mdpi.com/2409-9287/4/3/49 https://www.mdpi.com/2409-9287/4/3/49 https://www.mdpi.com/2409-9287/4/3/49 microsoft word r.m.5 cap.10.doc microsoft word cap7.doc ratio mathematica vol. 35, 2018, pp. 75-85 issn: 1592-7415 eissn: 2282-8214 the sum of the series of reciprocals of the quadratic polynomials with complex conjugate roots radovan potůček ∗ received: 07-06-2018. accepted: 10-10-2018. published: 31-12-2018 doi:10.23755/rm.v35i0.427 c©radovan potůček abstract this contribution is a follow-up to author’s papers [1], [2], [3], [4], [5], [6], [7], and in particular [8] dealing with the sums of the series of reciprocals of quadratic polynomials with different positive integer roots, with double non-positive integer root, with different negative integer roots, with double positive integer root, with one negative and one positive integer root, with purely imaginary conjugate roots, with integer roots, and with the sum of the finite series of reciprocals of the quadratic polynomials with integer purely imaginary conjugate roots respectively. we deal with the sum of the series of reciprocals of the quadratic polynomials with complex conjugate roots, derive the formula for the sum of these series and verify it by some examples evaluated using the basic programming language of the computer algebra system maple 16. this contribution can be an inspiration for teachers of mathematics who are teaching the topic infinite series or as a subject matter for work with talented students. keywords: sum of the series, harmonic number, imaginary conjugate roots, hyperbolic cotangent, computer algebra system maple. 2010 ams subject classifications: 40a05, 65b10. ∗department of mathematics and physics, faculty of military technology, university of defence in brno, brno, czech republic; radovan.potucek@unob.cz 75 radovan potůček 1 introduction let us recall the basic terms. for any sequence {ak} of numbers the associated series is defined as the sum ∞∑ k=1 ak = a1 + a2 + a3 + · · · . the sequence of partial sums {sn} associated to a series ∞∑ k=1 ak is defined for each n as the sum sn = n∑ k=1 ak = a1 + a2 + · · ·+ an. the series ∞∑ k=1 ak converges to a limit s if and only if the sequence of partial sums {sn} converges to s, i.e. lim n→∞ sn = s. we say that the series ∞∑ k=1 ak has a sum s and write ∞∑ k=1 ak = s. the sum of the reciprocals of some positive integers is generally the sum of unit fractions. for example the sum of the reciprocals of the square numbers (the basel problem) is π2/6: ∞∑ k=1 1 k2 = 1 12 + 1 22 + 1 32 + · · · = π2 6 . = 1.644934. the nth harmonic number is the sum of the reciprocals of the first n natural numbers: hn = n∑ k=1 1 k = 1 + 1 2 + 1 3 + · · ·+ 1 n , h0 being defined as 0. the generalized harmonic numbers of order n in power r is the sum hn,r = n∑ k=1 1 kr , where hn,1 = hn are harmonic numbers. generalized harmonic number of order n in power 2 can be written as a function of harmonic numbers using formula (see [9]) hn,2 = n−1∑ k=1 hk k(k + 1) + hn n . 76 series of reciprocals of the quadratic polynomials with complex conjugate roots from formulas for hn,r, where r = 1,2 and n = 1,2, . . . ,9, we get the following table: n 1 2 3 4 5 6 7 8 9 hn 1 3 2 11 6 25 12 137 60 49 20 363 140 761 280 7129 2520 hn,2 1 5 4 49 36 205 144 5269 3600 5369 3600 266681 176400 1077749 705600 771817 352800 table 1: nine first harmonic numbers hn and generalized harmonic numbers hn,2 the hyperbolic cotangent is defined as a ratio of hyperbolic cosine and hyperbolic sine cothx = coshx sinhx , x 6= 0. because hyperbolic cosine and hyperbolic sine can be defined in terms of the exponential function coshx = ex + e−x 2 = e2x + 1 2 ex , sinhx = ex − e−x 2 = e2x −1 2 ex , we get cothx = coshx sinhx = ex + e−x ex − e−x = e2x + 1 e2x −1 , x 6= 0. 2 the sum of the series of reciprocals of the quadratic polynomials with integer roots as regards the sum of the series of reciprocals of the quadratic polynomials with different positive integer roots a and b, a < b, i.e. of the series ∞∑ k=1 k 6=a,b 1 (k −a)(k − b) , in the paper [1] it was derived that the sum s(a,b)++ is given by the following formula using the nth harmonic numbers hn s(a,b)++ = 1 b−a (ha−1 −hb−1 + 2hb−a −2hb−a−1) . (1) in the paper [2] it was shown that the sum s(a,b)−− of the series of reciprocals of the quadratic polynomials with different negative integer roots a and b, a < b, 77 radovan potůček i.e. of the series ∞∑ k=1 1 (k −a)(k − b) , is given by the simple formula s(a,b)−− = 1 b−a (h−a −h−b) . (2) the sum of the series ∞∑ k=1 k 6=b 1 (k −a)(k − b) , of reciprocals of the quadratic polynomials with integer roots a < 0, b > 0 was derived in the paper [4]. this sum s(a,b)−+ is given by the formula s(a,b)−+ = (b−a) (h−a −hb−1) + 1 (b−a)2 . (3) in the paper [3] it was derived that the sum s(a,a)−− of the series of reciprocals of the quadratic polynomials with double non-positive integer root a, i.e. of the series ∞∑ k=1 1 (k −a)2 , is given by the following formula using the generalized harmonic number h−a,2 of order −a in power 2 s(a,a)−− = π2 6 −h−a,2. (4) the sum of the series ∞∑ k=1 k 6=a 1 (k −a)2 , of reciprocals of the quadratic polynomials with double positive integer root a, was derived in the paper [5]. this sum s(a,a)++ is given by the formula with the generalized harmonic number in power 2 s(a,a)++ = π 2 + ha−1,2. (5) the formula for the sum s(a,0)−0 of the series ∞∑ k=1 1 k(k −a) 78 series of reciprocals of the quadratic polynomials with complex conjugate roots of reciprocals of the quadratic polynomials with one zero and one negative integer root a and also the formula for the sum s(0,b)0+ of the series ∞∑ k=1 k 6=b 1 k(k − b) of reciprocals of the quadratic polynomials with one zero and one positive integer root b were derived in the paper [7]. these sums are given by the simple formulas s(a,0)−0 = h−a −a , (6) s(0,b)0+ = 1− bhb−1 b2 . (7) 3 the sum of the series of reciprocals of the quadratic polynomials with complex conjugate roots we deal with the problem to determine the sum s(a,b), where a,b are nonzero integers, of the infinite series of reciprocals of the quadratic polynomials with complex conjugate roots a± bi, i.e. the series of the form ∞∑ k=1 1 (k −a)2 + b2 . (8) the quadratic trinomial (k−a)2 +b2 = k2−2ak +a2 +b2 can be in the complex domain written in the product form [ k−(a+bi) ] · [ k−(a−bi) ] , so the quadratic trinomial k2−2ak+a2 +b2 has complex conjugate roots k1 = a+bi, k2 = a−bi. the series (8) is convergent because ∞∑ k=1 1 (k −a)2 + b2 ≤ ∞∑ k=1 1 (k −a)2 . for non-positive integer a we get by (4) an equality s(a,b) < π2/6 . = 1.6449 and for positive integer a we have by (5) s(a,b) < π/2 + π2/6 . = 3.2157 (see [5]). because it obviously does not matter the sign of an imaginary part b, let us deal further with two cases of the series (8) – with a positive real part a and with a negative one. if the integer real part a > 0, then the sum s(a,b) has the form ∞∑ k=1 1 (k −a)2 + b2 = ∞∑ k=1 1 (a−k)2 + b2 = = 1 (a−1)2 + b2 + 1 (a−2)2 + b2 + · · ·+ 1 12 + b2 + + 1 b2 + 1 12 + b2 + 1 22 + b2 + 1 32 + b2 + · · · = s(a−1,b) + 1 b2 + s(b), 79 radovan potůček where s(b) = ∞∑ k=1 1 k2 + b2 is the sum that was derived in the paper [6] and which is given by the formula s(b) = π 2b · e2πb + 1 e2πb −1 − 1 2b2 = π 2b cothπb− 1 2b2 (9) and where s(a−1,b) = a−1∑ k=1 1 k2 + b2 (10) is the sum of the finite series, which was in the paper [8] derived by means of the trapezoidal rule and which is given by the approximate formula s(a−1,b) .= 1 b arctan a b − 1 2 ( 1 b2 + 1 a2 + b2 ) . (11) in this paper it was shown that this approximate formula is a suitable approximation of the sum s(a−1,b), because one hundred results obtained by means of this formula when modelling in maple 16 have very small relative errors (in the range of 0.60% to 0.05%). in total, we get s(a,b) = s(a−1,b) + 1 b2 + s(b) . = . = 1 b arctan a b − 1 2 ( 1 b2 + 1 a2 + b2 ) + 1 b2 + π 2b cothπb− 1 2b2 , so after simple arrangement we have the following result: s(a,b) . = 1 b arctan a b − 1 2(a2 + b2) + π 2b cothπb, a > 0. if the integer real part a < 0, then the sum s(a,b) has for a = −a > 0 the form ∞∑ k=1 1 (k −a)2 + b2 = ∞∑ k=1 1 (k + a)2 + b2 = = 1 (1 + a)2 + b2 + 1 (2 + a)2 + b2 + 1 (3 + a)2 + b2 + · · · = = 1 12 + b2 + 1 22 + b2 + · · ·+ 1 a2 + b2 + 1 (1 + a)2 + b2 + 1 (2 + a)2 + b2 + · · · · · ·− ( 1 12 + b2 + 1 22 + b2 + · · ·+ 1 (a−1)2 + b2 ) − 1 a2 + b2 = = s(b)−s(a−1,b)− 1 a2 + b2 = s(b)−s(−a−1,b)− 1 a2 + b2 . 80 series of reciprocals of the quadratic polynomials with complex conjugate roots so we have s(a,b) = s(b)−s(−a−1,b)− 1 a2 + b2 . = . = π 2b cothπb− 1 2b2 − [ 1 b arctan −a b − 1 2 ( 1 b2 + 1 a2 + b2 )] − 1 a2 + b2 and after simple arrangement we get the same result as above for a > 0: s(a,b) . = 1 b arctan a b − 1 2(a2 + b2) + π 2b cothπb, a < 0. therefore for every integer a including zero we get the main result s(a,b) . = 1 b arctan a b − 1 2(a2 + b2) + π 2b cothπb. (12) 4 numerical verification we solve the problem to determine the values of the sum s(a,b) of the series ∞∑ k=1 1 (k −a)2 + b2 for a = −5,−4, . . . ,5 and for b = 1,2, . . . ,10. we use on the one hand an approximative direct evaluation of the sum s(a,b,t) = t∑ k=1 1 (k −a)2 + b2 , where t = 105, using the basic programming language of maple 16, and on the other hand the formula (12) for evaluation the sum s(a,b). we compare 110 pairs of these two ways obtained sums s(a,b,105) and s(a,b) to verify the formula (12). we use following procedure sumsab and succeeding for-loop statement: sumsab=proc(a,b,t) local k,s,s; s:=0; for k from 1 to t do s:=s+1/((k-a)*(k-a)+b*b); end do; print("s(",a,b,t,")=",evalf[6](s); s:=evalf[6]((1/b)*arctan(a/b)-1/(2*(a*a+b*b)) +(pi/(2*b))*coth(pi*b)); print("s(",a,b,")=",evalf[6](s)); print("relerr(s)=",evalf[10](100*abs(s-s)/s),"%"); end proc: 81 radovan potůček for a from -5 to 5 do for b from 1 to 10 do sumsab(100000,a,b); end do; end do; forty of these one hundred and ten approximative values of the sums s(a,b,105) and s(a,b) rounded to four decimals obtained by these procedure and the relative quantification accuracies r(a,b) = |s(a,b,105)−s(a,b)| s(a,b,105) of the sums s(a,b,105) (expressed as a percentage) are written into table 2 below. let us note that the computation of 110 values s(a,b,105) (abbreviated in table 2 as s(a,b)) and s(a,b) took over 5 hours 24 minutes. the relative quantification accuracies are approximately between 6.14% and 0.0006%, 96 of these 110 approximative values have the relative quantification accuracy smaller than 0.5%. s |s |r a =−3 a =−2 a =−1 a = 0 a = 1 a = 2 a = 3 a = 4 s(a,1) 0.2766 0.3767 0.5767 1.0767 2.0767 2.5767 2.7767 2.8767 s(a,1) 0.2776 0.3695 0.5413 1.0767 2.1121 2.5838 2.7757 2.8731 r(a,1) 0.34 1.90 6.14 0.0006 1.70 0.28 0.04 0.12 s(a,2) 0.2585 0.3354 0.4604 0.6604 0.9104 1.1104 1.2354 1.3123 s(a,2) 0.2555 0.3302 0.4536 0.6604 0.9172 1.1156 1.2383 1.3140 r(a,2) 1.13 1.55 1.48 0.002 0.75 0.47 0.24 0.13 s(a,3) 0.2356 0.2911 0.3680 0.4680 0.5791 0.6791 0.7561 0.8116 s(a,3) 0.2340 0.2891 0.3663 0.4680 0.5808 0.6811 0.7576 0.8127 r(a,3) 0.65 0.38 0.46 0.002 0.29 0.29 0.21 0.13 s(a,4) 0.2126 0.2526 0.3026 0.3614 0.4239 0.4828 0.5328 0.5728 s(a,4) 0.2118 0.2518 0.3020 0.3614 0.4245 0.4836 0.5335 0.5734 r(a,4) 0.37 0.33 0.19 0.003 0.14 0.18 0.15 0.12 s(a,5) 0.1918 0.2212 0.2557 0.2941 0.3341 0.3726 0.4071 0.4365 s(a,5) 0.1914 0.2208 0.2554 0.2942 0.3344 0.3730 0.4075 0.4369 r(a,5) 0.22 0.18 0.09 0.003 0.08 0.11 0.11 0.09 table 2: some approximate values of the sums s(a,b,105), s(a,b) and the relative quantification accuracies r(a,b) of the sums s(a,b,105) for some values of a and b 82 series of reciprocals of the quadratic polynomials with complex conjugate roots 5 conclusions we dealt with the problem to determine the sum s(a,b), where a,b are nonzero integers, of the series ∞∑ k=1 1 (k −a)2 + b2 of reciprocals of the quadratic polynomials with complex conjugate roots a± bi. we derived that the sum s(a,b) is given by the approximate formula s(a,b) . = 1 b arctan a b − 1 2(a2 + b2) + π 2b cothπb. we verified this result by computing 110 various sums by using the computer algebra system maple 16. this result also includes a special case, when b = a. in this case we get the approximate formula s(a,a) . = π 4a − 1 4a2 + π 2a cothπa. because for integer a ≥ 1 it holds cothπa → 1 (e.g. cothπ .= 1.004, coth 2π .= 1.000007, coth 3π . = 1.00000001), we have the simple aproximate formula s(a,a) . = 3πa−1 4a2 . let us note that this consequence of the main result corresponds to the numeric values in table 2: s(1,1) . = (3π − 1)/4 .= 2.1062, s(2,2) .= (6π − 1)/16 .= 1.1156, s(3,3) . = (9π −1)/36 .= 0.7576, s(4,4) .= (12π −1)/64 .= 0.5734. the series of the quadratic polynomials with complex conjugate roots a± bi so belong to special types of convergent infinite series, such as geometric and telescoping series, which sum can be found analytically and also presented by means of a simple numerical expression. 6 acknowledgements the work presented in this paper has been supported by the project ”rozvoj oblastı́ základnı́ho a aplikovaného výzkumu dlouhodobě rozvı́jených na katedrách teoretického a aplikovaného základu fvt (k215, k217)” výzkumfvt (dzro k-217). 83 radovan potůček references [1] r. potůček, the sum of the series of reciprocals of the quadratic polynomial with different positive integer roots. in: mathematics, information technologies and applied sciences (mitav 2016). university of defence, brno, 2016, p. 32-43. isbn 978-80-7231-464-5. [2] r. potůček, the sum of the series of reciprocals of the quadratic polynomial with different negative integer roots. in: ratio mathematica journal of foundations and applications of mathematics, 2016, vol. 30, no. 1/2016, p. 59-66. issn 1592-7415. [3] r. potůček, the sum of the series of reciprocals of the quadratic polynomials with double non-positive integer root. in: proceedings of the 15th conference on applied mathematics aplimat 2016. faculty of mechanical engineering, slovak university of technology in bratislava, 2016, p. 919-925. isbn 978-802274531-4. [4] r. potůček, the sum of the series of reciprocals of the quadratic polynomials with one negative and one positive integer root. in: proceedings of the 16th conference on applied mathematics aplimat 2017. faculty of mechanical engineering, slovak university of technology in bratislava, 2017, p. 1252-1253. isbn 978-80-227-4650-2. [5] r. potůček, the sum of the series of reciprocals of the quadratic polynomials with double positive integer root. in: meraa – mathematics in education, research and applications, 2016, vol. 2, no. 1. slovak university of agriculture in nitra, 2016, p. 15-21. issn 2453-6881. [6] r. potůček, the sum of the series of reciprocals of the quadratic polynomials with purely imaginary conjugate roots. in: meraa – mathematics in education, research and applications 2017, vol. 3, no. 1. slovak university of agriculture in nitra, 2017, p. 17-23. issn 2453-6881, [online], [cit. 201804-30]. available from: http://dx.doi.org/10.15414/meraa.2017.03.01.17-23. [7] r. potůček, the sum of the series of reciprocals of the quadratic polynomials with integer roots. in: proceedings of the 17th conference on applied mathematics aplimat 2018. faculty of mechanical engineering, slovak university of technology in bratislava, 2018, p. 853-860, isbn 978-80227-4765-3. [8] r. potůček, the sum of the finite series of reciprocals of the quadratic polynomial with integer purely imaginary conjugate roots. in: mathematics, in84 series of reciprocals of the quadratic polynomials with complex conjugate roots formation technologies and applied sciences (mitav 2018). university of defence in brno, brno, 2018, p. 101-107, isbn 978-80-7582-040-2. [9] wikipedia contributors, harmonic number. wikipedia, the free encyclopedia, [online], [cit. 2018-04-30]. available from: https://en.wikipedia.org/wiki/harmonic number. 85 microsoft word r.m.7 cap.17.doc microsoft word cap3.doc ratio mathematica vol. 34, 2018, pp. 67–76 issn: 1592-7415 eissn: 2282-8214 algebraic spaces and set decompositions. jan chvalina∗, bedřich smetana† received: 13-06-2018. accepted: 24-06-2018. published: 30-06-2018 doi:10.23755/rm.v34i0.415 c©jan chvalina et al. abstract the contribution is growing up from certain parts of scientific work by professor borůvka in several ways. main focus is on the decomposition theory, especially algebraized decompositions of groups. professor borůvka in his excellent and well-known book [3] has developped the decomposition (partition) theory, where the fundamental role belongs to so called generating decompositions. furthermore, the contribution is also devoted to hypergroups, to algebraic spaces called also quasi-automata or automata without outputs. there is attempt to develop more fresh view point on this topic. keywords: algebraic space; decomposition; join space; 2010 ams subject classifications: 20n20, 93a10, 20m35. ∗brno university of technology, brno, czech republic. chvalina@feec.vutbr.cz †university of defence, brno, czech republic. bedrichsmetana@unob.cz 67 jan chvalina and bedřich smetana 1 introduction the present contribution is growing up from certain parts of scientific work by professor borůvka in several ways. first of all is the decomposition theory, especially algebraized decompositions of groups. professor borůvka in his excellent and well-known book [3] has developped the decomposition (partition) theory, in (and on) sets which is applied to decompositions on groupoids and groups where the fundamental role belongs to so called generating decompositions. it is to be noted that a decomposition a in a groupoid (g, ·) is called generating if there exists, to any two-membered sequence of the elements ā, b̄ ∈ a an element c̄ ∈ a such that āb̄ ⊂ c̄. with the decomposition a in a groupoid (g, ·) there can be uniquely associate a groupoid denoted (in the mentioned book) by u and defined such a way that the carrier set of u is the decomposition a and the multiplication is defined by ā ◦ b̄ = c̄, where ā, b̄, c̄ ∈ a are such elements (i. e. cosets) that ā · b̄ ⊂ c̄ in the groupoid (g, ·). a special and important case of generating decompositions on a group (g, ·) created by left on right cosets of an invariant (normal) subgroup (h, ·) of (g, ·) is the carrier of a factor-group g/h which is a factoroid created by cosets of the form a · h (or which is the same h · a ) for an invariant subgroup h of g. on the other hand if left or right decompositions generated by a subgroup h which is not invariant in a noncommutative group g are algebraized in a similar way as above, we get multivalued binary operations on these decompositions which determine a structure called a multigroups or a hypergroup by the latest terminology. this one has been done by marty in 1934 and since the time these structures were investigated by many mathematicians in france, italy, greece, roumania, usa, canada , czechoslovakia and elsewhere. 2 preliminaries a hypergroup in the sense of marty is a pair (h, ·) where h is a non-empty set and · : h ×h →p (́h) (the system of all non-empty subsets off h) is an associative multioperation (called also a hyperoperation) satisfying the reproduction axiom: a ·h = h = h ·a for any a ∈ h [11, 12]. a commutative hypergroup (h, ·) is called a join hypergroup or a join space if it satisfies the exchange condition: for any quadruple a,b,c,d ∈ h such that a/b∩ c/d 6= ∅ (where a/b = {x ∈ h; a ∈ x · b} and similarly for c/d) we have (a ·d) ∩ (b · c) 6= ∅. in the last years investigations of hypergroups which are determined by binary relations (i.e. the binary hyperoperation · is derived by a certain standard way from a given relation on its carrier set) are of certain interests in investigations on this 68 algebraic spaces and set decompositions. field. the notion of a join space has been introduced by w. prenowitz and used by him and afterwards together with j. jantosciak to built again several branches of geometry. in the opinion of professor p. corsini which is one of present leading personalities in the hypergroup theory the presentation and development of geometry in the context of join spaces is an important moment in the recent history of mathematics. there are also close connections of the, mentioned structure to ternary spaces, especially formed by sets endowed by ternary betweenness relations, here. it is to be noted that any abelian group is a join space with single-valued operations. a simple example of a non-trivial join space or a join hypergmup can be constructed from arbitrary (non-extremal) decomposition of a set: let a be a decomposition on a non-empty set a. for any pair of elements x,y ∈ a let us define x ·y = ā∪ b̄, where ā, b̄ ∈ a are blocks of the given decomposition such that x ∈ ā,y ∈ b̄. then it is easy to see that (a, ·) is a join hypergroup (a join space) in which for a pair x,y ∈ a the fraction x/y is either a block of a containing x or x/y = a whenever x,y belong to the same block of a. the algebraic theory of automata is widely elaborated classical discipline; the golden age or which can be designated from the beginning of sixties up to the end of tne last century. nevertheless fundamental publications from the earlier time due to n. wiener, j. von neumann, s. ginsburg, m. a. arbib, v. m. gluškov, r. e. kálmán, m. o. rabin, d. scott, s. greibach, k. b. krohn, j. l. rhodes, e. f. more and others, have massive influence on the development of the automata and artificial languages theory. in spite of studies devoted to finite automata also infinite automata and their generalizations have been of some interests (cf. ferenc gécseg, istván peák nad others). it is to be noted that various concepts of a product of automata (the basic of which has been introduced and studied by m. v. gluškov in 1961 as an abstract model of electronic cirquits) are treated in a large collection of studies devoted to this topics. during the years of investigations of the mentioned thema, there occure various modifications; most of them can be generalized to the case of multiautomata or to actions of multistructures. investigations of automata in connection with multistructures yield more new impulses. it is evident that infinite antomata without outputs called also quasi-automata are in fact discrete modifications or “algebraic skelets” of dynamical systems. objects of investigations of the mentioned theories can be also considered as special general systems and they are close to the control theory. the other connection of this contribution to the research of professor borůvka consists in investigations of group and semigroup actions on sets which are substantial parts of the algebraic concept of an automaton, namely if we concentrate on changes of states rather than outputs which has been used by professor borůvka in his two-parted paper [4]. automata without outputs are termed also algebraic spaces (according to dubreil, dubreil jacobin and borůvka). so, we can use 69 jan chvalina and bedřich smetana also this terminology. in accordance with [4] we define an algebraic space with operators as a triad e = (e,g,α), where e 6= ∅ (a state set or a phase set), g is a monoid the identity e (in a special case g is supposed to be a group) called also an input or phase monoid and α : g × e → e is an action (called also a transition function) wich satisfies two conditions: 1. identity condition α(e,x) = x for any x ∈ e, 2. condition of mixed associativity α(b, (α(a,x)) = α(ab,x) for any a,b ∈ g, x ∈ e. an algebraic space e = (e,g,α) is said to be homogenous if g is acting on the set e transitively, i.e. for any pair of elements x,y ∈ e there exists a ∈ g such that α(a,x) = y. usually an algebraic space e is called homogenous if g is a group transitively acting on e, which we can called strong homogeneous or shortly s-homogeneous. 3 algebraic spaces and hypergroups we assign to every algebraic space e = (e,g,α) a commutative hypergroup h(e) = (e,•) in this way: for any pair x,y ∈ e we define x•y = α(g,x) ∪α(g,y), where α(g,x) = {α(a,x); a ∈ g} is the trajectory of the element x over the monoid g. then the hypergroup h(e) is called a state hypergroup of the algebraic space e. it is clear that on the state set of any algebraic space e = (e,g,α) there are defined two totally additive closure operations: s+, s− : p(e) →p(e) in this way: s+(x) = α(g,x), s−(x) = {x ∈ x; α(a,x) ∈ x for some a ∈ g} if x is a non-empty subset of the set e and s+(∅) = s−(∅) = ∅ (caled a source and an successor closure operation, respectively). the above defined transfer can be extended into functorial if we consider suitable morphism between hypergroups (where we use mostly homomorphisms and good homomorphisms ). by [18] a hypergroup h is said to be cyclic if for some h ∈ h we have h = ⋃ k∈n hk and it is called single-power cyclic (more exactly n-singie-power cyclic) if there exist h ∈ h, n ∈ n such that h = hn . in this case the element his called n-generating. from the above definition of a state hypergroup we get: 70 algebraic spaces and set decompositions. proposition 1. an algebraic space e is homogeneous if and only if its state hypergroup h(e) is 2-single-power cyclic and each element x ∈ e is a 2generating element of this hypergroup.2 the following theorem gives necessary and sufficient conditions under which the state hypergroup of an algebraic space is a join hypergroup: theorem 1. let (e,•) be a state hypergroup of an algebraic space (e,g,α). then the following conditions are equivalent: 1. (e,•) is a join hypergroup. 2. for any pair (x,y) ∈ e × e such that x • y ⊆ u2 for a suitable element u ∈ e, there exists an element v ∈ e with the property v2 ⊆ x2 ∩y2. 3. for any pair (x,y) ∈ e×e such that there exists a pair (a,b) ∈ g×g and an element u ∈ e with α(a,u) = x, α(b,u) = y, we have α(c,x) = α(d,y) for some pair (c,d) ∈ g×g. 2 on the contrary to the case of algebraic structures with single-valued operations in the case of hypergroups there are possible various modifications of the concept of generating decomposition of the carrier set of a hypergroup. it depends on the various approaches to the congruence concept for hyperstructures. one of them is the following notion: definition. let (g, ·) be a hypergroupoid (i. e. · : g×g →p(g) is an arbitrary mapping) let g be such a decomposition on the set g that for any quadruple a,b,c,d ∈ g with the property a,c ∈ ā, b,d ∈ b̄ for some a,b ∈ g we have (a · b [ g ) = (c · d [ g ) ; here x [ ḡ denotes the closure of the set x in the decomposition g ([3], 2. 3). then the decomposition g is called generating (on the hypergroupoid (g, ·)) or h-generating. example 1. let x be a nonempty set, f : x → x be a mapping. for x,y ∈ x we put x ·y = {fn(u); u ∈{x,y},n ∈ n0}, where fn is the n-th iteration of the mapping f. then it is easy to verify that (x, ·) is a commutative hypergroup in the above considered sense. then the decomposition xf corresponding to a kw-equivalence (kuratowski -whyburn equivalence) r on x is defined by x r y iff fm(x) = fn(x) for some pair m,n ∈ n0 (the set of all non-negative integers) then the decomposition xf is generating on the 71 jan chvalina and bedřich smetana hypergroup (x, ·). example 2. by a deformation of one hypergroupoid (g, ·) onto another one hypergroupoid (h, ·) we mean a good (also called strong) homomorphism f : (g, ·) → (h, ·), i.e. for any pair x,y ∈ g we have f(x · y) = f(x) ·f(y). then the decomposition g of the hypergroupoid (g, ·) corresponding to deformation f (i.e. elements x,y ∈ g belong to some element ā ∈ g if an only if f(x) = f(y) ) i.e. the decomposition corresponding to f is h-generating. 4 h-genenerating and levine‘s decompositions now we define a hyperoperation on an h-genenerating decomposition g on a hypergroupoid (g, ·). for arbitrary pair of elements ā, b̄ ∈ g we put ā · b̄ = (x.y)[g, where (x,y) ∈ ā× b̄ is an arbitrary pair. it is easy to prove that then (g, ·) is a hypergroupoid and that the definition is correct ( it is independent on the choice of elements x,y). the hypergroupoid (g, ·) is then called a factor hypergroupoid on (g, ·) or a hyperfactoroid on (g, ·) or a hyperfactoroid of (g, ·). moreover we have: theorem 2. let g be an h-generating decomposition on a hypergroup (g, ·). then the hyperfactoroid (g, ·) of (g, ·) is a hypergroup. 2 now consider an algebraic space with operators e = (e,g,α) with a monoid g of operators. on the system p(e) of all subsets of e, i.e. the power set of e, we define a decomposition in this way: denote s(e) = {k ∈p(e); s + k) = k}, i.e. k ∈ s(e) whenever α(g,k) = k. now suppose p(e) is a decomposition of p(e) such that sets x,y,∈ p(e) belong to some element of p(e) if for any set m ∈ p(e) such that m = e \ k(a complement) for some k ∈ s(e) we have x ⊆ m if and only if y ⊆ m. then the decomposition p(e) is called a decomposition of the levine‘s type or a levine‘s decomposition of the power set p(e). proposition 2. let e = (e,g,α) be an algebraic space with operators, p(e) be the levine‘s decomposition of power set p(e). then sets x,y ∈p(e) belong to the same element of p(e) if and only if x ∈ x implies α(g,x) ∩y 6= ∅ and y ∈ y implies α(g,y) ∩x 6= ∅. 72 algebraic spaces and set decompositions. denote by cs(e) = {m; m ⊆ e,e \m ∈ s(e)} and ue(x) = {m; x ⊆ m,m ∈cs(e)} for any x ∈p(e). then we get: theorem 3. let e = (e,g,α) be an algebraic space with operators. for any pair of sets a,b ∈ p(e) we define a • b = ue(a) ∪ue(b) ∪{a,b}. then (p(e),•) is a commutative extensive join hypergroup and the levine‘s decomposition p(e) is h-generating on (p(e),•). let f : x → y be a mapping. we denote by f+ : p(x) → p(y ) its lifing into power sets, i.e. we define f+(a) = f(a) = {f(a); a ∈ a} for any nonempty set a ∈p(x) and f+(∅) = ∅ then we have theorem 4. let ei = (ei,gi,αi), i = 1, 2, be algebraic spaces with operators, f : e1 → e2. be a mapping preserving cs systems of spaces ei, i.e. x ∈ cs(e1) implies f(x) ∈ cs(e2). then f+ is a homomorphism of the hypergroup (p(e1),•) into the hypergroup (p(e2),•). if moreover the mapping f is surjective and reflects cssystems, ie. y ∈ cs(e2) implies f−(y ) ∈ cs(e1) (where f−(y ) is the preimage of the set y ) we have f+ : (p(e1),•) → (p(e2),•) is a deformation, i. e. a good homomorphism of hypergroups and determines a homomorphism f++ of corresponding factor hypergroups f++ = (p(e1),•) → (p(e2). remark. the closure operations s+,s− : p(e) → p(e) determine a quasidiscrete or alexandroff discrete topologies on the state set e of the algebraic space e, thus some of the above constructions can be expressed in terms of the topological spaces theory with the use of their special morphisms. language of the decomposition theory is in certain sense parallel to algebra of equivalence relations, however the first approach is useful in the context with coverings of spaces and with non-associative hyperstructures which are determined by the mentioned coverings of sets. there are many papers devoted to hyperstructures hypergroups and some of their generalizations in connection with automata and multiautomata. we mention at least papers [6,7,8,9,10] and [12, 13, 14, 15, 16, 17] from references of this contribution. the mentioned papers contain investigation of transposition hypergroups and application of these multistructures for the constructing of actions and multiactions in connection with some other mathematical concepts. 73 jan chvalina and bedřich smetana 5 conclusion considering the class of all quasiautomata (algebraic spaces) with pointed monoids as input alphabets (i.e. monoids with distinguished elements) we can construct multiautomata in such a way that input alphabets are centralizers of distinguished elements within the given monoids. hyperoperations on mentioned alphabets are defined by products of elements using powers of distinguished elements. then we obtain a class of multiautomata, where the mentioned construction described exactly e. g. in paper [10], page 5 is functorial, which means that it preserves homomorphisms; more precisely homomorphisms of quasiautomata (of algebraic spaces with input monoids) turn out into good homomorphisms of multiautomata. it is to be noted that multiautomata are serving as suitable tools for modelling of various processes concernig important mathematical objects and structures. 74 algebraic spaces and set decompositions. references [1] z. bavel : the source as a tool in automata . inform. control 18 (1971), pp. 140 155. [2] o. borůvka : ber zerlegungen von mengen. mitteilungen. tschech akad. wiss. liii, 23 (1943), 14 pp. [3] o. borůvka :foundations of the theory of groupoids and groups. veb deutscher verlag der wissenschaften, berlin 1974. [4] o. borůvka :algebraic spaces with operators and their realization by differential equations i, ii (czech). text of the seminar on differential equations. brno 1988, 35 pp. [5] j. chvalina : functional graphs, quasi-ordered sets and commutative hypergroups (czech). masaryk university brno 1995. [6] j. chvalina l. chvalinová : state hypergroups of automata. acta math. et inform. univ. ostrav. 4, no. 1 (1996), pp. 105 119. [7] j. chvalina, š. křehlı́k and m. novák: cartesian composition and the problem of generalizing the mac condition to quasimultiautomata. analele stiintifice ale universitatii ovidius constanta, seria matematica, 2016, vol. xxiv, no. 3, pp. 79-100. [8] j. chvalina š. mayerová: on certain proximities and preorderings on the transposition hypergroups of linear first-order partial differential operators. analele stiintifice ale universitatii ovidius constanta, seria matematica, 2014, vol. 2014, no. 22, pp. 85-103. [9] j. chvalina š. mayerová: general omega-hyperstructures and certain applications of those. ratio mathematica, 2013, vol. 2012, no. 23, pp. 3-20. [10] j. chvalina, j. moučka and r. vémolová: functorial passage from quasiautomata to multiautomata. in xxiv international colloquium on the acquisition process management, cdrom. brno: unob brno, 2006. pp. 1 8. [11] p. corsini : prolegomena of hypergroup theory. aviani edittore, tricesimo 1993. [12] p. corsini and v. leoreanu: application of hyperstructure theory, dordrecht, kluwer academic pub., 2003. 75 jan chvalina and bedřich smetana [13] š. hošková j. chvalina: discrete transformation hypergroups and transformation hypergroups with phase tolerance space, discrete mathematics, 2008, vol. 2008, no. 308, pp. 4133-4143. [14] š. hošková, j. chvalina and p. račková: transposition hypergroups of fredholm integral operators and related hyperstructures i. journal of basic science, 2008, vol. 4(2008), no. 1, pp. 43-54. [15] š. hošková, j. chvalina and p. račková: transposition hypergroups of fredholm integral operators and related hyperstructures ii. journal of basic science, 2008, vol. 4(2008), no. 1, pp. 55-60. [16] n. levine: an equivalence relation in tapology. math. j. okayama univ. 15(1971-72), pp. 113 123. [17] b. mikolajczak (ed.) : algebraic and structural automota theory. annals of discretc math. 44, north holland amsterdam, new york, oxford, tokyo 1991. [18] t. vougiouklis : cyclicity in a special class of hypergroups. acta univ. carol. math phys. 22, 1 (1981), pp. 3 6. 76 ratio mathematica volume 42, 2022 vague positive implicative and associative wimplicative ideals of lattice wajsberg algebras a. ibrahim * m. mohamed rajik ** abstract in the present paper, we introduce the notions of vague positive implicative, and vague associative w-implicative ideals of lattice wajsberg algebra. further, we investigate some relevant properties. moreover, we obtain some relationship between the vague associative w-implicative ideal, and the vague w-implicative ideal. keywords: wajsberg algebra; lattice wajsberg algebra; w-implicative ideal; vague set; vague w-implicative ideal; vague positive implicative w-implicative ideal; vague associative w-implicative ideal. ams mathematical subject classification 2020: 06b10, 06d35, 06d72.* * p.g. and research department of mathematics, h. h. the rajah’s college, pudukkottai, affiliated to bharathidasan university, tamilnadu, india. email: dribra@hhrc.ac.in; dribrahimaadhil@gmail.com ** department of mathematics, jansons institute of technology, coimbatore, research scholar, p.g. and research department of mathematics, h.h. the rajah’s college, pudukkottai, affiliated to bharathidasan university, tamilnadu, india.email: rajik4u@gmail.com. * received on april 2nd, 2022. accepted on june 12nd, 2022. published on june 30th, 2021. doi: 10.23755/rm.v39i0.747. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 183 a. ibrahim and m. mohamed rajik 1. introduction the concept of wajsberg algebra was first proposed by mordchaj wajsberg[10] in 1935, and analysed by font, rodriguez and torrens[4] in 1984. zadeh introduced the notion of fuzzy set in 1965. fuzzy logic has been applied to many fields, from control theory to artificial intelligence. in 1993, the idea of vague set was introduced by gau and buehrer[5]. vague set as an extension of fuzzy sets, the idea of vague set is that the membership of every element can be divided into two aspects including supporting and opposing. it is the new extension not only provides a significant addition to existing theories for handling uncertainties, but it leads to potential areas of further field research and pertinent applications. the authors [9], introduced the notions of vague w-implicative ideals, vague implicative w-implicative ideals of lattice wajsberg algebra, and investigated some properties. in this paper, we introduce the definitions of the vague positive implicative w-implicative ideal, and the vague associative w-implicative ideal of lattice wajsberg algebra. also, we discuss the relationship between the vague positive implicative w-implicative ideal, and the vague w-implicative ideal. 2. preliminaries in this section, we recall some basic definitions and results that are helpful in developing our main results. definition 2.1[4] let (𝕎, →, ∗ ,1) be an algebra with a binary operation “→” and a quasi complement “∗ ”. then it is called wajsberg algebra, if the following axioms are satisfied for all 𝑥, 𝑦, 𝑧 ∈ 𝕎, i 1 → 𝑥 = 𝑥 ii (𝑥 → 𝑦) → ((𝑦 → 𝑧) → (𝑥 → 𝑧)) = 1 iii (𝑥 → 𝑦) → 𝑦 = (𝑦 → 𝑥) → 𝑥 iv (𝑥∗ → 𝑦∗) → (𝑦 → 𝑥) = 1. proposition 2.2[4] a wajsberg algebra (𝕎, →, ∗ ,1)is satisfied the following axioms for all 𝑥, 𝑦, 𝑧 ∈ 𝕎, i 𝑥 → 𝑥 = 1 ii if (𝑥 → 𝑦) = (𝑦 → 𝑥) = 1 then 𝑥 = 𝑦 iii 𝑥 → 1 = 1 iv (𝑥 → (𝑦 → 𝑥)) = 1 v if (𝑥 → 𝑦) = (𝑦 → 𝑧) = 1 then 𝑥 → 𝑧 = 1 vi (𝑥 → 𝑦) → ((𝑧 → 𝑥) → (𝑧 → 𝑦)) = 1 vii 𝑥 → (𝑦 → 𝑧) = 𝑦 → (𝑥 → 𝑧) viii 𝑥 → 0 = 𝑥 → 1∗ = 𝑥 ∗ ix (𝑥∗)∗ = 𝑥 184 vague positive implicative and associative wimplicative ideals of lattice wajsberg algebra x (𝑥∗ → 𝑦∗) = 𝑦 → 𝑥. definition 2.3[4] a wajsberg algebra (𝕎, →, ∗ ,1) is called a lattice wajsberg algebra, if the following conditions are satisfied for all 𝑥, 𝑦 ∈ 𝕎, i the partial ordering " ≤ "on a wajsberg algebra 𝕎, such that 𝑥 ≤ 𝑦 if and only if 𝑥 → 𝑦 = 1 ii (𝑥 ∨ 𝑦) = (𝑥 → 𝑦) → 𝑦 iii (𝑥 ∧ 𝑦) = ((𝑥∗ → 𝑦∗) → 𝑦∗)∗ . thus, (𝕎, ∨, ∧, ∗, 0, 1) is a lattice wajsberg algebra with lower bound 0 and upper bound 1. proposition 2.4[4] let (𝕎, →, ∗ ,1) be lattice wajsberg algebra. then the following axioms hold for all 𝑥, 𝑦, 𝑧 ∈ 𝕎, i if 𝑥 ≤ 𝑦 then 𝑥 → 𝑧 ≥ 𝑦 → 𝑧 and 𝑧 → 𝑥 ≤ 𝑧 → 𝑦 ii 𝑥 ≤ 𝑦 → 𝑧 if and only if 𝑦 ≤ 𝑥 → 𝑧 iii (𝑥 ∨ 𝑦)∗ = (𝑥 ∗ ∧ 𝑦∗) iv (𝑥 ∧ 𝑦)∗ = (𝑥∗ ∨ 𝑦∗) v (𝑥 ∨ 𝑦) → 𝑧 = (𝑥 → 𝑧) ∧ (𝑦 → 𝑧) vi 𝑥 → (𝑦 ∧ 𝑧) = (𝑥 → 𝑦) ∧ (𝑥 → 𝑧) vii (𝑥 → 𝑦) ∨ (𝑦 → 𝑥) = 1 viii 𝑥 → (𝑦 ∨ 𝑧) = (𝑥 → 𝑦) ∨ (𝑥 → 𝑧) ix (𝑥 ∧ 𝑦) → 𝑧 = (𝑥 → 𝑧) ∨ (𝑦 → 𝑧) x (𝑥 ∧ 𝑦) ∨ 𝑧 = (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧) xi (𝑥 ∧ 𝑦) → 𝑧 = (𝑥 → 𝑦) ∧ (𝑥 → 𝑧). definition 2.5[6] let(𝕎, →, ∗ ,1)be a lattice wajsberg algebra. then it is called lattice h-wajsberg algebra, if it satisfied (𝑥 ∨ 𝑦) ∨ (((𝑥 ∧ 𝑦) → 𝑧) = 1 for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. note. in a lattice h-wajsberg algebra 𝕎, the following hold: i 𝑥 → (𝑥 → 𝑦) = (𝑥 → 𝑦) ii 𝑥 → (𝑦 → 𝑧) = (𝑥 → 𝑦) → (𝑥 → 𝑧)for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. definition 2.6[6] let 𝐿 be a lattice. an ideal 𝐼 of 𝐿 is a non-empty subset of 𝐿 is called a lattice ideal, if the following axioms are satisfied for all 𝑥, 𝑦 ∈ 𝐼, i 𝑥 ∈ 𝐼and𝑦 ≤ 𝑥 imply 𝑦 ∈ 𝐼 185 a. ibrahim and m. mohamed rajik ii 𝑥, 𝑦 ∈ 𝐼implies 𝑥 ∨ 𝑦 ∈ 𝐼. definition 2.7[6] let (𝕎, →, ∗ ,1) be a lattice wajsberg algebra. let 𝐼 be a non-empty subset of 𝕎. then 𝐼 is called a w-implicative ideal of 𝕎, if the following axioms are satisfied for all 𝑥, 𝑦 𝜖𝕎, i 0 ∈ 𝐼 ii (𝑥 → 𝑦)∗ ∈ 𝐼 and 𝑦 ∈ 𝐼 imply 𝑥 ∈ 𝐼. definition 2.8[5] a vague set 𝐴 in the universal of discourse 𝕎 is characterized by two membership functions given by: i a truth membership function 𝑡𝐴: 𝑋 → [0,1] and ii a false membership function 𝑓𝐴: 𝑋 → [0,1]. where 𝑡𝐴(𝑥) is a lower bound of the grade of membership of 𝑥 derived from the “evidence for x”, and 𝑓𝐴 (𝑥) is a lower bound on the negation of 𝑥 derived from the “evidence against x” and 𝑡𝐴(𝑥) + 𝑓𝐴 (𝑥) ≤ 1. thus the grade of membership of x in the vague set 𝐴 is bounded by subinterval [𝑡𝐴(𝑥), 1 − 𝑓𝐴 (𝑥)] of [0, 1]. the vague set 𝐴 is written as 𝐴 = {〈𝑥, [𝑡𝐴(𝑥), 𝑓𝐴(𝑥)]〉/ 𝑥 ∈ 𝕎}. where the interval [𝑡𝐴(𝑥), 1 − 𝑓𝐴(𝑥)] is called the value of 𝑥 in the vague set 𝐴 and denoted by 𝑉𝐴(𝑥). definition 2.9[5] a vague set 𝐴 of a universe 𝑋 with 𝑡𝐴(𝑥) = 0 and 𝑓𝐴(𝑥) = 1 for all 𝑥 ∈ 𝕎, is called the zero vague set of 𝕎. definition 2.10[5] a vague set 𝐴 of a universe 𝑋 with 𝑡𝐴(𝑥) = 1 and 𝑓𝐴(𝑥) = 0 for all 𝑥 ∈ 𝕎 is called the zero vague set of 𝕎. definition 2.11[5] let 𝐴 be a vague set of a universe 𝑋 with the truth membership function 𝑡𝐴 and the false membership function𝑓𝐴. for any 𝛼, 𝛽 ∈ [0,1]with 𝛼 ≤ 𝛽, the (𝛼, 𝛽) − cut of a vague set a is a crisp subset 𝐴(𝛼,𝛽)of the set 𝑋 given by 𝐴(𝛼,𝛽) = {𝑥 ∈ 𝒲/ 𝑉𝐴(𝑥) ≥ [𝛼, 𝛽]}. definition 2.12[5] the 𝛼-cut, 𝐴𝛼 of the vague set is the (𝛼, 𝛼)-cut of 𝐴 and hence given by 𝐴𝛼 = {𝑥 ∈ 𝕎/ 𝑡𝐴(𝑥) ≥ 𝛼}. definition 2.13[4] let 𝐼 = [0,1] denote the family of all closed subintervals of [0,1]. if 𝐼1 = [𝑎1, 𝑏1], 𝐼2 = [𝑎2, 𝑏2] are two elements of 𝐼[0,1], we call 𝐼1 ≥ 𝐼2if 𝑎1 ≥ 𝑎2 and 𝑏1 ≥ 𝑏2. we define the term rmax to mean the maximum of two intervals as 𝑟𝑚𝑎𝑥[𝐼1, 𝐼2] = [max{𝑎1, 𝑎2} , max{𝑏1, 𝑏2}]. similarly, we can define the term rmin of any two intervals. definition 2.14[5] the intersection of two vague sets 𝐴 and 𝐵 with respective truth membership functions and the false membership functions 𝑡𝐴, 𝑡𝐵 , 𝑓𝐴 𝑎𝑛𝑑 𝑓𝐵 is a vague set 𝐶 = 𝐴 ∩ 𝐵, whose truth membership function and false membership functions are related to those of 𝐴 and 𝐵 by 𝑡𝐶 = min{𝑡𝐴, 𝑡𝐵 } , 1 − 𝑓𝐶 = min{1 − 𝑓𝐴, 1 − 𝑓𝐵 } = 1 − max {𝑓𝐴,, 𝑓𝐵,}. 186 vague positive implicative and associative wimplicative ideals of lattice wajsberg algebra definition 2.15[9] let 𝐴 be a vague set of lattice wajsberg algebra 𝕎. then 𝐴 is called a vague wi-ideal of 𝕎, if the following axioms are satisfied for all 𝑥, 𝑦 𝜖 𝕎, i 𝑉𝐴(0) ≥ 𝑉𝐴(𝑥), ii 𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(𝑥 → 𝑦) ∗, 𝑉𝐴(𝑦)}. 3.1. vague positive implicative w-implicative ideal in this section, we introduce the definition of vague positive implicative w-implicative ideal of lattice wajsberg algebra, and investigate some related properties. definition 3.1.1 a vague set 𝐴 of lattice wajsberg algebra 𝕎 is called a vague positive implicative w-implicative ideal of 𝕎, if for all 𝑥, 𝑦, 𝑧 ∈ 𝕎, i 𝑉𝐴(0) ≥ 𝑉𝐴(𝑥) ii 𝑉𝐴(𝑦) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦) ∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)}. example 3.1.2 consider a set 𝕎 = {0, 𝑎, 𝑏, 𝑐, 1} with partial ordering as in figure 3.1.1. defining a binary operation ′ → ′ and a quasi complement ‘∗’ on 𝕎 as given in tables 3.1.1 and 3.1.2. figure: 3.1.1 table: 3.1.1 table: 3.1.2 lattice diagram complement implication define " ∨ " and " ∧ " operations on 𝕎 as follows: (𝑥 ∨ 𝑦) = (𝑥 → 𝑦) → 𝑦 (𝑥 ∧ 𝑦) = ((𝑥∗ → 𝑦∗) → 𝑦∗)∗ for all 𝑥, 𝑦 ∈ 𝕎. then, (𝕎, ∨, ∧, ∗, 0, 1) is a lattice wajsberg algebra. let 𝐴 be a vague set of 𝕎 defined by 𝐴 = {〈0, [0.6,0.3]〉 , 〈𝑎, [0.5,0.2]〉 , 〈𝑏, [0.6,0.2]〉 , 〈𝑐, [0.7,0.2]〉 , 〈1, [0.7,0.3]〉} then, 𝐴 is is vague positive implicative w-implicative ideal of 𝕎. theorem 3.1.3 every vague positive implicative w-implicative ideal of lattice wajsberg algebra 𝕎 is a vague w-implicative ideal of 𝕎. proof: let a be a vague positive implicative w-implicative ideal of 𝕎. 𝑥 𝑥 ∗ 0 1 a b b a c c 1 0 → 0 a b c 1 0 1 1 1 1 1 a a a c 1 1 b c b 1 1 1 c b 1 1 1 1 1 0 a b c 1 1 b a c 0 187 a. ibrahim and m. mohamed rajik then from (ii) of definition 3.1.1, we have 𝑉𝐴(𝑦) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦) ∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)} for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. (3.1.1) taking 𝑥 = 𝑦, 𝑦 = 𝑥, and 𝑧 = 𝑥 in (3.1.1), we get 𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛 {𝑉𝐴(((𝑥 → (𝑥 → 𝑥) ∗)∗ → 𝑦)∗), 𝑉𝐴(𝑦)} = 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑥 → 1) ∗ →)∗)∗, 𝑉𝐴(𝑦)} = 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑥 → 0) ∗ →)∗)∗, 𝑉𝐴(𝑦)} = 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → 𝑦)) ∗, 𝑉𝐴(𝑦)} thus, 𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → 𝑦)) ∗, 𝑉𝐴(𝑦)}, and 𝑉𝐴(0) ≥ 𝑉𝐴(𝑥). ∎ note. the converse of the above proposition may not be true. proposition 3.1.4 let 𝑉𝐴 be a vague implicative w-implicative ideal of lattice wajsberg algebra 𝕎. 𝑉𝐴 is a vague positive implicative w-implicative ideal of 𝕎 if and only if 𝑉𝐴(𝑥) ≥ 𝑉𝐴(((𝑥 → (𝑦 → 𝑥) ∗)∗ for all 𝑥, 𝑦 ∈ 𝕎. proof: let 𝑉𝐴 be a vague positive implicative w-implicative ideal of 𝕎, then from (ii) of definition 3.1.1 we have 𝑉𝐴(𝑦) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦) ∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)} for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. (3.1.2) substituting 𝑥 = 0, 𝑦 = 𝑥 and 𝑧 = 𝑦 in (3.1.2) we get 𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑥 → (𝑦 → 𝑥) ∗)∗ → 0)∗)∗, 𝑉𝐴(0)} = 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → (𝑦 → 𝑥) ∗), 𝑉𝐴(0)} = 𝑉𝐴((𝑥 → (𝑦 → 𝑥) ∗)∗) conversely, suppose 𝑉𝐴 is a vague w-implicative ideal and it satisfies the inequality, 𝑉𝐴(𝑥) ≥ 𝑉𝐴((𝑥 → (𝑦 → 𝑥) ∗)∗) for all 𝑥, 𝑦, 𝑧 ∈ 𝕎 (3.1.3) put 𝑥 = 𝑦 in (3.1.3) then, we have 𝑉𝐴(𝑦) ≥ 𝑉𝐴((𝑦 → (𝑧 → 𝑦) ∗)∗) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦) ∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)} thus, we have 𝑉𝐴(𝑦) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦) ∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)}, and 𝑉𝐴(0) ≥ 𝑉𝐴(𝑥) [from (i) of definition 3.1.1] hence, 𝑉𝐴 is a vague positive implicative w-implicative ideal of 𝕎. ∎ proposition 3.1.5 if 𝑉𝐴 is a vague positive implicative w-implicative ideal of lattice wajsberg algebra 𝕎 then, 𝐼 = {𝑥 ∈ 𝐴/𝑉𝐴(𝑥) = 𝑉𝐴(0)} is a positive implicative w-implicative ideal of 𝕎. 188 vague positive implicative and associative wimplicative ideals of lattice wajsberg algebra proof: let 𝑉𝐴 be a vague positive implicative w-implicative ideal of 𝕎 and 𝐼 = {𝑥 ∈ 𝐴/𝑉𝐴(𝑥) = 𝑉𝐴(0)}. obviously, 0 ∈ 𝐴. let ((𝑦 → (𝑧 → 𝑦)∗)∗ → 𝑥)∗) ∈ 𝐼, 𝑥 ∈ 𝐼 for all 𝑥, 𝑦, 𝑧 ∈ 𝕎 then, we have 𝑉𝐴(((𝑦 → (𝑧 → 𝑦) ∗)∗ → 𝑥)∗) = 𝑉𝐴(0) and 𝑉𝐴(𝑥) = 𝑉𝐴(0) (3.1.4) since 𝑉𝐴 is a vague positive implicative w-implicative ideal, we have 𝑉𝐴(𝑦) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦) ∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)} [from (ii) of definition 3.1.1] = 𝑉𝐴(0) [from 3.1.4] and 𝑉𝐴(0) ≥ 𝑉𝐴(𝑦) [from (i) of definition 3.1.1] then, we get 𝑉𝐴(𝑦) = 𝑉𝐴(0) thus, 𝑦 ∈ 𝐼 it follows that 𝐼 is a positive implicative w-implicative ideal of 𝕎. ∎ theorem 3.1.6 let 𝑉𝐴 be a vague subset of lattice wajsberg algebra 𝕎. 𝑉𝐴 is a vague positive implicative w-implicative ideal of 𝕎 if and only if 𝑉𝐴(𝛼, 𝛽) ≠ ∅; 𝛼, 𝛽 ∈ [0,1]. proof: let 𝑉𝐴 is a vague positive implicative w-implicative ideal of 𝕎 and 𝛼, 𝛽 ∈ [0,1] such that 𝑉𝐴(𝛼, 𝛽) ≠ ∅. clearly 0 ∈ 𝑉𝐴(𝛼, 𝛽). let ((𝑦 → (𝑧 → 𝑦)∗)∗ → 𝑥)∗) ∈ 𝑉𝐴(𝛼, 𝛽) and 𝑥 ∈ 𝑉𝐴(𝛼, 𝛽) for all 𝑥, 𝑦, 𝑧 ∈ 𝕎 then, we have 𝑉𝐴(((𝑦 → (𝑧 → 𝑦) ∗)∗ → 𝑥)∗ ≥ [𝛼, 𝛽], 𝑉𝐴(𝑥) ≥ [𝛼, 𝛽]. it follows that, 𝑉𝐴(𝑦) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦) ∗)∗ → 𝑥)∗), 𝑉𝐴(𝑥)} ≥ [𝛼, 𝛽]. thus, 𝑦 ∈ 𝑉𝐴[𝛼, 𝛽]. hence, [𝛼, 𝛽] is a positive implicative w-implicative ideal of 𝕎. conversely, if 𝑉𝐴(𝛼, 𝛽) ≠ ∅ is a positive implicative w-implicative ideal of 𝕎, where 𝛼, 𝛽 ∈ [0,1]. for any 𝑥 ∈ 𝕎 and 𝑥 ∈ 𝑉𝐴(𝐴), it follows that 𝑉𝐴(𝐴)(𝑥) is a positive implicative w-implicative ideal of 𝕎. thus, 0 ∈ 𝑉𝐴(𝐴)(𝑥). that is, 𝑉𝐴(0) ≥ 𝑉𝐴(𝑥) for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. let [𝛼, 𝛽] = 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦) ∗)∗ → 𝑥)∗)∗, 𝑉𝐴(𝑥)}, it follows that 𝑉𝐴(𝛼, 𝛽) is a positive implicative w-implicative ideal and ((𝑦 → (𝑧 → 𝑦)∗)∗ → 𝑥)∗)∗ ∈ 𝑉𝐴[𝛼, 𝛽] and 𝑥 ∈ 𝑉𝐴[𝛼, 𝛽]. this implies that 𝑦 ∈ 𝑉𝐴[𝛼, 𝛽]. so, 𝑉𝐴(𝑦) ≥ [𝛼, 𝛽] = 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑦 → (𝑧 → 𝑦) ∗)∗ → 𝑥)∗, 𝑉𝐴(𝑥)} thus, 𝑉𝐴 is a vague positive implicative w-implicative ideal of 𝕎. ∎ 189 a. ibrahim and m. mohamed rajik corollary 3.1.7 a vague subset 𝑉𝐴 of lattice wajsberg algebra 𝕎 is a vague positive implicative w-implicative ideal of 𝕎 if and only if 𝑉𝛼 is a positive implicative w-implicative ideal of 𝕎, when 𝑉𝛼 ≠ ∅, 𝛼 ∈ [0,1]. proposition 3.1.8 let 𝑀 and 𝑁 be implicative w-implicative ideals of lattice wajsberg algebra 𝕎, such that 𝑀 ⊆ 𝑁. if 𝑉𝐴 is a vague positive implicative w-implicative ideal of 𝑀. then so on 𝑁. proof: let 𝑀 and 𝑁 be implicative w-implicative ideals of lattice wajsberg algebra 𝕎. let 𝑉𝐴 be a vague positive implicative w-implicative ideal of m. since 𝑀 ⊆ 𝑁, 𝑉𝑀 (𝑥) ≤ 𝑉𝑁(𝑥) for all 𝑥 ∈ 𝕎. then, clearly 𝑀𝛼 ≤ 𝑁𝛼 for every 𝛼 ∈ [0,1]. if 𝑉𝑀 is a vague positive implicative w-implicative ideal of 𝕎. hence, we get 𝑀𝛼 is a positive implicative w-implicative ideal of 𝕎. [from corollary 3.1.7] then, 𝑁𝛼 is a positive implicative w-implicative ideal of 𝕎. [from proposition 2.10] thus, 𝑉𝑁 is a positive implicative w-implicative ideal. hence 𝑉𝐴 is a vague positive implicative w-implicative ideal of n. ∎ 3.2. vague associative w-implicative ideal in this section,we introduce an notation of vague associative w-implicative ideal of lattice wajsberg algebra 𝕎 and examine its properties. definition 3.2.1 a vague subset 𝑉𝐴 of lattice wajsberg algebra 𝕎 is said to be a vague associative w-implicative ideal of 𝕎 with respect to 𝑥, where 𝑥 is a fixed element of 𝕎, if it satisfies, i 𝑉𝐴(0) ≥ 𝑉𝐴(𝑦) ii 𝑉𝐴(𝑧) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑧 → 𝑦) ∗ → 𝑥)∗, 𝑉𝐴((𝑦 → 𝑥) ∗)} for all 𝑥, 𝑦, 𝑧 ∈ 𝕎 note. a vague associative w-implicative ideal with respect to all 𝑥 ≠ 1 is called a vague associative w-implicative ideal. vague associative w-implicative ideal with respect to 1 is constant. example 3.2.2 consider a set 𝐴 = {0, 𝑓, 𝑔, 1} with partial ordering as in figure 3.2.1. define ‘*’ and ‘→’ on 𝕎 as given in tables 3.2.1 and 3.2.2. → 0 f g 1 0 1 1 1 1 190 vague positive implicative and associative wimplicative ideals of lattice wajsberg algebra figure 3.2.1 table 3.2.1 table 3.2.2 lattice diagram complement implication here, 𝕎 is a lattice wajsberg algebra. a vague subset 𝑉𝐴 of 𝕎 is defined by, 𝐴 = {〈0, [0.7,0.2]〉 , 〈𝑓, [0.7,0.2]〉 , 〈𝑔, [0.5,0.3]〉 , 〈1, [0.5,0.3]〉}. then, 𝑉𝐴 is a vague associative w-implicative ideal of 𝕎. proposition 3.2.3 if 𝑉𝐴 is a vague associative w-implicative ideal of 𝕎 with respect to 𝑥 then 𝑉𝐴(0) = 𝑉𝐴(𝑦). proof: let 𝑉𝐴 be a vague associative w-implicative ideal of 𝕎 with respect to 𝑥 if 𝑥 = (0, 1). then it is trivial. if 𝑥 is neither 0 nor 1. then, 𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → 0) ∗ → 𝑥)∗, 𝑉𝐴((𝑦 → 𝑥) ∗)} [from (ii) of definition 3.2.1] thus, 𝑉𝐴(𝑥) = 𝑉𝐴(0). ∎ proposition 3.2.4 every vague associative w-implicative ideal of lattice wajsberg algebra 𝕎 with respect to 0 is a vague w-implicative ideal of 𝕎. proof: let 𝑉𝐴 be a vague associative w-implicative ideal of 𝕎 with respect to 0. then, we have 𝑉𝐴(𝑥) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → 𝑦) ∗ → 0)∗, 𝑉𝐴((𝑦 → 0) ∗)} for all 𝑥, 𝑦 ∈ 𝕎 [from (ii) of definition 3.2.1] = 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑥 → 𝑦) ∗), 𝑉𝐴(𝑦)} thus, 𝑉𝐴 is a vague w-implicative ideal of 𝕎. ∎ theorem 3.2.5 let 𝑉𝐴 be a vague w-implicative ideal of lattice wajsberg algebra of 𝕎. 𝑉𝐴 is a vague associative w-implicative ideal of 𝕎 if and only if it satisfies, 𝑉𝐴((𝑧 → (𝑦 → 𝑥)∗)∗ ≥ 𝑉𝐴((𝑧 → 𝑦) ∗ → 𝑥)∗ for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. proof: let 𝑉𝐴 be a vague w-implicative ideal of 𝕎 satisfying f f 1 1 1 g f g 1 1 1 0 f g 1 𝑥 𝑥 ∗ 0 1 f g g f 1 0 0 g f 1 191 a. ibrahim and m. mohamed rajik 𝑉𝐴((𝑧 → (𝑦 → 𝑥) ∗)∗ ≥ 𝑉𝐴((𝑧 → 𝑦) ∗ → 𝑥)∗ for all 𝑥, 𝑦, 𝑧 ∈ 𝕎 then, 𝑉𝐴(𝑧) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑧 → (𝑦 → 𝑥) ∗)∗), 𝑉𝐴((𝑦 → 𝑥) ∗)} = 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑧 → 𝑦) ∗ → 𝑥)∗), 𝑉𝐴((𝑦 → 𝑥) ∗)} thus, 𝑉𝐴 is a vague associative w-implicative ideal of 𝕎. conversely, if 𝑉𝐴 be a vague associative w-implicative ideal of 𝕎. then, 𝑉𝐴((𝑧 → (𝑦 → 𝑥) ∗)∗) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑧 → (𝑦 → 𝑥) ∗)∗ → (𝑧 → 𝑦)∗)∗ → 𝑥), 𝑉𝐴((𝑧 → (𝑦 → 𝑥)∗)∗)} let us consider, (((𝑧 → (𝑦 → 𝑥)∗)∗ → (𝑧 → 𝑦)∗)∗ → 𝑥) = 𝑥 ∗ → ((𝑧 → (𝑦 → 𝑥)∗)∗ → (𝑧 → 𝑦)∗) = 𝑥∗ → ((𝑧 → 𝑦) → (𝑧 → (𝑦 → 𝑥)∗)) = (𝑥 → 𝑦) → (𝑥∗ → ((𝑦 → 𝑥) → 𝑧∗) = (𝑧 → 𝑦) → ((𝑦 → 𝑥) → (𝑥∗ → 𝑧∗)) = (𝑧 → 𝑦) → ((𝑧 → 𝑦) → (𝑧 → 𝑦)) = 1 it follows that 𝑉𝐴((𝑧 → (𝑦 → 𝑥) ∗)∗) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(0), 𝑉𝐴(((𝑧 → 𝑦) ∗ → 𝑥)∗)} = 𝑉𝐴(((𝑧 → 𝑦) ∗ → 𝑥 ∗) thus, 𝑉𝐴((𝑧 → (𝑦 → 𝑥) ∗)∗) ≥ 𝑉𝐴(((𝑧 → 𝑦) ∗ → 𝑥 ∗). ∎ theorem 3.2.6 let 𝑉𝐴 be a vague w-implicative ideal of lattice wajsberg algebra 𝕎. 𝑉𝐴 is a vague associative w-implicative ideal of 𝕎 if and only if 𝑉𝐴(𝑧) ≥ 𝑉𝐴(((𝑧 → 𝑥)∗ → 𝑥 ∗) for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. proof: let 𝑉𝐴 be a vague associative w-implicative ideal of 𝕎. then, 𝑉𝐴(𝑧) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑧 → 𝑦) ∗ → 𝑥)∗, 𝑉𝐴((𝑦 → 𝑥) ∗)} for all 𝑥, 𝑦, 𝑧 ∈ 𝕎. [from (ii) of definition 3.2.1] taking 𝑦 = 𝑥 we get, 𝑉𝐴(𝑧) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑧 → 𝑥) ∗ → 𝑥)∗, 𝑉𝐴((𝑥 → 𝑥) ∗)} = 𝑟𝑚𝑖𝑛{𝑉𝐴((𝑧 → 𝑦) ∗ → 𝑥)∗, 𝑉𝐴(0)} = 𝑉𝐴(((𝑧 → 𝑦) ∗ → 𝑥∗) conversely, if 𝑉𝐴 is a vague w-implicative ideal and satisfies 𝑉𝐴(𝑧) ≥ 𝑉𝐴(((𝑧 → 𝑦) ∗ → 𝑥)∗) for all 𝑥, 𝑦, 𝑧 ∈ 𝕎 clearly, (((𝑧 → 𝑥)∗) → (𝑦 → 𝑥)∗)∗ → (𝑧 → 𝑦)∗)∗ = 0 192 vague positive implicative and associative wimplicative ideals of lattice wajsberg algebra and ((𝑧 → 𝑦)∗ → (𝑧 → 𝑥)∗)∗ ≤ (𝑥 → 𝑦)∗ it follows that, (((𝑧 → (𝑦 → 𝑥)∗)∗ → 𝑥)∗ → ((𝑧 → 𝑦)∗ → 𝑥)∗)∗ = 0 𝑉𝐴((𝑧 → (𝑦 → 𝑥) ∗)∗ ≥ 𝑉𝐴(((𝑧 → 𝑦 → 𝑥) ∗) → 𝑥)∗ → 𝑥)∗) ≥ 𝑟𝑚𝑖𝑛{𝑉𝐴(((𝑧 → (𝑦 → 𝑥) ∗ → 𝑥)∗, 𝑉𝐴((𝑧 → 𝑦) ∗ → 𝑥)∗)∗} 𝑉𝐴(((𝑧 → 𝑦) ∗ → 𝑥)∗) = 𝑟𝑚𝑖𝑛{𝑉𝐴(0), 𝑉𝐴(((𝑧 → 𝑦) ∗ → 𝑥)∗) = 𝑉𝐴(((𝑧 → 𝑦) ∗ → 𝑥)∗) from the proposition 3.2.3, 𝑉𝐴 is a vague associative w-implicative ideal of 𝕎. ∎ 4. conclusions in this paper, we have introduced the notions of vague positive implicative w-implicative ideal and vague associative w-implicative ideal of lattice wajsberg algebras. further, we have discussed the relationship between vague positive implicative w-implicative ideal, and vague w-implicative ideal. moreover, we have given some of the characterization of the vague associative w-implicative ideal. references [1] anitha, t., and amarendra babu, v., vague positive implicative and vague associative li-ideals of lattice implication algebras, international journal of pure and applied mathematics volume 105, no. 1 (2015), 39-57. [2] anitha, t., and amarendra babu, v., vague li ideals on lattice implication algebras, j. emerging trends in computing and information sciences, 5, (2014), 788-793. [3] anitha, t., and amarendra babu,v., vague implicative li ideals of lattice implication algebras, mathematics and statistics, 3, no. 3, (2015), 53-57. [4] font, j. m., rodriguez, a. j., and torrens, a., wajsberg algebras, stochastica volume 8, number 1, (1984), 5-31. [5] gau, w. l., buehrer, d. j., vague sets, ieee transactions on systems, man and cybernetics, 23(20)(1993), 610-614. [6] ibrahim, a., and shajitha begum, c., on wi-ideals of lattice wajsberg algebras, global journal of pure and applied mathematics, volume 13, number 10 (2017), 7237-7254. [6] ibrahim, a., and shajitha begum, c., ideals and implicative wi-ideals of lattice wajsberg algebras, ipasj international journal of computer science, volume 6, issue 4 (2018), 30-38. [7] ibrahim, a., and shajitha begum, c., fuzzy implicative wi-ideals of lattice wajsberg algebras, journal of computer and mathematical science, volume 9, number 8 (2018), 1026 -1035. [8] ibrahim, a., and mohamed rajik, m., vwi-ideals of lattice wajsberg algebras, journal of xidian university, volume 14, number 3(2020), 1284-1289. [9] wajsberg, m., beitragezum metaaussagenkalkull,monat. mat. phys. 42 (1935), 221-242. [10] zadeh, l. a., fuzzy sets, information and control 8 (1965), 338-353. 193 ratio mathematica issn: 1592-7415 vol. 35, 2018, pp. 111 125 eissn: 2282-8214 111 an analysis of loan repayment plans according to the bank customer profile salvador cruz rambaud1 and maría de los ángeles del pino álvarez2 received: 04-05-2018. accepted: 15-07-2018. published: 31-12-2018. doi: 10.23755/rm.v35i0.430 ©cruz et al. abstract it has been demonstrated that there exists a general preference for improvement in loan repayment plans in the way that people prefer decreasing sequences of installments as tested by hoelzl et al. (2011). moreover, they also demonstrated that there exists a positive correlation between financial capability and financial literacy when it is given the possibility of having a gain by investing a part of the available money. in these cases, the most financial literate consumers showed a preference for increasing loan plans instead of decreasing ones. in this vein, independently of the level of the borrowers’ risk profile, we suggest that an ad hoc offer should be made to the customers taking into account these two characteristics by distinguishing three different levels for both personal traits: low, medium and high. thus, we have analyzed the interest rate which makes both the decreasing and the increasing loan plans indifferent when considering that the option to invest part of the money in savings products is given. moreover, the analysis has been carried out by considering that the loan repaid principal is variable either in arithmetic progression or in geometric progression. thus, regarding the main repayment plans offered by banks we have analyzed which one fits better to the defined customer’s profile. keywords: loan plan, sequence of payments, financial literacy, financial capability, customer’s profile. 1departamento de economía y empresa, universidad de almería (spain); scruz@ual.es. 2departamento de economía y empresa, universidad de almería (spain); mda058@inlumine.ual.es. salvador cruz rambaud and maría de los ángeles del pino álvarez 112 1. introduction traditionally, the offer of loans by banks has been focused on the analysis and description of the main repayment schedules and the calculation of the effective interest rate of the corresponding operation. thus, given the rate of interest, the client may indistinctly choose a loan with constant installments, constant repaid principal or payments variable in arithmetic progression, among others. from a theoretical point of view, all these repayment methods are equivalent because the interest rate (which is the price of the operation) is the same. however, in our opinion, neither the loan offer nor the choice of a repayment schedule should be indifferent for banks and costumers, respectively. in effect, it is necessary to previously analyze the costumer profile and the needs of the bank in order to try to satisfy both requirements. accordingly, we are going to first examine both the main elements defining the costumer profile and the variables of interest for the management of banking institutions. this paper starts from the conclusions obtained by hoelzl et al. (2011) consisting in the proposal of some suitable advises to different consumption groups based on their preferences on loan repayment plans considered as sequences of installments. to do that, this paper is focused on the results obtained from the three studies implemented by these scholars, especially in the second one. thus, in order to make an ad hoc offer to customers, the financial capability and the level of financial literacy could be the main two inputs which determine if borrowers choose falling, constant or rising installments, according to their preferences. in particular, as pointed out by hoelzl et al. (2011), there is a positive correlation between the rising profile and the financial education level when a part of the monthly available cash can be saved. in spite of the fact that financial security and consumer self-protection also require to have a good understanding of financial topics (kozup and hogarth, 2008), we are going to only focus on the financial literacy. thus, with respect to the concept of financial literacy, the following definition has been considered: “financial literacy is a measure of the degree to which one understands key financial concepts and possesses the ability and confidence to manage personal finances through appropriate, short-term decision-making and sound, long-range financial planning, while mindful of life events and changing economic conditionsˮ (remund, 2010). therefore, it has been demonstrated that financial literacy is a factor which determines the financial decisions taken by customers when they have to define their credit portfolio (disney and gathergood, 2013). in this regard, a low level of financial literacy has direct repercussions in selfcontrol and over-indebtedness (gathergood, 2012). that is the reason whereby it an analysis of loan repayment plans according to the bank customer profile 113 has been analyzed how this characteristic could be measured (huston, 2010). moreover, there is a relationship between financial satisfaction and financial capability for those consumers who have certain level of financial education and avoid risky decisions (xiao et al., 2014). in this vein, the effects of financial education in financial capability have been analyzed (xiao and o’neill, 2016). thus, considering that financial capability is “the ability to manage and take control of their financeˮ, the age and the unemployment status are the two factors with most impact on the financial capability (taylor, 2011). different analyses show that younger adults have more financial difficulties and make worse financial plans (atkinson et al., 2006; kempson et al., 2004). therefore, with respect to the customer profile, there are three significant variables of interest for banks: 1. the payment capability which can be defined as the financial potential to face the installments derived from the financial operation. usually, it is stated that the sum of all installments corresponding to the same client must be between 30% and 40% of his/her overall incomes. 2. the risk which is the probability of default in the operation. the possibility of failure in payments can be covered by personal or real guarantees. 3. the financial literacy which implies a financial knowledge sufficient to adequately invest the extra money and an understanding of the meaning of the effective rate of the operation. on the contrary, a costumer with a low financial training does not have enough financial knowledge to invest and will analyze the total amount of interests instead of the effective rate of the operation. in our opinion, these three variables are related with the loan schedule which is most suitable for a specific borrower. table 1 shows the relationship of variables #1 and #3 with the loan repayment plan, where the symbol  means that the principal repaid must be increasing in arithmetic progression, whilst  denotes that the plan must be decreasing. the variable #2 will be incorporated to the analysis in section 2. financial literacy high medium low payment capability high    medium    low    table 1. different customer profiles. salvador cruz rambaud and maría de los ángeles del pino álvarez 114 figure 1 helps to understand the existing relationship between variables #1 and #3 and the three identified profiles (decreasing, constant and increasing). figure 1. relationship between variables #1 and #3 and the three considered profiles: decreasing, constant and increasing. the organization of this paper is as follows. section 2 shows an analysis of the interest rate according to the loan repayment plan. in section 3 and section 4 the main repayment plans offered by banks are analyzed by considering that the principal is repaid in arithmetic progression and geometric progression, respectively. in section 5, the relationship between variables #1 and #3 and the three profiles is remade by considering the repayment plans obtained in section 3 and 4. finally, section 6 summarizes and concludes. 2. an analysis of the interest rate according to the loan repayment plan in general, in a loan with a principal 0c to be repaid in n periods, the repayment plan can follow either a constant or a variable repaid principal (increasing or decreasing arithmetic progression). figure 2 displays these three possible situations for a loan to be repaid in four periods where the principal is €20. preference over sequences in loan repayment plans financial capability financial literacy increasing profile decreasing profile constant profile an analysis of loan repayment plans according to the bank customer profile 115 figure 2. evolution of the repaid principal variable in arithmetic progression: constant (in blue), increasing (in red) or decreasing (in green). in general, the following identity holds for the three repayment plans: 0 1 ca n s s = = . although the financial cost of these three alternatives is the same (more specifically, the interest rate i), it has been experimentally shown that people prefer the third option in which the repaid principal is a decreasing arithmetic progression, all terms amounting 0c . maybe this is because people prefer to leave the best for last (kahneman et al., 1993) or because the total burden in this option is lower than the corresponding one to other plans. this is consistent with a borrower characterized by a low financial training or a high financial capability. moreover, this type of clients supposes a small risk for banking institutions (see variable #2 in section 1) since the most part of the principal is repaid at the beginning of the operation. therefore, a good initiative of banks could be to offer a smaller rate of interest because these clients pay more attention to the total amount of interests instead of the effective rate of the operation. the opposite can be said for clients with a high financial capability or a high financial training because they prefer a repaid principal increasing in arithmetic progression which implies a higher risk for banks and the possibility to invest the extra money. therefore, this type of clients can support a greater rate of interest. 0 1 2 3 4 5 6 7 8 9 period 1 period 2 period 3 period 4 plan 1 plan 2 plan 3 salvador cruz rambaud and maría de los ángeles del pino álvarez 116 therefore, the following question could be addressed: what is the new interest rate, say i ( ii  ), such that plan 2 is indifferent to the plan 3? if the difference involved in plan 3 is d (d < 0) and the common difference of the second plan is d ( 0=d or 0d ), then we can require that the sum of payments corresponding to plan 3 must coincide with the sum of payments corresponding to plan 2 minus the amount a earned in the investments. as the aggregated principal repaid in both plans must coincide with 0 c , we can propose the following equality between the aggregated interest due in both plans: aii n s s n s s −= == 11 , or equivalently, ad ss ascid ss asci n s n s −       −− −−−=      −− −−−  == 1 0 1 0 2 )2)(1( )1( 2 )2)(1( )1( . as d n n c a 2 1 0 − −= and d n n c a  − −= 2 1 0 , then we can write:  =       −− − −− +−− n s d ss d ns csci 1 00 2 )2)(1( 2 )1)(1( )1( ad ss d ns csci n s −       −− − −− +−−=  =1 00 2 )2)(1( 2 )1)(1( )1( . finally, as 21 −− sn , for every s = 1, 2, …, n, we can deduce that ii  . as indicated in sections 1 and 2, the loan repayment schedule is conditioned by the customer profile and the risk supported by the banking entity. therefore, section 3 will be devoted to deduce the different repayment plans of a loan starting from the different variability of the repaid principal. 3. the principal repaid is variable in arithmetic progression in this case, the principal repaid is an arithmetic progression whose first term is a and whose common difference is d: an analysis of loan repayment plans according to the bank customer profile 117 • aa =1 , • daa +=2 , • daa 2 3 += ,  • dnaa n )1( −+= . let us analyze the structure of the periodic payments: • icaa 01 += , • iacdaa )()( 02 −++= )()( 0 aidica −++= aida −+= 1 , • idacdaa )2()2( 03 −−++= idada )( 2 +−+= iada 22 −+= , • idacdaa )33()3( 04 −−++= idada )2( 3 +−+= iada 33 −+= ,  • in general, iadaa sss 11 −− −+= . this series is an arithmetic progression of second order. in effect, • iadaad kkkk 111 : −−− −=−= , and • iadaad kkkk −=−= +1: . therefore, didd kk −=− −1 . the general solution of an arithmetic progression of second order is: ),( 2 3 2 111 2 adrs r ds r a s +−+      −+= (1) where r is the common difference of 1−− kk dd , where didd kk −=− −1 . by applying equation (1) to our concrete case, it would remain: salvador cruz rambaud and maría de los ángeles del pino álvarez 118 ).( 2 3 2 0 2 icaaiddis di aids di a s +++−−+      +−+−= (2) the outstanding principal at time s is: d ss sacd ss sacc s 2 )1( 2 )1)(11( 00 − −−= −−+ −−= . therefore, the structure of the interest due is the following: id ss ascici ss       −− −−−== − 2 )2)(1( )1( 01 )23( 2 )1( 2 0 +−−−−= nn di ainic )( 2 3 2 0 2 diaiicn di ain di −++      +−+−= . as expected, observe that sss iaa += . finally, table 2 shows the repayment plan of this loan category. period payment s a interest due si principal repaid sa outstanding principal sc 0 0c 1 icaa 01 += 101 ici = aa =1 acc −= 01 2 aidaa −+= 12 212 ici = daa +=2 dacc −−= 02      n iadaa nnn 11 −− −+= nnn ici 1−= dnaan )1( −+= 0=nc table 2. repayment plan of a loan where the principal repaid varies in arithmetic progression. a special case is when 0=d in whose case a constant repayment plan is obtained (see table 3). an analysis of loan repayment plans according to the bank customer profile 119 period payment s a interest due s i principal repaid s a outstanding principal s c 0 0c 1 icaa 01 += 101 ici = aa =1 acc −= 01 2 aiaa −= 12 212 ici = aa =2 acc −= 02      n iaaa nnn 11 −− −= nnn ici 1−= aan = 0=nc table 3. repayment plan of a loan where the principal repaid varies in arithmetic progression where 0=d . moreover, if 0=a , the american repayment plan can be obtained (see table 4). period payment s a interest due s i principal repaid sa outstanding principal sc 0 0c 1 11 ia = 101 ici = 01 =a 01 cc = 2 22 ia = 202 ici = 02 =a 02 cc =      n nn ia = nn ici 0= 0can = 0=nc table 4. repayment plan of a loan where the principal repaid varies in arithmetic progression where 0=d and 0=a . 4. the principal repaid is variable in geometric progression in this case, the principal repaid is a geometric progression whose first term is a and whose common ratio is r: • aa =1 , • ara =2 , • 2 3 ara = ,  • 1− = n n ara . salvador cruz rambaud and maría de los ángeles del pino álvarez 120 let us analyze the structure of the periodic payments: • icaa 01 += , • iacara )( 02 −+= iacircircar )( 000 −+−+= ircacra )( 001 −−+= , • iaracara )( 0 2 3 −−+= iaraciraciracar )()()( 000 2 −−+−−−+= ircacra )( 002 −−+= , • iararacara )( 2 0 3 4 −−−+= iararaciraraciraracar )()()( 2 000 3 −−−+−−−−−+= ircacra )( 003 −−+= ,  • in general, ircacraa ss )( 001 −−+= − . this series is an arithmetic-geometric sequence whose general solution is: , 1 1 1 1 1 r r draa n s s − − += − − (3) where r is the common ratio and d is the difference of the progression. by applying equation (3) to our concrete case, it would remain: r r ircacricaa n n s − − −−++= − − 1 1 )()( 1 00 1 0 r r airicrica n nn − − −−++= − −− 1 1 )1()( 1 1 0 1 0 r r aiicar n n − − −+= − − 1 1 1 0 1 (4) the outstanding principal at time s is: r r acacc ss k ks − − −=−=  = 1 1 0 1 0 . an analysis of loan repayment plans according to the bank customer profile 121 therefore, the structure of the interest due is the following: i r r acici s ss         − − −== − − 1 1 1 01 i r r aic s − − −= − 1 1 1 0 . as expected, observe that sss iaa += . finally, table 5 shows the repayment plan of this loan category. period payment s a interest due si principal repaid sa outstanding principal sc 0 0c 1 icaa 01 += 101 ici = aa =1 acc −= 01 2 ircacraa )( 0012 −−+= 212 ici = ara =2 aracc −−= 02      n ircacraa nn )( 001 −−+= − nnn ici 1−= 1− = n n ara 0=nc table 5. repayment plan of a loan where the principal repaid varies according to a geometric progression. thus, when ir += 1 , the french repayment plan can be obtained (see table 6), where naaa === 21 and niii === 21 . period payment s a interest due si principal repaid sa outstanding principal s c 0 0c 1 aicaa =+= 01 101 ici = aa =1 acc −= 01 2 aiaciaa =−++= )()1( 02 212 ici = )1(2 iaa += )1(02 iaacc +−−=      n a nnn ici 1−= 1 )1( − += n n iaa 0=nc table 6. repayment plan of a loan where the principal repaid varies according to a geometric progression and ir += 1 . salvador cruz rambaud and maría de los ángeles del pino álvarez 122 5. general discussion in sections 3 and 4 we have obtained the main repayment plans offered by banks: the american plan, the constant principal repaid plan and the french plan. thus, we can observe that, the same characteristics being considered, the amount of interests in the constant principal repaid plan and the french plan is less than in the american plan (figure 3). therefore, the american plan will be the best choice for those customers with a high level of financial literacy and a high level of financial capability since they could invest the available money and repay the principal at the end of the loan. finally, for a low financial capability profile, the choice would be the french plan independently of the financial literacy profile. figure 3. amount of interests for a loan with a principal of €10,000 to be repaid in 5 years at a 5% interest rate. regarding now the relationship between variables #1 and #3 (shown in table 1), we can remake this relationship by considering, in this case, that the borrowers have to choose one of the aforementioned plans (see table 7). it is necessary to point out that the choice of the american plan and the constant principal repaid plan will depend on the possibility to invest the available money with a suitable profitability. 0 500 1.000 1.500 2.000 2.500 3.000 constant repaid principal french repayment plan american repayment plan e u ro s amount of interests amount of interests an analysis of loan repayment plans according to the bank customer profile 123 financial literacy high medium low payment capability high american constant constant medium american/constant constant french low french french french table 7. customer profiles considering the three main repayment plans (american, constant principal repaid and french). we would like also to remark that the use of a repayment plan where the payments are distributed in different years allows combining this yearly schedule with other repayment plans inside each year. in that way, the payment schedule will be adapted according to the borrower’s financial availability inside each year. 6. conclusions a preference for improvement has been demonstrated for sequences of both incomes (loewenstein and sicherman, 1991) and outcomes (loewenstein and prelec, 1993). nevertheless, focusing on loan repayment plans as sequences of installments, we have to take into account the conclusions obtained by hoelzl et al. (2011) about the positive correlation found between the preferences for increasing installments and a high level of financial literacy, by considering the same financial capability. in this regard, we have suggested that banks can evaluate the level of each borrower by considering two main categories: financial literacy and financial capability. in that way, they could make a classification for each of them in three different levels: low, medium and high. thus, according to the level of risk, we have to consider that it is higher if the loan schedule is based on increasing installments than if it is based on decreasing ones since most of principal is repaid at the beginning when considering the falling profile. taking into account the aforementioned statement, table 1 shows the suitability of each plan (falling or rising) according to the borrower’s level in each category. it seems that those consumers with a low level of financial literacy and high level of financial capability (low-high profile) prefer a falling plan. in those cases, banks could offer low interest rates since borrowers focus on the total amount of interests and the risk is lower. however, it is likely that customers with a high level of financial literacy and high level of financial capability (high-high profile) prefer rising plans since they are interested in investing part of their available money. thus, in this case a higher interest rate could be offered by banks. in that way, we have obtained the salvador cruz rambaud and maría de los ángeles del pino álvarez 124 interest rate which makes equivalent both offers by considering the amount saved by the consumer with a high-high profile. on the other hand, by considering that the principal is repaid in arithmetic progression, we have analyzed the evolution experienced by each parameter of the repayment plan during the loan lifetime. thus, when 0=d , the constant principal repaid plan is obtained. moreover, if 0=d and 0=a , the american repayment plan is obtained. the same analysis has been done by considering that the principal is repaid in geometric progression. here, the special case is when ir += 1 , by obtaining the french repayment plan. in that way, we have considered that these three plans are the main ones offered by banks. thus, taking into account the amount of interests of the three plans for a loan with the same characteristics, we obtained that the corresponding to the american repayment plan is quite higher than for the other plans. this leads us to conclude that this plan will be the best choice for those customers with a high-high profile since they could invest the available money and repay the principal at the end of the loan. finally, for a low financial capability profile, the choice would be the french plan independently of the financial literacy profile. funding this paper has been partially supported by the project “la sostenibilidad del sistema nacional de salud: reformas, estrategias y propuestas”, reference: der2016-76053-r, ministerio de economía y competitividad (spain). references 1. atkinson, a., mckay, s., kempson, e. and collard, s. 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(2014) “consumer financial capability and financial satisfaction”. social indicators research, vol. 118, no. 1, pp. 415-432. approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 39, 2020, pp. 261-268 261 thermodynamic behavior of the polytropic gas in cosmology prasanta das* kangujam priyokumar singh† abstract in this paper, we investigate on the thermodynamic behavior of polytropic gas as a candidate for dark energy by considering the relation 𝑃 = 𝐾𝜌 1+ 1 𝑛 , where 𝐾 and 𝑛 are the polytropic constant and polytropic index respectively. furthermore, 𝑃 indicates the pressure and 𝜌 is the energy density of the fluid such that 𝜌 = 𝑈 𝑉 where 𝑈and 𝑉 represent the internal energy and volume, respectively. at first, we find an exact expression for the energy density of the polytropic gas using thermodynamics and later on, discuss different physical parameters. finally our study shows that the polytropic gas may be used to describe the expansion history of the universe from the dust dominated era to the current accelerated era and it is thermodynamically stable. keywords: cosmology; dark energy; polytropic gas; thermodynamics. 2010 ams subject classification: 83f05, 37d35, 82b30.‡ *department of mathematical sciences, bodoland university, kokrajhar, btr, assam783370, india; prasantadasp4@gmail.com. †department of mathematical sciences, bodoland university, kokrajhar, btr, assam783370, india; pkangujam18@gmail.com. ‡received on october 4th, 2020. accepted on december 17th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.564. issn: 1592-7415. eissn: 2282-8214. © prasanta das et al. this paper is published under the cc-by licence agreement. mailto:prasantadasp4@gmail.com mailto:pkangujam18@gmail.com prasanta das and kangujam priyokumar singh 262 1. introduction cosmologists suggest that our universe expands under an accelerated expansion [1]-[7]. in the standard friedman lemaitre robertson walker (flrw) cosmology, a new energy with negative pressure, called dark energy (de) is responsible for this expansion [8]. the nature of the de is still unknown and various problems have been proposed by the researchers in this field. about 70% of the present energy of the universe is contained in the de. the cosmological constant with the time independent equation of state is the earliest, simplest and most traditional candidate for the dark energy which can be taken into account as a perfect fluid satisfying the relation 𝜌 + 𝑃 = 0. but it has some problems like fine-tuning and cosmic coincidence puzzles [9], [10]. besides the cosmological constant, the other dark energy models are quintessence [11], phantom [12], tachyon [13], holographic dark energy [14] [15], k-essence [16] and chaplygin gas models with various equation of state. polytropic gas is one of the dynamical dark energy models [17]. in the present study, we want to investigate the thermodynamic behavior of the polytropic gas. k. karami et al. investigated the interaction between the polytropic gas and cold dark matter and found that the polytropic gas behaves as the phantom dark energy [18]. k. karami and s. ghaffari showed that the generalized second law of thermodynamics is always satisfied by a universe filled with a polytropic gas and a cold dark matter [19]. k. kleidis and n.k. spyron used the first law of thermodynamics in the polytropic gas model and they show that the polytropic gas behaves as dark energy and this model leads to a suitable fitting with the observational data about the current expanding era [20]. h. moradpour, a. abri and h. ebadi, investigated the thermo dynamical behavior and stability of the polytropic gas [21]. m. salti et al. discussed validity of the first and generalized second law of thermodynamics in locally rotationally symmetric bianchi-type ii space time which is dominated by a combination of polytropic gas and baryonic matter[22]. moreover, muzaffer askin et al. studied the cosmological scenarios of the polytropic gas dark matter-energy proposal in a friedmann robertsonwalker universe and they found an exact expression for the energy density of the polytropic gas model according to the thermo dynamical point of views and a relationship between a homogeneous minimally coupled scalar field and the polytropic gas [23].this paper is organized as follows: in section 2 we construct the basic thermodynamic formalism of the polytropic gas model and discuss the thermodynamic behavior of this model. finally in section 3 we provide a brief discussion. thermodynamic behavior of the polytropic gas in cosmology 263 2. basic formalism in this work, we consider the following equation of state which is well known as polytropic gas equation of state 𝑃 = 𝐾𝜌 1+ 1 𝑛 (1) here 𝐾(> 0) and 𝑛(< 0) are polytropic constant and polytropic index respectively. moreover, 𝑃 is the pressure and 𝜌 is the energy density of the fluid such that 𝜌 = 𝑈 𝑉 (2) where 𝑈and 𝑉 are the internal energy and volume filled by the fluid respectively. first of all, we try to find the internal energy 𝑈 and energy density 𝜌 of the polytropic gas as a function of its volume 𝑉 and entropy 𝑆. from the general thermodynamics, we have ( 𝜕𝑈 𝜕𝑉 ) 𝑆 = −𝑃 (3) from the equations (1), (2) and (3), we get ( 𝜕𝑈 𝜕𝑉 ) 𝑆 = −𝐾 ( 𝑈 𝑉 ) 1+ 1 𝑛 (4) integrating the equation (4), we get 𝑈 = (−1)−𝑛 (𝐾𝑉 − 1 𝑛 + ξ) −𝑛 (5) where the parameter ξ is the constant of integration which may be a universal constant or a function of entropy 𝑆 only the equation (5) also can rewrite in the following form 𝑈 = (−1)−𝑛𝐾−𝑛𝑉 (1 + ( v ε ) 1 n ) −𝑛 (6) where 𝜀 = ( 𝐾 ξ ) 𝑛 (7) and it has a dimension of volume. therefore, the energy density 𝜌 of the polytropic gas is 𝜌 = 𝑈 𝑉 = (−1)−𝑛𝐾−𝑛 (1 + ( v ϵ ) 1 n ) −𝑛 (8) when 𝑛 < 0 then equation (8) gives 𝜌 ∼ (−1)−𝑛𝐾−𝑛 𝜀 𝑉 (9) now we will use these equations to discuss different physical parameters. prasanta das and kangujam priyokumar singh 264 a) pressure: using the equation (8) in the equation (1) we get the pressure of the polytropic gas as a function of entropy 𝑆 and volume 𝑉 in the following form 𝑃 = (−1)𝑛+1𝐾−𝑛 (1 + ( v ϵ ) 1 n ) −(𝑛+1) (10) we can rewrite the equation (10) in the following form 𝑃 = − 𝜌 1+( v ϵ ) 1 n (11) when 𝑛 < 0 and 𝜀 does not diverge then for small volume i.e. at early stage of universe, 𝑉 ≪ 𝜀 ie 𝑉 𝜀 ≪ 1 , we get p ≃ 0 , which represents a dust dominated universe. when 𝑛 < 0 and 𝜀 does not diverge then for large volume i.e. at late stage of universe, 𝑉 ≫ 𝜀 ie 𝑉 𝜀 ≫ 1, we get p ≃ −𝜌, which indicates an accelerated expansion of the universe. b) caloric equation of state: now from the equations (8) and (10) we get the caloric equation of state parameter as 𝜔 = 𝑃 𝜌 = − 1 1+( v ϵ ) 1 n (12) when 𝑛 < 0 and 𝜀 does not diverge then for small volume 𝑉 ≪ 𝜀 ie 𝑉 𝜀 ≪ 1 , we get 𝜔 ≃ 0 (dust dominated) when 𝑛 < 0 and 𝜀 does not diverge then for large volume 𝑉 ≫ 𝜀 ie 𝑉 𝜀 ≫ 1 , we get 𝜔 ≃ −1 (cosmological constant) thus the equation of state parameter ( 𝜔) of the polytropic gas with 𝑛 < 0 is decreased from 𝜔 ≃ 0 (for small volume) to 𝜔 ≃ −1 (for large volume). it indicates that the universe expands from the dust dominated era to the current accelerating era. c) deceleration parameter: we get the deceleration parameter of the polytropic gas with the help of equation (12) thermodynamic behavior of the polytropic gas in cosmology 265 𝑞 = 1 2 + 3 2 𝑃 𝜌 = 1 2 − 3 2 1 1+( v ϵ ) 1 n (13) when 𝑛 < 0 and 𝜀 does not diverge then for small volume 𝑉 ≪ 𝜀 ie 𝑉 𝜀 ≪ 1 , we get 𝑞 > 0, which correspond to the deceleration universe. when 𝑛 < 0 and 𝜀 does not diverge then for large volume 𝑉 ≫ 𝜀 ie 𝑉 𝜀 ≫ 1 , we get 𝑞 < 0, which correspond to the accelerated universe. d) square velocity of sound: from the equation (11) we get the velocity of sound (𝑉𝑠 ) as 𝑉𝑠 2 = ( 𝜕𝑃 𝜕𝜌 ) 𝑆 = − 1 1+( v ϵ ) 1 n (14) when 𝑛 < 0 and 𝜀 does not diverge then for small volume 𝑉 ≪ 𝜀 ie 𝑉 𝜀 ≪ 1, we get 𝑉𝑠 2 ≃ 0 .since velocity of sound is zero in vacuum. therefore the polytropic gas behaves like a pressure less fluid at the early stage of the universe. when 𝑛 < 0 and 𝜀 does not diverge then for large volume 𝑉 ≫ 𝜀 ie 𝑉 𝜀 ≫ 1, we get 𝑉𝑠 2 ≃ −1, which gives an imaginary speed of sound leading to a perturbation cosmology. e) thermodynamic stability: the conditions of the thermodynamic stability of a fluid are ( 𝜕𝑃 𝜕𝑉 ) 𝑆 < 0 (15) and 𝐶𝑉 > 0 (16) here 𝐶𝑉 is the thermal capacity at constant volume. from the equation (10) we have ( 𝜕𝑃 𝜕𝑉 ) 𝑆 = − (1 + 1 𝑛 ) 𝑃 𝑉 1 1+( v ϵ ) − 1 n (17) if −1 < 𝑛 < 0 and 𝜀 < 0 then from (17), we have ( 𝜕𝑃 𝜕𝑉 ) 𝑆 < 0 thus the stability condition (15) of thermodynamics is satisfied. now we have to verify the positivity of the thermal capacity at constant volume 𝐶𝑉 where 𝐶𝑉 = 𝑇 ( 𝜕𝑆 𝜕𝑇 ) 𝑉 (18) prasanta das and kangujam priyokumar singh 266 now we determine the temperature 𝑇 of the polytropic gas as a function of its entropy 𝑆 and its volume 𝑉. the temperature 𝑇 of the polytropic gas is determined from the relation 𝑇 = ( 𝜕𝑈 𝜕𝑆 ) 𝑉 (19) using (6) in (19) we get 𝑇 = (−1)𝑛+1𝑉 1+ 1 𝑛 (𝐾 + 𝜉𝑉 1 𝑛) −(𝑛+1) 𝑑𝜉 𝑑𝑆 (20) this gives the temperature of the polytropic gas. we can rewrite the equation (20) in the following form 𝑇 = −𝑛 𝜌𝑉 1+ 1 𝑛 1+( v ϵ ) 1 n 𝑑𝜉 𝑑𝑆 (21) from (5) we have [𝜉]−𝑛 = [𝑈] (22) since [𝑈] = [𝑇𝑆] (23) therefore from the equations (22) & (23) we get 𝜉 = [𝑈] − 1 𝑛 = [𝑇∗𝑆] − 1 𝑛 (24) where 𝑇∗ (> 0) is a universal constant with temperature dimension. differentiating (24) with respect to ‘s’ we get 𝑑𝜉 𝑑𝑆 = − 1 𝑛 𝑇∗ − 1 𝑛𝑆 − 1 𝑛 −1 (25) using (8) & (24) in (25) we get 𝑇 = (−1)𝑛𝑉 1+ 1 𝑛 (𝑇∗ − 1 𝑛𝑆 − 1 𝑛 −1 ) [𝐾 + 𝑇∗ − 1 𝑛𝑆 − 1 𝑛𝑉 1 𝑛] −(𝑛+1) (26) this leads to the entropy of the polytropic gas as 𝑆 = [(−1) 𝑛 𝑛+1 ( 𝑇∗ 𝑇 ) 1 𝑛+1 − 1] 𝑛 𝑉 𝐾𝑛𝑇∗ (27) we know that entropy (𝑆) of a thermo dynamical system should be positive ie 𝑆 > 0 [24] here 𝑆 > 0 if 𝐾𝑛𝑇∗ > 0 now the thermal capacity at constant volume is 𝐶𝑉 = 𝑇 ( 𝜕𝑆 𝜕𝑇 ) 𝑉 = (−1) 2𝑛+1 𝑛+1 ( 𝑛 𝑛+1 ) 𝑆 [(−1) 𝑛 𝑛+1( 𝑇∗ 𝑇 ) 𝑛 𝑛+1 −1] ( 𝑇∗ 𝑇 ) 1 𝑛+1 (28) therefore, the condition 𝐶𝑉 > 0 is satisfied if 𝐾 𝑛 𝑇∗ > 0. thus both the conditions of thermo dynamic stability are satisfied. so the polytropic gas is thermo dynamically stable. thermodynamic behavior of the polytropic gas in cosmology 267 3. discussion we have studied the thermo dynamical behavior of the polytropic gas. here, we have considered the value of 𝑛 < 0 to study the whole work done in this article. some important results are given below: (i) as we have considered 𝑛 < 0 , the pressure goes more and more negative as volume increases. 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[24] h.b.callen. thermo dynamics and thermo statics. newwork. john wiley and sons, 1985. https://www.worldscientific.com/worldscinet/ijmpd microsoft word capitolo intero n 8.doc ratio mathematica 29 (2015) 3-14 issn:1592-7415 rough sets applied in sublattices and ideals of lattices r. ameri1, h. hedayati2,z. bandpey3 1school of mathematics, statistics and computer science, college of sciences, university of tehran, p.o.box 14155-6455, teheran, iran rameri@ut.ac.ir 2department of mathematics, faculty of basic science, university of mazandaran, babolsar, iran zeinab bandpey@yahoo.com 3department of mathematics, faculty of basic science, babol university of technology, babol, iran h.hedayati@nit.ac.ir abstract the purpose of this paper is the study of rough hyperlattice. in this regards we introduce rough sublattice and rough ideals of lattices. we will proceed by obtaining lower and upper approximations in these lattices. keywords: rough set, lower approximation, upper approximation, rough sublattice, rough ideal doi: 10.23755/rm.v29i1.18 1 introduction never in the history of mathematics has a mathematical theory been the object of such vociferous vituperation as lattice theory (for more details see[3, 13]). lattices are partially ordered sets in which least upper bounds and greatest lower bounds of any two elements exist. a lattice is a set on which two operations are defined, called join and meet and denoted by ∨ 3 r. ameri, h. hedayati and z. bandpey and ∧, which satisfy the idempotent, commutative and associative laws, as well as the absorption laws: a∨ (b∧a) = a, a∧ (b∨a) = a. lattices are better behaved than partially ordered sets lacking upper or lower bounds. the concept of rough set was originally proposed by pawlak [21, 22] as a formal tool for modeling and processing incomplete information in information systems. since then the subject has been investigated in many papers (see [20, 23, 24]). the theory of rough set is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. a key notion in pawlak rough set model is an equivalence relation. the equivalence classes are the building blocks for the construction of the lower and upper approximations. the lower approximation of a given set is the union of all the equivalence classes which are subsets of the set, and the upper approximation is the union of all the equivalence classes which have a non-empty intersection with the set. some authors, for example, bonikowaski [5], iwinski [15], and pomykala and pomykala [24] studied algebraic properties of rough sets. the lattice theoretical approach has been suggested by iwinski [15]. in this paper we concentrates on the relationship between rough sets and lattice theory. we introduce the notion of rough sublattices (resp. ideals) of lattices, and investigate some properties of lower and upper approximations in lattices. 2 preliminaries suppose that u is a non-empty set. a partition or classification of u is a family p of non-empty subsets of u such that each element of u is contained in exactly one element of p . recall that an equivalence relation on a set u is a reflexive, symmetric, and transitive binary relation on u. each partition p induces an equivalence relation θ on u by setting: xθy ⇔ x and y are in the same class of p . conversely, each equivalence relation θ on u induces a partition p of u whose classes have the form [x]θ = {y ∈ u | xθy}. given a non-empty universe u, by p(u) we will denote the power set on u. if θ is an equivalence relation on u then for every x ∈ u, [x]θ denotes the equivalence class of θ determined by x. for any x ⊆ u, we write xc to denote the complementation of x in u, that is the set u \x. definition 2.1. [8] a pair(u,θ); where u 6= ∅ and θ is an equivalence 4 rough sets applied in sublattices and ideals of lattices relation on u, is called an approximation space. definition 2.2. [8] for an approximation space (u,θ), by a rough approximation in (u,θ) we mean a mapping a : p(u) → p(u) × p(u) defined by for every x ∈ p(u), a(x) = (a(x),a(x)) where a(x) = {x ∈ x | [x]θ ⊆ x}, a(x) = {x ∈ x | [x]θ ∩ x 6= ∅}. a(x) is called a lower rough approximation of x in (u,θ), where as a(x) is called upper rough approximation of x in (u,θ). definition 2.3. [8] given an approximation space (u,θ) a pair (a,b) ∈ p(u) × p(u) is called a rough set in (u,θ) iff (a,b) = a(x) for some x ∈ p(u). for the sake of illustration, let (u,θ) is an approximation space, where: u = {x1,x2,x3,x4,x5,x6,x7,x8}, and an equivalence relation θ with the following equivalence classes: e1 = {x1,x4,x8}, e2 = {x2,x5,x7}, e3 = {x3}, e4 = {x6}, let x = {x3,x5}, then a(x) = {x3} and a(x) = {x2,x3,x5,x7} and so ({x3},{x2,x3,x5,x7}) = a(x) is a rough set. the reader will find in [18,21-25] a deep study of rough set theory. definition 2.4. [7] a subset xof u is called definable if a(x) = a(x). if x ⊆ u given by a predicate p and x ∈ u, then: 1. x ∈ a(x) means that x certainly has property p , 2. x ∈ a(x) means that x possibly has property p , 3. x ∈ u \a(x) means that x definitely does not have property p . when a(a) v a(b), we say that a(a) is a rough subset of a(b). thus in the case of rough sets a(a) and a(b), a(a) v a(b) if and only if a(a) ⊆ a(b) and a(a) ⊆ a(b). this property of rough inclusion has all the properties of set inclusion. the rough complement of a(a) denoted by ac(a) is defined by: ac(a) = (u \ a(a),u \ a(a)). also, we can define a(a) \a(b) as follows: a(a) \a(b) = a(a) uac(b) = (a(a) \a(b),a(a) \a(b)). let l be a lattice and s ⊆ l, if s is a lattice, then s is called a sublattice of l. a sublattice i is called an ideal of l, if a ∈ l and x ∈ i imply a∧x ∈ l 5 r. ameri, h. hedayati and z. bandpey (see[2]). let ρ be an equivalence relation on l and x, y, z ∈ l. (1) ρ is called a congruence relation if xρy implies (x ∨ z)ρ(y ∨ z) and (x∧z)ρ(y ∧z). (2) ρ is called a complete congruence relation if [x]ρ∨ [y]ρ = [x∨y]ρ, and [x]ρ ∧ [y]ρ = [x∧y]ρ. if ρ is a congruence relation on l, then it is easy to verify that [x]ρ∨[y]ρ ⊆ [x∨y]ρ, [x]ρ ∧ [y]ρ ⊆ [x∧y]ρ. 3 rough ideals of lattices throughout this paper l denotes a lattice. let ρ be an equivalence relation on l and x be a non-empty subset of l. when u = l and θ is the above equivalence relation, then we use the pair (l,ρ) instead of the approximation space (u,θ). also, in this case we use the symbols aρ(x) and aρ(x) instead of a(x) and a(x). proposition 3.1. for every approximation space (l,ρ), where ρ is an equivalence relation, and every subsets a, b ⊆ l, we have: (1) aρ(a) ⊆ a ⊆ aρ(a); (2) aρ(∅) = ∅ = aρ(∅); (3) aρ(l) = l = aρ(l); (4) if a ⊆ b, then aρ(a) ⊆ aρ(b), and aρ(a) ⊆ aρ(b); (5) aρ(aρ(a)) = aρ(a); (6) aρ(aρ(a)) = aρ(a); (7) aρ(aρ(a)) = aρ(a); (8) aρ(aρ(a)) = aρ(a); (9) aρ(a) = (aρ(a c))c; (10)aρ(a) = (aρ(a c))c; (11)aρ(a∩b) = aρ(a) ∩aρ(b); (12)aρ(a∩b) ⊆ aρ(a) ∩aρ(b) ; (13)aρ(a∪b) ⊇ aρ(a) ∪aρ(b); (14)aρ(a∪b) = aρ(a) ∪aρ(b); (15)aρ([x]ρ) = aρ([x]ρ) for all x ∈ l; proof. (15) aρ([x]ρ) = {y ∈ l | [y]ρ ⊆ [x]ρ} = [x]ρ, and aρ([x]ρ) = {y ∈ l | [y]ρ ∩ [x]ρ 6= ∅} = [x]ρ. hence aρ([x]ρ) = aρ([x]ρ). 6 rough sets applied in sublattices and ideals of lattices the other parts of the proof is similar to the [17, theorem 2.1] and [7, proposition 4.1]. 2 the following example shows that the converse of (12) and (13) in proposition 3.1 are not true. example 3.2. let l = {1, 2, ..., 8}, then (l,∧,∨) is a lattice, where ∀a,b ∈ l, a ∧ b = min{a,b}, a ∨ b = max{a,b}. let ρ be an equivalence relation on l with the following equivalence classes: [1]ρ = {1, 4, 8}, [2]ρ = {2, 5, 7}, [3]ρ = {3}, [6]ρ = {6}, and a = {3, 5, 7}, b = {2, 6}. then: aρ(a) = {3}, aρ(b) = {6}, aρ(a∪b) = {2, 3, 5, 6, 7}, aρ(a) = {2, 3, 5, 7}, aρ(b) = {2, 5, 6, 7}, aρ(a∩b) = ∅, and so aρ(a) ∩aρ(b) * aρ(a∩b), aρ(a∪b) * aρ(a) ∪aρ(b). corollary 3.3. for every approximation space (l,ρ), (i) for every a ⊆ l, aρ(a) and aρ(a) are definable sets, (ii) for every x ∈ l, [x]ρ is definable set. proof. it is immediately by proposition 3.1 (parts (5), (6), (7), (8) and (15)). 2 if a and b are non-empty subsets of l, let a∧b and a∨b denotes the following sets: a∧b = {a∧ b | a ∈ a,b ∈ b}, a∨b = {a∨ b | a ∈ a,b ∈ b}. proposition 3.4. let ρ be a complete congruence relation on l, and a, b non-empty subsets of l, then aρ(a) ∧aρ(b) = aρ(a∧b). proof. suppose z be any element of aρ(a) ∧ aρ(b), then z = a ∧ b for some a ∈ aρ(a), b ∈ aρ(b), hence [a]ρ ∩ a 6= ∅ and [b]ρ ∩ b 6= ∅ and so there exist x ∈ [a]ρ ∩ a and y ∈ [b]ρ ∩ b. therefore x ∧ y ∈ a ∧ b 7 r. ameri, h. hedayati and z. bandpey and x ∧ y ∈ [a]ρ ∧ [b]ρ = [a ∧ b]ρ hence [a ∧ b]ρ ∩ (a ∧ b) 6= ∅ and so aρ(a) ∧aρ(b) ⊆ aρ(a∧b). conversely, let x ∈ aρ(a∧b) then [x]ρ ∩ (a∧b) 6= ∅ hence there exists y ∈ [x]ρ and y ∈ a ∧ b and so y = a ∧ b for some a ∈ a and b ∈ b. now we have x ∈ [y]ρ = [a ∧ b]ρ = [a]ρ ∧ [b]ρ. then there exist x′ ∈ [a]ρ and y′ ∈ [b]ρ such that x = x′ ∧ y′. since a ∈ [x′]ρ ∩a and b ∈ [y′]ρ ∩b, hence x′ ∈ aρ(a) and y′ ∈ aρ(b), which yields that x = x′ ∧ y′ ∈ aρ(a) ∧ aρ(b) and so aρ(a∧b) ⊆ aρ(a) ∧aρ(b). 2 proposition 3.5. let ρ be a complete congruence relation on l, and a, b non-empty subsets of l, then aρ(a) ∨aρ(b) = aρ(a∨b). proof. the proof is similar to the proof of proposition 3.4, by considering the suitable modification by using the definition of a∨b. 2 proposition 3.6. let ρ be a complete congruence relation on l, and a, b non-empty subsets of l, then aρ(a) ∧aρ(b) ⊆ aρ(a∧b). proof. suppose x be any element of aρ(a) ∧ aρ(b) then x = a ∧ b for some a ∈ aρ(a) and b ∈ aρ(b). hence [a]ρ ⊆ a and [b]ρ ⊆ b. since [a∧b]ρ = [a]ρ∧[b]ρ ⊆ a∧b, we get a∧b ∈ aρ(a∧b) and so x ∈ aρ(a∧b). 2 the following example shows that the converse of proposition 3.6 is not true. example 3.7. let l = {0, 1, 2, ..., 11}, then (l,∧,∨) is a lattice, where ∀a,b ∈ l, a ∧ b = min{a,b}, a ∨ b = max{a,b}. let ρ be a complete congruence relation on l with the following equivalence classes: [0]ρ = {0, 1, 2}, [3]ρ = {3, 4, 5}, [6]ρ = {6, 7, 8}, [9]ρ = {9, 10, 11}, and a = {1, 3, 4, 5}, b = {0, 1, 2, 6, 8}. then: aρ(a) = {3, 4, 5}, aρ(b) = {0, 1, 2}, a∧b = {0, 1, 2, 3, 4, 5} aρ(a∧b) = {0, 1, 2, 3, 4, 5}, aρ(a) ∧aρ(b) = {0, 1, 2} and so aρ(a∧b) * aρ(a) ∧aρ(b). 8 rough sets applied in sublattices and ideals of lattices proposition 3.8. let ρ be a complete congruence relation on l, and a, b non-empty subsets of l, then aρ(a) ∨aρ(b) ⊆ aρ(a∨b). proof. the proof is similar to the proof of proposition 3.6, by considering the suitable modification by using the definition of a∨b. 2 the following example shows that aρ(a∨b) ⊆ aρ(a) ∨aρ(b) does not hold in general. example 3.9. let l = {0, 1, 2, ..., 8}, then (l,∧,∨) is a lattice, where ∀a,b ∈ l, a ∧ b = min{a,b}, a ∨ b = max{a,b}. let ρ be a complete congruence relation on l with the following equivalence classes: [0]ρ = {0, 1, 2}, [3]ρ = {3, 4}, [5]ρ = {5, 6, 7, 8}, and a = {3, 4, 5, 7}, b = {0, 1, 2, 3, 6, 8}. then: aρ(a) = {3, 4}, aρ(b) = {0, 1, 2}, a∨b = {3, 4, 5, 6, 7, 8}, aρ(a∨b) = {3, 4, 5, 6, 7, 8}, aρ(a) ∨aρ(b) = {3, 4}, and so aρ(a∨b) * aρ(a) ∨aρ(b) lemma 3.10. let ρ1 and ρ2 be two complete congruence relations on l such that ρ1 ⊆ ρ2 and let a be a non-empty subset of l, then: (i) aρ2 (a) ⊆ aρ1 (a), (ii) aρ1 (a) ⊆ aρ2 (a). proof. it is straightforward. 2 the following corollary follows from lemma 3.10. corollary 3.11. let ρ1 and ρ2 be two complete congruence relations on l and a a non-empty subset of l, then: (i) aρ1 (a) ∩aρ2 (a) ⊆ a(ρ1∩ρ2)(a), (ii) a(ρ1∩ρ2)(a) ⊆ aρ1 (a) ∩aρ2 (a). proposition 3.12. let ρ be a congruence relation on l, and j be an ideal of l, then aρ(j) is an ideal of l. 9 r. ameri, h. hedayati and z. bandpey proof. suppose a,b ∈ aρ(j) and r ∈ l, then [a]ρ∩j 6= ∅ and [b]ρ∩j 6= ∅. so there exist x ∈ [a]ρ ∩j and y ∈ [a]ρ ∩j. since j is an ideal of l, we have x∨y ∈ j and x∨y ∈ [a]ρ∨[b]ρ ⊆ [a∨b]ρ. hence [a∨b]ρ∩j 6= ∅ which implies a∨ b ∈ aρ(j). also, we have r ∧x ∈ j and r ∧x ∈ [r]ρ ∧ [a]ρ ⊆ [r ∧a]ρ. so [r∧a]ρ∩j 6= ∅ which implies r∧a ∈ aρ(j). therefore aρ(j) is an ideal of l. 2 similarly, if ρ is a congruence relation on l and j is a sublattice of l, then aρ(j) is a sublattice of l. proposition 3.13. let ρ be a complete congruence relation on l, and j be an ideal of l, then aρ(j) is an ideal of l. proof. suppose a, b ∈ aρ(j) and r ∈ l, then [a]ρ ⊆ j and [b]ρ ⊆ j. so [a∨ b]ρ = [a]ρ ∨ [b]ρ ⊆ j, and [r ∧a]ρ = [a]ρ ∧ [b]ρ ⊆ j. hence a∨ b ∈ aρ(j) and r ∧a ∈ aρ(j). 2 similarly, if ρ is a complete congruence relation on l and j is a sublattice of l, then aρ(j) is a sublattice of l. definition 3.14. let ρ be a congruence relation on l and aρ(a) = (aρ(a),aρ(a)) a rough set in the approximation space (l,ρ). if aρ(a) and aρ(a) are ideals (resp. sublattice) of l, then we call aρ(a) a rough ideal (resp. sublattice). note that a rough sublattice also is called a rough lattice. corollary 3.15. (i) let ρ, be a congruence relation on l, and i an ideal of l then aρ(i) is a rough ideals. (ii) let ρ be a complete congruence relation on l and j a sublattice of l, then aρ(j) is a rough lattice. proof. it is obtained by 3.12 and 3.13. 2 let l and l′ be two lattices, a map f : l → l′ is said to be homomor− phism or (lattice homomorphism) if for all a, b ∈ l, f(a∧ b) = f(a) ∧f(b), and f(a∨ b) = f(a) ∨f(b). now, let l and l′ be two lattices and f : l → l′ a homomorphism. it is well known, θ = {(a,b) ∈ l × l | f(a) = f(b)} ⊆ l × l is a congruence relation on l. because if aθb then f(a) = f(b) and for all z ∈ l, we have f(a∧ z) = f(a) ∧f(z) = f(b) ∧f(z) = f(b∧ z). therefor (a∧ z) θ (b∧ z), and similarly (a∨z) θ (b∨z). 10 rough sets applied in sublattices and ideals of lattices theorem 3.16. let l and l′ be two lattices and f : l → l′ a homomorphism. if a is a non-empty subset of l, then f(aθ(a)) = f(a). proof. since a ⊆ aθ(a) it follows that f(a) ⊆ f(aθ(a)). conversely, let y ∈ f(aθ(a)). then there exists an element x ∈ aθ(a), such that f(x) = y, so we have [x]θ ∩a 6= ∅. thus there exists an element a ∈ [x]θ∩a. then a ∈ [x]θ, hence xθa, and so f(x) = f(a) ∈ f(a), therefore f(aθ(a)) ⊆ f(a). 2 let f : l → l′ be a homomorphism and a a subset of l, since aθ(a) ⊆ a it follows that f(aθ(a)) ⊆ f(a). but the following example shows that, in general, f(aθ(a)) 6= f(a). example 3.17. let (l,∧,∨) and (l′,∧,∨) be two lattices where l = {1, 2, 3, 4}; and l′ = {5, 6, 7}; and for all s, t in l or l′, s∧t= min{s,t} and s∨ t = max{s,t}. the map f : l → l′ given by f(4) = f(3)= 7, f(2) = 6, f(1) = 5, is a homomorphism. we have θ = {3, 4}. suppose a = {1, 2}, then f(a) = {5, 6}, aθ(a) = ∅ and f(aθ(a)) = ∅, and so f(aθ(a)) 6= f(a). the lower and upper approximations can be presented in an equivalent form as follows: let l be a lattice, ρ a congruence relation on l, and a a non-empty subset of l. then we define ∨ and ∧ on l/ρ = {[x]ρ | x ∈ l}, by [x]ρ∨[y]ρ = [x∨y]ρ, [x]ρ∧[y]ρ = [x∧y]ρ. this relation is well-defined, since if [x1]ρ = [x2]ρ and [y1]ρ = [y2]ρ, then x1ρx2 and y1ρy2. since ρ is a congruence relation we have (x1 ∨y1)ρ(x2 ∨y1) and (x2 ∨ y1)ρ(x2 ∨ y2). then (x1 ∨ y1)ρ(x2 ∨ y2), so [x1 ∨ y1]ρ = [x2 ∨ y2]ρ. therefore [x1]ρ∨[y1]ρ = [x2]ρ∨[y2]ρ. it is easy to see that (l/ρ,∨,∧), is a lattice. also if a 6= ∅, and a ⊆ l put a ρ (a) = {[x]ρ ∈ l/ρ | [x]ρ ⊆ a} and aρ(a) = {[x]ρ ∈ l/ρ | [x]ρ ∩a 6= ∅}. proposition 3.18. let ρ be a congruence relation on l and j be an ideal of l, then aρ(j) is an ideal of l/ρ. proof. assume that [a]ρ, [b]ρ ∈ aρ(j) and [r]ρ ∈ l/ρ. then [a]ρ∩j 6= ∅ and [b]ρ∩j 6= ∅, so there exist x ∈ [a]ρ∩j and y ∈ [b]ρ∩j. since j is an ideal of l, we have x∨y ∈ j and r∧x ∈ j. also, we have x∨y ∈ [a]ρ∨[b]ρ ⊆ [a∨b]ρ, and r∧x ∈ [r]ρ∧[a]ρ ⊆ [r∧a]ρ. therefore [a∨b]ρ∩j 6= ∅ and [r∧a]ρ∩j 6= ∅, 11 r. ameri, h. hedayati and z. bandpey which imply [a]ρ ∨ [b]ρ ∈ aρ(j) and [r]ρ ∧ [a]ρ ∈ aρ(j). therefore aρ(j) is an ideal of l/ρ. 2 proposition 3.19. let ρ be a complete congruence relation on l and j be an ideal of l, then a ρ (j) is an ideal of l/ρ. proof. assume that [a]ρ, [b]ρ ∈ aρ(j) and [r]ρ ∈ l/ρ. then [a]ρ ⊆ j and [b]ρ ⊆ j. since j is an ideal of l, we have a ∨ b ∈ j and r ∧ a ∈ j therefore [a]ρ ∨ [b]ρ = [a ∨ b]ρ ⊆ j ∨ j = j, and [r]ρ ∧ [a]ρ = [r ∧ a]ρ ⊆ j, which imply [a]ρ ∨ [b]ρ ∈ aρ(j) and [r]ρ ∧ [a]ρ ∈ aρ(j). therefore aρ(j) is an ideal of l/ρ. 2 proposition 3.20. (i) let ρ be a congruence relation on l and j a sublattice of l, then aρ(j) is a sublattice of l/ρ. (ii) let ρ be a complete congruence relation on l and j a sublattice of l, then a ρ (j) is a sublattice of l/ρ. proof. similar to the proof of propositions 3.13, 3.18 and 3.19. 2 acknowledgment the first author partially has been supported by the ”algebraic hyperstructures excellence, tarbiat modares university, tehran, iran” and ”research center in algebraic hyperstructures and fuzzy mathematics, university of mazandaran, babolsar, iran”. references [1] r. ameri, approximations in (bi-)hyperideals of semihypergroups, iranian journal of science and technology, ijst, 37a4 (2013) 527-532. [2] r. balbes, p. dwingel, distributive lattices, university of missouri press, columbia, (1974). 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[11] b. davvaz, fuzzy sets and probabilistic rough sets, int. j. sci. technol. univ. kashan 1 (1)(2000) 23-29. [12] d. dubois, h. prade, rough fuzzy sets and fuzzy rough sets, int. j. general syst. 17 (2-3)(1990) 191-209. [13] g. grätzer, general lattice theory, academic prees, new york, (1978). [14] j. hashimoto, ideal theory of lattices, math. japon, 2, (1952), 149-186. [15] t. iwinski, algebraic approach to rough sets, bull. polish acad. sci. math. 35 (1987) 673-683. [16] y. b. jun, roughness of ideals in bck-algebras, sci. math. jpn. 57 (1) (2003) 165-169. [17] n. kuroki, rough ideals in semigroups, inform. sci. 100 (1997) 139-163. [18] n. kuroki, j. n. mordeson, structure of rough sets and rough groups, j. fuzzy math. 5 (1)(1997) 183-191. [19] n. kuroki, p. p. wang, the lower and upper approximations in a fuzzy group, inform. sci. 90 (1996) 203-220. 13 r. ameri, h. hedayati and z. bandpey [20] s. nanda, s. majumdar, fuzzy rough sets, fuzzy sets syst. 45 (1992) 157-160. 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[25] j. a. pomykala, the stone algebra of rough sets, bull. polish acad. sci. math. 36(1988) 495-508. 14 ratio mathematica volume 44, 2022 connected 𝟐 − dominating sets and connected 𝟐 − domination polynomials of the complete bipartite graph 𝒌𝟐,𝒎. y. a. shiny1 t. anithababy2 abstract let 𝐺 = (𝑉, 𝐸) be a simple graph. let 𝐷𝑐2 (𝐺, 𝑗) be the family of connected 2−dominating sets in 𝐺 with cardinality 𝑗 and 𝑑𝑐2 (𝐺, 𝑗) = |𝐷𝑐2 (𝐺, 𝑗)|. then the polynomial 𝐷𝑐2 (𝐺, 𝑥) = ∑ 𝑑𝑐2 (𝐺, 𝑗)𝑥 𝑗 , |𝑉(𝐺)| 𝑗=𝛾𝑐2 (𝐺) is called the 2−domination polynomial of 𝐺 where 𝛾𝑐2 (𝐺) is the connected 2− domination number of 𝐺.let 𝐷𝑐2 (𝑘2,𝑚,𝑗) be the family of connected 2−dominating sets of the complete bipartite graph 𝑘2,𝑚 with cardinality 𝑗 and let 𝑑𝑐2 (𝑘2,𝑚,𝑗) = |𝐷𝑐2 (𝑘2,𝑚,𝑗) |. then the connected 2− domination polynomial 𝐷𝑐2 (𝑘2,𝑚,𝑥) of 𝑘2,𝑚 is defined as 𝐷𝑐2 (𝑘2,𝑚,𝑥) = ∑ 𝑑𝑐2 (𝑘2,𝑚,𝑗)𝑥 𝑗 , |𝑉(𝑘2,𝑚)| 𝑗=𝛾𝑐2 (𝑘2,𝑚) where 𝛾𝑐2 (𝑘2,𝑚,𝑗) is the connected 2 – domination number of 𝑘2,𝑚,. in this paper, we obtain a recursive formula for 𝑑𝑐2 (𝑘2,𝑚,𝑗).using this recursive formula, we construct the connected 2−domination polynomial 𝐷𝑐2 (𝑘2,𝑚,𝑥) = ∑ 𝑑𝑐2 (𝑘2,𝑚,𝑗)𝑥 𝑗 , where𝑚+2𝑗=3 𝑑𝑐2 (𝑘2,𝑚,𝑗) is the number of connected 2−dominating sets of 𝑘2,𝑚 of cardinality 𝑗 and some properties of this polynomial have been studied. keywords: dominating, connected and cardinality 2010 mathematical classification number: 05c69, 54d053. 1reg. no.: 19213042092006, research scholar (full time), research department of mathematics, women’s christian college, nagercoil. affiliated by manonmaniam sundaranar university, tamil nadu, india, shinyjebalin@gmail.com 2assistant professor, research department of mathematics, women’s christian college, nagercoil. affiliated by manonmaniam sundaranar university, tamil nadu, india. anithasteve@gmail.com 3received on june 7th, 2022. accepted on aug 10th, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.889. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 51 mailto:anithasteve@gmail.com y. a. shiny and t. anithababy 1. introduction let g = (v, e) be a simple graph of order, |v| = m. for any vertex vϵv, the open neighbourhood of v is the set n(v) = {uϵv/uvϵe} and the closed neighbourhood of v is the set 𝑁[𝑣] = 𝑁(𝑣) ∪ {𝑣}. for a set 𝑆 ⊆ 𝑉, the open neighbourhood of 𝑆 is 𝑁(𝑆) = 𝑈𝑣𝜖𝑠𝑁(𝑣) and the closed neighbourhood of s is 𝑁(𝑆) ∪ 𝑆.a set 𝐷 ⊆ 𝑉 is a dominating set of 𝐺, if 𝑁[𝐷] = 𝑉 or equivalently, every vertex in 𝑉 − 𝐷 is adjacent to atleast one vertex in 𝐷.the domination number of a graph 𝐺 is defined as the cardinality of a minimum dominating set 𝐷 of vertices in 𝐺 and is denoted by 𝛾(𝐺). a dominating set 𝐷 of 𝐺 is called a connected dominating set if the induced sub-graph < 𝐷 > is connected. the connected domination number of a graph 𝐺 is defined as the cardinality of a minimum connected dominating set 𝐷 of vertices in 𝐺 and is denoted by 𝛾𝑐 (𝐺). a graph 𝐺 = (𝑉, 𝐸)is called a bipartite graph if its vertices 𝑉 can be partitioned into two subsets 𝑉1 and 𝑉2 such that each edge of 𝐺 connects a vertex of 𝑉 1to a vertex of 𝑉2. if 𝐺 contains every edge joining a vertex of 𝑉1 and a vertex of 𝑉2 then 𝐺 is called a complete bipartite graph. it is denoted by 𝑘𝑚,𝑛, where 𝑚 and 𝑛 are the numbers of vertices in 𝑉1 and 𝑉2 respectively. let 𝑘2,𝑚 be the complete bipartite graph with 𝑚 + 2 vertices. throughout this paper let us take 𝑉(𝑘2,𝑚) = {𝑣1,𝑣2,𝑣3,...,𝑣𝑚+1,𝑣𝑚+2} and 𝐸(𝑘2,𝑚) = {(𝑣1, 𝑣3),( 𝑣1, 𝑣4), (𝑣1,𝑣5),…, (𝑣1, 𝑣𝑚+1), (𝑣1, 𝑣𝑚+2), (𝑣2, 𝑣3),( 𝑣2, 𝑣4), ( 𝑣2, 𝑣5),…, (𝑣2, 𝑣𝑚+1), (𝑣2, 𝑣𝑚+2). as usual we use ( 𝑚 𝑗 ) for the combination 𝑚 to 𝑗. also, we denote the set {1, 2, … … … ,2𝑚 − 1, 2𝑚} by [2𝑚], throughout this paper. 2. connected 2 – dominating sets of the complete bipartite graph 𝒌𝟐,𝒎 in this section, we state the connected 2 – domination number of the complete bipartite graph k2,m and some of its properties. definition 2.1. let 𝐺 be a simple graph of order 𝑚 with no isolated vertices. a subset 𝐷 ⊆ 𝑉 is a 2− dominating set of the graph 𝐺 if every vertex 𝑣 𝜖 𝑉 − 𝐷 is adjacent to atleast two vertices in 𝐷. a 2− dominating set is called a connected 2− dominating set if the induced subgraph <𝐷> is connected. definition 2.2. the cardinality of a minimum connected 2 – dominating sets of 𝐺 is called the connected 2 – domination number of 𝐺 and is denoted by 𝛾𝐶2 (𝐺). lemma 2.3 for all 𝑚 ∈ 𝑧+, ( 𝑚 𝑗 ) = 0 if 𝑗 > 𝑚 or 𝑗 < 0. theorem 2.4 𝑑𝑐2(𝑘2,𝑚, 𝑗) = {( 𝑚 + 2 𝑗 ) − ( 𝑚 + 1 𝑗 ) − ( 𝑚 𝑗 − 1) for 3 ≤ 𝑗 ≤ 𝑚 + 2 52 connected 2 − dominating sets and connected 2 − domination polynomials of the complete bipartite graph 𝑘2,𝑚. proof: let the partite sets of 𝑘2,𝑚 be 𝑉1 ={𝑣1,𝑣2,} and 𝑉2 = {𝑣3,𝑣4,,...,𝑣𝑚+1,𝑣𝑚+2}. since the subgraph induced by the vertex set as {𝑣1,𝑣2, } is not connected, every connected 2 −dominating set of 𝑘2,𝑚 must contain the vertex {𝑣1} or {𝑣2} or {𝑣1,𝑣2,}. when 3≤ 𝑗 ≤ 𝑚, every connected 2 −dominating set must contain {𝑣1,𝑣2,}. since, |𝑉(𝑘2,𝑚) | = 𝑚 + 2, 𝑘2,𝑚 contains ( 𝑚 + 2 𝑗 ) number of subsets of cardinality 𝑗. since, the subgraphs induced by {𝑣1,𝑣2,} and {𝑣3,𝑣4,,...,𝑣𝑚+1,𝑣𝑚+2} are not connected, each time ( 𝑚 + 1 𝑗 ) number of subsets of 𝑘2,𝑚 of cardinality j and ( 𝑚 𝑗 − 1) number of subsets of 𝑘2,𝑚 of cardinality 𝑗 − 1 are not connected 2 −dominating sets. hence, 𝑘2,𝑚 contains ( 𝑚 + 2 𝑗 ) − ( 𝑚 + 1 𝑗 ) − ( 𝑚 𝑗 − 1) number of subsets of connected 2 −dominating sets, when 3≤ 𝑗 ≤ 𝑚. therefore, 𝑑𝑐2(𝑘2,𝑚, 𝑗) = ( 𝑚 + 2 𝑗 ) − ( 𝑚 + 1 𝑗 ) − ( 𝑚 𝑗 − 1) for all 3≤ 𝑗 ≤ 𝑚. when the cardinality is 𝑚 + 1, every subset of 𝑘2,𝑚 containing {𝑣1} or {𝑣2} are connected 2 −dominating sets. therefore, two more sets are connected 2 −dominating sets when the cardinality is 𝑚 + 1 . hence, 𝑑𝑐2(𝑘2,𝑚, 𝑗) = ( 𝑚 + 2 𝑗 ) − ( 𝑚 + 1 𝑗 ) − ( 𝑚 𝑗 − 1) + 2 , when 𝑗 = 𝑚 + 1. since, there is only one subset of 𝑘2,𝑚 with cardinality 𝑚 + 2 and that set is a connected 2 −dominating set. we get 𝑑𝑐2(𝑘2,𝑚, 𝑗) = ( 𝑚 + 2 𝑗 ) when 𝑗 = 𝑚 + 2. theorem 2.5. let 𝑘2,𝑚 be the complete bipartite graph with 𝑚 ≥ 3. then (i) 𝑑𝑐2(𝑘2,𝑚, 𝑗) = 𝑑𝑐2(𝑘2,𝑚−1, 𝑗) + 𝑑𝑐2(𝑘2,𝑚−1, 𝑗 − 1) (ii) 𝑑𝑐2(𝑘2,𝑚, 𝑗) = 𝑑𝑐2(𝑘2,𝑚−1, 𝑗) + 1 if 𝑗 = 3. (iii) 𝑑𝑐2(𝑘2,𝑚, 𝑗) = 𝑑𝑐2(𝑘2,𝑚−1, 𝑗) + 𝑑𝑐2(𝑘2,𝑚−1, 𝑗 − 1) − 2 if 𝑗 = 𝑚. proof: (i) by theorem 2.4, we have, 𝑑𝑐2(𝑘2,𝑚, 𝑗) = ( 𝑚 + 2 𝑗 ) − ( 𝑚 + 1 𝑗 ) − ( 𝑚 𝑗 − 1) for all 3 ≤ 𝑗 ≤ 𝑚 + 2. 𝑑𝑐2(𝑘2,𝑚−1, 𝑗) = ( 𝑚 + 1 𝑗 ) − ( 𝑚 𝑗 ) − ( 𝑚 − 1 𝑗 − 1 ) 𝑑𝑐2(𝑘2,𝑚−1, 𝑗 − 1) = ( 𝑚 + 1 𝑗 − 1 ) − ( 𝑚 𝑗 − 1) − ( 𝑚 − 1 𝑗 − 2 ) . consider, 𝑑𝑐2(𝑘2,𝑚−1, 𝑗) + 𝑑𝑐2(𝑘2,𝑚−1, 𝑗 − 1) = ( 𝑚 + 1 𝑗 ) − ( 𝑚 𝑗 ) − ( 𝑚 − 1 𝑗 − 1 ) + ( 𝑚 + 1 𝑗 − 1 ) − ( 𝑚 𝑗 − 1) − ( 𝑚 − 1 𝑗 − 2 ) . = ( 𝑚 + 1 𝑗 ) + ( 𝑚 + 1 𝑗 − 1 ) − [( 𝑚 𝑗 ) + ( 𝑚 𝑗 − 1)] – [( 𝑚 − 1 𝑗 − 1 ) + ( 𝑚 − 1 𝑗 − 2 )] 53 y. a. shiny and t. anithababy = ( 𝑚 + 2 𝑗 ) − ( 𝑚 + 1 𝑗 ) − ( 𝑚 𝑗 − 1). = 𝑑𝑐2(𝑘2,𝑚, 𝑗) . therefore, 𝑑𝑐2(𝑘2,𝑚, 𝑗) = 𝑑𝑐2(𝑘2,𝑚−1, 𝑗) + 𝑑𝑐2(𝑘2,𝑚−1, 𝑗 − 1) for all 4≤ 𝑗 ≤ 𝑚 + 2 and 𝑗 ≠ 𝑚. (i) when 𝑗 = 3, 𝑑𝑐2(𝑘2,𝑚, 3) = ( 𝑚 + 2 3 ) − ( 𝑚 + 1 3 ) − ( 𝑚 2 ) by theorem 2.4 = ( 𝑚 + 1 2 ) − ( 𝑚 2 ) = ( 𝑚 1 ) consider, 𝑑𝑐2(𝑘2,𝑚−1, 3) = ( 𝑚 + 1 3 ) − ( 𝑚 3 ) − ( 𝑚 − 1 2 ) = ( 𝑚 2 ) − ( 𝑚 − 1 2 ) = ( 𝑚 − 1 1 ) = 𝑚 − 1. that is, 𝑑𝑐2(𝑘2,𝑚−1, 3) = 𝑑𝑐2(𝑘2,𝑚, 3) − 1. therefore, 𝑑𝑐2(𝑘2,𝑚 , 3) = 𝑑𝑐2(𝑘2,𝑚−1, 3) + 1. hence, 𝑑𝑐2(𝑘2,𝑚, 𝑗) = 𝑑𝑐2(𝑘2,𝑚−1, 𝑗) + 1 𝑖𝑓 𝑗 = 3. (ii) when 𝑗 = 𝑚, 𝑑𝑐2(𝑘2,𝑚, 𝑚) = ( 𝑚 + 2 𝑚 ) − ( 𝑚 + 1 𝑚 ) − ( 𝑚 𝑚 − 1 ), by theorem 2.2. = ( 𝑚 + 1 𝑚 − 1 ) − ( 𝑚 𝑚 − 1 ). = ( 𝑚 𝑚 − 2 ). consider, 𝑑𝑐2(𝑘2,𝑚−1, 𝑚) + 𝑑𝑐2(𝑘2,𝑚−1, 𝑚 − 1) = ( 𝑚 + 1 𝑚 ) − ( 𝑚 𝑚 ) − ( 𝑚 − 1 𝑚 − 1 ) + 2 + ( 𝑚 + 1 𝑚 − 1 ) − ( 𝑚 𝑚 − 1 ) − ( 𝑚 − 1 𝑚 − 2 ) = ( 𝑚 + 1 𝑚 ) − ( 𝑚 + 1 𝑚 − 1 ) − [( 𝑚 𝑚 ) + ( 𝑚 𝑚 − 1 )] – [( 𝑚 − 1 𝑚 − 1 ) + ( 𝑚 − 1 𝑚 − 2 )] +2 = ( 𝑚 + 2 𝑚 ) − ( 𝑚 + 1 𝑚 ) − ( 𝑚 𝑚 − 1 ) + 2 = ( 𝑚 + 1 𝑚 − 1 ) − ( 𝑚 𝑚 − 1 ) + 2 = ( 𝑚 𝑚 − 2 ) + 2. = 𝑑𝑐2(𝑘2,𝑚, 𝑚) + 2. therefore, 𝑑𝑐2(𝑘2,𝑚 , 𝑚) = 𝑑𝑐2(𝑘2,𝑚−1, 𝑚) + 𝑑𝑐2(𝑘2,𝑚−1, 𝑚 − 1) + 2. hence, 𝑑𝑐2(𝑘2,𝑚, 𝑗) = 𝑑𝑐2(𝑘2,𝑚−1, 𝑗) + 𝑑𝑐2(𝑘2,𝑚−1, 𝑗 − 1) − 2 when 𝑗 = 𝑚. 3. connected 2 −domination polynomials of the complete bipartite graph 𝒌𝟐,𝒎. definition 3.1. let 𝑑𝑐2(𝑘2,𝑚, 𝑗) be the number of connected 2 –dominating sets of the complete bipartite graph 𝑘2,𝑚 with cardinality 𝑗.then, the connected 2 − domination 54 connected 2 − dominating sets and connected 2 − domination polynomials of the complete bipartite graph 𝑘2,𝑚. polynomial of 𝑘2,𝑚 is defined as 𝐷𝑐2 (𝑘2,𝑚,𝑥) = ∑ 𝑑𝑐2 (𝑘2,𝑚,j)𝑥 𝑗 , |𝑉(𝑘2,𝑚)| 𝑗=𝛾𝑐2 (𝑘2,𝑚) where 𝛾𝑐2 (𝑘2,𝑚) is the connected 2 – domination number of 𝑘2,𝑚. remark 3.2 𝛾𝑐2 (𝑘2,𝑚) =3. proof. let 𝑘2,𝑚 be the complete bipartite graph with partite sets 𝑉1 ={𝑣1,𝑣2} and 𝑉2 = {𝑣3,𝑣4,,...,𝑣𝑚+1,𝑣𝑚+2}. let 𝑣1,𝑣2𝜖 v (𝑘2,𝑚) and 𝑣1,𝑣2 are adjacent to all the other vertices 𝑣3,𝑣4,...,𝑣𝑚+1,𝑣𝑚+2 of 𝑘2,𝑚. also since, 𝑣1 and 𝑣2 are not connected, every connected 2 –dominating set must contain the vertices 𝑣1,𝑣2 and one more vertex from {𝑣3, 𝑣4,...,𝑣𝑚+1,𝑣𝑚+2}. therefore, the minimum cardinality is 3. hence, 𝛾𝑐2 (𝑘2,𝑚) =3. theorem 3.3 let 𝑘2,𝑚 be the complete bipartite graph with 𝑚 ≥ 3. then 𝐷𝑐2(𝑘2,𝑚,𝑥)=(1 + 𝑥) 𝐷𝑐2(𝑘2,𝑚−1,𝑥)+𝑥 3 − 2𝑥𝑚. proof: from the definition of connected 2 − domination polynomial, we have, 𝐷𝑐2(𝑘2,𝑚,𝑥) = ∑ 𝑑𝑐2 (𝑘2,𝑚,j)𝑥 𝑗𝑚+2 𝑗=3 . = 𝑑𝑐2(𝑘2,𝑚, 3)𝑥 3 + ∑ 𝑑𝑐2 (𝑘2,𝑚,j)𝑥 𝑗𝑚−1 𝑗=4 + 𝑑𝑐2 (𝑘2,𝑚,𝑚)𝑥 𝑚 + ∑ 𝑑𝑐2 (𝑘2,𝑚,j)𝑥 𝑗𝑚+2 𝑗=𝑚+1 . = [𝑑𝑐2 (𝑘2,𝑚−1,3) + 1]𝑥 3 + ∑ [𝑑𝑐2 (𝑘2,𝑚−1,j) 𝑚+2 𝑗=4 + 𝑑𝑐2 𝑘2,𝑚−1, 𝑗 − 1]𝑥 𝑗 +[𝑑𝑐2 (𝑘2,𝑚−1,𝑚) + 𝑑𝑐2 (𝑘2,𝑚−1,𝑚 − 1)−2]𝑥 𝑚 , by theorem 2.5 = 𝑑𝑐2(𝑘2,𝑚−1, 3)𝑥 3 + 𝑥3 + ∑ 𝑑𝑐2 (𝑘2,𝑚−1,j) 𝑚+2 𝑗=4 𝑥 𝑗 + ∑ 𝑑𝑐2 (𝑘2,𝑚−1,j − 1) 𝑚+2 𝑗=4 𝑥 𝑗 − 2𝑥𝑚. = ∑ 𝑑𝑐2 (𝑘2,𝑚−1,j) 𝑚+2 𝑗=3 𝑥 𝑗 +𝑥 ∑ 𝑑𝑐2 (𝑘2,𝑚−1,j − 1) 𝑚+2 𝑗=4 𝑥 𝑗−1 + 𝑥3 − 2𝑥𝑚. = 𝐷𝑐2(𝑘2,𝑚−1,𝑥) +𝑥𝐷𝑐2(𝑘2,𝑚−1,𝑥)+ 𝑥 3 − 2𝑥𝑚 . hence, 𝐷𝑐2(𝑘2,𝑚,𝑥) = (1 + 𝑥)𝐷𝑐2(𝑘2,𝑚−1,𝑥) +𝑥 3 − 2𝑥𝑚, for every 𝑚 ≥ 3. example 3.4 let 𝑘2,7 be the complete bipartite graph with order 9 as given in figure 2.1. 𝑘2,7 : figure 2.1 𝐷𝑐2 (𝐾2,6,𝑥) = 6𝑥 3 + 15𝑥4+20𝑥5 + 15𝑥6 + 8𝑥7 + 𝑥8. 55 y. a. shiny and t. anithababy by theorem 3.3, we have, 𝐷𝑐2 (𝐾2,7,𝑥) = (1 + 𝑥)( 6𝑥 3+15𝑥4+20𝑥5+15𝑥6+8𝑥7+𝑥8)+ 𝑥3 − 2𝑥7. = 6𝑥3+15𝑥4+20𝑥5+15𝑥6+8𝑥7+𝑥8 +6𝑥4+15𝑥5+20𝑥6+15𝑥7+8𝑥8+𝑥9 + 𝑥3 − 2𝑥7. = 7𝑥3+21𝑥4+35𝑥5+35𝑥6+21𝑥7+9𝑥8+𝑥9 we obtain 𝑑𝑐2 (𝑘2,𝑚,𝑗) for 3≤ 𝑚 ≤ 15 and 3≤ 𝑗 ≤ 15 as shown in table 1. 𝑗 𝑚 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 3 0 3 5 1 4 0 4 6 6 1 5 0 5 10 10 7 1 6 0 6 15 20 15 8 1 7 0 7 21 35 35 21 9 1 8 0 8 28 56 70 56 28 10 1 9 0 9 36 84 126 126 84 36 11 1 10 0 10 45 120 210 252 210 120 45 12 1 11 0 11 55 165 330 462 462 330 165 55 13 1 12 0 12 66 220 495 792 924 792 495 220 66 14 1 13 0 13 78 286 715 1287 1716 1716 1287 715 286 78 15 1 14 0 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 16 1 15 0 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 107 17 1 table 1.𝑑𝑐2 (𝑘2,𝑚,𝑗), the number of connected 2− dominating sets of 𝑘2,𝑚 with cardinality 𝑗. in the following theorem, we obtain some properties of 𝑑𝑐2 (𝐾2,𝑚,𝑗). theorem: 3.5 the following properties hold for the coefficients of 𝐷𝑐2 (𝐾2,𝑚,𝑥) for all m. (i) 𝑑𝑐2 (𝑘2,𝑚,3) = 𝑚, for all 𝑚 ≥ 3. (ii) 𝑑𝑐2 (𝑘2,𝑚,𝑚 + 2) = 1, for all 𝑚 ≥ 3. (iii) 𝑑𝑐2 (𝑘2,𝑚,𝑚 + 1) = 𝑚 + 2 , for all 𝑚 ≥ 3. (iv) 𝑑𝑐2 (𝑘2,𝑚,𝑚) = ( 𝑚 + 2 2 ) − ( 𝑚 + 1 1 ) − 𝑚. (v) 𝑑𝑐2 (𝑘2,𝑚,𝑚 − 1) = ( 𝑚 + 2 3 ) − ( 𝑚 + 1 2 ) − ( 𝑚 2 ) , for all 𝑚 ≥ 4. (vi) 𝑑𝑐2 (𝑘2,𝑚,𝑚 − 2) = ( 𝑚 + 2 4 ) − ( 𝑚 + 1 3 ) − ( 𝑚 3 ) , for all 𝑚 ≥ 5. (vii) 𝑑𝑐2 (𝑘2,𝑚,𝑚 − 3) = ( 𝑚 + 2 5 ) − ( 𝑚 + 1 4 ) − ( 𝑚 4 ) , for all 𝑚 ≥ 6. 56 connected 2 − dominating sets and connected 2 − domination polynomials of the complete bipartite graph 𝑘2,𝑚. (viii) 𝑑𝑐2 (𝑘2,𝑚,𝑚 − 𝑖) = ( 𝑚 + 2 𝑖 + 2 ) − ( 𝑚 + 1 𝑖 + 1` ) − ( 𝑚 𝑖 ) , for all 𝑚 ≥ 4 and 𝑖 ≥ 1. proof: (i) 𝑑𝑐2 (𝑘2,𝑚,3) = 𝑚. we prove this by induction on m. when 𝑚 = 3 , 𝑑𝑐2 (𝑘2,𝑚,3) = 3. therefore, the result is true for 𝑚 = 3. now, suppose that the result is true for all numbers less than ‘m’ and we prove it for m. by theorem 2.6, 𝑑𝑐2 (𝑘2,𝑚,3) = 𝑑𝑐2 (𝑘2,𝑚−1,3) + 1 = 𝑚 − 1 + 1 = 𝑚. (ii) 𝑑𝑐2 (𝑘2,𝑚,𝑚 + 2) = 1, for all 𝑚 ≥ 3. since, there is only one connected 2− dominating set of cardinalities 𝑚 + 2, 𝑑𝑐2 (𝑘2,𝑚,𝑚 + 2) = 1. (iii) 𝑑𝑐2 (𝑘2,𝑚,𝑚 + 1) = 𝑚 + 2 , for all m≥4. since, 𝑑𝑐2 (𝑘2,𝑚,𝑚 + 1) = {[𝑚 + 2]−𝑥/𝑥𝜀[𝑚 + 2]},we have the result. (iv), (v), (vi), (vii) and (viii) follows from theorem 2.4. 4. conclusion in this paper, the connected 2− domination polynomials of the complete bipartite graph 𝐾2,𝑚 has been derived by identifying its connected 2− dominating sets. it also helps us to characterize the connected. connected 2− dominating sets of cardinality j. we can generalize this study to any of complete bipartite graph 𝐾𝑛,𝑚 and some interesting properties can be obtained. references [1] alikhani. s, peng. y. h.” introduction to domination polynomial of a graph,” ars combinatoria, available as arxiv:0905.2251|v| [math.co]14 may 2009. [2] alikhani. s, and hamzeh torabi,” on domination polynomials of complete partite graphs” world applied sciences journal,9(1):23−24,2010. [3] sahib. sh. kahat. abdul jalil m. khalaf and roslan hasni,” dominating sets and domination polynomials of wheels”. asian journal of applied sciences (issn:2321− 0893), volume 02− issue 03, june 2014. [4] sahib shayyal kahat, abdul jalil m. khalaf and roslan hasni,” dominating sets and domination polynomials of stars”. australian journal of basics and applied science,8(6) june 2014, pp 383−386. [5] a. vijayan, t. anithababy, g. edwin,” connected total dominating sets and 57 y. a. shiny and t. anithababy connected total domination polynomials of stars and wheels”, iosr journal of mathematics, volume ii, pp 112−121. [6] a. vijayan, t. anithababy, g. edwin, “connected total dominating sets and connected total domination polynomials of fan graphs 𝐹2,𝑛". international journal of mathematical sciences and engineering applications (ijmsea), vol.10, no.1(april 2016), pp.135−146, issn:0473 −9424. 58 ratio mathematica volume 37,2019, pp. 85-109 legendre wavelet expansion of functions and their approximations shyam lal∗ indra bhan † abstract in this paper, nine new legendre wavelet estimators of functions having bounded third and fourth derivatives have been obtained.these estimators are new and best approximation in wavelet analysis. legendre wavelet estimator of a function f of bounded higher order derivatives is better and sharper than the estimator of a function f of bounded less order derivative. keywords : legendre wavelet, legendre wavelet expansion, orthonormal basis,legendre wavelet approximation . mathematics subject classification:42c40, 65t60, 65l10, 65l60, 65r20. 1 1 introduction several researchers have determined the approximation of a functions by trigonometric polynomials in fourier analysis. in fourier analysis, a function can be represented generally in one fourier series. in wavelet analysis, a function can be expanded in many wavelet series corresponding to different wavelets. this is an advantage of wavelet analysis. there is no such advantage in fourier analysis.thus a signal can be represented by several wavelet series. hence wavelet analysis is superior to fourier analysis and has so many applications in engineering and technology. the wavelet approximation of a functions by its haar wavelet series and related approximations have been studied by devore[7], debnath[5], meyer[9] , morlet[3], mhaskar[2], sablonnière[6] and lal & kumar[8]. the purpose of this paper is to discuss the legendre wavelet series of function having bounded third and fourth derivatives, i.e. 0 ≤ |f ′′′ (x)| < ∞ ∀x ∈ [0, 1] and 0 ≤ |fiv(x)| < ∞ ∀x ∈ [0, 1] and to obtain legendre wavelet estimators of these functions. this is a significant observation of this research paper that estimate of a function is better and the sharper than the estimate having less order bounded derivative.therefore comparison of estimated approximations has very importance in wavelet analysis. ∗shyam lal, department of mathematics, institute of science, banaras hindu university, varanasi221005, india;shyam lal@rediffmail.com †indra bhan, department of mathematics, institute of science, banaras hindu university, varanasi221005, india;indrabhanmsc@gmail.com 1received on september 21st, 2019. accepted on december 20rd, 2019. published on december 31st, 2019. doi:10.23755/rm.v37i0.491. issn: 1592-7415. eissn: 2282-8214. c©shyam lal and indra bhan 85 shyam lal and indra bhan 2 definitions and preliminaries 2.1 legendre wavelet wavelets constitute a family of functions constructed from dilation and translation of a single function ψ ∈ l2(r) , called mother wavelet. we write ψb,a(x) = |a| −1 2 ψ ( x− b a ) , a 6= 0. if we restrict the values of dilation and translation parameter to a = a−n0 , b = mb0a0 −n,a0 > 1,b0 > 0 respectively, the following family of discrete wavelets are constructed: ψn,m(x) = |a0| n 2 ψ(an0x−mb0) the legendre wavelet over the interval [0,1) is defined as ψn,m(x) = { √ m + 1 2 2 k 2 pm(2 kx− n̂), n̂− 1 2k ≤ x < n̂ + 1 2k 0 , otherwise, where n = 1, 2, ..., 2k−1 and m = 0, 1, 2, 3, ..., n̂ = 2n − 1 and k is the positive integer. in this definition,the polynomials pm are legendre polynomials of degree m over the interval [-1,1] defined as follows: p0(x) = 1,p1(x) = x (m + 1)pm+1(x) = (2m + 1)xpm(x) −mpm−1(x) , m = 1, 2, 3, ... the set of {pm(x) : m = 1, 2, 3, ...} in the hilbert space l2[−1, 1] is a complete orthogonal set. orthogonality of legendre polynomial on the interval [-1,1] implies that 〈pm,pn〉 = ∫ 1 −1 pm(x)pn(x)dx = { 2 2m + 1 ,m = n 0 , otherwise. for m,n = 0, 1, 2, 3... furthermore, the set of wavelets ψn,m makes an orthonormal basis in l2[0, 1),i.e.∫ 1 0 ψn,m(x)ψn′m′ (x)dx = δn,n′δm,m′ in which δ denotes kronecker delta function defined by δn,m = { 1, n=m 0, otherwise. 86 legendre wavelet expansion of functions and their approximations the function f(x) ∈ l2[0, 1) is expressed in the legendre wavelet series as : f(x) = ∞∑ n=1 ∞∑ m=0 cn,mψn,m(x) where cn,m = 〈f,ψn,m〉. the (2k−1,m)th partial sums of above series are given by s2k−1,m (f)(x) = 2k−1∑ n=1 m∑ m=0 cn,mψn,m(x) = c tψ(x) in which c and ψ(x) are 2k−1(m + 1) vectors of the form ct = [c1,0,c1,1, ...c1,m,c2,0,c2,1, ...c2,m, ...,c2k−1,0, ...c2k−1,m ] and ψ(x) = [ψ1,0,ψ1,1, ...ψ1,m,ψ2,0,ψ2,1, ...ψ2,m, ...,ψ2k−1,0, ...ψ2k−1,m ] t 2.2 legendre wavelet approximation let s2k−1,m (f)(x) denote the (2 k−1,m)th partial sums of the series ∞∑ n=1 ∞∑ m=0 cn,mψn,m(x) i.e. s2k−1,m (f)(x) = 2k−1∑ n=1 m∑ m=0 cn,mψn,m(x) the legendre wavelet approximation e2k−1,m (f) of a function f ∈ l2[0, 1) by (2k−1,m)th partial sums s2k−1,m (f) of its legendre wavelet series is given by e2k−1,m (f) = min‖f −s2k−1,m (f)‖2, (zygmund[1],pp.115) where ‖f‖2 = (∫ 1 0 |f(x)|2dx )1 2 . if e2k−1,m (f) → 0 as k → ∞, m → ∞. then e2k−1,m (f) is called the best approximation of f of order (2k−1,m) (zygmund[1],pp.115) 3 example express the following function in the legendre wavelet series : f(t) = t3 ∀t ∈ [0, 1) 87 shyam lal and indra bhan proof: f(t) = 2k−1∑ n=1 ∞∑ m=0 cn,mψn,m(t) cn,m = n̂+1 2k∫ n̂−1 2k f(t)ψn,m(t)dt = n̂+1 2k∫ n̂−1 2k t3 ( 2m + 1 2 )1 2 2 k 2 pm(2 kt− n̂)dt = ( 2m + 1 2 )1 2 2 k 2 1∫ −1 ( v + n̂ 2k )3 pm(v) dv 2k , v = 2kt− n̂ cn,m = ( 2m + 1 27k+1 )1 2 1∫ −1 (n̂3 + v3 + 3n̂2v + 3n̂v2)pm(v)dv by above expression cn,0 = ( 1 27k+1 )1 2 1∫ −1 (n̂3 + v3 + 3n̂2v + 3n̂v2)p0(v)dv = ( 1 27k+1 )1 2 (2n̂3 + 2n̂) cn,1 = ( √ 3 27k+1 )1 2 1∫ −1 (n̂3 + v3 + 3n̂2v + 3n̂v2)p1(v)dv = ( √ 3 27k+1 )1 2 ( 2 5 + 2n̂2 ) cn,2 = ( √ 5 27k+1 )1 2 1∫ −1 (n̂3 + v3 + 3n̂2v + 3n̂v2)p2(v)dv = ( √ 5 27k+1 )1 2 ( 4n̂ 5 ) cn,3 = ( √ 7 27k+1 )1 2 1∫ −1 (n̂3 + v3 + 3n̂2v + 3n̂v2)p3(v)dv 88 legendre wavelet expansion of functions and their approximations cn,3 = ( 4 35 )( √ 7 27k+1 )1 2 cn,m = 0, for m ≥ 4 then, f(t) = 2k−1∑ n=1 cn,0ψn,0(t) + 2k−1∑ n=1 cn,1ψn,1(t) + 2k−1∑ n=1 cn,2ψn,2(t) + 2k−1∑ n=1 cn,3ψn,3(t) now, ||f||22 = 1 7 = 2k−1∑ n=1 c2n,0||ψn,0|| 2 2 + 2k−1∑ n=1 c2n,1||ψn,1|| 2 2 + 2k−1∑ n=1 c2n,2||ψn,2|| 2 2 + 2k−1∑ n=1 c2n,3||ψn,3|| 2 2 = 2k−1∑ n=1 c2n,0 + 2k−1∑ n=1 c2n,1 + 2k−1∑ n=1 c2n,2 + 2k−1∑ n=1 c2n,3 = 2k−1∑ n=1 [( 1 27k+1 )1 2 (2n̂3 + 2n̂) ]2 + 2k−1∑ n=1  ( √3 27k+1 )1 2 ( 2 5 + 2n̂2 )2 + 2k−1∑ n=1  ( √5 27k+1 )1 2 ( 4n̂ 5 )2 + 2k−1∑ n=1  ( 4 35 )( √ 7 27k+1 )1 2  2 = 1 7 . 4 theorems in this paper, we prove following new theorems: theorem (4.1) let a function f ∈ l2[0, 1) such that its third derivative be bounded ,i.e. 0 ≤ |f ′′′ (x)| < ∞∀ x ∈ [0, 1). then the legendre wavelet approximations of f satisfy : (i)e (1) 2k−1,0 (f) = ||f − 2k−1∑ n=1 cn,0ψn,0||2 = o ( 1 2k ) (ii)e (2) 2k−1,1 (f) = ||f − 2k−1∑ n=1 1∑ m=0 cn,mψn,m||2 = o ( 1 22k ) (iii)e (3) 2k−1,2 (f) = ||f − 2k−1∑ n=1 2∑ m=0 cn,mψn,m||2 = o ( 1 23k ) (iv)for f(x) = ∞∑ n=1 ∞∑ m=0 cn,mψn,m, e (4) 2k−1,m (f) = ||f − 2k−1∑ n=1 m∑ m=0 cn,mψn,m||2 89 shyam lal and indra bhan =  2k−1∑ n=1 ∞∑ m=m+1 c2n,m   1 2 = o ( 1 (2m − 3) 5 2 1 23k ) ,∀m ≥ 2. theorem (4.2) if a function f ∈ l2[0, 1) having bounded fourth derivative ,i.e. 0 ≤ |fiv(x)| < ∞∀ x ∈ [0, 1). then its legendre wavelet approximations are given by (i)e (5) 2k−1,0 (f) = ||f − 2k−1∑ n=1 cn,0ψn,0||2 = o ( 1 2k ) (ii)e (6) 2k−1,1 (f) = ||f − 2k−1∑ n=1 1∑ m=0 cn,mψn,m||2 = o ( 1 22k ) (iii)e (7) 2k−1,2 (f) = ||f − 2k−1∑ n=1 2∑ m=0 cn,mψn,m||2 = o ( 1 23k ) (iv)e (8) 2k−1,3 (f) = ||f − 2k−1∑ n=1 3∑ m=0 cn,mψn,m||2 = o ( 1 24k ) (v)for f(x) = ∞∑ n=1 ∞∑ m=0 cn,mψn,m , e (9) 2k−1,m (f) = ||f − 2k−1∑ n=1 m∑ m=0 cn,mψn,m||2 =  2k−1∑ n=1 ∞∑ m=m+1 c2n,m   1 2 = o ( 1 (2m − 5) 7 2 1 24k ) , ∀ m ≥ 3. 5 proofs 5.1 proof of the theorem (4.1) (i) the error e(0)n (x) between f(x) and its expression over any subinterval is defined as e (0) n (x) = cn,0ψn,0(x) −f(x) ,x ∈ [ n̂−1 2k , n̂+1 2k ) ,n = 1, 2, 3, ...2k−1 ||e(0)n || 2 2 = n̂+1 2k∫ n̂−1 2k (e(0)n (x)) 2dx = n̂+1 2k∫ n̂−1 2k (c2n,0ψ 2 n,m(x) + (f(x)) 2 − 2cn,0ψn,0(x)f(x))dx 90 legendre wavelet expansion of functions and their approximations = c2n,0 n̂+1 2k∫ n̂−1 2k ψ2n,0(x)dx + n̂+1 2k∫ n̂−1 2k (f(x))2dx− 2cn,0 n̂+1 2k∫ n̂−1 2k f(x)ψn,0(x)dx = n̂+1 2k∫ n̂−1 2k (f(x))2dx− c2n,0. (5.1) now, n̂+1 2k∫ n̂−1 2k (f(x))2dx = 1 2k−1∫ 0 ( f ( n̂− 1 2k + h ))2 dh,x = n̂− 1 2k + h = 1 2k−1∫ 0 [ f ( n̂− 1 2k ) + hf ′ ( n̂− 1 2k ) + h2 2 f ′′ ( n̂− 1 2k ) + h3 6 f ′′′ ( n̂− 1 2k + θh )]2 , 0 < θ < 1 by taylor′s expansion = 1 2k−1∫ 0 ( f ( n̂− 1 2k ))2 dh + 1 2k−1∫ 0 h2 ( f ′ ( n̂− 1 2k ))2 dh + 1 2k−1∫ 0 h4 4 ( f ′′ ( n̂− 1 2k ))2 dh + 1 2k−1∫ 0 h6 36 ( f ′′′ ( n̂− 1 2k + θh ))2 dh + 1 2k−1∫ 0 2hf ( n̂− 1 2k ) f ′ ( n̂− 1 2k ) dh + 1 2k−1∫ 0 h2f ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) dh + 1 2k−1∫ 0 h3 3 f ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh + 1 2k−1∫ 0 h3f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) dh + 1 2k−1∫ 0 h4 3 f ′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh + 1 2k−1∫ 0 h5 6 f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh = 2 2k ( f ( n̂− 1 2k ))2 + 8 3 1 23k ( f ′ ( n̂− 1 2k ))2 + 8 5 1 25k ( f ′′ ( n̂− 1 2k ))2 + 1 36 1 2k−1∫ 0 h6 ( f ′′′ ( n̂− 1 2k + θh ))2 dh + 4 22k f ( n̂− 1 2k ) f ′ ( n̂− 1 2k ) 91 shyam lal and indra bhan + 8 3 1 23k f ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 1 3 1 2k−1∫ 0 h3f ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh + 4 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) dh + 1 3 1 2k−1∫ 0 h4f ′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh + 1 6 1 2k−1∫ 0 h5f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh. (5.2) now, cn,0 = < f(x),ψn,0(x) > = n̂+1 2k∫ n̂−1 2k f(x)ψn,0(x)dx = 2 k−1 2 n̂+1 2k∫ n̂−1 2k f(x)dx = 2 k−1 2 1 2k−1∫ 0 f ( n̂− 1 2k + h ) dh,x = n̂− 1 2k + h = 2 k−1 2 1 2k−1∫ 0 [ f ( n̂− 1 2k ) + hf ′ ( n̂− 1 2k ) + h2 2 f ′′ ( n̂− 1 2k ) + h3 6 f ′′′ ( n̂− 1 2k + θh )] dh = 2 k−1 2   2 2k f ( n̂− 1 2k ) + 2 22k f ′ ( n̂− 1 2k ) + 4 3 1 23k f ′′ ( n̂− 1 2k ) + 1 6 1 2k−1∫ 0 h3f ′′′ ( n̂− 1 2k + θh ) dh   . next, c2n,0 = 2 2k ( f ( n̂− 1 2k ))2 + 2 23k ( f ′ ( n̂− 1 2k ))2 + 8 9 1 25k ( f ′′ ( n̂− 1 2k ))2 + 2k 2   1 2k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh   2 + 4 22k f ( n̂− 1 2k ) f ′ ( n̂− 1 2k ) + 8 3 1 23k f ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 1 3 1 2k−1∫ 0 h3f ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k + θh ) dh 92 legendre wavelet expansion of functions and their approximations + 8 3 1 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 2 2k f ′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh + 4 3 1 22k f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh. (5.3) now, by using equations (5.1), (5.2) and (5.3) we have ||e(0)n || 2 2 = 2 3 1 23k ( f ′ ( n̂− 1 2k ))2 + 32 45 1 25k ( f ′′ ( n̂− 1 2k ))2 + 1 36 1 2k−1∫ 0 h6 ( f ′′′ ( n̂− 1 2k + θh ))2 dh − 2k 2   1 2k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh   2 + 4 3 1 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 1 3 f ′ ( n̂− 1 2k ) 12k−1∫ 0 h4f ′′′ ( n̂− 1 2k + θh ) dh− 2 2k f ′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh + 1 6 f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h5f ′′′ ( n̂− 1 2k + θh ) dh− 4 3 1 22k f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh = i1 + i2 + i3 − i4 + i5 + i6 − i7 + i8 − i9, say. since |f ′ (x)| ≤ m1, |f ′′ (x)| ≤ m2, |f ′′′ (x)| ≤ m3,∀x ∈ [0, 1), therefore |i1| ≤ 2 3 1 23k m21 |i2| ≤ 32 45 1 25k m22 |i3| ≤ 32 63 1 27k m23 |i4| ≤ 2 9 1 27k m23 |i5| ≤ 4 3 1 24k m1m2 |i6| ≤ 32 15 1 25k m1m3 |i7| ≤ 4 3 1 25k m1m3 |i8| ≤ 16 9 1 26k m2m3 |i9| ≤ 8 9 1 26k m2m3. 93 shyam lal and indra bhan therefore, ||e(0)n || 2 2 ≤ |i1| + |i2| + |i3| + |i4| + |i5| + |i6| + |i7| + |i8| + |i9| ≤ 2 3 1 23k m21 + 32 45 1 25k m22 + 32 63 1 27k m23 + 2 9 1 27k m23 + 4 3 1 24k m1m2 + 32 15 1 25k m1m3 + 4 3 1 25k m1m3 + 16 9 1 26k m2m3 + 8 9 1 26k m2m3 = 2 3 1 23k m21 + 32 45 1 25k m22 + 56 63 1 27k m23 + 4 3 1 24k m1m2 + 52 15 1 25k m1m3 + 24 9 1 26k m2m3 < 2 23k [ m21 + ( m2 2k )2 + ( m3 22k )2 + 2m1m2 2k + 2m1m3 22k + 2m2m3 23k ] = 2 23k ( m1 + m2 2k + m3 22k )2 = 2m2 23k ( 1 + 1 2k + 1 22k )2 ,m = max[m1,m2,m3]. next, (e (1) 2k−1,0 (f))2 = 1∫ 0  2k−1∑ n=1 e(0)n (x)  2 dx = 1∫ 0 2k−1∑ n=1 (e(0)n (x)) 2dx + 2 2k−1∑ n=1 2k−1∑ n6=n′ 1∫ 0 e(0)n (x)e (0′) n (x)dx = 2k−1∑ n=1 1∫ 0 (en(x)) 2dx, due to disjoint supports of en and e ′ n = 2k−1∑ n=1 ||e(0)n || 2 2 ≤ (2k−1) 2m2 23k ( 1 + 1 2k + 1 22k )2 = m2 22k ( 1 + 1 2k + 1 22k )2 . then, e (1) 2k−1,0 (f) ≤ m 2k ( 1 + 1 2k + 1 22k ) ≤ m ( 1 2k + 1 2k + 1 2k ) = 3m ( 1 2k ) = o ( 1 2k ) . 94 legendre wavelet expansion of functions and their approximations (ii)e(1)n (x) = cn,0ψn,0(x) + cn,1ψn,1(x) −f(x) , x ∈ [ n̂− 1 2k , n̂ + 1 2k ) ||e(1)n || 2 2 = n̂+1 2k∫ n̂−1 2k (f(x))2dx− c2n,0 − c 2 n,1. (5.4) now, consider cn,1 = < f(x),ψn,1(x) > = n̂+1 2k∫ n̂−1 2k f(x)ψn,1(x)dx = √ 3 2 2 k 2 n̂+1 2k∫ n̂−1 2k f(x)p1(2 kx− n̂)dx = √ 3 2 2 k 2 1 2k−1∫ 0 f ( n̂− 1 2k + h ) p1(2 kh− 1)dh, x = n̂− 1 2k + h = √ 3 2 2 k 2 1 2k−1∫ 0 f ( n̂− 1 2k + h ) (2kh− 1)dh = √ 3 2 2 k 2 1 2k−1∫ 0 f ( n̂− 1 2k ) (2kh− 1)dh + √ 3 2 2 k 2 1 2k−1∫ 0 f ′ ( n̂− 1 2k ) h(2kh− 1)dh + √ 3 2 2 k 2 1 2k−1∫ 0 f ′′ ( n̂− 1 2k ) h2 2 (2kh− 1)dh + √ 3 2 2 k 2 1 2k−1∫ 0 f ′′′ ( n̂− 1 2k + θh ) h3 6 (2kh− 1)dh cn,1 = √ 3 2 2 k 2 [ 2 3 1 22k f ′ ( n̂− 1 2k ) + 2 3 1 23k f ′′ ( n̂− 1 2k )] + √ 3 2 2 k 2 1 2k−1∫ 0 f ′′′ ( n̂− 1 2k + θh ) h3 6 (2kh− 1)dh. now, c2n,1 = 2 3 1 23k ( f ′ ( n̂− 1 2k ))2 + 2 3 1 25k ( f ′′ ( n̂− 1 2k ))2 + 3 2 2k   1 2k−1∫ 0 h3 6 (2kh− 1)f ′′′ ( n̂− 1 2k ) dh   2 + 4 3 1 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) 95 shyam lal and indra bhan + 2 2k f ′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 (2kh− 1)f ′′′ ( n̂− 1 2k ) dh + 2 22k f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 (2kh− 1)f ′′′ ( n̂− 1 2k ) dh. (5.5) by using equations (5.2), (5.3), (5.4) and (5.5), we have ||e(1)n || 2 2 = 2 45 1 25k ( f ′′ ( n̂− 1 2k ))2 + 1 36 1 2k−1∫ 0 h6 ( f ′′′ ( n̂− 1 2k + θh ))2 dh − 4 3 1 22k f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k ) dh− 2k 2   1 2k−1∫ 0 h3 6 f ′′′ ( n̂− 1 2k + θh ) dh   2 + 1 6 f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h5f ′′′ ( n̂− 1 2k + θh ) dh− 3 2 2k   1 2k−1∫ 0 h3 6 (2kh− 1)f ′′′ ( n̂− 1 2k ) dh   2 − 2 22k f ′′ ( n̂− 1 2k ) 12k−1∫ 0 h3 6 (2kh− 1)f ′′′ ( n̂− 1 2k ) dh = i1 + i2 + i3 + i4 + i5 + i6 + i7,say. therefore |i1| ≤ 2 45 1 25k m22 |i2| ≤ 32 63 1 27k m23 |i3| ≤ 8 9 1 26k m2m3 |i4| ≤ 2 9 1 27k m23 |i5| ≤ 16 9 1 26k m2m3 |i6| ≤ 18 75 1 27k m23 |i7| ≤ 12 15 1 26k m2m3 ||e(1)n || 2 2 ≤ |i1| + |i2| + |i3| + |i4| + |i5| + |i6| + |i7| ≤ 2 45 1 25k m22 + 32 63 1 27k m23 + 8 9 1 26k m2m3 + 2 9 1 27k m23 + 16 9 1 26k m2m3 + 18 75 1 27k m23 + 12 15 1 26k m2m3 96 legendre wavelet expansion of functions and their approximations = 2 45 1 25k m22 + 1528 1575 1 27k m23 + 144 45 1 26k m2m3 < 2 25k ( m22 + ( m3 2k )2 + 2m2m3 2k ) = 2 25k m2 ( 1 + 1 2k )2 ,m = max[m2,m3]. next, (e (2) 2k−1,1 (f))2 = 2k−1∑ n=1 ||e(1)n || 2 2 ≤ (2k−1) 2 25k m2 ( 1 + 1 2k )2 = m2 24k ( 1 + 1 2k )2 . then, e (2) 2k−1,1 (f) ≤ m 22k ( 1 + 1 2k ) = o ( 1 22k ) . (iii) e(2)n (x) = cn,0ψn,0(x) + cn,1ψn,1(x) + cn,2ψn,2(x) −f(x) , x ∈ [ n̂−1 2k , n̂+1 2k ) similarly, it can be proved that e (3) 2k−1,2 (f) = o ( 1 23k ) . (iv) 0 ≤ |f ′′′ (x)| < m1 ,∀x ∈ [0, 1) cn,m = 1∫ 0 f(x)ψn,m(x)dx = n̂+1 2k∫ n̂−1 2k f(x) √ 2m + 1 2 2 k 2 pm(2 kx− n̂)dx = √ 2m + 1 2k+1 1∫ −1 f ( n̂ + t 2k ) pm(t)dt 97 shyam lal and indra bhan = √ 2m + 1 2k+1 1∫ −1 f ( n̂ + t 2k ) d(pm+1(t) −pm−1(t)) 2m + 1 = ( 1 2k+1(2m + 1) )1 2 ×  {f (n̂ + t 2k ) (pm+1(t) −pm−1(t)) }1 −1 − 1∫ −1 1 2k f ′ ( n̂ + t 2k ) (pm+1(t) −pm−1(t))dt   = ( 1 23k+1(2m + 1) )1 2   1∫ −1 f ′ ( n̂ + t 2k ) (pm−1(t) −pm+1(t))dt   = ( 1 23k+1(2m + 1) )1 2   1∫ −1 f ′ ( n̂ + t 2k ) (pm−1(t))dt− 1∫ −1 f ′ ( n̂ + t 2k ) (pm+1(t))dt   = ( 1 23k+1(2m + 1) )1 2 1∫ −1 [ f ′ ( n̂ + t 2k ) d(pm(t) −pm−2(t)) (2m− 1) −f ′ ( n̂ + t 2k ) d(pm+2(t) −pm(t)) (2m + 3) ] = ( 1 25k+1(2m + 1) )1 2 1∫ −1 [ f ′′ ( n̂ + t 2k ) (pm+2(t) −pm(t)) (2m + 3) −f ′′ ( n̂ + t 2k ) d(pm(t) −pm−2(t)) (2m− 1) ] = ( 1 25k+1(2m + 1) )1 2 1 (2m + 3) 1∫ −1 [ f ′′ ( n̂ + t 2k ) d(pm+3(t) −pm+1(t)) (2m + 5) ] − ( 1 25k+1(2m + 1) )1 2 1 (2m + 3) 1∫ −1 [ f ′′ ( n̂ + t 2k ) d(pm+1(t) −pm−1(t)) (2m + 1) ] + ( 1 25k+1(2m + 1) )1 2 1 (2m− 1) 1∫ −1 [ f ′′ ( n̂ + t 2k ) d(pm−1(t) −pm−3(t)) (2m− 3) ] − ( 1 25k+1(2m + 1) )1 2 1 (2m− 1) 1∫ −1 [ f ′′ ( n̂ + t 2k ) d(pm+1(t) −pm−1(t)) (2m + 1) ] = ( 1 27k+1(2m + 1) )1 2 1 (2m + 3) 1∫ −1 f ′′′ ( n̂ + t 2k )[ (pm+1(t) −pm−1(t)) (2m + 1) − (pm+3(t) −pm+1(t)) (2m + 5) ] dt − ( 1 27k+1(2m + 1) )1 2 1 (2m− 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ (pm−1(t) −pm−3(t)) (2m− 3) − (pm+1(t) −pm−1(t)) (2m + 1) ] dt 98 legendre wavelet expansion of functions and their approximations = ( 1 27k+1(2m + 1) )1 2 × 1∫ −1 f ′′′ ( n̂ + t 2k )[ 2(2m + 3)pm+1(t) − (2m + 5)pm−1(t) − (2m + 1)pm+3(t) (2m + 1)(2m + 5)(2m + 3) ] dt − ( 1 27k+1(2m + 1) )1 2 × 1∫ −1 f ′′′ ( n̂ + t 2k )[ 2(2m− 1)pm−1(t) − (2m + 1)pm−3(t) − (2m− 3)pm+1(t) (2m + 1)(2m− 1)(2m− 3) ] dt. let τ1(t) = 2(2m + 3)pm+1(t) − (2m + 5)pm−1(t) − (2m + 1)pm+3(t) τ2(t) = 2(2m− 1)pm−1(t) − (2m + 1)pm−3(t) − (2m− 3)pm+1(t) then, cn,m = ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m + 3)(2m + 5)   1∫ −1 f ′′′ ( n̂ + t 2k ) τ1(t)dt   − ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m− 1)(2m− 3)   1∫ −1 f ′′′ ( n̂ + t 2k ) τ2(t)dt   |cn,m| ≤ ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m + 3)(2m + 5)   1∫ −1 ∣∣∣∣f′′′ ( n̂ + t 2k )∣∣∣∣ |τ1(t)|dt   + ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m− 1)(2m− 3)   1∫ −1 ∣∣∣∣f′′′ ( n̂ + t 2k )∣∣∣∣ |τ2(t)dt|   ≤ m1 ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m + 3)(2m + 5) 1∫ −1 |τ1(t)|dt + m1 ( 1 27k+1(2m + 1) )1 2 1 (2m + 1)(2m− 1)(2m− 3) 1∫ −1 |τ2(t)|dt. (5.6) consider, 1∫ −1 |τ1(t)|dt = 1∫ −1 1.|τ1(t)|dt ≤   1∫ −1 12.dt   1 2   1∫ −1 |τ1(t)|2dt   1 2 99 shyam lal and indra bhan = √ 2   1∫ −1 (2(2m + 3)pm+1(t) − (2m + 5)pm−1(t) − (2m + 1)pm+3(t)) 2 dt   1 2 = √ 2   1∫ −1 [ 4(2m + 3)2p2m+1(t) + (2m + 5) 2p2m−1(t) + (2m + 1) 2p2m+3(t) ] dt   1 2 = √ 2 [ 4(2m + 3)2 2 2m + 3 + (2m + 5)2 2 2m− 1 + (2m + 1)2 2 2m + 7 ]1 2 by orthogonality condition on pm = 2 [ 4(2m + 3) + (2m + 5)2 2m− 1 + (2m + 1)2 2m + 7 ]1 2 ≤ 2 [ 4(2m + 3)(2m− 1) + (2m + 5)2 + (2m + 1)2 (2m− 1) ]1 2 = 2 [ 24m2 + 40m + 14 2m− 1 ]1 2 = 2 √ 2 [ (2m + 1)(6m + 7) 2m− 1 ]1 2 ≤ 2 √ 6 [ (2m + 1)(2m + 3) (2m− 1) ]1 2 . (5.7) now , 1∫ −1 |τ2(t)|dt = 1∫ −1 1.|τ2(t)|dt = √ 2   1∫ −1 [2(2m− 1)pm−1(t) − (2m + 1)pm−3(t) − (2m− 3)pm+1(t)]2dt   1 2 = √ 2   1∫ −1 [ (2m− 3)2pm+1(t) + (2m + 1)2p2m−3(t) + 4(2m− 1) 2p2m−1(t) ] dt   1 2 = √ 2 [ (2m− 3)2 2 (2m + 3) + (2m + 1)2 2 2m− 5 + 4(2m− 1)2 2 2m− 1 ]1 2 by orthogonality condition on pm = 2 [ (2m− 3)2 (2m + 3) + (2m + 1) (2m− 5) + 4(2m− 1) ]1 2 ≤ 2 [ (2m− 3)2 + (2m + 1)2 + 4(2m− 1)(2m− 5) 2m− 5 ]1 2 100 legendre wavelet expansion of functions and their approximations = 2 [ 24m2 − 56m + 30 2m− 5 ]1 2 = 2 √ 2 [ (2m− 3)(6m− 5) 2m− 5 ]1 2 ≤ 2 √ 6 [ (2m− 3)(2m− 1) (2m− 5) ]1 2 . (5.8) now , by using equations (5.6), (5.7) and (5.8) we have |cn,m| ≤ m1 ( 1 27k+1(2m + 1) )1 2 [ 2 √ 6 (2m− 3) 5 2 + 2 √ 6 (2m− 5) 5 2 ] ≤ m1 ( 1 27k+1(2m + 1) )1 2 [ 4 √ 6 (2m− 5) 5 2 ] ≤ 4 √ 6m1 2 7k+1 2 1 (2m− 5)3 . therefore, |cn,m| ≤ 4 √ 6m1 2 7k+1 2 1 (2m− 5)3 ,∀m ≥ 3. (5.9) s2k−1,m (f)(x) = 2k−1∑ n=1 m∑ m=0 cn,mψn,m(x) f(x) −s2k−1,m (f)(x) = 2k−1∑ n=1 ∞∑ m=0 cn,mψn,m(x) − 2k−1∑ n=1 m∑ m=0 cn,mψn,m(x) = 2k−1∑ n=1 m∑ m=0 cn,mψn,m(x) + 2k−1∑ n=1 ∞∑ m=m+1 cn,mψn,m(x) − 2k−1∑ n=1 m∑ m=0 cn,mψn,m(x) = 2k−1∑ n=1 ∞∑ m=m+1 cn,mψn,m(x). then, ||f −s2k−1,m (f)|| 2 2 = 1∫ 0  2k−1∑ n=1 ∞∑ m=m+1 cn,mψn,m(x)  2 dx = 2k−1∑ n=1 ∞∑ m=m+1 c2n,m, by orthogonality property of ψn,m 101 shyam lal and indra bhan ≤ 2k−1∑ n=1 ∞∑ m=m+1 ( 4 √ 6m1 2 7k+1 2 1 (2m− 5)3 )2 , by (5.9) = 96m21 2k−1∑ n=1 1 27k+1 ∞∑ m=m+1 1 (2m− 5)6 = 96m21 4 1 26k ∞∫ m+1 1 (2m− 5)6 dm = 12m21 5 1 26k 1 (2m − 3)5 ∴ e (4) 2k−1,m (f) ≤ 2 √ 3m1√ 5 1 23k(2m − 3) 5 2 = o ( 1 (2m − 3) 5 2 23k ) , m ≥ 2. 5.2 proof of the theorem(4.2) (i) the error e∗(0)n (x) between f(x) and its expression over any subinterval is defined as e∗(0)n (x) = cn,0ψn,0(x) − f(x) ,x ∈ [ n̂−1 2k , n̂+1 2k ) ,n = 1, 2, 3, ...2k−1 now consider, n̂+1 2k∫ n̂−1 2k (f(x))2dx = 1 2k−1∫ 0 ( f ( n̂− 1 2k + h ))2 dh,x = n̂− 1 2k + h = 1 2k−1∫ 0 [ f ( n̂− 1 2k ) + hf ′ ( n̂− 1 2k ) + h2 2 f ′′ ( n̂− 1 2k ) + h3 6 f ′′′ ( n̂− 1 2k ) + h4 24 fiv ( n̂− 1 2k + θh )]2 dh = 2 2k ( f ( n̂− 1 2k ))2 + 8 3 1 23k ( f ′ ( n̂− 1 2k ))2 + 8 5 1 25k ( f ′′ ( n̂− 1 2k ))2 + 32 63 1 27k ( f ′′′ ( n̂− 1 2k ))2 + 1 2k−1∫ 0 h8 576 ( fiv ( n̂− 1 2k + θh ))2 dh + 4 22k f ( n̂− 1 2k ) f ′ ( n̂− 1 2k ) + 8 3 1 23k f ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) 102 legendre wavelet expansion of functions and their approximations + 1 2k−1∫ 0 h4 12 f ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 4 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 32 15 1 25k f ′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 1 2k−1∫ 0 h5 12 f ′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 16 9 1 26k f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 1 2k−1∫ 0 h6 24 f ′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 1 2k−1∫ 0 h7 72 f ′′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 4 3 1 24k f ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) . now, cn,0 = 2 k−1 2 [ 2 2k f ( n̂− 1 2k ) + 2 22k f ′ ( n̂− 1 2k ) + 4 3 1 23k f ′′ ( n̂− 1 2k ) + 2 3 1 24k f ′′′ ( n̂− 1 2k )] + 2 k−1 2   1 2k−1∫ 0 h4 24 fiv ( n̂− 1 2k + θh ) dh   . next, c2n,0 = 2 2k ( f ( n̂− 1 2k ))2 + 2 23k ( f ′ ( n̂− 1 2k ))2 + 8 9 1 25k ( f ′′ ( n̂− 1 2k ))2 + 2 9 1 27k ( f ′′′ ( n̂− 1 2k ))2 + 2k 2   1 2k−1∫ 0 h4 24 fiv ( n̂− 1 2k + θh ) dh   2 + 4 22k f ( n̂− 1 2k ) f ′ ( n̂− 1 2k ) + 8 3 1 23k f ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 4 3 1 24k f ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 1 2k−1∫ 0 h4 12 f ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 8 3 1 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 4 3 1 25k f ′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 1 2k 1 2k−1∫ 0 h4 12 f ′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 8 9 1 26k f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 4 3 1 22k 1 2k−1∫ 0 h4 24 f ′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 2 3 1 23k 1 2k−1∫ 0 h4 24 f ′′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh. 103 shyam lal and indra bhan since |f ′ (x)| ≤ m1, |f ′′ (x)| ≤ m2, |f ′′′ (x)| ≤ m3 and |fiv(x)| ≤ m4, ∀x ∈ [0, 1) therefore, ||e∗(0)n || 2 2 ≤ 2 23k ( m1 + m2 2k + m3 22k + m4 23k )2 . ∴ e (5) 2k−1,0 (f) = o ( 1 2k ) . (ii) the error e∗(1)n (x) between f(x) and its expression over any subinterval is defined as e∗ (1) n (x) = cn,0ψn,0(x) + cn,1ψn,1(x) −f(x) ,x ∈ [ n̂−1 2k , n̂+1 2k ) ,n = 1, 2, 3, ...2k−1 ||e∗(1)n ||22 = n̂+1 2k∫ n̂−1 2k (e∗ (1) n (x)) 2dx = n̂+1 2k∫ n̂−1 2k (f(x))2dx− c2n,0 − c 2 n,1. now, cn,1 = √ 3 2 2 k 2 [ 2 3 1 22k f ′ ( n̂− 1 2k ) + 2 3 1 23k f ′′ ( n̂− 1 2k ) + 2 5 1 24k f ′′′ ( n̂− 1 2k )] + √ 3 2 2 k 2 1 2k−1∫ 0 h4 24 (2kh− 1)fiv ( n̂− 1 2k + θh ) dh. next, c2n,1 = 2 3 1 23k ( f ′ ( n̂− 1 2k ))2 + 2 3 1 25k ( f ′′ ( n̂− 1 2k ))2 + 6 25 1 27k ( f ′′′ ( n̂− 1 2k ))2 + 3 2 2k   1 2k−1∫ 0 h4 24 (2kh− 1)fiv ( n̂− 1 2k + θh ) dh   2 + 2 2k 1 2k−1∫ 0 h4 24 (2kh− 1)f ′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 4 3 1 24k f ′ ( n̂− 1 2k ) f ′′ ( n̂− 1 2k ) + 4 5 1 25k f ′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) + 4 5 1 26k f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) 104 legendre wavelet expansion of functions and their approximations + 2 22k 1 2k−1∫ 0 h4 24 (2kh− 1)f ′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 6 5 1 23k 1 2k−1∫ 0 h4 24 (2kh− 1)f ′′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh therefore, ||e∗(1)n || 2 2 ≤ 2 25k ( m2 + m3 2k + m4 22k )2 . then, e (6) 2k−1,1 (f) = o ( 1 22k ) . (iii) the error e∗(2)n (x) between f(x) and its expression over any subinterval is defined as e∗ (2) n (x) = cn,0ψn,0(x) + cn,1ψn,1(x) + cn,2ψn,2(x) −f(x) ,x ∈ [ n̂−1 2k , n̂+1 2k ) , n = 1, 2, 3, ...2k−1 ||e∗(2)n || 2 2 = n̂+1 2k∫ n̂−1 2k (e(2)n (x)) 2dx = n̂+1 2k∫ n̂−1 2k (f(x))2dx− c2n,0 − c 2 n,1 − c 2 n,2. now, cn,2 = < f(x),ψn,2(x) > = √ 5 2 2 k 2 2 15 [ 1 23k f ′′ ( n̂− 1 2k ) + 1 24k f ′′′ ( n̂− 1 2k )] + √ 5 2 2 k 2 1 2   1 2k−1∫ 0 h4 24 (3h222k − 6h2k + 2)fiv ( n̂− 1 2k + θh ) dh   . next, c2n,2 = 2 45 1 25k ( f ′′ ( n̂− 1 2k ))2 + 2 45 1 27k ( f ′′′ ( n̂− 1 2k ))2 +   1 2k−1∫ 0 h4 24 (3h222k − 6h2k + 2)fiv ( n̂− 1 2k + θh ) dh   2 + 4 45 1 26k f ′′ ( n̂− 1 2k ) f ′′′ ( n̂− 1 2k ) 105 shyam lal and indra bhan + 1 3 1 22k 1 2k−1∫ 0 h4 24 (3h222k − 6h2k + 2)f ′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh + 1 3 1 23k 1 2k−1∫ 0 h4 24 (3h222k − 6h2k + 2)f ′′′ ( n̂− 1 2k ) fiv ( n̂− 1 2k + θh ) dh. therefore, ||e∗(2)n || 2 2 ≤ 2 27k ( m3 + m4 2k )2 . then, e (7) 2k−1,2 (f) = o ( 1 23k ) . (iv) the error e∗(3)n (x) between f(x) and its expression over any subinterval is defined as e∗ (3) n (x) = cn,0ψn,0(x) + cn,1ψn,1(x) + cn,2ψn,2(x) + cn,3ψn,3(x) − f(x) , x ∈ [ n̂−1 2k , n̂+1 2k ) , n = 1, 2, 3, ...2k−1 similarly , it can be proved that e (8) 2k−1,3 (f) = o ( 1 24k ) . (v) following the proof of theorem (4.1)(iv) we have cn,m = ( 1 27k+1(2m + 1) )1 2 1 (2m + 3) 1∫ −1 f ′′′ ( n̂ + t 2k )[ (pm+1(t) −pm−1(t)) (2m + 1) − (pm+3(t) −pm+1(t)) (2m + 5) ] dt − ( 1 27k+1(2m + 1) )1 2 1 (2m− 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ (pm−1(t) −pm−3(t)) (2m− 3) − (pm+1(t) −pm−1(t)) (2m + 1) ] dt = ( 1 27k+1(2m + 1) )1 2 1 (2m + 3)(2m + 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(pm+2(t) −pm(t)) (2m + 3) ] − ( 1 27k+1(2m + 1) )1 2 1 (2m + 3)(2m + 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(pm(t) −pm−2(t)) (2m− 1) ] − ( 1 27k+1(2m + 1) )1 2 1 (2m + 3)(2m + 5) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(pm+4(t) −pm+2(t)) (2m + 7) ] 106 legendre wavelet expansion of functions and their approximations + ( 1 27k+1(2m + 1) )1 2 1 (2m + 3)(2m + 5) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(pm+2(t) −pm(t)) (2m + 3) ] − ( 1 27k+1(2m + 1) )1 2 1 (2m− 1)(2m− 3) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(pm(t) −pm−2(t)) (2m− 1) ] + ( 1 27k+1(2m + 1) )1 2 1 (2m− 1)(2m− 3) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(pm−2(t) −pm−4(t)) (2m− 5) ] + ( 1 27k+1(2m + 1) )1 2 1 (2m− 1)(2m + 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(pm+2(t) −pm(t)) (2m + 3) ] − ( 1 27k+1(2m + 1) )1 2 1 (2m− 1)(2m + 1) 1∫ −1 f ′′′ ( n̂ + t 2k )[ d(pm(t) −pm−2(t)) (2m− 1) ] = ( 1 29k+1(2m + 1) )1 2 1 (2m + 1)(2m + 3) × 1∫ −1 fiv ( n̂ + t 2k )[ (pm(t) −pm−2(t)) (2m− 1) − (pm+2(t) −pm(t)) (2m + 3) ] dt + ( 1 29k+1(2m + 1) )1 2 1 (2m + 3)(2m + 5) × 1∫ −1 fiv ( n̂ + t 2k )[ (pm+4(t) −pm+2(t)) (2m + 7) − (pm+2(t) −pm(t)) (2m + 3) ] dt + ( 1 29k+1(2m + 1) )1 2 1 (2m− 1)(2m + 1) × 1∫ −1 fiv ( n̂ + t 2k )[ (pm(t) −pm−2(t)) (2m− 1) − (pm+2(t) −pm(t)) (2m + 3) ] dt + ( 1 29k+1(2m + 1) )1 2 1 (2m− 1)(2m− 3) × 1∫ −1 fiv ( n̂ + t 2k )[ (pm(t) −pm−2(t)) (2m− 1) − (pm−2(t) −pm−4(t)) (2m− 5) ] dt. |cn,m| ≤ ( 1 29k )1 2 8 √ 6m2 (2m− 7)4 , (∵ |fiv(x)| ≤ m2 ∀ x ∈ [0, 1)). 107 shyam lal and indra bhan next, ||f −s2k−1,m (f)|| 2 2 = 2k−1∑ n=1 ∞∑ m=m+1 c2n,m ≤ 2k−1∑ n=1 ∞∑ m=m+1 (( 1 29k )1 2 8 √ 6m2 (2m− 7)4 )2 = 48m22 7 1 28k 1 (2m − 5)7 ∴ e (9) 2k−1,m (f) = √ 48 7 m2 24k 1 (2m − 5) 7 2 = o ( 1 (2m − 5) 7 2 1 24k ) , ∀ m ≥ 3. 6 conclusions (1) after discussing the legendre wavelet approximation of a function f with bounded third and fourth derivatives ,it is trivial to find out the wavelet estimators of a function f of bounded first and second derivatives . (2)the estimates of the theorems (4.1) an(4.2) are obtained as following: (i)e (1) 2k−1,0 (f) = o ( 1 2k ) → 0 as k →∞ (ii)e (2) 2k−1,1 (f) = o ( 1 22k ) → 0 as k →∞ (iii)e (3) 2k−1,2 (f) = o ( 1 23k ) → 0 as k →∞ (iv)e (4) 2k−1,m (f) = o ( 1 (2m−3) 5 2 1 23k ) → 0 as k →∞,m →∞ (v)e (5) 2k−1,0 (f) = o ( 1 2k ) → 0 as k →∞ (vi)e (6) 2k−1,1 (f) = o ( 1 22k ) → 0 as k →∞ (vii)e (7) 2k−1,2 (f) = o ( 1 23k ) → 0 as k →∞ (viii)e (8) 2k−1,3 (f) = o ( 1 24k ) → 0 as k →∞ (ix)e (9) 2k−1,m (f) = o ( 1 (2m−5) 7 2 1 24k ) → 0 as k →∞,m →∞ then e (1) 2k−1,0 (f),e (2) 2k−1,1 (f),e (3) 2k−1,2 (f),e (4) 2k−1,m (f),e (5) 2k−1,1 (f),e (6) 2k−1,1 (f),e (7) 2k−1,2 (f), e (8) 2k−1,3 (f),e (9) 2k−1,m (f) are best possible legendre wavelet approximation in wavelet analysis. (3)legendre wavelet estimators of a function f with bounded fourth order derivative is better and sharper than the estimator of a function f of bounded third order derivative. (4) legendre wavelet estimator of a function f of bounded higher order derivatives is better and sharper than the estimator of a function f of bounded less order derivatives. 108 legendre wavelet expansion of functions and their approximations 7 acknowledgments shyam lal, one of the authors, is thankful to dst cims for encouragement to this work. indra bhan, one of the authors, is grateful to c.s.i.r. (india) for providing financial assistance in the form of junior research fellowship vide ref. no. 18/12/2016 (ii) eu-v dated:01-07-2017 for this research work. references [1] a. zygmund , trigonometric series volume i, cambridge university press, 1959. [2] h. n. mhaskar, “ polynomial operators and local smoothness classes on the unit interval, ii,” jaen j. approx., vol. 1, no. 1(2009), pp. 1-25. [3] j. morlet, g. arens, e. fourgeau and d. giard, wave propagation and sampling theory, part i; complex signal and scattering in multilayer media, geophysics 47(1982) no. 2, 203-221. [4] j. morlet, g. arens, e. fourgeau and d. giard, wave propagation and sampling theory, part ii; sampling theory complex waves, geophysics 47(1982) no. 2, 222-236. [5] l. debnath, wavelet transform and their applications, birkhauser bostoon, massachusetts-2002. [6] p. sablonnière, “ rational bernstein and spline approximation. a new approach, ” jaen j. approx., vol. 1, no. 1(2009), pp. 37-53. [7] r. a. devore, nonlinear approximation, acta numerica, vol. 7, cambridge university press,cambridge(1998), pp. 51-150. [8] shyam lal and susheel kumar “best wavelet approximation of function belonging to generalized lipschitz class using haar scaling function,” thai journal of mathematics, vol. 15(2017), no. 2, pp. 409-419. [9] y. meyer (1993)(toulouse(1992))(y. meyer and s. roques , eds) frontieres, gif-sur-yvette, wavelets their post and their future, progress in wavelet analysis and applications, pp. 9-18. [10] lal, shyam, and indra bhan. ”approximation of functions belonging to generalized hölder’s class h(ω)α [0, 1) by first kind chebyshev wavelets and its applications in the solution of linear and nonlinear differential equations.” international journal of applied and computational mathematics 5.6 (2019): 155. [11] lal, shyam, and rakesh. ”the approximations of a function belonging hölder classhα[0, 1)by second kind chebyshev wavelet method and applications in solutions of differential equation.” international journal of wavelets, multiresolution and information processing 17.01 (2019): 1850062. 109 introduction definitions and preliminaries legendre wavelet legendre wavelet approximation example theorems proofs proof of the theorem (4.1) proof of the theorem(4.2) conclusions acknowledgments ratio mathematica 25 (2013), 95–104 issn:1592-7415 the sum of the reduced harmonic series generated by four primes determined analytically and computed by using cas maple radovan pot̊uček department of mathematics and physics, faculty of military technology, university of defence, kounicova 65, 662 10 brno, czech republic radovan.potucek@unob.cz abstract the paper deals with the reduced harmonic series generated by four primes. a formula for the sum of these convergent reduced harmonic series is derived. these sums (concretely 42 from all 12650 sums generated by four different primes smaller than 100) are computed by using the computer algebra system maple 15 and its programming language, although the formula is valid not only for four arbitrary primes, but also for four integers. we can say that the reduced harmonic series generated by four primes (or by four integers) belong to special types of convergent infinite series, such as geometric and telescoping series, which sum can be found analytically by means of a simple formula. key words: reduced harmonic series, sum of convergent infinite series, computer algebra system maple. msc2010: 40a05, 65b10. 1 introduction this paper is inspired by a small study material from the berkeley math circle (see [5]) and it is a free continuation of the papers [2], [3], and [4]. 95 radovan pot̊uček in two last mentioned papers the sum s(a, b) of the convergent reduced harmonic series g(a, b) = 1 a + 1 b + ( 1 a2 + 1 ab + 1 b2 ) + ( 1 a3 + 1 a2b + 1 ab2 + 1 b3 ) + + ( 1 a4 + 1 a3b + 1 a2b2 + 1 ab3 + 1 b4 ) + ( 1 a5 + 1 a4b + · · · + 1 b5 ) + · · · , (1) generated by two primes a and b, and the sum s(a, b, c) of the convergent reduced harmonic series g(a, b, c) = 1 a + 1 b + 1 c + ( 1 a2 + 1 b2 + 1 c2 + 1 ab + 1 ac + 1 bc ) + + ( 1 a3 + 1 b3 + 1 c3 + 1 a2b + 1 a2c + 1 b2a + 1 b2c + 1 c2a + 1 c2b + 1 abc ) + + ( 1 a4 + 1 b4 + 1 c4 + 1 a3b + 1 a3c + 1 b3a + 1 b3c + 1 c3a + 1 c3b + + 1 a2b2 + 1 a2c2 + 1 b2c2 + 1 a2bc + 1 b2ac + 1 c2ab ) + · · · , (2) generated by three primes a, b, and c, were derived and also computed for primes less than 100. it was shown that for arbitrary two primes (or integers) a and b it holds the formula s(a, b) = a + b − 1 (a − 1)(b − 1) (3) and for arbitrary three primes (or integers) a, b, and c it holds the formula s(a, b, c) = (a + b − 1)(c − 1) + ab (a − 1)(b − 1)(c − 1) . (4) in the paper [2] the sum s of all the unit fractions that have denominators with only factors from the set {2, 7, 11, 13} was determined. this sum was calculated by using numeric method based on the programming language in the computer algebra system maple 15 and also by analytical method. by these both attempts was obtained the same result: s = 1.7805. in this paper we shall deal with a certain variant of these two problems – the determination of the sum of the reduced harmonic series generated by four primes. let us recall the basic terms and notions. the harmonic series is the sum of reciprocals of all natural numbers (except zero), so this is the series 96 the sum of the reduced harmonic series generated by four primes in the form ∞∑ n=1 1 n = 1 + 1 2 + 1 3 + · · · + 1 n + · · · . the divergence of this series can be easily proved e.g. by using the integral test or the comparison test of convergence. the reduced harmonic series is defined as the subseries of the harmonic series, which arises by omitting some its terms. as an example of the reduced harmonic series we can take the series formed by reciprocals of primes and number one 1 + 1 2 + 1 3 + 1 5 + 1 7 + 1 11 + 1 13 + · · · . this reduced harmonic series is divergent. the first proof of its divergence was made by leonhard euler (15. 4. 1707– 18. 9. 1783) in 1737 (see e.g. [1]). 2 reduced harmonic series generated by four primes now, let us consider the reduced harmonic series g(a, b, c, d) below, generated by four primes a, b, c, d: g(a, b,c, d) = 1 a + 1 b + 1 c + 1 d + ( 1 a2 + 1 b2 + 1 c2 + 1 d2 + 1 ab + 1 ac + 1 ad + + 1 bc + 1 bd + 1 cd ) + ( 1 a3 + 1 b3 + 1 c3 + 1 d3 + 1 a2b + 1 a2c + 1 a2d + + 1 b2a + 1 b2c + 1 b2d + 1 c2a + 1 c2b + 1 c2d + 1 d2a + 1 d2b + 1 d2c + + 1 abc + 1 abd + 1 acd + 1 bcd ) + ( 1 a4 + 1 b4 + 1 c4 + 1 d4 + + 1 a3b + 1 a3c + 1 a3d + 1 b3a + 1 b3c + 1 b3d + 1 c3a + 1 c3b + 1 c3d + + 1 d3a + 1 d3b + 1 d3c + 1 a2bc + 1 a2bd + 1 a2cd + 1 b2ac + 1 b2ad + + 1 b2cd + 1 c2ab + 1 c2ad + 1 c2bd + 1 d2ab + 1 d2ac + 1 d2bc + + 1 a2b2 + 1 a2c2 + 1 a2d2 + 1 b2c2 + 1 b2d2 + 1 c2d2 + 1 abcd ) + · · · . (5) analogously as in the cases of the reduced harmonic series generated by two and three primes, we assume that its sum s(a, b, c, d) is finite, so the 97 radovan pot̊uček series (5) converges. because all its terms are positive, then the series (5) converges absolutely and so we can rearrange it. for easier determining the sum s(a, b, c, d) of the series g(a, b, c, d) it is necessary to rearrange it and divide it into ten subseries g(a), g(b), g(c), g(d), g(ab), g(ac), g(ad), g(bc), g(bd), and g(cd), where g(a) = 1 a + 1 a2 + 1 a3 + 1 a4 + · · · = 1 a ( 1 + 1 a + 1 a2 + 1 a3 + · · · ) , (6) g(b) = 1 b + 1 b2 + 1 b3 + 1 b4 + · · · = 1 b ( 1 + 1 b + 1 b2 + 1 b3 + · · · ) , (7) g(c) = 1 c + 1 c2 + 1 c3 + 1 c4 + · · · = 1 c ( 1 + 1 c + 1 c2 + 1 c3 + · · · ) , (8) g(d) = 1 d + 1 d2 + 1 d3 + 1 d4 + · · · = 1 d ( 1 + 1 d + 1 d2 + 1 d3 + · · · ) , (9) g(ab) = 1 ab + 1 a2b + 1 b2a + 1 abc + 1 abd + 1 a3b + 1 b3a + 1 a2bc + 1 a2bd + + 1 b2ac + 1 b2ad + 1 c2ab + 1 d2ab + 1 a2b2 + 1 abcd + · · · = = 1 ab ( 1 + 1 a + 1 b + 1 c + 1 d + 1 a2 + 1 b2 + 1 c2 + 1 d2 + 1 ab + 1 ac + 1 ad + + 1 bc + 1 bd + 1 cd + · · · ) , (10) g(ac) = 1 ac + 1 a2c + 1 c2a + 1 acd + 1 a3c + + 1 c3a + 1 a2cd + 1 c2ad + 1 d2ac + 1 a2c2 + · · · = = 1 ac ( 1 + 1 a + 1 c + 1 d + 1 a2 + 1 c2 + 1 d2 + 1 ac + 1 ad + 1 cd + · · · ) , (11) g(ad) = 1 ad + 1 a2d + 1 d2a + 1 a3d + 1 d3a + 1 a2d2 + · · · = = 1 ad ( 1 + 1 a + 1 d + 1 a2 + 1 d2 + 1 ad + · · · ) , (12) 98 the sum of the reduced harmonic series generated by four primes g(bc) = 1 bc + 1 b2c + 1 c2b + 1 bcd + 1 b3c + + 1 c3b + 1 b2cd + 1 c2bd + 1 d2bc + 1 b2c2 + · · · = = 1 bc ( 1 + 1 b + 1 c + 1 d + 1 b2 + 1 c2 + 1 d2 + 1 bc + 1 bd + 1 cd + · · · ) , (13) g(bd) = 1 bd + 1 b2d + 1 d2b + 1 b3d + 1 d3b + 1 b2d2 + · · · = = 1 bd ( 1 + 1 b + 1 d + 1 b2 + 1 d2 + 1 bd + · · · ) , (14) g(cd) = 1 cd + 1 c2d + 1 d2c + 1 c3d + 1 d3c + 1 c2d2 + · · · = = 1 cd ( 1 + 1 c + 1 d + 1 c2 + 1 d2 + 1 cd + · · · ) . (15) 3 analytic solution now, we determine by the analytic way the unknown sum s(a, b, c, d) by means of the sums of the series (6) – (15). by the formula s = a1 1 − q , for the sum s of the convergent infinite geometric series with the first term a1 and with the ratio q, |q| < 1, we get the sums s(a), s(b), s(c), and s(d) of the series (6) – (9): s(a) = 1 a · 1 1 − 1/a = 1 a · a a − 1 = 1 a − 1 (16) and, analogously s(b) = 1 b − 1 , s(c) = 1 c − 1 , s(d) = 1 d − 1 . (17) it is clear that the sum s(ab) of the series (10) we can write in the form s(ab) = 1 ab [ 1 + s(a, b, c, d) ] . (18) 99 radovan pot̊uček the sum s(ac) of the series (11) is the product of the fraction 1/(ac) and the sum of number one and the reduced harmonic series generated by three primes a, c, and d. so, by the formula (4) above, we can write s(ac) = 1 ac ( 1 + (a + c − 1)(d − 1) + ac (a − 1)(c − 1)(d − 1) ) = = (d − 1) [ (a − 1)(c − 1) + (a + c − 1) ] + ac ac(a − 1)(c − 1)(d − 1) = = (d − 1)(ac − a − c + 1 + a + c − 1) + ac ac(a − 1)(c − 1)(d − 1) = = (d − 1)ac + ac ac(a − 1)(c − 1)(d − 1) = acd ac(a − 1)(c − 1)(d − 1) = = d (a − 1)(c − 1)(d − 1) . (19) because the sum s(bc) of the series (13) is the product of the fraction 1/(bc) and the sum of number one and the reduced harmonic series generated by three primes b, c, and d, we can analogously write s(bc) = d (b − 1)(c − 1)(d − 1) . (20) obviously, the sum s(ad) of the series (12) is the product of the fraction 1/(ad) and the sum of number one and the reduced harmonic series generated by two primes a and d. so, by the formula (3) above, we can write s(ad) = 1 ad ( 1 + a + d − 1 (a − 1)(d − 1) ) = (a − 1)(d − 1) + a + d − 1 ad(a − 1)(d − 1) = = ad − a − d + 1 + a + d − 1 ad(a − 1)(d − 1) = 1 (a − 1)(d − 1) (21) and, analogously for the sums s(bd) and s(cd) of the series (14) and (15), we get s(bd) = 1 (b − 1)(d − 1) , s(cd) = 1 (c − 1)(d − 1) . (22) by the assumption of the absolute convergence of the series (5) we can write its sum s(a, b, c, d) in the form s(a) + s(b) + s(c) + s(d) + s(ab) + s(ac) + s(ad) + s(bc) + s(bd) + s(cd). 100 the sum of the reduced harmonic series generated by four primes according to (16) – (22) we get the equation s(a, b, c, d) = 1 a − 1 + 1 b − 1 + 1 c − 1 + 1 d − 1 + 1 + s(a, b, c, d) ab + + d (a − 1)(c − 1)(d − 1) + d (b − 1)(c − 1)(d − 1) + + 1 (a − 1)(d − 1) + 1 (b − 1)(d − 1) + 1 (c − 1)(d − 1) . multiplying both sides of this equation by ab(a− 1)(b− 1)(c− 1)(d− 1), we obtain the equation (ab − 1)(a − 1)(b − 1)(c − 1)(d − 1)s(a, b, c, d) = ab [ (b − 1)(c − 1)(d − 1) + + (a − 1)(c − 1)(d − 1) + (a − 1)(b − 1)(d − 1) + (a − 1)(b − 1)(c − 1) ] + + (a − 1)(b − 1)(c − 1)(d − 1) + abd [ (b − 1) + (a − 1) ] + + ab [ (b − 1)(c − 1) + (a − 1)(c − 1) + (a − 1)(b − 1) ] . it is easy to derive (e.g. by means of the computer algebra system maple 15 and its simplify and factor statements) that it holds s(a, b, c, d) = = abc + abd + acd + bcd − ab − ac − ad − bc − bd − cd + a + b + c + d − 1 (a − 1)(b − 1)(c − 1)(d − 1) , i.e. s(a, b, c, d) = [ (a + b − 1)(c − 1) + ab ] (d − 1) + abc (a − 1)(b − 1)(c − 1)(d − 1) . (23) this formula can be also written in another two equivalent forms: s(a, b, c, d) = [ (a + c − 1)(b − 1) + ac ] (d − 1) + abc (a − 1)(b − 1)(c − 1)(d − 1) and s(a, b, c, d) = [ (b + c − 1)(a − 1) + bc ] (d − 1) + abc (a − 1)(b − 1)(c − 1)(d − 1) . 4 numeric solution for approximate calculation of the sums s(a, b, c, d) for the primes a, b, c, d < 100, i.e. for 25 primes a, b, c, d ∈{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}, we use the computer algebra system maple 15. the sums s(a, b, c, d) we calculate for concrete four primes by the following for statements and by the procedure partabcd: 101 radovan pot̊uček partabcd:=proc(a,b,c,d) local s; s:=(((a+b-1)*(c-1)+a*b)*(d-1)+a*b*c)/((a-1)*(b-1)*(c-1)*(d-1)); print("s(a,b,c,d) for a=",a,"b=",b,"c=",c,"d=",d,"is",evalf[8](s)); end proc: p:=[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79, 83,89,97]: for i in p do for j in p do for k in p do for l in p do if (i < j and j < k and k < l) then partabcd(i,j,k,l); end if; end do; end do; end do; end do; fourty-two representative sums of these 12650 sums s(a, b, c, d), where 12650 is the number of combinations of size 4 from a collection of size 25, i.e.( 25 4 ) = 25! (25 − 4)! 4! = 25 · 24 · 23 · 22 4! = 12650, are presented in two following tables. there are 27 sums with a finite decimal expansion (called regular numbers) with at most 4 decimals in the table 1 and another 15 sums, including the sum s(2, 7, 11, 13) = 1.7805 calculated in the paper [2] and mentioned above, rounded to 6 decimals, in the table 2: (a, b, c, d) s(a, b, c, d) (a, b, c, d) s(a, b, c, d) (a, b, c, d) s(a, b, c, d) (2, 3, 5, 7) 3.375 (2, 3, 11, 13) 2.575 (2, 5, 11, 23) 1.875 (2, 3, 5, 11) 3.125 (2, 3, 11, 23) 2.45 (2, 11, 23, 41) 1.3575 (2, 3, 5, 13) 3.0625 (2, 3, 11, 31) 2.41 (2, 11, 23, 47) 1.35 (2, 3, 5, 31) 2.875 (2, 3, 11, 61) 2.355 (2, 11, 41, 83) 1.2825 (2, 3, 5, 61) 2.8125 (2, 3, 11, 67) 2.35 (2, 67, 79, 89) 1.0797 (2, 3, 7, 11) 2.85 (2, 3, 11, 89) 2.3375 (3, 7, 11, 23) 1.0125 (2, 3, 7, 29) 2.625 (2, 3, 13, 53) 2.3125 (3, 7, 11, 71) 0.9525 (2, 3, 7, 41) 2.5875 (2, 3, 31, 41) 2.1775 (3, 11, 23, 31) 0.7825 (2, 3, 7, 71) 2.55 (2, 3, 41, 83) 2.1125 (3, 11, 23, 43) 0.7625 table 1: the table with some values of the sums s(a, b, c, d) 102 the sum of the reduced harmonic series generated by four primes (a, b, c, d) s(a, b, c, d) (a, b, c, d) s(a, b, c, d) (a, b, c, d) s(a, b, c, d) (2, 3, 5, 17) 2.984375 (2, 5, 7, 31) 2.013889 (2, 23, 37, 43) 1.200156 (2, 3, 5, 73) 2.802083 (2, 5, 11, 53) 1.802885 (3, 7, 13, 19) 1.001157 (2, 3, 7, 31) 2.616667 (2, 7, 11, 13) 1.780556 (3, 7, 59, 89) 0.800402 (2, 3, 11, 29) 2.417857 (2, 5, 37, 83) 1.600779 (3, 31, 53, 79) 0.600002 (2, 3, 19, 89) 2.202652 (2, 7, 59, 89) 1.400536 (79, 83, 89, 97) 0.047622 table 2: the table with another values of the sums s(a, b, c, d) 5 conclusion in this paper the sums s(a, b, c, d) of the convergent reduced harmonic series g(a, b, c, d) generated by four primes a, b, c and d were derived. these sums were computed for a, b, c, d < 100, although the formula s(a, b, c, d) = [ (a + b − 1)(c − 1) + ab ] (d − 1) + abc (a − 1)(b − 1)(c − 1)(d − 1) derived above gives results for arbitrary four different primes a, b, c, d. so that, for example s(101,103,107,109) = (203·106 + 101·103)·108 + 101·103·107 100·102·106·108 . = 0.039056. it is clear that this formula is valid not only for four primes, but also for four integers. for example the sum of the series 1 2 + 1 4 + 1 6 + 1 8 + 1 22 + 1 42 + 1 62 + 1 82 + 1 2·4 + 1 2·6 + 1 2·8 + 1 4·6 + 1 4·8 + 1 6·8 + 1 23 + 1 43 +· · · is s(2, 4, 6, 8) = (5 · 5 + 2 · 4) · 7 + 2 · 4 · 6 1 · 3 · 5 · 7 . = 2.657143. we can say that the reduced harmonic series g(a, b, c, d) generated by four primes (or by four integers) belong to special types of convergent infinite series, such as geometric and telescoping series, which sum can be found analytically by means of a simple formula. references [1] g. h. hardy and e. m. wright, an introduction to the theory of numbers, 4th edition. oxford university press, london, 1975. isbn 978-0-19-853310-7. 103 radovan pot̊uček [2] r. pot̊uček, solving one infinite series problem using cas maple and analytically. proceedings of international conference presentation of mathematics ’11. liberec: faculty of science, humanities and education of the technical university of liberec, liberec, 2011, 107-112. isbn 978-80-7372-773-4. [3] r. pot̊uček, the sum of the reduced harmonic series generated by two primes determined analytically and computed by using cas maple. zborńık vedeckých prác ”aplikované úlohy v modernom vyučovańı matematiky”. slovenská pol’nohospodárska univerzita v nitre, 2012, 25-30. isbn 978-80-552-0823-7. [4] r. pot̊uček, the sum of the reduced harmonic series generated by three primes determined analytically and computed by using cas maple. zborńık vedeckých prác ”aplikácie matematiky – vstupná brána rozvoja matematických kompetencíı”. slovenská pol’nohospodárska univerzita v nitre, 2013, 84-89. isbn 978-80-552-1047-6. [5] t. rike, infinite series (berkeley seminary). berkeley math circle, march 24, 2002, 6 pp. 104 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 40, 2021, pp. 67-75 67 certain results on metric and norm in fuzzy multiset setting johnson mobolaji agbetayo* paul augustine ejegwa† abstract fuzzy multiset is an extension of fuzzy set in multiset framework. in this paper, we review the concept of fuzzy multisets and study the notions of metric and norm on fuzzy multiset. some results on metric and norm are established in fuzzy multiset context. keywords: fuzzy set; fuzzy multiset; metric; norm. 2010 ams subject classification: 03e72, 08a72, 20n25. *department of mathematics/statistics/computer science, federal university of agriculture, p.m.b. 2373, makurdi, nigeria; agbetayojohnson@gmail.com. † department of mathematics/statistics/computer science, federal university of agriculture, p.m.b. 2373, makurdi, nigeria; ocholohi@gmail.com. § received on january 12th, 2021. accepted on may 12th, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.582. issn: 1592-7415. eissn: 2282-8214. ©j. m. agbetayo and p. a. ejegwa. this paper is published under the cc-by licence agreement. j. m. agbetayo and p. a. ejegwa 68 1. introduction the theory of set proposed by cantor in 1915, is a collection of well-defined objects of thought and intuition. the limitation of set theory is its inability to deal with the vague properties of its member or element, and likewise its distinctness property which does not allows repetition in the collection. in other to handle the vague property of a set, zadeh [19] proposed a mathematical model that deal with vagueness of a set known as fuzzy sets. the distinct property of crisp set has been violated by allowing repetition of an element in a collection. this gave birth to a set called multiset. the term multiset was first suggested by de bruijn to knuth in a private correspondence as noted in [13]. the theory of multisets has been studied [3, 4, 10, 11, 12, 16]. lake [14] presented an abridge account on sets, fuzzy sets, multisets and functions. by synthesizing the concepts of fuzzy sets and multisets, yager [18] introduced the concept of fuzzy multiset (fms) that deal with vagueness property of a set and allowed the repetition of its membership function. in fact, fuzzy bag or fuzzy multiset generalizes fuzzy sets in such a way that the membership degree of a fuzzy set is allowed to repeat. some fundamentals properties of fuzzy bags have been studied [6, 15]. the concept of fuzzy bags has been applied in multi-criteria decision-making [1, 2], sequences [5] and computational science [17]. metric is a function that defines a concept of distance between any members of the set, which are usually called points. the notions of metric and norm have been extended to the environment of fuzzy sets [7, 8, 9]. in this work, we present the notions of norm and metric in fuzzy multiset context. 2. preliminaries in this section, we review some definitions and result that are important for the main work. certain results on metric and norm in fuzzy multiset setting 69 definition 2.1 [18]. assume is a set of elements. then, a fuzzy bag/multiset, drawn from can be characterized by a count membership function such that where is the set of all crisp bags or multisets from the unit interval, according to syropoulos [17], a fuzzy multiset can also be characterized by a high-order function. in particular, a fuzzy multiset can be characterized by a function or where and the count membership degrees, for is given as where ,… ∈ [0,1] such that ( ) ≥ ( ) ≥ ( ), ≥ … ≥ (x) ≥ …, whereas in a finite case, we write = { ( ), ( ),…, ( )} for ( ) ≥ ( )≥ … ≥ ( ). a fuzzy multiset can be represented in the form or in a simple term, a fuzzy multiset, of is characterized by the count membership function, for , that takes the value of a multiset of a unit interval . we denote the set of all fuzzy multisets by example 2.2. assume that is a set. then for ={0.5,0.4,0.2}, a is a fuzzy multiset of written as definition 2.3 [15]. let then, is called a fuzzy submultiset of written as if . also, if and , then a is called a proper fuzzy submultiset of and denoted as . j. m. agbetayo and p. a. ejegwa 70 definition 2.4 [15]. let then, and are comparable to each other if and only if definition 2.5 [17]. let .then, the intersection and union of , denoted by and are defined by (i) (ii) definition 2.6 [17]. let . then, the sum of and , denoted by , is defined by the addition operation in for crisp multiset. that is, the addition operation is carry out by merging the membership degree in a decreasing order. definition 2.7 [6]. let . then, the difference of from is a fuzzy multiset such that . definition 2.8 [6]. let . then, the complement of is a fuzzy multiset such that metric and norm defined over fuzzy multisets 3. metric and norm defined over fuzzy multisets in this section, we present metrics and norm defined over fuzzy multiset. definition 3.1. let be an arbitrary non-empty set and let . a metric or distance function between a and b on is a function with the following properties: (i) . (ii) iff . (iii) . (iv) certain results on metric and norm in fuzzy multiset setting 71 if note: (i) the distance is a non-negative function and only zero at a single point. (ii) the distance is a symmetric function. (iii) the distance satisfy triangle. proposition 3.2. let then is a metric defined on proof. we use definition 3.1: axiom (i) . axiom (ii) if . conversely, if . axiom (iii) . axiom (iv) the following are distances between fuzzy multisets: hamming distance; euclidean distance; normalized hamming distance; . normalized euclidean distance; j. m. agbetayo and p. a. ejegwa 72 . theorem 3.3. let be non-empty set and then . proof. we show that or . but and . thus, = hence corollary 3.4. if is a distance of fuzzy multiset of and then pr oof. clearly, . proposition 3.5. if is a metric of fuzzy multiset and then proof. by definition 3.1, if , so it follows that . proposition 3.6. let and is a metric defined on then is also a metric. certain results on metric and norm in fuzzy multiset setting 73 proof. the proof is obvious, since hence is a metric. corollary 3.7. if λ then . proof. the proof is straightforward. corollary 3.8. if then . proof. the proof is straightforward. corollary 3.9. if then proof. the proof is straightforward. definition 3.10. let be a non-empty set and be a fuzzy multiset of x. a non-negative real-valued function defined on is called a norm if the following properties are satisfied: (i) iff that is, iff (ii) which implies that (iii) which implies that . the fuzzy multiset equipped with a norm is called normed fuzzy multiset. proposition 3.11. let then proof. we show that . now, . proposition 3.12. let and a norm define over . proof. (i) . (ii) j. m. agbetayo and p. a. ejegwa 74 (iii) hence is a norm defined over fuzzy multiset corollary 3.13. if then and if , then . proof. the proof is obvious. 4 conclusions we have presented a brief review on the concept of fuzzy multisets and explored metric and norm in fuzzy multiset context. a number of results on metric and norm were established, respectively. references [1] l. baowen. fuzzy bags and applications. fuzzy sets and systems, 34, 61 71, 1990. [2] r. biswas. an application of yager’s bag theory in multicriteria based decision-making problems. international journal of intelligent systems, 14, 1231-1238, 1999. [3] w.d. blizard. multiset theory. notre dame journal of formal logic, 30(1), 36-66, 1989. [4] w.d. blizard. dedekind multisets and function shells. theoretical computer science, 110, 79-98, 1993. [5] p.a. ejegwa. correspondence between fuzzy multisets and sequences. global journal of science frontier research: mathematics and decision science, 14(7), 61-66, 2014. [6] p.a. ejegwa. synopsis of the notions of multiset and fuzzy multiset. annals of communications in mathematics, 2(2), 101-120, 2019. [7] p.a. ejegwa and j.m. agbetayo. norm and metric defined over fuzzy set. annals of pure and applied mathematics, 10(1), 59-64, 2015. certain results on metric and norm in fuzzy multiset setting 75 [8] c. felbin. finite dimensional fuzzy normed linear spaces. fuzzy sets and systems, 48:239-248, 1992. [9] g. gebray and b.k. reddy. fuzzy metric on fuzzy linear space. international journal of science and research, 3(6), 2286-2287, 2014. [10] j.l. hickman. a note on the concept of multiset. bulletin of the australian mathematical society, 22:211-217, 1980. [11] a.m. ibrahim, d. singh and j.n. singh. an outline of multiset space algebra. international journal of algebra, 5(31), 1515-1525, 2011. [12] s.p. jena, s.k. ghosh and b.k. tripathy. on the theory of bags and lists. information sciences, 132, 241-254, 2001. [13] d.e. knuth. the art of computer programming. vol. 2: seminumerical algorithms, addison-wesley, 1981. [14] j. lake. sets, fuzzy sets, multisets and functions. journal of london mathematical society, 12(2), 32-326, 1976. [15] s. miyamoto. basic operation of fuzzy multiset. journal of japan society fuzzy theory system, 8(4), 639-645, 1996. [16] s. miyamoto. multisets and their generalizations. institute of engineering mechanics and system university of tsukuba, japan, 2014. [17] a. syropoulos. on generalized fuzzy multisets and their use in computation, iranian journal of fuzzy system, 9(2), 113-125, 2012. [18] r.r. yager. on the theory of bags. international journal of general systems, 13, 23-37, 1986. [19] l.a. zadeh. fuzzy sets. information and control, 8, 338-353, 1965. microsoft word cap11.doc ratio mathematica volume 39, 2020, pp. 55-67 new structure of norms on rn and their relations with the curvature of the plane curves amir veisi* ali delbaznasab† abstract let f1,f2, . . . ,fn be fixed nonzero real-valued functions on r, the real numbers. let ϕn(xn) = ( x21f 2 1 + x 2 2f 2 2 + . . . + x 2 nf 2 n )1 2 , where xn = (x1,x2, . . . ,xn) ∈ rn. we show that ϕn has properties similar to a norm function on the normed linear space. although ϕn is not a norm on rn in general, it induces a norm on rn. for the nonzero function f : r2 → r, a curvature formula for the implicit curve g(x,y) = f2(x,y) = c 6= 0 at any regular point is given. a similar result is presented when f is a nonzero function from r3 to r. in continued, we concentrate on f(x,y) = ∫ b a ϕ2(x,y)dt. it is shown that the curvature of f(x,y) = c, where c > 0 is a positive multiple of c2. particularly, we observe that f(x,y) = ∫ π 2 0 √ x2 cos2 t + y2 sin2 tdt is an elliptic integral of the second kind. keywords: norm; curvature; homogeneous function; elliptic integral. 2010 ams subject classifications: 53a10. 2010 ams subject classifications: 53a10. 1 *faculty of petroleum and gas, yasouj university, gachsaran, iran; aveisi@yu.ac.ir †farhangian university, kohgiluyeh and boyer-ahmad province, yasouj, iran; delbaznasab@gmail.com 1received on october 31st, 2020. accepted on december 17th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.552. issn: 1592-7415. eissn: 2282-8214. ©amir veisi et al. this paper is published under the cc-by licence agreement. 55 amir veisi and ali delbaznasab 1 introduction a normed linear space is a real linear space x such that a number ‖x‖, the norm of x, is associated with each x ∈ x, satisfying: ‖x‖≥ 0 and ‖x‖ = 0 if and only if x = 0; ‖λx‖ = |λ|‖x‖ for all λ ∈ r and ‖x + y‖≤‖x‖+‖y‖. for example, let x be a tychonoff space, c∗(x) the ring of all bounded realvalued continuous functions on x. then c∗(x) is a normed linear space with the norm ‖f‖ = sup{|f(x)| : x ∈ x} and pointwise addition and scalar multiplication. this is called the supremum-norm on c∗(x). the associated metric is defined by d(f,g) = ‖f − g‖. a non-empty set c ⊆ rn is called a convex set if whenever p and q belong to c, the segment joining p and q belongs to c. analytically the definition can be formulated in this way: if p is represented by the vector x, and q by the vector y, then c is a convex set if with p and q it contains also every point with a vector of form λx + (1 −λ)y, where 0 ≤ λ ≤ 1. a point p is an interior point of a set s contained in rn, if there exists an n-dimensional ball, with center at p , all of whose points lie in s. an open set is a set containing only interior points. a subset c ⊆ rn is centrally symmetric (or 0-symmetric) if for every point q ∈ rn contained in c, −q ∈ c, where −q is the reflection of q through the origin, that is c = −c. definition 1.1. ([siegel, 1989, page 5]) a convex body is a bounded, centrally symmetric convex open set in rn. example 1.1. the interior of an n-dimensional ball, defined by x21 + x 2 2 + · · · + x2n < a 2 provides an example of a convex body. one of the many important ideas introduced by minkowski into the study of convex bodies was that of gauge function. roughly, the gauge function is the equation of a convex body. minkowski showed that the gauge function could be defined in a purely geometric way and that it must have certain properties analogous to those possessed by the distance of a point from the origin. he also showed that conversely given any function possessing these properties, there exists a convex body with the given function as its gauge function. definition 1.2. ([siegel, 1989, page 6]) given a convex body b ⊆ rn containing the origin o, we define a function f : rn → [0,∞) as follows. f(x) =   1 if x ∈ ∂b, 0 if x = 0, λ if 0 6= x = λy, where λ is the unique positive real number such that the ray through o and the point (whose vector is) x intersects the surface ∂b ( the boundary of b) in a point y. the function f so defined is the gauge function of the convex body b. 56 new structure of norms on rn and their relations with the curvature of the plane curves example 1.2. let f : r → [0,∞) defined by f(x) = max{|x1|, |x2|, . . . , |xn|}, where x = (x1,x2, . . . ,xn) ∈ rn. then intb, the interior of the cubic b = {(x1,x2, . . . ,xn) : |xi| ≤ 1} is a convex body and f is a gauge function of it. it is shown in [siegel, 1989, theorems 4-7] that a function f : r → [0,∞) is a gauge function if and only if the following conditions hold: f(x) ≥ 0 for x 6= 0, f(0) = 0; f(λx) = λf(x), for 0 ≤ λ ∈ r; and f(x + y) ≤ f(x) + f(y). moreover, f is continuous and the convex body of f is b = {x : f(x) < 1}. a brief outline of this paper is as follows. in section 2, we introduce a function ϕn on rn, by the formula ϕn(xn) = √ x21f 2 1 + x 2 2f 2 2 + · · ·+ x2nf2n, when n fixed nonzero real-valued functions f1,f2, . . . ,fn on r are given. we show that the mappings ϕn have similar properties such as norm functions within difference the ranges of these functions lie in rr while the range of a norm function is in the [0,∞). this definition allows us to define a norm and hence a gauge function on rn. so it turns rn into a metric space. in section 3, we focus on n = 2, ϕ2 and the induced norm on r2. first, we show that if f : r2 → r is a nonzero function, then k, the curvature of the implicit g(x,y) = f2(x,y) = c 6= 0 at every regular point is calculated by this formula: k = |hg|−4f2|hf| 4f ( f2x + f 2 y ) 3 2 , where hf and hg are the hessian matrices of f and g respectively. it is also shown if f(x,y) = ∫ b a √ x2f2(t) + y2g2(t)dt, then |hf| = 0 and the eigenvalues of hf and hg, where g = f2 are nonnegative. particularly, when f(t) = cost and g(t) = sint, we prove that ∫ π 2 0 √ x2f2(t) + y2g2(t)dt is an elliptical integral of the second type. 2 a norm on rn made by the real valued functions on r we begin with the following notation. notation 2.1. suppose that f1,f2, . . . ,fn are nonzero real-valued functions on r and define ϕn : rn → rr with ϕn(xn) = √ x21f 2 1 + x 2 2f 2 2 + · · ·+ x2nf2n, (∗) 57 amir veisi and ali delbaznasab where xn = (x1,x2, . . . ,xn) and rr is the set (in fact, ring) of all real-valued functions on r. the following statement is a key lemma. however, its proof is straightforward and elementary, it will be used in the proof of the triangle inequality in the next results. lemma 2.1. let a,b,c and d are nonnegative real numbers. then √ ac + √ bd ≤ √ (a + b)(c + d). proposition 2.1. let xn,yn ∈ rn, n = 1,2 or 3. then ϕn(xn+yn) ≤ ϕn(xn)+ ϕn(yn). proof. the inequality clearly holds when n = 1. next, we do the proof for n = 2. take x2 = (x1,y1), y2 = (x2,y2) ∈ r2 and suppose that f and g are nonzero elements of rr. then ϕ2(x2 + y2) = √ (x1 + x2)2f2 + (y1 + y2)2g2 ≤ √ x21f 2 + y21g 2 + √ x22f 2 + y22g 2 = ϕ2(x2) + ϕ2(y2) if and only if x1x2f 2 + y1y2g 2 ≤ √[ x21f 2 + y21g 2 ][ x22f 2 + y22g 2 ] = ϕ2(x2)ϕ2(y2). (?) now, if we let b := x1x2f2 + y1y2g2 and suppose that b ≥ 0, then (?) holds if and only if f2g2(x1y2 −x2y1)2 ≥ 0, which is always true (note, (?) trivially holds if b ≤ 0). hence, in this case, the proof is complete. here, we prove the proposition for n = 3. let x3 = (x1,y1,z1) = (x2,z1) and y3 = (x2,y2,z2) = (y2,z2), where x2 = (x1,y1), y2 = (x2,y2) and let f,g,h be nonzero elements of rr. then ϕ3(x3 + y3) = √ (x1 + x2)2f2 + (y1 + y2)2g2 + (z1 + z2)2h2 ≤ √ x21f 2 + y21g 2 + z21h 2 + √ x22f 2 + y22g 2 + z22h 2 = ϕ3(x3) + ϕ3(y3) 58 new structure of norms on rn and their relations with the curvature of the plane curves if and only if x1x2f 2 + y1y2g 2 + z1z2h 2 ≤ √ [x21f 2 + y21g 2 + z21h 2][x22f 2 + y22g 2 + z22h 2] = √[ ϕ22(x2) + z 2 1h 2 ][ ϕ22(y2) + z 2 2h 2 ] now, if we let a = ϕ22(x2),b = z 2 1h 2,c = ϕ22(y2) and d = z 2 2h 2, then by (∗) in notation 2.1, we have x1x2f 2 + y1y2g 2 ≤ √ ac. moreover, it is clear that z1z2h2 ≤ √ bd. therefore, x1x2f 2 + y1y2g 2 + z1z2h 2 ≤ √ ac + √ bd. in view of lemma 2.1, the proof is now complete. next, we state the general case of proposition 2.1. theorem 2.1. let xn = (x1,x2, . . . ,xn),yn = (y1,y2, . . . ,yn) ∈ rn,λ ∈ r and ϕn be as defined in notation 2.1. then the following statements hold. (i) ϕn(xn) = 0 if and only if xn = 0, (ii) ϕn(λxn) = |λ|ϕn(xn), (iii) ϕn(xn + yn) ≤ ϕn(xn) + ϕn(yn) (triangle inequality). proof. (i) and (ii) are evident. (iii). the proof is done by induction on n, see proposition 2.1. if we set xn−1 = (x1,x2, . . . ,xn−1) and yn−1 = (y1,y2, . . . ,yn−1) then xn and yn can be substituted by (xn−1,xn) and (yn−1,yn) respectively. therefore, ϕn(xn + yn) ≤ ϕn(xn) + ϕn(yn) if and only if x1y1f 2 1 + · · ·+ xnynf 2 n ≤ ϕn(xn)ϕn(yn) = √[ ϕ2n−1(xn−1) + x 2 nf 2 n ][ ϕ2n−1(yn−1) + y 2 nf 2 n ] . now, let a = ϕ2n−1(xn−1),b = x 2 nf 2 n,c = ϕ 2 n−1(yn−1) and d = y 2 nf 2 n plus the assumption of induction, we have x1y1f 2 1 + · · ·+ xn−1yn−1f 2 n−1 ≤ √ ac. moreover, it is obvious that xnynf2n ≤ √ bd. thus, x1y1f21 + · · · + xnynf2n ≤√ ac + √ bd. lemma 2.1 now yields the result. 59 amir veisi and ali delbaznasab corollary 2.1. if f1,f2, . . . ,fn are nonzero constant functions, then ϕn is a norm (and hence a gauge function) on rn. by theorem 2.1, we obtain the following result. proposition 2.2. let a,b be real numbers, f1,f2, . . . , and fn the restrictions of some non-zero elements of rr on [a,b] such that each of them is nonzero on this set, and let ϕn be as defined in the previous parts (notation 2.1). then the mapping ψn : rn → [0,∞) defined by ψn(xn) = ∫ b a ϕn(xn)dt is a norm on rn, and hence d(xn,yn) = ψ(xn − yn) turns rn into a metric space. corollary 2.2. the mapping ψn is a gauge function on rn with the convex body cn = {xn ∈ rn : ψn(xn) < 1}. 3 f(x,y) = ∫ b a ϕ2(x,y)dt as a norm on r2 and the curvature in the plane proposition 3.1. ([goldman, 2005, proposition 3.1]) for a curve defined by the implicit equation f(x,y) = 0, the curvature of f (denoted by κ) at a regular point (x0,y0) (i.e., the first partial derivatives fx and fy at this point are not both equal to 0) is given by the formula κ = |f2y fxx −2fxfyfxy + f2xfyy|( f2x + f 2 y )3 2 , where fx denotes the first partial derivative with respect to x, fy, fxx denotes the second partial derivative with respect to x, fyy, and fxy denotes the mixed second partial derivative (for readability of the above formulas, the argument (x0,y0) has been omitted). we recall that the hessian matrix of z = f(x,y) and w = f(x,y,z) are defined to be hz = [ fxx fxy fyx fyy ] and hw =  fxx fxy fxzfyx fyy fyz fzx fzy fzz   at any point at which all the second partial derivatives of f exist. 60 new structure of norms on rn and their relations with the curvature of the plane curves theorem 3.1. let f : r2 → r be a nonzero function and (x0,y0) ∈ r2 a regular point. suppose that the second partial derivatives of f at (x0,y0) exist and further fxy = fyx at this point. let hf and hg be the hessian matrices of f and f2 respectively (we assume that g = f2) and let k be the curvature of g(x,y) = f2(x,y) = c 6= 0 at (x0,y0). then we have k = |hg|−4f2|hf| 4f ( f2x + f 2 y ) 3 2 . proof. for simplicity, we do the proof without (x0,y0). the partial derivatives of g = f2 are as follows: gx = 2ffx, gxx = 2(fx 2 + ffxx), gy = 2ffy, gyy = 2(f 2 y + ffyy), and g 2 xy = 4(fxfy + ffxy) 2. therefore, |hg| = gxxgyy −g2xy = 4 ( fx 2 + ffxx )( f2y + ffyy ) −4 ( fxfy + ffxy )2 = 4 [ f2xf 2 y + ff 2 xfyy + ff 2 y fxx + f 2fxxfyy −f2xf 2 y −2ffxfyfxy −f2f2xy ] = 4 [ f2 ( fxxfyy −f2xy ) + f ( f2xfyy −2fxfyfxy + f 2yfxx )] = 4 [ f2|hf|+ f ( f2xfyy −2fxfyfxy + f 2yfxx )] . in view of proposition 3.1, we have |hg| = 4 [ f2|hf|+ f ( f2xfyy −2fxfyfxy + f 2yfxx )] = 4 [ f2|hf|+ fk ( f2x + f 2 y ) 3 2 ] therefore, k = |hg|−4f2|hf| 4f ( f2x + f 2 y ) 3 2 , and we are done. the next result is a similar consequence for the implicit surface. theorem 3.2. let f : r3 → r be a nonzero function and (x0,y0,z0) ∈ r3 a regular point. suppose that the second partial derivatives of f at (x0,y0,z0) exist 61 amir veisi and ali delbaznasab and further the mixed partial derivatives at this point are equivalent. if k is the curvature of g(x,y,z) = f2(x,y,z) = c 6= 0 at (x0,y0,z0), then we have k = |hg|−8f3|hf| 8f2 ( f2x + f 2 y + f 2 z ) 3 2 , where hf and hg are the hessian matrices of f and f2 respectively (we assume that g = f2). proof. as we did in the previous theorem, the proof is done without (x0,y0,z0). let k =   fxx fxy fxz fx fxy fyy fyz fy fxz fyz fzz fz fx fy fz 0  . it is known that the curvature k of the implicit surface f(x,y,z) = 0 is k = |k| at every regular point in which the second partial derivatives of f exist. we first calculate the partial derivatives of g and in continued we obtain determinant of hg. gx = 2ffx, gxx = 2(fx 2 + ffxx), g 2 xy = 4(fxfy + ffxy) 2 gy = 2ffy, gyy = 2(f 2 y + ffyy), g 2 xz = 4(fxfz + ffxz) 2 gz = 2ffz, gzz = 2(f 2 z + ffzz), g 2 yz = 4(fyfz + ffyz) 2. recall that the hessian matrices of f and g are hf =  fxx fxy fxzfxy fyy fyz fxz fyz fzz  , and hg =  gxx gxy gxzgxy gyy gyz gxz gyz gzz  . here, we compute the determinant of hg. 1/8|hg| = fxx ( fyyfzz −f2yz ) −fxy ( fxyfzz −fxzfyz ) + fxz ( fxyfyz −fxzfyy ) = fxxfyyfzz −fxxf2yz −fyyf 2 xz −fzzf 2 xy + 2fxyfyzfxz = ( f2x + ffxx )( f2y + ffyy )( f2z + ffzz ) − ( f2x + ffxx )( fyfz + ffyz )2 − ( f2y + ffyy )( fxfz + ffxz )2 −(f2z + ffzz)(fxfy + ffxy)2 + ( fxfz + ffxz )( fyfz + ffyz )( fxfy + ffxy ) = f3 [ fxxfyyfzz −fxxf2yz −fyyf 2 xz −fxxf 2 xy + 2fxyfyzfxz ] + f2 [ fxxfyyf 2 z + fxxfzzf 2 y + fyyfzzf 2 x −2fxyfxzfyfz −2fxyfyzfxfz −2fxzfyzfxfy + f2xyf 2 z + f 2 xzf 2 y + f 2 yzf 2 x ] + f [ 0 ] . 62 new structure of norms on rn and their relations with the curvature of the plane curves therefore, we have 1/8|hg| = f3|hf| + f2k(f2x + f2y + f2z ) 3 2 . so the result is obtained, i.e., k = |hg|−8f3|hf| 8f2 ( f2x + f 2 y + f 2 z ) 3 2 . theorem 3.3. let f,g be nonzero real-valued functions on r, a,b ∈ r and f : r2 → r defined by f(x,y) = ∫ b a √ x2f2(t) + y2g2(t)d(t). then (i) the curvature of f(x,y) = c, where c > 0 at any point of the curve is positive multiple of c2. (ii) tr(hf) = fxx + fyy ≥ 0. proof. (i). first, we note that f ≥ 0. the surface f meets the plane z = 0 at the origin only. but the intersection of f with the plane z = c (where c > 0) is the curve f(x,y) = c. here the partial derivatives of f are calculated (see [rudin, 1976, theorem 9.42]). fx = ∫ b a xf2(t)√ x2f2(t) + y2g2(t) d(t), fy = ∫ b a yg2(t)√ x2f2(t) + y2g2(t) d(t), fxx = ∫ b a y2f2(t)g2(t)( x2f2(t) + y2g2(t) )3 2 d(t), fyy = ∫ b a x2f2(t)g2(t)( x2f2(t) + y2g2(t) )3 2 d(t), and fxy = − ∫ b a xyf2(t)g2(t)( x2f2(t) + y2g2(t) )3 2 d(t) = fyx. let us put ϕ := √ x2f2(t) + y2g2(t). for the simplicity, we set fx = ∫ xf2 ϕ , fy = ∫ yg2 ϕ , and so on . . . 63 amir veisi and ali delbaznasab by formula of the curvature k in proposition 3.1, we obtain k = 1( f2x + f 2 y )3 2 [( y2 ∫ f2g2 ϕ3 )( y ∫ g2 ϕ )2 + 2 ∫ xyf2g2 ϕ3 ∫ xf2 ϕ ∫ yg2 ϕ + ( x2 ∫ f2g2 ϕ3 )( x ∫ f2 ϕ )2] = ∫ f2g2 ϕ3( f2x + f 2 y )3 2 [ y4 (∫ g2 ϕ )2 + 2x2y2 ∫ f2 ϕ ∫ g2 ϕ + x4 (∫ f2 ϕ )2] = ∫ f2g2 ϕ3( f2x + f 2 y )3 2 [∫ x2f2 ϕ + ∫ y2g2 ϕ ]2 = ∫ f2g2 ϕ3( f2x + f 2 y )3 2 [∫ x2f2 + y2g2 ϕ ]2 = ∫ f2g2 ϕ3( f2x + f 2 y )3 2 [∫ ϕ ]2 = ∫ f2g2 ϕ3( f2x + f 2 y )3 2 f2(x,y). hence, we observe that the curvature of f(x,y) = c at (x0,y0) is a positive multiple of f2(x0,y0) = c2, and we are done. (ii). since f2g2(x2 + y2) ϕ3 ≥ 0, it is clear that fxx + fyy ≥ 0. so the result holds. lemma 3.1. let f : r2 → r be a homogeneous function of degree one. suppose that the second derivatives of f at (a,b) ∈ r2 exist. moreover, fxy = fyx at this point. then (i) |hf|(a,b) = 0. (ii) the eigenvalues of hf are 0 and tr(hf) at (a,b). proof. (i). first, we note that f(λx,λy) = λf(x,y), for all (x,y) ∈ r2 and λ ∈ r. also, we remind the reader of the following fact, which is known as euler’s property, xfx + yfy = f(x,y). 64 new structure of norms on rn and their relations with the curvature of the plane curves therefore, xfxx + fx + yfxy = fx, and xfxy + fy + yfyy = fy. consequently, xfxx = −yfxy and xfxy = −yfyy. now, consider the hessian matrix hf = [ fxx fxy fxy fyy ] of f . for the point (0,b), where b 6= 0, we have fyy(0,b) = 0 = fxy(0,b). this implies that |hf| = 0. also, considering the point (a,0), where a 6= 0 gives fxy(a,0) = 0 = fxx(a,0), this again yields |hf| = 0. now, let (a,b) such that a 6= 0 and b 6= 0. then fxx(a,b) = −b a fxy(a,b) and fyy(a,b) = −a b fxy(a,b). hence, |hf| = 0. so we always have |hf| = 0. the proof of (i) is now complete. (ii). recall that the characteristic equation of hf is λ2 − (tr(hf) = fxx + fyy)λ + (|hf| = fxxfyy −f2xy) = 0. so λ2−(fxx+fyy)λ = 0. therefore, λ = 0 or λ = tr(hf), and we are done. proposition 3.2. let f,g be nonzero real-valued functions on r and f : r2 → r defined by f(x,y) = ∫ b a √ x2f2(t) + y2g2(t)dt and let g(x,y) = f2(x,y). then the eigenvalues of hf and hg at any point except the origin are nonnegative. (in fact, the eigenvalues of hf are zero and tr(hf) at that point). proof. we observe that f is a homogeneous function of degree one. so lemma 3.1 and theorem 3.3 (ii) yield the result. for the matrix hg, we look to the theorem 3.1. since, f2|hf| = 0, we have |hg| = 4fk ( f2x + f 2 y )3 2 . we notice that f,k ≥ 0 gives |hg| ≥ 0. on the other hand, tr(hg) = gxx + gyy ≥ 0. therefore, the roots of λ2 − tr(hg)λ + |hg| = 0, which are the eigenvalues of hg, are nonnegative. the proof is finished. in the following result, we present a norm on r2 which is an elliptic integral of the second kind. corollary 3.1. let f(t) = cost, g(t) = sint and let f : r2 → r is given by f(x,y) = ∫ π 2 0 √ x2 cos2 t + y2 sin2 tdt. then the following statements hold. (i) the eigenvalues of hf and hg, where g = f2 at every point except the origin are nonnegative. 65 amir veisi and ali delbaznasab (ii) f(x,y) is an elliptic integral of the second kind. proof. (i). it follows from proposition 3.2. (ii). notice that f(x,y) = ∫ π 2 0 √ x2(1− sin2 θ) + y2 sin2 θdθ = |x| ∫ π 2 0 √ 1−k2 sin2 θdθ, where k = √ x2−y2 |x| and |x| ≥ |y|. so this gives f(x,y) is an elliptic integral of the second kind and we are done. corollary 3.2. there are ordered pairs (x,y) with rational coordinates (other than the origin) which satisfy the inequality ∫ π 2 0 √ x2 cos2 θ + y2 sin2 θdθ ≤ r, when 0 < r ∈ q. also, if r /∈ q then (x,y) has irrational coordinates. proof. it is sufficient to take the pairs (r,0),(0,r),(−r,0) and (0,−r). we end this article with the next results. proposition 3.3. let 0 ≤ x,y ∈ r. then∫ π 2 0 √ x2 cos2 t + y2 sin2 tdt ≤ x + y. proof. first, note that x2 cos2 t + y2 sin2 t = (xcost + y sint)2 −2xy sintcost, and take 0 ≤ φ ≤ π 2 such that tanφ = y x (if x > 0). now, (xcost + y sint)2 = x2(cost + y x sint)2 = x2(cost + sinφ cosφ sint)2 = x2(costcosφ + sintsinφ)2 cos2 φ = x2 cos2(t−φ) cos2 φ = (x2 + y2) cos2(t−φ) (note, cos2 φ = x2 x2 + y2 ). hence, x2 cos2 t + y2 sin2 t ≤ (x2 + y2) cos2(t−φ). therefore,∫ π 2 0 √ x2 cos2 t + y2 sin2 tdt ≤ ∫ π 2 0 √ (x2 + y2) cos2(t−φ)dt = √ x2 + y2 ∫ π 2 0 |cos(t−φ)|dt = √ x2 + y2 ∫ π 2 −φ −φ costdt (t = t−φ) = x + y. 66 new structure of norms on rn and their relations with the curvature of the plane curves remark 3.1. we find 4 ∫ π 2 0 √ x2 cos2 t + y2 sin2 tdt ≤ 2(2x+2y). the left phrase is the length of the ellipse x′ = xcost and y′ = y sint, while 2x and 2y are the major axis and minor axis of this ellipse. references r. goldman. curvature formulas for implicit curves and surfaces. computer aided geometric design, 22(7):632–658, 2005. c. g. lekkerkerker. geometry of numbers north-holland publishing company, amsterdam, 1969. w. rudin. principles of mathematical analysis, mcgraw-hill, international book company, ltd, 1976. c. l. siegel. lecturcs on the geometry of numbers springer-verlag berlin heidelberg, 1989. s. willard. general topology, addison-wesley, 1970. 67 microsoft word capitolo intero n 12.doc ratio mathematica volume 38, 2020, pp. 313-328 minimal hv-fields thomas vougiouklis* abstract hyperstructures have applications in mathematics and in other sciences, which range from biology, hadronic physics, leptons, linguistics, sociology, to mention but a few. for this, the largest class of the hyperstructures, the hv-structures, is used. they satisfy the weak axioms where the non-empty intersection replaces equality. the fundamental relations connect, by quotients, the hv-structures with the classical ones. hv-numbers are elements of hv-field, and they are used in representation theory. we focus on minimal hv-fields. keywords: hyperstructure, hv-structure, hope, hypernumbers, iso-numbers. 2010 ams subject classifications: 20n20,16y99.1 1 introduction the class of hyperstructures called hv-structures introduced in 1990 [vougiouklis, 1991a], [vougiouklis, 1994] by vougiouklis, satisfy the weak axioms where the non-empty intersection replaces equality. algebraic hyperstructure (h, ·) is a set h equipped with a hyperoperation (abbreviated: hope) · : h ×h → p(h) −{∅} . we abbreviate by wass the weak associativity: (xy)z ∩x(yz) 6= ∅,∀x,y,z ∈ h and by cow the weak commutativity: xy∩yx 6= ∅,∀x,y ∈ h. (h, ·) is an hv-semigroup if it is wass, it is called hv-group if it is reproductive hv-semigroup, i.e., xh = hx = h, ∀x ∈ h. motivation. the quotient of a group by an invariant subgroup, is a group. the quotient of a group by a subgroup is a hypergroup, marty 1934. the quotient of a group by any partition (equivalence) is an hv-group, vougiouklis 1990. *1emeritus professor (democritus university of thrace, neapoli 14-6, xanthi 67100, greece; tvougiou@eled.duth.gr. 1received on june 3rd, 2020. accepted on june 23rd, 2020. published on june 30th, 2020. doi: 10.23755/rm.v38i0.522. issn: 1592-7415. eissn: 2282-8214. ©t. vougiouklis this paper is published under the cc-by licence agreement. 313 thomas vougiouklis in an hv-semigroup the powers are: h1 = {h},h2 = h·h,...,hn = h◦h◦...◦h, where (◦) is the n-ary circle hope, i.e. take the union of hyperproducts, n times, with all possible patterns of parentheses put on them. an (h, ·) is called cyclic of period s, if there exists an element h, called generator, and the minimum s, such that h = h1 ∪h2...∪hs. analogously the cyclicity for the infinite period is defined. if thereare h and s, the minimum one, such that h = hs, then (h, ·) is a single-power cyclic of period s. (r,+, ·) is called hv-ring if (+) and (·) are wass, the reproduction axiom is valid for (+) and (·) is weak distributive to (+): x(y + z)∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅, ∀x,y,z ∈ r. let (r,+, ·) be an hv-ring,a cow hv-group (m,+) is called hv-module over r, if there is an external hope · : r×m → p(m) : (a,x) → ax such that ∀a,b ∈ r and ∀x,y ∈ m we have a(x + y)∩ (ax + ay) 6= ∅, (a + b)x∩ (ax + bx) 6= ∅, (ab)x∩a(bx) 6= ∅, for more definitions and applications on hv-structures one can see in books and papers as [corsini, 1993],[corsini and leoreanu, 2003],[davvaz, 2003],[davvaz and leoreanu, 2007],[davvaz and vougiouklis, 2018],[vougiouklis, 1994],[vougiouklis, 1995],[vougiouklis, 1999b]. let (h, ·),(h,∗) hv-semigroups, the hope (·) is smaller than (∗), and (∗) greater than (·), iff there exists an automorphism f ∈ aut(h,∗) such that xy ⊂ f(x∗y), ∀x,y ∈ h. we write · ≤ ∗ and say that (h,∗) contains (h, ·). if (h, ·) is a structure then it is basic structure and (h,∗) is hb −structure. minimal is called an hv-group which contains no other hv-group defined on the same set. we extend this definition to any hv-structures with any more properties. theorem 1.1. (the little theorem). greater hopes than the ones which are wass or cow, are also wass or cow, respectively. the little theorem leads to a partial order on hv-structures and to posets. let (h, ·) be hypergroupoid. we remove h ∈ h, if we take the restriction of (·) in h −{h}. h ∈ h absorbs h ∈ h if we replace h by h. h ∈ h merges with h ∈ h, if we take as product of any x ∈ h by h, the union of the results of x with both h, h, and consider h and h a class with representative h. 314 minimal hv-fields m. koskas in 1970, introduced in hypergroups the relation β∗, which connects hypergroups with groups and it is defined in hv-groups as well. vougiouklis [vougiouklis, 1985], [vougiouklis, 1988], [vougiouklis, 1991a], [vougiouklis, 1994], [vougiouklis, 1995], [vougiouklis, 2016] introduced the γ* and �* relations, which are defined, in hv-rings and hv-vector spaces, respectively. he also named all these relations, fundamental. definition 1.1. the fundamental relations β*, γ* and �*, are defined, in hvgroups, hv-rings and hv-vector space, respectively, as the smallest equivalences so that the quotient would be group, ring and vector spaces, respectively. remark: let (g, ·) be group and r a partition in g, then (g/r, ·) is an hvgroup, therefore the quotient (g/r, ·)/β* is a group, the fundamental one. theorem 1.2. let (h, ·) be an hv-group and denote by u the set of all finite products of elements of h. define the relation β in h by: xβy iff {x,y} ⊂ u where u ∈ u. then β* is the transitive closure of β. analogous theorems are for hv-rings, hv-vector spaces and so on [vougiouklis, 1994]. theorem 1.3. let (r,+, ·) be hv-ring. denote u all finite polynomials of elements of r. define the relation γ in r by: xγy iff {x,y}⊂ u where u ∈ u. then the relation γ* is the transitive closure of the relation γ. proof. let γ be the transitive closure of γ, and denote by γ(a) the class of a. first, we prove that the quotient set r/γ is a ring. in r/γ the sum (⊕) and the product (⊗) are defined in the usual manner: γ(a)⊕γ(b) = {γ(c) : c ∈ γ(a) + γ(b)}, γ∗(a)⊗γ(b) = {γ(d) : d ∈ γ∗(a) ·γ(b)}, ∀a,b ∈ r. take a′ ∈ γ(a), b′ ∈ γ(b). then we have a′γa iff ∃x1, ...,xm+1 with x1 = a′,xm+1 = a and u1, ...,um ∈ u such that {xi,xi+1}⊂ ui, i = 1, ...,m, and b′γb iff ∃y1, ...,yn+1 with y1 = b′,yn+1 = b and v1, ...,vn ∈ u such that {yj,yj+1}⊂ vj, i = 1, ...,n. 315 thomas vougiouklis from the above we obtain {xi,xi+1}+ y1 ⊂ ui + v1, i = 1, ...,m−1, xm+1 +{yj,yj+1}⊂ um + vj, j = 1, ...,n. the sums ui + v1 = ti, i = 1, ...m−1 and um + vj = tim+j−1, j = 1, ...,n are also polynomials, thus, tk ∈ u∀k ∈{1, ...,m + n−1}. now, pick up z1, ...,zm+n such that zi ∈ xi + y1, i = 1, ...,n and zm+j ∈ xm+1 + yj+1, j = 1, ...,n, therefore, using the above relations we obtain {zk,zk+1}⊂ tk, k = 1, ...,m+n− 1. thus, every z1 ∈ x1 + y1 = a′ + b′ is γ equivalent to every zm+n ∈ xm+1 + yn+1 = a + b. so γ(a)⊕γ(b) is a singleton so we can write γ(a)⊕γ(b) = γ(c),∀c ∈ γ(a) + γ(b) in a similar way we prove that γ(a)⊗γ(b) = γ(d),∀d ∈ γ(a) ·γ(b) the wass and the weak distributivity on r guarantee that associativity and distributivity are valid for r/γ*. therefore r/γ* is a ring. let σ be an equivalence relation in r such that r/σ is a ring and σ(a) the class of a. then σ(a)⊕σ(b) and σ(a)⊗σ(b) are singletons ∀a,b ∈ r, i.e. σ(a)⊕σ(b) = σ(c),∀c ∈ σ(a) + σ(b), σ(a)⊗σ(b) = σ(d),∀d ∈ σ(a) ·σ(b). therefore we write, for every a,b ∈ r and a ⊂ σ(a), b ⊂ σ(b), σ(a)⊕σ(b) = σ(a + b) = σ(a + b), σ(a)⊗σ(b) = σ(ab) = σ(a ·b) by induction, we extend these relations on finite sums and products. thus, ∀u ∈ u, we have the relation σ(x) = σ(u) ∀x ∈ u. consequently x ∈ γ(a) => x ∈ σ(a),∀x ∈ r. but σ is transitively closed, so we obtain: x ∈ γ(x) => x ∈ σ(a). that γ is the smallest equivalence relation in r such that r/γ is a ring, i.e. γ = γ*. an element is called single if its fundamental class is singleton [vougiouklis, 1994]. general structures can be defined using fundamental structures. from 1990 there is the following [vougiouklis, 1991a], [vougiouklis, 1994]: 316 minimal hv-fields definition 1.2. an hv-ring (r,+, ·) is called hv-field if r/γ* is a field. an hv-module over an hv-field f, it is called hv-vector space. the analogous to theorem 1.3, on hv-vector spaces, can be proved: theorem 1.4. let (v,+) be hv-vector space over the hv-field f. denote u the set of all expressions of finite hopes either on f and v or the external hope applied on finite sets of elements of f and v. define the relation � in v as follows: x�y iff {x,y}⊂ u where u ∈ u. then �* is the transitive closure of the relation �. definition 1.3. let (l,+) be hv-vector space over the hv-field f, φ : f → f/γ* canonical; ωf = {x ∈ f : φ(x) = 0}, the core, 0 is the zero of f/γ. let ωl be the core of φ′ : l → l/�* and denote by 0 the zero of l/�*, as well. take the bracket (commutator) hope: [, ] : l×l → p(l) : (x,y) → [x,y] then l is an hv-lie algebra over f if the following axioms are satisfied: (l1) the bracket hope is bilinear, i.e. [λ1x1 + λ2x2,y]∩ (λ1[x1,y] + λ2[x2,y]) 6= ∅ [x,λ1y1 + λ2y2]∩ (λ1[x,y1] + λ2[x,y2]) 6= ∅, ∀x,x1,x2,y,y1,y2 ∈ l,λ1,λ2 ∈ f (l2) [x,x]∩ωl 6= ∅, ∀x ∈ l (l3) ([x, [y,z]] + [y, [z,x]] + [z, [x,y]])∩ωl 6= ∅, ∀x,y,z ∈ l definition 1.4. the hv-semigroup (h·) is h/v-group if h/β* is a group [vougiouklis, 2003]. the h/v-group is a generalization of hv-group where a reproductivity of classes is valid: if σ(x), ∀x ∈ h equivalence classes then xσ(y) = σ(xy) = σ(x)y,∀x,y ∈ h. similarly h/v-rings, h/v-fields, h/v-vector spaces etc, are defined. the uniting elements method introduced by corsini & vougiouklis in 1989, is the following [corsini and vougiouklis, 1989]: let g be a structure and a not valid property d, described by a set of equations. take the partition in g for which put in the same class, all pairs of elements that causes the non-validity of d. the quotient by this partition g/d is an hv-structure. then, the quotient out g/d by β*, is a stricter structure (g/d)β* for which d is valid. theorem 1.5. let (r,+, ·) be a ring, and f = {f1, ...,fm,fm+1, ...,fm+n} be a system of equations on r consisting of subsystems fm = {f1, ...,fm} and fn = {fm+1, ...,fm+n}. let σ, σm be the equivalence relations defined by the uniting elements procedure using f and fm respectively, and σn the equivalence defined on fn on the ring rm = (r/σm)/γ*. then (r/σ)/γ* ∼= (rm/σn)/γ* 317 thomas vougiouklis 2 large classes of hopes a class of hv-structures, introduced in [vougiouklis, 1991b], [vougiouklis, 1994], [vougiouklis, 2014b], is the following: definition 2.1. an hv-structure is called very thin if there exists a pair (a,b) ∈ h2 for which ab = a, with carda > 1, and all the other products are singletons. from the very thin hopes the attach construction is obtained [vougiouklis, 1999a], [vougiouklis, 2014b], [vougiouklis, 2017]: let (h, ·) be an hv-semigroup and v /∈ h. we extend the hope (·) into h = h ∪{v} by: x ·v = v ·x = v,∀x ∈ h, and v ·v = h. the (h, ·) is an hv-group, where (h, ·)/β∗ ∼= z2 and v is a single. let (h, ·) hv-semigroup, and [x] the fundamental class of ∀x ∈ h. unit class is [e] if ([e] · [x])∩ [x] 6= ∅ and ([x] · [e])∩ [x] 6= ∅,∀x ∈ h, and ∀x ∈ h, we call inverse class of [x], the class [x]−1, if ([x] · [x]−1)∩ [e] 6= ∅ and ([x]−1 · [x])∩ [e] 6= ∅. enlarged hopes are the ones where a new element appears in one result. the useful cases are those h/v-structures with the same fundamental structure. construction 2.1. (a) let (h, ·) be an hv-semigroup, v /∈ h. we extend (·) into h = h ∪{v} by: x ·v = v ·x = v,∀x ∈ h, and v ·v = h. the (h, ·) is an h/v-group, called attach, where (h, ·)/β∗ ∼= z2 and v is single. scalars and units of (h, ·) are scalars and units in (h, ·). if (h, ·) is cow then (h, ·) is cow. (b) (h, ·) hv-semigroup, v /∈ h, (h, ·) its attached h/v-group. take 0 /∈ h and define in h◦ = h ∪{v,0} two hopes: hypersum(+) : 0 + 0 = x + v = v + x = 0, 0 + v = v + 0 = x + y = v,0 + x = x + 0 = v + v = h,∀x,y ∈ h hyperproduct(·) : remains the same as in h moreover 0 ·0 = v ·x = x ·0 = 0,∀x ∈ h 318 minimal hv-fields then (ho,+, ·) is h/v-field with (ho,+, ·)/γ*∼= z3. (+) is associative, (·) is wass and weak distributive to (+). 0 is zero absorbing in (+). (ho,+, ·) is the attached h/v-field of (h, ·). definition 2.2. [vougiouklis, 2008], [vougiouklis, 2016] let (g, ·) be groupoid and f : g → g be a map. we define a hope (∂), called theta-hope, we write ∂-hope, on g as follows x∂y = {f(x) ·y,x ·f(y)}, ∀x,y ∈ g. if (·) is commutative then ∂ is commutative. if (·) is cow, then ∂ is cow. if (g, ·) is groupoid and f : g → p(g)−{∅} be multivalued map. we define the ∂-hope on g as follows x∂y = (f(x) ·y)∪ (x ·f(y)), ∀x,y ∈ g . motivation for the ∂-hope is the derivative where only the product of functions is used. basic property: if (g, ·) is semigroup then ∀f, the ∂-hope is wass. examples (a) in integers (z,+, ·) fix n 6= 0, a natural number. consider the map f such that f(0) = n and f(x) = x,∀x ∈ z −{0}. then (z,∂+,∂·), where ∂+ and ∂· are the ∂-hopes refereed to the addition and the multiplication respectively, is an hv-near-ring, with (z,∂+,∂·)/γ ∗ ∼= zn. (b) in (z,+, ·) with n 6= 0, take f such that f(n) = 0 and f(x) = x,∀x ∈ z−{n}. then (z,∂+,∂·) is an hv-ring, moreover, (z,∂+,∂·)/γ∗ ∼= zn. special case of the above is for n=p, prime, then (z,∂+,∂·) is an hv-field. combining the uniting elements procedure with the enlarging theory or the ∂-theory, we can obtain analogous results [vougiouklis, 1999a], [vougiouklis, 2014b], [vougiouklis, 2017]. theorem 2.1. in the ring (zn,+, ·), with n = ms we enlarge the multiplication only in the product of the elements 0 ·m by setting 0 ⊗m = {0,m} and the rest results remain the same. then (zn,+,⊗)/γ* ∼= (zm,+, ·) remark that we can enlarge other products as well, for example 2·m by setting 2 ⊗ m = {2,m + 2}, then the result remains the same. in this case 0 and 1 are scalars. 319 thomas vougiouklis corolary 2.1. in the ring (zn,+, ·), with n = ps, where p is prime , we enlarge only the product 0 ·p by 0⊕p = {0,p} and the rest results remain the same. then (zn,+,⊕) is a very thin hv-field. now we focus on very thin minimal hv-fields obtained by a classical field. theorem 2.2. in a field (f,+, ·), we enlarge only in the product of the special elements a and b, by setting a ⊗ b = {ab,c}, where c 6= ab, and the rest results remain the same. then we obtain the degenerate, minimal very thin, hv-field (f,+,⊗)/γ∗ ∼= {0} thus, there is no non-degenerate hv-field obtained by a field by enlarging any product. proof. take any x ∈ f −{0}, then from a⊗ b = {ab,c} we obtain (a⊗ b)−ab = {0,c−ab} and then (x(c−ab)−1)⊗ ((a⊗ b)−ab) = {0,x} thus, 0γx,x ∈ f −{0}. which means that every x is in the same fundamental class with the element 0. thus, (f,+,⊗)/γ∗ ∼= {0}. theorem 2.3. in a field (f,+, ·), we enlarge only in the sum of the special elements a and b, by setting a ⊕ b = {a + b,c}, where c 6= a + b, and the rest results remain the same. then we obtain the degenerate, minimal very thin, hvfield (f,+,⊕)/γ∗ ∼= {0}. thus, there is no non-degenerate hv-field obtained by a field by enlarging any sum. proof. take any x ∈ f −{0}, then from a⊕ b = {a + b,c} we obtain (a⊕b)−a+b = {0,c−a+b} and then (x(c−a+b)−1)·((a⊕b)−a+b) = {0,x} thus, 0γx,x ∈ f −{0}. which means that every x is in the same fundamental class with the element 0. thus, (f,+,⊕)/γ∗ ∼= {0}. the above two theorems state that there is no non-degenerate hv-field obtained by a field by enlarging any sum or product. hopes defined on classical structures are the following [corsini, 1993], [corsini and leoreanu, 2003], [vougiouklis, 1987], [vougiouklis, 1994] : 320 minimal hv-fields definition 2.3. let (g, ·) be groupoid then for every p ⊂ g, p 6= ∅, we define the following hopes called p-hopes: ∀x,y ∈ g p : xpy = (xp)y ∪x(py), pr : xpry = (xy)p ∪x(yp), p l : xp ly = (px)y ∪p(xy). the (g,p),(g,pr) and (g,p l) are called p-hyperstructures. if (g, ·) is semigroup, then xpy = (xp)y ∪x(py) = xpy and (g,p) is a semihypergroup. 3 representations and applications hv-structures used in representation (abbr. rep) theory of hv-groups can be achieved by generalized permutations [vougiouklis, 1992] or by hv-matrices [vougiouklis, 1985], [vougiouklis, 1994], [vougiouklis, 1999b]. hv-matrix is called a matrix if has entries from an hv-ring. the hyperproduct of hv-matrices (aij) and (bij), of type m × n and n × r, respectively, is defined in the usual manner, and it is a set of m× r hv-matrices. the sum of products of elements of the hv-ring is the n-ary circle hope on the hyper-sum. the problem of the hv-matrix (or h/v-group) reps is the following: definition 3.1. let (h, ·) be hv-group. find an hv-ring (r,+, ·), a set mr={(aij)|aij∈r}, and a map t : h → mr : h 7→ t(h) such that t(h1h2)∩t(h1)t(h2) 6= ∅,∀h1,h2 ∈ h. t is an hv-matrix rep. if t(h1h2) ⊂ t(h1)t(h2),∀h1,h2 ∈ h, then t is an inclusion rep. if t(h1h2) = t(h1)t(h2),∀h1,h2 ∈ h, then t is a good rep. if t is a good rep and one to one then it is a faithful rep. the rep problem is simplified in cases such as if the h/v-rings have scalars 0 and 1. the main theorem of the theory of reps is the following: theorem 3.1. a necessary condition in order to have an inclusion rep t of an h/v-group (h, ·) by n×n h/v-matrices over the h/v-ring (r,+, ·) is the following: ∀β*(x), x ∈ h there must exist elements aij ∈ h,i,j ∈{1, ...,n} such that t(β*(a)) ⊂{a = (a′ij)|a ′ ij ∈ γ*(aij), i,j ∈{1, ...,n}} the inclusion rep t : h → mr : a 7→ t(a) = (aij) induces an homomorphic t * of h/β* on r/γ* by t *(β*(a)) = [γ*(aij)],β*(a)h/β*, where γ*(aij)r/γ* is the ij entry of t *(β*(a)). t * is called fundamental induced rep of t . 321 thomas vougiouklis in reps we need small hv-fields with results of few elements. an important hope on non-square matrices is defined [vougiouklis, 2009], [vougiouklis and vougiouklis, 2005]: definition 3.2. let a = (aij) ∈ mm×n and s,t ∈ n, such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. define a mod-like map st from mm×n to ms×t by corresponding to a the matrix ast = (aij) with entries the sets aij = {ai+κs,j+λt|1 ≤ i ≤ s,1 ≤ j ≤ t and κ,λ ∈ n,i + κs ≤ m,j + λt ≤ n}. the map st : mm×nms×t : a → ast(aij), is called helix-projection of type st. ast is a set of s × t-matrices x = (xij) such that xij ∈ aij,∀i,j. obviously amn = a. leta = (aij) ∈ mm×n and s,t ∈ n, 1 ≤ s ≤ m, 1 ≤ t ≤ n. we apply the helix-projection first on the columns and then on the rows and the result is the same: (asn)st = (amt)st = ast. definition 3.3. let a = (aij) ∈ mm×n and b = (bij) ∈ mu×v be matrices. denote s=min(m,u), t=min(n,u), then we define the helix-sum by ⊕ : mm×nmu×vp(ms×t) : (a,b) → a⊕b = ast + bst = (aij) + (bij) ⊂ ms×t, where (aij) + (bij) = {(cij) = (aij + bij)|aij ∈ aij and bij ∈ bij}. denote s=min(n,u), then we define the helix-product by ⊗ : mm×nmu×vp(ms×t) : (a,b) → a⊗b = ams + bsv = (aij) + (bij) ⊂ mm×v, where (aij) · (bij) = {(cij) = ∑ (aij + bij)|aij ∈ aij and bij ∈ bij}.. remark. the definition of the lie-bracket is immediate, therefore the helixlie algebra is defined, as well. last decades hv-structures have applications in mathematics and in other sciences. applications range from biology and hadronic physics or leptons to mention but a few. the hyperstructure theory is related to fuzzy one; consequently, can be widely applicable in industry and production, too [corsini and leoreanu, 2003], [davvaz and leoreanu, 2007], [davvaz et al., 2015], [davvaz and vougiouklis, 2018], [santilli and vougiouklis, 1996], [vougiouklis, 2014a], [vougiouklis, 2020], [vougiouklis and kambaki-vougioukli, 2013]. an application, which combines hv-hyperstructures and fuzzy theory, is to replace in questionnaires the scale of likert by the bar of vougiouklis & vougiouklis (v & v bar) [vougiouklis and kambaki-vougioukli, 2013]. they suggest the following: 322 minimal hv-fields definition 3.4. in every question substitute the likert scale with ’the bar’ whose poles are defined with ’0’ on the left end, and ’1’ on the right end: 0 1 the subjects/participants are asked instead of deciding and checking a specific grade on the scale, to cut the bar at any point s/he feels expresses her/his answer to the specific question. the use of v & v bar bar instead of a likert scale has several advantages during both the filling-in and the research processing. the final suggested length of the bar, according to the golden ratio, is 6.2cm. the lie-santilli theory on isotopies was born to solve hadronic mechanics problems. santilli proposed a ’lifting’ of the n-dimensional trivial unit matrix into an appropriate new matrix. the original theory is reconstructed to admit the new matrix as left and right unit. the isofields needed in this theory correspond into the hyperstructures called e-hyperfields, introduced by [santilli and vougiouklis, 1996, davvaz et al., 2015]. definition 3.5. a hyperstructure (h, ·) which contain a unique scalar unit e, is called e-hyperstructure. in an e-hyperstructure, we assume that for every element x, there exists an inverse x−1, i.e. e ∈ x ·x−1 ∩x−1 ·x. definition 3.6. a hyperstructure (f,+, ·), where (+) is an operation and (·) is a hope, is called e-hyperfield if the following axioms are valid: (f,+) is an abelian group with the additive unit 0, (·) is wass, (·) is weak distributive with respect to (+), 0 is absorbing element: 0·x = x·0 = 0,∀x ∈ f , there exist a multiplicative scalar unit 1, i.e. 1 · x = x · 1 = x,∀x ∈ f , and for all x ∈ f there exists a unique inverse x−1, such that 1 ∈ x ·x−1 ∩x−1 ·x. the elements of an e-hyperfield are called e-hypernumbers. in the case that the relation: 1 = x · x−1 = x−1 · x, is valid, then we say that we have a strong e-hyperfield. definition 3.7. the main e-construction. given a group (g, ·), where e is the unit, then we define in g, a large number of hopes (⊗) as follows: x⊗y = {xy,g1,g2, ...},∀x,y ∈ g−{e}, and g1,g2, ... ∈ g−{e} g1,g2,... are not necessarily the same for each pair (x,y). then (g,⊗) becomes an hv-group, actually is an hb-group which contains the (g, ·). the hv-group (g,⊗) is an e-hypergroup. 323 thomas vougiouklis example. consider the quaternions q = {1,−1, i,−i,j,−j,k,−k} with i2 = j2 = −1, ij = −ji = k and denote i = {i,−i},j = {j,−j},k = {k,−k}. we define a lot of hopes (∗) by enlarging few products. for example, (−1)∗k = k,k ∗ i = j and i∗ j = k. then (q,∗) is strong e-hypergroup. a generalization of p-hopes used in santilli’s isotheory, is [davvaz et al., 2015], [vougiouklis, 2016] : let (g, ·) be abelian group, p ⊂ g with #p < 1. we define the hope ×p as follows: x×p y = { x ·p ·y = {x ·h ·y|h ∈ p} if x 6= e and c 6= e x ·y if x = e and y = e we call this hope pe-hope. the hyperstructure (g,×p) is abelian hv-group. 4 small hypernumbers. minimal h/v-fields the small non-degenerate h/v-fields on (zn,+, ·) in iso-theory, satisfy the following: 1. very thin minimal, 2. cow (non-commutative), 3. they have the elements 0 and 1, scalars, 4. if an element has inverse element, this is unique. therefore, we cannot enlarge the result if it is 1 and we cannot put 1 in enlargement. theorem 4.1. [vougiouklis, 2017] all multiplicative h/v-fields defined on (z4,+, ·), with non-degenerate fundamental field, satisfying the above 4 conditions, are the following isomorphic cases: the only product which is set is 2 ⊗ 3 = {0,2} or 3⊗2 = {0,2}. fundamental classes: [0]=0,2, [1]=1,3 and we have (z4,+,⊗)/γ∗ ∼= (z2,+, ·) example. denote eij the matrix with 1 in the ij-entry and zero in the rest entries. take the 2×2 upper triangular h/v-matrices on the above h/v-field (z4,+,⊗) of the case that only 2⊗3={0,2} is a hyperproduct: i = e11 +e22,a = e11 +e12 +e22,b = e11 +2e12 +e22,c = e11 +3e12 +e22, d = e11+3e22,e = e11+e12+3e22,f = e11+2e12+3e22,g = e11+3e12+3e22, 324 minimal hv-fields then, we obtain for x={i,a,b,c,d,e,f,g}, that (x,⊗) is non-cow, hv-group where the fundamental classes are a = {a,c},d = {d,f},e = {e,g} and the fundamental group is isomorphic to (z2 ×z2,+). there is only one unit and every element has unique double inverse. only f has one more right inverse element d, since f ⊗d = {i,b}. (x,⊗) is not cyclic. theorem 4.2. all multiplicative h/v-fields on (z6,+, ·), with non-degenerate fundamental field, satisfying the above 4 conditions, are the following isomorphic cases: we have the only one hyperproduct, (i) 2⊗3 = {0,3},2⊗4 = {2,5},3⊗4 = {0,3},3⊗5 = {0,3},4⊗5 = {2,5}. the fundamental classes are [0]=0,3, [1]=1,4, [2]=2,5 and we have (z6,+,⊗)/γ∗ ∼= (z3,+, ·). (ii) 2⊗3 = {0,2} or 2⊗3 = {0,4}, 2⊗4 = {0,2} or {2,4},2⊗5 = {0,4} or 2⊗5 = {2,4}, 3⊗4 = {0,2} or {0,4},3⊗5 = {3,5},4⊗5 = {0,2} or {2,4}. in all these cases the fundamental classes are [0]=0,2,4, [1]=1,3,5 and we have (z6,+,⊗)/γ∗ ∼= (z2,+, ·). example. in the h/v-field (z6,+,⊗) where only the hyperproduct is 2 ⊗ 4 = {2,5} take the h/v-matrices of type i = e11 +ie12 + 4e22, where i=0,1,...,5, then the multiplicative table of the hyperproduct of those h/v-matrices is ⊗ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 4 5 0 1 2 3 2 2 0,3 1,4 2,5 0,3 1,4 3 0 1 2 3 4 5 4 4 5 0 1 2 3 5 2 3 4 5 0 1 the fundamental classes are (0) = 0,3,(1) = 1,4,(2) = 2,5 and the fundamental group is isomorphic to (z3,+). the (z6,⊗) is h/v-group which is cyclic where 2 and 4 are generators of period 4. example. consider the h/v-field (z10,+,⊗) where only 3 × 8 = {4,9} is a hyperproduct. let us take the h/v-matrix a = 3e11 + e22 + 2e33 + 6e12 + 2e13 + 9e23 then from the above formulas we obtain that the set of inverse h/v-matrices is a−1 = [2]e11 + [1]e22 + [3]e33 + [3]e12 + [2]e13 + [3]e23 325 thomas vougiouklis so, for example, if we take the h/v-matrix a−1 = 7e11 + 6e22 + 8e33 + 8e12 + 2e13 + 3e23 we obtain that a ·a−1 = e11 + e22 + e33 +{0,5}e12 + 5e23 therefore, it contains a unit h/v-matrix. theorem 4.3. all multiplicative h/v-fields defined on (z9,+, ·), which have nondegenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: we have the only one hyperproduct, 2⊗3 = 0,6 or 3,6,2⊗4 = 2,8 or 5,8,2⊗6 = 0,3 or 3,6,2⊗7 = 2,5 or 5,8,2⊗8 = 1,7 or 4,7,3⊗4 = 0,3 or 3,6,3⊗5 = 0,6 or 3,6,3⊗6 = 0,3 or 0,6,3⊗7 = 0,3 or 3,6,3⊗8 = 0,6 or 3,6,4⊗5 = 2,5 or 2,8,4⊗6 = 0,6 or 3,6,4⊗8 = 2,5 or 5,8,5⊗6 = 0,3 or 3,6,5⊗7 = 2,8 or 5,8,5⊗8 = 1,4 or 4,7,6⊗7 = 0,6 or 3,6,6⊗8 = 0,3 or 3,6,7⊗8 = 2,5 or 2,8. in all the above cases the fundamental classes are [0] = {0,3,6}, [1] = {1,4,7}, [2] = {2,5,8},andwehave(z9,+,⊗)/γ∗ ∼= (z3,+, ·). theorem 4.4. all multiplicative h/v-fields on (z10,+, ·), which have non-degenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: (i) we have the only one hyperproduct, 2⊗4 = {3,8},2⊗5 = {0,5},2⊗6 = {2,7},2⊗7 = {4,9}, 2⊗9 = {3,8},3⊗4 = {2,7},3⊗5 = {0,5},3⊗6 = {3,8},3⊗8 = {4,9}, 3⊗9 = {2,7},4⊗5 = {0,5},4⊗6 = {4,9}, 4⊗7 = {3,8},4⊗8 = {2,7},5⊗6 = {0,5},5⊗7 = {0,5}, 5⊗8 = {0,5},5⊗9 = {0,5},6⊗7 = {2,7},6⊗8 = {3,8}, 6⊗9 = {4,9},7⊗9 = {3,8},8⊗9 = {2,7}. in all the above cases the fundamental classes are [0] = {0,3,6}, [1] = {1,4,7}, [2] = {2,5,8}, and we have (z9,+,⊗)/γ∗ ∼= (z3,+, ·). (ii) the cases with classes [0] = {0,2,4,6,8} and [1] = {1,3,5,7,9}, and fundamental field (z10,+,⊗)/γ∗ ∼= (z2,+, ·). are described as follows: in the multiplicative table only the results above the diagonal, we enlarge each of the products by putting one element of the same class of the results. we do not enlarge setting 1, and we cannot enlarge only the 3⊗7 = 1. the number of those h/v-fields is 103. 326 minimal hv-fields references p. corsini. prolegomena of hypergroup theory. aviani editore, 1993. p. corsini and v. leoreanu. application of hyperstructure theory. klower acad. publ, 2003. p. corsini and t. vougiouklis. from groupoids to groups through hypergroups. klower rendiconti mat. s., vii, 9,:173–181, 1989. b. davvaz. a brief survey of the theory of hv-structures. 8th aha, greece,, pages 39–70, 2003. b. davvaz and v. leoreanu. hyperring theory and applications. int. academic press,, 2007. b. davvaz and t. vougiouklis. a walk through weak hyperstructures,hvstructures. world scientific, 2018. b. davvaz, r.m. santilli, and t. vougiouklis. algebra, hyperalgebra and liesantilli theory. j. generalized lie theory and appl., 9:2:1–5, 2015. r.m. santilli and t. vougiouklis. isotopies, genotopies, hyperstructures and their applications. new frontiers hyperstr. related algebras, hadronic, pages 177– 188, 1996. s. vougiouklis. hv-vector spaces from helix hyperoperations. int. j. math. anal. 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shyam lal@rediffmail.com. †department of mathematics, institute of science, banaras hindu university, varanasi221005, india); satishkumar3102@gmail.com. 1received on june 21st, 2020. accepted on december 15th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.549. issn: 1592-7415. eissn: 2282-8214. ©lal and satish kumar. this paper is published under the cc-by licence agreement. 187 shyam lal and satish kumar 1 introduction wavelet is a very recent and powerful tool in pure as well as applied mathematical research area. it has wide range applications in engineering, science and technology, signal analysis, time-frequency analysis, fast numerical algorithm. several problems of physics, engineering, science and technology are found in the form of integral equations. in some cases, integral equations are reformulated into ordinary differential equations and partial differential equations. in many cases, it is very difficult to solve integral equations analytically and hence there is a need of approximate solution of integral equations. in recents years, the approximate solutions of integral equations have been obtained by orthogonal basis functions as well as orthogonal wavelets. the main advantage of using orthonormal basis is that it converts the mathematical problems to a system of algebraic equations. working in same direction, several researchers like [2], sahu [3] etc. have been solved integral equations. it is known that wavelets are considerably useful in the solution of integral equations. in science and technology, some problems are available in the form of fredholm integral equations of second kind: u(x) = f(x) + ∫ 1 0 k(x,y)u(y)dy (1) where f ∈ l2[0, 1) and k ∈ l2[0, 1) × l2[0, 1) are known functions and u is unknown function to be determined (ray and sahu [3]). in best of our knowledge, there is no work associated with the solution of fredholm integral eqn (1) by cas wavelet method. the main objectives of the research paper are as follows: 1. to estimate the approximation of functions belonging to hölder’s class hα[0, 1) of order 0 < α 6 1 by cas wavelet method. 2. to develop a procedure to solve fredholm integral equation of second kind by using cas wavelet approximation. 3. to compare the solutions of fredholm integral eqn (1) obtained by cas wavelet, legendre wavelet and haar wavelet method with their exact solutions. it is remarkable to note that the solution of fredholm integral eqn (1) obtained by cas wavelet method and its exact solution are almost same. the solution of fredholm integral eqn (1) obtained by cas wavelet method is better and more closed to its exact solution than the solutions obtained by legendre wavelet and haar wavelet method. it is observed in numerical 188 cas wavelet approximation of functions of hölder’s class hα[0, 1)... comparison of these solutions. it is a significant achievement of the proposed method. 2 definitions and preliminaries 2.1 basic wavelets and cas wavelets let ψ ∈ l2(ir). ψ is called a basic wavelet if it satisfies the admissibility condition: cψ = ∫ ∞ −∞ | ψ̂ |2 | w | dw < ∞ (chui [1]) (2) the integral wavelet transform, relative to a basic wavelet ψ, is defined by (wψf)(b,a) = |a|−1/2 ∫ ∞ −∞ f(t)ψ( b−a a )dt ,f ∈ l2(ir) (3) where a,b ∈ ir,a 6= 0 . set ψb,a(t) = |a|−1/2ψ( b−a a ). (4) this is a family of wavelets. if we restrict the parameters a and b to discrete values a = a−k0 ,b = nb0a −k 0 ,a0 > 1,b0 > 0 where n and k are positive integers, then ψb,a(t) = ψn,k(t) = |a0|k/2ψ(ak0t−nb0). (5) taking a0 = 2,b0 = 1 in eqn (5), ψn,k(t) = 2 k/2ψ(2kt−n). (6) if ψ(2kt−n) = cos(2mπ(2kt−n + 1)) + sin(2mπ(2kt−n + 1)) (7) = casm(2 kt−n + 1). (8) using eqn(7), eqn (6) becomes ψn,m(t) = { 2 k 2{cos(2mπ(2kt−n + 1)) + sin(2mπ(2kt−n + 1))}, if n−1 2k 6 t < n 2k , 0, otherwise. {ψn,m}n,m∈z are orthonormal cas wavelets defined on [0,1) . 189 shyam lal and satish kumar 3 function belonging to hölder’s class hα[0,1) a function f is said to belong to hölder’s class hα[0, 1) of order 0 < α 6 1 if f satifies the following condition : |f(x) −f(y)| 6 a|x−y|α, ∀x,y ∈ ir (9) for some positive constant a (zheng, wei [4]). 3.1 proposition let f be a function such that its second derivative f ′′ is in hα[0, 1), then its first derivative f ′ is in hα[0, 1). proof : let φ ′′ ∈ hα[0, 1) . f(x) = ∫ xα 0 φ ′ (t) dt f ′ (x) = ∫ xα 0 φ ′′ (t) dt and f ′ (y) = ∫ yα 0 φ ′′ (t) dt |f ′ (x) −f ′ (y)| = | ∫ xα 0 φ ′′ (t) dt− ∫ yα 0 φ ′′ (t) dt| = | ∫ xα yα φ ′′ (t) dt| ≤ m|xα −yα| ≤ m|x−y|α, m = sup t∈[0,1) {φ ′′ (t)} converse is not true. consider the example f(x) = x α+1 α+1 0 < α < 1.then, f ′ (x) = xα and f ′′ (x) = αxα−1. for x = 1 n 1 1−α , y = 1 (1+n) 1 1−α , we have |x−y| ≤ 1 n 1 1−α − 1 (1+n) 1 1−α ≤ 1 n 1 1−α = δ. and |f ′′(x) −f ′′(y)| = α(1 + n −n) = α if 0 < � < α, then |f ′′(x) − f ′′(y)| � � whenever |x− y| ≤ δ = 1 n 1 1−α . hence, f ′ ∈ hα[0, 1) but f ′′ 6∈ hα[0, 1). 3.2 difference between hölder’s class and lipschitz class 1. consider the function f(x) = √ x2 + 5 ∀x ∈ [0, 1]. then |f(x) −f(y)| ≤ | √ x2 + 5 − √ y2 + 5| ≤ | √ x2 −y2| ≤ √ 2|x−y| 1 2 (10) eqn(10) shows that f ∈ h 1 2 [0, 1). and also, we have |f ′ (x)| ≤ | x x2 + 5 | ≤ 1, ∀ x ∈ [0, 1] (11) 190 cas wavelet approximation of functions of hölder’s class hα[0, 1)... eqn(10) and eqn(11) shows that f ∈ lip1 2 [0, 1). 2. define the function f(x) = √ x ∀x ∈ [0, 1], then we have |f(x) −f(y)| ≤ | √ x− √ y| ≤ |x−y| 1 2 =⇒ f ∈ h 1 2 [0, 1). and since, f ′ (x) = 1 2 √ x →∞ as x → 0+. hence, f is not bounded. ∴ f 6∈ lip1 2 [0, 1). hence, we conclude that lipα[0, 1] ⊂ hα[0, 1]. 4 approximation of function since {ψn,m}n,m∈z forms an orthonormal basis for l2[0, 1] , therefore a function f ∈ l2[0, 1) can be expressed into cas wavelet series as: f(t) = ∞∑ n=1 ∞∑ m=−∞ cn,mψn,m(t) (12) where the coefficients cn,m are given by cn,m =< f,ψn,m > (13) (2k, 2m + 1)th partial sum s2k,2m+1(f)(t) of (12) is given by s2k,2m+1(f)(t) = 2k∑ n=1 m∑ m=−m cn,mψn,m(t) = c t ψ(t) (14) where c and ψ(t) are given by c = [c1,(−m),c1,(−m+1), ...,c1,m,c2,(−m), ...,c2,m, ...,c2k,(−m), ...,c2k,m ] t and ψ(t) = [ψ1,(−m)(t),ψ1,(−m+1)(t), ...,ψ1,m (t),ψ2,(−m)(t), ...,ψ2,m (t), ..., ψ2k,(−m)(t), ...,ψ2k,m (t)] t . extended legendre wavelet expansion of function f ∈ l2[0, 1) is f(x) = ∞∑ n=1 ∞∑ m=0 cn,mψ (µ) n,m(x), and its (µk,m)th partial sum is sµk,m (f)(x) = µk∑ n=1 m∑ m=0 cn,mψ (µ) n,m(x). 191 shyam lal and satish kumar the extended legendre wavelet approximation eµk,m (f) of f by (µ k,m)thpartial sum sµk,m (f) is defined by eµk,m (f) = min s µk,m (f) ||f −sµk,m (f)||2 . in our case, the cas wavelet approximation e2k,2m+1(f) of f by (2 k, 2m + 1)th partial sum s2k,2m+1(f) of series (12) is defined by e2k,2m+1(f) = min s 2k,2m+1 (f) ||f −s2k,2m+1(f)||2 . (15) 5 theorems in this paper, we prove the following theorems: theorem 5.1. if f ∈ l2[0, 1) is a function such that f ′ ∈ hα[0, 1) and its cas wavelet expansion is f(t) = ∞∑ n=1 ∞∑ m=−∞ cn,mψn,m(t) (16) then the approximation error e(1) 2k,2m+1 (f) of f by (2k, 2m + 1)th partial sum s2k,2m+1(f)(t) = 2k∑ n=1 m∑ m=−m cn,mψn,m(t) (17) of expansion 16 is given by e (1) 2k,2m+1 (f) = min s 2k,2m+1 (f) ||f − (s2k,2m+1f)||2 = o( 1 √ m + 1 2k(α+1) ) (18) theorem 5.2. if f ∈ l2[0, 1) is a function such that f ′′ ∈ hα[0, 1) and its cas wavelet expansion is given by the series (16) , then the approximation error e (2) 2k,2m+1 (f) of f by (2k, 2m + 1)th partial sum s2k,2m+1(f)(t) of series (16) is given by e (2) 2k,2m+1 (f) = min s 2k,2m+1 (f) ||f − (s2k,2m+1f)||2 = o( 1 (m + 1) 3 2 2k(α+2) ) (19) 192 cas wavelet approximation of functions of hölder’s class hα[0, 1)... proof of theorem (5.1) since f(t) = ∞∑ n=1 ∞∑ m=−∞ cn,mψn,m(t) and s2k,2m+1(f)(t) = 2k∑ n=1 m∑ m=−m cn,mψn,m(t) ∴ f(t) −s2k,2m+1(f)(t) = ∞∑ n=1 ∞∑ m=−∞ cn,mψn,m(t) − 2k∑ n=1 m∑ m=−m cn,mψn,m(t) = ( 2k∑ n=1 + ∞∑ n=2k+1 )( −m−1∑ m=−∞ + m∑ m=−m + ∞∑ m=m+1 )cn,mψn,m(t) − 2k∑ n=1 m∑ m=−m cn,mψn,m(t) = 2k∑ n=1 −m−1∑ m=−∞ cn,mψn,m(t) + 2k∑ n=1 ∞∑ m=m+1 cn,mψn,m(t) (f(t) −s2k,2m+1(f)(t))2 = 2k∑ n=1 −m−1∑ m=−∞ c2n,mψ 2 n,m(t) + 2k∑ n=1 ∞∑ m=m+1 c2n,mψ 2 n,m(t) +2 ∑∑ 16 n6=n′≤ 2k ∑∑ −∞≤m6=m′≤−m−1 cn,mcn′,m′ψ t n,m(t)ψn′,m′(t) +2 ∑∑ 16 n6=n′≤ 2k ∑∑ m+1≤m 6=m′≤∞ cn,mcn′,m′ψ t n,m(t)ψn′,m′(t) ||f −s2k,2m+1(f)||22 = ∫ 1 0 |f(t) −s2k,2m+1(f)(t)|2dt 6 2k∑ n=1 −m−1∑ m=−∞ |cn,m|2 ∫ 1 0 |ψn,m(t)|2dt + 2k∑ n=1 ∞∑ m=m+1 |cn,m|2 ∫ 1 0 |ψn,m(t)|2dt + 2 ∑∑ 16 n6=n′≤ 2k ∑∑ −∞≤m 6=m′≤−m−1 |cn,m||cn′,m′| ∫ 1 0 |ψtn,m(t)ψn′,m′(t)|dt 193 shyam lal and satish kumar +2 ∑∑ 16 n6=n′≤ 2k ∑∑ m+1≤m6=m′≤∞ |cn,m||cn′,m′| ∫ 1 0 |ψtn,m(t)ψn′,m′(t)|dt = 2k∑ n=1 −m−1∑ m=−∞ |cn,m|2 + 2k∑ n=1 ∞∑ m=m+1 |cn,m|2 , by orthonormality of {ψn,m}n,m∈z ||f −s2k,2m+1(f)||22 ≤ 2k∑ n=1 ( −m−1∑ m=−∞ + ∞∑ m=m+1 )|cn,m|2 (20) cn,m = < f,ψn,m > = ∫ n 2k n−1 2k f(t) 2 k 2{cos(2mπ(2kt−n + 1)) + sin(2mπ(2kt−n + 1))} dt = 1 2 k 2 ∫ 1 0 f( x + n− 1 2k ) (cos(2mπx) + sin(2mπx)) dx, 2kt−n + 1 = x = 1 (2mπ) 2 3k 2 ∫ 1 0 f ′ ( x + n− 1 2k )(cos(2mπx) − sin(2mπx))dx, integrating by part = 1 (2mπ) 2 3k 2 [ ∫ 1 0 {f ′ ( x + n− 1 2k ) −f ′ ( n− 1 2k )}(cos(2mπx) − sin(2mπx))dx −f ′ ( n− 1 2k ) ∫ 1 0 (cos(2mπx) − sin(2mπx))dx] = 1 (2mπ) 2 3k 2 ∫ 1 0 {f ′ ( x + n− 1 2k ) −f ′ ( n− 1 2k )}(cos(2mπx) − sin(2mπx))dx |cn,m| 6 1 (2mπ) 2 3k 2 ∫ 1 0 |f ′ ( x + n− 1 2k ) −f ′ ( n− 1 2k )| |cos(2mπx) − sin(2mπx)|dx 6 a (2mπ) 2 3k 2 ∫ 1 0 | x 2k |α |cos(2mπx) − sin(2mπx)| dx, since f ′ ∈ hα[0, 1) now by cauchy schwarz inequality, we have |cn,m| 6 a (2mπ) 2 3k 2 { ∫ 1 0 | x 2k |2α dx} 1 2 { ∫ 1 0 |cos(2mπx) − sin(2mπx)|2dx} 1 2 = a (2mπ) 2( 3k 2 +kα) { ∫ 1 0 |x|2α dx} 1 2 = a (2mπ) 2( 3 2 +α)k 1 √ 2α + 1 |cn,m| 6 a 2mπ √ 2α + 1 2( 3 2 +α)k 194 cas wavelet approximation of functions of hölder’s class hα[0, 1)... by eqn (20) and (21) , we have ||f −s2k,2m+1(f)||22 ≤ 2k∑ n=1 ( −m−1∑ m=−∞ + ∞∑ m=m+1 ) a2 4m2π2(2α + 1) 2(3+2α)k , = a2 4π2(2α + 1) ( −m−1∑ m=−∞ + ∞∑ m=m+1 ) 2k 2(3+2α)k m2 = a2 4π2(2α + 1) 1 2(2+2α)k ( −m−1∑ m=−∞ 1 m2 + ∞∑ m=m+1 1 m2 ) = a2 4π2(2α + 1) 1 2(1+α)2k ( 1 m + 1 + 1 m + 1 ) = a2 2π2(2α + 1) 1 2(1+α)2k 1 m + 1 ∴ min s 2k,2m+1 (f) ||f −s2k,2m+1(f)||2 6 a π √ 2(2α + 1) 1 2k(α+1) 1 √ m + 1 ∴ e (1) 2k,2m+1 (f) = min s 2k,2m+1 (f) ||f −s2k,2m+1(f)||2 = o( 1 √ m + 1 2k(α+1) ) thus, theorem (5.1) is completely established. proof of theorem (5.2) following the steps of the proof of theorem ( 5.1) cn,m = 1 (2mπ) 2 3k 2 ∫ 1 0 f ′ ( x + n− 1 2k )(cos(2mπx) − sin(2mπx))dx = −1 (4m2π2) 2 5k 2 ∫ 1 0 f ′′ ( x + n− 1 2k )(cos(2mπx) + sin(2mπx))dx, = −1 (4m2π2) 2 5k 2 [ ∫ 1 0 {f ′′ ( x + n− 1 2k ) −f ′′ ( n− 1 2k )}(cos(2mπx) + sin(2mπx))dx −f ′′ ( n− 1 2k ) ∫ 1 0 (cos(2mπx) − sin(2mπx))dx] |cn,m| 6 1 (4m2π2) 2 5k 2 ∫ 1 0 |f ′′ ( x + n− 1 2k ) −f ′′ ( n− 1 2k )| |cos(2mπx) + sin(2mπx)|dx 6 b (4m2π2) 2 5k 2 ∫ 1 0 | x 2k |α |cos(2mπx) + sin(2mπx)|dx, sincef ′′ ∈ hα[0, 1) now by cauchy schwarz inequality, we have |cn,m| 6 b (4m2π2) 2( 5k 2 +kα) { ∫ 1 0 | x 2k |2αdx} 1 2{ ∫ 1 0 |cos(2mπx) + sin(2mπx)|2dx} 1 2 195 shyam lal and satish kumar |cn,m| 6 b (4m2π2) 2( 5 2 +α)k ( 1 √ 2α + 1 ) |cn,m| 6 b 4m2π2 √ 2α + 1 2( 5 2 +α)k (21) from eqn (20) and (21), we have ||f −s2k,2m+1(f)||22 = 2k∑ n=1 ( −m−1∑ m=−∞ + ∞∑ m=m+1 )|cn,m|2 6 2k∑ n=1 ( −m−1∑ m=−∞ + ∞∑ m=m+1 ) b2 16m4π4 (2α + 1) 2(5+2α)k , = b2 16π4 (2α + 1) 2(5+2α)k ( −m−1∑ m=−∞ + ∞∑ m=m+1 ) 2k m4 = b2 16π4 (2α + 1) 2(4+2α)k ( −m−1∑ m=−∞ 1 m4 + ∞∑ m=m+1 1 m4 ) = b2 16π4 (2α + 1) 2(4+2α)k ( 1 3(m + 1)3 + 1 3(m + 1)3 ) = b2 24π4 (2α + 1) 22k(α+2) 1 (m + 1)3 ∴ min s 2k,m (f) ||f −s2k,2m+1(f)||2 6 b 2 √ 6π2 √ (2α + 1) 2k(α+2) 1 (m + 1) 3 2 ∴ e (2) 2k,2m+1 (f) = min s 2k,2m+1 (f) ||f −s2k,2m+1(f)||2 = o( 1 (m + 1) 3 2 2k(α+2) ) hence, theorem (5.2) has been proved. 6 solution of the fredholm integral equation of second kind consider the fredholm integral equation of second kind given by eqn (1). using cas wavelet approximations, u(x) = ut ψ(x) = ψt (x)u , (22) f(x) = ft ψ(x) = ψt (x)f , and k(x,y) = ψt (x)kψ(y) , 196 cas wavelet approximation of functions of hölder’s class hα[0, 1)... where k is a square matrix of order 2k(2m + 1), which is calculated as follows∫ 1 0 ∫ 1 0 ψn,m(x)ψn′,m′(y)k(x,y)dxdy , (23) where 1 6 n,n ′ 6 2k and −m 6 m,m′ 6 m , equation (1) becomes ψt(x)u = ψt(x)f + ψt(x)k ∫ 1 0 ψ(y)ψt(y)udy (24) by orthonormality of cas wavelets, equation (24) reduces to u = (i − k)−1f (25) where i is identity matrix of order 2k(2m + 1) . subtituting the value of u from eqn (25) in eqn (22) , the solution u(x) of fredholm integral equation of second kind (1) can be obtained. 6.1 solution of integral eqn (1) by haar wavelet method let haar wavelet solution of intgral eqn (1) be of the form u(x) = 2m∑ i=1 aihi(x) (26) subtituting the eqn (26) in eqn (1) , we have 2m∑ i=1 ai(hi(x) −gi(x)) = f(x) (27) where gi(x) = ∫ 1 0 k(x,y)hi(y)dy (28) taking the collocation points xk = k−1 2 2m , k = 1, 2, ..., 2m, in eqns (27) and (26), we obtain 2m∑ i=1 ai(hi(xk) −gi(xk)) = f(xk) (29) 197 shyam lal and satish kumar and u(xk) = 2m∑ i=1 aihi(xk) (30) the wavelet coefficients ai, i = 1, 2, ..., 2m are obtained by solving 2m system of equations in (29). subtituting these coefficients in the eqn( 30) we can obtain the haar wavelet solution of the integral eqn (1). 7 illustrated numerical examples two fredholm integral equations have been solved by proposed method ie. cas wavelet method discussed in this paper. exact solutions of considered integral eqn are compared with their approximate solutions obtained by cas wavelet, legendre wavelet and haar wavelet method. the graphs of these solutions are plotted. it is observed that exact solution and approximate solutions of fredholm integral equations obtained by cas wavelet method are almost equal. the solutions of fredholm integral equation derived by the help of cas wavelet method are more closed than the solutions of this integral equation obtained by legendre wavelet and haar wavelet method. this comparison shows the advantages of proposed method of this paper. this is illustrated in following two examples. example 1 subtituting f(x) = sin(8πx) and k(x,y) = y2 , in the fredholm integral equation (1), it reduces to u(x) = sin(8πx) + ∫ 1 0 y2u(y)dy (31) the exact solution of integral eqn (31) is given by u(x) = sin(8πx) − 3 16π (32) cas wavelet solution for cas wavelet solution, take k = 2,m = 1 in the eqn (14) . in this case, ψ(x) = [ψ1,−1(x),ψ1,0(x),ψ1,1(x),ψ2,−1(x),ψ2,0(x),ψ2,1(x), ψ3,−1(x),ψ3,0(x),ψ3,1(x),ψ4,−1(x),ψ4,0(x),ψ4,1(x)] t (33) 198 cas wavelet approximation of functions of hölder’s class hα[0, 1)... where ψ1,−1(x) = 2(cos(8πx) − sin(8πx)) ψ1,0(x) = 2 ψ1,1(x) = 2(cos(8πx) + sin(8πx))   0 6 x < 14 , ψ2,−1(x) = 2(cos(8πx) − sin(8πx)) ψ2,0(x) = 2 ψ2,1(x) = 2(cos(8πx) + sin(8πx))   14 6 x < 12 , ψ3,−1(x) = 2(cos(8πx) − sin(8πx)) ψ3,0(x) = 2 ψ3,1(x) = 2(cos(8πx) + sin(8πx))   12 6 x < 34 , and ψ4,−1(x) = 2(cos(8πx) − sin(8πx)) ψ4,0(x) = 2 ψ4,1(x) = 2(cos(8πx) + sin(8πx))   34 6 x < 1 . f = [ −1 4 , 0, 1 4 , −1 4 , 0, 1 4 , −1 4 , 0, 1 4 , −1 4 , 0, 1 4 ]t , the matrix k is calculated as follows: 199 shyam lal and satish kumar ki,j = ∫ 1 0 ∫ 1 0 ψi(x)k(x,y)ψj(y)dydx = ∫ 1 0 ψi(x) ( ∫ 1 0 y2ψj(y)dy) dx = ( ∫ 1 0 ψi(x)dx) ( ∫ 1 0 y2ψj(y)dy) k =   π+1 64π2 1 96 −π+1 64π2 3π+1 64π2 7 96 −3π+1 64π2 5π+1 64π2 19 96 −5π+1 64π2 7π+1 64π2 37 96 −7π+1 96π2   [ 0 1 2 0 0 1 2 0 0 1 2 0 0 1 2 0 ] 200 cas wavelet approximation of functions of hölder’s class hα[0, 1)... k =   0 π+1 128π2 0 0 π+1 128π2 0 0 π+1 128π2 0 0 π+1 128π2 0 0 1 192 0 0 1 192 0 0 1 192 0 0 1 192 0 0 −π+1 128π2 0 0 −π+1 128π2 0 0 −π+1 128π2 0 0 −π+1 128π2 0 0 3π+1 128π2 0 0 3π+1 128π2 0 0 3π+1 128π2 0 0 3π+1 128π2 0 0 7 192 0 0 7 192 0 0 7 192 0 0 7 192 0 0 −3π+1 128π2 0 0 −3π+1 128π2 0 0 −3π+1 128π2 0 0 −3π+1 128π2 0 0 5π+1 128π2 0 0 5π+1 128π2 0 0 5π+1 128π2 0 0 5π+1 128π2 0 0 19 192 0 0 19 192 0 0 19 192 0 0 19 192 0 0 −5π+1 128π2 0 0 −5π+1 128π2 0 0 −5π+1 128π2 0 0 −5π+1 128π2 0 0 7π+1 128π2 0 0 7π+1 128π2 0 0 7π+1 128π2 0 0 7π+1 128π2 0 0 37 192 0 0 37 192 0 0 37 192 0 0 37 192 0 0 −7π+1 128π2 0 0 −7π+1 128π2 0 0 −7π+1 128π2 0 0 −7π+1 128π2 0   ∴ u = (i − k)−1f = [ −1 4 , 0, 1 4 , −1 4 , 0, 1 4 , −1 4 , 0, 1 4 , −1 4 , 0, 1 4 ]t (34) putting the values of ψ(x) and u from eqns (33) and (34) in eqn (22), we have u(x) = sin(8πx) (35) which is the cas wavelet solution of the integral equation (31) . 201 shyam lal and satish kumar legendre wavelet solution legendre wavelets ψ(l)n,m(t) = ψ(l)(k,n,m,t) having four arguments; k = 2, 3, ..., 2n − 1 , n = 1, 2, 3, ..., 2k−1, m is the order of the legendre polynomial and t is the normalised time, are defined by : ψ(l)n,m(t) = { (m + 1 2 ) 1 2 2 k 2 pm(2 kt− 2n + 1), if n−1 2k−1 6 t < n 2k−1 , 0, otherwise. (36) where pm(t) are legendre ploynomials of order m (rehman and khan [7]). the set {ψ(l)n,m}n,m∈z of legendre wavelets forms an orthonormal set. a function f ∈ l2[0, 1) may be expanded into legendre wavelet series as: f(t) = ∞∑ n=1 ∞∑ m=0 cn,mψ (l) n,m(t), (37) where cn,m =< f,ψ (l) n,m > .the series (37) may be truncated as: (f)(t) ≈ 2k−1∑ n=1 m−1∑ m=0 cn,mψ (l) n,m(t) = c t ψ(l)(t) (38) where c and ψ(l)(t) are 2k−1m × 1 matrices given by: c = [c1,0,c1,1, ...,c1,m−1,c2,0, ...,c2,m−1, ..., c2k−1,0, ...,c2k−1,m−1] t and ψ(l)(t) = [ψ (l) 1,0 (t),ψ (l) 1,1 (t), ...,ψ (l) 1,m−1(t),ψ (l) 2,0 (t), ...,ψ (l) 2,m−1(t), ..., ψ (l) 2k−1,0 (t), ...,ψ (l) 2k−1,m−1(t)] t similarly, a function k ∈ l2[0, 1) ×l2[0, 1) may be approximated as: k(x,y) ≈ (ψ(l))t (x)k(l)ψ(l)(y), where k(l) is 2k−1m × 2k−1m matrix, whose entries are given by k(l)i,j =< ψ (l) i (x),< k(x,y),ψ (l) j (y) >> . (39) for legendre wavelet solution, take m = 3,k = 3 in eqn (38), then twelve basis functions are given by 202 cas wavelet approximation of functions of hölder’s class hα[0, 1)... ψ(l)(x) = [ψ (l) 1,0 (x),ψ (l) 1,1 (x),ψ (l) 1,2 (x),ψ (l) 2,0 (x),ψ (l) 2,1 (x),ψ (l) 2,2 (x), ψ (l) 3,0 (x),ψ (l) 3,1 (x),ψ (l) 3,2 (x),ψ (l) 4,0 (x),ψ (l) 4,1 (x),ψ (l) 4,2 (x)] t (40) where ψ (l) 1,0 (x) = 2 ψ (l) 1,1 (x) = 2 √ 3(8x− 1) ψ (l) 1,2 (x) = √ 5(3(8x− 1)2 − 1)   0 6 x < 1 4 , ψ (l) 2,0 (x) = 2 ψ (l) 2,1 (x) = 2 √ 3(8x− 3) ψ (l) 2,2 (x) = √ 5(3(8x− 3)2 − 1)   1 4 6 x < 1 2 , ψ (l) 3,0 (x) = 2 ψ (l) 3,1 (x) = 2 √ 3(8x− 5) ψ (l) 3,2 (x) = √ 5(3(8x− 5)2 − 1)   1 2 6 x < 3 4 , and ψ (l) 4,0 (x) = 2 ψ (l) 4,1 (x) = 2 √ 3(8x− 7) ψ (l) 4,2 (x) = √ 5(3(8x− 7)2 − 1)   3 4 6 x < 1 . 203 shyam lal and satish kumar k(l) =   1 192 0 0 1 192 0 0 1 192 0 0 1 192 0 0 √ 3 384 0 0 √ 3 384 0 0 √ 3 384 0 0 √ 3 384 0 0 √ 5 1920 0 0 √ 5 1920 0 0 √ 5 1920 0 0 √ 5 1920 0 0 7 192 0 0 7 192 0 0 7 192 0 0 7 192 0 0 √ 3 128 0 0 √ 3 128 0 0 √ 3 128 0 0 √ 3 128 0 0 √ 5 1920 0 0 √ 5 1920 0 0 √ 5 1920 0 0 √ 5 1920 0 0 19 192 0 0 19 192 0 0 19 192 0 0 19 192 0 0 5 √ 3 384 0 0 5 √ 3 384 0 0 5 √ 3 384 0 0 5 √ 3 384 0 0 √ 5 1920 0 0 √ 5 1920 0 0 √ 5 1920 0 0 √ 5 1920 0 0 37 92 0 0 37 92 0 0 37 92 0 0 37 92 0 0 7 √ 3 384 0 0 7 √ 3 384 0 0 7 √ 3 384 0 0 7 √ 3 384 0 0 √ 5 1920 0 0 √ 5 1920 0 0 √ 5 1920 0 0 √ 5 1920 0 0   f(l) = [0, − √ 3 2π , 0, 0, − √ 3 2π , 0, 0, − √ 3 2π , 0, 0, − √ 3 2π , 0]t , u(l) = (i − k(l))−1f(l) = [0, − √ 3 2π , 0, 0, − √ 3 2π , 0, 0, − √ 3 2π , 0, 0, − √ 3 2π , 0]t . (41) putting the values of ψ(l)(x) and u(l) from eqns (40) and (41) in eqn (22), we get the legendre wavelet solution of the integral equation (31) as: u(x) = − √ 3 2π ψ (l) 1,1 (x) − √ 3 2π ψ (l) 2,1 (x) − √ 3 2π ψ (l) 3,1 (x) − √ 3 2π ψ (l) 4,1 (x) (42) 204 cas wavelet approximation of functions of hölder’s class hα[0, 1)... haar wavelet solution the haar wavelet family for x ∈ [0, 1] is defined as follows: hi(x) =   1 if x ∈ [ k m , k+ 1 2 m ), −1 ifx ∈ [k+ 1 2 m , k+1 m ), 0, otherwise (43) where m = 2b, b = 0, 1, ...,j is the level of wavelet; k = 0, 1, ...,m − 1 is the translation parameter. j is the maximum level of resulution. i is calculated by i = m + k + 1. the minimum value of i for m = 1,k = 0 is 2. the maximum value of i is i = 2m = 2j+1 (arbabi and darvishi [6]). for i = 1, h1(x) is taken to be scaling function which is defined as follows: h1(x) = { 1 if x ∈ [0, 1), 0, otherwise any function f(x) can be expressed in terms of haar wavelets as follows: f(x) = 2m∑ i=1 aihi(x), (44) where the wavelet coefficients ai, i = 1, 2, ..., 2m are to be determined. for haar wavelet solution take j = 3 in eqn (43), b = 0, 1, 2, 3 , then m = 2b = 1, 2, 4, 8. by eqns (28) the haar wavelet coefficients ai, i = 1, 2, ..., 16 are given by [−0.008071, 0.001459, 0.002497, 0.001447, 0.000485, 0.006380, 0.000488,−0.000476, 1.000010, 1, 1, 0.988178, 1, 1, 0.999039, 1] (45) putting these values of ai in the eqn (26), we get the solution of integral equation (31) by haar wavelet method. the haar wavelet solutions of integral eqn 31 are shown in the table (1). the exact solution and approximate solutions of fredholm integral equation (31) obtained by cas wavelet, legendre wavelet and haar wavelet method for different values of x are given in the table (1). 205 shyam lal and satish kumar table (1) x exact soln cas wavelet soln legendre wavelet soln haar wavelet soln by eqn 32 by eqn (35) by eqn (42) by eqn (26) 0 -0.059680 0 0.954930 0.996370 0.1 0.528105 0.587785 0.190986 -1.003630 0.2 -1.010736 -0.951056 -0.572958 0.995399 0.3 0.891376 0.951056 0.572958 0.995399 0.4 -0.647465 -0.587785 -0.190986 0.972689 0.5 -0.059680 0 -0.954930 0.992404 0.6 0.528105 0.587785 0.190986 -1.008083 0.7 -1.010736 -0.951056 -0.572958 -1.007595 0.8 0.891376 0.951056 0.572958 0.987585 0.9 -0.647465 -0.587785 -0.190986 0.989498 the graphs of the exact solution and approximate solutions of integral equation (31) obtained by cas wavelet, legendre wavelet and haar wavelet method are shown in the fig.(1). fig.(1) by numerical comparison in table(1) and graphs shown in fig.(1), it is clear that the solution of fredholm integral equation (31) by cas wavelet method is better than solutions obtained by legendre wavelet and haar wavelet methods. 206 cas wavelet approximation of functions of hölder’s class hα[0, 1)... example 2 consider the fredholm integral equation: u(x) = sin(4πx) + ∫ 1 0 xyu(y)dy . (46) it is obtained by subtituting f(x) = sin(4πx) and k(x,y) = xy , in the fredholm integral equation (1). the exact solution of fredholm integral equation (46) is given by u(x) = sin(4πx) − 3x 8π (47) cas wavelet solution for cas wavelet solution, take k = 1,m = 1 in eqn (14), then following the procedure of example (31), we have f∗ = [ −1 2 √ 2 , 0, 1 2 √ 2 , −1 2 √ 2 , 0, 1 2 √ 2 ]t , the matrix k∗ is calculated as follows: k∗i,j = ∫ 1 0 ∫ 1 0 ψi(x)k(x,y)ψj(y)dydx = ∫ 1 0 ψi(x) ( ∫ 1 0 xyψj(y)dy) dx = ( ∫ 1 0 xψi(x)dx) ( ∫ 1 0 yψj(y)dy) k =   √ 2 8π √ 2 8 − √ 2 8π √ 2 8π √ 2 8 − √ 2 8π   [ √ 2 8π √ 2 8 − √ 2 8π √ 2 8π √ 2 8 − √ 2 8π ] 207 shyam lal and satish kumar k∗ =   1 32π2 1 32π −1 32π2 1 32π2 1 32π −1 32π2 1 32π 1 32 −1 32π 1 32π 1 32 −1 32π −1 32π2 −1 32π 1 32π2 −1 32π2 −1 32π 1 32π2 1 32π2 1 32π −1 32π2 1 32π2 1 32π −1 32π2 3 32π 3 32 −3 32π 3 32π 3 32 −3 32π −1 32π2 −1 32π 1 32π2 −1 32π2 −1 32π 1 32π2   and u∗ = [ −1 2 √ 2 , 0, 1 2 √ 2 , −1 2 √ 2 , 0, 1 2 √ 2 ]t . u(x) = 1.0188 sin(4πx) − 0.0294 (48) this is the approximate solution of the integral equation (46) by cas wavelet method. legendre wavelet solution for legendre wavelet solution, take m = 3,k = 2 in eqn (38), then we have ψ(l)(x) = [ψ (l) 1,0 (x),ψ (l) 1,1 (x),ψ (l) 1,2 (x),ψ (l) 2,0 (x),ψ (l) 2,1 (x),ψ (l) 2,2 (x)]. (49) following the procedure of the example (1), we have (f∗)(l) = [0, − √ 6 2π , 0, 0, − √ 6 2π , 0]t , (u∗)(l) = [−0.0211,−0.4020, 0,−0.0633,−0.4020, 0]t (50) putting the values of ψ(l)(x) and (u∗)(l) from eqns (49) and (50) in eqn (22), we get the solution of the integral equation (46) by legendre wavelet method as u(x) = −0.0211ψ(l)1,0 (x)−0.4020ψ (l) 1,1 (x)−0.0633ψ (l) 2,0 (x)−0.4020ψ (l) 2,1 (x) (51) 208 cas wavelet approximation of functions of hölder’s class hα[0, 1)... (k∗)(l) =   1 32 √ 3 96 0 3 32 √ 3 96 0 √ 3 96 1 96 0 √ 3 32 1 96 0 0 0 0 0 0 0 3 32 √ 3 32 0 9 32 √ 3 32 0 √ 3 96 1 96 0 √ 3 32 1 96 0 0 0 0 0 0 0   haar wavelet solution for haar wavelet solution, take j = 2 in eqn (43),b = 0, 1, 2 then m = 2b = 1, 2, 4. the haar wavelet coefficients ai, i = 1, 2, ..., 8 are given by [0.061361, 0.027885, 0.616015, 0.670955, 0.000922, 1.465270, 0.000264, 0.004906] putting these values of ai in the eqn (26), we get the solution of integral equation (46) by haar wavelet method. the haar wavelet solutions of integral eqn 46 are given in the table (2). the exact solution and approximate solutions of fredholm integral equation (46) obtained by cas wavelet, legendre wavelet and haar wavelet method for different values of x are given in the table (2). table (2) x exact soln cas wavelet soln legendre wavelet soln haar wavelet soln by eqn (47) by eqn (48) by eqn (51) by eqn (26) 0 0 -0.0294 0.9549 0.6955 0.1 0.9391 0.9395 0.5610 0.6955 0.2 0.5639 0.5694 0.1671 0.6722 0.3 -0.6236 -0.6282 -0.2268 -0.7701 0.4 -0.9988 -0.9983 -0.6207 -1.7239 0.5 -0.0597 -0.0294 0.8952 0.6194 0.6 0.8794 0.9395 0.5013 0.6194 0.7 0.5042 0.5694 0.1074 -0.3672 0.8 -0.6833 -0.6282 -0.2865 -0.8337 0.9 -1.0585 -0.9983 -0.6803 -0.8532 209 shyam lal and satish kumar the graphs of the exact solution and approximate solutions of integral equation (46) obtained by cas wavelet, legendre wavelet and haar wavelet method are shown in the fig.(2). fig.(2) by numerical comparison in table(2) and graphs shown in fig.(2), it is observed that the solution of fredholm integral equation (46) by cas wavelet method is more accurate than solutions obtained by legendre wavelet and haar wavelet methods. note: the solutions of fredholm integral equations in examples (1) and (2) by cas wavelet method propoesd in this research paper and their numerical comparison with legendre wavelet and haar wavelet methods show the advantages of cas wavelet method than legendre wavelet and haar wavelet methods. 8 remarks 1. cas wavelet approximation of theorem (5.1) is given by e (1) 2k,2m+1 (f) = o( 1√ m+1 2k(α+1) ) . e(1) 2k,2m+1 (f) → 0 as m →∞, k →∞ . cas wavelet approximation of theorem (5.2) is given by e (2) 2k,2m+1 (f) = o( 1 (m+1) 3 2 2k(α+2) ) . e(2) 2k,2m+1 (f) → 0 as m →∞, k →∞ . therefore, estimators e(1) 2k,2m+1 (f) and e(2) 2k,2m+1 (f) are best possible in wavelet 210 cas wavelet approximation of functions of hölder’s class hα[0, 1)... analysis (zygmund [5]). 2. ∵ (m + 1) 3 2 2k(α+2) > (m + 1) 1 2 2k(α+1), m > 1,k > 1 ∴ 1 (m + 1) 3 2 2k(α+2) 6 1 (m + 1) 1 2 2k(α+1) ie. e (2) 2k,2m+1 (f) 6 e (1) 2k,2m+1 (f). hence, estimator e(2) 2k,2m+1 (f) is sharper than estimator e(1) 2k,2m+1 (f) . this shows that the estimator of a function f having f ′′ ∈ hα[0, 1) is sharper than the estimator of f having f ′ ∈ hα[0, 1). 3. cas wavelet method is more effective than legendre wavelet and haar wavelet method in finding the solution of fredholm integral equations (31) and (46). 4. fredholm integral equation of first kind,∫ 1 0 k(x,t)y(t)dt = f(x) can be solved by cas wavelet method as follows:∫ 1 0 ψ(x)kψt (t)ψ(t)y = ψ(x)f using orthonormality of cas wavelet, we get ky = f . by finding the matrix k and f as in the case of fredholm integral of second kind, we can find y and hence the solution y(x). 9 acknowledgments shyam lal, one of the authors, is thankful to dst cims for encouragement to this work. satish kumar, one of the authors, is grateful to c.s.i.r. (india) for providing financial assistance in the form of junior research fellowship vide ref. no. 17/12/2017 (ii) eu-v dated:13-02-2019 for his research work. authors are greatful to the referee for his valuable comments and suggestions, to improve the quality of the research paper. 211 shyam lal and satish kumar references [1] c.k.chui, wavelets: a mathematical tool for signal analysis, siam, philadelphia pa,(1997). [2] anichini, conti and trotta: some results for volterra integradifferential equations depending on derivative in unbounded domains, ration mathematica, 37(2019) 55-38. [3] s. saha ray and p.k. sahu :numerical methods for solving fredholm integral equations of second kind. hindawi publishing corporation, 426916(2013). [4] xiaoyang zheng and zhengyuan wei : estimates of approximation error by legendre wavelet. applied mathematics. 694-700(2016). [5] zygmund a.: trigonometric series, vol.i. cambridge university press,cambridge (1959). [6] arbabi,nazari,darvishi : a two dimensional haar wavelets method for solving systems of pdes. applied mathematics and computation 292 (2017) 33-77. [7] rehman and khan : the legendre wavelet method for solving fractional differential equations. commun nonlinear sci numer simulat 16(2011) 4163-4173 212 microsoft word cap1.doc cooperative games and finite geometries 57 cooperative games, finite geometries and hyperstructures antonio maturo università di chieti pescara abstract in this paper some relations between finite geometric spaces and cooperative games are considered. in particular by some recent results on blocking sets we have new results on blocking coalitions. finally we introduce a new research field on the possible relations between quasihypergroups and cooperative games. keywords cooperative games, finite geometries, blocking sets, quasihypergoups 1. cooperative games let m = {1, 2, …, n} be a finite non-empty set, called the set of players. a function v: ℘(m) → r such that: (c1) v(∅) = 0; (c2) (superadditivity) ∀a, b∈℘(m), (a∩b = ∅) ⇒ v(a∪b) ≥ v(a) + v(b); is called characteristic function on m. the pair (m, v) is called cooperative game with n players and the subsets of m are called coalitions. for every a∈℘(m) the number v(a) is the total gain that the players of a can have certainly forming a coalition, independently on the actions of the players not belonging to a. we assume the condition of “side payment”, that is in every coalition a any player can transfer an amount of his gain to another player belonging to a and so it is important only the total gain of the coalition. the condition (c2) is a consequence of the fact that the total gain obtained with an alliance between two disjoint coalitions is not inferior to the one without cooperation. we write v(i) to denote v({i}). by (c2) it follows that in a cooperative game (m, v) we have v(m) ≥ σi∈m v(i). if v(m) > σi∈m v(i) the game (m, v) is said to be essential, if the equality holds (m, v) is inessential. it is easy to prove that a cooperative game is inessential if and only if: (ad) (additivity) ∀a, b∈℘(m), (a∩b = ∅) ⇒ (v(a∪b) = v(a) + v(b)) and so there are no advantages by the cooperation. 58 two cooperative games (m, v) and (m, v’), with the same set of n players, are called strategically equivalent, we write (m, v) ≈ (m, v’) if there exist n+1 real numbers k>0 and c1, c2,…, cn such that: (se) ∀a∈℘(m), v’(a) = k v(a) + σi∈a ci. we obtain the game (m, v’) by the game (m, v) with an initial payment cr to any player r and by multiplying the total gain of any coalition by the scale factor k. then we can assume the same strategies to solve (m, v) or (m, v’). proposition 1.1. let (m, v) a cooperative game. the system with n+1 equations and n+1 unknowns k > 0 and c1, c2,…, cn: (esi) k v(m) + σi∈m ci = 1, k v(i) + ci = 0, i∈m has determinant v(m) σi∈m v(i) and so has solutions if and only if (m, v) is essential. in this case k = 1/(v(m) σi∈m v(i)) and so k > 0. the system: (nes) k v(m) + σi∈m ci = 0, k v(i) + ci = 0, i∈m has not trivial solution if and only if v(m) σi∈m v(i) = 0. the relation ≈ is an equivalence relation among the cooperative games with the same set of players m. by proposition 1.1 we have that, for any equivalence class k with respect to ≈, we have a unique cooperative game (m, v)∈k, called normal element of k or normal form of the elements of k, such that v(i) = 0, ∀i∈m and v(m)∈{0, 1}. precisely, v(m) = 1 if the game is essential and v(m) = 0 if it is inessential. the inessential games are in the same equivalence class and the normal form is such that v(a) = 0, ∀a∈℘(m). on the contrary, for n > 2, the essential games are in different classes. 2. simple cooperative games and projective spaces let (m, v) be an essential cooperative game in normal form. then v(i) = 0, ∀i∈m, and v(m) = 1. we say that (m, v) is a simple game if, ∀a∈℘(m), v(a)∈{0, 1}. by (c2), for any coalition a, if ac = m-a, we have three possibility: (a) v(a) = 1 and v(ac) = 0; (b) v(a) = 0 and v(ac) = 1; (c) v(a) = 0 and v(ac) = 0. 59 the set a is called winning coalition if (a) holds and losing coalition if (b) holds. it is evident that m is a winning coalition and the complement of a winning coalition is a losing coalition. so the number of winning coalitions is equal to the number of losing coalitions. the set a is said to be a blocking coalition if (c) holds. if a is a blocking coalition then also ac is a blocking coalition. so, if there exist blocking coalitions, their number is even. we have the following: proposition 2.1 let w be a subset of a set ℘(m), with m set of players. then w is the set of the winning coalitions of a simple cooperative game (m, v) if and only if satisfy the following properties, called the “axioms of shapley” (see [32], [34]): (w1) m∈w; (w2) ∀a, b∈℘(m), (a∈w, a⊆b) ⇒ b∈w; (w3) ∀a∈℘(m), a∈w ⇒ ac∉w. proof. let (m, v) be a simple cooperative game in normal form and let w be the set of winning coalitions. then (w1) and (w2) are trivial and (w3) follows by (c2). on the converse, let w be a subset of ℘(m) satisfying the axioms of shapley. we put, for any a∈℘(m), v(a) = 1 if a∈w and v(a) = 0 otherwise. the pair (m, v) is a simple cooperative game and w is the set of winning coalitions. by previous proposition, in the sequel we consider a simple cooperative game indifferently as the pair (m, v) or the pair (m, w). all the properties of the losing coalitions are obtained from the ones of the winning coalitions by replacing any coalition a with ac and ⊆ with ⊇ and vice versa. we can prove the following: proposition 2.2 let (m, w) be a simple cooperative game. a family θ of subsets of m is the set of blocking coalitions of (m, w) if and only if: (bc) ∀x∈θ, ∀a∈w, x∩a≠∅, xc∩a≠∅. now we introduce some useful definitions. definition 2.1 let m be a non empty set and let ℑ be a family of subsets of m. we say that ℑ has the “intersection property” if we have: (ip) ∀a, b∈ℑ, a∩b ≠ ∅. we say that ℑ has the “non-inclusion property” if we have: (ni) ∀a, b∈ℑ, a∩bc ≠ ∅ and ac∩b ≠ ∅. 60 definition 2.2 let m be a non empty set and let φ and ℑ be two families of subsets of m. we say that “ℑ is a generator of φ” or “φ is the closure of ℑ”, and we write φ = k(ℑ), if (gk) φ = {a∈℘(m): ∃b∈ℑ / b⊆a}. we say that “ℑ is a minimal generator of φ” if ℑ has the non-inclusion property and is a generator of φ. by (w3) it follows that, in a simple cooperative game (m, w), w satisfies the intersection property. the following proposition shows that any family ℑ of subsets of m that has the intersection property generates the winning coalitions of a simple cooperative game. proposition 2. 3 let m be a n-set, whose elements are called players, and let ℑ be a family of subsets non-void of m satisfying the intersection property. then if w is the closure of ℑ, the pair (m, w) is a simple cooperative game with w set of winning coalitions, called “the game generated by (m, ℑ)”. proof. let w be the closure of ℑ. then (w1) and (w2) are evident. if a∈w, then there exists b∈ℑ such that a⊇b and so ac∩b = ∅. then, ∀c∈ℑ, ac don’t contains c. otherwise ac must contain c∩b ≠ ∅, a contradiction. it follows that ac∉w. now we examine some relations between the cooperative simple games and the geometric spaces. definition 2. 3 a geometric space is a pair (m, ∆), with m a non-empty set, called the support and ∆ a non-empty family of subsets of m. the elements of m are called points and the ones of ∆ are called blocks. if any block has at least two points and any two blocks have at most one point in common (m, ∆) is called “space of lines” and the blocks are called also lines. (m, ∆) is non-degenerate if there are at least two blocks. definition 2.4 a projective space is a geometric space (m, ∆) such that (see [7]): (ps1) ∀p, q∈m, p ≠ q, there is exactly one block containing {p, q}, called the line pq; (ps2) (veblen-young axiom) let a, b, c, d four distinct points such that ab intersects cd. then ac intersects bd. (ps3) any line contains at least three points. a non-degenerate projective space is a projective plane (or projective space with dimension 2) if the axiom (ps2) is replaced by the stronger axiom: 61 (ps2s) two lines have at least a point in common. if a non-degenerate projective space (m, ∆) is not a projective plane, for any a, b, c∈m, distinct and such that c not belongs to the line ab, we define “plane abc”, or “2-dimensional subspace abc” of m, the union of the lines cx, with x∈ab. we say that (m, ∆) has dimension 3 if: (psd3) a line and a plane have at least a point in common. for recurrence we can consider projective spaces and subspaces with greater dimensions. if (m, ∆) is a projective plane we have that ∆ satisfies both the intersection property and the non-inclusion property. so, by proposition 2.3, we have the following: proposition 2.4 let (m, ∆) be a finite projective plane and let w be the closure of ∆. then (m, w) is a simple cooperative game, with w set of winner coalitions, and ∆ is a minimal generator of w. if (m, ∆) is a projective space of dimension 3 or 4 the planes have both the intersection property and the non-inclusion property. then we have: proposition 2. 5 let (m, ∆) be a finite projective n-dimensional space with n∈{3, 4} and let ∆* be the set of all the planes of m. if w is the closure of ∆* then (m, w) is a simple cooperative game, with w set of winner coalitions, and ∆* is a minimal generator of w. definition 2.5 let (m, ∆) be a geometric space and let ℑ be a family of subsets of m. a subset x of m is called a blocking set with respect to ℑ if. (bs) ∀a∈ℑ, x∩a ≠ ∅ and xc∩a ≠ ∅. if c is a subset of m containing a a∈ℑ, by (bs) we have: x∩c ≠ ∅ and xc∩c ≠ ∅. then it follows the: proposition 2. 6 let (m, ∆) be a geometric space, ℑ be a family of subsets of m and φ be the closure of ℑ. then x is a blocking set with respect to ℑ if and only if it is a blocking set with respect to φ. some corollaries of the previous propositions are: 62 proposition 2. 7 let (m, ∆) be a geometric space such that ∆ has the intersection property. then the blocking sets with respect to ∆ are the blocking coalitions of the simple cooperative game (m, w), with w closure of ∆. proposition 2. 8 in a finite projective plane (m, ∆) the blocking sets with respect the lines are the blocking coalitions of the simple cooperative game (m, w), with w closure of ∆. proposition 2. 9 in a finite 3-dimensional or 4-dimensional projective space (m, ∆) the blocking sets with respect the planes are the blocking coalitions of the simple cooperative game (m, w), with w closure of ∆*, set of the planes. the previous propositions show the importance of the research of blocking sets in a finite projective space. in particular we have the fundamental problems to find: (a) the minimal or maximal blocking sets; (b) the spectrum of the minimal blocking sets, that is the set of all the possible cardinalities of the minimal blocking coalitions; (c) the minimal winning coalitions; (d) the winning coalitions containing blocking coalitions. by (bs) it follows that the complement of a blocking set is also a blocking set, so to find the maximal blocking sets is equivalent to find the minimal ones. now we show some results in the particular case of projective planes. it is well known that, in a non-degenerate finite projective plane, all the lines have the same number of points. if q+1 is such number, the projective plane is said to be of order q and is noted πq. moreover, the lines through a fixed point p are also q+1 and the points of πq are q 2 + q + 1. by (ps3) we have q ≥ 2. it is well known (see [7], [17]) that there exists a desarguesian projective plane if and only if q is a prime or a power of a prime and such plane is unique. the first value of q with non-desarguesian planes is q = 9. for small values of q we have: • in π2 there are not blocking sets; • in π3 there are exactly two blocking sets; • the blocking sets on π4 and π5 are classified, respectively, in papers of berardi eugeni ([2]) and berardi innamorati ([5]); • the blocking sets on π7 are classified in papers of innamorati and maturo (see [23], [24], [25]). if k is the cardinality of a minimal blocking set on π7 we have 12≤k≤19. in particular there are, up to isomorphism, only two 63 minimal blocking sets of order 12 and there is only a minimal blocking set with 19 points. in the general case there are the following results (innamorati – maturo, [23], [25]): proposition 2.10 let s(q) the spectrum of the minimal blocking sets in πq. then, if q≥4, s(q) ⊇ [2q-1, 3q-5]∪{3q-3} and, if πq is desarguesian, s(q) ⊇ [2q-1, 3q-3]. proposition 2.11 a sufficient condition for the existence of a minimal blocking set with 3q–4 points on a non-desarguesian plane πq is that πq contains a proper subplane of order two. in [29] h. newmann conjectured that any finite non-desarguesian plane contains a proper subplane of order two. by previous proposition, if the conjecture is true, we have that also for the non-desarguesian plane of order q there exists a blocking set with 3q – 4 points. 3. cooperative games and finite geometric spaces we introduce the following: definition 3.1 let m be a non-void set and let ψ and ℑ be two families of subsets of m. we say that “ℑ is a intersection-generator of ψ” or “ψ is the intersectionclosure of ℑ”, and we write ψ = ik(ℑ), if (ik) ψ = {a∈k(ℑ): ∀b∈ℑ, a∩b ≠ ∅}. let (m, ∆) be a geometric space. if ∆ has not the intersection property, and w* is the closure of ∆, the pair (m, w*) is not a simple cooperative game because (w3) is not valid. but we have the following proposition, that generalizes proposition 2.3: proposition 3.1 let (m, ∆) be a finite geometric space and let w be the intersectionclosure of ∆. then (m, w) is a simple cooperative game, called “the game generated by (m, ∆)”. proof. (w1) is evident. if a∈w and a⊆b⊆m, then ∀c∈∆, c∩a≠∅ ⇒ c∩b≠∅ and (w2) holds. if a∈w then there exists c∈∆: c⊆a and so c∩ac=∅ and ac∉w. the game (m, w) generated by a geometric space (m, ∆) has two types of blocking coalitions: (t1) the blocking sets with respect ∆; 64 (t2) the subsets of m containing at least a block and with intersection void with at least a block. the losing coalitions are the subsets y of m non containing blocks and having intersection void with at least a block. example 3.1 let m be a n-set, whose elements are called players, and let ℑ be a family of subsets non-void of m, called companies. by an economic point of view, we assume that a player belonging to a company has a power of veto and a coalition containing a company has the control of such company. then a winner coalition of the game generated by the geometric space (m, ℑ) has the control of at least a company and a right of veto on all the companies, a losing coalition don’t have a power of veto on at least one company and don’t control any company. finally a blocking coalition of type (t1) has veto for any company but don’t control anyone, and a blocking coalition of type (t2) control at least a company but has not veto for all the companies. we can construct a simple cooperative game by a finite geometric space (m, ∆) also with a “geometric” procedure different from the one of proposition 3.1, by assigning the set ∆* of minimal winner coalitions. precisely, we consider a set ∆* with the following properties: (ds1) any a∈∆* is a union of elements of ∆; (ds2) any element of ∆ is contained in at least an element of ∆*; (ds3) ∆* has the intersection and non-inclusion properties; and we assume w equal to the closure of ∆*. we have the following: proposition 3.2 let (m, ∆) be a finite geometric space and let ∆* be a family of subsets of m satisfying (ds1), (ds2) and (ds3). if w is the closure of ∆* then: (dw1) (m, w) is a simple cooperative game; (dw2) the blocking sets of (m, ∆) are blocking coalitions of (m, w). proof. property (dw1) follows by (ds3). let x be a blocking set of (m, ∆). then x and xc intersect any block and so, by (ds1), any element of ∆*. by proposition 2.7 it follows that x is a blocking coalition of (m, w). proposition 2.5 is a particular case of the proposition 3.2. another important particular case is concerning the affine planes. 65 definition 3.2 a geometric space (m, ∆) is an affine plane if: (ap1) through any two distinct points there is exactly one line; (ap2) (parallel axiom) if g is a line and p is a point outside g then there is exactly one line through p that has no points in common with g; (ap3) there exist three points that are not on a common line. let (m, ∆) be a finite affine plane. it is well known that all the lines have the same number q ≥ 2 of points. the plane is said to be of order q and is noted αq. the number of elements of αq is q 2 and the lines through a fixed point are q+1. let ∆* be a set whose elements are union of two non parallel lines and such that any line of ∆ is contained in at least one element of ∆*. we say that ∆* is “a set of paired lines”. the set ∆* has the intersection and non-inclusion properties and so, by proposition 3.2, we have the following: proposition 3.3 let (m, ∆) be a finite affine plane and let ∆* be a set of paired lines. if w is the closure of ∆* then (m, w) is a simple cooperative game. moreover, the blocking sets of (m, ∆) are blocking coalitions of (m, w). in general we can obtain simple cooperative games from block designs, in particular from steiner systems. definition 3.3 let t, k, v be natural numbers such that 2≤k≤v. a finite geometric space (m, ∆) is a steiner system with parameter t, k, v, noted s(t, k, v), if: (ss1) through any t distinct points there is exactly one block; (ss2) any block has exactly k points; (ss3) m has v points. it is well known that necessary conditions for the existence of a s(t, k, v) is the existence of natural positive numbers br, r∈{0, 1, …, t-1} such that: br       − − rt rk =       − − rt rv , r = 0, 1, …, t-1. (3.1) for any r∈{0, 1, …, t-1}, br is the number of blocks through r fixed points. in particular bo is the number of all the blocks. for t = k the blocks are the subsets of m with k elements and for k = v there is only a block. we say that s(t, k, v) is nondegenerate if t<k<v. for t=2 the blocks are called lines. if r is a line and p is a point not incident r, d=b1-k is the number of lines through p non intersecting r. if d = 0, s(2, k, v) is a projective plane and, if d = 1, it is an affine plane. we have the following: 66 proposition 3.4 let (m, ∆) be a s(2, k, v) and let d = b1–k. then: • k divides d-d2; • if r and s are incident lines, the number of lines not incident to r∪s is α = d2-d (d2-d)/k. (3.2) proof. by (3.1) we have: v = (k+d)(k-1) + 1, b0 = k 2+(2d-1)k+(d-1)2+(d-d2)/k. (3.3) so b0 is integer if and only if k divides d 2-d. if r and s are two incident lines for each of the k points of r pass k+d-1 lines different from r, for each of the k-1 points of s not belonging to r pass exactly d lines not intersecting r. so the number of lines intersecting r∪s is δ = k(k+d-1) + d(k-1) + 1= k2 + (2d-1)k d +1 it follows that the lines not intersecting r∪s are α = b0 δ = d 2-d + (d-d2)/k. if d=0 (projective plane) or d=1 (affine plane) we have α=0. then we assume d>1. let i[x] be the minimum integer not inferior to x. by previous proposition, we can find a set ρrs union of at most i[(d 2-d)(k-1)/(2k)] = i[α/2] lines, such that any line of s(2, k, v) intersects r∪s∪ρrs. then we have the following: proposition 3.5 let (m, ∆) be a s(2, k, v) with d>1. for any pair (r, s) of incident lines let ρrs be the union of a minimal set l of lines such that l intersects all the lines not incident to r∪s. let ∆* be the family of the sets r∪s∪ρrs, with r, s∈∆. then ∆* satisfies (ds1), (ds2) and (ds3) and so generates a set w such that (m, w) is a simple cooperative game. therefore every element of ∆* is the union of at most i[α/2] + 2 lines. example 3.2 for k = 2, a s(2, k, v) is the trivial case of a graph complete with v elements and d = v-3. any element of ∆* is the union of exactly i[(d2-d)/4] + 2 lines. for v = k a s(2, k, v) has only a line and d = 0. now we consider the non-degenerate steiner systems with small values of d>1. for d = 2, by proposition 3.4, is k =2 and so we don’t have non-degenerate steiner systems. for d = 3, k is a divisor of 6 different from 2. if k = 3 we have a s(2, 3, 13). it is proved (see [17]) that there exists two non isomorphic s(2, 3, 13). in this case we have α = 4 and the elements of ∆* are the union of at most 4 lines. if k = 6 we have a s(2, 6, 46) and α = 5. then there are at most 5 lines in any element of ∆*. 67 6. cooperative games and hyperstructures in this paragraph we introduce some ideas on the possible relations between cooperative games and some particular commutative weak associative quasihypergroups, called “geometric hypergroupoids”. we think that it is a very interesting argument of research. definition 4.1 let m be a non-empty set and let ℘*(m) be the family of non-empty subsets of m. a hyperoperation on m is a function σ: m×m → ℘*(m), such that to every ordered pair (x, y) of elements of m associates a non-empty subset of m, noted xσy. the pair (m, σ) is called hypergroupoid with support m and hyperoperation σ. if a and b are non-empty subsets of m, we put aσb = ∪{aσb: a∈a, b∈b}. moreover, ∀a, b∈m, we put, aσb = {a}σb and aσb = aσ{b}. definition 4.2 a hypergroupoid (m, σ) is said to be: (si) a semihypergroup, if ∀x, y, z∈m, xσ(yσz) = (xσy)σz (associativity); (qi) a quasihypergroup, if ∀x∈m, xσm = m = mσx (riproducibility); (hy) a hypergroup if it is both a semihypergroup and a quasihypergroup; (co) commutative, if ∀x, y∈m, xσy = yσx; (wa) weak associative, if ∀x, y, z∈m, xσ(yσz) ∩ (xσy)σz ≠ ∅; (cl) closed, if ∀x, y∈m, {x, y}⊆xσy; (ip) idempotent, if ∀x∈m, xσx={x}. definition 4.3 we say that a hypergroupoid (m, σ) is geometric if it is commutative, closed and idempotent. a geometric hypergroupoid (m, σ) is said to be a join space if the following incidence axiom holds: (ia) ∀a, b, c, d∈m, (∃x∈m: a∈bσx, c∈dσx) ⇒ (∃y∈m: y∈aσd∩bσc). definition 4.4 let (m, σ) be a geometric hypergroupoid. a geometric space (m, ∆) is said to be “associated to (m, σ)” if ∆ is the set of the hyperproducts aσb with a≠b. proposition 4.1 let (m, ∆) be a space of lines. then there exists only a geometric hypergroupoid (m, σ) with (m, ∆) associated geometric space. precisely we have: (gha) ∀x∈m, xσx = {x}, ∀x, y∈m, x≠y, xσy is the line containing {x, y}. example 4.1 let m be the support of a projective space and, for any x, y∈m, with x ≠ y, put xσx = {x} and xσy equal to the line xy. the hypergroupoid (m, σ) is 68 geometric and is a hypergroup. it is also a join space. the incidence axiom is the veblen-young axiom. example 4.2 let m be the support of an euclidean space and, for any x, y∈m, with x ≠ y, put xσx = {x} and xσy equal to the segment xy. the hypergroupoid (m, σ) is geometric. it is a hypergroup and a join space, but not a space of lines, and the incidence axiom is the pasch axiom. example 4.3 let m be the support of an affine space and, for any x, y∈m, with x ≠ y, put xσx = {x} and xσy equal to the line xy. (m, σ) is a geometric hypergroupoid and a space of lines but not a hypergroup. the incidence property is not valid. in the sequel we don’t distinguish from a geometric hypergroupoid and the geometric space associated. by previous considerations we have that the concept of geometric hypergroupoid generalizes the one of space of lines and so projective spaces, affine spaces and steiner systems s(2, k, v) are particular cases. the space of lines that are hypergroups or join spaces, e. g. projective spaces, have very interesting properties. also join spaces that are not spaces of lines such as the one of example 4.2, have important properties. let (m, σ) be a geometric hypergroupoid. we call blocks of order 1 the singletons {a}, a∈m, blocks of order 2 the hyperproducts aσb with a≠b and, for n≥3, we call blocks of order n the hyperproducts hσk, with h block of order h<n and k block of order n-h, that are not blocks of order less than n. a block of order n generalizes the concept of subspace of dimension n-1 of a space of lines and so we can generalize the results of the previous paragraphs. we denote by ∆n the set of hyperproducts of order n and by λn the set of hyperproducts of order h≤n. moreover we put λ0 = ∪n∈n λn. then by a geometric hypergroupoid (m, σ) we obtain the geometric spaces (m, ∆n), with n belonging to a subset of n, finite if m is finite. we have also the geometric spaces (m, λn), n∈n0. in particular ∆ = ∆2. suppose m is a finite set of players. from each of the geometric spaces (m, ∆n), n>1, we can obtain a cooperative game with particular properties dependent on the algebraic structure of (m, σ). a possible economic interpretation of a block b of order n is as the set of the players disposed to form a coalition because influenced by the set of players {a1, a2, …, an} that generates the block. if (m, σ) is not a hypergroup such coalition depend on the process of aggregation of the n players. another possible interpretation of the block b is as a company controlled by {a1, a2, …, an}. finally, we think that many other economic interpretations and geometric properties (e. g. blocking coalitions) depends on the algebraic structure of (m, σ) and we intend study these questions in a very near paper. 69 bibliography [1] k. j. arrow, social choose and individual values, j. wiley and sons, new york, 1951, 2nd ed 1963. 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[40] j. von neumann, o. morgenstern, theory of games and economic behaviour, princeton, 1947 ratio mathematica volume 40, 2021, pp. 77-85 recognizable hexagonal picture languages and xyz domino tiles anitha pancratius * rosini babu rajendran pillai† abstract in this paper we introduced xyz local hexagonal picture languages, where the usual notion of hexagonal tiles of size (2, 2, 2) are replaced by xyz dominoes, motivated by the studies of xyz domino systems. this new formalism is used for checking recognizability of hexagonal pictures. it is noticed that nonregular hexagonal pictures can also be studied in the place of regular pictures. recognizability of xyz local hexagonal picture were studied and the fact that every recognizable hexagonal p[icture languages can be obtained as a projection of xyz local hexagonal picture languages. keywords: xyz domino system, local hexagonal picture languages, recognizability of xyz domino system, mapping, hexagonal pictures, x domino, y domino, z domino.1 *department of mathematics, b j m government college, chavara; anitabenson321@gmail.com, †department of mathematics, f m n college, kollam;brosini@gmail.com 1received on january 12th, 2021. accepted on may 16th, 2021. published on june 30th, 2021. doi:10.23755/rm.v40i1.610. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 77 anitha p, rosini b. 1 introduction a picture is used to understand things in a better way. hexagonal pictures and tiles has got many significances. a lot of technologies are there to compute pictures with the help of computers. this resulted in the introduction of picture generating models. in [d.giammarresi, 1992] giammerresi et.al proposed the recognizability of picture languages. the languages in recognizable pictures were defined using tiling systems. hexagonal pictures have got various uses particularly in picture processing and image analysis. hexagonal arrays on triangular grid are viewed as two dimensional representation of three dimensional blocks and perceptual twins of pictures of a given set of blocks[ks, 2005]. since late seventies, formal models to generate or recognize hexagonal pictures has been found in literature in the frame work of pattern recognition and image analysis. recently searching for a new method for defining hexagonal pictures has moved towards the new definition for recognizable languages generated by hexagonal pictures which inherits many properties from existing cases, in [giammarresi, 1966],. local and recognizable hexagonal picture languages in terms of hexagonal tiling system were introduced and studied in [ks, 2005]. in [dersanambika, 2004] k s dersanambika et.al. define xyzdomino systems and charecterised hexagonal pictures using this. subsequently hexagonal hv-local picture languages via hexagonal domino systems were introduced in the light of two dimensional domino system introduced by latturex et.al [latteurx, 1997]. hexagonal arrays and hexagonal patterns are found in picture processing and image analysis [dersanambika, 2004]. it is very natural to consider hexagonal tiles on triangular grid, we require certain hexagonal tiles only to present in each hexagonal pictures of a hexagonal picture languages. this leads to recognizable hexagonal pictures and the hexagonal tiling systems. the xyz domino tiling characterize the hexagonal picture languages. so we define xyz local hexagonal picture languages over the usual notion of a hexagonal picture of size (2, 2, 2) and proved that every recognizable hexagonal picture language can be obtained as some projection of these languages. 2 recognizability of hexagonal pictures in this part we review the notions of formal language theory and some of the basic concepts on hexagonal pictures and hexagonal picture languages [anitha, 2011]. let σ be a finite alphabet of symbols. a hexagonal picture p over σ is a hexagonal array of symbols of σ. the set of all hexagonal arrays of the alphabet σ is denoted by σ∗∗h . a hexagonal picture over the alphabet a, b, c d is shown in figure 1. 78 recognizable hexagonal picture languages and xyz domino tiles a a b a b f c b c c a c d a a d c c a figure 1 with respect to a triad of triangular axes (x, y, z) the co-ordinates of each element of the hexagonal picture in figure 2(a) and figure 2(b) respectively are given below [ks, 2005]. figure2(a) figure2(b) 79 anitha p, rosini b. if p ∈ σ∗∗h , then p̂ is the hexagonal picture obtained by surrounding p with a special boundary # is called a bordered hexagonal picture which is shown in figure 3. # # # # # a b c # # b b a b # # c a b c f # # d b e d # # c f a # # # # # figure 3 let l1(p) = l, l2(p) = m,l3(p) = n be the size of the hexagonal arrays. we writep = (l,m,n), the size of a picture. for a picture p of size (l,m,n) we have the bordered picture p̂ is of size(l + 1,m + 1,n + 1). now we see the projections of hexagonal picture and projections of a language. γ and σ be two finite alphabets and π : γ → σ be a mapping, this mapping π is called a projection. a hexagonal tile is of the form as shown in figure 4. figure 4 given a hexagonal picture p of size (l,m,n) we denote the set of hexagonal subpicture of p of size (2, 2, 2) is called a hexagonal tile of size (2, 2, 2). figure 4 denote a hexagonal tile of size (2, 2, 2). a hexagonal tiling system [dersanambika, 2004] t is a 4-tuple (σ, γ,π,θ) where σ and γ are two finite set of symbols. π : γ −→ σ is a projection and θ is the set of hexagonal tiles over the alphabet γ∪{#}. definition 2.1. a hexagonal sub picture p̂′ is a picture which is a hexagonal sub array of the picture p̂. given a hexagonal picture p̂ then bl,m,n(p̂) denotes the set of hexagonal sub pictures of size l,m,n. for hexagonal pictures there are three types of concatenations namely type 1, type 2, and type 3 refer [anitha, 2011]. a hexagonal picture language is recognizable if there exist a local language l′ over an alphabet σ and a mapping π : γ −→ σ such that l ⊆ π(l′). 80 recognizable hexagonal picture languages and xyz domino tiles 3 xyz local hexagonal picture languages in this section we introduce the notion of xyz local hexagonal picture languages where the hexagonal tiles of size (2, 2, 2) are replaced by xyz dominos. definition 3.1. l be a hexagonal picture language included in σ∗∗h . l is said to be xyz local if there exist a set ∆ of x,y,z dominoes over σ ∪{#} such that l = {q ∈ σ∗∗h|t1,1,2(q) ∪t1,2,1(q) ∪t2,1,1(q) ⊆ ∆} example 3.1. if we consider the hexagonal picture langauages l over the alphabet σ = {0, 1} then all the hexagonal picture of can be obtained by the concatenation of xy,yz,xz dominoes. figure 5 the hexgonal picture so obtained is local as we can associate a set of xyz dominoes ∆ as follows which generates the whole picture. here figure 6 show the picture generated by x dominoes, figure 7 shows the y dominoes, while figure 8 shows the z dominoes. figure 6 figure 7 figure 8 81 anitha p, rosini b. theorem 3.1. let l ⊆ σ∗∗h be a hexagonal picture language. if l is xyz local, then l is local. proof. proof let l ⊆ σ∗∗h be a xyz local hexagonal picture language. for provingt l is local, construct a local hexagonal picture language l′ and show that l = l′. we know that there exist a set ∆ of xyz dominoes over σ ∪{#}(∆ ⊆ (σ ⊆{#})(1,1,2) ∪ (σ ⊆{#})(1,2,1) ∪ (σ ⊆{#})(2,1,1)) such that l = {p ∈ σ∗∗h|t1,1,2(p̂) ∪t1,2,1(p̂) ∪t2,1,1(p̂) ⊆ ∆}. we define the set of hexagonal pictures ∆ ′ , ∆ ′ = q ∈ (σ ∪{#})(2,2,2)|t1,2,1(q̂) ∪t1,1,2(q̂) ∪t2,1,1(q̂) ⊆ ∆}. let l′ = {p ∈ σ∗∗h|t2,2,2(p̂) ⊆ ∆ ′ }. clearly l′ is local. now let p ∈ l′. then t2,2,2(p̂) ⊆ ∆ ′ and t1,1,2(p̂) ⊆ t1,1,2(t2,2,2(p̂)) ⊆ t1,1,2(∆ ′ ) ⊆ ∆. similarly t2,1,1(p̂) ⊆ ∆ and t1,2,1(p̂) ⊆ ∆. hence p ∈ l. therefore l′ ∈ l. to show that l ∈ l′. let p ∈ l, let q ∈ l and a ∈ t2,2,2(q̂). then t1,2,1(a) ⊆ t1,2,1(q̂) ⊆ ∆, t2,1,1(a) ⊆ t2,1,1(q̂) ⊆ ∆, and t1,1,2(a) ⊆ t1,1,2(q̂) ⊆ ∆. so a ∈ ∆′ , and q ∈ l′. therefore l ∈ l′. hence l = l′. that is if l is an xyz local hexagonal picture language then l is a local hexagonal picture language. example 3.2. for instance, the hexagonal picture language defined above is local with, figure 9 theorem 3.2. let l ⊆ σ∗∗h be a hexagonal picture language. if l is local there exists a xyz local hexagonal picture language l′ over σ ′ and a mapping π : σ ′ → σ such that l = π(l ′ ). proof. we define an extended alphabet from σ. we denote this alphabet e(σ) = (σ ∪{#})3,3,3. now we define a mapping π, π : σ∗∗h → e(σ∗∗h ) 82 recognizable hexagonal picture languages and xyz domino tiles p → pe ∈ e(σ∗∗h ). we define p and pe with same size and for all 1 6 i 6 l + 1, 1 6 j 6 m + 1, 1 6 k 6 n + 1 where (l,m,n) be the size ofp. figure 10 it can be verified that every hexagonal tile of size (2, 2, 2) in pe where p ∈ σ∗∗h appear in π(p) and vice versa. t2,2,2(p e) = ∪t2,2,2(a). if p ∈ σ∗∗h where σ = {0, 1} then figure 11 where figure 12 83 anitha p, rosini b. we also define a mapping φ from e(σ∗∗h ) onto σ∗∗h by π(a) = a(2, 2, 2) for a ∈ e(σ). by using figure 10 it is clear that for all p ∈ σ∗∗h we have p = φ(π(p)). hence we conclude that l = π(l′) theorem 3.3. let l ∪ σ∗∗h be a hexagonal picture language l is recognizable if and only if there exist a xyz local hexagonal picture language l′ over σ ′ and a mapping π : σ ′ → σ such that l = π(l′). proof. let l be a recognizable hexagonal picture language over σ. by definition of recognizable picture languages we know that there exist a local hexagonal picture language l′ over an alphabet σ ′ and a mapping π : σ ′ → σ such that l = π(l′). according to theorem 1 there exist a xyz local hexagonal picture language l′′ over an alphabet σ ′′ and a mapping φ : σ ′′ → σ′ such that l′ = φ(l′′). from the above two results we get l = π(l′′) = π(φ(l′′)) where l′′ is xyz local. now let l′ be a xyz local hexagonal picture language over σ ′ , then π : σ ′ → σ be a mapping. applying theorem 2 it follows that l′ is local and hence the hexagonal picture π(l′) is recognizable. 4 conclusion xyz local recognizable hexagonal picture languages provides a new formalism of using xyz dominos instead of usual notation of hexagonal tile.we tried to prove that a hexagonal picture language l is recognizable and is the projection of a xyz local hexagonal picture language. in a similar way we can extend the various other properties of recognizable rectangular picture to recognizable hexagonal picture. references p. anitha,k. sujathakumari and k. s. dersanambika. hexagonal picture languages and its recognizability journal of science, technology and management, 4, 2011. k. s. dersanambika k. krithivasan,c. martin-vide and k. g. subramanian. hexagonal pattern languages lecture notes on computer science, 3322, 2004. k. s. dersanambika k. krithivasan,c. martin-vide and k. g. subramanian. local and recognizable hexagonal picture languages international journal of pattern recognition and artificial intelligence, 19(7), 2005. d. giammarresi and a. restivo. two dimensional finite state recognizability fundamenta informatica, 25(3,4), 1966. 84 recognizable hexagonal picture languages and xyz domino tiles d. giammarresi and a. restivo. recognizable picture languages in parallel image processing world scientific, 1992. m. latteurx and d. simplot. recognizable picture languages and domino tiling theoretical computer science,178, 1997. k. g. subramanian. hexagonal array grammars computer graphics and image processing,10, 1979. 85 ratio mathematica, 21, 2011, pp. 3-26 3 history and new possible research directions of hyperstructures piergiulio corsini abstract we present a summary of the origins and current developments of the theory of algebraic hyperstructures. we also sketch some possible lines of research . keywords. hypergroupoid, hypergroups, fuzzy sets. msc2010: 20n20, 68q70, 51m05 1 the origins of the theory of hyperstructures hyperstructure theory was born in 1934, when marty at the 8 th congress of scandinavian mathematiciens, gave the definition of hypergroup and illustrated some applications and showed its utility in the study of groups, algebraic functions and rational fractions. in the following years, around the 40’s, several others worked on this subject: in france, the same marty, krasner, kuntzman, croisot, in usa dresher and ore, prenowitz, eaton, griffith, wall (who introduced a generalization of hypergroups, where the hyperproduct is a multiset, i.e. a set in which every element has a certain multiplicity); in japan utumi, in spain san juan, in russia vikhrov, in uzbekistan dietzman, in italy zappa. in the 50’s and 60’s they worked on hyperstructures, in romania benado, in czech republic drbohlav, in france koskas, sureau, in greece mittas, stratigopoulos, in italy orsatti, boccioni, in usa prenowitz, graetzer , pickett, mcalister, in japan nakano, in yugoslavia dacic. 4 but it is above all since 70’ that a more luxuriant flourishing of hyperstructures has been and is seen in europe, asia, america, australia. 2. the most important definitions definition 1 let h be a nonempty set and p*(h) the family of the nonempty subsets of h. a multivalued operation (said also hyperoperation) < o > on h is a function which associates with every pair (x, y) h 2 a non empty subset of h denoted x o y. an algebraic hyperstructure or simply a hyperstructure is a non empty set h, endowed with one or more hyperoperations. a nonempty set h endowed with an hyperoperation < o > is called hypergroupoid and is denoted <h ; o >. if {a, b}  p*(h) , a o b denotes the set aa, bb a o b. definition 2 a hypergroupoid <h ; o> is called semi-hypergroup if (i)  (x, y, z)  h 3 , (x o y) o z = x o (y o z). a hypergroupoid <h ; o> is called quasi-hypergroup if (ii)  (a, b)  h 2 ,  (x, y)  h 2 such that a  b o x , a  y o b. definition 3 a hyperoperation is said weak associative if (iii)  (x, y, z)  h 3 , (x o y) o z  x o (y o z)   (see [141]). definition 4 a hypergroupoid <h ; o> is called hypergroup if satisfies both (i) and (ii). definition 5 a hyperoperation < o> is said commutative if (iv) (a, b)  h 2 , a o b = b o a. definition 6 a hyperoperation is said weak commutative if 5 (v)  (x, y)  h 2 , x o y  y o x  . definition 7 a hv – group is a quasi-hypergroup <h ; o > such that the hyperoperation < o > is weak associative. let <h ; o> be a commutative hypergroup. we denote with a / b the set {x  a  x o b }. definition 8 a hypergroupoid <h ; o> is called join space if it is a commutative hypergroup such that the following implication is satisfied (vi) (a, b, c, d)  h 4 , a / b  c / d    a o d  b o c  . 3. the recent history of the theory currently one works successfully in hyperstructures, in several continents, i shall remember only some names of hyperstructure scientists since 1970. around the 70’s and 80’s, hyperstructures where cultivated especially:  in greece by mittas and his school (canonical hypergroups and their applications, vougiouklis and his school (hv – groups);  in italy: by corsini (homomorphisms, join spaces, quasicanonical hypergroups, complete hypergroups, 1hypergroups, cyclic hypergroups etc.) and his school, tallini g. (hypergroups associated with projective planes), rota, procesi ciampi (hyperrings);  in usa: by prenowitz and jantoshak (join spaces and geometries, homomorphisms), roth (character of hypergroups, canonical hypergroups), comer (polygroups, representations of hypergroups);  in france by krasner and sureau, (structure of hypergroups), koskas, (semihypergroups associated with groupoids). deza; 6  in canada rosenberg (hypergroups associated with graphs, binary relations). around the 90’s and more recently, many papers appeared, made in europe, asia, asia , america and australia. europe:  in greece o at thessaloniki (aristotle univ.), mittas, konstantinidou, serafimidis kehagias, ioulidis, yatras, synefaki, o at alexandroupolis (democritus univ. of thrace), t. vougiouklis, dramalidis, s.vougiouklis, o at patras (patras univ.), stratigopoulos, o at orestiada (democritus univ. of thrace), spartalis, o at athens , ch. massouros, g. massouros, g. pinotsis;  in romania o at iasi (cuza univ.), v. leoreanu, cristea, tofan, gontineac., volf, l. leoreanu, o at cluj napoca babes bolyai univ.), purdea, pelea, calugareanu, o at constanta (ovidius univ.) stefanescu;  in czech republic o at praha (karlos univ.) kepka, jezek, drbohlav, (agricultural univ.) nemec, o at brno (brno univ. of technology) j. chvalina, (military academy of brno) hoskova, (technical univ. of brno) l. chvalinova, (masaryk univ.) novotny, (university of defence) rackova, at olomouc, (palacky univ.) hort, o at vyskov (military univ. of ground forces) moucka;  in montenegro o at podgorica (univ. of podgorica) dasic, rasovic;  in slovakia 7 o at bratislava, (comenius university), kolibiar, (slovak techn. univ.) jasem, o at kosice, (matematickz ustav sav), jakubik, (safarik univ.), lihova, repasky, csontoova;  in italy o at udine (udine univ.) corsini, o at messina (messina univ.) de salvo, migliorato, lo faro, gentile, o at rome ( universita’ la sapienza) g. tallini, m. scafati-tallini, rota, procesi ciampi, peroni, o at pescara (g. d’annunzio univ.) a. maturo, s. doria, b. ferri, o at teramo (univ. di teramo) eugeni, o at l’aquila (univ. dell’aquila) innamorati, l. berardi, o at brescia (universita’ cattolica del sacro cuore), marchi, o at lecce ( universita’ di lecce), letizia, lenzi, o at palermo, (univ. di palermo), falcone, o at milano, (politecnico), mercanti, cerritelli, gelsomini;  in france o at clermont-ferrand (universite’ des math. pures et appl.) sureau, m. gutan, c. gutan, o at lyon, (universite’ lyon 1), bayon, lygeros;  in spain o at malaga, (malaga univ.) martinez, gutierrez, de guzman, cordero;  in finland o at oulu, (univ. of oulu), nieminen, niemenmaa. america  in usa o at charleston (the citadel) comer, o at new york (brooklyn college, cuny), jantosciak, o at cleveland, ohio, (john carroll univ.), olson, ward, 8  in canada o at montreal, (universite’ de montreal), rosenberg, foldes, asia  in thailand o at bangkok, (chulalongkorn univ), kemprasit, punkla , chaopraknoi, triphop, tumsoun, o at samutprakarn, (hauchievw chalermprakiet univ.), juntakharajorn, o at phitsanulok, (naresuan univ.), c. namnak.  in iran o at babolsar (mazandaran univ.) ameri, razieh mahjoob, moghani, hedayati, alimohammadi, o at yazd (yazd univ.) davvaz, koushky, o at kerman, (shahid bahonar univ.) zahedi, molaei, torkzadeh, khorashadi zadeh, hosseini, mousavi, (islamic azad univ.) borumand saeid o at kashan (univ. of kashan) ashrafi, ali hossein zadeh, o at tehran (tehran univ.) darafsheh, morteza yavary, (tarbiat modarres univ.) iranmanesh, iradmusa, madanshekaf, (iran univ. of sci. and technology) ghorbany, alaeyan, (shahid beheshti univ.) mehdi ebrahimi, karimi, mahmoudi, o at zahedan (sistan and baluchestan univ.) borzooei, hasankhani, rezaei, o at zanjan (institute for advanced studies in basic sciences) barghi o at sari-branch, (islamic azad univ.), roohi.  in korea o at chinju (gyeongsang national univ.) young bae jun, e. h. roh, o at taejon, (chungnam national univ.) sang cho chung, 9 o (taejon univ.) byung-mun choi, o at chungju (chungju national univ.) k.h. kim.  in india o at kolkata, (uni. of calcutta), m.k. sen, dasgupta,chowdhury, o at tiruchendur, tamilnadu (adinatar college of arts and sciences), asokkumar,velrajan,  in china o at chongqing, (chongqing three gorges univ.) yuming feng, o at xi’an, (northwest univ.), xiao long xin, o at enshi, hubei province (hubei institute for nationalities), janming zhan, xueling ma;  in japon o at tokyo, (hitotsubashi univ. kunitachi), machida, o at tagajo, miyagi, (tohoku gakuin univ.), shoji kyuno;  in jordan o at karak, (mu’tah univ.) m.i. al ali;  in israel o at ramat gan, (bar ilan univ.), feigelstock. 4. join spaces, fuzzy sets, rough sets the join spaces were introduced by prenowitz in the 40’s and were utilized by him and later by him together with jantosciak, to construct again several kinds of geometry. join spaces had already many other applications, as to graphs, (nieminen, rosenberg, bandelt, mulder, corsini), to median algebras (bandelt-hedlikova) to hypergraphs (corsini, leoreanu), to binary relations (chvalina, rosenberg, corsini, corsini, leoreanu, de salvo-lo faro). fuzzy sets were introduced in the 60’s by an iranian scientist who lives in usa, zadeh [144]. he and others, in the following decades, found surprising applications to almost every field of science and 10 knowledge: from engineering to sociology, from agronomy to linguistic, from biology to computer science, from medicine to economy. from psychology to statistics and so on. they are now cultivated in all the world. let’s remember what is a fuzzy set. we know that a subset a of an universe h, can be represented as a function, the characteristic functions a from h to the set {0,1}. the notion of fuzzy subset generalizes that one of characteristic function. one considers instead of the functions a, functions a from h to the closed real interval [0,1]. these functions, called “membership functions” express the degree of belonging of an element x  h to a subset a of h. to consider in a problem, a fuzzy subset instead of an usual (cantor) subset, corresponds usually to think according to a multivalued logic instead of a bivalent logic. the reply to many questions in the science, and in the life, often is not possible in a dichotomic form, but it has a great variety of nuances. a cantor subset a of the universe h, can be represented as the class of objects which satisfy a certain property p , so an element x does not belong to a if it does not satisfy p . but in the reality an object can satisfy p in a certain measure. whence the advisability to size the satisfaction of p by a real number a (x)  [0,1]. rough sets, which have been proved to be a particular case of fuzzy sets (see [8]) are they also an important instrument for studying in depth some subjects of applied science. the first idea of rough set appears in the book by shaefer [134] as pair of “inner and outer reductions” (see pages 117-119), in the context of probability and scientific inference, but they became a well known subject of research in pure and applied mathematics, since pawlak [121] considered them again and proved their utility in some topics of artificial intelligence as decision making, data analysis, learning machines, switching circuits. connections between fuzzy sets and algebraic hyperstructures were considered for the first time by rosenfeld 130. many others worked in the same direction, that is studied algebraic structures endowed also with a fuzzy structure (see 1 2, 3 54,…, 59. 11 hyperstructures endowed with a fuzzy structure were considered by ameri and zahedi, tofan, davvaz, borzooei, hasankhani, bolurian and others. definition 9 a fuzzy subset of a set h is a pair (h; a ) where a is a function a : h 0,1, a is the set { x  h  a (x) = 1 }. corsini proved in 1993, 17 that to every fuzzy subset of a set h one can associate a join space, where the hyperoperation is defined as follows: (i) (x, y)h 2 , x  y = {z  min{(x), (y)}(z)max{(x), (y)}}. moreover in 2003 corsini proved [29] that to every hypergroupoid <h, o> one can associate a fuzzy subset. see (ii) : (ii) u  h, let q(u) = {(x, y)h 2  u  x o y }, q(u) = q(u), a(u) = x,yq(u) (1 /x o y) , a’(u) = a(u)/q(u). if we have a hypergroupoid <h, o”> and <o”> is a weak hyperoperation, we can associate with h the following fuzzy subset (ii”) set mx,y(u) the multiplicity of the element u in the hyperproduct x o y . set x,y(u) = mx,y(u) /  ( mx,y(v) v h, mx,y(v)  0 ) q(u) = (a,b)h 2  ma,b(u)  0 , q(u) = q(u) a(u) = x,yq(u) x,y(u) , ’(u) = a(u)/q(u). weak hyperstructures were introduced by vougiouklis (1981) and were studied by many people especially by vougiouklis and spartalis. so every fuzzy subset (and every hypergroupoid) determines a sequence of join spaces and of fuzzy subsets connections between hyperstructures and fuzzy sets have been considered by many people. in particular (i) and (ii) opened research lines studied in deep by several scientists. in this context several 12 papers have been made in italy, romania, greece, iran, for example by corsini, leoreanu, cristea, serafimidis, kehagias, konstantinidou, rosenberg. hyperstructures endowed also with a fuzzy structure have been considered especially in iran by ameri, zahedi, davvaz and many others. from (i) and (ii) follows clearly that every hypergroupod (o fuzzy subset) determines a sequence of fuzzy subsets and hypergroupoids or of hypergroupoids and fuzzy subset) which is obtained applying consecutevely (ii) e (i) (oppure (i) e (ii)) the fuzzy grade (minimum lengh of such sequences) has been calculated for several classes of hyperstructures: corsini-cristea for i.p,s, hypergroups (a particular case of canonical hypergroups), and 1 hypergroups (hypergroups such that if  is the heart of the hypergroup, = 1). corsini leoreanu for hypergroups associated with hypergraphs, leoreanu for hypergroups associated with rough sets. corsini and cristea for complete hypergroups. let h be a set, r an equivalence relation in h and x  h, we denote the equivalence class of x by r(x). it is known that with every binary relation r defined in a set h, a partial hypergroupoid corresponds defined  (x,y)  h 2 , x ·r x = u  xru  , x ·r y = x ·r x  y ·r y this structure that under certain conditions is a hypergroup was introduced by rosenberg in 1996, see [102] and afterwards studied also by corsini in multiple logic and applications (1997) and by corsini leoreanu in algebra universalis n. 43 (2000). hypergroupoids associated with multivalued functions, have been analyzed by corsini and razieh majoob in 2010, see [40]. 13 5. new lines of research 1) a research line could be to calculate the fuzzy grade of the hypergroupoid associated with a binary relation 2) in bull. greek math society, corsini has associated in different ways hypergropoids with an ordered set. it could be interesting to study the sequences of join spaces determined by these hypergroupoids. 3) another research line could be to study the sequence of join spaces determined by a hypergroupoid endowed with a weak hyperoperation. 4) it would be interesting also to consider the sequence of fuzzy sets and join spaces determined by a chinese hypergroupoid (see [ 24]) 5) in [25], [26] one has associated a hypergroupoid with a factor space, that is , given a function f from an universe u to a set of states x(f), and and a fuzzy subset of u, f called the extension of f, one has considered the hyperoperation in u, < o f f > . defined: x o f f y = w  f (w)  sup. f (z) f (z) = f(x), sup.f (v) f (v) = f(y)  one proposes to determine the fuzzy grade of the hypergroupoid <u; o f f >. 6) set a a non empty set and f the set of functions f: a p*(a) such that  xa f(x) = a . one considers the following hyperoperations in f(a), see [31] (i) f o1 g = hf   x  a, h(x)  f(g(x)) , (ii) f o2 g = hf   x  a, h(x)  f(x)  g(x) , (iii) let’s suppose now <a ; ò > to be a hypergroupoid. then set for every (f.g) f x f f o3 g = hf   x  a, h(x)  f(x) ò g(x) , problems: let’s suppose  a i* determine the fuzzy grade of the hypergroupoid (i) ii* determine the fuzzy grade of the hypergroupoid (ii) iii * determine the fuzzy grade of the hypergroupoid (iii) , for some hypergroupoid <a ; ò > 14 (7) it is known that every hypergraph < ; ai  > determines a sequence of quasi-hypergroups q0, q1,……., qm (see  18 , th. 6 ). set, for every k, mk the membership function associated with qk and j(qk) the corresponding join space. set * the sequence < q0 , 0 , j(qo) , 1 , j(q1) ,….., m , j(qm) > , and set  the sequence of membership functions and join spaces determined by the hypergroupoid q0 after (i) and (ii) it would be interesting to compare the two sequences  and * . references   1  ameri r: and zahedi m.m. – hypergroup and join space induced by a fuzzy subset, pu.m.a. vol. 8, (2-3-4) (1997)  2  ameri r and shafiyan n. – fuzzy prime and primary hyperideals in hyperrings, advances in fuzzy mathematics n. 1-2, (2007)  3  ameri r. and nozari t. – a nonnection betweem categories oh (fuzzy) multialgebras and (fuzzy) alebras, italian journal of pure and applied mathematics, n. 27, (2010)  4  ashrafi r. – about some join spaces and hyperlattices, italian journal of pure and applied mathematics, n. 10, (2001)  5  ashrafi r. , a. h. zadeh, m. yavari – hypergraphs and join spaces, italian journal of pure and applied mathematics, n. 12, (2002)  6  asokkumar a., velrajan m. – hyperring of matrices over regular hyperring, italian journal of pure and applied mathematics, n. 23, (2008)  7 asokkumar a., velrajan m. – characterization of regular hyperrings, italian journal of pure and applied mathematics, n. 22, (2007)  8  biswas r. – rough sets and fuzzy sets, busefal 83, (2000) 15  9  borzooei r.a., zahedi m.m, jun y.hasankhani a. b. – some results on hyper k-algebras, mathematicae, vol. 3, n.1, (2000)  10  borzooei r.a., zahedi m.m – fuzzy structures on hyperk-algebras, j. of uncertainty fuzziness and knowledge basedsystems, vol.10, n.1, (2002)  11  borzooei r.a., corsini , p. zahedi m.m. – some kinds of positive implicative hyperk-ideals, journal of discrete mathematical sciences and cryptography, vol 6 , n. 1. 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(1994) ratio mathematica volume 44, 2022 connected hub sets and connected hub polynomials of the lollipop graph 𝑳𝒑,𝟏 t. angelinshiny1 t. anitha baby2 abstract let 𝐺 be a graph with vertex set 𝑉(𝐺). the number of vertices in 𝐺 is the order of 𝐺 and is denoted by |𝑉(𝐺)|. the connected hub polynomial of g denoted by 𝐻𝑐 (𝐺, 𝑦) is defined as 𝐻𝑐 (𝐺, 𝑦) = ∑ ℎ𝑐 (𝐺, 𝑘)𝑦 𝑘|𝑉(𝐺)| 𝑘=𝒽𝑐(𝐺) where ℎ𝑐 (𝐺, 𝑘) denotes the number of connected hub sets of 𝐺 of cardinality 𝑘 and 𝒽𝑐 (𝐺) denotes the connected hub number of 𝐺. let 𝐿𝑝,1 denotes the lollipop graph with 𝑝 + 1 vertices. the connected hub polynomial of 𝐿𝑝,1 denoted by 𝐻𝑐 (𝐿𝑝,1, 𝑦) is defined as,𝐻𝑐 (𝐿𝑝,1, 𝑦) = ∑ ℎ𝑐 (𝐿𝑝,1, 𝑘) |𝑉(𝐿𝑝,1)| 𝑘=𝒽𝑐(𝐿𝑝,1) 𝑦𝑘where ℎ𝑐 (𝐿𝑝,1, 𝑘) denotes the number of connected hub sets of 𝐿𝑝,1 of cardinality 𝑘, and 𝒽𝑐 (𝐿𝑝,1) denotes the connected hub number of 𝐿𝑝,1.in this paper, we derive a recursive formula for hc(lp,1, k). from this recursive formula, we construct the connected hub polynomial of lp,1 as,hc(lp,1, y) = ∑ hc(lp,1, k) p+1 k=1 ykalso we study some properties of this polynomial. keywords: lollipop graph, connected hub set, connected hub number, connected hub polynomial. mathematics subject classification code: 05c31, 05c993 1research scholar (reg. no. 20213282092009), department of mathematics, women’s christian college, nagercoil, tamil nadu, india. affiliated by manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012. mail id: angelinshinyt@gmail.com 2 assistant professor, department of mathematics, women’s christian college, nagercoil, tamil nadu, india. affiliated by manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012. mail id: anithasteve@gmail.com 3received on june 19th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.886. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 29 mailto:angelinshinyt@gmail.com mailto:anithasteve@gmail.com t. angelinshiny and t. anitha baby 1. introduction if any two distinct vertices of a graph 𝐺 are adjacent, then 𝐺 is a complete graph. if a tree has two nodes of vertex degree 1 and other nodes of vertex degree 2, then it is a path graph. a complete graph of order 𝑝 is denoted by 𝐾𝑝 and a path graph of order 𝑞 is denoted by 𝑃𝑞 . join a complete graph 𝐾𝑝 to a path graph 𝑃𝑞 with a bridge. the resulting graph is a lollipop graph 𝐿𝑝,𝑞. 2. connected hub sets of the lollipop graph 𝐋𝐩,𝟏 in this section, we give the connected hub number of the lollipop graph 𝐿𝑝,1 and some of the properties of the connected hub sets of the lollipop graph 𝐿𝑝,1. definition 2.1 join a complete graph 𝐾𝑝 to a path graph 𝑃1 with a bridge. the resulting graph is a lollipop graph 𝐿𝑝,1. definition 2.2 let 𝐺 = (𝑉, 𝐸) be a connected graph. a subset 𝐻 of 𝑉 is called a hub set of 𝐺 if for any two distinct vertices 𝑢, 𝑣 ∈ 𝑉 − 𝐻, there exists a 𝑢 − 𝑣 path 𝑃 in 𝐺, such that all the internal vertices of 𝑃 are in 𝐻. the minimum cardinality of a hub set of 𝐺 is called the hub number of 𝐺 and is denoted by 𝒽(𝐺). definition 2.3 a hub set 𝐻 of 𝐺 is called a connected hub set if the induced subgraph < 𝐻 > is connected. the minimum cardinality of a connected hub set of 𝐺 is called connected hub number of 𝐺 and is denoted by 𝒽𝑐 (𝐺). theorem 2.4 ℎ𝑐 (𝐿𝑝,1,𝑘) = { ( 𝑝 + 1 𝑘 ) − ( 𝑝 𝑘 ) + 1 𝑤ℎ𝑒𝑛 𝑘 = 1 𝑎𝑛𝑑 𝑝 − 1 ( 𝑝 + 1 𝑘 ) − ( 𝑝 𝑘 ) 𝑖𝑓 2 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑 𝑘 ≠ 𝑝 − 1 proof. let 𝐿𝑝,1 be the lollipop graph with 𝑝 + 1 vertices and 𝑝 ≥ 4. let 𝑣1, 𝑣2, 𝑣3 … 𝑣𝑝, 𝑣𝑝+1 be the vertices of 𝐿𝑝,1, in which the degree of the vertices 𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑝−1 is 𝑝 − 1, the degree of the vertex 𝑣𝑝 is 𝑝 and the degree of the vertex 𝑣𝑝+1 is 1. since, 𝐿𝑝,1 contains 𝑝 + 1 vertices, the number of subsets of 𝐿𝑝,1 with cardinality 𝑘 is ( 𝑝 + 1 𝑘 ). also, since, the subgraph with vertex set {𝑣1, 𝑣2, 𝑣3 … 𝑣𝑝−1} is not adjacent to 𝑣𝑝+1 every hub set must contain the vertices 𝑣𝑝 or 𝑣𝑝+1. therefore, every time ( 𝑝 𝑘 ) number of subsets of 𝐿𝑝,1 of cardinality 𝑘 are not connected hub sets. thus, 𝐿𝑝,1 have ( 𝑝 + 1 𝑘 ) − ( 𝑝 𝑘 ) number of connected hubs sets of cardinalities 𝑘. when the cardinality is 𝑝 − 1, the set which contains 𝑣𝑝−1 is also a connected hub set. when the cardinality is 1, {𝑣𝑝} and {𝑣𝑝+1} are the only connected hub sets. 30 connected hub sets and connected hub polynomials of the lollipop graph 𝐿𝑝,1 hence, ℎ𝑐 (𝐿𝑝,1,𝑘) = { ( 𝑝 + 1 𝑘 ) − ( 𝑝 𝑘 ) + 1 𝑤ℎ𝑒𝑛 𝑘 = 1, 𝑝 − 1 ( 𝑝 + 1 𝑘 ) − ( 𝑝 𝑘 ) 𝑖𝑓 2 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑 𝑘 ≠ 𝑝 − 1 theorem 2.5 let 𝐿𝑝,1 be the lollipop graph with 𝑝 ≥ 4.then (i) ℎ𝑐 (𝐿𝑝,1,𝑘) = ( 𝑝 𝑘 − 1 ) for all 2 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑 𝑘 ≠ 𝑝 − 1 (ii) ℎ𝑐 (𝐿𝑝,1,𝑘) = ( 𝑝 𝑘 − 1 ) + 1 when 𝑘 = 1 𝑎𝑛𝑑 𝑝 − 1. (iii) ℎ𝑐 (𝐿𝑝,1,𝑘) = { ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−1,1,𝑘 − 1) 𝑖𝑓 1 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑 𝑘 ≠ 2 , 𝑝 − 2 ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−1,1,𝑘 − 1) − 1 𝑖𝑓 𝑘 = 2 𝑎𝑛𝑑 𝑝 − 2 proof: (i) from theorem 2.4, we have ℎ𝑐 (𝐿𝑝,1,𝑘) = ( 𝑝 + 1 𝑘 ) − ( 𝑝 𝑘 ) we know that, ( 𝑝 + 1 𝑘 ) − ( 𝑝 𝑘 ) = ( 𝑝 𝑘 − 1 ) therefore, ℎ𝑐 (𝐿𝑝,1,𝑘) = ( 𝑝 𝑘 − 1 ) for all 2 ≤ 𝑘 ≤ 𝑝 + 1 and 𝑘 ≠ 𝑝 − 1. (ii) from theorem 2.4, we have ℎ𝑐 (𝐿𝑝,1, 𝑘) = ( 𝑝 + 1 𝑘 ) − ( 𝑝 𝑘 ) + 1 𝑤ℎ𝑒𝑛 𝑘 = 1 𝑎𝑛𝑑 𝑝 − 1. we know that, ( 𝑝 + 1 𝑘 ) − ( 𝑝 𝑘 ) = ( 𝑝 𝑘 − 1 ) therefore, ℎ𝑐 (𝐿𝑝,1,𝑘) = ( 𝑝 𝑘 − 1 ) + 1 𝑤ℎ𝑒𝑛 𝑘 = 1 𝑎𝑛𝑑 𝑝 − 1. (iii) from (i) ℎ𝑐 (𝐿𝑝,1,𝑘) = ( 𝑝 𝑘 − 1 ) for all 2 ≤ 𝑘 ≤ 𝑝 + 1 and 𝑘 ≠ 𝑝 − 1. ℎ𝑐 (𝐿𝑝−1,1, 𝑘) = ( 𝑝 − 1 𝑘 − 1 ) and ℎ𝑐 (𝐿𝑝−,1,𝑘 − 1) = ( 𝑝 − 1 𝑘 − 2 ) consider, ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−,1,𝑘 − 1) = ( 𝑝 − 1 𝑘 − 1 ) + ( 𝑝 − 1 𝑘 − 2 ) = ( 𝑝 𝑘 − 1 ) = ℎ𝑐 (𝐿𝑝,1,𝑘) therefore, ℎ𝑐 (𝐿𝑝,1,𝑘) = ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−,1,𝑘 − 1) for 1 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑 𝑘 ≠ 2, 𝑝 − 2 when 𝑘 = 2, ℎ𝑐 (𝐿𝑝,1, 2) = ( 𝑝 1 ) ℎ𝑐 (𝐿𝑝−1,1, 2) = ( 𝑝 − 1 1 ) and ℎ𝑐 (𝐿𝑝−1,1, 1) = ( 𝑝 − 1 0 ) + 1, by (ii) consider, ℎ𝑐 (𝐿𝑝−1,1, 2) + ℎ𝑐 (𝐿𝑝−1,1, 1) = ( 𝑝 − 1 1 ) + ( 𝑝 − 1 0 ) + 1 = ( 𝑝 1 ) + 1 31 t. angelinshiny and t. anitha baby = ℎ𝑐 (𝐿𝑝,1, 2) + 1 that is, ℎ𝑐 (𝐿𝑝,1, 2) = ℎ𝑐 (𝐿𝑝−1,1, 2) + ℎ𝑐 (𝐿𝑝−1,1, 1) − 1. therefore, ℎ𝑐 (𝐿𝑝,1,𝑘) = ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−,1,𝑘 − 1) − 1 when 𝑘 = 2. when 𝑘 = 𝑝 − 2, ℎ𝑐 (𝐿𝑝,1, 𝑝 − 2) = ( 𝑝 𝑝 − 3 ) ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 2) = ( 𝑝 − 1 𝑝 − 3 ) and ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 3) = ( 𝑝 − 1 𝑝 − 4 ) + 1, by (ii) consider, ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 2) + ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 3) = ( 𝑝 − 1 𝑝 − 3 ) + ( 𝑝 − 1 𝑝 − 4 ) + 1 = ( 𝑝 𝑝 − 3) + 1 = ℎ𝑐 (𝐿𝑝,1, 𝑝 − 2) + 1 that is, ℎ𝑐 (𝐿𝑝,1, 𝑝 − 2) = ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 2) + ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 3) − 1. therefore, ℎ𝑐 (𝐿𝑝,1,𝑘) = ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−,1,𝑘 − 1) − 1 when 𝑘 = 𝑝 − 2. hence, ℎ𝑐 (𝐿𝑝,1,𝑘) = { ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−1,1,𝑘 − 1) 𝑖𝑓 1 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑 𝑘 ≠ 2 , 𝑝 − 2 ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−1,1,𝑘 − 1) − 1 𝑖𝑓 𝑘 = 2 𝑎𝑛𝑑 𝑝 − 2 3. connected hub polynomials of the lollipop graph 𝑳𝒑,𝟏. definition 3.1 let 𝐻𝑐 (𝐿𝑝,1,, 𝑘) denotes the family of connected hub sets of the lollipop graph 𝐿𝑝,1, of cardinality 𝑘 and ℎ𝑐 (𝐿𝑝,1,, 𝑘) = |𝐻𝑐 (𝐿𝑝,1,, 𝑘)|. then, the connected hub polynomial of 𝐿𝑝,1 denoted by 𝐻𝑐 (𝐿𝑝,1,, 𝑦) is defined as 𝐻𝑐 (𝐿𝑝,1,, 𝑦) = ∑ ℎ𝑐 (𝐿𝑝,1,, 𝑘) 𝑝+1 𝑘=𝒽𝑐(𝐿𝑝,1,) 𝑦𝑘 where 𝒽𝑐 (𝐿𝑝,1,) is connected hub number of 𝐿𝑝,1,. remark 3.2 𝒽𝑐 (𝐿𝑝,1,) = 1. proof: label the vertices of 𝐿𝑝,1 by 𝑣1, 𝑣2,𝑣3, … , 𝑣𝑝, 𝑣𝑝+1 in which the degree of the vertices 𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑝−1 is 𝑝 − 1, the degree of the vertex 𝑣𝑝 is 𝑝 and the degree of the vertex 𝑣𝑝+1 is 1. since, any two vertices of 𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑝 are adjacent there is a path between any two vertices of 𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑝. also, 𝑣𝑝 is the internal vertex for all the path between the vertices of {𝑣1, 𝑣2,𝑣3, … , 𝑣𝑝−1} and 𝑣𝑝+1. hence {𝑣𝑝} and {𝑣𝑝+1} are two connected hub sets of cardinalities 1. hence, 𝒽𝑐 (𝐿𝑝,1) = 1. 32 connected hub sets and connected hub polynomials of the lollipop graph 𝐿𝑝,1 theorem 3.3 𝐻𝑐 (𝐿𝑝,1,𝑦) = (1 + 𝑦)𝐻𝑐 (𝐿𝑝−1,1, 𝑦) − 𝑦 2 − 𝑦𝑝−2 with initial value 𝐻𝑐 (𝐿4,1,𝑦) = 2𝑦 + 4𝑦 2 + 7𝑦3 + 4𝑦4 + 𝑦5. proof. we have, 𝐻𝑐 (𝐿𝑝,1,𝑦) = ∑ ℎ𝑐 (𝐿𝑝,1, 𝑘)𝑦 𝑘𝑝+1 𝑘=1 𝐻𝑐 (𝐿𝑝,1,𝑦) = ∑ ℎ𝑐 (𝐿𝑝,1, 𝑘)𝑦 𝑘 + 𝑝+1 𝑘=1 𝑘≠2,𝑝−2 ℎ𝑐 (𝐿𝑝,1, 2)𝑦 2 + ℎ𝑐 (𝐿𝑝,1, 𝑝 − 2)𝑦 𝑝−2 = ∑ [ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−1,1, 𝑘 − 1)]𝑦 𝑘 + 𝑝+1 𝑘=1 𝑘≠2,𝑝−2 [ℎ𝑐 (𝐿𝑝−1,1, 2) + ℎ𝑐 (𝐿𝑝−1,1, 1) − 1]𝑦2 + [ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 2) + ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 3) − 1]𝑦 𝑝−2 = ∑ [ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−1,1, 𝑘 − 1)]𝑦 𝑘 − 𝑦2 − 𝑦𝑝−2 𝑝+1 𝑘=1 = ∑ ℎ𝑐 (𝐿𝑝−1,1, 𝑘)𝑦 𝑘 + ∑ ℎ𝑐 (𝐿𝑝−1,1, 𝑘 − 1) 𝑝+1 𝑘=1 𝑦 𝑘 − 𝑦2 − 𝑦𝑝−2 𝑝+1 𝑘=1 = ∑ ℎ𝑐 (𝐿𝑝−1,1, 𝑘)𝑦 𝑘 + 𝑦 ∑ ℎ𝑐 (𝐿𝑝−1,1, 𝑘 − 1) 𝑝+1 𝑘=1 𝑦 𝑘−1 − 𝑦2 − 𝑦𝑝−2 𝑝+1 𝑘=1 = 𝐻𝑐 (𝐿𝑝−1,1, 𝑦) + 𝑦𝐻𝑐 (𝐿𝑝−1,1, 𝑦) − 𝑦 2 − 𝑦𝑝−2 = (1 + 𝑦)𝐻𝑐 (𝐿𝑝−1,1, 𝑦) − 𝑦 2 − 𝑦𝑝−2 hence, 𝐻𝑐 (𝐿𝑝,1,𝑦) = (1 + 𝑦)𝐻𝑐 (𝐿𝑝−1,1, 𝑦) − 𝑦 2 − 𝑦𝑝−2 with initial value 𝐻𝑐 (𝐿4,1,𝑥) = 2𝑦 + 4𝑦 2 + 7𝑦3 + 4𝑦4 + 𝑦5. example 3.4 consider the lollipop graph 𝐿5,1 be with order 6 given in figure 1. figure 1 𝐻𝑐 (𝐿5,1,𝑦) = 2𝑦 + 5𝑦 2 + 10𝑦3 + 11𝑦4 + 5𝑦5 + 𝑦6 by theorem 3.3, we have, 𝐻𝑐 (𝐿5,1,𝑦) = (1 + 𝑦)𝐻𝑐 (𝐿4,1, 𝑦) − 𝑦 2 − 𝑦3 = (1 + 𝑦)(2𝑦 + 4𝑦2 + 7𝑦3 + 4𝑦4 + 𝑦5) −𝑦2 − 𝑦3 = 2𝑦 + 5𝑦2 + 10𝑦3 + 11𝑦4 + 5𝑦5 + 𝑦6 theorem 3.5 let 𝐿𝑝,1 be the lollipop graph with 𝑝 ≥ 4.then (𝑖) 𝐻𝑐 (𝐿𝑝,1,𝑦) = ∑ ( 𝑝 + 1 𝑘 ) 𝑝+1 𝑘=1 𝑦 𝑘 − ∑ ( 𝑝 𝑘 ) 𝑝+1 𝑘=1 𝑦 𝑘 − 𝑦2 − 𝑦𝑝−2. 33 t. angelinshiny and t. anitha baby (ii) 𝐻𝑐 (𝐿𝑝,1, 𝑦) = ∑ ( 𝑝 𝑘 − 1 ) 𝑝+1 𝑘=1 𝑦 𝑘 − 𝑦2 − 𝑦𝑝−2. proof. proof is obvious. ℎ𝑐 (𝐿𝑝,1, 𝑘) for 4 ≤ 𝑝 ≤ 14 and 1 ≤ 𝑘 ≤ 15. 𝑘 𝑝 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 4 2 4 7 4 1 5 2 5 10 11 5 1 6 2 6 15 20 16 6 1 7 2 7 21 35 35 22 7 1 8 2 8 28 56 70 56 29 8 1 9 2 9 36 84 126 126 84 37 9 1 10 2 10 45 120 210 252 210 120 46 10 1 11 2 11 55 165 330 462 462 330 165 56 11 1 12 2 12 66 220 495 792 924 792 495 220 67 12 1 13 2 13 78 286 715 1287 1716 1716 1287 715 286 79 13 1 14 2 14 91 364 100 2002 3003 3432 3003 2002 1001 364 92 14 1 table 1 theorem 3.6 the coefficients of 𝐻𝑐 (𝐿𝑝,1, 𝑦) satisfy the following properties. (i) ℎ𝑐 (𝐿𝑝,1, 𝑝 + 1) = 1, for every 𝑝 ≥ 4. (ii) ℎ𝑐 (𝐿𝑝,1, 𝑝) = 𝑝, for every 𝑝 ≥ 4. (iii) ℎ𝑐 (𝐿𝑝,1, 𝑝 − 1) = 1 2 (𝑝2 − 𝑝 + 2), for every 𝑝 ≥ 4. (iv) ℎ𝑐 (𝐿𝑝,1, 𝑝 − 2) = 1 6 (𝑝3 − 3𝑝2 + 2𝑝), for every 𝑝 ≥ 4. (v) ℎ𝑐 (𝐿𝑝,1, 𝑝 − 3) = 1 24 (𝑝4 − 6𝑝3 + 11𝑝2 − 6𝑝), for every 𝑝 ≥ 6. (vi) ℎ𝑐 (𝐿𝑝,1, 1) = 2, for every 𝑝 ≥ 4. (vii) ℎ𝑐 (𝐿𝑝,1, 2) = 𝑝, for every 𝑝 ≥ 4. 4. conclusion in this paper, we identified the connected hub sets of 𝐿𝑝,1 and using the connected hub sets we derived the connected hub polynomial of 𝐿𝑝,1. we can generalize this study to derive the connected hub polynomial of any lollipop graph 𝐿𝑝,𝑞 . 34 connected hub sets and connected hub polynomials of the lollipop graph 𝐿𝑝,1 references [1] walsh, matthew, "the hub number of a graph", int. j. math. comput. sci 1, no. 1 (2006): 117-124. [2] veettil, ragi puthan, and t. v. ramakrishnan, "introduction to hub polynomial of graphs", malaya journal of matematik (mjm) 8, no. 4, 2020 (2020): 1592-1596. [3] s. alikhani and y.h. peng, "dominating sets and domination polynomials of paths", international journal of mathematics and mathematical sciences, 2009. [4] s. alikhani and y.h. peng, "introduction to domination polynomial of a graph ", arxiv preprint arxiv:0905.2251, 2009. [5] sahib.sh. kahat, abdul jalil khalaf and roslan hasni, "dominating sets and domination polynomials of wheels", asian journal of applied sciences (issn:2321 0893), volume 02issue 03, june 2014. [6] sahib.sh. kahat, abdul jalil m. khalaf and roslan hasni, "dominating sets and domination polynomials of stars", australian journal of basics and applied science,8(6) june 2014, 383-386. [7] a. vijayan, t. anitha baby, g. edwin, "connected total dominating sets and connected total domination polynomials of stars and wheels", iosr journal of mathematics, volume ii, 112-121. 35 microsoft word r.m. 5 cap.6.doc ratio mathematica volume 40, 2021, pp. 247-257 mixed and non-mixed normal subgroups of dihedral groups using conjugacy classes vinod s∗ biju g.s† abstract in this paper, we characterize and compute the mixed and non-mixed basis of dihedral groups. also, by computing the conjugacy classes, we describe all the mixed and non-mixed normal subgroups of dihedral groups. keywords: group; dihedral group; mixed and non-mixed basis; normal subgroups; conjugacy classes; 2010 ams subject classifications: 08a05. 1 ∗department of mathematics, government college for women, thiruvananthapuram, kerala, india; wenod76@gmail.com. †department of mathematics, college of engineering, thiruvananthapuram-695016, kerala, india; gsbiju@cet.ac.in. 1received on february 09th, 2021. accepted on april 28th, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.604. issn: 1592-7415. eissn: 2282-8214. c©vinod et al. this paper is published under the cc-by licence agreement. 247 vinod s, biju g.s 1 introduction there are many interesting functions from the family of dihedral groups to set of natural numbers. for the dihedral group dn of order 2n, cavior [1975] proved that the number of subgroups is d(n) + σ(n) where σ(n) is the sum of positive divisors of n and d(n) denote number of positive divisors of n. for elementary facts about dihedral groups see conrad [retrieveda]. conrad [retrievedb] describes the subgroups of dn, including the normal subgroups. using characterization of dihedral groups in terms of generators and relations. calugareanu [2004] presents a formula for the total number of subgroups of a finite abelian group. in tărnăuceanu [2010] an arithmetic method is developed to count the number of some types of subgroups of finite abelian groups. subgroups of groups of smaller sizes are widely studied because their group properties can be easily verified and larger groups are usually studied in terms of their subgroups (see miller [1940]). in this paper we characterize and compute the different basis of dihedral groups. also we describe all mixed and non-mixed normal subgroups of dihedral groups via conjugacy classes. 2 notations and basic results most of the notations, definitions and results we mentioned here are standard and can be found in gallian [1994] and dummit and foote [2003]. for any given natural number n denote: d(n) = the number of positive divisors of n. σ(n) = the sum of positive divisors of n. ϕ(n) = the number of nonnegative integers less than n and relatively prime to n. also, the greatest common divisor of m and n is denoted by (m,n). let g be a group and a1,a2, . . . ,ap ∈ g. then the subgroup generated by a1,a2, . . . ,ap is denoted by < a1,a2, . . . ,ap >. definition 2.1. a group generated by two elements r and s with orders n and 2 such that srs−1 = r−1 is said to be the nth dihedral group and is denoted by dn. theorem 2.1. for each divisor d of n, the group zn has a unique subgroup of order d, namely 〈n d 〉 . theorem 2.2. for each divisor d of n, the group zn has exactly ϕ(d) elements of order d, namely {k n d : 0 ≤ k ≤ d−1, (k,d) = 1}. 248 mixed and non-mixed normal subgroups of dihedral groups using conjugacy classes theorem 2.3. the number of subgroups of zn is d(n), namely 〈n d 〉 where d is a divisor of n. theorem 2.4. let g be a group generated by a and b such that an = e, b2 = e and bab−1 = a−1. if the size of g is 2n then g is isomorphic to dn. by theorem 2.4, we make an abstract definition for dihedral groups. definition 2.2. for n ≥ 3, let rn = {r0,r1, . . . ,rn−1} and sn = {s0,s1, . . . ,sn−1}. define a binary operation on gn = rn ∪sn by the following relations: ri · rj = ri+j mod(n) ri ·sj = si+j mod(n) si ·sj = ri−j mod(n) si · rj = si−j mod(n) for all 0 ≤ i,j ≤ n−1. then (gn, ·) is a group of order 2n. note that in the group (gn, ·), the identity element is r0, ri = rj if and only if i = j mod(n), si = sj if and only if i = j mod(n), the inverse of ri is rn−i and the inverse of si is si for all 0 ≤ i,j ≤ n − 1. it is also clear that ri1 = ri and rj · s0 = sj for all 0 ≤ i,j ≤ n− 1. since gn is a group of order 2n and can be generated by r1 and s0 such that: rn1 = rn = r0, s 2 0 = r0 and s0r1s −1 0 = s0r1s0 = s−1s0 = r−1 = rn−1 = r −1 1 . therefore the group gn is isomorphic to dn =< r1,s0 >. the elements of rn are called rotations and that of sn are called reflections. a subgroup of dn which contain both rotations and reflections is called a mixed subgroup and subgroups contain rotations only is called non-mixed subgroup. from the group dn, we have the following. theorem 2.5. rn is a subgroup of dn and is isomorphic to zn. theorem 2.6. if n is even, the number of elements of order 2 in dn is n + 1, namely {rn/2,sj : 0 ≤ j ≤ n−1}. theorem 2.7. if n is odd, the number of elements of order 2 in dn is n, namely {sj : 0 ≤ j ≤ n−1}. theorem 2.8. if d divide n and d 6= 2, the number of elements of order d in dn is ϕ(d) namely {rkn/d : 0 ≤ k ≤ d−1, (k,d) = 1}. theorem 2.9. if a and b are two elements in dn, then < a,b >= {akbm : 0 ≤ k,m ≤ n−1} definition 2.3. let g be a finite group. an element y ∈ g is said to be a conjugate of x ∈ g iff y = gxg−1, for some g in g. 249 vinod s, biju g.s this relation conjugacy in a group g is an equivalence relation on g. the equivalence class determined by the element x is denoted by cl(x). thus cl(x) = {gxg−1 : g ∈ g}. the summation, ∑ x∈g |cl(x)|, where summation runs over one element from each conjugacy class of x is called the class equation of g. definition 2.4. a subgroup h of the group g is said to be a normal subgroup if ghg−1 ∈ h for all g ∈ g and h ∈ h. a normal subgroup which contain rotations alone is called a nonmixed normal subgroup and normal subgroups which contains both reflections and rotations is called mixed normal subgroup. theorem 2.10. every normal subgroup is a union of conjugacy classes. theorem 2.11. every subgroup of a cyclic normal subgroup of the group g is also normal in g. 3 subgroups of dn theorem 3.1. the number of non-mixed subgroups of dn is d(n), namely {< rn/d >: d is a divisor of n}. proof. the non-mixed subgroups of dn are subgroups of rn. since rn is isomorphic to zn, for each divisor d of n, the group rn has a unique subgroup of order d, namely < rn/d >. hence the number of non-mixed subgroups of dn is d(n), namely {< rn/d >: d is a divisor of n}. 2 theorem 3.2. every mixed subgroup of dn has even order of which half of them are rotation and half of them are reflection. proof. let h be a mixed subgroup of dn containing a reflection s. let a denote the set of rotations of h and b denote the set of all reflections of h. define a map ψ : a → b by ψ(r) = r · s for all r ∈ a. if sj is an element in b then sj · s is an element of a and ψ(sj · s) = sjss = sj. hence ψ is onto. also ψ(r) = ψ(r′) =⇒ rs = r′s =⇒ r = r′ and hence ψ is one-one. 2 theorem 3.3. every mixed subgroup of dn is dihedral. proof. let h be a mixed subgroup of dn. by theorem 3.2 , |h| = 2d for some d and h ∩ rn =< rn/d >. since order of h is 2d and < rn/d > is its subgroup of order d, we have h =< rn/d > ∪ < rn/d > s =< rn/d,s >, for some s in h. since (rn/d)d = ro,s2 = r0 and srn/ds−1 = (rn/d)−1, we have h ≡ dd and hence the proof. 2 250 mixed and non-mixed normal subgroups of dihedral groups using conjugacy classes corolary 3.1. if h is a mixed subgroup of dn then, 1. |h| = 2d, for some d which divides n. 2. h ≡ dn =< rn/d,s > for some s ∈ h. here we have a usual question: if d divides n, does there exist a subgroup of order 2d? if it exists, how many? theorem 3.4. if d divides n, the number of mixed subgroups of order 2d is n d . proof. by the corollary 3.1, it is clear that the mixed subgroups dn of order 2d are {< rn/d,sj >: 0 ≤ j ≤ n− 1}, all of them need not be distinct. suppose < rn/d,si >=< rn/d,sj > for some 0 ≤ i,j ≤ n−1. < rn/d,si > =< rn/d,sj > ⇐⇒ < rn/d > ∪ < rn/d > si =< rn/d > ∪ < rn/d > sj ⇐⇒ < rn/d > si =< rn/d > sj ⇐⇒ sis−1j ∈< rn/d > ⇐⇒ sis−1j = rkn/d for some 0 ≤ k ≤ d−1 ⇐⇒ sisj = rkn/d ⇐⇒ ri−j = rkn/d ⇐⇒ i− j ≡ kn d mod(n) for some 0 ≤ k ≤ d−1 ⇐⇒ d(i− j) ≡ 0mod(n) ⇐⇒ i− j ≡ 0mod (n d ) ⇐⇒ i ≡ j mod (n d ) hence the number of distinct mixed subgroups of order 2d in dn is n d , namely {< rn/d,si >: 0 ≤ i < n d }. 2 theorem 3.5. the number of mixed subgroups of dn is σ(n). proof. by theorem 3.4, the mixed subgroups of dn is ∑ d/n n d = ∑ d/n d = σ(n). they are ∪d/n{< rn/d,si >: 0 ≤ i ≤ n d −1}. 2 from theorem 3.1 and theorem3.5 we have, theorem 3.6. the number of subgroups of dn is σ(n) + d(n). 251 vinod s, biju g.s theorem 3.7. the number of abelian subgroups of dn is d(n) +n if n is odd and d(n) + n + n 2 if n is even. proof. all non-mixed subgroups of dn are cyclic and hence abelian. so by theorem 3.1, there are d(n) nonmixed abelian subgroups for dn. if n is odd, by theorem 3.3 and corollary 3.1, the mixed abelian subgroups of dn are of order 2 and hence there are n such subgroups. thus if n is odd, the number of abelian subgroups of dn is d(n) + n. if n is even, by theorem 3.3 and corollary 3.1, the mixed abelian subgroups of dn are of order 2 and 4, and hence there are n + n 2 such subgroups. thus if n is even, the number of abelian subgroups of dn is d(n) + n + n 2 . 2 theorem 3.8. the number of cyclic subgroups of dn is d(n) + n. proof. by theorem 3.1, the number of non-mixed cyclic subgroups of dn is d(n). also by theorem 3.3 and corollary 3.1,the mixed cyclic subgroups of dn is n. hence the number of cyclic subgroups of dn is d(n) + n. 2 4 basis of dn a basis of dn which contain both rotation and reflection is called a mixed basis and other basis is called non-mixed basis. by the definition 2.2, it is obvious that two rotations cannot generate dn. hence non-mixed basis of dn are basis consisting of two reflections. theorem 4.1. for n ≥ 3, the number of mixed basis of dn is nϕ(n). proof. let sj(0 ≤ j ≤ n − 1) be a reflection in dn. then for any 0 ≤ i ≤ n−1, < ri,sj > = {rmi s t j : 0 ≤ m,t ≤ n−1} ; by theorem 2.9 = {rmi sj, r m i r0 : 0 ≤ m ≤ n−1} ; since s t j = sj or r0 = {rmi sj, r m i : 0 ≤ m ≤ n−1} = {rmi sj : 0 ≤ m ≤ n−1}∪{r m i : 0 ≤ m ≤ n−1} =< ri > sj∪ < ri >= dn if and only if (i,n) = 1 hence corresponding to each reflection sj(0 ≤ j ≤ n− 1) there are ϕ(n) mixed bases, namely {{sj,ri} : 0 ≤ i ≤ n−1 and (i,n) = 1}. so the number of mixed basis for dn(n ≥ 3) is nϕ(n). 2 theorem 4.2. for n ≥ 3, the number of non-mixed basis of dn is nϕ(n) 2 . 252 mixed and non-mixed normal subgroups of dihedral groups using conjugacy classes proof. since the dimension of dn is 2, any basis of dn contain exactly two elements. the subgroup generated by two rotations always lies in rn and hence cannot form a basis. therefore any nonmixed basis of dn contain exactly two reflections. : let sj(0 ≤ j ≤ n − 1) be a reflection in dn. then for any 0 ≤ i ≤ n−1, < si,sj > =< ri−jsj,sj >=< ri−j,sj > ∼= dn if and only if i− j ≡ k mod(n) and (k,n) = 1 hence corresponding to each reflection sj(0 ≤ j ≤ n − 1) there are ϕ(n) nonmixed basis for dn namely {{si+j,sj} : 0 ≤ i ≤ n − 1 and (i,n) = 1}. if {si,sj} is a mixed basis corresponding to the reflection si, then it is also a basis corresponding to the reflection sj. hence the number of non-mixed basis for dn(n ≥ 3) is nϕ(n) 2 . 2 theorem 4.3. for n ≥ 3, the number of different basis for dn is 3n 2 ϕ(n). proof. the collection of all different bases of dn(n ≥ 3) is the union of all mixed and non-mixed bases. hence the different bases of dn(n ≥ 3) is nϕ(n) 2 + nϕ(n) = 3n 2 ϕ(n). 2 5 congugacy classes of dn in this section we will compute all conjugacy classes and class equation of dihedral groups. theorem 5.1. if n is odd, the number of conjugacy classes in dn is n + 3 2 . proof. let ri(0 ≤ i ≤ n−1) be a rotation in dn. then cl(ri) = {rjrir−1j ,sjris −1 j : 0 ≤ j ≤ n−1} = {rjrir−j,sjrisj : 0 ≤ j ≤ n−1} = {ri,sjrisj : 0 ≤ j ≤ n−1} = {ri,sj−isj : 0 ≤ j ≤ n−1} = {ri,r−i} since n is odd, ri = r−i if and only if i = 0. therefore cl(r0) = {r0} and cl(ri) = {ri,r−i}, a two element set, for all 1 ≤ i ≤ n−1. 253 vinod s, biju g.s also, cl(s0) = {rjs0r−1j ,sjs0s −1 j : 0 ≤ j ≤ n−1} = {rjs0r−1j ,sjs0sj : 0 ≤ j ≤ n−1} = {rjs0r−j,sjs0sj : 0 ≤ j ≤ n−1} = {s2j : 0 ≤ j ≤ n−1} = {sj : 0 ≤ j ≤ n−1}, since n odd. hence, if n is odd, {{sj : 0 ≤ j ≤ n−1},{r0},{ri,r−i } : 1 ≤ i ≤ (n−1)/2} are the conjugacy classes of dn. thus if n is odd, the number of conjugacy class of dn is (n−1) 2 + 2 = (n + 3) 2 . 2 corolary 5.1. the class equation of dn(n odd ) is 1 + 2 + 2 + . . . + 2 + n = 2n, the summation runs over (n−1)/2 times. theorem 5.2. if n is even, the number of conjugacy classes in dn is n + 6 2 . proof. let ri(0 ≤ i ≤ n−1) be a rotation in dn. then cl(ri) = {rjrir−1j ,sjris −1 j : 0 ≤ j ≤ n−1} = {rjrir−j,sjrisj : 0 ≤ j ≤ n−1} = {ri,sjrisj : 0 ≤ j ≤ n−1} = {ri,sj−isj : 0 ≤ j ≤ n−1} = {ri,r−i} since n is even ri = r−i if and only if i = 0 or n 2 . therefore cl(r0) = {r0},cl(rn/2) = {rn/2} and cl(ri) = {ri,r−i}, a two element set, for all 1 ≤ i ≤ n−1 and i 6= n 2 . also, cl(s0) = {rjs0r−1j ,sjs0s −1 j : 0 ≤ j ≤ n−1} = {rjs0r−1j ,sjs0sj : 0 ≤ j ≤ n−1} = {rjs0r−j,sjs0sj : 0 ≤ j ≤ n−1} = {s2j : 0 ≤ j ≤ n/2−1} 254 mixed and non-mixed normal subgroups of dihedral groups using conjugacy classes again, cl(s1) = {rjs1r−1j ,sjs1s −1 j : 0 ≤ j ≤ n−1} = {rjs1r−1j ,sjs1sj : 0 ≤ j ≤ n−1} = {rjs1r−j,sjs1sj : 0 ≤ j ≤ n−1} = {s2j+1 : 0 ≤ j ≤ n−1} = {s2j+1 : 0 ≤ j ≤ n/2−1} hence, if n is even,{ {s2j : 0 ≤ j < n/2},{s2j+1 : 0 ≤ j < n/2},{r0},{rn/2}, {ri,r−i } : 1 ≤ i ≤ (n−2)/2 } are the conjugacy classes of dn. thus if n is even, the number of conjugacy class of dn is (n−2) 2 + 4 = (n + 6) 2 . 2 corolary 5.2. the class equation of dn(n even ) is 1 + 1 + 2 + 2 + . . . + 2 + n/2 + n/2 = 2n, the summation runs over (n−2)/2 times. corolary 5.3. each conjugacy class of dn contains either rotations alone or reflections alone. corolary 5.4. the number of conjugacy classes of dn which contain rotations alone is (n + 1) 2 if n is odd and (n + 2) 2 if n is even. corolary 5.5. the number of conjugacy classes of dn which contain reflections alone is 1, namely dn, if n is odd and is 2, namely { {s2j : 0 ≤ j < n/2},{s2j+1 : 0 ≤ j < n/2} } , if n is even. 6 normal subgroups of dn in this section we will describe all mixed and non-mixed normal subgroups of dn. theorem 6.1. the number of non-mixed normal subgroups of dn is d(n). proof. since rn is a cyclic normal subgroup of dn, by theorem 2.11, the non-mixed subgroups and non-mixed normal subgroup of dn are same. hence the number of non-mixed normal subgroups of dn is d(n). 2 255 vinod s, biju g.s theorem 6.2. the number of mixed normal subgroups of dn is 1 if n odd and 3 if n even. proof. since normal subgroups are union of conjugacy classes, a mixed normal subgroup contain at least one conjugacy class having reflection. if n is odd, there is only one conjugacy class having reflection, namely {sj : 0 ≤ j ≤ n−1}. therefore dn is the only mixed normal subgroup of dn if n is odd. if n even, {s2j : 0 ≤ j < n/2} and {s2j+1 : 0 ≤ j < n/2} are the only conjugacy classes having reflection. therefore {s2j,r2j : 0 ≤ j < n/2},{s2j+1,r2j : 0 ≤ j < n/2} and dn are the only mixed normal subgroups of dn if n is even. therefore the number of mixed normal subgroups of dn is 3 if n is even. 2 corolary 6.1. the number of normal subgroups of dn is d(n) + 1 if n odd and d(n) + 3 if n even. 7 conclusion in this paper, it is proved that the number of mixed basis and non-mixed basis for dn(n ≥ 3) are nϕ(n) and nϕ(n) 2 respectively, where ϕ(n)is the number of nonnegative integers less than n and relatively prime to n. also it is shown that the number of different bases for dn(n ≥ 3) is 3n 2 ϕ(n). if n is odd, the number of conjugacy classes in dn is n + 3 2 and if n is even, the number of conjugacy classes in dn is n + 6 2 . finally we have shown that the number of non-mixed normal subgroups of dn is d(n) and the number of mixed normal subgroups of dn is 1 if n odd and 3 if n even. references grigore calugareanu. the total number of subgroups of a finite abelian group. scientiae mathematicae japonicae, 60(1):157–168, 2004. stephan r cavior. the subgroups of the dihedral group. mathematics magazine, 48(2):107–107, 1975. keith conrad. dihedral groups. spletni vir: http://www. math. uconn. edu/˜ kconrad/blurbs/grouptheory/dihedral. pdf, retrieveda. 256 mixed and non-mixed normal subgroups of dihedral groups using conjugacy classes keith conrad. dihedral groups ii. http://www.math.uconn.edu/ kconrad/blurbs/grouptheory/dihedral2.pdf, retrievedb. david steven dummit and richard m foote. abstract algebra. wiley hoboken, 2003. joseph a gallian. contemporary abstract algebra. d.c.heath and company, 1994. ga miller. subgroups of the groups whose orders are below thirty. proceedings of the national academy of sciences of the united states of america, 26(8): 500–502, 1940. marius tărnăuceanu. an arithmetic method of counting the subgroups of a finite abelian group. bulletin mathématique de la société des sciences mathématiques de roumanie, pages 373–386, 2010. 257 ratio mathematica vol. 35, 2018, pp. 5-27 issn: 1592-7415 eissn: 2282-8214 quasi-uniformity on bl-algebras r.a. borzooei ∗, n. kouhestani † received: 05-06-2018 accepted: 15-10-2018. published: 18-12-2018 doi:10.23755/rm.v35i0.423 c©borzooei, kouhestani abstract in this paper, by using the notation of filter in a bl-algebra a, we introduce the quasi-uniformity q and uniformity q∗ on a. then we make the topologies t (q) and t (q∗) on a and show that (a,∧,∨,�, t (q)) is a compact connected topological bl-algebra and (a, t (q∗)) is a topological bl-algebra. also we study q∗-cauchy filters and minimal q∗-filters on bl-algebra a and prove that the bicompletion (ã, q̃) of quasi-uniform bl-algebra (a, q) is a topological bl-algebra. 2010 msc: 06b10, 03g10. keywords : bl-algebra, (semi)topological bl-algebra, filter, quasiuniforme space, bicompletion 1 introduction bl-algebras have been introduced by hájek [11] in order to investigate manyvalued logic by algebraic means. his motivations for introducing bl-algebras ∗department of mathematics, shahid beheshti university, tehran, iran; borzooei@sbu.ac.ir †department of mathematics, sistan and balouchestan university, zahedan, iran; kouhestani@math.usb.ac.ir 5 r. a. borzooei, n. kouhestan were of two kinds. the first one was providing an algebraic counterpart of a propositional logic, called basic logic, which embodies a fragment common to some of the most important many-valued logics, namely lukasiewicz logic, gödel logic and product logic. this basic logic (bl for short) is proposed as ”the most general” many-valued logic with truth values in [0,1] and bl-algebras are the corresponding lindenbaum-tarski algebras. the second one was to provide an algebraic mean for the study of continuous t-norms (or triangular norms) on [0,1]. in 1973, andré weil [24] introduced the concept of a uniform space as a generalization of the concept of a metric space in which many non-topological invariant can be defined. this concept of uniformity fits naturally in the study of topological groups. the study of quasi-uniformities started in 1948 with nachbin’s investigations on uniform preordered spaces. in 1960, á. csaszar introduced quasi-uniform spaces and showed that every topological space is quasi-uniformizable. this result established an interesting analogy between metrizable spaces and general topological spaces. just as a metrizable space can be studied with reference to particular compatible metric(s), a topological space can be studied with reference to particular compatible quasi-uniformity(ies). in this and some other respects, a quasi-uniformity is a more natural generalization of a metric than is a uniformity. quasi-uniform structures were also studied in algebraic structures. in particular the study of paratopological groups and asymmetrically normed linear spaces with the help of quasi-uniformities is well known. see for example, [17], [18], [19], [20]. in the last ten years many mathematicians have studied properties of bl-algebras endowed with a topology. for example a. di nola and l. leustean [9] studied compact representations of bl-algebras, l. c. ciungu [7] investigated some concepts of convergence in the class of perfect bl-algebras, j. mi ko and y. c. kim [21] studied relationships between closure operators and bl-algebras. in [2] and [4] we study (semi)topological bl-algebras and metrizability on bl-algebras. we showed that continuity the operations � and → imply continuity ∧ and ∨. also, we found some conditions under which a locally compact topological bl-algebra become metrizable. but in there we can not answer some questions, for example: (i) is there a topology u on bl-algebra a such that (a,u) be a (semi)topological bl-algebra? (ii) is there a topology u on a bl-algebra a such that (a,u) be a compact connected topological bl-algebra? (iii) is there a topological bl-algebra (a,u) such that t0,t1 and t2 spaces be equivalent? (iv) if (a,→,u) is a semitopological bl-algebra, is there a topology v coarsere than u or finer than u such that (a,v) be a (semi)topological 6 quasi-uniformity on bl-algebras bl-algebra? now in this paper, we answer to some above questions and get some interesting results as mentioned in abstract. 2 preliminary recall that a set x with a family u = {uα}α∈i of its subsets is called a topological space, denoted by (x,u), if x,∅ ∈ u, the intersection of any finite numbers of members of u is in u and the arbitrary union of members of u is in u. the members of u are called open sets of x and the complement of x ∈u, that is x \u, is said to be a closed set. if b is a subset of x, the smallest closed set containing b is called the closure of b and denoted by b (or club). a subset p of x is said to be a neighborhood of x ∈ x, if there exists an open set u such that x ∈ u ⊆ p . a subfamily {uα : α ∈ j} of u is said to be a base of u if for each x ∈ u ∈u there exists an α ∈ j such that x ∈ uα ⊆ u, or equivalently, each u in u is a union of members of {uα}. let ux denote the totality of all neighborhoods of x in x. then a subfamily vx of ux is said to form a fundamental system of neighborhoods of x, if for each ux in ux, there exists a vx in vx such that vx ⊆ ux. (x,u) is said to be compact, if each open covering of x is reducible to a finite open covering. also (x,u) is said to be disconnected if there are two nonempty, disjoint, open subsets u,v ⊆ x such that x = u ∪v , and connected otherwise. the maximal connected subset containing a point of x is called the component of that point. topological space (x,u) is said to be: (i) t0 if for each x 6= y ∈ x, there is one in an open set excluding the other, (ii) t1 if for each x 6= y ∈ x, each are in an open set not containing the other, (iii) t2 if for each x 6= y ∈ x, both are in two disjoint open set.(see [1]) definition 2.1. [1] let (a,∗) be an algebra of type 2 and u be a topology on a. then a = (a,∗,u) is called a (i) left (right) topological algebra if for all a ∈ a, the map ∗a : a → a is defined by x → a∗x ( x → x∗a) is continuous, or equivalently, for any x in a and any open set u of a∗x (x∗a), there exists an open set v of x such that a∗v ⊆ u (v ∗a ⊆ u). (ii) semitopological algebra if a is a right and left topological algebra. (iii) topological algebra if the operation ∗ is continuous, or equivalently, if for any x,y in a and any open set (neighborhood) w of x ∗ y, there exist two open sets (neighborhoods) u and v of x and y, respectively, such that u ∗v ⊆ w. 7 r. a. borzooei, n. kouhestan proposition 2.2. [1] let (a,∗) be a commutative algebra of type 2 and u be a topology on a. then right and left topological algebras are equivalent. moreover, (a,∗,u) is a semitopological algebra if and only if it is right or left topological algebra. definition 2.3. [1] let a be a nonempty set and {∗i}i∈i be a family of operations of type 2 on a and u be a topology on a. then (i) (a,{∗i}i∈i,u) is a right(left) topological algebra if for any i ∈ i, (a,∗i,u) is a right (left) topological algebra. (ii) (a,{∗i}i∈i,u) is a semitopological (topological) algebra if for all i ∈ i, (a,∗i,u) is a semitopological (topological) algebra. definition 2.4. [11] a bl-algebra is an algebra a = (a,∧,∨,�,→, 0, 1) of type (2, 2, 2, 2, 0, 0) such that (a,∧,∨, 0, 1) is a bounded lattice, (a,�, 1) is a commutative monoid and for any a,b,c ∈ a, c ≤ a → b ⇔ a� c ≤ b, a∧ b = a� (a → b), (a → b) ∨ (b → a) = 1. let a be a bl-algebra. we define a′ = a → 0 and denote (a′)′ by a′′. the map c : a → a by c(a) = a′, for any a ∈ a, is called the negation map. also, we define a0 = 1 and an = an−1 �a, for all natural numbers n. example 2.5. [11] (i) let “�” and “→” on the real unit interval i = [0, 1] be defined as follows: x�y = min{x,y} x → y = { 1 , x ≤ y, y , otherwise. then i = (i, min, max,�,→, 0, 1) is a bl-algebra. (ii) let � be the usual multiplication of real numbers on the unit interval i = [0, 1] and x → y = 1 iff, x ≤ y and y/x otherwise. then i = (i, min, max,�,→, 0, 1) is a bl-algebra. proposition 2.6. [11] let a be a bl-algebra. the following properties hold. (b1) x�y ≤ x,y and x� 0 = 0, (b2) x ≤ y implies x�z ≤ y �z, (b3) x ≤ y iff x → y = 1, (b4) 1 → x = x, 1 �x = x, (b5) y ≤ x → y, (b6) x → (y → z) = (x�y) → z = y → (x → z), (b7) x∨y = ((x → y) → y) ∧ ((y → x) → x), (b8) x ≤ y ⇒ x → z ≥ y → z, z → x ≤ z → y, 8 quasi-uniformity on bl-algebras (b9) x → y ≤ (z → x) → (z → y), (b10) x → y ≤ (y → z) → (x → z), (b11) x → (y ∧z) = (x → y) ∧ (x → z), (b12) (y ∧z) → x = (y → x) ∨ (z → x), (b13) (y ∨z) → x = (y → x) ∧ (z → x), (b14) x → y ≤ x�z → y �z, (b15) (x → y) � (y → z) ≤ x → z, (b16) (x → y) � (a → z) ≤ (x∨a) → (y ∨z), (b17) (x → y) � (a → z) ≤ (x∧a) → (y ∧z), (b18) (x → y) � (a → z) ≤ (x�a) → (y �z). definition 2.7. [11] a filter of a bl-algebra a is a nonempty set f ⊆ a such that x,y ∈ f implies x�y ∈ f and if x ∈ f and x ≤ y imply y ∈ f , for any x,y ∈ a. it is easy to prove that if f is a filter of a bl-algebra a, then for each x,y ∈ f, x∧y, x∨y and x → y are in f proposition 2.8. [11] let f be a subset of bl-algebra a such that 1 ∈ f . then the following conditions are equivalent. (i) f is a filter. (ii) x ∈ f and x → y ∈ f imply y ∈ f. (iii) x → y ∈ f and y → z ∈ f imply x → z ∈ f . proposition 2.9. [11] let f be a filter of a bl-algebra a. define x ≡f y ⇔ x → y,y → x ∈ f. then ≡f is a congruence relation on a. moreover, if x/f = {y ∈ a : y ≡f x}, then (i) x/f = y/f ⇔ y ≡f x, (ii) x/f = 1/f ⇔ x ∈ f. definition 2.10. [2] (i) let a be a bl-algebra and (a,{∗i},u) be a semitopological (topological) algebra, where {∗i}⊆{∧,∨,�,→}, then (a,{∗i},u) is called a semitopological (topological) bl-algebra. remark 2.11. if {∗i} = {∧,∨,�,→}, we consider a = (a,u) instead of (a,{∧,∨,�,→},u), for simplicity. proposition 2.12. [2] let (a,{�,→},u) be a topological bl-algebra. then (a,u) is a topological bl-algebra. notation. from now on, in this paper, we use of bl-filter instead of filter in bl-algebras. 9 r. a. borzooei, n. kouhestan definition 2.13. [10] let x be a non-empty set. a family f of nonempty subsets of x is called a filter on x if (i) x ∈ f, (ii) for each f1,f2 of elements of f, f1 ∩f2 ∈f and, (iii) if f ∈f and f ⊆ g, then g ∈f. a subset b of a filter f on x is said to be a base of f if every set of f contains a set of b. if f is a family of nonempty subsets of x, then there exists the smallest filter on x containing f, denoted with fil(f) and called generated filter by f. definition 2.14. [10] a quasi-uniformity on a set x is a filter q on x such that (i) 4 = {(x,x) ∈ x ×x : x ∈ a}⊆ q, for each q ∈ q, (ii) for each q ∈ q, there is a p ∈ q such that p◦p ⊆ q, where p◦p = {(x,y) ∈ x ×x : ∃z ∈ a s.t (x,z), (z,y) ∈ p}. the pair (x,q) is called a quasi-uniform space. if q is a quasi-uniformity on a set x, q ∈ q and q−1 = {(x,y) : (y,x) ∈ q}, then q−1 = {q−1 : q ∈ q} is also a quasi-uniformity on x called the conjugate of q. it is well-known that if q satisfies condition: q ∈ q implies q−1 ∈ q, then q is a uniformity. furthermore, q∗ = q∨q−1 is a uniformity on x. if q and r are quasi-uniformities on x and q ⊆ r, then q is called coarser than r. a subfamily b of quasi-uniformity q is said to be a base for q if each q ∈ q contains some member of b.(see [10]) proposition 2.15. [22] let b be a family of subsetes of x ×x such that (i) 4⊆ q, for each q ∈b, (ii) for q1,q2 ∈b, there exists a q3 ∈b such that q3 ⊆ q1 ∩ q2, (iii) for each q ∈b, there is a p ∈b such that p◦p ⊆ q. then, there is the unique quasiuniformity q = {q ⊆ x ×x : for some p ∈ b,p ⊆ q} on x for which b is a base. the topology t(q) = {g ⊆ x : ∀x ∈ g ∃q ∈ q s.t q(x) ⊆ g} is called the topology induced by the quasi-uniformity q. definition 2.16. [10] (i) a filter g on quasi-uniform space (x,q) is called q∗-cauchy filter if for each u ∈ q, there is a g ∈g such that g×g ⊆ u. (ii) a quasi-uniform space (x,q) is called bicomplete if each q∗-cauchy filter converges with respect to the topology t(q∗). (iii) a bicompletion of a quasi-uniform space (x,q) is a bicomplete quasiuniform space (y,v) that has a t(v∗)-dense subspace quasi-unimorphic to 10 quasi-uniformity on bl-algebras (x,q). (iv) a q∗-cauchy filter on a quasi-uniform space (x,q) is minimal provided that it contains no q∗-cauchy filter other than itself. lemma 2.17. [10] let g be a q∗-cauchy filter on a quasi-uniform space (x,q). then, there is exactly one minimal q∗-cauchy filter coarser than g. furthermore, if b is a base for g, then {q(b) : b ∈ b and q is a symetric member of q∗} is a base for the minimal q∗-cauchy filter coarser than g. lemma 2.18. [10] let (x,q) be a t0 quasi-uniform space and x̃ be the family of all minimal q∗-cauchy filters on (a,q). for each q ∈ q, let q̃ = {(g,h) ∈ x̃ × x̃ : ∃g ∈g and h ∈h s.t g×h ⊆ q}, and q̃ = fil{q̃ : q ∈ q}. then the following statements hold: (i) (x̃,q̃) is a t0 bicomplete quasi-uniform space and (x,q) is a quasiuniformly embedded as a t((̃q∗))-dense subspace of (x̃,q̃) by the map i : x → x̃ such that, for each x ∈ x, i(x) is the t(q∗)-neighborhood filter at x. furthermore, the uniformities q̃∗ and (̃q∗) coincide. notation. from now on, in this paper we let a be a bl-algebra and f be a family of bl-filters in a which is closed under intersection , unless otherwise state. 3 quasi-uniformity on bl-algebras in this section, by using of bl-filters we introduce a quasi-uniformity q on bl-algebra a and stay some properties it. we show that (a,q) is not a t1 and t2 quasi-uniform space but it is a t0 quasi-uniform space. also we study q∗-cauchy filters, minimal q∗-cauchy filters and we make a quasiuniform space (ã,q̃) of minimal q∗-cauchy filters of (a,q) which admits the structure of a bl-algebra. lemma 3.1. let f be a bl-filter of bl-algebra a and f?(x) = {y : y → x ∈ f}, for each x ∈ a. then for each x,y ∈ a, the following properties hold. (i) x ≤ y implies f?(x) ⊆ f?(y), (ii) f?(x) ∧f?(y) = f?(x∧y) = f?(x) ∩f?(y), (iii) f?(x) ∨f?(y) ⊆ f?(x∨y), (iv) f?(x) �f?(y) ⊆ f?(x�y), (v) if for each a ∈ a, a�a = a, then f?(x) �f?(y) = f?(x�y), 11 r. a. borzooei, n. kouhestan (vi) x ∈ f ⇔ 1 ∈ f?(x) ⇔ f?(x) = a, (vii) for a,b ∈ a, if a∨ b ∈ f?(x), then a,b ∈ f?(x), (viii) if y ∈ f?(x), then f?(y) ⊆ f?(x). proof. (i) let x,y ∈ a, such that x ≤ y and z ∈ f?(x). then by (b8), z → x ≤ z → y. since f is a bl-filter and z → x ∈ f , z → y is in f and so z ∈ f?(y). (ii) let x,y ∈ a, such that a ∈ f?(x) and b ∈ f?(y). then a → x ∈ f and b → y ∈ f and so (a → x) � (b → y) ∈ f . since by (b17), (a → x) � (b → y) ≤ (a ∧ b) → (x ∧ y), we get (a ∧ b) → (x ∧ y) ∈ f. thus, a ∧ b ∈ f?(x ∧ y). now, if a ∈ f?(x ∧ y), since a → (x ∧ y) ∈ f and by (b11), a → (x∧ y) = (a → x) ∧ (a → y), we conclude that a → x ∈ f and a → y ∈ f. hence a ∈ f?(x) ∩ f?(y). finally, let a ∈ f?(x) ∩ f?(y). since a = a∧a, then a ∈ f?(x) ∧f?(y). (iii), (iv) the proof is similar to the proof of (ii), by some modification. (v) let x,y ∈ a such that z ∈ f?(x�y). then z → (x�y) ∈ f . by (b8), z → (x�y) ≤ z → x and z → (x�y) ≤ z → y which imply that z → x,z → y ∈ f. hence z is in both f?(x) and f?(y) and so z = z�z ∈ f?(x)�f?(y). (vi) the proof is clear. (vii), (viii) the proof come from by (b13) and (b15). lemma 3.2. let f be a bl-filter of bl-algebra a. define f? = {(x,y) ∈ a×a : y ∈ f?(x)} and f∗? = f? ∩f−1? . then (i) f−1? = {(x,y) ∈ a×a : x → y ∈ f}, (ii) f∗? = {(x,y) ∈ a×a : x ≡f y} = f∗ −1 ? , (iii) f∗? (x) = {y : x ≡f y}, (iv) f−1? (x) → y ⊆ f?(x → y), (v) if •∈ {∧,∨,�,→}, then f∗? (x) •f∗? (y) ⊆ f∗? (x•y). proof. the proof of (i), (ii) and (iii) are clear. (iv) let a ∈ f−1? (x) → y. then there exists a z ∈ f−1? (x) such that a = z → y and x → z ∈ f. by (b10), (z → y) → (x → y) ≥ x → z. since f is a filter, (z → y) → (x → y) ∈ f. hence a = z → y ∈ f?(x → y). (v) let a ∈ f∗? (x) and b ∈ f∗? (y). then by (iii), a ≡f x and b ≡f y. by proposition 2.9, a• b ≡f x•y. therefore, a• b ∈ f∗? (x•y). theorem 3.3. let f be a family of bl-filters of bl-algebra a which is closed under finite intersection. then the set b = {f? : f ∈ f} is a base for the unique quasi-uniformity q = {q ⊆ a × a : ∃f ∈ f s.t f? ⊆ q}. moreover, q∗ = {q ⊆ a×a : ∃f ∈f s.t f∗? ⊆ q}. proof. we prove that b satisfies in conditions (i), (ii) and (iii) of proposition 2.15. for (i), it is easy to see that for each f ∈f, 4⊆ f?. let f1,f2 ∈f 12 quasi-uniformity on bl-algebras and f = f1 ∩ f2. if (x,y) ∈ f?, then y → x ∈ f = f1 ∩ f2. hence (x,y) ∈ f1? ∩ f2?. this concludes that f? ⊆ f1? ∩ f2? and so (ii) is true. finally for (iii), let f ∈f and (x,y) ∈ f? ◦f?. then there is a z ∈ a such that (x,z) and (z,y) are both in f?. hence z → x and y → z are in f. since f is a filter and by (b15), (y → z) � (z → x) ≤ y → x, we conclude that y → x ∈ f. hence f?◦f? ⊆ f? and so (iii) is true. therefore, by proposition 2.15, q is a unique quasi-uniformity on a for which b is a base. now, we prove that q∗ = {q ⊆ a×a : ∃f ∈f s.t f∗? ⊆ q}. first we prove that p = {q ⊆ a×a : ∃f ∈ f s.t f∗? ⊆ q} is a uniformity on a. with a similar argument as above, we get {f∗? : f ∈f} is a base for the quasi-uniformity p = {q ⊆ a×a : ∃f ∈ f s.t f∗? ⊆ q}. to prove that p is a uniformity we have to show that for each q ∈p, q−1 is in p. suppose q ∈ p. then there exists a f ∈ f, such that f∗? ⊆ q. by lemma 3.2(ii), f∗? = f ∗−1 ? . hence f ∗ ? ⊆ q−1 and so q−1 ∈ p. thus p is a uniformity on a which contains q. since q∗ = q∨q−1, then q∗ ⊆p. on the other hand, if q ∈p, then there is a f ∈f such that f∗? ⊆ q. since f∗? = f? ∩f−1? ∈ q∗, we get that q ∈ q∗. therefore, q∗ = p. in theorem 3.3, we call q is quasi-uniformity induced by f, the pair (a,q) is quasi-uniform bl-algebra and the pair (a,q∗) is uniform blalgebra. notation. from now on, f, q and q∗ are as in theorem 3.3. example 3.4. let i be the bl-algebra in example 2.5 (i), and for each a ∈ [0, 1), fa = (a, 1]. then fa is a bl-filter in i and easily proved that for each a,b ∈ [0, 1), fa ∩ fb = fa∧b. hence f = {fa}a∈[0,1) is a family of bl-filters which is closed under intersection. for each a ∈ [0, 1), fa? = (a, 1] × [0, 1], f−1a? = [0, 1] × (a, 1] and f ∗ a? = (a, 1] × (a, 1]. by theorem 3.3, q = {q : ∃a ∈ [0, 1) s.t (a, 1] × [0, 1] ⊆ q} and q∗ = {q : ∃a ∈ [0, 1) s.t (a, 1] × (a, 1] ⊆ q}. recall that a map f from a (quasi)uniform space (x,q) into a (quasi)uniform space (y,r) is (quasi) uniformly continuous, if for each v ∈ r, there exists a u ∈ q such that (x,y) ∈ u implies (f(x),f(y)) ∈ v. if f : (x,q) ↪→ (y,r) is a quasi-uniform continuous map between quasi-uniform spaces, then f : (x,q∗) ↪→ (y,r∗) is a uniform continuous map. (see [10]) 13 r. a. borzooei, n. kouhestan proposition 3.5. in bl-algebra a, for each a ∈ a, the mappings ta(x) = a ∧ x, ra(x) = a ∨ x, la(x) = a � x and la(x) = a → x of quasi-uniform bl-algebra (a,q) into quasi-uniform bl-algebra (a,q) are quasi-uniformly continuous. moreover, they are uniformly continuous mappings of uniform bl-algebra (a,q∗) into uniform bl-algebra (a,q∗). proof. let q ∈ q. then, there is a f ∈ f such that f? ⊆ q. if (x,y) ∈ f?, then y → x ∈ f. by (b10) (a ∧ y) → (a ∧ x) ≥ y → x which implies that (a∧y) → (a∧x) ∈ f ⊆ q. hence ta is quasi-uniform continuous. moreover, ta : (a,q ∗) ↪→ (a,q∗) is uniform continuous. in a similar fashion and by use of (b16), (b14) and (b9), we can prove that, respectively, ra, la and la are quasi-uniform continuous of (a,q) ↪→ (a,q) and are uniform continuous of (a,q∗) ↪→ (a,q∗). let (x,q) be a (quasi)uniform space and b be a base for it. recall (x,q) is (i) t0 quasi-uniform if (x,y) and (y,x) are in ⋂ u∈b u, then x = y, for each x,y ∈ x, (ii) t1 quasi-uniform if 4 = ⋂ u∈b u, (iii) t2 quasi-uniform if 4 = ⋂ u∈b u −1 ◦u. (see [10]) theorem 3.6. quasi-uniform bl-algebra (a,q) is not t1 and t2 quasiuniform. if {1} ∈ f, then (a,q) is a t0 quasi-uniform space and uniform bl-algebra (a,q∗) is t0, t1 and t2 quasi-uniform space. proof. let x,y ∈ a and f ∈ f. since y → 1 = 1 ∈ f, we get that (1,y) ∈⋂ f∈f f?. hence (a,q) is not t0 quasi-uniform. also since x → 1 = y → 1 ∈ f, we conclude that (1,x), (1,y) ∈ f?. hence (x,y) ∈ f−1? ◦f? which implies that 4 6= ⋂ f∈f f −1 ? ◦f?. so (a,q) is not t2 quasi-unifom let {1} ∈ f and (x,y) and (y,x) be in ⋂ f∈f f?. then for each f ∈ f, x → y and y → x are in f. hence x ≡{1} y, which implies that x = y. therefore, (a,q) is t0 quasi-uniform. with a similar argument as above, we can prove that (a,q∗) is a t0 and t1 quasi-uniform space. to verify t2 quasi-uniformity, let (x,y) ∈ ⋂ f∈f f ∗−1 ? ◦f∗? . then for each f ∈f there is a z ∈ a such that (x,z) ∈ f∗−1? and (z,y) ∈ f∗? . by lemma 3.2(ii), x ≡f y. since {1} ∈ f, we get that x = y. therefore, (a,q∗) is a t2 quasi-uniform space. proposition 3.7. let b be a base for a q∗-cauchy filter g on quasi-uniform bl-algebra (a,q). then the set {f∗? (b) : f ∈ f, b ∈ b} is a base for the uniqe minimal q∗-cauchy filter coarser than g. 14 quasi-uniformity on bl-algebras proof. by lemma 2.17, the set {q(b) : b ∈ b, q−1 = q ∈ q∗} is a base for the unique minimal q∗-cauchy filter g0 coarser than g. let q−1 = q ∈ q∗ and b ∈ b. then for some f ∈ f, f∗? ⊆ q. so, f∗? (b) ⊆ q(b). now, it is easy to prove that the set {f∗? (b) : f ∈f, b ∈b} is a base for g0. proposition 3.8. f is a base for a minimal q∗-cauchy filter on quasiuniform bl-algebra (a,q). proof. let c = {s ⊆ a : ∃f ∈f s.t f ⊆ s}. it is easy to prove that c is a filter and f is a base for it. we prove that c is a q∗-cauchy filter. for this, let q ∈ q. there is a f ∈ f such that f? ⊆ q. since f is a filter, clearly f × f ⊆ f? ⊆ q. hence c is a q∗-cauchy filter. now, by proposition 3.7, the set {f∗? (f1) : f,f1 ∈ f} is a base for the unique minimal q∗-cauchy filter f0 coarser than c. to complete proof we show that for each f,f1 ∈f, f∗? (f1) = f1. let f,f1 ∈ f. if y ∈ f∗? (f1), then for some x ∈ f1, x ≡f y. by proposition 2.9, y ∈ f1. hence f∗? (f1) ⊆ f1. clearly, f1 ⊆ f∗? (f1). therefore, f1 = f ∗ ? (f1). thus proved that f is a base for f0. proposition 3.9. the set b = {f∗? (0) : f ∈ f} is a base for a minimal q∗-cauchy filter on quasi-uniform bl-algebra (a,q). proof. let c = {s ⊆ a : ∃f ∈ f s.t f∗? (0) ⊆ s}. it is easy to prove that c is a filter and the set b = {f∗? (0) : f ∈ f} is a base for it. to prove that c is a q∗-cauchy filter, let q ∈ q. there is a f ∈ f such that f? ⊆ q. if x,y ∈ f∗? (0), then x ≡f y and so (x,y) ∈ f∗? ⊆ f? ⊆ q. this prove that f∗? (0) × f∗? (0) ⊆ q. hence c is a q∗-cauchy filter. by proposition 3.7, the set {f∗? (f∗? (0)) : f ∈ f} is a base for the uniqe minimal q∗-cauchy filter i coarser than c. but it is easy to pove that fo each f ∈f, f∗? (f∗? (0)) = f∗? (0). therefore, b is a base for i. lemma 3.10. let g and h be q∗-cauchy filters on quasi-uniform bl-algebra (a,q). if • ∈ {∧,∨,�,→}, then g •h = {g • h : g ∈ g, h ∈ h} is a q∗-cauchy filter base on quasi-uniform bl-algebra (a,q). proof. let c = {s ⊆ a : ∃g, h s.t g ∈g, h ∈h, g•h ⊆ s}. it is easy to prove that c is a filter and the set b = {g•h : g ∈g, h ∈h} is a base for it. we prove that c is a q∗-cauchy filter. for this, let q ∈ q. then for some a f ∈f, f? ⊆ q. since g,h are q∗-cauchy filters, there are g ∈g and h ∈h such that g×g ⊆ f? and h×h ⊆ f?. we show that g•h×g•h ⊆ f? ⊆ q. let g1,g2 ∈ g and h1,h2 ∈ h. then (g1,g2), (g2,g1), (h1,h2), (h2,h1) are in f?. so g1 ≡f g2 and h1 ≡f h2. by proposition 2.9, g1 •h1 ≡f g2 •h2, which implies that (g1 •h1,g2 •h2) ∈ f?. 15 r. a. borzooei, n. kouhestan theorem 3.11. there is a quasi-uniform space (ã,q̃) of minimal q∗-cauchy filters of quasi-uniform bl-algebra (a,q) that admits a bl-algebra structure. proof. let ã be the family of all minimal q∗-cauchy filters on (a,q). let for each q ∈ q, q̃ = {(g,h) ∈ ã× ã : ∃g ∈g,h ∈h s.t g×h ⊆ q}. if q̃ = fil{q̃ : q ∈ q}, then (ã,q̃) is a quasi-uniform space of minimal q∗-cauchy filters of (a,q). let g,h∈ ã. since g,h are minimal q∗-cauchy filters on a, then by lemma 3.10, g∧h, g∨h, g�h and g →h are q∗cauchy filter bases on a. now, we define g fh, g gh, g }h and g ↪→h as the minimal q∗-cauchy filters contained g∧h, g∨h, g�h and g →h, respectively. thus, g fh, g gh, g }h and g ↪→h are in ã. now, we will prove that (ã,f,g,}, ↪→,i,f0) is a bl-algebra, where i is minimal q∗-cauchy filter in proposition 3.9 and f0 is minimal q∗-cauchy filter in proposition 3.8. for this, we consider the following steps: (1) (ã,f,g) is a bounded lattice. let g,h,k∈ ã. we consider the following cases: case 1.1: g fg = g, g gg = g by proposition 3.7, s1 = {f∗? (g) : g ∈g,f ∈f} and s2 = {f∗? (g1 ∧g2) : g1,g2 ∈g,f ∈f} are bases of the minimal q∗-cauchy filters g and g fg, respectively. first, we show that s2 ⊆ s1. let f∗? (g1 ∧ g2) ∈ s2. put g = g1 ∩ g2, then g ∈ g. let y ∈ f∗? (g). then there is a x ∈ g such that (x,y) ∈ f∗? . since x ∧ x = x, it follows that (x ∧ x,y) ∈ f∗? and so y ∈ f∗? (g1 ∧g2). hence s2 ⊆ s1. therefore, g fg ⊆g. by the minimality of g, g fg = g. the proof of the other case is similar. case 1.2: g fh = h fg, g gh = h gg by proposition 3.7, s1 = {f∗? (g ∧ h) : g ∈ g,h ∈ h,f ∈ f} and s2 = {f∗? (h ∧ g) : g ∈ g,h ∈ h,f ∈ f} are bases of g fh and h fg, respectively. for each g ∈ g and h ∈ h, since g ∧ h = h ∧ g, for each f ∈f, f∗? (g∧h) = f∗? (h ∧g). hence g fh = h fg. the proof of the other case is similar. case 1.3: g f (h fk) = (g fh) fk, g g (h gk) = (g gh) gk by proposition 3.7, the families s1 = {f∗1?(f ∗ 2?(g∧h) ∧k) : g ∈g,h ∈h,k ∈k,f1,f2 ∈f}, s2 = {f∗1?(g∧f ∗ 2?(h ∧k) : g ∈g,h ∈h,k ∈k,f1,f2 ∈f} are bases for the minimal q∗-cauchy filters (g f h) f k and g f (h f k), respectively. let f∗1?(f ∗ 2?g∧ (h ∧k) ∈ s2 and f = f1 ∩f2. then f ∈ f. 16 quasi-uniformity on bl-algebras now, we show that f∗? (f ∗ ? (g ∧ h) ∧ k) ⊆ f∗1?(g ∧ f∗2?(h ∧ k). let y ∈ f∗? (f ∗ ? (g ∧ h) ∧ k). then there are x ∈ f∗? (g ∧ h), k ∈ k, g ∈ g and h ∈ h such that y ≡f x ∧ k and x ≡f g ∧ h. hence y ≡f (g ∧ h) ∧ k = g∧(h∧k), which implies that y ∈ f∗? (g∧f∗? (h∧k) ⊆ f∗1?(g∧f∗2?(h∧k). therefore, g f (hfk) ⊆ (g fh) fk. by the minimality of (g fh) fk, g f (h fk) = (g fh) fk. the proof of the other case is similar. case 1.4: g f (g gh) = g, g g (g fh) = g it is enough to prove that g f (g g h) = g. the proof of the other case is similar. by proposition 3.7, the families s1 = {f∗? (g) : g ∈ g,f ∈ f} and s2 = {f∗1?(g1 ∧ f∗2?(g2 ∨ h) : g1,g2 ∈ g,h ∈ h,f1,f2 ∈ f} are bases for the minimal q∗-cauchy filters g and gf (ggh) , respectively. let f∗1?(g1 ∧f∗2?(g2 ∨h) ∈ s2. put g = g1 ∩g2 and f = f1 ∩f2. we prove that f∗? (g) ⊆ f∗1?(g1 ∧f∗2?(g2 ∨h). let y ∈ f∗? (g). then there is a g ∈ g such that y ≡f g. if h ∈ h, since g = g∧ (g∨h), then y ≡f g∧ (g∨h) and so y ∈ f∗1?(g1 ∧f∗2?(g2 ∨h). hence g f (g gh) ⊆g. by the minimality of g, we conclude that g f (g gh) = g. now the cases 1.1,1.2,1.3,1.4 imply that (ã,f,g) is a lattice. case 1.5: the lattice (ã,f,g) is bounded. for this, for each g,h ∈ ã, define g ≤h ⇔ g fh = g. it is clear that (ã,≤) is a partial ordered. now, we prove that for each g ∈ ã, i ≤g ≤f0. first, we show that i ≤g. let s ∈i. then for some a f ∈f, f∗? (0) ⊆ s. since g is a minimal q∗-cauchy filter, there is a g ∈g such that g×g ⊆ f?. we show that f∗? (g ∧ f∗? (0)) ⊆ s. let y ∈ f∗? (g ∧ f∗? (0)). then there are g ∈ g and x ∈ f∗? (0) such that y ≡f g∧x. on the other hand, since x ≡f 0, we get g ∧x ≡f 0. hence y ≡f 0 which implies that y ∈ f∗? (0) ⊆ s. since f∗? (g∧f∗? (0)) ∈gfi, then s ∈gfi. by the minimality of gfi, gfi = i. now, we prove that g ≤f0. by proposition 3.7, the set s1 = {f∗? (g∧f1) : g ∈ g, f,f1 ∈ f} is a base for g f f0. let f∗? (g ∧ f1) ∈ s1. we prove that f∗? (g) ⊆ f∗? (g∧f1). let y ∈ f∗? (g). then, there is a g ∈ g such that y ≡f g = g∧1. hence y ∈ f∗? (g∧f1). by the minimality of g, gff0 = g. (2) (ã,}) is a commutative monoid case 2.1: (ã,}) is a commutative semigroup. we will prove that g } (h }k) = (g }h) }k. by proposition 3.7, the sets s1 = {f∗1?(g�f ∗ 2?(h �k)) : g ∈g,h ∈h,k ∈k,f1,f2 ∈f}, s2 = {f∗1?(f ∗ 2?(g�h) �k)) : g ∈g,h ∈h,k ∈k,f1,f2 ∈f} are bases from g } (h }k) and (g }h) }k, respectively. let f∗1?(f∗2?(g� h)�k)) ∈ s2, f = f1∩f2 and y ∈ f∗? (g�f∗? (h�k). then there are g ∈ g, x ∈ f∗? (h�k), h ∈ h and k ∈ k such that y f ≡ g�x and x f ≡ h�k. hence 17 r. a. borzooei, n. kouhestan y f ≡ g�(h�k) = (g�h)�k and so y ∈ f∗? (f∗? (g�h)�k) ⊆ f∗1?(f∗2?(g�h)� k)). therefore, s2 ⊆ s1 which implies that (g }h) }k ⊆ g } (h }k). now, by the minimality of g } (h }k), g } (h }k) = (g }h) }k. finally, it is easy to prove that g }h = h }g. case 2.2: (ã,}) is a monoid we prove that g}f0 = g. by proposition 3.7, the set s2 = {f∗? (g�f1) : g ∈ g,f,f1 ∈f} is a base for g }f0. it is clear that for each f∗? (g�f1) ∈ s2, f∗? (g) ⊆ f∗? (g�f1) and this implies that g } f0 ⊆ g. by the minimality of g, g }f0 = g. (3) g } (g ↪→h) = g fh by proposition 3.7, the families s1 = {f∗? (g∧h) : g ∈g,h ∈h,f ∈f}, s2 = {f∗1?(g1 �f ∗ 2?(g2 → h)) : g1,g2 ∈g,h ∈h,f1,f2 ∈f} are bases for g fh and g } (g ↪→h), respectively. let f∗1?(g1 �f∗2?(g2 → h)) ∈ s2, g = g1 ∩g2 and f = f1 ∩f2. we will prove that f∗? (g∧h) ⊆ f∗1?(g1 � f∗2?(g2 → h)). let y ∈ f∗? (g ∧ h). then there are g ∈ g and h ∈ h such that y ≡f g ∧ h. it follows from g ∧ h = g � (g → h) which y ∈ f∗1?(g1 �f∗2?(g2 → h)). hence f∗? (g∧h) ⊆ f∗1?(g1 �f∗2?(g2 → h)) which implies that g}(g ↪→h) ⊆g fh. now, by the minimality of g fh, we get g } (g ↪→h) = g fh. (4) g ≤h ↪→k⇔g }h≤k first, we prove the following statements: (a) g ≤h⇔g ↪→h = f0 (b) g ↪→ (h ↪→k) = g }h ↪→k. (a) to prove it, let g ↪→h = f0. then g} (g ↪→h) = g}f0 = g. by (3), g fh = g and so g ≤h. conversely, let g ≤ h. by proposition 3.7, the set s = {f∗? (g → h) : g ∈ g,h ∈ h,f ∈ f} is a base for g ↪→ h. let f∗? (g → h) ∈ s. we prove that 1 ∈ f∗? (g → h). since by lemma 3.10, g → h is a q∗-cauchy filter base, there are g1 ∈g and h1 ∈h such that (g1 → h1)×(g1 → h1) ⊆ f?. put g2 = g1 ∩ g and h2 = h1 ∩ h. it is easy to see that g2 ∧ h2 ⊆ f∗? (g2∧h2) ∈gfh. since gfh = g, there is a g3 ∈g such that g3 ⊆ g1 and g3 ⊆ g2 ∧ h2. since g3 6= φ, there are g3 ∈ g3, g ∈ g2 and h ∈ h2 such that g3 = g ∧ h. since (g3 → h,g → h) and (g → h,g3 → h) both are in (g1 → h1) × (g1 → h1) ⊆ f?, we get g → h ≡f g3 → h = 1 and so 1 ∈ f∗? (g → h). hence f∗? (1) ⊆ f∗? (g → h). this implies that g ↪→ h ⊆ f0. by the minimality of f0, g ↪→ h = f0. therefore, we have (a). 18 quasi-uniformity on bl-algebras (b) by proposition 3.7, the families s1 = {f∗1?(g → f ∗ 2?(h → k)) : g ∈g,h ∈h,k ∈k,f1,f2 ∈f}, s2 = {f∗1?(f ∗ 2?(g�h) → k) : g ∈g,h ∈h,k ∈k,f1,f2 ∈f} are bases of g ↪→ (h ↪→k) and (g}h) ↪→k, respectively. let f∗1?(f∗2?(g� h) → k) ∈ s2, f = f1 ∩f2 and y ∈ f∗? (g → f∗? (h → k)). then there are g ∈ g and x ∈ f∗? (h → k) such that y ≡f g → x. also there are h ∈ h and k ∈ k such that x ≡f h → k. hence y ≡f g → x ≡f g → (h → k) = (g �h) → k. therefore, y ∈ f∗1?(f∗2?(g�h) → k). this implies that (g } h) ↪→ k ⊆ g ↪→ (h ↪→ k). by the minimality of g ↪→ (h ↪→ k), we get g ↪→ (h ↪→k) = g }h ↪→k. hence we have (b). now, by (a) and (b), we have g ≤h ↪→k⇔g ↪→ (h ↪→k) = f0 ⇔ (g }h) ↪→k = f0 ⇔g }h≤k. so g ≤h ↪→k⇔g }h≤k. (5) (g ↪→h) g (h ↪→g) = f0 by proposition 3.7, the set s = {f∗1?(f ∗ 2?(g1 → h1)∨f ∗ 3?(h2 → g2)) : g1,g2 ∈g,h1,h2 ∈h,f1,f2,f3 ∈f} is a base for (g ↪→ h) g (h ↪→ g). let f∗1?(f∗2?(g1 → h1) ∨ f∗3?(h2 → g2)) ∈ s, g = g1 ∩g2, h = h1 ∩h2 and f = f1 ∩f2 ∩f3. we show that 1 ∈ f∗? (f∗? (g → h) ∨ f∗? (h → g)). let g ∈ g and h ∈ h. since a is a bl-algebra, we have (g → h) ∨ (h → g) = 1. since g → h ∈ f∗? (g → h) and h → g ∈ f∗? (h → g), we have (g → h) ∨ (h → g) ∈ f∗? (f∗? (g → h)∨f∗? (h → g)) and so 1 ∈ f∗? (f∗? (g → h)∨f∗? (h → g)). hence f∗? (1) ⊆ f∗? (f ∗ ? (g → h)∨f∗? (h → g)) which implies that (g ↪→h)g(h ↪→g) ⊆f0. by the minimality of f0, (g ↪→h) g (h ↪→g) = f0. 4 some topological properties on quasi-unifom bl-algebra (a,q) let t(q) and t(q∗) be topologies induced by q and q∗, respectively. our goal in this section is to study (semi)topological bl-algebras (a,t(q)) and (a,t(q∗)). we prove that (a,∧,∨,�,t(q)) is a compact connected topological bl-algebra and (a,t(q∗)) is a regular topological bl-algebra. we study separation axioms on (a,t(q)) and (a,t(q∗)). also we stay conditions under which (a,q) becomes totally bounded. finally, we show that if 19 r. a. borzooei, n. kouhestan (a,q) is a t0 quasi-uniform space, then the bl-algebra (ã,q̃) in theorem 3.11 is the bicomplition topological bl-algebra of (a,q). theorem 4.1. the set t(q) = {g ⊆ a : ∀x ∈ g ∃f ∈ f s.t f?(x) ⊆ g} is the topology induced by q on a such that (a,{∧,∨,�},t(q)) is a topological bl-algebras. also (a,→,t(q)) is a left topological bl-algebra. furthermore, if the negation map c(x) = x′ is one to one, then (a,t(q)) is a topological bl-algebra. proof. first we prove that t(q) is a nonempty set. for this, we prove that for each f ∈ f and each x ∈ a, f?(x) ∈ t(q). let f ∈ f, x ∈ a and y ∈ f?(x). if z is an arbitrary element of f?(y), then z → y ∈ f. since y → x ∈ f, by (b15), we get z → x ∈ f. hence f?(y) ⊆ f?(x) which implies that f?(x) ∈ t(q). now we prove that t(q) is a topology on a. clearly, φ,a ∈ t(q). also it is easy to prove that the arbitrary union of members of t(q) is in t(q). let g1, ...,gn be in t(q) and x ∈ ⋂i=n i=1 gi. there are f1, ...,fn ∈ f such that fi?(x) ⊆ gi, for 1 ≤ i ≤ n. let f = f1 ∩ ... ∩ fn. then f ∈ f and f?(x) ⊆ f1?(x) ∩ ...∩fn?(x) ⊆ ⋂i=n i=1 gi. hence t(q) is a topology. since for each f ∈f, f? belongs to q, then t(q) is the topology induced by q. now, by lemmas 3.1, it is clear that (a,{∧,∨,�},t(q)) is a topological bl-algebra. in continue, we prove that (a,→,t(q)) is a left topological bl-algebra. let x,y,z ∈ a, and z ∈ f?(y). by (b9), (x → z) → (x → y) ≥ z → y which implies that (x → z) → (x → y) ∈ f. so x → z ∈ f?(x → y). hence x → f?(y) ⊆ f?(x → y) and so (a,→,t(q)) is a left topological bl-algebra. to complete the proof, suppose that the negation map c is one to one. since (a,→,t(q)) is a topological bl-algebra, c is continuous. now by [[2], theorem(3.15)], (a,t(q)) is a topological bl-algebra. theorem 4.2. bl-algebra (a,t(q)) is a connected and compact space and each f ∈f, is a closed compact set in (a,t(q)). proof. first we prove that if {gi : i ∈ i} is an open cover of a in t(q), then for some i ∈ i, a = gi. let a = ⋃ i∈i gi, where gi ∈ t(q). then, there are i ∈ i and f ∈f such that 1 ∈ gi and f?(1) ⊆ gi. by lemma 3.1 (vi), a = f?(1). hence a = gi. now, it is easy to show that (a,t(q)) is connected and compact. in continue we prove that each f ∈f, is a closed, compact set in (a,t(q)). for this, let f ∈ f and x ∈ f. then, there is a y ∈ f?(x) ∩ f. since y ∈ f and y → x ∈ f , we get x ∈ f. hence f = f. now, since (a,t(q)) is compact, f is compact. theorem 4.3. (i) bl-algebra (a,t(q)) is not a t1 and t2 topological space. (ii) bl-algebra (a,t(q)) is a t0 topological space iff, for each 1 6= x ∈ a, there is a f ∈f such that x 6∈ f. 20 quasi-uniformity on bl-algebras proof. (i) (a,t(q)) is not a t1 and t2 topological space because for each g ∈ t(q), 1 ∈ g if and only if g = a. (ii) suppose for each 1 6= x ∈ a, there is a f ∈f such that x 6∈ f. we prove that (a,t(q)) is a t0 topological space. for this, let 1 6= x ∈ a. then for some f ∈f, x 6∈ f. since 1 → x = x, then 1 6∈ f?(x). moreover, since (a,→ ,t(q)) is a left topological bl-algebra, by [[2], proposition(4.2)], (a,t(q)) is a t0 topological space. conversely, let (a,t(q)) is a t0 topological space and 1 6= x ∈ a. then for some f ∈f, 1 6∈ f?(x). hence x = 1 → x 6∈ f. theorem 4.4. the set t(q∗) = {g ⊆ a : ∀x ∈ g ∃f ∈f s.t f∗? (x) ⊆ g} is the topology induced by q∗ on bl-algebra a such that (a,t(q∗)) is a topological bl-algebras. proof. by the similar argument as theorem 4.1, we can prove that t(q∗) is the topology induced by q∗ on a. by lemma 3.2(v), (a,t(q∗)) is a topological bl-algebra. theorem 4.5. (i) bl-algebra (a,t(q∗)) is connected iff, f = {a}, (ii) f has only a proper filter iff, each f ∈f is a component. proof. (i) let f = {a}. then it is easy to prove that t(q∗) = {φ,a}. hence (a,t(q∗)) is connected. conversely, let f 6= {a}. then, there is a filter f ∈ f such that f 6= a. since for each x ∈ f, f∗? (x) ⊆ f, we conclude that f ∈ t(q∗). let y ∈ f. then there is a z ∈ f∗? (y) ∩ f. this proves that y ∈ f. hence f is closed. now, since f is a closed and open subset of a, then a is not connected. (ii) let f has a proper filter f. by the similar argument as (i), we get that f is closed and open. we show that f is connected. let g1 and g2 be in t(q∗) and f = (f ∩ g1) ∪ (f ∩ g2). without loss of generality, suppose that 1 ∈ f ∩ g1, then f ⊆ f∗? (1) ⊆ g1. hence f ∩ g1 = f, which implies that f is connected. therefore, f is a component. conversely, suppose each f ∈f is a component. if f1 and f2 are in f, then f1 ∩f2 is in f and is component. hence f1 = f1 ∩f2 = f2. recall that a topological space (x,u) is regular if for each x ∈ g ∈ u there is a u ∈u such that x ∈ u ⊆ u ⊆ g. theorem 4.6. bl-algebra (a,t(q∗)) is a regular space. proof. first we prove that for each f ∈ f and x ∈ a, f∗? (x) = f∗? (x). let y ∈ f∗? (x). then there is a z ∈ f∗? (y) ∩ f∗? (x). hence y ≡f z ≡f x which implies that y ∈ f∗? (x). therefore, f∗? (x) = f∗? (x). now if x ∈ g ∈ t(q∗), then for some a f ∈ f, x ∈ f∗? (x) = f∗? (x) ⊆ g. hence (a,t(q∗)) is a regular space. 21 r. a. borzooei, n. kouhestan theorem 4.7. on bl-algebra (a,t(q∗)) the follwing statements are equivalent. (i) (a,t(q∗)) is a t0 space, (ii) ⋂ f∈f f ∗ ? (1) = {1}, (iii) (a,t(q∗)) is a t1 space, (iv) (a,t(q∗)) is a t2 space. proof. (i ⇒ ii) let (a,t(q∗)) be a t0 space and 1 6= x ∈ a. by [[2], proposition(4.2)], there is a f ∈f such that 1 6∈ f∗? (x). hence x 6∈ f. this implies that x 6∈ f∗? (1). therefore, x 6∈ ⋂ f∈f f ∗ ? (1). (ii ⇒ i) let ⋂ f∈f f ∗ ? (1) = {1} and 1 6= x ∈ a. then for some a f ∈ f, x 6∈ f. hence 1 6∈ f∗? (x). now by [[2], proposition(4.2)], (a,t(q∗)) is a t0 space. by theorems 4.4 and 4.6, (a,t(q∗)) is a regular topological bl-algebra. hence by [[2], theorem(4.7)], the statements (ii), (iii) and (iv) are equivalent. example 4.8. in example 3.4, for each a ∈ [0, 1) and x ∈ [0, 1] fa∗(x) = { [0,x] , x ≤ a, [0,1] , x > a. f−1a∗ (x) = { [x,1] , x ≤ a, (a,1] , x > a. f∗a∗(x) =   x , x < a, a , x = a (a,1] , x > a. if t(q) is the induced topology by q and g ∈ t(q), then for each x ∈ g, there is a a ∈ [0, 1) such that f∗a?(x) ⊆ g. hence [0,x] ⊆ g or g = [0, 1]. if g ∈ t(q) and g 6= [0, 1], then for each x ∈ g, [0,x] ⊆ g. if g = supg, then g = [0,g] or [0,g). therefore t(q) = {[0,x] : x ∈ [0, 1]}∪{[0,x) : x ∈ [0, 1]}. also if t(q∗) is topology induced by q∗ and g ∈ t(q∗), then for each x ∈ g, there is a a ∈ [0, 1) such that f∗a?(x) ⊆ g. hence if g ∈ t(q∗), then for some a ∈ [0, 1), a ∈ g or (a, 1] ⊆ g. now since for each a ∈ [0, 1), f∗a?(1) = (a, 1], we get that ⋂ a∈[0,1) f ∗ a?(1) = {1}. hence by theorems 4.4, 4.6 and 4.7, (a,t(q∗)) is a ti regular topological bl-algebra, when 0 ≤ i ≤ 2. theorem 4.9. let (a,→,u) be a semitopological bl-algebra and f0 be an open proper bl-filter in a. then, there exists a nontrivial topology v on a such that v ⊆u and (a,v) is a topological bl-algebra. proof. let f be a collection of bl-open filters in a which closed under finite intersection and f0 ∈f. let q be the quasi-uniformity induced by f. since 22 quasi-uniformity on bl-algebras f0 6= a, by lemma 3.1(vi), there is a x ∈ a such that f∗0?(x) 6= a. so t(q∗) is a nontrivial topology. we prove that t(q∗) ⊆ u. let x ∈ g ∈ t(q∗). then, there is a f ∈ f such that f∗? (x) ⊆ g. since x → x = 1 ∈ f ∈ u, there is a u ∈ u such that x ∈ u and u → x ⊆ f and x → u ⊆ f . if z ∈ u, then z → x,x → z ∈ f and so z ∈ f∗? (x). hence x ∈ u ⊆ g. therefore, t(q∗) is a nontrivial topology coaser than u and so by theorem 4.4, (a,t(q∗)) is a topological bl-algebra. example 4.10. let i be the bl-algebra in example 2.5(ii), and u be a topology on i with the base s = {(a,b] ∩ i : a,b ∈ r}. we prove that (i,→,u) is a semitopological bl-algebra. let x,y ∈ i, and x → y ∈ (a,b]. if x ≤ y, then [0,x] and (ax,y] are two open neighborhoods of x and y, respectively, such that (0,x] → y ⊆ (a, 1] and x → (ax,y] ⊆ (a, 1]. if x > y and y = 0, then (0,x] and {0} are two open neighborhoods of x and 0, respectively, such that (0,x] → 0 ⊆ [0,b] and x → {0} ⊆ [0,b]. if x > y and y 6= 0, then (y/b,y/a] and (ax,bx] are two open sets of x,y, respectively, such that (y/b,y/a] → y ⊆ (a,b] and x → (ax,bx] ⊆ (a,b]. it is easy to prove that f = {(0, 1],a} is a collection of bl-filters which is closed under intersection. now since for each x ∈ a, a∗?(x) = a and (0, 1]∗?(x) = (0, 1], we conclude t(q∗) = {φ, (0, 1],a}. by theorem 4.9, (a,t(q∗)) is a topological bl-algebra. recall a quasi-uniform space (x,q) is totally-bounded if for each q ∈ q, there exist sets a1, ...,an such that x = ⋃i=n i=1 ai and for each 1 ≤ i ≤ n, ai ×ai ⊆ q.( see [10]) theorem 4.11. the following conditions on bl-algebra (a,t(q∗)) are equivalent. (i) for each f ∈f, a/f is finite, (ii) (a,q) is totally bounded, (iii) (a,t(q∗)) is compact. proof. (i ⇒ ii) let for each f ∈f, a/f be finite. we prove that (a,q) is totally bounded. for this it is enough to prove that, for each f ∈ f, there are a1, ...,an ∈ a, such that for each 1 ≤ i ≤ n, ai/f × ai/f ⊆ f?. let f ∈f. since a/f is finite, there are a1, ...,an ∈ a, such that a = ∪ni=1ai/f. for each 1 ≤ i ≤ n, ai/f ×ai/f ⊆ f? because if (x,y) ∈ ai/f ×ai/f, then x ≡f ai ≡f y and so (x,y) ∈ f?. this proves that (a,q) is totally bounded. (ii ⇒ iii) let (a,q) be totally bounded and f ∈ f. there exist sets a1, ...,an, such that ⋃i=n i=1 ai = a and for each 1 ≤ i ≤ n, ai ×ai ⊆ f?. let 1 ≤ i ≤ n and x,y ∈ ai. since (x,y) and (y,x) are in f?, we get x ≡f y. this proves that ai = ai/f, for some ai ∈ ai. 23 r. a. borzooei, n. kouhestan now to prove that (a,t(q∗)) is compact let a = ⋃ i∈i gi, where each gi is in t(q∗). then there are h1, ...,hn ∈ {gi : i ∈ i}, such that ai ∈ hi, for each 1 ≤ i ≤ n. now suppose x ∈ a, then x ∈ ai/f, for some 1 ≤ i ≤ n, and so x ∈ f∗? (ai) ⊆ hi. therefore, a ⊆ ⋃n i=1 hi, which shows that (a,t(q ∗)) is compact. (iii ⇒ i) let f ∈f. since {f∗? (x) : x ∈ a} is an open cover of a in t(q∗), then there are a1, ...,an ∈ a, such that a ⊆ ⋃n i=1 f ∗ ? (ai). now, it is easy to see that a/f = {a1/f,...,an/f}. in the end, we prove that the quasi-uniform bl-algeba (ã,q̃) in theorem 3.11, is t0 bicomplition quasi-uniform of bl-algebra (a,q). theorem 4.12. if quasi-uniform bl-algebra (a,q) is t0, then (i) (ã,q̃) is the bicompletion of (a,q). (ii) (ã,t(q̃)) is a topological bl-algebra. (iii) a is a sub bl-algebra of ã. (iv) (ã,t(q̃∗)) is a topological bl-algebra. proof. (i) by theorem 3.11 and lemma 2.18, (ã,q̃) is an unique t0-bicompletion quasi-uniform of (a,q) and the mapping i : a → ã by i(x) = {w ⊆ a : w is a t(q∗) − neighborhood of x} is a quasi-uniform embedded and clt(q∗)i(a) = ã. (ii) it is clear that t(q̃) = {s ⊆ ã : ∀g ∈ s ∃f ∈f s.t f̃?(g) ⊆ s}. let • ∈ {∧,∨,�} and •̃ ∈ {f,g,}}. we have to prove that for each g,h∈ ã, f̃?(g)•̃f̃?(h) ⊆ f̃?(g•̃h). let g1 ∈ f̃?(g) and h1 ∈ f̃?(h). then, there are g ∈ g, g1 ∈ g1, h ∈ h and h1 ∈ h1 such that g × g1 ⊆ f?, h×h1 ⊆ f?. by proposition 3.7, s1 = {f∗? (g•h) : g ∈g,h ∈h,f ∈f} and s2 = {f∗? (g1 • h1) : g1 ∈ g1,h1 ∈ h1,f ∈ f} are bases of g•̃h and g1•̃h1, respectively. we show that g1•̃h1 ∈ f̃?(g•̃h). for this, it is enough to show that f∗? (g • h) × f∗? (g1 • h1) ⊆ f?. let (y,y1) ∈ f∗? (g • h) × f∗? (g1 • h1) ⊆ f?. then, there are g ∈ g, g1 ∈ g1, h ∈ h and h1 ∈ h1 such that y ≡f g • h and y1 ≡f g1 • h1. by (b17), (b18) and (b19), we have (g1 → g) � (h1 → h) ≤ (g1 • h1) → (g • h). it follows from (g,g1) ∈ g×g1 ⊆ f? and (h,h1) ∈ h×h1 ⊆ f? that g1 → g and h1 → h are in f . hence g1 •h1 → g •h ∈ f. therefore, y1 → y ∈ f and so (y,y1) ∈ f?. thus we proved that f̃?(g)•̃f̃?(h) ⊆ f̃?(g•̃h). (iii) let • ∈ {∧,∨,�,→}, •̃ ∈ {f,g,}, ↪→} and a,b ∈ a. we shall prove 24 quasi-uniformity on bl-algebras that i(a)•̃i(b) = i(a • b). by proposition 3.7, the set s = {f∗? (wa • wb) : f ∈ f, wa,wb are t(q∗) −neighborhoods of a,b} is a base for i(a)•̃i(b). since f∗? (a•b) ⊆ f∗? (wa •wb) and f∗? (a•b) ∈ i(a•b), we deduce that filter i(a)•̃i(b) is contained in the filter i(a•b). since they are minimal q∗-cauchy filters, i(a)•̃i(b) = i(a• b). hence a is a sub-bl-algebra of ã. (iv) by lemma 2.18, q̃∗ = (q̃)∗. hence t(q̃∗) = {s ⊆ ã : ∀g ∈ s ∃f ∈f s.t f̃∗? (g) ⊆ s}. we prove that (ã,t(q̃∗)) is a topological bl-algebra. let •∈ {∧,∨,�,→} and •̃ ∈ {f,g,}, ↪→} and let g•̃h∈ f̃∗? (g•̃h). we show that f̃∗? (g)•̃f̃∗? (h) ⊆ f̃∗? (g•̃h). let g1 ∈ f̃∗? (g) and h1 ∈ f̃∗? (h). then, there are g ∈ g, g1 ∈ g1, h ∈ h and h1 ∈ h1 such that g × g1 ⊆ f∗? and h × h1 ⊆ f∗? . by proposition 3.7, f∗? (g1 • h1) ∈ g1•̃h1 and f∗? (g • h) ∈ g•̃h. we have to prove that g1•̃h1 ∈ f̃∗? (g•̃h). for this, it is enough to show that f∗? (g•h) ×f∗? (g1 •h1) ⊆ f∗? . let y ∈ f∗? (g•h) and y1 ∈ f∗? (g1 •h1). then y ≡f g • h and y1 ≡f g1 • h1 for some g ∈ g, g1 ∈ g1, h ∈ h and h1 ∈ h1. since (g,g1), (h,h1) are in f∗? , we get g • h ≡f g1 • h1. hence (y,y1) ∈ f∗? . 5 conclusions the aim of this paper is to study in [2] and [4] we study (semi)topological bl-algebras and metrizability on bl-algebras. we showed that continuity the operations � and → imply continuity ∧ and ∨. also, we found some conditions under which a locally compact topological bl-algebra become metrizable. but in there we can not answer some questions, for example: (i) is there a topology u on bl-algera a such that (a,u) be a (semi)topological bl-algebra? (ii) is there a topology u on a bl-algebra a such that (a,u) be a compact connected topological bl-algebra? (iii) is there a topological bl-algebra (a,u) such that t0,t1 and t2 spaces be equivalent? (iv) if (a,→,u) is a semitopological bl-algebra, is there a topology v coarsere than u or finer than u such that (a,v) be a (semi)topological bl-algebra? now in this paper, we answered to some above questions and got some interesting results as mentioned in abstract. 25 r. a. borzooei, n. kouhestan references [1] a. arhangel’skii, m. tkachenko, topological groups and related structures, atlantis press, 2008. [2] r. a. borzooei, g. r. rezaei, n. kouhestani, on (semi)topological blalgebra, iranian journal of mathematical sciences and informatics 6(1) (2011), 59-77. [3] r. a. borzooei, g. r. rezaei, n. kouhestani, metrizability on (semi)topological bl-algebra, soft computing, 16 (2012), 1681-1690. [4] r. a. borzooei, g. r. rezaei, n. kouhestani, separation axioms in (semi)topological quotient bl-algebras, soft computing, 16 (2012), 12191227. [5] n. bourbaki, elements of mathematics general topology, addison-wesley publishing company, 1966. 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[24] a. weil, sur les espaces a structure uniforme et sur la topologiebgeneral, gauthier-villars, paris, 1973. 27 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica 29 (2015) 15-24 issn: 1592-7415 15 application of point method in risk evaluation for railway transport oľga becherová 1 university of defence, fml, kounicova 65/662 10, brno, czech republic filarskaolga@yahoo.com abstract the paper is dealing with risk assessment affecting the hazardous substances shipping by rail; there are identified and assessed risks during the work process. the point method is applied to evaluate how serious risks are. in conclusion, there are suggested particular measures to reduce or eliminate the risks. the main priority of the system should consist in providing a safe workplace, or minimizing and eliminating undesirable factors. keywords: transport, accident, emergency, hazardous substance, railway, risks assessment doi: 10.23755/rm.v29i1.19 1 introduction safety belongs to basic prerequisites in the transport process; therefore, the emergence of rail accidents as well as emergencies cannot be passed over in the transport process particularly in cases of shipping hazardous substances. every responsible person involved in transporting hazardous substances is obliged to comply with the relevant rules and regulations so that risks could be prevented as much as possible. there are a number of methods able to anticipate and mitigate the impacts of accidents. all of these methods follow their purpose and are limited by restrictions. this paper is presenting the point method application. the risk assessment is a highly complex process considering various criteria. having mailto:filarskaolga@yahoo.com http://dx.doi.org/10.23755/rm.v29i1.19 becherová o. 16 identified threatening sources of risks and factors, assessment and subsequent managing risk can follow. 2 current situation occurrence and consequences of emergencies and accidents is a worldwide problem. an accident is such an activity of transport participants occurring in case of conflict with legal standards and regulations. there is an incorrect movement of means of transport, interaction with one another or collision with other traffic participants with consequences resulting in damage, destruction or deterioration of means, vehicles, communications and further damage. this fact is accompanied by damage to health or fatalities caused to participants of accidents. [1] thorough cooperation of stakeholders as well as institutions can support significantly the smooth railway operation. therefore, it becomes necessary to evaluate the situation and take measures while considering both the complex and partial situation solution processes. the available statistical data characterized the situations as follows: in the czech republic, a total of 1,100 accidents with 1,083 fatalities happened on the railways within the period 2006-2014. table 1 number of rail accidents in the czech republic and number of relevant casualties [2] year number of accidents number of casualties 2006 233 141 2007 115 126 2008 133 183 2009 113 118 2010 125 155 2011 99 103 2012 97 92 2013 91 76 2014 104 89 application of point method in risk evaluation for railway transport 17 graph 1 number of rail accidents in the czech republic [2] graf 2 number of relevant casualties in the czech republic [2] although the shipment by rail seems comparatively safe, it is not entirely without risk. the accidents occurrence is affected by aspects such as human factor, technical condition of the train, technical condition of railway superstructure natural conditions as well as the transported goods. the risk and affects are much higher in case of shipping hazardous substances. 2.1 risk assessment nowadays, there are high requirements for performance and work effort of employees; they dominate the threat resulting in working environment safety. employers often do not realize that safe workplace can improve the quality of the entire work process. considering all the factors affecting the safe working environment is the basis for risks assessment in the work process. risk analysis is a method for identifying and assessing factors, which may threaten individual activities and objectives of the organization. we can use it for the risks identification, to which the enterprise is exposed to in terms of external and internal perspectives. it is based on identification of risks factors, developing scenarios, assessing the likelihood and consequences, and, finally, 0 100 200 300 number of rail accidents situation of rail accidents 2006 2007 2008 2009 2010 2011 2012 2013 2014 0 200 number of casualties situation of casualties 2006 2007 2008 2009 2010 2011 2012 2013 2014 becherová o. 18 financial costs, in case that the emergency occurs. it is the basis for risk management and prevention of crisis situations in the enterprise. [2] the point method, extended risk definition, was selected to assess risk in our case. the point method is classified as one of most frequently used methods for risks assessment. the level of risk is expressed by combining the value of the likelihood of risks, possible consequence and the effect of the occupational safety and health (osh); having assessed, it is assigned to the relevant group of final risk. this method is focused on the protecting human life. r (risk) = p (probability) x d (consequence) x v (effect of osh level), [3] p – probability establishes the option estimation that the undesirable event occurs. it is expressed by assigning specific numbers 1 5 (table 2), d – consequence expresses the seriousness of the consequence of the emergency occurrence; it is defined by five stages with assigned values from 1 to 5 (table 3), v – osh level impact: this parameter comprises consideration of management level, the time of action period of threats, staff qualification, work ethic, the level of prevention, condition and age of technical equipment, maintenance level, the effect of work environment, workplace detachability, etc. (table 4). point value verbal expression 1 improbable 2 random 3 probable 4 highly probable 5 permanent table 2 probability estimation [4] table 3 consequence estimation [4] point value verbal expression 1 negligible effect on probability and injury consequences 2 little effect on probability and injury consequences 3 considerable effect 4 significant, big effect 5 more significant effects application of point method in risk evaluation for railway transport 19 table 4 osh impact estimation [4] risk – final indicator, which is the product of all three parameters of the risk value. the lowest value can reach 1 and the highest 125. according to point range, the risk is classified into five categories. (table 4). table 5 final risk range [4] point value verbal expression 1 damage to health and work activity 2 injury followed by sick leave 3 more serious injury resulting in hospitalization 4 severe occupational injury with permanent consequences 5 fatal occupational injury risk risk category point range safety assessment safety measures requirement negligibl e i 1-4 acceptable safety taking measures not required moderate ii 5-10 acceptable risk at increased attention system is classified as safe; improvement can be achieved, redress can be planned critical iii 11-50 risk cannot be accepted without taking protective measures safety measures should be taken undesira ble iv 51-100 inadequate safety, high possibility of injuries immediate corrective measures or short-term measures have to be taken unaccept able v 100-125 dangerous system, permanent threat of injury immediate cessation of activity, exclusion from operation becherová o. 20 2 point method application while transporting hazardous substances by rail the carriage of hazardous substances by rail accounts for a significant share of total rail freight. emergencies as well as accidents occur at shipping process resulting from hazardous substances characteristics. number and scope of rail accidents is affected by many factors, which can be called causes resulting in consequences of various extents. each hazardous substance has its characteristics, according to which the material should be packed, loaded and stowed, shipped via adequate route and unloaded. the employees are frequently a significant element at giving rise to an accident: it is caused by activities, either intentional non-compliance with regulations and rules or by ignorance. these accidents affect the smooth flow of work process and shipping hazardous substances and threaten the very persons involved as well as people around. they may also affect significantly the property of residents within the accident as well as the environment (soil and water contamination, air toxic pollution). therefore, all the time it is necessary to inspect and train the staff being focused on preventing accidents. the following table highlights the possible threats, which might arise during the rail-transport work process. number responsible action profession possible threat due to non-compliance with regulations 1. goods loading loader goods loading, which may react together, omitting the tank failure (rupture), failure to comply with test date, improper packaging (certified package of i, ii, iii group) and goods labelling, damage to goods, incorrectly completed waybill, inappropriate use of wagon for the particular goods 2. tank labelling shipper assigning wrong un code, excessive number of pieces, overload, assigning improper parameters to a particular category of hazardous substances, different data in waybill in terms of labelling tanks/wagons or particular content 3. tank cleaning tank cleaner failure to comply with the rules on safety equipment, sparking at a soiled tank threat to health state of an employee (fatality) application of point method in risk evaluation for railway transport 21 4. train dispatch conductor and chief guard faulty inspection of the train formation and brakes, improper connecting and disconnecting rail vehicles 5. maintenance of train-set and tanks train maintenance worker tank leaks, unclosed dome lid, cracks, bulging, violent damage, improper securing the bottom valve, missing protective caps, blind fastening screws (leakage of hazardous substance, fire, explosion) 6. track maintenance track engineer neglected maintenance of tracks, sliding rails, not removing snow, icing, vegetation, outdated track (derailment) 7. security devices inspection railway transport workerspecialist (shunter, train dispatcher, switchman, switch supervisor, signalman, announcer, levelcrossing operator) improper position of sliding rail/derailer, forgotten stop (derailment), faulty signalling (collision with a car, person) 8. shipping process engine driver demanding route (steep descent, sharp bends), collisions with objects, cars, people, gases leakage into the environment 9. loading inspection security advisor improper purchase of vehicles, faulty testing of means of transport, inadequately trained staff table 6 application of point method for expressing threat and risk identification becherová o. 22 table 7 results of point method 4 proposal of measures to reduce risks risks presented in table 6, to a greater or lesser extent, affect the occurrence of accidents and emergencies. knowledge of possible threats can result in taking measures, which might encourage risk reduction or elimination. rail transport brings risks of different levels. some risks are determined by illegal action of a third party (terroristic attack, criminality); therefore, these threats cannot be controlled properly. list of threats resulting from the assessment of risks in terms of transporting hazardous substances by rail: rigorous assessment of the particular goods characteristics and safe loading, modernization and inspection of used wagons and security devices, improvement and checking used packaging/containers, inspection of proper filling and pumping tanks, thorough inspection of labelling and marking wagons, data checking in a waybill and wagon labelling/marking, observing number of loaded units, not overloading wagons, applying adequate protective equipment and compliance with regulations at tank cleaning, number risk value p x d x v risk category 1. 2 x 2 x 3 12 iii 2. 3 x 3 x 2 18 iii 3. 2 x 4 x 5 40 iii 4. 3 x 3 x 2 18 iii 5. 4 x 5 x 4 80 iv 6. 2 x 3 x 2 12 iii 7. 2 x 3 x 4 24 iii 8. 2 x 3 x 3 12 iii 9. 2 x 2 x 1 4 i application of point method in risk evaluation for railway transport 23 regular and complex inspection of the technical condition of the train, tanks and brakes, observing time-period checks, rigorous track inspection, tracks modernization, regular removing obstacles from the railway track (vegetation, snow, icing), weather forecasting and thorough evaluation of transport options, assessing and selecting route that is appropriate for shipping, goods inspection while transported, timely reporting in case of a terrorist attack or other unlawful entry of a third party, assessment and investigation of accidents and their causes so that recurrence of accidents due to same causes could be avoided, proper planning of work process, responsible performing work by employees, creating friendly work environment by superiors, regular training: acquainting employees with risks, which might affect their work, work process knowledgeability, knowledge and compliance with relevant legislation, compliance with osh, knowledge to provide first aid help. 5 conclusions nowadays, risk identification belongs to a significant and inseparable prevention component leading to higher quality and safer working environment. the point method application does not have to provide objective assessment, and final risk determination does not result in accurate values. however, its benefit consists in identification of risks, which threaten the smooth transport by rail. risks assessment results are highly significant for taking suggested measures encouraging occupational health. becherová o. 24 bibliography [[1] hečko, i. (1999) teória a prax služby dopravnej polície, akadémia policajného zboru, bratislava isbn 80-8054-125-6. [2] štatistika eurostat. [3] hollá, k. (2008) vybrané metódy a techniky využívané v procese identifikácie a analýzy rizík. risk-management.cz, issn 1802-0496. [4] seňová, a., antošová, m. (2007) hodnotenie rizík možného ohrozenia bezpečnosti a zdravia zamestnancov ako súčasť kvality pracovného života v podniku. in: manažment v teórii a praxi, roč. 3, č.1-2, issn1336-7137. [5] zákon č.124/2006 z. z. v znení zákona 309/2007 z. z. o bezpečnosti a ochrane zdravia pri práci. [6] rosická, z. (2007) trained and educated employees – crucial assets to an organization. krízový manažment, žilinská univerzita, fakulta špeciálneho inžinierstva. issn 1336-0019. microsoft word cap2.doc approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica issn: 1592-7415 vol. 34, 2018, pp. 49-65 eissn: 2282-8214 49 a new provably secure cryptosystem using dedekind domain direct product approach amir hassani karbasi1 received: 27-02-2018. accepted: 01-06-2018. published: 30-06-2018 doi: 10.23755/rm.v34i0.404 ©amir hassani karbasi abstract we would like to prevent, detect, and protect communication and information systems' attacks, which include unauthorized reading of a message of file and traffic analysis or active attacks, such as modification of messages or files, and denial of service by providing cryptographic techniques. if we prove that an encryption algorithm is based on mathematical np-hard problems, we can prove its security. in this paper, we present a new ntru-like public-key cryptosystem with security provably based on the worst-case hardness of the approximate lattice problems (np-hard problems) in some structured lattices (ideal lattices) in order to attain the applicable objectives of preserving the confidentiality of communication and information system resources (includes hardware, software, firmware, information/data, and telecommunications). our proposed scheme is an improvement of etru cryptosystem. etru is an ntru-like public-key cryptosystem based on the eisenstein integers 1 department of mathematics, university of guilan, rasht, iran. karbasi@phd.guilan.ac.ir amir hassani karbasi 50 where is a primitive cube root of unity. etru has heuristic security and it has no proof of security. we show that our cryptosystem has security stronger than that of etru, over cartesian product of dedekind domains and extended cyclotomic polynomials. we prove the security for our main algorithm from the r-sis and r-lwe problems as np-hard problems. keywords: lattice-based cryptography; ideal lattices; etru; provable security; dedekind domain. 2010 subject classification: 94a60; 11t71; 14g50; 68p25. 1. introduction public-key cryptography has many exciting applications for web browsers, e-mail programs, cell phones, bank cards, rfid tags, smart cards, government communications, banking systems, and so on. the users to communicate over non-secure channels without any prior communication can use public-key cryptography. the idea of public-key cryptography was first proposed by diffie and hellman in 1976 [1]. lattice-based cryptography as a field of public-key cryptography has attracted considerable interest and it has been categorized into post-quantum cryptography [6]. lattice-based cryptography enjoys efficient implementations, very strong security proofs based on worst-case hardness, as well as great simplicity. our focus here will be mainly on the theoretical aspects of lattice-based cryptography. the ntru cryptosystem which is a famous lattice-based crypto scheme devised by hoffstein, pipher and silverman, was first presented at the crypto’96 rump session [2]. although its structure relies on arithmetic over the quotient polynomial ring [ ]/ 1 n q x x − z for n prime and q a small integer, it was quickly shown that breaking it could be reflected as a problem over euclidean lattices [3]. at the ants’98 conference, the ntru authors presented an improved variant including a thorough assessment of its practical security against lattice attacks [4]. the ntru cryptosystem standard number and version is ieee p1363.1 [5]. the ntru encryption (ntruencrypt) system is also often considered as the most practical post-quantum public-key crypto scheme [6] and this scheme uses the properties of structured lattices to achieve high efficiency but its security remains heuristic and it was an important open challenge to provide a provably secure scheme with comparable efficiency. for example, an 8-dimensional lattice in 2d view is shown in figure 1. by rising number of attacks and practical variants of ntru, provable security in lattice-based cryptography is developed. the first provably secure lattice-based cryptosystem and its variant of gapsvp in arbitrary lattices were presented by ajtai and dwork [8, 9]. ajtai’s average-case problem is now reflected to as the small integer solution problem (sis). another major a new provably secure cryptosystem using dedekind domain direct product approach 51 achievement in this field was the introduction in 2005 of the learning with errors problem (lwe) by regev [13]. micciancio [10] presented an alternative based on the worst-case hardness of the restriction of poly(n)-svp to cyclic lattices and succeeded in restricting sis to structured matrices while preserving a worst-case to average-case reduction, which correspond to ideals in polynomial ring [ ]/ 1 n x x − z . subsequently, lyubashevsky and micciancio [11] and independently peikert and rosen [12] showed how to modify micciancio's function to construct an efficient and provably secure collision resistant hash function. so, they introduced the more general class of ideal lattices, which correspond to ideals in polynomial rings [ ]/ q x   z with a  that is irreducible cyclotomic polynomial, also is sparse (e.g., 1 n x = + for n a power of 2). their system relies on the hardness of the restriction of poly(n)svp to ideal lattices (called poly(n)-ideal-svp). the average-case collisionfinding problem is a natural computational problem called ideal-sis, which has been reflected to be as hard as the worst-case instances of ideal-svp. in 2011, stehlé and steinfeld [14] proposed a structured variant of the ntru, which they proved as hard as cpa security from the hardness of a variant of r-sis and rlwe (ring learning with errors problem). r-lwe has great efficiency and provides more natural and flexible cryptographic constructions. the current paper was motivated by [14], in which the integers were replaced with the ring of cartesian product of eisenstein integers. figure 1. an 8-dimensional lattice in 2d view. the etru is obtained from the ntru by replacing z with the ring of eisenstein integers [7]. it is faster and has smaller size of keys for the same or better level of security than that of ntru. both division algorithm for eisenstein integers and the choice of lattice embedding are integral, thus significantly improving their efficiency over the complex-valued versions amir hassani karbasi 52 proposed in [15]. note that the etru security is based on both svp and then cvp so its security remains heuristic. the other author's lattice-based schemes are [20 – 28] which are suitable for application to wsns and iot [29-31]. in this paper, our proposed cryptosystem based on extended ideal lattices over 3 3 : ( [ ] [ ])[ ]/r xz z= ´ < f >z z (for 1 (1,1,1,1) (1,1,1,1) ... (1,1,1,1) (1,1,1,1) nn x x x −  = + + + +  with n+1 a prime) exploits the properties of the etru structured lattice to achieve high efficiency and it has ind-cpa security based on ideal lattices with established hardness of r-sis and r-lwe problems. we prove that our modification of etru is provably secure, assuming the quantum hardness of standard worst-case problems over extended ideal lattices. the rest of this paper is structured as follows: in section 2, we shortly review the etru system and explain the security related to the computational problems. in section 3, we study ideal lattices, r-sis and r-lwe problems. in section 4, we suggest a key generation algorithm, where the generated public key follows a distribution for which ideal-svp reduces to r-lwe. in section 5, we make our modified etru cryptosystem as secure as worst-case problems over ideal lattices. finally, the paper concludes in section 6. 2. etru cryptosystem 2.1. parameters creation we denote by 3  a complex cube root of unity, that is 3 3 1 = where 3 31 / 2( 1 )i = − + since 3 3 2 3 33 1 ( 1)( 1) 0   − = − + + = , we have 2 3 3 1 0 + + = and hence 2 3 3 1 = − − . the ring of eisenstein integers, denoted 3 [ ]z , is the set of complex numbers of the form 3 a b = + with ,a b  z . for 3 a b = + we will define 22 ( )d a b ab = = + − which is the square of the usual analytic complex norm | | . note that ( )d  is a positive integer for 0  since ( )d  is the square of a norm and ,a b  z . for any complex numbers ,  we have that | | | | . | |  = hence it follows that ( ) ( ). ( )d d d  = . the eisenstein integers 3 [ ]z form a lattice in  generated by the basis 3 {1, }b = . note that the two basis vectors 1 and 3  , represented by the vectors (1, 0) and ( 1 / 2, 3 / 2)− in 2, have 120 degrees with equal length. let  be an eisenstein integer. we define the ideal 3 ( ) { }| ,l a b a b  = +  z . therefore ( )l  is a lattice generated by the basis 3 { , }  . according to [7], we deduce that the eisenstein integers are an euclidean domain that the units and eisenstein primes exist. for each matrix b with entries that are eisenstein integers we will set < b > to be the 2n by 2n matrix. we choose an prime n and set 3 [ 1, ]/ n r x x= − z , we also choose p a new provably secure cryptosystem using dedekind domain direct product approach 53 and q in 3 [ ]z relatively prime, with |q| much larger than |p|. since each etru coefficient is a pair of integers, an element of etru at degree n is comparable with an element of ntru of degree n' = 2n. 2.2. key generation private key consists of two randomly chosen polynomials f, g in r. we define the inverses fq = f -1 in rq and fp = f -1 in rp. hence public key is generated by h = fq * g. the public key h is a polynomial with n coefficients which are reduced modulo q. each coefficient consists of two integers which by theorem 3 in [7] can be stored as binary strings of length 2 log (4 | | /3)q   , hence the size of the etru public key is 2 2 log (4 | | /3)k n q=    . an ntru public key, corresponding to polynomials with n' = 2n coefficients reduced modulo an integer q', has size 2 ' ' log ( ')k n q=    . therefore to maintain the same key size as ntru with n' = 2n and q' = 2k , we should choose | | (3 / 4) 'q q so that 2 2 log (4 | | /3) log ( ')q q       . 2.3. encryption each encryption requires the user to compute * mod e ph m q= + where m is a plaintext and  is a ephemeral key. in total one counts 22 ' ' ~ 4 2n n n n+ + operations for ntru encryption at ' ~ 2n n in contrast to only 2 3 27n n+ operations for etru encryption. 2.4. decryption each decryption requires the user to compute both * mod a f e q= and * mod p m f a p= . for decryption, we have 22 2 ' 2 ' ~ 8 4n n n n++ operations for ntru and only 2 6 29n n+ operations for etru. 2.5. decryption failure and security in [7] is shown that in fact | |~ (3 / 8) 'q q is an optimal choice in view of security against decryption failure and lattice attacks. based on this choice the public key size for etru will be smaller than that of the ntru public key. 3. ideal lattices and their hard problems our results are restricted to the sequence of rings 3 3 : ( [ ] [ ])[ ]/r xz z= ´ < f >z z with 1(1,1,1,1) (1,1,1,1) ... (1,1,1,1) (1,1,1,1) nn x x x −  = + + + +  where n+1 is a prime amir hassani karbasi 54 that  is irreducible cyclotomic polynomial. we can refer to [19] for irreducibility of cyclotomic polynomials n f in 3 [ ][ ]xzz where n is prime in 3 [ ]zz the r-lwe problem is known to be hard when  is cyclotomic [16]. the security analysis for our proposed scheme allows encrypting and decrypting ( )n plaintext bits for ( )o n bit operations, while achieving security against ( ) 2 g n -time attacks, for any g(n) that is (log )n and o(n), assuming the worst-case hardness of poly(n)-ideal-svp against ( ( )) 2 o g n -time quantum algorithms for each element component-wise in complex pair-wise system because note that each polynomial in r has its coefficients of the form 3 3 (( , ), ( , )) i i i i a b c dz z (ai, bi 3 ) where , , , i i ii a b c d  z , so in this paper, all operations execute for ai's, bi's, ci's and di's separately, that is,    2 component-wise. the latter assumption is believed to be valid for any g(n)=o(n). also we can define £ and ³ as poset orders. 3.1. notation similar to [14] we denote by 1 2 3 4( , , , ) 1 2 3 4 ( ), , ,x x x x      (respectively 1 2 3 4( , , , )     ) the standard n-dimensional gaussian function (respectively distribution) with center (0,0,0,0) and variance 1 2 3 4 ( ), , ,    . we denote by ( )exp  the exponential distribution on  4n with mean  and by u(e) the uniform distribution over a finite set e . if d1 and d2 are two distributions on discrete oracle e, their statistical distance is 2 1 21 1 2 3 4 1 2 3 4 ( ; ) 1 / 2 | ( ) ( ) |, , , , , , x e d d d x x x x d x x x x   = − . we write z d when the random variable z is chosen from the distribution d. the integer n is called the lattice dimension. note that in our proposed scheme with pairwise components and coefficients in vectors, the dimension increases four times without increasing n. the minimum 1 ( )l (respectively 1 ( )l  ) is the euclidean (respectively infinity) norm of any shortest vector of l \ (0,0,0,0). the dual of lattice l is the lattice 1 2 3 4 4 4 1 2 3 4 1 2 3 4 ˆ {( ) }: , ( ), (, , , , , , , , , ) i i i i n l c c c c r c c c ci b b b b=    z where the bij’s are a basis of l. for a lattice l, 1 2 3 4 ( ) (0, 0, 0, 0), , ,     and (c1,c2,c3,c4) 4n, we define the lattice gaussian distribution of support l, deviation 1 2 3 4 ( ), , ,    and center (c1,c2,c3,c4) by 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 ( , , , ),( , , , ) ( , , , ),( , , , )1 2 3 4 1 2 3 4,( , , , ),( , , , ) 1 2 3 4 1 2 3 4 ( ) ( ) / ( ), , , , , , c c c c c c c cl c c c c d b b b b b b b b l             = , for any 1 2 3 4 ( ), , ,b b b b l . we extend the definition of 1 2 3 4 1 2 3 4,( , , , ),( , , , )l c c c c d     to any m l (not necessarily a sub-lattice), by setting a new provably secure cryptosystem using dedekind domain direct product approach 55 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 ( , , , ),( , , , ) ( , , , ),( , , , )1 2 3 4 1 2 3 4,( , , , ),( , , , ) 1 2 3 4 1 2 3 4 ( ) ( ( )) / ( ( )), , , , , , c c c c c c c cm c c c c d b b b b b b b b m             = and for 1 2 3 4 ( ) (0, 0, 0, 0), , ,     , we denote the smoothing parameter 1 2 3 4( , , , ) ( )l      as the smallest 1 2 3 4 ( ) (0, 0, 0, 0), , ,     such that 1 2 3 4(1,1,1,1)/( , , , ) 1 2 3 4 ˆ( \ (0, 0, 0, 0)) ( ), , ,l          . it quantifies how large 1 2 3 4 ( ), , ,    needs to be for 1 2 4 1 2 3 4,( , , 3, ),( , , , )l c c c c d     to behave like a continuous gaussian. we will typically consider 2 n i  − = . 3.2. definition let n+1 be a prime and 1(1,1,1,1) (1,1,1,1) ... (1,1,1,1) (1,1,1,1) nn x x x −  = + + + +  which is irreducible over 3 3 [ ] [ ]z z´q q also let 3 3 : ( [ ] [ ])[ ]/r xz z= ´ < f >z z . an (integral) ideal i of r is a subset of r closed under addition and multiplication by arbitrary elements of r. by mapping polynomials to the vectors of their coefficients, we see that an ideal (0, 0, 0, 0)i  corresponds to a full-rank sublattice of 4n. thus we can view i as both a lattice and an ideal. an ideal lattice for  is a sub-lattice of (*)2n that corresponds to a non-zero ideal i r . the algebraic norm n(i) is equal to det i, where i is regarded as a lattice. in the following, an ideal lattice will implicitly refer to a  -ideal lattice. by restricting svp (respectively  -svp) to instances that are ideal lattices, we obtain ideal-svp (respectively  -ideal-svp). the latter is implicitly parameterized by the polynomial 1 (1,1,1,1) (1,1,1,1) ... (1,1,1,1) (1,1,1,1) nn x x x −  = + + + +  . no algorithm is known to perform non-negligibly better for  -ideal-svp than for  -svp [14]. 3.3. properties of the ring of cartesian product for 1 2 3 4( ), , ,v v v v r we define by ||(v1, v2, v3, v4)|| its euclidean norm. we denote the multiplicative expansion factor by , 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 ( ) max (|| ( ) ( ) ||) / (|| ( ) || . || ( ) ||), , , , , , , , , , , , u v ri i r u u u u v v v v u u u u v v v v  =  . since  is the n+1-th cyclotomic polynomial, the ring r is exactly the maximal order of the cyclotomic field 3 3( [ ] [ ])[ ]: [ , '] x k z z z z ´ = @ f q q q . we denote by 1 2 3 4 ( ), , , i i i i i n      the complex embeddings. we can choose 2 1 2 1 2 1 2 1 1 2 3 4 1 2 3 4 ( : ( ), , , ) , , , i i i i i i i i k k        + + + + → for i n . lemma 3.1. the norm of  as an element in  3 ( ) is a2 + b2 ab. this is also 2 | | , where is denoted as an element of . amir hassani karbasi 56 proof. the minimal polynomial of 3  over  is the cyclotomic polynomial 2 3 1x x = + + . thus, there exist exactly two monomorphisms (isomorphisms in this case) from  to  fixing  and permuting the roots of 3  . since 3  has two roots 3  and 2 3  , the embeddings are 3 31 ( )a b a b  + = + and 3 2 2 3 ( )a b a b  + = + , where ,a b . by definition, the algebraic norm of 3 a b = + is 2 2 3 1 3 ( ) ( ) ( ) ( )( ) n a b a b        = = + + note that 3 2 3  = and 33 1  = −+ . so we have 2 2 3 3 3 3 2 2 ( ) ( )( ) ( ) n a b a b a b ab a b ab     = + + = + + + = + − now we show that 2 ( ) ( ) | |d n  = = . 2 3 2 2 2 2 2 2 1 3 2 3 2 2 | | | | | ( ) | | | 3 2 2 b b a b i a b i a b b a a b ab   − + = + = + = − + = − + = + −            □ in rest of the paper, all of computations are done component-wise for each complex element as an integer. we define t2-norm by 2 2 2 22 1 1 2 2 3 3 4 42 1 2 3 4 ( ) ( | ( ) | | ( ) | | ( ) | | ( ) |, , , , , , ) i i i i i n i n i n i n t                 =     . we also use the fact that for any 1 2 3 4( ), , , r     , we have |n 1 2 3 4( ), , ,    | = det < 1 2 3 4 ( ), , ,    >, where < 1 2 3 4( ), , ,    > is the ideal of r generated by 1 2 3 4 ( ), , ,    . let (q1, q2, q3, q4) be a prime element such that  has n distinct linear factors modulo (q1, q2, q3, q4), that is, 1 2 3 41 2 3 4 1 2 3 4 ( (( ) ( )) mod ( ), , , , , , , , , i i i i i n x x x x q q q q      = − where i 's are primitive n+1-th root of unity modulo (q1, q2, q3, q4) component-wise. also we know that 1 2 3 4 ( , , , ) 1 2 3 4 / ( ) / ( ) / ( ) / ( ) q q q q r r q r r q r r q r r q r= ´ ´ ´ . a new provably secure cryptosystem using dedekind domain direct product approach 57 3.4. adaptation of ideal lattice problems definition 3.1. the ring small integer solution problem with parameters 1 2 3 4 1 2 3 4 ( , , , ), , ( , , , ),q q q q m b b b b f is: given m polynomials 11 21 31 41 1 1 1 1 ( , , , ), ..., ( , , , ) m m m m a a a a a a a a chosen uniformly and independently in 1 2 3 4( , , , )q q q q r , find 1 2 3 4 ( , , , )t t t t in assumed r-module such that 1 2 3 4 1 2 3 4 || ( , , , ) || ( , , , )t t t t b b b b£ . in [14] is shown that r-sis and r-lwe are dual. for 1 2 3 41 2 3 4 ( , , , ) ( ), , , q q q q s s s s r and 1 2 3 4( ), , ,    some distributions in 1 2 3 4( , , , )q q q qr , we have 1 2 3 4 1 2 3 4( , , , ),( , , , )s s s s a     as the distribution obtained by sampling the pair 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 (( ), ( )( ) ( )), , , , , , , , , , , ,a a a a a a a a s s s s e e e e+ with 1 2 3 41 2 3 4 1 2 3 4 ( , , , ) 1 2 3 4 (( ), ( )) ( ) ( ), , , , , , , , , q q q q a a a a e e e e u r      . the ring learning with errors problem (r-lwe) was introduced by lyubashevsky et al.[16] and shown hard for specific error distributions  . the error distributions 1 2 3 4( ), , ,    that we use are an adaptation of those introduced in [16]. definition 3.2. 1 2 3 4 1 2 3 4( , , , ),( , , , ) ( ) q q q q r lwe      − : let 1 2 3 41 2 3 4 ( , , , ) ( ), , ,           and ( )1 2 3 41 2 3 4 , , , ( ) ( ), , , q q q q s s s s u r where 1 2 3 4( , , , )     is a family of distributions. given access to an oracle o that produces samples in 1 2 3 4 ( , , , )1 2 3 4( , , , ) q q q qq q q q r r , distinguish whether o outputs samples from 1 2 3 4 1 2 3 4( , , , ),( , , , )s s s s a     or from 1 2 3 4 ( , , , )1 2 3 4( , , , ) ( ) q q q qq q q q u r r . the distinguishing advantage should be ( ) 1 / ( ) ( . 2 ) o n poly n resp − over the randomness of the input, the randomness of the samples and the internal randomness of the algorithm, component-wise [14]. theorem 1 in [14] indicates that r-lwe is hard, assuming that the worstcase  -ideal-svp cannot be efficiently solved using quantum computers, for small  . it was recently improved by lyubashevsky et al. [18] if the number of samples that can be chosen to the oracle o is bounded by a constant (which is the case in our application), then the result also holds with simpler errors than 1 2 3 41 2 3 4 1 2 3 4 ( , , , ) ( ) ( ), , , , , ,e e e e           , and with an even smaller ideal-svp approximation factor  . this should allow to both simplify the proposed scheme and to strengthen its security guarantee. 3.5. our proposed variants of r-lwe for 1 2 3 41 2 3 4 ( , , , ) ( ), , , q q q q s s s s r and 1 2 3 4 ( ), , ,    some distributions in 1 2 3 4 ( , , , )q q q q r , we denote 1 2 3 4 1 2 3 4, , , , , ,( ),( )s s s s a      as the distribution obtained by sampling amir hassani karbasi 58 the pair 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4(( ), ( )( ) ( )), , , , , , , , , , , ,a a a a a a a a s s s s e e e e+ with 1 2 3 41 2 3 4 1 2 3 4 ( , , , ) 1 2 3 4 (( ), ( )) ( ) ( ), , , , , , , , , q q q q a a a a e e e e u r        , where 1 2 3 4( , , , )q q q q r  is the set of invertible elements of 1 2 3 4( , , , )q q q q r . this variant is hard and called r lwe  ++ − as [14]. furthermore, as explained in [18], the nonce 1 2 3 4( , , , )s s s s can also be sampled from the error distribution without incurring any security loss. we call this variant hnf r lwe  ++ − . according to adaptation of lemmas 7, 8 and 9 as well as theorem 2 in [14] the problems r lwe  ++ − and hnf r lwe  ++ − are dual to  -ideal-svp and are defined some families of r-modules for i, an arbitrary ideal of 1 2 3 4( , , , )q q q q r as a lattice, also short vectors exist in ideal and statistical distance (regularity bound) is exactly appropriate and reliable. 4. the proposed key generation algorithm we now use the results of the previous section on modular ideal lattice to derive a key generation algorithm for the etru for each component in vectors, where the generated public key follows a distribution for which ideal-svp reduces to r-lwe. algorithm 1 is as follows. input: 1 2 3 41 2 3 4 1 2 3 4 ( , , , ) 1 2 3 4 , , , ( ), , , , , , , , , q q q q n q q q q p p p p r        z r . output: 1 2 3 4( , , , ) a key pair ( , ) q q q q sk pk r r    . sample 1 2 3 4 ( , , , ) 'f f f f from 4 1 2 3 4,( , , , ) nd    z ; let 1 2 3 4 1 2 3 4 1 2 3 4 ( , ) ( , ).( , ) ' (1,1,1,1), , , , , ,f f f f p p p p f f f f= + ; if ( 1 2 3 4 ( , , , )f f f f mod 1 2 3 4 ( , , , )q q q q ) 1 2 3 4( , , , )q q q q r   , resample. sample 1 2 3 4 ( , , , )g g g g from 4 1 2 3 4,( , , , ) nd    z ; if ( 1 2 3 4 ( , , , )g g g g mod 1 2 3 4 ( , , , )q q q q ) 1 2 3 4( , , , )q q q q r   , resample. return secret key sk = 1 2 3 4 ( , , , )f f f f and public key pk = 1 2 3 4 ( , , , )h h h h = 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 ( , , , ) ( , , , )( , , , ) / ( , , , ) q q q q p p p p g g g g f f f f r ´ î . the following theorem ensures that for some appropriate choice of parameters, the key generation algorithm terminates in expected polynomial time. theorem 4.1[adapted from 14]. let 8n  and n+1 be a prime such that 1 (1,1,1,1) (1,1,1,1) ... (1,1,1,1) (1,1,1,1) nn x x x −  = + + + +  splits into n linear factors modulo prime 1 2 3 4( ) (5, 5, 5, 5), , ,q q q q  component-wise. let a new provably secure cryptosystem using dedekind domain direct product approach 59 1/ ln(2 (1 1 / )) / . n i i i n n q   + , or an arbitrary (0,1 / 2)i  . let 1 2 3 4( , ), ,a a a a r and 1 2 3 41 2 3 4 ( , , , ) ( , ), , q q q q p p p p r   then component-wise. the following lemma ensures that the generated secret key is small. lemma 4.1[adapted from 14]. let 8n  and n+1 be a prime such that 1 (1,1,1,1) (1,1,1,1) ... (1,1,1,1) (1,1,1,1) nn x x x −  = + + + +  splits into n linear factors modulo prime 1 2 3 4( ) (8, 8, 8, 8), , ,q q q q n . let 1/ 2 ln(6 ) / . n i i n n q  . the secret key polynomials 1 2 3 4 1 2 3 4 ( , , , ), ( , , , )f f f f g g g g returned by the algorithm 1 satisfy, with probability 3 1 2 n− +  − : 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 || ( , ) || (2, 2, 2, 2) || ( , ) || ( ) || ( , ) || ( ) , , , , , , , , , , , ,f f f f n p p p p and g g g g n         . if deg 1 2 3 4 ( , , , )p p p p (1,1,1,1) , then 1 2 3 4 1 2 3 4 1 2 3 4 || ( , ) || (4, 4, 4, 4) || ( , ) || ( ), , , , , , ,f f f f n p p p p     with probability 3 1 2 n− +  − component-wise. theorem 3 in [14] shows that the public key can be uniformly distributed in the whole ring and this satisfy cryptographic pseudo randomness for our algorithm 1, which seems necessary for exploiting the established hardness of r-lwe (and r-sis). now we can construct the proposed cryptosystem over ideal lattices with high efficiency and provable security (cpa-secure). 5. the proposed new cryptosystem using our new results above, we describe our proposed cryptosystem for which we can provide a security proof under a worst-case hardness assumption. 5.1. decryption failure the correctness condition for each pairwise coefficient in the proposed cryptosystem is as follows. lemma 5.1 [adapted from 14]. if 21.5 ( log ) deg(( )) || ( ) || (1,1,1,1) i i i i n n p p    (resp. 20.5 ( log ) || ( ) || (1,1,1,1) deg( ) (1,1,1,1) i i i i n n p if p     ) and 0.5 i i q n  , then the decryption algorithm of the proposed cryptosystem recovers 1 2 3 4 ( , , , )m m m m with probability (1) 1 n − − over the choice of si, ei, fi and gi component-wise. amir hassani karbasi 60 proof. in the decryption algorithm, we have and let computed in r (not modulo 1 2 3 4 ( , , , )q q q q ). if 1 2 3 4 1 2 3 4 || ( , ) " || ( ) / 2, , , , ,c c c c q q q q   then we have 1 2 3 4 1 2 3 4 ( , ) ' ( , ) ", , , ,c c c c c c c c= in r and hence, since ( ) (1,1,1,1) mod ( ), ( ) ' mod ( ) ( ) " mod ( ) ( ) mod ( ) i i i i i i i i f p c p c p m p = = , i.e., the decryption algorithm succeeds. it thus suffices to give an upper bound on the probability that 1 2 3 4 1 2 3 4|| ( , ) " || ( ) / 2, , , , ,c c c c q q q q  . from lemma 2, we know that with probability 3 1 2 n− +  − both 1 2 3 4 ( , , , )f f f f and 1 2 3 4 ( , , , )g g g g have euclidean norms 2 || ( ) || ( . (4, 4, 4, 4) || ( ) || deg( ) (1,1,1,1)) i i i i i n p resp n p if p   this implies that, 2 21.5|| ( )( ) ||, || ( )( ) || (2, 2, 2, 2) || ( ) || ( . (8, 8, 8, 8) || ( ) || )n i i i i i i i i p f p g n p resp p  with probability 3 1 2 n− +  − . from lemma 6 in [14], both 1 2 3 4 1 2 3 4 1 2 3 4 ( , , , )( , , , )( , , , )p p p p f f f f e e e e and 1 2 3 4 1 2 3 4 1 2 3 4 ( , , , )( , , , )( , , , )p p p p g g g g s s s s have infinity norm (resp. 21.5 (2, 2, 2, 2) (log ). || ( ) || i i i i q n n p   2 (8, 8, 8, 8) (log ). || ( ) || i i i i q n n p   ), with probability (1) 1 n − − . independently: a new provably secure cryptosystem using dedekind domain direct product approach 61 proposed encryption scheme parameters creation: 1. we use 1 (1,1,1,1) (1,1,1,1) ... (1,1,1,1) (1,1,1,1) nn x x x −  = + + + +  with 8n  and n+1 a prime, 3 3 : ( [ ] [ ])[ ]/r xz z= ´ < f >z z and 1 2 3 4( , , , ) 1 2 3 4 / ( ) / ( ) / ( ) / ( ) q q q q r r q r r q r r q r r q r=    with 1 2 3 4 ( ) (5, 5, 5, 5), , ,q q q q  prime such that 1 n k k  =  =  in 1 2 3 4( , , , )q q q q r with distinct k  's component-wise. key generation: 2. we use the algorithm 1 and return 1 2 3 41 2 3 4 ( , , , ) ( , ), , q q q q sk f f f f r  =  with 1 2 3 4 1 2 3 4 ( , ) (1,1,1,1) mod ( , ), , , ,f f f f p p p p , and pk = 1 2 3 4 ( , , , )h h h h = 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 ( , , , ) ( , , , )( , , , ) / ( , , , ) q q q q p p p p g g g g f f f f r ´ î , component-wise. encryption: 3. given message 1 2 3 4( , ), ,m m m m p , set 1 2 3 41 2 3 4 1 2 3 4 ( , , , ) ( ), ( ), , , , , ,s s s s e e e e       and return ciphertext 1 2 3 41 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 ( , , , ) ( , ) ( , )( ) ( , )( ) ( , ), , , , , , , , , , , , , , q q q q c c c c h h h h s s s s p p p p e e e e m m m m r= + +  decryption: 4. given ciphertext 1 2 3 4 ( , , , )c c c c and secret key 1 2 3 4 ( , , , )f f f f , compute 1 2 3 41 2 3 4 1 2 3 4 1 2 3 4 ( , , , ) ( , ) ' ( , ).( , ), , , , , , q q q q c c c c f f f f c c c c r=  and return 1 2 3 4 ( , , , ) 'c c c c mod 1 2 3 4 ( , , , )p p p p . 2 2 || ( )( ) || || ( )( ) || || ( ) || . || ( ) || (2, 2, 2, 2).(deg( ) (1,1,1,1). || ( ) || i i i i i i i i i nf m f m f m p n p      + (resp. 2 (8, 8, 8, 8) || ( ) || i i n p  ). since i iq n  , we conclude that 1.5 2 || ( ) " || ((6, 6, 6, 6) (2, 2, 2, 2) deg( )). (log ). || ( ) || i i i i i i c p q n n p     + (resp. 20.5 (24, 24, 24, 24) (log ). || ( ) || i i i i q n n p   ), with probability (1) 1 n − − , component-wise. □ amir hassani karbasi 62 5.2. security the security of the proposed cryptosystem follows by an elementary reduction from the decisional hnf r lwe  ++ − , exploiting the uniformity of the public key in 1 2 3 4( , , , )q q q q r  (adaptation of theorem 3 in [14]), and the invertibility of 1 2 3 4 ( , , , )p p p p in 1 2 3 4( , , , )q q q q r . lemma 5.2 [adapted from 7]. suppose n+1 is a prime such that 1 (1,1,1,1) (1,1,1,1) ... (1,1,1,1) (1,1,1,1) nn x x x −  = + + + +  splits into n linear factors modulo prime (1)iq = . let (1/ 2,1/ 2,1/ 2,1/ 2) (2, 2, 2, 2) ln(8 ). i i i i n nq q   +  and 1 2 3 41 2 3 4 1 2 3 4 1 2 3 4 ( , , , ) ( ), ( ) (0, 0, 0, 0), ( ), , , , , , , , , q q q q p p p p r           . if there exists an ind-cpa attack against the proposed cryptosystem that runs in time t and has success probability (1 / 2,1 / 2,1 / 2,1 / 2) i+ with parameters  i and qi, then there exists an algorithm solving hnf r lwe  ++ − that runs in time t' = t + o(n) and has success probability ( ) ' n i i i q  − = − . proof. let a denote the given ind-cpa attack algorithm. we construct an algorithm b against hnf r lwe  ++ − that runs as follows, given oracle o that samples from either ( ) qi iq u r r   or ,i is a   for some previously chosen i is  and ii     .algorithm b first calls o to get a sample ((hi)', (ci)') from qi iq r r   . then, algorithm b runs a with public key ( ) ( ).( ) ' ii i i q h p h r=  . when a outputs challenge messages 10 ( , () ) ii m m p , algorithm b picks ({0,1})b u , computes the challenge ciphertext ( ) ( ).( ) ' ( ) qii i i bi c p c m r= +  , and returns (ci) to a. eventually, when a outputs its guess b' for b, algorithm b outputs 1 if b' = b and 0 otherwise. the (hi)' used by b is uniformly random in iq r  and therefore so is the public key (hi) given to a, thanks to the invertibility of (pi) modulo (qi). thus, by theorem 3 in [14], the public key given to a is within statistical distance ( )n q − of the public key distribution in the genuine attack, component-wise. moreover, since ( ) ' ( ).i i i ic h s e= + with ,i i is e  , the ciphertext (ci) given to a has the right distribution as in the ind-cpa attack. overall, if o outputs samples from ,i is a   then a succeeds and b returns 1 with probability ( ) (1 / 2,1 / 2,1 / 2,1 / 2) n i i q −  + − . now, if o outputs samples from ( ) qi iq u r r   , then, since ii q p r   , the value of (pi)(ci)' and hence (ci), is uniformly random in rqi and independent of b. it follows that b outputs 1 with probability 1/2, component-wise. the claimed advantage of b follows. □ a new provably secure cryptosystem using dedekind domain direct product approach 63 by combining lemmata 3 and 4 (with adaptation of theorem 1 in [14]) we obtain main result. 6. conclusions in this paper, we provided a new cryptosystem that uses the properties of the etru cryptosystem and its structured lattice to achieve high efficiency by providing a provable security (cpa-secure) based on ideal lattices and a variant of r-lwe problem. also we showed that each polynomial in 1 3 3 ( [ ] [ ])[ ]/ (1,1,1,1) (1,1,1,1) ... 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[31] s. tahouri, r.e. atani, a.h. karbasi, and y. deldjou, "application of connected dominating sets in wildfire detection based on wireless sensor networks," international journal of information technology, communications and convergence, (2015), 3, 2, 139-160. microsoft word capitolo intero n 2.doc ratio mathematica volume 47, 2023 weak and weak* ik-convergence in normed spaces amar kumar banerjee* mahendranath paul† abstract the main object of this paper is to study the concept of weak ik convergence, a generalization of weak i∗-convergence of sequences in a normed space, introducing the idea of weak* ik -convergence of sequences of functionals where i, k are two ideals on n, the set of all positive integers. also we study the ideas of weak ik and weak* ik -limit points to investigate the properties in the same space. keywords: weak ik -convergence, weak* ik -convergence, condition ap(i, k), weak ik -limit points, weak* ik -limit points. 2020 ams subject classifications: 40a35; 40h05. 1 *department of mathematics, the university of burdwan, purba burdwan -713104, w.b., india; akbanerjee1971@gmail.com, akbanerjee@math.buruniv.ac.in. †department of mathematics, the university of burdwan, purba burdwan -713104, w.b., india; mahendrabktpp@gmail.com. 1received on october 19, 2022. accepted on june 18, 2023. published on june 30, 2023. doi: 10.23755/rm.v39i0.882. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 309 a. k. banerjee, m. paul 1 introduction the idea of statistical convergence, an extended form of ordinary convergence, based on the concept of natural density of subsets of n, was introduced independently by steinhaus [27] and by fast [15]. over the years and under different forms of statistical convergence turned out to be one of the most active research areas in the summability theory after the remarkable works of friday [16, 17] and salat [21, 26]. in 2000, cannor et al.[10] introduced the idea of weak statistical convergence which has been used to characterize banach spaces with seperable duals. last few years some basic properties of this concept were studied by many authors in [8, 24]. recently the concept of weak* statistical convergence of sequence of functionals has been given by bala [1]. in 2001, kostyrko et al. [18] extended the idea of statistical convergence into i and i∗-convergence which depends on the structure of the ideals i of n. the mutual relation between i and i∗-convergence was given in [19] using the condition ap of the ideals.(such ideals are often called p-ideals [2]). later many works on ideal convergence have been done in [3, 4, 5, 11, 12, 13]. in 2010, pehlivan et al.[25] introduced the idea of weak i and i∗-convergence in a normed space and using the condition ap, they established a relation between such types convergence. in 2012, bhardwaj et al. [9] extended the idea of weak* statistical convergence to weak* ideal convergence of sequence of functionals and shew that the ideas of weak ideal convergence and weak* ideal convergence are identical in a reflexive banach space. in 2010, macaj et al. [22] introduced the idea of ik -convergence which is a generalization of all types of i∗-convergence. they have shown that if the ideal i has additive property with respect to an another ideal k (i.e. if condition ap(i, k) holds) then i-convergence implies ik -convergence. very recently more results and applications of ik -convergence have been carried out [23, 7, 10]. it seems therefore reasonable to think if we extend the idea of weak and weak* convergence using double ideals in a normed space and in that case we intend to investigate how far several basic properties are affected. in our paper, we have studied the idea of weak ik -convergence of sequences which is a generalization of weak i∗-convergent sequences as defined in [25] and we have presented an interrelation between weak i and weak ik -convergence using the condition ap(i, k). next we have introduced the concept of weak* ik convergence for a sequence of functionals and observed that the ideas of weak and weak* ik -convergence of sequences of functionals are same in a reflexive banach space. in the last section of this paper we have discussed the notion of weak ik limit point of sequences and weak* ik -limit points of sequences of functionals. since the importance of the notion of weak and weak* convergence in functional analysis is very significant, we have realized that the ideas of weak and weak* ik 310 weak and weak* ik -convergence in normed spaces convergence in a normed space give more general frame for functional analysis to study summability theory as well. 2 basic definitions and notations throughout the paper, we use n to denote the set of all positive integers and x for a normed linear space and x∗ for dual of x. first recall that a subset a of n is said to have natural density d(a) if d(a) = lim n 1 n n∑ k=1 χa(k), provided the limit exists where χa is characteristic function of a ⊂ n. definition 2.1. [15] a sequence {xn} in x is said to be statistically convergent to l if for every ϵ > 0 the set k(ϵ) = {k ∈ n : ||xk − l|| ≥ ϵ} has natural density zero. definition 2.2. [10] let x be a normed linear space then a sequence {xn}n∈n in x is said to be weak statistically convergent to x ∈ x provided that for any f ∈ x∗, the sequence {f(xn − x)}n∈n is statistically convergent to 0. in this case we write w-stlim n→∞ xn = x. definition 2.3. let s be a non empty set and a class i ⊂ 2s of subset of s is said to an ideal if (i)a, b ∈ i implies a ∪ b ∈ i and (ii)a ∈ i, b ⊂ a implies b ∈ i. i is said nontrivial ideal if s /∈ i and i ̸= {ϕ}. in view of condition (ii) ϕ ∈ i. if i ⫋ 2s we say that i is proper ideal on s. a nontrivial ideal i is said admissible if it contains all the singletons of s. a nontrivial ideal i is said non-admissible if it is not admissible. definition 2.4. let f be a class of subsets of non-empty set s. then f is said to be a filter in s if (i) ϕ /∈ f , (ii) a, b ∈ f implies a ∩ b ∈ f and (iii) a ∈ f, a ⊂ b implies b ∈ f . if i is a non-trivial on a non-void set s then f = f(i) = {a ⊂ s : s\a ∈ i} is clearly a filter on s and conversely. again f(i) is said associated filter with respect to ideal i. definition 2.5. [18] a sequence {xn}n∈n in x is said to be i-convergent to x if for any ϵ > 0 the set a(ϵ) = {n ∈ n : ||xn − x|| ≥ ϵ} ∈ i. in this case we write i − lim n→∞ xn = x. 311 a. k. banerjee, m. paul definition 2.6. [25] a sequence {xn}n∈n in x is said to be weak i-convergent to x ∈ x if for any ϵ > 0 and for any f ∈ x∗ the set a(f, ϵ) = {n ∈ n : |f(xn) − f(x)| ≥ ϵ} ∈ i. in this case we write w − i − lim n→∞ xn = x. note 2.1. it is easy to observe that weak i-limit of a weak i-convergent sequence is unique and moreover for an admissible ideal i, weak convergence implies weak i-convergence with the same limit point but converse part is not true which has been shown in paper [25] by an interesting examples. note 2.2. it is obvious that if two ideals i1, i2 on n such that i1 ⊆ i2 then for a sequence {xn} w-i1-lim xn = x implies w-i2-lim xn = x. 3 weak ik-convergence we have already mentioned that our aim to generalize the notion of weak i∗convergence of sequences. we need to modify this definition introduced in [25]. definition 3.1. (cf.[25]) a sequence {xn}n∈n in x is said to be weak i∗-convergent to x ∈ x if there exists a set m ∈ f(i) such that the sequence {yn}n∈n ⊂ x defined by yn = { xn if n ∈ m x if n /∈ m is weak-convergent to x. we denote it by the notation w-i∗-lim xn = x. definition 3.2. (cf.[22]) let i, k be two ideals on the set n. a sequence {xn}n∈n in x is said to be weak ik -convergent to x ∈ x if there exists a set m ∈ f(i) such that the sequence {yn}n∈n ⊂ x defined by yn = { xn if n ∈ m x if n /∈ m is weak k-convergent to x. we denote it by the notation w-ik -lim xn = x. remark 3.1. we can give an equivalent definition of weak-ik -convergence in the following way: if there exists an m ∈ f(i) such that the sequence {xn}n∈m is weak-k|m -convergent to x where k|m = {b ∩ m : b ∈ k}. lemma 3.1. if i and k are ideals on n, the set of all positive integers and if {xn}n∈n is a sequence in x such that w-k-lim{xn} = x, then w-ik -lim{xn} = x. the proof follows from the definition of weak k-convergence taking m = n ∈ f(i) and yn = xn. 312 weak and weak* ik -convergence in normed spaces proposition 3.1. let x be normed space and i be an ideal on n. a sequence {xn}n∈n ⊂ x is weak ii -convergent to x if and only if it is weak i-convergent to x. proof. let {xn} be weak ii -convergent to x then there exists an m ∈ f(i) such that the sequence {xn}n∈m is weak-i|m -convergent to x. so there exists g ∈ f(i) such that {n ∈ n : |f(xn) − f(x)| < ϵ} ∩ m = g ∩ m. clearly g ∩ m ∈ f(i) and {n ∈ n : |f(xn) − f(x)| < ϵ} ⊇ g ∩ m. therefore {n ∈ n : |f(xn) − f(x)| < ϵ} ∈ f(i) i.e. {xn} is weak i-convergent to x. converse part follows from lemma 3.1 taking k = i. proposition 3.2. let x be a normed space and i, i1, k and k1 be ideals on n such that i ⊆ i1 and k ⊆ k1. then for any sequence {xn}n∈n, we have (i) w-ik -lim xn = x ⇒ w-ik1 -lim xn = x and (ii) w-ik -lim xn = x ⇒ w-ik1 -lim xn = x. proof. (i) now as w-ik -lim xn = x so there exists an m ∈ f(i) such that the sequence {xn}n∈m is weak-k|m -convergent to x where k|m = {b ∩ m : b ∈ k}. here m ∈ f(i) ⊆ f(i1) as i ⊆ i1. so obviously w-ik1 -lim xn = x. (ii) again w-ik -lim xn = x then there exists a set m ∈ f(i) such that the {yn} ∈ x given by yn = { xn if n ∈ m x if n /∈ m is weak k-convergent to x. since k ⊆ k1 and from the note 2.2 we get {yn} is weak k-convergent to x. hence w-ik1 -lim xn = x. theorem 3.1. let i and k be ideals on n and {xn}n∈n be a sequence in x then (i) w-i-lim xn = x ⇒ w-ik -lim xn = x if i ⊆ k. (ii) w-ik -lim xn = x ⇒ w-i-lim xn = x if k ⊆ i. proof. (i) since {xn} is weak i-convergent to x ∈ x then for any ϵ > 0 and f ∈ x∗ the set a(f, ϵ) = {n ∈ n : |f(xn) − f(x)| ≥ ϵ} ∈ i. again i ⊆ k so a(f, ϵ) ∈ k. therefore the sequence {xn} is weak k-convergent to x. so from the lemma 3.1 we get {xn} is weak ik -convergent to x. (ii) now w-ik -lim xn = x then there exists a set m ∈ f(i) such that the sequence {yn} given by yn = { xn if n ∈ m x if n /∈ m is weak k-convergent to x. so a(f, ϵ) = {n ∈ n : |f(yn) − f(x)| ≥ ϵ} = {n ∈ n : |f(xn)−f(x)| ≥ ϵ}∩m ∈ k ⊆ i. consequently {n ∈ n : |f(xn)−f(x)| ≥ ϵ} ⊆ (n \ m) ∪ a(f, ϵ) ∈ i. so w-i-lim xn = x. 313 a. k. banerjee, m. paul note 3.1. if k ̸⊂ i and i ̸⊂ k then none of these implications in theorem3.1 may not be true. to support this we cite an example which is weak-ik -convergence but not weak-i-convergence. example 3.1. let i and k be two ideals on n such that k ̸⊂ i and i ̸⊂ k, but k ∩ i ̸= ϕ. let x, y ∈ x,x ̸= y and consider a set m ∈ k \ i. let us now consider a sequence {xn} with define by xn = { x if n /∈ m y if n ∈ m then for every ϵ > 0 and f ∈ x∗ we get {n ∈ n : |f(xn)−f(x)| ≥ ϵ} ⊂ m ∈ k. so w − k − lim xn = x. but, since x − y ̸= θ so from hanh banach theorem there exist a f ∈ x∗ such that f(x − y) = ||x − y||. choose an ϵ = ||x−y|| 2 . then {n ∈ n : |f(xn) − f(x)| ≥ ϵ}={n ∈ m : |f(xn) − f(x)| ≥ ϵ} ∪ {n ∈ n \ m : |f(xn) − f(x)| ≥ ϵ} = {n ∈ m : |f(y) − f(x)| ≥ ||x−y|| 2 } = {n ∈ m : ||x − y|| ≥ ||x−y|| 2 }m /∈ i. so w − i − lim xn ̸= x. note 3.2. consider any two ideals i and k on n then we can construct a new ideal i ∨ k = {a ∪ b : a ∈ i, b ∈ k} containing both i, k.the dual filter of i ∨k is f(i ∨k) = {g∩h : g ∈ f(i), h ∈ f(k)}, when i ∨k is non-trivial. it should be noted that if i ∨ k is non-trivial ideal and i, k are proper subsets of i ∨ k then both i and k are non-trivial. but converse part may or may not be true always. to establish this, following examples are given. example 3.2. let the two sets p = {5p : p ∈ n} and s = {5s − 1 : s ∈ n} now it is clear that 2p , 2s and 2p ∨ 2s all ideals are non-trivial on n. example 3.3. now let p be set of all odd integers and s be set of all even integers. then i = 2p , k = 2s both are non-trivial on the whole set n but i ∨ k is not a non-trivial ideal on n. theorem 3.2. if i ∨k is non-trivial ideal on n and x is normed space then weak ik -limit of a sequence {xn}n∈n in x is unique. proof. if possible let sequence {xn}n∈n has two distinct weak ik -limits say x and y. since x ̸= y i.e. (x − y) ̸= θ then by a consequence of hahn banach theorem there exists f such that f(x − y) = ||x − y|| ̸= θ then f(x) ̸= f(y) and let ϵ = |f(x)−f(y)| 3 > 0. since {xn}n∈n has weak ik -limit x then there exists a set a1 ∈ f(i) such that the {yn} ∈ x given by yn = { xn if n ∈ a1 x if n /∈ a1 314 weak and weak* ik -convergence in normed spaces is weak k-convergent to x. so,{n ∈ n : |f(yn) − f(x)| ≥ ϵ} ∈ k i.e. {n ∈ n : |f(yn) − f(x)| < ϵ} ∈ f(k) which implies that {n ∈ a1 : |f(yn) − f(x)| < ϵ} ∪ {n ∈ n \ a1 : |f(yn) − f(x)| < ϵ} ∈ f(k) i.e. (n \ a1) ∪ {n ∈ a1 : |f(yn)−f(x)| < ϵ} ∈ f(k) i.e. n\(a1\{n ∈ a1 : |f(yn)−f(x)| < ϵ}) ∈ f(k) so a1 \ b1 ∈ k where b1 = {n ∈ a1 : |f(xn) − f(x)| < ϵ}. similarly as {xn} has weak ik -limit y, so there exists a set a2 ∈ f(i) such that a2 \ b2 ∈ k where b2 = {n ∈ a2 : |f(xn) − f(y)| < ϵ}. so, (a1 \ b1) ∪ (a2 \ b2) ∈ k then (a1∩a2)∩(b1∩b2)c ⊂ (a1∩bc1)∪(a2∩bc2) ∈ k. thus (a1∩a2)∩(b1∩b2)c ∈ k i.e. (a1 ∩ a2) \ (b1 ∩ b2) ∈ k. now by our construction we get b1 ∩ b2 = ϕ. for if b1∩b2 ̸= ϕ, let n ∈ b1∩b2 then |f(xn)−f(x)| < ϵ and |f(xn)−f(y)| < ϵ. therefore, 3ϵ = |f(x) − f(y)| ≤ |f(x) − f(xn)| + |f(xn) − f(y)| < 2ϵ, which is a contradiction. so a1 ∩ a2 ∈ k i.e. n \ (a1 ∩ a2) ∈ f(k) −→ (i). since a1, a2 ∈ f(i) so a1 ∩ a2 ∈ f(i) −→ (ii). since i ∨ k is non-trivial so the dual filter f(i ∨ k) exits. now from (i) and (ii) we get ϕ ∈ f(i ∨ k), which is a contradiction. hence the weak ik -limit is unique. theorem 3.3. let x be normed space and i, k be two ideals on n. a sequence {xn}n∈n ∈ x is weak ik -convergent to x if and only if it is weak (i ∨ k)k convergent to x. proof. suppose that {xn} is weak ik -convergent to x then there exists an m ∈ f(i) such that the sequence {xn}n∈m is weak-k|m -convergent to x. since m ∈ f(i) so it is clear that m ∈ f(i ∨ k). therefore {xn} is also weak (i ∨ k)k convergent to x. conversely, let {xn} is weak (i ∨ k)k -convergent to x then there exists an m ∈ f(i ∨ k) such that the sequence {xn}n∈m is weak-k|m -convergent to x. so for any ϵ(> 0) and for every f ∈ x∗ there exists g ∈ f(k) such that a(f, ϵ) ∩ m = g ∩ m where a(f, ϵ) = n ∈ n : |f(xn) − f(x)| < ϵ. since m ∈ f(i ∨ k) then m = m1 ∩ m2 for some m1 ∈ f(i) and m2 ∈ f(k). now we have a(f, ϵ) ∩ m1 ⊇ a(f, ϵ) ∩ m = (g ∩ m2) ∩ m1. since g ∩ m2 ∈ f(k), this shows that a(f, ϵ) ∩ m1 ∈ f(k|m1) i.e. {xn} is weak ik -convergent to x. in the rest of this section, using additive property of ideals we will investigate the relationship between weak-i and ik -convergence. now we recall the definition of k-pseudo intersection and then ap(i, k)-condition. definition 3.3. [21] let k be an ideal on n. we denote a ⊂k b whenever a \ b ∈ k. if a ⊂k b and b ⊂k a then we denote a ∼k b. clearly a ∼k b ⇔ a △ b ∈ k. if a ⊂k an holds for each n ∈ n then we say that a set a is k-pseudo intersection of a system {an : n ∈ n}. 315 a. k. banerjee, m. paul definition 3.4. [21] let i, k be ideals on the set x. we say that i has additive property with respect to k or that the condition ap(i, k) holds if any one of the equivalent condition of following holds: (a) for every sequence (an)n∈n of sets from i there is a ∈ i such that an ⊂k a for every n′s. (b) any sequence (fn)n∈n of sets from f(i) has k-pseudo intersection in f(i). (c) for every sequence (an)n∈n of sets from the ideal i there exists a sequence (bn)n∈n ⊂ i such that aj ∼k bj for j ∈ n and b = ∪j∈nbj ∈ i. (d) for every sequence of mutually disjoint sets (an)n∈n ⊂ i there exists a sequence (bn)n∈n ⊂ i such that aj ∼k bj for j ∈ n and b = ∪j∈nbj ∈ i. (e) for every non-decreasing sequence a1 ⊆ a2 ⊆ · · · ⊆ an · · · of sets from i ∃ a sequence (bn)n∈n ⊂ i such that aj ∼k bj for j ∈ n and b = ∪j∈nbj ∈ i. (f) in the boolean algebra 2s/k the ideal i corresponds to a σ-directed subset, i.e. every countable subset has an upper bound. note that the proof of the conditions (a) to (f) in the definition 3.4 are equivalent has been given in [21][lemma 3.9]. above definition is reformulation of the definition given below: definition 3.5. [14] let i, k be ideals on the non-empty set s. we say that i has additive property with respect to k or that the condition ap(i, k) holds if for every sequence of pairwise disjoint sets an ∈ i, there exists a sequence bn ∈ i such that an △ bn ∈ k for each n and ∪n∈nbn ∈ i. theorem 3.4. if the condition ap(i, k) holds then weak-i-convergence implies weak-ik -convergence, where i, k are two ideals on n. proof. let {xn} be weak i-convergent sequence to x ∈ x. let f ∈ x∗ and choose a sequence of rationals {ϵi : i ∈ n} so that {(f(x)−ϵi, f(x)+ϵi) : i ∈ n} be a countable base for r at the point f(x). by weak i-convergence of {xn} we have bi = {n : |f(xn) − f(x)| < ϵi} ∈ f(i) for each i, thus by definition 3.4(b) there exists a set a ∈ f(i) with a ⊂k bi i.e. a \ bi ∈ k for all i’s. now it suffices to show that the sequence {yn} ∈ x given by yn = { xn if n ∈ a x if n /∈ a is weak k-convergent to x. now {n ∈ n : |f(yn) − f(x)| < ϵi} = {n ∈ a : |f(yn) − f(x)| < ϵi} ∪ {n ∈ n \ a : |f(yn) − f(x)| < ϵi} = (n \ a) ∪ {n ∈ a : |f(xn) − f(x)| < ϵi} = (n \ a) ∪ (bi ∩ a) = n \ (a \ bi). as a \ bi ∈ k then n \ (a \ bi) ∈ f(k). thus {n ∈ n : |f(yn) − f(x)| < ϵi} ∈ f(k) for each i and every f ∈ x∗. thus {yn} is weak k-convergent to x. hence {xn} is weak ik -convergent to x. 316 weak and weak* ik -convergence in normed spaces theorem 3.5. let i, k be ideals on n. if for any sequence {xn}n∈n in x weak i-convergence implies weak ik -convergence then the condition ap(i, k) holds. proof. let {yn} be a sequence in x which is weak i-convergent to x, since x is first countable and f(x) is not isolated point in r then there exists a sequence {zn} of points from x \ {x} which weak convergent to x. let {an : n ∈ n} be a system of mutually disjoint sets from i. let us define a sequence {xn} as xn = { zj if n ∈ aj yn if n /∈ ∪aj let f ∈ x∗ be arbitrary. now {n ∈ n : |f(xn) − f(x)| ≥ ϵ} ⊂ {n ∈ n : |f(yn) − f(x)| ≥ ϵ} ∪ ∪nj=1aj implies {n ∈ n : |f(xn) − f(x)| ≥ ϵ} ∈ i. this shows that {xn} is weak i-convergent to x. by our assumption this implies {xn} is weak ik -convergent to x i.e. there exists a set m ∈ f(i) such that {xn}n∈m is weak k|m -convergent to x i.e. {n ∈ n : |f(xn) − f(x)| ≥ ϵ} ∩ m = a ∩ m for some a ∈ k. this implies that {n ∈ n : |f(xn) − f(x)| ≥ ϵ} ∩ m ∈ k. let us define bi = ai \ m we have ∪i∈nbi ⊆ n \ m ∈ i. at the same time, for the set bi △ai = ai ∩m we have ai ∩m ⊆ {n ∈ n : |f(xn)−f(x)| ≥ ϵ}∩m for any ϵ > 0. consequently bi △ ai ∈ k. hence the condition ap(i, k) holds. 4 weak* ik-convergence in this section, following bala [1] and bhardwaj et al. [9], now we introduce the concept of weak* ik -convergence of sequence of functionals and present some result. definition 4.1. [9] a sequence {fn}n∈n in x∗ is said to be weak* i-convergent to f ∈ x∗ if for any ϵ > 0 and for each x ∈ x the set a(x, ϵ) = {n ∈ n : |fn(x) − f(x)| ≥ ϵ} ∈ i. in this case we write w∗-ilim n→∞ fn = f. definition 4.2. a sequence {fn}n∈n in x∗ is said to be weak* i∗-convergent to f ∈ x∗ if there exists a set m = {m1 < m2 < ... < mk < ...} ∈ f(i) such that lim k→∞ fmk(x) = f(x) for each x ∈ x. in this case we write w ∗-i∗lim n→∞ fn = f. theorem 4.1. let x be a normed space and {fn}n∈n be a sequence in x∗. if {fn} is weak* i∗-convergent to f ∈ x∗ then it is weak* i-convergent to f. proof. by assumption, there exists a set h ∈ i such that for m = n\h = {m1 < m2 < ... < mk < ...} we have lim k→∞ fmk(x) = f(x) for each x ∈ x. now let ϵ > 0 and for this there exists an n(ϵ, x) ∈ n such that |fmk(x)−f(x)| < ϵ for each k > 317 a. k. banerjee, m. paul n(ϵ, x). then we have {n ∈ n : |fn(x) − f(x)| ≥ ϵ} ⊂ h ∪ m1, m2, ..., mn(ϵ,x). since i is an admissible ideal so right-hand side of the above relation belongs to i. hence the result. remark 4.1. we can reformulate the definition4.2 in the following way: if there exists a set m ∈ f(i) such that the sequence {gn} ∈ x∗ given by gn = { fn if n ∈ m f if n /∈ m is weak* convergent to f. definition 4.3. let x be a normed space with a separable dual x∗ and i, k be two ideals on n. a sequence {fn}n∈n in x∗ is said to be weak* ik -convergent to f ∈ x∗ if there exists a set m ∈ f(i) such that the sequence {gn} ∈ x∗ given by gn = { fn if n ∈ m f if n /∈ m is weak* k-convergent to f and we write w∗-ik lim n→∞ fn = f theorem 4.2. if i ∨ k is a non-trivial ideal on n and x is normed space with dual x∗ then weak* ik -limit of a sequence {fn}n∈n in x∗ is unique. the proof is parallel to proof of theorem3.2 with slight modification. theorem 4.3. let x be a normed space. if a sequence {fk} in x∗ is weak ik convergent to f ∈ x∗ then it is weak* ik -convergent. proof. by our assumption, w −ik −lim fk = f then there exists a set m ∈ f(i) such that the sequence {gk} ∈ x∗ given by gk = { fk if k ∈ m f if k /∈ m is weak k-convergent to f. then for every h ∈ x∗∗ and ϵ > 0, we have {k : |h(gk) − h(f)| ≥ ϵ} ∈ k. let x ∈ x and fx = c(x) where c : x → x∗∗ is the canonical mapping we have fx(gk) = gk(x) and fx(f) = f(x) for every x ∈ x. so in particular for each x ∈ x,{k : |fx(gk) − fx(f)| ≥ ϵ} ∈ k i.e. {k : |gk(x) − f(x)| ≥ ϵ} ∈ k. so the sequence {gk} is weak* k-convergent to f. hence the result. theorem 4.4. let x be a reflexive normed space with dual x∗. if a sequence {fk} in x∗ is weak* ik -convergent to f ∈ x∗ then it is weak ik -convergent to f. 318 weak and weak* ik -convergence in normed spaces proof. by our assumption, w∗−ik −lim fk = f. so there exists a set m ∈ f(i) such that the sequence {gk} ∈ x∗ given by gk = { fk if k ∈ m f if k /∈ m is weak* k-convergent to f. then for each x ∈ x and ϵ(> 0) the set {k ∈ n : |gk(x) − f(x)| ≥ ϵ} ∈ k. let f ∈ x∗∗ then f = c(x0) for some x0 ∈ x where c : x → x∗∗ is the canonical mapping. we have in particular {k ∈ n : |gk(x0) − f(x0)| ≥ ϵ} ∈ k sincef(gk) = gk(x0) and f(f) = f(x0). we have {k ∈ n : |f(gk) − f(f)| ≥ ϵ} ∈ k for each ϵ(> 0) and f ∈ x∗∗. so the sequence {gk} is weak k-convergent to f. hence the result. 5 weak and weak* ik-limit points in this last part, we introduce weak and weak* ik -limit points of sequences and sequence of functionals respectively. first we define weak i-limit point of a sequence. definition 5.1. (cf. [17]) let x be a normed space and a sequence {xn} be a sequence in x. then y ∈ x is called an weak i-limit point of {xn} if there exists a set m /∈ i such that the sequence {yn}n∈n ∈ x defined by yn = { xn if n ∈ m y if n /∈ m is weak-convergent to y. definition 5.2. let x be a normed space and i, k be two ideals on n. then y ∈ x is called an weak ik -limit point of a sequence {xn} if there exists a set m /∈ i, k such that the sequence {yn}n∈n ∈ x defined by yn = { xn if n ∈ m y if n /∈ m is weak k-convergent to y. we denote i(lw) and ik(lw) the collection of all weak i and weak ik -limit points of xn ∈ x. theorem 5.1. if k is an admissible ideal and k ⊂ i then i(lw) ⊂ ik(lw). 319 a. k. banerjee, m. paul proof. let y ∈ i(lw), so there exists a set m /∈ i such that the sequence {yn} give by yn = { xn if n ∈ m y if n /∈ m is weak-convergent to y. then the sequence of scalars {f(yn)} converges to f(y) for all f ∈ x∗ i.e. {n : |f(yn)−f(y)| ≥ ϵ} is a finite set. so {n : |f(yn)−f(y)| ≥ ϵ} ∈ k as k is an admissible ideal. therefore {yn} is weak k-convergent sequence. again m /∈ i and k ⊂ i, so m /∈ i, k. thus y is weak ik -limit point of xn. hence the theorem. in the similar way we can set the definition of weak* ik -limit points for the sequence of functionals. definition 5.3. let x be a normed space with its dual x∗ and {fn} be a sequence in x∗. then h ∈ x∗ is called an weak* i-limit point of {fn} if there exists a set m /∈ i such that the sequence {gn}n∈n ∈ x∗ defined by gn = { fn if n ∈ m h if n /∈ m is weak*-convergent to h. definition 5.4. let x be a normed space with its dual x∗ and i, k be two ideals on n. then h ∈ x∗ is called an weak* ik -limit point of {fn} ⊂ x∗ if there exists a set m /∈ i, k such that the sequence {gn}n∈n ∈ x∗ defined by gn = { fn if n ∈ m h if n /∈ m is weak* k-convergent to h. we denote i(lw∗) and ik(lw∗) the collection of all weak* i and ik -limit points of the sequence fn ∈ x∗. theorem 5.2. if k is an admissible ideal and k ⊂ i then i(lw∗) ⊂ ik(lw∗). the proof is parallel to proof of the theorem 5.1. theorem 5.3. let x be a normed space with its dual x∗ . if h ∈ x∗ be weak ik -limit point of a sequence {fn} ⊂ x∗ then h is also weak* ik -limit point. proof. let y be weak ik -limit point of {fn} ∈ x∗ then there exists a set m /∈ i such that the sequence {gn}n∈n ∈ x∗ defined by gn = { fn if n ∈ m h if n /∈ m is weak k-convergent to y. again by theorem 4.3 we get {gn} is weak* kconvergent to h. hence h is weak* ik -limit point. 320 weak and weak* ik -convergence in normed spaces remark 5.1. by the theorem 4.4 we get weak* k-convergence implies weak kconvergence when x is reflexive normed space. therefore converse of above theorem holds when x is a reflexive normed space. 6 conclusions the study of weak and weak* convergence plays an important role in functional analysis. so investigations on weak and weak* convergence using single ideal or double ideals take place a great position in the study of summability theory. in [9] and [1] weak and weak* convergence have been studied using single ideal respectively. so it quite natural to investigate whether such these results hold for double ideals. so it would be worthwhile to consider extending the notion of weak and weak* convergence by incorporating double ideals in a normed space, and to examine how various fundamental properties are impacted as a result. in case of double ideals, weak i-convergence implies weak ik -convergence if the modified ap condition i.e. ap(i, k) condition holds where as in case of single ideal ap condition is sufficient to hold this result. it is observed citing suitable example that weak ik -convergence may not imply weak i-convergence in general although weak i∗-convergence always imply weak i-convergence. besides, it is verified that where x is a reflexive banach space the idea of weak ik -convergence and weak* ik -convergence coincide. due to the inherent importance of concept of weak and weak* convergence in function analysis, we recognize that the concept of weak and weak* convergence via double ideals in normed spaces would provide a more comprehensive framework for function analysis. this extension would offer a broader scope and enhance our understanding of convergence in normed spaces. acknowledgment the authors are grateful to dst, govt. of india for providing fist project in te department of mathematics , burdwan university. the second author is grateful to the university of burdwan, w.b., india for providing him state fund fellowship during the preparation of this work. authors are grateful to the referee for their valuable comments and suggestions which have improved the quality of paper. references [1] i. bala, on weak* statistical convergence of sequences of functionals, international j. pure appl. math., 70(5) (2011), 647-653. 321 a. k. banerjee, m. paul [2] m. balcerzak, k. dems, a. komisarski, statistical convergence and ideal convergence for sequences of functions, j. math anal. appl. 328(1) 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[27] h. steinhaus, sur la convergence ordinaire et la convergence asymptotique, colloq. math. 2 (1951) 73-74. 323 ratio mathematica volume 40, 2021, pp. 27-46 integrity of generalized transformation graphs bommanahal basavanagoud∗ shruti policepatil† abstract the values of vulnerability helps the network designers to construct such a communication network which remains stable after some of its nodes or communication links are damaged. the transformation graphs considered in this paper are taken as model of the network system and it reveals that, how network can be made more stable and strong. for this purpose the new nodes are inserted in the network. this construction of new network is done by using the definition of generalized transformation graphs of a graphs. integrity is one of the best vulnerability parameter. in this paper, we investigate the integrity of generalized transformation graphs and their complements. also, we find integrity of semitotal point graph of combinations of basic graphs. finally, we characterize few graphs having equal integrity values as that of generalized transformation graphs of same structured graphs. keywords: vulnerability; connectivity; integrity; generalized transformation graphs; semitotal point graph. 2020 ams subject classifications: 05c40, 90c35.1 ∗department of mathematics, karnatak university, dharwad 580 003, karnataka, india; b.basavanagoud@gmail.com. †department of mathematics, karnatak university, dharwad 580 003, karnataka, india; shrutipatil300@gmail.com. 1received on february 25th, 2021. accepted on june 23th, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.576. issn: 1592-7415. eissn: 2282-8214. c©the authors. this paper is published under the cc-by licence agreement. 27 bommanahal basavanagoud, shruti policepatil 1 introduction the stability of a communication network composed of processing nodes and communication links are of prime importance to network designers. as the network begins losing links or nodes, eventually there will be a decrease to certain extent in its effectiveness. thus, communication networks must be constructed as stable as possible; not only with respect to the initial disruption, but also with respect to the possible reconstruction of the network. in the analysis of vulnerability of a communication network we often consider the following quantities: 1. the number of members of the largest remaining group within mutual communication can still occur, 2. the number of elements that are not functioning. the communication network can be represented as an undirected graph. consequently, a number of other parameters have recently been introduced in order to attempt to cope up with this difficulty. tree, mesh, hypercube and star graphs are popular communication networks. if we think of graph as modeling a network, there are many graph theoretical parameters used in the past to describe the stability of communication networks. most notably, the vertex-connectivity and edge-connectivity have been frequently used. the best known measure of reliability of a graph is its vertex-connectivity. the difficulty with these parameters is that they do not take into account what remains after the graph has been disconnected. to estimate these quantities, the concept of integrity was introduced by barefoot et al. in [6] as a measure of the stability of a graph. the integrity of a graph g is defined in [6] as i(g) = min s⊂v (g) {|s|+ m(g−s)}, where m(g − s) denotes the order of the largest component of g − s. in [6], the authors have compared integrity, connectivity, toughness and binding number for several classes of graphs. in 1987, barefoot et al. [7] have investigated the integrity of trees and powers of cycles. in 1988, goddard et al. [14] have obtained integrity of the join, union, product and composition of two graphs. the integrity of a small class of regular graphs was studied by atici et al. [1].the authors in [3, 20] have studied the integrity of cubic graphs. for more details on integrity of a graph refer to [2, 4, 5, 11–13, 15]. in this paper, we are concerned with nontrivial, simple, finite, undirected graphs. let g be a graph with a vertex set v (g) and an edge set e(g) such that |v (g)| = n and |e(g)| = m. the degree of a vertex dg(v) is the number of edges incident to it in g. the symbol dxe denotes the smallest integer that is greater than or equal to x and bxc denotes the greatest integer smaller than or equal to x. for undefined graph theoretic terminologies and notations refer to [16] or [17]. 28 integrity of generalized transformation graphs 2 preliminaries 2.1 basic results on integrity in this subsection, we review some of the known results about integrity of graphs. theorem 2.1. [5] the integrity of (i) complete graph kn, i(kn) = n, (ii) null graph kn, i(kn) = 1, (iii) star k1,b, i(k1,b) = 2, (iv) path pn, i(pn) = d2 √ n + 1e−2, (v) cycle cn, i(cn) = d2 √ ne−1, (vi) complete bipartite graph ka,b, i(ka,b) = 1 + min{a,b}, (vii) wheel wn, i(wn) = d2 √ n−1e. 3 generalized transformation graphs sampathkumar and chikkodimath [19] defined the semitotal-point graph t2(g) as the graph whose vertex set is v (g)∪e(g), and where two vertices are adjacent if and only if (i) they are adjacent vertices of g or (ii) one is a vertex of g and other is an edge of g incident with it. inspired by this definition, basavanagoud et al. [9] introduced some new graphical transformations. these generalize the concept of semitotal-point graph. let g = (v,e) be a graph, and let α, β be two elements of v (g) ∪ e(g). we say that the associativity of α and β is + if they are adjacent or incident in g, otherwise is −. let xy be a 2-permutation of the set {+,}. we say that α and β correspond to the first term x of xy if both α and β are in v (g), whereas α and β correspond to the second term y of xy if one of α and β is in v (g) and the other is in e(g). the generalized transformation graph gxy of g is defined on the vertex set v (g)∪e(g). two vertices α and β of gxy are joined by an edge if and only if their associativity in g is consistent with the corresponding term of xy. we denote the complement of the generalized transformation graph gxy by gxy. in view of above, one can obtain four graphical transformations of graphs, since there are four distinct 2-permutations of {+,−}. note that g++ is just the semitotalpoint graph t2(g) of g, whereas the other generalized transformation graphs are g+−, g−+ and g−−. 29 bommanahal basavanagoud, shruti policepatil b b b b bc bc bc bc bc bc bc bc b b b b b b bbbc bc bc bc bc b b b b b b b b b b b b b bc bc bc bc bc bc bc bc bc bc bc bc b b b b b b b b b b b b bb b bc bc bc bc bc bc bc bcbc bc bc bc b b b b b b b b b b b b b b b b bc bc bc bc g++ g+− g−+ g −− g++ g+− g−+ g −− b b b b g : v1 v2 v3 v4 v1 v1 v1 v1 v1v1 v1 v1 e1 e2 e3 e4 v2 v2 v2 v2 v3 v4 v3 v4v3 v4v3 v4 v2 v3 v4 v2 v2 v2 v3 v4v3 v4v3 v4 e1 e2 e3 e4 e3 e1 e4 e2 e1 e3 e4 e2 e3 e2 e1 e4 e4 e3 e1 e2 e1 e2 e3 e4 e4e3 e2 e1 e1 e2 e3 e4 figure 1: graph g, its generalized transformation gxy and their complements gxy the generalized transformation graph gxy, introduced by basavanagoud et al. [9], is a graph whose vertex set is v (g) ∪ e(g), and α,β ∈ v (gxy). the vertices α and β are adjacent in gxy if and only if (a) and (b) holds: (a) α,β ∈ v (g), α,β are adjacent in g if x = + and α,β are nonadjacent in g if x = − (b) α ∈ v (g) and β ∈ e(g), α,β are incident in g if y = + and α,β are nonincident in g if y = − an example of generalized transformation graphs and their complements are shown in figure 1. the vertex v of gxy corresponding to a vertex v of g is referred to as a point vertex. the vertex e of gxy corresponding to an edge e of g is referred to as a line vertex. for more details on generalized transformation graphs, refer to [8–10, 17–19]. 4 main results in this section, we determine the integrity of semitotal point graph(g++) of some standard families of graphs. also, the integrity of generalized transforma30 integrity of generalized transformation graphs tion graphs g+−, g−+, g−−, g++, g+−, g−+ and g−− are obtained. then, we calculate integrity of semitotal point graph of cartesian product and composition of some graphs 4.1 integrity of generalized transformation graphs theorem 4.1. for a graph pn (n ≥ 4), i(p++n ) = { d2 √ 2ne−2, if n is odd, d2 √ 2n−1e−1, if n is even. proof. let s be a subset of v (p++n ). the number of remaining components after removing |s| = r vertices is given in table 1 and table 2. case 1. suppose n is even. the number of vertices in p++n is 2n−1. if r vertices are removed from graph p++n , then one of the connected components has at least 2n−1−r r vertices. so, the order of the largest component is m(p++n −s) ≥ 2n−1−rr . so i(p++n ) ≥ min { r + 2n−1− r r } . the function r + 2n−1−r r takes its minimum value at r = √ 2n−1. if we substitute the minimum value in the function, then we have i(p++n ) = 2 √ 2n−1 − 1. since the integrity is integer valued, we round this up to get a lower bound. so the integrity of p++n is, i(p ++ n ) = d2 √ 2n−1e−1. number of removing vertices 1 2 3 ... r number of remaining components 1 2 3 ... r table 1: n is even number of removing vertices 1 2 3 ... r number of remaining components 2 3 4 ... r + 1 table 2: n is odd case 2. suppose n is odd. since the number of vertices in p++n is 2n − 1 and m(p++n −s) ≥ 2n−1−rr+1 , we have i(p ++ n ) ≥ min{r+ 2n−1−rr+1 }. after the required elementary arithmetical operations, we get i(p++n ) = d2 √ 2ne−2. 31 bommanahal basavanagoud, shruti policepatil example 4.1. consider a graph p5 and its semitotal point graph p++5 . let s = {a,b}⊂ v (p++5 ) (see figure 2) such that |s| = 2 and m(p++5 −s) = 3. so, i(p++5 ) = 5. b b bb b b b b bbc bc bc bcbc bc bc bc p ++5 : p ++5 − s : b b b b b b bc bc bcbc a b figure 2: graph p++5 −s. theorem 4.2. for a cycle cn of length n ≥ 4, i(c++n ) = { ⌈ n 2 ⌉ + 3, if n(≤ 7) is odd and n(≤ 16) is even,⌈ n 3 ⌉ + 5, if n(≥ 9) is odd and n(≥ 18) is even. proof. c++n has 2n vertices and 3n edges. let s ⊂ v (c++n ). case 1. suppose n(≤ 7) is odd and n(≥ 16) is even. choose a set s in such a way that it is an independent set of vertices of cn. it is clear that |s| = ⌈ n 2 ⌉ = β0(cn) and m(c++n −s) = 3. so, i(c++n ) = ⌈ n 2 ⌉ + 3. case 2. suppose n(≥ 9) is odd and n(≤ 18) is even. choose a set s in such a way that it is an independent set of vertices of cn having distance 3 in between them. it is clear that |s| = ⌈ n 3 ⌉ and m(c++n −s) = 5. so, i(c++n ) = ⌈ n 3 ⌉ + 5. example 4.2. consider a graph c6 and its semitotal point graph c++6 . let s = {a,b,c}⊂ v (c++6 ) (see figure 3) such that |s| = 3 and m(c++6 −s) = 3. so, i(c++6 ) = 6. theorem 4.3. for a complete graph kn of order n ≥ 2, i(k++n ) = n + 1. proof. k++n has n(n+1) 2 vertices and 3n(n−1) 2 edges. let s ⊂ v (k++n ) be a set containing all the vertices of kn. so, |s| = n. the removal of vertices from k++n leaves a totally disconnected graph with n(n−1) 2 vertices. hence, m(k++n −s) = 1. therefore, |s| + m(k++n − s) = n + 1 is minimum for above set s. then it is clear that, i(k++n ) = n + 1. 32 integrity of generalized transformation graphs b b b b b b b b b b bbbc bc bc bc bc bc c++6 : a bc b b b b b b b b b bc bc bc bc bc bc c++6 − s: figure 3: graph c++6 −s. example 4.3. consider a graph k4 and its semitotal point graph k++4 . let s = {a,b,c,d}⊂ v (k++4 ) (see figure 4). it is clear that m(k++4 −s) = 1. so, i(k++4 ) = 5. b b b b b b b b b b b k++4 : bc bc bc bc bc bc k++4 − s: b b b b b b bc bc bc bc bc bc a b c d figure 4: graph k++4 −s. corolary 4.1. i(kp) = i(k++q ) if and only if p = q + 1. theorem 4.4. for a complete bipartite graph ka,b of order a + b, i(k++a,b ) = 2min{a,b}+ 1. proof. k++a,b has 2ab vertices and 3ab edges. let us select s in such a way that it should contain minimum number of vertices among two partite sets of ka,b. so, |s| = min{a,b}. the deletion of vertices of s from k++a,b results in union of stars k1,min{a,b}. hence, m(k ++ a,b −s) = min{a,b}+1. the value of |s|+m(k++a,b −s) whose sum is minimum for chosen s. therefore, i(k++a,b ) = 2min{a,b}+1. example 4.4. consider a graph k2,3. let s = {a,b}⊂ v (k++2,3 ) (see figure 5). it is clear to write |s| = 2 and m(k++2,3 −s) = 3. so, i(k++2,3 ) = 5. corolary 4.2. i(ka1,b1) = i(k ++ a2,b2 ) if and only if min{a1,b1} = 2min{a2,b2}. k2,2 and k1,2 are the smallest graphs satisfying above condition such that i(k2,2) = i(k++1,2 ). 33 bommanahal basavanagoud, shruti policepatil b b b b b b b bb b b bc bc bc bc bc bc k++2,3 : b b b b b b b b b bc bc bc bc bc bc k++2,3 − s: a b figure 5: graph k++2,3 −s. theorem 4.5. for a star k1,b of order b + 1, i(k++1,b ) = 3. proof. k++1,b has 2b + 1 vertices and 3b edges. let s ⊂ v (k++1,b ) containing a central vertex of k1,b. so, |s| = 1. the removal of a vertex of set s from k++1,b results in graph bk2. hence, m(k ++ 1,b −s) = 2. the value |s|+ m(k++1,b −s) is minimum for the chosen s. therefore, i(k++1,b ) = 3. remark 4.1. the values of integrity of star graph and integrity of semitotal point graph of star graph are never same, since i(k1,b) = 2 and i(k ++ 1,b ) = 3. example 4.5. consider a graph k1,3. let s = {a}⊂ v (k++1,3 ) (see figure 6). it is clear to write |s| = 1 and m(k++1,3 −s) = 2 so, i(k++1,3 ) = 3 b b b b b b bbc bc bc k++1,3 : k ++ 1,3 − s: b b b b b b bc bc bc a figure 6: graph k++1,3 −s. theorem 4.6. for a wheel wn of order n ≥ 5, i(w++n ) = ⌈ n−1 2 ⌉ + 5. 34 integrity of generalized transformation graphs proof. w++n has 3n−2 vertices and 6(n−1) edges. let s ⊂ v (w++n ). case 1. suppose n is odd clearly, the order of an outer cycle of wheel is n−1, which is even. choose a set s1 in such a way that it is an independent set of vertices of cn−1. it is clear that |s1| = n−12 = β0(cn−1). case 2. suppose n is even clearly, the order of an outer cycle of wheel is n − 1, which is odd. let s2 be an independent set of vertices of cn−1 such that |s2| = n−22 . let v1 be a vertex of v (cn−1) \ s2 such that v1 is adjacent to a vertex of s2 as well as to a vertex v (cn)\s2. let us take s1 = s2 ∪{v} and hence |s1| = n2 . combining the above two cases we get, |s1| = ⌈ n−1 2 ⌉ , for all n, let v2 be a central vertex of wn. let us define a set s in such a manner that s = s1 ∪{v2}. it is to be noted that |s| = ⌈ n−1 2 ⌉ + 1. the deletion of vertices of set s from w++n gives a graph whose components are p4’s and k1’s. hence, m(w++n −s) = 4. the set s defined in this manner gives minimum value of |s|+m(w++n −s). therefore, i(w++n ) = ⌈ n−1 2 ⌉ + 5. corolary 4.3. i(wp) = i(w++q ) if and only if d2 √ p−1e = ⌈ q−1 2 ⌉ + 5. w11 and w5 are the smallest graphs which satisfy the above condition such that i(w11) = i(w ++ 5 ). example 4.6. consider a graph w7. let s = {a,b,c,d}⊂ v (w++7 ) (see figure 7). it is clear to write |s| = 4 and m(w++7 −s) = 4. so, i(w++7 ) = 8 b b b b b b b b bb b b bb b b b b b b bc bc bc bc bcbc bc bc bc bc bc bc w ++7 : w ++ 7 − s: b b b b b b b b b b b b b bb bc bc bc bc bc bc bc bc bc bc bc bc a c d b figure 7: graph w++7 −s. theorem 4.7. for a connected graph g � k1,b of order n and size m, i(g+−) = n + 1. proof. for an (n,m) graph g, g+− has n + m vertices and m(n−1) edges. the n vertices have degree m and m vertices have n− 2 in g+−. let s ⊂ v (g+−). consider a set s consisting a vertices of g+− which corresponds to the vertices of 35 bommanahal basavanagoud, shruti policepatil a graph g. then it is clear that |s| = n. the removal of the vertices of set s from g+− results in a null graph km. hence, m(g+−−s) = 1. |s|+ m(g+−−s) is minimum for above chosen s. therefore, i(g+−) = n + 1.. theorem 4.8. for a star k1,b (b ≥ 3), i(k+−1,b ) = b + 1. proof. for a star k1,b of order b+ 1 and size b, the graph k +− 1,b has 2b+ 1 vertices and b2 edges. let s ⊂ v (k+−1,b ). choose as set s in such a way that it should contain the pendant vertices of k1,b. so, |s| = b. the deletion of the vertices of set s from k+−1,b results in null graph kb+1. so, m(k +− 1,b −s) = 1. |s|+m(k+−1,b −s) is minimum for above chosen s. therefore, i(k+−1,b ) = b + 1. the column 2 and 4 of table table 3 shows integrity of basic graphs and integrity of transformation graph g+− of graphs with same structure. g i(g) g+− i(g+−) g−+ i(g−+) g−− i(g−−) p6 4 p +− 3 4 p −+ 3 4 p −− 3 4 p10 5 p +− 10 11 p −+ 10 8 p −− 10 11 c5 4 c +− 3 4 c −+ 3 4 c −− 3 4 c7 5 c +− 7 8 c −+ 7 8 c −− 7 8 kn n k +− n n + 1 k −+ n n + 1 k −− n n + 1 k5,5 6 k +− 2,3 6 k −+ 2,3 6 k −− 2,3 6 k5,6 6 k +− 5,6 12 k −+ 5,6 12 k −− 5,6 12 k1,b 2 k +− 1,b b + 1 k −+ 1,b b + 2 k −− 1,b b + 1 w8 6 w +− 5 6 w −+ 5 6 w −− 5 6 w9 6 w +− 9 10 w −+ 9 10 w −− 9 10 table 3: theorem 4.9. for a connected graph g � k1,b of order n and size m, i(g−+) = n + 1. proof. for an (n,m) graph g, g−+ has n+m vertices and n(n−1) 2 +m edges. the n vertices have degree 2 in g−+. let s ⊂ v (g−+). consider a set s consisting a vertices of g−+ which corresponds to the vertices of a graph g. then it is clear that |s| = n. the removal of the vertices of set s from g−+ results in a null graph km. hence, m(g−+ − s) = 1. the value |s| + m(g−+ − s) is minimum for above chosen s. therefore, i(g−+) = n + 1. 36 integrity of generalized transformation graphs the column 2 and 6 of table 3 shows integrity of basic graphs and integrity of transformation graph g−+ of graphs with same structure. theorem 4.10. for a star k1,b (b ≥ 3), i(k−+1,b ) = b + 1. proof. the proof is similar to that of theorem 4.8. theorem 4.11. for a connected graph g � k1,b of order n and size m, i(g−−) = n + 1. proof. for an (n,m) graph g, g−− has n + m vertices and n(n−1) 2 + m(n − 3) edges. let s ⊂ v (g−−). consider a set s consisting a vertices of g−− which corresponds to the vertices of a graph g. then it is clear that |s| = n. deleting the vertices of set s from g−− results in a null graph km. hence, m(g−−−s) = 1. |s| + m(g−− − s) is minimum for above chosen s. therefore, i(g−−) = n + 1. theorem 4.12. for a star k1,b(b ≥ 3), i(k−−1,b ) = b + 1. proof. for a star k1,b of order b+1 and size b, the graph k −− 1,b has 2b+1 vertices. let s ⊂ v (k−−1,b ). choose as set s in such a way that it should contain the pendant vertices of k1,b. so, |s| = b. the deletion of the vertices of set s from k−−1,b results in null graph kb+1. so, m(k −− 1,b − s) = 1. |s| + m(k−−1,b − s) is minimum for above chosen s. therefore, i(k−−1,b ) = b + 1. the column 2 and 8 of table 3 shows integrity of basic graphs and integrity of transformation graph g−− of graphs with same structure. theorem 4.13. for any connected graph g of order n and size m ≥ 2, i(g++) = n + m−2. proof. for a connected graph g of order n and size m ≥ 2, g++ has order n + m. let s ⊂ v (g++). consider a set s containing all the vertices and edges of g except one edge and its incident vertices. so, |s| = n + m− 3. the removal of the vertices of set s from g++ gives 3k1. hence, m(g++ −s) = 1. the value |s| + m(g++ − s) is minimum for the selected subset s. therefore, i(g++) = n + m−2.. 37 bommanahal basavanagoud, shruti policepatil theorem 4.14. for any connected graph g of order n and size m ≥ 2, i(g+−) = n + m−2. proof. for a connected graph g of order n and size m ≥ 2, g+− has order n + m. let s ⊂ v (g+−). consider a set s containing all the vertices and edges of g except one edge and two nonincident vertices(adjacent vertices). so, |s| = n + m − 3. the removal of the vertices of set s from g+− gives 3k1. hence, m(g+− − s) = 1. the value of |s| + m(g+− − s) is minimum for the selected subset s. therefore, i(g+−) = n + m−2.. corolary 4.4. the integrity of (i) path pn, i(p++n ) = i(p+−n ) = 2n−3, (ii) cycle cn, i(c++n ) = i(c+−n ) = 2n−2, (iii) complete graph kn, i(k++n ) = i(k+−n ) = n(n+1) 2 −2, (iv) complete bipartite graph ka,b, i(k ++ a,b ) = i(k +− a,b ) = a + b + ab−2, (v) star k1,b, i(k ++ 1,b ) = i(k +− 1,b ) = 2b−1, (vi) wheel wn, i(w++n ) = i(w+−n ) = 3n−4. the table 4 shows integrity of basic graphs and integrity of transformation graphs g++ and g+− of graphs with same structure. g i(g) g++ and g+− i(g++) = i(g+−) p4 3 p ++ 3 and p +− 3 3 p5 3 p ++ 5 and p +− 5 7 c5 4 c ++ 3 and c +− 3 4 c6 4 c ++ 6 and c +− 6 10 kn n k++n and k+−n n(n+1) 2 −2 k8,8 9 k ++ 2,3 and k +− 2,3 9 k8,9 9 k ++ 8,9 and k +− 8,9 87 k1,b 2 k ++ 1,b and k +− 1,b 2b−1 w27 11 w ++ 5 and w +− 5 11 w28 11 w ++ 28 and w +− 28 80 table 4: 38 integrity of generalized transformation graphs theorem 4.15. for any connected graph g of order n and size m, i(g−+) = min{n + m−1,m + i(g)}. proof. for a connected graph g of order n and size m, g−+ has order n + m. let s1 ⊂ v (g−+). choose a set s1 containing the edges of g. so, |s1| = |e(g)| = m. the removal of elements of set s1 from a graph g−+ gives a graph g. consider the value |s1|+ i(g) = m + i(g). choose a set s2 ⊂ v (g−+) consisting of all the elements of g−+ except an edge and two incident vertices. so, |s2| = n + m − 3. the removal of elements of set s2 from g−+ gives k2 ∪ k1. hence, m(g−+ − s2) = 2. consider, |s2| + m(g−+ −s2) = n + m−1. the minimum value among m + i(g) and n + m − 1 gives integrity of g−+. therefore, i(g−+) = min{n + m−1,m + i(g)}. corolary 4.5. the integrity of (i) path pn(n ≥ 3), i(p−+n ) = n +d2 √ n + 1e−3, (ii) cycle cn(n ≥ 4), i(c−+n ) = n + 2dne−1, (iii) complete graph kn, i(k−+n ) = n(n+1) 2 −1, (iv) complete bipartite graph ka,b(a,b ≥ 2), i(k−+a,b ) = ab + 1 + min{a,b}, (v) star k1,b(b ≥ 2), i(k−+1,b ) = b + 2, (vi) wheel wn(n ≥ 5), i(w−+n ) = 2n +d2 √ n−1e−2. the column 2 and 4 table 5 shows integrity of basic graphs and integrity of transformation graphs g−+ of graphs with same structure. theorem 4.16. for any connected graph g of order n and size m, i(g−−) = m + i(g). proof. let g be an (n,m) graph. then g−− is a graph of order n + m. let s1 ⊂ v (g−−). consider a set s1 containing all the edges of g. so, |s| = |e| = m. the removal of the vertices of set s1 from g−− gives a graph g. therefore, i(g−−) = m + i(g). corolary 4.6. the integrity of (i) path pn, i(p−−n ) = n−3 + d2 √ n + 1e, 39 bommanahal basavanagoud, shruti policepatil g i(g) g−+ i(g−+) g−− i(g−−) p6 4 p −+ 3 4 p −− 3 4 p7 4 p −+ 7 10 p −− 7 10 c13 7 c −+ 4 6 c −− 4 7 c14 7 c −+ 14 21 c −− 14 21 kn n k−+n n(n+1) 2 −1 k−−n n(n+1) 2 k8,9 9 k −+ 2,3 9 k −− 2,3 9 k9,9 9 k −+ 9,9 91 k −− 9,9 91 k1,b 2 k −+ 1,b b + 2 k −− 1,b 2b + 1 w32 12 w −+ 5 12 w −− 5 12 w33 12 w −+ 33 76 w −− 33 76 table 5: (ii) cycle cn, i(c−−n ) = n−1 + d2 √ ne, (iii) complete graph kn, i(k−−n ) = n(n+1) 2 , (iv) complete bipartite graph ka,b, i(k −− a,b ) = ab + 1 + min{a,b}, (v) star k1,b, i(k −− 1,b ) = 2b + 1, (vi) wheel wn, i(w−−n ) = 2n−2 + d2 √ n−1e. the column 2 and 6 of table 5 shows integrity of basic graphs and integrity of transformation graphs g−− of graphs with same structure. 4.2 integrity of semitotal point graph of combination of basic graphs definition 4.1. [16] the product g × h of two graphs g and h is defined as follows: consider any two points u = (u1,u2) and v = (v1,v2) in v = (v1,v2). then u and v are adjacent in g×h whenever [u1 = v1 and u2 adj v2] or [u2 = v2 and u1 adj v1]. if g and h are (n1,m1) and (n2,m2) graphs respectively. then, g × h is (n1n2,n1m2 + n2m1) graph. 40 integrity of generalized transformation graphs theorem 4.17. for a graph k2 ×pn (n ≥ 3), i ( (k2 ×pn)++ ) =   7, if n = 3, 11, if n = 5, 5n−7 2 , if n is odd and n ≥ 7, 5n−2 2 , if n is even. proof. the graph (k2 ×pn)++ has 5n−2 vertices and 3(3n−2) edges. let s ⊂ v ( (k2 ×pn)++ ) . the proof includes the following cases. case 1. suppose n is odd and n ≥ 7. choose a set s containing the two internal vertices adjacent to corresponding central vertices of each of two pn’s in k2 × pn. so, |s| = 4. the deletion of vertices of set s from (k2 ×pn)++ results in a graph with components of orders 1, 7, 5(n−3) 2 . hence, m ( (k2 × pn)++ − s ) = 5(n−3) 2 , since n ≥ 7. the value of |s|+ m ( (k2 ×pn)++ −s ) for this set s is 5n−7 2 and it is minimum. therefore, i ( (k2 ×pn)++ ) = 5n−7 2 . case 2. suppose n is even. let s be a set containing the two internal vertices which are central vertices of each of two pn’s in k2 ×pn. so, |s| = 4. the removal of vertices of set s from (k2 × pn)++ results in a graph with components of orders 1, 5(n−2)2 . hence, we can write m ( (k2×pn)++−s ) = 5(n−2) 2 . the value of |s|+m ( (k2×pn)++−s ) for the above set s is minimum. therefore, i ( (k2 ×pn)++ ) = 5n−2 2 . case 3. suppose n = 3,5. by direct calculation using the definition of integrity, the result follows. theorem 4.18. for a graph k2 ×cn (n ≥ 4), i ( (k2 ×cn)++ ) =   2 ⌈ n 2 ⌉ + 7, if n is even, 2 ⌈ n 2 ⌉ + 7, if n is odd and n ≤ 7, 2 ⌈ n 3 ⌉ + 12, if n is odd and n ≥ 9. proof. the graph (k2 ×cn)++ has 5n vertices and 9n edges. let s ⊂ v ( (k2 ×cn)++ ) . case 1. suppose n is even. let s1 be an independent set of vertices of cn such that |s1| = β0(cn) = n2 . case 2. suppose n is odd. let s′ be an independent set cn such that |s′| = β0(cn) = n−12 . let v1 be a vertex of v (cn) \ s′ such that v1 is adjacent to a vertex of s′ as well as to a vertex of v (cn)\s′. let s1 = s′ ∪{v1}. combining the above two cases we get, s1 = ⌈ n 2 ⌉ . choose a set s consisting of vertices of two cn’s of k2 ×cn such that |s| = 2|s1| = 2 ⌈ n 2 ⌉ . the removal of 41 bommanahal basavanagoud, shruti policepatil vertices of set s from (k2 ×cn)++ results in a graph with components of orders 1, 7. hence, m ( (k2×cn)++−s ) = 7. the value of |s|+m ( (k2×cn)++−s ) for the above set s is minimum. therefore, i((k2 ×cn)++) = 2 ⌈ n 2 ⌉ + 7. case 3. suppose n(≥ 9) is odd. let s be an independent set of vertices of two cn’s in a manner that the distance between the selected vertices is 3. then, |s| = 2 ⌈ n 3 ⌉ . the removal of vertices of set s from (k2 × cn)++ results in a disconnected graph with components of orders 1, 12. hence, m ( (k2 ×cn)++ −s ) = 12. therefore, i ( (k2 ×cn)++ ) = 2 ⌈ n 3 ⌉ + 12. theorem 4.19. for a graph k2 ×k1,b (b ≥ 2), i ( (k2 ×k1,b)++ ) = 7. proof. the semitotal point graph (k2×k1,b)++ has 5b+3 vertices and 3(3b+1) edges. let s be a subset of v ( (k2 × k1,b)++ ) . choose s such that it contains the two vertices corresponding to central vertices of each of two stars of k2 × k1,b. clearly, |s| = 2. the removal of vertices of s from (k2 × k1,b)++ results in a graph with components of orders 1, 5. hence, m ( (k2×k1,b)++−s ) = 5. this set s gives least value of |s|+m ( (k2×k1,b)++−s ) . therefore, i ( (k2×k1,b)++ ) = 7. theorem 4.20. for a graph kp ×kq (p = q ≥ 2), i ( (kp ×kq)++ ) = pq + 1. proof. the semitotal point graph (kp×kq)++ has pq(p+q)2 vertices and 3pq(p+q−2) 2 edges. select a set s such that the elements of it correspond to all the vertices of kp×kq. so, |s| = pq. the removal of vertices of s from (kp ×kq)++ results in a totally disconnected graph with 3pq(p+q−2) 2 vertices. clearly, m ( (kp ×kq)++ −s ) = 1. therefore, i ( (kp ×kq)++ ) = pq + 1. definition 4.2. [16] the corona g ◦ h of graphs g and h is a graph obtained from g and h by taking one copy of g and |v (g)| copies of h and then joining by an edge each vertex of the i’th copy of h is named (h,i) with the i’th vertex of g. if g and h are (n1,m1) and (n2,m2) graphs respectively. then, g◦h is (n1(1 + n2),m1 + n1m2 + n1n2) graph. 42 integrity of generalized transformation graphs theorem 4.21. for a graph k2 ◦pn (n ≥ 3), i ( (k2 ◦pn)++ ) =   7, if n = 3, 10, if n = 5, 3(n+1) 2 , if n is odd and n ≥ 7, 3(n 2 + 1), if n is even. proof. the graph (k2 ◦pn)++ has 6n + 1 vertices and 3(4n−1) edges. let s ⊂ v ( (k2 ◦pn)++ ) . the proof includes the following cases. case 1. suppose n is odd and n ≥ 7. choose a set s containing the two internal vertices adjacent to corresponding central vertices of each of two pn’s and the two vertices of k2 in k2 ◦ pn. so, |s| = 6. the removal of vertices of s results in a graph with components of orders 1, 4, 3(n−1) 2 . hence, m ( (k2 ◦ pn)++ − s ) = 3(n−1) 2 , since n ≥ 7. the value of |s| + m ( (k2 ◦ pn)++ − s ) for this set s is 5n−7 2 is least. therefore, i ( (k2 ◦pn)++ ) = 5n−7 2 . case 2. suppose n is even. choose a set s containing the two internal vertices which are central vertices of each of two pn’s and the two vertices of k2 in k2 ◦ pn. so, |s| = 6. the removal of vertices of s from (k2 ◦ pn)++ results in a graph with components of orders 1, 3(n−2) 2 . clearly, we can write m ( (k2 ◦ pn)++ − s ) = 3(n−2) 2 . the value of |s| + m ( (k2 ◦pn)++ −s ) for the above set s is minimum. therefore, i ( (k2 ◦pn)++ ) = 3(n 2 + 1). case 3. suppose n = 3,5. by direct calculation using the definition of integrity, we can obtain the result. theorem 4.22. for a graph k2 ◦cn (n ≥ 4), i ( (k2 ◦cn)++ ) = 2 ⌈n 2 ⌉ + 6. proof. the graph (k2 ◦cn)++ has 7n + 2 vertices and 15n edges. let s ⊂ v ( (k2 ◦cn)++ ) . case 1. suppose n is even. let s1 be an independent set of vertices of cn such that |s1| = β0(cn) = n2 . case 2. suppose n is odd. let s′ be an independent set of vertices of cn such that |s′| = β0(cn) = n−12 . let v1 be a vertex of v (cn)\s′ such that v1 is adjacent to a vertex of s′ as well as to a vertex of v (cn)\s′. let s1 = s′ ∪{v1} and |s1| = n+12 . combining the above two cases we get, s1 = ⌈ n 2 ⌉ . choose a set s2 consisting of vertices of two cn’s of k2 ◦ cn such that |s2| = 2|s1| = 2dn2e. select a set s3 consisting of the vertices of k2 of k2 ◦ cn. so, s = s2 ∪ s3. hence, 43 bommanahal basavanagoud, shruti policepatil |s| = 2 ⌈ n 2 ⌉ + 2. the removal of vertices of set s from (k2 ◦cn)++ results in a graph with components of orders 1, 4. hence, m ( (k2 ◦ cn)++ − s ) = 4. the value of |s| + m ( (k2 ◦cn)++ −s ) for the above set s is minimum. therefore, i ( (k2 ◦cn)++ ) = 2 ⌈ n 2 ⌉ + 6. theorem 4.23. for a graph kp ◦kq, i ( (kp ◦kq)++ ) = pq + p + 1. proof. the semitotal point graph of kp◦kq has p(q+1) vertices and p2[p+q(q+ 1)−1] edges. let s be a subset of v ( (kp◦kq)++ ) . choose s such that it contains vertices of kp and vertices of p copies of kq of kp ◦kq. so |s| = p(q + 1). the removal of vertices of set s from (kp ◦ kq)++ results in a totally disconnected graph with p 2 [p + q(q + 1) − 1] vertices. clearly, m ( (kp ◦ kq)++ − s ) = 1. therefore, i ( (kp ◦kq)++ ) = pq + p + 1. 5 conclusion in this paper, we have computed the integrity of generalized transformation graphs in terms of elements of a graph g. also, integrity of semitotal point graph of combinations of basic graphs are obtained. finally, we have established the relation between integrity of basic graphs and integrity of their generalized transformation graphs. we conclude that integrity of generalized transformation graphs are greater than or equal to integrity of graphs that have same structure. 6 acknowledgement ∗this work is partially supported by the university grants commission (ugc), new delhi, through ugc-sap drs-iii for 2016-2021: f.510/3/drs-iii/2016(sapi). references [1] m. atici and r. crawford, the integrity of small cage graphs, australas. j. combin., 43, 2009, 39–55. 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[20] a. vince, the integrity of a cubic graph, discrete appl. math., 140, 2004, 223–239. 46 ratio mathematica issn: 1592-7415 vol. 35, 2018, pp. 8799 eissn: 2282-8214 87 preparation and application of mind maps in mathematics teaching and analysis of their advantages in relation to classical teaching methods andrej vanko* received: 04-08-2018 accepted: 30-11-2018 published: 31-12-2018 doi: 10.23755/rm.v35i0.428 ©andrej vanko abstract in this article, we are dealing with mind maps and describing the experiment with the application of mind maps in teaching mathematics at secondary schools. the experiment is aiming at comparing classical teaching and learning with mind maps. in the past, we created two groups of students (25 students per group), an experimental and a control group. we have set up a pre-test consisting of tasks not related to the subject that will be taken through mind maps. by the end of the experiment, we apply a post-test with tasks directly focused on the subject that we will teach through mind maps. we will then evaluate the individual tests and then we will evaluate the effectiveness of the mind maps in the teaching process compared to the traditional methods. keywords: mind maps, experiment, mathematics, application, pretest, post-test, comparing * constantine the philosopher university in nitra, trieda a. hlinku 1, 949 74 nitra, slovak republic; email: andrej.vanko@ukf.sk. andrej vanko 88 1. mind maps mind maps are diagrams that express essential relationships between terms in the form of claims. the statements are represented by briefly characterized combinations of terms that describe relationship information and describe the interrelation of concepts. the mind map illustrates the structure, hierarchy, and the relationship between the terms. it enhances the learning process efficiency and promotes creativity. they are very economical in expressing a very complex content, helpful to memorize, allowing a view of the same thing from multiple angles, allowing see the relationships between ideas in a complex way [8]. they also help to see paradoxes and opposites, which motivates students to ask new questions. it is very important to determine the central idea, from which lead the main and the secondary branches, which gradually form certain relations. we use different colors, shortcuts, diagrams, symbols, equations, and images in the map. mind maps as a schematic expression of thoughts, ideas or notes are not inventions of the 21st century. in addition, the use of learning methods built on the creation and presentation, which are logically arranged and the links of the conjugated terms are not the discovery of current educational trends. in the past, teachers structured the learned curriculum by using key concepts placed on a magnetic board, notice board, or complemented the prepared schemes with cutout images and characters. many important artists and geniuses, such as leonardo da vinci, michelangelo, isaac newton, pablo picasso, thomas edison, galileo, marie curie and others, brought certain schemes into their own ideas in the past. they tried to highlight their ideas, not just linear, using lines and words, but also with a strong language of images, drawings, schemes, codes, symbols, and graphs [1]. in current professional, as well as popular-learning literature, we can encounter several names of the linear and nonlinear layout of concepts, data and main themes in graphically integrated structures. the authors present conceptual maps, mental maps, thought maps, cognitive maps, semantic maps, knowledge maps, webs, mind maps, and so on. some of these terms do not distinguish individually and call their collectively mental maps or cognitive schemes [2]. j. d. novak [3] considered as a founder of conceptual map theories and their construction speaks of mind maps as a hierarchically arranged, graphical representation of relations between selected concepts. there are general terms at the top of the map that are associated with terms that are more specific in the lower tree level. from the central concept, the "branches" are connecting with the preparation and application of mind maps 89 concept in the lower parts of the map, from which the "branches" are connecting again with the concept at the lower levels of the map. psychologists veselský [4] and stewart [5] talk about conceptual maps as graphical imaging systems, whose basic building unit are concepts. they are represented by frames with inscribed notional names and the relationships are expressed by marked orientated lines linking the respective conceptual expressions. focus on the non-linear abstract representation of the structure of the subject and notes an opposite to the written, printed, projected, or otherwise presented text followed by the sentences one after the other [6], stresses mareš. according to him, it is based on the idea of organizing the best and the most transparent key concepts and relationships by "visualizing" them and creating a sketch, a schematic of an easily accessible abstract "outer" memory. although a learner learns to organize the key elements of the curriculum on paper first. he has to begin with organizing them in the head. thus, he is forced to consciously construct and reconstruct a network of concepts and relationships in his "mental space" [7]. it follows from the above that there is no terminological unity and consensus among the experts in understanding the different concepts of capturing concept ideas in the graphical structure of related concepts with the designation of relationships and links between them. the fisher’s definition of mind maps is probably the most precise according to them. a conceptual, thought-based, or otherwise called mental map is a diagram that illustrates the context and relationship between knowledge, serves to organize them. we understand the conceptual, mental or idea map as synonyms. we do this in particular because some of the used conceptual maps do not have a typical structure of mental maps, they consist of several levels, and there are significant links ̶ relationships between some terms [2]. 2. preparation, application and evaluation of the pre-test if we wanted to compare the two different teaching methods, we needed to have the experimental and control groups at the same level of knowledge before the comparisons began. to test knowledge of these groups, we created a pre-test that tested both groups before applying the mind maps. students in both groups wrote this pre-test on the same day. the pre-test included three tasks from the previous non-geometry related lesson. tasks aimed at adjusting the fractions, creating, and andrej vanko 90 solving the equation. in each assignment, we identified several characters, which we took into consideration during the evaluation and allowed us to compare the two groups of students more objectively. pre-test, task 1: determine when the expression is meaningful and adjust it to the simplest form. 𝑎 + 𝑏 𝑎 − 𝑏 − 𝑎 − 𝑏 𝑎 + 𝑏 1 − 𝑎2 + 𝑏2 𝑎2 − 𝑏2 ∶ 1 𝑏2 − 2 𝑏 + 1 2 − 1 + 𝑏2 𝑏 rated characters in task # 1: • a ≠ b • b ≠ 0,1 • transcription for multiplication • common denominator • modifying a fraction • cutting fractions • excluding -1 from the second fraction • result pre-test, task 2: if we enlarge one side of the square by 4 units and at the same time reduce the other side by 2 units; we create a rectangle whose content is 12% larger than the square. specify the square size of the square. rated characters in task # 2: • picture • content of a square • rectangle content • increase content by 12% • equality of contents • edit quadratic equation • result pre-test, task 3: if we increase the unknown number by 7 and if we create the square root of this enlarged number, we get a number that is by 5 smaller than the original number. specify an unknown number. rated characters in task # 3: • enlarged number • root • reduced number • equality • squaring • writing of quadratic equation • modification of quadratic equation • writing results preparation and application of mind maps 91 the selected characters represented the various conditions within the given task, important for its solvability, mathematical operations, mathematical entries, various comparisons, adjustments of equations, fractions and results of individual tasks. the choice of characters within the assignments helped us evaluate the solution of these tasks objectively without any external influence. picture 1: selected sample of students in pre-test, experimental group, task 1 picture 2: selected sample of students in pre-test, experimental group, task 2 picture 3: selected sample of students in pre-test, experimental group, task 3 picture 4: selected sample of students in pre-test, control group, task 1 picture 5: selected sample of students in pre-test, control group, task 2 picture 6: selected sample of students in pre-test, control group, task 3 a ≠ b b ≠ 0,1 transcription for multiplication common denominator modifying a fraction cutting fractions excluding -1 from the second fraction result n n n n n n n n n n n n n n n n n n y y n n n n n n y y n n n n picture content of a square rectangle content increase content by 12% equality of contents edit quadratic equation result y n y n n n n n n n n n n n y y y n n n n y y y n n n n enlarged number root reuced number equality squaring writting of quadratic equation modification of quadratic equation writting results y y y y n n n n y y y y n n n n y y y y n n n n y y n y n n n n a ≠ b b ≠ 0,1 transcription for multiplication common denominator modifying a fraction cutting fractions excluding -1 from the second fraction result n y n n n n n n n n y y n n n n n n y y n n n n n n n n n n n n picture content of a square rectangle content increase content by 12% equality of contents edit quadratic equation result n y y n n y n y n n n n n n n n n n n n n n n n n n n n enlarged number root reuced number equality squaring writting of quadratic equation modification of quadratic equation writting results y y y y n n n n y y y y n n n n y y y y n n n n n n n n n n n n andrej vanko 92 we evaluated the pre-test as the ratio of the characters in the resolution (y means, that character was in resolution, n means, that character was missing in solution) from all students within the given task to the total number of characters within the given task. in table 1 there is shown percentage of students’ success rate in each group in a particular task. table 1: comparison of pre-test results in experimental and control groups (%) task / group experimental control task 1 10.2 5.7 task 2 14.3 15.6 task 3 33 34 even though the students did not properly calculate the tasks, it was clear from the pre-test that both groups of students were about the same level of knowledge, what was essential for our experiment and we could move to the next stage, the application of mind maps in the teaching process. 3. application of conceptual maps in the teaching process in the subject of mathematics after agreement with the mathematics teacher in the experimental group, we had three lessons available, during which we presented the curriculum of geometry dealing with the mutual positions of lines and planes. we used a computer and a projector for this activity. the curriculum was processed using conceptual maps and inserted into the presentation. we divided the curriculum for lessons into individual groups as follows: 1st lesson: mutual position of lines (parallel, parallel identical, concurrent), 2nd lesson: mutual position of lines (skew), mutual position of lines and planes (parallel, parallel identical), 3rd lesson: mutual position of lines and planes (concurrent), mutual position of planes (parallel, parallel identical, concurrent). preparation and application of mind maps 93 we informed students about the content for next three lessons. picture 7: introductory division of the curriculum picture 8: detailed division of the mutual positions of the two lines in picture 8, we have explored in more detail the possible mutual positions of the two lines. picture 9: detailed division of the parallel positions of the two lines picture 9 focused on the case of two parallel lines and the possible representation of these lines. andrej vanko 94 picture 10: detailed parametric representation of mutual parallel positions of two lines in picture 10 there is an illustrative and detailed description of a branch of parametric representation. picture 11: an example of the mutual position of two parallel lines in parametric representation in picture 11 there is task, which the students were trying to solve after the theoretical part was completed. on the map there were marked intermediate results that served to students for check. preparation and application of mind maps 95 as can be seen in picture 7 ̶ 11, the principles of the mind map were retained. the deeper we got into the mind map, the more specific terms were in that part. our task was to explain these concepts to the students so they can join these concepts together alone and can apply them in solving different problems. after these three lessons, during which we were teaching using mind maps, we moved into the final phase of our experiment. this part consisted of the post-test we gave to the students. the post-test consisted of tasks that focused directly on the subject discussed at our three lessons. 4. preparation, application and evaluation of the post-test after completing the pre-test, which showed us that the students are about the same level of knowledge, following the use of mind maps in three teaching lessons, we have reached the final stage of our experiment. this final phase consisted of two phases: application of post-test and evaluation of results from post-test. the tasks in the post-test were, this time, directly focused on the mutual positions of planes and lines, in order to compare the effectiveness of this method in the experimental group against the classical way of teaching in the control group. in the given tasks, we have re-selected the characters that represented the key elements in the solving of the task. post-test, task 1: show that the planes α and β are concurrent and write the parametric representation of the intersection of these planes. α: 5x – 3y + 2z – 5 = 0 β: 2x – y – z – 1 = 0 rated characters in task # 1: • normal vector p • normal vector q • vector products • parameter at point p • parametric representation of intersection andrej vanko 96 post-test, task 2: determine the mutual position of plane β and line p. β: x – 5y + 4z – 6 = 0 p: x = 2 – t, y = 3t, z = 3 + 4t, t ∈ r rated characters in task # 2: • placing p to β, scalar product • place p into equation β • adjusting equation after placing p into β • determine final position post-test, task 3: determine the mutual positions of p, q. if p = (𝑨𝑩) ⃡ , q = (𝑪𝑫) ⃡ a = [7, 6] b = [6, 8] c = [6, -5] d = [4, -1] rated characters in task # 3: • line p • line q • expression of p • expression of q • vector comparison • computation and comparison of parameters • result picture 12: selected sample of students in post-test, experimental group, task 1 picture 13: selected sample of students in post-test, experimental group, task 2 picture 14: selected sample of students in post-test, experimental group, task 3 normal vector p normal vector q vector products parameter at point p parametric expression of intersection y y y y y y y y y y y y y y y y y y y n placing p to β, scalar product placing p to β, scalar product adjusting equation after placing p into β determine final position y y y y y y y y y y y y y y y y line p line q expression of p expression of q vector comparison computation and comparison of parameters result y y n n y n n y y y y y y y y y y y y y y y y y y y y y preparation and application of mind maps 97 picture 15: selected sample of students in post-test, control group, task 1 picture 16: selected sample of students in post-test, control group, task 2 picture 17: selected sample of students in post-test, control group, task 3 the post-test was evaluated in the same way as the pre-test and the results from both tests were subsequently recorded in table 2 table 2: comparison of the results of the post-test in the experimental and control group (%) task / group experimental control task 1 88.3 78.3 task 2 91.7 68.8 task 3 82.1 53.6 as we can see from table 2, the results compared to the pre-test are much better. the students were able to apply the acquired knowledge in solving of the given tasks. for our experiment is much more important that the results achieved in the experimental group, in the group where we were teaching with the help of mind maps, are obviously better than in the control group where the classic teaching methods were used. normal vector p normal vector q vector products parameter at point p parametric expression of intersection y y n n n y y y y n y y y y n y y y y y placing p to β, scalar product placing p to β, scalar product adjusting equation after placing p into β determine final position y n n n n n n n y y y y y y y y line p line q expression of p expression of q vector comparison computation and comparison of parameters result y y n n y n n y n n n n n n y y n n y n n y n n n y n n andrej vanko 98 5. conclusion this article was focused on the application of mind maps in the teaching process and the comparison of the mind map's effectiveness with the classical way of teaching. this comparison consisted of three important steps: 1st pre-test and evaluation 2nd application of conceptual maps in the teaching process 3rd post-test and evaluation in both tests, pre-test and post-test, we chose the rated characters which we were looking for during the correction of students’ tests. we subsequently evaluated and compared these rated characters. the students wrote the tests on the same day to prevent the possible influence and improvement of the results in one or the other group. in the first step, we gave the students a pre-test, in which the balance or imbalance of students’ knowledge in the experimental and control group should be demonstrated. the results of the pre-test showed that the students were about the same level of knowledge. in the next step, we had three lessons from the geometry. subject of these lessons was the mutual positioning of the lines and the planes. during these lessons we were using mind maps. in the last step, the students wrote a post-test with tasks related to the mutual positions of the lines and planes. the post-test was then evaluated and the results from both groups were compared in table 2. according to the values, we can see that success in solving problems is higher in the experimental group. these results are better in the range of 10% to almost 30% compared to the control group, which is not negligible. only for comparison, the results of the pre-test in both groups varied from 1% to less than 5%. at the end, we can conclude that the mind maps in our presentation with our teaching are more effective compared to the traditional classical teaching method. preparation and application of mind maps 99 references 1. mind mapping blog. software for mindmapping and information organization [online]. 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[cit. 2013-13-02] dostupné na internete: http://onlinelibrary.wiley.com/doi/10.1002/sce.3730640212 6. čáp, j., mareš, j. psychologie pro učitele. praha: portál, 2001. isbn 807178-463-x. strana. 441-505 7. mareš, j. styly učení žáků a studentů. praha: portál s.r.o., 1998. isbn 807178-246-7. strana 142-170 8. baralis, g. h.: the views of primary education teachers on the verification of multiplication. in: ratio mathematica journal of mathematics, statistics and application. vol 27 (2014), issn 2282-8214. p. 49-59 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 40, 2021, pp. 151-162 151 regular generalized fuzzy b-separation axioms in fuzzy topology varsha joshi * jenifer j.karnel † abstract regular generalized fuzzy b-closure and regular generalized fuzzy b interior are stated and their characteristics are examined, also regular generalized fuzzy b -𝜏𝑖 separation axioms have been introduced and their interrelations are examined. the characterization of regular generalized fuzzy b –separation axioms are analyzed. keywords: rgfbcs; rgfbos; rgfbcl; rgfbint; rgfbt0; rgfbt1; rgfbt2; rgfbt2 1 2 and fuzzy topological spaces x (in short fts). 2020 ams subject classification: 54a40 * mathematics department, sdm college of engineering & technology, dharwad-580 003. karnataka, india.e-mail: varshajoshi2012@gmail.com * mathematics department, sdm college of engineering & technology, dharwad-580 003. karnataka, india.e-mail: jeniferjk17@gmail.com †received on january 12th, 2021. accepted on may 12th, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.624. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. varsha joshi and dr.jenifer j.karnel 152 1. introduction the fundamental theory of fuzzy sets were introduced by zadeh [16] and chang [9] studied the theory of fuzzy topology. after this ghanim.et.al [10] introduced separation axioms, regular spaces and fuzzy normal spaces in fuzzy topology. the theory of regular generalized fuzzy b-closed set (open set) presented by jenifer et. al [11]. in this study we define rgfb-closure, rgfbinterior and rgfb-separation axioms and their implications are proved. effectiveness nature of the various concepts of fuzzy separation ideas are carried out. characterizations are obtained. 2. preliminary (x1, τ),( x2, σ) (or simply x1, x2 ) states fuzzy topological spaces(in short, fts) in this article. definition 2.1[1, 3]: in fts x1, 𝛼 be fuzzy set. (i) if 𝛼 =intcl(𝛼)then 𝛼 is fuzzy regular open(precisely, fros). (ii) if 𝛼 =clint(𝛼)then 𝛼 is fuzzy regular closed (precisely, frcs). (iii) if ( ) ( ) clintintcl  then 𝛼 is f b-open set (precisely, fbos). (iv) if ( ) ( ) clintintcl  then 𝛼 is f b-closed set (precisely, fbcs). remark 2.2 [1]: in a fuzzy topological space x, the following implication holds good figure1. interrelations between some fuzzy open sets definition 2.3[3]: let  be a fuzzy set in a fts x1. then , (i)   = , )fbcs(x a is : )bcl( 1 . (ii)   = ),fbos(x a is :)bint( 1 . definition 2.4[11]: in a fts x1, if bcl( ) ≤ 𝛽 , at any time when  ≤ 𝛽, then fuzzy set  is named as regular generalized fuzzy b-closed (rgfbcs).where 𝛽 is fropen. . remark 2.5[11]: in a fts x1, if 1- is rgfbcs(x1 ) then fuzzy set  is rgfbos. fros f-open fbos regular generalized fuzzy b-separation axioms in fuzzy topology 153 definition 2.6[11]: in a fts x1, if bint(  )≥ 𝛽 , at any time when  ≥ 𝛽, then fuzzy set  is named as regular generalized fuzzy b-open (rgfbos).where 𝛽 is frclosed. definition 2.7[13]: let (x1, τ), (x2, σ) be two fuzzy topological spaces. let f : x1 → x2 be mapping, (i) if f-1() is rgfbcs(x1), for each closed fuzzy set  in x2, then f is said to be regular generalized fuzzy b-continuous (briefly, rgfbcontinuous). (ii) if f-1() is open fuzzy in x1, for each rgfbos  in x2, then f is called strongly rgfb-continuous. (iii) if f-1() is rgfbcs in x1, for each rgfbcs  in x2, then f is called rgfb-irresolute. definition 2.8[10]: x1 is a fts which is named as (i) fuzzy t0(in short, ft0) if and only if for each pair of fuzzy singletons p1 and p2 with various supports there occurs open fuzzy set u such that either p1≤ u ≤ 1p2 or p2≤ u ≤ 1-p1 . (ii) fuzzy t1(in short ft1)if and only if for each pair of fuzzy singletons p1 and p2 with various supports, there occurs open fuzzy sets u and v such that p1≤ u≤1p2 and p2≤ v ≤ 1p1 . (iii) fuzzy t2(in short, ft2) or f-hausdorff if and only if for each pair of fuzzy singletons p1and p2 with various supports ,there occurs open fuzzy sets u and v such that p1≤u≤ 1p2 , p2≤v≤ 1p1 and u≤ 1v. (iv) fuzzy t2 1 2 (in short, ft2 1 2 ) or f-urysohn if and only if for each pair of fuzzy singletons p1 and p2 with various supports, there occurs open fuzzy sets u and v such that p1≤u≤ 1p2 , p2≤v≤ 1p1 and clu≤ 1-cl v. 3. regular generalized fuzzy b-closure (rgfbcl) and regular generalized fuzzy b-interior (rgfbint) definition 3.1:the regular generalized fuzzy b-closure is denoted and defined by, rgfbcl ( ) = λ {  :  is a rgfbcs( x1), ≥  }. where  be a fuzzy set in fts x1. theorem 3.2:let x1 be fts, then the properties that follows are occurs for rgfbcl of a set varsha joshi and dr.jenifer j.karnel 154 i. rgfbcl(0) = 0 ii. rgfbcl(1) = 1 iii. rgfbcl( ) is rgfbcs in x1 iv. rgfbcl[rgfbcl( )] = rgfbcl( ) definition 3.3:let  and 𝛽 be fuzzy sets in fuzzy topological space x1. then regular generalized fuzzy b-closure of (  v 𝛽) and regular generalized fuzzy b-closure of (  /\ 𝛽) are denoted and defined as follows i. rgfbcl ( v 𝛽 ) = /\ {  :  is a rgfbcs(x1) , where  ≥ ( v 𝛽 ) } ii. rgfbcl ( /\𝛽 ) = /\ {  :  is a rgfbcs(x1), where  ≥ ( /\ 𝛽) } theorem 3.4: let  and 𝛽 be fuzzy sets in fts x1, then the following relations occurs i. rgfbcl( ) v rgfbcl(𝛽) ≤ rgfbcl( v 𝛽) ii. rgfbcl( ) /\ rgfbcl(𝛽) ≥ rgfbcl( /\ 𝛽) proof: (i) we know that  ≤ ( v 𝛽) or 𝛽 ≤ ( v 𝛽)  rgfbcl ( ) ≤rgfbcl ( v 𝛽) orrgfbcl(𝛽) ≤rgfbcl ( v 𝛽) hence, rgfbcl (  ) v rgfbcl (𝛽) ≤rgfbcl ( v 𝛽). (ii) we know that  ≥ ( /\𝛽) or 𝛽 ≥ ( /\𝛽)  rgfbcl( ) ≥rgfbcl ( /\𝛽) orrgfbcl(𝛽) ≥rgfbcl ( /\𝛽) hence, rgfbcl(  ) /\ rgfbcl(𝛽) ≥rgfbcl ( /\𝛽). theorem 3.5:  is rgfbcs in a fts x1, if and only if  =rgfbcl( ). proof: suppose  is rgfbcs. since    and   { 𝛽: 𝛽 is rgfbcs(x1) and   𝛽 },  is the smallest and contained in 𝛽,therefore  =λ{ 𝛽: 𝛽 is rgfbcs( x1 )and   𝛽}=rgfbcl( ). hence,  =rgfbcl ( ). on the other hand, suppose  =rgfbcl ( ), then  = λ{ 𝛽: 𝛽 is rgfbcs,   𝛽 }    λ { 𝛽: 𝛽 is rgfbos,   𝛽 }. hence,  is rgfbcs. definition 3.6: the regular generalized fuzzy b-interior is denoted and defined by, rgfbint(  ) = v { 𝛿: 𝛿 is a rgfbos(x1),   }. where  be a fuzzy set in fts x1. theorem 3.7: let x1 be fts, then the properties that follows are occurs for rgfbint of a set i. rgfbint(0) = 0 ii. rgfbint(1) = 1 iii. rgfbint(  ) is rgfbos in x1 iv. rgfbint[rgfbint(  )] = rgfbint( ) . regular generalized fuzzy b-separation axioms in fuzzy topology 155 definition 3.8: let  and 𝛽 are fuzzy sets in fts x1. then regular generalized fuzzy b-interior of ( v 𝛽) and regular generalized fuzzy b-interior of ( /\ 𝛽) are denoted and defined as follows i. rgfbint (  v 𝛽 ) = v {𝛿: 𝛿 is a rgfbos( x1), where 𝛿 ≤ ( v 𝛽 ) }. ii. rgfbint(  /\ 𝛽 ) = v { 𝛿: 𝛿 is a rgfbos(x1), where 𝛿 ≤ ( /\ 𝛽 )}. theorem 3.9:let  and 𝛽 are fuzzy sets in fts x1, then the following relations occurs i. rgfbint(  ) v rgfbint(𝛽) ≤ rgfbint( v 𝛽) ii. rgfbint(  ) /\ rgfbint(𝛽) ≥ rgfbint( /\𝛽) proof: (i) we know that,  ≤ ( v 𝛽) or 𝛽 ≤ ( v 𝛽)  rgfbint(  ) ≤ rgfbint ( v 𝛽) or rgfbint(𝛽) ≤ rgfbint ( v 𝛽) hence, rgfbint(  ) v rgfbint(𝛽) ≤ rgfbint( v 𝛽). (ii)we know that  ≥ ( /\𝛽) or 𝛽 ≥ ( /\𝛽)  rgfbint( ) ≥ rgfbint ( /\𝛽) or rgfbint(𝛽) ≥ rgfbint ( /\𝛽) hence, rgfbint(  ) /\ rgfbint(𝛽) ≥ rgfbint ( /\𝛽). theorem 3.10: let x1 be fts,  is rgfbos if and only if  =rgfbint( ). proof: suppose  is rgfbos. since    ,   { 𝛿: 𝛿 is rgfbos and 𝛿  } since biggest  contains 𝛿. therefore,  = v{ 𝛿: 𝛿 is rgfbos 𝛿   } = rgfbint (  ). hence,  =rgfbint( ). on the other hand, suppose  =rgfbint ( ).then,  =v{ 𝛿: 𝛿 is rgfbos, 𝛿   }    v { 𝛿: 𝛿 is rgfbos 𝛿   }. hence,  is rgfbos. theorem 3.11: let  be a fuzzy set in a fts x1, in that case following relations holds good i. rgfbint(1- ) = 1-rgfbcl( ) ii. rgfbcl(1- ) = 1rgfbint( ) proof: (i) let  be a fuzzy set in fts x1. then we have rgfbcl ( ) = λ {  :  is a rgfbcs( x1), ≥  }. where  be a fuzzy set in fts x1. 1-rgfbcl (  ) = 1λ {  :  is a rgfbcs( x1), ≥  }. = v { 1 −  :  is a rgfbcs( x1), ≥  }. = v{ 1 −  : 1 −  is a rgfbos( x1), ≤ 1- }. = rgfbint (1- ) hence, 1-rgfbcl(  ) = rgfbint (1- ). (ii) let  be a fuzzy set in fts x1. then we have rgfbint(  ) = v { 𝛿: 𝛿 is a rgfbos(x1),   }. where  be a fuzzy set in fts x1. varsha joshi and dr.jenifer j.karnel 156 1-rgfbint (  ) = 1-v { 𝛿 : 𝛿   and 𝛿 is rgfbos (x1)} = λ{1𝛿: 𝛿   and 𝛿 is rgfbos(x1)} = λ {1𝛿 : 1-  1𝛿 and 1𝛿 is rgfbcs( x1)} = rgfbcl (1- ) hence 1-rgfbint (  ) = rgfbcl (1- ). 4. rgfb-separation axioms definition 4.1:a fts is known as rgfbt0, that is regular generalized fuzzy bt0, iff for each pair of fuzzy singletons q1 and q2 with various supports, there occurs rgfbos 𝛿 such that either q1≤ 𝛿 ≤ 1q2 or q2≤ 𝛿 ≤ 1q1. theorem 4.2: a fts is rgfbt0,that is regular generalized fuzzy bt0, if and only if rgfbcl of crisp fuzzy singletons q1 and q2 with various supports are different. proof: to prove the necessary condition: let a fuzzy topological space be rgfbt0 and two crisp fuzzy singletons be q1 & q2 with various supports x1 & x2 respectively i.e. x1 ≠ x2. since fts is rgfbt0 ,there exist a rgfbos 𝛿 such that, q1≤ 𝛿 ≤ 1q2  q2 ≤ 1𝛿, but q2≤ rgfbcl(q2) ≤ 1𝛿, where q1≤ rgfbcl(q2)  q1≤ 1𝛿 where 1𝛿 is rgfbcs. but, q1≤ rgfbcl(q1). this shows that, rgfbcl(q1) ≠ rgfbcl(q2). to prove the sufficiency: let p1 & p2 be fuzzy singletons with various supports x1 & x2 respectively, q1 & q2 be crisp fuzzy singletons such that q1(x1)=1, q2(x2)=1. but, q1≤ rgfbcl(q1)  1-rgfbcl(q1) ≤ 1-q1≤1-p1. as each crisp fuzzy singleton is rgfbcs, 1rgfbcl(q1) is rgfbos and p2≤ 1rgfbcl(q1) ≤ 1 p1.this proves, fts is rgfbt0 space. definition 4.3: a fts is known as rgfbt1,that is regular generalized fuzzy bt1, iff for each pair of fuzzy singletons q1 & q2 with various supports x1 & x2 respectively, there occurs rgfboss 𝛿1 & 𝛿2 such that, q1≤ 𝛿1≤ 1q2 and q2≤ 𝛿2≤ 1q1. theorem 4.4: a fts is rgfbt1, that is regular generalized fuzzy bt1, if and only if each crisp fuzzy singleton is rgfbcs. proof: to prove the necessary condition: let rgfbt1 be fts and crisp fuzzy singleton with supports x0 be q0 .there occurs, rgfboss 𝛿1 and 𝛿2 for any fuzzy singleton q with supports x (≠ x0), such that, q0≤ 𝛿1≤ 1q and q ≤ 𝛿2≤ 1q0. since, it includes each fuzzy set as the collection of fuzzy singletons. so that, 1-q0 = 01 qq v − q = 0. thus, 1-q0 is rgfbos. this shows that, q0 (crisp fuzzy singleton) is rgfbcs. regular generalized fuzzy b-separation axioms in fuzzy topology 157 to prove the sufficiency: assume p1 and p2 be pair of fuzzy singletons with various supports x1& x2 .further on fuzzy singletons with various supports x1 & x2 be q1 & q2, such that q1(x1) = 1 and q2(x2)=1. as each crisp fuzzy singleton is rgfbcs, the fuzzy sets 1-q1 & 1-q2 are rgfboss such that, p1≤ 1q1≤ 1p2 and p2≤1-q2≤ 1p1.this proves, fts is rgfbt1 space. remark 4.5:in a fts x1, each rgfbt1 space is rgfbt0 space. proof: it follows the above definition. the opposite of this theorem is in correct. this is shown as follows – example 4.6:let x1={a, b},p1={(a,0),(b,1)} and p2={(a,0.4),(b,0)} are fuzzy singletons. u= {(a, 0.5),(b, 1)} be rgfbos . let 𝜏= { 0,p1 , p2 , u,1 }. the space is rgfbt0 and it is not rgfbt1. definition 4.7: a fts is known as rgfbt2 , that is regular generalized fuzzy bt2 or rgfb-hausdorff iff, for each pair of fuzzy singletons q1 & q2 with various supports x1 & x2 respectively, there occurs, rgfbos 𝛿1 & 𝛿2 such that, q1≤ 𝛿1≤ 1q2 , q2≤ 𝛿2≤ 1q1 and 𝛿1≤ 1𝛿2 . theorem 4.8: a fts is known as rgfbt2, that is regular generalized fuzzy bt2 or rgfb-hausdorff if and only if for each pair of fuzzy singletons q1 & q2 with various supports x1 & x2 respectively, there occurs an rgfbos 𝛿1 such that, q1≤ 𝛿1≤ rgfbcl 𝛿1≤ 1q2 . proof: to prove the necessary condition: let rgfbt1 be fts and fuzzy singletons q1 & q2 with various supports .let 𝛿1 & 𝛿2 be rgfbos such that, q1≤ 𝛿1≤ 1q2 , q2≤ 𝛿2≤ 1q1 and 𝛿1≤ 1𝛿2 where 1𝛿2 is rgfbcs. we have by definition, rgfbcl(𝛿1)=/\ {(1𝛿2) : (1𝛿2) rgfbcs} where 𝛿 1≤ 1𝛿2 .also rgfbcl(𝛿1)≥ 𝛿1.this shows that, q1≤ 𝛿1≤ rgfbcl (𝛿)1≤ 1𝛿2≤ 1q2  q1≤ 𝛿1≤ rgfbcl (𝛿1)≤ 1q2 . to prove the sufficiency: assume q1 and q2 are pair of fuzzy singletons with various supports and 𝛿1 be rgfbos. since, q1≤ 𝛿1≤ rgfbcl (𝛿1)≤ 1q2  q1≤ 𝛿1≤1q2. also q1≤rgfbcl( 𝛿1)≤ 1q2  q2≤ 1rgfbcl (𝛿1) ≤1-q1. this shows that, 1rgfbcl (𝛿1) is rgfbos. also rgfbcl (𝛿1) ≤ 1rgfbcl (𝛿2) . this proves that, fts is rgfbt2 space. remark 4.9:in a fts x1,each rgfbt2 space is rgfbt1 space. proof: it follows the above definition. the opposite of this theorem is in correct. this is shown as follows – example 4.10: let x1={a,b}. q1={(a, 0.2),(b, 0)} and q2={(a,0), (b,0.4)} are fuzzy singletons, o1= {(a,0.3),(b,0.4)} and o2= {(a,0.8),(b,0.7)} are rgfbos .let 𝜏 = { 0, p1 , p2 , o1 , o2, 1}. the space is rgfbt1 and it's not rgfbt2. varsha joshi and dr.jenifer j.karnel 158 definition 4.11: a fts is known as rgfbt2 1 2 , that is regular generalized fuzzy bt2 1 2 or rgfb-urysohn iff for each pair of fuzzy singletons q1 & q2 with various supports x1 & x2 respectively, there occurs, rgfboss 𝛿1 & 𝛿2 such that, q1≤ 𝛿1≤ 1q2 , q2≤ 𝛿2≤ 1q1 and rgfbcl (𝛿1)≤ 1-rgfbcl (𝛿2) . remark 4.12:in a fts x1,each rgfbt2 1 2 space is rgfbt2 space. proof: it follows from the above definition. the opposite of this theorem is in correct. this is shown as follows – example 4.13: let x1={a, b}. q1={(a, 0.1),(b,0)} and q2={(a,0),(b,0.3)} are fuzzy singletons, o1= {(a,0.2),(b,0.3)} and o2= {(a,0.7),(b,0.5)} are rgfboss. let 𝜏 ={ 0, p1 , p2 , o1 , o2, 1 }. the space is rgfbt2 and it's not rgfbt2 1 2 . figure2. from the above definition and examples one can notice that the above chains of implication. theorem 4.14: an injective function f: x1 → x2 is rgfb-continuous, and x2 is ft0, then x1 is rgfbt0. proof: assume  & β be fuzzy singletons in x1 with various support then f () & f (β) belongs to x2, as f is injective and f ()≠ f (β). as x2 is ft0, there occurs, a open set o in x2 such that, 𝑓() ≤ 𝑂 ≤ 1 − 𝑓(𝛽)or 𝑓(𝛽) ≤ 𝑂 ≤ 1 − 𝑓(),   ≤ 𝑓 −1(𝑂) ≤ 1 − 𝛽 or 𝛽 ≤ 𝑓 −1(𝑂) ≤ 1 − . since, f : x1 → x2 is rgfb-continuous, 𝑓 −1(𝑂) is rgfbos in x1. this shows that, x1 is rgfbt0space[ 4.1]. theorem 4.15: an injective function f : x1 → x2 is rgfb-irresolute, and x2 is rgfbt0, then x1 is rgfbt0. proof: assume  & β be fuzzy singletons in x1 with various support. as f is injective 𝑓() & 𝑓(𝛽) belongs to x2 and 𝑓() ≠ 𝑓(𝛽). as, x2 is rgfbt0, there occurs rgfbos o in x2 so that 𝑓() ≤ 𝑂 ≤ 1 − 𝑓(𝛽) or 𝑓(𝛽) ≤ 𝑂 ≤ 1 − 𝑓()   ≤ 𝑓 −1(𝑂) ≤ 1 − 𝛽 or 𝛽 ≤ 𝑓 −1(𝑂) ≤ 1 − . as, f is rgfbirresolute 𝑓 −1(𝑂) is rgfbos(x1). this shows that, x1 is rgfbt0 space[4.1 ]. theorem 4.16:an injective function f : x1 → x2 is strongly rgfb-continuous, and x2 is rgfbt0, then x1 is ft0. rgfbt2 𝟏 𝟐 rgfbt2 rgfbt1 rgfbt0 regular generalized fuzzy b-separation axioms in fuzzy topology 159 proof: assume  & β be fuzzy singletons in x1 with various support. since f is injective f () & f (β) belongs to x2 and 𝑓() ≠ 𝑓(𝛽). as, x2 is rgfbt0, there occurs rgfbos o in x2 so that, 𝑓() ≤ 𝑂 ≤ 1 − 𝑓(𝛽) or 𝑓(𝛽) ≤ 𝑂 ≤ 1 − 𝑓(),   ≤ 𝑓 −1(𝑂) ≤ 1 − 𝛽 or 𝛽 ≤ 𝑓 −1(𝑂) ≤ 1 − . since, f is strongly rgfb-continuous, 𝑓 −1(𝑂) is fuzzy-open in x1. this shows that, x1 is ft0-space[ 2.8]. theorem 4.17:an injective function f : x1 → x2 is rgfb-continuous, and x2 is ft1, then x1 is rgfbt1. proof: assume  and β be fuzzy singletons in x1 with various supports. 𝑓() and 𝑓(𝛽) belongs to x2, since, f is injective. as, x2 is ft1 space hence, by the statement there occurs fuzzy-open sets o1 & o2 in x2 such that, 𝑓() ≤ 𝑂1 ≤ 1 − 𝑓(𝛽) and 𝑓(𝛽) ≤ 𝑂2 ≤ 1 − 𝑓( )   ≤ 𝑓 −1(𝑂1) ≤ 1 − 𝛽 and 𝛽 ≤ 𝑓 −1(𝑂2) ≤ 1 − . since, f is rgfb-continuous 𝑓 −1(𝑂1) and 𝑓 −1(𝑂2) are rgfb-open in x1. this shows that, x1 is rgfbt1 space[4.3 ]. theorem 4.18: an injective function f : x1 → x2 is rgfb-irresolute, and x2 is rgfbt1, then x1 is rgfbt1. proof: assume  & β be fuzzy singletons in with various supports. since f is injective, 𝑓() & 𝑓(𝛽) belongs to x2. as x2 is rgfbt1, there occurs two rgfbos o1& o2 in x2 so that 𝑓() ≤ 𝑂1 ≤ 1 − 𝑓(𝛽) and 𝑓(𝛽) ≤ 𝑂2 ≤ 1 − 𝑓()   ≤ 𝑓 −1(𝑂1) ≤ 1 − 𝛽 𝑎𝑛𝑑 𝛽 ≤ 𝑓 −1(𝑂2) ≤ 1 − . since, f is rgfbirresolute, then 𝑓 −1(𝑂1) 𝑎𝑛𝑑 𝑓 −1(𝑂2) are rgfbos(x1). this shows that, x1 is rgfbt1 space[ 4.3]. theorem 4.19:if f : x1 → x2 is strongly rgfb-continuous and x2 is rgfbt1, then x1 is ft1. proof: assume  & β be fuzzy singletons in x1 with various supports. since, f is injective, 𝑓() &𝑓(𝛽) belong to x2. as, x2 is rgfbt1, there occurs two rgfboss o1 and o2 in x2 so that, 𝑓() ≤ 𝑂1 ≤ 1 − 𝑓(𝛽) and 𝑓(𝛽) ≤ 𝑂2 ≤ 1 − 𝑓()   ≤ 𝑓 −1(𝑂1) ≤ 1 − 𝛽 and 𝛽 ≤ 𝑓 −1(𝑂2) ≤ 1 − . since, f is strongly rgfb-continuous, therefore 𝑓 −1(𝑂1) & 𝑓 −1(𝑂2) are fuzzy-open in x1. this shows that, x1 is ft1 space[2.8 ]. theorem 4.20: an injective function f : x1 → x2 is rgfb-continuous, and x2 is ft2, then x1 is rgfbt2. proof: assume  & β be fuzzy singletons in x1 with various supports. since, f is injective, so 𝑓() & 𝑓(𝛽) belongs to x2 and 𝑓() ≠ (𝛽). since, x2 is ft2, therefore there occurs open fuzzy set o in x2 so that, 𝑓() ≤ 𝑂 ≤ varsha joshi and dr.jenifer j.karnel 160 cl(𝑂) ≤ 1 − 𝑓(𝛽)   ≤ 𝑓 −1(𝑂) ≤ 𝑓 −1[𝐶𝑙(𝑂)] ≤ 1 − 𝛽. since, f is rgfbcontinuous 𝑓 −1(𝑂) is rgfbcs(x1). hence,  ≤ 𝑓 −1(𝑂) ≤ 𝑓 −1[cl(𝑂)] ≤ 𝑓 −1[rgfbcl(𝑂)] ≤ rgfbcl[𝑓 −1[(𝑂)] ≤ 1 − 𝛽. that is,  ≤ 𝑓 −1(𝑂) ≤ rgfbcl[𝑓 −1[(𝑂)] ≤ 1 − 𝛽. this shows that, x1 is rgfbt2[4.7]. theorem 4.21: an injective function f : x1 → x2 is rgfb-irresolute, and x2 is rgfbt2. then, x1 is rgfbt2. proof: obvious. theorem 4.22: an injective function f : x1 → x2 is strongly rgfb-continuous, and x2 is rgfbt2. then, x1 is ft2. proof: obvious. theorem 4.23: an injective function f : x1 → x2 is rgfb-continuous, and x2 is ft2 1 2 . then, x1 is rgfbt2 1 2 . proof: assume  & β be fuzzy singletons in x1 with various supports. since, f is injective, then 𝑓() and 𝑓(𝛽) belongs to x2 and 𝑓 () ≠ 𝑓(𝛽). since, x2 is ft2 1 2 , then there occurs open fuzzy sets o1 and o2 in x2 such that, 𝑓() ≤ 𝑂1 ≤ 1 − 𝑓(𝛽), 𝑓(𝛽) ≤ 𝑂2 ≤ 1 − 𝑓()and cl𝑂1 ≤ 1 − cl𝑂2   ≤ 𝑓 −1(𝑂1) ≤ 1 − 𝛽 ,𝛽 ≤ 𝑓 −1(𝑂2) ≤ 1 −  and cl𝑓 −1(𝑂1) ≤ 1 − cl𝑓 −1(𝑂2). since, 𝑓 is rgfb-continuous 𝑓 −1(𝑂1) and 𝑓 −1(𝑂2) are rgfbos(x1). cl(𝑓 −1(𝑂1)) ≤ rgfbcl(𝑓 −1(𝑂1)) and1 − 𝐶l(𝑓 −1(𝑂2)) ≤ 1 − rgfbcl(𝑓 −1(𝑂2)). hence, rgfbcl(𝑓 −1(𝑂1)) ≤ 1 − rgfbcl(𝑓 −1(𝑂2)). this shows that, x1 is rgfbt2 1 2 [ 4.11]. acknowledgements the authors are grateful to principal of sdmcet, dharwad and management sdm society for their support. regular generalized fuzzy b-separation axioms in fuzzy topology 161 references [1] azad,k. 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(1965). fuzzy sets, information and control,8, pp.338-353. microsoft word r.m.7 cap.12.doc ratio mathematica 27 (2014) 27-36 issn:1592-7415 some properties of residual mapping and convexity in ∧-hyperlattices reza ameria, mohsen amiri-bideshkib, a. borumand saeidc aschool of mathematics, statistics and computer science, college of sciences, university of tehran, p.o. box 14155-6455, teheran, iran, rameri@ut.ac.ir bdepartment of mathematics, payame-noor university, tehran, iran, amirimohsen61@yahoo.com cdepartment of pure mathematics, faculty of mathematics and computer, shahid bahonar university of kerman, kerman, iran, arsham@uk.ac.ir abstract the aime of this paper is the study of residual mappings and convexity in hyperlattices. to get this point, we study principal down set in hyperlattices and we give some conditions for a mapping between two hyperlattices to be equivalent with a residual maping. also, we investigate convex subsets in ∧-hyperlattices. key words: residuated map, convex, down-set, hyperideal, hyperfilter. 2000 ams: 06f35, 03g25. 1 introduction hyperalgebras (multialgebra) are generalization of classical algebras that are introduced by f. marty in the eighth congress of scandinavian in 1934 [11]. in [4], ameri and m. m. zahedi introduced and studied notion of hyperalgebraic systems. in [2], ameri and nozari studied relationship between the 27 reza ameri, mohsen amiri-bideshki, a. borumand saeid categories of multialgebra and algebra. c. pelea and i. purdea have been proved that complete hyperalgebra can be obtained from a universal algebra and a appropriate congruence on it. also, pelea and others studied multialgebra, direct limit, and identities, for more details see [16, 17, 18, 19]. hyperalgebras (multialgebra) are generalization of classical algebras that are introduced by f. marty in the eighth congress of scandinavian in 1934 [11]. in [4], ameri and m. m. zahedi introduced and studied notion of hyperalgebraic systems. in [2], ameri and nozari studied relationship between the categories of multialgebra and algebra. c. pelea and i. purdea have been proved that complete hyperalgebra can be obtained from a universal algebra and a appropriate congruence on it. also, pelea and others studied multialgebra, direct limit, and identities, for more details see [16, 17, 18, 19]. theory of hyperlattices introduced by konstantinidou and j. mittas in 1977[9]. in [10], g. a. moghani and a. r. ashrafi proved that in some cases the set of all subhypergroups g has a hyperlattice structure . in [24], x. l. xin and x. g. li studied hyperlattices and quotient hyperlattices. in [5], a. asokkumar in 2007 proved that under certain conditions, the idempotent elements of a hyperring form a hyperlattice and the orthogonal idempotent elements form a quassi-distributive hyperboolean algebra. in [1], r. ameri, m. amiri bideshki, and a. borumand said studied prime hyperfilters (hyperideals) in hyperlattices. also, they gave some examples of ∧-hyperlattices and dual distributive ∧-hyperlattices. in section 3, down set and residual maps in hyperlattices are studied and some properties of them are given. in section 4, convex subsets of a hyperlattice and some properties of them are given. 2 preliminary in this section we give some results of hyperlattices that we need to develop our paper. definition 2.1. [1] let l be a nonempty set. l is called a ∧− hyperlattice if (i) a ∈ a∧a,a∨a = a, (ii) a∧ b = b∧a,a∨ b = b∨a, (iii) a∧ (b∧ c) = (a∧ b) ∧ c,a∨ (b∨ c) = (a∨ b) ∨ c, 28 some properties of residual mapping and convexity in ∧-hyperlattices (iv) a ∈ (a∧ (a∨ b)) ∩ (a∨ (a∧ b)), (v) a ∈ a∧ b =⇒ a∨ b = b, for all a,b,c ∈ l. let a,b ⊆ l. then: a∧b = ∪{a∧ b|a ∈ a,b ∈ b}; a∨b = {a∨ b|a ∈ a,b ∈ b}. example 2.2. let (l,∨,∧) be a lattice and define a⊕ b = {x | x ≤ a∧ b}. then (l,∨,⊕) is a ∧− hyperlattice. definition 2.3. [1] let l be a ∧−hyperlattice. we say that l is bounded if there exist 0, 1 ∈ l, such that 0 ≤ x ≤ 1, for all x ∈ l. we say that 0 is the least element of l and 1 is the greatest element of l. example 2.4. let l = {0,a, 1}, and define ∧-hyper operation and ∨operation on l with tables 3. then (l,∧,∨) is a bounded ∧-hyperlattice. ∧ 0 a 1 0 {0} {0} {0} a {0} {a, 0} {a, 0} 1 {0} {a, 0} l (a) ∨ 0 a 1 0 0 a 1 a a a 1 1 1 1 1 (b) table 1 definition 2.5. [1] let i and f are nonempty subsets of l. then: (i) i is called hyperideal if the following conditions hold. (a) if x,y ∈ i, then x∨y ∈ i, (b) if x ∈ i and a ∈ l, such that a ≤ x, then a ∈ i. (ii) f is called hyperfilter if the following conditions hold. (a) if x,y ∈ f , then x∧y ⊆ f, (b) if x ∈ f and a ∈ l, such that x ≤ a, then a ∈ f . (iii) a hyperideal i is called prime if x∧y ∈ i, then x ∈ i or y ∈ i, for all x,y ∈ l. (iv) a hyperfilter f is called prime if x ∈ f or y ∈ f , where (x∧y)∩f 6= ∅, for all x,y ∈ l. 29 reza ameri, mohsen amiri-bideshki, a. borumand saeid 3 resedual mappings in ∧-hyperlattices in this section, we are going to introduce down-set and resedual mapping in ∧-hyperlattice. let l be a ∧-hyperlattice. definition 3.1. let ∅ 6= a ⊆ l. a is called a down-set, if x ∈ a and y ≤ x, then y ∈ a. example 3.2. every hyperideal of l is a down-set that is called principal down-set. example 3.3. let l = {0,a,b, 1}. ∧ and ∨ are given by table 2 and 3. ∧ 0 a b 1 0 {0} {0} {0} {0} a {0} {0,a} {0} {0,a} b {0} {0} {0,b} {0,b} 1 {0} {0,a} {0,b} {1} table 2: ∨ 0 a b 1 0 0 a b 1 a a a 1 1 b b 1 b 1 1 1 1 1 1 table 3: i = {0,a,b} is a down-set, but it is not a hyperideal. we have a,b ∈ i and a∨ b = 1 /∈ i. let x ∈ l and x↓ = {y ∈ l|y ∈ x∧y}. proposition 3.4. ∀x ∈ l,x↓ is a down set. x↓ is called a principal down-set. proposition 3.5. let l be a dual distributive ∧-hyperlattice. then every principal down-set is a hyperideal. proof. if a ⊆ l and a∨ b ⊆ a, for all a,b ∈ l, then a is called join-closed. 30 some properties of residual mapping and convexity in ∧-hyperlattices corollary 3.6. let i ⊆ l. then i is an ideal if and only if i is a down-set and it is a join-closed set. proposition 3.7. let l and k be hyperlattice. if f : l −→ k is a isotone map and a ⊆ l is a down-set, then f(a) is a down-set. proof. since a is a down-set, there exists x ∈ l such that a = x↓. it is sufficient set f(a) = f(x)↓. let l and k be hyperlattices and f : l −→ k is a mapping. we define two map f→ and f← that f→ is called direct image map and f← is called inverse image map. f→ : p(l) −→ p(k) is defined by f→(x) = {f(x)|x ∈ l}, for all x ⊆ l, and f← : p(k) −→ p(l) is defined by f←(y ) = {x{∈ l|f(x) ∈ y} for all y ⊆ k. definition 3.8. a mapping f : l −→ k is called residuated if the inverse image under f of every principal down-set of k is a principal down-set of l. example 3.9. let l be a ∧-hyperlattice and a ⊆ l. we define fa : p(l) −→ p(l) by fa(b) = a∩b, for all b ∈ p(l). then fa is a residuated and residual g is given by ga(c) = c ∪a′, where that a′ = l\a. example 3.10. let l be a ∧-hyperlattice. mapping f : p(l) −→ p(l) that is defined by f(a) = a, for all a ∈ p(l), is a residuated mapping. theorem 3.11. let l and k be two hyperlattices. a mapping f : l −→ k is a residuated iff f is a is isotone and there exists an isotone mapping g : k −→ l such that gof ≥ idl and fog ≤ idk . proof. for all x ∈ l, x ∈ f←[f(x)↓]. if y ≤ x, then y ∈ f←[f(x)↓]. we have: f(x)↓ = {y|y ≤ f(x)} and f←[f(x)↓] = {t ∈ l|f(t) ∈ f(x)↓}. y ∈ f←[f(x)↓], so f(y) ≤ f(x). then f is isotone. by assumption we have (∀y ∈ k)(∃x ∈ l) such that f←(y↓) = x↓. now, for every given y ∈ k, this element x is clearly unique. so we can define a mapping g : k −→ l by g(y) = x. since f← is isotone, it follow that so is g. for this mapping g, we have: g(y) ∈ g(y)↓ = x↓ = f←(y↓). so, f[g(y)] ≤ y, for all y ∈ k and therefore fog ≤ idk. also, x ∈ f←[f(x)↓] = g[f(x)]↓, so that x ≤ g[f(x)], for all x ∈ l, and therefore gof ≥ id l . conversely, since g is isotone, we have: f(x) ≤ y =⇒ x ≤ g[f(x)]. 31 reza ameri, mohsen amiri-bideshki, a. borumand saeid also, we have: x ≤ g(y) =⇒ f(x) ≤ f[g(x)] ≤ y. it follows from these observations that f(x) ≤ y iff x ≤ g(y) and therefore f←(y↓) = g(y)↓. proposition 3.12. the residual of f is unique. proof. suppose that g and g′ are residual of f. then we have: g = idlog ≤ (g′of)og = g′o(fog) ≤ g′oidk = g′. similarly, g′ ≤ g, then g = g′. we shall denote residual of f, by f+. proposition 3.13. mapping f : l =⇒ k is residuated iff for every y ∈ k, there exists g(y) = maxf←(y↓) = max{x ∈ l|f(x) ≤ y}. moreover, f+of ≥ idl and fof + ≤ idk . definition 3.14. let f : l −→ k be a residuated mapping. then f is called range closed if im(f) is a down-set of k. example 3.15. let l be a ∧-hyperlattice with a top element 1. given a ∈ l, consider the mapping fa : l −→ l given by: fa(x) = fa is residuated. clearly, im(fa) is the down-set a ↓ of l then fa is a range closed. remark 3.16. in example 3.15, l must have top element 1. example 3.17. let n be the set of natural numbers. we define ∧-hyperoperation and ∨ operation by: a∧ b = {m ∈ n|m ≤ min{a,b}}; a∨ b = max{a,b},foralla,b ∈ n. then (l,∧,∨) is a ∧-hyperlattice. consider f : n −→ n by f(x) = x, for all x ∈ n. f is a residated mapping, but it is not range closed. theorem 3.18. let f : l −→ k be a residuated mapping. then f = f+ iff f2 = idl. proof. =⇒ it is obvious. ⇐= since f is residuated, then f2 = idl. by f2 = idl, we have fof ≤ idl and fof ≥ idl. so f = f+. theorem 3.19. let l and k be two ∧-hyperlattices and let l has a top element 1. if f : l −→ k be a residuated mapping, then the following statements are equivalent. 32 some properties of residual mapping and convexity in ∧-hyperlattices (i) f is range closed. (ii) forally ∈ k inf{y,f(1)} there exists and it equal to ff+(y). proof. (i → ii):we have f+(y) ≤ 1, for all y ∈ l and by isotonic f, ff+(y) ≤ f(1). also ff+(y) ≤ y, for all y ∈ k. so ff+(y) is a lower bound of f(1) and y. we must show that ff+(y) is the greatest lower bound of f(1) and y. suppose that x ∈ k is such that x ≤ y and x ≤ f(1). by (i), we have x = f(z), for some z ∈ l and f(z) ≤ y; since f+ is isotone, f+f(x) ≤ f+(y). we have z ≤ f+f(x), so z ≤ f+(y). by isotonic f, f(z) ≤ ff+(y), then x ≤ ff+(y). thus inf{y,f(1)} = ff+(y). (ii → i): we claim that im(f) = f(1)↓. we have x ≤ 1, for all x ∈ l, then f(x) ≤ f(1), for all x ∈ l. so im(f) ⊆ f(1)↓. let y ∈ k be such that y ≤ f(1). then by (ii), ff+(y) = inf{y,f(1)} = y. we know ff+(y) ∈ im(f), so y ∈ im(f). thus f(1)↓ ⊆ im(f). therefore im(f) = f(1)↓. proposition 3.20. let f : l −→ k and g : k −→−→ m be residual map. then gof so is, also (gof)+ = f+og+. 4 convexity in ∧-hyperlattice in this section, we are going to introduce convex subsets in ∧-hyperlattices and we are going to give some properties of convex subsets. proposition 4.1. let f ⊆ l. then f is a hyperfilter of l, if and only if (i) a,b ∈ f implies that a∧ b ∈ f . (ii) ∀a ∈ f and ∀x ∈ l, a∨x ∈ f . proof. since f is a filter, ∀a,b ∈ f , a ∧ b ∈ f . we know a ≤ a ∨ x, then a∨x ∈ f. so (i) and (ii) hold. conversely, let a ∈ f and a ≤ x. so, a ∨ x = x, by (ii) a ∨ x ∈ f , then x ∈ f . proposition 4.2. every hyperfilter of a ∧-hyperlattice l is a ∧-subhyperlattice. remark 4.3. converse of the above proposition does not hold. consider hyperlattice in the example 3.2. a = {0,a} is a subhyperlattice. we have a ≤ 1 and 1 /∈ a, then a is not a filter. remark 4.4. every hyperideal of l is not a subhyperlattice. also, every subhyperlattice is not an ideal. 33 reza ameri, mohsen amiri-bideshki, a. borumand saeid definition 4.5. let ∅ 6= k ⊆ l. we say k to be convex subset, if a,b ∈ k and c ∈ l such that a ≤ c ≤ b, then c ∈ k. example 4.6. consider hyperlattice l in example 3.2. then a = {0,a} is a convex subset, but b = {0, 1} is not a convex subset. we have 0 ≤ a ≤ 1 and a /∈ b. proposition 4.7. every hyperideal (hyperfilter) of l is a convex subset of l. remark 4.8. every convex subset of l is not a hyperideal (a filter). consider hyperlattice l in example 3.2. then k = {a,b, 1} is a convex subset, but it is not a hyperideal (0 /∈ k). also, k is not a hyperfilter (a∧ b = {0} and 0 /∈ k). theorem 4.9. let l has a bottom element 0 and let k be a convex subset of l. if k is a chain and 0 ∈ k, then k is a hyperideal of l. remark 4.10. in example 4.9, k must be a chain; also k must contain bottom element 0. example 4.11. let l be hyperlattice in example3.2 (i) k1 = {a,b, 0} is a convex subset, but it is a not chain(a,b are not comparable). since a∨ b /∈ k1, k1 is not a hyperideal. (ii) k2 = {a,b, 1} is a convex subset, but it is not a hyperideal (0 /∈ k2). example 4.12. consider hyperlattice l in example 3.17. thenk = {2, 3, 4, ..., 10} is a convex subset; since k does not has bottom element 1, it is not a hyperideal. proposition 4.13. every principal down-set of l is a convex subset. theorem 4.14. let i be a hyperideal and f be a hyperfiler of l, such that i ∩ f 6= ∅, then i ∩ f is a convex sub-hyperlattice if and only if for all a,b ∈ i ∩f , a∧ b ⊆ i. proposition 4.15. if ki, ∀i ∈ i is a convex sub-hyperlattice of l, then ∩i∈iki is so. theorem 4.16. let k1 and k2 be convex sub-hyperlattices of l and let 0 ∈ k1 ∩ k2. then k1 ∪ k2 is a convex sub-hyperlattice if and only if k1 ⊆ k2 or k2 ⊆ k1. 34 some properties of residual mapping and convexity in ∧-hyperlattices proof. let k1 ∪k2 be a convex subhyperlattice, but k1 * k2 or k2 * k1. so, there exist a,b ∈ l, such that a ∈ k1 \ k2 and k1 \ k2. since a,b ∈ k1 ∪k2 and k1 ∪k2 is a sub-hyperlattice, a∨ b ∈ k1 ∪k2; it implies that a∨ b ∈ k1 or a∨ b ∈ k2. if a∨ b ∈ k1, 0 ≤ a ≤ a∨ b, then a ∈ k2, which is a contradiction; if a ∨ b ∈ k2, then we conclude that b ∈ k1, which is a contradiction. the converse is obvious. acknowledgements the first author partially has been supported by ”algebraic hyperstructure excellence, tarbiat modares university, tehran, iran” and ”research center in algebraic hyperstructures and fuzzy mathematics, university of mazandaran, babolsar, iran”. references [1] r. ameri, m. amiri bideshki and a. borumand saeid, on prime hyperfilter (hyperideal) in ∧-hyperlattices, submitted [2] r. ameri and t. nozari, ”a connection between categories of multialgebras and algebra”, italian journal of pure and applied mathematics, vol. 27, 201-208, 2010. [3] r. ameri and i. g. rosenberg, ”congruences of multialgebras”, j. of multi-valued logic & soft computing, vol. 00, 1-12, 2009. [4] r. ameri and m. m. zahedi, ”hyperalgebraic system”, italian journal of pure and applied mathematics, vol. 6, 21-39, 1999. [5] a. asokkumar, ”hyperlattice formed by the idempotents of a hyperring”, international journal of mathematics, vol. 38, 209-215, 2007. [6] b. b. n. koguep, c. nkuimi and c. lele, ”on fuzzy ideals of hyperlattice”, international journal of algebra, vol. 2, 739-750. 2008. [7] m. konstantinidou serafimidou, ”modular hyperlattices”, γ ranktika tes akademias athenon, vol. 53, 202-218, 1978. [8] m. konstantinidou serafimidou, ”distributive and complemented hyperlattices”, praktika tes akademias athenon, vol. 56, 339-360, 1981. 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[24] x. l. xin and x. g. li, ”on hyperlattice and quotient hyperlattice”. 36 ratio mathematica vol.34, 2018, pp. 85–93 issn : 1592-7415 eissn : 2282-8214 some kinds of homomorphisms on hypervector spaces elham zangiabadi∗, zohreh nazari† received: 24-05-2018 . accepted: 24-06-2018. published: 30-06-2018 doi: 10.23755/rm.v34i0.416 abstract in this paper, we introduce the concepts of homomorphism of type 1 , 2 and 3 and good homomorphism . then we investigate some properties of them. keywords : hypervector space , homomorphism , homomorphism of type 1 , 2 and 3 , good homomorphism . 2010 ams subject classifications : 20n20 , 22a30 . 1 introduction and preliminaries the concept of hyperstructure was first introduced by marty [13] in 1934 . he defined hypergroups and began to analysis their properties and applied them to groups and rational algebraic functions . tallini introduced the notion of hypervector spaces [14] , [15] and studied basic properties of them . homomorphisms of hypergroups are studied by several authers ([2] [12]) . since some kinds of homomorphisms on hypergroup were defined , we encourage to define them on hypervector spaces . in this paper , we introduce the concept of homomorphism of type 1 , 2 and 3. and give an example of a homomorphism that is not a homomorphism of type 1, 2 and 3. we show that if f be a homomorphism of type 1, 2 and 3, then f is a homomorphism and every homomorphism of type 2 or 3 ∗department of mathematics, vali-e-asr university, rafsangan, iran. e.zangiabadi@vru.ac.ir †department of mathematics, vali-e-asr university, rafsangan, iran. z.nazari@vru.ac.ir 85 elham zangiabadi and zohreh nazari is a homomorphism of type 1. also, we define a good homomorphism and obtain that every homomorphism of type 2 is a good homomorphism and every good homomorphism is a homomorphism. finally, w e prove that every onto strong homomorphism is a good homomorphism. let us recall some definitions which are useful in our results . definition 1.1. a hypervector space over a field k is a quadruplet (v, +,◦, k) such that (v, +) is an abelian group and ◦ : k ×v → p∗(v ) is a mapping of k × v into the power set of v (deprived of the empty set) , such that (a + b)◦x ⊆ (a◦x) + (b◦x), ∀a, b ∈ k, ∀x ∈ v, (1) a◦ (x + y) ⊆ (a◦x) + (a◦y), ∀a ∈ k, ∀x, y ∈ v, (2) a◦ (b◦x) = (ab)◦x, ∀a, b ∈ k, ∀ x ∈ v, (3) x ∈ 1◦x, ∀x ∈ v, (4) a◦ (−x) = −a◦x, ∀a ∈ k, ∀x ∈ v. (5) definition 1.2. let (v, +,◦, k) be a hypervector space . then h ⊆ v is a subspace of v , if 1) the zero vector , 0 , is in h , 2) u, v ∈ h, then u + v ∈ h , 3) u ∈ h, r ∈ k, then r ◦u ⊆ h . definition 1.3. let (v, +,◦, k) and (w,⊕,∗, k) be two hypervector spaces . a mapping f : v → w is called 1) a homomorphism , if ∀r ∈ k, ∀x, y ∈ v : f(x + y) = f(x)⊕f(y), (6) f(r ◦x) ⊆ r ∗f(x). (7) 2) a strong homomorphism, if ∀r ∈ k, ∀x, y ∈ v : f(x + y) = f(x)⊕f(y), (8) f(r ◦x) = r ∗f(x). (9) 86 some kinds of homomorphisms on hypervector spaces 2 the main results in this paper, the ground field of a hypervector space v is presented with k, this field is usually considered by r or c. let (v, +,◦) and (w,⊕,∗) be two hypervector spaces and f : v → w be a mapping. we employ for simplicity of notation xf = f−1(f(x)) and for a subset a of v , af = f−1(f(a)) = ⋃ {xf : x ∈ a}. lemma 2.1. let r ∈ k and x ∈ v . then the following statements are valid: i) r ◦x ⊆ (r ◦x)f, ii) r ◦x ⊆ r ◦xf, iii) (r ◦x)f ⊆ (r ◦xf)f, iv) r ◦xf ⊆ (r ◦xf)f. definition 2.1. let (v, +,◦, k) and (w,⊕,∗, k) be two hypervector spaces and f : v → w be a map such that f(x + y) = f(x) ⊕ f(y), for all a, b ∈ v . then, for any r ∈ k and x, y ∈ v, f is called a homomorphism of i) type 1, if f−1(r ∗f(x)) = (r ◦xf)f, ii) type 2, if f−1(r ∗f(x)) = (r ◦x)f, iii) type 3, if f−1(r ∗f(x)) = (r ◦xf). theorem 2.1. let (v, +,◦, k) and (w,⊕,∗, k) be two hypervector spaces, a be a non-empty subset of v and f : v → w be a map such that f(a + b) = f(a)⊕f(b), for all a, b ∈ v . then, f is a homomorphism of i) type 1 implies f−1(r ∗f(a)) = (r ◦af)f, ii) type 2 implies f−1(r ∗f(a)) = (r ◦a)f, iii) type 3 implies f−1(r ∗f(a)) = (r ◦af). proof. each part is established by a straightforward set theoretic argument. example 2.1. let (w, +, ·, k) be a classical vector space, p be a proper subspace of w , w1 = (w, +, ·, k) and w2 = (w,⊕,◦, k) that r◦a = r ·a + p for r ∈ k and a ∈ w. then w1 and w2 are hypervector spaces. let f : w1 → w2 be the function defined by f(x) = k ·x, where k ∈ k. we show 87 elham zangiabadi and zohreh nazari that f is a homomorphism, but not a homomorphism of type 1, 2 and 3. for every r ∈ k and x ∈ w1 we have f(r ·x) = rk ·x rk ·x + p = r ◦f(x). thus f is a homomorphism. since f is one to one, we obtain xf = x, for x ∈ w. it followes that (r ·xf)f = (r ·x)f = (r ·xf) = (r ·x). on the other hand, f−1(r ◦f(x)) = f−1(kr ·x + p) = {t ∈ w1 : f(t) ∈ kr ·x + p} = {t ∈ w1 : k · t ∈ kr ·x + p} = {t ∈ w1 : k · t−kr ·x ∈ p}. hence, f−1(r ◦f(x)) 6= r ·x. therefore, f is not a homomorphism of type 1, 2 and 3. theorem 2.2. let (v, +,◦, k) and (w,⊕,∗, k) be two hypervector spaces and f : v → w be a homomorphism of type n, for n=1,2,3. then f is a homomorphism map. proof. if f be a homomorphism of type 1. then by using lemma 2.1, we have f(r ◦x) ⊆ f(r ◦xf) ⊆ f((r ◦xf)f) = f(f−1(r ∗f(x)) ⊆ r ∗f(x). suppose f is a homomorphism of type 2. then f(r ◦x) ⊆ f((r ◦x)f) = f(f−1(r ∗f(x)) ⊆ r ∗f(x). similarly, if f is a homomorphism of type 3, then f(r ◦x) ⊆ f(r ◦xf) = f(f−1(r ∗f(x))) ⊆ r ∗f(x). lemma 2.2. let f be a homomorphism. then (r ◦xf)f ⊆ f−1(r ∗f(x)). 88 some kinds of homomorphisms on hypervector spaces proof. since f is a homomorphism, for all r ∈ k and x ∈ v, we have f(r ◦xf) ⊆ r ∗f(xf). since r ∗ f(xf) = r ∗ f(f−1(f(x)) ⊆ r ∗ f(x), hence, f(r ◦ xf) ⊆ r ∗ f(x). therefore, (r ◦xf)f ⊆ f−1(r ∗f(x)). proposition 2.1. let (v, +,◦, k) and (w,⊕,∗, k) be two hypervector spaces and f : v → w be a homomorphism of type 2 or 3. then f is a homomorphism of type 1. proof. suppose that r ∈ k, x ∈ v and f : v → w be a homomorphism of type 2, then by lemma 2.2 we have (r ◦x)f ⊆ (r ◦xf)f ⊆ f−1(r ∗f(x)) = (r ◦x)f. similarly, if f is a homomorphism of type 3, then r ◦xf ⊆ (r ◦xf)f ⊆ f−1(r ∗f(x)) = r ◦xf. proposition 2.2. let (v, +,◦, k) and (w, +⊕,∗, k) be two hypervector spaces and f : v → w be an onto mapping. then, given r ∈ k and x ∈ v , f is a homomorphism of i) type 1 if and only if f(r ◦xf) = r ∗f(x), ii) type 2 if and only if f(r ◦x) = r ∗f(x). proof. since f is onto, we obtain ff−1(r ∗f(x)) = r ∗f(x). thus, (i) and (ii) are trivial. corolary 2.1. let (v, +,◦, k) and (w,⊕,∗, k) be two hypervector spaces, a and b be non-empty subsets of v and f : v → w be an onto mapping.then, f is homomorphism of i) type 1 implies f(r ◦af) = r ∗f(a), ii) type 2 implies f(r ◦a) = r ∗f(a). 89 elham zangiabadi and zohreh nazari remark 2.1. on onto homomorphisms between hypervector spaces, a homomorphism of type 2 is equivalent with a strong homomorphism. theorem 2.3. let (v1, +1,◦1, k), (v2, +2,◦2, k) and (v3, +3,◦3, k) be hypervector spaces. for n = 1, 2, 3, let f be a homomorphism of type n of v1 onto v2 and g be a homomorphism of type n of v2 onto v3. then, gf is a homomorphism of type n of v1 onto v3. proof. let x, y ∈ v . we have gf(x+1 y) = g(f(x)+2 f(y)) = gf(x)+3 gf(y)). one can easily seen that xgf = f−1(f(x)g). let n = 1. by above relation, we obtain gf(r ◦xgf) = gf(r ◦f−1(f(x)g)). since f is onto, there exists a subset a of v such that f(a) = f−1(f(x)g). by corollary 2.1, we obtain gf(r ◦1 f−1(f(x)g)) = g(r ◦2 f(x)g). then, by proposition 2.2, we have g(r ◦2 f(x)g) = r ◦3 gf(x). let n = 2. similar to the previous case, but simpler. let n = 3. since g is of type 3, (gf)−1(r ◦3 (gf)(x)) = f−1g−1r ◦3 (gf)(x)) = f−1(r ∗f(x)g). since f is onto, the item (iii) of theorem 2.1 implies f−1(r ◦2 f(x)g) = r ◦1 f−1(f(x)g) = r ◦1 xgf. definition 2.2. let a ∈ v and r ∈ k. we define a/r = {x ∈ v : a ∈ r ◦x}. proposition 2.3. let (v1, +,◦, k) and (v2,⊕,∗, k) be two hypervector spaces. if f : v1 → v2 be an onto mapping. then we have 1) f(a/r) = f(a)/r, if f is a homomorphism of type 2. 2) f(a)/r ⊆ f(af)/r, if f is a homomorphism of type 3. 90 some kinds of homomorphisms on hypervector spaces proof. 1) we know that an onto homomorphism of type 2 is a strong homomorphism. suppose that y ∈ f(a/r). then, there exists t ∈ a/r such that f(t) = y, so a ∈ r ◦ t and f(a) ∈ r ∗ f(t). it implies that y = f(t) ∈ f(a)/r. therefore, f(a/r) ⊂ f(a)/r. note that the inverse inclusion is always true. 2) if y ∈ f(a)/r, there is t ∈ v1 such that f(t) = y. since f is homomorphism of type 3, we have af ∈ r ◦ tf , which means that tf ∈ af/r, therefore y ∈ f(af)/r. definition 2.3. let (v, +,◦, k) and (w,∗,⊕, k) be two hypervector spaces and f : v → w be a map such that f(a + b) = f(a)⊕f(b). then f is called a good homomorphism if f(a/r) = f(a)/r, for any a, b ∈ v and r ∈ k. remark 2.2. according to proposition 2.3, if f is a homomorphism of type 2, then f is a good homomorphism. theorem 2.4. let (v, +,◦, k) and (w,⊕,∗, k) be two hypervector spaces. if f : v → w be a good homomorphism then, f is a homomorphism. proof. let r ∈ k and a ∈ v1. if y ∈ f(r◦a), then, there exists t ∈ r◦a such that y = f(t). hence, f(a) ∈ f(t/r) = f(t)/r. abviously, y = f(t) ∈ r ∗f(a). theorem 2.5. let (v1, +1,◦1, k), (v2, +2,◦2, k), and (v3, +3,◦3, k) be hypervector spaces. let f be a good homomorphism of v1 to v2 and g be a good homomorphism of v2 to v3. then, gf is a good homomorphism of v1 to v3. proof. for every r ∈ k and a ∈ v1, we have gf(a/r) = g(f(a)/r) = gf(a)/r. proposition 2.4. let v and w be two hypervector spaces over k and f : v → w be a good homomorphism. then f(a/k) = f(a)/k, where a ⊆ v and a/k = ⋃ {a/r : a ∈ a, r ∈ k}. proof. let y ∈ f(a/k). there exist r ∈ k and a ∈ a such that y ∈ f(a/r) = f(a)/r ⊆ f(a)/k. conversely, let y ∈ f(a)/k. then, there exist r ∈ k and a ∈ v such that y ∈ f(a)/r = f(a/r) and so y ∈ f(a/k). theorem 2.6. let (v, +,◦, k) and (w,⊕,∗, k) be two hypervector spaces, f be onto strong homomorphism from v to w . then f is a good homomorphism. 91 elham zangiabadi and zohreh nazari proof. let f(t) ∈ f(x/r). so x ∈ r◦t. it followes that f(t) ∈ f(x)/r. therefore f(x/r) ⊆ f(x)/r. on the other hand, let y ∈ f(x)/r. since f is an onto mapping, there exists a t ∈ v such that y = f(t). hence, f(x) ∈ r ∗ f(t) = f(r ◦ t). thus x ∈ r ◦ t and then we have t ∈ x/r and y = f(t) ∈ f(x/r). therefore f(x)/r ⊆ f(x/r). this implies that f(x/r) = f(x)/r. references [1] r. ameri, o. r. dehghan, on dimension of hypervector spaces, eur. j. pure appl. math, 1 (2008), 32-50. [2] p. corsini, recent results in the theory of hypergroups, boll. unione mat. ital. (9), 2 (1983), 133-138. [3] p. corsini, v. leoreanu, applications of hyperstructure theory, kluwer academic publishers, advances in mathematics, 2003. [4] b. davvaz, isomorphism theorems of polygroups, bull. malays. math. sci. soc. (2), 33 (2010), 385-392. [5] b. davvaz, groups in polygroups, iran. j. math. sci. inform., 1 (2006), 2531. [6] j. e. eaton, theory of cogroups,duke math. j., 6 (1940), 101-107. [7] d. freni, strongly transitive geometric spaces : applications to hypergroups and semigroups theory, comm. algebra, 32 (2004), 969-988. [8] d. freni, a new characterization of the derived hypergroup via strongly regular equivalences, comm. algebra, 30 (2002), 3977-3989. [9] t. w. hungerford, algebra, graduate texts in mathematics, 73. springerverlag, new york-berlin, 1980. [10] j. jantosciak, homomorphisms, equivalences and reductions in hypergroups, riv. mat., 9 (1991), 23-47. [11] m. koskas, groupoides, demi-hypergroupes et hypergroupes. (french), j. math. pures appl. (9), 49 (1970), 155-192. [12] m. krasner, a class of hyperrings and hyperfiels, internat. j. math. math. sci., 6 (1983), 307-311. 92 some kinds of homomorphisms on hypervector spaces [13] f. marty, sur nue generalization de la notion do group, 8nd congress of the scandinavic mathematics, stockholm, (1934), 45-49. [14] m. s. tallini, weak hypervector space and norms in such spaces, algebraic hyper structurs and applications, jast, rumania, hadronic press, (1994), 199-206. [15] m. s. tallini, hypervector spaces, proceedings of the fourth international congress of algebraic hyperstructures and applications, xanthi, greece, ( 1990), 167-174. 93 ratio mathematica volume 40, 2021, pp. 113-121 on characterization of δ-topological vector space shallu sharma * tsering landol† sahil billawria‡ abstract the main objective of this paper is to present the study of δ-topological vector space. δ-topological vector space is defined by using δ-open sets and δ-continuous mapping which was introduced by j.h.h. bayati[3] in 2019. in this paper, along with basic inherent properties of the space, δ-closure and δ-interior operators are discussed in detail. we characterize some important properties like translation, dilation of the δ-topological vector space and an example of δ-topological vector space is also established. keywords: regular open set, δ-open set, δ-closed set, δ-continuous mapping and δ-topological vector space. 2020 ams subject classifications:57n17, 57n99, 54a05. 1 *department of mathematics, university of jammu jk-180006, india; shallujamwal09@gmail.com. †department of mathematics, university of jammu, jk-180006, india; tseringlandol09@gmail.com. ‡department of mathematics, university of jammu, jk-180006, india; sahilbillawria2@gmail.com. 1received on january 16th, 2021. accepted on june 23th, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.569. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 113 s. sharma, t. landol, s. billawria 1 introduction in functional analysis, topological vector space is one of the fundamental space being investigated by mathematicians due to the significant role played by it in other branches of mathematics such as operator theory, fixed point theory, variational inequality etc. the formalism of topological vector space belongs to kolmogroff [4] who was the first to introduce a well structured notion of topological vector space in his pioneering work done in 1934. since then, it has evolved further and many mathematicians have developed different generalizations of topological vector space. in 2015, the notion of s-topological vector space is being developed by m. khan et.al.[5], which is one of the generalization of topological vector space. later, many other significant generalizations of topological vector space are being introduced such as irresolute topological vector space [2], β-topological vector space [11], strongly preirresolute topological vector space [9], almost s-topological vector spaces [10], etc. 2 preliminaries in this paper, (x,τ)(or simply x) always means topological space on which no separation axioms are assumed unless stated explicitly. for a subset d of a space x , we denote closure and interior by cl(d) and int(d) respectively and neighborhood and δ-neighborhood of an element x in any topological space x is denoted by n(x) and nδ(x). definition 2.1. let b be a subset of a topological space (x,τ). then b is said to be (a) regular open [12] if b = int(cl(b)) (b) pre-open [6] if b ⊆ int(cl(b)) (c) β-open [1] if b ⊆ cl(int(cl(b))). definition 2.2. a subset c of a topological space x is called (a) regular closed if x \c is open i.e. c = cl(int(c)) (b) pre-closed if cl(int(c)) ⊆ c (e) β-closed if int(cl(int(c))) ⊆ c. definition 2.3. a subset d of a topological space x is said to be δ-open [13] if for each x ∈ d, there exist a regular open set r such that x ∈ r ⊆ d. remark 2.1. every regular open set is open and every open set is pre-open, while the converse need not be true. 114 on characterization of δ-topological vector space example 2.1. let r be a set of real numbers with usual topology. then int(cl(z)) = ∅, which implies z is not regular in topological space (r,τu). also, set of rational number denoted by q is pre-open but neither regular open nor open set in topological space (r,τu). the complement of δ-open set is δ-closed. the concept of δ-closure and δinterior are introduced by velicko [13] in 1968. the intersection of all δ-closed sets in x containing a subset d ⊆ x is called δ-closure of d and is denoted by clδ(d). a point x ∈ clδ(d) if and only if d ∩r 6= ∅, for a regular open set r in x containing x. a subset c of x is δ-closed if and only if c = clδ(c). the union of all δ-open sets in x that are contained in d ⊆ x is called δ-interior of d and is denoted by intδ(d). a point x ∈ x is called δ-interior of d ⊆ x if there exist a δ-open set u in x such that x ∈ u ⊆ d. definition 2.4. [4] let x be a vector space over the field f(r or c). let τ be a topology on x such that 1) for each x,y ∈ x, and for each open neighborhood w of x+y in x, there exist open neighborhoods u and v of x and y respectively in x such that u + v ⊆ w 2) for each λ ∈ f, x ∈ x and for each open neighborhood w of λ ·x in x, there exist open neighborhoods u of λ in f and v of x in x such that u ·v ⊆ w . then, the (x(f),τ) is called topological vector space. 3 δ-topological vector space in this section, we give an examples of δ-topological vector space and further illustrate the properties of this space. definition 3.1. [3] let x be a vector space over field f(r or c) with standard topology. let τ be a topology over x such that the following conditions hold: (a) for each x,y ∈ x and each open set r containing x + y, there exist δ-open set p and q containing x and y respectively, such that p + q ⊆ r; (b) for each λ ∈ f, x ∈ x and each open set r containing λx ∈ x, there exist δ-open set p and q containing λ and x respectively such that p.q ⊆ r. then the pair (x(f),τ) is said to be δ-topological vector space. following are some examples of δ-topological vector. example 3.1. let k = r with usual topology. let x = r with base b = {(a,b) : a,b ∈ r}. we shall show that (x(k),τ) is δ-topological vector space. for, we will check the following: (i) let x,y ∈ x. consider open set r = (x + y − �,x + y + �) in x containing 115 s. sharma, t. landol, s. billawria x + y. then we can choose δ-open sets p = (x−η,x+ η) and q = (y−η,y + η) in x containing x and y respectively such that p + q ⊆ r, for each η < � 2 . this establish the first condition of the definition of δ-topological vector space. (ii) let λ ∈ r and x ∈ x. consider an open set r = (λx − �,λx + �) in x containing λx. then, we have the following cases: case i: if λ > 0 and x > 0, then we can choose δ-open set p = (λ−η,λ + η) in r containing λ and q = (x−η,x + η) in x containing x such that p.q ⊆ r, for each η < � λ+x+1 . case ii: if λ < 0 and x < 0, then λx > 0. we choose δ-open set p = (λ−η,λ+η) of λ in r and q = (x−η,x + η) of x in r such that p.q ⊆ r, for η ≤ −� λ+x−1 . case iii: if λ > 0 and x = 0, (λ = 0 and x > o). we can choose δ-open sets as p = (λ − η,λ + η) (resp.(−η,η)) containing λ in k and q = (η,η) (resp.(x − η,x + η)) containing x in r such that p.q ⊆ r, for each η < � λ+1 (resp.(η < � x+1 )). case iv: if λ = 0 and x < 0, (λ = 0 and x > 0). we can choose δ-open sets as p = (η,η)(resp.(λ − η,λ + η)) containing λ in k and q = (x − η,x + η)(resp.(−η,η)) containing x in r, we have p.q ⊆ r, for every η < � 1−x(resp.(η < � 1−λ)). case v: if λ = 0 and x = 0. then, for δ-open set p = (η,η) and q = (η,η) of λ and x respectively such that p.q ⊆ r, for each η < √ �. this proves that the pair (x(k),τ) is δ-tvs. example 3.2. consider a vector space x = r of real number over the field k with the topology τ = {φ,qc,r}, where qc denotes the set of irrational numbers and the field k is endowed with standard topology. then (x(k),τ) is not δ-topological vector space. for x,y ∈ qc and open neighborhood qc of x + y in x, there doesn’t exist any δ-open sets p and q containing x and y respectively such that p + q ⊆ qc. theorem 3.1. [3] let d be any open subset of δ-topological vector space x. then (a) x + d ∈ δo(x), for each x ∈ x; (b) λd ∈ δo(x), for each non-zero scalar λ. theorem 3.2. let c be any closed subset of a δ-topological vector space x, then (a) x + c ∈ δc(x), for each x ∈ x; (b) λc ∈ δc(x), for each non-zero scalar λ. proof: (a) let y ∈ clδ(x + c), z = −x + y and r be an open set in x containing z. then there exist δ-open set p and q containing −x and y respectively, such that p + q ⊆ r. also, y ∈ clδ(x + c), y ∈ q and q is δ-open implies there exist regular open set q′ such that y ∈ q′ ⊆ q. so (x + c) ∩q′ 6= ∅. let a ∈ (x + c) ∩q′ ⇒ −x + a ∈ c ∩ (p ′ + q′) ⊆ c ∩ (p + q) ⊆ c ∩r 6= ∅. 116 on characterization of δ-topological vector space hence z ∈ cl(c) = c ⇒ −x + y ∈ c ⇒ y ∈ x + c. thus, x + c ∈ δc(x), for each x ∈ x. (b) assume that x ∈ clδ(λc) and r be open neighborhood of y = 1λx ∈ x. since x is δ−tv s, there exist δ-open neighborhood p of 1 λ in f and q of x in x such that p.q ⊆ r. by hypothesis, (λc)∩q′ 6= ∅, for regular open set q′ subset of q containing x. let a ∈ (λc)∩q′. now 1 λ a ∈ c∩p ′.q′ ⊆ c∩p.q ⊆ c∩r ⇒ c ∩r 6= ∅ i.e. y is a limit point of c and so y = 1 λ x ∈ cl(c) = c, since c is closed subset of x. hence y ∈ λc. since the inclusion λc ⊆ clδ(λc) holds generally, so clδ(λc) = λc. therefore, λc is δ-closed set in x. this completes the proof. theorem 3.3. for any subset d of δ-topological vector space x, (a) clδ(x + d) ⊆ x + cl(d), for each x ∈ x. (b) x + clδ(d) ⊆ cl(x + d), for each x ∈ x. proof: (a) let y ∈ clδ(x + d) and consider z = −x + y in x. let r be open neighborhood of z. by hypothesis, there exist δ-open set p and q containing -x and y respectively such that p + q ⊆ r. existence of δ-open set confirms the existence of regular open set p ′ and q′ such that −x ∈ p ′ ⊆ p and y ∈ q′ ⊆ q. since y ∈ clδ(x + d), (x + d) ∩ q′ 6= ∅. let a ∈ (x + d) ∩ q′. now, −x+a ∈ d∩(p ′ +q′) ⊆ d∩(p +q) ⊆ d∩r 6= ∅, which implies z ∈ cl(d). hence y ∈ x + cl(d). this completes the proof. (b) let z ∈ x + clδ(d). then z = x + y, for some y ∈ clδ(d). let r be any open neighborhood of z in x, then there exist δ-open neighborhood p and q of x and y respectively such that p + q ⊆ r. also, d ∩ q′ 6= ∅, for regular open set q′ ⊆ q containing y which implies d ∩ q 6= ∅. let a ∈ d ∩ q. then x + a ∈ (x + d) ∩ (p + q) ⊆ (x + d) ∩ r 6= ∅, which implies z is a limit point of x+d i.e z ∈ cl(x+d). hence the inclusion holds for each x ∈ x. theorem 3.4. for a subset d of δ-topological vector space x, the following are valid: (a) x + int(d) ⊆ intδ(x + d), for each x ∈ x. (b) int(x + d) ⊆ x + intδ(d), for each x ∈ x. proof: (a) assume y ∈ x + int(d). then,−x + y ∈ int(d). since x is δ-topological vector space, there exist δ-open sets p containing -x and q containing y in x such that q ⊆ int(d). also, δ-openness of p and q implies the existence of regular open set p ′ and q′ such that −x ∈ p ′ ⊆ p and y ∈ q′ ⊆ q satisfying p ′ + q′ ⊆ p + q ⊆ int(d). in particular, 117 s. sharma, t. landol, s. billawria −x + q′ ⊆ int(d) ⊆ d ⇒ q′ ⊆ x + d. thus there exist regular open set q′ containing y such that y ∈ q′ ⊆ x + d, which implies y is δ-interior point of x + d i.e. y ∈ intδ(x + d). hence the proof. (b) let y ∈ int(x + d), then y = x + a, for some a ∈ d. by definition of δ-topological vector space, there exist δ-open set p and q such that x ∈ p,a ∈ q satisfying p +q ⊆ int(x+d). hence, x+a ∈ p ′ +q′ ⊆ p +q ⊆ int(x+d), for each regular open set p ′ and q′ such that x ∈ p ′ ⊆ p and a ∈ q′ ⊆ q. now x + q′ ⊆ x + q ⊆ int(x + d) ⊆ x + d, which implies y ∈ x + intδ(d). hence the inclusion int(x + d) ⊆ x + intδ(d) holds. theorem 3.5. let d be any subset of δ-topological vector space x. then the following holds: (a) λclδ(d) ⊆ cl(λd), for every non-zero scalar λ. (b) clδ(λd) ⊆ λcl(d), for every non-zero scalar λ. proof the proof is trivial, omitted. theorem 3.6. let x be a δ-topological vector space and d be any subset of x. then the following holds: (a) int(λd) ⊆ λintδ(d), for every non-zero scalar λ. (b) λint(d) ⊆ intδ(λd), for every non-zero scalar λ. proof the proof is trivial, omitted. theorem 3.7. let c and d be any subset of a δ-topological vector space x. then clδ(c) + clδ(d) ⊆ cl(c + d). proof: let z ∈ clδ(c) + clδ(d). then z = x + y, where x ∈ clδ(c) and y ∈ clδ(d). let r be an open neighborhood of z in x. by definition of δ-topological vector space, there exist δ-open neighborhood p and q of x and y respectively such that p + q ⊆ r. since x ∈ clδ(c), c ∩ p ′ 6= ∅ for regular open set p ′ such that x ∈ p ′ ⊆ p and also y ∈ clδ(d), d ∩q′ 6= ∅ for regular open set q′ satisfying y ∈ q′ ⊆ q. let a ∈ c ∩ p ′ and b ∈ d ∩ q′ ⇒ (a + b) ∈ (c + d) ∩ (p ′ + q′) ⊆ (c + d) ∩ (p + q) ⊆ (c + d) ∩ r ⇒ (c + d) ∩ r 6= ∅. thus z is a closure point of (c + d) i.e. z ∈ cl(c + d). hence the inclusion holds. theorem 3.8. for any subsets c and d of δ-topological vector space x. then c + int(d) ⊆ intδ(c + d). 118 on characterization of δ-topological vector space proof: let z ∈ c + int(d) be arbitrary. then z = x + y, for some x ∈ c,y ∈ int(d), which results in −x + z ∈ int(d). by definition of δ-tvs, there exist δ-open neighborhood p and q containing -x and z respectively such that p + q ⊆ int(d). hence, there exist regular open sets p ′ and q′ containing -x and z respectively satisfying p ′ ⊆ p , q′ ⊆ q and p ′+q′ ⊆ p +q ⊆ int(d). in particular, −x + q′ ⊆ int(d) ⇒ q′ ⊆ x + int(d) ⊆ c + d. hence, there exist regular open set q′ containing z such that z ∈ q′ ⊆ c + d. therefore, z is δ-interior point of a + b. hence the proof. definition 3.2. [8] a function f : x → y is called δ-continuous if for each x ∈ x and each open neighborhood q of f(x), there exist open neighborhood p of x such that f(int(cl(p)) ⊆ int(cl(q)). lemma 3.1. [8] for a function f : x → x, the following are equivalent: (a) f is δ-continuous. (b) for each x ∈ x and each regular open set v containing f(x), there exist a regular open set u containing x such that f(u) ⊆ v . (c) f([a]δ) ⊂ [f(a)]δ, for every a ⊂ x. (d) [f−1(b)]δ ⊂ f−1([b]δ), for every b ⊂ x. (e) for every regular closed set f of y, f−1(f) is δ-closed in x. (f) for every δ-closed set v of y, f−1(v ) is δ-closed in x. (g) for every δ-open set v of y, f−1(v ) is δ-open in x. (h) for every regular-open set v of y, f−1(v ) is δ-open in x. theorem 3.9. [3] let x be δ-topological vector space, then the following are true: (a) the translation mapping ga : x → x defined by ga(b) = a + b, ∀b ∈ x is δ-continuous. (b) the mapping gλ : x → x defined by gλ(a) = λa, ∀a ∈ x is δ-continuous, where λ is a fixed scalar. theorem 3.10. for a δ-topological vector space x, the mapping φ : x×x → x defined by φ(x,y) = x + y, ∀x ∈ x ×x is δ-continuous. proof: take arbitrary elements x, y in x and let r be regular open neighborhood of x + y which implies r is open neighborhood of x+y. then by hypothesis, there exist δ-open neighborhood p and q of x and y respectively such that p + q ⊆ r. also, by definition of δ-open set, there exist regular open neighborhood p ′ and q′ such that x ∈ p ′ ⊆ p and y ∈ q′ ⊆ q. this implies that φ(p ′ × q′) = p ′ + q′ ⊆ p + q ⊆ r. since p ×q is regular open in x ×x(with respect to product topology), it follows that φ is δ-continuous. 119 s. sharma, t. landol, s. billawria theorem 3.11. for δ-topological vector space x, the mapping ψ : k ×x → x defined by ψ(λ,x) = λx, ∀(λ,x) ∈ k×x is δ-continuous. proof: let λ ∈ k and x ∈ x and r be a regular open neighborhood of λx in x. then there exist δ-open neighborhood p of λ in k and δ-open neighborhood q of x in x such that p.q ⊆ r. also, p ′.q′ ⊆ p.q ⊆ r, for regular open set p ′ and q′ contained in p and q containing λ and x respectively. since p ×q is regular in k × x, ψ(p ′.q′) ⊆ r. hence, it follows that ψ is δ-continuous for arbitrary element λ ∈ k and x ∈ x. theorem 3.12. let x be δ-topological vector space and y be topological vector space over the same field k. let f : d1 → d2 be a linear map such that f is continuous at 0. then f is δ-continuous. proof: let 0 6= x ∈ x and v be regular open set and hence open in y containing f(x). since translation of an open set is open in topological vector space, which implies v − f(x) is open in y containing 0. since f is continuous at 0, there exist open set u in x containing 0 such that f(u) ⊆ v − f(x). also, by linearity of f implies that f(x + u) ⊆ v . by theorem 3.1, x + u is δ-open and hence there exist regular open set q such that q ⊆ x + u. hence, f(q) ⊆ v . 4 conclusions δ-topological vector space is an extension of topological vector space and this paper give an insight into this space. we presented the space with new examples and inherent properties. moreover, important characterization of the space is studied in this paper. m. e. abd.el-monsef , s. n. el-deeb and r. a. mahmoud, β−open sets and β−continuous mappings, bull. fac. sci. assiut univ., 12 (1983), 77-90. t. al-hawary and a. al-nayef, on irresolute-topological vector spaces, math. sci. res. hot-line, 5 (2001), 49-53. j.h.h. bayati, certain types of topological vector space, second international conference of applied and pure mathematics 2019 (sicapm), university of baghdad, march 2019. kolmogroff, zur normierbarkeit eines topologischen linearen raumes, studia math.,5 (1934), 29-33. m. d. khan, s. azam and m. s. bosan, s-topological vector space, journal of linear and topological algebra, 04(02) (2015), 153-158, 120 on characterization of δ-topological vector space a. s. mashhour, m. e. abd el-monsef and s. n. el-deep, on pre-continuous and weak pre-continuous mappings, proceedings of the mathematical and physical society of egypt, 53 (1982), 47-53. o. njastad, on some classes of nearly open sets, pacific j. math., 15 (1965), 961-970. t. noiri, on δ-continuous functions, j. korean math. soc., 16(2) (1980), 1610166. n. rajesh, thanjavur, v. vijayabharathi and tiruchirappalli, on strongly preirresolute topological vector space, mathematica bohemica, 138 (2013), 372. m. ram, s. sharma, s. billawria and a. hussain, on almost s-topological vector spaces, jour. adv. stud. topol., 9(2) (2018), 139-146. s. sharma and m. ram, on β-topological vector spaces, journal of linear and topological algebra, 8(01) (2019), 63-70. m. stone, application of the theory of boolean rings to general topology, trans. amer. math. soc., 41 (1937), 374-481. n.v. velicko, h-closed topological spaces, amer. math. soc. transl., 78 (1968), 103-118. 121 ratio mathematica volume 39, 2020, pp. 229-236 on homomorphism of fuzzy multigroups j. a. awolola* m. a. ibrahim† abstract in this paper, the homomorphism of fuzzy multigroups is briefly delineated and some related results are shown. in particular, we consider the corresponding isomorphism theorems of fuzzy multigroups. keywords: fuzzy multiset, fuzzy multigroup, homomorphism of fuzzy multigroups. 2010 ams subject classifications: 54a40, 03e72, 20n25, 06d72. 1 *department of mathematics/statistics/computer science, university of agriculture, p.m.b. 2373, makurdi, nigeria; awolola.johnson@uam.edu.ng. †department of mathematics, ahmadu bello university, zaria, nigeria; amibrahim@abu.edu.ng. 1received on november 2nd, 2020. accepted on december 17rd, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.553. issn: 1592-7415. eissn: 2282-8214. ©j. a. awolola and a. m. ibrahim. this paper is published under the cc-by licence agreement. 229 j. a. awolola and a. m. ibrahim 1 introduction since the inception of the theory fuzzy multisets introduced by yager (1986), the subject has become an interesting area for researchers in algebra. the foundation of algebraic structures of fuzzy multisets was laid by shinoj et al. (2015); ibrahim and awolola (2015) discussed further some new results which will bring new openings and development of fuzzy multigroup concept. some group concepts like subgroups, abelian groups, normal subgroups and direct product of groups have been established (ejegwa, 2018a,b,d, 2019). the idea of homomorphism of fuzzy multigroups and their alpha-cuts have also been discussed (ejegwa, 2018c, 2020). in this paper, more results on homomorphism of fuzzy multigroups are established and the corresponding isomorphism theorems of fuzzy multigroups which analogously exist in group setting are discussed. 2 preliminaries we recall here some basic definitions and results used in the sequel. we refer the reader to (miyamoto, 2001; shinoj et al., 2015; ibrahim and awolola, 2015). definition 2.1. (miyamoto, 2001) let x be a nonempty set. a fuzzy multiset u over x is characterized by count membership function cmu : x → [0, 1] (giving a multiset of the unit interval [0, 1]). an expedient notation for a fuzzy multiset u over x is u = {(cmu (a)/a) | a ∈ x} with cmu (a) = {µ1u (a) ,µ 2 u (a), ...,µ m u (a), ...}, where µ 1 u (a) ,µ 2 u (a), ...,µ m u (x), ... ∈ [0, 1] such that (µ1u (x) ≥ µ 2 u (a) ≥, ...,≥ µ m u (a), ...). if the fuzzy multiset u is finite, then cmu (a) = {µ1u (a) ,µ 2 u (a), ...,µ m u (a)}, where µ1u (a) ,µ 2 u (a), ...,µ m u (a) ∈ [0, 1] such that µ 1 u (a) ≥ µ 2 u (a) ≥, ...,≥ µmu (a). the set of all fuzzy multisets over x is denoted by fms(x). throughout this paper fuzzy multisets are considered finite. the usual set operations can be carried over to fuzzy multisets. for instance, let u,v ∈ fms(x), then u ⊆ v ⇐⇒ cmu (a) ≤ cmv (a),∀ a ∈ x, u ∩v = {cmu (a) ∧cmv (a)/a | a ∈ x}, u ∪v = {cmu (a) ∨cmv (a)/x | a ∈ x}. 230 on homomorphism of fuzzy multigroups definition 2.2 (shinoj et al., 2015) let p and q be two nonempty sets such that ϕ : p → q is a mapping. consider the fuzzy multisets u ∈ fms(p) and v ∈ fms(q). then, (i) the image of u under ϕ is denoted by ϕ(u) has the count membership function cmϕ(u) (b) = { ∨ ϕ(a)=bcmu (a) , ϕ −1 (b) 6= ∅ 0, ϕ−1 (b) = ∅ (ii) the inverse image of v under ϕ denoted by ϕ−1 (v ) has the count membership function cmϕ−1(v ) (a) = cmv (ϕ (a)). definition 2.4 (shinoj et al., 2015) let x be a group. a fuzzy multiset u over x is called a fuzzy multigroup if (i) cmu (ab) ≥ cmu (a) ∧ cmu (b) , ∀ a,b ∈ x, and (ii) cmu (a−1) = cmu (a) , ∀ a ∈ x. the immediate consequence is that cmu (e) ≥ cmu (a) ∀ a ∈ x, where e is the identity element of x. the set all fuzzy multigroups is denoted by fmg(x). the next definition can be found in shinoj et al. (2015) . definition 2.5 let u ∈ fmg(x). then u is called an abelian fuzzy multigroup over x if cmu (ab) = cmu (ba) , ∀ a,b ∈ x. the set afmg (x) is the set of all abelian fuzzy multigroups over x. definition 2.6 let u ∈ fms(x). then u∗ = {x ∈ x | cu (a) = cu (e)} remark 2.1 for a fuzzy multigroup over a group x, u∗ is a group, certainly a subgroup of x shinoj et al. (2015). proposition 2.1 (ibrahim and awolola, 2015) let u ∈ fmg(x), then xu = yu ⇐⇒ x−1y ∈ u∗. the following propositions are shown in (ibrahim and awolola, 2015) . proposition 2.2 let u ∈ fmg(x). then the following assertions are equivalent: (i) cmu (ab) = cmu (ba), ∀ a,b ∈ x, 231 j. a. awolola and a. m. ibrahim (ii) cmu (aba−1) = cmu (b), ∀ a,b ∈ x, (iii) cmu (aba−1) ≥ cma(b), ∀ a,b ∈ x, (iv) cmu (aba−1) ≤ cmu (b), ∀ a,b ∈ x. proposition 2.3 let u ∈ fmg(x). then cmu (ab−1) = cmu (e) implies cmu (a) = cmu (b). as to the converse problem whether cmu (a) = cmu (b) implies cmu (ab−1) = cmu (e), we give a counter example. let x = {1,s,t,r} be a klein’s 4-group and u = {(1, 0.7, 0.6, 0.5, 0.5)/1, (0.6, 0.4, 0.2)/s}. we see that u is an abelian fuzzy multigroup over x. then, while cmu (t) = cmu (r) = 0, we have cmu (tr −1) = cmu (tr) = cmu (s) = (0.6, 0.4, 0.2) 6= (1, 0.7, 0.6, 0.5, 0.5) = cmu (1). thus the converse problem above does not hold. 3 main results proposition 3.1 let x be a group such that ϕ : x → x is an automorphism. if u ∈ fmg(x), then ϕ(u) = u if and only if ϕ−1(u) = u. proof. let a ∈ x. then ϕ(a) = a. now cmϕ−1(u)(a) = cmu (ϕ(a)) = cmu (a) =⇒ ϕ−1(u) = u conversely, let ϕ−1(u) = u. since ϕ is an automorphism, then cmϕ(u)(a) = ∨ {cmu (a ′ ) | a ′ ∈ x, ϕ(a ′ ) = ϕ(a)} = cmu (ϕ(a)) = cu(ϕ−1(u))(a) = cmu (a) hence, the proof. proposition 3.2 let ϕ : x → y be a homomorphism of groups such that u,v ∈ fmg(y ). if u is a constant on kerϕ, then ϕ−1(ϕ(u)) = u. proof. let ϕ(a) = b. then we have cmϕ−1(ϕ(u))(a) = cmϕ(u)ϕ(a) = cmϕ(u)(b) = ∨ {cmu (a) | a ∈ x, ϕ(a) = 232 on homomorphism of fuzzy multigroups b}. since ϕ(a−1c) = ϕ(a−1)ϕ(c) = (ϕ(a))−1ϕ(c) = b−1b = e′, ∀ c ∈ x, such that ϕ(c) = b, which implies that a−1c ∈ kerϕ. moreover, since u is constant on kerϕ, then cmu (a−1c) = cmu (e). therefore, cmu (a) = cmu (c). this completes the proof. proposition 3.3 let u ∈ afmg(x) such that a map ϕ : x → x/u is defined by ϕ(a) = au. then ϕ is a homomorphism with kerϕ = {a ∈ x | cmu (a) = cmu (e)}. proof. clearly, ϕ is a homomorphism. also, kerϕ = {a ∈ x : ϕ(a) = eu} = {a ∈ x : au = eu} = {a ∈ x : cmu (a−1b) = cmu (b) ∀ b ∈ x} = {a ∈ x : cmu (a−1) = cmu (e)} = {a ∈ x : cmu (a) = cmh (e)} = u∗ proposition 3.4 let ϕ : x → y be an epimorphism of groups and u ∈ afmg(x), then x/u∗ ∼= y . proof. define ψ : x/u∗ → y by ψ(xu∗) = ϕ(a) ∀ a ∈ x. let au = bu such that cmu (a−1b) = cmu (e). this implies that a−1b ∈ u∗. it is easy to show that ψ is well-defined, homomorphism and epimorphism. moreover, ϕ(a) = ϕ(b) =⇒ ϕ(a)−1ϕ(b) = ϕ(e) =⇒ ϕ(a−1)ϕ(b) = ϕ(a−1b) = ϕ(e) =⇒ a−1b ∈ u∗ =⇒ cmu (a−1b) = cmu (e) =⇒ au = bu this shows that ψ is an isomorphism. proposition 3.5 if u,v ∈ afmg(x) with cmu (e) = cmv (e), then u∗v∗/v ∼= u∗/u ∩v . 233 j. a. awolola and a. m. ibrahim proof. clearly, for some x ∈ u∗v∗, a = uv such that u ∈ u∗ and v ∈ v∗. define ϕ : u∗v∗/v → u∗/u ∩v by ϕ(av ) = u(u ∩v ). if av = bv with b = u1v1, u1 ∈ u∗ and v1 ∈ v∗, then cmv (a −1b) = cmv ((uv) −1u1v1) = cmv (v −1u−1u1v1) = cmv (u −1u1v −1v1) = cmv (e). hence, cmv (u−1u1) = cmv (v−1v1) = cmv (e). thus, cmu∩v (u −1u1) = cmu (u −1u1) ∧cmv (u−1u1) = cmu (e) ∧cmv (e) = cmu∩v (e) that is, u(u ∩v ) = u1(u ∩v ). therefore, ϕ is well-defined. if av,bv ∈ u∗v∗/v , then ab = uvu1v1. since u ∈ afmg(x), then cmu (vu1v1) = cmu (u1) =⇒ vu1v1 ∈ u∗. hence, ϕ(av bv ) = ϕ(abv ) = u(vu1v1)(u ∩v ) = u(u ∩v )vu1v1(u ∩v ) and cmu∩v (u −1 1 (vu1u1)) ≥ cmu (u −1 1 vu1v1) ∧cmv (u −1 1 vu1v1) = cmu (u −1 1 (vu1v1)) ∧cmv (v(u −1 1 u1v1)) = cmu (e) ∧cmv (e) = cmu∩v (e). hence, vu1v1(u ∩v ) = u1(u ∩v ) that is, ϕ(av bv ) = u(u ∩v )u1(u ∩v ) = ϕ(av )ϕ(bv ), and this shows that ϕ is a homomorphism. undeniably, it is also epimorphism. furthermore, if a,b ∈ u∗v∗ with a = uv and b = u1v1, u,u1 ∈ u∗ and v,v1 ∈ v∗ and u(u ∩v ) = u1(u ∩v ), then cmu∩v (u−1u1) = cmu∩v (e) that is, cmu (u−1u1) ∧cmv (u−1u1) = cmu (e) ∧cmv (e). however, cmu (e) = cmv (e) and cmu (u−1u1) = cmu (e) =⇒ cmv (u−1u1) = cmv (e). 234 on homomorphism of fuzzy multigroups therefore, cmv (a −1b) = cmv ((uv) −1u1v1) = cmv (v −1u−1u1u1) = cmv (u −1u1v −1v1) ≥ cmv (u−1u1) ∧cmv (v−1v1) = cmv (e) ∧cmv (e) = cmv (e) =⇒ cmv (a−1b) = cmv (e) thus, av = bv . hence, u∗v∗/v ∼= u∗/u ∩v . proposition 3.6 let u,v ∈ afmg(x) such that u ⊆ v and cmu (e) = cmv (e). then x/v ∼= (x/u)/(v∗/u). proof. define ϕ : x/u → x/v by ϕ(au) = av ∀a ∈ x such that cmu (a−1b) = cmu (e) = cmv (e) ∀ au = bu. since u ⊆ v , we have cmv (a−1b) ≥ cmu (a −1b) = cmv (e) and thus cmv (a−1b) = cmv (e), that is, av = bv , which implies that ϕ is well-defined. it is homomorphism and epimorphism too. moreover, kerϕ = {au ∈ x/u : ϕ(au) = ev} = {au ∈ x/u : av = ev} = {au ∈ x/u : cmv (a) = cmv (e)} = {au ∈ x/u : a ∈ v∗} = v∗/u. thus, kerϕ = v∗/u and so x/v ∼= (x/u)/(v∗/u). references p.a. ejegwa. on fuzzy multigroups and fuzzy submultigroups. journal of fuzzy mathematics, 26(3):641—-654, 2018a. p.a. ejegwa. on abelian fuzzy multigroups. journal of fuzzy mathematics, 26 (3):655—-668, 2018b. p.a. ejegwa. homomorphism of fuzzy multigroups. applications and applied mathematics, 13(1):114—-129, 2018c. 235 j. a. awolola and a. m. ibrahim p.a. ejegwa. on normal fuzzy submultigroups of a fuzzy multigroup. theory and applications of mathematics and computer science, 8(1):64—-92, 2018d. p.a. ejegwa. direct product of fuzzy multigroups. journal of new theory, 28: 62—-73, 2019. p.a. ejegwa. on alpha-cuts homomorphism of fuzzy multigroups. annals of fuzzy mathematics and informatics, 19(1):73—-87, 2020. a.m. ibrahim and j.a. awolola. fuzzification of some results on multigroups. journal of the nigerian association of mathematical physics, 32:119–124, 2015. s. miyamoto. fuzzy multisets and their generalizations, in: c.s. calude, et al. (eds.). multiset processing, lecture notes in computer science, springer, berlin, 2235:225—-235, 2001. t.k. shinoj, b. anagha, and j.j. sunil. on some algebraic structures of fuzzy multisets. annals of fuzzy mathematics and informatics, 9:77–90, 2015. r.r. yager. on the theory of bags. international journal of general systems, 13: 23–37, 1986. 236 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 39, 2020, pp. 253-259 253 determine the value d(m(g)) for non-abelian p-groups of order q = pnk of nilpotency c behnam razzaghmaneshi* abstract in this paper we prove that if n, k and t be positive integer numbers such that t < k < n and g is a non abelian p-group of order pnk with derived subgroup of order pkt and nilpotency class c, then the minimal number of generators of g is at most p1 2 ((nt+kt−2)(2c−1)(nt−kt−1)+n. in particular, |m(g)| _ p1 2 (n(k+1)−2)(n(k−1)−1)+n, and the equality holds in this last bound if and only if n = 1 and g = h ×z, where h is extra special p-group of order p3n and exponent p, and z is an elementary abelian p-group. keywords: schur multiplier, elementary abelian, p-group, extra special 2010 ams subject classification: 20f45; 20f05, 11y35.† * assistant professor of mathematics, algebra, islamic azad university, talesh branch, talesh, iran. b_razzagh@yahoo.com. † received on october 22nd, 2020. accepted on december 17th, 2020. published on december 31th, 2020. doi: 10.23755/rm.v39i0.560. issn: 1592-7415. eissn: 2282-8214. ©razzaghmaneshi. this paper is published under the cc-by licence agreement. behnam razzaghmaneshi 254 1. introduction let g be a finite group and g = fr a presentation for g as a factor group of the free group f. then schur in [11], show that m(g) = (f0\r) [f,r] . (1.1) recall that, for two finite groups a and b, ab _= ( aa0 )( bb0 ). michael r. jones in years 1973 and 1974 for the finite group g, get some inequalities for d(m(g)) and e(m(g)), which d(m(g)) and e(m(g)) the minimal number of generators and exponent of finite group g, respectively. now in current paper we generalized and compute the value d(m(g)) and e(m(g)) for non-abelian pgroups of order q = pnk and nilpotency c. notation: the notation used in this paper is as follows: (i) if g is a finite group then e(g) denotes exponent of g and d(g) denotes the minimal number of generators of g. (ii) the the lower central series of a group g is denoted by g = g1(g) _ g2(g) = g0 _ g3(g) _ ..., where for j _ 1, gi+1(g) = [gi(g),g]. and the upper central series of a group g is denoted by 1 = z0(g) _ z1(g) = g0 _ z2(g) _ ..., where for i _ 0, zi+1zi_ z( gzi(g) ). the main theorem of this paper as follows. main theorem: let n, k and t be positive integer numbers such that t < k < n and g is a non abelian p-group of order pnk with derived subgroup of order pkt and nilpotency class c, then the minimal number of generators of g, (d|m(g)|) is p12 ((2c−1)n2−k(k−1)−3n+4. 2. some definition, lemma and theorems the results of this section are several lemma and theorems, where the proofs of their in references [6], [7] and [8], and so we will be omitted. 2.1. lemma: let g be a finite group and b a normal subgroup. set a = gb . let g = f r be a presentation for g as a factor group of the free group f and suppose b = sr so that a = fs . then [f,s] [f,r][f,s,f]s0 is isomorphic with a factor group of ab. proof. see to ([6], lemma 2.1). 2.2. corollary. further to the notation and assumptions of lemma 2.1, let b 2 be a central subgroup of g. then [f,r] [f,r]s0 is an epimorphic image of ab. proof. see to ([6]). 2.3. definition. let g be a finite group. we say that g has (special) rank r(g) if every subgroup of g may be generated by r(g) elements and there is at least one subgroup that cannot be generated by fewer than r(g) elements. let g = f r be a presentation for the finite p-group g as a factor group of a free group f. let i+1 = gi+1(f) for all i. since g0 = f0r r we have by (1.1), that determine the value d(m(g)) for non-abelian p-groups of order q = pnk of nilpotency c 255 m( g g0 ) _= (f0\f0r) [f,f0r] = f0 [f,f0r] . with this notation we have: 2.4. theorem: let g be a finite p-group of nilpotency class c and qi = g gi(g) for 2 _ i _ c. then (i) |g0||m(g)| _ |m( g g0 )c−1 i=1 |qi+1gi+1(g)|, (ii)d(m(g)) _ d(m( g g0 ))+c−1 i=1 d(qi+1gi+1(g)), (iii) e(m(g)) _ e(m( g g0 ))c−1 i=1 e(qi+1gi+1(g)). (i) in the above notation, |g0||m(g)| = | f0 [f,r] | = |m( g g0 )| |[f,f0r] [f,r]| = |m( g g0 )| |[f,fi+2r] [f,r]| i k=1 | [f,k+1r| [f,k+2r, for all i _ 1. now, 1 = gc+1(g) = c+1r r so that c+1 _ r and [f,fc+1r] = [f,r]. next, gi(g) = ir r for all i _ 2. thus [f,r](ir)0[f,ir,f] = [f,r]i+2 = [f,i+1r] and (i) follows by lemma 2.1. (ii) we have, r( f0[f,r] ) _ r(m( g g0 ))+r([f,2r] [f,r] so that d(m(g)) _ d(m( g g0 ))+c−1 i=1 r([f,i+1r] [f,i+2r] ), and (ii) again follows by lemma 2.1. (iii) this follows as for (i) and (ii). 3. the proof of main theorem in this section we show that, let n, k and t be positive integer numbers such that t < k < n and g is a non abelian p-group of order pnk with derived subgroup of order pkt and nilpotency class c, then the minimal number of generators of g, (d|m(g)|) is p1 2 ((2c−1)n2−k(k−1)−3n+4. for proof of this work we action as follows: proof. let n, k and t be positive integer numbers such that t < k < n and g is a non abelian p-group of order pnk with derived subgroup of order pkt and nilpotency class c. then by using of theorem 2.4(ii),we have d(m(g)) _ d(m( g g0 ))+c−1 i=1 d(qi+1gi+1(g)). if d(m(g)) = n then the above relation will coming as follows: d(m(g)) _ 12 ((n+k−2)(n−k−1)+1)+n(c−1 i=1 gi+1(g)). = 12((n+k−2)(n−k−1)+1)+n2(c−1). which the result now follows. in 1904, schur [11,12] prove that for every finite groups h and k, then m(h × k) = m(h)×m(k)× h h0 k k0 . in 1957, green [5] show that if g be a p-group of order pn, then |m(g)| _ p1 2 n(n−1). in 1967, gaschatz el al [4] prove that if g be a d-generator p-group of order pn, g0 has order pc and g z(g) is a dgenerator group, then |m(g)|_ p12 d(2n−2c−d−1)+2(d−1)c. in 1973, jones [4-6] show that if g be a p-group of order pn and |g0| = pk, then |m(g)| _ p1 2 n(n−1)−k. in 1982, byel and tappe [2] shown that if g be a extra especial p-group of order p2m+1, then behnam razzaghmaneshi 256 (i) if m _ n, than |m(g)| = p2m2−m−1. (ii) if m = 1, then the order of schur multiplier of d8,q8,e1 and e2 are equal 2, 1, p2 and 1, respectively. in 1991, berkovich [1] show that if g be a p-group of order pn, then t(g) = 0 if and only if g _= z(n) p , and also t(g) = 1 if and only if g _= z(2) or g _= e1. in 1994, zhou [14]prove that if g be a p-group of order pn, then t(g) = 2 if and only if g _= z×zp2 or g _= d8, g _= e1×zp. in 1999, ellis [3]show that if g be a p-group of order pn, then t(g)=3 if and only if g _= zp3 , g _= z(2) p ×zp2 or g _= q8, g _= e2, g _= d8×z2 or g _= e1×z(2) p . in 2009, p.niroomand [10] show that if g be a non-abelian finite p-group of order pn and |g0| = pk, then |m(g)| is p1 2 ((n+k−2)(n−k−1)+1. in particular, |m(g)| _ p1 2 (n−2)(n−1)+1, and the equality holds in this last bound if and only if g = e1×z, where z is an elementary abelian p-group. the schur multiplier of abelian groups may be calculated easily by a result [12] which was obtained by schur. so in this paper, we focus on non-abelianpgroups. this paper is devoted to the derivation of certain upper bound for the schur multiplier of non-abelian p-groups of order pnk with derived subgroup of order pk. we prove that |m(g)| _ p12 (nk+nt−2)(nk−nt−1)+n . in particular, if |m(g)| = p1 2 (n(k+1)−2)(n(k−1)−1)+n, we characterize the structure of the group g. if g is a p-group of order pn, jones [4] proved that |m(g)||g0| _ p1 2 n(n−1) which shows that |m(g)| _ p1 2 n(n−1)+1 when g is a non-abelian p-group of order pn. so, the general bound given above is better than joness bound unless |g| = p3, in which case the two bounds are the same.the principal result of this paper is presented in the following theorem. main theorem. let g be a non-abelian finite p-group of order pnk. if |g0| = pnt , then we have m(g) _ p1 2 (nk+nt−2)(nk−nt−1)+n. in particular m(g) _ p12 (n(k+1)−2)(n(k−1)−1)+n, and the equality holds in this last bound if and only if n−1 and g = h×z, where h is an extra special p-group of order p3n and exponent p, and z is an elementary abelian p-group. preliminaries and elementary theorems. in this section, we want to several theorems and lemmas whose proved in references [1-14]. at first we list the following theorems, which are used in our proofs. our method for the proof is similar to p. niroomand (2009) and berkovich, ya.g. (1991), which we compute for groups of order pnk. determine the value d(m(g)) for non-abelian p-groups of order q = pnk of nilpotency c 257 theorem 2.1.(see [7,theorem 3.1 and theorem 4.1].) let g be a finite pgroup and let n be a central subgroup of g. then |m(g n | _ |m(g)||g0 \n| _ |m(g n ||m(n)||g n n| . theorem 2.2.(see[9, theorem 3.3.6].) let g be an extra special p-group of order p2m+1. then: (i) if m _ 2, then m(g) = p2m2−m−1. (ii) if m=1, then m(g) _ p2, and the equality holds if and only if g is of exponent p. theorem 2.3.(see [9, theorem 2.2.10].) for every finite groups h and k, we have m(h ×k _= m(h)×m(k)× h h0 k k0 . corollary 2.4. if g _= cm1 ×cm2 ×...×cmk , where mi+1 divides mi for all i, 1 _ i _ k, then m(g) _=cm2 ×c(2) m3 ×...×c(k−1) mk . proof of the main theorem in this section we want to prove our result. the following technical lemmas shorten the proof of our main theorem. lemma 3.1. let g be a finite p-group of order pn such that g g0 is elementary 6 of order pn−1, then g is a central product of an extra special p-group h and z(g) such that h \z(g) = g0. proof. let h g0 be the complement of z(g) g0 in g g0 . then g = hz(g), so g0 = h0 and z(h) = z(g) \h. on the other hand, 1 6= z(g) \h _ g0, and the result follows. lemma 3.2. let g be an abelian p-group of order pn which is elementary abelian. then m(g) _ p1 2 (n−1)(n−2). proof. the result is obtained obviously if g is cyclic. so, let g _=cpm1×cpm2× ...×cpmk such that k i=1mi = n and m1 _ m2 _ ... _ mk. we know that m1 _ 2, and then, by using corollary2.4, |m(g)| = pm2+2m3+...+(k−1)mk _ p(m2+m3+...+mk)+(m3+...+mk)+...+mk _ p1 2 (n−1)(n−2). lemma 3.3. let g be a nonabelian p-group of order pnk with derived subgroup of order p such that g g0 is not elementary abelian, then m(g) < p12 (nk−1)(nk−2)+1. proof. by using theorem 2.1 and lemma 3.2, m(g) _ p−1|m( g g0 )|| g g0 g0| _ p−1p1 2 (nk−2)(nk−3)p(nk−1) < p1 2 (nk−1)(nk−2)+1. which completes the proof. lemma 3.4. let g be a nonabelian p-group of order pnk, such that g g0 is elementary abelian of order pnk−1, then m(g) _ p1 2 (nk−1)(nk−2)+1 and the equality holds if and only if g = h×z, where h is extra special pgroup of order p3n and exponent p, and z is elementary abelian p-group. proof. by lemma 3.1, g is central product of h and z(g), and theorem 2.2, 7 we may assume that |z(g)| _ p2. let |h| = p2m+1, so |z(g)| = pn−2m. behnam razzaghmaneshi 258 suppose first that m _ 2. if z(g) is elementary abelian, let t be a group such that z(g) _= g0×t. by using theorems 2.2 and 2.3, we have |m(g)| = |m(h ×t)| = |m(h)||m(t)|| h h0 t| = p2m2−m−1p (n−2m−1)(n−2m−2) 2 2m(n−2m−1) = p1 2 (n2−3m) < p12 (n−1)(n−2)+1. now assume that z(g) is not elementary abelian. theorems 2.1 and 2.3 imply that |m(g)| _ p|m(h ×z(g)| = p|m(h)||m(z(g))|| h h0 z(g)|. hence by using theorem 2.2 and lemma 3.2, we have |m(g)| _ pp2m2−m−1p12 (n−2m−1)(n−2m−2)p2m(n−2m−1) < p1 2 (n−1)(n−2)+1. if h is extra special of order p3n and z(g) is not elementary abelian, then theorem 2.1 implies that |m(g)| _ p−1|m( g z(g) ||m(z(g))|| g z(g) z(g)| _ p12 nk(nk−3)+1 < p1 2 (nk−1)(nk−2)+1. by theorem 2.2, it is easy to see that if z(g) is elementary abelian, then |m(g)|= p1 2 (nk−1)(nk−2)+1 if h is extra special of order p3n and exponent p; and in other cases |m(g)| < p1 2 (nk−1)(nk−2)+1. proof of the main theorem we prove the theorem by induction on t. if t = 1 the result is obtained by lemma 3.2 and 3.4. let g be a non-abelian p-group of order pnk with derived subgroup of order pnt(t _ 2). choose k in g0 \z(g) of order p−1. by using induction hypothesis, we have |m(gk )| _ p12 nk+nt−4)(nk−nt−1)+n. on the other hand, by using theorem 2.1, implies that |m(g)| _ p−1|m(gk ||m(k)||( g g0 k)| _ p−1p12 (nk+nt−4)(nk−nt−1)pn−1p(nk−nt) _ p12 (nk+nt−4)(nk−nt−1)pn−1p(nk−nt) p12 (nk+nt−2)(nk−nt−1)+n. now let g be a p-group of order pnk such that |m(g)| = p1 2 (nk−1)(nk−2)+n. if |g0| _ p2k, then |m(g)| _ p1 2 (n(k−1)−1)(n(k+1)−2), which is a contradiction. since |g0| = pk, lemma 3.3 implies that g /g0 is elementary abelian. hence lemma 3.4 shows that g = h ×z, where h is an extra special pgroup of order p3n and exponent p, and z is an elementary abelian p-group, so the result follows. determine the value d(m(g)) for non-abelian p-groups of order q = pnk of nilpotency c 259 references [1] berkovich, ya.g., on the order of the commutator subgroups and the schur multiplier of a finite p-group,, j. algebra, 144. (1991) 269272. [2] beyl, f.r., and j. tappe, group extensions, representations and the schur multiplicator,, vol. 958, springerverlag, berlin/heidelberg/new york, 1972. [3] ellis, g., on the schur multiplier of p-group,, comm. algebra. 27(9), (1999), 4173-4177. [4] gaschutz, w., neubu¨ser, j. and yen. t., u¨ ber den multiplikator von p gruppen, math. z. 100 (1967), 93-96. [5] green, j.a., on the number of automorphisms of a finite group,, proc. roy. soc. a 237 (1956) 574581. [6] jones, m.r., multiplicators of p-groups,, math. z. 127 (1972) 165166. [7] jones, m.r., some inequalities for the multiplicator of a finite group, , proc. amer. math. soc. 39 (1973) 450456. [8] jones, m.r., some inequalities for the multiplicator of a finite group ii,, proc. amer. math. soc. 45 (1974) 167172. [9] karpilovsky, g.,the schur multiplier, london math. soc. monogr. (n.s.) (1987). [10] niroomand. p., on the order of schur multiplier of non-abelian p-groups, j. algebra 322 (2009), 4479 4482. [11] schur, i., ber die darstellung der endlichen gruppen durch gebrochene lineare substitutionen, , j. reine angew. math. 127 (1904) 2050. [12] schur, i., untersuchungen ber die darstellung der endlichen gruppen durchgebrochene lineare substitutionen,, j. reine angew. math. 132 (1907) 85137. [13] wiegold, j., the schur multiplier of p-groups with large derived subgroup groups,, arch. math. 95 (2010), 101-103. [14] zhou, x., on the order of schur multipliers of finite p-groups, comm. algebra. 22(1), (1994), 1-8. e:\uziv\sarka\clanky\rm_24\rm_23_8.dvi ratio mathematica 24 (2013), 41–52 issn: 1592-7415 visualization of algebraic properties of special hv -structures achilles dramalidis democritus university of thrace, school of education, 681 00 alexandroupolis, greece adramali@psed.duth.gr abstract the paper looks at visualization as it relates to special hv -structures, focusing upon how it can be used to improve the perception and understanding of abstract algebraic concepts. using position vectors into the plane ir2, abstract algebraic properties of hv -structures are gradually transformed into geometrical shapes. key words: hyperstructures, hv -structures, visualization. msc2010: 20n20, 16y99. 1 introduction in most branches of mathematical research, visualization has been an area of interest for mathematicians [1], [6], [9], specifying that visual thinking can be an alternative and powerful resource, as well as a serious tool, not only for specialists but also, for students doing mathematics. mathematicians have always used their ”mind’s eye” to visualize the abstract objects and processes that arise in mathematical research. but it is only in recent years that remarkable improvements in computer technology have made it easy to externalize these vague and subjective pictures that we see in our heads, replacing them with precise and objective visualizations that can be shared with others [7]. the subject is of such recent research that searching the literature, in preparation for this paper, it was surprising to discover that no papers were specifically focused on visualization in hyperstructures. visualization of algebraic properties . . . according to [10], the term visualization has been used in various ways in the research literature, so it is necessary to clarify how it is used in this paper. thus visualization is taken to include processes of constructing and transforming both mental imagery and abstract algebraic concepts. this paper, looks at visualization as it relates to special hv -structures, focusing upon how it can be used to improve the perception and understanding of abstract algebraic concepts, since, being able to ”see” something in a geometrical shape, is a common metaphor for understanding it. according to bruner [2], to understand a specific concept (algebraic), the first approach has to be intuitive. so, geometry or linear algebra into a two-dimensional real vector space, with constant references to the fundamental intuitively understood principles, are teaching and educative tools. using position vectors into the plane ir2, abstract algebraic properties of hv -structures are gradually transformed into geometrical shapes, which operate, not only as a translation of the algebraic concept but also, as a teaching process. 2 basic definitions on hyperstructures in 1934, f. marty introduced the definitions of the hyperoperation and of the hypergroup as a generalization of the operation and the group respectively. definition 2.1 in a set h 6= ∅, a hyperoperation is a map, such that: ◦ : h × h → p(h) − {∅} : (x, y) 7→ x ◦ y ⊂ h also, if a, b ⊂ h,then a ◦ b = ∪a∈a,b∈b(a ◦ b). properties of hyperoperations [3], [4], [12]: i) a hyperoperation (◦) in a set h is called associative, if (x ◦ y) ◦ z = x ◦ (y ◦ z), ∀x, y, z ∈ h ii) a hyperoperation (◦) in a set h is called commutative, if x ◦ y = y ◦ x, ∀x, y ∈ h 42 visualization of algebraic properties . . . iii) a hyperoperation (◦), in a set h, is having an identity or unit element if there exists e ∈ h, such that x ∈ x ◦ e and x ∈ e ◦ x, ∀x ∈ h iv) a hyperoperation (◦), in a set h, with a unit element e, is having an inverse element, if for every x ∈ h, there exists an element x′ ∈ h, such that e ∈ x ◦ x′ande ∈ x′ ◦ x, ∀x ∈ h v) in a set h, equipped with two hyperoperations (◦) and (∗), the (∗) is called distributive with respect to (◦), if x ∗ (y ◦ z) = (x ∗ y) ◦ (x ∗ z), ∀x, y, z ∈ h an algebraic hyperstructure (h, ◦), i.e. a set h equipped with a hyperoperation (◦), is called hypergroupoid. if this hyperoperation is associative, then the hyperstructure is called semihypergroup. the semihypergroup (h, ◦), is called hypergroup if it satisfies the reproduction axiom: x ◦ h = h ◦ x, ∀x ∈ h. one more complicated hyperstructure, is that (h, ◦, ∗), which is called hyperring, where (h, ◦) is a commutative hypergroup, the (∗) is associative and distributive with respect to (◦). one of the topics of great interest, in the last years, is the hv-stuctures, which was introduced by t. vougiouklis in 1990 [11]. the class of hvstuctures is the largest class of algebraic hyperstructures. these structures satisfy weak axioms, where the non-empty intersection replaces the equality, as bellow [12]: let h be a set and ◦ : h × h → p(h) − {∅} be a hyperoperation. i) the (◦)in h is called weak associative, we write wass, if (x ◦ y) ◦ z ∩ x ◦ (y ◦ z) 6= ∅, ∀x, y, z ∈ h ii) the (◦) is called weak commutative, we write cow, if (x ◦ y) ∩ (y ◦ x) 6= ∅, ∀x, y ∈ h iii) if h is equipped with two hyperoperations (◦) and (∗), then (∗)is called weak distributive with respect to (◦), if [x ∗ (y ◦ z)] ∩ [(x ∗ y) ◦ (x ∗ z)] 6= ∅, ∀x, y, z ∈ h 43 visualization of algebraic properties . . . the hyperstructure (h, ◦) is called hv-semigroup if it is wass and it is called hv-group if it is reproductive hv-semigroup. it is called commutative hv-group if (◦) is commutative and it is called hv-commutative group if(◦) is weak commutative. the hyperstructure (h, ◦, ∗) is called hv-ring if both hyperstructures (◦) and (∗) are wass, the reproduction axiom is valid for (◦) and (∗) is weak distributive with respect to (◦). what it follows to the end of the paragraph comes from [5]: definition 2.2 an hv-ring (r, +, •) is called dual hv-ring, if (r, •, +) is an hv-ring, too. definition 2.3 let v be a vector space over a field k. then, define two hyperoperations in v as follows: for all x, y ∈ v and r ∈ k, x ◦ y = {z/z = x + r(y − x), r ∈ [0, 1]} x • y = {z/z = x + ry, r ∈ [0, 1]} remark 2.1 into the plane ir2 : x◦y = [x, y], it is known as join operation [8] and x•y = [x, x+y]. the [α, β] denotes the line segment which is bounded by the two end points α and β. then, for the four hyperstructures occur, we get the following: proposition 2.1 the hyperstructure (v, ∗, �), where ∗, � ∈ {◦, •}, is a weak commutative dual hv-ring. let: e∗ be the set of the unit elements with respect to (∗). er ∗ be the set of the right unit elements with respect to (∗). el ∗ be the set of the left unit elements with respect to (∗). i∗(x, e) be the set of the inverse elements of x associated with the unit e (left or right), with respect to (∗). ir ∗ (x, e) be the set of the right inverse elements of x associated with the right unit e, with respect to (∗). il ∗ (x, e) be the set of the left inverse elements of x associated with the left unit e, with respect to (∗). proposition 2.2 i) e◦ = v , ii)i◦(x, e) = {z/z = (1 − r)x + re, r ≥ 1} proposition 2.3 i)er • = v , ii)ir • (x, e) = {z/z = r(e − x), r ≥ 1, e ∈ er • }, iii)el • = {o} ⊂ e•, iv)i l • (x, e) = [e, e − x], e ∈ er • . 44 visualization of algebraic properties . . . 3 visualization in hv-groups now, let us introduce a coordinate system into the ir2. we place a given vector p so that its initial point p determines an ordered pair (a1, a2). conversely, a point p with coordinates (a1, a2) determines the vector p = op , where o the origin of the coordinate system. we shall refer to the elements x, y, z,... of the set ir2 , as vectors whose initial point is the origin. these vectors are very well known as position vectors. i) the hyperoperation: x • y = {z/z = x + ry, r ∈ [0, 1]} = [x, x + y] in figure 3.1, to every point x and y of the plane, i.e. to every ordered pair (x, y) we map an infinite number of points (hyperstructure) instead of one point (operation). the infinite number of points is the line segment [x, x + y] which is bounded by the two end points x and x + y. graphically, having the points o, x, y, draw the parallelogram with vertices o, x, y, x + y. then, the side [x, x + y] is the hyperoperation x • y. o x y x+y fig.3.1 ii) reproduction : x • ir2 = ∪r∈ir2 (x • r) = ir 2 o r1 r2rn x+r1 x+r2 x+rn x fig.3.2 in figure 3.2, take any point x of the plane. for any of the infinite points ri of the plane, draw the parallelogram with vertices o, x, ri, x + ri. unite all these infinite line segments [x, x + ri], then all these segments cover the plane. 45 visualization of algebraic properties . . . iii) weak associativity: x • (y • z) ∩ (x • y) • z 6= ∅ in figure 3.3a, take three points x, y, z of the plane, then the side [y, y+ z] of the parallelogram with vertices o, y, z, y + z is the hyperoperation y•z. with the points o, x and every point ri of the line segment [y, y+z] draw, each time, the parallelogram with vertices o, x, ri, x + ri. all these infinite line segments [x, x + ri], create the triangle with vertices x, x + y, x + y + z. then the area of this triangle is the first part of the above intercection, i.e. x • (y • z). similarly, in figure 3.3b, the side [x, x + y] of the parallelogram with vertices o, x, y, x + y is the hyperoperation x • y. with the points o, z and every point ri of the line segment [x, x + y] draw, each time, the parallelogram with vertices o, z, ri, ri + z. all these infinite line segments [ri, ri + z] create the parallelogram with vertices x, x + y, x + y + z, x + z. then the area of this parallelogram is the second part of the above intercection, i.e. (x • y) • z. notice that the triangle with vertices x, x + y, x + y + z is part of the parallelogram with vertices x, x + y, x + y + z, x + z, i.e. the intersection of these two figures is not equal to the empty set. o x y x+y x+y+z z y+z o x y x+y x+y+z z x+z fig.3.3a fig.3.3b iv) weak commutativity: (x • y) ∩ (y • x) 6= ∅ in figure 3.4, take two o x y x+y fig.3.4 points x and y of the plane. then draw the parallelogram with vertices 46 visualization of algebraic properties . . . o, x, x + y, y. the side [x, x + y] is the hyperoperation x • y and the side [y, x + y] is the hyperoperation y • x. notice that the only common point of these two sides is the point x + y, i.e. the intersection of x • y and y • x is not equal to the empty set. v) the set of the right unit elements: (x ∈ x • e, ∀x ∈ ir2) er • = ir2 o e1 e2en x+e1 x+e2 x+en x fig.3.5 in figure 3.5, take any point x of the plane. then draw the parallelograms with vertices o, x, x + ei, ei, where ei any point of the plane. the side [x, x + ei] is the hyperoperation x • ei. notice that x belongs to every line segment [x, x + ei], i.e. x belongs to every x • ei. since all these ei’s, having the above property, are infinite, we get that the set er • of the right unit elements with respect to (•) is equal to ir2. vi) the set of the right inverse elements: ir • (x, e) = {z/z = r(e − x), r ≥ 1}, e ∈ er • having any point x of the plane, we want to find at least one point x′ of the plane, such that, for a right unit point e of the plane (i.e. any point of the plane) the following to be valid: e ∈ x • x′, i.e. we want e to be point of the line segment [x, x + x′]. in figure 3.6a, notice that all the infinite points x′ of the half-line [e − x, +∞) have the above property. indeed, in figure 3.6b, for a given x and e, take any x′ belonging to the half-line [e − x, +∞). draw the parallelogram with vertices o, x, x + x′, x′. then e belongs to the line segment [x, x + x′], i.e., e belongs to the hyperoperation x • x′. 47 visualization of algebraic properties . . . o o x x -x e e-x e e-x x+x' x' fig.3.6 o x -x e e-x o x e e-x x' x'+x fig.3.7a fig.3.7b vii) the set of the left inverse elements: il • (x, e) = [e, e − x], e ∈ er • take any point x of the plane, we want to find at least one point x′ of the plane, such that, for a right unit point e of the plane (i.e. any point of the plane) the following to be valid: e ∈ x′ • x, i.e. we want e to be point of the line segment [x′, x′ + x]. in figure 3.7b, notice that the points x′ of the line segment [e, e − x] have the above property. indeed, in figure 3.7b, for a given x and e, take any x′ belonging to the line segment [e, e − x]. draw the parallelogram with vertices o, x, x′ + x, x′. then e belongs to the line segment [x′, x′ + x], i.e., e belongs to the hyperoperation x′ • x. since, el • = {o} ⊂ e• (that means that the origin o of the coordinate system is simultaneously left and right unit element), set o ≡ e, then ie • (x, o) = [o, o − x]. remark 3.1 into the plane ir2, the hyperoperation (i), together with the axioms (ii) and (iii) are giving the concept of hv-group. furthermore, by putting together the axiom (iv) we get the concept of hv-commutative group. 48 visualization of algebraic properties . . . 4 visualization in hv-rings i) distributivity of (◦) with respect to (◦): x ◦ (y ◦ z) = (x ◦ y) ◦ (x ◦ z) o x z y o z x y fig.4.1a fig.4.1b in figure 4.1a, take three points x, y, z of the plane, then the line segment [y, z] is the hyperoperation y ◦ z. join the point x to each point of the segment [y, z]. then the area of the triangle with vertices x, y, z is the first part of the above equality, i.e. x ◦ (y ◦ z). similarly, in figure 4.1b, the line segment [x, y] is the hyperoperation x ◦ y and the line segment [x, z] is the hyperoperation x ◦ z. join every point of the segment [x, y] to every point of the segment [x, z]. then the area of the triangle with vertices x, y, z is the second part of the above equality, i.e. (x ◦ y) ◦ (x ◦ z). ii) weak distributivity of (•) with respect to (•): x • (y • z) ∩ (x • y) • (x • z) 6= ∅ in figure 4.2a, take three points x, y, z of the plane, then the side [y, y+ z] of the parallelogram with vertices o, y, z, y + z is the hyperoperation y•z. with the points o, x and every point ri of the line segment [y, y+z] draw, each time, the parallelogram with vertices o, x, ri, x + ri. all these infinite line segments [x, x + ri] create the triangle with vertices x, x + y, x + y + z. then the area of this triangle is the first part of the above inersection, i.e. x • (y • z). in figure 4.2b, the side [x, x + y] of the parallelogram with vertices o, x, y, x + y is the hyperoperation x • y and the side [x, x + z] of the parallelogram with vertices o, x, z, x + z is the hyperoperation x • z. 49 visualization of algebraic properties . . . with the points: o, every point ri of the side [x, x + y] and every point ti of the side [x, x + z] draw, each time, the parallelogram with vertices o, ri, ti, ri + ti. all these infinite line segments [ri, ri + ti] create the pentagon with vertices x, 2x, 2x + y, 2x + y + z, x + y. then the area of this pentagon is the second part of the above intersection, i.e. (x • y) • (x • z). notice that the line segment [x, x + y] is the common part of the triangle area with vertices x, x+y, x+y+z and the pentagon area with vertices x, 2x, 2x + y, 2x + y + z, x + y, i.e. the intersection of these two figures is not equal to the empty set. o x y x+y x+y+z z o x y x+y x+y+z z x+z y+z 2x 2x+y 2x+y+z fig.4.2a fig.4.2b iii) weak distributivity of (◦) with respect to (•): x ◦ (y • z) ∩ (x ◦ y) • (x ◦ z) 6= ∅ o z x y y+z o z x y y+z 2x x+y x+z fig.4.3a fig.4.3b in figure 4.3a, take three points x, y, z of the plane, then the side [y, y+ z] of the parallelogram with vertices o, y, z, y + z is the hyperoperation y • z. join the point x to each point of the segment [y, y + z]. then 50 visualization of algebraic properties . . . the area of the triangle with vertices x, y, y + z is the first part of the above inersection, i.e. x ◦ (y • z). in figure 4.3b, take three points x, y, z of the plane, then the line segment [x, y] is the hyperoperation x ◦ y and the line segment [x, z] is the hyperoperation x ◦ z. with the points: o, every point ri of the line segment [x, y] and every point ti of the line segment [x, z] draw, each time, the parallelogram with vertices o, ri, ti, ri + ti. all these infinite line segments [ri, ri + ti] create the pentagon with vertices x, 2x, x + y, y + z, x + z. then the area of this pentagon is the second part of the above intersection, i.e. (x◦y)•(x◦z). notice that the triangle with vertices x, y, y + z is part of the pentagon with vertices x, 2x, x + y, y + z, x + z, i.e. the intersection of these two figures is not equal to the empty set. iv) distributivity of (•) with respect to (◦): x • (y ◦ z) = (x • y) ◦ (x • z) o x x+y y x+z z x x+y y x+z z o fig.4.4a fig.4.4b in figure 4.4a, take three points x, y, z of the plane, then the line segment [y, z] is the hyperoperation y ◦ z. with the points o, x and every point ri of the line segment [y, z] draw, each time, the parallelogram with vertices o, x, ri, x + ri. all these infinite line segments [x, x + ri] create the triangle with vertices x, x + y, x + z. then the area of this triangle is the first part of the above equality, i.e. x • (y ◦ z). in figure 4.4b, the side [x, x + y] of the parallelogram with vertices o, x, y, x + y is the hyperoperation x • y and the side [x, x + z] of the parallelogram with vertices o, x, z, x + z is the hyperoperation x • z. join every point of the side [x, x+ y] to every point of the side [x, x+ z]. 51 visualization of algebraic properties . . . then the area of the triangle with vertices x, x + y, x + z is the second part of the above equality, i.e. (x • y) ◦ (x • z). remark 4.1 it is known that (ir2, ◦) is a commutative hypergroup. into the plane ir2, the hyperoperations (◦) and (•) together with the axioms 3ii), 3iii), 4i), 4ii), 4iii) and 4iv) are giving the concepts of hyperring, hv-ring and dual hv-ring. references [1] a.j. bishop, a review of research on visualization in mathematics education, in: a. borbás (ed.), proceedings of the 12th pme international conference, (1988), 170-176. [2] j.s bruner,the course of cognitive growth, american psychologist, 19,(1964), 1-15. [3] p.corsini, v.leoreanu, applications of hypergroup theory,kluwer academic publishers, 2003 [4] b.davvaz, v.leoreanu, hyperring theory and applications,, international academic press, 2007 [5] a. dramalidis, dual hv-rings, rivista di matematica pura ed applicata, 17,(1996), 55-62. [6] a. gutirrez, visualization in 3-dimensional geometry: in search of a framework, in: proceedings of the 20th pme international conference, l. puig & a. gutierrez (eds.), 3-19,(1996), 3-19. [7] r.palais, the visualization of mathematics: towards a mathematical exploratorium, notices of the ams, 6,(1999), 647-658 [8] prenowitz w., jantosciak j., join geometries. a theory of convex sets and linear geometry, springer, n.y,1979 [9] n.c presmeg, visualization and mathematical giftedness, educational studies in mathematics, 17, (1986), 297-311. [10] n.c presmeg, research on visualization in learning an teaching mathematics,in: handbook of research on the psychology of mathematics education: past, present and future. pme 19762006, ed sense publishers, (2006),205-235. [11] t. vougiouklis, the fundamental relation in hyperrings. the general hyperfield, in: 4th aha congress, world scientific,(1991), 203211. [12] t.vougiouklis, hyperstructures and their representations, monographs in mathematics, hadronic press,1994 52 ratio mathematica volume 40, 2021, pp. 191-212 191 fuzzy homotopy analysis method for solving fuzzy autonomous differential equation hadeer a. sabr* basim n. abood† mazin h. suhhiem‡ abstract in this paper, we have presented the theory and the applications of the fuzzy homotopy analysis method to find the fuzzy semi-analytical solutions of the second order fuzzy autonomous ordinary differential equation. this method allows for the solution of the fuzzy initial value problems to be calculated in the form of an infinite fuzzy series in which the fuzzy components can be easily calculated. some numerical results have been given to illustrate the used method. the obtained numerical results have been compared with the fuzzy exact-analytical solutions. keywords: fuzzy homotopy analysis method; fuzzy autonomous differential equation; fuzzy series solution. 2010 ams subject classification: applied mathematics. ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ * department of mathematics, university of wasit, alkut, iraq; haheelmath2@gmail.com † department of mathematics, university of wasit, alkut, iraq; basim.nasih@yahoo.com ‡ department of statistics, university of sumer, alrifaee, iraq; mazin.suhhiem@yahoo.com 1 received on january 12th, 2021. accepted on may 12th, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.589. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. h. sabr, b. abood, and m. suhhiem 192 1. introduction the topic of the fuzzy semi-analytical methods(fuzzy series method) for solving the fuzzy initial value problems(fivps) has been rapidly growing in recent years, whereas the fuzzy series solutions of fivp have been studied by several authors during the past few years. several fuzzy semi-analytical methods have been proposed to obtain the fuzzy series solution of the linear and non-linear fivb which are mostly first order problems. some of these methods have been proposed to obtain the fuzzy series solutions of the high order fivb. fuzzy homotopy analysis method was used for the first time to solve the fuzzy differential equations in 2012. researchers and scientists are continuing to develop this method for solving various types of the fuzzy initial value problems because it represents an efficient and effective technique. in the following we will review some of the findings of the researchers regarding this method. in 2012, hashemi, malekinagad and marasi[4] suggested and applied the fuzzy homotopy analysis method for solving a system of fuzzy differential equations with fuzzy initial conditions. in 2013, abu-arqub, el-ajou1 and momani[6] studied and developed the fuzzy homotopy analysis method to obtain the analytical solutions of the fuzzy initial value problems. in 2014, jameel, ghoreishi and ismail[8] introduced and applied the fuzzy homotopy analysis method to obtain the approximateanalytical solutions of the high order fuzzy initial value problems. in 2015, aljassar[9] introduced and presented fuzzy semi-analytical methods (including the fuzzy homotopy analysis method) to obtain the numerical and approximate-analytical solutions of the linear and non-linear fuzzy initial value problems. in 2016, otadi and mosleh[12] studied and developed the fuzzy homotopy analysis method to obtain numerical and approximate-analytical solutions of the hybrid fuzzy ordinary differential equations with the fuzzy initial conditions. as well, in 2016, lee, kumaresan and ratnavelu[11] suggested a solution of the fuzzy fractional differential equations with fuzzy initial conditions by using the fuzzy homotopy analysis method. in 2017, padma and kaliyappan[14] introduced and presented fuzzy semi-analytical methods(including the fuzzy homotopy analysis transform method) to obtain the numerical and approximate-analytical solutions of the fuzzy fractional initial value problems. in 2018, sevindir, cetinkaya and tabak[16] introduced and presented fuzzy semi-analytical methods(including the fuzzy homotopy analysis method) to obtain the numerical and approximate-analytical solutions of the first order fuzzy initial value problems. also, in 2018, jameel, saaban and altaie[15] suggested and applied a new concepts for solving the first order non-linear fuzzy initial value problems by using the fuzzy optimal homotopy fuzzy homotopy analysis method for solving fuzzy autonomous differential equation 193 asymptotic method. in 2020, nematallah and najafi[18] introduced and applied the fuzzy homotopy analysis method to obtain the fuzzy semianalytical solution of the fuzzy fractional initial value problems based on the concepts of generalized hukuhara differentiability. as well, in 2020, ali and ibraheem[17] studied and developed some fuzzy analytical and numerical solutions of the linear first order fuzzy initial value problems by using fuzzy homotopy analysis method based on the padè approximate method. in this work, we have studied and applied the fuzzy homotopy analysis method to find the fuzzy series solution(fuzzy approximate-analytical solution) of the second order fuzzy autonomous ordinary differential equation with real variable coefficients(real-valued function coefficients). the fuzzy semianalytical solutions that we have obtained during this work are accurate solutions and very close to the fuzzy exact-analytical solutions, based on the comparison that we have introduced between the results that we have obtained and the fuzzy exact-analytical solutions. 2. basic definitions in fuzzy set theory in this section, we will present some of the fundamental definitions and the primitive concepts related to the fuzzy set theory, which are very necessary for understanding this subject. definition (𝟏), [𝟏] (fuzzy set) the fuzzy set ã can be defined as: ã = {( x , µã(x) ) ∶ x ∈ x ; 0 ≤ µã(x) ≤ 1} (1) where x is the universal set and µã(x) is the grade of membership of x in ã. definition (𝟐), [𝟕] (α – level set) the α level ( or α cut ) set of a fuzzy set ã can be defined as: aα = {x ∈ x ∶ µã(x) ≥ α ; α ∈ [0,1]} . (2) definition (𝟑), [𝟗] (fuzzy number) a fuzzy number ũ is an ordered pair of functions ( u (α) , u (α) ) , 0 ≤ α ≤ 1 , with the following conditions: 1) u (α) is a bounded left continuous and non-decreasing function on [0,1] . 2) u (α) is a bounded left continuous and non-increasing function on [0,1] . h. sabr, b. abood, and m. suhhiem 194 3) u (α) ≤ u (α) , 0 ≤ α ≤ 1. (3) remark (1), [9] : 1) the crisp number u is simply represented by : u (α) = u (α) = u , 0 ≤ α ≤ 1 . (4) 2) the set of all the fuzzy numbers is denoted by e1. remark (𝟐), [𝟏𝟑]: the distance between two arbitrary fuzzy numbers ũ = (u , u) and ṽ = (v , v) can be defined as: d (ũ , ṽ) = [ ∫ ( u (α) v (α) 1 0 ) 2 dα + ∫ ( u (α) v (α) 1 0 ) 2 dα ] 1 2 (5) remark (𝟑), [𝟏𝟑]: (e1,d) is a complete metric space. definition (𝟒), [𝟗] (fuzzy function) a mapping f ∶ t → e1 for some interval t ⊆ e1 is called a fuzzy function or fuzzy process with non-fuzzy variable (crisp variable), and we denote α level sets by: [ f(t) ]α = [ f (t ; α) , f (t ; α)] (6) where t ∈ t , α ∈ [0,1]. we refer to f and f as the lower and upper branches on f. definition (5), [𝟗] (h-difference) let u , v ∈ e1 . if there exist w ∈ e1 such that u = v + w then w is called the h-difference (hukuhara-difference) of u and v and it is denoted by w= u ⊝ v, where u ⊝ v ≠ u + (-1) v. definition (6), [𝟏𝟑] (fuzzy derivative) let f : t → e1 for some interval t ⊆ e1 and t0 ∈ t. we say that f is hdifferential(hukuhara-differential) at t0 , if there exists an element fˊ(t0) ∈ e1 such that for all h> 0 (sufficiently small) , ∃ f (t0 +h)⊝f(t0) , f(t0) ⊝ f (t0 h) and the limits(in the metric d) fuzzy homotopy analysis method for solving fuzzy autonomous differential equation 195 lim h→0 f(t0 + h) ⊝ f(t0) h = lim h→0 f(t0) ⊝ f(t0 − h) h = fˊ(t0) (7) then fˊ(t0) is called the fuzzy derivative(h-derivative) of f at t0 . where d is the distance between two fuzzy numbers. definition (7), [𝟗] (nth order fuzzy derivative) let f′ : t → e1 for some interval t ⊆ e1 and t0 ∈ t. we say that f ′ is h differential(hukuhara-differential) at t0, if there exists an element f (n)(t0) ∈ e1 such that for all h > 0 (sufficientlysmall), ∃ f(n−1)(t0 + h) ⊝ f(n−1)(t0) , f (n−1)(t0)⊝f (n−1) (t0 h) and the limits(in the metric d) lim h→0 f(n−1)(t0+h) ⊝ f (n−1)(t0) h = lim h→0 f(n−1)(t0)⊝f (n−1)(t0−h) h = f(n)(t0) (8) then f(n)(t0) is called the nth order fuzzy derivative (h-derivative of order n) of f at t0. theorem(1), [9]: let f ∶ t → e1 for some interval t ⊆ e1 be an nth order hukuhara differentiable functions at t ∈ t and denote [ f(t) ]α = [ f (t ; α) , f (t ; α)], ∀ α ∈ [0,1]. then the boundary functions f (t ; α) , f (t ; α) are both nth order hukuhara differentiable functions and [ f(n)(t) ]α = [ f (n)(t ; α) , f (n)(t ; α)], ∀ α ∈ [0,1]. (9) 3. fuzzy autonomous differential equation a fuzzy ordinary differential equation is said to be autonomous if it is independent of it's independent crisp variable t. this is to say an explicit nth order fuzzy autonomous differential equation is of the following form[13] : x(n)(t) = f ( x (t) , x′(t) , x′′(t) , … . , x(n−1)(t)) , t ∈ [t0 , h] (10) with the fuzzy initial conditions : x(t0) = x0 , x ′(t0) = x0 ′ , x′′(t0) = x0 ′′ , … , x(n−1)(t0) = x0 (n−1) h. sabr, b. abood, and m. suhhiem 196 where : x is a fuzzy function of the crisp variable t , f (x (t) , x′(t) , x′′(t) , … . , x(n−1)(t)) is a fuzzy function of the crisp variable t and the fuzzy variable x , x(n)(t) is the fuzzy derivative of the x (t) , x′(t) , x′′(t) , … ., x(n−1)(t) , and x(t0) , x ′(t0) , x ′′(t0) , … , x (n−1)(t0) are fuzzy numbers. the fuzzy differential equations that are dependent on t are called nonautonomous, and a system of fuzzy autonomous differential equations is called a fuzzy autonomous system. the main idea in solving the fuzzy autonomous differential equation is to convert it into a system of non-fuzzy(crisp) differential equations, and then solve this system by the known and commonly used methods of solving the non-fuzzy differential equations. now it is possible to replace (10) by the following equivalent system of the nth order crisp ordinary differential equations: x(n)(t) = f ( x , x′, x′′ , … , x(n−1)) = f( x , x′ , x′′ , … , x(n−1) , x , x ′ , x ′′ , … , x (n−1) ) ; x(t0) = x 0 , x ′(t0) = x 0 ′ , x′′(t0) = x 0 ′′ , … , x(n−1)(t0) = x 0 (n−1) , (11) x (n) (t) = f ( x , x′, x′′ , … , x(n−1)) = g( x , x′ , x′′ , … , x(n−1) , x , x ′ , x ′′ , … , x (n−1) ) ; x (t0) = x0 , x ′ (t0) = x0 ′ , x ′′ (t0) = x0 ′′ , … , x (n−1) (t0) = x0 (n−1) (12) where f( x , x′ , x′′ , … , x(n−1) , x , x ′ , x ′′ , … , x (n−1) )= min{ f (t , u) ∶ u ∈ [x , x′ , x′′ , … , x(n−1) , x , x ′ , x ′′ , … , x (n−1) ] }, (13) g( x , x′ , x′′ , … , x(n−1) , x , x ′ , x ′′ , … , x (n−1) )= max{ f (t , u) ∶ u ∈ [x , x′ , x′′ , … , x(n−1) , x , x ′ , x ′′ , … , x (n−1) ] } . (14) fuzzy homotopy analysis method for solving fuzzy autonomous differential equation 197 the parametric form of system (13-14) is given by: x(n)(t , α) = f( x(t , α) , x′(t , α) , x′′(t , α) , … , x(n−1)(t , α) , x (t , α) , x ′ (t , α) , x ′′ (t , α) , … , x (n−1) (t , α) ) x(t0 , α) = x 0(α) , x ′(t0 , α) = x 0 ′ (α) , x′′(t0 , α) = x 0 ′′ (α) , … , x(n−1)(t0 , α) = x 0 (n−1) (α) (15) x (n) (t , α) = g( x(t , α) , x′(t , α) , x′′(t , α) , … , x(n−1)(t , α) , x (t , α) , x ′ (t , α) , x ′′ (t , α) , … , x (n−1) (t , α) ) x (t0 , α) = x0(α) , x ′ (t0 , α) = x0 ′ (α) , x ′′ (t0 , α) = x0 ′′ (α) , … , x (n−1) (t0 ,α) = x0 (n−1) (α) (16) where t ∈ [t0 , h] and α ∈ [0 ,1]. the following theorem ensures the existence and uniqueness of the fuzzy solution of the nth order fuzzy autonomous differential equation. theorem(2), [13] : if we return to problem (10), x(n)(t) = f ( x (t) , x′(t) , x′′(t) , … . , x(n−1)(t)) , t ∈ [t0 , h] let fi : t → e 1 , 1 ≤ i ≤ n be a continuous fuzzy functions, t= [t0 , h ] and assume that there exist a real numbers ki > 0 such that d( fi( t , zi) , fi(t , wi )) ≤ ki d(zi , wi) (17) for all t ∈ t and all zi , wi ∈ e 1 . then the above nth order fivb has a unique fuzzy solution on t in each case. 4. fuzzy homotopy analysis method a fuzzy homotopy analysis method is one of the fuzzy semianalytical methods used to obtain the fuzzy series solution(fuzzy approximate-analytical solution) of the fivbs. this technique utilizes homotopy in order to generate a convergent fuzzy series of fuzzy linear equations from fuzzy non-linear ones. this means that this technique is based on generating a convergent fuzzy series of fuzzy solutions to approximate the fuzzy analytical solution of the fivb. h. sabr, b. abood, and m. suhhiem 198 the basic mathematical concepts of the fuzzy homotopy analysis method are the same as the basic mathematical concepts of the homotopy analysis method, but with the use of the concepts of the fuzzy set theory. this means that solving any fivb by using fuzzy homotopy analysis method is based on converting the fivb into a system of non-fuzzy(crisp) initial value problems by using the steps that we explained in section(3), and then using the homotopy analyss method to solve this system. the fuzzy homotopy analysis method provides us with both the freedom to choose proper base fuzzy functions for approximating a non-linear fuzzy problem and a simple way to ensure the convergence of the fuzzy series solution. 5. description of the method to describe the basic mathematical ideas of the fuzzy homotopy analysis method, we consider the following nth order fivb : [n(x(t)]α = 0 , (18) where n is the fuzzy non-linear operator , t denotes the independent crisp variable , x(t) is an unknown fuzzy function . by the concepts of section(3), we can conclude that: [n(x(t)]α = [ [n(x(t)]α l , [n(x(t)]α u ] (19) since 0 = [ 0 , 0], we can get: [n(x(t)]α l = 0 (20i) [n(x(t)]α u = 0 (20ii) now, we construct the zero-order fuzzy deformation equation: [(1 − w)l (θ(t ; w) − x0(t))]α = [w h n(θ(t ; w))]α , (21) where w ∈ [0 , 1] is the homotopy embedding parameter, h ∈ [−1 , 0) is the convergence control parameter , l is the fuzzy linear operator , x0(t) is the fuzzy initial guess of x(t) and θ(t; w) is a fuzzy function. by the concepts of section(3), we can get: (1 − w)l( [θ(t ; w)]α l − [x0(t)]α l ) = w h( [n(θ(t ; w))]α l ) (22i) (1 − w)l( [θ(t ; w)]α u − [x0(t)]α u ) = w h( [n(θ(t ; w))]α u ) (22ii) fuzzy homotopy analysis method for solving fuzzy autonomous differential equation 199 obviously , when w = 0 and w = 1 , both [θ(t ; 0)]α = [x0(t)]α , (23) [θ(t ; 1)]α = [x(t)]α (24) hold, therefore when w is increasing from 0 to 1 , the fuzzy solutions [θ(t ; w)]α l and [θ(t ; w)]α u varies from the fuzzy initial guess [x0(t)]α to the fuzzy solution[x(t)]α . thus, we have: [θ(t ; 0)]α l = [x0(t)]α l (25i) [θ(t ; 0)]α u = [x0(t)]α u (25ii) [θ(t ; 1)]α l = [x(t)]α l (26i) [θ(t ; 1)]α u = [x(t)]α u (26ii) by expanding [θ(t ; w) ]α in taylor series with respect to w , one has: [θ(t ; w)]α = [x0(t)]α + ∑ [xm(t)w m]α ∞ m=1 (27) where [xm(t)]α = 1 m! ∂m [θ(t ;w)]α ∂wm | w=0 (28) by the concepts of parametric form in section(3), we can conclude that: [θ(t ; w)]α l = [x0(t)]α l + ∑ [xm(t)]α l∞ m=1 w m (29i) [θ(t ; w)]α u = [x0(t)]α u + ∑ [xm(t)]α u∞ m=1 w m (29ii) where [x0(t)]α l = 1 m! ∂m [θ(t ; w)]α l ∂wm | w=0 (30i) [x0(t)]α u = 1 m! ∂m [θ(t ; w)]α u ∂wm | w=0 (30ii) if the fuzzy linear operator , the fuzzy initial guess , the auxiliary parameter h , and the auxiliary fuzzy function are so properly chosen , then the fuzzy series (27) converges at w = 1 , and one has: [θ(t ; 1)]α = [x(t)]α = [x0(t)]α + ∑ [xm(t)]α ∞ m=1 (31) h. sabr, b. abood, and m. suhhiem 200 where [θ(t ; 1)]α l = [x(t)]α l = [x0(t)]α l + ∑ [xm(t)]α l∞ m=1 (32i) [θ(t ; 1)]α u = [x(t)]α u = [x0(t)]α u + ∑ [xm(t)]α u∞ m=1 (32ii) which must be one of the fuzzy solutions of the problem(18). if h = −1 , (21) becomes [(1 − w)l (θ(t ; w) − x0(t))]α + [w n(θ(t ; w))]α = 0 , (33) where (1 − w)l( [θ(t ; w)]α l − [x0(t)]α l ) + w ( [n(θ(t ; w))]α l ) =0 (34i) (1 − w)l( [θ(t ; w)]α u − [x0(t)]α u ) + w ( [n(θ(t ; w))]α u ) =0 (34ii) which is used mostly in the fuzzy homotopy analysis method. we define the fuzzy vectors [ x⃗ i ]α = { [ x0(t)]α , [x1(t)]α , [ x2(t)]α , … , [ xi(t)]α } (35) where [ x⃗ i ]α l = { [ x0(t)]α l , [ x1(t)]α l , [ x2(t)]α l , … , [ xi(t)]α l } (36i) [ x⃗ i ]α u = { [ x0(t)]α u , [ x1(t)]α u , [ x2(t)]α u , … , [ xi(t)]α u } (36ii) now, by differentiating (21) mtimes with respect to the parameter w and then setting w = 0 and finally dividing them by m! , we have the mth-order fuzzy deformation equation: l( [xm(t)]α − χm[xm−1(t)]α ) = h( [rm(x⃗ m−1) ]α ) (37) where [ rm(x⃗ m−1) ]α = 1 (m−1)! ∂m−1[n(θ(t ; w))]α ∂wm−1 | w=0 (38) by the concepts of parametric form in section(3), we get: l( [xm(t)]α l − χm[xm−1(t)]α l ) = h ( [rm(x⃗ m−1)]α l ) (39i) l( [xm(t)]α u − χm[xm−1(t)]α u ) = h ( [rm(x⃗ m−1)]α u ) (39ii) where fuzzy homotopy analysis method for solving fuzzy autonomous differential equation 201 [rm(x⃗ m−1)]α l = 1 (m−1)! ∂m−1( [n(θ(t ; w))]α l ) ∂wm−1 | w=0 (40i) [rm(x⃗ m−1)]α u = 1 (m−1)! ∂m−1( [n(θ(t ; w))]α u ) ∂wm−1 | w=0 (40ii) χm = { 0 , m ≤ 1 , 1 , m > 1 . (41) 6. applied example in this section, one fuzzy problem has been solved in order to clarify the efficiency and the accuracy of the method. according to liao's book [3], the optimal value of h was found to be approximately −1 ≤ h < 0. in addition, the practical examples in[3,5,10] showed that the optimal value of h can be determined while solving the problem by experimenting with a number of different values of h. the optimal value of h depends greatly on the nature of the problem, but still h = −1 is an optimal value and achieves a rapid convergence. example 1: consider the second order fuzzy autonomous differential equation x′′(t) + x(t) = 0 , (42) subject to the fuzzy initial conditions : [x(0)]α = [0,0] , [x ′(0)]α = [0.01α + 0.02 , −0.01α + 0.04] , α ∈ [0,1] . solution: the fuzzy linear operator is : [l(θ(t; w))]α = [ [l(θ(t; w))]α l , [l(θ(t; w))]α u ] (43) where [l(θ(t; w))]α l = [ ∂2θ(t;w) ∂t2 ] α l (44i) [l(θ(t; w))]α u = [ ∂2θ(t;w) ∂t2 ] α u (44ii) we define the fuzzy non-linear operator as : [n(θ(x; w))]α = [ [n(θ(x; w))]α l , [n(θ(x; w))]α u ] (45) where h. sabr, b. abood, and m. suhhiem 202 [n(θ(x; w))]α l = ∂2[θ(t;w)]α l ∂t2 + [θ(t; w)]α l (46i) [n(θ(x; w))]α u = ∂2[θ(t;w)]α u ∂t2 + [θ(t; w)]α u (46ii) the fuzzy series solution is : [x(t)]α = [ [x(t)]α l , [x(t)]α u ] (47) where [x(t)]α l = [x0(t)]α l + [x1(t)]α l + [x2(t)]α l + [x3(t)]α l + ⋯ (48i) [x(t)]α u = [x0(t)]α u + [x1(t)]α u + [x2(t)]α u + [x3(t)]α u + ⋯ (48ii) by taylor series expansion , the fuzzy initial approximation is: [x0(t)]α = [ [x0(t)]α l , [x0(t)]α u ] (49) where [x0(t)]α l = 0.01 αt + 0.02 t (50i) [x0(t)]α u = −0.01 αt + 0.04 t (50ii) to find [x1(t)]α = [ [x1(t)]α l , [x1(t)]α u ] from(29), we can find [θ(t; w)]α l = [x0(t)]α l + w[x1(t)]α l (51i) [θ(t; w)]α u = [x0(t)]α u + w[x1(t)]α u (51ii) from(37), we can find l([x1(t)]α l − 0) = h[r1]α l (52i) l([x1(t)]α u − 0) = h[r1]α u (52ii) then from(38), we can get : [r1]α l = [n(θ(t; w))]α l|w=0 (53i) [r1]α u = [n(θ(t; w))]α u|w=0 (53ii) then , we apply the following steps : fuzzy homotopy analysis method for solving fuzzy autonomous differential equation 203 ∂[θ(t;w)]α l ∂t = ∂[x0(t)]α l ∂t + w ∂[x1(t)]α l ∂t (54i) ∂[θ(t;w)]α u ∂t = ∂[x0(t)]α u ∂t + w ∂[x1(t)]α u ∂t (54ii) 𝜕2[θ(t;w)]α l ∂t2 = ∂2[x0(t)]α l ∂t2 + w ∂2[x1(t)]α l ∂t2 (55i) 𝜕2[θ(t;w)]α u ∂t2 = ∂2[x0(t)]α u ∂t2 + w ∂2[x1(t)]α u ∂t2 (55ii) [n(θ(x; w))]α l = ∂2[x0(t)]α l ∂t2 + w ∂2[x1(t)]α l ∂t2 + [x0(t)]α l + w[x1(t)]α l (56i) [n(θ(x; w))]α u = ∂2[x0(t)]α u ∂t2 + w ∂2[x1(t)]α u ∂t2 + [x0(t)]α u + w[x1(t)]α u (56ii) [r1]α l = ∂2[x0(t)]α l ∂t2 + [x0(t)]α l (57i) [r1]α u = ∂2[x0(t)]α u ∂t2 + [x0(t)]α u (57ii) [r1]α l = 0.01 αt + 0.02 t (58i) [r1]α u = −0.01 αt + 0.04 t (58ii) l([x1(t)]α l) = 0.01 αht + 0.02 ht (59i) l([x1(t)]α u) = −0.01 αht + 0.04 ht (59ii) [x1(t)]α l = ∬(0.01 αht + 0.02 ht ) dt dt (60i) [x1(t)]α u = ∬(−0.01 αht + 0.04 ht ) dt dt (60ii) [x1(t)]α l = 0.001667 αht3 + 0.003333 ht3 (61i) [x1(t)]α u = −0.001667 αht3 + 0.006667 ht3 (61ii) now, to find [x2(t)]α = [ [x2(t)]α l , [x2(t)]α u ] from(29 ), we can find [θ(t; w)]α l = [x0(t)]α l + w[x1(t)]α l + w2[x2(t)]α l (62i) [θ(t; w)]α u = [x0(t)]α u + w[x1(t)]α u + w2[x2(t)]α u (62ii) from(37 ), we can find h. sabr, b. abood, and m. suhhiem 204 l([x2(t)]α l − [x1(t)]α l) = h[r2]α l (63i) l([x2(t)]α u − [x1(t)]α u) = h[r2]α u (63ii) then from (38 ) , we can get : [r2]α l = ∂[n(θ(t;w))]α l ∂w | w=0 (64i) [r2]α u = ∂[n(θ(t;w))]α u ∂w | w=0 (64ii) then , we apply the following steps : ∂[θ(t;w)]α l ∂t = ∂[x0(t)]α l ∂t + w ∂[x1(t)]α l ∂t + w2 ∂[x2(t)]α l ∂t (65i) ∂[θ(t;w)]α u ∂t = ∂[x0(t)]α u ∂t + w ∂[x1(t)]α u ∂t + w2 ∂[x2(t)]α u ∂t (65ii) 𝜕2[θ(t;w)]α l ∂𝑡2 = 𝜕2[x0(t)]α l ∂𝑡2 + w 𝜕2[x1(t)]α l ∂𝑡2 + w2 𝜕2[x2(t)]α l ∂𝑡2 (66i) 𝜕2[θ(t;w)]α u ∂𝑡2 = 𝜕2[x0(t)]α u ∂𝑡2 + w 𝜕2[x1(t)]α u ∂𝑡2 + w2 𝜕2[x2(t)]α u ∂𝑡2 (66ii) [n(θ(x; w))]α l = ∂2[x0(t)]α l ∂t2 + w ∂2[x1(t)]α l ∂t2 + w2 ∂2[x2(t)]α l ∂t2 + [x0(t)]α l + w[x1(t)]α l + w2[x2(t)]α l (67i) [n(θ(x; w))]α u = ∂2[x0(t)]α u ∂t2 + w ∂2[x1(t)]α u ∂t2 + w2 ∂2[x2(t)]α u ∂t2 + [x0(t)]α u + w[x1(t)]α u + w2[x2(t)]α u (67ii) ∂[n(θ(t;w))]α l ∂w = ∂2[x1(t)]α l ∂t2 + 2w ∂2[x2(t)]α l ∂t2 + [x1(t)]α l + 2w[x2(t)]α l (68i) ∂[n(θ(t;w))]α u ∂w = ∂2[x1(t)]α u ∂t2 + 2w ∂2[x2(t)]α u ∂t2 + [x1(t)]α u + 2w[x2(t)]α u (68ii) [r2]α l = ∂2[x1(t)]α l ∂t2 + [x1(t)]α l (69i) [r2]α u = ∂2[x1(t)]α u ∂t2 + [x1(t)]α u (69ii) [r2]α l = 0.010002 αht + 0.019998 ht + 0.001667 αht3 + 0.003333 ht3 , (70i) [r2]α u = −0.010002 αht + 0.040002 ht − 0.001667 αht3 + 0.006667 ht3 . (70ii) fuzzy homotopy analysis method for solving fuzzy autonomous differential equation 205 l([x2(t)]α l − [x1(t)]α l) = 0.010002 αh2t + 0.019998 h2t + 0.001667 αh2t3 + 0.003333 h2t3 (71i) l([x2(t)]α u − [x1(t)]α u) = −0.010002 αh2t + 0.040002 h2t − 0.001667 αh2t3 + 0.006667 h2t3 (71ii) [x2(t)]α l − [x1(t)]α l = ∬(0.010002 αh2t + 0.019998 h2t + 0.001667 αh2t3 + 0.003333 h2t3) dtdt , (72i) [x2(t)]α u − [x1(t)]α u = ∬(−0.010002αh2t + 0.040002h2t − 0.001667αh2t3 + 0.006667h2t3) dtdt . (72ii) [x2(t)]α l − [x1(t)]α l = 0.001667 αh2t3 + 0.003333 h2t3 + 0.000083 αh2t5 + 0.000167 h2t5 , (73i) [x2(t)]α u − [x1(t)]α u = −0.001667 αh2t3 + 0.006667 h2t3 − 0.000083 αh2t5 + 0.000333 h2t5 . (73ii) [x2(t)]α l = 0.001667 αh2t3 + 0.003333 h2t3 + 0.000083 αh2t5 + 0.000167 h2t5 + 0.001667 αht3 + 0.003333 ht3 (74i) [x2(t)]α u = −0.001667 αh2t3 + 0.006667 h2t3 − 0.000083 αh2t5 + 0.000333 h2t5 − 0.001667 αht3 + 0.006667 ht3 (74ii) now, to find [x3(t)]α = [ [x3(t)]α l , [x3(t)]α u ] : from(29), we can find [θ(t; w)]α l = [x0(t)]α l + w[x1(t)]α l + w2[x2(t)]α l + w3[x3(t)]α l (75i) [θ(t; w)]α u = [x0(t)]α u + w[x1(t)]α u + w2[x2(t)]α u + w3[x3(t)]α u (75ii) from(37), we can find l([x3(t)]α l − [x2(t)]α l) = h[r3]α l (76i) l([x3(t)]α u − [x2(t)]α u) = h[r3]α u (76ii) then from(38), we can get : h. sabr, b. abood, and m. suhhiem 206 [r3]α l = 1 2 ∂2[n(θ(t;w))]α l ∂w2 | w=0 (77i) [r3]α u = 1 2 ∂2[n(θ(t;w))]α u ∂w2 | w=0 (77ii) then , we apply the following steps : ∂[θ(t;w)]α l ∂t = ∂[x0(t)]α l ∂t + w ∂[x1(t)]α l ∂t + w2 ∂[x2(t)]α l ∂t + w3 ∂[x3(t)]α l ∂t (78i) ∂[θ(t;w)]α u ∂t = ∂[x0(t)]α u ∂t + w ∂[x1(t)]α u ∂t + w2 ∂[x2(t)]α u ∂t + w3 ∂[x3(t)]α u ∂t (78ii) ∂2[θ(t;w)]α l ∂t2 = ∂2[x0(t)]α l ∂t2 + w ∂2[x1(t)]α l ∂t2 + w2 ∂2[x2(t)]α l ∂t2 + w3 ∂2[x3(t)]α l ∂t2 (79i) ∂2[θ(t;w)]α u ∂t2 = ∂2[x0(t)]α u ∂t2 + w ∂2[x1(t)]α u ∂t2 + w2 ∂2[x2(t)]α u ∂t2 + w3 ∂2[x3(t)]α u ∂t2 (79ii) [n(θ(x; w))]α l = ∂2[x0(t)]α l ∂t2 + w ∂2[x1(t)]α l ∂t2 + w2 ∂2[x2(t)]α l ∂t2 + w3 ∂3[x3(t)]α l ∂t2 + [x0(t)]α l + w[x1(t)]α l + w2[x2(t)]α l + w3[x3(t)]α l (80i) [n(θ(x; w))]α u = ∂2[x0(t)]α u ∂t2 + w ∂2[x1(t)]α u ∂t2 + w2 ∂2[x2(t)]α u ∂t2 + w3 ∂3[x3(t)]α u ∂t2 + [x0(t)]α u + w[x1(t)]α u + w2[x2(t)]α u + w3[x3(t)]α u (80ii) ∂[n(θ(t;w))]α l ∂w = ∂2[x1(t)]α l ∂t2 + 2w ∂2[x2(t)]α l ∂t2 + 3w2 ∂3[x3(t)]α l ∂t2 + [x1(t)]α l + 2w[x2(t)]α l + 3w2[x3(t)]α l (81i) ∂[n(θ(t;w))]α u ∂w = ∂2[x1(t)]α u ∂t2 + 2w ∂2[x2(t)]α u ∂t2 + 3w2 ∂3[x3(t)]α u ∂t2 + [x1(t)]α u + 2w[x2(t)]α u + 3w2[x3(t)]α u (81ii) ∂2[n(θ(t;w))]α l ∂w2 = 2 ∂2[x2(t)]α l ∂t2 + 6w ∂3[x3(t)]α l ∂t2 + 2[x2(t)]α l + 6w[x3(t)]α l (82i) ∂2[n(θ(t;w))]α u ∂w2 = 2 ∂2[x2(t)]α u ∂t2 + 6w ∂3[x3(t)]α u ∂t2 + 2[x2(t)]α u + 6w[x3(t)]α u (82ii) [r3]α l = ∂2[x2(t)]α l ∂t2 + [x2(t)]α l (83i) [r3]α u = ∂2[x2(t)]α u ∂t2 + [x2(t)]α u (83ii) [r3]α l = 0.010002 αh2t + 0.019998 h2t + 0.003327 αh2t3 + 0.006673 h2t3 + 0.010002 αht + 0.019998 ht + 0.000083 αh2t5 + fuzzy homotopy analysis method for solving fuzzy autonomous differential equation 207 0.000167 h2t5 + 0.001667 αht3 + 0.003333 ht3 (84i) [r3]α u = −0.010002 αh2t + 0.040002 h2t − 0.003327 αh2t3 + 0.013327 h2t3 − 0.010002 αht + 0.040002 ht − 0.000083 αh2t5 + 0.000333 h2t5 − 0.001667 αht3 + 0.006667 ht3 (84ii) l([x3(t)]α l − [x2(t)]α l) = 0.010002 αh3t + 0.019998 h3t + 0.003327 αh3t3 + 0.006673 h3t3 + 0.010002 αh2t + 0.019998 h2t + 0.000083 αh3t5 + 0.000167 h3t5 + 0.001667 αh2t3 + 0.003333 h2t3 (85i) l([x3(t)]α u − [x2(t)]α u) = −0.010002 αh3t + 0.040002 h3t − 0.003327 αh3t3 + 0.013327 h3t3 − 0.010002 αh2t + 040002 h2t − 0.000083 αh3t5 + 0.000333 h3t5 − 0.001667 αh2t3 + 0.006667 h2t3 (85ii) [x3(t)]α l − [x2(t)]α l = ∬(0.010002 αh3t + 0.019998 h3t + 0.003327 αh3t3 + 0.00667 h3t3 + 0.010002 αh2t + 0.019998 h2t + 0.000083 αh3t5 + 0.000167 h3t5 + 0.001667 αh2t3 + 0.003333 h2t3) dt dt (86i) [x3(t)]α u − [x2(t)]α u = ∬(−0.010002 αh3t + 0.040002 h3t − 0.003327 αh3t3 + 0.013327h3t3 − 0.010002 αh2t + 0.040002 h2t − 0.000083 αh3t5 + 0.000333 h3t5 − 0.001667 αh2t3 + 0.006667 h2t3) dt dt (86ii) [x3(t)]α l − [x2(t)]α l = 0.001667 αh3t3 + 0.000333 h3t3 + 0.000166 αh3t5 + 0.000334 h3t5 + 0.001667 αh2t3 + 0.000333 h2t3 + 0.000002 αh3t7 + 0.000004 h3t7 + 0.000083 αh2t5 + 0.000167 h2t5 (87i) [x3(t)]α u − [x2(t)]α u = −0.001667 αh3t3 + 0.006667 h3t3 − 0.000166 αh3t5 + 0.000666 h3t5 − 0.001667 αh2t3 + 0.006667 h2t3 − 0.000002 αh3t7 + 0.000008 h3t7 − 0.000083 αh2t5 + 0.000333 h2t5 (87ii) [x3(t)]α l = 0.001667 αh3t3 + 0.000333 h3t3 + 0.000166 αh3t5 + 0.000334 h3t5 + 0.003334 αh2t3 + 0.003666 h2t3 + 0.000002 αh3t7 + 0.000004 h3t7 + 0.000334 h2t5 + 0.000166 αh2t5 + 0.001667 αht3 + 0.003333 ht3 (88i) [x3(t)]α u = −0.001667 αh3t3 + 0.006667 h3t3 − 0.000166 αh3t5 + 0.000666 h3t5 − 0.003334 αh2t3 + 0.013334 h2t3 − 0.000002 αh3t7 + h. sabr, b. abood, and m. suhhiem 208 0.000008 h3t7 − 0.000166 αh2t5 + 0.000666 h2t5 − 0.001667 αht3 + 0.006667 ht3 (88ii) then, the fuzzy series solution is: [x(t)]α = [ [x(t)]α l , [x(t)]α u] (89) where [x(t)]α l = 0.01 αt + 0.02 t + 0.005001 αht3 + 0.009999 ht3 + 0.005001 αh2t3 + 0.006999 h2t3 + 0.000249 αh2t5 + 0.000501 h2t5 + 0.001667 αh3t3 + 0.000333 h3t3 + 0.000166 αh3t5 + 0.000334 h3t5 + 0.000002 αh3t7 + 0.000004 h3t7 + ⋯ (90i) [x(t)]α u = −0.01 αt + 0.04 t − 0.005001 αht3 + 0.020001 ht3 − 0.005001 αh2t3 + 0.020001 h2t3 − 0.000249 αh2t5 + 0.000999 h2t5 − 0.001667 αh3t3 + 0.006667 h3t3 − 0.000166 αh3t5 + 0.000666 h3t5 − 0.000002 αh3t7 + 0.000008 h3t7 + ⋯ (90ii) the fuzzy series solution at h = −1, will be [x(t)]α = [ [x(t)]α l , [x(t)]α u] (91) where [x(t)]α l = 0.01 αt + 0.02 t − 0.001667 αt3 − 0.003333 t3 + 0.000083 αt5 + 0.000167 t5 − 0.000002 αt7 − 0.000004 t7 + ⋯ (92i) [x(t)]α u = −0.01 αt + 0.04 t + 0.001667 αt3 − 0.006667 t3 − 0.000083 αt5 + 0.000333 t5 + 0.000002 αt7 − 0.000008 t7 + ⋯ (92ii) 7. discussion when solving a fuzzy autonomous differential equation by using the fuzzy homotopy analysis method , the accuracy of the results depends greatly on the value of the parameter h, other factors also affect, including : the number of terms of the solution series, the value of the constant α and the period to which the variable t belongs. the fuzzy semi-analytical solutions that we obtained during this work are accurate solutions and very close to the fuzzy exactanalytical solutions, based on the comparison that we will make between the results that we obtained and the fuzzy exact-analytical solutions to the chosen problem. fuzzy homotopy analysis method for solving fuzzy autonomous differential equation 209 if we go back to example(1) : x′′(t) + x(t) = 0 , t ∈ [0 , 0.5] (93) the fuzzy exact-analytical solution for this problem is : [x(t)]α = [ [x(t)]α l , [x(t)]α u ] (94) where [x(t)]α l = ( 0.02 + 0.01α )sint (95i) [x(t)]α u = ( 0.04 − 0.01α )sint (95ii) while the fuzzy semi-analytical solution that we got(at h = −1, α = 0.3) is : [x(t)]α = [ [x(t)]α l , [x(t)]α u ] (96) where [x(t)]α l = 0.023 t − 0.003833 t3 + 0.000122 t5 − 0.000005 t7 + ⋯ , (97i) [x(t)]α u = 0.037 t − 0.006167 t3 + 0.000308 t5 − 0.000007 t7 + ⋯ . (97ii) also, the fuzzy semi-analytical solution that we got(at h = −1, α = 0.4) is : [x(t)]α = [ [x(t)]α l , [x(t)]α u ] (98) where [x(t)]α l = 0.024 t − 0.004 t3 + 0.0002 t5 − 0.000005 t7 + ⋯ (99i) [x(t)]α u = 0.036 t − 0.006 t3 + 0.0003 t5 − 0.000007 t7 + ⋯ (99ii) we test the accuracy of the obtained solutions by computing the absolute errors [error]α l = | [xexact(t)]α l − [xseries(t)]α l | (100i) [error]α u = | [xexact(t)]α u − [xseries(t)]α u | (100ii) the following tables provides a comparison between the fuzzy exact-analytical solution and the fuzzy semianalytical solution for this problem. t [xseries(t)]α l [error]r l [xseries(t)]α u [error]r u h. sabr, b. abood, and m. suhhiem 210 table 1. compar ison of the results of exampl e(1), α = 0.3. table 2. comparison of the results of example(1), α = 0.4. 8. conclusion in this work, we have studied the fuzzy approximate-analytical solutions of the second order fuzzy autonomous differential equation. obviously the accuracy of the results that can be obtained when solving using the fuzzy homotopy analysis method, these results may improve further when increasing the number of terms of the solution series or using another value for the parameter h. the value of the variable t greatly affects the accuracy of the results, if the value of the variable t is close to the initial value, the results will be more accurate. also, the value of the constant α greatly affects the accuracy of the results. certainly, the best value of the constant α cannot be determined, as it changes from one problem to another. references 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0 0.001149520 0.002296168 0.003437072 0.004569374 0.005690228 0.006796804 0.007886297 0.008955929 0.010002950 0.011024648 0 1.99 e-11 3.63 e-10 4.00 e-9 1.90 e-8 6.20 e-8 1.60 e-7 3.51 e-7 6.92 e-7 1.26 e-6 2.14 e-6 0 0.001849229 0.003693836 0.005529209 0.007350762 0.009153940 0.010934237 0.012687203 0.014408454 0.016093689 0.017738695 0 4.18 e-11 3.37 e-10 1.00 e-9 2.00 e-9 5.00 e-9 9.00 e-9 1.50 e-8 2.40 e-8 3.50 e-8 4.90 e-8 t [xseries(t)]α l [error]r l [xseries(t)]α u [error]r u 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0 0.001199500 0.002396002 0.003586515 0.004768063 0.005937695 0.007092484 0.008229547 0.009346039 0.010439171 0.011506210 0 0 0 4.20 e-13 3.08 e-12 1.48 e-11 5.34 e-11 1.58 e-10 4.07 e-10 9.40 e-10 1.00 e-9 0 0.001799250 0.003594002 0.005379772 0.007152095 0.008906542 0.010638727 0.012344321 0.014019060 0.015658759 0.017259320 0 0 0 2.30 e-13 1.78 e-12 8.32 e-12 2.93 e-11 8.41 e-11 2.08 e-10 4.59 e-10 9.23 e-10 fuzzy homotopy analysis method for solving fuzzy autonomous differential equation 211 [1] shokri j. , " numerical solution of fuzzy differential equations ", journal of applied mathematical sciences, vol. 1, no. 45, 2231-2246, 2007. 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[18] najafi n. , " homotopy analysis method to solve fuzzy impulsive fractional differential equations ", international journal of modern mathematical sciences,vol. 18, no. 1, 58-75, 2020. microsoft word r.m.7 cap.15.doc ratio mathematica issue n. 30 (2016) pp. 67-81 issn (print): 1592-7415 issn (online): 2282-8214 on hyper hoop-algebras rajabali borzooei(a), hamidreza varasteh(b), keivan borna(c) (a)department of mathematics, shahid beheshti university, tehran, iran borzooei@sbu.ac.ir (b) department of research and technology, kharazmi university, tehran, iran varastehhamid@gmail.com (c)faculty of mathematics and computer science, kharazmi university, tehran, iran borna@khu.ac.ir abstract in this paper, we apply the hyper structure theory to hoop-algebras and introduce the notion of (quasi) hyper hoop-algebra which is a generalization of hoop-algebra and investigate some related properties. we also introduce the notion of (weak)filters on hyper hoop-algebras, and give several properties of them. finally, we characterize the (weak) filter generated by a non-empty subset of a hyper hoop-algebra. keywords: hoop-algebra, (quasi) hyper hoop-algebra, (weak) filter. 2010 ams subject classifications: 20n20, 14l17, 97h50, 03g25 doi: 10.23755/rm.v30i1.14 1 introduction hoop-algebras or hoops are naturally ordered commutative residuated integral monoids were originally introduced by bosbach in [7] under the name of complementary semigroups. it was proved that a hoop is a meet-semilattice. hoopalgbras then investigated by büchi and owens in an unpublished manuscript [8] of 1975, and they have been studied by blok and ferreirim [2],[3], and aglianò et.al. [1], among others. the study of hoops is motivated by their occurrence both in universal algebra and algebraic logic. typical examples of hoops include both brouwerian semilattices and the positive cones of lattice ordered abelian groups, 67 rajabali borzooei, hamidreza varasteh and keivan borna while hoops structurally enriched with normal multiplicative operators naturally generalize the normal boolean algebras with operators. in recent years, hoop theory was enriched with deep structure theorems. many of these results have a strong impact with fuzzy logic. particularly, from the structure theorem of finite basic hoops one obtains an elegant short proof of the completeness theorem for propositional basic logic introduced by hájek in [12]. the algebraic structures corresponding to hájek’s propositional (fuzzy) basic logic, bl-algebras, are particular cases of hoops and mv-algebras, product algebras and gödel algebras are the most known classes of bl-algebras. hypersructure theory was introduced in 1934[13], when marty at the 8th congress of scandinavian mathematicians, gave the definition of hypergroup and illustrated some applications and showed its utility in the study of groups, algebraic functions, and rational fraction. till now, the hyperstructures have been studied from the theoretical point of view for their applications to many subject of pure and applied mathematics. some fields of applications of the mentioned structures are lattices, graphs, coding, ordered sets, median algebra, automata, and cryptography[9]. many researchers have worked on this area. r.a.borzooei et al. introduced and studied hyper residuated lattices and hyper k-algebras in [4],[6] and s.ghorbani et al.[11], applied the hyper structures to mv-algebras and introduced the concept of hyper mv-algebra, which is a generalization of mv-algebra. in this paper we construct and introduce the notion of (quasi) hyper hoop-algebra which is a generalization of hoop-algebra. then we study some properties of this structure. we also introduce the notion of (weak)filters on hyper hoop-algebras, and give several properties of them. finally, we characterize the (weak) filter generated by a non-empty subset of a hyper hoop-algebra. 2 preliminaries in this section, we recall some definitions and theorems in hoop algebras which will be needed in this paper. definition 2.1. [1] a hoop-algebra or a hoop is an algebra (a,∗,→, 1) of the type (2, 2, 0) such that, for all x, y, z ∈ a: (h1) (a,∗, 1) is a commutative monoid, (h2) x → x = 1, (h3) (x → y)∗x = (y → x)∗y, (h4) x → (y → z) = (x∗y) → z. on the hoop a, if we define x ≤ y iff x → y = 1, for any x, y ∈ a, it is proved that ≤ is a partial order on a. a hoop a is bounded if there is an element 68 on hyper hoop-algebras 0 ∈ a such that 0 ≤ x for all x ∈ a. proposition 2.2. [1] let a be a hoop-algebra. then for every a, b, c ∈ a the following hold: (i) (a,≤) is a ∧-semilattice and a∧ b = a∗ (a → b), (ii) a ≤ b → c iff a∗ b ≤ c, (iii) 1 → a = a, (iv) a → 1 = 1, i.e. a ≤ 1, (v) a → b ≤ (c → a) → (c → b), (vi) a ≤ b → a, (vii) a ≤ (a → b) → b, (viii) a → (b → c) = b → (a → c), (ix) a → b ≤ (b → c) → (a → c), (x) a ≤ b implies b → c ≤ a → c and c → a ≤ c → b. now, we recall some basic notions of the hypergroup theory from [9]: let h be a non-empty set. a hypergroupoid is a pair (h,�), where � : h × h −→ p(h) \∅ is a binary hyperoperation on h. if a � (b � c) = (a � b) � c holds, for all a, b, c ∈ h then (h,�) is called a semihypergroup, and it is said to be commutative if � is commutative. an element 1 ∈ h is called a unit, if a ∈ 1�a∩a�1, for all a ∈ h and is called a scaler unit, if {a} = 1�a = a�1, for all a ∈ a. if the reproduction axiom a � h = h = h � a, for any element a ∈ h is satisfied, then the pair (h,�) is called a hypergroup. note that if a, b ⊆ h, then a�b = ⋃ a∈a,b∈b(a� b). 3 hyper hoop-algebras definition 3.1. aquasi hyper hoop-algebra or briefly, a quasi hyper hoop is a non-empty set a endowed with two binary hyperoperations �,→ and a constant 1 such that, for all x, y, z ∈ a satisfying the following conditions: (hha1) (a,�, 1) is a commutative semihypergroup with 1 as the unit, (hha2) 1 ∈ x → x, (hha3) (x → y)�x = (y → x)�y, (hha4) x → (y → z) = (x�y) → z, a quasi hyper hoop (a,�,→, 1) is called a hyper hoop if the following hold; (hha5) 1 ∈ x → 1, (hha6) if 1 ∈ x → y and 1 ∈ y → x then x = y, (hha7) if 1 ∈ x → y and 1 ∈ y → z then 1 ∈ x → z. in the sequel we will refer to the (quasi) hyper hoop (a,�,→, 1) by its universe a. on (quasi) hyper hoop a, for any x, y ∈ a, we define x ≤ y if and 69 rajabali borzooei, hamidreza varasteh and keivan borna only if 1 ∈ x → y. if a is a hyper hoop, it is easy to see that ≤ is a partial order relation on a. moreover, for all b, c ⊆ a we define b � c iff there exist b ∈ b and c ∈ c such that b ≤ c and define b ≤ c iff for any b ∈ b there exists c ∈ c such that b ≤ c. a (quasi) hyper hoop a is bounded if there is an element 0 ∈ a such that 0 ≤ x, for all x ∈ a. in the following examples, we will show that the conditions (hha5), (hha6), and (hha7) are independent from the other conditions. example 3.2. (i) let a = {1, a, b}. define the hyperoperations �, and → on a as follows: � 1 a b 1 {1} {a} {a, b} a {a} {a} {a, b} b {a, b} {a, b} {b} → 1 a b 1 {1} {a, b} {b} a {b} {1, a, b} {b} b {1, a, b} {1, a, b} {1, a, b} then (a,�,→, 1) is a quasi hyper hoop, but doesn’t satisfy the condition (hha5). since 1 /∈ a → 1. (ii) let a = {1, a, b}. define the hyperoperations � and → on a as follows: � 1 a b 1 {1} {a} {b} a {a} {a} {a} b {b} {a} {1} → 1 a b 1 {1, b} {a} {1, b} a {1, b} {1, b} {1, b} b {1, b} {a} {1, b} then (a,�,→, 1) is a quasi hyper hoop, but doesn’t satisfy the condition (hha6). since 1 ∈ b → 1 and 1 ∈ 1 → b, but 1 6= b. (iii) let a = {1, a, b, c}. define hyperoperations � and → on a as follows: � 1 a b c 1 {1} {a} {b} {c} a {a} {a} {a, b} {a, b} b {b} {a, b} {b} {b} c {c} {a, b} {b} {c} → 1 a b c 1 {1} {a} {b} {c} a {1} {a, 1} {1, b, c} {c} b {1} {a} {1, b, c} {1, b, c} c {1} {a} {b} {1, b, c} then (a,�,→, 1) is a quasi hyper hoop, but doesn’t satisfy the condition (hha7). because 1 ∈ a → b and 1 ∈ b → c but 1 /∈ a → c. 70 on hyper hoop-algebras in the following, we give some examples of (quasi) hyper hoop algebras. example 3.3. (i) in any (quasi) hyper hoop (a,�,→, 1), if x � y and x → y are singletons, for any x, y ∈ a, then (a,�,→, 1) is a hoop. then (quasi) hyper hoops are generalizations of hoops. (ii) let a = {1}. if we consider 1 → 1 = {1}, 1 � 1 = {1}, then it is clear that a = (a,�,→, 1) is a (quasi) hyper hoop. (iii) let a = {1, a}. define the hyperoperations � and → on a as follows: � 1 a 1 {1} {1, a} a {1, a} {a} → 1 a 1 {1, a} {a} a {1} {1, a} then (a,�,→, 1) is a bounded (quasi) hyper hoop. (iv) let a = {1, a, b}. define the hyperoperations � and → on a as follows, � 1 a b 1 {1} {a} {b} a {a} {a, b} {a, b} b {b} {a, b} {b} → 1 a b 1 {1} {a} {b} a {1} {1, a, b} {1, b} b {1} {a} {1, b} then (a,�,→, 1) is a bounded (quasi) hyper hoop. (v) let a = {1, a, b, c}. define the hyperoperations � and → on a as follows: � 1 a b c 1 {1} {a} {b} {c} a {a} {a} {a, b, c} {a, c} b {b} {a, b, c} {b, c} {b, c} c {c} {a, c} {b, c} {c} → 1 a b c 1 {1} {a} {b} {c} a {1} {1, a} {1, b, c} {1, c} b {1} {a} {1, b, c} {b, c} c {1} {a} {b} {1, b, c} 71 rajabali borzooei, hamidreza varasteh and keivan borna then (a,�,→, 1) is a bounded (quasi) hyper hoop. (vi) let a = {1, a, b, c}. define the hyperoperations � and → on a as follows: � 1 a b c 1 {1} {a} {b} {c} a {a} {a} {a, b, c} {a, c} b {b} {a, b, c} {b, c} {b, c} c {c} {a, c} {b, c} {c} → 1 a b c 1 {1} {a} {b} {c} a {1} {1, a} {b} {1, c} b {1} {a} {1, b, c} {b, c} c {1} {a} {b} {1, b, c} then (a,�,→, 1) is an unbounded (quasi) hyper hoop. hence, (quasi) hyper hoops may not be bounded, in general. (vii) let a = [0, 1]. define the hyperoperations � and → on a as follows: x�y = {1, x, y} x → y = { {1, y} , if x ≤ y, {y} , otherwise. then (a,�,→, 1) is an infinite (quasi) hyper hoop. proposition 3.4. let a be a quasi hyper hoop. then the following hold, for all x, y, z ∈ a and b, c, d ⊆ a: (hha8) b � c ⇔ 1 ∈ b → c, (hha9) (b �c) → d = b → (c → d), (hha10) x�y �{z}⇔{x}≤ y → z, (hha11) b �c � d ⇔ b � c → d, (hha12) x → (y → z) = y → (x → z), (hha13) {x}≤ y → z ⇔{y}≤ x → z, (hha14) {x}≤ (x → y) → y, (hha15) x� (x → y) �{y}. proof. let x, y, z ∈ a and b, c, d ⊆ a. then, (hha8): b � c ⇔ there exist b ∈ b and c ∈ c such that b ≤ c i.e. 1 ∈ b → c ⇔ 1 ∈ b → c. 72 on hyper hoop-algebras (hha9): by (hha4), the proof is clear. (hha10): x � y � {z} ⇔ by (hha8), 1 ∈ (x � y) → z ⇔ by (hha4), 1 ∈ x → (y → z) ⇔ by (hha8), {x}≤ y → z. (hha11): the proof is similar to the proof of (hha10). (hha12): by (hha4) and (hha1), x → (y → z) = (x�y) → z = (y �x) → z = y → (x → z). (hha13): {x}≤ y → z ⇔ by (hha10), x�y �{z}⇔ by (hha1), y �x � {z}⇔ by (hha10), {y}≤ x → z. (hha14): since x → y � x → y, by (hha1)and (hha11), x�(x → y) �{y} and so by (hha11), {x}≤ (x → y) → y. (hha15): by (hha10) and (hha14), the proof is clear. proposition 3.5. let a be a hyper hoop. then the following hold, for all x, y, z, t ∈ a and b, c, d ⊆ a, (hha16) x�y �{x},{y}, (hha17) {y}≤ x → y, (hha18) if 1 ∈ 1 → x, then x = 1, (hha19) x ∈ 1 → x, and x is the maximum element of 1 → x, (hha20) 1�1 = {1}, (hha21) if a is bounded, then 0 ∈ x�0, (hha22) if b � c ≤ d, then b � d, and {x}≤ b ≤{y} implies x ≤ y, (hha23) if b ≤ c ≤ d, then b ≤ d, and {x}≤{y}≤ b implies {x}≤ b, (hha24) if b �{x}� c, then b � c, and b �{x}≤ c implies b � c, (hha25) if x ≤ y, then z → x ≤ z → y, (hha26) if x ≤ y, then y → z ≤ x → z, (hha27) z → y ≤ (y → x) → (z → x), (hha28) z → y � (x → z) → (x → y), (hha29) if x ≤ y, then x�z � y �z, (hha30) if x ≤ y and z ≤ t, then x�z � y � t, (hha31) (x → y)�z � x → (y �z). proof. (hha16): by (hha2)and(hha5), {y} ≤ x → x and so by (hha10), x � y � {x}. moreover by (hha5), {x} ≤ y → y and so by (hha10), x�y �{y}. (hha17): by (hha16) and (hha10), the proof is clear. (hha18): let 1 ∈ 1 → x. since by (hha5), 1 ∈ x → 1, by (hha6), 1 = x. (hha19): for all u ∈ 1 → x by (hha2), 1 ∈ u → (1 → x). then by (hha12), 1 ∈ 1 → (u → x) and so there exists v ∈ u → x such that 1 ∈ 1 → v. then by (hha18), v = 1. hence 1 ∈ u → x and so u ≤ x. on the other hand, by (hha17), {x} � 1 → x. then there exists a t ∈ 1 → x such that x ≤ t. since for all u ∈ 1 → x we have u ≤ x, by considering u = t, we have t ≤ x ≤ t and 73 rajabali borzooei, hamidreza varasteh and keivan borna so by (hha6), x = t. hence x ∈ 1 → x and so x is the maximum element of 1 → x. (hha20): by (hha1), 1 is the unit and so 1 ∈ 1 � 1. let 1 6= a ∈ 1 � 1. then 1 � 1 � a and so by (hha10), 1 ≤ 1 → a. hence 1 ∈ 1 → a and by (hha18), a = 1. then 1�1 = {1}. (hha21): let a be bounded. since by (hha2), 1 ∈ 0 → 0, we get {x}≤ 0 → 0, for all x ∈ a. then by (hha10), x � 0 � {0}. hence since a is bounded, we get 0 ∈ x�0. (hha22): straightforward, by (hha7). (hha23): straightforward, by (hha7). (hha24): straightforward, by (hha7). (hha25): let x ≤ y. for all u ∈ z → x we have {u} ≤ (z → x) and so by (hha10), u � z � {x}. since x ≤ y, by (hha24), u � z � {y} and so by (hha10), {u}≤ z → y. hence z → x ≤ z → y. (hha26): let x ≤ y. for all u ∈ y → z we have {u} � (y → z) and so by (hha13), {y}� u → z. since x ≤ y, by (hha23), {x}� (u → z). hence by (hha13), {u}� (x → z) and so y → z ≤ x → z. (hha27): for all u ∈ z → y we have {u} � z → y and so by (hha10) and (hha14), u � z � {y} � (y → x) → x. hence by (hha24) and (hha10), {u} � z → ((y → x) → x) and so by (hha12), {u} � (y → x) → (z → x). therefore, z → y ≤ (y → x) → (z → x). (hha28): by (hha27), (x → z) � (z → y) → (x → y). hence by (hha13), (z → y) � (x → z) → (x → y). (hha29): let x ≤ y. since y � z � y � z, by (hha10), {y} ≤ z → (y � z). hence by (hha23), {x}� z → (y�z) and so by (hha10), (x�z) � (y�z). (hha30): let x ≤ y and z ≤ t. since z ≤ t, by (hha29), y � z � y � t. then by (hha10), {y} ≤ z → (y � t). hence by (hha23), {x} ≤ z → (y � t) and so by (hha10), x�z � y � t. (hha31): since x → y � x → y, by (hha10), (x → y) � x �{y}. hence by (hha29), (x → y) � x � z � y � z. therefore, by (hha10), (x → y) � z � x → (y �z). notation: let a be a bounded (quasi) hyper hoop. then for any x ∈ a, we consider x′ = x → 0. proposition 3.6. let a be a bounded quasi hyper hoop. then 1 ∈ 0′ and for any x ∈ a,{x}≤ x′′. proof. by (hha2), 1 ∈ 0 → 0. then 1 ∈ 0′. since by (hha12), (x → 0) → (x → 0) = x → ((x → 0) → 0) = x → x′′ and by (hha2), 1 ∈ (x → 0) → (x → 0). then 1 ∈ x → x′′ and so, {x}≤ x′′. 74 on hyper hoop-algebras proposition 3.7. let a be a bounded hyper hoop. then the following hold, for any x, y ∈ a, (i) x ≤ y, implies that y′ ≤ x′, (ii) x′ ≤ x → y, (iii) x → y ≤ y′ → x′. proof. (i) if x ≤ y, then by (hha26), y → 0 ≤ x → 0. hence y′ ≤ x′ . (ii) since 0 ≤ y, by (hha25), x → 0 ≤ x → y. hence x′ ≤ x → y. (iii) by proposition 3.6 , y ≤ y′′. then by (hha25) and (hha12), x → y ≤ x → y′′ = x → ((y → 0) → 0) = (y → 0) → (x → 0) = y′ → x′. theorem 3.8. any (quasi) hyper hoop of order n, can be extend to a (quasi) hyper hoop of order n + 1, for any n ∈ n. proof. let a be a (quasi) hyper hoop of order n ∈ n, e be an element such that e /∈ a and a1 = a ∪{e}. then we define two hyperoperations �′ and →′ on a1 by: a�′ b=   a� b if a, b ∈ a, {a} if a ∈ a, b = e, {b} if b ∈ a, a = e a →′ b =   a → b∪{e} if a, b ∈ a, 1 ∈ a → b, a → b if a, b ∈ a, 1 /∈ a → b, {e} if b = e, {b} if a = e by some modification we can prove that (a1,�′, e) is a commutative semihypergroup with e as the unit and satisfies the conditions (hha2), (hha3), (hha4), (hha5), (hha6), and (hha7). therefore, (a1,�′,→′, e) is a (quasi) hyper hoop and e is the unit element of it. corollary 3.9. there exist at least one (quasi) hyper hoop of order n, for any n ∈ n proof. by theorem 3.8 and example 3.3 (ii), the proof is clear. note: from now on, we let a be a hyper hoop, unless otherwise is stated. 75 rajabali borzooei, hamidreza varasteh and keivan borna 4 some filters on hyper hoop-algebras in this section we define the concepts of some filters on hyper hoops and we get some properties. definition 4.1. let f be a non-empty subset of a. then f is called an upset of a, if x ∈ f and x ≤ y imply y ∈ f , for all x, y ∈ a, definition 4.2. let f be a non-empty subset of a. then: (i) f is called a weak filter of a, if f is an upset and for all x, y ∈ f , x�y∩f 6= ∅. (ii) f is called a filter of a, if f is an upset and for all x, y ∈ f , x�y ⊆ f . note: let f be a (weak) filter of a and x ∈ f . since f is an upset and x ≤ 1, we get 1 ∈ f . example 4.3. (i) in example 3.3(iv), f = {b, 1} is a filter. (ii) in example 3.3(v), f = {b, 1} is a weak filter. example 4.4. it is clear that a is a (weak) filter of a. by (hha20), {1} = 1�1 and so 1�1 ⊆{1}. then {1} is a (weak)filter of a. proposition 4.5. any filter of a is a weak filter. proof. let f be a filter of a. then f is an upset and x�y ⊆ f , for all x, y ∈ f . hence (x�y)∩f 6= ∅, for all x, y ∈ f . then f is a weak filter. note: any weak filter is not a filter, in general. it can be verified by the following example. example 4.6. in example 3.3(vi), f = {b, 1} is a weak filter, but it is not a filter. theorem 4.7. let f be a non-empty subset of a. then f is a weak filter of a if and only if f is an upset and f � x�y , for all x, y ∈ f . proof. (⇒) straightforward. (⇐) let f be an upset and f � x�y, for all x, y ∈ f . hence there exist u ∈ f and v ∈ x�y such that u ≤ v. since f is an upset and u ∈ f , then v ∈ f and so x�y ∩f 6= ∅. hence f is a weak filter of a. theorem 4.8. let f be a filter of a. then for all x, y, z ∈ a, (i) if x → y ⊆ f and x ∈ f , then y ∈ f , (ii) if x → y ⊆ f and x�z ⊆ f , then y �z ⊆ f , (iii) if x, y ∈ f and x � y → z, then z ∈ f . 76 on hyper hoop-algebras proof. (i) let x ∈ f and x → y ⊆ f , for x, y ∈ a. then x � (x → y) =⋃ u∈x→y x � u ⊆ f . on the other hand, since x → y � x → y, by (hha11), (x → y) � x � y. therefore, there is v ∈ (x → y) � x such that v ≤ y. since v ∈ f , we get y ∈ f . (ii) by (hha16), x � z � x, z. then there exists u ∈ x � z ⊆ f such that u ≤ x, z. since u ∈ f and f is a filter, we get x, z ∈ f . now, since x ∈ f and x → y ⊆ f , by (i) y ∈ f . finally, since y, z ∈ f and f is a filter, y �z ⊆ f . (iii) let x, y ∈ f . since f is a filter, x � y ⊆ f and since x � y → z, by (hha10), x � y � z. then there exists u ∈ x � y ⊆ f such that u ≤ z. since f is a filter and u ∈ f , we get z ∈ f . theorem 4.9. let f be a non-empty subset of a. then f is a filter of a if and only if 1 ∈ f and f � x → y and x ∈ f implies y ∈ f , for any x, y ∈ a. proof. (⇒) let f be a filter, f � x → y and x ∈ f , for x, y ∈ a. hence there exist u ∈ f and v ∈ x → y such that u ≤ v. since u ∈ f and f is an upset, we get v ∈ f and since f is a filter, we get x � v ⊆ f . by v ∈ x → y we have {v} ≤ x → y. then by (hha10), v � x � y and so there exists t ∈ v � x ⊆ f such that t ≤ y. since f is an upset, we get y ∈ f . (⇐) let x ≤ y and x ∈ f , for x, y ∈ a. then 1 ∈ x → y and since 1 ∈ f , we get f � x → y. then, by hypothesis y ∈ f and so f is an upset. now, let x, y ∈ f and u ∈ x � y. then x � y � u and so by (hha10), {y} ≤ x → u. since y ∈ f , we get f � x → u and so by hypothesis, u ∈ f . hence x�y ⊆ f and so f is a filter of a. definition 4.10. let s be a non-empty subset of a. if s is a hyper hoop with respect to the hyperoperations � and → on a, we say that s is a hyper hoopsubalgebra of a. theorem 4.11. let s be a non-empty subset of a. then s is a hyper hoopsubalgebra of a iff x�y ⊆ s and x → y ⊆ s, for all x, y ∈ s. proof. (⇒) the proof is clear. (⇐) let x ∈ s. by (hha2), 1 ∈ x → x and by assumption, x → x ⊆ s. hence 1 ∈ s. it is easy to show that (s,�,→, 1) is a hyper hoop. then s is a hyper hoop-subalgebra of a. example 4.12. (i) in example 3.3(iv), f = {b, 1} is a hyper hoop-subalgebra. (ii) in example 3.3(iii), f = {1} is a (weak)filter, but it is not a hyper hoopsubalgebra. (iii) in example 3.3(vi), f = {a, 1} is a hyper hoop-subalgebra, but it is not a 77 rajabali borzooei, hamidreza varasteh and keivan borna (weak)filter. since a ≤ c and a ∈ f , but c /∈ f and so f is not an upset. theorem 4.13. if {fi} is a finite family of filters of a, then ∩{fi} is a filter of a. proof. the proof is easy. definition 4.14. let d be a subset of a. the intersection of all (weak) filters of a containing d is called the (weak) filter generated by d. the filter generated by d denoted by [d) and the weak filter generated by d denoted by [d)w. it is trivial to verify that [d) is the least filter containing d and [d)w is the least weak filter containing d. theorem 4.15. if ∅ 6= d ⊆ a, then [d)w ⊆{x ∈ a|∃ a1, ..., an ∈ d, s.t. a1 � ....�an �{x}} proof. let f = {x ∈ a|∃ a1, ..., an ∈ d, s.t. a1 �a2 � ......�an �{x}} it is sufficient to show that f is a weak filter containing d. let x ≤ y and x ∈ f , for x, y ∈ a . then there exist a1, ..., an ∈ d, such that, a1 � ...... � an � {x}. since x ≤ y, by (hha23), a1 � ...... � an � {y} and so y ∈ f . hence f is an upset. now, let x, y ∈ f . then there exist a1, ..., an, b1, ..., bm ∈ d, such that, a1�......�an �{x} and b1�......�bm �{y}. hence there exist u ∈ a1�.....�an and v ∈ b1 � .....�bm, such that u ≤ x and v ≤ y. by (hha30) u�v � x�y. then a1� ......�an�b1� ......�bm � x�y. hence there exists s ∈ x�y such that a1 � ...... � an � b1 � ...... � bm �{s} and so x � y ∩ f 6= ∅. thus f is a weak filter of a. for all d ∈ d we have {d} � {d}, and so d ∈ f . therefore f is a weak filter of a containing d. note: in the following example we will show that the equation, [d)w = f is not true, in general, where f = {x ∈ a|∃ a1, ..., an ∈ d, s.t. a1 � ....�an �{x}} example 4.16. in example 3.3(v), if we take d = {b} then it follows that f = {1, b, c}, that is a weak filter containing d, but [d)w = {1, b}. hence in this example [d)w 6= f . theorem 4.17. if ∅ 6= d ⊆ a, then [d) = {x ∈ a|∃ a1, ..., an ∈ d, s.t. a1 � ....�an �{x}} 78 on hyper hoop-algebras proof. let f = {x ∈ a|∃ a1, ..., an ∈ d, s.t. a1 �a2 � ......�an �{x}} let x ≤ y and x ∈ f , for x, y ∈ a. then there exist a1, ..., an ∈ d, such that, a1 � ......�an �{x} since x ≤ y, by (hha24), a1 � ...... � an � {y} and so y ∈ f . hence f is an upset. now, let x, y ∈ f . then there exist a1, ..., an, b1, ..., bm ∈ d, such that, a1 � ......�an � x and b1 � ......�bm �{y}. for all u ∈ x�y, x�y �{u}. then by(hha10), {x}≤ y → u. since a1 � ......�an �{x} and {x}≤ y → u by (hha24), a1 � ...... � an � y → u. since b1 � ...... � bm � y by (hha26), y → u ≤ (b1 � ......� bm) → u. hence a1 � ......�an � y → u ≤ (b1 � ......� bm) → u and so by (hha22), a1 � ......�an � (b1 � ......�bm) → u. then by (hha11), (a1 � ......�an)� (b1 � ......� bm) �{u} and so u ∈ f . therefore x�y ⊆ f and so f is a filter. since d � d, for all d ∈ d, we have d ∈ f and so f is a filter of a containing d. let d ⊆ c and c be a filter of a. for all x ∈ f , there exist a1, ..., an ∈ d, such that a1 � ....�an �{x} then there exists v ∈ a1 � .... � an, such that v ≤ x. by a1, ..., an ∈ d ⊆ c and c is a filter, it follows that a1 � .....�an ⊆ c and so v ∈ c. since c is an upset we have x ∈ c and so f ⊆ c. therefore [d) = f . definition 4.18. let a be bounded. then d ⊆ a is said to have the finite intersection property if a1 �a2......�an ∩{0} = ∅, for all a1, ...., an ∈ d. theorem 4.19. let a be bounded and d ⊆ a. then [d) is a proper filter of a if and only if d has the finite intersection property. proof. let [d) be a proper filter of a and d has not the finite intersection property, by the contrary. then there exist a1, ...., an ∈ d such that 0 ∈ a1 �a2......� an. hence a1 � a2...... � an � {0} and so by theorem 4.17, 0 ∈ [d). since 0 ≤ x, for all x ∈ a and [d) is a filter, we have x ∈ [d) and so [d) = a, which is a contradiction. hence d has the finite intersection property. conversely, let d has the finite intersection property and [d) is not a proper filter, by the contrary. then [d) = a and so 0 ∈ [d). then by theorem 4.17, there exist a1, ...., an ∈ d such that a1�a2......�an �{0} and so 0 ∈ a1�a2......�an. then d has not the finite intersection property, which is a contradiction. hence [d) is a proper filter. 79 rajabali borzooei, hamidreza varasteh and keivan borna theorem 4.20. if f is a filter of a and a ∈ a, then [f ∪{a}) = {x|x ∈ a,∃n ∈ n, s.t., an → x∩f 6= ∅} proof. suppose that x ∈ [f ∪{a}). by theorem 4.17, there exist b1, ...., bm ∈ f and n ∈ n such that b1 � .....� bm �an �{x} by (hha11), we have b1 � ..... � bm � an → x. then there exists u ∈ b1 � .....� bm and v ∈ an → x such that u ≤ v. since f is a filter and b1, ...., bm ∈ f , we get b1 � ..... � bm ⊆ f and so u ∈ f . now, since f is a filter, we get v ∈ f . hence an → x∩f 6= ∅. conversely, let there exists n ∈ n such that an → x∩f 6= ∅. if s ∈ an → x∩f , then 1 ∈ s → (an → x). hence by (hha4), 1 ∈ (s � an) → x. therefore, s�an �{x} and so by theorem 4.17, x ∈ [f ∪{a}). 5 conclusion in this paper, we applied the hyper structure theory to the hoop algebras and introduced the notion of (quasi) hyper hoop algebra which is a generalization of hoop-algebra. then we studied some properties and filter theory of this structure. topological and categorical properties, quotient structures and relation with the other hyperstructures can be studied for the future researches. references [1] p. aglianò, i. m. a. ferreirim, f. montagna, basic hoops: an algebraic study of continuous t-norms, studia logica, 87(1),2007, 73 98. [2] w. j. blok, i. m. a. ferreirim, hoops and their implicational reducts (abstract), algebraic methods in logic and computer science, banach center publications, 28, 1993, 219 230. [3] w.j. blok, i.m.a. ferreirim, on the structure of hoops, algebra universalis, 43, 2000, 233 257. [4] r. a. borzooei, m. bakhshi, o. zahiri, filter theory on hyper residuated lattices, quasigroups and related systems, 22 (2014),33-50. 80 on hyper hoop-algebras [5] r. a. borzooei, w. a. dudek, o. zahiri, a. radfar, some remarks on hyper mv-algebras, journal of intelligent and fuzzy systems, to appear doi:10.3233/ifs-141258. [6] r. a. borzooei, a. hasankhani, m. m. zahedi,y.b. jun, on hyper kalgebras, mathematica japonica, 52 (2000), 113-121. [7] b. bosbach, komplementare halbgruppen. axiomatik und arithmetik, fundamenta mathematicae, vol. 64 (1969), 257-287. [8] j.r. büchi, t.m. owens, complemented monoids and hoops, unpublished manuscript. [9] p.corsini, v.leoreanu, applications of hyperstructure theory, advances in mathematics, kluwer academic publishers, dordrecht, 2003. [10] g. georgescu, l. leustean, v. preoteasa, pseudo-hoops, journal of multiple-valued logic and soft computing, 11 (2005), 153-184. [11] sh. ghorbani, a. hasankhani, e. eslami, hyper mv-algebras, set-valued math, appl, 1, 2008, 205-222. [12] p. hájek, metamathematics of fuzzy logic, trends in logic-studia logica library, dordrecht/boston/london,(1998). [13] f. marty, sur une generalization de la notion de groupe, 8th congress mathematiciens scandinaves, stockholm, (1934), 45-49. 81 introduction preliminaries hyper hoop-algebras some filters on hyper hoop-algebras conclusion microsoft word capitolo intero n 3.doc ratio mathematica volume 44, 2022 markov chain model and its application yearly rainfall data in nagapattinam district dr. s. santha1 t. subasini2 abstract a stochastic model expresses a sequence of possible events in which the possible event of each event depends on the previous event and is called a markov chain. this paper has analyzed yearly rainfall in the nagapattinam district and formulated three-state models. the first-order markov chain to determine the long-term probability of rainfall in the following years and the steady-state. it can be used to make a forecast of the annual rainfall pattern. this model can give some information about rainfall to farmers and the government to plan strategies for high crop production in the nagapattinam district. keywords: markov chain, yearly rainfall, transition probability ams classification 2010: 37a30, 47d073. 1 assistant professor and head, department of mathematics, government arts and science college, nagercoil, affiliated by manonmaniam sundaranar university, tirunelveli tamil nadu, india. santhawilliam14@gmail.com. 2 reg no.18121172092018, research scholar, rani anna government arts and science college for women, tirunelveli, affiliated by manonmaniam sundaranar university, tirunelveli tamil nadu, india. subasinit@gmail.com. 3 received on june 18th 20th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.887. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 36 mailto:santhawilliam14@gmail.com mailto:subasinit@gmail.com dr. s. santha and t. subasini 1. introduction forecasting is a science of the future. here based on the knowledge of the previous, predictions are made on the future. this involves a knowledge of all forecasting methods. the importance of forecasting is becoming necessary for prosperity. preparation of a forecast needs mathematical formula and historical data into the future. the most effective forecasting methods is to use mathematical techniques to routinely forecast demands. by combining mathematical techniques with informed judgement, they can serve checks on each other and tend to eliminate gross errors. agriculture largely depends on water resources. the variation and quantity of rainfall have two extreme impacts – bumper harvest or lean year. rainfall modelling plays a prominent role for rainfall prediction. apart from data generation, the application of rainfall modelling is vital for water resource management and in the field of hydrology and agriculture. data relating to climate and rainfall needs a wide range of models in which time and spatial scales are involved. rainfall, its arrival, intensity, duration, was generally decided by the atmospheric factors which are available just before the onset. hence, for forecasting the future rainfall it is difficult to estimate the probable environmental factors which are the causes of the rainfall. in view of this the researchers are left with the option of predicting it with the previous data set. abubaker et al [10] have formulated a, fourstate markov model of annual rainfall in minna with respect to crop production in the region. the present study emphasizes analyzing the annual rainfall data in a three-state model. in this paper the forecasting of annual rainfall pattern for the period of 20 years (2001 to 2020) in nagapattinam district. 2. methodology nagapattinam district is one of the 38 districts (a coastal district) of tamilnadu state in southern india. the district lies between northern latitude 10.7906 degrees and 79.8428 degrees eastern longitude. since the study was forecasting of rainfall, the annual rainfall data was collected from the statistical department in nagapattinam from the year 2001 to 2020. figure: 1 annual rainfall in nagapattinam district from 2001-2020. 0 500 1000 1500 2000 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 2 0 1 6 2 0 1 7 2 0 1 8 2 0 1 9 2 0 2 0 37 markov chain model and its application yearly rainfall data in nagapattinam district from figure 1 time-series analysis of yearly rainfall is converted into rainfall states prepared by a suitable frequency distribution table. the frequency distribution of each class is specified states of rainfall and is denoted by s1, s2, and s3 in table 1. table: 1 3. markov chain modeling definition: 3.1 a markov chain is a random sequence (xn, n є n) for all xo, x1 ……. xn-1 є i such that p (xn = jn / x0 = j0, x1 = j1, x2 = j2………xn-1 = jn-1) = p (xn = jn / xn-1 = jn-1) (1) definition: 3.2 if a markov chain (jn, n ≥ 0) is homogeneous. we consider p (xn =j / xn-1 = i) = pij and we put the matrix p i.e., p = [pij] the markov process x has steady state transition probabilities if for any pair of states i, j: the first step transition probabilities pij, pij n denote the n step transition probability. that is, pij n = p {xn+m = j / xm = i} n ≥ 0 all i, j ≥ 0 we have pij 1 = pij. pij n+m = ∑ 𝑃∞𝑘= 0 ik n pkj m for all n, m ≥ 0 (2) from (2) p n+m = p(n)×p(m) p(2 )= p(1+1) = p2 hence by induction p(n) = p(n-1+1) = p(n-1) p1 = pn we have pn = p0 pn (3) here p0 denote the initial state vector of transition matrix pn denote limiting state probability. now, let pn = [pn1, p2 n, pn3]. also let p0 = [ p01 p 0 2 p 0 3]. limiting state probabilities: 3.3 the probability distribution π = [π1 π2………. πn] is called the limiting distribution of the continuous time markov chain x (t) if π = (π1π2 π3) howard [8] let n→ ∞ in equation (3) we get π = πp (4) also, π = ∑ 𝜋𝑖 = 13𝑖=1 abubakar et al [10] these equations will be used to find the limiting state equilibrium probabilities. sl. no class interval frequency distribution states 1. below 1100 8 s1 2 1100-1600 9 s2 3 above 1600 3 s3 38 dr. s. santha and t. subasini 4. result and discussion let the model for yearly rainfall is s1: less than 1100 s2: in between 1100-1600 s3: greater value of 1600. therefore, the transition probability matrix p = [ 𝑃11 𝑃12 𝑃13 𝑃21 𝑃22 0 𝑃31 0 0 ] the probability of transition matrix p =| 4 3 1 4 5 0 3 0 0 | (5) i, e. pij = 𝑓𝑖𝑗 ∑ 𝑓𝑖𝑗 i, j = 1,2,3. tamil and samuel (9) where fij → transition frequency from state i to state j, 0 ≤ pij ≤ 1. we get the probability matrix p = | 4/8 3/8 1/8 4/9 5/9 0 3/3 0 0 |, p = | 0.5 0.375 0.125 0.44 0.56 0 1 0 0 | n-step transition probability we have p2 = [ 0.5415 0.396 0.0625 0.4689 0.4756 0.0555 0.5 0.375 0.0625 ] p3 = [ 0.5091 0.4232 0.0442 0.5011 0.4402 0.0586 0.5415 0.396 0.0625 ] p4 = [ 0.5101 0.4262 0.0519 0.5046 0.4326 0.0521 0.5091 0.4232 0.0442 ] 39 markov chain model and its application yearly rainfall data in nagapattinam district p5 = [ 0.5079 0.4282 0.0509 0.5071 0.4297 0.0519 0.5101 0.4262 0.0519 ] p6 = [ 0.5079 0.4284 0.0515 0.5074 0.4290 0.0514 0.5079 0.4282 0.0509 ] p7 = [ 0.5078 0.4286 0.0515 0.5076 0.4287 0.0515 0.5079 0.4284 0.0515 ] p8 = [ 0.5078 0.4286 0.0515 0.5077 0.4286 0.0515 0.5078 0.4286 0.0515 ] p10 = [ 0.5078 0.4286 0.0515 0.5078 0.4286 0.0515 0.5078 0.4286 0.0515 ] (6) p10 = [ 0.51 0.43 0.05 0.51 0.43 0.05 0.51 0.43 0.05 ] corrected to 2 decimal places limiting state probabilities after n steps pn gets the fixed value (6) i, e. n ≥10 let us take p0 = (1 0 0 ) p0. p n = (100)[ 0.507 0.4286 0.0515 0.5078 0.4286 0.0515 0.5078 0.4286 0.0515 ]= = (0.5078 0.4286 0.0515) = (0.51 0.43 0.05) from equation (4) and (6), n = np = (0.51 0.43 0.05). this result shows that the probabilities of yearly rainfall after ten years. in the firstyear probability is (0.5 0.38 0.13) respectively. by the comparison of the probabilities, state 3 dropped slowly and the probability of state 1 and 2 increased in the above ten years. this shows that in the 51% annual rainfall in nagapattinam will be state 1, 43%will be state 2 and 5% will be state 3. 5. conclusion this article show analyzes the yearly rainfall data used by the first-order markov chain model. yearly rainfall forecasting of the current year can be used to make a 40 dr. s. santha and t. subasini forecast for the following year and in the long run. a yearly rainfall forecasting pattern used to give details about the production of the crops in the nagapattinam district. references [1] akintunde a.a. asiribo o.e adelakun a.a agwuegbo s.o.n (2008). stochastic modelling of daily precipitation in abeokuta, proceedings of the third conference on science and national development. 108-118 www.unaab.edu.ng/journal/index.php/colna/article........./150/153 accessed 15/10/2012. [2] lawal adamu, hakimi danladi, laminuidris ''stochastic modelling of annual rainfall at new-bussu'' iosr journal of mathematics, volume 10, issue3 ver ii. [3] ratna singham srikanthan and tom mcmahon (2000) stochastic generation of climate data, a review: technical report 00/16 cooperative research centre for catchment, hydrology, australia. [4] gabriel k. r and neumann j. (1962) a markov chain model for daily rainfall occurrences at tel aviv. quart j. roy. met soc 88: 90 – 95. [5] kottegoda n. t. natale. l and raiteri. e (2004) some consideration of periodicity and persistence in daily rainfall j. hydrol.296:23-37. [6] howard r.a (1971), dynamic probabilistic systems, vols. 1 and 2, john wiley, new york. [7] abubakar usman yusuf, lawal addamu, muhammed abdullahi (2014). markov chain model and it's application to annual rainfall distribution for crop production, american journal of theoretical and applied statistics vol3, no.2 pp 39 – 43. [8] abubakar usman yusuf, laval a. muhammed "the use of markov model in continuous time for the prediction of rainfall for crop production, international organization of scientific research (iosr) vol7, issue 1, pp 38 – 45. [9] tamil s. and samuel. s. (2011) stochastic modelling of annual rainfall at tamil nadu. 41 http://www.unaab.edu.ng/journal/index.php/colna/article........./150/153accessed%2015/10/2012 http://www.unaab.edu.ng/journal/index.php/colna/article........./150/153accessed%2015/10/2012 ratio mathematica volume 45, 2023 inverse domination parameters of jump graph s. santha1 g.t. krishna veni2 abstract let 𝐺 = (𝑉, 𝐸) be a connected graph. let 𝐷 be a minimum dominating set in 𝐺. if 𝑉 − 𝐷 contains a dominating set 𝐷′ of 𝐺, then 𝐷′ is called an inverse dominating set with respect to 𝐷. theminimum cardinality of an inverse dominating set of 𝐺 is called inverse domination number of 𝐺. in this article, we determine inverse domination parameters of jump graph of a graph. keywords: domination number, inverse domination number, non-split inverse domination number, connected inverse domination number, jump graph. 2010 ams subject classification: 05c693. 1assistant professor, department of mathematics, government arts and science, konam, nagercoil629004, kanyakumari dt, tamil nadu, india. email: santhawilliam14@gmail.com. 2 register number: 18221172092026, research scholar, pg & research dept. of mathematics, rani anna govt. college for women, tirunelveli-8, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india. email: krishnavenisiva55@gmail.com. 3received on july 10, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm. v45i0.1001. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 135 s. santha and g.t. krishna veni 1.introduction the undirected graph𝐺 = (𝑉, 𝐸) discussed in this paper is simple and connected. the order and size are denoted by 𝑛 and 𝑚 respectively. for basic graph theoretical reference, we refer [2]. if 𝑢𝑣 is an edge of 𝐺, then two vertices 𝑢 and 𝑣 are said to be adjacent. if 𝑢𝑣 ∈ 𝐸(𝐺), then 𝑢 is 𝑣′𝑠 neighbour, and the set of 𝑣′𝑠 neighbours is denoted by 𝑁(𝑣). vertex 𝑣 ∈ 𝑉 has a degree of 𝑑𝑒𝑔(𝑣) = |𝑁(𝑣)|. if 𝑑𝑒𝑔(𝑣) = 𝑛 − 1, a vertex v is referred to as a universal vertex. globally, the domination conception in graphs start off its root in 1850s along the concern of certain chess players. domination has diverse function consists of the morphological analysis, social network theory, cctv installation and the most generally argued is the computer network communication. the aforementioned network comprises communication links among a firm number of slots. the trouble is to prime a least number of location where transmitters are installed such that a network as a whole united by a direct communication link to the transmitter site. in alternative words, the issue is to locate the least dominant set in the graph that corresponds to this network. v.r. kulli and c. sigarkanthi [10] were the first to propose the idea of an inverse domination number. a set 𝐷 of vertices in a graph 𝐺 = (𝑉, 𝐸) is a dominating set if every vertex in 𝑉 − 𝐷is adjacent to some vertex in 𝐷. the domination number 𝛾(𝐺)of 𝐺is the minimum cardinality of a dominating set of𝐺. if 𝑉 − 𝐷 contains a dominating set 𝐷′ of 𝐺, then 𝐷′ is called an inverse dominating set with respect to 𝐷. the inverse domination number 𝛾 ′(𝐺)of 𝐺is the minimum cardinality of an inverse dominating set of 𝐺. the dominating set 𝑆 of 𝐺 is connected dominating set of 𝐺 if induced sub graph 〈𝑆〉 is connected. the connected domination number 𝛾𝑐 (𝐺) of 𝐺 is referred to as a minimum cardinality of connected dominating set. the dominating set 𝑆 of 𝐺 is a non-split dominating set of 𝐺 if induced sub graph 〈𝑉 − 𝑆〉 is connected. the non-split domination number 𝛾𝑛𝑠(𝐺) of 𝐺 is referred to as a minimum cardinality of non-split dominating set.[7] the n-sunlet graph is a graph on 2𝑛 vertices isobtained by attaching 𝑛-pendant edges to the cycle 𝐶𝑛 and it is denoted by 𝑆𝑛. let 𝑃𝑛 be a path graph in 𝑛 vertices. the comb graph is defined as 𝑃𝑛 ʘ𝐾1.it has 2𝑛 vertices and 2𝑛 − 1 edges. fan graph 𝐹𝑛n ≥ 2 determined by joining all vertices of a path 𝑃𝑛 to a different vertex, called centre. thus 𝐹𝑛has 𝑛 + 1 vertices, such as 𝑢, 𝑢1, 𝑢2, 𝑢3, … . 𝑢𝑛 and (2𝑛 − 1) edges, such as 𝑢𝑢𝑖 , 1 ≤ 𝑖 ≤ 𝑛 – 1. [12] the line graph 𝐿(𝐺) of 𝐺 has the edges of 𝐺 as its vertices which are adjacent in 𝐿(𝐺) if and only if the corresponding edges are adjacent in 𝐺. we call the complement of line graph 𝐿(𝐺)as the jump graph 𝐽(𝐺) of 𝐺, found in [11]. the jump graph 𝐽(𝐺) of a graph 𝐺 is the graph defined on𝐸(𝐺) and in which two vertices are adjacent if and only if they are not adjacent in 𝐺. since both 𝐿(𝐺) and 𝐽(𝐺) are defined on the edge set of a graph 𝐺. 136 inverse domination parameters of jump 2. main results inverse domination number of jump graph of 𝒏–sunlet graph theorem 2.1. for the graph 𝐺 = 𝐽(𝑆𝑛) (𝑛 ≥ 4), 𝛾ꞌ (𝐺)) = 2. proof. for 𝑛 ≥ 4, the number of vertices of 𝑛-sunlet graph is 2𝑛. then it has 2𝑛 edges. let 𝐺 = 𝐽(𝑆𝑛). the number of vertices of jump graph of the 𝑛-sunlet graph is 2𝑛. let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑢1, 𝑢2, 𝑢3, … . 𝑢𝑛}, since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝐷 = {𝑢1, 𝑢3}. then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺)) = 2. let 𝐷′ = {𝑢2, 𝑢4}. then 𝐷 ′ is a inverse dominating set of 𝐺 so that 𝛾 ′(𝐺) = 2. ∎ theorem 2.2. for the graph 𝐺 = 𝐽(𝑆𝑛) (𝑛 ≥ 4), 𝛾𝑛𝑠 ′ (𝐺)) = 2. proof. for 𝑛 ≥ 4, the number of vertices of 𝑛-sunlet graph is 2𝑛. then it has 2𝑛 edges. let 𝐺 = 𝐽(𝑆𝑛). the number of vertices of jump graph of the 𝑛-sunlet graph is 2𝑛. let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑢1, 𝑢2, 𝑢3, … . 𝑢𝑛}, since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝐷 = {𝑢1, 𝑢3}. then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺) = 2. let 𝐷′ = {𝑢2, 𝑢4}. then 𝐷 ′ is a non-split inverse dominating set of 𝐺 so that 𝛾𝑛𝑠 ′ (𝐺) = 2. ∎ theorem 2.3. for the graph 𝐺 = 𝐽(𝑆𝑛) (𝑛 ≥ 4), 𝛾𝑐 ′ (𝐺) = 2. proof. for 𝑛 ≥ 4, the number of vertices of 𝑛-sunlet graph is 2𝑛. then it has 2𝑛 edges. let 𝐺 = 𝐽(𝑆𝑛). the number of vertices of jump graph of the 𝑛-sunlet graph is 2𝑛. let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑢1, 𝑢2, 𝑢3, … . 𝑢𝑛}, since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝐷 = {𝑢1, 𝑢3}. then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺)) = 2. let 𝐷′ = {𝑢2, 𝑢4}. then 𝐷 ′ is a connected inverse dominating set of 𝐺 so that 𝛾𝑐 ′(𝐺) = 2. ∎ inverse domination number of jump graph of comb graph theorem 2.4. for the graph 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾1) (𝑛 ≥ 4), 𝛾 ′(𝐺) = 2. proof. for 𝑛 ≥ 4, the number of vertices of combgraph is 2𝑛. then it has 2𝑛 − 1 edges. let 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾1). the number of vertices of jump graph of the comb graph is 2𝑛 − 1. let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , 𝑣𝑛−1, 𝑢1, 𝑢2, 𝑢3, … . 𝑢𝑛}, since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝐷 = {𝑢1, 𝑣1}. then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺) = 2. let 𝐷′ = {𝑢2, 𝑢4}. then 𝐷 ′ is a inverse dominating set of 𝐺 so that 𝛾 ′(𝐺) = 2. ∎ theorem 2.5. for the graph 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾1) (𝑛 ≥ 4), 𝛾𝑛𝑠 ′ (𝐺) = 2. proof. for 𝑛 ≥ 4, the number of vertices of comb graph is 2𝑛. then it has 2𝑛 − 1 edges. let 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾1). the number of vertices of jump graph of the comb graph is 2𝑛 − 1. let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , 𝑣𝑛−1, 𝑢1, 𝑢2, … , 𝑢𝑛}. since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝐷 = {𝑢1, 𝑣1}. then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺) = 2. let 𝐷′ = {𝑢2, 𝑢4}. then 𝐷 ′ is a non-split inverse dominating 137 s. santha and g.t. krishna veni set of 𝐺 so that 𝛾𝑛𝑠 ′ (𝐺) = 2. ∎ theorem 2.6. for the graph 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾1) (𝑛 ≥ 4), 𝛾𝑐 ′(𝐺) = 2. proof. for 𝑛 ≥ 4, the number of vertices of comb graph is 2𝑛. then it has 2𝑛 − 1 edges. let 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾1). the number of vertices of jump graph of the comb graph is 2𝑛 − 1. let the vertices of the graph is labeled as {𝑣1, 𝑣2, … , 𝑣𝑛−1, 𝑢1, 𝑢2, … , 𝑢𝑛}, since 𝐺 contains no universal vertices, 𝛾(𝐺)) ≥ 2. let 𝐷 = {𝑢1, 𝑣1}. then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺) = 2. let 𝐷′ = {𝑢2, 𝑢4}. then 𝐷 ′ is a connected inverse dominating set of 𝐺 so that 𝛾𝑐 ′(𝐺) = 2. ∎ inverse domination number of jump graph of fan graph theorem 2.7. for fan graph 𝐺 = 𝐽(𝐹𝑛) (𝑛 ≥ 5), 𝛾 ′(𝐺) = { 3 𝑖𝑓 𝑛 = 3,4 2 𝑖𝑓 𝑛 ≥ 5 . proof. the number of vertices of fan graph is 𝑛 + 1. then it has 2𝑛 − 1 edges. let 𝐺 = 𝐽(𝐹𝑛 ). the number of vertices of jump graph of the fan graph is 2𝑛 − 1. let the vertices of the graph is labeled as {𝑢1, 𝑢2, … , 𝑢𝑛, 𝑣1, 𝑣2, … , 𝑣𝑛−1}, since 𝐺 contains no universal vertices, 𝛾(𝐺)) ≥ 2. let 𝑛 = 3. it is easily verified that no two element subsets of𝐽(𝐹𝑛 ) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) ≥ 3.let 𝐷 = {𝑣1, 𝑣3, 𝑣4}. then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺)) = 3. let 𝐷′ = {𝑣2, 𝑢2, 𝑢3}. then 𝐷 ′ is a 𝛾 ′-set of 𝐺. since 𝛾(𝐺) = 3, we have𝛾 ′(𝐺) = 3. let 𝑛 = 4.let 𝐷 = {𝑣1, 𝑣4}. then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺)) = 2. let 𝐷 ′ = {𝑣2, 𝑣3, 𝑢3}. then 𝐷 ′ is a 𝛾 ′-set of 𝐺 and so𝛾 ′(𝐺) ≤ 3. it is easily observed that no two element subsets of 𝐺 is not a 𝛾 ′-set of 𝐺. therefore 𝛾 ′(𝐺)) = 3. let 𝑛 ≥ 5. let 𝐷 = {𝑣1, 𝑣4}. then 𝐷 is a dominatingset of 𝐺 and so 𝛾(𝐺) = 2. let 𝐷′ = {𝑣2, 𝑣5}. then 𝐷 ′ is a inverse dominating set of 𝐺 and so 𝛾 ′(𝐺) = 2. ∎ theorem 2.8. for the graph 𝐺 = 𝐽(𝐹𝑛 ) (𝑛 ≥ 5), 𝛾𝑛𝑠 ′ (𝐺) = { 3 𝑖𝑓 𝑛 = 3 2 𝑖𝑓 𝑛 ≥ 5 . proof. the number of vertices of fan graph is 𝑛 + 1. then it has 2𝑛 − 1 edges. let 𝐺 = 𝐽(𝐹𝑛 ). the number of vertices of jump graph of the fan graph is 2𝑛 − 1. let the vertices of the graph is labeled as {𝑢1, 𝑢2, … , 𝑢𝑛, 𝑣1, 𝑣2, … , 𝑣𝑛−1}, since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝑛 = 3. it is easily verified that no two element subsets of 𝐽(𝐹𝑛 ) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) ≥ 3. let 𝐷 = {𝑣2, 𝑢2, 𝑢3}. then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 3. let 𝐷′ = {𝑣1, 𝑣3, 𝑢4}. then 𝐷 ′ is a non-split inverse dominating set of 𝐺 so that 𝛾𝑛𝑠 ′ (𝐺) = 3. let 𝑛 = 4. let 𝐷 = {𝑣1, 𝑣4}. then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 2. let 𝐷 ′ = {𝑣2, 𝑣3, 𝑢3}. then 𝐷 ′ is a not a non-split inverse dominating set of 𝐺. let 𝑛 ≥ 5. let 𝐷 = {𝑣1, 𝑣4}. then 𝐷 is a dominatingset of 𝐺 and so 𝛾(𝐺) = 2. let 𝐷′ = {𝑣2, 𝑣5}. then 𝐷 ′ is a non-split inverse dominating set of 𝐺 and so 𝛾𝑛𝑠 ′ (𝐺) = 2. ∎ theorem 2.9. for the graph 𝐺 = 𝐽(𝐹𝑛 ) (𝑛 ≥ 5), 𝛾𝑐 ′(𝐺) = { 3 𝑖𝑓 𝑛 = 3 2 𝑖𝑓 𝑛 ≥ 5 . proof. the number of vertices of fan graph is 𝑛 + 1. then it has 2𝑛 − 1 edges. let 𝐺 = 138 inverse domination parameters of jump 𝐽(𝐹𝑛 ). the number of vertices of jump graph of the fan graph is 2𝑛 − 1. let the vertices of the graph is labeled as {𝑢1, 𝑢2, … , 𝑢𝑛, 𝑣1, 𝑣2, … , 𝑣𝑛−1}, since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝑛 = 3. it is easily verified that no two element subsets of 𝐽(𝐹𝑛 ) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) ≥ 3. let 𝐷 = {𝑣2, 𝑢2, 𝑢3}. then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 3. let 𝐷′ = {𝑣1, 𝑣3, 𝑢4}. then 𝐷 ′ is a connected inverse dominating set of 𝐺 so that 𝛾𝑐 ′(𝐺) = 3. let 𝑛 = 4. let 𝐷 = {𝑣1, 𝑣4}. then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 2. let 𝐷 ′ = {𝑣2, 𝑣3, 𝑢3}. then 𝐷 ′ is a not a connected inverse dominating set of 𝐺. let 𝑛 ≥ 5. let 𝐷 = {𝑣1, 𝑣4}. then 𝐷 is a dominatingset of 𝐺 and so 𝛾(𝐺) = 2. let 𝐷′ = {𝑣2, 𝑣5}. then 𝐷 ′ is a connected inverse dominating set of 𝐺so that𝛾𝑐 ′(𝐺) = 2. ∎ inverse domination number of jump graph of 𝑪𝒏 ⊙ 𝑲𝟐 theorem 2.10. for the graph 𝐺 = 𝐽(𝐶𝑛 ⊙ 𝐾2) (𝑛 ≥ 4), 𝛾 ′(𝐺) = { 3 𝑖𝑓 𝑛 = 3 2 𝑖𝑓 𝑛 ≥ 4 . proof. the number of vertices of 𝐶𝑛 ⊙ 𝐾2is 3𝑛. then it has 3𝑛 edges. let 𝐺 = 𝐽(𝐶𝑛 ⊙ 𝐾2). the number of vertices of jump graph of the 𝐶𝑛 ⊙ 𝐾2 is 3𝑛. let the vertices of the graph is labeled as {𝑢1, 𝑢2, … , 𝑢𝑛, 𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑤1, 𝑤2, … , 𝑤𝑛}. since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝑛 = 3. it is easily verified that no two element subsets of 𝐽(𝐶𝑛 ⊙ 𝐾2) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) ≥ 3. let 𝐷 = {𝑣2, 𝑤2, 𝑤3}. then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 3. let 𝐷′ = {𝑢1, 𝑢2, 𝑢3}. then 𝐷 ′ is a 𝛾 ′-set of 𝐺. since 𝛾(𝐺) = 3, we have 𝛾 ′(𝐺) = 3. let 𝑛 ≥ 4. let 𝐷 = {𝑢2, 𝑢4}. then 𝐷 is a dominatingset of 𝐺 and so 𝛾(𝐺) = 2. let 𝐷′ = {𝑤2, 𝑤4}. then 𝐷 ′ is a inverse dominating set of 𝐺 so that𝛾 ′(𝐺) = 2. ∎ theorem 2.11. for the graph 𝐺 = 𝐽(𝐶𝑛 ⊙ 𝐾2) (𝑛 ≥ 4), 𝛾𝑛𝑠 ′ (𝐺) = { 3 𝑖𝑓 𝑛 = 3 2 𝑖𝑓 𝑛 ≥ 4 . proof. the number of vertices of 𝐶𝑛 ⊙ 𝐾2 is 3𝑛. then it has 3𝑛 edges. let 𝐺 = 𝐽(𝐶𝑛 ⊙ 𝐾2). the number of vertices of jump graph of the 𝐶𝑛 ⊙ 𝐾2 is 3𝑛. let the vertices of the graph is labeled as {𝑢1, 𝑢2, … , 𝑢𝑛, 𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑤1, 𝑤2, … , 𝑤𝑛}. since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝑛 = 3. it is easily verified that no two element subsets of 𝐽(𝐶𝑛 ⊙ 𝐾2) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) ≥ 3. let 𝐷 = {𝑣2, 𝑤2, 𝑤3}. then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 3. let 𝐷′ = {𝑢1, 𝑢2, 𝑢3}. then 𝐷 ′ is a non-split inverse dominating set of 𝐺 so that 𝛾𝑛𝑠 ′ (𝐺) = 3. let 𝑛 ≥ 4. let 𝐷 = {𝑢2, 𝑢4}. then𝐷 is a dominating set of 𝐺 so that 𝛾(𝐺) = 2. let𝐷 ′= {𝑤2, 𝑢4}. then 𝐷 ′ is a non-split inverse dominating set of 𝐺 so that 𝛾𝑛𝑠 ′ (𝐺) = 2. theorem 2.12. for the graph 𝐺 = 𝐽(𝐶𝑛 ⊙ 𝐾2) (𝑛 ≥ 4), 𝛾𝑐 ′(𝐺) = { 3 𝑖𝑓 𝑛 = 3 2 𝑖𝑓 𝑛 ≥ 4 . proof. the number of vertices of 𝐶𝑛 ⊙ 𝐾2 is 3𝑛. then it has 3𝑛 edges. let 𝐺 = 139 s. santha and g.t. krishna veni 𝐽(𝐶𝑛 ⊙ 𝐾2). the number of vertices of jump graph of the 𝐶𝑛 ⊙ 𝐾2 is 3𝑛. let the vertices of the graph is labeled as {𝑢1, 𝑢2, … , 𝑢𝑛, 𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑤1, 𝑤2, … , 𝑤𝑛}. since 𝐺 contains no universal vertices, 𝛾𝑐 ′(𝐺) ≥ 2. let 𝑛 = 3. it is easily verified that no two element subsets of 𝐽(𝐶𝑛 ⊙ 𝐾2) is not a 𝛾-set of 𝐺 and so 𝛾(𝐺) ≥ 3. let 𝐷 = {𝑣2, 𝑤2, 𝑤3}. then 𝐷 is a 𝛾-set of 𝐺 so that 𝛾(𝐺) = 3. let 𝐷′ = {𝑢1, 𝑢2, 𝑢3}. then 𝐷 ′ is a non-split inverse dominating set of 𝐺 so that 𝛾𝑐 ′(𝐺) = 2. let 𝑛 ≥ 4. let 𝐷 = {𝑢2, 𝑢4}. then 𝐷 is a dominating set of 𝐺 so that 𝛾(𝐺) = 2. let 𝐷′ = {𝑤2, 𝑤4}. then 𝐷 ′ is a connected inverse dominating set of 𝐺 so that 𝛾𝑐 ′(𝐺) = 2. ∎ inverse domination number of jump graph of 𝑷𝒏 ⊙ 𝑲𝟐 theorem 2.13. for the graph 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾2) (𝑛 ≥ 3), 𝛾 ′(𝐺) = 2. proof. for𝑛 ≥ 3,the number of vertices of 𝑃𝑛 ⊙ 𝐾2is 3𝑛. then it has 3𝑛 − 1 edges. let 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾2). the number of vertices of jump graph of the 𝑃𝑛 ⊙ 𝐾2 is 3𝑛 − 1. let the vertices of the graph is labeled as {𝑢1, 𝑢2, … , 𝑢𝑛 , 𝑣1, 𝑣2, … 𝑣𝑛−1, 𝑤1, 𝑤2, …, 𝑤𝑛}. since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝐷 = {𝑢1, 𝑢3}. then 𝐷 is a 𝛾set of 𝐺 so that 𝛾(𝐺) = 2. let 𝐷′ = {𝑤1, 𝑤3}. then 𝐷 ′ is a inverse dominating set of 𝐺 so that𝛾 ′(𝐺) = 2. ∎ theorem 2.14. for the graph 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾2) (𝑛 ≥ 3), 𝛾𝑛𝑠 ′ (𝐺) = 2. proof. for𝑛 ≥ 3,the number of vertices of 𝑃𝑛 ⊙ 𝐾2is 3𝑛. then it has 3𝑛 − 1 edges. let 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾2). the number of vertices of jump graph of the 𝑃𝑛 ⊙ 𝐾2 is 3𝑛 − 1. let the vertices of the graph is labeled as {𝑢1, 𝑢2, … , 𝑢𝑛 , 𝑣1, 𝑣2, … , 𝑣𝑛−1, 𝑤1, 𝑤2, … , 𝑤𝑛}. since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝐷 = {𝑢1, 𝑢3}. then 𝐷 is a dominating set of 𝐺and so 𝛾(𝐺) = 2. let 𝐷′ = {𝑤1, 𝑤3}. then 𝐷 ′ is a non-split inverse dominating set of 𝐺 so that 𝛾𝑛𝑠 ′ (𝐺) = 2. ∎ theorem 2.15. for the graph 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾2) (𝑛 ≥ 3), 𝛾𝑐 ′(𝐺) = 2. proof. for𝑛 ≥ 3,the number of vertices of 𝑃𝑛 ⊙ 𝐾2is 3𝑛. then it has 3𝑛 − 1 edges. let 𝐺 = 𝐽(𝑃𝑛 ⊙ 𝐾2). the number of vertices of jump graph of the 𝑃𝑛 ⊙ 𝐾2 is 3𝑛 − 1. let the vertices of the graph is labeled as{𝑢1, 𝑢2, … , 𝑢𝑛 , 𝑣1, 𝑣2, … , 𝑣𝑛−1, 𝑤1, 𝑤2, …, , 𝑤𝑛}. since 𝐺 contains no universal vertices, 𝛾(𝐺) ≥ 2. let 𝐷 = {𝑢1, 𝑢3}. then 𝐷 is a dominating set of 𝐺 and so 𝛾(𝐺) = 2. let 𝐷′ = {𝑤1, 𝑤3}. then 𝐷 ′ is a connected inverse dominating set of 𝐺 so that 𝛾𝑐 ′(𝐺) = 2. ∎ 3. conclusions in this article, we determined some inverse domination parameter for jump graph of some special graphs. we will determine some more inverse domination parameters for jump graph of some special graph in future work. 140 inverse domination parameters of jump references [1] allan r.b. and r. c. laskar, on domination inverse domination numbers of a graph, discrete math.,23, ,73-76, 1978. [2] j.a. bondy, u.s.r. murty, graph theory with applications, north holland, new york. [3] g. chatrand, h. galvas, k.c, vandell and f. harary, the forcing domination number of a graph, j. combin. math, combin. comput., (1997), 25: 161-174. [4] e.j. cockayne and hedetniemi., towards a theory of domination in graphs, networks, 7, 247-261, 1997. [5] t.w. haynes and s.t. hedetniemi, p.j. slater, fundamentals of domination in graph, marcel dekkar, new york,1998. [6] j. john, the forcing monophonic and the forcing geodetic numbers of a graph, indonesian journal of combinatorics, 4(2), (2020),114-125. [7] j. john and malchijah raj, the forcing non-split domination number of a graph, korean journal of mathematics, 29(1) (2021),1-12. 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[12] dr. s. meena and k. vaithilingan, prime labeling for some fan related graphs, international journal of engineering research and technology (ijert), 9, 141 ratio mathematica vol. 31, 2016, pp. 79–92 issn: 1592-7415 eissn: 2282-8214 a delayed mathematical model to break the life cycle of anopheles mosquito muhammad a. yau1, bootan rahman2,3 1department of mathematical sciences, nasarawa state university keffi, nigeria muhammadyau@nsuk.edu.ng 2department of mathematics, university of sussex, brighton, uk 3department of mathematics, college of science, salahaddin university erbil, kurdistan, iraq bootan.rahman@su.edu.krd received on: 22-12-2016. accepted on: 22-01-2017. published on: 27-02-2017 doi: 10.23755/rm.v31i0.319 c©muhammad a. yau and bootan rahman abstract in this paper, we propose a delayed mathematical model to break the life cycle of anopheles mosquito at the larva stage by incorporating a time delay τ at the larva stage that accounts for the period of growth or development to pupa. we prove local stability of the system’s equilibrium and find the critical values for hopf bifurcation to occur. also, we find that the system’s equilibrium undergoes stability switching from stable to periodic to unstable and vice versa when the time delay τ crosses the imaginary axis from the left half plane to the right half plane in the (re,im) plane. finally, we perform some numerical simulations and the results agree well with the analytical analysis. this is the first time such a model is proposed. keywords: delayed model; anopheles mosquito; malaria control; hopf bifurcation; larva; stability analysis 2010 ams subject classifications: 97u99. 79 muhammad a. yau and bootan rahman 1 introduction every year, one to three million deaths is attributed to malaria parasite in subsaharan africa out of which one third are children. much work has been done to genetically modify mosquitoes in the laboratory to hinder the parasite from transmission thus, making the mosquitoes refractory. this can be achieved by inserting of genes at appropriate site to create stable germline. the progress in this area is fairly recent. malaria is a killer disease, is one of the leading causes of death in many parts of the world. its devastating effect has persisted for many decades. despite the longevity of the disease, malaria, which has been brought under control in some developed countries, still constitutes a major health menace in many developing countries, where most areas of high endemic reside. some african countries, especially countries within sub-saharan africa, still feature among the leading areas of high malaria endemic in the world [21]. according to world health organization report [34], an estimated of about 225 million malaria clinical cases occurred in 2009, with an estimated 781,000 malaria mortalities. although these statistics reflect a reduction compared to an estimated 243 million malaria cases, with an estimated 863,000 malaria deaths, 89% of which occurred in africa in 2008 [35], the reduction is not sufficient. generally, susceptibility to malaria is universal, that is, any person living in a country where malaria is prevalent is at risk of contracting the disease. however, the impact of malaria is greatest amongst children below five [36], where one in every five childhood deaths is due to the effects of the disease, among pregnant women, and among people from non-malarious regions. temperature is known to affect the life stages of the mosquito parasite [3]. there is a general consensus that future changes in climate may alter the prevalence and incidence of malaria; however, there are conflicting views among authors [20], [39], [40], [11]. however, some authors argued that climate and ecology are the main factors the severity of malaria and the difficulty in controlling it [12]. other factors that have led to difficulties in controlling malaria are socioeconomic conditions, population growth, urbanization, drug resistance, deficiencies in health care systems, poor sanitation, lack of information and education, water storage, garbage disposal, unpaved roads, and drainage systems that generate good breeding stagess for malaria transmission close to human settlements [14], [23], [32]. thus, research in malaria that integrates the disease dynamics with breeding sites/life cycle properties of the vector and the different developmental stages of the parasite may provide novel insights toward disease control and eradication. although malaria is deadly, it can be cured by administering anti-malaria drugs. however, in endemic regions, the malaria parasite develops resistance to 80 a delayed mathematical model to break the life cycle of anopheles mosquito such drugs and there is no effective vaccine for malaria. consequently, prevention is the only other option. prevention can be achieved through the use of prophylactic drugs and vector control strategies. to advance, plan, design, and implement effective or better vector control measures, a clear understanding of mosquito population dynamics, the disease dynamics, and mosquito interaction with the human population is necessary. we introduce a new approach to the development of models for malaria transmission, wherein the mosquito vector is placed at the centre of the transmission process. our objective is to develop a mathematical model for the dynamics of malaria transmission that takes into consideration the population dynamics of the malaria vector and how these vectors interact with the human population. to do that, an understanding of the vector population demography and dynamics is needed. the malaria vector undergoes a complete metamorphosis, as it passes through four different life stages in its cycle: egg, larva, pupa and adult. the egg, larva and pupa stages are aquatic, while the adult stage is terrestrial. the entire cycle from egg laying to the emergence of the adult mosquito takes approximately 7-20 days, with 2-3 days spent in the egg stage, 4-10 days spent in the larva stage, and 2-4 days spent in the pupa stage [14]. while the average life span of the adult female mosquito ranges from 2-3 weeks, that of the males is approximately one week. as for the first three life stages, the life span of the adult mosquito depends on the species and ambient temperature. in addition to natural factors, survival of the adult female anopheles mosquito also depends on its success in acquiring blood meals from humans. therefore, in this research we propose a delayed model to break the life cycle at larva stage. to this end, we introduce a time delay τ at the larva compartment to account for the control measures (this can be bio-organism eg copecods or chemical substances). this is the firt such a delayed model is proposed. 2 model derivation in this section, we derive the delayed model from the life cycle of anopheles mosquito following the approached used in the paper by [22]. we make the following assumptions: the total population of anopheles mosquito is sub-divided into four compartments (adults, eggs, larva, and pupa). the birth rate b is constant and proportional to the total population b, there is a time delay τ in the growth or development to pupa at the larva stage cause by the introduction of control measures (can be natural enemy e.g bio-organisms or chemical substances) that can slow the growth process. anopheles mosquito are assumed to transmit malaria only through direct contact. 81 muhammad a. yau and bootan rahman pupa x4(t) µ ρ ν adult x1(t) µ bn η larva x3(t) µ γ egg x2(t) µ figure 1: a flow chart of the life cycle of a mosquito from the model assumptions and the flow chart in figure (1) above, we derive the following model. let x1(t),x2(t),x3(t),x4(t) be the number of adult mosquitoes, eggs, larva, and pupa at time t respectively. then, the life cycle of anopheles mosquito is represented by the following model: ẋ1(t) = bn − (η + µ)x1(t) + ρx4(t) ẋ2(t) = ηx1(t)− (γ + µ)x2(t) ẋ3(t) = γx2(t)−νx3(t− τ)−µx3(t) ẋ4(t) = νx3(t− τ)− (ρ + µ)x4(t) (1) where b is the natural birth rate, η is the rate at which adult mosquitoes oviposit, µ is the natural death rate, γ is the rate at which the eggs hatch, ν is the rate at which larva develops to pupa, ρ is the rate at which pupa develops to adult mosquitoes. the initial data are x1(θ) = φ1(θ),x2(θ) = φ2(θ),x3(θ) = φ3(θ),x4(θ) = φ4(θ)) for τ ∈ [−τ,0], where φ = (φ1,φ2,φ3,φ4)t ∈ c([−τ,0],r4) such that φi ≥ 0, i = 1,2,3,4. 82 a delayed mathematical model to break the life cycle of anopheles mosquito 3 local stability analysis it is obvious that model (1) has a trivial equilibrium e0 = (bn/µ,0,0,0) and a unique positive non-trivial equilibrium e∗ = (x∗1,x ∗ 2,x ∗ 3,x ∗ 4), where x∗1 = b[(ρ + µ)(ν + µ)n + ρν] (η + µ)(ρ + µ)(ν + µ)−ρνγη , x∗2 = ηx∗1 γ + µ , x∗3 = γηx∗1 + b ν + µ , x∗4 = ν(γηx∗1 + b) (ν + µ)(ρ + µ) . the characteristic polynomial equation for the linearised system 1 is λ4 + p0λ 3 + p1λ 2 + p2λ + p3 + (q0λ 3 + q1λ 2 + q2λ + q3)e −λτ = 0, (2) where p0 = 4µ + ρ + γ + η, p1 = µ (ρ + γ + η + 3µ) + 2η µ + η γ + 3µ 2 + η ρ + 2µρ + γ ρ + 2µγ, p2 = µ (2η µ + η γ + 3µ 2 + η ρ + 2µρ + γ ρ + 2µγ) + η γ ρ + µ3 +µγ ρ + µ2ρ + η µ2 + µ2γ + η µρ + η γ µ, p3 = µ (η γ ρ + µ 3 + µγ ρ + µ2ρ + η µ2 + µ2γ + η µρ + η γ µ) , q0 = ν, q1 = ν (ρ + γ + η + 3µ) , q2 = ν (µ (ρ + γ + η + 2µ) + γ ρ + µγ + η µ + η ρ + µ 2 + µρ + η γ) , q3 = ν µ (γ ρ + µγ + η µ + η ρ + µ 2 + µρ + η γ) . (3) if τ = 0 the characteristic equation 2 becomes λ4 + (p0 + q0)λ 3 + (p1 + q1)λ 2 + (p2 + q2)λ + (p3 + q3) = 0. (4) by routh-hurwitz condition, we have the following necessary and sufficient conditions for 4 to have roots with negative real part h1 = p0 + q0 > 0 h2 = (p0 + q0)(p1 + q1)− (p2 + q2) > 0 h3 = (p0 + q0)[(p1 + q1)(p2 + q2)− (p0 + q0)(p3 + q3)]− (p2 + q1)2 > 0 h4 = p3 + q3 > 0. (5) hi > 0, i = 1,2,3,4. (a2) 83 muhammad a. yau and bootan rahman lemma 3.1. if a2 is satisfied, then the characteristic equation 4 have roots with negative real part. the above result is true only when τ = 0. now if τ > 0, we let λ = iξ (ξ > 0) be a root of the characteristic equation 2, then ξ4 − ip0ξ3 −p1ξ2 + ip2ξ + p3 + (−iq0ξ3 −2q1ξ2 + iq2ξ + q3)(cos(ξτ)− isin(ξτ)) = 0. (6) separating equation 6 into real and imaginary parts we have ξ4 −p1ξ2 + p3 = (q1ξ2 − q3) cos (ξ (τ)) + (q0ξ3 − q2ξ) sin (ξ (τ)) , −p0ξ3 + p2ξ+ = (q0ξ3 − q2ξ) cos (ξ (τ))− (q1ξ2 − q3) sin (ξ (τ)) . (7) squaring both sides of 7 and adding we have the following ξ8 + s0ξ 6 + s1ξ 4 + s22 + s3 = 0, (8) where s0 = p02 − q02 −2p1, s1 = 2p3 + p12 + 2q0q2 − q12 −2p0p2, s2 = −2p1p3 + 2q1q3 + p22 − q22, s3 = p32 − q32. let z = ξ2, then z4 + s0z 3 + s1z 2 + s2z + s3 = h(z). (9) from 9 dh(z) dz = 4z3 + 3s0z 2 + 2s1z + s2 = g(z). (10) let y = z + s0 4 then g(z) = 0, ⇒ y3 + ay + b = 0, (11) where a = 8s0−3s 2 0 16 , b = s30−4s0s1+8s2 32 . by cardano’s theorem, we have q = 24s1−9s20 144 r = 216s0s1−432s2−54s30 3456 d = q3 + r2 k1 = 3 √ r + √ d k2 = 3 √ r− √ d (12) and then z1 = k1 + k2 − s04 z2 = −k1+k22 − 3s0 12 + i √ 3 2 (k1 −k2) z3 = −k1+k22 − 3s0 12 − i √ 3 2 (k1 −k2) (13) 84 a delayed mathematical model to break the life cycle of anopheles mosquito assume that d > 0, then the equation g(z) = 0 has one real root namely; z∗1 = z1 and two complex conjugates, if d = 0, then all roots of g(z) = 0 are real and at least two are equal, namely; z1,z2 = z3, where z∗2 = max{z1,z2}, if d < 0, then all roots of g(z) = 0 are real and distinct, namely; z1,z2,z3, where z∗3 = max{z1,z2,z3}. according to lemma 2.2 in li and hu [38], we have the following lemma 3.2. 1. if s3 < 0, then equation 9 has at least one positive root. 2. if s3 ≥ 0, then equation 9 has no positive root if and only if one of these conditions holds: (a) d > 0 and z∗1 ≤ 0; (b) d = 0 and z ∗ 2 ≤ 0; (c) d < 0 and z ∗ 3 ≤ 0. 3. if s3 ≥ 0, then equation 9 has at least a positive root if and only if one of these conditions holds: (a) d > 0, z∗1 > 0, and h(z ∗ 1) < 0; (b) d = 0, z ∗ 2 > 0 and h(z ∗ 2) < 0; (c) d < 0 z∗3 ≤ 0 and h(z∗3) < 0. now, suppose that equation 9 have four postive real roots, given by z1, z2, z3, z4, then equation 8 also have positive real roots, namely; ξ1 = √ z1, ξ2 = √ z2, ξ3 =√ z3, ξ4 = √ z4. from 2, we find the critical time delay τ0 as follows τ j n = 1 ξ [ arctan { − ω ( q0ω 6+(−q0p1−q2+p0q1)ω4+(−p0q3+q2p1−p2q1+q0p3)ω2+p2q3−q2p3 ) (q1−q0p0)ω6+(−q3+q2p0+q0p2−q1p1)ω4+(q3p1+q1p3−q2p2)ω2−q3p3 } +jπ ] , (14) where n = 1,2,3,4, j = 0,1,2, .... then (τjn) are solutions of 6 and λ = ±iξn are a pair of purely imaginary roots of 2 with τ = τjn. we define τ0 = τ 0 n0 = min 1≤n≤4 {τ0n}, ξ0 = ξn0, where n0 ∈ {1,2,3,4}. then τ0 is the first value of τ such that 2 have purely imaginary roots. let λ(τ) = α(τ) ± iξ(τ) be the root of 2, around τ = τjn satisfying α(τjn) = 0, ξ(τjn) = ξ0(n = 1,2,3,4, j = 0,1,2...). 85 muhammad a. yau and bootan rahman lemma 3.3. suppose h ′ (zn) 6= 0 (n = 1,2,3,4), where h(z) is defined by 9, then the following transversality condition holds: dre{λ(τ)} dτ ∣∣∣ τ=τ j n 6= 0. (15) moreover, the sign of dre{λ(τ)} dτ ∣∣∣ τ=τ j n is consistent with that of h ′ (zn). theorem 3.1. suppose that a2 holds, we have the following: 1. the quasi-polynomial 2 have roots with negative real parts and the steady state solution of system 1 is stable if s3 ≥ 0 and one of these conditions holds: (a) d > 0 and z∗1 ≤ 0; (b) d = 0 and z ∗ 2 ≤ 0; (c) d < 0 and z ∗ 3 ≤ 0. 2. the quasi-polynomial 2 have roots with negative real parts and the steady state solution of system 1 is asymptotically stable if τ ∈ [0,τ0)0 for s3 < 0 or s3 ≥ 0 and one of these conditions holds: (a) d > 0, z∗1 > 0, and h(z ∗ 1) < 0; (b) d = 0, z ∗ 2 > 0 and h(z ∗ 2) < 0; (c) d < 0 z∗3 ≤ 0 and h(z∗3) < 0. 3. if the conditions in (2.) hold and also h ′ (zn) 6= 0, then the system 1 have periodic solutions arising from the hopf bifurcation at τ = τjn(n = 1,2,3,4,j = 0,1,2...). in the figures below, we illustrate the above stability results and also numerically compute real part of the leading eigenvalue of the characteristic equation using tracedde suite in matlab and plot the results in gnuplot. by varying the natural clearance rate µ, we investigate how stability changes in the τ,ν plane, and also the effects on the dynamical behaviour of the system. 86 a delayed mathematical model to break the life cycle of anopheles mosquito figure 2: stability charts: (a) µ = 0.1, (b) µ = 0.3, (c) µ = 0.5, (d) µ = 0.7. the color in the figures corresponds to the real part of the leading eigenvalue of the characteristic quasi-polynomial from the figures above, we can see that µ played an important role in the dynamical behaviour of the system (1). the colors in the figures stand for: yellow “most stable region”, red “more stable region”, dark-violet “stable region”, black “critical line or hopf region” and the remaining area (white) corresponds to “unstable region”. as µ increased, the dynamics of the system also increased in the (τ,ν) plane. again, we observed that change in γ has similar system dynamics as above. therefore, in general in the (τ,ν) plane, the overall dynamical behaviour of the system is determined by the parameters µ and γ. 4 numerical simulation in this section, we present some numerical simulations using dde23 suit in matlab. we will show stable, periodic and unstable solutions as τ is varied. we have stability switches from stable to periodic to unstable and to stable as τ takes on the critical values τ0 or as τ crosses the imaginary axis. in the first simulation, we take τ < τ0, and we have stable solutions (τ = 0.45,τ0 = 0.85). 87 muhammad a. yau and bootan rahman 0 10 20 30 40 50 60 70 80 90 100 −1 −0.5 0 0.5 1 1.5 2 2.5 3 time(t) u i (a) −1 −0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 (b) figure 3: stable solutions and phase portrait of system 1 for τ = 0.45 0 10 20 30 40 50 60 70 80 90 100 −1 −0.5 0 0.5 1 1.5 2 2.5 3 time(t) u i (a) −1 −0.5 0 0.5 1 1.5 −1 0 1 2 3 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 (b) figure 4: periodic solutions and phase portrait of system 1 for τ = 0.85 88 a delayed mathematical model to break the life cycle of anopheles mosquito 0 10 20 30 40 50 60 70 80 90 100 −20 −15 −10 −5 0 5 10 15 20 time(t) u i (a) −20 −10 0 10 20 −20 −10 0 10 20 0 0.5 1 1.5 2 2.5 (b) figure 5: unstable solutions and phase portrait of system 1 for τ = 0.95 5 conclusion in this paper, we derived a mathematical model to break the life cycle of a mosquito that incorporate a time delay at the larva stage that accounts for the period of growth and development to pupa. we prove the local stability of the system’s equilibrium and the critical values for hopf bifurcation to occur. we find that the model undergoes stability switching from stable to periodic and to unstable when the time delay τ crosses the imaginary axis from the left half plane to the right half plane in the (re,im) plane. that is, the system’s equilibrium e∗ is stable if τ < τ0 (see figure 3), if τ = τ0, e∗ loses its stability and a hopf bifurcation occurs which means, a family of periodic solutions bifurcate from e∗ (see figure 4). and if τ > τ0, then e∗ is unstable as seen in figure 5. references [1] j. l. aron, mathematical modeling of immunity to malaria, mathematical biosciences 90, 385-396 (1988). 32 [2] t. j. n. bailey, the mathematical theory of infectious diseases and its application, (griffin, london, 1975), 2nd edition. 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[32] m. i. teboh-ewungkem, malaria control: the role of local communities as seen through a mathematical model in a changing populationcameroon, chapter 4, 103-140, in advances in disease epidemiology (nova science publishers, 2009). 31 [33] j. n. wilford, malaria is a likely killer in king tuts post-mortem, technical report 16, the new york times, accessed march 2011. [34] world health organisation, the world malaria report, who press, acessed march 2011 (2010). [35] world health organisation, the world health report, who press (2009). [36] world health organization, 10 facts on malaria, who press (2009). [37] s. wyborny, parasites: the malaria parasite, (kidhaven press, 2005), 1st edition. [38] xiaoling li, guangping hu, stability and hopf bifurcation on a neuron network with discrete and distributed delays, appl. math. sci., 2077-2084 (42), 2011. 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[40] j. a. patz and s. h. olson, malaria risk and temperature: influences from global climate change and local land use practices, proceedings of the national academy of sciences 103, 5635-5636 (2006). 92 ratio mathematica 26(2014), 113–128 issn:1592-7415 the lie-santilli admissible hyperalgebras of type an pipina nikolaidou, thomas vougiouklis democritus university of thrace, school of education, 68 100 alexandroupolis, greece pnikolai@eled.duth.gr,tvougiou@eled.duth.gr abstract the largest class of hyperstructures is the one which satisfy the weak properties. these are called hv-structures introduced in 1990 and they proved to have a lot of applications on several applied sciences. in this paper we present a construction of the hyperstructures used in the lie-santilli admissible theory on square matrices. key words: hyperstructures, hv-structures, hopes, weak hopes, ∂-hopes, e-hyperstructures, admissible lie-algebras. msc 2010: 20n20, 17b67, 17b70, 17d25. 1 introduction we deal with hyperstructures called hv-structures introduced in 1990 [30], which satisfy the weak axioms where the non-empty intersection replaces the equality. some basic definitions are the following: in a set h equipped with a hyperoperation (abbreviation hyperoperation = hope) · : h ×h → p(h) −{∅}, we abbreviate by wass the weak associativity : (xy)z ∩x(yz) 6= ∅,∀x,y,z ∈ h and by cow the weak commutativity : xy ∩yx 6= ∅,∀x,y ∈ h. p. nikolaidou, th. vougiouklis the hyperstructure (h, ·) is called an hv-semigroup if it is wass, it is called hv-group if it is reproductive hv-semigroup, i.e., xh = hx = h,∀x ∈ h. the hyperstructure (r, +, ·) is called an hv-ring if (+) and (·) are wass, the reproduction axiom is valid for (+) and (·) is weak distributive with respect to (+): x(y + z) ∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅, ∀x,y,z ∈ r. motivations. the motivation for hv-structures is the following: we know that the quotient of a group with respect to an invariant subgroup is a group. f. marty from 1934, states that, the quotient of a group with respect to any subgroup is a hypergroup. finally, the quotient of a group with respect to any partition (or equivalently to any equivalence relation) is an hv-group. this is the motivation to introduce the hv-structures [24]. in an hv-semigroup the powers of an element h ∈ h are defined as follows: h1 = {h},h2 = h ·h,...,hn = h◦h◦ ...◦h, where (◦) denotes the n-ary circle hope, i.e. take the union of hyperproducts, n times, with all possible patterns of parentheses put on them. an hvsemigroup (h, ·) is called cyclic of period s, if there exists an element h, called generator, and a natural number s, the minimum one, such that h = h1 ∪h2...∪hs. analogously the cyclicity for the infinite period is defined [23]. if there is an element h and a natural number s, the minimum one, such that h = hs, then (h, ·) is called single-power cyclic of period s. for more definitions and applications on hv-structures, see the books [2],[8],[24],[4],[1] and papers as [3],[28],[21],[22],[26],[9],[14],[13]. the main tool to study hyperstructures are the fundamental relations β*, γ* and �*, which are defined in hv-groups, hv-rings and hv-vector spaces, resp., as the smallest equivalences so that the quotient would be group, ring and vector space, resp. these relations were introduced by t. vougiouklis [30],[24],[29]. a way to find the fundamental classes is given by theorems as the following [24],[21],[25],[22],[7],[9],[20]: theorem 1.1. let (h, ·) be an hv-group and denote by u the set of all finite products of elements of h. we define the relation β in h by setting xβy iff {x,y}⊂ u where u ∈ u. then β* is the transitive closure of β. 114 the lie-santilli admissible hyperalgebras of type an analogous theorems for the relations γ* in hv-rings, �* in hv-modules and hv-vector spaces, are also proved. an element is called single if its fundamental class is singleton [24]. fundamental relations are used for general definitions. thus, an hv-ring (r, +, ·) is called hv-field if r/γ* is a field. let (h, ·), (h,∗) be hv-semigroups defined on the same set h. the hope (·) is called smaller than the hope (∗), and (∗) greater than (·), iff there exists an f ∈ aut(h,∗) such that xy ⊂ f(x∗y), ∀x,y ∈ h. then we write · ≤ ∗ and we say that (h,∗) contains (h, ·). if (h, ·) is a structure then it is called basic structure and (h,∗) is called hb-structure and (∗) is called b-hope. theorem 1.2. (the little theorem). greater hopes than the ones which are wass or cow, are also wass or cow, respectively. definition 1.1. [20],[25] let (h, ·) be hypergroupoid. we remove h ∈ h, if we consider the restriction of (·) in the set h −{h}. h ∈ h absorbs h ∈ h if we replace h by h and h does not appear in the structure. h ∈ h merges with h ∈ h, if we take as product of any x ∈ h by h, the union of the results of x with both h, h, and consider h and h as one class with representative h, therefore, h does not appear in the hyperstructure. for several definitions and applications of hyperstructures in mathematics or in sciences and social sciences one can see [11],[15],[13],[3]. 2 the theta (∂) hopes in [19],[32],[11],[15] a hope, in a groupoid with a map f on it, denoted ∂f , is introduced. since there is no confusion, we write simply theta ∂. the symbol ”∂” appears in greek papyrus to represent the letter ”theta ”usually in middle rather than the beginning of the words. definition 2.1. let h be a set equipped with n operations (or hopes) ⊗1, ...,⊗n and a map (or multivalued map) f : h → h (or f : h → p(h) −∅, respectively), then n hopes ∂1,∂2,...,∂n on h can be defined, called theta-operations (we rename here theta-hopes and we write ∂-hope) by putting x∂iy = {f(x) ⊗i y,x⊗i f(y)},∀x,y ∈ h and i ∈{1, 2, ...,n} 115 p. nikolaidou, th. vougiouklis or, in case where ⊗i is hope or f is multivalued map, we have x∂iy = (f(x) ⊗i y) ∪ (x⊗i f(y)),∀x,y ∈ h and i ∈{1, 2, ...,n} if ⊗i is associative then ∂i is wass. analogously one can use several maps f, instead than only one. let (g, ·) be a groupoid and fi : g → g,i ∈ i, be a set of maps on g. take the map f∪ : g → p(g) such that f∪(x) = {fi(x)|i ∈ i} and we call it the union of the fi(x). we call union ∂-hopes, on g if we consider the map f∪(x). a special case is to take the union of f with the identity, i.e. f = f ∪ (id), so f(x) = {x,f(x)},∀x ∈ g, which is called b-∂-hope. we denote the b-∂-hope by (∂), so x∂y = {xy,f(x) ·y,x ·f(y)},∀x,y ∈ g this hope contains the operation (·) so it is a b-hope. if f : g → p(g)− {∅}, then the b-∂-hope is defined by using the map f(x) = {x}∪f(x),∀x ∈ g. motivation for the definition of the theta-hope is the map derivative where only the multiplication of functions can be used. therefore, in these terms, for two functions s(x), t(x), we have s∂t = {s′t,st′} where (′) denotes the derivative. for several results one can see [19],[32]. examples. (a) taking the application on the derivative, consider all polynomials of up to first degree gi(x) = aix + bi. we have g1∂g2 = {a1a2x + a1b2,a1a2x + b1a2}, so this is a hope in the first degree polynomials. remark that all polynomials x+c, where c be a constant, are units. (b) the constant map. let (g, ·) be group and f(x) = a, thus x∂y = {ay,xa},∀x,y ∈ g. if f(x) = e, then we obtain x∂y = {x,y}, the smallest incidence hope. properties. if (g, ·) is a semigroup then: (a) for every f, the ∂-hope is wass. (b) for every f, the b-∂-hope (∂) is wass. (c) if f is homomorphism and projection, then (∂) is associative. 116 the lie-santilli admissible hyperalgebras of type an properties. reproductivity. if (·) is reproductive then (∂) is also reproductive. commutativity. if (·) is commutative then (∂) is commutative. if f is into the centre of g, then (∂) is commutative. if (·) is cow then, (∂) is cow. unit elements. the elements of the kernel of f, are the units of (g,∂). inverse elements. for given x, the elements x′ = (f(x))−1u and x′ = u(f(x))−1, are the right and left inverses, respectively. we have two-sided inverses iff f(x)u = uf(x). proposition. let (g, ·) be a group then, for all maps f : g → g, the hyperstructure (g,∂) is an hv-group. definition 2.2. let (r, +, ·) be a ring and f : r → r, g : r → r be two maps. we define two hopes (∂+) and (∂−), called both theta-hopes, on r as follows x∂+y = {f(x) + y,x + f(y)} and x∂·y = {g(x) ·y,x ·g(y)},∀x,y ∈ g. a hyperstructure (r, +, ·), where (+), (·) are hopes which satisfy all hvring axioms, except the weak distributivity, will be called hv-near-ring. propositions. (a) let (r, +, ·) be a ring and f : r → r, g : r → r be maps. the (r,∂)+,∂·), called theta, is an hv-near-ring. moreover (∂+) is commutative. (b) let (r, +, ·) be a ring and f : r → r, g : r → r maps, then (r,∂+,∂·), is an hv-ring. properties.(special classes). the theta hyperstructure (r,∂+,∂·) takes a new form and has some properties in several cases as the following ones: (a) if f is a homomorphism and projection, then x∂·(y∂+z)∩(x∂·y)∂+(x∂·z) = {f(x)f(y)+f(x)z,f(x)y+f(x)f(z)} 6= ∅. therefore, (r,∂)+,∂·) is an hv-ring. (b) if f(x) = x,∀x ∈ r, then (r, +,∂·) becomes a multiplicative hv-ring: x∂·(y + z) ∩ (x∂·y) + (x∂·z) = {g(x)y + g(x)z} 6= ∅. if, moreover, f is a homomorphism, then we have a ”more” strong distributivity: x∂·(y + z) ∩ ((x∂·y) + (x∂·z)) = {g(x)y + g(x)z,xg(y) + xg(z)} 6= ∅. 117 p. nikolaidou, th. vougiouklis now we can see theta hopes in hv-vector spaces and hv-lie algebras: theorem 2.1. let (v, +, ·) be an algebra over the field (f, +, ·) and f : v → v be a map. consider the ∂-hope defined only on the multiplication of the vectors (·), then (v, +,∂) is an hv-algebra over f, where the related properties are weak. if, moreover f is linear then we have λ(x∂y) = (λx)∂y = x∂(λy). another well known and large class of hopes is given as follows [23],[24]: let (g, ·) be a groupoid then for every p ⊂ g, p 6= ∅, we define the following hopes called p-hopes : for all x,y ∈ g p : xpy = (xp)y ∪x(py), p r : xp ry = (xy)p ∪x(yp), p l : xp ly = (px)y ∪p(xy). the (g,p),(g,p r) and (g,p l) are called p-hyperstructures. the most usual case is if (g, ·) is semigroup, then xpy = (xp)y ∪x(py) = xpy and (g,p) is a semihypergroup but we do not know about (g,p r) and (g,p l). in some cases, depending on the choice of p, the (g,p r) and (g,p l) can be associative or wass. a generalization of p-hopes, introduced by davvaz, santilli, vougiouklis in [7],[6] is the following: construction 2.1. let (g, ·) be an abelian group and p any subset of g with more than one elements. we define the hope ×p as follows: x×p y = { x×p y = x ·p ·y = {x ·h ·y|h ∈ p} if x 6= e and c 6= e x ·y if x = e and y = e we call this hope pe-hope. the hyperstructure (g,×p) is an abelian hv-group. matrix representations hv-structures are used in representation theory of hv-groups which can be achieved either by generalized permutations or by hv-matrices [28],[24]. representations by generalized permutations can be faced by translations. in this theory the single elements are playing a crucial role. hv-matrix is called a matrix if has entries from an hv-ring. the hyperproduct of hvmatrices is defined in a usual manner. in representations of hv-groups by hv-matrices, there are two difficulties: to find an hv-ring and an appropriate set of hv-matrices. 118 the lie-santilli admissible hyperalgebras of type an most of hv-structures are used in representation (abbreviate by rep) theory. reps of hv-groups can be considered either by generalized permutations or by hv-matrices [24]. reps by generalized permutations can be achieved by using translations. in the rep theory the singles are playing a crucial role. the rep problem by hv-matrices is the following: hv-matrix is called a matrix if has entries from an hv-ring. the hyperproduct of hv-matrices a= (aij ) and b= (bij ), of type m × n and n × r, respectively, is a set of m× r hv-matrices, defined in a usual manner: a ·b = (aij ) · (bij ) = {c = (cij )|(cij ) ∈⊕ ∑ aik · bkj}, where (⊕) denotes the n-ary circle hope on the hyperaddition. definition 2.3. let (h, ·) be an hv-group,(r, +, ·) be an hv-ring r and consider a set mr = {(aij )|aij ∈ r} then any map t : h → mr : h 7→ t(h) with t(h1h2) ∩t(h1)t(h2) 6= ∅,∀h1,h2 ∈ h. is called hv-matrix rep. if t(h1h2) ⊂ t(h1)t(h2), then t is an inclusion rep, if t(h1h2) = t(h1)t(h2), then t is a good rep. 3 the general hv-lie algebra definition 3.1. let (f, +, ·) be an hv-field, (v, +) be a cow hv-group and there exists an external hope · : f ×v → p(v ) −{∅} : (a,x) → zx such that, for all a,b in f and x,y in v we have a(x + y) ∩ (ax + ay) 6= ∅, (a + b)x∩ (ax + bx) 6= ∅, (ab)x∩a(bx) 6= ∅, then v is called an hv-vector space over f. in the case of an hv-ring instead of an hv-field then the hv-modulo is defined. in these cases the fundamental relation �* is the smallest equivalence relation such that the quotient v/�* is a vector space over the fundamental field f/γ*. the general definition of an hv-lie algebra was given in [31] as follows: definition 3.2. let (l, +) be an hv-vector space over the hv-field (f, +, ·), φ : f → f/γ* the canonical map and ωf = {x ∈ f : φ(x) = 0}, where 0 is the zero of the fundamental field f/γ. similarly, let ωl be the core of the 119 p. nikolaidou, th. vougiouklis canonical map φ′ : l → l/�* and denote by the same symbol 0 the zero of l/�*. consider the bracket (commutator) hope: [, ] : l×l → p(l) : (x,y) → [x,y] then l is an hv-lie algebra over f if the following axioms are satisfied: (l1) the bracket hope is bilinear, i.e. ∀x,x1,x2,y,y1,y2 ∈ l,λ1,λ2 ∈ f [λ1x1 + λ2x2,y] ∩ (λ1[x1,y] + λ2[x2,y]) 6= ∅ [x,λ1y1 + λ2y2] ∩ (λ1[x,y1] + λ2[x,y2]) 6= ∅, (l2) [x,x] ∩ωl 6= ∅, ∀x ∈ l (l3) ([x, [y,z]] + [y, [z,x]] + [z, [x,y]]) ∩ωl 6= ∅, ∀x,y ∈ l definition 3.3. let (a, +, ·) be an algebra over the field f. take any map f : a → a, then the ∂-hope on the lie bracket [x,y] = xy−yx, is defined as follows x∂y = {f(x)y −f(y)x,f(x)y −yf(x),xf(y) −f(y)x,xf(y) −yf(x)}. remark that if we take the identity map f(x) = x,∀x ∈ a, then x∂y = {xy −yx}, thus we have not a hope and remains the same operation. proposition. let (a, +, ·) be an algebra f and f : a → a be a linear map. consider the ∂−hope defined only on the multiplication of the vectors (·), then (a, +, ·) is an hv-algebra over f, with respect to the ∂-hopes on lie bracket, where the weak anti-commutativity and the inclusion linearity is valid. proposition. let (a, +, ·) be an algebra and f : a → a : f(x) = a be a constant map. consider the ∂-hope defined only on the multiplication of the vectors (·), then (a, +,∂) is an hv-lie algebra over f. in the above theorem if one take a=e, the unit element of the multiplication, then the properties become more strong. 4 santilli’s admissibility the lie-santilli isotopies born to solve hadronic mechanics problems. santilli proposed [16] a ”lifting” of the trivial unit matrix of a normal theory into a nowhere singular, symmetric, real-valued, new matrix. the original theory is reconstructed such as to admit the new matrix as left and right unit. 120 the lie-santilli admissible hyperalgebras of type an the isofields needed correspond to hv-structures called e-hyperfields which are used in physics or biology. definition: let (ho, +, ·) be the attached hv-field of the hv-semigroup (h, ·). if (h, ·) has a left and right scalar unit e then (ho, +, ·) is e-hyperfield, the attached hv-field of (h, ·). the lie-santilli theory on isotopies was born in 1970’s to solve hadronic mechanics problems. santilli proposed a ”lifting” of the n-dimensional trivial unit matrix of a normal theory into a nowhere singular, symmetric, realvalued, positive-defined, n-dimensional new matrix. the original theory is reconstructed such as to admit the new matrix as left and right unit. the isofields needed in this theory correspond into the hyperstructures were introduced by santilli and vougiouklis in 1996 [5],[17] and they are called e-hyperfields. the hv-fields can give e-hyperfields which can be used in the isotopy theory in applications as in physics or biology. we present in the following the main definitions and results restricted in the hv-structures. definition 4.1. a hyperstructure (h, ·) which contain a unique scalar unit e, is called e-hyperstructure. in an e-hyperstructure, we assume that for every element x, there exists an inverse x−1, i.e. e ∈ x ·x−1 ∩x−1 ·x. remark that the inverses are not necessarily unique. definition 4.2. a hyperstructure (f, +, ·), where (+) is an operation and (·) is a hope, is called e-hyperfield if the following axioms are valid: 1. (f, +) is an abelian group with the additive unit 0, 2. (·) is wass, 3. (·) is weak distributive with respect to (+), 4. 0 is absorbing element: 0 ·x = x · 0 = 0,∀x ∈ f , 5. exist a multiplicative scalar unit 1, i.e. 1 ·x = x · 1 = x,∀x ∈ f , 6. for every x ∈ f there exists a unique inverse x−1, such that 1 ∈ x · x−1 ∩x−1 ·x. the elements of an e-hyperfield are called e-hypernumbers. in the case that the relation: 1 = x ·x−1 = x−1 ·x, is valid, then we say that we have a strong e-hyperfield. now we present a general construction which is based on the partial ordering of the hv-structures and on the little theorem. 121 p. nikolaidou, th. vougiouklis definition 4.3. the main e-construction. given a group (g, ·), where e is the unit, then we define in g, a large number of hopes (⊗) as follows: x⊗y = {xy,g1,g2, ...},∀x,y ∈ g−{e}, and g1,g2, ... ∈ g−{e} g1,g2,... are not necessarily the same for each pair (x,y). then (g,⊗) becomes an hv-group, actually is an hb-group which contains the (g, ·). the hv-group (g,⊗) is an e-hypergroup. moreover, if for each x,y such that xy = e, so we have x⊗y = xy, then (g,⊗) becomes a strong e-hypergroup the proof is immediate since we enlarge the results of the group by putting elements from g and applying the little theorem. moreover one can see that the unit e is a unique scalar and for each x in g, there exists a unique inverse x−1, such that 1 ∈ x ·x−1 ∩x−1 ·x and if this condition is valid then we have 1 = x ·x−1 = x−1 ·x. so the hyperstructure (g,⊗) is a strong e-hypergroup. 5 mathematical realisation of type an the representation theory by matrices gives to researchers a flexible tool to see and handle algebraic structures. this is the reason to see lie-santilli’s admissibility using matrices or hypermatrices to study the multivalued (hyper) case. using the well known p-hyperoperations we extend the liesantilli’s admissibility into the hyperstructure case. we present the problem and we give the basic definitions on the topic which cover the four following cases: construction 5.1. [18] suppose r, s be sets of square matrices (or hypermatrices). we can define the hyper-lie bracket in one of the following ways: 1. [x,y]rs = xry −ysx (general case) 2. [x,y]r = xry −yx 3. [x,y]s = xy −ysx 4. [x,y]rr = xry −yrx the question is when the conditions, for all square matrices (or hypermatrices) x, y, z, [x,x]rs 3 0 [x, [y,z]rs ]rs + [y, [z,x]rs ]rs + [z, [x,y]rs ]rs 3 0 of a hyper-lie algebra are satisfied [18]. we apply this generalization on the lie algebras of the type an. 122 the lie-santilli admissible hyperalgebras of type an we deal with lie-algebra of type an, of traceless matrices m (tr(m)=0), which is a graded algebra, using the principal realisation used in infinite dimensional kac moody lie algebras introduced in 1981[10] by lepowsky and wilson, kac [12]. in this special algebra examples on the above described hyperstructure theory are being presented. denote as eij (i,j = 1, ...,n) the n×n matrix which is 1 in the ij-entry and 0 everywhere else and by ei = eii −ei+1,i+1, i = 1, ...,n− 1 the simple base of the above type is the following: base of level 0 : ei, i = 1, 2, ...,n− 1 base of level 1 : ei,i+1, i = 1, 2, ...,n base of level 2 : ei,i+2, i = 1, 2, ...,n ... base of level n-1 : ei,i+(n−1), i = 1, 2, ...,n denote that all the subscripts are mod n. therefore the levels are in bold as follows: level 0 :   a11 0 0 . . . 0 0 a22 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . 0 0 0 0 . . . ann   level 1 :   0 a12 0 . . . 0 0 0 a23 . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . an−1,n an1 0 0 . . . 0   level 2 : 123 p. nikolaidou, th. vougiouklis   0 0 a13 . . . 0 0 0 0 . . . a2n . . . . . . . . . . . . . . . . . . . . . . . . . . an−1,1 0 0 . . . 0 0 an2 0 . . . 0   .................................................. level n-1 :   0 0 0 . . . a1n a21 0 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . 0 0 0 . . . an,n−1 0   for our examples the konstant’s cyclic element e is being used as the sum of first level’s simple base [10]. e = e12 + e23 + e34 + ... + en−1,n + en1 this element is shifting every element of level l to the next level l + 1 [10],[27]. the base of the first level as well as for every level, except zero, has n elements. level 0 has a n − 1 dimension because of the limitation of the zero trace. the cyclic element gets different element from the base and goes to different ellement of the next level, creating an 1-1 correspondance. the element e shifts level n − 1 to the level 0 and because, as already remarked, level 0 has n − 1 elements, contrary with every other level, the 1-1 correspondance is being corrupted. to summarize, according to the related theory, removing from every level (except level-0), all the powers of e until n− 1 (e,e2, ...,en−1), an one to one complete correspondance between all levels, level-0 included, is being created. we denote the first power : [e,en1] 1 = e ·en1 −en1 ·e = a1 the second power: [e,en1] 2 = [e,a1] = a2 ............................... and inductively by the n-power: [e,en1] n = [e,an−1] = an one can prove the following: 124 the lie-santilli admissible hyperalgebras of type an theorem 5.1. [e,en1] n = = diag( ( n−1 0 ) , (−1)1 ( n−1 1 ) , (−1)2 ( n−1 2 ) , ..., (−1)n−2 ( n−1 n−2 ) , (−1)n−1 ( n−1 n−1 ) ) the above theorem helps as to find the basic element of first level’s base and based on this theorem all the nth powers of the elements of the first level can also be found. theorem 5.2. based on this theory and p-hyperstructures a set p with two elements can be used, either from zero or first level, but only with two elements. in this case the shift is depending on the level, so if we take p from level-0, the result will not change, although the result will be multivalued. in case of different level insted, the shift will be analogous to the level of p. in the general case in construction 5.1(1), one can notice the possible cardinality of the result, checking the jacoby identity is very big. even in the small case when |r| = |s| = |p| = 2 in the anticommutativity xpx−xpx could have cardinality 4 and the left side of the jacoby identity is (xp(ypz −zpy) − (ypz −zpy)px) + (yp(zpx−xpz)− −(zpx−xpz)py) + (zp (xpy −ypx) − (xpy −ypx)pz) could have cardinality 218. the number is reduced in special cases. theorem 5.3. in the case of the lie-algebra of type an, of traceless matrices m, we can define a hyper-lie-santilli-admissible bracket hope as follows: [xy]p = xpy −ypx where p = {p,q}, with p,q elements of the zero level. then we obtain a hyper-lie-santilli-algebra. proof we need only to proof the anticommutativity and the jacobi identity as in the hyperstructure case. therefore we have (a) [xy]p = xpy − ypx = {0,xpx − xqx,xqx − xpx} 3 0, so the ”weak” anticommutativity is valid, and (b) [x, [y,z]p]p + [y, [z,x]p]p + [z, [x,y]p]p = (xp(ypz −zpy) − (ypz −zpy)px) + (yp(zpx−xpz)− −(zpx−xpz)py) + (zp (xpy −ypx) − (xpy −ypx)pz). but this set contains the element xpypz −xpzpy −ypzpx + zpypx + ypzpx−ypxpz− −zpxpy + xpzpy + zpxpy −zpypx−xpypz + ypxpz = 0 125 p. nikolaidou, th. vougiouklis so the ”weak” jacobi identity is valid. thus, zero belongs to the above results, as it has to be, but there are more elements because it is a multivalued operation. references [1] p. corsini, prolegomena of hypergroup theory, aviani, 1994. 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[32] t. vougiouklis, ∂-operations and hv-fields, acta math. sinica, (engl. ser.), vol. 24, n.7 (2008), 1067-1078. 128 ratio mathematica volume 45, 2023 analytical study of mixed convective flow and heat transfer in vertical channel filled with immiscible viscous fluids daimi syeda mariya begum* sharad kumar jagtap† abstract in this paper investigation of mixed convective flow and heat transfer in vertical channel filled with immiscible viscous fluids has been carried out. the governing differential equations are solved analytically by regular perturbation method. the impact of governing parameters on velocity and temperature fields namely grash of number, brinkman number, perturbation parameter, viscosity ratio, width ratio, conductivity ratio, nusselt number are investigated and represented graphically. keywords: mixed convective flow, heat transfer, perturbation method. 2010 ams subject classification: 76d05, 35q30‡ * research student at department of mathematics, srtm university, maharashtra 431606, india; daimimariya@gmail.com. † hod mathematics, shivaji college, udgir, maharashtra, india 413517; sharadvjagtap@gmail.com. ‡ received on july 10th, 2022. accepted on october 15th, 2022. published on january 30, 2023.doi: 10.23755/rm. v45i0.1015. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 194 mailto:daimimariya@gmail.com mailto:sharadvjagtap@gmail.com mariya daimi1, s. jagtap2 1. introduction mixed convective flow and heat transfer has importance from researchers due to their diverse application in engineering, automobile sector and various technical fields. this includes geothermal mining, nuclear reactors, heat and cold storage, wielding equipment, aircrafts desai and vafai [5]. the study of laminar fully developed mixed convection in a vertical channel with uniform wall temperature was one of the first attempt of tao [10]. recently hamadah and wirtz [11], bratetta [3], aung and worku [1] assumed symmetric and asymmetric heating of walls of the vertical channel. prathap kumar et al [7] studied the chemical reaction effects on mixed convection flow in vertical channel with immiscible fluids analytical as well as numerically. umavati and chamkha [6] analyzed mixed convection in presence of heat source or heat sink in a vertical channel. jha and oni [2] assumed temperature dependent viscosity to study the mixed convection flow in a vertical channel where they found that increase in viscosity parameter increases fluid velocity. with hall and ion-slip effects srinivasacharya and shafeeurrahaman [4] analyzed mixed convection flow in a vertical channel filled with nanofluid where they found with the increment in magnetic parameter decrease in temperature, velocity, nanoparticle concentration were occurred. prathap kumar et al [8] analyzed and found impact of different governing parameter on mixed convective flow in a vertical channel using third kind of boundary condition where channel is filled with porous media using differential transfer method as well as perturbation method. in fluid dynamics regular perturbation method is most preferably used, rashidi and ganji [9]. keeping in view of several applications of mixed convective flow in vertical channel, the aim of this paper is to investigate laminar fully developed mixed convection flow and heat transfer in vertical channel with immiscible viscous fluid analytically to extend the studies available in the literature. the governing differential equations are solved using regular perturbation method valid for small values of perturbation parameter. the thermal buoyancy force, viscous dissipation, viscosity ratio, width ratio, conductivity ratio, nusselt number are considered to investigate their impact on flow field. notation: bratio of thermal expansion coefficients (β2/β1) br brinkman number (μ1 u0 (1)2/k1 δt) gacceleration due to gravity gr grashof number (gβ1d13δt/𝜈12) gr mixed convective parameter (gr/re) d width ratio (d2/d1) d1, d2 width of regions k thermal conductivities (k1/k2) m ratio of viscosities (μ1/μ2) n ratio of densities (ρ2/ρ1) p pressure (assuming p1=p2=p) p=p+ρ0gx, difference between the pressure and hydrostatic pressure 195 analytical study of mixed convective flow and heat transfer… re reynolds number (d1u0(1)/𝜈1) u0(i) reference velocity greek symbols α1, α2thermal diffusivities β1, β2coefficients of thermal expansion δtdifference in temperature (t2 ̶ t1) ɛdimensionless parameter/perturbation parameter (gr br) θdimensionless temperature θ1, θ2temperatures μ1, μ2viscosities 𝜈1, 𝜈2kinematics viscosities ρ1, ρ2 densities subscripts i= 1,2 corresponding to region-i and region-ii respectively. 2. preliminaries consider a steady two dimensional laminar fully developed mixed convection flow in open ended vertical channel filled with immiscible viscous fluids the x axis is taken upward and parallel to the walls and y axis is normal on it, shown in fig 1. we consider fluid to be incompressible, and temperature between the plate and fluid is small, so that the fluid properties taken as constant except the density in the buoyancy term of equation of motion. fig.1. physical configuration region −i gβ1(t1 − t0) − 1 ρ1 ⅆp ⅆx + μ1 ρ1 ⅆ2u1 ⅆy2 = 0 (2.1) α1 ⅆ2t1 ⅆy2 + 𝜈1 cp ( ⅆu1 ⅆy ) 2 = 0 (2.2) y = d2 2 y = − d1 2 𝑔 y region-i region-ii 196 mariya daimi1, s. jagtap2 region −ii gβ2(t2 − t0) − 1 ρ2 ⅆp ⅆx + μ2 ρ2 ⅆ2u2 ⅆy2 = 0 (2.3) α2 ⅆ2t2 ⅆy2 + 𝜈2 cp ( ⅆu2 ⅆy ) 2 = 0 (2.4) p depends only on x ⅆp ⅆy = 0 (2.5) in the presence of viscous dissipation, the energy balance equation can be written as region −i ⅆ2t1 ⅆy2 = −𝜈1 β1g ⅆ4u1 ⅆy4 (2.6) from eq. (2.2) and eq. (2.6) ⅆ4u1 ⅆy4 = β1gρ1 k1 ( ⅆu1 ⅆy ) 2 (2.7) region −ii ⅆ2t2 ⅆy2 = −𝜈2 β2g ⅆ4u2 ⅆy4 (2.8) from eq. (2.4) and eq. (2.8) ⅆ4u2 ⅆy4 = β2gρ2 k2 ( ⅆu2 ⅆy ) 2 (2.9) on account of eq. (2.1) and eq. (2.3) there exists a constant a such that ⅆp ⅆx = a (2.10) the boundary and interface conditions are ui ( −d1 2 ) = 0 = u2 ( −d2 2 ) , u1(0) = u2(0), t0 = t1 + t2 2 , ⅆ2u1 ⅆy2 | y=− d1 2 = a μ1 + β1g[t2−t1] 2𝜈1 , ⅆ2u2 ⅆy2 | y= d2 2 = a μ2 − β2g[t2−t1] 2𝜈2 , μ1 ⅆu1 ⅆy = μ2 ⅆu2 ⅆy , at y = 0 , ⅆ3u1 ⅆy3 = 1 mnbk ⅆ3u2 ⅆy3 , at y = 0 ⅆ2u1 ⅆy2 = 1 mnb ⅆ2u2 ⅆy2 + a μ1 [1 − 1 nb ] at y = 0 𝑇1(0) = 𝑇2(0), k1 ⅆt1 ⅆy = k2 ⅆt2 ⅆy , at y = 0 (2.11) equations (2.7),(2.9) and (2.11) determine the velocity distribution. they can be written in a non-dimensional form by means of following dimensionless variables u1 = u1 u0 (1) ; u2 = u2 u0 (2) ; y1 = y1 d1 ; y2 = y2 d2 ; gr = gβiδtd1 3 ν1 2 ; re = u0 (1) d1 ν1 br = μ1u0 (1)2 k1δt ; u0 (1) = −ad1 2 48𝜇1 ; u0 (2) = − ad2 2 48𝜇2 ; 197 analytical study of mixed convective flow and heat transfer… gr = gr re ; θ1 = t1 − t0 δt ; θ2 = t2 − t0 δt ; rt = t2 − t1 δt (2.12) eqs. (2.7), (2.9) becomes region −i ⅆ4u1 ⅆy4 = grbr ( ⅆu1 ⅆy ) 2 (2.13) region −ii ⅆ4u2 ⅆy4 = grbr mnbkd 4 ( ⅆu2 ⅆy ) 2 (2.14) the boundary and interface conditions are u1u0 (1) = 0 , at y = − 1 4 ; u1 (− 1 4 ) = 0 = u2 ( 1 4 ) ; u1(0) = md 2u2(0) ; ⅆ2u1 ⅆy2 = −48 + grrt 2 , at y = − 1 4 ; ⅆ2u2 ⅆy2 = −48 − gr 𝑛𝑏 rt 2 , at y = 1 4 ; ⅆu1 ⅆy = d ⅆu2 ⅆy , at y = 0 ; ⅆ2u1 ⅆy2 = 1 nb [ ⅆ2u2 ⅆy2 + 48(1 − nb)] , at y = 0 ; ⅆ3u1 ⅆy3 = 1 nbkd ⅆ3u2 ⅆy3 , at y = 0 where d = d2 d1 , m = μ1 μ2 , n = ρ2 ρ1 , b = β2 β1 , k = k1 k2 (2.15) solutions case of negligible of viscous dissipation (𝐁𝐫 = 𝟎) the solution of eqs. (2.13) and (2.14) can be obtained using eq. (2.15) in the absence of viscous dissipation, that is, when the parameter, (𝐁𝐫 = 𝟎) is given by region −i u1 = e1 + e2y + e3y 2 + e4y 3 (3.1) region −ii u2 = e5 + e6y + e7y 2 + e8y 3 (3.2) using eq. (2.12) in eqs. (2.1) and (2.3) the energy balance equations are region −i θ1 = − 1 gr [48 + ⅆ2u1 ⅆy2 ] (3.3) region −ii θ2 = − 1 nbgr [48 + ⅆ2u2 ⅆy2 ] (3.4) using the expressions obtained in eqs. (3.1) and (3.2) the energy balance eqs. (3.3) and (3.4) becomes region −i 198 mariya daimi1, s. jagtap2 θ1 = − 1 gr [48 + 2e3 + 6e4y] (3.5) region −ii θ2 = − 1 nbgr [48 + 2e7 + 6e8y] (3.6) case of negligible buoyancy force (gr = 0) when the buoyancy forces are negligible (gr = 0)and viscous dissipation is dominating (br ≠ 0),so that purely forced convection occurs. for this case, the solutions of eqs. (2.13) and (2.14) can be obtained using the eq. (2.15), the velocities are given by region −i u1 = f1 + f2y + f3y 2 + f4y 3 (3.7) region −ii u2 = f5 + f6y + f7y 2 + f8y 3 (3.8) the energy balance eqs. (2.6) and (2.8) in nondimensional form can also be written as region −i ⅆ2θ1 ⅆy2 = −br ( ⅆu1 ⅆy ) 2 (3.9) region −ii ⅆ2θ2 ⅆy2 = −br m k d 4 ( ⅆu2 ⅆy ) 2 (3.10) the boundary and interface conditions are θ1 (− 1 4 ) = − rt 2 ; θ2 ( 1 4 ) = rt 2 ; θ1(0) = θ2(0) ; ⅆθ1 ⅆy = 1 kd ⅆθ2 ⅆy , at y = 0 ; (3.11) solving eqs. (3.9) and (3.10) ,using eqs (3.7) and (3.8) we obtain region −i θ1 = −br ( g3y 2 + g4y 3 + g5y 4 g6y 5 + g7y 6 ) + g2y + g1 (3.12) region −ii θ2 = −m k 𝐷 4 br ( g10y 2+g11y 3+g12y 4 + g13y 5+g14y 6 ) +g9y + g8 (3.13) combine effects of buoyancy force and viscous dissipation by using the perturbation method, we solve eqs. (2.13) and (2.14) with a dimensionless parameter |𝜀| (< 1) defined as ε = grbr (3.14) which is independent of the reference temperature difference δt. the solutions are assumed in the form u(y) = u0(y) + εu1(y) + ε 2u2(y) + ⋯ = ∑ ε nun(y) ∞ n=0 (3.15) 199 analytical study of mixed convective flow and heat transfer… substituting eq. (3.15) in eqs. (2.13) and (2.14) and the coefficients of like powers of ɛ to obtain the zeroth and first order equations as follows region −i (zeroth -order equation) ⅆ4u10 ⅆy4 = 0 (3.16) first-order equation ⅆ4u11 ⅆy4 = ( ⅆu10 ⅆy ) 2 (3.17) region −ii (zeroth-order equation) ⅆ4u20 ⅆy4 = 0 (3.18) first-order equation ⅆ4u21 ⅆy4 = mnbk𝐷4 ( ⅆu20 ⅆy ) 2 (3.19) the corresponding boundary and interface conditions for the zeroth and first order by eq. (2.15) reduces to u10 (− 1 4 ) = 0 = u20 ( 1 4 ) ; u11 (− 1 4 ) = 0 = u21 ( 1 4 ) ; u10(0) = md 2u20(0) ; u11(0) = md 2u21(0) ; ⅆ2u10 ⅆy2 = −48 + grrt 2 , at y = − 1 4 ; ⅆ2u11 ⅆy2 = 0 , at y = − 1 4 ; ⅆ2u20 ⅆy2 = −48 − nbgrrt 2 , at y = 1 4 ; ⅆ2u21 ⅆy2 = 0, at y = 1 4 ; ⅆu10 ⅆy = d ⅆu20 ⅆy , at y = 0 ; ⅆu11 ⅆy = d ⅆu21 ⅆy , at y = 0 ; ⅆ2u10 ⅆy2 = 1 nb [ ⅆ2u20 ⅆy2 + 48(1 − nb)] , at y = 0 ; ⅆ2u11 ⅆy2 = 1 nb [ ⅆ2u21 ⅆy2 ] , at y = 0 ; ⅆ3u10 ⅆy3 = 1 nbkd ⅆ3u20 ⅆy3 , at y = 0 ; ⅆ3u11 ⅆy3 = 1 nbkd ⅆ3u21 ⅆy3 , at y = 0 ; (3.20) solutions of zeroth-order eqs. (3.16) and (3.18) using eq. (3.20) are u10 = c1 + c2y + c3y 2 + c4y 3 (3.21) u20 = b1 + b2y + b3y 2 + b4y 3 (3.22) solutions of first-order eqs. (3.17) and (3.19) using eq. (3.20) are u11 = p5y 8 + p6y 7 + p7y 6 + p8y 5 +p9y 4 + p1 6 y3 + p2 2 y2 + p3y + p4 (3.23) u21 = q5y 8 + q6y 7 + q7y 6 + q8y 5 +q9y 4 + q1 6 y3 + q2 2 y2 + q3y + q4 (3.24) using the velocities given by eqs. (3.21) (3.24) the energy balance eqs. (3.3) and (3.4) becomes 200 mariya daimi1, s. jagtap2 region −i θ1 = − 1 gr [ 48 + 2c3 + 6c4y +ε ( 56p5y 6 + 42p6y 5 + 30p7y 4 +20p8y 3 + 12p9y 2 + p1 y + p2 ) ] (3.25) region −ii θ2 = − 1 nbgr [ 48 + 2b3 + 6b4y +ε ( 56q5y 6 + 42q6y 5 + 30q7y 4 +20q8y 3 + 12q9y 2 + q1 y + q2 ) ] (3.26) heat transfer the wall heat transfer expression in terms of the nusselt number is nu− = (1 + d) ⅆθ1 ⅆy , at y = − 1 4 nu+ = (1 + 1 d ) ⅆθ1 ⅆy , at y = 1 4 nu− = − (1+d) gr [6c4 − ε( 21 64 p5 − 105 128 p6 + 15 8 p7 − 15 4 p8 + 6p9 − p1)] (3.27) nu+ = − (1+ 1 d ) nbgr [6b4 + ε( 21 64 q5 + 105 128 q6 + 15 8 q7 + 15 4 q8 + 6q9 − q1)] (3.28) 3. results and discussions investigation of laminar mixed convection flow in vertical channel filled with immiscible viscous fluid has been done analytically by using a regular perturbation method taking the product of the thermal grashof number (gr=gr/re) and brinkman number br as perturbation parameter. and solution are valid only for small values of perturbation parameter ɛ(<1). viscous dissipation term is also included in the energy equations. the flow fields are evaluated in case of asymmetric heating (rt=1) and are represented graphically in fig 2-8. the velocity and temperature fields in the absence of brinkman number (br=0) are obtained for different values of thermal grashof number (gr) and are depicted in fig.2a, fig.2b respectively. for negative value of gr velocity field increases in region-i and decreases in region ii whereas for positive values of gr velocity decreases in region i and increases in region ii. one can also observe that flow was an increasing function for value gr (>0) and decreasing function for gr (<0). but temperature field decreases in both the regions for all different values of gr. 201 analytical study of mixed convective flow and heat transfer… the dimensionless temperature field θ is obtained and is shown in fig.3 for different values of brinkman number br in case of negligible buoyancy force (gr=0). as brinkman number increases temperature field is also increases in both regions. the velocity and temperature fields are obtained at gr=±500 for different values of ɛ and are shown in fig.4a, fig,4b respectively. the velocity field is an increasing function of ɛ for upward flow ɛ (>0) and decreasing function of ɛ for downward flow ɛ(<0). whereas temperature field increases for both ɛ (>0) and ɛ (<0). the perturbation parameter ɛ is more effective on velocity field than temperature field. from fig.4a it can also pointed out that at cold (left) and hot (right) walls reversal flow occurs for upward and downward flow respectively. the effect of viscosity ratio m, width ratio d and conductivity ratio k on flow field evaluated for the values of gr =100, and ɛ=0.01 in case of asymmetric heating (rt=1). the effect of viscosity ratio m on the velocity and temperature fields are shown in fig.5a, fig.5b respectively. as the viscosity ratio m increases the flow field increases in both regions. temperature increases from cold to hot walls for all values of m. the effect of width ratio d on the velocity and temperature fields are shown in fig.6a, fig.6b respectively. effect of the width ratio is similar to effect of viscosity ratio on flow field as d increases both velocity and temperature fields increases. the effect of conductivity ratio k on the fields of velocity and temperature are depicted in fig.7a, fig.7b respectively. as k increases, the velocity and temperature fields reduce in both the regions, this means larger the conductivity ratio smaller the flow rate is. the nusselt number at the cold wall (nu-) and hot wall (nu+) for |ɛ| is shown in fig.8.the nuis an increasing function of |ɛ| and nu+ is a decreasing function of |ɛ| fig. 2a. velocity profiles for different values of gr 202 mariya daimi1, s. jagtap2 fig. 2b. temperature profiles for different values of gr fig. 3. temperature profile for different values of br 203 analytical study of mixed convective flow and heat transfer… fig. 4a. velocity profile for different values of ɛ. fig. 4b. temperature profile for different values of ɛ. 204 mariya daimi1, s. jagtap2 fig. 5a. velocity profile for different values of m. fig. 5b. temperature profile for different values of m. 205 analytical study of mixed convective flow and heat transfer… fig. 6a. velocity profile for different values of d. fig. 6b. temperature profile for different values of d. 206 mariya daimi1, s. jagtap2 fig. 7a. velocity profile for different values of k fig. 7b. temperature profile for different values of k 207 analytical study of mixed convective flow and heat transfer… fig. 8. nusselt number vs ∣ɛ∣. 4. conclusions the problem of mixed convective flow and heat transfer in vertical channel filled with immiscible viscous fluids was analyzed analytically by regular perturbation method and represented graphically. the conclusions made are, the flow was an increasing function of perturbation parameter ɛ for upward flow and decreasing function of ɛ for downward flow, viscosity ration m and width ratio d enhance the velocity and temperature fields where as the larger the value of conductivity ratio k, smaller the fields of velocity and temperature. references [1] aung w., worku g., developing flow and flow reversal in mixed convection in vertical channel with asymmetric wall temperatures, j. heat transfer, 108, 485-488, 1986. [2] basant k. jha, michael o. oni., mixed convection flow in vertical channel with temperature dependent viscosity and flow reversal: an exact solution, international journal of heat and technology, 36,607-613,2018. [3] barletta a., laminar mixed convection with viscous dissipation in a vertical channel, international journal of heat mass transfer, 41,3501-3513, 1998. 208 mariya daimi1, s. jagtap2 [4] d. srinivasacharya, md. shafeeurrahaman, mixed convention flow of nanofluid in a vertical channel with hall and ion-slip effects, fhmt, 8-11,2017. [5] desai c., vafai k., three-dimensional buoyancy induced flow and heat transfer around the wheel outboard of an aircraft, international journal of heat fluid flow,13,50-64,1992. [6] j.c. umavati, ali j. chamkha, fully developed mixed convection in a vertical channel in the presence of heat source or heat sink, international journal of energy and technology,3(24),1-9,2011. [7] j. prathap kumar, j.u. umavati and shreedevi kalyan, chemical reaction effects on mixed convection flow of two immiscible viscous fluid in a vertical channel, hmmt, 2(2), 28-46.2014. [8] j. prathap kumar, j.u. umavati and y. ramarao, mixed convective heat transfer of immiscible fluids in a vertical channel with boundary conditions of the third kind, computational thermal science, 9(5),447-465,2017. [9] m. rashidi, d. ganji, homotopy perturbation combined with padeꞌ approximation for solving two-dimensional viscous flow in the extrusion process, international journal of nonlinear science,7,387-394,2009. [10] tao, combined free and force convection in channels, asme journal of heat transfer,82,233-238,1980. [11] t.t hamadah, r. a. wirtz, analysis of laminar fully developed mixed convection in a vertical channel with opposing buoyancy, asme journal of heat transfer,113,507510,1991. 209 ratio mathematica 28 (2015) 31-43 issn:1592-7415 influence of information on behavioral effects in decision processes angelarosa longo, viviana ventre dipartimento di diritto, economia, management e metodi quantitativi università degli studi del sannio, benevento, italy e-mail address ventre@unisannio.it abstract rational models in decision processes are marked out by many anomalies, caused by behavioral issues. we point out the importance of information in causing inconsistent preferences in a decision process. in a single or multi agent decision process each mental model is influenced by the presence, the absence or false information about the problem or about other members of the decision making group. the difficulty in modeling these effects increases because behavioral biases influence also the modeler. behavioral operational research (bor) studies these influences to create efficient models to define choices in similar decision processes. keywords: behavioral operational research, intertemporal choice, information in decision processes. 2000 ams subject classifications: 91b16, 91b08, 91e10. doi: 10.23755/rm.v28i1.26 1 introduction some failures occur when dealing with traditional discounted utility (du) model from both normative and descriptive setting. indeed, some studies, especially in psychology and neuroeconomics (a more specialized field of decision neuroscience), point out anomalies that violate some axioms in the traditional model (sec. 2). bechara et al. [1] show positive effects of anomalies in intertemporal choices and the use of hyperbolic delay discounting (declining as the length of the delay 31 a. longo and v. ventre increases) to represent inconsistent preferences. on the other hand, the negative effects of emotions mainly stem from impulsivity. to properly describe individual differences in intertemporal choices, derived from impulsivity and inconsistency, behavioral economists have proposed the q-exponential delay discount function and a multiple selves model (quasi-hyperbolic discount model) (sec. 3). mental models of each person, based on different assumptions and preferences, influence the effects of emotions (positives and negatives) and impulsivity. to control impulsivity strotz [11] proposed two strategies that might be employed by a person who foresees how his preferences will change over time, and thaler and shefrin [12] proposed a model in which the individual is treated as if he contained two distinct psyches denoted as planner and doer (sec. 4). the information held by the agent plays an important role on the delineation of his mental model. in a multi agent decision context all people involved have their mental model and influence other mental models. hence, in these strategic decisions information about others and about the problem definitely influence the final choice (sec. 5). if there is no information about other players, as shown in an experiment of engelmann and strobel [4], people weight their own decisions more heavily than that of a randomly selected person from the same population (false consensus effect). this happens in non-cooperative decision problems, not properly modeled in or. indeed, because false consensus effect and impulsivity not always lead each agent towards the best strategy according to the theory of games, so obtaining a common decision is only a chance (sec. 6). on the contrary, when all information is explicit people can consider the choices of others as more informative then their own (excess of consensus or overconfidence). an example is a cooperative decision problem, modeled by or with cooperative games, in which final decision is based on mental models of the participants and their tendency to overconfidence (sec. 7). however, the final decision is influenced not only by intrinsic characteristics of every one, but also by the way in which information is passed: misunderstandings and manipulations (above all for self-interest) change people’s reactions (sec. 8). at last, also the modeler is influenced by his mental models: creating a model to predict a decision making process is itself a decision making process. a new branch of research (bor, behavioral operational research) studies human impacts of using or models in decisional processes (sec. 9). 32 influence of information on behavioral effects in decision processes 2 effects of behavioral aspects: violations of traditional discount utility model operational research (or) has modeled human behavior in intertemporal choice in terms of du model, which assumes an exponential temporal discounting function and a constant discount rate: this represents the individual’s pure rate of time preference. an important implication of constant discount rate and exponential discounting function is that a person’s intertemporal preference is timeconsistent. however, decision neuroscience, whose goal is to integrate research in neuroscience and behavioral decision-making, highlights that there are a number of behavior patterns that violate rational choice theory. several empirical studies on individual behavior, when discounting real or hypothetical rewards, stress the existence of violations of the traditional discounting model [2]. theory and algorithms of or models are free of behavioral effects but as soon as we use them in real life problem solving behavioral effects will be present. hence behavioral perspective is essential in decision analysis [6]. research in psychology has reported many types of cognitive and motivational biases as well as heuristics which relate to human behavior and may significantly distort the decision analysis generating inconsistent preferences in intertemporal choices. delay effect, magnitude effect and sign effect are among the relevant anomalies in intertemporal choice, we will deal with (see also [13]). the delay effect. as waiting time increases, the discount rates tend to be higher in short intervals than in longer ones. we can set out this effect as follows: (x,s) ∼ (y,s) but (x,s + h) < (y,t + h), for y > x,s < t and h > 0 the magnitude effect. larger outcomes are discounted at a lower rate than smaller outcomes. this effect can be formulated as follows: (x,s) ∼ (y,s) implies (ax,s) < (ay,t), for y > x > 0,s < t and (−x,s) ∼ (−y,s) implies (−ax,s) > (−ay,t) the sign effect. gains are discounted at a higher rate than losses of the same magnitude. this anomaly implying that, changing the sign of an amount from gains to losses, the weight of this amount increases: (x,s) ∼ (y,s) implies (−x,s) > (−y,t) for y > x > 0 and s < t 33 a. longo and v. ventre 3 effects of emotions: violations of traditional discount utility model in a series of studies (see, e. g., ([1], [3], [8])) using a gambling task, it emerges that individuals with emotional dysfunction tend to perform poorly compared with those who are endowed with intact emotional processes. bechara et al. [1] demonstrated that normal people possess anticipatory scrs (skin conductance response) – indices of somatic states – which represent unconscious biases that are linked to prior experiences with reward or punishment and produce inconsistent preferences. these biases alarm the normal subject about selecting a disadvantageous course of action, even before the subject becomes aware of the goodness or badness of the choice he is about to make. as a consequence there is considerable agreement among psychologists and economists that the notion of exponential discounting should be replaced by some form of hyperbolic discounting, which can point out the delay effect (or present bias), that is the tendency of the individuals to increasingly choose a smaller-sooner reward over a larger-later reward as the delay occurs sooner in time. many authors proposed different hyperbolic discount functions, in which temporal discount function increases with the delay to an outcome. one of these proposed functions has the following form: d(t) = ( 1 1 + αt )β/α where β > 0 is the degree of discounting and α > 0 is the departure from exponential discounting. hyperbolic discounting has been applied to a wide range of phenomena, including consumption-saving behavior. consistent with hyperbolic discounting, people’s investment behavior exhibits patience in the long run and impatience in the short run [13]. a second type of empirical support for hyperbolic discounting comes from experiments on dynamic inconsistency. studies and empirical evidences show that delay effect can derive in preference reversal between two rewards as the timedistance to these rewards diminishes. a hyperbolic discount model can clarify this; in fact, hyperbolic time-preference curves can cross [11] and consequently the preference for one future reward over another may change with time [13]. however, in some contexts individuals deprived of normal emotional reactions might actually make better decisions than normal individuals, because of the loss of self-control, as damasio found when studying behavior of people with ventromedial prefrontal damage [3]. temptations are manifestations of loss of self-control and in many cases induce disadvantageous behavior. indeed, as far as temptation increases the best long run interest of the problem solver conflicts 34 influence of information on behavioral effects in decision processes with his short run desires, moreover impulsive behavior may fail to evaluate the consequences of his behavior appropriately [13]. other evidences suggest that even relatively mild negative emotions that do not result in a loss of self-control can play a counterproductive role among normal individuals in some situations. when gambles that involve some possible loss are presented one at a time, most people display extreme levels of risk aversion toward the gambles, a condition known as myopic loss aversion. shiv et al. [9] show that individuals deprived of normal emotional reactions might, in certain situations, make more advantageous decisions than those not deprived of such reactions; so the lack of emotional reactions may lead to more advantageous decisions. inconsistent preference is the greatest contradiction of rational theory in intertemporal choice. this behavior can be typically seen in psychiatric disorders (alcoholism, drug abuse), but also in more ordinary phenomena (overeating, credit card debt) [13]. neuroeconomics has found that addicts are more myopic (have large time-discount rates) in comparison with non-addicted population. however, the preference for more immediate rewards per se is not always irrational or inconsistent; addicts’ behavior is clinically problematic, but economically rational when their choices are time-consistent (if they have large discount rates with an exponential discount function). but addicts also discount delayed outcomes hyperbolically, suggesting the intertemporal choices of addicts are timeinconsistent, resulting in a loss of self-control: they act more impulsively at the moment of the choice, against their own previously intended plan. moreover if large discount rates are due to habitual drug intake, it is expected that discount rates decreased after long-term abstinence. behavioral neuroeconomics and econophysical studies have proposed two discount models, in order to better describe the neural and behavioral correlates of impulsivity and inconsistency in intertemporal choice. q-exponential discount model. this function has been proposed and examined for subjective value v (d) of delayed reward: v (d) = a expq(kqd) = a/[1 + (1 − q)kq]d 1 1−q where d denotes a delay until receipt of a reward, a the value of a reward at d = 0, and kq a parameter of impulsivity at delay d = 0 (q-exponential discount rate) and the q-exponential function is defined as: expq(x) = (1 + (1 − q)) 1 1−q the function can distinctly parametrize impulsivity and inconsistency [13]. 35 a. longo and v. ventre quasi-hyperbolic discount model. behavioral economists have proposed that the inconsistency in intertemporal choice is attributable to an internal conflict between “multiple selves” within a decision maker. as a consequence, there are (at least) two exponential discounting selves (with two exponential discount rates) in a single human individual; and when delayed rewards are at the distant future (> 1 year), the self with a smaller discount rate wins, while delayed rewards approach to the near future (within a year), the self with a larger discount rate wins, resulting in preference reversal over time. this intertemporal choice behavior can be parametrized in a quasi-hyperbolic discount model (also as a β − δ model). for discrete time τ (the unit assumed is one year) it is defined as: f(τ) = βδt (for τ = 1,2,3, . . .) and f(0) = 1 (0 < β < δ < 1) a discount factor between the present and one-time period later (β) is smaller than that between two future time-periods (δ). in the continuous time, the proposed model is equivalent to the linearly-weighted two-exponential functions (generalized quasi-hyperbolic discounting): v (d) = a[w exp(−k1d) + (1 − w) exp(−k2d)] where w, 0 < w < 1, is a weighting parameter and k1 and k2 are two exponential discount rates (k1 < k2). note that the larger exponential discount rate of the two k2, corresponds to an impulsive self, while the smaller discount rate k1 corresponds to a patient self [13]. 4 mental models: self-control against impulsivity behavioral issues fit in each phase of the problem solving process, both if it is a single agent decision process or a multi agent one. every individual choice is influenced by impulsivity and by all positive and negative biases derived from it. the impulsive choices derived from mental models, which are informal models, quickly constructed by problem solvers, which go on constantly during problem solving. mental models help us to relate cause and effect, but often in a highly simplified and incomplete way. they are always influenced by our preferences and our personal experiences. so they can be extremely limiting. this explains why emotions do not have always positive or negative effects on decision process and why impulsivity generates sometimes positive and sometimes negative effects. strotz proposed two strategies that might be employed by a person who foresees how his preferences will change over time [11]: 36 influence of information on behavioral effects in decision processes 1) the “strategy of precommitment”: a person can commits to some plan of action; 2) the “strategy of consistent planning”: an individual take into account future changes in the utility function and reject any plan that he will not follow through. his problem is then to find the best plan among those she will actually follow. hyperbolic discounting predicted a number of mechanisms of self-control. however, the hyperbolic model, as well as the exponential one, is only a special case of interpreting reality. common sense highlights how people, when are in front of identical short term opportunities, perform only sometimes self-control, independently of the use of one’s strotz strategy. in the setting of multiple selves models, to control impulsivity, thaler and shefrin proposed a “planner-doer” model which draws upon principal agent theory [12]. they deal with an individual as if he contained two distinct psyches: one planner, which pursue longer-run results; and multiple doers, that are concerned only with short-term satisfactions, so they care only about their own immediate gratification (and have no affinity for future or past doers). for example, consider an individual with a fixed income stream, where which has to be allocated over the finite interval (0,t). the planner would choose a consumption plan to maximize his utility function v (z1,z2, . . . ,zt) subject to t∑ t=1 ct ≤ y in which is a utility function of consumption level in t. on the other hand, an unrestrained doer 1 would borrow y − y1 on the capital market and therefore choose c1 = y ; the resulting consequence is naturally c2 = c3 = · · · = ct = 0. such action would suggest a complete absence of psychic integration. the model proposes two instruments that the planner can use to control the behavior of the doers: (a) he can impose rules on the doers’ behavior, which operate by altering the constraints imposed on any given doer; or (b) he can use discretion accompanied by some method of altering the incentives or rewards to the doer without any self-imposed constraints [13]. 5 role of information in decision process in many decision processes the information held by the agent and the way in which they are represented play an important role, above all in multi agent 37 a. longo and v. ventre decision problems, in which all the people involved have their intrinsic mental models, intentions, expectations and cultural habits, and emotions of each agent can be contagious and influence group behavior, modifying their mental models. in this process the way the interaction and communication is carried out becomes important and has an effect on the dynamics of the problem solving process. an or process can get opposite results depending on the way the phenomenon is described and how the questions are phrased and graphs used. this can influence the behavior and preferences of the participants. as a result, we need to pay attention to the way we communicate. in a multi agent decision model the influence of communication depends, first of all, on whether the information is absent or present. 6 false consensus effect for lack of information in a non-cooperative decision problem as observed in social psychology, people with a certain preference tend to make higher judgments of the popularity of that preference in others, compared to the judgments of those with different preferences. this empirical result has been termed the false consensus effect. consequently, as pointed out in several experiments, in a multi agent decision problem each decision maker overestimates his own opinion. however, this effect becomes more pervasive when people lack necessary data to base their judgments [10] about the choice of other members of their own group, there are influences in opposite direction to a false consensus effect, while results of experiment are in line with a false consensus effect in all groups in which the information were implicit. this shows that most subjects are unwilling or unable to use information that is not handed to them on a silver platter [4]. as a consequence, in multi agent decision problem without information about others members and about the problem, the false consensus effect produces partial objectivity and incomplete impartiality [10]. mathematical instrument used to describe strategic interactions, as a multi agent decision problem, is the theory of games, and a non-cooperative game can be assimilated to situations in which information about decision of other members of decision group is absent, so implicit. in this kind of interaction it is not possible to implement some precommitment to control the doer’s actions (the impulsive part that represents the effects of emotions), as a consequence it is not possible to recognize the best choice on a rational base [7]. if we analyze a non-cooperative multi agent decision problem like the traditional prisoner’s dilemma, on one temporal interval and with only two alternatives, 38 influence of information on behavioral effects in decision processes we note that the agents achieve common decision, and this is the best strategy, because each doer wants to obtain the higher advantage which is the same and, for the false consensus effect, each one thinks that other make the same. the doer of each prisoner will choose the strategy of “do not confess”. in the traditional version of the game, the police arrest two suspects (a and b) and interrogate them in separate rooms. each can either confess, thereby implicating the other, or keep silent. in terms of years in prison the payoff for each strategy are these: agent a confess (c) do not confess (nc) agent b confess (c) 5,5 10,0 do not confess (nc) 0,10 1,1 according to the theory of games, given this set of payoffs, in absence of information there is a strong tendency for each to confess (optimal decision in terms of pareto), implying two rational players with consistent preferences. this creates the paradoxical situation that rational players lead to a poorer outcome than irrational players. actually, when each player has to choose the best strategy every doer drives his agent to make decision that leads him a greater advantage, believing that the other will do the same due to the effect of the false consensus. consequently, the decision made by each leads to optimal decision in terms of pareto, because both have the same utility function and both doers choose the only action that is the best strategy. however, it is just a coincidence that the two players have achieved a common strategy. in other types of non-cooperative problems this can not happen, with the result that you will never achieve a joint decision without a prior agreement, if there is no information. consider, for example, a multi agent decision problem in which the agents set to save money to realize a common purchase. even agent has a fixed income, ya and yb and a nonnegative level of saving, sa and sb. the planner of each agent choose the best strategy which maximize his utility function of saving (thinking for future), but the present doer of each agent wants to obtain the highest advantage now, so it would consume y and therefore choose= 0, with a degree= 1. indeed, the doers are impulsive, each one assigns weight= 1 to one preference and weight= 0 to all the others, thinking that everybody will make in the same way for effect of false consensus. in this case it is not possible obtain a common decision. the plan made in advance by a group of agents (to realize a common purchase) is not feasible if they don’t set some rule or some method to alter the incentives for the doers. this type of problem can be represented in the following scheme: 39 a. longo and v. ventre agent a save (s) do not save (ns) agent b save (s) 10,10 5,5 do not save (ns) 5,5 -10,-10 where the payoff represent the utility of each agent for each strategy. according to the rational choice, the nash equilibrium coincides with the best strategy (s,s). however false consensus effect and impulsivity lead each agent to the worst equilibrium, because utility functions of the agents are different among them (each agent prefers consumptions to savings). this causes the lack of consensus on a common decision. in conclusion, in a non-cooperative multi agent decision problem, there are two situations: 1) the doers of each agent have the same preference and they will reach a common decision that is given by the unanimous choice (doers don’t affect), 2) the doers have different preferences and do not assign any weight to the other preferences, so it is not possible to aggregate the preferences. hence, we can affirm that in a non-cooperative decision problem it is only a chance obtaining a common decision. 7 excess of consensus in a cooperative decision problem according to engelmann and strobel’s experiment there is no false consensus effect if representative information is highly prominent and retrievable without any effort. on the contrary, there is a significant effect in the opposite direction, indicating that subjects consider others’ choices more informative than their own [4]. this is the overconfidence o “groupthink”, a psychological phenomenon which can occur in highly trained cohesive groups. hence, in the extreme case in which all is known in decision making process, the interplay between different subjects involves anyway other behavioral effects, as the excess of consensus, apart from influence of mental models and all behavioral effects of each individual. in the or field this kind of decision making process can be modeled with cooperative games where the rationality of the equilibrium choice is saved by the possibility of making an agreement among agents, which represents a pure rule to maintain self-control at later time in thaler and shefrin’s model. moreover with an arrangement the agents have explicit information about the choices of 40 influence of information on behavioral effects in decision processes other members, so the lack of false consensus effect is in line with the result of engelmann and strobel’s experiment. however, only the decision of one member will prevail, and this is influenced by the strength of each mental model. an example of cooperative game is a coordination game, when players choose the strategies by a consensus decision making process [7]. consider the classic example of coordination game: the “battle-of-the sexes”. in this game an engaged couple must choose what to do in the evening: the man prefers to attend a baseball game and the woman prefers to attend an opera. in terms of utility the payoff for each strategy is as follows: man opera (o) baseball (b) woman opera (o) 3,1 0,0 baseball (b) 0,0 1,3 in this example there are multiple outcomes that are equilibriums: (b,b) and (o,o). however both players would rather do something together than go to separate events, so no single individual has an incentive to deviate if others are conforming to an outcome. in this context, a consensus decision making process can be considered as an instrument to choose the best strategy in a coordination game. a common final decision is achievable only if the man and the woman have explicit information, then only if there is cooperation [7]. if we follow the thaler and shefrin’s model, we can analyze choices in a cooperative game in this way: at period-one the planner of each agent states his preference, which is the best strategy because the planner wants to maximize his utility function. then, the influence of doers, that want to obtain an immediate gratification, can be avoided, because agents can enforce contracts through parties at period-one, what eliminate the problem of loss of self-control, because they eliminate all choices. however, only one will maximize his utility function and this is not known in advance because it depends on the different strength of each mental models. 8 strategic communication modify behavioral effects in a multi agent decision problem, information held by the participants can be wrong for two causes, not independent of each other: 1) misunderstandings and 2) manipulations. 41 a. longo and v. ventre in the first case, the different reaction of people at the same problem with the same information do not reflect people’s lack of cognitive abilities but the way the situation is described in the communication. in the second case, there is the will to misrepresent the problem for self-interest, from one member of the group. human cognitive processes relate strongly to motivational issues which interplay between people in social contexts. self-interest is the primary cause of biases especially in participatory processes with multiple stakeholders. self-interest is the driver of strategic behavior, which produce above all priming and framing effects. as a result, some biases can be unintentional consequences of cognitive limitations, others can be motivated by omissions or over or under emphasization of aspects, strategically or not [6]. 9 bor: behavioral operational research in considering the behavioral effects we should take a humble approach and accept the fact that we are not likely be able to produce a “perfect” model but still could find one that is useful. however, modeling is not about models only, but it matters how we choose the models and how we work with the models [6]. creating a model to manage or solve problems is a process composed of many phases and human behavior moderates each stage of the process and mediates the progression through stages [5]. hence the behavioral lens needs to be integrated in the practice of or as an additional perspective. behavioral operational research (bor) considers the human impact on the process of using operational research (or) methods in problem solving and decision support as well as using or methods to model human behavior [6]. not only decision makers but also modelers are subject to cognitive and motivational biases and the way the decision problems are framed. moreover, the cumulative effects of biases in a modeling process can also result in path dependency (a phenomenon where the order in which steps are taken in the modeling process can have an impact on the resulting model). in large models the initial modeling choices can be very hard to change later and these can have a crucial impact on the path the modeling process will proceed. the loss aversion effect in decision making can also have an effect on modeling in general. theoretically it can be equivalent to use and label variables as losses or gains but in the interpretation of the model results there can be a difference. a somewhat related effect is the so called action bias where people choose to foster improvement rather than prevent deterioration [6]. also communication is an important part of modeling. the modeler should not only be focused only on the perfection of the accuracy of the model, the process and communication counts a lot too. modelers should recognize the possibility of strategic behavior of the participants. such behavior 42 influence of information on behavioral effects in decision processes can mean, for example, the misrepresentation of preferences or data in an environmental participation process. finally, also modelers are guided by self-interest. the purpose for which the model is developed is reflected in the parameters and scales as well as in the level of detail used. there is not a single valid model fitting every purpose [6]. hence biases exist in each aspect of a problem solving process and in each phase of modeling of these processes, however finding ways to avoid them is an open research field. references [1] a. bechara, h. damasio, d. tranel and a. r. damasio, deciding advantageously before knowing the advantageous strategy, science 275, (1997),1293-1295. [2] a. bechara, the role of emotion in decision-making: evidence from neurological patients with orbitofrontal damage, brain and cognition 55, (2004), 30-40. [3] a. r. damasio, descartes’ error: emotion, reason, and the human brain, grosset/putnam, new york, 1994. [4] d. engelmann and m. strobel, the false consensus effect: deconstruction and reconstruction of an anomaly, cerge-ei working paper 233, (2004). [5] r. p. hamalainen, j. luoma and e. saarinen, on the importance of behavioral operational research: the case of understanding and communicating about dynamic systems, european journal of operational research 228, (2013), 623-634. [6] r. p. hamalainen, behavioral issues in environmental modeling the missing perspective, environmental modelling and software 73, (2015), 244253. [7] a. longo, m. squillante, a. g. s. ventre and v. ventre, the intertemporal choice behaviour the role of emotions in a multi-agent decision problem, atti accad. pelorit. pericol. cl. sci. fis. mat. nat., 93 (2), c* (2015) [14 pages]. [8] m. c. nussbaum, upheavals of thought. the intelligence of emotions, cambridge university press, cambridge, 2001. [9] b. shiv, g. loewenstein, a. bechara, h. damasio and a. r. damasio, investment behavior and the negative side of emotion, psychological science 16, (2005), 435-439. 43 a. longo and v. ventre [10] m. squillante and v. ventre, assessing false consensus effect in a consensus enhancing procedure, international journal of intelligent system 25, (2010), 274-285. [11] r. h. strotz, myopia and inconsistency in dynamic utililty maximization, review of economic studies 23(3), (1955-1956), 165-80. [12] r. thaler and h. m. shefrin, an economic theory of self-control, journal of political economy 89(2), (1981), 392-406. [13] a. g. s. ventre and v. ventre, the intertemporal choice behavior: classical and alternative delay discounting models and control techniques, atti accad. pelorit. pericol. cl. sci. fis. mat. nat. 90 (1), c3, (2012) [21 pages]. 44 ratio mathematica volume 47, 2023 detour global domination for degree splitting graphs of some graphs c. jayasekaran* s.v. ashwin prakash† abstract in this paper, we introduced the new concept detour global domination number for degree splitting graph of standard graphs. a set s is called a detour global dominating set of g = (v, e) if s is both detour and global dominating set of g. the detour global domination number is the minimum cardinality of a detour global dominating set in g. let v (g) be s1 ∪ s2 ∪ · · · ∪ st ∪ t, where si is the set having at least two vertices of same degree and t = v (g) − ∪si, where 1 ≤ i ≤ t. the degree splitting graph ds(g) is obtained from g by adding vertices w1, w2, · · · , wt and joining wi to each vertex of si for i = 1, 2, · · · , t. in this article we recollect the concept of degree splitting graph of a graph and we produced some results based on the detour global domination number for degree splitting graph of path graph, cycle graph, star graph, bistar graph, complete bipartite graph and complete graph. keywords: detour set, dominating set, detour domination, global domination, detour global domination, degree splitting graphs. 2020 ams subject classifications: 05c12, 05c69.1 *associate professor, department of mathematics, pioneer kumaraswamy college, nagercoil 629003, tamil nadu, india; jayacpkc@gmail.com. †research scholar, department of mathematics, pioneer kumaraswamy college, nagercoil 629003, tamil nadu, india; ashwinprakash00@gmail.com. affliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627012, tamil nadu, india. 1received on october 10, 2022. accepted on may 1, 2023. published online on may 10, 2023. doi: 10.23755/rm.v39i0.871. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 290 c. jayasekaran, s.v. ashwin prakash 1 introduction by a graph g = (v, e) we mean a finite, connected, undirected graph with neither loops nor multiple edges. the order |v | and size |e| of g are denoted by p and q respectively. for graph theoretic terminology we refer to west[1]. for vertices x and y in a connected graph g, the detour distance d(x, y) is the length of a longest x − y path in g[2]. an x − y path of length d(x, y) is called an x − y detour. the closed interval id[x, y] consists of all vertices lying on some x − y detour of g. for s ⊆ v, id[s] = ∪x,y∈sid[x, y]. a set s of vertices is a detour set if id[s] = v , and the minimum cardinality of a detour set is the detour number dn(g). a detour set of cardinality dn(g) is called a minimum detour set [3]. a set s ⊆ v (g) in a graph g is a dominating set of g if for every vertex v in v -s, there exists a vertex u ∈ s such that v is adjacent to u. the domination number of g, denoted by γ(g), is the minimum cardinality of a dominating set of g[4]. the complement g of a graph g also has v (g) as its point set, but two points are adjacent in g if and only if they are not adjacent in g. a set s ⊆ v (g) is called a global dominating set of g if it is a dominating set of both g and g[5]. a vertex of degree 0 is called an isolated vertex and a vertex of degree 1 is called an end vertex or a pendant vertex. a vertex that is adjacent to a pendant vertex is called a support vertex. definition 1.1. let g = (v, e) be a connected graph with atleast two vertices. a set s ⊆ v (g) is said to be a detour global dominating set of g if s is both detour and global dominating set of g. the detour global domination number, denoted by γ̄d(g) is the minimum cardinality of a detour global dominating set of g and the detour global dominating set with cardinality γ̄d(g) is called the γ̄d-set of g or γ̄d(g)-set.[6] in [7], r. ponraj and s. somasundaram have initiated a study on degree spliting graph ds(g) of a graph g which is defined as follows: definition 1.2. let g = (v, e) be a graph with v (g) = s1 ∪ s2 ∪ · · · ∪ st ∪ t, where si is the set having at least two vertices of same degree and t = v (g) − ∪si, where 1 ≤ i ≤ t. the degree splitting graph ds(g) is obtained from g by adding vertices w1, w2, · · · , wt and joining wi to each vertex of si for i = 1, 2, · · · , t. example 1.1. in figure 1.1, a graph g and its degree splitting graph ds(g) are shown. 291 detour global domination for degree splitting graphs of some graphs g ds(g) figure 1.1 v1 v2 v3 v4 v5 v6 v8 v7 v1 v2 v3 v4 v5 v6 v8 v7 w3 w1 w2 here, s1 = {v1, v7, v8}, s2 = {v2, v6}, s3 = {v4, v5} and t = {v3}. remark 1.1. if v (g) = ∪si, 1 ≤ i ≤ t, then t = ϕ. 2 some basic results in this section, we recall some basic results of detour global domination number of a graph which will be used throughout the paper. theorem 2.1. [6] every isolated vertex of g belongs to every detour global dominating set of g. theorem 2.2. [6] every full vertex of a connected graph g of order p belongs to every detour global dominating set of g. theorem 2.3. [6] for the path graph pp, (p ≥ 4), γ̄d(pp) = γd(pp) = ⌈p+23 ⌉. theorem 2.4. [6] for any star graph k1,n−1, (n ≥ 2), γ̄d(k1,n−1) = n. theorem 2.5. [6] for any bistar graph bm,n, m, n ≥ 1, γ̄d(bm,n) = m + n. theorem 2.6. [6] for m, n ≥ 2, γ̄d(km,n) = 2. 292 c. jayasekaran, s.v. ashwin prakash 3 degree splitting graphs of known graphs and their detour global domination number let us find detour global domination number of degree splitting graph ds(g) of the graphs path, cycle, star, bistar, complete bipartite and complete graph. theorem 3.1. for any integer n ≥ 3, γ̄d(ds(pn)) = 2. proof. let v1v2 · · · vn be the path pn with partitions s1 = {v2, v3, · · · , vn−1} and s2 = {v1, vn}. to obtain ds(p3) from p3 we add x, which corresponds to s2 also p3 is isomorphic to c4 and to obtain ds(pn) for n ≥ 4 we add x1 and x2, which corresponds to s1 and s2, respectively. as a result, v (ds(p3)) = {x, v1, v2, v3} and v (ds(pn)) = {x1, x2, v1, v2, · · · , vn}, where |v (ds(pn))| = n + 2 for n ≥ 4. pn ds(pn) figure 3.1 v1 v2 v3 vn−1 vn v1 v2 v3 vn−1 vn x2 x1 consider id[x, v1] for n = 3, which has only one x − v1 detour path of length 3 that contains all the vertices of ds(p3). as a result, s = {x, v1} is a minimum cardinality detour set. also, x dominates s2 and v2 is dominated by v1. thus, s is a minimum detour dominating set of ds(p3). moreover in ds(p3), x dominates v2 and v3 is dominated by v1. hence, s is a minimum detour global dominating set of ds(p3) and so γ̄d(ds(p3)) = 2. consider id[x1, x2] for n ≥ 4, which has two distinct x1 − x2 detour path of length n that contains all the vertices of ds(pn). as a result, s = {x1, x2} is a minimum cardinality detour set. also, x1 dominates s2 and x2 dominates 293 detour global domination for degree splitting graphs of some graphs s1. thus, s is a detour dominating set of ds(pn). moreover in ds(pn), x1 dominates s1 and x2 dominates s2 and hence s is a detour global dominating set ds(pn). hence, we conclude that for n ≥ 3, γ̄d(ds(pn) = 2. theorem 3.2. for any integer n ≥ 3, γ̄d(ds(cn)) = 3. proof. let v1v2 · · · vn be the cycle cn. to obtain ds(cn) for n ≥ 3 we add a vertex x which is adjacent to every vertices in cn. as a result, v (ds(cn)) = {x, v1, v2, · · · , vn}, where |v (ds(cn))| = n + 1 for n ≥ 3. clearly, ds(cn) is isomorphic to the wheel graph wn. cn ds(cn) figure 3.2 vn v1v5 vn v1v5 v2 v3 v4 v2 v3 v4 x since, ds(cn) is a connected graph, then γ̄d(ds(cn)) ≥ 2. consider the set s = {x, vi}, where 1 ≤ i ≤ n. clearly, the x − vi detour path of ds(cn) contains all the vertices of ds(cn) and also, x dominates every vertices of cn. hence, s is a minimum detour dominating set of ds(cn). but in ds(cn), the neighbours of vi are not dominated by any vertices of s. so, choose vi+1 from 294 c. jayasekaran, s.v. ashwin prakash ds(cn) then vi, vi+1 and x dominates every vertices from ds(cn). therefore, s1 = s ∪ {vi+1} = {v, vi, vi+1} is a minimum detour global dominating set of ds(cn) and hence γ̄d(ds(cn)) = 3. theorem 3.3. for any integer n ≥ 2, γ̄d(ds(k1,n)) = 2. proof. let v1, v2, · · · , vn−1 are the end vertices and v is the full vertex of the star k1,n−1 and x be the corresponding vertex which is added to obtain the graph ds(k1,n). then v (ds(k1,n)) = {v, v1, v2, · · · , vn, x}. clearly, |v (ds(k1,n−1)| = n + 2. k1,n ds(k1,n) figure 3.3 v v1 v2 v3 vn v v1 v2 v3 vn x since, ds(k1,n) is connected, then 2 ≤ γ̄d(ds(k1,n)) ≤ n + 1. consider s = {v, vi} for some i, 1 ≤ i ≤ n. then there are n− detour path which travels between v and vi that includes all the vertices of ds(k1,n). therefore, s = {v, vi} is a detour set of minimum cardinality. moreover, v dominates every vi and v itself in ds(k1,n) and vi dominates x in ds(k1,n) for 1 ≤ i ≤ n. now consider in ds(k1,n) where v dominates x and vi dominates all other vj for 1 ≤ j ̸= i ≤ n. this concludes that s is a minimum detour global dominating set of ds(k1,n) and hence γ̄d(ds(k1,n)) = 2. theorem 3.4. for the bistar graph bm,n, γ̄d(ds(bm,n) = 2. 295 detour global domination for degree splitting graphs of some graphs proof. consider the bistar graph bm,n with v (bm,n) = {u, v, ui, vj/1 ≤ i ≤ m; 1 ≤ j ≤ n}. here, ui and vj are the vertices adjacent with u and v respectively. let x1 and x2 be the corresponding vertices which are added to obtain ds(bm,n). then v (ds(bm,n)) = {u, v, ui, vj, x1, x2/1 ≤ i ≤ m; 1 ≤ j ≤ n} and so |v (s′(bm,n)| = m + n + 4. bm,n ds(bm,n) figure 3.4 u v u1 u2 um v1 v2 vn u v u1 u2 um v1 v2 vn x1 x2 since g is a connected graph, γ̄d(ds(bm,n)) ≥ 2. consider id[x1, x2] which has m + n transversal detour path of length four between x1 and x2 which include all the vertices of ds(bm,n). therefore, s = {x1, x2} is a detour set of minimum cardinality. also, x1 dominates u and v and x2 dominates all ui and vj for 1 ≤ i ≤ n and 1 ≤ j ≤ n in ds(bm,n). it follows that s is a detour dominating set of ds(bm,n). moreover, in ds(bm,n), x2 dominates x1, u and v; x1 dominates u1, u2, · · · , um, v1, v2, · · · , vm. clearly, s is a minimum detour global dominating set of ds(bm,n) and hence γ̄d(ds(bm,n)) = 2. theorem 3.5. for any integer m, n ≥ 2, γ̄d(ds(km,n)) = { 3 if m = n 2 if m ̸= n proof. consider km,n with v (km,n) = {ui, vj/1 ≤ i ≤ m, 1 ≤ j ≤ n} with partition v1 = {v1, v2, · · · , vm} and v2 = {u1, u2, · · · , un}. now we consider the following two cases. case (1) m = n 296 c. jayasekaran, s.v. ashwin prakash in this case each vertex is of same degree and so let x be the added vertex which is adjacent to every ui and vj, 1 ≤ i ≤ m and 1 ≤ j ≤ n. thus, we obtain the graph ds(km,n). then v (ds(km,n)) = {ui, vj, x/1 ≤ i ≤ m; 1 ≤ j ≤ n} and so |v (ds(km,n))| = m + n + 1. ds(km,m) figure 3.5 v1 u1 unu2 u3 v2 v3 vm x consider the set s = {vi, uj} for some i, j where 1 ≤ i ≤ m and 1 ≤ j ≤ n. then id[vi, uj] = v (ds(km,n)) and also vi dominates every uj and x; uj dominates every vi. hence, s = {vi, uj} is a minimum detour dominating set. now consider ds(km,n), where x is an isolated vertex and not dominated by any vertex of s. this shows that s is not a detour global dominating set of ds(km,n). hence, we include x in s such that s = {ui, vj, x} is a detour global dominating set of ds(km,n) for 1 ≤ i ≤ m; 1 ≤ j ≤ n. therefore, for m = n, γ̄d(ds(km,n)) = 3. case (2) m ̸= n in this case each vertex ui is of same degree and each vertex vj is of same degree where deg(ui) ̸= deg(vj), 1 ≤ i ≤ m; 1 ≤ j ≤ n so let x1 and x2 be the added vertex where x1 is adjacent to every ui and x2 is adjacent to every vj. thus, we obtain the graph ds(km,n). then v (ds(km,n)) = {ui, vj, x1, x2/1 ≤ i ≤ m; 1 ≤ j ≤ n} and so |v (ds(km,n))| = m + n + 2. 297 detour global domination for degree splitting graphs of some graphs ds(km,m) figure 3.6 v1 u1 un u2 u3 v2 v3 vm x1 x2 consider the set s = {x1, x2}, where id[x1, x2] = v (ds(km,n)) and also x1 dominates every vi and x2 dominates every uj. hence, s = {x1, x2} is a minimum detour dominating set. now consider ds(km,n), where x1 dominates every ui for 1 ≤ i ≤ n and x2 dominates every vj for 1 ≤ j ≤ n. clearly, s = {x1, x2} is a minimum detour global dominating set and so for m ̸= n, γ̄d(ds(km,n)) = 2. theorem 3.6. for any integer n ≥ 2, γ̄d(ds(kn)) = n. proof. here, ds(kn) is isomorphic to kn+1. we know that all the vertices are isolated vertices in the complement graph of ds(kn). therefore, the detour global dominating set must contain all the vertices of ds(kn) and so, for n ≥ 2, γ̄d(ds(kn)) = n. 4 conclusion inspired by the global dominating set and detour set we introduce the detour global dominating set for degree splitting graph. we have determined the detour global domination number for degree splitting graph of path graph, cycle graph, star graph, bistar graph, complete bipartite graph and complete graph. furthermore our results are also justified with suitable examples. the detour global domination number can also be obtained for many more graphs. 298 c. jayasekaran, s.v. ashwin prakash acknowledgements the authors express their gratitude to the managementratio mathematica for their constant support towards the successful completion of this work. we wish to thank the anonymous reviewers for the valuable suggestions and comments. references [1] d.b. west. introduction to graph theory. prentice-hall, upper saddle river, nj, 2001. [2] f. harary f. buckley. distance in graphs. addison-wesley,redwood city, 1990. [3] p. zang g. chartrand, n. johns. detour number of a graph. util. math., 64, 2003. [4] p.j. slater t.w. haynes, s.t. hedetniemi. fundamentals of domination in graphs. marcel dekker, inc., new york, 1998. [5] e. sampath kumar. the global domination number of a graph. journal of mathematical and physical sciences, 23(5), 1989. [6] s.v. ashwin prakash c. jayasekaran. detour global domination number of some graphs. malaya journal of matematik, s(1), 2020. [7] s. somasundaram r. ponraj. on the degree splitting graph of a graph. national academy science letters, 27(7-8), 2004. 299 ratio mathematica volume 45, 2023 near mean labeling in dicyclic snakes palani k* shunmugapriya a† abstract k. palani et al. defined the concept of near mean labeling in digraphs. let 𝐷 = (𝑉, 𝐴) be a digraph where 𝑉the vertex is set and 𝐴 is the arc set. let 𝑓: 𝑉 → {0, 1, 2, … , 𝑞} be a 1-1 map. define 𝑓∗: 𝐴 → {1, 2, … , 𝑞} by𝑓∗(𝑒 = 𝑢𝑣⃗⃗⃗⃗ ) = ⌈ 𝑓(𝑢)+𝑓(𝑣) 2 ⌉. let𝑓∗(𝑣) = |∑ 𝑓∗(𝑣𝑤⃗⃗⃗⃗ ⃗)𝑤∈𝑉 − ∑ 𝑓 ∗(𝑤𝑣⃗⃗⃗⃗ ⃗)𝑤∈𝑉 |. if𝑓 ∗(𝑣) ≤ 2 ∀ 𝑣 ∈ 𝐴(𝐷), then 𝑓 is said to be a near mean labeling of d and 𝐷 is said to be a near mean digraph. in this paper, different dicyclic snakes are defined and the existence of near mean labeling in them is checked. keywords: near mean labeling, digraphs, di-cyclic, di-quadrilateral, di-pentagonal, snake. 2010 ams subject classification: 05c78‡ *pg & research department of mathematics, a.p.c. mahalaxmi college for women, thoothukudi-628 002, tamil nadu, india); palani@apcmcollege.ac.in. † department of mathematics, sri sarada college for women (autonomous), tirunelveli-627 011. (research scholar-19122012092005, a.p.c. mahalaxmi college for women, thoothukudi-628 002, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamil nadu, india); priyaarichandran@gmail.com. ‡ received on july 16, 2022. accepted on september 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.1022. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 238 mailto:palani@apcmcollege.ac.in mailto:priyaarichandran@gmail.com palani k and shunmugapriya a 1. introduction graph theory has applications in many areas of the computing, social and natural science. the theory is also intimately related to many branches of mathematics, including matrix theory, numerical analysis, probability, topology and combinatory. over the last 50 year graph theory has evolved into an important mathematical tool in the solution of a wide variety of problems in many areas of society. a graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions have been motivated by practical problems, labelled graphs serve useful mathematical models for a broad range of applications such as: coding theory, including the design of good types codes, synch-set codes, missile guidance codes and convolutional codes with optimal auto correlation properties. the concept of graph labeling was introduced by rosa in 1967 [6]. a useful survey on graph labeling by j.a. gallian (2014) can be found in [2]. somasundaram and ponraj [4] have introduced the notion of mean labeling of graphs. a directed graph or digraph 𝐷 consists of a finite set 𝑉 of vertices and a collection of ordered pairs of distinct vertices. any such pair (𝑢, 𝑣) is called an arc or directed line and will usually be denoted by𝑢𝑣⃗⃗⃗⃗ . a digraph 𝐷 with 𝑝 vertices and 𝑞 arcs is denoted by𝐷 (𝑝, 𝑞). the indegree 𝑑−(𝑣) of a vertex 𝑣 in a digraph 𝐷 is the number of arcs having 𝑣 as its terminal vertex. the outdegree 𝑑+(𝑣) of 𝑣 is the number of arcs having 𝑣 as its initial vertex [1]. k. palani et al. introduced the concepts of mean and near mean digraphs in [3]. in this paper, different di-cyclic snakes are introduced and the existence of near mean labeling is investigated. 2. preliminaries the following definition and theorem are basics which are needed for the subsequent section. definition 2.1: [3] let 𝑓: 𝑉 → {0, 1, 2, … , 𝑞} be a 1-1 map. define 𝑓∗: 𝐴 → {1, 2, … , 𝑞} by𝑓∗(𝑒 = 𝑢𝑣⃗⃗⃗⃗ ) = ⌈ 𝑓(𝑢)+𝑓(𝑣) 2 ⌉. let𝑓∗(𝑣) = |∑ 𝑓∗(𝑣𝑤⃗⃗⃗⃗ ⃗)𝑤∈𝑉 − ∑ 𝑓 ∗(𝑤𝑣⃗⃗⃗⃗ ⃗)𝑤∈𝑉 |. if𝑓 ∗(𝑣) ≤ 2 ∀ 𝑣 ∈ 𝐴(𝐷), then 𝑓 is said to be a near mean labeling of d and 𝐷 is said to be a near mean digraph. definition 2.2: [5] a cyclic snake 𝑚𝐶𝑛 is obtained by replacing every edge of 𝑃𝑚 by𝐶𝑛. theorem 2.3: [3] the directed cycle 𝐶𝑛⃗⃗⃗⃗ is a near mean digraph. 3. main results in this section, the different dicyclic snakes are defined and the near mean labeling existence is verified. 239 near mean labeling in dicyclic snakes definition 3.1: in cyclic snake𝑚𝐶𝑛, orient the edges of each cycle clockwise. the resulting graph is called di-cyclic snake and it is denoted as𝑚𝐶𝑛⃗⃗⃗⃗ . for𝑛 = 3, 4, 5, dicyclic snakes are called di-triangular snake 𝑇𝑆𝑛⃗⃗⃗⃗⃗⃗ ⃗, di-quadrilateral snake 𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗ and dipentagonal snake 𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗ ⃗ respectively. definition 3.2: in 𝑚𝐶𝑛 when 𝑛 is even, orient the edges of the cycle alternately and call resulting graph as alternating di-cyclic snake. denote it as𝐴𝑚𝐶𝑛⃗⃗⃗⃗ . theorem 3.3: di-quadrilateral snake 𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗ admits near mean labeling. proof: let 𝑉(𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗) = {𝑢𝑖|1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑤𝑖|1 ≤ 𝑖 ≤ 𝑛 − 1} be the vertex set and let 𝐴(𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗) = {{𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗ ⃗} ∪ {𝑣𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗ } ∪ {𝑢𝑖+1𝑤𝑖⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗} ∪ {𝑤𝑖𝑢𝑖⃗⃗⃗⃗⃗⃗⃗⃗ }|1 ≤ 𝑖 ≤ 𝑛 − 1} be the arc set. define 𝑓: 𝑉(𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗) → {0, 1, 2, … , (4𝑛 − 4)} by 𝑓(𝑢𝑖) = 4(𝑖 − 1) for 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑣𝑖) = 4𝑖 − 2 for 1 ≤ 𝑖 ≤ 𝑛 − 1 figure 3.1: the labeling of a di-quadrilateral snake for 𝑖 = 1 to 𝑛 − 1 𝑓∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗ ⃗) = ⌈ 𝑓(𝑢𝑖)+𝑓(𝑣𝑖) 2 ⌉ = ⌈ [4(𝑖−1)]+[4𝑖−2] 2 ⌉ = ⌈ 8𝑖−6 2 ⌉ = 4𝑖 − 3. (3.3.1) 𝑓∗(𝑣𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗ ) = ⌈ 𝑓(𝑣𝑖)+𝑓(𝑢𝑖+1) 2 ⌉ = ⌈ [4𝑖−2]+[4(𝑖+1−1)] 2 ⌉ = ⌈ 8𝑖−2 2 ⌉ = 4𝑖 − 1. (3.3.2) 𝑓∗(𝑢𝑖+1𝑤𝑖⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗) = ⌈ 𝑓(𝑢𝑖+1)+𝑓(𝑤𝑖) 2 ⌉ = ⌈ [4(𝑖+1−1)]+[4𝑖−1] 2 ⌉ = ⌈ 8𝑖−1 2 ⌉ = 8𝑖 2 = 4𝑖. (3.3.3) 𝑓∗(𝑤𝑖𝑢𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ) = ⌈ 𝑓(𝑤𝑖)+𝑓(𝑢𝑖) 2 ⌉ = ⌈ [4𝑖−1]+[4(𝑖−1)] 2 ⌉ = ⌈ 8𝑖−5 2 ⌉ = 8𝑖−4 2 = 4𝑖 − 2. (3.3.4) now 𝑓∗(𝑢1) = |𝑓 ∗(𝑢1𝑣1⃗⃗⃗⃗⃗⃗⃗⃗ ⃗) − 𝑓 ∗(𝑤1𝑢1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ )| = |[4(1) − 3] − [4(1) − 2]| [by (3.3.1) & (3.3.4)] = |1 − 2| = |−1| < 2 therefore, 𝑓∗(𝑢1) < 2 (3.3.5) for 𝑖 = 2 to 𝑛 − 1 𝑓∗(𝑢𝑖) = |[𝑓 ∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗ ⃗) + 𝑓 ∗(𝑢𝑖𝑤𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗)] − [𝑓 ∗(𝑤𝑖𝑢𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ) + 𝑓 ∗(𝑣𝑖−1𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )]| = |[4𝑖 − 3 + 4(𝑖 − 1)] − [4𝑖 − 2 + 4(𝑖 − 1) − 1]| [by (1), (3), (4) & (2)] = |8𝑖 − 7 − 8𝑖 + 7| = |0| < 2 therefore, 𝑓∗(𝑢𝑖) < 2 for 𝑖 = 2 to 𝑛 − 1 (3.3.6) 𝑓∗(𝑢𝑛) = |𝑓 ∗(𝑢𝑛𝑤𝑛−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) − 𝑓 ∗(𝑣𝑛−1𝑢𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )| = |4(𝑛 − 1) − [4(𝑛 − 1) − 1| [by (3.3.3) & (3.3.2)] 240 palani k and shunmugapriya a = |1| < 2 therefore, 𝑓∗(𝑢𝑛) < 2 (3.3.7) for 𝑖 = 1 to 𝑛 − 1 𝑓∗(𝑣𝑖) = |𝑓 ∗(𝑣𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗ ) − 𝑓 ∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗ ⃗)| = |(4𝑖 − 1) − (4𝑖 − 3)| [by (3.3.2) & (3.3.1)] = |4𝑖 − 1 − 4𝑖 + 3| = |2| = 2 therefore, 𝑓∗(𝑣𝑖) = 2 for 𝑖 = 1 to 𝑛 − 1 (3.3.8) for 𝑖 = 1 to 𝑛 − 1 𝑓∗(𝑤𝑖) = |𝑓 ∗(𝑤𝑖𝑢𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ) − 𝑓 ∗(𝑢𝑖+1𝑤𝑖⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗)| = |(4𝑖 − 2) − (4𝑖)| [by (3.3.4) & (3.3.3)] = |−2| ≤ 2 therefore,𝑓∗(𝑤𝑖) ≤ 2 for 𝑖 = 1 to 𝑛 − 1 (3.3.9) from equations (5) to (9), 𝑓∗(𝑢) ≤ 2 ∀ 𝑢 ∈ 𝑉(𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗) hence di-quadrilateral snake 𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗ is a near mean digraph. theorem 3.4. di-pentagonal snake 𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗ ⃗ is a near mean digraph. proof: let 𝑉(𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗ ⃗) = {𝑢𝑖|1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑤𝑖|1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑥𝑖|1 ≤ 𝑖 ≤ 𝑛 − 1} be the vertex set and let 𝐴(𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗ ⃗) = {{𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗ ⃗} ∪ {𝑣𝑖𝑤𝑖⃗⃗⃗⃗⃗⃗⃗⃗ } ∪ {𝑤𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗} ∪ {𝑢𝑖+1𝑥𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ } ∪ {𝑥𝑖𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗}|1 ≤ 𝑖 ≤ 𝑛 − 1} be the arc set. define 𝑓: 𝑉(𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗ ⃗) → {0, 1, 2, … , (5𝑛 − 5)} by 𝑓(𝑢𝑖) = 5(𝑖 − 1) for 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑣𝑖) = 5𝑖 − 3 for 1 ≤ 𝑖 ≤ 𝑛 − 1 𝑓(𝑤𝑖) = 5𝑖 − 1 for 1 ≤ 𝑖 ≤ 𝑛 − 1 𝑓(𝑥𝑖) = 5𝑖 − 2 for 1 ≤ 𝑖 ≤ 𝑛 − 1 figure 3.2. the labeling of a di-pentagonal snake for 𝑖 = 1 to 𝑛 − 1 𝑓∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗ ⃗) = ⌈ 𝑓(𝑢𝑖)+𝑓(𝑣𝑖) 2 ⌉ = ⌈ [5(𝑖−1)]+[5𝑖−3] 2 ⌉ = ⌈ 10𝑖−8 2 ⌉ = 5𝑖 − 4 (3.4.1) 𝑓∗(𝑣𝑖𝑤𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ) = ⌈ 𝑓(𝑣𝑖)+𝑓(𝑤𝑖) 2 ⌉ = ⌈ [5𝑖−3]+[5𝑖−1] 2 ⌉ = ⌈ 10𝑖−4 2 ⌉ = 5𝑖 − 2 (3.4.2) 𝑓∗(𝑤𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗) = ⌈ 𝑓(𝑤𝑖)+𝑓(𝑢𝑖+1) 2 ⌉ = ⌈ [5𝑖−1]+[5(𝑖+1−1)] 2 ⌉ = ⌈ 10𝑖−1 2 ⌉ = 10𝑖 2 = 5𝑖 (3.4.3) 𝑓∗(𝑢𝑖+1𝑥𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) = ⌈ 𝑓(𝑢𝑖+1)+𝑓(𝑥𝑖) 2 ⌉ = ⌈ [5(𝑖+1−1)]+[5𝑖−2] 2 ⌉ = ⌈ 10𝑖−2 2 ⌉ = 5𝑖 − 1 (3.4.4) 241 near mean labeling in dicyclic snakes 𝑓∗(𝑥𝑖𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗) = ⌈ 𝑓(𝑥𝑖)+𝑓(𝑢𝑖) 2 ⌉ = ⌈ [5𝑖−2]+[5(𝑖−1)] 2 ⌉ = ⌈ 10𝑖−7 2 ⌉ = 10𝑖−6 2 = 5𝑖 − 3 (3.4.5) next to find 𝑓∗(𝑢𝑖) now 𝑓∗(𝑢1) = |𝑓 ∗(𝑢1𝑣1⃗⃗⃗⃗⃗⃗⃗⃗ ⃗) − 𝑓 ∗(𝑥1𝑢1⃗⃗⃗⃗⃗⃗⃗⃗ ⃗)| = |[5(1) − 4] − [5(1) − 3]| [by (3.4.1) & (3.4.5)] = |1 − 2| = |−1| < 2 therefore, 𝑓∗(𝑢1) < 2 (3.4.6) for 𝑖 = 2 to 𝑛 − 1 𝑓∗(𝑢𝑖) = |[𝑓 ∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗ ⃗) + 𝑓 ∗(𝑢𝑖𝑥𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )] − [𝑓 ∗(𝑥𝑖𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗) + 𝑓 ∗(𝑤𝑖−1𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗)]| = |[5𝑖 − 4 + 5(𝑖 − 1) − 1] − [5𝑖 − 3 + 5(𝑖 − 1)]| [by (3.4.1), (3.4.4), (3.4.5) & (3.4.3)] = |10𝑖 − 10 − 10𝑖 + 8| = |−2| ≤ 2 therefore, 𝑓∗(𝑢𝑖) ≤ 2 for 𝑖 = 2 to 𝑛 − 1 (3.4.7) 𝑓∗(𝑢𝑛) = |𝑓 ∗(𝑢𝑛𝑥𝑛−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) − 𝑓 ∗(𝑤𝑛−1𝑢𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗)| = |5(𝑛 − 1) − 1 − 5(𝑛 − 1)| [by (3.4.4) & (3.4.3)] = |−1| < 2 therefore, 𝑓∗(𝑢𝑛) < 2 (3.4.8) for 𝑖 = 1 to 𝑛 − 1 𝑓∗(𝑣𝑖) = |𝑓 ∗(𝑣𝑖𝑤𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ) − 𝑓 ∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗ ⃗)| = |(5𝑖 − 2) − (5𝑖 − 4)| [by (3.4.2) & (3.4.1)] = |2| = 2 therefore, 𝑓∗(𝑣𝑖) = 2 for 𝑖 = 1 to 𝑛 − 1 (3.4.9) for 𝑖 = 1 to 𝑛 − 1 𝑓∗(𝑤𝑖) = |𝑓 ∗(𝑤𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗) − 𝑓 ∗(𝑣𝑖𝑤𝑖⃗⃗⃗⃗⃗⃗⃗⃗ )| = |(5𝑖) − (5𝑖 − 2)| [by (3.4.3) & (3.4.2)] = |2| = 2 therefore, 𝑓∗(𝑤𝑖) = 2 for 𝑖 = 1 to 𝑛 − 1 (3.4.10) for 𝑖 = 1 to 𝑛 − 1 𝑓∗(𝑥𝑖) = |𝑓 ∗(𝑥𝑖𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗) − 𝑓 ∗(𝑢𝑖+1𝑥𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )| = |(5𝑖 − 3) − (5𝑖 − 1)| [by (3.4.3) & (3.4.2)] = |−2| ≤ 2 therefore, 𝑓∗(𝑥𝑖) ≤ 2 for 𝑖 = 1 to 𝑛 − 1 (3.4.11) from equations (6) to (11), 𝑓∗(𝑢) ≤ 2 ∀ 𝑢 ∈ 𝑉(𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗ ⃗) hence, di-pentagonal snake 𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗ ⃗ is a near mean digraph. theorem 3.5. di-cyclic snake 𝑚𝐶𝑛⃗⃗⃗⃗ is a near mean digraph for 𝑚 ≥ 1 and 𝑛 ≥ 3. proof: let 𝑢𝑖𝑗 denote the 𝑗th vertex in the 𝑖th copy of 𝐶𝑛⃗⃗⃗⃗ here, 𝑉(𝑚𝐶𝑛⃗⃗⃗⃗ ) = {𝑢𝑖𝑗|1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛} and 𝐴(𝑚𝐶𝑛⃗⃗⃗⃗ ) = {𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛 − 1} ∪ {𝑢𝑖𝑛𝑢𝑖1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑚} following the procedure of near mean labeling of 𝐶𝑛⃗⃗⃗⃗ in [3], define 𝑓: 𝑉(𝑚𝐶𝑛⃗⃗⃗⃗ ) → {0,1,2, … , 𝑚𝑛} as below 242 palani k and shunmugapriya a 𝑓(𝑢𝑖𝑗) = { (𝑖 − 1)𝑛 + 2(𝑗 − 1) for 1 ≤ 𝑗 ≤ ⌊ 𝑛 2 ⌋ + 1, 1 ≤ 𝑖 ≤ 𝑚 (𝑖 − 1)𝑛 + 2(𝑛 − (𝑗 − 1)) + 1 for ⌊ 𝑛 2 ⌋ + 2 ≤ 𝑗 ≤ 𝑛, 1 ≤ 𝑖 ≤ 𝑚 where |𝑉(𝑚𝐶𝑛⃗⃗⃗⃗ )| = 𝑚𝑛 − 𝑚 + 1 and |𝐴(𝑚𝐶𝑛⃗⃗⃗⃗ )| = 𝑚𝑛 figure 3.3: the labeling of the di-cyclic snake 𝑚𝐶𝑛⃗⃗⃗⃗ when 𝑛 is even figure 3.4: the labeling of the di-cyclic snake 𝑚𝐶𝑛⃗⃗⃗⃗ when 𝑛 is odd now to find 𝑓∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) for 𝑖 = 1 to 𝑚, 𝑗 = 1 to ⌊ 𝑛 2 ⌋ 𝑓∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) = ⌈ 𝑓(𝑢𝑖𝑗) + 𝑓(𝑢𝑖(𝑗+1)) 2 ⌉ = ⌈ [(𝑖−1)𝑛+2(𝑗−1)]+[(𝑖−1)𝑛+2((𝑗+1)−1)] 2 ⌉ = ⌈ 2(𝑖−1)𝑛+4𝑗−2 2 ⌉ = (𝑖 − 1)𝑛 + 2𝑗 − 1 = 𝑛𝑖 + 2𝑗 − 𝑛 − 1. therefore, 𝑓∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) = 𝑛𝑖 + 2𝑗 − 𝑛 − 1 for 𝑖 = 1 to 𝑚, 𝑗 = 1 to ⌊ 𝑛 2 ⌋ (3.5.1) 243 near mean labeling in dicyclic snakes 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) 𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) = ⌈ 𝑓 (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) ) + 𝑓 (𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ) 2 ⌉ = ⌈ [(𝑖−1)𝑛+2(⌊ 𝑛 2 ⌋+1−1)]+[(𝑖−1)𝑛+2(𝑛−(⌊ 𝑛 2 ⌋+2−1))+1] 2 ⌉ = ⌈ 2(𝑖−1)𝑛+2𝑛−1 2 ⌉ = ⌈ 2(𝑖−1+1)𝑛−1 2 ⌉ = ⌈ 2𝑖𝑛−1 2 ⌉ = 2𝑖𝑛 2 = 𝑛𝑖 therefore, 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) 𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) = 𝑛𝑖 for 𝑖 = 1 to 𝑚 and 𝑛 ≥ 3 (3.5.2) for 𝑖 = 1 to 𝑚, 𝑗 = ⌊ 𝑛 2 ⌋ + 2 to 𝑛 − 1 𝑓∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) = ⌈ 𝑓(𝑢𝑖𝑗) + 𝑓(𝑢𝑖(𝑗+1)) 2 ⌉ = ⌈ [(𝑖−1)𝑛+2(𝑛−(𝑗−1))+1]+[(𝑖−1)𝑛+2(𝑛−((𝑗+1)−1))+1] 2 ⌉ = ⌈ 2(𝑖−1)𝑛+4𝑛−4𝑗+4 2 ⌉ = (𝑖 + 1)𝑛 − 2𝑗 + 2 = 𝑛𝑖 − 2𝑗 + 𝑛 + 2 𝑓∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) = 𝑛𝑖 − 2𝑗 + 𝑛 + 2 for 𝑖 = 1 to 𝑚, 𝑗 = ⌊ 𝑛 2 ⌋ + 2 to 𝑛 − 1 (3.5.3) 𝑓∗(𝑢𝑖𝑛𝑢𝑖1⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) = ⌈ 𝑓(𝑢𝑖𝑛) + 𝑓(𝑢𝑖1) 2 ⌉ = ⌈ [(𝑖−1)𝑛+2(𝑛−(𝑛−1))+1]+[(𝑖−1)𝑛+2(1−1)] 2 ⌉ = ⌈ 2(𝑖−1)𝑛+2(𝑛−𝑛+1)+1 2 ⌉ = ⌈ 2(𝑖−1)𝑛+3 2 ⌉ = 2(𝑖−1)𝑛+4 2 = (𝑖 − 1)𝑛 + 2 = 𝑛𝑖 − 𝑛 + 2. therefore, 𝑓∗(𝑢𝑖𝑛𝑢𝑖1⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) = 𝑛𝑖 − 𝑛 + 2 for 𝑖 = 1 to 𝑚 and 𝑛 ≥ 3 (3.5.4) next to find 𝑓∗(𝑢𝑖𝑗) 𝑓∗(𝑢11) = |𝑓 ∗(𝑢11𝑢12⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) − 𝑓 ∗(𝑢1𝑛𝑢11⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗)| = |[𝑛(1) + 2(1) − 𝑛 − 1] − [𝑛(1) − 𝑛 + 2]| [by (3.5.1) & (3.5.4)] = |1 − 2| = |−1| < 2 therefore, 𝑓∗(𝑢11) < 2 (3.5.5) for 𝑖 = 1 to 𝑚, 𝑗 = 2 to ⌊ 𝑛 2 ⌋ 𝑓∗(𝑢𝑖𝑗) = |𝑓 ∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) − 𝑓 ∗(𝑢𝑖(𝑗−1)𝑢𝑖𝑗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗)| = |[𝑛𝑖 + 2𝑗 − 𝑛 − 1] − [𝑛𝑖 + 2(𝑗 − 1) − 𝑛 − 1]| [by (3.5.1)] = |−1 + 3| = |2| = 2 therefore, 𝑓∗(𝑢𝑖𝑗) = 2 for 𝑖 = 1 to 𝑚, 𝑗 = 2 to ⌊ 𝑛 2 ⌋ (3.5.6) for 𝑖 = 1 to 𝑚 − 1, two cases for 𝑓∗(𝑢𝑖𝑗) when 𝑗 = ⌊ 𝑛 2 ⌋ + 1 & ⌊ 𝑛 2 ⌋ + 2 case (i): 𝑛 is odd. 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) ) = |𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) 𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) − 𝑓∗ (𝑢 𝑖⌊ 𝑛 2 ⌋ 𝑢 𝑖(⌊ 𝑛 2 ⌋+1) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )| = |𝑛𝑖 − (𝑛𝑖 + 2 ⌊ 𝑛 2 ⌋ − 𝑛 − 1)| [by (3.5.2) & (3.5.1)] 244 palani k and shunmugapriya a = |𝑛 − 2 ⌊ 𝑛 2 ⌋ + 1| = |𝑛 − 2 ( 𝑛−1 2 ) + 1| = |2| = 2 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) ) = 2 when 𝑛 is odd (3.5.7) 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ) = |(𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+2) 𝑢 𝑖(⌊ 𝑛 2 ⌋+3) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) + 𝑓∗(𝑢(𝑖+1)1𝑢(𝑖+1)2⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗)) − (𝑓∗ (𝑢 𝑖⌊ 𝑛 2 ⌋+1 𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) + 𝑓∗(𝑢(𝑖+1)𝑛𝑢(𝑖+1)1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ))| = |[(𝑛𝑖 − 2 (⌊ 𝑛 2 ⌋ + 2) + 𝑛 + 2) + 𝑛(𝑖 + 1) + 2 − 𝑛 − 1] − [𝑛𝑖 + 𝑛(𝑖 + 1) − 𝑛 + 2]| [by (3.5.3), (3.5.1), (3.5.2) & (3.5.4)] = |[(𝑛𝑖 − 2 (⌊ 𝑛 2 ⌋) − 4 + 𝑛 + 2) + (𝑛𝑖 + 𝑛 + 2 − 𝑛 − 1)] − [𝑛𝑖 + 𝑛𝑖 + 2]| = |[2𝑛𝑖 − 2 ( 𝑛−1 2 ) + 𝑛 − 1] − [2𝑛𝑖 + 2]| = |−2| ≤ 2 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ) ≤ 2 when 𝑛 is odd. (3.5.8) case (ii): 𝑛 is even. 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) ) = |[𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) 𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) + 𝑓∗(𝑢(𝑖+1)1𝑢(𝑖+1)2⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗)] − [𝑓∗ (𝑢 𝑖⌊ 𝑛 2 ⌋ 𝑢 𝑖(⌊ 𝑛 2 ⌋+1) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) + 𝑓∗(𝑢(𝑖+1)𝑛𝑢(𝑖+1)1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )]| = |[𝑛𝑖 + (𝑛(𝑖 + 1) + 2(1) − 𝑛 − 1)] − [(𝑛𝑖 + 2 ⌊ 𝑛 2 ⌋ − 𝑛 − 1) + (𝑛(𝑖 + 1) − 𝑛 + 2)]| [by (3.5.2), (3.5.1) & (3.5.4) ] = |[𝑛𝑖 + 𝑛𝑖 + 𝑛 + 2 − 𝑛 − 1] − [𝑛𝑖 + 2 ( 𝑛 2 ) − 𝑛 − 1 + 𝑛𝑖 + 2]| = |0| < 2 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) ) < 2 when 𝑛 is even (3.5.9) 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ) = |𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+2) 𝑢 𝑖(⌊ 𝑛 2 ⌋+3) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) − 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) 𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )| = |(𝑛𝑖 − 2 (⌊ 𝑛 2 ⌋ + 2) + 𝑛 + 2) − 𝑛𝑖| [by (3) & (4)] = |𝑛 − 2 ⌊ 𝑛 2 ⌋ − 4 + 2| = |𝑛 − 2 ( 𝑛 2 ) − 2| = |−2| ≤ 2 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ) ≤ 2 when 𝑛 is even (3.5.10) cases (i) and (ii) imply 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) ) ≤ 2 for 𝑖 = 1 to 𝑚 − 1 (3.5.11) and 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ) ≤ 2 for 𝑖 = 1 to 𝑚 − 1 (3.5.12) 245 near mean labeling in dicyclic snakes now to find 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+1) ) and 𝑓∗ (𝑢 𝑖(⌊ 𝑛 2 ⌋+2) ) for 𝑖 = 𝑚 𝑓∗ (𝑢 𝑚(⌊ 𝑛 2 ⌋+1) ) = |𝑓∗ (𝑢 𝑚(⌊ 𝑛 2 ⌋+1) 𝑢 𝑚(⌊ 𝑛 2 ⌋+2) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) − 𝑓∗ (𝑢 𝑚⌊ 𝑛 2 ⌋ 𝑢 𝑚(⌊ 𝑛 2 ⌋+1) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗)| = |𝑛𝑚 − (𝑛𝑚 + 2 ⌊ 𝑛 2 ⌋ − 𝑛 − 1)| [by (2) & (1)] = |𝑛 − 2 ⌊ 𝑛 2 ⌋ + 1| = { |𝑛 − 2 ( 𝑛−1 2 ) + 1| if 𝑛 is odd |𝑛 − 2 ( 𝑛 2 ) + 1| if 𝑛 is even = { |2| if 𝑛 is odd |1| if 𝑛 is even therefore, 𝑓∗ (𝑢 𝑚(⌊ 𝑛 2 ⌋+1) ) ≤ 2 (3.5.13) 𝑓∗ (𝑢 𝑚(⌊ 𝑛 2 ⌋+2) ) = |𝑓∗ (𝑢 𝑚(⌊ 𝑛 2 ⌋+2) 𝑢 𝑚(⌊ 𝑛 2 ⌋+3) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) − 𝑓∗ (𝑢 𝑚(⌊ 𝑛 2 ⌋+1) 𝑢 𝑚(⌊ 𝑛 2 ⌋+2) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗)| = |(𝑛𝑚 − 2 (⌊ 𝑛 2 ⌋ + 2) + 𝑛 + 2) − 𝑛𝑚| [by (3.5.3) & (3.5.2)] = |𝑛 − 2 ⌊ 𝑛 2 ⌋ − 2| = { |𝑛 − 2 ( 𝑛−1 2 ) − 2| if 𝑛 is odd |𝑛 − 2 ( 𝑛 2 ) − 2| if 𝑛 is even = { |−1| if 𝑛 is odd |−2| if 𝑛 is even therefore, 𝑓∗ (𝑢 𝑚(⌊ 𝑛 2 ⌋+2) ) ≤ 2 (3.5.14) for 𝑖 = 1 to 𝑚, 𝑗 = ⌊ 𝑛 2 ⌋ + 3 to 𝑛 − 1 𝑓∗(𝑢𝑖𝑗) = |𝑓 ∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) − 𝑓 ∗(𝑢𝑖(𝑗−1)𝑢𝑖𝑗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗)| = |[𝑛𝑖 − 2𝑗 + 𝑛 + 2] − [𝑛𝑖 − 2(𝑗 − 1) + 𝑛 + 2]| [by (3)] = |−2| therefore, 𝑓∗(𝑢𝑖𝑗) ≤ 2 for 𝑖 = 1 to 𝑚, 𝑗 = ⌊ 𝑛 2 ⌋ + 3 to 𝑛 − 1 (3.5.15) for 𝑖 = 1 to 𝑚 and 𝑗 = 𝑛 𝑓∗(𝑢𝑖𝑛) = |𝑓 ∗(𝑢𝑖𝑛𝑢𝑖1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) − 𝑓 ∗(𝑢𝑖(𝑛−1)𝑢𝑖𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )| = |[𝑛𝑖 − 𝑛 + 2] − [𝑛𝑖 − 2(𝑛 − 1) + 𝑛 + 2]| [by (4)] = |𝑛𝑖 − 𝑛 + 2 − 𝑛𝑖 + 2𝑛 − 2 − 𝑛 − 2| = |−2| therefore, 𝑓∗(𝑢𝑖𝑛) ≤ 2 (3.5.16) equations (3.5.5), (3.5.6) and (3.5.11) to (3.5.16) imply 𝑓∗(𝑢𝑖𝑗) ≤ 2 for1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛. thus, di-cyclic snake 𝑚𝐶𝑛⃗⃗⃗⃗ is a near mean digraph for 𝑚 ≥ 1 and𝑛 ≥ 3. theorem 3.6. alternating di-cyclic snakes are non near mean digraphs. 246 palani k and shunmugapriya a proof: in an alternating di-cyclic snake, either 𝑑+(𝑢21) = 0 and 𝑑 −(𝑢21) = 4 (or) 𝑑+(𝑢21) = 4 and𝑑 −(𝑢21) = 0. therefore, corresponding to every𝑓: 𝑉 → {0, 1, 2, … , 𝑞}, ∑ 𝑓∗(𝑢21𝑤⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ )𝑤∈𝑉 = 0 and ∑ 𝑓∗(𝑤𝑢21⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗ )𝑤∈𝑉 is a sum of at least three positive integers (or) ∑ 𝑓 ∗(𝑤𝑢21⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗ )𝑤∈𝑉 = 0 and ∑ 𝑓∗(𝑢21𝑤⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ )𝑤∈𝑉 is a sum of at least three positive integers. therefore, 𝑓∗(𝑢21) > 2. therefore, no function 𝑓: 𝑉 → {0, 1, 2, … , 𝑞} is a near mean labeling. thus, an alternating di-cyclic snake is a non near mean digraph. 4. conclusions in this article, different dicyclic snakes are introduced. also, existence of near mean labeling is verified to dicyclic snakes and its generalisation. most of the labeling are proved only for graphs. in this way, we develop the concept of labeling into digraphs references [1] gallian j a, a dynamic survey of graph labeling, the electronic journal of combinatorics, 17, 2014. [2] harary f, graph theory, addition wesley, massachusetts, 1972. [3] palani k and shunmugapriya a, near mean labeling in dragon digraphs, journal of xidian university, 14(3): 1298-1307, 2020. [4] ponraj r and somasundaram s, mean labeling of graphs, national academy of science letters, 26: 210-213, 2003. [5] raval k k and prajapati u m, vertex even and odd mean labeling in the context of some cyclic snake graphs, journal of emerging technologies and innovative research (jetir), 4(6), 2017. [6] rosa a, 1967. on certain valuations of the vertices of a graph, theory of graphs, gordon and breach, dunod, paris, 349-355, 1966. 247 ratio mathematica volume 44, 2022 polygonal graceful labeling of some simple graphs a. rama lakshmi 1 m.p. syed ali nisaya2 abstract let 𝐺 = (𝑉, 𝐸) be a graph with 𝑝 vertices and 𝑞 edges. let 𝑉(𝐺) and𝐸(𝐺) be the vertex set and edge set of 𝐺respectively. a polygonal graceful labeling of a graph 𝐺 is an injective function 𝜂: 𝑉(𝐺) → 𝑍+, where 𝑍+ is a set of all non-negative integers that induces a bijection 𝜂*: 𝐸(𝐺) → {𝑃𝑠(1),𝑃𝑠(2), . . . , 𝑃𝑠(𝑞)}, where 𝑃𝑠(𝑞) is the 𝑞 𝑡ℎ polygonal number such that 𝜂*(𝑢𝑣) = |𝜂(𝑢) − 𝜂(𝑣)| for every edge 𝑒 = 𝑢𝑣 ∈ 𝐸(𝐺). a graph which admits such labeling is called a polygonal graceful graph. for 𝑠 = 3, the above labeling gives triangular graceful labeling. for 𝑠 = 4, the above labeling gives tetragonal graceful labeling and so on. in this paper, polygonal graceful labeling is introduced and polygonal graceful labeling of some simple graphs is studied. keywords: polygonal numbers, polygonal graceful labeling, polygonal graceful graph. ams classification: 05c123 1research scholar (part time-internal), department of mathematics, the m.d.t hindu college, affiliated to manonmaniam sundaranar university, tirunelveli–627010, tamil nadu, india. mail id: guruji.ae@gmail.com 2 assistant professor, department of mathematics, the m.d.t hindu college, affiliated to manonmaniam sundaranar university, tirunelveli–627010, tamil nadu, india. mail id: syedalinisaya@mdthinducollege.org 3received on june 9th, 2022. accepted on sep 5st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.883. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement 9 mailto:guruji.ae@gmail.com mailto:syedalinisaya@mdthinducollege.org a. rama lakshmi and m.p. syed ali nisaya 1. introduction we shall consider a simple, undirected and finite graph 𝐺 = (𝑉, 𝐸) on 𝑝 = |𝑉|vertices and 𝑞 = |𝐸|edges. for all standard terminology, notations and basic definitions, we follow harary [2] and for number theory, we follow apostal [1]. a graph labeling is an assignment of integers to the vertices or the edges or both subject to certain conditions. if the domain of the mapping is the set of vertices (edges/both) then the labeling is called a vertex (edge/ total) labeling. rosa [6] introduced –valuation of a graph. golomb [4] called it as graceful labeling. for a detailed survey of graph labeling, one can refer gallian [3]. ramesh and syed ali nisaya [5] introduced some more polygonal graceful labeling of path. here, we shall recall some definitions which are used in this paper. 2. preliminaries definition 2.1. the star graph 𝐾1,𝑛of order 𝑛 + 1 is a tree on 𝑛 edges with one vertex having degree 𝑛 and other vertices having degree 1. definition 2.2. a graph 𝑆(𝐺) which can be obtained from a given graph by breaking up each edge into one (or) more segment by inserting intermediate vertices between its two ends is called subdivision graph. it is denoted by 𝑆(𝐺). definition 2.3. a path 𝑃𝑛is obtained by joining𝑢𝑖to the consecutive vertices𝑢𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 − 1. definition 2.4. a coconut tree 𝐶𝑇(𝑛, 𝑚) is a graph obtained from the path 𝑃𝑛 by appending m new pendant edges at an end vertex. definition 2.5. let 𝑃2 be a path on two vertices and let 𝑢 and 𝑣 be the vertices of 𝑃2. from 𝑢, there are m pendant vertices say 𝑢1, 𝑢2, . . . , 𝑢𝑚and from 𝑣, there are 𝑛 pendant vertices say𝑣1, 𝑣2, . . . , 𝑣𝑛. the resulting graph is a bistar𝐵𝑚,𝑛. definition 2.6. a graceful labeling of a graph 𝐺 is an injective function 𝑓: 𝑉(𝐺) → {0,1,2, . . . , 𝑞} that induces a bijection 𝑓*: 𝐸(𝐺) → {1,2, . . . , 𝑞} of the edges of 𝐺 defined by 𝑓*(𝑒) = |𝑓(𝑢) − 𝑓(𝑣)|,∀𝑒 = 𝑢𝑣 ∈ 𝐸(𝐺). the graph which admits such a labeling is called a graceful graph. definition 2.7. a polygonal number is a number represented as dots (or) pebbles arranged in the shape of regular polygon. if 𝑠 is the number of sides in a polygon, the formula for the 𝑛𝑡ℎ𝑠–gonal number 𝑃𝑠(𝑛) is 𝑃𝑠 (𝑛) = (𝑠−2)(𝑛)2−(𝑠−4)(𝑛) 2 . for 𝑠 = 3 gives triangular numbers. for 𝑠 = 4 gives tetragonal numbers and so on. 3. main results definition 3.1: let 𝐺 = (𝑉, 𝐸) be a graph with 𝑝 vertices and 𝑞 edges. let 𝑉(𝐺) and𝐸(𝐺) be the vertex set and edge set of 𝐺respectively. a polygonal graceful labeling 10 polygonal graceful labeling of some simple graphs of a graph 𝐺 is an injective function 𝜂: 𝑉(𝐺) → 𝑍+, where 𝑍+ is a set of all non-negative integers that induces a bijection 𝜂*: 𝐸(𝐺) → {𝑃𝑠(1), 𝑃𝑠(2), . . . , 𝑃𝑠(𝑞)}, where 𝑃𝑠(𝑞) is the 𝑞𝑡ℎ polygonal number such that 𝜂*(𝑢𝑣) = |𝜂(𝑢) − 𝜂(𝑣)| for every edge 𝑒 = 𝑢𝑣 ∈ 𝐸(𝐺). a graph which admits such labeling is called a polygonal graceful graph. for 𝑠 = 3, the above labeling gives triangular graceful labeling. for 𝑠 = 4, the above labeling gives tetragonal graceful labeling and so on. example 3.2: the hexagonal graceful labeling of 𝐾1,5is given in figure (1) figure (1) theorem 3.3: the star graph 𝐾1,𝑛 admits polygonal graceful labeling. proof: consider the star graph 𝐾1,𝑛. let 𝑉(𝑘1,𝑛) = {𝑣, 𝑣𝑖: 1 ≤ 𝑖 ≤ 𝑛} and𝐸(𝑘1,𝑛) = {𝑣𝑣𝑖: 1 ≤ 𝑖 ≤ 𝑛}. then 𝐾1,𝑛 has 𝑛 + 1 vertices and 𝑛 edges. define 𝜂: 𝑉(𝐾1,𝑛) → {0,1,2, . . . , 𝑃𝑠(𝑛)} as follows. 𝜂(𝑣) = 0 = 𝑃𝑠(𝑖),1 ≤ 𝑖 ≤ 𝑛 clearly 𝜂 is injective and the induced edge labels are the first 𝑛 polygonal numbers. hence the star graph 𝐾1,𝑛 admits polygonal graceful graph. example 3.4: the pentagonal graceful labeling of 𝐾1,7 is shown in the following figure (2) . figure (2) theorem 3.5: 𝑆(𝐾1,𝑛), the subdivision of the star 𝐾1,𝑛 admits polygonal graceful labeling. proof: consider the subdivision of the star graph 𝑆(𝐾1,𝑛). let 𝑉(𝑆(𝐾1,𝑛)) = {𝑣, 𝑣𝑖, 𝑢𝑖: 1 ≤ 𝑖 ≤ 𝑛} and 𝐸(𝑆(𝐾1,𝑛)) = {𝑣𝑣𝑖, 𝑣𝑖𝑢𝑖: 1 ≤ 𝑖 ≤ 𝑛}. thus 𝑆(𝐾1,𝑛) has 2𝑛 + 1 vertices and 2𝑛 edges. define 𝜂: 𝑉(𝑆(𝐾1,𝑛)) → {0,1,2, . . . , 𝑃𝑠(2𝑛)} as follows 𝜂(𝑣) = 0 ni isis v i  −−− = 1, 2 )4()2( )( 2  11 a. rama lakshmi and m.p. syed ali nisaya 𝜂(𝑣𝑖) = (𝑠 − 2)(2𝑛 − 𝑖 + 1)2 − (𝑠 − 4)(2𝑛 − 𝑖 + 1) 2 , 1 ≤ 𝑖 ≤ 𝑛 = 𝑃𝑠(2𝑛 − 𝑖 + 1),1 ≤ 𝑖 ≤ 𝑛 𝜂(𝑢𝑖) = 𝜂(𝑣𝑖) + (𝑠 − 2)(𝑛 − 𝑖 + 1)2 − (𝑠 − 4)(𝑛 − 𝑖 + 1) 2 , 1 ≤ 𝑖 ≤ 𝑛 = 𝜂(𝑣𝑖) + 𝑃𝑠(𝑛 − 𝑖 + 1), 1 ≤ 𝑖 ≤ 𝑛 clearly 𝜂 is injective and the induced edge labels are the first 2𝑛 polygonal numbers. hence 𝑆(𝐾1,𝑛) admits polygonal graceful graph. example 3.6: the triangular graceful labeling of 𝑆(𝐾1,3) is shown in the following figure (3). figure (3) theorem 3.7: the path of length 𝑛 is a polygonal graceful graph for all 𝑛 ≥ 3. proof. let 𝐺 = (𝑉, 𝐸) be the path of length 𝑛 with the vertex set 𝑉 = {𝑣1, 𝑣2, . . . , 𝑣𝑛}and the edge set 𝐸 = {𝑣𝑖𝑣𝑖+1/1 ≤ 𝑖 ≤ 𝑛 − 1}. then 𝐺 has 𝑛 vertices and 𝑛 − 1 edges. define 𝜂: 𝑉(𝐺) → {0,1,2, . . . , 𝑃𝑠(𝑛 − 1)} as follows. 𝜂(𝑣2𝑖−1) = (𝑖 − 1)((𝑠 − 2)𝑛 − (𝑠 − 2)𝑖 + 1),𝑖 = 1,2, . . , ⌊ 𝑛 + 1 2 ⌋ 𝜂(𝑣2𝑖) = 1 2 (𝑛 − 1)[(𝑠 − 2)𝑛 − 2𝑠 + 6] − [(𝑠 − 2)𝑛 − (𝑠 − 2)𝑖 − (𝑠 − 3)](𝑖 − 1),𝑖 = 1,2, . . , ⌊ 𝑛 2 ⌋ clearly 𝜂 is injective and the induced edge labels are the first 𝑛 − 1 polygonal numbers. hence the path of length 𝑛 is a polygonal graceful graph for all 𝑛 ≥ 3. example 3.8: the heptagonal graceful labeling of path of length 6 is given in figure (4). figure (4) theorem 3.9: coconut tree 𝐶𝑇(𝑛, 𝑚) is a polygonal graceful labeling∀𝑛 ≥ 1, 𝑚 ≥ 2. proof: consider the graph coconut tree 𝐶𝑇(𝑛, 𝑚). let 𝑉(𝐶𝑇(𝑛, 𝑚)) = {𝑣, 𝑣𝑖, 𝑢𝑗: 1 ≤ 𝑖 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑚 − 1} and 𝐸(𝐶𝑇(𝑛, 𝑚)) = {𝑣𝑣𝑖, 𝑣𝑢1, 𝑢𝑗𝑢𝑗+1: 1 ≤ 𝑖 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑚 − 1} . then 𝐶𝑇(𝑛, 𝑚) has 𝑚 + 𝑛 vertices and 𝑚 + 𝑛 − 1 edges. define𝜂: 𝑉(𝐶𝑇(𝑛, 𝑚)) → {0,1,2, . . . , 𝑃𝑠(𝑚 + 𝑛 − 1)} as follows. 𝜂(𝑣) = 0 𝜂(𝑣𝑖) = (𝑠 − 2)(𝑚 + 𝑛 − 𝑖)2 − (𝑠 − 4)(𝑚 + 𝑛 − 𝑖) 2 , 1 ≤ 𝑖 ≤ 𝑛 12 polygonal graceful labeling of some simple graphs = 𝑃𝑠(𝑚 + 𝑛 − 𝑖),1 ≤ 𝑖 ≤ 𝑛 𝜂(𝑢1) = (𝑠 − 2)(𝑚 − 1)2 − (𝑠 − 4)(𝑚 − 1) 2 = 𝑃𝑠(𝑚 − 1) 𝜂(𝑢𝑗) = 𝜂(𝑢𝑗−1) + (𝑠−2)(𝑚−𝑗)2−(𝑠−4)(𝑚−𝑗) 2 , 𝑖𝑓𝑗𝑖𝑠𝑜𝑑𝑑𝑎𝑛𝑑2 ≤ 𝑗 ≤ 𝑚 − 1 𝜂(𝑢𝑗−1) − (𝑠 − 2)(𝑚 − 𝑗)2 − (𝑠 − 4)(𝑚 − 𝑗) 2 , 𝑖𝑓𝑗𝑖𝑠𝑒𝑣𝑒𝑛𝑎𝑛𝑑2 ≤ 𝑗 ≤ 𝑚 − 1 clearly 𝜂 is an injective function. let 𝜂 ⥂* be the induced edge labeling of 𝜂. then 𝜂*(𝑣𝑣𝑖) = (𝑠 − 2)(𝑛 + 𝑚 − 𝑖)2 − (𝑠 − 4)(𝑛 + 𝑚 − 𝑖) 2 , 1 ≤ 𝑖 ≤ 𝑛 = 𝑃𝑠(𝑛 + 𝑚 − 𝑖),1 ≤ 𝑖 ≤ 𝑛 𝜂*(𝑣𝑢1) = (𝑠 − 2)(𝑚 − 1)2 − (𝑠 − 4)(𝑚 − 1) 2 = 𝑃𝑠(𝑚 − 1) 𝜂*(𝑢𝑗𝑢𝑗+1) = (𝑠 − 2)(𝑚 − 𝑗 − 1)2 − (𝑠 − 4)(𝑚 − 𝑗 − 1) 2 , 1 ≤ 𝑗 ≤ 𝑚 − 2 = 𝑃𝑠(𝑚 − 𝑗 − 1),1 ≤ 𝑗 ≤ 𝑚 − 2 hence the induced edge labels are the first 𝑚 + 𝑛 − 1polygonal numbers. hence 𝐶𝑇(𝑛, 𝑚) is a polygonal graceful graph. example 3.10. the octagonal graceful labeling of coconut tree 𝐶𝑇(5,6) is given in the following figure (5). figure (5) theorem 3.11. the bistar 𝐵𝑚,𝑛 admits polygonal graceful labeling. proof.: consider the graph 𝐵𝑚,𝑛. let 𝑉(𝐵𝑚,𝑛) = {𝑢, 𝑣, 𝑢𝑖, 𝑣𝑗/1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛} and 𝐸(𝐵𝑚,𝑛) = {𝑢𝑣, 𝑢𝑢𝑖, 𝑣𝑣𝑗/1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛}. then 𝐵𝑚,𝑛 has 𝑚 + 𝑛 + 2 vertices and 𝑚 + 𝑛 + 1 edges. define𝜂: 𝑉(𝐵𝑚,𝑛) → {0,1,2, . . . , 𝑃𝑠(𝑚 + 𝑛 + 1)} by 𝜂(𝑢) = 0 12),()( 12),()()( 1 1 −−− −−+= − − mjandevenisjifjmpu mjandoddisjifjmpuu sj sjj   13 a. rama lakshmi and m.p. syed ali nisaya 𝜂(𝑣) = (𝑠 − 2)(𝑚 + 𝑛 + 1)2 − (𝑠 − 4)(𝑚 + 𝑛 + 1) 2 = 𝑃𝑠(𝑚 + 𝑛 + 1)𝜂(𝑢𝑖) = (𝑠 − 2)(𝑚 + 𝑛 − 𝑖 + 1)2 − (𝑠 − 4)(𝑚 + 𝑛 − 𝑖 + 1) 2 , 1 ≤ 𝑖 ≤ 𝑚 = 𝑃𝑠(𝑚 + 𝑛 − 𝑖 + 1),1 ≤ 𝑖 ≤ 𝑚 𝜂(𝑣𝑗) = 𝜂(𝑣) − (𝑠 − 2)(𝑛 − 𝑗 + 2)2 − (𝑠 − 4)(𝑛 − 𝑗 + 2) 2 , 1 ≤ 𝑗 ≤ 𝑛 = 𝜂(𝑣) − 𝑃𝑠(𝑛 − 𝑗 + 2),1 ≤ 𝑗 ≤ 𝑛 clearly 𝜂 is injective and the induced edge labels are the first 𝑚 + 𝑛 + 1polygonal numbers. hence 𝐵𝑚,𝑛admits polygonal graceful graph. example 3.12: the hexagonal graceful labeling of 𝐵5,4 is given in the following figure (6). figure (6) 4. conclusion in this paper, polygonal graceful labeling is introduced and polygonal graceful labeling of some simple graphs is studied. this work contributes several new results to the theory of graph labeling. references [1] m. apostal, introduction to analytic number theory, narosa publishing house, second edition – (1991). [2] frank harary, graph theory, narosa publishing house – (2001). [3] j.a. gallian, a dynamic survey of graph labeling, the electronic journal of combinatorics, 16(2013), #ds6. [4] s.w. golomb, how to number a graph, graph theory and computing, r.c. read, academic press, new york (1972), 23-37. [5] d. s. t. ramesh and m.p. syed ali nisaya, some more polygonal graceful labeling of path, international journal of imaging science and engineering, vol. 6, no.1, january 2014, 901-905. [6] a. rosa, on certain valuations of the vertices of a graph, theory of graphs, (proc. internet symposium, rome, 1966, gordon and breach n.y. and dunod paris (1967), 349-355. 14 ratio mathematica vol. 34, 2018, pp. 5–13 issn: 1592-7415 eissn: 2282-8214 sums of generalized harmonic series with periodically repeated numerators (a, b) and (a, a, b, b) radovan potůček∗ received: 01-15-2018. accepted: 05-04-2018. published: 30-06-2018. doi:10.23755/rm.v34i0.405 c©radovan potůček abstract this paper deals with certain generalization of the alternating harmonic series – the generalized convergent harmonic series with periodically repeated numerators (a, b) and (a, a, b, b). firstly, we find out the value of the numerators b of the first series, for which the series converges, and determine the formula for the sum s(a) of this series. then we determine the value of the numerators b of the second series, for which this series converges, and derive the formula for the sum s(a, a) of this second series. finally, we verify these analytically obtained results and compute the sums of these series by using the computer algebra system maple 16 and its basic programming language. keywords: harmonic series, alternating harmonic series, sequence of partial sums, computer algebra system maple. 2010 ams subject classifications: 40a05, 65b10. ∗department of mathematics and physics, faculty of military technology, university of defence in brno, brno, czech republic; radovan.potucek@unob.cz 5 radovan potůček 1 introduction let us recall the basic terms, concepts and notions. for any sequence {ak} of numbers the associated series is defined as the sum ∞∑ k=1 ak = a1 + a2 + a3 + · · · . the sequence of partial sums {sn} associated to a series ∞∑ k=1 ak is defined for each n as the sum of the sequence {ak} from a1 to an, i.e. sn = a1 + a2 + · · · + an . the series ∞∑ k=1 ak converges to a limit s if and only if the sequence of partial sums {sn} converges to s, i.e. lim n→∞ sn = s. we say that the series ∞∑ k=1 ak has a sum s and write ∞∑ k=1 ak = s. the harmonic series is the sum of reciprocals of all natural numbers (except zero), so this is the series ∞∑ k=1 1 k = 1 + 1 2 + 1 3 + · · ·+ 1 k + · · · . the divergence of this series can be easily proved e.g. by using the integral test or the comparison test of convergence. in this paper we will deal with the series of the form ∞∑ k=1 ( a 2k −1 + b 2k ) and ∞∑ k=1 ( a 4k −3 + a 4k −2 + b 4k −1 + b 4k ) , where a,b are such numbers that these series converge. if we know, these two types of infinite series has not yet been studied in the literature. the author has previously in the papers [1], [2], and [3] dealt with the series of the form ∞∑ k=1 ( 1 2k −1 + a 2k ) , ∞∑ k=1 ( 1 3k −2 + 1 3k −1 + a 3k ) , ∞∑ k=1 ( 1 3k −2 + a 3k −1 + b 3k ) , and ∞∑ k=1 ( 1 4k−3 + a 4k−2 + b 4k−1 + c 4k ) , so that this contribution is a free follow-up to these three papers. let us note that in the previous issues of ratio mathematica, for example, papers [4] and [5] also deal with the topic infinite series and their convergence. 6 sums of generalized harmonic series with periodically repeated numerators 2 the sum of generalized harmonic series with periodically repeated numerators (a, b) we deal with the numerical series of the form ∞∑ k=1 ( a 2k −1 + b 2k ) = a 1 + b 2 + a 3 + b 4 + a 5 + b 6 + · · · , (1) where a,b ∈ r are appropriate constants for which the series (1) converges. this series we shall call generalized harmonic series with periodically repeated numerators (a,b). we determine the value of the numerators b, for which the series (1) converges, and the sum s(a) of this series. the power series corresponding to the series (1) has evidently the form ∞∑ k=1 ( ax2k−1 2k −1 + bx2k 2k ) = ax 1 + bx2 2 + ax3 3 + bx4 4 + ax5 5 + bx6 6 + · · · . (2) we denote its sum by s(x). the series (2) is for x ∈ (−1,1) absolutely convergent, so we can rearrange it and rewrite it in the form s(x) = a ∞∑ k=1 x2k−1 2k −1 + b ∞∑ k=1 x2k 2k . (3) if we differentiate the series (3) term-by-term, where x ∈ (−1,1), we get s′(x) = a ∞∑ k=1 x2k−2 + b ∞∑ k=1 x2k−1. (4) after reindexing and fine arrangement the series (4) for x ∈ (−1,1) we obtain s′(x) = a ∞∑ k=0 x2k + bx ∞∑ k=0 x2k, that is s′(x) = (a + bx) ∞∑ k=0 ( x2 )k . (5) when we summate the convergent geometric series on the right-hand side of (5) with the first term 1 and the ratio x2, where ∣∣x2∣∣ < 1, i.e. for x ∈ (−1,1), we get s′(x) = a + bx 1−x2 . 7 radovan potůček we convert this fraction using the cas maple 16 to partial fractions and get s′(x) = a− b 2(x + 1) − a + b 2(x−1) = a− b 2(1 + x) + a + b 2(1−x) , where x ∈ (−1,1). the sum s(x) of the series (2) we obtain by integration in the form s(x) = ∫ ( a− b 2(1 + x) + a + b 2(1−x) ) dx = a− b 2 ln(1 + x)− a + b 2 ln(1−x) + c. from the condition s(0) = 0 we obtain c = 0, hence s(x) = a− b 2 ln(1 + x)− a + b 2 ln(1−x) . (6) now, we will deal with the convergence of the series (2) in the right point x = 1. after substitution x = 1 to the power series (2) – it can be done by the extended version of abel’s theorem (see [6], p. 23) – we get the numerical series (1). by the integral test we can prove that the series (1) converges if and only if b + a = 0. after simplification the equation (6), where b = −a, we have s(x) = a + a 2 ln(1 + x) = a ln(1 + x) . for x = 1 and after re-mark s(1) as s(a), we obtain a very simple formula s(a) = a ln 2 , (7) which is consistent with the well-known fact that the sum of the alternating harmonic series ∞∑ k=1 (−1)k+1 k = 1− 1 2 + 1 3 − 1 4 + 1 5 − 1 6 + · · · is equal to ln 2. 3 the sum of generalized harmonic series with periodically repeated numerators (a, a, b, b) now, we deal with the numerical series of the form ∞∑ k=1 ( a 4k −3 + a 4k −2 + b 4k −1 + b 4k ) = a 1 + a 2 + b 3 + b 4 + · · · , (8) where a,b ∈ r are appropriate constants for which the series (8) converges. this series we shall call generalized harmonic series with periodically repeated numerators (a,a,b,b). we determine the value of the numerators b, for which the series (8) converges, and the sum s(a,a) of this series. 8 sums of generalized harmonic series with periodically repeated numerators the power series corresponding to the series (8) has evidently the form ∞∑ k=1 ( ax4k−3 4k −3 + ax4k−2 4k −2 + bx4k−1 4k −1 + bx4k 4k ) = ax 1 + ax2 2 + bx3 3 + bx4 4 + · · · . (9) we denote its sum by s(x). the series (9) is for x ∈ (−1,1) absolutely convergent, so we can rearrange it and rewrite it in the form s(x) = a ∞∑ k=1 x4k−3 4k −3 + a ∞∑ k=1 x4k−2 4k −2 + b ∞∑ k=1 x4k−1 4k −1 + b ∞∑ k=1 x4k 4k . (10) if we differentiate the series (10) term-by-term, where x ∈ (−1,1), we get s′(x) = a ∞∑ k=1 x4k−4 + a ∞∑ k=1 x4k−3 + b ∞∑ k=1 x4k−2 + b ∞∑ k=1 x4k−1. (11) after reindexing and fine arrangement the series (11) for x ∈ (−1,1) we obtain s′(x) = a ∞∑ k=0 x4k + ax ∞∑ k=0 x4k + bx2 ∞∑ k=0 x4k + bx3 ∞∑ k=0 x4k, that is s′(x) = (a + ax + bx2 + bx3) ∞∑ k=0 ( x4 )k . (12) after summation the convergent geometric series on the right-hand side of (12) with the first term 1 and the ratio x4, where ∣∣x4∣∣ < 1, i.e. for x ∈ (−1,1), we get s′(x) = a + ax + bx2 + bx3 1−x4 = (a + bx2)(1 + x) (1 + x2)(1−x)(1 + x) = a + bx2 (1 + x2)(1−x) . we convert this fraction using the cas maple 16 to partial fractions and get s′(x) = a + b 2(1−x) + a− b + ax− bx 2(1 + x2) , where x ∈ (−1,1). the sum s(x) of the series (9) we obtain by integration: s(x) = ∫ ( a + b 2(1−x) + a− b 2(1 + x2) + (a− b)x 2(1 + x2) ) dx = = − a + b 2 ln(1−x) + a− b 2 arctanx + a− b 4 ln(1 + x2) + c. 9 radovan potůček from the condition s(0) = 0 we obtain c = 0, hence s(x) = − a + b 2 ln(1−x) + a− b 2 arctanx + a− b 4 ln(1 + x2). (13) now, we will deal with the convergence of the series (9) in the right point x = 1. after substitution x = 1 to the power series (9) we get the numerical series (8). by the integral test we can prove that the series (8) converges if and only if a + b = 0. after simplification the equation (13), where b = −a, we have s(x) = aarctanx + a 2 ln(1 + x2) = a 2 [ 2 arctanx + ln(1 + x2) ] . for x = 1 and after re-mark s(1) as s(a,a), we obtain a simple formula s(a,a) = a 4 (π + 2 ln 2) . (14) 4 numerical verification we have solved the problem to determine the sums s(a) and (a,a) above of the convergent numerical series (1) and (8) for several values of a (and for b = −a) by using the basic programming language of the computer algebra system maple 16. they were used two following very simple procedures sumab and sumaabb: sumab=proc(t,a) local r,k,s; s:=0; r:=0; for k from 1 to t do r:=a*(1/(2*k-1)1/(2*k)); s:=s+r; end do; print("s(",a,")=",evalf[9](s), "f=",evalf[9](a*ln(2))); end proc: sumaabb:=proc(t,a) local r,k,s; s:=0; r:=0; for k from 1 to t do r:=a*(1/(4*k-3)+ 1/(4*k-2)1/(4*k-1)1/(4*k)); s:=s+r; end do; print("s(",a,a,")=",evalf[9](s), "f=",evalf[9](a*(pi+2*ln(2))/4); end proc: 10 sums of generalized harmonic series with periodically repeated numerators for evaluation the sums s(106,a) and s(106,a,a) and the corresponding values s(a) and s(a,a) defined by the formulas (7) and (14) it was used this for-loop statement: for a from 1 to 10 do sumab(1000000,a); sumaabb(1000000,a); end do; the approximative values of the sums s(106,a), s(a), s(106,a,a), and s(a,a) rounded to seven decimals obtained by these procedures are written into the following table 1. let us note that the computation of 20 values s(a) and s(a,a) took almost 51 hours 26 minutes. the relative quantification accuracies r(a) = |s(106,a)−s(a)| s(106,a) of the sum s(a,106) and r(a,a) = |s(106,a,a)−s(a,a)| s(106,a,a) of the sum s(a,a,106) are stated in the fourth and eighth columns of table 1. these relative quantification accuracies are approximately between 4 ·10−7 and 2 ·10−7. a s(106,a) s(a) r(a) a s(106,a,a) s(a,a) r(a,a) 1 0.6931469 0.6931472 4·10−7 1 1.1319715 1.1319718 3·10−7 2 1.3862939 1.3862944 4·10−7 2 2.2639430 2.2639435 2·10−7 3 2.0794408 2.0794415 3·10−7 3 3.3959145 3.3959153 2·10−7 4 2.7725877 2.7725887 4·10−7 4 4.5278860 4.5278870 2·10−7 5 3.4657347 3.4657359 3·10−7 5 5.6598575 5.6598588 2·10−7 6 4.1588816 4.1588831 4·10−7 6 6.7918290 6.7918305 2·10−7 7 4.8520285 4.8520303 4·10−7 7 7.9238005 7.9268023 2·10−7 8 5.5451754 5.5451774 4·10−7 8 9.0557720 9.0557740 2·10−7 9 6.2383224 6.2383246 4·10−7 9 10.1877435 10.1877458 2·10−7 10 6.9314693 6.9314718 4·10−7 10 11.3197150 11.3197175 2·10−7 table 1: the approximate values of the sums of the generalized harmonic series with periodically repeating numerators (a,−a) and (a, a,−a,−a) for a = 1, 2, . . . , 10 5 conclusions in this paper we dealt with the generalized harmonic series with periodically repeated numerators (a,b) and (a,a,b,b), i.e. with the series ∞∑ k=1 ( a 2k −1 + b 2k ) = a 1 + b 2 + a 3 + b 4 + a 5 + b 6 + · · · 11 radovan potůček with the sum s(a) and with series ∞∑ k=1 ( a 4k −3 + a 4k −2 + b 4k −1 + b 4k ) = a 1 + a 2 + b 3 + b 4 + · · · with the sum s(a,a), where a,b ∈ r. we derived that the only value of the numerators b ∈ r, for which these series converge, are b = −a, and we also derived that the sums of these series are determined by the formulas s(a) = a ln 2 and s(a,a) = a 4 (π + 2 ln 2) . so, for example, the series ∞∑ k=1 ( 5 2k −1 − 5 2k ) = 5 2 ∞∑ k=1 1 (2k −1)k has the sum s(5) . = 3.4657 and the series ∞∑ k=1 ( 5 4k −3 + 5 4k −2 − 5 4k −1 − 5 4k ) = 5 4 ∞∑ k=1 32k2 −24k + 3 (4k −3)(2k −1)(4k −1)k has the sum s(5,5) . = 5.6599. finally, we verified these two main results by computing some sums by using the cas maple 16 and its basic programming language. these generalized harmonic series so belong to special types of convergent infinite series, such as geometric and telescoping series, which sum can be found analytically and also presented by means of a simple numerical expression. from the derived formulas for s(a) and s(a,a) above it follows that a = s(a) ln 2 and a = 4s(a,a) π + 2 ln 2 . these relations allow calculate the value of the numerators a for a given sum s(a) or s(a,a), as illustrates the following table 2: s(a) a s(a,a) a 1 1.4427 1 0.8834 ln 0.5 . = −0.6391 −1 (−π −2 ln 2)/4 .= −1.1320 −1 ln 2 . = 0.6391 1 (π + 2 ln 2)/4 . = 1.1320 1 table 2: the approximate values of the numerators a for some sums s(a) and s(a, a) 12 sums of generalized harmonic series with periodically repeated numerators 6 acknowledgements the work presented in this paper has been supported by the project ”rozvoj oblastı́ základnı́ho a aplikovaného výzkumu dlouhodobě rozvı́jených na katedrách teoretického a aplikovaného základu fvt (k215, k217)” výzkumfvt (dzro k-217). references [1] r. potůček, sums of generalized alternating harmonic series with periodically repeated numerators (1,a) and (1,1,a). in: mathematics, information technologies and applied sciences 2014 post-conference proceedings of selected papers extended versions. university of defence, brno, (2014), 83-88. isbn 978-8-7231-978-7. [2] r. potůček, sum of generalized alternating harmonic series with three periodically repeated numerators. mathematics in education, research and applications, vol. 1, no. 2, (2015), 42-48. issn 2453-6881. [3] r. potůček, sum of generalized alternating harmonic series with four periodically repeated numerators. in: proceedings of the 14th conference on applied mathematics aplimat 2015. slovak university of technology in bratislava, publishing house of stu, slovak republic, (2015), 638-643. isbn 978-80-227-4314-3. [4] a. fedullo, kalman filters and arma models. in: ratio mathematica, vol. 14 (2003), 41-46. issn 1592-7415. available at: http://eiris.it/ojs/index.php/ratiomathematica/article/view/11. accessed july 8, 2017. [5] f. bubenı́k and p. mayer, a recursive variant of schwarz type domain decomposition methods. in: ratio mathematica, vol. 30 (2016), 35-43. issn 1592-7415. available at: http://eiris.it/ojs/index.php/ratiomathematica/article/view/4. accessed july 8, 2017. [6] m. hušek and p. pyrih, matematická analýza. matematicko-fyzikálnı́ fakulta, univerzita karlova v praze, (2000), 36 pp. available at: http://matematika.cuni.cz/dl/analyza/25-raf/lekce25-raf-pmax.pdf. accessed july 8, 2017. 13 ratio mathematica volume 40, 2021, pp. 87-111 on odd integers and their couples of divisors giuseppe buffoni* abstract a composite odd integer can be expressed as the product of two odd integers. possibly, this decomposition is not unique. from 2n + 1 = (2i + 1)(2j + 1) it follows that n = i + j + 2ij. this form of n characterizes the composite odd integers. it allows the formulation of simple algorithms to compute all the couples of divisors of odd integers and to identify the odd integers with the same number of couples of divisors (including the primes, with the number of non trivial divisors equal to zero). the distributions of odd integers ≤ 2n+1 vs. the number of their couples of divisors have been computed up to n ' 5 107, and the main features are illustrated. keywords: divisor computation; odd integer distribution vs. divisor number. 2020 ams subject classifications: 11axx, 11yxx. 1 1 introduction: characterization of composite odd and prime numbers let n be the set of positive integers and p the set of prime numbers with the exception of 2. composite odd integers 2n + 1, n ∈ n, may be expressed as product of two odd integers, 2n + 1 = (2i + 1)(2j + 1), i,j ∈ n, (1) or of more than two odd integers, i.e. as product of two odd integers in different ways. the decomposition (1) implies that *cnr-imati “enrico magenes”, via a. corti 12, 20133 milano, italy; giuseppe.buffoni9av2@alice.it. 1received on june 8th, 2021. accepted on june 23rd, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.618. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 87 g.buffoni n = kij = i + j + 2ij, (2) which may also be rewritten in the form n = kij = i(j + 1) + (i + 1)j. (3) either equation (2) or (3) specifies the structure of a composite odd integer 2n+ 1. let k ⊂ n be the set of the integers kij ∀i,j ∈ n. since any odd integer 2n + 1 greater than one is either a composite or a prime number, it follows that n ∈ k ⇐⇒ 2n + 1 ∈ n\p, or, equivalently, n ∈ n\k ⇐⇒ 2n + 1 ∈ p. remark. more involved characterizations of prime numbers can be formulated. they are obtained starting from the observation that all prime numbers greater than c ∈ n, are of the form c#h + ι, where c# represents c primorial, h,ι ∈ n, and ι < c# is coprime to c#, i.e. gcd(ι,c#) = 1. as an example, let c = 4, c# = 6; thus, all prime numbers > 4 may be expressed as 6h + ι with ι = 1, 5. since 6h + 5 = 6(h + 1) − 1, then all prime numbers may be expressed in the form 6h± 1, with the exception of 2 and 3. let the odd integer 2n + 1 be written as 2n + 1 = 6h± 1, so that either n = 3h or n = 3h− 1. for composite integers n = kij, and consequently 3 should be a dvisor of either kij or kij + 1. the paper is organized as follows. in section 2 varios formulations of the relationship between n and the pair (i,j) are viewed. an algorithm to compute the divisors of an odd integer is described; it can also be used as a primality test. in sections 3 and 4 it is shown how odd integers with the same number of couples of divisors can be identified. moreover, the distributions of odd integers ≤ 2n+ 1 vs. the number of their couples of divisors are computed up to n = 5 107 and illustrated. some concluding remarks can be found in section 5. details of calculations are reported in appendix. 2 the relationship between n and the pair (i,j) the functional relationship between a composite integer 2n+ 1 and the factors 2i + 1, 2j + 1 of its decompositions, or between n, i, j, can be written in different forms. the decomposition (1) is an inverse proportional relationship (hyperbolic relation) between 2i + 1 and 2j + 1. here and in the following it is assumed that i ≤ j, so that 2i + 1 ≤ √ 2n + 1 ≤ 2j + 1 (equality holds iff i = j), or equivalently 88 on odd integers and their couples of divisors i ≤ in = 1 2 (−1 + √ 2n + 1) ≤ j. (4) the relation (1) has been written in the forms (2) and (3). these equations define the entries of the matrix k = {kij}, used for the computation of the distribution of odd integers vs. the number of their couples of divisors. properties of k can be found in appendix 1. by making explicit the variable j, (2) can be written in the form of an homographic function j = φn(i) = n− i 2i + 1 , 1 ≤ i ≤ in. (5) thus, 2i + 1 is a divisor of both 2n + 1 and n − i. from (12) in appendix 1, it follows that 2i + 1 is also a divisor of n−kii. equation (5) can be used to compute the couples of divisors of an integer 2n+1 by means of the following algorithm: given n, compute φn(i) for i = 1, 2, ..., [in], where [·] is the integer part of the real argument; if for some i = iq we obtain that jq = φn(iq) ∈ n, then 2iq + 1 ≤ √ 2n + 1 ≤ 2jq + 1 is a couple of divisors of 2n + 1. the order of the number of operations is √ n/2. the algorithm can also be used as a primality test: if the computed φn(i) /∈ n ∀i, then 2n + 1 is a prime. the functions y = φn(x), x+y, xy, y−x, of the real variable x, are monotone for 0 ≤ x ≤ in (figure 1). in, defined in (4), is the unique positive solution to the equation φn(x) = x, i.e. 2x2 + 2x−n = 0. the point x = in corresponds to the minimum of x + y, to the maximum of xy, and, obviously, to y −x = 0. by means of a change of variables, the relationship (1) can be put in the form 2n + 1 = (s + t)(s− t) = s2 − t2, with s = i + j + 1, t = j − i, (6) while (2) and (3), representing partitions of the integer n in two sections, can be put in linear forms n = s + 2t, with s = i + j, t = ij, (7) n = s + t, with s = i(j + 1), t = (i + 1)j. (8) equation (6) shows the well known fact that composite odd integers can be written as a difference of two squares in different ways, while for a prime only holds the decomposition 2n + 1 = (n + 1)2 −n2. 89 g.buffoni figure 1: top left y = φn(x), top right x + y, bottom left xy, bottom right y−x. circle: point x = in on the x axis, and corresponding points on the curves. n = 50, in = 4.52. 90 on odd integers and their couples of divisors given n,s ∈ n, it is possible to prove when s and t = n−s can be expressed as either in (7) or in (8). the details are reported in appendix 2: it is shown that i,j are solutions to second order equations, and they are integer satisfying either (7) or (8), iff the square root of a quadratic form in n and s is an integer, 3 identification of odd integers ≤ 2n + 1 with the same number of couples of divisors let 2m + 1 be a composite integer and let ψ(m) = number of couples of divisors of 2m + 1. obviously, ψ(m) is also equal to the number of divisors of 2m + 1 ≤ √ 2m + 1. if ψ(m) = ν, then the entry m = kij, with i ≤ j, appears ν times in the matrix k = {kij}. composite integers 2m + 1 with m ≤ n are identified by the pairs (i,j) such that 4 ≤ m = kij ≤ n. (9) by assuming i ≤ j, it follows that (9) holds for the pairs (i,j) ∈ ω(4,n) = {i,j ∈ n : i = 1, 2, ..., [in]; j = i, i + 1, ..., [φn(i)]}. an estimation of the number of these pairs as n −→ +∞ is given by κ∗n ' n( 1 4 ln(n) + c). (10) with c = −0.4415. the details can be found at the end of appendix 1. in doing so we do not consider the couple (0,n), corresponding to the couple of trivial divisors (1, 2n + 1). the odd integers 2m+ 1, m ≤ n, with the same number of couples of divisors can be identified by means of the following algorithm: let ψ(m) = 0, m = 1, ...,n; compute kij, ∀(i,j) ∈ ω(4,n); for kij = m let ψ(m) = ψ(m) + 1. when ψ(m) = 0, then the integer 2m + 1 is a prime. all the integers 2m + 1, with ν couples of divisors, are identified by the values of m for which ψ(m) = ν. 91 g.buffoni furthermore, let πn(ν) = number of odd integers ≤ 2n + 1 with ν couples of divisors. πn(0) is the number of primes ≤ 2n + 1, except 2. πn(ν) is estimated as follows: for ν = 0 : πn(0) = number of ψ(m) = 0, for ν > 0 : πn(ν) = 1 ν ∑ ψ(m)=ν ψ(m). this approach, used to identify the prime numbers, is an equivalent formulation of the common implementation of the eratostene’s sieve (see for example the c program source in (2), section 6.3). in this case ψ(m) could be a logical variable. the algorithm may be easily applied to the integers in a generic set [2a + 1, 2n + 1], with 4 < a < n, to identify either the odd integers in this interval with the same number of couples of divisors or the primes. the inequalities identifying these integers, a ≤ kij ≤ n, with i ≤ j, hold for the pairs (i,j) ∈ ω(a,n) = {i,j ∈ n : i = 1, 2, ..., [in]; j = ja(i),ja(i)+1, ..., [φn(i, )]}, where: when i ≤ [ia] : either ja(i) = [φa(i)] + 1, φa(i) /∈ n, or ja(i) = φa(i) ∈ n; when i > [ia] : ja(i) = i. the set of the points (i,j) ∈ ω(a,n), with integer coordinates, is contained in a closed and convex set ω∗(a,n) of a plane. see figure 2, where the boundaries of this set are plain defined. some remarks on the case with large n and n − a << n can be found in appendix 3. 92 on odd integers and their couples of divisors figure 2: set ω∗(a,n) in the plane (x,y). continuous lines: y = φa(x) < y = φn(x); dotted line: y = x; asterisk: points (0,a), (0,n); circle: points (ia, 0), (in, 0), and corresponding points on the curves. different scales for x and y. 93 g.buffoni 4 distributions of odd numbers vs. the number of their couples of divisors the computation of the distributions πn(ν) has been performed, by means of the algorithm described in the previous section, for n ≤ 5 107, i.e. for odd integers 2n + 1 ≤ 108 + 1 (see the tables 1, 2 for some values of n). n = 5 n = 50 n = 5 102 n = 5 103 n = 5 104 ν 0 4 25 167 1228 9591 1 1 20 207 1964 18259 2 5 52 382 2824 3 56 925 11380 4 5 50 308 5 12 264 3200 6 0 4 32 7 1 128 2785 8 22 265 9 9 188 10 0 3 11 24 826 12 1 13 18 14 27 15 195 16 0 17 66 18 0 19 13 20 0 21 0 22 1 23 18 tot. 4 1 30 20 224 276 1686 3314 13052 36948 table 1: distribution πn(ν) of odd integers ≤ 2n + 1, with ν couples of divisors, for n = 5, 50, 5 102, 5 103, 5 104. let ν∗n = maximum number of couples of divisors of odd integers ≤ 2n + 1. 94 on odd integers and their couples of divisors n = 5 105 n = 5 106 n = 5 105 n = 5 106 ν ν 0 78497 664578 1 168522 1555858 2 21711 174188 3 126518 1336044 4 2030 14919 5 32314 309137 6 236 1758 7 42022 542740 8 2228 17481 9 2171 21649 10 16 74 11 13521 172181 12 2 12 13 238 2343 14 295 2376 15 5733 105676 16 0 4 17 1403 17487 18 0 0 19 545 8847 20 24 268 21 0 15 22 11 60 23 1537 34648 24 6 57 25 0 0 26 50 705 27 17 566 28 0 0 29 67 1503 30 0 0 31 179 8098 32 0 0 33 0 0 34 0 7 35 88 3589 36 0 0 37 0 4 38 0 0 39 13 922 40 0 11 41 0 69 42 0 0 43 0 0 44 0 65 45 0 0 46 0 0 47 6 1693 48 0 49 7 50 0 51 0 52 0 53 118 54 0 55 16 56 0 57 0 58 0 59 86 60 0 61 0 62 1 63 91 64 0 65 0 66 0 67 0 68 0 69 0 70 0 71 46 72 0 73 0 .. .. .. .. 78 0 79 3 tot. 105106 876564 tot. 394894 4123436 table 2: distribution πn(ν) of odd integers ≤ 2n + 1, with ν couples of divisors, for n = 5 105, 5 106. (since ν∗n = 143 for n = 5 10 7, this case is not reported here). 95 g.buffoni thus, πn(ν) = 0 for ν > ν∗n. it has been estimated (table 3, figure 3) that ν ∗ n increases as a power of n. the following approximation has been found by a fitting procedure ν∗n = µn λ, µ = e0.3992±0.1050, λ = 0.2586 ± 0.0083, 5 102 ≤ n ≤ 5 107. n 5 102 5 103 5 104 5 105 5 106 5 107 ν∗n 8 12 24 48 80 144 µ nλ 7.43 13.48 24.46 44.37 80.48 145.99 table 3: computed values of ν∗n and those produced by ν ∗ n = µn λ for some values of n. a visual inspection of the patterns of πn(ν) (the scattered plots of ln(πn(ν)) vs. ν are shown in the figures 4, 5) suggests that the odd integers with even and odd numbers of couples of divisors should belong to different populations. this view has to be considered only as a guess of the author, trying to interpret special features of πn(ν). anyhow, to avoid repetitions, we nickname these integers as ravens the odd integers with 2ν couples of divisors, cods the odd integers with 2ν + 1 couples of divisors. the primes, identified by ν = 0, are included in the ravens. we have that πn(2ν) = number of ravens ≤ 2n + 1, πn(2ν + 1) = number of cods ≤ 2n + 1. for n > 150, in general πn(2ν) < πn(2ν + 1). (11) only for few values of ν this inequality is not satisfied in the computed distributions (table 4). the number of ravens is less large than that of cods (see the last row in the tables 1, 2). 96 on odd integers and their couples of divisors figure 3: ν∗n vs. ln(n). circles: computed values, continuous line: approximation ν∗n = µ exp(λ ln(n)) for 2 10 2 ≤ n ≤ 5 107. n 5 102 5 103 5 104 5 105 5 106 5 107 8-9 8-9 8-9 20-21 20-21 24-25 20-21 24-25 24-25 26-27 44-45 32-33 44-45 74-75 80-81 n. couples 0 1 1 4 3 6 table 4: couples of ν for which inequality (11) is not satisfied. 97 g.buffoni figure 4: distributions ln(πn(ν) vs. ν for n = 5 102, 5 103 5 104, 5 105. circle: ravens, asterisk: cods. 98 on odd integers and their couples of divisors figure 5: distribution ln(πn(ν)) vs. ν for n = 5 106, n = 5 107. circle: ravens, asterisk: cods. 99 g.buffoni for ' 5 102 ≤ n ≤' 5 104 both the points πn(2ν) and πn(2ν + 1) show well distinct decreasing trends with ν (figure 4). however, points not belonging to the initial trends begin to appear for n ' 5 103. indeed, new branches (generally decreasing with ν) grow for increasing n, beginning at ν values not detected in the previous branches (figure 5). a branch may be roughly defined as a sequence of points in the plane (ν, ln π) which approximately lay on a straight line. for example, in the plot for n = 5 105 in figure 4, we can recognize two raven branches: the initial at the points ν = 0, 2, 4, 6, 8 and a second branch at ν = 8, 14, 20, 22, 24 (the point at ν = 8 is already present in the distribution for n = 5 103), and a single point at ν = 26. moreover, four cod branches: the initial at the points ν = 1, 3, 7, 11, 15, 23, 31, 35, and then at ν = 5, 9, 13, at ν = 17, 19, 27, and at ν = 29, 39. the attribution of a point to a branch is sometimes uncertain. indeed, the interpretation of the evolution of the distributions πn(ν) with n in terms of growing branches is arbitrary. the straight lines in the plane (ν, ln π) approximating the initial trends of both ravens and cods are estimated by a fitting procedure (table 5, figure 6). n α±σ β ±σ σ ravens 5 105 11.3131 ± 0.2663 −0.8846 ± 0.0377 0.3901 5 106 13.4700 ± 0.2292 −0.9257 ± 0.0324 0.3358 5 107 15.7156 ± 0.1925 −0.9823 ± 0.0272 0.2820 cods 5 105 12.2196 ± 0.1155 −0.2234 ± 0.0059 0.1971 5 106 14.3920 ± 0.1244 −0.1770 ± 0.0063 0.2122 5 107 16.4599 ± 0.2084 −0.1484 ± 0.0106 0.3557 table 5: initial branches of πn(ν): coefficients of the linear relationship ln(πn(ν)) = α+βν and their standard deviations σ. last column: σ of ln(πn(ν)). all the points (ν, ln(πn(ν)) are contained in a bounded region of the plane (ν, ln π) (figures 4, 5). this region is bounded from the bottom by the initial branch of ravens, starting from the number of primes ln(πn(0)) and ending in ν ' 20, and then by the axis ln π = 0. from the top by the initial branch of cods, starting from ln(πn(1)) and ending in ν ' 40, and then by sparse decreasing cod points, belonging to different branches. the upper boundary can be approximated by a straight line with ' the same slope of the initial cod trend. a guess about the description of the evolution of πn(ν) with n has been sug100 on odd integers and their couples of divisors figure 6: top: initial branch of cods at ν = 1, 3, 7, 11, 15, 23, 31, 35. bottom: initial branch of ravens at ν = 0, 2, 4, 6, 10, 12. triangle: n = 5 105; square: n = 5 106; circle: n = 5 107. continuous line: linear approximation. 101 g.buffoni gested by the most simple formula ((1), p. 8) approximating the number of primes ≤ 2n + 1: πn(0) = 2n + 1 ln(2n + 1) . taking into account that ln(ln(n)) ' 1.2334 + 0.0992 ln(n), 102 ≤ n ≤ 107, this relationship can be approximated by ln(πn(0)) ' −0.5403 + 0.9008 ln(n). fitting of ln(πn(ν)) to the linear expression α + β ln(n) has been carried out for ν = 0, 1, 2, 3 (table 6, figure 7). the straight lines are ' parallel for the ravens ν = 0, 2, while the lines for the codes ν = 1, 3 show different slopes (the line for ν = 3 is not shown in figure 7 for clearness of the figure). it is worth here to remind that a logarithmic approximation of a quantity may lead to a rough estimation of the quantity. ν α±σ β ±σ σ 0 −0.4902 ± 0.0755 0.8993 ± 0.0068 0.0847 1 −0.7130 ± 0.0282 0.9714 ± 0.0025 0.0317 2 −1.7076 ± 0.0550 0.8944 ± 0.0049 0.0617 3 −2.3972 ± 0.1632 1.0721 ± 0.0140 0.1528 table 6: ln(πn(ν)) vs. ln(n): coefficients of the linear relationship ln(πn(ν)) = α + β ln(n) and their standard deviations σ. last column: σ of ln(πn(ν)). ten n−points, from n = 100 to n = 5 107 are used in fitting ln(πn(ν)) for ν = 0, 1, 2. since πn(3) is very small for n = 100, this point is not included for ν = 3. the ”regularity” of some relationships between πn(ν) (figure 8) may arouse some surprise. we have performed a survey on the ratios between πn(ν) with ν = 0, 1, 2, 3. the trends of the ratios πn(1)/πn(0), πn(3)/πn(0), πn(3)/πn(1) (figure 8 top), increasing with n, seem to be reasonable. on the other hand, the trends of the ratios πn(2)/πn(ν), ν = 0, 1, 3, (figure 8 bottom), are disturbing. this might be due to the shortage of ravens with ν = 2 detected. obviously, computations with n greater than n = 5 107, the maximum value here considered, should be carried out to confirm the results, and to try to explain the trends. a careful analysis to produce a thorough knowledge has to be hoped for. 102 on odd integers and their couples of divisors figure 7: distributions ln(πn(ν)) vs. ln(n). circle: πn(0), asterisk: πn(1), square: πn(2). continuous line: linear approximation. 5 concluding remarks we have focused our attention on the computation of the couples of divisors of odd integers. indeed, any even integer can be written in the form 2m (2n + 1), with m ≥ 1, n ≥ 0. thus, it is characterized by the power of two, and possibly by an odd integer with its divisors. the following considerations hold for the computed distributions for n up to 5 107. for small n the following inequalities hold: πn(0) > πn(1) > πn(2) > πn(ν), ν > 2, n < 149. πn(0) is the number of primes, πn(1) is the number of cods either products of two primes or primes cubed, while πn(2) is the number of ravens either products of primes by primes squared or primes to the fourth. the previous inequalities can be explained by the following reasonings: for small n, (1) the density of primes is high, and (2) the prime factors in the divisors of both cods and ravens should be small. as an example, the possible decompositions of cods ≤ 101 with ν = 1 are reported here: 103 g.buffoni figure 8: ratios of distributions πn(ν) vs. ln(n). top: πn(1)/πn(0) asterisk, πn(3)/πn(0) circle, πn(3)/πn(1) square. bottom: πn(2)/πn(0) asterisk, πn(2)/πn(1) circle, πn(2)/πn(3) square. 104 on odd integers and their couples of divisors p1pi, i = 1, ..., 10; p2pi, i = 2, 3, ..., 7; p3pi, i = 3, 4, 5; p1p 2 1; where p1 = 3, p2 = 5, ... are the primes. thus, for small n, few factors produce cod integers ≤ 2n + 1. for increasing n, the inequality πn(1) > πn(ν), ν 6= 1, hold. for n = 149 we have πn(0) = πn(1) (table 7. n πn(0) πn(1) πn(2) 50 25 20 5 100 45 40 10 148 61 60 16 149 61 61 16 150 61 62 16 200 78 83 21 250 94 104 25 table 7: the transition from πn(0) > πn(1) to πn(0) < πn(1). the distributions πn(ν) have been obtained by identifying all the couples (2i+ 1, 2j + 1) of divisors of the integers 2m + 1 with m = kij ≤ n. for large n the number κ∗n of kij ≤ n is n(ln(n)/4 + c) (10). the number k∗a+1n of kij such that a + 1 ≤ kij ≤ n can be estimated by k∗a+1n = k ∗ n −k ∗ a = 1 4 (n ln(n) −a ln(a)) + c(n−a). under the assumption n−a << a < n it follows that 1 < n a = 1 + ( n a − 1) << 2, so that 0 < n a − 1 << 1. thus, k∗a+1n = k ∗ n−k ∗ a = (n−a)( 1 4 ln(n) + c) + 1 4 a ln( n a ) = (n−a)( 1 4 ln(n) + c + 1 4 ). some explanations on the numerical computations are given. the algorithms described in sections 2 and 3 can be easily implemented in fortran language. the algorithm (in section 2) for the computation of the couples of divisors of a given integer 2n + 1 does not require the storage of large dimension vectors. it has been successfully used to determine the couples of divisors of odd composite integers (and whether a number is prime or composite), up to input numbers of 105 g.buffoni order 260 ' 1018. note that quadruple precision for floating point operations is necessary for numbers of order 260. we do not have recourse to computer algebra systems, with numbers of variable length (3; 4). the algorithm (in section 3) for the computaion of the distributions πn(ν) requires the storage of an integer*8 vector; the computation has been carried out up to the limit of the storage carrying capacity of the available computer (about a vector of 6 107 of integers*8 entries). the computation of the primes in a given interval [2a+ 1, 2n+ 1] has been performed either with small n−a ∈ [5, 50] and a up to 1018, or with large n−a ∈ [102, 4 107] and a up to 109. acknowledgments i am very grateful to fabrizio luccio (un. pisa) and alessandro zaccagnini (un. parma) for reading an old version of the work, and for their precise criticism, and to andrea cappelletti (enea pisa) and sara pasquali (cnr imati milano) for their prompt replies to my requests. references [1] t. m. apostol, introduction to analytic number theory. springer-verlag, berlin, 1976. [2] a. languasco and a. zaccagnini, introduzione alla crittografia. u. hoepli editore, milano, 2004. [3] pari/gp computer algebra system develoiped by h. cohen et al., 1985. available from: http://pari.math.u.bordeaux.fr/ [4] the gnu mp bignum library. available from: http:77gmplib.org/ appendix 1. the matrix k = {kij} obviously, the symmetry property holds for the elements kij of k: kij = kji,∀ i,j ∈ n. thus, they can be represented by means of a symmetric matrix. (see the table 8). since some composite odd numbers 2n + 1 may be expressed as product of two odd numbers in different ways, it follows that 2n + 1 = (2i1 + 1)(2j1 + 1) = (2i2 + 1)(2j2 + 1) =⇒ ki1j1 = ki2j2, 106 on odd integers and their couples of divisors as it can be observed in the matrix k (table 8). the number of couples (i,j) such that n = kij, if they exist, is the number of decompositions of 2n + 1 in two factors. −↓ i j → 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 2 12 17 22 27 32 37 42 47 52 57 62 67 72 77 3 24 31 38 45 52 59 66 73 80 87 94 101 108 4 40 49 58 67 76 85 94 103 112 121 130 139 5 60 71 82 93 104 115 126 137 148 159 170 6 84 97 110 123 136 149 162 175 188 201 7 112 127 142 157 172 187 202 217 232 8 144 161 178 195 212 229 246 263 9 180 199 218 237 256 275 294 10 220 241 262 283 304 325 11 264 287 310 333 356 12 312 337 362 387 13 364 391 418 14 420 449 15 480 table 8: matrix k = {kij} for 1 ≤ i ≤ j ≤ 15. i=row and j=column index. besides the symmetry identity, the elements kij satisfy other combinatorial properties, obtained from the equation (2n + 1)(2kij + 1) = 2knkij + 1. the identities kpkqr = kqkrp = krkpq, p,q,r ∈ n follow from the product of three odd numbers (2p + 1)(2q + 1)(2r + 1), while the identities kkpqkrs = kkprkqs = ... = kpkqkrs = kqkpkrs = ..., p,q,r,s ∈ n follow from the product of four odd numbers (2p + 1)(2q + 1)(2r + 1)(2s + 1). obviously, more involved identities are obtained from products of more than four odd numbers. the quantities kij − i = (2i + 1)j and kij − j = i(2j + 1) are divisible by 2i + 1, j and by i, 2j + 1, respectively. moreover, kij can be written in the form 107 g.buffoni kij = kii+(2i+1)(j−i), i = 1, 2, ..., j = i, i+1, ... with kii = 2i2 +2i, (12) which leads to the following recurrence formula for the computation (by means of additions) of the entries of the i− th row, with i ≤ j, of the matrix k: kij = ki (j−1) + 2i + 1, j = i + 1, i + 2, ... now we estimate κ∗n = number of pairs (i,j) with 1 ≤ i ≤ j such that 4 ≤ kij ≤ n. it is given by κ∗n = in∑ i=1 [qn(i)]. where qn(i) = φn(i) − i + 1 = 1 2 ( 2n + 1 2i + 1 − (2i− 1)), and here in denotes the integer part of the quantity defined in (4). qn(i) are decreasing with i, and qn(in) = 1 ≤ qn(i) ≤ qn(1) = n− 1 3 . by direct calculation we have that qn = in∑ i=1 qn(i) = = (n + 1 2 ) in∑ i=1 1 2i + 1 − 1 2 i2n = (n + 1 2 ) in∑ i=1 1 2i + 1 − 1 4 (1 + n− √ 2n + 1). taking into account the logarthmic growth of the harmonic series, we have for n large enough in∑ i=1 1 2i + 1 ' ln( 2in + 1√ in ) + γ 2 − 1, where γ ' 0.5772 is the euler-mascheroni constant. it follows that 108 on odd integers and their couples of divisors qn n −→ ln(2 √ in + 1 √ in ) + γ 2 − 5 4 ' 1 4 ln(n) + c as n −→ +∞, with c = 0.5(1.5 ln(2) + γ − 2.5) = −0.4415. since the following inequalities qn − in ≤ k∗n ≤ qn. hold, and in/n −→ 0 as n −→ +∞, for the increasing function κ∗n/n we have that κ∗n n −→ 1 4 ln(n) + c as n −→ +∞. a linear fit of κ∗n/n vs. ln(n), for 10 ≤ n ≤ 1017.5, produces the line κ∗n/n ' (−0.4017 ± 0.0102) + (0.2486 ± 0.0004) ln(n). appendix 2. the partitions n = s + 2t and n = s + t equation (2) represents a partition of the integer n in two sections s = i + j and n−s = 2ij, with 2s ≤ n (2s = n iff i = j = 1). (13) since i ∈ [1,in] and j = φn(i), the bounds for s in the partition (13) are given by in + φn(in) = 2in and 1 + φn(1) (see figure 1, plot of x + y). therefore, the set of admissible values for s is ω1 = [2in, n + 2 3 ]. (14) from (13) it follows that i and j are the positive integer solutions,if they exist, to the equation x2 −sx + n−s 2 = 0. (15) the solutions to (15) are x± = 1 2 (s± √ ∆, with ∆ = s2 + 2s− 2n. (16) since ∆ and s have the same parity, positive integer solutions exist iff 109 g.buffoni ∃s ∈ ω1 : √ ∆ ∈ n. equation (3) represents another partition of the integer n in two sections s = i(j+1) and n−s = (i+1)j, with 2s ≤ n (2s = n iff i = j). (17) the set of admissible values for s is ω2 = [ n + 2 3 , n 2 ]. (18) from (17) it follows that i is solution to the following equation i2 + (n− 2s + 1)i−s = 0, and j = i + n− 2s. the results are given by i = 1 2 [−1 − (n− 2s) + √ ∆], j = 1 2 [−1 + (n− 2s) + √ ∆], where ∆ = 2n + 1 + (n− 2s)2. since ∆ and s have the same parity, positive integer solutions to the system (17) exist iff ∃s ∈ ω2 : √ ∆ ∈ n. we can summarize the reasonings on the partitions of n in the following implications: n ∈ k, ∆ = s2 + 2s− 2n ⇐⇒∃s ∈ ω1 : √ ∆ ∈ n, n ∈ k, ∆ = 4s2 − 4ns + 2n + 1 ⇐⇒∃s ∈ ω2 : √ ∆ ∈ n. 110 on odd integers and their couples of divisors appendix 3. remarks on the sets ω(a,n) and ω∗(a,n) here we consider the case n−a << ia = min(a,n,ia,in). (19) for example, this situation happens when we are looking for very few primes in [2a + 1, 2n + 1] with large a. in virtue of the prime distribution ((1), p. 8) we should choose n−a ' ι 2 ln(2a + 1), with 1 ≤ ι ≤ 10. the difference between the top and bottom boundary lines of ω∗(a,n) (figure 2) is φn(i) −φa(i) = n−a 2i + 1 . it is decreasing with i, and φn(i) −φa(i) < 1 for i ≥ [i0], i0 = n−a− 1 2 + 1 < ia. (20) when (20) holds, at most only one integer is in [φa(i),φn(i)]. it follows that for i ≥ [i0] the points of ω∗(a,n) with integer coordinates are the points (i,j) with j = [φa(i) + 1] = [φn(i)]. furthermore, since ia < 2 √ n + a, from (19) we have that also n − a < 2 √ n + a, which implies in−ia < 1. thus, the boundary of ω∗(a,n) between the points (ia,ia) and (in,in) (figure 2) does not contain points with integer coordinates. in the limit case a = n, the set ω∗(a,n) (figure 2) reduces to the curve y = φn(x) for 0 ≤ x ≤ in, and ω(n,n) is then defined by (i,j) ∈ ω(n,n) = {i = 1, 2, ..., [in] : φn(i) ∈ n, j = φn(i)}. the algorithm described in section 2 identifies the points of the set ω(n,n). 111 ratio mathematica vol. 35, 2018, pp. 101-109 issn: 1592-7415 eissn: 2282-8214 note on heisenberg characters of heisenberg groups alieh zolfi ∗and ali reza ashrafi † received: 08-08-2018. accepted: 21-10-2018. published: 31-12-2018 doi:10.23755/rm.v35i0.429 c©alieh zolfi and ali reza ashrafi abstract an irreducible character χ of a group g is called a heisenberg character, if kerχ ⊇ [g, [g,g]]. in this paper, the heisenberg characters of the quaternion heisenberg, generalized heisenberg, polarised heisenberg and three other types of infinite heisenberg groups are computed. keywords: heisenberg character, heisenberg group. 1 introduction suppose g is a finite group and v is a vector space over the complex field c. a representation of g is a homomorphism ϕ : g −→ gl(v ), where gl(v ) denotes the group of all invertible linear transformations v −→ v equipped with ∗department of pure mathematics, faculty of mathematical sciences, university of kashan, kashan 87317-53153, i. r. iran †department of pure mathematics, faculty of mathematical sciences, university of kashan, kashan 87317-53153, i. r. iran; ashrafi@kashanu.ac.ir 101 alieh zolfi and ali reza ashrafi . composition of functions. the commutator subgroup [g,g] is the subgroup generated by all the commutators [x,y] = xyx−1y−1 of the group g. an irreducible character χ of a group g is called a heisenberg character, if kerχ ⊇ [g, [g,g]] [1]. suppose ϕ : g −→ gl(v ) is an irreducible representation with irreducible character χ. since [g,g′] ≤ kerχ, ϕ : g [g,g′] −→ gl(v ) is an irreducible representation of g [g,g′] . conversely, we assume that χ ∈ irr ( g [g,g′] ) and δ : g [g,g′] −→ gl(v ) affords the irreducible character χ. if γ : g −→ g [g,g′] denotes the canonical homomorphism then δoγ : g −→ g [g,g′] is an irreducible representation for g and ker δoγ ⊇ [g,g′]. this proves that there is a one to one correspondence between heisenberg characters of g and irreducible characters of g [g,g′] , see [2, 8] for details. marberg [8] in his interesting paper proved that the number of heisenberg characters of the group un(q) is a polynomial in q − 1 with nonnegative integer coefficients, with degree n − 1, and whose leading coefficient is the (n − 1)−th fibonacci number. the present authors [1], characterized groups with at most five heisenberg characters. the aim of this paper is to compute all heisenberg characters of five classes of infinite heisenberg groups. these are as follows: 1. suppose t denotes the set of all complex numbers of unit modulus and h = rn × rn × t. define (y1,x1,z1)(y2,x2,z2) = (y1 + y2,x1 + x2,e −2πiy2.x1z1z2). it is easy to see that h is a group under this operation. this group is called the heisenberg group of second type [5]. 2. the polarised heisenberg group h3n is defined as the set of all triples in rn ×rn ×r under the multiplication (x,y,z)(a,b,c) = (x + a,y + b,z + c + 1 2 (x.b−y.a)), see [3] for details. 3. suppose a = (a1,a2, . . . ,an) is an n−tuple in rn, where ai’s are positive real constants, 1 ≤ i ≤ n. following tianwu and jianxun [10], we define a group operation on han = r n ×rn ×r given by (x,y,z)(r,s,t) = (x + r,y + s,z + t + 1 2 n∑ j=1 aj(rjyj −sjxj)). 102 note on heisenberg characters of heisenberg groups this group is called the generalized heisenberg group. in the mentioned paper, the authors proved that the group operation of the generalized heisenberg group can be simplified in the following way: suppose x = (x1,x2, . . . ,xn) and y = (y1,y2, . . . ,yn). define x ∗ y = (x1y1,x2y2, . . . ,xnyn) and ab = ∑n j=1 ajbj. if c = (c1,c2, . . . ,cn) and λ ∈ r then we can see that (i) x ∗ (y + c) = x ∗ y + x ∗ c; (ii) (x ∗ y)c = x(y ∗ c); and (iii) (λx) ∗ y = λ(x ∗ y). therefore, the group operation of the generalized heisenberg group can be written as (x,y,z)(r,s,t) = (x + r,y + s,z + t + 1 2 ((a∗ r)y − (a∗s)x)). 4. suppose h denotes the set of all of quaternion numbers with three imaginary units i,j and k such that i2 = j2 = k2 = ijk = −1. following liu and wang [7], we define the quaternion heisenberg group n as a nilpotent lie group with underlying manifold r4 ×r3. the group structure is given by (q,t)(p,s) = (q + p,t + s + 1 2 im(pq)), where p,q ∈ r4 and t,s ∈ r3. 5. following qingyan and zunwei [9], the heisenberg group hn of third type is a non-commutative nilpotent lie group, with the underlying manifold r2n ×r. the group operation can be given as: (x1,x2, . . . ,x2n,x2n+1)(x ′ 1,x ′ 2, . . . ,x ′ 2n,x ′ 2n+1) = (x1 + x ′ 1,x2 + x ′ 2, . . . ,x2n + x ′ 2n,x2n+1 + x ′ 2n+1 + 2 n∑ j=1 (x ′ jxn+j −xjx ′ n+j). 6. suppose hn = cn×r with group law defined by (z,t)·(w,s) = (z+w,t+ s + 2im(z.w)). this is our sixth class of heisenberg groups. following chang et al. [4], this group can be realized as the boundary of the siegel upper half-space un+1 in cn+1, where the group operation gives a group action on the hypersurface. throughout this paper our notation is standard and can be taken from the famous book of isaacs [6]. suppose g is a group and {{e} = a0,a1, . . . ,an = g} is a set of normal subgroups of g such that a0 � a1 � . . . � an = g. (1) 103 alieh zolfi and ali reza ashrafi . the sequence (1) is called a central series for g, if [g,ai+1] ≤ ai in which [g,h] denotes the subgroup of g generated by all commutators ghg−1h−1, where g ∈ g,h ∈ h. the group g is called nilpotent, if it has a central series. the nilpotency class of g, nc(g), is the length of its central series. the set of all irreducible characters of g is denoted by irr(g) and the trivial character of g is denoted by 1g. 2 main results the aim of this section is to compute the heisenberg characters of five different types of heisenberg groups. to do this, we first note that every linear character of a group g is heisenberg. this proves that all irreducible characters of abelian groups are heisenberg. lemma 2.1. all irreducible characters of a group g are heisenberg if and only if g is nilpotent of class two. proof. suppose nc(g) = 2. then [g,g′] = 1 and so all irreducible characters are heisenberg. if all irreducible characters are heisenberg then [g,g′] ≤ ∩χ∈irr(g) = {e}, as desired. theorem 2.2. all irreducible characters of the heisenberg groups h, h3n, h a n, n , hn and hn are heisenberg. proof. apply lemma 2.1. our main proof will consider five separate cases as follows: 1. the heisenberg group h. we first compute the derived subgroup h′. we have, h′ = 〈 [(x,y,z),(x ′ ,y ′ ,z ′ )] | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ h,z = eiθ1,z ′ = eiθ2 〉 = 〈(x + x ′ ,y + y ′ ,ei(θ1+θ2−2πx ′ y))(−x−x ′ ,−y −y ′ , ei(θ1+θ2+2πxy+2πx ′ y ′ +2πx ′ y)) | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ h,z = eiθ1,z ′ = eiθ2〉 = 〈 (0,0,e2πi(x.y ′ −x ′ .y) | x,y,x′,y′ ∈ rn 〉 . 104 note on heisenberg characters of heisenberg groups therefore, [h,h′] = 〈 (x,y,eiθ1)(0,0,eiθ2)(−x,−y,e−iθ1−2πixy)(0,0,e−iθ2) | (x,y,eiθ1) ∈ h,(0,0,eiθ2) ∈ h′ 〉 = 〈(x,y,eiθ1+iθ2)(−x,−y,e−i(θ1+θ2+2πixy) | (x,y,eiθ1) ∈ h,(0,0,eiθ2) ∈ h′〉 = {(0,0,1)}. so, all irreducible characters of h are heisenberg. 2. the heisenberg group h3n. the commutator subgroup of h 3 n can be computed as follows: (h3n) ′ = 〈[(x,y,z),(x ′ ,y ′ ,z ′ )] | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ h3n〉 = 〈(x,y,z)(x ′ ,y ′ ,z ′ )(x,y,z)−1(x ′ ,y ′ ,z ′ )−1 | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ h3n〉 = 〈(0,0,(x.y ′ −y.x ′ ) | x,y,x′,y′ ∈ rn〉. on the other hand, [h3n,(h 3 n) ′] = {(0,0,0)} and so all irreducible characters of this group are heisenberg. 3. the heisenberg group han. again, we first compute the commutator subgroup (han) ′. we have, (han) ′ = 〈[(x,y,z),(x ′ ,y ′ ,z ′ )] | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ han,a ∈ r n +〉 = 〈 (x + x ′ ,y + y ′ , 1 2 ((a∗x ′ )y − (a∗y ′ )x))(−x−x ′ ,−y −y ′ , + 1 2 ( (a∗−x ′ )(−y)− (a∗−y ′ )(−x) ) | (x,y,z),(x ′ ,y ′ ,z ′ ) ∈ han,a ∈ r n +〉 = 〈 (0,0,(a∗x ′ )y − (a∗y ′ )x) | x,y,x′,y′ ∈ rn,a ∈ rn+ 〉 . therefore, [han,(h a n) ′] = {(0,0,0)}. this shows that all irreducible characters are heisenberg. 4. the heisenberg group n . by definition of this group, we have n ′ = 〈[(p,t),(q,s)] | (p,t),(q,s) ∈n〉 = 〈(p + q,t + s + 1 2 im(qp))(−p− q,−t−s + 1 2 im(qp)) | p,q ∈ r4, t,s ∈ r3〉 = 〈(0,im(qp)) | p,q ∈ r4〉. therefore, we have again [n ,n ′] = {(0,0)}. now apply lemma 2.1 to deduce that all irreducible characters of this group are heisenberg. 105 alieh zolfi and ali reza ashrafi . 5. the heisenberg group hn. by definition of this group, (hn)′ = 〈 [(x1, . . . ,x2n,x2n+1),(x ′ 1, . . . ,x ′ 2n,x ′ 2n+1)] | (x1, . . . ,x2n,x2n+1),(x ′ 1, . . . ,x ′ 2n,x ′ 2n+1) ∈ h n 〉 = 〈(x1 + x ′ 1, . . . ,x2n + x ′ 2n,x2n+1 + x ′ 2n+1 + 2 ( n∑ j=1 (x ′ jxn+j −xjx ′ n+j) ) (−x1 −x ′ 1, . . . ,−x2n −x ′ 2n,−x2n+1 −x ′ 2n+1 + 2 ( n∑ j=1 (x ′ jxn+j −xjx ′ n+j) ) | (x1, . . . ,x2n,x2n+1),(x ′ 1, . . . ,x ′ 2n,x ′ 2n+1) ∈ h n〉 = 〈 (0, . . . ,4 ( n∑ j=1 (x ′ jxn+j −xjx ′ n+j) ) | x1, . . . ,x2n,x2n+1,x ′ 1, . . . ,x ′ 2n,x ′ 2n+1 ∈ r 〉 . therefore, [hn,(hn)′] = {(0, . . . ,0,0)} and by lemma 2.1 all irreducible characters of this group are heisenberg. 6. the heisenberg group hn. the derived subgroup of this group can be computed as follows: (hn)′ = 〈[(z,t),(w,s)] | (z,t),(w,s) ∈hn〉 = 〈(z,t)(w,s)(−z,−t)(−w,−s) | (z,t),(w,s) ∈hn〉 = 〈(z + w,t + s + 2im(zw))(−z −w,−t−s + 2im(zw)) | z,w ∈ cn, t,s ∈ r〉 = 〈(0,2im(zw −wz)) | z,w ∈ cn〉. therefore, [hn,(hn)′] = {(0,0,0)} and by lemma 2.1, all irreducible characters of this group are again heisenberg. this completes our argument. in the end of this paper we compute the factor groups of six types of heisenberg groups modulus their centers. theorem 2.3. the factor groups of all heisenberg groups modulus their centers 106 note on heisenberg characters of heisenberg groups can be computed as: h z(h) ∼= rn ×rn, h3n z(h3n) ∼= rn ×rn, han z(han) ∼= rn ×rn n z(n) ∼= r4, hn z(hn) ∼= rn ×rn, h z(h) ∼= cn. proof. an easy calculations show that z(h) = {(0,0,z)|z ∈ t}, z(h3n) = {(0,0,s)|s ∈ r}, z(han) = {(0,0,s)|s ∈ r}, z(hn) = {(0, . . . ,0,x2n+1)| x2n+1 ∈ r} and z(hn) ∼= r. therefore, hz(h) ∼= rn × rn, h 3 n z(h3n) ∼= rn × rn, han z(han) ∼= rn ×rn, h n z(hn) ∼= rn ×rn and h z(h) ∼= cn. so, it is enough to compute n z(n) ∼= r4. to do this, we assume that (p,t) ∈ z(n) is arbitrary. hence for each pair (q,s), (p,t)(q,s) = (q,s)(p,t). this proves that (p + q,s + t + 1 2 im(qp)) = (q + p,s + t + 1 2 im(pq)) and so im(pq) = 0. suppose p = p0 + ip1 + jp2 + kp3. then by considering three different values q = (1,0,0,0),(0,1,0,0),(0,0,1,0), we will have the following system of equations:  p1 + p2 + p3 = 0, p0 + p2 −p3 = 0, p0 −p1 + p3 = 0. hence z(n)∼= r3 and n z(n) ∼= r4 that completes the proof. 3 concluding remarks in this paper the heisenberg characters of six classes of heisenberg groups were computed. it is proved that all irreducible characters of these heisenberg groups are heisenberg. we also compute all factor groups of these heisenberg groups which show these factor groups are abelian and so all irreducible characters of these factor groups are again heisenberg. acknowledgment we are very thankful from the referee for his/her comments and corrections. the authors also would like to express our sincere gratitude to professor alireza abdollahi for discussion on the exact form of lemma 1 during the 48th annual 107 alieh zolfi and ali reza ashrafi . iranian mathematics conference in that was held in the university of hamedan. the research of the authors are partially supported by the university of kashan under grant no 364988/127. references [1] a. r. ashrafi and a. zolfi, on the number of heisenberg characters of finite groups, submitted. [2] m. boyarchenko and v. drinfeld, a motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic, arxiv:math/0609769v2. [3] a. brodlie, the representation theory of the heisenberg group and beyond, xi-th international conference symmetry methods in physics, prague, czech republic, june 21-24, 2004. [4] d.-c. chang, w. eby and e. grinberg, deconvolution for the pompeiu problem on the heisenberg group i, i. sabadini, d. c. struppa (eds.), the mathematical legacy of leon ehrenpreis, springer proceedings in mathematics 16, springer-verlag italia, 2012. [5] r. howe, on the role of the heisenberg group in harmonic analysis, american mathematical society. bulletin. new series 3 (2) (1980) 821–843. [6] i. m. isaacs, character theory of finite groups, ams chelsea publishing, providence, ri, 2006. [7] h.-p. liu and y.-z. wang, a restriction theorem for the quaternion heisenberg group, applied mathematics. a journal of chinese universities. ser. b 26 (1) (2011) 86–92. [8] e. marberg, heisenberg characters, unitriangular groups, and fibonacci numbers, journal of combinatorial theory, ser. a 119 (2012) 882–903. [9] w. qingyan and f. zunwei, sharp estimates for hardy operators on heisenberg group, frontiers of mathematics in china 11 (1) (2016) 155–172. 108 note on heisenberg characters of heisenberg groups [10] l. tianwu and h. jianxun, the radon transforms on the generalized heisenberg group, isrn mathematical analysis (2014) art. id 490601, 7 pp. 109 ratio mathematica volume 46, 2023 bounds on fuzzy dominator chromatic number of fuzzy soft bipartite graphs jahir hussain rasheed* afya farhana mohammed shaik† abstract an fsg gs(t,v) fuzzy’s soft dominator colouring (fsdc) is a suitable fuzzy soft colouring (fsc) where every node of a colour group is dominated by a vertex of gs(t,v). in the current work, we characterize the sharp bounds for the fuzzy dominator chromatic number (fdcn) of fuzzy soft bipartite graphs and we present limits on the fdcn of fuzzy soft bipartite graph in terms of the γe(gs(t,v )). furthermore, we classify fuzzy soft bipartite graphs into three classes based on fdcn. keywords:fuzzy soft bipartite graph, fuzzy dominator chromatic number, fuzzy soft path, fuzzy soft cycle, strong arcs. 2020 ams subject classifications: 05c72; 05c15. 1 *pg and research department of mathematics, jamal mohamed college (autonomous), affiliated to bharathidasan university, tiruchirappalli 620 024, tamil nadu, india. hssn jhr@yahoo.com. †pg and research department of mathematics, jamal mohamed college (autonomous), affiliated to bharathidasan university, tiruchirappalli 620 024, tamil nadu, india. afyafarhana@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1063. issn: 1592-7415. eissn: 2282-8214. ©r. jahir hussain et al. this paper is published under the cc-by licence agreement. 109 r. jahir hussain and m.s. afya farhana 1 introduction fuzzy soft graphs are a useful mathematical tool for simulating the ambiguity of the actual world in view of parameters. fuzzy soft graphs, which are often utilised in many different disciplines, including decision-making issues, combine fuzzy soft sets and the graph model. scheduling problems are just one of many practical issues involving the allocation of scarce resources for which graph colouring is used as a model. graph colouring also has an important place in discrete mathematics and combinatorial optimization. to address ambiguous issues in the fields of engineering, social science, economics, medical research, and environment, molodstov [1999] created the idea of soft set theory. fuzzy soft sets, a blend of a fuzzy set and a soft set, were first introduced by p.k. maji and biswas [2001]. thumkara and george introduced the idea of a soft graph in 2014. rosenfeld first proposed the idea of fuzzy graph theory in 1975. fuzzy soft graphs were independently introduced in 2015 by mohinta and samanta as well. akram and nawaz [2015] presented fuzzy soft graphs and examined their operations as well as several other graph theoretical ideas. domination is a fast growing area of graph theory study, and the numerous ways it is used to networks, distributed computers, social networks, and online graphs helps to explain why there is more interest in this subject. the dominator colouring problem in graphs was first described by gera [2007]. an appropriate colouring of a graph g with the extra characteristic that each vertex in the graph dominates an entire class is known as a dominator colouring of g. the smallest number of colour classes in a graph’s dominator colouring is known as the dominator chromatic number. we want to reduce the number of colour categories. in 2007,gera [2007] explored the dominator chromatic number for the hypercube and more broadly for bipartite graphs. he also presented dominator colorings in bipartite graphs. chellali and maffary [2012] studied dominator colorings in some classes of graphs. the idea of fuzzy dominator colouring in fuzzy graphs was created by hussain and fathima [2015]. they investigated the fuzzy dominator chromatic number for a number of fuzzy graphs and explored its boundaries. the fdcn of bipartite, middle, and subdivision fuzzy graphs was created by hussain and fathima [2015], and established its limits. domination in fuzzy soft graphs was the subject of study done by hussain and hussain [2017]. fuzzy dominator coloring applied to fuzzy soft graph yields fuzzy soft dominator coloring which concentrates on minimizing the number of color classes of fsg. a soft, fuzzy dominator fsg’s colouring should be done in such a way that every node of a colour group is dominated by a vertex of gs(t,v ). the proposed method concentrates on strength of connectedness, strong arc and strong neighbor of fuzzy soft bipartite graph in view of parameters from the existing method. 110 bounds on fuzzy dominator chromatic number of fuzzy soft bipartite graphs in this paper, we introduced bounds of fuzzy soft bipartite graphs with fuzzy dominator chromatic number. 2 preliminaries definition 2.1. let r be a parameter set and t is a subset of r, let v = {x1,x2,x3, ·· ·,xn} is a non-empty set. as well (i) α: t → f(v ) (v’s collection of all fuzzy subsets) e a α(e) = αe(say) αe : v → [0,1] (t,α) : fuzzy soft vertex (ii) β: t → f(v× v) (v× v’s collection of all fuzzy subsets) e a β(e) = βe (say) βe : v× v → [0,1] (t,β) : fuzzy soft edge followed by ((t,α),(t,β)) iff βe(x,y) ≤ αe(x) ∧ αe(y) for all e in t and this fsgs are written by gs(t,v ), it is referred to as a fuzzy soft graph (fsg). additionally, a fsg is a parametrized family unit of fuzzy graphs. definition 2.2. a path is intended to be a set of different points x1,x2, · · ·xn in an fsg such that for all e in t and βe (xi−1, xi) > 0, for all i = 1 to n. definition 2.3. if an fsg gs(t,v ) contains more than 1 smallest arc, ∀ e ∈ t it is referred to as a fuzzy soft cycle. definition 2.4. a fsg gs(t,v ) is supposed to be a fuzzy soft bipartite if the node set v can be divided into 2 non-empty sets v e1 and v e 2 such that v e 1 and v e 2 are fuzzy independent sets. these sets are called fuzzy bipartition of v, thus each efficient arc of fsg has one end in v e1 and other end in v e 2 , ∀ e ∈ t. definition 2.5. a fuzzy soft bipartite graph is complete if for an individual node v e1 , each single node of v e 2 is an efficient neighbour, e ∈ t. 3 fuzzy soft graphs with fuzzy dominator colouring definition 3.1. if βe(x,y) = β∞e (x,y), e ∈ t iff the arc (x,y) in fsg is said to be a strong arc, where β∞e (x,y), e ∈ t is the maximum strength of all pathways among x and y. definition 3.2. a soft, fuzzy dominator a fsg’s colouring should be done in such a way that every node of a colour group is dominated by a vertex of gs(t,v). 111 r. jahir hussain and m.s. afya farhana definition 3.3. in a fsdc of fsg, an fdcn is the least number of colour groups, and it is denoted as χefd (g s(t,v)), e ∈ t. 4 bounds on fuzzy soft bipartite graphs with fuzzy dominator chromatic number theorem 4.1. let gs(t,v) be a connected fuzzy soft bipartite graph. soon after 2 ≤ χefd(g s(t,v )) ≤bn 2 c+1, e ∈ t and these limits are precise. proof. take into account a connected fuzzy soft bipartite graph gs(t,v). every acceptable fuzzy soft colour must also be present in every fuzzy soft dominator colour, χef(g s(t,v )) = 2, hussain and farhana [2020], as a result χefd(g s(t,v )) ≥ 2, e ∈ t. to acquire the upper bound, let v e1 and v e 2 be the 2 bipartite sets of g s(t,v) with the condition |v e1 | ≤ |v e2 |. after that, allocate colours 1,2,...,|v e1 | to the nodes of v e1 and colour |v e1 | +1 to the nodes of v e2 , is a least fsdc. hence χefd(g s(t,v )) ≤|v e1 | + 1,since |v e1 |≤ b n 2 c, which implies χefd(g s(t,v )) ≤bn 2 c+ 1, e ∈ t.2 definition 4.1. consider a fsg gs(t,v) with n nodes. connect each node of gs(t,v) onto any one of the n isolated nodes, where n is the total number of nodes in the fsg.the resultant graph is called corona of gs(t,v) and is symbolized as cor(gs(t,v))= ((t,α1),(t,β1))where αe1(u) = αe(u),u ∈v, e∈t and αe(u) = αe1(u)∈ (0,1], if u is isolated, e∈t. βe1(u,v)= βe(u,v)∈ e, e∈t and βe1(u,v)= α e 1(u) ∧ αe(v), if u ∈ v and v is isolated, e∈t. remark: 1. obviously cor(gs(t,v)) is a fsg if gs(t,v) be a fsg. 2. if gs(t,v) has ’n’ nodes and ’e’ edges then cor(gs(t,v)) has 2n nodes and e+n edges. 3. cor(gs(t,v)) has k+n strong arcs if the no. of strong nodes in gs(t,v) is ’k’. now we discuss about the sharpness of lower bound and upper bound of theorem:4.1. since the lower bound is sharp, it may be inferred that the entire fuzzy soft bipartite graph’s fdcn is 2. the subsequent theorem proves the sharpness of the upper bound. theorem 4.2. having a ’n’-node fuzzy soft path gs(t,v) and n’ is the number of nodes in cor(gs(t,v )). then χefd(cor(g s(t,v )) = bn ′ 2 c+1, e ∈ t . proof. consider a fuzzy soft path having v1,v2,...,vn as nodes. noticeably gs(t,v) is a fuzzy soft bipartite graph, hussain and farhana [2020] and so 112 bounds on fuzzy dominator chromatic number of fuzzy soft bipartite graphs cor(gs(t,v )) is also a fuzzy soft bipartite graph. this implies every arc is a strong arc in cor(gs(t,v )). the number of nodes in gs(t,v) is n, then n’ = 2n nodes in cor(gs(t,v )). at this instant, we have to colour the nodes of cor(gs(t,v )). to obtain the least fuzzy soft dominator colouring, every node in minimum dominating set be full of distinctive colour. the nodes in gs(t,v)is strong adjoining to just one node of cor(gs(t,v )). this implies that every node of gs(t,v) is in a minimum dominating set. therefore γe(cor(gs(t,v )))=n, e∈t. color 1,2,...,n is distributed to the nodes of gs(t,v), and colour n + 1 is distributed to the left over nodes. this is the least fsdc of cor(gs(t,v )). hence we prove χefd(cor(g s(t,v ))) = bn ′ 2 c+1, e ∈ t which is the upper bound and it is sharp.2 now we prove that the values in the middle of 2 and bn ′ 2 c + 1 can be achieved as the fdcn of some fuzzy soft bipartite graphs of order n. theorem 4.3. let k be an integer with 2 ≤ k ≤bn 2 c+1, if so, a connected fuzzy soft bipartite graph gs(t,v) with nnodes and its fdcn is k. proof. consider a fuzzy soft path pek =((t,α1),(t,β1)) with k ≥ 2 nodes and it is assumed as v1,v2,...vk. now put up a fsg from fuzzy soft path by extending k nodes u1,u2,...,uk in order that uivi ∈ e,(1≤i≤k) and by adding n-2k nodes xj to vk (1≤j≤n-2k). we have n-2k ≥ 1 and k ≥2, it follows that n ≥5, where αe(v1) =αe1(vi),e ∈ t βe(u,v) = βe1(u,v), (u,v)∈ e, e ∈ t βe(ui,vj)= αe(ui) ∧ αe(vj), for every i, e ∈ t. βe(vk, xj)= αe(vk) ∧ αe(xj), for every j, e ∈ t. clearly, gs(t,v) is a fuzzy soft tree because there are no fuzzy soft cycles of odd length, implying that it is a fuzzy soft bipartite graph, hussain and farhana [2020]. at the moment, the sets {v1,v2, ...,vk} and {u1,u2, ...uk−1,vk} have the lowest dominance. as a result,γe(gs(t,v)) = k, e ∈ t . since γe(gs(t,v)) = k, we require no less than k distinctive colours for least fsdc. allocate colours 1,2,...,k to v1,v2,...,vk respectively and k+1 more colour to u1,u2,...,uk,x1, x2,...,xn−2k. so each node has k+1 colours offered in gs(t,v) dominate no less than one colour group. at last, the colouring is least fsdc. alternatively, we allocate k unique colours to the nodes of the set {u1,u2, ...uk−1,vk} then we have a fsdc of ’k’ colours, since we can colour the 113 r. jahir hussain and m.s. afya farhana remaining nodes using k colours. thus χefd(g s(t,v )) = k e ∈ t.2 we have a necessary condition for the upper bound. theorem 4.4. define gs(t,v) a soft fuzzy bipartite graph with double-edged (u,v). following this, corona of gs(t,v) is a fuzzy soft bipartite graph whose χefd is b |cor(gs(t,v ))| 2 c+1, e∈ t where |cor(gs(t,v ))| is the number of nodes in cor(gs(t,v )). proof. assume a soft fuzzy bipartite graph with bipartition (u,v) and |u| + |v | = number of nodes in gs(t,v). label gs(t,v)’ be the corona of gs(t,v). undoubtedly gs(t,v)’ is a fuzzy soft bipartite graph. the points of soft fuzzy bipartite graph allocated with colours 1,2,...|v | and 1+|v | to the left behind nodes, which is a fsdc of gs(t,v)’. seeing as each node of gs(t,v) dominate itself and the left over nodes dominates its nearest nodes. this colouring is a least fsdc. this implies that each node in minimum dominating set gets an exclusive colour. hence χefd(cor(g s(t,v ))) = b |cor(g s(t,v ))| 2 c+1, e∈ t . 5 bounds on fuzzy dominator chromatic number in terms of domination number theorem 5.1. define a soft fuzzy bipartite graph gs(t,v). in that case γe(gs(t,v)) ≤ χefd(g s(t,v )) ≤ 2 + γe (gs(t,v)), e∈ t . proof. allow gs(t,v) to be a soft fuzzy bipartite graph and c to be the least fsdc with colours 1,2,...,χefd(g s(t,v )). for every colour group of gs(t,v), allow an be a point in the colour group n having 1 ≤ n ≤ χefd(g s(t,v )),e∈ t . we have to prove that s = {an:1 ≤ n ≤χefd(g s(t,v )), e∈ t} is a dominating set. by the definition of fsdc, all nodes of gs(t,v) will dominate all nodes of some colour class. since s contains a node of each colour group, each node of gs(t,v) dominate some node in s. as a result, s is a fuzzy dominating set, and every least fsdc of gs(t,v) yields a fuzzy dominating set of gs(t,v). hence γe(gs(t,v)) ≤ χefd(g s(t,v )), e∈ t . we must now validate the upper bound. because the fcn of fuzzy soft bipartite graph is 2, hussain and farhana [2020], we can colour the nodes of gs(t,v) using 2 colours 1 and 2. allot colours 3,4,...,γe(gs(t,v))+2 to the nodes of s and offer colours 1 and 2 to the lasting nodes of gs(t,v) such that 2 strong adjoining 114 bounds on fuzzy dominator chromatic number of fuzzy soft bipartite graphs nodes be given dissimilar colours. this colouring fall out in fuzzy soft dominator colouring since it is an appropriate fuzzy soft colouring and each node in gs(t,v) dominate all nodes of at least one colour group. hence χefd(g s(t,v )) ≤ 2 + γe (gs(t,v)), e∈ t .2 observation: fuzzy soft bipartite graphs gs(t,v) on the basis of the limits, may be divided into three categories on χefd(g s(t,v )). 1. gs(t,v) is of class 0 if χefd(g s(t,v )) = γe(gs(t,v)), e∈ t . 2. gs(t,v) is of class 1 if χefd(g s(t,v )) = γe(gs(t,v)) + 1, e∈ t . 3. gs(t,v) is of class 2 if χefd(g s(t,v )) = γe(gs(t,v)) + 2, e∈ t . example:class 0 consider a fuzzy soft path of k ≥ 2 nodes. now form a fsg as in theorem:4.3. then fsg is of class 0. example:class 1 consider a complete fuzzy soft bipartite graph k1,n. the fdcn of k1,n is two and its γe(gs(t,v)) is one, therefore χefd(g s(t,v )) = γe(gs(t,v)) + 1. hence gs(t,v) is of class 1. example:class 2 the fuzzy soft cycle of length n≥ 5 has fsdc dn 3 e+2 and its γe(gs(t,v)) is dn 3 e, therefore χefd(g s(t,v )) = γe(gs(t,v)) + 2. hence gs(t,v) is of class 2. 6 conclusions in this work, we characterized the bounds on fdcn of fuzzy soft bipartite graphs and also presented bounds in respect of γe(gs(t,v)). additionally, based on the fuzzy dominator chromatic number (fdcn) that was found, the fuzzy soft bipartite graphs are divided into three categories as fuzzy soft path in class 0, complete fuzzy soft bipartite graph in class 1 and fuzzy soft cycle in class 2. we suggest this study on a few distinct classes of fuzzy soft graphs. references m. akram and s. nawaz. on fuzzy soft graphs. italian journal of pure and applied mathematics, 34:497– 514, 2015. m. chellali and f. maffary. dominator colorings in some classes of graphs. graphs combin, 28:97–107, 2012. 115 r. jahir hussain and m.s. afya farhana r. gera. on the dominator colorings in bipartite graphs. proceedings of the 4th international conference on information technology.new generations, pages 947– 952, 2007. r. j. hussain and m. a. farhana. fuzzy chromatic number of fuzzy soft graphs. advances and applications in mathematical sciences, pages 1125 – 1131, 2020. r. j. hussain and k. k. fathima. on fuzzy dominator colouring in fuzzy graphs. applied mathematical sciences, 9:1131–1137, 2015. r. j. hussain and s. s. hussain. domination in fuzzy soft graphs. international journal of fuzzy mathematical archieve, 14(2):243 – 252, 2017. d. molodstov. soft set theory – first results. computers and mathematics with applications, 37:19–31, 1999. a. r. p.k. maji and r. biswas. fuzzy soft sets. journal of fuzzy mathematics, 9(3):589– 602, 2001. 116 ratio mathematica volume 40, 2021, pp. 47-66 47 analysis of classical retrial queue with differentiated vacation and state dependent arrival rate poonam gupta* naveen kumar# abstract in the present paper, we have introduced the concept of differentiated vacations in a retrial queueing model with state dependent arrival rates of customers. the arrival rate of customers is different in various states of the server. the vacation types are differentiated by means of their durations as well as the previous state of the server. in type i vacation, the server goes just after providing service to at least one customer whereas in type ii, it comes after remaining free for some time. in a steady state, we have obtained the system size probabilities and other system performance measures. finally, sensitivity and cost analysis of the proposed model is also performed. the probability generating function technique, parabolic method and matlab is used for this purpose. keywords: retrial queue; markov process; differentiated vacations; exponential distribution etc. 2010 ams subject classification:60k25, 60k30 _________________________________ * baba mastnath university, asthal bohar rohtak india; poonammittal2207@gmail.com # baba mastnath university, asthal bohar rohtak india; naveenkapilrtk@gmail.com 1received on may 4th 2021. accepted on june 24th 2021. published on june 30th 2021. doi: 10.23755/rm.v40i1.619. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. poonam gupta, naveen kumar 48 1. introduction retrial queues have wide applications in communication system, production system, computer networking system, telecommunication etc. retrial queues are characterized by the fact that arriving customers on finding the server busy, leave the system and join the retrial group to complete their request for service after a random time period. a good survey on retrial queues have been done by falin,templeton [5] and artalejo,gomez-coral [1]. in queueing theory, many situations occur where arrival rate of customers depends upon the different states of server such as busy state, idle state or on vacation state etc. singh et al.[14] studied m/g/1 queueing model with state dependent arrival of customers. batch arrival queueing system under retrial policy with state dependent admission is analysed by bagyam and chandrika [2]. niranjan et al. [12] did the pioneer work on state dependent arrival in bulk retrial queues with bernoulli feedback and multiple vacations. nowadays, retrial queueing system with server vacation has become increasingly important due to wide applications in research area. in queueing system with vacation, server becomes unavailable from service station for random period of time due to some reasons like server breakdown, maintenance of server, service provided by server in secondary service station when primary station is empty or simply going for break etc. the time period during which the server is not available for primary customers is known as vacation. in single vacation queueing model, server goes for vacation of random time duration whenever there is no customer in the system and returns to the system after vacation completion. the idea of queueing system with server vacation was first discussed by levy and yechiali [9]. doshi [3] had performed good survey on queueing model with vacation. later on takagi [16], tian and zhang [18] did the pioneer work on vacation queueing system. in multiple vacation system, if server finds no customer in system on returning from vacation, then server immediately goes for another vacation otherwise server will serve the customers. servi and finn [13] introduced the concept of working vacation queueing system in which server works at slow rate during vacation period rather than completely stopping the service during vacation. in queueing literature, lot of work have been done on queueing model with working vacation by many researchers [8,23]. li and tian [10] analysed m/m/1 queueing model with working vacation and interruption. retrial queueing model with working vacation was first studied by do [4]. later on li analysis of classical retrial queue with differentiated vacations and state dependent arrival rate 49 et al. and tao et al. [11,17] did pioneer work on retrial queueing model with working vacation and interruption. in differentiated vacation queueing model, server takes vacation i i.e. vacation of longer duration after serving all the customers in system and vacation ii i.e. vacation of shorter duration will be taken by server if there is no customer in system after completing the type i vacation. the concept of differentiated vacations in queuing literature was first introduced by ibe and isijola [6]. in this paper they considered two types of vacations with different durations. further they extended their model by introducing the concept of vacation interruption [7]. m/m/1 single server queue with m kinds of differentiated working vacations was analyzed by zhang and zhou [22]. vijayashree and janani [21] performed transient solution of m/m/1 queueing system with differentiated vacation. suranga sampath and liu [15] studied the customer’s impatience behaviour on m/m/1 queueing system subject to differentiated vacation. unni and mary [19] studied queueing system with multiple servers under differentiated vacations. further they extended their work by introducing differentiated working vacation [20]. in this paper, we have extended the concept of differentiated vacations to queueing system under classical retrial policy considering the state dependent arrival of customers. the organization of rest of the paper into different sections is as follows. the model description is given in section 2. section 3 is devoted to steady state equations and solutions. the closed form expressions for some of the performance measures are derived in section 4. section 5 represents the effect of various parameters on some important system performance measures graphically. conclusion and future scope is discussed in section 6. 2. model description the model is outlined as follows. 1. customers arrive according to poisson process but with different rates depending on the present state of the server. the different arrival rates of customers are λ, α, γ, δ in busy, free, vacation i, vacation ii states of the server, respectively. 2. the arriving customers are served on fcfs basis. if server is free in active period, the arriving customer is immediately served otherwise due to unavailability of waiting space in service area, he has to join a free pool of poonam gupta, naveen kumar 50 infinite capacity known as orbit to wait for the service. from the orbit, customers retry for their turn with classical rate β. for convenience, the service time is supposed to follow exponential distribution with parameter μ. 3. as soon as the last customer is served i.e. system gets empty, the server leaves for type i vacation. at the end of type i vacation, if system is still empty, the server goes on type ii vacation otherwise returns to active state to serve the waiting customers. on completion of vacation ii, if there is a customer waiting in the system, server returns to free state in normal active period otherwise again goes on vacation ii repeatedly. the vacation i is assumed to be of longer duration than vacation ii. the time period of both vacations is assumed to follow exponential distribution with parameters 𝑣1,𝑣2 respectively. 3. steady state equations and solution denoting the probability of n customers in state k of the server by 𝑝𝑛 𝑘 and server states at time t by s(t) were s(t)= { 1, 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑏𝑢𝑠𝑦 𝑖𝑛 𝑎𝑐𝑡𝑖𝑣𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 2, 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑓𝑟𝑒𝑒 𝑖𝑛 𝑎𝑐𝑡𝑖𝑣𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 3, 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑜𝑛 𝑡𝑦𝑝𝑒 𝐼 𝑣𝑎𝑐𝑎𝑡𝑖𝑜𝑛 4, 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑜𝑛 𝑡𝑦𝑝𝑒 𝐼𝐼 𝑣𝑎𝑐𝑎𝑡𝑖𝑜𝑛 let n(t) be the number of customers in the orbit at time t. then the quasi birthdeath process is a markovian process represented by {n(t),s(t)} with state space {(n, k), n ≥ 0, k=1,3,4} u {(n, 2), n ≥1}. using markov process, the differential difference equations for the proposed model are 𝑑 𝑑𝑡 𝑝0 1(𝑡) = 𝛽𝑝1 2(𝑡) − (𝜆 + 𝜇)𝑝0 1(𝑡) (1) 𝑑 𝑑𝑡 𝑝𝑛 1(𝑡) = 𝜆𝑝𝑛−1 1(𝑡) + (𝑛 + 1)𝛽𝑝𝑛+1 2(𝑡) + 𝛼𝑝𝑛 2(𝑡) − (𝜆 + 𝜇)𝑝𝑛 1(𝑡), 𝑛 ≥ 1 (2) 𝑑 𝑑𝑡 𝑝𝑛 2(𝑡) = 𝑣1𝑝𝑛 3(𝑡) + 𝑣2𝑝𝑛 4(𝑡) + 𝜇𝑝𝑛 1(𝑡) − (𝛼 + 𝑛𝛽)𝑝𝑛 2(𝑡), 𝑛 ≥ 1 (3) analysis of classical retrial queue with differentiated vacations and state dependent arrival rate 51 𝑑 𝑑𝑡 𝑝0 3(𝑡) = 𝜇𝑝0 1(𝑡) − (𝛾 + 𝑣1)𝑝0 3(𝑡) (4) 𝑑 𝑑𝑡 𝑝𝑛 3(𝑡) = 𝛾𝑝𝑛−1 3(𝑡) − (𝛾 + 𝑣1)𝑝𝑛 3(𝑡), 𝑛 ≥ 1 (5) 𝑑 𝑑𝑡 𝑝0 4(𝑡) = 𝑣1𝑝0 3(𝑡) − 𝛿𝑝0 4(𝑡) (6) 𝑑 𝑑𝑡 𝑝𝑛 4(𝑡) = 𝛿𝑝𝑛−1 4(𝑡) − (𝛿 + 𝑣2)𝑝𝑛 4(𝑡), 𝑛 ≥ 1 (7) to obtain steady state equations, taking limit t → ∞ and using lim 𝑡→∞ 𝑝 𝑛 𝑖 (t) = 𝑝 𝑛 𝑖 lim 𝑡→∞ 𝑑 𝑑𝑡 𝑝 𝑛 𝑖 (𝑡) = 0 } 𝑖 = 1, 2, 3, 4 the steady state equations are (𝜆 + 𝜇)𝑝0 1 = 𝛽𝑝1 2 (8) (𝜆 + 𝜇)𝑝𝑛 1 = 𝜆𝑝𝑛−1 1 + (𝑛 + 1)𝛽𝑝𝑛+1 2 + 𝛼𝑝𝑛 2, 𝑛 ≥ 1 (9) (𝛼 + 𝑛𝛽)𝑝𝑛 2 = 𝑣1𝑝𝑛 3 + 𝑣2𝑝𝑛 4 + 𝜇𝑝𝑛 1, 𝑛 ≥ 1 (10) (𝛾 + 𝑣1)𝑝0 3 = 𝜇𝑝0 1 (11) (𝛾 + 𝑣1)𝑝𝑛 3 = 𝛾𝑝𝑛−1 3, 𝑛 ≥ 1 (12) 𝛿𝑝0 4 = 𝑣1𝑝0 3 (13) (𝛿 + 𝑣2)𝑝𝑛 4 = 𝛿𝑝𝑛−1 4 , 𝑛 ≥ 1 (14) defining the probability generating functions as 𝑃𝑖 (𝑧) = ∑ 𝑝𝑛 𝑖 𝑧 𝑛 ∞ 𝑛=0 , 𝑖 = 1,3,4 (15) 𝑃2(𝑧) = ∑ 𝑝𝑛 2𝑧 𝑛 ∞ 𝑛=1 (16) using equations (10), (11), (13)and p.g.fs defined in(15) and (16), we get 𝑧𝛽𝑃2 ′ (𝑧) + 𝛼𝑃2(𝑧) = 𝑣1𝑃3(𝑧) + 𝑣2𝑃4(𝑧) + 𝜇𝑃1(𝑧) − (𝛾 + 2𝑣1 + 𝑣1𝑣2 𝛿 ) 𝑝0 3 (17) poonam gupta, naveen kumar 52 from equations (8), (9), (15) and (16) we obtain (𝜆 + 𝜇 − 𝜆𝑧)𝑃1(𝑧) = 𝛽𝑃2 ′ (𝑧) + 𝛼𝑃2(𝑧) (18) similarly using equations (11) and (12) along with (15), we get (𝛾 + 𝑣1 − 𝛾𝑧)𝑃3(𝑧) = (𝛾+𝑣1)𝑝0 3 𝑃3(𝑧) = (𝛾+𝑣1)𝑝0 3 (𝛾 + 𝑣1 − 𝛾𝑧) (19) on similar steps from equations (13), (14) and (15) we obtain p4(z) = v1(δ + v2) δ(δ + v2 − δz) p0 3 (20) taking z=1 in equation (20),we obtain 𝑃4(1) = 𝑣1(𝛿 + 𝑣2) 𝛿𝑣2 𝑝0 3 (21) from equation (17) 𝑧𝛽𝑃2 ′ (𝑧) + 𝛼𝑃2(𝑧) = 𝑣1𝑃3(𝑧) + 𝑣2𝑃4(𝑧) + 𝜇𝑃1(𝑧) − 𝐴𝑝0 3 (22) where a = (𝛾 + 2𝑣1 + 𝑣1𝑣2 𝛿 ) using equations (18), (22) together, after some rearrangement of terms we obtain 𝑃2 ′ (𝑧) + 𝛼𝜆 𝛽(𝜆𝑧 − 𝜇) 𝑃2(𝑧) = (𝜆 + 𝜇 − 𝜆𝑧) 𝛽(1 − 𝑧)(𝜆𝑧 − 𝜇) (𝑣1𝑃3(𝑧) + 𝑣2𝑃4(𝑧) − 𝐴𝑝0 3) (23) to solve the differential equation (23) taking i. f = (λz − μ) α β p2(z) = (λz − μ) −α β ∫(λx − μ) α β (λ + μ − λx) β(1 − x)(λx − μ) (v1p3(x) + v2p4(x) z 0 − ap0 3) dx (24) substituting value of 𝑃2 ′ (𝑧) in equation (18) and solving, we get analysis of classical retrial queue with differentiated vacations and state dependent arrival rate 53 𝑃1(𝑧) = (𝛼 − 𝛼𝑧)𝑃2(𝑧) − 𝑣1𝑃3(𝑧) − 𝑣2𝑃4(𝑧) + 𝐴𝑝0 3 (𝜇 − 𝑧(𝜆 + 𝜇 − 𝜆𝑧)) (25) on differentiating equations (19),(20) we get 𝑃3 ′ (𝑧) = 𝛾(𝛾+𝑣1)𝑝0 3 (𝛾 + 𝑣1 − 𝛾𝑧) 2 (26) 𝑃4 ′(𝑧) = 𝑣1(𝛿 + 𝑣2) 𝛿(𝛿 + 𝑣2 − 𝛿𝑧) 2 𝑝0 3 (27) again, differentiating equations (26) and (27), we get 𝑃3 ′′(𝑧) = 2𝛾 2(𝛾+𝑣1)𝑝0 3 (𝛾 + 𝑣1 − 𝛾𝑧) 3 (28) 𝑃4 ′′(𝑧) = 2𝛿𝑣1(𝛿 + 𝑣2) (𝛿 + 𝑣2 − 𝛿𝑧) 3 𝑝0 3 (29) also, from equation (18), we get 𝑃2 ′ (𝑧) = (𝜆 + 𝜇 − 𝜆𝑧)𝑃1(𝑧) − 𝛼𝑃2(𝑧) 𝛽 (30) taking limit 𝑧 → 1in equations (19), (20), (24), (26), (27), (28) and (29) we get 𝑃3(1) = (𝛾+𝑣1)𝑝0 3 𝑣1 (31) 𝑃4(1) = 𝑣1(𝛿 + 𝑣2) 𝛿𝑣2 𝑝0 3 (32) 𝑃2(1) = (𝜆 − 𝜇) −𝛼 𝛽 ∫(𝜆𝑥 − 𝜇) 𝛼 𝛽 (𝜆 + 𝜇 − 𝜆𝑥) 𝛽(1 − 𝑥)(𝜆𝑥 − 𝜇) (𝑣1𝑃3(𝑥) + 𝑣2𝑃4(𝑥) 1 0 − 𝐴𝑝0 3) 𝑑𝑥 (33) 𝑃3 ′ (1) = 𝛾(𝛾+𝑣1) (𝑣1) 2 𝑝0 3 (34) 𝑃4 ′(1) = 𝑣1(𝛿 + 𝑣2) 𝑣2 2 𝑝0 3 (35) poonam gupta, naveen kumar 54 𝑃3 ′′(1) = 2𝛾 2(𝛾+𝑣1) 𝑣1 3 𝑝0 3 (36) 𝑃4 ′′(1) = 2𝛿𝑣1(𝛿 + 𝑣2) 𝑣2 3 𝑝0 3 (37) taking limit 𝑧 → 1 in equation (25) and using l-hospital rule, we get 𝑃1(1) = 𝛼𝑃2(1) + 𝑣1𝑃3 ′ (1) + 𝑣2𝑃4 ′(1) 𝜇 − 𝜆 (38) taking limit 𝑧 → 1 in equation (30) 𝑃2 ′ (1) = 𝜇𝑃1(1) − 𝛼𝑃2(1) 𝛽 (39) on differentiating equation (25) and taking limit 𝑧 → 1 we get 𝑃1 ′(1) = (2𝛼𝑃2 ′ (1) + 𝑣1𝑃3 ′′(1) + 𝑣2𝑃4 ′′(1))(𝜇 − 𝜆) + 2𝜆(𝛼𝑃2(1) + 𝑣1𝑃3 ′ (1) + 𝑣2𝑃4 ′(1)) 2(𝜇 − 𝜆)2 (40) all the p.g. f’s are expressed in terms of 𝑝0 3 which is obtained by using normalization condition ∑ 𝑃𝑖 (1) 4 𝑖=1 = 1 (41) it follows that, 𝑝0 3 [( 𝛼 + 𝜇 − 𝜆 𝜇 − 𝜆 ) (𝜆 − 𝜇) −𝛼 𝛽 ∫ (𝜆 − 𝜇) 𝛼 𝛽 (λ + μ − λz) 𝛽(1 − 𝑧)(𝜆𝑧 − 𝜇) {𝑣1 ( 𝛾 + 𝑣1 𝛾 + 𝑣1 − 𝛾𝑧 ) 1 0 + 𝑣1𝑣2 𝛿 ( 𝛿 + 𝑣2 𝛿 + 𝑣2 − 𝛿𝑧 ) − 𝐴} 𝑑𝑧 + 𝛾 + 𝑣1 𝑣1 + 𝛾(𝛾 + 𝑣1) 𝑣1(𝜇 − 𝜆) + 𝑣1(𝛿 + 𝑣2) 𝑣2(𝜇 − 𝜆) + 𝑣1(𝛿 + 𝑣2) 𝑣2𝛿 ] = 1 (42) analysis of classical retrial queue with differentiated vacations and state dependent arrival rate 55 𝑝0 3 = [( 𝛼 + 𝜇 − 𝜆 𝜇 − 𝜆 ) (𝜆 − 𝜇) −𝛼 𝛽 ∫ (𝜆 − 𝜇) 𝛼 𝛽 (λ + μ − λz) 𝛽(1 − 𝑧)(𝜆𝑧 − 𝜇) {𝑣1 ( 𝛾 + 𝑣1 𝛾 + 𝑣1 − 𝛾𝑧 ) 1 0 + 𝑣1𝑣2 𝛿 ( 𝛿 + 𝑣2 𝛿 + 𝑣2 − 𝛿𝑧 ) − 𝐴} 𝑑𝑧 + 𝛾 + 𝑣1 𝑣1 + 𝛾(𝛾 + 𝑣1) 𝑣1(𝜇 − 𝜆) + 𝑣1(𝛿 + 𝑣2) 𝑣2(𝜇 − 𝜆) + 𝑣1(𝛿 + 𝑣2) 𝑣2𝛿 ] −1 (43) 4. important performance measures in this section, we present some of the important performance measures of the system as follows. the expected number of customers in the orbit is e[𝐿0] = ∑ 𝑃𝑖 ′(1) 4 𝑖=1 (44) the expected number of customers in the system is e[𝐿𝑠] = e[𝐿0] + 𝑃1(1) (45) probability of server in type i vacation 𝑃𝑟𝑉1 = 𝑃3(1) = ∑ 𝑝𝑛 3 ∞ 𝑛=0 = (𝛾+𝑣1)𝑝0 3 𝑣1 (46) probability of server in type ii vacation 𝑃𝑟𝑉2 = 𝑃4(1) poonam gupta, naveen kumar 56 = ∑ 𝑝𝑛 4 ∞ 𝑛=0 = 𝑣1(𝛿 + 𝑣2) 𝛿𝑣2 𝑝0 3 (47) probability of server on vacations 𝑃𝑟𝑉 = 𝑃𝑟𝑉1 + 𝑃𝑟𝑉2 = (𝛾+𝑣1)𝑝0 3 𝑣1 + 𝑣1(𝛿 + 𝑣2) 𝛿𝑣2 𝑝0 3 (48) probability of server in working (active) state 𝑃𝑟𝑁 = 𝑃1(1) + 𝑃2(1) = ∑ 𝑝𝑛 1 ∞ 𝑛=0 + ∑ 𝑝𝑛 2 ∞ 𝑛=1 (49) 5. graphical results in this section, we illustrate the effect of various parameters on some of the performance measures of system. we have also optimized the cost with respect to service rate. in the below graphs, we have set λ=1.2, μ=3, β=2, γ=0.6, α=1, 𝑣1 = 0.6, 𝑣2 = 1, δ=0.8 unless they are varied in the graphs. 5.1 sensitivity analysis for qualitative analysis of the proposed model, we represent some of the numerical results graphically. analysis of classical retrial queue with differentiated vacations and state dependent arrival rate 57 figure1. effect of active state arrival rate (λ) on system performance measures. from figure1, we observe that with the increase in arrival rate λ, expected orbit length, system length and probability of normal state increase, whereas the probability of vacation decreases. this is explained by the fact that with the increase in arrival rate, the number of customers increases in orbit and in system. hence, the probability of normal state increases and thereby, the probability of vacation decreases. poonam gupta, naveen kumar 58 figure 2. effect of service rate (μ) on system performance measures. figure 2 reveals that the expected length of orbit, system and probability of normal (active) state decrease, but the probability of vacation state increases with an increase in service rate μ. the reason being that with the increase in μ, the customers will be served fasterand this reduces the number of customers in orbit and hence in the system. also, due to faster service, the probability of normal period decreases and this increases the probability of a vacation period. analysis of classical retrial queue with differentiated vacations and state dependent arrival rate 59 figure3.effect of rate of type i vacation (𝑣1) on system performance measures. from figure 3, we see that as the type i vacation rate increases, the expected length of orbit, expected length of system and probability of type i vacation decrease but the probability of type ii vacation and probability of normal (active) state increase. the fact behind the observation is that with the increase in type i vacation rate, the duration of type i vacation decreases and this causes increase in probability of normal state and the probability of type ii vacation. due to which the expected number of customers in orbit and that in the system decrease. poonam gupta, naveen kumar 60 figure 4. effect of variation in retrial rate (β) on system performance measures. figure 4 shows the effect of change in retrial rate on expected orbit length, system length, probability of vacation and active server states. the graphical results obtained here matches the intuitive expectations. analysis of classical retrial queue with differentiated vacations and state dependent arrival rate 61 figure 5. effect of variation in rate of type ii vacation (𝑣2) on system performance measures. figure5 represents that expected orbit length,expected system length, probability of server in normal state and probability of type i vacation decrease as the rate of type ii vacation increase. as the type ii vacation rate increases, the duration of type ii vacation decreases: hence, the expected queue length and system length decrease. 5.2 cost analysis in this subsection, we optimize the operating cost function with respect to service rate in working state. to obtain the optimal value of𝜇, some cost elements are taken as 𝐶𝐿 = cost per unit time for each customer present in the orbit. 𝐶𝜇 = cost per unit time for service in working state. poonam gupta, naveen kumar 62 𝐶𝑣1= cost per unit time in type i vacation. 𝐶𝑣2= cost per unit time in type ii vacation. the corresponding cost function per unit time is defined as f(μ) = 𝐶𝐿 𝐸[𝐿0] + μ𝐶𝜇 + 𝑣1𝐶𝑣1+ 𝑣2𝐶𝑣2 we take𝐶𝐿 = 20, 𝐶𝜇 =28, 𝐶𝜃=10, 𝐶𝜙=8 in the parabolic method for obtaining optimal cost f(x) and the corresponding value of x. parabolic-method works by generating quadratic function through calculated points in every iteration to which the function f(x) can be approximated. the point at which f(x) is optimum in threepoint pattern {𝑥1, 𝑥2, 𝑥3} is given by 𝑥𝐿 = 0.5(𝐹(𝑥1)(𝑥2 2 − 𝑥3 2) + 𝐹(𝑥2)(𝑥3 2 − 𝑥1 2) + 𝐹(𝑥3)(𝑥1 2 − 𝑥2 2)) 𝐹(𝑥1)(𝑥2 − 𝑥3) + 𝐹(𝑥2)(𝑥3 − 𝑥1) + 𝐹(𝑥3)(𝑥1 − 𝑥2) the new value obtained replaces one of the three points to improve the current three-point pattern. the process is repeatedly applied until optimum value is obtained up to the desired degree of accuracy. table 1 shows that optimum value 𝐹(𝜇) =112.83101corresponding to μ= 2.15566 with the permissible error of 10−4, which is verified by figure 6. table 1. optimization of cost by parabolic method 𝒙𝟏 𝒙𝟐 𝒙𝟑 𝑭(𝒙𝟏) 𝑭(𝒙𝟐) 𝑭(𝒙𝟑) 𝒙𝑳 𝟏. 𝟕𝟎 𝟐. 𝟎𝟎 𝟐. 𝟓𝟎 𝟏𝟐𝟕. 𝟑𝟕𝟐𝟓𝟑 𝟏𝟏𝟑. 𝟖𝟓𝟎𝟐𝟒 𝟏𝟏𝟓. 𝟕𝟕𝟓𝟕𝟏 𝟐. 𝟐𝟏𝟖𝟓𝟐 𝟐. 𝟎𝟎 𝟐. 𝟐𝟏𝟖𝟓𝟐 𝟐. 𝟓𝟎 𝟏𝟏𝟑. 𝟖𝟓𝟎𝟐𝟒 𝟏𝟏𝟐. 𝟗𝟓𝟗𝟎𝟓 𝟏𝟏𝟓. 𝟕𝟕𝟓𝟕𝟏 𝟐. 𝟏𝟖𝟏𝟔𝟓 𝟐. 𝟎𝟎 𝟐. 𝟏𝟖𝟏𝟔𝟓 𝟐. 𝟐𝟏𝟖𝟓𝟐 𝟏𝟏𝟑. 𝟖𝟓𝟎𝟐𝟒 𝟏𝟏𝟐. 𝟖𝟓𝟑𝟖𝟑 𝟏𝟏𝟐. 𝟗𝟓𝟗𝟎𝟓 𝟐. 𝟏𝟔𝟐𝟔𝟗 𝟐. 𝟎𝟎 𝟐. 𝟏𝟔𝟐𝟔𝟗 𝟐. 𝟏𝟖𝟏𝟔𝟓 𝟏𝟏𝟑. 𝟖𝟓𝟎𝟐𝟒 𝟏𝟏𝟐. 𝟖𝟑𝟐𝟕𝟑 𝟏𝟏𝟐. 𝟖𝟓𝟑𝟖𝟑 𝟐. 𝟏𝟓𝟖𝟒𝟒 𝟐. 𝟎𝟎 𝟐. 𝟏𝟓𝟖𝟒𝟒 𝟐. 𝟏𝟔𝟐𝟔𝟗 𝟏𝟏𝟑. 𝟖𝟓𝟎𝟐𝟒 𝟏𝟏𝟐. 𝟖𝟑𝟏𝟐𝟖 𝟏𝟏𝟐. 𝟖𝟑𝟐𝟕𝟑 𝟐. 𝟏𝟓𝟔𝟒𝟖 𝟐. 𝟎𝟎 𝟐. 𝟏𝟓𝟔𝟒𝟖 𝟐. 𝟏𝟓𝟖𝟒𝟒 𝟏𝟏𝟑. 𝟖𝟓𝟎𝟐𝟒 𝟏𝟏𝟐. 𝟖𝟑𝟏𝟎𝟑 𝟏𝟏𝟐. 𝟖𝟑𝟏𝟐𝟖 𝟐. 𝟏𝟓𝟓𝟗𝟒 𝟐. 𝟎𝟎 𝟐. 𝟏𝟓𝟓𝟗𝟒 𝟐. 𝟏𝟓𝟔𝟒𝟖 𝟏𝟏𝟑. 𝟖𝟓𝟎𝟐𝟒 𝟏𝟏𝟐. 𝟖𝟑𝟏𝟎𝟏 𝟏𝟏𝟐. 𝟖𝟑𝟏𝟎𝟑 𝟐. 𝟏𝟓𝟓𝟕𝟐 𝟐. 𝟎𝟎 𝟐. 𝟏𝟓𝟓𝟕𝟐 𝟐. 𝟏𝟓𝟓𝟗𝟒 𝟏𝟏𝟑. 𝟖𝟓𝟎𝟐𝟒 𝟏𝟏𝟐. 𝟖𝟑𝟏𝟎𝟎 𝟏𝟏𝟐. 𝟖𝟑𝟏𝟎𝟏 𝟐. 𝟏𝟓𝟓𝟔𝟔 analysis of classical retrial queue with differentiated vacations and state dependent arrival rate 63 figure 6. variation in expected operating cost per unit time with service rate (μ) 6. conclusion and future scope in this paper, we have analyzed single server markovian queueing model with state dependent arrival rates of customers under differentiated vacations and classical retrial policy. the closed form expressions for various performance measures are derived with the help of probability generating functions. the performance of the proposed model is represented graphically using matlab software. the operating cost of the queueing system is optimized with respect to service rate of the server. the model can be extended to multiple servers. conflicts of interests the authors declare that there is no conflict of interests. poonam gupta, naveen kumar 64 references [1] artalejo, j.r. and gómez-corral, a. 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(2010).performance analysis of m/g/1 queue with working vacations and vacation interruption. journal of computational and applied mathematics, 234(10), 2977–2985.available at: https://doi.org/10.1016/j.cam.2010.04.010 ratio mathematica volume 40, 2021, pp. 225-246 225 about two countable families in the finite sets of the collatz conjecture michele ventrone* abstract with 𝒕 ∈ ℕ we define the sets 𝑲𝒕 and 𝑲𝒕 ∗ containing all positive integers that converge to 1 in t iterations in the form of collatz algorithm. the following are the properties of the {𝑲𝒕}𝒕∈ℕ and {𝑲𝒕 ∗}𝒕∈ℕ: countability, empty intersection between the elements of the same family, and at the end of the work we conjecture that both of the two families are a partition of ℕ𝟎. we demonstrate also that each set 𝑲𝒕 and 𝑲𝒕 ∗ is the union of two sets, a set includes even positive integers, the other, if it is non-empty, includes odd positive integers different from 1 and we go on proving that the maximum of each set 𝑲𝒕 and 𝑲𝒕 ∗ is 2t and that 𝑲𝒕 ∩ 𝑲𝒕 ∗ = {𝟐𝒕}. keywords: collatz conjecture, syracuse problem, 3n+1 problem, hailstone numbers. 2010 mathematics subject classification: 11a25, 11n56, 11m06. * received on 2021-03-30. accepted on 2021-05-30, 2021. published on 2021-06-30. doi: 10.23755/rm.v39i0.5xx. issn: 1592-7415. eissn: 2282-8214. doi: 10.23755/rm.v40i1.585 ©the authors. this paper is published under the cc-by licence agreement. micheleventrone57@gmail.com. about two countable families in the finite sets of the collatz conjecture 226 1 introduction let us consider the collatz conjecture (leggerini, 2004), also known as the 3𝑛 + 1 problem. we start from a positive integer n, if it is even we divide it by two, if it is odd we multiply it by three and add one to it, then we start over by applying the same rules on the number obtained. for example, starting from 3 the sequence is generated: 3, 10, 5, 16, 8, 4, 2, 1. in the second form of the algorithm of 3𝑛 + 1 we calculate 3𝑛+1 2 if n is odd. with 3 we obtain the sequence 3, 5, 8, 4, 2, 1. it is conjectured that, from any positive integer we start, the sequences always arrive at 1 in a finite number of steps. it seems that all trajectories fall into the banal cycle 4, 2, 1 if n > 2. the conjecture has not yet been proven and many mathematicians believe the question be undecidable (conway, j. h, 1972). by applying the algorithm to a positive integer n, a sequence of integers is generated which we will call a sequence or trajectory of n which we will denote with 𝑇(𝑛) (оr 𝑇∗(𝑛) with the second form of the algorithm). for example 𝑇(5) = {5,16,8,4,2,1} and 𝑇∗(3) = {3,5,8,4,2,1}. let ℕ = {0, 1, 2 … } and ℕ0 = {1, 2, 3 … }. if 𝑖 ∈ ℕ and 𝑛 ∈ ℕ0, we denote by 𝑇𝑖 (𝑛) the element of place i in the trajectory 𝑇(𝑛). if 𝑖 = 0 we set 𝑇0(𝑛) = 𝑛. the same meaning will have 𝑇𝑖 ∗(𝑛). for example 𝑇0(5) = 5, 𝑇3(5) = 4, 𝑇2 ∗(3) = 8. we define convergent a trajectory that contains the number 1. in any trajectory containing 1 we will ignore the terms subsequent. if the trajectory generated by the integer n converges we will say that the number n converges. any number of a trajectory will be treated as a positive integer. the term ”t-convergent” will be equivalent to ”convergent in t iterations”. we will call the number t the convergence time. the notation 𝑘𝑡 will indicate that the positive integer k is tconvergent. in the following tc will be the set of convergence times of the converging positive integers. 2 the two forms of collatz conjecture first form. with 𝑛 ∈ ℕ0 e 𝑖 ∈ ℕ0 the algorithm is the iteration of the function: 𝑇𝑖 (𝑛) = { 𝑇𝑖−1(𝑛) 2 if 𝑇𝑖−1(𝑛) ≡ 0(𝑚𝑜𝑑2) 3 ⋅ 𝑇𝑖−1(𝑛) + 1 if 𝑇𝑖−1(𝑛) ≡ 1(𝑚𝑜𝑑2) (2.1) with 𝑇0(𝑛) = 𝑛 if 𝑖 = 0. second form. with 𝑛 ∈ ℕ0 and 𝑖 ∈ ℕ0, the algorithm is the iteration of the function: michele ventrone 227 𝑇𝑖 ∗(𝑛) = { 𝑇𝑖−1 ∗ (𝑛) 2 if 𝑇𝑖−1 ∗ (𝑛) ≡ 0(𝑚𝑜𝑑 2) 3𝑇𝑖−1 ∗ (𝑛)+1 2 if 𝑇𝑖−1 ∗ (𝑛) ≡ 1(𝑚𝑜𝑑 2) (2.2) with 𝑇0 ∗(𝑛) = 𝑛 𝑖𝑓 𝑖 = 0. 3 construction of the sets k let us put in the same set 𝐾𝑡 the totality of positive integers t-convergent with the algorithm in the first form: ∀𝑡 ∈ ℕ, 𝐾𝑡 = {𝑘 ∈ 𝑁0: 𝑘 = 𝑘𝑡 }. (3.1) for example, applying the algorithm in the first form: 𝐾0 = {1} because 1 converges to 1 in zero iterations; 𝐾1 = {2} because 2 converges to 1 in an iteration; 𝐾2 = {4} because 4 converges to 1 in two iterations; … … … … if the collatz algorithm is used in the second form, in (3.1) we will add to 𝐾𝑡 and its elements the symbol ∗, that is: ∀𝑡 ∈ ℕ, 𝐾𝑡 ∗ = {𝑘∗ ∈ ℕ0: 𝑘 ∗ = 𝑘𝑡 ∗}. (3.2) proposition 3.1. (basic) ∀𝑡 ∈ ℕ, 𝐾𝑡 e 𝐾𝑡 ∗ are non-empty. proof. trivially: whatever 𝑡 ∈ ℕ, the number 2𝑡 converges to 1 in t iterations, hence in 𝐾𝑡 there is at least 2 𝑡 . for the same reason 𝑘𝑡 ∗ is also non-empty. □ we consider the set tc of all times of convergence. since each 𝑡 ∈ ℕ can be associated with a 𝐾𝑡 and a 𝐾𝑡 ∗ by means of 2𝑡 and vice versa, we can state that 𝑇𝐶 = ℕ and that the families {𝐾𝑡 }𝑡∈ℕ and {𝐾𝑡 ∗}𝑡∈ℕ are countable. the following corollaries then hold. corollary 3.2. any positive or null integer is a time of convergence. about two countable families in the finite sets of the collatz conjecture 228 corollary 3.3. each of the families {𝐾𝑡 }𝑡∈ℕ and {𝐾𝑡 ∗}𝑡∈ℕ is countable. proposition 3.4. if 𝑡1 𝑎𝑛𝑑 𝑡2, with 𝑡1 ≠ 𝑡2, are in the set tc, then it results: i) 𝐾𝑡1 ∩ 𝐾𝑡2 = ∅ i*) 𝐾𝑡1 ∗ ∩ 𝐾𝑡2 ∗ = ∅ . proof. i) algorithm in the first form. by proposition 3.1, 𝐾𝑡1 and 𝐾𝑡2 are non-empty. assume that 𝐾𝑡1 ∩ 𝐾𝑡2 ≠ ∅ , with 𝑡1 ≠ 𝑡2. if 𝑘 ∈ 𝐾𝑡1 ∩ 𝐾𝑡2 then k must converge in the same number of iterations, so 𝑡1 = 𝑡2, against the hypothesis. therefore 𝐾𝑡1 ∩ 𝐾𝑡2 = ∅ . • i*) algorithm in the second form. the proof is similar to the previous one: just insert the asterisk to the sets 𝐾𝑡 . □ each family {𝐾𝑡 }𝑡∈ℕ and {𝐾𝑡 ∗}𝑡∈ℕ divides ℕ0 into classes that we cannot consider at the moment of equivalence. 4 decomposition of sets k let 𝑡 ∈ ℕ0. applying the first form of the collatz algorithm we will prove that each set 𝐾𝑡 is formed by a set 𝐴𝑡 and a set 𝐵𝑡, that is 𝐾𝑡 = 𝐴𝑡 ∪ 𝐵𝑡 with 𝐴𝑡 containing only even numbers and 𝐵𝑡 empty or containing only odd numbers different from 1. applying the second form of the collatz algorithm we will prove that 𝐾𝑡 ∗ = 𝐴𝑡 ∗ ∪ 𝐵𝑡 ∗ with 𝐴𝑡 ∗ containing only even numbers and 𝐵𝑡 ∗ empty or containing only odd numbers different from 1. we will also prove that the elements of 𝐾𝑡 can be obtained from all the elements of 𝐾𝑡−1 and the elements of 𝐾𝑡 ∗ can be obtained from all elements of 𝐾𝑡−1 ∗ . if t = 0 it is 𝐾0 = 𝐾0 ∗ = {1} and therefore 2𝐾0 = 2𝐾0 ∗ = {2} = 𝐾1 = 𝐾1 ∗. some b sets are empty such as sets 𝐵1, 𝐵1 ∗, 𝐵2, 𝐵2 ∗, 𝐵3, 𝐵3 ∗, 𝐵4, 𝐵4 ∗, 𝐵6, 𝐵8, 𝐵10. i don't know if there are other empty b sets. in this study 𝐴0 = {1} , 𝐴0 ∗ = {1} , 𝐵0 = ∅ and 𝐵0 ∗ = ∅ . let t ∈ ℕ. here we will assume that 𝐾𝑡+1 is made up of two sets of numbers: 1) by the doubles of the numbers of 𝐾𝑡; 2) from the integers 𝑏 ≠ 1 which are odd solutions in ℕ0 of the equation 3𝑏 + 1 = 𝑘𝑡 , with 𝑘𝑡 ∈ 𝐾𝑡 , 𝑘𝑡 even and 𝑘𝑡 ≠ 4; and that 𝐾𝑡+1 ∗ is formed by two sets of numbers: michele ventrone 229 1*) by the doubles of the numbers of 𝐾𝑡 ∗; 2*) from the integers 𝑏∗ ≠ 1 which are odd solutions in ℕ0 of the equation 3𝑏∗+1 2 = 𝑘𝑡 ∗ , with 𝑘𝑡 ∗ ∈ 𝐾𝑡 ∗ and 𝑘𝑡 ∗ ≠ 2. called p the set of even positive integers, we denote by 2𝐾𝑡 (set of even derivatives of the first type or set of even derivatives of 𝐾𝑡 or set of doubles of the first type) the set obtained by doubling all the numbers of 𝐾𝑡: ∀𝑡 ∈ ℕ, 2𝐾𝑡 = {𝑎 ∈ 𝑃: 𝑎 = 2𝑘𝑡 , 𝑘𝑡 ∈ 𝐾𝑡 }. (4.1) we denote by 2𝐾𝑡 ∗ (set of even derivatives of the second type or set of even derivatives of 𝐾𝑡 ∗ or set of doubles of the second type) the set obtained by doubling all the numbers of 𝐾𝑡 ∗: ∀𝑡 ∈ ℕ, 2𝐾𝑡 ∗ = {𝑎∗ ∈ 𝑃: 𝑎∗ = 2𝑘𝑡 ∗, 𝑘𝑡 ∗ ∈ 𝐾𝑡 ∗} . (4.2) we denote by 𝐵𝑡+1 (set of odd derivatives of 𝐾𝑡 or set of odd derivatives of the first type) the numbers with the property 2) and by 𝐵𝑡+1 ∗ (set of odd derivatives of 𝐾𝑡 ∗ to set of odd derivatives of second type) numbers with the property 2∗). called d the set of integers odd positive, the set of odd derivatives of 𝐾𝑡 we have: ∀𝑡 ∈ ℕ, 𝐵𝑡+1 = {𝑏 ∈ 𝐷 − {1}: 3𝑏 + 1 = 𝑘𝑡 , 𝑘𝑡 ∈ 𝐾𝑡 ∩ 𝑃, 𝑘𝑡 ≠ 4} (4.3) while the set of odd derivatives of 𝐾𝑡 ∗ is ∀𝑡 ∈ ℕ, 𝐵𝑡+1 ∗ = {𝑏∗ ∈ 𝐷 − {1}: 3𝑏∗+1 2 = 𝑘𝑡 ∗, 𝑘𝑡 ∗ ∈ 𝐾𝑡 ∗, 𝑘𝑡 ∗ ≠ 2} . (4.4) theorem 4.1. (theorem of the inclusion of doubles) the even derivative of 𝐾𝑡 (𝐾𝑡 ∗) is contained in 𝐾𝑡+1(𝐾𝑡+1 ∗ ), that is: a) ∀𝑡 ∈ ℕ, 2𝐾𝑡 ⊆ 𝐾𝑡+1 b) ∀𝑡 ∈ ℕ, 2𝐾𝑡 ∗ ⊆ 𝐾𝑡+1 ∗ . (4.5) proof. by corollary 3.2 every t is a time of convergence. given 𝑡 ∈ ℕ, we consider 𝐾𝑡 (which is non-empty by proposition 3.1). trivially: ∀𝑘𝑡 ∈ 𝐾𝑡, the trajectory 𝑇(𝑘𝑡 ) = {𝑘𝑡 , … ,4,2,1} is contained in the trajectory 𝑇(2𝑘𝑡 ) = {2𝑘𝑡 , 𝑘𝑡 , … ,4,2,1}. this means that 2𝑘𝑡 is (t+1)-convergent, so 2𝑘𝑡 ∈ 𝐾𝑡+1. • about two countable families in the finite sets of the collatz conjecture 230 if 𝐾𝑡+1 is devoid of odd numbers, only the sign of equality holds. to prove it, let's suppose that 𝐾𝑡+1 is devoid of odd numbers and that, absurdly, it contains an even number 𝑎𝑡+1 which does not is double of any number of 𝐾𝑡. since the even 𝑎𝑡+1 is also is (t+1)-convergent, the trajectory 𝑇(𝑎𝑡+1) = {𝑎𝑡+1, 𝑎𝑡+1 2 , … ,4,2,1} will contain the trajectory 𝑇 ( 𝑎𝑡+1 2 , ) = { 𝑎𝑡+1 2 , … ,4,2,1} so 𝑎𝑡+1 2 is t-convergent, that is 𝑎𝑡+1 2 ∈ 𝐾𝑡 , against our hypothesis. it follows that 2𝐾𝑡 coincides with 𝐾𝑡+1 if this is devoid of odd. then the relation a) of (4.5) holds for the arbitrariness of t. • in the case of 2𝐾𝑡 ∗ proceed in the same way, mutatis mutandis. □ theorem 4.2. (odd derivative theorem of the first type) let 𝑘𝑡 be even and 𝑘𝑡 ≠ 4 . if there is a positive integer b satisfying the equation 3𝑏 + 1 = 𝑘𝑡 (4.6) then 𝑏 = 𝑘𝑡−1 3 (4.7) belongs to 𝐵𝑡+1. proof. let b and 𝐾𝑡 satisfy the hypotheses. since 𝑏 is odd and different from 1, its successor is 𝑘𝑡, because to b is applied (2.1), so the trajectory 𝑇(𝑏) = 𝑇 ( 𝑘𝑡−1 3 ) = { 𝑘𝑡−1 3 , 𝑘𝑡 , … ,4,2,1} contains the trajectory 𝑇(𝑘𝑡 ) = {𝑘𝑡 , … ,4,2,1}. this means that b converges in t + 1 iterations, that is 𝑏 ∈ 𝐵𝑡+1. □ recall that an odd derivative 𝐵𝑡 either is empty or is formed only by odd positive different from 1. theorem 4.3. (theorem of strict inclusion of odd derivatives of the first type) the odd derivative of 𝐾𝑡 (𝐾𝑡 ∗) is strictly contained in 𝐾𝑡+1(𝐾𝑡+1 ∗ ), that is: a) ∀𝑡 ∈ ℕ, 𝐵𝑡+1 ⊂ 𝐾𝑡+1 b) ∀𝑡 ∈ ℕ, 𝐵𝑡+1 ∗ ⊂ 𝐾𝑡+1 ∗ . (4.8) proof. by proposition 3.1 every 𝐾𝑡+1(𝐾𝑡+1 ∗ ) is non-empty because it contains at least the even number 2𝑡+1, therefore 𝐵𝑡+1 even if it were empty could not coincide with 𝐾𝑡+1(𝐾𝑡+1 ∗ ). □ michele ventrone 231 theorem 4.4. (theorem of the union of even and odd derivatives of the first type) the set 𝐾𝑡+1 is the union of the set of doubles of 𝐾𝑡 and of the odd derivative of 𝐾𝑡, that is: ∀𝑡 ∈ ℕ, 𝐾𝑡+1 = 2𝐾𝑡 ∪ 𝐵𝑡+1. (4.9) (remarkable equality, algorithm in first form) proof. let us consider 𝐾𝑡, with 𝑡 ∈ ℕ. it is necessary to demonstrate that 1) there are no other even integers (t+1)-convergent beyond those of 2𝐾𝑡; 2) the odd numbers (t+1)-convergent are only those of 𝐵𝑡+1. we prove 1). we denote by 𝐴𝑡+1 the totality of even positive integers converging in t + 1 iterations that we know to be non-empty (each 𝐴𝑡 contains at least 2𝑡 ). it immediately turns out that ∀𝑡 ∈ ℕ, 2𝐾𝑡 ⊆ 𝐴𝑡+1. we show that ∀𝑡 ∈ ℕ, 2𝐾𝑡 = 𝐴𝑡+1. (4.10) if for a fixed 𝑡 ∈ ℕ there were an even 𝑎𝑡+1 ∈ 𝐴𝑡+1 that was not double of any positive integer of 𝐾𝑡, it would be absurd because the trajectory 𝑇(𝑎𝑡+1) = {𝑎𝑡+1, 𝑎𝑡+1 2 , … 4,2,1} would contain the trajectory 𝑇 ( 𝑎𝑡+1 2 ) = { 𝑎𝑡+1 2 , … ,4,2,1} whose seed at 𝑎𝑡+1 2 ∈ 𝐾𝑡 and whose double 𝑎𝑡+1 is in 𝐴𝑡+1, against the hypothesis. hence the strict inclusion cannot hold and, by the arbitrariness of t, (4.10) is true. • we prove 2). with the same fixed 𝑡 ∈ ℕ, we denote by 𝛽𝑡+1 the totality of the odd positive integers converging in t + 1 iterations . obviously we have 𝐵𝑡+1 ⊆ 𝛽𝑡+1. we show that ∀𝑡 ∈ ℕ, 𝐵𝑡+1 = 𝛽𝑡+1 . (4.11) if for the fixed t, 𝛽𝑡+1 = ∅, then also 𝐵𝑡+1 = ∅ and therefore 𝐾𝑡+1 = 2𝐾𝑡, that is (4.9) for the arbitrariness of t. otherwise, for fixed t, let 𝛽𝑡+1 ≠ ∅ . if there was an 𝑏𝑡+1 ∈ 𝛽𝑡+1 not coming by any even of 𝐾𝑡, that is such that 𝑏𝑡+1 ∉ 𝐵𝑡+1 , then an absurdity would follow because the trajectory 𝑇(𝑏𝑡+1) = {𝑏𝑡+1, 3𝑏𝑡+1 + 1, … ,4,2,1} would contain the trajectory 𝑇(3𝑏𝑡+1 + 1) = {3𝑏𝑡+1 + 1, … ,4,2,1} whose even seed 3𝑏𝑡+1 + 1 = 𝑘𝑡 ∈ 𝐾𝑡 , therefore, by theorem 4.2, 𝑏𝑡+1 ∈ 𝐵𝑡+1 against the hypothesis. for this reason strict inclusion cannot be valid and, due to the arbitrariness of t (4.11) is true. • about two countable families in the finite sets of the collatz conjecture 232 from 1) and 2) follows the remarkable equality (4.9). □ by (4.10), (4.9) becomes ∀𝑡 ∈ ℕ, 𝐾𝑡+1 = 𝐴𝑡+1 ∪ 𝐵𝑡+1 . (4.12) (remarkable equality, algorithm in the first form) if for a given t the derivative 𝐵𝑡+1 of 𝐾𝑡 is empty, we have 𝐾𝑡+1 = 𝐴𝑡+1. (4.13) we now find the numbers of 𝐵𝑡+1 ∗ . theorem 4.5. (theorem of the odd derivative of the second type) let 𝑘𝑡 ∗ ∈ ℕ0, 𝑘𝑡 ∗ ≠ 2 . if there exists the positive integer b satisfying the equation 3𝑏∗ + 1 = 2𝑘𝑡 ∗ (4.14) then 𝑏∗ = 2𝑘𝑡 ∗−1 3 (4.15) belongs to 𝐵𝑡+1 ∗ . proof. let 𝑘𝑡 ∗ and 𝑏∗ satisfy the hypotheses. since 𝑏∗ is odd and different from 1, its successor is 𝑘𝑡 ∗, because (2.2) is applied to 𝑏∗, so the trajectory 𝑇(𝑏∗) = 𝑇 ( 2𝑘𝑡 ∗−1 3 ) = { 2𝑘𝑡 ∗−1 3 , 𝑘𝑡 ∗, … ,4,2,1} contains the trajectory 𝑇(𝑘𝑡 ∗) = {𝑘𝑡 ∗, … ,4,2,1} . this means that 𝑏∗ converges in t + 1 iterations, that is 𝑏∗ ∈ 𝐵𝑡+1 ∗ . □ recall that an odd derivative 𝐵𝑡 ∗ o is either empty or is formed only by odd positive integers different from 1. as shown for (4.10) it results ∀𝑡 ∈ ℕ, 2𝐾𝑡 ∗ = 𝐴𝑡+1 ∗ (4.16) michele ventrone 233 where 𝐴𝑡+1 ∗ is the totality of the even positive integers (t+1)-convergent, that is of the doubles of the numbers of 𝐾𝑡 ∗. equation (4.16) is demonstrated how it is done for the first part of the proof of the theorem 4.4 by adding the asterisk ∗ to the 2𝐾𝑡 and 𝐴𝑡 sets. equation (4.16) will occur in the proof of first part of theorem 4.6. theorem 4.6. (theorem of the union of even and odd derivatives of the second type) the set 𝐾𝑡+1 ∗ is the union of the set of doubles of 𝐾𝑡 ∗ and the odd derivative of 𝐾𝑡 ∗, that is: ∀𝑡 ∈ ℕ, 𝐾𝑡+1 ∗ = 2𝐾𝑡 ∗ ∪ 𝐵𝑡+1 ∗ . (4.17) (remarkable equality, algorithm in the second form) proof. we proceed as in the proof of theorem 4.4 adding the asterisk ∗ to all the sets and considering, in the second part, ( 3𝑏𝑡+1 ∗ +1 2 ) as successor of 𝑏𝑡+1 ∗ ∈ 𝛽𝑡+1 ∗ . □ by (4.16), (4.17) can be written ∀𝑡 ∈ ℕ, 𝐾𝑡+1 ∗ = 𝐴𝑡+1 ∗ ∪ 𝐵𝑡+1 ∗ (4.18) (remarkable equality, algorithm in the second form) and if, for a certain t, the derivative 𝐵𝑡+1 ∗ of 𝐾𝑡+1 ∗ it is empty, then 𝐾𝑡+1 ∗ = 𝐴𝑡+1 ∗ . (4.19) 5 examples to obtain the set 𝐾𝑡+1 it will be necessary to double all the numbers 𝑘𝑡 of 𝐾𝑡 in order to have 𝐴𝑡+1 and it will be necessary to determine all the numbers 𝑏 ∈ 𝐵𝑡+1 starting from the even numbers 𝑘𝑡 of 𝐾𝑡, that is, it will be necessary to verify if 𝑘𝑡 − 1 is divisible by three when 𝑘𝑡 is even with 𝑘𝑡 ≠ 4 (theorem 4.2 and definition of 𝐵𝑡+1 in (4.3)). ► we determine the sets 𝐾8 and 𝐾9. 𝐾8 we use the set 𝐾7 = {3,20,21,128} . we have 𝐴8 = 2𝐾7 = {6,40,42,256} . it turns out 𝐵8 = ∅ since none of the equations about two countable families in the finite sets of the collatz conjecture 234 (1) 3b + 1 = 20 (2) 3b + 1 = 128 has solutions in ℕ0. hence 𝐾8 = 𝐴8 ∪ ∅ = {6,40,42,256}. 𝐾9 we use the set 𝐾8 = {6,40,42,256}. we have 𝐴9= 2𝐾8 = {12, 80, 84, 512}. we solve in ℕ0 the following equations: (1) 3b + 1 = 6 (2) 3b + 1 = 40 (3) 3b + 1 = 42 (4) 3b + 1 = 256. the first and third equations have no solutions in ℕ0. the second and fourth equations have as solutions in ℕ0 13 and 85 respectively, therefore 𝐵9 = {13, 85}. thus 𝐾9 = 𝐴9 ∪ 𝐵9 = {12,80,84,512} ∪ {13,85} = {12,13,80,84,85,512} . in the same way they are obtained 𝐾10 = {4 − 26 − 160 − 168 − 170 − 1024} 𝐾11 = {48 − 52 − 53 − 320 − 336 − 340 − 341 − 2048} 𝐾12 = {17 − 96 − 104 − 106 − 113 − 640 − 672 − 680 − 682 − 4096} … the underlined numbers are the odd derivatives of the previous set. to obtain the set 𝐾𝑡+1 ∗ it will be necessary to double all the numbers 𝑘𝑡 ∗ of 𝐾𝑡 ∗ in order to have 𝐴𝑡+1 ∗ and it will be necessary to determine all the numbers 𝑏∗ ∈ 𝐵𝑡+1 ∗ starting from each 𝑘𝑡 ∗ of 𝐾𝑡 ∗, that is, it will be necessary to verify whether 2𝑘𝑡 ∗ − 1 is divisible by three when 𝑘𝑡 ∗ ≠ 2 (theorem 4.5 and definition of 𝐵𝑡+1 ∗ in (4.4)). ► we determine the sets 𝐾5 ∗ and 𝐾6 ∗. 𝐾5 ∗ we consider 𝐾4 ∗ = {5,16} . its even derivative is 𝐴5 ∗ = {10,32}. of the two equations michele ventrone 235 (1) 3𝑏∗+1 2 = 5 (2) 3𝑏∗+1 2 = 16 only the first admits in ℕ0 the solution 𝑏 ∗= 3 therefore 𝐵5 ∗ = {3} е 𝐾5 ∗ = 𝐴5 ∗ ∪ 𝐵5 ∗ = {3,10,32}. 𝐾6 ∗ we consider 𝐾5 ∗ = {3,10,32} . its even derivative is 𝐴6 ∗ = {6,20,64} . of the three equations (1) 3𝑏∗+1 2 = 3 (2) 3𝑏∗+1 2 = 10 (3) 3𝑏∗+1 2 = 32 only the third has solution 𝑏∗ = 21 in ℕ0. hence 𝐵6 ∗ = {21} е 𝐾6 ∗ = 𝐴6 ∗ ∪ 𝐵6 ∗ = {6,20,21,64} . in the same way they are obtained 𝐾7 ∗ = {12 − 13∗ − 40 − 42 − 128} 𝐾8 ∗ = {24 − 26 − 80 − 84 − 85∗ − 256} 𝐾9 ∗ = {17∗ − 48 − 52 − 53∗ − 160 − 168 − 170 − 512} 𝐾10 ∗ = {11∗ − 34 − 35∗ − 96 − 104 − 106 − 113∗ − 320 − 336 − 340 − 341∗ − 1024} ... the numbers with an asterisk are the odd derivatives of the previous set. about two countable families in the finite sets of the collatz conjecture 236 6 the maxima of 𝑲𝒕 and 𝑲𝒕 ∗ by examining the sets k, we can suppose that the number 2𝑡 is the maximum of every set 𝐾𝑡 and of every 𝐾𝑡 ∗. this is confirmed by the subsequent theorem 6.2. the following lemma 6.1 contains some obvious conclusions. lemma 6.1. i) if 𝑘𝑡 ∈ 𝐾𝑡 then 2𝑘𝑡 ∈ 𝐴𝑡+1, ∀𝑡 ∈ ℕ ii) if 𝑎𝑡 ∈ 𝐴𝑡 then 2𝑎𝑡 ∈ 𝐴𝑡+1, ∀𝑡 ∈ ℕ0 i∗) if 𝑘𝑡 ∗ ∈ 𝐾𝑡 ∗ then 2𝑘𝑡 ∗ ∈ 𝐴𝑡+1 ∗ , ∀𝑡 ∈ ℕ ii∗) if 𝑎𝑡 ∗ ∈ 𝐴𝑡 ∗ then 2𝑎𝑡 ∗ ∈ 𝐴𝑡+1 ∗ , ∀𝑡 ∈ ℕ0 . proof. recall that (4.10) and (4.16) hold. i) let 𝑘𝑡 ∈ 𝐾𝑡, with 𝑡 ∈ ℕ. the trajectory 𝑇(𝑘𝑡 ) is contained in the trajectory 𝑇(2𝑘𝑡 ) = {2𝑘𝑡 , 𝑘𝑡 , … ,4,2,1} because 2𝑘𝑡 is an even that converges in t + 1 iterations, that is 2𝑘𝑡 ∈ 𝐴𝑡+1. • ii) let 𝑎𝑡 ∈ 𝐴𝑡, with t ∈ ℕ0. since 𝐴𝑡 ⊆ 𝐾𝑡 is also 𝑎𝑡 ∈ 𝐾𝑡. applying i) it follows that 2𝑎𝑡 ∈ 𝐴𝑡+1 ∀𝑡 ∈ ℕ0. • the i∗) and ii∗) prove to be the i) and ii) respectively, just asterisking the sets 𝐾𝑡, 𝐴𝑡 and their elements. □ theorem 6.2. (maxima theorem of 𝐾𝑡 and 𝐾𝑡 ∗) 𝑖) ∀𝑡 ∈ ℕ0, 𝑚𝑎𝑥(𝐾𝑡 ) = 2 𝑡 i*) ∀𝑡 ∈ ℕ0, 𝑚𝑎𝑥(𝐾𝑡 ∗) = 2𝑡 . proof. i) we will proceed by induction using the remarkable equality (4.12). if 𝑡 = 1 then max 𝑚𝑎𝑥(𝐾1) = 2 1 = 2. let us fix a 𝑡 > 1 and let, by inductive hypothesis 𝑚𝑎𝑥(𝐾𝑡 ) = 2 𝑡 . (6.1) we will prove that it is also 𝑚𝑎𝑥(𝐾𝑡+1) = 2 𝑡+1. to do this, it will be necessary to prove that 1) max 𝑚𝑎𝑥(𝐴𝑡+1) = 2 𝑡+1 and 2) every number of 𝐵𝑡+1 is less than 2 𝑡+1. michele ventrone 237 first part 1) we show that every number of 𝐴𝑡+1 is less than or equal to 2 𝑡+1 and that 2𝑡+1is in 𝐴𝑡+1. let 𝑘𝑡 ∈ 𝐾𝑡. then, by hypothesis (6.1) ∀𝑘𝑡 ∈ 𝐾𝑡 , 𝑘𝑡 ≤ 2 𝑡 . (6.2) by the i) of lemma 6.1 2𝑘𝑡 ∈ 𝐴𝑡+1. (6.3) from (6.2) it follows that ∀𝑘𝑡 ∈ 𝐾𝑡 , 2𝑘𝑡 ≤ 2 𝑡+1. (6.4) since for the inductive hypothesis (6.1) it is 2𝑡 ∈ 𝐾𝑡 , then, for the remarkable equality (4.12), we have 2𝑡 ∈ 𝐴𝑡, from which, for the ii) of lemma 6.1, it follows that 2𝑡+1 ∈ 𝐴𝑡+1. (6.5) from (6.3), (6.4) and (6.5) we obtain that 𝑚𝑎𝑥(𝐴𝑡+1) = 2 𝑡+1. • second part 2) if 𝐵𝑡+1 = ∅ from (4.12) it follows that 𝐾𝑡+1 = 𝐴𝑡+1 and from 𝑚𝑎𝑥(𝐴𝑡+1) = 2𝑡+1 (first part) it follows that 𝑚𝑎𝑥(𝐾𝑡+1) = 2 𝑡+1. let 𝐵𝑡+1 ≠ ∅. we show that every element 𝑏𝑡+1 of 𝐵𝑡+1 is less than 2 𝑡+1. the numbers of 𝐵𝑡+1 are the odd numbers of the form (4.7): 𝑏𝑡+1 = 𝑘𝑡−1 3 𝑐𝑜𝑛 𝑘𝑡 ∈ 𝐾𝑡 𝑎𝑛𝑑 𝑘𝑡 even (6.6) but, from 𝑘𝑡 − 1 < 𝑘𝑡 we get that 𝑘𝑡−1 3 < 𝑘𝑡 (6.7) then from (6.6), (6.7) and (6.1) it follows that 𝑏𝑡+1 < 𝑘𝑡 ≤ 2 𝑡 (6.8) about two countable families in the finite sets of the collatz conjecture 238 and therefore: ∀𝑏𝑡+1 ∈ 𝐵𝑡+1, 𝑏𝑡+1 < 2 𝑡+1, that is 2). from the first and the second part it follows that all the numbers of 𝐾𝑡+1 are less than or equal to 2𝑡+1and this proves the i). • i*) we will proceed by induction using the remarkable equality (4.18). if 𝑡 = 1 then 𝑚𝑎𝑥(𝐾1 ∗) = 21=2. let, by inductive hypothesis, be 𝑚𝑎𝑥(𝐾𝑡 ∗) = 2𝑡 con 𝑡 > 1 . (6.9) we will prove that it is also 𝑚𝑎𝑥(𝐾𝑡+1 ∗ ) = 2𝑡+1. to do this, it will be necessary to prove that 1*) 𝑚𝑎𝑥(𝐴𝑡+1 ∗ ) = 2𝑡+1 and 2*) every number of 𝐵𝑡+1 ∗ is less than 2𝑡+1 . first part ∗ 1*) the proof is similar to that of the first part of i), just adding the asterisk ∗ to the sets 𝐴𝑡+1, 𝐾𝑡+1 and their elements. therefore 2 𝑡+1 is the maximum of 𝐴𝑡+1 ∗ and 1*) is proved. • second part ∗ 2*) if 𝐵𝑡+1 ∗ ≠ ∅ from 4.18) it follows that 𝐾𝑡+1 ∗ = 𝐴𝑡+1 ∗ and from 𝑚𝑎𝑥(𝐴𝑡+1 ∗ ) = 2𝑡+1 (first part*) it follows that 𝑚𝑎𝑥(𝐾𝑡+1 ∗ ) = 2𝑡+1. let 𝐵𝑡+1 ∗ ≠ ∅. we show that every 𝑏𝑡+1 ∗ of 𝐵𝑡+1 ∗ is less than 2𝑡+1. the numbers of 𝐵𝑡+1 ∗ are the odd numbers of the form (4.15): 𝑏𝑡+1 ∗ = 2𝑘𝑡 ∗−1 3 , with 𝑘𝑡 ∗ ∈ 𝐾𝑡 ∗ (6.10) but, from 2𝑘𝑡 ∗ − 1 < 2𝑘𝑡 ∗ we get that 2𝑘𝑡 ∗−1 3 < 2𝑘𝑡 ∗ (6.11) and for the inductive hypothesis (6.9) we also have that ∀𝑘𝑡 ∗ ∈ 𝐾𝑡 ∗, 2𝑘𝑡 ∗ ≤ 2𝑡+1 . (6.12) michele ventrone 239 finally for (6.10), (6.11), (6.12) we can write that 𝑏𝑡+1 ∗ < 2𝑘𝑡 ∗ ≤ 2𝑡+1, then ∀𝑡 > 1 all numbers 𝑏𝑡+1 ∗ di 𝐵𝑡+1 ∗ are less than 2𝑡+1. the 2*) is thus proved. • from 1*) and from 2*) it follows that all integers of 𝐾𝑡+1 ∗ are less than or equal to 2𝑡+1 and so i*) is also proved. □ the following corollaries immediately follow from theorem 6.2. corollary 6.3. i) ∀𝑡 ∈ ℕ, 𝑚𝑎𝑥(2𝐾𝑡 ) = 2 𝑡+1 i∗) ∀𝑡 ∈ ℕ, 𝑚𝑎𝑥(2𝐾𝑡 ∗) = 2𝑡+1 . corollary 6.4. i) ∀𝑡 ∈ ℕ0, 𝑚𝑎𝑥(𝐴𝑡 ) = 2 𝑡 i∗) ∀𝑡 ∈ ℕ0, 𝑚𝑎𝑥(𝐴𝑡 ∗) = 2𝑡 . theorem 6.2 provides indications on the type of numbers contained in the sets k: either there is only 2𝑡 or there are positive integers less than or equal to 2𝑡 and this means that each set k is finite. therefore, the following corollary can also be stated. corollary 6.5. ∀𝑡 ∈ ℕ, 𝐾𝑡 𝑎𝑛𝑑 𝐾𝑡 ∗ are finite. each set k is formed by the finite numerical sets a and b. it follows that if b is non-empty then it has a maximum. therefore the following corollary holds. corollary 6.6. i) if for 𝑡 ∈ ℕ0 is 𝐵𝑡 ≠ ∅ then ∃ 𝑚𝑎𝑥(𝐵𝑡 ) i*) if for 𝑡 ∈ ℕ0 is 𝐵𝑡 ∗ ≠ ∅ then ∃ 𝑚𝑎𝑥(𝐵𝑡 ∗) . in the following paragraph 7 we will investigate the maxima of the sets b. about two countable families in the finite sets of the collatz conjecture 240 7 on the maxima of the sets b we give a strict increase of the maxima of the sets b. proposition 7.1. i) if 𝐵𝑡+1 ≠ ∅ then ∃𝑘𝑡 ∈ 𝐾𝑡 , 𝑘𝑡 ≠ 4, 𝑘𝑡 even : 𝑚𝑎𝑥(𝐵𝑡+1) < 𝑘𝑡−1 2 i*) if 𝐵𝑡+1 ∗ ≠ ∅ then ∃𝑘𝑡 ∗ ∈ 𝐾𝑡 ∗, 𝑘𝑡 ∗ ≠ 2: 𝑚𝑎𝑥(𝐵𝑡+1 ∗ ) < 2𝑘𝑡 ∗−1 2 . proof. i) if 𝐵𝑡+1 ≠ ∅, then by definition of 𝐵𝑡+1 in correspondence of every odd 𝑏𝑡+1 ∊ 𝐵𝑡+1 will exist an even number 𝑘𝑡 ∈ 𝐾𝑡 with 𝑘𝑡 ≠ 4 such that 𝑏𝑡+1 = 𝑘𝑡−1 3 but 𝑘𝑡−1 3 < 𝑘𝑡−1 2 , then 𝑏𝑡+1 < 𝑘𝑡−1 2 . then, in particular, i) holds also for the maximum of 𝐵𝑡+1. • i*) if 𝐵𝑡+1 ∗ ≠ ∅, then by definition of 𝐵𝑡+1 ∗ in correspondence of every odd 𝑏𝑡+1 ∗ ∈ 𝐵𝑡+1 ∗ will exist an even number 𝑘𝑡 ∗ ∈ 𝐾𝑡 ∗ with 𝑘𝑡 ∗ ≠ 2 such that 𝑏𝑡+1 ∗ = 2𝑘𝑡 ∗−1 3 but 2𝑘𝑡 ∗−1 3 < 2 𝑘𝑡 ∗−1 2 , then 𝑏𝑡+1 ∗ < 2 𝑘𝑡 ∗−1 2 . then, in particular, also for the maximum of 𝐵𝑡+1 ∗ holds i*). □ from proposition 7.1 follows the following corollary which gives a plus a bit more large of the maxima of the sets b. corollary 7.2. i) if 𝐵𝑡+1 ≠ ∅, then 𝑚𝑎𝑥(𝐵𝑡+1) < 2 𝑡 i*) if 𝐵𝑡+1 ∗ ≠ ∅, then 𝑚𝑎𝑥(𝐵𝑡+1 ∗ ) < 2𝑡 . proof. i) if 𝐵𝑡+1 ≠ ∅, then the inequality i) of proposition 7.1 holds and also 𝑘𝑡−1 2 < 𝑘𝑡 but, by theorem 6.2, the maximum of 𝐾𝑡 is 2 𝑡 , so 𝑚𝑎𝑥(𝐵𝑡+1 ∗ ) < 2𝑡 . • i*) if 𝐵𝑡+1 ∗ ≠ ∅, then the inequality i*) of proposition 7.1 holds and also 2𝑘𝑡 ∗−1 2 < 𝑘𝑡 ∗ but, by theorem 6.2, the maximum of 𝐾𝑡 ∗ is 2𝑡 , so 𝑚𝑎𝑥(𝐵𝑡+1 ∗ ) < 2𝑡 . □ michele ventrone 241 in some cases it is possible to determine the maximum of the sets b. let's see how. the numbers of 𝐵𝑡+1and of 𝐵𝑡+1 ∗ come from the integer solutions, if they exist, of the equations 𝑏𝑡+1 = 𝑘𝑡−1 3 with 𝑘𝑡 ∈ 𝐾𝑡 , 𝑘𝑡 even and 𝑘𝑡 ≠ 4 (7.1) 𝑏𝑡+1 ∗ = 2𝑘𝑡 ∗−1 3 with 𝑘𝑡 ∗ ∈ 𝐾𝑡 ∗ and 𝑘𝑡 ∗ ≠ 2 (7.2) by the theorems, respectively, 4.2 and 4.5. in fact, the largest odd integer that can be obtained from (7.1), if we substitute the maximum of 𝐾𝑡 for 𝑘𝑡, is 2𝑡−1 3 , which is integer if 2𝑡 − 1 is divisible by three. likewise, the largest odd integer which can be obtained from (7.2), if we replace 𝑘𝑡 ∗ by the maximum of 𝐾𝑡 ∗, is 2𝑡+1−1 3 , which is integer if 2𝑡+1 − 1 is divisible by 3. we can therefore state the following theorem. theorem 7.3. i) if 2𝑡 − 1 ≡ 0(𝑚𝑜𝑑 3), with 𝑡 ∈ ℕ0 and 𝑡 > 2, then 𝑚𝑎𝑥(𝐵𝑡+1) = 2𝑡−1 3 i*) if 2𝑡+1 − 1 ≡ 0(𝑚𝑜𝑑 3), with 𝑡 ∈ ℕ0 and 𝑡 > 1, then 𝑚𝑎𝑥(𝐵𝑡+1 ∗ ) = 2𝑡+1−1 3 . second demonstration of the theorem 7.3 proof. i) by hypothesis the number 2𝑡 is t-convergent and the equation 3𝑏 + 1 = 2𝑡 is satisfied by 𝑏 = 2𝑡−1 3 which is different from 1 because 𝑡 > 2, therefore, by theorem 4.2 it is 𝑏 ∈ 𝐵𝑡+1. assume that ∃𝛽 ∈ 𝐵𝑡+1: 𝑏 < 𝛽 that is, taking into account the form of b and β, we suppose that it is 2𝑡−1 3 < 𝑘𝑡−1 3 with 𝑘𝑡 ∈ 𝐾𝑡 e 𝑘𝑡 even; from this it follows that 2 𝑡 < 𝑘𝑡 , absurd thing because the maximum of 𝐾𝑡 is 2 𝑡 . then it must turn out ∀𝛽 ∈ 𝐵𝑡+1: 𝛽 ≤ 𝑏, that is the thesis. • about two countable families in the finite sets of the collatz conjecture 242 i*) by hypothesis the number 2𝑡 is (t+1)-convergent and the equation 3𝑏 + 1 = 2𝑡+1 is satisfied by 𝑏 = 2𝑡+1−1 3 which is different from 1 because 𝑡 > 1, therefore, by theorem 4.5 it is 𝑏∗ ∈ 𝐵𝑡+1 ∗ . assume that ∃𝛽∗ ∈ 𝐵𝑡+1 ∗ : 𝑏∗ < 𝛽∗ that is, taking into account the form of 𝑏∗ and 𝛽∗, supposing it is 2𝑡+1−1 3 < 2𝑘𝑡 ∗−1 3 with 𝑘𝑡 ∗ ∈ 𝐾𝑡 ∗, from this it follows that 2𝑡 < 𝑘𝑡 ∗, which is absurd because the maximum of 𝐾𝑡 ∗ is 2𝑡 . then it must turn out ∀𝛽∗ ∈ 𝐵𝑡+1 ∗ : 𝛽∗ ≤ 𝑏∗ that is the thesis. □ for example: • ... • for 𝑡 = 14 risults 214 − 1 ≡ 0(𝑚𝑜𝑑3), then 𝑚𝑎𝑥(𝐵15) = 𝑚𝑎𝑥(𝐵14 ∗ ) = 5461 • for 𝑡 = 16 risults 216 − 1 ≡ 0(𝑚𝑜𝑑3), then 𝑚𝑎𝑥(𝐵17) = 𝑚𝑎𝑥(𝐵16 ∗ ) = 21845 • for 𝑡 = 18 risults 218 − 1 ≡ 0(𝑚𝑜𝑑3), then 𝑚𝑎𝑥(𝐵19) = 𝑚𝑎𝑥(𝐵18 ∗ ) = 87381 • .... 8 on the intersection of 𝑲𝒕 and 𝑲𝒕 ∗ in this paragraph we will prove that the intersection of the sets 𝐾𝑡 and 𝐾𝑡 ∗ is {2𝑡 }. lemma 8.1. the intersection of the odd derivatives of the first type t-convergent and of the even derivatives of the second type t-convergent is empty, that is ∀𝑡 ∈ ℕ0, 𝐵𝑡 ∩ 𝐴𝑡 ∗ = ∅. (8.1) proof. obviously, because an odd derivative either is empty or is made up of odd integers different from 1 and an even derivative contains only even numbers. □ lemma 8.2. the intersection of the odd derivatives of the second type t-convergent and of the even derivatives of the first type t-convergent is empty, that is ∀𝑡 ∈ ℕ0, 𝐵𝑡 ∗ ∩ 𝐴𝑡 = ∅. (8.2) michele ventrone 243 proof. obviously, because an odd derivative either is empty or is made up of odd integers different from 1 and an even derivative contains only even numbers. □ lemma 8.3. the intersection of the odd derivatives of the first type t-convergent and of the odd derivatives of the second type t-convergent is empty, that is ∀𝑡 ∈ ℕ0, 𝐵𝑡 ∩ 𝐵𝑡 ∗ = ∅. (8.3) proof. trivially, if 𝑡 = 1 the sets 𝐵1 and 𝐵1 ∗ are both empty. assume absurdly that for 𝑡 > 1 it results 𝐵𝑡 ∩ 𝐵𝑡 ∗ ≠ ∅ and consider every 𝑛𝑡 ∈ 𝐵𝑡 ∩ 𝐵𝑡 ∗. from 𝑛𝑡 ∈ 𝐵𝑡 = {𝑛𝑡 ∈ 𝐷 − {1}: 3𝑛𝑡 + 1 = 𝑘𝑡−1, 𝑘𝑡−1 ∈ 𝐾𝑡−1 ⋂ 𝑃 , 𝑘𝑡−1 ≠ 4, 𝑘𝑡−1 − 1 ≡ 0(𝑚𝑜𝑑 3)} follows that nt is an odd integer of the form (4.7), that is 𝑛𝑡 = 𝑘𝑡−1−1 3 . from 𝑛𝑡 ∈ 𝐵𝑡 ∗ = {𝑛𝑡 ∈ 𝐷 − {1}: 3𝑛𝑡 + 1 2 = 𝑘𝑡−1 ∗ , 𝑘𝑡−1 ∗ ∈ 𝐾𝑡−1 ∗ , 𝑘𝑡−1 ∗ ≠ 2, 2𝑘𝑡−1 ∗ − 1 ≡ 0(𝑚𝑜𝑑 3)} it follows that nt is an odd integer of the form (4.15), that is 𝑛𝑡 = 2𝑘𝑡−1 ∗ −1 3 . by equating the two expressions of 𝑛𝑡 we have 𝑘𝑡−1−1 3 = 2𝑘𝑡−1 ∗ −1 3 and therefore 𝑘𝑡−1 = 2𝑘𝑡−1 ∗ . (8.4) equality (8.4) is manifestly absurd because 𝑘𝑡−1 is (t-1)-convergent and 2𝑘𝑡−1 ∗ is t-convergent. therefore it makes no sense to suppose that the intersection 𝐵𝑡 ∩ 𝐵𝑡 ∗ for 𝑡 > 1 is non-empty and (8.3) is proved. □ lemma 8.4. the intersection of 𝐾𝑡 and 𝐾𝑡 ∗ is equal to the intersection of the even derivatives of 𝐾𝑡−1 and of the derivatives even of 𝐾𝑡 ∗, that is ∀𝑡 ∈ ℕ0, 𝐾𝑡 ∩ 𝐾𝑡 ∗ = 𝐴𝑡 ∩ 𝐴𝑡 ∗. (8.5) proof. we will use the notable equalities 4.12) and 4.18). we have ∀𝑡 ∈ ℕ0: about two countable families in the finite sets of the collatz conjecture 244 𝐾𝑡 ∩ 𝐾𝑡 ∗ = (𝐴𝑡 ∪ 𝐵𝑡 ) ∩ (𝐴𝑡 ∗ ∪ 𝐵𝑡 ∗) = = ((𝐴𝑡 ∪ 𝐵𝑡 ) ∩ 𝐴𝑡 ∗) ∪ ((𝐴𝑡 ∪ 𝐵𝑡 ) ∩ 𝐵𝑡 ∗) = = (𝐴𝑡 ∩ 𝐴𝑡 ∗) ∪ (𝐵𝑡 ∩ 𝐴𝑡 ∗) ∪ (𝐴𝑡 ∩ 𝐵𝑡 ∗) ∪ (𝐵𝑡 ∩ 𝐵𝑡 ∗). (8.6) the thesis follows by applying, in order, lemmas 8.1, 8.2 and 8.3 to the second, third and fourth intersection in the last line of (8.6). □ lemma 8.5. the intersection of the even derivatives of the first and second type t-convergent is {2𝑡 }, that is: ∀𝑡 ∈ ℕ0, 𝐴𝑡 ∩ 𝐴𝑡 ∗ = {2𝑡 }. (8.7) proof. applying the equalities (4.10) and (4.16) to the intersection 𝐴𝑡 ∩ 𝐴𝑡 ∗ we have: ∀𝑡 ∈ ℕ0, 𝐴𝑡 ∩ 𝐴𝑡 ∗ = 2𝐾𝑡−1 ∩ 2𝐾𝑡−1 ∗ = 2(𝐾𝑡−1 ∩ 𝐾𝑡−1 ∗ ). (8.8) applying lemma 8.4 to the intersection in the last parenthesis of (8.8) we have ∀𝑡 ∈ ℕ0, 2(𝐾𝑡−1 ∩ 𝐾𝑡−1 ∗ ) = 2(𝐴𝑡−1 ∩ 𝐴𝑡−1 ∗ ) = = 2(2𝐾𝑡−2 ∩ 2𝐾𝑡−2 ∗ ) = 22(𝐾𝑡−2 ∩ 𝐾𝑡−2 ∗ ). (8.9) applying lemma 8.4 again to the intersection in the last parenthesis of (8.9) and iterating, we obtain ∀𝑡 ∈ ℕ0, 2 2(𝐾𝑡−2 ∩ 𝐾𝑡−2 ∗ ) = 22(𝐴𝑡−2 ∩ 𝐴𝑡−2 ∗ ) = ⋯ = 2𝑡−1(𝐾1 ∩ 𝐾1 ∗). (8.10) finally, applying lemma 8.4 again to the intersection in the last parenthesis of (8.10), we have ∀𝑡 ∈ ℕ0, 2 𝑡−1(𝐾1 ∩ 𝐾1 ∗) = 2𝑡−1(𝐴1 ∩ 𝐴1 ∗ ) = 2𝑡−1(2𝐾0 ∩ 2𝐾0 ∗) = = 2𝑡 (𝐾0 ∩ 𝐾0 ∗) = 2𝑡 ({1} ∩ {1}) = {2𝑡 } . □ (8.11) theorem 8.6. the intersection between 𝐾𝑡 and 𝐾𝑡 ∗ is equal to {2𝑡 }, that is ∀𝑡 ∈ ℕ0, 𝐾𝑡 ∩ 𝐾𝑡 ∗ = {2𝑡 }. (8.12) michele ventrone 245 proof. applying lemma 8.4 to the intersection 𝐾𝑡 ∩ 𝐾𝑡 ∗, we have (8.5). applying the lemma 8.5 at the intersection 𝐴𝑡 ∩ 𝐴𝑡 ∗ we obtain (8.12). □ 9 conclusions collatz's conjecture can be re-proposed using the sets k and their first properties. we have seen that the sets 𝐾𝑡 and 𝐾𝑡 ∗ are non-empty (basic 3.1) and they are also two by two disjoint (corollary 3.4). so, if the following coverage equalities of ℕ0 were also true: a) ⋃ 𝐾𝑡 +∞ 𝑡=0 = ℕ0 , 𝑡 ∈ ℕ b) ⋃ 𝐾𝑡 ∗+∞ 𝑡=0 = ℕ0 , 𝑡 ∈ ℕ we could say that each of the families {𝐾𝑡 }𝑡∈ℕ and {𝐾𝑡 ∗}𝑡∈ℕ is a partition of ℕ0. in this case the collatz conjecture would be proved. about two countable families in the finite sets of the collatz conjecture 246 references [1] leggerini, s. (2004). “l’enigmatico pari e dispari che da cinquant’anni non fa dormire i matematici”. newton, n.7, 2004. [2] conway, j. h. "unpredictable iterations." proc. 1972 number th. conf., university of colorado, boulder, colorado, pp. 49-52, 1972. ratio mathematica volume 46, 2023 anti q-m-fuzzy normal subgroups s.palaniyandi * r.jahir hussain† abstract the fuzzy set has been applied in wide area by many researchers. we define the concept of anti-homomorphism in q-fuzzy subgroups and q-fuzzy normal subgroups and establish some result in this research article and develop some theory of antihomomorphism in q-fuzzy subgroups, normal subgroups and also extend results on qfuzzy abelian subgroup and qfuzzy normal subgroup. many research scholars completed their research in field of fuzzy subgroup, anti fuzzy subgroup, q-fuzzy subgroup, anti q-fuzzy subgroup, homomorphism, anti homomorphism etc. keywords: q-m-fuzzy subgroup, q-mfuzzy normal subgroups, anti q-mfuzzy normal subgroups, group q-m-homomorphism and group anti q-m-homomorphism. ams subject classification: 03e72, 03e75, 08a72 1 *pg & research department of mathematics, jamal mohamed college, affiliated to bharathidasan university, tiruchirappalli, tamilnadu, india. palanijmc85@gmail.com. †pg & research department of mathematics, jamal mohamed college, affiliated to bharathidasan university, tiruchirappalli, tamilnadu, india. hssn jhr@yahoo.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1071. issn: 1592-7415. eissn: 2282-8214. ©s.palaniyandi et al. this paper is published under the cc-by licence agreement. 147 s.palaniyandi and r.jahir hussain 1 introduction according to zadeh.l.a rosenfeld [1971] was introduce fuzzy sets. it has subsequently been employed in a variety of scientific domains, including engineering, social science, medicine, and pure and applied mathematics. rosenfeld developed the concept of fuzzy subgroups asaad [1991]. biswas.r proposed antifuzzy subgroups in biswas [1990]. solairaju.a and nagarajan.r pioneered the structure of q-fuzzy groups palaniappan and muthuraj [2004]. jacobson.n was the first to use the words m-group and m-subgroup in jacobson [1951]. in this study, we present and discuss the concepts of group qm homomorphism, group anti qm homomorphism, and anti qm-fuzzy normal subgroup of m-group. 2 preliminaries definition 2.1. let x 6= φ. let x is fuzzy θ ⊆ θ : x ∈ [0,1]. definition 2.2. zadeh [1965] a fuzzy ⊆ θ ≤ n. it is satisfying the axioms, (i) θ (αβ) ≥ lower{θ (α) , θ (β)} (ii) θ (α−1) = θ(α),∀α, β ∈ n. definition 2.3. biswas [1990] afuzzy ⊆ θ of a group g is said to be anti fuzzy ⊆ g if it is satisfying the following conditions, (i) θ (uv) ≤ lower{θ (u) , θ (v)} (ii) θ (u−1) = θ(u),∀u, v ∈ g. definition 2.4. biswas [1990] let g: m and θ ⊆ g. then θ is called m-fuzzy ⊆ g if ∀u ∈ g and m ∈ m, then θ(mx) ≤ θ(u) definition 2.5. jacobson [1951] a q-fuzzy set θ is q−fuzzy ≤ g if ∀u,v ∈ g, and ρ ∈ q (i) θ (uv, ρ) ≥ lower{θ (u, ρ) , θ (v, ρ)} (ii) θ (u−1, ρ) = θ (u, ρ) definition 2.6. solairaju and nagarajan [2009] let fuzzy λ ⊆ x. for t�[0,1] ⊆ θ is denoted by [θt = {u ∈ u : θλ(u) ≥ t}] definition 2.7. sithar selvam et al. [2014] a q-fuzzy set θ is called q ≤ g if ∀u,v ∈ g, and ρ ∈ q in anti q fuzzy. 148 anti q-m-fuzzy normal subgroups (i) θ (uv,ρ) ≤ lower{θ (u,ρ) ,θ (v,ρ)} (ii) θ (u−1,ρ) = θ (u,ρ) definition 2.8. sithar selvam et al. [2014] an antifuzzy normal q ≤ g. then g ↗ θ of g if ∀x,y ∈ g and ρ ∈ q, θ(uyx−1,ρ) = θ(v,ρ). 3 anti q-mfuzzy normal subgroups and its level subsets definition 3.1. let θ be anti fuzzy q−m−≤ m −groupg, then θ ≤ m(g) if ∀u,v ∈ g, ρ ∈ q, and m ∈ m such that θ(m(uvu−1),ρ) = θ(m(v),ρ) (or) θ(m(uv),ρ) = θ(m(vu),ρ). definition 3.2. let θ be anti fuzzy q−m ≤ m −groupg. for any t ∈ [0,1], the subset θt is defined by θt = {u ∈ g,ρ ∈ q,m ∈ mθ(m(u),ρ) ≤ t} and it is the subset of θ. theorem 3.1. if θ is a fuzzy q−m− ⊆ of a m-group g, then θ is an anti fuzzy q−m−≤ m −groupg iff the level subset θt, t ∈ [0,1] is subgroup of m-group g. proof. let us assume that θ is an antifuzzy −q−m−≤ m −groupg. the level subset θt = {u ∈ g,ρ ∈ q,m ∈ mθ(m(u),ρ) ≤ t, t ∈ [0,1]}. let u,v ∈ θt, then θ(mx,ρ) ≤ t and θ(my,ρ) ≤ t now θ(m(uy−1),ρ) ≤ upper{θ((mu),ρ),θ(m(v−1),ρ)} = upper{θ(mu,ρ),θ(mv,ρ)} ≤ upper{t, t} thus θ(m(uv−1),ρ) ≤ t hence xy−1 ∈ θt. therefore θt ≤ m(g). conversely, let θt be a subgroup of a m-group g. let u,v ∈ θt. then θ(mu,ρ) ≤ t and θ(mv,ρ) ≤ t. ⇒ θ(m(uv−1),ρ) ≤ t , because{uv−1 ∈ θt} = upper{t, t} = upper{θ(mu,ρ),θ(mv,ρ)} 149 s.palaniyandi and r.jahir hussain therefore θ(m(uv−1),ρ) ≤ upper{θ(mu,ρ),θ(mv,ρ)} hence θ is an anti fuzzy q ≤ m −groupg. definition 3.3. let θ be a anti fuzzy q − m ≤ m − groupg. the set n(θ) is defined by n(θ) = {α ∈ gθ(m(αua−1),ρ) = θ(m(u),ρ)} ∀u ∈ g and ρ ∈ q, m ∈ m. and it is called an anti fuzzy q-m-normalizer of θ. theorem 3.2. if θ is a fuzzy q − m ≤ m − groupg. then θ is an anti fuzzy q−m −fuzzy ≤ m −groupg iff the level subsets θt, t ∈ [0,1] ≤ m(g). proof. let us assume that θ ≤ q−m −antifuzzynormalsubgroupofam(g) and the level subsets θt, t ∈ [0,1] is a subgroup of a m-group g. we take u ∈ g and α ∈ θt, then θ(ma,ρ) ≤ t now θ(m(αxa−1),ρ) = θ(ma,ρ) ≤ t. since θ is an anti fuzzy normal q-m ≤ m(g), θ(m(uau−1),ρ) ≤ t therefore uαu−1 ∈ θt, hence θt ≤ m(g). theorem 3.3. if θ is an ≤ q−m −antifuzzynormalsubgroupofam(g) then (i) n(θ) ≤ m(g). (ii) θ is an normal anti fuzzy -q-m ≤ iff n(θ) = g. (iii) θ is an normal fuzzy anti−q−m ≤ n(θ). proof. let α,β ∈ n(θ). (i) then θ(m(αua−1),ρ) = θ(mu,ρ)∀u ∈ g,ρ ∈ q,m ∈ m and θ(m(βxβ−1),ρ) = θ(mu,ρ)∀u ∈ g,ρ ∈ q,m ∈ m. now θ(m(αβu(αβ)−1),ρ) = θ(m(αβuβ−1α−1),ρ) = θ(m(βuβ−1),ρ) = θ(mu,ρ) then we have, θ(m(αβu(αβ)−1),ρ) = θ(mu,ρ) ⇒ αβ ∈ n(θ) therefore n(θ) ≤ m(g). (ii) we know that nθ ⊆ g, (1) θ is an normal fuzzy anti-q−m ≤ g. let α ∈ g, then θ(m(αuα−1),ρ) = θ(mu,ρ) ∀ u ∈ g, ρ ∈ q, m ∈ m. then α ∈ n(θ) ⇒ g ⊆ n(θ) (2) 150 anti q-m-fuzzy normal subgroups from (1)&(2), we get n(θ) = g conversely, assume that n(θ) = g we have, θ(m(αxα−1),ρ) = θ(mx,ρ)∀α,x ∈ g,ρ ∈ q,m ∈ m. therefore θ is an fuzzy normal anti q−m ≤ m(g). (iii) let θ be an fuzzy normal anti q−m ≤ m we take α ∈ g, then we have θ(m(αua−1),ρ) = θ(mu,ρ)∀u ∈ g,ρ ∈ q,m ∈ m. therefore α ∈ n(θ) ⇒ g ⊆ n(θ). hence θ is an fuzzy normal anti q−m ≤ n(θ) theorem 3.4. let θ be an fuzzy normal anti q−m ≤ m(g), then hθh−1 is also an fuzzy normal anti q−m ≤ m(g) ∀ h ∈ g, ρ ∈ q, m ∈ m. proof. given θ ≤ m(g) ≤ m(g) (i)(hθh−1)(m(uv),ρ) = θ(m(h−1(uv)h),ρ) = θ(m(h−1(uhh−1v)h),ρ) = θ(m((h−1uh)(h−1vh)),ρ) ≤ upper{θ(m(h−1uh),ρ),θ(m(h−1vh),ρ)} ≤ upper{hθh−1(mu,ρ),hθh−1(mv,ρ)} ∀ u,v ∈ g, ρ ∈ q and m ∈ m. (ii)hθh−1(mu,ρ) = θ(m(h−1uh),ρ) = θ(m(h−1uh)−1,ρ) = θ(m(h−1u−1h),ρ) = hθh−1(mu−1,ρ) ∀ u,v ∈ g, ρ ∈ q , m ∈ m. therefore hθh−1 is an fuzzy anti q−m ≤ m(g). theorem 3.5. let θ fuzzy antiq−m ≤ m(g), then hθh−1 fuzzy antiq−m ≤ m(g), ∀h ∈ g, ρ ∈ q, m ∈ m. proof. given θ is an anti-q-m-fuzzy normal subgroup of m-group g. then hθh−1 ≤ g. now hθh−1(m(uvu−1),ρ) = θ(m(h−1(uvu−1)h),ρ) = θ(m(uvu−1),ρ) = θ(mv,ρ) = θ(m(hvh−1),ρ) = hθh−1(mv,ρ) 151 s.palaniyandi and r.jahir hussain therefore hθh−1 is also an fuzzy normal anti q−m ≤ m(g). theorem 3.6. the disjoint two fuzzy normal anti q−m ≤ m(g) is also an anti fuzzy anti q−m ≤ m(g)g. proof. let α and β be two anti-q-m-fuzzy subgroups of a m-group g. then (αβ)(m(uv−1),ρ) = lower{α(m(uv−1),ρ),β(m(uv−1),ρ)} ≤ lower{upper{α(c(u),ρ),α(m(v−1),ρ)} ,upper{β(c(u),ρ),β(m(v−1),ρ)}} ≤ lower{upper{α(cu,ρ),α(mv,ρ)}, upper{β(cu,ρ),β(mv,ρ)}} ≤ upper{lower{α(cu,ρ),α(mv,ρ)}, lower{β(cu,ρ),β(mv,ρ)}} therefore {(αβ)(c(uv−1),ρ)}≤ upper {(αβ)(cu,ρ),(αβ)(cv,ρ)} hence αβ is an fuzzy normal anti q−m ≤ m(g). theorem 3.7. if c and d are an anti fuzzy normal q-m ≤ m(g).then a∩b is anti fuzzy normal q-m ≤ m(g). proof. for any x,y ∈ g, q ∈ q, m ∈ m we have (c ∩d)(m(xyx−1),q) = upper{c(m(xyx−1),q),d(m(xyx−1),q)} = upper{c(my,q),d(my,q)} = (c ∩d)(my,q) hence c∩ is an anti fuzzy normal q-m ≤ m(g). 4 group q-mhomomorphism and group anti qmhomomorphism definition 4.1. the function f : g×q → h ×q is homorphism group q-m (i) f : g → h is a homomorphism group and (ii) f(m(uv),ρ) = (f(mv).f(mu),ρ) ∀ u,v ∈ g,ρ ∈ q,m ∈ m. where g and h are m-groups. definition 4.2. the function f : g×q → h ×q is anti homomorphism group of q-m if 152 anti q-m-fuzzy normal subgroups (i) f : g → h is homomorphism group (ii) f(m(uv),ρ) = (f(mx).f(mv),ρ) ∀ u,v ∈ g,ρ ∈ q,m ∈ m. theorem 4.1. if the function f : g × q → h × q is a group anti q-mhomomorphism (i) if θ is an anti q-m-fuzzy normal subgroup of h, then f−1(θ) is an fuzzy normal anti q−m ≤ m(g). (ii) if f is an epimorphism and θ is an fuzzy normal anti q−m ≤ m(g) then f(θ) is an anti normal fuzzy q-m ≤ h. where g and h are m-groups. proof. (i) given the function f : g × q → h × q is a group anti-q-mhomomorphism and θ is an anti normal fuzzy q-m ≤ h. for all u,v ∈ g,ρ ∈ q,m ∈ m we have, f−1(θ)(m(uvu−1),ρ) = θ(f(m(uvu−1)),ρ) = θ(fm(u−1).f(mv).f(mu),ρ) = θ(f(mv),ρ) = f−1(mv,ρ) hence f−1(θ) is an fuzzy normal anti q−m ≤ m(g). (ii) given θ is an fuzzy normal anti q − m ≤ m(g). then f(θ) is an anti q-m-fuzzy subgroup of h. for any ℵ,β ∈ h, we have f(θ)(m(αβℵ−1),ρ) = inf θ(mv,ρ) = infθ(m(uvu−1),ρ) f(v) = αβα−1 = infθ(mv,ρ) f(u) = a,f(v) = β = f(θ)(mb,ρ) (since f is an epimorphism) therefore f(θ) is an fuzzy normal anti q−m ≤ h. definition 4.3. let a and b be two fuzzy anti q−m ≤ m(g). the product of a and b is defined by ab(m(u),ρ) = infupper(a(mv,ρ),vz = u,b(mz,ρ))u ∈ g,ρ ∈ q,m ∈ m. theorem 4.2. if a and b are fuzzy normal anti q−m ≤ m(g), then ab is an fuzzy normal anti q−m ≤ g. 153 s.palaniyandi and r.jahir hussain proof. given a and b are two fuzzy normal anti q−m ≤ m(g). (i) ab(m(uv),ρ) = inf upper{a(m(u1y1),ρ),b(m(u2y2),ρ)} where u = u1y1 and v = u2y2 ≤ inf upper{upper{a(mu1,ρ),a(mv1,ρ)}, upper{b(mu2,ρ),b(mv2,ρ)}} ≤ upper{inf upper{a(mu1,ρ),a(mv1,ρ)}, inf upper{b(mu2,ρ),b(mv2,ρ)}} i.e.,ab(m(uv),ρ) ≤ upper{ab(m(u1y1),ρ),ab(m(u2y2),ρ)} (ii) ab(m(u−1),ρ) = infupper{b(m(z−1),ρ),a(m(v−1),ρ)} where(vz)−1 = u−1 = inf upper{b(mz,ρ),a(mv,ρ)} = inf upper{a(mv,ρ),b(mv,ρ)} = ab(mv,ρ). ab(mv−1,ρ) = ab(mv,ρ) hence ab is anti fuzzy normal q-m ≤ m(g). 5 conclusions in this research article, we gave some results of anti q-m-fuzzy normal subgroup, group q-m homomorphism and group anti q-m homomorphism. this article used to further research in fuzzy algebra. references m. asaad. groups and fuzzy subgroups. fuzzy sets and systems, 39(3):323–328, 1991. r. biswas. fuzzy subgroups and anti fuzzy subgroups. fuzzy sets and systems, 35(1):121–124, 1990. n. jacobson. lectures in abstract algebra. ebook, 1951. n. palaniappan and r. muthuraj. anti fuzzy group and lower level subgroups. antartica j.math, 1(1):71–76, 2004. 154 anti q-m-fuzzy normal subgroups a. rosenfeld. fuzzy groups. j. math. anal.appl., 35(3):512–517, 1971. p. sithar selvam, t. priya, k. nagalakshmi, and t. ramachandran. on some properties of anti-q-fuzzy normal subgroups. gen. math. notes, 22(1):1–10, 2014. a. solairaju and r. nagarajan. a new structure and construction of q-fuzzy groups. advances in fuzzy mathematics, 4(1):23–29, 2009. l. zadeh. fuzzy setl. information and control, 8(3):338–353, 1965. 155 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 39, 2020, pp. 69-77 69 parameter estimation of p-dimensional rayleigh distribution under weighted loss function arun kumar rao* himanshu pandey † abstract in this paper, p-dimensional rayleigh distribution is considered. the classical maximum likelihood estimator has been obtained. bayesian method of estimation is employed in order to estimate the scale parameter of p-dimensional rayleigh distribution by using quasi and inverted gamma priors. the bayes estimators of the scale parameter have been obtained under squared error and weighted loss functions. keywords: bayesian method, p-dimensional rayleigh distribution, quasi and inverted gamma priors, squared error and weighted loss functions. 2010 ams subject classification: 60e05, 62e15, 62h10, 62h12.‡ * department of statistics, maharana pratap p.g. college, jungle dhusan, gorakhpur, india e-mail: arunrao1972@gmail.com † department of mathematics & statistics, ddu gorakhpur university, gorakhpur, india e-mail: himanshu_pandey62@yahoo.com ‡ received on october 26th, 2020. accepted on december 17th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.561. issn: 1592-7415. eissn: 2282-8214. ©rao and pandey. this paper is published under the cc-by licence agreement. arun kumar rao and himanshu pandey 70 1 introduction the probability density function (pdf) of p-dimensional rayleigh distribution is given by ( ) ( ) ( )2 2 1 2 0 0 2 p x p f x; e ; x , . p x    − − − =    (1) (cohen and whitten [1]). the distribution with pdf (1), in which p=1, sometimes called the folded gaussian, the folded normal, or the half normal distribution. with p=2, the pdf of (1) is reduced to two-dimensional rayleigh distribution. with p=3, the pdf of (1) is reduced to maxwell-boltzmann distribution. let 1 2 n x , x ,.........., x be a random sample of size n having probability density function (1), then the likelihood function of (1) is given by (rao and pandey [2]) ( ) ( ) 2 1 1 2 1 1 2 2 n i i n n x np p i i f x; x e p    = − − − =     =          (2) the log likelihood function is given by ( ) ( ) 1 2 11 1 2 2 2 n n p i i ii np log f x; nlog nlog p log log x x   − == = −  − + −  (3) differentiating (3) with respect to θ and equating to zero, we get 2 1 2 n i i x np   ==  (4) 2 bayesian method of estimation in bayesian analysis the fundamental problem is that of the choice of prior distribution g (θ) and a loss function l ,        . the squared error loss function for the scale parameter θ are defined as 2 l ,         = −        (5) parameter estimation of p-dimensional rayleigh distribution under weighted loss function 71 the bayes estimator under the above loss function, say, s  is the posterior mean, i.e, ( )s e   = (6) this loss function is often used because it does not lead to extensive numerical computations but several authors ( zellner [3], basu and ebrahimi [4]) have recognized that the inappropriateness of using symmetric loss function. j.g.norstrom [5] introduced an alternative asymmetric precautionary loss function. and also presented a general class of precautionary loss functions with quadratic loss function as a special case. weighted loss function (ahamad et al. [6]) is given a 2 l ,          −      =    (7) the bayes estimator under weighted loss function is denoted by w  and is obtained as 1 1 w e  −     =       (8) let us consider two prior distributions of θ to obtain the bayes estimators. (i) quasi-prior: for the situation where the experimenter has no prior information about the parameter θ, one may use the quasi density as given by ( )1 1 0 0 d g ; , d ,   =   (9) where d = 0 leads to a diffuse prior and d = 1, a non-informative prior. (ii) inverted gamma prior: the most widely used prior distribution of θ is the inverted gamma distribution with parameters  and ( )0  with probability density function given by ( ) ( ) ( )1 2 0g e ; .         − + − =   (10) arun kumar rao and himanshu pandey 72 3 bayes estimators under ( )1g  the posterior density of θ under ( )1g  , on using (2), is given by ( ) ( ) ( ) 2 1 2 1 1 2 1 1 1 2 1 10 2 2 2 2 n i i n i i n n x np p d i i n n x np p d i i x e p f x x e d p         = = − − − − =  − − − − =             =               2 1 2 1 1 2 1 2 0 n i i n i i np xd np xd e e d      = =   −− +      −− +     =   2 1 1 2 2 1 1 2 1 2 n i i np d n npi xd i x e np d   =   + −      −− +  =         =    + −     (11) theorem 1. assuming the squared error loss function, the bayes estimate of the scale parameter θ, is of the form 2 1 2 2 n i i s x np d   ==   + −     (12) proof. from equation (6), on using (11), ( ) ( )s e f x d      = =  2 1 1 2 2 1 1 1 2 01 2 n i i np d n npi xd i x e d np d   =   + −      −− + −  =         =    + −      parameter estimation of p-dimensional rayleigh distribution under weighted loss function 73 1 2 2 1 2 2 2 1 2 2 1 2 np d n i i np d n i i np x d np d x + − = + − =      + −        =    + −            or, 2 1 2 2 n i i s x np d   ==   + −     . theorem 2. assuming the weighted loss function, the bayes estimate of the scale parameter θ, is of the form 2 1 1 2 n i i w x np d   ==   + −     (13) proof. from equation (8), on using (11), ( ) 1 1 1 1 w e f x d     − −       = =           2 1 1 1 2 2 1 1 1 2 01 2 n i i np d n npi xd i x e d np d   = − + −   −− + +  =             =      + −          1 1 2 2 1 2 2 1 2 1 2 np d n i i np d n i i np x d np d x − + − = + =         +        =      + −               arun kumar rao and himanshu pandey 74 1 2 1 1 2 n i i np d x − =   + −   =        2 1 1 2 n i i w x np d   = =   + −     4 bayes estimators under ( )2g  under ( )2g  , the posterior density of θ, using equation (2), is obtained as ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 1 12 1 1 1 12 1 10 2 2 2 2 n i i n i i n n x np p i i n n x np p i i x e e p f x x e e d p                     = = − − +− − − =  − − +− − − =             =               2 1 2 1 1 1 2 1 1 2 0 n i i n i i np x np x e e d        = =     − + − + +            − + − + +         =   2 1 1 1 2 2 2 12 n i i np x np n i i e np x       =     − + − + +        + =  =     + +         2 1 2 2 1 1 1 2 2 n i i np n npi x i x e np       = +     − + − + +    =       +     =    +     (14) parameter estimation of p-dimensional rayleigh distribution under weighted loss function 75 theorem 3. assuming the squared error loss function, the bayes estimate of the scale parameter θ, is of the form 2 1 1 2 n i i s x np     = + = + −  (15) proof. from equation (6), on using (14), ( ) ( )s e f x d      = =  2 1 2 2 1 1 2 0 2 n i i np n npi x i x e d np        = +     − + − +    =       +     =    +      2 2 1 1 2 2 1 1 2 2 np n i i np n i i np x np x       + = + − =     +  + −        =    +    +        or, 2 1 1 2 n i i s x np     = + = + −  . theorem 4. assuming the weighted loss function, the bayes estimate of the scale parameter θ, is of the form 2 1 2 n i i w x np     = + = +  (16) proof. from equation (8), on using (14), ( ) 1 1 1 1 w e f x d     − −       = =           arun kumar rao and himanshu pandey 76 2 1 1 2 2 1 2 1 2 0 2 n i i np n npi x i x e d np        = − +     − + − + +    =         +     =      +          1 2 2 1 1 2 2 1 1 2 2 np n i i np n i i np x np x       − + = + + =       +  + +        =      +    +           1 2 1 2 n i i np x   − =    +  =    +       or, 2 1 2 n i i w x np     = + = +  . 5 conclusion in this paper, we have obtained a number of estimators of parameter. in equation (4) we have obtained the maximum likelihood estimator of the parameter. in equation (12) and (13) we have obtained the bayes estimators under squared error and weighted loss function using quasi prior. in equation (15) and (16) we have obtained the bayes estimators under squared error and weighted loss function using inverted gamma prior. in the above equation, it is clear that the bayes estimators depend upon the parameters of the prior distribution. parameter estimation of p-dimensional rayleigh distribution under weighted loss function 77 references [1] cohen,a.c.,whitten,b.j., “parameter estimation in reliability and life span models”, dekker, new york, isbn 0-8247-7980-0, 182. 1988. [2] rao, a.k. and pandey, h., “bayesian estimation of scale parameterof p-dimensional rayleigh distribution using precautionary loss function”, varahmihir journal of mathematical sciences, vol. 8, no. 1, 1-8. 2008. [3] zellner, a., “bayesian estimation and prediction using asymmetric loss functions,” jour. amer. stat. assoc., 91, 446-451. 1986. [4] basu, a. p. and ebrahimi, n., “bayesian approach to life testing and reliability estimation using asymmetric loss function,” jour. stat. plann. infer., 29, 21-31. 1991. [5] norstrom, j. g., “the use of precautionary loss functions in risk analysis,” ieee trans. reliab., 45(3), 400-403. 1996. [6] ahmad, s. p., et al., “bayesian estimation for the class of life-time distributions under different loss functions,” statistics and applications, vol. 14, nos. 1&2, pp. 75-91. 2016. ratio mathematica volume 46, 2023 ifg#α-cs in intuitionistic fuzzy topological spaces christy jenifer j* kokilavani v † abstract the primary aim of this prospectus is to introduce and study the basic properties of intuitionistic fuzzy generalized #α-closed sets, intuitionistic fuzzy generalized #α-open sets. here we, compare the g #αclosed sets with the existing closed sets with proper examples given. keywords: ifs, ift, ifg#αcs, ifg #αos. 2020 ams subject classifications: 54a40. 1 *kongunadu arts and science college, coimbatore, tamil nadu, india; christijeni94@gmail.com. †kongunadu arts and science college, coimbatore, tamil nadu,india; vanikasc@yahoo.co.in. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1058. issn: 1592-7415. eissn: 2282-8214. ©christy jenifer j et al. this paper is published under the cc-by licence agreement. 63 christy jenifer j and kokilavani v 1 introduction earlier g alpha closed sets has been introduced in general topology, fuzzy topology, supra topology, nano topology, we wish to introduce g #α in intuitionistic fuzzy topological spaces. following authors motivated me to further continue my research in intuitionistic fuzzy topology. a. zadeh a.zadeh [1965] initiated the concept of fuzzy sets in which the values are taken between 0 and 1. further atanassov [1986] established the idea of ifs by generalizing fuzzy sets, here similar to fuzzy topology values are taken between 0 and 1 but it is defined as membership value and non-membership value. later on ifts was initiated using the notion of ifss which was proposed by coker coker [1997], using the membership and non-membership values it was applied in general topological axioms. in continuation of above we initiate ifg #α -closed sets and ifg alpha-open sets and establish its characterization and find the weaker and stronger forms of topology by comparing it to other existing sets and also find whether it satisfies the topological axioms as union and finite intersection properties 2 preliminaries in this segment, few basic definitions and results are reviewed. definition 2.1. thakur and chaturvedi [2008]let a be an ifs in (x,τ), is proposed to be ifgcs if cl(a) ⊆ u whenever a ⊆ u and u is an ifos in x. similarly, ifαgcs and ifgαcs kalamani et al. [2012], , ifgspcs santhi and jayanthi [2010] , ifgscs santhi and sakthivel [2009], ifgsrcs anitha and mohana [2018], were introduced. with the help of above closed sets we, initiate new set ifg#αclosed set. 3 ifg#αclosed sets definition 3.1. an ifs c in (x, τ) is proposed to be an ifg#α-closed set if αcl(c) ⊆ c, whenever c ⊆ u and u is an ifgos in (e, τ). the family of all ifg#αcs of an ifts (e, τ) is defined by ifg#αc(x). example 3.1. consider e = {p, q}, τ = {0∼, j, 1∼} is ift on e, in that j =< e, (0.3, 0.2), (0.5, 0.6) >. in this the only α-open sets are 0∼, 1∼, j. at that point ifs, c =< e, (0.1, 0), (0.6, 0.8) > an ifg#αcs in (e, τ). theorem 3.1. every ifcs is ifg#αcs, but reverse implication is not possible. 64 ifg#α-cs in intuitionistic fuzzy topological spaces proof. considerc is ifcs in (e, τ). suppose an ifs, c ⊆ u where u is ifgos. considering αcl(c) ⊆ cl(c) and c is an ifcs in e, αcl(c) ⊆ cl(c) = c ⊆ u and u is ifgos. that is αcl(c) ⊆ u. consequently c is ifg#αcs in e. example 3.2. let e = {p, q} and let τ = {0∼, j, 1∼} is an ift on e, where j =< e, (0.4, 0.1), (0.5, 0.6) >. let c =< e, (0.3, 0.1), (0.7, 0.9) > be an ifs in e. here c is an ifg#αcs but not ifcs in (e, τ). theorem 3.2. every ifαcs is ifg#αcs but, reverse implication is not possible. proof. consider c is a ifαcs in e. suppose an ifs, c ⊆ u, where u is ifgos. considering c is ifαcs, α cl(c) = c. hence αcl(c) ⊆ u once c ⊆ u, u is ifgos. consequently ifg#αcs in e. example 3.3. let e = {p, q} and let τ = {0∼, j, 1∼} be ift, we have j =< e, (0.2, 0.4), (0.6, 0.5) >. consider c =< e, (0.2, 0.4), (0.6, 0.5) > be an ifs on e. then c is ifg#αcs but not ifα cs in (e, τ). theorem 3.3. every ifrcs is ifg#αcs but, reverse implication is not possible. proof. consider c is an ifrcs. we know that c=cl(int(c)) using definition. this signifies cl(c)=cl(int(c)). consequently cl(c)=c. by which c is ifcs in e. c is an ifg#αcs in e. example 3.4. consider e = {p, q}, τ = {0∼, j, 1∼} be ift, we have j =< e, (0.4, 0.4), (0.5, 0.5) >. here an ifs,c =< e, (0.2, 0.2), (0.7, 0.8) > is an ifg#αcs but not ifrcs in (e, τ). theorem 3.4. every ifg#αcs is ifsgcs but, reverse implication is not possible. proof. consider c an ifg#αcs. suppose an ifs, c ⊆ u where u is ifso set. since, every ifso set is ifgo set and c be an ifg#αcs. we have scl(c) ⊆ αcl(c) ⊆ u. therefore c is ifsgcs. example 3.5. let e = {p, q} τ = {0∼, j, 1∼} be ift we have j =< e, (0.5, 0.7), (0.6, 0.7) >. let c =< e, (0.5, 0.6), (0.8, 0.9) >. then c is an ifsgcs but not ifg#αcs in (e, τ). theorem 3.5. every ifg#αcs is ifgscs but, reverse implication is not possible. proof. consider c an ifg#αcs in (e, τ). suppose an ifs, c ⊆ u where u is ifos. since, every ifos set is an ifgos and a be an ifg#αcs. we have scl(c) ⊆ αcl(c) ⊆ u. therefore, c is ifgscs set. 65 christy jenifer j and kokilavani v example 3.6. let e = {p, q}, τ = {0∼, j, 1∼} be ift, we have j =< e, (0.1, 0.2), (0.6, 0.6) >. let, c =< e, (0, 0.2), (0.9, 0.6) >. here c is an ifgscs but not an ifg#αcs. theorem 3.6. every ifg#αcs is ifgsrcs in (e, τ) but, reverse implication is not possible. proof. consider c an ifg#αcs in (e, τ). suppose an ifs, c ⊆ u where u is an ifros. since, every ifros set is an ifgsros and c be an ifg#αcs. we have scl(c) ⊆ αcl(c) ⊆ u. therefore, c is ifgsrcs set. example 3.7. let e = {p, q}, τ = {0∼, j, 1∼}, here j =< e, (0.3, 0.6), (0.7, 0.4) >. consider, c =< e, (0.3, 0.4), (0.7, 0.6) >. here c is an ifgsrcs but not an ifg#αcs. remark 3.1. for any two ifg#αcs intersection is also ifg#αcs. proof. consider c and d any two ifg#αcs. that is iαcl(c) ⊆ g. once c ⊆ g and g is ifgos and is iαcl(d) ⊆ g whenever d ⊆ g and g is ifgos. now, iαcl(c ⊆ d) = iαcl(c) ∩ iαcl(d) ⊆ g, where (c ∩ d) ⊆ g and g is ifgos. thus, intersection of any two ifg#αclosed set is ifg#αcs. theorem 3.7. let (e, τ) be ifts. then for every c ⊆ ifg#αcs(e) and for every d ∈ ifs(e), c ⊆ d ⊆ αcl(c) implies d ∈ ifg#αcs(e). proof. consider ifs d ⊆ u and u be ifgos, considering c ⊆ d, c ⊆ u and c is ifg#αcs, αcl(c) ⊆ u. by assumption, d ⊆ αcl(c), α cl(d) ⊆ α cl(c) ⊆ u. consequently αcl(d) ⊆ u. thus c is ifg#αcs of e. theorem 3.8. consider e an ifts. then ifgo(e) = ifgc(e) if and only if every ifs in e an ifg#αcs in e. proof. necessity : assume ifgo(e) = ifgc(e). consider c ⊆ g, g an ifgos. this signifies αcl(c) ⊆ αcl(g). considering g an ifgos in e, by assumption g is ifgcs in e, αcl(c) ⊆ g. this signifies αcl(c) ⊆ g. consequently c is ifg#αcs in e. sufficiency : assume, every ifs is ifg#αcs. consider g∈ ifo(e), we have g∈ ifgo(e) and so c ⊆ g also g is ifos in e, by assumption αcl(c) ⊆ g. that is g∈ ifgc(e). accordingly ifgo(e)⊆ ifgcs(e). consider c∈ ifgc(e), we have cc is ifgos in e. but ifgo(e)⊆ ifgc(e). consequently cc ∈ ifgc(e). here c∈ ifgo(e). consequently ifgc(e)⊆ ifgo(e). we know that ifgo(e) ⊆ ifgc(e). theorem 3.9. let c be ifg#αcs of e, then αcl(c)-c contains no non-empty ifgcs. 66 ifg#α-cs in intuitionistic fuzzy topological spaces proof. suppose c is ifg#αcs of e and consider f to be non-empty ifgcs of e, and so f ⊆ αcl(c)-c. we have a ⊆ e f. considering c is ifg#αcs and e-f is ifgos, and so αcl(c) ⊆ e − f . this signifies f ⊆ e − αcl(c). also f ⊆ (e −αcl(c))∩(αcl(c)−c) ⊆ (e −αcl(c))∩αcl(c) = ϕ. consequently f is empty. theorem 3.10. let c ⊆ d ⊆ e and assume that c is ifg#αcs in e then c is an ifg#αcs relative to d. proof. here we have, c ⊆ d ⊆ e also c an ifg#αcs. considering c ⊆ d ∩ f where f is ifgos in e. since c is an ifg#αcs in e, c ⊆ f implies, αcl(c) ⊆ f . it follows that d ∩ αcl(c) ⊆ d ∩ f = f . thus c is an ifg#αcs relative to d. 4 ifg#α-open sets in this segment, we define and establish the idea of ifg#α –open sets (briefly ifg#αos) in ifts and establish its characterizations. definition 4.1. a subset b of ifts e is proposed to be an ifg#α-open if ac is ifg#α-open set. theorem 4.1. consider e an ifts we have, (i) . every if-open set is ifg#αos. (ii) . every ifα-open set is ifg#αos. (iii) . every ifr-open set is ifg#αos. proof. proof is obvious theorem 4.2. let (e, τ)be the ifts then, (i) every ifg#α-open set is ifsgos. (ii) every ifg#α-open set is ifgspos. (iii) every ifg#α-open set is ifgsos. (iv) every ifg#α-open set is ifgsros. proof. proof is obvious theorem 4.3. an ifs c of ifts e is ifg#αos on the condition that d ⊆ αint(c) at any moment d is ifgcs in e also d ⊆ f . 67 christy jenifer j and kokilavani v proof. necessity: let c is ifg#αos in e. consider d be ifgcs in e also d ⊆ c. we have dc is ifgos in e in this extent cc ⊆ dc. considering cc is ifg#αcs, then αcl(cc) ⊆ dc. so (αcl(c))c ⊆ dc. consequently d ⊆ αcl(c). sufficiency: consider d ⊆ αint(c) at any moment d is ifgcs also d ⊆ c. we have cc ⊆ dc also dc an ifgos. by assumption, (αcl(c))c ⊆ dc.therefore cc is ifg#αcs of e. consequently c is ifg#αos. theorem 4.4. consider e an ifts. we have for all c ⊆ ifg#αos also probably d ∈ ifs(e), αint(c) ⊆ d ⊆ c signifies d∈ ifg#αos. proof. here αint(c) ⊆ d ⊆ c implies cc ⊆ dc ⊆ (αint(c))c. consider dc ⊆ f also f is ifgos in e. considering cc ⊆ dc, cc ⊆ f . since cc is ifg#αcs, αcl(cc) ⊆ f and so dc ⊆ (αint(c))c = αcl(cc). consequently αcl(dc) ⊆ αcl(cc) ⊆ f . accordingly dc is ifg#αcs in e. this signifies d is ifg#αos in e. therefore d ∈ ifg#αcs. 5 conclusions here we have derived a new concept of closed set called ifg #α cs. we have proved that ifg#αcs is stronger than ifcs, ifcs, ifrcs. also, ifgscs, ifsgcs and ifgsrcs is stronger than ifg#αcs. further, it satisfies the intersection axiom but it does not satisfy union property. thus we can conclude that ifg#αcs does not form a topology. this concept can further be extended to continuous functions, irresolute functions, various forms of continuous functions such as completely continuous, perfectly continuous, contra continuous, almost continuous, slightly continuous and spaces can be introduced which are normal space and regular space, also connectedness can be described. application can be done based on membership and non-membership values and find the mcdm problems. references s. anitha and k. mohana. ifgsr – closed sets in intuitionistic fuzzy topological spaces. international journal of innovative research in technology, 5(2):365– 369, 2018. l. a.zadeh. fuzzy sets. information and control, 8:338–353, 1965. d. coker. an introduction to fuzzy topological space. fuzzy sets and systems, 88: 81–89, 1997. 68 ifg#α-cs in intuitionistic fuzzy topological spaces d. kalamani, k. sakthivel, and c. s. gowri. generalized alpha closed sets in intuitionistic fuzzy topological spaces. applied mathematical sciences, 6(94): 4691–4700, 2012. r. santhi and d. jayanthi. intuitionistic fuzzy generalized semipreclosed mappings. notes on intuitionistic fuzzy sets, 16:28–39, 2010. r. santhi and k. sakthivel. intuitionistic fuzzy generalized semicontinuous mappings. advances in theoretical and applied mathematics, 5:73–82, 2009. s. s. thakur and r. chaturvedi. generalized closed sets in intuitionistic fuzzy topology. the jornal of fuzzy mathematics, 16:559–572, 2008. 69 ratio mathematica volume 45, 2023 radio mean labeling of digraphs palani k* sabarina subi s s† abstract let 𝐷 be a strong digraph and let 𝑑(𝑢, 𝑣) denote the distance between any two vertices in 𝐷. a radio mean labeling is a one-to-one mapping 𝑓 from 𝑉(𝐷) to 𝑁 satisfying the condition 𝑑(𝑢, 𝑣) +⌈ 𝑓(𝑢)+𝑓(𝑣) 2 ⌉ ≥ 1 + 𝑑𝑖𝑎𝑚(𝐷) for every 𝑢, 𝑣 ∈ 𝑉(𝐷). the span of a labeling 𝑓 is the maximum integer that 𝑓 maps to a vertex of𝐷. the radio mean number of 𝐷, 𝑟𝑚𝑛 (𝐷) is the lowest span taken over all radio mean labelings of the graph 𝐷. in this paper, we analyze radio mean labeling for some newly defined digraphs. keywords: radio mean, radio mean number, radio mean labeling, digraphs. ams subject classification: 05c78‡. *associate professor, pg & research department of mathematics (a.p.c. mahalaxmi college for women, thoothukudi-628 002, tamilnadu, india); palani@apcmcollege.ac.in. †research scholar (reg no.20112012092001), a.p.c. mahalaxmi college for women, thoothukudi-628 002, tamilnadu, india. sabarin203@gmail.com. (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamil nadu, india). ‡received on july 29, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.1023. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 248 mailto:palani@apcmcollege.ac.in palani k and sabarina subi s s 1. introduction the graph labeling problem is one of the recent developing area in graph theory. alex rosa first introduced this problem in 1967[10]. radio labeling is motivated by the channel assignment problem introduced by w. k. hale in 1980[4].in 2001, gary chartrand defined the concept of radio labeling of 𝐺[2].liu and zhu first determined the radio number in 2005[5].ponraj et al.[8] introduced the notion of radio mean labeling of graphs and investigated radio mean number of some graphs [9]. in this paper, we introduce a new definition for radio mean labeling of digraphs and also we study radio mean number of some newly defined digraphs. radio labeling is used for x-ray, crystallography, coding theory, network security, network addressing, channel assignment process, social network analysis such as connectivity, scalability, routing, computing, cell biology etc., the following results are used in the subsequent section. definition 1.1.let 𝐷 be a strong digraph and let 𝑑(𝑢, 𝑣) denote the distance between any two vertices in𝐷. a radio mean labeling is a one-to-one mapping 𝑓 from 𝑉(𝐷) to 𝑁 satisfying the condition 𝑑(𝑢, 𝑣) +⌈ 𝑓(𝑢)+𝑓(𝑣) 2 ⌉ ≥ 1 + 𝑑𝑖𝑎𝑚(𝐷) for every 𝑢, 𝑣 ∈ 𝑉(𝐷). the span of a labeling 𝑓 is the maximum integer that 𝑓 maps to a vertex of𝐷. the radio mean number of𝐷, 𝑟𝑚𝑛 (𝐷) is the lowest span taken over all radio mean labelings of the graph𝐷. definition 1.2. consider a globe. let 𝑢, 𝑣 be the vertices of degree𝑛. orient the edges of all but one 𝑢 − 𝑣 path of length two in same direction. orient the left out 𝑢 − 𝑣 path in opposite direction. it is strongly connected and is called a diglobe. it is denoted as 𝐺𝑙(𝑛).⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ figure 1.1. diglobe definition 1.3. if the edges of all 𝑢 − 𝑣 paths are in a single common direction it is not a strong digraph. we name it as sole diglobe (𝑆𝐺𝑙(𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗). remark 1.4. if there are atleast two paths are oriented in opposite directions, the diglobe becomes strongly connected. 249 radio mean labeling of digraphs remark 1.5. orient the edges of the globe in such a way that the two edges in each 𝑢 − 𝑣 path of length 2 get opposite directions. the resulting digraph is called alternate diglobe and is denoted as 𝐴𝐺𝑙(𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗. 𝐴𝐺𝑙(𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ is not a strong digraph. definition 1.6. consider a book with 𝑛 pages sharing a common edge. the common edge is called the spine or base of the book. orient all the edges except the spine in the one single direction and the spine in opposite direction. the resulting digraph is strongly connected and is called as directed book. the directed book is denoted as 𝐵(𝑚, 𝑛).⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ remark 1.7. the ditriangular book with n-pages is defined as n copies of cycle c3 sharing a common edge in a directed book. the ditriangular book is denoted as 𝐵(3, 𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ . figure 1.2. ditriangular book remark 1.8. the diquadrilateral book with n-pages is defined as 𝑛 copies of cycle 𝐶4 sharing a common edge in a directed book. the diquadrilateral book is denoted as 𝐵(4, 𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗. figure 1.3. diquadrilateral book 250 palani k and sabarina subi s s 2. main results theorem 2.1.the radio mean number of diglobe(𝐺𝑙(𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) is less than or equal to 𝑛 + 3 for3 ≤ 𝑛 ≤ 4. proof. let 𝐷 be a diglobe. let 𝑉(𝐷) = {𝑣1, 𝑣2, 𝑣3, … … . , 𝑣𝑛 , 𝑢, 𝑣} and 𝐴(𝐷) = {𝑢𝑣𝑖⃗⃗⃗⃗⃗⃗⃗/1 ≤ 𝑖 ≤ ⌊ 𝑛 2 ⌋} ∪ {𝑢𝑣𝑖⃗⃗⃗⃗⃗⃗⃗/ ⌊ 𝑛 2 ⌋ + 2 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣 ⌊ 𝑛 2 ⌋+1 𝑢⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗} ∪ {𝑣𝑖𝑣⃗⃗ ⃗⃗ ⃗⃗ /1 ≤ 𝑖 ≤ ⌊ 𝑛 2 ⌋} ∪ {𝑣𝑖𝑣⃗⃗ ⃗⃗ ⃗⃗ / ⌊ 𝑛 2 ⌋ + 2 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑣 ⌊ 𝑛 2 ⌋+1 }⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ the diameter of diglobe is 4. define 𝑓: 𝑉(𝐷) → 𝑁 by 𝑓(𝑣𝑖 ) = 𝑖 + 1 , 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑢) = 𝑛 + 2 𝑓(𝑣) = 𝑛 + 3 claim. 𝑓 is a valid radio mean labeling. since the diameter is 4, to prove 𝑓 is a radio mean labeling, it is enough to prove that, 𝑑(𝑥, 𝑦) + ⌈ 𝑓(𝑥)+𝑓(𝑦) 2 ⌉ ≥ 5 … … … … (1)for every pair of vertices (𝑥, 𝑦) where 𝑥 ≠ 𝑦. equivalently, it is enough to prove (1) for pair of vertices with minimum 𝑓 values and minimum 𝑑(𝑥, 𝑦)values. hence, the proof involves the following cases case a. consider the pairs(𝑢, 𝑣𝑖 ). here, 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣𝑖 ) ≥ 1 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣𝑖 ) + ⌈ 𝑓(𝑢) + 𝑓(𝑣𝑖 ) 2 ⌉ ≥ 1 + ⌈ 𝑛 + 2 + 𝑖 + 1 2 ⌉ ≥ 5 case b. consider the pairs(𝑣𝑖 , 𝑣). here, 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣) ≥ 1 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣) + ⌈ 𝑓(𝑣𝑖 ) + 𝑓(𝑣) 2 ⌉ ≥ 1 + ⌈ 𝑖 + 1 + 𝑛 + 3 2 ⌉ ≥ 5 case c. consider the pairs (𝑢, 𝑣) 𝑎𝑛𝑑 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣) = 2 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣) + ⌈ 𝑓(𝑢) + 𝑓(𝑣) 2 ⌉ ≥ 2 + ⌈ 𝑛 + 2 + 𝑛 + 3 2 ⌉ > 5 case d. consider the pairs (𝑣𝑖 , 𝑣𝑖+1) and 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣𝑖+1) = 2, then, 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣𝑖+1) + ⌈ 𝑓(𝑣𝑖 ) + 𝑓(𝑣𝑖+1) 2 ⌉ ≥ 2 + ⌈ 𝑖 + 1 + 𝑖 + 2 2 ⌉ > 5 hence, by all the above cases, the radio mean condition is satisfied by 𝑓. further, 𝑓 attains its maximum corresponding to 𝑣 and therefore 𝑟𝑚𝑛(𝐺𝑙(𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) ≤ 𝑛 + 3 for 3 ≤ 𝑛 ≤ 4. theorem 2.2. the radio mean number of diglobe(𝐺𝑙(𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) is 𝑛 + 2 for n > 4. proof. let 𝐷 be a diglobe. let 𝑉(𝐷) = {𝑣1, 𝑣2, 𝑣3, … … . , 𝑣𝑛 , 𝑢, 𝑣} and 𝐴(𝐷) = {𝑢𝑣𝑖⃗⃗⃗⃗⃗⃗⃗/1 ≤ 𝑖 ≤ ⌊ 𝑛 2 ⌋} ∪ {𝑢𝑣𝑖⃗⃗⃗⃗⃗⃗⃗/ ⌊ 𝑛 2 ⌋ + 2 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣 ⌊ 𝑛 2 ⌋+1 𝑢⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗} ∪ {𝑣𝑖𝑣⃗⃗ ⃗⃗ ⃗⃗ /1 ≤ 𝑖 251 radio mean labeling of digraphs ≤ ⌊ 𝑛 2 ⌋} ∪ {𝑣𝑖𝑣⃗⃗ ⃗⃗ ⃗⃗ / ⌊ 𝑛 2 ⌋ + 2 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑣 ⌊ 𝑛 2 ⌋+1 }⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ the diameter of diglobe is 4. define 𝑓: 𝑉(𝐷) → 𝑁 by 𝑓(𝑣𝑖 ) = 𝑖 , 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑢) = 𝑛 + 1 𝑓(𝑣) = 𝑛 + 2 claim.𝑓 is a valid radio mean labeling. since the diameter is 4, to prove 𝑓 is a radio mean labeling, it is enough to prove that, 𝑑(𝑥, 𝑦) + ⌈ 𝑓(𝑥)+𝑓(𝑦) 2 ⌉ ≥ 5 … … … … (1) for every pair of vertices (𝑥, 𝑦) where 𝑥 ≠ 𝑦. equivalently, it is enough to prove (1) for pair of vertices with minimum 𝑓 values and minimum 𝑑(𝑥, 𝑦)values. hence, the proof involves the following cases case a. consider the pairs (𝑢, 𝑣𝑖 ). here, 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣𝑖 ) ≥ 1 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣𝑖 ) + ⌈ 𝑓(𝑢) + 𝑓(𝑣𝑖 ) 2 ⌉ ≥ 1 + ⌈ 𝑛 + 1 + 𝑖 2 ⌉ ≥ 5 case b. consider the pairs (𝑣𝑖 , 𝑣). here, 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣) ≥ 1 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣) + ⌈ 𝑓(𝑣𝑖 ) + 𝑓(𝑣) 2 ⌉ ≥ 1 + ⌈ 𝑖 + 𝑛 + 2 2 ⌉ ≥ 5 case c. consider the pairs (𝑢, 𝑣) 𝑎𝑛𝑑 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣) = 2 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣) + ⌈ 𝑓(𝑢) + 𝑓(𝑣) 2 ⌉ ≥ 2 + ⌈ 𝑛 + 1 + 𝑛 + 2 2 ⌉ > 5 case d. consider the pairs (𝑣𝑖 , 𝑣𝑖+1) and 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣𝑖+1) = 2, then, 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣𝑖+1) + ⌈ 𝑓(𝑣𝑖 ) + 𝑓(𝑣𝑖+1) 2 ⌉ ≥ 2 + ⌈ 𝑖 + 𝑖 + 1 2 ⌉ > 5 hence, by all the above cases, the radio mean condition is satisfied by 𝑓. further, 𝑓 attains its maximum corresponding to 𝑣 and is 𝑛 + 2 for 𝑛 > 4. since 𝐷 contains only 𝑛 + 2 vertices, 𝑛 + 2 is the minimum of the maximum integer that could be assigned to the vertices of 𝐷. hence 𝑟𝑚𝑛(𝐷) = 𝑛 + 2 for𝑛 > 4. theorem 2.3. the radio mean number of ditriangular book(𝐵(3, 𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) is less than or equal to 𝑛 + 3 for𝑛 = 2. proof. let 𝐷 be a ditriangular book. let 𝑉(𝐷) = {𝑣1, 𝑣2, 𝑣3, … … . , 𝑣𝑛 , 𝑢, 𝑣} and 𝐴(𝐷) = {𝑢𝑣𝑖⃗⃗⃗⃗⃗⃗⃗/1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑖𝑣⃗⃗ ⃗⃗ ⃗⃗ /1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑢}⃗⃗ ⃗⃗ ⃗⃗ ⃗ the diameter of ditriangular book is 3. define 𝑓: 𝑉(𝐷) → 𝑁 by 𝑓(𝑣𝑖 ) = 𝑖 + 1 , 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑢) = 𝑛 + 2 𝑓(𝑣) = 𝑛 + 3 claim.𝑓 is a valid radio mean labeling. 252 palani k and sabarina subi s s since the diameter is 3, to prove 𝑓 is a radio mean labeling, it is enough to prove that, 𝑑(𝑥, 𝑦) + ⌈ 𝑓(𝑥)+𝑓(𝑦) 2 ⌉ ≥ 4 … … … … (1)for every pair of vertices (𝑥, 𝑦) where 𝑥 ≠ 𝑦. equivalently, it is enough to prove (1) for pair of vertices with minimum 𝑓 values and minimum 𝑑(𝑥, 𝑦)values. hence, the proof involves the following cases case a. consider the pairs (𝑢, 𝑣𝑖 ). here, 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣𝑖 ) ≥ 1 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣𝑖 ) + ⌈ 𝑓(𝑢) + 𝑓(𝑣𝑖 ) 2 ⌉ ≥ 1 + ⌈ 𝑛 + 2 + 𝑖 + 1 2 ⌉ ≥ 4 case b. consider the pairs (𝑣𝑖 , 𝑣). here, 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣) ≥ 1 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣) + ⌈ 𝑓(𝑣𝑖 ) + 𝑓(𝑣) 2 ⌉ ≥ 1 + ⌈ 𝑖 + 1 + 𝑛 + 3 2 ⌉ > 4 case c. consider the pairs (𝑢, 𝑣) 𝑎𝑛𝑑 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣) = 1 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣) + ⌈ 𝑓(𝑢) + 𝑓(𝑣) 2 ⌉ ≥ 1 + ⌈ 𝑛 + 2 + 𝑛 + 3 2 ⌉ > 4 case d. consider the pairs (𝑣𝑖 , 𝑣𝑖+1) and 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣𝑖+1) = 3, then, 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣𝑖+1) + ⌈ 𝑓(𝑣𝑖 ) + 𝑓(𝑣𝑖+1) 2 ⌉ ≥ 3 + ⌈ 𝑖 + 1 + 𝑖 + 2 2 ⌉ > 4 hence, by all the above cases, the radio mean condition is satisfied by 𝑓. further, 𝑓 attains its maximum corresponding to 𝑣 and therefore 𝑟𝑚𝑛(𝐵(3, 𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) ≤ 𝑛 + 3 for 𝑛 = 2. theorem 2.4.the radio mean number of ditriangular book(𝐵(3, 𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) is 𝑛 + 2 for 𝑛 > 2. proof. let 𝐷 be a ditriangular book. let 𝑉(𝐷) = {𝑣1, 𝑣2, 𝑣3, … … . , 𝑣𝑛 , 𝑢, 𝑣} and 𝐴(𝐷) = {𝑢𝑣𝑖⃗⃗⃗⃗⃗⃗⃗/1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑖𝑣⃗⃗ ⃗⃗ ⃗⃗ /1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑢}⃗⃗ ⃗⃗ ⃗⃗ ⃗ the diameter of ditriangular book is 3. define 𝑓: 𝑉(𝐷) → 𝑁 by 𝑓(𝑣𝑖 ) = 𝑖 , 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑢) = 𝑛 + 1 𝑓(𝑣) = 𝑛 + 2 claim. 𝑓 is a valid radio mean labeling. since the diameter is 3, to prove 𝑓 is a radio mean labeling, it is enough to prove that, 𝑑(𝑥, 𝑦) + ⌈ 𝑓(𝑥)+𝑓(𝑦) 2 ⌉ ≥ 4 … … … … (1)for every pair of vertices (𝑥, 𝑦) where 𝑥 ≠ 𝑦. equivalently, it is enough to prove (1) for pair of vertices with minimum 𝑓 values and minimum 𝑑(𝑥, 𝑦)values. hence, the proof involves the following cases case a. consider the pairs (𝑢, 𝑣𝑖 ). here, 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣𝑖 ) ≥ 1 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣𝑖 ) + ⌈ 𝑓(𝑢) + 𝑓(𝑣𝑖 ) 2 ⌉ ≥ 1 + ⌈ 𝑛 + 1 + 𝑖 2 ⌉ ≥ 4 case b. consider the pairs (𝑣𝑖 , 𝑣). here, 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣) ≥ 1 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣) + ⌈ 𝑓(𝑣𝑖 ) + 𝑓(𝑣) 2 ⌉ ≥ 1 + ⌈ 𝑖 + 𝑛 + 2 2 ⌉ ≥ 4 253 radio mean labeling of digraphs case c. consider the pairs (𝑢, 𝑣) 𝑎𝑛𝑑 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣) = 1 𝑑⃗⃗⃗ ⃗(𝑢, 𝑣) + ⌈ 𝑓(𝑢) + 𝑓(𝑣) 2 ⌉ ≥ 1 + ⌈ 𝑛 + 1 + 𝑛 + 2 2 ⌉ > 4 case d. consider the pairs (𝑣𝑖 , 𝑣𝑖+1) and 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣𝑖+1) = 3, then, 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣𝑖+1) + ⌈ 𝑓(𝑣𝑖 ) + 𝑓(𝑣𝑖+1) 2 ⌉ ≥ 3 + ⌈ 𝑖 + 𝑖 + 1 2 ⌉ > 4 hence, by all the above cases, the radio mean condition is satisfied by𝑓. further, 𝑓 attains its maximum corresponding to 𝑣 and is 𝑛 + 2 for 𝑛 > 2. since 𝐷 contains only 𝑛 + 2 vertices, 𝑛 + 2 is the minimum of the maximum integer that could be assigned to the vertices of𝐷. hence 𝑟𝑚𝑛(𝐷) = 𝑛 + 2 for𝑛 > 2. theorem 2.5. the radio mean number of diquadrilateral book(𝐵(4, 𝑛)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗) is 2𝑛 + 2 for 𝑛 > 3. proof. let 𝐷 be a diquadrilateral book. let 𝑉(𝐷) = {𝑣1, 𝑣2, 𝑣3, … … . , 𝑣𝑛 , 𝑢1, 𝑢2, 𝑢3, … … . , 𝑢𝑛 , 𝑢, 𝑣} and 𝐴(𝐷) = {𝑢𝑢𝑖⃗⃗⃗⃗⃗⃗ ⃗/1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑢𝑖𝑣𝑖⃗⃗ ⃗⃗ ⃗⃗⃗⃗ /1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑖𝑣⃗⃗ ⃗⃗ ⃗⃗ /1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑢}⃗⃗ ⃗⃗ ⃗⃗ ⃗ the diameter of diquadrilateral book is 5. define 𝑓: 𝑉(𝐷) → 𝑁 by 𝑓(𝑢𝑖 ) = 𝑖 , 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑣𝑖 ) = 2𝑛 − 𝑖 + 1 , 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑢) = 2𝑛 + 1 𝑓(𝑣) = 2𝑛 + 2 claim. 𝑓 is a valid radio mean labeling. since the diameter is5, to prove 𝑓 is a radio mean labeling, it is enough to prove that, 𝑑(𝑥, 𝑦) + ⌈ 𝑓(𝑥)+𝑓(𝑦) 2 ⌉ ≥ 6 … … … … (1)for every pair of vertices (𝑥, 𝑦) where 𝑥 ≠ 𝑦. equivalently, it is enough to prove (1) for pair of vertices with minimum 𝑓 values and minimum 𝑑(𝑥, 𝑦)values. hence, the proof involves the following cases case a. consider the pairs (𝑢, 𝑢𝑖 ). here, 𝑑⃗⃗⃗ ⃗(𝑢, 𝑢𝑖 ) ≥ 1 𝑑⃗⃗⃗ ⃗(𝑢, 𝑢𝑖 ) + ⌈ 𝑓(𝑢) + 𝑓(𝑢𝑖 ) 2 ⌉ ≥ 1 + ⌈ 2𝑛 + 1 + 𝑖 2 ⌉ ≥ 6 case b. consider the pairs (𝑣𝑖 , 𝑣). here, 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣) ≥ 1 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣) + ⌈ 𝑓(𝑣𝑖 ) + 𝑓(𝑣) 2 ⌉ ≥ 1 + ⌈ 2𝑛 − 𝑖 + 1 + 2𝑛 + 2 2 ⌉ > 6 case c. consider the pairs (𝑢𝑖 , 𝑣𝑖 ) 𝑎𝑛𝑑 𝑑⃗⃗⃗ ⃗(𝑢𝑖 , 𝑣𝑖 ) = 1 𝑑⃗⃗⃗ ⃗(𝑢𝑖 , 𝑣𝑖 ) + ⌈ 𝑓(𝑢𝑖 ) + 𝑓(𝑣𝑖 ) 2 ⌉ ≥ 1 + ⌈ 𝑖 + 2𝑛 − 𝑖 + 1 2 ⌉ > 6 case d. consider the pairs (𝑢𝑖 , 𝑢𝑖+1) and 𝑑⃗⃗⃗ ⃗(𝑢𝑖 , 𝑢𝑖+1) = 4, then, 𝑑⃗⃗⃗ ⃗(𝑢𝑖 , 𝑢𝑖+1) + ⌈ 𝑓(𝑢𝑖 ) + 𝑓(𝑢𝑖+1) 2 ⌉ ≥ 4 + ⌈ 𝑖 + 𝑖 + 1 2 ⌉ ≥ 6 254 palani k and sabarina subi s s case e. consider the pairs (𝑣𝑖 , 𝑣𝑖+1) and 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣𝑖+1) = 4, then, 𝑑⃗⃗⃗ ⃗(𝑣𝑖 , 𝑣𝑖+1) + ⌈ 𝑓(𝑣𝑖 ) + 𝑓(𝑣𝑖+1) 2 ⌉ ≥ 4 + ⌈ 2𝑛 − 𝑖 + 1 + 2𝑛 − 𝑖 2 ⌉ > 6 hence, by all the above cases, the radio mean condition is satisfied by 𝑓. further, 𝑓 attains its maximum corresponding to 𝑣 and is 2𝑛 + 2 for 𝑛 > 3. since 𝐷 contains only 2𝑛 + 2 vertices, 2𝑛 + 2 is the minimum of the maximum integer that could be assigned to the vertices of 𝐷. hence, 𝑟𝑚𝑛(𝐷) = 2𝑛 + 2 for 𝑛 > 3. 3. conclusions in this papers we compute some types of newly defined digraphs. in future we will find that digraphs (radio labeling) is used for x-ray, crystallography, coding theory, network security, network addressing, channel assignment process, social network analysis such as connectivity, scalability, routing, computing, cell biology etc., references [1] bang-jensen, jorgen, gutin gregory, “digraphs: theory, algorithms and applications”, springer, 2000. [2] chatrand g, erwin d, zhang p & harary f, radio labeling of graphs, bull, inst.combin.appl.33(2001), pp.77-85. [3] dunbar j e, haynes t w, domination in inflated graphs, congr.numer.118 (1996), pp.143-154. [4] hale w k, frequency assignment theory and applications, proceedings of ieee, vol.68, no.12, 1980, 1497-1514. [5] liu d & zhu x, multi-level distance labeling for paths and cycles, siam j. discrete math., vol 19(3), 2005, pp.610-621. [6] manimekalai k & thirusangu k, pair sum labeling of some special graphs, international journal of computer applications, volume 69, no.8, may 2013, pp.34-38. [7] palani k & s.s. sabarina subi, radio labeling of some splitting graphs, advances and applications in mathematical sciences, vol 21, no.3, january 2022, pp.1217-1228. [8] ponraj r, sathish narayanan s & kala r, radio mean labeling of a graph, akce international journal of graphs and combinatorics 12(2015), pp.224-228. [9] ponraj r & sathish narayanan s, on radio mean number of some graphs, international j. math. combin. vol.3 (2014), pp.41-48. [10] rosa a, on certain valuations of the vertices of a graph, theory of graphs (internet, symposium, rome, july 1966), gordan and dunod paris (1967). 255 radio mean labeling of digraphs [11] stephen john b & jovit vinish melma g, radio labeling of complete related graphs, compliance engineering journal, volume 11, issue 2, 2020, pp.173-187. [12] sunitha k, david raj c & subramanian a, radio mean labeling of path and cycle related graphs, global journal of mathematical sciences: theory and practical, volume 9, no.3(2017), pp.337-345. 256 ratio mathematica vol. 35, 2018, pp. 29-45 issn: 1592-7415 eissn: 2282-8214 case studies on the application of fuzzy linear programming in decision-making miracle k. eze∗and babatunde o. onasanya† received: 10-09-2018. accepted: 11-12-2018. published: 31-12-2018. doi:10.23755/rm.v35i0.424 c©miracle k. eze et al. abstract this study demonstrated the effectiveness of fuzzy method in decision-making and recommends the integration of fuzzy methods in decision-making in production, transportation, power production and distribution and utility maintenance in nigeria companies. keywords: fuzzy set, fuzzy linear programming, fuzzy constraints, fuzzy optimization. 1 introduction linear programming (lp), an important tool in operations research, has developed over the years in solving management problems [13]. it is in two forms: classical and fuzzy linear programming. it takes various linear inequalities relating to the situation being considered and finds the best value obtainable under that situation. ∗centre for petroleum economics, energy and law (university of ibadan, ibadan, nigeria), ezemiraclekasie@gmail.com. †department of mathematics, faculty of science (university of ibadan, ibadan, nigeria), babtu2001@yahoo.com. 29 m.k. eze and b.o. onasanya let a′is be the constraint functions and b ′ is the available resources. generally, a linear programming problem can be written as min(max) z = cx (1) subject to ai(x) ≤ bi, where x ≥ 0. in practice, all of the needed information such as c, a′is, bis are not completely available or determined; these parameters are uncertain and are said to be fuzzy variables [10]. a typical mathematical programming problem is to optimise an objective function subject to some constraints. usually, the classes of objects encountered in the real world do not have clearly defined criteria of membership. hence, constraints and objective functions could be fuzzy [25]. in production processes, hardly does the firm utilize the exact resources available to meet a proposed target. this may be due to waste in the cause of production and/or machine wear and tear over time or some other factors due to exigency. thus, a firm is required to optimally plan around its available resources. having recognized the shortcomings of traditional mathematical models in some areas of real life application, zadeh (1965) proposed the notion of a fuzzy set. it began as an effort to use mathematics to define such concepts as “slightly” or “tall” or “fast” or “beautiful” or any other concept that has ambiguous boundaries. the fuzzy set theory was developed to improve the over simplified model, thereby developing a more robust and flexible model in order to solve real-world complex systems involving human aspects. fuzziness was modeled by membership functions which might be described as an extension of the usual characteristic function in the setting of mathematical sets [16]. fuzzification offers superior expressive power, greater generality and an improved capability to model complete problems at a low solution cost. the application of fuzzy set theory is claimed to be effective in decision making and coordinating multiple system requirements [18],[11]. thus, it is an excellent method for planning and making decision under uncertainty. 2 preliminaries definition 2.1. [23] a fuzzy set a in x is a set of ordered pairs a = {(x,µa(x)) : x ∈ x}, where µa(x) is the grade of membership of x ∈ a and µa : x −→ [0,1]. example 2.1. [25] let x = {10,20,30,40,50,60,70,80,90,100,110} be possible speeds(mph) at which cars can cruise over long distances. then the fuzzy set a of “uncomfortable speeds for long distances” may be defined by a certain individual as: a = {(30,0.7),(40,0.75),(50,0.8),(60,0.8),(70,1.0),(80,1.0),(90,1.0)}, 30 fuzzy linear programming in decision-making where 0.7, 0.75, 0.8 and 1.0 are the degree of uncomfortability, attaining ”certainly uncomfortable” at ≥ 70 mph. definition 2.2. [14] the support of a fuzzy set a, s(a) = {x ∈ a : µa(x) > 0}. definition 2.3. [23] a fuzzy set a is empty if and only if µa(x) = 0,∀x ∈ x. definition 2.4. [23] two fuzzy sets a and b are equal if and only if µa(x) = µb(x), ∀x ∈ x. definition 2.5. [23] a fuzzy set a is contained in a fuzzy set b, written as a ⊆ b, if and only if µa(x) ≤ µb(x). definition 2.6. [23] the intersection of two fuzzy sets a and b is denoted by a∩b and is defined as the largest fuzzy set contained in both a and b. the membership function of a∩b is given by µa(x)∧µb(x) = min{µa(x),µb(x),∀x ∈ x}. example 2.2. consider the following set of cars, x = {mercedez, camry, chevrolet, accord}. suppose a is the fuzzy subset of “durable cars” and b is the fuzzy subset of “fast cars”. a = {0.8/me, 0.6/ac, 0.5/ca, 0.3/ch} and b = {0.3/me, 0.8/ac, 0.6/ca, 1.0/ch}. the intersection of a and b, µa(x)∧µb(x) = {0.3/me, 0.6/ac, 0.5/ca, 0.3/ch}, is the fuzzy subset of the degree of compatibility of the quality of the cars being “durable and fast”. definition 2.7. [23] the union of a and b, denoted as a ∪ b, is defined as the smallest fuzzy set containing both a and b. the membership function of a∪b is given by µa(x)∨µb(x) = max{µa(x),µb(x),∀x ∈ x}. example 2.3. consider the following set of cars, x = {mercedez, camry, chevrolet, accord}. suppose a is the fuzzy subset of “durable cars” and b is the fuzzy subset of “fast cars”. consider a and b as in example 2.2. the union of a and b, µa(x)∨µb(x) = {0.8/me, 0.8/ac, 0.6/ca, 1.0/ch}, is the fuzzy subset of the degree of the quality of either “durable or fast or both”. 31 m.k. eze and b.o. onasanya definition 2.8. [23] if a is a fuzzy subset of x, then an α-level set of a is a nonfuzzy set aα which comprises all elements of x whose grade of membership is greater than or equal to α. it is denoted by aα = {x ∈ x : µa(x) ≥ α ∀ x ∈ x}. example 2.4. the intelligence quotient of students were tested and some were discovered to possess high intelligence quotient while some very low. let fsiq be fuzzy set of intelligence quotient. fsiq = {(c,0.9),(m,0.7),(b,0.5),(s,0.4),(p,0.3)}. then, a0.5 = (b,m,c). 3 methodology the data were collected from two places: the production data for two products from a water venture and the value-added services provided by an institute of economic and law, both in oyo state, nigeria. 3.1 fuzzy linear programming models in fuzzy linear programming, the fuzziness of available resources is characterised by the membership function over the tolerance range. the general model of linear programming with fuzzy resources is: max(min)z = cx, (2) subject to (s.t.) ai(x) ≤ b̃i, i = 1,2, ...,m,x ≥ 0, where, for each i, ai(x)′s are the m constraints, b̃i ∈ [bi,bi +pi] are the real numbers representing the quantities of each fuzzy resources and p′is are the tolerance levels of the decision-maker for each of the resources. the fuzzy linear programming may also be considered as: max(min) z = cx, (3) subject to (s.t.) ai(x) / bi, i = 1,2, ...,m,x ≥ 0, where / is called “fuzzy less than or equal to”. if the tolerance pi is known for each fuzzy constraint, ai(x) / bi could be seen as ai(x) ≤ (bi + θpi), for all i, where θ ∈ [0,1]. 32 fuzzy linear programming in decision-making 3.2 verdegay’s approacha nonsymmetric model verdegay [21] considered that if the membership functions of the fuzzy constraints. µi(x) =   1, if ai(x) < bi 1− ai(x)−bi pi , bi ≤ ai(x) ≤ bi + pi, i = 1, ...,m + 1 0, ai(x) > bi + pi (4) are continuous and monotonic functions, and trade-off between those fuzzy constraints are allowed, the general model of linear programming with fuzzy resources will be equivalent to: max cx, s.t x ∈ xα, (5) where xα = {x : µ(x) ≥ α,x ≥ 0, for each α ∈ [0,1]}. the α-level concept is based on the work of [20]. it is indicated in the membership function that if ai(x) ≤ bi then the i− th constraint is satisfied and µi(x) = 1. but, on the other hand, if ai(x) ≥ bi + pi, where pi is the maximum tolerance from bi, (which is always determined by the decision-maker), then the i − th constraint is violated at this point and µi(x) = 0. finally, if ai(x) ∈ (bi,bi + pi), then the membership function is monotonically decreasing and, the less satisfied the decision-maker becomes. using parametric programming, where α = 1 − θ, we can substitute membership function of equation (4) into (5) and the problem below is obtained: max cx, s.t (ax)i ≤ bi + (1−α)pi, ∀i, (6) for x ≥ 0 and α ∈ [0,1]. 4 result analysis and discussions in this section, fuzzy linear programming method is applied to some cases to optimize the decisions. these are the cases of a water venture and an institute. 4.1 the water ventures the study was based on two different bottles of water which the venture produces : 75cl and 50cl. it makes 134.62ngn per carton of 50cl and 150.26ngn per carton of 75cl as profits. the firm employs machine for 7 hours in a day, with 33 m.k. eze and b.o. onasanya basic variables x1 x2 g1 g2 b x1 1 1.189 23.681 0 166.573 g2 0 0.032 -1.003 1 1.003 p 0 10.002 3,204.03 0 22,437.129 table 1: final solution to equation (7) by simplex method tolerance level of 2 hours and labor for 8 hours with tolerance level of 1 hour. the classical linear programming problem is constructed thus: max p = 134.62x1 + 150.26x2, (7) s.t. g1(x) = 0.042x1 + 0.05x2 ≤ 7,g2(x) = 0.042x1 + 0.082x2 ≤ 8. where g1 is machine time, g2 is labour time, x1 is the 50cl bottle water and x2 is the 75cl bottle water. the final result of the simplex method is in table 1. the fuzzy membership function of the machine time: µ1(x) =   1, if g1(x) ≤ 7 1− g1(x)−7 2 , 7 < g1(x) < 9 0, g1(x) ≥ 9 (8) the membership function of the labour time: µ2(x) =   1, if g2(x) ≤ 8 1− g2(x)−8 1 , 8 < g2(x) < 9 0, g2(x) ≥ 9 (9) the fuzzy linear programming problem associated with equation (7) is max p = 134.62x1 + 150.26x2, (10) s.t. µ1(x) ≥ α, µ2(x) ≥ α, where α ∈ [0,1] and x1,x2 ≥ 0. the fuzzy linear programming problem is expanded thus: max p = 134.62x1 + 150.26x2, (11) s.t. g1 = 0.042x1 + 0.05x2 ≤ 7 + 2(1 −α), and g2(x) = 0.042x1 + 0.082x2 ≤ 8 + (1−α), where x1,x2 ≥ 0 and α ∈ [0,1]. 34 fuzzy linear programming in decision-making basic variables x1 x2 g1 g2 b x1 1 1.189 23.681 0 166.573 + 47.362θ g2 0 0.032 -1.003 1 1.003 1.006θ p 0 10.002 3,204.03 0 22,437.129 + 6,408.06θ table 2: solution to the fuzzy linear programming equation (12) θ p∗ x∗1 g1 g2 0.0 22,437.13 166.573 6.996 6.663 0.1 22,077.94 171.309 7.195 6.852 0.2 23,718.74 176.045 7.394 7.042 0.3 24,359.55 180.782 7.593 7.231 0.4 25,000.35 185.518 7.792 7.421 0.5 25,641.16 190.254 7.991 7.610 0.6 26,281.97 194.990 8.189 7.799 0.7 26,922.77 199.726 8.389 7.989 0.8 27,563.58 204.463 8.587 8.178 0.9 28,204.38 209.199 8.786 8.368 1.0 28,845.19 213.935 8.925 8.557 table 3: result of the parametric problem setting θ = 1−α, the programming problem above becomes max p = 134.62x1 + 150.26x2 (12) s.t. g1 = 0.042x1 + 0.05x2 ≤ 7 + 2θ, g2(x) = 0.042x1 + 0.082x2 ≤ 8 + θ, x1,x2 ≥ 0, where θ ∈ [0,1] is a parameter determining the tolerance level. using the parametric technique and final result of simplex method, table 2 was obtained. the optimal solution is (x∗1,x ∗ 2) = (166.573 + 47.362θ,0) and p∗ = 22,424.06 + 6,375.87θ. therefore, the solution of the parametric programming problem is in table 3. from the analysis above, it is observed that the water venture could make more profit by producing more of 50cl bottles than producing 75cl bottles. in essence, it will be more profitable for the firm to scale up its production of 75cl bottle water and cut down the production of 50cl. 35 m.k. eze and b.o. onasanya basis e l g1 g2 g3 b g1 0 2 3 1 0 −1 1,440 710 3 g2 0 2 3 0 1 −1 1,440 4,496 3 e 1 1 3 0 0 1 1,440 4 3 p 0 259 3 0 0 235 1,440 940 3 table 4: final result of the simplex method 4.2 the institute of energy law and energy economics this section seeks to maximise profit and minimise cost in the sessional operation of the institute based on tuition alone. annually, the institute admits law and energy studies students. on each law student, the institute makes a loss of approximately 8,000ngn and on each energy studies student, a profit of approximately 235,000ngn. for both energy law and energy economics, if the institute is willing to spend 238,000ngn with tolerance of 70,000ngn on internet, 1,500,000ngn with tolerance of 500,000ngn on conference support, and 3 graduate assistant for energy study, 1 graduate assistant for energy law, with tolerance of 2 additional graduate assistants, the following will be the linear programming problem. max p(e,l) = 235e −8l, (13) s.t. g1(e,l) = e+l ≤ 238(internet), g2(e,l) = e+l ≤ 1,500(coference support) and g3(e,l) = 1,440e + 480l ≤ 1,920(graduateassistants). where e is energy studies, l is energy law, g1 is internet, g2 is conference support and g3 is graduate assistants. using the simplex method, table 4 was obtained. the membership function of internet µ1(e,l) =   1, if g1(e,l) ≤ 238 1− g1(e,l)−238 70 , 238 < g1(e,l) < 308 0, g1(e,l) ≥ 308 (14) 36 fuzzy linear programming in decision-making basis e l g1 g2 g3 b g1 0 2 3 1 0 −1 1,440 710 3 + 208θ 3 g2 0 2 3 0 1 −1 1,440 4,496 3 + 1,498θ 3 e 1 1 3 0 0 1 1,440 4 3 + 2θ 3 p 0 259 3 0 0 235 1,440 940 3 + 470θ 3 table 5: matrix multiplication of the simplex method solution and the tolerance level the membership function of conference support µ2(e,l) =   1, if g2(e,l) ≤ 1,500 1− g2(e,l)−1,500 500 , 1,500 < g2(e,l) < 2,000 0, g2(e,l) ≥ 2,000 (15) the membership function of graduate assistants µ3(e,l) =   1, if g3(e,l) ≤ 1,920 1− g3(e,l)−1,440 960 , 1,920 < g3(e,l) < 2,880 0, g3(e,l) ≥ 2,880 (16) the fuzzy linear programming is max p(e,l) = 235e −8l, (17) s.t. g1(e,l) = e +l ≤ 238 + 70(1−α) g2(e,l) = e +l ≤ 1,500 + 500(1− α) and g3(e,l) = 1,440e + 480l ≤ 1,920 + 960(1−α). setting θ = 1−α, the following is the parametric problem: max p = 235e −8l, (18) s.t. g1(e,l) = e + l ≤ 238 + 70θ, g2(e,l) = e + l ≤ 1,500 + 500θ and g3(e,l) = 1,440e + 480l ≤ 1,920 + 960θ, where θ ∈ [0,1] is a parameter given the tolerance level. using the parametric technique and final result of simplex method, table 5 was obtained. 37 m.k. eze and b.o. onasanya θ e∗ p∗ internet conf. supp. g.a 0.0 1.33 313.33 1.33 1.33 1,920.00 0.1 1.40 329.00 1.40 1.40 2,016.00 0.2 1.47 344.67 1.47 1.47 2,112.00 0.3 1.53 360.33 1.53 1.53 2,208.00 0.4 1.60 376.00 1.60 1.60 2,304.00 0.5 1.67 391.67 1.67 1.67 2,400.00 0.6 1.73 407.33 1.73 1.73 2,496.00 0.7 1.80 423.00 1.80 1.80 2,592.00 0.8 1.87 438.67 1.87 1.87 2,688.00 0.9 1.93 454.33 1.93 1.93 2,784.00 1.00 2.00 470.00 2.00 2.00 2,880.00 table 6: result of the parametric problem the optimal solution is p∗ = ( 940 3 + 470θ 3 )ngn and x∗ = (e∗,l∗) = (4 3 + 2θ 3 ,0). therefore, the final result for the parametric problem is in table 6. from the above analysis, it is observed that (under varying resources) the profit gotten by the institute comes from the energy study program. it is observed that the energy law program is not adding to the institute, instead they run at loss to keep the program. the researcher also observed that the random allocation of conference support to both program is not profiting the institute, but will rather jeopardise its continuity. 4.3 minimisation of cost minimising the cost of operation of the institute, the classical linear programming problem becomes min c = 125e + 368l, (19) s.t. g1(e,l) = e + l ≤ 238, g2(e,l) = e + l ≤ 1,500 and g3(e,l) = 1,440e + 480l ≤ 1,920. using the simplex method, table 7 was obtained. 38 fuzzy linear programming in decision-making basis e l g1 g2 g3 b g1 -2 0 1 0 −1 480 234 g2 -2 0 0 1 −1 480 1,496 l 3 1 0 0 1 480 4 p 979 0 0 0 368 480 1,472 table 7: final result of the simplex method the membership functions of the constraints, respectively internet, conference support and graduate assistants are: µ1(e,l) =   1, if g1(e,l) ≤ 238 1− g1(e,l)−238 70 , 238 < g1(e,l) < 308 0, g1(e,l) ≥ 308 (20) µ2(e,l) =   1, if (g2(e,l)) ≤ 1,500 1− g2(e,l)−1,500 500 , 1,500 < g2(e,l) < 2,000 0, g2(e,l) ≥ 2,000 (21) µ3(e,l) =   1, if g3(e,l) ≤ 1,920 1− g3(e,l)−1,920 960 , 1,920 < g3(e,l) < 2,880 0, g3(e,l) ≥ 2,880 (22) the required fuzzy linear programming is min c = 125e + 368l, (23) s.t. g1(e,l) = e +l ≤ 238+70(1−α), g2(e,l) = e +l ≤ 1,500+500(1− α) and g3(e,l) = 1,440e + 480l ≤ 1,920 + 960(1−α). setting θ = 1−α, the following is the parametric programming problem: min c = 125e + 368l, (24) s.t. g1(e,l) = e + l ≤ 238 + 70θ g2(e,l) = e + l ≤ 1,500 + 500θ and g3(e,l) = 1,440e + 480l ≤ 1,920 + 960θ, where θ ∈ [0,1] is a parameter. using the parametric technique and final result of simplex method, table 8 was obtained. 39 m.k. eze and b.o. onasanya basis e l g1 g2 g3 b g1 -2 0 1 0 −1 480 234 + 68θ g2 -2 0 0 1 −1 480 1,496 + 498θ e 3 1 0 0 1 480 4 + 2θ c 979 0 0 0 368 480 1,472 + 736θ table 8: matrix multiplication of the simplex method and the tolerance level θ c∗ internet conf. supp. g.a energy law 0.0 1,472.00 4.00 4.00 1,920.00 4.00 0.1 1,545.60 4.20 4.20 2,016.00 4.20 0.2 1,619.20 4.40 4.40 2,112.00 4.40 0.3 1,692.80 4.60 4.60 2,208.00 4.60 0.4 1,766.40 4.80 4.80 2,304.00 4.80 0.5 1,840.00 5.00 5.00 2,400.00 5.00 0.6 1,913.60 5.20 5.20 2,496.00 5.20 0.7 1,987.20 5.40 5.40 2,592.00 5.40 0.8 2,060.80 5.60 5.60 2,688.00 5.60 0.9 2,134.40 5.80 5.80 2,784.00 5.80 1.0 2,208.00 6.00 6.00 2,880.00 6.00 table 9: result of the parametric problem the optimal solution is c∗ = (1,472 + 736θ)ngn and x∗ = (e∗,l∗) = (0,4 + 2θ). therefore, the final result for the parametric problem is in table 9. from the analysis above on cost minimisation, the energy law program viably increases the cost of running the institute. 4.4 proposed model from the results above, the institute is discovered not to be making optimal profit running both energy law and energy studies’ program. therefore, the researcher proposes that the fees of the energy law should be increased in such a way that it contributes meaningfully to the institute. suppose the law student and energy student contribute 230,000ngn and 235,000ngn respectively, and the conference support is given in the ratio 742 to 758 (from contribution made by 40 fuzzy linear programming in decision-making basis e l g1 g2 g3 b g1 0 0 1 −238 1,468 90202 1,056,960 0 l 0 1 0 3 1,468 −758 704,640 1 e 1 0 0 −1 1,468 2,226 2,113,920 1 p 0 0 0 455 1,468 1 23,488 1,395 3 table 10: final result of the simplex method each program), then the new linear programming problem becomes: max p(e,l) = 235e + 230l, (25) s.t. g1(e,l) = 119e + 119l ≤ 238 g2(e,l) = 758e + 742l ≤ 1,500 and g3(e,l) = 1,440e +480l ≤ 1,920, where e is energy studies and l is energy law. using the simplex method, table 10 was obtained. the membership functions of the constraints, respectively internet, conference support and graduate assistants are: µ1(e,l) =   1, if g1(e,l) ≤ 238 1− g1(e,l)−238 70 , 238 < g1(e,l) < 308 0, g1(e,l) ≥ 308 (26) µ2(e,l) =   1, if g2(e,l) ≤ 1,500 1− g2(e,l)−1,500 500 , 1,500 < g2(e,l) < 2,000 0, g2(e,l) ≥ 2,000 (27) µ3(e,l) =   1, if g3(e,l) ≤ 1,920 1− g3(e,l)−1,920 960 , 1,920 < g3(e,l) < 2,880 0, g3(e,l) ≥ 2,880 (28) 41 m.k. eze and b.o. onasanya basis e l g1 g2 g3 b g1 0 0 1 −230 1,468 90202 1,056,960 74,901,120θ 1,056,960 l 0 1 0 3 1,468 −758 704,640 1 11,520θ 1,056,960 e 1 0 0 −1 1,468 2,226 2,113,920 1 + 708,480θ 1,056,960 p 0 0 0 455 1,468 1 23,488 1,395 3 + 163,939,200θ 1,056,960 table 11: matrix multiplication of the simplex method and the tolerance level hence, max p = 235e + 230l, (29) s.t. g1(e,l) = e + l ≤ 238 + 70(1−α) g2(e,l) = 758e + 742l ≤ 1,500 + 500(1−α) and g3(e,l) = 1,440e + 480l ≤ 1,920 + 960(1−α). setting θ = 1−α, the following is the parametric problem: max p = 235e + 230l, (30) s.t. g1(e,l) = e + l ≤ 238 + 70θ g2(e,l) = 758e + 742l ≤ 1,500 + 500θ and g3(e,l) = 1,440e + 480l ≤ 1,920 + 960θ, where θ ∈ [0,1] is a parameter. using the parametric technique and final result of simplex method, table 11 was obtained. the optimal solution is p∗ = ( 1,395 3 + 163,939,200θ 1,056,960 )ngn = 465 + 155.10445ngn and x∗ = (e∗,l∗) = (1 + 708,480θ 1,056,960 ,1− 11,520θ 1,056,960 ). therefore, the final result for the parametric problem is given in table 12. from the analysis above, the profit of the institute increased greatly as a result of the viable contribution from both programs. 5 conclusions the potency of fuzzy set theory, fuzzy logic and so on in decision-making cannot be over-emphasized. its use has proved very efficient from the above analysis, and gives the decision-maker the opportunity to make decision in a robust and flexible environment. 42 fuzzy linear programming in decision-making θ e l internet conf. supp. g.a. p 0.0 1.000 1.000 238.00 1500.00 1920.00 465.00 0.1 1.070 1.001 246.09 1551.53 2016.96 480.51 0.2 1.130 1.002 254.18 1603.06 2113.92 496.02 0.3 1.200 1.003 262.28 1654.58 2210.88 511.53 0.4 1.270 1.004 270.37 1706.11 2307.84 527.04 0.5 1.340 1.005 278.46 1757.64 2404.80 542.55 0.6 1.400 1.006 286.55 1809.17 2501.76 558.06 0.7 1.470 1.007 294.64 1860.70 2598.72 573.57 0.8 1.540 1.008 302.74 1912.22 2695.68 589.08 0.9 1.600 1.009 310.83 1963.75 2792.64 604.59 1.0 1.670 1.010 318.92 2015.28 2889.60 620.10 table 12: result of the parametric problem 6 acknowledgements the authors acknowledge the support of the members of staff of the centre for petroleum economics, energy and law and those of the water venture where the data used in this study were obtained. references [1] d. ali, m. yohanna, m.i. puwu and b.m. 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[26] h-j. zimmermann, fuzzy set theory and its applications, springer science + business media, new york, 2001. 45 ratio mathematica 24 (2013), 53–62 issn: 1592-7415 some algebraic properties of fuzzy s-acts m. haddadi department of mathematics, statistic and computer science, semnan university, semnan, iran. haddadi 1360@yahoo.com abstract s-acts, a useful and important algebraic tool, have always been interest to mathematicians, specially to computer scientists. when a. zadeh introduced the notion of the fuzzy subset in 1965, his idea opened a new direction to reserchers to provide tools in the various fields of mathematics. here we are going to investigate some algebraic properties of fuzzy s-acts. we first make an s-act from the fuzzy subsets of an s-act a. then we use this tool to give a characterization for fuzzy s-acts. we then introduce the notion of generated fuzzy s-act by a fuzzy subset of an s-act and give a characterization for the fuzzy actions. and then we define the notion of indecomposable fuzzy s-act and find some indecomposable fuzzy actions. key words: fuzzy set, fuzzy acts over fuzzy semigroups. msc2010: 08a72, 20m30. 1 introduction and preliminaries no need to mention the importance of the prominent and well established fuzzy set theory, introduced by zadeh in 1965 [7], which offered tools and a new approach to model imprecision and uncertainty. since then, very many researchers have worked on this concept and its applications to logic, set theory, algebra, analysis, topology, computer science, control engineering, information science, etc [1, 2, 3]. actions of a semigroup (monoid or group) s on a set a have always been interest to mathematicians, specially to computer m. haddadi scientists and logicians. the algebraic structures so obtained are called ssets, s-acts, and by some other terminologies [4, 6]. in [5] we have used the fuzzy concept and introduced the notion of the actions of a (fuzzy) semigroup on a fuzzy set (fuzzy s-act) and studied the relation between this structure and sheaves. here we are going to study some of algebraic details of this structure. but first we recall that: a set x together with a function µ : x → [0, 1] is called a fuzzy set (over x) and is denoted by (x,µ) or x(µ). we call x the underlying set and µ the membership function of the fuzzy set x(µ), and µ(x) ∈ [0, 1] is the grade of membership of x in x(µ). if µ is a constant function with value a ∈ [0, 1], x(µ) is denoted by x(a). the fuzzy set x(1) is called a crisp set and may sometimes simply be denoted by x. for a fuzzy set x(µ) and α ∈ [0, 1], x(µ)α := {x ∈ x | µ(x) ≥ α} is called the α-cut or the α-level set of the fuzzy set x(µ). a fuzzy function from x(µ) to y (η), written as f : x(µ) → y (η), is an ordinary function f : x → y such that the following is a fuzzy triangle: x µ // f �� [0, 1] y η ==zzzzzzzz meaning that µ ≤ ηf (that is, µ(x) ≤ ηf(x) for all x ∈ x). the set of all fuzzy sets with a fixed underlying set x is called the fuzzy power or the set of fuzzy subsets of x and is denoted by fsubx. clearly fuzzy sets together with fuzzy functions between them form a category denoted by fset. to define the actions of a (fuzzy) semigroup on a fuzzy set first we note that: definition 1.1 a semigroup s together with a function ν : s → [0, 1] is called a fuzzy semigroup if its multiplication is a fuzzy function: for every s,r ∈ s, ν(s) ∧ν(r) ≤ ν(sr); that is, the following is a fuzzy triangle: s ×s ν∧ν // λs �� [0, 1] s ν ::vvvvvvvvvv if s has an identity 1, one usually add the condition ν(1) = 1. now, recall from [5] that, for a (crisp) semigroup s, a (crisp) set a can be made into an (ordinary) s-act in the following two equivalent ways: 54 some algebraic properties of fuzzy s-acts universal algebraic way: the set a together with a family (λs : a → a)s∈s of unary operations satisfying (st)a = s(ta) (and 1a = a, if s has an identity) where sa = λs(a). common way: the set a together with a function λ : s × a → a satisfying (st)a = s(ta) (and 1a = a, if s has an identity) where sa = λ(s,a). now, having these two, so called, universal algebraic and common actions of s on a, we get the following two, not necessarily equivalent, definitions for a fuzzy act over a fuzzy monoid. definition 1.2 let s(ν) be a fuzzy semigroup and a(µ) be a fuzzy set such that a is an s-act, as defined above. then, a(µ) is called: (universal algebraic) a fuzzy s-act (or fuzzy s(1)-act, to emphasize fuzziness) if each λs is a fuzzy function; that is µ(a) ≤ µ(sa), for every s ∈ s and a ∈ a (with no mention of ν). that is, for every s ∈ s, the following triangle is fuzzy: a µ // λs �� [0, 1] a µ =={{{{{{{{ (common) a fuzzy s(ν)-act if λ : s × a → a is a fuzzy function; that is, ν(s) ∧ µ(a) ≤ µ(sa), for every s ∈ s and a ∈ a. that is, the following triangle is fuzzy: s ×a ν∧µ // λ �� [0, 1] s µ ::uuuuuuuuuu corolary 1.1 (1) note that universal algebraic definition implies common definition, and if s(1) = s is a (crisp) semigroup, then ν(s) ∧ µ(a) = 1 ∧ µ(a) = µ(a), and so the above two definitions are equivalent. (2) every fuzzy semigroup s(ν) is naturally a fuzzy s(ν)-act and s(1) = s is a fuzzy s(1) = s-act (universal algebraicly, and hence commonly). also, if s(ν) is a fuzzy left ideal, then it is a fuzzy s-act (universal algebraicly, and hence commonly). a morphism between fuzzy s(ν)-acts (with both definitions), also called an s(ν)-map is simply an s-map as well as a fuzzy function. the set of all (fuzzy) s(ν)-acts with a fixed a is denoted by s(ν)-fsuba, and the category of all fuzzy s(ν)-acts is denoted by s(ν)-fact. since an s-act a is naturally a (unary) universal algebra, the universal algebraic definition of fuzzy acts, being compatible with the definition of 55 m. haddadi other fuzzy algebraic structures, may be considered to be more natural than the second one. thus, from now on we consider the universal algebraic definition of fuzzy acts and we recall that: theorem 1.1 [5] an s-act a with µ : a → [0, 1] is a fuzzy s = s(1)-act if and only if for every α ∈ [0, 1], a(µ)α is an ordinary s-subact of a. 2 fuzzy subsets of an s-act as a fuzzy s-act in this short section we make an s-action from the fuzzy subsets of an s-act a which is used thorough the paper and give a characterization of fuzzy s-acts defined in preliminary. lemma 2.1 let s be an commutative monoid and a be an s-act. then fuzzy subsets of a form an s-act. proof. to prove, for each fuzzy subset a(µ) and each m ∈ s we define: mµ : a → [0, 1] a ∨ {µ(x) | mx = a} first we note that mµ is a fuzzy s-act, because mµ(na) =∨ {µ(x) | mx = na}, for every n ∈ s, and mµ(a) = ∨ {µ(x) | mx = a}. but if mx = a, then nmx = na, and since s is commutative, mnx = nmx = na. also µ(x) ≤ µ(nx). so ∨ {µ(x) | mx = a}≤ ∨ {µ(x) | mx = na}. now we check the s-act properties. (m1m2)µ(a) = ∨ x∈a {µ(x) | (m1m2)x = a} = ∨ x∈a {µ(x) | m2x = y, m1y = a} = ∨ x∈a { ∨ y∈a µ(y) | m2x = y, m1y = a} = ∨ y∈a {m2µ(y) | m1y = a} = m1(m2µ(x))(a) and 1sµ(a) = ∨ {µ(x) | 1sx = a} = µ(a). also if ν ≤ µ, then (mν)(a) =∨ {ν(x) | mx = a}≤ ∨ {µ(x) | mx = a} = (mµ)(a).2 theorem 2.1 let µ : a → [0, 1] be a fuzzy subset. then a(µ) is an fuzzy s-act if and only if mµ ≤ µ for every m ∈ s. 56 some algebraic properties of fuzzy s-acts proof. (⇒) let a(µ) be a fuzzy s-act and xa = {x ∈ a | mx = a}. then µ(x) ≤ µ(mx) = µ(a), for every x ∈ xa implies that ∨ {µ(x) | mx = a}≤ µ(a). that is mµ ≤ µ. (⇐) let a(µ) be a fuzzy subset. to prove we show that µ(a) ≤ µ(ma), for every m ∈ s and a ∈ a. but we know that mµ ≤ µ and hence we have mµ(ma) ≤ µ(ma). now since mµ(ma) = ∨ x∈xma µ(x) and a ∈ xna, we have µ(a) ≤ mµ(ma) ≤ µ(ma). 2 3 cyclic fuzzy s-acts in this section we define a generated fuzzy s-act by a fuzzy subset of an action and then we characterize the generated fuzzy s-actions by the action introduced in lemma 2.1. we then define the cyclic fuzzy s-acts which are a useful class of fuzzy s-acts and infact every fuzzy s-act is made of a class of cyclic ones. lemma 3.1 intersection and union of fuzzy s-acts of an s-set a is an fuzzy s-act. proof. let {a(µi)}i∈i be a family of fuzzy s-act. then ( ⋃ i∈i µi)(ma) =∨ i∈i µi(ma) ≥ ∨ i∈i µi(a) = ( ⋃ i∈i µi)(a) and ( ⋂ i∈i µi)(ma) = ∧ µi(ma) ≥∧ µi(a) = ( ⋂ i∈i µi)(a). 2 theorem 3.1 let µ : a → [0, 1] be a fuzzy s-act and {µi}i∈i⊆[0,1] be family of i-cuts of µ. then ⋃ i∈i µi and ⋂ i∈i µi are fuzzy s-acts of the form α-cut. proof. by lemma 3.1, it is enough to show that ⋃ i∈i µi = µ w i∈i i and⋂ i∈i µi = µ v i∈i . but since ( ⋃ i∈i µi)(a) = ∨ µi(a) ≥ i, for every i ∈ i and a ∈ a, hence ∨ µi(a) ≥ ∨ i. also ( ⋂ i∈i µi)(a) = ∧ µi(a) ≥ ∧ i. so ⋃ i∈i µi and ⋂ i∈i µi are fuzzy s-acts of the form α-cut. 2 now by the above lemma having the following definition is natural. definition 3.1 let µ : a → [0, 1] be a fuzzy s-act. then we take < µ > to be ⋂ {ν : a → [0, 1] | µ ≤ ν and ν is a fuzzy s-act}. the fuzzy s-act < µ > is called the generated fuzzy s-act by µ. theorem 3.2 let s be an commutative semigroup and a(µ) be a fuzzy s-set of a. then < µ >= ⋃ m∈s mµ. 57 m. haddadi proof. first we prove that ⋃ m∈s mµ is an fuzzy s-act. to do we show that ⋃ m∈s mµ(a) ≤ ⋃ m∈s mµ(na), for each n ∈ s. but ⋃ m∈s mµ(a) =∨ m∈s( ∨ mx=a µ(x)) and ⋃ m∈s mµ(na) = ∨ m∈s( ∨ mx=na µ(x)). but if mx = a, then mnx = nmx = na. since µ(x) ≤ µ(nx), for every x ∈ a,⋃ m∈s mµ(a) ≤ ⋃ m∈s mµ(na). now let a(ν) be a fuzzy s-act such that µ ≤ ν. then for every m ∈ s we have mµ ≤ µ ≤ ν, see theorem 2.1 for the first inequality, and hence⋃ m∈s mµ(a) ≤ ν(a), for every a ∈ a. 2 in the following we have some morte propeties about generated fuzzy s acts. theorem 3.3 (1) << µ >>=< µ >. (2)< ⋃ i∈i µi >= ⋃ i∈i < µi > . proof. (1) it is trivial by definition of generated fuzzy s-act. (2) by theorem 3.2 we have < ⋃ i∈i µi > (a) = ∨ { ⋃ µi(x) | mx = a for some m ∈ m} = ∨ i∈i ∨ mx=a µi(x) = ⋃ i∈i < µi > (a) (1) for every a ∈ a. 2 definition 3.2 let a be an s-act and α ∈ [0, 1] and x ∈ a. then by cyclic fuzzy s-act < xα > we mean: < xα > (a) = { α if a ∈ sx 0 othrewise for every a ∈ a. corolary 3.1 let µ be a fuzzy s-act of an s-act a and x ∈ a. then < xµ(x) > ≤ µ. theorem 3.4 let s be a monoid and µ be a fuzzy s-act of an s-act a. then µ = ⋃ x∈a < xµ(x) >. proof. ⋃ x∈a < xµ(x) > (a) = ∨ x∈a < xµ(x) > (a) = ∨ {µ(x) | a = mx for some m ∈ s}. but since 1sa = a, µ(a) ≤ ∨ {µ(x) | a = mx for some m ∈ m}. also since µ is a fuzzy s-act, µ(x) ≤ µ(mx). hence we have µ(a) ≤ ∨ mx=a µ(x) ≤ µ(a), for every a ∈ a. that is⋃ x∈a < xµ(x) >= µ 2 58 some algebraic properties of fuzzy s-acts theorem 3.5 for every m ∈ s and every cyclic fuzzy s-act < xα > of a, m < xα >=< xα >. proof. m < xα > (a) = ∨ {< xα > (y) | my = a} = { α if a ∈ sx 0 otherwise =< xα > (a). 2 4 decomposable and indecomposable fuzzy s-act here we give a definition of indecomposable fuzzy s-act and show that the cyclic fuzzy s-acts are indecomposable. we also see some properties of indecomposable fuzzy s-acts in this section. definition 4.1 an fuzzy s-act µ 6= 0 of a is called decomposable whenever there exist two fuzzy s-acts ν,η 6= 0 of a such that ν,η ≤ µ and η ∨ν = µ, and η ∧ν = 0. otherwise µ is called indecomposable. theorem 4.1 let s be a commutative monoid. then every cyclic fuzzy s-act < xi > of a is indecomposable. proof. let < xi > be decomposable. then there are fuzzy s-acts ν and η of a such that ν,η ≤< xi > and η ∨ ν =< xi >, and η ∧ ν = 0. so for every a ∈ a, η(a) ∧ ν(a) = 0 and η(a) ∨ ν(a) = { i ifa = mx 0 otherwise . now let ν(m0x) = i. then we claim that for every m ∈ s, ν(mx) = i and η(mx) = 0. because if there exists m1 ∈ s such that η(m1x) = i, then η(m1m0x) = i and ν(m1m0x) = i, so η(m1m0x) ∧ν(m1m0x) = i 6= 0. 2 definition 4.2 a fuzzy s-act a(µ) is called finitely generated whenever µ =< ⋃n i=1(xi)αi >, where αi ∈ [0, 1]. theorem 4.2 let f : a → b be an s-act homomorphism and a(µ) be an fuzzy s-act. then bf(µ) is finitely generated, if so is µ. 59 m. haddadi proof. to proof, we show that f(µ) =< ⋃n i=1 f((xi)αi ) >, where µ =<⋃n i=1(xi)αi >. for f(µ)(b) = ∨ {µ(a) | f(a) = b} = { ∨ j∈j⊆{1,...,n} αj f(xj) = b 0 otherwise = ⋃ i∈i f(< (xi)αi >)(b). 2 lemma 4.1 let g be a group. then for every finitely generated fuzzy g-act a(µ), there exists a finite subsets {a1, . . . ,an} ⊆ [0, 1] and {x1, . . . ,xn} ⊆ a such that µ(o(xi)) = ai and o(xi) ∩ o(xj) = ø, if i 6= j, where o(xi) is a notation for orbit of xi that is the set {gxi | g ∈ g. proof. since µ is finitely generated, µ =< ⋃n i=1(xi)ai ) > and since g is a group, o(xi) ∩o(xj) = ø, if i 6= j. so µ(x) = { ai if x ∈ o(xi) 0 otherwise. 2 theorem 4.3 let f : a → b be an s-act homomorphism with commutative s, and a(µ) be an fuzzy s-act. then f(µ) =< f(ν) >, if µ =< ν >. proof. f(µ)(b) = ∨ {µ(a) | f(a) = b} = ∨ {(mν)(a) | m ∈ s, f(a) = b} by theorem 3.2 = ∨ {ν(x) | m ∈ m,mx = a, f(a) = b} = ∨ {f(ν)(y) | m ∈ m, my = b} = ∨ m∈m mf(ν)(b) =< f(ν) > (b). 2 theorem 4.4 let {a(νi)}i∈i be a family of fuzzy s-act in which there is i0 ∈ i such that µi0 is indecomposable. then ∨ µi is indecomposable. proof. let ∨ i∈i µi be decomposable. so there are ν1,ν2 ≤ ∨ i∈i µi such that ∨ i∈i µi = ν1 ∨ ν2 and ν1 ∧ ν2 = 0. then µi0 = µi0 ∧ ( ∨ i∈i µi) = (µi0 ∧ν1) ∨ (µi0 ∧ν2) also (µi0 ∧ν1) ∧ (µi0 ∧ν2) = µi0 ∧ (ν1 ∧ν2) = 0. 2 corolary 4.1 the union of indecomposable fuzzy s-acts is indecomposable. 60 some algebraic properties of fuzzy s-acts references [1] belohlavek, birkhoff variety theorem and fuzzy logic arch. math. logic 42 (2003), 781790. [2] belohlavek, r., vychodil, v., algebras with fuzzy equalities fuzzy sets and systems 157 (2006), 161.201. [3] bosnjak. i, r.madarasz, g.vojvodic algebra of fuzzy sets, fuzzy sets and systems 160 (2009), 29792988. [4] ebrahimi, m.m., m. mahmoudi, the category of m-sets. ital. j. pure appl. math. 9 (2001), 123-132. [5] m. haddadi fuzzy acts over fuzzy semigroups and summited [6] kilp, m., u. knauer, a. mikhalev, monoids, acts and categories, walter de gruyter, berlin, new york, 2000. [7] lotfi a. zadeh fuzzy sets. information and control. 8 (1965), 338353. 61 62 ratio mathematica 22 (2012) 13-35 issn:1592-7415 classifications of hyper pseudo bck-algebras of order 3 r. a. borzooei, a. rezazadeh, r. ameri department of mathematics, shahid beheshti university, tehran, iran department of mathematics, payam noor university, tehran, iran department of mathematics, tehran university, tehran, iran borzooei@sbu.ac.ir, rezazade2008@gmail.com, rez−ameri@yahoo.com abstract in this paper by considering the notion of hyper pseudo bckalgebra, we classify the set of all non-isomorphic hyper pseudo bckalgebras of order 3. for this, we define the notion of simple and normal condition and we characterize the all of hyper pseudo bck-algebras of order 3 that satisfies these conditions. key words: hyper pseudo bck-algebra, simple condition, normal condition. 2000 ams subject classifications: 97u99. 1 introduction the study of bck-algebras was initiated at 1966 by y. imai and k. iséki in [5] as a generalization of the concept of set-theoretic difference and propositional calculi. in order to extend bck-algebras in a noncommutative form, georgescu and iorgulescu [4] introduced the notion of pseudo bckalgebras and studied their properties. the hyperstructure theory (called also multialgebra) was introduced in 1934 by f. marty [10] at the 8th congress of scandinavian mathematicians. since then many researchers have worked on algebraic hyperstructures and developed it. a recent book [3] contains a wealth of applications. via this book, corsini and leoreanu presented some of the numerous applications of algebraic hyperstructures, especially those from the last fifteen years, to the following subjects: geometry, hypergraphs, 13 r. a. borzooei, a. rezazadeh and r. ameri binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, codes, median algebras, relation algebras, artificial intelligence and probabilities. hyperstructures have many applications to several sectors of both pure and applied sciences. in [1, 9], r. a. borzooei et al. applied the hyperstructures to (pseudo) bck-algebras and introduced the notion of hyper (pseudo) bck-algebra which is a generalization of (pseudo) bck-algebra and investigated some related properties. in [2], r. a. borzooei et al. classified all hyper bck-algebras of order 3. now, in this paper we classify the set of all non-isomorphic hyper pseudo bck-algebras of order 3. 2 preliminaries definition 2.1. [9] by a hyper bck-algebra we mean a nonempty set h endowed with a hyperoperation ”◦” and a constant 0 satisfy the following axioms: (hk1) (x ◦ z) ◦ (y ◦ z) ≪ x ◦ y, (hk2) (x ◦ y) ◦ z = (x ◦ z) ◦ y, (hk3) x ◦ h ≪ {x}, (hk4) x ≪ y and y ≪ x imply x = y. for all x, y, z ∈ h, where x ≪ y is defined by 0 ∈ x ◦ y and for every a, b ⊆ h, a ≪ b is defined by ∀a ∈ a, ∃b ∈ b such that a ≪ b. in such case, we call ” ≪ ” the hyperorder in h. definition 2.2. [1] a hyper pseudo bck-algebra is a structure (h, ◦, ∗, 0) where ”∗” and ”◦” are hyper operations on h and ”0” is a constant element, that satisfies the following: for all x, y, z ∈ h, (phk1) (x ◦ z) ◦ (y ◦ z) ≪, x ◦ y, (x ∗ z) ∗ (y ∗ z) ≪ x ∗ y, (phk2) (x ◦ y) ∗ z = (x ∗ z) ◦ y, (phk3) x ◦ h ≪ {x}, x ∗ h ≪ {x}, (phk4) x ≪ y and y ≪ x imply x = y. where x ≪ y ⇔ 0 ∈ x ◦ y ⇔ 0 ∈ x ∗ y and for every a, b ⊆ h, a ≪ b is defined by ∀a ∈ a, ∃b ∈ b such that a ≪ b. theorem 2.1. [2, 9] any bck-algebra and hyper bck-algebra is a hyper pseudo bck-algebra. 14 classifications of hyper pseudo bck-algebras of order 3 proposition 2.1. [1] in any hyper pseudo bck-algebra h, the following hold: (i) 0 ◦ 0 = {0}, 0 ∗ 0 = {0}, x ◦ 0 = {x}, x ∗ 0 = {x}, (ii) 0 ≪ x, x ≪ x, a ≪ a, (iii) 0 ◦ x = {0}, 0 ∗ x = {0}, 0 ◦ a = {0}, 0 ∗ a = {0}. for all x, y, z ∈ h and for all nonempty subsets a and b of h. theorem 2.2. [2] there are 19 non-isomorphic hyper bck-algebras of order 3. note: from now on, in this paper h = {0, a, b} is a hyper pseudo bckalgebra of order 3, unless otherwise state. 3 characterization of hyper pseudo bckalgebras of order 3 definition 3.1. [2] we say that h satisfies the normal condition if one of the conditions a ≪ b or b ≪ a holds. if no one of these conditions hold, then we say that h satisfies the simple condition. definition 3.2. let (h1, ◦1, ∗1, 01) and (h2, ◦2, ∗2, 02) be two hyper pseudo bck-algebras and f : h1 → h2 be a function. then f is said to be a homomorphism iff (i) f(01) = 02 (ii) f(x ◦1 y) = f(x) ◦2 f(y), ∀x, y ∈ h1 (iii) f(x ∗1 y) = f(x) ∗2 f(y), ∀x, y ∈ h1. if f is one to one (onto) we say that f is a monomorphism (epimorphism) and if f is both one to one and onto, we say that f is an isomorphism. definition 3.3. let i ⊆ h. then we say that i is a proper subset of h if i ̸= {0} and i ̸= h. 15 r. a. borzooei, a. rezazadeh and r. ameri 3.1 characterization of hyper pseudo bck-algebras of order 3 that satisfy the simple condition theorem 3.1. there are only 10 hyper pseudo bck-algebras of order 3, that satisfy the simple condition. proof. let h satisfy the simple condition. now, we prove the following statements: (i) for all x, y ∈ h which x ̸= y, then x ̸∈ y ◦ x and x ̸∈ y ∗ x. (ii) a ◦ b = a ∗ b = {a} and b ◦ a = b ∗ a = {b}. (iii) a ◦ a and a ∗ a are equal to {0} or {0, a} and b ◦ b and b ∗ b are equal to {0} or {0, b}. for the proof of (i), let x ̸= y and x ∈ y ◦ x, by the contrary. clearly x ̸= 0. because if x = 0, then y ̸= 0 and 0 ∈ y ◦ 0 = {y}, which is impossible. moreover, since y ◦ x ≤ y, then x ≤ y which is impossible by the simplicity of h. by the similar way, we can prove that x ̸∈ y ∗ x. (ii) since a ̸≤ b, then 0 ̸∈ a ◦ b and 0 ̸∈ a ∗ b. hence a ◦ b and a ∗ b can not be equal to {0}, {0, a}, {0, b} or {0, a, b}. since by (i), we have b ̸∈ a ◦ b and b ̸∈ a ∗ b, we conclude that a ◦ b and a ∗ b can not be equal to {b} or {a, b}. thus a ◦ b = a ∗ b = {a}. by the similar way, we can prove that b ◦ a = b ∗ a = {b}. (iii) since a ≪ a, the only cases for a◦a and a∗a are {0}, {0, a}, {0, b} or {0, a, b}. also we have a ◦ a ≤ a and a ∗ a ≤ a. thus b ̸∈ a ◦ a and b ̸∈ a ∗ a. hence the only cases for a ◦ a and a ∗ a are {0} or {0, a}. by the similar way, we can prove that b ◦ b and b ∗ b are equal to {0} or {0, b}. therefore, by (i), (ii) and (iii) we conclude that there are 16 hyper pseudo bck-algebras of order 3, which satisfy the simple condition. but some of them are isomorphic under the map f : h → h which is defined by f(0) = 0, f(a) = b and f(b) = a. hence there are 10 hyper pseudo bck-algebras of order 3, that satisfy the simple condition. now, we give these hyper pseudo bck-algebras: ◦1 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0} ∗1 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0} ◦2 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0} ∗2 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0,b} 16 classifications of hyper pseudo bck-algebras of order 3 ◦3 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0} ∗3 0 a b 0 {0} {0} {0} a {a} {0,a} {a} b {b} {b} {0,b} ◦4 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0,b} ∗4 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0} ◦5 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0,b} ∗5 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0,b} ◦6 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0,b} ∗6 0 a b 0 {0} {0} {0} a {a} {0,a} {a} b {b} {b} {0,b} ◦7 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0,b} ∗7 0 a b 0 {0} {0} {0} a {a} {0,a} {a} b {b} {b} {0} ◦8 0 a b 0 {0} {0} {0} a {a} {0,a} {a} b {b} {b} {0,b} ∗8 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0} ◦9 0 a b 0 {0} {0} {0} a {a} {0,a} {a} b {b} {b} {0,b} ∗9 0 a b 0 {0} {0} {0} a {a} {0} {a} b {b} {b} {0,b} ◦10 0 a b 0 {0} {0} {0} a {a} {0,a} {a} b {b} {b} {0,b} ∗10 0 a b 0 {0} {0} {0} a {a} {0,a} {a} b {b} {b} {0,b} 17 r. a. borzooei, a. rezazadeh and r. ameri 3.2 characterization of hyper pseudo bck-algebras of order 3 that satisfy the normal condition note: from now on, in this section we let h = {0, a, b} satisfies the normal condition. since in this condition, a ≤ b or b ≤ a, so without loss of generality we let a ≤ b and b ̸≤ a i.e., 0 ∈ a ◦ b ∩ a ∗ b, 0 ̸∈ b ◦ a and 0 ̸∈ b ∗ a. lemma 3.1. only one of the following cases hold for b ∗ a and b ◦ a. (nphb1) b ◦ a = b ∗ a = {a}, (nphb2) b ◦ a = {a}, b ∗ a = {a, b}, (nphb3) b ◦ a = b ∗ a = {b}, (nphb4) b ◦ a = {a, b}, b ∗ a = {a}, (nphb5) b ◦ a = b ∗ a = {a, b}. proof. since 0 ̸∈ b ◦ a and 0 ̸∈ b ∗ a, then b ◦ a and b ∗ a are equal to one of the sets {a}, {b} or {a, b}. if b◦a = {a}, then b∗a ̸= {b}. since if b∗a = {b}, then (b◦a)∗a ̸= (b∗a)◦a. hence b ∗ a = {a} or {a, b}. if b ◦ a = {b}, then b ∗ a ̸= {a, b} and {a}. since if b ∗ a = {a, b} or {a}, then (b ◦ a) ∗ a ̸= (b ∗ a) ◦ a. hence b ∗ a = {b}. if b ◦ a = {a, b}, then b ∗ a ̸= {b}. since if b ∗ a = {b}, then (b ◦ a) ∗ a ̸= (b ∗ a) ◦ a. hence b ∗ a = {a} or {a, b}. therefore, we have the above cases. lemma 3.2. only one of the following cases hold for a ∗ b and a ◦ b. (npha1) a ◦ b = a ∗ b = {0}. (npha2) a ◦ b = a ∗ b = {0, a}. (npha3) a ◦ b = {0}, a ∗ b = {0, a}. (npha4) a ◦ b = {0, a}, a ∗ b = {0}. proof. since 0 ∈ a ◦ b ∩ a ∗ b, then a ◦ b and a ∗ b are equal to one of the sets {0}, {0, a}, {0, b} or {0, a, b}. moreover, since a ◦ b ≤ a and a ∗ b ≤ a, then b ̸∈ a ◦ b and b ̸∈ a ∗ b. hence the only cases for a ◦ b and a ∗ b are {0} or {0, a}. therefore, we have the above cases. lemma 3.3. only one of the following cases hold for a ∗ a and a ◦ a. (i) a ◦ a = a ∗ a = {0}, (ii) a ◦ a = {0}, a ∗ a = {0, a}, 18 classifications of hyper pseudo bck-algebras of order 3 (iii) a ◦ a = {0, a}, a ∗ a = {0}, (iv) a ◦ a = a ∗ a = {0, a}. proof. since 0 ∈ a ◦ a ∩ a ∗ a, then a ◦ a and a ∗ a can be equal to the one of cases {0}, {0, a}, {0, b} or {0, a, b}. moreover, since a ◦ a ≤ a and a ∗ a ≤ a, then b ̸∈ a ◦ a and b ̸∈ a ∗ a. hence the only cases for a ◦ a and a ∗ a are {0} or {0, a}. therefore, we have the above cases. theorem 3.4. there are only 5 non-isomorphic hyper pseudo bck-algebras of order 3, that satisfy the normal condition and condition (nphb1). proof. since h satisfies the condition (nphb1), then b◦a = b∗a = {a}. case (npha1): we have a ∗ b = a ◦ b = {0}. if a ◦ a = {0, a} or a ∗ a = {0, a}, then (a ◦ a) ◦ (b ◦ a) ̸≪ a ◦ b and (a ∗ a) ∗ (b ∗ a) ̸≪ a ∗ b. therefore, a◦a = a∗a = {0}. moreover, if b◦b and b∗b are equal to the one of sets {0, b} or {0, a, b}, then (b◦b)◦(a◦b) ̸≪ b◦a and (b∗b)∗(a∗b) ̸≪ b∗a. therefore, in this case b◦b and b∗b are equal to the one of sets {0} or {0, a}. now, we consider the following cases: (1) b ◦ b = b ∗ b = {0}. thus in this case, we have the following tables: ∗1 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a} {0} ◦1 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a} {0} (2) b◦b = {0} and b∗b = {0, a}. thus in this case, we have the following tables: ∗2 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a} {0} ◦2 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a} {0,a} (3) b ◦ b = {0, a} and b ∗ b = {0}. thus similar to (2), we have one hyper pseudo bck-algebra in this case. (4) b ◦ b = b ∗ b = {0, a}. thus in this case, we have the following tables: ∗4 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a} {0,a} ◦4 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a} {0,a} case (npha2): we have a ∗ b = a ◦ b = {0, a}. if a ◦ a = {0} or a ∗ a = {0}, then (a ◦ b) ◦ (a ◦ b) ̸≪ a ◦ a and (a ∗ b) ∗ (a ∗ b) ̸≪ a ∗ a. therefore, a ◦ a = a ∗ a = {0, a}. moreover, if b ◦ b and b ∗ b are equal to the one of sets {0, b} 19 r. a. borzooei, a. rezazadeh and r. ameri or {0, a, b}, then (b ◦ b) ◦ (a ◦ b) ̸≪ b ◦ a and (b ∗ b) ∗ (a ∗ b) ̸≪ b ∗ a and if b◦b = {0} or b∗b = {0}, then (b◦a)◦(b◦a) ̸≪ b◦b and (b∗a)∗(b∗a) ̸≪ b∗b. therefore, b ◦ b = b ∗ b = {0, a}. thus we have the following case: ∗5 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a} {0,a} ◦5 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a} {0,a} case (npha3): we have a ∗ b = {0, a} and a ◦ b = {0}. since (a ∗ b) ∗ (a ∗ b) ≪ a ∗ a, then a ∗ a = {0, a}. moreover, since a ◦ b = {0}, then a ◦ a ̸= {0, a}, because (a ◦ a) ◦ (b ◦ a) ̸≪ a ◦ b. thus a ◦ a = {0} and in this case (b ∗ a) ◦ a ̸= (b ◦ a) ∗ a. case (npha4): we have a ∗ b = {0} and a ◦ b = {0, a}. in this case we can show that (b ∗ a) ◦ a ̸= (b ◦ a) ∗ a. therefore, we have not any hyper pseudo bck-algebras. we can check that all of the these 5 cases are hyper pseudo bck-algebras and each of them are not isomorphic together. theorem 3.5. there are only 6 non-isomorphic hyper pseudo bck-algebras of order 3, that satisfy the normal condition and condition (nphb2). proof. since h satisfies the condition (nphb2), then b ◦ a = {a} and b ∗ a = {a, b}. case (npha1): we have a ∗ b = a ◦ b = {0}. if one of the a ◦ a or a ∗ a are equal to {0, a}, then (a ◦ a) ◦ (b ◦ a) ̸≪ a ◦ b and (a ∗ a) ∗ (b ∗ a) ̸≪ a ∗ a. hence we have a ∗ a = a ◦ a = {0}. but in this case (b ∗ a) ◦ a ̸= (b ◦ a) ∗ a. therefore, we have not any hyper pseudo bck-algebras in this case. case (npha2): we have a ∗ b = a ◦ b = {0, a}. if a ◦ a = {0} or a∗a = {0}, then (a◦b)◦(a◦b) ̸≪ a◦a and (a∗b)∗(a∗b) ̸≪ a∗a. therefore, a ◦ a = a ∗ a = {0, a}. moreover, if b ∗ b is equal to the one of sets {0} or {0, a}, then (b∗a)∗(b∗a) ̸≪ b∗b. therefore, in this case b∗b is equal to the one of sets {0, b} or {0, a, b} and if b ◦ b is equal to the one of sets {0, b} or {0, a, b}, then (b ◦ b) ◦ (a ◦ b) ̸≪ b ◦ a. hence in this case b ◦ b is equal to the one of sets {0} or {0, a}. moreover, since a ◦ a = {0, a}, if b ◦ b = {0}, then (b ◦ a) ◦ (b ◦ a) ̸≪ b ◦ b. thus b ◦ b = {0, a}. now, we consider the following cases: (1) b ◦ b = {0, a} and b ∗ b = {0, b}. thus in this case, we have the following tables: ◦1 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a} {0,a} ∗1 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,b} 20 classifications of hyper pseudo bck-algebras of order 3 (2) b ◦ b = {0, a} and b ∗ b = {0, a, b}. thus in this case, we have the following tables: ◦2 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a} {0,a} ∗2 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,a,b} case (npha3): we have a ∗ b = {0, a} and a ◦ b = {0}. if a ∗ a = {0} and a ◦ a is equal to {0} or {0, a}, then (a ∗ b) ∗ (a ∗ b) ̸≪ a ∗ a. thus a ∗ a = {0, a}. moreover, if b ∗ b is equal to the one of sets {0} or {0, a}, then (b ∗ a) ∗ (b ∗ a) ̸≪ b ∗ b. therefore, in this case b ∗ b is equal to the one of sets {0, b} or {0, a, b} and if b ◦ b is equal to the one of sets {0, b} or {0, a, b}, then (b ◦ b) ◦ (a ◦ b) ̸≪ b ◦ a. thus in this case b ◦ b is equal to the one of sets {0} or {0, a}. now, we consider the following cases: (1) b◦b = {0} and b∗b = {0, b}. if a◦a = {0, a}, then (b◦a)◦(b◦a) ̸≪ b◦b. hence a ◦ a = {0}. thus in this case, we have the following tables: ◦3 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a} {0} ∗3 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,b} (2) b◦b = {0} and b∗b = {0, a, b}. if a◦a = {0, a}, then (b◦a)◦(b◦a) ̸≪ b ◦ b. hence a ◦ a = {0}. thus in this case, we have the following tables: ◦4 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a} {0} ∗4 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,a,b} (3) b◦b = {0, a} and b∗b = {0, b}. if a◦a = {0, a}, then (a◦a)◦(b◦a) ̸≪ a ◦ b. hence a ◦ a = {0}. thus in this case, we have the following tables: ◦5 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a} {0,a} ∗5 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,b} (4) b◦b = {0, a} and b∗b = {0, a, b}. if a◦a = {0, a}, then (a◦a)◦(b◦a) ̸≪ a ◦ b. hence a ◦ a = {0}. thus in this case, we have the following tables: ◦6 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a} {0,a} ∗6 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,a,b} 21 r. a. borzooei, a. rezazadeh and r. ameri case (npha4): we have a ∗ b = {0} and a ◦ b = {0, a}. if a ◦ a = {0} and a ∗ a is equal to {0} or {0, a}, then (a ◦ b) ◦ (a ◦ b) ̸≪ a ◦ a. thus a ◦a = {0, a}. moreover, if b ∗b is equal to the one of sets {0} or {0, a}, then (b ∗ a) ∗ (b ∗ a) ̸≪ b ∗ b. therefore, in this case b ∗ b is equal to the one of sets {0, b} or {0, a, b}. moreover, if a ∗ a = {0, a}, then (a ∗ a) ∗ (b ∗ a) ̸≪ a ∗ b. hence a ∗ a = {0}. but in this case (b ◦ a) ∗ b ̸= (b ∗ b) ◦ a. therefore, we have not any hyper pseudo bck-algebras. we can check that all of the these 6 cases are hyper pseudo bck-algebras and each of them are not isomorphic together. theorem 3.6. there are only 70 non-isomorphic hyper pseudo bckalgebras of order 3, that satisfy the normal condition and condition (nphb3). proof. since h satisfies the condition (nphb3), then b◦a = b∗a = {b}. case (npha1): we have a ∗ b = a ◦ b = {0}. now, we consider the following cases: (1) b ◦ b = b ∗ b = {0}. in this case, we have the following tables: ◦1 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0} ∗1 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0} ◦2 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0} ∗2 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ◦3 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗3 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0} ◦4 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗4 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} (2) b ◦ b = {0} and b ∗ b = {0, b}. in this case, we have the following tables: ◦5 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0} ∗5 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,b} 22 classifications of hyper pseudo bck-algebras of order 3 ◦6 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0} ∗6 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,b} ◦7 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗7 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,b} ◦8 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗8 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,b} (3) b ∗ b = {0} and b ◦ b = {0, b}. similar to (2), we have four hyper pseudo bck-algebras in this case. (4) b◦b = {0} and b∗b = {0, a}. if a◦a = {0}, then (b◦a)∗b ̸= (b∗b)◦a. thus in this case, we have the following tables: ◦13 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗13 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,a} ◦14 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗14 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a} (5) b∗b = {0} and b◦b = {0, a}. if a∗a = {0}, then (b∗a)◦b ̸= (b◦b)∗a. thus similar to (4), we have two hyper pseudo bck-algebras in this case. (6) b◦b = {0} and b∗b = {0, a, b}. if a◦a = {0}, then (b◦a)∗b ̸= (b∗b)◦a. thus in this case, we have the following tables: ◦17 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗17 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,a,b} ◦18 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗18 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a,b} 23 r. a. borzooei, a. rezazadeh and r. ameri (7) b∗b = {0} and b◦b = {0, a, b}. if a∗a = {0}, then (b∗a)◦b ̸= (b◦b)∗a. thus similar to (6), we have two hyper pseudo bck-algebras in this case. (8) b ◦ b = b ∗ b = {0, b}. thus in this case, we have the following tables: ◦21 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,b} ∗21 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,b} ◦22 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,b} ∗22 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,b} ◦23 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,b} ∗23 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,b} ◦24 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,b} ∗24 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,b} (9) b ◦ b = {0, b} and b ∗ b = {0, a}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (10) b ∗ b = {0, b} and b ◦ b = {0, a}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (11) b ◦ b = {0, b} and b ∗ b = {0, a, b}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (12) b ∗ b = {0, b} and b ◦ b = {0, a, b}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (13) b ◦ b = b ∗ b = {0, a}. if a ◦ a = {0}, then (b ◦ a) ∗ b ̸= (b ∗ b) ◦ a and if a ∗ a = {0}, then (b ∗ a) ◦ b ̸= (b ◦ b) ∗ a. therefore, a ◦ a = a ∗ a = {0, a}. thus in this case, we have the following tables: ◦25 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a} ∗25 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a} (14) b ◦ b = {0, a} and b ∗ b = {0, a, b}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (15) b ∗ b = {0, a} and b ◦ b = {0, a, b}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (16) b ◦ b = b ∗ b = {0, a, b}. if a ◦ a = {0}, then (b ◦ a) ∗ b ̸= (b ∗ b) ◦ a and if a ∗ a = {0}, then (b ∗ a) ◦ b ̸= (b ◦ b) ∗ a. therefore, a ◦ a = a ∗ a = {0, a}. thus in this case, we have the following tables: 24 classifications of hyper pseudo bck-algebras of order 3 ◦26 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a,b} ∗26 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a,b} case (npha2): we have a ∗ b = a ◦ b = {0, a}. if one of the a ∗ a or a ◦ a are equal to {0}, then (a∗b)∗(a∗b) ̸≪ a∗a or (a◦b)◦(a◦b) ̸≪ a◦a. hence a ∗ a = a ◦ a = {0, a}. (1) b ◦ b = b ∗ b = {0}. thus in this case, we have the following tables: ◦27 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0} ∗27 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0} (2) b ◦ b = {0} and b ∗ b = {0, b}. thus in this case, we have the following tables: ◦28 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0} ∗28 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} (3) b ∗ b = {0} and b ◦ b = {0, b}. thus similar to (2), we have one hyper pseudo bck-algebra in this case. (4) b ◦ b = {0} and b ∗ b = {0, a}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (5) b ∗ b = {0} and b ◦ b = {0, a}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (6) b ◦ b = {0} and b ∗ b = {0, a, b}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (7) b ∗ b = {0} and b ◦ b = {0, a, b}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (8) b ◦ b = b ∗ b = {0, b}. thus in this case, we have the following tables: ◦30 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ∗30 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} (9) b ◦ b = {0, b} and b ∗ b = {0, a}. thus in this case, we have the following tables: ◦31 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ∗31 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a} 25 r. a. borzooei, a. rezazadeh and r. ameri (10) let b ∗ b = {0, b} and b ◦ b = {0, a}. thus similar to (9), we have one hyper pseudo bck-algebra in this case. (11) b ◦ b = {0, b} and b ∗ b = {0, a, b}. thus in this case, we have the following tables: ◦33 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ∗33 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a,b} (12) b ∗ b = {0, b} and b ◦ b = {0, a, b}. thus similar to (11), we have one hyper pseudo bck-algebra in this case. (13) b ◦ b = b ∗ b = {0, a}. thus in this case, we have the following tables: ◦35 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a} ∗35 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a} (14) b ◦ b = {0, a} and b ∗ b = {0, a, b}. thus in this case, we have the following tables: ◦36 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a} ∗36 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a,b} (15) b ∗ b = {0, a} and b ◦ b = {0, a, b}. thus similar to (14), we have one hyper pseudo bck-algebra in this case. (16) b ◦ b = b ∗ b = {0, a, b}. thus in this case, we have the following tables: ◦38 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a,b} ∗38 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a,b} case (npha3): we have a∗b = {0, a} and a◦b = {0}. if a∗a = {0} and a◦a is equal to {0} or {0, a}, then (a∗b)∗(a∗b) ̸≪ a∗a. thus a∗a = {0, a}. (1) b ◦ b = b ∗ b = {0}. thus in this case, we have the following tables: ◦39 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0} ∗39 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0} 26 classifications of hyper pseudo bck-algebras of order 3 ◦40 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗40 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0} (2) b ◦ b = {0} and b ∗ b = {0, b}. thus in this case, we have the following tables: ◦41 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0} ∗41 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ◦42 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗42 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} (3) b ∗ b = {0} and b ◦ b = {0, b}. thus similar to (2), we have two hyper pseudo bck-algebras in this case. (4) b◦b = {0} and b∗b = {0, a}. if a◦a = {0} then (b◦a)∗b ̸= (b∗b)◦a. thus in this case, we have the following tables: ◦45 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗45 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a} (5) b ∗ b = {0} and b ◦ b = {0, a}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (6) b◦b = {0} and b∗b = {0, a, b}. if a◦a = {0}, then (b◦a)∗b ̸= (b∗b)◦a. thus in this case, we have the following tables: ◦46 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} ∗46 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a,b} (7) b ∗ b = {0} and b ◦ b = {0, a, b}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (8) b ◦ b = b ∗ b = {0, b}. thus in this case, we have the following tables: ◦47 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,b} ∗47 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} 27 r. a. borzooei, a. rezazadeh and r. ameri ◦48 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,b} ∗48 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} (9) b ◦ b = {0, b} and b ∗ b = {0, a}. then (b ∗ b) ◦ b ̸= (b ◦ b) ∗ b. (10) b ∗ b = {0, b} and b ◦ b = {0, a}. thus in this case, we have the following tables: ◦49 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,a} ∗49 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ◦50 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a} ∗50 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} (11) b ◦ b = {0, b} and b ∗ b = {0, a, b}. then (b ∗ b) ◦ b ̸= (b ◦ b) ∗ b. (12) b ∗ b = {0, b} and b ◦ b = {0, a, b}. thus in this case, we have the following tables: ◦51 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,a,b} ∗51 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ◦52 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a,b} ∗52 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} (13) b ◦ b = b ∗ b = {0, a}. then (b ∗ b) ◦ b ̸= (b ◦ b) ∗ b. (14) b ◦ b = {0, a} and b ∗ b = {0, a, b}. if a ◦ a = {0}, then (b ◦ a) ∗ b ̸= (b ∗ b) ◦ a. thus in this case, we have the following tables: ◦53 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a} ∗53 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a,b} 28 classifications of hyper pseudo bck-algebras of order 3 (15) b ∗ b = {0, a} and b ◦ b = {0, a, b}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (16) b ◦ b = b ∗ b = {0, a, b}. if a ◦ a = {0}, then (b ◦ a) ∗ b ̸= (b ∗ b) ◦ a. thus in this case, we have the following tables: ◦54 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a,b} ∗54 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a,b} case (npha4): we have a◦b = {0, a} and a∗b = {0}. if a◦a = {0} and a∗a is equal to {0} or {0, a}, then (a◦b)◦(a◦b) ̸≪ a◦a. thus a◦a = {0, a}. (1) b ◦ b = b ∗ b = {0, b}. thus in this case, we have the following tables: ◦55 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0} ∗55 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0} ◦56 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0} ∗56 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} (2) b ◦ b = {0} and b ∗ b = {0, b}. thus in this case, we have the following tables: ◦57 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0} ∗57 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,b} ◦58 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0} ∗58 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,b} (3) b ∗ b = {0} and b ◦ b = {0, b}. thus similar to (2), we have two hyper pseudo bck-algebras in this case. (4) b ◦ b = {0} and b ∗ b = {0, a}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (5) b∗b = {0} and b◦b = {0, a}. if a∗a = {0}, then (b∗a)◦b ̸= (b◦b)∗a. thus in this case, we have the following tables: 29 r. a. borzooei, a. rezazadeh and r. ameri ◦61 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a} ∗61 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} (6) b ◦ b = {0} and b ∗ b = {0, a, b}. then (b ◦ b) ∗ b ̸= (b ∗ b) ◦ b. (7) b∗b = {0} and b◦b = {0, a, b}. if a∗a = {0}, then (b∗a)◦b ̸= (b◦b)∗a. thus in this case, we have the following tables: ◦62 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a,b} ∗62 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0} (8) b ◦ b = b ∗ b = {0, b}. thus in this case, we have the following tables: ◦63 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ∗63 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,b} ◦64 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ∗64 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,b} (9) b ◦ b = {0, b} and b ∗ b = {0, a}. thus in this case, we have the following tables: ◦65 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ∗65 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,a} ◦66 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ∗66 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a} (10) b ∗ b = {0, b} and b ◦ b = {0, a}. then (b ∗ b) ◦ b ̸= (b ◦ b) ∗ b. (11) b ◦ b = {0, b} and b ∗ b = {0, a, b}. thus in this case, we have the following tables: 30 classifications of hyper pseudo bck-algebras of order 3 ◦67 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ∗67 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {b} {0,a,b} ◦68 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,b} ∗68 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a,b} (12) b ∗ b = {0, b} and b ◦ b = {0, a, b}. then (b ∗ b) ◦ b ̸= (b ◦ b) ∗ b. (13) b ◦ b = b ∗ b = {0, a}. then (b ∗ b) ◦ b ̸= (b ◦ b) ∗ b. (14) b ◦ b = {0, a} and b ∗ b = {0, a, b}. then (b ∗ b) ◦ b ̸= (b ◦ b) ∗ b. (15) b ∗ b = {0, a} and b ◦ b = {0, a, b}. if a ∗ a = {0}, then (b ∗ a) ◦ b ̸= (b ◦ b) ∗ a. thus in this case, we have the following tables: ◦69 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a,b} ∗69 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a} (16) b ◦ b = b ∗ b = {0, a, b}. if a ∗ a = {0}, then (b ∗ a) ◦ b ̸= (b ◦ b) ∗ a. thus in this case, we have the following tables: ◦70 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {b} {0,a,b} ∗70 0 a b 0 {0} {0} {0} a {a} {0,a} {0} b {b} {b} {0,a,b} we can check that all of the these 70 cases are hyper pseudo bckalgebras and each of them are not isomorphic together. theorem 3.7. there are only 6 non-isomorphic hyper pseudo bck-algebras of order 3, that satisfy the normal condition and condition (nphb4). proof. the proof is the similar to the proof of theorem 3.5, by the some modification. theorem 3.8. there are only 9 non-isomorphic hyper pseudo bck-algebras of order 3, that satisfy the normal condition and condition (nphb5). proof. since h satisfies the condition (nphb5), then b ◦ a = b ∗ a = {a, b}. case (npha1): we have a∗b = a◦b = {0}. if one of the b◦b or b∗b are equal to {0}, {0, a} or {0, b}, then (b◦a)◦(b◦a) ̸≪ b◦b or (b∗a)∗(b∗a) ̸≪ b∗b. therefore, we have only the following case: 31 r. a. borzooei, a. rezazadeh and r. ameri (1) b◦b = b∗b = {0, a, b}. if a◦a = {0, a}, then (a◦a)◦(b◦a) ̸≪ a◦b and if a ∗ a = {0, a}, then (a ∗ a) ∗ (b ∗ a) ̸≪ a ∗ b. therefore, a ◦ a = a ∗ a = {0}. thus in this case, we have the following tables: ◦1 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a,b} {0,a,b} ∗1 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a,b} {0,a,b} case (npha2): we have a ∗ b = a ◦ b = {0, a}. if one of the a ∗ a or a ◦ a are equal to {0}, then (a∗b)∗(a∗b) ̸≪ a∗a or (a◦b)◦(a◦b) ̸≪ a◦a. hence a ∗ a = a ◦ a = {0, a}. if b◦b = {0}, but b∗b = {0}, {0,b}, {0,a} or {0,a,b}, then (b◦a)◦(b◦a) ̸≪ b ◦ b. if b ◦ b = {0, b}, but b ∗ b = {0} or {0, a}, then (b ∗ a) ∗ (b ∗ a) ̸≪ b ∗ b. hence b ∗ b = {0, b} or {0, a, b}. if b ◦ b = {0, a}, but b ∗ b = {0}, {0, b}, {0, a} or {0, a, b}, then (b ◦ a) ◦ (b ◦ a) ̸≪ b ◦ b. if b ◦ b = {0, a, b}, but b ∗ b = {0} or {0, a}, then (b ∗ a) ∗ (b ∗ a) ̸≪ b ∗ b. hence b ∗ b = {0, b} or {0, a, b}. therefore, we have the following cases: (1) b ◦ b = b ∗ b = {0, b}. thus we have the following tables: ◦2 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,b} ∗2 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,b} (2) b ◦ b = {0, b} and b ∗ b = {0, a, b}. thus we have the following tables: ◦3 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,b} ∗3 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,a,b} (3) b ◦ b = {0, a, b} and b ∗ b = {0, b}. thus similar to (2), we have one hyper pseudo bck-algebra in this case. (4) let b ◦ b = b ∗ b = {0, a, b}. thus we have the following tables: ◦5 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,a,b} ∗5 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,a,b} 32 classifications of hyper pseudo bck-algebras of order 3 case (npha3): we have a∗b = {0, a} and a◦b = {0}. if a∗a = {0} and a◦a is equal to {0} or {0, a}, then (a∗b)∗(a∗b) ̸≪ a∗a. thus a∗a = {0, a}. if b◦b = {0}, but b∗b = {0}, {0,b}, {0,a} or {0,a,b}, then (b◦a)◦(b◦a) ̸≪ b ◦ b. if b ◦ b = {0, b}, but b ∗ b = {0} or {0, a}, then (b ∗ a) ∗ (b ∗ a) ̸≪ b ∗ b. if b ◦ b = b ∗ b = {0, b}, then (b ∗ a) ◦ b ̸= (b ◦ b) ∗ a. if b ◦ b = {0, b} and b ∗ b = {0, a, b}, then (b ∗ b) ◦ b ̸= (b ◦ b) ∗ b. if b◦b = {0, a}, but b∗b = {0}, {0, b} or {0, a, b}, then (b◦a)◦(b◦a) ̸≪ b◦b. if b ◦ b = b ∗ b = {0, a}, then (b ∗ b) ◦ b ̸= (b ◦ b) ∗ b. if b ◦ b = {0, a, b}, but b ∗ b = {0} or {0, a}, then (b ∗ a) ∗ (b ∗ a) ̸≪ b ∗ b. hence b ∗ b = {0, b} or {0, a, b}. therefore, we have the following cases: (1) b◦b = {0, a, b} and b∗b = {0, b}. if a◦a = {0, a}, then (a◦a)◦(b◦a) ̸≪ a ◦ b. thus in this case, we have the following tables: ◦6 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a,b} {0,a,b} ∗6 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,b} (2) b ◦ b = b ∗ b = {0, a, b}. if a ◦ a = {0, a}, then (a ◦ a) ◦ (b ◦ a) ̸≪ a ◦ b. thus in this case, we have the following tables: ◦7 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a,b} {0,a,b} ∗7 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,a,b} case (npha4): we have a◦b = {0, a} and a∗b = {0}. if a◦a = {0} and a∗a is equal to {0} or {0, a}, then (a◦b)◦(a◦b) ̸≪ a◦a. thus a◦a = {0, a}. if b◦b = {0}, but b∗b = {0}, {0,b}, {0,a} or {0,a,b}, then (b◦a)◦(b◦a) ̸≪ b ◦ b. if b ◦ b = {0, b}, but b ∗ b = {0} or {0, a}, then (b ∗ a) ∗ (b ∗ a) ̸≪ b ∗ b. if b ◦ b = b ∗ b = {0, b}, then (b ◦ a) ∗ b ̸= (b ∗ b) ◦ a. hence b ∗ b = {0, a, b}. if b◦b = {0, a}, but b∗b = {0}, {0, b} or {0, a, b}, then (b◦a)◦(b◦a) ̸≪ b◦b. if b ◦ b = b ∗ b = {0, a}, then (b ∗ b) ◦ b ̸= (b ◦ b) ∗ b. if b ◦ b = {0, a, b}, but b ∗ b = {0} or {0, a}, then (b ∗ a) ∗ (b ∗ a) ̸≪ b ∗ b. if b ◦ b = {0, a, b}, but b ∗ b = {0, b}, then (b ∗ b) ◦ b ̸= (b ◦ b) ∗ b. hence b ∗ b = {0, a, b}. therefore, we have the following cases: (1) b◦b = {0, b} and b∗b = {0, a, b}. if a∗a = {0, a}, then (a∗a)∗(b∗a) ̸≪ a ∗ b. thus in this step, we have the following case: 33 r. a. borzooei, a. rezazadeh and r. ameri ◦8 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,b} ∗8 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a,b} {0,a,b} (2) b ◦ b = b ∗ b = {0, a, b}. if a ∗ a = {0, a}, then (a ∗ a) ∗ (b ∗ a) ̸≪ a ∗ b. thus in this step, we have the following case: ◦9 0 a b 0 {0} {0} {0} a {a} {0,a} {0,a} b {b} {a,b} {0,a,b} ∗9 0 a b 0 {0} {0} {0} a {a} {0} {0} b {b} {a,b} {0,a,b} we can check that all of the these 9 cases are hyper pseudo bck-algebras and each of them are not isomorphic together. theorem 3.9. there are 96 hyper pseudo bck-algebras of order 3, that satisfies the normal condition. proof. by theorems 3.4, 3.5, 3.6, 3.7 and 3.8, the proof is clear. 4 conclusion theorem 4.1. there are 106 hyper pseudo bck-algebras of order 3 up to isomorphism. proof. the proof follows by theorems 3.1 and 3.9. definition 4.1. we say that h is a proper hyper pseudo bck-algebra if h is not a hyper bck-algebra. theorem 4.2. there are 87 proper hyper pseudo bck-algebras of order 3 up to isomorphism. proof. the proof follows by theorems 2.5 and 4.1. references [1] r. a. borzooei, a. rezazadeh and r. ameri, on hyper pseudo bckalgebras, submitted. [2] r. a. borzooei, m. m. zahedi and h. rezaei, classification of hyper bck-algebras of order 3, ital. j. pure appl. math., 12(2002), 175-184. [3] p. corsini and v. leoreanu, applications of hyperstructure theory, kluwer academic publications, (2003). 34 classifications of hyper pseudo bck-algebras of order 3 [4] g. georgescu and a. iorgulescu, pseudo bck-algebras: an extension of bck-algebras, proceedings of dmtcs 01: combinatorics, computability and logic, springer, london, (2001), 97-114. [5] y. imai and k. iseki, on axiom systems of prepositional calculi, xiv. proc japan acad, 42(1996), 26-29. [6] a. iorgulescu, classes of pseudo-bck algebras part i, journal of multiple-valued logic and soft computing, 12(2006), 71-130. [7] a. iorgulescu, classes of pseudo-bck algebras part ii, journal of multiple-valued logic and soft computing, 12(2006), 575-629. [8] y. b. jun, m. kondo and k. h. kim, pseudo-ideals of pseudo-bck algebras, scientiae mathematicae japonicae, 8(2003), 87-91. [9] y. b. jun, m. m. zahedi, x. l. xin and r. a. borzooei, on hyper bckalgebras, italian journal of pure and applied mathematics, 10(2000), 127-136. [10] f. marty, sur une generalization de la notion degroups, 8th congress math. scandinaves, stockholm, (1934), 45-49. 35 microsoft word documento1 ratio mathematica volume 46, 2023 influence of chemical reaction, radiation and heat source on separation of binary fluid mixture with soret and dufour effects on mhd mixed convective flow past a vertical plate krishnandan verma* sanjit basfor abstract the motive of the current study is to analyse the consequences of soret, dufour and thermal radiation on chemically reactive mhd mixed convective flow on separation of binary fluid mixture. numerical techniques are applied to obtain the results by using matlab solver bvp4c. results are acquired for velocity, temperature and concentration distributions for various parameters and are portrayed graphically.the study discloses some interesting results where different parameters like soret number, dufour number, chemical reaction parameter helps in restricting the temperature near the plate and at the same time raising the lightweight ingredient’s concentration close to the plate, ultimately assisting the separation process. numerical data are computed for coefficient of skin-friction coefficient, local nusselt number as well as sherwood number to analyse the effects of dimensional shear stress at the surface of the plate, heat and mass transmission process. keywords: binary fluid mixture; bvp4c; dufour effect; mhd; soret effect 2020 ams subject classifications: 76d05; 76s05. 1 *krishnandan verma and sanjit basfor are with department of mathematics, dibrugarh university (dibrugarh, india-786004); verma.kisu@gmail.com, verma.kisu@gmail.com 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1040. issn: 1592-7415. eissn: 2282-8214. ©krishnandan verma. this paper is published under the cc-by licence agreement. 7 krishnandan verma and sanjit basfor 1 introduction separation of the constituents of a mixture is important in the field of chemical engineering. hydrocarbon mixtures are separated and analyzed by chemist in laboratories whereas chemical engineers uses the process of distillation. often it is seen that some component, which is present in a binary fluid mixture in a very small amount, is very important in engineering and industrial use. the isotope of heavy water that is found in sea water is present in the ratio 1:5000. though, the amount of the isotope is negligible but in atomic power plant, it is used as a coolant in nuclear reactor. therefore, in a binary fluid composition the lightweight or the fewer constituents that is contained in a small amount can have many practical uses. in certain engineering areas, e.g. nuclear reactors and operations that involve metals in liquefied form, the necessity of the process involving detachment of fluid mixtures play a remarkable role. in a given volume, the configuration of binary fluid mixture is indicated by c, which represent concentration as the fraction of the light weight and fewer ingredient’s mass to the entire fluid mixture’s mass, and the configuration of the heavy weight and significant ingredient is indicated by c′ = 1 −c. separation process involving liquid mixture is affected by pressure, temperature, and concentration gradients. the impact of pressure gradient on the separation of the liquid mixture is very less as compared to temperature gradient. landau [1987] have given the expression for diffusion flux density as. ~i = −ρd(∇c + kp∇p + kt∇t) (1) where d,kpd and ktd represent diffusion coefficient, pressure diffusion coefficient and thermal diffusion coefficient respectively. many researchers studied the separation process of binary fluid mixture on various geometries. kim et al. [2007] studied theoretically instabilities in binary nanofluids with soret and dufour effects in convective conditions. elhajjar et al. [2008] investigated the detachment of the constituents of a binary liquid combination in a pervious environment in a horizontal cavity using rayleigh-bernard configuration. sharma and singh [2008] carried out analytical study on the detachment of binary fluid mixture with soret effect and baro-diffusion confined between two parallel disks. they found that the lighter constituent of the fluid is concentrated near the rotating disks for larger values of reynolds number. sharma and singh [2010a] investigated separation of constituents of binary fluid mixture under the existence of strong radial magnetic field that was restricted in between two concentric cylinders rotating about a fixed axis. sharma and singh [2010b] investigated analytically the dissociation of binary fluid ingredients through a rotating disk with thermal diffusion in a weak magnetic environment. they found that the detachment of the binary 8 inf. ch. react., rad. and h. source on sep. bfm sr and df effects fluid ingredients was affected significantly by thermal diffusion and axial magnetic field. sharma et al. [2014] studied numerically the consequences of soret and diffusion thermo effect on the operation of detachment of constituents of a binary ingredient composition in pervious environment considering heat source and chemical reaction. cates and tjhung [2018] investigated theoretically binary fluid mixture with kinetics of phase separation and active emulsions. soret effect has an active role in the dissociation of binary fluid ingredients under the influence of temperature difference in a convection free fluid mixture accompanied by the transfer of heat. many works have been reported in literature on soret effect as well as its counter effect i.e. dufour effect on heat and mass transmission process. chamkha and khaled [2000] analysed thermal transmission on hydromagnetic hybrid nanofluid flow between two parallel plates where one plate is stretchable while other is penetrable and both the plates and fluid rotates simultaneously under ohmic heating and thermal effects. sharma and aich [2015] studied numerically thermal and mass transmission near stagnation point of a circular cylinder on a free convective flow with radiation and thermal diffusion effects. ashraf et al. [2018] examined thermal and mass transmission past a semi-infinite erect plate on convective flow with soret and dufour effects. verma et al. [2020a] probed numerically soret and dufour effects past a stretching sheet on mhd chemically reacting fluid and discovered that the fluid concentration increases near the sheet due to soret effect while dufour effect increases the fluid temperature. verma et al. [2020b] studied numerically the action of thermal and mass transmission on mhd fluid flow in porous medium on darcy-forchheimer model on a rotating infinite disk. krishna et al. [2021] investigated mhd convective heat generating/absorbing second grade rotating unsteady fluid flow past a semi-infinite pervious surface with ion slip and hall effects. they found that the ion slip and hall parameter boosts the fluid velocity while the heat source parameter decreases the temperature profile. other important research on heat and mass transmission are reported by sharma [2011], bhattacharyya et al. [2014], rashidi et al. [2015], reddy and chamkha [2016], ullah et al. [2017], jain and choudhary [2018], verma et al. [2021] and verma and sharma [2022]. the above list of literature reviews motivate us to investigate numerically the separation of the lightweight ingredient of the binary fluid in the existence of radiation,heat source, soret, dufour and chemical effects over an upright plane surface insertedin a porous medium with darcy-boussinesq model. the current work will focus to examine the separation process numerically using bvp4c scheme, which is an inbuilt solver in matlab that uses finite difference code to obtain the approximate solution. numerical method in solving the current problem will enable us investigate the effects of different parameter deeply asbvp4c is able to obtain the results will ease for all the parameters which is sometimes difficult to attain by analytic method depending upon the complexity of the problem. the results 9 krishnandan verma and sanjit basfor figure 1: geometric configuration will be demonstrated by graphs for various parameters representing the changes in velocity, temperature and concentration of the fluid. 2 construction of the problem consider a stationary impenetrable vertical plate in a binary fluid mixture in a semi-infinite porous medium in a two-dimensional system. the x-axis is alongside the plate and y-axis perpendicular to it. the flow is steady and the fluid is newtonian. an unvarying magnetic field of strength h0 is exerted in a direction normal to the plate. let tw be the constant wall temperature of the plate whereas t∞ is the ambient fluid temperature and tw > t∞. let c be the light weight or the fewer ingredient’s concentration of the binary fluid close to the surface and c∞ outlying the plate. the governing equation of the problem together with boussinesq approximations are given by: ∂u ∂x + ∂v ∂y = 0, (2) u ∂u ∂x +v ∂u ∂y = υ ∂2u ∂y2 − υ k (1+ kh20σµe 2 µ )u+gβt(t−t∞)+gβc(c−c∞), (3) u ∂t ∂x + v ∂t ∂y = α ∂2t ∂y2 + dkt cscp ∂2c ∂y2 + 1 ρ q cp (t −t∞)− 1 cpρ ∂qr ∂y , (4) u ∂c ∂x + v ∂c ∂y = d ∂2c ∂y2 + dkt tm ∂2t ∂y2 −k1(c −c∞). (5) 10 inf. ch. react., rad. and h. source on sep. bfm sr and df effects the border restrictions are u = 0,v = 0,t = tw, ∂c ∂y + kt tm ∂t ∂y at y = 0 (6) u → u∞,t → t∞,c → c∞ as y →∞ (7) where υ,g,βt ,βc,k,µe, σ,t,α,d,kt ,cs,cp,ρ,q,c,tm and k1 represent kinematic viscosity, coefficient of thermal expansion, coefficient of concentration expansion, permeability, magnetic permeability, electrical conductivity, fluid temperature, thermal diffusivity, mass diffusion rate, heat diffusion ratio, concentration susceptibility, specific heat at constant pressure, density, heat source, concentration of the rarer constituent, mean fluid temperature and rate of chemical reaction respectively. heat flow by radiation is expressed as, qr = − 4σ∗ 3k∗ ∂t4 ∂y (8) where t4 = 4t3∞t −3t4∞ introducing stream function ψ given by u = ∂ψ ∂y and v = ∂ψ ∂x , it is found that equation (2) is automatically satisfied. the dimensionless variables given below are introduced to reduce equations (2)(5) to non-dimensional form: η = y √ u∞ xυ , ψ = √ υxu∞f(η),θ(η) = t−t∞ tw−t∞ ,φ(η) = c−c∞ cw−c∞ the non-dimensional form of (2)-(5) are f ′′′ + 1 2 ff ′ + λ1θ + λ2φ− 1 dare (1 + m2)f ′ = 0, (9) (1 + 4 3 r)θ′′ + prfθ′ + dfprφ ′′ + prδθ = 0, (10) φ′′ + 1 2 scfφ′+ srscθ′′ −sckcφ = 0. (11) where, gr = gβt (tw−t∞)x2 υ2 ,gm = gβcc∞x 2 υ2 , λ1 = gr re2 , λ2 = gm re2 , da = k x2 , re = u∞x υ , pr = υ α , sc = υ d , sr = dkt (tw−t∞) υtm(cw−c∞) , df = dkt (cw−c∞) υcscp(tw−t∞) , m = (kσµe 2h0 2 υ )2, kc = xk1 u∞ , r = 4σ ∗t3∞ k∗k and δ = qx u∞ represents local grashof number, local modified grashof number, temperature buoyancy parameter, concentration buoyancy parameter, local darcy number, local reynolds number, prandtl number, schmidt number, soret number, dufour number, magnetic parameter, chemical reaction 11 krishnandan verma and sanjit basfor parameter, radiation parameter and heat generation parameter respectively. the dimensionless border restrictions are f = 0,f ′ = 0,θ = 1,φ′ + scsrθ′ = 0,at η = 0, (12) f ′ → 1,θ → 0,φ → 0 as η →∞ (13) local skin-friction coefficient, nusselt number and sherwood number are equivalent to f ′′(0),−θ′(0) and φ′(0) respectively in the current problem and their numerical values are calculated to analyse the heat and mass transfer process. 3 results and discussion matlab inbuilt solver bvp4c has been used to solve equations (9)-(11) with boundary restrictions (12) and (13). numerical results are obtained for various parameters in the form of graph and tables using matlab solver bvp4c. for computational purpose, the parameters are assigned the given values sr = 0.4,sc = 0.22,pr = 0.71,λ1 = 1,λ2 = 0.1,da = 0.5,re = 400,m = 1,df = 0.15,r = 0.5,δ = 0.2 and kc = 0.2. numerical data of f ′′(0),−θ′(0) and φ′(0) are obtained for sr,df and m to gain some insight of skin friction coefficient, thermal and mass transmission rate. it is observed from fig. 2 and fig. 3 that the velocity and temperature of the fluid decrease for larger values of soret number. soret number depends directly upon (tw −t∞) and tw > t∞. therefore, increase in soret number increases the temperature difference and hence heat diffuses away from the sheet which results in lowering of fluid temperature. from fig. 4, it is spotted that the hike in soret number leads to accumulation of larger concentration of the rarer constituent nearby the plate. the gathering of the light weight ingredient of the binary fluid is high in between 0.5 ≤ η ≤ 3, so soret effect contributes highly in the dissociation of the binary ingredients of the fluid by limiting the temperature and increasing the assemblage of the rarer constituent alongside the plate. it is spotted from fig. 5 and fig. 6 that velocity as well as temperature decreases with the growth in the values of dufour number. fig. 7 disclose us that the hike in dufour number decreases the concentration of the rarer constituent but the figure also informs us that the rarer constituent is accumulated near the surface i.e., 1 ≤ η ≤ 3.5 thereby throwing the denser component away from surface of the plate. thus dufour number helps in the dissociation of the binary ingredients of the fluid. the action of heat generation parameter on fluid velocity is illustrated in fig. 8. it is perceived that the rise in heat generation parameter augments the fluid velocity. heat generation parameter provides heat in the flow field because of which the 12 inf. ch. react., rad. and h. source on sep. bfm sr and df effects temperature increases as seen in the fig. 9. due to addition of heat by heat generation effect, the molecules of the fluid get excited thereby increasing the velocity of the binary fluid mixture. fig. 10 reveals us that the rise in heat generation parameter raises the concentration of the lightweight constituent but the increase is quite negligible. in fig. 11 and fig. 12. it is clearly perceived that temperature and concentration rises with the hike in radiation parameter. the rarer constituent of the fluid are concentrated near the plate i.e.,1 ≤ η ≤ 4, thus helping in the separation process. the velocity and temperature profiles due to chemical reaction parameter, kc are illustrated in fig. 13 and fig. 14. there is a minor decrement in fluid velocity and temperature with the hike in the values of kc. from fig. 15, it is perceived that the concentration of the lightweight ingredient of the fluid hikes with the growth in the values of kc. the rarer constituent is concentrated near the plate i.e.,1 ≤ η ≤ 4.5, thereby throwing the denser component far away from the plate utimately aiding the separation process. figure 2: distribution of velocity due to sr 4 conclusions the prime motive of the current study is to analyse soret and dufour effectsalong with different parameters on the dissociation of the binary fluid mixture. industrial importance of separation of species of fluid mixture e.g., isotopes of heavy water that is present in a negligible amount in sea is very useful in nuclear reactor and so many researchers have reported several works on dissociation of 13 krishnandan verma and sanjit basfor figure 3: changes in temperature as a consequence of sr figure 4: changes in concentration as a consequence of sr 14 inf. ch. react., rad. and h. source on sep. bfm sr and df effects figure 5: distribution velocity due to df figure 6: distribution temperature due to df 15 krishnandan verma and sanjit basfor figure 7: distribution concentration due to df figure 8: distribution velocity due to δ 16 inf. ch. react., rad. and h. source on sep. bfm sr and df effects figure 9: distribution temperature due to δ figure 10: distribution concentration due to δ 17 krishnandan verma and sanjit basfor figure 11: changes in temperature as a consequence of r figure 12: changes in concentration as a consequence of r 18 inf. ch. react., rad. and h. source on sep. bfm sr and df effects figure 13: distribution velocity due to kc figure 14: distribution temperature due to kc 19 krishnandan verma and sanjit basfor figure 15: distribution concentration due to kc table 1: numerical data of f ′′(0),−θ′(0) and φ′(0) for sr,df and m when sc = 0.22,pr = 0.71,λ1 = 1,λ2 = 0.1,da = 0.5,re = 400,r = 0.5,δ = 0.2 and kc = 0.2. sr df m f ′′(0) −θ′(0) φ′(0) 0.4 0.15 1 1.7398 0.5946 0.0523 3 0.15 1 1.7885 0.6215 0.4102 6 0.15 1 1.8400 0.6542 0.8638 9 0.15 1 1.9035 0.6880 1.3622 0.4 0.15 1 1.7398 0.5936 0.0523 0.4 3 1 1.6073 0.6257 0.0551 0.4 6 1 1.4951 0.6742 0.0593 0.4 10 1 1.2346 0.7298 0.0634 0.4 0.15 1 1.7398 0.5946 0.0523 0.4 0.15 4 1.7364 0.5950 0.0524 0.4 0.15 7 1.7304 0.5960 0.0524 0.4 0.15 10 1.7245 0.5977 0.0526 20 inf. ch. react., rad. and h. source on sep. bfm sr and df effects the fluid species using many analytical methods that are complicated, lengthy and time-consuming method used to obtain approximate solution. numerical solution has provided a cost and time effective process to obtain the solution. the following problem can also be solved using other numerical method apart from bvp4c and can also be extended to different geometry like wedge, cone etc. the problem can also be extended using nanoparticles to see the necessary changes in heat transmission process which is an important future development of the current work. in the present work numerical solution presents some interesting results where different parametershelp in restricting the temperature and raising the rarer ingredient’s concentrationnearby the plate that are highlighted below: 1. heat generation parameter increases the fluid velocity while increase in soret number, dufour number and chemical reaction parameter decreases the fluid motion. 2. temperature of the fluid decreases for increasing values of soret number, dufour number and chemical reaction parameter while heat generation and radiation parameter enhance the temperature of the fluid. 3. all the parameters help in raising the concentration of the lightweight ingredient of the binary fluid mixture near the plate and thus helping in the separation process by throwing the denser and heavier component away from the plate. 4. soret number, dufour number and magnetic parameter boost the heat transfer as well as the mass transfer rate of the fluid. soret number increases the skin friction coefficient 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2yildiz technical university, department of mathematics, istanbul, turkey ssanemy@gmail.com 3yildiz technical university, department of mathematics, istanbul, turkey ersoya@gmail.com received on: 14-04-2017. accepted on: 28-06-2017. published on: 30-06-2017 doi:10.23755/rm.v32i0.325 c©serkan onar et al. abstract in this paper, the definitions of soft γ -module, soft γ module homomorphism and soft γ -exactness are introduced with the aid of the concept of soft set theory introduced by molodtsov. in the meantime, some of their properties and structural characteristics are investigated and discussed. thereafter, several illustrative examples are given. keywords: soft set; soft module; γring; γ -module; soft γmodule; soft γ -module homomorphism; soft γ -module isomorphism; soft γ -exactness. 2010 ams subject classifications: 03e72; 08a72. 45 s. onar, s. yavuz and b. a. ersoy 1 introduction in the real world, there are some various uncertainties but classical mathematical tools is not convenient for modeling these. uncertain and unclear data which are contained by economy, engineering, environmental science, social science, medical science, business administration and many other fields are common. although many diverse theories such as probability theory, soft set theory, intuitionistic fuzzy soft set theory and rough set theory are known and these present advantageous mathematical approaches for modeling of uncertainties, each of these theories have their inherent diffuculties. in 1999, molodstov [1] developed soft set theory which is considered a mathematical tool for working with uncertainties. since the emergence of soft set theory attracts attention and especially recently works on the soft set theory is progressing rapidly. maji et al. [2] described some operations on soft sets and these operations are used soft sets of decision making problems. chen et al. [3] offered a new definition for decrease of parametrerization on soft sets. they made comparasion between this definition and concept of restriction of property in the rough set theory. in theory, maji et al. [4] worked various operator on soft set. kong et al. [5] developed definition of parametrerization reduction on soft set. zou and xiao suggested some approach of data analysis in case of insufficent information on soft set. jiang et al. presented a unique approach of the semantic decision making by means of ontological thinking and ontology-based soft sets. besides studies on classic module theory have continued and interesting results have been discovered recently. macias diaz et al. [6] studied on modules which are isomorphic to relatively divisible or pure submodules of each other. abuhlail et al. [7] presented on topological lattices and their applications to module theory. on the other hand, ameri et al. [8] investigated gamma module and davvaz et al. [9] studied tensor product of gamma modules. as for soft module theory, sun et al. [10] presented the notion of soft set and soft module. xiang [11] worked soft module theory. t.shah et al. [12] defined the notion of primary decomposition in a soft ring and soft module, and derived some related properties. erami et al. [13] gave the concept of a soft mvmodule and soft mvsubmodule. in these days, there are some studies reletad with soft sets. ali et al. [14] investigated some new operations in soft set theory and pei et al. [15] studied from soft sets to information systems. xiao et al. [16] presented research on synthetically evaluating method for business competitive capacity based on soft set. aktaş et al. [17] showed soft sets and soft groups and acar et al. [18] also showed soft sets and soft rings. the main purpose of this paper is to deal with algebraic structure of γ− module by applying soft set theory. the concept of soft γ− module is introduced, their characterization and algebraic properties are investigated by giving some several 46 soft γmodules examples. in addition to this, soft γ− homomorphism , soft γ− isomorphism and their properties are introduced. after all, we make inferences that images of soft γ− homomorphisms and inverse images of soft γ− homomorphisms are soft γ− homomorphisms. furthermore soft γ− exactness is investigated and illustrated with a related example. 2 preliminaries in this section, preliminary informations will be required to soft γ− modules. first of all we give basic concepts of soft set theory. definition 2.1. [18] let x denotes an initial universe set and e is a set of parameters. the power set of x is denoted by p (x). a pair of (f,e) is called a soft set over x if and only if f is a mapping from e into the set of all subsets of x, i.e, f : e → p (x). definition 2.2. [18] let (f,a) and (g,b) be two soft sets over a common universe x. i) if a⊆̃ b and f (a)⊆̃ g (a) for all a ∈ a then we say that (f,a) is a soft subset of (g,b), denoted by (f,a) ⊆̃(g,b). ii) if (f,a) is a soft subset of (g,b) and (g,b) is a soft subset of (f,a), then we say that (f,a) is a soft equal to (g,b), denoted by (f,a) =̃ (g,b) . example 2.1. let x = m2(z3) denotes an initial universe set, i.e, 2 × 2 matrices with z3 terms and e = { [ 0 0 0 0 ] , [ 1 0 0 1 ] }is a set of parameters. then f : e → p(x) where f( [ 0 0 0 0 ] ) = { [ 0 1 1 1 ] , [ 2 1 0 2 ] },f( [ 1 0 0 1 ] ) = { [ 2 0 0 2 ] }. clearly,(f,e) is called a soft set over x. definition 2.3. [18] let (f,a) and (g,b) be two soft sets over a common universe x. the intersection of (f,a) and (g,b) is defined as the soft set (h,c) satisfying the following conditions: i) c = a∩b. ii) for all c ∈ c, h (c) = f (c) or g (c) . in this case, we write (f,a)∩̃(g,b) = (h,c) . definition 2.4. [18] let (f,a) and (g,b) be two soft sets over a common universe x. the union of (f,a) and (g,b) is defined as the soft set (h,c) satisfying the following conditions: i) c = a∪b. 47 s. onar, s. yavuz and b. a. ersoy ii) for all c ∈ c, h (c) =   f(c) if c ∈ a−b, g(c) if c ∈ b −a, f(c) ∪g(c) if c ∈ a∩b.   this is denoted by (f,a)∪̃(g,b) = (h,c) . definition 2.5. [18] if (f,a) and (g,b) are two soft sets over a common universe x, then (f,a) and (g,b) denoted by (f,a)∧̃(g,b) is defined as (f,a)∧̃(g,b) = (h,c), where c = a × b and h (x,y) = f (x) ∩ g (y), for all (x,y) ∈ c. definition 2.6. let {(fi,ai) : i ∈ i} be a nonempty family soft sets. the ∧−intersection of a non-empty family soft sets is defined by (ψ,y ) = ∧̃i∈i (fi,ai) where (ψ,y ) is a soft set, y = ∏ i∈i ai and ψ(y) = ∩i∈ifi(y) for every y = (yi)i∈i ∈ y. definition 2.7. [18] if (f,a) and (g,b) are two soft sets over a common universe x, then (f,a) or (g,b) denoted by (f,a)∨̃(g,b) is defined as (f,a)∨̃(g,b) = (h,c), where c = a×b and h (x,y) = f (x) ∪g (y), for all (x,y) ∈ c. definition 2.8. let {(fi,ai) : i ∈ i} be a nonempty family soft sets. the ∨−union of a non-empty family soft sets is defined by (ψ,y ) = ∨̃i∈i (fi,ai) where (ψ,y ) is a soft set, y = ∏ i∈i ai and ψ(y) = ∪i∈ifi(y) for every y = (yi)i∈i ∈ y. on the other hand we will introduce modules and soft modules, then we will study some properties and theories of soft modules such as trivial soft module, whole soft module, the concepts of soft submodule and soft module homomorphisms. definition 2.9. [10] let r be a ring with identity. m is said to be a left rmodule if left scalar multiplication λ : r × m → m via (a,x) 7→ ax satisfying the axioms ∀r,r1,r2, 1 ∈ r; m,m1,m2 ∈ m : i) m is an abelian group, ii) r(m1 + m2) = rm1 + rm2, (r1 + r2)m = r1m + r2m, iii) (r1r2)m = r1(r2m), iv) 1m = m. left r−module is denoted by rm or m for short. similarly we can define right rmodule and denote it by mr. 48 soft γmodules example 2.2. let r = m2(z) and m = { [ a b ] |a,b ∈ z}. then m is module on r. definition 2.10. [10] let m be a left rmodule, a be a any nonempty set and (f,a) is a soft set over m. (f,a) is said to be a soft module over m if and only if f (x) is submodule over m, for all x ∈ a. definition 2.11. [10] let (f,a) be a soft module over m then i) (f,a) is said to be a trivial soft module over m if f(x) = 0 for all x ∈ a,where 0 is zero element of m. ii) (f,a) is said to be an whole soft module over m if f(x) = m for all x ∈ a. proposition 2.1. [10] let (f,a) and (g,b) be two soft modules over m. 1) (f,a)∩̃(g,b) is a soft module over m. 2) (f,a)∪̃(g,b) is a soft module over m if a∩b = ∅. definition 2.12. [10] if (f,a) and (g,b) be two soft modules over m, then (f,a) + (g,b) is defined as (h,a×b), where h (x,y) = f (x) + g (y) for all (x,y) ∈ a×b. proposition 2.2. [10] assume that (f,a) and (g,b) are two soft modules over m.then (f,a) + (g,b) is soft module over m. definition 2.13. [10] suppose that (f,a) and (g,b) be two soft modules over m and n respectively. then (f,a)×(g,b) = (h,a×b) is defined as h(x,y) = f(x) ×g(y) for all (x,y) ∈ a×b. proposition 2.3. [10] let (f,a) and (g,b) be two soft modules over m and n respectively. then (f,a) × (g,b) is soft module over m ×n. definition 2.14. [10] let (f,a) and (g,b) be two soft modules over m.then (g,b) is soft submodule of (f,a) if i) b ⊂ a, ii) g(x) < f(x),∀ x ∈ b. this is denoted by (g,b)<̃(f,a). proposition 2.4. [10] let (f,a) and (g,b) be two soft modules over m.we say that (g,b) is soft submodule of (f,a) if g(x) ⊆ f(x),∀x ∈ a. definition 2.15. [10] assume that e = {e}, where e is unit of a.then every soft module (f,a) over m at least have two soft modules (f,a) and (f,e) called trivial soft submodule. 49 s. onar, s. yavuz and b. a. ersoy proposition 2.5. [10] let (f,a) and (g,b) are two soft modules over m and (g,b) is soft submodule of (f,a). if f : m → n is a homomorphism of module, then (f(f),a) and (f(g),b) are all soft modules over n and (f(g),b) is soft submodule of (f(f),a). definition 2.16. [10] let (f,a) and (g,b) be two soft modules over m and n respectively, f : m → n,g : a → b be two functions. then we say that (f,g) is a soft homomorphism if the following conditions are satisfied: i) f : m → n is a homomorphism of module, ii) g : a → b is a mapping, iii) for all x ∈ a, f(f(x)) = g(g(x)). we say that (f,a) is a soft homomorphic to (g,b) which denoted by (f,a)−̃(g,b). in this definition, if f is an isomorphism from m to n and g is a one-to-one mapping from a onto b, then we say that (f,a) is a soft isomorphism and that (f,a) is a soft isomorphic to (g,b), this is denoted by (f,a) =̃ (g,b) . finally, we will define γring and γmodule and their homomorphisms which are basic definitions for soft γmodule. definition 2.17. [8] let r and γ be additive abelian groups. then we say that r is a γring if there exists a mapping: . : r× γ ×r → r (r1,γ,r2) → r1γr2 such that for every a,b,c ∈ r and α,β ∈ γ,the following hold: i) (a + b)αc = aαc + bαc, ii) a(α + β)c = aαc + aβc, iii) aα(b + c) = aαb + aαc, iv) (aαb)βc = aα(bβc). definition 2.18. [8] a subset a of a γring r is said to be a right ideal of r if a is an additive subgroup of r and aγr ⊆ a,where aγr = {aαc| a ∈ a,α ∈ γ,r ∈ r}. a left ideal of r is defined in a similar way. if a is both right and left ideal, we say that a is an ideal of r. definition 2.19. [8] if r and s are γrings, then a pair (θ,ϕ) of maps from r into s is called a homomorphism from r into s if i) θ(x + y) = θ(x) + θ(y), ii) ϕ is an isomorphism on γ, iii) θ(xγy) = θ(x)ϕ(γ)θ(y). definition 2.20. [8] let r be a γring. a left γmodule r is an additive abelian group m together with a mapping . : r × γ × m → m such that for all m,m1,m2 ∈ m and γ,γ1,γ2 ∈ γ,r,r1,r2 ∈ r the following hold: 50 soft γmodules i) rγ(m1 + m2) = rγm1 + rγm2, ii) (r1 + r2)γm = r1γm + r2γm, iii) r(γ1 + γ2)m = rγ1m + rγ2m, iv) r1γ1(r2γ2m) = (r1γ1r2)γ2m. a right γ module r is defined in analogous manner. example 2.3. let r = { [ k m ] |k,m ∈ z2}, i.e, 1 × 2 matrices and γ = { [ 0 0 ] , [ 1 0 ] } ∈ z2, where γis 2 × 1 matrices. then we say that r is a γring. similarly, r and γ are same if we choose m = { [ 0 0 ] , [ 1 1 ] }, then m is γ module r. definition 2.21. [8] presume that (m, +) be an γ module r . a nonempty subset n of (m, +) is said to be a left γ submodule r of m if n is a subgroup of m and rγn ⊆ n,where rγn = {rγn |γ ∈ γ,r ∈ r,n ∈ n}, that is for all n1,n2 ∈ n and for all γ ∈ γ,r ∈ r; n1 −n2 ∈ n and rγn ∈ n. in this case we write n ≤ m. example 2.4. in previous example, let n = { [ 0 0 ] }⊂ m and h : n → p(m) be a set valued function defined by h(a) = {b ∈ m|r(a,α,b) ⇔ aαb ∈ [ 0 0 ] } for all a ∈ n.h is clear that h( [ 0 0 ] ) = ( [ 0 0 ] ) is γ submodule r of m.. definition 2.22. [8] let m and n be arbitrary γ module r . a mapping f : m → n is a homomorphism of γ module r if for all x,y ∈ m and ∀r ∈ r,∀γ ∈ γ we have i) f(x + y) = f(x) + f(y), ii) f(rγx) = rγf(x). a homomorphism f is monomorphism if f is one-to-one and f is epimorphism if f is onto. f is called isomorphism if f is both monomorphism and epimorphism. we denote the set of all rγhomomorphisms from m into n by homrγ (m,n) or shortly by homrγ (m,n). in particular m = n we denote hom(m,m) by end(m). definition 2.23. [18] let m be a nonempty set and a γ−module. the pair (f,a) is a soft set over m. the set supp(f,a) = {x ∈ a : f(x) 6= ∅} is called a support of the soft set (f,a). the soft set (f,a) is non-null if supp(f,a) 6= ∅. 3 soft γmodules in this section, firstly we will define soft γ− modules, then we will give some operations on this modules.throughout the section, m is a γ−module. 51 s. onar, s. yavuz and b. a. ersoy definition 3.1. let (f,a) be a non-null soft set over m. then, (f,a) is said to be a soft γ−module over m if f(a) is a γ−submodule m such that f : a → p(m), (i.e. a → f(a)) for all a ∈ a,y ∈ supp(f,a). example 3.1. for consider the additively abelian groups z6 = {0, 1, 2, 3, 4, 5} and γ = {0, 2}. let . : z6 ×γ×z6 → z6, (m1, γ,m2) = m1γm2. hence z6 is a γ− module. let a = z6 and f : a → p(m) be a set valued function defined by f(0) = f(2) = f(4) = z6, f(1) = f(3) = f(5) = {0, 3} are γ−submodule of z6. hence (f,a, ) is a soft γ−module over z6. example 3.2. let m is a γ− module and (f,a) be a soft set over m.f : a → p(m) is defined by f(x) = {y ∈ m| xαy = 0} for all x ∈ a,α ∈ γ. it is clear that (f,a) is a soft γ− module. example 3.3. for consider the additively abelian groups m = r = { [ 0 0 ] , [ 1 0 ] , [ 0 1 ] , [ 1 1 ] }⊆ (z2)1×2 and γ = { [ 0 0 ] , [ 1 0 ] }⊆ (z2)2×1 with addition defined as matrice addition. it is trivial that r is a γ− ring. also m is a γ− module over r. let n = { [ 0 0 ] } ⊆ m and h : n → p(m) be a set valued function defined by h(a) = {b ∈ m| r(a,α,b) ↔ aαb ∈ [ 0 0 ] ,∀α ∈ γ} for all a ∈ n. it is clear that h( [ 0 0 ] ) = { [ 0 0 ] } are sub γ− module of m. hence (h,n) is soft γ− module of m. theorem 3.1. let (f,a) and (g,b) are two soft γ−modules over m. then (f,a)∩̃(g,b) is a soft γ−module over m if it is non-null. proof. by definition, we have that (f,a)∩̃(g,b) = (h,c) where h(c) = f(x) ∩g(y) for all c ∈ c. we assume that (h,c) is a non-null soft set over m. if c ∈ supp(h,c), then h(c) = f(x)∩g(y) 6= ∅. we know that (f,a) , (g,b) are both soft γ−module over m, and so, the nonempty sets f(x) and g(y) are both γ−submodule over m. thus, h(c) is a γ−submodule over m for all c ∈ supp(h,c). in this position, (h,c) = (f,a)∩̃(g,b) is a soft γ−module over m. 2 theorem 3.2. let (f,a) and (g,b) are two soft γ−modules over m. then (f,a)∪̃(g,b) is a soft γ−module over m if a∩b = ∅. 52 soft γmodules proof. by definition, we have that (f,a)∪̃(g,b) = (h,c) where h(c) = f(x) ∩ g(y) for all c ∈ c. note first that (h,c) is a non-null owing to the fact that supp(h,c) = supp (f,a)∪̃(g,b) . suppose that c ∈ supp(h,c).then h(c) 6= ∅ so we have f(x),g(y) 6= ∅. from the hypothesis a∩b = ∅, we follow that h(c) = f(x)∩g(y). on the other hand f(x)∩g(y) is a soft γ−module over m, we conclude that (h,c) is a soft γ−module over m for all c ∈ supp(h,c). consequently (f,a)∪̃(g,b) = (h,c) is a soft γ−module over m. 2 on the other hand, union of two soft γ− modules is not always soft γ− module. we will explain this situation with following example. example 3.4. let m = z6 = {0, 1, 2, 3, 4, 5} is a mγ−module, γ = {0, 1},a = z2 = {0, 1} and b = z3 = {0, 1, 2} such that f(0) = f(1) = {0, 2, 4},g(0) = g(1) = g(2) = {0, 3}a∩b = {0, 1}. if this condition is hold, then (f,a)∪̃(g,b) is not a soft γ−module over m. indeed, h(1) = {0, 2, 3, 4} /∈ p(m). definition 3.2. if (f,a) and (g,b) are two soft γ−modules over m, then (f,a) and (g,b) denoted by (f,a)∧̃(g,b) is defined as (f,a)∧̃(g,b) = (h,c), where c = a×b and h (x,y) = f (x)∩̃g (y), for all (x,y) ∈ c. theorem 3.3. suppose that (f,a) and (g,b) are two soft γ−modules over m. then (f,a)∧̃(g,b) is soft γ−module over m if it is non-null. proof. using definition, we have that (f,a)∧̃(g,b) = (h,c) where c = a×b and h (x,y) = f (x)∩̃g (y), for all (x,y) ∈ c. then the hypothesis, (h,c) is a non-null soft set over m. since (h,c) is a non-null, supp (h,c) 6= ∅ and so, for (x,y) ∈ supp (h,c) ,h (x,y) = f (x)∩̃g (y) 6= ∅. we assume that t1, t2 ∈ f (x)∩̃g (y) . in this position i) if t1, t2 ∈ f (x) = {y : r(x,y)} we have that xt1 ∈ a,xt2 ∈ a. this implies that x(t1 + t2) ∈ a. ii) if t1, t2 ∈ g (y) = {y1 : r(y,y1)} we have that yt1 ∈ b, yt2 ∈ b. this implies that y(t1 + t2) ∈ b. hence f (x)∩̃g (y) is a γ− submodule. by the definition of soft γ− module, (f,a) and (g,b) are soft γ−modules over m. f (x) ,g (y) are also γ− submodule over m. furthermore h (x,y) = f (x)∩̃g (y) is a γ− submodule over m for all (x,y) ∈ (h,c) = (f,a)∧̃(g,b) . hence (f,a)∧̃(g,b) is soft γ−module over m. 2 definition 3.3. if (f,a) and (g,b) are two soft γ−modules over m, then (f,a) or (g,b) denoted by (f,a)∨̃(g,b) is defined as (f,a)∨̃(g,b) = (h,c), where c = a×b and h (x,y) = f (x)∪̃g (y), for all (x,y) ∈ c. theorem 3.4. suppose that (f,a) and (g,b) are two soft γ−modules over m. then (f,a)∨̃(g,b) is soft γ−module over m. 53 s. onar, s. yavuz and b. a. ersoy proof. using definition, we have that (f,a)∨̃(g,b) = (h,c), where c = a × b and h (x,y) = f (x)∪̃g (y), for all (x,y) ∈ c. assume that c ∈ supp(h,c). then h(c) 6= ∅ and so we have that f(x) 6= ∅,g(y) 6= ∅. by assumption, f (x)∪̃g (y) is a soft γ− module of m for all c ∈ supp(h,c). consequently (f,a)∨̃(g,b) = (h,c) is a soft γ−module over m. 2 definition 3.4. let (f,a) and (g,b) are two soft γ−modules over m. then (f,a) +̃ (g,b) = (h,a × b) is defined as h(x,y) = f(x) + g(y) for all (x,y) ∈ a×b. theorem 3.5. suppose that (f,a) and (g,b) are two soft γ−modules over m. then (f,a) +̃ (g,b) is soft γ−module over m. proof. by the definition we write (f,a) +̃ (g,b) = (h,a × b) and h(x,y) = f(x) + g(y) for all (x,y) ∈ a × b. let (x,y) ∈ supp(h,a × b).then, h(x,y) 6= ∅ and so we have f(x) 6= ∅, g(y) 6= ∅. by taking into account, (f,a) and (g,b) are two soft γ−modules over m, it follows that f(x) + g(y) is a soft γ−module over m for all (x,y) ∈ supp(h,a × b). hence (f,a) +̃ (g,b) is soft γ−module over m. 2 definition 3.5. let (f,a) and (g,b) are two soft γ−modules over m. then (f,a)×̃(g,b) = (h,a × b) is defined as h(x,y) = f(x) × g(y) for all (x,y) ∈ a×b. theorem 3.6. suppose that (f,a) and (g,b) are two soft γ−modules over m. then (f,a)×̃(g,b) is soft γ−module over m. proof. by the definition we write (f,a)×̃(g,b) = (h,a × b) and h(x,y) = f(x) × g(y) for all (x,y) ∈ a × b. let (x,y) ∈ supp(h,a× b). then, h(x,y) 6= ∅ and so we have f(x) 6= ∅, g(y) 6= ∅. by taking into account, (f,a) and (g,b) are two soft γ−modules over m, it follows that f(x)×g(y) is a soft γ−module over m for all (x,y) ∈ supp(h,a×b). hence (f,a)×̃(g,b) is soft γ−module over m. 2 definition 3.6. let (f,a) and (g,b) are two soft γ−modules over m. then (g,b) is called a soft γ−submodule of (f,a) if i) b ⊆ a, ii) ∀b ∈ supp(g,b),g(b) is a γ−submodule of f (b) . this denoted by (g,b) ⊂ (f,a) . from the definition, it is easily deduced that if (g,b) is a soft γ−submodule of (f,a) , then supp(g,b) ⊂ supp(f,a). theorem 3.7. let (f,a) and (g,b) be two soft γ−modules over m and (f,a)⊆̃ (g,b) . then (g,b) ⊂ (f,a) . 54 soft γmodules proof. straight forward. 2 corolary 3.1. let (f,a) be a soft γ−module over m and {(fi,ai) : i ∈ i} be a nonempty family of soft γ−submodules of (f,a) .then, i) ∩̃i∈i (fi,ai) is a soft γ−submodule of (f,a) if it is non-null. ii) ∪̃i∈i (fi,ai) is a soft γ−submodule of (f,a) , if ai∩aj = ∅ for all i,j ∈ i and if it is non-null. iii) if fi(ai) ⊆ fj(aj) or fj(aj) ⊆ fi(ai) for all i,j ∈ i,ai ∈ ai, then ∨̃i∈i (fi,ai) is a soft γ−submodule of ∨̃i∈i (f,a) . iv) ∧̃i∈i (fi,ai) is a soft γ−submodule of ∧̃i∈i (f,a) . v) the cartesian product of the family ∏̃ i∈i (fi,ai) is a soft γ−submodule of∏̃ i∈i (f,a) . vi) ∑̃ i∈i (fi,ai) is a soft γ−submodule of ∑̃ i∈i (f,a) . proof. similar to the proof of theorems 3.5, 3.6, 3.9, 3.11, 3.13 and 3.15. 2 4 soft γ− module homomorphism in this section, firstly we will define trivial and whole soft γ−modules over γ−module m, homomorphism of γ−modules and their properties. moreover we will study soft γ−module homomorphism and soft γ−module isomorphism. throughout the section, m is a γ−module. definition 4.1. let (ρ,a) and (σ,b) be two soft γ−modules over γ−module m and γ−module m1 respectively. let f : m → m1 and g : a → b be two functions. the following conditions: i) f is an epimorphism of γ−module, ii) g is a surjective mapping, iii) f(ρ(y)) = σ(ρ(y)) for all y ∈ a, were satisfied by the pair (f,g), then (f,g) is called soft γ−module homomorphism. if there exists a soft γ−module homomorphism between (ρ,a) and (σ,b), we say that (ρ,a) is soft homomorphic to (σ,b), and is denoted by (ρ,a) ∼ (σ,b). if there exists a soft γ−module isomorphism between (ρ,a) and (σ,b), we say that(ρ,a) is soft isomorphic to (σ,b), and is denoted by (ρ,a)−̃(σ,b). definition 4.2. let (f,a) be soft γ−module over m. i) (f,a) is called the trivial soft γ−module over m if f(a) = {0} for all a ∈ a. 55 s. onar, s. yavuz and b. a. ersoy ii) (f,a) is called the whole soft γ−module over m if f(a) = m for all a ∈ a. definition 4.3. let m and m1 be two γ−modules and m : m → m1 a mapping of γ−module. if (f,a) and (g,b) are soft sets over m and m1 respectively, then i) (m(f),a) is a soft set over m1 where m(f) : a → p(m1), m(f)(a) = m(f(a)) for all a ∈ a. ii) (m−1(g),b) is a soft set over m where m−1(g) : b → p(m),m−1(g)(b) = m−1(g(b)) for all b ∈ b. corolary 4.1. let m : m → m1 be an onto homomorphism of γ−module. then following statements can be given. i) (f,a) be soft γ−module over m, then (m(f),a) is a soft γ−module over γ−module m1. ii) (g,b) be soft γ−module over γ−module m1, then (m−1(g),b) is a soft γ−module over m. proof. i) since (f,a) is a soft γ−module over m, it is clear that (m(f),a) is a non-null soft set over m1. for every y ∈ supp(m(f),a) we have m(f)(y) = m(f(y)) 6= ∅. hence m(f(y)) which is the onto homomorphic image of γ−module f(y) is a γ−module of m1 for all y ∈ supp(f(m),a). that is (m(f),a) is a soft γ−module over γ−module m1. ii) it is easy to see that supp(m−1(g),b) ⊆ supp(g,b). by this way let y ∈ supp(m−1(g),b).then, g(y) 6= ∅. hence m−1(g(y)) which is homomorphic inverse image of γ−module g(y), is a soft γ−module over m for all y ∈ b. 2 theorem 4.1. let m : m → m1 be a homomorphism of γ−module and (f,a), (g,b) be two soft γ−modules over γ−module m and γ−module m1 respectively. then following statements can be given. i) if f(a) = ker (m) for all a ∈ a,then (m(f),a) is the trivial soft γ−module over m1. ii) if m is onto and (f,a) is whole, then (m(f),a) is the whole soft γ−module over m1. iii) if g(b) = m(m) for all b ∈ b,then (m−1(g),b) is the whole soft γ−module over m. iv) if m is injective and (g,b) is trivial, then (m−1(g),b) is the trivial soft γ−module over m. proof. i) by using f(a) = ker (m) for all a ∈ a. then m(f)(a) = m(f(a)) = {0m1} for all a ∈ a. hence (m(f),a) is soft γ−module over m1. 56 soft γmodules ii) suppose that m is onto and (f,a) is whole. then f(a) = m for all a ∈ a and so m(f)(a) = m(f(a)) = m(m) = m1 for all a ∈ a. hence (m(f),a) is whole soft γ−module over m1. iii) if we use hypothesis g(b) = m(m) for all b ∈ b, we can write m−1(g)(b) = m−1(g(b)) = m−1(m(m)) = m for all b ∈ b. it is clear that, (m−1(g),b) is the whole soft γ−module over m. iv) suppose that m is injective and (g,b) is trivial. then, g(b) = {0} for all b ∈ b,so m−1(g)(b) = m−1(g(b)) = m−1({0}) = ker m = {0m} for all b ∈ b. consequently, (m−1(g),b) is the trivial soft γ−module over m. 2 theorem 4.2. let m : m → m1 be a homomorphism of γ−module and (f,a), (g,b) be two soft γ−modules over m. if (g,b) is soft γ−submodule of (f,a), then (m(g),b) is soft γ−submodule of (m(f),a). proof. suppose that y ∈ supp (g,b). then y ∈ supp (f,a) .we know that b ⊆ a and g(y) is a γ−submodule f(y) for all y ∈ supp (g,b). from the expression hypothesis m is a homomorphism, m(g)(y) = m(g(y)) is a γ−submodule of m(f)(y) = m(f(y)) and therefore (m(g),b) is soft γ−submodule of (m(f),a). 2 theorem 4.3. let m : m → m1 be a homomorphism of γ−module and (f,a), (g,b) be two soft γ−modules over m. if (g,b) is soft γ−submodule of (f,a), then (m−1(g),b) is soft γ−submodule of (m−1(f),a). proof. let y ∈ supp(m−1(g),b). b ⊆ a and g(y) is a γ−submodule of f(y) for all y ∈ b. since m is a homomorphism, m−1(g)(y) = m−1(g(y)) is a γ−submodule of m−1(g(y)) = m(g)(y) for all y ∈ supp(m−1(g),b). hence (m−1(g),b) is soft γ−submodule of (m−1(f),a). 2 5 soft γ− exactness in this section, we will introduce maximal and minimal soft γ−submodules. then, we will investigate short exact and exact sequence of γ−modules. finally, we will explain soft γ−exactness and some their basic theories. throughout this section m is γ−module. definition 5.1. let (f,a) and (g,b) be two soft γ−modules over m and (g,b) be soft γ−submodule of (f,a) . we say (g,b) is maximal soft γ−submodule of (f,a) if g(x) is a maximal γ−submodule of f(x) for all x ∈ b. we say (g,b) is minimal soft γ−submodule of (f,a) if g(x) is a minimal γ−submodule of f(x) for all x ∈ b. 57 s. onar, s. yavuz and b. a. ersoy proposition 5.1. let (f,a) be a soft γ−module over m. i) if {(gi,bi) |i ∈ i} is a nonempty family of maximal soft γ−submodules of (f,a) , then ⋂ i∈i (gi,bi) is maximal soft γ−submodule of (f,a) . ii) if {(gi,bi) |i ∈ i} is a nonempty family of minimal soft γ−submodules of (f,a) , then ∑ i∈i (gi,bi) is minimal soft γ−submodule of (f,a) . proof. straight forward. 2 corolary 5.1. let (f,a) be a soft γ−module over m and f : m → n be a homomorphism if f(x) = ker f for all x ∈ a, then (f(f),a) is the rivial soft γ−module over n. similarly, let (f,a) be an whole soft γ−module over m and f : m → n be an epimorphism, then (f(f),a) is a whole soft γ−module over n. definition 5.2. the homomorphism sequence of γ−modules ... → mn−1 →fn−1 mn →fn mn+1 → ... is called exact sequence of γ−modules if imfn−1 = kerfn for all n ∈ n and we call the exact sequence of γ−modules form as 0 → m1 →f m →g m2 → 0 the short exact sequence of γ−modules. proposition 5.2. let (f,a) be a trivial soft γ−module over γ−module m1 and (g,b) be a whole soft γ−module over γ−module m2 if 0 → m1 →f→ m →g→ m2 → 0 is a short exact sequence, then 0 → f(x) →f̃ m →g̃→ g(y) → 0 is a short exact sequence for all x ∈ a,y ∈ b. proof. f(x) = 0,∀x ∈ a since (f,a) is a trivial soft γ−module over γ−module m1,so f̃ is a monomorphism. g(y) = m2,∀y ∈ b since (g,b) is a whole soft γ−module over γ−module m2.g : m → m2 is an epimorphism as 0 → m1 →f→ m →g→ m2 → 0 is a short exact sequence, so g̃ is an epimorphism. 2 proposition 5.3. let (f,a) be a trivial soft γ−module over γ−module m1 and (g,b) be a whole soft γ−module over γ−module m if 0 → m1 →f→ m →g→ m2 → 0 is a short exact sequence, then 0 → f(f)(x) →f̃ m →g̃ g(g)(y) → 0 is a short exact sequence for all x ∈ a,y ∈ b. proof. f(x) = 0,∀x ∈ a since (f,a) is a trivial soft γ−module over γ−module m1.kerf = 0, so kerf = f(x),∀x ∈ a,consequently (f(f),a) is trivial soft γ−module over m. (g,b) is a whole soft γ−module over m and g : m → m2 is an epimorphism, so (g(g),b) is a whole soft γ−module over m2, thus 0 → f(f)(x) →f̃ m →g̃ g(g)(y) → 0 is a short exact sequence for all x ∈ a,y ∈ b. 2 58 soft γmodules definition 5.3. let (f,a), (g,b) and (h,c) are three soft γ−modules over γ−modules m,n and k respectively. then we say soft γ− exactness at (g,b) , if the following conditions are satisfied: i) m →f1 n →f2 k is exact, ii) a →g1 b →g2 c is exact, iii) f1(f(x)) = g(g1(x)) for all x ∈ a, iv) f2(g(x)) = h(g2(x)) for all x ∈ b, which is denoted by (f,a) →(f1,g1) (g,b) →(f2,g2) (h,c). in this definition, if every (fi,ai), i ∈ i is soft γ− exact, then we say that (fi,ai)i∈i is soft γ− exact. proposition 5.4. let (f,a) and (g,b) are two soft γ−modules over γ−modules m and n respectively. if (f,a) →(f,g) (g,b) → 0 is soft γ− exact, then (f,g) is soft γ− homomorphism. in particular, if 0 → (f,a) →(f,g) (g,b) → 0 is soft γ− exact, then (f,g) is soft γ−isomorphism. proof. since (f,a) →(f,g) (g,b) → 0 is soft γ− exact, we have m →f n → 0 and a →g b → 0 are exact. thus f and g are epimorphisms, it is clear that (f,g) is homomorphism. if 0 → (f,a) →(f,g) (g,b) → 0 is soft γ− exact, then 0 → m →f n → 0 and 0 → a →g b → 0 are exact. thus f and g are isomorphisms, it is clear that (f,g) is soft γ−isomorphism. 2 definition 5.4. let m = 0 and a = 0, then (f,a) = 0. we call (f,a) is a zero-soft γ− module. proposition 5.5. let (f,a), (g,b) and (h,c) are three soft γ−modules over γ−modules m,n and k respectively. if (f,a) →(f1,g1) (g,b) →(f2,g2) (h,c) is soft γ− exact with f1,g1 epimorphism and f2,g2 monomorphism, then (g,b) is a zero-soft γ− module. proof. since (f,a) →(f1,g1) (g,b) →(f2,g2)→ (h,c) is soft γ− exact with f1,g1 epimorphism and f2,g2 monomorphism, we have m →f1 n →f2 k and a →g1 b →g2 c, hence n = 0 and b = 0, it is clear that (g,b) is zero-soft γ− module. 2 theorem 5.1. let (f,a) and (h,b) are two soft γ−modules over γ−modules m and n respectively. for any m ⊂ n,a ⊂ b and m ⊂ h(x) where x ∈ b. if (f,a) →(f,g) (h,b) is soft γ−homomorphism, then 0 → (f,a) →(f,g) (h,b) →(f1,g1) (i,b/a) → 0 is soft γ− exact, where i(x + a) = h(x)/m for all x ∈ b. 59 s. onar, s. yavuz and b. a. ersoy proof. we know that 0 → m →f n →f1 n/m → 0 and 0 → a →g b →g1 b/a → 0 are exact. it is clear that m is a γ−submodule of n, so that n/m is a γ−module and m is a γ−submodule of h(x) and h(x)/m is always a γ−submodule of n/m. this shows that (i,b/a) is a soft γ−module over n/m. for all x ∈ b/a. define f1 : n → n/m by f1(n) = n + m, for all n ∈ n. meanwhile, we define g1 : b → b/a by g1(b) = b + a, for all b ∈ b. therefore, it gives that f1(h(x)) = h(x) + m,i(g1(x)) = i(x + a) = h(x) + m for all x ∈ b, and hence f1(h(x)) = i(g1(x)).this implies 0 → (f,a) →(f,g) (h,b) →(f1,g1) (i,b/a) → 0 is soft γ− exact. 2 theorem 5.2. let (f,a2), (g,a1) and (h,a) are three soft γ−modules over γ−modules m2,m1 and m respectively. if m1 and m2 are γ−submodules of m with m2 ⊂ m1,a1 and a2 are γ−submodules of a with a2 ⊂ a1, where m1 ⊂ h(x), for all x ∈ a and m2 ⊂ g(x) for all x ∈ a1. then 0 → (i,a1/a2) →(f1,g1) (j,a/a1) →(f2,g2) (p,a/a1) → 0 is soft γ− exact, where i(x + a2) = g(x)/m2, for all x ∈ a1, j(x + a2) = h(x)/m2, for all x ∈ a,p(x + a1) = h(x)/m1, for all x ∈ a. proof. since m1 and m2 are γ−submodules of m with m2 ⊂ m1, we have a short exact sequence 0 → m1/m2 →f1 m/m2 →f2 m/m1 → 0. since a1 and a2 are γ−submodules of a with a2 ⊂ a1, there is a short exact sequence 0 → a1/a2 →g1 a/a2 →g2 a/a1 → 0. it is clear that m2 is a γ−submodule of m1, so that m1/m2 is a γ−module. it gives that g(x)/m2 is a γ−module for all x ∈ a1 from m2 is a γ−submodule of g(x). however g(x)/m2 is always a γ−submodule of m1/m2. this shows that (i,a1/a2) is a soft γ− module over m1/m2 for all x ∈ a1/a2. it is clear that (j,a/a2) and (p,a/a1) be a soft γ− module over m/m2 and m/m1 respectively. define f1 : m1/m2 → m/m2 by f1(m1 + m1) = m + m2, for all m1 ∈ m1. meanwhile, we define g1 : a1/a2 → a/a2 by g1(a1 + a2) = a + a2, for all a1 ∈ a1. therefore, we have f1(i(x)) = f1(g((x)/m2) = h(x) + m2,j(g1(x)) = j(x + a2) = h(x) + m2 for all x ∈ a1/a2, so f1(i(x)) = j(g1(x)) for all x ∈ a1/a2. define f2 : m/m2 → m/m1 by f2(m + m2) = m + m1, for all m ∈ m. let g2 : a/a2 → a/a1 be defined by g2(a + a2) = a + a1, for all a ∈ a. also, we have f2(j(x)) = f2(h((x)/m2) = h(x) + m1 for all x ∈ a/a2, so f2(j(x)) = p(g2(x)) for all x ∈ a/a2. hence 0 → (i,a1/a2) →(f1,g1) (j,a/a1) →(f2,g2) (p,a/a1) → 0 is soft γ− exact. 2 60 soft γmodules theorem 5.3. let (fi,ai), i = 1, 2, 3, 4, 5 be a soft γ−module over γ−module mi, i = 1, 2, 3, 4, 5 respectively. if 0 → (f1,a1) →(f1,g1) (f2,a2) →(f2,g2) (f3,a3) → 0 and 0 → (f3,a3) →(f3,g3) (f4,a4) →(f4,g4) (f5,a5) → 0 are soft γ− exact. then 0 → (f1,a1) →(f1,g1) (f2,a2) →(f3 f2,g3 g2) (f4,a4) →(f4,g4) (f5,a5) → 0 is soft γ− exact. proof. since 0 → (f1,a1) →(f1,g1) (f2,a2) →(f2,g2) (f3,a3) → 0 and 0 → (f3,a3) →(f3,g3) (f4,a4) →(f4,g4) (f5,a5) → 0 are soft γ− exact, we have 0 → m1 →f1 m2 →f2 m3 → 0 and 0 → m3 →f3 m4 →f4 m5 → 0 are exact. it is clear that 0 → m1 →f1 m2 →f3 f2 m4 →f4 m5 → 0 is exact. since 0 → a1 →g1 a2 →g2 a3 → 0 and 0 → a3 →g3 a4 →g4 a5 → 0 are exact. it is clear that 0 → a1 →g1 a2 →g3 g2 a4 →g4 a5 → 0 is exact. since f2(f2(x)) = f3(g2(x)) for all x ∈ a2 and f3(f3(x)) = f4(g3(x)) for all x ∈ a3. we have f3f2(f2(x)) = f3(f3(g2(x))) = f4(g3g2(x)) for all x ∈ a2. this implies 0 → (f1,a1) →(f1,g1) (f2,a2) →(f3 f2,g3 g2) (f4,a4) →(f4,g4) (f5,a5) → 0 is soft γ− exactness. 2 6 conclusion in this work the theoretical point of view of soft γ− module is discussed. the work is focused on soft γ− module, soft γ− module homomorphism and soft γ− exactness. by using these concepts, we studied the algebraic properties of soft sets in γ− module structure. one could extend this work by studying other algebraic structures. references [1] d. molodtsov, soft set theory-first results, computers mathematics applications, 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[18] u. acar, f. koyuncu, b. tanay, soft sets and soft rings, computers and mathematics with applications, 59 (11), 2010, 34583463. 62 ratio mathematica volume 40, 2021, pp. 7-25 a survey on fuzzy semigroups johnson aderemi awolola∗ musa adeku ibrahim† xxx abstract fuzzy semigroup is an algebraic extension of semigroup. it has found application in fuzzy coding theory, fuzzy finite state machines and fuzzy languages. in this paper, a comprehensive literature review on fuzzy semigroup theory is realized. we will begin with a review of fuzzy groups which heavily inspired the notion of fuzzy semigroups put forth by the algebraists. subsequently, a brief tour of semigroup theory is considered as a precursor to the emerging subject. keywords: fuzzy group; semigroup; fuzzy semigroup. 2020 ams subject classifications: 20n25; 03e72. 1 ∗johnson awolola (federal university of agriculture, makurdi, nigeria); remsonjay@yahoo.com, awolola.johnson@uam.edu.ng. †musa ibrahim (ahmadu bello university, zaria, nigeria); amibrahim@abu.edu.ng. 1received on january 12th, 2021. accepted on may 12th, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.557. issn: 1592-7415. eissn: 2282-8214. c©the authors. this paper is published under the cc-by licence agreement. 7 guest freehand j. a. awolola, m. a. ibrahim 1 introduction semigroup theory is a thriving field of modern abstract algebra. as the name suggests, a semigroup is a generalization of a group; because a semigroup need not in general have an element which has an inverse. the earliest major contributions to the theory of semigroups are strongly motivated by comparisons with groups and rings. semigroup theory can be considered as one of the most successful off-springs of ring theory in the sense that the ring theory gives a clue how to develop the ideal theory of semigroups. the algebraic structure enjoyed by a semigroup is a non-empty set together with an associative binary operation. however, the fuzzy algebraic structures and their extensions are very important. nowadays, a lot of extensions of fuzzy algebraic structures have been introduced by many authors and have been applied to real life problems in different fields of science. many crisp concepts of algebraic structures have been extended to the nonclassical structures. fuzzy groups were first considered by rosenfeld (1971). in 1971, he defined fuzzy subgroup and established some of its properties. his definition of fuzzy group is a turning point for pure mathematicians. since then, the study of fuzzy algebraic structure has been pursued in many directions such as groups, rings, modules, vector space and so on. aktas and cagman (2007) gave a definition of soft groups and derived their basic properties. rough groups were defined by biswas and nanda (1994), and some other authors have studied the algebraic properties of rough set as well. demirci (2001) introduced the concept of smooth groups by using fuzzy binary operation. multigroups were first described by marty and several scholars put forth different definitions in an attempt to generalize group concept (see dresher and ore (1938), griffiths (1938), schein (1987) barlotti and strambach (1991), nazmul et al. (2013), tella and daniel (2013)). moreover, the algebraic extensions of a semigroup have been been studied by many authors. among others are the notion of ternary semigroups known to banach (cf. los (1955)) who is credited with an example of a ternary semigroup which does not reduce to a semigroup. kazim and naseeruddin (1972) introduced left almost semigroups (la-semigroups). the structure is also known as ag-groupoid and modular groupoid and has a variety of applications in topology, matrices, flock theory, finite mathematics and geometry. sen (1981) introduced the concept of γ-semigroups as a generalization of semigroups. the purpose of this paper is to promote research and disseminate fuzzy proficiency by presenting a comprehensive and up to date literature review of the fuzzy semigroup theory. 8 a survey on fuzzy semigroups 2 a brief review of fuzzy groups the important concept of a fuzzy set put forth by zadeh (1965) has opened up keen insights and applications in a wide range of scientific fields. since then, many papers on fuzzy sets appeared showing the importance of the concept and its applications to logic, set theory, group theory, groupoids and topology. the study of fuzzy algebraic structures started in the pioneering paper of rosenfeld (1971). rosenfeld introduced the notion of fuzzy groups and successfully extended many results from groups to fuzzy groups. though some other definitions of fuzzy groups are available in the literature (for example, anthony and sherwood (1979) redefined the fuzzy groups in terms of a t-norm which replaced the minimum operation), rosenfeld’s definition seems to be the most conventional and accepted one. most of the recent contributions in the field are the validations of rosenfeld’s definition where a fuzzy subset a of a group x is called a fuzzy subgroup of x if and only if µa(xy) ≥ min{µa(x),µa(y)} and µa(x−1) ≥ µa(x). das (2014) defined a level subgroup of a fuzzy subgroup a of a group x as an ordinary subgroup at of x, where t ∈ [0, 1]. wu (1981) studied fuzzy normal subgroup. also, fuzzy normal subgroups were studied by liu (1982) and kumar et al. (1992). in line with this, ajmal and jahan (2012) introduced the notion of a characteristic fuzzy subgroup of a group and related results. mukherjee and bhattacharya (1984) introduced the concept of fuzzy cosets and their relation with fuzzy normal subgroups. moreover, the authors proved fuzzy generalizations of some important theorems like lagranges and cayleys theorems. also, the authors initiated the notions of a fuzzy normalizer of a fuzzy subgroup and fuzzy solvable in mukherjee and bhattacharya (1986) and mukherjee and bhattacharya (1987). the effect of group homomorphism on fuzzy groups was studied by rosenfeld rosenfeld (1971) and proved that a homomorphic image of a fuzzy subgroup is a fuzzy subgroup provided the fuzzy subgroup has ∨ -property, while a homomorphic pre image of a fuzzy subgroup is always a fuzzy subgroup. anthony and sherwood anthony and sherwood (1979) later proved that even without the ∨ property the homomorphic image of a fuzzy subgroup is a fuzzy subgroup. sidky and mishref (1990) proved that if f : x −→ y is a group homomorphism and a is a fuzzy subgroup of x ”with respect to a continuous t-norm t , then f(a) 9 j. a. awolola, m. a. ibrahim is a fuzzy subgroup of y with respect to t ”. since ∧ is a continuous t-norm (anthony and sherwood), it follows that f(a) ∈ fg(y ) whenever a ∈ x. it was proved by akgul (1988) that f−1(b) is a fuzzy subgroup of x whenever b is a fuzzy subgroup of y . fang (1994) introduced the concepts of fuzzy homomorphism and fuzzy isomorphism by a natural way, and study some of their properties. ajmal (1994) defined a notion of ’containment’ of an ordinary kernel of a group homomorphism in a fuzzy subgroup and provided the long-awaited solution of the problem of showing a one-to-one correspondence between the family of fuzzy subgroups of a group, containing the kernel of a given homomorphism, and the family of fuzzy subgroups of the homomorphic image of the given group. yong (2004) constructed a quotient group induced by a fuzzy normal subgroup and proved the corresponding isomorphism theorems. demirci and racasens (2004) initiated fuzzy equivalence relation associated with a fuzzy subgroup and showed that a fuzzy subgroup is normal if only if the operation of the group is compatible with its associated fuzzy equivalence relation. kondo (2004) modified the idea of demirci and recasens and defined a fuzzy congruence on a group. ngcibi et al. (2010) obtained a formula for the group zpm × zpn when n = 1, 2, 3 and sehgal et al. (2016) extended the concept for all values of n. 3 a tour of semigroup theory the term semigroup was first coined in a french group theory textbook (de seguier (1904)) with a more stringent definition than the modern one, before being introduced to the english-speaking mathematical world by leonard dickson the following year dickson (1905). three decades after, the only semigroup theory being done was that done in near-obscurity (at least from the western perspective) by a russian mathematician, anton kazimirovich suschkewitsch. suschkewitsch (1928) was essentially doing semigroup theory before the rest of the world knew that there was such a thing, thus many of his results were rediscovered by later researchers who were unaware of his achievements. the study of semigroups exploded after the publication of a series of highly influential papers in the early 1940s. ree (1940) obtained the structure of finite simple semigroups and proved that the minimal ideal (green’s relation) of a finite semigroup is simple. clifford (1941) introduced semigroups admitting relative inverses. dubreil (1941) studied semigroup theory from the concept of lattice of equivalence relations on sets. preston (1954) defined and developed the concept of 10 a survey on fuzzy semigroups inverse semigroups. furthermore, preston described congruences on completely 0-simple semigroup and free inverse semigroups were also studied by the author (see preston (1961) and preston (1973)). munn (1955), having carried out research in a different direction, introduced the notion of semigroup algebras. kimura (1957) studied semigroups very widely and vividly and carried studies on idempotent semigroups. he further researched idempotent semigroups which satisfies some identities. moreover, idempotent semigroups was earlier studied by mclean (1954). yamada (1997) analyzed idempotent semigroups. green (1951) authored a classical paper on the structure of semigroups and with rees to study those semigroups in which xr = x. (green and rees, 1952) for over three decades, howie (1976)-howie (1995) worked on embedding theorems for semigroups in his book howie (1976). he collaborated with munn and weirert and edited a proceeding of the conference on semigroups and their applications. howie et al. (1992). his contributions to semigroup theory is very significant. in this period of three decades, semigroup theorists like petrich (1973)petrich (1984), mcalister and mcfadden (1974)-mcalister (1974), alan (1998), lawson (1998) and lajos (1971) have done lots of research on special class of semigroups in the vein of inverse semigroups, free semigroups, etc. and their properties. okninski (1998) published a book on semigroups of matrices. several researchers have worked on the types of semigroup mentioned earlier and developed more properties with applications across a broad spectrum of areas (see eilenberg (1974), eilenberg (1976), hopcropt and ullman (1979), howie (1991), lallement (1979), straubing (1994)). in a conference, meakin (2005) delivered a lecture on groups and semigroups exploring their connections and contrasts. he clearly acknowledged that in the past several decades, group theory and semigroup theory have developed in different directions. cayley’s theorem enables one to view groups as groups of permutations of some set while semigroups are represented as semigroups of functions from a set to itself. however, significant research has been carried out both in group theory and semigroup theory beyond the early viewpoints. in reality, several concepts in modern semigroup theory are closely related to group theory. for instance, automata theory and formal language theory turn out to be related (see hopcropt and ullman (1979), howie (1991)). very recently, gould and yang (2014) presented a piece of research work ti11 j. a. awolola, m. a. ibrahim tled ”every group is a maximal subgroup of a naturally occurring free idempotent generated semigroup”. the structures of generalized inverse semigroups by kudryavtseva and lausa (2014) is also a recent work on inverse semigroups. haggins (1992) carried out a research on permutations of a semigroup that maps to inverses. the variety of unary semigroups with associate inverse subsemigroup by billhardt et al. (2014) is however an additional view on inverse subsemigroups. thus, semigroup theory has developed rapidly to become the extremely prolific area of research for scholars. 4 development of fuzzy semigroup theory in this section, we systematically provides research work done on fuzzy semigroup analogue of some basic notions from semigroup theory as well as record some elementary properties and applications of fuzzy semigroups. 4.1 fuzzy semigroup the study of fuzzy algebraic structures started with the introduction of the concepts of fuzzy subgroup (subgroupoid) and fuzzy (left, right) ideal in the pioneering paper by rosenfeld rosenfeld (1971). in 1979, fuzzy semigroups were introduced by ((kuroki (1981), kuroki (1982)), which is a generalization of classical semigroups. he had published a series of papers kuroki (1981)-kuroki (1997), in which he laid the foundation of an algebraic theory of semigroup in the fuzzy framework. in literature, many related works vis-á-vis fuzzy ideals of semigroups can be found in (lajos (1979), mclean and kummer (1992), xueping et al. (1992), ahsan et al. (1995), zhi-wen and xue-ping (1995), dib and galhum (1997), xiang-yun (1999b), das (1999), xiang-jun (2001b)-xiang-jun (2002), ahsan et al. (2001), lee and shun (2001), ahsan et al. (2002), jun and seok-zun (2016a)-jun and seok-zun (2016b), kazanci and yamak (2008), zhan and jun (2010), khan et al.). shen (1990) initiated the concepts of fuzzy regular subsemigroups, fuzzy weakly regular subsemigroups fuzzy completely regular subsemigroups, fuzzy weakly completely regular subsemigroup and investigated some of their algebraic properties. based on the definition of fuzzy regular subsemigroup given by shen (1990), xue-ping and wang-jin (1993) defined fuzzy (left, intra-) regular subsemigroup in semigroups and studied some related properties. furthermore, point-wise depiction of fuzzy regularity of semigroups was introduced by zhi12 a survey on fuzzy semigroups wen and xue-ping (1993). they also proposed the concept of a fuzzy weakly left (right, intra-) regular subsemigroup and exhibited some algebraic properties. shabir et al. (2010) in the twentieth century characterized regular semigroups by (α,β) fuzzy ideals. samhan (1993) discussed the fuzzy congruence relation generated by a given fuzzy relation on a semigroup. he also studied the lattice of fuzzy congruence relation on a semigroup and gave some lattice theoretic properties. kuroki (1997) introduced the notion of a quotient semigroup induced by a fuzzy congruence relation on a semigroup and obtained homomorphism theorems with respect to the fuzzy congruence. earlier before his published paper in kuroki (1997), kuroki (1995) studied fuzzy congruences on t∗pure semigroups. moreover, he had earlier proposed the concept of an idempotent-separating fuzzy congruence on inverse semigroups before das (1997) developed fuzzy congruences in an inverse semigroup and established some important results. the notions of fuzzy kernel and fuzzy trace of a fuzzy congruence on an inverse semigroup were introduced by al-thurkair (1993). he established a one-to-one correspondence between fuzzy congruence pair and fuzzy congruences on an inverse semigroup. xiangyun (1999a) introduced fuzzy rees congruences on semigroups and obtained that a homomorphic image of a fuzzy rees congruences semigroup is a fuzzy rees congruences semigroup. tan (2001) studied fuzzy congruences on a regular semigroup. zhang (2000) introduced the concept of fuzzy group congruences on a semigroup and investigated some of its properties. two years after, he examined fuzzy congruences on completely 0-simple semigroups. recently, ma and tian (2011) introduced the notion of fuzzy congruence triple on a completely simple semigroup and used it to characterize fuzzy congruence on a completely simple semigroup. the concept of fuzzy semiprimality in a semigroup as an extension of semiprimality in a semigroup was introduced by kuroki (1982). he described a semigroup that is a semilattice of simple semigroups in terms of fuzzy semiprimality. kuroki (1993) characterized a completely regular semigroup and a semigroup that is semilattice of groups in terms of fuzzy semiprime quasi-ideals. xiang-jun (2000) defined and studied prime fuzzy ideals of a semigroup. subsequently, xiang-jun (2001a) introduced and studied the quasi-prime and weakly quasi-prime fuzzy left ideals of a semigroup. kehayopulu et al. (2001) worked on characterization of prime and semiprime ideals of semigroups in terms of fuzzy subsets. shabir (2015) characterized semigroups in which each fuzzy ideal is prime. kazanci and yamak (2009) defined ϕ-semiprime fuzzy ideals of a fuzzy semigroup and described all of ϕ-semigroups in which every ϕ-fuzzy ideal is ϕ-semiprime. manikantan and peter (2015) proposed some new kind of fuzzy 13 j. a. awolola, m. a. ibrahim subsets of a semigroup by using fuzzy magnified translation, fuzzy translation, fuzzy multiplication and extension of a fuzzy subset and obtained some results on fuzzy semiprime ideals of semigroups. among other authors who reported work done on fuzzy semigroup are lizasoain and gomez (2017) who showed that the direct of two fuzzy transformation is again a fuzzy transformation semigroup if and only if the lattice is distributive. budimirovic et al. (2014) introduced fuzzy semigroups with respect to a fuzzy equality. sen and choudhury (2006) studied the intersection graphs of fuzzy semigroups and showed related results. 4.2 elemantary properties of fuzzy semigroup here, we refer readers to mordeson et al. (2003) for more details. 4.2.1 fuzzy set let s be a non empty set. a fuzzy set in s is a function f : s −→ [0, 1]. 4.2.2 semigroup a semigroup is an algebraic structure (s,.) consisting of a non empty set together with an associative binary operation ” · ”. 4.2.3 fuzzy ideals in semigroups let s be a semigroup and f,g be two fuzzy subsets of s. the product of f ◦g is defined by f ◦g (x) = { ∨ x=yz{f(y) ∧g(z)}, if ∃ y,z ∈ s such that x = yz, 0, otherwise. for all x ∈ s. a fuzzy subset f of s is called a fuzzy subsemigroup of s if f(ab) ≥ f(a) ∧ f(b) for all a,b ∈ s, and is called a fuzzy left (right) ideal of s if f(ab) ≥ f(b) (f(ab) ≥ f(a)) for all a,b ∈ s. a fuzzy subset f of s is called a fuzzy two-sided ideal (or a fuzzy ideal) of s if it is both a fuzzy left and a fuzzy right of s. 14 a survey on fuzzy semigroups lemma 4.1. let f be a fuzzy subset of a semigroup s. then the following properties hold. (i) f is a fuzzy subsemigroup of s if and only if f ◦f ⊆ f. (ii) f is a left ideal of s if and only if s ◦f ⊆ f. (iii) f is a right ideal of s if and only if f ◦s ⊆ f. (iv) f is a two-sided ideal of s if and only if s ◦f ⊆ f and f ◦s ⊆ f. proof. see mordeson et al. (2003) lemma 4.2. let s be a semigroup. then the following properties hold. (i) let f and g be two fuzzy subsemigroups of s. then f∩g is a fuzzy subsemigroup of s. (ii) let f and g be (left, two-sided) ideal of s. then f ∩ g is also a fuzzy left (right two-sided) ideal of s. proof. see mordeson et al. (2003) lemma 4.3. if f is a fuzzy left (right) ideal of s. then f ∪ (s ◦f) (f ∪ (f ◦s)) is a fuzzy two-sided ideal of s. proof. see mordeson et al. (2003) 4.2.4 fuzzy regular subsemigroup and homomorphism if f is a fuzzy subsemigroup of s and ∀ x ∈ s, there exists x′ ∈ rx such that f(x ′ ) ≥ f(x) provided f(x) 6= 0, then f is called a fuzzy regular subsemigroup of s. proposition 4.1. f is a fuzzy regular subsemigroup of s if and only if ∀ t ∈ (0, 1], ft is a regular subsemigroup of s provided ft 6= ∅. proof. see mordeson et al. (2003) proposition 4.2. if f is a fuzzy regular subsemigroup of s, then f ◦f = f. 15 j. a. awolola, m. a. ibrahim proof. see mordeson et al. (2003) proposition 4.3. let α be a semigroup surjection homomorphism from s onto t . (i) if f is a fuzzy regular subsemigroup of s, then α(f) is a fuzzy regular subsemigroup of t . (ii) if g is a fuzzy regular subsemigroup of t , then α−1(g) is a fuzzy regular subsemigroup of s. proof. see mordeson et al. (2003) 4.2.5 fuzzy congruences on semigroups and fuzzy factor semigroups a fuzzy equivalence relation on a semigroup s which is compatible is called a fuzzy congruence relation on s. theorem 4.1. let µ and ν be fuzzy congruences on semigroup s. then the following conditions are equivalent. (i) µ◦ν is a fuzzy congruence. (ii) µ◦ν is a fuzzy equivalence. (iii) µ◦ν is fuzzy symmetric. (iv) µ◦ν = ν ◦µ. proof. see mordeson et al. (2003) let µ be a fuzzy congruence on s. then s/µ = {µa | a ∈ s}, where µa = µ(a,x) for all x ∈ s. theorem 4.2. the binary relation ∗ on s/µ is well-defined. proof. see mordeson et al. (2003) theorem 4.3. let µ be a fuzzy congruence on a semigroup s. then µ−1(1) = {(a,b) ∈ s ×s | µ(a,b) = 1} is a congruence on s. proof. see mordeson et al. (2003) 16 a survey on fuzzy semigroups 4.3 applications of fuzzy semigroups there are some important areas in which the fuzzy semigroup-theoretic approach is quite substantial and more completely utilized. the most significant such areas are the theories of fuzzy codes, fuzzy finite state machines and fuzzy languages. for greater details on the subject, the readers are also directed to the monograph by mordeson et al. (2003). 4.3.1 fuzzy codes let x be an alphabet with 1 ≤| x |< ∞ and x∗(x+) is the free monoid (semigroup) generated by x with operation of concatenation. if a is a fuzzy submonoid of x∗ and b ∈ fp(x∗) such that b ⊆ a, then b is its fuzzy base with b(e) = 0 and (b1) ∀ x ∈ supp(a)\e, b∗(x) ≥ a(x); (b2) ∀ x ∈ supp(a)\e, xiyj ∈ x∗, i = 1, ...,n; j = 1, ...,m and x = xi...xn = y1...ym, ∧ {b(x1), ...,b(xn),b(y1), ...,b(yn)}∝ ∧ {[m = n], [x1 = y1], ..., [xn = yn} ≥ a(x), where e and fp(x∗) denote the empty string and the class of all fuzzy subsets of x∗. this explains the origin of the concept. a fuzzy code a over x+ is such that a 6= ∅ and a is a fuzzy base of a∗. 4.3.2 fuzzy finite state machine a fuzzy finite state machine is an ordered triple m = (q,x,µ), where q and x are non-empty finite sets and µ : q × x × x −→ [0, 1]. the elements of q are called states and those of x are called inputs. however, a fuzzy finite state machine can be regarded as a finite state machine when m ⊆{0, 1}. fuzzy finite state machines can be divided into four categories: (i) m is called a deterministic fuzzy finite state if µ is a partial fuzzy function. (ii) m is called a non-deterministic fuzzy finite state if µ is a fuzzy relation. (iii) m is called a complete deterministic fuzzy finite state if µ is a complete partial fuzzy function. (iv) m is called a complete non-deterministic fuzzy finite state if µ is a complete fuzzy relation. 17 j. a. awolola, m. a. ibrahim 4.3.3 fuzzy languages a fuzzy formal language or a fuzzy language µ : t∗ −→ [0, 1] can serve to indicate the degree of meaningfulness of each string in t∗, namely, for x ∈ t∗, µ(x) near 1 implies that x is meaningful and µ(x) near 0 implies that x is not meaningful. a language l is defined to be a sequential fuzzy language if there is a finite fuzzy automata af and a cut-point t such that l is the set of coded words that yield at least one path from the initial state to a final state of af whose fuzzy measure is greater than t. 5 conclusion we have presented a comprehensive literature survey on the concept of fuzzy semigroups with some basic properties outlined and significant notable applications highlighted. for future research, we can hybridize non-classical structures to study their algebraic structures in semigroups. 6 acknowledgements the authors are 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set. inform. sci., 86:203–210, 1995. 25 microsoft word documento1 microsoft word documento1 ratio mathematica volume 40, 2021, pp. 163-177 generalized double fibonomial numbers shah mansi s.* shah devbhadra v.† abstract from the beginning of 20th century, generalization of binomial coefficient has been deliberated broadly. one of the most famous generalized binomial coefficients are fibonomial coefficients, obtained by substituting fibonacci numbers in place of natural numbers in the binomial coefficients. in this paper, we further generalize the concept of fibonomial coefficient and called it generalized double fibonomial number and obtain interesting properties of it. we also discuss its special case, double fibonomial number along with the situation in which they give integer values. other properties of it have also been discussed along with its upper and lower bounds. keywords: fibonacci numbers, lucas numbers, fibonomial numbers, binomial coefficient, double factorial. 2020 ams subject classifications: 11b39,05a10,11b65. 1 *veer narmad south gujarat university, surat, india; mansi.shah 88@yahoo.co.in. †veer narmad south gujarat university, surat, india; drdvshah@yahoo.com. 1received on january 12th, 2021. accepted on may 12th, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.625. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 163 m. shah, d. shah 1 introduction in combinatorics, the factorial of a positive integer n, denoted by n!, is defined by n! = n×n−1×···×2×1;n ≥ 1 and 0! = 1. whereas the double factorial of a positive integer n, usually denoted by n!! is defined as n!! =   n×n−2×···×3×1; n is odd n×n−2×···×4×2; n is even 1; n = 0 note that n!! is not the same as the iterated factorial (n!)!, which grows much faster. we do not know precisely when, where, or by whom, the double factorial notation was devised. it was used by meserve [6] in 1948, and it is not mentioned by cajori in his very detailed work in history of mathematical notations during 1928 – 1929 [2]. thus, we summaries that the notation was introduced at some times during the period 1928 – 1948. in this definition of n!!, if we replace the natural numbers by the terms of the generalized fibonacci numbers wn defined by the recurrence relation wn = pwn−1 + qwn−2, for n ≥ 2; w0 = a and w1 = b, where a,b,p and q are any integers, then what we get will be called generalized double fibonorial n!!w and is defined as n!!w ≡   wn ×wn−2 ×···×w3 ×w1 n > 0 is odd wn ×wn−2 ×···×w4 ×w2 n > 0 is even 1 n = 0 (1) here note that when we substitute p = q = b = 1 and a = 0 in the definition of wn, we get regular fibonacci numbers fn. the definition of n!!w helps to express the generalized double fibonorial in terms of regular generalized fibonorial as shown in the following lemma. lemma 1.1. n!!w = n!w(n−1)!!w = (n+1)!w (n+1)!!w ;n ≥ 1. in 1964 fontene [3] generalized the notion of binomial coefficients and introduce the new concept of fibonomial coefficients. in the definition of binomial coefficients ( m k ) , he replaced the natural numbers by the terms of an arbitrary sequence {an} of real or complex numbers. since then there has been an accelerated interest in the study of fibonomial coefficients. when the sequence {an} is considered as the sequence {fn} of fibonacci numbers, the fibonomial coefficients [ m k ] f , for 1 ≥ k ≥ m, is defined as [ m k ] f = m!f k!f (m−k)!f . the elaborated study on the generalized fibonomial coefficients can be found in literature. (see [5]) 164 generalized double fibonomial numbers this quantity will always be an integer, which can be shown by an induction argument by replacing fm in the numerator with fkfm−k+1+fk−1fm−k, resulting in [ m k ] f = fm−k+1 [ m−1 k −1 ] f + fk−1 [ m−1 k ] f (2) we use the concept of generalized double fibonorial to further generalize the concept of generalized fibonomial coefficients. we define the generalized double fibonomial numbers [[ m k ]] w as [[ m k ]] w = m!!w k!!w (m−k)!!w (3) it is easy to observe that[[ m 0 ]] w = 1, [[ m 2 ]] w = wmand [[ m k ]] w = [[ m m−k ]] w (4) 2 generalized double fibonomial numbers: 2.1 some properties of generalized double fibonomial numbers: the following results are now easy consequences from this definition: lemma 2.1. for any positive integers k,m and n, 1. (iterative rule) [[ n k ]] w [[ k m ]] w = [[ n m ]] w [[ n−m k −m ]] w . 2. wm−k [[ m k ]] w = wm [[ m−2 k ]] w . 3. wk [[ m k ]] w = wm−k+2 [[ m k −2 ]] w . 4. wk [[ m k ]] w = wm [[ m−2 k −2 ]] w . lemma 2.2. (m−1)!!w [[ m k ]] w will always give an integer value. 165 m. shah, d. shah this result is an easy derived from the definition of generalized fibonorial and generalized double fibonomial numbers. the basic recurrence relations for the generalized double fibonomial numbers is as follows: lemma 2.3. [[ m k ]] w − [[ m−2 k ]] w = [[ m−2 k −2 ]] w { wm−wm−k wk } . by changing k to m−k and using (4), we get lemma 2.4. [[ m k ]] w − [[ m−2 k −2 ]] w = [[ m−2 k ]] w { wm−wk wm−k } . the following result can be easily obtained when we apply the sum on both sides with respect to the upper index such that m and k have the same parity. lemma 2.5. [[ m k ]] w = ∑m j=k { wj−wj−k wk }[[ j −2 k −2 ]] w ; where the sum is taken over integers starting from k with spacing of 2 up to m. 2.2 star of david theorem: in 1972, gould gave a result related to one interesting arithmetic property of binomial coefficients which was named as the star of david theorem, which was stated as “the greatest common divisors of the binomial coefficients forming each of the two triangles in the star of david shape in pascal’s triangle are equal: gcd {( n−1 k−1 ) , ( n k+1 ) , ( n+1 k )} . the two sets of three numbers, which the star of david theorem says, have equal greatest common divisors and equal products. interestingly, gould’s result can be imitated for generalized double fibonomial numbers too as shown in the following result. theorem 2.1. [[ m−a k − b ]] w [[ m k + b ]] w [[ m + b k ]] w = [[ m−a k ]] w [[ m + b k + a ]] w [[ m k − b ]] w ; where a,b are positive integers. proof. using the definition of generalized double fibonomial numbers, the left side of the result becomes[[ m−a k − b ]] w [[ m k + b ]] w [[ m + b k ]] w = (m−a)!!w (k−b)!!w(m−k−a+b)!!w × (m)!!w (k+a)!!w(m−k−a)!!w × (m+b)!!w (k)!!w(m−k+b)!!w 166 generalized double fibonomial numbers = (m−a)!!w (k)!!w(m−k−a)!!w × (m+b)!!w (k+a)!!w(m−k−a+b)!!w × (m)!!w (k−b)!!w(m−k+b)!!w = [[ m−a k ]] w [[ m + b k + a ]] w [[ m k − b ]] w , as required.2 corolary 2.1. if a = b = 1, we get the product of six generalized double fibonomial numbers, which are equally spaced around [[ m k ]] w . 2.3 generalized double multinomial numbers: let m = k1 +k2 +· · ·+kr then we can define generalized double multinomial number as [[ m k1,k2, · · · ,kr ]] w = m!!w k1!!wk2!!w···kr!!w following result expresses generalized double multinomial numbers as the multiplication of generalized double fibonomial numbers. lemma 2.6. generalized double multinomial numbers can be expressed as the multiplication of generalized double fibonomial numbers. proof. in the definition of generalized double multinomial numbers, consider r = 2, then we have [[ m k1,k2 ]] w = [[ m k1 ]] w ; where k1 + k2 = m. for r = 3 and m = k1 + k2 + k3, [[ m k1,k2,k3 ]] w = [[ m k1 ]] w [[ m−k1 k2 ]] w . let us now consider r = n and m = k1 + k2 + · · ·+ kn. thus[[ m k1,k2, · · · ,kr ]] w = m!!w k1!!wk2!!w···kn!!w = m!!w k1!!wk2!!w···kn−2!!w × 1 kn−1!!wkn!!w = m!!w k1!!wk2!!w···kn−2!!w×(m−k1−k2−···−kn−2)!!w × (m−k1−k2−···−kn−2)!!w kn−1!!w(m−k1−k2−···−kn−2−kn−1)!!w = [[ m k1 ]] w [[ m−k1 k2 ]] w · · · [[ m−k1 −k2 −···−kn−2 kn−1 ]] w . hence, by the principle of mathematical induction, we get the required result.2 it is obvious that all the above results related to generalized double fibonorials and generalized double fibonomial numbers are also true for double fibonomials n!!f and double fibonomial coefficients [[ m k ]] f . but there are some additional results related to them, which are discussed in the following article. 167 m. shah, d. shah 3 double fibonomial numbers: 3.1 definition and some properties of double fibonomial numbers: using the definitions (1) and (3), double fibonorials and double fibonomial numbers can be respectively expressed as n!!f ≡   fn ×fn−2 ×···×f3 ×f1 n > 0 is odd fn ×fn−2 ×···×f4 ×f2 n > 0 is even 1 n = 0 and [[ m k ]] f = m!!f k!!f (m−k)!!f . the following table shows first few terms of double fibonorials for some initial values of n. n 0 1 2 3 4 5 6 7 8 9 10 n!!f 1 1 1 2 3 10 24 130 504 4420 27720 table 1: double fibonorial numbers also by (4), double fibonomial numbers have the symmetry property. thus table 2 shows the first few terms of double fibonomial numbers of one side only. 1 1 1 1 1 2 1 3/2 3 1 10/3 5 1 24/10 8 6 1 65/12 13 65/3 1 252/65 21 126/5 56 table 2: double fibonomial numbers we further show how double fibonorial and double fibonomial numbers are connected with the sequence {ln} of lucas numbers. this sequence is famously known as the twin sequence of fibonacci sequence, which can be obtained by 168 generalized double fibonomial numbers substituting p = q = b = 1 and a = 2 in the definition of wn. that is ln = ln−1 + ln−2;l0 = 2 and l1 = 1. it is easy to observe that f2n = fnln. if we define n!l = ln×ln−1×···×l2×l1, then the following lemma follows easily. lemma 3.1. n!!f = k!f ×k!l, for even positive integer n = 2k. if we consider [ m k ] l = m!l k!l(m−k)!l ,then the following is an easy consequence of lemma 3.1. lemma 3.2. [[ 2m 2k ]] f = [ m k ] f × [ m k ] l from the table 2 it is clear that double fibonomial numbers are not always an integer. obviously, for any integer m, [[ m 0 ]] f = [[ m m ]] f = 1, will always have an integer value. also [[ m 2 ]] f = [[ m m−2 ]] f = fm will be integers. these two will serve as the trivial cases. following theorem speaks about when double fibonomial numbers attain integer values. theorem 3.1. the nontrivial double fibonomial numbers [[ m k ]] f are integers only when either m and k both are even integers together or [[ m k ]] f = [[ 6 3 ]] f . proof. we prove the result in four cases depending on the parity of m and k. case 1: when m and k both are even integers, we have[[ m k ]] f = [[ 2n 2l ]] f = (2n)!!f (2l)!!f (2n−2l)!!f = f2n×f2n−2×···×f2n−2l+2 f2l×···×f4×f2 note that number of elements in numerator and denominator are same. also, they are fibonacci numbers with even subscripts, such that in the denominator we have first l even subscripted fibonacci numbers. since these numbers always divide multiplication of any l consecutive even subscripted fibonacci numbers, it follows that [[ m k ]] f will always be an integer. case 2: when m and k both are odd integers, we have in this case, we have[[ m k ]] f = [[ 2n + 1 2l + 1 ]] f = (2n+1)!!f (2l+1)!!f (2n−2l)!!f = f2n+1×f2n−1×···×f2l+3 f2n−2l×···×f4×f2 169 m. shah, d. shah in the numerator, every fibonacci number is with odd subscript. consequently, none of them will be divisible by f4 = 3. thus [[ m k ]] f will not be an integer in this case. case 3: when m is odd integer and k is even integer, we have in this case, we have[[ m k ]] f = [[ 2n + 1 2l ]] f = (2n+1)!!f (2l)!!f (2n−2l+1)!!f = f2n+1×f2n−1×···×f2n−2l+3 f2l×···×f4×f2 here again in the numerator, every fibonacci number is with odd subscript, so none of them will be divisible by f4 = 3. and therefore, in this case [[ m k ]] f will not be an integer. case 4: when m is even integer and k is odd integer, we have[[ m k ]] f = [[ 2n 2l + 1 ]] f = (2n)!!f (2l+1)!!f (2n−2l−1)!!f = f2n×f2n−2×···×f4×f2 (f2l+1×···×f3×f1)(f2n−2l−1×···×f3×f1) here number of terms in the numerator and denominator are same. also, the fibonacci numbers in the numerator are with only even subscripts and in the denominator with only odd subscripts. but, for any fibonacci number fn, there exists a prime p such that if p | fn, then p will only divide fmn; for every m ≥ 1. since [[ m k ]] f = [[ m m−k ]] f , for convenience we take k > m−k, that is, 2k > m. then there will not be the same fibonacci numbers in the numerator and denominator. also, there will not be any multiple subscripts of k in the numerator. thus, there will exist a prime p in the denominator such that p | fk which will not divide any of the fibonacci number in the numerator. likewise, when k = m−k, then except for k = 3, there will be a prime p such that p | fk as well as p | fm, but it will appear in the denominator only once where as in the numerator twice. thus in this case, except for [[ 6 3 ]] f = 6, [[ m k ]] f will not be an integer.2 in the following theorem we obtain the recurrence relation for the double fibonomial numbers. theorem 3.2. [[ m k ]] f = fk−1 [[ m−2 k ]] f + fm−k+1 [[ m−2 k −2 ]] f . proof. from [4], we observe that the fibonomial coefficients [ m k ] f has the recurrence relation 170 generalized double fibonomial numbers [ m k ] f = fk−1 [ m−1 k ] f + fm−k+1 [ m−1 k −1 ] f now, using this relation and lemma 3.2, we get[[ m k ]] f = m!f (m−1)!!f × (k−1)!!f k!f × (m−k−1)!!f (m−k)!f = [ m k ] f × (k−1)!!f×(m−k−1)!!f (m−1)!!f . = { fk−1 [ m−1 k ] f + fm−k+1 [ m−1 k −1 ] f } × (k−1)!!f×(m−k−1)!!f (m−1)!!f . = { fk−1(m−1)!f (m−1)!!f × (k−1)!!f k!f × (n−k−1)!!f (n−k−1)!f } + { fn−k+1(m−1)!f (m−1)!!f × (k−1)!!f (k−1)!f × (n−k−1)!!f (n−k−1)!f } = fk−1(n−2)!!f k!!f×(n−k−2)!!f + fn−k+1(n−2)!!f (k−2)!!f×(n−k)!!f[[ m k ]] f = fk−1 [[ m−2 k ]] f + fm−k+1 [[ m−2 k −2 ]] f , as required. 2 lemma 3.3. [[ m k ]] f = ∑|m−k2 | j=1 f j−1 k−1fm−k+1−2(j−1) [[ m−2j k −2 ]] f + f |m−k2 | k−1 a; where a =   1;when m and k both are even or both are odd integers[[ k + 1 k ]] f ;otherwise proof. from above theorem, we have[[ m k ]] f = fm−k+1 [[ m−2 k −2 ]] f + fk−1 [[ m−2 k ]] f = fm−k+1 [[ m−2 k −2 ]] f + fk−1 { fm−k−1 [[ m−4 k −2 ]] f + fk−1 [[ m−4 k ]] f } = fm−k+1 [[ m−2 k −2 ]] f + fk−1fm−k−1 [[ m−4 k −2 ]] f . + f2k−1 { fm−k−3 [[ m−6 k −2 ]] f + fk−1 [[ m−6 k ]] f } continuing this process, we get [[ m k ]] f =   ∑|m−k2 | j=1 f j−1 k−1fm−k+1−2(j−1) [[ m−2j k −2 ]] f + f |m−k2 | k−1 [[ k k ]] f ; when n and k both are even or odd∑|m−k2 | j=1 f j−1 k−1fm−k+1−2(j−1) [[ m−2j k −2 ]] f + f |m−k2 | k−1 [[ k + 1 k ]] f ; otherwise ,as required. 2 to illustrate the result, we consider m = 9 and k = 5. then[[ m k ]] f = ∑|m−k2 | j=1 f j−1 k−1fm−k+1−2(j−1) [[ m−2j k −2 ]] f + f |m−k2 | k−1 a = ∑2 j=1 f j−1 4 f5−2(j−1) [[ 9−2j 3 ]] f + f24 [[ 5 5 ]] f 171 m. shah, d. shah = f5 [[ 7 3 ]] f + f4f3 [[ 5 3 ]] f + f24 = ( 5× 65 3 ) + (3×2×5) + (32) = 442 3 = [[ 9 5 ]] f , as expected. the following result is an easy consequence from the definition of double fibonomial numbers and the basic identity fmln +fnlm = 2fm+n relating both fibonacci numbers and lucas numbers. lemma 3.4. [[ m k ]] f = 1 2 ( lk [[ m−2 k ]] f + lm−k [[ m−2 k −2 ]] f ) . proof. since 2fm = fklm−k + fm−klk , we have 2 [[ m k ]] f fm =[[ m k ]] f fklm−k + [[ m k ]] f fm−klk = [[ m−2 k −2 ]] f fmlm−k + [[ m−2 k ]] f fmlk. thus 2 [[ m k ]] f = lk [[ m−2 k ]] f + lm−k [[ m−2 k −2 ]] f , as required. 2 using lemma 3.4 and applying the same logic of lemma 3.3, the following result can be proved easily. lemma 3.5. [[ m k ]] f = ∑bm−k2 c j=1 l j−1 k lm−k−2(j−1) 2j [[ m−2j k −2 ]] f + ( lk 2 )bm−k2 ca; where a =   1;when m and k both are even or odd integers[[ k + 1 k ]] f ;otherwise to illustrate the result, we consider m = 10 and k = 3. then [[ m k ]] f = ∑bm−k2 c j=1 l j−1 k lm−k−2(j−1) 2j [[ m−2j k −2 ]] f + ( lk 2 )bm−k2 ca = ∑3 j=1 l j−1 3 l7−2(j−1) 2j [[ 10−2j 3 ]] f + ( l3 2 )3 [[4 3 ]] f = l7 2 [[ 8 1 ]] f + l3l5 22 [[ 6 1 ]] f + l23l3 23 [[ 4 1 ]] f + l33 23 [[ 4 3 ]] f = ( 29 2 × 252 65 ) + ( 4×11 22 × 24 10 ) + ( 43 23 )( 3 2 + 3 2 ) = 1386 13 = [[ 10 3 ]] f , as expected. in the following section we find the bounds of these numbers. 172 generalized double fibonomial numbers 3.2 bounds of double fibonomial numbers: the binet formula for the fibonacci number is given by fn = αn−βn α−β ; where α = 1+ √ 5 2 and β = 1− √ 5 2 . the following theorem gives us the bounds of double fibonomial numbers in terms of α. theorem 3.3. for χ(n) = { 0;when n is even 1;when n is odd , α (k−χ(k))(m−k−χ(m(m−k−1)−1)) 2 ≤ [[ m k ]] f ≤ α (k+χ(k))(m−k+χ(m(m−k−1)−1)) 2 . proof. it is well-known that αn−2 ≤ fn ≤ αn−1; for all n ≥ 1. then it is easy to observe that fm−2t f2t+2 ≤ αm−4t−1 (5) and fm−2t f2t+2 ≥ αm−4t−3 (6) here we consider the four cases depending on the parity of m and k. when both m and k are even, using the definition of double fibonomial numbers and (5), we have[[ m k ]] f = m!!f k!!f×(m−k)!!f = fm×fm−2×···×fm−k+2 f2×f4×···×fk ≤ αm−1 ×αm−5 ×···×α(m−2k+3) = α k(m−k+1) 2 thus [[ m k ]] f ≤ α k(m−k+1) 2 . again using (6) in the definition of double fibonomial number, we get[[ m k ]] f ≥ αm−3 ×αm−7 ×···×αm−2k+1 = α k(m−k−1) 2 . this shows that [[ m k ]] f ≥ α k(m−k−1) 2 . thus when m and n both are even, we have α k(m−k−1) 2 ≤ [[ m k ]] f ≤ α k(m−k+1) 2 . considering χ(n) = { 0;when n is even 1;when n is odd , this result can be written as α (k−χ(k))(m−k−χ(m(m−k−1)−1)) 2 ≤ [[ m k ]] f ≤ α (k+χ(k))(m−k+χ(m(m−k−1)−1)) 2 . 173 m. shah, d. shah the required result can be proved using the similar technique for all the remaining cases. to illustrate it, we consider m = 9 and k = 4. then [[ m k ]] f = 442 3 . also, α (k−χ(k))(m−k−χ(m(m−k−1)−1)) 2 = α k(m−k−1) 2 = α8 = 46.97 and α (k+χ(k))(m−k+χ(m(m−k−1)−1)) 2 = α k(m−k+1) 2 = α12 = 321, which shows that α (k−χ(k))(m−k−χ(m(m−k−1)−1)) 2 ≤ [[ m k ]] f ≤ α (k+χ(k))(m−k+χ(m(m−k−1)−1)) 2 . 4 double fibonomial numbers and fibonacci numbers: by [1], it is known that a primitive divisor of a fibonacci number fn is any prime integer p such that p | fn but p fm; where m < n. also, primitive divisor theorem says that for n ≥ 13, every fn has a primitive divisor. we use this result to prove many interesting relations between generalized double fibonomial numbers and fibonacci numbers. 4.1 double fibonomial number as a power of fibonacci number: in literature, there are many results involving fibonomial numbers and fibonacci numbers. from (4), it is clear that [[ m k ]] f = fm for k = 2. thus, the diophantine equation [[ m k ]] f = fn will always have a trivial solution (m,k,n) = (m,2,m). following result claims that there is no other solution for the considered diophantine equation. lemma 4.1. the diophantine equation [[ m k ]] f = fn has no solution for k > 2. proof. we know that except for [[ 6 3 ]] f = 6, which is not a fibonacci number, and trivial cases, [[ m k ]] f is an integer only when both m and k are even integers. thus, [[ m k ]] f = fn implies fm ×fm−2 ×···×fm−k+2 fk ×fk−2 ×···×f2 = fn (7) 174 generalized double fibonomial numbers if we consider n ≥ 13 and n > m, then by the primitive divisor theorem, there exists a prime p such that p | fn but p fm. that is, (7) has no solution possible. similarly, for m ≥ 13 and m > n, primitive divisor theorem implies that (7) has no solution. thus, we can narrow down the range of m and n as max(m,n). a quick look at the table 2 reveals that for k > 2, the diophantine equation [[ m k ]] f = fn has no solution. 2 the following result can be proved through the similar arguments. theorem 4.1. for any positive integer t, the diophantine equation [[ m k ]] f = f tn has no solution for k > 2. though the double fibonomial numbers do not possess the value of a fibonacci number except for the trivial cases, they do stand in the neighborhood of fibonacci number. we present this fact in the following final result. theorem 4.2. the only solutions of the diophantine equation [[ m k ]] f ±1 = fn are (m,k,n) = (3,1,2) ,(3,2,2) ,(4,2,3) ,(6,3,5) ,(8,4,10) for ′+′ case and (3,1,4) ,(3,2,4) for ′−′ case. poof. from the table 2, it is easy to observe that the diophantine equation[[ m k ]] f ± 1 = fn has solution (m,k,n) = (3,1,2) ,(3,2,2) ,(4,2,3) for ′+′ case and (m,k,n) = (3,1,4) ,(3,2,4) for the ′−′ case for m ≤ 5. now for m > 5, when m is an odd integer, double fibonomial number will not be an integer. and when m is an even integer such that k is an odd integer, [[ 6 3 ]] f = 6 is the only possibility integer value of double fibonomial. thus (m,k,n) = (6,3,5) will be a solution of the given diophantine equation for ′+′ case. now, we can narrow down our possible solution to the even integers for both m and k. since falb = fa+b + (−1) b fa−b, the different factorizations for fn±1 depending on the class of nmodulo4 can be written as: f4l + 1 = f2l−1l2l+1 f4l −1 = f2l+1l2l−1 f4l+1 + 1 = f2l+1l2l f4l+1 −1 = f2ll2l+1 f4l+2 + 1 = f2l+2l2l f4l+2 −1 = f2ll2l+2 f4l+3 + 1 = f2l+1l2l+2 f4l+3 −1 = f2l+2l2l+1 175 m. shah, d. shah therefore, the considered diophantine equation, which can also be written as[[ m k ]] f = fn ∓1, can be factorized for the ′+′ case as[[ m k ]] f = f2l+1l2l−1 [[ m k ]] f = f2ll2l+1 [[ m k ]] f = f2ll2l+2[[ m k ]] f = f2l+2l2l+1; and for the ′−′ case as[[ m k ]] f = f2l−1l2l+1 [[ m k ]] f = f2l+1l2l [[ m k ]] f = f2l+2l2l[[ m k ]] f = f2l+1l2l+2; it is obvious that all these eight cases can be handled in the similar manner. thus, we shall only focus on the proof of the first case. now, [[ m k ]] f = f2l+1l2l−1 implies fm×fm−2×···×fm−k+2 fk×fk−2×···×f2 = f2l+1l2l−1. thus, we have fm ×fm−2 ×···×fm−k+2 = f2l+1 ×l2l−1 ×fk ×fk−2 ×···×f2 since f2n = fnln, we write l2l−1 = f4l−2 f2l−1 . thus fm ×fm−2 ×···×fm−k+2 ×f2l−1 = f2l+1 ×f4l−2 ×fk ×fk−2 ×···×f2. since l = ⌊ n 4 ⌋ > 2, we have4l − 2 > 2l + 1. therefore, from primitive divisor theorem, we can write m = 4l−2. thus, fm−2 ×···×fm−k+2 ×f2l−1 = f2l+1 ×fk ×fk−2 ×···×f2 (8) if we assume that m ≥ max{14,k + 1}, we have m − 2 ≥ 12. so, again by primitive divisor theorem, we get m − 2 = max{2l + 1,k}. but m − 2 = 4l − 4 > 2l + 1, which implies m− 2 = k and from (8), we get f2l−1 = f2l+1, which is not possible. thus, we only need to consider the range 4 ≤ k ≤ 10 and k + 2 ≤ m ≤ 12. again, from table 2, we can easily claim that the only solution of the diophantine equation [[ m k ]] f ±1 = fn are (m,k,n) = (3,1,2) ,(3,2,2) ,(4,2,3) , (6,3,5) ,(8,4,10) for ′+′ case and (3,1,4) ,(3,2,4) for ′−′ case. 2 5 conclusion: in this paper, we have defined double fibonorial numbers and double fibonomial numbers. we have proved many properties for these numbers including recursive equations in terms of fibonacci numbers and lucas numbers. we have 176 generalized double fibonomial numbers extended the star of david theorem for double fibonomial numbers and also discussed various diophantine equations related to double fibonomial numbers and fibonacci numbers. 6 acknowledgement: the authors are thankful to the department of science and technology for providing financial support under wos – a fellowship. references [1] bilu, y., hanrot, g. and voutier, p. m. 2011. existence of primitive divisors of lucas and lehmer numbers, j. reine angew. math. [2] cajori f. 1993. history of mathematical notations, reprinted 1993, courier dover publications, 1928–1929. [3] fontene g. 1915. generalization d’une formule connue, nouv. ann. math., 4(15), 112. [4] gould h. w. 1969. the bracket function and fountené–ward generalized binomial coefficients with application to fibinomial coefficients, fibonacci quarterly, 7, 23 – 40. [5] kilic m. 2010. the generalized fibonomial matrix, european journal of combinatorics, 31(1), 193 – 209. [6] meserve b. e. 1948. double factorials, amer. math. monthly, 55, 425 426. 177 ratio mathematica volume 39, 2020, pp. 213-228 uniqueness of an entire function sharing a polynomial with its linear differential polynomial imrul kaish* nasir uddin gazi† abstract in this paper we consider an entire function when it shares a polynomial with its linear differential polynomial. our result is an improvement of a result of p.li. keywords: uniqueness; entire function; differential polynomial; sharing. 2010 ams subject classifications: 30d35. 1 *department of mathematics and statistics, aliah university, kolkata, west bengal 700160, india; imrulksh3@gmail.com. †department of mathematics and statistics, aliah university, kolkata, west bengal 700160, india; nsrgazi@gmail.com 1received on november 2nd, 2020. accepted on december 17th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.554. issn: 1592-7415. eissn: 2282-8214. ©kaish et al. this paper is published under the cc-by licence agreement. 213 i. kaish and n. gazi 1 introduction, definitions and results let f be a non-constant meromorphic function defined in the open complex plane c and a = a(z) be a polynomial. we denote by e(a; f) the set of zeros of f−a, counted with multiplicities and by e(a; f) the set of distinct zeros of f−a. if for two non-constant meromorphic functions f and g, we have e(a; f) = e(a; g), we say that f and g share a cm and if e(a; f) = e(a; g), we say that f and g share a im. we denote by s(r,f) any function satisfying s(r,f) = o{t(r,f)}, as r → ∞, possibly outside of a set with finite measure. for an entire function f, we define deg(f) in the following way: deg(f) = ∞, if f is a transcendental entire function and deg(f) is the degree of the polynomial, if f is a polynomial. the investigation of uniqueness of an entire function sharing two values introduced by l. a. rubel and c. c. yang [rubel and yang, 1977] in 1977. following is their result. theorem a. [rubel and yang, 1977] let f be a non-constant entire function. if e(a; f) = e(a; f(1)) and e(b; f) = e(b; f(1)), for distinct finite complex numbers a and b, then f ≡ f(1). in 1979 e. mues and n. steinmetz [mues and steinmetz, 1979] tried to improve theorem a by considering im sharing of values. they proved the following theorem. theorem b. [mues and steinmetz, 1979]. let f be a non-constant entire function and a, b be two distinct finite complex values. if e(a; f) = e(a; f(1)) and e(b; f) = e(b; f(1)), then f ≡ f(1). in 1986 g. jank, e. mues and l. volkmann [jank et al., 1986] considered an entire function sharing a nonzero value with its derivatives and they proved the following result. theorem c. [jank et al., 1986] let f be a non-constant entire function and a be a non-zero finite value. if e(a; f) = e(a; f(1)) ⊂ e(a; f(2)), then f ≡ f(1). h. zhong [zhong, 1995] tried to improve theorem c by taking higher order derivatives. by the following example he concluded that in theorem c the second derivative cannot be straight way replaced by any higher order derivatives. example 1.1. [zhong, 1995] let k(≥ 3) be a positive integer and ω( 6= 1) be a (k − 1)th root of unity. if f = eωz + ω − 1, then f, f(1), and f(k) share the value ω cm, but f 6≡ f(1). 214 uniqueness of an entire function sharing a polynomial with its linear differential polynomial considering two consecutive higher order derivatives h. zhong [zhong, 1995] improved theorem c in another direction. the following is the improved result. theorem d. [zhong, 1995] let f be a non-constant entire function and a be a non-zero finite value. if e(a; f) = e(a; f(1)) and e(a; f) ⊂ e(a; f(n)) ∩ e(a; f(n+1)) for n(≥ 1), then f ≡ f(n). for further discussion we need the following notation. let f be a non-constant meromorphic function, a = a(z) be a polynomial and a be a set of complex numbers. we denote by na(t,a; f), the number of zeros of f−a, counted according to their multiplicities which lie in a∩{z : |z|≤r}. the integrated counting function na(r,a; f) of the zeros of f −a which lie in a∩{z : |z|≤r} is defined as na(r,a; f) = ∫ r 0 na(t,a; f) −na(0,a; f) t dt + na(0,a; f) log r, where na(0,a; f) denotes the multiplicity of zeros of f−a at origin. na(r,a; f) be the reduced counting function of zeros of f −a in a∩{z : |z|≤r}. clearly if a = c then na(r,a; f) = n(r,a; f) and na(r,a; f) = n(r,a; f). for standard definitions and notations of the value distribution theory we refer the reader to [hayman, 1964] and [yang and yi, 2003]. recently i. lahiri and i. kaish [lahiri and kaish, 2017] improved theorem d by considering a shared polynomial. they proved the following result. theorem e. [lahiri and kaish, 2017] let f be a non-constant entire function and a = a(z)(6≡ 0) be a polynomial with deg(a) 6= deg(f). suppose that a = e(a; f)∆e(a; f(1)) and b = e(a,f(1))\{e(a,f(n)) ∩ e(a,f(n+1))}, where 4 denotes the symmetric difference of sets and n(≥ 1) is an integer. if (i) na(r,a; f) + na(r,a; f(1)) = o{logt(r,f)}, (ii) nb(r,a; f(1)) = s(r,f) and (iii) each common zero of f −a and f(1) −a has the same multiplicity, then f = λez, where λ( 6= 0) is a constant. throughout the paper we denote by l = l(f) a nonconstant linear differential polynomial generated by f of the form l = l(f) = a1f (1) + a2f (2) + ......... + anf (n), (1) where a1,a2, .......,an(6= 0) are constants. considering linear differential polynomial p.li [li, 1999] improved theorem d in the following way. 215 i. kaish and n. gazi theorem f. [li, 1999]. let f be a non-constant entire function and l be defined in (1) and a be a non-zero finite complex number. if e(a; f) = e(a; f(1)) ⊂ e(a; l) ∩e(a; l(1)) then f = f(1) = l. in this paper we extend theorem d and theorem f in the following way theorem 1.1. let f be a non-constant entire function, l be defined in (1) and a = a(z)(6≡ 0) be a polynomial with deg(a) 6= deg(f). suppose that a = e(a; f)∆e(a; f(1)) and b = e(a,f(1))\{e(a,l(p))∩e(a,l(q))} where p,q are integers satisfying q > p ≥ deg(a). if (i) na(r,a; f) + na(r,a; f(1)) = o{log t(r,f)}, (ii) nb(r,a; f(1)) = s(r,f) and (iii) each common zero of f −a and f(1) −a has the same multiplicity, then f = l = λez, where λ( 6= 0) is a constant. putting a = b = ∅ we get the following corollary. corolary 1.1. let f be a non-constant entire function, l be defined in (1) and a = a(z)(6≡ 0) be a polynomial with deg(a) 6= deg(f). if e(a; f) = e(a; f(1)) and e(a,f(1)) ⊂ e(a,l(p)) ∩ e(a,l(q)) where p,q are integers satisfying q > p ≥ deg(a), then f = l = λez, where λ( 6= 0) is a constant. remark 1.1. if in corollary 1.1, a is a non-zero constant and p = deg(a) = 0,q = p + 1 then it is a particular form of theorem f. remark 1.2. if in (1), a1 = a2 = .......an−1 = 0 and an = 1 then l = f(n) and if in corollary 1.1, a is a non-zero constant and p = deg(a),q = p + 1, then corollary 1.1 is the theorem d. remark 1.3. it is an open problem whether the theorem 1.1 is valid or not if we omit the condition p ≥ deg(a). 2 lemmas in this section we present some necessary lemmas. lemma 2.1. [lahiri and kaish, 2017]. let f be a transcendental entire function of finite order and a = a(z)(6≡ 0) be a polynomial and a = e(a; f)∆e(a; f(1)). if 216 uniqueness of an entire function sharing a polynomial with its linear differential polynomial (i) na(r,a; f) + na(r,a; f(1)) = o{log t(r,f)}, (ii) each common zero of f −a and f(1) −a has the same multiplicity, then m(r,a; f) = s(r,f). lemma 2.2. [lain, 1993]. suppose f be an entire function, a0,a1, .....an are polynomials and a0,an are not identically zero. then each solution of the linear differential equation anf(n) + an−1f(n−1) + ...... + a0f = 0 is of finite order. lemma 2.3. [hayman, 1964]. let f be a non-constant meromorphic function and a1,a2,a3 be three distinct meromorphic functions satisfying t(r,aν) = s(r,f) for ν = 1, 2, 3 then t(r,f) ≤ n(r, 0; f −a1) + n(r, 0; f −a2) + n(r, 0; f −a3) + s(r,f). lemma 2.4. let f be a transcendental entire function and a = a(z)(6≡ 0) be a polynomial. also let l(f), l(a) be the linear differential polynomials generated by f and a respectively. suppose h = (a−a(1))(l(p)(f) −l(p)(a)) − (a−l(p)(a))(f(1) −a(1)) f −a , a = e(a; f)\e(a; f(1)) and b = e(a,f(1))\{e(a,l(p)) ∩ e(a,l(q))}, where p,q are integers satisfying 0 ≤ p < q. if (i) na(r,a; f) + nb(r,a; f(1)) = s(r,f), (ii) each common zero of f −a and f(1) −a has the same multiplicity, (iii) h is a transcendental entire or meromorphic, then m(r,a,f(1)) = s(r,f). proof. since a−a(1) = (f(1)−a(1))−(f(1)−a), if z0 be a common zero of f−a and f(1)−a with multiplicity r(≥ 2), then z0 is a zero of a−a(1) with multiplicity r − 1. so n(2(r,a; f) ≤ 2n(r, 0; a−a(1)) + na(r,a; f) = s(r,f), (2) where n(2(r,a; f) be the counting function of multiple zeros of f −a. using (2) and from the hypothesis we get n(r,h) ≤ na(r,a; f) + nb(r,a; f(1)) + n(2(r,a; f) + s(r,f) = s(r,f) 217 i. kaish and n. gazi since m(r,h) = s(r,f), we have t(r,h) = s(r,f) from h = (a−a(1))(l(p)(f) −l(p)(a)) − (a−l(p)(a))(f(1) −a(1)) f −a , we get f = a + 1 h {(a−a(1))(l(p)(f) −l(p)(a)) − (a−l(p)(a))(f(1) −a(1))} = a + 1 h {(a−a(1))(l(p)(f) −a) − (a−l(p)(a))(f(1) −a)}. (3) case 1. let p > 0 . differentiating (3) we get f(1) = a(1) + ( 1 h )(1){(a−a(1))(l(p)(f) −a) − (a−l(p)(a))(f(1) −a)} + 1 h {(a(1) −a(2))(l(p)(f) −a) + (a−a(1))(l(p+1) −a(1))}− 1 h {(a(1) −l(p+1)(a))(f(1) −a) + (a−l(p)(a))(f(2) −a(1))}. this implies (f(1) −a){1 + ( 1 h )(1)(a−l(p)(a)) + 1 h (a(1) −l(p+1)(a))} = a(1) −a + ( 1 h )(1)(a−a(1))(l(p)(f)−a) + 1 h (a(1) −a(2))(l(p)(f)−a) + 1 h (a− a(1))(l(p+1)(f) −a(1)) − 1 h (a−l(p)(a))(f(2) −a(1)) = a(1) − a + (a−a (1) h )(1)(l(p)(f) − l(p−1)(a)) + (a−a (1) h )(1)(l(p−1)(a) − a) + a−a(1) h (l(p+1)(f)−l(p)(a)) + a−a (1) h (l(p)(a)−a(1))− 1 h (a−l(p)(a))(f(2)−a(1)), or, (f(1) −a){1 + (a−l (p)(a) h ))(1)} = (a(1) −a) + {(a−a (1) h )(l(p−1)(a) −a)}(1) + (a−a (1) h )(1)(l(p)(f)−l(p−1)(a))+a−a (1) h (l(p+1)(f)−l(p)(a))−1 h (a−l(p)(a))(f(2)− a(1)), or 1 f(1) −a = h1 h2 − 1 h2 ( a−a(1) h )(1)( l(p)(f) −l(p−1)(a) f(1) −a ) +( a−a(1) hh2 )( l(p+1)(f) −l(p)(a) f(1) −a ) − 1 hh2 (a−l(p)(a))( f(2) −a(1) f(1) −a ), (4) where h1 = 1 + ( a−l(p)(a) h )(1), h2 = a (1) −a + {(a−a (1) h )(l(p−1)(a) −a)}(1). we now verify that h1 6≡ 0,h2 6≡ 0. 218 uniqueness of an entire function sharing a polynomial with its linear differential polynomial if h1 ≡ 0, then 1 + ( a−l(p)(a) h )(1) ≡ 0. integrating we get 1 h = c1−z a−l(p)(a) , where c1 is a constant. this is a contradiction, because h is transcendental. if h2 ≡ 0, then a(1) − a + {(a−a (1) h )(l(p−1)(a) −a)}(1) ≡ 0. integrating we get h = (a−a (1))(l(p−1)(a)−a) p(z) , where p(z) is a polynomial. this is again a contradiction. therefore h1 6≡ 0,h2 6≡ 0. again t(r,h1) + t(r,h1) = s(r,f), since t(r,h) = s(r,f). now from (4) and using lemma of logarithmic derivative we get m(r,a; f(1)) = m(r, 1 f(1)−a) = s(r,f). case 2. let p = 0. then l(p)(f) = l(f). suppose l(f) = a1f(1) + a2f(2) + ......... + anf(n) and l(a) = a1a(1) + a2a(2) + ......... + ana(n), where a1,a2, .......,an( 6= 0) are constant, n(≥ 1) be an integer. from the definition of h we get f = a + 1 h {(a−a(1))(l(f) −a) − (a−l(a))(f(1) −a)} differentiating we get f(1) = a(1) + ( 1 h )(1){(a−a(1))(l(f) −a) − (a−l(a))(f(1) −a)} + 1 h {(a(1) −a(2))(l(f) −a) + (a−a(1))(l(1)(f) −a(1))} − 1 h {(a(1) −l(1)(a))(f(1) −a) − (a−l(a))(f(2) −a(1))}. this implies (f(1)−a){1 + (a−l(a) h )(1)} = (a(1)−a)+(a−a (1) h )(1)(l(f)−a)+a−a (1) h (l(1)(f)− a(1))−a−l(a) h (f(2)−a(1)) = (a(1)−a)+(a−a (1) h )(1)(l(f)−l1(a))+(a−a (1) h )(1)(l1(a)− a)+(a−a (1) h )(l(1)(f)−l(a))+(a−a (1) h )(l(a)−a(1))−a−l(a) h (f(2)−a(1)) = (a(1)− a) + {(a−a (1) h )(l1(a) −a)}(1) + (a−a (1) h )(1)(l(f) − l1(a)) + (a−a (1) h )(l(1)(f) − l(a)) − a−l(a) h (f(2) −a(1)) or, 1 f(1) −a = h3 h4 − 1 h4 ( a−a(1) h )(1)( l(f) −l1(a) f(1) −a ) +( a−a(1) hh4 )( l(1)(f) −l(a) f(1) −a ) − ( a−l(a) hh4 )( f(2) −a(1) f(1) −a ), (5) where l1(a) = a1a + a2a (1) + ..... + ana (n−1), h3 = 1 + ( a−l(a) h )(1) and h4 = a (1) −a + {(a−a (1) h )(l1(a) −a)}(1) 219 i. kaish and n. gazi similarly as in case 1, h3 6≡ 0, h4 6≡ 0. also t(r,h3) + t(r,h4) = s(r,f). therefore from (5) and using lemma of logarithmic derivative we get m(r,a; f(1)) = m(r, 1 f(1)−a) = s(r,f). this completes the proof of the lemma. lemma 2.5. let f be a transcendental entire function, a = a(z)(6≡ 0) be a polynomial and l = l(f) be define in (1). suppose (i) na(r,a; f) + na(r,a; f(1)) = s(r,f), where a = e(a; f)∆e(a; f(1)) (ii) nb(r,a; f(1))) = s(r,f), where b = e(a,f(1))\{e(a,l(p)) ∩e(a,l(q))} p,q are integers satisfying q > p ≥ deg(a), (iii) each common zero of f −a and f(1) −a has the same multiplicity, (iv) m(r, a; f) = s(r, f), then f = l = λez, where λ(6= 0) is a constant. proof. let α = f(1) −a f −a , (6) from the hypothesis we get, n(r,α) ≤ na(r,a; f) + s(r,f) = s(r,f) and m(r,α) = m(r, f(1) −a f −a ) = m(r, f(1) −a(1) + a(1) −a f −a ) ≤ m(r,a; f) + s(r,f) = s(r,f). therefore t(r,α) = s(r,f). from (6) we get f(1) = αf + a(1 −α) = α1f + β1, where α1 = α and β1 = a(1 −α) differentiating we get, f(2) = α2f + β2, 220 uniqueness of an entire function sharing a polynomial with its linear differential polynomial where α2 = α (1) 1 + α1α1 and β2 = β (1) 1 + α1β1. similarly, f(k) = αkf + βk, where αk+1 = α (1) k + α1αk and βk+1 = β (1) k + αkβ1. clearly t(r,αk) + t(r,βk) = s(r,f), because t(r,α) = s(r,f). now l(p) = n∑ k=1 akf (p+k) = ( n∑ k=1 akαp+k)f + ( n∑ k=1 akβp+k) = µ1f + ν1, (7) where µ1 = n∑ k=1 akαp+k, ν1 = n∑ k=1 akβp+k l(q) = n∑ k=1 akf (q+k) = ( n∑ k=1 akαq+k)f + ( n∑ k=1 akβq+k) = µ2f + ν2, (8) where µ2 = n∑ k=1 akαq+k, ν2 = n∑ k=1 akβq+k. clearly t(r,µi) + t(r,νi) = s(r,f), i = 1, 2. let d = e(a; f) ∩e(a; f(1)) ∩e(a; l(p)) ∩e(a; l(q)). note that d 6= ∅, because otherwise, n(r,a; f) = s(r,f). then from the hypothesis t(r,f) = s(r,f), a contradiction. let z1 ∈ d then f(z1) = f(1)(z1) = l(p)(z1) = l(q)(z1) = a(z1). now from (7) and (8) we get a(z1) = µ1(z1)a(z1) + ν1(z1) and a(z1) = µ2(z1)a(z1) + ν2(z1) if µ1a + ν1 −a 6≡ 0, then n(r,a; f) ≤ na(r,a; f) + nb(r,a; f(1)) + nd(r,a; f) + s(r,f) ≤ na(r, 0; µ1a + ν1 −a) + s(r,f) = s(r,f), a contradiction. therefore µ1a + ν1 −a ≡ 0. (9) 221 i. kaish and n. gazi similarly µ2a + ν2 −a ≡ 0. (10) from (9) and (10) we get µ1 ≡ µ2 ≡ 1 and ν1 ≡ 0 ≡ ν2. then from (7) l(p) ≡ f. (11) also µ1 ≡ 1 implies n∑ k=1 akαp+k ≡ 1. (12) from (12) we see that α has no pole. because if α has a pole of order d(≥ 1) then the left hand side of (12) has a pole of order (p + k)d but the right hand side is a constant. again by simple calculation from (12) we get anα n+p + p [α] ≡ 0. (13) where p [α] is a differential polynomial in α with degree not exceeding (n + p− 1). if α is transcendental entire, then by clunie’s lemma we have m(r,α) = s(r,α), a contradiction. if α is a nonconstant polynomial then left hand side of (13) is also a nonconstant polynomial, which is again a contradiction. therefore α is a constant. now from f (1)−a f−a = α, we get f (1) −αf = a(1 −α). integrating we get e−αzf = (1 −α) ∫ ae−αzdz = (1 −α)p(z)e−αz + λ, where λ(6= 0) is a constant and p(z) is a polynomial of degree atmost deg(a), or, f = (1 −α)p(z) + λeαz. now f(r+1) = λαr+1eαz, if r = deg(a) 222 uniqueness of an entire function sharing a polynomial with its linear differential polynomial therefore l(p) = n∑ k=1 akf (p+k) = ( n∑ k=1 akα p+k)λeαz = λeαz = f(1) α − 1 −α α p(1)(z), (14) suppose α 6= 1. since d = e(a; f) ∩e(a; f(1)) ∩e(a; l(p)) ∩e(a; l(q)) 6= ∅, we have f(z2) = f(1)(z2) = l(p)(z2) = l(q)(z2) = a(z2), for some z2 ∈ d. from (14) we get a(z2) = a(z2) α − 1 −α α p(1)(z2) or, a(z2)(1 − 1 α ) + 1 −α α p(1)(z2) = 0 or, (α− 1){a(z2) −p(1)(z2)} = 0 or, a(z2) −p(1)(z2) = 0. clearly a(z) −p(1)(z) 6≡ 0, because deg(p(1)(z)) is less than deg(a). n(r,a; f) ≤ na(r,a; f) + nb(r,a; f(1)) + nd(r,a; f) + s(r,f) ≤ n(r, 0; a−p(1)) + s(r,f) = s(r,f). then from the hypothesis t(r,f) = s(r,f), a contradiction. therefore α = 1, so f = λez. again l = n∑ k=1 akf (k) = ( n∑ k=1 akα k)λeαz = λez. 223 i. kaish and n. gazi therefore f = l = λez. this completes the lemma. 3 proof of the main theorem proof. first we claim that f is a transcendental entire function. if f is a polynomial, then t(r,f) = o(log r) and na(r,a; f) + na(r,a; f(1)) = o(log r). then from the hypothesis we get o(log r) = o(log t(r,f)) = s(r,f), which implies t(r,f) = s(r,f), a contradiction. therefore a = ∅. similarly nb(r,a; f(1)) = s(r,f) implies b = ∅. therefore e(a; f) = e(a; f(1)) and e(a; f(1)) ⊂ e(a; l(p)) ∩e(a; l(q)). let deg(f) = m and deg(a) = r . if m ≥ r + 1 then deg(f − a) = m and deg(f(1) −a) ≤ m− 1 which contradicts that e(a,f) = e(a,f(1)). if m ≤ r − 1 , then deg(f − a) = deg(f(1) − a) = r. since e(a,f) = e(a,f(1)), (f −a) = t(f(1) −a), where t(6= 0) is a constant. if t = 1, then f = f(1), which is a contradiction because f is a polynomial. if t 6= 1 then tf(1)−f ≡ (t−1)a, which is impossible because deg((t−1)a) = r and deg(tf(1) −f) = m and m < r. therefore our claim ” f is transcendental entire function ” is established. now we prove the result into two cases. case 1. let f ≡ l(p). then m(r,a; f) = m(r, a f −a 1 a ) ≤ m(r, a f −a ) + s(r,f) = m(r, a f −a + 1 − 1) + s(r,f) ≤ m(r, a f −a + 1) + s(r,f) ≤ m(r, f f −a ) + s(r,f) = m(r, l(p) f −a ) + s(r,f), (15) since p ≥ deg(a), by lemma of logarithmic derivative, m(r, l (p) f−a) = s(r,f). so from (15) m(r,a; f) = s(r,f). therefore by lemma 5, f = l = λez,λ(6= 0) is a constant. case 2. let f 6≡ l(p). this case can be divided into two subcases. 224 uniqueness of an entire function sharing a polynomial with its linear differential polynomial subcase 2.1. let f(1) 6≡ l(p). since a−a(1) = (f(1)−a(1))−(f(1)−a), a common zero of f−a and f(1)−a of multiplicity s(≥ 2) is a zero of a−a(1) with multiplicity s− 1(≥ 1). therefore n(2(r,a; f(1) | f = a) ≤ 2n(r, 0; a−a(1)) = s(r,f), where n(2(r,a; f(1) | f = a) denotes the counting function (counted with multiplicities) of those multiple zeros of f(1) −a which are also zeros of f −a. now n(2(r,a; f (1)) ≤ na(r,a; f(1)) + nb(r,a; f(1)) + n(2(r,a; f(1) | f = a) + s(r,f) = s(r,f). (16) using (16) and from the hypothesis we get n(r,a; f(1)) ≤ nb(r,a; f(1)) + n(r, a−l(p)(a) a−a(1) ; l(p)(f) −l(p)(a) f(1) −a(1) ) + s(r,f) ≤ t(r, a−l(p)(a) a−a(1) ; l(p)(f) −l(p)(a) f(1) −a(1) ) + s(r,f) = n(r, l(p)(f) −l(p)(a) f(1) −a(1) ) + s(r,f) ≤ n(r,a(1); f(1)) + s(r,f). (17) again m(r,a; f) = m(r, f(1) −a(1) f −a 1 f(1) −a(1) ) ≤ m(r,a(1); f(1)) + s(r,f) = t(r,f(1)) −n(r,a(1); f(1)) + s(r,f) = m(r,f(1)) −n(r,a(1); f(1)) + s(r,f) ≤ m(r,f) −n(r,a(1); f(1)) + s(r,f) = t(r,f) −n(r,a(1); f(1)) + s(r,f), i.e n(r,a(1); f(1)) ≤ n(r,a; f) + s(r,f). so from (17) we get n(r,a; f(1)) ≤ n(r,a; f) + s(r,f). (18) also n(r,a; f) ≤ na(r,a; f) + n(r,a; f | f(1) = a) ≤ n(r,a; f(1)) + s(r,f). (19) 225 i. kaish and n. gazi from (18) and (19) we get n(r,a; f(1)) = n(r,a; f) + s(r,f). (20) let h = (a−a(1))(l(p)(f) −l(p)(a)) − (a−l(p)(a))(f(1) −a(1)) f −a , which is defined in lemma 2.4. clearly t(r,h) = s(r,h). now t(r,f) = m(r,f) = m(r,a + 1 h {(a−a(1))(l(p)(f) −l(p)(a)) − (a−l(p))(f(1) −a(1))} ≤ m(r, (a−a(1))l(p)(f) − (a−l(p))f(1)) + s(r,f) ≤ m(r,f(1)) + s(r,f) = t(r,f(1)) + s(r,f) = m(r,f(1)) + s(r,f) ≤ m(r,f) + s(r,f) = t(r,f) + s(r,f). therefore t(r,f(1)) = t(r,f) + s(r,f). (21) if h is transcendental, then by lemma 2.4, m(r,a; f(1)) = s(r,f) and from (20) and (21) m(r,a; f) = s(r,f). so from lemma 2.5, f = l = λez, λ( 6= 0), is a constant. if h is rational, then by lemma 2.2 we see that f is of finite order. so by lemma 2.1 we get m(r,a; f) = s(r,f). therefore from lemma 2.5, f = l = λez, λ(6= 0) is a constant. 226 uniqueness of an entire function sharing a polynomial with its linear differential polynomial subcase 2.2. let f(1) ≡ l(p). now m(r,a; f) = m(r, a(1) f −a 1 a(1) ) ≤ m(r, a(1) f −a ) + s(r,f) = m(r, f(1) − (f(1) −a(1)) f −a + s(r,f) ≤ m(r, f(1) f −a ) + s(r,f) = m(r, l(p) f −a ) + s(r,f). (22) since p ≥ deg(a), by lemma of logarithmic derivative, m(r, l (p) f−a) = s(r,f), so from (22) m(r,a; f) = s(r,f). therefore from lemma 2.5, we get f = l = λez, λ(6= 0), is a constant. this completes the proof of the main theorem. 4 conclusions finally we arrive at the conclusion that a non-constant entire function sharing a polynomial with its linear differential polynomial with some conditions defined in theorem (1.1) belongs to the class of functions f = {λez : λ ∈ c\{0}}. 5 acknowledgements authors are thankful to all the referees for their remarkable suggestions and all the authors of various papers and books which have been consulted to built the work. references w. k. hayman. meromorphic functions. the clarendon press, oxford, 1964. g. jank, e. mues, and l. volkmann. meromorphe funktionen, die mit ihrer ersten und zweiten ableitung einen endlichen wert teilen. complex var. theory appl., 6:51–71, 1986. 227 i. kaish and n. gazi i. lahiri and i. kaish. an entire function sharing a polynomial with its derivatives. boll. unione mat. ital., 10:229–240, 2017. i. lain. nevanlinna theory and complex differential equations. watter de gruyter, new york, 1993. p. li. entire functions that share one value with their linear differential polynomials. kodai math. j., 22:446–457, 1999. e. mues and n. steinmetz. meromorphe functionen, die mit ihrer ableitung werte teilen. manuscripta math., 29:195–206, 1979. l. a. rubel and c. c. yang. values shared by an entire function and its derivative. lecture notes in math.(springer), 599:101–103, 1977. c. c. yang and h. x. yi. uniqueness theory of meromorphic functions. science press and kluwer academic publishers, new york, 2003. h. zhong. entire functions that share one value with their derivatives. kodai math. j., 18:250–259, 1995. 228 ratio mathematica 27 (2014) 37-47 issn:1592-7415 multivalued linear transformations of hyperspaces r. ameria, r. a. borzooeib, k. ghadimic a school of mathematics, statistic and computer sciences, college of science, university of tehran,tehran, iran rameri@ut.ac.ir b department of mathematics, shahid beheshti university, tehran, iran, borzooei@sbu.ac.ir c department of mathematics, payame noor university, tehran, iran, ghadimi@phd.pnu.ac.ir abstract the purpose of this paper is the study of multivalued linear transformations of hypervector spaces (or hyperspaces) in the sense of tallini. in this regards first we introduce and study various multivalued linear transformations of hyperspaces and then constitute the categories of hyperspaces with respect the different linear transformations of hyperspaces as the morphisms in these categories. also, we construct some algebraic hyperoperations on homk (v, w ), the set of all multivalued linear transformations from a hyperspace v into hyperspaces w , and obtaine their basic properties. finally, we construct the fundamental functor f from hvk , category of hyperspaces over field k into vk , the category of clasical vector space over k. key words: hypervector space, multivalued linear transformation, category,fundamental relation 2000 ams: 20n20 37 r. ameri, r. a. borzooei and k. ghadimi 1 introduction the theory of algebraic hyperstructures is a well-established branch of classical algebraic theory. hyperstructure theory was first proposed in 1934 by marty, who defined hypergroups and began to investigate their properties with applications to groups, rational fractions and algebraic functions [15]. it was later observed that the theory of hyperstructures has many applications in both pure and applied sciences; for example, semi-hypergroups are the simplest algebraic hyperstructures that possess the properties of closure and associativity. the theory of hyperstructures has been widely reviewed ([11], [12], [13],[14] and [20])( for more see [2, 3, 5, 6, 7, 8, 9]). m.s. tallini introduced the notion of hyperspaces(hypervector spaces) ([17], [18] and [19]) and studied basic properties of them. r. ameri and o. dehghan [2] introduced and studied dimension of hyperspaces and in [16] m. motameni et. el. studied hypermatrix. r. ameri in [1] introduced and studied categories of hypermodules. in this paper we introduce and study various types of multivalued linear transformations of hyperspaces. we will proceed by constructing various categories of hyperspaces based on various multilinear linear transformations of hyperspaces. also, we construct some hyperalgebraic structures on (homk(v, w ). finally, we construct the fundumental functor from category of hyperspaces and multilinear transformations, as morphisms into the category of vectorspces. 2 preliminaries the concept of hyperspace, which is a generalization of the concept of ordinary vector space. definition 2.1. let h be a set. a map . : h × h −→ p∗(h) is called hyperoperation or join operation, where p∗(h) is the set of all non-empty subsets of h. the join operation is extended to subsets of h in natural way, so that a.b is given by a.b = ⋃ {a.b : a ∈ a and b ∈ b}. the notations a.a and a.a are used for {a}.a and a.{a} respectively. generally, the singleton {a} is identified by its element a. definition 2.2. [17] let k be a field and (v, +) be an abelian group. we define a hyperspace over k to be the quadrupled (v, +,◦, k), where ◦ is a 38 multivalued linear transformations of hyperspaces mapping ◦ : k ×v −→ p∗(v ), such that the following conditions hold: (h1) ∀ a ∈ k, ∀ x, y ∈ v, a◦ (x + y) ⊆ a◦x + a◦y, right distributive law, (h2) ∀ a, b ∈ k, ∀ x ∈ v, (a + b) ◦x ⊆ a◦x + b◦x, left distributive law, (h3) ∀ a, b ∈ k, ∀ x ∈ v, a◦ (b◦x) = (ab) ◦x, associative law, (h4) ∀ a ∈ k, ∀ x ∈ v, a◦ (−x) = (−a) ◦x = −(a◦x), (h5) ∀ x ∈ v, x ∈ 1 ◦x. remark 2.3. (i) in the right hand side of (h1) the sum is meant in the sense of frobenius, that is we consider the set of all sums of an element of a◦x with an element of a◦y. similarly we have in (h2). (ii) we say that (v, +,◦, k) is anti-left distributive, if ∀ a, b ∈ k, ∀ x ∈ v, (a + b) ◦x ⊇ a◦x + b◦x, and strongly left distributive, if ∀ a, b ∈ k, ∀ x ∈ v, (a + b) ◦x = a◦x + b◦x, in a similar way we define the anti-right distributive and strongly right distributive hyperspaces, respectvely. v is called strongly distributive if it is both strongly left and strongly right distributive. (iii) the left hand side of (h3) means the set-theoretical union of all the sets a◦y, where y runs over the set b◦x, i.e. a◦ (b◦x) = ⋃ y∈b◦x a◦y. (iv) let ωv = 0 ◦ 0v , where 0v is the zero of (v, +), in [17] it is shown if v is either strongly right or left distributive, then ωv is a subgroup of (v, +). let v be a hyperspace over a field k. w ⊆ v is a subhyperspace of v , if w 6= ∅, w −w ⊆ w, ∀a ∈ k, a◦w ⊆ w. example 2.4. [2] consider abelian group (r2, +). define hyper-compositions{ ◦ : r×r2 −→ p∗(r2) a◦ (x, y) = ax×r and { � : r×r2 −→ p∗(r2) a� (x, y) = r×ay. then (r2, +,◦, r) and (r2, +,�, r) are a strongly distributive hyperspaces. 39 r. ameri, r. a. borzooei and k. ghadimi example 2.5. [2] let (v, +, ., k) be a classical vector space and p be a subspace of v . define the hyper-composition{ ◦ : k ×v −→ p∗(v ) a◦x = a.x + p. then it is easy to verify that (v, +,◦, k) is a strongly distributive hyperspace. example 2.6. [?] in (r2, +) define the hyper-composition ◦ as follows: ∀a ∈ r,∀x ∈ r2 : a◦x = { line ōx if x 6= 0v {0v} if x = 0v , where 0v = (0, 0). then (r2, +,◦, r) is a strongly left, but not right distributive hyperspace. proposition 2.7. [?] every strongly right distributive hyperspace is strongly left distributive hyperspace. let (v, +) be an abelian group, ω a subgroup of v and k a field such that w = v/ω is a classical vector space over k. if p : v −→ w is the canonical projection of (v, +) onto (w, +) and set:{ ◦ : k ×v −→ p∗(v ) a◦x = p−1(a.p(x)). then (v, +,◦, k) is a strongly distributive hyperspace over k. moreover every strongly distributive hyperspace can be obtained in such a way. proposition 2.8. [?] if (v, +,◦, k) be a left distributive hyperspace, then for all a ∈ k and x ∈ v 1) 0 ◦x is a subgroup of (v, +); 2) ωv is a subgroup of (v, +); 3) a◦ 0v = ωv = a◦ ωv ; 4) ωv ⊆ 0 ◦x; 5) x ∈ 0 ◦x ⇐⇒ 1 ◦x = 0 ◦x ⇐⇒ a◦x = 0 ◦x, ∀a ∈ k. remark 2.9. let (v, +,◦, k) be a hyperspace and w be a subhyperspace of v . consider the quotient abelian group (v/w, +). define the rule{ ∗ : k ×v/w −→ p∗(v/w ) (a, x + w ) 7−→ a◦x + w. then it is easy to verify that (v/w, +,∗, k) is a hyperspace over k and it is called the quotient hyperspace of v over w . 40 multivalued linear transformations of hyperspaces 3 multivalued linear transformations definition 3.1. let v and w be two hyperspaces over a field k. a multivalued linear transformation (mlt ) t : v −→ p∗(w ) is a mapping such that : ∀x, y ∈ v,∀a ∈ k 1) t (x + y) ⊆ t (x) + t (y); 2) t (a◦x) ⊆ a◦t (x); 3) t (−a) = −t (a). remark 3.2. (i) in definition 3.1(1) and (2), if the equality holds, then t is called a strong multivalued linear transformation (smlt ). (ii) in definition 3.1, if we consider t as a mapping t : v −→ w , then is it is called a linear transformation. here we consider only inclusion and equality cases. (iii) if t is a mlt , then 0 ∈ t (x), since t (x) 6= ∅, so ∃y ∈ t (x); 0 = y −y ∈ t (x) −t (x) = t (x) + t (−x) = t (x + (−x)) = t (x−x) = t (0). definition 3.3. [1] let v and w be two hyperspaces over a field k and t : v −→ p∗(w ) be a smlt . then multivalued kernel and multivalued image of t , denoted by kert and imt , respectively, are defined as follows: kert = {x ∈ v | 0w ∈ t (x)}; and imt = {y ∈ w | y ∈ t (x) for some x ∈ v}. remark 3.4. (i) note that kert 6= ∅, by remark 3.2(iii). (ii) for hyperspaces v and w over a field k, by homk(v, w ) and homsk(v, w ), we mean the set of all mlt and smlt , respectively and sometimes we use morphism instead multivalued linear transformation, respectively. also, by homk(v, w ) and hom s k(v, w ), we mean the set of all linear transformation lt and strong linear transformation slt respectively and sometimes we use morphism instead multivalued , respectively. in the following we briefly introduced the categories of hyperspaces and study the relationship between monomorphism, epimorphism, isomorphism andmonic, epic and iso objects in these category. definition 3.5. the category of hyperspaces over a field k denoted by hvk is defined as follows: 1) the objects of hvk are all hyperspaces over k; 41 ratio mathematica volume 39, 2020, pp. 165-186 kcd indices and coindices of graphs keerthi g. mirajkar* akshata morajkar† abstract the relationship between vertices of a graph is always an interesting fact, but the relations of vertices and edges also seeks attention. motivation of the relationship between vertices and edges of a graph has helped to arise with a set of new degree based topological indices and coindices named kcd indices and coindices. these indices and coindices are elaborated by establishing a set of properties. more fascinating results of some graph operations using kcd indices are developed in this article. keywords: kcd indices, kcd coindices, graph operations. 2010 ams subject classifications: 05c07, 05c76. 1 *department of mathematics, karnatak university’s karnatak arts college, dharwad 580 001, karnataka, india; keerthi.mirajkar@gmail.com †department of mathematics, karnatak university’s karnatak arts college, dharwad 580 001, karnataka, india; akmorajkar@gmail.com 1received on october 26th, 2020. accepted on december 17th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.550. issn: 1592-7415. eissn: 2282-8214. ©mirajkar et al. this paper is published under the cc-by licence agreement. 165 keerthi g. mirajkar and akshata morajkar 1 introduction graph theory plays a vital role in the quantification of chemical structures through topological indices. topological indices are molecular descriptors which characterize the topology of a graph through numerical parameters. abundant number of topological indices are identified these days. amongst these the first degree based topological indices are zagreb indices [gutman and trinajstić, 1972]. recently along with zagreb indices zagreb coindices is also gaining much attention for research. this has put forward versitile forms of zagreb indices of graphs. the present work aims to establish some new form of topological indices of graphs. this paper considers the graph to be simple, finite and undirected. the graph is denoted as g = (v, e) with |v (g)| = n as the vertex set and |e(g)| = m as the edge set. the set of vertices are also referred to as the order of the graph g and the edge set as the size of the graph g. the edge connecting the two vertices u and v is denoted as e = uv. the degree of the vertex u in a graph g is denoted as dg(u) and defined as the number of edges of a graph g incident with the vertex u. the degree of edge dg(e) of a graph g is defined as dg(e) = dg(u) + dg(v)−2. the complement g of a graph g is one in which two vertices are adjacent if and only if they are not adjacent in g. for g, |v (g)| = n, |e(g)| = m = ( n 2 ) − m [alwardi et al., 2018]. also uv ∈ e(g) ⇐⇒ uv /∈ e(g). the degree of a vertex u in g is denoted as dg(u) and defined as dg(u) = n − 1 − dg(u) [alwardi et al., 2018]. the degree of edge of g is represented as dg(e), defined as dg(e) = dg(u) + dg(v)−2. for undefined terminologies refer [harary, 1969]. the zagreb indices were defined by gutman and trinajstić [gutman and trinajstić, 1972] as m1(g) = ∑ u∈v (g) dg(u) 2 (1) m2(g) = ∑ uv∈e(g) dg(u)dg(v) . (2) here m1(g) refers first zagreb index and m2(g) refers second zagreb index. first zagreb index is also expressed as [došlić, 2008, došlic et al., 2011] m1(g) = ∑ uv∈e(g) ( dg(u) + dg(v) ) . (3) for properties and information on zagreb indices refer [gutman and das, 2004, zhou and gutman, 2005, zhou, 2004]. 166 kcd indices and coindices of graphs further, zagreb coindices were introduced by došlić [došlić, 2008] as m1(g) = ∑ uv /∈e(g) ( dg(u) + dg(v) ) (4) m2(g) = ∑ uv /∈e(g) dg(u)dg(v). (5) the detailed study on zagreb coindices is reported in [ashrafi et al., 2010, 2011], the association between zagreb indices and coindices is encountered in [das et al., 2012, gutman et al., 2015]. shirdel et al. [shirdel et al., 2013] defined hyper zagreb index as hm(g) = ∑ uv∈e(g) ( dg(u) + dg(v) )2 . (6) further, hyper zagreb coindex was introduced as hm(g) = ∑ uv /∈e(g) ( dg(u) + dg(v) )2 . (7) these graph invariants were studied in [pattabiraman and vijayaragavan, 2017, veylaki et al., 2016]. relationship between hyper zagreb index and coindex is established in [gutman, 2017]. now, we introduce a set of new degree-based topological indices and coindices named as karnatak college dharwad indices and coindices or kcd indices and coindices in short, which is dedicated to karnatak college dharwad as the college has completed hundred years of its service in education to the society in the year 2017. further the research supervisior and research scholar belong to the same college. i.e., the first and second kcd indices of a graph g are respectively kcd1(g) = ∑ e=uv∈e(g) (( dg(u) + dg(v) ) + dg(e) ) (8) kcd2(g) = ∑ e=uv∈e(g) ( dg(u) + dg(v) ) dg(e). (9) we proceed further to define kcd coindices as follows kcd1(g) = ∑ e=uv /∈e(g) (( dg(u) + dg(v) ) + dg(e) ) (10) kcd2(g) = ∑ e=uv /∈e(g) ( dg(u) + dg(v) ) dg(e). (11) 167 keerthi g. mirajkar and akshata morajkar here kcd1(g) and kcd2(g) are first and second kcd coindices of a graph g respectively. the remaining paper is distributed as follows. section 2 expresses the properties of first kcd indices and coindices of a graph and its complement. section 3 concentrates on properties of second kcd indices and coindices of a graph and its complement, while section 4 is devoted for the study of kcd indices of certain graph operations. the following previously known results are considered for present investigation. theorem 1.1. [gutman et al., 2015] let g be a graph with n vertices and m edges. then, m1(g) = m1(g) + n(n−1)2 −4m(n−1) (12) m1(g) = 2m(n−1)−m1(g). (13) corollary 1.2. [gutman et al., 2015] let g be any graph and g its complement. then m1(g) = m1(g). (14) theorem 1.3. [gutman, 2017] let g be a graph with n vertices and m edges. then, hm(g) = 4m2 + (n−2)m1(g)−hm(g) (15) hm(g) = 2n(n−1)3 −12m(n−1)2 + 4m2 (16) +(5n−6)m1(g)−hm(g) hm(g) = 4m(n−1)2 + 4(n−1)m1(g) + hm(g). (17) 2 basic properties of first kcd indices and coindices theorem 2.1. let g be a graph with n vertices and m edges. then, kcd1(g) = (4n−6)m−4m(n−1) + 2m1(g) (18) kcd1(g) = 4n(m−m)−6m + 4m + 2m1(g) (19) kcd1(g) = 4m(n−1)−2 ( m + m1(g) ) (20) kcd1(g) = (4n−6)m−2m1(g). (21) proof. proof of eq. (18): 168 kcd indices and coindices of graphs for any vertex u of g, dg(u) = n−1−dg(u). (22) and for any edge e = uv of g, dg(e) = 2n−4− ( dg(u) + dg(v) ) . (23) thus by eqs. (8), (22) and (23), we have kcd1(g) = ∑ e=uv∈e(g) (( dg(u) + dg(v) ) + dg(e) ) = ∑ e=uv /∈e(g) (( n−1−dg(u) + n−1−dg(v) ) + ( 2n−4− (dg(u) + dg(v) )) = ∑ e=uv /∈e(g) ( 4n−6−2 ( dg(u) + dg(v) )) = (4n−6)m−2 ∑ e=uv /∈e(g) ( dg(u) + dg(v) ) . according to eq. (4) m1(g) = ∑ uv /∈e(g) ( dg(u) + dg(v) ) . hence, kcd1(g) = (4n−6)m−2m1(g) (24) substitution of eqs. (13) and (14) in (24) results into eq. (18). proof of eq. (19): for any vertex u of the complement g, dg(u) = n−1−dg(u). (25) 169 keerthi g. mirajkar and akshata morajkar and for any edge e = uv of the complement g, dg(e) = 2n−4− ( dg(u) + dg(v) ) . (26) bearing in mind eqs. (8), (25) and (26), we get kcd1(g) = ∑ e=uv∈e(g) (( dg(u) + dg(v) ) + dg(e) ) = ∑ e=uv /∈e(g) (( n−1−dg(u) + n−1−dg(v) ) + ( 2n−4− (dg(u) + dg(v) )) = ∑ e=uv /∈e(g) ( 4n−6−2 ( dg(u) + dg(v) )) = (4n−6)m−2 ∑ e=uv /∈e(g) ( dg(u) + dg(v) ) . thus by eq. (4), kcd1(g) = (4n−6)m−2m1(g) (27) employing eq. (13) in (27) generates eq. (19). proof of eq. (20): using eqs. (10), (22) and (23), we have kcd1(g) = ∑ e=uv /∈e(g) (( dg(u) + dg(v) ) + dg(e) ) = ∑ e=uv∈e(g) (( n−1−dg(u) + n−1−dg(v) ) + ( 2n−4− (dg(u) + dg(v) )) = ∑ e=uv∈e(g) ( 4n−6−2 ( dg(u) + dg(v) )) . 170 kcd indices and coindices of graphs by eq. (3) m1(g) = ∑ uv∈e(g) ( dg(u) + dg(v) ) . thus, kcd1(g) = (4n−6)m−2m1(g) (28) substitution of eq. (12) in (28) gives eq. (20). proof of eq. (21): in view of eq. (10), (25) and (26), we get kcd1(g) = ∑ e=uv /∈e(g) (( dg(u) + dg(v) ) + dg(e) ) = ∑ e=uv∈e(g) (( n−1−dg(u) + n−1−dg(v) ) + ( 2n−4− (dg(u) + dg(v) )) = ∑ e=uv∈e(g) ( 4n−6−2 ( dg(u) + dg(v) )) = (4n−6)m−2 ∑ e=uv∈e(g) ( dg(u) + dg(v) ) considering eq. (3) we directly arrive at eq. (21). 2 171 keerthi g. mirajkar and akshata morajkar 3 basic properties of second kcd indices and coindices theorem 3.1. let g be a graph with n vertices and m edges. then, kcd2(g) = hm(g)−2m1(g) (29) kcd2(g) = 4(n−1) ( m(n−2)−m(2n−3) ) + 4m2 (30) +(5n−8)m1(g)−hm(g) kcd2(g) = 4(n−1)(n−2)m− (4n−6)(n−1) ( n(n−1)−4m ) (31) +2(n−1)2 ( n(n−1)−6m ) + 4m2 + nm1(g)−hm(g) kcd2(g) = 4(n−1)(n−2)m− (4n−6)m1(g) + hm(g). (32) proof. proof of eq. (29): considering eqs. (9), (22) and (23), we have kcd2(g) = ∑ e=uv∈e(g) ( dg(u) + dg(v) ) dg(e) = ∑ e=uv /∈e(g) ( n−1−dg(u) + n−1−dg(v) )( 2n−4− (dg(u) + dg(v) ) = ∑ e=uv /∈e(g) 4(n−1)(n−2)− (4n−6) ∑ e=uv /∈e(g) ( dg(u) + dg(v) ) + ∑ e=uv /∈e(g) ( dg(u) + dg(v) )2 . by an analogous reasoning, m1(g) = ∑ uv /∈e(g) ( dg(u) + dg(v) ) and hm(g) = ∑ uv /∈e(g) ( dg(u) + dg(v) )2 . thus, kcd2(g) = 4m(n−1)(n−2)− (4n−6)m1(g) + hm(g). in view of eq. (14) kcd2(g) = 4m(n−1)(n−2)− (4n−6)m1(g) + hm(g) (33) 172 kcd indices and coindices of graphs taking into account eqs. (13) and (17), eq. (33) results into eq. (29). proof of eq. (30): in view of eqs. (9), (25) and (26), we get kcd2(g) = ∑ e=uv∈e(g) ( dg(u) + dg(v) ) dg(e) = ∑ e=uv /∈e(g) ( n−1−dg(u) + n−1−dg(v) )( 2n−4− (dg(u) + dg(v) ) = ∑ e=uv /∈e(g) 4(n−1)(n−2)− (4n−6) ∑ e=uv /∈e(g) ( dg(u) + dg(v) ) + ∑ e=uv /∈e(g) ( dg(u) + dg(v) )2 . by eqs. (4) and (7), it directly follows kcd2(g) = 4m(n−1)(n−2)− (4n−6)m1(g) + hm(g) (34) application of eqs. (13) and (15) to eq. (34) yields eq. (30). proof of eq. (31): using eqs. (11), (22) and (23), we have kcd2(g) = ∑ e=uv /∈e(g) ( dg(u) + dg(v) ) dg(e) = ∑ e=uv∈e(g) ( n−1−dg(u) + n−1−dg(v) ) ( 2n−4− (dg(u) + dg(v) ) = ∑ e=uv∈e(g) 4(n−1)(n−2)− (4n−6) ∑ e=uv∈e(g) ( dg(u) + dg(v) ) + ∑ e=uv∈e(g) ( dg(u) + dg(v) )2 . by reasoning, m1(g) = ∑ uv∈e(g) ( dg(u) + dg(v) ) and hm(g) = ∑ uv∈e(g) ( dg(u) + dg(v) )2 . 173 keerthi g. mirajkar and akshata morajkar hence kcd2(g) = 4m(n−1)(n−2)− (4n−6)m1(g) + hm(g) (35) substituting eqs. (12) and (16) in eq. (35), simple calculation yields eq. (31). proof of eq. (32): with the help of eqs. (11), (25) and (26), we get kcd2(g) = ∑ e=uv /∈e(g) ( dg(u) + dg(v) ) dg(e) = ∑ e=uv∈e(g) ( n−1−dg(u) + n−1−dg(v) )( 2n−4− (dg(u) + dg(v) ) = ∑ e=uv∈e(g) 4(n−1)(n−2)− (4n−6) ∑ e=uv∈e(g) ( dg(u) + dg(v) ) + ∑ e=uv∈e(g) ( dg(u) + dg(v) )2 eq. (32) immediately follows. 2 4 kcd indices of some graph operations in this section, we study the graph operations using kcd indices. the well-known graph operations sum(join), cartesian product and composition of graphs are considered. all operations considered under the context are binary, with finite and simple graphs g and h. for the graphs g and h vertex and edge sets are denoted by v (g) and v (h), e(g) and e(h) respectively. the detailed information on sum(join) of graphs is refered in[khalifeh et al., 2008a], cartesian product of graphs studied in[khalifeh et al., 2008b] and composition of graphs is reported in [imrich and klavzar, 2000, khalifeh et al., 2008a]. we refer [khalifeh et al., 2009] for detailed information about graph operations. sum(join): the sum(join) g + h of two graphs g and h with disjoint vertex sets |v (g)| and |v (h)| is the graph on the vertex set v (g) ∪ v (h) and the edge set e(g) ∪ e(h) ∪{uv : u ∈ v (g) and v ∈ v (h)}. for the graph g + h, |v (g+h)| = |v (g)|+v (h)|, |e(g+h)| = |e(g)|+|e(h)|+|v (g)||v (h)|, 174 kcd indices and coindices of graphs the degree of any vertex u ∈ g + h is dg+h(u) = { dg(u) + |v (h)| u ∈ v (g) dh(u) + |v (g)| u ∈ v (h). theorem 4.1. let g and h be graphs. then kcd1(g + h) = 2 ( m1(g) + m1(h) + |e(h)| ( 4|v (g)|−1 ) +|e(g)| ( 4|v (h)|−1 ) +|v (g)||v (h)| ( |v (g)|+ |v (h)|−1 )) . proof: by definition of sum(join) g + h of two graphs g, h and eq. (8), we have kcd1(g + h) = ∑ e=uv∈e(g+h) (( dg+h(u) + dg+h(v) ) + dg+h(e) ) . since, dg+h(e) = dg+h(u) + dg+h(v)−2. kcd1(g + h) = 2 ∑ e=uv∈e(g+h) ( dg+h(u) + dg+h(v)−1 ) . kcd1(g + h) = 2 ∑ e=uv∈e(h) ( dg+h(u) + dg+h(v)−1 ) (36) +2 ∑ e=uv∈e(g) ( dg+h(u) + dg+h(v)−1 ) +2 ∑ e=uv∈{uv:u∈v (g),v∈v (h)} ( dg+h(u) + dg+h(v)−1 ) . observe that,∑ e=uv∈e(h) ( dg+h(u) + dg+h(v)−1 ) = ∑ e=uv∈e(h) ( dh(u) + |v (g)| +dh(v) + |v (g)|−1 ) = ∑ e=uv∈e(h) ( dh(u) + dh(v) + 2|v (g)|−1 ) . 175 keerthi g. mirajkar and akshata morajkar thus, ∑ e=uv∈e(h) ( dg+h(u) + dg+h(v)−1 ) = m1(h) + 2|v (g)||e(h)| (37) −|e(h)|. similarly,∑ e=uv∈e(g) ( dg+h(u) + dg+h(v)−1 ) = m1(g) + 2|v (h)||e(g)| (38) −|e(g)|. in the same way,∑ u∈v (g),v∈v (h) ( dg+h(u) + dg+h(v)−1 ) = 2|v (h)||e(g)|+ |v (h)|2|v (g)| +2|e(h)||v (g)|+ |v (g)|2|v (h)|− |v (g)||v (h)|. (39) substituting eqs. (37), (38) and (39) in eq. (36) completes the proof. 2 theorem 4.2. let g and h be graphs. then kcd2(g + h) = hm(g) + hm(h) + ( 5|v (h)|−2 ) m1(g) + ( 5|v (g)|−2 ) m1(h) +8 ( |v (g)||e(h)| ( |v (g)|−1 ) + |v (h)||e(g)| ( |v (h)|−1 ) + |e(g)||e(h)| ) +|v (g)||v (h)| (( |v (g)|+ |v (h)| )2 + 4 ( |e(g)|+ |e(h)| ) −2 ( |v (g)|+ |v (h)| )) . proof. with the knowledge of sum(join) g+h of two graphs g, h and eq. (9), we have kcd2(g + h) = ∑ e=uv∈e(g+h) ( dg+h(u) + dg+h(v) ) dg+h(e). as, dg+h(e) = dg+h(u) + dg+h(v)−2. 176 kcd indices and coindices of graphs this implies, kcd2(g + h) = ∑ e=uv∈e(g+h) ( dg+h(u) + dg+h(v) )2 −2 ( dg+h(u) + dg+h(v) ) = ∑ e=uv∈e(h) ( dg+h(u) + dg+h(v) )2 −2 ( dg+h(u) + dg+h(v) ) + ∑ e=uv∈e(g) ( dg+h(u) + dg+h(v) )2 −2 ( dg+h(u) + dg+h(v) ) + ∑ e=uv∈{uv:u∈v (g),v∈v (h)} ( dg+h(u) + dg+h(v) )2 −2 ( dg+h(u) + dg+h(v) ) . it follows that, ∑ e=uv∈e(h) ( dg+h(u) + dg+h(v) )2 −2 ( dg+h(u) + dg+h(v) ) = ∑ e=uv∈e(h) (( dh(u) +|v (g)|+ dh(v) + |v (g)| )2 −2 ( dh(u) + |v (g)|+ dh(v) + |v (g)| )) = ∑ e=uv∈e(h) (( dh(u) + dh(v) )2 + 4|v (g)|2 + 4|v (g)| ( dh(u) + dh(v) ) −2 ( dh(u) +dh(v) ) −4|v (g)| ) . ∑ e=uv∈e(h) ( dg+h(u) + dg+h(v) )2 −2 ( dg+h(u) + dg+h(v) ) = hm(h) +4|v (g)|2|e(h)|+ 4|v (g)|m1(h)−2m1(h)−4|v (g)||e(h)|. (40) similarly,∑ e=uv∈e(g) ( dg+h(u) + dg+h(v) )2 −2 ( dg+h(u) + dg+h(v) ) = hm(g) +4|v (h)|2|e(g)|+ 4|v (h)|m1(g)−2m1(g)−4|v (h)||e(g)|. (41) 177 keerthi g. mirajkar and akshata morajkar in the same way ∑ u∈v (g),v∈v (h) ( dg+h(u) + dg+h(v) )2 −2 ( dg+h(u) + dg+h(v) ) = m1(g)|v (h)| +m1(h)|v (g)|+ 8|e(g)||e(h)|+ |v (g)||v (h)| ( |v (g)|+ |v (h)| )2 +4|e(g)||v (h)|2 + 4|e(g)||v (g)||v (h)|+ 4|e(h)||v (g)||v (h)| +4|e(h)||v (g)|2 −4|e(g)||v (h)|−4|e(h)||v (g)| −2|v (g)||v (h)| ( |v (g)|+ |v (h)| ) . (42) finally, the summaton of eqs. (40), (41) and (42) gives the desired result. 2 cartesian product: the cartesian product g × h of two graphs g and h has the vertex set v (g × h) = v (g) × v (h) and e = (a, x)(b, y) is an edge of g × h if a = b and xy ∈ e(h), or ab ∈ e(h) and x = y. for the graph g×h, |v (g×h)| = |v (g)|v (h)|, |e(g×h)| = |e(g)||v (h)|+|v (g)||e(h)|, the degree of any vertex (a, x) ∈ g×h is dg×h((a, x)) = dg(a) + dh(x). theorem 4.3. let g and h be graphs. then kcd1(g×h) = 2 ( |v (g)|m1(h) + |v (h)|m1(g) + 8|e(g)||e(h)|− ( |v (g)||e(h)|+ |v (h)||e(g)| )) . proof. in the view of definition of cartesian product g × h of two graphs g, h and eq. (8), we have kcd1(g×h) = ∑ e=(a,x)(b,y)∈e(g×h) (( dg×h((a, x)) + dg×h((b, y)) ) + dg×h((e)) ) . it is known that, dg×h((e)) = dg×h((a, x)) + dg×h((b, y))−2. 178 kcd indices and coindices of graphs thus, kcd1(g×h) = 2 ∑ e=(a,x)(b,y)∈e(g×h) ( dg×h((a, x)) + dg×h((b, y))−1 ) = 2 ∑ a∈v (g) ∑ xy∈e(h) ( dg(a) + dh(x) + dg(a) + dh(y)−1 ) +2 ∑ x∈v (h) ∑ ab∈e(g) ( dh(x) + dg(a) + dh(x) + dg(b)−1 ) = 2 ∑ a∈v (g) ∑ xy∈e(h) ( 2dg(a) + ( dh(x) + dh(y) ) −1 ) +2 ∑ x∈v (h) ∑ ab∈e(g) ( 2dh(x) + ( dg(a) + dg(b) ) −1 ) by simple reasoning we straightforwardly obtain the required result. 2 theorem 4.4. let g and h be graphs. then kcd2(g×h) = |v (g)|hm(h) + |v (h)|hm(g) + ( 12|e(h)|−2|v (h)| ) m1(g) + ( 12|e(g)|−2|v (g)| ) m1(h)−16|e(g)||e(h)|. proof. taking into account the definition of cartesian product g × h of two graphs g and h, start with eq. (9) as kcd2(g×h) = ∑ e=(a,x)(b,y)∈e(g×h) ( dg×h((a, x)) + dg×h((b, y)) ) dg×h((e)). since dg×h((e)) = dg×h((a, x)) + dg×h((b, y))−2. 179 keerthi g. mirajkar and akshata morajkar we have kcd2(g×h) = ∑ e=(a,x)(b,y)∈e(g×h) (( dg×h((a, x)) + dg×h((b, y)) )2 −2 ( dg×h((a, x)) + dg×h((b, y)) )) = ∑ a∈v (g) ∑ xy∈e(h) (( dg(a) + dh(x) + dg(a) + dh(y) )2 −2 ( dg(a) + dh(x) + dg(a) + dh(y) )) + ∑ x∈v (h) ∑ ab∈e(g) (( dh(x) +dg(a) + dh(x) + dg(b) )2 −2 ( dh(x) + dg(a) + dh(x) + dg(b) )) = ∑ a∈v (g) ∑ xy∈e(h) (( 2dg(a) + dh(x) + dh(y) )2 −2 ( 2dg(a) + dh(x) +dh(y) )) + ∑ x∈v (h) ∑ ab∈e(g) (( 2dh(x) + dg(a) + dg(b) )2 −2 ( 2dh(x) + dg(a) + dg(b) )) kcd2(g×h) = ∑ a∈v (g) ∑ xy∈e(h) ( 4 ( dg(a) )2 + ( dh(x) + dh(y) )2 +4dg(a) ( dh(x) + dh(y) ) −2 ( 2dg(a) + ( dh(x) + dh(y) ))) + ∑ x∈v (h) ∑ ab∈e(g) ( 4 ( dh(x) )2 + ( dg(a) + dg(b) )2 +4dh(x) ( dg(a) + dg(b) ) −2 ( 2dh(x) + ( dg(a) + dg(b) ))) and the required result immediately follows. 180 kcd indices and coindices of graphs 2 composition: the composition g[h] of two graphs g and h with disjoint vertex sets v (g) and v (h), edge sets e(g) and e(h) is the graph with vertex set v (g) × v (h) and (a,x) is adjacent to (b,y) whenever a is adjacent to b, or a = b and x is adjacent to y. for the graph g[h], |v (g[h])| = |v (g)||v (h)|, |e(g[h])| = |e(g)||v (h)|2 + |e(h)||v (g)|, the degree of any vertex (a, x) ∈ g[h] is dg[h]((a, x)) = |v (h)|dg(a) + dh(x). theorem 4.5. let g and h be graphs. then kcd1(g[h]) = 2 ( |v (h)|3m1(g) + |v (g)|m1(h) + 8|v (h)||e(g)||e(h)| −|v (h)|2|e(g)|− |e(h)||v (g)| ) . proof. using the definition of composition g[h] of two graphs g, h and eq. (8), we have kcd1(g[h]) = ∑ e=(a,x)(b,y)∈e(g[h]) (( dg[h]((a, x)) + dg[h]((b, y)) ) + dg[h]((e)) ) . but dg[h]((e)) = dg[h]((a, x)) + dg[h]((b, y))−2. this implies, kcd1(g[h]) = 2 ∑ e=(a,x)(b,y)∈e(g[h]) ( dg[h]((a, x)) + dg[h]((b, y))−1 ) . kcd1(g[h]) = 2 ∑ x∈v (h) ∑ y∈v (h) ∑ ab∈e(g) ( |v (h)|dg(a) + dh(x) + |v (h)|dg(b) +dh(y)−1 ) + 2 ∑ a∈v (g) ∑ xy∈e(h) ( |v (h)|dg(a) + dh(x) +|v (h)|dg(a) + dh(y)−1 ) . (43) we start with 181 keerthi g. mirajkar and akshata morajkar ∑ x∈v (h) ∑ y∈v (h) ∑ ab∈e(g) ( |v (h)|dg(a) + dh(x) + |v (h)|dg(b) + dh(y)−1 ) = ∑ x∈v (h) ∑ y∈v (h) ∑ ab∈e(g) ( |v (h)| ( dg(a) + dg(b) ) + ( dh(x) + dh(y) ) −1 ) thus,∑ x∈v (h) ∑ y∈v (h) ∑ ab∈e(g) ( |v (h)|dg(a) + dh(x) + |v (h)|dg(b) + dh(y)−1 ) = |v (h)|3m1(g) + 4|v (h)||e(g)||e(h)|− |v (h)|2|e(g)|. (44) similarly,∑ a∈v (g) ∑ xy∈e(h) ( |v (h)|dg(a) + dh(x) + |v (h)|dg(a) + dh(y)−1 ) = 4|v (h)||e(g)||e(h)|+ |v (g)|m1(h)−|v (g)||e(h)|. (45) substituting eqs. (44) and (45) in eq. (43) generates the desired result. 2 theorem 4.6. let g and h be graphs. then kcd2(g[h]) = |v (h)|4hm(g) + |v (g)|hm(h) +2|v (h)|2m1(g) ( 6|e(h)|− |v (h)| ) +2m1(h) ( 5|v (h)||e(g)|− |v (g)| ) +8|e(g)||e(h)| ( |e(h)|−2|v (h)| ) . proof. in view of definition of composition g[h] of two graphs g, h and eq. (9), we start with kcd2(g[h]) = ∑ e=(a,x)(b,y)∈e(g[h]) ( dg[h]((a, x)) + dg[h]((b, y)) ) dg[h]((e)). it is known that, dg[h]((e)) = dg[h]((a, x)) + dg[h]((b, y))−2. 182 kcd indices and coindices of graphs we get, kcd2(g[h]) = ∑ e=(a,x)(b,y)∈e(g[h]) (( dg[h]((a, x)) + dg[h]((b, y)) )2 −2 ( dg[h]((a, x)) + dg[h]((b, y)) )) . kcd2(g[h]) = ∑ x∈v (h) ∑ y∈v (h) ∑ ab∈e(g) (( |v (h)|dg(a) + dh(x) + |v (h)|dg(b) + dh(y) )2 −2 ( |v (h)|dg(a) + dh(x) + |v (h)|dg(b) + dh(y) )) + ∑ a∈v (g) ∑ xy∈e(h) (( |v (h)|dg(a) + dh(x) + |v (h)|dg(a) + dh(y) )2 −2 ( |v (h)|dg(a) + dh(x) + |v (h)|dg(a) + dh(y) )) . thus, kcd2(g[h]) = ∑ x∈v (h) ∑ y∈v (h) ∑ ab∈e(g) (( |v (h)| ( dg(a) + dg(b) ) + dh(x) + dh(y) )2 −2 ( |v (h)| ( dg(a) + dg(b) ) + dh(x) + dh(y) )) + ∑ a∈v (g) ∑ xy∈e(h) (( 2|v (h)|dg(a) + dh(x) + dh(y) )2 −2 ( 2|v (h)|dg(a) + dh(x) + dh(y) )) . 183 keerthi g. mirajkar and akshata morajkar it follows that, ∑ x∈v (h) ∑ y∈v (h) ∑ ab∈e(g) (( |v (h)| ( dg(a) + dg(b) ) + dh(x) + dh(y) )2 −2 ( |v (h)| ( dg(a) + dg(b) ) + dh(x) + dh(y) )) = ∑ x∈v (h) ∑ y∈v (h) ∑ ab∈e(g) ( |v (h)|2 ( dg(a) + dg(b) )2 + ( dh(x) + dh(y) )2 +2|v (h)| ( dg(a) + dg(b) )( dh(x) + dh(y) ) −2|v (h)| ( dg(a) + dg(b) ) −2|v (h)|dh(x)−2|v (h)|dh(y) ) hence, ∑ x∈v (h) ∑ y∈v (h) ∑ ab∈e(g) (( |v (h)| ( dg(a) + dg(b) ) + dh(x) + dh(y) )2 −2 ( |v (h)| ( dg(a) + dg(b) ) + dh(x) + dh(y) )) = |v (h)|4hm(g) +2|v (h)||e(g)|m1(h) + 8|e(h)|2|e(g)|+ 8|v (h)|2|e(h)|m1(g) −2|v (h)|3m1(g)−8|e(h)||e(g)||v (h)|. (46) similarly, ∑ a∈v (g) ∑ xy∈e(h) (( 2|v (h)|dg(a) + dh(x) + dh(y) )2 −2 ( 2|v (h)|dg(a) +dh(x) + dh(y) )) = 4|v (h)|2|e(h)|m1(g) + |v (g)|hm(h) +8|v (h)||e(g)|m1(h)−8|v (h)||e(g)||e(h)|−2|v (g)|m1(h). (47) finally, summation of eqs. 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reshmikm@gmail.com. †university of calicut, malappuram, india-673365; rajipilakkat@gmail.com. 1received on september 24, 2022. accepted on june 18, 2023. published on june 30, 2023. doi: 10.23755/rm.v39i0.860. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 260 reshmi k m, raji pilakkat 1 introduction topological indices and centrality measures are graph invariant. numerous studies have been carried out in these areas. the first notable topological index was the wiener index, named after harry wiener, a pioneer in chemical graph theory. it is defined as the sum of the lengths of the shortest paths between all pairs of vertices in the chemical graph representing the non-hydrogen atoms in the molecule. wiener made fundamental contributions to the study of topological indices and established a correlation between the wiener index and boiling points (hence viscosity and surface tension) of the paraffin. he could establish relationships with many chemical properties of alkanes with the wiener index. centrality measures are a vital tool for understanding graphs. each measure has its own definition of importance. some are based on the degree of a vertex while others take the closeness to other vertices as the score of significance. in the paper [shimbel and alfonso], they introduced the concept of the stress of a vertex. it is the number of shortest paths on which a vertex lies. this was further modified to produce measures of centrality. it found applications in social networking, for analyzing communication dynamics. keeping in mind the above two concepts, we introduced a new index, called the transit index of a graph. it considers the distances in the graph as well as the degree of vertices. in computing the stress of a vertex, we only take into account the number of shortest paths through it; the length of the paths is not considered. be it in data transmission or in the measure of closeness, the length of the paths also matters. hence, in the computation of transit we account for the number of shortest paths as well as their length. graph products are binary operations. two graphs g1 and g2 are combined to produce a new graph h. in this paper, we study the transit of vertices in the corona product of graphs. information on individual graphs is used to compute the transit of vertices in their corona products. thus the transit of vertices in large graphs and networks, which can be viewed as corona products of simple graphs, can be computed more efficiently. 2 preliminaries in this section, we come across certain definitions and terminologies employed in developing results in the latter sections. throughout this paper we only consider simple connected and finite graphs. definition 2.1 (k.m.reshmi and pilakkat. raji [2020]). let v ∈ v . then the transit of v denoted by t(v) is “the sum of the lengths of all shortest paths with v 261 transit in corona product of graphs as an internal vertex” and the transit index of g denoted by ti(g) is ti(g) = ∑ v∈v t(v) lemma 2.2 (k.m.reshmi and pilakkat. raji [2020]). t(v) = 0 iff ⟨n[v]⟩ is a clique. theorem 2.3 (k.m.reshmi and pilakkat. raji [2020]). for a path pn, transit index is ti(pn) = n(n + 1)(n2 − 3n + 2) 12 theorem 2.4. for a cycle, the transit of any vertex v is, t(v) = (n 2−4)n 24 and i) ti(cn) = n2(n2−4) 24 ,n even. ii) ti(cn) = n(n2−1)(n−3) 24 , n odd definition 2.5. two vertices of a graph are called transit identical if the shortest paths passing through it are same in number and length. we use the following terminologies. the order of a graph g, denoted by |g| is the number of vertices in v (g). the distance between two vertices u, v ∈ v is the length of any shortest u − v path in g. a shortest path from u to v is also called a u − v geodesic. the number of shortest u − v paths is denoted by σ(u, v) and the number of shortest u − v path with ’a’ as an internal vertex is denoted by σ(u, v/a). it can be noted that a vertex ’a’ lies on a shortest u − v path iff d(u, v) = d(u, a) + d(a, v). the number of shortest u − v path with ’a’ as an internal vertex can be computed as σ(u, v/a) = σ(u, a) × σ(a, v). number of shortest paths in g with ’a’ as an internal vertex is denoted by σg(a). clearly σg(a) = ∑ (u,v) σ(u, v/a) 3 transit of vertices in corona product of graphs 3.1 corona product of graphs definition 3.1. (frucht and harary [1970]) let g1 and g2 be two graphs. the corona product g1 ◦ g2, is obtained by taking one copy of g1 and |v (g1)| copies of g2; and by joining each vertex of the i-th copy of g2 to the i-th vertex of g1, where 1 ≤ i ≤ |v (g1)| 262 reshmi k m, raji pilakkat whenever we consider g1 ◦ g2, we use the following notations. 1. gi2 the ith copy of g2 in g1 ◦ g2 2. v (g1) = {u1, u2, . . . , un1}, |e(g1)| = m1 3. v (gi2) = {vi1, vi2, . . . , vin2}, |e(g i 2)| = m2, ∀i lemma 3.2. (agnes [2015]) let g1 and g2 be two arbitrary graphs. then, • dg1◦g2(ui, up) = dg1(ui, up), 0 ≤ i, p ≤ n1 • dg1◦g2(ui, v p q) = dg1(ui, up) + 1, 0 ≤ i, p ≤ n1, 0 ≤ q ≤ n2 • dg1◦g2(v i j, v p q) =   dg1(ui, up) + 2, i ̸= p 1, if i = p, vjvq ∈ e(g2) 2, if i = p, vjvq /∈ e(g2) proposition 3.3. for any two graphs g1 and g2, 1. tg2(a) = 0 iff tg1◦g2(a) = 0, a ∈ g2 2. tg2(a) = tg1◦g2(a), for a ∈ g2 iff every shortest path in g2 with ’a’ as an internal vertex is of length 2. proof. note that as the result 2 is obvious, we prove only 1. 1) tg2(a) = 0 ⇐⇒ ⟨ng2[a]⟩ is a clique. ⇐⇒ ⟨ng1◦g2[a]⟩ is a clique⇐⇒ tg1◦g2(a) = 0. proposition 3.4. let g1 and g2 be arbitrary graphs, 1)for any up in g1, σg1◦g2(up) = (n2 + 1) [ (n2 + 1)σg1(up) + n2 ∑n1 p̸=k=1 σg1(up, uk) ] 2) σg1◦g2(v i k) = number of geodesic of length 2 in g2 with vk as an internal vertex. proof. 1)let up be any vertex of g1. every geodesic in g1 with up as an internal vertex will be counted in σg1◦g2(up). for k ̸= l, geodesics connecting gk2 ∪{uk} to gl2 ∪{ul} will have ul −uk geodesic as its part. let p1 be one of the ul − uk geodesic with up as an internal vertex. then p1 will be part of a geodesic connecting vertices of gk2 ∪ {uk} to vertices of gl2 ∪ {ul}. there will be n22 geodesics connecting g k 2 to g l 2, n2 geodesics connecting gk2 to ul and 263 transit in corona product of graphs n2 geodesics connecting gl2 to uk, with p1 as its part. hence for the pair of vertices (uk, ul), there will be (n22 + 2n2 + 1)σg1(up) geodesics with up as an internal vertex. the geodesics connecting vertices of gp2 to other vertices of g1 ◦ g2 will have up − uk geodesic as a part for some k. if p2 is one of the up − uk geodesic, it will be part of n22 geodesics connecting g k 2 to g p 2 and n2 geodesics connecting g p 2 to uk. hence for every up − uk geodesic in g1 there will be σg1(up, uk)[n2(n2 +1)] geodesics in g1 ◦g2 with up as an internal vertex. considering every pair uk − up the result follows. 2) since every vertex of gi2 are joined to ui, the maximum distance between vertices of gi2 is 2. hence the proof. next we find an expression for the transit of a vertex, up in g1 ◦ g2, where g1 and g2 are arbitrary. let (uk, ul) be a pair of vertices in g1 such that uk − ul geodesic has up as an internal vertex. let tkl(up) denote the contribution to transit of up, due to geodesic connecting vertices of gk2 ∪ {uk} to gl2 ∪ {ul}. also we denote the contribution of vertices in gp2 to t(up) by tp(up). lemma 3.5. for arbitrary graphs g1 and g2, tkl(up) = σg1(uk, ul/up) [ (n2 + 1) 2d(uk, ul) + 2n2(n2 + 1) ] proof. table 1 gives the length and number of geodesics through up vertices connected length number gk2 to g l 2 2 + d(uk, ul) n 2 2σ(uk, ul/up) uk to gl2 1 + d(uk, ul) n2σ(uk, ul/up) gk2 to ul 1 + d(uk, ul) n2σ(uk, ul/up) uk to ul d(uk, ul) σ(uk, ul/up) table 1: table detailing geodesics through up the result follows. lemma 3.6. tp(up) = ∑n1 p ̸=k=1 [ σg1(up, uk) [ n2(n2+1)d(up, uk)+n2(1+2n2) ]] + 2 [( n2 2 ) − m2 ] proof. table 2 gives the contribution of geodesics through up to tp(up). considering every vertex uk, k ̸= p, the result follows. theorem 3.7. 1) t(up) = tp(up) + ∑ kl tkl(up) 2) t(vpi ) = 2× number of geodesics in g2 of length 2 through vi. 264 reshmi k m, raji pilakkat vertices connected length number uk to g p 2 1 + d(uk, up) n2σ(uk, up) gk2 to g p 2 2 + d(uk, up) n 2 2σ(uk, up) g p 2 to g p 2 2 ( n2 2 ) − m2 table 2: geodesics through up proof. 1) geodesics through up are either considered in tp(up) or in tkl(up). hence the result is evident. 2) follows from proposition 3.4. in the remaining sections we consider g2 as arbitrary, while g1 is replaced by various graph classes like pn, cn, kn, km,n and sm+1 3.2 path graphs let pn be the path graph with vertices 1, 2, . . . , n. we give an expression for transit of k using theorem 3.7 in pn ◦ g2. theorem 3.8. t(k) = (k − 1)(n2 + 1)(n − k) 2 [ (n2 + 1)(n + 1) + 4n2 ] + n2(n2+1) [(k − 1)k 2 + (n − k)(n − k + 1) 2 ] +n2(2n2+1)(n−1)+2 [(n2 2 ) −m2 ] proof. let 1 ≤ l < k < m ≤ n. since g1 is a path, we have σg1(l, m/k) = 1. hence tlm(k) = (n2 + 1)2(m − l) + 2n2(n2 + 1) ∴ ∑ l,m tlm(k) = (n2 + 1) 2 k−1∑ l=1 n∑ m=k−1 (m − l) + k−1∑ l=1 n∑ m=k−1 2n2(n2 + 1) = (n2 + 1) 2tg1(k) + (k − 1)(n − k − 1)2n2(n2 + 1) = (n2 + 1) 2 (n + 1)(k − 1)(n − k) 2 + (k − 1)(n − k − 1)2n2(n2 + 1) = (k − 1)(n2 + 1)(n − k) 2 [ (n2 + 1)(n + 1) + 4n2 ] 265 transit in corona product of graphs in a similar manner we compute tk(k) tk(k) = n∑ k ̸=i=1 [ (d(k, i) + 1)n2 + (d(k, i) + 2)n 2 2 ] +2 [(n2 2 ) − m2 ] = (n2 + n 2 2) [ (k − 1)k 2 + (n − k)(n − k + 1) 2 ] + n2(2n2 + 1)(n − 1) +2 [(n2 2 ) − m2 ] the result follows. in the following examples we compute transit for the vertices in various corona product of pn. from theorem 3.8, we have t(k) = (k − 1)(n2 + 1)(n − k) 2 [ (n2 + 1)(n + 1) + 4n2 ] +n2(n2+1) [(k − 1)k 2 + (n − k)(n − k + 1) 2 ] +n2(2n2+1)(n−1)+2 [(n2 2 ) −m2 ] = t1 + t2, say where t2 = 2 [( n2 2 ) − m2 ] examples 1. g2 = pm. here n2 = m, m2 = m−1. hence t(k) = t1 +(m−1)(m−2). for pendant vertices of p im, the transit is 0 and 2 for others, ∀i. 2. g2 = cm. here n2 = m2 = m. for every vertex in cim, there exist only one geodesic of length 2 through it. here t(k) = t1 + m(m − 3) and t(vik) = 2, ∀k, i. 3. g2 = km. then n2 = m, m2 = ( m 2 ) . thus, t(k) = t1 and t(vik) = 0, ∀k, i. 4. g2 = sm. here n2 = m, m2 = m−1. hence t(k) = t1 +(m−1)(m−2). t(vik) = 0, for pendant vertices and for central vertex of s − m, t(v i k) = (m − 1)(m − 2) 5. g2 = kl1,l2 . n2 = m = l1 + l2 and m = l1l2 t(k) = t1 + (m − 1)m − 2l1l2 and t(vik) = tkl1,l2 (vk) 266 reshmi k m, raji pilakkat 3.3 cycle in this section we consider g1 to be a cycle. we have already seen that transit of vertices in cycles with order 2n and 2n + 1 are the same. hence we consider g1 = c2n1+1. we represent the vertices by 0, 1, . . . , 2n1. also every vertex in the cycle being transit identical, it is enough we compute the transit for n1. theorem 3.9. if a is any vertex of the cycle c2n or c2n+1, its transit in the corona product c2n ◦ g2 or c2n+1 ◦ g2 is given by t(a) = (n2+1)(n1−1)n1 3 [n2n1 + 4n2 + n1 + 1] + n2n1 [ (n2 + 1)(n1 + 1) + 2(1 + 2n2) ] + 2 [( n2 2 ) − m2 ] proof. for any k, l we know that σg1(k, l/n1) = 1. tk,l(n1) = [ (n2 + 1) 2d(k, l) + 2n2(n2 + 1) ] ∴ ∑ k,l tk,l(n1) = (n2 + 1) 2 ∑ k,l d(k, l) + n1(n1 − 1) 2 2n2(n2 + 1) = (n2 + 1) 2tg1(n1) + n1(n1 − 1) 2 2n2(n2 + 1) = (n2 + 1) 2 (n 2 − 1)n 24 + n1(n1 − 1) 2 2n2(n2 + 1) = (n2 + 1)(n1 − 1)n1 3 [n2n1 + 4n2 + n1 + 1] next we compute tn1(n1) tn1(n1) = 2n1∑ n1 ̸=i=0 [ n2(n2 + 1)d(n, i) + n2(1 + 2n2) ] +2 [( n2 2 ) − m2 ] , σg1(n1, i) being 1 = n2n1 [ (n2 + 1)(n1 + 1) + 2(1 + 2n2) ] + 2 [( n2 2 ) − m2 ] and tg1◦g2(n1) = ∑ k,l tk,l(n1) + tn1(n1). hence the proof. 3.4 star let g1 = sn+1. in a star there are n pendant vertices and one central vertex. all pendant vertices are transit identical. hence we need to compute transit of one of the pendant vertex and the central vertex in sn+1 ◦g2. let us name the vertices as 1, 2, . . . , n + 1, where n + 1 is the central vertex. 267 transit in corona product of graphs theorem 3.10. in sn+1 ◦ g2, t(n + 1) = n [ (n − 1)(n2 + 1)(2n2 + 1) + n2(3n2 + 2) ] + 2 [( n2 2 ) − m2 ] and t(i) = n2 [ (n2 + 1)(2n − 1) + n(2n2 + 1) ] + 2 [( n2 2 ) − m2 ] , i ̸= n + 1 proof. consider n + 1. we have σ(k, l/(n + 1)) = 1 and d(k, l) = 2 thus tk,l(n + 1) = 2(n2 + 1)(2n2 + 1) (1) ∴ ∑ k,l tk,l(n + 1) = ( n 2 ) 2(n2 + 1)(2n2 + 1) (2) = n(n − 1)(n2 + 1)(2n2 + 1) (3) while computing tn+1(n + 1), we see that σsn+1(n + 1, i) = 1 and d(n + 1, i) = 1, ∀i. thus we get tn+1(n+1) = nn2(3n2 +2)+2 [( n2 2 ) − m2 ] , which completes the computation for t(n + 1) now consider the vertex i ̸= n + 1. it can easily be verified that σ(k, l/i) = 0, ∀k, l. hence ∑ k,l tk,l(i) = 0. for a fixed i, σ(i, k) = 1∀k and d(i, n + 1) = 1 and d(i, k) = 2, k ̸= n + 1 ∴ ti(i) = n2 [ 4nn2 + 3n − n2 − 1 ] + 2 [( n2 2 ) − m2 ] . hence the proof. 3.5 complete graph and complete bipartite graph theorem 3.11. in the corona product kn ◦ g2, the transit of any vertex of kn is (n − 1)n2(3n2 + 2) + 2 [( n2 2 ) − m2 ] proof. since every vertex of kn is transit identical, we consider one of them. let ui be any vertex of kn. σ(uk, ul/ui) = 0 =⇒ ∑ tk,l(ui) = 0 again σ(ui, uk) = 1, ∀k ̸= i and d(ui, uk) = 1 ∴ ti(ui) = ∑ k ̸=i [ n2(n2 + 1) + n2(2n2 + 1) ] + 2 [( n2 2 ) − m2 ] . hence the result. next we consider a complete bipartite graph kl1,l2 with bipartition v1, v2. let v1 = a1, a2, . . . , al1 and v2 = b1, b2, . . . , bl2 . then all ai are transit identical. similarly all bi are also transit identical. computation of t(ai) and t(bi) are similar. hence we compute t(ai) only. theorem 3.12. in kl1,l2 ◦ g2, the transit of ai, t(ai) = ( l2 2 ) 2(n2 + 1)(2n2 + 1) + l2n2(4n2 + 3)(l1 − 1) + l2n2(3n2 + 2) + 2 [( n2 2 ) − m2 ] proof. the shortest path in kl1,l2 through ai are those connecting vertices of v2. ∴ tk,l(ai) = σkl1,l2 (bk, bl/ai) [ (n2 + 1) 2d(uk, ul) + 2n2(n2 + 1) ] . thus∑ k,l tk,l(ai) = ( l2 2 ) 2(n2 + 1)(2n2 + 1). 268 reshmi k m, raji pilakkat while computing ti(ai), we see that vertices in v1 and v2 behaves differently. hence we split the summation as follows. ti(ai) = ∑ aj σ(ai, aj) [ n2(n2 + 1)d(ai, aj) + n2(2n2 + 1) ] + ∑ bj σ(ai, bj) [ n2(n2 + 1)d(ai, bj) + n2(2n2 + 1) ] + 2 [( n2 2 ) − m2 ] = l2n2(4n2 + 3)(l1 − 1) + l2n2(3n2 + 2) + 2 [( n2 2 ) − m2 ] 4 conclusion in this paper, we first considered arbitrary graphs g1 and g2. we could give an expression for the transit of vertices in their corona product. this result was applied to compute the transit of vertices in g1/circg2, where g1 refers to a particular graph. it should be noted that we could express the transit of vertices in g1 ◦ g2 in terms of individual graph parameters. thus, the computation of transit in huge networks, which are corona products, is now much easier. references v. agnes. degree distance and gutman index of corona product of graphs. transactions on combinatorics, 4, 2015. j. bondy and u. murty. graph theory. springer, 2019. r. frucht and f. harary. on the corona of two graphs. aeq. math., 4:322–325, 1970. doi: https://doi.org/10.1007/bf01844162. p. s. gajendra, a. borah, and s. ray. a review paper on corona product of graphs. advances and applications in mathematical sciences, 19:1047–1054, 2020. h. harary. graph theory. addison wesley, 1969. k.m.reshmi and pilakkat. raji. transit index of a graph and its correlation with mon of octane isomers. advances in mathematics: scientific journal, 9, 2020. doi: https://doi.org/10.37418/amsj.9.4.39. 269 transit in corona product of graphs shimbel and alfonso. structural parameters of communication networks. bulletin of mathematical biophysics, 15:501–507. wagner.s and wang.h. introduction to chemical graph theory. taylor and francis group, boca raton, fl : crc press, 2019. 270 microsoft word documento1 ratio mathematica volume 45, 2023 tri𝒃𝒈 ̂closed sets in tritopological spaces l. jeyasudha* k. bala deepa arasi† abstract in this paper, we introduce a new class of sets called tri𝑏𝑔 ̂closed sets and tri𝑏𝑔 ̂open sets via the concept of tri𝑔 ̂closed sets in tri topological spaces. also, we investigate the relationship with other existing closed sets in tri-topological space. keywords: tri𝑏𝑔 ̂closed sets, tri𝑏𝑔 ̂open sets, tri𝑏𝑔 ̂closure, tri𝑏𝑔 ̂interior. 2010 ams subject classification: 54a40‡. *research scholar, reg. no: 20122012092004, pg & research department of mathematics, a.p.c. mahalaxmi college for women, thoothukudi, tn, india. affiliated to manonmaniam sundaranar university, tirunelveli, tn, india. e. mail: jeyasudha555@gmail.com. †assistant professor of mathematics, a.p.c. mahalaxmi college for women, thoothukudi, tamilnadu, india. e. mail: baladeepa85@gmail.com ‡ received on july 18, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.1002. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 142 mailto:jeyasudha555@gmail.com mailto:baladeepa85@gmail.com l. jeyasudha, k. bala deepa arasi 1. introduction the concept of tritopological space was first initiated by m. kovar [6] in 2000, in 2003, r. subasree and m. maria singam [10] defined 𝑏𝑔 ̂closed sets in topological spaces. in [3], we introduced triĝ closed sets in tritopological spaces and studied their properties. in this paper, we define tri𝑏𝑔 ̂closed sets and tri𝑏𝑔 ̂open sets via the concept of tri𝑔 ̂closed sets. also, we investigate the relationship with other existing closed sets in tritopological space. 2. preliminaries throughout this paper (x, τ1, τ2, τ3) (or simply x) represents tritopological spaces on which no separation axioms are assumed unless other wised mentioned. for a subset a of (x,τ1,τ2,τ3), tricl(a), triint(a) and a c denote the triclosure of a, triinterior of a and compliment of a respectively. definition 2.1 let x be a non-empty set. a family τ of subsets of x is said to be a topology on x, if τ satisfies the following axioms. a) ф, x ∈τ, b) if ai∈τ for i = 1,2,…..,n, then ⋂ 𝐴 𝑛 𝑖=1 i ∈τ, c) if aα∈τ for α ∈ i, then ⋃ 𝐴𝛼 α∈ τ. the pair (x, τ) is called a topological space and any set a in 𝜏 is called an open set. the complement of an open set a is called closed set. definition 2.2 let x be a non-empty set. a family g of subsets of x is said to be a generalized topology on x, if g satisfies the followings axioms. a) ф ∈ g, b) if aα∈g for α ∈i, then ⋃ 𝐴𝛼 α∈ g. the pair (x, g) is called a generalized topological space. definition 2.3 let x be a non-empty set. a family τ* of subsets of x is said to be a supra topology on x, if τ* satisfies the following axioms. a) ф, x ∈ τ*, b) if aα∈ τ* for α ∈ i, then ⋃ 𝐴𝛼 α∈ τ*. the pair (x, τ*) is called a supra topological space. definition 2.4 let x be a non-empty set. a family τix of subsets of x is said to be a infra topology on x, if τix satisfies the following axioms. a) ф, x ∈ τix, b) if ai∈ τix for i = 1, 2… n, then ⋂ 𝐴 𝑛 𝑖=1 i ∈ τix. the pair (x, τix) is called infra topological space. 143 tri – gb ˆ closed sets in tritopological spaces definition 2.5 let (x, τ) be a topological space then τ is said to be indiscrete topology if τ is a collection of only x and ф. indiscrete topology is also known as trivial topology. definition 2.6 let (x, τ) be a topological space then τ is said to be discrete topology if τ is a collection of all subsets of x. definition 2.7 let (x, τ) be a topological space then a subset a of x is said to be 𝑏𝑔 ̂closed set if bcl (a) ⊆ u whenever a ⊆ u, u is ĝopen in x. definition 2.8 let x be a nonempty set and τ1, τ2 and τ3 are topologies on x. then a subset a of x is said to be triopen set if a ∈ τ1∪τ2∪τ3 and its complement is said to be triclosed set and x with three topologies called tritopological spaces (x, τ1, τ2, τ3). definition 2.9 let (x, τ1, τ2, τ3) be a tritopological space and let a ⊆ x. the union of all triopen sets contained in a is called the triinterior of a. the intersection of all tri closed sets containing a is called the triclosure of a. definition 2.10 let (x, τ1, τ2, τ3) be a tritopological space. a ⊆ x is said to be 1) a triα open set if a ⊆ triint (tricl (triint (a))). 2) a trib open set if a ⊆ [tricl (triint (a))] ∪ [triint (tricl (a))]. 3) a trisemi closed set if triint (tricl (a)) ⊆ a. 4) a trig closed set if tricl (a) ⊆ u whenever a ⊆ u and u is triopen set in x. 5) a trigs closed set if triscl (a) ⊆ u whenever a ⊆ u and u is triopen set in x. 6) a tribτ closed set if triclb (a) ⊆ u whenever a ⊆ u and u is triopen set in x. 7) a trig*bw closed set if tribcl (a) ⊆ u whenever a ⊆ u, u is trigs open in x. 8) a triĝ closed set if tricl (a) ⊆ u whenever a ⊆u, u is trisemi open in x. the complement of triα open set, trib open set, trisemi closed set, trig closed set, trigs closed set, tribτ closed set, trig*bw closed set and triĝ closed setis called tri α closed set, trib closed set, trisemi open set, trig open set, trigs open set, tribτ open set, trig*bw open set and triĝ open set respectively. theorems 2.11 1) every triclosed set is trisemi closed. 2) every triclosed set is trib closed. 3) every triclosed set is trigs closed. 4) every triclosed set is tribτ closed. 5) every triclosed set is trig*bω closed. 6) every triclosed set is triĝclosed set. 7) every trisemi closed set is trigs closed. 144 l. jeyasudha, k. bala deepa arasi 8) every trisemi closed set is trib closed. 9) every trisemi closed set is trig*bω closed. 10) every trib closed set is tribτ closed. 11) every trisemi closed set is tribτ closed. 12) every triα closed set is trib closed set. 13) every trig*bω closed set is tribτ closed. 14) every triĝ closed set is trig closed. 15) every triĝ closed set is trigs closed. 3. tri𝑏𝑔 ̂closed sets in tritopological space we introduce the following definitions definition 3.1 let (x,τ1,τ2,τ3) be a tritopological space then a subset a of x is said to be tri𝑏𝑔 ̂closed set if tribcl (a) ⊆ u whenever a ⊆ u, u is triĝ open in x. the family of all tri𝑏𝑔 ̂closed sets of x is denoted by tri𝑏𝑔 ̂ c(x). example 3.2 let x = {a, b, c} with the topologies τ1 = {x, ф, {a, b}}, τ2 = {x, ф, {b, c}}, τ3 = {x, ф, {a, c}}, open sets in tritopological spaces are union of all three topologies. τ1 ∪ τ2 ∪ τ3 = {x, ф, {a, b}, {b, c}, {a, c}}; triĝo(x) = {x, ф, {a, b}, {b, c}, {a, c}}; hence tri𝑏𝑔 ̂c(x) = {x, ф, {a}, {b}, {c}}. remark 3.3 ф and x are always tri𝑏𝑔 ̂closed set. remark 3.4 intersection of tri𝑏𝑔 ̂closed sets need not be tri𝑏𝑔 ̂closed set. example 3.5 let x = {a, b, c}, τ1 = {x, ф}, τ2 = τ3 = {x, ф, {a}}, trigb ˆ c(x) = {x, ф, {b}, {c}, {a, b}, {b, c}, {a, c}}. here, {a, b}, {a, c} are tri𝑏𝑔 ̂closed sets but {a, b} ∩ {a, c} = {a} is not a tri𝑏𝑔 ̂closed set. remark 3.6 union of tri𝑏𝑔 ̂closed sets need not be tri𝑏𝑔 ̂ closed set. example 3.7 let x = {a, b, c}, τ1 = {x, ф, {a, c}}, τ2 = {x, ф, {b}, {b, c}}, τ3 = {x, ф, {c}, {a, b}}, tri𝑏𝑔 ̂c(x) = {x, ф, {a}, {b}, {c}, {a, b}, {a, c}}. here, {b}, {c} are tri 𝑏𝑔 ̂closed sets but {b} ∪ {c} = {b, c} ∉ tri𝑏𝑔 ̂c(x). remark 3.8 difference of two tri𝑏𝑔 ̂closed sets need not be tri𝑏𝑔 ̂closed set. example 3.9 in previous example – 3.7, tri𝑏𝑔 ̂c(x) = {x, ф, {a}, {b}, {c}, {a, b}, {a, c}}. let a = x and b = {a}, also a and b are tri𝑏𝑔 ̂closed sets. but a \ b = x \ {a} = {b, c} is not a tri𝑏𝑔 ̂closed set. 145 tri – gb ˆ closed sets in tritopological spaces remark 3.10 1) (x, tri𝑏𝑔 ̂c(x)) need not be topological space. 2) (x, tri𝑏𝑔 ̂c(x)) need not be generalized topological space. 3) (x, tri𝑏𝑔 ̂c(x)) need not be supra topological space. 4) (x, tri𝑏𝑔 ̂c(x)) need not be infra topological space. example 3.11 in examples – 3.5, 3.7 we get the results. definition 3.12 let (x, τ1, τ2, τ3) be a tritopological space. the intersection of all tri 𝑏𝑔 ̂closed sets of x containing a subset a of x is called tri𝑏𝑔 ̂closure of a and is denoted by tri𝑏𝑔 ̂cl(a). (i.e) tri𝑏𝑔 ̂cl (a) = ∩ {b ⊆ x: b ⊇ a and b is tri𝑏𝑔 ̂ closed set}. remark 3.13 1) tri𝑏𝑔 ̂cl(ф) = ф, 2) tri𝑏𝑔 ̂cl(x) = x, 3) a ⊆ tri𝑏𝑔 ̂cl(a), 4) tri𝑏𝑔 ̂cl(a) = tri𝑏𝑔 ̂cl(tri𝑏𝑔 ̂cl(a)). proposition 3.14 let (x,τ1,τ2,τ3) be a tritopological space. let a ⊆ x, then a = tri 𝑏𝑔 ̂ cl (a) if a is tri𝑏𝑔 ̂closed set. proof. suppose a is a tri𝑏𝑔 ̂closed set in x then, tribcl (a) ⊆ u whenever a⊆u, u is triĝ open in x. since, a⊇a and a is tri𝑏𝑔 ̂closed set. let b ⊆ x then a ∈{b ⊆ x : b ⊇ a and b is tri𝑏𝑔 ̂closed} ⇒ a = ∩ {b ⊆ x : b ⊇ a and b is tri𝑏𝑔 ̂closed}. hence a= tri𝑏𝑔 ̂cl(a). remark 3.15 the tri𝑏𝑔 ̂closure of a set a is not always tri𝑏𝑔 ̂closed set. example 3.16 let x = {a, b, c}, τ1 = {x, ф}, τ2 = τ3 = {x, ф,{a}}, tri𝑏𝑔 ̂c(x) = {x, ф, {b},{c},{a, b},{b, c}, {a, c}}.here, tri𝑏𝑔 ̂cl({a}) = {a} is not a tri𝑏𝑔 ̂closed set. proposition 3.17 every trib closed set is tri𝑏𝑔 ̂closed set. proof: let a be any trib closed set in x and u be any triĝ open set in x such that a⊆u. since, a is trib closed then tribcl (a) =a for every subset a of x. tribcl(a) = a ⊆ u. hence a is tri𝑏𝑔 ̂closed set. converse of the above proposition need not be true as seen in the following example. example 3.18 let x = {a, b, c}, τ1 = {x, ф}, τ2 = τ3 = {x, ф,{a}}, trib c(x) = {x, ф,{b}, {c},{b, c}; tri𝑏𝑔 ̂c(x) = {x, ф, {b}, {c}, {a, b}, {b, c}, {a, c}}; here {a, b}, {a, c} are tri𝑏𝑔 ̂ closed sets but not a trib closed set. 146 l. jeyasudha, k. bala deepa arasi proposition 3.19 every triclosed set is tri𝑏𝑔 ̂closed set. proof: let a be any triclosed set in x. since every triclosed set is trib closed set. therefore, a is trib closed set in x. by proposition 3.17, a is tri𝑏𝑔 ̂closed set. converse of the above proposition need not be true as seen in the following example. example 3.20 let x = {a, b, c}, τ1 = {x, ф, {a}}, τ2 = {x, ф, {b}}, τ3 = {x, ф, {a, c}}, tric(x) = {x, ф, {b}, {a, c}, {b, c}}; tri𝑏𝑔 ̂c(x) = {x, ф, {b}, {c}, {b, c}, {a, c}}; here {c} is tri𝑏𝑔 ̂closed set but not a triclosed set. proposition 3.21 every trisemi closed set is tri𝑏𝑔 ̂closed set. proof: let a be any trisemi closed set in x. since every trisemi closed set is trib closed set. therefore, a is trib closed set in x. by proposition 3.17, a is tri𝑏𝑔 ̂closed converse of the above proposition need not be true as seen in the following example. example 3.22 let x = {a, b, c}, τ1 = {x, ф, {a}}, τ2 = {x, ф, {a, b}}, τ3 = {x, ф, {b, c}}, trisc(x) = {x, ф, {a}, {c}, {b, c}}; tri𝑏𝑔 ̂c(x) = {x, ф, {a}, {b}, {c}, {b, c}, {a, c}}; here {b},{a, c} are tri𝑏𝑔 ̂closed sets but not a tri-semi closed set. proposition 3.23 every triα closed set is tri𝑏𝑔 ̂closed set. proof: let a be any triα closed set in x. since every triα closed set is trib closed set. therefore, a is trib closed set in x. by proposition 3.17, a is tri𝑏𝑔 ̂closed set. converse of the above proposition need not be true as seen in the following example. example 3.24 let x = {a, b, c}, τ1 = τ2 = {x, ф, {a}}, τ3 = {x, ф, {b, c}}, triα c(x) = {x, ф, {a}, {b, c}}; tri𝑏𝑔 ̂c(x) = {x, ф, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}}; here {b}, {c}, {a, b}, {b, c} are tri𝑏𝑔 ̂closed sets but not a tri-α closed set. proposition 3.25 every tri-g*bω closed set is tri𝑏𝑔 ̂closed set. proof: let a be any tri-g*bω closed set in x and a ⊆ u, where u is triĝ open set in x. since, every triĝ open set is trigs open. therefore, u is trigs open in x. since, a is trig*bω closed set in x then tribcl(a) ⊆ u. hence a is tri𝑏𝑔 ̂closed set in x. converse of the above proposition need not be true as seen in the following example. example 3.26 let x = {a, b, c}, τ1 = {x, ф}, τ2 = {x, ф, {a}}, τ3 = {x, ф, {a, b}}, trig*bω c(x) = {x, ф, {b}, {c}, {b, c}}; tri𝑏𝑔 ̂c(x) = {x, ф, {b}, {c}, {b, c}, {a, c}}; here {a, c} is tri𝑏𝑔 ̂closed set but not a trig*bω closed set. proposition 3.27 every tri𝑏𝑔 ̂closed set is tribτ closed set. proof: let a be any tri𝑏𝑔 ̂closed set in x and a ⊆ u, where u is triopen set in x. since, every triopen set is triĝ open. therefore, u is triĝ open in x. since, a is tri 𝑏𝑔 ̂closed set in x then tribcl(a) ⊆ u. hence a is tribτ closed set in x. converse of the above proposition need not be true as seen in the following example. 147 tri – gb ˆ closed sets in tritopological spaces example 3.28 let x = {a, b, c}, τ1 = {x, ф, {a}}, τ2 = {x, ф, {b}}, τ3 = {x, ф, {a, c}}, tri𝑏𝑔 ̂c(x) = {x, ф, {b}, {c}, {b, c}, {a, c}}; tribτ c(x) = {x, ф, {b}, {c}, {a, b}, {b, c}, {a, c}}; here {a, b} is tribτ closed set but not a tri𝑏𝑔 ̂closed set. remark 3.29 trig closed sets and tri𝑏𝑔 ̂closed sets are independent. example 3.30 let x = {a, b, c}, τ1 = {x, ф, {a}}, τ2 = {x, ф, {b}}, τ3 = {x, ф, {a, b}}, trigc(x) = {x, ф, {c}, {b, c}, {a, c}}; tri𝑏𝑔 ̂c(x) = {x, ф, {a}, {b}, {c}, {b, c}, {a, c}}; here {a} and {b} are tri𝑏𝑔 ̂closed sets but not a tri-g closed sets. example 3.31 let x = {a, b, c}, τ1 = {x, ф, {a}}, τ2 = {x, ф, {b}}, τ3 = {x, ф, {a, c}}, tri𝑏𝑔 ̂c(x) ={x, ф, {b}, {c}, {b, c}, {a, c}}; trigc(x) ={x, ф, {b}, {c}, {a, b}, {b, c}, {a, c}}; here {a, b} is trig closed set but not a tri𝑏𝑔 ̂closed set. remark 3.32 trigs closed sets and tri𝑏𝑔 ̂closed sets are independent. example 3.33 let x = {a, b, c}, τ1 = {x, ф}, τ2 = {x, ф, {a, b}}, τ3 = {x, ф, {b, c}}, trigsc(x) = {x, ф, {a}, {c}, {a, c}}; tri𝑏𝑔 ̂c(x) = {x, ф, {a}, {b}, {c}, {a, c}}; here {b} is tri𝑏𝑔 ̂closed set but not a trigs closed set. example 3.34 let x ={a, b, c}, τ1 = {x, ф}, τ2 = {x, ф, {a}}, τ3 = {x, ф, {b}}, tri 𝑏𝑔 ̂c(x) = {x, ф, {a}, {b}, {c}, {b, c}, {a, c}}; trigsc(x) = p(x); here {a, b} is tri gs closed set but not a tri𝑏𝑔 ̂closed set. remark 3.35 the following diagram shows the relationship of tri𝑏𝑔 ̂closed sets with other known existing closed sets in tritopological space. c b d h i e f g a 148 l. jeyasudha, k. bala deepa arasi a → tri𝑏𝑔 ̂closed set b → triclosed set c → trib closed set d → trig closed set e → triα closed set f → trigs closed set g → trig*bw closed set h → tribτ closed set i → trisemi closed set remark 3.36 if (x, tric(x)) is indiscrete topology then (x, tri𝑏𝑔 ̂c(x)) is discrete topology but converse part need not be true. example 3.37 let x = {a, b, c}, τ1 = {x, ф, {a}}, τ2 = τ3 = {x, ф, {b, c}; tric(x) = {x, ф, {a}, {b, c}}; tri𝑏𝑔 ̂c(x) = {x, ф, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}} = p(x). here, (x, tri𝑏𝑔 ̂c(x)) is discrete topology but (x, tric(x)) is not an indiscrete topology. remark 3.38 if (x, tric(x)) is discrete topology then (x, tri𝑏𝑔 ̂c(x)) is discrete topology but converse part need not be true. example 3.39 in example – 3.7, (x, tri𝑏𝑔 ̂c(x)) is discrete topology but (x, tri c(x)) is not a discrete topology. remark 3.40 if (x, tric(x)) is indiscrete topology then, 1) every tri𝑏𝑔 ̂closed set is trib closed set. 2) every tri𝑏𝑔 ̂closed set is trig closed set. 3) every tri𝑏𝑔 ̂closed set is trigs closed set. 4) every tri𝑏𝑔 ̂closed set is trig*bω closed set. 5) every trig closed set is tri𝑏𝑔 ̂closed set. 6) every trigs closed set is tri𝑏𝑔 ̂ closed set. 7) every tribτ closed set is tri𝑏𝑔 ̂closed set. example 3.41 let x be any non-empty set, τ1 = τ2 = τ3 = {x, ф} are topologies of x. tric(x) ={x, ф}; tribc (x) = trigc(x) = trigsc(x) = tribτc(x) = tri g*bωc(x) = tri𝑏𝑔 ̂c(x) = p(x). remark 3.42 if (x, tric(x)) is discrete topology then, 1) every tri𝑏𝑔 ̂closed set is triclosed set. 2) every tri𝑏𝑔 ̂closed set is trisemi closed set. 3) every tri𝑏𝑔 ̂closed set is triα closed set. 4) every tri𝑏𝑔 ̂closed set is trib closed set. 5) every tri𝑏𝑔 ̂closed set is trig closed set. 6) every tri𝑏𝑔 ̂closed set is trigs closed set. 7) every tri𝑏𝑔 ̂closed set is trig*bω closed set. 8) every trig closed set is tri𝑏𝑔 ̂closed set. 9) every trigs closed set is tri𝑏𝑔 ̂closed set. 10) every tribτ closed set is tri𝑏𝑔 ̂closed set. 149 tri – gb ˆ closed sets in tritopological spaces example 3.43 let x be any non-empty set, τ1 = τ2 = τ3 = p(x) are topologies of x. tric(x) = trisc(x) = triαc(x) = tribc(x) = trigc(x) = trigsc(x) = tribτc(x) = trig*bωc(x) = tri𝑏𝑔 ̂c(x) = p(x). 4. tri𝒃𝒈 ̂open sets in tritopological space definition 4.1 the complement of a tri𝑏𝑔 ̂closed set is called the tri𝑏𝑔 ̂open set. the family of all tri𝑏𝑔 ̂open sets of x is denoted by tri𝑏𝑔 ̂o(x). example 4.2 in example 3.2, tri𝑏𝑔 ̂o(x) = {x, ф, {a, b}, {b, c}, {a, c}}. remark 4.3 ф and x are always tri𝑏𝑔 ̂open set. remark 4.4 intersection of tri𝑏𝑔 ̂open sets need not be tri𝑏𝑔 ̂open set. example 4.5 in example – 3.2, tri𝑏𝑔 ̂o(x) = {x, ф, {a, b}, {b, c}, {a, c}}. here, {a, b}, {b, c} are tri𝑏𝑔 ̂open sets but {a, b} ∩ {b, c} = {b} ∉ tri𝑏𝑔 ̂o(x). remark 4.6 union of tri𝑏𝑔 ̂open sets need not be tri𝑏𝑔 ̂open set. example 4.7 in example – 3.16, tri𝑏𝑔 ̂o(x) = {x, ф, {a}, {b}, {c}, {a, b}, {a, c}}. here, {b} and {c} are tri𝑏𝑔 ̂open sets but {b} ∪ {c} = {b, c} ∉ tri𝑏𝑔 ̂o(x). remark 4.8 difference of two tri𝑏𝑔 ̂open sets need not be tri𝑏𝑔 ̂open set. example 4.9 in previous example – 4.7, tri𝑏𝑔 ̂o(x) = {x, ф, {a}, {b}, {c}, {a, b}, {a, c}}. let a = x and b = {a}, also a and b are tri𝑏𝑔 ̂open sets. but a\b = x\{a} = {b, c} is not a tri𝑏𝑔 ̂open set. definition 4.10 let (x, τ1, τ2, τ3) be a tritopological space. the union of all tri 𝑏𝑔 ̂open sets of x contained in a is called the tri𝑏𝑔 ̂interior of a and is denoted by tri 𝑏𝑔 ̂int(a). (i.e) tri𝑏𝑔 ̂(a) = ∪ {b ⊆ x / b ⊆ a and a is tri𝑏𝑔 ̂open set}. remark 4.11 1) tri𝑏𝑔 ̂int(ф) = ф, 2) tri𝑏𝑔 ̂int(x) = x, 3) tri𝑏𝑔 ̂int(a) ⊆ a, 4) tri𝑏𝑔 ̂int(a) = tri𝑏𝑔 ̂int(tri𝑏𝑔 ̂int(a)). proposition 4.12 for any a ⊆x, (tri𝑏𝑔 ̂int(a))c = tri 𝑏𝑔 ̂cl(ac). 150 l. jeyasudha, k. bala deepa arasi proof: (tri𝑏𝑔 ̂int(a))c = [∪ {g / g ⊆ a & g is tri𝑏𝑔 ̂open set}]c = ∩ { gc / gc ⊇ ac & gc is tri𝑏𝑔 ̂closed set} = ∩{ f / f ⊇ ac& f is tri𝑏𝑔 ̂closed set} where f = gc. hence, (tri𝑏𝑔 ̂ int(a))c = tri𝑏𝑔 ̂cl(ac). proposition 4.13 let (x,τ1,τ2,τ3) be a tri-topological space. let a ⊆ x. then tri 𝑏𝑔 ̂int(a) = a if a is tri𝑏𝑔 ̂open set. proof: suppose a is a tri𝑏𝑔 ̂open set in x, then ac is tri𝑏𝑔 ̂closed set in x. (i.e) tri 𝑏𝑔 ̂cl (ac) ⊆ ac. by the definition, ac ⊆ tri𝑏𝑔 ̂cl(ac). therefore tri𝑏𝑔 ̂cl(ac) = ac ⇒ (tri𝑏𝑔 ̂int(a))c = ac ⇒ tri𝑏𝑔 ̂int(a) = a. remark 4.14 the tri𝑏𝑔 ̂interior of a set a is not always tri𝑏𝑔 ̂open set. example 4.15 let x = {a, b, c}, τ1 = {x, ф}, τ2 = τ3 = {x, ф, {a}}, tri𝑏𝑔 ̂c(x) = {x, ф, {b}, {c}, {a, b}, {b, c}, {a, c}}; tri𝑏𝑔 ̂o(x) = {x, ф, {a}, {b}, {c}, {a, b}, {a, c}}. here, tri𝑏𝑔 ̂int ({b, c}) = {b, c} is not a tri𝑏𝑔 ̂open set. proposition 4.16 1) every triopen set is tri𝑏𝑔 ̂open set. 2) every trib open set is tri𝑏𝑔 ̂open set. 3) every trisemi open set is tri𝑏𝑔 ̂open set. 4) every triα open set is tri𝑏𝑔 ̂open set. 5) every trig*bω open set is tri𝑏𝑔 ̂open set. 6) every tri𝑏𝑔 ̂open set is tribτ open set. proof: by proposition – 3.17, 3.19, 3.21, 3.23, 3.25, 3.27 we get the results. 5. conclusions in this paper, we dealt with tri𝑏𝑔 ̂closed sets and tri𝑏𝑔 ̂open sets. in future we wish to do our research work in tri𝑏𝑔 ̂continuous functions, tri𝑏𝑔 ̂separated, tri 𝑏𝑔 ̂connected sets, tri𝑏𝑔 ̂ compact and so on. references [1] d. andrijievic, on bopen sets, mat. vesnik, 48, no. 1-2, 59-64, 1996. [2] j. c. kelly, bitopological spaces, proc. london math. soc., 3, 17-89, 1963. [3] k. bala deepa arasi and l. jeyasudha, on triĝ closed sets in tritopological spaces, 2021. [4] k. bala deepa arasi and l. jeyasudha, on triĝ continuous functions in tri topological spaces, journal of physics, vol. 19947, issue 1, 2021. 151 tri – gb ˆ closed sets in tritopological spaces [5] m. k. r. s. veerakumar, ĝclosed sets in topological space, bull. allahabad. math. soc., vol.18, 99-112, 2003. [6] m. kovar, on 3topological version of thetregularity, internet. j. matj, sci., 23(6), 393-398, 2000. [7] n. f. hameed & mohammed yahya abid, certain types of separation axioms in tri topological spaces, iraqi journal of science, vol 52,(2), 212-217, 2011. [8] n. levine, generalized closed sets in topology rend. circ. mat. palermo, 19, 89-96, 1970. [9] p. priyadharshini and a. parvathi, tribcontinuous function in tri topological spaces, international journal of mathematics and its applications, vol. 5, issue 4f, 959962, 2017. [10] r. subasree and m. maria singam, on 𝑏𝑔 ̂closed sets in topological spaces, ijma, 4(7), 68-173, 2003. [11] s. palaniammal, study of tri topological spaces, ph. d thesis, 2011. [12] u. d. tapi, r. sharma and b. deole, semi open sets and preopen sets in tri topological space, i-manager’s journals, on mathematics, 5(3), 2016. 152 microsoft word documento1 ratio mathematica 23 (2012), 3–20 issn: 1592-7415 general ω-hyperstructures and certain applications of those jan chvalina, šárka hošková-mayerová brno university of technology, faculty of electrical engineering and communication, department of mathematics, czech republic university of defence, brno, faculty of military technology, department of mathematics and physics, czech republic chvalina@feec.vutbr.cz, sarka.mayerova@unob.cz abstract the aim of this paper is to investigate general hyperstructures construction of which is based on ideas of a. d. nezhad and r. s. hashemi. concept of general hyperstructures considered by the above mentioned authors is generalized on the case of hyperstructures with hyperoperations of countable arity. specifications of treated concepts to examples from various fields of the mathematical sturctures theory are also included. key words: action of a hyperstructure on a set, general nhyperstructure, transformation hypergroup, fredholm integral operator, ordinary and partial differential operator msc2010: primary: 20n20; secondary: 37l99, 68q70 1 preliminaries a resent book [7] contains a wealth of applications of hyperstructure theory developed since 1934, see [20]. there are applications to the following subjects: geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, combinatoric, codes, artificial intelligence, and probability. in section 2 and 3 we give some basic definition and then, we consider three types of actions. in section 4 we present some applications. in section 5 there are described some applications of formerly j. chvalina, š. hošková-mayerová investigated hyperstructures and corrected certain mistake from [14]. there is a connection between automata and dynamical systems realized by infinite automata without outputs and discrete dynamical systems.they are also shortly called as action of semigroups or groups on a given phase (or state) set. in connection with non-deterministic automata or with multifunctions (relations) on algebraic structures and topological spaces seems to be natural to investigate actions of multistructures on sets of various objects. various generalizations of the above mentioned classical concepts are possible. some motivating factors are coming from the general system theory [11, 23]; one illustrating example below is just based on the concept of a general time system. in [8, 9] there are investigated various binary relation and hyperstructures. notice, the binary relation on a binary hyperstructure, e.g. on a semihypergroup, are quite natural created by inner translations. we use [1, 3, 7, 10, 21, 22, 26] for terminology and notations which are not define here. we suppose that the reader is familiar with some well-known notation such as presentation in hyperstructure theory. the following facts are some definitions and propositions in the theory of hyperstructure which we need for formulation of our results and in the proofs of the main results. for x from an ordered set h we denote by [x)≤ = {y ∈ h | x ≤ y} the upper end generated by x. the following lemma is called ends lemma. lemma 1.1. [3, 6, 24, 25] let (h,◦,≤) be an ordered semigroup. let a?b = [a◦ b)≤ for any a,b ∈ h. the following conditions are equivalent: 1) for any pair a,b ∈ h there exists a pair c,d ∈ h such that b ◦ c ≤ a, c ◦ d ≤ a. 2) a hypergroupoid (h,?) associated with (h,◦,≤) satisfies the associativity law and the reproduction axioms, i.e., (h,?) is a hypergroup. dually we can define the beginnings lemma: lemma 1.2. [6] let (h,◦,≤) be an ordered semigroup. let a ? b = (a◦ b]≤ for any a,b ∈ h. the following conditions are equivalent: 1) for any pair a,b ∈ h there exists a pair c,d ∈ h such that b ◦ c ≥ a, c ◦ d ≥ a. 2) a hypergroupoid (h,?) associated with (h,◦,≤) satisfies the associativity law and the reproduction axioms, i.e., (h,?) is a hypergroup. 4 general ω-hyperstructures and certain applications of those quasi-order hypergroups have been introduced and studied by j. chvalina. the following definition can be found e.g. in [7, 12, 24]. definition 1.1. a hypergroup (h,?) such that the following condition are satisfied: 1) a ∈ a2 = a3 for any a ∈ h, 2) a?b = a2 ∪b2 for any pair a,b ∈ h is called a quasi-order hypergroup. if moreover the unique square root condition: 3) a,b ∈ h, a2 = b2 implies a = b is satisfied then (h,?) is called an order hypergroup. definition 1.2. [19] a hypergroup (g,?) is called a transposition hypergroup if it satisfies the transposition axiom: for all a,b,c,d ∈ g the relation b\a∩ c/d 6= ∅ implies a ? d∩ b ? c 6= ∅. the sets b \ a = {x ∈ g|a ∈ b ? x},c/d = {x ∈ g|c ∈ x ? d} are called left and right extensions, respectively. definition 1.3. [15, 16, 17] let x be a set, (g,•) be a (semi)hypergroup and π : x ×g → x a mapping such that π(π(x,t),s) ∈ π(x,t•s), where π(x,t•s) = {π(x,u); u ∈ t•s)} (1) for each x ∈ x, s,t ∈ g. then (x,g,π) is called a discrete transformation (semi)hypergroup or an action of the (semi)hypergroup g on the phase set x. the mapping π is usually said to be simply an action. remark 1.1. [16] the condition (1) used above is called generalized mixed associativity condition, shortly gmac. 2 general hyperstuctures and ω-hyperstructures throughout this paper, the symbol x,y will denote two non-empty sets, where p∗(x ∪y ) denotes the set of all non-empty subsets of x ∪y . a general hyperstructure is formed by two non-empty sets x,y together with a hyperoperation ∗ : x ×y −→ p∗(x ∪y ), (x,y) 7→ x∗y ⊆ (x ∪y ) r ∅. 5 j. chvalina, š. hošková-mayerová remark 2.1. a general hyperoperation ∗ : x ×y −→ p∗(x ∪y ) yields a map of powersets determined by this hyperoperation. thus the map ⊗ : p∗(x) ×p∗(y ) −→ p∗(x ∪y ) is defined by a⊗b = ⋃ a∈a,b∈b a∗ b. conversely an general hyperoperation on p∗(x)×p∗(y ) yields a general hyperoperation on x ×y , defined by x∗y = {x}⊗{y}. in the above definition if a ⊆ x, b ⊆ y, x ∈ x, y ∈ y, then we define, a∗y = a∗{y} = ⋃ a∈a a∗y, x∗b = {x}∗b = ⋃ b∈b x∗ b, a⊗b = ⋃ a∈a,b∈b a∗ b. remark 2.2. if x = y = h, then we obtain the classical hyperstructure theory. the concept of general hyperstructure with a hyperoperation which is a mapping ∗ : x ×y −→ p∗(x ∪y ) mentioned above (used by a. d. nezhad and r. s. hashemi) allows straightforward generalization onto case of “ hyperoperation of an arbitrary finite arity” or direct generalization to hyperoperation of countable arity. let us define the general ω-hyperstructure: definition 2.1. let ω be the smallest infinite countable ordinal. as usually, we consider the smallest infinite ordinal ω as the set of all smaller ordinals, i.e. as the domain of all finite ordinals (non-negative integers). let {xk; k ∈ ω} be a system of non-empty sets. by an general ω-hyperstructure we mean the pair ({xk; k ∈ ω},∗ω), where ∗ω : ∏ k∈ω xk → p∗ ( ⋃ k∈ω xk ) is a mapping assigning to any sequence {xk}k∈ω ∈ ∏ k∈ω xk a non-empty subset ∗ω({xk}k∈ω) ⊂ ⋃ k∈ω xk. similarly as above, with this hyperoperation there is associated a mapping of power sets ⊗ω : ∏ k∈ω p∗(xk) →p∗ (⋃ k∈ω xk ) defined by ⊗ω ( {ak}k∈ω ) = ⋃{ ∗ω ( {xk}k∈ω ) ; ( {xk}k∈ω ) ∈ ∏ k∈ω ak } . let us formulate the special case: 6 general ω-hyperstructures and certain applications of those definition 2.2. [13] let n ∈ ω be an arbitrary positive integer, n ≥ 1. let {xk; k = 1, . . . ,n} be a system of non-empty sets. by a general n-hyperstructure we mean the pair ({xk; k = 1, . . . ,n},∗n), where ∗n : n∏ k=1 xk → p∗ ( n⋃ k=1 xk ) is a mapping assigning to any n-tuple (x1, . . . ,xn) ∈ n∏ k=1 xk a non-empty subset ∗n(x1, . . . ,xn) ⊂ n⋃ k=1 xk. similarly as above, with this hyperoperation there is associated a mapping of power sets ⊗n : n∏ k=1 p∗(xk) →p∗ ( n⋃ k=1 xk ) defined by ⊗n(a1, . . . ,an) = ⋃{ ∗n(x1, . . . ,xn); (x1, . . . ,xn) ∈ n∏ k=1 ak } . this construction is based on an idea of nezhad and hashemi for n = 2. hyperstructures with n-ary hyperoperations are investigated among others in [2, 27]. definition 2.3. let g1(ω) = ( {xk; k ∈ ω},∗ω ) , g2 (ω) = ( {yk; k ∈ ω},•ω ) , be a pair of general ω-hyperstructures. by a good homomorphism h : g1(ω) → g2 (ω) we mean any system of mappings h = {hk : xk → yk} such that the following diagram is commutative: ∏ xk ∗ω−−−→ p∗ ( ⋃ k∈ω xk ) q k∈ω hk y yϕ]∏ yk •ω−−−→ p∗ ( ⋃ k∈ω yk) (d1) here ∏ k∈ω hk({xk}k∈ω) = {hk(xk)}k∈ω for any sequence {xk}k∈ω and ϕ] : p∗ ( ⋃ k∈ω xk ) →p∗ ( ⋃ k∈ω yk ) is the lifting of a mapping ϕ: ⋃ k∈ω xk → ⋃ k∈ω yk defined by the mathematical induction. for x ∈ x0 we put ϕ(x) = h0(x). suppose ϕ: k⋃ j=0 xj → k⋃ j=0 yj is well-defined. then for any x ∈ xk+1\ k⋃ j=0 xj 7 j. chvalina, š. hošková-mayerová we put ϕ(x) = hk+1(x). then using mathematical induction the mapping ϕ: ⋃ k∈ω xk → ⋃ k∈ω yk is well-defined. if all mappings hk ∈ h are bijections (or isomorphism if all hk,yk are endowed with some structures) we call the h the isomorphism of ωhyperstructures g1(ω), g2(ω). 3 general ω-hyperstructures created by ordered sets and by differential operators as a certain generalization of the general n-hyperstructure from [13], example 3.2, we will construct the following structure: example 3.1. consider a countable system of pairwise disjoint ordered sets (xk,≤k), k ∈ ω and for x ∈ xk let us denote [x)k = {y ∈ xk; x ≤k y}, i.e. [x)k is the principal end generated by the element x within the ordered set (xk,≤k). further, put ∗ω ( {xk}k∈ω ) = ⋃ k∈ω [xk)k for any sequence ∗ω ( {xk}k∈ω ) ∈ ∏ kω xk. then ∗ω ( {xk}k∈ω ) ⊆ ⋃ k∈ω xk, thus g(ω) = ( {xk; k ∈ ω},∗ω ) is a general ω-hyperstructure in the sense of the above definition. if h(ω) = ( {yk; k ∈ ω},•ω ) is a general ω-hyperstructure such that (yk,�k), k ∈ ω are pairwise disjoint ordered sets and •ω({yk}k∈ω) = ⋃ k∈ω [yk)k ⊆ ⋃ k∈ω yk for any sequence {yk}k∈ω ∈ ∏ k∈ω yk we consider a system hk : (xk,≤k) → (yk,�k), k ∈ ω, of strongly isotone mappings, i.e. for any x ∈ xk there holds hk ( [xk)k ) = [ hk(xk) ) k . then denoting h = {hk : xk → yk; k ∈ ω} we obtain that h is a good homomorphism of the general ω-hyperstructure; g(ω) into the general ω-hyperstructure h(ω). indeed, consider an arbitrary sequence {xk}k∈ω ∈ ∏ k∈ω xk. as above denote by ϕ: p∗ ( ⋃ k∈ω xk ) →p∗( ⋃ k∈ω yk) the lifting of the mapping ϕ: ⋃ k∈ω xk → ⋃ k∈ω yk 8 general ω-hyperstructures and certain applications of those induced by the system {hk : xk → yk; k ∈ ω}—here in such a way that ϕ|xk = hk. then for any sequence {xk}k∈ω ∈ ∏ k∈ω xk we have ϕ ( ∗ω({xk}k∈ω) ) = ϕ (⋃ k∈ω [xk)k ) = ⋃ k∈ω ϕ ( [xk)k ) = ⋃ k∈ω hk ( [xk)k ) = ⋃ k∈ω [ hk(xk)k ) = •ω ( {hk(xk)}k∈ω ) = •ω (∏ k∈ω hk ( {xk}k∈ω )) , i.e. ϕ◦∗ω = •ω ◦ ∏ k∈ω hk, thus the diagram ∏ xk ∗ω−−−→ p∗ ( ⋃ k∈ω xk ) q k∈ω hk y yϕ]∏ yk •ω−−−→ p∗ ( ⋃ k∈ω yk ) (d2) is commutative. from the above example there follows immediately the following assertion. proposition 3.1. let (xk,≤k), (yk,�k), k ∈ ω, be two countable collections of pairwise disjoint ordered sets and g(ω), h(ω) be the corresponding ωgeneral hyperstructures. suppose (xk,≤k) ∼= (yk,�k) for each k ∈ ω and hk : (xk,≤k) → (yk,�k) are corresponding order-isomorphisms. then we have g(ω) ∼= h(ω). example 3.2. let j ⊂ r be an open interval, cn(j) be the ring (with respect to usual addition and multiplication of functions) of all real functions f : j → r with continuous derivatives up to the order n ≥ 0 including. denote l(p0,p1, . . . ,pn−1) : c n(j) → cn(j) the linear differential operator defined by l(p0,p1, . . . ,pn−1)(y) = dny(x) dxn + n−1∑ s=0 ps(x) dsy(x) dxs where y ∈ cn(j) and ps ∈ cn(j), s = 0, 1, . . . ,n − 1. in accordance with [4, 5] we put lan(j) = {l(p0, . . . ,pn−1); pk ∈ cn(j)}. 9 j. chvalina, š. hošková-mayerová instead of l(p1,0,p1,1, . . . ,p1,n−1) we write l(~p1). we put l(~p1) ≤ l(~p2 ) whenever l(~pj) = l(pj,0, . . . ,pj,n−1), j = 1, 2, p1,s(x) ≤ p2,s(x), s = 0, 1, . . . ,n − 1, x ∈ j and p1,0(x) ≡ p2,0(x). defining ∗n ( l(~p1),l(~p2 ), . . . ,l(~pn) ) = n⋃ k=1 {l(~p) ∈ lak(j); l(~pk) ≤ l(~p)} for any n-tuple ( l(~p1),l(~p2 ), . . . ,l(~pn) ) ∈ n∏ k=1 lak(j) we obtain that l(n) = ( {lak(j); k = 1, 2, . . . ,n},∗n ) is a general n-hyperstructure. of course, la1(j) is the set of all first-order linear differential operators of the form l(p0 )(y) = y ′(x) + p0 (x)y, where p0 ∈ c(j) and y ∈ c1(j). evidently laj(j) ∩ lak(j) = ∅ whenever j 6= k. it is to be noted that if k,m ∈{1, 2, . . . ,n} are fixed different integers then setting x = lak(j), y = lam(j) we obtain from the above construction an example of a general hyperstructure in sense of nezhad and hashemi. if, moreover x = y = lan(j) then the resulting general hyperstructure is an order hypergroup of linear differential n-order operators in the sense of [3], chap. iv, or [4, 5]. theorem 3.1. let ω be the smallest infinite countable ordinal, j ⊆ r be an open interval. if l(j; ω) = (∏ k∈ω lak(j),∗ω,p∗ (⋃ k∈ω lak(j) )) is the general ω-hyperstructure of ordinary linear differential operators and s(j; ω) = (∏ k∈ω vak(j),•ω,p∗ (⋃ k∈ω vak(j) )) is the general ω-hyperstructure of solution spaces of linear ordinary homogeneous differential equations associated with l(j; ω). then we have l(j; ω) ∼= s(j; ω), i.e. in the commutative diagram∏ k∈ω lak(j) ∗ω−−−→ p∗ ( ⋃ k∈ω lak(j) ) q k∈ω φk y yϕ]∏ k∈ω vak(j) •ω−−−→ p∗ ( ⋃ k∈ω vak(j) ) (d3) arrows ∏ k∈ω φk, ϕ ] are bijections. 10 general ω-hyperstructures and certain applications of those proof. by [4, 5] we have φk : lak(j) → vak(j) is a group-isomorphism for any k ∈ ω thus ∏ k∈ω φk : ∏ k∈ω lak(j) → ∏ k∈ω vak(j) is a bijection. since {lak(j); k ∈ ω},{vak(j); k ∈ ω} are pairwise disjoint families we have that the mapping ϕ: ⋃ k∈ω lak(j) → ⋃ k∈ω vak(j) such that ϕ|lak(j) = φk, k ∈ ω is a well-defined bijection hence the bijection p∗ ( ⋃ k∈ω lak(j) ) → p∗ ( ⋃ k∈ω vak(j) ) is also well-defined. now, for an arbitrary sequence {ln}n∈ω ∈ ∏ k∈ω lak(j) we obtain that •ω ((∏ k∈ω φk ) {ln}n∈ω ) = •ω { φn(ln) } n∈ω = •ω{vn}n∈ω = = ϕ ( ∗ω { φ−1n (ln) } n∈ω ) = ϕ] ( ∗ω{ln}n∈ω ) , since the hyperoperation “•ω” is associated with the hyperoperation “∗ω” . therefore the diagram d3 in the theorem 3.1 is commutative. let { (sk, ·,≤k); k ∈ ω } be a system of quasi-ordered semigroups. define a mapping �ω : ∏ k∈ω sk →p∗ ( ⋃ k∈ω sk ) by the rule �ω(x1, . . . ,xn) = ⋃ k∈ω [x2k)≤k for any sequence {xk}k∈omega ∈ ∏ k∈ω sk. then the general ω-hyperstructure is called the general ω-hyperstructure determined by the ends lemma or shortly el-determined general ω-hyperstructure. corollary of theorem 3.1 let ω be the smallest infinite countable ordinal, ω 6= 0, j ⊆ r be an open interval. let lel(j; n) = (∏ k∈ω lak(j),∗n,p∗ ( ⋃ k∈ω lak(j) )) be the el-determined general ω-hyperstructure of all linear ordinary differential operators of all orders k ∈ ω. let sel(j; n) = (∏ k∈ω vak(j),∗n,p∗ ( ⋃ k∈ω vak(j) )) be the el-determined general ω-hyperstructure of solutions of homogeneous linear ordinary differential equations ly = 0, l ∈ ⋃ k∈ω lak(j). then lel(j; n) ∼= sel(j; n). in the above construction we can use a finite sequence of positive integers {m1,m2, . . . ,mn} and then define the ω-hyperoperation �ω ( {xk}k∈ω ) =⋃ k∈ω [x mk k )≤k for any ω sequence {xk}k∈ω ∈ ∏ k∈ω sk. 11 j. chvalina, š. hošková-mayerová 4 general r-hyperstuctures (or l-hyperstuctures) in the paper [13], due to ideas of nezhad and hashemi, there are considered general r-hyperstuctures (or l-hyperstuctures) in the following sense. a general right hyperstructure (or left-hyperstructure) consist of two nonempty sets x,y together with a hyperoperation: ∗r : x ×y −→ p∗(x) or ∗l : x ×y −→ p∗(y ) (x,y) 7→ x∗r y ⊆ x, (x,y) 7→ x∗l y ⊆ y. to be more precise a general right hyperstructure (or left hyperstructure) is the quadruple (x,y,p∗(x),∗r) or (x,y,p∗(x),∗l), shortly general rhyperstructure or general l-hyperstructure. the set of points yrx = {x∗y : y ∈ y} that can be reached from a given point x ∈ x by the r-hyperoperation of two non-empty sets x,y , is called the r-hyperorbit of x. if yrx = x for all x ∈ x. then the set y is said to be r-hypertransitive on y . if yrx = {x}. then x is called a r-hyperfixed point to the rhyperoperation. the set {y ∈ y : x ∗r y = {x}}, is called r-hyperisotopy set at x. example 4.1. let x 6= ∅ be an arbitrary set, f : x → x be a mapping, i.e. the pair (x,f) is a monounary algebra. put y = n (the set of all positive integers) and define ∗fr : x × y → p ∗(x) by the rule x ∗fr n = {fk(x); k ∈ n,n ≤ k}. then the quadruple (x,y,p∗(x),∗fr) is a general right hyperstructure, i.e. r-hyperstructure. (here, fk is the k-th iteration of f). example 4.2. let t be a linearly ordered set (i.e. a chain) with the least element. then t is called a time scale or time axis. suppose a 6= ∅ 6= b are arbitrary sets and s is a binary relation between sets of mappings (impulses) at , bt , i.e. s ⊂ at ×bt . then the triad (at ,bt ,s) is called a general time system with input space at , the output space bt and with input-output relation (or the transition relation) s—cf.[23]. now, denote x = at , y = bt and define ∗sl : x × y → p ∗(y ) by x ∗sl y = s(x) = {u ∈ y ; xs u} for any pair of time-impulses x: t → a,y : t → b. then we obtain the quadruple (x,y,p∗(x),∗sl) which is a general left hyperstructure, i.e. a general l-hyperstructure. in the above mentioned classical monography [23] as a general system is considered a relation s ⊆ ∏ i∈i vi on non-void abstract sets. however—in 12 general ω-hyperstructures and certain applications of those detail—the index set i is decomposed into two subsets ix, iy and by inputoutput system is ment a relation s ⊆ x × y , where x = ∏ i∈ix vi is called the input object and y = ∏ i∈iy vi is termed as the output object. (cf. [23], definition 1.2). in the connection with general ω-hyperstructures we introduce inputoutput systems with inner evaluated input and output objects. denote k = {1, 2, . . . ,n}. by an ieo-general system, i.e. an input-output general system with inner evaluated objects we mean a qudruple ( ∏ k∈k xk, ∏ k∈k yk, r,fr), where xk, yk are non-empty sets, r ⊆ ∏ k∈k xk × ∏ k∈k yk is a domain full binary relation, i.e. dom r = ∏ k∈k xk = {x ∈ ∏ k∈k xk;∃y ∈ ∏ k∈k : x r y}, ∗x : ∏ k∈k xk → p( ⋃ k∈k xk), ∗y : ∏ k∈k yk → p( ⋃ k∈k yk) are mappings (i.e. nary multioperations)—thus ( ∏ k∈k xk,∗x),( ∏ k∈k yk,∗y) are general n-ary hyperstructures and fr : p( ⋃ k∈k xk) → p( ⋃ k∈k yk) is a mapping defined in this way: for a non-empty set m ⊂ ⋃ k∈k xk, i.e. m ∈p( ⋃ k∈k xk) we denote m − x ={ [x1, . . . , (x1, . . . ,xn)n] ∈ ∏ k∈k xk;∗x(x1, . . . , (x1, . . . ,xn)n) ∩ m 6= ∅ } . then we put fr(m) = ∗y ( r(m−x ) ⊂ ⋃ k∈k yk. if m − x = ∅ we define fr(m) = ∅. (this definition includes the case fr(∅) = ∅). hence the diagram ∏ k∈k xk ∗x−−−→ p ( ⋃ k∈k xk ) r y yfr∏ k∈k yk ∗y−−−→ p (⋃ k∈k yk) (d4) is commutative. now, in the case when xk = a, yk = b for all k ∈ k, then denoting the index set k by t (the time set-continuous, which means a linearly ordered continuum with the minimal element, or discrete n0) we obtain ∏ k∈k xk = a t ,∏ k∈k yk = b t and ⋃ k∈k xk = a ⋃ k∈k yk = b. thus the above ieo-general system turns to the above mentioned general time system (here with an inner evaluation) (at ,bt ,r,∗a,∗b), where fr ◦∗a = ∗b ◦r. 13 j. chvalina, š. hošková-mayerová by a homomorphism of an ieo-general system ( ∏ k∈k xk, ∏ k∈k yk, r,∗x,∗y ) into another ieo-general system ( ∏ k∈k uk, ∏ k∈k vk, s,∗u,∗v ) we mean a quadruple of mappings h = [φxu, φy v , φpxu, φpy v ], φxu : ∏ k xk → ∏ k uk; φy v : ∏ k yk → ∏ k vk; φpxu : p( ⋃ k xk →p( ⋃ k uk); φpy v : p( ⋃ k yk) →p( ⋃ k vk) such that the diagram ∏ k∈k uk p (⋃ k∈k uk ) ∏ k∈k xk p (⋃ k∈k xk ) ∏ k∈k yk p (⋃ k∈k yk ) ∏ k∈k vk p (⋃ k∈k vk ) // ∗ u �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � s �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � fs __?????????????? φxu // ∗ x �� � � � � � � � � � � r ??�������������� φp xu �� � � � � � � � � � � fr ���� �� �� �� �� �� �� φy u // ∗ y �� ?? ?? ?? ?? ?? ?? ?? φp y v // ∗ v (d5) is commutative. in concrete cases of modelling special systems, the used objects and their mappings take a concrete interpretation. 4.1 l-hyperoperation (or r-hyperoperation) of a hyperstucture on a non-empty set in this paragraph, we recall two definitions of a l-hyperoperation (or rhyperoperation) of a hyperstucture on a non-empty set. let us make our point clear with some examples. definition 4.1. [13] let (g,?) be a hyperstructure and x be a non-empty set. 14 general ω-hyperstructures and certain applications of those a generalized l-hyperaction of g on x is a l-hyperoperation ψ : g×x −→ p∗(x) such that the following axioms are satisfied: 1) for all g,h ∈ g and x ∈ x, ψ(g ? h,x) ⊆ ψ(g,ψ(h,x)), 2) for all g ∈ g, ψ(g,x) = x. for any g ∈ g and a ⊆ x, ψ(g,a) = ⋃ x∈a ψ(g,x), also for any x ∈ x and b ⊆ g, ψ(b,x) = ⋃ b∈b ψ(b,x). if in the axiom 1) of definition the equality holds, the generalized r-hyperaction is called strong. as application of the above concepts we mention the classical interval binary hyperoperation on a linearly ordered group. see [18]. in detail if (g, ·,≤) is a linearly ordered group then we define a binary hyperoperation ∗: g×g →p∗(g) by a∗ b = [ min{a,b} ) ≤ ∩ ( max{a,b} ] ≤ = [ min{a,b}, max{a,b} ] ≤ = { x ∈ g; min{a,b}≤ x ≤ max{a,b} } (which is a closed interval) where min{a,b}, max{a,b} is the least element, the greatest element of the set {a,b}, respectively. it is easy to verify that the obtained hypergroupoid (g,∗) is an extensive commutative hypergroup. this hypergroup we obtain even in the case if we restrict ourselves onto the set g+ of all positive elements of the linearly ordered group (g, ·,≤), (cf. the proof of proposition 4.1). proposition 4.1. let (g, ·,≤) be a linearly ordered group, g+ be its subset of all positive elements (i.e. the positive cone) endowed with the interval binary hyperoperation “∗l”. define a mapping ψg : g+ ×g →p ∗(g) by ψg(a,b) = (a + b]≤ = {x ∈ g; x ≤ a + b} for all pairs (a,b) ∈ g+ ×g. then the quadruple ( g+,g,p∗(g),ψg ) is the generalized l-hyperoperation of the commutative extensive hypergroup (g,∗l) on the group (g, +,≤). proof. for the proof see [13]. 15 j. chvalina, š. hošková-mayerová 5 homomorphism of transformation semihypergroups definition 5.1. let (x,g,ψ), (y,h,η) be two generalized transformation semihypergroups (gts). a pair of mappings φ = [µ,ϕ] such that µ: g → h is a homomorphism of semihypergroups and ϕ: x → y is a mapping, is said to be a homomorphism of gts (x,g,ψ) into gts (y,h,η) if for any pair [g,x] ∈ g×x the equality η ( µ(g),ϕ(x) ) = ϕ ( ψ(g,x) ) is satisfied, i.e. the diagram, where ϕ∗ : p∗(x) →p∗(y ) is the corresponding liftation of the mapping ϕ: x → y , g×x ψ−−−→ p∗(x) µ×ϕ y yϕ∗ h ×y ω−−−→ p∗(y ) (d6) commutes. example 5.1. let x,y be equivalent non-empty sets and f : x → x, h: y → y be mappings such that mono-unary algebras (x,f) ∼= (y,h). denote g = {fn; n ∈ n0}, h = {hn,n ∈ n0} and define binary hyperoperations ?: g×g →p∗(g), •: h ×h →p∗(h), by fn ?fm = {fk; k ∈ n0,m + n ≤ k} and hn•hm = {hk; k ∈ n0,m + n ≤ k}. define mappings ψ : g × x → p∗(x), η : h × y → p∗(y ), by the same rule ψ(fn,x) = { fk(x); k ∈ {0,n,n + 1,n + 2, . . .} } , ω(hn,y) = { hk(y); k ∈ {0,n,n + 1,n + 2, . . .} } . suppose ξ : (x,f) → (y,h) is an isomorphism and ϕ: (x,f) → (y,h) a homomorphism of the mono-unary algebra (x,f) onto the mono-unary algebra (y,h). denote φ = [µ,ϕ] the pair of mappings such that µ(fn) = ξ ◦ fn ◦ ξ−1. then φ is a ii-homomorphism of the generalized transformation semihypergroup (x,g,ψ) into the gts (y,h,ω). indeed, for an arbitrary pair [fn,x] ∈ g×x we have ϕ ( ψ(fn,x) ) = ϕ { x,fn(x),fn+1(x), . . . } = { ϕ(x),ϕ ( fn(x) ) ,ϕ ( fn+1(x) ) , . . . } = { ϕ(x),ϕ ( hn(x) ) ,ϕ ( hn+1(x) ) , . . . } = ω ( hn,ϕ(x) ) = ω ( ξ ◦fn ◦ ξ−1,ϕ(x) ) = ω ( µ(fn),ϕ(x) ) = ω ( µ×ϕ)[fn,x] ) . 16 general ω-hyperstructures and certain applications of those the following example of generalized transformation hypergroup is based on consideration published in [4, 5]. example 5.2. let j ⊂ r be an open interval and denote c∞(j) the ring of all infinitely differentiable functions on j. let us consider the set lan(j), n ∈ n, of linear differential operators of the n-th order in the form l(p0, . . . ,pn−1) = dn dxn + n−1∑ k=0 pk(x) dk dxk . where pk ∈ c∞(j), k = 0, 1, . . . ,n−1; l(p0, . . . ,pn−1) : c∞(j) −→ c∞(j), thus l(p0, . . . ,pn−1)(f) = f (n)(x)+pn−1(x)f (n−1)(x)+· · ·+p0 (x)f(x), f ∈ c∞(j). let δij stand for the kronecker symbol δ. for any but fixed m ∈{0, 1, . . . ,n− 1} we denote by lan(j)m = { l(p0, . . . ,pn−1)|pk ∈ c∞(j),pm > 0 } . shortly we put p = (p0 (x), . . . ,pn−1(x)),x ∈ j and on the set lan(j)m we define a binary operation “◦m” and a binary relation ≤m in this way: l(p) ◦m l(q) = l(u) where uk(x) = pm(x)qk(x) + (1 − δkm)pk(x),x ∈ j, 0 ≤ k ≤ n− 1, and l(p) ≤m l(q) whenever pk(x) ≤ qk(x),k 6= m,k ∈{0, 1, . . . ,n− 1},pm(x) = qm(x),x ∈ j. it is easy to verify that (lan(j)m,◦m,≤m) is an ordered noncommutative group with the neutral element d(w), where d(w) = (w0, . . . ,wn−1), wk(x) = δkm. an inverse to any d(q) is d −1(q) = ( −q0 qm , . . . , 1 qm , . . . , −qn−1 qm ) . let (z, +,≤) be an additive group of all integers with an usual ordering “≤”. then by lemma 1.1 the structure (z,?), where ?: z × z −→p∗(z) was defined by k ? l = [k + l)≤ is a hypergroup. for fixed d(q) ∈ lan(j)m we define an action ψq : z × lan(j)m −→ p∗(lan(j)m) as follows, ψq(k,l(p)) = {lt(q) ◦m l(p)|t ≤ k}. so (lan(j)m, z,ϕq) is a generalized transformation hypergroup. 17 j. chvalina, š. hošková-mayerová references [1] ashrafi, a. r., hossein zadeh, a. and yavari, m. : hypergraphs and join spaces, italian journal pure appl. math 12, 185–196 (2002) [2] bošnjak, i., madrász, r. : on power structures, algebra and discrete math. 2, 14–35 (2003) [3] chvalina, j. : function graphs, quasi-ordered sets and commutative hypergroups, mu brno (1995) (czech) [4] chvalina, j., chvalinová, l. : actions of centralizer hypergroups of n-th order linear differential operators on rings of smooth functions, aplimat, journal of appl. math. i, no. i, 45–53 (2008) [5] chvalina, j., chvalinová, l. : realizability of the endomorphism monoid of a semi-cascade formed by solution spaces of linear ordinary n–th order differential equations, aplimat, journal of appl. math. iii, no. ii, 211– 223 (2010) [6] chvalina, j., hošková, š. : abelizations of weakly associative hyperstructures based on their direct squares, acta math. inform. univ. ostraviensis 11, no. 1, 11–23 (2003) [7] corsini. p., leoreanu.v. : application of hyperstructure theory, kluwer academic publishers, (2003) [8] cristea, i., stefanescu, m., angheluta, c.: about the fundamental relations defined on the hypergroupoids associated with binary relations. eur. j. comb. 32(1): 72-81 (2011) [9] cristea, i., stefanescu, m. : binary relations and reduced hypergroups. discrete mathematics 308(16): 3537-3544 (2008) [10] davvaz, b. : polygroups with hyperoperators, the journal of fuzzy mathematics 9, no. 4, 815–823 (2004) [11] dehghan nezhad a., davvaz, b. : universal hyperdynamical systems, bulletin of the korean mathematical society, 47(3), 513–526 (2010) [12] hošková, š. : order hypergroups—the unique square root condition for quasi-order hypergroups, set-valued mathematics and applications 1, no.1, global research publications, india, 1–7 (2008) 18 general ω-hyperstructures and certain applications of those [13] hošková-mayerová, š., chvalina, j., dehghan nezhad a.: general actions of hyperstuctures and some applications, analele stiintifice ale universitatii ‘’ovidius”, constanta, romania, in print [14] hošková, š., chvalina, j., račková, p. : transposition hypergroups of fredholm integral operators and related hyperstructuress—part 2, journal of basic science, iran, 4, 55–60 (2008) [15] hošková, š., chvalina, j. : action of hypergroups of the first–order partial differential operators, in: aplimat 2007, bratislava, slovakia, 177–185 (2007) [16] hošková, š., chvalina, j. : discrete transformation hypergroup and transformation hypergroups with phase tolerance space, discrete mathematics 308, 4133–4143 (2008) [17] hošková, š. : discrete transformation hypergroups, in: 4th international conference on aplimat 2005, bratislava, slovakia, pt ii, 275–278 (2005) [18] iwasava, k. : on linearly ordered groups, journal of the math. soc. japan, 1, no.1, 1–9 (1948) [19] jantosciak, j. : transposition hypergroups: noncommutative join spaces, j. algebra 187, 97–19 (1997) [20] marty, f. : sur une generalization de la notion de groupe, in: 8th congress math. scandenaves, stockholm, 45–49 (1934) [21] massouros, ch. g. mittas, j. : on the theory of generalized mpolysymmetric hypergroups, proceedings of the 10th international congress on algebraic hyperstructures and applications, brno, czech republic 2009, pp. 217-228. 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[2002]. accordingly a cfs is an extension of the fuzzy set whose range is extended from [0,1] to a disc of unit radius in complex plane. xueling et al. [2019] introduced some basic operations on complex fuzzy set. they have developed a new algorithm in signals and system by using complex fuzzy sets. after that some results have been given by p.bhattacharya [1987] about fuzzy graphs. thirunavukarasu et al. [2016] extended the fuzzy graph to complex fuzzy graph. fuzzy graph were narrated by mordeson and peng [1994]. nagoorgani and latha [2015] gave some operations such as cartesian product, conjuction, disjunction in fuzzy graph. shannon and atanassov [1994] defined intuitionistic fuzzy graphs. after that many authors added their ideas to intuitionistic fuzzy graph. naveed et al. [2019] introduced the complex intuitionistic fuzzy graphs with certain notions of union, join and composition. for crisp graph we can refer graph theory by harary [1969]. in this paper, we introduced some types of complex fuzzy graphs with examples. also some operations on complex fuzzy graphs such as union, intersection, composition and cartesian product with suitable examples. 2 premilinaries definition 2.1. a complex fuzzy graph gc = (σc,µc) is defined on a graph g = (v,e) is a pair of complex functions σc : v → r(z)eiθ(z),µc : e ⊆ v × v → r(e)eiϕ(e) such that µc(z1,z2) = r(e)eiϕ(e), where r(e) ≤ min{r(z1),r(z2)} and ϕ(e) ≤ min{θ(z1),θ(z2)} for all z1,z2 ∈ v and 0 ≤ r(z1),r(z2) ≤ 1,0 ≤ θ(z1),θ(z2) ≤ 2π. example 2.1. consider the complex fuzzy graph gc = (σc,µc) where σc = {z1/0.2eiπ,z2/0.5ei0.5π,z3/0.7}, µc = {(z1,z2)/0.1ei0.5π,(z1,z3)/0.1,(z2,z3)/0.5} definition 2.2. a complex fuzzy graph gc = (σc,µc) is said to be a complement of cfg gc if i) σc(z) = σc(z) and ii) µc(z1,z2) = r(e)eiϕ(e), where r(e) = min {r(z1),r(z2)} − r(e) and ϕ(e) = min {θ(z1),θ(z2)} − ϕ(e),∀z1,z2 ∈ v . 80 some operations on complex fuzzy graphs z2(0.5e i0.5π) z3(0.7) e3(0.5) z1(0.2e iπ) e2(0.1) z1(0.2e iπ) e1(0.1e i0.5π) figure 1: complex fuzzy graph example 2.2. consider the example 2.1, the complement of a cfg given in the figure 1 is given by σc = {z1/0.2eiπ,z2/0.5ei0.5π,z3/0.7} , µc = {(z1,z2)/0.1,(z1,z3)/0.1} definition 2.3. the order p and size q of a cfg gc = (σc,µc) defined on g = (v,e) are defined by p = ∑ z∈v r(z).e i ∑ z∈v θ(z) ; q = ∑ e=(zi,zj)∈e r(e).e i ∑ e=(zi,zj)∈e ϕ(e) , where r(e) ≤ min{r(zi),r(zj)} and ϕ(e) ≤ min{θ(zi),θ(zj)} for all zi,zj ∈ v . example 2.3. consider the example 2.1, the order of gc is p = 1.4ei1.5π, the size of gc is q = 0.7ei0.5π definition 2.4. the degree of a vertex zi in a cfg gc = (σc,µc) defined on g = (v,e) is defined by d(zi) = ∑ e=(zi,zj)∈µc r(e) . e i ∑ e=(zi,zj)∈µc ϕ(e) such that µc(zi,zj) = r(e) . eiϕ(e), for all zj ∈ σc. example 2.4. consider the example 2.1, d(z1) = 0.2ei0.5π;d(z2) = 0.6ei0.5π; d(z3) = 0.6 definition 2.5. a cfg is said to be complete for every pair of vertices µc(z1,z2) = r(e)eiϕ(e) where r(e) = min {r(z1),r(z2)} and ϕ(e) = min {θ(z1),θ(z2)} for all z1,z2 ∈ v example 2.5. let gc = (σc,µc) be a cfg,where σc = {z1/0.5ei0.7π,z2/0.8eiπ,z3/0.6eiπ} µc = {(z1,z2)/0.5ei0.7π,(z1,z3)/0.5ei0.7π,(z2,z3)/0.6eiπ} definition 2.6. a cfg is regular if d(zi) = d(zj) for all zi,zj ∈ σc. definition 2.7. in a cfg gc = (σc,µc) for all zi,zj ∈ σc, the neighbourhood of zi is defined by n(zi) = {zj ∈ σc/(zi,zj) ∈ µc} 81 n.azhagendran and a.mohamed ismayil definition 2.8. a path p in a cfg gc = (σc,µc) is a sequence of distinct vertices z0,z1,z2, · · ·zn ∈ v (except possibly z0andzn) such that µc(zi−1,zi)= r(ei)e iϕ(ei),r(ei) > 0,ϕ(ei) ≥ 0, i = 1,2, ...,n. here n is called the length of the path. the consecutive pairs are called edges of the path. the strength of the path in a cfg is defined by µc(zi−1,zi) = min {r(ei)}eiminϕ(ei), i = 1,2,3, ...,n. it is denoted by s(p). definition 2.9. the strength of connectedness between two vertices zi abd zj which is defined as the maximum amplitude and maximum phase term values of the strength of all paths between zi and zj. in symbol we denote it as µ∞c (zi,zj) = conngc(zi,zj), µ ∞ c (zi,zj) = t(e)e iψ(e),0 ≤ t(e) ≤ 1,0 ≤ ψ(e) ≤ 2π, where t(e) is maximum amplitude value of all paths between zi and zj and ψ(e) is maximum phase term value of all paths between zi and zj. for any arc (zi,zj), if r(e) ≥ t(e) and ϕ(e) ≥ ψ(e) then the arc (zi,zj) is said to be strong. definition 2.10. the strong degree of a vertex z in a cfg is defined by sum of membership values of strongarcs incident at z ,and it is denoted by ds(z). definition 2.11. the strong neighbourhood of zi in a cfg is defined by ns(zi)={zj ∈ v/(zi,zj)is a strong arc}. definition 2.12. a cfg gc = (σc,µc) is said to be bipartite if the vertex σc can be partitioned into two non-empty sets σc1 and σc2 such that µc(zi,zj) = 0 if zi,zj ∈ σc1 and zi,zj ∈ σc2. definition 2.13. a cfg gc = (σc,µc) is said to be complete bipartite if the vertex σc can be partitioned into two non-empty sets σc1 and σc2 such that µc(zi,zj) = r(e).eiϕ(e), where r(e) = min {r(zi),r(zj)} ,ϕ(e) = min {θ(zi),θ(zj)} for zi ∈ σc1 and zj ∈ σc2. definition 2.14. a vertex zi of a complex fuzzy graph gc = (σc,µc) is said to be an isolated vertex if µc(zi,zj) = 0,∀zj ∈ v − {zi} ,(i.e)n(zi) = ∅. 3 operations on complex fuzzy graph in this section the operations union, intersection, composition and cartesian product of cfgs are defined with examples. some propositions based on the above operations are stated and proved. definition 3.1. let the two complex fuzzy graphs gc1 = (σc1,µc1), and gc2 = (σc2,µc2) defined on two graphs g1 = (v1,e1) and g2 = (v2,e2) respectively. let gc1 be a pair of complex functions σc1 : v1 → r1(z)eiθ1(z),µc1 : e1 ⊆ 82 some operations on complex fuzzy graphs v1 × v1 → r1(e).eiϕ1(e) such that µc(z1,z2) = r1(e)eiϕ1(e) where r1(e) ≤ min {r1(z1),r1(z2)} ,ϕ1(e) ≤ min {θ1(z1),θ1(z2)}. also gc2 is a pair of complex functions σc2 : v2 → r2(z)eiθ2(z),µc2 : e2 ⊆ v2 × v2 → r2(e).eiϕ2(e) such that µc(z1,z2) = r2(e)eiϕ2(e) where r2(e) ≤ min {r2(z1),r2(z2)} ,ϕ2(e) ≤ min {θ2(z1),θ2(z2)}. then the union of two complex fuzzy graphs gc = (σc = σc1 ∪ σc2,µc = µc1 ∪ µc2) on g = (v = v1 ∪ v2,e = e1 ∪ e2) is defined as follows (i) σc(z) = (σc1∪σc2)(z) =   r1(z)e iθ1(z),∀z ∈ v1and z /∈ v2 r2(z)e iθ2(z),∀z ∈ v2and z /∈ v1 max {r1(z),r2(z)}eimax{θ1(z),θ2(z)},z ∈ v1 ∩ v2 (ii) µc(z1,z2) = (µc1 ∪ µc2)(z1,z2) =  r1(e)e iϕ1(e),∀ (z1,z2) ∈ e1 and(z1,z2) /∈ e2 r2(e)e iϕ2(e),∀ (z1,z2) ∈ e2 and (z1,z2) /∈ e1 max {r1(e),r2(e)}eimax{ϕ1(e),ϕ2(e)},(z1,z2) ∈ e1 ∩ e2 proposition 3.1. prove that the union of two complex fuzzy graph is also a complex fuzzy graph. proof. let gc1 and gc2 be two cfgs, then the union gc = gc1 ∪gc2 is discussed in three different case. for vertices, i) suppose that z ∈ v1 and z /∈ v2 then (σc1 ∪ σc2)(z) = max {r1(z),r2(z)} .eimax{θ1(z),θ2(z)} = max {r1(z),0} .eimax{θ1(z),0} = r1(z).e iθ1(z),∀z ∈ v1and z /∈ v2. ii) similarly we can prove, for z /∈ v1 and z ∈ v2 iii) suppose that z ∈ v1 ∩ v2, then (σc1 ∪ σc2)(z) = max {σc1,σc2} = max { r1(z)e iθ1(z).r2(z)e iθ2(z) } = max {r1(z),r2(z)} .eimax{θ1(z),θ2(z)},∀z ∈ v1 ∩ v2. for edges, i) suppose that (z1,z2) ∈ e1,(z1,z2) /∈ e2 (µc1 ∪ µc2)(z1,z2) = max {µc1(z1,z2),µc2(z1,z2)} ≤ max {min {r1(z1),r1(z2),min {r2(z1),r2(z2)}}} . eimax{min{θ1(z1),θ1(z2)},min{θ2(z1),θ2(z2)}} = max {r1(e),r2(e)} .eimax{ϕ1(e),ϕ2(e)} = max {r1(e),0} .eimax{ϕ1(e),0} = r1(e).e iϕ1(e),∀(z1,z2) ∈ e1,(z1,z2) /∈ e2 83 n.azhagendran and a.mohamed ismayil ii) similarly, we can prove that, (z1,z2) /∈ e1,(z1,z2) ∈ e2 iii) suppose that (z1,z2) ∈ e1 ∩ e2 (µc1 ∪ µc2)(z1,z2) = max {µc1(z1,z2),µc2(z1,z2)} ≤ max {min {r1(z1),r1(z2)} ,min {r2(z1),r2(z2)}} . eimax{min{θ1(z1),θ1(z2)},min{θ2(z1),θ2(z2)}} = max {r1(e),r2(e)} .eimax{ϕ1(e),ϕe(z)} for (z1,z2) ∈ e1 ∩ e2 therefore the union of two cfgs is also a cfg. example 3.1. consider the two complex fuzzy graphs gc1 = (σc1,µc1) and gc2 = (σc2,µc2) given below σc1 = {z1/0.4ei0.8π,z2/0.7ei2π,z3/0.5ei1.2π} , µc1 = {(z1,z2)/0.4ei0.8π,(z2,z3)/0.4eiπ} and σc2 = {z1/0.5eiπ,z3/0.5ei1.4π, z4/0.8e i2π, µc2 = {(z1,z4)/0.4ei0.6π,(z3,z4)/0.5ei1.4π} then the union gc = (σc1 ∪ σc2 = σc,µc1 ∪ µc2 = µc) on g = (v1 ∪ v2,e1 ∪ e2) can be written as σc = {z1/0.5eiπ,z2/0.7ei2π,z3/0.5ei1.4π,z4/0.8ei2π} µc = {(z1,z2)/0.4ei0.8π,(z1,z4)/0.4ei0.6π,(z3,z4)/0.5ei1.4π,(z2,z4)/0.4eiπ} z1(0.4e i0.8π) z2(0.7e i2π) 0.4ei0.8π z3(0.5e i1.2π) 0.4eiπ figure 2: gc1 z1(0.5e iπ) z4(0.8e i2π) 0.4ei0.6π z3(0.5e i1.4π) 0.5ei1.4π figure 3: gc2 z2(0.7e i2π) z1(0.5e iπ) 0.4ei0.8π z3(0.5e i1.4π) 0.4eiπ z4(0.8e i2π) z1(0.5e iπ) z4(0.8e i2π) z3(0.5e i1.4π) 0.4ei0.6π 0.5ei1.4π figure 4: gc1 ∪ gc2 remark 3.1. union of two strong cfg need not be a strong cfg. 84 some operations on complex fuzzy graphs example 3.2. consider the strong cfgs gc1 = (σc1,µc1) and gc2 = (σc2,µc2) where σc1 = {z1/0.4ei2π,z2/0.6eiπ}, µc1 = {(z1,z2)/0.4eiπ} and σc2 = {z1/0.8ei0.2π,z2/0.2eiπ}, µc2 = {(z1,z2)/0.2ei0.2π} then the union is given by σc = {z1/0.8ei2π,z2/0.6eiπ}, µc = {(z1,z2)/0.4ei0.2π}. however this is not a strong cfg. definition 3.2. let the two complex fuzzy graphs gc1 = (σc1,µc1), and gc2 = (σc2,µc2) defined on two graphs g1 = (v1,e1) and g2 = (v2,e2) respectively. let gc1 be a pair of complex functions σc1 : v1 → r1(z)eiθ1(z),µc1 : e1 ⊆ v1 × v1 → r1(e).eiϕ1(e) such that µc(z1,z2) = r1(e)eiϕ1(e) where r1(e) ≤ min {r1(z1),r1(z2)} ,ϕ1(e) ≤ min {θ1(z1),θ1(z2)}. also gc2 is a pair of complex functions σc2 : v2 → r2(z)eiθ2(z),µc2 : e2 ⊆ v2 × v2 → r2(e).eiϕ2(e) such that µc(z1,z2) = r2(e)eiϕ2(e) where r2(e) ≤ min {r2(z1),r2(z2)} ,ϕ2(e) ≤ min {θ2(z1),θ2(z2)}. then the intersection of two complex fuzzy graphs gc = (σc = σc1 ∩ σc2,µc = µc1 ∩ µc2) is defined as follows (i) σc(z) = (σc1 ∩ σc2)(z) = min {r1(z),r2(z)}eimin{θ1(z),θ2(z)},z ∈ v1 ∩ v2 (ii) µc(z1,z2) = (µc1 ∩ µc2)(z1,z2) ={ min {r1(e),r2(e)}eimin{ϕ1(e),ϕ2(e)},(z1,z2) ∈ e1 ∩ e2 0,otherwise proposition 3.2. prove that the intersection of two complex fuzzy graphs is also a complex fuzzy graph. proof. let gc1 and gc2 be two complex fuzzy graphs, then i) (σc1 ∩ σc2)(z) = min {r1(z),r2(z)} .eimin{θ1(z),θ2(z)} itistrivial. ii) for (z1,z2) ∈ e1 ∩ e2 (µc1 ∩ µc2)(z1,z2) = min {µc1(z1,z2),µc2(z1,z2)} .eimin{ϕ1(z),ϕ2(z)} ≤ min {min {r1(z1),r1(z2)} ,min {r2(z1),r2(z2)}} . eimin{min{θ1(z1),θ1(z2)},min{θ2(z1),θ2(z2)}} = min {min {r1(z1),r2(z1)} ,min {r1(z2),r2(z2)}} . eimin{min{θ1(z1),θ1(z2)},min{θ2(z1),θ2(z2)}} = min {r1(z),r2(z)} .eimin{ϕ1(z),ϕ2(z)} where r1(z) ≤ min {r1(z1),r2(z1)} ,r2(z) ≤ min {r1(z2),r2(z2)} , ϕ(z1) ≤ min {θ1(z1),θ2(z1)} , ϕ(z2) ≤ min {θ2(z1),θ2(z2)} hence the gc1 ∩ gc2 is a cfg. 85 n.azhagendran and a.mohamed ismayil example 3.3. consider the two complex fuzzy graphs gc1 = (σc1,µc1) where σc1 = {z1/0.5ei0.7π,z2/0.8,z3/0.7ei1.5π} ;µc1 = {(z1,z2)/0.5,(z2,z3)/0.7} and gc2 = (σc2,µc2) where σc2 = {z1/0.4ei2π,z2/0.6ei0.8π,z4/0.7eiπ} ; µc2 = {(z1,z4)/0.4eiπ,(z1,z2)/0.4ei0.8π} then the intersection of gc1 and gc2 on a pair g = (v1 ∩v2,e1 ∩e2) is given by v = {z1/0.4ei0.7π,z2/0.6} ,e = {(z1,z2)/0.4} where v = v1∩v2;e = e1∩e2. remark 3.2. intersection of two strong cfgs is also a strong cfg. definition 3.3. let the two complex fuzzy graphs gc1 = (σc1,µc1), and gc2 = (σc2,µc2) defined on two graphs g1 = (v1,e1) and g2 = (v2,e2) respectively. let gc1 be a pair of complex functions σc1 : v1 → r1(a)eiθ1(a),µc1 : e1 ⊆ v1 × v1 → r1(e).eiϕ1(e) such that µc(a1,a2) = r1(e)eiϕ1(e) where r1(e) ≤ min {r1(a1),r1(a2)} ,ϕ1(e) ≤ min {θ1(a1),θ1(a2)}. also gc2 is a pair of complex functions σc2 : v2 → r2(a)eiθ2(a),µc2 : e2 ⊆ v2 × v2 → r2(e).eiϕ2(e) such that µc(a1,a2) = r2(e)eiϕ2(e) where r2(e) ≤ min {r2(a1),r2(a2)} ,ϕ2(e) ≤ min {θ2(a1),θ2(a2)}. then the composition of two complex fuzzy graphs is defined as follows (i) (σc1 ◦ σc2)(a1,a2) = min {r1(a1),r2(a2)}eimin{θ1(a1),θ2(a2)},∀a1,a2 ∈ v = v1 ◦ v2 (ii) (µc1 ◦ µc2)((a,a2),(a,b2)) = min {r1(a),r2(a)}eimin{θ1(a),ϕ2(a)} where r2(a) ≤ min {r2(a2),r2(b2)} ;ϕ2(a) ≤ min {θ2(a2),θ2(b2)} ,∀a ∈ v1,(a2,b2) ∈ e2 (iii) (µc1 ◦ µc2)((a1,a),(b1,a)) = min {r1(a),r2(a)}eimin{ϕ1(a),θ2(a)} wherer1(a) ≤ min {r1(a1),r1(b1)} ;ϕ1(a) ≤ min {θ1(a1),θ1(b1)} ,∀a ∈ v2,(a1,b1) ∈ e1 (iv) (µc1 ◦ µc2)((a1,a2),(b1,b2)) = min {r2(a2),r2(b2),r1(a)} . eimin{θ2(a2),θ2(b2),ϕ1(a)} ,∀a2,b2 ∈ v2,a2 ̸= b2,(a1,b1) ∈ e1, wherer1(a) ≤ min {r1(a1),r1(b1)} ;ϕ2(a) ≤ min {θ1(a1),θ1(b1)} , proposition 3.3. prove that the composition of two complex fuzzy graphs is also a complex fuzzy graph. proof. let gc1 and gc2 be two cfgs then we prove that gc1 ◦ gc2 is a cfg. (i) (σc1 ◦ σc2) = (a1,a2) = min {r1(a1),r2(a2)}eimin{θ1(a1),θ2(a2)},∀a1,a2 ∈ v . it is trivial. 86 some operations on complex fuzzy graphs (ii) (µc1 ◦ µc2)((a,a2)(a,b2)) = min {r1(a),r2(a)}eimin{θ1(a),θ2(a)} ≤ min {r1(a),min {r2(a2),r2(b2)}}eimin{θ1(a),min{θ2(a2),θ2(b2)}} = min {min {r1(a),r2(a2)} ,min {r1(a),r2(b2)}} . eimin{min{θ1(a),θ2(a2)},min{θ1(a),θ2(b2)}} = min {r(a1),r(b1)}eimin{θ(a1),θ(b1)} , where r(a1) = min {r1(a),r2(a2)} , r(b1) = min {r1(a),r2(b2)} ,θ(a1) = min {θ1(a),θ2(a2)} , θ(b1) = min {θ1(a),θ2(b2)} for all a ∈ v1and(a2,b2) ∈ e2. (iii) (µc1 ◦ µc2)((a1,b)(b1,b)) = min {r1(a),r2(b)}eimin{θ1(a),θ2(b)} ≤ min {min {r1(a1),r1(b1)} ,r2(b)}eimin{min{θ1(a1),θ1(b1)},θ2(b)} = min {min {r1(a1),r2(b)} ,min {r1(b1),r2(b)}}. eimin{min{θ1(a1),θ2(b)},min{θ1(b1),θ2(b)}} = min {r(a2),r(b2)}eimin{θ(a2),θ(b2)}, where r(a2) = min {r1(a1),r2(b)} , r(b2) = min {r1(b1),r2(b)} ,θ(a2) = min {θ1(a1),θ2(b)} , θ(b2) = min {θ1(b1),θ2(b)} for all b ∈ v2 and (a1,b1) ∈ e1. (iv) for all a2,b2 ∈ v2,a2 ̸= b2,(a1,b1) ∈ e1 (µc1 ◦ µc2)((a1,a2),(b1,b2)) = min {r2(a2),r2(b2),r1(a)} . eimin{θ2(a2),θ2(b2),ϕ(a)} ≤ min {r2(a2),r2(b2),min {r1(a1),r1(b1)}} . eimin{θ2(a2),θ2(b2),min{θ1(a1),θ1(b1)}} = min {r2(a2),r2(b2),r1(a1),r1(b1)} .eimin{θ2(a2),θ2(b2),θ1(a1),θ1(b1)} = min {r(a),r(b)} .eimin{θ(a),θ(b)}. where r(a) = min {r1(a1),r2(a2)} ,r(b) = min {r1(b1),r2(b2)} θ(a) = min {θ1(a1),θ2(a2)} ,θ(b) = min {θ1(b1),θ2(b2)}. hence gc1 ◦ gc2 is a cfg. example 3.4. consider the cfgs, gc1 = (σc1,µc1) where σc1 = {z1/0.4ei0.7π,z2/0.6ei2π} ;µc1 = {(z1,z2)/0.4ei0.5π} and gc2 = (σc2,µc2) where σc2 = {z3/0.5ei1.2π,z4/0.6ei0.7π} ;µc2 = {(z3,z4)/0.5ei0.7π} . then the composition of gc1 and gc2 is given by gc = (σc,µc) where σc = {z1z3/0.4ei0.7π,z1z4/0.4ei0.7π,z2z3/0.5ei1.2π,z2z4/0.6ei0.7π} , µc = {(z1z3,z1z4)/0.4ei0.7π,(z1z4,z2z4)/0.4ei0.5π,(z2z3,z2z4)/0.5ei0.7π, (z1z3,z2z3)/0.4e i0.5π,(z1z3,z2z4)/0.4e i0.5π,(z1z4,z2z3)/0.4e i0.5π } remark 3.3. the composition of two strong cfg is need not be a strong cfg. definition 3.4. let the two complex fuzzy graphs gc1 = (σc1,µc1), and gc2 = (σc2,µc2) defined on two graphs g1 = (v1,e1) and g2 = (v2,e2) respectively. let gc1 be a pair of complex functions σc1 : v1 → r1(a)eiθ1(a),µc1 : e1 ⊆ v1 × v1 → r1(e).eiϕ1(e) such that µc(a1,a2) = r1(e)eiϕ1(e) where r1(e) ≤ 87 n.azhagendran and a.mohamed ismayil min {r1(a1),r1(a2)} ,ϕ1(e) ≤ min {θ1(a1),θ1(a2)}. also gc2 is a pair of complex functions σc2 : v2 → r2(a)eiθ2(a),µc2 : e2 ⊆ v2 × v2 → r2(e).eiϕ2(e) such that µc(a1,a2) = r2(e)eiϕ2(e) where r2(e) ≤ min {r2(a1),r2(a2)} ,ϕ2(e) ≤ min {θ2(a1),θ2(a2)}. then the cartesian product of two complex fuzzy graphs is defined as follows (i) (σc1 × σc2)(a1,a2) = min {r1(a1),r2(a2)}emin{θ1(a1),θ2(a2)},fora1,a2 ∈ v (ii) (µc1 × µc2)((a,a2),(a,b2)) = min {r1(a),r2(a)} .emin{θ1(a),ϕ2(a)} where r2(a) ≤ min {r2(a2),r2(b2)} ;ϕ2(a) ≤ {θ2(a2),θ2(b2)} for all a ∈ v1 and (a2,b2) ∈ e2 (iii) (µc1 × µc2)((a1,a),(b1,a)) = min {r1(a),r2(a)} .emin{ϕ1(a),θ2(a)} where r1(a) ≤ min {r1(a1),r1(b1)} ;ϕ2(a) ≤ {θ1(a1),θ1(b1)} for all a ∈ v2 and (a1,b1) ∈ e1 proposition 3.4. the cartesian product of two complex fuzzy graphs is also a complex fuzzy graph. proof. let gc1 and gc2 be two complex fuzzy graphs, then we prove that gc1 × gc2 is a cfg. (i) σc1 × σc2(a1,a2) = min {r1(a1),r2(a2)} .eimin{θ1(a1,θ2(a2))},∀a1,a2 ∈ v1 × v2. it is trivial, we have to verify the conditions only for e1 × e2 (ii) (µc1 × µc2)((a,a2),(a,b2)) = min {r1(a),r2(a)} .eimin{θ1(a),ϕ2(a)} ≤ min {r1(a),min {r2(a2),r2(b2)}} .eimin{θ1(a),min{θ2(a2),θ2(b2)}} = min {min {r1(a),r2(a2)} ,min {r1(a),r2(b2)}} . eimin{min{θ1(a),θ2(a2)},min{θ1(a),θ2(b2)}} = min {r(a1),r(b1)} .eimin{θ(a1),θ(b1)} where r(a1) = min {r1(a),r2(a2)} ,r(b1) = min {r1(a),r2(b2)} , θ(a1) = min {θ1(a),θ2(a2)} ,θ(b1) = min {θ1(a),θ2(b2)} forall a ∈ v1,(a2,b2) ∈ e2 (iii) (µc1 × µc2)((a1,a),(b1,a)) = min {r1(a),r2(a)} .eimin{ϕ1(a),θ2(a)} ≤ min {min {r1(a1),r1(b1)} ,r2(a)} .eimin{min{θ1(a1),θ1(b1)},θ2(a)} = min {min {r1(a1),r2(a)} ,min {r1(b1),r2)}} . eimin{min{θ1(a1),θ2(a)},min{θ1(b1),θ2(a)}} = min {r(a2),r(b2)} .eimin{θ(a2),θ(b2)} where r(a2) = min {r1(a1),r2(a)} ,r(b2) = min {r1(b1),r2(a)} , θ(a2) = min {θ1(a1),θ2(a)} ,θ(b2) = min {θ1(b1),θ2(a)} forall a ∈ v2,(a1,b1) ∈ e1 hence gc1 × gc2 is a cfg. 88 some operations on complex fuzzy graphs example 3.5. let gc1 and gc2 be two complex fuzzy graphs. σc1 = {z1/0.4ei2π,z2/0.5ei1.2π} , µc1 = {(z1,z2)/0.4ei1.2π} , σc2 = {z3/0.5ei0.7π,z4/0.8eiπ, z5/0.4ei1.7π, µc2 = {(z3,z4)/0.4ei0.7π,(z4,z5)/0.4eiπ}. then the cartesian product is given by σc1 × σc2 = {z1z3/0.4ei0.7π,z1z4/0.4eiπ,z1z5/0.4ei1.7π, z2z3/0.5e i0.7π,z2z4/0.5e iπ,z2z5/0.4e i1.2π } µc1 × µc2 = {(z1z3,z1z4)/0.4ei0.7π, (z1z4,z1z5)/0.4e iπ,(z1z3,z2z3)/0.4e i0.7π,(z2z3,z2z4)/0.5e i0.7π, (z2z4,z2z5)/0.4e i1.2π,(z1z4,z2z4)/0.4e iπ,(z1z5,z2z5)/0.4e i1.2π } z1(0.4e i1.2π) z2(0.5e i1.2π) 0.4ei1.2π figure 5: gc1 z3(0.5e i0.7π) z4(0.8e iπ) 0.4ei0.7π z5(0.4e i1.7π) 0.4eiπ figure 6: gc2 (z1,z3)(0.4e i0.7π) (z1,z4)(0.4e iπ) 0.4ei0.7π (z1,z5)(0.4e i1.7π) 0.4eiπ (z2,z3)(0.5e i0.7π) (z2,z4)(0.5e iπ) 0.4ei0.7π (z2,z5)(0.4e i1.2π) 0.4eiπ 0.4ei0.7π 0.4eiπ 0.4ei1.2π figure 7: gc1 × gc2 remark 3.4. cartesian product of two strong cfg is also a strong cfg. 4 conclusions in this paper, we discussed about some operations on complex fuzzy graphs such as union, intersection, composition and cartesian product with examples. 89 n.azhagendran and a.mohamed ismayil complex fuzzy graph is an extension of a fuzzy graph. we are working to extend the algorithms applied to complex fuzzy graphs are i) domination on complex fuzzy graphs ii) connectivity of complex fuzzy graphs and more operations on complex fuzzy graphs references harary. graph theory. addition wesley, london, 1969. l.a.zadeh. fuzzy sets. information and control, 8(3):338–353, 1965. j. mordeson and c. peng. operations on fuzzy graphs. information sci., 79: 159–170, 1994. a. nagoorgani and r. latha. some properties on operations of fuzzy graphs. advances in fuzzy sets and systems, 19(1):1–24, 2015. y. naveed, g. muhammad, k. seifedine, and a. w. hafiz. complex intuitionistic fuzzy graphs with applications in cellular network provider companies. mathematics, 7(35):1–18, 2019. p.bhattacharya. some remarks on fuzzy graphs. pattern recognition lett.6, pages 297–302, 1987. d. ramot, m.friedma, g.langholz, r. milo, and k. a. complex fuzzy sets. ieee transactions on fuzzy systems, 10(2):171–186, 2002. a. shannon and k. atanassov. a first step to a theory of the intuitionistic fuzzy graph. in the first workshop on fuzzy based expert system ( d.lakov, ed.), pages 59–61, 1994. p. thirunavukarasu, r. suresh, and k. viswanathan. energy of a complex fuzzy graph. international journal of math. science and engg. appls (ijmsea), 10 (1):243–248, 2016. m. xueling, z. jianming, k. madad, z. muhammad, a. saima, and s. a. abdul. complex fuzzy sets with applications in signals. computational and applied mathematics, 38(150):1–34, 2019. 90 introduction premilinaries operations on complex fuzzy graph conclusions approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica issn: 1592-7415 vol. 35, 2018, pp. 47-52 eissn: 2282-8214 47 a proof of descartes’ rule of signs antonio fontana1 received: 20-11-2018. accepted: 20-12-2018. published: 31-12-2018. doi: 10.23755/rm.v35i0.425 ©antonio fontana abstract in 1637 descartes, in his famous géométrie, gave the rule of the signs without a proof. later many different proofs appeared of algebraic and analytic nature. among them in 1828 the algebraic proof of gauss. in this note, we present a proof of descartes’ rule of signs that use the roots of the first derivative of a polynomial and that can be presented to the students of the last year of a secondary school. keywords: roots of a polynomial; derivative of a polynomial. 2010 ams subject classification: 12d10; 26c10. 1 introduction descartes’ rule of signs first appeared in 1637 in descartes’ famous géométrie [1], where also analytic geometry was given for the first time. descartes gave the rule without a proof. later several discussions appear trying to understand which one was the first proof of the rule. it seems that a first proof of the rule was given in segner’s degree thesis in 1728 and it is contained in a letter that segner sent to hamberger [3]. in 1828 gauss [2] gave a purely algebraic and very simple proof. many other proofs, both 1 liceo scientifico “g. mercalli”, naples, italy; a.fontana2004@libero.it. antonio fontana 48 algebraic and analytic in nature have been given later. one of the possible statements giving descartes’ rule of signs is the following: theorem 1.1 if a polynomial with real coefficients in one unknown has all of its roots being real number, then the number of positive roots, counted with their multiplicity, equals the number of variations of signs among the ordered sequence of his coefficients. a more general statement covers the case when one does not know if all roots are real and it is given in the following: theorem 1.2 the number of variations of sign is the maximum number of positive roots of a polynomial with real coefficients. the number of positive roots equals either the maximum or the maximum minus an even number. the previous theorems do not give results on the number of negative roots. the negative roots of p(x) are in number equal to the number of positive roots of p(-x) and hence in order to count the number of negative roots of p(x) one can count the number of positive roots of p(-x)by applying descartes’ rule of signs. let p(x) be a polynomial whose monomials are given either in increasing or in decreasing order. consider the sequence of its coefficients in the same order. one says that there is a “change of sign” if two consecutive terms have opposite sign. for example if p(x) = x 6 3x 5 + 4x 3 + x 2 5x+ 9, then the sequence of its coefficients is: 1, −3,4,1, −5,9 and the number of variations is 4. hence the number of positive roots of p(x) is either 4 or 2 or 0. observe that if the number of variations is even, then the rule cannot say that the polynomial has a positive root. if the number of variations is odd, the rule says that there is at least a positive root. a proof of descartes’ rule of signs 49 2 the proof 2.1 the derivative of a polynomial we first start with some results giving information between the roots of a polynomial p(x) and the roots of its derivative p'(x). lemma 2.1 a roots of p(x) is also a root of p'(x) with multiplicity one less. proof. let p(x) = (x-a) k q(x) with (x -a) that does not divide q(x). hence q(a) ¹ 0 . then k is the multiplicity of 𝑎 as root of p(x). it is p'(x) = k(x-a) k-1 q(x) + (x-a) k q'(x) = = (𝑥 − 𝑎)𝑘−1[𝑘𝑞(𝑥) + (𝑥 − 𝑎)𝑞′(𝑥)] = (𝑥 − 𝑎)𝑘−1𝐹(𝑥). since 𝐹(𝑎) = 𝑘𝑞(𝑎) ≠ 0, (𝑥 − 𝑎) does not divide the polynomial 𝐹(𝑥). hence 𝑘 − 1 is the multiplicity of 𝑎 as root of 𝑝′(𝑥). next results follows from rolle's theorem. lemma 2.2 if all roots of a polynomial 𝑝(𝑥) are real numbers, then also all roots of 𝑝′(𝑥) are real numbers. moreover between to consecutive roots of 𝑝(𝑥) there is a simple (multiplicity 1) root of 𝑝′(𝑥). proof. let 𝑥1 < 𝑥2 <∙∙∙∙∙∙∙∙∙∙∙∙∙< 𝑥𝑘 be the roots of 𝑝(𝑥) with multiplicity 𝑚1, 𝑚2, … … … , 𝑚𝑘, respectively. since all roots are real numbers we have that 𝑚1 + 𝑚2 +∙∙∙∙∙∙∙∙ +𝑚𝑘 = 𝑛 = deg⁡(𝑝(𝑥)). from the previous lemma 𝑝′(𝑥) has roots 𝑥1 < 𝑥2 <∙∙∙∙∙∙∙∙∙∙∙∙∙< 𝑥𝑘 with multiplicity 𝑚1 − 1, 𝑚2 − 1, … …… , 𝑚𝑘 − 1. moreover, from rolle's theorem between two real roots of 𝑝(𝑥) there is at least a real root of 𝑝′(𝑥). hence 𝑝′(𝑥) has at least other 𝑘 − 1 real roots. from (⁡𝑚1 − 1) + (𝑚2 − 1) +⁡∙∙∙∙ +(𝑚𝑘 − 1) + 𝑘 − 1 = 𝑛 − 1 = deg⁡(𝑝′(𝑥)), it follows that 𝑝′(𝑥) cannot have other roots. the assertion follows. antonio fontana 50 lemma 2.3 if all roots of a polynomial 𝑝(𝑥) are real numbers and 𝑘 of them are positive numbers, then 𝑝′(𝑥) has either 𝑘 or 𝑘 − 1 positive roots. proof. let 𝑥1 < 𝑥2 <∙∙∙∙∙∙∙∙∙∙∙∙∙< 𝑥𝑠 be the positive roots of 𝑝(𝑥) with multiplicity 𝑚1, 𝑚2, … …… , 𝑚𝑠, respectively. from the hypothesis we have 𝑚1 + 𝑚2 +∙∙∙∙∙∙∙∙ +𝑚𝑠 = 𝑘. the derivate 𝑝′(𝑥) will have as positive roots 𝑥1 < 𝑥2 <∙∙∙∙∙∙∙∙∙∙∙∙∙< 𝑥𝑠 with multiplicity 𝑚1 − 1, 𝑚2 − 1, …… … , 𝑚𝑠 − 1, the simple roots 𝑦1, 𝑦2, … … … , 𝑦𝑠−1 between consecutive positive roots and, possibly, another simple root 𝑦0 between the maximum negative root and 𝑥1. so the total number of positive roots of 𝑝′(𝑥) is either (⁡𝑚1 − 1) + (𝑚2 − 1) +∙∙∙∙∙∙∙∙ +(𝑚𝑠 − 1) + 𝑠 − 1 = 𝑘 − 1 if 𝑦0 is not a positive number or (⁡𝑚1 − 1) + (𝑚2 − 1) +∙∙∙∙∙∙∙∙ +(𝑚𝑠 − 1) + 𝑠 − 1 + 1 = 𝑘 if 𝑦0 a positive number. 2.2 proof of theorem 1.1 let 𝑝(𝑥) = 𝑎𝑛𝑥 𝑛 + 𝑎𝑛−1𝑥 𝑛−1 +⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ +𝑎0 be a degree 𝑛⁡polynomial. hence 𝑎𝑛 ≠ 0. we may assume that 𝑎𝑛 > 0. in what follows we assume that all roots of 𝑝(𝑥) are real numbers. lemma 2.4 if 𝑝(𝑥) has 𝑘 positive roots, then the sign of the last non zero coefficient of 𝑝(𝑥) is⁡(−1)𝑘. proof. let 𝑎ℎ be the last non zero coefficient. since all roots of 𝑝(𝑥) are real numbers we can factorize the polynomial as 𝑝(𝑥) = 𝑎𝑛𝑥 𝑛 + 𝑎𝑛−1𝑥 𝑛−1 +⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ +𝑎ℎ𝑥 ℎ = = 𝑎𝑛𝑥 ℎ(𝑥 − 𝑥1) ⋅⋅⋅⋅⋅ (𝑥 − 𝑥𝑘)(𝑥 − 𝑥𝑘+1) ⋅⋅⋅⋅⋅ (𝑥 − 𝑥𝑛−ℎ) a proof of descartes’ rule of signs 51 where 𝑥1, …… … , 𝑥𝑘 are the positive roots and 𝑥𝑘+1, … … … , 𝑥𝑛−𝑘⁡are the negative roots. it follows that 𝑎ℎ = 𝑎𝑛 ⋅ (−1) 𝑘 ⋅ 𝑥1𝑥2 ⋅⋅⋅⋅ 𝑥𝑘 ⋅ (−𝑥𝑘+1) ⋅⋅⋅ (−𝑥𝑛−ℎ ) and since all numbers are positive the sign is ⁡(−1)𝑘. we will now give the proof of the theorem 1.1 by induction on 𝑛 = deg⁡(𝑝(𝑥)). if 𝑛 = 1, the assertion holds. indeed 𝑝(𝑥) = 𝑎1𝑥 + 𝑎0 has a unique root 𝑥1 = −𝑎0/𝑎1. it is a positive root if and only if 𝑎1 and 𝑎0 have opposite sign, that is there is a variation. suppose the assertion holds for all polynomials of degree 𝑛 − 1 with all real roots. let 𝑝(𝑥) = 𝑎𝑛𝑥 𝑛 + 𝑎𝑛−1𝑥 𝑛−1 +⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ +𝑎0 be a polynomial of degree 𝑛. if 𝑎0 = 0, then 𝑝(𝑥) = 𝑥𝑞(𝑥) and the polynomials 𝑝(𝑥) and 𝑞(𝑥) have the same number of positive roots and the same number of variations of sign. since deg(𝑞(𝑥)) = 𝑛 − 1 and the assertion holds for 𝑞(𝑥), then it also holds for 𝑝(𝑥). if 𝑎0 ≠ 0, then 𝑝′(𝑥) = 𝑛𝑎𝑛𝑥 𝑛−1 + (𝑛 − 1)𝑎𝑛−1𝑥 𝑛−2 +⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ +𝑎1. the last non zero coefficient in 𝑝′(𝑥) is the non zero coefficient consecutive to 𝑎0 in 𝑝(𝑥). if the sign of 𝑎0 and the last non zero coefficient in 𝑝′(𝑥) coincide, then 𝑝(𝑥) and 𝑝′(𝑥) have the same number of variations of sign, otherwise 𝑝′(𝑥) has one less variations of sign compared with 𝑝(𝑥). since the sign of the last non zero coefficient determines the parity of the number of positive roots (lemma 2.4) in the first case the parity of the number of roots of 𝑝(𝑥) and 𝑝′(𝑥) is the same in the second case it is different. on the other hand from lemma 2.3 the number of roots of 𝑝(𝑥) and 𝑝′(𝑥) is different for at most 1. hence either 𝑝(𝑥) and 𝑝′(𝑥) have the same number of positive roots or this number is different for 1. antonio fontana 52 from the inductive hypothesis 𝑝′(𝑥) has the same number of positive roots as the number of variations of sign. since from 𝑝(𝑥) to 𝑝′(𝑥) the number of positive roots and the number of variations of sign either remains the same for both or it is 1 more for both, the assertion follows also for 𝑝(𝑥). 3 conclusions the proof of descartes rule of signs is a good example of math reasoning and it should be taught to the students of last year of secondary schools. contrary to this in many schools it is given the rule without a proof. in particular it is a good example for understanding the relation between the roots of a polynomials and its first derivative. it also uses rolle’s theorem, that is one of the most important result shown to the students of last year of secondary schools. moreover descartes’ rule of signs is one of the math results that puts together analysis and algebra and it doesn’t happen so often in curricula of secondary school. in math, except for axioms, everything should be demonstrated. references [1] descartes, r. (1637). la géometrie (discours de la méthode, third part), ed. of leyde, 373. [2] gauss, c. f. (1828). beweis eines algebraischen lehrsatzes, crelle’s journal fur die reine und ange-wandte mathematik, 3(1). [3] segner, j. a. (1728) dissertatio epistolica, qua regulam harriotti, university of jena. ratio mathematica volume 46, 2023 strong interval – valued pythagorean fuzzy soft graphs mohammed jabarulla mohamed* sivasamy rajamanickam † abstract a strong interval – valued pythagorean fuzzy soft sets (sivpfss) an extending the theory of interval-valued pythagorean fuzzy soft set (ivpfss). then we propose strong interval valued pythagorean fuzzy soft graphs (sivpfsgs). we also present several different types of operations on strong intervalvalued pythagorean fuzzy soft graphs and explore of their analysis. keywords: strong interval-valued pythagorean fuzzy graph; strong interval-valued pythagorean fuzzy soft graph; 2020 ams subject classifications: 05c72, 06d72, 12d15. 1 *pg and research department of mathematics, jamal mohamed college(autonomous), (affiliated to bharathidasan university), tiruchirappalli, tamil nadu, india. m.md.jabarulla@gmail.com. †pg and research department of mathematics, jamal mohamed college(autonomous), (affiliated to bharathidasan university), tiruchirappalli, tamil nadu, india; sivasamyr1998@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1067. issn: 1592-7415. eissn: 2282-8214. ©m. mohammed jabarulla et al. this paper is published under the cc-by licence agreement. 127 m. mohammed jabarulla and r. sivasamy 1 introduction fuzzy set is a analytical imitation to grips the exciting and insufficient details. consider a differentiating that uncertainty is also independently, fs was continued to intuitionistic fuzzy set (ifs) by atanassov and gargov [1989]. if assigned a membership value α and a non membership value β to the conditions, satisflying this results α + β ≤ 1 and uncertainty elements, γ = 1 − α − β. in decision-making problems, the membership value 0.7 and non membership value 0.4 for some information, then if fails in this situation because 0.7 + 0.4 > 1, but (0.7)2 + (0.4)2 ≤ 1. to overcome this situation, the notion of pythagorean fuzzy set (pfs) was satisfying the condition α2 + β2 ≤ 1. a pfs has more potential as compared to ifs is solving decision-making problems. the pythagorean fuzzy number (pfg) was determinate by zhang ( see s.shahzadi and akram [2020]). zhang provided the pythagorean fuzzy weighted averaging operator. the theory of ivfs was introduced by zadeh [1965] as a perpetuation of fuzzy sets. because they present more adequate description for uncertainty, intervalvalued fuzzy sets more useful than conventional fuzzy sets. soft set theory was started by molodstov [1999] for the parameterized point of view for uncertainty modeling and soft computing. the iterpretation of ifsgs was given by akram [2011]. the explanation of novel intuitionistic fuzzy soft multiple – decisionmaking methods of grips by akram. pythagorean fuzzy soft graphs with applications was proposed by s.shahzadi and akram [2020].the sivpfsg is defined and some results on sivpfsg are studied. also explore of their analysis. 2 preliminaries definition 2.1. an ivfsg over the set v is given by ordered 4 tuple ξ̃ = (ξ∗,x,y,a) such that (i) a is of parameters. (ii) (x,a) is an ivfss over v . (iii) (y,a) is an ivfss over e. (iv) (x(e),y (e)) is an ivfsg for all e ∈ a. that is, α− y (e) ((pq)) ≤ min(α− x(e) (p),α− x(e) (q)) and α+ y (e) ((pq)) ≤ min(α+ x(e) (p),α+ x(e) (q))forall pq ∈ e. we denote ξ∗ = (v,e) a crisp graph h(e) = (x(e),y (e)) an ivfsg and ξ̃ = (ξ∗,x,y,a) an ivfsg. definition 2.2. an ivfsg over the set v is defined to be a pair ξ = (x,y ) where 1) the conditions α̃x : v → d[0, 1] and β̃x : v → d[0, 1] denote the degree of 128 strong interval – valued pythagorean fuzzy soft graphs membership and non membership of the element p ∈ v . such that 0 ≤ α̃x (p) + β̃x (p) ≤ 1∀(p,q) ∈ v. 2) the conditions α̃y : e ⊆ v ×v → d[0, 1] and β̃y : e ⊆ v ×v → d[0, 1] defined by α−y l((p,q)) ≤ min(α − xl(p),α − xl(q)) and β − y l((p,q)) ≥ max(β − xl(p),α − xl(q)), α+y u ((p,q)) ≤ min(α + xu (p),α + xu (q)) and β + y u ((p,q)) ≥ max(β + xu (p),α + xu (q)), such that 0 ≤ α2y u (p,q) + β 2 y u (p,q) ≤ 1∀(p,q) ∈ e. we the notation pq for (p,q) an element of e. definition 2.3. an ivpfsg over the set v is given by ξ̃ = (ξ∗,x,y,a) such that 1) the conditions α̃x : v → d[0, 1] and β̃x : v → d[0, 1] standered for the degree of membership and non membership of the element p ∈ v . such that 0 ≤ α̃x (p,q) + β̃x (p,q) ≤ 1∀(p,q) ∈ v. 2)(i) a is set of parameters (ii) (x,a) is an ivpfss over v . (iii) (y,a) is an ivpfss over e. (iv) (x(e),y (e)) is an ivpfsg for all e ∈ a. the conditions α̃y : e ⊆ v ×v → d[0, 1] and β̃y : e ⊆ v ×v → d[0, 1] defined by α+y u ((p,q)) ≤ min(α + xu (p),β + xu (q)) and β + y u ((p,q)) ≥ max(β + xu (p),β + xu (q)), α−y l((p,q)) ≤ min(α − xl(p),β − xl(q)) and β − y l((p,q)) ≥ max(β − xl(p),β − xl(q)), such that 0 ≤ α2y u (p,q) + β 2 y u (p,q) ≤ 1∀(p,q) ∈ e. 3 strong intervel-valued pythagorean fuzzy graphs definition 3.1. an sivpfsg over the set v is given by ξ̃ = (ξ∗,x,y,a) such that 1) the conditions α̃x : v → d[0, 1] and β̃x : v → d[0, 1] denote the degree of membership and non membership of the element x ∈ v . such that 0 ≤ α̃x (p,q) + β̃x (p,q) ≤ 1∀(p,q) ∈ v. 2)(i) a is set of parameters (ii) (x,a) is an sivpfss over v . (iii) (y,a) is an sivpfss over e. (iv) (x(e),y (e)) is an sivpfsg for all e ∈ a. the conditions α̃y : e ⊆ v ×v → d[0, 1] and β̃y : e ⊆ v ×v → d[0, 1] defined by 129 m. mohammed jabarulla and r. sivasamy α+y u ((p,q)) = min(α + xu (p),β + xu (q)) and β + y u ((p,q)) = max(β + xu (p),β + xu (q)), α−y l((p,q)) = min(α − xl(p),α − xl(q)) and β − y l((p,q)) = max(α − xl(p),β − xl(q)), such that 0 ≤ α2y u (p,q) + β 2 y u (p,q) ≤ 1∀(p,q) ∈ e. example 3.1. if ξ∗ = (x,y ) is a simple graph with x = {a,b,c,d} and y = {ab,bc,cd,ad}. let a = {e1,e2} be a parameter set and (x,a) be an sivpfss v determine x1(e) = { 〈a, [0.3, 0.4][0.2, 0.7]〉,〈b, [0.2, 0.5][0.3, 0.7]〉,〈c, [0.1, 0.6][0.2, 0.5]〉, and〈d[0.2, 0.7][0.3, 0.5]〉 } x2(e) = { 〈a, [0.2, 0.7][0.3, 0.5]〉,〈b[0.1, 0.6][0.2, 0.5]〉,〈c, [0.3, 0.4][0.2, 0.7]〉 } take (y,a) be an sivpfss e determine y1(e) = { 〈ab[0.2, 0.5][0.3, 0.7]〉,〈bc[0.1, 0.6][0.3, 0.7]〉,〈ad[0.2, 0.7][0.3, 0.7]〉, and〈cd[0.1, 0.6][0.3, 0.5]〉 } y2(e) = { 〈ab[0.1, 0.5][0.4, 0.6]〉,〈bc[0.1, 04][0.4, 0.8]〉,〈ac[0.1, 0.3][0.4, 0.8]〉 } it is clearly seen that h(e1) = (x(e1),y (e1)) and h(e2) = (x(e2),y (e2)) are sivpfsgs comparable to the parameters e1 and e2 accordingly, by figure 1. hence ξ̃ = (ξ∗,x,y,a) sivpfsgs. figure 1: sivpfsgs ǧ. definition 3.2. if ξ̃1 = (ξ∗1,x1,y1,a) and ξ̃2 = (ξ ∗ 2,x2,y2,b) be double sivpfsgs of ξ∗1 = (x1,y1) and ξ ∗ 2 = (x2,y2) accordingly. the cross product of ξ̃1 and 130 strong interval – valued pythagorean fuzzy soft graphs ξ̃2 is denoted by ξ̃1 × ξ̃2 = (x1 ×x2,y1 ×y2) and is defined by 1) (αx1l ×αx2l)(p1,p2) = min(αx1l(p1),βx2l(p2)), (αx1u ×αx2u )(p1,p2) = min(αx1u (p1),αx2u (p2)), (βx1l ×βx2l)(p1,p2) = min(βx1l(p1),βx2l(p2)), (βx1u ×βx2u )(p1,p2) = max(βx1u (p1),βx2u (p2)),∀p1 ∈ v1,p2 ∈ v2. 2) (αy1l ×αy2l)(p,p2)(p,q2) = min(αy1l(p),αy2l(p2,q2)), (αy1u ×αy2u )(p,p2)(p,p2) = min(αy1u (p),αy2u (p2,q2)), (βy1l ×βy2l)(p,p2)(p,q2) = max(βy1l(p),βy2l(p2,q2)), (βy1u ×βy2u )(p,p2)(p,q2) = max(βy1u (p),βy2u (p2,q2)),∀p ∈ v1,p2q2 ∈ e2. 3) (αy1l ×αy2l)(p1,r)(q1,r) = min(αy1l(p1q1),αy2l(r)), (αy1u ×αy2u )(p1,r)(q1,r) = min(αy1u (p1q1),αy2u (r)), (βy1l ×βy2l)(p1,r)(q1,r) = max(βy1l(p1q1),βy2l(r)), (βy1u ×βy2u )(p1,r)(q1,r) = max(βy1u (p1q1),βy2u (r)),∀r ∈ v2,p1q1 ∈ e1. example 3.2. let consider a graph ξ∗1 = (x1,y1) and ξ ∗ 2 = (x2,y2) be two graphs such that x1 = {a1,b1,c1,d1},y1 = {a1b1,c1d1} and x2 = {a2,b2,c2,d2}, y2 = {a2b2,c2d2}. let a = e1 be a set of parameters and let (x1,a) and (y1,a) be two sivpfsss over x1 and y1 accordingly, defined by figure 2: sivpfsgs ξ̃1 and ξ̃2. x1(e) = { 〈a1[0.2, 0.5][0.4, 0.8]〉,〈b1[0.1, 0.4][0.4, 0.5]〉,〈c1[0.2, 0.6][0.3, 0.5]〉, and〈d1[0.1, 0.5][0.4, 0.6]〉 } y1(e) = { 〈a1b1[0.1, 0.4][0.4, 0.8]〉,〈c1d1[0.1, 0.5][0.4, 0.6]〉 } 131 m. mohammed jabarulla and r. sivasamy figure 3: cross product of ξ̃1 and ξ̃2. take b = e2 be a set of parameters and let (x2,b) and (y2,b) be two sivpfsss over x2 and y2 accordingly, find out x2(e) = { 〈a2[0.1, 0.4][0.2, 0.6]〉,〈b2[0.3, 0.3][0.7, 0.3]〉,〈c2[0.3, 0.7][0.4, 0.5]〉, and〈d2[0.3, 0.4][0.1, 0.6]〉 } y2(e) = { 〈a2b2[0.1, 0.4][0.7, 0.6]〉,〈c2d2[0.3, 0.7][0.4, 0.6]〉 } clearly h(e1) = (x(e1),y (e1)) and h(e2) = (x(e2)),y (e2)) are sivpfsgs. hence ξ̃1 = (ξ∗1,x1,y1,a) and ξ̃2 = (ξ ∗ 2,x2,y2,b) are sivpfsgs ξ ∗ 1 and ξ ∗ 2 , accordingly, as shown in the figure 2. definition 3.3. if ξ̃1 = (ξ∗1,x1,y1,a) and g̃2 = (ξ ∗ 2,x2,y2,b) be two sivpfsgs of ξ∗1 = (x1,y1) and ξ ∗ 2 = (x2,y2) accordingly. the composition of ξ̃1 and ξ̃2 is standed by ξ̃1 ◦ ξ̃2 = (x1 ◦x2,y1 ◦y2) and is defined by 1) (αx1l ◦αx2l)(p1,p2) = min(αx1l(p1),βx2l(p2)), (αx1u ◦αx2u )(p1,p2) = min(αx1u (p1),αx2u (p2)), (βx1l ◦βx2l)(p1,p2) = min(βx1l(p1),βx2l(p2)), (βx1u ◦βx2u )(p1,p2) = max(βx1u (p1),βx2u (p2)),∀p1 ∈ v1,p2 ∈ v2. 2) (αy1l ◦αy2l)(p,p2) = min(αy1l(p),αy2l(p2,q2)), (αy1u ◦αy2u )(p,p2) = min(αy1u (p),αy2u (p2,q2)), (βy1l ◦βy2l)(p,q2) = max(βy1l(p),βy2l(p2,q2)), (βy1u ◦βy2u )(p,q2) = max(βy1u (p),βy2u (p2,q2)),∀p1 ∈ v1,p2q2 ∈ e2. 132 strong interval – valued pythagorean fuzzy soft graphs 3) (αy1l ◦αy2l)(p1,r)(q1,r) = min(αy1l(p1q1),αy2l(r)), (αy1u ◦αy2u )(p1,r)(q1,r) = min(αy1u (p1q1),αy2u (r)), (βy1l ◦βy2l)(p1,r)(q1,r) = max(βy1l(p1q1),βy2l(r)), (βy1u ◦βy2u )(p1,r)(q1,r) = max(βy1u (p1q1),βy2u (r)),∀r ∈ v2,p1q1 ∈ e1. 4) (αy1l ◦αy2l)(p1,p2)(q1,q2) = min(αx2l(p2),αx2l,αx1l(p1,q1)), (αy1u ◦αy2u )(p1,r)(q1,r) = min(αx2u (p2),αx2u (q2),αy1u (p1,q1)), (βy1l ◦βy2l)(p1,r)(q1,r) = max(βx2l(p2),βx2l(q2),βy1l(p1,q1)), (βy1u ◦βy2u )(p1,r)(q1,r) = max(βx2u (p2),βx2u (q2),βy1u )(p1,p2)(q1,q2)), ∀(p1,p2)(q1,q2) ∈ e◦ −e. where e◦ = e ∪{(p1,p2)(q1,q2)|p1q1 ∈ e1,p2 6= q2}. definition 3.4. let ξ̃1 = (ξ∗1,x1,y1,a) and ξ̃2 = (ξ ∗ 2,x2,y2,b) be two sivpfsgs of ξ∗1 = (x1,y1) and ξ ∗ 2 = (x2,y2) accordingly. if ξ̃1 and ξ̃2 is standed by ξ̃1 ∪ ξ̃2 = (g∗,x,y,a∪b) where (x1 ∪x2,y1 ∪y2) and is replace 1) (i) (αx1l ∪αx2l)(p) = max(αx1l)(p),αx2l)(p))ifp ∈ v1 ∩v2 (αx1u ∪αx2u )(p) = max(αx1u )(p),αx2u )(p))ifp ∈ v1 ∩v2 (ii) (βx1l ∪βx2l)(p) = max(βx1u )(p),βx2l)(p))ifp ∈ v1 ∩v2 (βx1u ∪βx2u )(p) = max(βx1u )(p),βx2u )(p))ifp ∈ v1 ∩v2 2) (i) (αy1l ∪αy2l)(p,q) = max(αx1l(p,q),αx2l(p,q))if pq ∈ e1 ∩e2 (αy1u ∪αy2u )(p,q) = max(αx1u (p,q),αx2u (p,q))if pq ∈ e1 ∩e2 (ii) (βy1l ∪βy2l)(p,q) = max(βx1l(p),βx2l(q))if pq ∈ e1 ∩e2 (βy1u ∪βy2u )(p,q) = max(βx1u (p),βy2u (q))if pq ∈ e1 ∩e2 definition 3.5. let g̃1 = (ξ∗1,x1,y1,a) and ξ̃2 = (ξ ∗ 2,x2,y2,b) be two sivpfsgs of ξ∗1 = (x1,y1) and ξ ∗ 2 = (x2,y2) accordingly. if ξ̃1 and ξ̃2 is standed by ξ̃1 + ξ̃2 = (ξ ∗ 1,x1,y1,a + b). where ξ ∗ = (x1 + x2,y1 + y2) and is defined by 1) (αx1l + αx2l)(p) = (αx1l ∪αx2l))(p) (αx1u + αx2u )(p) = (αx1u ∪αx2u )(p)if p ∈ v1 ∪v2 (βx1l + βx2l)(p) = (βx1l ∪βx2l)(p) (βx1u + βx2u )(p) = (βx1u ∪βx2u )(p)if p ∈ v1 ∪v2 2)(αy1l + αy2l)(p,q) = (αy1l ∪αy2l)(p,q) (αy1u + αy2u )(p,q) = (αy1u ∪αy2u )(p,q)if p ∈ e1 ∩e2 (βy1l + βy2l)(p,q) = (βy1l ∪βy2l)(p,q) (βy1u + βy2u )(p,q) = (βy1u ∪βy2u )(p,q)if (p,q) ∈ e1 ∩e2. 3)(αy1l + αy2l)(p,q) = min(αx1l(p),αx2l(q)) (αy1u + αy2u )(p,q) = min(αx1u (p),αx2u (q)) (βy l + βy2l)(p,q) = max(βx1l(p),βx2l(q)) (βy1u + βy2u )(p,q) = max(βx1u (p),βx2u (q))if pq ∈ e where e is the set of all edges joining the vertices of v1 and v2. theorem 3.1. if ξ̃1 and ξ̃2 are sivpfsgs, then so is ξ̃1 × ξ̃2. 133 m. mohammed jabarulla and r. sivasamy proof let ξ̃1 = (ξ∗1,x1,y1,a) and ξ̃2 = (ξ ∗ 1,x1,y1,b) be two sivpfsgs of simple graphs ξ∗1 = (x1,y1) and ξ ∗ 2 = (x2,y2) accordingly. for all e1 ∈ a and e2 ∈ b, there are some results. let ξ1 and ξ2 be sivpfsgs let e = {(p,p2)(p,q2)/p ∈ v1,p2q2 ∈ e2}∪{(p1,r)(q1,r)/r ∈ v2,p1q1 ∈ e1}. consider (p,p2)(p,q2) ∈ e, we have (αy1l ×αy2l)(p,p2)(p,q2) = min(αx1l(p),αy2l(p2q2)) =min(αx1l(p),αx2l(p2).αx2l(q2)) =min(min(αx1l(p),αx2l(p2))min(αx1l(p),αx2l(q2))) (αy1l ×αy2l)(p,p2)(p,q2) = min((αx1l ×αx2l)(p,p2), (αx1l ×αx2l)(p,q2)) similarly, (αy1u ×αy2u )(p,p2)(p,q2) = min((αx1u ×αx2u )(p,p2), (αy1u ×αy2u )(p,q2)) now, (βy1l ×βy2l)(p,p2)(p,q2) = max((βx1l ×βx2l)(p,p2), (βx1l ×βx2l)(p,q2)) similarly, (βy1u ×βy2u )(p,p2)(p,q2) = max((βx1u ×βx2u )(p,p2), (βx1u ×βx2u )(p,q2)) consider, (p1,r)(q1,r) ∈ e, we have (αy1l ×αy2l)(p1,r)(q1,r) = min(αy1l(p1q1), (αx2l(r)) =min(αx1l(p1),αx2l(q1).αx2l(r)) =min(min(αx1l(p1),αx2l(r))min(αx1l(y1),αx2l(r))) (αy1l ×αy2l)(p1,r)(q1,r) = min((αx1l ×αx2l)(p1,r), (αx1l ×αx2l)(q1,r)) similarly, (αy1u ×αy2u )(p1,r)(q1,r) = min((αx1u ×αx2u )(p1,r), (αx1u ×αx2u )(q1,r)) now, (βy1l ×βy1u )(p1,r)(q1,r) = max((βx1l ×βx2l)(p1,r), (βx1l ×βx2l)(q1,r)) similarly, (βy1u ×βy2u )(p1,r)(q1,r) = max((βx1u ×βx2u )(p1,r), (βx1u ×βx2u )(q1,r)) hence ξ1 × ξ2 is an sivpfsgs. 134 strong interval – valued pythagorean fuzzy soft graphs theorem 3.2. if ξ̃1[ξ̃2] be sivpfsgs ξ̃1 and ξ̃2 of ξ∗1 and ξ ∗ 2 is an sivpfsgs. proof take (p,p2)(p,q2) ∈ e, we get (αy1l ◦αy2l)((p,p2)(p,q2)) = min((αx1l(p),αy2l)(p2q2) =min(αx1l(p),αx2l(p2),αx2l(q2)) =min(min(αx1l(p),αx2l(p2)),min(αx1l(p),αx2l(q2))) (αy1l ◦αy2l)((p,p2)(p,q2) = min(αx1l ◦αx2l)(p,p2), (αx1l ◦αx2l)(p,q2)). similarly, (αy1u ◦αy2u )((p,p2)(p,q2) = min(αx1u ◦αx2u )(p,p2), (αx1u ◦αx2u )(p,q2)) consider (p1,r)(q1,r) ∈ e, (αy1l ◦αy2l)((p1,r)(q1,r)) = min(αy1l(p1,q1),αx2l(r)) =min(αx1l(p1),αx1l(q1),αx2l(r)) =min(min(αx1l(p1),αx2l(r)),min(αx1l(q1),αx2l(r))) (αy1l ◦αy2l)((p1,r)(q1,r)) = min(αx1l ◦αx2l)(p1,r), (αx1l ◦αx2l)((q1,r)) similarly, (αx1u ◦αx2u )((p1,r)(q1,r) = min(αx1u ◦αx2u )(p1,r), (αx1uαx2u )((q1,r)) consider (p1,p2)(q1,q2) ∈ e, (αy1l ◦αy2l)((p1,p2)(q1,q2)) = min(αx2l(p2),αx2l(q2),αy1l(p1q1)) =min(αx2l(p2),αx2l(q2)),min(αx1l(p1),αx1l(q1)) =min(min(αx1l(p1),αx2l(p2)),min(αx1l(q1),αx2l(q2))) (αy1l ◦αy2l)((p1,p2)(q1,q2)) = min(αx1l ◦αx2l)(p1,p2), (αx1l ◦αx2l) ((q1,q2)) hence ξ̃1[ξ̃2] be sivpfsg . theorem 3.3. if ξ̃1 ∪ ξ̃2 be sivpfsgs ξ̃1 and ξ̃2 of ξ∗1 and ξ∗2 is an sivpfsgs. proof take ξ̃1 and ξ̃2 be the sivpfsgs of ξ̃1 and ξ̃2 accordingly. since all conditions for x1∪x2 are obviously satisfied. it is enough to verify the conditions for y1 ∪y2, consider (p,q) ∈ e1 ∪e2. then (αy1l ∪αy2l)(p,q) = max(αy1l(p,q),αy2l(p,q)) =max(min(αx1l(p),αx1l(q)), (min(αx2l)(p),αx2l(q)) =min(max(αx1l(p),αx2l(p)), (max(αx1l(p),αx2l(q)))) =min((αy1l ∪αy2l)(p), (αy1l ∪αy2l)(q)) (αy1l ∪αy2l)(p,q) = min((αy1l ∪αy2l)(p), (αy1l ∪αy2l)(q)). 135 m. mohammed jabarulla and r. sivasamy similarly, (αy1u ∪αy2u )(p,q) = min((αy1u ∪αy2u )(p), (αy1u ∪αy2u )(q)) if (x,y) ∈ e1 and (x,y) /∈ e2, (αy1l ∪αy2l)(p,q) =min((αy1l ∪αy2l)(p), (αy1l ∪αy2l)(q)) (αy1u ∪αy2u )(p,q) =min((αy1u ∪αy2u )(p), (αy1u ∪αy2u )(q)). if (p,q) ∈ e2 and (p,q) ∈ e1, (αy1l ∪αy2l)(p,q) =min((αy1l ∪αy2l)(p), (αy1l ∪αy2l)(q)) (αy1u ∪αy2u )(p,q) =min((αy1u ∪αy2u )(p), (αy1u ∪αy2u )(q)). theorem 3.4. if ξ̃1 + ξ̃2 be sivpfsgs ξ̃1 and ξ̃2 of ξ∗1 and ξ ∗ 2 is an sivpfsgs. proof take ξ̃1 + ξ̃2 be the sivpfsgs of ξ∗1 and ξ ∗ 2 accordingly. , it is enough to find that ξ̃1 + ξ̃2 = (x1 + x2,y1 + y2) is an sivpfsgs. then let (p,q) ∈ e (αy1l + αy2l)(p,q) =min(αx1l(p),αx2l(q)) =min((αx1l ∪αx2l)(p), ((αx1l ∪αx2l)(q))) (αy1l + αy2l)(p,q) =min((αx1l + αx2l)(p), ((αx1l + αx2l)(q))). similarly, (αy1u + αy2u )(p,q) = min((αx1u + αx2u )(p), ((αx1u + αx2u )(q))). 4 conclusions graph theory is a very helpful mathematical tool for tackling challenging issues in a variety of disciplines. the ivpfss model is appropriate for modeling issues involving uncertainty and inconsistent data when human understanding and evaluation are required. in contrast to ivfs models, ivifs models, and, ivpfs models provide systems with sensitivity, flexibility, and conformance. sivpfsgs are a novel idea that is introduced in this work. we also defined for the cartesian product as well as some information about its composition on sivpfsgs. we plan to use this data to create some algorithms and models shortly soon. references m. akram. bipolar fuzzy graphs. inform.sci., 181:5548–5564, 2011. 136 strong interval – valued pythagorean fuzzy soft graphs k. atanassov and g. gargov. intervalvalued intuitionistic fuzzy sets, fuzzy sets and systems. neural computing and applications, 31:343–349, 1989. d. molodstov. soft set theoryfirst results. journal of fuzzy mathematics, 37: 19–31, 1999. s.shahzadi and m. akram. pythagorean fuzzy soft graphs with applications. journal of intelligence and fuzzy systems, 5(10):4977–4991, 2020. l. zadeh. fuzzy sets. infrom.control, 8(3):338–353, 1965. 137 ratio mathematica issn: 1592-7415 vol. 31, 2016, pp. 3--24 eissn: 2282-8214 3 information and intertemporal choices in multi-agent decision problems 1mariagrazia olivieri, 2massimo squillante, 3viviana ventre 1 demm, università of sannio, benevento, italy mgolivieri@unisannio.it 2 demm, università of sannio, italy prorettore@unisannio.it 3 demm, università of sannio, italy ventre@unisannio.it received on: 15-12-2016. accepted on: 15-01-2017. published on: 28-02-2017 doi: 10.23755/rm.v31i0.316 © olivieri et al. abstract psychological evidences of impulsivity and false consensus effect lead results far from rationality. it is shown that impulsivity modifies the discount function of each individual, and false consensus effect increases the degree of consensus in a multi-agent decision problem. analyzing them together we note that in strategic interactions these two human factors involve choices which change equilibriums expected by rational individuals. keywords: consensus, intertemporal choice, decisions 2010 ams subject classification: 90b50,91b06,91b08 m. olivieri, m. squillante, v. ventre 4 1. introduction in 1937, to compare future alternatives, samuelson introduced the discounted utility model (du model), which assumes an exponential delay discount function, with a constant discount rate that implies dynamic consistency and stationary intertemporal preferences. contrary to this normative economic theory, it has been established that human and animal intertemporal choice behaviors are not rational (i.e., inconsistent). for this reason, recent behavioral decision theory on intertemporal choice has adopted a hyperbolic discount model, in which result preference reversal as time passes (takahashi, 2009) (section 2). neurobiological and psychological factors have determined individual differences in intertemporal choice and have been explored in recent neuroeconomic and econophysical studies. takahashi (2007) attempts to dissociate impulsivity and inconsistency in their econophysical studies proposing the q-exponential delay discount function. other behavioral economists propose multiple selves models attempting to measure the strength of the internal conflict within the decision maker, best known as quasihyperbolic discount model first introduced by laibson (1997) (section 3). thaler and shefrin (1981), in the field of multiple selves models, consider that the concept of self-control is incorporated in a theory of individual intertemporal choice by modeling the individual as an organization. the individual is treated as if he contained two distinct psyches denoted as planner and doer. this model can be compared with the principal-agent problem present in any organization, so the individual may adopt many of the same strategies to solve self-control problems in intertemporal choice (section 4). in a multi-agent decision context the objective for a group decision is to choose a common decision, among each choice, that is to say an alternative which is judged the best by the majority of the decision makers. so in most strategic decisions, it is important to be able to estimate the characteristics and behavior of others. if the characteristics of other players are unknown, estimating them is a critical task. moreover, psychological evidence suggests people’s own beliefs, values, and habits tend to bias their perceptions of how widely they are shared (false consensus effect). this effect demonstrates an inability of individuals to process information rationally (section 5). therefore when we use the aggregation of the agents’ preferences to assess consensus, we obtain a coefficient which includes the false consensus effect that information and intertemporal choices in multi-agent decision problems 5 depends on the subjectivity and also increases the degree of consensus. to eliminate this aspect of human judgment vagueness we can use a model defined by ordered weighted averaging (owa) operators introduced in yager (1988) (section 6). many decision problems are characterized by interplay between intertemporal considerations and strategic interactions. two or more agents could have to take a common decision for a future time, in that process they are influenced by false consensus effect and by impulsivity that reveals inconsistency. finally in order to consider intertemporal choices in a multi-agent decision process needs to study the problem of each agent and the influence of false consensus effect (section 7). a strategic interaction is mathematical developed with the use of the theory of games, then it is possible to demonstrate the difference of psychological influence between a cooperative interaction (section 8) and non-cooperative one (section 9). 2. intertemporal discounting standard discount model. the standard economic model of discounted utility (du model) assumes that economic agents make intertemporal choices over consumption profiles (𝑐𝑡 , … , 𝑐𝑇 ) and such preferences can be represented by an intertemporal utility function 𝑈𝑡 (𝑐𝑡 , … , 𝑐𝑇 ), which can be described by the following form: 𝑈𝑡 (𝑐𝑡 , … , 𝑐𝑇 ) = ∑ 𝐷(𝑘)𝑢(𝑐𝑡+𝑘 𝑇−𝑡 𝑘=0 ) where 𝐷(𝑘) = ( 1 1+𝜌 ) 𝑘 so the du model assumes an exponential temporal discounting function and a constant discount rate (𝜌). an important implication of these two features is that a person’s intertemporal preferences are time-consistent: if in period t a person prefers 𝑐2 at t+2 to 𝑐1 at t+1, then in period t+1 she must prefer 𝑐2 at t+2 to 𝑐1 instantly. however, several empirical studies, mainly arisen from the field of psychology, have documented various inadequacies of the du model as a descriptive model of behavior. the first anomaly found to contradict discounted utility was that, instead of remaining constant over time, observed discount rates appear to decline with m. olivieri, m. squillante, v. ventre 6 time, this reveal decreasing impatience, or hyperbolic discounting: a later outcome is discounted less per unit of time than an earlier one (delay effect). furthermore, other anomalies derive from the fact that, even for a given delay, discount rates vary across different types of intertemporal choices: larger outcomes are discounted at a lower rate than smaller outcomes (magnitude effect); gains are discounted at a higher rate than losses of the same magnitude (sign effect); increasing sequences of consumption are preferred over decreasing ones even if the total amount is the same (improving sequence effect). hyperbolic discount model. a hyperbolic discount model can represent the tendency of the individuals to increasingly choose a smaller-sooner reward over a larger-later reward as the delay occurs sooner in time (delay effect). many authors proposed different hyperbolic discount functions, in which δ (temporal discount function) increases with the delay to an outcome. in 1992 loewenstein and prelec proposed this form: 𝑑(𝑡) = ( 1 1 + 𝛼𝑡 ) 𝛽 𝛼⁄ where β > 0 is the degree of discounting and α > 0 is the departure from exponential discounting. a second type of empirical support for hyperbolic discounting comes from experiments on dynamic inconsistency. several studies report systematic preference reversals between two rewards as the time-distance to these rewards diminishes. a hyperbolic discount model can demonstrate this; in fact, nonexponential time-preference curves can cross (strotz, 1955/56) and consequently the preference for one future reward over another may change with time. information and intertemporal choices in multi-agent decision problems 7 3. neuroeconomics: two model to consider impulsivity and inconsistency in intertemporal choice behavioral economist have found that there is a number of behavior patterns that violate the rational choice theory (kahneman et al., 1982; thaler, 1991); the most important is inconsistent preference, which represent behavior typically seen in psychiatric disorders (alcoholism, drug abuse), but also in more ordinary phenomena (overeating, credit card debt). neuroeconomics has found that addicts are more myopic (have large timediscount rates) in comparison to non-addicted populations (ainslie, 1975; bickel, et al. 1999), so hyperbolic discounting may explain various human problematic behaviors (laibson, 1997): loss of self-control, failure in planned abstinence from addictive drugs, etc. recently, behavioral neuroeconomic and econophysical studies have proposed two discount models, in order to better describe the neural and behavioral correlates of impulsivity and inconsistency in intertemporal choice. q-exponential discount model. takahashi et al. (2007) have proposed and examined this function for subjective value v(d) of delayed reward: 𝑉(𝐷) = 𝐴 𝑒𝑥𝑝𝑞 (𝑘𝑞 𝐷) = 𝐴/[1 + (1 − 𝑞)𝑘𝑞 𝐷] 1 1−𝑞 where d denotes a delay until receipt of a reward, a the value of a reward at d = 0, and kq a parameter of impulsivity at delay d = 0 (q-exponential discount rate) and the q-exponential function is defined as: 𝑒𝑥𝑝𝑞 (𝑥) = (1 + (1 − 𝑞)) 1 1−𝑞 this function can distinctly parametrized impulsivity and inconsistency. if q < 0, the intertemporal choice behavior is more inconsistent than hyperbolic discounting (ventre and ventre, 2012). m. olivieri, m. squillante, v. ventre 8 quasi-hyperbolic discount model. behavioral economists have proposed that the inconsistency in intertemporal choice may be attributable to an internal conflict between “multiple selves” within a decision maker. as a consequence, there are (at least) two exponential discounting selves (with two exponential discount rates) in a single human individual; and when delayed rewards are at the distant future (>1 year), the self with a smaller discount rate wins, while delayed rewards approach to the near future (within a year), the self with a larger discount rate wins, resulting in preference reversal over time. this intertemporal choice behavior can be parametrized in a quasi-hyperbolic discount model (also as a β-δ model) (laibson 1997; o’donoghue and rabin, 1999). for discrete time τ (the unit assumed is one year) it is defined as (laibson, 1997): 𝐹(𝜏) = 𝛽𝛿 𝑡 (for τ=1,2,3,…) and 𝐹(0) = 1 (0 < 𝛽 < 𝛿 < 1). a discount factor between the present and one-time period later (β) is smaller than that between two future time-periods (δ). in the continuous time, the proposed model is equivalent to the linearlyweighted two-exponential functions (generalized quasi-hyperbolic discounting): 𝑉(𝐷) = 𝐴[𝑤 exp(−𝑘1𝐷) + (1 − 𝑤) exp(−𝑘2𝐷)] where w, 0 < w < 1, is a weighting parameter and k1 and k2 are two exponential discount rates (k1 < k2). note that the larger exponential discount rate of the two k2, corresponds to an impulsive self, while the smaller discount rate k1 corresponds to a patient self (ventre and ventre, 2012). these economists proposed different multiple self models, which often draw analogies between intertemporal choice and a variety of different models of interpersonal strategic interactions. 4. self-control in intertemporal choices in many cases a dynamic inconsistent behavior is attributed to the existence of contingent “temptations” that increase impulsivity and induce a deviation from the desirable behavior. what the person knows to be his best long run interest conflict with his short run desires. information and intertemporal choices in multi-agent decision problems 9 stroz’s model. to represent this incoherent purpose, strotz (1955) proposed two strategies that might be employed by a person who foresees how her preferences will change over time. the “strategy of pre-commitment”: a person can commits to some plan of action. for example, consider a consumer with an initial endowment k0 of consumer goods which has to be allocated over the finite interval (0, t). at time period t he wishes to maximize his utility function: 𝐽0 = ∫ 𝜆(𝑡 − 0)𝑈[ 𝑇 0 𝑐̅(𝑡), 𝑡]𝑑𝑡 subject to ∫ 𝑐(𝑡)𝑑𝑡 = 𝐾0 𝑇 0 where [𝑐̅(𝑡), 𝑡], is the instantaneous rate of consumption at time period t, and λ(t − 0) is a discount factor, the value of which depends upon the elapse of time between a past or future date and present. and this implies that the discounted marginal utility of consumption should be the same for all periods. but, at a later date, the consumer may reconsider his consumption plan. the problem then is to maximize 𝐽0 = ∫ 𝜆(𝑡 − 𝜏)𝑈[ 𝑇 0 𝑐(𝑡), 𝑡]𝑑𝑡 subject to ∫ 𝑐(𝑡)𝑑𝑡 = 𝐾𝜏 = 𝐾0 − 𝑇 𝜏 ∫ 𝑐(𝑡)𝑑𝑡 𝜏 0 the optimal pattern of consumption will change with changes in τ and if the original plan is altered, the individual is said to display dynamic inconsistency. strotz showed that individuals will not alter the original plan only if 𝜆(𝑡, 𝜏) is exponential in |t − τ|. the “strategy of consistent planning”: since pre-commitment is not always a feasible solution to the problem of intertemporal conflict, an individual may adopt a different strategy: take into account future changes in the utility function and reject any plan that he will not follow through. his problem is then to find the best plan among those he will actually follow. thaler and shefrin’s model. in the setting of multiple selves models, to control impulsivity, thaler and shefrin (1981) proposed a “planner-doer” model which draws upon principal-agent theory. they treat an individual as if he contained two distinct psyches: one planner, which pursue longer-run results, and multiple doers, which are concerned only with short-term satisfactions, so m. olivieri, m. squillante, v. ventre 10 they care only about their own immediate gratification (and have no affinity for future or past doers). for example, consider an individual with a fixed income stream 𝑦 = [𝑦1, 𝑦2, … , 𝑦𝑇 ],where ∑ 𝑦𝑡 = 𝑌𝑡 which has to be allocated over the finite interval (0, t). the planner would choose a consumption plan to maximize his utility function 𝑉(𝑍1, 𝑍2, … , 𝑍𝑇 ) subject to ∑ 𝑐𝑡 ≤ 𝑌 𝑡 𝑡=1 in which such 𝑍𝑡 is a function of utility of level consumption in t (𝑐𝑡). on the other hand, an unrestrained doer 1 would borrow 𝑌 − 𝑦1 on the capital market and therefore choose c1 = y; the resulting consequence is naturally 𝑐2 = 𝑐3 = ⋯ = 𝑐𝑇 = 0. such action would suggest a complete absence of psychic integration. then the model focuses on the strategies employed by the planner to control the behavior of the doers, and it proposes two instruments he can use. (a) he can impose rules on the doers’ behavior, which operate by altering the constraints imposed on any given doer. pure rules, like pre-commitment, can be a very effective self-control strategy because they eliminate all choice. the advantage of these strategies is that once in place they require little or no self-enforcement. however, they may be unavailable or too expensive. (b) he can use discretion accompanied by some method of altering the incentives or rewards to the doer without any self-imposed constraints. one planner can alter the doer’s utility function directly introducing a modification parameter 𝜃 = 𝜃1, 𝜃2, … , 𝜃𝑇 . z is assumed to be a function of two arguments, ct and θt. if θt = 0, then the doer is completely unrestrained. as θt increases, both z and (δzt)/(δct) are reduced. θ might be thought of as a guilt parameter. the higher is θt , the more guilt the doer feels for any level of ct (ventre and ventre, 2012). in conclusion, the essential insight that multi selves model capture is that, much like cooperation in a social dilemma, self-control often requires the cooperation of a series of temporally situated selves. when one “self” defects by opting for immediate gratification, the consequence can be a kind of unraveling or “falling off the wagon” whereby subsequent selves follow the precedent (frederick, loewenstein, and o’donoghue, 2002). information and intertemporal choices in multi-agent decision problems 11 5. multi-agent decision problem: consensus and false consensus effect in a multi-agent decision problem an individual needs to take his intertemporal choice considering others’ preferences, to the purpose of achieving a consensus on a common decision. group decision problems, indeed, consist in finding the best alternative(s) from a set of feasible alternatives 𝐴 = {𝑎1, … , 𝑎𝑛} according to the preferences provided by a group of agents 𝐸 = {𝑒1, … , 𝑒𝑚}. the objective is to obtain the maximum degree of agreement among the agents’ overall performance judgements on the alternatives. once the alternatives have been evaluated, the main problem is to compare agents’ judgements to verify the consensus among them; in the case of unanimous consensus, the evaluation process ends with the selection of the best alternative(s). however, in real situations humans rarely come to a unanimous agreement: this has led to evaluate not only crisp degrees of consensus (degree 1 for fully and unanimous agreement) but also intermediate degrees between 0 and 1 corresponding to partial agreement among all agents. furthermore, full consensus (degree = 1) can be considered not necessarily as a result of unanimous agreement, but it can be obtained ever in the case of agreement among a fuzzy majority of agents (fedrizzi m, kacprzyk j, nurmi h., 1992/1993). the judgements of each agent are frequently based, in part, on intuition or subjective beliefs, rather than detailed data on the preferences of the people being predicted. such intuitive judgements become more pervasive judgements when people lack necessary data to base their judgements. research in others areas of social judgement has revealed that people are egocentric: they judge others in the same way that they judge themselves. consequently, as pointed out in several experiments, each decision maker overestimates his own opinion. social psychology has founded that people with a certain preference tend to make higher judgements of the popularity of that preference in others, compared to the judgements of those with different preferences. this empirical result has been termed the false consensus effect (ross et al., 1977; mullen, et al., 1985). it states that individuals overestimate the number of the people who possess the same attributes as they do. people often believe that others are more like themselves than they really are. thus, their predictions about others’ beliefs or behaviors, based on casual observation, m. olivieri, m. squillante, v. ventre 12 are very likely to err in the direction of their own beliefs or behavior. for example, college students who preferred brown bread estimated that over 50% of all other college students preferred brown bread, while white-bread eaters estimated that 37% showed brown bread preference (ross et al., 1977). as the consequence, in multi-agent decision problem we often have to deal with different opinions, different importance of criteria and agents, who are not fully impartial objective. in this sense, the false consensus effect produces partial objectivity and incomplete impartiality, which perturbs the agreements over the evaluation. 6. assessment of consensus and false consensus effect in the literature, different methods to compute a degree of a consensus in fuzzy environments have been defined, and some approaches have been proposed to measure consensus in the context of fuzzy preference relations (fedrizzi, kacprzyk, nurmi, 1992-1993). but, as we have seen, the false consensus effect can lead to an absence of objectivity in the evaluation process. indeed, there may be cases where an agent would not be able to objectively express any kind of preference degree between two or more of the available options caused by the presence of the false consensus effect. then just a numerical indication seems not to be sufficient to synthesize the degree of consensus of agents. to put in evidence the lack of objectivity and, consequently, synthesized judgements, a description of the individual opinion should incorporate both the true knowledge generated agent opinion and the subjective component that produces false consensus outputs. the opinion of each agent is decomposed into two components: a vector, made of the ranking of the alternatives, built by means of a classical procedure, e.g., a hierarchical procedure, and a fuzzy component that represents the contribution of the false consensus effect, which we assume to be fuzzy in nature. this allows us to consider aggregation operators, such as owa operators, useful when synthesis among fuzzy variables is to be built (squillante and ventre, 2010). the formal model considers the set 𝑁 of decision makers, the set 𝐴 of the alternatives, and the set 𝐶 of the criteria. let any decision maker 𝐼 ∈ 𝑁 be able to assess the relevance of each criterion. precisely, for every 𝑖, a function ℎ𝑖 : 𝐶 → [0,1] with ∑ ℎ𝑖 (𝑐) = 1𝑐∈𝐶 information and intertemporal choices in multi-agent decision problems 13 denoting the evaluation or weight that the decision maker assigns to the criterion 𝑐, is defined. furthermore, the function 𝑔𝑖 : 𝐴×𝐶 → [0,1] is defined, such that 𝑔𝑖 (𝑎, 𝑐) is the value of the alternative 𝑎 with respect to the criterion 𝑐, in the perspective of 𝑖. let 𝑛, 𝑝,and 𝑚 denote the (positive integer) numbers of the elements of the sets 𝑁, 𝐶, and 𝐴, respectively. the value ℎ𝑖 (𝑐)𝑐∈𝐶 denotes the evaluation of the 𝑝-tuple of the criteria by the decision maker 𝑖 and the value 𝑔𝑖 (𝑐, 𝑎)𝑐∈𝐶,𝑎∈𝐴 denotes the matrix 𝑝×𝑚 whose elements are the evaluations, made by 𝑖, of the alternatives with respect to each criterion in 𝐶. function: 𝐴 → [0,1] , defined by (𝑓𝑖 (𝑎))𝑎∈𝐴 = ℎ𝑖 (𝑐)𝑐∈𝐶 ⋅ 𝑔𝑖 (𝑐, 𝑎)𝑐∈𝐶,𝑎∈𝐴 is the evaluation, made by 𝑖, of the alternative 𝑎 ∈ 𝐴. an euclidean metric that acts between couples of decision makers 𝑖 and 𝑗, i.e., between individual rankings of alternatives, is defined by 𝑑(𝑓𝑖 , 𝑓𝑗 ) = √ 1 ǀ𝐴ǀ ∑(𝑓𝑖 (𝑎) − 𝑓𝑗 (𝑎)) 2 𝑎∈𝐴 if the functions ℎ𝑖 , 𝑔𝑖 range in [0, 1], then also 0 ≤ 𝑑(𝑓𝑖 , 𝑓𝑗 ) ≤ 1. if we set 𝑑∗ = 𝑚𝑎𝑥{𝑑(𝑓𝑖 , 𝑓𝑗 )ǀ𝑖, 𝑗 ∈ 𝑁}, then a degree of consensus 𝛿 ∗ can be defined as the complement to one of the maximum distance between two positions of the agents: 𝛿 ∗ = 1 − 𝛿 ∗ = 1 − 𝑚𝑎𝑥{𝑑(𝑓𝑖 , 𝑓𝑗 )ǀ𝑖, 𝑗 ∈ 𝑁}. now to identify the portion of the false consensus effect internal to the consensus-reaching process we have to consider a vector that represents the components of the consensus = 𝑝(𝑎)𝑃 + 𝑞(𝑎)𝑄 . this polynomial representation of the measure of the effect is composed by a numeric component m. olivieri, m. squillante, v. ventre 14 𝑝(𝑎)𝑃 that contains all quantitative information available derived from the consensus-reaching process, and 𝑞(𝑎)𝑄 that reflects the false consensus effect. then the measure of the effect is: 𝑞(𝑎) = 1 𝑁(𝑑∗)2 ∑(𝑓𝑖 − 𝑓𝑗 ) 2 𝑁 𝑖=1 with 0 ≤ 𝑞(𝑎) ≤ 1, ∀𝑖, 𝑗 ∈ 𝑁 . this component can be estimate with owa operators (a large class of decision support tools for providing heuristic solution to situations where several trade-offs should be taken into consideration). in yager (1988) is introduced an approach for multiple criteria aggregation, based on ordered weighted averaging (owa) operators. by ranking the alternatives, the operators provide an enhanced methodology for evaluating actions on a qualitative basis. 7. false consensus effect and intertemporal choice in a multi-agent context many decisions are made in condition of strategic interaction, i.e. situations in which consequences of our choices depend on decisions of others interactive. for example, in bidding in auctions or in a bargaining the choice depends not only on one’s evaluation of the good but also on the evaluation of other individuals. mathematical instrument used to describe these situations is the theory of games. indeed, a strategic game is considered as an interactive situation where two or more rivals interact and try to obtain an advantage from this interdependence. in this perspective, the theory of games can be considered as a tool for understanding and forecasting the decision-making processes; according to this theory the outcome of the game coincides with the decision of equilibrium, it occurs when each agent adopts the best strategy, which is the one selected on the basis of rational choice. rationality is one of the most important assumptions made in theory of games. it implies that every player always maximizes his utility, thus being able to perfectly calculate the probabilistic result of every action. so they have http://www.gametheory.net/dictionary/utility.html information and intertemporal choices in multi-agent decision problems 15 consistent preferences on the final outcome of the decision-making process and their aim is to maximize these preferences. however, first of all we have showed that intertemporal choices of each individual are influenced by impulsivity and show inconsistency; furthermore we have seen that in a group decision problem each individual tends to overestimate the extent to which other people share one’s beliefs, attitudes and behaviors. this means that in a strategic interaction people are not rationales; their choices are not solely a function of the objective response but of their subjective structure. the consequence is that in a strategic interaction, the equilibrium of the decision is the result of an internal process (which not reveals rationality). rational choice and equilibrium decision coincide only if decision makers (alone or in group) succeed to fight loss of self-control and to keep out false consensus effect. so these psychological evidences involve new equilibriums in strategic games, which are not justified with rational behaviors. the consequences are different according to the nature of the interactions; indeed, in theory of games the basic classification of interactions is between non-cooperative games and cooperative ones, consequently we have noncooperative decision problems and cooperative decision problems too. the first group summarizes the dynamics by which each person pursues his own interests without regard to gains / losses of others. in the second group, subjects form a coalition and assume mutual commitments to share the surplus generated by cooperation. psychological aspects of impulsivity and false consensus effect influence in different way these two kinds of interactions. a way to analyze these effects is to identify the portion of the false consensus effect in the equilibrium point (section 6), and to consider influence of doers in each individual choice (thaler and shefrin, 1981). 8. cooperative decision problems in a cooperative game a group of players (coalitions) may enforce cooperative behavior; hence the game is a competition between coalitions of players, rather than between individual players. an example is a coordination game, when players choose the strategies by a consensus decision-making process. indeed, coordination games are a class of games with multiple pure strategy nash equilibria in which players choose the http://en.wikipedia.org/wiki/coordination_game http://en.wikipedia.org/wiki/consensus_decision-making http://en.wikipedia.org/wiki/pure_strategy http://en.wikipedia.org/wiki/nash_equilibrium m. olivieri, m. squillante, v. ventre 16 same or corresponding strategies. the classic example of coordination game is the “battle-of-the sexes”, where an engaged couple must choose what to do in the evening: the man prefers to attend a baseball game and the women prefers to attend an opera. in term of utility the payoff for each strategy is: man opera (o) baseball (b) w o m a n opera (o) 3, 1 0, 0 baseball (b) 0, 0 1, 3 in this example there are multiple outcomes that are equilibriums: (b,b) and (o,o). however both players would rather do something together than go to separate events, so no single individual has an incentive to deviate if others are conforming to an outcome: the man would attend the opera if he thinks the woman will be there even though he prefers the other equilibrium outcome in which both attend the baseball game. one of the most commonly suggested criteria for the analysis of games with multiple equilibria is to select the one with the highest payoffs for all, if such a “paretodominant” outcome exists. in this context, a consensus decision-making process can be considered as an instrument to choose the best strategy in a coordination game. the final decision is often not the first preference of each individual in the group and they may not even like the final result. but it is a decision to which they all consent because it is the best for the group. if we follow the thaler and shefrin’s model, we can analyze choices in a cooperative game in this way: at period-one the planner of each agent states his preference, which is the best strategy because the planner wants maximize his utility function; indeed planners are rational part of each player. however, the period-one doers of each agent want obtain an immediate gratification, so they drive each agent to act differently from rational program of own planner, thinking that the others make the same by effect of false consensus. but each agent have a different utility function, so each one will select a different choice with degree = 1, and this make impossible the aggregation of the preferences with owa operators to obtain a common decision. in fact according the model to measure consensus proposed in section http://en.wikipedia.org/wiki/bijection http://en.wikipedia.org/wiki/strategy information and intertemporal choices in multi-agent decision problems 17 6 a certain consensus degree 𝛿 ∈ (0,1] is required in advance, consensus is reached if the constraint 𝛿 ∗ ≥ 𝛿 is satisfied. nevertheless, in cooperative decision problem, the influence of doers can be avoid, indeed agents can enforce contracts through parties at period-one, which eliminates the problem of loss of self-control, because it eliminates all choices. as a consequence the consensus is obtained with the aggregation of preferences of each planner. the planners are rationales, so the final common choice is the best strategy according to the theory of games. however, the result of this aggregation includes a part of the coefficient called the false consensus effect that depends on the subjectivity and also increases the degree of the opinions (squillante and ventre, 2010): with cooperation the group utility is higher than real utility of each one derived from strategy chosen. so they have to extract from the degree of consensus the measure of false consensus effect according the model analyzed in section 6. this means that at the best solution corresponds an improvement in terms of utility that is overrated as a result of the false consensus. then in a cooperative decision problem the influence of false consensus effect is present at period-one, while the loss of self-control of each agent is fought by the imposition of a rule (thaler and shefrin, 1981). the rationality of the equilibrium choice of the game is saved by the possibility of making an arrangement among agents, which represents a pure rule to control the behavior of the doers and maintain self-control at later time (section 4); nevertheless the final decision has a higher consensus degree because it is influenced by the false consensus effect. however this effect acts only on planners, so we can eliminate it in planners’ utility functions: the false consensus effect directly influence the discount function of each agent. for example, consider two person who live together and put in common a part of their monthly income to do the common expenses, this part of each salary form a fixed income stream 𝑦 = [𝑦1, 𝑦2, … , 𝑦𝑇 ], where ∑ 𝑦𝑡 = 𝑌𝑡 which has to be allocated over the finite interval (0, t). the two agents must agree on how to spend this money. we can eliminate the influence of the doers because both are obliged to deposit in common fund a fixed amount of money, and also because they made the plan of consumption of common expenses at period-one, so they can not use this money for other http://en.wikipedia.org/wiki/contract m. olivieri, m. squillante, v. ventre 18 purpose. in this way we can take into account only each planner and get the consensus about the common choice through the process of evaluation of a multi-agent decision problem. the planner’s preferences are represented by a utility function 𝑉(𝑍1, 𝑍2, … , 𝑍𝑇 ) , in which such 𝑍𝑡 is a function of utility of level consumption in t (𝑐𝑡). then the planner would choose a consumption common plan to maximize 𝑉(𝑍1, 𝑍2, … , 𝑍𝑇 ), subject to their fixed income stream ∑ 𝑐𝑡 ≤ 𝑌 𝑡 𝑡=1 . the consumption plan chosen by each agent will provide different degrees of preference for different types of consumption according to their preferences, then to reach an agreement it simply suffices aggregate the preferences of each planner (section 6). however, the consensual choice obtained will have a greater degree due to the false consensus effect established in the preferences of each planner. so the utility function of each planner may be released in advance of the false consensus effect by reducing the degree of preference of favorite choices. the function to maximize will always be 𝑉(𝑍1, 𝑍2, … , 𝑍𝑇 ), but each z will represent a degree of utility lower for each type of preferred consume. this example can be analyzed according to the theory of repeated games. the choice of “what we consume with the common fund” can be seen as a choice that is repeated over time. the repeated games study the repetition of the strategic choices over time. according to the theory of games, if in a repeated game, finitely or infinitely, there are multiple nash equilibria, then there are many subgame perfect equilibria. some of these involve the play of strategies that are collectively more profitable for players than the one-shot game nash equilibria. the economic reasoning that supports this balance is as follows: the players will agree to maximize their utility in the first period, while the actions to be taken in the second period are of two types: a punishment if the rival does not maintain the agreement and a prize (the best nash equilibrium of the single game) if it is fair. in this case the strategies take into account the history of the game, which makes possible the cooperation. when the agents interact only once, they often have an incentive to deviate from cooperation, but in a repeated interaction, any mutually beneficial outcome can be sustained in an equilibrium. the deviation information and intertemporal choices in multi-agent decision problems 19 is not convenient in the long run, since players can make retaliation and this operates especially when the game is repeated infinitely. according to our theory, the end result is the same: repeating a cooperative game make possible to obtain a common result which is not achievable in a oneperiod situation (see the battle of the sexes). however, this happens not because the rational player has more convenience to cooperate in the long run, but because through the agreements made at first period they eliminate any temptation to deviate, which is then made impossible. it is necessary set the impossibility to divert, otherwise, in later games, the doer of each player push his agent to deviate, also believing that the others will do the same as a result of the false consensus. 9. non-cooperative decision problems in non-cooperative games, also called competitive games, players can not stipulate binding agreements, regardless of their goals. so in a non-cooperative decision problem each agent makes decisions independently, without collaboration or communication with any of the others (j. nash, 1951), an example is the daily trading on the stock exchange. in this category the solution is given by nash equilibrium. consequently in this kind of interaction is not possible to implement some pre-commitment to control the doer’s actions, as a consequence is not possible recognize the best choice on a rational base. if we analyze a non-cooperative multi-agent decision problem like the traditional prisoner’s dilemma, on one temporal interval and with only two alternatives, we see that the agents achieve common decision, and this is the best strategy, because each doer wants obtain the higher advantage which is the same and, for the false consensus effect, each one thinks that other make the same. the doer of each prisoner will choose the strategy of “do not confess”. in the traditional version of the game, the police arrest two suspects (a and b) and interrogate them in separate rooms. each can either confess, thereby implicating the other, or keep silent. in terms of years in prison the payoff for each strategy are these: m. olivieri, m. squillante, v. ventre 20 agent a confess (c) do not confess (nc) a g e n t b confess (c) 5, 5 0, 10 do not confess (nc) 10, 0 1, 1 according to the theory of games, given this set of payoffs, there is a strong tendency for each to confess. if prisoner a remains silent, prisoner b is better off confessing (because 0 is better than 1 year in jail). however, b is also better off confessing if a confesses (because 5 years is better than 10). hence, b will tend to confess regardless of what a will do; and by an identical argument, a will also tend to confess. this line of reasoning implies two rational players with consistent preferences. actually, when each player has to choose the best strategy every doer drives his agent to make decision that leads him a greater advantage, believing that the other will do the same due to the effect of the false consensus. consequently, the decision made by each leads to optimal decision in terms of pareto, because both have the same utility function and both doers choose the only action that is the best strategy. this creates the paradoxical situation that rational players lead to a poorer outcome than irrational players. however, it is just a coincidence that the two players have achieved a common strategy. in other types of non-cooperative problems this can not happen, with the result that you will never achieve a joint decision without a prior agreement. consider, for example, a multi-agent decision problem in which the agents set to save money to realize a common purchase. even agent has a fixed income, 𝑌𝐴 and 𝑌𝐵 , and a nonnegative level of saving, 𝑆𝐴 and 𝑆𝐵. as in cooperative games, the planner of each agent choose the best strategy which maximize his function utility of saving (thinking for future), but the doer of each agent want obtain the highest advantage now, so it would consume 𝑌 and therefore choose = 0 , with a degree =1. indeed, the doers are impulsives, each one assigns weight=1 at one preference and weight=0 at all the others, thinking that everybody will make in the same way for effect of false consensus. in this case, as we see in cooperative game, is not possible to aggregate the preferences to obtain a common decision. information and intertemporal choices in multi-agent decision problems 21 the plan made in advance by group of agent (to realize a common purchase) is not feasible if they don’t set some rule or some method to alter the incentives for the doers. this type of problem can be represented in the following way: agent a save (s) do not save (ns) a g e n t b save (s) 10, 10 5, 5 do not save (ns) 5, 5 -10, -10 where the payoff represent the utility of each agent for each strategy. according rational choice we note the nash equilibrium coincides with the best strategy (s,s). however false consensus effect and impulsivity lead each agent to the worst equilibrium, because utility functions of the agents are different among them (each agent prefers consumptions to savings). this causes the lack of consensus on a common decision. in conclusion, in a non-cooperative multi-agent decision problem, there are two situations: 1) the doers of each agent have the same preference and they will reach a common decision that is given by the unanimous choice, 2) the doers have different preferences and do not assign any weight to the other preferences, so it is not possible to aggregate the preferences. then the influence of doers don’t affect if their choices are unanimous, and in this case the final decision will be also the best decision in term of pareto, but if this does not happen is impossible to achieve a common strategy without arresting impulsivity, and when the number of agents increases unanimity becomes increasingly difficult to obtain. analyzing this type of decision problem in long time, we note that the influence of the psychological aspects leads to the same conclusion of the theory of games, namely the impossibility of obtaining cooperation over time, but in a different way: according to the theory of games because the dominant strategy prevails, according to our analysis because the doers will divert to their preferences. indeed, according to the theory of games a repeated game with a unique nash equilibrium has the same subgame perfect equilibrium outcome, because in the last stage the strategy which will be played by each player does not depend on the history of the game, that is the strategies of the last stage of game are history m. olivieri, m. squillante, v. ventre 22 independent: every player in last round probably choose the equilibrium of dominant strategy so he betray (playing the last time is like playing a single time). thus, in finitely repeated games, if you fail to cooperate in the last game you can not do in any other round. however analyzing the situation according our theory we obtain the same conclusion but for different causes. we can reconsider the example of the two agents who save for common expenses, and continue the game for several years: in the same way, in subsequent periods, the doer of each agent will push to consume all what he has saved. if we consider two periods, at the first the payoffs are the same, in the second they are the sum: agent a save (s) do not save (ns) a g e n t b save (s) 20, 20 10, 10 do not save (ns) 10, 10 -20, -20 the doers of the second period will want to consume everything and choose 𝑆2 = 0, with the result that is not possible achieve the plan and the equilibrium is the worst solution (ns,ns). the planners will establish a consumption plan by discounting the expected future payoff and so smearing the savings over the years, but in every period the doers will deviate their agents for the temptation to consume everything today and save tomorrow, this impulsiveness is psychologically justified by the effect of the false consensus. in conclusion, even in the long time psychological influence of the doers can not lead to cooperation and to achievement of rational results. we can affirm that in a non-cooperative decision problem is only a chance obtaining a common 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[21] yager r.r. “on ordered weighted averaging aggregation operators in multi criteria decision making”, ieee trans syst man cybern, 18(1):183–190 (1988). microsoft word documento1 ratio mathematica volume 39, 2020, pp. 137-145 on homomorphisms from cn to cm vinod s* biju g.s† abstract in this paper, using elementary algebra and analysis, we characterize and compute all continuous homomorphism from cn to cm. also we prove that the cardinality of the set of all non-continous group homomorphism from cn to cm is at least the cardinality of the continuum. keywords: homomorphism; continuous function; hamel basis; 2010 ams subject classifications: 97u99. 1 *department of mathematics, government college for women, thiruvananthapuram, kerala, india; wenod76@gmail.com. †department of mathematics, college of engineering, thiruvananthapuram-695016, kerala, india; gsbiju@cet.ac.in. 1received on june 27th, 2020. accepted on december 15th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.562. issn: 1592-7415. eissn: 2282-8214. ©vinod et al. this paper is published under the cc-by licence agreement. 137 vinod s, biju g.s 1 introduction hamel [1905] introduced the concept of basis for real numbers and proved its existence in 1905 by exploring functions which satisfy cauchy’s functional equation f(x + y) = f(x) + f(y) for all x,y ∈ r. using the existence of such a basis, he described all solutions of cauchy’s functional equation and established the existence of discontinuous solutions. cauchy demonstrated that any additive function is rationally homogeneous. he also proved that the only continuous additive functions are real homogeneous and thus linear, and that an additive function with a discontinuity is discontinuous throughout. further restrictions were placed on a non-linear additive function by darboux [1875] who showed in 1875 that an additive function bounded above or below on some interval is continuous, hence linear. a survey of the research concerning additive functions can be found in green and gustin [1950] the continuous ring homomorphisms from c to c are trivial map, identity map and complex conjugation. since c is a field, all non-trivial ring homomorphisms are automorphisms on c. thus identity map and complex conjugation are the only continuous automorphisms on c. any automorphisms on c other than identity and complex conjugation is called a ”wild” automorphism on c. kestelman [1951] proved the existence of so-called wild automorphism on c and the showed that the set of such automorphisms on c has cardinality 2c. many properties of wild automorphism on c are still open. calculating the number of homomorphisms between two groups or two rings is a fundamental problem in abstract algebra. it is not easy to determine the number of distinct homomorphism between any two given groups or rings. most of the current results in this area are limited to groups or specific types of rings. for example, chigira et al. [2000] studied the number of homomorphisms from a finite group to a general linear group over a finite field. in a later study bate [2007] furnished the upper and lower limits for the number of completely reducible homomorphisms from a finite group γ to general linear and unitary groups over arbitrary finite fields and to orthogonal and symplectic groups over finite fields of odd characteristics. matei and suciu [2005] discusses a method for calculating the number of epimorphisms from a finitely presented group g to a finite solvable group γ. further discussion on homomorphisms on certain finite groups can be found in mal’cev [1983], riley [1971], hyers and rassias [1992], but the solution to the general problem is still elusive. hence the purpose of the paper is to characterize and compute all continuous group homomorphisms from cn to cm. 138 on homomorphisms from cn to cm 2 notations and basic results most of the notations, functions and terms we mentioned in this paper can be find in jacobson [2013], gallian [1994] and kestelman [1951]. we can interpret hamel’s concept as follows. the set r of real numbers is a linear space over the field q of rational numbers. this linear space has a basis. namely, there exists a subset h ⊂ r such that every non-zero x ∈ r can uniquely be written as a linear combination of the elements of h with rational coefficients. that is, there exist distinct elements h1,h2, . . . ,hk of h and non-zero rational numbers wh1 (x),wh2 (x), . . . ,whk (x) such that x = k∑ i=1 whi (x)hi (1) thus for x ∈ r, by adding the terms of the form 0 · hj in the representation (1), we can write x = ∑ h∈h wh(x)h (2) where wh(x) ∈ q and wh(x) = 0 for all h except for a finite number of values of h. hamel based his argument on zermelo’s fundamental result which states that every set can be well ordered. hamel’s argument is valid for an arbitrary linear space l 6= {0} over a field. for this reason, recently such a basis is called a hamel basis(see also cohn and cohn [1981], halpern [1966], jacobson [2013], kharazishvili [2017]). if f : r → r is additive, then it is easy to derive f(rx) = rf(x) for every r ∈ q and x ∈ r. thus, if h ⊂ r is a hamel basis and x is a real number, we obtain f(x) = f (∑ h∈h wh(x)h ) = ∑ h∈h f ( wh(x)h ) = ∑ h∈h wh(x)f(h). (3) observing that the hamel bases of a linear space l coincide with the maximal linearly independent subsets of l the existence of a hamel basis is established with the aid of zorn’s maximum principle. theorem 2.1. let l be a vector space over the field f . then l has a hamel basis. 139 vinod s, biju g.s theorem 2.2. any continuous function f : cn → c which assume only rational values is constant. halbeisen and hungerbühler [2000] showed that in an infinite dimensional banach space, every hamel base has the cardinality of the banach space, which is at least the cardinality of the continuum. theorem 2.3. if k ⊂ c is a field and e is a banach space over k such that dim(e) = ∞, then every hamel base of e has cardinality |e|. 3 homomorphisms from cn to cm first we will characterize all continuous group homomorphisms from cn to cm . theorem 3.1. the cardinality of the set of continuous group homomorphisms from cn to cm is equal to the cardinality of the continuum. proof. let φ : cn → cm be a continuous group homomorphism. for 1 ≤ j ≤ n; denote ej for the n-tuple whose jth component is 1 and 0’s elsewhere, and denote êj for the n-tuple whose jth component is i and 0’s elsewhere. we will complete the proof by the following steps. step 1: φ(nej) = nφ(ej) and φ(nêj) = nφ(êj) for all n ∈ z and for all j (1 ≤ j ≤ n). for n ∈ n, the argument is clear since φ is a group homomorphism. since φ is a group homomorphis, φ(−nej) = −φ(nej) = −nφ(ej) and φ(0ej) = 0φ(ej) therefore φ(nej) = nφ(ej) for all n ∈ z and for all j (1 ≤ j ≤ n). similarly we can prove φ(nêj) = nφ(êj) for all n ∈ z and for all j (1 ≤ j ≤ n). step 2: φ(rej) = rφ(ej) and φ(rêj) = rφ(êj) for all r ∈ q and for all j (1 ≤ j ≤ n). 140 on homomorphisms from cn to cm let r = p q , where p ∈ z, q ∈ n. then rq = p and hence rqej = pej. so φ(rqej) = φ(pej) =⇒ qφ(rej) = pφ(ej) =⇒ φ(rej) = p q φ(ej) =⇒ φ(rej) = rφ(ej), for all r ∈ q and for all j (1 ≤ j ≤ n). similarly, φ(rêj) = rφ(êj) for all r ∈ q and for all j (1 ≤ j ≤ n). step 3: φ(xej) = xφ(ej) and φ(xêj) = xφ(êj) for all x ∈ r and for all j (1 ≤ j ≤ n). let x ∈ r and 1 ≤ j ≤ n. then there is a sequence (rm) of rational numbers such that rm → x in r. then rmej → xej as m → ∞. since φ is continuous at xej, we have φ(xej) = lim m→∞ φ(rmej) = ( lim m→∞ rm)φ(ej) ; by step 2 = xφ(ej) similarly, φ(xêj) = xφ(êj) for all x ∈ r and for all j (1 ≤ j ≤ n). step 4: characterization of continuous homomorphisms from cn to cm. let z = (z1,z2, . . . ,zn) ∈ cn. for 1 ≤ j ≤ n, let xj = re(zj) and yj = im(zj). then z = (x1,x2, . . . ,xn) + (iy1, iy2, . . . , iyn) = n∑ j=1 xjej + n∑ j=1 yjêj 141 vinod s, biju g.s so φ(z) = φ ( n∑ j=1 xjej + n∑ j=1 yjêj ) = n∑ j=1 φ(xjej) + n∑ j=1 φ(yjêj) = n∑ j=1 xjφ(ej) + n∑ j=1 yjφ(êj) = n∑ j=1 re(zj)φ(ej) + n∑ j=1 im(zj)φ(êj). conversly, if aj(1 ≤ j ≤ n) and bj(1 ≤ j ≤ n) be 2n complex numbers , then the map φ given by φ(z1,z2, . . . ,zn) = n∑ j=1 re(zj)aj + n∑ j=1 im(zj)bj is a continuous group homomorphism from cn to cm. hence the cardinality of the set of continuous group homomorphisms from cn to cm is same as the cardinality of c2nm, which is the cardinality of the continuum. 2 now, we provide a proof to the existence of non-continuous group homomorphism from cn to cm. theorem 3.2. the cardinality of the set of all non-continous group homomorphism from cn to cm is at least the cardinality of the continuum. proof. consider cn as a vector space over the field q of rational numbers and h be a hamel basis of cn over q. then every vector z ∈ cn can be uniquely expressed z = ∑ h∈h wh(z)h (4) where wh(z) ∈ q and wh(x) = 0 for all h except for a finite number of values of h. let e0 and ê0 are the zero elements in cn and cm respectively. let e1 and ê1 are the n−tuple and m−tuple respectively such that first component is 1 and all other components are 0. let h′ be a fixed element in h. define a map ψh′ : cn → cm by ψh′ (z) = ψh′ (∑ h∈h wh(z)h ) = wh′ (z)ê1. 142 on homomorphisms from cn to cm let z = ∑ h∈h wh(z)h and z′ = ∑ h∈h wh(z ′)h be two elements in cn. then ψh′ (z + z ′) = ψh′ (∑ h∈h wh(z)h + ∑ h∈h wh(z ′)h ) = ψh′ (∑ h∈h [wh(z) + wh(z ′)]h ) = [wh′ (z) + wh′ (z ′)]ê1 = wh′ (z)ê1 + wh′ (z ′)ê1 = ψh′ (z) + ψh′ (z ′). hence ψh′ : cn → cm is a group homomorphism. for z = (z1,z2, . . . ,zm) ∈ cm, define φ : cm → c by φ(z) = z1. then φ is a continuous function. define g : cn → c by g(z) = φ◦ψh′ (z) for all z ∈ cn. then for z = ∑ h∈h wh(z)h ∈ cn, g(z) = φ◦ψh′ (∑ h∈h wh(z)h ) = φ(wh′ (z)ê1) = wh′ (z) ∈ q, g(h′) = g ( 1 ·h′ + ∑ h∈h,h 6=h′ 0h ) = φ◦ψh′ ( 1 ·h′ + ∑ h∈h,h 6=h′ 0h ) = φ(1 · ê1) = 1 and g(e0) = φ◦ψh′ (e0) = φ◦ψh′ ( 0 ·h′ + ∑ h∈h,h 6=h′ 0h ) = φ(0 · ê1) = 0. hence g is a non-constant function from cn to c which assumes only rational values. therefore g is not continuous and which gives the function ψh′ is discontinuous. let h′ and h′′ be two distinct elements in h. then ψh′ (h ′) = ψh′ (1 ·h′) = 1 · ê1 = ê1 and ψh′′ (h ′) = ψh′′ (0 ·h′′ + 1 ·h′) = 0 · ê1 = ê0. therefore ψh′ and ψh′′ are distinct. then the cardinality of set of all non-continuous group homomorphism from cn to cm is at least |h| = |cn| = the cardinality of the continuum. 2 143 vinod s, biju g.s 4 conclusions in this paper, we characterized all continuous group homomorphisms from cn to cm . also we proved that the cardinality of the set of all non-continous group homomorphism from cn to cm is at least the cardinality of the continuum. references michael bate. the number of homomorphisms from finite groups to classical groups. journal of algebra, 308(2):612–628, 2007. naoki chigira, yugen takegahara, and tomoyuki yoshida. on the number of homomorphisms from a finite group to a general linear group. journal of algebra, 232(1):236–254, 2000. paul moritz cohn and paul moritz cohn. universal algebra, volume 159. reidel dordrecht, 1981. gaston darboux. sur la composition des forces en statique. bulletin des sciences mathématiques et astronomiques, 9:281–288, 1875. joseph a gallian. contemporary abstract 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functions in real analysis. crc press, 2017. anatoly i mal’cev. on homomorphisms onto finite groups. american mathematical society translations, series, 2(119):67–79, 1983. daniel matei and alexander i suciu. counting homomorphisms onto finite solvable groups. journal of algebra, 286(1):161–186, 2005. robert riley. homomorphisms of knot groups on finite groups. mathematics of computation, 25(115):603–619, 1971. 145 paper1.dvi the general solutions of a functional equation related to information theory prem nath and dhiraj kumar singh department of mathematics, university of delhi delhi – 110007, india e-mail: dksingh@maths.du.ac.in dhiraj426@rediffmail.com abstract. the general solutions of a functional equation, containing two unknown functions, and related to a functional equation characterizing the shannon entropy and the entropy of degree α, are obtained. keywords: functional equations, continuous solutions, lebesgue measurable solutions, the shannon entropy, the nonadditive entropy of degree α, multiplicative functions, additive functions. 2 prem nath and dhiraj kumar singh 1. introduction for n = 1, 2, 3, . . ., let γn = { (p1, . . . , pn) : pi ≥ 0, i = 1, . . . , n; n ∑ i=1 pi = 1 } denote the set of all n-component complete discrete probability distributions with nonnegative elements and let f : i → r, r denoting the set of all real numbers and i = { x ∈ r : 0 ≤ x ≤ 1 } , the unit closed interval. the functional equation k ∑ i=1 ∑̀ j=1 f (piqj) = k ∑ i=1 f (pi) + ∑̀ j=1 f (qj)(1.1) with (p1, . . . , pk) ∈ γk and (q1, . . . , q`) ∈ γ` was first studied by t.w. chaundy and j.b. mcleod [4]. they proved that if (1.1) holds for integers k = 2, 3, . . . and ` = 2, 3, . . . and f is continuous on i , then f is of the form f (p) = c p log 2 p , 0 ≤ p ≤ 1(1.2) where c is an arbitrary real constant and 0 log 2 0 = 0. later on, j. aczél and z. daróczy [1] proved the same by assuming k = ` = 2, 3, . . .. z. daróczy [5] obtained the lebesgue measurable solutions of (1.1) by fixing k = 3, ` = 2 and assuming f (1) = 0. gy. maksa [13] obtained the solutions of (1.1) by fixing k = 3, ` = 2 but assuming f to be bounded on a subset, of i , of positive lebesgue measure. if f ( 1 2 ) = 1 2 , then (1.2) gives c = −1 and then (1.2) reduces to f (p) = − p log 2 p(1.3) for all p ∈ i . the general solutions of a functional equation ... 3 for any probability distribution (p1, . . . , pm) ∈ γm , hm(p1, . . . , pm) = − m ∑ i=1 pi log2 pi(1.4) is known as the shannon entropy [15] of the probability distribution (p1, . . . , pm) ∈ γm and the sequence hm : γm → r, m = 1, 2, . . . is known as the sequence of the shannon entropies. a generalization of the shannon entropy with which we shall be concerned in this paper is (with h αm : γm → r, m = 1, 2, 3, . . .) hαm(p1, . . . , pm) = (1−2 1−α)−1 ( 1− m ∑ i=1 pαi ) , α > 0, α 6= 1, 0α := 0, α∈r.(1.5) the entropies (1.5) are due to j. havrda and f. charvat [7]. the axiomatic characterization of the entropies (1.5) leads to the study of the functional equation k ∑ i=1 ∑̀ j=1 f (piqj) = k ∑ i=1 f (pi) + ∑̀ j=1 f (qj) + λ k ∑ i=1 f (pi) ∑̀ j=1 f (qj)(1.6) where (p1, . . . , pk) ∈ γk , (q1, . . . , q`) ∈ γ` and λ = 2 1−α − 1, α ∈ r. clearly, (1.6) reduces to (1.1) if λ = 0. by taking λ = 21−α − 1, α 6= 1, α ∈ r, 0α := 0, the continuous solutions of (1.6) were obtained by m. behara and p. nath [3] for all positive integers k = 2, 3, . . . ; ` = 2, 3, . . . . later on pl. kannappan [10] and d.p. mittal [14] also obtained the continuous solutions of (1.6) for λ 6= 0 and k = 2, 3, . . .; ` = 2, 3, . . . . for fixed integers k ≥ 3 and ` ≥ 2, l. losonczi [11] obtained the measurable solutions of (1.6). also, pl. kannappan [8] obtained the lebesgue measurable solutions of both (1.1) and (1.6) for fixed integers k ≥ 3, ` ≥ 3. 4 prem nath and dhiraj kumar singh it seems that l. losonczi and gy. maksa [12] are the first to obtain the general solutions of (1.6) in both cases, namely λ 6= 0 and λ = 0, by fixing integers k and `, k ≥ 3 and ` ≥ 3. there are several generalizations of (1.6), with λ ∈ r, containing at least two unknown functions. below we list only three important generalizations of (1.6), namely, k ∑ i=1 ∑̀ j=1 f (piqj) = k ∑ i=1 h(pi) + ∑̀ j=1 h(qj) + λ k ∑ i=1 h(pi) ∑̀ j=1 h(qj)(1.7) k ∑ i=1 ∑̀ j=1 f (piqj) = k ∑ i=1 f (pi) + ∑̀ j=1 h(qj) + λ k ∑ i=1 f (pi) ∑̀ j=1 h(qj)(1.8) and k ∑ i=1 ∑̀ j=1 f (piqj) = k ∑ i=1 g(pi) + ∑̀ j=1 h(qj) + λ k ∑ i=1 g(pi) ∑̀ j=1 h(qj).(1.9) the object of this paper is to investigate the general solutions of the functional equation (1.7) for fixed integers k ≥ 3 and ` ≥ 3. the corresponding results for the functional equations (1.8) and (1.9) have also been investigated by the authors and shall be presented elsewhere in our subsequent research work. the process of finding the general solutions of (1.7) requires a detailed study of the following two functional equations : k ∑ i=1 ∑̀ j=1 g(piqj) = k ∑ i=1 g(pi) ∑̀ j=1 g(qj) + `(k − 1) g(0)(1.10) and k ∑ i=1 ∑̀ j=1 f (piqj) = k ∑ i=1 h(pi) ∑̀ j=1 h(qj)(1.11) where f : [0, 1] → r, g : [0, 1] → r and h : [0, 1] → r. the general solutions of a functional equation ... 5 the functional equation (1.10) is, indeed, a generalization of the multiplicativetype functional equation k ∑ i=1 ∑̀ j=1 g(piqj) = k ∑ i=1 g(pi) ∑̀ j=1 g(qj)(1.12) whose importance in information theory is well-known (see l. losonczi and gy. maksa [12]). the functional equation (1.6), for λ 6= 0, can be written in the multiplicative form (1.12) by defining g : i → r as g(x) = λ f (x) + x for all x ∈ i . likewise, each of the functional equations (1.7), (1.8) and (1.9), for λ 6= 0, can also be written in the corresponding multiplicative forms. this is precisely the reason for paying attention to the functional equations (1.7) to (1.9). 2. the general solutions of functional equation (1.10) before investigating the general solutions of (1.10) for fixed integers k and `, k ≥ 3, ` ≥ 3, we need some definitions and results already existing in the literature (see [12]). let ∆ = { (x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ x + y ≤ 1 } . in other words, ∆ denotes the unit closed triangle in r 2 = r × r = { (x, y) : x ∈ r, y ∈ r } . a mapping a : r → r is said to be additive if it satisfies the equation a(x + y) = a(x) + a(y)(2.1) for all x ∈ r, y ∈ r. a mapping a : i → r, i = [0, 1] is said to be additive on the triangle ∆ if it satisfies (2.1) for all (x, y) ∈ ∆. 6 prem nath and dhiraj kumar singh a mapping m : [0, 1] → r is said to be multiplicative if m(0) = 0, m(1) = 1 and m(xy) = m(x) m(y) for all x ∈ ]0, 1[, y ∈ ]0, 1[. now we state : lemma 1. let ψ : i → r be a mapping which satisfies the functional equation n ∑ i=1 ψ(pi) = c(2.2) for all (p1, . . . , pn) ∈ γn ; c a given constant and n ≥ 3 a fixed integer. then there exists an additive mapping a : r → r such that ψ(p) = a(p) + ψ(0) , 0 ≤ p ≤ 1(2.3) where a(1) = c − n ψ(0) .(2.4) conversely, if (2.4) holds, then the mapping ψ : i → r, defined by (2.3), satisfies the functional equation (2.2). this lemma appears on p-74 in [12]. lemma 2. every mapping a : i → r, i = [0, 1], additive on the unit triangle ∆, has a unique additive extension to the whole of r. note. this unique additive extension to the whole of r will also be denoted by the symbol a but now a : r → r. for lemma 2, see theorem (0.3.7) on p-8 in [2] or z. daróczy and l. losonczi [6]. now we prove : the general solutions of a functional equation ... 7 theorem 1. let k ≥ 3, ` ≥ 3 be fixed integers and g : [0, 1] → r be a mapping which satisfies the functional equation (1.10) for all (p1, . . . , pk) ∈ γk and (q1, . . . , q`) ∈ γ` . then g is of the form g(p) = a(p) + g(0)(2.5) where a : r → r is an additive function such that a(1) satisfies the equation a(1) + k` g(0) = [a(1) + k g(0)][a(1) + ` g(0)] + `(k − 1) g(0)(2.6) or g(p) = m (p) − a(p) + g(0)(2.7) where a : r → r is an additive function with a(1) = ` g(0)(2.8) and m : [0, 1] → r is a mapping such that m (0) = 0(2.9) m (1) = g(1) + (` − 1) g(0)(2.10) and m (pq) = m (p) m (q) for all p ∈ ]0, 1[ , q ∈ ]0, 1[ .(2.11) proof. let us put p1 = 1, p2 = . . . = pk = 0 in (1.10). we obtain [1 − g(1) − (k − 1)g(0)] ∑̀ j=1 g(qj) = 0.(2.12) 8 prem nath and dhiraj kumar singh case 1. 1 − g(1) − (k − 1) g(0) 6= 0. then (2.12) reduces to ∑̀ j=1 g(qj) = 0.(2.13) hence, by lemma 1, g is of the form (2.5) in which a : r → r is an additive mapping such that a(1) = − ` g(0) satisfies the equation (2.6). case 2. 1 − g(1) − (k − 1) g(0) = 0. the functional equation (1.10) may be written in the form ∑̀ j=1 [ k ∑ i=1 g(piqj) − g(qj) k ∑ i=1 g(pi) ] = `(k − 1) g(0). hence, by lemma 1, k ∑ i=1 g(piq) − g(q) k ∑ i=1 g(pi)(2.14) = a1(p1, . . . , pk, q) − 1 ` a1(p1, . . . , pk, 1) + (k − 1) g(0) where a1 : γk ×r → r is additive in the second variable. the substitution q = 0 in (2.14) gives a1(p1, . . . , pk, 1) = ` g(0) [ k ∑ i=1 g(pi) − 1 ] .(2.15) let x ∈ [0, 1], (r1, . . . , rk) ∈ γk . put q = xrt , t = 1, . . . , k in (2.14); add the resulting k equations; and use the additivity of a1 . we get k ∑ i=1 k ∑ t=1 g(pirtx) − k ∑ i=1 g(pi) k ∑ t=1 g(xrt)(2.16) = a1(p1, . . . , pk, x) − k ` a1(p1, . . . , pk, 1) + k(k − 1) g(0). the general solutions of a functional equation ... 9 now put q = x, p1 = r1, . . . , pk = rk in (2.14). we obtain k ∑ t=1 g(xrt) = g(x) k ∑ t=1 g(rt) + a1(r1, . . . , rk, x)(2.17) − 1 ` a1(r1, . . . , rk, 1) + (k − 1) g(0). from (2.16) and (2.17), it follows that k ∑ i=1 k ∑ t=1 g(pirtx) − g(x) k ∑ i=1 g(pi) k ∑ t=1 g(rt) − k(k − 1) g(0)(2.18) = (k − 1) g(0) k ∑ i=1 g(pi) + a1(r1, . . . , rk, x) k ∑ i=1 g(pi) − 1 ` a1(r1, . . . , rk, 1) k ∑ i=1 g(pi) + a1(p1, . . . , pk, x) − k ` a1(p1, . . . , pk, 1). the left hand side of (2.18) does not undergo any change if we interchange pi and ri, i = 1, . . . , k . so, the right hand side of (2.18) must also remain unchanged on interchanging pi and ri , i = 1, . . . , k . consequently, we obtain a1(p1, . . . , pk, x) [ k ∑ t=1 g(rt)− 1 ] − 1 ` a1(p1, . . . , pk, 1) [ k ∑ t=1 g(rt) − k ] (2.19) + (k − 1) g(0) k ∑ t=1 g(rt) = a1(r1, . . . , rk, x) [ k ∑ i=1 g(pi)−1 ] − 1 ` a1(r1, . . . , rk, 1) [ k ∑ i=1 g(pi)−k ] + (k − 1) g(0) k ∑ i=1 g(pi). now we divide our discussion into two cases depending upon whether k ∑ t=1 g(rt)−1 vanishes identically on γk or does not vanish identically on γk . 10 prem nath and dhiraj kumar singh case 2.1. k ∑ t=1 g(rt) − 1 vanishes identically on γk . then k ∑ t=1 g(rt) = 1 for all (r1, . . . , rk) ∈ γk . by using lemma 1, it follows that g is of the form (2.5) in which a(1) = 1 − k g(0) satisfies the equation (2.6). case 2.2. k ∑ t=1 g(rt) − 1 does not vanish identically on γk . in this case, there exists a probability distribution (r∗ 1 , . . . , r∗k) ∈ γk such that k ∑ t=1 g(r∗t ) − 1 6= 0.(2.20) putting r1 = r ∗ 1 , . . . , rk = r ∗ k in (2.19), making use of (2.20) and (2.15); and performing necessary calculations, it follows that a1(p1, . . . , pk, x) = a(x) [ k ∑ i=1 g(pi) − 1 ] (2.21) where a : r → r is such that a(x) = [ k ∑ t=1 g(r∗t ) − 1 ] −1 a1(r ∗ 1 , . . . , r∗k, x)(2.22) from (2.22) it is easy to conclude that a : r → r is additive as the mapping x 7→ a1(r ∗ 1 , . . . , r∗k, x) is additive. also, putting x = 1 in (2.22) and making use of (2.15) by taking pi = r ∗ i , i = 1, . . . , k ; (2.8) follows. also, from (2.14), (2.15), (2.21) and (2.8), it follows that k ∑ i=1 [g(piq) + a(piq) − g(0)] − [g(q) + a(q) − g(0)](2.23) × k ∑ i=1 [g(pi)+a(pi) − g(0)] + [g(q)+a(q) − g(0)](` − k) g(0) = 0. the general solutions of a functional equation ... 11 define a mapping m : i → r, i = [0, 1], as m (p) = g(p) + a(p) − g(0)(2.24) for all p ∈ i . then, (2.23) reduces to the equation k ∑ i=1 [m (piq) − m (q) m (pi) + (` − k) g(0) m (q) pi] = 0.(2.25) hence, by lemma 1, m (pq) − m (q) m (p) + (` − k) g(0) m (q) p = e1(p, q) − 1 k e1(1, q)(2.26) where e1 : r × [0, 1] → r is additive in its first variable. since a(0) = 0 and a(1) = ` g(0), (2.9) and (2.10) follow from (2.24). also, putting p = 0 in (2.26) and making use of (2.9), it follows that e1(0, q) = 0(2.27) for all q , 0 ≤ q ≤ 1. consequently, e1(1, q) = 0(2.28) for all q , 0 ≤ q ≤ 1. now, (2.26) reduces to m (pq) − m (p) m (q) = e1(p, q) − (` − k) g(0) m (q) p(2.29) for all p ∈ [0, 1] and q ∈ [0, 1]. since m (1) = g(1) + (` − 1) g(0), from now onwards, we divide our discussion into two subcases, depending upon whether g(1) + (` − 1) g(0) = 1 or g(1) + (` − 1) g(0) 6= 1. case 2.2.1. g(1) + (` − 1) g(0) = 1. in this case, 1 = g(1) + (` − 1) g(0) = g(1) + (k − 1) g(0) + (` − k) g(0). 12 prem nath and dhiraj kumar singh since g(1)+(k −1)g(0) = 1, it follows that (`−k) g(0) = 0. then, (2.29) reduces to m (pq) − m (p) m (q) = e1(p, q)(2.30) where e1 : r × [0, 1] → r is additive in the first variable and 0 ≤ p ≤ 1, 0 ≤ q ≤ 1. the left hand side of (2.30) is symmetric in p and q . hence, e1(p, q) = e1(q, p) for all p ∈ [0, 1], q ∈ [0, 1]. consequently, e1 is also additive in second variable. also, we may suppose that e1(p, ·) has been extended additively to the whole of r and this extension is unique by lemma 2. from (2.30), as on p-77 in [12], it follows that m (pqr) − m (p) m (q) m (r) = e1(pq, r) + m (r) e1(p, q)(2.30a) = e1(qr, p) + m (p) e1(q, r) for all p, q , r in [0,1]. now, we prove that e1(p, q) = 0 for all p, q , 0 ≤ p ≤ 1, 0 ≤ q ≤ 1. if possible, suppose there exist p∗ and q∗ , 0 ≤ p∗ ≤ 1, 0 ≤ q∗ ≤ 1, such that e1(p ∗, q∗) 6= 0. then, from (2.30a), m (r) = [ e1(p ∗ , q ∗) ] −1 [ e1(q ∗ r, p ∗) + m (p∗) e1(q ∗ , r) − e1(p ∗ q ∗ , r) ] from which it is easy to conclude that m is additive. now, making use of (2.8), (2.10), (2.20), (2.24), the condition g(1) + (` − 1) g(0) = 1; and the additivity of a and m , we have 1 6= k ∑ t=1 g(r∗t ) = m (1) − a(1) + k g(0) = 1 a contradiction. hence e1(p, q) = 0 for all p and q , 0 ≤ p ≤ 1, 0 ≤ q ≤ 1. thus, (2.30) reduces to m (pq) = m (p) m (q) for all p and q , 0 ≤ p ≤ 1, 0 ≤ q ≤ 1. so, m is a nonconstant multiplicative function. hence, from (2.24), it follows that g is of the form (2.7). the general solutions of a functional equation ... 13 case 2.2.2. g(1) + (` − 1) g(0) 6= 1. since the values of m at 0 and 1 are given by (2.9) and (2.10), our next task is to get some information about m (r) when 0 < r < 1. for this purpose, we proceed as follows: let p, q , r be in ]0,1[. now, from (2.29), one can derive m (pqr) − m (p) m (q) m (r)(2.31) = e1(r, pq)−(`−k) g(0) m (pq) r+m (r) [ e1(p, q)−(`−k) g(0) m (q) p ] = e1(rq, p)−(`−k) g(0) m (p) rq+m (p) [ e1(r, q)−(`−k) g(0) m (q) r ] . now, we prove that e1(p, q) − (` − k) g(0) m (q) p = 0 for all p, q , 0 < p < 1, 0 < q < 1. if possible, suppose there exist p∗ ∈ ]0, 1[ and q∗ ∈ ]0, 1[ such that e1(p ∗, q∗) − (` − k) g(0) m (q∗) p∗ 6= 0. then, from (2.31), it follows that for all r ∈ ]0, 1[, m (r) = [ e1(p ∗ , q ∗) − (` − k) g(0) m (q∗) p∗ ] −1 (2.32) × [ e1(rq ∗ , p ∗) − (` − k) g(0) m (p∗) rq∗ + m (p∗) { e1(r, q ∗) − (` − k) g(0) m (q∗) r } − e1(r, p ∗q∗) + (` − k) g(0) m (p∗q∗) r ] . now we prove that m : [0, 1] → r is additive on ∆, that is, m (x + y) = m (x) + m (y)(2.33) for all 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ x + y ≤ 1. if x = 0, 0 ≤ y ≤ 1 or y = 0, 0 ≤ x ≤ 1, then (2.33) holds trivially. if 0 < x < 1, 0 < y < 1, 0 < x + y < 1, then (2.33) follows from (2.32). now consider the case when 0 < x < 1, 0 < y < 1 but x + y = 1. in this case, let us choose q = 1 and p = x + y in (2.29) and use the additivity of e1 14 prem nath and dhiraj kumar singh with respect to first variable. we obtain m (x + y) { 1 − g(1) − (` − 1) g(0) } = { m (x) + m (y) }{ 1 − g(1) − (` − 1) g(0) } . since g(1) + (` − 1) g(0) 6= 1, (2.33) follows. thus, m is additive on the triangle ∆. now, making use of (2.8), (2.10), (2.20), (2.24), the condition g(1) + (k − 1) g(0) = 1; and the additivity of a and m , we have 1 6= k ∑ t=1 g(r∗t ) = m (1) − a(1) + k g(0) = 1 a contradiction. hence e1(p, q) − (` − k) g(0) m (q) p = 0 for all p, q, 0 < p < 1, 0 < q < 1. thus, (2.29) reduces to (2.11). but in this case m is not multiplicative because m (1) = g(1) + (` − 1) g(0) 6= 1. hence from (2.24), the solution (2.7) follows. note. it is easy to verify that (2.5); subject to the condition (2.6), satisfies (1.10). however (2.7) also satisfies (1.10). in this case we need to use (2.25) in addition to (2.8) to (2.11). 3. the general solutions of functional equation (1.11) now we prove : theorem 2. let k ≥ 3, ` ≥ 3 be fixed integers and f : i → r, h : i → r, i = [0, 1], be mappings which satisfy the functional equation (1.11) for all (p1, . . . , pk) ∈ γk and (q1, . . . , q`) ∈ γ` . then any general solution of (1.11) is of the form { f (p) = b(p) + f (0) h(p) = b(p) + h(0) (3.1) the general solutions of a functional equation ... 15 subject to the condition b(1) + k` f (0) = [b(1) + k h(0)][b(1) + ` h(0)];(3.2) or    f (p) = [h(1) + (k − 1) h(0)]2 a(p) + a∗(p) + f (0) h(p) = [h(1) + (k − 1) h(0)] a(p) + h(0) (3.3) subject to the condition [h(1) + (k − 1)h(0)]2a(1) + a∗(1) + k` f (0)(3.3a) = { [h(1)+(k−1)h(0)]a(1)+k h(0) }{ [h(1)+(k−1)h(0)]a(1)+` h(0) } or                    f (p) = [h(1) + (k − 1) h(0)]2[m (p) − a(p)] + a∗(p) + f (0) h(p) = [h(1) + (k − 1) h(0)][m (p) − a(p)] + h(0) a(1) = ` h(0) h(1) + (k − 1) h(0) , a∗(1) = ` { [h(1) + (k − 1) h(0)] h(0) − k f (0) } (3.4) where a∗ : r → r, a : r → r, b : r → r , a : r → r, b : r → r are additive functions; f (0) and h(0) are arbitrary constants; and m : [0, 1] → r is a mapping which satisfies (2.9), (2.11) and m (1) = h(1) + (` − 1) h(0) h(1) + (k − 1) h(0) (3.5) with h(1) + (k − 1) h(0) 6= 0 in (3.3), (3.3a), (3.4) and (3.5). to prove this theorem, we need to prove some lemmas : lemma 3. if a mapping f : i → r satisfies the functional equation k ∑ i=1 ∑̀ j=1 f (piqj) = 0(3.6) 16 prem nath and dhiraj kumar singh for all (p1, . . . , pk) ∈ γk , (q1, . . . , q`) ∈ γ` , k ≥ 3, ` ≥ 3 fixed integers; then f (p) = b(p) + f (0)(3.7) where b : r → r is an additive function with b(1) = − k` f (0). proof. choose q1 = 1, q2 = . . . = q` = 0. then, equation (3.6) reduces to k ∑ i=1 f (pi) = − k(` − 1) f (0). hence, by lemma 1, f (p) = b(p) − 1 k b(1) − (` − 1) f (0)(3.8) for all p, 0 ≤ p ≤ 1, b : r → r being any additive function with b(1) = − k` f (0). putting this value of b(1) in (3.8), (3.7) readily follows. lemma 4. under the assumptions stated in the statement of theorem 2, the following conclusions hold : f (p) = [h(1)+(k−1)h(0)]h(p)+a∗(p)−[h(1)+(k−1)h(0)]h(0)+f (0)(3.9) [h(1) + (k − 1) h(0)] k ∑ i=1 ∑̀ j=1 h(piqj) − k ∑ i=1 h(pi) ∑̀ j=1 h(qj)(3.10) = `(k − 1) h(0)[h(1) + (k − 1) h(0)] [h(1) + (` − 1) h(0)] k ∑ i=1 ∑̀ j=1 h(piqj) − k ∑ i=1 h(pi) ∑̀ j=1 h(qj)(3.11) = k(` − 1) h(0)[h(1) + (` − 1) h(0)] where a∗ : r → r is an additive function. the general solutions of a functional equation ... 17 proof. putting p1 = 1, p2 = . . . = pk = 0 in (1.11), we obtain ∑̀ j=1 { f (qj) − [h(1) + (k − 1) h(0)] h(qj) } = − `(k − 1) f (0).(3.12) hence, by lemma 1 (changing q to p), f (p) = [h(1) + (k − 1) h(0)] h(p) + a∗(p) − 1 ` a∗(1) − (k − 1) f (0)(3.13) for all p, 0 ≤ p ≤ 1, a∗ : r → r being an additive function with a∗(1) = ` { [h(1) + (k − 1) h(0)] h(0) − k f (0) } .(3.14) from equations (3.13) and (3.14), equation (3.9) follows. from (3.9) and (3.14), it is easy to see that k ∑ i=1 ∑̀ j=1 f (piqj) = [h(1) + (k − 1) h(0)] k ∑ i=1 ∑̀ j=1 h(piqj)(3.15) − `(k − 1) [h(1) + (k − 1) h(0)] h(0). from (1.11) and (3.15), we get (3.10). the proof of (3.11) is similar and hence omitted. proof of theorem 2. we divide our discussion into three cases : case 1. k ∑ i=1 h(pi) vanishes identically on γk , that is, k ∑ i=1 h(pi) = 0(3.16) for all (p1, . . . , pk) ∈ γk . then, (1.11) reduces to (3.6). so, f is of the form (3.7) for all p, 0 ≤ p ≤ 1. also applying lemma 1 to (3.16), we obtain h(p) = b(p) − 1 k b(1)(3.17) for all p, 0 ≤ p ≤ 1, b : r → r being an additive function with b(1) = − k h(0). 18 prem nath and dhiraj kumar singh now (3.17) reduces to h(p) = b(p) + h(0).(3.18) equations (3.7), (3.18), together with the condition (3.2), constitute the solution (3.1) of (1.11). case 2. ∑̀ j=1 h(qj) vanishes identically on γ` . in this case, we also get the solution (3.1), subject to the condition (3.2); of (1.11). the proof is omitted as it is similar to that in case 1. case 3. neither k ∑ i=1 h(pi) vanishes identically on γk nor ∑̀ j=1 h(qj) vanishes identically on γ` . then, there exist a (p ∗ 1 , . . . , p∗k) ∈ γk and a (q ∗ 1 , . . . , q∗` ) ∈ γ` such that k ∑ i=1 h(p∗i ) 6= 0 and ∑̀ j=1 h(q∗j ) 6= 0; and consequently k ∑ i=1 h(p∗i ) ∑̀ j=1 h(q∗j ) 6= 0 .(3.19) now, we prove that h(1) + (k − 1) h(0) 6= 0. if possible, suppose h(1) + (k − 1) h(0) = 0. then (3.10) reduces to the equation k ∑ i=1 h(pi) ∑̀ j=1 h(qj) = 0 valid for all (p1, . . . , pk) ∈ γk and (q1, . . . , q`) ∈ γ` . in particular, k ∑ i=1 h(p∗i ) ∑̀ j=1 h(q∗j ) = 0 contradicting (3.19). hence h(1) + (k − 1) h(0) 6= 0. similarly, making use of(3.11), we can prove that h(1) + (` − 1) h(0) 6= 0. let us consider the case when h(1) + (k − 1) h(0) 6= 0. in this case, let us define a mapping g : [0, 1] → r as g(x) = [h(1) + (k − 1) h(0)]−1h(x)(3.20) for all x ∈ [0, 1]. then, with the aid of (3.20), (3.10) reduces to the functional the general solutions of a functional equation ... 19 equation (1.10). also, from (3.20), it is easy to see that g(1) + (k − 1) g(0) = 1. consequently, from the discussion, carried out under this case, in the proof of theorem 1, it follows that g is of the form (2.5), subject to the condition (2.6); and (2.7). from equations (2.5), (2.7), (3.9) and (3.20), the solutions (3.3) subject to the condition (3.3a); and (3.4) of functional equation (1.11) follow. the details are omitted for the sake of brevity. 4. the general solutions of functional equation (1.7) when λ 6= 0 in this section we prove the following : theorem 3. let k ≥ 3, ` ≥ 3 be fixed integers and f : i → r, h : i → r, i = [0, 1], be mappings which satisfy the functional equation (1.7) for all (p1, . . . , pk) ∈ γk and (q1, . . . , q`) ∈ γ` . then, any general solution of (1.7) is of the form f (p) = b(p) + λ f (0) − p λ , h(p) = b(p) + λ h(0) − p λ (4.1) subject to the condition b(1) + λk` f (0) = [b(1) + λk h(0)][b(1) + λ` h(0)](4.2) or        f (p) = [λ (h(1) + (k − 1) h(0)) + 1]2 a(p) + a∗(p) + λ f (0) − p λ h(p) = [λ (h(1) + (k − 1) h(0)) + 1] a(p) + λ h(0) − p λ (4.3) subject to the condition [λ (h(1) + (k − 1) h(0)) + 1]2 a(1) + a∗(1) + λk` f (0)(4.3a) = { [λ (h(1) + (k − 1) h(0)) + 1] a(1) + λk h(0) } × { [λ (h(1) + (k − 1) h(0)) + 1] a(1) + λ` h(0) } 20 prem nath and dhiraj kumar singh or                                      f (p) =   [λ (h(1)+(k − 1) h(0))+1]2[m (p)−a(p)] + a∗(p)+λ f (0)−p   λ h(p) = [λ(h(1) + (k − 1) h(0)) + 1][m (p)−a(p)] + λ h(0)−p λ a(1) = λ` h(0) [λ (h(1) + (k − 1) h(0))] , a∗(1) = λ` { [λ (h(1) + (k − 1) h(0)) + 1] h(0) − k f (0) } (4.4) where a∗ : r → r, a : r → r, b : r → r, a : r → r, b : r → r are additive functions; m : [0, 1] → r satisfies (2.9), (2.11) and m (1) = λ (h(1) + (` − 1) h(0)) + 1 λ (h(1) + (k − 1) h(0)) + 1 (4.5) with [λ (h(1) + (k − 1) h(0)) + 1] 6= 0 in (4.3), (4.3a), (4.4) and (4.5). proof. let us write (1.7) in the multiplicative form k ∑ i=1 ∑̀ j=1 [λ f (piqj) + piqj] = k ∑ i=1 [λ h(pi) + pi] ∑̀ j=1 [λ h(qj) + qj].(4.6) define the mappings f : i → r, h : i → r as f (x) = λ f (x) + x, h(x) = λ h(x) + x(4.7) for all x ∈ i . then, (4.6) reduces to the functional equation (1.11) whose solutions are given by (3.1) subject to the condition (3.2); (3.3) subject to (3.3a); and (3.4) in which a∗ : r → r, a : r → r, b : r → r, a : r → r , b : r → r are additive functions; and m : [0, 1] → r is a mapping which satisfies (2.9), (2.11) and (3.5). now making use of (4.7) and (3.1) subject to the condition (3.2); (3.3) subject to the condition (3.3a); and (3.4); the required solutions (4.1) the general solutions of a functional equation ... 21 subject to the condition (4.2); (4.3) subject to the condition (4.3a) and (4.4) follow. the details are omitted. 5. the general solutions of functional equation (1.7) when λ = 0 if λ = 0, then (1.7) reduces to the functional equation k ∑ i=1 ∑̀ j=1 f (piqj) = k ∑ i=1 h(pi) + ∑̀ j=1 h(qj)(5.1) where k ≥ 3, ` ≥ 3 are fixed integers and (p1, . . . , pk) ∈ γk , (q1, . . . , q`) ∈ γ` . the substitutions p1 = 1, p2 = . . . = pk = 0 in (5.1) yield ∑̀ j=1 [f (qj) − h(qj)] = h(1) + (k − 1) h(0) − `(k − 1) f (0).(5.2) hence, by lemma 1, f (p) = h(p)+a∗ 1 (p)− 1 ` a∗ 1 (1)+ 1 ` { h(1)+(k−1)h(0)−`(k−1)f (0) } (5.3) where a∗ 1 : r → r is additive with a∗ 1 (1) = h(1) + (k + ` − 1) h(0) − k` f (0).(5.4) from (5.3) and (5.4), we obtain k ∑ i=1 ∑̀ j=1 f (piqj) = k ∑ i=1 ∑̀ j=1 h(piqj) + h(1) − (k − 1)(` − 1) h(0).(5.5) from (5.1) and (5.5), we obtain k ∑ i=1 ∑̀ j=1 h(piqj) = k ∑ i=1 h(pi)+ ∑̀ j=1 h(qj)− { h(1)−(k−1)(`−1)h(0) } .(5.6) define h1 : [0, 1] → r as h1(x) = h(x) − { h(1) − (k − 1)(` − 1) h(0) } x(5.7) 22 prem nath and dhiraj kumar singh for all x ∈ [0, 1]. then, equation (5.6) reduces to k ∑ i=1 ∑̀ j=1 h1(piqj) = k ∑ i=1 h1(pi) + ∑̀ j=1 h1(qj).(5.8) putting p1 = q1 = 1 and p2 = . . . = pk = q2 = . . . = q` = 0 in (5.8), we obtain h1(1) = (k − 1)(` − 1) h1(0). define h2 : [0, 1] → r as h2(x) = h1(x) − h1(0) − [h1(1) − h1(0)] x(5.9) for all x ∈ [0, 1]. then k ∑ i=1 ∑̀ j=1 h2(piqj) = k ∑ i=1 h2(pi) + ∑̀ j=1 h2(qj)(5.10) where h2(1) = h2(0) = 0, and (p1, . . . , pk) ∈ γk , (q1, . . . , q`) ∈ γ` , k ≥ 3, ` ≥ 3 fixed integers. theorem 2 (p-78 in [12]) may now be written as : theorem 4. let k ≥ 3, ` ≥ 3 be fixed integers. the mapping h2 : [0, 1] → r with h2(1) = 0, h2(0) = 0, defined in (5.9) is a solution of (5.10) if and only if h2(p) = { a(p) + d(p, p) if 0 < p ≤ 1 0 if p = 0 (5.11) where a : r → r is additive; d : r × ]0, 1] → r is additive in the first variable and there exists a function e : r × r → r, additive in both variables such that e(1, 1) = a(1) and, moreover, d(pq, pq) − d(pq, p) − d(pq, q) = e(p, q) if 0 < p ≤ 1, 0 < q ≤ 1.(5.12) making use of corollary 3 on p-81 in [12], it follows that h1(p) =    c + c (k` − k − `) p + a(p) + d(p, p) if 0 < p ≤ 1 c if p = 0 (5.13) where c = h1(0) is an arbitrary real constant, a : r → r is additive, d : r × ]0, 1] → r is as described above in theorem 4. now from (5.7) and the general solutions of a functional equation ... 23 (5.13), we obtain h(p) =    c (1 − p) + d1 p + a(p) + d(p, p) if 0 < p ≤ 1 c if p = 0 (5.14) where c = h(0) and d1 = h(1) are arbitrary real constants, a : r → r is additive function; d : r × ]0, 1] → r as described above in theorem 4. now from (5.3), (5.4) and (5.14), we obtain f (p) =    d0 + d1 p − c p + a(p) + a ∗ 1 (p) + d(p, p) if 0 < p ≤ 1 d0 if p = 0 (5.15) where c = h(0), d0 = f (0), d1 = h(1) are arbitrary real constants; a : r → r, a∗ 1 : r → r are additive functions with a∗ 1 (1) given by (5.4); d : r × ]0, 1] → r as described above in theorem 4. thus, we have proved the following: theorem 5. let k ≥3, `≥3 be fixed integers. the mappings f : [0, 1] → r, h : [0, 1] → r satisfy the equation (5.1) if and only if f and h are respectively of the forms (5.15) and (5.14) with a∗ 1 (1) given by (5.4) and d as described above in theorem 4. references [1] j. aczél and z. daróczy, characterisierung der entropien positiver ordnung und der shannonschen entropie, acta math. acad. sci. hungar., 14 (1963), 95–121. [2] j. aczél and z. daróczy, on measures of information and their characterizations, academic press, new york-san francisco-london, 1975. [3] m. behara and p. nath, additive and non-additive entropies of finite measurable partitions, probability and information theory ii, lecture notes in math., vol. 296, berlin. heidelberg-new york, 1973, 102–138. [4] t.w. chaundy and j.b. mcleod, on a functional equation, edinburgh math. notes, 43 (1960), 7–8. 24 prem nath and dhiraj kumar singh [5] z. daróczy, on the measurable solutions of a functional equation, acta math. acad. sci. hungar., 22 (1971), 11–14. [6] z. daróczy and l. losonczi, über die erweiterung der auf einer punktmenge additiven funktionen, publ. math. (debrecen), 14 (1967), 239–245. [7] j. havrda and f. charvat, quantification method of classification process, concept of structural α-entropy, kybernetika (prague), 3 (1967), 30–35. [8] pl. kannappan, on some functional equations from additive and nonadditive measures-i, proc. of the edin. mathematical society, 23 (1980), 145–150. [9] pl. kannappan, on some functional equations from additive and nonadditive measures-ii, advances in communication; second internat. conf. on information sciences and system (university of patras, patras, 1979), 1, 45–50, reidel, dordrecht-boston-mass., 1980. [10] pl. kannappan, on a generalization of some measures in information theory, glasnik mat., 9 (29) (1974), 81–93. [11] l. losonczi, a characterization of entropies of degree α, metrika, 28 (1981), 237–244. [12] l. losonczi and gy. maksa, on some functional equations of the information theory, acta math. acad. sci. hung., 39 (1982), 73–82. [13] gy. maksa, on the bounded solutions of a functional equation, acta math. acad. sci. hung., 37 (1981), 445–450. [14] d.p. mittal, on continuous solutions of a functional equation, metrika, 22 (1970), 31–40. [15] c.e. shannon, a mathematical theory of communication, bell syst. tech. jour., 27 (1948), 378–423, 623–656. ratio mathematica volume 45, 2023 weakly �̆�-𝓘-closed sets and weakly �̆�-𝓘continuous functions with respect to an ideal topological spaces m. vijayasankari1 g. ramkumar2 abstract in this paper, we introduce �̆�-ℐ-closed sets, �̆�-ℐ-closed sets, �̆�-ℐ-continuous functions and �̆�-ℐ-continuous functions and investigate their properties and its characterizations. after that we introduce weakly �̆�-ℐ-continuous functions and study the relationship between other types of continuous functions with suitable examples. keywords: �̆�-ℐ-cld, w�̆�-ℐ-cld, w�̆�-ℐ-openfunction, w�̆�-ℐ-continuous. 2010 mathematics subject classification: 54a053 1research scholar department of mathematics, madurai kamaraj university, madurai. tamil nadu, india. e-mail: vijayasankariumarani1985@gmail.com. 2assistant professor, department of mathematics, arul anandar college, karumathur, madurai, tamil nadu, india. e-mail: ramg.phd@gmail.com. 3received on july 10, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i01016. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by license agreement. 210 mailto:vijayasankariumarani1985@gmail.com mailto:ramg.phd@gmail.com m. vijayasankari and g. ramkumar 1. introduction the advent of generalized closed sets, a new trend was created by sundaram and sheik john. accordingly, a new notion called weakly closed sets were introduced by them. in this direction, many modifications of weakly closed sets are being introduced by the modern topologists due to their requirements. in this way, ravi et al. [18] introduced weakly g-closed sets and sundaram and nagaveni [24] introduced weakly g-closed sets in topological spaces. in 1961 levine [14] obtained a decomposition of continuity. later professor rose improved levine’s decomposition. in 1986 tong [26] obtained a decomposition of continuity and proved that his decomposition is independent of levine’s. in 1989, tong [27] improved upon his earlier decomposition and obtained yet another decomposition of continuity. in 1990, ganster and reilly [8] obtained a decomposition of continuity improving the first result of tong. a. acikgoz and et al. [1], introduced on α-i-continuous and α-i-open functions. j. antony rex rodrigo and et al. [2], the introduced the mildly-i-locally closed sets and decompositions of ⋆-continuity . k. kuratowski [11], introduced topology. s. jafari and n. rajesh [12], introduced the generalized closed sets with respect to an ideal. n. levine [13], introduced the generalized closed sets in topology. o. njastad [17], introduced the on some classes of nearly open sets. r. devi, k. balachandran and h. maki [4], introduced the semigeneralized closed maps and generalized semi-closed maps. devi, r., balachandran, k. and maki, h.[5], introduced the on generalized  -continuous maps and  -generalized continuous maps. devi, r., balachandran, k. and maki, h.[6], introduced the semigeneralized homeomorphisms and generalized semi-homeomorphisms in topological spaces. dontchev, j.[7], introduced the on generalizing semi-preopen sets. levine, n.[15], introduced the semi-open sets and semi-continuity in topological spaces. a. s. mashhour, et al. [16], introduced the  -continuous and  -open mappings. rajamani, m. and viswanathan, k.[19], introduced the on gs-continuous maps in topological spaces. v. renukadevi [20], introduced the note on ir-closed and air-sets. sundaram, p. and et al. [22], introduced the semi-generalized continuous maps and semi-t1/2spaces. sundaram, p.[23], introduced the study on generalizations of continuous maps in topological spaces. p. sundaram and m. rajamani [25], introduced some decompositions of regular generalized continuous maps in topological spaces. veera kumar, m. k. r. s.[28], introduced the between semi-closed sets and semi pre-closed sets. in this paper, we introduce �̆�-ℐ-closed sets, �̆�-ℐ-closed sets, �̆�-ℐ-continuous functions and �̆�-ℐ-continuous functions and investigate their properties and its characterizations. after that we introduce weakly �̆�-ℐ-continuous functions and study the relationship between other types of continuous functions with suitable examples. 211 weakly�̆�-𝓘-closed sets and weakly �̆�-𝓘-continuous functions with respect to an ideal topological spaces 2. preliminaries an ideal i on a topological space (briefly, tps) (x, τ) is a nonempty collection of subsets of x which satisfies (1) a∈i and b⊆a⇒b∈i and (2) a∈i and b∈i⇒a∪b∈i. given a topological space (x, τ) with an ideal ion x if ℘(x) is the set of all subsets of x, a set operator ( •)⋆: ℘(x)→ ℘(x), called a local function [10] of a with respect to τ and i is defined as follows: for a ⊆x, a⋆(i, τ)={ x ∈ x : u∩a i for every u ∈ τ(x)} where τ(x)={u ∈ τ : x ∈ u}. a kuratowski closure operator cl⋆( •) for a topology τ⋆(i, τ ), called the ⋆-topology and finer than τ, is defined by cl⋆(a) = a ∪a⋆(i, τ) [10]. we will simply write a⋆ for a⋆(i, τ) and τ⋆ for τ⋆(i, τ ). if i is an ideal on x, then (x, τ, i) is called an ideal topological space(briefly, itps). a subset a of an ideal topological space (x, τ, i) is ⋆-closed (briefly, ⋆-cld) [10] if a⋆⊆a. the interior of a subset a in (x, τ⋆(i)) is denoted by int⋆(a). definition 2.1 a subset k of a tps x is called: (i) semi-open set [9] if k cl(int(k)); (ii) regular open set [21] if k = int(cl(k)); the complements of the above mentioned open sets are called their respective closed sets. definition 2.2 a subset k of a tps x is called (i) 𝑔 -closed set (briefly, 𝑔-cld) [13] if cl(k)  v whenever k  v and v is open. (ii) semi-generalized closed (briefly, sg-cld)[8] if scl(k) v whenever kv and v is semi-open. (iii) generalized semi-closed (briefly, gs-cld)[29] if scl(k) v whenever kv and v is open. the complements of the above mentioned closed sets are called their respective open sets. definition 2.3 a subset k of a itps x is called (i) ig-closed (briefly, ig-cld) set [9] if k*  v whenever k  v and v is open. the complements of the above-mentioned closed sets are called their respective open sets. definition 2.4 a subset a of a topological space x is called: (i) a weakly g-closed (briefly,wg-cld) set [24] if cl(int(a))  u whenever a  u and u is open in x. (ii) a weakly g-closed (briefly, wg-cld) set [18] if cl(int(a))  u whenever a  u and u is -open in x. 212 m. vijayasankari and g. ramkumar (iii) a regular weakly generalized closed (briefly, rwg-cld) set [18] if cl(int(a))  u whenever a  u and u is regular open in x. 3. weakly �̆�-𝓘-closed sets we introduce the following definition: definition 3.1 a subset k of x is called (i) �̆�-ℐ-closed (briefly, �̆�-ℐ-cld) if k*  v whenever k  v and v is sg-open. the complement of �̆�-ℐ-cld is called �̆�-ℐ-open. the family of all �̆�-ℐ-cld in x is denoted by �̆�-ℐc(x). (ii) �̆�-ℐ-closed (briefly, �̆�-ℐ-cld) if  cl(k*)  v whenever k  v and v is sgopen. the complement of �̆�-ℐ-cld is called �̆�-ℐ-open. (iii) a function f : (x, , ℐ) → (y, ) is called �̆�-ℐ-continuous if the inverse image of every closed set in y is �̆�-ℐ-cld set in x. (iv) ⋆-continuous if f-1(v) is a ⋆-cld set in x, for every closed set in y. (v) a function f : (x, , ℐ) → (y, ) is called �̆�-ℐ-continuous if f-1(v) is a �̆�ℐ-cld set in x, for every closed set in y. (vi) a subset a of an ideal topological space (x, , ℐ) is called a weakly �̆�-ℐclosed (briefly, w�̆�-ℐ-cld) set if (int(a))*v whenever av and v is sg-open in x. theorem 3.2 every �̆�-𝓘-cld set is w�̆�-𝓘-cld but not conversely. example 3.3 let x = {p, q, r} and  = {, {p, q}, x} with 𝓘 = {}. then the set {p} is w�̆�-𝓘-cld set but it is not a �̆�-𝓘-cld in x. theorem 3.4 every w�̆�-𝓘-cld set is wg-cld but not conversely. proof let h be any w�̆�-ℐ-cld set and v be any open set containing h. then v is a sg-open set containing h. we have (int(h))*v. thus, h is wg-cld. example 3.5 let x = {p, q, r} and  = {, {p}, x} with 𝓘 = {}. then the set {p, q} is wg-cld but it is not a w�̆�-𝓘-cld. theorem 3.6 every w�̆�-𝓘-cld set is wg-cld but not conversely. proof let h be any w�̆�-ℐ-cld set and v be any -open set containing h. then v is a sg-open set containing h. we have (int(h))*v. thus, h is wg-cld. 213 weakly�̆�-𝓘-closed sets and weakly �̆�-𝓘-continuous functions with respect to an ideal topological spaces example 3.7 in example 3.5, the set {p, r} is wg-cld but it is not a w�̆�-𝓘-cld. theorem 3.8 every w�̆�-𝓘-cld set is rwg-cld but not conversely. proof let h be any w�̆�-ℐ-cld set and v be any regular open set containing h. then v is a sgopen set containing h. we have (int(h))*v. thus, h is rwg-cld. example 3.9 in example 3.5, the set {p} is rwg-cld but it is not a w�̆�-𝓘-cld. theorem 3.10 if a subset h of an ideal topological space x is both ⋆-cld and  g-cld, then it is w�̆�-𝓘-cld in x. proof let h be a g-cld set in x and v be any open set containing h. then v cl(h) = h (int(h*))*. since h is⋆-cld, v (int(h))*and hence w�̆�-ℐ-closed in x. theorem 3.11 if a subset h of an ideal topological space x is both open and w�̆�-𝓘-cld, then it is ⋆-cld. proof since h is both open and w�̆�-ℐ-cld, h (int(h))* = h* and hence h is ⋆-cld in x. corollary 3.12 if a subset h of an ideal topological space x is both open and w�̆�-𝓘-cld, then it is both regular open and regular closed in x. theorem 3.13 let x be an ideal topological space and h x be open. then, h is w�̆�𝓘-cld if and only if h is �̆�-𝓘-cld. proof let h be �̆�-𝓘-cld. by proposition 3.2, it is w�̆�-𝓘-cld. conversely, let h be w�̆�-ℐ-cld. since h is open, by theorem 3.11, h is ⋆-cld. hence h is �̆�-ℐ-cld. theorem 3.14 a set h is w⋆-cld if and only if (int(h))*−h contains no non-empty sgcld set. proof necessity. let g be a sg-cld set such that g (int(h))*−h. since gc is sg-open and hgc, from the definition of w�̆�-ℐ-closedness it follows that (int(h))*gc. i.e., g ((int(h))*)c. this implies that g ((int(h))*)  ((int(h))*)c = . sufficiency. let hj, where j is *-cld and sg-open set in x. if (int(h))* is not contained in j, then (int(h))*jc is a non-empty sg-cld subset of (int(h))*−h, we obtain a contradiction. this proves the sufficiency and hence the theorem. theorem 3.15 let x be an ideal topological space and h y x . if h is w�̆�-𝓘-cld in x, then h is w�̆�-𝓘-cld relative to y. 214 m. vijayasankari and g. ramkumar proof let hy j where j is sg-open in x. since h is w�̆�-ℐ-cld in x, hj implies (int(h))*j. that is y  ((int(h))*)c yj where y  (int(h))* is closure of interior of h in y. thus, h is w�̆�-ℐ-cld relative toy. theorem 3.16 if a subset h of an ideal topological space x is nowhere dense, then it is w�̆�-𝓘-cld. proof since int(h)  int(h*) and h is nowhere dense, int(h) = . therefore (int(h))* =  and hence h is w�̆�-ℐ-cld in x. the converse of theorem 3.16 need not be true as seen in the following example. example 3.17 let x = {p, q, r} and  = {, {p}, {q, r}, x} with 𝓘 = {}. then the set {p} is w�̆�-𝓘-cldset but not nowhere dense in x. remark 3.18 the following examples show that w�̆�-ℐ-closedness and semi-closedness are independent. example 3.19 in example 3.3, we have the set {p, r} is w�̆�-ℐ-cldset but not semi-cld in x. example 3.20 let x = {p, q, r} and  = {, {p}, {q}, {p, q}, x} with ℐ = {}. then the set {p} is semi-cld set but not w�̆�-ℐ-cld in x. remark 3.21 from the above discussions and known results in [18]. we obtain the following diagram, where a→b represents a implies b but not conversely. diagram ⋆-cld→w�̆�-ℐ-closed →wg-closed → wg-closed →rwg-closed definition 3.22 a subset h of an ideal topological space x is called w�̆�-𝓘-open set if hc is w�̆�-𝓘-cld in x. proposition3.23 (i) every �̆�-ℐ-open set is w�̆�-ℐ-open but not conversely. (ii) every g-open set is w�̆�-ℐ-open but not conversely. theorem 3.24 a subset h of an ideal topological space x is w�̆�-𝓘-open if j  int(h*) whenever j  h and j is sg-cld. 215 weakly�̆�-𝓘-closed sets and weakly �̆�-𝓘-continuous functions with respect to an ideal topological spaces proof let h be any w�̆�-ℐ-open. then hc is w�̆�-ℐ-cld. let j be a sg-cld set contained in h. then jc is a sg-open set containing hc. since hc is w�̆�-ℐ-cld, we have (int(hc))* jc. therefore j  int(h*). conversely, we suppose that j int(h*) whenever j  h and j is sg-cld. then jc is a sgopen set containing hc and jc (int(h*))c. it follows that jc  (int(hc))*. hence hc is w�̆�-ℐ-cld and so a is w�̆�-ℐ-open. 4. weakly �̆�-𝓘-continuous functions definition 4.1 let x and y be two an ideal topological space. a function f : x → y is called weakly �̆�-𝓘-continuous (briefly, w�̆�-𝓘-continuous) if f-1(v) is a w�̆�-𝓘-open set in x for each open set vof y. example 4.2 let x = y = {p, q, r},  = {, {p}, {q, r},x} with 𝓘 = {} and  = {, {p}, y}. the function f : (x, , 𝓘) → (y, ) defined by f(p) = q, f(q) = r and f(r) = p is w�̆�𝓘-continuous, because every subset of y is w�̆�-𝓘-cld in x. theorem 4.3 every �̆�-𝓘-continuous function is w�̆�-𝓘-continuous. proof it follows from proposition 3.23 (i). the converse of theorem 4.3 need not be true as seen in the following example. example 4.4 let x = y = {p, q, r},  = {, {p}, {q, r}, x} with 𝓘 = {} and  = {, {q}, y}. let f : (x, , 𝓘) → (y, ) be the identity function. then f is w�̆�-𝓘-continuous but not �̆�-𝓘-continuous. theorem 4.5 a function f : x → y is w�̆�-𝓘-continuous if and only if f-1(v) is a w�̆�-𝓘cld set in x for each closed set vof y. proof let v be any closed set of y. according to the assumption f-1(vc)=x \ f-1(v) is w�̆�-ℐopen in x, so f-1(v) is w�̆�-ℐ-cld in x. the converse can be proved in a similar manner. definition 4.6 an ideal topological space x is said to be locally �̆�-𝓘-indiscrete if every �̆�-𝓘-open set of x is ⋆-cld in x. theorem 4.7 let f : x → y be a function. if f is �̆�-𝓘-continuous and x is locally �̆�-𝓘indiscrete, then f is ⋆-continuous. proof. let w be an open in y. since f is �̆�-ℐ-continuous, f-1(w) is �̆�-ℐ-open in x. since x is locally �̆�-ℐ-indiscrete, f-1(w) is ⋆-cld in x. hence f is ⋆-continuous. 216 m. vijayasankari and g. ramkumar theorem 4.8 let f : x → y be a function. if f is contra �̆�-𝓘-continuous and x is locally �̆�-𝓘-indiscrete, then f is w�̆�-𝓘-continuous. proof let f : x → y be contra �̆�-ℐ-continuous and x is locally �̆�-ℐ-indiscrete. by theorem 4.7, f is ⋆-continuous, then f is w�̆�-ℐ-continuous. proposition 4.9 if f : x → y is perfectly continuous and w�̆�-𝓘-continuous, then it is rmap. proof let w be any regular open subset of y. according to the assumption, f-1(w) is both open and ⋆-cld in x. since f-1(w) is⋆-cld, it is wg-closed. we have f-1(v) is both open and wg-closed. hence, it is regular open in x, so f is r-map. definition 4.10 an ideal topological space x is called �̆�-𝓘-compact if every cover of x by �̆�-𝓘-open sets has finite subcover. definition 4.11 an ideal topological space x is weakly �̆�-𝓘-compact (briefly, w�̆�-𝓘compact) if every w�̆�-𝓘-open cover of x has a finite subcover. remark 4.12 every w�̆�-𝓘-compact space is �̆�-𝓘-compact. theorem 4.13 let f : x → y be surjective w�̆�-𝓘-continuous function. if x is w�̆�-𝓘compact, then y is compact. proof let {ai :ii} be an open cover of y. then {f -1(ai) : ii} is a w�̆�-ℐ-open cover in x. since x is w�̆�-ℐ-compact, it has a finite subcover, say{f-1(a1), f -1(a2),…., f -1(an)}. since f is surjective {a1, a2, ….., an} is a finite subcover of y and hence y is compact. definition 4.14 an ideal topological space x is weakly �̆�-𝓘-connected (briefly, w�̆�-𝓘connected) if x cannot be written as the disjoint union of two non-empty w�̆�-𝓘-open sets. theorem 4.15 if an ideal topological space x is w�̆�-𝓘-connected, then x is almost connected (resp. �̆�-𝓘-connected). proof it follows from the fact that each regular open set (resp. �̆�-ℐ-open set) is w�̆�-ℐ-open. theorem 4.16 for an ideal topological space x, the following statements are equivalent: i.x is w�̆�-ℐ-connected. ii.the empty set  and x are only subsets which are both w�̆�-ℐ-open and w�̆�-ℐ-cld. 217 weakly�̆�-𝓘-closed sets and weakly �̆�-𝓘-continuous functions with respect to an ideal topological spaces iii.each w�̆�-ℐ-continuous function from x into a discrete space y which has at least two points is a constant function. proof (i)  (ii). let s  x be any proper subset, which is both w�̆�-ℐ-open and w�̆�-ℐ-cld. its complement x \ s is also w�̆�-ℐ-open and w�̆�-ℐ-cld. then x = s  (x \ s) is a disjoint union of two non-empty w�̆�-ℐ-open sets which is a contradiction with the fact that x is w�̆�-ℐ-connected. hence, s =  or x. (ii)  (i). let x = a  b where a  b = , a , b  and a, b are w�̆�-ℐ-open. since a = x \ b, a is w�̆�-ℐ-closed. according to the assumption a = , which is a contradiction. (ii)  (iii). let f : x → y be a w�̆�-ℐ-continuous function where y is a discrete space with at least two points. then f-1({y}) is w�̆�-ℐ-closed and w�̆�-ℐ-open for each y y and x = {f-1({y})  y  y}. according to the assumption, f-1({y}) =  or f-1({y}) = x. if f1({y}) =  for all y y, f will not be a function. also there is no exist more than one y y such thatf-1({y}) = x. hence, there exists only one y y such that f-1({y}) = x and f-1({y1}) =  where y  y1 y. this shows that f is a constant function. (iii)  (ii). let s be both wθ̆-ℐ-open and wθ̆-ℐ-closed in x. let f : x → y be a wθ̆ℐ-continuous function defined by f(s) = {a} and f(x \ s) = {b} where a  b. since f is constant function,we get s = x. theorem 4.17 let f : x → y be a w�̆�-𝓘-continuous surjective function. if x is w�̆�-𝓘connected, then y is connected. proof we suppose that y is not connected. then y = a  b where a  b = , a , b  and a, b are open sets in y. since f is w�̆�-ℐ-continuous surjective function, x = f-1(a) f-1(b) are disjoint union of two non-empty w�̆�-ℐ-open subsets. this is contradiction with the fact that x is w�̆�-ℐ-connected. 5. weakly �̆�-𝓘-open functions and weakly �̆�-𝓘-closed functions definition 5.1 let x and y be an ideal topological space. a function f : x → y is called weakly �̆�-𝓘-open (briefly, w�̆�-𝓘-open) if f(v) is a w�̆�-𝓘-open set in y for each open set v of x. remark 5.2 every �̆�-𝓘-open function is w�̆�-𝓘-open but not conversely. example 5.3 let x = y = {p, q, r, s},  = {, {p}, {p, q, s}, x} with 𝓘 = {} and  = {, {p}, {q, r}, {p, q, r}, y}. let f : (x, , 𝓘) → (y, ) be the identity function. then f is w�̆�-𝓘-open but not �̆�-𝓘-open. 218 m. vijayasankari and g. ramkumar definition5.4 let x and y be an ideal topological space. a function f : x → y is called weakly �̆�-𝓘-closed (briefly, w�̆�-𝓘-cld) if f(v) is a w�̆�-𝓘-cld set in y for each closed set v of x. it is clear that an open function is w�̆�-ℐ-open and a closed function is w�̆�-ℐ-cld. theorem 5.5 let x and y be an ideal topological space. a function f : x → y is w�̆�𝓘-closed if and only if for each subset b of y and for each open set g containing f-1(b) there exists a w�̆�-𝓘-open set f of y such that b  f andf-1(f)  g. proof let b be any subset of y and let g be an open subset of x such thatf-1(b)  g. then f = y \ f(x \ g) is w�̆�-ℐ-open set containing b and f-1(f)  g. conversely, let u be any closed subset of x. then f-1(y \ f(u))  x \ u and x \ u is open. according to the assumption, there exists a w�̆�-ℐ-open set f of y such that y \ f(u)  f and f-1(f)  x \ u. then u  x \ f-1(f). from y \ f  f(u) f(x \f-1(f)) y \ f it follows that f(u) = y \ f, so f(u) is w�̆�-ℐ-cld in y. therefore, f is a w�̆�-ℐcldfunction. remark 5.6 the composition of two w�̆�-𝓘-cld functions need not be a w�̆�-𝓘-cld as we can see from the following example. example 5.7 let x = y = z = {p, q, r},  = {, {p}, {p, q}, x} with 𝓘= {} and  = {, {p}, {q, r}, y} and  = {, {p, q}, z} with j = {}. we define f : (x, , 𝓘) → (y, ) by f(p) = r, f(q) = q and f(r) = p and let g : (y, ) → (z, , j) be the identity function. hence both f and g are w�̆�-𝓘-cld functions. hence the composition of two w�̆�𝓘-cld functions need not be a w�̆�-𝓘-cld. theorem 5.8 let x, y and z be an ideal topological space. if f : x → y is a ⋆-cld function and g : y → z is a w�̆�-𝓘-cld function, then gf : x → z is a w�̆�-𝓘-cld function. theorem 5.9 a set k of x is �̆�-𝓘-open if and only if f  int(k) whenever f is sg-cld and f  k. proof suppose that f  int(k) such that f is sg-cld and f  k. let kc  g where g is sgopen. then gc  k and gc is sg-cld. therefore gc  int(k) by hypothesis. since gc  int(k), we have (int(k))c g. i.e., (kc)*  g, since (kc)* = (int(k))c. thus, kc is �̆�-ℐcld. i.e., k is �̆�-ℐ-open. conversely, suppose that k is �̆�-ℐ-open such that f  k and f is sg-cld. then fc is sgopen and kc fc. therefore, (kc)*  fc by definition of �̆�-ℐ-cld and so f int(k), since (kc)* = (int(k))c. 219 weakly�̆�-𝓘-closed sets and weakly �̆�-𝓘-continuous functions with respect to an ideal topological spaces lemma 5.10 for an x x, x �̆�-𝓘-cl(k) if and only if v  k  for every �̆�-𝓘-open set v containing x. proof let x �̆�-ℐ-cl(k) for any x x. to prove v  k  for every �̆�-ℐ-open set v containing x. prove the result by contradiction. suppose there exists a �̆�-ℐ-open set v containing x such that v  k = . then k vc and vc is �̆�-ℐ-cld. we have �̆�-ℐ-cl(k) vc. this shows that x �̆�-ℐ-cl(k) which is a contradiction. hence v  k  for every �̆�-ℐ-open set v containing x. conversely, let v  k  for every �̆�-ℐ-open set v containing x. to prove x �̆�-ℐcl(k). we prove the result by contradiction. suppose x �̆�-ℐ-cl(k). then there exists a �̆�-ℐ-cld set f containing k such that x  f. then x fc and fc is �̆�-ℐ-open. also, fc k = , which is a contradiction to the hypothesis. hence x �̆�-ℐ-cl(k). proposition 5.11 a function f : (x, , i) → (y, ) is �̆�-ℐ-continuous if and only if f-1(u) is �̆�-ℐ-open in x, for every open set u in y. proof let f : (x, , i) → (y, ) be �̆�-ℐ-continuous and u be an open set in y. then uc is closed in y and since f is �̆�-ℐ-continuous, f-1(uc) is �̆�-ℐ-cld in x. but f-1(uc) = (f-1(u))c and so f-1(u) is �̆�-ℐ-open in x. conversely, assume that f-1(u) is �̆�-ℐ-open in x, for each open set u in y. let f be a closed set in y. then fc is open in y and by assumption, f-1(fc) is �̆�-ℐ-open in x. since f-1(fc) = (f-1(f))c, we have f-1(f) is �̆�-ℐ-cld in x and so f is �̆�-ℐ-continuous. theorem 5.12 if f : (x, , i) → (y, ) is �̆�-𝓘-continuous and pre-sg-closed and if a is an �̆�-𝓘-open (or �̆�-𝓘-cld) subset of y, then f-1(h) is �̆�-𝓘-open (or �̆�-𝓘-cld) in x. proof let h be an �̆�-ℐ-open set in y and f be any sg-closed set in x such that f  f-1(h). then f(f)  h. by hypothesis, f(f) is sg-closed and h is �̆�-ℐ-open in y. therefore, f(f)  int(h) and so f  f-1(int(h)). since f is �̆�-ℐ-continuous and int(h) is open in y, f-1(int(h)) is �̆�-ℐ-open in x. thus f  int(f-1(int(h))) int( f-1(h)). i.e., f  int(f-1(h)) and f-1(h) is �̆�-ℐ-open in x. by taking complements, we can show that if h is �̆�-ℐ-cld in y, f-1(h) is �̆�-ℐ-cld in x. 6. conclusions the new class of generalized closed sets called weakly �̆�-𝓘-closed sets and w�̆�-𝓘continuous functions are very useful for new research in topological spaces. this may leads some new applications in real life problems. 220 m. vijayasankari and g. ramkumar references 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[30] k. viswanathan and j. jayasutha, some decompositions of continuity in ideal topological spaces, eur. j. math. sci., 1(1)(2012), 131-141. 222 ratio mathematica volume 41, 2021, pp.146-161 146 study of feedback queueing system with unreliable waiting server under multiple differentiated vacation policy rajni gupta* sangeeta malik# abstract this manuscript analyses a queueing system with bernoulli schedule feedback of customers, unreliable waiting server under differentiated vacations. the unsatisfied customer may again join the queue with probability α, following the bernoulli schedule. the stationary solution is obtained for the model with aid of the probability generating function technique. some important system performance measures are derived and the graphical behavior of these measures with some parameters is analysed. finally, to obtain the optimal value of service rate for the model, cost optimization is performed through the quadratic fit approach. keywords: feedback; differentiated vacations; optimization; bernoulli schedule; unreliable waiting server. 2010 ams subject classification: 60k25, 60k30. ________________________________ *baba mastnath university, asthal bohar, rohtak, india; goelrajni0101@gmail.com *hindu college sonipat, india. #baba mastnath university, asthal bohar, rohtak, india; sangeetastat@gmail.com 1received on june 22, 2021. accepted on december 19, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.619issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. study of feedback queueing system with unreliable waiting server under multiple d.v. policy 147 1. introduction queueing system with server vacations and feedback is a powerful tool and is successfully applied in many real-life congestion problems such as traffic systems, operating systems, communication systems, manufacturing and production lines. queueing systems with feedback assume that unsatisfied customers may join the queue again to repeat their request before leaving the system. this concept was first introduced by takacs [16]. disney and konig [4] analysed bernoulli feedback queueing system. sharma and kumar [14] studied the markovian feedback queueing model with the impatient behaviour of customers. later on, kalidass and kasthuri [6] did the pioneering work on m/g/1 queueing model with immediate feedback. sunder et al. [15] analysed the feedback queueing model with different service stations and vacation policies under reneging. the vacation queueing model is another remarkable and unavoidable feature due to its widespread applications in real-life situations. the server can go on vacations for many reasons like insufficient workload, server breakdown and some secondary tasks assigned to him, etc. the assumption of the service station being a hundred percent reliable is not a feasible one. the server may suffer a breakdown at any instant of time. ke, j.c. [7, 8, 9] analysed different queueing models with an unreliable server. later on, ke et al. [10] performed pioneer work on finite buffer m/m/c queueing system with server breakdown. kim et al. [11] analysed m/g/1 queueing model with a working breakdown. levy and yechiali [12] first analysed vacation queuing systems. later on doshi [3], tian and zhang [17] performed a comprehensive survey on vacation queueing models. researchers analysed queues with different vacation strategies like single vacations, multiple vacations and working vacations. servi and finn [13] first introduced the concept of working vacation in which the server remains engaged in some auxiliary works and thereby provides service at a relatively slower rate. banik et al. [2] analysed multiple working vacation queueing systems. a new class of differentiated vacations is introduced by ibe and isijola [5]. zhang and zhou [20] studied queueing model with m kinds of differentiated working vacations. the transient solution of a differentiated vacation queueing system was carried out by vijyashree and janani [19]. vyshna unni and rose mary [18] considered multi-server queues with differentiated vacations. amar aissani et al. [1] studied differentiated vacation queues under general service times. r. gupta, s. malik 148 in this paper, a queue with waiting and unreliable servers under differentiated vacation policy and bernoulli feedback of customers with different arrival rates in various states is considered. 2. mathematical description of model the queueing model is analysed under the following assumptions: 1. the customers arrive in accordance with the poisson process. the customers are served on an fcfs basis by the server, the service time is assumed to be exponentially distributed with a mean of 1/μ. 2. on getting the service, the customers may decide to leave the queue with probability β or unsatisfied customers may re-join the queue for another service with complementary probability 1-β (=α). 3. when the system gets empty on serving all the customers, the server keeps on waiting for customers for a random period exponentially distributed with a mean 1/ w and leaves for vacation only if none arrives in that duration. on returning from type i vacation, if it finds customers waiting for the service, it resumes to active state otherwise it goes for type ii vacation. in the same way, on returning from vacation ii, it switches to vacation i or resumes to active state depending on whether the system is empty or customers are waiting in the system respectively. this process continues in the same manner. the period of both the vacations are exponentially distributed but with different means 1/𝜃1 and 1/𝜃2 respectively. 4. the server breaks down at any point of time in the active service state. the breakdowns occur according to poisson distribution with parameter γ. the server is immediately sent for repair in such a situation. the repair time is also assumed to follow an exponential distribution with a mean of 1/δ. the customers have to wait for their service until the server gets repaired. 5. the arrival rates of customers in two types of vacations, downstate of server and active state are taken to be λ1, λ2, λ0, 𝜆3 respectively. 3. steady state equations denoting the number of customers in the system at any time t by c(t) and the server state at time t by s(t), we observe that {c(t), s(t)} is a continuous markov chain. study of feedback queueing system with unreliable waiting server under multiple d.v. policy 149 the different possible server states are s(t) = { 0, 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑛 𝑣𝑎𝑐𝑎𝑡𝑖𝑜𝑛 𝐼 1, 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑛 𝑣𝑎𝑐𝑎𝑡𝑖𝑜𝑛 𝐼𝐼 2, 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑛 𝑎𝑐𝑡𝑖𝑣𝑒 𝑠𝑡𝑎𝑡𝑒 3, 𝑠𝑒𝑟𝑣𝑒𝑟 𝑢𝑛𝑑𝑒𝑟 𝑟𝑒𝑝𝑎𝑖𝑟 denoting the probability of n customers in the ith state of server by 𝑝𝑛 𝑖, the steady-state equations governing the proposed quasi birth-death model using the markov process are (𝜆2 + 𝜃2)𝑝0 1 = 𝜃1𝑝0 0 (1) (𝜆2 + 𝜃2)𝑝𝑛 1 = 𝜆2𝑝𝑛−1 1, 𝑛 ≥ 1 (2) (𝜆3 + 𝑤)𝑝0 2 = 𝜇𝛽𝑝1 2 (3) (𝜆3 + 𝜇𝛽 + 𝛾)𝑝𝑛 2 = 𝜆3𝑝𝑛−1 2 + 𝜇𝛽𝑝𝑛+1 2 + 𝛿𝑝𝑛 3 + 𝜃1𝑝𝑛 0 + 𝜃2𝑝𝑛 1 , 𝑛 ≥ 1 (4) (𝜆0 + 𝛿)𝑝1 3 = 𝛾𝑝1 2 (5) (𝜆0 + 𝛿)𝑝𝑛 3 = 𝛾𝑝𝑛 2 + 𝜆0𝑝𝑛−1 3, 𝑛 ≥ 2 (6) (𝜆1 + 𝜃1)𝑝0 0 = 𝜃2𝑝0 1 + 𝑤𝑝0 2 (7) (𝜆1 + 𝜃1)𝑝𝑛 0 = 𝜆1𝑝𝑛−1 0, 𝑛 ≥ 1 (8) defining probability generating functions as 𝐻𝑖 (𝑧) = ∑ 𝑝𝑛 𝑖 𝑧 𝑛 ∞ 𝑛=0 , 𝑓𝑜𝑟 𝑖 = 0, 1, 2. (9) 𝐻3(𝑧) = ∑ 𝑝𝑛 3 𝑧 𝑛 ∞ 𝑛=1 (10) multiplying system of equations (1) and (2) by appropriate powers of z and summing over n from 1to ∞. 𝐻1(𝑧) = 𝜃1 (𝜆2 + 𝜃2 − 𝜆2𝑧) 𝑝0 0 (11) similarly, equations (7) and (8) yield r. gupta, s. malik 150 𝐻0(𝑧) = (𝜆1 + 𝜃1) (𝜆1 + 𝜃1 − 𝜆1𝑧) 𝑝0 0 (12) on similar steps from equations (5) and (6) we have (𝜆0 + 𝛿 − 𝜆0𝑧)𝐻3(𝑧) = 𝛾𝐻2(𝑧) − 𝛾𝑝0 2 after some rearrangement of terms, we obtain 𝐻3(𝑧) = 𝛾(𝐻2(𝑧) − 𝑝0 2) 𝜆0(1 − 𝑧) + 𝛿 (13) where, 𝑝0 2 = 1 𝑤 (𝜆1 + 𝜃1 − 𝜃1𝜃2 𝜆2+𝜃2 ) 𝑝0 0, 𝑈𝑠𝑖𝑛𝑔 (7) (14) multiplying equations (3) and (4) by an appropriate power of z and taking summation over n, together with the use of equations (9) and (10) we obtain (𝜆3 + 𝜇𝛽 + 𝛾 − 𝜆3𝑧 − 𝜇𝛽 𝑧 ) 𝐻2(𝑧) = 𝜃1𝐻0(𝑧) + 𝜃2𝐻1(𝑧) + 𝛿𝐻3(𝑧) − (𝜆1 + 2𝜃1)𝑝0 0 + (𝜇𝛽 + 𝛾 − 𝜇𝛽 𝑧 ) 𝑝0 2 (15) considering equations (13) and (15) simultaneously, we obtain 𝐻2(𝑧) = 𝜃1𝐻0(𝑧) + 𝜃2𝐻1(𝑧) − (𝜆1 + 2𝜃1)𝑝0 0 + 𝑔2(𝑧)𝑝0 2 𝑔1(𝑧) (16) where, 𝑔1(𝑧) = 𝜆3(1 − 𝑧) + 𝜇𝛽 (1 − 1 𝑧 ) − 𝛾𝛿 𝜆0(1 − 𝑧) + 𝛿 + 𝛾 (17) 𝑔2(𝑧) = 𝜇𝛽 (1 − 1 𝑧 ) − 𝛾𝛿 𝜆0(1 − 𝑧) + 𝛿 + 𝛾 (18) taking limits 𝑧 → 1 in equation (11) and (12) the closed-form expressions for p.g.f’s are 𝐻0(1) = (𝜆1 + 𝜃1) 𝜃1 𝑝0 0 (19) study of feedback queueing system with unreliable waiting server under multiple d.v. policy 151 𝐻1(1) = 𝜃1 𝜃2 𝑝0 0 (20) differentiating equation (11) and (12) and taking limits 𝑧 → 1 𝐻0 ′ (1) = 𝜆1(𝜆1 + 𝜃1) 𝜃1 2 𝑝0 0 (21) 𝐻1 ′ (1) = 𝜆2𝜃1 𝜃2 2 𝑝0 0 (22) 𝐻0 ′′(1) = 2𝜆1 2 (𝜆1 + 𝜃1) 𝜃1 3 𝑝0 0 (23) 𝐻1 ′′(1) = 2𝜆2 2 𝜃1 𝜃2 3 𝑝0 0 (24) similarly, taking limits in equation (16) and using the l-hospital rule 𝐻2(1) = ( 𝜆1(𝜆1 + 𝜃1) 𝜃1 + 𝜃1𝜆2 𝜃2 ) 𝑝0 0 + (𝜇𝛽 − 𝜆0𝛾 𝛿 ) 𝑝0 2 𝛽𝜇 − 𝜆3 − 𝜆0𝛾 𝛿 (25) on taking limit in equation (13), 𝐻3(1) = 𝛾(𝐻2(1) − 𝑝0 2) 𝛿 (26) now, differentiating 𝐻2(𝑧), taking limits 𝑧 → 1 and applying l’ hospital rule twice, 𝐻2 ′ (1) = (𝜃1𝐻0 ′′(1) + 𝜃2𝐻1 ′′(1) + 𝑔2 ′′(1)𝑝0 2)𝑔1 ′ (1) 2(𝑔1 ′ (1))2 − (𝜃1𝐻0 ′ (1) + 𝜃2𝐻1 ′ (1) + 𝑔2 ′ (1)𝑝0 2)𝑔1 ′′(1) 2(𝑔1 ′ (1))2 (27) where, 𝑔1 ′ (1) = 𝜇𝛽 − 𝜆0𝛾 𝛿 − 𝜆3 𝑔2 ′ (1) = 𝜇𝛽 − 𝜆0𝛾 𝛿 r. gupta, s. malik 152 𝑔1 ′′(1) = 𝑔2 ′′(1) = −2 (𝜇𝛽 + 𝛾 ( 𝜆0 𝛿 ) 2 ) similarly, after differentiating equation (11) and taking limits 𝑧 → 1, we get 𝐻3 ′ (1) = 𝛾(𝛿𝐻2 ′ (1) + 𝜆0(𝐻2(1) − 𝑝0 2)) 𝛿 2 (28) the closed-form expressions for all the p.g.f’s are implicitly expressed in terms of only one probability 𝑝0 0. to obtain 𝑝0 0 we use the following normalization condition. ∑ 𝐻𝑖 (1) 3 𝑖=0 = 1 (29) after some rearrangements of terms, we get: 𝑝0 0 ( 𝜆1 + 𝜃1 𝜃1 + 𝜃1 𝜃2 + ( 𝛾 + 𝛿 𝛿 ) (𝐴 + 𝐵𝐶) − 𝛾𝐶 𝛿 ) = 1 (30) where, 𝐴 = 𝜆1𝜃2(𝜆1 + 𝜃1) + 𝜆2𝜃1 2 𝜃1𝜃2 (𝜇𝛽 − 𝜆3 − 𝜆0𝛾 𝛿 ) 𝐵 = 𝜇𝛽𝛿 − 𝜆0𝛾 𝛿 (𝜇𝛽 − 𝜆3 − 𝜆0𝛾 𝛿 ) 𝐶 = 𝜆1𝜆2 + 𝜆1𝜃2 + 𝜆2𝜃1 𝑤(𝜆2 + 𝜃2) 𝑝0 0 = ( 𝜆1 + 𝜃1 𝜃1 + 𝜃1 𝜃2 + ( 𝛾 + 𝛿 𝛿 ) (𝐴 + 𝐵𝐶) − 𝛾𝐶 𝛿 ) −1 (31) 4. system performance measures expected system length = mean number of customers in the system = 𝐸𝐿𝑠 = ∑ ∑ 𝑛𝑝𝑛 𝑖 ∞ 𝑛=1 3 𝑖=0 expected system length in breakdown state = mean number of customers in the system in down-state of the server (repair state) = el[b] study of feedback queueing system with unreliable waiting server under multiple d.v. policy 153 = ∑ 𝑛𝑝𝑛 3 ∞ 𝑛=1 probability of server being on vacations 𝑃𝑣 = ∑ ∑ 𝑝𝑛 𝑖 1 𝑖=0 ∞ 𝑛=1 probability of server in the active (normal) state 𝑃𝑤 = ∑ 𝑝𝑛 2 ∞ 𝑛=1 probability of server being under repair 𝑃𝑟 = ∑ 𝑝𝑛 3 ∞ 𝑛=1 5. numerical results in this section, the sensitivity of different performance measures of the queueing model towards the system parameters is analysed and the observed numerical results are graphically represented with aid of matlab software. for the purpose, the parameters are fixed as 𝜆3 = 2.4, , 𝜆1= 2, 𝜆0= 1.2, μ = 7, w = 0.3, γ = 0.7, 𝜆2 = 2.2, δ = 0.6, β = 0.8, 𝜃1= 0.6, 𝜃2= 0.8 , unless they are changed in graphs as shown. figure 1: effect of service rate μ on mean system length r. gupta, s. malik 154 figure 1 shows variation in mean (expected) system length with μ. the mean system length decreases as μ increases. this is due to a decrease in mean service time with increasing μ. on increasing the probability of leaving the system β by unsatisfied customers on service completion, this system length goes on decreasing further, the reason being that with increasing β, the feedback probability decreases and hence the length of the system. figure 2: variation in mean system length versus leaving probability β figure 2 illustrates how the expected system length varies with change in leaving probability of customers for different values of w. as w increases, the mean waiting time of the server in normal state decreases and this results in a corresponding increase in mean system length. figure 3 reveals the effect of variation in leaving probability β on the probability of the server being on vacation. as we observe from the figure, with an increase in leaving probability, the probability of the server being on vacations increases. this increase is more obvious with increasing values of repair rate δ; this is due to the reason that as δ increases, the mean repair time decreases, thereby increasing the probability of servers being on vacation. study of feedback queueing system with unreliable waiting server under multiple d.v. policy 155 figure 3: variation in the probability of server in vacations versus β for different repair rates δ figure 4: effect of rate of type i vacation on expected system length the effect of change in the rate of type i vacation on expected system length is depicted in figure 4. the expected system length decreases as 𝜃1 increases. this is due to an increase in the duration of type i vacation with decreasing 𝜃1. the expected system length further increases as the parameter of waiting for the server w increases. the reason is that with an increase in w, the waiting time of the server in normal state decreases. r. gupta, s. malik 156 figure 5: variation in the probability of server in vacations versus service rate μ the variation in the probability of the server being on vacation with service rate μ is shown in figure 5. as μ increases the mean service time decreases. this results in faster service thereby increasing the probability of the server being on vacation. this increase is more obvious with increasing the waiting time parameter w. this is because of the corresponding decrease in the mean waiting time of the server in the active (normal) state. figure 6: probability of waiting server versus leaving probability β study of feedback queueing system with unreliable waiting server under multiple d.v. policy 157 figure 6 represents the effect of leaving probability β of the customer after being served on the probability of waiting server 𝑝0 2. the probability of waiting server increases with an increase in the probability of leaving the system b. as w increases, the probability of waiting for server 𝑝0 2 decreases. this is due to a decrease in mean waiting time with an increase in w. 6. cost optimization in this section, the operating cost function is optimized relative to 𝜇. here we define some cost elements as 𝐶𝐿𝑆 = cost per unit time for the customer present in the system. 𝐶𝜇 = cost per unit time for service in the active state. 𝐶𝜃1 = cost per unit time in the period of type i vacation. 𝐶𝜃2 = cost per unit time in the period of type ii vacation. 𝐶𝛾= cost per unit time in the breakdown state 𝐶𝛿 = cost per unit time for repair the cost function per unit time is defined as f(μ) = 𝐸𝐿𝑆 𝐶𝐿𝑆 + μ𝐶𝜇 + 𝜃1𝐶𝜃1 + 𝜃2𝐶𝜃2 + 𝛾𝐶𝛾+ 𝛿𝐶𝛿 fix 𝐶𝐿𝑆 = 14, 𝐶𝜇 = 20, 𝐶𝜃1 = 10,𝐶𝜃2 = 8, 𝐶𝛾= 7, 𝐶𝛿 = 10 in parabolic method to find the optimal cost f(x) and corresponding value of x. this method starts by generating the quadratic function through calculated points in every iteration. the point at which f(x) is optimum in three-point pattern {𝑥1, 𝑥2, 𝑥3} is given by 𝑥𝐿 = 0.5(𝐹(𝑥1)(𝑥2 2 − 𝑥3 2) + 𝐹(𝑥2)(𝑥3 2 − 𝑥1 2) + 𝐹(𝑥3)(𝑥1 2 − 𝑥2 2)) 𝐹(𝑥1)(𝑥2 − 𝑥3) + 𝐹(𝑥2)(𝑥3 − 𝑥1) + 𝐹(𝑥3)(𝑥1 − 𝑥2) this value obtained improves the current three-point pattern by replacing one of the three points. optimum value up to the desired degree of accuracy is obtained by recursively using the process. table 1 shows that optimum value 𝐹(𝜇) =310.19155 with the permissible error of 10−4 for μ= 7.36987. this value verifies the results of figure 7. r. gupta, s. malik 158 s.no 𝝁𝟏 𝝁𝟐 𝝁𝟑 f(𝝁𝟏) f(𝝁𝟐) f(𝝁𝟑) 𝝁𝑳 1 7.0 7.50000 8.00000 311.66873 310.34146 313.16952 7.40971 2 7.0 7.40971 7.50000 311.66873 310.20608 310.34146 7.38091 3 7.0 7.38091 7.40971 311.66873 310.19268 310.20608 7.37334 4 7.0 7.37334 7.38091 311.66873 310.19166 310.19268 7.37086 5 7.0 7.37086 7.37334 311.66873 310.19155 310.19166 7.37016 6 7.0 7.37016 7.37086 311.66873 310.19155 310.19155 7.36994 7 7.0 7.36994 7.37016 311.66873 310.19154 310.19155 7.36987 table 1: optimal service rate for operating cost by quadratic fit approach figure 7: expected operating cost per unit time versus service rate μ study of feedback queueing system with unreliable waiting server under multiple d.v. policy 159 7. conclusion the queueing model under differentiated vacations with an unreliable waiting server and bernoulli schedule feedback of customers is considered. the impact of state-dependent arrival of customers is studied on the queueing model in steady-state. the sensitivity of some important system measures towards feedback probability, waiting parameter of server, service rate, and duration of vacations is illustrated graphically. cost is also optimized for the model using the parabolic method. the model can be extended to bulk arrival, general service times for future research. conflicts of interests the authors declare that there is no conflict of interest. references [1] aissani, a., braham, k. a. and taleb, s. 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(2017). m/m/1 queue with m kinds of differentiated working vacations. journal of applied mathematics and computing, 54, 213-227. available at: doi10.1007/s12190-016-1005-z ratio mathematica volume 40, 2021, pp. 213-224 on δ-preregular e∗-open sets in topological spaces jagadeesh b.toranagatti∗ abstract in this paper, we introduce a new class of sets called, δ-preregular e∗open sets and investigate their properties and characterizations. by using δ-preregular e∗-open sets, we obtain decompositions of complete continuity and decompositions of perfect continuity. keywords: δ-preopen;e∗-open;e∗-closed;δpe∗-open;δpe∗-closed;δpe∗continuity. 2020 ams subject classifications: 54a05,54c08. 1 ∗department of mathematics, karnatak university’s karnatak college, dharwad-580001, karnataka, india; jagadeeshbt2000@gmail.com 1received on may 5th, 2021. accepted on june 22nd, 2021. published on june 30th, 2021.doi: 10.23755/rm.v40i1.609. issn: 1592-7415. eissn: 2282-8214. c©toranagatti this paper is published under the cc-by licence agreement. 213 jagadeesh b.toranagatti 1 introduction the study of δ-open sets was initiated by veličko[velicko, 1968] in 1968. following this raychaudhuri and mukherjee[raychaudhuri and mukherjee, 1993] established the concept of δ-preopen sets. later, ekici[ekici, 2009] introduced the concept of e∗-open sets as a generalization of e-open sets. the aim of this paper is to introduce and study a new class of sets called, δ-preregular e∗-open sets using δ-preinterior and e∗-closure operators. the notion of δpe∗-continuity is also introduced which is stronger than δ-precontinuity.finally, we obtain decompositions of complete continuity and decompositions of perfect continuity. throughout this paper, (u,τ) and (v,η)(or simply u and v ) represent topological spaces on which no separation axioms are assumed unless explicitly stated and f:(u,τ)→(v,η) or simply f:u → v denotes a function f of a topological space u into a topological space v. let n ⊆ u, then cl(n) = ∩{f: n ⊆ f and fc ∈ τ} is the closure of n and int(n) = ∪{o: o ⊆ n and o ∈ τ} is the interior of n. 2 preliminaries definition 2.1. a set m ⊆ u is called δ-closed[velicko, 1968] if m = δ-cl(m) where δ-cl(m)={p∈u:int(cl(g))∩m 6=φ,g∈τ and p∈g }. definition 2.2. a set m ⊆ u is called (1) e-open[ekici, 2008c] if m ⊆ cl(δ-int(m))∪int(δ-cl(m)) and e-closed if cl(δint(m))∩int(δ-cl(m))⊆m. (2) a-open[ekici, 2008d] if m ⊆ int(cl(δ-int(m))) and a-closed if cl(int(δ-cl(m)))⊆m. (3)e∗-open[ekici, 2009] if m ⊆ cl(int(δ-cl(m))) and e∗-closed if int(cl(δ-int(m)))⊆m . (4)δ-semiopen[park et al., 1997] if m⊆cl(δ-int(m))) and δ-semiclosed if int(δcl(m))⊆m). (5)δ-preopen[raychaudhuri and mukherjee, 1993] if m⊆int(δ-cl(m)) and δ-preclosed if cl(δ-int(m))⊆m. (6)regular-open[stone, 1937] if m = int(cl(m)) and regular-closed if m=cl(int(m)). definition 2.3. [ekici, 2008b] a subet m of a space u is said to be a δ-dense set if δ-cl(m)=u. the class of open(resp,closed, regular open,δ-preopen, δ-semiopen, e∗-open and clopen) sets of (u,τ) is denoted by o(u) (resp,c(u), ro(u), δpo(u), δso(u), e∗o(u) and co(u)). 214 on δ-preregular e∗-open sets in topological spaces theorem 2.1. [raychaudhuri and mukherjee, 1993] let m be a subset of a space (u,τ), then δ-pcl(m)=m∪cl(δ-int(m)) and δ-pint(m)=m∩int(δ-cl(m)). theorem 2.2. [ekici, 2009]let m be a subset of a space (u,τ),then: (i) e∗-cl(m) = m∪ int(cl(δ-int(m)) and e∗-int(m) = m∩cl(int(δ-cl(m)) (ii) int(cl(δ-int(m))=e∗-cl(δ-int(m))=δ-int(e∗-cl(m)). theorem 2.3. let m be a subset of a space (u,τ),then: (i)δ-pint(e∗-cl(m))=e∗-cl(m)∩ int(δ-cl(m)). (ii)δ-pint(e∗-cl(m))=δ-pint(m)∪ int(cl(δ-int(m)). (iii)δ-pint(e∗-cl(m))=δ-pint(m)∪ e∗-cl(δ-int(m)) (iv)δ-pint(e∗-cl(m))=δ-pint(m)∪ δ-int(e∗-cl(m)). (v) δ-pint(e∗-cl(m)) = (m∩int(δ-cl(m))∪ int(cl(δ-int(m)) lemma 2.1. [benchalli et al., 2017]for a subset m of a space (u,τ),the following are equivalent: (a)m is clopen; (b)m is δ-open and δ-closed; (c)m is regular-open and regular-closed. definition 2.4. [kohli and singh, 2009] a space (u,τ) is called δ-partition if δo(u)=c(u). definition 2.5. [caldas and jafari, 2016] a space (u,τ) is a δ-door space if every subset of u is δ-open or δ-closed. theorem 2.4. [caldas and jafari, 2016] if (u,τ) is a δ-door space, then every δ-preopen set in (u,τ) is δ-open. 3 δ-preregular e∗-open sets in topological spaces definition 3.1. a subset n of a space (u,τ) is said to be δ-preregular e∗-open(briefly δpe∗-open) if n =δ-pint(e∗-cl(n)). the complement of a δ-preregular e∗-open is called a δ-preregular e∗-closed(briefly δpe∗-closed) set. clearly, n is δpe∗-closed if and only if n = δ-pcl(e∗-int(n)) the class of δpe∗-open (resp,δpe∗-closed) sets of (u,τ) will be denoted by δpe∗o(u)(resp,δpe∗c(u)). theorem 3.1. let (u,τ) be a topological space and m, n ⊆ u. then the following hold: (i) if m ⊆ n, then δ-pint(e∗-cl(m) ⊆ δ-pint(e∗-cl(n)). (ii) if m ∈δpo(u), then m ⊆ δ-pint(e∗-cl(m)). (iii) if m ∈e∗c(u), then e∗-cl(δ-pint(m)) ⊆ m. 215 jagadeesh b.toranagatti (iv) δ-pint(e∗-cl(n)) is δpe∗-open (v) if m ∈e∗c(u), then δ-pint(m) is δpe∗-open.. proof:(i)obvious. (ii) let m ∈ δpo(u). as m ⊆ e∗-cl(m),then m ⊆ δ-pint(e∗-pcl(m). (iii) let m ∈ e∗c(u). since δ-pint(m) ⊆ m, then e∗-cl(δ-pint(m)) ⊆ m. (iv) we have δ-pint(e∗-cl(δ-pint(e∗-cl(m)) ⊆ δ-pint(e∗-cl(e∗-cl(m)) = δ-pint(e∗-cl(m) and δ-pint(e∗-cl(δ-pint(e∗-cl(m))) ⊇ δ-pint(δ-pint(e∗-cl(m)) = δ-pint(e∗-cl(m). hence δ-pint(e∗-cl(δ-pint(e∗-cl(m))) = δ-pint(e∗-cl(m). (v) suppose that m ∈ e∗c(u). by (i), δ-pint(e∗-cl(δ-pint(m))⊆ δ-pint(e∗-cl(m)=δ-pint(m). on the other hand, we have δ-pint(m) ⊆ e∗-cl(δ-pint(m) so that δ-pint(m) ⊆ δ-pint(e∗-cl(δ-pint(m)). therefore δ-pint(e∗-cl(δ-pint(m))=δ-pint(m). this shows that δ-pint(m) is δpe∗-open. theorem 3.2. (i)every δpe∗-open set is δ-preopen(hence e-open,e∗-open). (ii)every δpe∗-open set is e∗-closed.. proof: (i)let m be δpe∗-open,then by theorem 2.3(i), δ-pint(e∗-cl(m))=e∗-cl(m)∩int(δ-cl(m). therefore, m ⊆int(δ-cl(m), m is δ-preopen. (ii)let n be δpe∗-open.by theorem 2.3(ii), n =δ-pint(n)∪int(cl(δ-int(n))). therefore,int(cl(δ-int(n)))⊆n. thus n is e∗-closed. remark 3.1. by the following example,we show that every δ-preopen(resp,e∗closed) set need not be a δpe∗-open set example 3.1. let u = {a,b,c,d} and τ = {u, φ, {a}, {b}, {a,b}, {a,c}, {a,b,c}}. then {a,b,c} is a δ-preopen set but {a,b,c} /∈ δpe∗o(u) and {d} is e∗-closed but {d} /∈ δpe∗o(u)it is not δpe∗-open corolary 3.1. for a topological space (u,τ), we have δ-po(u)∩δ-pc(u) ⊆ δpe∗o(u)⊆ e∗o(u)∩e∗c(u). proof: obvious. the converse inclusions in the above corollary need not be true as seen from the following example example 3.2. let (u,τ) as in example 3.1,then {b} is δpe∗-open but it is not δ-preclopen. moreover, {a,d} is e∗-clopen but not δpe∗-open remark 3.2. the notions of δpe∗-open sets and δ-open sets (hence a-open sets, δ-semiopen sets, δ∗-sets) are independent of each other. 216 on δ-preregular e∗-open sets in topological spaces example 3.3. consider (u,τ) as in example 3.1.the set {a} is δpe∗-open but it is not δ∗-set. moreover, {a,b,c} is δ-open but not δpe∗-open theorem 3.3. in a δ-partition space (u,τ), a subset m of u is δpe∗-open if and only if it is δ-preopen. proof: necessity:it follows from theorem 3.2(i) . sufficiency:let n be δ-preopen. since (u,τ) is δ-partition and by theorem 2.3(ii), we have δ-pint(e∗-cl(m)) = δ-pint(m)∪ int(cl(δ-int(m)) = m ∪ int(cl(cl(m)) = m ∪int(cl(m) = m ∪δ-int(cl(m) = m ∪δ-int(δ-int(m) = m ∪δ-int(m) = m therefore, δ-pint(e∗-cl(m)) = m.hence m is δpe∗-open. theorem 3.4. a subset n ⊆ u is δpe∗-open if and only if n is e∗-closed and δpreopen. proof: necessity:it follows from theorem 3.2. sufficiency:let n be both e∗-closed and δ-preopen. then n = e∗-cl(n) and n = δ-pint(n). therefore, δ-pint(e∗-cl(n)) = δ-pint(n) = n. hence n is δpe∗-open. remark 3.3. the class of δpe∗-open sets is not closed under finite union as well as finite intersection. it will be shown in the following example. example 3.4. consider (u,τ) as in example 3.1. let a = {a,c} and b = {b,c},the a and b are δpe∗-open sets but a ∪ b = {a,b,c} /∈ δpe∗o(u). moreover,c = {a,b,d} and d = {b,c,d} are δpe∗-open sets but c ∩ d = {b,d} /∈ δpe∗o(u). theorem 3.5. for a subset m of a space (u,τ),the following are equivalent: (i) m is δpe∗-open. (ii) m = e∗-cl(m)∩ int(δ-cl(m)). (iii) m = δ-pint(m)∪ int(cl(δ-int(m)). (iv)m = δ-pint(m)∪ e∗-cl(δ-int(m)) (v) m = δ-pint(m)∪ δ-int(e∗-cl(m)). (vi) m = (m∩int(δ-cl(m))∪ int(cl(δ-int(m)). proof:it follows from theorem 2.3 theorem 3.6. in any space (u,τ) , the empty set is the only subset which is nowhere δ-dense and δpe∗-open. proof: suppose m is nowhere δ-dense and δpe∗-open. then by theorem 2.3(i), m = δ-pint(e∗-cl(m)) =e∗-cl(m)∩ int(δ-cl(m)= e∗-cl(m)∩φ = φ. 217 jagadeesh b.toranagatti lemma 3.1. if (u,τ) is a δ-door space, then any finite intersection of δ-preopen sets is δ-preopen. proof:obvious since δo(x) is closed under finite intersection. theorem 3.7. if (u,τ) is a δ-door space, then any finite intersection of δpe∗-open sets is δpe∗-open. proof:let {ai:i=1,2,...,n} be a finite family of δpe∗-open. since the space (u,τ) is δ-door, then by lemma 3.1, we have n⋂ i=n ai∈δpo(u). by theorem 3.1(ii), n⋂ i=n ai⊆ δ-pint(e∗-cl( n⋂ i=n ai). for each i, we have n⋂ i=n ai⊆ ai and thus δ-pint(e∗-cl( n⋂ i=n ai)⊆ δ-pint(e∗-cl(ai) = ai. therefore, δ-pint(e∗-cl( n⋂ i=n ai)⊆ n⋂ i=n ai. lemma 3.2. if a subset m of a space (u,τ) is regular open,then m = int(cl(m)=int(δ-cl(m)). theorem 3.8. every regular open set is δpe∗-open. proof: let m be regular open. then m=int(cl(m))=int(δ-cl(m)). by theorem 2.6(i), δ-pint(e∗-cl(m)) = e∗-cl(m)∩int(δ-cl(m))=e∗-cl(m)∩m=m. this shows that m is δpe∗-open. definition 3.2. a subset m of a space (u,τ) is called δ∗-set if int(δ-cl(m))⊆cl(δ-int(m)) theorem 3.9. (i) every δ-semiopen set is δ∗-set. (ii)every δ-semiclosed set is δ∗-set. proof:clear definition 3.3. a subset m of a space (u,τ) is called b∗-open if m = cl(δ-int(m))∪ int(δ-cl(m)). b∗-closed if m = cl(δ-int(m))∩ int(δ-cl(m)) theorem 3.10. a subset m of a space (u,τ) is regular open if and only if it is b∗-closed. proof:let m be regular open. then by lemma 3.2, m = int(cl(m)=int(δ-cl(m)). since every regular open set is δ-open, we have cl(δ-int(m))∩ int(δ-cl(m)) = cl(m)∩ m = m. hence a is b∗-closed. conversely, let m be b∗-closed.then int(cl(δ-int(m))⊆int(δ-cl(δ-int(m))⊆ cl(δint(m))∩ int(δ-cl(m))=m. by definition 3.3, we have m ⊆ int(δ-cl(m)) ⊆ int(δcl(cl(δ-int(m))) = int(cl(cl(δ-int(m))) = int(cl(δ-int(m))). therefore, m = int(cl(δ-int(m)). now, int(cl(m)) = int(cl(int(cl(δ-int(m))) = int(cl(δint(m)) = m. hence m is regular open. 218 on δ-preregular e∗-open sets in topological spaces theorem 3.11. (i) every b∗-closed set is δ-preopen. (ii)every b∗-closed set is δ-semiopen. (iii)every b∗-closed set is δpe∗-open. proof:(i) and (ii) are obvious (iii)let m be b∗-closed,then we have m = int(cl(δ-int(m)). then δ-pint(e∗-cl(m)) = δ-pint(m) ∪ int(cl(δ-int(m)) = δ-pint(m) ∪ m = m.hence m is δpe∗-open remark 3.4. the above discussions can be summarized in the following diagram: diagram regular open −→ δ-open −→ a-open −→ δ-semiopen −→ δ∗-set l ↓ ↓ b∗-closed −→ δpe∗-open −→ δ-preopen −→ e-open −→ e∗-open theorem 3.12. for a subset m of a space (u,τ), the following are equivalent: (i) m is regular open; (ii) m is δpe∗-open and δ-open; (iii) m is δpe∗-open and a-open; (iv) m is δpe∗-open and δ-semiopen; (v) m is δpe∗-open and δ∗-set. proof: (i) −→(ii)−→(iii)−→(iv)−→(v):follows from the above diagram (v)−→(i):let m be δpe∗-open and δ∗-set.then int(δ-cl(m))⊆cl(δ-int(m)) and int(δ-cl(m))⊆ int(cl(δ-int(m))⊆int(δ-cl(δ-int(m))⊆int(δ-cl(m)). therefore we have int(δ-cl(m))=int(cl(δ-int(m)). since m is δpe∗-open, m = δ-pint(δ-pcl(m)) =(m∪int(cl(δ-int(m))∩int(δ-cl(m)) =int(δ-cl(m)∩int(δ-cl(m)) =int(δ-cl(m)). therefore m =int(δ-cl(m))=int(cl(m)) and hence m is regular open. theorem 3.13. for a subset m of a space (u,τ), the following are equivalent: (i) m is regular open. (ii) m is δpe∗-open and δ-semiclosed. (iii) m is e∗-closed and a-open. proof: (i)−→(ii):it follows from theorem 3.8 (ii)−→(i):let m be δpe∗-open and δ-semiclosed. since every δ-semiclosed set is δ∗-set. hence by theorem 3.12(v), m is regular open. (ii) −→(iii):clear (i)←→(iii):it is shown in theorem 3 [ekici, 2008b] corolary 3.2. for a subset m of a space (u,τ), the following are equivalent: (i) m is regular open; (ii) m is δpe∗-open and δ-open; 219 jagadeesh b.toranagatti (iii) m is δpe∗-open and a-open; (iv) m is δpe∗-open and δ-semiopen; (v) m is δpe∗-open and δ∗-set;. (vi) m is δpe∗-open and δ-semiclosed; (vii) m is e∗-closed and a-open; (viii) m is b∗-closed. theorem 3.14. for a subset m of a space (u,τ), the following are equivalent: (i) m is clopen; (ii) m is δ-open and δ-closed; (iii) m is regular open and regular closed; (iv) m is δpe∗-open and δ-closed. proof: (i) ←→(ii)←→(iii):follows from lemma 2.1 (iii)−→(iv). it follows from theorem 3.8 (iv)−→(ii)let m be δpe∗-open and δ-closed.by theorem 2.3(i) , we have n = e∗-cl(n) ∩ int(δ-cl(n)) = e∗-cl(n) ∩δ-int(δ-cl(n))=δ-pcl(n)∩δ-int(n)=δ-int(n). therefore m is δ-open. 4 decompositions of complete continuity in this section, the notion of regular δ-preopen continuity is introduced and the decompositions of complete continuity are discussed. definition 4.1. a function f:(u,τ)→(v,σ) is said to be (i) δpe∗-continuous if the inverse image of every open subset of (v,σ) is δpe∗-open set in (u,τ). (ii)perfectly continuous[noiri, 1984] (resp,e-continuous[ekici, 2008c], e∗-continuous[ekici, 2009], δ-almost continuous[raychaudhuri and mukherjee, 1993], δ∗-continuous, contra-super-continuous[jafari and noiri, 1999], completely continuous[arya and gupta, 1974], rc-continuous[dontchev and noiri, 1998], super-continuous[munshi and bassan, 1982], contra continuous[dontchev, 1996], a-continuous[ekici, 2008d], δ-semicontinuous[noiri, 2003], contra e∗-continuous[ekici, 2008a], contra δsemicontinuous[ekici, 2004], contra b∗-continuous) if the inverse image of every open subset of (v,σ) is clopen (resp,e-open,e∗-open,δ-preopen,δ∗-set, δ-closed, regular open, regular closed, δ-open, closed, a-open, δ-semiopen, e∗-closed, δsemiclosed, b∗-closed) set in (u,τ) by theorems 3.9 and 3.11, we obtain the following theorem. theorem 4.1. (i) every contra b∗-continuous set is δ-almost continuous. (ii)every contra b∗-continuous set is δ-semicontinuous 220 on δ-preregular e∗-open sets in topological spaces (iii)every contra b∗-continuous set is δpe∗-continuous. (iv) every δ-semicontinuous set is δ∗-continuous. (v)every contra δ-semicontinuous is δ∗-continuous. remark 4.1. by diagram i, we have the following diagram: diagram ii c.cont. −→ s.cont. −→ a.cont. −→ δs.cont. −→ δ∗.cont. l ↓ ↓ cb∗.cont.−→δpe∗.cont.−→δp.cont. −→ e.cont. −→ e∗.cont. where c.cont.=completely continuity, s.cont.=super continuity, a.cont.=a-continuity, δs.cont.=δ-semicontinuity, δ∗.cont.=δ∗-continuity, cb∗.cont.=contra b∗-continuity, δpe∗.cont.=δ-preregular e∗-continuity, δp.cont.=δ-precontinuity, e.cont.=e-continuity, e∗.cont.=e∗-continuity theorem 4.2. for a function f:(u,τ)→(v,η), the following are equivalent: (i) f is completely continuous; (ii)f is δpe∗-continuous and super continuous; (iii)f is δpe∗-continuous and a-continuous; (iv) f is contra e∗-continuous and a-continuous; (v)f is δpe∗-continuous and δ-semicontinuous; (vi)f is δpe∗-continuous and contra δ-semicontinuous; (vii)f is δpe∗-continuous and δ∗-continuous; (viii) f is contra b∗-continuous. remark 4.2. (i) δpe∗-continuity and super-continuity(hence a-continuity,δ-semicontinuity, δ∗∗-continuity) are independent notions. (ii) δpe∗-continuity and contra δ-semicontinuity are independent notions. example 4.1. let (u,τ) be a space as in example 3.1 and let η = {u, φ, {a}, {b}, {a,b}, {a,b,c}} (i) define f:(u,τ) → (u,η) by f(a) = f(c) = a , f(b) = b and f(d) = d. clearly f is super-continuous but for {a,b}∈ o(v), f−1({a,b}) = {a,b,c} /∈ δpe∗o(u). therefore f is not δpe∗-continuous. define g:(u,τ) → (u,η) by g(a) = b, g(b) = g(c) = g(d) = a.then g is δpe∗continuous but for {a}∈ o(v), g−1({a}) = {b,c,d} /∈ q∗o(u). therefore g is not q∗-continuous. (ii)define f:(u,τ) → (u,η) by f(a) = f(c) = f(d) = b and f(b) = a. clearly f is δ-semiregular-continuous but for {b}∈ o(v), f−1({b}) = {a,c,d} /∈ δpe∗po(u). therefore f is not δpe∗-continuous. define g:(u,τ) → (u,η) by g(a) = g(b)=g(d)=a,g(c) = b.then g is δpe∗-continuous 221 jagadeesh b.toranagatti but for {a} ∈ o(v), g−1({a}) = {a,b,d} /∈ δsc(u).therefore g is not contra δsemicontinuous. 5 decompositions of perfectly continuity in this section, the decompositions of perfectly continuity are obtained. theorem 5.1. for a function f:(u,τ) → (u,η), the following are equivalent: (i) f is perfectly continuous; (ii) f is super continuous and contra super continuous; (iii) f is completely continuous and rc-continuous; (iv) f is δpe∗-continuous and contra super continuous. proof: it is a direct consequence of theorem 3.14 remark 5.1. as shown by the following examples,δpe∗-continuity and contra super continuity are independent of each other. example 5.1. consider (u,τ) as in example 3.1 and (u,η) as in example 4.1. define f: (u,τ) → (u,η) by f(a) = f(c) = f(d) = a and f(b) = c. then f is contra super continuous but it is not δpe∗-continuous since {a}∈ o(v), f−1({a}) = {a,c,d} /∈ δpe∗o(u). define g: (u,τ) → (u,η) by g(a) =b, g(b) = g(c) = g(d) = a.then g is δpe∗-continuous but it is not contra super continuous since {a}∈ o(v), g−1({a}) = {b,c,d} /∈ δc(u). 6 conclusions: the notions of sets and functions in topological spaces and fuzzy topological spaces are extensively developed and used in many engineering problems, information systems, particle physics, computational topology and mathematical sciences. by researching generalizations of closed sets, some new continuity have been founded and they turn out to be useful in the study of digital topology. therefore, δpe∗-continuous functions defined by δpe∗-open sets will have many possibilities of applications in digital topology and computer graphics. references shashi prabha arya and ranjana gupta. on strongly continuous 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han park, bu young lee, and mj son. on δ-semiopen sets in topological space. j. indian acad. math, 19(1):59–67, 1997. s. raychaudhuri and m. n. mukherjee. on δ-almost continuity and δ-preopen sets. bull. inst. math. acad. sinica, 21:357–366, 1993. marshall harvey stone. applications of the theory of boolean rings to general topology. transactions of the american mathematical society, 41(3):375–481, 1937. n. v. velicko. h-closed topological spaces. amer. math. soc.transl., 78:103–118, 1968. 224 ratio mathematica volume 48, 2023 approximation and moduli of continuity for a function belonging to hölder’s class hα[0,1) and solving lane-emden differential equation by boubaker wavelet technique shyam lal* swatantra yadav† abstract in this paper, boubaker wavelet is considered. the boubaker wavelets are orthonormal. the series of this wavelet is verified for the function f(t) = t ∀ t ∈[0,1). the convergence analysis of solution function of lane-emden differential equation has been studied. new boubaker wavelet estimator e2k,m(f) for the approximation of solution function f belong to hölder’s class hα[0,1) of order 0 < α ≤ 1, has been developed. furthermore, the moduli of continuity of ( f − s2k,m(f) ) of solution function f of lane-emden differential equation has been introduced and it has been estimated for solution function f∈ hα[0,1) class. these estimator and moduli of continuity are new and best possible in wavelet analysis. boubaker wavelet collocation method has been proposed to solve lane-emden differential equations with unknown boubaker coefficients. in this process, lane-emden differential equations are reduced into a system of algebraic equations and these equations are solved by collocation method. three lane-emden type equations are solved to demonstrate the applicability of the proposed method. the solutions obtained by the proposed method are compared with their exact solutions. the *department of mathematics, institute of science, banaras hindu university, varanasi 221005, india; shyam lal@rediffmail.com. †department of mathematics, institute of science, banaras hindu university, varanasi 221005, india; swatantrayadavbhu@gmail.com. shyam lal and swatantra yadav absolute errors are negligible. thus, this shows that the method described in this paper is applicable and accurate. keywords: boubaker wavelet, boubaker polynomial, boubaker wavelet approximation, moduli of continuity, convergence analysis, collocation method, lane-emden differential equations. 2020 ams subject classifications:42c40, 65t60, 45g10, 45b05.1 1 introduction wavelet theory is a newly emerging area of research in a mathematical sciences. it has applications in engineering disciplines; such as signal analysis for wave representation and segmentation etc. wavelets allow the accurate representation of a variety of signals and operators. wavelets are assumed as a basis function {ψn,m(·)} continuously in time domain. special feature of wavelets basis is that all functions {ψn,m(·)} are constructed from a single mother wavelet ψ(·) which is small pulse. many practical and physical problems in the field of science and engineering are formulated as intial and boundry value problems. approximation of functions by the wavelet method has been discussed by many researchers like devorce[2], morlet[5], meyer[4], debnath[3], lal and satish[11]. the wavelet functions have been applied for finding approximate solutions for some problems arising in numerous branches of science and engineering. in this paper, boubaker wavelet has been studied. this wavelet is defined by the orthogonal boubaker polynomials. it has several interesting and useful properties. the main aims of present paper are as follows: (i) to define boubaker wavelet and to verify boubaker wavelet series by examples. (ii) to study the properties of boubaker wavelet coefficient in expansion of characteristic function. (iii) to estimate the approximation of solution function f of lane-emden differential equations belonging to hölder’s class hα[0,1) by the boubaker wavelet series. (iv) to estimate the moduli of continuity of ( f − s2k,m(f) ) of solution function f of lane-emden differential equations belonging to hölder’s class hα[0,1) by the boubaker wavelet series. (v) to solve lane-emden differential equation by boubaker wavelet series by 1received on january 28, 2023. accepted on july 15, 2023. published on august 1, 2023. doi: 10.23755/rm.v39i0.1092. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. approximation and moduli of continuity... collocation method. the paper is organized as follows: in section-2, boubaker wavelet,boubaker wavelet approximation and moduli of continuity of function of hölder’s class hα[0,1) are defined. the boubaker wavelet series is verified by example. in section-3 ,theorem concerning the convergence analysis of boubaker wavelet has been discussed. in section-4, theorem concerning boubaker wavelet coefficient in the expansion of characteristic function and approximation in hα[0,1) have been obtained. in section-5, theorem concerning the moduli of continuity of ( f − s2k,m(f) ) have been determined. in section-6, boubaker wavelet method for solution of differential equation has been discussed . in section-7, lane-emden differential equations have been solved using boubaker wavelet series by collocation method. finally, the main conclusions are summarized in section-8. 2 definitions and preliminaries 2.1 boubaker wavelet wavelet functions are constructed from dilation and translation of a definite function, named mother wavelet ψ . ψb,a may be defined as ψb,a(t) = |a|−1/2ψ ( t − b a ) a,b ∈ ir,a ̸= 0 (daubechies[6]) (1) where a and b are dilation and translation parametres respectively while t is normalized time. by taking a = 1 2k , b = n 2k and ψ(t) = √ (2m + 1) (2m!) (m!)2 bm(t), where bm(t) is boubaker polynomial of degree m, in equation (1), it reduces into form ψ(b)n,m(t) = √ (2m + 1) (2m!) (m!)2 2 k 2 bm(2 kt − n). in precise, ψ(b)n,m(t) = {√ (2m + 1) (2m!) (m!)2 2 k 2 bm(2 kt − n), if n 2k ⩽ t < n+1 2k , 0, otherwise. where n = 0,1,2...,2k − 1, k = 0,1,2,3, ..., while m represents the order of orthogonal boubaker polynomial (shiralashetti et al.[8]). it has four parameters m,n,k & t. the orthogonal boubaker polynomial bm(t) of order m satisfies the shyam lal and swatantra yadav following conditions: b0(t) = 1, b1(t) = 1 2 (2t − 1),b2(t) = 1 6 (6t2 − 6t + 1) ; b3(t) = 1 20 (20t3 − 30t2 + 12t − 1), b4(t) = 1 70 (70t4 − 140t3 + 90t2 − 20t + 1) ; b5(t) = 1 252 (252t5 − 630t4 + 560t3 − 210t2 + 30t − 1); b6(t) = 1 924 (924t6 − 2772t5 + 3150t4 − 1680t3 + 420t2 − 42t + 1); 2.2 boubaker wavelet approximation a function f ∈ l2[0,1) may be expanded in boubaker wavelet series as f(t) = ∞∑ n=0 ∞∑ m=0 cn,mψ (b) n,m(t), (2) where the coefficients cn,m are given by cn,m =< f(t),ψ (b) n,m(t) > . (3) if the series (2) is truncated, then (s2k,mf)(t) = 2k−1∑ n=0 ∞∑ m=0 cn,mψ (b) n,m(t) = c t ψ(t) (4) where (s2k,mf) is the (2 k,m)th partial sum of series (2) and c, ψ(t) are 2km ×1 matrices given by c = [c0,0,c0,1, ...,c0,m − 1,c1,0, ...,c1,m−1, ...,c2k−1,0, ...,c2k−1,m−1]t (5) and ψ(t) = [ψ0,0(t),ψ0,1(t), ...,ψ0,m−1(t),ψ1,0(t),ψ1,m−1(t), ...,ψ2k−1,0(t), ...,ψ2k−1,m−1] t . (6) the boubaker wavelet approximation of f by (s2k,mf) under norm || ||2, denoted by e2k,m(f), is defined by e2k,m(f) = min||f − (s2k,m)||2 (zygmund)[10]). if e2k,m(f) → 0 as k → ∞, m→∞,then e2k,m(f) is best approximation of f order (2k,m) (zygmund[10]). approximation and moduli of continuity... 2.3 moduli of continuity the moduli of continuity of a function f ∈ l2[0,1) is defined as w(f,δ) = sup 0≤h≤δ ||f(· + h) − f(·)||2 = sup 0≤h≤δ ( ∫ 1 0 |f(t + h) − f(t)|2dt )1 2 it is remarkable to note that w(f,δ) is a non-decreasing function of δ and w(f,δ) → 0 as δ → 0 , (chui [1]). 2.4 function of hölder’s class hα[0,1) a function f ∈ hα[0,1) if f is continuous and satisfies the inequality f(x) − f(y) = o(|x − y|α),∀x,y ∈ [0,1) and 0 < α ≤ 1 (das)[9]). 2.5 example the example of this section illustrates the validity of the boubaker wavelet series as follows: consider the function f : [0,1) → r defined by f(t) = t ∀ t ∈[0,1). let f(t) = ∞∑ n=0 ∞∑ m=0 cn,mψ (b) n,m(t). (7) cn,m = < f(t),ψ (b) n,m(t) >= ∫ n+1 2k n 2k f(t)ψ(b)n,m(t)dt = ∫ n+1 2k n 2k t √ (2m + 1) (2m!) (m!)2 2 k 2 bm(2 kt − n) dt = √ (2m + 1) (2m!) (m!)2 2 k 2 ∫ 1 0 v + n 2k bm(v) dv 2k , 2kt − n = v = √ (2m + 1) (2m!) (m!)2 1 2 3k 2 ∫ 1 0 (v + n)bm(v)dv. by above expansion ,taking m = 0 , cn,0 = 1 2 3k 2 ∫ 1 0 (v + n)b0(v)dv = 1 2 3k 2 ∫ 1 0 (v + n)dv = 2n + 1 2 3k+2 2 . (8) shyam lal and swatantra yadav next cn,1 = 2 √ 3 2 3k 2 ∫ 1 0 (v + n)b1(v)dv = √ 3 2 3k 2 ∫ 1 0 (v + n)(2v − 1)dv = √ 3 3.2 3k+2 2 . (9) cn,2 = 6 √ 5 2 3k 2 ∫ 1 0 (v + n)b2(v)dv = √ 5 2 3k 2 ∫ 1 0 (v + n)(6v2 − 6v + 1)dv = 0 for, m ≥ 2 cn,m = √ (2m + 1) (2m!) (m!)2 1 2 3k 2 ∫ 1 0 (v + n)bm(v)dv = √ (2m + 1) (2m!) (m!)2 1 2 3k 2 ( ∫ 1 0 vbm(v)dv + ∫ 1 0 nbm(v)dv ) = √ (2m + 1) (2m!) (m!)2 1 2 3k 2 ( ∫ 1 0 1 2 (2v − 1)bm(v)dv + ∫ 1 0 1 2 bm(v)dv + ∫ 1 0 nbm(v)dv ) = √ (2m + 1) (2m!) (m!)2 1 2 3k 2 ( ∫ 1 0 b1(v)bm(v)dv + 1 2 ∫ 1 0 b0(v)bm(v)dv + n ∫ 1 0 b0(v)bm(v)dv ) = √ (2m + 1) (2m!) (m!)2 1 2 3k 2 (0 + 0 + 0) = 0 ,bm(v) is orthogonal . cn,m = 0 ∀n ≥ 2k, by defintion of ψ(b)n,m. then , f(t) = 2k−1∑ n=0 cn,0ψ (b) n,0 (t) + 2k−1∑ n=0 cn,1ψ (b) n,1 (t) + ∞∑ n=2k cn,mψ (b) n,m(t), (10) next, ||f||22 = < f,f > = < ∞∑ n=0 ∞∑ m=0 cn,mψ (b) n,m(t), ∞∑ n=0 ∞∑ m=0 cn,mψ (b) n,m(t) > = 〈 2k−1∑ n=0 cn,0ψ (b) n,0 (t) + 2k−1∑ n=0 cn,1ψ (b) n,1 (t) + ∞∑ n=2k cn,mψ (b) n,m(t), 2k−1∑ n=0 cn,0ψ (b) n,0 (t) + 2k−1∑ n=0 cn,1ψ (b) n,1 (t) + ∞∑ n=2k cn,mψ (b) n,m(t) 〉 = 2k−1∑ n=0 c2n,0||ψ (b) n,0 || 2 2 + 2k−1∑ n=0 c2n,1||ψ (b) n,1 || 2 2 + ∞∑ n=2k c2n,m||ψ (b) n,m|| 2 2, approximation and moduli of continuity... = 2k−1∑ n=0 c2n,0 + 2k−1∑ n=0 c2n,1 + 0,{ψn,m}n,m∈z being orthonormal = 2k−1∑ n=0 (2n + 1)2 2(3k+2) + 2k−1∑ n=0 1 3.23k+2 = 1 3 , by eqns (8) and (9). also, ||f||22 = < f,f >= ∫ 1 0 |f(t)|2dt = ∫ 1 0 t2dt = 1 3 . hence, the boubaker wavelet expansion (7) is verified for f(t) = t. 3 convergence analysis in this section, the convergence analysis of solution of lane-emden differential equation has been studied. 3.1 theorem if f be a exact solution of of lane-emden differential equation and its boubaker wavelet series is f(·) = ∞∑ n=0 ∞∑ m=0 cn,mψ (b) n,m(·) (11) then its (2k,m)th partial sums (s2k,mf)(·) = ∑2k−1 n=0 ∑m−1 m=0 cn,mψ (b) n,m(·) converges to f(·) as m→∞, k→ ∞. proof of theorem 3.1 now, < f,f > = 〈 ∞∑ n=0 ∞∑ m=0 cn,mψ (b) n,m, ∞∑ n ′ =0 ∞∑ m ′ =0 cn′,m′ψ (b) n ′ ,m ′ 〉 = ∞∑ n=0 ∞∑ m=0 ∞∑ n ′ =0 ∞∑ m ′ =0 cn,mcn′,m′ < ψ (b) n,m,ψ (b) n ′ ,m ′ > = ∞∑ n=0 ∞∑ m=0 cn,mcn,m < ψ (b) n,m,ψ (b) n,m > = ∞∑ n=0 ∞∑ m=0 |cn,m|2||ψ(b)n,m|| 2 shyam lal and swatantra yadav = 2k−1∑ n=0 ∞∑ m=0 |cn,m|2||ψ(b)n,m|| 2 + ∞∑ n=2k ∞∑ m=0 |cn,m|2||ψ(b)n,m|| 2 = 2k−1∑ n=0 ∞∑ m=0 |cn,m|2 + 0, by defintion of {ψn,m}n,m∈z 2k−1∑ n=0 ∞∑ m=0 |cn,m|2 = ||f||22 < ∞,f ∈ l 2[0,1). (12) for m > n, using eqn (12), ||(s2k,mf) − (s2k,nf)||22 = ∣∣∣∣ ∣∣∣∣ 2 k−1∑ n=0 m−1∑ m=0 cn,mψ (b) n,m(t) − 2k−1∑ n=0 n−1∑ m=0 cn,mψ (b) n,m(t) ∣∣∣∣ ∣∣∣∣2 2 = ∣∣∣∣ ∣∣∣∣ 2 k−1∑ n=0 m−1∑ m=n cn,mψ (b) n,m(t) ∣∣∣∣ ∣∣∣∣2 2 = 〈 2k−1∑ n=0 m−1∑ m=n cn,mψ (b) n,m(t), 2k−1∑ n=0 m−1∑ m=n cn,mψ (b) n,m(t) 〉 = 2k−1∑ n=0 m−1∑ m=n cn,mcn,m < ψ (b) n,m(t),ψ (b) n,m(t) > = 2k−1∑ n=0 ∞∑ m=0 |cn,m|2||ψ(b)n,m(t)|| 2 = 2k−1∑ n=0 m−1∑ m=n |cn,m|2 → 0 as m → ∞,n → ∞. hence, { (s2k,mf) } m∈n is a cauchy sequence in l 2[0,1), l2[0,1) is a banach space and hence { (s2k,mf) } m∈n converges to a function g(t) ∈ l 2[0,1). now we need to prove that g(t) = f(t). for this < g(t) − f(t),ψ(b)n0,m0(t) > = < g(t),ψ (b) n0,m0 (t) > − < f(t),ψ(b)n0,m0(t) > = < lim m→∞ (s2k,mf)(t), ,ψ (b) n0,m0 (t) > −cn0,m0 = lim m→∞ 2k−1∑ n=0 m−1∑ m=0 cn,m < ψ (b) n,m(t),ψ (b) n0,m0 (t) > −cn0,m0 = cn0,m0 < ψn0,m0(t),ψn0,m0(t) > −cn0,m0 = cn0,m0 − cn0,m0 = 0. approximation and moduli of continuity... thus < g(t) − f(t),ψ(b)n,m(t) > = 0 ∀ n ⩾ n0,m ⩾ m0. then g(t) = f(t). hence, ∑2k−1 n=0 ∑m−1 m=0 cn,mψ (b) n,m(t) converges to f(t) as k → ∞,m → ∞ . 4 approximation analysis in this section, approximation f by (s2k,mf) is estimated as follows. 4.1 theorem let a function f = χ[n0 2k , n0+1 2k ), where n0 is positive integer less than equal to 2k and boubaker wavelet expansion of f is f(t) = 2k−1∑ n=0 ∞∑ m=0 cn,mψ (b) n,m(t) (13) then the coefficients cn,m satisfy cn,m =  o ( ((2m)! √ (2m+1) 2−k/2) (m!)2 ) , if n = n0, 0, n ̸= n0, 4.2 theorem let the solution function f of lane-emden differential equation be a uniformly continuous defined in [0,1) such that |f(t1) − f(t2)| ≤ |t1 − t2|α 1 2m−1m 3 2 , ∀t1, t2 ∈ [0,1),m ≥ 1 (14) and its boubaker wavelet series f(t) = ∞∑ n=0 ∞∑ m=0 cn,mψ (b) n,m(t) (15) having (2k,m)th partial sums (s2k,mf)(t) = 2k−1∑ n=0 m−1∑ m=0 cn,mψ (b) n,m(t) (16) shyam lal and swatantra yadav then boubaker wavelet approximation e2k,m(f) satisfies e2k,m(f) = min||f − (s2k,mf)||2 = o ( 1 2kα √ m ) . proof of theorem 4.1 for f = χ[n0 2k , n0+1 2k ), and cn0,m = < f(t),ψ (b) n0,m (t) > = ∫ n0+1 2k n0 2k f(t)ψ(b)n0,m(t)dt = ∫ n0+1 2k n0 2k χ[n0 2k , n0+1 2k )(t) √ (2m + 1) (2m)! (m!)2 2 k 2 bm(2 kt − n0) dt = √ (2m + 1) (2m)! (m!)2 2 k 2 ∫ n0+1 2k n0 2k bm(2 kt − n0) dt = √ (2m + 1) (2m)! (m!)2 2 k 2 ∫ 1 0 bm(v) dv 2k , 2kt − n0 = v = √ (2m + 1) (2m)! (m!)2 1 2 k 2 ∫ 1 0 bm(v) dv |cn0,m| ≤ √ (2m + 1) (2m)! (m!)2 2− k 2 ∫ 1 0 |bm(v)| dv ≤ √ (2m + 1) (2m)! (m!)2 2− k 2 , ∫ 1 0 |bm(v)|dv ≤ 1. (17) then cn,m =  o ( ((2m)! √ (2m+1) 2−k/2) (m!)2 ) , if n = n0, 0, n ̸= n0, thus, theorem 4.1 is completely established. approximation and moduli of continuity... proof of theorem 4.2 f(t) − (s2k,mf)(t) = 2k−1∑ n=0 ∞∑ m=0 cn,mψ (b) n,m(t) − 2k−1∑ n=0 m−1∑ m=0 cn,mψ (b) n,m(t) = 2k−1∑ n=0 ( m−1∑ m=0 + ∞∑ m=m )cn,mψ (b) n,m(t) − 2k−1∑ n=0 m−1∑ m=0 cn,mψ (b) n,m(t) = 2k−1∑ n=0 m−1∑ m=0 cn,mψ (b) n,m(t) + 2k−1∑ n=0 ∞∑ m=m cn,mψ (b) n,m(t) − 2k−1∑ n=0 m−1∑ m=0 cn,mψ (b) n,m(t) = 2k−1∑ n=0 ∞∑ m=m cn,mψ (b) n,m(t). next cn,m = < f(t),ψ (b) n,m(t) > = ∫ n+1 2k n 2k f(t)ψ(b)n,m(t)dt = ∫ n+1 2k n 2k {f(t) − f( n 2k )}ψ(b)n,m(t)dt + ∫ n+1 2k n 2k f( n 2k )ψ(b)n,m(t)dt = ∫ n+1 2k n 2k {f(t) − f( n 2k )}ψ(b)n,m(t)dt + f( n 2k ) ∫ n+1 2k n 2k ψ(b)n,m(t)dt = ∫ n+1 2k n 2k {f(t) − f( n 2k )}ψ(b)n,m(t)dt , ∫ n+1 2k n 2k ψ(b)n,m(t)dt = 0,m ≥ 1. then |cn,m| ≤ ∫ n+1 2k n 2k |f(t) − f( n 2k )| |ψ(b)n,m(t)|dt = √ (2m + 1) (2m)! (m!)2 2 k 2 ∫ n+1 2k n 2k |f(t) − f( n 2k )| |bm(2kt − n)|dt = √ (2m + 1) (2m)! (m!)2 1 2 k 2 ∫ 1 0 |f( u + n 2k ) − f( n 2k )||bm(u)|du,2kt − n = u ≤ √ (2m + 1) (2m)! (m!)2 1 2 k 2 1 2kα 1 2(m−1)m 3 2 ∫ 1 0 |bm(u)|du shyam lal and swatantra yadav ≤ √ (2m + 1) (2m)! (m!)2 1 2kα+ k 2 1 2(m−1)m 3 2 , ∫ 1 0 |bm(u)|du ≤ 1 ≤ 1 2k(α+ 1 2 )m , (m!)2 (2m)! ⩽ 1 2m−1 . (18) next, ||f − (s2k,mf)||22 = ∫ 1 0 |f(t) − (s2k,mf)(t)|2dt = ∫ 1 0 ( 2k−1∑ n=0 ∞∑ m=m cn,mψ (b) n,m(t) )2 dt = ∫ 1 0 ( 2k−1∑ n=0 ∞∑ m=m c2n,m(ψ (b) n,m(t)) 2 + ∑ ∑ 0⩽ n̸=n′≤ 2k−1 ∑ ∑ m⩽ m ̸=m′≤∞ cn,mcn′,m′ψ (b) n,m(t)ψ (b) n′,m′(t) ) dt = 2k−1∑ n=0 ∞∑ m=m c2n,m ∫ 1 0 (ψ(b)n,m(t)) 2 dt + ∑ ∑ 0⩽ n̸=n′≤ 2k−1 ∑ ∑ m⩽ m̸=m′≤∞ cn,mcn′,m′ ∫ 1 0 ψ(b)n,m(t)ψ (b) n′,m′(t)dt = 2k−1∑ n=0 ∞∑ m=m |cn,m|2||ψ(b)n,m(t)|| 2 2 = 2k−1∑ n=0 ∞∑ m=m |cn,m|2 , {ψn,m}n,m∈z being orthonormal in [0,1) ≤ 2k−1∑ n=0 ∞∑ m=m 1 2k(2α+1)m2 , by eqn (18). = 2k 2k(2α+1) ∞∑ m=m 1 m2 ≤ 2k 2k(2α+1) ( 1 m2 + ∫ ∞ m dm m2 ), by cauchy’s intergal test = 1 22kα ( 1 m2 + 1 m ) ≤ 2 22kαm then e2k,m(f) = min||f − (s2k,mf)||2 ⩽ √ 2 2kα √ m = o( 1 2kα √ m ) approximation and moduli of continuity... thus, theorem 4.2 is completely established. 5 moduli of continuity the moduli of continuity of ( f − (s2k,mf) ) have been determined in this section as follows : 5.1 theorem if the solution function f of lane-emden differential equation satisfies eqns (14), (15) & (16), then moduli of continuity of ( f − (s2k,mf) ) is given by w (( f − (s2k,mf) ) , 1 2k ) = sup 0≤h≤ 1 2k || ( f − (s2k,mf))(· + h) − ( f − (s2k,mf))(·)||2 = o ( 1 2kα √ m ) proof of theorem (5.1) following the proof of theorem (4.2) , ||f − (s2k,mf)||2 = o ( 1 2kα √ m ) . then w (( f − (s2k,mf) ) , 1 2k ) = sup 0≤h≤ 1 2k || ( f − (s2k,mf))(t + h) − ( f − (s2k,mf))(t)||2 ≤ || ( f − (s2k,mf))||2 + || ( f − (s2k,mf))||2 = 2|| ( f − (s2k,mf))||2 = 2.o ( 1 2kα √ m ) . w (( f − (s2k,mf) ) , 1 2k ) = o ( 1 2kα √ m ) 6 boubaker wavelet method for solution of differential equations in this section, the solution of lane-emden differential equations are obtained by applying boubaker wavelet collocation method. shyam lal and swatantra yadav consider the lane-emden differential of the form f ′′ (t) + α t f ′ (t) + f(t) = h(t), where t ∈[0,1) ( wazwaz)[7]). (19) f(0) = a,f ′ (0) = b (20) the solution of any differential equation can be expanded as boubaker wavelet series as follows f(t) = ∞∑ n=0 ∞∑ m=0 cn,mψ (b) n,m(t) now f(t) is approximated by truncated series (s2k,mf)(t) = 2k−1∑ n=0 m−1∑ m=0 cn,mψ (b) n,m(t) (21) then the following residual is obtained by substituting (s2k,mf) from eqn(21) into eqn(19) r(t) = t 2k−1∑ n=0 m−1∑ m=0 cn,mψ ′′(b) n,m (t) + α 2k−1∑ n=0 m−1∑ m=0 cn,mψ ′(b) n,m (t) + t 2k−1∑ n=0 m−1∑ m=0 cn,mψ (b) n,m(t) − t × h(t). the collocation method yields r(ti) = 0, i = 1,2,3, ...,2 km − 2. moreover using the intial conditions eqn (20), 2k−1∑ n=0 m−1∑ m=0 cn,mψ (b) n,m(0) = a, 2k−1∑ n=0 m−1∑ m=0 cn,mψ ′(b) n,m (0) = b. (22) hence 2km system of equations are derived in the unknown coefficients cn,m which can be computed. this procedure is applied for differential equations of heigher order. 7 illustrated examples in this section, three lane-emden differential equations have been solved by using the procedure discussed in previous section-6. illustrated examples are as follows: approximation and moduli of continuity... example (1) consider the following lane-emden differential equation f ′′ (t) + 2 t f ′ (t) + f(t) = 1 + 12t + t3, f(0) = 1, f ′ (0) = 0, 0 ≤ t < 1 (23) the exact solution of eqn (23) is f(t) = t3 + 1. now the differential equation has been solved by applying the procedure described in section-6, using boubaker wavelet method by taking m = 5,k = 0. consider f(t) = 4∑ m=0 c0,mψ (b) 0,m(t) = c0,0ψ (b) 0,0 + c0,1ψ (b) 0,1 + c0,2ψ (b) 0,2 c0,3ψ (b) 0,3 + c0,4ψ (b) 0,4 (24) f(t) = c0,0 + c0,1 √ 3(2t − 1) + c0,2 √ 5(6t2 − 6t + 1) + c0,3 √ 7(20t3 − 30t2 + 12t − 1) + c0,43(70t 4 − 140t3 + 90t2 − 20t + 1) (25) differentiate eqn (25) with respect to t, f ′ (t) = c0,1(2 √ 3) + c0,2 √ 5(12t − 6) + c0,3 √ 7(60t2 − 60t + 12) + c0,43(280t 3 − 420t3 + 180t − 20) (26) f ′′ (t) = c0,2(12 √ 5) + c0,3 √ 7(120t − 60) + c0,43(840t2 − 840t + 180) substitute these values of f(t),f ′ (t) and f ′′ (t) in given differential eqn (23) c0,0 + c0,1 [4√3 t + √ 3(2t − 1) ] + c0,2 [ 12 √ 5 + 2 √ 5 t (12t − 6) + √ 5(6t2 − 6t + 1) ] + c0,3 [√ 7(120t − 60) + 2 √ 7 t (60t2 − 60t + 12) + √ 7(20t3 − 30t2 + 12t − 1) ] + c0,4 [ 3(840t2 − 840t + 180) + 6 t (280t3 − 420t2 + 180t − 20) + 3(70t4 − 140t3 + 90t2 − 20t) + 1 ] = 1 + 12t + t3 (27) using intial condition, f(0) = 1 and f ′ (0) = 0 in eqns (25) and (26) c0,0 − √ 3c0,1 + √ 5c0,2 − √ 7c0,3 + 3c0,4 = 1 (28) 2 √ 3c0,1 − 6 √ 5c0,2 + 12 √ 7c0,3 − 60c0,4 = 0 (29) shyam lal and swatantra yadav now collocate the eqn (27) at t1 = 0.5,t2 = 0.7 and t3 = 0.9, which are obtained by xi = i− 1 2 2km = i− 1 2 5 ,i = 2,4,5 respectively. a system of three linear equations are derived. c0,0 + 13.8564c0,1 + 25.7147c0,2 − 31.7490c0,3 − 88.875c0,4 = 7.125 (30) c0,0 + 10.5902c0,1 + 41.5844c0,2 + 57.79832c0,3 − 21.7675c0,4 = 9.743 (31) c0,0 + 9.0836c0,1 + 51.7127c0,2 + 166.0120c0,3 + 351.9676c0,4 = 12.529 (32) solving these eqns (30),(31) and (32) with (28) and (29) c0,0 = 1.2499999999, c0,1 = 0.2598076211, c0,2 = 0.1118033988 c0,3 = 0.0188982236, c0,4 = −0.0000000000. substitute all these values of c0,0,c0,1,c0,2,c0,3,c0,4 in eqn (25) f(t) = 1.2499999999 + 0.2598076211 √ 3(2t − 1) + 0.1118033988 √ 5(6t2 − 6t + 1) + 0.0188982236 √ 7(20t3 − 30t2 + 12t − 1) − 0.0000000000(70t4 − 140t3 + 90t2 − 20t + 1) (33) comparison of exact and boubaker wavelet solutions are given in table (1) for k = 0,m = 5 . table (1) t exact solution approximate solution absolute error(×10−15) 0.1 1.001000000000000 1.001000000000000 0 0.2 1.008000000000000 1.0080000000000000 0 0.3 1.027000000000000 1.0270000000000000 0 0.4 1.064000000000000 1.0640000000000000 0 0.5 1.125000000000000 1.1250000000000000 0 0.6 1.216000000000000 1.2160000000000000 0 0.7 1.343000000000000 1.3430000000000000 0 0.8 1.512000000000000 1.5120000000000000 0 0.9 1.729000000000000 1.7290000000000000 0.222044604925031 table(1):comparison table of exact and boubaker wavelet solutions. approximation and moduli of continuity... 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 f( t) exact solution boubaker wavelet solution fig.(1):the graphs of boubaker wavelet and exact solutions. example (2) consider the following lane-emden differential equation f ′′ (t)+ 2 t f ′ (t)+f(t) = 6+12t+t2 +t3, y(0) = 0, y ′ (0) = 0, 0 ≤ t < 1 (34) the exact solution of eqn (34) is f(t) = t3 + t2. following the procedure adopted in example (1); f(t) = 4∑ m=0 c0,mψ (b) 0,m(t) = c0,0ψ (b) 0,0 + c0,1ψ (b) 0,1 + c0,2ψ (b) 0,2 + c0,3ψ (b) 0,3 + c0,4ψ (b) 0,4 f(t) = c0,0 + c0,1 √ 3(2t − 1) + c0,2 √ 5(6t2 − 6t + 1) + c0,3 √ 7(20t3 − 30t212t − 1) + c0,43(70t 4 − 140t3 + 90t2 − 20t + 1) (35) f ′ (t) = c0,1(2 √ 3) + c0,2 √ 5(12t − 6) + c0,3 √ 7(60t2 − 60t + 12) + c0,43(280t 3 − 420t3 + 180t − 20) (36) f ′′ (t) = c0,2(12 √ 5) + c0,3 √ 7(120t − 60) + c0,43(840t2 − 840t + 180) shyam lal and swatantra yadav substitute these values of f(t),f ′ (t) and f ′′ (t) in given differential eqn (34) c0,0 + c0,1 [4√3 t + √ 3(2t − 1) ] + c0,2 [ 12 √ 5 + 2 √ 5 t (12t − 6) √ 5(6t2 − 6t + 1) ] + c0,3 [√ 7(120t − 60) + 2 √ 7 t (60t2 − 60t + 12) √ 7(20t3 − 30t2 + 12t − 1) ] + c0,4 [ 3(840t2 − 840t + 180) + 6 t (280t3 − 420t2 + 180t − 20) + 3(70t4 − 140t3 + 90t2 − 20t) + 1 ] = 6 + 12t + t2 + t3 (37) using intial condition, f(0) = 0 and f ′ (0) = 0 in eqns (35) and (36) c0,0 − √ 3c0,1 + √ 5c0,2 − √ 7c0,3 + 3c0,4 = 0 (38) 2 √ 3c0,1 − 6 √ 5c0,2 + 12 √ 7c0,3 − 60c0,4 = 0 (39) now collocating the equations (37) at t1 = 0.5,t2 = 0.7 and t3 = 0.9 , c0,0 + 13.856405c0,1 + 25.71478c0,2 − 31.74901c0,3 − 88.875c0,4 = 12.375 (40) c0,0 + 10.59025c0,1 + 41.58447c0,2 + 57.79832c0,3 − 21.76757c0,4 = 15.233 (41) c0,0 + 9.08364c0,1 + 51.71279c0,2 + 166.01207c0,3 + 351.96766c0,4 = 18.339 (42) solving these eqns (40),(41) and (42) with (38) and (39) , c0,0 = 0.5833333333, c0,1 = 0.5484827557, c0,2 = 0.1863389981 c0,3 = 0.0188982236, c0,4 = −0.0000000000. substitute all these values of c0,0,c0,1,c0,2,c0,3,c0,4 in eqn (35). f(t) = 0.5833333333 + 0.5484827557 √ 3(2t − 1) + 0.1863389981 √ 5(6t2 − 6t + 1) + 0.0188982236 √ 7(20t3 − 30t2 + 12t − 1) − 0.0000000000(70t4 − 140t3 + 90t2 − 20t + 1) (43) comparison of exact and boubaker wavelet solutions are given in table (2) for k = 0,m = 5 . table (2) t exact solution approximate solution absolute error (× 10−15) 0.1 0.011000000000000 0.011000000000000 0.017347234759768 0.2 0.048000000000000 0.048000000000000 0.020816681711722 0.3 0.117000000000000 0.117000000000000 0.027755575615629 0.4 0.224000000000000 0.224000000000000 0 0.5 0.375000000000000 0.375000000000000 0 0.6 0.576000000000000 0.576000000000000 0.222044604925031 0.7 0.833000000000000 0.833000000000000 0.111022302462516 0.8 1.152000000000000 1.152000000000000 -0.222044604925031 0.9 1.539000000000000 1.539000000000000 -0.222044604925031 approximation and moduli of continuity... 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 f( t) exact solution boubaker wavelet solution fig.(2):the graphs of boubaker wavelet and exact solutions. example 3 consider the following lane-emden differential equation f ′′ (t) + 2 t f ′ (t) = 2(2t2 + 3)f(t), f(0) = 1, f ′ (0) = 0, 0 ≤ t < 1 (44) the exact solution of eqn (44) is f(t) = et 2 . following the procedure adopted in example(1); f(t) = 4∑ m=0 c0,mψ (b) 0,m(t) = c0,0ψ (b) 0,0 + c0,1ψ (b) 0,1 + c0,2ψ (b) 0,2 + c0,3ψ (b) 0,3 + c0,4ψ (b) 0,4 f(t) = c0,0 + c0,1 √ 3(2t − 1) + c0,2 √ 5(6t2 − 6t + 1) + c0,3 √ 7(20t3 − 30t2 + 12t − 1) + c0,43(70t 4 − 140t3 + 90t2 − 20t + 1) (45) f ′ (t) = c0,1(2 √ 3) + c0,2 √ 5(12t − 6) + c0,3 √ 7(60t2 − 60t + 12) + c0,43(280t 3 − 420t3 + 180t − 20) (46) f ′′ (t) = c0,2(12 √ 5) + c0,3 √ 7(120t − 60) + c0,43(840t2 − 840t + 180) shyam lal and swatantra yadav substitute these values of f(t),f ′ (t) and f ′′ (t) in given differential eqn (44) − 2(2t2 + 3)c0,0 + c0,1 [4√3 t − 2(2t2 + 3) √ 3(2t − 1) ] + c0,2 [ 12 √ 5 + 2 √ 5 t (12t − 6) − 2(2t2 + 3) √ 5(6t2 − 6t + 1) ] + c0,3 [√ 7(120t − 60) + 2 √ 7 t (60t2 − 60t + 12) + 2(2t2 + 3) √ 7(20t3 − 30t2 + 12t − 1) ] + c0,4 [ 3(840t2 − 840t + 180) + 6 t (280t3 − 420t2 + 180t − 20) − 6(2t2 + 3)(70t4 − 140t3 + 90t2 − 20t) + 1 ] = 0 (47) using intial condition, f(0) = 1 and f ′ (0) = 0 in eqns (45) and (46) c0,0 − √ 3c0,1 + √ 5c0,2 − √ 7c0,3 + 3c0,4 = 1 (48) 2 √ 3c0,1 − 6 √ 5c0,2 + 12 √ 7c0,3 − 60c0,4 = 0 (49) now collocating the equation (47) at t1 = 0.5,t2 = 0.7 and t3 = 0.9, −7c0,0 + 13.85640c0,1 + 34.65905c0,2 − 31.74901c0,3 − 97.8750c0,4 = 0 (50) −7.96c0,0 + 4.38258c0,1 + 46.79361c0,2 + 68.22893c0,3 − 18.73013c0,4 = 0 (51) −9.24c0,0−5.10531c0,1+41.18002c0,2+163.84467+02c0,3+359.12542c0,4 = 0 (52) solving these eqns (50), (51) and (52) with (48) and (49), c0,0 = 1.38821917722, c0,1 = 0.388372764863, c0,2 = 0.158389887721 c0,3 = 0.02903271870, c0,4 = 0.002368327161 substitute all these values of c0,0,c0,1,c0,2,c0,3,c0,4 in eqn (45) f(t) = 1.3882191 + 0.38837276 √ 3(2t − 1) + 0.1583898 √ 5(6t2 − 6t + 1) + 0.02903271870 √ 7(20t3 − 30t2 + 12t − 1) + 0.002368327161(70t4 − 140t3 + 90t2 − 20t + 1) (53) comparison of exact and boubaker wavelet solutions are given in table (3) for k = 0,m = 5 table (3) t exact solution approximate solution absolute error 0.1 1.010050167084168 1.005192015148452 0.004858151935716 0.2 1.040810774192388 1.023531157694065 0.017279616498324 0.3 1.094174283705210 1.060057300954230 0.034116982750980 0.4 1.173510870991810 1.121003955135684 0.052506915856126 0.5 1.284025416687741 1.213798267334502 0.070227149353240 0.6 1.433329414560340 1.347061021536102 0.086268393024238 0.7 1.632316219955379 1.530606638615246 0.101709581340133 0.8 1.896480879304952 1.775443176336035 0.121037702968916 0.9 2.247907986676472 2.093772329351916 0.154135657324556 approximation and moduli of continuity... table(3):comparision table of exact and boubaker wavelet solutions. comparison of exact and boubaker wavelet solutions are given in table (4) for k = 0,m = 6 . table (4) t exact solution approximate solution absolute error 0.1 1.010050167084168 1.000547638755526 0.009502528328642 0.2 1.040810774192388 1.011676778238292 0.029133995954096 0.3 1.094174283705210 1.043773805588921 0.050400478116290 0.4 1.173510870991810 1.103730511655493 0.069780359336317 0.5 1.284025416687741 1.196980758433975 0.087044658253766 0.6 1.433329414560340 1.329537146508637 0.103792268051703 0.7 1.632316219955379 1.510027682492480 0.122288537462899 0.8 1.896480879304952 1.751732446467655 0.144748432837297 0.9 2.247907986676472 2.074620259425891 0.173287727250581 table(4):comparison table of exact and boubaker wavelet solutions. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 1 1.2 1.4 1.6 1.8 2 2.2 2.4 f( t) exact solution boubaker wavelet solution for k=0, m=5 boubaker wavelet solution for k=0, m=6 fig.(3):the graphs of boubaker wavelet and exact solutions. 8 discussion and conclusion 1. boubaker wavelets have applications in approximation theory, moduli of continuity and solution of lane-emden differential equations. the boubaker wavelet expansion of solution of lane-emden differential equation is verified and its shyam lal and swatantra yadav convergence analysis has been studied. the estimator e2k,m(f) of ( f − s2k,m(f) ) has been developed. furthermore, moduli of continuity w ( (f − (s2k,mf)), 1 2k ) ≤ 2e2k,m(f) this shows that moduli of continuity w (( f − (s2k,mf) ) , 1 2k ) is sharper and better than the approximation e2k,m(f) of ( f − s2k,m(f) ) . hence the moduli of continuity w (( f − (s2k,mf) ) , 1 2k ) has been also estimated in this research paper. 2. a method has been proposed to solve lane-emden differential equation by boubaker wavelet collocation method. to illustrate the effectiveness and accuracy of the proposed method, three lane-emden differential equations have been solved by proposed method, it is observed that the exact solutions of considered differential equations are atmost same to their solutions obtained by proposed method. this is a significant achivement of the research paper in wavelet analysis. 3. our results are concerned with boubaker wavelet estimator e2k,m(f), moduli of continuity w (( f − (s2k,mf) ) , 1 2k ) and the solutions of lane-emden diffential equations by this method. 4. (i) by theorem 4.2, e2k,m(f) = o ( 1 2kα √ m ) → 0 as k → ∞,m → ∞. . (ii) as per theorem 5.1, w (( f − s2k,m(f) ) , 1 2k ) = o ( 1 2kα √ m ) → 0 as k → ∞,m → ∞. thus e2k,m(f) and w (( f − s2k,m(f) ) , 1 2k ) are best possible estimation in wavelet analysis 5. solution of lane-emden differential equation by boubaker wavelet series by collocation method is approximately same as exact solution of lane-emden differential equation. only a few number of boubaker wavelet basis is needed to achieve the heigh accuracy. this is significant achivement in wavelet analysis. 6. limitations and possible future development: (i) a non-linear lane-emden equation can not solved by boubaker wavelets without using collocation method (ii) in general, boubaker wavelets in one variable are ineffective to solve a problem expressed in partial differential equations of two or more variables. (iii) it is known that hα[0,1) ⊈ hα2 [0,1).to find the approximate solution of lane-emden differential equation in class hα2 [0,1). approximation and moduli of continuity... (iv) to define two dimensional boubaker wavelets and to find the solution of the partial differential equation by this method. acknowledgments shyam lal, one of the authors, is thankful to dst cims for encouragement to this work. swatantra yadav, one of the authors, is grateful to u.g.c (india) for providing financial assistance in the form of junior research fellowship vide nta ref. no:211610150821 dated:24-03-2022 for his research work. authors are greatful to the referee for his valuable comments and suggestions, to improve the quality of the research paper. references [1] c.k.chui, wavelets: a mathematical tool for signal analysis, siam, philadelphia pa,(1997). [2] devore,r.a.: nonlinear approximation, acta numerica, vol.7, pp. 51-150, cambridge university press, cambridge (1998). [3] debnath, l.: wavelet transform and their applications, birkhauser bostoon, massachusetts (2002). [4] meyer,y.: wavelets; their past and their future, progress in wavelet analaysis and (applications) (toulouse, 1992) (meyer,y.and roques,s. eds) frontieres, gif-sur-yvette,pp. 9-18 (1998) [5] morlet,j.; arens,g; fourgeau,e.; and giard,d.: wave propagation and sampling theory, part ii. sampling theory and complex waves, geophysics 47(2), 222-236 (1982) [6] daubechies,i.: ten lectures on wavelets, siam, philadelphia, pa (1992). [7] wazwaz, a.-m.: a new algorithm for solving differential equations of lane–emden type. applied mathematics and computation, 118(2), 287–310 (2001). [8] shiralashetti, siddu & lamani, lata. (2020). boubaker wavelet based numerical method for the solution of abel’s integral equations. 28. 114-124. [9] das, g.; ghosh, t.; ray, b.k.: degree of approximation of functions by their fourier series in the generalized h¨older metric. proc. indian acad. sci. math. sci. vol.106, no.2, pp. 139–153 (1996) [10] zygmund a.: trigonometric series, vol.i. cambridge university press,cambridge (1959). shyam lal and swatantra yadav [11] lal, shyam, & satish kumar. ”cas wavelet approximation of functions of holder class and solutions of fredholm integral equation.” ratio mathematica [online], 39 (2020): 187-212. microsoft word documento1 microsoft word documento1 ratio mathematica volume 46, 2023 odd prime labeling for some arrow related graphs g. gajalakshmi* s. meena† abstract in a graph g a mapping g is known as odd prime labeling , if g is a bijection from v to {1,3,5, ....,2|v|−1} satisfying the condition that for each line xy in g the gcd of the labels of end points (g(x),g(y)) is one. in this article we prove that some new arrow related graphs such as a2y, a 3 y,a 5 y, are all odd prime graphs. also we prove that double arrow graphs, da2y and da3y are odd prime graphs. keywords: prime graph, odd prime graph, arrow graphs. 2020 ams subject classifications: 05c78 1 *department of mathematics, govt, arts & science college, chidambaram; gaja61904@gmail.com. †department of mathematics, govt, arts & science college, chidambaram 608 102, india; meenasaravanan14@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1059. issn: 1592-7415. eissn: 2282-8214. ©g. gajalakshmi et al. this paper is published under the cc-by licence agreement. 70 g. gajalakshmi and s. meena 1 introduction in this article by a graph g = 〈v (g),e(g)〉 we mean a simple graph. for graph theoretical notations we refer j.a.bondy and u .s. r.murthy [1976] . graph labeling has been introduced in mid 1960. for entire survey of graph labeling we refer gallian [2015]. the concept of prime labeling was established by roger entringer and was discussed in a article by deretsky et al. [1991], tout et al. [1982]. a graph g of order p is known as prime graph if it’s points can be labeled with distinct positive integers {1,2,3, ·,p} such that the labels of any two adjacent points are relatively prime meena and vaithilingam [2013]. meena and kavitha [2014] investigated prime labeling for some butterfly related graphs. meena et al. [2021] investigated odd prime labeling for some new classes of graph. the notion of odd prime labeling was established by prajapati and shah [2018] and many researchers. arrow graph was introduced by kaneria et al. [2015]. motivated by this study, in this article investigate the existence of odd prime labeling of some graphs related to arrow graphs. definition 1.1. let h = 〈v(h),e(h)〉 be a graph. a bijection g : v(h) → o|v | is know as odd prime labeling if for each line xy ∈ e, greatest common divisor 〈g(x),g(y)〉 = 1. a graph is know as odd prime graph if its admits odd prime labeling . definition 1.2. let h1 = (p1,q1) and h2 = (p2,q2) be two graphs with p1 ∩ p2 = φ. the cartesian product h1 × h2 is defined as a graph having p = p1 × p2 and x = (x1,x2) and y = (y1,y2) are adjacent if x1 = y1 and x2 is adjacent to y2 in h2 or x1 is adjacent to y1 in h1 and x2 = y2. the cartesian product of two paths pm and pn denoted as pm ×pn is known as a grid graph on nm points and 2nm− (n + m). definition 1.3. in rectangular grid pm ×pn on mn points the n points v1,1,v2,1,v3,1....vm,n and points v1,n,v2,n,v3,n....vm,n are called an superior points from both the ends. definition 1.4. an arrow graph axy with width x and length y is got by connecting a point v with superior points of px ×py by new edges from one end. definition 1.5. a double arrow graph daxy with width x and length y is got by conecting two points v and w with superior points of pm×py by x+x new edges from both the end. 71 odd prime labeling for some arrow related graphs 2 main results theorem 2.1. a2y is an odd prime graph where y ≥ 2. proof. let g = a2y be an arrow graph got by connecting a point g(u0) with superior points of p2 ×py by new lines. let v(g) = {ul/0 ≤ l ≤ y}∪{vl/1 ≤ l ≤ y} e(g) = {ulul+1/1 ≤ l ≤ y −1}∪{u0v1}∪{u0u1} ∪{vlvl+1/1 ≤ l ≤ y −1}∪{ulvl/1 ≤ l ≤ y}. now |v(g)| = 2y+1 and |e(g)| = 3y define a mapping f : v → o2y as follows g(u0) = 1 g(ul) = 4l−1 for 1 ≤ l ≤ y g(vl) = 4l + 1 for 1 ≤ l ≤ y clearly point labels are distinct. for each e ∈ e, if gcd(g(u),g(v)) = 1 (i) e = u0ul, gcd(g(u0),g(ul)) = gcd(1,3) = 1 (ii) e = u0v1, gcd(g(u0),g(vl)) = gcd(1,5) = 1 (iii)e = ulvl, gcd(g(ul),g(vl)) = gcd(4l−1,4l + 1) = 1 for 1 ≤ l ≤ y (iv) e = ulul+1, gcd(g(ul),g(ul+1))= gcd(4l−1,4l + 3) = 1 for 1 ≤ l ≤ y −1 (v) e = vivl+1, gcd(g(vl)),g(vl+1) = gcd(4l + 1,4l + 5) = 1 for 1 ≤ l ≤ y −1 hence a2y is an odd prime graph. figure 1: arrow graph a2y and its odd prime labeling theorem 2.2. a3y is an odd prime graph where y ≥ 2. proof. let g = a3y be an arrow graph got by connecting a point g(u0) with superior points of p3 ×p2 by 3 new lines. v(g) = {ul,vl,wl,/1 ≤ l ≤ y}∪{u0} e(g) = {ulul+1,vlvl+1,wlwl+1/1 ≤ l ≤ y −1}∪{vlwl,ulvl/1 ≤ l ≤ y} ∪{uou1}∪{u0v1}∪{u0w1} 72 g. gajalakshmi and s. meena now |v(g)| = 3y + 1 and |e(g)| = 5y −1 define a mapping f : v → o2y as follows g(u0) = 1 g(ul) = 6l−3 for 1 ≤ l ≤ y, l is odd g(ul) = 6l−1 for 1 ≤ l ≤ y, l is even g(vl) = 6l−1 for 1 ≤ l ≤ y, l is odd g(vl) = 6l−3 for 1 ≤ l ≤ y, l is even g(wl) = 6l + 1 for 1 ≤ l ≤ y clearly all the point labels are distinct. with this labeling for each e = uv ∈ e if (i) e = u0u1,gcd(g(u0),g(u1)) = gcd(1,3) = 1 for 1 ≤ l ≤ y (ii) e = u0w1,gcd(g(u0),g(w1)) = gcd(1,7) = 1 for 1 ≤ l ≤ y (iii)e = u0v1,gcd(g(u0),g(v1)) = gcd(1,5) = 1 for 1 ≤ l ≤ y (iv)e = ulvl,gcd(g(ul),g(vl) = gcd(6l − 3,6l − 1) = 1 for 1 ≤ l ≤ y l 6≡ 0(mod2) (v)e = ulvl,gcd(g(ul),g(vl)) = gcd(6l − 3,6l − 1) = 1 for 1 ≤ l ≤ y l ≡ 0(mod2) (vi)e = vlwl,gcd(g(vl),g(wl)) = gcd(6l − 1,6l + 1) = 1 for 1 ≤ l ≤ y l 6≡ 0(mod2) (vii)e = vlwl,gcd(g(ul),g(vl)) = gcd(6l − 1,6l − 3) = 1 for 1 ≤ l ≤ y l ≡ 0(mod2) (viii)e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l−3,6l−5) = 1 for 1 ≤ l ≤ y l 6≡ 0(mod2) (ix)e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l−1,6l + 3) = 1 for 1 ≤ l ≤ y l ≡ 0(mod2) (x)e = vlvl + 1,gcd(g(vl),g(vl+1)) = gcd(6l−3,6l−1) = 1 for 1 ≤ l ≤ y l ≡ 0(mod2) (xi) e = vlvl + 1,gcd(g(vl),g(vl+1)) = gcd(6l−1,6l−3) = 1 for 1 ≤ l ≤ y l 6≡ 0(mod2) (xii) e = wlwl + 1,gcd(g(wl),g(wl+1)) = gcd(6l + 1,6l + 7) = 1 for 1 ≤ l ≤ y hence a3y is an odd prime graph . figure 2: arrow graph a3y and its odd prime labeling theorem 2.3. a5y is an odd prime graph where y ≥ 5. 73 odd prime labeling for some arrow related graphs proof. let g = a5y be an arrow graph got by connecting a point v with superior points p5 ×py by 5 new lines. v(g) = {ul,vl,wl/1 ≤ l ≤ y}∪{u0} e(g) = {ulvl,vlwl/1 ≤ l ≤ y}∪{(ulul+1),(vlvi+1),(wlwl+1/1 ≤ l ≤ y −1} now |v(g)| = 5y + 1 and |e(g)| = 9y define a mapping f : v → oy as follows g(u0) = 1 g(ul) = 6l−3 for 1 ≤ l ≤ l, l is odd g(ul) = 6l−1 for 1 ≤ l ≤ l, l is even g(vl) = 6l−1 for 1 ≤ l ≤ l, l is odd g(vl) = 6l−3 for 1 ≤ l ≤ l, l is even g(wl) = 6l + 1 for 1 ≤ l ≤ l, clearly all the point labels are distinct. with this labeling for each e ∈ e if gcd(g(u),g(v)) = 1 (i) e = u0u1,gcd(g(u0),g(u1)) = gcd(1,3) = 1 (ii) e = u0ul+1,gcd(g(u0),g(ul+1)) = gcd(1,6l−3) = 1 for 1 ≤ l ≤ y (iii)e = ulvl,gcd(g(ul),g(vl) = gcd(6l − 3,6l − 1) = 1 for 1 ≤ l ≤ y l 6≡ 0(mod2) (iv) e = ulvl,gcd(g(ul),g(vl)) = gcd(6l − 1,6l − 3) = 1 for 1 ≤ l ≤ y l ≡ 0(mod2) (v) e = vlwl,gcd(g(vl),g(wl)) = gcd(6l − 1,6l + 1) = 1 for 1 ≤ l ≤ y l 6≡ 0(mod2) (vi) e = vlwl,gcd(g(vl),g(wl)) = gcd(6l − 3,6l + 1) = 1 for 1 ≤ l ≤ y l ≡ 0(mod2) (vii) e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l − 3,6l + 5) = 1 for 1 ≤ l ≤ y l 6≡ 0(mod2) (viii) e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l−1,6l−3) = 1 for 1 ≤ l ≤ y l ≡ 0(mod2) (ix) e = vlvl+1,gcd(g(vl),g(vl+1)) = gcd(6l − 1,6l + 3) = 1 for 1 ≤ l ≤ y l 6≡ 0(mod2) (x) e = vivi+1,gcd(g(vl),g(vl+1)) = gcd(6l − 3,6l + 5) = 1 for 1 ≤ l ≤ y l ≡ 0(mod2) (xi) e = wlwl+1,gcd(g(wl),g(wl+1)) = gcd(6l + 1,6l + 7) = 1 for 1 ≤ l ≤ y hence a5y is an odd prime graph . 74 g. gajalakshmi and s. meena figure 3: arrow graph a5y and its odd prime labeling theorem 2.4. da2y is an odd prime graph where y ≥ 2. proof. let g = da2y be a double arrow graph got by connecting two points u,v with superior points from both the ends of p2 ×py by 2+2 new lines. let v(g) = {ulvl/1 ≤ l ≤ y}∪{v,v0} e(g) = {(ulul+1),(vlvl+1),1 ≤ l ≤ y −1}∪{vlul/1 ≤ l ≤ y}∪{vv1}∪{vu1} ∪{uyv0}∪{vyv0} now |v(g)| = 2y+2 and |e(g)| = 3y+4 define a mapping f : v → o2y as follows g(v) = 1 g(ui) = 4l−1 for 1 ≤ l ≤ y g(vi) = 4l + 1 for 1 ≤ l ≤ y g(v0)= 4y + 3 clearly point labels are distinct. for every e = uv ∈ e, if gcd(g(u),g(v)) = 1 (i) e = vu1, gcd(g(v),g(u1)) = gcd(1,3) = 1 (ii) e = vv1, gcd(g(v),g(v1)) = gcd(1,5) = 1 (iii) e = ulul+1, gcd(g(ul),g(ul+1))= gcd(4l−1,4l + 3) = 1 for 1 ≤ l ≤ y −1 (iv) e = vlvl+1, gcd(g(vl)),g(vl+1) = gcd(4l + 1,4l + 5) = 1 for 1 ≤ l ≤ y −1 (v) e = vlul, gcd(g(vl),g(ul)) = gcd(4l + 1,4l−1) = 1 for 1 ≤ l ≤ y (vi) e = vyw, gcd(g(vy),g(w)) = gcd(4y + 1,4y + 3) = 1 (vii)e = uyw, gcd(g(uy),g(w)) = gcd(4y −1,4y + 3) = 1 hence da2y is an odd prime graph. 75 odd prime labeling for some arrow related graphs figure 4: arrow graph da2y and its odd prime labeling theorem 2.5. da3y is an odd prime graph where y ≥ 3. proof. let d = da3y be an arrow graph got by connecting two point set u0 and z0 with superior pointss from both the ends of p3 ×p2 by 3+3 new lines. v(g) = {ul,vl,wl,/1 ≤ l ≤ y}∪{u0}∪{z0} e(g) = {ulul+1,vlvl+1,wlwl+1/1 ≤ l ≤ y −1}∪{wlvl,vlul/1 ≤ l ≤ y}∪ {u0u1,u0v1,u0w1,z0uy,z0vy,z0wy} now |v(g)| = 3y + 2 and |e(g)| = 5y + 3 define a mapping f : v → oy as follows g(u0) = 1 g(ul) = 6l−3 for 1 ≤ l ≤ y, l is odd g(ul) = 6l−1 for 1 ≤ l ≤ y, l is even g(vl) = 6l−1 for 1 ≤ l ≤ y, l is odd g(vl) = 6l−3 for 1 ≤ l ≤ y, l is even g(wl) = 6l + 1 for 1 ≤ l ≤ y g(z0) = 6y + 3 for 1 ≤ i ≤ y clearly all the point values are different. with this labeling for each e ∈ e if (i) e = u0u1,gcd(g(u0),g(u1)) = gcd(1,3) = 1 (ii) e = u0v1,gcd(g(u0),g(v1)) = gcd(1,5) = 1 (iii) e = u0w1,gcd(g(u0),g(w1) = gcd(1,7) = 1 (iv) e = ulvl,gcd(g(ul),g(vl)) = gcd(6l − 3,6l − 1) = 1 for 1 ≤ l ≤ y l 6≡ 0(mod2) (v) e = ulvl,gcd(g(ul),g(vl)) = gcd(6l − 1,6l − 3) = 1 for 1 ≤ l ≤ y l ≡ 0(mod2); (vi) e = vlwl,gcd(g(vl),g(wl)) = gcd(6l − 1,6l + 1) = 1 for 1 ≤ l ≤ y l 6≡ 0(mod2) (vii) e = vlwl,gcd(g(vl),g(wl)) = gcd(6l − 3,6l − 1) = 1 for 1 ≤ l ≤ y l ≡ 0(mod2); (viii) e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l−3,6l + 5) = 1 for 1 ≤ l ≤ y−1 l 6≡ 0(mod2); (ix) e = ulul+1,gcd(g(ul),g(ul+1)) = gcd(6l − 1,6l + 3) = 1 for 1 ≤ l ≤ y − 1 l ≡ 0(mod2); hence da3y is an odd prime graph . 76 g. gajalakshmi and s. meena figure 5: arrow graph da3n and its odd prime labeling 3 conclusions the odd prime labeling of various classes of graphs such as a2y where y ∈ n, a3y, a 5 y, where y ≥ 2 are odd prime graph and double arrow graphs da2y,da3y are proved. to derive similar results for other graph families is an open area of research. references t. deretsky, s. lee, and j. mitchem. on vertex prime labelings of graphs, in graph theory, combinatorics and applications, j. alavi, g. chartrand, o. oellerman, and a. schwenk, eds.,. proceedings 6th international conference theory and applications of graphs (wiley, new york),, 1:359 – 369, 1991. j. gallian. a dynamic survey of graph labeling. ds6, 2015. j.a.bondy and u .s. r.murthy. graph theory and applications. (north-holland), new york, 1976. v. kaneria, m. jariya, and h. makadia. graceful of arrow graphs and double arrow graph. malaya journal of math., 3:382 – 386, 2015. s. meena and p. kavitha. prime labeling for some butterfly related graphs. international journal of mathematical archive, 5:15 – 25, 2014. s. meena and k. vaithilingam. prime labeling for some crown related graphs. international journal of scientific & technology research, 2:92 – 95, 2013. s. meena, g. gajalakshmi, and p. kavitha. odd prime labeling for some new classes of graph (communicated). seajm, 2021. 77 odd prime labeling for some arrow related graphs u. prajapati and k. shah. on odd prime labeling. international journal of research and analytical reviews, 5:284 – 294, 2018. a. tout, a. dabboucy, and k. howalla. prime labeling of graphs. nat. acad. sci letters, 11:365 – 368, 1982. 78 microsoft word documento1 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 40, 2021, pp. 179-190 179 intuitionistic fwi-ideals of residuated lattice wajsberg algebras r. shanmugapriya* a. ibrahim† abstract the notions of intuitionistic fuzzy wajsberg implicative ideal (𝐹𝑊𝐼 –ideal) and intuitionistic fuzzy lattice ideal of residuated wajsberg algebras are introduced. also, we show that every intuitionistic 𝐹𝑊𝐼ideal of residuated lattice wajsberg algebra is an intuitionistic fuzzy lattice ideal of residuated lattice wajsberg algebra. further, we discussed its converse part. keywords: wajsberg algebra; lattice wajsberg algebra; residuated lattice wajsberg algebra; 𝑊𝐼–ideal; 𝐹𝑊𝐼 –ideal; intuitionistic 𝐹𝑊𝐼 –ideal; intuitionistic fuzzy lattice ideal. 2010 ams subject classification: 06b10, 03e72, 03g10. *research scholar, p. g. and research department of mathematics, h. h. the rajah’s college, pudukkotai, affiliated to bharathidasan university, trichirappalli, tamilnadu, india; priyasanmu@gmail.com †assistant professor, p.g. and research department of mathematics, h. h. the rajah’s college, pudukkotai, affiliated to bharathidasan university, trichirappalli, tamilnadu, india; ibrahimaadhil@yahoo.com; dribra@hhrc.ac.in †received on january 12th, 2021. accepted on may 12th, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.587. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. shanmugapriya and ibrahim 180 1. introduction the concept of fuzzy set was introduced by zadeh [13] in 1935. the concept of intuitionistic fuzzy set was introduced by atanassov [1, 2]. the idea of wajsberg algebra was introduced by mordchaj wajsberg [10]. the author [8] introduced the notions of fwi-ideals and investigated their properties with illustrations. in the present paper, we introduce the notions of intuitionistic 𝐹𝑊𝐼 –ideal and intuitionistic fuzzy lattice ideal of residuated lattice wajsberg algebras. also, we show that every intuitionistic 𝐹𝑊𝐼–ideal of residuated lattice wajsberg algebra is an intuitionistic fuzzy lattice ideal of residuated lattice wajsberg algebra. further, we verify its converse part. 2. preliminaries in this section, we recall some basic definitions and properties which are helpful to develop our main results. definition 2.1[3]. let (𝐴, →,∗ ,1) bean algebra with a binary operation “ → " and a quasi-complement “ ∗ ”. then it is called a wajsberg algebra, if the following axioms are satisfied for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, (i) 1 → 𝑥 = 𝑥 (ii) (𝑥 → 𝑦) → 𝑦 = ((𝑦 → 𝑧) → (𝑥 → 𝑧)) = 1 (iii) (𝑥 → 𝑦) → 𝑦 = (𝑦 → 𝑥) → 𝑥 (iv) (𝑥∗ → 𝑦∗) → (𝑦 → 𝑥) = 1. definition 2.2[3].let(𝐴, →,∗ ,1) be a wajsberg algebra. then the following axioms are satisfied for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, (i) 𝑥 → 𝑥 = 1 (ii) if (𝑥 → 𝑦) = (𝑦 → 𝑥) = 1 then 𝑥 = 𝑦 (iii) 𝑥 → 1 = 1 (iv) (𝑥 → (𝑦 → 𝑥)) = 1 (v) if (𝑥 → 𝑦) = (𝑦 → 𝑧) = 1 then 𝑥 → 𝑧 = 1 (vi) (𝑥 → 𝑦) → ((𝑧 → 𝑥) → (𝑧 → 𝑦)) = 1 (vii) 𝑥 → (𝑦 → 𝑧) = 𝑦 → (𝑥 → 𝑧) (viii) 𝑥 → 0 = 𝑥 → 1∗ = 𝑥∗ (ix) (𝑥∗)∗ = 𝑥 (x) (𝑥∗ → 𝑦∗) = 𝑦 → 𝑥. definition 2.3[3]. let (𝐴, →,∗ ,1) be a wajsberg algebra. then it is called a lattice wajsberg algebra, if the following axioms are satisfied for all 𝑥, 𝑦 ∈ 𝐴, intuitionistic fwi-ideals of residuated lattice wajsberg algebras 181 (i) the partial ordering " ≤ " on a wajsberg algebra such that 𝑥 ≤ 𝑦 if and only if 𝑥 → 𝑦 = 1 (ii) 𝑥 ∨ 𝑦 = (𝑥 → 𝑦) → 𝑦 (iii) 𝑥 ∧ 𝑦 = ((𝑥∗ → 𝑦∗) → 𝑦∗)∗. thus, (𝐴, ∨, ∧, ∗ ,0 , 1) is a lattice wajsberg algebra with lower bound 0 and upper bound 1. proposition 2.4[3].let(𝐴, →, ∗ ,1) be a lattice wajsberg algebra. then the following axioms are satisfied for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, (i) if 𝑥 ≤ 𝑦 then 𝑥 → 𝑧 ≥ 𝑦 → 𝑧 and 𝑧 → 𝑥 ≤ 𝑧 → 𝑦 (ii) 𝑥 ≤ 𝑦 → 𝑧 if and only 𝑖𝑓 𝑦 ≤ 𝑥 → 𝑧 (iii) (𝑥 ∨ 𝑦)∗ = (𝑥∗˄ 𝑦∗) (iv) (𝑥 ∧ 𝑦)∗ = (𝑥∗ ∨ 𝑦∗) (v) (𝑥 ∨ 𝑦) → 𝑧 = (𝑥 → 𝑧) ∧ (𝑦 → 𝑧) (vi) 𝑥 → (𝑦 ∧ 𝑧) = (𝑥 → 𝑦) ∧ (𝑥 → 𝑧) (vii) (𝑥 → 𝑦) ∨ (𝑦 → 𝑥) = 1 (viii) 𝑥 → (𝑦 ∨ 𝑧) = (𝑥 → 𝑦) ∨ (𝑥 → 𝑧) (ix) (𝑥 ∧ 𝑦) → 𝑧 = (𝑥 → 𝑧) ∨ (𝑦 → 𝑧) (x) (𝑥 ∧ 𝑦) ∨ 𝑧 = (𝑥 ∨ 𝑧) ∧ (𝑦 ∨ 𝑧) (xi) (𝑥 ∧ 𝑦) → 𝑧 = (𝑥 → 𝑦) → (𝑥 → 𝑧). definition 2.5[11]. let(𝐴, ∨, ∧, ⊗, →, 0, 1) be an algebra of type (2, 2, 2, 2, 0, 0). then it is called a residuated lattice, if the following axioms are satisfied for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, (i) (𝐴, ∨, ∧, 0, 1) is a bounded lattice, (ii) (𝐴,⊗, 1) is commutative monoid, (iii) 𝑥 ⊗ y ≤ 𝑧 if and only if 𝑥 ≤ 𝑦 → 𝑧. definition 2.6[3]. let (𝐴, ∨, ∧, ∗, →, 1) be a lattice wajsberg algebra. if a binary operation “ ⊗ " on 𝐴 satisfies 𝑥 ⊗ 𝑦 = (𝑥 → 𝑦∗)∗ for all 𝑥, 𝑦 ∈ 𝐴. then (𝐴, ∨, ∧, ⊗, →, ∗, 0, 1) is called a residuated lattice wajsberg algebra. definition 2.7[4].let 𝐴 be a lattice wajsberg algebra. let 𝐼 be a non-empty subset of 𝐴, then 𝐼 is called awi-ideal of lattice wajsberg algebra 𝐴, if the following axioms are satisfied for all𝑥, 𝑦 ∈ 𝐴, (i) 0 ∈ 𝐼 (ii) (𝑥 → 𝑦)∗ ∈ 𝐼and 𝑦 ∈ 𝐼 imply 𝑥 ∈ 𝐼 . definition 2.8[4].let 𝐿 be a lattice. an ideal 𝐼 of 𝐿 is a non-empty subset of 𝐿 is called a lattice ideal, if the following axioms are satisfied for all 𝑥, 𝑦 ∈ 𝐴, (i) 𝑥 ∈ 𝐼, 𝑦 ∈ 𝐿 and 𝑦 ≤ 𝑥 imply 𝑦 ∈ 𝐼 shanmugapriya and ibrahim 182 (ii) 𝑥, 𝑦 ∈ 𝐼 implies 𝑥 ∨ 𝑦 ∈ 𝐼. definition 2.9[7]. let 𝐴 be a residuated lattice wajsberg algebra and 𝐼 be a non-empty subset of 𝐴.then 𝐼 is called a 𝑊𝐼-ideal of residuated lattice wajsberg algebra 𝐴, if the following axioms are satisfiedfor all 𝑥, 𝑦 ∈ 𝐴, (i) 0 ∈ 𝐼 (ii) 𝑥 ⊗ 𝑦 ∈ 𝐼 and 𝑦 ∈ 𝐼 imply 𝑥 ∈ 𝐼 (iii) (𝑥 → 𝑦)∗ ∈ 𝐼 and 𝑦 ∈ 𝐼 imply 𝑥 ∈ 𝐼. definition 2.10[13].let 𝐴 be a set. a function 𝜇: 𝐴 → [0, 1] is called a fuzzy subset on 𝐴 for each 𝑥 ∈ 𝐴, the value of 𝜇(𝑥) describes a degree of membership of 𝑥 in 𝜇. definition 2.11[5].let 𝐴 be a lattice wajsberg algebra. then the fuzzy subset 𝜇 of 𝐴 is called a fuzzy 𝑊𝐼-ideal of 𝐴, if the following axioms are satisfied for all 𝑥, 𝑦 ∈ 𝐴, (i) 𝜇(0) ≥ 𝜇(𝑥) (ii) 𝜇(𝑥) ≥ min{ 𝜇((𝑥 → 𝑦)∗), 𝜇(𝑦)}. definition 2.12[5].a fuzzy subset𝜇 of a lattice wajsberg algebra 𝐴 is called a fuzzy lattice ideal if for all 𝑥, 𝑦 ∈ 𝐴, (i) if 𝑦 ≤ 𝑥 then 𝜇(𝑦) ≥ 𝜇(𝑥) (ii) 𝜇(𝑥 ∨ 𝑦) ≥ min{𝜇(𝑥), 𝜇(𝑦)}. definition 2.13[8]. let 𝐴 be a residuated lattice wajsberg algebra. then the fuzzy subset 𝜇 of 𝐴 is called a 𝐹𝑊𝐼-ideal of residuated lattice wajsberg algebra 𝐴, if the following axioms are satisfied for all 𝑥, 𝑦 ∈ 𝐴, (i) 𝜇(0) ≥ 𝜇(𝑥) (ii) 𝜇(𝑥) ≥ min{ 𝜇(𝑥 ⊗ 𝑦), 𝜇(𝑦)} (iii) 𝜇(𝑥) ≥ min{ 𝜇((𝑥 → 𝑦)∗), 𝜇(𝑦)}. definition 2.14[2]. an intuitionistic fuzzy subset 𝑆 is a non-empty set 𝑋 is an object having the form 𝑆 = {(𝑥, 𝜇𝑠(𝑥), 𝛾𝑠(𝑥))|𝑥 ∈ 𝑋} = (𝜇𝑠, 𝛾𝑠)where the functions 𝜇𝑠(𝑥): 𝑋 → [0, 1]denote the degree of membership and the degree of non-membership respectively and 0 ≤ 𝜇𝑠(𝑥) + 𝛾𝑠(𝑥) ≤ 1 for any 𝑥 ∈ 𝑋. definition 2.15[13]. if 𝜇 and ѵ are fuzzy sets in 𝐴, define 𝜇 ≤ ѵ if and only if 𝜇(𝑥) ≤ ѵ(𝑥) for all 𝑥 ∈ 𝐴. definition 2.16[13]. the level set 𝜇𝑡 defined by 𝜇𝑡 = {𝑥 ∈ 𝐴/𝜇(𝑥) ≥ 𝑡}, where𝑡 ∈ [0, 1], then 𝜇𝑡 is also denoted by 𝑈(𝜇; 𝑡). intuitionistic fwi-ideals of residuated lattice wajsberg algebras 183 3. properties of intuitionistic fwi-ideal of a residuated lattice wajsberg algebra in this section, we introduce the concept of an intuitionistic fwi-ideal and intuitionistic fuzzy lattice ideals. also, we obtain some properties of an intuitionistic 𝐹𝑊𝐼-ideal. definition 3.1. let 𝐴 be a residuated lattice wajsberg algebra. an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) of 𝐴 is called an intuitionistic fwi-ideal of residuated lattice wajsberg algebra 𝐴 if it satisfies the following inequalities for all 𝑥, 𝑦 ∈ 𝐴, (i) 𝜇𝑠(0) ≥ 𝜇𝑠(𝑥) and 𝛾𝑠(0) ≤ 𝛾𝑠(𝑥) (ii) 𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)} (iii) 𝛾𝑠(𝑥) ≤ max {𝛾𝑠(𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦)} (iv) 𝜇𝑠(𝑥) ≥ min {𝜇𝑠((𝑥 → 𝑦) ∗, 𝜇𝑠(𝑦) (v) 𝛾𝑠(𝑥) ≤ max {𝛾𝑠((𝑥 → 𝑦) ∗, 𝛾𝑠(𝑦)} . example 3.2. consider a set 𝐴={0, 𝑎, 𝑏, 𝑐, 𝑑,𝑟, 𝑠, 𝑡, 1}. define a partial ordering “≤” on 𝐴, such that 0 ≤ 𝑎 ≤ 𝑏 ≤ 𝑐 ≤ 𝑑 ≤ 𝑟 ≤ 𝑠 ≤ 𝑡 ≤ 1 with a binary operations“ ⊗ ”and" → ”and a quasi-complement " ∗ "on 𝐴 as in following tables 3.1 and 3.2. table 3.1: complement table 3.2: implication define ∨ and ∧ operations on 𝐴 as follows: (𝑥 ∨ 𝑦) = (𝑥 → 𝑦) → 𝑦, (𝑥 ∧ 𝑦) = (𝑥∗ → 𝑦∗) → 𝑦∗)∗, 𝑥 𝑥 ∗ 0 1 𝑎 𝑡 𝑏 𝑏 𝑐 𝑟 𝑑 𝑑 𝑟 𝑐 𝑠 𝑏 𝑡 𝑎 1 0 → 0 𝑎 𝑏 𝑐 𝑑 𝑟 𝑠 𝑡 1 0 1 1 1 1 1 1 1 1 1 𝑎 𝑡 1 1 𝑡 1 1 𝑡 1 1 𝑏 𝑏 𝑡 1 𝑠 𝑡 1 𝑠 𝑡 1 𝑐 𝑟 𝑟 𝑟 1 1 1 1 1 1 𝑑 𝑑 𝑟 𝑟 𝑡 1 1 𝑡 1 1 𝑟 𝑐 𝑑 𝑟 𝑠 𝑡 1 𝑠 𝑡 1 𝑠 𝑏 𝑏 𝑏 𝑟 𝑟 𝑟 1 1 1 𝑡 𝑎 𝑏 𝑏 𝑑 𝑟 𝑟 𝑡 1 1 1 0 𝑎 𝑏 𝑐 𝑑 𝑟 𝑠 𝑡 1 shanmugapriya and ibrahim 184 𝑥 ⊗ 𝑦 = (𝑥 → 𝑦∗)∗ for all 𝑥, 𝑦 ∈ 𝐴. then, 𝐴 is a residuated lattice wajsberg algebra. consider an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) on 𝐴 as, 𝜇𝑠(𝑥) = { 1 if 𝑥 ∈ (0, 𝑞) for all 𝑥 ∈ 𝐴 0.54 otherwise for all 𝑥 ∈ 𝐴 ; 𝛾𝑠(𝑥) = { 0 if 𝑥 ∈ (0, 𝑞) for all 𝑥 ∈ 𝐴 0.36 otherwise for all 𝑥 ∈ 𝐴 then, 𝑆 is an intuitionistic fwi-ideal of 𝐴. in the same example 3.2, we consider an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) on 𝐴 as, 𝜇𝑠(𝑥) = { 1 if 𝑥 ∈ {0, 𝑎, 𝑏} for all 𝑥 ∈ 𝐴 0.55 otherwise for all 𝑥 ∈ 𝐴 ; 𝛾𝑠(𝑥) = { 0 if 𝑥 ∈ {0, 𝑎, 𝑏} for all 𝑥 ∈ 𝐴 0.42 otherwise for all 𝑥 ∈ 𝐴 then, 𝑆 is not an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. since 𝜇𝑠(𝑥) ≱ min {𝜇𝑠(𝑠 ⊗ 𝑏), 𝜇𝑠(𝑏)} and 𝛾𝑠(𝑥) ≰ max{𝛾𝑠(𝑠 ⊗ 𝑏), 𝛾𝑠(𝑏)}. proposition 3.3. every intuitionistic 𝐹𝑊𝐼-ideal 𝑆 = (𝜇𝑠, 𝛾𝑠) of residuated lattice wajsberg algebra 𝐴 is an intuitionistic monotonic. that is, if 𝑥 ≤ 𝑦, then 𝜇𝑠(𝑥) ≥ 𝜇𝑠(𝑦) and 𝛾𝑠(𝑥) ≤ 𝛾𝑠(𝑦). proof. let 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. let 𝑥, 𝑦 ∈ 𝐴, 𝑥 ≤ 𝑦. then 𝑥 ⊗ 𝑦 = (𝑥 → 𝑦∗)∗ [from the definition 2.6] = (𝑥 → 𝑥)∗ = 1∗ = 0 [from (i) of definition 2.2] 𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)} [from (ii) of definition 3.1] we have 𝜇𝑠(𝑥) ≥ 𝜇𝑠(𝑦) now,𝛾𝑠(𝑥) ≤ max{𝛾𝑠(𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦)} [from (iii) of definition 3.1] = max{𝛾𝑠(0), 𝛾𝑠(𝑦)} = 𝛾𝑠 (𝑦) [from the definition 2.6] hence 𝛾𝑠(𝑥) ≤ 𝛾𝑠(𝑦) and 𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑥 → 𝑦) ∗, 𝜇𝑠(𝑦)} [from (iv) of definition 3.1] = min{𝜇𝑠(0), 𝜇𝑠(𝑦)} = 𝜇𝑠(𝑦) [from (ii) of definition 2.7] we have 𝜇𝑠(𝑥) ≥ 𝜇𝑠(𝑦) now, 𝛾𝑠(𝑥) ≤ max{𝛾𝑠(𝑥 → 𝑦) ∗, 𝛾𝑠(𝑦)} [from (v) of definition 3.1] = max{𝛾𝑠(0), 𝛾𝑠(𝑦)} = 𝛾𝑠 (𝑦) [from (ii) of definition 2.7] therefore, 𝛾𝑠(𝑥) ≤ 𝛾𝑠(𝑦). ∎ example 3.4. let 𝐴 be a residuated lattice wajsberg algebra defined in example 3.2, define an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) of 𝐴 as follows, intuitionistic fwi-ideals of residuated lattice wajsberg algebras 185 (i) 𝜇𝑠(0) = 𝜇𝑠(𝑐) = 1 (ii) 𝜇𝑠(𝑥) = 𝑚 for any 𝑥 ∈ {𝑎, 𝑏, 𝑐, 𝑑, 𝑟, 𝑠, 𝑡, 1} (iii) 𝛾𝑠(0) = 𝛾𝑠(𝑐) = 0 (iv) 𝛾𝑠(𝑥) = 𝑛 for any 𝑥 ∈ {𝑎, 𝑏, 𝑐, 𝑑, 𝑟, 𝑠, 𝑡, 1}. where 𝑚, 𝑛 ∈ [0, 1] and 𝑚 + 𝑛 ≤ 1. then 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. example 3.5. consider a set𝐴 = {𝑎, 𝑏, 𝑝, 𝑞, 𝑐, 𝑑, 1}. define a partial ordering “≤” on 𝐴, such that 0 ≤ 𝑎 ≤ 𝑏 ≤ 𝑝 ≤ 𝑞 ≤ 𝑐 ≤ 𝑑 ≤ 1 with a binary operations“ ⊗ ”and " → ”and a quasi-complement " ∗ "on 𝐴 as in following tables 3.3 and 3.4. table 3.3: complement table 3.4: implication define ∨ and ∧ operations on 𝐴 as follows: (𝑥 ∨ 𝑦) = (𝑥 → 𝑦) → 𝑦, (𝑥 ∧ 𝑦) = (𝑥∗ → 𝑦∗) → 𝑦∗)∗, 𝑥 ⊗ 𝑦 = (𝑥 → 𝑦∗)∗ for all 𝑥, 𝑦 ∈ 𝐴. then, 𝐴 is a residuated lattice wajsberg algebra. consider an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) on 𝐴 as, 𝜇𝑠(𝑥) = { 1 if 𝑥 ∈ (0, 𝑞) for all 𝑥 ∈ 𝐴 0.54 otherwise for all 𝑥 ∈ 𝐴 ; 𝛾𝑠(𝑥) = { 0 if 𝑥 ∈ (0, 𝑞) for all 𝑥 ∈ 𝐴 0.36 otherwise for all 𝑥 ∈ 𝐴 then, 𝑆 is an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. in the same example 3.5, we consider an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) on 𝐴 as, 𝑥 𝑥 ∗ 0 1 𝑎 𝑏 𝑏 𝑎 𝑝 0 𝑞 0 𝑐 0 𝑑 0 1 0 → 0 𝑎 𝑏 𝑝 𝑞 𝑐 𝑑 1 0 1 1 1 1 1 1 1 1 𝑎 𝑏 1 𝑏 1 1 1 1 1 𝑏 𝑎 𝑎 1 1 1 1 1 1 𝑝 0 𝑎 𝑏 1 1 1 1 1 𝑞 0 𝑎 𝑏 𝑝 1 1 1 1 𝑐 0 𝑎 𝑏 𝑝 𝑑 1 𝑑 1 𝑑 0 𝑎 𝑏 𝑝 𝑐 𝑐 1 1 1 0 𝑎 𝑏 𝑝 𝑞 𝑐 𝑑 1 shanmugapriya and ibrahim 186 𝜇𝑠(𝑥) = { 1 if 𝑥 ∈ {0, 𝑎, 𝑏} for all 𝑥 ∈ 𝐴 0.55 otherwise for all 𝑥 ∈ 𝐴 ; 𝛾𝑠(𝑥) = { 0 if 𝑥 ∈ {0, 𝑎, 𝑏} for all 𝑥 ∈ 𝐴 0.42 otherwise for all 𝑥 ∈ 𝐴 then, 𝑆 is not an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. since 𝜇𝑠(𝑥) ≱ min {𝜇𝑠(𝑐 ⊗ 𝑎), 𝜇𝑠(𝑎)} and 𝛾𝑠(𝑥) ≰ max{𝛾𝑠(𝑐 ⊗ 𝑎), 𝛾𝑠(𝑎)}. proposition 3.6. let 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic 𝐹𝑊𝐼-ideal of residuated lattice wajsberg algebra 𝐴. for any 𝑥, 𝑦, 𝑧 ∈ 𝐴 which satisfies 𝑥 ≤ 𝑦∗ → 𝑧 then 𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑦), 𝜇𝑠(𝑧)} and 𝛾𝑠(𝑥) ≤ max{𝛾𝑠(𝑦), 𝛾𝑠(𝑧)}. proof. let𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. if 𝑥 ≤ 𝑦 ∗ → 𝑧 then, we have 1 = 𝑥 → (𝑦∗ → 𝑧) = 𝑧∗ → (𝑥 → 𝑦) = (𝑥 → 𝑦)∗ → 𝑧 for all 𝑥, 𝑦, 𝑧 ∈ 𝐴 [from (x) of definition 2.2] and ((𝑥 → 𝑦)∗ → 𝑧)∗) = 0. it follows that, 𝜇𝑠(𝑥) ≥ min{𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)} [from (ii) of definition 3.1] ≥ min {min {𝜇𝑠((𝑥 ⊗ 𝑦) → 𝑧), 𝜇𝑠(𝑧)}, 𝜇𝑠(𝑦)} = min{min{𝜇𝑠((0) → 𝑧), 𝜇𝑠(𝑧)} , 𝜇𝑠(𝑦)} [from the definition 2.6] = min{min{𝜇𝑠(0), 𝜇𝑠(𝑧)} , 𝜇𝑠(𝑦)} = min{𝜇𝑠(𝑦), 𝜇𝑠(𝑧)} [from (ii) of definition 3.1] we have 𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑦), 𝜇𝑠(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴 now,𝛾𝑠(𝑥) ≤ max {max{𝛾𝑠((𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦))} ≤ max {max{ 𝛾𝑠 (((𝑥 ⊗ 𝑦) → 𝑧), 𝛾𝑠(𝑧)} , 𝛾𝑠(𝑦)} = max{max{𝛾𝑠((0) → 𝑧), 𝛾𝑠(𝑧)} , 𝛾𝑠(𝑦)} [from the definition 2.6] = max {max{𝛾𝑠(0), 𝛾𝑠(𝑧)} , 𝛾𝑠 (𝑦)} = max {𝛾𝑠(𝑦), 𝛾𝑠(𝑧)} [from (iii) of definition 3.1] hence 𝛾𝑠(𝑥) ≤ max {𝛾𝑠(𝑦), 𝛾𝑠(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴 now, 𝜇𝑠(𝑥) ≥ min{𝜇𝑠((𝑥 → 𝑦) ∗), 𝜇𝑠(𝑦)} [from (iv) of definition 3.1] ≥ min{min{𝜇𝑠(𝑥 → 𝑦) ∗ → 𝑧)∗) , 𝜇𝑠(𝑧)} , 𝜇𝑠(𝑦)} = min {min{𝜇𝑠(0), 𝜇𝑠(𝑧)} , 𝜇𝑠(𝑦)} = min {𝜇𝑠(𝑦), 𝜇𝑠(𝑧)} [from (ii) of definition 3.1] we have 𝜇𝑠(𝑥) ≥ min {𝜇𝑠(𝑦), 𝜇𝑠(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴 and 𝛾𝑠(𝑥) ≤ max{𝛾𝑠((𝑥 → 𝑦 ∗), 𝛾𝑠(𝑦))} [from (v) of definition 3.1] ≤ max {max{𝛾𝑠((𝑥 → 𝑦 ∗) → 𝑧)∗), 𝛾𝑠(𝑧)} , 𝛾𝑠(𝑦)} = max {max{𝛾𝑠(0), 𝛾𝑠(𝑧)} , 𝛾𝑠(𝑦)} = max {𝛾𝑠(𝑦), 𝛾𝑠(𝑧)} [from (iii) of definition 3.1] hence, 𝛾𝑠(𝑥) ≤ max {𝛾𝑠(𝑦), 𝛾𝑠(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴.∎ intuitionistic fwi-ideals of residuated lattice wajsberg algebras 187 definition 3.7. an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) of residuated lattice wajsberg algebra 𝐴 is called an intuitionistic fuzzy lattice ideal of 𝐴 if it satisfies the following axioms for all𝑥, 𝑦 ∈ 𝐴, (i) 𝑆 = (𝜇𝑠, 𝛾𝑠) is intuitionistic monotonic (ii) 𝜇𝑠(𝑥 ∨ 𝑦) ≥ min{𝜇𝑠(𝑥), 𝜇𝑠(𝑦)} (iii) 𝛾𝑠(𝑥 ∨ 𝑦) ≤ max{𝛾𝑠(𝑥), 𝛾𝑠(𝑦)}. remark 3.8. in the definition 3.7(ii) and (iii) can be equivalently replaced by 𝜇𝑠(𝑥 ∨ 𝑦) = min{𝜇𝑠(𝑥), 𝜇𝑠(𝑦)} and 𝛾𝑠(𝑥 ∨ 𝑦) = max {𝛾𝑠(𝑥), 𝛾𝑠(𝑦)} respectively by 𝛾. example 3.9. let 𝐴 be a residuated lattice wajsberg algebra defined in the example 3.2 and 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic fuzzy set of 𝐴 defined by 𝜇𝑠(𝑥) = { 1 if 𝑥 ∈ (0, 𝑑) for all 𝑥 ∈ 𝐴 𝑚 otherwise for all 𝑥 ∈ 𝐴 ; 𝛾𝑠(𝑥) = { 0 if 𝑥 ∈ (0, 𝑑) for all 𝑥 ∈ 𝐴 𝑛 otherwise for all 𝑥 ∈ 𝐴 where 𝑚, 𝑛 ∈ [0, 1] and 𝑚 + 𝑛 ≤ 1. [from the definition 3.11] then, 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitionistic fuzzy lattice ideal of residuated lattice wajsberg algebra 𝐴. proposition 3.10. let𝐴be a residuated lattice wajsberg algebra. every intuitionistic 𝐹𝑊𝐼-ideal of 𝐴 is an intuitionistic fuzzy lattice ideal of 𝐴. proof. let 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic fuzzy lattice ideal of 𝐴. then we have 𝑆 = (𝜇𝑠, 𝛾𝑠) is intuitionistic monotonic. [from proposition 3.6] now ((𝑥 ∨ 𝑦) → 𝑦)∗ = (((𝑥 → 𝑦) → 𝑦)) → 𝑦)∗ from (ii) of definition 2.3] = (𝑥 → 𝑦)∗ ≤ (𝑥∗)∗ for all 𝑥, 𝑦 ∈ 𝐴 [from (ix) of proposition 2.2] it follows that 𝜇𝑠(𝑥 ∨ 𝑦) ≥ min{𝜇𝑠(𝑥 ∨ 𝑦) ⊗ 𝑦, 𝜇𝑠(𝑦)} [from definition 3.1 and definition 3.7] ≥ min{𝜇𝑠(𝑥 → 𝑦) → 𝑦) ⊗ 𝑦, 𝜇𝑠(𝑦)} [from (ii) of definition 2.3] ≥ min {𝜇𝑠(0), 𝜇𝑠(𝑦)} ≥ min{𝜇𝑠(𝑥), 𝜇𝑠(𝑦)}for all 𝑥, 𝑦 ∈ 𝐴 [from (i) of proposition 2.10] 𝛾𝑠(𝑥) ≤ max {𝛾𝑠((𝑥 ∨ 𝑦) ⊗ 𝑦), 𝛾𝑠(𝑦)} ≤ max{𝛾𝑠((𝑥 → 𝑦) → 𝑦) ⊗ 𝑦) , 𝛾𝑠(𝑦)} [from (ii) of definition 2.3] ≤ max {𝛾𝑠(0), 𝛾𝑠(𝑦)} shanmugapriya and ibrahim 188 ≤ max{𝛾𝑠(𝑥), 𝛾𝑠(𝑦)}for all 𝑥, 𝑦 ∈ 𝐴 [from (ii) of definition 2.10] and we have 𝜇𝑠(𝑥 ∨ 𝑦) ≥ min{𝜇𝑠(𝑥 ∨ 𝑦) → 𝑦) ∗) , 𝜇𝑠(𝑦)} ≥ min {𝜇𝑠(𝑥), 𝜇𝑠(𝑦)} 𝛾𝑠(𝑥) ≤ max {𝛾𝑠((𝑥 ∨ 𝑦) → 𝑦) ∗), 𝛾𝑠(𝑦)} ≤ max{𝛾𝑠(𝑥), 𝛾𝑠(𝑦)} for all 𝑥, 𝑦 ∈ 𝐴. hence, we have 𝑆 = (𝜇𝑠, 𝛾𝑠)is an intuitionistic fuzzy lattice ideal of residuated lattice wajsberg algebra 𝐴. ∎ proposition 3.11. let 𝐴 be a residuated lattice wajsberg algebra. an intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitionistic fwi-ideal of 𝐴 if and only if the fuzzy subsets 𝜇𝑠 and 𝛾𝑠 𝑐 are 𝐹𝑊𝐼-ideal of 𝐴, where 𝛾𝑠 𝑐 (𝑥) = 1 − 𝛾𝑠(𝑥)for all 𝑥 ∈ 𝐴. proof. let 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic fwi-ideal of 𝐴. then 𝜇𝑠 is a fwi-ideal of 𝐴. now, we have 𝛾𝑠 𝑐 = 1 − 𝛾𝑠(0) ≥ 1 − 𝛾𝑠(𝑥) [from (i) of proposition 2.10] 𝛾𝑠 𝑐 (0) = 𝛾𝑠 𝑐 (𝑥) for all 𝑥, 𝑦 ∈ 𝐴 and 𝛾𝑠 𝑐 (𝑥) = 1 − 𝛾𝑠(𝑥) ≥ 1 − max {𝛾𝑠(𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦)} = min{ 1 − 𝛾𝑠(𝑥 ⊗ 𝑦), 1 − 𝛾𝑠(𝑦)} = min{ 𝛾𝑠 𝑐 (𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦)} 𝛾𝑠 𝑐 (𝑥) = 1 − 𝛾𝑠(𝑥) ≥ 1 − max {𝛾𝑠((𝑥 → 𝑦) ∗), 𝛾𝑠(𝑦)} = min{ 1 − 𝛾𝑠((𝑥 → 𝑦) ∗), 1 − 𝛾𝑠(𝑦)} 𝛾𝑠 𝑐 (𝑥) = min{ 𝛾𝑠 𝑐 ((𝑥 → 𝑦)∗), 𝛾𝑠(𝑦)}for all 𝑥, 𝑦 ∈ 𝐴 hence, we have 𝛾𝑠 𝑐 is a fwi-ideal of 𝐴. conversely, assume that 𝜇𝑠 and 𝛾𝑠 𝑐 are fwi-ideal of 𝐴. then, we have 𝜇𝑠(0) ≥ 𝜇𝑠(𝑥)and 1 − 𝛾𝑠(0) = 𝛾𝑠 𝑐 (0) ≥ 𝛾𝑠 𝑐 (𝑥) = 1 − 𝛾𝑠(𝑥) 𝛾𝑠(0) ≤ 𝛾𝑠(𝑥) for all 𝑥, 𝑦 ∈ 𝐴 now, 𝜇𝑠(𝑥) ≥ min {𝜇𝑠 𝑐 (𝑥 ⊗ 𝑦), 𝜇𝑠 𝑐 (𝑦)} = min {1 − 𝜇𝑠(𝑥 ⊗ 𝑦), 1 − 𝜇𝑠(𝑦)} = 1 − max {𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)} 𝛾𝑠(𝑥) ≤ max {𝛾𝑠(𝑥 ⊗ 𝑦), 𝛾𝑠(𝑦)} for all 𝑥, 𝑦 ∈ 𝐴 𝜇𝑠(𝑥) ≥ min {𝜇𝑠 𝑐 (𝑥 → 𝑦)∗, 𝜇𝑠 𝑐 (𝑦)} = min {1 − 𝜇𝑠((𝑥 → 𝑦) ∗), 1 − 𝜇𝑠(𝑦)} = 1 − max {𝜇𝑠((𝑥 → 𝑦) ∗), 𝜇𝑠(𝑦)} 𝛾𝑠(𝑥) ≤ max{𝛾((𝑥 → 𝑦) ∗), 𝛾𝑠(𝑦)}for all 𝑥, 𝑦 ∈ 𝐴 hence, we have 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴.∎ intuitionistic fwi-ideals of residuated lattice wajsberg algebras 189 proposition 3.12. let 𝐴 be a residuated lattice wajsberg algebra and 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitionistic 𝐹𝑊𝐼-ideal of𝐴. then 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitionistic fwi-ideal of 𝐴 if and only if (𝜇𝑠, 𝜇𝑠 𝑐 ) and (𝛾𝑠 𝑐 , 𝛾𝑠) are intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. proof. let 𝑆 = (𝜇𝑠, 𝛾𝑠) be an intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. then, 𝜇𝑠 and 𝛾𝑠 𝑐 are 𝐹𝑊𝐼-ideal of 𝐴[from proposition 3.11] hence, we have (𝜇𝑠, 𝜇𝑠 𝑐 ) and (𝛾𝑠 𝑐 , 𝛾𝑠) are intuitionistic 𝐹𝑊𝐼-ideal of 𝐴. conversely, if (𝜇𝑠, 𝜇𝑠 𝑐 ) and (𝛾𝑠 𝑐 , 𝛾𝑠) are intuitionistic 𝐹𝑊𝐼-idealof 𝐴 [from proposition 3.11] then, the fuzzy sets 𝜇𝑠 and 𝛾𝑠 𝑐 are 𝐹𝑊𝐼-ideal of 𝐴 hence, 𝑆 = (𝜇𝑠, 𝛾𝑠) is an intuitonistic 𝐹𝑊𝐼-ideal of 𝐴. ∎ proposition 3.13. let 𝐴 be residuated lattice wajsberg algebra, 𝑉 a non-empty subset of [0, 1] and {𝐼𝑡 / 𝑡 ∈ 𝑉} a collection of 𝐹𝑊𝐼 -ideal of 𝐴 such that (i) 𝐴 = 𝐼𝑡 𝑡∈𝑣 ⋃ (ii) 𝑟 > 𝑡 if and only if 𝐼𝑟 ⊆ 𝐼𝑡 for any 𝑟, 𝑡 ∈ 𝑉 then the intuitionistic fuzzy set 𝑆 = (𝜇𝑠, 𝛾𝑠) of 𝐴 defined by 𝜇𝑠 = sup{𝑡 ∈ 𝑉/𝑥 ∈ 𝐼𝑡 } and𝛾𝑠 = inf{𝑡 ∈ 𝑉/𝑥 ∈ 𝐼𝑡 } for any 𝑥 ∈ 𝐴 is intuitionistic 𝐹𝑊𝐼 -ideal of 𝐴. proof. according to proposition 3.10, it is sufficient to show that 𝜇𝑠 and 𝛾𝑠 𝑐 are 𝐹𝑊𝐼–idealof 𝐴 for all𝑥 ∈ 𝐴. 𝜇𝑠(0) = sup {𝑡 ∈ 𝑉/0 ∈ 𝐼𝑡 } = sup𝑉 ≥ 𝜇𝑠(𝑥) [from (i) of definition 3.1] if there exists 𝑥, 𝑦 ∈ 𝐴 such that 𝜇𝑠(𝑥) < min {𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)} and 𝜇𝑠(𝑥) < min {𝜇𝑠((𝑥 → 𝑦) ∗), 𝜇𝑠(𝑦)}. there exists 𝑡1 such that 𝜇𝑠(𝑥) < 𝑡1 < min {𝜇𝑠(𝑥 ⊗ 𝑦), 𝜇𝑠(𝑦)} and 𝜇𝑠(𝑥) < 𝑡1 < min {𝜇𝑠((𝑥 → 𝑦) ∗), 𝜇𝑠(𝑦)} it follows that 𝑡1 such that𝑡1 < 𝜇𝑠(𝑥 ⊗ 𝑦),𝑡1 < 𝜇𝑠((𝑥 → 𝑦) ∗),𝑡1 < 𝜇𝑠(𝑦) and hence, there exist 𝑡2, 𝑡3 ∈ 𝑉, 𝑡2 > 𝑡1, 𝑡3 > 𝑡1, (𝑥 ⊗ 𝑦) ∈ 𝐼𝑡2 , (𝑥 → 𝑦) ∗) ∈ 𝐼𝑡2 and 𝑦 ∈ 𝐼𝑡3 it follows that (𝑥 ⊗ 𝑦) ∈ 𝐼𝑡2⋀𝑡3 , (𝑥 → 𝑦) ∗) ∈ 𝐼𝑡2⋀𝑡3 and 𝑦 ∈ 𝐼𝑡2⋀𝑡3 now, we have 𝑥 ∈ 𝐼𝑡2⋀𝑡3 that is, 𝜇𝑠(𝑥) = sup {𝑡 ∈ 𝑉 𝑥 ∈ 𝐼𝑡 } ≥ 𝑡2⋀𝑡3 > 𝑡1 [from definition 2.16] therefore, 𝜇𝑠(𝑥) > 𝑡1 this is a contradiction. hence, we have 𝜇𝑠 is a 𝐹𝑊𝐼 -ideal of 𝐴. 𝛾𝑠 𝑐 is a𝐹𝑊𝐼 -ideal, which can be proved by similar method. ∎ shanmugapriya and ibrahim 190 4. conclusions in this paper, we have introduced the notions of intuitionistic 𝐹𝑊𝐼 –ideal and intuitionistic fuzzy lattice ideal of residuated wajsberg algebras. also, we have shown that every intuitionistic 𝐹𝑊𝐼ideal of residuated lattice wajsberg algebra is an intuitionistic fuzzy lattice ideal of residuated lattice wajsberg algebra. further, we have discussed its converse part. references [1] k. t. atanassav, intuitionistic fuzzy sets, fuzzy sets and systems, 20(1):87-96,1986. 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[13] l. a. zadeh, fuzzy sets, information and control, 8(3):338-353, 1965. ratio mathematica volume 46, 2023 a dynamical analysis of a mathematical model on type-2 diabetic from obestiy gnana priya g. * a.sabarmathi † naga soundarya lakshmi v.s.v‡ abstract a model for type-2 diabetic from obesity with two control variables diet with physical activity and medication is formulated.the disease free and endemic equilibrium points of the model are obtained. the existence of optimal controls is verified through pontryagainaximum principle. the local stability is analyzed using routh-hurwitz criteria. the global stability is studied using lyapunov function. the parameters are chosen based on the female population in india. the aim of this research is to construct a model for type-2 diabetic from obesity using parameters based on the female population in india. we have introduced two control variables as diet with physical activity and medication. the positive endemic equilibrium is obtained. the local and global stability of the model are analyzed with some specific conditions. numerical simulations are carried out to exhibit the flow of variables with controls. our study mainly highlights the awareness of metabolic risk by healthy diet, physical activities and medications. keywords: obesity, diabetic, pontryagain’s maximum principle, lyapunov function, stability. 2020 ams subject classifications: 34d20, 34d23,81t80 1 *department of mathematics, auxilium college, (affiliated to thiruvalluvar university,serkadu,vellore-632115) ; priya.prakasam51@gmail.com †department of mathematics, auxilium college,(affiliated to thiruvalluvar university,serkadu,vellore-632115. ‡department of mathematics, auxilium college, (affiliated to thiruvalluvar university,serkadu,vellore-632115. san9sak14@gmail.com 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1072. issn: 1592-7415. eissn: 2282-8214. ©g. gnana priya et al. this paper is published under the cc-by licence agreement. 156 g. gnana priya, a.sabarmathi and v.s.v. naga soundarya lakshmi 1 introduction obesity and diabetes are two epidemics that are wreaking havoc around the world. obesity puts you at a much increased chance of having diabetes throughout the course of your life. obesity is more than a cosmetic issue. it is a long-term medical condition that can progress to diabetes. obese people account for over 90 percent of people with type 2 diabetes which is characterised by the body’s failure to regulate and control blood sugar levels. obesity allows diabetes to take up residence in the body. obesity increases the risk of developing diabetes which is characterised by an excess of glucose in the bloodstream. obesity is linked to a higher risk of non communicable diseases such as diabetes, heart disease and hypertension. obesity is one of the primary causes of early death and 4.7 million deaths worldwide was recorded in 2017 due to obesity. the key to understanding the link between obesity and diabetes is to look at insulin resistance. physical inactivity, overeating, heredity, medications and psychological factors are the most prevalent causes of obesity. type 2 diabetes has no known treatment however, it can be in control. inactivity, family history, high blood pressure and obese are the most common causes of type 2 diabetics. obesity reduction is a significant objective for the treatment and prevention of type 2 diabetes. many authors studied a mathematical models for obesity and diabetic. a boutayed boutayed et al. [2010] developed a mathematical model for the burdern of diabetes and its complications. wiam boutayeb et al. [2014] introduced a mathematical model for the impact of obesity on predisposed people to type 2 diabetes. nikhil kumar et al. [2017] studied a diabetsity an epidemic with its causes, prevention and control with special focus on dietary regime. miguel a. ortega ortega et al. [2020] illustrated a technique for type 2 diabetes mellitus associated with obesity. abdelfatah kouidere k.abdelfatah et al. [2020] developed a mathematical model for the dynamics of population of diabetics and its complications with effect of behavioral factors. wellars banzi et al. [2021] studied a mathematical model for glucose-insulin system and test of abnormalities of type 2 diabetic patients. in this paper, we formulated and anlaysed a mathematical model for type-2 diabetic from obesity with control variables. 2 formulation of model the system of ordinary differential equation represents the obesity and diabetic model as follows:: 157 a dynamical analysis of a mathematical model on type-2 diabetic ds dt = ωn − (β + µ)s do dt = βs − (α + µ + u)o dd dt = αo − (µ + u + v)d dr dt = (u + v)d + uo −µr (1) with the initial conditions as s(t),o(t),d(t),r(t) ≥ 0. also β,µ,ω,α,u,v > 0. where s(t),o(t),d(t),r(t) are the susceptible, infected by obesity, infected by diabetic and recovery state respectively and ω average birth rate of female, µ average death rate of female, β obesity rate , α diabetic rate, n(t) female population. the control variables are u diet with physical activity and v medication. the following figure shows the considered obesity and diabetic model: figure 1: obesity and diabetic model 3 equilibrium analysis the steady states are g0(0, 0, 0, 0), g1(s, 0, 0, 0), g2(s̃, õ, 0, 0), g3(s′,o′,d′, 0) and g4(s∗,o∗,d∗,r∗). 158 g. gnana priya, a.sabarmathi and v.s.v. naga soundarya lakshmi case(1): g0(0, 0, 0, 0) exists always. case(2): in g1(s, 0, 0, 0), let s be the positive solution of ds dt = 0 from (1), s = ωn (β + µ) > 0 s = ωn (β + µ) > 0 ∴ g1(s, 0, 0, 0) = ( ωn (β+µ) , 0, 0, 0) case(3): in g2(s̃, õ, 0, 0) , let s̃, õ be the positive solutions of ds dt = 0,do dt = 0. from (1), s̃ = ωn (β + µ) > 0 õ = βωn (β + µ)(α + µ + u) > 0 ∴ g2(s̃, õ, 0, 0) = g2 ( ωn (β+µ) , βωn (β+µ)(α+µ+u) , 0, 0 ) case(4): in g3(s′,o′,d′, 0), let s′,o′ and d′ be the positive solutions of ds dt = 0,do dt = 0,dd dt = 0 from (1), s′ = ωn (β + µ) o′ = βωn (β + µ)(α + µ + u) d′ = αβωn (β + µ)(α + µ + u)(µ + u + v) ∴ g3(s ′,o′,d′, 0) = ( ωn β+µ , βωn (β+µ)(α+µ+u) , αβωn (β+µ)(α+µ+u)(µ+u+v) , 0 ) case(5): for g4(s∗,o∗,d∗,r∗), let s∗, o∗, d∗ and r∗ be the positive solutions of ds dt = 0,do dt = 0,dd dt = 0, dr dt = 0 from (1), s∗ = ωn ( 1 β + µ ) o∗ = ωn ( β (α + µ + u)(β + µ) ) 159 a dynamical analysis of a mathematical model on type-2 diabetic d∗ = ωn ( αβ (α + µ + u)(β + µ)(µ + u + v) ) r∗ = ωn [ 1 µ ( αβ(u + v) (α + µ + u)(β + µ)(µ + u + v) + βu (α + µ + u)(β + µ) )] ∴ the diabesity equilibrium is given by g4(s ∗,o∗,d∗,r∗) = (ωn ( 1 β + µ ) ,ωn ( β (α + µ + u)(β + µ) ) , ωn ( αβ (α + µ + u)(β + µ)(µ + u + v) ) , ωn [ 1 µ ( αβ(u + v) (α + µ + u)(β + µ)(µ + u + v) + βu (α + µ + u)(β + µ) )] ) 4 pontryagain’s maximum principle we compute the optimal controls of the control variables by using pontryagain’s maximum principle. we prove the following four properties : i: the set of controls and state variables that correspond to them is non empty. ii: u is a closed and convex control set. iii: the state system’s r.h.s is bounded by a linear function in the state and control variables. iv: the integrand of the objective functional is convex on u and is bounded below by k2 + k1|u| η, with k1 > 0, k2 > 0, η > 1. to prove i: from (1), f1 = ωn − (β + µ)s f2 = βs − (α + µ + u)o f3 = αo − (µ + u + v)d f4 = (u + v)d + uo −µr (2) from (2) ∣∣∂f1 ∂s ∣∣ < ∞, ∣∣∂f1 ∂o ∣∣ < ∞,∣∣∂f1 ∂d ∣∣ < ∞, ∣∣∂f1 ∂r ∣∣ < ∞∣∣∂f2 ∂s ∣∣ < ∞, ∣∣∂f2 ∂o ∣∣ < ∞,∣∣∂f2 ∂d ∣∣ < ∞, ∣∣∂f2 ∂r ∣∣ < ∞∣∣∂f3 ∂s ∣∣ < ∞, ∣∣∂f3 ∂o ∣∣ < ∞,∣∣∂f3 ∂d ∣∣ < ∞, ∣∣∂f3 ∂r ∣∣ < ∞∣∣∂f4 ∂s ∣∣ < ∞, ∣∣∂f4 ∂o ∣∣ < ∞,∣∣∂f4 ∂d ∣∣ < ∞, ∣∣∂f4 ∂r ∣∣ < ∞ 160 g. gnana priya, a.sabarmathi and v.s.v. naga soundarya lakshmi as it is continuous and bounded then there exist a unique solution of (1) by existence and uniqueness theorem. hence, i is satisfied. to prove ii: the control set is given by, u = {(u(t),v(t))/0 ≤ u ≤ 1, 0 ≤ v ≤ 1, t ∈ [0, 1]} u is closed, from our definition. let u,v ∈ u, α ∈ [0, 1] thenαu + (1 −α)v ≥ 0 as u ≤ 1 and v≤ 1, 0 ≤ αu + (1 −α)v ≤ (1 −α), for all u,v ∈ u hence ii is satisfied. to prove iii: from (2), f1 ≤ ωn f2 ≤ βs −uo f3 ≤ αo − (u + v)d f4 ≤ (u + v)d + uo it can be rewritten as follows: f(t,x,u) ≤   0 0 0 0 β 0 0 0 0 α 0 0 0 0 0 0   .   s o d r   +   0 −o −d d + o   [ u v ] which can be written as the linear combination of controls∣∣f(t,x,u)∣∣ ≤‖m̄‖ ∣∣x̄∣∣ + ‖d + o‖|u(t),v(t)| ≤ k [∣∣x̄∣∣ + |u(t),v(t)|] as s,o,d are bounded and k is upper bound. ∴ iii is satisfied. to prove iv: the control variables and state variables are non negative u,v ∈ u. u,v is convex and closed. then from (3), j = ai(t) + ω1u 2 + ω2v 2 2 161 a dynamical analysis of a mathematical model on type-2 diabetic ≥−ai(t) + ω 2 (u2 + v2) = −k2 + k1(u,v)2 where ω = ω1 + ω2, k1 = ω 2 . here k2 > 0 , k1 > 0 and η = 2 > 1 hence iv is satisfied. hence there exists an optimal control variables which is found in the following section. 5 existence of optimal control the objective functional is defined as in (2) as follows: j = minai(t) + 1 2 ∫ t 0 (ω1u 2 + ω2v 2)dt (3) where ai(t) is the number of individuals, ω1, ω2 are the weight parameters for the cost of diet with physical activity and the cost of medication respectively. from previous section, our system (1) converts into a problem of minimizing. an hamiltonian h is defined as, h(s,o,d,r) = ai(t) + ω1u 2 2 + ω2v 2 2 + λs (ωn − (β + µ)s) + λo (βs − (α + µ + u)o) + λd (αo − (µ + u + v)d) + λr ((u + v)d + uo −µr) (4) the adjoint system is given by dλs dt = λs(β + µ) −λoβ dλo dt = λo(α + µ + u) −αλd −λru dλd dt = λd(µ + u + v) −λr(u + v) dλr dt = λrµ with the final condition λs(t) = λo(t) = λd(t) = λr(t)=0 for free problem. the optimality is found from ∂h ∂u = 0 and ∂h ∂v = 0. ∴ the optimal controls are, u∗ = min ( 1,max ( 0, d(λd−λr)−oλr ω1 )) , v∗ = min ( 1,max ( 0, d(λd−λr) ω2 )) here u∗, v∗ are positive when λd > λr and d(λd −λr) > oλr. 162 g. gnana priya, a.sabarmathi and v.s.v. naga soundarya lakshmi 6 local stability the jacobian matrix for the system (1) is   −(β + µ) 0 0 δ β −(α + µ + u) 0 0 0 α −(µ + u + v) 0 0 u u + v −µ   (5) at the interior equlilibrium (5) becomes   −ωn s 0 0 0 β −βs o 0 0 0 α −αo d 0 0 u u + v −(u+v)d+uo r   (6) the characteristic equation of (6) is given by ∣∣∣∣∣∣∣∣ −ωn s −λ 0 0 0 β −βs o −λ 0 0 0 α −αo d −λ 0 0 u (u + v) −(u+v)d+uo r −λ ∣∣∣∣∣∣∣∣ = 0 λ4 + ( ωn s + βs o + αs d + uo r + (u + v) r ) λ3 + ( ωnβ o + (u + v)dωn sr + uoωn sr + ωnα d + (u + v)dβs r + uβs r + βαs2 do + (u + v)αs r + uoαs rd )λ2 + ( (u + v)αωndβ r + uβωn r + ωnβαs do + (u + v)αωn r + ωnuoα rd + (u + v)αβs2 ro + uαβs2 rd )λ + (u + v)αβωns ro + uαβωns rd = 0 (7) 163 a dynamical analysis of a mathematical model on type-2 diabetic comparing (4) with s4 + as3 + bs2 + cs + d = 0 where a = ωn s + βs o + αs d + uo r + (u + v) r , b = ωnβ o + (u + v)dωn sr + uoωn sr + ωnα d + (u + v)dβs r + uβs r + βαs2 do + (u + v)αs r + uoαs rd , c = (u + v)αωndβ r + uβωn r + ωnβαs do + (u + v)αωn r + ωnuoα rd + (u + v)αβs2 ro + uαβs2 rd , d = (u + v)αβωns ro + uαβωns rd here a > 0; d > 0; ab −c > 0; c(ab −c) −a2d > 0 when 1 −s > 0 and the system (1) is locally stable by the routh-hurwitz criteria. 7 global stability we establish the lyapunov function given below, v (s,o,d,r) = ( (s −s∗) −s∗ln s s∗ ) + l1 ( (o −o∗) −o∗ln o o∗ ) +l2 ( (d −d∗) −d∗ln d d∗ ) + l3 ( (r−r∗) −r∗ln r r∗ ) (8) differentiate (8) with respect to t, dv dt = ( s −s∗ s ) ds dt + ( o −o∗ o ) do dt + ( d −d∗ d ) dd dt + ( r−r∗ r ) dr dt from the model equations (1), dv dt = ( s −s∗ s ) (ωn − (β + µ) s) + ( o −o∗ o ) (βs − (α + µ + u)o) +( d −d∗ d ) (αo − (µ + u + v)d) + ( r−r∗ r ) ((u + v)d + uo −µr) = (s −s∗) ( ωn s − (β + µ) ) + l1(o −o∗) ( βs o − (α + µ + u) ) 164 g. gnana priya, a.sabarmathi and v.s.v. naga soundarya lakshmi +l2(d −d∗) ( αo d − (µ + u + v) ) + l3(r−r∗) ( (u + v)d + uo r −µ ) at (s∗,o∗,d∗,r∗), we have dv dt = (s −s∗)[ ωn s − ( ωn s∗ )] + l1(o −o∗)[ βs o − ( βs∗ o∗ )] +l2(d−d∗)[ αo d −( αo∗ d∗ )]+l3(r−r∗)[(u+v)( d + uo r −( (u + v)d∗ + uo∗ r∗ )] = (s −s∗)ωn( 1 s − 1 s∗ ) + l1(o −o∗)β( s o − s∗ o∗ ) +l2(d −d∗)α( o d − o∗ d∗ ) + l3(r−r∗)[(u + v)( d r − d∗ r∗ ) + u( o r − o∗ r∗ )] choosing l1 = 1 β , l2 = 1 α , l3 = 1 (u+v)u dv dt = (s −s∗)ωn ( 1 s − 1 s∗ ) + (o −o∗)β β ( s o∗ − s∗ o∗ ) + (d −d∗)α α( o d − o∗ d∗ ) + (r−r∗) (u + v)u ( (u + v)d r − d∗ r∗ ) + (r−r∗) (u + v)u ( u( d r − d∗ r∗ ) ) = − ωn(s −s∗)2 ss∗ + 1 oo∗ [soo∗ −s∗o2 −so∗2 + s∗o∗o] + 1 dd∗ [od∗d −d2o∗ −od∗2 + d∗o∗d] + 1 urr∗ [dr∗r−r2d∗ −r∗2d + r∗rd∗] + 1 (u + v)rr∗ [dr∗r−r2d∗ −r∗2d + r∗rd∗] = −(s −s∗)ωn ( s −s∗ ss∗ ) + (o −o∗) ( so∗ −s∗o oo∗ ) + (d −d∗)( od∗ −do∗ dd∗ ) + r−r∗ u ( dr∗ −rd∗ rr∗ ) + r−r∗ u + v ( dr∗ −rd∗ rr∗ ) = −ωn(s −s∗)2 ss∗ + ( s − so∗ o − s∗o o∗ + s∗ ) + ( o − od∗ d + do∗ d∗ + o∗ ) + 1 u ( r− dr∗ r + d∗r r∗ + r∗ ) + 1 u + v ( r− dr∗ r + d∗r r∗ + r∗ ) ∴ dv dt < 0, when r s < r ∗ s∗ , s o < s ∗ o∗ , o d < o ∗ d∗ ,d r < d ∗ r∗ ,o r < o ∗ r∗ here the system (1) is globally asymptotically stable by lyapunov theorem. 165 a dynamical analysis of a mathematical model on type-2 diabetic 8 numerical simulations we have choose the values of parameters as ω = 0.0234, µ = 0.008, α = 0.2792, β = 0.23 based on a female population. figure (2) shows the flow of variables with control parameters. the obesity rate figure 2: obesity and diabetic with control parameters figure 3: obesity and diabetic without control parameters and diabetic rate are stable with high recovery rate with respect to time . figure (3) shows the flow of variables without control parameters. the diabetic rate is high due to no recovery rate. figure (4) shows the flow of individuals of susceptible class for various values of β. if β values are increasing, the number of individuals of susceptible class move 166 g. gnana priya, a.sabarmathi and v.s.v. naga soundarya lakshmi figure 4: flow of individuals in susceptible class for various values of β towards the infected by obesity class which becomes stable with respect to time. figure 5: infected by obesity class for different values of α with control parameters figure(5) and figure(6) show the flow of individuals of infected by obesity class for various values of α with and without control parameters respectively. if we use the control parameters by giving different values to α, then the number of individuals in obesity class will be stable with respect to time. otherwise it will increase the diabetic rate. while comparing, figures (5) and (6) the obesity rate of figure (5) is higher than the obesity rate of figure(6). so we can understand that the obesity rate can be managed with control parameters. which gives that the 167 a dynamical analysis of a mathematical model on type-2 diabetic figure 6: infected by obesity class for different values of α without control parameters obesity rate is controlled with the control parameters. figure 7: infected by diabetic class for different values of α with control parameters figure(7) and figure(8) show the flow of individuals in infected by diabetic class for various values of α with and without control parameters respectively. if we use the control parameters by giving different values to α, then the number of individuals in diabetic class will be stable with respect to time. otherwise it will reduce the recovery rate. while comparing, figures (7) and (8) the diabetic rate of figure (7) is higher than the diabetic rate of figure(8). so we can understand that 168 g. gnana priya, a.sabarmathi and v.s.v. naga soundarya lakshmi figure 8: infected by diabetic class for different values of α without control parameters the diabetic rate can be managed with control parameters. which gives that the diabetic rate is controlled with the control parameters. figure 9: recovered class for various values of control parameters figure(9) shows the flow of individuals in recovered state for the various values of control parameters. we can observe the following cases: case 1 : the recovery rate increases when both the control parameters exist. case 2 : the recovery rate is slightly decrease when the control parameter diet with physical activity is involved. casse 3 : the recovery rate decreases from case 2, when the control parameter medication is involved. case 4 : there is no recovery rate, when no control parameters are involved. 169 a dynamical analysis of a mathematical model on type-2 diabetic hence, both the control parameters are required to control the obesity and diabetic rates and to increase the recovery rate. 9 conclusions this paper presents a mathematical model for type 2 diabetic from obesity with controls diet with physical activity and medication. by the pontryagin’s maximum principle, the optimal levels of two control variables are attained with some specific conditions. an optimal control approach is proposed in order to reduce the burden of obesity to diabetic. the positive equilibrium points have been found. the system is locally asymptotically stable with a condition on susceptible population. the system is globally asymptotically state in the identified parametric domain. the numerical simulations show the effectiveness of proposed control strategies. in the numerical simulations, we can observe that the flow of diabetic state was dropped down, when the two control parameters are involved. we can also observe that the obesity rate can be managed at moderate levels when the controls parameters are at optimal levels. hence the two introduced control parameters healthy diet with physical activity and medication are effectively reduce the rates of obesity and diabetic and increase the recovery rate. references w. banzi, i. kambutse, v. dusabejambo, e. rutaganda, f. minani, j. niyobuhungiro, l. mpinganzima, and j. m. ntaganda. mathematical modelling of glucose-insulin system and test of abnormalities of type 2 diabetic patients,. international journal of mathematics and mathematical sciences, 2021, 2021. w. boutayeb, m. e. lamlili, a. boutayeb, and m. derouich. mathematical modelling and simulation of β-cell mass, insulin and glucose dynamics: effect of genetic predisposition to diabetes. journal of biomedical science and engineering, 2014, 2014. a. boutayed, e. twizell, k.achouayb, and a.chetouani. on the solution of fourth-order rational recursive sequence. advanced studies in contemporary mathematics, 20:525– 545, 2010. k.abdelfatah, l. abderrahim, f. hanane, b.omar, and r.mostafa. a new mathematical modeling with optimal control strategy for the dynamics of population of diabetics and its complications with effect of behavioral factors. journal of applied mathematics, hindawi, 2020. 170 g. gnana priya, a.sabarmathi and v.s.v. naga soundarya lakshmi n. kumar, n. puri, f. marotta, t. dhewa, s. calabrò, m. puniya, and j. carter. new insights into the noise reduction wiener filter. audio, speech, and language processing. functional foods in health and disease, 7:1 –16, 2017. m. a. ortega, o. fraile-martı́nez, i. naya, n. garcı́a-honduvilla, m. álvarezmon, j. buján, á. asúnsolo, and b. de la torre. type 2 diabetes mellitus associated with obesity (diabesity). the central role of gut microbiota and its translational applications. nutrients, 12:2749, 2020. 171 ratio mathematica volume 48, 2023 results on binary soft topological spaces p. g. patil* asha g. adaki† abstract the baire space is used in the proof of results in many areas of analysis and geometry, including some of the fundamental theorems of functional analysis. the concept of baire spaces has been studied extensively in general topology in [5, 8, 9, 10, 11, 15]. thangaraj and anjalmose [35] studied baire spaces in the context of fuzzy theory. in this paper, we discussed the notions of the binary soft nowhere dense, binary soft dense, binary soft gδ-set, binary soft first and second category sets, binary soft baire spaces. many of their properties are revealed and different characterizations of each are given. in conclusion, we determined some conditions under which the subspace property of a baire space is preserved. keywords: binary soft set, binary soft nowhere dense set, binary soft dense set, binary soft gδ-set, binary soft baire space. 2020 ams subject classifications: 54a05, 54e52. 1 1 introduction the idea of soft sets was introduced by molodtsov [14] in 1999. soft set theory allows researchers to choose the type of parameters they need, which greatly simplifies decision-making and makes the method more productive in the absence of partial data. later, shabir et al. [19] started researching on soft topological spaces. many researchers continued their work on soft topology, including *department of mathematics, karnatak university, dharwad 580 003, india. pgpatil@kud.ac.in. †department of mathematics, karnatak university, dharwad 580 003, india. asha.adaki@kud.ac.in 1received on february 18, 2023. accepted on july 20, 2023. published on august 1, 2023. doi: 10.23755/rm.v42i0.1136. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. p. g. patil and asha g. adaki aygunoghu [4], ahmad [2], maji [13] and hussain [12]. in 2016, acikgöz et al. [1] presented the idea of binary soft set theory on two universal sets and investigated various features. later, patil et al. [16] introduced the separation axioms in binary soft topological spaces such as n − t0, n − t1 and n − t2 spaces. baire space is a space through which applications of completeness are made in analysis. also, it is used in well-known theorems as the closed graph theorem, the open mapping theorem and the uniform boundedness theorem. the baire category theorem (bct), which specifies sufficient conditions for a topological space to be a baire space, is a significant result in general topology and functional analysis. further, bct is used to prove hartogs’s theorem, a fundamental result in the theory of several complex variables. the baire space was initially introduced in bourbaki’s [7] topologie generale chapter lx and was named in honour of rene louis baire. baire spaces have been extensively studied in classical topology [5, 8, 9, 10, 11, 15]. the first and second category sets were first established by rene louis baire in 1899 [6]. thangaraj and anjalmose introduced and studied the concept of baire spaces in fuzzy topology [20]. riaz and fatima [18] studied the concept of soft baire spaces in soft metric spaces. ameen and khalaf investigated the soft baire invariance using soft semicontinuous, soft somewhat continuous and soft somewhat open functions in [3]. in this article, we introduced a class of binary soft topological space called binary soft baire space and studied some topological properties that are preserved by proper subspaces such as binary soft nowhere dense, binary soft dense, binary soft first and second category sets and binary soft baire spaces. we obtained some conditions under which subspace property is preserved. later, we shall show that binary soft compact, n − t2 space falls in the class of binary soft baire space. we have referred the paper [1] for the definition of binary soft (briefly bs) set, binary absolute soft set, binary null soft set, union of two bs sets, intersection of two bs sets and [16, 17] for bs topology, binary soft topological space (briefly bsts), bs subspace and bs relative topology. 2 binary soft nowhere dense set definition 2.1. a bs subset (f, e) of a bsts (x1, x2, τb, e) is called a bs dense set (resp. bs co-dense set) if (f, e) = ˜̃e (resp. (f, e)⊚ = ˜̃ϕ). definition 2.2. a bs subset (f, e) of a bsts (x1, x2, τb, e) is called a bs nowhere dense set if ( (f, e) )⊚ = ˜̃ ϕ. definition 2.3. a bs subset (f, e) of a bsts (x1, x2, τb, e) is called a bs gδ-set if it is countable bs intersection of bs open sets. results on binary soft topological spaces example 2.1. let x1 = r and x2 = q be two initial universal sets and e = {e1, e2} be a set of parameters and τb = { ˜̃ e, ˜̃ ϕ, (f1, e), (f2, e), (f3, e)}, where (f1, e) = {(e1, (n, n)), (e2, (q, q))}, (f2, e) = {(e1, (q ′ , ϕ)), (e2, (ϕ, ϕ))}, (f3, e) = {(e1, (n ∪ q ′ , n)), (e2, (q, q))} then bs closed sets are ˜̃e, ˜̃ϕ (f1, e) ′ ={(e1, (r \ n, q \ n)), (e2, (q ′ , ϕ))}, (f2, e) ′ ={(e1, (q, q)), (e2, (r, q))}, (f3, e) ′ ={(e1, (q \ n, q \ n)), (e2, (q ′ , ϕ))}. consider the bs sets (f3, e) = {(e1, (q ′ ∪ n, n)), (e2, (q, q))} then, (f3, e) = ˜̃ e. therefore, (f3, e) is a bs dense set in (r, q, τb, e). (g1, e) = {(e1, (q−, q−)), (e2, (q ′ , ϕ))} then, (g1, e)⊚ = ˜̃ ϕ and ( (g1, e) )⊚ = ˜̃ ϕ. therefore, (g2, e) bs co-dense and bs nowhere dense set (r, q, τb, e). (g2, e) = {(e1, (q+ \n, q+ \n)), (e2, (q ′ , ϕ))} then, (g2, e)⊚ = ˜̃ ϕ. therefore (g2, e) is also a bs nowhere dense set (r, q, τb, e). theorem 2.1. let (f, e) be a bs subset of a bsts (x1, x2, τb, e) then the following assertions are interchangeable: 1. (f, e) is bs nowhere dense in (x1, x2, τb, e). 2. ˜̃e − (f, e) is bs dense in (x1, x2, τb, e). 3. for each non-empty bs open set (g, e) in (x1, x2, τb, e), there exist a non-empty bs open set (h, e) in (x1, x2, τb, e) such that (h, e) ˜̃⊆(g, e) and (h, e)˜̃∩(f, e) = ϕ. proof. (1) ⇒ (2), let (f, e) is bs nowhere dense set in (x1, x2, τb, e) then ( (f, e) )⊚ = ˜̃ ϕ. let (w, e) be a bs open subset of (x1, x2, τb, e). since,( (f, e) )⊚ = ˜̃ ϕ therefore (w, e) intersects ˜̃e − ((f, e)) this holds for all bs open subsets of (x1, x2, τb, e). therefore, ˜̃ e − ((f, e)) is bs dense set in (x1, x2, τb, e). (2) ⇒ (3), let ˜̃e − ((f, e)) be a bs dense set in (x1, x2, τb, e) and (g, e) be non-empty bs open subset of (x1, x2, τb, e) then, (g, e)˜̃∩( ˜̃ e−(f, e)) ̸= ϕ. let (h, e) = (g, e)˜̃∩( ˜̃e−(f, e)) then clearly (h, e)˜̃⊆(g, e) and (h, e)˜̃∩(f, e) = ϕ. (3) ⇒ (1), suppose (f, e) is not bs nowhere dense set in (x1, x2, τb, e). that p. g. patil and asha g. adaki is ( (f, e) )⊚ ̸= ˜̃ϕ then, any non-empty bs open subset of ( (f, e) )⊚ would intersect (f, e), which is a contradiction. therefore, (f, e) is bs nowhere dense in (x1, x2, τb, e). the following theorem gives the relation between bs dense and bs nowhere dense sets. theorem 2.2. if (f, e) is a bs open and dense subset of (x1, x2, τb, e) then, (f, e)c is bs nowhere dense set. proof. let (f, e) is a bs open and dense subset of (x1, x2, τb, e) then, (f, e) = ˜̃ e, this implies ˜̃e − (f, e) = ˜̃ϕ, ( ˜̃e − (f, e))⊚ = ˜̃ϕ, ((f, e)c)⊚ = ˜̃ϕ. since, (f, e)c is bs closed subset. therefore, (f, e)c is bs nowhere dense set. remark 2.1. the converse of the preceding theorem is generally untrue. it can be demonstrated using the example below. example 2.2. let x1 = {a1, a2, a3}, x2 = {b1, b2, b3}, e = {e1, e2} and τb = { ˜̃ e, ˜̃ ϕ, (f1, e), (f2, e), (f3, e), (f4, e), (f5, e), (f6, e), (f7, e), (f8, e)} where (f1, e) = {(e1, ({a1}, {b1})), (e2, ({a2}, {b2}))}, (f2, e) = {(e1, ({a2}, {b2})), (e2, ({a3}, {b1}))}, (f3, e) = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a2, a3}, {b1, b2}))}, (f4, e) = {(e1, ({a1}, {b2})), (e2, ({a2}, {b1}))}, (f5, e) = {(e1, ({a1}, {b1, b2})), (e2, ({a2}, {b1, b2}))}, (f6, e) = {(e1, ({a1}, ϕ)), (e2, ({a2}, ϕ))}, (f7, e) = {(e1, ({a1, a2}, {b2})), (e2, ({a2, a3}, {b1}))}, (f8, e) = {(e1, (ϕ, {b2})), (e2, (ϕ, {b1}))} then binary soft closed sets are ˜̃e, ˜̃ϕ (f1, e) ′ ={(e1, ({a2, a3}, {b2, b3})), (e2, ({a1, a3}, {b1, b3}))}, (f2, e) ′ ={(e1, ({a1, a3}, {b1, b3})), (e2, ({a1, a2}, {b2, b3}))}, (f3, e) ′ ={(e1, ({a3}, {b3})), (e2, ({a1}, {b3}))}, (f4, e) ′ ={(e1, ({a2, a3}, {b1, b3})), (e2, ({a1, a3}, {b2, b3}))}, (f5, e) ′ ={(e1, ({a2, a3}, {b3})), (e2, ({a1, a3}, {b3}))}, (f6, e) ′ ={(e1, ({a2, a3}, {b1, b2, b3})), (e2, ({a1, a3}, {b1, b2, b3}))}, (f7, e) ′ ={(e1, ({a3}, {b1, b3})), (e2, ({a1}, {b2, b3}))}, (f8, e) ′ ={(e1, ({a1, a2, a3}, {b1, b3})), (e2, ({a1, a2, a3}, {b2, b3}))}. let (h, e) = {(e1, ({a3}, ϕ)), (e2, ({a3}, {b3}))} be a binary soft subset of (x1, x2, τb, e). clearly, (h, e) = {(e1, ({a2, a3}, {b3})), (e2, ({a1, a3}, {b3}))} then ( (h, e) )⊚ = ˜̃ ϕ. therefore, (h, e) is bs nowhere dense set in (x1, x2, τb, e) but, (h, e)c = {(e1, ({a1, a2}, {b1, b2, b3})), (e2, ({a1, a2}, {b1, b2}))} is not bs open set. results on binary soft topological spaces theorem 2.3. if (f, e) is a bs closed and nowhere dense subset of (x1, x2, τb, e) then, (f, e)c is bs dense subset of (x1, x2, τb, e). proof: let (f, e) is a bs closed and nowhere dense subset of (x1, x2, τb, e) then, ( (f, e) )⊚ = ˜̃ ϕ, this implies (f, e)⊚ = ˜̃ϕ, ˜̃e−(f, e)⊚ = ˜̃e, ( ˜̃e − (f, e)) = ˜̃ e, ((f, e)c) = ˜̃e. therefore, (f, e)c is bs dense set. remark 2.2. the previous theorem’s converse is generally not true. the example below can be used to illustrate it. example 2.3. in the example 2.2 consider, (g, e) = {(e1, ({a1, a2, a3}, {b1, b2, b3})), (e2, ({a1, a2}, {b1, b2, b3}))} be a bs dense subset of (x1, x2, τb, e) but, (g, e)c = {(e1, (ϕ, ϕ)), (e2, ({a3}, ϕ))} is not bs closed set. theorem 2.4. let (y, τby, e) be a bs subspace of (x1, x2, τb, e) and (f, e) be a bs subset of ˜̃y . if (f, e) is bs nowhere dense set in ˜̃y , then (f, e) is bs nowhere dense set in (x1, x2, τb, e). proof: suppose (f, e) is a bs nowhere dense subset of ˜̃y , let (g, e) be a bs open subset of (x1, x2, τb, e) then, there exists a non-empty bs open set (h, e) in ˜̃y such that (h, e)˜̃⊆(g, e)˜̃∩(y, e) and (h, e)˜̃∩(f, e) = ˜̃ϕ. now there exists a bs open set (w, e) in (x1, x2, τb, e) such that (h, e) = (w, e)˜̃∩(y, e) thus (w, e)˜̃⊆(g, e) and (w, e)˜̃∩(f, e) = ˜̃ϕ. therefore, (f, e) is bs nowhere dense set in (x1, x2, τb, e). remark 2.3. the previous theorem’s converse is generally not true. the example below can be used to illustrate it. example 2.4. in the example 2.2, consider the bs subspace ˜̃ y = {(e1, ({a3}, {b3})), (e2, ({a3}, {b3}))} with bs subspace topology τby = { ˜̃ y, ˜̃ ϕ, (g1, e)}, where (g1, e) = {(e1, (ϕ, ϕ)), (e2, ({a3}, ϕ))} and (g1, e) ′ ={(e1, ({a1, a2, a3}, {b1, b2, b3})), (e2, ({a1, a2}, {b1, b2, b3}))}. let (h, e) = {(e1, ({a3}, ϕ)), (e2, ({a3}, {b3}))} be a binary soft subset of (x1, x2, τb, e). clearly, (h, e) is a bs nowhere dense set in (x1, x2, τb, e). in bs subspace (y, τby, e), (h, e) y = ˜̃ y then ( (h, e) y)⊚y = ( ˜̃ y )⊚y ̸= ˜̃ϕ. therefore, (h, e) is not bs nowhere dense set in (˜̃y, τby, e). theorem 2.5. let (˜̃y, τby, e) is bs open or bs dense subspace of (x1, x2, τb, e) and (f, e) is bs nowhere dense in (x1, x2, τb, e) then, (f, e) is bs nowhere dense in (˜̃y, τby, e). p. g. patil and asha g. adaki proof: let (˜̃y, τby, e) be a bs open subspace of (x1, x2, τb, e) and (f, e) is bs nowhere dense in (x1, x2, τb, e). let (g, e) is bs open subset of ( ˜̃ y, τby, e) then (g, e) is bs open subset of (x1, x2, τb, e). therefore, there exists a nonempty bs open subset (h, e) of (g, e) such that (h, e)˜̃∩(f, e) = ˜̃ϕ, (h, e) is also bs open subset of (˜̃y, τby, e). therefore, (f, e) is also bs nowhere dense in ( ˜̃ y, τby, e). the proof is similar in the case of (˜̃y, τby, e) is bs dense subspace of (x1, x2, τb, e). corolary 2.1. the above theorem need not be true for bs closed subspaces. that is, if (˜̃y, τby, e) is a bs closed subspace of a bs topological space (x1, x2, τb, e) and (f, e) is bs nowhere dense set in (x1, x2, τb, e) then, (f, e) need not to be bs nowhere dense set in (˜̃y, τby, e). example 2.5. let x1 = {a1, a2, a3}, x2 = {b1, b2, b3}, e = {e1, e2} and τb = { ˜̃ e, ˜̃ ϕ, (f1, e), (f2, e), (f3, e), (f4, e), (f5, e), (f6, e), (f7, e), (f8, e), (f9, e), (f10, e), (f11, e)} where (f1, e) = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a1, a2}, {b1, b2}))}, (f2, e) = {(e1, ({a3}, {b3})), (e2, ({a3}, {b3}))}, (f3, e) = {(e1, ({a2}, {b1})), (e2, ({a1}, {b2}))}, (f4, e) = {(e1, ({a2, a3}, {b1, b3})), (e2, ({a1, a3}, {b2, b3}))}, (f5, e) = {(e1, ({a3}, {b1})), (e2, ({a3}, {b2}))}, (f6, e) = {(e1, ({a1, a2, a3}, {b1, b2})), (e2, ({a1, a2, a3}, {b1, b2}))}, (f7, e) = {(e1, (ϕ, {b1})), (e2, (ϕ, ϕ))}, (f8, e) = {(e1, (ϕ, {b2})), (e2, (ϕ, {b1}))}, (f9, e) = {(e1, ({a3}, {b1, b3})), (e2, ({a3}, {b2, b3}))}, (f10, e) = {(e1, ({a2, a3}, {b1})), (e2, ({a1, a3}, {b2}))}, (f11, e) = {(e1, (ϕ, {b1})), (e2, (ϕ, {b2}))} then binary soft closed sets are ˜̃e, ˜̃ϕ (f1, e) ′ = {(e1, ({a3}, {b3})), (e2, ({a3}, {b3}))}, (f2, e) ′ = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a1, a2}, {b1, b2}, (f3, e) ′ = {(e1, ({a1, a3}, {b2, b3})), (e2, ({a2, a3}, {b1, b3}))}, (f4, e) ′ = {(e1, ({a1}, {b2})), (e2, ({a2}, {b1}))}, (f5, e) ′ = {(e1, ({a1, a2}, {b2, b3})), (e2, ({a1, a2}, {b1, b3}))}, (f6, e) ′ = {(e1, (ϕ, {b3})), (e2, (ϕ, {b3}))}, (f7, e) ′ = {(e1, ({a1, a2, a3}, {b2, b3})), (e2, ({a1, a2, a3}, {b1, b2, b3}))}, (f8, e) ′ = {(e1, ({a1, a2}, {b2})), (e2, ({a1, a2}, {b1}))}, (f9, e) ′ = {(e1, ({a1, a2}, {b1, b2, b3})), (e2, ({a1, a2}, {b1, b2, b3}))}, (f10, e) ′ = {(e1, ({a1}, {b2, b3})), (e2, ({a2}, {b1, b3}))}, (f11, e) ′ = {(e1, ({a1, a2, a3}, {b2, b3})), (e2, ({a1, a2, a3}, {b1, b3}))}. results on binary soft topological spaces let ˜̃y = {(e1, ({a1, a2}, {b1, b2, b3})), (e2, ({a1, a2}, {b1, b2, b3}))} be a bs closed subset of (x1, x2, τb, e) then the bs subspace topology is τy = { ˜̃ y, ˜̃ ϕ, (g1, e), (g2, e), (g3, e), (g4, e), (g5, e), (g6, e), (g7, e)} where (g1, e) = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a1, a2}, {b1, b2}))}, (g2, e) = {(e1, (ϕ, {b3})), (e2, (ϕ, {b3}))}, (g3, e) = {(e1, ({a2}, {b1})), (e2, ({a1}, {b2}))}, (g4, e) = {(e1, ({a2}, {b1, b3})), (e2, ({a1}, {b2, b3}))}, (g5, e) = {(e1, (ϕ, {b1})), (e2, (ϕ, {b2}))}, (g6, e) = {(e1, (ϕ, {b1})), (e2, (ϕ, ϕ))}, (g7, e) = {(e1, (ϕ, {b1, b3})), (e2, (ϕ, {b2, b3}))} then bs closed sets are ˜̃y, ˜̃ϕ (g1, e) ′ = {(e1, (ϕ, {b3})), (e2, (ϕ, {b3}))}, (g2, e) ′ = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a1, a2}, {b1, b2}))}, (g3, e) ′ = {(e1, ({a1}, {b2, b3})), (e2, ({a2}, {b1, b3}))}, (g4, e) ′ = {(e1, ({a1}, {b2})), (e2, ({a2}, {b1}))}, (g5, e) ′ = {(e1, ({a1, a2}, {b2, b3})), (e2, ({a1, a2}, {b1, b3}))}, (g6, e) ′ = {(e1, ({a1, a2}, {b2, b3})), (e2, ({a1, a2}, {b1, b2, b3}))}, (g7, e) ′ = {(e1, ({a1, a2}, {b2})), (e2, ({a1, a2}, {b1}))}. consider the bs set {(e1, (ϕ, {b3})), (e2, (ϕ, {b3}))} which is bs nowhere dense set in (x1, x2, τb, e), but it is not bs nowhere dense set in the bs closed subspace ( ˜̃ y, τby, e). 3 binary soft baire spaces definition 3.1. a bs subset (f, e) of a bsts (x1, x2, τb, e) is called bs first category set if it is the bs union of countable family of bs nowhere dense sets. definition 3.2. a bs subset (f, e) of a bsts (x1, x2, τb, e) is called bs second category set if it is not bs first category set. definition 3.3. a bs baire space is a bsts such that every non-empty bs open subset is bs second category. example 3.1. consider the example 2.1, the bs set (h, e) = {(e1, (q \ n, q \ n)), (e2, (q ′ , ϕ))} is a bs first category set, because (h, e) = (g1, e) ˜̃∪(g2, e). also, (r, q, τb, e) is a bs baire space. theorem 3.1. the following assertions are interchangeable for a bsts (x1, x2, τb, e): 1. (x1, x2, τb, e) is a bs baire space. p. g. patil and asha g. adaki 2. the bs intersection of any sequence of bs dense open sets is bs dense in (x1, x2, τb, e). 3. the bs complement of any bs first category set in (x1, x2, τb, e) is bs dense in (x1, x2, τb, e). 4. every countable bs union of bs closed sets with empty bs interior in (x1, x2, τb, e) has empty bs interior in (x1, x2, τb, e). proof:(1) ⇒ (2), suppose (x1, x2, τb, e) is a bs baire space. let {(gi, e)} be a sequence of bs dense open sets in (x1, x2, τb, e). let us assume that their bs intersection ˜̃∩(gi, e) is not bs dense in (x1, x2, τb, e), then there exists a bs open set (h, e) such that it doesnot intersect ˜̃∩(gi, e). that is (h, e) = ˜̃ e − ˜̃∩(gi, e), (h, e) = ˜̃∪( ˜̃ e − (gi, e)) where each ˜̃ e − (gi, e)) is bs nowhere dense set in (x1, x2, τb, e). therefore, (h, e) is of bs first category set. which is contradiction to (x1, x2, τb, e) is a bs baire space. therefore, ˜̃∩(gi, e) is a bs dense set in (x1, x2, τb, e). (2) ⇒ (3), let {(fi, e)} be a sequence of bs closed nowhere dense subsets of (x1, x2, τb, e), then their bs union ˜̃∪(fi, e) is a bs first category set in (x1, x2, τb, e), we have to prove ˜̃ e− ˜̃∪(fi, e) is bs dense in (x1, x2, τb, e). let us assume that it is not dense in (x1, x2, τb, e). consider a bs open set (g, e) in (x1, x2, τb, e) such that it does not intersect with ˜̃ e−˜̃∪(fi, e) = ˜̃∩( ˜̃ e−(fi, e)), where each ( ˜̃e − (fi, e)) is bs dense openset in (x1, x2, τb, e) then their binary soft intersection is not bs dense in (x1, x2, τb, e), which is contradiction. therefore, ˜̃e − ˜̃∪(fi, e) is bs dense in (x1, x2, τb, e). (3) ⇒ (4), let {(fi, e)} be a sequence of bs closed sets with empty interiors. suppose ˜̃∪(fi, e) does not has empty bs interior, then ˜̃ e − ˜̃∪(fi, e) would not be bs dense in (x1, x2, τb, e), which is a contradiction. therefore, ˜̃∪(fi, e) has empty bs interior. (4) ⇒ (1), let {(fi, e)} be a sequence of bs nowhere dense subsets in (x1, x2, τb, e). suppose ˜̃∪(fi, e) is a bs open set, then each cl(fi, e) has no bs interior point, that is ( (fi, e) )⊚ = ˜̃ e , then ˜̃∪(fi, e) = (˜̃∪(fi, e)), since ˜̃∪(fi, e) is bs open, this implies it contains all its interior points, therefore ˜̃∪(fi, e) has bs interior points, which is contradiction to (4), therefore ˜̃∪(fi, e) is not bs openset. therefore, bs opensets cannot be written as bs union of bs nowhere dense sets. therefore, (x1, x2, τb, e) is a bs baire space. theorem 3.2. the following assertions are interchangeable for a bsts (x1, x2, τb, e). results on binary soft topological spaces 1. (x1, x2, τb, e) is a bs baire space. 2. (g, e)⊚ = ˜̃ϕ for every bs first category set (g, e) in (x1, x2, τb, e). 3. (h, e) = ˜̃e for every bs residual set (h, e) in (x1, x2, τb, e). proof: (1) ⇒ (2), let (x1, x2, τb, e) be a bs baire space, let (g, e) be a bs first category set then (g, e) = ˜̃∪(gi, e) where (gi, e)’s are bs nowhere dense sets then (g, e)⊚ = ( ˜̃∪(gi, e) )⊚ = ˜̃ ϕ. therefore, (g, e)⊚ = ˜̃ϕ (2) ⇒ (3), let (h, e) be a bs residual set then (h, e)′ is bs first category set i.e. ˜̃ e−(h, e) is bs first category set, then ( ˜̃e−(h, e))⊚ = ˜̃ϕ, then ˜̃e−(h, e) = ˜̃ϕ. therefore, (h, e) = ˜̃e. (3) ⇒ (1), let (f, e) be a bs first category set then ˜̃e − (f, e) is bs residual set then ( ˜̃e − (f, e)) = ˜̃e, then ˜̃e − (f, e)⊚ = ˜̃e, (f, e)⊚ = ˜̃ϕ therefore ( ˜̃∪(fi, e) )⊚ = ˜̃ ϕ where (fi, e)’s are bs nowhere dense sets. therefore, (x1, x2, τb, e) is a bs baire space. theorem 3.3. every bs open or bs dense subspace of a bs baire space is a bs baire space. proof: let (x1, x2, τb, e) be a bs baire space, let ( ˜̃ y, τby, e) be a bs open subspace of (x1, x2, τb, e), consider (fi, e) be a sequence of bs nowhere dense sets in (˜̃y, τby, e), by theorem 2.4 (fi, e) is also a sequence of bs nowhere dense sets in (x1, x2, τb, e). since (x1, x2, τb, e) is a bs baire space then ˜̃∪(fi, e) is also bs nowhere dense set in (x1, x2, τb, e). let us assume that ˜̃∪(fi, e) is not bs nowhere dense set in (˜̃y, τby, e), then ( (˜̃∪(fi, e)) y)⊚y ̸= ˜̃ϕ then there exists a bs open set (g, e) in (˜̃y, τby, e) such that (g, e) ˜̃∩(˜̃∪(fi, e)) ̸= ˜̃ ϕ, since ( ˜̃ y, τby, e) is a bs open subspace of (x1, x2, τb, e). thus (g, e) is a bs open set in (x1, x2, τb, e) then (g, e)˜̃∩(˜̃∪(fi, e)) ̸= ˜̃ ϕ, which is contradiction that (x1, x2, τb, e) is a binary soft baire space, therefore ( (˜̃∪(fi, e)) y)⊚y = ˜̃ ϕ, thus ˜̃∪(fi, e) is bs nowhere dense set in ( ˜̃ y, τby, e). therefore ( ˜̃ y, τby, e) is also a bs baire space. the proof is similar for a bs dense subspace. example 3.2. consider the example 2.5, let ˜̃y = {(e1, ({a1, a2}, {b1, b2})), (e2, ({a1, a2}, {b1, b2}))} be a binary soft open p. g. patil and asha g. adaki subset of a bs baire space (x1, x2, τb, e) then the bs subspace topology is τby = { ˜̃ y, ˜̃ ϕ, (g1, e), (g2, e), (g3, e)} where (g1, e) = {(e1, ({a2}, {b1})), (e2, ({a1}, {b2}))}, (g2, e) = {(e1, (ϕ, {b1})), (e2, (ϕ, {b2}))}, (g3, e) = {(e1, (ϕ, {b1})), (e2, (ϕ, ϕ))} then bs closed sets are ˜̃y, ˜̃ϕ, (g1, e) ′ = {(e1, ({a1}, {b2})), (e2, ({a2}, {b1}))}, (g2, e) ′ = {(e1, ({a1, a2}, {b2})), (e2, ({a1, a2}, {b1}))}, (g3, e) ′ = {(e1, ({a1, a2}, {b1})), (e2, ({a1, a2}, {b1, b2}))}. since, (x1, x2, τb, e) is a bs baire space and ( ˜̃ y, τby, e) is a bs open subspace of (x1, x2, τb, e), then ( ˜̃ y, τby, e) is also bs baire space. corolary 3.1. a bs compact and bs n − t2 space is bs regular. theorem 3.4. if (x1, x2, τb, e) is a bs compact and bs n − t2 space then (x1, x2, τb, e) is a bs baire space. proof: let (x1, x2, τb, e) is a bs compact and bs n−t2 space, let {(fi, e)} be a countable collection of closed sets of (x1, x2, τb, e) having empty bs interiors, to prove ˜̃∪(fi, e) has empty bs interior, that is ˜̃∪(fi, e) does not contain any bs open set. consider a bs openset (g1, e) of (x1, x2, τb, e), then we must find a pair of points (x0, y0) ∈ (g1, e) but does not lie in any of the sets (fi, e). let us first consider the set (f1, e), by hypothesis (f1, e) does not contain (g1, e), that is there exist a point (x1, y1) ∈ (g1, e) but (x1, y1) /∈ s (f1, e), since (x1, x2, τb, e) is a bs compact and bs n − t2 space then by corollary 3.1 it is a bs n-regular space, thus for any (x1, y1) ∈ (g1, e) there exist a bs open set (h1, e) such that (x1, y1) ∈ (h1, e)˜̃⊆(h1, e)˜̃⊆(g1, e) also for any (x1, y1) ∈ x1 × x2 and (x1, y1) /∈ s (f1, e) then, there is a bs open set (h1, e) such that (x1, y1) ∈ (h1, e) and (h1, e)˜̃∩(f1, e) = ˜̃ ϕ [16], (h1, e) ˜̃⊆(g1, e). there fore given any nonempty binary soft open set gn, e, we choose a point (xn, yn) ∈ (gn, e) and it does not lie in the bs closed set (fi, e) for all i, then we choose a bs open set (hn, e) such that (hn, e)˜̃∩(fn, e) = ˜̃ ϕ and (hn, e) ˜̃⊆(gn, e). since (x1, x2, τb, e) is a bs compact space then consider {(hn, e)} be a family of bs closed sets, by finite intersection property of bs closed sets [16], we have ˜̃∩(hi, e) ̸= ˜̃ ϕ, let (x0, y0) ∈ ˜̃∩(hi, e), that is (x0, y0) ∈ (hi, e) for all i, this implies (x0, y0) ∈ (gi, e) also (x0, y0) /∈ s (fi, e) for all i, thus (x0, y0) /∈ s ˜̃∪(fi, e), that is ˜̃∪(fi, e) is also has empty bs interior, there fore (x1, x2, τb, e) is a bs baire space. remark 3.1. the converse of the preceding theorem is generally untrue. it can be demonstrated using the example below. results on binary soft topological spaces example 3.3. let x1 = {a1, a2}, x2 = {b1, b2}, e = {i, ii} and τb = { ˜̃ e, ˜̃ ϕ, (g1, e), (g2, e), (g3, e), (g4, e), (g5, e), (g6, e), (g7, e)} where (g1, e) = {(i, ({a1, a2}, {b1})), (ii, ({a1, a2}, {b1}))}, (g2, e) = {(i, ({a2}, {b1, b2})), (ii, ({a2}, {b1, b2}))}, (g3, e) = {(i, ({a1, a2}, {b2})), (ii, ({a1, a2}, {b2}))}, (g4, e) = {(i, ({a2}, {b1})), (ii, ({a2}, {b1}))}, (g5, e) = {(i, ({a2}, {b2})), (ii, ({a2}, {b2}))}, (g6, e) = {(i, ({a1, a2}, ϕ)), (ii, ({a1, a2}, ϕ))}, (g7, e) = {(i, ({a2}, ϕ)), (ii, ({a2}, ϕ))}. then, (x1, x2, τb, e) is a bs baire space and bs compact space as it is a finite space. but, it is not n−t2 space because, (a1, b1) ̸= (a2, b2) ∈ x1 ×x2 and there does not exist bs disjoint open sets (g, e) and (f, e) such that (a1, b1) ∈ (g, e) and (a2, b2) ∈ (f, e). 4 discussion and conclusion this paper contributes to the area of baire spaces in the bs topological spaces. we defined bs nowhere dense, bs dense, bs first, second category sets and obtained their properties. also, we have introduced bs baire space and obtained their characterizations. we determined some conditions under which the subspace property of a baire space is preserved and established a relation between bs compact and bs baire space. to support the obtained result and relations we have provided examples. in the future, we intend to investigate the further properties of bs topological spaces with baire property. acknowledgment the second author is grateful to karnataka science and technology promotion society(ksteps), karnataka, india for the financial support to research work. further, the authors thank the anonymous referees for the helpful remarks and suggestions for the improvement of this paper. references [1] a. acikgöz and n. tas. binary soft set theory. eur.jl. of pure. and appl. math, 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[20] g. thangaraj and s. anjalmose. on fuzzy baire spaces. j. fuzzy. math., 21(3):667–676, 2013. microsoft word documento1 microsoft word documento1 ratio mathematica 27 (2014) 49-59 issn:1592-7415 the views of primary education teachers on the verification of multiplication george h. baralis assistant professor of mathematics, faculty of primary education, university of athens, greece gmparalis@primedu.uoa.gr abstract learning and using the four mathematical operations -addition, subtraction, multiplication and divisionare very important in the primary school syllabus curriculum. the verifications for the correctness of the operations are simple since they can be justified with the use of basic mathematical properties. however, this is not the case for one of the verifications of multiplication which seems to be preferred by most of the elementary school teachers in their practice. with this verification, the control of multiplication’s correctness is only a necessary but not sufficient condition and it is based on the numbers’ theory. in this paper we present the findings of a study on the views of a group consisted of twenty four elementary school teachers using activities related to the operation of the multiplication and its verification. key words: multiplication and its verifications; elementary school teachers. 2000 ams: 40a05, 65b10. 1 introduction university education and continuous training of primary school teachers must include the essential scientific and technological knowledge which will enable them to contribute to education promotion taking into account the 49 george h. baralis added value of their pedagogical role and practices. the scientific knowledge of the cognitive objects taught in elementary school and of the way that teaching is performed are essential prerequisites in order for the teachers to be successful in their work. during their continuous education practicing teachers can face a variety of issues such as completion of their basic education, introduction of new methods of teaching or even reforms of the educational system. in the framework of the continuous education in mathematics of a team consisted of twenty four primary school teachers, we introduced a curriculum which included three hours of teaching multiplication and its verifications. in the beginning we analyzed the conceptual field of multiplicative structures. as it is well known the theory of conceptual fields according to vergnaud [1], has two aims: ”to describe and analyze the progressive complexity of competences that students develop in mathematics inside and outside the school and to establish better connections between the operational form of knowledge, which consists in action in the physical and social world, and the predicative form of knowledge, which consists in the linguistic and symbolic expressions of this knowledge.” the conceptual field is ”a set of problems and situations for the treatment of which concepts, procedures, and representations of different but narrowly interconnected types are necessary” [2]. the multiplication structures are ”a conceptual field of multiplicative type, as a system of different but interrelated concepts, operations, and problems such as multiplication, division, fractions, ratios, similarity” [2] . ”a single concept does not refer to only one type of situation and a single situation cannot be analyzed with only one concept” [3]. in addition, conceptual field is ”a set of situations, the mastering of which requires mastery of several concepts of different natures” [3]. ”concepts-in-action serve to categorize and select information whereas theorems-in-action serve to infer appropriate goals and rules from the available and relevant information”[4]. even the most complicated concepts, in order to be meaningful and functional should be placed in a framework and be explained via examples. thus a concept is simultaneously a set of situations, a set of operational constants and a set of linguistic and symbolic representations. the use of the framework of conceptual fields is necessary for the analysis of the continuities and the discontinuities of development in mathematics and for the invention of situations that will prompt and help students to move along the multifaceted complexity of conceptual field” [1] . the multiplicative structures constitute a part of the field of the additive structures, if multiplication is considered as repeated addition. however, due 50 the views of primary education teachers on the verification of multiplication to the fact that multiplication has its own internal structure and organization, it can also be considered as an independent operation. the approach of multiplication as a repeated addition has a lot of limitations. some of them are that ”it does not easily generalize to rationals, it does not demonstrate commutativity and it emphasizes grouping over sharing approaches to division” [5]. the difficulties in learning multiplicative reasoning are often due to the different ways in which students think about multiplication problems and the models they use. the standard algorithm for teaching the multiplication of larger numbers requires memorization of the basic multiplication facts. however, a wide variety of efficient, alternative algorithms exists based on the history of mathematics, such as finger multiplication, multiplication’s area model, lattice multiplication, line, circle/radius, paper strip, egyptian, russian peasant, etc. [6]. despite the fact that the development of procedural techniques for multiplication in the greek school textbooks is performed mainly via the ”grid method”, the area model does address some of the limitations of repeated addition, even if it does not easily relate to rate. however, most applications of multiplicative reasoning include the rates, therefore, certain researchers propose the use of double number line [5]. the main types of multiplicative structures are: 1. isomorphism of measures 2. multiplication factor, or an area of measures 3. product of measures or cartesian product 4. multiple proportion [2] for the multiplication’s problems most researchers identify four different categories of multiplicative structures. two of them, the equal groups (repeated addition) and the multiplicative comparison, are the most prevalent in the elementary school. the two others, combinations (cartesian products) and problems with product of measures (length on width equal acreage), are used less frequently [7]. the difference of multiplicative problems from the problems of addition or abstraction is due to the fact that their numbers represent different types of things. a number or a factor counts how many sets, groups, or parts of equal size are involved (multiplier) and the other tells the size of each set or part (multiplicand) while the third number is the whole or the total of all the parts [7]. kindergarten and first-grade children can solve multiplication and divisions problems, even if division involves remainders. the strategies they 51 george h. baralis follow are not reflection of multiplicative reasoning, but their involvement in all four operations improves their level of understanding and guides them early enough to the development of multiplicative strategies. strategies used for the algorithm of multiplication are more complex than these for addition and subtraction. in addition the ability to break numbers apart in flexible ways is even more important in multiplication [7]. 2 brief discussion on the verifications of the operations during our group discussions a teacher proposed to also refer to the other operations and theirs verifications. in particular, the teachers suggested that in the set of natural numbers the verifications of the operations of addition, subtraction and division are based on their simple properties. thus, for the addition α+β = γ the verification is β+α = γ, meaning the use of the commutative property. certainly we can also check the correctness of the addition through subtraction, if from the sum we subtract one of the two addends, so we will find the other, that is to say α + β = γ ⇔ γ−α = β or γ − β = α. the second method, as it was pointed out by the teachers, can be used only if the students have already been taught subtraction. for the subtraction α−β = γ the verification is β +γ = α . moreover the correctness of the subtraction can also be checked using the relation α/γ = β. for the division ∆ : δ where ∆ δ = π + ν δ the verification is ∆ = δ ·π + ν with 0 ≤ ν < δ. in each one of the previous cases verification is a necessary and sufficient cond for an operation to be correct. 3 the discussion on multiplication and its verification in order to study the multiplication’s verification the teachers were assigned the following activity: the numbers 4789 and 635 were given and they were asked to: 1. find their product. 2. perform the verification of multiplication and interpret it. 52 the views of primary education teachers on the verification of multiplication before moving to algorithms of multiplication we mentioned various useful representations (material pieces of decimal base, area model, etc), which subsequently were excluded because the numbers used were large. one teacher (teacher 1) was then asked to perform the multiplication using the known traditional algorithm which is also considered as the most difficult (figure 1). if the students fail to understand it, they place -as it was reported by the teachersthe numbers in error columns, they add the carries before they multiply and in this way they make a lot of errors. 4789 × 635 23945 14367 + 28734 3041015 (1) after extensive discussion the following ways of multiplication’s performance were presented such as: 1. (teacher 2) analyze (635) to (600+30+5) and then multiply the multiplicand with 5, 30 and 600 meaning: perform three multiplications and then add their products (figure 2).(teacher 3) this method is correct and it is based on the distributive property of multiplication in regard to addition , which is an important concept for multiplication. (teacher 1) in this method the final products are the partial products of the initial multiplication. 4789 × 5 23945 4789 × 30 143670 4789 × 600 2873400 23945 143670 + 2873400 3041015 (2) 2. (teacher 4) another method is to change the position of the multiplicand by the multiplier, for example to perform the multiplication. this method is based on the use of the commutative property. the resulting partial products are different from the partial products of the multiplication(initial case).(figure 3) 53 george h. baralis 635 × 4789 5715 5080 4445 + 2540 3041015 (3) 3. (teacher 5) we could also perform the multiplication without using carries, as shown below, but in this way we would have 12 products and a ”great” addition afterwards. (figure 4) in that case, emphasis should be given to the proper placement of the obtained products and to the following addition. 4789 × 635 45 40 35 20 27 24 21 12 54 48 42 + 24 3041015 (4) 4. (teacher 6) because the multiplication can be performed as follows and has twelve partial products (figure 5). 54 the views of primary education teachers on the verification of multiplication 4789 × 635 2400000 420000 48000 5400 120000 21000 270 20000 3500 400 + 45 3041015 (5) 5. other types of strategies for multiplication’s performance were also mentioned but were later excluded by the same teachers because the numbers were large. in particular the teachers reported the following: 1) strategies without breaking numbers into parts (they usually use successive additions in different ways (teacher 7), 2) partitioning strategies (breaking the numbers in a variety of ways and subsequently use the distributive property teacher 7) and 3) compensation strategies (breaking the numbers in a variety of ways so that the calculations are easier and result to partial products which are then added, teacher 8) [7]. the other issue that was discussed included whether the previous ways constitute verification of the standardized multiplication’s algorithm. the view of most schoolteachers was that they represent a different way of finding the product by which the correctness of the result of standardized algorithm can be checked, without them constituting verification. as it was found by the discussion that followed, the verification that teachers chose in their practice wasn’t based on the usual properties but followed a special method, that one of the cross [8]. a detailed report of this method was presented and followed by an attempt to highlight the teachers’ views regarding its validity. educator (e): who would want to perform the multiplication’s verification? teacher (t1): (he made the cross and began to supplement it explaining every step that he followed). 1 5 5 5 55 george h. baralis top left corner: we add the digits of the first factor until one-digit number results: 4+7+8+9=29, 2+8=10, 1+0=1. we write this number on this corner. top right corner: we add the digits of the second factor until one-digit number results: 6+3+5=13 and 1+4=5. we write number 5 on this corner. bottom left corner: we calculate the product of the two numbers and we find 1 · 5 = 5. we write it in the bottom left corner. bottom right corner: we calculate the sum of the digits of the two numbers, we find 3+0+4+1+0+1+5=14 and 1+4=5 . we write it on the bottom right corner. educator (e): is the multiplication correct? teacher (t10): yes! educator (e): when a multiplicationif checked with this methodis correct? teacher (t11): the multiplication is correct if the numbers on the second line of the cross are the same. educator (e): is this always true? the following discussion took place: teacher (t12): this method does not always ensure that the multiplication is correct. educator (e): why? teachers (t): for many reasons most of the teachers answered simultaneously. educator (e): who would like to discuss some and then try to analyze them? teacher (t13): one possibility is that the digits that are presented are not placed in the correct position. that is, instead of the number 3041015 that is the correct result, the number 3041015 is written which results by reversing two of its digits. teacher (t14): another possibility is that the digits that are presented in the last product are different (due to an error in the addition of the partial sums) resulting in the same sum. that is to say, instead of the number 3041015 which expresses the correct result and number 5 being the final one-digit sum of digits, the number written is 3041915 which has also the same sum of digits. teacher (t15): another case is when an additional 0 is interposed between the digits of the correct number, meaning instead of the number 3041015, the number 30410015 is written. teacher (t16): or a 0 is added at the end of the number. thus, instead of the number 3041015, the number 30410150 is written. teacher (t17): a 0 is skipped either between the digits of the number or before its end, so for example instead of the number 3041015 we have number 56 the views of primary education teachers on the verification of multiplication 304115. teacher (t18): a badly written 0 can be considered as 9 or vice versa, for example instead of the number 3041015 we have number 3941015. teacher (t2): with this method many errors can occur. educator (e): based on the above cases or others, by this verification the multiplication seems ”correct”, but it is not. so, what do you think about this verification, does it always show if the multiplication is correct or not? teacher (t9): if the multiplication is correctly performed this method verifies it. if, however, the multiplication is not correctly performed, it is not certain that this will be shown by this verification. teacher (t9): why does this happen and how is it explained? educator (e): do you think that this verification is similar to the verifications of the other three operations? teacher (t9): does this verification is a ”necessary condition” for the multiplication to be correct but not sufficient? educator (e): a condition can only be necessary as it happens with the cross verification. afterwards the educator presented the basics from the equal remainder numbers’ theory. he proved the method of the particular verification of multiplication and it was applied to the example that was previously discussed. if two numbers x,y ∈ r, then sx.y ≡ sx ·sy(mod9) proof we know that: x ≡ sx(mod9) and y ≡ sy(mod9), therefore x · y ≡ sx · sy(mod9). however x · y ≡ sx.y(mod9), therefore sx.y ≡ sx · sy(mod9). example of the above proof: consider the numbers x = 4789 and y = 635. then sx = 4+7+8+9 = 28 ≡ 1 (mod9) and sy = 6 + 3 + 5 = 14 ≡ 5 (mod9), therefore sx.y ≡ 1.5(mod9) ≡ 5(mod9). however x.y = 4789.635 = 3041015, sx.y = 3 + 0 + 4 + 1 + 0 + 1 + 5 = 14 ≡ 5. hence, sx.y ≡ sx ·sy(mod9). with this proposal it is proved that the necessary but not the sufficient condition exists in order for the multiplication to be correct. this method, as it was shown from the discussion, is used by all the teachers in their practice, without being as easy and understandable as the verifications of the other three operations. this resulted from teachers’ comments during the interpretation of the cross method, which monopolized the discussion. regarding the clarification of the phrase ’necessary and sufficient condition’ 57 george h. baralis it was shown that this was not always clear to teachers. however this was not the subject of this study. 4 conclusion the standard algorithm for the multiplication of large numbers is brought to europe by the arabic-speaking people of africa and requires the memorization of the basic multiplication facts. multiplication is an important tool not only for constructing a firm foundation for proportional reasoning and the algebraic thinking, but also for solving real-life problems [6]. teachers’ mathematical knowledge includes not only mathematics but also their teaching. the framework of the conceptual field may help them organize appropriate didactic situations and interventions [1]. it is essential that the improvement of the teaching of mathematics regarding teachers’ explanations, the representations and the examples they use and also of the method with which all the above are developed in addition to the way they themselves interact with their students, something that is achieved with their continuous training [9]. in the framework of this particular training it was found that schoolteachers use different ways in order to verify if a multiplication is correct. in conclusion: a. they reverse the multiplier with the multiplicand using the commutative property, b. they calculate partial products and they sum them up afterwards, c. they perform multiplication without using held, d. they calculate the partial products writing analytical numbers in thousands, hundreds, tens and ones using the distributive property, e. they use various informal forms for the execution of multiplication and g. they use the cross method. the justification of the first four ways of verification was complete and understandable by most of the teachers; however, the third way was not used as much as the others. for the verification of the cross method the results showed that all teachers use it in their daily practice at school, even if it is not included in the school textbooks. most of the teachers that participated in this study they used but didn’t empirically consider it reliable since they could not explain it adequately. it is probable that the use of this method of multiplication’s verification is related to teachers’ age, which in our case teachers had at least ten years of professional experience. further research which will include candidate schoolteachers with different curriculum of undergraduate studies would be of special interest. 58 the views of primary education teachers on the verification of multiplication references [1] g. vergnaud, the theory of conceptual fields . human development, s. karger ag (ed.), basel (2009), v.52, 83-94. [2] g. vergnaud, multiplicative structures, acquisition of mathematics concepts and processes , r.a. lesh m. landau (eds), london academic press (1983), p.127-174. [3] g. vergnaud, multiplicative structures, number concepts and operations in the middle grades hiebert j. & behr m. (eds), hillsdale, nj: lawrence erlbaum associates (1988), p.141-161. [4] g. vergnaud, towards a cognitive theory of practice, mathematics education as a research domain: a search for identity, an icmi study sierpinska a. & kilpatrick j. (eds), the netherlands: kluwer academic publishers, dordrecht (1997), v.2, 227-240. [5] d. kchemann : j. hodgen : m. brown models and representations for the learning of multiplicative reasoning: making sense using the double number line proceedings of the british society for research into learning mathematics (march 2011), 31, no.1: 85-90. [6] l. west an introduction to various multiplication strategies thesis for the master of arts in teaching with a specialization in the teaching of middle level mathematics in the department of mathematics, lewis j. advisor, bellevue, ne (2011), 1-22. [7] j. a. van de walle elementary and middle school mathematics teaching developmentally boston: pearson education inc., 6th edition (2007) [8] o. oystein invitation to number theory the mathematical association of america (1967). [9] h. c. hill: b. rowan: d. loewenberg ball effects of teachers’ mathematical knowledge for teaching on student achievement american educational research journal, 42 (2005), no. 2: 371-406. 59 ratio mathematica 28 (2015) 3-14 issn:1592-7415 sums of generalized convergent harmonic series with eight periodically repeated numerators radovan potůček department of mathematics and physics, faculty of military technology, university of defence, brno, czech republic radovan.potucek@unob.cz abstract this contribution deals with the generalized convergent harmonic series with eight periodically repeated numerators and it is a follow-up to author’s papers dealing with the generalized alternating harmonic series with two up to seven periodically repeated numerators. it is derived the only expression of the last numerator depending on preceding numerators for which this series converges. then the formula for the sum of this series is analytically derived. this analytical result is numerically verified by using the cas maple 16. keywords: alternating harmonic series, geometric series, sum of the series, cas maple. 2000 ams subject classifications: 40a05, 65b10. doi:10.23755/rm.v28i1.24 1 introduction and basic notions let us recall the basic terms and notions. the harmonic series is the sum of reciprocals of all natural numbers (except zero), so this is the series ∞∑ n=1 1 n = 1 + 1 2 + 1 3 + · · ·+ 1 n + · · · . 3 radovan potůček the divergence of this series can be proved e.g. by using the integral test or the comparison test of convergence. the series ∞∑ n=1 (−1)n+1 n = 1− 1 2 + 1 3 − 1 4 + 1 5 −··· is known as the alternating harmonic series. this series converges by the alternating series test. in particular, the sum is equal to the natural logarithm of 2: 1− 1 2 + 1 3 − 1 4 + 1 5 −··· = ln 2. 2 sum of generalized alternating harmonic series with two up to seven periodically repeated numerators this paper is a continuation of the author’s contributions [1], [2], [3], [4], [5] and [6]. the paper [1] deals, among others, with the generalized alternating harmonic series with two periodically repeated numerators (1,a), i.e with the series of the form ∞∑ n=1 ( 1 2n−1 + a 2n ) = 1 1 + a 2 + 1 3 + a 4 + 1 5 + a 6 + 1 7 + a 8 + 1 9 + a 10 + · · · , where a ∈ r. in entire agreement with the well-known fact it was derived that the only one value of the coefficient a, for which this series converges, is a = −1 and that the sum of this series is s = ln 2. the paper [2] deals with the generalized alternating harmonic series with three periodically repeated numerators (1,a,b), i.e. with the series ∞∑ n=1 ( 1 3n−2 + a 3n−1 + b 3n ) = 1 1 + a 2 + b 3 + 1 4 + a 5 + b 6 + 1 7 + a 8 + b 9 + · · · . it was derived that the only value of the coefficient b ∈ r, for which this series converges, is b = −a−1, and that the sum of this series is given by the formula s(a) = a + 1 2 ln 3− a−1 6 √ 3 π. the contribution [3] deals with the generalized alternating harmonic series with four periodically repeated numerators (1,a,b,c), i.e. with the series ∞∑ n=1 ( 1 4n−3 + a 4n−2 + b 4n−1 + c 4n ) = 1 1 + a 2 + b 3 + c 4 + 1 5 + a 6 + b 7 + c 8 +· · · . 4 sums of generalized convergent harmonic series with eight numerators it was derived that the only value of the coefficient c ∈ r, for which this series converges, is c = −a− b−1 and it was also derived that the sum of this series is s(a,b) = 2a + 3b + 3 4 ln 2− b−1 8 π. the paper [4] is about the generalized alternating harmonic series with five periodically repeated numerators (1,a,b,c,d), i.e. with the series ∞∑ n=1 ( 1 5n−4 + a 5n−3 + b 5n−2 + c 5n−1 + d 5n ) = = 1 1 + a 2 + b 3 + c 4 + d 5 + 1 6 + a 7 + b 8 + c 9 + d 10 + · · · . it was derived that the only value of the coefficient d ∈ r, for which this series converges, is d = −a−b−c−1. it was also derived that the sum of this series is s(a,b,c) = 1 + a + b + c 4 ln 5 + √ 5(1−a− b + c) 20 ln 3 + √ 5 2 + + √ 5(1− c) + 1 + 2a−2b− c √ 10 √ 5 + √ 5 arctan √ 2 √ 5 + √ 5 5− √ 5 + + √ 5(1− c)− (1 + 2a−2b− c) √ 10 √ 5− √ 5 arctan √ 2 √ 5− √ 5 5 + √ 5 . the contribution [5] is about the generalized alternating harmonic series with six periodically repeated numerators (1,a,b,c,d,e), i.e. with the series ∞∑ n=1 ( 1 6n−5 + a 6n−4 + b 6n−3 + c 6n−2 + d 6n−1 + e 6n ) = = 1 1 + a 2 + b 3 + c 4 + d 5 + e 6 + · · · . it was derived that the only value of the coefficient e ∈ r, for which this series converges, is e = −a−b−c−d−1. it was also derived that the sum of this series is s(a,b,c,d) = 1 + b + d 3 ln 2 + 1 + a + c + d 4 ln 3 + 3 + a− c−3d 12 √ 3 π. the paper [6] deals with the generalized alternating harmonic series with seven periodically repeated numerators (1,a,b,c,d,e,f), i.e. with the series ∞∑ n=1 ( 1 7n−6 + a 7n−5 + b 7n−4 + c 7n−3 + d 7n−2 + e 7n−1 + f 7n ) . 5 radovan potůček it was derived that the only value of the coefficient f ∈ r, for which this series converges, is f = −a−b−c−d−e−1. it was also derived that the sum of this series is s(a,b,c,d,e) = 0.4440431881a + 0.2555147273b + 0.1530792957c + + 0.0861379681d + 0.0375971731e + 0.9695377967. 3 sum of generalized alternating harmonic series with eight periodically repeated numerators now, we deal with the numerical series of the form ∞∑ n=1 ( 1 8n−7 + a 8n−6 + b 8n−5 + c 8n−4 + d 8n−3 + e 8n−2 + + f 8n−1 + g 8n ) = 1 1 + a 2 + b 3 + c 4 + d 5 + e 6 + f 7 + g 8 + + 1 9 + a 10 + b 11 + c 12 + d 13 + e 14 + f 15 + g 16 + · · · , (1) where a,b,c,d,e,f,g ∈ r. this series we shall call generalized alternating harmonic series with eight periodically repeated numerators (1,a,b,c,d,e,f,g). we express the numerator g, for which the series (1) converges, as a function of the numerators a,b,c,d,e,f, and determine the sum of this series. the power series corresponding to the series (1) has evidently the form ∞∑ n=1 ( x8n−7 8n−7 + ax8n−6 8n−6 + bx8n−5 8n−5 + cx8n−4 8−4 + dx8n−3 8n−3 + ex8n−2 8n−2 + + fx8n−1 8n−1 + gx8n 8n ) . (2) we denote its sum by s(x). the series (2) is for x ∈ (−1,1) absolutely convergent, so we can rearrange it and rewrite it in the form s(x) = ∞∑ n=1 x8n−7 8n−7 + a ∞∑ n=1 x8n−6 8n−6 + b ∞∑ n=1 x8n−5 8n−5 + c ∞∑ n=1 x8n−4 8n−4 + + d ∞∑ n=1 x8n−3 8n−3 + e ∞∑ n=1 x8n−2 8n−2 + f ∞∑ n=1 x8n−1 8n−1 + g ∞∑ n=1 x8n 8n . (3) 6 sums of generalized convergent harmonic series with eight numerators if we differentiate the series (3) term-by-term, where x ∈ (−1,1), we get s′(x) = ∞∑ n=1 x8n−8 + a ∞∑ n=1 x8n−7 + b ∞∑ n=1 x8n−6 + c ∞∑ n=1 x8n−5 + + d ∞∑ n=1 x8n−4 + e ∞∑ n=1 x8n−3 + f ∞∑ n=1 x8n−2 + g ∞∑ n=1 x8n−1. (4) after reindexing and fine arrangement the series (4) we obtain s′(x) = ∞∑ n=0 x8n + ax ∞∑ n=0 x8n + bx2 ∞∑ n=0 x8n + cx3 ∞∑ n=0 x8n + + dx4 ∞∑ n=0 x8n + ex5 ∞∑ n=0 x8n + fx6 ∞∑ n=0 x8n + gx7 ∞∑ n=0 x8n, that is s′(x) = ( 1 + ax + bx2 + cx3 + dx4 + ex5 + fx6 + gx7 ) ∞∑ n=0 ( x8 )n . (5) when we summate the convergent geometric series on the righthand side of (5) with the first term 1 and the ratio x8, where ∣∣x8∣∣ < 1, i.e. for x ∈ (−1,1), we get s′(x) = 1 + ax + bx2 + cx3 + dx4 + ex5 + fx6 + gx7 1−x8 . we convert this fraction using the cas maple 16 to partial fractions and get s′(x) = ax + b x2 + 1 + cx + d x2 − √ 2x + 1 + ex + f x2 + √ 2x + 1 + g x + 1 + h x−1 , where x ∈ (−1,1) and a = a− c + e−g 4 , b = 1− b + d−f 4 , c = −1+b+ √ 2c+d−f − √ 2g 4 √ 2 , d = √ 2 + a− c− √ 2d−e + g 4 √ 2 , e = 1− b + √ 2c−d + f − √ 2g 4 √ 2 , f = √ 2−a + c− √ 2d + e−g 4 √ 2 , g = 1−a+b−c+d−e+f −g 8 , h = −1−a−b−c−d−e−f −g 8 . (6) 7 radovan potůček the sum s(x) of the series (2) we obtain by integration in the form s(x) = ∫ ( ax+b x2 +1 + cx+d x2− √ 2x+1 + ex+f x2 + √ 2x+1 + g x+1 + h x−1 ) dx = = a 2 ∫ 2x x2 +1 dx + b ∫ 1 x2 +1 dx + ∫ c(2x− √ 2)/2+d+c √ 2/2 x2− √ 2x+1 dx + + ∫ e(2x+ √ 2)/2+f −e √ 2/2 x2 + √ 2x+1 dx + g ln |x+1|+ h ln |x−1|+ k, so s(x) = a 2 ln(x2 +1) + b arctanx + c 2 ln(x2− √ 2x+1) + + 2d+c √ 2 2 ∫ dx (x− √ 2/2)2 +( √ 2/2)2 + e 2 ln(x2 + √ 2x+1) + + 2f −e √ 2 2 ∫ dx (x+ √ 2/2)2 +( √ 2/2)2 + g ln |x+1|+ h ln |x−1|+ k = = a 2 ln(x2 +1) + b arctanx + c 2 ln(x2− √ 2x+1) + 2d+c √ 2 √ 2 × ×arctan 2x− √ 2 √ 2 + e 2 ln(x2 + √ 2x+1) + 2f −e √ 2 √ 2 arctan 2x+ √ 2 √ 2 + + g ln |x+1|+ h ln |x−1|+ k, where k is the constant of integration and where we used the formulas∫ f ′(t) f(t) dt = ln |f(t)|+ k and ∫ dt t2 + α2 = 1 α arctan t α + k. from the condition s(0) = 0, and because we have ln 1 = 0, arctan 0 = 0, arctan(±1) = ± π 4 , we obtain 2d + c √ 2 √ 2 · −π 4 + 2f −e √ 2 √ 2 · π 4 + k = 0 , hence k = π 4 √ 2 ( 2d + c √ 2−2f + e √ 2 ) . because 2(d−f)+(c+e) √ 2 = a−e √ 2 , we get k = π 4 √ 2 · a−e √ 2 = (a−e)π 8 . after application the relations (6), where √ 2d + c = 1 + √ 2a + b−d− √ 2e−f 4 √ 2 , 8 sums of generalized convergent harmonic series with eight numerators √ 2f −e = 1− √ 2a + b−d + √ 2e−f 4 √ 2 , we get s(x) = a− c + e−g 8 ln(x2 + 1) + 1− b + d−f 4 arctanx + + −1 + b + √ 2c + d−f − √ 2g 8 √ 2 ln(x2 − √ 2x + 1) + + 1 + √ 2a + b−d− √ 2e−f 4 √ 2 arctan( √ 2x−1) + + 1− b + √ 2c−d + f − √ 2g 8 √ 2 ln(x2 + √ 2x + 1) + + 1− √ 2a + b−d + √ 2e−f 4 √ 2 arctan( √ 2x + 1) + + 1−a + b− c + d−e + f −g 8 ln |x + 1| − − 1 + a + b + c + d + e + f + g 8 ln |x−1|+ (a−e)π 8 . now, we will deal with the convergence of the power series (2) in the point x = 1. substituting x = 1 to the series (2) – it can be done by the extended version of abel’s theorem (see [7], p. 23) – we get the numerical series (1). by the integral test we can prove that the series (1) converges if and only if h = 0, i.e. for g = −a−b−c−d−e−f −1. simplifying the formula for s(x) above, where g = −a− b− c−d−e−f −1, and for x = 1 we get s(1) = 1 + 2a + b + d + 2e + f 8 ln 2 + 1− b + d−f 4 arctan 1 + + √ 2−1 + √ 2a + ( √ 2 + 1)b + 2 √ 2c + ( √ 2 + 1)d + √ 2e + ( √ 2−1)f 8 √ 2 × × ln(2− √ 2) + 1 + √ 2a + b−d− √ 2e−f 4 √ 2 arctan( √ 2−1) + + √ 2 + 1 + √ 2a + ( √ 2−1)b + 2 √ 2c + ( √ 2−1)d + √ 2e + ( √ 2 + 1)f 8 √ 2 × × ln(2 + √ 2) + 1− √ 2a + b−d + √ 2e−f 4 √ 2 arctan( √ 2 + 1) + + 1 + b + d + f 4 ln 2−0 + (a−e)π 8 . 9 radovan potůček because ln 1 = 0, arctan 1 = π 4 , arctan( √ 2 − 1) = π 8 , arctan( √ 2 + 1) = 3π 8 , we have s(1) = 1−b+d−f 16 π + −1+b+d−f + √ 2(1+a+b+2c+d+e+f) 8 √ 2 × × ln(2− √ 2) + 1 + b−d−f + √ 2(a−e) 32 √ 2 π + + 1− b−d + f + √ 2(1 + a + b + 2c + d + e + f) 8 √ 2 ln(2 + √ 2) + + 3(1 + b−d−f)−3 √ 2(a−e) 32 √ 2 π + + 3(1 + b + d + f) + 2(a + e) 8 ln 2 + a−e 8 π. after simplification and after re-mark s(1) as s(a,b,c,d,e,f) we obtain s(a,b,c,d,e,f) = 1 + 2a− b + d−2e−f 16 π + + √ 2−a + √ 2b− √ 2d + e− √ 2f 16 π + 1− b−d + f 8 √ 2 ln(3 + 2 √ 2) + + 1 + a + b + 2c + d + e + f + 3 + 2a + 3b + 3d + 2e + 3f 8 ln 2. finally, we get the required formula s(a,b,c,d,e,f) = √ 2(1 + b−d−f) + 1 + a− b + d−e−f 16 π + + 1− b−d + f 8 √ 2 ln(3 + 2 √ 2) + 4(1 + b + d + f) + 3a + 2c + 3e 8 ln 2. (7) 4 numerical verification we have solved the problem to determine the sum s(a,b,c,d,e,f) above for several values of a, b, c, d, e, f by using the basic programming language of the computer algebra system maple 16. it was used the following simple procedure sumgenhar1abcdefg. as a sample of the hexads (a,b,c,d,e,f) we took 12 hexads (1,0,0,0,0,0), (0,1,0,0,0,0), (0,0,1,0,0,0), (0,0,0,1,0,0), 10 sums of generalized convergent harmonic series with eight numerators (0,0,0,0,1,0), (0,0,0,0,0,1), (0,0,0,0,0,0), (−1000,0,0,0,0,0), (6,−1,0,1,−6,−1), (−1,1,−1,1,−1,1), (−2,1,−2,1,−2,1), and (1/2,1/4,1/8,1/16,1/32,1/64). it was chosen t = 106 summands with 8 terms 1 8n−7 + a 8n−6 + b 8n−5 + c 8n−4 + d 8n−3 + + e 8n−2 + f 8n−1 − a + b + c + d + e + f + 1 8n for the computations whose results will be compared with the results obtained by the formula (7). the procedure sumgenhar1abcdefg consists of the following commands: sumgenhar1abcdefg:=proc(t,a,b,c,d,e,f) local g,r,k,s,w; s:=0; r:=0; g:=-a-b-c-d-e-f-1; for k from 1 to t do r:=1/(8*k-7)+a/(8*k-6)+b/(8*k-5)+c/(8*k-4)+d/(8*k-3) +e/(8*k-2)+f/(8*k-1)+g/(8*k); s:=s+r; end do; print("t=",k-1,"s(",a,b,c,d,e,f,")=",evalf[20](s)); w:=pi*(sqrt(2)*(1+b-d-f)+1+a-b+d-e-f)/16 +ln(3+2*sqrt(2))*(1-b-d+f)/(8*sqrt(2)) +ln(2)*(4*(1+b+d+f)+3*a+2*c+3*e)/8 print("s(",a,b,c,d,e,f")=",evalf[20](w)); end proc: computation of the twelve sums s(106,a,b,c,d,e,f) took about 47 hours and 39 minutes. the relative quantification accuracies of the twelve sums s(106,a,b,c,d,e,f), that is the ratio ∣∣∣∣s(106,a,b,c,d,e,f)−s(a,b,c,d,e,f)s(106,a,b,c,d,e,f) ∣∣∣∣ , have here place value about 10−7. the results of the procedure above are presented in the table 1, where the computed sums are denoted briefly s(106) instead of s(106,a,b,c,d,e,f) and the sums s(a,b,c,d,e,f) are denoted as s(abcdef) and are evaluated by means of the formula (7): 11 radovan potůček a b c d e f s(106) s(abcdef) 1 0 0 0 0 0 0.9591518 0.9591520 0 1 0 0 0 0 1.4326892 1.4326894 0 0 1 0 0 0 1.1496962 1.1496964 0 0 0 1 0 0 1.0858461 1.0858463 0 0 0 0 1 0 1.0399901 1.0399903 0 0 0 0 0 1 1.0047597 1.0047598 0 0 0 0 0 0 0.9764095 0.9764096 −103 0 0 0 0 0 −455.30323 −455.30332 6 −1 0 1 −6 −1 3.1415922 3.1415927 −1 1 −1 1 −1 1 0.6931471 0.6931472 −2 1 −2 1 −2 1 0.0000001 0 1/2 1/4 1/8 1/16 1/32 1/64 1.3035043 1.3035045 table 1: the approximate values of the sums of the generalized harmonic series with periodically repeating numerators (1, a, b, c, d, e, f,−a−b−c−d−e−f−1) for 12 hexads (a, b, c, d, e, f) 5 conclusion we dealt with the generalized convergent harmonic series with eight periodically repeated numerators (1,a,b,c,d,e,f,g), where a,b,c,d,e,f,g ∈ r, i.e. with the series ∞∑ n=1 ( 1 8n−7 + a 8n−6 + b 8n−5 + c 8n−4 + d 8n−3 + e 8n−2 + f 8n−1 + g 8n ) . we derived that the only value of the numerator g, for which this series converges, is g = −a− b− c−d− e−f − 1, and we derived that the sum of this series is given by the formula s(a,b,c,d,e,f) = √ 2(1 + b−d−f) + 1 + a− b + d−e−f 16 π + + 1− b−d + f 8 √ 2 ln(3 + 2 √ 2) + 4(1 + b + d + f) + 3a + 2c + 3e 8 ln 2. this formula allows to determine another sums whose periodically repeated numerators need not be (1,a,b,c,d,e,f,−a − b − c − d − e − f − 1), but also (k,`,m,n,p,q,r,−k−`−m−n−p−q−r), for k,`,m,n,p,q,r ∈ r, at least 12 sums of generalized convergent harmonic series with eight numerators one nonzero. for example, the series ∞∑ n=1 ( 64 8n−7 + 32 8n−6 + 16 8n−5 + 8 8n−4 + 4 8n−3 + 2 8n−2 + 1 8n−1 − 127 8n ) has the sum s(64,32,16,8,4,2,1,−127) = = 64 ·s ( 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , 1 64 ) . = 64 ·1.303 504 .= 83.424. there are special series with sums expressed only by one summand or with the special or the null sum. it can be easily derived that s(a,b,0,−b,−a,−1) = 1 + a− b + √ 2(1 + b) 8 π, so e.g. s(6,−1,0,1,−6,−1) = π, s(a,1,c,1,a,1) = 8 + 3a + c 4 ln 2, so e.g. s(−1,1,−1,1,−1,1) = ln 2, and s(−2,−1,0,1,2,−1) = 0, s(−2,1,−2,1,−2,1) = 0. we verified the main result (7) by computing 12 sums by using the cas maple 16. the generalized convergent harmonic series with eight periodically repeated numerators so belong to special types of infinite series, such as geometric and telescoping series, which sums are given analytically by means of a relatively simple formula. 13 radovan potůček references [1] potůček r.: sum of generalized alternating harmonic series with periodically repeated numerators (1,a) and (1,1,a). in: proceedings of conference matematika, informačnı́ technologie a aplikované vědy mitav 2014, brno: university of defence, 2014, p. 171-177. isbn 978-80-7231-961-9. [2] potůček r.: sum of generalized alternating harmonic series with three periodically repeated numerators. mathematics in education, research and applications, slovenská pol’nohospodárska univerzita v nitre, 2015, p. 42-48. issn 2453-6881. [3] potůček r.: sum of generalized alternating harmonic series with four periodically repeated numerators. in: proceedings of the 14th conference on applied mathematics aplimat 2015. bratislava, slovak university of technology in bratislava, 2015, p. 638-643. isbn 978-80-2274-314-3. [4] potůček r.: sum of generalized alternating harmonic series with five periodically repeated numerators. in: proceedings of reviewed contributions on cd-rom, xxxiii international colloquium on the management of educational process, university of defence, brno, 2015, p. 33-40. isbn 978-80-7231-995-4. [5] potůček r.: sum of generalized alternating harmonic series with six periodically repeated numerators. in: proceedings of conference matematika, informačnı́ technologie a aplikované vědy mitav 2015, university of defence, brno, 2015, p. 42-49. isbn 978-80-7231-998-5. [6] potůček r.: sum of generalized alternating harmonic series with seven periodically repeated numerators. in: 9. didaktická konference s mezinárodnı́ účastı́ – sbornı́k abstraktů a přı́spěvků. masarykova univerzita, brno, 2015, p. 43-52. isbn 978-80-2107-814-7. [7] hušek m., pyrih p.: matematická analýza [online]. [cit. 2015-03-21]. available from: http://matematika.cuni.cz/dl/analyza/25-raf/lekce25-rafpmax.pdf. 14 ratio mathematica volume 41, 2021, pp. 207-213 on the traversability of near common neighborhood graph of a graph keerthi g. mirajkar * anuradha v. deshpande† abstract the near common neighborhood graph of a graph g, denoted by ncn(g) is defined as the graph on the same vertices of g, two vertices are adjacent in ncn(g), if there is at least one vertex in g not adjacent to both the vertices. in this research paper, the conditions for ncn(g) to be disconnected are discussed and characterization for graph ncn(g) to be hamiltonian and eulerian are obtained. keywords: near common neighborhood graph; hamiltonian graph; eulerian graph. 2020 ams subject classifications: 05c07, 05c10, 05c38, 05c60, 05c76. 1 *department of mathematics, karnatak university’s karnatak arts college, dharwad 580001, karnataka, india; keerthi.mirajkar@gmail.com. †department of mathematics, karnatak university’s karnatak arts college, dharwad 580001, karnataka, india; anudesh08@gmail.com. 1received on july 9, 2021. accepted on november 23, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.626 . issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 207 keerthi g. mirajkar, anuradha v. deshpande 1 introduction let g be a graph. the near common neighborhood graph of g denoted by ncn(g) is a graph with the same vertices as g in which two vertices u and v are adjacent if there exists at least one vertex w ∈ v (g) not adjacent to both of u and v [alkenani et al., 2016]. a cycle in a graph g that contains every vertex of g is called spanning cycle of g. thus a hamiltonian cycle of g is a spanning cycle of g. a hamiltonian graph is a graph that contains a hamiltonian cycle. an euler trail in a graph g is a trail that contains every edge of that graph. an euler tour is a closed euler trail. an eulerian graph is a graph that has an euler tour. the graphs cosiderered in this paper are simple, undirected and connected with vertex set vi ∈ v (g), i = 1, 2, 3, ...,n. let deg(vi) be the degree of vertices of g. basic terminologies are referred from [harary, 1969]. the common neighborhood graph (congraph) of g [zadeh et al., 2014] which is exactly the opposite of near common neighborhood is denoted by con(g) is a graph with vertex set, in which two vertices are adjacent if and only if they have at least one common neighbor in the graph g. here the common neighborhood of some composite graphs are computed and also the relation between hamiltonicity of graph g and con(g) is investigated. [hamzeh et al., 2018] computed the congraphs of some composite graphs and also results have been calculated for graph valued functions. [sedghi et al., 2020] obtained the characteristics of congraphs under graph operations and relations between cayley graphs and its congraphs. [al-kenani et al., 2016] studied near common neighborhood of a graph and obtained results for paths, cycles and complete graphs. motivated by the results on [zadeh et al., 2014], [hamzeh et al., 2018] and [sedghi et al., 2020], in this paper, the conditions for ncn(g) to be disconnected are discussed and also theorems stating necessary and sufficient conditions for a graph ncn(g) to possess hamiltonian and eulerian cycle are studied. 2 prelimnaries below mentioned some important results are used through out the paper for proving the theorems. proposition 2.1. [al-kenani et al., 2016] for any path pn, ncn(pn) =   kn, if n = 2, 3 2k2, if n = 4 kn, if n ≥ 7. 208 on the traversability of near common neighborhood graph of a graph proposition 2.2. [al-kenani et al., 2016] for any path cn, ncn(cn) =   kn, if n = 3, 4 c5, if n = 5 kn, if n ≥ 7. proposition 2.3. [al-kenani et al., 2016]. for any complete graph kn and totally disconnected graph kn, we have 1.ncn(kn) = kn 2.ncn(kn) = kn,n ≥ 3 theorem 2.1. [singh, 2010]. let g be a simple graph with p ≥ 3 and δ ≥ p/2. then g is hamiltonian. theorem 2.2. [singh, 2010] a nonempty connected graph is eulerian if and only if it has no vertices of odd degree. remark 2.1. [singh, 2010] the complete graph kp, for p ≥ 3, is always hamiltonian. 3 results theorem 3.1. if g is a graph with n vertices, then ncn(g) is disconnected if any one of the following conditions holds 1. g is pn,cn, n ≤ 4 2. g is kn, n ≥ 3 3. g has ∆(g) = n− 1 4. g is a graph with two complete graphs connected by bridge 5. g is kn •pt, n ≥ 3 , t ≤ 3 proof. the proof of the theorem is constructed by considering the following cases. case 1. suppose g=pn or cn, n ≤ 4. we consider the following two subcases. subcase 1.1. suppose g=pn , n ≤ 4. if n = 2, 3, 4, then by the proposition 2.1, ncn(g) is disconnected. subcase 1.2. suppose g=cn , n ≤ 4. if n = 3, 4, then from the proposition 2.2, ncn(g) is disconnected. case 2. suppose g=kn, n ≥ 3. then from the proposition 2.3, ncn(g) is disconnected. case 3. let g be a graph with vertex set v (g) = {vi|i ∈ n} and ∆(g) = n−1. near common neighbourhood graph ncn(g) is a graph with same vertices vi as 209 keerthi g. mirajkar, anuradha v. deshpande g. the vertices vi and vj, j = 1, 2, 3, ...,n,i 6= j of ncn(g) are adjacent if there is at least one vertex w ∈ v (g) not adjacent to both vi and vj. since ∆(g) = n− 1 in g ( that is vi is adjacent to all other vertices of g), there does not exists any nonadjacent vertex for vi and thus vi cannot be connected to any vertex of ncn(g). this results ncn(g) into disconnected. case 4. let g be a graph with two complete graphs km and kn connected by bridge. let vi ∈ v (g), i = 1, 2, 3, ...,m be the vertex set of km and vj ∈ v (g), j = m + 1,m + 2,m + 3, ...,n be the vertex set of kn, where m and n are the vertices of bridge. as g consists of two complete graphs, vertices vi of km and vj of kn are respectively mutually adjacent. nonadjacent vertices for all the vertices vi ∈ km exists in kn and for vj ∈ kn exists in km. thus the vertices vi of km and vj of kn are mutually connected in ncn(g). this produces the disconnected graph with two components. further, the end points of bridge are also mutually adjacent to all the vertices of km and kn respectively. hence nonadjacent vertex does not exists for end points of bridge. this produces the graph ncn(g) into disconnected. case 5. let g be a kn •pt, n ≥ 3 and t ≤ 3. ncn(g) has the same vertices as g. two vertices of ncn(g) are adjacent if there is at least one vertex w ∈ v (g) not adjacent to both the vertices. let vi, i = 1, 2, 3, ...,n,n + 1,n + 2 be the vertex set of kn •pt .the vertices of kn are v1,v2,v3, ...,vn and vertices of pt are vn,vn+1,vn+2. the vertex vn is the common vertex which connects kn and pt. we consider the following subcases. subcase 5.1 suppose t = 2 that is pt is path with two vertices, then g = kn•p2. since ∆(g) = n − 1, there exists at least one vertex which is adjacent to all the other vertices of g. from theorem 3.1 (case 3), ncn(g) is disconnected. subcase 5.2 suppose t = 3 that is t = n,n + 1,n + 2. in g, all the pairs of vertices of kn have the nonadjacent vertex as vn+2 and can be mutually connected in ncn(g). also the vertices of pt, vn+1 and vn+2 have the nonadjacent vertices in kn and can be connected in ncn(g). as there does not exist any nonadjacent vertices for the pair of common vertex n and the vertices of pt in g, they cannot be connected in ncn(g). this produces the the graph ncn(g) with two components. thus ncn(g) is disconnected. 2. theorem 3.2. for any graph g, ncn(g) is hamiltonian if and only if 1. g contains all the vertices of deg(vi) < n − 1 except for c4 • p2 and g is a graph with two complete graphs connected by bridge. 2. g = pn or cn, n ≥ 5. 3. g is kn •pt, n ≥ 3, t ≥ 4. proof. let g be graph with vertex set v (g). suppose ncn(g) is hamiltonian. 210 on the traversability of near common neighborhood graph of a graph in light of the above theorem 3.1 that ncn(g) is disconnected if g is pn,cn, n ≤ 4, kn, n ≥ 3, g has ∆(g) = n− 1, g is a graph with two complete graphs connected by bridge and g is kn •pt, n ≥ 3, t ≤ 3. now we consider the graphs for which ncn(g) is connected. case 1. suppose g contains all the vertices of deg(vi) < n− 1. let g be a graph vi ∈ v (g) vertices with degree(vi) < n− 1 (vi is not adjacent to all the vertices) and ncn(g) be the graph with same set of vertices as g. as deg(vi) < n− 1 in g, there exists at least one nonadjacent vertex for any pair of vertices of g. hence by definition of ncn(g), those vertices in ncn(g) can be connected which produces connected ncn(g) graph. further, since for each pair of vertices of g there exists a nonadjacent vertex, ncn(g) contains a cycle and δ ≥ n/2. from the theorem 2.1 ncn(g) is hamiltonian. next suppose g = c4 • p2. let vi, i = 1, 2, 3, 4, 5 be the vertex set of c4 • p2 with one common vertex between c4 and p2. among the four vertices of c4 of g, three vertices (except the common vertex) can be connected mutually adjacent as they have nonadjacent vertex (not common vertex) in p2. similarly, a vertex of p2 which is not common can be connected with only three vertices of c4 in ncn(g) as there exists a nonadjacent vertex for these each pair of vertices. whereas the common vertex can be connected only with a vertex of p2 in ncn(g) which is not common, since there exists a nonadjacent vertex in c4 for this pair and there does not exist nonadjacent vertex for the pair of vertices with c4. this results ncn(g) into connected graph with one pendent vertex and consequently does not contain hamiltonian cycle. thus, ncn(c4 •p2) is connected but not hamiltonian. case 2. suppose g is pn or cn, n ≥ 5. let g be a pn or cn, n ≥ 5. from the propositions 2.1 and 2.2, ncn(pn) and ncn(cn), n = 5, 6 are connected which contains a cycle and δ ≥ n/2. for n ≥ 7, ncn(g) is kn. from the theorem 2.1 and remark 2.1, ncn(g) is hamiltonian. case 3. suppose g is kn •pt, n ≥ 3, t ≥ 4. let g be a kn •pt,n ≥ 3, t ≥ 4, where n is the number of vertices of kn and t is the number of vertices of pt. let vi ∈ v (g), i = 1, 2, 3, ...,n be the vertex set of kn and vj ∈ v (g), j = n,n + 1,n + 2, ..., t be the vertex set of pt, where n is the common vertex of kn and pt. as there is increase in the number of vertices (path length) in pt of g, there exists a nonadjacent vertex for each pair of vertices of kn and vertices of pt, which produces connected graph ncn(g) with cycles and also δ ≥ n/2. from the theorem 2.1, ncn(g) is hamiltonian. converse is obvious. 211 keerthi g. mirajkar, anuradha v. deshpande 2 theorem 3.3. for any graph g, ncn(g) is eulerian if only if g is 1. pn,n ≥ 7. 2. kn •pt, n ≥ 3, t ≥ 5. 3. g = cn, n = 5, 6. proof. let g is a graph with vertex set vi ∈ g, i = 1, 2, 3, ...,n. suppose ncn(g) is eulerian, then degree of each vi of ncn(g) is even. from the theorems 3.1 and 3.2, ncn(g) is disconnected if g is pn,cn, n ≤ 4, kn, n ≥ 3, g has ∆(g) = n−1, g is a graph with two complete graphs connected by bridge, g is kn •pt, n ≥ 3 , t ≤ 3 and is connected only if g contains all the vertices of deg(vi) < n− 1 , g = pn or cn, n ≥ 5 and g is kn •pt, n ≥ 3, t ≥ 4. from the proposition 2.1, ncn(g = pn) is kn with even degree; n ≥ 7, where n = 2s + 1, s = 2, 3, 4, .... from the proposition 2.2, ncn(g = cn), n = 5, 6 is kn of even degree. from the theorem 3.2, ncn(g = kn •pt) is kn with even degree; n ≥ 3, t ≥ 5, where n = 2s + 1, s = 1, 2, 3, .... hence from the theorem 2.2, ncn(g) is eulerian. conversely, ncn(g) is a graph with same vertices as g. from theorem 3.1, ncn(g) is disconnected if g is pn,cn; n ≤ 4, kn; n ≥ 3, g has ∆(g) = n − 1, g is a graph with two complete graphs connected by bridge and g is kn •pt; n ≥ 3, t ≤ 3 in all other cases it is connected. we consider the following cases. case 1. suppose ncn(g) is kn, n = 1, 2, 3, ...,n, then from the theorem 3.2, if ncn(g) is connected and it is kn for g = pn,cn, n ≥ 5 and kn • pt; n ≥ 3, t ≥ 5. subcase 1.1 suppose g = pn or cn, then from the propositions 2.1 and 2.2, ncn(g) is kn, n ≥ 7. the degree of each vertex of kn is even and n = 2s + 1, s = 1, 2, 3, ...,n. from theorem 2.2, ncn(g) is eulerian. subcase 1.2. suppose g = kn •pt, n ≥ 3, t ≥ 5, then from the theorem 3.2, if ncn(g) is connected and it is kn for n ≥ 3, t ≥ 5. the degree of kn is even if n is odd. from the theorem 2.2, ncn(g) is eulerian. case 2. suppose g = cn, n = 5, 6. subcase 2.1 suppose g = cn, n = 5, then from the proposition 2.2, ncn(g) is c5 or 2-regular graph. from the theorem 2.2 ncn(g) is eulerian. subcase 2.2 suppose g = cn, n = 6, then from the proposition 2.2, ncn(g) is 2-regular graph. hence from the theorem 2.2, ncn(g) is eulerian. case 3. suppose g = kn •pt, n ≥ 3, t ≥ 5. from the theorem 3.2, if ncn(g) is connected and it is kn for n ≥ 3, t ≥ 5. the degree of kn is even if n is odd. from the theorem 2.2, ncn(g) is eulerian. 212 on the traversability of near common neighborhood graph of a graph 4 conclusions in this paper the study on near common neighborhood graph of a graph is extended and various general conditions for which ncn(g) to be disconnected are discussed. it is disconnected for the graphs pn,cn; n ≤ 4, kn; n ≥ 3, g has ∆(g) = n− 1, g is a graph with two complete graphs connected by bridge and g is kn • pt; n ≥ 3 , t ≤ 3 in all other cases it is connected. we have also obtained the necessary and sufficient condition for ncn(g) to possess hamiltonian and eulerian cycles. it contains hamiltonian cycle whenever it is connected except for c4 •p2. it contains eulerian cycle if ncn(g) is kn with odd vertices and g is cn, n = 5, 6. references a.n al-kenani, a. alwardi, and o.a. al-attas. on the near-common neighborhood graph of a graph. international journal of computer applications, 146 (1):1–4, 2016. a. hamzeh, a. iranmanesh, s.h. zadeh, m.a. hosseinzadeh, and i. gutman. on common neighborhood graphs ii. iranian journal of mathematical chemistry, 9(1):37–46, 2018. f. harary. graph theory. addison-wesley mass, reading, new york, 1969. s. sedghi, d.w. lee, and n. shobe. characteristics of common neighborhood graph under graph operations and on cayley graphs. iranian journal of mathematical sciences and informatics, 15(2):13–20, 2020. g.s. singh. graph theory. phi learning private limited, new york, 2010. s.h zadeh, a. iranmanesh, a. hamzeh, and m.a hosseinzadeh. on the common neighborhood graphs. electronic notes in discrete mathematics, 45:51–56, 2014. 213 ratio mathematica volume 46, 2023 transversal eccentric domination in graphs riyaz ur rehman a* a mohamed ismayil† abstract eccentricity of a vertex vis a maximum among the shortest distances between the vertex v and all other vertices. a set dis called eccentric dominating if every vertex in its compliment has an eccentric vertex in the set d.a dominating set is transversal if the intersection of the set with all the minimum dominating sets is non-empty. inspired by both the concepts we introduce transversal eccentric dominating(ted) set. an eccentric dominating set d is called a ted-set if it intersects with every minimum eccentric dominating set d’. we find the ted-number γted (g) of family of graphs, theorems related to their properties are stated and proved. keywords: eccentricity, ted-set, ted-number. 2020 ams subject classifications: 05c69. 1 *jamal mohamed college (affiliated to bharathidasan university), tiruchirappalli, india. fouzanriyaz@gmail.com. †jamal mohamed college (affiliated to bharathidasan university), tiruchirappalli, india. amismayil1973@yahoo.co.in. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1043. issn: 1592-7415. eissn: 2282-8214. ©riyaz ur rehman et al. this paper is published under the cc-by licence agreement. 24 riyaz ur rehman a and a mohamed ismayil 1 introduction the classical queens problem in chess or the study of networks in electronics domination finds its application everywhere and plays a pivotal role in modern day science and technology. domination is a vast arena in graph theory which is just not limited to adjacency between vertices belonging to the dominating set and its compliment. for a graph g(v,e), a set s ⊆ v is said to be a dominating set, if every vertex in v-s is adjacent to some vertex in s. the domination number γd (g) of a graph g equals the minimum cardinality of an dominating set. there are many different invariants of domination. the concept of transversal domination in graphs was introduced by nayaka s.r, anwar alwardi and puttaswamy in 2018. a dominating set dwhich intersects every minimum dominating set in g is called a transversal dominating set. the minimum cardinality of a transversal dominating set is called the transversal domination number denoted by γtd (g). geodesic being the shortest distance between any two vertices. the concept of shortest path has always intrigued the researchers in graph theory, operation research, computer science and other fields.there are many different types of distances in graphs, one such distance is eccentricity. the concept of eccentricity incorporated with a dominating set yields an eccentric dominating set.eccentric domination was introduced by t. n. janakiraman et al in 2010. the eccentricitye (v) of v is the distance to a vertex farthest from v. thus, e(v) = maxd(u,v) : u ∈ v . for a vertex v, each vertex at a distancee (v) from v is an eccentric vertex. eccentric set of a vertex v is defined as e(v) = u ∈ (g) : d(u,v) = e(v). a set d ⊆ v (g) is an eccentric dominating set if d is a dominating set of g and for every v ∈ v − d, there exists at least one eccentric vertex of v in d. the eccentric domination number γed (g) of a graph g equals the minimum cardinality of an eccentric dominating set. the main motive of this paper is to hybrid two different types of dominations and define a new domination variant. inspired by this idea we combine transversal domination with eccentric domination. in this paper, we introduce transversal eccentric domination and calculate the ted-number of different graphs. results related to ted-number of family of complete, star, path, cycle and wheel graphs are discussed. the upper ted-set, upper ted-number, lower ted-set and lower ted-number of different standard graphs are tabulated. for undefined terminologies refer the book graph theory by frank harary. 2 transversal eccentric domination in graphs definition 2.1. an eccentric dominating(ed) set s ⊆ v (g) is called a transversal eccentric dominating set(ted-set) if it intersects with every minimum ed-set d′ ie s ⋂ d′ 6= ∅. 25 transversal eccentric domination in graphs definition 2.2. a ted-set s is called a minimal ted-set if no proper subset of s is ted-set. definition 2.3. the ted-number γted(g) of a graph g is the minimum cardinality among the minimal ted-sets of g. definition 2.4. the upper ted-number γted(g) of a graph g is the maximum cardinality among the minimal ted-sets of g. example 2.1. . ℘2 ℘3 ℘4 ℘1 ℘5 ℘6 figure 2.1: graph g consider the above example where the graph g consists of 6 vertices and 9 edges. (i) the dominating sets are {℘1,℘2}, {℘1,℘3}, {℘1,℘4}, {℘2,℘5}, {℘2,℘6}, {℘3,℘5}, {℘3,℘6}, {℘4,℘5}, {℘4,℘6}. (ii) the minimum ed-sets are {℘1,℘2}, {℘3,℘5}, {℘4,℘6}. (iii) the ted-sets are {℘1,℘5,℘6}, {℘2,℘3,℘4}. observation 2.1. for any graph g, 1. γ(g) ≤ γed(g) ≤ γted(g) ≤ γted(g). 2. γted(g) ≤ n and γted(g) ≤ n. 3. v (g) is also a ted-set. theorem 2.1. for complete graph kn, γted(kn) = n, ∀n ≥ 2. proof: let v (kn) = {℘1,℘2, . . .℘n}. since deg(℘i) = n− 1 ∀℘i ∈ v (kn) the eccentric vertex of ℘i is given by e(℘i) = v −{℘i} and every single vertex dominates all other vertices. since every vertex ℘i ∈ v forms an ed-set of the form d1 = {℘1}, d2 = {℘2}, d3 = {℘3}, . . .dn = {℘n}. the vertex set v is the only set which forms a ted-set, since v (kn)∩di 6= ∅ where i = 1, 2, 3, . . .n and di is any ed-set. 26 riyaz ur rehman a and a mohamed ismayil theorem 2.2. for star graph sn, γted(sn) = 2 ∀ n ≥ 3. proof: let v (sn) = {℘1, . . .℘i, . . .℘n} where deg(℘i) = n− 1 where ℘i is the central vertex and deg(℘j) = 1 where ℘j is a pendant vertex of star graph sn. e(℘i) = v −{℘i} and e(℘j) = v −{℘i,℘j}. the central vertex ℘i forms a dominating set {℘i} but it is not an ed-set for any ℘j ∈ v − d, e(℘j) /∈ d. but d = {℘i,℘j} forms an ed-set, then for s3 we have 3 ed-sets which forms the minimum ed-sets and for any star graph sn, ∀ n ≥ 4, we have (n − 1) edsets which forms the minimum ed-sets d1 = {℘i,℘1}, d2 = {℘i,℘2}, d3 = {℘i,℘3}, . . .dn = {℘i,℘n}. any minimum ed-set d = {℘i,℘j} also forms a ted-set, since d ∩{℘i,℘j} = {℘i} 6= ∅. therefore γted(sn) = 2 ∀n ≥ 3. theorem 2.3. for path graph pn, γted(pn) = bn+13 c + 1, ∀ n ≥ 2. proof: let the vertices of pn be given by v (pn) = {℘1,℘2, . . .℘n}. every path pn contains two pendant vertices {℘1,℘n}. for any vertex ℘i ∈ v (pn) the eccentric vertex of ℘i is e(℘i) = {℘1} or {℘n} where n is even. if n is odd then e(℘i) = {℘1} or {℘n} but if ℘i is a vertex equidistant from both the pendant vertices then ℘i = ℘n+1 2 , e(℘n+1 2 ) = {℘1,℘n}. for any path pn, dn3e set of vertices can dominate all the vertices of pn. similarly a set d whose cardinality is bn+1 3 c + 1 will eccentric dominate all the vertices of pn. by the definition of ted-set, a set d should intersect all the minimum ed-set. an ed-set d will intersect all the minimum ed-sets. therefore every minimum ed-set is a tedset. therefore γed(pn) = γted = bn+13 c + 1 theorem 2.4. for cycle graph cn where n ≥ 3 γted(cn) = { 5, for n = 8 dn+1 3 e + 1, otherwise proof: case(i): for c8, the set d = {℘i,℘j,℘k,℘l} whose cardinality is dn+1 3 e + 1 = 4 does not form a ted-set which is an exception from case(i). adding a vertex to d is of the form {℘i,℘j,℘k,℘l,℘m} whose cardinality is five will increasing the cardinality of d. here every vertex in v (c8) − d has an eccentric vertex in d and d is also dominating set which intersects all the minimum dominating sets of c8. therefore γted(c8) = 5. case(ii): for a cycle graph cn, if n is even and n 6= 8 then every vertex ℘i ∈ v (cn) has a unique eccentric vertex ie, e(℘i) = {℘j |℘j ∈ v (cn)}. e(℘i) is at a distance of n 2 edges from ℘i for an even cycle. if n is odd then every vertex ℘i has two eccentric vertices. e(℘i) = {℘j,℘k |℘j,℘k ∈ v (cn)}. e(℘i) is at a distance of bn 2 c edges from ℘i for odd cycle. every single vertex ℘i can dominate itself and two vertices adjacent to it. therefore for any cycle cn, dn3e set of vertices forms the dominating set. here we see that any set d = {℘1,℘2, . . .℘i} which has the cardinality of the form dn+1 3 e + 1 forms a dominating set as well 27 transversal eccentric domination in graphs as an ed-set. since d whose cardinality is dn+1 3 e + 1 intersects every minimum ed-set of cardinality γed(cn) = { n 2 , if n is even dn 3 eordn 3 e + 1, if n is odd d forms a ted-set. hence γted(cn) = dn+13 e + 1. theorem 2.5. for wheel graph wn where n ≥ 4, a ≥ 1 γted(wn) = { 3, for n = (6a− 1), (6a) or (6a + 1) 4, for n = (6a− 2), (6a + 2) or (6a + 3) proof: case(i): if n = 6a − 1, 6a and 6a + 1, the wheel graphs are of the form w5,w6,w7,w11,w12,w13,w17,w18,w19, . . .w6a−1,w6a,w6a+1. let ℘c be the central vertex of wheel graph, deg(℘c) = n − 1. therefore ℘c has n − 1 eccentric vertices, |e(℘c)| = n−1. let ℘i be the non-central vertex, deg(℘i) = 3. then closed neighbourhood of ℘i ie, n[℘i] = 4. therefore ℘i has n−4 eccentric vertices, |e(℘i)| = n−4. d = {℘c} forms the only dominating set of cardinality one, but not an ed-set. other than w5 and w7 every other wheel graph has an ed-set d = {℘c,℘x,℘y} where ℘c,℘x,℘y ∈ v (wn) forms an ed-set and for every v ∈ v (wn)−d there exists a vertex ℘c,℘x or ℘y in d such that e(v) = ℘c or ℘x or ℘y and d = {℘c,℘x,℘y} forms a ted-set, since d intersects every minimum ed-set. therefore |d| = 3, γted(wn) = 3 for n = 6a− 1, 6a, 6a + 1. case(ii): if (6a−2), (6a + 2) and (6a + 3), then the wheel graphs are of the form w4,w8,w9,w10,w14,w15,w16, . . .w6a−2,w6a+2,w6a+3. for w4, γted(w4) = 4. since w4 is k4 which is complete graph (by theorem-2.1). similar to case(i), ℘c is the central vertex of wheel graph and ℘j is the non-central vertex, |e(℘c)| = n− 1 and |e(℘i)| = n−4. similar to case(i) the only unique dominating set d = {℘c} whose cardinality is one does not form an ed-set. but a set d = {℘c,℘x,℘y} containing three vertices forms an ed-set, since every vertex ℘i ∈ v (wn) − d has an eccentric vertex in d ie, e(℘i) = ℘c,℘x or ℘y. but d = {℘c,℘x,℘y} whose cardinality is three does not form a ted-set since it does not intersect every minimum ed-set. but an addition of vertex ℘z to the same set gives us a set d = {℘c,℘x,℘y,℘z} whose cardinality is four forms an ed-set and it intersects every minimum ed-set of cardinality three, thus becoming ted-set. therefore γted(wn) = 4 for n = (6a− 2), (6a + 2) and (6a + 3). proposition 2.1. for any graph g, 1. γted(g) ≥b (2n−q) 4 c. 2. γted(g) ≥ diam(g)+1 3 . 3. γted(g) ≤b p ∆(g) δ c. 28 riyaz ur rehman a and a mohamed ismayil 4. γted(g) ≥d p1+∆(g)e. 5. γted(g) ≤dn + ∆(g) − √ 2qe. the transversal eccentric dominating set, γted(g), upper transversal eccentric dominating set and γted(g) of standard graphs are tabulated. graph figure d minimum ted set. |d| = γted(g) γted(g) s upper ted set. |s| = γted(g) γted(g) diamond graph ℘1 ℘4 ℘2 ℘3 {℘2,℘3}. 2 {℘2,℘3}. 2 tetrahedral graph ℘2 ℘1 ℘3 ℘4 {℘1,℘2,℘3,℘4}. 4 {℘1,℘2,℘3,℘4}. 4 claw graph ℘2 ℘3 ℘1 ℘4 {℘1,℘3}, {℘2,℘3}, {℘3,℘4}. 2 {℘1,℘2,℘4}. 3 (2,3)-king graph ℘2 ℘3℘1 ℘5℘4 ℘6 {℘1,℘2,℘4}, {℘1,℘3,℘4}, {℘1,℘3,℘6}, {℘1,℘4,℘5}, {℘1,℘4,℘6}, {℘2,℘3,℘6}, {℘3,℘4,℘6}, {℘3,℘5,℘6}. 3 {℘1,℘2,℘4}, {℘1,℘3,℘4}, {℘1,℘3,℘6}, {℘1,℘4,℘5}, {℘1,℘4,℘6}, {℘2,℘3,℘6}, {℘3,℘4,℘6}, {℘3,℘5,℘6}. 3 antenna graph ℘2 ℘1 ℘3 ℘4 ℘5 ℘6 {℘1,℘2,℘5}, {℘1,℘2,℘6}, {℘1,℘3,℘5}, {℘1,℘3,℘6}, {℘1,℘4,℘5}, {℘1,℘4,℘6}, {℘1,℘5,℘6}, {℘2,℘5,℘6}. 3 {℘1,℘2,℘3,℘4}. 4 29 transversal eccentric domination in graphs graph figure d minimum ted set. |d| = γted(g) γted(g) s upper ted set. |s| = γted(g) γted(g) paw graph ℘2 ℘3 ℘1 ℘4 {℘1,℘3}, {℘2,℘3} {℘3,℘4}. 2 {℘1,℘2,℘4}. 3 bull graph ℘3 ℘4 ℘5 ℘2℘1 {℘1,℘2,℘3}, {℘1,℘2,℘4}, {℘1,℘2,℘5}, {℘1,℘3,℘4}, {℘2,℘3,℘4}. 3 {℘1,℘2,℘3}, {℘1,℘2,℘4}, {℘1,℘2,℘5}, {℘1,℘3,℘4}, {℘2,℘3,℘4}. 3 butterfly graph ℘3 ℘2 ℘5 ℘1 ℘4 {℘1,℘2,℘4}, {℘1,℘2,℘5}, {℘1,℘3,℘4}, {℘1,℘4,℘5}, {℘2,℘3,℘5}, {℘2,℘4,℘5}. 3 {℘1,℘2,℘4}, {℘1,℘2,℘5}, {℘1,℘3,℘4}, {℘1,℘4,℘5}, {℘2,℘3,℘5}, {℘2,℘4,℘5}. 3 banner graph ℘3 ℘4 ℘1 ℘2 ℘5 {℘2,℘5}. 2 {℘2,℘5}. 2 fork graph ℘2 ℘3 ℘1 ℘4 ℘5 {℘1,℘2,℘5}, {℘1,℘4,℘5}, {℘2,℘3,℘5}, {℘2,℘4,℘5}. 3 {℘1,℘2,℘3,℘4}, {℘1,℘3,℘4,℘5}. 4 (3,2)-tadpole graph ℘2 ℘3 ℘4 ℘1 ℘5 {℘1,℘4}, {℘4,℘5}. 2 {℘1,℘2,℘3,℘5}. 4 kite graph ℘3 ℘4 ℘1 ℘5 ℘2 {℘2,℘4}. 2 {℘1,℘2,℘3,℘5}. 4 (4,1)-lollipop graph ℘3 ℘4 ℘1 ℘5 ℘2 {℘1,℘4}, {℘2,℘4}, {℘3,℘4}, {℘4,℘5}. 2 {℘1,℘2,℘3,℘5}. 4 30 riyaz ur rehman a and a mohamed ismayil graph figure d minimum ted set. |d| = γted(g) γted(g) s upper ted set. |s| = γted(g) γted(g) house graph ℘2 ℘3 ℘1 ℘4 ℘5 {℘1,℘2,℘3}, {℘1,℘4,℘5}, {℘2,℘3,℘4}, {℘2,℘3,℘5}, {℘2,℘4,℘5}, {℘3,℘4,℘5}. 3 {℘1,℘2,℘3}, {℘1,℘4,℘5}, {℘2,℘3,℘4}, {℘2,℘3,℘5}, {℘2,℘4,℘5}, {℘3,℘4,℘5}. 3 house x graph ℘2 ℘3 ℘1 ℘4 ℘5 {℘1,℘2}, {℘1,℘3}, {℘1,℘4}, {℘1,℘5}. 2 {℘2,℘3,℘4,℘5}. 4 gem graph ℘1 ℘2 ℘5 ℘3 ℘4 {℘1,℘2,℘3}, {℘1,℘2,℘4}, {℘1,℘3,℘4}, {℘1,℘3,℘5} {℘2,℘3,℘4} {℘2,℘4,℘5}. 3 {℘1,℘2,℘3}, {℘1,℘2,℘4}, {℘1,℘3,℘4}, {℘1,℘3,℘5}, {℘2,℘3,℘4}, {℘2,℘4,℘5}. 3 dart graph ℘3 ℘4 ℘1 ℘5 ℘2 {℘2,℘3}, {℘2,℘4}. 2 {℘1,℘2,℘5}, {℘1,℘3,℘4}, {℘3,℘4,℘5}. 3 cricket graph ℘4 ℘5℘3 ℘1 ℘2 {℘3,℘4}, {℘4,℘5}. 2 {℘1,℘2,℘4}, {℘1,℘3,℘5}, {℘2,℘3,℘5}. 3 pentatope graph ℘1 ℘4 ℘5 ℘2 ℘3 {℘1,℘2,℘3,℘4,℘5}. 5 {℘1,℘2,℘3,℘4,℘5}. 5 johnson solid skeleton 12 graph ℘2 ℘1 ℘3 ℘4 ℘5 {℘1,℘3}. 2 {℘1,℘2,℘4,℘5}, {℘2,℘3,℘4,℘5}. 4 cross graph ℘3 ℘1 ℘2 ℘4 ℘5 ℘6 {℘1,℘3,℘6}, {℘2,℘3,℘6}, {℘3,℘4,℘6}, {℘3,℘5,℘6}. 3 {℘1,℘2,℘3,℘4,℘5}, {℘1,℘2,℘4,℘5,℘6}. 5 31 transversal eccentric domination in graphs graph figure d minimum ted set. |d| = γted(g) γted(g) s upper ted set. |s| = γted(g) γted(g) net graph ℘5 ℘6 ℘3 ℘4 ℘1 ℘2 {℘1,℘2,℘5}, {℘1,℘2,℘6}, {℘1,℘4,℘6}, {℘2,℘3,℘6}. 3 {℘1,℘3,℘4,℘5}, {℘2,℘3,℘4,℘5}, {℘3,℘4,℘5,℘6}. 4 fish graph ℘4 ℘2 ℘5 ℘1 ℘6 ℘3 {℘2,℘3}, {℘3,℘5}. 2 {℘1,℘2,℘4,℘5,℘6}. 5 a graph ℘3 ℘4 ℘1 ℘5 ℘2 ℘6 {℘1,℘5,℘6}, {℘2,℘5,℘6}. 3 {℘1,℘2,℘3,℘4,℘5}, {℘1,℘2,℘3,℘4,℘6}. 5 r graph ℘3 ℘4 ℘1 ℘5 ℘2 ℘6 {℘1,℘2,℘3}, {℘2,℘3,℘4}, {℘2,℘3,℘5}, {℘2,℘3,℘6}, {℘2,℘5,℘6}. 3 {℘1,℘3,℘4,℘5,℘6}. 5 4-polynomial graph ℘2 ℘3℘1 ℘5℘4 ℘6 {℘3,℘4}. 2 {℘1,℘2,℘3,℘5,℘6}, {℘1,℘2,℘4,℘5,℘6}. 5 octahedral graph ℘4 ℘3℘2 ℘1 ℘5 ℘6 {℘1,℘2,℘3,℘4}, {℘1,℘2,℘3,℘5}, {℘1,℘2,℘3,℘6}, {℘1,℘2,℘4,℘5}, {℘1,℘2,℘5,℘6}, {℘1,℘3,℘4,℘6}, {℘1,℘3,℘5,℘6}, {℘1,℘4,℘5,℘6}, {℘2,℘3,℘4,℘5}, {℘2,℘3,℘4,℘6}, {℘2,℘4,℘5,℘6}, {℘3,℘4,℘5,℘6}. 4 {℘1,℘2,℘3,℘4}, {℘1,℘2,℘3,℘5}, {℘1,℘2,℘3,℘6}, {℘1,℘2,℘4,℘5}, {℘1,℘2,℘5,℘6}, {℘1,℘3,℘4,℘6}, {℘1,℘3,℘5,℘6}, {℘1,℘4,℘5,℘6}, {℘2,℘3,℘4,℘5}, {℘2,℘3,℘4,℘6}, {℘2,℘4,℘5,℘6}, {℘3,℘4,℘5,℘6}. 4 3 conclusions in this paper ted-set of a graph is defined. theorems related to find the tednumber of different family of graphs are stated and proved. the upper and lower 32 riyaz ur rehman a and a mohamed ismayil graph figure d minimum ted set. |d| = γted(g) γted(g) s upper ted set. |s| = γted(g) γted(g) 3-prism graph ℘2 ℘3 ℘4 ℘1 ℘5 ℘6 {℘1,℘5,℘6}, {℘2,℘3,℘4}. 3 {℘1,℘2,℘3,℘6}, {℘1,℘2,℘4,℘5}, {℘1,℘3,℘4,℘5}, {℘1,℘3,℘4,℘6}, {℘2,℘3,℘5,℘6}, {℘2,℘4,℘5,℘6}. 4 ted-number along with their respective sets of different standard graphs are tabulated. in future the comparative study of ted-set with eccentric dominating set will be done. the properties of a ted-set related to graph operations such as union, intersection, join and product of graphs will be explored. references a. alwardi, s. nayaka, et al. transversal domination in graphs. gulf journal of mathematics, 6(2), 2018. e. j. cockayne and s. t. hedetniemi. towards a theory of domination in graphs. networks, 7(3):247–261, 1977. f. harary. graph theory. narosa publishing house, new delhi, 2001. a. m. ismayil and a. r. u. rehman. accurate eccentric domination in graphs. our heritage, 68(4)(1, 2020):209–216, 2020a. a. m. ismayil and a. r. u. rehman. equal eccentric domination in graphs. malaya journal of matematik (mjm), 8(1, 2020):159–162, 2020b. t. janakiraman, m. bhanumathi, and s. muthammai. eccentric domination in graphs. international journal of engineering science, computing and biotechnology, 1(2):1–16, 2010. o. ore. theory of graphs. amer. math. soc. colloq. publ., 38 (amer. math. soc., providence, ri), 38, 1996. 33 microsoft word documento1 microsoft word documento1 microsoft word 2006_geononarch_ratiomath.doc ratio mathematica nr. 17 (2006) la geometria non-archimedea. dalle premesse agli infiniti modelli attuali raffaele mascella dipartimento di scienze della comunicazione università degli studi di teramo rmascella@unite.it la geometria non-archimedea sembra essere un’ipotesi astratta e fantasiosa, accettabile nella matematica ma non per la rappresentazione spaziale, perché usa concetti a lungo esplorati nel pensiero scientifico e filosofico e spesso rigettati, quali l’infinito e l’infinitesimo attuali. l’articolo analizza gli aspetti storici, epistemologici, filosofici e matematici legati a questa geometria ed alle sue radici, considerando l’impostazione dell’inventore giuseppe veronese, della formalizzazione analitica di levi-civita e di altri matematici e filosofici che sul tema hanno fornito risultati e dibattuto, quali cantor, hilbert e hahn, per terminare con gli infiniti modelli che oggi conosciamo. questa geometria appare sottovalutata, ma si presta ad un ruolo non meno importante di alcune alternative non-euclidee, spesso semplicisticamente considerate come uniche varianti astratte dei modelli euclideo e riemanniano. 1. introduzione dati due segmenti rettilinei, oppure due pesi o altri due rappresentanti di una stessa grandezza fisica, un qualche multiplo del primo sarà capace di superare il secondo. questa affermazione rispecchia talmente in profondità la nostra esperienza del mondo, che ci risulta difficile immaginare un universo ed una geometria che non la rispettino. anche in ambito cosmologico, la stella conosciuta a noi più distante sarà lontana tanti anni luce, ed anche prendendo il millimetro come segmento da moltiplicare o come unità di misura, se ne prendiamo un numero molto, molto grande, riusciremo a raggiungere la stella ed anche a superarla. dunque, se consideriamo la geometria una scienza osservativa ed empirica che si preoccupa di rappresentare il mondo che conosciamo, sembra inevitabile che una tale affermazione debba esserne posta alla base come principio di verità assoluta. anche se le nostre teorie dello spazio e dell’universo includono, in modo esplicito o sottinteso, questa affermazione sulla vicendevole raggiungibilità delle grandezze, da quelle più affermate a quelle formulate ancora in chiave ipotetica. ma la storia della scienza, e quella della geometria in particolare, ci insegna che non sono i secoli di affermate ed inossidabili convinzioni sulla natura del mondo a rendere definitive le teorie: che il mondo non è piatto, che lo spazio non è esattamente euclideo, che le tre dimensioni spaziali (o le quattro spazio-temporali) potrebbero benissimo essere in numero maggiore, e così via. nella matematica astratta, invece, dove una geometria è un sistema teorico che rispetta un certo numero di assiomi, possiamo invece considerare e costruire una geometria in cui una tale r. mascella 2 affermazione non valga, ed è una geometria che in questo ambito può esistere legittimamente, con dignità di appartenenza pari, se non superiore, alle alternative non-euclidee, nondesarguesiane, non-pascaliane, e via dicendo, per le notevoli implicazioni filosofiche che una geometria di tal specie porta con sé. per una tale costruzione, peraltro, proprio per la sua natura “perlustrativa”, non è necessario far riferimento ad enti di base semplici, che siano numeri o segmenti; siamo nell’ambito dei modelli astratti, ed allo scopo tornano utili anche una serie di enti e concetti più complessi, algebrici ed analitici; in primis i campi ordinati, non importa se reali o astratti. nel 1891, nel libro fondamenti di geometria ed a seguire in una serie di articoli fino al 1909, giuseppe veronese pose per la prima volta in modo chiaro alla comunità matematica e filosofica il problema di una geometria che non rispondesse necessariamente al postulato di archimede, tanto nelle forme astratte, tanto nella rappresentazione spaziale. in tutti i modelli concepiti fino a quel punto, ed in tutte le varianti fino allora indagate, esso non era stato mai tirato in causa geometricamente. alla questione sollevata dal matematico italiano, la comunità scientifico-filosofica rispose in parte con il rifiuto, in parte con l’accettazione e con una serie di importanti lavori che la completarono e ne sancirono la fondatezza, tra cui quelli di du bois-reymond, levi-civita, hahn e hilbert. ma la geometria non-archimedea è rimasta sempre in secondo piano, trascurata rispetto alla più classica forma euclidea ed alle quasi contemporanee varianti non-euclidee, rese oggetto d’attenzione poco prima ma rese più importanti dalla fisica di einstein e dalla connessione con i dati osservativi ed empirici. l’interesse per la geometria non-euclidea, per queste ragioni innanzitutto, si è rivelata da subito più incisiva ed accettata, in quanto destabilizzava più in profondità le apparentemente solide conoscenze dello spazio. la geometria non-archimedea, anch’essa in un certo senso non-euclidea, nel senso di “diversa” dalla geometria di euclide, non ha invece goduto dello stesso fascino. certamente il suo impatto è stato meno devastante rispetto alla geometria “noneuclidea” propriamente detta, perché mette in discussione una proprietà del continuo rettilineo che, dopo essere passata attraverso l’identificazione con i numeri reali, sembra essere una variante molto più fantasiosa ed artificiosa. in realtà, proprio l’assioma archimedeo è un’architrave di passaggio dalla geometria all’aritmetica ed alla teoria dei numeri e viceversa, perché enuncia una proprietà che utilizziamo sia per caratterizzare lo spazio, sia per caratterizzare gli insiemi attuali (in senso aristotelico) di cui possiamo disporre. è, insomma, uno dei principali collanti tra la geometria rettilinea e la teoria dei numeri. russell (1897) ad esempio, fa riferimento continuo all’opera di veronese (1891) per quanto riguarda l’assiomatica, riferendosi all’italiano per il primo assioma che russell vede nella geometria proiettiva a priori, ma anche per chiarire concetti e sviluppi storici, nel solco del quadro storico-critico della geometria tracciato da veronese, ma non fa riferimento alla variante non-archimedea, pur avendo scritto il suo saggio nel pieno della polemica nata su tale questione. russell concentra la sua attenzione sulla geometria a priori e sugli assiomi di base che dovrebbero porre la geometria proiettiva come preliminare alla geometria metrica, e poi sulle argomentazioni filosofiche che intendono stabilire la necessità empirica anche se non logica dell’euclideo rispetto alle sue varianti iperbolico ed ellittico. tiene in gran conto veronese, ma dell’archimedeo, o della sua negazione, russell non fa menzione. ad ogni modo, questa “nuova” geometria richiama già in prima istanza alcuni dei concetti più dibattuti della storia della scienza matematica, per comprendere la natura del mondo la geometria non-archimedea. da veronese agli infiniti modelli attuali 3 empirico e dei nostri modelli concettuali, che fanno capo alla continuità e all’infinito. e se la continuità è stata appannaggio quasi esclusivo dello scienziato e del filosofo di scienza, l’infinito con le sue problematiche, ha portato da sempre fascino e curiosità anche nello scrittore e nel poeta, nel filosofo metafisico e nell’uomo di strada. poi, sulla questione dell’infinito e dell’infinitesimo, il mondo è stato da sempre diviso in una perenne controversia tra spazio osservabile e non, in particolare su come le grandezze che conosciamo empiricamente nello spazio osservabile si possano, o debbano, comportarsi nello spazio esterno alla nostra osservazione. al centro della problematica vi è la validità, o la non validità, del cosiddetto postulato di archimede (o di eudosso-archimede). pur essendo attribuito ad archimede, il postulato così come oggi lo conosciamo è ricondotto ad euclide che lo esplicita nella definizione 4 del libro v degli elementi stabilendo che (boyer 1980: p. 134): si dice che due grandezze stanno in rapporto l’una con l’altra, quando, se moltiplicate, sono in grado l’una di superare l’altra. e’ un postulato che oggi si ritrova in tutti i pilastri dell’edificio matematico, a cominciare dalla geometria, dall’aritmetica e dall’analisi. è infatti con esso che possiamo riconoscere che la successione 1, 2, 3, … è divergente o che la successione 1/2, 1/3, 1/4, e così via, è infinitesima. in origine, esso è probabilmente dovuto ad eudosso (iv sec. a.c.), così come tutta la teoria delle grandezze. fu stolz a proporne questa denominazione ritenendo che euclide ne facesse solo un uso implicito e che invece fosse stato archimede a darne una definizione precisa e ad usarlo in modo esplicito anche se nel contesto di aree e volumi. archimede, nello specifico, lo usa nell’opera sulla sfera e sul cilindro per dimostrare che i rapporti tra le rette possono tendere ad uno, purché sia opportunamente definito la modalità con cui formare i rapporti successivi tra loro. per archimede, si possono trovare coppie di segmenti di rette disuguali, tali che il loro rapporto sia minore di qualunque rapporto prefissato maggiore dell’unità, ma vicino all’unità quanto si voglia. in altri termini, le grandezze si possono avvicinare tra loro quanto si vuole. e' interessante notare, inoltre, come il postulato di archimede formalizzi matematicamente il pensiero filosofico di anassagora (v secolo a.c.): «rispetto al piccolo non c'è un minimo, ma c'è sempre un ancor più piccolo, ma anche rispetto al grande c'è anche un ancor più grande». anche aristotele lo accettava, confermandone l’uso fatto dai matematici, e nella fisica chiarisce: «aggiungendo continuamente a una grandezza finita oltrepasserò qualsiasi grandezza limitata e sottraendo ne lascerò similmente una minore di qualsiasi altra». con la scuola d’elea si apre il dibattito sull’infinito attuale e l'infinito potenziale, che nel caso non-archimedeo ritorna in campo prepotentemente, accompagnati dall’altro grande problema dei greci, l’incommensurabilità. la direzione più comune che si può storicamente riscontrare è quella della messa al bando dell’infinito attuale, ovvero l'infinito visto come ente a se stante (ad esempio la totalità dei numeri interi). l’infinito potenziale rimane l’unico e vero infinito trattabile (una grandezza è finita, ma può diventare arbitrariamente grande, o arbitrariamente piccola). questa è stata, peraltro, la chiave concettuale con cui fu proposto ed utilizzato il primo metodo infinitesimale propriamente detto della matematica classica, ovvero r. mascella 4 il metodo di esaustione. questo è in fondo un metodo di integrazione, ma non uno strumento di calcolo, bensì un metodo per dimostrare in modo rigoroso la validità di certe uguaglianze di aree o volumi, facendo uso, appunto, del postulato archimedeo: “date due grandezze omogenee a e b, con a < b, esiste un numero n tale che na > b”. in questo postulato si avverte la presenza dell'infinito potenziale: comunque sia grande o piccola la grandezza a rispetto alla grandezza b, ci sarà sempre un multiplo di a che supera b, poiché ripetendo a per un numero arbitrario di volte (ovvero iterando il processo di auto-addizione) è possibile superare qualunque grandezza assegnata. in potenza, qualsiasi quantità è raggiungibile. archimede ne dà un’enunciazione diversa, ed invece di richiedere l’esistenza di un multiplo della quantità minore che superi la maggiore, impone la condizione che per due grandezze a e b ci sia un multiplo della differenza che superi una terza grandezza c, omogenea con le precedenti. il postulato di eudosso-archimede è dunque un postulato vero e proprio: esso non vale per tutte le grandezze, ma vale solo per certe grandezze d’ora in poi denominate, per definizione, archimedee. archimede è anche consapevole che questo postulato non è sostanzialmente una novità. infatti nella lettera a dositeo, prefazione al testo quadratura della parabola, scrive: «[…] avendo assunto il seguente lemma per la sua dimostrazione: date due aree disuguali è possibile, aggiungendo a se stesso l’eccesso di cui la maggiore supera la minore, superare ogni area limitata data. anche i geometri anteriori a noi si son serviti di questo lemma: infatti se ne sono serviti per dimostrare che i cerchi stanno fra loro in ragione duplicata dei diametri, e che le sfere stanno in ragione triplicata dei diametri, e ancora che ogni piramide è la terza parte del prisma avente la stessa base della piramide ed uguale altezza, e che qualunque cono è la terza parte del cilindro avente la stessa base del cono ed altezza uguale, ciò assumendo un lemma simile a quello suddetto […]». ad ogni modo euclide non assunse mai esplicitamente la validità della proprietà archimedea per una grandezza geometrica. ciò che egli assunse era soltanto che la linea retta si comportava in modo archimedeo, suggerendo o che le rette infinite si comportano come quelle finite, o che quelle infinite sono escluse dalle proposizioni riguardanti le misure. un’ultima questione da sottoporre in questo paragrafo introduttivo, riguarda la linearità del postulato ed il suo legame principalmente con il rettilineo geometrico. in tutta la letteratura geometrica, dagli scritti di euclide fino ai nostri giorni, poca considerazione è stata generalmente data all’idea della retta come struttura assiomatica a se stante, ovvero come struttura non indotta da un’assiomatica più ampia, ad esempio quella del piano o dello spazio. d’altronde la concezione prevalente della geometria è stata principalmente quella di scienza della rappresentazione spaziale, quindi con una natura empirica sottostante alla formalizzazione matematica ed alle procedure logiche deduttive. in altre parole, la scarsa considerazione probabilmente dipende dal fatto che la retta è un sistema “povero”, che non permette una rappresentazione adeguata, se vogliamo “completa”, dell’ambiente fisico in cui viviamo. nel caso della geometria non-archimedea, come anticipato, maggiore attenzione va posta proprio al continuo rettilineo, perché l’assioma di archimede tira in ballo una proprietà lineare delle grandezze e questa trova la sua naturale e principale applicazione nella geometria unodimensionale. ciò non esclude, tuttavia, che l’idea non sia trasportabile anche in modelli geometrici con più di una dimensione. la geometria non-archimedea. da veronese agli infiniti modelli attuali 5 una serie di modelli sia lineari che a più dimensioni sono perciò ripercorsi, dalla pionieristica retta di veronese, con l’evoluzione in chiave algebrica di levi-civita, passando per la caratterizzazione data da hahn (1907) e, per terminare, sui modelli da me discussi in precedenti lavori (eugeni-mascella 2002, 2005). in questi ultimi, abbiamo costruito un sistema di assiomi contenente gli assiomi di ordinamento di peano da cui, dopo aver dimostrato l'isomorfismo tra la retta euclidea e l’insieme ordinato dei reali, abbiamo presentato un nuovo modo di caratterizzare le rette non-archimedee con diversi esempi (infiniti) non isomorfi di rette non-archimedee. 2. l’infinito e l’infinitesimo “ciò che è conoscibile è finito”. potrebbe essere questa, una delle massime che sintetizzano secoli di pensiero matematico e filosofico. ma potremmo anche aggiungere, “… e con poca precisione”, visto che nella finitezza la nostra lente di osservazione ha chiari limiti percettivi, peraltro scientificamente provati. l’infinitamente grande e l’infinitamente piccolo hanno giocato un ruolo centrale nel storia del nostro pensiero. talvolta nel senso dell’inconoscibilità, talvolta nel ricorso all’extraumano ed all’assoluto, talvolta nel senso della sua negazione. ciò che ci interessa in questa sede è però l’approccio intuitivamente più semplice, quello che assegna dei limiti ai processi conoscitivi, senza ulteriori riferimenti sovrastrutturali e senza eliminazioni artificiose delle idee indotte sugli infiniti e sugli infinitesimi. in tutto il periodo storico dal mondo classico alla prima metà dell’ottocento, vi è stata la concezione che, sebbene ogni grandezza possa essere concepita come limitata o illimitata, quelle che si possono effettivamente utilizzare, quelle che esistono concretamente, nonché i processi che si possono “effettivamente” eseguire, sono solo quelli finiti. anche se possiamo concepire nozioni che coinvolgono l’infinito, non è stata generalmente ammessa la loro esistenza reale. in breve, nella forma aristotelica, con l’infinito si tratta un concetto potenziale, non attuale. seguendo geymonat (1970: i, p. 58): «si dice che una grandezza variabile costituisce un ‘infinito potenziale’ quando, pur assumendo sempre valori finiti, essa può crescere al di là di ogni limite; se per esempio immaginiamo di suddividere un segmento con successivi dimezzamenti... il numero delle parti a cui perveniamo, pur essendo in ogni caso finito, può crescere ad arbitrio. si parla invece di ‘infinito attuale’ quando ci si riferisce ad un ben determinato insieme, effettivamente costituito da un numero illimitato di elementi; se per esempio immaginiamo di aver scomposto un segmento in tutti i suoi punti, ci troveremo di fronte a un infinito attuale, perché non esiste alcun numero finito che riesca a misurare la totalità di questi punti». l’infinito è potenziale e può essere avvicinato solo attraverso iterazioni indefinite, cioè eseguite un certo numero di volte via via più lungo. dunque una finzione, ma necessaria per poter risolvere alcuni problemi posti dal nostro intelletto. più in dettaglio, aristotele distingue tre diversi modi di considerare l’infinito: (a) per composizione, come nel caso dei numeri che possono generare numeri sempre più grandi, (b) per divisione, come nel caso della materia, divisibile fino a raggiungere elementi sempre più piccoli e infine (c) per composizione e divisione, come il tempo che non ha né inizio né fine. in aristotele l’infinito attuale è respinto, associato all’idea di imperfezione. è sulla stessa scia che euclide non fa riferimento a rette illimitate o infinite, ma a segmenti prolungabili a r. mascella 6 lunghezze arbitrarie o, analogamente, non all’esistenza di infiniti numeri primi, ma all’esistenza di numeri primi in numero sempre maggiore (luminet-lachièze rey, 2005: p. 69). così anche nell’assioma delle parallele: le rette non sono infinitamente estese, bensì si possono estendere indefinitamente. l’infinito attuale viene evitato definendo un infinito minimale, giusto il necessario per permettere la costruzione dell’edificio teorico, ma senza per questo introdurre un “nuovo” numero. dunque, con tale approccio, i numeri interi sono potenzialmente infiniti, perché possiamo costruirne sempre di più grandi, ma il loro insieme infinito, come tale, non esiste. l’infinito è cioè im-perfetto, in-terminabile e im-pensabile. sulle orme di questi classici, folte schiere di scienziati e filosofi hanno mostrato una dura resistenza al concetto di infinito attuale, anche oltre i limiti di un atteggiamento razionale. da un lato, l’infinito pensato come attributo attuale è resistito solo in riferimento al divino religioso; dall’altro, ha mostrato segni di rinascita come elemento essenziale nella matematica e nella filosofia naturale, a partire dalla geometria descrittiva, per finire agli studi infinitesimali (integrazione e derivazione su tutti) e cosmologici. peraltro, gli infiniti hanno costituito e continuano a costituire un terreno di paradossi molto fertile, ostacolando nei secoli la formazione di una teoria compiuta per la loro manipolazione. e ad ogni buon conto, in ogni momento della sua biografia, «lo statuto fisico dell’infinito è legato inestricabilmente con il suo statuto metafisico» (luminet-lachièze rey, 2005: p. xii). il paradosso che più ha minato la nozione di infinito attuale è stato quello della riflessività, che riguarda l’infinitamente grande: in un insieme infinito esiste la possibilità di mettere in relazione biunivoca il tutto con una parte propria. d’altro canto è proprio questa apparente contraddizione a fare da sfondo negli sviluppi successivi della teoria dell’infinito, a cominciare da bolzano, dedekind e cantor. è infatti bolzano che segna una tappa decisiva sostenendo convintamente l’esistenza dell’infinito attuale, avente uno statuto ontologico come quello che spetta ai numeri finiti. ma per far ciò occorre abbandonare il carattere paradossale delle sue proprietà; anzi, tali proprietà vanno utilizzate per la definizione stessa di infinito. a dare forma alle idee di bolzano provvedettero dedekind e cantor, con la teoria e la gerarchia degli infiniti tuttora unanimemente accettati. cantor, in particolare, enucleò per primo i cosiddetti numeri “transfiniti”, che rappresentano gli infiniti matematici sia in senso cardinale che ordinale. dunque esistono tanti infiniti, a partire dal più semplice, quello numerabile, per continuare con il continuo, e per proseguire, anche qui all’infinito, con potenze successive sempre più grandi, e senza limiti. il tutto nell’ambito di una precisa gerarchia, definita dal carattere costruttivo che ne prova l’esistenza. i transfiniti sono numeri infiniti che possiamo manipolare ed utilizzare per fare calcoli, anche se hanno una natura diversa dagli usuali numeri interi e reali. ciò è stato evidenziato dalla teoria dei numeri non-standard, introdotti negli anni sessanta da abraham robinson. il modello di robinson interessa l’aritmetica, nell’idea di contenere sia i numeri interi, sia i numeri infinitamente grandi con i loro inversi, che dunque si comportano come infiniti ed infinitesimi attuali. se da un lato questo modello pone interrogativi sullo statuto entro cui la nuova matematica si muove, dall’altro legittima il calcolo su queste quantità ma l’infinitamente piccolo e l’infinitamente grande, seppure appaiono simmetrici nella concezione matematica (se a diventa molto grande, 1/a diventa molto piccolo) e fin da aristotele, sembrano avere una storia completamente differente. il problema la geometria non-archimedea. da veronese agli infiniti modelli attuali 7 dell’infinitamente piccolo nasce dal fatto che una grandezza finita possa essere, almeno al livello del pensiero astratto, divisa in sottoelementi sempre più piccoli. la divisibilità “indefinita” è connessa con la nostra percezione della natura continua delle cose e del mondo, che ci appare stabile e permanente. per l’infinitamente grande, invece, possiamo certamente pensare all’estensione del mondo, ma per padroneggiarlo abbiamo bisogno di un apparato concettuale relativamente più complesso. e la situazione paradossale messa in risalto dalla tartaruga di zenone è rappresentativa delle difficoltà storiche. la tartaruga, seppur lentamente, si muove sempre ed il suo inseguitore, achille, può correre alla velocità che vuole, comunque finita, e non potrà mai raggiungerla. per john stuart mill queste difficoltà sull’infinito derivano da un errore di ragionamento, generato dalla confusione che nel nostro intelletto si crea tra tempo divisibile in modo indefinito e tempo infinito. ora, sebbene la matematica abbia imparato a padroneggiare tali situazioni ed a risolvere circostanze in apparenza irragionevoli (per la tartaruga di zenone vi è la confutazione di russell, per cui si possono mettere in corrispondenza biunivoca le due successioni di punti che identificano i luoghi occupati dalla tartaruga ed achille. con questa posizione le due successioni raggiungono lo stesso punto finale), questo genere di paradossi, in definitiva, può emergere su tutte le grandezze, nel momento in cui ammettiamo la possibilità della divisione infinita. la presenza di grandezze infinitesime è chiara anche ad euclide. il matematico tolemaico di solito rivolge la sua attenzione agli angoli rettilinei, ma nel libro iii (alla proposizione 16) ottiene un risultato interessante per i nostri scopi, relativo all’angolo formato da una circonferenza e dalla tangente in un punto, angolo successivamente chiamato di contingenza da g. nemorario. il risultato complessivo vive su tre proposizioni strettamente collegate, di cui l’ultima riguarda la quantità infinitesimale: (i) se dal diametro di un cerchio consideriamo la perpendicolare all’estremo del diametro, questa cade fuor dal cerchio; (ii) fra la circonferenza e la retta perpendicolare al diametro non esiste nessuna altra retta; infine (iii) l’angolo compreso fra l’arco di circonferenza e la perpendicolare è minore di qualsiasi angolo rettilineo. si comprende come l’ultima proposizione sia una “banale” conseguenza della seconda, in quanto per l’esistenza di un angolo rettilineo minore dell’angolo di contingenza è richiesta l’esistenza di una retta compresa tra la circonferenza e la perpendicolare al diametro, ma ciò è appunto dichiarato impossibile. dunque si profila una difficoltà che appare insormontabile, se consideriamo gli angoli rettilinei, curvilinei e mistilinei come grandezze della stessa specie. per tali grandezze non vale il postulato di archimede, cioè a dire non esiste un multiplo del minore che supera il maggiore o, parimenti, esiste un sottomultiplo del maggiore più piccolo del minore, ammettendo la divisibilità delle grandezze. e questo è il caso, visto che nessun sottomultiplo di un angolo rettilineo può diventare più piccolo dell’angolo di contingenza. euclide, pur ammettendo la possibilità teorica di grandezze infinitesime, le bandisce con la definizione 4 del libro v. in tal modo esclude esplicitamente le grandezze che non soddisfano il postulato di archimede, perché grandezze che non sono in rapporto creano inevitabili problemi alla sua teoria delle proporzioni. anche l’atteggiamento di archimede, nei confronti di tali grandezze è della stessa specie di euclide. pur non escludendo la loro possibilità logica, r. mascella 8 al momento opportuno le lascia fuori dalla sua costruzione razionale. le grandezze che considera non sono le uniche effettivamente presenti in natura o che si possono studiare, sono soltanto quelle che vengono trattate per definire il campo di validità delle sue teorie. e dunque il postulato di archimede non esprime una verità necessaria, seppure indiscutibile nella nostra esperienza, ma soltanto più “comoda” per la nostra visione del mondo. inevitabile, in questo caso, il richiamo a posizioni di tipo convenzionalistico, che peraltro esprime lo stesso archimede nella quadratura della parabola: «accade ora che dei suddetti teoremi ciascuno è considerato non meno degno di fiducia di quelli dimostrati senza quel lemma: a noi basta che venga concessa simile fiducia ai teoremi da noi qui dati.» dunque l’infinito che viene preso in considerazione e che interessa è soltanto quello potenziale, mentre quello attuale non viene considerato o interpretato concettualmente come non esistente. la discussione sugli angoli di contingenza rimane però aperta: è lecito considerarle grandezze non nulle? è logicamente possibile l’infinitesimo matematico attuale? nei pensatori del cinquecento e seicento si trova un interesse per l’infinito matematico attuale che non trova riscontro nella matematica classica. a fungere da stimolo in tale direzione è stata certamente la prospettiva nella pittura, con la considerazione dei punti di fuga come corrispondenti ai punti all’infinito, anche se lo sviluppo della geometria cartesiana, affermatasi per l’influenza del pensiero cartesiano, fece passare in seconda linea i metodi sintetici della geometria descrittiva e proiettiva. ma un altro stimolo è anche la concezione filosofica dell’infinito nel pensiero divino, come si riscontra negli scritti di s. agostino o nella posizione speculativa di cartesio, che è agnostica nei confronti dell’infinito fisico e matematico, dunque con atteggiamento negativo sulle posizioni galileiane dell’infinito e infinitesimo attuale, e l’arrivare a possederlo sarebbe un assurdo tentativo di raggiungere l’infinito divino. infine acquista senso la decomposizione delle figure in infinitesimi, come in leonardo da vinci che nel codice atlantico (arrigo-d’amore 1992: p. 72) dice: «[…] tal prova resta persuasiva immaginando esser diviso il circolo in strettissimi paralleli, a modo di sottilissimi capelli in continuo contatto fra loro». è l’apripista all’analisi infinitesimale sviluppata da galileo e keplero. lo stesso galileo, pur non arrivando alla considerazione delle diverse tipologie di infinità, mostra diffidenza ed avversione all’infinito potenziale, con un atteggiamento vicino a quello di george cantor, ed usa teorie sugli infinitesimi attuali che ben si collegano alle concezioni sulla struttura della materia. il problema dell’infinito attuale si è riproposto varie volte nel corso della storia della matematica. si inquadra anche in questo contesto la nascita del calcolo infinitesimale in era più moderna con contribuzioni di vari matematici, da newton a leibniz, da pascal a cartesio e torricelli. il calcolo infinitesimale, sviluppato a partire dal seicento, mostra come l’infinitesimo sia uno strumento matematico molto utile, anche nelle altre discipline scientifiche, a partire dalla fisica. ammettendo la divisibilità infinita, si arriva a manipolare numeri infinitamente piccoli e per questo delle nozioni puramente ideali, senza realtà ontologica. ed è curioso come matematici e fisici abbiano messo assieme un’ampia serie di tecniche di calcolo per utilizzare tale strumento, e contemporaneamente ne contestavano il fondamento filosofico. dunque gli infinitesimi hanno una lunga tradizione. apparsi nella matematica classica, ma poi bandita dalla teoria delle grandezze e dalla geometria ufficiale, sono riapparsi nel tardo medioevo in varie forme ancor più problematiche, ad esempio come indivisibili per giocare un la geometria non-archimedea. da veronese agli infiniti modelli attuali 9 ruolo fondamentale nell’evoluzione dell’analisi. da sempre hanno avuto uno statuto logico e filosofico dubbio, che ha portato al loro abbandono nell’ottocento, per essere rimpiazzati dal concetto di limite. oggi tendiamo a rappresentare lo spazio ed il tempo, e molti altri processi naturali che ci circondano, facendo uso del concetto di continuità, che nasce dall’idea dell’assenza di salti immotivati ed ingiustificati, come testimonia il «natura non facit saltus» di leibniz. da un lato il continuo è visto come indiviso, ininterrotto, banalmente consequenziale, dall’altro ammette la possibilità di processi di divisione infiniti. leibniz stesso fu molto occupato dalle riflessioni sul “labirinto del continuo”, tanto che il monadismo, il suo sistema filosofico, traeva origine dal problema di capire se e come le entità continue potessero essere costituite di elementi indivisibili. leibniz si chiedeva: se ogni entità reale è un’unità ed una molteplicità, e la molteplicità è un’aggregazione di unità, come possiamo classificare il continuo geometrico? una linea è estesa, e l’estensione è una forma di ripetizione, allora la linea essendo divisibile non può essere un’unità. allora è una molteplicità, o un’aggregazione di unità. ma di quali unità parliamo? i punti possono essere solo estremità di parti estese, ed in ogni caso, rifacendosi all’argomento aristotelico, nessun continuo può essere costituito da punti. dunque un continuo non è né un’unità, né un’aggregazione di unità, esso è piuttosto formato da entità ideali. in questo modo leibniz introdusse l’uso degli infinitesimi, in modo strumentale, per snellire le argomentazioni e per aiutare l’intuito. il suo calcolo infinitesimale si fondava in modo essenziale sul concetto di infinitesimo. e il concetto di infinitesimo è stato sempre legato a quello di continuo, come sua ultima parte. così come nel caso discreto vi sono delle unità individuali che compongono il tutto, così nel continuo è l’ infinitesimo che è suscettibile di comporre l’entità compessiva. nella matematica del settecento era diffusa l’idea che, se il continuo ammette la divisione infinita, allora esso non può essere costituito da punti, che per definizione non sono divisibili, ed anzi devono esserci delle grandezze infinitesimali non puntiformi, a partecipare nella costruzione e nella divisione del continuo. eulero usa gli infinitesimi non come elementi ideali, ma come elementi più o meno concreti in grado di preservare la forma rettilinea. berkeley finì con il pensarli come finzioni reali, dopo averne inizialmente combattuto duramente l’uso definendoli “il fantasma delle quantità svanite” e affermando: «concepire una quantità infinitamente minore di ogni sensibile o immaginabile quantità oltrepassa, lo confesso, ogni mia capacità. ma concepire una parte di questa quantità infinitesima, tale che sia ancora infinitamente minore di essa, questa è un’infinita difficoltà per qualunque uomo» (arrigo-d’amore 1992: p. 123). cauchy caratterizzò la continuità con una definizione rigorosa di infinitesimale, come “quantità variabile che decresce indefinitamente in modo da convergere al limite 0”. in questa affermazione giace proprio la definizione di continuità di una funzione. dunque le quantità infinitesimali, sebbene non nulle, sono più piccole di ogni altro numero finito. questa concezione, che ha portato anche a risultati molto utili nella storia della matematica, è rimasta comunque difficile da sostenere filosoficamente e ad un’indagine di tipo logico. in questa direzione, le ricerche di dedekind e cantor, se da un lato aiutarono a comprendere l’infinito, dall’altro colpirono “quasi” mortalmente gli infinitesimali. infatti la concezione dei numeri reali di cantor e dedekind, fondata sulla teoria degli insiemi, soppiantò le altre concezioni di grandezza in competizione: quella geometrico-euclidea in cui venivano r. mascella 10 utilizzati segmenti rettilinei infinitesimali, e quella analitica che tirava in causa quantità infinitesimali di una qualche specie. d’altronde, uno degli obiettivi di dedekind era proprio quello di caratterizzare i numeri reali prescindendo dall’evidenza geometrica e dall’intuito spaziale, cercando di liberare i fondamenti dell’aritmetica dall’intuizione geometrica. in breve, divenne un’ortodossia nei fondamenti della matematica, come testimonianoanche le successive generalizzazioni dei numeri reali del xx secolo, che da essa generalmente partono. cantor, che esplorò i numeri transfiniti, antenne per tutta la vita un atteggiamento ostile nei confronti degli infinitesimi, attaccando tutti gli sforzi matematici in questa direzione. il suo rigetto nasceva dal fatto dalla credenza che la sua teoria fosse esauriente per descrivere e comprendere la realtà dei numeri, e nessuna altra generalizzazione del concetto di numero, in particolare quelle che includevano gli infinitesimali, potesse essere ammissibile. ad ogni modo cantor ruppe l’idea kantiana, secondo cui oltre il finito esisteva l’assoluto, ovvero un’unica forma di infinito. egli invece dimostrò che esistevano tanti infiniti, e che il limite del finito è il transfinito, ciò che possiamo davvero aggiungere. per cui esistono le cardinalità del numerabile, del continuo, e altre cardinalità via via più grandi. dimostrando che non tutti gli infiniti sono numerabili, cantor fece fare un balzo in avanti al pensiero matematico e filosofico, e provò l’esistenza dell’infinito attuale transfinito, non assoluto, e sempre accrescibile. in una lettera a weierstrass nel 1887 cantor tuttavia scriveva (abeles 2001: p. 8): «i numeri lineari non nulli (in breve, numeri che possono essere pensati come lunghezze limitate, continue di una linea retta) che sarebbero più piccoli di qualsiasi numero finito arbitrariamente piccolo non esistono, cioè, essi contraddicono il concetto di numeri lineari». cantor li considerò “il bacillo del colera” che infetta la matematica, con particolare riferimento ai matematici italiani che la promuovevano (veronese e levi-civita su tutti). nei fondamenti della geometria (1891), infatti, in una grande opera di revisione fondazionale della geometria e di approccio storico-critico ai concetti geometrici, veronese elaborò una teoria che ammetteva segmenti infiniti ed infinitesimi come elementi di un campo ordinato nonarchimedeo. questo sviluppo fu ripreso in chiave aritmetica dal suo studente levi-civita che, generalizzando la costruzione di veronese, introdusse gli infinitesimi in modo consistente. in tal modo, nel 1898, era chiaro a tutti che era possibile la costruzione di un campo totalmente ordinato e non-archimedeo, con elementi costituiti da serie di potenze formali. i concetti di infinito ed infinitesimo attuale trattati nella scuola italiana da veronese, levicivita e bindoni affondano le radici nelle opere di leibniz e cavalieri, col metodo degli indivisibili. salvo poi aver avuto un periodo di appannamento a favore di altri approcci concettuali, con le parole di levi-civita (1893: p. 1765) «nell’analisi perché, fissate le basi del calcolo sul concetto di limite, nessun’altra teoria ne faceva sentire il bisogno, nella geometria, per l’influenza dell’empirismo ad oggi dominante.» l’avversione era comunque diffusa. anche russell li condannò definendoli non neessari, sbagliati e auto-contraddittori (russell 1938: p. 345). quando il continuo ebbe delle solide fondamenta date dalla teoria degli insiemi, l’uso degli infinitesimi fu abbandonato, anche se l’idea non fu completamente estirpata, neanche dopo le dure critiche rivolte a veronese e la noncuranza riservata alla questione nei primi del novecento. poche voci si levavano che non contrastavano le ricerche su di essi. tra queste, vi era la concezione di peirce del continuo numerico che comprendeva anche gli infinitesimali, perché i metodi basati su di essi erano la geometria non-archimedea. da veronese agli infiniti modelli attuali 11 utili ed efficienti, e perché li vedeva come la “colla” che fa perdere l’identità ai punti sulla linea continua. lo stesso poincaré accettò gli infinitesimi, anche se non li riteneva particolarmente utili. il primo segnale di rinascita ci fu con i lavori di laugwitz e schmieden, ma la definitiva ripresa si ebbe con abraham robinson negli anni sessanta. robinson creò la matematica nonstandard, un’estensione dell’analisi matematica che incorporava al suo interno numeri finiti, infinitamente grandi e infinitesimi, in cui le usuali leggi dell’aritmetica continuano a valere, in cui si possono tradurre facilmente in un nuovo linguaggio tutte le proposizioni dell’analisi ordinaria dei reali. ed in cui la proprietà archimedea non è soddisfatta. 3. la retta di veronese ed i monosemii di levi-civita a partire dalla seconda metà dell’ottocento, un grosso contributo alla comprensione e caratterizzazione matematica dell’infinito, nonché al suo statuto filosofico, venne dato dagli studi di dedekind e cantor, con i numeri transfiniti. per la prima volta, l’infinito non veniva trattato ontologicamente, dal di fuori, come ente da comprendere nella sua apparenza esterna, ma veniva indagato dal suo interno, cercando di comprenderne la natura e le sue possibili alternative. per la teoria cantoriana esistono infiniti “infiniti”, peraltro considerabili in modo gerarchico, in cui il primo è l’infinito numerabile e, da qualunque ordine di infinito, è possibile ottenerne uno di ordine più grande prendendo l’insieme delle sue parti. non sappiamo se tra questi due ordini ve ne sono altri in mezzo, ad oggi questa possibilità viene scartata ma è ancora soltanto un’ipotesi (del continuo). tanto basta, comunque, per poter avere una prima chiave di lettura sul fatto che all’infinito non tutto deve procedere sempre nello stesso identico modo. nello stesso periodo si assiste ad una rivisitazione dei concetti e dei fondamenti della geometria, che portano a guardare alla possibilità logica di altre geometrie non soddisfacenti l’assioma delle parallele. un postulato euclideo la cui affermazione, pur nascendo dall’intuizione nello spazio osservabile, fa un’ipotesi per il suo esterno non controllabile, così come tale sembra essere la proprietà delle grandezze di potersi raggiungere l’un l’altra sempre ed inequivocabilmente, siano esse segmenti o angoli o altro ancora. ed è intanto il concetto di continuità ad aver acquisito molta rilevanza in ambito geometrico. sulla base degli studi di dedekind e cantor, il sistema dei numeri reali viene denominato continuo aritmetico perché si ritiene sia adeguato a rappresentare tutti i tipi di fenomeni continui. in accordo a ciò, il continuo lineare geometrico viene pensato isomorfo e provvisto di un’assiomatizzazione coerente con i numeri reali. cantor e dedekind sono i primi a proporre questa tesi matematico-filosofica, ed il presunto isomorfismo tra le strutture è talvolta chiamato assioma di dedekind-cantor (ehrlich 2006). in altri termini, se i numeri reali godono di una serie di proprietà, e se la geometria uno-dimensionale è intuitivamente in rapporto strettissimo con essi, come possiamo trasferire tali proprietà assiomaticamente? sotto la forma data da dedekind, la continuità si esprime nella forma (postulato di dedekind): se un segmento di retta ab è diviso in due parti in modo che: (1) ogni punto del segmento ab appartiene ad una sola delle due parti; (2) a appartiene alla prima parte, b alla seconda; r. mascella 12 (3) un punto qualunque della prima parte precede un punto qualunque della seconda, nell’ordine ab del segmento; allora esiste un punto c del segmento ab, che può appartenere all’una come all’altra parte, tale che ogni punto di ab che precede c appartiene alla prima parte, ed ogni punto che lo segue appartiene alla seconda parte. questo postulato può essere enunciato in maniera analoga considerando segmenti in luogo di punti (segmenti aventi un estremo nel punto a o nel punto b, e l’altro estremo negli altri punti considerati). invece, il postulato della continuità nella forma di cantor, espresso in modo armonico con la sua teoria dei numeri, è il seguente (postulato di cantor): se due classi di segmenti di retta sono tali che: (1’) nessun segmento della prima classe sia maggiore di qualche segmento della seconda; (2’) prefissato un segmento s piccolo a piacere, esistono un segmento della prima classe ed uno della seconda la cui differenza è minore di s allora esiste un segmento che non è minore di alcun segmento della prima classe né maggiore di alcun segmento della seconda. anche in questo caso il postulato si può enunciare in modo analogo, riferendosi ai punti estremi dei segmenti. può sembrare che i due enunciati siano equivalenti, invece esiste una differenza sostanziale data proprio dall’assioma archimedeo: dal postulato di dedekind è possibile dedurre il postulato di archimede, non altrettanto da quello di cantor. ovvero, si possono costruire delle grandezze che soddisfano il postulato di cantor ma non quello di archimede. ad una prima occhiata, ciò sembra derivare dal fatto che il postulato di dedekind è fondamentalmente più condizionante e vincolante, nel senso che non si limita ad enunciare l’esistenza di una entità che soddisfa una certa proprietà. nel caso cantoriano si parla genericamente di un segmento che soddisfa una proprietà, senza alcuna precisazione sulla tipologia del segmento, ad esempio se il segmento è finito o infinito. invece nella formulazione di dedekind si parla di un punto preciso c, o se vogliamo di segmenti ac e bc, e dunque si ferma l’attenzione su segmenti che hanno un inizio ed una fine ben specificati. dunque, ammettendo il postulato di dedekind, è possibile dimostrare che “dati due segmenti esiste sempre un multiplo dell’uno maggiore dell’altro”, ovvero il postulato archimedeo, e perciò, una volta che l’assioma di cantor-dedekind è stato adottato dal sistema numerico, ne viene fuori la classica formulazione geometrica rettilinea e la prospettiva filosoficomatematica che gli infinitesimi siano superflui all’analisi della struttura di una linea retta continua. si può dimostrare, infine, che i due postulati di cantor ed archimede equivalgono logicamente al postulato di dedekind. a notare questa particolare connotazione semantica del postulato di dedekind è stato giuseppe veronese che evidenzia la possibilità di strutture geometriche non-archimedee, che ammettono la presenza di infiniti ed infinitesimi attuali, ovvero di grandezze irraggiungibili nelle direzioni dell’indefinitamente grande e dell’indefinitamente piccolo. veronese a tal proposito scrive (1891: p. xxvii): «sosteniamo l’infinitesimo attuale perché ne abbiamo dimostrato non solo la possibilità ma anche l’utilità nel campo geometrico; ché anzi per quanto possa essere per sé interessante una tale teoria non l’avremmo forse qui trattata senza la geometria non-archimedea. da veronese agli infiniti modelli attuali 13 le applicazioni geometriche che ne abbiamo fatte.» ed ancora (veronese 1891: p. xxx): «[…] volendo trattare il problema scientifico in tutta la sua generalità, abbiamo date queste ipotesi per stabilire una geometria indipendente dall’assioma v di archimede». egli fornì anche l’esempio pionieristico di una tale specie di grandezze, noto come retta di veronese, che soddisfa tutti i postulati dell’ordine, della congruenza, e di cantor, ma non quello di archimede, e dunque neppure quello di dedekind. veronese (1891) affronta il problema della continuità dando due ipotesi per stabilire sulla forma fondamentale rettilinea la continuità relativa o assoluta, a seconda che i segmenti considerabili nella forma soddisfino il postulato di archimede o se invece la forma ammetta l’esistenza di segmenti infiniti ed infinitesimi attuali. il primo concetto di continuo nasce con l’intuizione, ma questo non si può definire; esso si può però rappresentare e determinare formalmente. emerge, dunque, un approccio che conserva la natura empirica della geometria e dell’interpretazione spaziale delle forme, ma emerge anche la natura logico-formale della scienza geometrica che definisce gli enti indipendentemente dalla comprensione profonda del risultato empirico, che al matematico ed al filosofo naturale appare solo in superficie. da un lato la geometria trae la sua origine dall’osservazione del mondo esterno, poi da tali osservazioni formalizzate permette di dedurre le prime verità che vengono postulate, ma è altresì necessario fare attenzione al reale contenuto dei postulati, per non introdurre, magari non intenzionalmente, più di quanto è realmente suggerito dall’osservazione. e la nostra osservazione si estende ad una parte limitata e finita dello spazio. veronese affronta la questione mettendosi, rispetto al postulato di archimede, da un punto di vista analogo a quello di bolyai e lobacevski rispetto al postulato delle parallele. con le parole di fano (1950: p. 500), veronese procede in questa maniera: «[…] dopo introdotto in una forma a una dimensione un segmento unità e la relativa scala, ottenuta portando quel segmento sulla forma, ripetutamente, a partire da una posizione iniziale qualsiasi e in ambo i sensi, egli postula l’esistenza, nella forma, di almeno un elemento esterno al campo raggiungibile colla scala, in un determinato verso; allora ogni segmento avente un estremo nel campo della scala e l’altro fuori di esso, nel detto verso, si dirà “maggiore” e “infinito attuale” rispetto a qualsiasi segmento avente entrambi gli estremi nel campo della scala. con questo concetto, eventualmente applicato più volte, si perviene a un continuo non archimedeo, pel quale risultano tuttavia verificati i soliti postulati lineari e della congruenza.» la retta non-archimedea di veronese può essere riguardata graficamente nel modo che segue. consideriamo un sistema di rette, in numero finito o infinito, che per semplicità possiamo immaginare parallele ed equidistanti. ora fissiamo uno dei due versi, ad esempio nella figura che segue prendiamo il verso da sinistra a destra. i punti su una singola retta si succedono su ogni singola retta nel modo consueto (dunque nell’esempio si ha a < b); inoltre ogni punto precede tutti i punti delle rette che seguono (nell’esempio a < c, ma anche b < c). in questa retta è evidente che il segmento ab, pur essendo ripetuto un numero indefinito di volte sulla destra di b, non riuscirà mai a portare il punto finale oltre il punto c. dunque abbiamo un segmento infinitesimo attuale, nello specifico ab, ed un segmento infinito rispetto ad ab, nello specifico ac. r. mascella 14 figura 1. poi costruisce il piano attraverso il fascio di rette che si ottiene congiungendo tutti i punti di questa retta con un punto fisso al suo esterno, quindi allo stesso modo genera lo spazio tridimensionale, ed a seguire anche spazi n-dimensionali. in altri termini veronese sviluppa una “geometria assoluta” che è indipendente dal postulato di archimede, con un numero di qualunque di dimensioni e a più unità rettilinee (come d’altronde recita il sottotitolo dell’opera), e con riferimento alle ipotesi che la retta si presenti come linea aperta (come nel caso euclideo) o come linea chiusa (come nel caso riemanniano). con le parole di veronese (1891: p. xxix-xxx): «[…] diamo alcune ipotesi le quali ci permettono di stabilire una geometria assoluta, indipendente cioè dall’assioma v di archimede, e di far scaturire da essa due sistemi generali nei quali sono compresi i sistemi particolari di euclide e riemann.» quello in fig. 1 è comunque l’esempio più semplice di continuo non-archimedeo. nelle opere di veronese, e in quelle di levi-civita, ci si muove nell’ambito di una scienza che non è puramente geometrica, né puramente algebrica o analitica. è un terreno misto, con influenze e conseguenze tanto per la geometria che per l’analisi. se veronese ne dà inizialmente un’introduzione geometrica tipicamente sintetica, e per questo forse poco chiara, levi-civita si preoccupa degli stessi oggetti da un punto di vista meramente analitico, pur ricononoscendone i pregi in ambito geometrico, che fa di tali oggetti entità direttamente applicabili ed utilizzabili. questi numeri, che sono denominati monosemii, o più precisamente, numeri monosemii del continuo numerico di seconda specie, costituiscono un sistema numerico non-archimedeo che si accompagna alla forma uno-dimensionale di veronese. levi-civita pensa a due numeri reali, a e , e considera dei numeri a (i monosemii, appunto) in cui a è la caratteristica, mentre  è l’indice. i “vecchi” numeri reali sono compresi tra questi numeri come numeri con indice zero. due numeri monosemii sono uguali se hanno uguali sia l’indice che la caratteristica, con la sola eccezione dei numeri con caratteristica nulla, che sono considerati uguali anche se hanno indice diverso. l’ordinamento su questi numeri, che oggi possiamo riguardare come coppie ordinate di numeri reali, è dato dall’ordinamento lessicografico, che nella notazione originale è espressa dalla seguente: a > b se  >  oppure  =  e a > b. e qui appare una generalizzazione compiuta da levi-civita, che parla di numeri di caratteristica reale con indice reale, laddove nella ricostruzione per via algebrica della retta di veronese sarebbe bastato considerare numeri di caratteristica reale e di indice intero. per la somma di monosemii con stesso indice abbiamo banalmente a + b = (a + b) ma per indici fra loro distinti abbiamo a che fare, in generale con somme del tipo      i ia . queste somme, le cui componenti sono denominate integranti, ora possono essere finite o infinite. in particolare levi-civita è interessato a quelle che risultano soddisfare alcune condizioni sul la geometria non-archimedea. da veronese agli infiniti modelli attuali 15 numero di addendi. tra queste, fissato un numero a piccolo quanto si vuole, interessa la proprietà di avere solo un numero finito di addendi con indice superiore ad a; o, analogamente, fissato a grande quanto si vuole, interessano quelle integranti che hanno un numero finito di addendi con indice inferiore ad a. i primi sono detti ellittici (o soddisfacenti la condizione e), i secondi iperbolici (o soddisfacenti la condizione i). proseguendo nella costruzione teorica, definendo le ulteriori operazioni di addizione, sottrazione, moltiplicazione e divisione, nonché limiti serie e funzioni, levi-civita precisa che (1893: p. 1777): «per tutte le operazioni aritmetiche sui numeri di seconda specie si troverà mantenuto l’algoritmo ordinario: questa caratteristica, comune ai numeri di veronese e ai sistemi ellittico ed iperbolico, li distingue essenzialmente dai numeri transfiniti di cantor, dagli ordini di infinità del du bois-reymond, dai momenti di stolz.» ed ancora rimarca che sia effettivamente possibile la definizione di un nuovo raggruppamento di numeri con tali operazioni «[…] questa possibilità di conservare tutte le leggi fondamentali dell’aritmetica non è in contraddizione col teorema di weierstrass, da cui discende che, all’infuori dei numeri reali e complessi ordinari, non esiste alcun altro sistema, con un numero finito n di unità indipendenti, per il quale valgano le ordinarie regole di calcolo. […] poiché ciascuno dei due sistemi contiene un numero infinito di unità 1 , dove l’indice  assume tutti i valori reali da  a +.» la definizione di numeri finiti, infiniti e infinitesimi di levi-civita (1898: pp. 113-114) è la seguente. sia  un elemento qualsiasi in m, che sia ordinato e sia un corpo rispetto a somma e sottrazione. potendo avere in m anche l’elemento  consideriamo il valore assoluto || per prendere il valore non negativo. allora due elementi  e ’ di m sono finiti tra loro se esiste un numero intero e positivo k tale che il maggiore dei valori assoluti, ad esempio || sia inferiore a k|’|. se un tale numero k non esiste,  è infinito rispetto a ’, ovvero ’ è infinitesimo rispetto a . nell’ultimo paragrafo dell’articolo del 1893, levi-civita si occupa dell’applicazione geometrica di tali concetti, considerando segmenti con un estremo in 00 e l’altro variabile. in tal modo i segmenti diventano infiniti, finiti o infinitesimi a seconda che l’indice dell’altro estremo sia maggiore, uguale o minore di zero. nel confronto tra due segmenti, il primo è infinito, finito o infinitesimo rispetto al secondo se la divisione tra i due estremi produce un monosemio con indice maggiore, uguale o minore di zero. emerge ancora una volta, almeno come approccio iniziale, la concezione identificatrice tra continuo geometrico e continuo numerico da cui levi-civita muove le sue considerazioni per proporre «una retta più generale, atta cioè a rappresentare non i soli numeri ordinari, ma tutti i numeri ellittici o tutti quelli almeno, che appartengono ad un dato intervallo.» l’idea nasce, come anticipato, sulla scorta della forma fondamentale di veronese, di cui peraltro levi-civita era allievo, ma con alcune importanti differenze. se da un lato levicivita considera elementi infiniti ed infinitesimi con indice finito, e dunque corrisponde ad una parte della forma di veronese, dall’altra appare più generale perché non si limita a considerare ordini di infinità con l’insieme di riferimento dei soli numeri interi, ma allargando il concetto ai numeri reali. ma è in questa ultima nota dell’articolo che levi-civita esprime l’idea più complessa ed interessante, sebbene la stessa caratterizzazione analitica sia di un certo interesse generale. ovvero la possibilità di non limitarsi alla presente costruzione per considerare elementi infiniti r. mascella 16 ed infinitesimi, ma di allargarne le possibilità ripetendo lo stesso ragionamento effettuato sui numeri reali per arrivare ai monosemii, stavolta partendo dai numeri monosemii per ottenere altri nuovi numeri con le stesse modalità operative, cioè considerando nuovi monosemii con indici tra i monosemii, per poi poter di nuovo applicare il procedimento tutte le volte che vogliamo. si tratterebbe di una generalizzazione che però non conserverebbe tutte le proprietà aritmetiche dei reali e dei monosemii. un’ulteriore generalizzazione che invece rimanga conforme a tali principi è quella che utilizza come caratteristica un monosemio e, come indice, un numero reale ordinario. l'opera fu oggetto di varie critiche, tra gli altri da parte di killing, cantor e schönflies. secondo cantor le quantità infinitamente piccole degli infinitesimali non potevano esistere, ed offrì una prova che la proprietà archimedea era una conseguenza necessaria del concetto di “quantità lineare” e dei suoi teoremi sulla teoria dei numeri transfiniti. per cantor, il postulato di archimede era in realtà un teorema. la prova che questa argomentazione non fosse valida per tutti i sistemi numerici che stavano germogliando fu data da stolz, che in particolare la provò per le sue quantità infinitesimali e per quelle di du bois-reymond (dauben 1979). in alcuni casi la critica fu molto accentuata, come nel caso della stroncatura da parte di peano, che concludeva la sua recensione indicando nella mancanza di precisione e di rigore di veronese un elemento che toglieva ogni valore allo studio. le controversie sulla geometria non archimedea durarono a lungo, con numerosi interventi dello stesso veronese e di numerosi matematici in italia e all’estero, e terminarono solo con la pubblicazione dei “grundlagen” di hilbert, che mostravano la possibilità logica di una tale geometria. la consacrazione definitiva avvenne nel 1908 al congresso internazionale di roma, dove veronese tenne una relazione generale dal titolo “la geometria non-archimedea”. ad ogni modo in queste due decadi a cavallo del secolo il dibattito è accesso e senza soluzione di continuità. già nel 1897 veronese ritorna sulla questione della continuità, in risposta alle critiche. veronese ribatte su tutta la linea, in particolare asserendo la possibilità della geometria proiettiva per segmenti infiniti ed infinitesimi, ribadendo l’equivalenza dei suoi numeri con quelli di levi-civita ed infine evidenziano i contrasti tra gli stessi suoi critici. ritorna sul concetto intuitivo di continuo, già evidenziato nel 1891, «[…] il continuo intuitivo non si definisce, ma che pel geometra basta definire il continuo come un gruppo di punti assoggettato a certe proprietà. in qual modo si formi in noi l’intuizione del continuo è un problema che spetta al psicologo risolvere, se pure può essere risolto; come si determini il continuo come un gruppo di punti spetta al geometra». il continuo intuitivo, la cui genesi ci è oscura dalle profondità del nostro intelletto, è però rappresentabile avvalendosi delle proposizioni fondamentali che vengono dall’esperienza e dall’osservazione, utilizzando fatti semplici su cui vi è unanime convergenza, facendo astrazione dalle qualità particolari degli oggetti, ma facendo sempre attenzione a non creare contraddizioni all’interno dell’edificio teorico. e veronese esprime la sua critica al fatto che il continuo rettilineo venga determinato univocamente dai numeri reali, perché a quel punto si introducono inevitabilmente nel continuo geometrico altri concetti che ad esso, in generale, non appartengono. ed il punto centrale di conflitto, in cui la subordinazione emerge in tutta la sua influenza, è il postulato di archimede che, seppur valido nei reali, non per questo deve valere nel continuo geometrico, ma con l’identificazione iniziale viene automaticamente trasportato su di esso. la geometria non-archimedea. da veronese agli infiniti modelli attuali 17 e d’altro canto, ammettendo come elementi i suoi segmenti infiniti ed infinitesimi, non si creano contraddizioni né in senso logico-fondazionale, né per ciò che riguarda l’aspetto empirico della realtà spaziale. lo stesso veronese dice (1897: p. 162) «[…] ho fatto vedere che ammettendo gl’infiniti e infinitesimi non si contraddice alle proposizioni ricavate direttamente dall’osservazione e necessarie per dimostrare la proprietà della figura corrispondente allo spazio fisico.» se di tali segmenti si ammette l’esistenza logica, non è nemmeno necessaria la loro esistenza attuale, in senso strettamente fisico, così come non è necessario nemmeno pensare ad uno spazio con più dimensioni per giustificare la riflessione logica su uno spazio generale con infinite dimensioni. a far maturare l’idea di una geometria non-archimedea è certamente la concezione generale della geometria che veronese ha, e che appare chiara nei suoi scritti. in essi, adottando una linea di indagine filosofica, emergono i presupposti espistemologici alla sua concezione, che spaziano dal ricorso all’intuizione per determinare le proprietà geometriche dello spazio, il ruolo degli assiomi nella costruzione delle teorie geometriche e matematiche, dunque un concetto di scienza mista tra l’aspetto empirico-intuitivo e logico-fondazionale, infine le nozioni di infiniti ed infinitesimi attuali, continuo, forma ed iperspazio. più nel dettaglio, la visione di veronese per annientare l’identificazione del continuo rettilineo geometrico con quello numerico reale risiede nella concezione filosofico-matematica del continuo come classi contigue, sulla scorta dei lavori di dedekind e cantor, piuttosto che come insieme di punti. veronese da un lato rivendica l’origine esperienziale della geometria, dall’altro utilizza un atteggiamento razionalistico per giustificare gli enti ideali assiomaticamente introdotti. nel progresso geometrico, la visione epistemologica non è un discorso a posteriori, come accade spesso negli uomini nell’indagine filosofico-scientifica, ma è pertinente allo stesso sviluppo della scienza geometrica, che emerge e trova linfa nelle pieghe stesse dell’indagine conoscitiva. in tal senso la riflessione non si limita alla legittimazione di risultati già acquisiti, ma indaga la natura degli oggetti concepiti o concepibili per promuovere le idee migliori o quelle che potrebbero ancora emergere. è una riflessione filosofica che è strettamente collegata, potremmo dire fondazionale, come spirito e come approccio, alla possibilità di nuove scoperte geometriche e matematiche. ma è tutto il dibattito di quegli anni che in effetti si realizza passo dopo passo sulle conquiste e sulle riflessioni contestuali che arrivano nella scienza dello spazio e in alcune parti della matematica, legati al tentativo di comprendere e formalizzare una parte sempre più consistente del mondo. sono artefici con veronese, peano, cantor e vivanti per ciò che riguarda gli infinitesimi attuali; gauss, poincaré, hilbert, helmholtz e grassman tra gli altri, per ciò che riguarda la scienza e l’intuizione geometrico-spaziale. ai lavori di veronese e levi-civita che fanno da apripista sulla questione, seguono come parti di un unico ideale programma di ricerca, i lavori di hilbert e hahn, che reinterpretano e sviluppano le conseguenze algebriche delle strutture ordinate non-archimedee. 4. altri modelli e classificazione del non-archimedeo tra i modelli di geometria non-archimedea più rilevanti, un posto ed un ruolo di primo piano spetta a quello di david hilbert. la costruzione di hilbert, che soddisfa tutti gli assiomi r. mascella 18 euclidei ma non quello di archimede, ha fatto ritenere chiara e risolta la questione della possibilità logica azzerando, nel contempo, anche le critiche filosofiche rivolte all’uso di infiniti ed infinitesimi attuali. poincaré, pur riconoscendo a veronese il ruolo di precursore, riconobbe la priorità ed il merito per la portata innovativa di questa geometria completamente ad hilbert. l’esposizione di hilbert, semplicemente, era più chiara e più elegante, perché usava un metodo che veronese aveva deliberatamente lasciato da parte, per non contaminare, come rimarcò più volte, il processo di acquisizione della conoscenza geometrica. veronese infatti afferma (1905: pp. 349-350): «[…] io non volli avere a mia disposizione le risorse dell’analisi […] partendo invece da alcuni fatti e operazioni semplicissime del pensiero logico, e scomponendo, più che mi fu possibile, i vari concetti, senza far uso di alcun algoritmo già noto.» e poi, ancora riferendosi a poincaré ed hilbert (veronese 1905: p. 350): «[…] trattando la geometria più con le vedute dell’analisi che con quelle dell’intuizione spaziale, sono portati a dare alla geometria un’estensione maggiore di quella che secondo l’intuizione essa possa e debba avere.» hilbert, insomma, sostenuto dalla una visione filosofica più ampia del complesso dei fondamenti geometrici, forse seppe trarre maggior profitto dall’idea non-archimedea e ciò servì a fare del non-archimedeo una sua scoperta. è stato lo stesso veronese, a convenire sul fatto che l’esposizione di hilbert appare più semplice, proprio per le tecniche analitiche utilizzate per costruire le sue forme matematiche, delle risorse decisamente più potenti che evitano le lungaggini dell’indagine sintetica. ad ogni buon conto, hilbert è stato tra i pochi matematici dell’epoca ad avere una reazione positiva ai lavori di veronese, così come anche l’algebrista hans hahn, con il primo che si diede il problema della consistenza di un tale sistema, il secondo che si diede quello della giustificazione algebrica al “continuo intuitivo” di veronese. nella elegante costruzione di hilbert (1899) sui fondamenti della scienza geometrica, com’è ampiamente noto, si riformula la struttura logico-deduttiva a partire da cinque gruppi di postulati, che semplificano e chiariscono le strutture che a mano a mano si delineano: postulati di collegamento (i), di ordinamento (ii), di congruenza (iii), delle parallele (iv) e di continuità (v). in particolare ci interessa il quinto gruppo, che nella prima edizione dei grundlagen prevedeva un solo assioma, quello appunto archimedeo denominato “assioma di continuità”, ma già nella seconda edizione (fatto, questo, rimarcato dallo stesso veronese per mettere in evidenza il cambiamento di prospettiva del matematico tedesco a seguito dei lavori di veronese stesso, e dunque per dar credito alla priorità del suo lavoro rispetto a quello di hilbert) esso viene integrato da un altro postulato. così, i due postulati che completano il quadro fondazionale, che si dimostrano equivalenti al postulato di dedekind ed alla coppia di postulati cantor-archimede, diventano per hilbert i seguenti: v1 (assioma della misura o archimedeo). se ab e cd sono due segmenti qualsiasi, c’è un numero n tale che il trasporto del segmento cd reiterato n volte da a sulla semiretta passante per b, porta al di là del punto b. v2 (assioma di completezza lineare). il sistema dei punti di una retta con le sue relazioni di ordinamento e congruenza non è suscettibile di un ampliamento per il quale rimangono inalterate le relazioni sussistenti tra gli elementi precedenti come pure le proprietà fondamentali di ordinamento lineare e congruenza che seguono dagli assiomi i-iii ed anche v 1. la geometria non-archimedea. da veronese agli infiniti modelli attuali 19 l’assioma di completezza lineare, non è una conseguenza dell’assioma archimedeo, infatti l’assioma v1 anche quando vale, può essere ancora “completato”, ovvero può rendere ancora possibile l’aggiunta di nuovi elementi senza alterare per questo la validità degli assiomi i-iii, ma ciò non deve esser fatto necessariamente (l’esempio tipico è la cosiddetta retta noncantoriana, in cui cioè non vale il postulato di cantor, ben rappresentata dalla retta dei razionali). se invece vale l’assioma di completezza, esso è comunque condizionato al fatto che tra le proprietà e relazioni da conservare ci sia anche l’assioma archimedeo v1. ovvero, l’assioma di completezza lineare è sufficiente per l’assioma archimedeo, l’assioma archimedeo è necessario per quello della completezza. nel caso di validità di entrambi, la geometria hilbertiana «è identica con l’usuale geometria analitica “cartesiana”» (hilbert 1899: p. 33). questa citazione proveniente direttamente dal matematico tedesco, vuole anche essere una conferma che gli strumenti da lui adottati sono analitici, ed il confronto viene fatto con la geometria cartesiana, non con quella sintetica, né di euclide, né per ciò che ci interessa, di veronese. l’assioma di completezza è una diversa ma equivalente formulazione, nel contesto assiomatico fino a quel punto enucleato, dell’assioma nella veste cantoriana, che prevede l’esistenza del limite nelle sezioni di dedekind. in tali considerazioni, l’assioma di archimede ha il ruolo di «preparare l’esigenza della continuità», che dunque rimane delegata all’assioma di completezza. hilbert, resosi conto dell’utilità dell’assioma archimedeo, da un lato comprende che la sua indipendenza va analizzata per suo conto; d’altra parte sottintende anche che l’accettazione della geometria non-archimedea si può compiere solo analizzando l’indipendenza dell’assioma v1 dagli altri postulati e chiarendone senza esitazioni i contorni fondazionali e le conseguenze. per far ciò hilbert si avvale di un insieme funzionale. egli considera il dominio (t) costituito dalle funzioni algebriche (t) reali di variabile reale, che si ottengono attraverso le quattro operazioni fondamentali – addizione, sottrazione, moltiplicazione e divisione – e con l’ulteriore operazione 21  (ovvero considerando anche l’estensione pitagorica). l’insieme (t) è chiaramente numerabile, e sulle sue funzioni algebriche è possibile un ordinamento, in analogia con quanto hilbert fa su sistemi complessi di numeri, ad esempio con i reali. l’ordinamento è possibile pensando a come dette funzioni si comportano per valori sufficiente grandi, ovverosia andando idealmente oltre gli zeri delle funzioni per valutare il comportamento asintotico. essendo algebriche, infatti, queste funzioni si annullano in un numero finito di punti, e dunque da un certo punto in poi hanno valori sempre positivi o sempre negativi e si può pertanto considerare come si comportano da quel punto in poi. possiamo quindi dire, ad esempio, che a > b (rispettivamente a < b) se la funzione differenza c(t) = a(t)  b(t) da quel punto in poi (oltre gli zeri di a e b) risulta sempre positiva (rispettivamente negativa). fatto l’ordinamento sulle grandezze, in analogia con i reali, e ammettendo la validità dei teoremi sulla conservazione delle disuguaglianze (addizionando o moltiplicando i membri con la stessa quantità), possiamo notare che le funzioni così costruite non soddisfano l’assioma archimedeo. se consideriamo un qualunque numero reale n e qualunque funzione (t) indefinitamente positiva, ad esempio (t) = t, vale sempre che n  r. mascella 20 (t) è negativa per t sufficientemente grande. hilbert enuncia la questione in questo modo (1899, p. 49): «i due numeri 1 e t del dominio (t), che sono tutti e due > 0, hanno la proprietà che qualsiasi multiplo del primo è sempre minore del secondo numero.» a questo punto si può fondare coerentemente una geometria non archimedea sui numeri complessi di (t) che spicca nella sua originale discussione anche per mettere a disposizione del lettore una geometria non solamente rettilinea, anche se i due assiomi della continuità si applicano a sistemi di punti lineari, ma addirittura planare. infatti si può considerare il punto del piano come un sistema di tre numeri (x, y, z) nel dominio (t)3, ed il piano come un rapporto di quattro punti (u : v : w : r) con u, v, w non tutti nulli, ovvero con l’insieme di punti che soddisfano l’equazione ux + vy + wz + r = 0. in questo piano una retta si identifica con una totalità di punti che appartiene a due piani diversi, ovvero con diversi u : v : w. per l’ordinamento su (t)3 possiamo ricorrere all’usuale ordinamento lessicografico, ovvero (x, y, z) < (x’, y’, z’) se x < x’, oppure x = x’ e y < y’, oppure x = x’ e y = y’ e z < z’. ovviamente l’ordinamento può essere scelto anche invertendo la priorità delle componenti. infine occorre avere uno strumento per la trasportabilità di segmenti ed angoli. ciò può avvenire usando le convenzionali trasformazioni x’ = x + a, y’ = y + b, z’ = z + c. a questo punto è nata una geometria non-archimedea, in cui sono soddisfatti tutti gli assiomi tranne quello di archimede. per convincersene basta prendere il segmento 1 ed il segmento t, e notare come, anche trasportando il segmento 1 un numero arbitrario di volte, mai si riesce a superare il segmento t, contraddicendo l’assioma archimedeo. hilbert stesso, come suffragato di lì a poco da poincaré, riconosce a questa geometria un alto status scientifico (hilbert 1999, p. 49): «anche la geometria non-archimedea, come la non-euclidea, è importante in linea di principio» e fa particolare riferimento al ruolo che l’assioma di archimede svolge nella dimostrazione del teorema di saccheri-legendre. su questo punto occorre una riflessione più approfondita, per arrivare alla ricerca di max dehn, intrapresa su suggerimento di hilbert, che chiarisce completamente la questione e delinea il rapporto esistente tra gli assiomi “della continuità” e “delle parallele”. consideriamo innanzitutto i primi tre gruppi di assiomi dati da hilbert nel suo classico grundlagen, ovvero il sistema assiomatico che comprende gli assiomi di incidenza, d’ordine e di congruenza. un modello di questo sistema è denominato h-piano (greenberg 1979: p. 757). se a questo punto adottiamo l’assioma euclideo delle parallele, allora avremo a che fare con un piano di tipo euclideo, se invece adottiamo l’assioma delle due parallele (come formulato da bolyai, lobacevski, klein, ecc.) ci troveremo ad aver a che fare con un piano di tipo iperbolico. in realtà questi due modelli risentono del fatto che già implicitamente si sta assumendo la continuità come proprietà dei modelli, ad esempio nella forma unificante di dedekind, infatti si può dimostrare che un h-piano continuo deve essere necessariamente euclideo o iperbolico. ma se consideriamo anche la possibilità che l’assioma archimedeo possa non valere allora la prospettiva cambia radicalmente e chiarisce anche meglio le idee sulle mutue relazioni dei postulati e sui possibili modelli di geometria con essi ottenibili. che l’assioma archimedeo è assunto implicitamente già nella usuale trattazione sulla sola geometria degli h-piani, emerge dal fatto che esso interviene nella dimostrazione classica del teorema di saccheri-legendre, con il quale si afferma che, dato un quadrilatero abcd con gli angoli in a ed in b retti e con i segmenti ab e cd congruenti, allora si possono presentare tre la geometria non-archimedea. da veronese agli infiniti modelli attuali 21 casi per le coppie di angoli rimanenti: sono entrambi ottusi, retti o acuti. com’è noto le tre differenti ipotesi corrispondono ai tre conclamati casi non-euclidei, tutti logicamente possibili: l’ipotesi dell’angolo retto corrisponde alla geometria euclidea; l’ipotesi dell’angolo acuto corrisponde alla geometria iperbolica; l’ipotesi dell’angolo ottuso, infine corrisponde ad una terza geometria in una regione limitata dello spazio, qualora si abbandoni l’ipotesi dell’infinità della retta, ovvero la geometria ellittica. dunque il risultato ottenuto dal teorema di saccherilegendre è indipendente dai postulati dell’h-piano, perché dalle sue premesse rimangono aperte due possibilità, escludendo il caso dell’angolo ottuso perché questa geometria non può essere costruita sugli h-piani, essendo incompatibile con le altre premesse euclidee (gli assiomi di ordinamento sono differenti). la costruzione di max dehn (1900), ma anche quella di bonola (1906), parte innanzitutto con il dimostrare il teorema di saccheri, nella formulazione per i triangoli, ovvero rispetto alla somma complessiva dei suoi angoli (che può essere maggiore, uguale o minore di due angoli retti), senza far uso del postulato di archimede. il risultato di dehn è che se si ammette il postulato archimedeo, le tre diverse ipotesi che si possono fare sulla somma degli angoli interni di un triangolo sono rispettivamente equivalenti alle ipotesi che si possono dare su un fascio di rette quando si cerchino le parallele (o meglio, quelle non secanti) ad una retta data (che possono essere nessuna, una o infinite), ma se il postulato non si ammette, questa equivalenza viene meno (fano 1950: p. 506). nello specifico, in questa situazione, l’ipotesi che due rette su un piano si incontrano sempre implica ancora che la somma degli angoli di un triangolo sia maggiore di due angoli retti, ma non è vero il viceversa, ovvero da questa ultima proprietà non discende la prima. ancora, l’ipotesi di unicità della retta parallela implica che la somma degli angoli di un triangolo sia di due retti, ma quest’ultima non implica l’unicità della parallela. infine, sempre ammettendo la non validità del postulato archimedeo, l’ipotesi di infinite rette di un fascio che non incontrano una retta data rimane compatibile con tutte le situazioni, ovvero con le tre diverse ipotesi che si possono considerare per gli angoli di un triangolo. dunque ci troviamo di fronte a due nuove geometrie, che dehn denomina geometria nonlegendriana e geometria semi-euclidea (fano 1950: p. 506). la geometria non-legendriana, che per fano sarebbe più giusto chiamare non-saccheriana, è una geometria non-archimedea a metrica ellittica, in quanto su di essa non vale il postulato di archimede, una retta ammette infinite non-secanti passanti per un punto dato e la somma degli angoli interni di un triangolo è sempre maggiore di due angoli retti. la geometria semi-euclidea è invece non-archimedea a metrica euclidea, in quanto su di essa non vale il postulato archimedeo, una retta ammette ancora infinite non-secanti passanti per un punto dato ma la somma degli angoli interni di un triangolo è stavolta uguale a due angoli retti. una completa classificazione in tal senso è stata data in modo diverso anche da pejas (1961). un’ulteriore situazione molto particolare si ha a proposito di uno dei postulati della congruenza, esattamente quello che asserisce l’uguaglianza dei triangoli nel caso siano uguali rispettivamente due coppie di lati e l’angolo compreso e del rapporto tra questo ed il teorema per cui un triangolo isoscele ha uguali gli angoli alla base. se il postulato viene formulato in maniera “ristretta”, ovvero ammettendo l’uguaglianza dei rispettivi lati destri, dei rispettivi lati sinistri e dei rispettivi angoli compresi, se non aggiungiamo il postulato di archimede, il predetto teorema non è dimostrabile. in questa particolare situazione non-archimedea, valgono ancora il teorema di talete con le sue proporzioni, nonché i teoremi desarguesiano sui r. mascella 22 triangoli omologici e pascaliano sull’esagono inscritto, ma si presentano risultati paradossali sconcertanti nelle nostre abitudini, ad esempio: in un triangolo isoscele i due angoli alla base non sono uguali; la somma dei due lati di un triangolo non è sempre maggiore del terzo lato, potendo essere inferiore di un infinitesimo; triangoli con basi ed altezze rispettivamente uguali non sono equivalenti, ma lo sono solo per differenza, analogamente triangoli equivalenti con stessa base non è detto che abbiano la stessa altezza e in un triangolo rettangolo la somma dei quadrati sui cateti è equivalente solo per differenza al quadrato sull’ipotenusa (fano 1950: p. 508) perché anche in questi risultati, nell’usuale geometria, si fa uso a fini dimostrativi del postulato archimedeo. per questa ultima ragione, questa geometria è anche denominata nonpitagorica (non valendo la relazione a2 + b2 = c2 per i lati del triangolo rettangolo). su tutta questa questione, peraltro, hilbert ha indagato a fondo (1899: pp. 156-157). in definitiva, tornando ad hilbert, pur riconoscendo a veronese di aver trattato la questione della geometria non-archimedea prima di lui in un «profondo lavoro» (hilbert 1999, p. 48) ne costruisce un nuovo modello di natura prettamente algebrica, e soprattutto preferisce investigare sull’indipendenza dell’assioma rchimedeo e sulle conseguenze della sua non validità, non mettendo a confronto la sua teoria ed il suo modello con quella dell’italiano. questo fatto, seppur apparentemente banale, non lo è per il riconoscimento attribuito ad hilbert di fondatore di tale geometria. infatti, hilbert prova sì elegantemente l’indipendenza dell’assioma archimedeo dal complesso restante di assiomi, ma oltre a non essere stato il primo ad introdurre la possibilità non-archimedea nell’alveo delle idee e dei modelli geometrici, il modello da lui fornito si colloca in secondo piano rispetto alla retta di veronese. il modello geometrico non archimedeo di veronese, infatti, comprende al suo interno il modello non-archimedeo di hilbert. su questo punto la spiegazione viene da bindoni (1902). bindoni considera (1902: pp. 207-209) il campo (x) degli elementi espressi nella indeterminata x che assumono la forma 11 11 ... ... 21 21 w m ww n xbxbxb xaxaxa mm nn       in cui a1, a2, …, an, b1, b2, …, bm sono numeri reali, mentre n ..., ,, 21 e m21 ..., ,,  sono numeri reali o espressioni ottenute da questa stessa espressione generale. questo campo, come si può verificare facilmente, è ordinabile, costituisce un corpo rispetto alle quattro operazioni algebriche fondamentali e valgono le proprietà fondamentali del calcolo algebrico. ma ha anche elementi che sono infiniti ed infinitesimi attuali rispetto ai numeri reali. ciò appare evidente confrontando, sulla scia di quanto fa hilbert, gli elementi x, …, xn, …, x1, a, x, …, xn, …, x. da questi numeri si possono ottenere con opportune limitazioni sia i numeri di veronese (con a1, a2, …, an interi, e con in particolare a1 positivo, poi con b1, b2, …, bm1 tutti nulli, ma con bm = 1, ed infine con 1, 2, …, n interi e positivi o espressioni ottenute in precedenza nello stesso modo) che quelli di hilbert, potendo peraltro dimostrare (bindoni 1902: p. 209) che «il campo dei numeri del sig. hilbert è compreso nel campo dei numeri del prof. veronese» e dunque potendo affermare che «il sistema geometrico di hilbert è compreso in quello di veronese» (il corsivo è dello stesso autore). la geometria non-archimedea. da veronese agli infiniti modelli attuali 23 è rilevante la posizione di poincaré che nella relazione ai “grundlagen” di hilbert esprime giudizi sulla sua teoria e sulla priorità delle geometrie non archimedee. poincaré ritiene innanzitutto che i numeri di veronese siano ottenuti semplicemente riprendendo in un contesto geometrico i numeri transfiniti di cantor, cosa che oggi appare inesatta in tutta la sua evidenza. in secondo luogo il matematico-filosofo francese assegna la priorità della scoperta non-archimedea ad hilbert, ritenendo che questa sia la scoperta più originale dell’intero suo libro. ma su questo punto, la visione di veronese è netta (veronese 1905: p. 349): «la priorità [...] debbo reclamarla intera.» di esempi non-archimedei, ad ogni buon conto ne esistono tanti. per meglio dire, infiniti. così come i gruppi archimedei sono relativamente pochi, quelli non-archimedei sono in maggioranza, ed anzi una struttura archimedea può essere vista come una sottostruttura di una struttura non-archimedea. la struttura più semplice è certamente (r2,+,), con r l’insieme dei reali, e prendendo come ordinamento della struttura quello lessicografico sulle coppie di reali. prendiamo ad esempio gli elementi (0,1) e (1,0), che sono positivi essendo (0,1) > (0,0) e (1,0) > (0,0). chiaramente, per l’ordine dato vale anche (0,1) < (1,0), e nessun multiplo di (0,1) riesce a superare (1,0), cioè per qualunque n naturale risulta n(0,1) = (0,n) < (1,0). un altro esempio semplice da visualizzare di gruppo ordinato non archimedeo è il gruppo additivo sostegno dell’anello dei polinomi r[x], dove ogni elemento è del tipo f(t) = f0 + f1t + … + fnt n. in questo caso la relazione d’ordine, che chiaramente deve essere compatibile con la struttura algebrica, può essere assegnata ancora una volta in maniera lessicografica sui coefficienti delle potenze, poste in ordine decrescente. in questo modo, tra l’altro, un polinomio è positivo se ha il coefficiente direttore positivo. chiaramente, anche in questo caso la struttura è nonarchimedea. ad esempio, i polinomi h(x) = 1 e g(x) = x sono positivi e vale h(x) < g(x), perché g(x)  h(x) = x  1 è positivo, inoltre ogni multiplo di 1, ovvero qualunque numero naturale n, non potrà mai raggiungere e superare x, perché g(x)  nh(x) = x  n è sempre, ancora, positivo. in realtà abbiamo a disposizione tanti ordini di infinito, quanti ne vogliamo, essendo anche x < x2 < x3 < x4 e così via. anche considerando il campo r(t) delle funzioni razionali di t, ovvero tutti i quozienti f(t)/g(t) dove f e g sono funzioni polinomiali a coefficienti reali e con g(t) non identicamente nulla. allora r(t) ha un ordine naturale che lo rende un campo ordinato non-archimedeo. per convincersene basta considerare che (i) ogni somma, differenza, prodotto e divisione di funzioni razionali è ancora una funzione razionale, (ii) le funzioni sono positive (f > 0) o negative (f < 0) a seconda se il coefficiente principale è positivo (f0 > 0) o negativo (f0 < 0) e ciò rispecchia il comportamento di f per valori di t sufficientemente grandi, infine (iii) per due elementi di f risulta f < g se esiste un valore di t, diciamo t0, tale che per valori di t più grandi di t0 vale sempre f(t) < g(t). a questo punto, prendendo ancora banalmente gli elementi f(t) = 1 e g(t) = t, entrambi maggiori di zero, risulta f(t) < g(t) e per qualunque multiplo del primo risulta sempre n < t. un altro esempio di campo euclideo non-archimedeo è ottenibile semplicemente anche a partire da un qualsiasi campo euclideo k0 (come l’insieme dei reali, oppure l’insieme dei reali costruibili con riga e compasso, e così via) considerando l’unione dei campi delle serie di potenze su nt 2 1 , cioè r. mascella 24            1 2 1 0 n n tk . con la presentazione di questi modelli e con l’idea che si potrebbero considerare molti altri sistemi consistenti, nel rispetto della logica fondazionale geometrica, particolare valore è da assegnare alle ricerche di hans hahn. nel 1907, infatti, hahn si fa carico del problema della classificazione dei modelli fino ad allora conosciuti e, attraverso una classificazione completa dei gruppo ordinati abeliani, dimostra che il reale è solo un caso particolare all’interno di una struttura più generale e non-archimedea,. peraltro, questo risultato è stato rivisitato da ehrlich (1997) e ripreso recentemente anche dall’autore (eugeni-mascella 2000a, 2005), partendo da un contesto algebrico ed utilizzando un linguaggio più moderno, ma utilizzando in fondo le stesse idee di base. hahn studia (ehrlich 1997) un gruppo abeliano ordinato g, con notazione additiva. considerati due elementi a e b in g, con i rispettivi valori assoluti (|a| è il numero più grande dell’insieme {a, –a}), questi sono archimedeo-equivalenti se esistono due numeri positivi n ed m tali che n|a| > |b| e m|b| > |a|. questa relazione di equivalenza permette un partizionamento di g – {0} in classi archimedee. se a e b adesso non sono archimedeo-equivalenti allora a è infinitesimo rispetto a b (in valore assoluto) se |a| < |b|. in tal modo 0, che non è membro di nessuna classe archimedea, è infinitesimo rispetto ad ogni elemento di g. dunque si considera il comportamento reciproco degli elementi in funzione del principio archimedeo, e su questa base il dominio di partenza viene partizionato o, se vogliamo, diviso in elementi di “rango” diverso. questo rango, rappresentato dalle classi di equivalenza, sintetizza i domini di validità della proprietà archimedea. inoltre, se g è ordinato e non-archimedeo, esistono almeno due classi differenti in g, e la relazione di essere di rango inferiore, se vale per un elemento di una classe rispetto ad un elemento della seconda, vale nello stesso modo per tutti gli altri elementi della prima classe con rispetto a quelli della seconda classe, dunque possiamo anche parlare di ordinamento delle classi. le classi di g, in tal modo formano un insieme semplicemente ordinato, l’insieme delle classi di g e viceversa hahn prova che se abbiamo un insieme semplicemente ordinato i, allora esiste sempre un gruppo abeliano g semplicemente ordinato tale che le il tipo delle sue classi è uguale al tipo di ordine dell’insieme di partenza i. ma dato un gruppo abeliano, possiamo anche ragionare sulla sua estendibilità con la preservazione della proprietà archimedea, o meglio, senza alterare le classi di equivalenza o i ranghi disponibili con g. dati due gruppi abeliani g e g’, con g  g’, g’ è un’estensione archimedea di g se e soltanto se per ogni y in g’ – {0} esiste un x in g – {0} tale che x ed y sono archimedeo-equivalenti. se g non ammette alcuna estensione archimedea propria, cioè con g’ diverso da g, allora g è archimedeo-completo. un campo ordinato f avente la struttura additiva archimedea–completa, si dice campo ordinato archimedeo-completo. naturalmente l’essere archimedeo-completo per un campo non vuol dire essere complessivamente archimedeo. hahn scopre che tali strutture esistono, ma introduce anche una costruzione che permette di isolarle a meno di isomorfismi (ehrlich 1997: p. 59). se r è il campo ordinato dei reali, e g è un gruppo abeliano ordinato, denotiamo con r(g) l’insieme di tutte le serie formali del tipo la geometria non-archimedea. da veronese agli infiniti modelli attuali 25     yr dove  è un ordinale, {y :  < } è una sequenza discendente di elementi di g, e gli r sono elementi di r – {0}. il campo di hahn ha dunque r(g) come dominio, è ordinato lessicograficamente, e le operazioni di addizione e moltiplicazione sono definite dalle regole     ggg y y yy y y y y y y baba                 ggg y y yy y y y y y baba si può notare facilmente che r(g) è isomorfo ad r soltanto quando g è identicamente {0}. le classi archimedee [a] formate sul campo ordinato f formano un gruppo abeliano ordinato a, prendendo l’ordinamento e la moltiplicazione con le seguenti definizioni: (i) [a] < [b] se [a] è infinitesimo rispetto a [b], e (ii) [a][b] = [ab]. in virtù di ciò, possiamo dire che f è un campo ordinato di tipo archimedeo g se il gruppo abeliano a delle classi archimedee di f è isomorfo ad un qualche gruppo abeliano g. a questo punto valgono i seguenti teoremi sui numeri reali:  (teorema di immersione) se f è un campo ordinato di tipo archimedeo g, allora r(g) è un’estensione archimedea di una campo isomorfo a f.  (teorema di completezza) a meno di isomorfismi, r(g) è l’unico campo ordinato archimedeo-completo di tipo archimedeo g. in definitiva il teorema di hahn, che occupava 27 pagine, senza contare i preliminari (clifford 1954), asserisce che ogni gruppo abeliano ordinato (se è ordinato, il gruppo è necessariamente infinito) è isomorfo ad un sottogruppo di un prodotto cartesiano ristretto di fattori tutti uguali a (r, +, ), cioè il gruppo additivo dei reali con l’ordinamento naturale. se infatti l’assioma di archimede non vale sul dominio generale, consideriamo la sua validità locale e, guardando alle classi di equivalenza che ne derivano, si può ottenere una partizione per cui in ciascuna classe vale archimede. ogni gruppo ordinato può essere incluso in questa categorizzazione, dando anche luogo ad una serie di modelli diversi, a seconda della classe del gruppo in questione. è particolarmente interessante il caso in cui il gruppo abeliano si riduce ad una sola componente, perché in tal caso abbiamo a che fare con gruppi archimedei. ma vi è di più. per il teorema di holder vale anche il viceversa, ovvero la validità del principio archimedeo è sufficiente per dimostrare che un gruppo ordinato (e dunque archimedeo, con la commutatività che ne segue automaticamente a posteriori) è isomorfo ad un sottogruppo di (r, +, ). ultimamente questi risultati sono stati riscoperti da herlich ed altri autori. in particolare, in articoli recenti (eugeni-mascella 2002°, 2005) la classificazione assume forma e linguaggio diversi nel caso specifico della geometria uno-dimensionale. in particolare sono stati provati due risultati. con il primo, ovvero r. mascella 26 (teorema 1) sia (l, , ) una retta soddisfacente agli assiomi a1, …, a8, a10, ma non soddisfacente al postulato di eudosso-archimede. allora esiste un gruppo abeliano ordinato g tale che (l, , ) risulta isomorfa alla struttura (g  r,  , ). si determina l’esistenza di strutture non-archimedee e se ne stabilisce forma e costruzione, per il fatto che detto postulato non viene incluso tra gli assiomi della retta e di una struttura così delineata si fornisce un isomorfismo con un prodotto cartesiano tra un opportuno gruppo g ed i reali (gli assiomi a1, …, a10 sono una rivisitazione di quelli hilbertiani nella chiave indicata da peano). con il secondo, cioè (teorema 2) siano (g1  r,  , ) e (g’  r,  , ) due rette non archimedee. queste rette sono isomorfe se, e solo se(g, +, ) e (g’, +, ) sono isomorfe. si stabilisce l’esistenza di molteplici modelli non-archimedei non isomorfi, che vengono generati a partire semplicemente da gruppi g non isomorfi. dunque un’infinità di modelli, uno per ogni gruppo abeliano ordinato g che possiamo prendere in considerazione. esempi di rette non-archimedee non isomorfe sono: z  r, q  r, r  r, (z  r)  r, (q  r)  r, (r  r)  r, (z  q  r)  r, e così via (dove z è il gruppo additivo degli interi, q il gruppo additivo dei razionali ed r è il consueto gruppo dei reali). infine possiamo considerare tra queste strutture le estensioni non-archimedee di hahn. dati i campi r  r, q  r e z  r, si può facilmente notare come il gruppo additivo di q  r è un’estensione archimedea di z  r, il gruppo additivo di r  r è un’estensione archimedea di q  r, il gruppo r  r è archimedeo-completo e dunque la struttura di campo su r  r rende tale dominio un campo ordinato archimedeo-completo. 5. conclusioni. l’infinitesimo e l’infinito hanno da sempre avuto uno statuto filosofico controverso, e le ricerche delle attività pionieristiche in tali direzioni, come quelle di veronese e levi-civita, hanno avuto riscontri altalenanti e suscitato dibattiti anche feroci. ma alla fine, con la sistemazione più accurata ed elegante di tali studi, anche le comunità dei ricercatori hanno finito con l’avallare ed accettare ricerche apparentemente fantasiose, ma con enormi risvolti di natura filosofica ed epistemologica. in tal senso ha certamente contribuito anche l’intervento di voci autorevoli. ma soprattutto, sembra che il concetto di infinitesimo nelle descrizioni formali sia diventata una concezione matura, tendenzialmente svincolata dall’ipotesi metafisica che nella riduzione di ciò che appare come continuo si arrivi a “monadi”, ad “atomi” o ad “ultimi elementi” (giorello 1972: p. 153). gli studi su tali argomenti che si sono succedute nei secoli, fino alle teorie di cantor ancora basilari nei fondamenti della matematica, appaiono oggi come piccole rivelazioni di una teoria del continuo molto più ricca, in cui sono ammessi gli infinitesimali e delle generalizzazioni della teoria cantoriana dell’infinito. una serie di ricerche portate avanti da conway e robinson in particolare ha fatto emergere una nuova realtà oltre quella classica dei la geometria non-archimedea. da veronese agli infiniti modelli attuali 27 numeri reali, e proposto nuove entità numeriche in tali sistemi che non soddisfano la proprietà archimedea. dunque, facendo un parallelo tra i modelli analitici e quelli geometrici, appare evidente che tale caratteristica, oltre ad essere vicendevolmente influenzata, così come d’altronde sempre è stato, apre le porte a indagini più profonde e fantasiose. in tal senso, la geometria non-archimedea di veronese ed i monosemii di levi-civita appaiono entità pionieristiche in questo senso appaiono centrali una serie di teorie numeriche che estendono i numeri reali in varie direzioni. tra queste, le teorie analitiche non-standard, che includono i numeri iperreali, i numeri surreali ottenuti con una generalizzazione dell’idea alla base della concezione di sezione di dedekind, nonché i numeri complessi ed ipercomplessi. nei numeri surreali, ad esempio, rappresentati da coppie di insiemi x = { l | r }, con la sola restrizione che ogni numero in l deve essere inferiore ad ogni numero in r, i numeri si ottengono con una regola costruttiva ripetuta iterativamente. ciò porta a considerare tra i numeri surreali anche veri e propri numeri infiniti ed infinitesimi. ed anche in questo caso, per classificare gli ordini dei numeri surreali infiniti, si fa ricorso alle classi archimedee. infatti si può parlare di numeri infinitamente vicini qualora due numeri differiscano di un infinitesimo e, con questa relazione d’equivalenza, si possono considerare tanto le cosiddette monadi, ovvero gli intorni infinitesimi del numeri reali, quanto i numeri infiniti, anch’essi comunque raggruppati all’interno di monadi. per i numeri iperreali abbiamo invece rappresentazioni della forma a+ε dove a è un numero reale ed ε un infinitesimo. le estensioni non-standard dei reali nella cornice dell’analisi non-standard sono un tipico esempio di campi non-archimedeo. ricordiamo che un’estensione non-standard *r di r è un’estensione propria cioè con r  *r ma r  *r, in cui esiste una funzione * dalla superstruttura (la superstruttura su r è ottenuta partendo da r e iternado l’operazione di aggiunta dell’insieme potenza di r, e prendendo l’unione della sequenza risultante) v(r) nell’altra v(*r) che mappa r in *r (il principio di estensione) e che mantiene inalterata la validità delle formule tra sottoinsiemi di r e le loro superstrutture (il principio del transfert). ed anzi sembra che «[…] la considerazione di alcune idee fondamentali della analisi nonstandard dia utili direttive per una valutazione del problema storico degli infinitesimi» (giorello 1972: p. 153) ma queste estensioni non-standard, proprio perché racchiudono numeri di tali particolari specie, sono esempi di ulteriori campi non-archimedei. ma il loro limite è chiaro. a. robinson, riferendosi ai fondamenti storici dell’analisi non standard, dice: «leibniz intuì che la teoria degli infinitesimi implica l’introduzione di numeri ideali che possono essere infinitamente piccoli [...] questi numeri ideali, governati dalle stesse leggi dei numeri ordinari, sono solo una comoda finzione, adottata per abbreviare l’argomentazione e per facilitare l’invenzione o la scoperta matematica» (robinson, 1974). rimane aperta un’ultima questione. se i nostri sensi ci consentono di cogliere ed osservare una parte del mondo, una parte che per definizione è raggiungibile e controllabile, siamo altresì sicuri che tutto il resto del mondo, la parte oltre la porzione osservabile, lo sia altrettanto? a tal proposito hartshorne (2000: p. 161) scrive: «per aiutare a visualizzare la geometria non-archimedea, immaginiamo per un momento di vivere in un universo nonarchimedeo. ciò che percepiamo con i nostri telescopi 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(1909), la geometria non archimedea, proceedings of iv mathematicians international congress, vol. i, 197-208. ratio mathematica volume 45, 2023 changing and unchanging strong efficient edge domination number of some standard graphs when a vertex is removed or an edge is added m. annapoopathi1 n. meena2 abstract let 𝐺 = (𝑉, 𝐸) be a simple graph. a subset s of e(g) is a strong (weak) efficient edge dominating set of g if │ns[e]  s│ = 1 for all e  e(g)(│nw[e]  s│ = 1 for all e  e(g)) where ns(e) ={f / f  e(g), f is adjacent to e & deg f ≥ deg e}(nw(e) ={f / f  e(g), f is adjacent to e & deg f ≤ deg e}) and ns[e]=ns(e) {e}(nw[e] = nw(e) {e}). the minimum cardinality of a strong efficient edge dominating set of g (weak efficient edge dominating set of g) is called a strong efficient edge domination number of g and is denoted by 𝛾′ 𝑠𝑒 (𝐺) (𝛾′ 𝑤𝑒 (𝐺)).when a vertex is removed or an edge is added to the graph, the strong efficient edge domination number may or may not be changed. in this paper the change or unchanged of the strong efficient edge domination number of some standard graphs are determined, when a vertex is removed or an edge is added. keywords: domination, edge domination, strong edge domination, efficient edge domination, strong efficient edge domination. ams subject classification: 05c693 1 reg.no:17231072092002, research scholar, department of mathematics, p.g. & research department of mathematics, the m.d.t. hindu college, tirunelveli. (affiliated to manonmaniam sundaranar university, tirunelveli – 627 012, tamil nadu, india) and assistant professor, national engineering college, kovilpatti 628503, tamil nadu, india. email: annapoopathi.nec@gmail.com 2 assistant professor, p.g. & research department of mathematics, the m.d.t. hindu college, tirunelveli. (affiliated to manonmaniam sundaranar university, tirunelveli – 627 012, tamil nadu, india). email: meena@mdthinducollege.org 3 received on july 10, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.1024. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 257 mailto:annapoopathi.nec@gmail.com mailto:meena@mdthinducollege.org m. annapoopathi and n. meena 1. introduction it is meant by the graph that it is a finite, undirected graph without loops and multiple edges. the concept of domination in graphs was introduced by ore. two volumes on domination have been published by t. w. haynes, s. t. hedetniemi and p. j. slater [4, 5]. let g be a graph with vertex set v and edge set e. a subset d of v (g) is called a strong dominating set of g if every vertex in vd is strongly dominated by at least one vertex in d. similarly, a set d is a subset of v (g) is called a weak dominating set of g if every vertex in vd is weakly dominated by at least one vertex in d. the strong (weak) domination number 𝛾𝑠(𝐺) (𝛾𝑤 (𝐺)) respectively of g is the minimum cardinality of a strong (weak) dominating set of g. a subset d of v (g) is called an efficient dominating set of g if for every vertex u ∈ v (g), |𝑁[𝑢] ∩ 𝐷| = 1 [1, 2]. edge dominating sets were also studied by mitchell and hedetniemi [6, 7]. a subset f of edges in a graph g = (v, e) is called an edge dominating set of g if every edge in e-f is adjacent to at least one edge in f. the edge domination number 𝛾′(𝐺) of a graph g is the smallest cardinality among all minimum edge dominating sets of g. the degree of an edge uv is defined to be deg u + deg v 2. an edge uv is called an isolated edge if deg uv = 0. a subset f of e is called an efficient edge dominating set if every edge in e is either in f or dominated by exactly one edge in f. the cardinality of minimum efficient edge dominating set is called the edge domination number of g. motivated by these definitions; strong efficient edge domination in graphs is defined as follows. a subset s of e(g) is a strong (weak) efficient edge dominating set of g if |𝑁𝑠[𝑒] ∩ 𝑆| = 1 for all 𝑒 ∈ 𝐸(𝐺) [ |𝑁𝑤 [𝑒] ∩ 𝑆| = 1 for all 𝑒 ∈ 𝐸(𝐺)] where 𝑁𝑠 (𝑒) = { 𝑓/𝑓 ∈ 𝐸(𝐺) & 𝑑𝑒𝑔 𝑓 ≥ deg 𝑒} ( 𝑁𝑤 (𝑒) = {𝑓/𝑓 ∈ 𝐸(𝐺) & 𝑑𝑒𝑔 𝑓 ≤ deg 𝑒} ) and 𝑁𝑠[𝑒] = 𝑁𝑠(𝑒) ∪ {𝑒} (𝑁𝑤 [𝑒] = 𝑁𝑤 (𝑒) ∪ {𝑒}) .the minimum cardinality of a strong efficient edge dominating set of g (weak efficient edge dominating set of g) is called as a strong efficient edge domination number of g (weak efficient edge domination number of g ) and also denoted by 𝛾′𝑠𝑒 (𝐺) (𝛾 ′ 𝑤𝑒 (𝐺)). definition 1.1. let 𝐺 = (𝑉, 𝐸) be a simple graph. let 𝐸(𝐺) = {𝑒1, 𝑒2, 𝑒3, 𝑒4, … … … … . 𝑒𝑛 }. an edge 𝑒𝑖 is said to be the full degree edge if and only if deg 𝑒𝑖 = n-1. observation 1.2. 𝛾 ′ 𝑠𝑒 (𝐺) = 1 if and only if g has a full degree edge. observation 1.3. 𝛾 ′ 𝑠𝑒 (𝐾1,𝑛) = 1, 𝑛 ≥ 1 and 𝛾 ′ 𝑠𝑒 (𝐷𝑟,𝑠) = 1, 𝑟, 𝑠 ≥ 1 theorem 1.4. for any path 𝑃𝑚 , 𝛾 ′ 𝑠𝑒 ( 𝑃𝑚 ) = { 𝑛, 𝑖𝑓 𝑚 = 3𝑛 + 1, 𝑛 ≥ 1 𝑛 + 1, 𝑖𝑓 𝑚 = 3𝑛, 𝑛 ≥ 2 𝑛 + 1, 𝑖𝑓 𝑚 = 3𝑛 + 2, 𝑛 ≥ 1 theorem 1.5. nnnc nse = ,)( 3 '  258 changing and unchanging strong efficient edge domination number of some standard graphs when a vertex is removed or an edge is added theorem 1.6. let wm be a wheel graph. then wm has a strong efficient edge dominating set if and only if m = 3n, n ≥ 2 and 𝛾′ 𝑠𝑒 (𝑊3𝑛) = 𝑛, n ≥ 2. 2. main results definition 2.1. )})(())(/()({)()( ''0' geggeege sesese  =+= )})(())((/)({)()( ''' geggeege sesese  += + , )})(())((/)({)()( ''' geggeege sesese  += − example 2.2. consider the following graph since e5 is the full degree edge of g, )( ' g se  =1. {e3, e6} is the unique strong efficient edge dominating set of g – e5. therefore ))((2))(( ' 5 ' geg sese  =− and e5 is the full degree edge of g – e3 and )(1)( ' 3 ' geg sese  ==− . hence ))(()( 3 '' egg sese −=  . definition 2.3. )}()(/)({)()( ''0' gvggvvgv sesese  =−= )}()(/)({)()( ''' gvggvvgv sesese  −= + , )}()(/)({)()( ''' gvggvvgv sesese  −= + example 2.4. consider the following graph s= {e4, e7} is the strong efficient edge dominating set of g and )( ' g se  =2. 3,1,)()( '' ==− igvg seise  and 3,1,)()( '' − igvg seise  e2 2 e1 2 e3 2 e5 2 e4 2 e6 2 e1 e4 e2 e3 e6 e5 e7 v1 v2 v3 v4 v5 v6 259 m. annapoopathi and n. meena theorem 2.5. let g = p3n, .1n then = + )()( ' gv se proof: case (1): let g = p3n, .1n let v be the end vertex of g. then g – v = p3n-1. nnpp nsense =+−== +−− 11)()( 2)1(3 ' 13 '  and 1)( ' += ng se  . therefore )()( '' gvg sese  − . hence )()( ' gvv sei +  . case (2): let 11, 3 −= nkvv k . thus g – v = p3k --1 ∪ p3n-3k and kp kse =− )( 13 '  , 1)()( )(3 ' 33 ' +−== −− knpp knseknse  . therefore )()()( 33 ' 13 '' knseksese ppvg −− +=−  = )(11 ' gnknk se =+=+−+ . hence )()( 0' gvv se  . case (3): let 11, 13 −= + nkvv k . thus g – v = p3k ∪p3n-3k-1 and 1)( 3 ' += kp kse  , knpp knseknse −== +−−−− )()( 2)1(3 ' 133 '  . therefore )()()( 133 ' 3 '' −− +=− knseksese ppvg  = )(11 ' gnknk se =+=−++ . hence )()( 0' gvv se  . case (4): let 21, 23 −= + nkvv k . thus g – v = p3k+1 ∪p3n-3k-2 and kp kse =+ )( 13 '  , 1)()( 1)1(3 ' 233 ' −−== +−−−− knpp knseknse  . therefore )()()( 233 ' 13 '' −−+ +=− knseksese ppvg  11 −=−−+= nknk )( ' g se  . hence )()( ' gvv se −  . case (5): when v = v2 or v3n-1. thus g-v = p3n-2 ∪ p1 having no strong efficient dominating set. from the above given the cases it is identified that, = + )()( ' gv se theorem 2.6. let g = p3n+1, .1n then = − )()( ' gv se proof: case (1): let g = p3n+1, .1n let v be the end vertex of g. then g – v = p3n. 1)( ' +=− nvg se  but ng se =)( '  . therefore )()( '' gvg sese  − . hence )()( ' gvv se +  case (2): let 11, 3 −= nkvv k . thus g – v = p3k --1 ∪ p3n+1-3k. therefore )()()( 313 ' 13 '' knseksese ppvg −+− +=−  = )()( 1)(3 ' 2)1(3 ' +−+− + knsekse pp  = )( ' gnknk se ==−+ . hence )()( 0' gvv se  . case (3): let 11,13 −= + nkvv k . thus g – v = p3k ∪ p3n-3k. therefore )()()( )(3 ' 3 '' knseksese ppvg − +=−  = )(211 ' gnnknk se =+=+−++ . hence )()( ' gvv se +  . case (4): let 21,23 −= + nkvv k . thus g – v = p3k+1 ∪p3n-3k-1 = p3k+1 ∪p3(n-k-1)+2 and, therefore )()()( 2)1(3 ' 13 '' +−−+ +=− knseksese ppvg  nknk =+−−+= 11 )( ' g se = . hence )()( 0' gvv se  . case (5): when v = v2 or v3n. g-v = p3n-1 ∪p1 which has no strong efficient dominating set. from the above all the cases, = − )()( ' gv se theorem 2.7. let g = p3n+2, .1n then = − )()( ' gv se 260 changing and unchanging strong efficient edge domination number of some standard graphs when a vertex is removed or an edge is added proof: case (1): let g = p3n+2, .1n let v be the end vertex of g. then g – v = p3n+1. npvg nsese ==− + )()( 13 ''  but 1)( ' += ng se  . therefore )()( '' gvg sese  − . hence )()( ' gvv se −  case (2): let 11, 3 −= nkvv k . thus g – v = p3k --1 ∪ p3n+2-3k. therefore )()()( 323 ' 13 '' knseksese ppvg −+− +=−  = )()( 2)(3 ' 2)1(3 ' +−+− + knsekse pp  = )(11 ' gnknk se =+=+−+ . hence )()( 0' gvv se  . case (3): let 11, 13 −= + nkvv k . thus g – v = p3k ∪ p3n-3k+1. therefore )()()( 1)(3 ' 3 '' +− +=− knseksese ppvg  = )(11 ' gnknk se =+=−++ . hence )()( 0' gvv se  . case (4): let 21, 23 −= + nkvv k . thus g – v = p3k+2 ∪p3n-3k-1 = p3k+2 ∪p3(n-k-1)+2 and, therefore )()()( 2)1(3 ' 23 '' +−−+ +=− knseksese ppvg  112 +=−−++= nknk )( ' g se = . hence )()( 0' gvv se  . case (5): when v = v2 or v3n+1. g-v = p3n-2 ∪p1 which has no strong efficient dominating set. from the above all the cases, = − )()( ' gv se theorem 2.8. let 1, 3 = ncg n . then ).()()( 0' gvgv se = proof: let 1, 3 = ncg n . let )(gvv  . then )( ' g se  =n, g – v = p3n-1 and )()( 2)1(3 ' 13 ' +−− = nsense pp  n= therefore )()( '' gvg sese  =− . hence ).()()( 0' gvgv se = theorem 2.9. let 2, ,1 = nkg n . then ).()()( 0' gvgv se = proof: let 2, ,1 = nkg n . let )(gvv  . then )( ' g se  =1, g – v = k1, n-1 and 1)( 1,1 ' = −nse k . therefore )()( '' gvg sese  =− . hence ).()()( 0' gvgv se = theorem 2.10. let 1,, , = srdg sr . then .2|)()(| 0' −+= srgv se proof: let 1,, , = srdg sr . let v (g) = }1,1/,,,{ sjrivuvu ji  ,  .1,1/,,)( sjrivvuuuvge ji = let v = ui or vj. then )( ' g se  =1, g – v = dr -1, s = dr, s-1 and 1)()( 1, ' ,1 ' == −− srsesrse dd  . therefore )()( '' gvg sese  =− . hence  .,)()()( 0' vugvgv se −= therefore ( ) 2|)(| 0' −+= srgv se theorem 2.11. let 2,3 = nwg n . then )()( 0' gvv se  if )( 1 kv  and )()( ' gvv se −  if ( ) 13 −  n cvv 261 m. annapoopathi and n. meena proof: let 2, 3 = nwg n . let v(g) = }31/,{ nivv i  ,  131,31/,,)( 131 −= + ninjvvvvvvge niij case (1): g – v = c3n. therefore ncvg nsese ==− )()( 3 ''  . hence )()( 0' gvv se  . case (2): let v = vi, ni 31  . g – v = f3n-1. therefore nffvg nsensese ===− +−− )()()( 2)1(3 ' 13 ''  . but ng se 2)( ' = . hence )()( '' gvg sese  − . therefore )()( ' gvv se −  theorem 2.12. let g = p3n, .2n let e = uv be any edge incident with any vertex of g and g’ = g+e. then 1)()( '' −=+ geg sese  if e is incident with u1 or 231,3 − niu i and )( ' eg se + has no strong efficient edge dominating set if e is incident with u2, u3n-1, 231, 23 − + niu i proof: let g = p3n, .2n v(g) = }31/{ niui  ,  131/)( 1 −== + niuuege iii . let e = uv be the new edge incident with any vertex of g and g’ = g+e. case 1: let e be an end edge of g’. then g’ = p3n+1. therefore npg nsese == + )()'( 13 ''  but .1)( ' += ng se  therefore ).()( '' geg sese  + hence ).()( ' gee se −  case 2: let the edge e be incident with the vertex u2. let s be a strong efficient edge dominating set of g’. suppose .2n among all the edges, the edge e2 have maximum degree. it must belong to s. it strongly efficiently dominates e, e3, e1. also the edges e5, e8, e11, …, e3n-4 belong to s. if the edge e3n-2 belongs to s, then |𝑁𝑆[𝑒3𝑛−3] ∩ 𝑆| = |{𝑒3𝑛−4, 𝑒3𝑛−2}| = 2 > 1 , a contradiction. hence g’ has no strong efficient edge dominating set. the proof is similar if the edge e is added at the vertex u3n-1. case 3: let the edge e be incident with the vertex u3. e2 and e3 are the only maximum degree edges. hence any strong efficient edge dominating set contains either e2 or e3 .then s1 = {e1, e3, e6, …., e3n-3, e3n-1} , s2 = {e2, e4, e7, …. e3n-2} are the strong efficient edge dominating sets of g’. therefore |s1|=n+1, |s2|=n. hence )()'( '' gng sese  = . therefore ).()( ' gee se −  the proof is similar if the edge e is incident with the vertex 12, 3 − niu i case 4: let the edge e be incident with the vertex u4. then s = {e2, e4, e7, …. e3n-2} is the unique strong efficient edge dominating set of g’ and ng se =)'( '  . therefore ).()( '' geg sese  + hence ).()( ' gee se −  the proof is similar if the edge e is incident with the vertex 12, 13 − + niu i case 5: let the edge e be incident with the vertex u5. let s be a strong efficient edge dominating set of g’. the edge e4 & e5 are the only maximum degree edges. if the edge e4 belongs to s then no edge in s to strongly efficiently dominate e2. if the edge e5 belongs to s then e2, e8, e11, …., e3n-4 belongs to s and there is no edge in s to strongly 262 changing and unchanging strong efficient edge domination number of some standard graphs when a vertex is removed or an edge is added efficiently dominate e3n-2. hence strong efficient edge dominating set does not exists. proof is similar if the edge e is incident with the vertex 22, 23 − + niu i . from all the above cases, ( ) =)(0' ge se remark: let n = 1, g’=k1, 3. ).(1)'( '' gg sese  == therefore ( ) )(0' gee se  theorem 2.13. let g = p3n+1, .1n let e = uv be any edge incident with any vertex of g and g’ = g+e. then )()( '' geg sese  =+ if e is incident with u2 or 11,3 − niu i , 21, 23 − + nju j and 1)()( '' +=+ geg sese  if e is incident with u1, u3n, 11, 13 − + niu i . proof: let g = p3n+1, .1n v(g) = }131/{ + niui ,  niuuege iii 31/)( 1 == + . let e = uv be the any edge incident with any vertex of g and g’ = g+e. case 1: let e be an end edge of g’. then g’ = p3n+2. therefore 1)()'( 23 '' +== + npg nsese  but .)( ' ng se = therefore ).()( '' geg sese  + hence ).()( ' gee se −  therefore 1)()( '' +=+ geg sese  case 2: let the edge e be incident with the vertex u2. suppose .1n s = {e2, e5, e8, …. e3n-1} is the unique strong efficient edge dominating sets of g’ and |s| = n. therefore ngg sese == )()'( ''  hence ).()( 0' gee se  the proof is similar if the edge e is incident with the vertex u3n. case 3: let the edge e be incident with the vertex u3. e2 and e3 are the only maximum degree edges. hence any strong efficient edge dominating set contains either e2 or e3 .if e2 belongs to s then s = {e2, e5, e8, …., e3n-3, e3n-1} is the unique strong efficient edge dominating sets of g’and |s| =n. hence )()'( '' gng sese  == . therefore = + )()( ' ge se . if the edge e3 belongs to s then there is no edge in s to strongly efficiently dominate e3. the proof is similar if the edge e is incident with the vertex .12, 3 − niu i case 4: let the edge e be incident with the vertex u4. then s1 = {e1, e3, e5, …. e3n-1} , s2 = {e2, e4, e7, …. e3n-2, e3n} are the strong efficient edge dominating sets of g’ and |s1|=|s2|=n+1. therefore 1)'( ' += ng se  but .)( ' ng se = therefore ).()( '' geg sese  + hence ).()( ' gee se +  the proof is similar if the edge e is incident with the vertex 12, 13 − + niu i case 5: let the edge e be incident with the vertex u5. let s be a strong efficient edge dominating set of g’. the edge e4 & e5 are the only maximum degree edges. if the edge e4 belongs to s then no edge in s to strongly efficiently dominate e2. if the edge e5 belongs to s then e2, e8, e11, …., e3n-4 belongs to s and there is no edge in s to strongly efficiently dominate e3n-2. hence strong efficient edge dominating set does not exists. 263 m. annapoopathi and n. meena proof is similar if the edge e is incident with the vertex 22, 23 − + niu i . from all the above cases, ( ) =)(0' ge se theorem 2.14. let g = p3n+2 .1n let e = uv be any edge incident with any vertex of g and g’ = g+e. then )()( '' geg sese  =+ if e is incident with all 231/ + niu i except u1, u3n and 1)()( '' +=+ geg sese  if e is incident with u1, u3n. proof: let g = p3n+2, .1n v(g) = }231/{ + niui ,  131/)( 1 +== + niuuege iii . let e = uv be the any edge incident with any vertex of g and g’ = g+e. case 1: let e be an end edge of g’. then g’ = p3n+3=p3(n+1). therefore 2)()'( )1(3 '' +== + npg nsese  but .1)( ' += ng se  therefore ).()( '' geg sese  + hence ).()( ' gee se +  therefore 1)()( '' +=+ geg sese  case 2: let the edge e be incident with the vertex u2. suppose .1n s = {e2, e5, e8, …. e3n-1, e3n+1} is the unique strong efficient edge dominating sets of g’ and |s| = n+1. therefore .1)()'( '' +== ngg sese  hence ).()( 0' gee se  the proof is similar if the edge e is incident with the vertex u3n. case 3: let the edge e be incident with the vertex u3. e2 and e3 are the only maximum degree edges. hence any strong efficient edge dominating set contains either e2 or e3 .if e2 belongs to s then s = {e2, e5, e8, …., e3n-3, e3n-1, e3n+1} is the unique strong efficient edge dominating set of g’ and |s| = n+1. hence )(1)'( '' gng sese  =+= . therefore ).()( 0' gee se  if e3 belongs to s then s = {e1, e3, e6, ….,e3n } is the unique strong efficient edge dominating set of g’ and |s| = n+1. hence )(1)'( '' gng sese  =+= . therefore ).()( 0' gee se  the proof is similar if the edge e is incident with the vertex .12, 3 − niu i case 4: let the edge e be incident with the vertex u4. the edge e3 & e4 are the only maximum degree edges. hence any strong efficient edge dominating set s contains either e3 or e4. if the edge e3 belongs to s then s = {e1, e3, e6, ….,e3n } is the unique strong efficient edge dominating set of g’ and |s| = n+1. hence )(1)'( '' gng sese  =+= . therefore ).()( 0' gee se  if the edge e4 belongs to s then there is no edge in s to strongly efficiently dominate e3n. hence strong efficient edge dominating set does not exist. the proof is similar if the edge e is incident with the vertex 12,13 −+ niu i case5: let the edge e be incident with the vertex u5. the edge e4 & e5 are the only maximum degree edges. hence any strong efficient edge dominating set s contains either e4 or e5. if the edge e4 belongs to s then there is no edge in s to strongly efficiently dominate e2. if the edge e5 belongs to s then {e2, e5, e8, …., e3n-3, e3n-1, e3n+1}} is the unique strong efficient edge dominating set of g’ and |s| = n+1. hence 264 changing and unchanging strong efficient edge domination number of some standard graphs when a vertex is removed or an edge is added )(1)'( '' gng sese  =+= . therefore ).()( 0' gee se  proof is similar if the edge e is incident with the vertex 22, 23 − + niu i . theorem 2.15. let g = c3n, .1n let e = uv be any edge incident with any vertex of g and g’ = g+e. then ).()( '' egg sese +=  proof: let g = c3n, .1n let e = uv be the new edge. v(g’) = }31/,{ niuui  ,  vueuueniuuege inniii ==−== + ,,131/)'( 1331 and g’ = g + e. let the edge e be incident with the vertex u1. e1 and e3n are the maximum degree edges and they are adjacent. any strong efficient dominating set contains e1 or e3n. then s1 = {e1, e4, e7, …. e3n-2}, s2 = {e3, e6, e9, …. e3n-3, e3n} are the strong efficient edge dominating sets of g’ and |s1|=|s2|= n, .1n .1,)'( ' = nng se  no other strong efficient edge dominating set exists without e1 and e3n. therefore .1,)()( '' =+= nnegg sese  the proof is similar if the edge e is with any niu i 32,  theorem 2.16. let g = k1, n, .1n let e be any edge incident with any vertex of g and g’ = g+e. then ).()( '' egg sese +=  proof: let g = k1, n, .1n let v(g) = },1/,{ niuui  e(g) = }.1/{ niuu i  v(g’) = }1/,,{ nivuui  . case 1: let e be the new edge incident with u. then g’ = k1, n+1. therefore 1)()'( '' == gg sese  case2: let uiv be the new edge incident with ui. then {uui} is the unique strong efficient edge dominating set of g’ and .1)'( ' =g se  therefore 1)()( '' =+= egg sese  . theorem 2.17. let g = dr, s, .1, sr let e = xy be any edge incident with any vertex of g and g’ = g+e. then     =+ = = wvorwueifg wvorwueifg g jise se se ,1)( ),( )'( ' ' '    . proof: g = dr, s, .1, sr let e = xy be the new edge. v(g) = },1,1/,,,{ sjrivuvu ji  v(g’) = }1,1/,,,,,{ sjriyxvuvu ji  ,  sjrixyevvfuvfuuege jjii ===== 1,1/,,,)'( . then g’ = g+e. case 1: if the edge e is incident with either the vertex u or the vertex v. then g’= dr+1, s or g’= dr, s+1 and 1)()'( '' == gg sese  265 m. annapoopathi and n. meena case 2: let the edge e be incident with the vertex ui, ri 1 or vj, sj 1 . then s = {uv, uiw} or s = {uv, vjw} is the strong efficient edge dominating set of g’ and |s|=2. .2)'( ' =g se  2)()( '' =+= egg sese  . 3. conclusions in this paper, the change or unchanged of the strong efficient edge domination number of some standard graphs are determined, when a vertex is removed or an edge is added. references [1] d.w. bange, a. e. barkauskas, l. h. host, and p. j. slater. generalized domination and efficient domination in graphs. discrete math., 159:1 – 11, 1996. [2] d.w. bange, a. e. barkauskas, and p. j. slater. efficient dominating sets in graphs. in r. d. ringeisen and f. s. roberts, editors, applications of discrete mathematics, pages 189 – 199. siam, philadelphia, pa, 1988. [3] dominngos m. cardoso, j. orestes cerdefra charles delorme, pedro c.silva , efficient edge domination in regular graphs, discrete applied mathematics 156 , 3060 3065(2008) [4] teresa w. haynes, stephen t. hedetniemi, peter j. slater (eds), domination in graphs: advanced topics, marcel decker, inc., new york 1998. [5] teresa w. haynes, stephen t. hedetniemi, peter j. slater, fundamentals of domination in graphs, marcel decker, inc., new york 1998. [6] c. l. lu, m-t. ko, c. y. tang, perfect edge domination and efficient edge domination in graphs, discrete appl. math. 119227-250(2002) [7] s. l. mitchell and s. t. hedetniemi, edge domination in trees. congr. number.19489-509 (1977) [8] e. sampath kumar and l. pushpalatha, strong weak domination and domination balance in a graph, discrete math., 161:235 242, 1996. [9] c. yen and r.c. t. lee., the weighted perfect domination problem and its variants, discrete applied mathematics, 66, p147-160, 1996. 266 ratio mathematica volume 40, 2021, pp. 123-137 123 reliability estimation of weibullexponential distribution via bayesian approach arun kumar rao* himanshu pandey† abstract weibull-exponential distribution is considered. bayesian method of estimation is employed in order to estimate the reliability function of weibull-exponential distribution by using non-informative and beta priors. in this paper, the bayes estimators of the reliability function have been obtained under squared error, precautionary and entropy loss functions. keywords: weibull-exponential distribution. reliability. bayesian method. non-informative and beta priors. squared error, precautionary and entropy loss functions. 2010 ams subject classification: 60e05, 62e15, 62h10, 62h12.§ * department of statistics, maharana pratap p.g. college, jungle dhusan, gorakhpur, india; arunrao1972@gmail.com. † department of mathematics & statistics, ddu gorakhpur university, gorakhpur, india; himanshu_pandey62@yahoo.com. §received on january 12th, 2021. accepted on may 12th, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.570. issn: 1592-7415. eissn: 2282-8214. ©rao and pandey. this paper is published under the cc-by licence agreement. a.k.rao and h.pandey 136 1. introduction the weibull-exponential distribution was proposed by oguntunde et al. [1]. they obtained some of its basic mathematical properties. this distribution is useful as a life testing model and is more flexible than the exponential distribution. the probability function ( )f x; and distribution function ( )f x; of weibullexponential distribution are respectively given by ( ) ( ) ( ) 1 1 1 0 a a x a x x f x; a e e exp e ; x .       − −  = − − −     (1) ( ) ( )1 1 0 0 a x e f x; e ; x , .     − − − = −   (2) let ( )r t denote the reliability function, that is, the probability that a system will survive a specified time t comes out to be ( ) ( )1 0 0 a t e r t e ; t , .    − − =   (3) and the instantaneous failure rate or hazard rate, h(t) is given by ( ) ( )1a t th t a e e .  −= − (4) from equation (1) and (3), we get ( )( ) ( ) ( ) ( ) ( ) ( ) 1 1 11 1 1 a x t a x e a x e a t a e f x; r t e log r t r t ; 0<r t . e        − −  −  − = − −        − (5) the joint density function or likelihood function of (5) is given by ( )( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 n i a n xi i ti i a x n en a nx e na t i a e f x | r t e log r t r t e       = =  − −  −  −  =     = − −         −  (6) the log likelihood function is given by reliability estimation of weibull-exponential distribution via bayesian approach 137 ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 i i n nn a x ina t i i n a x a t i a log f x | r t log a x log e e n log log r t e log r t e       − − = = =     = + + −    −   + − + −       −    (7) differentiating (7) with respect to r (t) and equating to zero, we get the maximum likelihood estimator of r (t) as ( ) ( ) ( ) 1 1 1i n aa xt i r t exp n e e   =    = − − −       . (8) 2. bayesian method of estimation the bayesian estimation procedure have been developed generally under squared error loss function ( ) ( ) ( ) ( ) 2 l r t , r t r t r t      = −        . (9) where ( )r t  is an estimate of ( )r t . the bayes estimator under the above loss function, say ( ) s r t  , is the posterior mean, i.e., ( ) ( ) s r t e r t  =    . (10) the squared error loss function is often used also because it does not lead extensive numerical computation but several authors (zellner [2], basu & ebrahimi [3]) have recognized the inappropriateness of using symmetric loss function. canfield [4] points out that the use of symmetric loss function may be inappropriate in the estimation of reliability function. norstrom [5] introduced an alternative asymmetric precautionary loss function and also presented a general class of precautionary loss function with quadratic loss function as a special case. a very useful and simple asymmetric precautionary loss function is a.k.rao and h.pandey 136 ( ) ( ) ( ) ( ) ( ) 2 r t r t l r t , r t r t      −      =    (11) the bayes estimator of ( )r t under precautionary loss function is denoted by ( ) p r t  , and is obtained by solving the following equation ( ) ( )( ) 1 22 p r t e r t   =   . (12) in many practical situations, it appears to be more realistic to express the loss in terms of the ratio ( ) ( ) r t r t  . in this case, calabria and pulcini [6] points out that a useful asymmetric loss function is the entropy loss ( ) ( ) 1p el p log    − −  , where ( ) ( ) r t , r t   = and whose minimum occurs at ( ) ( )r t r t  = when 0p , a positive error ( ) ( )r t r t       causes more serious consequences than negative error, and vice-versa. for small | p | value, the function is almost symmetric when both ( ) ( )r t and r t  are measured in a logarithmic scale, and approximately ( ) ( ) ( ) 22 2 e e p l log r t log r t .    −     also, the loss function ( )l  has been used in dey et al. [7] and dey and liu [8], in the original form having 1p .= thus ( )l  can be written as ( ) ( ) 1 0el b log ; b .  = − −    (13) the bayes estimator of ( )r t under entropy loss function is denoted by e  and is obtained as reliability estimation of weibull-exponential distribution via bayesian approach 137 ( ) ( ) 1 1 e r t e . r t −     =         (14) for the situation where we have no prior information about ( )r t , we may use non-informative prior distribution ( )( ) ( ) ( ) ( )1 1 0 1h r t ; r t . r t logr t =   (15) the most widely used prior distribution for ( )r t is a beta distribution with parameters 0, ,   given by ( )( ) ( ) ( ) ( ) ( ) 1 1 2 1 1 0 1h r t r t r t ; r t . b ,     − − = −         (16) 3. bayes estimators of ( )r t under ( )( )1h r t under ( )( )1h r t , the posterior distribution is defined by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) 1 1 1 0 f x | r t h r t f r t | x f x | r t h r t dr t =  (17) substituting the values of ( )( )1h r t and ( )( )f x | r t from equations (15) and (6) in (17), we get a.k.rao and h.pandey 136 ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n i i i a n xi t i n i i i x n na x a nx na t i e e n na x a nx na t i e a e e log r t e r t r t log r t f r t | x a e e log r t e r t            = = = − − =  −    −  − − =      − −     −            =    − −     −      ( ) ( ) ( ) 1 1 0 1 1 1 a n i t i e dr t r t log r t  =  −    −                   ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 0 a n xi t i a n xi t i e n e e n e r t -log r t r t -log r t dr t     = =  − − −   −   − − −   −          =         or, ( )( ) ( ) ( ) ( )1 11 1 1 1 1 1 i a n xi t i n a xn t ei n e e e f r t | x r t -log r t n     =  −= − −   −    −     −    =          (18) theorem 1. assuming the squared error loss function, the bayes estimate of ( )r t , is of the form ( ) ( ) ( ) 1 1 1 1i n a t ns a x i e r t e   −  =    −  = +   −     (19) proof. from equation (10), on using (18), ( ) ( ) s r t e r t  =    reliability estimation of weibull-exponential distribution via bayesian approach 137 ( ) ( )( ) ( ) 1 0 r t f r t | x dr t=  ( ) ( ) ( ) ( ) ( )1 1 11 1 1 1 0 1 1 i a n xi t i n a xn t ei n e e e r t r t -log r t dr t n     =  −= − −   −    −     −    =           ( ) ( ) ( ) ( )1 1 11 1 1 0 1 1 i a n xi t i n a xn t ei n e e e = r t -log r t dr t n     =  −= −   −    −     −              ( ) ( )1 1 1 1 1 1 1 i i n a xn t i n a xn t i e e n = n e e     = =   −     −        −   +   −       or, ( ) ( ) ( ) 1 1 1 1i n a t ns a x i e r t e   −  =    −  = +   −     . theorem 2. assuming the precautionary loss function, the bayes estimate of ( )r t , is of the form ( ) ( ) ( ) 2 1 2 1 1 1 n i a t np a x i e r t = e   −  =    −  +   −      . (20) proof. from equation (12), on using (18), ( ) ( )( ) 1 22 p r t e r t   =   a.k.rao and h.pandey 136 ( )( ) ( )( ) ( ) 1 21 2 0 = r t f r t | x dr t        ( )( ) ( ) ( ) ( ) ( ) 1 2 1 1 11 1 12 1 0 1 1 i a n xi t i n a xn t ei n e e e = r t r t -log r t dr t n     =  −= − −   −     −      −                     ( ) ( ) ( ) ( ) 1 2 1 1 11 1 1 1 0 1 1 i a n xi t i n a xn t ei n e e e = r t -log r t dr t n     =  −= + −   −     −      −                     ( ) ( ) 1 2 1 1 1 1 1 2 1 i i n a xn t i n a xn t i e e n = n e e     = =     −     −          −   +   −         1 2 1 1 1 1 1 2 1 i i n a xn t i n a xn t i e e = e e     = =     −     −        −   +   −         or, ( ) ( ) ( ) 2 1 2 1 1 1 n i a t np a x i e r t = e   −  =    −  +   −      . reliability estimation of weibull-exponential distribution via bayesian approach 137 theorem 3. assuming the entropy loss function, the bayes estimate of ( )r t , is of the form ( ) ( ) ( ) 1 1 1 1i n a t ne a x i e r t = e    =    −  −   −      (21) proof. from equation (14), on using (18), ( ) ( ) ( ) ( )( ) ( ) 1 1 1 0 1 1 e r t e r t = f r t | x dr t r t −  −    =                ( ) ( ) ( ) ( ) ( )1 1 1 11 1 1 1 0 1 11 i a n xi t i n a xn t ei n e e e = r t -log r t dr t r t n     = −  −= − −   −     −      −                     ( ) ( ) ( ) ( )1 1 1 11 2 1 1 0 1 1 i a n xi t i n a xn t ei n e e e = r t -log r t dr t n     = −  −= − −   −     −      −                     ( ) ( ) 1 1 1 1 1 1 1 1 i i n a xn t i n a xn t i e e n = n e e     − = =     −     −          −   −   −         a.k.rao and h.pandey 136 1 1 1 1 1 1 1 1 i i n a xn t i n a xn t i e e = e e     − = =     −     −        −   −   −         or, ( ) ( ) ( ) 1 1 1 1i n a t ne a x i e r t = e    =    −  −   −      . 4. bayes estimators of ( )r t under ( )( )2h r t under ( )( )2h r t , the posterior distribution is defined by ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) 2 1 2 0 f x | r t h r t f r t | x f x | r t h r t dr t =  (22) substituting the values of ( )( )2h r t and ( )( )f x | r t from equations (16) and (6) in (22), we get ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n i i i a n xi t i n i i i xi t n na x a nx na t i e e n na x a nx na t i e e a e e log r t e r t r t r t b , f r t | x a e e log r t e r t                 = = = − − =  − − −   −  − − =  − −      − −     −      −              =    − −     −      ( ) ( ) ( ) ( ) 1 1 0 1 11 1 a n i dr t r t r t b ,     =  − −                 −           reliability estimation of weibull-exponential distribution via bayesian approach 137 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 0 1 1 a n xi t i a n xi t i e n e e n e r t -log r t r t r t -log r t r t dr t         = =  − + − −   −   − + − −   −   −           =  −           or, ( )( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 a n xi t i i e n e n a xn k t k i r t -log r t r t f r t | x e n k ek         =  − + − −   −  + − = =  −           =    −   −    + − + +      −           (23) theorem 4. assuming the squared error loss function, the bayes estimate of ( )r t , is of the form ( ) ( ) ( ) 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 i i n a xn k t k i ns a xn k t k i e k ek r t e k ek         + −  = = + − = =    −   −   − + + +      −       =    −   −   − + +       −            (24) proof. from equation (10), on using (23), ( ) ( ) s r t e r t  =    ( ) ( )( ) ( ) 1 0 r t f r t | x dr t=  ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 10 0 1 1 1 1 1 1 1 1 a n xi t i i e n e n a xn k t k i r t -log r t r t r t dr t e n k ek         =   − + − −    −   + − = =  −           =    −   −    + − + +      −            a.k.rao and h.pandey 136 ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 a n xi t i i e n e n a xn k t k i r t -log r t r t dr t = e n k ek         =   − + −    −   + − = =  −              −   −    + − + +      −            or, ( ) ( ) ( ) 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 i i n a xn k t k i ns a xn k t k i e k ek r t e k ek         + −  = = + − = =    −   −   − + + +      −       =    −   −   − + +       −            . theorem 5. assuming the precautionary loss function, the bayes estimate of ( )r t , is of the form ( ) ( ) ( ) 1 21 1 0 1 1 1 0 1 1 1 1 1 2 1 1 1 1 1 1 i i n a xn k t k i np a xn k t k i e k ek r t e k ek         + −  = = + − = =    −   −   − + + +      −       =    −   −   − + +       −            (25) proof. from equation (12), on using (23), ( ) ( )( ) 1 22 p r t e r t   =   ( )( ) ( )( ) ( ) 1 21 2 0 = r t f r t | x dr t        reliability estimation of weibull-exponential distribution via bayesian approach 137 ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 1 1 1 1 1 2 1 10 0 1 1 1 1 1 1 1 1 a n xi t i i e n e n a xn k t k i r t -log r t r t = r t dr t e n k ek         =   − + − −    −   + − = =       −                  −   −    + − + +       −              ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 a n xi t i i e n e n a xn k t k i r t -log r t r t dr t = e n k ek         =   − + + −    −   + − = =      −                   −   −    + − + +       −              or, ( ) ( ) ( ) 1 21 1 0 1 1 1 0 1 1 1 1 1 2 1 1 1 1 1 1 i i n a xn k t k i np a xn k t k i e k ek r t e k ek         + −  = = + − = =    −   −   − + + +      −       =    −   −   − + +       −            . theorem 6. assuming the entropy loss function, the bayes estimate of ( )r t , is of the form ( ) ( ) ( ) 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 i i n a xn k t k i ne a xn k t k i e k ek r t e k ek         + −  = = + − = =    −   −   − + +      −       =    −   −   − + − +       −            (26) proof. from equation (14), on using (23), a.k.rao and h.pandey 136 ( ) ( ) 1 1 e r t e r t −     =         ( ) ( )( ) ( ) 1 1 0 1 = f r t | x dr t r t −        ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 10 0 1 11 1 1 1 1 1 1 a n xi t i i e n e n a xn k t k i r t -log r t r t = dr t r t e n k ek         = −   − + − −    −   + − = =       −                  −   −    + − + +       −              ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 2 1 1 0 1 1 0 1 1 1 1 1 1 1 1 a n xi t i i e n e n a xn k t k i r t -log r t r t dr t = e n k ek         = −   − + − −    −   + − = =      −                   −   −    + − + +       −              ( ) ( ) 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 i i n a xn k t k i n a xn k t k i e k ek e k ek         − + − = = + − = =    −   −   − + − +      −       =    −   −   − + +       −            or, ( ) ( ) ( ) 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 i i n a xn k t k i ne a xn k t k i e k ek r t e k ek         + −  = = + − = =    −   −   − + +      −       =    −   −   − + − +       −            . reliability estimation of weibull-exponential distribution via bayesian approach 137 5. conclusion we have obtained a number of bayes estimators of reliability function ( )r t of weibull-exponential distribution. in equations (19), (20), and (21), we have obtained the bayes estimators by using non-informative prior and in equations (24), (25), and (26), under beta prior. from the above said equation, it is clear that the bayes estimators of ( )r t depend upon the parameters of the prior distribution. in this case the risk function and corresponding bayes risks do not exist. references [1] oguntunde, p.e., balogun, o.s., okagbue, h.i. and bishop, s.a., (2015): “weibull-exponential distribution: its properties and applications”. journal of applied sciences, 15(11): 1305-1311. [2] zellner, a., (1986): “bayesian estimation and prediction using asymmetric loss functions”. jour. amer. stat. assoc., 91, 446-451. [3] basu, a. p. and ebrahimi, n., (1991): “bayesian approach to life testing and reliability estimation using asymmetric loss function”. jour. stat. plann. infer., 29, 21-31. [4] canfield, r.v.: a bayesian approach to reliability estimation using a loss function. ieee trans. rel., r-19, 13-16, (1970). [5] norstrom, j. g., (1996): “the use of precautionary loss functions in risk analysis”. ieee trans. reliab., 45(3), 400-403. [6] calabria, r., and pulcini, g. (1994): “point estimation under asymmetric loss functions for left truncated exponential samples”. comm. statist. theory & methods, 25 (3), 585-600. [7] d.k. dey, m. ghosh and c. srinivasan (1987): “simultaneous estimation of parameters under entropy loss”. jour. statist. plan. and infer., 347-363. [8] d.k. dey, and pei-san liao liu (1992): “on comparison of estimators in a generalized life model”. microelectron. reliab. 32 (1/2), 207-221. microsoft word r.m.13 cap.2.doc ratio mathematica volume 40, 2021, pp. 139-149 gpαkuratowski closure operators in topological spaces p. g. patil* bhadramma pattanashetti† abstract in this paper, we introduce and study topological properties of gpαlimit points, gpα-derived sets, gpα-interior and gpα-closure using the concept of gpα-open set. further, the relationships between these concepts are investigated. also, kuratowski axioms are discussed. keywords: gpα-open set, gpα-closed set, gpα-limit point, gpα-derived set, gpα-closure, gpα-interior points. 2020 ams subject classifications: 54a05, 54c08. 1 *department of mathematics (karnatak university, dharwad, karnataka, india.); pgpatil@kud.ac.in. †department of mathematics (karnatak university, dharwad, karnataka, india.); geetagma@gmail.com. 1received on january 12th, 2021. accepted on may 12th, 2021. published on june 30th, 2021. doi: 10.23755/rm.v40i1.578. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 139 p.g. patil and bhadramma pattanashetti 1 introduction topology is an indispensable object of study in mathematics with open sets as well as closed sets being the most fundamental concepts in topological spaces. general topology plays an important role in many branches of mathematics as well as many fields of applied sciences. introduction to the concept of pre-open sets and pre-closed sets were made by mashhour et al. (7) and the idea of α-open sets was introduced by njastad (6).the concepts and characterizations of semi open and semi pre open sets are studied in (4) and (1) repectively. the concept, generalized closed sets of levine (5) opened the flood gates of research in generalizations of closed sets in general topology. many researchers (3), (8), (10), (11) worked on weaker forms of closed sets. recently, benchalli et al.(2) and patil et al. (9) introduced and studied the concept of ωα-open sets and gpα-open sets in topological spaces. the present authors continued the study of gpα-closed sets and their properties. we study the gpα-closure, gpα-interior, gpα-neighbourhood, gpα-limit points and gpα-derived sets by using the concept of gpα-open sets and their topological properties. we provide the relationship between gpα-derived set (resp. gpα-limit points, gpα-interior) and pre-derived set (resp. pre-limit points and preinterior). also, we studied kuratowski closure axioms with respect to gpα-open sets. 2 preliminaries throught the paper, let x and y (resp. (x,τ) and (y,σ)) always denotes non-empty topological spaces on which no separation axioms are assumed unless explicitly mentioned. the following definitions are useful in the sequel: definition 2.1. a subset a of x is said to be a (a) ωα-closed (2) if α-cl(a) ⊆ u whenever a ⊆ u and u is ω-open in x. (b) gpα-closed (9) if p-cl(a) ⊆ u whenever a ⊆ u and u is α-open in x. 3 gpα-interior and gpα-closure in topological spaces this section deals with gpα-interior and gpα-closure and some of their properties. 140 gpαkuratowski closure operators in topological spaces definition 3.1. (9) let a ⊂ x, then gpα-interior of a is denoted by gpα-int(a), and is defined as gpα-int(a) = ∪{g : a ⊆ g : g is gpα-open in x }. definition 3.2. (9) let a ⊂ x, then gpα-closure of a is denoted by gpα-cl(a), and is defined as gpα-cl(a) = ∩{g : a ⊆ g : g is gpα-closed in x }. theorem 3.1. if a is gpα-closed then gpα-cl(a) = a. proof: let a be gpα-closed in x. since a ⊆ a, and a is gpα-closed in x. then a ∈ {g : a ⊆ g and g is gpα-closed in x }, that is a = ∩{g : a ⊆ g and g is gpα-closed }. hence gpα-cl(a) ⊆ a. but a ⊆ gpα-cl(a) is always true. therefore gpα-cl(a) = a. in general the converse of theorem 3.1 is not true. example 3.1. let x = {a,b,c,d,e} and τ = {x,φ,{a},{c,d},{a,c,d}}. let a = {a,b}. then we can observe that gpα-cl(a) = {a,b}. therefore gpα-cl(a) = a. but a is not gpα-closed in x. remark 3.1. (9) let a ⊆ x and a is gpα-closed in x. then gpα-cl(a) is the smallest gpα-closed set containing a. however, the converse of the remark 3.1 is not true in general. example 3.2. let x = {a,b,c} and τ = {x,φ,{a},{a,c}}. consider a = {a,c}, then gpα-cl(a) = x , which is the smallest gpα-closed set containing a. but a is not gpα-closed in x. remark 3.2. (9) for subsets a,b of x, then gpα-cl(a∩b)⊆gpα-cl(a) ∩ gpαcl(b). remark 3.3. (9) for subsets a,b of x, then gpα-cl(a∪b) = gpα-cl(a) ∪gpαcl(b). remark 3.4. for any a,b ⊆ x then we have the following properties: (i) gpα-int(x) = x and gpα-int(φ) = φ. (9) (ii) gpα-int(a) ⊆ a.(9) (iii) if b is any gpα-open set contained in a then b ⊆ gpα-int(a).(9) (iv) if a ⊆ b, gpα-int(a) ⊆ gpα-int(b). (v) gpα-int(gpα-int(a)) = gpα-int(a). remark 3.5. for any a,b ⊆ x, gpα-int(a∪b) = gpα-int(a) ∪ gpα-int(b). 141 p.g. patil and bhadramma pattanashetti 4 gpα-neighbourhood points in topological spaces this section deals with the properties of gpα-neighbourhood points in topological spaces. definition 4.1. (9) a subset n of x is said to be gpα-neighbourhood of a point x ∈ x, if there exists an gpα-open set g containing x such that x ∈ g ⊆ n. theorem 4.1. every neighbourhood n of x is a gpα-neighbourhood of x. proof: follows from the definition 4.1 and every open set is gpα-open (9). from the following example converse of the theorem 4.1is not true. example 4.1. let x = {a,b,c,d,e} and τ = {x,φ,{a,b},{a,b,c}}. let a ∈ x. consider a = {a,d}. since a is gpα-neighbourhood of the point a, but a is not a neighbourhood of the point a. remark 4.1. if n ⊆ x is gpα-open then, n is gpα-neighbourhood of each of its points. converse of the theorem 4.1 need not true in general, as seen from the following example. example 4.2. let x = {a,b,c,d} and τ = {x,φ,{a},{c,d},{a,c,d}}. here the gpα-open sets are: x, φ, {a}, {b}, {c}, {c,d}, {a,c,d}. then the set a = {a,b} is gpα-neighbourhod of the points a and b, but the set a = {a,b} is not gpα-open in x. theorem 4.2. let a be gpα-closed set in x and x ∈ ac. then there exists gpαneighbourhood n of x such that n ∩a = φ. proof: let a be gpα-closed in x and x ∈ ac. then ac = x \ a which is gpαopen in x. then from remark 4.1, ac is gpα-neighbourhood of each of its points. hence, for every point x ∈ ac, there exists gpα-neighbourhood n of x such that n ⊆ ac. hence n ∩a = φ. theorem 4.3. let x ∈ x and gpαn(x) be the collection of all gpα-neighbourhoods of x. then the following results holds: (i) gpαn(x) 6= φ, ∀ x ∈ x. (ii) n ∈ gpαn(x) implies x ∈ n. (iii) let n ∈ gpαn(x) and n ⊆ m, then m ∈ gpαn(x). (iv) n ∈ gpαn(x) and m ∈ gpαn(x) then n ∩m ∈ gpαn(x). proof: (i) we have x is always gpα-open in x. hence x is in gpαn(x) of every point x ∈ x. therefore gpα-n(x) 6= φ for every point x ∈ x. (ii) from definition of n ∈ gpαn(x), it follows that x ∈ n. 142 gpαkuratowski closure operators in topological spaces (iii) let n ∈ gpαn(x) and n ⊆ m. then there exists gpα-open set g such that x ∈ g ⊆ n. since n ⊆ m, then x ∈ g ⊆ m. so by definition of 4.1, m is a gpα-neighbourhood point of x. hence m ∈ gpαn(x). (iv) let n ∈ gpαn(x) and m ∈ gpαn(x). then, there exist gpα-open sets u and v such that x ∈ u ⊆ n and x ∈ v ⊆ m. that is x ∈ u ∩ v ⊆ n ∩ m. therefore, for every point x ∈ x, there exists gpα-open set u ∩ v such that x ∈ u ∩v ⊆ n ∩m. so, n ∩m is a gpα-neighbourhood of a point x. hence, intersection of two gpα-neighbourhood of a point is again a gpα-neighbourhood of point. corollary 4.1. for any subset a of x, every α-interior point of a is gpα-interior point of a. proof: it follows from the fact that every α-open set is gpα-open in x (9). theorem 4.4. for any subset a of x, every pre-interior point of a is gpα-interior point of a. proof: for any pre-interior point x of a. then there exists pre-open set g containing x such that g ⊆ a. since every pre-open set is gpα-open (9), then g is gpα-open in x. hence x is a gpα-interior point of a. 5 gpα-kuratowski closure operators in topological spaces theorem 5.1. if p-c(x,τ) is closed under finite union, then gpα-c(x,τ) is closed under finite union, where p-c(x,τ) and gpα-c(x,τ) are the families of pre-closed sets and gpα-closed sets in (x,τ) respectively. proof: let a and b are gpα-closed sets in x and a∪b ⊆ g, where g is ωα-open in x. then a ⊆ g and b ⊆ g. since a and b are gpα-closed, then pcl(a) ⊆ g and pcl(b) ⊆ g. then pcl(a)∪pcl(b) = pcl(a∪b) ⊆ g from (7). thus, from hypothesis, pcl(a∪b) ⊆ g. hence a∪b is gpα-closed in x. definition 5.1. let τ∗gpα be the topology on x generated by gpα-closure in the usual manner, τ∗gpα = {g ⊂ x : gpα-cl(x \g) = x \g}. definition 5.2. let τ∗g∗p be the topology on x generated by g ∗p-closure in the usual manner, that is τ∗g∗p = {g ⊂ x : g∗p-cl(x \g) = x \g} theorem 5.2. let a ⊆ x. then the following statements holds: (i) τ ⊆ τ∗gpα (ii) τ ⊆ τp ⊆ τ∗gpα (τp is family of pre-open sets.(12)) 143 p.g. patil and bhadramma pattanashetti proof: (i) let a ∈ τ, then ac is closed in x. we have ac ⊆ gpα-cl(ac) ⊆ cl(ac). since, ac is closed, then cl(ac) is also closed in x. hence ac ⊆ gpα-cl(ac) ⊆ ac. therefore gpα-cl(ac) ⊆ ac. but ac ⊆ gpα-cl(ac) is always true. thus gpαcl(ac) = ac. hence a ∈ τ∗gpα. (ii) since, every pre-closed set is gpα-closed in x, proof follows. theorem 5.3. for any subset a of a topological space x, τ ⊆ τ∗gpα ⊆ τ∗g∗p. proof: let us consider a ∈ τ, then ac is closed in x. then ac ⊆ gpα-cl(ac) ⊆ cl(ac). since, ac is closed, then cl(ac) = ac. therefore ac ⊆ gpα-cl(ac) ⊆ ac. hence, gpα-cl(ac) ⊆ ac. but (ac) ⊆ gpα-cl(ac) is always true. hence (ac) = gpα-cl(ac). thus a ∈ τ∗gpα and hence τ ⊆ τ∗gpα. let a ∈ τ∗gpα. then gpα-cl(ac) = ac. but ac ⊆ g∗p-cl(ac) ⊆ gpα-cl(ac) = ac from (9). hence g∗p-cl(ac) = ac. hence a ∈ τ∗g∗p. thus a ∈ τ∗gpα implies that a ∈ τ∗g∗p. hence τ ⊆ τ∗gpα ⊆ τ∗g∗p. theorem 5.4. the following statements are equal for the space x: (i) every gpα-closed set is pre-closed. (ii) τp = τ∗gpα. (iii) for each x ∈ x, {x} is ωα-open or pre-open. proof: (i) → (ii) let g ∈ τ∗gpα. then from (9) and by the theorem 3.1, we have gpα-cl(a) = a. hence x \g = gpα-cl(x \g) = p-cl(x \g). therefore x \g is pre-closed and so g is pre-open. therefore τ∗gpα ⊆ τp and from (9), τp ⊆ τ∗gpα. hence τp = τ∗gpα. (ii) → (iii) let {x} ∈ x. by (9), we have x \{x} = gpα-cl(x \{x}) is true only when {x} is not ωα-closed. hence {x}∈ ωα-c(x,τ) or x ∈ τp. (iii) → (i) let a be gpα-closed in x and x ∈ p− cl(a). then, we have x ∈ a. case i: if {x} is ωα-closed. suppose x /∈ a, then p−cl(a)\a contains ωα-closed set {x}, which is contradiction. hence x ∈ a. case ii: if {x} is pre-open. since x ∈ pcl(a), then {x}∩a = φ. hence, we have pcl(a) = a and thus a is pre-closed. therefore gpα-c(x,τ) ⊂ p-c(x,τ). theorem 5.5. every gpα-closed set closed if and only if τ = τ∗gpα. proof: suppose every gpα-closed set is closed. let a be gpα-closed then, gpαcl(a) = cl(a). thus τ = τ∗gpα. conversely, let a be gpα-closed then from (9), a = gpα-cl(a). hence x \ a ∈ τ∗gpα. hence, a is closed in x. theorem 5.6. every gpα-closed set is pre-closed if and only if τp = τ∗gpα. proof: suppose that every gpα-closed set is pre-closed. let a be gpα-closed in x. then from hypothesis, gpα-cl(a) = pcl(a). thus τp = τ∗gpα. conversely, let a be gpα-closed in x. then a = gpα-cl(a). thus x \ a ∈ τ∗gpα. hence, a is pre-closed in x. 144 gpαkuratowski closure operators in topological spaces remark 5.1. let a be any subset of x. then gpα-int(a) is the largest gpα-open set contained in a if a is gpα-open. theorem 5.7. gpα-closure is a kuratowski closure operator on x. proof:follows from the definition 3.2 and (9) 6 characterizations of gpα-closed sets in topological spaces definition 6.1. (9) a point x ∈ x is a gpα-limit point of a subset a of x, if and only if every gpα-neighbourhood of x contains a point of a distinct from x. that is, [n \{x}]∩a 6= φ for each gpα-neighbourhood n of x. definition 6.2. (9) the set of all gpα-limit points of a is a gpα-derived set of a and is denoted by gpα-d(a). example 6.1. let x = {a,b,c} and τ = {x,φ,{a}}. let a = {c}. then the only limit point with respect to the set a = c is point b. therefore d(a) = {b}. but gpα-limit point with respect to the set a is φ. therefore gpα-d(a) = φ. theorem 6.1. let a be any subset of x, then,a is gpα-closed if and only if gpαd(a)⊆ a. proof: let a be gpα-closed in x, then ac is gpα-open in x such that x ∈ ac. then for each point x ∈ x and from definition 4.1 , there exist gpα-open set g such that x ∈ g ⊆ x \a. then a∩ (x \a) = φ. therefore, gpα-neighbourhood of g contains no points of a. hence, x is not a gpα-limit point of a. thus, no point of x \ a is a gpα-limit point of a, that is a contains all the gpα-limit points. therefore a contains the gpα-derived points. hence gpα-d(a) ⊆ a. conversely, suppose gpα-d(a)⊆ a and let x ∈ ac. so x /∈ a. hence x /∈ gpαd(a). therefore x is not a limit point of a. then, there exists gpα-open set g such that g∩ (a\{x}) = φ, that is g ⊆ x \a. therefore for each x ∈ x \a, there exists gpα-open set g such that x ∈ g ⊆ x \a. therefore x \a is gpα-open in x and hence a is gpα-closed. theorem 6.2. let τ1 and τ2 be any two topologies on a set x such that gpαo(x,τ1) ⊆ gpα-o(x,τ2). then for every subset a of x, every gpα-limit point of a with respect to τ2 is gpα-limit point of a with respect to τ1. proof: let x be a gpα-limit point of a with respect to τ2. then by definition of gpαlimit point (g∩a)\{x} 6= φ, this is true for every g ∈ gpα-o(x,τ2) and x ∈ g. 145 p.g. patil and bhadramma pattanashetti but by hypothesis, gpα-o(x,τ1) ⊆ gpα-o(x,τ2). hence (g∩a)\{x} 6= φ for every g ∈ gpα-o(x,τ1) such that x ∈ g. hence x is a gpα-limit point of a with respect to the topology τ1. theorem 6.3. let a and b be any two subsets of (x,τ). then the following assertions are valid: (i) gpα-d(a) ⊆ dp(a), where dp is a pre-derived set (12). (ii) gpα-d(a∪gpα-d(a)) ⊆ a∪gpα-d(a). proof: (i) it clearly observed from the fact that every pre-open set is gpα-open in x. then y ∈ g and y ∈ gpα-d(a) \{x}. that is y ∈ g and y ∈ gpα-d(a). hence g ∩ (a \ {y}) 6= φ. let z ∈ g ∩ (a \ {y}), then x 6= z as x /∈ a. thus g∩ (a\{x}) 6= φ. (ii) let x ∈ gpα-d(a ∪ gpα-d(a)). if x ∈ a, then x ∈ gpα-d(a). therefore x ∈ a∪gpα-d(a). on the contrary assume that x /∈ a. then g∩(a∪gpα-d(a)) \{x}) 6= φ, is true for all g ∈ gpα-d(a) and x ∈ g. therefore (g∩a)\{x} 6= φ or g∩ (gpα-d(a)) \{x}) 6= φ. thus x ∈ gpα-d(a). if g ∩ (gpα-d(a)) \{x} 6= φ, then will get x ∈ gpα-d(gpα-d(a)). since x /∈ a, then x ∈ gpα-d(gpα-d(a)) \a. therefore gpα-d(a∪gpα-d(a)) ⊆ a∪gpα-d(a). remark 6.1. we can see the following implification with respect to gpα-open sets. example 6.2. let x = {a,b,c,d,e} and τ = {x,φ,{a},{c,d},{a,c,d}}. in (x,τ) we have, pre-open sets are: x, φ, {a},{c},{d},{a,c},{b,c},{c,d},{d,e},{a,c,d}, {a,c,e},{a,d,e},{a,b,c,d},{a,b,c,e},{a,b,d,e},{a,c,d,e}. gpα-open sets are: x, φ, {a},{c},{d},{a,b},{b,d},{c,d},{a,c}, {b,c},{d,e},{a,b,d},{a,c,d},{b,c,d},{a,b,c},{a,c,e},{a,d,e}, {a,b,c,d},{a,b,c,e},{a,b,d,e},{a,c,d,e} . (i)let a = {b,c,d}. then pre-limit point of the set a is {b,e} and gpα-limit point of a is {e}. hence dp(a) = {b,e} and gpα-d(a) = {e}. hence, gpα-d(a) ⊆ dp(a). (ii)let a = {a,c,d}, then gpα-d(a) = {b,e}. consider a∪gpα-d(a) = {a,c,d}∪{b,e} = x. but gpα-d(x) = {b,e}. therefore gpα-d(a∪gpα-d(a)) = gpα-d(x) = {b,e}. now consider a∪gpα-d(a) = {a,c,d}∪{b,e} = x. hence gpα-d(a ∪ gpα-d(a)) 6= a ∪ gpα-d(a), that is {b,e} 6= x, but gpαd(a∪gpα-d(a) ⊂ a∪gpα-d(a). 146 gpαkuratowski closure operators in topological spaces remark 6.2. if a ⊆ b, then gpα-d(a) ⊆ gpα-d(b). example 6.3. consider the example 6.2, let a = {b,c,d}. then we have gpα-d(a) = {e}. let b = {a,e}. then gpα-limit point of the set b is φ, that is gpα-d(b) = φ. thus we can observe that gpα-d(b) ⊂ gpα-d(a) but b * a. remark 6.3. gpα-d(a∩b) ⊆ gpα-d(a) ∩ gpα-d(b). example 6.4. from the example 6.2, let a = {b,c,d} and b = {b,c} are any two subsets of x. then gpα-d(a) = {e} and gpα-d(b) = φ. also gpα-d(a∩b) = φ. therefore gpα-d(a∩b) ⊆ gpα-d(a) ∩ gpα-d(b) theorem 6.4. let a be any subset of x and x ∈ x. then the following statements are equal: (i) for each x ∈ x, a∩g 6= φ where g is gpα-open in x. (ii) x ∈ gpα-cl(a). proof: let a be any subset of x. (i) → (ii): on the contrary assume that x /∈ gpα-cl(a). then there exists gpαclosed set f such that a ⊆ f and x /∈ f . then x\f is gpα-open in x containing a point x. hence a∩ (x \f) ⊆ a∩ (x \a) = φ, which is contradiction to the assumption. hence x ∈ gpα-cl(a). (ii) → (i): follows from the definition 3.2. corollary 6.1. for any subset a of a space x, gpα-d(a) ⊆ gpα-cl(a). theorem 6.5. let a be any subset of x, then gpα-cl(a) = a∪gpα-d(a). proof: let x ∈ gpα-cl(a). on the contrary assume that x /∈ a. let g be any gpαopen set containing a point x. then (g\{x}) ∩a 6= φ. therefore x is gpα-limit point of a and hence x is gpα-derived set of a, that is x ∈ gpα-d(a). hence gpα-cl(a) ⊆ a∪gpα-d(a). from the corollary 6.1, we have gpα-d(a) ⊆ gpα-cl(a) and a ⊆ gpα-cl(a) is always true. hence a∪gpα-d(a) ⊆ gpα-cl(a). therefore gpα-cl(a) = a∪gpα-d(a). theorem 6.6. let a be gpα-open set in x and b be any subset of x. then a∩gpαcl(b) ⊆ gpα-cl(a∩b). proof: let x ∈ a∩gpα-cl(b). then x ∈ a and x ∈ gpα-cl(b). from the theorem 6.5, we have gpα-cl(b) = b ∪gpα-d(b). if x ∈ b then x ∈ a∩b. then a∩b ⊆ gpα-cl(a∩b). if x /∈ b then x ∈ gpα-d(b). from the definition of gpα-limit point, we have 147 p.g. patil and bhadramma pattanashetti g ∩ b 6= φ for every gpα-open set g containing x. therefore g ∩ (a ∩ b) = (g ∩ a) ∩ b 6= φ. hence x ∈ gpα-d(a ∩ b) ⊆ gpα-cl(a ∩ b). therefore a∩gpα-cl(b) ⊆ gpα-cl(a∩b). however the equality does not holds in general example 6.5. let x = {a,b,c,d,e} and τ = {x,φ,{a}}. let a = {a,b} and b = {a,c} are two subsets of x. then a∩gpα-cl(b) = {a,b} and gpα-cl(a∩b) = x this implies a∩gpα-cl(b) 6= gpα-cl(a∩b). but , a∩gpα-cl(b) ⊂ gpα-cl(a∩b). 7 conclusions in this present work, we have analyzed the notion of generalized pre α-closed sets in topological spaces. we have established the results of gpα-closure, gpαinterior, gpα-neighbourhood and gpα-limit points. moreover, we have characterized these concepts with suitable examples. finally, we apply gpα-open sets for kuratowski closure axioms. there is a scope to study and extend these newly defined concepts. 8 acknowledgment the first author is grateful to the university grants commission, new delhi, india for financial support under ugc sap drs-iii:f-510/3/drs-iii/2016(sapi) dated 29th feb. 2016 to the department of mathematics, karnatak university, dharwad, india. the second author is grateful to karnatak university dharwad for the financial support to research work under urs scheme. authors are very thankful to the anonymous reviewer/s for their suggestions in the improvement of this paper. references [1] d. andrijevic,(1986), semi-pre open sets, mat. vensik, 38(1), 24-32. [2] s. s. benchalli, p. g. patil and t. d. rayanagoudar, (2009), ωα-closed sets in topological spaces, the global jl. of appl. maths and mth. sciences, (2), 53-63. 148 gpαkuratowski closure operators in topological spaces [3] s. jafari, s. s. benchalli, p.g. patil and t. d. rayanagoudar, (2012), pre g∗closed sets in topological spaces, jl. of adv.studies in topology, (3), 55-59. [4] n. levine,(1963), semi-open and semi-continuity in topological space, amer. math. monthly 70, 36-41. [5] n. levine, (1970), generalized closed sets in topology, rend. circ. math. palermo, 19, (2), 89-96. [6] o. njastad, (1965), on some classes of nearly open sets, pacific jl. of math., 15, 961-970. [7] a.s. mashhour, m.e.abd el-monsef and s.n.deeb, (1982), on precontinuous and weak pre continuous mappings, proc. math. phys. soc. egypt, 53, 47-53. [8] p.g. patil ,s. s. benchalli, pallavi s. mirajakar, (2015), generalized star ωα-closed sets in topological spaces, jl. of new results in sci, (9), 37-45. [9] praveen h. patil and p.g. patil (2018), generalized pre α-closed sets in topological spaces, jl. of new theory, (20), 48-56. [10] p. sundaram and m. sheik john, (2002),on ω-closed sets, acta. ciencia indica, 4 389-392. [11] m.k.r.s. veerakumar, (2002), g∗pre-closed sets, acta. cienia indica, 28, 51-60 [12] young bae jun, seong w. jeong, hyeon j. lee and joon woo lee, (2008), applications of pre-open sets, appl. general topology, 9(2), 213-228. 149 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica 28 (2015) 45-64 issn: 1592-7415 45 the impact of the financial crisis on earnings management: empirical evidence from the top 5,000 non-listed stock italian companies francesco paolone 1 , francesco de luca 2 , jenice prather-kinsey 3 1 department of law, parthenope university of naples, italy, fra.paolone@gmail.com 2 department of business administration, university “g. d’annunzio” of chieti-pescara, italy, fdeluca@unich.it 3 department of accounting and finance school of business, university of alabama at birmingham, usa, prahterkinsey@uab.edu abstract account manipulation has been the subject of accounting discussions not only in the u.s. but across the world, especially during times of financial crises. this paper investigates the impact of the recent financial crisis on account manipulation probability by adopting the beneish model (1999, 2013) of eight performance ratios. the analysis has been conducted using the top 5,000 non-listed stock italian companies (società per azioni) ranked by revenues during the time period 2005-2012. we use the aida bureau van dijk database. we test the existence of earnings management (em) within the top non-listed stock italian companies through a comparison between the pre-crisis period (2005-2008) and the crisis period (2009-2012). our findings show that the number of firms with a higher likelihood of earnings manipulation decreases by 4.53% from the pre-crisis to crisis period. as a consequence, we argue that em increases when the crisis is weak while em decreases during the crisis period. keywords: financial crisis, earnings management, earnings manipulation, transparency, italian stock companies. 2010 ams subject classification: 97m30. doi: 10.23755/rm.v28i1.27 http://dx.doi.org/10.23755/rm.v28i1.27 paolone f., de luca f., prather-kinsey j. 46 1 introduction company managers engage in account manipulation, including earnings management, to meet stakeholders’ expectations resulting in financial reporting that may not fairly present the firms’ operations. moreover, stolowy and breton [28] contend that account manipulation can lead to inefficient capital markets. extant accounting research [8], [3], and [16] states that executives’ acknowledge the importance of meeting earnings to achieve targets (i.e. loss avoidance or analysts’ forecasts) as well as recognize that earnings attainment represents a relevant motivation for accounting manipulation [29]. stolowy and breton [28] define account manipulation as management’s discretionary decision to make accounting choices that may affect the transfer of wealth between companies, the company and capital providers, the company and managers or managers. one form of account manipulation is earnings management (em). the objective of this paper is to assess whether managers do manipulate accounts more often during the time of financial crisis than otherwise. to this end we study the group of about top 5,000 non-listed stock italian companies, and we compute the eight ratios as defined by beneish [5]. beneish [4] finds that his eight ratios capture financial statement distortions and provide timely assessments of the likelihood of distortions 1 especially when considered in conjunction with management incentives. so, for each firm-year from 2005 to 2012, we compute the beneish ratios and consider management’s incentive. then we group these observations as pre-crisis or crisis-period in order to assess whether companies have a high probability of em or with a low probability for em. that is, we compare the final scores across two different time periods: precrisis (2005-2008) and crisis-period (2009-2012), assuming 2009 as the year of financial crisis in the u.s. and worldwide. findings show that within the top 5,000 stock italian companies (nonlisted on the italian financial markets), the number of firms with a higher likelihood of earnings manipulation decreased by 4.53% from pre-crisis to crisis periods. this means that financial crisis has had a positive impact by lessening earnings manipulation of the top stock italian companies. we believe that italian firms have a greater propensity to manipulate and hide wealth creation during non-crisis periods to obtain tax savings and restrain the distribution of wealth. from the opposite point of view, it does not make sense for firms to 1 we intend distortions as financial statement distortions which capture unusual accumulations in receivables (dsri, indicative of revenue inflation), unusual growth of sales (sgi), unusual growth of selling, general and administrative expenses (sgai), unusual capitalization and declines in depreciation (aqi and depi, both indicative of expense deflation), unusual propensity to borrow money (lvgi), deterioration of gross margin (gmi) and the extent to which reported accounting profits are supported by cash profits (tata). the impact of the financial crisis on earnings management 47 manipulate earnings in times of financial crisis, because there is less earnings in general. our analysis is conducted by adopting a reliable model of the likelihood of manipulation of accounts in order to assess the impact of the financial crisis on non-listed italian stock companies’ accounts. moreover, this study is useful in assessing the reliability of the financial statements of italian stock companies. this analysis could also be helpful to banks and other lending and investing entities as it represents an additional tool useful to detect account manipulation and accounting fraud, and to reduce information asymmetry during the period of financial crisis. finally, the results have implications for future researchers that study managements’ incentives concurrently with security offerings. we assess the impact of the financial crisis (by assuming year 2009 as the trigger point) on em for the top non-listed italian companies sample ranked by sales revenues. we use the beneish model [5] of eight performance ratios to predict the probability of fraud cases of these italian companies. in explaining our analyses, the remainder of this paper proceeds as follow. next we present a literature review of em studies during the financial crisis followed by an identification of the performance indicators used to determine em probability as developed by beneish. then we present our empirical analyses results of the top stock italian companies ranked by sales revenues and tests of these probabilities pre-crisis and during the financial crisis. we conclude with comments on our main findings and provide suggestions for further research. 2 prior literature and hypothesis 2.1 earnings manipulation many accounting scholars have defined and associated earnings manipulation with accrual accounting. earnings management (em) has been defined by schipper [27] as “a purposeful intervention in the external financial reporting process, with the intent of obtaining some private gain.” many scholars have debated the role of em as resulting in misleading stakeholders about a firm’s performance [16]. in this context, em is an active manipulation of earnings towards a predetermined target [22]. however, our objective is not to argue the merits of accrual accounting. rather we study em as a means of achieving a target during non-financial crisis vs financial crisis times. according to the prior literature, “accruals” are used as a means for em adjustments that may result in adverse consequences. accruals may be explained as the difference between cash flows and operating income and is computed as follows [15], [9]: paolone f., de luca f., prather-kinsey j. 48 accruals = reported earnings – cash flows from operations healy [15] and de angelo [28] have used the above model to find evidence of income manipulation in a different setting, adopting non-discretionary accruals. many accounting scholars have analyzed the relation between em and accruals estimates driven by the advent of readily calculable em metrics [18], [11], and policy concerns raised by influential accounting standard setters [24]. the relevant contribution provided by jones [18] is based on a linear regression approach that uses non-discretionary accrual variables including sales revenue and property, plant and equipment. many studies have improved upon em measurement models. dechow et al. [11] updated the jones model by providing the modified jones model which has become one of the most widely used models in earnings management research. the modified jones model includes an adjustment to sales based on the change in receivables. peek et al. [23] have recently contributed by comparing abnormal accruals across different countries. by using the two accruals estimation models, the modified jones model and the dechow and dichev [10], they found that the accruals models exhibit considerable cross-country variation in predictive accuracy and power to detect earnings management. other authors stated that em can be achieved by using accounting methods and estimates (i.e., an accrual-based manipulation) [1] or by undertaking transactions that make reported income closer to some target numbers, rather than maximizing the firm’s discounted expected cash flows [26]. in addition, several studies have explored real earnings manipulation in the context of early debt retirements [14]. some [25], [12], [30] have contributed to this literature by showing that em can be undertaken through asset sales. in this context, beneish [5] provides a contribution by concentrating on eight financial indicators (performance ratios), and demonstrating their ability to categorize companies in two different groups: potential and non-potential earnings manipulators. 2.2 financial crisis and earnings manipulation one issue of the financial crises in general is the increase of uncertainty among lenders and investors about fundamental values of assets, which leads to a greater volatility in the market prices of assets [29]. according to trombetta and imperatore [29], a financial crisis can be defined as a sudden or gradual interruption in the ongoing functioning of financial markets. this situation of uncertainty increases the asymmetry of information and lenders progressively lose confidence in the accuracy of the information they have about borrowers [21], [13]. the impact of the financial crisis on earnings management 49 under the conditions of financial crises, financial and capital market participants are more skeptical and the investors are willing to sell off their securities, sending a negative signal to the markets as well as to new potential investors who may be reluctant to invest. these investors could also require a higher return as a consequence of the higher levels of capital market risks. both investors and creditors might have a less propensity to invest or lend money because of the higher probability of the counterpart’s default. many scholars have discussed the impact of the financial crises on em. kasznik and mcnichols [19] and matsumoto [20] have provided a significant contribution by analyzing how executives carry out earnings manipulation policies in order to attain firms’ targets and avoid, at the same time, the communication of bad earnings news to markets. bartov et al. [2] described how managers manage earnings in order to alter market’s evaluation of firm’s likelihood to survive and, hence, reduce the average cost of capital. willekens and bauwhede [31] and huijgen and lubberink [17] state that managers are less likely to manipulate earnings in a situation of stronger litigation risk in order to reduce the external exposure of the litigation. these results imply that during times of financial crisis, regulatory bodies may be more likely to closely regulate firms than in times of nonfinancial crisis. therefore firms may be more likely to not manage earnings in financial crisis periods. in considering extant accounting literature, several possibilities are equally likely and we could expect either more or less em during a financial crisis. consequently, we consider it relevant for this debate to conduct an analysis of this relationship specifically within the italian market. we apply the reclassified beneish model, also known as manipulation score [4], [5], [6], [7], in order to verify whether the impact of the financial crisis on em is positive or negative during the time-period from 2005 to 2012. hypothesis for our empirical analysis is stated as follows: h1: on the one hand, on average, more firms will have a high probability of em manipulation before the financial crisis: 2005-2008 than otherwise: 2009-2012. on the other hand, on average, fewer firms will have a high probability of manipulation during and immediately after the financial crisis: 2009-2012 than otherwise: 2005-2008. 3. the beneish model the manipulation score [4], [5], [6], [7], is a mathematical model based on eight financial ratios used to identify whether a company has a significant likelihood of managing and manipulating its earnings. the variables are obtained from the firms’ financial statements and linked together within a score paolone f., de luca f., prather-kinsey j. 50 that describes the rate of earnings manipulation and, consequently, the profile of a company as a “potential earnings manipulator.” beneish suggests using the value of -1.78 as a threshold to distinguish which firms have manipulated their earnings. the variables of the model follow (see the respective extended formulas in appendix 1): 1. dsri (days sales in receivables index). it is the indicator of revenue inflation that measures the days’ sales in receivables compared to the prior year. a significant increase in days' sales in receivables means a disproportionate increase in receivables relative to sales that suggest revenue inflation. the higher increase in the dsri the greater likelihood that revenues and earnings are overstated. 2. gmi (gross margin index). the decrease of gross margin value can be a negative signal about a company's health and future incomes. a value higher than 1 suggests a deterioration of gross margin and can force managers to manipulate earnings. to sum up, the gross margin is related to the change in inventories and other production that can increase the likelihood of manipulation. thus, beneish assumes this variable specifically related to production costs and changes in inventory, which can cause earnings manipulation practices. 3. aqi (asset quality index). the asset quality indicator is the ratio of noncurrent assets other than property, plant, and equipment (ppe) to total asset and measures the proportion of total assets for which future benefits are less certain. beneish expects a positive relationship between aqi and earnings manipulation practices. the higher value of aqi the greater the propensity in deferring and capitalizing costs in order to increase earnings. 4. sgi (sales growth index). “if growth companies face large stock price losses at the first indication of a slowdown, they may have greater incentives than non-growth companies to manipulate earnings” [5]. there would be a strong positive relationship between the growth of sales and the likelihood of em because managers may be more incentivized to manipulate earnings. 5. depi (depreciation index). the depi measures the ratio of the depreciation rate in year t-1 to the corresponding rate in year t. if the index is greater than 1, it indicates that the tangible assets are being depreciated at a slower rate. this suggests that the firm might be revising useful asset life assumptions upwards in a way to increase income. there would be a positive correlation between depi and the earnings manipulation. 6. sgai (sales, general and administrative expenses index). this ratio shows the sga expenses in year t relative to the previous year. if there is a disproportioned increase in selling, general and administrative expenses compared to sales revenues, there would be a negative signal about a the impact of the financial crisis on earnings management 51 company´s prospects. beneish expects a strong positive association between the index and the likelihood of manipulation. 7. lvgi (leverage index). this ratio shows the total debt (current and long-term) in year t relative to the previous year. beneish stated that lvgi was included to capture incentives in debt covenants for earnings manipulation. 8. tata (total accruals to total assets). the value of total accruals, normalized by total assets, is a proxy used to assess the discretionary accounting choices undertaken by managers in order to practice manipulations. there would be thus a positive correlation between accruals and the em. in summary, these ratios have a predictive function and focus on financial statement distortions which capture unusual accumulations in receivables (dsri, indicative of revenue inflation), unusual growth of sales (sgi), unusual growth of selling, general and administrative expenses (sgai), unusual capitalization and declines in depreciation (aqi and depi, both indicative of expense deflation), unusual propensity to borrow money (lvgi), deterioration of gross margin (gmi) and the extent to which reported accounting profits are supported by cash profits (tata). 4. data collection and model reclassification the analysis was conducted using the top 5,000 non-listed stock italian companies ranked by sales revenues during the time period 2005-2012. these companies have been selected based on meeting a sales revenue threshold of € 50 million and the resulting sample is exactly made of 4,898 companies. stock italian companies (with sales revenues > € 50 mln) 4,898 100% companies in liquidation sale and companies with no more than 2 missing values 1,126 23% observed companies 3,772 77% table 1: top stock italian companies with available data 2005-2012 table 1 illustrates the sample selection process. we gathered accounting data from the aida bureau van dijk database of firm-year observations from 2005 to 2012. since several financial data variables were not available from this database and some companies were in liquidation sale during the observation period, we eliminated all the firms with more than two years of missing values and those in liquidation sale during the above mentioned period. then we paolone f., de luca f., prather-kinsey j. 52 attained the coverage percentage by dividing the number of companies included in the study (3,772) by the number of the entire sample (4,898) with sales revenue of at least € 50 million. the coverage is shown as shown in table 1 is about 77%. beneish model has been developed within the us environment and given that there are many differences between u.s. gaap and italian accounting standards, we propose a reclassification of the beneish model by adapting the financial accounting data to the italian scenario (see appendix 2 indicators legend and reclassification). according to the italian accounting principles, “selling, general and administrative expenses” do not appear separately on financial statements, since their value would result from a classification of expenses by function (as provided for by u.s. gaap), while italian financial statements, according to the civil code, classify expenses and revenues by nature. for this reason, in this analysis, we use the neutral value equal to 1 for sgai index since the income statement reclassification, which follows the italian gaap, does not show selling, general and administrative expenses . we use the “full version” of the reclassified beneish model (8m-score) in order to monitor the impact of the financial crisis on em before and after the financial crisis periods. therefore, we expect for italian firms a negative correlation between the financial crisis and the number of non-listed stock companies with a high probability of being manipulated. the eight diagnostic tools have been reclassified according italian gaap (see appendix 2) into the m-score formula in order to achieve the final score that will be later compared to the threshold of -1.78 [7]. by applying the reclassified model, it is possible to categorize companies into two different groups: firms with a low probability of em, and firms with a high probability of em. manipulation score = -4.840 + 0.920*dsri + 0.528*gmi + 0.404*aqi + 0,892*sgi + 0.115*depi – 0.172*sgai – 0.327*lvgi + 4.679*tata the final manipulation score for each firm is obtained by computing the average scores separately between the pre-crisis period (2005-2008) and the crisis period (2009-2012). 5 main findings using on the list of available companies from the aida bureau van dijk database, the top 5,000 italian stock companies ranked by sales revenues (see the impact of the financial crisis on earnings management 53 table 1) are 4,898 and among them 3,772 report the variables needed to develop the manipulation score. pre-crisis (2005-2008) crisis (2009-2012) high probability of em (n° of companies) 1,929 1,758 low probability of em (n° of companies) 1,843 2,014 total companies 3,772 3,772 high probability of em (% of companies) 51.14% 46.61% low probability of em (% of companies) 48.86% 53.39% table 2: probability of em pre-crisis and crisis periods table 2 illustrates the probability of em during the pre-crisis and crisis periods. using a threshold of -1.78 [7], 51.14% of companies have a high probability of manipulating earnings while the 48.86% have a low probability of em in the pre-crisis period. with the starting of the financial crisis in 2009, there is a decrease in the percentage of companies with a high probability of em (from 51.14 % to 46.61%) and an increased percentage of companies with low probability of em (from 48.86% to 53.39%). that is the number of firms with a higher likelihood of earnings manipulation decreased from the pre-crisis to crisis period similar to our overall findings. both pre-crisis and crisis periods % of total companies companies with low probability of em 1,426 37.80% companies with high probability of em 1,341 35.55% table 3: number of companies with the same probability (high or low) before and after crisis table 3 highlights the number of companies with the same probability of em consistently (either high or low) throughout the database period. within the observed sample (3,772 companies) there are 1,426 companies that always have a low probability of em both in the pre-crises and crisis periods (a percentage of 37.80% of the total companies), and 1,341 companies which have high probability of em in both periods (a percentage of 37.80% of the total companies). this means that for these companies the financial crisis had no impact on increasing or decreasing their probability of em. paolone f., de luca f., prather-kinsey j. 54 pre-crisis 2005-2008 crisis 2009-2012 % of total companies companies which manipulate only in pre-crisis period 588 15.59% companies which manipulate only in post-crisis period 417 11.06% table 4: number of companies which changes probability (from high to low and vice-versa) from pre-crisis to crisis period table 4 illustrates em results for those other companies of the sample that consistently manipulate accounts in the pre-crisis period different from the crisis period. table 4 highlights that there are 588 companies which have a high probability of performing manipulated accounting data but only in the pre-crisis period (15.59% of the total companies) while 417 with a high probability of account manipulation only in the crisis period (11.06%). this means that for these 1,005 (588+417) or 26.64% of the companies studied, hypothesis h1 is confirmed. appendix 3 and 4 show the range of em scores which is vast. therefore we provide additional descriptive statistics both for the set of top 100 firms ranked by sales revenues and for the set of worst 100 firms based on sales revenues. 5.1 top 100 and worst 100 firms ranked by sales revenues in the pre-crisis and crisis periods. pre-crisis 2005-2008 crisis 2009-2012 number of top 100 firms with a high probability of em 42 36 number of top 100 firms with a low probability of em 58 64 total number of top 100 firms 100 100 table 5: manipulation scores on top 100 by sales pre-crisis 2005-2008 crisis 2009-2012 number of worst 100 firms with a high probability of em 51 52 number of worst 100 firms with a low probability of em 49 48 total number of worst firms 100 100 table 6: manipulation scores on worst 100 by sales the impact of the financial crisis on earnings management 55 table 5 presents the em scores of the top 100 firms ranked by sales revenues and table 6 presents the manipulation scores of the worst 100 firms ranked by sales revenues (included to the top 5,000) in the pre-crisis and crisis periods. see appendix 3 and 4 for more details on tables 5 and 6. the top 100 firms included in table 5 show a decrease in the number of potential manipulators by 14.29% (from 42 to 36) but regarding the worst 100 firms (table h) the number of companies with a high probability of being manipulated increases by 1.96% (from 51 to 52). these findings show that the average reducing percentage of potential manipulators between pre-crisis and crisis period has been impacted by financial crisis stronger for companies with higher level of revenues than for companies performing lower revenues. we believe and discussed previously that during the pre-crisis period (20052008), there is a greater propensity for manipulating earnings in the italian market which has a tendency to hide the wealth creation through the income boost years to obtain tax savings and to restrain the distribution of wealth. from the opposite point of view, the em policy has a tendency to decrease because the tax burden tends to decrease based on the natural reduction of earnings as a result of the crisis itself. this is to say that it does not make sense to manipulate earnings in times of financial crisis, because there are less earnings in general. on the other hand, while the results of the specific analysis on the top 100 companies (by sales revenues) confirms our hypothesis, the results regarding the worst 100 companies showing a slight increase of the likelihood of em during the crisis period, could be explained as a necessity of those firms to keep constant values of their main performance indicators, compared with those of previous periods, after that the crisis may have impacted too negatively on firm revenues and financial equilibrium. 6 suggestion for further research suggestions for future contributions are based on expanding the data in terms of number of companies. for example, future studies could include analyses of all the limited italian companies (società a responsabilità limitata) as well as partnerships (società di persone), and assessing the difference in em between the two types of companies. it would be useful to focus on a multiple country-setting (eu-nations as well as no eu countries) in order to analyze the impact of crisis on em in different contexts. furthermore, it would be useful to consider other parameters in addition to sales revenues and ranking firms. for example, the sample could be analyzed base on differences in legal 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(2003). “auditor reporting conservatism as a defense mechanism against increased post-enron litigation risk”. working paper. the impact of the financial crisis on earnings management 59 appendix 1: the eight indicators of beneish model dsri = gmi = aqi = sgi = depi = sgai = lvgi = tata = paolone f., de luca f., prather-kinsey j. 60 appendix 2: indicators legend and reclassification receivables consist of a series of short and long-term accounting transactions dealing with the billing of a customer for goods and services they have ordered. in aida they named as “crediti vs clienti entro 12 mesi ed oltre 12 mesi”. sales are the act of selling a product or service in return for money or other compensation. in aida they named as “ricavi di vendite e prestazioni”. cost of goods sold is computed as “cost of beginning inventory + cost of goods purchased (net of any returns or allowances) – cost of ending inventory”. in aida they named as “costo del venduto = rimanenze iniziali + costo delle materie prime – rimanenze finali” current assets consists of any asset reasonably expected to be sold, consumed, or exhausted through the normal operations of a business within the current fiscal year or operating cycle. in aida, they named as “attivo circolante”. ppe (property, plant and equipment) consists of “tangible assets” that are included in fixed assets. in aida they named as “immobilizzazioni materiali”. total assets is computed as the sum of current assets and fixed assets. in aida, they named as “totale attivo”. depreciation is the decrease in value of tangible assets (property, plant and equipment) while “amortization” is the decrease of intangible assets. in aida, they named as “ammortamento dei beni materiali”. sga expenses (selling, general and administrative expenses) is the sum of all direct and indirect selling expenses and all general and administrative expenses of a company. aida doesn’t show this cost category. we assume the value of 1. ltd (long term debts) is the sum of all long term borrowings of a company. aida doesn´t show this cost category. in aida, the named as “totale debiti oltre l’esercizio”. current liabilities consists of all debts or obligations that are due within one year. in aida, they named as “passivo corrente”. cash consists of legal tender or coins that can be used in exchange goods, debt, or services. in aida, they named as “totale disponibilitá liquide”. current maturity of ltd consists of the amount of ltd that expired within one year. this item is included in the general area of “passivo corrente”. so that, “passivo corrente = current liabilities + current maturity of ltd”. income tax payable comprised of taxes that must be paid to the government within one year. in aida, this is computed as “imposte correnti + imposte differite – imposte anticipate”. depr.&amort. are decrease in value of both tangible and intangible assets. from aida, this is computed as “ammortamento beni materiali + ammortamenti beni immateriali”. http://en.wikipedia.org/wiki/selling http://en.wikipedia.org/wiki/asset http://en.wikipedia.org/wiki/fiscal_year the impact of the financial crisis on earnings management 61 appendix 3: ranking of top 100 stock italian companies based on sales revenue average average 2012 2011 2010 2009 2008 2007 2006 2005 1 gestore dei mercati energetici s.p.a. 8299 -2,54 -2,41 -2,74 -2,80 -2,62 0,46 -2,36 -2,32 5,26 0,26 2 gse s.p.a. 3510 -2,35 -2,20 -2,79 -2,54 -2,47 -3,93 -2,17 5,24 -3,89 -1,19 3 kuwait petroleum italia s.p.a. 1920 -3,19 -2,10 -3,33 -3,04 -2,92 -3,70 -3,45 -3,11 -3,01 -3,31 4 enel energia s.p.a. 3511 -1,65 -1,68 -1,87 -2,49 -1,92 . -2,06 -2,09 -4,68 -2,95 5 au s.p.a. 3510 -2,28 -2,49 -2,14 -2,33 -2,31 -2,80 -2,75 -2,57 -1,90 -2,50 6 enel produzione s.p.a. 3511 -3,46 -3,27 -3,52 -3,49 -3,43 -3,18 -3,36 -2,71 -2,58 -2,96 7 enel distribuzione s.p.a. 3510 -2,25 -2,96 -2,74 -2,87 -2,71 -4,23 -2,94 . . -3,58 8 esselunga spa 4711 -3,16 -3,15 -3,65 -4,17 -3,53 -4,18 -3,80 -2,64 -2,09 -3,18 9 kri s.p.a. 1920 -2,32 -3,08 -3,41 -3,60 -3,10 -3,22 -3,44 -3,86 -1,53 -3,01 10 gdf suez energia italia s.p.a. 7112 -2,81 3,44 -0,59 227,40 56,86 -3,84 -3,48 . . -3,66 11 tamoil italia s.p.a. 1920 -3,37 -3,71 -3,92 -3,62 -3,66 -3,42 -3,21 -2,65 -2,18 -2,87 12 trenitalia s.p.a. 4900 -2,90 -3,37 -2,64 -2,21 -2,78 -2,83 -2,98 -1,89 -2,40 -2,53 13 wind telecomunicazioni s.p.a. 6100 -3,28 -3,13 -2,98 -3,15 -3,13 . -3,01 -2,90 -2,45 -2,79 14 versalis s.p.a. 2010 -2,19 -1,97 -0,87 -1,54 -1,64 -2,35 -1,80 -1,44 -1,59 -1,79 15 enoi s.p.a. 3510 . -1,69 -0,33 -1,75 -1,26 0,62 -0,15 -0,52 -0,67 -0,18 16 iren mercato s.p.a. 3510 . . -2,17 -2,67 -2,42 -2,13 -1,26 3,08 -0,57 -0,22 17 autostrade per l'italia s.p.a. 5221 -3,09 -2,82 -3,00 -2,43 -2,84 -3,11 -2,89 -2,60 -2,73 -2,83 18 gs spa 4711 . . -3,51 -4,10 -3,80 -3,86 -4,19 -4,19 -3,53 -3,94 19 marcegaglia s.p.a 2420 . -1,45 -2,32 -1,89 -1,82 -0,99 -0,63 -1,11 -1,14 20 nuovo pignone s.p.a. 2829 -2,24 -2,71 -1,93 -1,80 -2,17 -2,93 -3,14 -3,14 -1,66 -2,72 21 ies italiana energia e servizi s.p.a. 1920 . . -3,32 -3,58 -3,45 -3,04 -2,74 -2,46 -2,33 -2,64 22 costa crociere s.p.a. 4669 -3,67 -3,88 -3,84 -2,99 -3,59 -2,07 -2,19 -2,50 -2,49 -2,31 23 logista italia s.p.a. 4635 -2,63 -2,75 -2,72 -2,55 -2,66 -2,63 -2,59 -3,16 -1,02 -2,35 24 auchan s.p.a. 4791 . -4,20 -3,74 -4,45 -4,13 -4,34 -3,07 -4,28 -3,70 -3,85 25 sevel-spa 2910 . 2,59 -4,08 -2,47 -1,32 -4,00 -3,63 -3,75 -3,63 -3,75 26 sorgenia s.p.a. 3510 . -2,43 -1,73 -2,05 -2,07 -1,27 -2,05 -2,14 -1,15 -1,65 27 samsung electronics italia s.p.a. 4643 -0,02 -0,80 -1,02 -0,76 -0,65 . . -0,86 -0,74 -0,80 28 ferrero societa' per azioni 1082 -2,89 -2,84 -3,05 -3,15 -2,98 -3,39 -3,05 -2,86 -1,88 -2,80 29 rai radiotelevisione italiana spa 6020 -3,13 -2,58 -3,43 -2,27 -2,85 -2,55 -2,75 -2,66 -2,35 -2,58 30 cnh industrial italia s.p.a. 2830 . -2,69 -3,32 -2,58 -2,86 -3,19 -2,59 -2,42 -3,23 -2,86 31 barilla g. e r. fratelli s.p.a. 1073 -2,22 -2,10 -2,86 -2,83 -2,50 -3,81 -2,77 -2,50 -2,89 -2,99 32 ibm italia s.p.a. 6201 -2,33 -2,35 -2,32 -2,03 -2,26 -2,53 -3,36 -3,12 -3,34 -3,09 33 rfi s .p.a. 4900 -1,99 -1,98 -2,63 -2,80 -2,35 -2,39 -2,80 -2,66 -2,20 -2,51 34 agustawestland s.p.a. 3030 -1,86 -2,08 -1,58 -2,06 -1,89 -1,51 . . -1,56 -1,53 35 mediamarket spa 4719 -3,15 -3,39 -3,58 -3,61 -3,43 -3,64 -4,49 -4,00 -4,27 -4,10 36 sma s.p.a. 4711 . -3,62 -4,11 11,63 1,30 3,16 -4,71 -4,24 -4,18 -2,49 37 acea energia spa 3510 -2,73 -2,18 -2,03 -2,39 -2,33 -1,98 9,18 4,77 -2,03 2,49 38 abb s.p.a. 2790 . . -1,71 -1,73 -1,72 58,43 2,89 1,05 0,07 15,61 39 mercedes-benz italia s.p.a. 4511 -2,65 -1,97 -2,15 -1,81 -2,15 -1,91 -1,72 -1,46 -1,51 -1,65 40 ferrari s.p.a. 2910 -0,83 -1,49 -1,32 -2,76 -1,60 -1,82 -2,15 -2,11 -1,85 -1,98 41 a2a energia s.p.a. 3514 . -0,48 -2,13 -1,32 -1,31 -0,84 -0,87 -1,62 -1,16 -1,12 42 comifar distribuzione s.p.a. 4646 . -1,66 -1,94 -1,87 -1,82 -1,96 -1,85 -1,90 1,65 -1,01 43 publitalia 80 s.p.a. 7312 -2,07 -1,58 -1,65 -2,21 -1,88 . . -1,23 -1,20 -1,22 44 pam panorama s.p.a. 4711 -3,16 -3,68 -3,78 -2,82 -3,36 -3,14 -2,80 -3,26 -3,01 -3,05 45 fiat powertrain s.p.a. 2932 . -4,36 -3,88 -3,68 -3,97 -5,23 -3,72 -3,16 -3,14 -3,81 46 burgo group s.p.a. 1712 -2,69 -2,52 -2,35 -2,51 -2,52 -2,40 -2,33 -2,98 -2,88 -2,65 47 bmw italia spa 4500 -1,78 -0,73 -1,45 -1,75 -1,43 -0,72 1,10 -0,21 0,01 0,04 48 iplom s.p.a. 1920 -2,63 -3,23 -2,04 -1,72 -2,41 -1,24 -2,62 -2,21 -2,30 -2,09 49 michelin italiana s.a.m.i. 2211 -3,41 -3,31 -2,14 -2,56 -2,86 -2,56 -2,70 -2,40 -2,13 -2,45 50 italpreziosi s.p.a. 4672 -3,12 -3,23 -3,01 -1,59 -2,74 -1,60 -5,08 -0,78 -3,17 -2,66 manipulation-score per year crisis period pre-crisis periodcode # rank sales companies' list (top 100 by sales revenues) paolone f., de luca f., prather-kinsey j. 62 specifications:  code values represent the industry in which each company operates according to the uk standard industrial classification of economic activities (sic) as updated in 2007;  values represents the beneish score for each year while the average value is introduced separately for the pre-crises period (2005-2008) and for the crises period (2009-2012);  score values expressed in red font represent those higher than the beneish threshold for high probability of em (-1.78) 51 iper montebello s.p.a. 4711 -2,81 -2,94 -2,93 -3,08 -2,94 -4,35 -4,79 -4,49 -4,22 -4,46 52 metro italia cash and carry s.p.a. 4690 -2,43 -2,13 -3,02 -3,54 -2,78 -4,35 7,68 -2,56 -2,26 -0,37 53 green network s.p.a. 3513 -2,71 -2,95 -1,04 -4,41 -2,78 -2,15 -3,42 -3,99 1,17 -2,10 54 chimet s.p.a.2440 0,53 0,51 1,08 0,44 0,64 0,18 -0,81 -0,63 -1,06 -0,58 55 acciaieria arvedi s.p.a. 2430 -4,01 -3,99 -3,35 -3,39 -3,69 -3,56 -2,91 -2,97 -3,26 -3,17 56 bennet s.p.a. 6810 -3,03 -3,02 -4,27 -3,89 -3,55 -3,08 -3,03 -3,39 -3,06 -3,14 57 fastweb spa 6100 -2,70 -2,67 -3,18 -2,78 -2,83 -2,94 -2,79 -3,58 -2,26 -2,89 58 giorgio armani s.p.a. 7410 -0,76 -1,24 -0,61 1,72 -0,22 -2,73 -2,55 -2,15 -2,30 -2,43 59 siemens s.p.a. 2562 0,94 -0,58 -0,96 -1,24 -0,46 -1,37 -2,01 47,06 28,70 18,09 60 merck serono s.p.a. 2120 . . -1,46 -1,94 -1,70 0,53 0,13 0,92 -0,30 0,32 61 gdf suez energie s.p.a. 3510 -2,70 -2,51 -2,90 -2,55 -2,67 -1,46 1,42 -0,94 -0,39 -0,34 62 repsol italia s.p.a. 4730 -1,81 -1,53 -1,26 0,84 -0,94 -2,25 -0,15 -1,36 . -1,25 63 alpha trading s.p.a. 4321 -2,84 -2,13 -2,76 -2,62 -2,59 -2,09 -2,54 -2,76 -2,23 -2,41 64 shell italia e&p s.p.a 0620 -0,10 1,35 -2,27 -1,68 -0,67 -1,05 -0,87 -0,98 -1,32 -1,05 65 renault italia s.p.a. 4511 -2,19 -2,05 3,44 -3,03 -0,96 -2,08 -0,96 -0,47 -1,94 -1,36 66 sasol italy s.p.a. 1920 -1,40 -0,72 0,02 -1,54 -0,91 5,62 -3,17 -2,61 -1,47 -0,41 67 carlo colombo s.p.a. 2400 -3,29 -3,28 -3,63 -0,90 -2,78 -3,19 -3,22 -1,93 -2,18 -2,63 68 sanofi-aventis s.p.a. 2120 -0,19 -0,12 -1,71 -1,17 -0,80 -2,37 0,49 0,90 1,06 0,02 69 alstom ferroviaria s.p.a. 3020 . -1,96 -1,83 -1,85 -1,88 -1,53 -2,09 -1,82 -0,75 -1,55 70 ericsson telecomunicazioni s.p.a. 2630 -1,78 -1,52 -1,74 -1,45 -1,62 -1,32 0,44 -0,10 0,21 -0,19 71 calzedonia s.p.a. 4642 -1,83 -1,66 -1,10 -1,83 -1,61 -1,31 -1,70 -1,53 -1,18 -1,43 72 spesa intelligente s.p.a. 4711 -4,59 -3,49 -5,30 -4,70 -4,52 -4,32 -4,50 -3,92 -4,47 -4,30 73 sata s.p.a. 2910 -1,61 -2,13 . . -1,87 -2,06 -2,23 -1,87 -1,15 -1,83 74 alpiq energia italia s.p.a. 3510 -0,82 -1,45 -1,36 -1,85 -1,37 -1,01 -1,18 -1,77 -0,90 -1,22 75 societa' italiana per il gas per azioni 3521 -2,56 -3,06 -0,26 -3,18 -2,27 . -3,82 -1,83 -2,18 -2,61 76 e.on energia s.p.a. 3523 -1,20 -2,41 -2,73 -0,72 -1,77 -2,25 -2,67 -1,94 . -2,29 77 italia marittima s.p.a. 5000 -3,11 -3,46 -3,83 -3,85 -3,56 -2,95 -3,84 -3,96 -3,53 -3,57 78 repower italia s.p.a. 3511 -1,10 -1,28 -1,77 -1,45 -1,40 -2,44 -0,69 . . -1,56 79 dalmine spa 2420 -1,14 -1,23 -1,29 -1,87 -1,38 -1,69 -1,48 -0,35 -1,10 -1,15 80 ford italia s.p.a. 4511 -0,55 0,18 -2,50 . -0,96 . -1,64 -0,23 -0,81 -0,89 81 ne.it. s.p.a. 1000 -2,23 . . -1,62 -1,93 -1,87 -1,95 -1,77 -2,45 -2,01 82 ti sparkle s.p.a. 6100 -2,30 -2,62 . . -2,46 -0,76 -0,64 -1,00 -1,36 -0,94 83 tecnimont s.p.a. 7110 . 1,84 -1,50 -2,30 -0,66 4,90 -0,61 65,43 6,12 18,96 84 accenture s.p.a. 6201 -1,18 0,16 -0,27 -0,52 -0,46 0,68 0,07 -0,02 0,09 0,21 85 e.on produzione s.p.a. 7010 -0,47 -2,03 -1,49 -3,03 -1,76 -2,25 -2,89 -2,76 -1,86 -2,44 86 arval service lease italia s.p.a. 7711 -4,16 -4,31 -4,35 -4,11 -4,23 -4,37 -4,36 -3,84 -4,07 -4,16 87 unico la farmacia dei farmacisti s.p.a. 4646 . -2,14 -2,01 -2,12 -2,09 -0,18 -1,68 -1,97 -1,86 -1,42 88 lavazza s.p.a. 1083 -1,79 -1,51 -1,65 -1,76 -1,68 -1,36 -1,50 -1,25 -0,61 -1,18 89 benind s.p.a. 5229 -2,01 -1,86 4,19 -2,48 -0,54 -2,27 -1,87 0,07 -2,17 -1,56 90 henkel italia s.p.a. 2010 -2,59 -2,11 -1,44 -2,00 -2,04 -1,01 -1,09 -1,39 -0,91 -1,10 91 brt s.p.a. 4941 -2,69 -2,62 -3,11 -3,06 -2,87 -3,02 -2,98 . . -3,00 92 p.a.i. s.p.a. 4511 . -3,56 1,49 -1,96 -1,34 -3,51 -1,21 -3,12 -2,59 -2,61 93 indesit company s.p.a. 2751 -2,87 -2,48 -2,41 -2,31 -2,52 -2,37 -2,92 -0,72 -3,05 -2,27 94 citroen italia s.p.a. 2910 -3,39 -2,12 -2,51 -2,21 -2,56 -2,89 -2,29 -2,59 -3,00 -2,69 95 confirmec s.p.a. 4614 -2,06 -2,37 -2,36 -2,50 -2,32 -1,69 -1,71 -2,02 -1,92 -1,83 96 societa' agricola la pellegrina s.p.a. 0147 0,04 -1,46 -1,69 -2,10 -1,30 8,52 -0,29 12,38 3,56 6,04 97 acea energia holding s.p.a. 3510 5,76 17,54 -2,26 -2,74 4,57 -3,60 -0,96 . . -2,28 98 novartis farma spa 2120 -1,74 -1,26 -0,83 -0,59 -1,10 -0,71 -1,11 -1,07 -1,17 -1,02 99 techint s.p.a. 7490 2,14 2,79 14,81 57,60 19,33 -4,41 -1,79 -1,95 -2,14 -2,57 100 magneti marelli s.p.a. 2931 -2,97 -2,63 -2,49 -2,51 -2,65 22,47 -2,54 -1,71 -2,43 3,95 the impact of the financial crisis on earnings management 63 appendix 4: ranking of worst 100 stock italian companies based on sales revenue average average 2012 2011 2010 2009 2008 2007 2006 2005 1 honda automobili italia s.p.a. 4511 . -2,81 48,07 -1,96 14,43 -2,10 -1,16 -2,33 -0,68 -1,57 2 abc acqua bene comune napoli 3600 . -1,81 -1,96 -2,07 -1,95 -2,03 -1,93 -2,29 -2,62 -2,22 3 sacer petroli s.p.a. 4671 . . -2,71 -2,62 -2,67 -2,19 -2,75 -1,74 -2,07 -2,19 4 v.ar.vit. vescovini aristide viterie bullonerie s.p.a. 4674 . . 0,50 -1,53 -0,51 0,06 -0,02 -1,56 -0,62 -0,54 5 durst phototechnik spa % durst phototechnik ag 2670 . -0,77 -0,11 -1,38 -0,75 -0,93 -0,39 -0,39 -0,16 -0,47 6 hospal s.p.a. 2660 . . -0,45 -1,09 -0,77 -1,13 -1,67 -1,70 -0,57 -1,27 7 alessi s.p.a. 2599 . -0,95 -1,51 -1,90 -1,45 -1,57 -0,75 -0,55 -1,46 -1,08 8 kerself s.p.a. 4674 . 7,01 -5,53 -3,71 -0,75 -2,36 -0,52 0,00 -1,71 -1,14 9 riva fire s.p.a. 7010 . -3,21 -3,37 -4,04 -3,54 -3,00 -4,02 -5,29 -2,46 -3,69 10 progetto s.p.a. 4520 . . -2,95 -3,23 -3,09 -2,52 -2,48 -2,91 -2,57 -2,62 11 sifer spa 4672 . -1,43 -0,78 -1,18 -1,13 -2,14 -2,40 -0,57 9,37 1,07 12 dott. formenti s.p.a. 2120 . . -1,84 -1,66 -1,75 -2,13 -1,77 -2,25 -2,22 -2,10 13 promatech s.p.a. 2894 . . -3,77 -3,98 -3,87 -3,55 -3,48 -3,06 -2,35 -3,11 14 d.g.s. s.p.a. 4771 . . -3,23 -3,26 -3,24 -4,79 -4,53 -2,38 -3,92 -3,90 15 abbott products spa 2120 . -3,08 -1,97 0,21 -1,61 -0,30 -0,67 5,19 -0,24 0,99 16 mondadori franchising s.p.a. 4649 . . -1,12 -0,63 -0,87 -0,64 -0,94 -0,05 -1,58 -0,80 17 omvp s.p.a 2815 . . -2,46 -2,10 -2,28 -2,58 -1,85 -3,06 -2,88 -2,59 18 fitt s.p.a. societa' unipersonale 2016 . -1,34 -1,31 -1,38 -1,34 -0,84 -0,36 -1,48 -1,90 -1,14 19 sonepar italia sud s.p.a. 4647 . . -2,40 -1,71 -2,05 -2,10 -1,72 -1,51 -1,50 -1,71 20 ykk italia s.p.a. 3299 . -0,56 1,20 2,02 0,89 -0,78 -0,03 -0,69 -0,50 -0,50 21 igap s.p.a. 3291 . -1,91 -1,77 -1,88 -1,85 -1,23 -1,84 -1,57 0,58 -1,01 22 centostazioni s.p.a. 5221 . -1,56 2,36 -2,05 -0,42 -2,83 -1,87 -1,69 -0,27 -1,66 23 otis spa 6420 . . -3,92 -2,48 -3,20 -2,48 -2,97 -2,94 -2,53 -2,73 24 ragall s.p.a. 2453 . . -1,25 -1,11 -1,18 -1,71 -1,58 -0,13 -0,20 -0,91 25 canessa spa 2562 . -4,16 -4,82 -0,24 -3,07 -3,77 -1,67 -1,79 -1,83 -2,26 26 salumificio fratelli riva s.p.a. 1013 . . -1,46 -1,62 -1,54 -1,72 -1,77 -1,84 -2,15 -1,87 27 mondolibri s.p.a. 4791 . . -1,17 -2,49 -1,83 -2,06 -1,73 -1,99 -1,25 -1,76 28 cellular italia s.p.a. 4652 . . -0,68 -1,17 -0,93 -1,53 -1,12 -0,71 -2,08 -1,36 29 oviesse franchising societa' per azioni 7740 . -1,53 -2,11 -1,58 -1,74 -2,41 -1,30 -1,24 -1,58 -1,63 30 calce s pellegrino spa 2352 . -2,41 -2,03 -2,19 -2,21 -2,18 -1,89 -1,80 -2,09 -1,99 31 pastificio guido ferrara spa 1073 . -2,16 -2,66 -2,49 -2,44 -2,49 -2,21 -2,86 -3,53 -2,77 32 fonderie e officine meccaniche tacconi s.p.a. 2450 . -1,63 -1,82 -1,28 -1,58 -1,67 0,23 -0,58 -1,60 -0,90 33 sumitomo corporation italia s.p.a. 4619 . . -0,25 -1,02 -0,64 -2,16 -1,85 -1,39 0,98 -1,11 34 estel office spa 3101 . -2,26 -1,85 -3,20 -2,44 -0,74 -1,24 -3,01 . -1,66 35 ingegneria biomedica santa lucia s.p.a. 2660 . -2,08 -2,43 -2,49 -2,33 -2,14 -0,45 -2,21 -1,13 -1,48 36 jaguar italia s.p.a 4511 . . -0,59 -1,65 -1,12 8,59 -0,49 -3,23 -3,03 0,46 37 bsl spa 5229 . -1,51 -1,87 -2,25 -1,88 -2,15 -2,76 -2,84 -1,58 -2,33 38 in.cam. s.p.a. 2592 . . -0,36 -0,79 -0,57 -0,35 -0,90 -1,37 -0,53 -0,79 39 ftm s.p.a. 4321 . -2,50 -3,16 -2,93 -2,86 -2,21 -0,36 -2,19 -2,83 -1,90 40 farmaceutici rinaldi spa 4646 . -0,03 -0,91 -0,79 -0,58 -0,22 6,28 -0,57 -0,26 1,31 41 saint gobain isover italia s.p.a. 2311 . . -1,83 -2,84 -2,33 -2,51 -1,48 -2,42 -1,98 -2,10 42 generali real estate s.p.a. 6832 . -2,90 -1,12 -2,91 -2,31 -1,04 -0,64 1,48 10,20 2,50 43 montebovi societa per azioni 4636 . -3,34 -2,72 -3,00 -3,02 -2,26 -2,43 -1,86 -1,88 -2,11 44 aluberg s.p.a. 2511 . -1,76 -1,63 -1,54 -1,64 -1,05 -0,83 -0,72 -0,62 -0,80 45 maquet italia societa per azioni 4646 . -1,60 -1,47 -0,88 -1,31 -1,30 -1,02 -0,98 1,19 -0,53 46 trevisanalat spa 1051 . -2,89 -1,87 -1,29 -2,01 -2,21 -2,91 -2,86 -2,52 -2,62 47 dasty italia s.p.a. 2041 . . -0,99 -1,46 -1,23 -1,59 -2,18 -2,64 -1,90 -2,08 48 papernet s.p.a. 1720 . . -4,60 -3,82 -4,21 -3,90 -3,58 -3,26 -4,13 -3,72 49 national can italiana (n.c.i.) s.p.a. 2592 . . -1,70 -1,62 -1,66 -1,72 -2,13 -2,03 -1,84 -1,93 50 vg holding s.p.a. 4778 . -2,12 -2,20 -2,99 -2,44 -2,49 -2,64 -1,77 -2,25 -2,29 # rank sales companies' list (top 100 by sales revenues) code manipulation-score per year crisis period pre-crisis period paolone f., de luca f., prather-kinsey j. 64 specifications:  code values represent the industry in which each company operates according to the uk standard industrial classification of economic activities (sic) as updated in 2007;  values represents the beneish score for each year while the average value is introduced separately for the pre-crises period (2005-2008) and for the crises period (2009-2012);  score values expressed in red font represent those higher than the beneish threshold for high probability of em (-1.78) 51 ykk mediterraneo s.p.a. 3299 . -1,41 -0,14 3,98 0,81 -3,87 -3,57 -2,98 -3,58 -3,50 52 concerta s.p.a. 5629 . -1,81 -1,26 -0,79 -1,29 -0,94 -1,66 -2,55 -2,23 -1,84 53 opera21 s.p.a. 6202 . -2,73 -2,52 0,03 -1,74 -2,05 -1,57 -2,31 -2,06 -2,00 54 aldinet spa 4643 . -0,32 -1,80 -2,31 -1,48 -2,27 -3,55 -1,51 -2,02 -2,34 55 innse-berardi s.p.a. 2840 . . -2,63 -0,55 -1,59 -2,95 -2,07 -2,18 -3,10 -2,57 56 isogas spa 3523 . -0,19 -0,93 -1,61 -0,91 -1,69 -2,02 -1,69 1,23 -1,04 57 carl zeiss spa 4643 . -1,38 -1,16 -1,39 -1,31 -1,22 -0,98 -1,33 -1,05 -1,15 58 beauty point s.p.a. 4775 . . -2,96 -2,85 -2,91 -2,97 -2,72 -2,63 -2,88 -2,80 59 saip&schyller spa 2220 . -3,09 -3,16 -4,51 -3,59 -1,12 -2,52 -3,52 -4,12 -2,82 60 autoitalia s.p.a. 4511 . -2,75 -16,64 -1,37 -6,92 -1,41 -1,69 -3,59 0,88 -1,45 61 alupress spa 2453 . -1,08 -0,66 -1,47 -1,07 -0,48 -0,88 -1,52 -1,63 -1,13 62 invensys systems italia s.p.a. 3320 . -2,87 -3,22 -1,57 -2,56 -2,69 -3,38 -3,39 -2,63 -3,02 63 gozzo impianti societa' per azioni 2790 . -2,29 -1,64 -2,66 -2,20 -1,97 -1,51 -1,71 -2,22 -1,85 64 sices spa 2820 . -2,77 -2,60 -2,88 -2,75 -2,91 -3,52 -2,40 . -2,95 65 casa dolce casa s.p.a 4673 . . -1,07 -1,35 -1,21 -1,82 -1,79 -1,35 -2,22 -1,80 66 officine ferroviarie veronesi s.p.a. 3020 . -1,78 -1,64 -3,00 -2,14 -1,63 -0,82 -1,54 1,75 -0,56 67 camar s.p.a. 4511 . . -0,18 -1,85 -1,02 -2,76 -2,14 -2,26 -1,58 -2,19 68 sepsa spa socper l esercizio di pubblici servizi 4910 . . -2,68 -3,02 -2,85 -2,16 -2,19 -2,09 -1,89 -2,08 69 biomasse italia s.p.a. 3511 . -4,04 -1,11 -6,03 -3,73 -4,29 -4,43 -2,89 -2,34 -3,49 70 marzoli s.p.a. 2894 . . -3,13 -3,48 -3,30 -4,03 -2,05 -2,48 -3,57 -3,03 71 reggiana alimentari spa abbreviabile in real spa 4711 . -4,64 -4,02 -4,05 -4,23 -4,06 -4,93 -4,50 -4,04 -4,38 72 malavolta spa 2511 . -1,85 -0,83 -2,13 -1,61 -1,80 -1,35 -1,42 -1,83 -1,60 73 migliore sonepar s.p.a. unipersonale 4669 . . -2,25 -2,63 -2,44 -2,66 -2,32 -2,42 -2,26 -2,41 74 fini s.p.a. 2813 . -2,43 -2,50 -2,18 -2,37 -2,80 -2,52 -2,46 -2,25 -2,51 75 fin. al s.p.a. 2442 . . -1,40 -2,29 -1,84 -1,46 -1,79 -1,42 -1,83 -1,62 76 antica farmaceutica modenese s.p.a. 6420 . 35,33 0,47 -0,31 11,83 -0,32 -0,09 -0,30 -0,44 -0,29 77 costruzioni dondi spa 4311 . -0,21 -2,05 -1,74 -1,33 -2,12 -0,60 -1,16 -1,28 -1,29 78 sada spa 4941 . -3,27 -2,99 -3,35 -3,20 -2,86 -2,37 -2,38 -2,63 -2,56 79 emerson industrial automation italy s.p.a. 2790 . 1,27 1,22 0,43 0,98 1,00 1,99 1,54 1,44 1,49 80 istituto gentili s.p.a. 2120 . . 4,41 -0,57 1,92 -2,70 -0,56 1,27 -2,12 -1,03 81 fep rimondi s.p.a. 4647 . . -1,01 -1,63 -1,32 -1,19 -1,53 -1,17 -1,31 -1,30 82 motia compagnia di navigazione s.p.a. 5000 . -2,98 -2,71 -0,55 -2,08 -2,94 -3,17 -2,46 -1,40 -2,49 83 pircher oberland s.p.a. 1610 . -1,25 0,02 0,54 -0,23 0,66 -0,07 0,41 0,08 0,27 84 desmet ballestra oleo s.p.a. 2562 . . -2,77 -3,18 -2,98 -0,73 2,78 -0,13 -0,05 0,47 85 italease gestione beni s.p.a. 6810 . -4,69 -1,77 -5,14 -3,87 -4,08 -5,87 -3,53 -1,45 -3,73 86 mair research s.p.a. 3320 . . -1,61 -1,83 -1,72 -3,15 -2,61 -1,64 -1,08 -2,12 87 parker hiross s.p.a. 2829 . . -0,38 -1,08 -0,73 -0,58 0,10 0,24 -1,24 -0,37 88 techint cimimontubi spa 2562 . . -0,06 -1,16 -0,61 -1,10 -1,66 -3,78 -1,46 -2,00 89 m & z rubinetterie s.p.a. abbreviabile in 2814 . -1,81 -1,93 -1,89 -1,88 -2,48 -2,29 -1,82 -1,98 -2,14 90 li.sit. s.p.a. 6201 . . -4,39 -4,36 -4,37 -5,33 -4,07 -2,83 0,39 -2,96 91 gruppo bonifaci spa 6810 . -4,62 -2,81 3,38 -1,35 -3,83 43,82 9,91 . 16,63 92 zimmerhofer spa % zimmerhofer ag 4100 . . -2,93 -2,71 -2,82 0,37 -2,17 -2,94 -0,94 -1,42 93 rodriquez cantieri navali spa 3011 . -1,20 -2,13 -2,92 -2,08 -2,78 -3,48 -3,54 -0,63 -2,61 94 siemens holding s.p.a. 7010 . . -4,47 -4,02 -4,25 -4,52 2,24 -4,01 -1,86 -2,04 95 semplice spa 7740 . -2,85 4,66 -1,47 0,11 -2,19 0,84 -0,86 -0,69 -0,72 96 i castellani s.p.a. 2331 . . 1,52 -3,13 -0,81 -1,33 -1,72 -1,84 -1,92 -1,70 97 orecchia s.p.a. 6499 . . -2,82 19,01 8,09 -0,39 -1,98 -2,14 -2,08 -1,65 98 cala container shipping s.p.a. 6820 . . -0,48 -1,23 -0,86 13,93 -1,39 -3,43 -4,90 1,05 99 orsi macchine tessili s.p.a. 4660 . -2,44 -3,90 4,30 -0,68 4,98 -3,32 -4,23 -0,35 -0,73 100 binda spa 7010 . . -3,31 -2,14 -2,73 -3,82 8,93 -1,33 -1,54 0,56 microsoft word capitolo intero n 4.doc microsoft word documento1 microsoft word documento1 microsoft word documento1 ratio mathematica volume 46, 2023 degree of an edge and platt number in signed networks diviya k d* anjaly kishore† abstract positive labelled edges play a vital role in network analysis.the degree of edges in signed graphs is introduced by giving importance to positive edges incident on the end vertices of that edge. the concept of platt number of a graph, which is the sum of degrees of its edges, is extended to signed graphs based on the degree defined. bounds of degree of an edge and platt number in certain classes of signed graphs are determined. some characterizations on platt number of signed graphs are also established. a model to analyse social networks using degree of edges and platt number is also proposed. keywords: signed graph, positive edges, negative edges, networks, information diffusion, degree of an edge, platt number 2020 ams subject classifications: 54a40. 1 *department of mathematics, sree narayana college nattika, thrissur, india. diviyarenju@gmail.com. †department of mathematics, vimala college thrissur, thrissur, india. anjalykishor@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1061. issn: 1592-7415. eissn: 2282-8214. ©diviya k d et al. this paper is published under the cc-by licence agreement. 91 diviya k d and anjaly kishore 1 introduction graph theory is a branch of mathematics that is extensively used to mathematically model social science which is discussed in harary and norman [1953]. according to cartwright and harary, in the paper cartwright and harary [1956], graphs can be used for representing relationships between persons whatever be the nature of relationships. the social relationships between individuals in society is not uniform. there may be a lot of variations in the type and intensity in the relationships between individuals. friendship, like etc. are examples of positive relationships between two persons and enmity, indifference etc. are examples of negative relationships between them. in graph theory, positive labelled edges can be used to represent the positive relationships and negative labelled edges can be used to represent negative relationships. if there does not exist any kind of relationships between persons, it can be represented by non adjacent vertices. such graphs that are used to represent networks, especially in social science, are called signed graphs by harary et al. [1965]. from acharya [1986], zaslavsky [2012], anchuri and magdon-ismail [2012] and tang et al. [2016] we can understand that signed graphs are well studied in literature because of their easiness to represent social systems in the real world. a signed graph is a graph ∑ (g,σ) with underlying graph g(v,e) and signature function σ : e → {1,−1}. degree of an edge in an unsigned graph is the number of edges incident on the end vertices of that edge excluding it. degree of an edge plays a vital role in networks such as communication networks, transportation networks etc. as it indicates the influence of that edge(type of connection) in that particular network. in platt [1947] the concept of platt number is discussed and it is mainly intended to study molecular structures in chemistry. several characterizations related to platt number of various graphs representing molecular structures are discussed in belavadi and mangam [2018]and mahadevi et al. [2018]. here we study these concepts in signed graphs. information diffusion in social networks is widely studied in literature which can be seen in guille et al. [2013], jalali et al. [2016] and tselykh et al. [2020]. various models also were proposed in literature to study information diffusion and influence maximisation in social networks which was discussed in rui et al. [2018] and li et al. [2019]. the motivation behind the study is that signed graphs can be effectively used to represent social networks as depicted in acharya [1985], gionis et al. [2020] and sun et al. [2020]. degree of an edge can be used to interpret the strength of influence of that particular edge which represents the specified relationship in the concerned network. hence information diffusion is possible through the edge with maximum degree. platt number can also be used to analyse the strength of connectedness of various networks. in this paper, we extend the definition of degree of an edge and platt number 92 degree of an edge and platt number in signed networks to signed graphs by giving more importance to positive labelled edges incident on the end vertices of that edge. this is done because of the extensive importance of positive edges in networks especially in social networks. further, we find bounds on degree of an edge and platt number in signed graphs and certain classes of signed graphs related to social networks. we then present some characerizations on the platt number of signed graphs. finally we propose a model to analyse networks based on degree of edges and platt number. 2 degree of an edge in a signed graph in this section the definition of degree of an edge, discussed in acharya et al. [2009], is extended to signed graphs and also some of its characterizations in signed graphs are investigated. 2.1 definition if xεv , then positive degree of x is defined as the number of positive edges incident at x and is denoted by d+(x) and negative degree of x is defined as the number of negative edges incident at x and is denoted by d−(x). the degree of x is denoted by d(x) and is given by d(x) = d+(x) +d−(x) in acharya et al. [2009]. 2.2 definition(degree of an edge) let ∑ (g,σ) be a signed graph with underlying graph g(v,e) and signature function σ : e → {1,−1}. let xyεe where x,yεv . the degree of an edge is defined as follows d(xy) = { d+(x) + d+(y) − 1 if σ(xy) = +1 d+(x) + d+(y) if σ(xy) = −1 (1) 2.3 example consider the following signed graph. 93 diviya k d and anjaly kishore x y z w u v -1+1 -1 -1 +1 -1 signed graph by the definition, d(xy) = 2, d(yz) = 1, d(zw) = 1, d(wx) = 1, d(xu) = 1 and d(uv) = 1. 2.4 bounds on degree of an edge now, bounds on degree of an edge in certain classes of graphs are determined. it is obvious that the lower bound of degree of an edge in any class of graphs is 0. the upperbound of degree of an edge in certain classes of graphs are given below class of graphs upperbound of edge degree d(e) pn-path graph 3 cn-cycle graph 3 wn-wheel graph n + 1 kn-complete graph 2n− 3 kp,q-complete bipartite graph p + q − 1 hn-helm graph n + 3 sn-sunflower graph n + 4 table 1: upper bound-edge degree the following result is on the bounds of the degree of an edge in any signed graph proposition 2.1. for a given signed graph ∑ (g,σ) with underlying graph g(v,e) with order n ≥ 3, 0 ≤ d(e) ≤ 2n− 3 proof. consider a connected signed graph ∑ (g,σ). by definition of degree of an edge it is clear that d(e) ≥ 0 in order to find the upperbound of degree of an edge consider an edge e = xy where x and y are vertices of maximum degree i.e. n − 1 each and all edges on 94 degree of an edge and platt number in signed networks x and y are positive. in order to determine the maximum possible value of degree of an edge we assume that e is labelled positive. now d(e) = 2(n− 1) − 1 = 2n− 3 hence 0 ≤ d(e) ≤ 2n− 3 for any edge e in ∑ (g,σ) 3 platt number of a signed graph the concept of platt number was discussed in platt [1947] and it is defined as the sum of degrees of edges of a graph as in mahadevi et al. [2018]. here we extend the definition of platt number to signed graphs. consider a signed graph ∑ (g,σ) where g(v,e) is the underlying graph and σ is the signature function. the platt number of σ is given by f(σ) = ∑ xyεe(g) d(xy) (2) 3.1 example consider the graph given in example 2.3. degree of edges are already calculated and its platt number is given by f(σ) = d(xy) + d(yz) + d(zw) + d(wx) + d(xu) + d(uv) = 7 4 bounds on platt number in this section bounds on platt number of certain graph classes are proposed. from the defintion of platt number itself it is clear that its lowerbound is 0 and is attained in the graph in which all edges are labelled negative. the upperbound of platt number of certain graph classes are given below. class of graphs upperbound of platt number pn-path graph 3n− 5 cn-cycle graph 3n wn-wheel graph (n− 1)(n + 6) kn-complete graph n(n−1)(2n−3) 2 kp,q-complete bipartite graph pq(p + q − 1) hn-helm graph n2 + 14n sn-sunflower graph(n ≥ 5) n2 + 25n table 2: upperbound-platt number 95 diviya k d and anjaly kishore theorem 4.1. in a connected signed graph ∑ (g,σ) with underlying graph g(v,e) and order n ≥ 2, 0 ≤ f(σ) ≤ n(n−1)(2n−3) 2 proof. for any signed graph ∑ (g,σ), f(σ) ≥ 0 since d(e) ≥ 0 for any edge e in σ. upperbound of the degree of an edge in a signed graph is 2n − 3 and the maximum number of edges in a signed graph of order n is n(n−1) 2 . so consider the maximum possible case which is the signed graph with maximum number of edges for a given order n and with all edges are labelled positive. here platt number is n(n−1)(2n−3) 2 . hence for any signed graph ∑ (g,σ) 0 ≤ f(σ) ≤ n(n− 1)(2n− 3) 2 remark 4.1. in the above theorem, the upper bound is attained by the complete graph with all edges labelled positive. theorem 4.2. for a connected signed graph ∑ (g,σ) of size m ≥ 2, 0 ≤ f(σ) ≤ m2. in particular if all edges are signed positive, then 3m− 2 ≤ f(σ) ≤ m2 proof. it is obvious that f(σ) ≥ 0 for any connected signed graph σ. now for a connected graph of given size m ≥ 2, platt number takes its maximum value in the case of a signed graph in which all edges are labelled positive and each edge takes its maximum degree which is m. in this case f(σ) = m2. hence 0 ≤ f(σ) ≤ m2 for any connected signed graph ∑ of size m ≥ 2 in the case of a connected signed graph with all edges are labelled positive and size m ≥ 2, degree of an edge takes its least value 2 which is by the edges of pendant vertices and for all other m− 2 edges the least value is 3. in this case platt number is 3m− 2. hence 3m− 2 ≤ f(σ) ≤ m2 for a connected signed graph of order m ≥ 2 with all edges labelled positive. remark 4.2. in the above theorem,the upper bound is attained by the graph k1,n−1 with all edges labelled positive and lower bound is attained by the path graph pn with all edges labelled positive. theorem 4.3. for a signed tree tn of order n ≥ 3, 0 ≤ f(tn) ≤ (n − 1)2. in particular if all edges are labelled positive, then 3n− 5 ≤ f(tn) ≤ (n− 1)2. 96 degree of an edge and platt number in signed networks proof. for a given signed tree on n vertces , the maximum degree for an edge is n − 1 which is attained when all edges are labelled positive and the maximum number of edges is n− 1. in this case the platt number is maximum and is given by (n− 1)2. hence 0 ≤ f(tn) ≤ (n− 1)2 for any signed tree tn of order n ≥ 3. when all edges are labelled positive, the minimum value of degree of an edge in a tree is 2 and as in the above theorem it is taken by edges incident on pendant vertices and for all other n − 3 edges, the minimum value is 3. in this case platt number is given by f(tn) = 3n−5. hence 3n−5 ≤ f(tn) ≤ (n−1)2 for any signed tree with all edges labelled positive. remark 4.3. in the above theorem also, when all edges are labelled positive the upper bound is attained by the tree k1,n−1 and lower bound is attained by the tree pn. 5 network analysis model using degree of edges and platt number   0 +1 −1 0 +1 −1 0 +1 0 +1 +1 0 +1 −1 0 +1 0 −1 +1 +1 −1 +1 0 +1 +1 +1 −1 +1 +1 0 0 −1 +1 0 +1 0 +1 −1 +1 +1 +1 0 +1 +1 0 +1 0 0 +1 0 −1 +1 +1 0 +1 0 +1 −1 0 +1 0 0 −1 +1 0 +1 0 −1 +1 0 +1 −1 +1 −1 0 −1 −1 0 −1 −1 0 +1 +1 +1 +1 0 +1 −1 0 +1 +1 +1 0 +1 0 +1 0 +1 +1 0   . first we calculate degree of edges in order to analyse the network as degree of an edge can be considered as a measure of the influence of that particular relationship in the given network. degree of edges are given in the following table. note that, since d(xy) = d(yx) we have filled the upper triangular part of the table. as we have mentioned in the introductory part of this paper, degree of an edge can be used to analyse social networks since degree of an edge indicates the influence of that particular relationship in the corresponding network. information diffusion in social networks is possible at its maximum through the edges, which represent relationships, with maximum influence. so information diffusion is fast through the edge with maximum degree. in this section we propose a model to analyse a network and to find the most influential relationship in that network through an example. consider a classroom consisting of ten students. suppose a 97 diviya k d and anjaly kishore v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v1 8 10 8 9 6 8 v2 10 10 9 8 10 9 v3 10 10 10 9 9 11 v4 9 7 7 10 9 v5 9 10 v6 7 8 9 v7 5 8 v8 8 7 v9 10 v10 table 3: degree of edges of the network teacher needs to identify the most influential student or relationship in the class in order to implement new methods of teaching effectively in the class. first we model that class using signed graphs. edges with signature function σ(e) = +1 represent the friendship between students and those with σ(e) = −1 represent the indifference between them and vertices without edges between them shows that there is neither a friendship nor an indifference between them. consider the following adjacency matrix of that particular signed graph which represents the above mentioned network of the class room. from the above table it is clear that edge with maximum degree is v3v9 and is 11. hence we can conclude that the most influential relationship in this network is between v3 and v9. so in order to make communications in that class more effective and fast or to implement new strategies effectively by the teacher it is more desirable to communicate it with either v3 or v9. platt number of this graph is 287. we can compare different networks using this platt number as platt number indicates the intensity of the connectedness between individuals of that particular network. moreover, by assigning degree of edges to corresponding edges, signed graphs can be converted to weighted graphs in which weight indicates the intensity of relationships between individuals. 6 conclusions in this paper the definition of degree of an edge and platt number is extended to signed graphs. some characterizations of degree of an edge and platt number are proposed. a 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12-01-2017. published on: 28-02-2017 doi: 10.23755/rm.v31i0.317 © salvador cruz rambaud and ana maría sánchez pérez abstract derivatives are products of different nature which are becoming increasingly common in financial markets. in certain cases, determining the assessment criteria can sometimes be a difficult task. specifically, this paper focuses on one type of exotic option: the barrier option. this option has to satisfy some conceptual conditions which are specified at the time of its purchase and define its characteristics. in order to analyze this type of option more deeply, in this paper we choose a specific one, the so-called barrier option cap, whose value is going to be derived by the binomial pricing model. keywords: barrier options; exotic options; barrier option cap; binomial model. 2010 ams subject classification: 62p05, 91g20, 97m30, 05a10. salvador cruz rambaud, ana maría sánchez pérez 26 1. introduction in the last years, the interest rate has reached historic low levels. as a consequence, the investment habits have changed and investors are interested in new and more profitable products. for this reason, derivatives have been selected as an alternative to traditional investment products. these are financial products whose price does not only vary according to parameters such as risk, but also depends on the market price of another asset, called the underlying asset (stock, foreign exchange, stock market index, etc.) (carr, 1998). the option holder is committed to the evolution, up or down, of a certain underlying asset in the securities market. there are different products: options, warrant, futures, etc. the main difference is the way in which the price is derived and the nature of the transaction to which this instrument gives rise, that is to say, how and when the delivery of the asset takes place. a derivative is a forward contract whose characteristics are established at the agreement moment, whilst the money exchange occurs at a future moment. derivatives, like financial options, are products with higher profits since the premium is lower than the corresponding to the underlying asset, whereby the results can be multiplied, either in the positive or negative sense, in relation to the premium. hence they are highly risky products. in order to make them more attractive, exotics options, specifically the barrier option, arise in order to allow taking more control of the operational risk by employing covertures. barrier options are very popular but there is a scarce economic research given its novelty and complexity (rich, 1994). we start with an analysis of the product from a theoretical point of view and by studying the analogies and differences of this type of exotic options with financial standard options (plain vanilla). they are options whose exercise will depend on whether the underlying asset reaches a pre-set barrier level during a certain period of time. if this occurs, the conditional option becomes a simple call or put option (knock-in options) or, on the other hand, it may cease to exist from the moment that the barrier level is reached (knock-out options). once this financial product has been introduced as an alternative to traditional investment products, we present this paper organization. in section 2, barrier options are described and studied from a mathematical point of view. in section 3, we focus on the barrier option cap which is a specific barrier option. then, in section 4, the financial analysis to derive the value of this type of financial option is presented. finally, section 5 summarizes and concludes. valuation of barrier options with the binomial pricing model 27 2. barrier options barrier options are derivatives which can be canceled or activated depending on the prices reached by the corresponding underlying asset (soltes and rusnakova, 2013). they are available for a predetermined period of time if, during this period, the underlying asset reaches a certain level, the conditional option is converted from that moment into a simple option (knock-in options) or, in another case, if the option already exists, it is canceled from that moment (knock-out options). these options are similar to a call or a put option with a specified barrier (called b). it ensures that the option has a fixed value (called l) if the maximum or minimum of the underlying asset price (called s) do not touch the barrier (rubinstein and reiner, 1991). below the different types of barrier options are explained. 2.1. knock-in options these options only arise if the underlying asset price reaches a certain level, known as barrier level (fernández, 1996). they can be classified into two types: 1) up-and-in options: the right to exercise the option is activated when the underlying asset price is above a certain level (b) during the option’s life. its price at maturity, if the strike price is denoted by k, is: -call up-and-in option -put up-and-in option 2) down-and-in options: the right to exercise the option at maturity appears if the underlying asset price falls below the pre-determined barrier (b). in this way, we can distinguish between: ),;0(max ks t  if bsss t ),,,(max 1  ,0 if bsss t ),,,(max 1  ),;0(max t sk  if bsss t ),,,(max 1  ,0 if bsss t ),,,(max 1  salvador cruz rambaud, ana maría sánchez pérez 28 -call down-and-in option -put down-and-in option 2.2. knock-out options these options only may be exercised if the underlying asset price does not reach the barrier, that is to say, the right to be exercised disappears if the underlying asset price intersects the barrier at any time of the option’s life; at this moment, the option acquires a fix price (l) (fernández, 1996). they can be classified into two types: 1) up-and-out options: they only make sense if the underlying asset price is above a pre-determined value during the option’s life: -call up-and-out option -put up-and-out option 2) down-and-out options: the right to exercise the option disappears if the underlying asset price is below the level established by the barrier. ),;0(max ks t  if bsss t ),,,(min 1  ,0 if bsss t ),,,(min 1  ),;0(max t sk  if bsss t ),,,(min 1  ,0 if bsss t ),,,(min 1  ),;0(max ks t  if bsss t ),,,(max 1  ,l if bsss t ),,,(max 1  ),;0(max t sk  if bsss t ),,,(max 1  ,l if bsss t ),,,(max 1  valuation of barrier options with the binomial pricing model 29 -call down-and-out option -put down-and-out option there is another type of option called “double barrier option” which disappears if the underlying asset does not stay within a certain interval (kunitomo and ikeda, 1992 and fernández and somalo, 2006). the main advantage of using barrier options is its lower price, compared to a vanilla equivalent option. the saving of using barrier options versus simple options depends on: -the proximity of the barrier to the current price of the underlying asset (with greater proximity to savings of the “out” type) and, conversely, to greater distance (in the “in” type). -the option’s life (the longer the time to maturity, the greater the probability of reaching the barrier and therefore the greater the savings in the “out” and inversely in the “in” type). -the greater the volatility (greater probability of touching the barrier and, therefore, greater savings in the “out” type, and inversely in the “in” type). barrier options can be very useful in hedging commodities providing protection at a lower price than traditional options for coverage of risks (crespo, 2001). 3. barrier option cap in this section, we are going to study a specific barrier option, the so-called barrier option cap. it guarantees a certain profitability called “option level” at maturity, i.e. it guarantees a final sale price independently of the share price, with the only condition that during the option’s life the underlying asset does not reach a certain lower level, called the “barrier”. this product was issue by pnb paribas bank in spain with the name “bonus cap”. it has been marketed for a short time since they were first issued in spain on june 16, 2010. ),;0(max ks t  if bsss t ),,,(min 1  ,l if bsss t ),,,(min 1  ),;0(max t sk  if bsss t ),,,(min 1  ,l if bsss t ),,,(min 1  salvador cruz rambaud, ana maría sánchez pérez 30 the barrier and option level are given by the issuer bank and they are known from the beginning, that is to say, from the issue date and during the barrier option cap life. in case that the underlying asset reaches the barrier, this does not imply that the option disappears but simply loses the guaranteed price at maturity (the “option level”), for which the barrier option cap will continue being traded with normality being able to give profits if the share has an upward tendency which allows the holder to sell above the level of purchase. the barrier option cap profit is limited to the “option level” so it should be clarified that in case that the underlying asset quotes above the “option level” at maturity, the holder will receive at most the profitability previously fixed corresponding to the “option level”. on the other hand, if the barrier option cap reaches the “option level” before maturity, the holder can get rid of his/her investment since it does not make sense to keep the investment when the highest allowed profitability has been already achieved. in this way, we would have achieved the maximum expected return without waiting to maturity. therefore, it can be said that the barrier option cap limits the profits which can be obtained, in exchange for ensuring a known profit provided that the underlying asset price is higher than the barrier value. 3.1. analogies and differences of barrier options cap with other derivatives -a future contract is an agreement whereby two persons (physical or legal) undertake to sell and to buy, respectively, an asset, called the underlying asset, at a price and at a future date according to the conditions fixed in advance by both parties. however, the holder of a barrier option cap will never be the owner of the underlying asset; he/she will receive the cash corresponding to the price of the underlying asset. the future is a compromise, whilst the purchase of a barrier option cap is an option to buy. -an option is an agreement granting the buyer, in return for payment of a price (premium), the right (not the obligation) to buy or sell an underlying asset at a price (strike price) and at a future date, in accordance with the conditions set forth in advance by both parties. as for the sale of the barrier option cap, it is a liquid product and can be sold at any time, so we could say that it keeps more similarities with the american options since it is not necessary to wait for the expiration to exercise the sale. barrier options cap present the following differences with respect to other derivative products which make them a new banking product: valuation of barrier options with the binomial pricing model 31 -they present a well-known and bounded return from the moment of contracting, as long as the underlying asset does not reach the barrier during the life of the barrier option cap. -the underlying asset is not acquired at any time. -at maturity, the option owner will receive in cash the traded price of the underlying asset in case it reaches the barrier and never exceeding the “option level”. -it is not necessary to wait until maturity to obtain liquidity. 4. assessment of a barrier option cap the methodology we are going to use in this paper is the binomial model, introduced by cox, ross and rubinstein (1973) to value financial options. it is a discrete-time model based on the binomial tree, with different possible trajectories. a barrier option cap is a derivative over an underlying asset (usually a share) which is defined by the following elements: b: barrier. l: option level. ks : price of the underlying asset at moment k (k = 1, 2, , n). the possible performances of the barrier option cap are the following ones: -if, at any time, the underlying asset is traded between the barrier and the option level, the barrier option cap guarantees the payment of the option level. figure 1. underlying asset between the barrier and the option level. source: own elaboration from bnp paribas bank data. salvador cruz rambaud, ana maría sánchez pérez 32 -if, at any moment, the underlying asset quotes above the option level, the option can be sold at that time obtaining, in advance, the maximum amount that could be reached with the option, i.e. the option level. figure 2. underlying asset above option level. source: own elaboration from bnp paribas bank data. -if, at maturity, the asset quotes below the pre-set barrier level, the holder of the option will receive at maturity the price of the share at that time, with limit the level of the option. figure 3. underlying asset below the barrier level. source: own elaboration from bnp paribas bank data. valuation of barrier options with the binomial pricing model 33 taking into account the given definition of the barrier option cap, the option price p at moment 0 is given by the mathematical expectation of the following random variable (boc) that represents the possible values of the option at that instant:                kbs kls klsb rls rl rl boc k k k n fn k f n f somefor ,if somefor ,if allfor ,if ,)1}(,min{ ,)1( ,)1( where f r is the risk-free interest rate. therefore, ][bocep  . in figure 4, we are going to describe a methodology to calculate the price of a barrier option cap assuming that the underlying asset follows a binomial process with a rising factor u and a downward factor d, starting from the price volatility of the underlying asset at time 0 )( 0s . to do this, we start from an example in which the option maturity is after five periods. in this case, the value of the barrier option cap is:                     555 0 4454 0 32532 0 43 4345 yprobabilitwith ,)1( 32yprobabilitwith ,)1( yprobabilitwith ,)1( yprobabilitwith ,)1( 31yprobabilitwith ,)1( qrds pqpqruds qprdus prn pqqprn boc f f f f f as previously indicated, ][bocep  . 5. conclusions taken into account the wide offer of financial products with different risks, profitability and liquidity, an accurate analysis of their characteristics and real values is completely necessary. despite barrier options increase the covertures of risks, they are cheaper than the equivalent standard option. specifically, a barrier option cap is a derivative with a given profitability provided that a certain condition is satisfied. in this way, a barrier option cap limits the benefit which can be obtained, in exchange of ensuring a known profit (the option level) if the price of the underlying asset salvador cruz rambaud, ana maría sánchez pérez 34 is higher than the barrier value. so, this paper aims to analyze the assessment of this option by employing the binomial options pricing model. figure 4: value of the barrier option cap within 5 periods (instants). in0 s 0 in1 in2 in3 in4 in5 prob 5 0 us  5 p 4 0 us   l 3 0 us   dus 4 0  qp 4 5 2 0 us   dus 3 0   us 0   dus 2 0   23 0 dus  23 10 qp 0 s   uds 0   22 0 dus   ds 0   2 0 uds   32 0 dus  32 10 qp 2 0 ds   3 0 uds   3 0 ds   4 0 uds  4 5 pq b 4 0 ds   5 0 ds  5 q source: own elaboration. valuation of barrier options with the binomial pricing model 35 references [1] bnp paribas website https://pi.bnpparibas.es/warrants/bonus-cap7. [2] carr, p., ellis k. and gupta v. (1998). “static hedging of exotic options”. journal of finance, 53 (3), pp. 1165-1191. [3] cox, j. c., ross, s. a. & rubinstein, m. (1979). “option pricing: a simplified approach”. journal of financial economics, 7(3), pp. 229-263. [4] crespo espert, j. l. (2001). “utilización práctica de las opciones exóticas: opciones asiáticas y opciones barrera”. boletín económico del ice, no. 2686, pp. i-viii. [5] fernández, p. (1996). derivados exóticos. documento de investigación del iese (308). [6] fernández, p. l. & somalo, m. p. (2006). opciones financieras y productos estructurados. mcgraw-hill, madrid. [7] kunitomo, n. & ikeda, m. (1992). “pricing options with curved boundaries”. mathematical finance, 2, pp. 275–298. [8] rich, d. r. (1994). “the mathematical foundations of barrier optionpricing theory”. advances in futures and options research (7), pp. 267– 311. [9] rubinstein, m. & reiner, e. (1991). “breaking down the barriers”. risk, 4(8), pp. 28-35. [10] soltes, v. & rusnakova, m. (2013). “hedging against a price drop using the inverse vertical ratio put spread strategy formed by barrier options”. engineering economics, 24(1), pp. 18-27. https://pi.bnpparibas.es/warrants/bonus-cap7 ratio mathematica issn: 1592-7415 vol. 31, 2016, pp. 65-78 eissn: 2282-8214 65 helix-hopes on finite hyperfields thomas vougiouklis1, souzana vougiouklis2 1 emeritus professor, democritus university of thrace, alexandroupolis, greece tvougiou@eled.duth.gr 2 researcher in maths and music, 17 oikonomou, exarheia, athens 10683, greece, elsouvou@gmail.com received on: 03-12-2016. accepted on: 14-01-2017. published on: 28-02-2017 doi: 10.23755/rm.v31i0.321 © thomas vougiouklis and souzana vougiouklis abstract hyperstructure theory can overcome restrictions which ordinary algebraic structures have. a hyperproduct on non-square ordinary matrices can be defined by using the so called helix-hyperoperations. we study the helixhyperstructures on the representations using ordinary fields. the related theory can be faced by defining the hyperproduct on the set of non square matrices. the main tools of the hyperstructure theory are the fundamental relations which connect the largest class of hyperstructures, the hvstructures, with the corresponding classical ones. we focus on finite dimensional helix-hyperstructures and on small hv-fields, as well. keywords: hyperstructures, hv-structures, h/v-structures, hope. 2010 ams subject classification: 20n20, 16y99. thomas vougiouklis, souzana vougiouklis 66 1 introduction we deal with the largest class of hyperstructures called hv-structures introduced in 1990 [10], [11], which satisfy the weak axioms where the nonempty intersection replaces the equality. definitions 1.1 in a set h equipped with a hyperoperation (which we abbreviate it by hope) ∙ : hhp (h)-{}: (x,y) x∙yh we abbreviate by wass the weak associativity: (xy)zx(yz), x,y,zh and by cow the weak commutativity: xyyx, x,yh. the hyperstructure (h,) is called hv-semigroup if it is wass and is called hvgroup if it is reproductive hv-semigroup: xh=hx=h, xh. (r,+,) is called hv-ring if (+) and () are wass, the reproduction axiom is valid for (+) and () is weak distributive with respect to (+): x(y+z)(xy+xz), (x+y)z(xz+yz), x,y,zr. for more definitions, results and applications on hv-structures, see books and the survey papers as [2], [3], [11], [1], [6], [15], [16], [20]. an extreme class is the following: an hv-structure is very thin iff all hopes are operations except one, with all hyperproducts singletons except only one, which is a subset of cardinality more than one. thus, in a very thin hv-structure in a set h there exists a hope () and a pair (a,b)h2 for which ab=a, with carda>1, and all the other products, with respect to any other hopes (so they are operations), are singletons. the fundamental relations β* and γ* are defined, in hv-groups and hv-rings, respectively, as the smallest equivalences so that the quotient would be group and ring, respectively [9], [10], [11], [12], [13]. the main theorem is the following: theorem 1.2 let (h,) be an hv-group and let us denote by u the set of all finite products of elements of h. we define the relation β in h as follows: xβy iff {x,y}u where uu. then the fundamental relation β* is the transitive closure of the relation β. an element is called single if its fundamental class is a singleton. motivation for hv-structures: the quotient of a group with respect to an invariant subgroup is a group. marty states that, the quotient of a group by any subgroup is a hypergroup. helix-hopes on finite hyperfields 67 now, the quotient of a group with respect to any partition is an hv-group. definition 1.3 let (h,), (h,) be hv-semigroups defined on the same h. () is smaller than (), and () greater than (), iff there exists automorphism faut(h,) such that xyf(xy), xh. then (h,) contains (h,) and write  . if (h,) is structure, then it is basic and (h,) is an hb-structure. the little theorem [11]. greater hopes of the ones which are wass or cow, are also wass and cow, respectively. fundamental relations are used for general definitions of hyperstructures. thus, to define the general hv-field one uses the fundamental relation γ*: definition 1.4 [10], [11]. the hv-ring (r,+,) is called hv-field if the quotient r/γ* is a field. let ω* be the kernel of the canonical map from r to r/γ*; then we call reproductive hv-field any hv-field (r,+,) if x(r-ω*)=(r-ω*)x=r-ω*,xr-ω*. from this definition, a new class is introduced [15]: definition 1.5 the hv-semigroup (h,) is h/v-group if the h/β* is a group. similarly h/v-rings, h/v-fields, h/v-modulus, h/v-vector spaces, are defined. the h/v-group is a generalization of the hv-group since the reproductivity is not necessarily valid. sometimes a kind of reproductivity of classes is valid, i.e. if h is partitioned into equivalence classes σ(x), then the quotient is reproductive xσ(y)=σ(xy)=σ(x)y, xh. an hv-group is called cyclic [11], if there is element, called generator, which the powers have union the underline set, the minimal power with this property is the period of the generator. if there exists an element and a special power, the minimum, is the underline set, then the hv-group is called single-power cyclic. definitions 1.6 [11], [14]. let (r,+,) be an hv-ring, (m,+) be cow hv-group and there exists an external hope : rmp(m):(a,x)ax, such that, a,br and x,ym we have a(x+y)(ax+ay), (a+b)x(ax+bx), (ab)xa(bx), then m is called an hv-module over r. in the case of an hv-field f instead of hvring r, then the hv-vector space is defined. definition 1.7 [17]. let (l,+) be hv-vector space on (f,+,), φ:ff/γ*, the canonical map and ωf={xf:φ(x)=0}, where 0 is the zero of the fundamental thomas vougiouklis, souzana vougiouklis 68 field f/γ*. similarly, let ωl be the core of the canonical map φ: ll/ε* and denote again 0 the zero of l/ε*. consider the bracket (commutator) hope: [ , ] : llp(l): (x,y)[x,y] then l is an hv-lie algebra over f if the following axioms are satisfied: (l1) the bracket hope is bilinear: [λ1x1+λ2x2,y](λ1[x1,y]+λ2[x2,y]) [x,λ1y1+λ2y](λ1[x,y1]+λ2[x,y2]), x,x1,x2,y,y1,y2l and λ1,λ2f (l2) [x,x]ωl, xl (l3) ([x,[y,z]]+[y,[z,x]]+[z,[x,y]])ωl, x,yl two well known and large classes of hopes are given as follows [11], [16]: definitions 1.8 let (g,) be a groupoid, then for every subset pg, p, we define the following hopes, called p-hopes: x,yg p: xpy = (xp)yx(py), pr: xpry= (xy)px(yp), pl: xply= (px)yp(xy). the (g,p), (g,pr) and (g,pl) are called p-hyperstructures. the usual case is for semigroup (g,), then xpy=(xp)yx(py)=xpy and (g,p) is a semihypergroup but we do not know about (g,pr) and (g,pl). in some cases, depending on the choice of p, the (g,pr) and (g,pl) can be associative or wass. a generalization of p-hopes: let (g,) be abelian group and p a subset of g with more than one elements. we define the hope p as follows: xpy = {xhy hp} if xe and ye xpy = xy if x=e or y=e we call this hope, pe-hope. the hyperstructure (g,p) is an abelian hv-group. definition 1.9 let (g,) be groupoid (resp., hypergroupoid) and f:gg be a map. we define a hope (), called theta-hope, we write -hope, on g as follows xy = {f(x)y, xf(y)} ( resp. xy = (f(x)y)(xf(y) ), x,yg. if () is commutative then  is commutative. if () is cow, then  is cow. helix-hopes on finite hyperfields 69 if (g,) is groupoid (or hypergroupoid) and f:gp(g)-{} multivalued map. we define the -hope on g as follows: xy = (f(x)y)(xf(y)), x,yg. motivation for the -hope is the map derivative where only the product of functions can be used. basic property: if (g,) is semigroup then f, the -hope is wass. 2 some applications of hv-structures last decades hv-structures have applications in other branches of mathematics and in other sciences. these applications range from biomathematics -conchology, inheritanceand hadronic physics or on leptons to mention but a few. the hyperstructure theory is closely related to fuzzy theory; consequently, hyperstructures can be widely applicable in industry and production, too [2], [3], [7], [18]. the lie-santilli theory on isotopies was born in 1970’s to solve hadronic mechanics problems. santilli proposed a ‘lifting’of the n-dimensional trivial unit matrix of a normal theory into a nowhere singular, symmetric, real-valued, positive-defined, n-dimensional new matrix. the original theory is reconstructed such as to admit the new matrix as left and right unit. the isofields needed correspond into the hyperstructures introduced by santilli & vougiouklis in 1999 [7] and they are called e-hyperfields. the hv-fields can give e-hyperfields which can be used in the isotopy theory in applications as in physics or biology. definition 2.1 a hyperstructure (h,) which contain a unique scalar unit e, is called e-hyperstructure. in an e-hyperstructure, we assume that for every element x, there exists an inverse x-1, i.e. exx-1x-1x. definition 2.2 a hyperstructure (f,+,), where (+) is an operation and () is a hope, is called e-hyperfield if the following axioms are valid: (f,+) is an abelian group with the additive unit 0, () is wass, () is weak distributive with respect to (+), 0 is absorbing element: 0x=x0=0, xf, there exist a multiplicative scalar unit 1, i.e. 1x=x1=x, xf, and xf there exists a unique inverse x-1, such that 1xx-1x-1x. the elements of an e-hyperfield are called e-hypernumbers. if the relation: 1=xx-1=x-1x, is valid, then we say that we have a strong e-hyperfield. definition 2.3 the main e-construction. given a group (g,), where e is the unit, then we define in g, a large number of hopes () as follows: xy = {xy, g1, g2,…}, x,yg-{e}, where g1, g2,…g-{e} thomas vougiouklis, souzana vougiouklis 70 g1, g2,… are not necessarily the same for each pair (x,y). (g,) is an hv-group, it is an hb-group which contains the (g,). (g,) is an e-hypergroup. moreover, if for each x,y such that xy=e, so we have xy=xy, then (g,) becomes a strong e-hypergroup. the main e-construction gives an extremely large number of e-hopes. example 2.4 consider the quaternion group q={1,-1, i,-i, j,-j, k,-k} with defining relations i2 = j2 = -1, ij = -ji = k. denoting i={i,-i}, j={j,-j}, k={k,-k} we may define a very large number () hopes by enlarging only few products. for example, (-1)k=k, ki=j and ij=k. then the hyperstructure (q,) is a strong e-hypergroup. mathematicalisation of a problem could make its results recognizable and comparable. this is because representing a research object or a phenomenon with numbers, figures or graphs might be simplest and in a recognizable way of reading the results. in questionnaires vougiouklis & vougiouklis proposed the substitution of likert scales with the bar [5], [18].this substitution makes things simpler and easier for both the subjects of an empirical research and the researcher, either at the stage of designing or that of results processing, because it is really flexible. moreover, the application of the bar opens a window towards the use of fuzzy sets in the whole procedure of empirical research, activating in this way more recent findings from different sciences, as well. the bar is closelly related with hyperstructure and fuzzy theories, as well. more specifically, the following was proposed: in every question, substitute the likert scale with the ‘bar’ whose poles are defined with ‘0’ on the left and ‘1’ on the right: 0 1 the subjects/participants are asked, instead of deciding and checking a specific grade on the scale, to cut the bar at any point they feel best expresses their answer to the specific question. the suggested length of the bar is approximately 6.18cm, or 6.2cm, following the golden ration on the well known length of 10cm. 3 small hv-numbers. hv-matrix representations in representations important role are playing the small hypernumbers. construction 3.1 on the ring (z4,+,∙) we will define all the multiplicative h/vfields which have non-degenerate fundamental field and, moreover they are, (a) very thin minimal, helix-hopes on finite hyperfields 71 (b) cow (non-commutative), (c) they have the elements 0 and 1, scalars. then, we have only the following isomorphic cases 23={0,2} or 32={0,2}. fundamental classes: [0]={0,2}, [1]={1,3} and we have (z4,+,)/γ*(z2,+,∙). thus it is isomorphic to (z2×z2,+). in this hv-group there is only one unit and every element has a unique double inverse. only f has one more right inverse element, the d, since fd={i,b}. moreover, the (x,) is not cyclic. construction 3.2 on (z6,+,∙) we define, up to isomorphism, all multiplicative h/v-fields which have non-degenerate fundamental field and, moreover they are: (a) very thin minimal (b) cow (non-commutative) (c) they have the elements 0 and 1, scalars then we have the following cases, by giving the only one hyperproduct, (i) 23={0,3} or 24={2,5} or 25={1,4} 34={0,3} or 35={0,3} or 45={2,5} in all 6 cases the fundamental classes are [0]={0,3}, [1]={1,4}, [2]={2,5} and we have (z6,+,)/γ*  (z3,+,∙). (ii) 23={0,2} or 23={0,4} or 24={0,2} or 24={2,4} or 25={0,4} or 25={2,4} or 34={0,2} or 34={0,4} or 35={1,3} or 35={3,5} or 45={0,2} or 45={2,4}. in all 12 cases the fundamental classes are [0]={0,2,4}, [1]={1,3,5} and we have (z6,+,)/γ*  (z2,+,∙). remark that if we need h/v-fields where the elements have at most one inverse element, then we must exclude the case of 25={1,4} from (i), and the case 35={1,3} from (ii). hv-structures are used in representation theory of hv-groups which can be achieved by generalized permutations or by hv-matrices [11], [12], [13], [14]. hv-matrix (or h/v-matrix) is a matrix with entries of an hv-ring or hv-field (or h/v-ring or h/v-field). the hyperproduct of two hv-matrices (aij) and (bij), of type mn and nr respectively, is defined in the usual manner and it is a set of mr hv-matrices. the sum of products of elements of the hv-ring is considered to be the n-ary circle hope on the hyperaddition. the hyperproduct of hvmatrices is not necessarily wass. the problem of the hv-matrix (or h/v-group) representations is the following: definition 3.3 let (h,) be hv-group (or h/v-group). find an hv-ring (or h/vring) (r,+,), a set mr={(aij)aijr} and a map t:hmr: h t(h) such that thomas vougiouklis, souzana vougiouklis 72 t(h1h2) t(h1)t(h2)  , h1,h2h. t is hv-matrix (or h/v-matrix) representation. if t(h1h2)t(h1)t(h2), h1,h2h, then t is called inclusion. if t(h1h2)=t(h1)t(h2)= {t(h)hh1h2}, h1,h2h, then t is good and then an induced representation t* for the hypergroup algebra is obtained. if t is one to one and good then it is faithful. the main theorem on representations is [13]: theorem 3.4 a necessary condition to have an inclusion representation t of an h/v-group (h,) by nn, h/v-matrices over the h/v-ring (r,+,) is the following: for all classes β*(x), xh must exist elements aijh, i,j{1,...,n} such that t(β*(a))  {a=(aij)aijγ*(aij), i,j{1,...,n}} inclusion t:hmr:a t(a)=(aij) induces homomorphic representation t* of h/β* on r/γ* by setting t*(β*(a))=[γ*(aij)], β*(a)h/β*, where γ*(aij)r/γ* is the ij entry of the matrix t*(β*(a)). t* is called fundamental induced of t. in representations, several new classes are used: definition 3.5 let m=mmn be the module of mn matrices over r and p={pi:ii}m. we define a p-hope p on m as follows p: mm  p(m): (a,b)  apb={aptib: ii } m where pt denotes the transpose of p. the hope p is bilinear map, is strong associative and inclusion distributive: ap(b+c)  apb+apc, a,b,cm definition 3.6 let m=mmn the mn matrices over r and let us take sets s={sk:kk}r, q={qi:jj}m, p={pi:ii}m. define three hopes as follows s: rmp(m): (r,a)rsa = {(rsk)a: kk} m q+: mmp(m): (a,b)aq+b = {a+qj+b: jj} m p: mmp(m): (a,b)apb = {aptib: ii} m then (m,s,q+,p) is hyperalgebra on r called general matrix p-hyperalgebra. helix-hopes on finite hyperfields 73 4 helix-hopes and applications recall some definitions from [19], [8], [20], [4]: definition 4.1 let a=(aij)mmn be mn matrix and s,tn be naturals such that 1sm, 1tn. we define the map cst from mmn to mst by corresponding to the matrix a, the matrix acst=(aij) where 1is, 1jt. we call this map cutprojection of type st. thus acst is matrix obtained from a by cutting the lines, with index greater than s, and columns, with index greater than t. we use cut-projections on all types of matrices to define sums and products. definitions 4.2 let a=(aij)mmn be an mn matrix and s,tn, 1sm, 1tn. we define the mod-like map st from mmn to mst by corresponding to a the matrix ast=(aij) which has as entries the sets aij = {ai+κs,j+λt 1is, 1jt. and κ,λn, i+κsm, j+λtn}. thus we have the map st: mmn  mst: a  ast = (aij). we call this multivalued map helix-projection of type st. ast is a set of stmatrices x=(xij) such that xijaij, i,j. obviously amn=a. let a=(aij)mmn be a matrix and s,tn such that 1sm, 1tn. then it is clear that we can apply the helix-projection first on the rows and then on the columns, the result is the same if we apply the helix-progection on both, rows and columns. therefore we have (asn)st = (amt)st = ast. let a=(aij)mmn be matrix and s,tn such that 1sm, 1tn. then if ast is not a set but one single matrix then we call a cut-helix matrix of type st. in other words the matrix a is a helix matrix of type st, if acst= ast. definitions 4.3 (a) let a=(aij)mmn , b=(bij)muv be matrices and s=min(m,u), t=min(n,u). we define a hope, called helix-addition or helix-sum, as follows: : mmnmuvp(mst): (a,b)ab=ast+bst=(aij)+(bij) mst, where (aij)+( bij)= {(cij)= (aij+bij) aijaij and bijbij}. (b) let a=(aij)mmn and b=(bij)muv be matrices and s=min(n,u). we define a hope, called helix-multiplication or helix-product, as follows: : mmnmuvp(mmv):(a,b)ab=amsbsv=(aij)(bij)mmv, thomas vougiouklis, souzana vougiouklis 74 where (aij)(bij)= {( cij)=(aitbtj) aijaij and bijbij}. the helix-sum is external hope since it is defined on different sets and the result is also in different set. the commutativity is valid in the helix-sum. for the helix-product we remark that we have ab=amsbsv so we have either ams=a or bsv=b, that means that the helix-projection was applied only in one matrix and only in the rows or in the columns. if the appropriate matrices in the helix-sum and in the helix-product are cut-helix, then the result is singleton. remark. in mmn the addition is ordinary operation, thus we are interested only in the ‘product’. from the fact that the helix-product on non square matrices is defined, the definition of the lie-bracket is immediate, therefore the helix-lie algebra is defined [17], as well. this algebra is an hv-lie algebra where the fundamental relation ε* gives, by a quotient, a lie algebra, from which a classification is obtained. in the following we restrict ourselves on the matrices mmn where m<n. we have analogous results if m>n and for m=n we have the classical theory. notation. for given κℕ-{0}, we denote by κ the remainder resulting from its division by m if the remainder is non zero, and κ=m if the remainder is zero. thus a matrix a=(aκλ)mmn, m<n is a cut-helix matrix if we have aκλ=aκλ, κ,λℕ-{0}. moreover let us denote by ic=(cκλ) the cut-helix unit matrix which the cut matrix is the unit matrix im. therefore, since im=(δκλ), where δκλ is the kronecker’s delta, we obtain that, κ,λ, we have cκλ=δκλ. proposition 4.4 for m<n in (mmn,) the cut-helix unit matrix ic=(cκλ), where cκλ=δκλ, is a left scalar unit and a right unit. it is the only one left scalar unit. proof. let a,bmmn then in the helix-multiplication, since m<n, we take helix projection of the matrix a, therefore, the result ab is singleton if the matrix a is a cut-helix matrix of type mm. moreover, in order to have ab=ammb=b, the matrix amm must be the unit matrix. consequently, ic=(cκλ), where cκλ=δκλ, κ,λℕ-{0}, is necessarily the left scalar unit. let a=(auv)mmn and consider the hyperproduct aic. in the entry κλ of this hyperproduct there are sets, for all 1κm, 1λn , of the form aκscsλ = aκsδsλ= aκλ aκλ. therefore aica, ammn. ■ in the following examples of the helix-hope, we denote eij any type of matrices which have the ij-entry 1 and in all the other entries we have 0. helix-hopes on finite hyperfields 75 example 4.5 consider the 23 matrices of the forms, aκλ = e11+e13+κe21+e22+λe23, κ,λℤ. then we obtain aκλast = {aκ+s,κ+t, aκ+s,λ+t, aλ+s,κ+t, aλ+s,λ+t}. moreover astaκλ={aκ+s,λ+s,aκ+s,λ+t,aκ+t,λ+s,aκ+t,λ+t}, so aκλastastaκλ={aκ+s,λ+t}, thus () is cow. the helix multiplication () is associative. example 4.6 consider all traceless matrices a=(aij)m23, in the sense that we have a11+ a22=0. the cardinality of the helix-product of any two matrices is 1, or 23, or 26. these correspond to the cases: a11=a13 and a21=a23, or only a11=a13 either only a21=a23, or if there is no restriction, respectively. proposition. the lie-bracket of two traceless matrices a=(aij), b=(bij)mmn, m<n, contain at least one traceless matrix. example 4.7 let us denote by eij the matrix with 1 in the ij-entry and zero in the rest entries. then take the following 2×2 upper triangular h/v-matrices on the above h/v-field (z4,+,), on the set z4={0,1,2,3}, of the case that only 23={0,2} is a hyperproduct: i=e11+e22, a=e11+e12+e22, b=e11+2e12+e22, c=e11+3e12+e22, d=e11+3e22, e=e11+e12+3e22, f=e11+2e12+3e22, g=e11+3e12+3e22, a hyper-matrix representation of four dimensional case with helix-hope: example 4.8 on the field of real or complex numbers we consider the four dimensional space of all 2×4 matrices of type, called helix-upper triangular matrices, a b a c a = 0 d 0 d this set is closed under the helix-hope. that means that the helix-product of two such matrices is a 2×4 matrix, of the same type. in fact we have a b a c a b a c aa = 0 d 0 d  0 d 0 d = a {b,c} a b a c = 0 d • 0 d 0 d = aa {ab+bd, ab+cd} aa {ac+bd, ac+cd} = 0 dd 0 dd thomas vougiouklis, souzana vougiouklis 76 therefore the result is a set with 4 matrices. examples 4.9 (a) on the same type of matrices using the construction 4.1, on (z4,+,∙) we take the small h/v-field (z4,+,), where only 23={0,2}, where we remind that the fundamental classes are {0,2}, {1,3}. we take from the set of all matrices a b a c a = 0 d 0 d the matrix 2 1 2 3 x = 0 1 0 1 then the powers of this matrix are 0 {1,3} 0 {1,3} x2 = 0 1 0 1 , 0 {1,3} 0 {1,3} x3 = 0 1 0 1 , we obtain that the generating set is the following 2 1 2 3 0 {1,3} 0 {1,3} 0 1 0 1  0 1 0 1 the classes remain the same. (b) if we take the matrix 2 1 2 2 y = 0 1 0 1 then the powers of this matrix are 0 {0,3} 0 {1,2} y2 = 0 1 0 1 , 0 z4 0 z4 y3 = 0 1 0 1 , we obtain that the generating set is the following 2 1 2 2 0 z4 0 z4 0 1 0 1  0 1 0 1 we have only one class. helix-hopes on finite hyperfields 77 references [1] chvalina j., hoskova s. (2007). modelling of join spaces with proximities by first-order linear partial differential operators. ital. j. pure appl. math. 21, 177–190. [2] corsini p., leoreanu v. (2003). applications of hypergroup theory, kluwer academic publishers. [3] davvaz b., leoreanu v. (2007). hyperring theory and applications. int. academic press. [4] davvaz b., vougioukli s., vougiouklis t. (2011) on the multiplicative hv-rings derived from helix hyperoperations, util. math., 84, 53-63. [5] kambaki-vougioukli p., vougiouklis t. (2008). bar instead of scale. ratio sociologica, 3, 49-56. [6] maturo a., sciarra e., tofan i. 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(2015). hyper-representations by non square matrices. helix-hopes, american j. modern physics, 4(5), 52-58. ratio mathematica volume 45, 2023 forcing vertex square free detour number of a graph g. priscilla pacifica ∗ k. christy rani† abstract let g be a connected graph and s a square free detour basis of g. a subset t ⊆ s is called a forcing subset for s if s is the unique square free detour basis of s containing t . a forcing subset for s of minimum order is a minimum forcing subset of g. the forcing square free detour number of g is fdn �fu (g) = min{fdn �fu (su)}, where the minimum is taken over all square free detour bases s in g. in this paper, we introduce the forcing vertex square free detour sets. the general properties satisfied by these forcing subsets are discussed and the forcing square free detour number for a certain class of standard graphs are determined. we show that the two parameters dn �fu (g) and fdn �fu (g) satisfy the relationship 0 ≤ fdn �fu (g) ≤ dn �fu (g). also, we prove the existence of a graph g with fdn �fu (g) = α and dn �fu (g) = β, where 0 ≤ α ≤ β and β ≥ 2 for some vertex u in g. keywords: forcing square free detour number; forcing vertex square free detour set; forcing vertex square free detour number. 2020 ams subject classifications: 05c12, 05c691 ∗assistant professor, department of mathematics, st. mary’s college(autonomous), thoothukudi 628 001; priscillamelwyn@gmail.com; e-mail@address. †research scholar, reg. no.: 20122212092002, department of mathematics, st. mary’s college(autonomous), thoothukudi 628 001, manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamilnadu, india ; christy.agnes@gmail.com 1received on june 5, 2022. accepted on august 25, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.1030. issn: 1592-7415. eissn: 2282-8214. c©the authors. this paper is published under the cc-by licence agreement. 286 g. priscilla pacifica and k. christy rani 1 introduction in a simple connected graph g = (v,e) of order at least two, a longest xy path is called an x− y detour path for any two vertices x and y in g. a set s of vertices of g is a detour set of g if each vertex v of g lies on an x − y detour path for some elements x and y in s. the minimum cardinality of a detour set of g is the detour number of g and is denoted by dn(g). a subset t of a minimum detour set s of g is a forcing detour subset for s if s is the unique detour basis containing t . a forcing detour subset for s of minimum cardinality is a minimum forcing detour subset of s. the forcing detour number fdn(s) of s, is the minimum cardinality of a forcing detour subset for s. the forcing detour number fdn(g) of g, is min{fdn(s)}, where the minimum is taken over all detour bases s in g. this notion of forcing detour number was initiated by chartrand et al. [2003]. from then onwards there have been many relevant studies about the forcing detour concept in the past few decades. the connected detour parameter was defined by santhakumaran and athisayanathan [2009]. also, santhakumaran and athisayanathan [2010] focussed on edge of a graph and introduced forcing edge detour number and the forcing weak edge detour number. further results on connected detour number was studied by ali and ali [2019]. titus and balakrishnan [2016] concentrated on a vertex of a graph and compiled the findings on forcing vertex detour monophonic number arising from the chordless path. ramalingam et al. [2016] extended this concept to triangle free and extracted forcing results. the square free detour concept was introduced by rani and pacifica [2022]. though there are several novel forcing parameters, this paper aims at filling some significant research gaps and brings forth the results on the forcing vertex square free detour number of a graph. we determine the bounds for it and compare the relationship with the vertex square free detour number of a graph. these ideas have interesting application in channel assignment problem and in radio technologies. moreover, this concept can be applied in security based community design. for the basic terminologies we refer to chartrand et al. [1993]. 2 preliminaries theorem 2.1. let u be any vertex of a connected graph g. (i) every end-vertex of g other than the vertex u (whether u is an end-vertex or not) belongs to every u-square free detour set. (ii) no cut-vertex of g belongs to any dn �fu -set. theorem 2.2. for any vertex u in a non-trivial connected graph g of order n,1 ≤ dn �fu (g) ≤ n−1. 287 forcing vertex square free detour number of a graph theorem 2.3. let g be a connected graph. (i) if g is the complete graph kn, then dn�fu (g) = 1 for every vertex u in kn. (ii) if g is the complete bipartite graph km,n(2 ≤ m ≤ n) with partitions x and y , then dn �fu (g) = { m−1 if u ∈ x n−1 if u ∈ y (iii) if g is the cycle cn, then dn�fu (g) = 1for every vertex u in cn. (iv) for a wheel wn(n ≥ 4), dn �fu (wn) =   ⌈ n−1 3 ⌉ if u ∈ k1. n ≥ 4 2 if u ∈ cn−1, n ≥ 6 1 if u ∈ cn−1, 4 ≤ n ≤ 5 3 forcing vertex square free detour number of a graph even though every connected graph contains a vertex square free detour set, some connected graph may contain several vertex square free detour sets. for each minimum vertex square free detour set su in a connected graph there is always some subset t of su that uniquely determines su as the minimum vertex square free detour set containing t such forcing subsets are considered in this section. definition 3.1. let u be any vertex of a connected graph g and su be a dn�fu -set of g. a subset fu ⊆ su is called a u-forcing subset for su if su is the unique dn �fu -set consisting of fu. the u-forcing subset for su of minimum cardinality is a minimum u-forcing subset of su. the forcing u-square free detour number of su, denoted by fdn�fu (su) is the order of a minimum u-forcing subset for su. the forcing u-square free detour number of g is fdn �fu (g) = minfdn �fu (su) where the minimum is considered over all dn �fu -sets su in g. example 3.1. for the graph g shown in figure 1, the only dn �fv1 -sets are {v2,v4}, {v2,v5},{v3,v4} and {v3,v5}. hence fdn�fu (g) = 2. also dn�fu (g) = 2 and fdn �fu (g) = 1 for u = v3 and v4 in g. moreover v5 and v4 are the unique vertex square free detour sets for vertices v2 and v5 respectively and so fdn�fu (g) = 0 for u = v2,v5. 288 g. priscilla pacifica and k. christy rani figure 1: g the following theorem follows from the definitions of vertex square free detour number and forcing vertex square free detour number of a graph g. theorem 3.1. for any vertex u in a connected graph g, 0 ≤ fdn �fu (g) ≤ dn �fu (g). proof. let u be any vertex of g. from the definition of fdn �fu (g), we find that fdn �fu (g) ≥ 0, consider a dn �fu -set su in g. we have fdn�fu (g) = min{fdn �fu (su) : su is a dn�fu -set in g } and so fdn�fu (g) ≤ dn�fu (g). hence 0 ≤ fdn �fu (g) ≤ dn �fu (g). now, we characterize the graph g for which the bounds in theorem 3.1 are reached and the graph for which fdn �fu (g) = 1. theorem 3.2. let u be any vertex of a connected graph g. then (i) fdn �fu (g) = 0 if and only if g has a unique dn �fu -set, (ii) fdn �fu (g) = 1 if and only if g has at least two dn �fu -sets one of which is a unique dn �fu -set containing any one of its elements, (iii) fdn �fu (g) = dn �fu (g) if and only if no dn �fu -set of g is the unique dn �fu set containing any of its proper subsets. proof. (i) let fdn �fu (g) = 0. then, fdn �fu (su) = 0 where su is any dn�fu set by definition and so the empty set is the minimum u-forcing subset for su. since the empty set φ is a subset of every set we have su is the unique minimum u-forcing subset of g. the converse of this theorem is obvious. 289 forcing vertex square free detour number of a graph (ii) let fdn �fu (g) = 1. then by (i),g has at least two fdn �fu -sets. also, since fdn �fu (g) = 1, there is a singleton subset f of a dn �fu -set su of g such that f is not a subset of any other dn �fu -set of g. thus su is the unique dn �fu -set containing one of its elements. the converse is obvious. (iii) let fdn �fu (g) = dn �fu . then fdn �fu (su) = dn�fu (g) for every dn�fu -set su in g. by theorem 2.1, dn�fu (g) ≥ 1 and so fdn�fu (g) = 1. also by (i), g has at least two dn �fu -sets and hence the empty set φ is not a u-forcing subset of any dn �fu -set su of g is unique dn�fu -set which consists of its proper subsets. theorem 3.3. let g = (v,e) be a connected graph and let s∗u be the set of all u-square free detour vertices of g. then fdn �fu (g) ≤ dn �fu (g)−|s∗u|. proof. let su be any square free detour basis of g. then dn�fu (su) = |su|,s ∗ u ⊆ su and su is the unique square free detour basis containing su − s∗u. thus fdn �fu (g) ≤ |su −s∗u| = |su|− |s∗u| = dn�fu (g)−|s ∗ u|. remark 3.1. the bound in theorem 3.3 is sharp. for the graph g given in figure 2, fdn �fv1 (g) = 0, |s∗v1| = 2 and dn�fv1 (g) = 2. also, the inequality in theorem 3.3, can be strict. for the graph g given in figure 1, fdn �fv3 (g) = 1, |s∗v3| = 0 and dn �fv3 (g) = 2. thus fdn �fv3 (g) < dn �fv3 (g)−|s∗v3|. figure 2: g theorem 3.4. let g be a connected graph and let = be the set of relative complements of the minimum u-forcing subsets in their respective minimum u-square free detour sets in g. then ∩f∈=f is the set of all u-square free detour vertices of g. proof. let s∗u be the set of all u-square free detour vertices of g. we claim that s∗u ⊆∩f∈=f . let x ∈ s∗u. then x is a u-square free detour vertex of g so that x belongs to every u-square free detour set su of g. let t ⊆ su be any minimum u-forcing subset for any u-square free detour basis su of g. we claim that x /∈ t . 290 g. priscilla pacifica and k. christy rani if x ∈ t , then t ′ = t −{x} is a proper subset of su is the unique u-square free detour containing t ′ so that t ′ is a u-forcing subset for su with |t ′| < |t|, which is a contradiction to t is a minimum u-forcing subset for su. thus x /∈ t and so x ∈ f , where f is the relative complement of t in su. hence x ∈∩f∈=f so that s∗u ⊆∩f∈=f . conversely, let x ∈ ∩f∈=f . then x belongs to the relative complement of t in su for every t for every su such that t ⊂ su, where t is a minimum u-forcing subset for su. since f is the relative complement of t in su,f ⊂ su and so x ∈ su for every su so that is a u-square free detour vertex of g. thus x ∈ s∗u and so ∩f∈=f ⊂ s∗u. hence s∗u ∩f∈= f . theorem 3.5. let u be any vertex of a connected graph g and let su be any dn�fu -set of g. then (i) no u-square free detour vertex belongs to any minimum u-forcing subset of su. (ii) no cut-vertex of g belongs to any minimum u-forcing subset of su. proof. (i) the proof follows from the first part of theorem 3.4. (ii) since any minimum u-forcing subset of su is a subset of su, the proof follows from theorem 2.1(ii). corolary 3.1. let u be any vertex of a connected graph g. if g contains l endvertices, then fdn �fu (g) ≤ dn �fu (g)− l + 1. proof. this follows from theorems 2.1(i) and 3.5(i). remark 3.2. the bound in corollary 3.1 is sharp. for a tree t with l endvertices, fdn �fu (g) = dn �fu (g)− l + 1 for any end-vertex u in t . theorem 3.6. let g be any connected graph of order n. then (i) if g is a tree with l end-vertices, then fdn �fu (g) = 0 for every vertex l in g. (ii) if g is the complete graph kn, then fdn �fu (g) = { 0 if n = 4 n otherwise 291 forcing vertex square free detour number of a graph (iii) if g is the complete bipartite graph km,n(2 ≤ m ≤ n), with partitions x and y , dn �fu (km,n) = 0 for every vertex u in km,n. (iv) if g is the cycle cn, then fdn �fu (g) = { 0 if n = 4 1 otherwise (v) if g is the wheel wn, then fdn �fu (wn) =   0 if n = 5,u ∈ k1 1 if n = 4,u ∈ wn and n ≥ 6,u ∈ cn−1⌈ n−1 3 ⌉ if n ≥ 6,u ∈ k1. proof. (i) by the fact that dn �fu (g) = l−1 or dn �fu (g) = l when u is an endvertex or not an end-vertex. since the set of all end-vertices of a tree is the unique dn �fu -set, the result follows from theorem 3.2(i) that fdn �fu (g) = 0. (ii) by theorem 2.3(i) for the complete graph k4, su consists of the antipodal vertex of u. since the set of antipodal vertex is unique for k4, the result follows from theorem 3.2(i) that fdn �fu (g) = 0. for kn(n 6= 4) it follows from theorem 2.3(i), that su consists of exactly one vertex of v −{u}. thus there exist n−1 distinct vertices other than u in kn. then the result follows from theorem 3.2(ii) that fdn �fu (g) = 1. (iii) by theorem 2.3(ii), for km,n(2 ≤ m ≤ n) with partitions x,y with |x| = m and |y | = n, we have dn �fu (g) = m−1 or dn �fu (g) = n−1 according to whether the vertex u lies in x or y . since the dn �fu -set su is unique in both the cases, the result follows from theorem 3.2(i) that fdn �fu (g) = 0. (iv) by theorem 2.3(iii) for c4,dn�fu -set su consists of the antipodal vertex of u. thus we observe that su is unique and so fdn�fu (g) = 0 by theorem 3.2(i). by theorem 2.3(iii), for an even cycle cn(n 6= 4),su consists of exactly one vertex which is antipodal or adjacent to u. also, for an odd cycle cn(n ≥ 3)dn�fu -set su contains exactly one vertex which is adjacent to u. since there exist two adjacent vertices for an odd cycle and in addition an antipodal vertex for an even cycle we have fdn �fu (g) = 1. 292 g. priscilla pacifica and k. christy rani (v) by theorem 2.3(iv), for wn = k1 + cn−1(n ≥ 5) a dn�fu -set su consists of ⌈ n−1 3 ⌉ vertices of the rim cn−1 of wn, where u is a central vertex of wn that is in k1 called as hub. thus there exist three different dn�fu -sets with ⌈ n−1 3 ⌉ vertices. therefore, from theorem 3.2(ii) the result follows that fdn �fu (g) = ⌈ n−1 3 ⌉ for the central vertex of wn(n ≥ 5). when u is a vertex on cn−1 of wn by theorem 2.3(iv), su contains exactly one adjacent or antipodal vertex on cn−1 with the hub of the wheel, according as n − 1 is odd or n − 1 is even. thus there are two adjacent vertices for a vertex on an odd cycle and an antipodal vertex besides two adjacent vertices for a vertex on even cycle. hence the result follows from theorem 3.2(ii) that fdn �fu (g) = 1 for u ∈ cn−1(n ≥ 6). by theorem 2.3(iv) for w4,su consists of exactly one adjacent vertex for every u ∈ w4. thus there are n− 1 such dn�fu -sets, for the vertices of w4 are adjacent to each other. hence by theorem 3.2(ii), we have fdn �fu (g) = 1. also, for w5,su contains two antipodal vertices of c4 where u is the central vertex of w5. since there exist two different su with distinct pair of antipodal vertices of c4, from theorem 3.2(iii), the result follows that fdn�fu (g) = dn �fu (g) = 2. furthermore, when u is a vertex on the rim of w5,su consists of the antipodal vertex of u. since there is only one antipodal vertex for any vertex u on c4 of w5, we have unique su for all vertices on the rim of w5. hence the result follows from theorem 3.2(i) that fdn�fu (g) = 0. theorem 3.7. for every pair α,β of positive integers with 0 ≤ α ≤ β and β ≥ 2, there exists a connected graph g with fdn �fu (g) = α and dn �fu (g) = β for some vertex u in g. proof. we consider two cases. case 1: let α = 0. let g be any tree with β + 1 end-vertices. then for an endvertex u in g,fdn �fu (g) = 0 by theorems 3.2(i) and 3.5(i). case 2: let α ≥ 1. for each i(1 ≤ i ≤ α), let di6 be a dutch windmill graph consisting of i copies of c6 : u,pi,qi,ri,si, ti,u. let h be a graph obtained by adding α new vertices r′1,r ′ 2, ...,r ′ α and joining each r ′ i(1 ≤ i ≤ α) to both the vertices qi and si of di6(1 ≤ i ≤ α). 293 forcing vertex square free detour number of a graph let k1,β−α be the star with common vertex wo and let w = {w1,w2, ...,wβ−α} be the set of all end-vertices of k1,β−α. let g be the required graph produced by identifying the vertex u of di6 with the common vertex wo of k1,β−α as pictured in figure 3. figure 3: g first we show that dn �fu (g) = β for some vertex u in g. by the fact that every dn �fu -set of g contains w and exactly one vertex from each c6 of di6(1 ≤ i ≤ α). then dn�fu (g)(β − α) + α and so dn�fu (g) = β. let su = w ∪ {x1,x2, ...,xα}, where xj = pj or tj of ci6(1 ≤ i ≤ α). clearly su is a dn�fu -set of g and so (g) ≤ |su| = (β−α)+α = β. thus dn�fu (g) = β. next we show that fdn �fu (g) = α. since dn �fu (g) = β, we find that every dn �fu -set of g consists of w and exactly one vertex from each c6 of di6(1 ≤ i ≤ α). let f ⊆ su be any minimum u-forcing subset of su. then by theorem 3.5(ii),f ⊆ su − w and so |f| ≤ α. if |f| < α then there is a vertex yj of ci6(1 ≤ i ≤ α) distinct from xj(1 ≤ i,j ≤ α). then s′u = (su − xj) ∪ yj is also a minimum u-forcing subset containing f such that xj /∈ f . thus su is not a unique u-square free detour set which consists of f and so f is not a minimum u-forcing subset of su. thus fdn�fu (g) = α. 294 g. priscilla pacifica and k. christy rani 4 conclusion in this paper, we computed the forcing vertex square free detour number of some standard graphs. we discussed the characteristics of the forcing vertex square free detour sets. also, the relationship between the vertex square free detour number and the forcing vertex square free detour number has been exhibited. in future, this concept can be extended to edge related parameter. to derive similar results in the context of some other variants of detour number is the open area of research. references a. m. ali and a. a. ali. the connected detour numbers of special classes of connected graphs. journal of mathematics, 2019, 2019. g. chartrand, g. l. johns, and s. tian. detour distance in graphs. in annals of discrete mathematics, volume 55, pages 127–136. elsevier, 1993. g. chartrand, g. l. johns, and p. zhang. the detour number of a graph. utilitas mathematica, 64, 2003. s. s. ramalingam, i. k. asir, and s. athisayanathan. vertex triangle free detour number of a graph. mapana journal of sciences, 15(3):9–24, 2016. k. c. rani and g. p. pacifica. outer independent square free detour number of a graph. ratio mathematica, 44:309–315, 2022. a. santhakumaran and s. athisayanathan. on the connected detour number of a graph. journal of prime research in mathematics, 5:149–170, 2009. a. santhakumaran and s. athisayanathan. forcing edge detour number of an edge detour graph. journal of prime research in mathematics, 6:13–21, 2010. p. titus and p. balakrishnan. the forcing vertex detour monophonic number of a graph. akce international journal of graphs and combinatorics, 13(1): 76–84, 2016. 295 microsoft word documento1 microsoft word documento1 ratio mathematica 29 (2015) 53-64 issn:1592-7415 solvable groups derived from fuzzy hypergroups e. mohammadzadeh 1, t. nozari 2 1 department of mathematics, faculty of science, payame noor university, p.o. box 19395-3697, tehran, iran, mohammadzadeh.e@pnurazavi.ac.ir 2 department of mathematics, golestan university, gorgan, iran t.nozari@gu.ac.ir abstract in this paper we introduce the smallest equivalence relation ξ∗ on a finite fuzzy hypergroup s such that the quotient group s/ξ∗, the set of all equivalence classes, is a solvable group. the characterization of solvable groups via strongly regular relation is investigated and several results on the topic are presented. key words: fuzzy hypergroups, strongly regular relation, solvable groups, fundamental relation. 2000 ams subject classifications: 8a72; 20n20, 20f18, 20f19. doi:10.23755/rm.v29i1.23 1 introduction in mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. all abelian groups are trivially solvable a subnormal series being given by just the group itself and the trivial group. but non-abelian groups may or may not be solvable. a small example of a solvable, non-nilpotent group is the symmetric group s3. in fact, as the smallest simple non-abelian group is a5, (the alternating 53 e. mohammadzadeh and t. nozari group of degree 5) it follows that every group with order less than 60 is solvable. the study of fuzzy hyperstructures is an interesting research topic for fuzzy sets. there are many works on the connections between fuzzy sets and hyperstructures. this can be considered into three groups. a first group of papers studies crisp hyperoperations defined through fuzzy sets. this study was initiated by corsini in [3, 4] and then continued by other researchers. a second group of papers concerns the fuzzy hyperalgebras. this is a direct extension of the concept of fuzzy algebras. this was initiated by zahedi in [12]. a third group was introduced by corsini and tofan in [5]. the basic idea in this group of papers is the following: a multioperation assigns to every pair of elements of s a non-empty subset of s, while a fuzzy multioperation assigns to every pair of elements of s a nonzero fuzzy set on s. this idea was continuated by sen, ameri and chowdhury in [10] where fuzzy semihypergroups are introduced. the fundamental relations are one of the most important and interesting concepts in fuzzy hyperstructures that ordinary algebraic structures are derived from fuzzy hyperstructures by them. fundamental relation α∗ on fuzzy hypersemigroups is studied in [1].also in [8], the smallest strongly regular equivalence relation γ∗ on a fuzzy hypersemigroup s such that s/γ∗ is a commutative semigroup is studied. in this paper, we introduce and study the fundamental relation ξ∗ of a finite fuzzy hypergroup s such that s/ξ∗ is a solvable group. finally, we introduce the concept of ξ-part of a fuzzy hypergroup and we determines necessary and sufficient conditions such that the relation ξ to be transitive. 2 preliminary recall that for a non-empty set s, a fuzzy subset µ of s is a function from s into the real unite interval [0, 1]. we denote the set of all nonzero fuzzy subsets of s by f∗(s). also for fuzzy subsets µ1 and µ2 of s, then µ1 is smaller than µ2 and write µ1 ≤ µ2 iff for all x ∈ s, we have µ1(x) ≤ µ2(x). define µ1 ∨µ2 and µ1 ∧µ2 as follows: ∀x ∈ s, (µ1 ∨µ2)(x) = max{µ1(x),µ2(x)} and (µ1 ∧µ2)(x) = min{µ1(x),µ2(x)}. a fuzzy hyperoperation on s is a mapping ◦ : s ×s 7→ f∗(s) written as (a,b) 7→ a◦ b = ab. the couple (s,◦) is called a fuzzy hypergropoid. definition 2.1. a fuzzy hypergropoid (s,◦) is called a fuzzy hypersemigroup if for all a,b,c ∈ s, (a◦b) ◦c = a◦ (b◦c), where for any fuzzy subset µ of s (a◦µ)(r) =   ∨ t∈s ((a◦ t)(r) ∧µ(t)), µ 6= 0 0, µ = 0 54 solvable groups derived from fuzzy hypergroups (µ◦a)(r) =   ∨ t∈s (µ(t) ∧ (t◦a)(r)), µ 6= 0 0, µ = 0 for all r ∈ s. definition 2.2. let µ,ν be two fuzzy subsets of a fuzzy hypergropoid (s,◦). then we define µ ◦ ν by (µ ◦ ν)(t) = ∨ p,q∈s (µ(p) ∧ (p ◦ q)(t) ∧ ν(q)), for all t ∈ s. definition 2.3. a fuzzy hypersemigroup (s,◦) is called fuzzy hypergroup if x◦s = s ◦x = χs, for all x ∈ s, where χs is characteristic function of s. example 2.1. consider a fuzzy hyperoperation ◦ on a non-empty set s by a ◦ b = χ{a,b}, for all a,b ∈ s. then (s,◦) is a fuzzy hypersemigroup and fuzzy hypergroup as well. theorem 2.1. let (s,◦) be a fuzzy hypersemigroup. then χa ◦χb = a◦ b, for all a,b ∈ s. definition 2.4. let ρ be an equivalence relation on a fuzzy hypersemigroup (s,◦), we define two relations ρ and ρ on f∗(s) as follows: for µ,ν ∈ f∗(s); µρν if µ(a) > 0 then there exists b ∈ s such that ν(b) > 0 and aρb, also if ν(x) > 0 then there exists y ∈ s, such that µ(y) > 0 and xρy. µρν if for all x ∈ s such that µ(x) > 0 and for all y ∈ s such that ν(y) > 0 , xρy. definition 2.5. an equivalence relation ρ on a fuzzy hypersemigroup (s,◦) is said to be (strongly) fuzzy regular if aρb,a′ρb′ implies a◦a′ ρ b◦b′(a◦a′ ρ b◦b′). if ρ is a equivalence relation on a fuzzy hypersemigroup (s,◦), then we consider the following hyperoperation on the quotient set s/ρ as follows: for every aρ,bρ ∈ s/ρ aρ⊕ bρ = {cρ : (a′ ◦ b′)(c) > 0,aρa′,bρb′} theorem 2.2. [2] let (s,◦) be a fuzzy hypersemigroup and ρ be an equivalence relation on s. then (i) the relation ρ is fuzzy regular on (s,◦) iff (s/ρ,⊕) is a hypersemigroup. (ii) the relation ρ is strongly fuzzy regular on (s,◦) iff (s/ρ,⊕) is a semigroup. 55 e. mohammadzadeh and t. nozari 3 new strongly regular relation ξ∗n now in this paper we introduce and analyze a new strongly regular relation ξ∗n on a fuzzy hypergroup s such that the quotient group s/ξ ∗ n is solvable. definition 3.1. let (s,o) be a fuzzy hypergroup. we define 1) l0(s) = s 2) lk+1(s) = {t ∈ s | (xy)(r) > 0, (tyx)(r) > 0, in which x,y ∈ lk(s), for some r ∈ s}. for all k ≥ 0. suppose that n ∈ n and ξn = ∪m≥1ξm,n, where ξ1,n is the diagonal relation and for every integer m > 1,ξm,n is the relation defined as follows: aξm,nb ⇐⇒ ∃x1, ...,xm ∈ h(m ∈ n),∃σ ∈ sm : σ(i) = i, if zi 6∈ ln(h) : (x1o...oxm)(a) > 0 and (xσ1o...oxσm )(b) > 0. it is clear that ξn is symmetric. define for any a ∈ s, a(a) = (χa)(a) = 1, thus ξn is reflexive. we take ξ ∗ n to be transitive closure of ξn. then it is an equivalence relation on h. corolary 3.1. for every n ∈ n, we have α∗ ⊆ ξ∗n ⊆ γ∗. theorem 3.1. for every n ∈ n, the relation ξ∗n is a strongly regular relation. proof. suppose n ∈ n. clearly, ξm,n is an equivalence relation. first we show that for each x,y,z ∈ s xξny ⇒ xzξnyz, zxξnzy (∗). if xξny, then there exists m ∈ n such that xξm,ny, and so there exist (z1, . . . ,zm) ∈ sm and σ ∈ sm such that if zi 6∈ ln(s) then m∏ i=1 zi(x) > 0, m∏ i=1 zσ(i)(y) > 0. let z ∈ s, for any r,s such that (xz)(r) > 0 and (yz)(s) > 0. we have (( ∏m i=1 zi)z)(r) = ∨ p{( m∏ i=1 zi)(p)∧(pz)(r)}. let p = x, then (( m∏ i=1 zi)(z))r > 0,σ(i) = i, if zi 6∈ ln(s), (( m∏ i=1 zσ(i))(z))(s) = ∨ q {( m∏ i=1 zσ(i))(q) ∧ (qz)(s)}. let q = y, then (( m∏ i=1 zσ(i))(z))(s) > 0, and σ(i) = i, if zi 6∈ ln(s). now suppose that zm+1 = z and we define 56 solvable groups derived from fuzzy hypergroups σ ′ ∈ sm + 1: σ′(i) = { σ(i), ∀i ∈{1, 2, . . . ,m} m + 1, i = m + 1. thus for all r,s ∈ s; ( m∏ i=1 zi)(r) > 0, ( m∏ i=1 z ′ σ)(s) > 0; σ ′ (i) = i if zi 6∈ ln(s). therefore xzξnyz. now if xξ∗ny, then there exists k ∈ n and u0 = x,u1, . . . ,uk = y ∈ s such that u0 = xξnu1ξnu2ξn . . .ξnum = y, by the above result we have u0z = xzξnu1zξnu2zξn . . .ξnukz = yz and so xzξnyz. similarly we can show that zxξnzy. therefore ξ ∗ n is a strongly regular relation on s. 2 proposition 3.1. for every n ∈ n, we have ξ∗n+1 ⊆ ξ∗n. proof. let xξn+1y so ∃(z1, ...,zm) ∈ sm;∃δ ∈ sm : δ(i) = i if zi 6∈ ln+1(s), such that ( m∏ i=1 zi)(x) > 0, ( m∏ i=1 zδ(i))(y) > 0. now let δ1 = δ, since ln+1(s) ⊆ ln(s) so xξny.2 the next result immediately follows from previous theorem. corolary 3.2. if s is a commutative hypergroup, then β∗ = ξ∗n. a group g is solvable if and only if g(n) = {e} for some n ≥ 1 in which, g(0) = g, g(1) = g ′ , commutator subgroup of g, and inductively g(i) = (g(i−1)) ′ . theorem 3.2. if s is a fuzzy hypergroup and ϕ is a strongly regular relation on s, then lk+1(s/ϕ)) = 〈t | t ∈ lk(s)〉 for k ∈ n. proof. suppose that g = s/ϕ and x = ϕ(x) for all x ∈ s. we prove the theorem by induction on k. for k = 0 we have l1(g) = 〈t | t ∈ l0(s)〉. now suppose that a = t where t ∈ lk+1(s) so there exist r1 ∈ s ; (xy)(r1) > 0, (tyx)(r1) > 0 in which x,y ∈ lk(s). then xy = z1; (xy)(z1) > 0 and so xy = r1. also tyx = z2; (tyx)(z2) > 0 and tyx = r1 = xy which implies that t = [x,y]. by hypotheses of induction we conclude that t ∈ lk+1(g). hence a = [t,s] ∈ lk+2(g). conversely, let a ∈ lk+2(g). then a = [x,y], where x,y ∈ lk+1(g), so by hypotheses of induction we have x = u and y = v, where u,v ∈ lk(s). let c ∈ s; (uv)(c) > 0 we show that there exists t ∈ s such that (tvu)(c) > 0. since s ◦u = χs and c ∈ s then there exists r ∈ s such that (ru)(c) > 0 and so by r ∈ s = s ◦v there exist t ∈ s; (tv)(r) > 0. therefore (tvu)(c) = ∨ n((tv)(n)∧(nu)(c)) ≥ (tv)(r)∧(ru)(c) > 0. thus (uv)(c) > 0, (tvu)(c) > 0 which implies that t ∈ lk+1(s). now since 57 e. mohammadzadeh and t. nozari uv = c = tvu, then t = [u,v] = [x,y] = a and t ∈ lk+1(s). therefore, a = t ∈ 〈t; t ∈ lk+1(s)〉.2 theorem 3.3. s/ξ∗n is a solvable group of class at most n + 1. proof. using theorem 3.2, lk(s/ξ ∗ n)is an abelian group and lk+1(s/ξ ∗ n) = {e}. 2 4 on solvable groups derived from finite fuzzy hypergroups in this section we introduce the smallest strongly relation ξ∗ on a finite fuzzy hypergroup s such that h/ξ∗ is a solvable group. definition 4.1. let s be a finite fuzzy hypergroup. then we define the relation ξ∗ on s by ξ∗ = ⋂ n≥1 ξ∗n. theorem 4.1. the relation ξ∗ is a strongly regular relation on a finite fuzzy hypergroup s such that s/ξ∗ is a solvable group. proof. since ξ∗ = ⋂ n≥1 ξ ∗ n, it is easy to see that ξ ∗ is a strongly regular relation on s. by using proposition 3.1, we conclude that there exists k ∈ n such that ξ∗k+1 = ξ ∗ k. thus ξ∗ = ξ ∗ k for some k ∈ n. 2 theorem 4.2. the relation ξ∗ is the smallest strongly regular relation on a finite fuzzy hypergroup s such that s/ξ∗ is a solvable group. proof. suppose ρ is a strongly regular relation on s such that k = s/ρ is a solvable group of class c. suppose that xξy. then xξny, for some n ∈ n and so there exists m ∈ n such that xξmny ⇐⇒ ∃(z1, ..zm) ∈ sm,∃δ ∈ sm : δ(i) = i if zi 6∈ ln(s) such that ( ∏m i=1 zi)(x) > 0, ( ∏m i=1 zδ(i))(y) > 0, lc+1(s/ρ) = 〈ρ(t); t ∈ lc(s)〉 = {ρ(e)}, and so ρ(zi) = ρ(e), for every zi ∈ lc(s). therefore ρ(x) = ρ(y), which implies that xρy.2 58 solvable groups derived from fuzzy hypergroups 5 transitivity of ξ∗ in this section we introduce the concept of ξ-part of a fuzzy hypergroup and we determine necessary and sufficient condition such that the relation ξ to be transitive. definition 5.1. let x be a non-empty subset of s. then we say that x is a ξ-part of s if the following condition holds: for every k ∈ n and (z1, ...,zm) ∈ hm and for every σ ∈ sk such that σ(i) = i if zi 6∈ ∪n≥1ln(s), and there exists x ∈ x such that ( m∏ i=1 zi)(x) > 0, then for all y ∈ s\x, ( m∏ i=1 zσ(i))(y) = 0. theorem 5.1. let x be a non-empty subset of a fuzzy hypergroup s. then the following conditions are equivalent: 1) x is a ξ-part of s, 2) x ∈ x, xξy =⇒ y ∈ x, 3) x ∈ x, xξ∗y =⇒ y ∈ x. proof. (1) =⇒ (2) if (x,y) ∈ s2 is a pair such that x ∈ x and xξy, then there exist (z1, ...,zi) ∈ sk; ( m∏ i=1 zi)(x) > 0, ( m∏ i=1 zσ(i))(y) > 0 and σ(i) = i if zi 6∈ ∪n≥1ln(s). since x is a ξ-part of s, we have y ∈ x. (2) =⇒ (3) suppose that (x,y) ∈ s2 is a part such that x ∈ x and xξ∗y. then there is (z1, ...,zi) ∈ sk such that x = z0ξz1ξ...ξzk = y. now by using (2) k-times we obtain y ∈ x. (3) =⇒ (1) for every k ∈ n and (z1, ...,zi) ∈ sk and for every σ ∈ sk such that σ(i) = i if zi 6∈ ∪n≥1ln(s), then there exists x ∈ x; ( m∏ i=1 zi)(x) > 0 and there exist y ∈ s\x ; ( ∏ i=1 zσ(i))(y) > 0, then xξny and so xξy. therefore by (3) we have y ∈ x which is a contradiction.2 theorem 5.2. the following conditions are equivalent: 1) for every a ∈ h, ξ(a) is a ξ-part of s, 2) ξ is transitive. proof. (1) =⇒ (2) suppose that xξ∗y. then there is (z1, ...,zi) ∈ sk such that x = z0ξz1ξ...ξzk = y, since ξ(zi) for all 0 ≤ i ≤ k, is a ξ-part, we have zi ∈ ξ(zi−1), for all 1 ≤ i ≤ k. thus y ∈ ξ(x), which means that xξy. (2) =⇒ (1) suppose that x ∈ s, z ∈ ξ(x) and zξy. by transitivity of ξ, we have y ∈ ξ(x). now according to the last theorem, ξ(x) is a ξ-part of s.2 59 e. mohammadzadeh and t. nozari definition 5.2. the intersection of all ξ-parts which contain a is called ξ-closure of a in s and it will be denoted by k(a). in what follows, we determine the set w(a), where a is a non-empty subset of s. we set 1) w1(a) = a and 2) wn+1(a) = {x ∈ s | ∃(z1, ...,zi) ∈ sk : ( m∏ i=1 z(i))(x) > 0,∃σ ∈ sk such that σ(i) = i, if zi 6∈ ∪n≥1ln(s) and there exists a ∈ wn(a); ( m∏ i=1 zσ(i))(a) > 0}. we denote w(a) = ⋃ n≥1 wn(a). theorem 5.3. for any non-empty subset of s, the following statements hold: 1) w(a) = k(a), 2) k(a) = ∪a∈ak(a). proof. 1) it is enough to prove: a) w(a) i a ξ-part, b) if a ⊆ b and b is a ξ-part, then w(a) ⊆ b. in order to prove (a), suppose that a ∈ w(a) such that ( ∏ i=1 zi)(a) > 0 and σ ∈ sk such that σ(i) = i, if zi 6∈ ∪n≥1ln(s). therefore, there exists n ∈ n such that ( m∏ i=1 zi)(a) > 0 a ∈ wn(a). now if there exists t ∈ s such that ( ∏ i=1 zσ(i))(t) > 0 we obtain t ∈ wn+1(a). therefore, t ∈ w(a) which is a contradiction. thus ( m∏ i=1 zσ(i))(t) = 0 and so w(a) is a ξ-part. now we prove (b) by induction on n. we have w1(a) = a ⊆ b. suppose that wn(a) ⊆ b. we prove that wn+1(a) ⊆ b. if z ∈ wn+1(a), then there exists k ∈ n; (z1, ...,zk) ∈ sk; ( m∏ i=1 zi)(z) > 0 and there exists σ ∈ sk such that σ(i) = i,if zi 6∈ ∪t≥1lt(s) and there exists t ∈ wn(a) ; ( m∏ i=1 zσi )(t) > 0, since wn(a) ⊆ b we have t ∈ b and ( m∏ i=1 zσi )(t) > 0. now since b is ξ-part , ( m∏ i=1 zi)(z) > 0 then z ∈ b. 60 solvable groups derived from fuzzy hypergroups 2) it is clear that for all a ∈ a, k(a) ⊆ k(a). by part 1), we have k(a) = ∪n≥1wn(a) and w1(a) = a = ∪a∈a{a}. it is enough to prove that wn(a) = ∪a∈awn(a), for all n ∈ n. we follow by induction on n. suppose it is true for n. we prove that wn+1(a) = ∪a∈awn+1(a). if z ∈ wn+1(a), then there exists k ∈ n, (z1, ...,zk) ∈ sk; ( m∏ i=1 zi)z > 0 and there exists σ ∈ sk such that σ(i) = i, if zi 6∈ ∪t≥1lt(s) and there exist a ∈ wn(a); ( m∏ i=1 zσ(i))(a) > 0. by the hypotheses of induction there exists a ∈ wn(a) = ∪b∈awn(b); ( m∏ i=1 zσ(i))(a ′ ) > 0 for some a ′ ∈ wn(b) in which b ∈ a. therefore, z ∈ wn+1(b), and so wn+1(a) ⊆∪b∈awn+1(b). hence k(a) = ∪a∈ak(a).2 theorem 5.4. the following relation is equivalence relation on h. xwy ⇐⇒ x ∈ w(y), for every (x,y) ∈ s2, where w(y) = w({y}). proof. it is easy to see that w is reflexive and transitive. we prove that w is symmetric. to this, we check that: 1) for all n ≥ 2 and x ∈ s, wn(w2(x)) = wn+1(x), 2) x ∈ wn(y) if and only if y ∈ wn(x). we prove (1) by induction on n. w2(w2(x)) = {z | ∃q ∈ n, (a1, ...,aq) ∈ sq; ( ∏ i=1 ai)(z) > 0 and ∃σ ∈ sk such that σ(i) = i, if zi 6∈ ∪s≥1ls(s) and ∃y ∈ w2(x); ( m∏ i=1 aσ(i))(y) > 0} = w3(x). now we proceed by induction on n. suppose wn(w2(x)) = wn+1(x) then wn+1(w2(x)) = {z | ∃q ∈ n, (a1, ...,aq) ∈ sq; ( m∏ i=1 ai)(z) > 0 and ∃σ ∈ sk such that σ(i) = i, if zi 6∈ ∪s≥1ls(s) and ∃t ∈ wn(w2(x)); ( ∏ i=1 aσ(i))(t) > 0} = wn+2(x). now we prove (2) by induction on n, too. it is clear that x ∈ w2(y) if and only if y ∈ w2(x). suppose x ∈ wn(y) if and only if y ∈ wn(x). let x ∈ wn+1(y), then there exists q ∈ n, (a1, ...,aq) ∈ sq; ( m∏ i=1 ai)(x) > 0 and ∃σ ∈ sk such that σ(i) = i, if ai 6∈ 61 e. mohammadzadeh and t. nozari ∪s≥1ls(s) and ∃t ∈ wn(y); ( m∏ i=1 aσ(i))t > 0. now, ( m∏ i=1 ai)(x) > 0, x ∈ w1(x) and ( m∏ i=1 aσ(i))(t) > 0 implies that t ∈ w2(x). since t ∈ wn(y), then by hypotheses of induction y ∈ wn(t) and we see that t ∈ w2(x), therefore y ∈ wn(w2(x)) = wn+1(x). 2 remark 5.1. if s is a fuzzy hypergroup, then s/ξ∗ is a group. we define ωs = φ −1(1s/ξ∗), in which φ : s → s/ξ∗ is the canonical projection. lemma 5.1. if s is a fuzzy hypergroup and m is a non-empty subset of s, then (i) φ−1(φ(m)) = {x ∈ s : (ωsm)(x) > 0} = {x ∈ s : (mωs)(x) > 0} (ii) if m is a ξ part of s, then φ−1(φ(m)) = m. proof. (i) let x ∈ s and (t,y) ∈ ωs × m such that (ty)(x) > 0, so φ(x) = φ(t) ⊕ φ(y) = 1s/ξ∗ ⊕ φ(y) = φ(y), therefore x ∈ φ−1(φ(y)) ⊂ φ−1(φ(m)). conversely, for every x ∈ φ−1(φ(m)), there exists b ∈ m such that φ(x) = φ(b). by reproducibility, a ∈ s exists such that (ab)(x) > 0, so φ(b) = φ(x) = φ(a) ⊕φ(b). this implies φ(a) = 1s/ξ∗ and a ∈ φ−1(1s/ξ∗) = ωs. therefore (ωsm)(x) > 0. in the same way, we can prove that φ−1(φ(m)) = {x ∈ s : (mωs)(x) > 0}. (ii) we know m ⊆ φ−1(φ(m)). if x ∈ φ−1(φ(m)), then there exists b ∈ m such that φ(x) = φ(b). therefore x ∈ ξ∗(x) = ξ∗(b). since m is a ξ part of s and b ∈ m, by lemma 5.1, we conclude ξ∗(b) ⊆ m and x ∈ m. 2 definition 5.3. let (s, ·) be a fuzzy hypergroup. k ⊆ s is called a fuzzy subhypergroup of s if i) (a · b) · c = a · (b · c), for all a,b,c ∈ s ii) a ·k = χk, for all a ∈ k. theorem 5.5. ωs is a fuzzy subhypergroup of s, which is also a ξ-part of s. proof. clearly, ωs ⊆ s and so (a·b)·c = a·(b·c), for all a,b,c ∈ ωs. now we show that ωsy = χωs for all y ∈ ωs. let x,y ∈ ωs, then there exists u ∈ s such that (uy)(x) > 0. therefore, uy = x, which implies that u = 1. thus u ∈ ωs. consequently, ωsy = χωs . hence, ωs is a fuzzy subhypergroup of s. now we prove that k(y) = φ−1(φ({y})) = {x ∈ s : (ωsy)(x) > 0} = ωs. 62 solvable groups derived from fuzzy hypergroups z ∈ φ−1(φ({y})) ⇐⇒ ϕ(z) = ϕ(y) ⇐⇒ ξ∗(z) = ξ∗(y) ⇐⇒ zξ∗y ⇐⇒ z ∈ ξ∗(z) = ω({y}) = k(y). also since y ∈ ωs, then {x ∈ s : (ωsy)(x) > 0} = {x ∈ s : (χωs )(x) > 0} = ωs. therefore k(y) = ωs and so ωs is ξ part. 2 acknowledgment the first outer was supported by a grant from payame noor university. 63 e. mohammadzadeh and t. nozari references [1] r. ameri, t. nozari, complete parts and fundamental relation on fuzzy hypersemigroups, j. of mult.-valued logic. soft computing, vol. 19 (2011) 451-460. 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[12] m. m. zahedi, m. bolurian, a. hasankhani, on polygroups and fuzzy subpolygroups, j. fuzzy math, 3 (1995) 1-15. 64 ratio mathematica volume 46, 2023 anti-homomorphism in q-fuzzy subgroups and normal subgroups r.jahir hussain* s.palaniyandi† abstract the fuzzy set has been applied in wide area by many researchers. we define the concept of anti-homomorphism in q-fuzzy subgroups and q-fuzzy normal subgroups and establish some result in this research article and develop some theory of antihomomorphism in q-fuzzy subgroups, normal subgroups and also extend results on qfuzzy abelian subgroup and qfuzzy normal subgroup. many researchers have explored the fuzzy set extensively. we propose the notion of anti-homomorphism in q is fuzzy subgroups and normal subgroups. it is establish some findings in this study article and build the theory of anti-homomorphism in q-fuzzy subgroups, normal subgroups. it is also extend results on q-fuzzy abelian subgroup and q-fuzzy normal subgroup. keywords: fuzzy, subgroup, q-fuzzy, fuzzy abelian, fuzzy normal subgroup, anti-homomorphism,. ams subject classification: 03e72, 03e75, 08a72 1 *pg and research department of mathematics, jamal mohamed college (autonomous), affiliated to bharathidasan university, tiruchirappalli, tamilnadu, india.; hssn jhr@yahoo.com. †pg and research department of mathematics, jamal mohamed college (autonomous), affiliated to bharathidasan university, tiruchirappalli, tamilnadu, india.; palanijmc85@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1073. issn: 1592-7415. eissn: 2282-8214. ©r.jahir hussain et al. this paper is published under the cc-by licence agreement. 172 r.jahir hussain and s.palaniyandi 1 introduction zadeh l.a. zadeh [1965] introduced the fuzzy set concept. numerous scholars have used the fuzzy set in several different contexts. fuzzy subgroups are first discussed by rosenfeld rosenfeld [1971]. biswas.r biswas [1990] was introduced, the anti-fuzzy subgroups. the novel structure of q-fuzzy subgroups was introduced by solairaju.a and nagarajan.r solairaju and nagarajan [2009]. fuzzy subgroups and fuzzy homomorphisms were defined by choudhury, f.p., chakraborty, a. b, and khare choudhury et al. [1988] sheik anti-homomorphism in fuzzy subgroups was defined by abdullah a. and jeyaraman k. sheik abdullah and jeyaraman [2010]. in this study, we demonstrate various results and define the notion of anti-homomorphism in q-fuzzy subgroups and fuzzy normal subgroups. 2 preliminaries definition 2.1. zadeh [1965] a function of fuzzy subset δ 6= s is δ : s → [0,1]. definition 2.2. rosenfeld [1971] a fuzzy subset δ of a group j = j (fuzzy subgroup) if it is satisfying the following conditions, (i) δ (%) ≥ min{δ (%) , δ (γ)} (ii) δ (%−1) = δ(%), ∀ %, γ ∈ j. definition 2.3. solairaju and nagarajan [2009] a q-fuzzy set δ = j if ∀ %, gammaf ∈ j, and κ ∈ q (i) δ (%γ, κ) ≥ min{δ (%, κ) , δ (γ, κ)} (ii) δ (%−1, κ) = δ (%, κ) definition 2.4. zadeh [1965] ν ⊆ s (fuzzy subset of a set) . for β ∈ [0,1], the level subset of δ is defined by δβ = {e ∈ s : δν(%) ≥ β definition 2.5. solairaju and nagarajan [2009] ν ⊆ s. for β ∈ [0,1], the set δβ = {e ∈ s, κ ∈ q : δν(%, κ) ≥ β} is called a q ⊆ δ. definition 2.6. palaniappan and muthuraj [2004] consider δ < j. the fuzzy subgroup δ is said to be fuzzy normal subgroup if δ (%γ) = δ (fe), ∀ %, γ ∈ j. definition 2.7. palaniappan and muthuraj [2004] a fuzzy subgroup δ of a group j is a q-fuzzy normal subgroup if δ (%γ, κ) = δ (γ%, κ), ∀ varrho γ ∈ j, and κ ∈ q. 173 anti-homomorphism in q-fuzzy subgroups and normal subgroups definition 2.8. choudhury et al. [1988] let (j1, •) and j2, •) be the function g : j1 → j2 is called a group homomorphism if g (%γ) = g (%) .g(γ), ∀%, γ ∈ j1. definition 2.9. sheik abdullah and jeyaraman [2010] let (j1, •) and j2, •) be the function g : j1 → j2 is called a group anti homomorphism if g (%γ) = g (γ) .g (%), ∀ %, γ ∈ j1. definition 2.10. sheik abdullah and jeyaraman [2010] let g : j1 → j2 is called anti automorphism if g (%γ) = g (f) .g(%) ∀ %, γ ∈ j1. definition 2.11. sheik abdullah and jeyaraman [2010] the function δ is a fuzzy characteristic subgroup of a group j if δ (h(%)) = δ(%). 3 some results on q -fuzzy subgroups in antihomomorphism theorem 3.1. let g : j → j∗ be an anti-homomorphism, if δ∗ is a q<j∗. then g−1(δ∗) is a q < j. proof. let e, γ ∈ j. then g−1 (δ∗) (%γ, κ) = δ∗(h(%γ, κ)) = δ∗{g (γ, κ) .g (%, κ)} ≥ min{δ∗ (h(γ, κ)) , δ∗(h(%, κ))} = min{(h−1δ ∗ )(γ, κ), (h −1 δ ∗ )(%, κ)} (1) and g−1 (δ∗) ( %−1, κ ) = δ∗(h ( %−1, κ ) ) = δ∗(h(%, κ)) = g−1 (δ∗) (%, κ) (2) from (1) and (2), g−1(δ∗) is a q < j. theorem 3.2. if δ is a q <j and g : s → s∗ is an anti-homomorphism, then g−1(δ) is a q-fuzzy normal subgroup of s∗. proof. for every %, γ ∈ s. we get, g−1 (δ) (%, κ) = δ(h(%, κ) = δ{g (γ, κ) , g (%, κ)} = δ(h(γ%, κ) = g−1 (δ) (γ%, κ) hence g−1(δ) is a q <j. 174 r.jahir hussain and s.palaniyandi theorem 3.3. a fuzzy characteristic subgroup of a q < q. it is a fuzzy normal subgroup. proof. given g is an anti automorphism of s. for all %, γ ∈ s, and κ ∈ q. then g (%γ) = g (γ) .g(%), for every e, γ ∈ s now, δ (%γ, κ) = δ(h(%γ, κ) = δ{g (f, κ) ,g (%, κ)} since δ is a characteristics q <s. then δ (%γ, κ) = δ{g (γ, κ) ,g (%, κ)} since g is anti automorphism of s, δ (%γ, κ) = δ(h(fe, κ)) = δ(γ%, κ), for all %, γ ∈ s, and κ ∈ q δ is a characteristics q <s. hence δ is a q <s. definition 3.1. the q-fuzzy subgroup δ of a group s is called a q -fuzzy abelian subgroup of s if h = {% ∈ s : δ (%, κ) = δ (i, κ)}, ∀ %, γ ∈ s, and κ ∈ q. theorem 3.4. the commutative property satisfies all anti-homomorphism preimages of a q-fuzzy commutative subgroup. proof. if δ is a q <s. to prove δ is a q=[s, s] let us consider ν is a q = [s∗, s∗] (q fuzzy commutative subgroup of s∗). since δ is a q = [s∗, s∗] and ν is q = [s∗, s∗] then v = {γ ∈ s∗, κ ∈ q : ν (γ, κ) = ν (i∗,κ)} is a q = [s∗, s∗], where i∗ is the identify element of s∗. let t = {% ∈ s, κ ∈ q : δ (%, κ) = δ (i, κ)} where i is the identify element of s. take %, γ ∈ t this implies %γ ∈ t ⊆ s. then δ (%γ, κ) = δ(i, κ) ν (h(%γ, κ)) = ν(h(i,κ)) = ν (i∗,κ) ν{g (γ, κ) .g (%, κ)} = ν (i∗,κ) 175 anti-homomorphism in q-fuzzy subgroups and normal subgroups since g (γ, κ) .g (%, κ) ∈ v and v is abelian, g (f, κ) .g (%, κ) = g (%, κ) .g (γ, κ) ν(h(γ, κ) .g (%, κ)) = ν(h(%, κ) .g (γ, κ)) ν (γ (γ%, κ)) = ν(γ (%γ, κ)), since g is anti−homomorphism. δ (%γ, κ) = δ (γ%, κ) δ (i, κ) = δ (γ%, κ) i.e (γ%, κ) = δ (i, κ) , this implies fe ∈ t and κ ∈ q. for all %, γ ∈ t , %γ ∈ t and γ% ∈ t . this implies %γ = γ% t satisfies commutative. therefore δ satisfies commutative property. theorem 3.5. anti-homomorphism image of a q -fuzzy commutative subgroup is also satisfies commutative level. proof. let ν be a q -fuzzy subgroup of s∗. to prove: ν is a q -fuzzy commutative subgroup of s∗. let g be an anti-homomorphism from s to s∗. since δ is a q=[s, s]. then t = {% ins, κ ∈ q : δ (%, κ) = δ (i, κ)} is an commutative q -fuzzy subgroup of s∗ where i is the identity element of s. let ν be the q fuzzy subgroup of s∗. let v = {x ∈ s∗, κ ∈ q : ν (%, κ) = δ (i∗, κ)} where i∗is the identity element s∗. let e, γ in v ⊆ s∗ ν (%γ, κ) = ν(i∗, κ) sup δ (r, κ) = sup δ (r, κ) r ∈ g−1(%γ) , r ∈ g−1(i∗) δ (%γ, κ) = δ(i, κ) then %γ ∈ t and t is an commutative q fuzzy subgroup. (%γ,κ) = (γ%, κ) δ (%γ,κ) = δ(γ%, κ) sup δ (r, κ) = sup δ (r, κ) r ∈ g−1(%γ) , r ∈ g−1(γ%) ν (%γ,κ) = ν(γ%, κ) ν (i∗,κ) = ν(γ%, κ) 176 r.jahir hussain and s.palaniyandi that is ν (%γ,κ) = ν (i∗,κ), this implies γ% ∈ v and κ ∈ q. for all %, γ ∈ v , %γ ∈ v this implies γ% ∈ v and %γ = γ%. then v = [s∗, s∗] (commutative subgroup of s∗). therefore ν = [s∗, s∗]. where q fuzzy commutative subgroup of s∗. 4 conclusions many results can be found from the research article. but, in this paper we found few concepts of anti-homomorphism in q-fuzzy subgroups. further this paper used to developing the concept of q-fuzzy abelian subgroup. there are so many concepts can be availed by future research work. references r. biswas. fuzzy subgroups and anti fuzzy subgroups. fuzzy sets and systems, 35(1):121–124, 1990. f. choudhury, a. chakraborty, and s. khare. a note on fuzzy subgroups and fuzzy homomorphism. j. math. anal. appl, 131(2):537–553, 1988. n. palaniappan and r. muthuraj. anti fuzzy group and lower level subgroups. antartica j.math, 1(1):71–76, 2004. a. rosenfeld. fuzzy groups. j. math. anal.appl., 35(3):512–517, 1971. a. sheik abdullah and k. jeyaraman. antihomomorphism in fuzzy subgroups. international journal of computer applications, 12(8):0975–8887, 2010. a. solairaju and r. nagarajan. a new structure and construction of q-fuzzy groups. advances in fuzzy mathematics, 4(1):23–29, 2009. l. zadeh. fuzzy setl. information and control, 8(3):338–353, 1965. 177 ratio mathematica volume 46, 2023 robust fuzzy graph a.sudha* p. sundararajan† abstract the idea of fuzzy graphs is crucial for dealing with uncertainty in day-to-day living situations. in this work, we have a present a work fuzzy graph associates with robustness model that’s easy, flexible, and simple. with this model, we have a tendency to address the examination planning downside below uncertainty supported. randomness and unclearness are two very separate types of uncertainty in knowledge. this study bridges resilient fuzzy graphs and addresses every type of uncertainty in higher cognitive processes. in this paper we introduce a type of fuzzy graph relating with robustness is called robust fuzzy graph and some of its properties. robust fuzzy graph becomes a best tool in evidence theory for calculating belief functions, plausibility functions, spanning functions etc. keywords: robust; fuzzy graph; belief measure; plausibility measure; spanning. 2020 ams subject classifications: 93b51, 03e72, 15a09, 58j52. 1 *the kavery arts and science college, mecheri, salem, tamilnadu, india. vikaashhkanishka@gmail.com. †department of mathematics, arignar anna govt. arts college, namakkal. tamilnadu, india. ponsundar03@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1089. issn: 1592-7415. eissn: 2282-8214. ©a.sudha et al. this paper is published under the cc-by licence agreement. 319 a.sudha and p.sundararajan 1 introduction graphs are commonly thought to be nothing more than relational representations. a graph can be used to express information about item connections. edges represent connections, whereas vertices represent things. it is common to need to develop a ”fuzzy graph model” when the description of an object, its relationships, or both are imprecise. particularly in the fields of cluster analysis, neural networks, computer networks, pattern recognition, decision making, and professional systems, the use of fuzzy connections is pervasive and essential. every one of them has a fuzzy graph as its fundamental mathematical structure. a graph is a symmetric binary relation on a nonempty set v, as we are well aware. a fuzzy graph is a symmetric binary fuzzy relation on a fuzzy subset. in 1973, a.kaufmann [1965] initially proposed the idea of a fuzzy graph, which was based on zadeh’s fuzzy relations zadeh [1980], zadeh [1968]. goguen is recognised with creating the idea of fuzzy graphs and taking into account fuzzy relations on fuzzy sets in 1968, however. ravi.j et.al.,ravi [2022] discovered several connection notions in fuzzy graphs at the same time. there are few high-quality papers on fuzzy graphs. according to our data, a strength-associated degreed fuzzy graph and a degreed fuzzy graph have never been combined in a programming task. the works m. blue and puckett [2002] and d.e.goldberg [1989], address programming flaws a.lim and wang [2005], sunitha [2001], sunitha and vijayakumar [2002],wang and xu [2013]and y´a˜nez and ramırez [2003] contain works on fuzzy graph colouring.the structure of the paper is presented below. the section 2 briefly reviews a few essential definitions of fuzzy graphs. section 3 describes the resilient fuzzy graph model and section 4 concludes our discussion. 2 preliminaries definition 2.1. the two functions: q → [0,1] and : q x q → [0,1] and a nonempty set v. such that the fuzzy graph g = (q) is formed for each x, y in v by ρ(.) ≤ µ(.) λ µ(.). fuzzy edge set and fuzzy vertex set for g are represented by and, respectively. definition 2.2. let g = (µ,ρ) to have ρ* = (u, v, w) and x. let (u,v) = 0.2, (v, w) = 0.2, (w, x) = 0.3, (x, u) = 0.5, and (u, w) = 0.4.. due to the presence of the two weakest arcs in g, arcs (u, v) and (v, w), c1 = u, v, w, x, u is a fuzzy cycle, but c2 = u, w, x, u is not.. definition 2.3. assume that the graph g=(µ,ρ) has more fuzziness. the word 320 robust fuzzy graph ”connectedness strength” refers to the total strength of all paths linking two vertices, x and y, and is denoted by the sign conng (.). the strongest x-y route is one in which p = conng (.).. definition 2.4. this is a definition of a mathematical objectif conng(.) > 0 for each pair of (.) in ρ*, an f-graph g = (.) is connected. when g is not present, components are maximally connected fuzzy graphs. if g is linked, any two vertices can be connected by a route. the arc (x,y) of a fuzzy graph g is considered normal if and only if (.) = conng(.) >0. definition 2.5. based on judgments of plausibility and belief, the evidence hypothesis is developed. the function bel.: (.) → [0,1] represents a belief measure for a finite universal set x such that bel(.) =0, bel(.) =1, and bel(.)j bel(.j) jk bel(.,.) +..+ (-1)n+1j bel(.) ⊆ x.. according to the data that is now available, bel understands for each a(.) the degree of confidence that a certain member of (.) belongs to the set a. belief measures are superadditive when (.) is infinite and continuous above.. definition 2.6. a plausibility measure is a function pi : (.) → [0,1] such that pi (.) =0, pi (.) =1, and pi (.) ≤ ∑ (. ) – ∑ < (.∪. )+..+ (-1)n+1 ∑ (.) ⊆ x. definition 2.7. a function m:(.)→[0,1] that has the properties m(.)=0 and m(.)=1 characterises belief and plausibility measurements. the term ”basic probability assignment” refers to this function.. the value m(.) for each a indicates the strength of the argument that a specific member of (.), whose categorization in terms of relevant attributes is lacking, belongs to the set a(.).. 3 robust fuzzy graph (rfg) the kth regular super fuzzy matrix is examined in this section. definition 3.1. let {x} and a q6= p(.) and two functions m: q→ [0,1] and : q x q→ [0,1] a,b∈ q, (.) ∈ δ , whenever a ⊆b and (.) =m(.) m are present, constitute a resilient fuzzy graph (.). also ∑ .∈q (.)=1. rfg is represented by the formula g=(.) where m is referred to as the assignment function and is referred to as the edge function. definition 3.2. if n(.) m(.) for all a and (.) (.) for all (.) such that the fg, h=(.) is referred to as a partial rfg of g=(. ). if p v, n(.) = m(.), (.) = (.) for any (.) such that the fg, h=(.) is referred to as a rfg of g = (.) induced by p. 321 a.sudha and p.sundararajan definition 3.3. the vertex in a rfg which is adjacent from every other vertex is called complete vertex. definition 3.4. consider the robust fuzzy matrix mg=(.) of g= (.), where mab= (.,.) (.)⊆ (.) 0 h. the matrix mg k such that mg k =mg k+1 , where k is a positive integer is called the evidence reachability matrix of g denoted by rg=(rab). definition 3.5. two vertices a and b of a rfg is said to be mutually disconnected if there is neither an edge (.,.) nor (.,.). the vertices a and b are mutually disconnected. the belief and plausibility. the measures of a complete vertex are always one. theorem 3.1. fuzzy graphs (fg) include robust fuzzy graphs (rfg). proof. proof is evident from the definition of rfg. theorem 3.2. number of vertices of a rfg corresponding to a crispest x with n elements is, 2n-1. proof. let g = (m,δ)be the rfg. then by the definition, v= ()\ψ is the vertex set and vertices is 2n -1. theorem 3.3. number of edges of a robust fuzzy graph corresponding to a crisp set with n elements is ( ∑−1( − 1) )−1+(∑−2( − 2) )−2+ + n. proof. proof is the consequence of set theory. let us start with singleton sets. every vertex corresponding to singleton sets is adjacent to all the vertices corresponding to their supersets 2-element sets, 3element sets etc. {.}→{l,m},{l,m,n},{l,m,n,o}........ this can be done in ∑−1( − 1) = ncn-1 so the total cases corresponding to singleton sets is ( ∑−1( − 1) )−1 the vertices corresponding to 2-element sets is adjacent to all vertices corresponding to all vertices corresponding to their super sets 3-element sets, 4-element sets etc. {.}→{l,m},{l,m,n},{l,m,n,o}......... 2 theorem 3.4. robust fuzzy graph is complete. proof. a complete fuzzy graph (cfg) is a fg, g=(.) such that δ(.)=m(.)λ m(.) for all (.,.). so by definition every rfg is complete.2 theorem 3.5. there does not exist an edge (.), such that δ (.) =1 in a rfg, g=(.). 322 robust fuzzy graph proof. if possible, suppose that there exist an edge (.) such that δ (.) =1 ⇒ m(.)=m(.)=1 by the definition of rfg. ⇒ ∑ (. ) 6=1, a contradiction.2 theorem 3.6. the partial rfs and rfs of an rfg need not be a rfg proof. for a partial rfg and rfg, ∑ ( ) equal to 1. but ∑ ( ) ≤ 1.2 theorem 3.7. the partial rfs of an rfg is an rfg if and only if m(.) = n(.) for all a and τ(.,.) = δ (.,.) for all (.). proof. by definition ∑ ( ) = 1. let h=(.) be a partial rfs of g. for a partial rfs, h=(n, τ) of g=(.), n(.) ≤ m(.) for all a and τ(.,.) ≤ δ (.,.). but ∑ (. ) 6= 1 if n(.) < m(.).so n(.) = m(.) for all a which implies τ(.,.) = δ (.,.) for all (.,.). converse is obvious.2 theorem 3.8. maximum length of a path p in a rfg with n vertices is n-1. proof. consider a rfg with n vertices a0,a1,.an. start from an arbitrary vertex ai. since there are only n-1 vertices remaining, choose a vertex aj such that ρ(ai,aj) >0, ai⊆aj, i 6= .similarly choose a vertex ar from the remaining n2 such that ρ(aj,ar) >0, aj⊆ar, and so on. since there are only n distinct vertices the process must terminate at a vertex ap such that, ρ(ap-1,ap) >0, ap-1⊆ap, p< n. we get the sequence ai,aj,—-ap which is of length less than n and equal to n-1 only if every vertex is ordered by the relation ⊆. 2 theorem 3.9. rfg does not contain cycles and so fuzzy cycles. proof. since in a rfg g = (.), for all (.,) ∈ v, (.,.) ∈ δ whenever (.)⊆ (.) there will not be an edge (.,.) and so a cycle.2 theorem 3.10. the rfg is always disconnected. proof. for x= {.,.} there does not exist a path between x and y.2 4 conclusions in this paper we introduce new type of fuzzy graph called robust fuzzy graph. we find some of its properties like completeness, paths, connectivity etc. we also present fuzzy graph’s application in robust theory for finding belief measure 323 a.sudha and p.sundararajan plausibility measure etc in uncertain situations. we can calculate belief measure using rfg as bel(.) = m(.) + ∑ (.), (., .) is an edge; where m(.) is the degree of the vertex (.) in rfg. our proposed method rfg perform is well even it is uncertainty situation. in future this concept is applying in computer vision concept. references a.kaufmann. introduction a la theorie des sous-ensembles flous. masson, paris. zadeh, loas angles, 1965. a.lim and f. wang. robust graph coloring for uncertain supply chain management. in: proceedings of the 38th annual hawaii international conference on system sciences, 33(3):211–221, 2005. e. a. d.e.goldberg. genetic algorithms in search, optimization, and machine learning. addison-wesley, reading, 412:678–687, 1989. b. b. m. blue and j. puckett. unified approach to fuzzy graph problems. fuzzy sets and systems, 125(3):355–368, 2002. j. ravi. fuzzy graph and their applications: a review. international journal for science and advance research in technology, 8(1):107–111, 2022. m. sunitha. studies on fuzzy graphs. ph.d. thesis, cochin university of science and technology, i:1–118, 2001. m. sunitha and a. vijayakumar. complement of a fuzzy graph. indian journal of pure and applied mathematics, 33(9):1451–1464, 2002. f. wang and z. xu. metaheuristics for robust graph coloring. journal of heuristics, 19(4):529–548, 2013. j. y´a˜nez and j. ramırez. the robust coloring problem. european journal of operational research, 148(3):546–558, 2003. l. zadeh. probability measures of fuzzy events. journal of mathematical analysis and applications, 23(2):421–441, 1968. l. zadeh. fuzzy sets versus probability. proceedings of the ieee, 68(3):421–441, 1980. 324 ratio mathematica 22 (2012) 3-12 issn:1592-7415 about a definition of metric over an abelian linearly ordered group bice cavallo, livia d’apuzzo university federico ii, naples, italy bice.cavallo@unina.it, liviadap@unina.it abstract a g-metric over an abelian linearly ordered group g = (g,⊙,≤) is a binary operation, dg, verifying suitable properties. we consider a particular g metric derived by the group operation ⊙ and the total weak order ≤, and show that it provides a base for the order topology associated to g. key words: g-metric, abelian linearly ordered group, multi-criteria decision making. 2000 ams subject classifications: 06f20, 06a05, 90b50. 1 introduction the object of the investigation in our previous papers have been the pairwise comparison matrices that, in a multicriteria decision making context, are a helpful tool to determine a weighted ranking on a set x of alternatives or criteria [1], [2], [3]. the pairwise comparison matrices play a basic role in the analytic hierarchy process (ahp), a procedure developed by t.l. saaty [17], [18], [19]. in [14], the authors propose an application of the ahp for reaching consensus in multiagent decision making problems; other consensus models are proposed in [6], [11], [15], [16]. the entry aij of a pairwise comparison matrix a = (aij) can assume different meanings: aij can be a preference ratio (multiplicative case) or a preference difference (additive case) or aij is a preference degree in [0, 1] (fuzzy case). in order to unify the different approaches and remove some drawbacks linked to the measure scale and a lack of an algebraic structure, 3 b. cavallo and l. d’apuzzo in [7] we consider pairwise comparison matrices over abelian linearly ordered groups (alo-groups). furthermore, we introduce a more general notion of metric over an alo-group g = (g,⊙,≤), that we call g-metric; it is a binary operation on g d : (a, b) ∈ g2 → d(a, b) ∈ g, verifying suitable conditions, in particular: a = b if and only if the value of d(a, b) coincides with the identity of g. in [7], [8], [9], [10] we consider a particular g-metric, based upon the group operation ⊙ and the total order ≤. this metric allows us to provide, for pairwise comparison matrices over a divisible alo-group, a consistency index that has a natural meaning and it is easy to compute in the additive and multiplicative cases. in this paper, we focus on a particular g-metric introduced in [7] looking for a topology over the alo-group in which the g-metric is defined. by introducing the notion of dg-neighborhood of an element in an alo-group g = (g,⊙,≤), we show that the above g-metric generates the order topology that is naturally induced in g by the total weak order ≤. 2 abelian linearly ordered groups let g be a non empty set, ⊙ : g × g → g a binary operation on g, ≤ a total weak order on g. then g = (g,⊙,≤) is an alo-group, if and only if (g,⊙) is an abelian group and a ≤ b ⇒ a⊙ c ≤ b⊙ c. (1) as an abelian group satisfies the cancellative law, that is a⊙c = b⊙c ⇔ a = b, (1) is equivalent to the strict monotonicity of ⊙ in each variable: a < b ⇔ a⊙ c < b⊙ c. (2) let g = (g,⊙,≤) be an alo-group. then, we will denote with: • e the identity of g; • x(−1) the symmetric of x ∈ g with respect to ⊙; • ÷ the inverse operation of ⊙ defined by a÷ b = a⊙ b(−1), • x(n), with n ∈ n0, the (n)-power of x ∈ g: x(n) = { e, if n = 0 x(n−1) ⊙x, if n ≥ 1; 4 about a definition of metric over an abelian linearly ordered group • < the strict simple order defined by x < y ⇔ x ≤ y and x ̸= y; • ≥ and > the opposite relations of ≤ and < respectively. then b(−1) = e÷ b, (a⊙ b)(−1) = a(−1) ⊙ b(−1), (a÷ b)(−1) = b÷a; (3) moreover, assuming that g is no trivial, that is g ̸= {e}, by (2) we get a < e ⇔ a(−1) > e, a < b ⇔ a(−1) > b(−1) (4) a⊙a > a ∀a > e, a⊙a < a ∀a < e. (5) by definition, an alo-group g is a lattice ordered group [4], that is there exists a ∨ b = max{a, b}, for each pair (a, b) ∈ g2. nevertheless, by (5), we get the following proposition. proposition 2.1. a no trivial alo-group g = (g,⊙,≤) has neither the greatest element nor the least element. order topology. if g = (g,⊙,≤) is an alo-group, then g is naturally equipped with the order topology induced by ≤ that we will denote with τg. an open set in τg is union of the following open intervals: • ]a, b[= {x ∈ g : a < x < b}; • ] ←, a[= {x ∈ g : x < a}; • ]b,→ [= {x ∈ g : x > b}; and a neighborhood of c ∈ g is an open set to which c belongs. then g×g is equipped with the related product topology. we say that g is a continuous alo-group if and only if ⊙ is continuous. isomorphisms between alogroups an isomorphism between two alogroups g = (g,⊙,≤) and g′ = (g′,◦,≤) is a bijection h : g → g′ that is both a lattice isomorphism and a group isomorphism, that is: x < y ⇔ h(x) < h(y) and h(x⊙y) = h(x)◦h(y). (6) thus, h(e) = e′, where e′ is the identity in g′, and h(x(−1)) = (h(x))(−1). (7) 5 b. cavallo and l. d’apuzzo by applying the inverse isomorphism h−1 : g′ → g, we get: h−1(x′ ◦y′) = h−1(x′)⊙h−1(y′), h−1(x′ (−1) ) = (h−1(x′))(−1). (8) by the associativity of the operations ⊙ and ◦, the equality in (6) can be extended by induction to the n-operation ⊙n i=1 xi, so that h( n⊙ i=1 xi) = ⃝ni=1h(xi), h(x (n)) = h(x)(n). (9) 3 g-metric following [5], we give the following definition of norm: definition 3.1. let g = (g,⊙,≤) be an alo-group. then, the function: || · || : a ∈ g →||a|| = a∨a(−1) ∈ g (10) is a g-norm, or a norm on g. the norm ||a|| of a ∈ g is also called absolute value of a in [4]. proposition 3.1. [7] the g-norm satisfies the properties: 1. ||a|| = ||a(−1)||; 2. a ≤ ||a||; 3. ||a|| ≥ e; 4. ||a|| = e ⇔ a = e; 5. ||a(n)|| = ||a||(n); 6. ||a⊙ b|| ≤ ||a||⊙ ||b||. (triangle inequality) definition 3.2. let g = (g,⊙,≤) be an alo-group. then, the operation d : (a, b) ∈ g2 → d(a, b) ∈ g is a g-metric or g-distance if and only if: 1. d(a, b) ≥ e; 2. d(a, b) = e ⇔ a = b; 6 about a definition of metric over an abelian linearly ordered group 3. d(a, b) = d(b, a); 4. d(a, b) ≤ d(a, c)⊙d(b, c). proposition 3.2. [7] let g = (g,⊙,≤) be an alo-group. then, the operation dg : (a, b) ∈ g2 → dg(a, b) = ||a÷ b|| ∈ g (11) is a g-distance. proposition 3.3. [7] let g = (g,⊙,≤) and g′ = (g′,◦,≤) be alo-groups, h : g → g′ an isomorphism between g and g′. then, for each choice of a, b ∈ g : dg′(h(a), h(b)) = h(dg(a, b)). (12) corolary 3.1. let h : g → g′ be an isomorphism between the alo-groups g = (g,⊙,≤) and g′ = (g′,◦,≤). if a′ = h(a), b′ = h(b), r′ = h(r) ∈ g′, then r > e ⇔ r′ > e′ and dg′(a ′, b′) < r′ ⇔ dg(a, b) < r. 4 examples of continuous alo-groups over a real interval an alo-group g = (g,⊙,≤) is a real alo-group if and only if g is a subset of the real line r and ≤ is the total order on g inherited from the usual order on r. if g is a proper interval of r then, by proposition 2.1, it is an open interval. examples of real divisible continuous alo-groups are the following (see [8] [9]): additive alo-group r = (r, +,≤), where + is the usual addition on r. then, e = 0 and for a, b ∈ r and n ∈ n: a(−1) = −a, a÷ b = a− b, a(n) = na. the norm ||a|| = |a| = a∨ (−a) generates the usual distance over r: dr(a, b) = |a− b| = (a− b)∨ (b−a). 7 b. cavallo and l. d’apuzzo multiplicative alo-group ]0,+∞[ = (]0, +∞[, ·,≤), where · is the usual multiplication on r. then, e = 1 and for a, b ∈]0, +∞[ and n ∈ n: a(−1) = 1/a, a÷ b = a b , a(n) = an. the norm ||a|| = |a| = a∨a−1 generates the following ]0,+∞[ distance d]0,+∞[(a, b) = a b ∨ b a . fuzzy alo-group ]0,1[= (]0, 1[,⊗,≤), where ⊗ is the binary operation in ]0, 1[: ⊗ : (a, b) ∈]0, 1[×]0, 1[ 7→ ab ab + (1−a)(1− b) ∈]0, 1[, (13) then, 0.5 is the identity element, 1 − a is the inverse of a ∈]0, 1[, a÷ b = a(1−b) a(1−b)+(1−a)b, a (0) = 0.5, a(n) = an an + (1−a)n ∀n ∈ n (14) and d]0,1[(a, b) = a(1− b) a(1− b) + (1−a)b ∨ b(1−a) b(1−a) + (1− b)a = a(1− b)∨ b(1−a) a(1− b) + b(1−a) . (15) remark 4.1. by proposition 2.1, the closed unit interval [0, 1] can not be structured as an alo-group; thus, in [7], the authors propose ⊗ as a suitable binary operation on ]0, 1[, satisfying the following requirements: 0.5 is the identity element with respect to ⊗; 1 − a is the inverse of a ∈]0, 1[ with respect to ⊗; (]0, 1[,⊗,≤) is an alo-group. the operation ⊗ is the restriction to ]0, 1[×]0, 1[ of the uninorm: u(a, b) = { 0, (a, b) ∈{(0, 1), (1, 0)}; ab ab+(1−a)(1−b), otherwise. the uninorms have been introduced in [12] as a generalization of t-norm and t-conorm [13] and are commutative and associative operations on [0, 1], verifying the monotonicity property (1). 8 about a definition of metric over an abelian linearly ordered group 5 dgneighborhoods and order topology in this section g = (g,⊙,≤) is an alo-group and dg the g-distance in (11). definition 5.1. let c, r ∈ g and r > e; then the dg-neighborhood of c with radius r is the set: ndg(c; r) = {x ∈ g : dg(x, c) < r}. (16) of course c ∈ ndg(c; r) for each r > e. then, ndg(c) will denote a dgneighborhood of c and ndg the set of the all dg-neighborhoods of the elements of g. proposition 5.1. let c, r ∈ g and r > e; then: ndg(c; r) =]c÷ r, c⊙ r[ proof. by properties (2), (3), (4) we get c ÷ r = c ⊙ r(−1) < c < c ⊙ r and: x ∈ ndg(c; r) ⇕  e ≤ x÷ c < r or e < c÷x < r ⇕  e ≤ x÷ c < r or r(−1) < x÷ c < e ⇕  c ≤ x < c⊙ r or c÷ r < x < c ⇕ x ∈]c÷ r, c⊙ r[. 2 9 b. cavallo and l. d’apuzzo proposition 5.2. let h : g → g′ be an isomorphism between the alo-group g = (g,⊙,≤) and the alo-group g′ = (g′,◦,≤). then, for each choice of c, r ∈ g and c′, r′ ∈ g such that c′ = h(c), r > e and r′ = h(r), the following equality holds: ndg′ (c ′; r′) = h(ndg(c; r)). (17) proof. by proposition 3.3 and corollary 3.1. 2 example 5.1. the neighborhoods related to the examples in section 4 are the following: • in the additive alo-group r = (r, +,≤), the neighborhood of c with radius r is the open interval ]c− r, c + r[; • in the multiplicative alo-group ]0,+∞[ = (]0, +∞[, ·,≤), the neighborhood of c with radius r is the interval ]c r , c · r[; • in the fuzzy alo-group ]0,1[= (]0, 1[,⊗,≤), the neighborhood of c with radius r is the open interval ] c(1−r) c(1−r)+(1−c)r, cr cr+(1−c)(1−r)[. by proposition 5.1, ndg(c; r) is a particular neighborhood of c in the order topology τg. we show by means of the following results that the set ndg generates the order topology associated to g. proposition 5.3. let a be an open set in the order topology τg. then for each c ∈ a there exists a dg-neighborhood of c included in a. proof. it is enough to prove the assertion in the case that a is an open interval ]a, b[. let c ∈]a, b[ and r = dg(a, c)∧dg(b, c) = (c÷a)∧ (b÷c). let us consider the cases: 1. r = c÷a ≤ b÷ c; 2. r = b÷ c < c÷a. in the first case, a = c ÷ r, c ⊙ r ≤ b and so ]c ÷ r, c ⊙ r[⊆ a =]a, b[; thus, by proposition 5.1, ndg(c; r) ⊆ a. in the second case, the inclusion ndg(c; r) ⊆ a can be proved by similar arguments. 2 corolary 5.1. the set ndg of the all dg-neighborhoods of the elements of g is a base for the order topology τg. 10 about a definition of metric over an abelian linearly ordered group references [1] j. barzilai, consistency measures for pairwise comparison matrices. journal of multi-criteria decision analysis 7(1998), 123–132. 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[19] t.l. saaty, decision making for leaders. university of pittsburgh (1988). 12 ratio mathematica volume 45, 2023 forecasting of annual rainfall using fuzzy logic interval based partitioning in different intervals rajan d* sugunthakunthalambigai r† abstract fuzzy time series models have been proposed by many researchers around the world for rainfall forecasting, but the forecasting has not been as accurate as existing methods. frequency density or ratio-based segmentation methods have been used to represent discourse segmentation. in this paper, to make such predictions, we used interval-based segmentation as the discourse segmentation and the urban mean rainfall in the trichy district as the discourse universe. fuzzy models are used for forecasting in many fields such as admissions prediction, stock price analysis, agricultural production, horticultural production, marine production, weather forecasting, and more. keywords. mean square error; fuzzy time series; average forecast error rate. ams subject classification: 05c78‡. *associate professor of mathematics, (tbml college, affiliated to bharathidasan university porayar-609 307, mayiladuthurai dist.); dan_rajan@rediffmail.com. †assistant professor of mathematics, hc&ri (w), trichy (tnau, tamilnadu, india); suguntha@tnau.ac.in. ‡ received on july 10, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm. v45i0.1006. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 153 rajan d and sugunthakunthalambigai r 1. introduction a forecasting process issued to predict future outcomes. relevant data and figures are carefully analyzed to make accurate forecasts and make optimal choices regarding the future. there are two main reasons for choosing time series forecasting. first, most of the data that exists in the real world, such as economics, business, and finance, are time series. second, evaluation of time series data is easy, and many techniques are available for evaluation of time series forecasts. if the future is in doubt, the forecasting process is mandatory. implement using the fuzzy time series method to forecast precipitation and compare results with other existing techniques. accurate rainfall information is essential for planning and managing water resources. moreover, in urban areas, rainfall has a strong impact on transportation, sewerage, n. q. hung (nguyenquang.hung@ait.ac.th) systems, and other human activities. nevertheless, precipitation is best understood and modelled for hydrological cycles because of the complexity of the atmospheric processes that generate precipitation and the tremendous range of variability over a wide range of scales in both space and time. it is one of the complex and difficult factors (french et al., 1992). accurate rainfall forecasting is thus one of the greatest challenges in operational hydrology, despite many advances in weather forecasting in recent decades (gwangseob and ana, 2001). the data for prediction has all the uncertainties. for this purpose, we generate rainfall simulations that reproduce in a distributional sense the set of key rainfall statistics obtained from the observational dataset (benoit and mariethoz, 2017). the practical interest of probabilistic rainfall models is, among other things, to complement numerical weather models for simulating rainfall heterogeneity at fine scales and to quantify uncertainties associated with rainfall reconstructions. . indeed, numerical weather models face the challenge of reproducing spatial and temporal rainfall heterogeneity, especially at fine scales (bauer et al., 2015; bony et al., 2015). some of the main applications of probabilistic rainfall models are therefore for local impact studies, e.g. related to hydrology (paschalis et al., 2014; caseri et al., 2016). fuzzy time series forecasting is a smart method in areas where information is explicit, imprecise, and approximate. fuzzy time series can also tackle situations that do not provide trend investigation and analysis, or visualization of time series patterns. indepth research work has been done on forecasting problems using this concept. vikas [1] proposed various techniques for predicting crop yields and used artificial neural networks to predict wheat yields. did. adesh [2] conducted a comparative study of various techniques, including neural networks and fuzzy models. askar [3] also tried to predict yield using time series models. sachin [4-5] specifically worked on rice yield prediction using fuzzy time series models. narendra [6] attempted to predict wheat yields. pankaj [7] used an adaptive neuro-fuzzy system for predicting wheat yield. w. qiu, x. liu, and h. li proposed a generalized method of forecasting based on fuzzy time series models [30]. the concept and definition of fuzzy time series was devised and published by song & chissom. they also delineated concepts and notions of variant and invariant time series [8-9]. first, the time-series data for the university of alabama were obtained, the 154 forecasting of annual rainfall using fuzzy logic interval based partitioning in different intervals admission prediction was performed, and after a few years [10] average auto, then chen [11-12] was the max-min configuration operation previously used by song &chissom. we depicted simplified arithmetic operations instead of using, and used higher-order fuzzy time series to organize our forecasting models. huarng [13-14], hwang and chen [15], lee wang and chen [16], li and kozma [17] all produced a number of fuzzy prediction methods with slight variations in each. lee et al. subsequently, a multivariate heuristic model was designed and implemented to obtain highly complex and complex matrix computations [20]. research work has been done to ascertain the interval length of fuzzy time series [21]. wong et al. (2003) used som and back propagation neural networks to build a fuzzy rule base and used the rule base to develop a forecast model for swiss rainfall using spatial interpolation. bardossy et al. (1995) implemented fuzzy logic for classification of atmospheric circulation patterns. ozerkan et al. (1996) compared the performance of regression analysis and fuzzy logic in studying the relationship between monthly atmospheric circulation patterns and precipitation. pesti et al. (1996) implemented fuzzy logic for drought assessment. baum et al. (1997) developed a cloud classification model using fuzzy logic. fujibe (1989) classified precipitation patterns in honshu using the fuzzy c-means method. garanboshi et al. (1999) using fuzzy logic, we investigated the effects of enso and macro circulation patterns on precipitation over arizona. vivekanandan et al. (1999) developed and implemented a fuzzy logic algorithm for water meteor particle identification that is simple and efficient enough to run in real time for operational use. wuwardi et al. (2006) use a neural fuzzy system to model tropical rainfall during the rainy season. the model results in lower rmse values, indicating that the forecast model is reliable in representing recent inter annual variability in tropical rainfall during the wet season. 2. basic concepts of fuzzy time series modeling definition 1.1. (song and chissom, 1993a, 1994; liaw, 1997). a fuzzy number on the real line  is a fuzzy subset of  that is normal and convex. definition 1.2. (song and chissom, 1993a, 1994). suppose that 𝑅1 = ⋃𝑖𝑗 𝑅1𝑖𝑗(𝑡, 𝑡 − 1) and 𝑅2(𝑡, 𝑡 − 1) = ⋃𝑖𝑗 𝑅2𝑖𝑗(𝑡, 𝑡 − 1) are two fuzzy relations between 𝐹(𝑡)and𝐹(𝑡 − 1). if, for any𝑓𝑗 (𝑡) ∈ 𝐹(𝑡), where𝑗 ∈ 𝐽, there exists𝑓𝑖 (𝑡 − 1) ∈ 𝐹(𝑡 − 1), where𝑖 ∈ 𝐼, and fuzzy relations 𝑅1 𝑖𝑗(𝑡, 𝑡 − 1) and 𝑅2 𝑖𝑗(𝑡, 𝑡 − 1) such that 𝑓𝑖 (𝑡) = 𝑓𝑖 (𝑡 − 1) ∘ 𝑅1 𝑖𝑗(𝑡, 𝑡 − 1) and𝑓𝑖 (𝑡) = 𝑓𝑖 (𝑡 − 1) ∘ 𝑅2 𝑖𝑗(𝑡, 𝑡 − 1), then define𝑅1(𝑡, 𝑡 − 1) = 𝑅2(𝑡, 𝑡 − 1). definition 1.3. (song and chissom, 1993a, 1994). suppose that 𝐹(𝑡) is only caused by 𝐹(𝑡 − 1), 𝐹(𝑡 − 2), ,or𝐹(𝑡 − 𝑚) (𝑚 > 0). this relation can be expressed as the following fuzzy relational equation: 𝐹(𝑡) = 𝐹(𝑡 − 1) ∘ 𝑅0(𝑡, 𝑡 − 𝑚), which is called the first-order model of 𝐹(𝑡). 155 rajan d and sugunthakunthalambigai r definition 1.4 (song and chissom, 1993a, 1994). suppose that 𝐹(𝑡) is simultaneously caused by 𝐹(𝑡 − 1), 𝐹(𝑡 − 2), , and𝐹(𝑡 − 𝑚) (𝑚 > 0). this relation can be expressed as the following fuzzy relational equation:𝐹(𝑡) = (𝐹(𝑡 − 1) × 𝐹(𝑡 − 2) ×  × 𝐹(𝑡 − 𝑚)) ∘ 𝑅𝑎(𝑡, 𝑡 − 𝑚)), which is called the m th-order model of𝐹(𝑡). definition 1.5 (chen, 1996). 𝐹(𝑡) is fuzzy time series if 𝐹(𝑡) is a fuzzy set. the transition is denoted as𝐹(𝑡 − 1) ⟶ 𝐹(𝑡). definition 1.6 (chou, 2011). let 𝑑(𝑡) be a set of real numbers𝑑(𝑡) ⊆ 𝑅. a lower interval 𝑑(𝑡) is a number b such that 𝑥 ≥ 𝑏 for all𝑥 ∈ 𝑑(𝑡). the set 𝑑(𝑡) is said to be an interval below if 𝑑(𝑡) has a lower interval. a number, min, is the minimum of 𝑑(𝑡) if min is a lower interval 𝑑(𝑡)andmin ∈ 𝑑(𝑡). definition 1.7 (chou, 2011). let 𝑑 (𝑡) be a set of real numbers 𝑑(𝑡) ⊆ 𝑅. an upper interval 𝑑(𝑡) is a number b such that 𝑥 ≤ 𝑏 for all𝑥 ∈ 𝑑 (𝑡). the set 𝑑 (𝑡) is said to be an interval higher if 𝑑 (𝑡) has an upper interval. a number, max, is the maximum of 𝑑 (𝑡)if max is an upper interval 𝑑 (𝑡)and max ∈ 𝑑(𝑡). 3. proposed method in this section, we use real-world rainfall as the universe of discourse and propose a method for forecasting using interval-based segmentation. relevant concepts and definitions regarding this can be found by referring to a previously published paper [29]. another method for predicting the values provided in this paper is clearly explained in the following lines. the forecasting process follows these steps: step 1: first, obviously analysis descriptive statistics. it helps in facilitating data visualization. next, we describe the discourse universe u and the parcel u in intervals of equal length. here, according to the data, 379.79 is the minimum value and 1163.59 is the maximum value. we need to specify the discourse universe, the intervals in which all given values of rainfall exist. so in this case the discourse universe would be [300, 1200]. descriptive statistics and block-by-block rainfall data are shown in tables i and ii. step 2: fuzzy partitioning is a methodology for generating fuzzy sets that represent the underlying data. the techniques can be classified into three categories: grid partitioning, tree partitioning, and distributed partitioning. among the various fuzzy partitioning methods, grid partitioning is the most commonly used in practice, especially in system control applications. grid partitioning forms partitions by dividing the input space into several fuzzy slices. next, divide universe of discourse in 6, 9 and 18 equal intervals these are as following. the discourse universe can be defined by 𝑈 = [300, 1200]. u is then divided into 6 equal length intervals and the midpoint of the 6th interval is calculated as shown below. 156 forecasting of annual rainfall using fuzzy logic interval based partitioning in different intervals table 3.1. annual rainfall in trichy district (from2004 to 2010) blocks rainfall(mm) andanallur 1085.23 lalgudi 1136.79 manachanallur 569.44 manapparai 837.43 manikandam 379.79 marungapuri 942.16 musiri 794.31 pullambadi 1163.59 t.pet 710.16 thiruverumbur 960.9 thottiam 866.73 thuraiur 942.11 uppiliyapuram 473.35 vaiyampatty 922.27 table 3.2. descriptive statistics minimum = 379.79 maximum = 1163.59 range = 783.8 count = 14 sum = 11784.26 mean = 841.733 median = 894.5 mode = no mode standard deviation = 237.65 variance = 56479.54 157 rajan d and sugunthakunthalambigai r table 3.3. a) 6 equal intervals here, u is partitioned into 9 equal length intervals and calculated mid points of 9th intervals given below: table 3.4. b) 9 equal intervals with midpoints here, u is partitioned into 18 equal length intervals and calculated mid points of 18th intervals given below. table 3.5. c) 18 equal intervals with midpoints u1 [300 − 450] 375 u2 [450 − 600] 525 u3 [600 − 750] 675 u4 [750 − 900] 825 u5 [900 − 1050] 975 u6 [1050 − 1200] 375 v1 [300 − 400] 350 v2 [400 −500] 450 v3 [500 − 600] 550 v4 [600 −700] 650 v5 [700 − 800] 750 v6 [800 − 900] 850 v7 [900 −1000] 950 v8 [1000 − 1100] 1050 v9 [1100 −1200] 1150 w1 [300 − 350] 325 w2 [350 − 400] 375 w3 [400 − 450] 425 w4 [450 − 500] 475 w5 [500 − 550] 525 w6 [550 − 600] 575 w7 [600 − 650] 625 w8 [650 − 700] 675 w9 [700 − 750] 725 w10 [750 − 800] 775 w11 [800 − 850] 825 w12 [850 − 900] 875 w13 [900 − 950] 925 w14 [950 − 1000] 975 158 forecasting of annual rainfall using fuzzy logic interval based partitioning in different intervals step 3. define a fuzzy set based on 6, 9, and 18 intervals to fuzz the historical data. a. 6 equal intervals let 𝑈 = {𝑢1, 𝑢2, 𝑢3, 𝑢4, 𝑢5, 𝑢6} be the world of discourse. the number of intervals depends on the number of considered linguistic variables (fuzzy sets) 𝐴1, 𝐴2, 𝐴3, 𝐴4, 𝐴5, 𝐴6.define 6 fuzzy sets 𝐴1, 𝐴2, … , 𝐴6 as linguistic variables in the discourse world u. these fuzzy variables are defined as: table 3.6. label linguistic value of enrolments fuzzified linguistic value a1 very few a2 very very few a3 moderate a4 high a5 very high a6 very very high b. 9 equal intervals let u be the universe of discourse, where𝑈 = {𝑣1, 𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣6 … 𝑣9}. the number of intervals will be in accordance with the number of linguistic variables (fuzzy sets) 𝐵1, 𝐵2, 𝐵3, 𝐵4, … 𝐵9, to be considered. define 9fuzzy sets 𝐵1, 𝐵2, 𝐵3, 𝐵4, … 𝐵9, as linguistic variables on the universe of discourse u. these fuzzy variables are being defined as: table 3.7. label linguistic value of enrolments w15 [1000 − 1050] 1025 w16 [1050 − 1100] 1075 w17 [1100 − 1150] 1125 w18 [1150 − 1200] 1175 fuzzified linguistic value b1 very 3 few b2 very 2 few b3 very 1 few b4 few b5 moderate b6 high 159 rajan d and sugunthakunthalambigai r c. 18 equal intervals let u be the universe of discourse, and let 𝑈 = {𝑤1, 𝑤2, 𝑤3, … 𝑤18}. the number of intervals depends on the number of considered linguistic variables (fuzzy sets) 𝐶1, 𝐶2, 𝐶3, … 𝐶18. define 18 fuzzy sets 𝐶1, 𝐶2, 𝐶3, … 𝐶18 as linguistic variables in the universe of discourse u. these fuzzy variables are defined as: table 3.8. label linguistic value of enrolments fuzzified linguistic value c1 very 8 few c2 very 7 few c3 very 6 few c4 very 5 few c5 very 4 few c6 very 3 few c7 very 2 few c8 very 1 few c9 few c10 high c11 very 1 high c12 very 2 high c13 very 3 high c14 very 4 high c15 very 5 high c16 very 6 high c17 very 7 high c18 very 8 high step 4: fuzzy set defined by u (all intervals). a fuzzy set 𝐴𝑖 is represented as 𝐴1 = 1 𝑢1⁄ + 0.5 𝑢2⁄ + 0 𝑢3⁄ + 0 𝑢4⁄ + 0 𝑢5⁄ + 0 𝑢6⁄ 𝐴2 = 0.5 𝑢1⁄ + 1 𝑢2⁄ + 0.5 𝑢3⁄ + 0 𝑢4⁄ + 0 𝑢5⁄ + 0 𝑢6⁄ ……………………………………………………… ………………………………………………………… ………………………………………………………… 𝐴10 = 0 𝑢1⁄ + 0 𝑢2⁄ + 0 𝑢3⁄ + 0 𝑢4⁄ + 0.5 𝑢5⁄ + 1 𝑢6⁄ b7 very 1 high b8 very 2 high b9 very 3 high 160 forecasting of annual rainfall using fuzzy logic interval based partitioning in different intervals a fuzzy set 𝐵𝑖 are expressed as follows: 𝐵1 = 1 𝑢1⁄ + 0.5 𝑢2⁄ + 0 𝑢3⁄ + ⋯ + 0 𝑢9⁄ 𝐵2 = 0.5 𝑢1⁄ + 1 𝑢2⁄ + ⋯ + 0 𝑢9⁄ ……………………………………… ………………………………… ………………………………………… 𝐵9 = 0 𝑢1⁄ + 0 𝑢2⁄ + ⋯ + 0.5 𝑢5⁄ + 1 𝑢9⁄ a fuzzy set 𝐶𝑖 are expressed as follows: 𝐶1 = 1 𝑢1⁄ + 0.5 𝑢2⁄ + 0 𝑢3⁄ + 0 𝑢4⁄ + 0 𝑢5⁄ + 0 𝑢6⁄ 𝐶2 = 0.5 𝑢1⁄ + 1 𝑢2⁄ + 0.5 𝑢3⁄ + 0 𝑢4⁄ + 0 𝑢5⁄ + 0 𝑢6⁄ ……………………………………………………… ………………………………………………………… ………………………………………………………… 𝐶18 = 0 𝑢1⁄ + 0 𝑢2⁄ + 0 𝑢3⁄ + ⋯ + 0.5 𝑢17⁄ + 1 𝑢18⁄ step 5: fuzzify historical data. table 3.9. linguistic values for the enrolments from 2004 to 2010 block rainfall (mm) linguistic value 6th interval linguistic value 9th interval linguistic value 18th interval andanallur 1085.23 c6 d8 e16 lalgudi 1136.79 c6 d9 e17 manachanallur 569.44 c2 d3 e6 manapparai 837.43 c4 d6 e12 manikandam 379.79 c1 d1 e2 marungapuri 942.16 c5 d7 e13 musiri 794.31 c4 d5 e10 pullambadi 1163.59 c6 d9 e18 t.pet 710.16 c4 d5 e9 thiruverumbur 960.9 c5 d7 e14 thottiam 866.73 c4 d6 e12 thuraiur 942.11 c5 d7 e13 uppiliyapuram 473.35 c2 d2 e4 vaiyampatty 922.27 c5 d7 e13 161 rajan d and sugunthakunthalambigai r step 6: calculate predicted registrations for 6, 9, and 18 intervals given below: table 3.10. forecasted value for all intervals step 7: calculate mse and afer values for 6, 9, and 18 intervals given below: mean squared error (mse) measures the amount of error in a statistical model. evaluate the mean squared difference between observed and predicted values. if the model has no errors, mse is equal to zero. mse formula = (1 𝑛⁄ ) ∗ σ(𝐴𝑐𝑡𝑢𝑎𝑙 – 𝐹𝑜𝑟𝑒𝑐𝑎𝑠𝑡)2 mean absolute percentage error (mape), also known as mean absolute percentage deviation (mapd), is a measure of the predictive accuracy of a forecasting method in statistics. accuracy is usually expressed as a ratio defined by the following formula: |𝐴𝑐𝑡𝑢𝑎𝑙−𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑| 𝐴𝑐𝑡𝑢𝑎𝑙 × 100% = average forecasting error rate (afer) blocks rainfall (mm) forecasted value (6 intervals) forecasted value (9 intervals) forecasted value (18 intervals) andanallur 1085.23 lalgudi 1136.79 manachanallur 569.44 manapparai 837.43 900 900 912.5 manikandam 379.79 712.5 725 737.5 marungapuri 942.16 675 675 687.5 musiri 794.31 750 725 737.5 pullambadi 1163.59 825 800 812.5 t.pet 710.16 937.5 900 900 thiruverumbur 960.9 937.5 900 912.5 thottiam 866.73 937.5 925 937.5 thuraiur 942.11 900 875 875 uppiliyapuram 473.35 825 800 812.5 vaiyampatty 922.27 825 800 800 162 forecasting of annual rainfall using fuzzy logic interval based partitioning in different intervals 4. performance evaluation and comparative studies a. performance rating: two parameters are used to compare the results of the proposed method with existing methods. these are mse &afer. mse & afer are the calculated values for intervals 6, 9 and 18 as shown in tables xi, xii and xiii. interval-based partitioning is calculated in table xi. mse indicates the deviation error from the actual value to the predicted value. the deviation is shown in figure 4.1 in the form of a graphical representation for better visualization. as we can see the proposed algorithm gives values very close to what is the actual rainfall value. the same is done for the 9th and 18th intervals, as shown in tables xii & xiii and fig. 4.2 and 4.3. table 3.11. mse and afer values (6 intervals) block 𝑨𝑖 (rainfall mm) 𝑭𝑖 mse (𝑨𝒊 − 𝑭𝒊) 𝟐 afer |𝑨𝒊 − 𝑭𝒊|/𝑨𝒊 andanallur 1085.23 lalgudi 1136.79 manachanallur 569.44 manapparai 837.43 900 3915.005 0.074717 manikandam 379.79 712.5 110695.9 0.876037 marungapuri 942.16 675 71374.47 0.283561 musiri 794.31 750 1963.376 0.055784 pullambadi 1163.59 825 114643.2 0.290987 t.pet 710.16 937.5 51683.48 0.320125 thiruverumbur 960.9 937.5 547.56 0.024352 thottiam 866.73 937.5 5008.393 0.081652 thuraiur 942.11 900 1773.252 0.044698 uppiliyapuram 473.35 825 123657.7 0.742896 vaiyampatty 922.27 825 9461.453 0.105468 mse = 35337.42 afer = 20.72% 163 rajan d and sugunthakunthalambigai r table 3.12. mse and afer values (9 intervals) block 𝑨𝑖 (rainfall mm) 𝑭𝑖 mse (𝑨𝒊 − 𝑭𝒊) 𝟐 afer |𝑨𝒊 − 𝑭𝒊|/𝑨𝒊 andanallur 1085.23 lalgudi 1136.79 manachanallur 569.44 manapparai 837.43 912.5 5635.505 0.089643 manikandam 379.79 737.5 127956.4 0.941863 marungapuri 942.16 687.5 64851.72 0.270294 musiri 794.31 737.5 3227.376 0.071521 pullambadi 1163.59 812.5 123264.2 0.30173 t.pet 710.16 900 36039.23 0.26732 thiruverumbur 960.9 912.5 2342.56 0.050369 thottiam 866.73 937.5 5008.393 0.081652 thuraiur 942.11 875 4503.752 0.071234 uppiliyapuram 473.35 812.5 115022.7 0.716489 vaiyampatty 922.27 800 14949.95 0.132575 mse=35914.42 afer=21.39% 164 forecasting of annual rainfall using fuzzy logic interval based partitioning in different intervals table 3.13. mse and afer values (18 intervals) the following figures (fig.4.1, fig 4.2 and fig 4.3) are compared in forecasted and actual rainfall for the corresponding intervals respectively 6,9&18.and also fig. 4.4 compares the mse for all intervals. fig. 4.1. forecasted vs. rainfall (6 intervals) block 𝑨𝑖 (rainfall mm) 𝑭𝑖 mse (𝑨𝒊 − 𝑭𝒊) 𝟐 afer |𝑨𝒊 − 𝑭𝒊|/𝑨𝒊 andanallur 1085.23 lalgudi 1136.79 manachanallur 569.44 manapparai 837.43 900 3915.005 0.074717 manikandam 379.79 725 119169.9 0.90895 marungapuri 942.16 675 71374.47 0.283561 musiri 794.31 725 4803.876 0.087258 pullambadi 1163.59 800 132197.7 0.312473 t.pet 710.16 900 36039.23 0.26732 thiruverumbur 960.9 900 3708.81 0.063378 thottiam 866.73 925 3395.393 0.06723 thuraiur 942.11 875 4503.752 0.071234 uppiliyapuram 473.35 800 106700.2 0.690081 vaiyampatty 922.27 800 14949.95 0.132575 mse=35768.45 afer=21.13% 165 rajan d and sugunthakunthalambigai r fig. 4.2. forecasted vs. rainfall (9 intervals) fig. 4.3. forecasted vs. rainfall (18 intervals) 166 forecasting of annual rainfall using fuzzy logic interval based partitioning in different intervals fig. 4.4. graph of mse 6, 9 and 18 intervals b. results and discussion. the mse and afer calculated in tables 3.11, 3.12 and 3.13 above were analyzed. the paper shows working with different intervals such as 6 , 9 and 18 . the majority of recently published papers work in one of these intervals. the focus of this paper was to propose a new algorithm and check its predicted variability over all these intervals. the results show that prediction works best on the 6th interval among all other intervals. all results are presented in an easy-to-understand bar chart format to reduce the complexity of this study and present it in a more understandable manner. to prove that this algorithm is efficient, a comparison is made with other existing methods proposed by 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http://www.sciencedirect.com/science/article/pii/s1568494608001270 http://www.sciencedirect.com/science/article/pii/s1568494608001270 http://www.sciencedirect.com/science/article/pii/s1568494608001270 http://www.sciencedirect.com/science/article/pii/s1568494608001270 http://www.sciencedirect.com/science/article/pii/s1568494608001270 construction of k-hyperideals by p -hyperoperations h. hedayati, r. ameri∗ department of mathematics, faculty of basic science, university of mazandaran, babolsar, iran e-mail : {h.hedayati, ameri}@umz.ac.ir abstract in this note we present a method to construction new k-hyperideals from given k-ideals of a semiring r by using of the p -hyperoperations. then we investigate the relationship between them. in particular, we describe all khyperideals of the semihyperring of the nonnegative integers. keywords: (semi)hyperring, k-(hyper)ideal, p -hyperoperation, weak distributive 1 1 introduction hyperstructures theory was born in 1934 when marty [12] defined hypergroups as a generalization of groups. also wall in 1937 defined the notion of cyclic hypergroup. this theory has been studied in the following decades and nowadays by many mathematicians. a short review of the theory of hypergroups appears in [2]. a recent books [2], [3] and [15] contain a wealth of applications. there are applications 1* correspondence author 75 to the following subjects: geometry, hypergraphs, binary relations, combinatorics, codes, cryptography, probability, groups, rational algebraic functions and etc. one of the several contexts which they arise is hyperring. first m. krasner studied hyperrings, which is a triple (r, +, .), where (r, +) is a canonical hypergroup and (r, .) is a semigroup, such that for all a, b, c ∈ r, a(b + c) = ab + ac, (b + c)a = ba + ca ([10]). the notion of k-ideals in ordinary semirings was introduced by d. r. latore in 1965 ([11]). also m. k. sen and others worked on one-sided k-ideals and maximal k-ideals of semirings ([14], [16]). the authors in [6] introduced the notion of k-hyperideals in the sense of krasner and obtained some related results about this notion. we now follow [6] to introduce a method to construct new k-hyperideals from given k-ideals. in section 2 of this paper, we gather all the preliminaries of (semi)hyperrings and k-(hyper)ideals which will be used in the next sections. in section 3, we represent some methods for construction semihyperrings from semirings by p -hyperoperations and then we investigate the relationship between their k-hyperideals and k-ideals. as an important result of this section, all k-hyperideals of the nonnegative integers n∗ as a semihyperring, constructed by p -hyperoperations, are described. in section 4, we characterize the k-hyperideals of product of semihyperrings which are made by p -hyperoperations and a family of semirings. 2 preliminaries a map ◦ : h × h −→ p∗(h) is called hyperoperation or join operation. a hypergroupoid is a set h with together a (binary) hyperoperation ◦. a hypergroupoid (h, ◦), which is associative, that is x ◦ (y ◦ z) = (x ◦ y) ◦ z, ∀x, y, z ∈ h is called a semihypergroup . a hypergroup is a semihypergroup such that ∀x ∈ h we have x◦h = h = h ◦x, which is called reproduction axiom (see [2]). let h be a hypergroup and k be a nonempty subset of h. then k is said to be 76 a subhypergroup of h if itself is a hypergroup under hyperoperation ”◦” restricted to k. hence it is clear that a subset k of h is a subhypergroup if and only if ak = ka = k, under the hyperoperation on h. definition 2.1. a hyperalgebra (r, +, .) is called a semihyperring if and only if (i) (r, +) is a semihypergroup; (ii) (r, .) is a semigroup; (iii) ∀a, b, c ∈ r, a.(a + b) = a.b + a.c and (b + c).a = b.a + c.a. remark. in definition 2.1, if we replace (iii) by ∀a, b, c ∈ r, a.(a + b) ⊆ a.b + a.c and (b + c).a ⊆ b.c + c.a, we say that r is a weak distributive semihyperring. a semihyperring r is called with zero element, if there exists an unique element 0 ∈ r such that 0 + x = x = x + 0 and 0x = 0 = x0 for all x ∈ r. a semihyperring r is called additive commutative, if x + y = y + x, ∀x, y ∈ r. a semihyperring (r, +, .) is called a hyperring provided (r, +) is a canonical hypergroup. definition 2.2. a hyperring (r, +, .) is called (i)commutative if a.b = b.a for all a, b ∈ r; (ii)with identity, if there exists an element, say 1 ∈ r, such that 1.x = x.1 = x for all x ∈ r. let (r, +, .) be a hyperring, a nonempty subset s of r is called a subhyperring of r if (s, +, .) is itself a hyperring. definition 2.3. a subhyperring i of a hyperring r is said to be a (resp. right) left hyperideal of r provided that ( resp. x.r ∈ i ) r.x ∈ i for all r ∈ r and for all x ∈ i. we say that i is a hyperideal if i is both a left and right hyperideal. definition 2.4.[11] let (r, +, .) be a semiring. a nonempty subset i of r is called a left k-ideal of r, if i is a left ideal of r and for a ∈ i and x ∈ r we have a + x ∈ i or x + a ∈ i =⇒ x ∈ i. 77 similarly a right k-ideal is defined. a two sided k-ideal or simply a k-ideal is both a left and right k-ideal. we denote i as k-ideal (resp. ideal) of r by i ck r (resp. i c r). in the sequel, by r we mean a semihyperring, unless otherwise specified. definition 2.5.[6] let (r, +, .) be a ( weak distributive ) semihyperring. a nonempty subset i of r is called (i) a left ( resp. right) hyperideal of r if and only if (a) (i, +) is a semihypergroup of (r, +); and (b) rx ∈ i (resp. xr ∈ i), for all r ∈ r and for all x ∈ i. (ii) a hyperideal of r if it is both left and right hyperideal of r. the hyperideal i of r is denoted by i ch r. (iii) a left k-hyperideal of r, if i is a left hyperideal of r and for a ∈ i and x ∈ r we have a + x ≈ i or x + a ≈ i =⇒ x ∈ i, where by a ≈ b we mean a ∩ b 6= ∅. (iv) similarly a right k-hyperideal is defined. a two sided k-hyperideal or simply a k-hyperideal is both a left and right k-hyperideal. we denote i as k-hyperideal of r by i ck.h r. 3 construction of k-hyperideals by p -hyperoperations in this section we apply three kinds of p -hyperoperations (which were introduced for hv-structures in [15]) to construct semihyperrings from semirings. then we investigate the relationship between their k-hyperideals and k-ideals . definition 3.1. let (r, +, .) be semiring and ∅ 6= p ⊆ r. we define two hyperoperations as follows x ⊕c y = {x + t + y | t ∈ p}, 78 x � y = x.y = xy, which ⊕c is called centre p -hyperoperation. proposition 3.2. let (r, +, .) be semiring and p ⊆ r be a nonempty such that p r ⊆ p and rp ⊆ p , then (r, ⊕c, �) is a weak distributive semihyperring. proof . first, we show (r, ⊕c) is a semihypergroup. for this we prove that (x ⊕c y) ⊕c z = x ⊕c (y ⊕c z). for x, y, z ∈ r we have a ∈ (x ⊕c y) ⊕c z =⇒ ∃a1 ∈ x ⊕c y, a ∈ a1 ⊕c z =⇒ ∃t1, t2 ∈ p, a = a1 + t1 + z, a1 = x + t2 + y =⇒ a = x + t2 + y + t1 + z =⇒ a = x + t2 + b, b = y + t1 + z ∈ y ⊕c z =⇒ a ∈ x ⊕c b, b ∈ y ⊕c z =⇒ a ∈ x ⊕c (y ⊕c z) =⇒ (x ⊕c y) ⊕c z ⊆ x ⊕c (y ⊕c z). similarly, we obtain that (x ⊕c y) ⊕c z ⊇ x ⊕c (y ⊕c z). clearly (r, �) is a semigroup, since (r, .) is a semigroup and x � y = xy. we now prove weak distributivity, that is x � (y ⊕c z) ⊆ (x � y) ⊕c (x � z) = xy ⊕c xz. for this we have a ∈ x � (y ⊕c z) =⇒ ∃a1 ∈ y ⊕c z, a = x � a1 = xa1 =⇒ ∃t ∈ p, a = xa1, a1 = y + t + z =⇒ a = x(y + t + z) = xy + xt + xz ∈ xy ⊕c xz ( rp ⊆ p ) =⇒ x � (y ⊕c z) ⊆ xy ⊕c xz. 79 similarly we conclude that (y ⊕c z) � x ⊆ yx ⊕c zx.� definition 3.3. let (r, +, .) be a semiring and ∅ 6= p ⊆ r. we define the following hyperoperations x ⊕r y = {x + y + t | t ∈ p}, x ⊕l y = {t + x + y | t ∈ p}, x � y = xy, which ⊕r and ⊕l are called right p -hyperoperation and left p -hyperoperation respectively. proposition 3.4. let (r, +, .) be a semiring and p ⊆ r be a nonempty such that p r ⊆ p and rp ⊆ p and x + p = p + x, for all x ∈ r. then (r, ⊕r, �) and (r, ⊕l, �) are weak distributive semihyperrings. proof. first, we prove that (x ⊕r y) ⊕r z = x ⊕r (y ⊕r z). for this we have a ∈ (x ⊕r y) ⊕r z =⇒ ∃a1 ∈ x ⊕r y, a ∈ a1 ⊕r z =⇒ ∃t1, t2 ∈ p, a1 = x + y + t1, a = a1 + z + t2 =⇒ ∃t1, t2 ∈ p, a = x + y + t1 + z + t2 (1) also we have b ∈ x ⊕r (y ⊕r z) =⇒ ∃b1 ∈ y ⊕r z, b ∈ x ⊕r b1 =⇒ ∃w1, w2 ∈ p, b1 = y + z + w1, b = x + b1 + w2 =⇒ ∃w1, w2 ∈ p, b = x + y + z + w1 + w2 (2) from (1) we have a = x + y + t1 + z + t2 = x + y + z + w1 + t2, ∃w1 ∈ p (z + p = p + z) =⇒ a ∈ x ⊕r (y ⊕r z) (by (2)) =⇒ (x ⊕r y) ⊕r z ⊆ x ⊕r (y ⊕r z). 80 similarly we can prove that (x ⊕r y) ⊕r z ⊇ x ⊕r (y ⊕r z). clearly (r, �) is semigroup, since (r, .) is a semigroup. in a similar way to the proposition 3.2 we can prove weak distributivity. therefore (r, ⊕r, �) is a weak distributive semihyperring. analogously we can prove that (r, ⊕l, �) is a weak distributive semihyperring. � remark. in propositions 3.2 and 3.4, if we replace the conditions rp ⊆ p and p r ⊆ p by rp = p = p r for all r ∈ r, then (r, ⊕c, �) and (r, ⊕r, �) and (r, ⊕l, �) become semihyperring. theorem 3.5. let (r, +, .) be a semiring with zero and p be the same as proposition 3.2 such that 0 ∈ p . then there is a one-to-one correspondence between the k-ideals of (r, +, .) containing p and k-hyperideals of (r, ⊕c, �). proof. let i be a k-ideal of (r, +, .) containing p . first we prove that i /h (r, ⊕c, �). suppose that x, y ∈ i, we prove x ⊕c y ⊆ i. for this we have z ∈ x ⊕c y =⇒ ∃t ∈ p ⊆ i, z = x + t + y =⇒ z = x + t + y ∈ i ( since x, t, y ∈ i ) =⇒ x ⊕c y ⊆ i. also if r ∈ r and x ∈ i, then r � x = rx ∈ i, since i / (r, +, .). thus i is a hyperideal of (r, ⊕c, �). we now prove that i /k.h (r, ⊕c, �). for r ∈ r and x ∈ i we have r ⊕c x ≈ i =⇒ ∃z ∈ r ⊕c x ≈ i =⇒ ∃t ∈ p, z = r + t + x, z ∈ i =⇒ r + t + x ∈ i, t + x ∈ i =⇒ r ∈ i ( since i ck (r, +, .) ) =⇒ i /k.h (r, ⊕c, �). 81 conversely, suppose that i /k.h (r, ⊕c, �). we prove that i is a k-ideal of (r, +, .) containing p . for this we have x, y ∈ i =⇒ x ⊕c y ⊆ i ( i ch (r, ⊕c, �) ) =⇒ ∀t ∈ p, x + t + y ∈ i =⇒ x + y ∈ i ( 0 ∈ p ) . on the other hand r ∈ r, x ∈ i =⇒ r � x ∈ i ( i ch (r, ⊕c, �) ) =⇒ rx ∈ i. also we have r + x ∈ i, x ∈ i =⇒ r + 0 + x ∈ i, x ∈ i (0 ∈ p ) =⇒ r ⊕c x ≈ i, x ∈ i =⇒ r ∈ i ( i ck.h (r, ⊕c, �) =⇒ i ck (r, +, .). we have 0 ⊕c 0 ⊆ i, then {0 + t + 0 | t ∈ p} ⊆ i, therefore p ⊆ i. � theorem 3.6. let (r, +, .) be a semiring with zero and p be the same as proposition 3.4 such that 0 ∈ p . then there is a one-to-one correspondence between k-ideals of (r, +, .) containing p and k-hyperideals of ( (r, ⊕l, �) ) (r, ⊕r, �). proof. the proof is similar to the proof of theorem 3.5 by some manipulation. � examples. (i) let n be the set of natural numbers and 2n = {2, 4, 6, 8, ...}. clearly (n, +, .) is a semiring and 2n is a k-ideal of (n, +, .). now if p = {4, 8, 12, 16, ...} ⊆ 2n, then it is easy to verify that (n, ⊕c, �) is a weak distributive semihyperring, where for all m, n ∈ n we have m ⊕c n = {m + k + n | k ∈ p} and m � n = mn. thus 2n is a k-hyperideal of (n, ⊕c, �). 82 (ii) let n∗ = n ∪ {0} and n∗[x] = {f (x) = n∑ i=1 aix i | ai ∈ n∗}. clearly (n∗[x], +, .) is a semiring and < x >= {f (x) ∈ n∗[x] | a0 = 0} is a k-ideal of (n∗[x], +, .) generated by x. set p =< xm > for m ∈ n. obviously, 0 ∈ p ⊆< x >. then by propositions 3.2 and 3.5, (n∗[x], ⊕c, �) is a weak distributive semihyperring and < x > is a k-hyperideal of (n∗[x], ⊕c, �). in the next theorem we describe all k-hyperideals of semihyperring of the natural numbers constructed by p -hyperoperation. for this we consider the semiring (n, +, .) of natural numbers by usual ordinary operations. theorem 3.7. let 0 ∈ p ⊆ n∗ and p n∗ ⊆ p and n∗p ⊆ p and p ⊆ i. then i is a k-hyperideal of (n∗, ⊕c, �) if and only if there exists a ∈ n∗ such that i = {na | n ∈ n∗}. proof. by theorem 3.5, i ck.h (n∗, ⊕c, �) if and only if i ck (n∗, +, .). also by proposition 4.1 [14], i ck (n∗, +, .) if and only if there exists a ∈ n∗ such that i = {na | n ∈ n∗}. � 4 product of k-hyperideals in the sequel by ∏ i∈i ri, we mean the cartesian product of the family {ri}i∈i . it means ∏ i∈i ri = {(xi)i∈i | xi ∈ ri}. proposition 4.1. let {ri}i∈i be a family of semirings and pi ⊆ ri be nonempty such that ripi ⊆ pi and piri ⊆ pi, for all i ∈ i. for (xi)i∈i , (yi)i∈i ∈ ∏ i∈i ri. define (xi)i∈i ⊕c (yi)i∈i = {(xi + ti + yi)i∈i | ti ∈ pi}, (xi)i∈i � (yi)i∈i = (xiyi)i∈i . then ( ∏ i∈i ri, ⊕c, �) is a weak distributive semihyperring . 83 proof. first we show that ( ∏ i∈i ri, ⊕c) is a semihypergroup. for this we prove that (xi)i∈i ⊕c [(yi)i∈i ⊕c (zi)i∈i ] = [(xi)i∈i ⊕c (yi)i∈i ] ⊕c (zi)i∈i . we have a ∈ (xi)i∈i ⊕c [(yi)i∈i ⊕c (zi)i∈i ] =⇒ ∃ti ∈ pi, a ∈ (xi)i∈i ⊕c (yi + ti + zi)i∈i =⇒ ∃t′i ∈ pi, a = (xi + t ′ i + yi + ti + zi)i∈i =⇒ a ∈ (xi + t′i + yi)i∈i ⊕c (zi)i∈i =⇒ a ∈ [(xi)i∈i ⊕c (yi)i∈i ] ⊕c (zi)i∈i =⇒ (xi)i∈i ⊕c [(yi)i∈i ⊕c (zi)i∈i ] ⊆ [(xi)i∈i ⊕c (yi)i∈i ] ⊕c (zi)i∈i . in a similar way, we can prove the reverse inclusion. therefore, ( ∏ i∈i ri, ⊕c) is a semihypergroup. clearly ( ∏ i∈i ri, �) is a semigroup. it is enough we prove weak distributivity. for this we should prove that (xi)i∈i � [(yi)i∈i ⊕c (zi)i∈i ] ⊆ (xiyi)i∈i ⊕c (xizi)i∈i . we have a ∈ (xi)i∈i � [(yi)i∈i ⊕c (zi)i∈i ] =⇒ ∃ti ∈ pi, a ∈ (xi)i∈i � (yi + ti + zi)i∈i =⇒ a = (xi(yi + ti + zi))i∈i = (xiyi + xiti + xizi)i∈i ∈ (xiyi)i∈i ⊕c (xizi)i∈i ( ripi ⊆ pi ). this completes the proof. � proposition 4.2. if {ri}i∈i is a family of semirings and for all i ∈ i, pi ⊆ ri is nonempty such that ripi ⊆ pi and piri ⊆ pi and xi + pi = pi + xi, for all xi ∈ ri, then ( ∏ i∈i ri, ⊕r, �) and ( ∏ i∈i ri, ⊕l, �) are weak distributive semihyperring where (xi)i∈i ⊕r (yi)i∈i = {(xi + yi + ti)i∈i | ti ∈ pi}, (xi)i∈i ⊕l (yi)i∈i = {(ti + xi + yi)i∈i | ti ∈ pi}, 84 (xi)i∈i � (yi)i∈i = (xiyi)i∈i . proof. first we prove that ( ∏ i∈i ri, ⊕r) is a semihypergroup. for this we prove that (xi)i∈i ⊕r [(yi)i∈i ⊕r (zi)i∈i ] = [(xi)i∈i ⊕r (yi)i∈i ] ⊕r (zi)i∈i . we have a ∈ (xi)i∈i ⊕r [(yi)i∈i ⊕r (zi)i∈i ] =⇒ ∃ti ∈ pi, a ∈ (xi)i∈i ⊕r (yi + zi + ti)i∈i =⇒ ∃t′i ∈ pi, a = (xi + yi + zi + ti + t ′ i)i∈i =⇒ ∃wi ∈ pi, a = (xi + yi + wi + zi + t ′ i)i∈i ( since zi + pi = pi + zi ) ∈ (xi + yi + wi)i∈i ⊕r (zi)i∈i ⊆ [(xi)i∈i ⊕r (yi)i∈i ] ⊕r (zi)i∈i =⇒ (xi)i∈i ⊕r [(yi)i∈i ⊕r (zi)i∈i ] ⊆ [(xi)i∈i ⊕r (yi)i∈i ] ⊕r (zi)i∈i . similarly, we can prove that the reverse inclusion. clearly ( ∏ i∈i ri, �) is a semigroup. also the weak distributivity is obtained similar to the proof of proposition 4.1. therefore ( ∏ i∈i ri, ⊕r, �) is a semihyperring. analogously we can prove that ( ∏ i∈i ri, ⊕l, �) is a weak distributive semihyperring. this completes the proof. � remark. in propositions 4.1 and 4.2, if we replace the conditions ripi ⊆ pi and piri ⊆ pi by the condition ripi = pi = piri, for all ri ∈ ri and for all i ∈ i, then ( ∏ i∈i ri, ⊕c, �), ( ∏ i∈i ri, ⊕r, �) and ( ∏ i∈i ri, ⊕l, �) will be semihyperrings. proposition 4.3. if {rj}j∈j is a family of semirings and for all j ∈ j, pj ⊆ rj is nonempty such that rjpj ⊆ pj and pjrj ⊆ pj. then i is a k-hyperideal of ( ∏ j∈j rj, ⊕c, �) if and only if i = ∏ j∈j ij such that ij /k.h (rj, ⊕cj , �j), where xj ⊕cj yj = {xj + tj + yj | tj ∈ pj}, xj �j yj = xjyj. 85 proof. (=⇒) for all j ∈ j define ij = {x ∈ rj | (xi)i∈j ∈ i, ∃xi ∈ ri, x = xj}. we have x, y ∈ i =⇒ ∃xi, yi ∈ ri, (xi)i∈j , (yi)i∈j ∈ i, x = xj, y = yj =⇒ (xi)i∈j ⊕c (yi)i∈j ⊆ i (i ch ( ∏ j∈j rj, ⊕c, �)) =⇒ ∀ti ∈ pi, (xi + ti + yi)i∈j ∈ i (∀i ∈ j) =⇒ ∀tj ∈ pj, x + tj + y ∈ ij =⇒ x ⊕cj y ⊆ ij. now suppose that rj ∈ rj, x ∈ ij =⇒ ∃ri ∈ ri, (ri)i∈j ∈ ∏ i∈j ri and ∃xi ∈ ri, (xi)i∈j ∈ i, x = xj =⇒ (ri)i∈j � (xi)i∈j ∈ i ( i ch ( ∏ i∈j ri, ⊕c, �) ) =⇒ (rixi)i∈j ∈ i =⇒ rjxj ∈ ij ( by definition of ij ) . therefore ij ch rj. we now show that ij /k.h rj for all j ∈ j. we have rj ∈ rj, xj ∈ ij, rj ⊕cj xj ≈ ij =⇒ ∃tj ∈ pj, rj + tj + xj ∈ ij =⇒ (rj)j∈j ⊕c (xj)j∈j ≈ i, where (rj)j∈j ∈ ∏ j∈j rj, (xj)j∈j ∈ ∏ j∈j ij. then since i ck.h ( ∏ j∈j rj, ⊕c, �) we have (rj)j∈j ∈ i =⇒ rj ∈ ij, ∀j ∈ j =⇒ ij /k.h rj. (⇐=) suppose that i = ∏ j∈j ij such that ij /k.h (rj, ⊕cj , �j). first we prove i /h ( ∏ j∈j rj, ⊕c, �). let (xj)j∈j , (yj)j∈j ∈ i, then (xj)j∈j ⊕c (yj)j∈j = {(xj + tj + yj)j∈j | tj ∈ pj} ⊆ ∏ j∈j ij; 86 also we have ij /h (rj, ⊕cj , �j) =⇒ ∀tj ∈ pj, xj + tj + yj ∈ ij =⇒ (xj)j∈j ⊕c (yj)j∈j ⊆ i. now if (rj)j∈j ∈ ∏ j∈j rj and (xj)j∈j ∈ i, then (rj)j∈j � (xj)j∈j = (rjxj)j∈j ∈ ∏ j∈j ij, since rjxj ∈ ij by hypothesis. we now prove that i /k.h ( ∏ j∈j rj, ⊕c, �). for this we have (rj)j∈j ∈ ∏ j∈j rj, (xj)j∈j ∈ i, (rj)j∈j ⊕c (x1, x2) ≈ i =⇒ ∃tj ∈ pj, (rj + tj + xj)j∈j ∈ i = ∏ j∈j ij =⇒ ∃tj ∈ pj, rj + tj + xj ∈ ij, ∀j ∈ j =⇒ rj ⊕cj xj ≈ ij, rj ∈ rj, xj ∈ ij =⇒ rj ∈ ij ( ij ck.h (rj, ⊕cj , �j) ) =⇒ (rj)j∈j ∈ ∏ j∈j ij. � proposition 4.4. let {rj}j∈j be a family of semirings. suppose that pj ⊆ rj be nonempty such that rjpj ⊆ pj and pjrj ⊆ pj and xj + pj = pj + xj, for all xj ∈ rj and for all j ∈ j. then i is a k-hyperideal of ( ∏ j∈j rj, ⊕r, �) ( resp. ( ∏ j∈j rj, ⊕l, �)) if and only if i = ∏ j∈j ij such that for all j ∈ j, ij /k.h (rj, ⊕rj , �j), (resp. ij /k.h (rj, ⊕lj , �j)), where xj ⊕rj yj = {xj + yj + tj | tj ∈ pj}, xj ⊕lj yj = {tj + xj + yj | tj ∈ pj}, xj �j yj = xjyj. proof. the proof is similar to the proof of proposition 4.3. � references 87 [1] r. ameri, and m. m. zahedi, ”hyperalgebraic system”, italian journal of pure and applied mathematics, no. 6 (1999) 21-32. [2] p. corsini, ”prolegomena of hypergroup theory”, second edition aviani editor, (1993). [3] p. corsini, and v. leoreanu, ”applications of hyperstructure theory”, kluwer academic publications (2003). [4] d. freni, ” a new characterization of the derived hypergroup via strongly regular equivalences ”, communication in algebra, vol. 30, no. 8 (2002), pp. 3977-3989 [5] h. hedayati, and r. ameri,” fuzzy k-hyperideals ”, int. j. pu. appl. math. sci., vol. 2, no. 2, (to appear). [6] h. hedayati, and r. ameri, ” on k-hyperideals of semihyperrings”, (to appear). 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[14] m. k. sen, and m. r. adhikari, ” on maximal k-ideals in semirings”, proceedings of the american mathematics society, vol. 118, no. 3, july 1993. [15] t. vougiuklis, ”hyperstructures and their representations”, hardonic press, inc. (1994). [16] h. j. weinert, and m. k. sen, and m. r. adhikari, ” one-sided k-ideals and h-ideals in semirings”, mathematica pannonica, 7/1 (1996), 147-162. 89 ratio mathematica 23 (2012), 21–38 issn: 1592-7415 (intuitionistic) fuzzy grade of a hypergroupoid: a survey of some recent researches irina cristea centre for systems and information technologies university of nova gorica, slovenia irinacri@yahoo.co.uk abstract this paper aims to present a short survey on two numerical functions determined by a hypergroupoid, called the fuzzy grade and the intuitionistic fuzzy grade of a hypergroupoid. it starts with the main construction of the sequences of join spaces and (intuitionistic) fuzzy sets associated with a hypergroupoid. after some computations of the above grades, we discuss some similarities and differences between the two grades for the complete hypergroups and for the i.p.s. hypergroups. we conclude with some open problems. key words: (intuitionistic) fuzzy grade, join space, i.p.s. hypergroup, complete hypergroup. msc2012: 20n20; 03e72. 1 introduction the study of hyperstructures connected with fuzzy sets represents a growing and new line of research spanning over the last twenty years. till now one can distinguish three principal approaches of this theme: the study of new crisp hyperoperations obtained by means of fuzzy sets; the study of fuzzy subhyperstructures (i.e. fuzzy sets those level sets are crisp hyperstructures); the study of structures endowed with fuzzy hyperoperations and called fuzzy hyperstructures. irina cristea we concentrate here on the first line of reserach. its origin is due to corsini [12] in 1993, when he defined a join space from a nonempty set endowed with a fuzzy set. another definition of a join space induced by a fuzzy set was given in 1997 by ameri and zahedi [1]. the corsini’s direction was then investigated by corsini and leoreanu [16]. ten years later, the same corsini [13] considered the inverse problem: to define a fuzzy set from a hypergroupoid. iterating both constructions he obtained a sequence of join spaces and fuzzy sets that stops when there exist two consecutive non-isomorphic join spaces. the length of the sequence, i.e. the number of the non-isomorphic join spaces in it, was called by corsini and cristea [14, 15] the fuzzy grade of a hypergroupoid. at the beginning the sequence was studied for some particular hypergroups: the complete hypergroups and the i.p.s. hypergroups. recently this grade has been investigated with a general and innovative method by cristea [22, 23], ştefănescu and cristea [30], angheluţă and cristea [2, 3] making use of an ordered n-tuple determined by an equivalence relation. we will see the details in the next section. one of the principal concerns of several researchers is that to obtain new hyperstructures using modern tools for investigation of uncertain problems, like intuitionistic fuzzy sets, rough sets, soft sets, interval-valued fuzzy sets. following corsini’s idea, cristea and davvaz [24] associated with any hypergroupoid a sequence of join spaces and intuitionistic fuzzy sets with the length called the intuitionistic fuzzy grade. although the fuzzy grade and the intuitionistic fuzzy grade have similar definitions, the corresponding associated sequences of join spaces have different properties. in order to highlight the differences between them, davvaz, hassani-sadrabadi and cristea [25, 26, 27] have investigated this problem for the complete hypergroups and i.p.s. hypergroups of order less than 8. in this survey our main goal is to realize a comparison between the two sequences. the second section of the paper gives a survey of the construction of the sequences of join spaces associated with a hypergroupoid or with a non-empty set endowed with an (intuitionistic) fuzzy set. section 3 is dedicated to some computations of the two grades: the fuzzy and the intuitionistic fuzzy grade of a hypergroupoid. in section 4 we focus our comparison between the two grades on the cases of the complete hypergroups and of the i.p.s. hypergroups of order less than 8. in the last section we conclude with the statement of some open problems. 22 fuzzy grade of a hypergroupoid: a survey of some recent researches 2 sequences of the join spaces associated with a hypergroupoid we briefly recall the construction of the sequences of join spaces and fuzzy sets/intuitionistic fuzzy sets associated with a hypergroupoid. for a comprehensive overview of hypergroups theory the reader is refereed to [11, 17, 31] and for fuzzy and intuitionistic fuzzy sets theory see [32, 5, 6, 7, 8]. we will use the terminology and the notations from [13, 23, 30]. let x be a nonempty set. a fuzzy set a in x is characterized by a membership function µa : x −→ [0, 1], where for any x ∈ x, the value µa(x) represents the grade of membership of x in a. more generally, any function µ : x −→ [0, 1] is called a fuzzy subset of x. we denote by fs(x) the set of all fuzzy subsets of x. an atanassov’s intuitionistic fuzzy set a in x is a triplet of the form a = {(x,µa(x),λa(x)) | x ∈ x}, where, for any x ∈ x, the degree of membership of x (namely µa(x)) and the degree of non-membership of x (namely λa(x)) verify the relation 0 ≤ µa(x) + λa(x) ≤ 1; for simplicity, we denote it by a = (µ,λ). we denote by ifs(x) the set of all intuitionistic fuzzy subsets of x. for any nonempty set h endowed with a fuzzy subset α ∈ fs(h), corsini[12] defined a join space (h,◦α), where the hyperproduct is defined as it follows: x◦α y = {z ∈ h | α(x) ∧α(y) ≤ α(z) ≤ α(x) ∨α(y)} (1) conversely, from any hypergroupoid (h,◦) corsini[13] defined a fuzzy subset µ̃ ∈ fs(h) by the following formula: µ̃(u) = ∑ (x,y)∈q(u) 1 |x◦y| q(u) (2) where q(u) = {(a,b) ∈ h2 | u ∈ a◦ b}, q(u) = |q(u)|. if q(u) = ∅, we set µ̃(u) = 0. in 2010 cristea and davvaz[24] defined in a similar way an intuitionistic 23 irina cristea fuzzy subset associated with a hypergroupoid (h,◦): µ(u) = ∑ (x,y)∈q(u) 1 |x◦y| n2 , λ(u) = ∑ (x,y)∈q(u) 1 |x◦y| n2 (3) iterating equations (1) and (2) one obtains a sequence (ih = (h,◦i), µ̃i−1)i≥1 of join spaces and fuzzy sets associated with h. the sequence stops when there exist two consecutive isomorphic join spaces. the length of this sequence, that is the number of the non-isomorphic join spaces in the sequence, is called the fuzzy grade of the hypergroupoid h. more exactly we have the following definition. definition 2.1. [14] a hypergroupoid h has the fuzzy grade m ∈ n∗, and we write f.g.(h) = m if, for any 1 ≤ i < m, the join spaces ih and i+1h associated with h are not isomorphic and, for any s > m, sh is isomorphic with mh. we say that the hypergroupoid h has the strong fuzzy grade m, and we write s.f.g.(h) = m, if f.g.(h) = m and, for all s,s > m, sh = mh. similarly, iterating equations (1) and (3) one obtains two sequences (ih = (h,◦µi∧λi ); ai = (µi,λi))i≥0 and ( ih = (h,◦µi∨λi ); ai = (µi,λi))i≥0 of join spaces and intuitionistic fuzzy sets associated with h. the sequences stop when there exist two consecutive isomorphic join spaces and their lengths are called the lower/upper intuitionistic fuzzy grade of the hypergroupoid h. let’s see better their definitions. definition 2.2. [24] a set h endowed with an intuitionistic fuzzy set a = (µ,λ) has the lower intuitionistic fuzzy grade m and we write l.i.f.g.(h) = m if, for any 0 ≤ i < m, the join spaces (h,◦µi∧λi ) and (h,◦µi+1∧λi+1 ) associated with h are not isomorphic and, for any s ≥ m, sh is isomorphic with m−1h. definition 2.3. [24] a set h endowed with an intuitionistic fuzzy set a = (µ,λ) has the upper intuitionistic fuzzy grade m and we write u.i.f.g.(h) = m if, for any 0 ≤ i < m, the join spaces (h,◦µi∨λi ) and (h,◦µi+1∨λi+1 ) associated with h are not isomorphic and, for any s ≥ m, sh is isomorphic with m−1h. the construction of the above sequences can start in two different ways: 1. from a hypergroupoid (h,◦); 24 fuzzy grade of a hypergroupoid: a survey of some recent researches 2. from a nonempty set h endowed with a fuzzy set or an intuitionistic fuzzy set. it is easy to prove that in both cases one obtains the same fuzzy grade. but what can we say about the other two sequences of join spaces and intuitionistic fuzzy sets? if we start the construction from a hypergroupoid (h,◦), we obtain only one sequence of join spaces, because the first join spaces 0h = (h,◦µ∧λ) and 0h = (h,◦µ∨λ) are always isomorphic. indeed, since, for any x ∈ h, the sum µ(x) + λ(x) is always constant, it follows that µ(x) = µ(y) if and only if λ(x) = λ(y). therefore in this case l.i.f.g.(h) = u.i.f.g.(h) and it is simply called the intuitionistic fuzzy grade of h (shortly i.f.g.(h)). on the other side, if we start the construction from a nonempty set h endowed with an intuitionistic fuzzy set, one may obtain two different sequences with different lengths. we only illustrate this case by the following example. example 2.1. [24] on h = {a,b,c,d} we define the atanassov’s intuitionistic fuzzy set: µ(a) = 0.25 µ(b) = 0.25 µ(c) = 0.30 µ(d) = 0.10 λ(a) = 0.40 λ(b) = 0.40 λ(c) = 0.50 λ(d) = 0.90. to start with, we construct the first sequence of join spaces and we determine its l.i.f.g.(h). for the associated join space ◦µ∧λ a b c d a {a,b} {a,b} {a,b,c} {a,b,d} b {a,b} {a,b} {a,b,c} {a,b,d} c {a,b,c} {a,b,c} c h d {a,b,d} {a,b,d} h d we calculate the atanassov’s intuitionistic fuzzy set associated with h as in (3) and we obtain the following values: µ1(a) = 31/96 µ1(b) = 31/96 µ1(c) = 17/96 µ1(d) = 17/96 λ1(a) = 12/96 λ1(b) = 12/96 λ1(c) = 26/96 λ1(d) = 26/96, therefore µ1 ∧ λ1(a) = µ1 ∧ λ1(b) < µ1 ∧ λ1(c) = µ1 ∧ λ1(d) and thereby it 25 irina cristea results the following join space ◦µ1∧λ1 a b c d a {a,b} {a,b} h h b {a,b} {a,b} h h c h h {c,d} {c,d} d h h {c,d} {c,d} for any x ∈ h, we compute µ2(x) = 4/16 and λ2(x) = 2/16; thus, for any x,y ∈ h, x◦µ2∧λ2 y = h. so the first sequence of join spaces associated with h has 3 elements, that is l.i.f.g.(h) = 3. now, in order to determine the u.i.f.g.(h), we start with the join space ◦µ∨λ a b c d a {a,b} {a,b} {a,b,c} h b {a,b} {a,b} {a,b,c} h c {a,b,c} {a,b,c} c {c,d} d h h {c,d} d note that 〈0h,◦µ∧λ〉 and 〈 0h,◦µ∨λ〉 are not isomorphic. since µ1(a) = 13/48 µ1(b) = 13/48 µ1(c) = 13/48 µ1(d) = 9/48 λ1(a) = 9/48 λ1(b) = 9/48 λ1(c) = 9/48 λ(d) = 13/48 it follows that µ1 ∨λ1(x) = 13/48, whenever x ∈ h, which means that, for any x,y ∈ h, x◦µ1∨λ1 y = h. therefore the second sequence of join spaces associated with h has 2 elements, that is u.i.f.g.(h) = 2. sometimes the computations of the values of the membership functions µ̃i,µi,λi, i ≥ 0 could take much time, thus it is useful to determine the (intuitionistic) fuzzy grade of a hypergroupoid without calculating all these values. in order to realize this we introduce some notations. with any join space ih in the sequence of join spaces and fuzzy sets corresponding to h, one may associate an ordered chain (ic1, ic2, ..., icr) and an ordered r-tuple (ik1, ik2, ..., ikr), where 1. ∀j ≥ 1: x,y ∈icj ⇐⇒ µ̃i−1(x) = µ̃i−1(y), 2. for x ∈icj and z ∈ick, if j < k then µ̃i−1(x) < µ̃i−1(z) 3. ikj = |icj|, for all j. we have similar notations for the sequence (ih = (h,◦µi∧λi ); ai = (µi,λi))i≥0. with any join space ih one may associate an ordered chain ( ic1, ic2, ..., icr) and an ordered r-tuple (ik1, ik2, ..., ikr), where 26 fuzzy grade of a hypergroupoid: a survey of some recent researches 1. ∀j ≥ 1: x,y ∈icj ⇐⇒ µi ∧λi(x) = µi ∧λi(y), 2. for x ∈icj and z ∈ick, if j < k then µi ∧λi(x) < µi ∧λi(z) 3. ikj = |icj|, for all j. one of the crucial questions we have to ask is: when two consecutive join spaces in these sequences are isomorphic? a first answer was given by corsini and leoreanu [16] in 1995. theorem 2.1. [16] let ih and i+1h be two consecutive join spaces associated with h determined by the membership functions µ̃i−1 and µ̃i, where ih = r1⋃ l=1 cl, i+1h = r2⋃ l=1 c′l and (k1,k2, ...,kr1 ) is the r1-tuple associated with ih, (k′1,k ′ 2, ...,k ′ r2 ) is the r2-tuple associated with i+1h. the join spaces ih and i+1h are isomorphic if and only if r1 = r2 and (k1, . . . ,kr1 ) = (k ′ 1, . . . ,k ′ r1 ) or (k1, . . . ,kr1 ) = (k ′ r1 , . . . ,k′1). a similar theorem can be formulated also for the join spaces in the sequence (ih = (h,◦µi∧λi ); ai = (µi,λi))i≥0. in order to use this result we need to determine both r-tuples associated with ih and i+1h. but ştefănescu and cristea [30] have given a sufficient condition such that two consecutive join spaces are not isomorphic, that is the sequence doesn’t stop, using only one r-tuple. theorem 2.2. [30] let (k1,k2, ...,kr) be the r-tuple associated with the join space ih. if (k1, . . . ,kr) = (kr,kr−1, . . . ,k1), then the join spaces ih and i+1h are not isomorphic. a second crucial problem arises: does there exist a hypergroupoid h such that s.f.g.(h)/i.f.g.(h) = n, whenever n is a natural number? the anser is yes, and an example is given in the following fundamental result. theorem 2.3. [30, 24] let h = {x1,x2, . . . ,xs}, where s = 2n, n ∈ n∗ \ {1, 2}, be the commutative hypergroupoid defined by the hyperproduct xi ◦xi = xi, 1 ≤ i ≤ n, xi ◦xj = {xi,xi+1, . . . ,xj}, 1 ≤ i < j ≤ n. then s.f.g.(h) = i.f.g.(h) = n. moreover, h is a join space. 27 irina cristea 3 computation of the fuzzy grade/intuitionistic fuzzy grade of a hypergroupoid let (h,◦) be a hypergroupoid. we define the following equivalence on h xreµy ⇐⇒ µ̃(x) = µ̃(y). we determine f.g.(h) when |h/reµ| ∈ {2, 3}. theorem 3.1. [2] let (h,◦) be a finite hypergroupoid. 1. if |h/reµ| = 2, that is (k1,k2) is the pair associated with h, then s.f.g.(h) = 2 whenever k1 = k2, and f.g.(h) = 1 otherwise. 2. if |h/reµ| = 3, that is (k1,k2,k3) is the triple associated with h, and (a) if k1 = k2 = k3, then f.g.(h) = 2 (b) if k1 = k3 6= k2, then f.g.(h) = 2 whenever k2 6= 2k3 and s.f.g.(h) = 3, otherwise (c) if k1 < k2 = k3, then f.g.(h) = 1 (d) if k1 = k2 < k3, then f.g.(h) = 1 whenever p = 2k 3 3 − 8k31 − k21k3 + 5k1k 2 3 > 0 and f.g.(h) = 3, otherwise (e) if k1 6= k2 6= k3, there is no precise order between µ̃1(x), µ̃1(y), µ̃1(z). a similar result we obtain for the quotient h/rµ∧λ. theorem 3.2. [3] let (h,◦) be a finite hypergroupoid. 1. if |h/rµ∧λ| = 2, that is (k1,k2) is the pair associated with h, then i.f.g.(h) = 2 whenever k1 = k2, and i.f.g.(h) = 1, otherwise. 2. if |h/rµ∧λ| = 3, that is (k1,k2,k3) is the triple associated with h, and (a) if k1 = k2 = k3, then i.f.g.(h) = 2 (b) if k1 = k3 6= k2, then i.f.g.(h) = 2 whenever k2 6= 2k3 and i.f.g.(h) = 3, otherwise (c) if k1 < k2 = k3, then i.f.g.(h) = 1 whenever 2k 2 2 > 3k1k2 + 3k 2 1 and i.f.g.(h) = 3, otherwise (d) if k1 = k2 < k3, then i.f.g.(h) = 1 whenever k3 6= 2k1 and i.f.g.(h) = 2, otherwise (e) if k1 6= k2 6= k3, then there is no precise order between µ1(x) ∧ λ1(x),µ1(y) ∧λ1(y),µ1(z) ∧λ1(z). 28 fuzzy grade of a hypergroupoid: a survey of some recent researches angheluţă and cristea [3] have studied the intuitionistic fuzzy grade of a hypergroupoid in the case of some particular ternaries. we integrate here this theorem with similar results for the fuzzy grade. in the following the ternary (k1,k2,k3) associated with h is intended respected with the equivalence reµ (when we talk about the fuzzy grade) and rµ∧λ, respectively, (when we talk about the intuitionistic fuzzy grade). proposition 3.1. let h be a hypergroupoid of cardinality 2k, k ≥ 3, with the ternary associated with h of the type (k,k − 1, 1). then s.f.g.(h) = i.f.g.(h) = 1. generalizing, we obtain the same result for the ternary (pk,p(k−1),p), with p,k ∈ n \{0, 1}. proof. we prove only that s.f.g.(h) = 1. for the proof of the fact that i.f.g.(h) = 1, see [3]. let h be a hypergroupoid such that |h| = 2k, k ≥ 3, and (k,k−1, 1) be the ternary associated with h determined by the equivalence reµ. thus h can be written as the union h = c1∪c2∪c3, with |c1| = k, |c2| = k−1, |c3| = 1. using equation (2), for any x ∈ c1,y ∈ c2,z ∈ c3, one obtains µ̃1(x) = 4k2 −k − 1 3k2(2k − 1) µ̃1(y) = 8k3 + 2k2 − 12k + 4 2k(2k − 1)(3k2 − 1) µ̃1(z) = 4k − 2 k(4k − 1) after some computations that we omit here for lack of space, it results the following relation µ̃1(x) ≤ µ̃1(y) ≤ µ̃1(z), which means that (k,k − 1, 1) is the ternary associated with the join space 1h and thereby s.f.g.(h) = 1. proposition 3.2. let (2k,k, 1), with k ≥ 2, be the ternary associated with a hypergroupoid h. then 1. s.f.g.(h) = 1. 2. i.f.g.(h) = 3 and the sequence of join spaces is cyclic, that is 3lh ' 3h, 3l+1h ' 1h, 3l+2h ' 2h, for any l ∈ n∗. 29 irina cristea proof. considering (2k,k, 1), with k ≥ 2, be the ternary associated with a hypergroupoid h respected with the equivalence reµ, we decompose h as the union h = c1 ∪ c2 ∪ c3, with |c1| = 2k, |c2| = k, |c3| = 1. using again equation (2), for any x ∈ c1,y ∈ c2,z ∈ c3, one obtains µ̃1(x) = 15k + 11 6(2k + 1)(3k + 1) µ̃1(y) = 21k2 + 58k + 25 3(k + 1)(3k + 1)(5k + 6) µ̃1(z) = 13k2 + 10k + 1 (k + 1)(3k + 1)(6k + 1) for any k ≥ 2, it follows that µ̃1(x) ≤ µ̃1(y) ≤ µ̃1(z), that is s.f.g.(h) = 1. for the second part of the theorem see [3]. proposition 3.3. let (h,k, 1), with 2 ≤ h ≤ k, be the ternary associated with a hypergroupoid h. 1. then s.f.g.(h) = 1. 2. if h = 2 and (a) k = 2, then i.f.g.(h) = 3; (b) k = 3, then i.f.g.(h) = 2; (c) k ≥ 4, then i.f.g.(h) = 3 and the sequence of join spaces associated with h is cyclic. 3. if h > 2, then i.f.g.(h) = 1, whenever k ≥ h. proof. since (h,k, 1), with 2 ≤ h ≤ k, is the ternary associated with a hypergroupoid h respected with the equivalence reµ, we write h as the union h = c1 ∪c2 ∪c3, with |c1| = h, |c2| = k, |c3| = 1. calculating the values of the membership function µ̃1 by using the formula (2), we obtain, 30 fuzzy grade of a hypergroupoid: a survey of some recent researches for any x ∈ c1,y ∈ c2,z ∈ c3, that µ̃1(x) = h2 + 3k2 + 4hk + 3h + 5k (h + k)(h + k + 1)(h + 2k + 2) µ̃1(y) = 3h2k2 + 7h2k + 4hk3 + 9hk2 + 7hk + k4 + 4k3 + 3k2 + 2h2 (h + k)(h + k + 1)(k + 1)(2hk + 2h + k2 + 2k) µ̃1(z) = 5hk + 3h + 3k2 + 4k + 1 (h + k + 1)(k + 1)(2h + 2k + 1) similarly as in the previous two lemmas, we prove that µ̃1(x) ≤ µ̃1(y) ≤ µ̃1(z) and thus s.f.g.(h) = 1. for the second part of the theorem see [3]. 4 (intuitionistic) fuzzy grade of special types of hypergroups in this section we deal with two classes of hypergroups: the complete hypergroups and the hypergroups with partial scalar identities, shortly called i.p.s. hypergroups. we determine the fuzzy and intuitionistic fuzzy grades of all these hypergroups of order less than 8. this part of the paper is based on the articles of corsini and cristea [14, 15], cristea [21], cristea and davvaz [24], angheluţă and cristea [2, 3], davvaz, hassani-sadrabadi and cristea [25, 26, 27]. 4.1 the complete hypergroups we start with the characterization of the complete hypergroups. a hypergroup (h,◦) is a complete hypergroup if it can be written as the union of its subsets h = ∪g∈gag, where 1. (g, ·) is a group; 2. for any (g1,g2) ∈ g2, g1 6= g2, we have ag1 ∩ag2 = ∅; 3. if (a,b) ∈ ag1 ×ag2 , then a◦ b = ag1g2. 31 irina cristea let g = {g1, . . . ,gm} be a finite group. then with h = m⋃ i=1 agi we may associate an m-tuple [k1, . . . ,km], where ki = |agi|. using these notations and based on the fact that, for any u ∈ h, ∃!gu ∈ g : u ∈ agu , corsini [13] has determined the formula for the membership function µ̃ associated with a complete hypergroup as the following one: µ̃(u) = 1 |agu| . similarly, the formula of the membership functions µ,λ for a complete hypergroup is simpler that those in the general case, as cristea and davvaz proved [24]. defining on h the following equivalence u ∼ v ⇐⇒∃g ∈ g : u,v ∈ ag, one obtains that: µ̄(u) = |q(u)| |agu| · 1 n2 , λ̄(u) = (∑ v/∈û |q(v)| |agv| ) · 1 n2 . we notice that, if g1 and g2 are non isomorphic groups of the same cardinality and h1, h2 are the complete hypergroups obtained by them, then f.g.(h1) = f.g.(h2), but their i.f.g. may be different. cristea [21] determined the fuzzy grade of all complete hypergroups of order less than 7. later on, davvaz together with hassani-sadrabadi and cristea [27] determined their intuitionistic fuzzy grade. we recall here these results. theorem 4.1. let h be a complete hypergroup of order n ≤ 6. 1. there are two non isomorphic complete hypergroups of order three that have s.f.g.(h) = i.f.g.(h) = 1. 2. among the five non isomorphic complete hypergroups of order four, three of them have s.f.g.(h) = i.f.g.(h) = 1 and two of them have s.f.g.(h) = i.f.g.(h) = 2. 3. there are 12 non isomorphic complete hypergroups of order 5. all of them have s.f.g.(h) = 1. nine of them have i.f.g.(h) = 1 and the other three have i.f.g.(h) = 3. 32 fuzzy grade of a hypergroupoid: a survey of some recent researches 4. there are 21 non isomorphic complete hypergroups of order 6: 17 with s.f.g.(h) = 1 and 4 with s.f.g.(h) = 2. 16 with i.f.g.(h) = 1, 3 with i.f.g.(h) = 2 and 2 with i.f.g.(h) = 3. angheluţă and cristea [2] have studied the fuzzy grade of some particular complete hypergroups. it remains an open problem, and not a very easy one, to determine their intuitionistic fuzzy grade. theorem 4.2. let h be a complete hypergroup of type 1. [p,p, ...,p︸ ︷︷ ︸ k times ,kp], where n = |h| = 2kp. then s.f.g.(h) = 2. 2. [p,p, . . . ,p︸ ︷︷ ︸ s times ,k,k, . . . ,k︸ ︷︷ ︸ t times ,ps], 2 ≤ p < k < ps, n = |h| = 2ps + kt. • for n = 4ps, s.f.g.(h) = 3, • for kt 6= 2ps, f.g.(h) = 2. 3. [k,k, . . . ,k︸ ︷︷ ︸ l times ,p,p, . . . ,p︸ ︷︷ ︸ s times ,s,s, . . . ,s︸ ︷︷ ︸ p times , l, l, . . . , l︸ ︷︷ ︸ k times ], 2 ≤ k < p < s < l, n = |h| = 2(ps + kl). • if kl = ps, then s.f.g.(h) = 3, • otherwise f.g.(h) = 2. 4. [1, . . . , 1︸ ︷︷ ︸ k times , 2, . . . , 2︸ ︷︷ ︸ k 2 times , 3, . . . , 3︸ ︷︷ ︸ k 3 times , . . . ,m− 1, . . . ,m− 1︸ ︷︷ ︸ k m−1 times ,k], k = m.c.m.(1, 2, . . . ,m− 1), n = |h| = mk = 2s−1k. then s.f.g.(h) = s. 4.2 the i.p.s. hypergroups an i.p.s. hypergroup (i.e. a hypergroup with partial scalar identities) is a particular type of canonical hypergroup. we recall that a canonical hypergroup is a commutative hypergroup (h,◦) with a scalar identity, where every element a ∈ h has a unique inverse a−1 such that it satisfies the following properties 1. if y ∈ a◦x, then x ∈ a−1 ◦y; 2. x ∈ x◦a =⇒ x = x◦a 33 irina cristea the canonical hypergroups were introduced for the first time by krasner [28] as the additive structures of the hyperfields and then mittas [29] studied them independently from the others operations. later on corsini determined all i.p.s. hypergroups of order less than 9, proving that they are strongly canonical. corsini and cristea [14, 15] calculated the fuzzy grade for all i.p.s. hypergroups of order less than 8 and their intuitionistic fuzzy grade have been determined by davvaz, hassan-sadrabadi and cristea [25, 26]. here we summarize these results. theorem 4.3. 1. there exists one i.p.s. hypergroup h of order 3 and f.g.(h) = i.f.g.(h) = 1. 2. there exist 3 i.p.s. hypergroups of order 4 with f.g.(h1) = i.f.g.(h1) = 1, f.g.(h2) = 1, i.f.g.(h2) = 2, f.g.(h3) = i.f.g.(h3) = 2. 3. there exist 8 i.p.s. hypergroups of order 5: one has f.g.(h) = 2, all the others have f.g.(h) = 1; four of them have i.f.g.(h) = 1, three have i.f.g.(h) = 2 and one is with i.f.g.(h) = 3. 4. there exist 19 i.p.s. hypergroups of order 6: 14 of them have f.g.(h) = 1, 4 have f.g.(h) = 2 and one has f.g.(h) = 3; ten of them have i.f.g.(h) = 1, eight of them have i.f.g.(h) = 2 and only for one of them we find i.f.g.(h) = 3. 5. there exist 36 i.p.s. hypergroups of order 6: 27 of them with f.g.(h) = 1, 8 with f.g.(h) = 2, 1 with f.g.(h) = 3; 10 of them with i.f.g.(h) = 1, 10 with i.f.g.(h) = 2, 9 with i.f.g.(h) = 3, 5 with i.f.g.(h) = 4, 2 with i.f.g.(h) = 5. moreover, for 18 of them we find that the associated sequences of join spaces and intuitionistic fuzzy sets are cyclic. 5 future work based on the definition of the intuitionistic fuzzy set associated with a hypergroupoid [24], ashgari-larimi and cristea [4] have defined the atanassov’s intuitionistic fuzzy index of a hypergroupoid. it would be interesting to obtain some relations between this index and the intuitionistic fuzzy grade. we think that, finding necessary and sufficient conditions in order that the sequences of join spaces (ih = (h,◦µi∧λi ); ai = (µi,λi))i≥0 and ( ih = (h,◦µi∨λi ); ai = (µi,λi))i≥0 associated with a nonempty set h endowed with 34 fuzzy grade of a hypergroupoid: a survey of some recent researches an intuitionistic fuzzy set (like in section 2) coincide, could be another line of research on this argument. we also intend to find in our future work some possible connection between hypergroups and automata. for example, what can we say about its (intuitionistic) fuzzy grade? does there exist an automaton with (intuitionistic) fuzzy grade of its state hypergroup equal to a given natural number? references [1] r. ameri, m.m. zahedi, hypergroup and join space induced by a fuzzy subset, pure math. appl., 8 (1997). 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[32] l.a. zadeh, fuzzy sets, inform and control, 8 (1965), 338-353. 37 irina cristea 38 ratio mathematica 23 (2012), 39–50 issn: 1592-7415 hv-semigroups as noise pollution models in urban areas achilles dramalidis, thomas vougiouklis democritus university of thrace, school of education, 681 00 alexandroupolis, greece adramali@psed.duth.gr, tvougiou@eled.duth.gr abstract poor urban planning may give rise to noise pollution, since side-byside industrial and residential buildings can result in noise pollution in the residential area. in this paper we represent the noise pollution with an hv-semigroup. more specific, we introduce the concept of right reproductive hv-semigroup which seems to be a useful tool to study the noise pollution problem in urban areas. key words: hv-structures, hv-group, hv-semigroup msc2010: 20n20. 1 introduction the hv-structures, introduced in the fourth aha congress [13], where the known axioms were replaced by weaker ones. more precisely in axioms like associativity, commutativity and so on, the equality was replaced by the non-empty intersection. in (h, ·) we abbreviate by wass, the weak associativity : (xy)z ∩x(yz) 6= ∅, ∀x,y,z ∈ h and by cow, the weak commutativity : xy ∩yx 6= ∅, ∀x,y ∈ h recall some basic definitions [14]: definition 1.1 let h be a non-empty set and · : h×h → p(h)−{∅} be a hyperoperation defined on h. also, we have abbreviated the ”hyperoperation” by ”hope”[16]. the (h, ·) is called hv-semigroup if it is wass and it is called a. dramalidis, t. vougiouklis hv-group if it is hv-semigroup and the reproduction axiom x · h = h · x = h,∀x ∈ h, is valid. the hyperstructure (r, +, ·) is called hv-ring if both hopes (+) and (·) are wass, the reproduction axiom is valid for (+) and (·) is weak distributive with respect to (+): x(y + z) ∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅,∀x,y,z ∈ r. an hv-ring (r, +, ·) is called dual hv-ring [7] if the hyperstructure (r, ·, +) is hv-ring, too. let (h, ·) be a hypergroupoid. an element e ∈ h is called right unit element if a ∈ a · e,∀a ∈ h and is called left unit element if a ∈ e ·a,∀a ∈ h. the element e ∈ h is called unit element if a ∈ a ·e∩e ·a,∀a ∈ h. an element e ∈ h is called right scalar unit element if a = a · e,∀a ∈ h and is called left scalar unit element if a = e ·a,∀a ∈ h. it is called scalar unit element if a = a · e = e ·a,∀a ∈ h. let (h, ·) be a hypergroupoid endowed with at least one unit element. an element a′ ∈ h is called an inverse element of the element a ∈ h, if there exists a unit element e ∈ h such that e ∈ a′ ·a∩a·a′. the element a ∈ h is called right simplifiable element (resp. left ) if ∀x,y ∈ h the following is valid: x ·a = y ·a ⇒ x = y (resp. a ·x = a ·y ⇒ x = y). moreover, let us define here: if x ∈ x ·y (resp. x ∈ y ·x ) ∀y ∈ h, then, x is called left absorbing-like element (resp. right absorbing-like element ).the nth power of an element h, denoted hs, is defined to be the union of all expressions of n times of h, in which the parentheses are put in all possible ways. an hv-group (h, ·) is called cyclic with finite period with respect to h ∈ h, if there exists a positive integer s, such that h = h1 ∪ h2 ∪ ... ∪ hs. the minimum such an s is called period of the generator h. if all generators have the same period, then h is cyclic with period. if there exists h ∈ h and s positive integer, the minimum one, such that h = hs then h is called singlepower cyclic and h is a generator with single-power period s. the cyclicity in the infinite case is defined similarly. thus, for example, the hv-group (h, ·) is called single-power cyclic with infinite period with generator h if every element of h belongs to a power of h and there exists s0 ≥ 1, such that ∀s ≥ s0 we have: h1 ∪h2 ∪ ... ∪hs−1 ⊂ hs . an element a ∈ h is called idempotent element if a2 = a. the main tool to study hv-groups, is the fundamental relation β*. the relation β* is defined in hv-groups, as the smallest equivalence relation on h, 40 hv-semigroups as noise pollution models in urban areas such that the quotient would be group. it is called the fundamental group and β* is called the fundamental equivalence relation on h. the relation β is defined on an hv-group in the same way as in a hypergroup. the basic theorem is the following [14]: let (h, ·) be an hv-group and denote by u the set of all finite products of elements of h. we define the relation β on h by setting xβy iff {x,y} ⊂ u where u ∈ u. then β* is the transitive closure of β. an element s ∈ h is called single if β*(s) = {s}. the set of all single elements is denoted by sh and if sh 6= ∅ then one can find easily the fundamental classes. a way to find the fundamental classes is given in [6],[13],[14]. for more definitions, results and applications onhv-structures, see also books [4],[5],[14]. 2 the noise problem noise pollution is displeasing human, animal or machine-created sound that disrupts the activity or balance of human or animal life. the source of most outdoor noise worldwide is not only transportation systems (including motor vehicle noise, aircraft noise and rail noise), but, noise caused by people as well (audio entertainment systems, electric megaphones and loud people) [9]. the fact that noise pollution is also a cause of annoyance, is that, a 2005 study by spanish researchers found that in urban areas households are willing to pay approximately 4 euros per decibel per year for noise reduction [2]. poor urban planning may give rise to noise pollution, since side-by-side industrial and residential buildings can result in noise pollution in the residential area. we set the following problem: the noise pollution that comes from a certain block of flats in urban areas, obviously annoys not only the block of flats itself but possibly neighboring blocks of flats or buildings, as well. if every city is considered as a set ω with elements the blocks of flats or the buildings, then, the above situation could be described with an algebraic hyperstructure and its properties. in this paper, we present the right reproductive hv-semigroup, as a tool to study the noise pollution problem in urban areas. one can find hyperstructures on related problems in several survey papers like [10], [11],[12] and papers with a wide variety of applications [1], [3], [15]. 41 a. dramalidis, t. vougiouklis definition 2.1 a hypergroupoid (h,*), such that, the weak associativity holds and ∀x ∈ h, h*x=h , is called right reproductive hv-semigroup. a hypergroupoid h,*), such that, the weak associativity holds and ∀x ∈ h, x*h=h , is called left reproductive hv-semigroup now we give the following definition: definition 2.2 let ω 6= ∅ and f : ω → p(ω) be a map, then we define a hyperoperation rl: ω × ω → p(ω), on ω as follows: ∀x,y ∈ ω, we set xrly = f(x) ∪{x} we call the above hyperoperation (rl) noise hyperoperation. remark that the noise hyperoperation, always contains the element x ∈ ω. that means that the element x ∈ ω could be considered as the representative of the elements of the set xrly. so, we symbolize: xrly = f(x) ∪{x} = [x] if, ∀x ∈ ω,x ∈ f(x) then the hyperoperation is simplified as xrly = f(x) = [x] therefore, the noise hyperoperation (rl) depends only on the left element. that means that if one composes an element x, on the left, with any other element y, on the right, then the result is always the same set [x]. example 2.1 consider a set ω = {x1,x2,x3,x4,x5,x6,x7,x8,x9} and a map f : ω → p(ω) such that: f(x1) = {x2},f(x2) = {x2,x3},f(x3) = {x2},f(x4) = {x4} f(x5) = {x5,x6,x7},f(x6) = {x6,x7},f(x7) = {x5}, f(x8) = {x8,x9},f(x9) = ∅. then, as in the defined above noise hyperoperation: [x1] = f(x1) ∪{x1} = {x1,x2}, [x2] = f(x2) = {x2,x3}, [x3] = f(x3) ∪{x3} = {x2,x3}, [x4] = f(x4) = {x4}, [x5] = f(x5) = {x5,x6,x7}, [x6] = f(x6) = {x6,x7} [x7] = f(x7) ∪{x7} = {x5,x7}, [x8] = f(x8) = {x8,x9}, [x9] = f(x9) ∪{x9} = {x9}. then, the ”multiplication” table of (rl) is given by: 42 hv-semigroups as noise pollution models in urban areas rl x1 x2 x3 x4 x5 x6 x7 x8 x9 x1 x1,x2 x1,x2 x1,x2 x1,x2 x1,x2 x1,x2 x1,x2 x1,x2 x1,x2 x2 x2,x3 x2,x3 x2,x3 x2,x3 x2,x3 x2,x3 x2,x2 x2,x3 x2,x3 x3 x2,x3 x2,x3 x2,x3 x2,x3 x2,x3 x2,x3 x2,x2 x2,x3 x2,x3 x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 x5 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x5,x6,x7 x6 x6,x7 x6,x7 x6,x7 x6,x7 x6,x7 x6,x7 x6,x7 x6,x7 x6,x7 x7 x5,x7 x5,x7 x5,x7 x5,x7 x5,x7 x5,x7 x5,x7 x5,x7 x5,x7 x8 x8,x9 x8,x9 x8,x9 x8,x9 x8,x9 x8,x9 x8,x9 x8,x9 x8,x9 x9 x9 x9 x9 x9 x9 x9 x9 x9 x9 example 2.2 let x 6= ∅ and m : x → [0, 1] be a fuzzy subset of x. we define the hyperoperation (◦) on x as follows: ∀x,y ∈ x, ◦ : x × x → p(x), such that x◦y = {z ∈ x/m(z) = m(x)} then, consider the map f(x) = {z ∈ x/m(z) = m(x)}. since x ∈ f(x),∀x ∈ x, as above we have the hyperoperation (rl) on x as follows : ∀x,y ∈ x, rl : x × x → p(x) , such that xrly = f(x). 3 the right reproductive hv-semigroup some properties of (rl) 1. xrlω = [x],∀x ∈ ω 2. [x]rly = [x]rl[y] ⊃ [x],∀y ∈ ω 3. x2 = xrlx = [x],∀x ∈ ω proposition 3.1 the hypergroupoid (ω, rl) is an hv-semigroup. proof. we have to prove that the weak associativity holds. indeed, ∀x,y,z ∈ ω xrl(yrlz) = ⋃ v∈yrlz (xrlv) = ⋃ v∈[y] (xrlv) = [x] (xrly)rlz = ⋃ w∈xrly (wrlz) = ⋃ w∈[x] (wrlz) = ⋃ w∈[x] [w] ⊃ [x] therefore (xrly)rlz ⊃ xrl(yrlz), so (xrly)rlz ∩xrl(yrlz) 6= ∅,∀x,y,z ∈ ω. 2 43 a. dramalidis, t. vougiouklis proposition 3.2 ∀x ∈ ω, ωrlx = ω and xrlω = [x]. proof. ∀x ∈ ω, ωrlx = ⋃ ω∈ω (ωrlx) = ⋃ ω∈ω [ω] = ω. on the other hand, ∀x ∈ ω,xrlω = ⋃ ω∈ω (xrlω) = [x].2 by propositions (3.1) and (3.2), we get that: proposition 3.3 the hypergoupoid (ω,rl) is a right reproductive hv-semigroup. remark that the right reproductive hv-semigroup (ω,rl) is an hv-group if, ∀x ∈ ω, we have xrlω = ω. proposition 3.4 the strong associativity of (rl) is valid iff we have⋃ w∈[x] (wrlz) = [x],∀x,z ∈ ω proof. let (x,y,z) ∈ ω3, such that (xrly)rlz = xrl(yrlz), then (xrly)rlz = xrl(yrlz) ⇒ [x]rlz = xrl[y] ⇒ [x]rlz = [x] ⇒ ⋃ w∈[x] (wrlz) = [x]. now,let (x,y,z) ∈ ω3, such that⋃ w∈[x] (wrlz) = [x] then, (xrly)rlz = [x]rlz = ⋃ w∈[x] (wrlz) = [x] xrl(yrlz) = xrl[y] = [x].2 for the hyperoperation (rl), we shall check conditions such that the strong or the weak commutativity is valid. proposition 3.5 . if y ∈ [x] and x ∈ [y],∀x,y ∈ ω, then the weak commutativity of (rl) is valid. the strong commutativity of (rl) is valid, iff [x] = [y],∀x,y ∈ ω. 44 hv-semigroups as noise pollution models in urban areas proof. let y ∈ [x] and x ∈ [y],∀x,y ∈ ω, then y ∈ [x] and x ∈ [y] ⇒ x,y ∈ [x] and x,y ∈ [y] ⇒ [x] ∩ [y] 6= ∅ ⇒ ⇒ (xrly) ∩ (yrlx) 6= ∅. the proof for the strong commutativity is straightforward.2 proposition 3.6 let (ω, +,rl) be an hv-ring. if xrlω = ω,∀x ∈ ω then the hyperstructure (ω, +,rl) is a dual hv-ring, i.e. both (ω, +,rl) and (ω,rl, +) are hv-rings. proof. from the remark of proposition 3.3, we have that the (ω,rl) is an hv-group. for the weak distributivity of (+) with respect to (rl) we have: ∀x,y,z ∈ ω x + (yrlz) ⊃ x + y and (x + y)rl(x + z) = ⋃ s∈x+y,t∈x+z (srlt) ⊃ ⋃ s∈x+y s = x + y so, [x + (yrlz)] ∩ [(x + y)rl(x + z)] 6= ∅,∀x,y,z ∈ ω similarly, the weak distributivity of (+) with respect to (rl) from the right side. 2 4 special elements since x ∈ xrly,∀x,y ∈ ω, the next proposition is obvious: proposition 4.1 (a) all the elements of ω are right unit elements with respect to (rl). (b) all the elements of ω are left absorbing-like elements with respect to (rl). proposition 4.2 the left scalar elements of the hv-semigroup (ω,rl), are left absorbing elements. proof. let u ∈ ω be a left scalar unit element, then urlx = x,∀x ∈ ω. but since u ∈ urlx,∀x ∈ ω, we get that urlx = u,∀x ∈ ω.2 proposition 4.3 the right scalar unit elements of the hv-semigroup (ω,rl), are idempotent elements. 45 a. dramalidis, t. vougiouklis proof. let α ∈ ω be a right scalar unit element, then xrlα = x,∀x ∈ ω. so, αrlα = α ⇒ α2 = α.2 proposition 4.4 if there exists x ∈ ω such that f(x)=x or [x]=x, then x is left absorbing element and every element of (ω,rl) is right scalar unit of x. proof. suppose there exist x ∈ ω such that [x]=x, then ∀y ∈ ω : xrly = [x] = x. that means that x is left absorbing element and every element of (ω,rl) is right scalar unit of x.2 since all the elements of the hv-semigroup (ω,rl) are right unit elements, let us denote [7] by ilrl (x,y) the set of the left inverses of the element x ∈ ω, associated with the right unit y ∈ ω , with respect to the hyperoperation (rl). the set of the right inverses of the element x ∈ ω , associated with the right unit y ∈ ω, with respect to the hyperoperation (rl), is denoted by irrl (x,y). proposition 4.5 y ∈ ilrl (x,y) proof. let x′ ∈ ω such that x′ ∈ ilrl (x,y) ⇒ y ∈ x ′rlx. but ∀x ∈ ω the relation y ∈ yrlx is valid. that means that y ∈ ilrl (x,y).2 proposition 4.6 irrl (x,y) = ω if and only if y ∈ [x]. proof. let y ∈ ω be right unit element and x ∈ ω,then y ∈ [x] ⇔ y ∈ xrlx′,∀x′ ∈ ω ⇔ x′ ∈ irrl (x,y),∀x ′ω ⇔ irrl (x,y) = ω.2 since x ∈ [x],∀x ∈ ω,the following is obvious. corollary 4.1 irrl (x,x) = ω remark 4.1 notice that, according to the example 2.1, the elements x4 and x9 are idempotent elements, since x 2 4 = x4 and x 2 9 = x9. they are, also, left absorbing elements, since x4rlx = x4 and x9rlx = x9,∀x ∈ ω. also, taking for example, the element x2 of ω, notice that i l rl (x,x2) = {x1,x2,x3}, ∀x ∈ ω. even more, since x2 ∈ [x1] we get that irrl (x1,x2) = ω and i r rl (x1,x1) = ω. 46 hv-semigroups as noise pollution models in urban areas 5 applications as we mentioned above, the noise pollution in urban areas coming from a spot, annoys a certain area in which the noisy spot belongs to. that was the motivation which led to the mathematical expression xrly = [x], ∀x,y ∈ ω. that means that if a city is considered as a set ω with elements its buildings (or spots which could produce noise pollution), then every building (or a spot) x, which is a source of noise pollution, together with any other building (or a spot) y of the city, will affect anyhow the noise pollution area [x], where x ∈ [x] and maybe y. it is clear, that the source of the noise pollution x, could not be seen as the center of a cyclic disk, but as any spot of a certain area which is affected by x. we shall try to explain some of the properties of the noise hyperoperation (rl) developed above, in terms of noise pollution problems in urban areas. the property x ∈ xrly,∀y ∈ ω means that the building x, as a source of noise pollution, first of all, annoys the residents of the building x. the property rl[y] = [x] means that the source of noise pollution x together with any region [y] is not only independent on the spots of the region [y] but the noise pollution region remains [x], as well. the property [x]rly = [x]rl[y] ⊃ [x] means that the noise pollution region that results when either the noise pollution region [x] operates with the spot y or with the region [y], is the same and anyhow this noise pollution region is bigger than [x]. the property xrly = xrlz means that [x] remains the noise pollution region when x as a source of noise pollution affects any other spot of the city ω. continuously, the relation xrlω 6= ω means that, the noise pollution region coming from spot x, can’t affect the whole city ω. the weak associativity which is expressed by the inclusion on the left parenthesis, i.e. (xrly)rlz ⊃ xrl(yrlz) actually means that, the noise pollution region coming from the noise pollution region [x] together with any spot, is not only bigger than that one which comes from the noise pollution spot x together with any other region but includes it, as well. an absorbing element, as in the relation αrlx = α, could be considered as a spot surrounded by a wall or a forest, which doesnt annoy any other spot of the city ω. since the weak associativity is valid, the concept of transitive closure can be applied here, in order to obtain the fundamental β* classes. the actual meaning of this situation is that the city ω can be divided, using the noise hyperoperation, in a partition, where every fundamental class does not annoy any other blocks of flats from other fundamental classes. the next example gives an idea: 47 a. dramalidis, t. vougiouklis example 5.1 according to the example 2.1, consider now that ω is a city where ω = {x1,x2,x3,x4,x5,x6,x7,x8,x9}. from ”multiplication” table of (rl),we obtain that β*(x1) = {x1,x2,x3}, β*(x4) = {x4},β*(x5) = {x5,x6,x7},β*(x8) = {x8,x9} . so,the fundamental semi-group ω/β* is: ω/β* = { {x1,x2,x3},{x4},{x5,x6,x7},{x8,x9} } and the ”multiplication” table is: ◦ x1 x4 x5 x8 x1 x1 x1 x1 x1 x4 x4 x4 x4 x4 x5 x5 x5 x5 x5 x8 x8 x8 x8 x8 in other words and beyond the mathematical content of the present example, the city ω was divided into four regions, where every region (fundamental class) does not annoy any other spot belonging to the rest regions. so, one could consider that among the four regions there exists a green park, full of trees, which absorbs the possible noise pollution caused by any of the four regions. since β*(x4) = {x4}, the element x4 ∈ ω (spot or building of the city) is a single element and that means that it doesn’t annoy any other spot of the city ω , so it can be considered as the remotest spot of the city. references [1] n. antampoufis, s1-hv-groups, s1-hypergroups and the ∂-operation, proceedings 10th aha congress 2008, brno czech republic, (2009), 99 112. 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[9] j. m. field, effect of personal and situational variables upon noise annoyance in residential areas, journal of the acoustical society of america, (1993), 93: 2753-2763. [10] a. maturo, alternative fuzzy operations and applications to social sciences. international journal of intelligent systems, doi: 10.1002/ int. 20383. issn 1098-111x, (2009), vol. 24, pp. 1243-1264. [11] a. maturo, b. ferri, on some applications of fuzzy sets and commutative hypergroups to evaluation in architecture and town-planning, ratio mathematica, 13, (1999), 51-60. [12] a. maturo, a. ventre, multilinear decision making, consensus and associated hyperstructures, 10th aha congress, brno, czech rep. 2008, (2009),181-190. [13] t. vougiouklis, the fundamental relation in hyperrings. the general hyperfield, proc. of the 4th aha, xanthi (1990), world scientific, 1991, 203-211. [14] t. vougiouklis, hyperstructures and their representations, monographs, hadronic press, usa, 1994. [15] t. vougiouklis, p. kambaki, algebraic models in applied research, jordan journal of mathematics and statistics, (jjms) 2008 1(1), 78-87. [16] t. vougiouklis, on the hyperstructures with ∂-hopes, 10th aha congress, brno, czech republic 2008, (2009), 281-296. 49 a. dramalidis, t. vougiouklis 50 ratio mathematica volume 44, 2022 medium domination decomposition of graphs ebin raja merly e1 saranya j2 abstract a set of vertices 𝑆 in a graph 𝐺 dominates 𝐺 if every vertex in 𝐺 is either in 𝑆 or adjacent to a vertex in 𝑆. the size of any smallest dominating set is called domination number of 𝐺. the concept of medium domination number was introduced by vargor and dunder which finds the total number of vertices that dominate all pairs of vertices and evaluate the average of this value. the medium domination number is a notation which uses neighbourhood of each pair of vertices. for g = (v, e) and ∀ u, v∈ v if u, v are adjacent they dominate each other, then atleast dom (u, v) = 1. the total number of vertices that dominate every pair of vertices is defined as tdv(g)=∑ dom (u, v), for every u, v∈ v(g). for any connected, undirected, loopless graph g of order p, the medium domination number md(g) = 𝑇𝐷𝑉(𝐺) 𝑝𝐶2 . in this paper we have introduced the new concept medium domination decomposition. a decomposition (𝐺1, 𝐺2, … , 𝐺𝑛 ) of a graph g is said to be medium domination decomposition (mdd) if ⌊𝑀𝐷(𝐺𝑖 )⌋ = 𝑖 − 1, 𝑖 = 1,2, . . . , 𝑛. keywords: domination number, medium domination number, medium domination decomposition. mathematical classification number: 05c69, 54d053 1 associate professor, department of mathematics, nesamony memorial christian college, marthandam. (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamilnadu, india), ebinmerly@gmail.com 2 research scholar (reg. no: 20113112092022), department of mathematics, nesamony memorial christian college, martha. (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamilnadu, india), saranyajohnrose@gmail.com 3 received on june 8th, 2022. accepted on aug 10th, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.890. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 59 mailto:ebinmerly@gmail.com mailto:saranyajohnrose@gmail.com ebin raja merly e and saranya j 1. introduction graph is a mathematical representation of a network and it describes the relationship between vertices and edges. let g = (v, e) be a simple, connected, undirected, loopless graph with p vertices and q edges and gi be the subgraph of g with pi vertices and qi edges, where 1 ≤ i ≤ n, n is the number of subgraphs of g. the length of a shortest 𝑢 − 𝑣 path in a connected graph g is called the distance from a vertex 𝑢 to a vertex 𝑣 . 𝑑(𝑢, 𝑣) denotes the distance between 𝑣 and 𝑢 . two 𝑢 − 𝑣 paths are internally disjoint if they have no vertices in common, other than 𝑢 and 𝑣. the degree of a vertex 𝑣 in a graph 𝐺 is the number of edges of incident with 𝑣 and denoted by deg(𝑣). the minimum degree among the vertices of a graph 𝐺 is denoted by 𝛿(𝐺). the maximum degree among vertices of a graph 𝐺 is denoted by ∆(𝐺). [1] the concept of medium domination number was introduced by vargor and dunder which finds the total number of vertices that dominate all pairs of vertices and evaluate the average of this value.[5] t.i. joel and e.e. merly introduced the concept of geodetic decomposition of graphs. motivated by the above we have introduced the new concept of medium domination decomposition of graphs. for basic terminologies in graph theorem, we refer [2], [3] and [4]. the following are the basic definitions and results needed for the main section. definition 1.1. [1] for 𝐺 = (𝑉, 𝐸) and ∀ 𝑢, 𝑣 ∈ 𝑉 , if 𝑢 and 𝑣 are adjacent they dominate each other, then atleast 𝑑𝑜𝑚 (𝑢, 𝑣) = 1. definition 1.2. [1] for 𝐺 = (𝑉, 𝐸) and ∀ 𝑢, 𝑣 ∈ 𝑉 , the total number of vertices that dominate every pair of vertices is defined as 𝑇𝐷𝑉(𝐺) = σ∀𝑢,𝑣∈𝑉(𝐺)𝑑𝑜𝑚 (𝑢, 𝑣). definition 1.3. [1] for any connected, undirected, loopless graph 𝐺 of order 𝑝 the medium domination number of 𝐺 is defined as 𝑀𝐷(𝐺) = 𝑇𝐷𝑉(𝐺) ( 𝑃 2 ) . 2. medium domination decomposition of graphs definition. 2.1 let 𝐺 be a simple connected (𝑝, 𝑞) graph. a decomposition (𝐺1, 𝐺2, … , 𝐺𝑛 ) of a graph 𝐺 is said to be a medium domination decomposition (𝑀𝐷𝐷) if ⌊𝑀𝐷(𝐺𝑖 )⌋ = 𝑖 − 1, 𝑖 = 1,2,3, … . , 𝑛. 60 medium domination decomposition of graphs example. 2.2 figure 2.3 here 𝑀𝐷(𝐺1) = 0.8, 𝑀𝐷(𝐺2) = 1 and 𝑀𝐷(𝐺3) = 2 that is ⌊𝑀𝐷(𝐺1)⌋ = 0, 𝑀𝐷(𝐺2) = 1 and 𝑀𝐷(𝐺3) = 2 remark. 2.4 i) star graph does not admit 𝑀𝐷𝐷 ii) 𝐾𝑝, 𝑝 ≤ 4, does not admit 𝑀𝐷𝐷 theorem. 2.5 if a graph 𝐺 admits 𝑀𝐷𝐷(𝐺1, 𝐺2, … , 𝐺𝑛), then 𝑝 ≥ 4 and 𝑞 ≥ 3. proof. since the medium domination number is one, when 𝑝 ≤ 3 and 𝑞 ≤ 2 and the medium domination number is two, when 𝑝 and 𝑞 is 3, we can’t get any subgraph with ⌊𝑀𝐷(𝐺𝑖 )⌋ = 0. note: 2.6 the converse of the above theorem is need not be true. for example, the complete graph 𝐾4. theorem: 2.7 let 𝐺 be a graph and 𝐺 admits 𝑀𝐷𝐷(𝐺1, 𝐺2, … , 𝐺𝑛). then (i) 𝑀𝐷(𝐺2) = 𝑝2 − ∆(𝐺2) if and only if 𝐺2 is 𝐾1,𝑚 for any 𝑚. (ii) 𝑀𝐷(𝐺𝑖 ) = ∆(𝐺𝑖) if and only if 𝐺𝑖 is a complete graph, where 𝑖 = 2,3, … . , 𝑛 proof: suppose 𝐺 admits 𝑀𝐷𝐷(𝐺1, 𝐺2, … , 𝐺𝑛). 𝑣1 𝑣2 𝑣3 𝑣4 𝑣5 𝑣6 𝐺 𝑣1 𝑣2 𝑣3 𝑣4 𝐺1 𝑣2 𝑣5 𝑣6 𝐺3 𝑣1 𝑣4 𝑣5 𝐺2 61 ebin raja merly e and saranya j (i) let 𝐺2 be 𝐾1,𝑚 for any 𝑚.  𝐺2 has 𝑝2 vertices and 𝑞2(= 𝑝2 − 1) edges and the maximum degree of 𝐺2 = 𝑝2 − 1  𝑀𝐷(𝐺2) = (𝑝2−1)+( 𝑝2−1 2 ) ( 𝑝2 2 )  𝑀𝐷𝐷(𝐺2) = 1  𝑀𝐷(𝐺2) = 𝑝2 − ∆(𝐺2) (ii) let 𝐺𝑖 be a complete graph, 𝑖 = 1,2,3, … , 𝑛  𝐺𝑖 has 𝑝𝑖 vertices and 𝑞𝑖 = 𝑝𝑖(𝑝𝑖−1) 2 edges and the maximum degree of 𝐺𝑖 = 𝑝𝑖 − 1.  𝑀𝐷(𝐺𝑖 ) = 𝑝𝑖(𝑝𝑖−1) 2 +𝑝𝑖[𝑝𝑖−1)𝐶2] ( 𝑝𝑖 2 )  𝑀𝐷(𝐺𝑖 ) = 𝑝𝑖 − 1  𝑀𝐷(𝐺𝑖 ) = ∆(𝐺𝑖 ), 𝑖 = 2,3, … , 𝑛 hence the proof. theorem: 2.8 let 𝐺 be a graph and 𝐺 admits 𝑀𝐷𝐷(𝐺1, 𝐺2, … , 𝐺𝑛 ) . then ∑ ⌊𝑀𝐷(𝐺𝑖 )⌋ < 𝑛(𝑛+1) 2 𝑛 𝑖=1 ⌈𝑀𝐷(𝐺)⌉, where 𝑛 is the number of decompositions of 𝐺. proof: we prove this theorem by induction on 𝑛. when 𝑛 = 1, then ⌊𝑀𝐷(𝐺1)⌋ < ⌈𝑀𝐷(𝐺)⌉, therefore, the result is true for 𝑛 = 1 assume that the theorem is true for 𝑛 − 1 that is, ∑ ⌊𝑀𝐷(𝐺𝑖 )⌋ < 𝑛(𝑛−1) 2 𝑛−1 𝑖=1 ⌈𝑀𝐷(𝐺)⌉ to prove: the theorem is true for 𝑛. that is, ∑ ⌊𝑀𝐷(𝐺𝑖 )⌋ < 𝑛(𝑛+1) 2 𝑛 𝑖=1 ⌈𝑀𝐷(𝐺)⌉. now, ∑ ⌊𝑀𝐷(𝐺𝑖 )⌋ = ⌊𝑀𝐷(𝐺1)⌋ + ⌊𝑀𝐷(𝐺2)⌋ + ⋯ + ⌊𝑀𝐷(𝐺𝑛)⌋ 𝑛−1 𝑖=1  ∑ ⌊𝑀𝐷(𝐺𝑖 )⌋ + ⌊𝑀𝐷(𝐺𝑛)⌋ < 𝑛(𝑛+1) 2 + ⌈𝑀𝐷(𝐺)⌉ + ⌊𝑀𝐷(𝐺𝑛)⌋ 𝑛−1 𝑖=1  ∑ ⌊𝑀𝐷(𝐺𝑛)⌋ < 𝑛2−𝑛 2 + ⌊𝑀𝐷(𝐺)⌋ + ⌈𝑀𝐷(𝐺𝑛)⌉ 𝑛 𝑖=1 = ( 𝑛2 − 𝑛 + 𝑛 − 𝑛 2 ) ⌈𝑀𝐷(𝐺)⌉ + ⌊𝑀𝐷(𝐺𝑛)⌋ = ( (𝑛2 + 𝑛) 2 − 2𝑛 2 ) ⌈𝑀𝐷(𝐺)⌉ + ⌊𝑀𝐷(𝐺𝑛)⌋ = ( 𝑛2 + 𝑛 2 ) ⌈𝑀𝐷(𝐺)⌉ − 𝑛⌈𝑀𝐷(𝐺)⌉ + ⌊𝑀𝐷(𝐺𝑛)⌋ < ( 𝑛2+𝑛 2 ) ⌈𝑀𝐷(𝐺)⌉, since the value of −𝑛⌈𝑀𝐷(𝐺)⌉ + ⌊𝑀𝐷(𝐺)⌋ is negative ∑ ⌊𝑀𝐷(𝐺𝑛)⌋ < ( 𝑛(𝑛+1) 2 ) ⌊𝑀𝐷(𝐺)⌋𝑛𝑖=1 62 medium domination decomposition of graphs the result is true for 𝑛. hence the proof. theorem: 2.8 if 𝐺 admits 𝑀𝐷𝐷(𝐺1, 𝐺2, … , 𝐺𝑛 ) and ∆(𝐺) = 𝑝 − 1 then i) 𝛾(𝐺) < ⌈𝑀𝐷(𝐺)⌉ ii) 𝛾(𝐺) ≤ 𝛾(𝐺𝑖 ) iii) 𝛾(𝐺) ≤ ⌈𝑀𝐷(𝐺𝑖 )⌉ proof: let 𝐺 be a graph with 𝑝 vertices and 𝑞 edges and ∆(𝐺) = 𝑝 − 1. since ∆(𝐺) = 𝑝 − 1, 𝛾(𝐺) = 1. but the minimum value of ⌈𝑀𝐷(𝐺)⌉ = 1. therefore 𝛾(𝐺) < ⌈𝑀𝐷(𝐺)⌉. the proof of (ii) and (iii) is obvious. remark: 2.9 the equality holds in theorem 2.8, (ii) and (iii) when 𝐺𝑖 is star. theorem: 2.10 if 𝐺 admits 𝑀𝐷𝐷(𝐺1, 𝐺2, … , 𝐺𝑛 ) then i) ⌈𝑀𝐷(𝐺)⌉ < 𝛾(𝐺) + ∆(𝐺) ii) 𝛾(𝐺𝑖 ) < 𝛾(𝐺) + ∆(𝐺) the proof is obvious. theorem: 2.11 if 𝐺 admits 𝑀𝐷𝐷(𝐺1, 𝐺2, … , 𝐺𝑛 ) and 𝐻 is a spanning subgraph of a graph 𝐺, then (i) 𝛾(𝐺) ≤ 𝛾(𝐻) (ii) ⌈𝑀𝐷(𝐺)⌉ > ⌊𝑀𝐷(𝐻)⌋ proof: let 𝐺 be a simple connected graph and 𝐺 admits 𝑀𝐷𝐷. case (i): 𝑛 = 1 then 𝛾(𝐺) = 𝛾(𝐻) and ⌈𝑀𝐷(𝐺)⌉ > ⌊𝑀𝐷(𝐻)⌋.hence the result is obvious. case (ii): 𝑛 > 1 then, let 𝐺1, 𝐺2, … 𝐺𝑖 , … , 𝐺𝑛 be the decomposition of 𝐺. let 𝐻 = 𝐺𝑖 be the spanning subgraph of 𝐺. since the number of decompositions is more than one, |𝐸(𝐻)| < |𝐸(𝐺)| also ∆(𝐺) ≥ ∆(𝐻) and 𝛿(𝐺) ≥ 𝛿(𝐻). therefore 𝛾(𝐻) ≥ 𝛾(𝐺). obviously, since 𝐻 is a spanning subgraph of 𝐺, the medium domination number of 𝐻 is less than the medium domination number of 𝐺. therefore ⌈𝑀𝐷(𝐺)⌉ > ⌊𝑀𝐷(𝐻)⌋. hence the proof. 3. conclusion in this paper, we calculated the number of vertices that are capable of dominating both of u and v. the total number of vertices that dominate every pair of vertices is examined and the average of this value is calculated which is called “the medium domination number” of graph. some theorems and results on the medium domination decomposition of a graph and basic graph classes are given. further this concept can be extended to some family of graphs. 63 ebin raja merly e and saranya j references [1] duygu vargor, pinar dundar, “the medium domination number of a graph”, international journal of pure and applied mathematics volume of no.3, 2011 297-306. [2] fairouz beggas, “decomposition and domination of some graphs” data structures and algorithms [cs.ds]. university claude bernard lyon 1,2017. [3] s. arumugan and s. ramachandran, “invitation to graph theory”, scitech publications (india) pvt. ltd. (2003). [4] teresa w. haynes, stephen t. hedetnimi and peter j. slater, “fundamentals of domination in graphs” marcel dekkar, inc., new york, 1998. [5] t.i. joel and e.e.r. merly, “geodetic decomposition of graphs”, journal of computer and mathematical sciences, 9(7) (2018), 829 833. [6] sr little femilin jana. d., jaya. r., arokia ranjithkumar, m., krishnakumar. s., “resolving sets and dimension in special graphs”, advances and applications in mathematical sciences 21 (7) (2022), 3709-3717. 64 https://scholar.google.com/citations?view_op=view_citation&hl=en&user=xwcp70yaaaaj&sortby=pubdate&authuser=1&citation_for_view=xwcp70yaaaaj:ijcspb-oge4c approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 41, 2021, pp. 162-172 162 perfect edge domination in vague graphs m. kaliraja* p. kanibose † a. ibrahim‡ abstract in this paper, we introduce the notions of perfect edge domination set and perfect edge domination number in vague graphs. also, we introduce the definitions of connected perfect edge domination set and connected perfect edge dominating number. moreover, we investigate some related properties of these concepts with comprehensive results and illustrations. keywords: vague graphs; edge dominating set; perfect edge domination set; perfect edge domination number; connected perfect edge domination set; connected perfect edge domination number. 2010 ams subject classification§: 03b60, 06b10, 06b20. * assistant professor, p.g. and research department of mathematics, h. h. the rajah’s college, pudukkotai, affiliated to bharathidasan university, trichirappalli, tamilnadu, india; mkr.maths009@gmail.com. †research scholar, p. g. and research department of mathematics, h. h. the rajah’s college, pudukkotai, affiliated to bharathidasan university, trichirappalli, tamilnadu, india; kanibose77@gmail.com. ‡ assistant professor, p.g. and research department of mathematics, h. h. the rajah’s college, pudukkotai, affiliated to bharathidasan university, trichirappalli, tamilnadu, india; ibrahimaadhil@yahoo.com; dribra@hhrc.ac.in § received on june 25, 2021. accepted on december 19, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.588. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. perfect edge domination in vague graphs 163 1. introduction edge domination sets in graphs are a phenomenon in research that has gained prominence due to the extreme scope of offering research solutions with varied dimensions. with the aid of perfect edge dominance and successful edge dominance in graphs, chin lung lua, ming-tat koa and chuan yi tangb [3] investigated perfect edge dominance in graphs. r. s. chitra and n. prabhavathi [2] performed an analysis of perfect edges, perfect edge covering, and perfect edge vertex domination sets in graphs. the concept of ideal domination sets was proposed by s. revathi, p.j. jayalakshmi, and c.v. r. harinarayanan [10].the notion of connected edge domination in fuzzy graphs was suggested by c.y. ponnappan, s. basheer ahamed, and p. surulinathan [7]. p. karpagam and v. revathi [6] introduced the idea of connected edge perfect domination in fuzzy graphs with the idea of the principle of connected edge dominance in fuzzy graphs. s. revathi, c.v.r. harinarayanan, and r. muthuraj [9] introduced an advanced idea of perfect domination in intuitionistic fuzzy graphs. w. l. gau and d. j. buehrer [4] proposed the concept of a vague set. r.a. borzooeiy and h. rashmanlou [1] introduced the notion of domination in vague graphs, and obtained strong domination numbers with applications. the definition of dominating sets in vague graphs was efficiently used in vague graphs by yahya talebi and hossein rashmanlou [11]. recently, the authors [5] explored the concepts of edge dominance, independent edge domination in vague graphs, and obtained its related properties. in this paper, we present the notion of a perfect edge domination set and the perfect edge domination number of the vague graphs. further, we introduce the connected perfect edge domination set and connected perfect edge dominating number. also, we obtained some relevant properties. 2. preliminaries in this section, we will show some basic definitions and properties that are helpful in developing our main results. definition 2.1[4] a vague set 𝑃 in the universe of discourse 𝑋 is characterized by two membership functions given by i. a truth membership function 𝑡𝑃 : 𝑋 → [0, 1], ii. a false membership function 𝑓𝑃 : 𝑋 → [0, 1]. where 𝑡𝑃 (𝑥) is lower bound of the grade of membership of x derived from the ‘evidence for x’, and 𝑓𝑃 (𝑥) is a lower bound of the negation of x derived from the ‘evidence against x’ and 𝑡𝑃 (𝑥) + 𝑓𝑃 (𝑥) ≤ 1. thus the grade m. kaliraja, p. kanibose, a. ibrahim 164 of membership of x in the vague set 𝑃 is bounded by a subinterval [ 𝑡𝑃 (𝑥), 1 − 𝑓𝑃 (𝑥)] of [0, 1]. the vague set 𝑃 is written as 𝑃 = {(𝑥, [ 𝑡𝑃 (𝑥), 𝑓𝑆 (𝑥)])/𝑥 ∈ 𝑋}, where the interval [ 𝑡𝑃 (𝑥), 1 − 𝑓𝑆 (𝑥)] is called the value of x in the vague set 𝑃. definition 2.2[1] a vague graph is of the form 𝐺 = (𝑃, 𝑄), where i. a sequence of distinct vertices 𝑃 = {𝑣1, 𝑣2, … 𝑣𝑛 }, such that 𝑡𝑃 : 𝑃 → [0,1] and 𝑓𝑃 : 𝑃 → [0, 1] are truth and false membership functions, respectively such that 0≤ 𝑡𝑃 (𝑥) + 𝑓𝑃 (𝑥) ≤ 1. for all 𝑥 ∈ 𝑃. ii. a vague relation of the vague subsets 𝑋 × 𝑌 is an expression 𝑅, defined by 𝑅 = {{(𝑥, 𝑦), 𝑡𝑅 (𝑥, 𝑦), 𝑓𝑅 (𝑥, 𝑦)}/𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌}, where 𝑡𝑅 : 𝑋 × 𝑌 → [0,1] are𝑓𝑅 : 𝑋 × 𝑌 → [0,1], which satisfies the condition 0 ≤ 𝑡𝑅 (𝑥, 𝑦) + 𝑓𝑅 (𝑥, 𝑦) ≤ 1 for all (𝑥, 𝑦) ∈ 𝑋 × 𝑌. definition 2.3[1] let 𝐺 = (𝑃, 𝑄) be a vague graph, where 𝑃 = (𝑡𝑃,𝑓𝑃 ) is a vague set of vertex and 𝑄 = (𝑡𝑄,𝑓𝑄 ) is a set edge in 𝐺, then for every 𝑢𝑣 ∈ 𝑄, such that 𝑡𝑄 (𝑢𝑣) ≤ min{𝑡𝑄 (𝑢), 𝑡𝑄 (𝑣)} and 𝑓𝑄 (𝑢𝑣) ≤ max {𝑓𝑄 (𝑢), 𝑓𝑄 (𝑣)}. definition 2.4 [1] let 𝐺 = (𝑃, 𝑄) be a vague graph. if cardinality of the arcs 𝑡𝑄 (𝑣𝑖 𝑣𝑗 ) = min{𝑡𝑃 (𝑣𝑖 ), 𝑓𝑃 (𝑣𝑗 )} and 𝑓𝑄 (𝑣𝑖 𝑣𝑗 ) = max {𝑡𝑃 (𝑣𝑖 ), 𝑓𝑃 (𝑣𝑗 )} for all 𝑣𝑖 𝑣𝑗 ∈, then 𝐺 is called a complete vague graph. definition 2.5[10] an arc of a vague graph 𝐺 = (𝑃, 𝑄) is said to be a strong edge in 𝐺. if 𝑡𝑄 (𝑢𝑣) ≥ (𝑡𝑄 ) ∞(𝑢𝑣) and 𝑓𝑄 (𝑢𝑣) ≤ (𝑓𝑄 ) ∞(𝑢𝑣). where i. (𝑡𝑄 ) ∞(𝑢𝑣) = 𝑠𝑢𝑝{(𝑡𝑄 ) 𝑟 (𝑢𝑣): 𝑟 = 1,2, … , 𝑛} ii . (𝑓𝑄 ) ∞(𝑢𝑣) = 𝑖𝑛𝑓{(𝑓𝑄 ) 𝑟 (𝑢𝑣): 𝑟 = 1,2, … , 𝑛}. definition 2.6[10] let 𝑒𝑖 be an edge in a vague graph 𝐺 = (𝑃, 𝑄). then the neighborhood of 𝑒𝑗 is representing by 𝑁(𝑒𝑖 ) = { 𝑒𝑗 ∈ 𝑄/(𝑢, 𝑣) is a strong arc} . definition 2.7[7] let 𝐺 = (𝑃, 𝑄) be a vague graph and 𝑒𝑖 , 𝑒𝑗 ∈ 𝑄. if a strong arc 𝑒𝑖 is adjacent to 𝑒𝑗 . then, we say that 𝑒𝑖, dominates 𝑒𝑗 . it is denoted by 𝐷. definition 2.8[1] an edge 𝑒𝑗 in a vague graph 𝐺 = (𝑃, 𝑄) is called a neighbor of 𝑒𝑖 ∈ 𝐷 with respect to 𝐷, if 𝑁(𝑒𝑗 ) ∩ 𝐷 = {𝑒𝑖 }. definition 2.9[1] let 𝐺 = (𝑃, 𝑄) be a vague graph. if neighborhood degree is defined by i. the minimum neighborhood degree of 𝐺 is 𝛿(𝐺) = 𝑚𝑖𝑛{𝑑𝑁 (𝑒), 𝑒 ∈ 𝑄} ii. the maximum neighborhood degree of 𝐺 is ∆(𝐺) = 𝑚𝑎𝑥{𝑑𝑁 (𝑒), 𝑒 ∈ 𝑄}. perfect edge domination in vague graphs 165 definition 2.10[4] two vertices 𝑣𝑖 and 𝑣𝑗 in a vague graph 𝐺 = (𝑃, 𝑄) are called a strong neighborhood 𝐺. if either one of the conditions are hold, i. 𝑡𝑄 (𝑣𝑖 𝑣𝑗 ) > 0, 𝑓𝑄 (𝑣𝑖 𝑣𝑗 ) > 0 ii. 𝑡𝑄 (𝑣𝑖 𝑣𝑗 ) = 0, 𝑓𝑄 (𝑣𝑖 𝑣𝑗 ) > 0 iii. 𝑡𝑄 (𝑣𝑖 𝑣𝑗 ) > 0, 𝑓𝑄 (𝑣𝑖 𝑣𝑗 ) = 0. definition 2.11[10] let 𝐺 = (𝑃, 𝑄) be a vague graph. then number of edge (the cardinality of 𝑄) is called the order size of a vague graph and is denoted by 𝑂(𝑆) = ∑ ( 1+𝑡𝑄(𝑣𝑖𝑣𝑗)−𝑓𝑄(𝑣𝑖𝑣𝑗) 2 )𝑣𝑖𝑣𝑗∈𝑄 for all 𝑣𝑖 𝑣𝑗 ∈ 𝑄. definition 2.12[4] two edges in a vague graph 𝐺 = (𝑃, 𝑄) is celled an independent if there is no any strong arcs between them. definition 2.13[4] let 𝐺 = (𝑃, 𝑄) be a vague graph. if the sub graph is induced by 𝐷 has an isolated edge. definition 2.14[4] a edge in vague graph 𝐺 = (𝑃, 𝑄)is an isolated edge, if it is not adjacent to any strong arc in 𝐺. 3. main results in this section, we introduce perfect edge domination set, perfect edge domination number, connected perfect edge domination set and connected perfect edge dominating number of vague graphs, and obtain some properties with illustrations. definition 3.1 let 𝐺 = (𝑃, 𝑄) be a vague graph and 𝐷 be an edge dominating set in 𝐺. if for every edge of 𝑄(𝐺) − 𝐷 is adjacent to exactly one edge in 𝐷, then 𝐷 is called a perfect edge domination set in 𝐺. example 3.2 let 𝐺 = (𝑃, 𝑄) be a vague graph as shown in the figure 3.1. consider the edge set 𝑄 = {𝑒1,𝑒2,𝑒3,𝑒4,𝑒5}. we have 𝑒1,𝑒4 and 𝑒5 are strong arcs in 𝐺. m. kaliraja, p. kanibose, a. ibrahim 166 here, {𝑒5} and {𝑒1,𝑒4} are edge dominating sets in 𝐺. now, 𝐷 = {𝑒5} is a perfect edge domination in 𝐺. since, {𝑒5} is dominates the all other neighbor edges in 𝐺. figure 3.1: perfect edge domination set definition 3.3 let 𝐺 = (𝑃, 𝑄) be a vague graph. if it has a minimum perfect edge dominating set of vague graph 𝐺. then it is called a perfect edge dominating number of 𝐺, and it is denoted by 𝛾𝑝(𝐷) in 𝐺. example 3.4 let 𝐺 = (𝑃, 𝑄) be a vague graph as shown in the figure 3.2. from the edge set 𝑄 = {𝑒1,𝑒2,𝑒3,𝑒4,}, we see that {𝑒1} and {𝑒4} are strong arc 𝐺. then {𝑒1} and {𝑒4} are a perfect edge dominating sets of 𝐺. then minimum perfect edge dominating number is 𝛾𝑝(𝐷) = 0.70. figure 3.2: minimum perfect edge dominating number. proposition 3.5 let 𝐺 = (𝑃, 𝑄) be a vague graph with at least one isolated edge, then perfect edge dominating set does not exist. proof: let 𝐷 be a minimal perfect edge dominating set and 𝑒𝑖 be the path of 𝐷, since 𝐺 has at least one isolated edge. incase 𝑄(𝐺) − 𝐷 is a perfect edge dominating set of vague graph, it has 𝑒𝑗 as neighborhood of perfect edge 𝑣3(0.2,0.6) 𝑣2(0.5,0.4) 𝑣1(0.3,0.4) 𝑒1(0.3,0.4) 𝑒3(0.2,0.8) 𝑣4 (0.5,0.3) 𝑒2(0.2,0.7) 𝑒4(0.3,0.5) 𝑒5(0.2,0.6) 𝑣1 (0.3,0.6) 𝑒4(0.2,0.5) (0.2,0.5) 𝑣4 (0.2,0.4) 𝑣3 (0.4,0.5) 𝑣2 (0.1,0.7) 𝑒1 (0.1,0.7) 𝑒3 (0.3,0.7) 𝑒2 (0.1,0.9) perfect edge domination in vague graphs 167 domination set 𝐷. there exist a complement path of vague graph is denoted by 𝑁(𝑒𝑗 ). from the definition 2.6, we have |𝑁(𝑒𝑗 ) ∩ 𝐷|=1, but in this vague graph |𝑁(𝑒𝑗 ) ∩ 𝐷| ≠1. this is a contradiction. therefore, 𝑄(𝐺) − 𝐷 not a perfect dominating set. ∎ proposition 3.6 let 𝐺 = (𝑃, 𝑄) be a complete vague graph. then, every edge in 𝐷 is a perfect edge dominating set. proof: let 𝐺 = (𝑃, 𝑄)be a complete vague graph and let 𝑒𝑖 ∈ 𝐷 be a edge dominating set in 𝐺. then every edges in the graph is 𝑡𝑄 (𝑣𝑖 𝑣𝑗 ) = min{𝑡𝑃 (𝑣𝑖 ), 𝑓𝑃 (𝑣𝑗 )} and 𝑓𝑄 (𝑣𝑖 𝑣𝑗 ) = max {𝑡𝑃 (𝑣𝑖 ), 𝑓𝑃 (𝑣𝑗 )} for all 𝑣𝑖 𝑣𝑗 ∈ 𝑄. therefore, every path in 𝐺 is a strong arc and complement edges are 𝑒𝑗 ∈ 𝑄(𝐺) − 𝐷 adjacent to exactly one edge in 𝐷 is said to be a perfect edge domination set in vague graph. here, 𝐷 is a perfect edge dominating set of vague graph 𝐺, and then every complete vague graph 𝐺 is a perfect edge dominating set. ∎ example 3.7 let 𝐺 = (𝑃, 𝑄) be a complete vague graph as shown in the figure 3.3. from the edge set 𝑄 = {𝑒1,𝑒2,𝑒3,𝑒4,𝑒5} is an edge dominating set. then, we have 𝑒1,𝑒4 and 𝑒5 which are strong arcs of the vague graph. here {𝑒5} and {𝑒1,𝑒4} are edge dominating set of 𝐺, for 𝐷 = {𝑒5} is a perfect edge domination in vague graph. figure 3.3: perfect edge domination set 𝑣2(0.3, 0.6) 𝑣4(0.1, 0.6) 𝑣3 (0.4,0.5) 𝑣1 (0.2, 0.7) 𝑒1(0.1, 0.8) 𝑒2(0.4, 0.5) 𝑒3 (0.1, 0.6) 𝑒4(0.1, 0.7) 𝑒5(0.2, 0.7) 𝑒4 (0.1,0.7) m. kaliraja, p. kanibose, a. ibrahim 168 proposition 3.8 let 𝐺 = (𝑃, 𝑄) be a vague graph, then 𝐷 is a minimal perfect edge dominating set. if for each 𝑄(𝐺) − 𝐷 is not a perfect edge dominating set. proof: let 𝐺 be a vague graph and has a minimal perfect edge domination set 𝐷. from the definition 3.1, if 𝐷 is minimum, the arcs must be strong. suppose 𝑒𝑖 and 𝑒𝑗 are any two edges adjacent in 𝐺, but 𝑒𝑖 ∈ 𝐷 is a minimum perfect edge domination set of a vague graph. then 𝑒𝑗 may or may not have any strong in this graph. thus, each edge in 𝑒𝑗 has no strong neighbor of edge in 𝑄(𝐺) − 𝐷. hence, 𝑄(𝐺) − 𝐷 is not perfect edge domination set of vague graph 𝐺. ∎ definition 3.9 let 𝐺 = (𝑃, 𝑄) be a vague graph and 𝑆 is an edge dominating set of 𝐺 is connected perfect edge domination set with〈𝑆〉 is connected. example 3.10 let 𝐺 = (𝑃, 𝑄) be a vague graph as shown in the figure3.4. from the edge set to 𝑄 = {𝑒1,𝑒2,𝑒3,𝑒4,𝑒5} , which are strong arcs of 𝐺. here {𝑒1, 𝑒2} and {𝑒3,𝑒4} are edge dominating sets of 𝐺. then 𝐷 = {𝑒1, 𝑒4} is connected perfect edge domination in 𝐺.since, {𝑒1, 𝑒4} is a dominating set in 𝐺. figure 3.4: connected perfect edge domination connected strong perfect edge dominating set 𝑄𝑐𝑠 = {𝑒1, 𝑒2} is connected, then 𝑄 − 𝑄𝑐𝑠 = {𝑒3, 𝑒4 , 𝑒5, 𝑒6} is connected 𝑁(𝑒3) = {𝑒2, 𝑒4 , 𝑒6 } ∩ {𝑒1, 𝑒2} = 𝑒2. 𝑁(𝑒5) = {𝑒1, 𝑒4 , 𝑒5 } ∩ {𝑒1, 𝑒2} = 𝑒1. 𝑣5(0.2,0.7) 𝑣1(0.3,0.6) 𝑣3(0.1,0.4) 𝑣2(0.1,0.8) 𝑒5(0.1,0.8) 𝑒6(0.1,0.8) 𝑒1(0.2,0.7) 𝑣4(0.4,0.5) 𝑒4(03,0.6) 𝑒3(0.1,0.5) 𝑒2(0.1,0.7) perfect edge domination in vague graphs 169 𝑁(𝑒6) = {𝑒1, 𝑒5 , 𝑒3 } ∩ {𝑒1, 𝑒2} = 𝑒2. 𝑁(𝑒3) = {𝑒1, 𝑒5 , 𝑒3 } ∩ {𝑒1, 𝑒2} = 𝑒1. definition 3.11 let 𝐺 = (𝑃, 𝑄) be a vague graph. then the smallest cardinality number of the arc in any edge connected perfect edge dominating set of 𝐺 is called connected perfect edge domination number, which is denoted as 𝛾𝑐𝑝. proposition3.12 let 𝐺 = (𝑃, 𝑄)be a vague graph and 〈𝑆〉 be a minimal connected edge perfect dominating set of 𝐺,then 𝑄(𝐺) − 〈𝑆〉 is also a connected edge perfect dominating set of 𝐺. proof: let {𝑒1, 𝑒2, 𝑒 3, … , 𝑒𝑛} be an edge set of vague graph 𝐺and 𝑀 be a minimal connected edge dominating set of 𝑄(𝐺). if 𝑄(𝐺) − 𝑀 is not a connected perfect edge dominating set. then from, the definition 3.1, we have 𝐷 has minimum set of arcs that should be strong. if 𝑒𝑖 and 𝑒𝑗 are any two edges adjacent in 𝐺, but 𝑒𝑖 ∈ 𝐷 is a minimum perfect edge domination set of a vague graph, then 𝑒𝑗 is may (or) may not have any strong arc in this graph, thus, every edge in 𝑒𝑗 has no strong neighbor of edge in 𝑄(𝐺) − 𝐷. hence 𝑄(𝐺) − 𝐷 is not a perfect edge domination set of vague graph 𝐺. ∎ example 3.13 let 𝐺 = (𝑃, 𝑄) be a vague graph as shown in the figure 3.5. from the edge set 𝑄 = {𝑒1, 𝑒2, 𝑒3, 𝑒4, 𝑒5, 𝑒6,}, we see that 𝑒1,𝑒2 and 𝑒5 are strong arcs in the graph 𝐺.then {𝑒1,𝑒5} are a connected perfect edge dominating set of 𝐺. thus, {𝑒1,𝑒5} is minimum connected perfect edge dominating number is 𝛾𝑐𝑝(𝐷) = 0.50. . figure 3.5: minimum perfect edge dominating number 𝑣1(0.2,0.6) 𝑒1(0.1,0.6) 𝑣2(0.4,0.6) 𝑒2(0.1,0.7) 𝑣3(0.1,0.7) 𝑒4(0.1,0.8) 𝑣4(0.3,0.4) 𝑒6(0.2,0.6) 𝑣5(0.2,0.5) 𝑒5(0.2,0.7) 𝑒3(0.5,0.4) m. kaliraja, p. kanibose, a. ibrahim 170 remark 3.14 let 𝐺 = (𝑃, 𝑄)be a vague graph and 〈𝑆〉 be a minimal connected edge perfect dominating set of 𝐺, then 𝑄(𝐺) − 〈𝑆〉 is a minimal connected edge perfect dominating set of 𝐺. proposition 3.15 let 𝐺 = (𝑃, 𝑄) be a vague graph and 𝐷 is a connected perfect edge domination set and it is without isolated edges, then 𝑟 ∆𝑐(𝐺)+1 ≥ 𝛾𝐶𝑃 (𝐺). proof: let 𝐺 be a vague graph and 𝐷 be a connected perfect edge domination set. from the definition 3.9 of maximum neighbourhood and minimum neighbourhood of connected perfect edge domination set, then |𝐷|∆𝑐 (𝐺) ≤ ∑ 𝑑𝑄 (𝑒) = ∑|𝑁(𝑒)| 𝑒∈𝐷𝑒∈𝐷 ≤ |⋃ 𝑁(𝑒) 𝑒∈𝐷 | ≤ |𝑄 − 𝐷| ≤ 𝑟 − |𝐷| |𝐷|∆𝑐 (𝐺) + |𝐷| ≤ 𝑟 therefore, 𝑟 ∆𝑐(𝐺)+1 ≥ 𝛾𝐶𝑃 (𝐺). ∎ proposition 3.16 every connected perfect edge dominating set of a vague graph 𝐺 = (𝑃, 𝑄) is not independent. proof: let 𝐺 be a vague graph and 𝐷 be a minimal connected perfect edge dominating set, then path of graphs have strong arcs. clearly, every edges in the graph are satisfied by the condition of 𝑡𝑄 (𝑢𝑣) ≥ (𝑡𝑄 ) ∞(𝑢𝑣) and 𝑓𝑄 (𝑢𝑣) ≤ (𝑓𝑄 ) ∞(𝑢𝑣). suppose, we assume that a vague graph 𝐷 contains an isolated edge 𝑒𝑖 in 𝐺. since 𝐺 is a connected perfect edge dominating set, 𝑒𝑖 is a strong neighborhood of at least one edge 𝐷 − {𝑒𝑗 }. thus, 𝐷 − {𝑒𝑗 } is an edge dominating set which contradicts in the minimum edge domination set 𝐷. hence a vague graph is not independent. ∎ proposition 3.17 let 𝐺 = (𝑃, 𝑄) be a vague graph and 𝑆𝑐 be a minimal connected perfect edge dominating set of 𝐺. proof: let 𝑆𝑐 be a minimal connected perfect edge dominating set of vague graph 𝐺 = (𝑃, 𝑄). if 𝑒𝑗 is not dominated by edge 𝑒𝑖 in 𝑆𝑐 and the induced vague graph < 𝑆𝑐 > is disconnected. we know that from the definition of perfect edge domination in vague graphs 171 connected perfect dominating set, if every edge 𝑒𝑗 ∈ 𝑄 − 𝑆𝑐 is perfect dominated by exactly one edge 𝑒𝑖 in 𝑆𝑐 and the induced vague sub graph < 𝑆𝑐 > is connected. so, we have every edge 𝑒𝑗 ∈ 𝑄 − 𝑆𝑐 is dominated by some edge 𝑒𝑖 in 𝑆𝑐 which is a contradiction. therefore, 𝑆𝑐 is connected dominating set. ∎ 4. conclusions in this study, we have introduced the notions of perfect edge domination set and perfect edge domination number of vague graphs. furthermore, we have investigated some related properties with suitable examples. moreover, we have introduced the notion of connected perfect domination set and connected perfect edge domination number of the vague graph. finally, we have obtained some properties. references [1] r. a. borzooei and h. rashmanlou, domination in vague graphs and its applications, journal of intelligent & fuzzy systems, 29, 1933 -1940, 2015. 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[12] l. a. zadeh, fuzzy sets, information and control, 8, 338 353, 1965. microsoft word capitolo intero n 6.doc ratio mathematica 29 (2015) 65-76 issn:1592-7415 neutrosophic filters in be-algebras akbar rezaei1, arsham borumand saeid2, florentin smarandache3 1department of mathematics, payame noor university, p.o.box. 19395-3697, tehran, iran. rezaei@pnu.ac.ir 2department of pure mathematics, faculty of mathematics and computer, shahid bahonar university of kerman, kerman, iran. arsham@uk.ac.ir 3florentin smarandache, university of new mexico, gallup, nm 87301, usa. smarand@unm.edu abstract in this paper, we introduce the notion of (implicative) neutrosophic filters in be-algebras. the relation between implicative neutrosophic filters and neutrosophic filters is investigated and we show that in self distributive bealgebras these notions are equivalent. keywords: be-algebra, neutrosophic set, (implicative) neutrosophic filter. 2010 ams subject classifications: 03b60, 06f35, 03g25. doi:10.23755/rm.v29i1.22 1 introduction neutrosophic set theory was introduced by smarandache in 1998 ([10]). neutrosophic sets are a new mathematical tool for dealing with uncertainties which are free from many difficulties that have troubled the usual theoretical approaches. research works on neutrosophic set theory for many applications such as information fussion, probability theory, control theory, decision making, measurement 65 a. rezaei, a. borumand saeid and f. smarandache theory, etc. kandasamy and smarandache introduced the concept of neutrosophic algebraic structures ([3, 4, 5]). since then many researchers worked in this area and lots of literatures had been produced about the theory of neutrosophic set. in the neutrosophic set one can have elements which have paraconsistent information (sum of components > 1), others incomplete information (sum of components < 1), others consistent information (in the case when the sum of components =1) and others interval-valued components (with no restriction on their superior or inferior sums). h.s. kim and y.h. kim introduced the notion of a be-algebra as a generalization of a dual bck-algebra ([6]). b.l. meng give a procedure which generated a filter by a subset in a transitive be-algebra ([7]). a. walendziak introduced the notion of a normal filter in be-algebras and showed that there is a bijection between congruence relations and filters in commutative be-algebras ([11]). a. borumand saeid and et al. defined some types of filters in be-algebras and showed the relationship between them ([1]). a. rezaei and et al. discussed on the relationship between be-algebras and hilbert algebras ([9]). recently, a. rezaei and et al. introduced the notion of hesitant fuzzy (implicative) filters and get some results on be-algebras ([8]). in this paper, we introduce the notion of (implicative) neutrosophic filters and study it in details. in fact, we show that in self distributive be-algebras concepts of implicative neutrosophic filter and neutrosophic filter are equivalent. 2 preliminaries in this section, we cite the fundamental definitions that will be used in the sequel: definition 2.1. [6] by a be-algebra we shall mean an algebra x = (x;∗, 1) of type (2, 0) satisfying the following axioms: (be1) x∗x = 1, (be2) x∗1 = 1, (be3) 1∗x = x, (be4) x∗ (y ∗z) = y ∗ (x∗z), for all x, y, z ∈ x. from now on, x is a be-algebra, unless otherwise is stated. we introduce a relation “≤” on x by x ≤ y if and only if x∗y = 1. a be-algebra x is said to be self distributive if x∗(y∗z) = (x∗y)∗(x∗z), for all x, y, z ∈ x. a be-algebra x is said to be commutative if satisfies: 66 neutrosophic filters in be-algebras (x∗y)∗y = (y ∗x)∗x, for all x, y ∈ x. proposition 2.1. [11] if x is a commutative be-algebra, then for all x, y ∈ x, x∗y = 1 and y ∗x = 1 imply x = y. we note that “≤” is reflexive by (be1). if x is self distributive then relation “≤” is a transitive ordered set on x, because if x ≤ y and y ≤ z, then x∗z = 1∗ (x∗z) = (x∗y)∗ (x∗z) = x∗ (y ∗z) = x∗1 = 1. hence x ≤ z. if x is commutative then by proposition 2.1, relation “≤” is antisymmetric. hence if x is a commutative self distributive be-algebra, then relation “≤” is a partial ordered set on x. proposition 2.2. [6] in a be-algebra x, the following hold: (i) x∗ (y ∗x) = 1, (ii) y ∗ ((y ∗x)∗x) = 1, for all x, y ∈ x. a subset f of x is called a filter of x if it satisfies: (f1) 1 ∈ f, (f2) x ∈ f and x∗y ∈ f imply y ∈ f . define a(x, y) = {z ∈ x : x∗ (y ∗z) = 1}, which is called an upper set of x and y. it is easy to see that 1, x, y ∈ a(x, y), for any x, y ∈ x. every upper set a(x, y) need not be a filter of x in general. definition 2.2. [1] a non-empty subset f of x is called an implicative filter if satisfies the following conditions: (if1) 1 ∈ f , (if2) x∗ (y ∗z) ∈ f and x∗y ∈ f imply that x∗z ∈ f , for all x, y, z ∈ x. if we replace x of the condition (if2) by the element 1, then it can be easily observed that every implicative filter is a filter. however, every filter is not an implicative filter as shown in the following example. 67 a. rezaei, a. borumand saeid and f. smarandache example 2.1. let x = {1, a, b} be a be-algebra with the following table: ∗ 1 a b 1 1 a b a 1 1 a b 1 a 1 then f = {1, a} is a filter of x, but it is not an implicative filter, since 1∗ (a∗ b) = 1∗a = a ∈ f and 1∗a = a ∈ f but 1∗ b = b /∈ f . definition 2.3. [10] let x be a set. a neutrosophic subset a of x is a triple (ta, ia, fa) where ta : x → [0, 1] is the membership function, ia : x → [0, 1] is the indeterminacy function and fa : x → [0, 1] is the nonmembership function. here for each x ∈ x, ta(x), ia(x) and fa(x) are all standard real numbers in [0, 1]. we note that 0 ≤ ta(x) + ia(x) + fa(x) ≤ 3, for all x ∈ x. the set of neutrosophic subset of x is denoted by ns(x). definition 2.4. [10] let a and b be two neutrosophic sets on x. define a ≤ b if and only if ta(x) ≤ tb(x), ia(x) ≥ ib(x), fa(x) ≥ fb(x), for all x ∈ x. definition 2.5. let x1 = (x1;∗, 1) and x2 = (x2;◦, 1′) be two be-algebras. then a mapping f : x1 → x2 is called a homomorphism if, for all x1, x2 ∈ x1 f(x1 ∗ x2) = f(x1) ◦ f(x2). it is clear that if f : x1 → x2 is a homomorphism, then f(1) = 1′. 3 neutrosophic filters definition 3.1. a neutrosophic set a of x is called a neutrosophic filter if satisfies the following conditions: (nf1) ta(x) ≤ ta(1), ia(x) ≥ ia(1) and fa(x) ≥ fa(1), (nf2) min{ta(x∗y), ta(x)}≤ ta(y), min{ia(x∗y), ia(x)}≥ ia(y) and min{fa(x∗y), fa(x)}≥ fa(y), for all x, y ∈ x. 68 neutrosophic filters in be-algebras the set of neutrosophic filter of x is denoted by nf(x). example 3.1. in example 2.1, put ta(1) = 0.9, ta(a) = ta(b) = 0.5, ia(1) = 0.2, ia(a) = ia(b) = 0.35 and fa(1) = 0.1, fa(a) = fa(b) = 0. then a = (ta, ia, fa) is a neutrosophic filter. proposition 3.1. let a ∈ nf(x). then (i) if x ≤ y, then ta(x) ≤ ta(y), ia(x) ≥ ia(y) and fa(x) ≥ fa(y), (ii) ta(x) ≤ ta(y ∗x), ia(x) ≥ ia(y ∗x) and fa(x) ≥ fa(y ∗x), (iii) min{ta(x), ta(y)}≤ ta(x∗y), min{ia(x), ia(y)}≥ ia(x∗y) and min{fa(x), fa(y)}≥ fa(x∗y), (iv) ta(x) ≤ ta((x∗y)∗y), ia(x) ≥ ia((x∗y)∗y) and fa(x) ≥ fa((x∗y)∗y), (v) min{ta(x), ta(y)}≤ ta((x∗ (y ∗z))∗z), min{ia(x), ia(y)}≥ ia((x∗ (y ∗z))∗z) and min{fa(x), fa(y)}≥ fa((x∗ (y ∗z))∗z), (vi) if min{ta(y), ta((x∗y)∗z)}≤ ta(z ∗x), then ta is order reversing and ia, fa are order (i.e. if x ≤ y, then ta(y) ≤ ta(x), ia(y) ≥ ia(x) and fa(y) ≥ fa(x)) (vii) if z ∈ a(x, y), then min{ta(x), ta(y)}≤ ta(z), min{ia(x), ia(y)}≥ ia(z) and min{fa(x), fa(y)}≥ fa(z) (viii) if n∏ i=1 ai ∗x = 1, then n∧ i=1 ta(ai) ≤ ta(x), n∧ i=1 ia(ai) ≥ ia(x) and n∧ i=1 fa(ai) ≥ fa(x) where n∏ i=1 ai ∗x = an ∗ (an−1 ∗ (. . . (a1 ∗x) . . . )). proof. (i). let x ≤ y. then x∗y = 1 and so ta(x) = min{ta(x), ta(1)} = min{ta(x), ta(x∗y)}≤ ta(y), ia(x) = min{ia(x), ia(1)} = min{ia(x), ia(x∗y)}≥ ia(y), fa(x) = min{fa(x), fa(1)} = min{fa(x), fa(x∗y)}≥ fa(y). (ii). since x ≤ y ∗x, by using (i) the proof is clear. 69 a. rezaei, a. borumand saeid and f. smarandache (iii). by using (ii) we have min{ta(x), ta(y)}≤ ta(y) ≤ ta(x∗y), min{ia(x), ia(y)}≥ ia(y) ≥ ia(x∗y), min{fa(x), fa(y)}≥ fa(y) ≥ fa(x∗y). (iv). it follows from definition 3.1, ta(x) = min{ta(x), ta(1)} = min{ta(x), ta((x∗y)∗ (x∗y))} = min{ta(x), ta(x∗ ((x∗y)∗y))} ≤ ta((x∗y)∗y). also, we have ia(x) = min{ia(x), ia(1)} = min{ia(x), ia((x∗y)∗ (x∗y))} = min{ia(x), ia(x∗ ((x∗y)∗y))} ≥ ia((x∗y)∗y) and fa(x) = min{fa(x), fa(1)} = min{fa(x), fa((x∗y)∗ (x∗y))} = min{fa(x), fa(x∗ ((x∗y)∗y))} ≥ fa((x∗y)∗y). (v). from (iv) we have min{ta(x), ta(y)} ≤ min{ta(x), ta((y ∗ (x∗z))∗ (x∗z))} = min{ta(x), ta((x∗ (y ∗z))∗ (x∗z))} = min{ta(x), ta(x∗ (x∗ (y ∗z))∗z))} ≤ ta((x∗ (y ∗z))∗z)), min{ia(x), ia(y)} ≥ min{ia(x), ia((y ∗ (x∗z))∗ (x∗z))} = min{ia(x), ia((x∗ (y ∗z))∗ (x∗z))} = min{ia(x), ia(x∗ (x∗ (y ∗z))∗z))} ≥ ia((x∗ (y ∗z))∗z)) 70 neutrosophic filters in be-algebras and min{fa(x), fa(y)} ≥ min{fa(x), fa((y ∗ (x∗z))∗ (x∗z))} = min{fa(x), fa((x∗ (y ∗z))∗ (x∗z))} = min{fa(x), fa(x∗ (x∗ (y ∗z))∗z))} ≥ fa((x∗ (y ∗z))∗z)). (vi). let x ≤ y, that is, x∗y = 1. ta(y) = min{ta(y), ta(1∗1)} = min{ta(y), ta((x∗y)∗1)}≤ ta(1∗x) = ta(x), ia(y) = min{ia(y), ia(1∗1)} = min{ia(y), ia((x∗y)∗1)}≥ ia(1∗x) = ia(x), fa(y) = min{fa(y), fa(1∗1)} = min{fa(y), fa((x∗y)∗1)}≥ fa(1∗x) = fa(x). (vii). let z ∈ a(x, y). then x∗ (y ∗z) = 1. hence min{ta(x), ta(y)} = min{ta(x), ta(y), ta(1)} = min{ta(x), ta(y), ta(x∗ (y ∗z))} ≤ min{ta(y), ta(y ∗z)} ≤ ta(z). also, we have min{ia(x), ia(y)} = min{ia(x), ia(y), ia(1)} = min{ia(x), ia(y), ia(x∗ (y ∗z))} ≥ min{ia(y), ia(y ∗z)} ≥ ia(z), and min{fa(x), fa(y)} = min{fa(x), fa(y), fa(1)} = min{fa(x), fa(y), fa(x∗ (y ∗z))} ≥ min{fa(y), fa(y ∗z)} ≥ fa(z). (viii). the proof is by induction on n. by (vii) it is true for n = 1, 2. assume that it satisfies for n = k, that is, 71 a. rezaei, a. borumand saeid and f. smarandache k∏ i=1 ai∗x = 1 ⇒ k∧ i=1 ta(ai) ≤ ta(x), k∧ i=1 ia(ai) ≥ ia(x) and k∧ i=1 fa(ai) ≥ fa(x) for all a1, . . . , ak, x ∈ x. suppose that k+1∏ i=1 ai ∗x = 1, for all a1, . . . , ak, ak+1, x ∈ x. then k+1∧ i=2 ta(ai) ≤ ta(a1 ∗x), k+1∧ i=2 ia(ai) ≥ ia(a1 ∗x), and k+1∧ i=2 fa(ai) ≥ fa(a1 ∗x). since a is a neutrosophic filter of x, we have k+1∧ i=1 ta(ai) = min{( k+1∧ i=2 ta(ai)), ta(a1)}≤ min{ta(a1 ∗x), ta(a1)}≤ ta(x), k+1∧ i=1 ia(ai) = min{( k+1∧ i=2 ia(ai)), ia(a1)}≥ min{ia(a1 ∗x), ia(a1)}≥ ia(x) and k+1∧ i=1 fa(ai) = min{( k+1∧ i=2 fa(ai)), fa(a1)}≥ min{fa(a1 ∗x), fa(a1)}≥ fa(x). 2 theorem 3.1. if {ai}i∈i is a family of neutrosophic filters in x, then ⋂ i∈i ai is too. theorem 3.2. let a ∈ nf(x). then the sets (i) xta = {x ∈ x : ta(x) = ta(1)}, (ii) xia = {x ∈ x : ia(x) = ia(1)}, (iii) xfa = {x ∈ x : fa(x) = fa(1)}, are filters of x. proof. (i). obviously, 1 ∈ xha. let x, x∗y ∈ xta . then ta(x) = ta(x∗y) = ta(1). now, by (nf1) and (nf2), we have ta(1) = min{ta(x), ta(x∗y)}≤ ta(y) ≤ ta(1). hence ta(y) = ta(1). therefore, y ∈ xta. the proofs of (ii) and (iii) are similar to (i).2 72 neutrosophic filters in be-algebras definition 3.2. a neutrosophic set a of x is called an implicative neutrosophic filter of x if satisfies the following conditions: (inf1) ta(1) ≥ ta(x), (inf2) ta(x∗z) ≥ min{ta(x∗ (y ∗z)), ta(x∗y)}, ia(x∗z) ≤ min{ia(x∗ (y ∗z)), ia(x∗y)} and fa(x∗z) ≤ min{fa(x∗ (y ∗z)), fa(x∗y)}, for all x, y, z ∈ x. the set of implicative neutrosophic filter of x is denoted by inf(x). if we replace x of the condition (inf2) by the element 1, then it can be easily observed that every implicative neutrosophic filter is a neutrosophic filter. however, every neutrosophic filter is not an implicative neutrosophic filter as shown in the following example. example 3.2. let x = {1, a, b, c, d} be a be-algebra with the following table: ∗ 1 a b c d 1 1 a b c d a 1 1 b c b b 1 a 1 b a c 1 a 1 1 a d 1 1 1 b 1 then x = (x;∗, 1) is a be-algebra. define a neutrosophic set a on x as follows: ta(x) = { 0.85 if x = 1, a 0.12 otherwise and ia(x) = fa(x) = 0.5, for all x ∈ x. then clearly a = (ta, ia, fa) is a neutrosophic filter of x, but it is not an implicative neutrosophic filter of x, since ta(b∗ c) 6≥ min{ta(b∗ (d∗ c)), ta(b∗d)}. theorem 3.3. let x be a self distributive be-algebra. then every neutrosophic filter is an implicative neutrosophic filter. proof. let a ∈ nf(x) and x ∈ x. obvious that ta(x) ≤ ta(1), ia(x) ≥ ia(1) and fa(x) ≥ fa(1). by self distributivity and (nf2), we have min{ta(x∗(y∗z)), ta(x∗y)} = min{ta((x∗y)∗(x∗z)), ta(x∗y)}≤ ta(x∗z), 73 a. rezaei, a. borumand saeid and f. smarandache min{ia(x∗(y∗z)), ia(x∗y)} = min{ia((x∗y)∗(x∗z)), ia(x∗y)}≥ ia(x∗z) and min{fa(x∗(y∗z)), fa(x∗y)} = min{fa((x∗y)∗(x∗z)), fa(x∗y)}≥ fa(x∗z). therefore a ∈ inf(x).2 let t ∈ [0, 1]. for a neutrosophic filter a of x, t-level subset which denoted by u(a; t) is defined as follows: u(a; t) := {x ∈ a : t ≤ ta(x), ia(x) ≤ t and fa(x) ≤ t} and strong t-level subset which denoted by u(a; t)> as u(a; t)> := {x ∈ a : t < ta(x), ia(x) < t and fa(x) < t}. theorem 3.4. let a ∈ ns(x). the following are equivalent: (i) a ∈ nf(x), (ii) (∀t ∈ [0, 1]) u(a; t) 6= ∅ imply u(a; t) is a filter of x. proof. (i)⇒(ii). let x, y ∈ x be such that x, x ∗ y ∈ u(a; t), for any t ∈ [0, 1]. then t ≤ ta(x) and t ≤ ta(x∗y). hence t ≤ min{ta(x), ta(x∗y)}≤ ta(y). also, ia(x) ≤ t and ia(x ∗ y) ≤ t and so t ≥ min{ia(x), ia(x ∗ y)} ≥ ia(y). by a similar argument we have t ≥ min{fa(x), fa(x∗y)}≥ fa(y). therefore, y ∈ u(a; t). (ii)⇒(i). let u(a; t) be a filter of x, for any t ∈ [0, 1] with u(a; t) 6= ∅. put ta(x) = ia(x) = fa(x) = t, for any x ∈ x. then x ∈ u(a; t). since u(a; t) is a filter of x, we have 1 ∈ u(a; t) and so ta(x) = t ≤ ta(1). now, for any x, y ∈ x, let ta(x∗y) = ia(x∗y) = fa(x∗y) = tx∗y and ta(x) = ia(x) = fa(x) = tx. put t = min{tx∗y, tx}. then x, x ∗ y ∈ u(a; t), so y ∈ u(a; t). hence t ≤ ta(y), t ≥ ia(y), t ≥ fa(y) and so min{ta(x∗y), ta(x)} = min{tx∗y, tx} = t ≤ ta(y), min{ia(x∗y), ia(x)} = min{tx∗y, tx} = t ≥ ia(y), and min{fa(x∗y), fa(x)} = min{tx∗y, tx} = t ≥ fa(y). therefore, a ∈ nf(x).2 74 neutrosophic filters in be-algebras theorem 3.5. let a ∈ nf(x). then we have (∀a, b ∈ x) (∀t ∈ [0, 1]) (a, b ∈ u(a; t) ⇒ a(a, b) ⊆ u(a; t)). proof. assume that a ∈ nf(x). let a, b ∈ x be such that a, b ∈ u(a; t). then t ≤ ta(a) and t ≤ ta(b). let c ∈ a(a, b). hence a ∗ (b ∗ c) = 1. now, by proposition 3.1(v) and (be3), we have t ≤ min{ta(a), ta(b)}≤ ta((a∗ (b∗ c)∗ c)) = ta(1∗ c) = ta(c), t ≥ min{ia(a), ia(b)}≥ ia((a∗ (b∗ c)∗ c)) = ia(1∗ c) = ia(c) and t ≥ min{fa(a), fa(b)}≥ fa((a∗ (b∗ c)∗ c)) = fa(1∗ c) = fa(c). then c ∈ u(a; t). therefore, a(a, b) ⊆ u(a; t)).2 corolary 3.1. let a ∈ nf(x). then (∀t ∈ [0, 1]) (u(a; t) 6= ∅ ⇒ u(a; t) = ⋃ a,b∈u(a;t) a(a, b)). proof. it is sufficient prove that u(a; t) ⊆ ⋃ a,b∈u(a;t) a(a, b). for this, assume that x ∈ u(a; t). since x∗ (1∗x) = 1, we have x ∈ a(x, 1). hence u(a; t) ⊆ a(x, 1) ⊆ ⋃ x∈u(a;t) a(x, 1) ⊆ ⋃ x,y∈u(a;t) a(x, y). 2 theorem 3.6. let x be a self distributive be-algebra and a ∈ nf(x). then the following conditions are equivalent: (i) a ∈ inf(x), (ii) ta(y ∗ (y ∗x)) ≤ ta(y ∗x), ia(y ∗ (y ∗x)) ≥ ia(y ∗x) and fa(y ∗ (y ∗x)) ≥ fa(y ∗x), (iii) min{ta((z ∗ (y ∗ (y ∗x))), ta(z)}≤ ta(y ∗x), min{ia((z ∗ (y ∗ (y ∗x))), ia(z)}≥ ia(y ∗x) and min{fa((z ∗ (y ∗ (y ∗x))), fa(z)}≥ fa(y ∗x). 75 a. rezaei, a. borumand saeid and f. smarandache proof. (i)⇒(ii). let a ∈ nf(x). by (inf1) and (be1) we have ta(y ∗ (y ∗x)) = min{ta(y ∗ (y ∗x)), ta(1)} = min{ta(y ∗ (y ∗x)), ta(y ∗y)} ≤ ta(y ∗x), ia(y ∗ (y ∗x)) = min{ia(y ∗ (y ∗x)), ia(1)} = min{ia(y ∗ (y ∗x)), ia(y ∗y)} ≥ ia(y ∗x) and fa(y ∗ (y ∗x)) = min{fa(y ∗ (y ∗x)), fa(1)} = min{fa(y ∗ (y ∗x)), fa(y ∗y)} ≥ fa(y ∗x). (ii)⇒(iii). let a be a neutrosophic filter of x satisfying the condition (ii). by using (nf2) and (ii) we have min{ta(z ∗ (y ∗ (y ∗x))), ta(z)} ≤ ta(y ∗ (y ∗x)) ≤ ta(y ∗x), min{ia(z ∗ (y ∗ (y ∗x))), ia(z)} ≥ ia(y ∗ (y ∗x)) ≥ ia(y ∗x) and min{fa(z ∗ (y ∗ (y ∗x))), fa(z)} ≥ fa(y ∗ (y ∗x)) ≥ fa(y ∗x). (iii)⇒(i). since x∗ (y ∗z) = y ∗ (x∗z) ≤ (x∗y)∗ (x∗ (x∗z)), we have ta(x∗ (y ∗z)) ≤ ta((x∗y)∗ (x∗ (x∗z))), ia(x∗ (y ∗z)) ≥ ia((x∗y)∗ (x∗ (x∗z))) and fa(x∗ (y ∗z)) ≥ fa((x∗y)∗ (x∗ (x∗z))), by proposition 3.1(i). thus min{ta(x∗ (y ∗z)), ta(x∗y)} ≤ min{ta((x∗y)∗ (x∗ (x∗z))), ta(x∗y)} ≤ ta(x∗z). 76 neutrosophic filters in be-algebras min{ia(x∗ (y ∗z)), ia(x∗y)} ≥ min{ia((x∗y)∗ (x∗ (x∗z))), ia(x∗y)} ≥ ia(x∗z) and min{fa(x∗ (y ∗z)), fa(x∗y)} ≥ min{fa((x∗y)∗ (x∗ (x∗z))), fa(x∗ y)} ≥ fa(x∗z). therefore, a ∈ inf(x). let f : x → y be a homomorphism of be-algebras and a ∈ ns(x). define tree maps taf : x → [0, 1] such that taf (x) = ta(f(x)), iaf : x → [0, 1] such that iaf (x) = ia(f(x)) and faf : x → [0, 1] such that faf (x) = fa(f(x)), for all x ∈ x. then taf , iaf and faf are well-define and af = (taf , iaf , faf ) ∈ ns(x).2 theorem 3.7. let f : x → y be an onto homomorphism of be-algebras and a ∈ ns(y). then a ∈ nf(y) (resp. a ∈ inf(y)) if and only if af ∈ nf(x) (resp. af ∈ inf(x)). proof. assume that a ∈ nf(y). for any x ∈ x, we have taf (x) = ta(f(x)) ≤ ta(1y ) = ta(f(1x)) = taf (1x), iaf (x) = ia(f(x)) ≥ ia(1y ) = ia(f(1x)) = iaf (1x) and faf (x) = fa(f(x)) ≥ fa(1y ) = fa(f(1x)) = faf (1x). hence (nf1) is valid. now, let x, y ∈ x. by (nf1) we have min{taf (x∗y), taf (x)} = min{ta(f(x∗y)), ta(f(x))} = min{ta(f(x)∗f(y)), ta(f(x))} ≤ ta(f(y)) = taf (y) also, min{iaf (x∗y), iaf (x)} = min{ia(f(x∗y)), ia(f(x))} = min{ia(f(x)∗f(y)), ia(f(x))} ≥ ia(f(y)) = iaf (y). 77 a. rezaei, a. borumand saeid and f. smarandache by a similar argument we have min{faf (x ∗ y), faf (x)} ≥ faf (y). therefore, af ∈ nf(x). conversely, assume that af ∈ nf(x). let y ∈ y . since f is onto, there exists x ∈ x such that f(x) = y. then ta(y) = ta(f(x)) = taf (x) ≤ taf (1x) = ta(f(1x)) = ta(1y ), ia(y) = ia(f(x)) = iaf (x) ≥ iaf (1x) = ia(f(1x)) = ia(1y ) and fa(y) = fa(f(x)) = faf (x) ≥ faf (1x) = fa(f(1x)) = fa(1y ), now, let x, y ∈ y . then there exists a, b ∈ x such that f(a) = x and f(b) = y. hence we have min{ta(x∗y), ta(x)} = min{ta(f(a)∗f(b)), ta(f(a))} = min{ta(f(a∗ b)), ta(f(a))} = min{taf (a∗ b), taf (a)} ≤ taf (b) = ta(f(b)) = ta(y). also, we have min{ia(x∗y), ia(x)} = min{ia(f(a)∗f(b)), ia(f(a))} = min{ia(f(a∗ b)), ia(f(a))} = min{iaf (a∗ b), iaf (a)} ≥ iaf (b) = ia(f(b)) = ia(y). by a similar argument we have min{fa(x∗y), fa(x)}≥ fa(y). therefore, a ∈ nf(y).2 4 conclusion f. smarandache as an extension of intuitionistic fuzzy logic introduced the concept of neutrosophic logic and then several researchers have studied of some neutrosophic algebraic structures. in this paper, we applied the theory of neutrosophic sets to be-algebras and introduced the notions of (implicative) neutrosophic filters and many related properties are investigated. 78 neutrosophic filters in be-algebras acknowledgment we thank the anonymous referees for the careful reading of the paper and the suggestions on improving its presentation. references [1] a. borumand saeid, a. rezaei, r. a. borzooei, some types of filters in bealgebras, math. comput. sci., 7(3) (2013), 341–352. 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[11] a. walendziak, on normal filters and congruence relations in be-algebras, commentationes mathematicae, 52(2) (2012), 199–205. 79 ratio mathematica 22 (2012) 37-43 issn:1592-7415 operators on weak hypervector spaces ali taghavi and roja hosseinzadeh department of mathematics, faculty of basic sciences, university of mazandaran, p. o. box 47416-1468, babolsar, iran. taghavi@umz.ac.ir, ro.hosseinzadeh@umz.ac.ir abstract let x and y be weak hypervector spaces and lw(x, y ) be the set of all weak linear operators from x into y . we prove some algebraic properties of lw(x, y ). key words: weak hypervector space, weak subhypervector space, normal weak hypervector space, weak linear operator 2000 ams subject classifications: 46j10, 47b48. 1 introduction the concept of hyperstructure was first introduced by marty [3] in 1934 and has attracted attention of many authors in last decades and has constructed some other structures such as hyperrings, hypergroups, hypermodules, hyperfields, and hypervector spaces. these constructions has been applied to many disciplines such as geometry, hypergraphs, binary relations, combinatorics, codes, cryptography, probability and etc. a wealth of applications of this concepts are given in [1 − 2] and [12]. in 1988 the concept of hypervector space was first introduced by scafatitallini. she studied more properties of this new structure in [11]. in [11], tallini introduced the concept of norm on weak hypervector spaces. we used this definition to extend some theorems of analysis from classic vector spaces to hypervector spaces. for example see [5−7, 9]. moreover, in [4] we defined the concept of dimension of weak hypervector spaces and also authors in [8] introduce the new concept hyperalgebra and quotient hyperalgebra. 37 a. taghavi and r. hosseinzadeh now we want to use some of our defined concepts and prove some algebraic properties of lw(x, y ), where lw(x, y ) is the set of all weak linear operators from the weak hypervector space x into the weak hypervector space y . note that the hypervector spaces used in this paper are the special case where there is only one hyperoperation, the external one, all the others are ordinary operations. the general hypervector spaces have all operations multivalued also in the hyperfield (see [12]). 2 preliminaries we need some preliminary definitions for to state our results. in this section we state them. definition 2.1. [11] a weak or weakly distributive hypervector space over a field f is a quadruple (x,+,o,f) such that (x,+) is an abelian group and o : f × x −→ p∗(x) is a multivalued product such that (i) ∀a ∈ f, ∀x, y ∈ x, [ao(x + y)] ∩ [aox + aoy] ̸= ∅, (ii) ∀a, b ∈ f, ∀x ∈ x, [(a + b)ox] ∩ [aox + box] ̸= ∅, (iii) ∀a, b ∈ f, ∀x ∈ x, ao(box) = (ab)ox, (iv) ∀a ∈ f, ∀x ∈ x, ao(−x) = (−a)ox = −(aox), (v) ∀x ∈ x, x ∈ 1ox. we call (i) and (ii) weak right and left distributive laws, respectively. note that the set ao(box) in (3) is of the form ∪y∈boxaoy. definition 2.2. [11] let (x, +, o, f) be a weak hypervector space over a field f, that is the field of real or complex numbers. we define a pseudonorm in x as a mapping ∥.∥ : x −→ r, of x into the reals such that: (i) ∥0∥ = 0, (ii) ∀x, y ∈ x, ∥x + y∥ ≤ ∥x∥ + ∥y∥, (iii) ∀a ∈ f, ∀x ∈ x, sup ∥aox∥ = |a|∥x∥. definition 2.3. let x and y be hypervector spaces over f. a map t : x −→ y is called (i) linear if and only if t(x + y) = t(x) + t(y), t(aox) ⊆ aot(x), ∀x, y ∈ x, a ∈ f 38 operators on weak hypervector spaces (ii) antilinear if and only if t(x + y) = t(x) + t(y), t(aox) ⊇ aot(x), ∀x, y ∈ x, a ∈ f (iii) strong linear if and only if t(x + y) = t(x) + t(y), t(aox) = aot(x), ∀x, y ∈ x, a ∈ f. 3 main results before to state our results we describe some fundamental concepts and lemmas from [4]. for more details see [4]. by lemma 3.1 in [4] we have the following definition. throughout paper, suppose that x and y are weak hypervector spaces over a field f . definition 3.1. [4] if a ∈ f and x ∈ x, then zaox for 0 ̸= a is that element of aox such that x ∈ a−1ozaox and for a = 0, we define zaox = 0. as the descriptions in [4], zaox is not unique, necessarily. so the set of all these elements denoted by zaox. in the mentioned paper we introduced a certain category of weak hypervector spaces. these weak hypervector spaces have been called ”normal”. we proved that zaox is singleton in a normal weak hypervector space. definition 3.2. [4] suppose x satisfy the following conditions: (i) (za1ox + za2ox) ∩ z(a1+a2)ox ̸= ∅, ∀x ∈ x, ∀a1, a2 ∈ f, (ii) (zaox1 + zaox2) ∩ zao(x1+x2) ̸= ∅, ∀x1, x2 ∈ x, ∀a ∈ f. then x is called a normal weak hypervector space. lemma 3.1. [4] if a ∈ f, 0 ̸= b ∈ f and x ∈ x, then the following properties hold: (i) x ∈ z1ox; (ii) aozbox = abox; (iii) z−aox = −zaox; (iv) if x is normal, then zaox is singleton. in [4], the following lemma stated a criterion for normality of a weak hypervector space. 39 a. taghavi and r. hosseinzadeh lemma 3.2. [4] x is normal if and only if (i) za1ox + za2ox = z(a1+a2)ox, ∀x ∈ x, ∀a1, a2 ∈ f, (ii) zaox1 + zaox2 = zao(x1+x2), ∀x1, x2 ∈ x, ∀a ∈ f. definition 3.3. [6] let t : x −→ y be an operator. t is said to be bounded if there exists a positive real number k such that we have ∥tx∥ ≤ k∥x∥ (∀x ∈ x). definition 3.4. [9] a map t : x −→ y is called weak linear operator if t is additive and satisfies t(zaox) ⊆ aotx, (a ∈ f, x ∈ x). denote the set of all weak linear operators and the set of all bounded weak linear operators from x into y by lw(x, y ) and bw(x, y ), respectively. theorem 3.1. [4] let x be normal. then x with the same defined sum and the following scalar product is a classical vector space: a.x = zaox, ∀a ∈ f, x ∈ x. lemma 3.3. let y be normal, t ∈ lw(x, y ) and a ∈ f. define at : x → y x 7→ a.tx then at is a weak linear operator, where the operation ’.’ is the defined scalar product in theorem 3.1. moreover, for all a, b ∈ f and t, s ∈ lw(x, y ) we have a(t + s) = at + as, (a + b)t = at + bt. proof. let a.u = zaou, where u ∈ y . from theorem ?, we know that y with this scalar product is a classical vector space. let x, y ∈ x and b ∈ f . by lemma 3.1 we have (at)(zbox) = a.t(zbox) ⊆ a.(botx) = {a.u : u ∈ botx} = {zaou : u ∈ botx} = zao(botx) = zabotx = bozaotx = bo(a.tx) = bo(at)x 40 operators on weak hypervector spaces and also from the normality of y , we obtain (at)(x + y) = a.t(x + y) ⊆ a.(tx + ty) = zao(tx+ty) = zaotx + zaoty = a.tx + a.ty = (at)x + (at)y. hence at is a weak linear operator. now let t, s ∈ lw(x, y ) and x ∈ x. the normality of y yields [a(t + s)]x = a.(t + s)x = zao(t+s)x = zao(tx+sx) = zaotx + zaosx = a.tx + a.sx = (at)x + (as)x = (at + as)x which implies that a(t + s) = at + as. the second relation is proved in a similar way. theorem 3.2. let y be normal. then lw(x, y ) with the following sum and product is a weak hypervector space over f. (t + s)x = tx + sx (t, s ∈ lw(x, y ), x ∈ x) aot = {s ∈ lw(x, y ) : sx ∈ aotx, ∀x ∈ x} (a ∈ f, t ∈ lw(x, y )). proof. first we show that aot is a nonempty subset of lw(x, y ). by lemma 3.3, at ∈ lw(x, y ) and for any x ∈ x we have (at)x = a.tx = zaotx ∈ aotx which imply that at ∈ aot . it is easy to check that (lw(x, y ), +) is an abelian group. we show the correctness the first property of scalar product, the rest properties are obtained in a similar way. by lemma 3.3, for all a ∈ f and t, s ∈ lw(x, y ) we have a(t + s) = at + as. this together with a(t + s) ∈ ao(t + s), at ∈ aot, as ∈ aos 41 a. taghavi and r. hosseinzadeh imply that [ao(t + s)] ∩ [aot + aos] ̸= ∅ and this completes the proof. theorem 3.3. let y be normal. then the following statements are hold. (i) for all a ∈ f and t ∈ lw(x, y ) we have zaot = at. (ii) lw(x, y ) is a normal weak hypervector space. proof. (i) by definition 3.1 we have zaot ∈ aot, t ∈ a−1ozaot which for all x ∈ x implies zaot x ∈ (aot)x, tx ∈ (a−1ozaot )x. since by theorem 3.2 we have (aot)x ⊆ aotx, (a−1ozaot )x ⊆ a−1ozaot x, we obtain zaot x ∈ aotx, tx ∈ a−1ozaot x. these relations, by definition 3.1 yield that zaot x = zaotx. so we obtain zaot x = (at)x, for all x ∈ x and hence zaot = at. (ii) the normality of lw(x, y ) can be concluded from lemma 3.3 and part (i). theorem 3.4. let y be normal. then bw(x, y ) with the defined sum and scalar product in theorem ? is a subhypervector space of lw(x, y ). proof. it is enough to show that t + s, aot ∈ bw(x, y ) for any a ∈ f and t, s ∈ bw(x, y ) it is easy to check that t +s ∈ bw(x, y ). let s ∈ aot . hence sx ∈ aotx and so ∥sx∥ ≤ |a|∥tx∥ ≤ |a|∥t∥∥x∥. this completes the proof. acknowledgements. this research is partially supported by the research center in algebraic hyperstructures and fuzzy mathematics, university of mazandaran, babolsar, iran. 42 operators on weak hypervector spaces references [1] p. corsini, prolegomena of hypergroup theory, aviani editore, (1993). [2] p. corsini and v. leoreanu, applications of hyperstructure theory, kluwer academic publishers, advances in mathematics (dordrecht), (2003). 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[12] t. vougiouklis, hyperstructures and their representations, hadronic press, (1994). 43 microsoft word documento1 microsoft word documento1 microsoft word documento1 ratio mathematica 24 (2013), 63–72 issn: 1592-7415 on some probability concepts in fuzzy framework i. tofan faculty of mathematics, ”al.i. cuza” university of iaşi, cefair, iaşi abstract in this paper some modalities in which the concept of probability can be fuzzified are investigated in order to obtain new tools useful in the modelization of the risks. some papers related to this approach are [1-6]. finally, some open problems are proposed to the reader. key words: probability, fuzzy set, fuzzy probability, fuzzy numbers msc2010: 60a86, 03e72. 1 classical case in the set theory the following operations are used: the intersection (a∩b); the union (a∪b); the complement (a); the difference (a\b = a∩b); the implication (a → b = a\b = a∪b); the symmetric difference (a4b = (a\b)∪(b\a) = (a∩b)∪(a∩b)); the equivalence (a ↔ b = (a → b) ∩ (b → a) = a4b), where a,b are sets. the empty set is denoted by ∅, and the set of subsets of a set s will be denoted by p(s). let a,b,c ∈p(s). remark 1. we have: i.tofan i) ∩,∪ are commutative and associative; ii) a∩a = a, a∪a = a; iii) a∩ (a∪b) = a, a∪ (a∩b) = a; iv) a∩ (b ∪c) = (a∩b) ∪ (a∩c); a∪ (b ∩c) = (a∪b) ∩ (a∪c); v) a∩a = ∅, a∪a = s; vi) a∩b = a∪b; a∪b = a∩b; vii) a = a. let ω 6= ∅. definition 2. by field of events (in relation with the space ω) one intend k ⊆p(ω) such that i) ω ∈ k; ii) a,b ∈ k ⇒ a∪b ∈ k; iii) a ∈ k ⇒ a ∈ k. remark 3. let k be a field of events. we have i) a,b ∈ k ⇒ a∩b ∈ k; ii) a,b ∈ k ⇒ a\b ∈ k; iii) a,b ∈ k ⇒ a → b ∈ k; iv) a,b ∈ k ⇒ a4b ∈ k; v) a,b ∈ k ⇒ a ↔ b ∈ k; vi) ∅∈ k. let k be a field of events. definition 4. by probability on k one intend p : k → [0, 1] such that: i) p(ω) = 1; ii) a∩b = ∅⇒ p(a∪b) = p(a) + p(b). 64 on some probability concepts in fuzzy framework remark 5. we have i) p(∅) = 0; ii) p(a) = 1 −p(a); iii) p(a\b) = p(a) −p(a∩b); iv) p(a∩b) + p(a∪b) = p(a) + p(b); v) p(a → b) = 1 −p(a) + p(a∩b); v) p(a) + p(a → b) = p(b) + p(b → a). remark 6. if p : k → [0, 1] is an application satisfying p(∅) = 0, p(ω) = 1, then the condition ii), from the definition and the condition vi) from the remark are equivalent. 2 fuzzy case for the construction which will be given in this case we need the concepts of t-norms and t-conorms. definition 7. a function t : [0, 1]× [0, 1] → [0, 1] will be called t-norm if the following conditions are satisfied: i) t(x, 1) = x, ∀x ∈ [0, 1]; ii) t(x,y) = t(y,x), for any x,y ∈ [0, 1]; iii) t(x,t(y,z)) = t(t(x,y),z), for any x,y,z ∈ [0, 1]; iv) x ≤ z ⇒ t(x,y) ≤ t(z,y), ∀y ∈ [0, 1]. remark 8. we have also: v) t(x, 0) = t(1, 0) = t(0, 1) = 0, ∀x ∈ [0, 1]. example 9. i) p : [0, 1] × [0, 1] → [0, 1], p(x,y) = xy; ii) min : [0, 1] × [0, 1] → [0, 1], min(x,y) = { x, if x ≤ y y, if x > y ii) tm : [0, 1] × [0, 1] → [0, 1], tm(x,y) = max{x + y − 1, 0}. definition 10. a function t∗ : [0, 1] × [0, 1] → [0, 1] will be called t-conorm if the following conditions are satisfied: 65 i.tofan i) t∗(x, 0) = x, ∀x ∈ [0, 1]; ii) t∗(x,y) = t∗(y,x), for any x,y ∈ [0, 1]; iii) t∗(x,t∗(y,z)) = t∗(t∗(x,y),z), for any x,y,z ∈ [0, 1]. iv) x ≤ z ⇒ t∗(x,y) ≤ t∗(z,y), ∀y ∈ [0, 1]. example 11. i) p∗ : [0, 1] × [0, 1] → [0, 1], p∗(x,y) = x + y −xy; ii) max : [0, 1] × [0, 1] → [0, 1], max(x,y) = { x, if x ≥ y y, if x < y ; iii) t∗m : [0, 1] → [0, 1] → [0, 1], t∗m(x,y) = min{x + y, 1}. definition 12. the t-norm t and the t-conorm t∗ are called dual each another if for any x,y ∈ [0, 1] t(x,y) = 1 − t∗(1 −x, 1 −y). for example, p,p∗ or min, max or tm, t ∗ m are such couples. definition 13. a couple (u,µ) where u 6= ∅ and µ : u → [0, 1] is an application will be called fuzzy set (on the universe u) or fuzzy subset of u. the empty fuzzy set is given by φ̃ : u → [0, 1], φ̃(x) = 0, ∀x ∈ u. we shall denote µ ⊆ η if µ(x) ≤ η(x), ∀x ∈ u. by ũ : u → [0, 1] one intend the application given by ũ(x) = 1, ∀x ∈ u. let f(u) be the family of fuzzy subsets of u. the operations with fuzzy subsets can be defined in the following way: for µ,η : f(u), µ ⋂ t η : u → [0, 1], (µ ⋂ t η)(x) = t(µ(x),η(x)) µ ⋃ t η → [0, 1], (µ ⋃ t η)(x) = t ∗(µ(x),η(x)). the complement µ : u → [0, 1] will be given by µ(x) = 1 − µ(x). in a similar way with the classical case one define µ t→ η, µ t→ η, etc. µ t −η : u → [0, 1], (µ t −η)(x) = t(µ(x), 1 −η(x)); and µ t→ η : u → [0, 1], (µ t→ η)(x) = t∗(1 −µ(x),η(x)). for the couples t-norm/conorm described above we obtain: �,⊕; ∩,∪; ... more precisely for µ,η : u → [0, 1] we have: a. µ�η : u → [0, 1], (µ�η)(x) = µ(x)η(x); µ⊕η : u → [0, 1], (µ⊕η)(x) = µ(x) + η(x) −µ(x)η(x). µ : u → [0, 1], µ(x) = 1 −µ(x); and µ η : u → [0, 1], (µ η)(x) = µ(x) −µ(x)η(x); µ©→ η : u → [0, 1], (µ©→ η)(x) = 1 −µ(x) + µ(x)η(x); 66 on some probability concepts in fuzzy framework remark 14. we have i) �,⊕ are commutative and associative; ii) µ�µ ⊆ µ, µ ⊆ µ⊕µ; iii) µ ⊇ µ� (µ⊕η); µ ⊆ µ⊕ (µ�η); iv) µ⊕ (η � τ) ⊇ (µ⊕η) � (µ⊕ τ); µ� (η ⊕ τ) ⊆ (µ�η) ⊕ (µ� τ); v) (µ�µ)(x) ≤ 1 4 , (µ⊕µ)(x) ≥ 3 4 , ∀x ∈ u; v) µ⊕η = µ�η; µ�η = µ⊕η. b. µ∩η : u → [0, 1], (µ∩η) = min{µ(x),η(x)}; µ∪η : u → [0, 1], (µ∪η)(x) = max{µ(x),η(x)}; µ : u → [0, 1], µ(x) = 1 −µ(x); µ−η : u → [0, 1], (µ−η)(x) = min{µ(x), 1 −η(x)}; µ → η : u → [0, 1], (µ → η)(x) = 1 − min{µ(x), 1 −η(x)}; remark 15. we have i) ∩,∪ are commutative and associative; ii) µ∩µ = µ, µ∪µ = µ iii) µ∪ (µ∩η) = µ; µ∩ (µ∪η) = µ; iv) µ∪ (η ∩ τ) = (µ∪η) ∩ (µ∪ τ) = (µ∩η) ∪ (µ∩ τ); v) (µ∩µ)(x) ≤ 1 2 , (µ∪µ)(x) ≥ 1 2 , ∀x ∈ u; vi) µ∪η = µ∩η; µ∩η = µ∪η. 67 i.tofan c. µ5η : u → [0, 1], (µ5η)(x) = max{µ(x) + η(x) − 1, 0}; µ4η : u → [0, 1], (µ4η)(x) = min{µ(x) + η(x), 1}; µ : u → [0, 1], µ(x) = 1 −µ(x); µ•η : u → [0, 1], (µ•η)(x) = max{µ(x) −η(x), 0}; µ ·→ η : u → [0, 1], (µ ·→ η)(x) = min{1 −µ(x) + η(x), 1}. remark 16. we have i) 5,4 are commutative and associative; ii) µ5η ⊆ µ, µ ⊆ µ4µ; iii) µ ⊆ µ4 (µ�η); µ ⊇ µ5 (µ4η); iv) (µ5µ)(x) = 0, (µ4µ)(x) = 1, ∀x ∈ u; v) µ5η = µ4η = µ5η. remark 17. we have µ5η ⊆ µ�η ⊆ µ∩η; µ∪η ⊆ µ⊕η ⊆ µ4η and µ = µ. 3 fuzzy numbers in the last section of the paper fuzzy number will be used. let r be the field of real numbers. definition 18. by triangular fuzzy number one intend a triple (a,b,c), where a,b,c ∈ r, a ≤ b ≤ c. we shall denote rt the set of triangular fuzzy numbers. for a = (a1,b1,c2), b = (a2,b2,c2) from rt, if c1 ≤ a2, or a2 ≤ c1 and a1+2b1+c14 < a2+2b2+c2 4 , or a2 ≤ c1, a1+2b1+c14 = a2+2b2+c2 4 and b1 < b2, or a2 ≤ c1, a1+2b1+c14 = a2+2b2+c2 4 , b1 = b2 and c1 −a1 < c2 −a2, we shall write a . b (a special kind of ”order” being obtained in this way). remark 19. a triangular fuzzy number (a,b,c) ∈ rt is uniquely determined by a triple (λ,b,ρ) where λ = b − a, ρ = c − b are positive reals called the left, respectively right tolerance. 68 on some probability concepts in fuzzy framework we will use the notation with the central value on the first place (b,λ,ρ). we consider the operations (these operations are introduced by the author and was presented for the first time at a conference given at the university of chieti in 2007 and was published in [6]): (a,λ,ρ) � (b,λ′,ρ′) = (a + b, max{λ,λ′}, max{ρ,ρ′}) (a,λ,ρ) � (b,λ′,ρ′) = (ab, max{λ,λ′}, max{ρ,ρ′}) and the relation ”∼” given by (a,λ,ρ) ∼ (b,λ′,ρ′)if { a = b λ−λ′ = ρ−ρ′. one obtains: remark 20. we have: i) �,� are commutative and associative; ii) � is distributive with respect to �; iii) (0, 0, 0) is neutral element for �, and (1, 0, 0) is neutral element for �; iv) (a,λ,ρ) � (−a,ρ,λ) ∼ (0, 0, 0); if a 6= 0 (a,λ,ρ) � ( 1 a ,ρ,λ) ∼ (1, 0, 0). v) ”∼” is an equivalence relation on rt. 4 fuzzy events let be ω 6= ∅ and f(ω). definition 21. by fuzzy field of events one intend k ⊆f(ω) such that: i) ω̃ ∈ k ii) µ,η ∈ k ⇒ µ ⋃ t η ∈ k; iii) µ ∈ k ⇒ µ ∈ k. remark 22. we have: i) φ̃ ∈ k; 69 i.tofan ii) µ,η ∈ k ⇒ µ ⋂ t η ∈ k; µ t −η ∈ k, µ t→ η ∈ k; iii) (µ ∈ k ⇒ µ ∈ k) ⇔ (µ,η ∈ k ⇒ µ t − η ∈ k)⇔ (µ,η ∈ k ⇒ µ t→ η ∈ k). let k be a fuzzy field of events. definition 23. by probability on k one intend p : k → [0, 1] such that i) p(ω̃) = 1 ii) µ ⋂ t η = φ ⇒ p(µ ⋃ t η) = p(µ) + p(η). remark 24. verify the following: i) p(φ̃) = 0; ii) p(µ) = 1 −p(µ); iii) µ ⊆ η ⇒ p(η t −µ) = p(η); iv) p(µ t −η) = p(µ) −p(µ ⋂ t η); v) p(µ ⋃ t η) + p(µ ⋂ t η) = p(µ) + p(η); vi) p(µ t→ η) = 1 −p(µ) + p(µ ⋂ t η); vii) p(µ) + p(µ t→ η) = p(η) + p(η t→ µ). in the case t = tm we suppose also that µ,η ∈ k ⇒ µ� η ∈ k. in this context we shall denote p(µ/η) = p(µ�η)/p(η)(p(η) 6= 0). proposition 25. in the above condition we have: p(µ/η) = p(µ)p(η/µ) p(µ)p(η/µ) + p(η)p(µ/η) . we have also proposition 26. if µ1, . . . ,µn ∈ k are such that µi ⋂ t µj = φ̃ for i 6= j, then p(µ ⋃ t, . . . , ⋃ t µn) = p(µ1) + . . . + p(µn). 70 on some probability concepts in fuzzy framework 5 fuzzy probability the next step is to substitute [0, 1] in the definition of the probability (in k) with the it = {(a,λ,ρ) ∈ rt/λ ≤ a,ρ ≤ 1 −a,a ∈ [0, 1]}. we have two possibilities: a. we shall use the operations and the equivalence relation given in iii. remark 27. if (a,λ,ρ) is such that a ∈ [0, 1] then there exists (a′,λ′,ρ′) ∈ it such that (a,λ,ρ) ∼ (a′,λ′,ρ′). let k be a fuzzy field of events. definition 28. by fuzzy probability on k one intend an application p : k → it such that i) p(φ̃) = 0; ii) µ ⋂ t η = φ ⇒ p(µ ⋃ t η) ∼ p(µ) �p(η); iii) if p(µ) = (α,λ,ρ) then p(µ) = (1 −a,ρ,λ). remark 29. in view to obtain more properties ii) can be replaced by ii′) p(µ) �p(η) ∼ p(µ ⋂ t η) �p(µ ⋃ t η). problem 30. in the case i), ii′), iii), verify the following: i) p(ω̃) = (1, 0, 0); ii) p(µ t \η) ∼ p(µ) −p(µ ⋂ t η); iii) p(µ t→ η) ∼ p(µ) + p(µ ⋂ t η); iv) p(µ) + p(µ t→ η) ∼ p(η) + p(η t→ µ). b. in the following we propose new operations: (a,λ,ρ)+̃(a′,λ′,ρ′) = (a+a′−aa′,a+a′−aa′−max{a+λ,a′+λ′}, min{a+ ρ + a′ + ρ′, 1}−a−a′ + aa′) (a,λ,ρ)̃·(a′,λ′,ρ) = (aa′,aa′−max{a−λ + a′−λ′−1, 0}min{a + ρ,a′ + ρ′}−aa′) when the numbers are written in the form (a,b,c) (a ≤ b ≤ c), the operation are defined by (a,b,c)+̃(a′,b′,c′) = (max{a,a′},b + b′ − bb′, min{c + c′, 1}) (a,b,c)̃·(a′,b′,c′) = (max{a + a′ − 1, 0},bb′, min{c,c′}). 71 i.tofan remark 31. the above operations are satisfying 0 ≤ max{a,a′}≤ b + b′ − bb′ ≤ min{c + c′, 1}≤ 1 0 ≤ max{a + a′ − 1, 0}≤ bb′ ≤ min{c,c′}≤ 11. in this frame using the form (a,b,c) we can propose the following definition 32. by fuzzy probability on k one intend p : k → it such that i) p(ω̃) = (1, 1, 1), p(φ̃) = (0, 0, 0); ii) p(µ)+̃p(η) = p(µ ⋂ t η)+̃p(µ ⋃ t η); iii) µ ≤ η, p(µ) . p(η). problem 33. verify the following: i) p(µ t −η) = p(µ) −p(µ ⋂ t η); ii) p(µ t→ η) = p(µ) + p(µ ⋂ t η); iii) p(µ) + p(µ t→ η) = p(η) + p(η t→ µ). references [1] bugajski, s., fundamentals of fuzzy probability theory, int. j. theor. phys., 35, 2229-2244, 1996. [2] georgescu, g., bosbach states on fuzzy structures, soft computing, 8, 217-230, 2004. [3] georgescu, g., probabilistic models for intuitionistic predicate logic, journal of logic and computation advances access, 2010. [4] gudder, s., fuzzy probability theory, demonstr. math., 31, 235-254, 1998. [5] gudder, s., what is fuzzy probability theory, foundations of physics, 30, 1663-1678, 2000. [6] tofan, i., some remarks about fuzzy numbers, international journal of risk theory, vol. 1, 2011, p. 87-92. 72 ratio mathematica volume 46, 2023 algebraic coding theory using pell equation x2 − 8y2 = 1 janaki g* gowri shankari a† abstract an interdisciplinary field with significant practical use is cryptography. the difficulty of specific mathematical computing tasks affects a public key cryptosystem’s security. the technique for coding and decoding the messages was described in this work, utilising the solutions of pell equation x2 − 8y2 = 1 and matrix q8∗ . it was noted that messages can be turned into an even size that is then divided into slabs. considering the information security has become a more serious issue in recent years, coding and decoding algorithms are essential in order to improve information security. in this article, a new matrix q8* and a decryption system based on the solutions of the pell equation x2 −8y2 = 1 are devised. the messages can be divided into even-sized slabs during encryption. this algorithm will not only improve information security but also has a high degree of accuracy. keywords: pell equation; encryption-decryption algorithm; q8 ∗ matrix; cryptography 2020 ams subject classifications: 11b37, 11c20, 11d09, 11t71. 1 *cauvery college for women (autonomous), affiliated to bharathidasan university, tiruchirappalli-18, tamil nadu, india. janakikarun@rediffmail.com. †cauvery college for women (autonomous), affiliated to bharathidasan university, tiruchirappalli-18, tamil nadu, india. gowrirajinikanth@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1062. issn: 1592-7415. eissn: 2282-8214. ©janaki g et al. this paper is published under the cc-by licence agreement. 101 janaki and gowri shankari a 1 introduction currently, almost all current coding methods can be supported by number theory carmichael [1950], disckson [1952], trappe and washington [2006]. samuel f.b. morse, an american inventor, developed the first coding technique that used just two symbols—a dot and a comma—to transmit the first cypher message in 1844 through an electric telegraph. he called it morse code. later, binary code encoding improved as a more secure method for encrypting messages, and it is still in use today saranya and janaki [2019], janaki and saranya [2020]. binary code encoding divides each coded word into blocks of ones and zeros. the twentieth century saw important advancements in coding. it is common knowledge that the fibonacci sequences are defined as fn + 1 = fn + fn − 1 with the initial parameters f0=0and f1=1. the definition of the fibonacci q-matrix is q = [(11@10)], and its nth power is of the form carmichael [1950], disckson [1952], trappe and washington [2006]. several approaches were used to study the fibonacci coding principle. for instance, utilising the key variability notion in symmetric key algorithm and the fibonacci q-matrix, a novel method for safe information transmission through communication channels was developed. by using blocking matrices and fibonacci numbers, a new cryptography algorithm was recently described in berges [1981], gould [1981]. using the help of the pell equation’s solutions x28y2 = 1 that were found in tas et al. [2018], sumeyra et al. [2019] and the new matrix q8*, we present new coding and decoding algorithms in this work. also, this method’s fundamental principle hinges on subdividing the message into a 2x2 slab matrix. the security of information will be improved by using this technology, which also has a high degree of accuracy in data transmission via communication channels saranya and janaki [2019], janaki and saranya [2020]. the connection of recurrence for the pell equation x2 − 8y2 = 1 is xn+1 = 3xn + 8yn yn+1 = xn + 3yn for n ≥ 1, where x1 = 3, y1 = 1. it is noted that q8 ∗ = ( xj 8yj yj xj ) the primary concept is to turn the messages into order 2 × 2 block matrices. the number of blocks and letter positions must be determined based on this. then encrypted matrix p is obtained and q8 ∗ utilized for decryption. 102 algebraic coding theory using pell equation x2 − 8y2 = 1 2 main results 2.1 representations: 1. g− a message-to-be-sent matrix constituted by an even order. 2. gj − jth block of g whose size is 2. 3. g− number of slabs gj of g. 4. m = { 3 if g ≥ 3 g if g > 3 5. dj = |gj|. 6. gj = ( gj1 gj2 gj3 gj4 ) 7. p− encrypted matrix defined by p = ( dj gjk ) k∈{1,2,4} 8. ( q8 ∗)m = ( q11 q12 q21 q22 ) 9. ∗− denotes the space between the words. 2.2 assignment of alphabets: a b c d e f g h i m m+1 m+2 m+3 m+4 m+5 m+6 m+7 m+8 j k l m n o p q r m+9 m+10 m+11 m+12 m+13 m+14 m+15 m+16 m+17 s t u v w x y z * m+18 m+19 m+20 m+21 m+22 m+23 m+24 m+25 m-1 2.3 algorithm for encryption: 1. create an even order matrix g for the provided message. 2. separate gj into slabs of size 2 and calculate g. 3. selecting m using g. 103 janaki and gowri shankari a 4. replace the alphabets with the provided numbers to get the elements of gj. 5. find dj. 6. create p . 2.4 algorithm for decryption: using p , one must find g. the goal is to locate gj3 as p contains gj1, gj2, gj4. 1. identify ( q8 ∗)m 2. identify its constituents as qij’s. 3. explore pj1 = q11gj1 + q21gj2 4. explore pj2 = q12gj1 + q22gj2 5. solve dj = pj1(q12sj + q22gj4) − pj2(q11sj + q21gj4) for sj. 6. replace sj = gj3 7. establish gj 8. establish g 2.5 encryption decryption algorithm using (x, y) such that x2 − 8y2 = 1 the following are a few pell equation x2 − 8y2 = 1 solutions: j 1 2 3 4 5 6 7 8 xj 3 17 99 577 3363 19601 114243 665857 yj 1 6 35 204 1189 6930 40391 235416 case (i): g=1 example: 1 take “army” as the encrypted message. encryption: g = ( a r m y ) 104 algebraic coding theory using pell equation x2 − 8y2 = 1 here there is only one block so g = 1. this implies m = 3. thus g1 = ( 3 20 15 27 ) and so g11 = 3, g12 = 20, g13 = 15, g14 = 27. ∴ d1 = −219 p = ( −219 3 20 27 ) decryption: ( q8 ∗)3 = ( x3 8y3 y3 x3 ) = ( 99 280 35 99 ) ⇒ q11 = 99, q12 = 280, q21 = 35, q22 = 99. apply pj1 and pj2, one can get p11 = 997 and p12 = 2820. d1 = p11(q12s1 + q22g14) − p12(q11s1 + q21g14) −219 = 997(280s1 + 2673) − 2820(99s1 + 945) ⇒ s1 = 15 thus g13 = s1 = 15. ∴ g1 = ( 3 20 15 27 ) from p hence g = ( a r m y ) . case (ii): g=4 example: 2 take “amount received” as the encrypted message. encryption: g =   a m o u n t ∗ r e c e i v e d ∗   here there are four blocks so g = 4. ∴ g1 = ( a m n t ) , g2 = ( o u ∗ r ) , g3 = ( e c v e ) , g4 = ( e i d ∗ ) 105 janaki and gowri shankari a choose m = 4 thus g1 = ( 4 16 17 23 ) , g2 = ( 18 24 3 21 ) , g3 = ( 8 6 25 8 ) , g4 = ( 8 12 7 3 ) hence d1 = −180, d2 = 306, d3 = −86, d4 = −60 p =   −180 4 216 23 306 18 24 21 −86 8 6 8 −60 8 12 3   decryption: ( q8 ∗)4 = ( x4 8y4 y4 x4 ) = ( 577 1632 204 577 ) ⇒ q11 = 577, q12 = 1632, q21 = 204, q22 = 577. ∴ one can find, p11 p12 p21 p22 p31 p32 p41 p42 5572 15760 15282 43224 5840 16518 7064 19980 on solving the equation dj = pj1(q12sj + q22gj4) − pj2(q11sj + q21gj4), one can get s1 = 17, s2 = 3, s3 = 25, s4 = 7. thus g13 = s1 = 17, g23 = s2 = 3, g33 = s3 = 25, g43 = s4 = 7. thus, g1 = ( 4 16 17 23 ) , g2 = ( 18 24 3 21 ) , g3 = ( 8 6 25 8 ) , g4 = ( 8 12 7 3 ) from p. hence g =   a m o u n t ∗ r e c e i v e d ∗   106 algebraic coding theory using pell equation x2 − 8y2 = 1 3 conclusions the paper identifies a few particular cryptographic applications that can be utilised to demonstrate and comprehend the fundamental idea of the fibonacci q matrix. fibonacci, often spelled bonacci, leonard of pisa, or leonardo bigollo pisano, was an italian mathematician who was born in the republic of pisa. he is regarded as ”the most talented western mathematician of the middle ages” and had a significant impact on number theory. moreover, pell’s equation refers to any diophantine equation of the type x2 − 8y2 = 1 , where n is a given positive non-square integer and integer solutions are sought for x and y. in this work, using the solutions of the pell equationx2 − 8y2 = 1 and qd∗ is defined. this contributes to the decryption algorithm. in encryption, the message must first be transformed into an even single matrix, then into slabs of size 2. the fact that the entries qd ∗ get bigger and bigger depends on the strict secrecy. one may search the decryption algorithm for the solutions of other well-known equations with the new matrix qd ∗ . on the basis of this scheme various more algorithms can be developed. references c. berges. a history of the fibonacci q-matrix and a higherdimensional problem. fibonacci quart, 19(3):250–257, 1981. r. carmichael. the theory of numbers and diophantine analysis. dover publications co., new york, 1950. l. disckson. history of the theory of numbers, volume ii. chelsia publishing co., new york, 1952. h. gould. a history of the fibonacci q-matrix and a higherdimensional problem. fibonacci quart, 19(3):250–257, 1981. g. janaki and c. saranya. observations on the binary quadratic diophantine equation x2−2xy−y2+2x+14y = 72. international journal of scientific research in mathematical and statistical sciences, 7(2):152–155, 2020. c. saranya and g. janaki. solutions of pell’s equation involving jarasandha numbers. international journal of scientific research in mathematical and statistical sciences, 6(1):234–236, 2019. u. sumeyra, t. nihal, and n. ozgur. new application to coding theory via fibonacci and lucas numbers. mathematical sciences and applications e-notes, 7(1):62–70, 2019. 107 janaki and gowri shankari a n. tas, s. ucar, n. ozgur, and o. kaymak. a new coding/ decoding algoirithm using fibonacci numbers. discrete mathematics, algorithms and applications, 10(2), 2018. w. trappe and l. washington. introduction to cryptography. prentice hall, 2006. 108 ratio mathematica vol. 32, 2017, pp. 3–19 issn: 1592-7415 eissn: 2282-8214 some improved mixed regression estimators and their comparison when disturbance terms follow multivariate t-distribution manoj kumar1, vikas bist2 and man inder kumar3,∗. 1department of statistics, panjab university, chandigarh, india mantiwa@gmail.com 2department of mathematics, panjab university, chandigarh, india bistvikas@gmail.com 3department of statistics, panjab university, chandigarh, india maninderbajarh@gmail.com received on: 27-05-2017. accepted on: 15-06-2017. published on: 30-06-2017 doi:10.23755/rm.v32i0.330 c©manoj kumar et al. abstract the mean square error matrices, bias vector and risk functions of proposed improved mixed regression estimators are obtained by employing the small disturbance approximation technique under the condition, when disturbance terms follows multivariate t-distribution. further, the risk function criterion is used to examine the efficiency of proposed improved mixed regression estimators. keywords: stochastic restrictions; mixed regression estimator; steinrule estimator; multivariate t-distribution etc. 2010 ams subject classifications: 62j05 3 manoj kumar, vikas bist and man inder kumar 1 introduction when incomplete prior information is expressible in the form of set of linear stochastic restrictions on the coefficients in a linear regression model, the method of mixed regression for the estimation of regression coefficients provides asymptotically a more efficient estimator than the least squares method that ignores the prior restrictions. stemming from the philosophy of stein-rule in this paper we proposed two families of improved estimators for the regression coefficients and study their properties when disturbances have multivariate t-distribution. for multivariate t distribution see, [12], [10] and [3]. in section 2, we discuss the framework and estimators. the properties of these estimators are presented in section 3 and the results are compared in section 4. simulation study is carried out to support theoretical finding in section 5. 2 model specification and the estimators let us postulate the linear regression model y = xβ + u (1) where, y is a n × 1 vector of dependent variables; x is a n × p column rank matrix of n-observations on p explanatory non-stochastic variables; β is a p × 1 non-null vector of regression coefficient and u is a n × 1 vector of disturbance following multivariate student t-distribution with probability density function as: f ( u v ,σ2 ) = γv/2γ ( v+n 2 ) π n 2 γ ( v 2 ) σ−n [v + u ′u σ2 ]−n+v 2 (2) where, v > 0,σ > 0 are respectively the degree of freedom and dispersion parameters; the vector u has its error components ui ∈ (−∞,∞), i = 1, 2, ...,n. here the error vector u has mean vector e(u) = 0 for v > 1, variance-covariance matrix e(u ′u) = σ2 ( v v−2 ) i, for v > 2, measure of skewness γ1 = 0 and measure of kurtosis γ2 = σ4 ( 6 v−4 ) i for v > 4. let the stochastic restrictions on β in (1) be r = rβ + v (3) where, r is a j × 1 vector of known elements, r is a j ×p full row rank matrix of known elements and v is a j × 1 vector of distribution such that e(v ) = 0 ; e(v ′v ) = ω (4) 4 some improved mr estimators & their comparison when disturbance terms follow multivariate t-distribution where, ω is a j ×j positive definite symmetric matrix of known elements. further, we assume that the errors associated with the stochastic restriction are independent with the distribution in model (1). the ordinary least square (ols) estimator of β that ignores the prior restrictions (3) is bo = (x ′x)−1x′y (5) if we consider the prior information (3), then the mixed regression (mr) estimator of β is given by bmr = [x ′x + s2r′ω−1r]−1[x′y + s2r′ω−1r] (6) where , s2 = 1 n−p ((y −xb)′(y −xb)) (7) the stein-rule estimator of β is bs = [ 1 −k (y −xb)′(y −xb) b′o(x ′x)b0 ] bo (8) where, k is a positive scalar characterizing the estimator. the stein-mixed regression (smr) estimator of β is given as bsmr = [ x′x + 1 n−p [(y −xbs)′(y −xbs)]r′ω−1r ]−1 [ x′y + 1 n−p [(y −xbs)′(y −xbs)]r′ω−1r ] (9) the mixed stein-regression (msr) estimator of β is bmsr = [ 1 −k (y −xbmr)′(y −xbmr) b′mr(x ′x)bmr ] bmr (10) 3 properties of the estimators px = x(x ′x)−1x′ (11) m = [i −px ] (12) 5 manoj kumar, vikas bist and man inder kumar mj = [px − jc−1xββ′x′] j = 1, 2, . (13) nj = [(x ′x)−1 − jc−1ββ′] j = 1, 2, . (14) c = β′x′xβ (15) µ = (x′x)−1r′ω−1r(x′x)−1 (16) the ols estimator defined in (5) is found to be unbiased if v > 1, with variance covariance matrix and risk function given by e[(b0 −β)(b0 −β)′] = σ2 ( v v − 2 ) (x′x)−1; v > 2 (17) risk(bo) = σ 2 ( v v − 2 ) tr(x′x)−1l; v > 2 (18) where, l is a positive definite symmetric loss matrix. the properties of the mr estimator are same as the smr estimator, so we consider only the smr estimator and present the results in the form of following theorems. theorem 3.1. the asymptotic expression for the bias vector, mean squared error matrix and risk function of smr estimator, up to order o(σ4) of approximations are given as b(bsmr) = 0 (19) m(bsmr) = σ 2 ( v v − 2 ) (x′x)−1 −σ4v1; v > 4 (20) where, v1 = [( 1 − 2 n−p − 6 v − 4 θ ) µ + 6 (v − 4)(n−p)( µx′(in ∗m)x(x′x)−1 + (x′x)−1x′(in ∗m)xµ )] (21) θ = trm(in ∗m) (n−p)2 (22) 6 some improved mr estimators & their comparison when disturbance terms follow multivariate t-distribution risk(bsmr) = σ 2 ( v v − 2 ) tr(x′x)−1l−σ4trv1l (23) proof 3.1: to employ small disturbances asymptotic approximations. let us write model (1) as y = xβ + σω (u = σω) (24) so that the i.i.d. elements of ω have multivariate-t distribution with mean zero for v > 1, variance ( v v−2 ) , for v > 2, measure of skewness γ1 = 0 and measure of kurtosis γ2 = ( 6 v−4 ) for v > 4. now, using (24) in (5), we find b0 = β + σ(x ′x)−1x′ω (25) so that y −xb0 = σmω (26) where m = [in −x(x′x)−1x′] (27) using (25), we find up to order o(σ) of approximations. 1 b′o(x ′x)b0 = c−1[1 − 2σc−1β′d′ω] (28) now, using (25), (26), and (28) in (8), we get up to order o(σ2) of approximations. bs −β = σ(x′x)−1x′ω −σ2kω′mωc−1β (29) and for the same order of approximation, we have y −xbs = σmω −σ2kω′mωc−1xβ (30) thus, using (30) and (3) in (9), we get bsmr −β = σh1 + σ2h2 + σ3h3 + σ4h4 (31) here, h1 = (x ′x)−1x′ω (32) h2 = ( ω′mω n−p ) (x′x)−1r′ω−1v (33) 7 manoj kumar, vikas bist and man inder kumar h3 = ( ω′mω t −g ) µx′ω (34) h4 = ( ω′mω n−p )2 [k2(n−p)c−1(x′x)−1r′ω−1v −µr′ω−1v ] (35) it is easy to see that e(h1) = e(h2) = e(h3) = e(h4) = 0 (36) utilizing (36) in (31), we obtain the result (19) of the theorem 1. now using (31), we get (bsmr −β)(bsmr −β)′ = σ2h1h′1 + σ 3(h1h ′ 2 + h2h ′ 1) + σ4(h1h ′ 3 + h2h ′ 2 + h3h ′ 1) (37) here, e(h1h ′ 1) = (x ′x)−1 (38) e(h1h ′ 2) = e(h2h ′ 1) = 0 (39) e(h1h ′ 3) = 1 n−p [ 6 v − 4 (x′x)−1x′(in ∗m)xµ + (n−p)µ ] (40) e(h2h ′ 2) = [( 6 v − 4 ) θ + ( n−p + 2 n−p )] µ (41) utilizing (38), (39), (40) and (41) in (37), we obtain the result (20) of the theorem 1. risk(bsmr) = trm(bsmr)l (42) thus, result (23) of the theorem 1 follows from (42). theorem 3.2. the asymptotic expression for bias vector, mean squared error matrix and risk function of msr estimator, up to order o(σ4) of approximations are given as b(bmsr) = −σ2 kv(n−p) v − 2 c−1β + σ4 [ 6k v − 4 c−2( (trm4(in ∗m))i + 2(x′x)−1x′(in ∗m)x −cθµ(x′x) ) β + kc−2 ( (n−p)(p− 2)i − n−p + 2 n−p cµ(x′x) ) β ] (43) 8 some improved mr estimators & their comparison when disturbance terms follow multivariate t-distribution where * denotes hadamard product. m(bmsr) = σ 2 ( v v − 2 ) (x′x)−1 −σ4 [ v1 + 12k v − 4 c−1[ (x′x)−1x′(in ∗m)x(x′x)−1 −c−1 ( (x′x)−1x′(in ∗m)xββ′ + ββ′x′(in ∗m)x(x′x)−1 + ( k 2 ) (trm(in ∗m))ββ′ )] + 2k(n−p)n(2+ k 2 (n−p+2)) ] (44) risk(bmsr) = σ 2 ( v v − 2 ) tr(x′x)−1l−σ4 [ trv1l + 12 k v − 4 c−1 ( tr(x′x)−1x′(in ∗m)x(x′x)−1l −c−1 ( 2β′x′(in ∗m)x(x′x)−1lβ + k 2 (trm(in ∗m))β′lβ )) + 2k(n−p)trn(2+ k 2 (n−p+2))l ] (45) proof 3.2: using (3), (24) and (26) in (6), we obtain up to order o(σ2) of approximations. bmr = β + σ(x ′x)−1x′ω + σ2 ( ω′mω n−p ) (x′x)−1r′ω−1v (46) thus, for the same order of approximation, we have 1 b′mr(x ′x)bmr = c−1 [ 1 − 2σc−1β′x′ω −σ2c−1 ( 2 n−p ω′mωv ′ω−1rβ + ω′mdω )] (47) using (46), we get up to order o(σ2)of approximations. y −xbmr = σmω −σ2 ( ω′mω n−p ) x(x′x)−1r′ω−1v (48) using (46), (47) and (48) in (10), we obtain up to order o(σ4), we get bmsr −β = σh∗1 + σ 2h∗2 + σ 3h∗3 + σ 4h∗4 (49) where h∗1 = (x ′x)−1x′ω (50) 9 manoj kumar, vikas bist and man inder kumar h∗2 = ( ω′mω n−p )[ (x′x)−1r′ω−1v −kc−1β ] (51) h∗3 = − ( ω′mω n−p )[ µ + k(n−p)c−1n2) ] x′ω (52) h∗4 = k(ω ′mω)c−2 ( 2 n−p ω′mωββ′r′ω−1v + ω′m4ωβ + 2(x′x)−1x′ωω′xβ ) − ( ω′mω n−p )2 [ µr′ω−1v + kc−1(β′v ′ω−1r + (n−p)i)(x′x)−1r′ω−1v ] (53) here, it is easy to verify that e(h∗1) = 0 (54) e(h∗2) = −k(n−p)c −1β (55) e(h∗3) = 0 (56) e(h∗4) = 6k v − 4 c−2 [ (trm4(in ∗m))i + 2(x′x)−1x′(in ∗m)x −cθµ(x′x) ] β + kc−2 [ (n−p)(p− 2)i − ( n−p + 2 n−p ) cµ(x′x) ] β (57) utilizing (54), (55), (56) and (57) in (53), we obtain the result (43) of the theorem 2. now, using (53) we get (bmsr −β)(bmsr −β)′ = σ2h∗1h ∗′ 1 + σ 3(h∗1h ∗′ 2 + h ∗ 2h ∗′ 1 ) + σ4(h∗1h ∗′ 1 + h ∗ 2h ∗′ 2 + h ∗ 3h ∗′ 1 ) (58) here, we see that e(h∗1h ∗′ 1 ) = ( v v − 2 ) (x′x)−1 (59) e(h∗1h ∗′ 2 ) = 0 (60) e(h∗1h ∗′ 3 ) = 6 (v − 4)(n−p) [ (x′x)−1x′(in ∗m)xµ + k(n−p)c−1(x′x)−1x′(in ∗m)xn2 ] −µ−k(n−p)c−1n2 (61) 10 some improved mr estimators & their comparison when disturbance terms follow multivariate t-distribution e(h∗2h ∗′ 2 ) = µ [ 6 v − 4 θ + ( n−p + 2 n−p ) i ] + k2c−2ββ′ [ 6 v − 4 trm(in ∗m) + (n−p)(n−p + 2) ] (62) utilizing (59), (60), (61) and (62) in (58), we obtain the result (44) of the theorem 2. similarly, we can obtain the result (45) of the theorem 2. 4 comparison of the estimators 4.1 the comparison the risk functions of ols and smr estimators on comparison the risk functions of ols and smr estimators. we observe that up to order o(σ2) of approximations, both the estimators have same risk and for higher order of approximation, we see that risk(b0) −risk(bsmr) = σ4 [ 6 v − 2 ( 2 n−p tr(x′x)−1x′(in ∗m)xµl−θtrµl ) + (n−p− 2 n−p ) trµl ] (63) if we choose l = (x′x), then expression (63) becomes risk(b0) −risk(bsmr) = σ4 [ 6 v − 2 ( 2 n−p tr(x′x)−1x′(in ∗m)x(x′x)−1r′ω−1r −θtr(x′x)−1r′ω−1r ) + ( n−p− 2 n−p ) tr(x′x)−1r′ω−1r ] (64) since, the expression (64) is positive semi-definite, so bsmr dominates b0 and as v →∞, expression (64) reduces to risk(b0) −risk(bsmr) = σ4 ( n−p− 2 n−p ) tr(x′x)−1r′ω−1r (65) which is positive semi-definite. thus, bsmr dominates b0, so long as n−p > 2. 11 manoj kumar, vikas bist and man inder kumar 4.2 the comparison the risk functions of ols and msr estimators on comparison the risks of ols and msr, we see that up to order o(σ2) of approximations, both the estimators have same risk and for higher order of approximations, we find that bmsr dominates bo so long as (65) is positive semidefinite and if we choose k to satisfy, 0 < k < 2(n−p) t c β′aβ [ tr(x′x)−1l− 2c−1β′lβ + 6 (n−p)(v − 4)( tr(x′x)−1x′(in ∗m)x(x′x)−1l− 2c−1β′x′(in ∗m)x(x′x)−1lβ )] (66) where t = [ 6 v − 4 (trm(in ∗m)) + (n−p)(n−p + 2) ] (67) if we choose l = (x′x), then the above condition of dominance becomes 0 < k < 2(n−p) t [ p− 2 + 6 (n−p)(v − 4) ( tr(x′x)−1x′(in ∗m)x − 2c−1β′x′(in ∗m)xβ )] (68) and as v →∞, condition (68) reduces to 0 < k < 2 n−p + 2 (p− 2); p > 2 (69) which is well known condition of dominance of stein-rule estimator over the least squares estimator. 4.3 the comparison the risk functions of smr and msr estimators on comparing the risk function associated with the estimators smr and msr respectively, we observe that the estimator msr dominates the estimator smr so long as (30 )holds and as v → ∞ and again by choosing l = (x′x), the condition of dominance becomes (69). 12 some improved mr estimators & their comparison when disturbance terms follow multivariate t-distribution 5 simulation results the proposed estimator bsmr is more efficient than olse under given linear model. although, theoretically the results are drawn in equation (65), the proposed stein-mixed regression (smr) estimator bsmr is more efficient than ordinary least square estimator b0 under condition n−p > 2. in this section, we perform simulations for exact equation (65) under conditions n−p > 2,n > p > j with sigma equal to one. each result is based on 100,000 simulations runs using matlab. the result shown for n = 10, 11, 12, 13, 14, 15 in table 1, 2, 3 & 4. the main finding of our numerical evaluation is following:1. the simulation results strongly support the theoretical findings. 2. the simulation result also explains the strength keep on increasing as we go for large value of n,p and j. 3. the results are independent of value of sigma. 4. hence, bsmr is more efficient than b0 under condition n−p > 2. 5. the simulation results also reveals that bmsr is also more efficient over b0 (as it also depends on (65) under condition at (69)). figure 1: average dominance condition for difference between n & p based on simulation study, the dominance of bsmr has been proven over b0 under certain set of conditions. further, the behavior of dominance is studied for various combination of different values of n,p and j. the average dominance is derived based on probability for different combination of n,p and j; when σ = 1. the figure 1 depicts average dominance keeps on decreasing with increase in gap 13 manoj kumar, vikas bist and man inder kumar figure 2: dominance behavior for different values of p; when n=10 & j=5 figure 3: average dominance condition for given value of p & j for n=20 between n and p. the figure 2 also depicts a decreasing trend with increase in value of p, when n = 10 and j = 5. similarly, figure 3 shows the behavior of dominance condition for different value of p and j for fixed value of n equal to 20. 14 some improved mr estimators & their comparison when disturbance terms follow multivariate t-distribution table 1: average value of dominance for different values of n and p for j = 2 sigma = 1. j=2 n=10 n=11 n=12 n=13 n=14 n=15 p=3 0.67823 0.67492 0.67793 0.67688 0.67390 0.67539 p=4 0.67300 0.67079 0.66877 0.66903 0.66816 0.66654 p=5 0.67187 0.66562 0.66558 0.66668 0.66416 0.66691 p=6 0.66660 0.66642 0.66755 0.66509 0.66370 0.66186 p=7 0.66755 0.66497 0.66408 0.65994 0.66305 0.65982 p=8 0.66816 0.66611 0.66010 0.66261 0.66156 p=9 0.66861 0.66352 0.66143 0.66036 remark: no value for dominance where n−p 6 2. 15 manoj kumar, vikas bist and man inder kumar table 2: average value of dominance for different values of n and p for j = 3 sigma = 1. j=3 n=10 n=11 n=12 n=13 n=14 n=15 p= 4 0.70809 0.70806 0.70722 0.70725 0.70806 0.70611 p= 5 0.70570 0.70449 0.70581 0.70385 0.70381 0.70162 p= 6 0.70612 0.70319 0.70252 0.69998 0.70281 0.70205 p= 7 0.70651 0.70266 0.70281 0.70253 0.69810 0.69816 p= 8 0.70735 0.70167 0.70011 0.70183 0.69960 p= 9 0.70592 0.70486 0.70001 0.70028 p=10 0.70784 0.70426 0.70033 p=11 0.70692 0.70500 remark: no value for dominance where n−p 6 2. 16 some improved mr estimators & their comparison when disturbance terms follow multivariate t-distribution table 3: average value of dominance for different values of n and p for j = 4 sigma = 1. j=4 n=10 n=11 n=12 n=13 n=14 n=15 p= 5 0.74738 0.74891 0.74713 0.74538 0.74603 0.74668 p= 6 0.74694 0.74586 0.74771 0.74337 0.74335 0.74346 p= 7 0.75060 0.74841 0.74539 0.74412 0.74366 0.74381 p= 8 0.74740 0.74561 0.74405 0.74213 0.74607 p= 9 0.74752 0.74623 0.74547 0.74207 p=10 0.74601 0.74774 0.74091 p=11 0.74832 0.74426 p=12 0.74612 remark: no value for dominance where n−p 6 2. 17 manoj kumar, vikas bist and man inder kumar table 4: average value of dominance for different values of n and p for j = 5 sigma = 1. j=5 n=20 n=25 n=30 n=35 n=40 n=45 n=50 p = 5 0.78748 0.78793 0.78713 0.78373 0.78562 0.78336 0.78686 p=10 0.78359 0.78228 0.78284 0.78158 0.77898 0.77644 0.78005 p=15 0.78586 0.78143 0.77689 0.78072 0.77643 0.77454 0.77655 p=20 0.78350 0.78017 0.77976 0.77657 0.77576 0.77628 p=25 0.78453 0.78062 0.77884 0.77852 0.77563 p=30 0.78408 0.78044 0.77743 0.77703 p=35 0.78478 0.77772 0.77486 p=40 0.78208 0.77728 remark: no value for dominance where n−p 6 2. references [1] chaturvedi, a. and shukla, g., stein-rule estimation in linear models with non-scalar error covariance matrix, sankhya, series b, 52, (1990), 293-304. [2] chaturvedi, a. ,wan, a. t. k. and singh, s. p., stein-rule restricted regres18 some improved mr estimators & their comparison when disturbance terms follow multivariate t-distribution sion estimator in a linear regression model with non-spherical disturbances, communications in statistics, theory and methods, 30, (2001), 55-68. [3] giles, a.j., pretesting for linear restriction in a regression model with spherically symmetric distributions, journal of econometrics, 50, (1991), 377398. [4] judge, g. g. and bock, m. e., the statistical implications of pre-test and stein-rule estimators in econometrics, north holland, amsterdam, 1978. [5] kadane, j.b.,comparison of k-class estimators when disturbance are small, econometrica, 39, (1971), 723-737. [6] ohtani, k. and wan, a. t. k., on the sampling performance of an improved stein inequality restricted estimator, australian and new zealand journal of statistics, 40, (1998), 181-187. [7] rao, c. r., linear statistical inference and its applications, 2nd edition. john wiley,new york, 1973 . [8] shalabh and wan, a. t. k., stein-rule estimation in mixed regression models, biometrical journal, 42, (2000) 203-214. [9] sutradhar, b.c. and ali, m.m., estimation of parameters of regression with a multivariate t-error variable, communication statistics theory and methods, a 15, (1986), 429-450. [10] sutradhar, b.c., testing linear hypothesis with t error variable, sankhya: the indian journal of statistics, series b (1960-2002), 50(2), (1988), 175180. [11] theil, h., principles of econometrics, vol. 1. wiley, new york, 1971. [12] zellner, a., bayesian and non-bayesian analysis of regression model with multivariate t-error terms, journal of the american statistical association, 71, (1976), 400-405. 19 ratio mathematica volume 45, 2023 some new �̆�-𝓘-locally closed sets with respect to an ideal topological spaces m. vijayasankari1 g. ramkumar2 abstract in this paper, we introduce the new notions called �̆�-𝓘-locally closed sets, �̆�-𝓘-locally closed sets and �̆�-𝓘-closed functions and investigated their properties and also we have studied their relations to the other types of locally closed sets with suitable examples. finally we introduce the notion �̆�-𝓘-submaximal spaces and also investigated the properties with examples. keywords: �̆�-ℐ-cld, �̆�-ℐ-lc ,�̆�-ℐlc , �̆�-ℐlc . 2010 mathematics subject classification: 54a053 1 research scholar department of mathematics, madurai kamaraj university, madurai. tamil nadu, india. e-mail: vijayasankariumarani1985@gmail.com. 2 assistant professor, department of mathematics, arul anandar college, karumathur, madurai, tamil nadu, india. e-mail: ramg.phd@gmail.com. 3received on july 10, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.1017. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 223 mailto:vijayasankariumarani1985@gmail.com mailto:ramg.phd@gmail.com m. vijayasankari &g. ramkumar 1. introduction in 1970, levine [11] introduced the generalized closed sets and after that many authors introduced and studied many types of generalized closed sets in topological spaces. in the continuity, locally closedness was done by bourbaki [4]. he defined a set a to be locally closed if it is the intersection of an open set and a closed set. in literature many general topologists introduced the studies of locally closed sets. extensive research on locally closedness and generalizing locally closedness were done in recent years. stone [16] used the term fg for a locally closed set. ganster and reilly used locally closed sets in [6] to define lc-continuity and lc-irresoluteness. balachandran et al [3] introduced the concept of generalized locally closed sets. veera kumar [18] (sheik john [15]) introduced ĝ-locally closed sets ( -locally closed sets) respectively. in this paper, we introduce the new notions called �̆�-𝓘-locally closed sets, �̆�-𝓘-locally closed sets and �̆�-𝓘-closed functions and investigated their properties and also we have studied their relations to the other types of locally closed sets with suitable examples. finally we introduce the notion �̆�-𝓘-submaximal spaces and also investigated the properties with examples. 2. preliminaries an ideal i on a topological space (briefly, tps) (x, τ) is a nonempty collection of subsets of x which satisfies (1) a∈i and b⊆a⇒b∈i and (2) a∈i and b∈i⇒a∪b∈i. given a topological space (x, τ) with an ideal i on x if ℘(x) is the set of all subsets of x, a set operator ( •)⋆: ℘(x)→ ℘(x), called a local function [10] of a with respect to τ and i is defined as follows: for a ⊆x, a⋆(i, τ)={ x ∈ x : u∩a i for every u ∈ τ(x)} where τ(x)={u ∈ τ : x ∈ u}. a kuratowski closure operator cl⋆( •) for a topology τ⋆(i, τ ), called the ⋆-topology and finer than τ, is defined by cl⋆(a) = a ∪a⋆(i, τ) [10]. we will simply write a⋆ for a⋆(i, τ) and τ⋆ for τ⋆(i, τ ). if i is an ideal on x, then (x, τ, i) is called an ideal topological space(briefly, itps). a subset a of an ideal topological space (x, τ, i) is ⋆-closed (briefly, ⋆-cld) [10] if a⋆⊆a. the interior of a subset a in (x, τ⋆( i)) is denoted by int⋆(a). definition 2.1 a subset k of a tps x is called: (i) semi-open set [9] if k cl(int(k)); (ii) 𝛼-open set [9] if k  int(cl(int(k))); (iii) regular open set [12] if k = int(cl(k)); the complements of the above mentioned open sets are called their respective closed sets. definition 2.2 a subset k of a tps x is called (i) 𝑔 -closed set (briefly, 𝑔 -cld) [11] if cl(k)  v whenever k  v and v is open. 224 some new �̆�-ℐ-locally closed sets with respect to an ideal topological spaces (ii) semi-generalized closed (briefly, sg-cld)[7] if scl(k) v whenever kv and v is semi-open. (iii) generalized semi-closed (briefly, gs-cld)[18] if scl(k) v whenever kv and v is open. (iv) ĝ-closed set [18] ( -cld set [18]) if cl(k)  v whenever k  v and v is semiopen in x. the complements of the above mentioned closed sets are called their respective open sets. definition 2.3 a subset k of a itps x is called (i) ig-closed (briefly, ig-cld) set [9] if k*  v whenever k  v and v is open. the complements of the above mentioned closed sets are called their respective open sets. definition 2.4 a subset k of a space x is called a regular generalized closed (briefly, rg-closed) set [12] if cl(k) v whenever kv and v is regular open in x. the complement of rg-closed set is called rg-open set; remark 2.5 the collection of all rg-closed sets in x is denoted by rg c(x). the collection of all rg-open sets in x is denoted by rg o(x). definition 2.6 a subset k of a space x is called i. generalized locally closed (briefly, glc) [2] if a = v f, where v is g-open and f is g-closed in x. ii. semi-generalized locally closed (briefly, sglc) [13] if k = v f, where v is sg-open and f is sg-closed in x. iii. regular-generalized locally closed (briefly, rg-lc) [1] if k = v f, where v is rg-open and f is rg-closed in x. iv. generalized locally semi-closed (briefly, glsc) [8] if k = v f, where v is g-open and f is semi-closed in x. v. locally semi-closed (briefly, lsc) [8] if k = v f, where v is open and f is semiclosed in x. vi. -locally closed (briefly,  -lc) [8] if k = v f, where v is  -open and f is  closed in x. vii. -locally closed (briefly,  -lc) [15] if k = v f, where v is  -open and f is  closed in x. the class of all generalized locally closed (resp. generalized locally semi-closed, locally semi-closed,  -locally closed) sets in x is denoted by glc (x) (resp. glsc (x), lsc (x),  -lc(x)). definition 2.7 a topological space x is called i.submaximal [5, 18] if every dense subset is open. 225 m. vijayasankari &g. ramkumar ii.ĝ (or )-submaximal [15, 18] if every dense subset is  -open. iii.g-submaximal [2] if every dense subset is g-open. iv.rg-submaximal [12] if every dense subset is rg-open. theorem 2.8 let x be a topological space i.if x is submaximal, then x is ĝ-submaximal. ii.if x is ĝ-submaximal, then x is g-submaximal. iii.if x is g-submaximal, then x is rg-submaximal. 3. �̆�-𝓘-locally closed sets we introduce the following definition. definition 3.1 a subset k of x is called (i) �̆�-ℐ-closed (briefly, �̆�-ℐ-cld) if k*  v whenever k  v and v is sg-open. the complement of �̆�-ℐ-cld is called �̆�-ℐ-open. the family of all �̆�-ℐ-cld in x is denoted by �̆�-ℐc(x). (ii) �̆�-ℐ-locally closed (briefly, �̆�-ℐ-lc) if k = s  g, where s is �̆�-ℐ-open and g is �̆�-ℐ-cld. (iii) a function f : (x, , ℐ) → (y, ) is called �̆�-ℐ-continuous if the inverse image of every closed set in y is �̆�-ℐ-cld set in x. the class of all �̆�-ℐ-locally closed sets in x is denoted by �̆�-ℐ-lc(x). proposition 3.2 every �̆�-𝓘-cld (resp. �̆�-𝓘-open) set is �̆�-𝓘-lc set but not conversely. proof it follows from definition 3.1 (i) and (ii). example 3.3 let x = {p, q, r} and  = {, {q}, x} with 𝓘 = {}. then the set {q} is �̆�𝓘-lc set but it is not �̆�-𝓘-closed and the set {p, r} is �̆�-𝓘-lc set but it is not �̆�-𝓘-open in x. proposition 3.4 every lc set is �̆�-𝓘-lc set but not conversely. proof it follows from proposition 3.2. example 3.5 let x = {p, q, r} and  = {, {q, r}, x} with 𝓘 = {}. then the set {q} is �̆�-𝓘-lc set but it is not lc set in x. proposition 3.6 every �̆�-𝓘-lc set is a (i)  -lc set, (ii) glc set and (iii) sglc set. however the separate converses are not true. proof 226 some new �̆�-ℐ-locally closed sets with respect to an ideal topological spaces it is obviously. example 3.7 let x = {p, q, r} and  = {, {p}, x} with 𝓘 = {}. then the set {b} is glc set but it is not �̆�-𝓘-lc set in x. moreover, the set {r} is sglc set but it is not �̆�-𝓘-lc set in x. example 3.8 let x = {p, q, r} and  = {, {q}, {p, r}, x} with 𝓘 = {}. then the set {p} is  -lc set but it is not �̆�-𝓘-lc set in x. remark 3.9 the concepts of  -lc sets and �̆�-𝓘-lc sets are independent of each other. example 3.10 the set {q, r} in example 3.3 is  -lc set but it is not a �̆�-𝓘-lc set in x and the set {p, q} in example 3.5 is �̆�-𝓘-lc set but it is not an  -lc set in x. remark 3.11 the concepts of lsc sets and �̆�-𝓘-lc sets are independent of each other. example 3.12 the set {p} in example 3.3 is lsc set but it is not a �̆�-𝓘-lc set in x and the set {p, q} in example 3.5 is �̆�-𝓘-lc set but it is not a lsc set in x. remark 3.13 the concepts of �̆�-𝓘-lc sets and glsc sets are independent of each other. example 3.14 the set {q, r} in example 3.3 is glsc set but it is not a �̆�-𝓘-lc set in x and the set {p, q} in example 3.5 is �̆�-𝓘-lc set but it is not a glsc set in x. remark 3.15 the concepts of �̆�-𝓘-lc sets and  sglc sets are independent of each other. example 3.16 the set {q, r} in example 3.3 is  sglc set but it is not a �̆�-𝓘-lc set in x and the set {p, q} in example 3.5 is �̆�-𝓘-lc set but it is not a  sglc set in x. theorem 3.17 for a t�̆�-𝓘-space x, the following properties hold (i) �̆�-ℐ-lc(x) = lc (x). (ii) �̆�-ℐ-lc(x) glc (x). (iii) �̆�-ℐ-lc(x) glsc (x). (iv) �̆�-ℐ-lc(x)  lc (x). proof (i) since every �̆�-ℐ-open set is open and every �̆�-ℐ-closed set is closed in (x, ), �̆�ℐ-lc(x)  lc (x) and hence �̆�-ℐ-lc(x) = lc (x). (ii), (iii) and (iv) follows from (i), since for any space x, lc (x) glc (x), lc (x)  glsc (x) and lc (x)  lc (x). corollary 3.18 if g o(x) = , then �̆�-𝓘-lc(x) glsc (x)  lsc (x). 227 m. vijayasankari &g. ramkumar proof g o(x) =  implies that x is a t�̆�-ℐ-space and hence by theorem 3.17, �̆�-ℐ-lc(x)  glsc (x). let kglsc (x). then k = v f, where v is g-open and f is semi-closed. by hypothesis, v is open and hence k is a lsc-set and so k lsc (x). definition 3.19 a subset k of a space x is called: (i) �̆�-ℐlc set if k= s  g, where s is �̆�-ℐ-open in x and g is closed in x. (ii) �̆�-ℐlc set if k= s  g, where s is open in x and g is �̆�-ℐ-closed in x. the class of all �̆�-ℐlc (resp. �̆�-ℐlc ) sets in a ideal topological space x is denoted by �̆�-ℐ-lc*(x) (resp. �̆�-ℐ-lc**(x)). proposition 3.20 every lc-set is �̆�-𝓘lc set but not conversely. proof it follows from definition 3.19 (i) and definition of locally closed set. example 3.21 the set {q} in example 3.5 is �̆�-𝓘lc set but it is not a lc set in x. proposition 3.22 every lc-set is �̆�-𝓘lc set but not conversely. proof it follows from definition 3.19 (ii) and definition of locally closed set. example 3.23 the set {p, r} in example 3.5 is �̆�-𝓘lc set but it is not a lc set in x. proposition 3.24 every �̆�-𝓘lc set is �̆�-𝓘-lc set but not conversely. proof it follows from definitions 3.1 and 3.19 (i). example 3.25 the set {p, q} in example 3.5 is �̆�-𝓘-lc set but it is not a �̆�-𝓘lc set in x. proposition 3.26 every �̆�-𝓘lc set is �̆�-𝓘-lc set but not conversely. proof it follows from definitions 3.1 and 3.19 (ii). remark 3.27 the concepts of �̆�-𝓘lc sets and lsc sets are independent of each other. example 3.28 the set {r} in example 3.5 is �̆�-𝓘lc set but it is not a lsc set in x and the set {p} in example 3.3 is lsc set but it is not a �̆�-𝓘lc set in x. remark 3.29 the concepts of �̆�-𝓘lc sets and  -lc sets are independent of each other. 228 some new �̆�-ℐ-locally closed sets with respect to an ideal topological spaces example 3.30 the set {p, q} in example 3.5 is �̆�-𝓘lc set but it is not a  -lc set in x and the set {p, q} in example 3.3 is  -lc set but it is not a �̆�-𝓘lc set in x. remark 3.31 from the above discussions we have the following implications where a → b (resp. a ≠→ b) represents a implies b but not conversely (resp. a and b are independent of each other). figure 1: relations between some of generalized closed sets proposition 3.32 if g o(x) = , then �̆�-𝓘-lc(x) = �̆�-𝓘-lc*(x) = �̆�-𝓘-lc**(x). proof for any space (x, ), �̆�-ℐ-o(x)g o(x). therefore by hypothesis, �̆�-ℐ-o(x) = . i.e., (x, ) is a t�̆�-ℐ-space and hence �̆�-ℐ-lc(x) = �̆�-ℐ-lc*(x) = �̆�-ℐ-lc**(x). remark 3.33 the converse of proposition 3.32 need not be true. for the ideal topological space x in example 3.3. �̆�-ℐ-lc(x) = �̆�-ℐ-lc*(x) = �̆�-ℐlc**(x). however, g o(x) = {, {p}, {q}, {r}, {p, q}, {q, r}, x} . proposition 3.34 let x be an ideal topological space. if g o(x)  lc (x), then �̆�-𝓘lc(x) = �̆�-𝓘-lc**(x). proof let k�̆�-ℐ-lc(x). then k = s  g where s is �̆�-ℐ-open and g is 𝜃-ℐ-closed. since �̆�ℐ-o(x)g o(x) and by hypothesis g o(x)  lc (x), s is locally closed. then s = p  q, where p is open and q is *-closed. therefore, k = p  (q  g). we have, q  g is �̆�-ℐ-closed and hence k�̆�-ℐ-lc**(x). i.e., �̆�-ℐ-lc(x) �̆�-ℐ-lc**(x). for any ideal topological space, 𝜃-ℐ-lc**(x) �̆�-ℐ-lc(x) and so �̆�-ℐ-lc(x) = �̆�-ℐ-lc**(x). remark 3.35 the converse of proposition 3.34 need not be true in general. 229 m. vijayasankari &g. ramkumar for the ideal topological space x in example 3.3, then �̆�-ℐ-lc(x) = �̆�-ℐlc**(x) = {, {q}, {p, r}, x}. but g o(x) = {, {p}, {q}, {r}, {p, q}, {q, r}, x}  lc (x) = {, {q}, {p, r}, x}. corollary 3.36 let x be an ideal topological space. if  o(x)  lc (x), then �̆�-𝓘lc(x) = �̆�-𝓘-lc**(x). proof it follows from the fact that  o(x) g o(x) and proposition 3.34. remark 3.37 the converse of corollary 3.36 need not be true in general. for the ideal topological space x in example 3.8, then �̆�-ℐ-lc(x) = �̆�-ℐlc**(x) = {, {q}, {p, r}, x}. but  o(x) = p(x)  lc (x) = {, {q}, {p, r}, x}. the following results are characterizations of �̆�-ℐ-lc sets, �̆�-ℐlc sets and �̆�-ℐ lc sets. theorem 3.38 assume that �̆�-𝓘-c(x) is closed under finite intersection. for a subset k of x, the following statements are equivalent: (i) k�̆�-ℐ-lc(x). (ii) k = s �̆�-ℐ-cl(k) for some �̆�-ℐ-open set s. (iii) �̆�-ℐ-cl(k) −k is �̆�-ℐ-closed. (iv) k (�̆�-ℐ-cl(k))c is �̆�-ℐ-open. (v) k�̆�-ℐ-int( k (�̆�-ℐ-cl(k))c). proof (i)  (ii). let k�̆�-ℐ-lc(x). then k = s  g where s is �̆�-ℐ-open and g is �̆�-ℐclosed. since k g, �̆�-ℐ-cl(k)  g and so s�̆�-ℐ-cl(k) k. also k s and k�̆�-ℐcl(k) implies k s �̆�-ℐ-cl(k) and therefore k = s�̆�-ℐ-cl(k). (ii)  (iii). k = s�̆�-ℐ-cl(k) implies �̆�-ℐ-cl(k)−k = �̆�-ℐ-cl(k)  sc which is �̆�-ℐ-closed since sc is �̆�-ℐ-closed and �̆�-ℐ-cl(k) is �̆�-ℐ-closed. (iii)  (iv). k (�̆�-ℐ-cl(k))c = (�̆�-ℐ-cl(k)−k)c and by assumption, (�̆�-ℐ-cl(k) −k)c is �̆�ℐ-open and so is k (�̆�-ℐ-cl(k))c. (iv)  (v). by assumption, k (�̆�-ℐ-cl(k))c = �̆�-ℐ-int(k(�̆�-ℐ-cl(k))c) and hence k�̆�ℐ-int(k (�̆�-ℐ-cl(k))c). (v) (i). by assumption and since k�̆�-ℐ-cl(k), k=�̆�-ℐ-int(k(�̆�-ℐ-cl(k))c) �̆�-ℐcl(k). therefore, k�̆�-ℐ-lc(x). theorem 3.39 for a subset k of x, the following statements are equivalent: i.k�̆�-ℐ-lc*(x). ii.k = s k* for some �̆�-ℐ-open set s. iii.k*−k is �̆�-ℐ-closed. iv.k (k*)c is �̆�-ℐ-open. proof 230 some new �̆�-ℐ-locally closed sets with respect to an ideal topological spaces (i)  (ii). let k𝜃-ℐ-lc*(x). there exist an �̆�-ℐ-open set s and a *-closed set g such that k = s  g. since k s and kk*, k s k*. also, since k* g, sk* s  g = k. therefore k = s k*. (ii)  (i). since s is �̆�-ℐ-open and k* is a *-closed set, k = s k*�̆�-ℐ-lc*(x). (ii)  (iii). since k*−k =k* sc, k*−k is �̆�-ℐ-closed. (iii)  (ii). let s = (k*−k)c. then by assumption s is �̆�-ℐ-open in x and k = s k*. (iii)  (iv). let g = k*−k. then gc = k (k*)c and k (k*)c is �̆�-ℐ-open. (iv)  (iii). let s =k (k*)c . then sc is �̆�-ℐ-closed and sc =k*−k and so k*−k is �̆�-ℐ-closed. theorem 3.40 let k be a subset of x. then k�̆�-𝓘-lc**(x) if and only if k = s �̆�𝓘-cl(k) for some open set s. proof let k�̆�-𝓘-lc**(x). then k = s  g where s is open and g is �̆�-𝓘-closed. since k g, �̆�-𝓘-cl(k)  g. we obtain k = k�̆�-𝓘-cl(k) = s  g �̆�-𝓘-cl(k) = s �̆�-𝓘-cl(k). converse part is trivial. corollary 3.41 let k be a subset of x. if k�̆�-𝓘-lc**(x), then �̆�-𝓘-cl(k)−k is �̆�-𝓘closed and k (�̆�-𝓘-cl(k))c is �̆�-𝓘-open. proof let k�̆�-ℐ-lc**(x). then by theorem 3.40, k = s�̆�-ℐ-cl(k) for some open set s and �̆�-ℐ-cl(k)−k = �̆�-ℐ-cl(k)  sc is �̆�-ℐ-closed in x. if g = �̆�-ℐ-cl(k) −k, then gc = k (�̆�-ℐ-cl(k))cand gc is �̆�-ℐ-open and so is k (�̆�-ℐ-cl(k))c. 4. �̆�-𝓘-submaximal spaces definition 4.1 i.a subset k of a space x is called i-dense if k* = x. ii.a subset k of a space x is called ig-submaximal if every i-dense subset is ig-open. proposition 4.2 every �̆�-𝓘-dense set is i-dense. proof let k be an �̆�-𝓘-dense set in x. then �̆�-𝓘-cl(k) = x. since �̆�-𝓘-cl(k)  cl(k), we have k* = x and so k is i-dense. the converse of proposition 4.2 need not be true as can be seen from the following example. example 4.3 the set {p, r} in example 3.5 is a i-dense in x but it is not �̆�-𝓘-dense in x. we introduce the following definition. definition 4.4 an ideal topological space x is called �̆�-𝓘-submaximal if every i-dense subset in it is �̆�-𝓘-open in x. 231 m. vijayasankari &g. ramkumar proposition 4.5 every submximal space is �̆�-𝓘-submaximal. proof let x be a submximal space and k be a i-dense subset of x. then k is open. but every open set is �̆�-𝓘-open and so k is �̆�-𝓘-open. therefore, x is �̆�-𝓘-submaximal. the converse of proposition 4.5 need not be true as can be seen from the following example. example 4.6 for the ideal topological space x of example 3.5, every i-dense subset is �̆�-𝓘-open and hence x is �̆�-𝓘-submaximal. however, the set k = {p, q} is i-dense in x, but it is not open in x. therefore, x is not submaximal. proposition 4.7 every �̆�-𝓘-submaximal space is  -submaximal. proof let x be an �̆�-𝓘-submaximal space and k be a i-dense subset of x. then k is �̆�𝓘-open. but every �̆�-𝓘-open set is  -open and so k is  -open. therefore, x is  submaximal. the converse of proposition 4.7 need not be true as can be seen from the following example. example 4.8 consider the ideal topological space x in example 3.8. then x is  submaximal but it is not �̆�-𝓘-submaximal, because the set k = {q, r} is a i-dense set in x but it is not �̆�-𝓘-open in x. remark 4.9 from propositions 4.5, 4.7, we have the following diagram: submaximal �̆�-ℐ-submaximal  -submaximal ig-submaximal rg-submaximal theorem 4.10 a space x is �̆�-𝓘-submaximal if and only if p(x) = �̆�-𝓘-lc*(x). proof necessity. let k p(x) and let v = k (k*)c. this imply that v* = k* (k*)c = x. hence v* = x. therefore, v is a dense subset of x. since x is �̆�-ℐ-submaximal, v is �̆�ℐ-open. thus k (k*)c is �̆�-ℐ-open and by theorem 3.39, we have k�̆�-ℐ-lc*(x). sufficiency. let k be a i-dense subset of x. this implies k (k*)c= kxc = k = k. now k�̆�-ℐ-lc*(x) implies that k = k (k*)c is �̆�-ℐ-open by theorem 3.39. hence x is �̆�-ℐ-submaximal. 232 some new �̆�-ℐ-locally closed sets with respect to an ideal topological spaces remark 4.11 union of two �̆�-𝓘-lc sets (resp. �̆�-𝓘lc sets, �̆�-𝓘lc sets) need not be an �̆�-𝓘-lc set (resp. �̆�-𝓘lc set, �̆�-𝓘lc set) as can be seen from the following examples. example 4.12 let x= {p, q, r} with  = {, {p}, {p, q}, x}. then �̆�-𝓘-lc(x) ={, {p}, {q}, {r}, {p, q}, {q, r}, x}. then the sets {p} and {r} are �̆�-𝓘-lc sets, but their union {p, r} �̆�-𝓘-lc(x). example 4.13 let x= {p, q, r} and  = {, {q}, {p, q}, x} with 𝓘 = {}. then �̆�-𝓘lc*(x) ={, {p}, {q}, {r}, {p, q}, {p, r}, x}. then the sets {q} and {r} are �̆�-𝓘lc sets, but their union {q, r} �̆�-𝓘-lc*(x). example 4.14 let x= {p, q, r} and  = {, {q}, {q, r}, x} with 𝓘 = {}. then �̆�-𝓘lc**(x) ={, {p}, {q}, {r}, {p, r}, {q, r}, x}. then the sets {p} and {q} are �̆�-𝓘lc sets, but their union {p, q} �̆�-𝓘-lc**(x). we introduce the following definition. definition 4.15 let k and b be subsets of x. then k and b are said to be �̆�-𝓘separated if k�̆�-𝓘-cl(b) =  and �̆�-𝓘-cl(k)  b = . example 4.16 for the ideal topological space x of example 3.5. let k = {q} and let b = {r}. then �̆�-𝓘-cl(k) = {p, q} and �̆�-𝓘-cl(b) = {p, r} and so the sets k and b are �̆�-𝓘separated. proposition 4.17 assume that �̆�-𝓘-o(x) forms an ideal topology. for the ideal topological space x, the followings are true i.let k, b �̆�-ℐ-lc(x). if k and b are �̆�-ℐ-separated then k b �̆�-ℐ-lc(x). ii.let k, b �̆�-ℐ-lc*(x). if a and b are separated (i.e., kb* =  and k* b = ), then k b �̆�-ℐ-lc*(x). iii.let k, b �̆�-ℐ-lc**(x). if k and b are �̆�-ℐ-separated then k b �̆�-ℐlc**(x). proof (i) since k, b �̆�-ℐ-lc(x), by theorem 3.38, there exist �̆�-ℐ-open sets u and v of x such that k = u�̆�-ℐ-cl(a) and b = v�̆�-ℐ-cl(b) . now g = u  (x −�̆�-ℐ-cl(b)) and h = v  (x −�̆�-ℐ-cl(k)) are �̆�-ℐ-open subsets of x. since k�̆�-ℐ-cl(b) = , k (�̆�-ℐcl(b))c. now k = u �̆�-ℐ-cl(k) becomes k (�̆�-ℐ-cl(b))c = g �̆�-ℐ-cl(k). then k = g �̆�-ℐ-cl(k). similarly b = h�̆�-ℐ-cl(b). moreover g�̆�-ℐ-cl(b) =  and h�̆�-ℐ-cl(k) = . since g and h are �̆�-ℐ-open sets of x, g  h is �̆�-ℐ-open. therefore k b = (g  h) �̆�-ℐ-cl(k b) and hence a  b �̆�-ℐ-lc(x). (ii) and (iii) are similar to (i), using theorems 3.39 and 3.40. 233 m. vijayasankari &g. ramkumar remark 4.18 the assumption that k and b are �̆�-𝓘-separated in (i) of proposition 4.17 cannot be removed. in the ideal topological space x in example 4.12, the sets {p} and {r} are not �̆�-𝓘-separated and their union {p, r} �̆�-𝓘-lc(x). lemma 4.19 for an x x, x �̆�-𝓘-cl(k) if and only if v  k  for every �̆�-𝓘-open set v containing x. proof let x�̆�-ℐ-cl(k) for any xx. to prove v  k  for every �̆�-ℐ-open set v containing x. prove the result by contradiction. suppose there exists a �̆�-ℐ-open set v containing x such that v  k = . then k vc and vc is �̆�-ℐ-cld. we have �̆�-ℐ-cl(k) vc. this shows that x �̆�-ℐ-cl(k) which is a contradiction. hence v  k  for every �̆�-ℐ-open set v containing x. conversely, let v k  for every �̆�-ℐ-open set v containing x. to prove x �̆�-ℐ-cl(k). we prove the result by contradiction. suppose x �̆�-ℐ-cl(k). then there exists a �̆�-ℐ-cld set f containing k such that xf. then xfc and fc is �̆�-ℐ-open. also, fck = , which is a contradiction to the hypothesis. hence x �̆�-ℐ-cl(k). theorem 4.20 suppose that a is �̆�-𝓘-open in x and that b is �̆�-𝓘-open in y. then a x b is �̆�-𝓘-open in x x y. proof suppose that f is ⋆-cld and hence sg-cld in x x y and that f  a x b. it suffices to show that f int(a x b). let (x, y)  f. then, for each (x, y)  f, ({x})* x ({y})* = ({x} x {y})* = ({x, y})*  f* = f  a x b. two ⋆-cld sets ({x})* and ({y})* are contained in a and b respectively. it follows from the assumption that ({x})* int(a) and that ({y})*  int(b). thus (x, y)  ({x})* x ({y})*  int(a) x int(b)  int(a x b). it means that, for each (x, y)  f, (x, y)int(a x b) and hence f  int(a x b). therefore, a x b is �̆�-ℐopen in x x y. theorem 4.21 the following are equivalent for a function f : (x, , i) → (y, ) i. f is �̆�-ℐ-continuous. ii. the inverse image of a regular closed set of y is �̆�-ℐ-open in x. iii. f-1(int(v*)) is �̆�-ℐ-closed in x for every open subset v of y. iv. f-1((int(f))*) is �̆�-ℐ-open in x for every closed subset f of y. v. f-1(u*) is �̆�-ℐ-open in x for every u o(y). vi. f-1(u*) is �̆�-ℐ-open in x for every u  so(y). vii. f-1(int(u*)) is �̆�-ℐ-closed in x for every u  po(y). proof (i)  (ii). obvious. (i)  (iii). let v be an open subset of y. since int(v*) is regular open, f-1(int(v*)) is �̆�ℐ-closed. the converse is similar. 234 some new �̆�-ℐ-locally closed sets with respect to an ideal topological spaces (ii)  (iv). similar to (i)  (iii). (ii)  (v). let u be any -open set of y. we have u* is regular closed. then by (ii) f-1(u*) is �̆�-ℐ-open in x. (v)  (vi). obvious from the fact that so(y) o(y). (vi)  (vii). let upo(y). then y\ int(u*) is regular closed and hence it is semi-open. then, we have x\f-1(int(u*)) = f-1(y\int(u*)) = f-1((y\int(u*))*) is �̆�-ℐ-open in x. hence f-1(int(u*)) is �̆�-ℐ-closed in x. (vii)  (i). let u be any regular open set of y. then upo(y) and hence f-1(u) = f-1(int(u*)) is θ̆-ℐ-closed in x. proposition 4.22 a function f : (x, , i) → (y, ) is �̆�-𝓘-continuous if and only if f-1(u) is �̆�-𝓘-open in x, for every open set u in y. proof let f : (x, , i) → (y, ) be �̆�-ℐ-continuous and u be an open set in y. then uc is closed in y and since f is �̆�-ℐ-continuous, f-1(uc) is �̆�-ℐ-cld in x. but f-1(uc) = (f-1(u))c and so f-1(u) is �̆�-ℐ-open in x. conversely, assume that f-1(u) is �̆�-ℐ-open in x, for each open set u in y. let f be a closed set in y. then fc is open in y and by assumption, f-1(fc) is �̆�-ℐ-open in x. since f-1(fc) = (f-1(f))c, we have f-1(f) is �̆�-ℐ-cld in x and so f is �̆�-ℐ-continuous. theorem 4.23 if f : (x, , i) → (y, ) is �̆�-𝓘-continuous and pre-sg-closed and if a is an �̆�-𝓘-open (or �̆�-𝓘-cld) subset of y, then f-1(h) is �̆�-𝓘-open (or �̆�-𝓘-cld) in x. proof let h be an �̆�-ℐ-open set in y and f be any sg-closed set in x such that f  f-1(h). then f(f)  h. by hypothesis, f(f) is sg-closed and h is �̆�-ℐ-open in y. therefore, f(f)  int(h) and so f  f-1(int(h)). since f is �̆�-ℐ-continuous and int(h) is open in y, f-1(int(h)) is �̆�-ℐ-open in x. thus f  int(f-1(int(h))) int(f-1(h)). i.e., f  int(f-1(h)) and f-1(h) is �̆�-ℐ-open in x. by taking complements, we can show that if h is �̆�-ℐ-cld in y, f-1(h) is �̆�-ℐ-cld in x. 5. conclusion the notions of sets and functions in ideal topological spaces and fuzzy topological spaces are extensively developed and used in many engineering problems, information systems, particle physics, computational topology and mathematical sciences. by researching generalizations of closed sets, some new separation axioms have been founded and they turn out to be useful in the study of digital topology. therefore, all topological functions defined in this thesis will have many possibilities of applications in digital topology and computer graphics. 235 m. vijayasankari &g. ramkumar references [1] arockiarani, i., balachandran, k. and ganster, m.: regular-generalized locally closed sets and rgl-continuous functions, indian j. pure. appl. math., 28 (1997), 661669. 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[18] veera kumar, m. k. r. s.: ĝ -locally closed sets and lcĝ -functions, indian j. math., 43 (2) (2001), 231-247. 237 microsoft word documento1 ratio mathematica volume 45, 2023 application of analytic hierarchy process in engineering education m. tirumala devi1 sameena afreen2 v. shyam prasad3 abdul majeed4 g. mahender reddy5 abstract analytic hierarchy process (ahp) provides a mathematical technique to formulate a problem as a hierarchical structure and believes in an amalgamation of quantitative and qualitative criteria. it is this uniqueness of ahp that makes it one of the important inclusive systems, considered to make decisions with multiple criteria. this paper focuses on conducting analytic hierarchy process, based on the data collected from several engineering colleges in the state of telangana. this paper aims to understand the reasons for removing the staple engineering streams such as mechanical engineering, production engineering, electronics and instrumentation engineering and introducing new and contemporary streams such as artificial intelligence and data science, artificial intelligence and machine learning and internet of things. the world economic forum’s latest “future of jobs” report highlights the impact of ‘double disruption’ of automation, followed by covid-19. the report indicates that while 85 million jobs will be displaced, 47% of core skills will change by 2025. the topic thus is of immense value since it looks closely at the paradigm shift mentioned above and its further consequences. the result of the present study would be helpful to indicate the exact rankings of the programming and non-programming branches in the engineering field and thus would be instrumental in gauging learners’ inclination towards studying specific branches. this paper aims to analyze the growing demand of programming branches over traditional, non-programming branches. keywords: analytic hierarchy process, pair-wise comparison, priority vector. ams mathematical classification: 03d55, 93a136 1 department of mathematics, kakatiya university, warangal, ts, india. email: oramdevi@yahoo.com, orcid id: https://orcid.org/0000-0002-4162-0084. 2 department of mathematics, kakatiya university, warangal, ts, india. email: afreensama82@gmail.com. 3 guru nanak institutions technical campus, hyderabad, telangana, india. mail:shyamnow4u@gmail.com, https://orcid.org/0000-0002-7966-1682. 4 muffakham jah college of engineering & technology, hyderabad, telangana, india. email: abdulmajeed.maths@mjcollege.ac.in. orcid id: https://orcid.org/0000-0002-0286-0042. 5 guru nanak institutions technical campus, hyderabad, telangana, india. email: mahender1563@yahoo.co.in, https://orcid.org/0000-0002-1387-0131 267 mailto:oramdevi@yahoo.com https://orcid.org/0000-0002-4162-0084 mailto:afreensama82@gmail.com mailto:shyamnow4u@gmail.com https://orcid.org/0000-0002-7966-1682 mailto:abdulmajeed.maths@mjcollege.ac.in https://orcid.org/0000-0002-0286-0042 mailto:mahender1563@yahoo.co.in https://orcid.org/0000-0002-1387-0131 m. tirumala devi et al. 1. introduction the process of choosing the optimal alternative among all potential options is classified as decision-making, however in practice, attaining an optimized result can be difficult because decision-makers are frequently challenged with diverse decision-making problems [1]. multi-criteria decision-making (mcdm) is one of the most significant fields of decision theory, and it is used to find the optimum solution out of all the possibilities [4]. mcdm has been improved by the development of several approaches, including: analytic hierarchy process (ahp) (saaty 1980)[8]; techniques for determining superiority and inferiority [10]; simos’ technique of ranking [6]; multiattribute utility theory (maut) [3]; elimination and selection in accordance with reality [7]; for enrichment evaluations, a preference ranking organization method is used [2]; and selecting based on benefits [9]. these mcdm techniques are commonly used to help solve real-world decision-making difficulties. saaty's (1980) ahp is a popular mcdm method that has gotten a lot of attention in the industry, including education and management. earlier many researchers have done a good amount of work on decision making [5]. students with technological capabilities will have excellent work possibilities in the post-covid global economic landscape. according to the world economic forum, 92 per cent of firms are speeding up their digitalization efforts, with more than 90 per cent using technology such as artificial intelligence, big data analytics, and cloud computing. students who graduate with specialized technical skills will have access to desirable job possibilities in fields such as it, cloud services, and healthcare. it is observed that, many engineering colleges have dropped the non-programming branches such as mechanical engineering, production engineering, electronics & instrumentation engineering etc. and these are being replaced by computer science engineering-emerging technology courses such as artificial intelligence & data science, artificial intelligence & machine learning, internet of things & cyber security etc. the authors of the paper are motivated by this scenario to present a paper on this topic. this paper tries to prove that, emerging technology courses are providing more employability skills required by the students in the current scenario. with this goal, variables were selected and study was done to observe the results. there are not too many discussions to understand this issue with the application of ahp method. in this context, the approach towards understanding the change in terms of choices of courses among engineering students, using ahp technique is not only a novel idea but also presents a vivid picture to understand the paradigm shift. 2. the physical importance of all criteria scope of employment (c1): the emerging technology courses have a wide scope of employability in many sectors like telecommunication, transportation, corporate sector, medical etc. these graduates progress more quickly in terms of developing the necessary abilities to obtain and 6received on july 6th, 2022. accepted on september 15th, 2022. published on january 30, 2023.doi: 10.23755/rm.v45i0.1025. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement 268 application of analytic hierarchy process in engineering education manage a job, resulting in better chances. however, while having such abilities increases these graduates’ chances of finding work, it does not guarantee it [3]. subjectspecific knowledge and abilities are essential for work. other talents, undoubtedly, play an important role in the lives of students, assisting them in obtaining not just a job, but also a profession of their choice. acquiring the right skills not only ensures sustenance but also promotes growth in the employment sector. subject-related skills are those that students gain throughout their degree program, whereas transferable skills are those that people learn on the job and may be used to other occupations inside or outside of their organization, allowing them to go further in their careers [5]. because information technology has such a strong influence on today's global marketplaces, this study focuses mostly on it workers. it firms are constantly developing and adopting new technology. employers also seek talented candidates that can adapt to today's fast-paced work environment and are eager to learn new skills. while working to improve employability, the international labor organization emphasized fast globalization, new working environments, and technological advancement. this will necessitate investing in their workforce's skill development and training. the indian economy has been moving toward digitalization since 2015, necessitating a highly qualified and capable workforce. through numerous open online learning platforms such as moocs, swayam, and others, where at least 350 online courses enable students to virtually attend courses taught by the best faculty, many jobs will be created in the future. right employability skills (c2): computer science and technology is more of an applied computer science than a completely theoretical discipline. it focuses on addressing user needs in an organizational and societal environment by performing the following tasks: selection, creation, application, integration, and administration of computing technologies. as a result, computer graduates must have particular abilities and knowledge in order to obtain relevant positions at work; they must be able to anticipate changes in the field of technology and express the value of the same to an individual or organization. every year, the software sector employs roughly 10 million people. in the $124–130 billion market, it contributes 67 percent. for the fiscal year 2016-17, it grew at a rate of 1214 percent. because the software business is growing, it generates strong cash flows and a high return on equity. nonetheless, rather than adding new staff, numerous it organizations are deploying automation to improve digitalization. employability skills are a continuous practice that can be included into the curriculum as well as the unpredictable work environment of an organization. this emphasizes the need of having the appropriate employment skills. higher pay package (c3): computer science engineering is a field that is in high demand in the industry. as a result, obtaining a professional degree or certification in computer science will make one a valued asset to a company, especially in the it area. furthermore, the pay packages in these companies are fairly high. wide career options and career stability (c4): according to data from popular study destinations, 269 m. tirumala devi et al. ➢ computer and information technology (it) occupations are expected to expand by 12% in the united states by 2028, according to the bureau of labor statistics. ➢ by 2023, the number of software and application programmers in australia is predicted to increase from 121,300 to 146,800. ➢ immigration.ca reports that “qualified software engineers are being hired by canadian employers as quickly as they become available.” freedom to work from anywhere (c5): computer science and information technology have an impact on everything from scientific research to health development, transportation, banking, and communications, among other things. microwave ovens, refrigerators, and door locks are now all connected to our wi-fi networks. applications in all the fields (c6): artificial intelligence applications have advanced tremendously in recent years. its applications can be found in virtually every industry. artificial intelligence applications in e-commerce ➢ personalized purchasing ➢ assistants with artificial intelligence ➢ preventing fraud applications of artificial intelligence in education ➢ educators will benefit from automated administrative tasks ➢ creating intelligent content ➢ virtual assistants ➢ personalized education applications of artificial intelligence in lifestyle ➢ vehicles that drive themselves ➢ filters for spam ➢ recognition of faces ➢ system of recommendation navigation, robotics, human resources, healthcare, agriculture, gaming, automobiles, social media, marketing, chat bots, finance, and many more fields have been transformed by ai applications. 3. significance of each alternative artificial intelligence & data science (ai&ds) (a1): the artificial intelligence and data science program teaches students how to do intelligent data analysis, which is a critical component in many real-world applications. data science has evolved as one of the most high-growth, dynamic, and rewarding occupations in technology during the last 10 years. this course intends to teach not only essential technologies like artificial intelligence, data mining, and data modeling, but also advanced topics like machine learning and big data analytics. students will gain cross-disciplinary skills in fields such as statistics, computer science, machine learning, and logic, as well as data scientists, and will have 270 application of analytic hierarchy process in engineering education career opportunities in healthcare, business, e-commerce, social networking companies, climatology, biotechnology, genetics, and other important fields by the end of this course. students will learn statistical, mathematical reasoning, machine learning, knowledge discovery, and visualization abilities as part of this program. artificial intelligence & machine learning (ai&ml) (a2): the student under b.tech artificial intelligence and machine learning is required to write the code of the said machine. this code in essence works as a guiding instruction for the machine, where it can perform tasks with less human intervention. internet of things/cyber security (iot/cs)(a3): b.tech cse (iot& cyber security including block chain technology), undergraduate program familiarizes students with the functional and operational aspects of iot, cyber security and block chain technology. cyber security is a specialist topic of information technology (it) that is considered a sub-discipline of computer science. students will gain the information and abilities needed to safeguard computer operating systems, networks, and data from cyber-attacks in cyber security courses. because of the rising occurrence of cybercrime, cyber security as a profession has evolved over time. any industry that transacts online or handles sensitive data requires a cyber security expert to protect its data from such criminals. because cyberspace is a global platform that anybody may access from anywhere in the world, the scope of cyber security is equally distributed. cyber security is a lucrative and rapidly expanding subject that focuses on safeguarding businesses from digital threats and keeping their data and networks secure. experts in cyber security identify flaws, offer software and hardware programs to limit risks, and create rules and processes to ensure security. the demand for qualified cyber security specialists is expected to expand as more firms transfer their activities online and cyberattacks become more common, particularly in healthcare and financial institutions. information security analysts, for example, are expected to expand by 40% between 2020 and 2038, according to the bureau of labor statistics. mechanical engineering (mech) (a4): students in this program will learn how to become mechanical engineers. the goal of this curriculum is to prepare students to use mechanical engineering principles in the design, manufacture, and maintenance of mechanical systems. production engineering (pe) (a5): production engineering is a branch of engineering that is closely related to mechanical engineering. production engineers are educated to increase the efficiency and effectiveness of manufacturing and service industries. manufacturing technology, which is a branch of mechanical engineering, is combined with management science in production engineering. a production engineer works in a variety of industries, dealing with engineering methods and management difficulties relating to manufacturing. electronics and instrumentation engineering (a6): electronics and instrumentation engineering is a program combining motor skills and academic skills which carve out a career in various fields of electronics, measurement, and complex process understanding. 271 m. tirumala devi et al. 4. methodology the ahp technique is broken down into the following steps. i) choosing criteria and structuring a decision-making problem ii) prioritization of criteria using pair wise comparison iii) on each criterion, compare options in pairs iv) calculating a relative score for each option 4.1 prioritization methods: there are a few methods for determining alternate priorities, like as i) geometric mean method, ii) additive normalization method, iii) stochastic vector method are available to find the priorities of alternatives. the geometric mean method (gmm) is employed in this paper. 4.1.1 geometric mean method (gmm): the weights for the criteria or alternatives are determined using this procedure. the alternate pair-wise comparison matrix is shown in table 1. here 𝐾1,𝐾2,…. . ,𝐾𝑛 represents the alternatives which are to be ranked and 𝑘11,𝑘12,…. . ,𝑘𝑛𝑛 represents expert opinions. the geometric mean method, which is used to calculate the priority weight vectors, is described below. 𝐾1 𝐾2 ……… 𝐾𝑛 𝐾1 𝑘11 𝑘12 ……… 𝑘1𝑛 𝐾2 𝑘21 𝑘22 ……… 𝑘2𝑛 . . . . . . . . . . . . . . . 𝐾𝑛 𝑘𝑛1 𝑘𝑛2 ……… 𝑘𝑛𝑛 table 1: pair-wise comparisons obtain the geometric row means of each row as priority vector 𝑘1 = (𝑘11 × 𝑘12 × …..× 𝑘1𝑛) 1 𝑛 priority vector 𝑘2 = (𝑘21 × 𝑘22 × …..× 𝑘2𝑛) 1 𝑛 priority vector 𝑘𝑛 = (𝑘𝑛1 × 𝑘𝑛2 × …..× 𝑘𝑛𝑛) 1 𝑛 the normalized vector of (𝑘1,𝑘2, …. . ,𝑘𝑛) becomes the solution vector. table 2 describes ahp measurement scale about the importance of saaty’s crisp numbers. intensity of importance definition explanation 1 same importance two elements contribute same to the property 3 moderate importance of one over another experience and judgment some favor one over the other 5 essential or high experience and judgment highly favor 272 application of analytic hierarchy process in engineering education importance one over another 7 very strong importance an element is highly favored and its dominance is demonstrated in practice 9 extreme importance one of the most possible orders of affirmation is evidence favoring one element over another. 2,4,6,8 lying between two adjacent judgments comprise is needed between two judgments reciprocals whenever activity i compared to j is assigned one of the above numbers, the activity j compared to i is assigned its reciprocal rational ratios occurring from forcing consistency of judgments table 2: ahp measurement scale table 3 lists the number of courses offered in selected engineering colleges. s.no name of the college ts eamcet code number of courses percentage 1. muffakhamjah college of engineering & technology mjct 10 15 2. cvr college of engineering cvrh 10 15 3. geetanjali college of engineering and technology gctc 9 13 4. guru nanak institute of technology gnit 8 12 5. guru nanak institutions technical campus guru 9 13 6. methodist college of engineering and technology meth 6 8 7. nalla narasimha reddy educational society group of institutions nnrg 8 12 8. lords institute of engineering & technology lrds 8 12 table 3: number of courses from selected engineering colleges table 4 exhibits program-wise intake in three consecutive academic years 2019-2020, 2020-2021 and 2021-2022. mjct cse inf ece civ mec h pe eee eie ai& ds ai& ml iot/ cs 2019-2020 120 120 120 120 120 60 60 60 0 0 -- 2020-2021 120 120 120 120 120 0 60 60 60 0 -- 2021-2022 120 120 120 120 120 0 60 0 60 60 -- cvrh 2019-2020 300 240 240 120 120 --120 60 0 0 0 2020-2021 300 240 240 60 60 --60 60 60 60 60 273 m. tirumala devi et al. 2021-2022 300 240 120 60 60 --60 60 120 120 60 gctc 2019-2020 240 60 240 120 120 --120 --0 0 0 2020-2021 240 60 240 60 60 --60 --60 60 120 2021-2022 240 60 240 60 60 --60 --60 180 120 gnit 2019-2020 180 60 120 120 120 --60 ----0 0 2020-2021 180 60 120 120 120 --60 ----60 60 2021-2022 180 60 120 120 120 --60 ----60 60 guru 2019-2020 300 60 300 180 300 --120 --0 0 0 2020-2021 300 60 300 180 300 --120 --60 60 60 2021-2022 300 60 300 180 180 --120 --60 60 60 meth 2019-2020 120 --120 120 120 --60 --0 ---- 2020-2021 120 --120 120 60 --60 --60 ---- 2021-2022 120 --120 60 60 --60 --120 ---- nnrg 2019-2020 180 0 180 60 120 --60 --0 0 -- 2020-2021 180 60 180 60 60 --60 --0 0 -- 2021-2022 180 60 180 30 30 --30 --60 60 -- lrds 2019-2020 180 120 120 180 180 --60 --0 0 -- 2020-2021 180 180 120 180 180 --60 --60 120 -- 2021-2022 180 180 120 180 120 --60 --60 180 -- table 4: program-wise intake in three consecutive academic years 5. weight vectors of each criteria to find the weight vectors of each criterion geometric mean method (gmm) has been applied. the data furnished below is fetched from inputs collected from the major stake holders like students, parents, alumni and employers. table 5 to table 10 shows the measurement of the weight vectors of criteria 1 to 6. table 11 displays the weight vectors of all criteria and table 12 presents average weights of each alternative with respect to all criteria. ai&ds ai&ml iot/cs mech pe eie weight vector ai&ds 1 1/2 2 3 5 7 2.1720 ai&ml 2 1 3 4 5 7 3.0717 iot/cs 1/2 1/3 1 2 3 5 1.3076 mech 1/3 1/4 1/2 1 2 3 0.7937 pe 1/5 1/5 1/3 1/2 1 2 0.4869 eie 1/7 1/7 1/5 1/3 1/2 1 0.2965 table 5: weights of c1 (scope of employment) 274 application of analytic hierarchy process in engineering education ai&ds ai&ml iot/cs mech pe eie weight vector ai&ds 1 1 2 4 3 5 2.2209 ai&ml 1 1 2 3 3 4 2.0396 iot/cs 1/2 1/2 1 2 2 2 1.1224 mech 1/4 1/3 1/2 1 1 2 0.6609 pe 1/3 1/3 1/2 1 1 2 0.6933 eie 1/5 1/4 1/2 1/2 1/2 1 0.4291 table 6: weights of c2 (right employability skills) ai&ds ai&ml iot/cs mech pe eie weight vector ai&ds 1 1/2 3 3 3 6 2.0800 ai&ml 2 1 2 4 5 7 2.8709 iot/cs 1/3 1/2 1 3 3 5 1.3990 mech 1/3 1/4 1/3 1 1 2 0.6177 pe 1/3 1/5 1/3 1 1 1/2 0.4723 eie 1/6 1/7 1/5 1/2 2 1 0.4101 table 7: weights of c3 (higher pay package) ai&ds ai&ml iot/cs mech pe eie weight vector ai&ds 1 1/2 2 4 4 7 2.1955 ai&ml 2 1 3 5 5 8 3.2598 iot/cs 1/2 1/3 1 2 2 4 1.1775 mech 1/4 1/5 1/2 1 1 3 0.6493 pe 1/4 1/5 1/2 1 1 1/3 0.4502 eie 1/7 1/8 1/4 1/3 3 1 0.4057 table 8: weights of c4 (wide career options and career stability) ai&ds ai&ml iot/cs mech pe eie weight vector ai&ds 1 1 1 5 5 7 2.3650 ai&ml 1 1 1 5 5 7 2.3650 iot/cs 1 1 1 5 5 7 2.3650 mech 1/5 1/5 1/5 1 1 1 0.4472 pe 1/5 1/5 1/5 1 1 1 0.4472 eie 1/7 1/7 1/7 1 1 1 0.3779 table 9: weights of c5 (freedom to work from anywhere) ai&ds ai&ml iot/cs mech pe eie weight vector ai&ds 1 1 1 5 5 7 2.3650 ai&ml 1 1 1 7 7 5 2.5014 275 m. tirumala devi et al. iot/cs 1 1 1 6 6 4 2.2894 mech 1/5 1/7 1/6 1 1 1 0.4101 pe 1/5 1/7 1/6 1 1 1 0.4101 eie 1/7 1/5 1/4 1 1 1 0.4388 table 10: weights of c6 (applications in all the fields) 𝑪𝟏 𝑪𝟐 𝑪𝟑 𝑪𝟒 𝑪𝟓 𝑪𝟔 priority vector 𝑪𝟏 1 2 2 1 1/2 1/3 0.9346 𝑪𝟐 1/2 1 1/2 1/2 1/3 1/4 0.4673 𝑪𝟑 1/2 2 1 1/2 1/3 1/4 0.5887 𝑪𝟒 1 2 2 1 1/3 1/3 0.8735 𝑪𝟓 2 3 3 3 1 1/2 1.7320 𝑪𝟔 3 4 4 3 2 1 2.5697 table 11: weight vectors of all criteria 𝑪𝟏 𝑪𝟐 𝑪𝟑 𝑪𝟒 𝑪𝟓 𝑪𝟔 average weight 𝑨𝟏 2.1720 2.2209 2.0800 2.1955 2.3650 2.3650 2.2307 𝑨𝟐 3.0717 2.0396 2.8709 3.2598 2.3650 2.5014 2.6506 𝑨𝟑 1.3076 1.1224 1.3990 1.1775 2.3650 2.2894 1.5351 𝑨𝟒 0.7937 0.6609 0.6177 0.6493 0.4472 0.4101 0.5813 𝑨𝟓 0.4869 0.6933 0.4723 0.4502 0.4472 0.4101 0.4859 𝑨𝟔 0.2965 0.4291 0.4101 0.4057 0.3779 0.4388 0.3898 table 12: average weights of each alternative with respect to all criteria following are the 3-d clustered column charts representing the weights of alternatives with respect to criteria. figure 1 to 6 will show the graphical representation of all the criteria i.e. criteria 1 to criteria 6. figure 1: weights of alternatives with respect to c1 figure 2: weights of alternatives with respect to c2 0 0,5 1 1,5 2 2,5 3 3,5 0 0,5 1 1,5 2 2,5 276 application of analytic hierarchy process in engineering education figure 3: weights of alternatives with respect to c3 figure 4: weights of alternatives with respect to c4 figure 5: weights of alternatives with respect to c5 figure 6: weights of alternatives with respect to c6 table 13 indicates the ranking and weights of alternatives which is further graphically presented through bar chart(figure 7). s.no alternatives weights ranks 1 ai&ds 2.2307 2 2 ai&ml 2.6506 1 3 iot/cs 1.5351 3 4 mech 0.5813 4 5 pe 0.4859 5 6 eie 0.3898 6 table 13: ranking and weights of alternatives 0 0,5 1 1,5 2 2,5 3 0 1 2 3 4 0 0,5 1 1,5 2 2,5 0 0,5 1 1,5 2 2,5 3 277 m. tirumala devi et al. figure 7: ranks of alternatives 6. conclusions according to the ranking of the alternative weights, artificial intelligence & machine learning branch is the most important branch in the field of engineering education, followed by artificial intelligence & data science, internet of things, mechanical engineering, production engineering and finally electronics and instrumentation engineering. the authors are of the opinion that the technological disruptions appear to be more powerful in the post-covid scenario. this is one of the potential reasons behind the current change in the employment scenario, affecting the equilibrium of the workforce. the stakeholders as well as the government policy makers must be aware of the quick transition in the educational field and thus make judicious plans where the boom of a particular field does not create a vacuum or dearth of skilled labourers in another field especially in the core, non-programming branches of engineering education. acknowledgement the authors are grateful to all of the editors and anonymous reviewers whose insightful comments and ideas significantly improved the paper's quality. references [1] angelis di, lee cy., strategic investment analysis using activity-based costing concepts and analytical hierarchy process techniques. int j prod res. 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[7] roy b., the outranking approach and the foundations of electre methods. theory and decision (1991) 31(1): pp.49–73. [8] saaty tl., the analytical hierarchy process. new york: mcgraw-hill (1980). [9] suhr j., the choosing by advantages decision making system. connecticut (ct): greenwood publishing group (1999). [10] xu x., the sir method: a superiority and inferiority ranking method for multiple criteria decision making. eur j oper res (2001) 131(3): pp.587–602. 279 introduction 27 dynamic programming approach to testing resource allocation problem for modular software p.k. kapur1 p.c. jha1 a.k. bardhan1 abstract testing phase of a software begins with module testing. during this period modules are tested independently to remove maximum possible number of faults within a specified time limit or testing resource budget. this gives rise to some interesting optimization problems, which are discussed in this paper. two optimization models are proposed for optimal allocation of testing resources among the modules of a software. in the first model, we maximize the total fault removal, subject to budgetary constraint. in the second model, additional constraint representing aspiration level for fault removals for each module of the software is added. these models are solved using dynamic programming technique. the methods have been illustrated through numerical examples. key words: software reliability, non homogeneous poisson process, resource allocation, dynamic programming 1. introduction growth in software engineering technology has led to production of software for highly complex situations occurring in industry, scientific research, defense and day to day life. consequently, the dependence of mankind on computers and computerbased systems is increasing day by day. any failure in these systems can cost heavily in terms of money and/or human lives. though high reliability of hardware part of these systems can be guaranteed, the same cannot be said for software. therefore a lot of importance is attached to the testing phase of the software development process, where around half the developmental resources are used [8]. essentially testing is a process of executing a program with the explicit intention of finding faults and it is this phase, which is amendable to mathematical modeling. it is always desirable to remove a substantial number of faults from the software. in fact the reliability of a software is directly proportional to the number of faults removed. hence the problem of maximization of software reliability is identical to 1 department of operational research, faculty of mathematical sciences, university of delhi, delhi 110007, india 28 that of maximization of fault removal. at the same time testing resource are not unlimited, and they need to be judiciously used. in this paper we discuss and solve such a management problem of allocation of testing resources among modules, through a software reliability growth model (srgm). a software reliability growth model (srgm) is a relationship between the number of faults removed from a software and the execution time/cpu time/calendar time. several attempts have been made to represent the actual testing environment through srgms [1,4,5,9]. these models have been used to predict the fault content, reliability and release time of a software. srgms have also been used to manage the testing phase. again large software consists of modules. often these modules are developed independently and each module may contain different number of faults and that of different severity. therefore distinct srgms should be used to represent the testing process of each module, as testing for these modules are done independently. an srgm with testing effort [9] has been chosen to represent the fault removal process for the two optimization problems discussed in this paper. the first optimization model (p1) maximizes the total number of faults expected to be removed, when available testing resource is known. the management normally aspires for some reliability level that can be translated in terms of number of faults removed. in the second optimization model (p2) we add a constraint in (p1) in terms of minimum number of faults aspired to be removed from each module. dynamic programming technique is used to solve these problems. this is the first time that this has been done in software engineering, according to our knowledge. dynamic programming approach, which is easy to solve and understand provides global optima for these problems. the methodology discussed in the paper has been illustrated through numerical examples. notations n : number of modules in the software (>1) ai : expected number of faults in the i th module (i=1,2,…,n) bi : proportionality constant for the i th module xi(t) : current testing effort expenditure at testing time t and ∫= t ii dwwxtx 0 )()( for i th module xi, z : the amount of testing resource to be allocated to the i th module and total testing resource available. mi(t) : number of faults removed in (0,t] the i th module, mean value function of nhpp, i = 1,…,n t : total testing time xi * : optimal value of xi , i = 1,…,n fn(z) : optimal number of faults removed upto n th modules (i.e. corresponding to nth stage in a dynamic programming algorithm) 29 aio : aspiration level of ith module (i.e. number of faults desired to be removed from ith module) pi : the minimum proportion of total faults to be removed from ith module. 2. mathematical modelling 2.1 resource allocation problem consider a software having n modules, which are being tested independently for removing faults lying dormant in them. the duration of module testing is often fixed when scheduling is done for the whole testing phase. hence limited resources are available, that need to be allocated judiciously. if mi faults are expected to be removed from the ith module with effort xi, the resulting testing resource allocation problem can be stated as follows [5,6]. max ∑ = n i im 1 subject to ∑ = = n i i zx 1 , 0≥ix , i = 1,…,n … … (p1) above optimization problem is the simplest one as it considers the resource constraint only. later in this paper, we incorporate additional constraints to the basic model. for solving (p1) a functional relationship between fault removal and resource consumption is required, which is discussed in the following section. 2.2 srgm for modules a software reliability growth model explains the time dependent behavior of fault removal. as modules are tested independently distinct srgms would represent their reliability growth. the influence of testing effort can also be included in the srgms [9]. in this paper we discuss the resource allocation problem using such a srgm for the modules. model assumptions 1. software consist of a finite number of modules and testing for each module is done independently 30 2. a module is subject to failures at random time caused by faults remaining in the software. 3. on a failure, the fault causing that failure is immediately removed and no new faults are introduced. 4. fault removal phenomenon is modelled by non homogeneous poisson process (nhpp). 5. the expected number of faults removed in ( )ttt ∆+, to the current testing resource is proportional to the expected remaining number of faults. under assumption 5, following differential equation may easily be written for ith module ))(( )( )( tmab tx tm dt d iii i i −= , i = 1,…,n …. … (1) solving equation (1) with the initial condition that, at t = 0, xi(t) = 0, mi(t) = 0 we get )1()( )(txbii iieatm −−= , i = 1,…,n … … (2) to describe the behaviour of testing effort, either exponential or rayleigh function has been used [5,9]. both can be derived form the assumption that, " the testing effort rate is proportional to the testing resource available". [ ])()()( txtc dt tdx iii i −= α , i = 1,…,n … … (3) where ci(t) is the time dependent rate at which testing resources are consumed, with respect to the remaining available resources. solving equation (3) under the initial condition 0)0( =x we get         −= ∫ t iii dkkctx 0 )(exp1)( α , i = 1,…, n … … (4) when β=)(tc , a constant )1()( tii ietx βα −−= , i = 1,…,n … … (5) if ttc .)( β= , (1) gives a rayleigh type curve )1()( 2 2t ii i etx β α − −= , i = 1,…,n … … (6) in this paper we have chosen exponential function (5) to represent testing effort in the optimization problems. 31 2.3 estimation of parameters the testing effort data are given in the form of testing effort )...( 21 nk xxxx <<< consumed in time ],0( it ; ni ,..,2,1= . the testing effort model parameters αi and βi can be estimated by the method of least squares as follows. minimize [ ]∑ = − n i i xx 1 ˆ subject to nn xx =ˆ (i.e. the estimated value of testing effort is equal to the actual value). once the estimates of αi and βi are known, the parameters of the srgms (2) for the modules can be estimated through maximum likelihood estimation method using the underlying stochastic process, which is described by a non homogeneous poisson process. during estimation, estimated values of αi and βi are kept fixed. if the fault removal data for a module is given in the form of cumulative number of faults removed yj in time (0,tj]. the likelihood function for that module is given as [ ] ∏ = −− − − − − − − − = n j tmtm jj yy jiji iiiii jiji jj e yy tmtm wybal 1 ))()(( 1 1 1 1 )!( )()( ),/(,( 3. optimal allocation of resources from the estimates of parameters of srgms for modules, the total fault content in the software ∑ = n i ia 1 is known. modules testing aims at removing maximum number of them, within available resources. hence (p1) can be restated as maximize ∑∑ = − = −= n i xb i n i ii iieaxm 11 )1()( subject to zx n i i ≤∑ =1 , ix 0≥ i = 1, … , n … (p1a) (p1a) can be solved using dynamic programming approach. from bellman's principle of optimality, we can write the following recursive equation [2]. 32 { })1(max)( 11 11 11 xb zx eazf − = −= { })()1(max)( 1 0 nn xb n zx n xzfeazf nn n −+−= − − ≤≤ , n = 2,…,n … (7) to index the modules, they can be arranged in descending order of their values of aibi i.e. nn bababa ≥≥≥ ...2211 . through this approach resources are allocated to the modules sequentially. but for some values of z (z < zr) one or more modules with higher index number i.e. having lower detectability may not get any allocation. we summarize this result in the following simple theorem. theorem 1 if for any n = 2,…,n; nn nnz ba v e n 1111 −−− ≤≤ − µµ , then values of nnn xxx ..., ,1+ are zero and problem reduces to (n-1) stage problem with             − + = −−− − rr rr r rr r ba v z b x 111 1 log 1 µ µ µ , r = 1,…,(n-1) … (8) where ∑ = = i j j i b1 ) 1 ( 1 µ and ∏ = = i j b jjii jibav 1 ( )/)( µ µ , i = 1,…,n proof of the theorem is given in appendix. as a result of the above allocation procedure, some modules may not be tested at all. this situation is not advisable. again management often aspires to achieve certain minimum reliability level for the software and that for each module of the software i.e. a certain percentages of the fault content in each module of the software is desired to be removed. hence (p1) needs to be suitably modified to maximize removal of faults in the software under resource constraint and minimum desired level of faults to be removed from each of the modules in the software. the resulting testing resource allocation problem can be stated as follows: ∑ ∑ = = −−= n i n i xb ii iieam 1 1 )1(max subject to 0 )1( iii xb ii aapeam ii =≥−= − , i = 1, … , n 33 ∑ = ≤ n i i zx 1 , 0≥ix , i = 1, … , n … (p2) (p2) can be solved using dynamic programming approach either by reducing the dimensionality of the problem through lagrange multiplier or converting to (p1) by substitution. we first consider the dimensionality reduction in dynamic programming approach [2] as follows. { }[ ]∑ = −− −−+−= n i i xb ii xb i x aeaeax iiii 1 0 )1()1(),(minmax ααφ α subject to zx n i i ≤∑ =1 0, ≥iix α i = 1, … , n (p3) where iα (i = 1, … , n) is lagrange multiplier for i th constraint corresponding to the ith module. the above problem can be solved by dynamic programming approach in which kuhn-tuckker optimality conditions are obtained at each stage [2]. at any stage αi (i = 1,…,n) can be zero or non-zero depending upon ineffectiveness or effectiveness of constraint respectively. hence each stage has two possibilities and corresponding to each possibility of preceding stage present stage has two possibilities. so at any stage i, total number of cases is 2i-1. infact, above problem reduces to that of finding an optimal path by searching for an optimal solution at each stage in which only one option could be chosen. this procedure does not provide a closed form solution. hence without further elaboration of the above method, the substitution method is adopted for converting the problem (p2) to the problem (p1) as follows: 0 )( iii axm ≥ implies 0 ii )1( i i xb aea ≥− − hence, ii za a b x =         −−≥ i i i 01log 1 (say), i = 1, … , n therefore (p2) can be restated as, maximize ∑∑ = − = −= n i xb i n i i iieam 11 )1( subject to ii zx ≥ i = 1, … , n ∑ = ≤ n i i zx 1 , 0≥ix , i = 1, … , n (p4) 34 let iii zxy −= (i = 1, … , n), then (p4) can be written as the problem (p1) given below ∑ ∑ = = −−= n i n i yb ii iieam 1 1 )1(maxmax subject to ∑ ∑ = = =−≤ n i n i ii zzzy 1 1 (say) 0≥iy , i = 1, … , n 0iii aaa −= , i = 1, … , n (p5) the problem (p5) is similar to the problem (p1) and hence using theorem-1 the problem ( p5 ) can also be solved. if for any i = 2, … , n ii iiz ba v e i 111 −−− ≤≤ µµ ,then nii yyy ,...,, 1+ are zeroes, then problem (p5) reduces to a ( 1−i ) stage problem and its solution is given as                 − + = −−− + nn nn n nn n ba v z b y 111 1 log 1 µ µ µ , n = 1,…,(i-1) … (9) ∑ − = −−= 1 1 )( i n z ni nevazf µ … … (10) through equation (9) optimal allocation of resources to the modules can be calculated. in the following section we numerically illustrate these results. 4. numerical example it is assumed that parameters ai and bi for the i th module (i=1,.....n) are already estimated using the software failure data. consider a software having 10 modules whose parameter estimates are as given in table-1. suppose the total resource available for testing is 97000. first the problem (p1) is solved and from the recursion equation (7) optimal allocation of resources (xi *) for the modules are computed. these are listed in table-1 along with the corresponding expected number of fault removed, percentages of faults removed and faults remaining for each module. the total number of faults that can be removed through this allocation is 152 (i.e. 60.6% of the fault content is removed from the software). it is observed 35 that in some modules (module-9,10) the remaining faults after allocation is high. this can lead to frequent failure during operational phase. obviously this will not satisfy the developer and he may desire that at least 50% of fault content from each of the modules of the software is removed (i.e. pi=0.5 for each i = 1…10). since faults in each module are integral values, nearest integer larger then 50% of the fault content in each module is taken as lower limit that has to be removed. the new allocation of resource along with expected number of fault removed, percentages of faults removed and faults remaining for each module after solving the problem (p2) through the problem (p5) is summarized in table-2. the total number of faults that can be removed through this allocation is 146.8(i.e. 58.4% of the fault content is removed from the software). in addition to the above if it is desired that a certain percentage of the total faults are to be removed then additional testing resources would be required. it is interesting to study this tradeoff and table-3 summarizes results, where the required percentage of faults removed is 60%. to achieve this, 3000 units of additional testing effort is required. the total number of faults that can be removed through this allocation is 150.8(i.e. 60.09% of the fault content is removed from the software). analysis given in tables-1, 2 and 3 help in providing the developer an insight into the resource allocation and the corresponding fault removal phenomenon and the objective can be set accordingly. table 1 module ai bi xi * mi* % of faults removed % of faults remaining 1 63 5.33e-05 25435 46.7689 74.24 25.76 2 13 0.000252 5280.7 9.56979 73.61 26.39 3 6 0.000526 2459.5 4.3553 72.59 27.41 4 51 5.17e-05 21549 34.2571 67.17 32.83 5 15 0.000171 6354.5 9.93004 66.2 33.8 6 39 5.72e-05 16554 23.8778 61.23 38.77 7 21 9.94e-05 8857.2 12.2916 58.53 41.47 8 9 0.000174 3412.3 4.03476 44.83 55.17 9 23 5.06e-05 5845.6 5.88626 25.59 74.41 10 11 8.78e-05 1251.9 1.14528 10.41 89.59 total 251 97000 152.117 60.6 39.4 36 table-2 module ai aio zi* yi* mi(yi) mi* % of faults removed % of faults remaining 1 63 32 13300 7495.5 10.2 42 67 33 2 13 7 3064.6 1235.6 1.61 8 66.21 33.79 3 6 3 1317.3 672.1 0.89 4 64.89 35.11 4 51 26 13793 2969.7 3.56 30 57.96 42.04 5 15 8 4464.8 440.4 0.51 8 56.71 43.29 6 39 20 12565 0 0 20 51.28 48.72 7 21 11 7465.7 0 0 11 52.38 47.62 8 9 5 4652.5 0 0 5 55.56 44.44 9 23 12 14586 0 0 12 52.17 47.83 10 11 6 8978.1 0 0 6 54.55 45.45 total 251 130 84187 12813.3 16.8 146 58.48 41.52 table-3 module ai aio zi* yi* mi(yi) xi* mi* % of faults removed % of faults remaining 1 63 32 13300 8624.7 11.74 21924.6 44 69.43 30.57 2 13 7 3064.6 1474.2 1.93 4538.82 9 68.7 31.3 3 6 3 1317.3 786.52 1.048 2103.79 4 67.47 32.53 4 51 26 13793 4134.5 5.13 17927.4 31 61.05 38.95 5 15 8 4464.8 793.16 0.984 5257.95 9 59.9 40.1 6 39 20 12565 0 0 12565.5 20 51.3 48.7 7 21 11 7465.7 0 0 7465.66 11 52.38 47.62 8 9 5 4652.5 0 0 4652.5 5 55.55 44.45 9 23 12 14586 0 0 14585.7 12 52.18 47.82 10 11 6 8978.1 0 0 8978.11 6 54.55 45.45 total 251 130 84187 15813 20.8 100000 151 60.09 39.91 37 5. conclusion in this paper we have discussed a couple of optimization problems occurring during module testing phase of software development life cycle. a dynamic programming approach for finding the optimal solution has been proposed. using simple recursion equations it is shown how fault removal for each module and that of the software can be maximized, by judicious allocation of resources. it is observed that after certain duration of testing, fault removal becomes difficult in the sense that greater effort will be required to remove each additional fault. as the reliability of software is of utmost importance scientific decision making is required while deciding the resource budget. the tradeoff as shown in section-4 can be useful in this regard. alternatively if the developer is not too keen on an optimal solution but is satisfied with an efficient solution, goal programming approach may be desirable in that case. we are further looking into this aspect. appendix: proof of the theorem-1 we have following recursion equations given in (7): { })1(max)( 11 1 11 xb zx eazf − = −= { })()1(max)( 1 0 nn xb n zx n xzfeazf nn n −+−= − − ≤≤ , n = 2, … , n the above problem can be solved through forward recursion in n stages as follows. stage-1: let n=1 then we have { })1(max)( 11 1 11 xb zx eazf − = −= = )1( 11 zbea −− stage-2: let n=2 then we have { })()1(max)( 212 20 2 22 xzfeazf xb zx −+−= − ≤≤ substituting )( 21 xzf − in above we have { })1()1(max)( )(12 20 2 2122 xzbxb zx eaeazf −−− ≤≤ −+−= 38 now let, { })1()1()( )(1222 2122 xzbxb eaeaxf −−− −+−= then { })(max)( 22 20 2 xfzf zx ≤≤ = the maxima can be found through calculus. )( 1122 2 22 2122)( xzbxb ebaeba dx xdf −−− −= the sufficiency condition can be checked through the second derivative condition: 0 )( )(2 11 2 222 2 22 2 2122 ≤−−= −−− xzbxb ebaeba dx xfd the following three situations can occur. (i) 0 )( 2 22 < dx xdf (ii) 0 )( 2 22 = dx xdf (iii) 0 )( 2 22 > dx xdf if 0 )( 2 22 < dx xdf , then x2 =0. at x2 =0 0 )( 1 1122 2 22 <−= − zbebaba dx xdf i.e. 22 1111 ba ba e zb <≤ which implies 2211 baba > ,in other words the detectability in module -1 is higher than module –2. similarly 0 )( 2 22 > dx xdf implies x2 = z and we have 12 11 22 ≥> zbe ba ba hence 1122 baba > , the testing resources would be allocated to module -2 first as the detectability is higher there. finally if 0 )( 2 22 = dx xdf       − + =∗ 22 11 1 21 2 log 1 ba ba zb bb x , i.e.       − + =∗ 22 11 1 21 2 log 1 ba v z b x µ µ µ , 39 and           +−= ++ − = ∑ 21 1 21 2 22 11 )()()( 11 22 1 22 11 2 2 1 2 bb b bb b z ba ba i i ba ba a ba ba aeazf i.e. 2 2 1 2 2)( veazf z i i µ− = −= ∑ where 1 1 1 1 1 b b ==µ , 11 av = , 21 21 21 2 11 1 bb bb bb + = + =µ           += 2 2 1 2 )()( 11 22 1 22 11 22 b v ba v ba v av µ µ µ µ µ now proceeding by induction it can be shown for nth stage,       − + = −−− − ∗ nn nn n nn n ba v z b x 111 1 log 1 µ µ µ and n z n i in veazf n µ− = −= ∑ 1 )( for n=1…n the proof is complete. references 1. goel a.l., software reliability models: assumptions, limitations and applicability, ieee trans. on software engineering, se-11, pp. 1411-1423, 1985. 2. hadley, g., nonlinear and dynamic programming, addision-wesley, reading mass, 1964. 3. ichimori, t, yamada, s. and nishiwaki m., optimal allocation policies for testing-resource based on a software reliability growth model, proceedings of the australia –japan workshop on stochastic models in engineering, technology and management, pp. 182-189, 1993. 4. kapur p.k. and garg r.b.; cost reliability optimum release policies for a software system with testing effort, opsearch, vol. 27, no. 2, pp. 109118, 1990. 5. kapur p.k., garg r.b. and kumar, s.; contributions to hardware and software reliability, world scientific, singapore, 1999. 40 6. kapur, p.k. and bardhan, a.k., modelling, allocation and control of resources: an interdisciplinary approach in software reliability and marketing, operations research, eds. m. agarwal and k. sen, narosa publishing house, new delhi 2001. 7. kubat p. and koch h.s., managing test procedures to achieve reliable software, ieee trans. on reliability, vol. r-32, pp. 299-303, 1983. 8. musa j.d., iannino a. and okumoto k, software reliabilitymeasurement, prediction and application, mc graw hill, 1987. 9. yamada s. and ohtera h. and narihisa h., software reliability growth model with testing effort, ieee trans. on reliability, vol. r-35, pp. 19 23, 1986. ratio mathematica volume 41, 2021, pp. 71-78 71 blocks within the period of lucas sequence rima p. patel* devbhadra v. shah† abstract in this paper, we consider the periodic nature of the sequence of lucas numbers 𝐿𝑛 defined by the recurrence relation 𝐿𝑛 = 𝐿𝑛−1 + 𝐿𝑛−2; for all 𝑛 ≥ 2; with initial condition 𝐿0 = 2 and 𝐿1 = 1. for any modulo 𝑚 > 1, we introduce the ‘blocks’ within this sequence by observing the distribution of residues within a single period of lucas sequence. we show that length of any one period of the lucas sequence contains either 1,2 or 4 blocks. keywords: fibonacci sequence, lucas sequence, periodicity of lucas sequence 2010 ams subject classification‡: 11b37, 11b39, 11b50 * rima p. patel (mahavir swami college of polytechnic, bhagwan mahavir university, surat, india); rimapatel25@gmail.com †devbhadra v. shah (department of mathematics, veer narmad south gujarat university, surat, india); drdvshah@yahoo.com ‡ received on september 21, 2021. accepted on december 5, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.664. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. r. patel and d. shah 72 1. introduction the fibonacci sequence and the lucas sequence are well-known sequences among all the integer sequences. the fibonacci sequence {𝐹𝑛} satisfies the recurrence relation 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2, with the initial conditions 𝐹0 = 1 and 𝐹1 = 1. lucas sequence {𝐿𝑛} is considered as the ‘twin sequence’ of fibonacci sequence which satisfies the similar recursive relation 𝐿𝑛 = 𝐿𝑛−1 + 𝐿𝑛−2, with the initial conditions 𝐿0 = 2 and 𝐿1 = 1. on the other hand, some researchers have conducted important research on the period of these two recursive sequences [1, 3, 4, 5, 6]. wall [1] defines the length of period of the fibonacci sequence by reducing it through the modulo any positive integer 𝑚 > 1. kramer and hoggatt jr. [3] also defined the length of the period of the lucas sequence obtained by reducing the sequence through modulo any positive integer 𝑚 > 1. in this paper, we take deep insight in to the periodic nature of lucas sequence and introduce the concept of ‘blocks’ by observing the distribution of residues within a single period of lucas sequence when considered modulo any positive integer 𝑚 > 1. we denote the sequence of least non-negative residues of the terms of {𝐿𝑛} taken modulo 𝑚 (𝑚 ≥ 2) by 𝐿(𝑚𝑜𝑑 𝑚). if we examine the sequence of final digits of {𝐿𝑛}, then we notice an interesting pattern that the sequence 𝐿(𝑚𝑜𝑑 10) = {2,1,3,4,7,1,8,9,7,6,3,9,2,1,3,…} repeats after the 12 terms again and again all the way. for any modulo 𝑚, it is easy to observe that the sequence {𝐿𝑛} is always periodic and it repeats from starting values 0 and 1. by 𝑘𝐿 = 𝑘𝐿(𝑚), we mean the lengths of the period of {𝐿𝑛} modulo any positive integer 𝑚. these leads us to the following easy consequences. lemma 𝟏.𝟏. (a) 𝐿𝑘𝐿(𝑚)−1 ≡ −1(𝑚𝑜𝑑𝑚) (b) 𝐿𝑘𝐿(𝑚) ≡ 2(𝑚𝑜𝑑𝑚) (c) 𝐿𝑘𝐿(𝑚)+1 ≡ 1(𝑚𝑜𝑑𝑚) (d) 𝐿𝑘𝐿(𝑚)+2 ≡ 3(𝑚𝑜𝑑𝑚) (e) 𝐿𝑘𝐿(𝑚)+𝑛𝑟 ≡ 𝐿𝑛(𝑚𝑜𝑑 𝑚), for all 𝑟 ∈ ℤ. the following is an important result which speaks about the divisibility property of 𝑘𝐿(𝑚). fact 𝟏.𝟐. for any 𝑚 > 1, since 𝐿(𝑚𝑜𝑑 𝑚) is always periodic, we conclude that if 𝐿𝑛 ≡ 2 (𝑚𝑜𝑑 𝑚) and 𝐿𝑛 + 1 ≡ 1(𝑚𝑜𝑑 𝑚), then 𝑘𝐿(𝑚) | 𝑛. blocks of lucas sequence 73 2. blocks within 𝑳(𝒎𝒐𝒅 𝒎) in this article, we restrict our attention to the behavior of the blocks within the residues for a given modulus and consequently some interesting relationships will be derived. definition 2.1. by 𝛼(𝑚) we mean the smallest positive value of the index 𝑛 of lucas numbers such that 𝐿𝑛 ≡ 2𝐿𝑛+1(𝑚𝑜𝑑 𝑚), when 𝑛 > 1. we call 𝛼(𝑚) the restricted period of 𝐿(𝑚𝑜𝑑 𝑚). equivalently, 𝛼(𝑚) is the position of the first repeated term in the sequence 𝐿(𝑚𝑜𝑑 𝑚). thus, 𝐿𝛼(𝑚) ≡ 2𝐿𝛼(𝑚)+1 when considered (𝑚𝑜𝑑 𝑚). as an illustration, if we consider 𝐿(𝑚𝑜𝑑 3) = {2,1,0,1,1,2,0,2,2,1,0,1,1…}, then it is appearent that 𝐿4 ≡ 1(𝑚𝑜𝑑 3) and 𝐿5 ≡ 2(𝑚𝑜𝑑 3). thus 𝐿4 ≡ 2𝐿5(𝑚𝑜𝑑 3), which gives 𝛼(3) = 4. we call the finite sequence 𝐿0, 𝐿1,… , 𝐿𝛼(𝑚)−1 to be the first block occurring in 𝐿(𝑚𝑜𝑑 𝑚). it may happen that 𝛼(𝑚) = 𝑘𝐿(𝑚). in such case, we call 𝐿(𝑚𝑜𝑑 𝑚) to be without restricted period. for 𝐿(𝑚𝑜𝑑 4) = {2,1,3,0,3,3,2,1,3…}, clearly 𝛼(4) = 𝑘𝐿(4) = 6, and thus 𝐿(𝑚𝑜𝑑 4) has no restricted period. definition 2.2. by 𝑠(𝑚) we mean the second positive residue ′𝑡′, which appears after the first block in 𝐿(𝑚𝑜𝑑 𝑚). this clearly means that 2𝑠(𝑚) ≡ 𝐿𝛼(𝑚) (𝑚𝑜𝑑 𝑚); 𝑠(𝑚) = 𝐿𝛼(𝑚)+1. using the definition of 𝐿𝑛, we now conclude that 𝐿𝛼(𝑚)+2 = 3𝑠(𝑚), 𝐿𝛼(𝑚)+3 = 4𝑠(𝑚), 𝐿𝛼(𝑚)+4 = 7𝑠(𝑚),… . also the first block ends with 𝑚 − 𝑠(𝑚). thus, (𝐿𝛼(𝑚), 𝐿𝛼(𝑚)+1, 𝐿𝛼(𝑚)+2, 𝐿𝛼(𝑚)+3,…) = 𝑠(𝑚)(2, 1, 3, 4, 7,…)(𝑚𝑜𝑑 𝑚). this implies that the successive terms in 𝐿(𝑚𝑜𝑑 𝑚) after the first block are the multiples of 𝑠(𝑚). we therefore call 𝑠(𝑚) to be a multiplier. again, in the sequence 𝐿(𝑚𝑜𝑑 𝑚), the blocks are of the form 2,1,…,𝑚 − 𝑠(𝑚),2𝑠(𝑚),𝑠(𝑚),… , 𝑚 − 𝑥,𝑥,𝑥,…; where 2,1,…,𝑚 − 𝑠(𝑚) is the first block, 2𝑠(𝑚),𝑠(𝑚),…,𝑚 − 𝑥 is the second block, and so on. the occurrence of 3 − 2𝑚,𝑚 − 1 in 𝐿(𝑚𝑜𝑑 𝑚) will indicate that the end of the period has been reached and there after repetition begins, since the next two terms will be 2,1. here we note that each block contains the same (that is 𝛼(𝑚)) number of terms and the subscripts are in arithmetic progression. thus, 𝐿𝛼(𝑚)𝑢 ≡ r. patel and d. shah 74 2𝐿𝛼(𝑚)𝑢+1(𝑚𝑜𝑑 𝑚), for each positive integer 𝑢. since 𝐿𝑘𝐿(𝑚) ≡ 2𝐿𝑘𝐿(𝑚)+1(𝑚𝑜𝑑 𝑚), we conclude that 𝛼(𝑚)𝑢 = 𝑘𝐿(𝑚), where 𝑢 is a positive integer, which implies that 𝛼(𝑚) | 𝑘𝐿(𝑚). later in the paper we show that the value of 𝑢 is either 1 or 2 or 4. definition 2.3. by 𝛽(𝑚) we mean the order of 𝑠(𝑚), when considered modulo 𝑚. that is, 𝑠(𝑚)𝛽(𝑚) ≡ 1 (𝑚𝑜𝑑 𝑚) and if 𝑛 < 𝛽(𝑚) then 𝑠(𝑚)𝑛 ≢ 1 (𝑚𝑜𝑑 𝑚). to illustrate above definitions, we consider the following three examples: (1) for 𝐿(𝑚𝑜𝑑 4) = {2,1,3,0,3,3,2,1, . . . }, clearly 𝑘𝐿(4) = 6. also, the restricted period 𝛼(4) is 6 and multiplier 𝑠(4) is 1. thus, the order of 𝑠(4) = 1 is 1 and hence 𝛽(4) = 1. (2) if we consider 𝐿(𝑚𝑜𝑑 6) = {2,1,3,4,1,5,0,5,5,4,3,1,4,5,3,2, 5,1,0,1,1,2,3,5,2,1,3,4,1,…} , then clearly 𝑘𝐿(6) is 24, 𝛼(6) is 12 and 𝑠(𝑚) is 5. since 52 ≡ 1 (𝑚𝑜𝑑 6), we get 𝛽(11) = 2. (3) if we consider 𝐿(𝑚𝑜𝑑 13) = 2,1,3,4,7,11,18,3,8,11,6,4,10,1, 11,12,10,9,6,2,8,10, 5,2,7,9,3,12,2,1,3,4,… , then 𝑘𝐿(13) = 28, 𝛼(13) = 7 and 𝑠(𝑚) = 8. since 84 ≡ 1 (𝑚𝑜𝑑 13), we have 𝛽(13) = 4. the following theorem ties together the three functions 𝑘𝐿(𝑚), 𝛼(𝑚) and 𝛽(𝑚). theorem 2.4. 𝑘𝐿(𝑚) = 𝛼(𝑚)× 𝛽(𝑚). proof: we first divide the single period of 𝐿(𝑚𝑜𝑑 𝑚) into smaller finite subsequences, say 𝑅0,𝑅1,𝑅2,…,𝑅𝑛,… as shown below: 2,1,… 3𝑠1 −𝑚,𝑚 − 𝑠1⏞ , 𝑅0 2𝑠1,𝑠1, . . . ,3𝑠2 − 𝑚,𝑚 − 𝑠2⏞ 𝑅1 ,2𝑠2,𝑠2, . . . ,3𝑠3 − 𝑚,𝑚 −𝑠3⏞ 𝑅2 … ..2𝑠𝑛,𝑠𝑛, . . . ,3− 𝑚,𝑚 −1⏞ 𝑅𝑛 ,2,1,…3𝑠1 − 𝑚,𝑚 − 𝑠1⏞ , 𝑅𝑛+1 … , (2.1) where 𝑠1 = 𝑠(𝑚). obviously each finite subsequence ‘𝑅𝑖’ has 𝛼(𝑚) terms and it contains exactly one block. hence every subsequence 𝑅𝑖(𝑖 ≥ 1) is a multiple of ‘𝑅0’. therefore, we have the following congruences modulo 𝑚: 𝑅1 = 𝑠1𝑅0 ; 𝑅2 = 𝑠2𝑅0 ; 𝑅3 = 𝑠3𝑅0 ; ⋯ ; 𝑅𝑛−1 = 𝑠𝑛−1𝑅0 ; 𝑅𝑛 = 𝑠𝑛𝑅0. since the first term of 𝑅1 is 𝑚 − 𝑠2 and that of 𝑅0 is 𝑚 − 𝑠1 and we also have 𝑅1 = 𝑠1𝑅0, we get 𝑚 − 𝑠2 = 𝑠1(𝑚 − 𝑠1). if we consider the modulo 𝑚, we get 𝑠2 = 𝑠1 × 𝑠1. according to similar arguments, when considering the modulo blocks of lucas sequence 75 𝑚, we have 𝑠3 = 𝑠2 × 𝑠1,𝑠4 = 𝑠3 × 𝑠1,𝑠5 = 𝑠4 ×𝑠1,⋯ ,𝑠𝑛 = 𝑠𝑛−1 × 𝑠1. therefore, we have 𝑠𝑛 = 𝑠𝑛−1 × 𝑠1 = (𝑠𝑛−2 ×𝑠1) × 𝑠1 = (𝑠𝑛−3 ×𝑠1) × 𝑠1 × 𝑠1 ⋮ = (𝑠𝑛−(𝑛−1) ×𝑠1) × 𝑠1 ×𝑠1 × …×𝑠1⏞ (𝑛−2) 𝑡𝑖𝑚𝑒𝑠 thus, 𝑠𝑛 = 𝑠1 𝑛. now since the order of 𝑠1 is 𝛽(𝑚), we can write single period of 𝐿(𝑚𝑜𝑑 𝑚) as follows: 2,1,3,4,7,…,3𝑠1 − 𝑚,𝑚 − 𝑠1,𝑠1,…,3𝑠1 2 − 𝑚,𝑚 − 𝑠1 2, 𝑠1 2,…,3𝑠1 3 − 𝑚, 𝑚 − 𝑠1 3,… ,3 − 𝑚,𝑚 − 1, 𝑠1 𝛽(𝑚)−1 ,…,2,1. therefore, 𝛽(𝑚) can be interpreted differently as the number of blocks in a single period of 𝐿(𝑚𝑜𝑑 𝑚). it now follows easily that 𝑘𝐿(𝑚) = 𝛼(𝑚)×𝛽(𝑚). following are some interesting consequences which follows from these results. corollary 2.5. 𝐿𝑛×𝛼(𝑚)+𝑟 ≡ (𝑠(𝑚)) 𝑛 × 𝐿𝑟 (𝑚𝑜𝑑 𝑚). proof: from the previous theorem, we have 𝑅𝑛 ≡ 𝑠𝑛𝑅0(𝑚𝑜𝑑 𝑚) and 𝑠𝑛 ≡ 𝑠1 𝑛(𝑚𝑜𝑑 𝑚) . thus, we have 𝑅𝑛 ≡ 𝑠1 𝑛𝑅0 (𝑚𝑜𝑑 𝑚). (2.2) this shows that the 𝑟𝑡ℎ term of 𝑅𝑛 is equal to 𝑠1 𝑛 times the 𝑟𝑡ℎ term of 𝑅0, when considering the modulo 𝑚. also, from the definition of 𝑠(𝑚), an immediate conclusion that would be drawn is 𝑠1 = 2𝐿𝛼(𝑚) when considering modulo 𝑚. therefore, from lemma 1.1 and above arguments, we can say that 𝐿𝑛×𝛼(𝑚)+𝑟 ≡ (𝐿𝛼(𝑚)) 𝑛 × 𝐿𝑟 (𝑚𝑜𝑑 𝑚). this finally gives 𝐿𝑛×𝛼(𝑚)+𝑟 ≡ (𝑠(𝑚)) 𝑛 × 𝐿𝑟 (𝑚𝑜𝑑 𝑚). corollary 2.6. 𝑔𝑐𝑑(𝑚,𝑠𝑖) = 1; for all 𝑖 ≥ 1. proof: from the definition, when considered modulo 𝑚 we have 𝑠𝑛 = 𝑠1 𝑛. therefore, we write 𝑠𝑖 𝛽(𝑚) ≡ (𝑠1 𝑖) 𝛽(𝑚) ≡ (𝑠1 𝛽(𝑚) ) 𝑖 (𝑚𝑜𝑑 𝑚). thus, since r. patel and d. shah 76 𝑠1 𝛽(𝑚) ≡ 1(𝑚𝑜𝑑 𝑚), we have (𝑠1 𝛽(𝑚) ) 𝑖 ≡ 1(𝑚𝑜𝑑 𝑚). this gives 𝑠𝑖 𝛽(𝑚) ≡ 1(𝑚𝑜𝑑 𝑚). now suppose 𝑔𝑐𝑑(𝑚,𝑠𝑖) = 𝑑. then, 𝑑 | 𝑚 and 𝑑 | 𝑠𝑖, which gives 𝑑 | 𝑠𝑖 𝛽(𝑚) . also, 𝑚 | (𝑠𝑖 𝛽(𝑚) − 1). using both together, we have 𝑑 | (𝑠𝑖 𝛽(𝑚) − (𝑠𝑖 𝛽(𝑚) − 1)). this gives, 𝑑 = 1. thus, 𝑔𝑐𝑑(𝑚,𝑠𝑖) = 1. corollary 2.7. 𝑠𝑛 𝑟 ≡ 𝑠𝑛×𝑟(𝑚𝑜𝑑 𝑚). proof: from the definition of 𝑠(𝑚), we have 𝑠𝑛 ≡ 𝑠1 𝑛(𝑚𝑜𝑑 𝑚). then we can write 𝑠𝑛 𝑟 ≡ (𝑠1 𝑛)𝑟 ≡ 𝑠1 𝑛×𝑟 ≡ 𝑠𝑛×𝑟(𝑚𝑜𝑑 𝑚). it now follows that 𝑠𝑛 𝑟 ≡ 𝑠𝑛×𝑟(𝑚𝑜𝑑 𝑚). the following theorem doesn’t seem to give us an immediate idea about 𝐿(𝑚𝑜𝑑 𝑚), but some good results follow. the evidence comes from robinson [2] but admits that morgan wood knew the result in the early 1930’s. theorem 2.8. 𝑘𝐿(𝑚) = gcd(2,𝛽(𝑚))× 𝑙𝑐𝑚[2,𝛼(𝑚)], for 𝑚 > 2. proof: koshy [5] proved that 𝐿𝑛 2 − 𝐿𝑛−1𝐿𝑛+1 = 5(−1) 𝑛. taking 𝑛 = 𝛼(𝑚), we get 𝐿𝛼(𝑚) 2 − 𝐿𝛼(𝑚)−1𝐿𝛼(𝑚)+1 = 5(−1) 𝛼(𝑚). (2.3) now, 𝐿𝛼(𝑚) ≡ 2𝑠(𝑚)(𝑚𝑜𝑑 𝑚), 𝐿𝛼(𝑚)−1 ≡ −𝑠(𝑚)(𝑚𝑜𝑑 𝑚) and 𝐿𝛼(𝑚)+1 ≡ 𝑠(𝑚)(𝑚𝑜𝑑 𝑚). therefore, by (2.2) we have (2𝑠(𝑚))2 −(−𝑠(𝑚))(𝑠(𝑚)) ≡ 5(−1)𝛼(𝑚)(𝑚𝑜𝑑 𝑚). this gives 5(𝑠(𝑚))2 ≡ 5(−1)𝛼(𝑚)(𝑚𝑜𝑑 𝑚). (2.4) thus (𝑠(𝑚)) 2 and (−1)𝛼(𝑚) has same order modulo 𝑚. but the order of −1 is 2 and the order of 𝑠(𝑚) is 𝛽(𝑚) modulo 𝑚. thus, 𝛽(𝑚) gcd(2,𝛽(𝑚)) = 2 gcd(2,𝛼(𝑚)) . thus, 𝑘𝐿(𝑚) = 𝛼(𝑚)𝛽(𝑚) = 𝛼(𝑚) 2gcd(2,𝛽(𝑚)) gcd(2,𝛼(𝑚)) = gcd(2,𝛽(𝑚))× 𝑙𝑐𝑚[2,𝛼(𝑚)], as required. blocks of lucas sequence 77 finally, we calculate the possible values of 𝛽(𝑚). theorem 2.9. 𝛽(𝑚) = 1 or 2 or 4; for any 𝑚 ≥ 2. proof: by above theorem we have 𝜇(𝑚) = gcd(2,𝛽(𝑚))× 𝑙𝑐𝑚[2,𝛼(𝑚)],= (1 𝑜𝑟 2) ×(𝛼(𝑚) 𝑜𝑟 2𝛼(𝑚)). therefore, 𝜇(𝑚) = 𝛼(𝑚) or 2𝛼(𝑚) or 4𝛼(𝑚). thus, we have 𝛽(𝑚) = 1 or 2 or 4; for any 𝑚 ≥ 2. we conclude the paper by noting the following obvious result which is a direct consequence of theorem 2.4 and theorem 2.9. corollary 2.10. 𝑘(𝑚) = 𝛼(𝑚) or 2𝛼(𝑚) or 4𝛼(𝑚). 3. conclusion in this article we had introduced the ‘blocks’ within the period of the lucas sequence and shown that length of any one period of the lucas sequence contains either 1,2 or 4 blocks. references [1] d. d. wall. fibonacci series modulo 𝑚. the american mathematical monthly, 67, 525 – 532, 1960. [2] d. w. robinson. the fibonacci matrix modulo 𝑚. the fibonacci quarterly, 1, 29 – 36, 1963. [3] j. kramer, v. e. jr. hoggatt. special cases of fibonacci periodicity. the fibonacci quarterly, 1:5, 519 – 522, 1972. [4] k. thomas. fibonacci and lucas numbers with applications. john wiley and sons,inc., new york, 2001. r. patel and d. shah 78 [5] r. marc. properties of the fibonacci sequence under various moduli. master’s thesis, wake forest university, 1996. [6] r. p. patel, d. v. shah. periodicity of generalized lucas numbers and the length of its period under modulo 2𝑒. the mathematics today, 33, 67 – 74, 2017. estimation of fuzzy metric spaces from metric spaces ratio mathematica 64 estimation of fuzzy metric spaces from metric spaces senthil kumar pichai* thiruveni packirisamy† abstract in this paper we estimate the new way for analyzing the fuzzy metric spaces from metric spaces using fuzzy fixed point theorem and vice versa. we derive some definitions and theorems for analyzing the metric spaces with new structure of fuzzy metric space using fixed point theorems. also, we have given new examples for fuzzy metric spaces using fixed point theorem. keywords: metric space, fuzzy logic, fixed point theorem. 2020 ams subject classifications: 54e35, 03b52, 47h10 * department of mathematics, rajah serfoji government college (autonomous) (affiliated to bharathidasan university), thanjavur, india; pskmaths@rsgc.ac.in. †department of mathematics, rajah serfoji government college (autonomous) (affiliated to bharathidasan university), thanjavur, india; smile.thiruveni@gmail.com. received on october 10, 2021. accepted on december 15, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.670. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. volume 41, 2021, pp. 64-70 mailto:pskmaths@rsgc.ac.in mailto:smile.thiruveni@gmail.com estimation of fuzzy metric spaces from metric spaces 65 1. introduction in earlier of 1906, maurice freechet provided the concept of metric spaces. in 1922, banach developed a reliable result called banach contraction principle on the basis of fixed point theory [15]. later in 1965, zadeh [10] defined the way of fuzzy set in metric spaces. this fuzzy concept used in various field of engineering, science and technology. this fuzzy concept is used to analyze the complex state as to easy by simple condition and conversion. then several mathematicians gave their various concept towards the concept of fuzzy metric space erceg [4] [5], diamond and kolden [7], george & veeramani [6], gregori and romaguera [11]. the analysis of fuzzy metric spaces was introduced in various way and its topologies developed by many researchers. on the line of this, we frame a new structure to analyze the fuzzy metric spaces from metric spaces using fuzzy fixed point theorem [13] [14]. in this paper, we discuss and give various definition and theorem to estimate the fuzzy metric space from metric space and converse also using fixed point theorem[1][9][12][16]. 2. prefatory in this part, we look back on some basic concepts and results in both metric and fuzzy based metric spaces. definition 2.1: [2] a metric space is given by a set 𝑋 and a distance function d̅ ∶ x × x → r defined on 𝑋 such that 𝑎,𝑏,𝑐 ∈ 𝑋 (𝑖) d̅(𝑎,𝑏) ≥ 0,d̅(𝑎,𝑏) = 0 ↔ 𝑎 = 𝑏 (𝑖𝑖) d̅(𝑎,𝑏) = d̅(𝑏,𝑎) (𝑖𝑖𝑖) d̅(𝑎,𝑐) ≤ d̅(𝑎,𝑏)+ d̅ (𝑏,𝑐) definition 2.2: [10] a fuzzy set 𝐴 in 𝑋 is a function with domain 𝑋 and values in [0, 1]. definition 2.3: [8] a binary operation ∗ ∶ [0,1] 2 → [0,1] is called a continuous triangular norm (called tnorm) if it satisfies the following conditions: (i) * is associative and commutative, (ii) * is continuous, (iii) 𝑝∗1 = 𝑝 for all 𝑝,𝑞,𝑟 ∈ [0,1], (iv) 𝑝∗𝑞 ≤ 𝑟 ∗𝑡 whenever 𝑝 ≤ 𝑟 and 𝑞 ≤ 𝑝 for all 𝑝,𝑞,𝑟,𝑡 ∈ [0,1] senthil kumar pichai and thiruveni packirisamy 66 examples of t-norm are 𝑝∗𝑞 = 𝑝𝑞,𝑝 ∗𝑞 = min{𝑝,𝑞} and 𝑝∗𝑞 = max {𝑝,𝑞}. definition 2.4: [3, 6] the 3-tuple (𝑋,𝐹�̅�,∗) is called a fuzzy metric space if 𝑋 is an arbitrary (non-empty) set, * is a continuous t-norm and 𝐹�̅� is a fuzzy set on 𝑋2 ×[0,∞) satisfying the following conditions, for all 𝑎,𝑏,𝑐 ∈ 𝑋, each t and 𝑢 > 0 (i) 𝐹�̅�(𝑎,𝑏,𝑡) > 0 (ii) 𝐹�̅�(𝑎,𝑏,𝑡) = 0 if and only if 𝑎 = 𝑏, (iii) 𝐹�̅�(𝑎,𝑏,𝑡) ∗𝐹�̅�(𝑏,𝑎,𝑡), (iv) 𝐹�̅�(𝑎,𝑏,𝑡) ∗𝐹�̅�(𝑏,𝑐,𝑢) ≤ 𝐹�̅�(𝑎,𝑐,𝑡 +𝑢), (v) 𝐹�̅�(𝑎,𝑏,°):(0,∞) → [0,1] is continuous. then is 𝐹�̅� called a fuzzy metric on 𝑋. then 𝐹�̅�(𝑎,𝑏,𝑡) denotes the degree of nearness between 𝑎 and 𝑏 with respect to 𝑡. lemma 2.5: (𝑋,𝐹�̅�,∗) is non-decreasing for all 𝑎,𝑏 ∈ 𝑋. 3. main results the main aim of this paper is to estimate the fuzzy metric spaces from any ordinary metric spaces and converse also and justify the banach fixed point. theorem 3.1: let �̅� and 𝐹�̅� are metric and fuzzy metric respectively, so the following diagram 𝑋 × 𝑋 × ℛ́+ 𝐹�̅� ⇒ 𝐼 𝑑𝑝𝑟 ↓ ↑ 𝛽 𝑋 × 𝑋 �̅� ⇒ ℛ́ + figure (1): commutative diagram is commutative. where, �̅�𝑥𝑦:(𝑎,𝑏,𝑡) → (𝑡𝑎,𝑡𝑏), �̅�(𝑡𝑎,𝑡𝑏) → 𝑡𝑦 for some metric �̅�(𝑎,𝑏) = 𝑦 > 0 and 𝛽:(𝑡𝑦) → 1−sin(𝑡𝑦) =: �̂�𝜖𝐼. further more 𝐹�̅� = 𝛽 ° �̅� ° �̅�𝑥𝑦. proof: according to the below equation it is very easy to check that 𝛽 is continuous. sin−1(𝑡𝑦) in ℛ́+ is continuous ⟹ 𝛽 is continuous. now we justify that �̅� ° �̅�𝑥𝑦 = 𝛽 −1 ° 𝐹�̅�. for (a,b,c) ∈ x×x×ℛ́ +, we have, �̅� ° �̅�𝑥𝑦(𝑎,𝑏,𝑡) = �̅�(𝑡𝑎,𝑡𝑎𝑏) = 𝑡𝑦 ≔ 𝑞 > 0. on the other hand, 𝛽−1 °𝐹�̅�(𝑎,𝑏,𝑡) = 𝛽 −1(�̂�) = 𝛽−1(1−sin𝑡𝑦)= sin−1{1−(1−sin𝑡𝑦)} = 𝑡𝑦 therefore, the above diagram (figure 1) is commutative. estimation of fuzzy metric spaces from metric spaces 67 lemma 3.2: let 𝑡1, 𝑡2 ∈ ℛ́ +, if 𝑡1 ≤ 𝑡2, then 𝛽(𝑡1) ≥ 𝛽(𝑡2) and 𝛽(𝑡1 +𝑡2) ≥ max { 𝛽(𝑡1),𝛽(𝑡2)}. proof: if 𝑡1 ≤ 𝑡2, this ⟹ sin𝑡1 ≤ sin𝑡2 ⟹ −sin𝑡1 ≥ −sin𝑡2 ⟹ 1− sin𝑡1 ≥ 1−sin𝑡2 hence, 𝛽(𝑡1) ≥ 𝛽(𝑡2). now, the second declaration is clear. theorem 3.3: let (𝑋, �̅�) be the metric space and (𝑋,𝐹�̅�,∗) is a fuzzy metric space with 𝑝∗𝑞 = max {𝑝,𝑞} for all 𝑝,𝑞 ∈ 𝐼. then for all 𝑎,𝑏,𝑐 ∈ 𝑋,𝑡,𝑞,𝑦 ∈ ℛ́+, we have (𝑋,𝛽 ° �̅� ° �̅�𝑥𝑦,∗) is a fuzzy metric space. proof: we verify the fuzzy metric space conditions (i), (ii), (iii) from the above definition (2.4), for (i) 𝛽 ° �̅� ° �̅�𝑥𝑦(𝑎,𝑏,𝑡) = 𝛽 ° �̅�(𝑡𝑎,𝑡𝑏) = 𝛽 { �̅�(𝑡𝑎,𝑡𝑏)} = �̂� > 0 . for (ii) 𝛽 ° �̅� ° �̅�𝑥𝑦(𝑎,𝑎,𝑡) = 𝛽 ° �̅�(𝑡𝑎,𝑡𝑎) = 𝛽 { �̅�(𝑡𝑎,𝑡𝑎)} = 𝛽(0) = 1 for (iii) 𝛽 ° �̅� ° �̅�𝑥𝑦(𝑎,𝑏,𝑡) = 𝛽 ° �̅�(𝑡𝑎,𝑡𝑏) = 𝛽 { �̅�(𝑡𝑎,𝑡𝑏)} = 𝛽(𝑡𝑦) = �̂�. another side, 𝛽 ° �̅� ° �̅�𝑥𝑦(𝑏,𝑎,𝑡) = 𝛽 ° �̅�(𝑡𝑏,𝑡𝑎) = 𝛽 { �̅�(𝑡𝑏,𝑡𝑎)} = 𝛽(𝑡𝑦) = �̂� > 0 therefore, symmetric condition satisfied. next, we check the condition (iv) 𝛽 ° �̅� ° �̅�𝑥𝑦(𝑎,𝑐,𝑡 +𝑢) = 𝛽 ° �̅� {(𝑡+𝑢)𝑎,(𝑡+𝑢)𝑐) =𝛽 {�̅�(𝑡+𝑢)𝑎,(𝑡+𝑢)𝑐) =𝛽 { �̅�(𝑡+𝑢)𝑟)} =𝛽 (𝑡𝑟 +𝑢𝑟) using the above lemma, 𝛽(𝑡𝑟 +𝑢𝑟) ≥ max{𝛽(𝑡𝑟),𝛽(𝑢𝑟)} =𝛽(𝑡𝑟)∗𝛽(𝑢𝑟) =𝛽{�̅�(𝑡𝑎,𝑡𝑏)}∗𝛽{�̅�(𝑢𝑎,𝑢𝑐)} =𝛽 ° �̅� ° �̅�𝑥𝑦(𝑎,𝑏,𝑡) ∗𝛽 ° �̅� ° �̅�𝑥𝑦(𝑏,𝑎,𝑢) for (v) is trivially true. therefore, (𝑋,𝛽 ° �̅� ° �̅�𝑥𝑦,∗) is a fuzzy metric space. comments: on the other hand we reach the metric space from the fuzzy metric space basis on the above commutative diagram (figure 1). so, if (𝑋,𝛽 ° �̅� ° �̅�𝑥𝑦,∗ ) is a fuzzy metric space, then the associative metric is (𝑋 ,𝛽 −1 ° 𝐹�̅� ° �̅�𝑥𝑦 −1 ). definition 3.4: let (x, �̅�) be a metric space on x, and {𝑎𝑛} be a sequence in x the n is {𝑎𝑛} called converge sequence to some fixed 𝑎 ∈ 𝑋 𝑖𝑓 𝜀 > 0,𝑁 𝜖 ℕ, �̅�(𝑎𝑛,𝑎) < 𝜖 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 > 𝑁 we represent 𝑎𝑛 → a if {𝑎𝑛} converge to 𝑎 ; and {𝑎𝑛} is called a cauchy senthil kumar pichai and thiruveni packirisamy 68 sequence. �̅�(𝑎𝑛,𝑎𝑚) < 𝜖 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛,𝑚 > 𝑁 definition 3.5: let (x, d̅)and(𝑋,𝐹�̅�,∗) are metric and fuzzy metric space on x, respectively. and {𝑎𝑛} is a sequence in 𝑋 then the following is equivalent. (i) {𝑎𝑛} is convergent in the metric space (𝑋,�̅�) (ii) �̅�(𝑎𝑛,𝑎) < 𝜖 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 > 𝑁 (iii) {𝑎𝑛} is convergent in the fuzzy metric space (𝑋,𝛽 ° �̅� ° �̅�𝑥𝑦,∗) (iv) for any 0 < 𝜀 < 1 and 𝑡 > 0 there exists 𝑛 > 𝑁 such that 𝛽 ° �̅� ° �̅�𝑥𝑦(𝑎𝑛,𝑎, 𝑡) > 1−𝜀 definition 3.6: a metric space (𝑋,𝑑̅) is complete if every cauchy sequence in 𝑋 is convergent. definition 3.7: a fuzzy metric space (𝑋,𝛽 ° �̅� ° �̅�𝑥𝑦,∗) is complete if and only if (𝑋,𝑑̅) is complete. in the below mentioned theorem we prove that if any self map has fixed point theorems in the metric space, then we induced fuzzy metric space for the same point and vice versa. we point to mihet & shen et al. [5] for fixed point theorems in the fuzzy metric spaces. theorem 3.8: let (𝑋,𝑑̅) be a complete metric space on 𝑋, suppose the mapping 𝑆:𝑋 → 𝑋 satisfy the contractive condition, thus �̅�(𝑆𝑎,𝑆𝑏) < 𝑘 �̅�(𝑎,𝑏), for all 𝑎,𝑏 ∈ 𝑋,𝑘 ∈ [0,1) is a constant. if 𝑇 has a unique fixed point in 𝑋 with respect to the metric (𝑋,𝑑̅), then 𝑇 has a unique fixed point with respect to the induced fuzzy metric (𝑋,𝛽 ° �̅� ° �̅�𝑥𝑦,∗). proof: suppose that 𝑇 has a unique fixed point in 𝑋 with respect to the metric space (𝑋,𝑑̅). so, we have 𝑑̅(𝑆𝑎,𝑎) = 0 for some 𝑥. therefore 𝐹�̅�(𝑆𝑎,𝑎,𝑡) = 𝛽 ° �̅� ° �̅�𝑥𝑦(𝑆𝑎,𝑎,𝑡) = 𝛽 ° {𝑠(𝑆𝑎),𝑠(𝑆𝑏)} = 𝛽(𝑡𝑦) = 𝛽(0) = 1 this represents that 𝑆𝑎 = 𝑎 with respect to the fuzzy metric space (𝑋,𝛽 ° �̅� ° �̅�𝑥𝑦,∗) if there is another fixed point 𝑏 ∈ 𝑋 then, 𝐹�̅�(𝑎,𝑏, 𝑡) = 𝛽 ° �̅� ° �̅�𝑥𝑦(𝑆𝑎,𝑆𝑏,𝑡) = 𝛽 ° {𝑠(𝑆𝑎),𝑠(𝑆𝑏)} = 𝛽(𝑡𝑦) = 𝛽(0) = 1 and therefore, 𝑎 = 𝑏. theorem 3.9: let �̅� and 𝐹�̅� are metric and fuzzy metric respectively and commutative. where, �̅�𝑥𝑦:(𝑎,𝑏,𝑡) → (𝑡𝑎,𝑡𝑏), �̅�(𝑡𝑎,𝑡𝑏) → 𝑡𝑦 for some metric �̅�(𝑎,𝑏) = 𝑦 > 0 and 𝛽:(𝑡𝑦) → 1−cos(𝑡𝑦) =:�̂�𝜖𝐼. further more 𝐹�̅� = 𝛽 ° �̅� ° �̅�𝑥𝑦. estimation of fuzzy metric spaces from metric spaces 69 proof for this theorem is similar to the proof of theorem 3.1. example 3.10: let �̅� and 𝐹�̅� are metric and fuzzy metric respectively. �̅�𝑥𝑦:(𝑎,𝑏,𝑡) → (𝑡𝑎,𝑡𝑏), �̅�(𝑡𝑎,𝑡𝑏) → 𝑡𝑦 for some metric �̅�(𝑎,𝑏) = 𝑦 > 0 and 𝛽:(𝑡𝑦) → 1−𝑠𝑖𝑛2(𝑡𝑦) =:�̂�𝜖𝐼. furthermore 𝐹�̅� = 𝛽 ° �̅� ° �̅�𝑥𝑦. and we prove it is commutative. solution: according to the below equation it is very easy to check that 𝛽 is continuous. sin2(𝑡𝑦) in ℛ́+ is continuous ⟹ 𝛽 is continuous. now we justify that �̅� ° �̅�𝑥𝑦 = 𝛽 −1 ° 𝐹�̅�. for (a,b,c) ∈ x×x×ℛ́ +, we have, �̅� ° �̅�𝑥𝑦(𝑎,𝑏,𝑡) = �̅�(𝑡𝑎,𝑡𝑎𝑏) = 𝑡𝑦 ≔ 𝑞 > 0. on the other hand, 𝛽−1 °𝐹�̅�(𝑎,𝑏,𝑡) = 𝛽 −1(�̂�) = 𝛽−1(1−𝑠𝑖𝑛2(𝑡𝑦))= sin−1{√1−(1−𝑠𝑖𝑛2(𝑡𝑦)) } = 𝑡𝑦 therefore, it is commutative. 4. conclusion in this article we design a new structure to estimate the fuzzy metric space with the help of metric space using fuzzy fixed point theorems and converse also. we discussed the various definitions and provided the proof for the same definitions with new structure using fuzzy fixed point theorems. we have also given some examples which satisfies the condition of our new structure basis on fixed point theorem. this paper makes away to analysis the applications of fuzzy metric space in various field of engineering. references [1] saleh omran, h.s. al-saadi (2017), some notes on metric and fuzzy metric spaces, international journal of advanced and applied sciences, 4(5); 4143. 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[15] banach s (1922), sur les operations dans les ensembles abstraitsetleur application aux equations integrals, fundamenta mathematicae, vol.3, pp.133-181 [16] krishnakumar r, damodharan d (2016), fixed point theorems in normal cone metric spaces, international journal of math.sci & engg.appls.10, 213-224. approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica 28 (2015) 15-30 issn: 1592-7415 15 approach of the value of an annuity when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions salvador cruz rambaud 1 , fabrizio maturo 2 , ana maría sánchez pérez 3 1 department of economics and business, university of almería, spain, scruz@ual.es 2 department of management and business administration, university of chieti-pescara, italy, f.maturo@unich.it 3 department of economics and business, university of almería, spain, amsanchez@ual.es (the authors are entered in alphabetical order by their last name) abstract this paper proposes an expression of the value of an annuity with payments of 1 unit each when the interest rate is random. in order to attain this objective, we proceed on the assumption that the non-central moments of the capitalization factor are known. specifically, to calculate the value of these annuities, we propose two different expressions. first, we suppose that the random interest rate is normally distributed; then, we assume that it follows the beta distribution. a practical application of these two methodologies is also implemented using the r statistical software. keywords: annuity; random interest rate; non-central moments. 2010 ams subject classification: 91g30; 46n30; 65c60; 91g70; 62p05. doi: 10.23755/rm.v28i1.25 mailto:scruz@ual.es mailto:f.maturo@unich.it http://dx.doi.org/10.23755/rm.v28i1.25 cruz rambaud s., maturo f. and sánchez pérez a.m. 16 1 introduction this study aims to determine an approximate expression for the present, or final, value of an annuity when the interest rate is random. in the context of annuities assessment, the interest rate has a great relevance because even small changes may cause major changes in the total annuity value. thus, the determination of the value of the interest rate should be carried out as accurately as possible. the traditional approach treats interest rates deterministically; indeed, in contexts of certainty, the use of a single possible value for each period may be enough [8]. however, for those operations developed in uncertain environments, it is more reasonable the formulation of potential scenarios, which are subsequently reduced to one by statistical treatment [2]. the determination of the interest rate value must be based on the current situation, as well as on its possible future evolution, of both companies and environment. in this way, if prospects are unfavorable, interest rates must be higher, compared to more favorable situations, and hence to reduce the operation value as a consequence of the risk attached to it. however, in most cases, determining the interest rate of a financial operation is subject to the propensity/aversion to risk of the agent to be responsible for the assessment. in this sense, the adopted interest rate would be affected by a degree of subjectivity that may over/undervalue the project [7]. in this paper, we consider the interest rate as a random variable that is represented as x. therefore, the capitalization factor, i1 , is also a random variable represented as u. obviously, it is verified that xu  1 , thus, the relationship between the mean and standard deviation of both variables is as follows: as a result, if x is defined in an interval ],[ ba , then u will be in the interval ]1,1[  ba . henceforth, when the mean and standard deviation are mentioned we will refer, unless otherwise specified, to the random variable u. in this case, the final value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate x, would be the following random variable: thus, its expected value is: xu   1 and xu   . 12 1 1    n un uuus  . (1) approach of the value of an annuity when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions 17 on the other hand, the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate x, would be: being )( r r ue the moment of order r, with respect to the origin, of the random variable u; hence, if the random variable is discrete, it adopts the following expression [1]: being i p the probability that the random variable takes the value iu . in the continuous case, the expression of the moment of order r is: for all values of r, being )(uf the density function of the random variable u. as indicated, this paper proposes a mathematical expression of the final value of an annuity, immediate or due; specifically, we compute it using a random interest rate and suppose that the non-central moments of the capitalization factor are known. section 2 shows the case of interest rates following the normal distribution. section 3 takes into account the beta distribution, as an example of distribution with finite range. section 4 shows a practical application using the r statistical software. lastly, the conclusions are presented. 2 the expression of the final value of an annuity when the interest rate follows a normal distribution the successive non-central moments of order r, with respect to the normal distribution, can be computed according to its mean  and variance 2  [4]: .1 )()()()1()( 12 12 1      n n un ueueueese    ,)()()()( 2 2 1 n n un ueueuese     (2)    k i r ii r r upue 1 )( , (3)  max min d)()( u u r r uuufue , (4) cruz rambaud s., maturo f. and sánchez pérez a.m. 18  1 0  ;   1 ;  22 2   ;  23 3 3  ;  4224 4 36   ;  4235 5 1510   ;  642246 6 154515   ;  643257 7 10510521   ;  86244268 8 10542021028   .  therefore, the final value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate) that is the sum of the n first non-central moments   1 0 n r r  , is composed of the following partial sums:  . 1 1 1 1 0 12           nn r rn this can be written as         1 0 0 0 n r r r  .  . 2 )28211510631( 1 2 22654322            n r r r   the coefficients of successive powers of , in parentheses, are the numbers in red in the following tartaglia’s triangle (figure 1): figure 1: tartaglia’s triangle. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 approach of the value of an annuity when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions 19  . 4 3)70351551(3 1 4 444324            n r r r   the coefficients of successive powers of , enclosed in the parentheses, are the numbers in green of the previous tartaglia’s triangle.  , 6 15)2871(15 1 6 6626            n r r r   whose coefficients are in blue.  and so forth. in short, the sum of the n first non-central moments is:                    )2/)1((e 1 1 2 22 1 0 2 )12(531 1 1 n k n kr krk nn r r k r k      , which can also be written as follows: . 2)!1(2 )!12( 1 1 )2/)1((e 1 1 2 22 1 1 0                       n k n kr krk k nn r r k r k k     (5) this method is used to calculate the final value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuityimmediate), with a random interest rate. whereas, the calculation of the final value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), has the following expression: . 2 )12(531 1 1 )2/(e 1 2 22 1                 n k n kr krk nn r r k r k      (6) in equations (5) and (6), the function )(e x represents the integer part of x. to carry out the calculations in a comfortable and orderly manner, we propose to refer to table 1. cruz rambaud s., maturo f. and sánchez pérez a.m. 20 3 the expression of the final value of an annuity when the interest rate follows a beta distribution the best known random variable with a bounded range is the beta distribution. the expression of the non-central moments of the standard beta distribution of parameters  and  is the following ( nr  ) [3]: . )1()1)(( )1()1( )()( )()(       r r r r r        (7) in this case, it is not feasible to give a closed expression of the sum of the n first non-central moments, but, having in mind that )()1(   , we can write the following recurrence relation [5]: rr r r        1 . (8) tartaglia’s triangle exponents of  0 1 2 3 4 1 0  1 1 1  1 2 1 2  0  1 3 3 1 3  1  1 4 6 4 1 4  2  0  1 5 10 10 5 1 5  3  1  1 6 15 20 15 6 1 6  4  2  0  1 7 21 35 35 21 7 1 7 5  3  1  1 8 28 56 70 56 28 8 1 8  6  4  2  0                      0 2  4  6  8  1 1 3 15 105      sum of all products approach of the value of an annuity when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions 21 table 1. tabular organization for calculations (the number that occupies the place ),( sr in tartaglia’s triangle is equal to the sum of those in places )1,1(  sr and ),1( sr  . however, it should be considered that the above mentioned moments refer to the standard beta distribution, z, of parameters,  and  , that is, with range ].1,0[ furthermore, to obtain the moments r corresponding to the distribution u without normalize, that is to say, the beta distribution of parameters  and , with range ],[ ba : ,)( zabau  (9) it is necessary to consider the relationship between its moment-generating functions [6]: ).)((e)()( )( tabmtmtm z a t zabau   (10) therefore, having in mind the expression of the nth derivative of a product of functions, we can write: .))(()(e ))(( d d e d d )( d d 0 ) 0                      r k kr z kra tk r k zkr kr a t k k ur r tabmaba k r tabm ttk r tm t therefore,             r k kr krk t ur r r aba k r tm t 00 )()( d d  . (11) 4 calculation of the value of an annuity, with payments of 1 unit each: an r application next, we are going to obtain the final value of an annuity, with payments of 1 unit each for five years through the different expressions developed in this work. in its calculus we consider that the payments are made at the end, or the cruz rambaud s., maturo f. and sánchez pérez a.m. 22 beginning, of each period. present value calculation has been omitted provided it can be carried out similarly. given that in this work it has been contemplate that non-central moments of the capitalization factor are known, two possible options have been considered, where discount rate, x, follows:  a normal distribution;  a beta distribution. discount rate with a normal distribution to estimate the mean and variance of the normal distribution, we consider euribor’s data containing the estimated annual euribor of different banks (table 2), available at http://www.emmi-benchmarks.eu/euribor-org/euribor-rates.html. specifically, the euribor at 12 months is considered on 27/07/2016. estimates euribor in 1 year (%) bnp-paribas 0.00 banca monte dei paschi di siena 0.05 banco bilbao vizcaya argentaria 0.05 banco santander 0.06 banque et caisse d'épargne de l'état 0.06 barclays bank 0.02 belfius 0.06 cecabank 0.05 caixa geral de depósitos 0.04 caixabank s.a. 0.05 crédit agricole s.a. 0.03 dz bank 0.06 deutsche bank 0.04 hsbc france 0.05 ing bank 0.08 intesa sanpaolo 0.05 table 2: euribor distribution at 27/07/2016. source: http://www.emmi-benchmarks.eu/euribor-org/euribor-rates.html. to enter the data in the r environment, it is necessary to create a vector as follows: http://www.emmi-benchmarks.eu/euribor-org/euribor-rates.html http://www.emmi-benchmarks.eu/euribor-org/euribor-rates.html approach of the value of an annuity when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions 23 >data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,-0.03,0.06,0.04,-0.05,-0.08,-0.05,-0.12,-0.03,-0.05,-0.04,-0.06) to check for normality of the data, we need the r package “tseries”; thus, the jarque-bera test is implemented: > library(tseries) > jarque.bera.test(data) we can accept the normality of the data because the p-value is greater than 0.05 (x-squared = 3.559, df = 2, p-value = 0.1687). following a preliminary analysis, we obtain 0.0442 and 0.0332 using the following scripts: > mean(data) > sd(data) the expressions formulated in section 2 allow computing the expected final value when the expression of the non-central moments of the capitalization factor is known. now, we suppose to compute the final value of an annuity, with payments of 1 unit each for five years using the estimated mean and variance. thus, if the payments of the annuity are at the end of each period, the final value is: 4.958541. 4 0    r r  it is possible to check this result building a function to compute the sum of the non-central moments of the normal distribution for annuities whose payments are at the end of each period, with a duration of k years (however, firstly, we need to load the “moments” library): >library(moments) >sum_k_moments_post=function(data,k){ app.moments_post=rep(na,k) for (i in 0:(k-1)) app.moments_post[i+1]=moment(data, central = false, absolute = false, order =i) sum_moments_post=sum(app.moments_post)+(k-1) return(sum_moments_post) } >sum_k_moments_post(data,5) cruz rambaud s., maturo f. and sánchez pérez a.m. 24 instead, in case the final value of an annuity, with payments of 1 unit each at the beginning of every year, for five years, we obtain: 4.958539. 5 1    r r  also in this case, we can check the result by creating a function to compute the sum of the non-central moments of the normal distribution for annuities whose payments are at the beginning of each period, with a duration of k years: > sum_k_moments_ant=function(data,k){ app.moments_ant=rep(na,k) for (i in 1:k) app.moments_ant[i]=moment(data, central = false, absolute = false, order =i) sum_moments_ant=sum(app.moments_ant)+k return(sum_moments_ant) } >sum_k_moments_ant(data,5) using the formulation proposed in equation 5 (annuities with payments at the end of each period) and equation 6 (annuities with payments at the beginning of each period), we reach the same results (replacing the values of the mean and the standard deviation, it is simple to demonstrate this identity). discount rate with a beta distribution because the beta distribution is suitable to approximate also the data of table 2, we refer to the same data of euribor with the aim to compare it with the results obtained in the previous paragraph. at this purpose we load the data of table 2 and the r packages “actuar” and “envstats” as follows: >library(actuar) >library(envstats) then, we create a function to normalize data and we apply it to the data of table 2 as follows: >nor=function(x){(x-min(x))/(max(x)-min(x))} >data2=nor(data) afterwards, we estimate the shapes of the beta distribution with the function “ebeta”: >ebeta(data2, method = "mle") approach of the value of an annuity when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions 25 following this approach, we get the shapes parameters a = 2.394501 and b = 2.66557. then, we build a function to compute the mean and the standard deviation: >mean_beta=function(a,b){a/(a+b)} >mean_beta(2.394501,2.665577) 0.4732142 >var_beta=function(a,b){a*b/((a+b)^2*(a+b+1))} >var_beta(2.394501,2.665577) 0.0411352 as known, a generic moment of the standard beta distribution is given by: ( ) ( ) r r r       (12) where ).1(1)()(  rr   to compute the moments of the standard beta distribution using this method, we need a function to calculate the factorial: >fattoriale_crescente=function(n,f){n*factorial(x=n+f-1)/factorial(x=n)} in this way, we can compute the moments as: >momento_beta=function(a,b,f){fattoriale_crescente(a,f)/fattoriale_cresce nte((a+b),f)} where a and b are the shapes parameters and f represents the order of the moment. using these codes, it is simple to calculate the sum of the moments of the standard beta distribution. however, we need the non-central moments of the original beta distribution (without normalize). at this purpose, we build a function to perform equation (11) and obtain the non-central moments of the original beta distribution. we set a as the lower bound of our interest rates, b as the upper bound of our interest rates, c and d as the parameters of the standard beta distribution, n as the number of years: >c=2.394501 >d=2.665577 cruz rambaud s., maturo f. and sánchez pérez a.m. 26 >a=-0.12 >b=0.04 >n=5 >momento_nn_normalizzato=function(n,a,b,c,d){for(k in 0:n) { m=mbeta(n-k, c, d) moment_nn_norm=sum( choose(n,k)*(a^k)*((b-a)^(n-k))*m )} return(moment_nn_norm)} afterwards, we present two functions: the first one computes the final value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate); the second one calculates the final value of an npayment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due). the first function is built as follows: >sum_n_moments_non_norm_beta_pag_anticip=function(n,a,b,c,d){ app=rep(na,n) for (i in 1:n) app[i]=(momento_nn_normalizzato(i,a,b,c,d)+1) return(sum(app))} >sum_n_moments_non_norm_beta_pag_anticip(5,a,b,c,d) thus, the final value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate) for five years is: 5 1 4.892854 r r     . the second function is provided by the following code: >sum_n_moments_non_norm_beta_pag_post=function(n,a,b,c,d){ app2=rep(na,n) for (i in 0:(n-1)) app2[i+1]=momento_nn_normalizzato(i,a,b,c,d) app2[2:n]=app2[2:n]+1 return(sum(app2))} >sum_n_moments_non_norm_beta_pag_post(5,a,b,c,d) therefore, with our data, the final value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuityimmediate) for five years is: 4 0 4.892879 r r     . approach of the value of an annuity when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions 27 5 conclusions in this paper we have presented two methodologies to obtain the value of an annuity whose discount rate is a variable known in terms of random. once the expected value of the discount rate has been analyzed through the non-central moments of the discount factor, the expression for determining the expected final value of an n-payment annuity has been deduced. specifically, the theoretical development of this methodology has been carried out in two different ways: by supposing that the interest rate follows a normal distribution, and considering that it follows a beta distribution. furthermore, we provided the code to reproduce our results with the r statistical software (available in appendix 1). our results show slight differences between the estimates of the same data, approximating these with different distributions. this shows how the choice of the distribution of the approximation of data is important for the calculation of the value of an annuity when interest rates are represented by random variables. cruz rambaud s., maturo f. and sánchez pérez a.m. 28 appendix 1: replication material data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,-0.03,0.06,0.04,-0.05,-0.08,-0.05,-0.12,-0.03,-0.05,-0.04,-0.06) library(tseries) jarque.bera.test(data) mean(data) sd(data) library(moments) sum_k_moments_post=function(data,k){ app.moments_post=rep(na,k) for (i in 0:(k-1)) app.moments_post[i+1]=moment(data, central = false, absolute = false, order =i) sum_moments_post=sum(app.moments_post)+(k-1) return(sum_moments_post)} sum_k_moments_post(data,5) sum_k_moments_ant=function(data,k){ app.moments_ant=rep(na,k) for (i in 1:k) app.moments_ant[i]=moment(data, central = false, absolute = false, order =i) sum_moments_ant=sum(app.moments_ant)+k return(sum_moments_ant)} sum_k_moments_ant(data,5) library(actuar) library(envstats) nor=function(x){(x-min(x))/(max(x)-min(x))} data2=nor(data) ebeta(data2, method = "mle") mean_beta=function(a,b){a/(a+b)} mean_beta(2.394501,2.665577) approach of the value of an annuity when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions 29 var_beta=function(a,b){a*b/((a+b)^2*(a+b+1))} var_beta(2.394501,2.665577) fattoriale_crescente=function(n,f){n*factorial(x=n+f-1)/factorial(x=n)} momento_beta=function(a,b,f){fattoriale_crescente(a,f)/fattoriale_crescente ((a+b),f)} momento_nn_normalizzato=function(n,a,b,c,d){ for(k in 0:n) { m=mbeta(n-k, c, d) moment_nn_norm=sum( choose(n,k)*(a^k)*((b-a)^(n-k))*m ) } return(moment_nn_norm)} sum_n_moments_non_norm_beta_pag_anticip=function(n,a,b,c,d){app=rep (na,n) for (i in 1:n) app[i]=(momento_nn_normalizzato(i,a,b,c,d)+1) return(sum(app))} sum_n_moments_non_norm_beta_pag_anticip(5,a,b,c,d) sum_n_moments_non_norm_beta_pag_post=function(n,a,b,c,d){ app2=rep(na,n) for (i in 0:(n-1)) app2[i+1]=momento_nn_normalizzato(i,a,b,c,d) app2[2:n]=app2[2:n]+1 return(sum(app2))} sum_n_moments_non_norm_beta_pag_post(5,a,b,c,d) cruz rambaud s., maturo f. and sánchez pérez a.m. 30 bibliography [1] calot, g. (1974). curso de estadística descriptiva. madrid: ed. paraninfo. [2] cruz rambaud, s. and valls martínez, m.c. (2002). “la determinación de la tasa de actualización para la valoración de empresas”. análisis financiero, 87, 72-85. [3] fisz, m. (1963). probability theory and mathematical statistics, 3rd edition. new york: john wiley and sons, inc. [4] mood, a.m.; graybill, f.a. and boes, d.c. (1974). introduction to the theory of statistics, 3rd edition. new york: mcgraw hill. [5] rice, j.a. (1995). mathematical statistics and data analysis (2nd ed.). california: ed. duxbury press. [6] spiegel, m.r. (1975). probability and statistics. united states of america: ed. mcgraw-hill. [7] suárez suárez, a.s. (2005). decisiones óptimas de inversión y financiación en la empresa. madrid: ed. pirámide. [8] villalón, j.g.; martínez barbeito, j. and seijas macías, j.a. (2009). “sobre la evolución de los tantos de interés”. xvii jornadas de asepuma y v encuentro internacional, 17, 1, 502. approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 45, 2023 pell even sum cordial labeling of graphs christina mercy a* tamizh chelvam t† abstract let𝐺 = (𝑉, 𝐸) be a simple graph and let 𝑃𝑖 be pell numbers. for a bijection𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃|𝑉|−1}, assign the label 1 for the edge 𝑒 = 𝑢𝑣 if 𝑓(𝑢) + 𝑓(𝑣) is even and label 0 otherwise. then 𝑓 is said to be a pell even sum cordial labeling of 𝐺 if |𝑒𝑓 (0) − 𝑒𝑓 (1)| ≤ 1 where 𝑒𝑓 (0) and 𝑒𝑓 (1) denote the number of edges labeled with 0 and 1 respectively. if any graph admits pell even sum cordial labeling, it is called pell even sum cordial graph. in this study, we show that star, comb, bistar, jewel, crown, bipartite graph 𝐾𝑚,𝑚, flower graph, helm, wheel, triangular book, 𝐾2 + 𝑚𝐾1 are pell even sum cordial. keywords: cordial labeling, pell numbers, pell even sum cordial labeling. 2010 ams subject classification: 05c78‡. *research scholar, department of mathematics, manonmaniam sundaranar university, tirunelveli-627 012, tamilnadu, india. e-mail: mercyudhayan@gmail.com. †csir emeritus scientist, department of mathematics, manonmaniam sundaranar university, tirunelveli-627 012, tamilnadu, india. ‡ received on july 10, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm. v45i0.1034. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 304 christina mercy a and tamizh chelvam t 1. introduction by a graph 𝐺 = (𝑉, 𝐸) we mean a finite, undirected simple graph. we refer to harary [5] for graph theory concepts and refer gallian [4] for the literature on graph labeling. the pell numbers are defined by the recurrence relation 𝑃𝑛 = 2𝑃𝑛−1 + 𝑃𝑛−2 where 𝑃0 = 0 and 𝑃1 = 1. the pell number sequence is given as 0,1,2,5,12, …cahit [2] is credited for inventing cordial labeling. shiama [11] defined pell labeling and shown that path, cycle, star, double star, coconut tree, bistar and 𝐵𝑚,𝑛,𝑘 are pell graphs. muthu ramakrishnan and sutha[8] proposed pell graceful labeling as an extension of fibonacci graceful labeling and demonstrated that cycle, path, olive tree, comb graph are pell graceful graphs whereas complete graph and wheel graphs are not pell graceful. indira et al. [6] suggested some algorithms for the existence of pell labeling in quadrilateral snake, extended duplicate graph of quadrilateral graph. sriram et al. [12] investigated the pell labeling for the joins of square of a path. muthu ramakrishnan and sutha [7] also demonstrated that bistar, subdivision of bistar, caterpillar graphs, jelly fish graph, star graph, coconut tree are pell graceful. sharon philomena and thirusangu [10] shown that < 𝐾1,𝑛 ∶ 2 > is a pell graph. avudainayaki and selvam [1] shown that the extended duplicate graph of arrow graph and splitting graph of path admits harmonious and pell labeling. celine mary et al. [3] demonstrated through an algorithm that inflation of alternate triangular snake graph of odd length in which the alternate block starts from the second vertex is a pell graph. muthu ramakrishnan and sutha [9] proposed the pell square graceful labeling and proved that subdivision of the edges of a path𝑃𝑛 in 𝑃𝑛 ⊙ 𝐾1, < 𝑆𝑛: 𝑚 >, olive tree, twig graph are pell square graceful. inspired by the concepts discussed above, the pell even sum cordial labeling is being introduced here. it is defined as follows. definition 1.1 let𝐺 = (𝑉, 𝐸) be a simple graph and let 𝑃𝑖 be pell numbers. for a bijection𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃|𝑉|−1}, assign the label 1 for the edge 𝑒 = 𝑢𝑣 if 𝑓(𝑢) + 𝑓(𝑣) is even and label 0 otherwise. then 𝑓 is said to be a pell even sum cordial labeling of 𝐺 if |𝑒𝑓 (0) − 𝑒𝑓 (1)| ≤ 1 where 𝑒𝑓 (0) and 𝑒𝑓 (1) denote the number of edges labeled with 0 and 1 respectively. if any graph admits pell even sum cordial labeling, it is called pell even sum cordial graph. definition 1.2 the comb 𝑃𝑛 ⊙ 𝐾1 is the graph created by adding a pendent edge to each vertex of a path. definition 1.3 the bistar 𝐵𝑛,𝑛 is the graph obtained by joining the apex vertices of two copies of 𝐾1,𝑛. definition 1.4 the jewel graph 𝐽𝑛 is a graph with vertex set 𝑉(𝐽𝑛 ) = {𝑢, 𝑥, 𝑣, 𝑦, 𝑣𝑖 : 1 ≤ 𝑖 ≤ 𝑛} and the edge set 𝐸(𝐽𝑛) = {𝑢𝑥, 𝑣𝑥, 𝑢𝑦, 𝑣𝑦, 𝑥𝑦, 𝑢𝑣𝑖 , 𝑣𝑣𝑖 : 1 ≤ 𝑖 ≤ 𝑛}. 305 pell even sum cordial labeling of graphs definition 1.5 the crown 𝐶𝑛 ⊙ 𝐾1 is the graph obtained from a cycle by attaching a pendent edge to each vertex of the cycle. definition 1.6 a complete bipartite is a graph whose vertices can be partitioned into two subsets 𝑉1 and 𝑉2 such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. definition 1.7 the helm is the graph obtained from a wheel graph by adjoining a pendent edge at each node of the cycle. definition 1.8 the flower 𝐹𝑙𝑛 is the graph obtained from a helm by attaching each pendent vertex to the apex of the helm. definition 1.9 the join of two graphs 𝐺1 and 𝐺2 is denoted by 𝐺1 + 𝐺2 and whose vertex set is 𝑉(𝐺1 + 𝐺2) = 𝑉(𝐺1) ∪ 𝑉(𝐺2) and edge set 𝐸(𝐺1 + 𝐺2) = 𝐸(𝐺1) ∪ 𝐸(𝐺2) ∪ {𝑢𝑣 ∶ 𝑢 ∈ 𝐺1, 𝑣 ∈ 𝐺2}. definition 1.10 one edge union of cycles of same length is called a book. the common edge is called the base of the book. if we consider 𝑡 copies of cycles of length 𝑛 ≥ 3, then the book is denoted by 𝐵𝑛 𝑡 . if 𝑛 = 3,4,5 or 6, the book 𝐵 is called book with triangular, rectangular, pentagonal or hexagonal pages respectively. definition 1.11 the wheel 𝑊𝑛 can be defined as the graph join 𝐾1 + 𝐶𝑛−1. definition 1.12 𝐾1,kis a tree with one internal node and 𝑘 leaves. 2. pell even sum cordial graphs in this section, we prove that star, comb, bistar, jewel, crown, bipartite graph 𝐾𝑛,𝑛, flower graph, helm, wheel, triangular book, 𝐾2 + 𝑚𝐾1 are pell even sum cordial. theorem 2.1. for 𝑛 ≥ 2, the comb is pell even sum cordial graph proof. recall that comb 𝐺 = (𝑉, 𝐸) is a graph obtained from the path with vertices 𝑢1, 𝑢2, … , 𝑢𝑛 by joining a vertex 𝑣𝑖 to each 𝑢𝑖 where 1 ≤ 𝑖 ≤ 𝑛. actually |𝑉(𝐺)| = 2𝑛 and |𝐸(𝐺)| = 2𝑛 − 1. consider 𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃2𝑛−1} defined by 𝑓(𝑢𝑖 ) = 𝑃2𝑖−1 for 1 ≤ 𝑖 ≤ 𝑛 and 𝑓(𝑣𝑖 ) = 𝑃2𝑖−2 for 1 ≤ 𝑖 ≤ 𝑛 . the induced edge labels are given 𝑓 ∗(𝑢𝑖 𝑢𝑖+1) = 1 for 1 ≤ 𝑖 ≤ 𝑛 − 1 and 𝑓 ∗(𝑢𝑖 𝑣𝑖 ) = 0 for 1 ≤ 𝑖 ≤ 𝑛. from this, 𝑒𝑓 (0) = 𝑛, 𝑒𝑓 (1) = 𝑛 − 1 and so |𝑒𝑓 (0) − 𝑒𝑓 (0)| ≤ 1. therefore, for 𝑛 ≥ 2, the comb is a pell even sum cordial graph. theorem 2.2.for 𝑛 ≥ 2, the bistar 𝐵𝑚,𝑚 is a pell even sum cordial. proof. consider the bistar 𝐺 = 𝐵𝑚,𝑚. then 𝑉(𝐺) = {𝑢, 𝑣, 𝑢𝑖 , 𝑣𝑖 : 1 ≤ 𝑖 ≤ 𝑚}, 𝐸(𝐺) = {𝑢𝑣, 𝑢𝑢𝑖 , 𝑣𝑣𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑚}, |𝑉(𝐺)| = 2𝑚 + 2 and |𝐸(𝐺)| = 2𝑚 + 1. consider 306 christina mercy a and tamizh chelvam t 𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃2𝑚+1} defined by 𝑓(𝑢) = 𝑃0, 𝑓(𝑣) = 𝑃1, 𝑓(𝑢𝑖 ) = 𝑃𝑖+1 for 1 ≤ 𝑖 ≤ 𝑚 − 1 and 𝑓(𝑣𝑖 ) = 𝑃𝑚+𝑖 for 1 ≤ 𝑖 ≤ 𝑚. the induced edge labels are given by 𝑓 ∗(𝑢𝑣) = 0. for 1 ≤ 𝑖 ≤ 𝑚 − 1, 𝑓 ∗(𝑢𝑢𝑖 ) = { 1 𝑖𝑓 𝑖 ≡ 1 (𝑚𝑜𝑑 2); 0 𝑖𝑓 𝑖 ≡ 0 (𝑚𝑜𝑑 2). for 1 ≤ 𝑖 ≤ 𝑚, 𝑓 ∗(𝑣𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≡ 1 (𝑚𝑜𝑑 2); 0 𝑖𝑓 𝑖 ≡ 0 (𝑚𝑜𝑑 2). when 𝑚 is odd, 𝑒𝑓 (1) = 𝑚 + 1, 𝑒𝑓 (0) = 𝑚 and on the otherhand, when 𝑚 is even, 𝑒𝑓 (1) = 𝑚 , 𝑒𝑓 (0) = 𝑚 + 1. therefore, for 𝑛 ≥ 2, bistar 𝐵𝑚,𝑚 is pell even sum cordial. theorem 2.3. for 𝑛 ≥ 1, the jewel graph 𝐽𝑛 is pell even sum cordial. proof. consider the jewel graph 𝐺 = 𝐽𝑛 . here 𝑉(𝐺) = {𝑢, 𝑣, 𝑥, 𝑦, 𝑢𝑖 : 1 ≤ 𝑖 ≤ 𝑛}, 𝐸(𝐺) = {𝑢𝑥, 𝑢𝑦, 𝑥𝑦, 𝑥𝑣, 𝑦𝑣, 𝑢𝑢𝑖 , 𝑣𝑣𝑖 : 1 ≤ 𝑖 ≤ 𝑛}, |𝑉(𝐺)| = 𝑛 + 4 and |𝐸(𝐺)| = 2𝑛 + 5. let 𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃𝑛+3} be defined by 𝑓(𝑢) = 𝑃0, 𝑓(𝑣) = 𝑃1, 𝑓(𝑥) = 𝑃2 and 𝑓(𝑢𝑖 ) = 𝑃𝑖+3 for 1 ≤ 𝑖 ≤ 𝑛. then the induced edge labels are given by,𝑓 ∗(𝑢𝑣) = 1, 𝑓 ∗(𝑢𝑦) = 1, 𝑓 ∗(𝑥𝑦) = 0, 𝑓 ∗(𝑣𝑥) = 0 and 𝑓 ∗(𝑣𝑦) = 1 for 1 ≤ 𝑖 ≤ 𝑛, 𝑓 ∗(𝑢𝑢𝑖 ) = { 1 𝑖𝑓 𝑖 ≡ 0 (𝑚𝑜𝑑 2); 0 𝑖𝑓 𝑖 ≡ 1 (𝑚𝑜𝑑 2). 𝑓 ∗(𝑣𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≡ 1 (𝑚𝑜𝑑 2); 0 𝑖𝑓 𝑖 ≡ 0 (𝑚𝑜𝑑 2). from the above, 𝑒𝑓 (0) = 𝑛 + 3, 𝑒𝑓 (1) = 𝑛 + 2 and so|𝑒𝑓 (0) − 𝑒𝑓 (0)| ≤ 1. therefore, for 𝑛 ≥ 1, the jewel graph 𝐽𝑛 is pell even sum cordial. theorem 2.4 for 𝑛 ≥ 3, the crown 𝐶𝑛 ⊙ 𝐾1 is pell even sum cordial. proof. consider the crown 𝐺 = 𝐶𝑛 ⊙ 𝐾1. here 𝑉(𝐺) = {𝑢𝑖 , 𝑣𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛} and 𝐸(𝐺) = {𝑢𝑖 𝑢𝑖+1: 1 ≤ 𝑖 ≤ 𝑛 − 1; 𝑢𝑛𝑢1, 𝑢𝑖 𝑣𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛}, |𝑉(𝐺)| = 2𝑛 = |𝐸(𝐺)|. consider 𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃2𝑛−1} defined by 𝑓(𝑢𝑖 ) = 𝑃2𝑖−2 for 1 ≤ 𝑖 ≤ 𝑛 and 𝑓(𝑣𝑖 ) = 𝑃2𝑖−1 for 1 ≤ 𝑖 ≤ 𝑛. then the induced edge labels are given by 𝑓 ∗(𝑢𝑖 𝑢𝑖+1) = 1 for 1 ≤ 𝑖 ≤ 𝑛, 𝑓 ∗(𝑢𝑛𝑢1) = 1 and 𝑓 ∗(𝑢𝑖 𝑣𝑖 ) = 0 for 1 ≤ 𝑖 ≤ 𝑛. from the above, 𝑒𝑓 (0) = 𝑛 + 3, 𝑒𝑓 (1) = 𝑛 + 2 and so |𝑒𝑓 (0) − 𝑒𝑓 (0)| ≤ 1. therefore, for 𝑛 ≥ 1, the crown 𝐶𝑛 ⊙ 𝐾1 is pell even sum cordial. theorem 2.5 for 𝑛 ≥ 2, the complete bipartite graph 𝐾𝑛,𝑛 is pell even sum cordial. proof. let 𝐺 = 𝐾𝑛,𝑛. let the partitions of vertex set be 𝑉1 = {𝑢1, 𝑢2, … , 𝑢𝑛 }and 𝑉2 = {𝑣1, 𝑣2, … , 𝑣𝑛 }. hence 𝑉(𝐺) = {𝑢1, 𝑢2, … , 𝑢𝑛 , 𝑣1, 𝑣2, … , 𝑣𝑛 }and 𝐸(𝐺) = {𝑢𝑖 𝑣𝒊 ∶ 1 ≤ 𝑖 ≤ 𝑛}. then |𝑉(𝐺)| = 2𝑛 , |𝐸(𝐺)| = 𝑛𝟐. consider 𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃2𝑛−1}defined by 𝑓(𝑢𝑖 ) = 𝑃𝑖−1for 1 ≤ 𝑖 ≤ 𝑛 and𝑓(𝑣𝑖 ) = 𝑃(𝑛−1)+𝑖 for 1 ≤ 𝑖 ≤ 𝑛. the induced edge labels are given by, 307 pell even sum cordial labeling of graphs for 1 ≤ 𝑖 ≤ 𝑛, 𝑓 ∗(𝑢𝑖 𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≡ 1,0 (𝑚𝑜𝑑 2); 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. when 𝑛 is odd, 𝑒𝑓 (0) = 𝑛2+1 2 , 𝑒𝑓 (1) = 𝑛2−1 2 and when, 𝑛 is even, 𝑒𝑓 (0) = 𝑒𝑓 (1) = 𝑛2 2 . therefore, for 𝑛 ≥ 2, the complete bipartite graph 𝐾𝑛,𝑛 is pell even sum cordial. theorem 2.6 for 𝑛 ≥ 3, the flower graph 𝐹𝑙𝑛 is pell even sum cordial. proof. let 𝐺 = 𝐹𝑙𝑛. then 𝑉(𝐺) = {𝑣, 𝑢𝑖 , 𝑣𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛} and 𝐸(𝐺) = {𝑣𝑣𝑖 , 𝑣𝑖 𝑢𝑖 , 𝑣𝑢𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛 ; 𝑣𝑛 𝑣1 ; 𝑣𝑖 𝑣𝑖+1 ∶ 1 ≤ 𝑖 ≤ 𝑛 − 1}. then,|𝑉(𝐺)| = 2𝑛 + 1, |𝐸(𝐺)| = 4𝑛. consider 𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … 𝑃2𝑛} defined by 𝑓(𝑣𝑛) = 𝑃2𝑛 for 1 ≤ 𝑖 ≤ 𝑛 and 𝑓(𝑣𝑖 ) = 𝑃2𝑖−2 for 1 ≤ 𝑖 ≤ 𝑛 and 𝑓(𝑢𝑖 ) = 𝑃2𝑖−1 for 1 ≤ 𝑖 ≤ 𝑛. the induced edge labels are given by 𝑓 ∗(𝑣𝑖 𝑣𝑖+1) = 1, 𝑓 ∗(𝑣𝑖 𝑢𝑖 ) = 0, 𝑓 ∗(𝑣𝑢𝑖 ) = 0, 𝑓 ∗(𝑣𝑣𝑖 ) = 1 and 𝑓 ∗(𝑣𝑛𝑣1) = 1 for 1 ≤ 𝑖 ≤ 𝑛. from the above, 𝑒𝑓 (0) = 𝑒𝑓 (1) = 2𝑛 and so |𝑒𝑓 (0) − 𝑒𝑓 (0)| ≤ 1. therefore, the flower graph 𝐹𝑙𝑛 is pell even sum cordial for 𝑛 ≥ 3. theorem 2.7 for 𝑛 ≥ 3 and 𝑛 is even, the helm 𝐻𝑛 is pell even sum cordial. proof. let 𝐺 = 𝐻𝑛. then 𝑉(𝐺) = {𝑣, 𝑢𝑖 , 𝑣𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛}, 𝐸(𝐺) = {𝑢𝑖 𝑢𝑖+1: 1 ≤ 𝑖 ≤ 𝑛 − 1 ; 𝑢𝑖 𝑣𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛 ; 𝑣𝑢𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛 − 1 ; 𝑢1𝑢𝑛}, |𝑉(𝐺)| = 2𝑛 + 1, |𝐸(𝐺)| = 3𝑛. consider 𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃2𝑛 } defined by 𝑓(𝑢𝑖 ) = 𝑃𝑖 for 1 ≤ 𝑖 ≤ 𝑛 − 1, 𝑓(𝑣𝑖 ) = 𝑃𝑛+𝑖 for 1 ≤ 𝑖 ≤ 𝑛 and 𝑓(𝑣) = 0. then the induced edge labels are given by, for 1 ≤ 𝑖 ≤ 𝑛 − 1, 𝑓 ∗(𝑢𝑖 𝑢𝑖+1) = 0, 𝑓 ∗(𝑣𝑖 𝑢𝑖 ) = 1, 𝑓 ∗(𝑢𝑛𝑢1) = 0 and 𝑓 ∗(𝑣𝑢𝑖 ) = { 0 𝑖𝑓 𝑖 ≡ 1 (𝑚𝑜𝑑 2); 1 𝑖𝑓 𝑖 ≡ 0 (𝑚𝑜𝑑 2). from the above, 𝑒𝑓 (0) = 𝑒𝑓 (1) = 3𝑛 2 and so |𝑒𝑓 (0) − 𝑒𝑓 (0)| ≤ 1. therefore, for an even integer ≥ 3 , the helm 𝐻𝑛 is pell even sum cordial. theorem 2.8 for 𝑛 ≥ 4, the wheel graph 𝑊𝑛 is pell even sum cordial. proof. let 𝐺 = 𝑊𝑛. then 𝑉(𝐺) = {𝑢0, 𝑢1, 𝑢𝟐, … , 𝑢𝑛 } and (𝐺) = {𝑢𝑖 𝑢𝑖+1: 1 ≤ 𝑖 ≤ 𝑛 − 1 ; 𝑢0𝑣𝑖 : 1 ≤ 𝑖 ≤ 𝑛 − 1}, |𝑉(𝐺)| = 𝑛 + 1 and |𝐸(𝐺)| = 2𝑛 − 2. consider 𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃2𝑛 } defined by 𝑓(𝑢𝑖 ) = 𝑃2𝑖−2 for 1 ≤ 𝑖 ≤ 𝑛 and 𝑓(𝑢0) = 𝑃1. then the induced edge labels are given by 𝑓 ∗(𝑢𝑖 𝑢𝑖+1) = 1 for 1 ≤ 𝑖 ≤ 𝑛 − 1, 𝑓 ∗(𝑢𝑛𝑢1) =1 and 𝑓 ∗(𝑢0𝑢𝑖 ) = 0 for 1 ≤ 𝑖 ≤ 𝑛. from this, 𝑒𝑓 (0) = 𝑒𝑓 (1) = 𝑛 − 1 and so |𝑒𝑓 (0) − 𝑒𝑓 (0)| ≤ 1. therefore, for 𝑛 ≥ 4, the wheel graph 𝑊𝑛 is pell even sum cordial. theorem 2.9 for 𝑛 ≥ 2, the star graph 𝐾1,𝑛 is pell even sum cordial. proof. let 𝐺 = 𝐾1,𝑛. then 𝑉(𝐺) = {𝑣, 𝑢1, 𝑢2, … , 𝑢𝑛} and 𝐸(𝐺) = {𝑣𝑣𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛}, |𝑉(𝐺)| = 𝑛 + 1 and |𝐸(𝐺)| = 𝑛. consider 𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃𝑛 } defined by 𝑓(𝑣) = 𝑃0, 𝑓(𝑢𝑖 ) = 𝑃𝑖 for 1 ≤ 𝑖 ≤ 𝑛 − 1. the induced edge labels are given by, for 1 ≤ 𝑖 ≤ 𝑛, 𝑓 ∗(𝑣𝑢𝑖 ) = { 1 𝑖𝑓 𝑖 ≡ 0 (𝑚𝑜𝑑 2); 0 𝑖𝑓 𝑖 ≡ 1 (𝑚𝑜𝑑 2). 308 christina mercy a and tamizh chelvam t when 𝑛 is odd, 𝑒𝑓 (0) = n+1 2 , 𝑒𝑓 (1) = 𝑛−1 2 whereas when 𝑛 is even,𝑒𝑓 (0) = n 2 = 𝑒𝑓 (1).therefore, for 𝑛 ≥ 2, the star graph 𝐾1,𝑛 is pell even sum cordial. theorem 2.10 for 𝑛 ≥ 3 , the triangular book 𝑇𝐵𝑛 is pell even sum cordial. proof. let 𝐺 = 𝑇𝐵𝑛 . then 𝑉(𝐺) = {𝑣1, 𝑣2, … , 𝑣𝑛−1, 𝑣𝑛}, 𝐸(𝐺) = {𝑣1𝑣𝑖 , 𝑣𝑛 𝑣𝑖 , 𝑣𝑛 𝑣1 ∶ 2 ≤ 𝑖 ≤ 𝑛 − 1} , |𝑉(𝐺)| = 𝑛 and |𝐸(𝐺)| = 2𝑛 − 3. consider 𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃𝑛−1} defined by 𝑓(𝑣1) = 𝑃0 , 𝑓(𝑣𝑛 ) = 𝑃1 and 𝑓(𝑣𝑖 ) = 𝑃𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 − 2. then the induced edge labels are given by 𝑓 ∗(𝑣1𝑣𝑛) = 0. for 1 ≤ 𝑖 ≤ 𝑛 − 2, 𝑓 ∗(𝑣1𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≡ 0 (𝑚𝑜𝑑 2); 0 𝑖𝑓 𝑖 ≡ 1 (𝑚𝑜𝑑 2). 𝑓 ∗(𝑣𝑛 𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≡ 1 (𝑚𝑜𝑑 2); 0 𝑖𝑓 𝑖 ≡ 0 (𝑚𝑜𝑑 2). from the above, 𝑒𝑓 (0) = 𝑛 − 1, 𝑒𝑓 (1) = 𝑛 − 2 and so |𝑒𝑓 (0) − 𝑒𝑓 (0)| ≤ 1. therefore, for 𝑛 ≥ 3 , the triangular book 𝑇𝐵𝑛 is pell even sum cordial. theorem 2.11. for 𝑚 ≥ 3, the graph 𝐾2 + 𝑚𝐾1 is pell even sum cordial. proof. let 𝐺 = 𝐾2 + 𝑚𝐾1. then (𝐺) = {𝑢1, 𝑢2, 𝑣1, 𝑣2 … , 𝑣𝑚} , 𝐸(𝐺) = {𝑢1𝑢2, 𝑢1𝑣𝑖 , 𝑢2𝑣𝑖 : 1 ≤ 𝑖 ≤ 𝑚} , |𝑉(𝐺)| = 𝑚 + 2 and |𝐸(𝐺)| = 2𝑚 + 1. consider 𝑓: 𝑉(𝐺) → {𝑃0, 𝑃1, … , 𝑃𝑚+1} defined by 𝑓(𝑢1) = 𝑃0, 𝑓(𝑢2) = 𝑃1 and 𝑓(𝑣𝑖 ) = 𝑃𝑖+1 for 1 ≤ 𝑖 ≤ 𝑚. then the induced edge labels are given by 𝑓 ∗(𝑢1𝑢2) = 0. for 1 ≤ 𝑖 ≤ 𝑚, 𝑓 ∗(𝑢1𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≡ 1 (𝑚𝑜𝑑 2); 0 𝑖𝑓 𝑖 ≡ 0 (𝑚𝑜𝑑 2). 𝑓 ∗(𝑢2𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≡ 0 (𝑚𝑜𝑑 2); 0 𝑖𝑓 𝑖 ≡ 1 (𝑚𝑜𝑑 2). from the above, 𝑒𝑓 (0) = 𝑚 + 1, 𝑒𝑓 (0) = 𝑚 and so |𝑒𝑓 (0) − 𝑒𝑓 (0)| ≤ 1. therefore, for 𝑚 ≥ 3, the graph 𝐾2 + 𝑚𝐾1 is pell even sum cordial. 3. conclusions this article introduces a novel graph labeling technique called pell even sum cordial labeling and it is demonstrated that several typical graphs are pell even sum cordial. more pell even sum graphs will be provided in the following article. references [1] ayudainayaki r, selvam b, harmonious and pell labeling for some extended duplicate graph, international journal of scientific research in mathematical and statistical sciences, 5(2)(2018), 152-156. [2] cahit i, cordial graphs: a weaker version of graceful and harmonious graphs, ars combinatoria, 23(1987), 201-207. 309 pell even sum cordial labeling of graphs [3] celine mary v, suresh d, thirusangu k, some graph labeling on the inflation of alternate triangular snake graph of odd length, international journal of mathematics trends and technology – special issue, nccfqet, may 2018, 134-139. [4] gallian j a, a dynamic survey on graph labelings, electronic journal of combinatorics, 24, #ds6 (2021), 1-576. [5] harary f, graph theory, addison wesley publishing company inc, usa (1969). [6] indira p, selvam b, thirusangu k, pell labeling and mean square sum labeling for the extended duplicate graph of quadrilateral snake, advances and applications in mathematical sciences, 20(9) (2021), 1709-1718. [7] muthu ramakrishnan d, sutha s, some pell graceful graphs, international journal of scientific research and review, 8(7) (2019), 255-262. [8] muthu ramakrishnan d, sutha s, pell graceful labeling of graphs, malaya journal of mathematik, 7(3) (2019), 508-512. [9] muthu ramakrishnan d, sutha s, some pell square graceful graphs, compliance engineering journal, 10(10) (2019), 99-117. [10] sharon philomena, thirusangu k, 3-total product cordial and pell labeling on tree < k1,n: 2 >, international journal of pure and applied mathematics, 101(5) (2015), 747-752. [11] shiama j, pell labeling for some graphs, asian journal of current engineering and maths, volume 2(2013), 267-272. [12] sriram s, govindarajan r, thirusangu k, pell labeling of square of path graph, international journal of engineering and advanced technology, 9(1s3) (2019) 310 ratio mathematica volume 41, 2021, pp. 45-52 on decomposition of multistars into multistars reji t* ruby r† abstract the multistar sw1,...,wn is the multigraph whose underlying graph is an n-star and the multiplicities of its n edges are w1, ...,wn. let g and h be two multigraphs. an h-decomposition of g is a set d of h-subgraphs of g, such that the sum of ω(e) over all graphs in d which include an edge e, equals the multiplicity of e in g, for all edges e in g. in this paper, we fully characterize s1,2,3,k1,m and sm l decomposable multistars, where ml is m repeated l times. keywords: decomposition; multigraph; multistar 2020 ams subject classifications: 05c51; 05c70; 05c75 1 *department of mathematics, government college, chittur, palakkad, kerala, india-678104; rejiaran@gmail.com. †department of mathematics, government college, chittur, palakkad, kerala, india-678104; rubymathpkd@gmail.com. 1received on october 24, 2021. accepted on november 24, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.681. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 45 reji t, ruby r 1 introduction if g and h are two simple graphs with out isolated vertices, then g is hdecomposable or h divides g if there exists a partition of the edge set of g into disjoint isomorphic copies of h. the above definition can be extended to mutigraphs also. let g and h be two multigraphs. then the corresponding h-decomposition problem is to decide for a fixed h and an input g, whether such a partition exists. we can formally define the concepts about multigraphs and the multigraph decomposition problems as follows. definition 1.1. a multigraph (v,e,w) consists of a simple underlying graph (v,e) and a multiplicity function w : e → n, where n is the set of natural numbers. the multigraph on an underlying graph g with constant multiplicity λ is denoted by λ.g and this is different from λg, denoting λ disjoint copies of g. when referring to a simple graph g as a multigraph we mean 1.g. an isomophism between multigraphs is an isomophism between their underlying simple graphs which peserves edge multiplicity. definition 1.2. a subgraph h of a multigraph g is a multigraph h whose underlying graph is a subgraph of that of g and its multiplicity function is dominated by the multiplicity function of g, i.e. the multiplicity of an edge in h does not exceed its multiplicity in g. definition 1.3. an h-subgraph of g is a subgraph of a multigraph g, isomorphic to a multigraph h. definition 1.4. let g and h be two multigraphs. an h-decomposition of g is a set d of h-subgraphs of g, such that the sum of ω(e) over all graphs in d which include an edge e, equals the multiplicity of e in g, for all edges e in g. definition 1.5. the multistar sw1,w2,··· ,wn is the multigraph whose underlying graph is an n-star and the multiplicities of its n edges are w1,w2, · · · ,wn. there are considerable number of papers dealing with an h-decomposition of g and some them are provided in the reference [shyu, 2013, lin and shyu, 1996, lin, 2010, lee and lin, 2005, lee et al., 2005, bryant et al., 2001, bialostocki and roditty, 1982]. priesler and tarsi [priesler and tarsi, 2004] showed that, for any multistar h (except a few cases), h-decomposition is np -complete. priesler and tarsi [priesler and tarsi, 2005] fully characterized s1,2-decomposable multistars in the following theorem. 46 on decomposition of multistars into multistars theorem 1.1. [priesler and tarsi, 2005] the multistar sw1,w2,··· ,wn , n ≥ 2 is s1,2-decomposable if and only if 1. σni=1wi ≡ 0(mod 3) 2. the number of odd multiplicities among the wi is at most 1 3 (σni=1wi) 3. the largest among the wi is at most twice the sum of all the others. in this paper we fully characterize those multistars which are s1,2,3-decomposable, k1,m-decomposable, s2 m -decomposable and sm l -decomposable where ml denotes m repeated l times. 2 main results 2.1 s1,2,3 decomposability of sw1,w2,w3 theorem 2.1. let w1 ≥ w2 ≥ w3 ≥ 2 be positive integers and n = w1+w2+w36 . then sw1,w2,w3 is s1,2,3-decomposable if 1. w1 + w2 + w3 ≡ 0(mod 6) 2. 2 ≤ w1 −n ≤ 2n, 2 ≤ w2 −n ≤ 2n 3. 5w2 ≤ w1 + 7w3 − 12 4. w1 ≤ w2 +w3−6 if m and l−( m−12 ) are odd where m = w1−n,l = w2−n. proof. consider the equations 3x1 + 2x2 + 1(n− (x1 + x2)) = w1 3y1 + 2y + 2 + 1(n− (y1 + y2)) = w2 3(n− (x1 + y1)) + 2(n− (x2 + y2)) + 1(n− (2n− (x1 + x2 + y1 + y2))) = w3 let the above three equations be called as (a). firstly we claim that under the given conditions (1),(2),(3) and (4) we can find non negative integers x1,x2,y1,y2 satisfying equations (a) such that n−(xl + x2) ≥ 0, n−(y1 + y2) ≥ 0, n−(xl + y1) ≥ 0, n− (x2 + y2) ≥ 0, n− [2n− (xl + x2 + y1 + y2)] ≥ 0. let these five inequalities be called as (b). the equations in (a) can be simplified as 2x1 + x2 = w1 −n (2.1.1) 2y1 + y2 = w2 −n (2.1.2) 2x1 + 2y1 + x2 + y2 = 4n−w3 (2.1.3) 47 reji t, ruby r since n = w1+w2+w3 6 , w1 −n + w2 −n = 4n−w3. thus to prove our claim we have to solve the equations ( 2.1.1), ( 2.1.2) such that all the inequalities in (b) are satisfied. observing ( 2.1.1) and ( 2.1.2), it is clear that they have a positive integral solution such that all the inequalities in (b) are satisfied if and only if w1 − n ≤ 2n and w2 − n ≤ 2n. by condition (2), these inequalities holds. let m = w1 −n, l = w2 −n. case 1: w2 −n ≤ n here we have to solve the equations 2x1 + x2 = m, 2y1 + y2 = l. subcase 1.1: m is even take x1 = m 2 , x2 = 0, y1 = 0, y2 = l. thus x1 +x2 = m 2 ≤ n, y1 +y2 = l ≤ n, xl + y1 = m 2 ≤ n, x2 + y2 = l ≤ n. so the only inequality in (b), which has to be verified is n− (2n− (xl + x2 + y1 + y2)) ≥ 0. since n− (xl + x2) ≥ 0 and n−(y1 +y2) ≥ 0, xl +x2 +y1 +y2 ≤ 2n. thus n−(2n−(xl +x2 +y1 +y2)) ≥ 0 ⇔xl + x2 + y1 + y2 ≥ n⇔ m2 + l ≥ n⇔ w1−n 2 + w2 −n ≥ n⇔w1 + 2w2 ≥ 5n ⇔ w1 + 2w2 ≥ 5( w1+w2+w36 ) ⇔ 5w3 ≤ w1 + 7w2, which is always true, since w1 ≥ w2 ≥ w3· thus in this subcase all the inequalities in (b) are satisfied. subcase 1.2: m is odd take x1 = m−1 2 , x2 = 1, y1 = 1, y2 = l − 2. here x1 + y1 = m−12 + 1 ≤ n [since m is odd and m ≤ 2n]. x2 + y2 = 1 + l − 2 = l − 1 ≤ n [since in this subcase l = w2−n ≤ n], y1 + y2 = l−1 ≤ n and x1 + x2 = m−12 + 1 ≤ n. as in the above subcase n− (2n− (x1 + x2 + y1 + y2)) ≥ 0 ⇔x1 + x2 + y1 + y2 ≥ n ⇔ m − 1 + 2l ≥ 2n ⇔ w1 + 2w2 − 1 ≥ 5n ⇔ 5w3 + 6 ≤ w1 + 7w2. since w3 ≥ 2,w1 > 2 + n, w2 ≥ 2 + n, we get w1 ≥ 3, w2 ≥ 3, w3 ≥ 2. also w1 ≥ w2 ≥ w3. thus 7w2 ≥ 5w3 + 2w3 ≥ 5w3 + 4. thus w1 + 7w2 ≥ 5w3 + 4 + w1 ≥ 5w3 + 7 > 5w3 + 6. hence x1 + x2 + y1 + y2 ≥ n and thus n− (2n− (x1 + x2 + y1 + y2)) ≥ 0. thus in this subcase also all the conditions in (b) are satisfied. case 2: w2 −n > n here also we have to solve the equations 2x1 + x2 = m, 2y1 + y2 = l. subcase 2.1: m is even and l− m 2 is even take x1 = m 2 , x2 = 0, y1 = l− m 2 2 , y2 = m 2 . here n− (xl + x2) = n− m2 ≥ 0, since m 2 ≤ n. n−(y1 +y2) = n−( 2l−m4 + m 2 ) = n− 2l+m 4 . thus n−(y1 +y2) ≥ 0 ⇔ 2l+m 4 ≤ n ⇔ 2(w2 −n) + w1 −n ≤ 4n ⇔ 5w2 ≤ w1 + 7w3, which is true by the given condition (3). similarly n−(x1 + y1) ≥ 0 and n−(x2 + y2) ≥ 0. as in the above case, n− (2n− (x1 + x2 + y1 + y2)) ≥ 0 ⇔x1 + x2 + y1 + y2 ≥ n⇔ m 2 + 2l−m 4 + m 2 ≥ n ⇔ 3m + 2l ≥ 4n ⇔ 3w1 + 2w2 ≥ 9n ⇔ 3w3 ≤ 3w1 + w2, which is always true since w1 ≥ w2 ≥ w3. subcase 2.2: m is even and l− m 2 is odd here take x1 = m 2 , x2 = 0, y1 = l− m 2 +1 2 , y2 = m 2 − 1. we can easily verify that n− (xl + x2) ≥ 0 ⇔ 5w2 ≤ w1 + 7w3 − 12, which is true by condition (3). 48 on decomposition of multistars into multistars similarly n − (y1 + y2) ≥ 0. also it easily follows that n − (x1 + y1) ≥ 0 and n− (x2 + y2) ≥ 0. n− (2n− (x1 + x2 + y1 + y2)) ≥ 0 ⇔ 3w1 + w2 ≥ 3w3 + 4. but w1 ≥ w2 ≥ w3 ≥ 2 and by condition (2), w1 ≥ n + 2 and n = w1+w2+w36 . thus w1 ≥ 4. also 3w1 + w2 = w1 + 2w1 + w2 ≥ w1 + 3w3 ≥ 3w3 + 4 (since w1 ≥ 4). thus n− (2n− (x1 + x2 + y1 + y2)) ≥ 0 and hence all the inequalities in (b) are satisfied. subcase 2.3: m is odd and l− m−1 2 is even take x1 = m−1 2 , x2 = 1, y1 = l− m−1 2 2 , y2 = m−1 2 . n−(x1 + x2) = n−( m−12 + 1) ≥ 0 ⇔ 1 + m−1 2 ≤ n. this is true since m ≤ 2n and m is odd. similarly n−(x2 + y2) ≥ 0. also n−(x1 + y1) ≥ 0 ⇔ 5w2 ≤ w1 + 7w3 + 6, which is true by condition (3). similarly n−(y1+y2) ≥ 0. also n−(2n−(x1+x2+y1+y2)) ≥ 0 ⇔ 3w1 + w2 + 2 ≥ 3w3, which is always true since w1 ≥ w2 ≥ w3. thus all the inequalities in (b) are satisfied. subcase 2.4: m is odd and l− m−1 2 is odd take x1 = m−1 2 , x2 = 1, y1 = l− m+1 2 2 , y2 = m+1 2 . n− (xl + x2) = n− m−12 + 1) = n − m+1 2 . since m is odd and m ≤ 2n, m+1 2 ≤ n. so n − (xl + x2) ≥ 0. n − (x2 + y2) = n − ( m+12 + 1) ≥ 0 ⇔ w1 ≤ w2 + w3 − 6, which is true by condition(4). also we can verify that n−(xl + y1) ≥ 0 and n−(y1 + y2) ≥ 0 by condition(3). similarly n−(2n−(xl +x2 +y1 +y2)) ≥ 0 ⇔ 3w1 +w2 + 6 ≥ 3w3, which is always true. hence all the conditions in (b) are satisfied in this subcase also. hence our claim is proved in both cases. thus using equations (a), we can properly partition w1 into x1 copies of 3, x2 copies of 2 and n− (x1 + x2) copies of 1’s. w2 can be partitioned into y1copies of 3, y2 copies of 2 and n− (y1 + y2) copies of 1. w3 can be partitioned into n − (x1 + y1) copies of 3, n − (x2 + y2) copies of 2 and n−(2n−(x1 + x2 + y1 + y2)) copies of 1. using these partitions of w1,w2,w3, we can decompose sw1,w2,w3 into copies of s1,2,3.2 2.2 k1,m decomposability of sw1,w2,··· ,wn theorem 2.2. the multistar sw1,w2,··· ,wn , w1 ≥ w2 ≥ ··· ≥ wn, is k1,mdecomposable (n ≥ m) if and only if 1. σni=1wi ≡ 0(mod m) 2. for each k = 1, 2, · · · ,m− 1, σki=1wi ≤ k m−k (wk+1 + · · · + wn) proof. suppose the multistar the multistar sw1,w2,··· ,wn , w1 ≥ w2 ≥ ···≥ wn, is k1,m-decomposable (n ≥ m). then clealy σni=1wi ≡ 0(mod m). to prove (2), assume the contrary. suppose that ∑k i=1 wi > k m−k (wk+1 +· · ·+ 49 reji t, ruby r wn), for some k with 1 ≤ k ≤ m− 1. this implies (m−k) k∑ i=1 wi > k(wk+1 + · · · + wn) ⇒ m k∑ i=1 wi > k n∑ i=1 wi ⇒ k∑ i=1 wi > k m ( n∑ i=1 wi). this is not possible, since 1 m ( ∑n i=1 wi) is the number of copies of k1,m to which sw1,w2,··· ,wn can be decomposed. each copy of k1,m can contribute at most k to ∑k i=1 wi. thus ∑k i=1 wi ≤ k m ( ∑n i=1 wi). we prove sufficiency by induction on w = ∑n i=1 wi. for w = m, the multistar is k1,m itself. if w ≥ 2m, one copy of k1,m is deleted from sw1,w2,··· ,wm by subtracting m number of 1’s from the largest m multiplicities. the multistar obtained after this process still satisfies conditions 1 and 2. hence by induction the proof follows.2 2.3 s2 m decomposability of sw1,w2,··· ,wn theorem 2.3. the multistar sw1,w2,··· ,wn , w1 ≥ w2 ≥ ···≥ wn, is s2 m -decomposable (n ≥ m) if and only if 1. σni=1wi ≡ 0(mod 2m) 2. for 1 ≤ i ≤ n, wi ≡ 0(mod 2) 3. for each k = 1, 2, · · · ,m− 1, σki=1wi ≤ k m−k (wk+1 + · · · + wn) proof. assume that the multistar sw1,w2,··· ,wm is s2 m -decomposable. then as in the above theorems conditions 1 and 3 hold. since sw1,w2,··· ,wm is s2 m decomposable, clearly wi ≡ 0(mod 2). we prove sufficiency by induction on w = ∑n i=1 wi. for w = 2m, the multistar is s2 m itself. if w ≥ 4m, delete one copy of s2m from sw1,w2,··· ,wn by subtracting m number of 2’s from the largest m multiplicities. the multistar obtained after this deletion still satisfies all the three conditions. hence by induction the proof follows.2 the above two theorems can be generalized to characterize sm l -decomposable multistars. 2.4 sm l decomposability of sw1,w2,··· ,wn theorem 2.4. the multistar sw1,w2,··· ,wn , w1 ≥ w2 ≥ ···≥ wn, is sm l -decomposable (n ≥ l) if and only if 50 on decomposition of multistars into multistars 1. σni=1wi ≡ 0(mod lm) 2. for 1 ≤ i ≤ n, wi ≡ 0(mod m) 3. for each k = 1, 2, · · · , l− 1, σki=1wi ≤ k l−k (wk+1 + · · · + wn) proof. assume that the multistar sw1,w2,··· ,wn is sm m -decomposable. then conditions 1,2 and 3 follows as in the above theorem. the sufficiency can similarly be proved using induction by deleting one copy of sm m from sw1,w2,··· ,wn by subtracting l number of m’s from the largest l multiplicities.2 3 conclusions in this paper we have characterized those multistars which are s1,2,3-decomposable, k1,m-decomposable, s2 m -decomposable and sm l -decomposable. 4 acknowledgement we are indebted to prof.m.i.jinnah, formerly university of kerala, for his guidance and valuable suggestions during the preparation of this manuscript. references a bialostocki and y roditty. 3k 2-decomposition of a graph. acta mathematica academiae scientiarum hungarica, 40(3-4):201–208, 1982. darryn e bryant, saad el-zanati, charles vanden eynden, and dean g hoffman. star decompositions of cubes. graphs and combinatorics, 17(1):55–59, 2001. hung-chih lee and chiang lin. balanced star decompositions of regular multigraphs and λ-fold complete bipartite graphs. discrete mathematics, 301(2-3): 195–206, 2005. hung chih lee, jeng jong lin, chiang lin, and tay woei shyu. multistar decomposition of complete multigraphs. ars combinatoria, 74:49–63, 2005. chiang lin and tay-woei shyu. a necessary and sufficient condition for the star decomposition of complete graphs. journal of graph theory, 23(4):361–364, 1996. jenq-jong lin. decomposition of balanced complete bipartite multigraphs into multistars. discrete mathematics, 310(5):1059–1065, 2010. 51 reji t, ruby r miri priesler and michael tarsi. on some multigraph decomposition problems and their computational complexity. discrete mathematics, 281(1-3):247–254, 2004. miri priesler and michael tarsi. multigraph decomposition into stars and into multistars. discrete mathematics, 296(2-3):235–244, 2005. tay-woei shyu. decomposition of complete bipartite graphs into paths and stars with same number of edges. discrete mathematics, 313(7):865–871, 2013. 52 ratio mathematica volume 39, 2020, pp. 111-136 111 teaching as a decision-making model: strategies in mathematics from a practical requirement viviana ventre* eva ferrara dentice roberta martino† abstract the need in the current social context to adopt teaching methods that can stimulate students and lead them towards autonomy, awareness and independence in studying could conflict with the needs of students with specific learning disorders, especially in higher education, where self-learning and self-orientation are required. in this sense, the choice of effective teaching strategies becomes a decision-making problem and must, therefore, be addressed as such. this article discusses some mathematical models for choosing effective methods in mathematics education for students with specific learning disorders. it moves from the case study of a student with specific reading and writing disorders enrolled in the mathematical analysis course 1 of the degree course in architecture and describes the personalised teaching strategy created for him. keywords: decision-making; inquiry model; social skills; personalised didactic strategy; analytic hierarchy process. 2010 ams subject classification: 91b06, 97d60.‡ *. corresponding author: viviana.ventre@unicampania.it †. all the authors belong to the department of mathematics and physics of the university of the study of campania “l. vanvitelli”, viale a. lincoln n.5, i-81100, caserta (ce), italy. ‡ received on november 20th, 2020. accepted on december 22th, 2020. published on december 31st, 2020. doi: 10.23755/rm.v39i0.559. issn: 1592-7415. eissn: 22828214. ©ventre et al. this paper is published under the cc-by licence agreement. viviana ventre, eva ferrara dentice, roberta martino 112 1 introduction in recent years the institutions have increased their interest in a very worrying phenomenon that concerns italy and generally speaking the countries in the western world: the growth of 'disaffection' towards mathematics due to a traditional didactic approach to the subject (piochi, 2008). young people coming out of secondary schools often have the idea that mathematics consists of mechanical processes, seeing it as an arid and pre-packaged discipline whose understanding and description seem impersonal. the mathematics one learns at school is very often a set of basic notions, axioms and definitions given by the teacher and practically impossible to discuss, causing the view of a subject that is “already done” and immutable (castelnuovo, 1963). the experience of mathematicians, on the other hand, is very different: mathematics is something extremely changeable whose results are the result of hard work, debate and controversy. so, axioms and definitions first presented in textbooks come into reality only at the end, when the whole structure of the problem is understood. then, the following question arises: what is mathematics? definitions such as “mathematics is the science of numbers and forms” accepted 200 years ago is now reductive and ineffective because mathematics has developed so rapidly and intensely that no definition can take into account all the multiple aspects (baccaglini-frank, di martino, natalini, rosolini, 2018 (a)). the list of applications of this discipline in daily life could be endless, and so could the list of motivations that could be given to pupils to convince them to study. about this matter it is really important the following statement: "no doubt, mathematical knowledge is crucial to produce and maintain the most important aspects of our present life. this does not imply that the majority of people should know mathematics." (vinner, 2000). mathematics can also cause terror in students (the phenomenon of “fear for mathematics” (bartilomo and favilli, 2005)) or a state of dissatisfaction with the common conception that “you have to be made for it” so much that even great professionals boast that they have never understood anything about mathematics. so, is it necessary to teach mathematics to everyone? the answer is simple: apart from the fact that having a basis in mathematics is a cultural question regardless of the future job, mathematics teaches to evaluate multiple aspects of a question, and provides knowledge and skills in order to consciously face a discussion defending one's own positions with responsibility and respect for the arguments of others (national indications, 2007). the key role of mathematics education in the development of rational thinking and with it the responsibilities of mathematics teachers at all levels is therefore underlined. already in 1958, the theme of the congress of the belgian teaching as a decision-making model: strategies in mathematics from a practical requirement 113 mathematical society was entitled “the human responsibility of the mathematics teacher” (castelnuovo, 1963). the most effective way to bring students closer to mathematics is, therefore, the image of a “method for dealing with problems, a language, a box of tools that allows us to strengthen our intuition” (baccaglini-frank, di martino, natalini, rosolini, 2018 (a)). 2 mathematical education: theories and models 2.1 the concept of error and the inquiry model: mathematics as a humanistic discipline mathematics is one of the disciplines in which many 'students' manifest difficulties that compromise the relationship with the subject. a student who comes out of secondary school has a long series of 'failures' accompanied by the conviction that she can never do mathematics because she is not good at it. the problem lies in identifying errors and difficulties in mathematical learning with the conviction that the absence of errors certifies the absence of difficulties and on the other hand the absence of difficulties guarantees the absence of errors (zan, 2007 (a)). this identification leads to the didactic objective of obtaining the greatest number of correct answers by nourishing the "compromise of correct answers" (gardner, 2002): on the one hand the teacher chooses activities that are not "too" difficult and on the other hand the students elaborate the answers expected by the teacher in a reproductive way. of course, this method does not guarantee any learning, revealing itself dangerous and counterproductive (di martino, 2017). moreover, with it the fear of making mistakes arise and also the conviction that mathematics is not for everyone (for instance: you can't study mathematics if you do not have a good memory!) (zan, di martino, 2004). in order to face the identification of difficulty-error, there is, therefore, a need to revolutionise the conception of error and to convey to students that ''making a mistake at school may not be perceived as something negative to avoid at all, because it could be an opportunity for new learning (and teaching) opportunities to be exploited'' (borasi,1996). the inquiry model is a teaching-learning model that proposes a positive and fundamental role of errors in mathematics teaching. this model sees knowledge as a dynamic process of investigation where cognitive conflict and doubt represent the motivations to continuously search for a more and more refined understanding. therefore, instead of eliminating ambiguities and contradictions to avoid confusion or errors, these elements must be highlighted to stimulate and give shape to ideas and discussions. questions such as "what would happen if this result were true?" or "under what circumstances could this error be corrected?" lead to a reformulation of the problem where the error is only the viviana ventre, eva ferrara dentice, roberta martino 114 starting point for a deeper understanding. communication in the classroom plays a fundamental role, and so does the conception of mathematics as a humanistic discipline: the teacher provides the necessary support for the student's autonomous search for understanding, who in turn is an active member of a research community (tematico, pasucci, 2014). learning turns out to be a process of constructing meanings, and in this way, the students also understand that what is written in textbooks is the result of debates and arguments and not simply something for its own sake ("falling from the sky"). 2.2 cooperative learning: development of disciplinary and social skills some studies highlight the need to build learning teaching models that take into account students' emotions, perceptions and culture based on the idea that human learning has a specific social character (radford, 2006). the collaborative group and peer tutoring are two models that take on both the disciplinary dimension and the affective and social dimension and facilitate discussion in the classroom. in fact, in most cases, the teacher cannot give everyone the opportunity to express themselves, nor is he able to solicit the interventions of those who are not used to intervene. collaborative learning, instead, sees the involvement of all the students in two successive moments: first within the individual group and then in the final discussion in class. the necessary conditions for such learning are positive interdependence and the assignment of roles: the first is reached when the members of the group understand that there can be no individual success without collective success; the second condition allows the distribution of social and disciplinary competences among the various members of the group favouring collaboration and interdependence. the recognition of roles also helps to overcome problems such as low self-esteem or a sense of ineffectiveness, allowing social skills to grow: knowing how to make decisions, how to express one's own opinions and listen to those of others, how to mediate and share, how to encourage, help and resolve conflicts are skills that the school must teach with the same care with which disciplinary skills are taught. dialogue among peers guarantees greater freedom and spontaneity: the majority of students identify that among peers there is no fear of expressing doubts and perplexities, the main motivation that justifies the effectiveness of such models (baldrighi, pesci, torresani, 2003; pesci, 2011). 2.3 recovery and enhancement interventions: breaking the educational contract the variety of possible processes, the fact that behind correct answers there can be difficulties and that some mistakes can come out of significant thought processes, brings important elements to support the criticism of the teaching as a decision-making model: strategies in mathematics from a practical requirement 115 identification between mistakes and difficulties. for example, the incorrect resolution of a problem is not necessarily due to the inability to manage the mathematical structure of the proposed situation but is probably due to a lack of understanding of the problem itself. the understanding of a text is not always immediate because it involves the student's personal knowledge of common words and scripts. understanding is, therefore reduced to a selective reading that aims to identify the numerical data and the right operations suggested by keywords (zan, 2012). recovery interventions must therefore be based on the analysis of the processes that led the student to make mistakes, shifting the attention from the observation of errors to the observation of failed behaviour with the sole objective of change. the student, in turn, must take responsibility for her own recovery and therefore there is a need for teaching that makes her feel that she is the protagonist of new situations and not simply the executor of procedures to be applied to repetitive exercises (zan, 2007 (b)). the teacher must propose exercises and problems that do not favour a mechanical approach but question the rules that pupils are used to use and that form part of the so-called teaching contract (d'amore, 2007; d'amore, gagatsis,1997). the idea of a didactic contract was born to explain the causes of elective failure in mathematics, that is, the kind of failure reserved only for mathematics by students who instead do well in other subjects. the didactic contract holds the interactions between student and teacher and is made up of "the set of teacher's behaviours expected by the student and the set of student's behaviours expected by the teacher" (brousseau, 1986). this explains the students' belief that a problem or exercise always has a solution because it is the teacher's job to make sure that there is only one answer to the proposed question and that all the data is necessary (baruk, 1985). in bagni (1997) the following goniometry test is proposed to fourth-year students in three classes of scientific high school (students aged 17 to 18). determine the values of x belonging to ℝ for which it results: a) sinx = 1 2τ b) cosx = 1 2τ c) sinx = 1 3τ d) tgx = 2 e) sinx = 𝜋 3τ f) cosx = 𝜋 2τ g) sinx = ξ3 h) cosx = ξ3 3τ table 2.1 experiment in bagni, 1997. remember that the goniometric functions are often introduced by making initial reference to the values they assume in correspondence to relatively common angles of use, so we have the well-known table shown in the next page (table 2.2.) viviana ventre, eva ferrara dentice, roberta martino 116 x 0 𝜋 6τ 𝜋 4τ 𝜋 3τ 𝜋 2τ 2 𝜋 3τ 3 𝜋 4τ 5 𝜋 6τ 𝜋 … sinx 0 1 2τ ξ2 2τ ξ3 2τ 1 ξ3 2τ ξ2 2τ 1 2τ 0 … cosx 1 ξ3 2τ ξ2 2τ 1 2τ 0 − 1 2τ − ξ2 2τ − ξ3 2τ -1 … tgx 0 ξ3 3τ 1 ξ3 n.d. −ξ3 -1 − ξ3 3τ 0 … … … … … … … … … … … … table 2.2 values assumed by common angles where “n.d.” is “not define”. let us now examine the test: agreed time 30 minutes and pupils were not allowed to use protractor tables nor scientific calculator. it has been conceived with: • two "traditional" questions (a), (b); • two possible questions, but with the results not included between the values of x "of common use" (c), (d); • two impossible questions (e), (f) but with values (of sinx and cosx) that recall the measurements in radians of "common use" angles (𝜋/3, 𝜋 /2); • two questions (g), (h) where the first impossible and the second possible. they propose instead values (of sinx, cosx) that are included in the table referred to the angles "of common use" but in relation to other goniometric functions (tgx, cotgx). well, as far as the answers to the questions (e), (f) are concerned, the didactic contract has led some pupils to look for solutions anyway; and the "solutions" that most spontaneously presented themselves to their mind are the ones that they see associated, in the case of the sinus function, the two values 𝜋/3 and ξ3 2 τ and, in the case of the cosine function, the two values 𝜋 /2 and 0. so we have, for instance, the following errors: sinx = 𝜋 /3 so x = ξ3 2 τ cosx = 𝜋 /2 so x = 0 as far as the answers to questions (g), (h) are concerned, the reference to the tangent function was clearly expressed in the answers of some students: also in this case, some students, not finding the proposed values among those corresponding to the most frequently used x values (for the sine and cosine functions, in the table above), were induced to look for another correspondence in which the proposed values are involved. we then find errors such as: if sinx = ξ3 , then x = 𝜋 /3 if sinx = ξ3 , then x = 𝜋 /3+k 𝜋 what has now been pointed out obliges us to conclude that the need that leads the student to always and in any case look for a result for each proposed exercise is unstoppable: breaking the teaching contract can be used as a teaching as a decision-making model: strategies in mathematics from a practical requirement 117 teaching strategy to overcome the mechanical approach used by the students and enhance knowledge (bagni, 1997). 3 concept image ad concept definition these notions were developed to analyse the learning processes of mathematical definitions (tall and vinner, 1981). concept image is the whole cognitive structure related to the concept and includes all mental images, the properties and processes of recall and manipulation associated with a concept, bringing into play its meaning and use. it is built through years of experience of all kinds, changing with the encounter of new stimuli and the growth of the individual. the concept definition is the set of words used to specify a concept and turns out to be personal and can often differ from the formal definition because it represents the reconstruction made by the student and the form of the words he uses to explain his concept image. it can change from time to time and for each individual the concept definition can generate its own concept image which can be called in this case concept definition image. the acquisition of a concept occurs when a good relationship is developed between the concept name, the concept image and the concept definition. students tend to learn definitions in a mechanical way and this can lead to conflict factors when concept image or concept definition are invoked at the same time which conflict with another part of the concept image or concept definition acquired on the same concept. to explore this topic a questionnaire was administered to 41 students with an a or b grade in mathematics. they were asked: "which of the following functions are continuous? if possible, give your reason for your answer." figure 3.1 images from tall and vinner, 1981. we see that the concept image of this topic comes from a variety of resources such as the colloquial use of the term “continuous” in phrases such as “it rained all day long”. so, often the use of the term “continuous function” implies the idea that the graph of the function can be drawn continuously. the answers are summarised in the table shown in figure 3.2. viviana ventre, eva ferrara dentice, roberta martino 118 figure 3.2 tables from tall and vinner, 1981. it summarises the results of the experiment. the reasons given to justify the discontinuity of f2(x) and f4(x) are of the type: “the graph is not in a single piece”, “there is no single formula”. in these answers, we see that many students invoked a concept image including a graph without any interruption or a function defined by a “single formula”. instead, there are many continuous functions that conflict with the concept images just mentioned as the following: 𝑓ሺ𝑥ሻ = ൜ 0 ሺ𝑥 < 0 𝑜𝑟 𝑥2 < 2ሻ 1ሺ𝑥 > 0 𝑜𝑟 𝑥2 > 2ሻ whose graph is: figure 3.3 image from tall and vinner, 1981. it represents the function defined above. the idea that emerges from similar issues is that mathematical concepts should be learned in the everyday, not technical, way of thinking, starting with many examples and non-examples through which the concept image is formed and then arriving at a formal definition. students should use the formal definition, but in order to internalise the concept it is necessary to aim at cognitive conflicts between concept image and concept definition. to do this it is necessary to give tasks that do not refer only to the concept image for a correct resolution, inducing the students to use the definition (baccaglini-frank, di martino, natalini and rosolini , 2018 (b)). 4 teaching as a decision problem today more than ever, the world of education has to work on the construction of personalities that can favour to all the students with freedom of choice and reactivity. the social context in which we live is complex because it comprehends factors of unpredictability and uncertainty: the educational systems have the job to provide a path that aims to thought and action autonomy. teaching as a decision-making model: strategies in mathematics from a practical requirement 119 the school, therefore, has an orientation character, where the term “orientation” indicates a continuous and personal process that involves awareness, learning and education in choice (biagioli, 2003). in particular, placing orientation as the main purpose of teaching "means developing strategies, methodologies and contents aimed from the acquisition of awareness to understanding the complex society and the mechanisms that govern the world of studies and work" (guerrini, 2017). to give the proper and necessary instruments to the student, in order to activate the auto-orientation processes, the teacher has to chose what the best didactic strategy is. therefore, on an operational point of view, the teaching is a decisional problem and has to be faced as it is. indeed, we can speak of decision when in a situation there are: alternatives (being able to act in several different ways), probability (the possibility that the results relating to each alternative will be achieved) and the consequences associated with the results. such factors are characteristic of the school world. so, to realise the best didactic strategy it is necessary to start with a representation of the problem: only through the calculation of the expectations and the evaluation of the results, it is possible to choose the right option. decisions can be studied in terms of absolute rationality or limited rationality. the first model ideally combines rationality and information by preferring the best alternative; the second recognises the objective narrowness of the human mind by proposing the selection of the most satisfactory alternative (lanciano, 2019-20). it is important to emphasise that the consequences of a decision are determined also by the context in which the decision-making process is developed. on the basis of the decision maker's knowledge of the state of nature. we distinguish various types of decisions: • decisions in a situation of certainty: when the decision-maker knows the state of nature; • decisions in risk situations: when the decision-maker does not directly know each state of nature, but has a probability measure for them; • decisions in situations of uncertainty: when the decision maker has neither information on the state of nature nor the probability associated with it. the decision maker can adopt two kinds of approaches: • normative approach. which bases the choice with reference to rational decision-making ideals; • descriptive approach which analyses how to make a decision based on the context. so, the teacher has to consider on the basis of the objectives and the context the various alternatives, and for each one of them, the possible consequences. for each pair (alternative, circumstance) the teacher obtains a result according to a utility function. viviana ventre, eva ferrara dentice, roberta martino 120 however, the decision is subjective: it is based on the criterion of obtaining a maximum value for the utility function. moreover, even if the choice is rational, it is made in terms of limited rationality because, in general, there are few alternatives, but it increases as the teacher expands his/her culture and experience (delli rocili, maturo, 2013; maturo, zappacosta, 2017). 4.1 a model for evaluating educational alternatives multi-criteria decision analysis (mcda) provides support to the decision maker, or a group of decision makers, when many conflicting assessments have to be considered, especially in data synthesis phase while working with complex and heterogeneous pieces of information. let a = {a1, a2, ..., am} be the set of the alternatives, i. e. the possible educational strategies. let o = {o1, o2, ..., on} be the set of the objectives that we want to achieve. let d = {d1, d2, ..., dk} be the set of the decision making processes. the first phase consists of the establishment of a procedure that is able to assign to each couple (alternative ai, objective oj) a pij score. in this way, the responsible for the decision measure the grade in which the alternative ai satisfies the objective oj. assume that pij is in [0, 1], where: • pij = 0 if the objective oj is not at all satisfied by ai; • pij = 1 if the objective oj is completely satisfied by ai. at the end of the procedure we obtain a matrix p = [pij] of the scores which is the starting point of the elaborations that lead to the choice of the alternative, or at least to their ordering, possibly even partial (maturo, ventre, 2009a, 2009b). there may be constraints: it could be necessary to establish for each objective oj a threshold j > 0, with the constraint pij ≥ j, for each i. furthermore, through a convex linear combinations of alternatives ai it is possible to take into consideration mixed strategies that will have the following form: a(h1, h2, ..., hm) = h1 a1 + h2 a2 + ... + hm am with: • h1, h2, ..., hm non-negative real numbers; • the hi’s are such that h1 + h2 + ... + hm = 1; the number hi can represent the fraction of time in which the teaching strategy ai is adopted. if we consider also the mixed strategies, then the single alternatives ai are called pure strategies. the mixed strategies are particularly considered in presence of “at risk” alternatives: these situations have high scores for certain objectives and low for others (possibly below the threshold). it is appropriate to construct a ranking of the alternative educational plans, i. e., a linear ordering of the alternatives that takes into account the objectives which contribute to the most suitable formation of the student. such a ranking can be usefully obtained by means of the application of the analytic hierarchy process, a procedure due to t. l. saaty (1980, 2008). teaching as a decision-making model: strategies in mathematics from a practical requirement 121 4.2 the analytic hierarchy process: attributions of weights and scores the analytic hierarchy process (ahp) is both a method and a technique that allows to compare alternatives of different qualitative and quantitative nature, not easily comparable in a direct way, through the assignment of numerical values that specify their priority. the first thing to do is represent the elements of the decision problem through the construction a hierarchical structure. indeed, the analytic hierarchy process is based on the representation of the problem in terms of a directed graph g = (v, a). let us recall that (knuth, 1973): • a directed graph, or digraph, is a pair g = (v, a), where v is a non-empty set whose elements are called vertices and a is a set of ordered pairs of vertices, called arcs; • a vertex is indicated with a latin letter; for every arc (u, v) u is called the initial vertex and v the final vertex or end vertex; • an ordered n-tuple of vertices (v1, v2, ..., vn), n > 1, is called a path with length n -1, if, and only if, every pair (vi, vi + 1), i = 1, 2, …, n-1, is an arc of g. furthermore, in our context, we assume the following conditions be satisfied from a directed graph: • the vertices are distributed in a fixed integer number n ≥ 2 of levels; each level is indexed from 1 to n; • there is only one vertex of level 1, called the root of the directed graph; • for every vertex v different from the root there is at least one path having the root as the initial vertex and v as the final vertex; • every vertex u of level i < n is the initial vertex of at least one arc and there are no arcs with the initial vertex of level n; • if an arc has the initial vertex of level i<n, then it has the end vertex in the level i+1. let us describe, considering for example n=3, functional aspects of each level: • the level 1 vertex is called the general objective and denoted go. it indicates the objective of the entire decision making process; • the vertices of level 2 are called criteria. with this level we indicate the parameters used to evaluate the alternatives; • vertices of level 3 are called alternatives that represent the various ways of reaching the go. so we have the structure shown in the next page (figure 4.1). viviana ventre, eva ferrara dentice, roberta martino 122 figure 4.1 the digraph of the functional aspects. for instance, to build up a decisional model for the mathematical didactic the elements of the hierarchy might be the following: • general objective: to guarantee a route that, through education, ensures to all students the formation of knowledge and the development of action and communication skills; • criterion 1: acquisition of the objectives set out in the educational plan; • criterion 2: ability to measure oneself against peers in a safe and rational way while respecting the ideas of others; • criterion 3: internalisation of the objectives set by the didactic plan. with the term “internalisation” we indicate the ability to re-elaborate knowledge from a critical and personal point of view; • criterion 4: ability to cope with trials, planned or not, without negative moods, anxiety and terror of judgement; • alternative 1: new didactics, inspired by the inquiry model that puts the pupil and her emotions at the center of the context with cooperative learning experiences; • alternative 2: traditional didactic, that is a model of theaching-learning that prefers frontal lessons. in terms of learning this alternative hypothesises that the acquisition and internalisation take place at the same time and that the error is the manifestation of the failure to complete one of the two processes. • alternative 3: distance learning, inspired by the inquiry model mediated through the use of technological tools characterised by a total absence of sharing the same physical space between student and teacher. a decision-maker assigns a score to each arc following the ahp procedure (saaty, 1980, 2008; see also: maturo, ventre, 2009a, 2009b). so, the second step consists in determining the ratios of preference of the elements of a level over teaching as a decision-making model: strategies in mathematics from a practical requirement 123 any of the higher, i.e. previous, level: we, therefore, compare alternatives a1, a2 and a3 with respect to each criterion and the individual criteria with respect to the general objective. in order to determine the value of each comparison, the following scale of evaluations is used: value aij interpretation 1 i and j are equally important 3 i is a little more important than j 5 i is quite more important than j 7 i is definitely more important than j 9 i is absolutely more important than j 1/3 i is a little less important than j 1/5 i is quite less important than j 1/7 i is definitely less important than j 1/9 i is absolutely less important than j table 4.1 scale of evaluations. matrices are called pairwise comparison matrices and they represent quantitative preferences between criteria or between alternatives and satisfy the followings: • if alternative i assumes the value x in comparison with alternative j with respect to a criterion, then the comparison of alternative j with alternative i with respect to the same criterion assumes the value 1/x. analogous is the procedure to assign values when comparing couples of criteria; • since equally important alternatives correspond to value 1, the diagonal of the matrices are composed entirely of unit values. in our case, we obtain the matrices shown in the next page (from table 4.2 to table 4.6) whose values have been assigned due to the following considerations: • acquisition and internalisation are two different processes and only through a mutual combination of them the full formation of the student can be guaranteed; • traditional didactic is far from the social character of human learning; • the relationship with others is a fundamental space for personal and social development, necessary for the student to learn to respect rules and roles; • distance learning offers insufficient physical interaction between studentteacher and student-student: expressions and gestures make the difference in the learning process; • exercising young people to face tests in a lucid way is a fundamental aspect for the construction of a personality that faces in a competitive way the working challenges of a competitive society. viviana ventre, eva ferrara dentice, roberta martino 124 o c1 c2 c3 c4 c1 1 5 1 1 c2 1/5 1 1/5 1/7 c3 1 5 1 1 c4 1 7 1 1 table 4.2 matrix m1: comparison of criteria by the ratio of preference with respect to the general objective c1 a1 a2 a3 a1 1 5 1 a2 1/5 1 1/3 a3 1 3 1 table 4.3 matrix m2: comparison of alternatives by the ratio of preference with respect to the criterion 1 c2 a1 a2 a3 a1 1 5 7 a2 1/5 1 3 a3 1/7 1/3 1 table 4.4 matrix m3: comparison of alternatives by the ratio of preference with respect to the criterion 2 c3 a1 a2 a3 a1 1 7 5 a2 1/5 1 3 a3 1/7 1/3 1 table 4.5 matrix m4: comparison of alternatives by the ratio of preference with respect to the criterion 3 teaching as a decision-making model: strategies in mathematics from a practical requirement 125 c4 a1 a2 a3 a1 1 3 7 a2 1/3 1 3 a3 1/7 1/3 1 table 4.6 matrix m5: comparison of alternatives by the ratio of preference with respect to the criterion 4 after constructing the pairwise comparison matrices, we have to fix the weights of the elements in each level. this step is fundamental to determine whether the matrices are relevant through a scale of values ranging from 0 to 1. these weights must meet the normality condition: w1+w2+…+wn =1 this procedure supposes that, if the decision maker knew all the actual weights of the elements of the pairwise comparison matrix, then it would be: 𝐴 = ቀ 𝑤𝑖 𝑤𝑗ൗ ቁ = ቌ 𝑤1 𝑤1ൗ ⋯ 𝑤1 𝑤𝑛ൗ ⋮ ⋱ ⋮ 𝑤𝑛 𝑤1ൗ ⋯ 𝑤𝑛 𝑤𝑛ൗ ቍ in this case the weights would be obtained from any of the rows which are all multiple of the same row ൫1 𝑤1ൗ , 1 𝑤2ൗ , … , 1 𝑤𝑛ൗ ൯. it follows that the matrix a has rank 1. being w= (w1, w2, …, wn) t we get: aw = nw. thus, from the equation above, n is an eigenvalue of a and w is one of the eigenvectors associated with n. since the elements on the diagonal are all 1, denoted with λ1, λ2, …, λn = n the eigenvalues of a, the value of the trace of a is: tr(a)= λ1 + λ2 + … + λn = n as n is an eigenvalue of a the other n-1 eigenvalues of a must be zero. the matrix a satisfies the condition: aijajk = aik for every i, j, k, called consistency condition, that implies transitivity of the preferences, and a is said to be consistent is said to be a. in practice the decision maker does not know the vector w: the aij values that he assigns according to his judgement may deviate from the unknown wi/wj. so the decision maker may produce inconsistent pairwise comparison matrices. however the closer the aij values are to wi/wj, the closer the maximum eigenvalue is to n and the closer the other eigenvalues are to zero. therefore the vector of the weights w't = (w'1, w'2, …, w'n) associated to the maximum eigenvalue (among the infinite w't we choose the one for which w'1 + w'2 + … + w'n =1) will be an estimate of the vector w t = (w1, w2, …, wn) the viviana ventre, eva ferrara dentice, roberta martino 126 more precise the more the maximum eigenvalue 𝜆max of a is close to n, what is due to the continuity of the involved operations (ventre, 2019). where: 𝐶𝐼 = 𝜆𝑚𝑎𝑥 − 𝑛 𝑛 − 1 is defined as the consistency index of a, and n is the order of the matrix itself. the consistency index reveals how far from consistency the matrix is. in our case with the help of matlab we obtain: m1 m2 m3 m4 m5 𝜆max 4,0142 3,0291 3,0649 3,0649 3,0070 ci 0,0047 0,0145 0,0324 0,0342 0,0035 table 4.7 ci values for each matrix. we can therefore write the local priorities obtained by proceeding with the last step of the ahp method which, through the aggregation of the relative weights of each level, provides a weighted ranking of the alternatives. the third step implements the estimation of local assessments, i.e. the weightings that express the relative importance of the elements of a hierarchical level over any element of the next higher level. figure 4.2 the digraph of the functional aspects, with weights. we observe that scores are nonnegative real numbers and such that the sum of the scores of the arcs coming out of the same vertex u is equal to 1. the score assigned to an arc (u, v) indicates the extent to which the final vertex v meets the initial vertex u: the score of a path is the product of the scores of the arcs that form the path. • for every vertex v different from go the score p(v) of v is the sum of the scores of all the paths that start from go and arrive in v. teaching as a decision-making model: strategies in mathematics from a practical requirement 127 • for every level, the sum of the points of the vertices of level i is equal to 1. the global properties determined will then be: • a1=0,42*0,29+0,43*0,06+0,44*0,29+0,65*0,36=0,51; • a2=0,28*0,29+0,42*0,06+0,15*0,29+0,26*0,36=0,24; • a3=0,3*0,29+0,15*0,06+0,41*0,29+0,09*0,36=0,25. we can conclude that, in order to achieve the objective, a new teaching strategy is preferred to a traditional one and direct communication and human contact are factors not to be overlooked. therefore, teaching cannot be reduced to an online practice because the presence of the student is not attributable to a "virtual presence". 5 mathematics and specific learning disorders 5.1 background students with specific learning disorders (sld) need a personalised learning plan that appropriately accommodates their difficulties. at the university a student with sld attending the course of mathematical analysis has succeeded, thanks to the semester tutoring activity dedicated to the subject, to face his difficulties by passing the exam at the first useful date. the course has evolved through a personalised didactic strategy mostly based on the use of mind maps and peer comparison activities. specific learning disorders sld usually involve reading, writing and calculation skills. in italy, two students with dyslexia out of three do not receive an adequate diagnosis of the disorder and therefore sld are one of the main causes of school difficulties with important negative repercussions also in the personal sphere of the individual. it is therefore clear the importance of a conscious environment able to respect the “different” way of learning of a student with sld whose cognitive abilities and physical characteristics are in the norm. in fact, although they have an intelligence appropriate to their age, unlike their peers, these subjects learn at a slower pace because during the study they dissipate most of their energy to compensate for their disorders. initiatives promoted by the miur, accompanied by individual schooling, support the right to study of students with sld, which since the beginning of the year two thousand is also protected at the legislative level (miur law no.170/2010). to this end, the regional school offices are committed to promote the issue of detailed certifications that allow as much as possible, together with parents and the figures who follow the student in school activities, a personalised educational plan that aims to achieve the same objectives of peers through compensatory tools and dispensative measures (miur, 2011). viviana ventre, eva ferrara dentice, roberta martino 128 5.2 slds in mathematics specific reading disorder on the whole, this disorder leads to difficulties in reading, which is slow and incorrect, making it difficult to understand the text and to distinguish useful parts from those containing additional information (icd f81.0, 2007). in mathematics, these deficits make even the simplest problems complex as the student may not be able to distinguish between hypotheses and data. in addition, to maximise learning, rather than studying theory from the book, we suggest the constant use of mind maps that allow you to correct and rework ideas slowly until the subject is mastered. this type of maps is the one that best suits the way of learning of students with specific reading disorders because, through images and colors, it aims to stimulate visual memory through the insertion of mental associations. specific writing disorder the specific writing disorder is called dysgraphia if it affects writing and dysorthography if it affects spelling. the former involves a deficit of a motor nature and refers to the graphic aspects of handwriting, the latter is a text encoding disorder that therefore involves the linguistic component. disgraphers therefore produce poorly readable texts (even by themselves) with words that are often misaligned and characterised by letters of different sizes, while disorthographers manifest errors such as inversion of syllables, arbitrary cuts of words and omissions of letters in words making the content unclear. in both cases the elaboration of a written text is a difficult and long process with serious repercussions also in the mathematical field. in fact, mathematics has its own language, characterised by symbols, signs and letters of the greek and latin alphabets: just think of the use of lowercase greek letters in the geometric field to indicate angles and uppercase latin letters to indicate vertices. in the set of symbols, the symbol of belonging “∈” may not be decoded, leading to confusion with “e” even though it does not denote, from a didactic point of view, a lack of understanding of the set meaning itself. inaccurate writing also causes errors in the resolution of algebraic expressions, for example by confusing the letter “s” and the number “5”, or in the resolution of a linear system which requires many transcription steps. another difficulty due to alignment is found in the case of powers where base and exponent are confused with a multi-digit number (34 instead of 34). along this line, the teacher should take into account the content rather than the form, in the process of evaluating the written texts. errors should not be penalised when the concept expressed is clear, whereas oral verifications should acquire more weight in the final evaluations. in order to deal with these problems, the teacher, in the process of evaluating the written tests, could take into account the teaching as a decision-making model: strategies in mathematics from a practical requirement 129 content rather than the form, not penalising errors when the concept expressed is clear and giving more weight to oral verifications. specific disturbance of arithmetic activities the specific disturbance of arithmetic activities disorder implies a deficit that concerns the mastery of fundamental calculation skills which generally can be divided into two different profiles according to the type of error. the first profile is defined as "weakness in the cognitive structuring of numerical cognition components" and summarises both the difficulty in comparing/quantifying the elements of one or more sets and the reduced counting capacity. this type of disorder is inspired by butterworth's studies (1999; 2005): he hypothesised the existence of a "mathematical brain" specialised in classifying the world in terms of numbers. the second profile is renamed with "difficulty in acquiring calculation procedures and algorithms" and includes the following three cases (temple, 1991): dyslexia for digits indicating an incorrect reading/writing of the numbers (the student sees the number 3 and pronounces 6); procedural dyscalculia indicating the difficulty in the choice, application and maintenance of procedures leading to errors in borrowing, carry-over or sticking; dyscalculia for arithmetic facts which leads to confusion between the rules of rapid access with the consequent compromise of the acquisition of numerical facts within the calculation system. for both profiles examined the tools to compensate could be tables, diagrams, calculators and an extensive collection of procedural examples. objective learning disorders are, therefore varied and create different deficits that require different compensatory instruments and dispensation measures. for this reason, it is important to experiment the different strategies for teaching in order to identify a scheme which, although it can never be universal, can be taken as a canvas and then refined according to the personal limits and objectives of the student. the aim is, therefore to encourage learning with the aim of making the student as autonomous as possible, also increasing the level of self-esteem and personal gratification. below is the strategy used for a student with specific reading and writing disorders during the tutoring activity of the mathematical analysis course 1. the case study the student, after presenting his certificate at the beginning of the course, immediately showed interest in possible remedial activities. although the student was initially autonomous, after almost a month, the first difficulties began to emerge. this situation prompted the student to make constant use of the tutoring hours in which a personalised teaching strategy was constructed. the construction of such a strategy was obtained through the procedure previously shown: alternatives, objectives and criteria were modified viviana ventre, eva ferrara dentice, roberta martino 130 considering the specific difficulties. the method proved to be successful: indeed, the student was able to acquire his study methodology, which proved to be effective. stuff and methods from a didactical point of view, it is particularly interesting that problems related to short-term memory are a common feature of the different slds. baddeley and hitch (1974) extended this concept, in the field of cognitive psychology, to the "working memory" which specifically indicates that set of notions necessary for written and oral productions that remain short in the student's mind. if this capacity is reduced, then temporary archiving and the first management/manipulation of data will be compromised with the following consequences: difficulty in taking notes, difficulty in maintaining attention, need for longer periods. to cope with this situation, it is necessary to reduce the information load by giving priority to fundamental concepts and using support tools that favour direct observation and experimentation. some indications for the didactic strategy are: • use of schemes and concept maps; • dispense with reading aloud and mnemonic study; • privileging learning from experience and laboratory teaching; • encourage students to self-assess their learning processes; • encourage peer tutoring and promote collaborative learning; • guarantee longer times for written tests and study; • take an encouraging attitude to improve self-esteem; • evaluate according to progress and difficulties; • use of calculator and digital devices. in our case, the implementation of the didactic strategy took place through afternoon meetings, lasting two hours, usually held after the lesson held by the teacher in the same morning. at first, the meeting was based on the study of the last topics explained by the teacher and already at this stage it was evident to what extent the characteristic features of dysgraphia hindered learning: the bare and confused notes, characterised by large empty spaces, were practically impossible to read and to study. in order to try to provide constructivist learning and to avoid the use of the book, since reading was slow and incorrect due to dyslexia, the theory was flanked by practice or questions to answer. once the new subject was finished, work was done on the previous ones. in order to prevent the pupil from distracting himself, while following me with interest, i often drew his attention by naming him and repeating the concept in different ways. difficulties began to come up from the first topics when, in the numerical set exercises, there was a lot of confusion between the parentheses used to describe the intervals. for example: • interval from 'a' to 'b' not including the extremes, ]a, b[; teaching as a decision-making model: strategies in mathematics from a practical requirement 131 • interval from 'a' to 'b' including the extremes, [a, b]; to cope with this difficulty, with a bit of imagination, i made him imagine drawing two arms 'embracing' the number if the number had to be present in the interval or, otherwise, they refused to do so. in this way, together with the use of graphic representations, the pupil acquired the competence to distinguish between the two types of parentheses and rarely found confusion until the end of the course. the mind maps, which we built together, were of great help especially in solving equations and inequalities with absolute values. initially, the student's difficulty consisted in not being able to “visualize” the writing of the systems that came out of the procedure. with the use of a map, similar to the one below, he was able, after several lessons, to carry out the simplest exercises correctly but still presenting difficulties for the more complex ones. the result was satisfactory, however, because i believe that these difficulties were linked not so much to a lack of understanding of the absolute value function, but rather to a lack of ability to concentrate for so long on the same procedure. example of a map for the absolute value function: figure 5.1 example of a map built during the activity. towards the end of the course, as the exam date was approaching, other students also started to attend tutoring lessons on a regular basis. for this motive, the last topics of the course, derived and function study, were addressed through a collective study during which the student with specific learning disorders discussed with his classmates to the point of realising, under my guidance, his mistakes. during the lessons we made extensive use of the calculator, reducing the material to be memorised as much as possible. for example, to study the domain of a function, initially the student made extensive use of the tables summarising the domains of elementary functions but, subsequently, using the calculator (we get "err" if the function is not applied to an elementary of the viviana ventre, eva ferrara dentice, roberta martino 132 domain) he learned what this procedure really represented by obtaining, on his own, the conditions for simple functions such as roots and logarithms. results the boy achieved the minimum objectives, and at the end of the course, he passed the exam. the fundamental concepts were acquired and therefore, the test was approached independently. the improvements were gradual: as the tutoring lessons increased, the student was able to follow more and more. the shared study experience with peers was certainly helpful as, through peer comparison, he gained such confidence that he could guide his peers in difficulty during the exercises. in this regard, it is important to stress that the objectives achieved are not only didactic in nature but also personal: the ability to compare oneself with peers while respecting the ideas of others has increased the level of self-esteem and gratification. 6 conclusion the purpose of this document is, therefore, to underline that by using saaty's hierarchical structure it is possible to choose the didactic strategies that best suit the various situations. in fact, all students and in particular students with sld can maximise their learning if followed correctly. in our case, the didactic plan allowed for peer learning. this strategy, having been applied to only one student, cannot be generalised because each case requires different objectives and tools depending on the disturbance and the problems that this entails. to conclude, we stress that the affective dimension has played a fundamental role: a serene and stimulating environment has been a necessary condition to motivate students to improve themselves. however, the suggested indications are a good start to deal with different cases. 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[4] miur, ministero dell’istruzione, dell’università e della ricerca (2011), linee guida per il diritto allo studio degli alunni e degli studenti con disturbi specifici di apprendimento. ratio mathematica volume 46, 2023 theoretical analysis on the growth kinetics of sars-cov (within host) pavithra sivasamy* vanthana ramesh kumar† abstract a mathematical model is investigated to analyze the biological interactions between the immune system and sars-cov (within host). homotopy perturbation method is executed to obtain an analytical solution to the non-linear system of ordinary differential equations. graphical illustration to these solutions is also presented. the reliability and the simplicity of the aforementioned method is studied through the comparison between the numerical and graphical results. this comparison aids the better understanding of the disease dynamics and also the establishment of probable strategies for the treatment of covid-19. keywords: mathematical modeling, covid-19, non -linear initial value problem, homotopy perturbation method. 2020 ams subject classifications: find your subject classification at https://mathscinet.ams.org/msnhtml/msc2020.pdf. 1 *the standard fireworks rajaratnam college for women, sivakasi – 626123, india. pavithramat@sfrcollege.edu.in †the standard fireworks rajaratnam college for women, sivakasi – 626123, india. vanthanamat@sfrcollege.edu.in 1received on september 15, 2022. accepted on december 15, 2022.published on march 20, 2023. doi: 10.23755/rm.v46i0.1074. issn: 1592-7415. eissn: 2282-8214. ©pavithra sivasamy et al. this paper is published under the cc-by licence agreement. 178 pavithra sivasamy and vanthana ramesh kumar 1 introduction the world health organization declared the sars-cov as a potential threat to the human society sharma et al. [2020]ramadan and shaib [2019]ouassou et al. [2020]. since then, researchers have been working really hard on various strategies to control the disease ahmed et al. [2020]nie et al. [2020]. a key factor to control the disease is by studying the disease severity and progression in the host. however, the quantitative analysis of the growth kinetics of the virus has not been able to study the severity in the host. it is therefore important to analyze the interactions within the host between sars-cov and the immune system du and yuan [2020]hernandez-vargas and velasco-hernandez [2020]wang et al. [2020]. and this can be achieved by developing a mathematical model as they serve as a tool for characterizing the disease dynamics and in the forecast of severity of the disease. a mathematical model is therefore developed by s.m.e.k chowdhury et.al chowdhury et al. [2022]in the context of immune surveillance. we study this model analytically. the goal of our work is to derive a closed form analytical solution for the covid-19 model for the susceptible, infected, recovered and exposed. this is done by implementing the technique of homotopy perturbation method [hpm] he [1999]he [2004a]he [2004b]he [2004c] which is the coupling of homotopy and perturbation technique. the derived analytical results are beneficial in two fronts: first, it would contribute to a better comprehension of the disease dynamics which assist in designing effective treatment strategies and secondly, would enable the scientist in modifying the covid-19 model to study the correlation between various model parameters and their impacts. this approach in its implementation is novel to this non-linear system of covid-19 model. 2 model formulation the mathematical framework of the developed model is built on the interplay between the lymphocytes and virus particles within the host. taking into account this interaction the model is proposed as chowdhury et al. [2022] de dt = a1 − a2e − a3ev (1) dei dt = a3ev − b1ei (2) dv dt = c1e i − c2v − −c3v l (3) dn dt = d1 − d2n (4) 179 theoretical analysis on the growth kinetics of sars-cov (within host) parameters physical meaning a1 regenration rate of epithelial cells a2 rate at which epithelial cells die a3 rate of infection of epithelial cells b1 rate at which the infected epithelial cells die c1 production rate of virus from infected epithelial cells c2 rate at which the virus dies e1 rate of proliferation of t-lymphocytes e2 rate at which the t-lymphocytes die e3 regeneration rate of t-lymphocytes c3 rate at which the natural killer cells kills the virus d1 external influx rate of natural killer cells d2 rate at which the natural killer cells die c4 rate at which the t-lymphocytes kills the virus n half saturation constant of t-lymphocytes table 1: nomenclature dl dt = e1lv (n + v ) − e2l + e3 (5) where denotes the count of susceptible epithelial cells e(t) , infected epithelial cells ei(t), viral load v (t) , natural killer cells n(t) and t-lymphocytes l(t) respectively. the production rate of the virus is c1 where as the infection rate is denoted by a3 . the natural death rate of susceptible epithelial cells, infected epithelial cells, viral load , natural killer cells and t-lymphocytes are a2, b1,c2,d2,e−2 respectively. a1 denotes the regeneration rate of the epithelial cells. the degeneration of virus particles occur when they are interacted with natural killer cells and t-lymphocytes at the rate d1 and d3 respectively. the nk cell’s constant external source is denoted as a2 . refers to the t-lymphocytes natural recruitment rate c4 . in the presence of virus particles, the t-lymphocytes proliferate at a rate n. 3 homotopy perturbation method epidemiological modeling of the diseases using nonlinear dynamical equations gives deeper insights into the behavioral patterns and the transmission dynamics of the disease. as solutions to these non-linear problems are a bit more complex, a substantial amount of work has been dedicated by researchers in developing a solution method to these models. such methods include adm , vim , ham , hpm he [1999]he [2004c]he [2005]abbasbandy [2006]rafei and ganji [2006]yıldırım and öziş [2007]sivasamy and kumar [2021] etc. in this paper, 180 pavithra sivasamy and vanthana ramesh kumar we execute hpm in finding an analytical solution to the covid-19 model. the advantage of hpm over other method is its ability to reduce a complex non-linear problem into a serious of linear equation and finding the solution the same. 3.1 basic idea of homotopy perturbtion method [hpm] consider the following function do(u) − f(r) = 0, rϵω (6) with the boundary conditions as bo(u, ∂u ∂n ) = 0, rϵγ (7) where d0 is a general differential operator, b0 is a boundary operator, f(r) is a known analytical function and γ is the boundary of the domain ω . in general, the operator d0 can be divided into a linear part l and a non-linear part n. we can rewrite the eqn (6) as l(u) + n(u) − f(r) = 0 (8) we now construct a homotopy as v(r, p) :→ ω × [0, 1] × ℜ by the homotopy technique which statisfies h(v, p) = (1 − p)[l(v) − l(v0)] + p[d0 − f(r)] = 0 (9) h(v, p) = l(v) − l(v0) + p[n(v) − f(r)] = 0 (10) here p is the embedding parameter and belongs to the interval [0, 1] and the initial approximation of eqn.(6) is u0 which satisfies the boundary condition. now,eqn (9) and (10) leads to h(v, 0) = l(v) − l(u0) = 0 (11) h(v, 1) = d0 − f(r) = 0 (12) setting p=0 makes the eqns. (9) and (10) as linear and seeting p=1 makes them non-linear.. this process is presented as l(v) − l(u0) = 0 to d0 − f(r) = 0. we use p, the embedding parameter as a small parameter and assume that the solutions of eqns. (9) and (10) can be written as a power series in p: v = v0 + pv1 + p 2v2 + ... (13) fixing p=1 leads to the approximation of eqn (6) as v = v0 + pv1 + p 2v2 + ... (14) this is the basic idea of the hpm. 181 theoretical analysis on the growth kinetics of sars-cov (within host) 4 analytical solution to the covid-19 model using homotopy perturbation method (1 − p) ( de dt = a1 − a2e ) + p ( de dt = a1 − a2e − a3ev ) = 0 (15) (1 − p) ( dei dt + b1e i ) = p ( ( dei dt = a3ev − b1ei ) = (16) (1 − p) ( dv dt + c2v ) + p ( ( dv dt = c1e i − c2v − −c3v l ) = 0 (17) (1 − p) ( dl dt + e2l − e3 ) + p ( ( dl dt = e1lv (n + v ) − e2l + e3 ) = 0 (18) supposing the approximate solutions of eq. (1-5) have the form e = e0 + pe1 + p 2e2 + ... (19) ei = ei0 + pe i 1 + p 2ei2 + ... (20) v = v0 + pv1 + p 2v2 + ... (21) l = l0 + pl1 + p 2l2 + ... (22) substituting the eq. (15-18) respectively into eq. (1-5) (1 − p) ( d(e0 + pe1 + p 2e2 + ...) dt − a1 + a2(e0 + pe1 + p2e2 + ...) ) + p ( d(e0 + pe1 + p 2e2 + ...) dt − a1 + a2(e0 + pe1 + p2e2 + ...) + a3(e0 + pe1 + p 2e2 + ...)(v0 + pv1 + p 2v2 + ...) = 0 (23) (1 − p) ( d(ei0 + pe i 1 + p 2ei2 + ...) dt + b1(e i 0 + pe i 1 + p 2ei2 + ...) ) + p ( d(ei0 + pe i 1 + p 2ei2 + ...) dt − a3(e0 + pe1 + p2e2 + ...)(v0 + pv1 + p2v2 + ...) − b1(ei0 + pe i 1 + p 2ei2 + ...) = 0 (24) 182 pavithra sivasamy and vanthana ramesh kumar (1 − p) ( d(v = v0 + pv1 + p 2v2 + ...) dt + c2(v = v0 + pv1 + p 2v2 + ...) ) + p ( ( d(v = v0 + pv1 + p 2v2 + ...) dt − c1(ei0 + pe i 1 + p 2ei2 + ...) − c2(v = v0 + pv1 + p2v2 + ...) − c3(v = v0 + pv1 + p2v2 + ...)(l0 + pl1 + p2l2 + ...) = 0 (25) (1 − p) ( d(l0 + pl1 + p 2l2 + ...) dt + e2(l0 + pl1 + p 2l2 + ...) − e3 ) + p ( ( dl dt = e1(l0 + pl1 + p 2l2 + ...)(v0 + pv1 + p 2v2 + ...) (n + (v0 + pv1 + p2v2 + ...)) − e2(l0 + pl1 + p2l2 + ...) + e3 = 0 (26) equating the terms of eq (23-26) with the identical powers of p, we obtain p0 : de0 dt + a2e − a1 = 0 (27) p0 : dei0 dt + b1e i 0 = 0 (28) p0 : dv0 dt + c2v0 = 0 (29) p0 : dl0 dt + e2l0 − e3 = 0 (30) de1 dt + a2e1 + a3e0v0 = 0 (31) dei1 dt − a3e0v0 + b1ei0 = 0 (32) dv1 dt − c1ei0 − c2v1 + c3v0l0 = 0 (33) dl dt − e1l0v0 (n + vi) + e2l1 = 0 (34) number of susceptible epithelial cells are given by e(t) = a1 a2 + (ei − a1 a2 )e−a2t − a3a1vie −c2t a2(a2 − c2) + a3(ei − a1a2 )vie (−a2+c2)t c2 + a3a1vie −a2t a2(a2 − c2) − a3(ei − a1a2 )vie −a2t c2 (35) 183 theoretical analysis on the growth kinetics of sars-cov (within host) infected epithelial cells are given by ei(t) = ei(t)eb1t + a3a1vie −c2t a2(b1 − c2) + a3(ei − a1a2 )vie (−a2−c2)t b1 − a2 − c2 + a3a1vie −b1t a2(b1 − c2) − a3(ei − a1a2 )vie −b1t b1 − a2 − c2 (36) viral load cells are given by v (t) = vi(t)e −c2t + c1eie −b1t c2b1 + ( c3d1vi d2 − c4e3vi e2 )te−c2t + c3(ni − d1d2 )vie (−(d2+c2)t d2 − c1eie −c2t c2(b1 − c2) − c3(ni − d1d2 )vie (−c2)t d2 + c3(ni − d1d2 )lie (−(e2+c2)t e2 − c3(ni − d1d2 )lie (−e2)t e2 (37) natural killer cells are given by n(t) = d1 d2 − ( ni − d1 d2 ) e−d2t (38) t-lymphocytes cells are given by l(t) = e3 e2 + (li − e3 e2 )e−e2t + e3e1vie −c2t e3(n + vi)(−c2 + e2) − e1(li − e3e2 )vie (−a2+e2)t c2(n + vi) − e3e1vie −e2t e3(n + vi)(−c2 + e2) + e1(li − e3e2 )vie (−e2+e2)t c2(n + vi) (39) 5 numerical simulation an analytical expression for the time-dependent non-linear covid-19 model is derived by executing the method of homotopy perturbation for the equations (1-6). the numerical solutions to these equations are obtained using matlab software. in order to check the efficiency of hpm in solving the covid-19 model, comparison between the numerical and analytical results are carried out which is illustrated in the figures 1-12. 184 pavithra sivasamy and vanthana ramesh kumar figure 1: illustration of numerical and analytical results for the populations of susceptible epithelial cells , infected epithelial cells , viral load , natural killer cells and t-lymphocytes against time t. figure 2: illustration of analytical and graphical results for the population of susceptible against time t. 185 theoretical analysis on the growth kinetics of sars-cov (within host) figure 3: illustration of analytical and graphical results for the population of susceptible against time t. figure 4: illustration of analytical and graphical results for the population of susceptible against time t. 186 pavithra sivasamy and vanthana ramesh kumar figure 5: illustration of analytical and graphical results for the population of susceptible against time t. figure 6: illustration of analytical and graphical results for the population of infected epithelial against time t. 187 theoretical analysis on the growth kinetics of sars-cov (within host) figure 7: illustration of analytical and graphical results for the population of infected epithelial against time t. figure 8: illustration of analytical and graphical results for the population of natural killer cells against time t. 188 pavithra sivasamy and vanthana ramesh kumar figure 9: illustration of analytical and graphical results for the population of natural killer cells against time t. figure 10: illustration of analytical and graphical results for the population of tlymphocytes against time t. 189 theoretical analysis on the growth kinetics of sars-cov (within host) figure 11: illustration of analytical and graphical results for the population of tlymphocytes against time t. figure 12: illustration of analytical and graphical results for the population of tlymphocytes against time t. 190 pavithra sivasamy and vanthana ramesh kumar 5.1 results and discussion figure 1 illustrates the comparison between the analytical and numerical results for the populations of susceptible epithelial cells , infected epithelial cells , viral load , natural killer cells and t-lymphocytes against time t for the parameter values n = 0.1, d1 = 5, a1 = 10, a2 = 0.02, a3 = 0.10, b1 = 0.10, c1 = 0.24, c2 = 5.36, c3 = 0.231, c4 = 0.431, e1 = 0.0041, e2 = 0.0796, e3 = 0.0146, d2 = 0.02. figure 2-4 presents the plot of susceptible epithelial cells against time t. figure 5-7 presents the plot of infected epithelial cells against time t. figure 8-9 indicated the plot of natural killer cells against time t. figure 10-12 represents the plot of t-lymphocytes against time t. figure 2 depicts that the ratio of susceptible cells increases with time as the regeneration rate increases. figure 3 indicates that the growth of epithelial cells is exponential when the rate of infection is higher. figure 4 presents that even when the death of the infected cells is higher the growth of susceptible cells increases as there is an increase in the production rate of infected cells. figure 5 depicts that the there is a steady decline in the rate of infected cells as the death rate of the infected cells increases. figure 6 presents that the rate of growth of infected cells decreases when the rate of infection of the epithelial cells is lower and also a steady decline in the death rate of infected epithelial cells. figure 7 indicates that the growth rate of the infected epithelial cells also depends on the death rate of the virus. figure 8 and 9 represents that rate of natural killer cells increases in spite of the infection when there is an increase in external influx. figure 10-12 presents that proliferation rate, regeneration are and the death rate of the t-lymphocytes has an impact on the growth of t-lymphocytes. 6 conclusions a theoretical model outling the interactions between the sars-cov and the immune system within the host has been investigated by combining the functions of nk cells and t-lymphocytes. homotopy perturbation method is executed to solve the non-linear equations and a closed form analytical solution is obtained.. graphical illustration of the analyical and numerical solution is performed. a sound agreement between these results is noted. this comparison shows that hpm is an effective approach for solving such models. references s. abbasbandy. application of he’s homotopy perturbation method for laplace transform. chaos, solitons & fractals, 30(5):1206–1212, 2006. 191 theoretical analysis on the growth kinetics of sars-cov (within host) s. ahmed, a. quadeer, and m. mckay. preliminary identification of potential vaccine targets for the covid-19 coronavirus (sars-cov-2) based on sars-cov. immunol stud, pages 1–15, 2020. s. chowdhury, j. chowdhury, s. f. ahmed, p. agarwal, i. a. badruddin, and s. kamangar. mathematical modelling of covid-19 disease dynamics: interaction between immune system and sars-cov-2 within host. aims mathematics, 7(2):2618–2633, 2022. s. q. du and w. yuan. mathematical modeling of interaction between innate and adaptive immune responses in covid-19 and implications for viral pathogenesis. journal of medical virology, 92(9):1615–1628, 2020. j.-h. he. homotopy perturbation technique. computer methods in applied mechanics and engineering, 178(3-4):257–262, 1999. j.-h. he. the homotopy perturbation method for nonlinear oscillators with discontinuities. applied mathematics and computation, 151(1):287–292, 2004a. j.-h. he. comparison of homotopy perturbation method and homotopy analysis method. applied mathematics and computation, 156(2):527–539, 2004b. j.-h. he. the homotopy perturbation method for nonlinear oscillators with discontinuities. applied mathematics and computation, 151(1):287–292, 2004c. j.-h. he. homotopy perturbation method for bifurcation of nonlinear problems. international journal of nonlinear sciences and numerical simulation, 6(2): 207–208, 2005. e. a. hernandez-vargas and j. x. velasco-hernandez. in-host modelling of covid19 kinetics in humans. medrxiv, page 20044487, 2020. q. nie, x. li, w. chen, d. liu, y. chen, h. li, d. li, m. tian, w. tan, and j. zai. 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relapsing remitting multiple sclerosis using homotopy perturbation method. nveo-natural volatiles & essential oils journal— nveo, pages 2613–2624, 2021. s. wang, y. pan, q. wang, h. miao, a. n. brown, and l. rong. modeling the viral dynamics of sars-cov-2 infection. mathematical biosciences, 328:108438, 2020. a. yıldırım and t. öziş. solutions of singular ivps of lane–emden type by homotopy perturbation method. physics letters a, 369(1-2):70–76, 2007. 193 ratio mathematica on δ-open sets in ideal nano topological spaces n. sekar ∗ r. asokan † i. rajasekaran ‡ abstract aim of this paper, the new notions of introduce δ-open sets in ideal nano topological spaces and investigate some of their properties. a comparison between these types of ideal nano continuity will be discussed. finally, we introduce application examples in ideal nano topological spaces. keywords: nano-open (resp. n-open), semi-n-open (resp. ns-open) semi-ni-open, pre-ni-open and strong β-ni-open and niδ-open. 2020 ams subject classifications: 54b05, 54c60, 54e55. 1 ∗research scholar, department of mathematics, school of mathematics, madurai kamaraj university, madurai, tamil nadu, india; sekar.skrss@gmail.com. †department of mathematics, school of mathematics, madurai kamaraj university, madurai, tamil nadu, india; rasoka mku@yahoo.co.in. ‡department of mathematics, tirunelveli dakshina mara nadar sangam college, t. kallikulam-627 113, tirunelveli district, tamil nadu, india; sekarmelakkal@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.1013. issn: 1592-7415. eissn: 2282-8214. c©the authors. this paper is published under the cc-by licence agreement. volume 45, 2023 185 n. sekar, r. asokan and i. rajasekaran 1 introduction an ideal i (16) on a space (x,τ) is a non-empty collection of subsets of x which satisfies the following conditions. 1. a∈ i and b ⊂a imply b ∈ i and 2. a∈ i and b ∈ i imply a∪b ∈ i. given a space (x,τ) with an ideal i on x if ℘(x) is the set of all subsets of x, a set operator (.)? : ℘(x) → ℘(x), called a local function of a with respect to τ and i is defined as follows: for a ⊂ x, a?(i,τ) = {x ∈ x : u ∩a /∈ i for every u ∈ τ(x)} where τ(x) = {u ∈ τ : x ∈ u} (2). the closure operator defined by cl?(a) = a∪a?(i,τ) (15) is a kuratowski closure operator which generates a topology τ?(i,τ) called the ?-topology which is finer then τ. we will simply write a? for a?(i,τ) and τ? for τ?(i,τ). if i is an ideal on x, then (x,τ,i) is called an ideal topological space or an ideal space. rajasekaran and nethaji, introduced pre-ni-open sets and α-ni-open sets in the concept of ideal nano topological spaces. in this paper, we introduce the notions of δ-open sets in ideal nano topological spaces and investigate some of their properties. a comparison between these types of ideal nano continuity will be discussed. finally, we introduce application examples in ideal nano topological spaces. 2 preliminaries definition 2.1. (9) let u be a non-empty finite set of objects called the universe and r be an equivalence relation on u named as the indiscernibility relation. elements belonging to the same equivalence class are said to be indiscernible with one another. the pair (u, r) is said to be the approximation space. let x ⊆u. 1. the lower approximation of x with respect to r is the set of all objects, which can be for certain classified as x with respect to r and it is denoted by lr(x). that is, lr(x) = ⋃ x∈u{r(x) : r(x) ⊆ x}, where r(x) denotes the equivalence class determined by x. 2. the upper approximation of x with respect to r is the set of all objects, which can be possibly classified as x with respect to r and it is denoted by ur(x). that is, ur(x)= ⋃ x∈u{r(x) :r(x)∩x 6=φ}. 3. the boundary region of x with respect to r is the set of all objects, which can be classified neither as x nor as not x with respect to r and it is denoted by br(x). that is, br(x)=ur(x)−lr(x). 186 on δ-open sets in ideal nano topological spaces definition 2.2. (3) let u be the universe, r be an equivalence relation on u and τr(x) = {u,φ,lr(x),ur(x),br(x)} where x ⊆ u. then r(x) satisfies the following axioms: 1. u and φ∈ τr(x), 2. the union of the elements of any sub collection of τr(x) is in τr(x), 3. the intersection of the elements of any finite subcollection of τr(x) is in τr(x). thus τr(x) is a topology on u called the nano topology with respect to x and (u,τr(x)) is called the nano topological space. the elements of τr(x) are called nano-open sets (briefly n-open sets). the complement of a n-open set is called n-closed. in the rest of the paper, we denote a nano topological space by (u,n), where n = τr(x). the nano-interior and nano-closure of a subset o of u are denoted by in(o) and cn(o), respectively. a nano topological space (u,n) with an ideal i on u is called (6) an ideal nano topological space and is denoted by (u,n ,i). gn(x)= {gn |x∈gn,gn ∈ n}, denotes (6) the family of nano open sets containing x. in future an ideal nano topological spaces (u,n ,i) is referred as a space. definition 2.3. (6) let (u,n ,i) be a space with an ideal i on u. let (.)?n be a set operator from ℘(u) to ℘(u) (℘(u) is the set of all subsets of u). for a subset o ⊆ u, o?n(i,n) = {x ∈ u : gn ∩ o /∈ i, for every gn ∈ gn(x)} is called the nano local function (briefly, n-local function) of a with respect to i and n . we will simply write o?n for o ? n(i,n). theorem 2.1. (6) let (u,n ,i) be a space and o and b be subsets of u. then 1. o ⊆b ⇒o?n ⊆b ? n, 2. o?n =cn(o ? n)⊆cn(o) (o ? n is a n-closed subset of cn(o)), 3. (o?n) ? n ⊆o ? n, 4. (o∪b)?n =o ? n ∪b ? n, 5. v ∈n ⇒v ∩o?n =v ∩ (v ∩o) ? n ⊆ (v ∩o) ? n, 6. j ∈ i ⇒ (o∪j)?n =o ? n =(o−j) ? n. theorem 2.2. (6) let (u,n ,i) be a space with an ideal i and o ⊆ o?n, then o?n =cn(o ? n)=cn(o). definition 2.4. (8) a subset a of a space (u,n ,i) is n?-dense in itself (resp. n?-perfect and n?-closed) if o ⊆o?n (resp. o =o ? n, o ? n ⊆o). the complement of a n?-closed set is said to be n?-open. 187 n. sekar, r. asokan and i. rajasekaran definition 2.5. (5) a subset o of u in a nano topological space (u,n) is called nano-codense (briefly n-codense) if u −o is n-dense. theorem 2.3. (6) let (u,n ,i) be an ideal nano space. then is i is n-codense ⇐⇒ o ⊆o? for every n-open set o. definition 2.6. (6) let (u,n ,i) be a space. the set operator c?n called a nano ?-closure is defined by c?n(o)=o∪o ? n for o ⊆u. it can be easily observed that c?n(o)⊆cn(o). theorem 2.4. (7) in a space (u,n ,i), if o and b are subsets of u, then the following results are true for the set operator n-cl?. 1. o ⊆c?n(o), 2. c?n(φ)=φ and c ? n(u)=u, 3. ifo ⊂b, then c?n(o)⊆c ? n(b), 4. c?n(o)∪c ? n(b)=c ? n(o∪b). 5. c?n(c ? n(o))=c ? n(o). definition 2.7. a subset o of a nano space (u,n), is called a 1. nano pre-open (resp. np-open) set (3) if o ⊆ in(cn(o)). 2. nano semi-open (resp. ns-open) set (3) if o ⊆cn(in(o)). 3. nano ε-open (resp. nε-open) set (13) if in(cn(o))⊆cn(in(o)). 4. nano nowhere dense (resp. n-nowhere dense) (4)if in(cn(o))=φ. definition 2.8. a subset o of an ideal nano space (u,n ,i), is called a 1. nano pre-i-open (resp. pre-ni-open) (10) if o ⊆ in(c ? n(o)). 2. nano semi-i-open (resp. semi-ni-open) (10) if o ⊆c?n(in(o)). 3. nano α-i-open (resp. α-ni-open) (10) if o ⊆ in(c ? n(in(o))). 4. strongly nano β-i-open (resp. sβ-ni-open) (11) if o ⊆c?n((in(c ? n(o))). theorem 2.5. (10) in a nano space (u,n ,i), if o is α-ni-open, then o is semini-open. definition 2.9. (1) a function f : (u,n ,i)−→ (v,n ′) is said to be 1. α-ni-continuous if f−1(h) is α-ni-open set in (u,n ,i) for every n-open set h in (v,n ′). 2. pre-ni-continuous if f−1(h) is pre-ni-open set in (u,n ,i) for every n-open set h in (v,n ′). 3. semi-ni-continuous if f−1(h) is semi-ni-open set in (u,n ,i) for every n-open set h in (v,n ′). 188 on δ-open sets in ideal nano topological spaces 3 on δ-open sets in ideal nano space definition 3.1. a subset o of an ideal nano space (u,n ,i), is called a nano iδ-open (resp. niδ-open) set if in(c ? n(o))⊆c ? n(in(o)). example 3.1. let u = {a1,a2,a3,a4} with u/r = {{a2},{a4},{a1,a3}} and x = {a3,a4}. then the nano topology n = {φ,{a4},{a1,a3},{a1,a3,a4},u} and i = {φ,{a3}}. clear that {φ,{a2},{a3},{a4},{a1,a3},{a2,a3},{a2,a4}, {a1,a2,a3},{a1,a3,a4},u} is niδ-open. proposition 3.1. let (u,n ,i) be an ideal nano space. then a subset of u is semi-ni-open ⇐⇒ if it is both niδ-open and sβ-ni-open. proof. necessity. let o be a semi-ni-open, then we have o ⊆ c?n(in(o)) ⊆ c?n(in(c ? n(o))). this show that o is sβ-ni-open. moreover, in(c ? n(o)) ⊆ c ? n(o) ⊆ c ? n(c ? n(in(o))) = c ? n(in(o)). therefore o is niδ-open. sufficiency. let o be niδ-open and sβ-ni-open, then we have in(c?n(o))⊆ c?n(in(o)). thus we obtain that c ? n(in(c ? n(o)))⊆c ? n(c ? n(in(o)))=c ? n(in(o)). since o is sβ-ni-open, we have o ⊆ c?n(in(c ? n(o))) ⊆ c ? n(in(o)) and o ⊆ c?n(in(o)). hence o is semi-ni-open. proposition 3.2. let (u,n ,i) be an ideal nano space. then a subset of u is α-ni-open ⇐⇒ if it is both niδ-open and pre-ni-open. proof. necessity. let o be a α-ni-open, since every α-ni-open set is semi-ni-open (10), by proposition 3.1 o is a niδ-open set. now we prove that o ⊆ in(c ? n(o)). since o is α-ni-open set, we have o ⊆ in(c ? n(in(o))) ⊆ in(c ? n(o)). hence o is a pre-ni-open set. sufficiency. let o be a niδ-open and pre-ni-open. then we have in(c?n(o))⊆ c?n(in(o)) and hence in(c ? n(o)) ⊆ in(c ? n(in(o))). since o is pre-ni-open, we have o ⊆ in(c ? n(o)). therefore we obtain that o ⊆ in(c ? n(in(o))) and hence o ia α-ni-open. remark 3.1. in (u,n ,i) ideal nano space. 1. sβ-ni-open and niδ-open are independent. 2. pre-ni-open and niδ-open are independent. example 3.2. in example 3.1, 189 n. sekar, r. asokan and i. rajasekaran 1. the set {a1} is sβ-ni-open set but not niδ-open. 2. the set {a2} is not sβ-ni-open but niδ-open. 3. the set {a1,a4} is pre-ni-open but not niδ-open. 4. the set {a2,a3} is not pre-ni-open but niδ-open. proposition 3.3. let o, p be subsets of an ideal nano space (u,n ,i). if o ⊆ p ⊆c?n(o) and o is niδ-open, then p is niδ-open. proof. suppose that o ⊆p ⊆c?n(o) and o is niδ-open. then, since o is niδ-open, we have in(c ? n(o)) ⊆ c ? n(in(o)). since, o ⊆ p , c ? n(in(o)) ⊆ c ? n(i(p)) and in(c ? n(o)) ⊆ c ? n(in(p)). since p ⊆ c ? n(o), we have c ? n(p) ⊆ c ? n(c ? n(o)) = c?n(o) and in(c ? n(p)) ⊆ in(c ? n(o)). therefore, we obtain that in(c ? n(p)) ⊆ c?n(in(p)). this show that p is a niδ-open. proposition 3.4. let o, p and q be subsets of an ideal nano space (u,n ,i). if o is niδ-open, then o = p ∪q, where p is α-ni-open, in(c ? n(q)) = φ and p ∩q=φ. proof. suppose that o is niδ-open. then we have in(c ? n(o)) ⊆ c ? n(in(o)) and in(c ? n(o))⊆ in(c ? n(ino))). now we have o = ( in(c ? n(o)) ∩ o ) ∪ ( o − in(c ? n(o)) ) . now, we set p = in(c ? n(o))∩o and q=o− in(c ? n(o)). we first show that p is α-ni-open, that is, p ⊆ in(c ? n(in(p)). now we have in(c ? n(in(p)) = in(c ? n(in(in(c ? n(o))∩o))) = in(c ? n(in(c ? n(o))∩ in(o))) = in(c ? n(in(o))). since o is niδ-open, in(c ? n(in(o)))⊇ in(c ? n(o))⊇p and thus p is α-ni-open. next we show that in(c ? n(q))=φ. since n ⊆n ?, c?n(k)⊆cn(k) for any subset k of u. therefore, we have in(c ? n(q))= in(c ? n(o∩(u−in(c ? n(o)))))⊆ in(c ? n(o))∩in(c ? n(u−in(c ? n(o))))⊆ in(c ? n(o))∩in(cn(u−in(c ? n(o))))⊆ in(c ? n(o))∩ (u − in(c ? n(o)))=φ. it is obvious that p ∩q=(in(c ? n(o))∩o)∩ (o− in(c ? n(o))) =φ. remark 3.2. in an ideal nano space (u,n ,i), nε-open and niδ-open sets are independent. example 3.3. in example 3.1, 1. the set {a1,a2} is nε-open but not niδ-open. 2. the set {a3} is not nε-open but niδ-open. 190 on δ-open sets in ideal nano topological spaces proposition 3.5. let (u,n ,i) be an ideal nano space and o ⊆ u. if o is both niδ-open and ns-closed, then o is a nε-open. proof. since o is ns-closed, in(cn(o)) ⊆ o and hence in(cn(o)) = in(o). thus, in(cn(o))⊆ in(o)⊆ in(c ? n(o))⊆c ? n(in(o))= in(o)∪(in(o)) ? n. (in(o)) ? n ⊆ cn(in(o)) and hence we obtain in(cn(o)) ⊆ cn(in(o)). this show that o is nε-open. proposition 3.6. let (u,n ,i) be an ideal nano space. let i = {φ} or i = h, where h is the ideal of n-nowhere dense. then a subset o of u is niδ-open ⇐⇒ o is a nε-open. proof. 1. let i = {φ}. then for every subset o of u, o?n = cn(o) and c ? n(o) = o∪o?n =o∪cn(o)=cn(o). therefore, the statement holds obviously. 2. let i =h, we have o?n =cn(in(cn(o))). first, let o be a nε-open. then in(cn(o)) ⊆ cn(in(o)) and in(c ? n(o)) = in(o ? n∪o)⊆ in(cn(o)∪o)= in(cn(o))⊆cn(in(o))=cn(in(cn(in(o))))= (in(o)) ? n ⊆c ? n(in(o)). therefore, o is niδ-open set. next, let o be a niδ-open set. then in(c ? n(o))⊆ c?n(in(o)). we have in(cn(o))= in(cn(in(cn(o))))= in(o ? n)⊆ in(o ? n ∪ o)= in(c ? n(o))⊆c ? n(in(o))= (in(o)) ? n∪in(o)=cn(in(cn(in(o))))∪ in(o)=cn(in(o)). therefore, nε-open. 4 on semi-δ-ni-continuous definition 4.1. a function f : (u,n ,i)−→ (v,n ′) is called a 1. semi-δ-ni-continuous if every h ∈n ′, f−1(h)∈niδ-open. 2. strong β-ni-continuous (resp. sβ-ni-continuous) if every h ∈n ′, f−1(h)∈ sβ-ni-open. theorem 4.1. for a function f : (u,n ,i)−→ (v,n ′), the following properties are equivalent. 1. f is semi-ni-continuous. 2. f is sβ-ni-continuous and semi-δ-ni-continuous. proof. the proof is obvious by proposition 3.1. 191 n. sekar, r. asokan and i. rajasekaran theorem 4.2. for a function f : (u,n ,i)−→ (v,n ′), the following properties are equivalent. 1. f is α-ni-continuous. 2. f is pre-ni-continuous and semi-ni-continuous. 3. f is pre-ni-continuous and semi-δ-ni-continuous. proof. the proof is obvious by proposition 3.1 and proposition 3.2. 5 conclusion because of the spaces is stripped of the geometric form and its is used to measure thing that are difficult to measure, such as intelligence, beauty and goodness. in this paper, different type of ideal nano continuity and ideal nano closed sets are introduced and studied. also, we introduce an applications example in ideal nano topology. some applications on them are given in some real life branches such as medicine and physics. references [1] v. inthumathi, r. abinprakash and m. parveen banu, some weaker form of continuous and irresolute mapping in nano ideal topological spaces, journal of new results in science, 8(1)(2019), 14-25. 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[16] r. vaidyanathaswamy, set topology, chelsea publishing company, new york, 1946. 193 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 41, 2021, pp. 206-222 206 on the solutions of pellian equation 𝑼𝟐 − 𝑫𝑽𝟐 = 𝒌𝟐𝑵 bilkis m. madni * devbhadra v. shah † abstract in this paper we consider a class of pell’s equation 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁, where 𝐷, 𝑁 are positive integers, 𝐷 is non-square and 𝑘 is any integer. when (𝑢, 𝑣) satisfy this equation, we define 𝑢+𝑣√𝐷 𝑘 to be its solution. we first introduce the class of solutions of this equation and call 𝑢+𝑣√𝐷 𝑘 to be the fundamental solution of the class, if 𝑣 is the smallest positive value which occurs in the solutions of that class. we first derive the necessary and sufficient condition for any two solutions of proposed equation to belong to the same class and the bounds for the values of 𝑢, 𝑣 occurring in the fundamental solution. we also derive an explicit formula which gives all the solutions of this equation. we further present some interesting recurrence relations connecting the values of 𝑢, 𝑣. finally, we obtain the results for total number of positive solutions of proposed equation not exceeding any given positive real number 𝑍. keywords: pell’s equation, solutions of pell’s equation, recurrence relations. 2010 ams subject classification‡: 11d09, 11d45. * bilkis madni (department of mathematics, veer narmad south gujarat university, surat, india); bilkis1110@gmail.com † devbhadra shah (department of mathematics, veer narmad south gujarat university, surat, india); drdvshah@yahoo.com ‡received on july 15, 2021. accepted on december 25, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.628. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. on the solutions of pellian equation 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 207 1. introduction in number theory, pell’s equation falls in the category of diophantine equations and it is considered to be one of the oldest diophantine equations. specifically, the term pell’s equation is used to refer any diophantine equation of the form 𝑢2 − 𝐷𝑣 2 = 𝑁, (1.1) where 𝐷 and 𝑁 are fixed non-zero integers and we wish to find integers 𝑢 and 𝑣 that satisfy the equation. throughout, we consider the integer 𝐷 to be positive and non-square. this condition is helpful because only it leaves open the possibility of infinitely many solutions in positive integers 𝑢, 𝑣. pellian equation 𝑥2 − 𝐷𝑦2 = 1, (1.2) where 𝐷 is fixed positive non-square integer, attracted attention of early mathematicians. it is known that (1.2) always has infinitude of solutions. we refer to stolt [11, 12] who nicely studied the equation 𝑥2 − 𝐷𝑦2 = ±4𝑁. in this paper we use the notions proposed by stolt. many authors considered some specific pell equations and discussed about their solutions. for completeness we recall that there are many papers which considered different types of pell’s equation. for extensive resources on pell’s equation, one can refer [1] – [14]. in the present paper we consider the pellian equation 𝑈2 − 𝐷𝑉 2 = 𝑘2𝑁, (1.3) where 𝐷, 𝑁 are positive integers, 𝐷 is non-square and 𝑘 is any integer. throughout we assume that (1.3) is solvable. if (𝑢, 𝑣) satisfy this equation, we define 𝑢+𝑣√𝐷 𝑘 to be the solution of (1.3). if 𝑢+𝑣√𝐷 𝑘 is any solution of (1.3), 𝑥 + 𝑦√𝐷 is any solution of (1.2) and 𝑥1 + 𝑦1√𝐷 is its smallest positive solution, then it is easy to observe that 𝑢+𝑣√𝐷 𝑘 × (𝑥 + 𝑦√𝐷) is also a solution of (1.3). this solution is said to be associated with the solution 𝑢+𝑣√𝐷 𝑘 . here we note that if two positive solutions 𝑢𝛼+𝑣𝛼√𝐷 𝑘 and 𝑢𝛽+𝑣𝛽√𝐷 𝑘 are associated then there exists an integer 𝑡 such that 𝑢𝛽+𝑣𝛽√𝐷 𝑘 = 𝑢𝛼+𝑣𝛼√𝐷 𝑘 × (𝑥1 + 𝑦1√𝐷) 𝑡 ; 𝑡 = 0, ±1, ± 2, … . (1.4) the set of all solutions associated with other forms a class of solutions of (1.3). among all the solutions 𝑢+𝑣√𝐷 𝑘 in any given class 𝐾 we now choose a solution 𝑢1+𝑣1√𝐷 𝑘 in the following way: let 𝑣1 be the least non-negative value of 𝑣 which occurs in 𝐾. then in this case, 𝑢1 is also uniquely determined. the solution 𝑢1+𝑣1√𝐷 𝑘 defined in this way is said to be fundamental solution of bilkis m. madni, devbhadra v. shah 208 the class. it can be observed that 𝑢1+𝑣1√𝐷 𝑘 is a fundamental solution of class 𝐾 if 𝑢1+𝑣1√𝐷 𝑘 × (𝑥1 − 𝑦1√𝐷) is not a positive solution of (1.3). thus if 𝑢𝛼+𝑣𝛼√𝐷 𝑘 is any fixed fundamental solution then all the positive solutions given by (1.4) are associated with each other. moreover, we observe that if (1.3) is solvable, then it has only finite number of classes of solutions. 2. the equation 𝑼𝟐 − 𝑫𝑽𝟐 = 𝒌𝟐𝑵 it is easy to observe that if 𝑢+𝑣√𝐷 𝑘 and 𝑢′+𝑣′√𝐷 𝑘 are any two integer solutions of (1.3) such that 𝑢 ≡ 𝑢′ and 𝑣 ≡ 𝑣′(𝑚𝑜𝑑 𝑁), then ( 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 , 𝑢𝑣′−𝑢′𝑣 𝑘𝑁 ) is the positive solution of the equation 𝑅2 − 𝐷𝑆2 = 𝑘2. we first present the necessary and sufficient condition for any two solutions of (1.3) to be associated with each other. theorem 2.1. if 𝑢+𝑣√𝐷 𝑘 and 𝑢′+𝑣′√𝐷 𝑘 are any two solutions of (1.3) then the necessary and sufficient condition for these two solutions to be associated with each other is that 𝑢𝑣′ −𝑢′𝑣 𝑘𝑁 is an integer. proof. since 𝑢+𝑣√𝐷 𝑘 and 𝑢′+𝑣′√𝐷 𝑘 are solutions of (1.3), we get 𝑢2 − 𝐷𝑣2 = 𝑘2𝑁 and 𝑢′ 2 − 𝐷𝑣′ 2 = 𝑘2𝑁. multiplying these two equations we get ( 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 ) 2 − 𝐷 ( 𝑢𝑣′−𝑢′𝑣 𝑘𝑁 ) 2 = 𝑘2. we first show that the solutions 𝑢+𝑣√𝐷 𝑘 and 𝑢′+𝑣′√𝐷 𝑘 of the pellian equation (1.3) are associated with each other if and only if both 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 and 𝑢𝑣′−𝑢′𝑣 𝑘𝑁 are integers. if two numbers 𝑢𝑢′ −𝐷𝑣𝑣′ 𝑘𝑁 and 𝑣𝑢′−𝑢𝑣′ 𝑘𝑁 are integers, then ( 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 ) 2 − 𝐷 ( 𝑢𝑣′−𝑢′𝑣 𝑘𝑁 ) 2 = 𝑘2. thus 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 + 𝑢𝑣′−𝑢′𝑣 𝑘𝑁 √𝐷 𝑘 is a solution of pellian equation 𝑅2 − 𝐷𝑆2 = 𝑘2. now 𝑢+𝑣√𝐷 𝑘 𝑢′+𝑣′√𝐷 𝑘 = 𝑢+𝑣√𝐷 𝑢′+𝑣′√𝐷 = (𝑢 + 𝑣√𝐷) × 𝑢′−𝑣′√𝐷 ±𝑘2𝑁 = 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘2𝑁 − 𝑢𝑣′−𝑢′𝑣 𝑘2𝑁 √𝐷. clearly this is a solution of (1.2). thus, both the solutions 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 and 𝑣𝑢′−𝑢𝑣′ 𝑘𝑁 of (1.3) are associated. conversely let two solutions 𝑢+𝑣√𝐷 𝑘 and 𝑢′+𝑣′√𝐷 𝑘 of (1.3) be associated with each other. then clearly, they should lie in the same solution class of (1.3). then on the solutions of pellian equation 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 209 we have 𝑢+𝑣√𝐷 𝑘 𝑢′+𝑣′√𝐷 𝑘 = 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘2𝑁 − 𝑢𝑣′−𝑢′𝑣 𝑘2𝑁 √𝐷, which should be some solution of (1.2). in this case we have 𝑢2 − 𝐷𝑣2 = 𝑘2𝑁 and 𝑢′ 2 − 𝐷𝑣′ 2 = 𝑘2𝑁. this gives ( 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 ) 2 − 𝐷 ( 𝑣𝑢′−𝑢𝑣′ 𝑘𝑁 ) 2 = 𝑘2. thus, 𝑘 × ( 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘2𝑁 , 𝑢𝑣′−𝑢′𝑣 𝑘2𝑁 ) = ( 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 , 𝑢𝑣′−𝑢′𝑣 𝑘𝑁 ) is an integer solution of 𝑅2 − 𝐷𝑆2 = 𝑘2. hence 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 and 𝑢𝑣′−𝑢′𝑣 𝑘𝑁 are integers. thus, it is now sufficient to show that 𝑢𝑢′ −𝐷𝑣𝑣′ 𝑘𝑁 is an integer when 𝑢𝑣′−𝑢′𝑣 𝑘𝑁 is an integer; and 𝑢𝑣′−𝑢′𝑣 𝑘𝑁 is not an integer when 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 is not an integer. since 𝑢+𝑣√𝐷 𝑘 and 𝑢′+𝑣′√𝐷 𝑘 are solutions of (1.3), we have 𝑢2 − 𝐷𝑣2 = ±𝑘2𝑁 , 𝑢′ 2 − 𝐷𝑣′ 2 = ±𝑘2𝑁. multiplying these two equations we get (𝑢𝑢′ − 𝑣𝑣′𝐷)2 − 𝐷(𝑢𝑣′ − 𝑣𝑢′)2 = 𝑘2(𝑘𝑁)2. (2.1) it is obvious from (2.1) that 𝑢𝑢′ − 𝐷𝑣𝑣 ′ is divisible by 𝑘𝑁 when 𝑢𝑣′ − 𝑢′𝑣 is divisible by 𝑘𝑁. that is, 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 is an integer when 𝑢𝑣′−𝑢′𝑣 𝑘𝑁 is an integer. conversely, if 𝑢𝑢′−𝐷𝑣𝑣′ 𝑘𝑁 is not an integer, then there exists an integer 𝑑 such that 𝑑 | 𝑘𝑁 but 𝑑 ∤ (𝑢𝑢′ − 𝐷𝑣𝑣′). now from (2.1) it is seen that if 𝑑 | (𝑢𝑣′ − 𝑢′𝑣), then 𝑑 | (𝑢𝑢′ − 𝑣𝑣′𝐷) too, which is contrary to the assumption. thus 𝑑 ∤ (𝑢𝑣′ − 𝑢′𝑣), that is 𝑢𝑣′−𝑢′𝑣 𝑘𝑁 is not an integer, as required. this proves the required result. we now derive the bounds for the values of 𝑢, 𝑣 occurring in the fundamental solutions. theorem 2.2. if 𝑢+𝑣√𝐷 𝑘 is the smallest fundamental solution of the class 𝐾 of the pellian equation (1.3) and if 𝑥1 + 𝑦1√𝐷 is the fundamental solution of (1.2), then 0 < |𝑢| ≤ 𝑘√ (𝑥1+1)𝑁 2 and 0 ≤ 𝑣 ≤ 𝑘𝑦1√ 𝑁 2(𝑥1+1) . proof. if both the inequalities to be proved are true for a class 𝐾, then they are also true for the conjugate class �̅�. thus, we can assume that 𝑢 is positive. note that 𝑢𝑥1−𝐷𝑣𝑦1 𝑘 = 𝑢𝑥1 𝑘 − (𝑦1√𝐷) ( 𝑣√𝐷 𝑘 ) = 𝑢𝑥1 𝑘 − √𝑥1 2 − 1 √ 𝑢2−𝑘2𝑁 𝑘2 = 𝑢𝑥1 𝑘2 − √(𝑥1 2 − 1) ( 𝑢2 𝑘2 − 𝑁) > 0. bilkis m. madni, devbhadra v. shah 210 now consider the solution ( 𝑢+𝑣√𝐷 𝑘 ) (𝑥1 − 𝑦1√𝐷) = 𝑢𝑥1−𝐷𝑣𝑦1+(𝑥1𝑣−𝑦1𝑢)√𝐷 𝑘 of (1.3) which belongs to the same class as 𝑢+𝑣√𝐷 𝑘 . but 𝑢+𝑣√𝐷 𝑘 is the fundamental solution of the class and by above 𝑢𝑥1−𝐷𝑣𝑦1 𝑘 is positive. thus, we must have 𝑢𝑥1−𝐷𝑣𝑦1 𝑘 ≥ 𝑢 𝑘 , as 𝑢 𝑘 occurs in fundamental solution of (1.3). from this inequality it now follows that 𝑢𝑥1 − 𝐷𝑣𝑦1 ≥ 𝑢. this gives 𝑢(𝑥1 − 1) ≥ 𝐷𝑣𝑦1, which gives 𝑢2(𝑥1 − 1) 2 ≥ 𝐷2𝑣2𝑦1 2 = (𝑢2 − 𝑘2𝑁)(𝑥1 2 − 1). this can be written as 𝑢2 𝑥1−1 𝑥1+1 ≥ 𝑢2 − 𝑘2𝑁, which eventually gives 𝑢2 ≤ (𝑥1+1)𝑘 2𝑁 2 . this proves the first of required inequality. again by (1.3) we have 𝐷𝑣2 = 𝑢2 − 𝑘2𝑁. using above inequality, we get 𝐷𝑣2 ≤ (𝑥1+1)𝑘 2𝑁 2 − 𝑘2𝑁 = (𝑥1−1)𝑘 2𝑁 2 . this gives 𝑣 ≤ 𝑘√ (𝑥1−1)𝑁 2𝐷 = 𝑘√ (𝑥1 2−1)𝑁 2𝐷(𝑥1+1) . since 𝑥1 + 𝑦1√𝐷 is the solution of (1.2), we thus get 0 ≤ 𝑣 ≤ 𝑘𝑦1√ 𝑁 2(𝑥1+1) , as required. we now present an illustration to demonstrate the results of this theorem. illustration. consider the pellian equation 𝑈2 − 7𝑉2 = 18. then clearly, 𝐷 = 7, 𝑘 = 3 and 𝑁 = 2. if we consider the equation 𝑥2 − 7𝑦2 = 1, then it is easy to see that 8 + 3√7 is its fundamental solution. this gives 𝑥1 = 8, 𝑦1 = 3. now if 𝑢+𝑣√7 3 is the smallest fundamental solution of equation 𝑈2 − 7𝑉2 = 18, then by above theorem we should have 0 < |𝑢| ≤ 3 × √ 9×2 2 and 0 ≤ 𝑣 ≤ 3 × 3√ 2 2×9 . this gives 0 < |𝑢| ≤ 9 and 0 ≤ 𝑣 ≤ 3. these are indeed true as 5+√7 3 is the smallest fundamental solution of the equation 𝑈2 − 7𝑉2 = 18. we now prove a very important result which produces all the fundamental solutions of (1.3). theorem 2.3. if 𝑢𝑖 + 𝑣𝑖 √𝐷 runs through all the fundamental solutions of (1.1) and 𝑟𝑗+𝑠𝑗√𝐷 𝑘 runs through all the fundamental solutions of 𝑅2 − 𝐷𝑆2 = 𝑘2, then all the fundamental solutions of 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 are covered by 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 = ( 𝑟𝑗+𝑠𝑗√𝐷 𝑘 ) (𝑢𝑖 ± 𝑣𝑖 √𝐷). (2.2) on the solutions of pellian equation 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 211 proof. if we multiply the surd conjugate of (2.2) with (2.2) then we get 𝐴𝑖𝑗 2 −𝐷𝐵𝑖𝑗 2 𝑘2 = ( 𝑟𝑗 2−𝐷𝑠𝑗 2 𝑘2 ) (𝑢𝑖 2 − 𝐷𝑣𝑖 2). since 𝑢𝑖 2 − 𝐷𝑣𝑖 2 = 𝑁 and 𝑟𝑗 2 − 𝐷𝑠𝑗 2 = 𝑘2, we get 𝐴𝑖𝑗 2 − 𝐷𝐵𝑖𝑗 2 = 𝑘2𝑁. this shows that 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 defined by (2.2) are the solutions of (1.3). we next show that the solutions 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 defined by (2.2) covers all the fundamental solutions of (1.3). on the contrary assume that there exists some positive solution, say 𝑋+𝑌√𝐷 𝑘 of (1.3) which is not covered by (2.2). then this solution will lie between any two successive solutions of (1.3) of some fixed class generated by 𝑟𝑗+𝑠𝑗√𝐷 𝑘 . this means for some fixed 𝑗, we have ( 𝑟𝑗+𝑠𝑗√𝐷 𝑘 ) (𝑢𝑖 ± 𝑣𝑖 √𝐷) ≤ 𝑋+𝑌√𝐷 𝑘 < ( 𝑟𝑗+𝑠𝑗√𝐷 𝑘 ) (𝑢𝑖+1 ± 𝑣𝑖+1√𝐷). then 𝑢𝑖 ± 𝑣𝑖 √𝐷 ≤ ( 𝑋+𝑌√𝐷 𝑘 ) ( 𝑘 𝑟𝑗+𝑠𝑗√𝐷 ) < 𝑢𝑖+1 ± 𝑣𝑖+1√𝐷. this gives 𝑢𝑖 ± 𝑣𝑖 √𝐷 ≤ ( 𝑋+𝑌√𝐷 𝑘 ) ( 𝑟𝑗−𝑠𝑗√𝐷 𝑘 ) < 𝑢𝑖+1 ± 𝑣𝑖+1√𝐷. we denote 𝜖 + 𝛿√𝐷 = ( 𝑋+𝑌√𝐷 𝑘 ) ( 𝑟𝑗−𝑠𝑗√𝐷 𝑘 ). (2.3) then 𝑢𝑖 ± 𝑣𝑖 √𝐷 ≤ 𝜖 + 𝛿√𝐷 < 𝑢𝑖+1 ± 𝑣𝑖+1√𝐷. (2.4) to prove the required result, it is sufficient to prove that (i) 𝜖 + 𝛿√𝐷 is a solution of (1.1), and (ii) 𝜖 > 0 and 𝛿 > 0 or 𝛿 < 0 depending on the sign of 𝑢𝑖 ± 𝑣𝑖 √𝐷. this will produce one positive solution of (1.1) between two consecutive fundamental solutions of (1.1) for any fixed class 𝑗, which is a contradiction. to prove (i), we multiply surd conjugate of (2.3) with (2.3). this gives 𝜖 2 − 𝐷𝛿 2 = ( 𝑋2−𝐷𝑌2 𝑘2 ) ( 𝑟𝑗 2−𝐷𝑠𝑗 2 𝑘2 ). since 𝑋2 − 𝐷𝑌2 = 𝑘2𝑁 and 𝑟𝑗 2 − 𝐷𝑠𝑗 2 = 𝑘2, we get 𝜖 2 − 𝐷𝛿 2 = 𝑁, which proves the first part. next, we show that 𝜖 and 𝛿 defined by (2.3) are positive. now 𝑟𝑗 2 − 𝐷𝑠𝑗 2 = 𝑘2 implies ( 𝑟𝑗+𝑠𝑗√𝐷 𝑘 ) ( 𝑟𝑗−𝑠𝑗√𝐷 𝑘 ) = 1 and 𝑋+𝑌√𝐷 𝑘 > 1. since 0 < 𝑟𝑗+𝑠𝑗√𝐷 𝑘 < ∞, clearly 0 < 𝑟𝑗−𝑠𝑗√𝐷 𝑘 < ∞. since 𝜖 + 𝛿√𝐷 = ( 𝑋+𝑌√𝐷 𝑘 ) ( 𝑟𝑗−𝑠𝑗√𝐷 𝑘 ), we get 0 < 𝜖 + 𝛿√𝐷 < ∞. also, since 𝜖 2 − 𝐷𝛿 2 = 𝑁, we get 0 < 𝑁 𝜖−𝛿√𝐷 < ∞, that is 0 < 𝜖−𝛿√𝐷 𝑁 < ∞. this gives 0 < 𝜖 − 𝛿√𝐷 < ∞. adding this result with 0 < 𝜖 + 𝛿√𝐷 < ∞ proves that 𝜖 > 0. bilkis m. madni, devbhadra v. shah 212 we further observe by (2.4) that 1 < 𝑢𝑖 ± 𝑣𝑖 √𝐷 ≤ 𝜖 + 𝛿√𝐷 < 𝑢𝑖+1 ± 𝑣𝑖+1√𝐷 < ∞. by considering the ‘+’ sign, we get 0 < 1 𝜖+𝛿√𝐷 ≤ 1 𝑢𝑖+𝑣𝑖√𝐷 < 1. then 0 < 𝜖−𝛿√𝐷 𝑁 ≤ 𝑢𝑖−𝑣𝑖√𝐷 𝑁 , which gives 0 < 𝜖 − 𝛿√𝐷 ≤ 𝑢𝑖 − 𝑣𝑖 √𝐷. subsequently we get 2𝛿√𝐷 = (𝜖 + 𝛿√𝐷) − (𝜖 − 𝛿√𝐷) ≥ (𝑢𝑖 + 𝑣𝑖 √𝐷) − (𝑢𝑖 − 𝑣𝑖 √𝐷) = 2𝑣𝑖 √𝐷. this gives 𝛿 ≥ 𝑣𝑖 > 0. also, if we select ‘–’ sign in 𝑢𝑖 ± 𝑣𝑖 √𝐷, then we have 𝜖 + 𝛿√𝐷 < 𝑢𝑖+1 − 𝑣𝑖+1√𝐷 < ∞, which implies that 0 < 1 𝑢𝑖+1−𝑣𝑖+1√𝐷 < 1 𝜖+𝛿√𝐷 . then 0 < 𝑢𝑖+1+𝑣𝑖+1√𝐷 𝑁 < 𝜖−𝛿√𝐷 𝑁 . this gives 𝜖 − 𝛿√𝐷 > 𝑢𝑖+1 + 𝑣𝑖+1√𝐷. thus, we get 2𝛿√𝐷 = (𝜖 + 𝛿√𝐷) − (𝜖 − 𝛿√𝐷) < (𝑢𝑖+1 − 𝑣𝑖+1√𝐷) − (𝑢𝑖+1 + 𝑣𝑖+1√𝐷) = −2𝑣𝑖+1√𝐷. this gives 𝛿 < −𝑣𝑖+1 < 0. hence 𝛿 > 0 or 𝛿 < 0 depends on the sign of 𝑢𝑖 ± 𝑣𝑖 √𝐷, which proves (ii). hence all the fundamental solutions 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 of (1.3) are covered by (2.2). we illustrate this by an example which justifies the meaning of “… covered by (2.2)”. illustration. as earlier we once again consider the equation 𝑈2 − 7𝑉2 = 18. then we have 𝐷 = 7, 𝑘 = 3 and 𝑁 = 2. next, we consider the pellian equation 𝑅2 − 7𝑆2 = 9 and if 𝑟𝑗+𝑠𝑗√7 3 runs through all of its fundamental solutions, then it can be observed that 𝑟1+𝑠1√7 3 = 4+√7 3 , 𝑟2+𝑠2√7 3 = 11+4√7 3 , 𝑟3+𝑠3√7 3 = 24+9√7 3 . also 𝑢𝑖 + 𝑣𝑖 √7 runs through all the fundamental solutions of 𝑢 2 − 7𝑣2 = 2, then we have 𝑢1 ± 𝑣1√7 = 3 ± √7. then above theorem claims that all the fundamental solutions of 𝑈2 − 7𝑉2 = 18 are covered by 𝐴1𝑗+𝐵1𝑗√2 3 = ( 𝑟𝑗+𝑠𝑗√𝐷 3 ) (3 ± √7); 𝑗 = 1, 2, 3. thus, 𝐴11+𝐵11√2 3 = ( 4+√7 3 ) (3 ± √7) = 19+7√7 3 , 5−√7 3 ; 𝐴12+𝐵12√2 3 = ( 11+4√7 3 ) (3 ± √7) = 61+23√7 3 , 𝟓+√𝟕 𝟑 and 𝐴13+𝐵13√2 3 = ( 24+9√7 3 ) (3 ± √7) = 135+51√7 3 , 𝟗+𝟑√𝟕 𝟑 . on the solutions of pellian equation 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 213 it can be observed that 5+√7 3 , 9+3√7 3 and 19+7√7 3 are the only three fundamental solutions of the equation 𝑈2 − 7𝑉2 = 18. thus 𝐴1𝑗+𝐵1𝑗√2 3 ; 𝑗 = 1,2,3 covers all the fundamental solutions of 𝑈2 − 7𝑉2 = 18. here the smallest fundamental solution of 𝑈2 − 7𝑉2 = 18 is 5+√7 3 and 𝑥1 + 𝑦1√𝐷 = 8 + 3√7 is the fundamental solution of 𝑥 2 − 7𝑦2 = 1. then every fundamental solution of 𝑈2 − 7𝑉2 = 18 should be smaller than 5+√7 3 × (8 + 3√7) = 61+23√7 3 . this is indeed true as the only fundamental solutions of 𝑈2 − 7𝑉2 = 18 are 𝐴11+𝐵11√2 3 = 19+7√7 3 , 𝐴12+𝐵12√2 3 = 5+√7 3 and 𝐴13+𝐵13√2 3 = 9+3√7 3 . we next derive an explicit formula which produces all the positive solutions of (1.3). theorem 2.4. if 𝑥1 + 𝑦1√𝐷 is the smallest positive solution of (1.2) and 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 defined by (2.2) runs through all the fundamentals solutions of (1.3), then all the integer solutions 𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 of 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 are given by 𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 = ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) (𝑥1 + 𝑦1√𝐷) 𝑛 ; 𝑛 ≥ 0. (2.5) proof. if we consider the surd conjugate of (2.5) and multiply with (2.5), we get 𝑢𝑖𝑗,𝑛 2 −𝐷𝑣𝑖𝑗,𝑛 2 𝑘2 = ( 𝐴𝑖𝑗 2 −𝐷𝐵𝑖𝑗 2 𝑘2 ) (𝑥1 2 − 𝐷𝑦1 2)𝑛. since 𝐴𝑖𝑗 2 − 𝐷𝐵𝑖𝑗 2 = 𝑘2𝑁 and 𝑥1 2 − 𝐷𝑦1 2 = 1, we get 𝑢𝑖𝑗,𝑛 2 − 𝐷𝑣𝑖𝑗,𝑛 2 = 𝑘2𝑁. thus 𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 defined by (2.5) are the solutions of (1.3). we next show that the solutions 𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 defined by (2.5) gives all the positive solutions of (1.3). on the contrary assume that there exists some positive solution, say 𝑋+𝑌√𝐷 𝑘 of (1.3) which is not covered by (2.5). then this solution will lie between any two successive solutions of (1.3) of some classes generated by 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 (for a fixed 𝑖, 𝑗). this means for some fixed 𝑖, 𝑗 and for some fixed 𝑚, we have ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) (𝑥1 + 𝑦1√𝐷) 𝑚 ≤ 𝑋+𝑌√𝐷 𝑘 < ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) (𝑥1 + 𝑦1√𝐷) 𝑚+1 . then ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) ≤ ( 𝑋+𝑌√𝐷 𝑘 ) (𝑥1 − 𝑦1√𝐷) 𝑚 < ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) (𝑥1 + 𝑦1√𝐷). we denote bilkis m. madni, devbhadra v. shah 214 𝜖+𝛿√𝐷 𝑘 = ( 𝑋+𝑌√𝐷 𝑘 ) (𝑥1 − 𝑦1√𝐷) 𝑚 . (2.6) thus, ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) ≤ 𝜖+𝛿√𝐷 𝑘 < ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) (𝑥1 + 𝑦1√𝐷). (2.7) to prove the required result, it is sufficient to prove that (i) 𝜖+𝛿√𝐷 𝑘 is a solution of (1.3) (ii) 𝜖 > 0, 𝛿 > 0. (iii) 𝜖+𝛿√𝐷 𝑘 is the solution of (1.3) smaller than all its fundamental solutions for the fixed classes 𝑖, 𝑗. this will contradict the fact that 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 runs through every fundamental solution of (1.3) for some fixed value of 𝑖, 𝑗. to prove (i), we take surd conjugate of (2.6) and multiply it with (2.6). this gives 𝜖2−𝐷𝛿2 𝑘2 = ( 𝑋2−𝐷𝑌2 𝑘2 ) (𝑥1 2 − 𝐷𝑦1 2)𝑚. since 𝑋2 − 𝐷𝑌2 = 𝑘2𝑁 and 𝑥1 2 − 𝐷𝑦1 2 = 1, we get 𝜖 2 − 𝐷𝛿 2 = 𝑘2𝑁, as required. again, since 1 < (𝑥1 + 𝑦1√𝐷) 𝑚 < ∞ and 𝑋+𝑌√𝐷 𝑘 > 1, we have 0 < (𝑥1 − 𝑦1√𝐷) 𝑚 < 1. hence, we get 0 < 𝜖+𝛿√𝐷 𝑘 < ∞, as 𝜖+𝛿√𝐷 𝑘 = ( 𝑋+𝑌√𝐷 𝑘 ) (𝑥1 − 𝑦1√𝐷) 𝑚 . this gives 0 < 𝜖−𝛿√𝐷 𝑘𝑁 < ∞, that is 0 < 𝜖−𝛿√𝐷 𝑘 < ∞, as 𝜖 2 − 𝐷𝛿 2 = 𝑘2𝑁. adding this result with 0 < 𝜖+𝛿√𝐷 𝑘 < ∞ proves that 𝜖 > 0. we further observe by (2.7) that 1 < 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ≤ 𝜖+𝛿√𝐷 𝑘 < ∞. taking reciprocal, we get 0 < 𝑘 𝜖+𝛿√𝐷 ≤ 𝑘 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 < 1. then 0 < 𝜖−𝛿√𝐷 𝑘𝑁 ≤ 𝐴𝑖𝑗−𝐵𝑖𝑗√𝐷 𝑘𝑁 . this gives 0 < 𝜖−𝛿√𝐷 𝑘 ≤ 𝐴𝑖𝑗−𝐵𝑖𝑗√𝐷 𝑘 . subsequently we get 2𝛿√𝐷 𝑘 = ( 𝜖+𝛿√𝐷 𝑘 ) − ( 𝜖−𝛿√𝐷 𝑘 ) ≥ ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) − ( 𝐴𝑖𝑗−𝐵𝑖𝑗√𝐷 𝑘 ) = 2𝐵𝑖𝑗√𝐷 𝑘 . this now gives 𝛿 ≥ 𝐵𝑖𝑗 > 0. hence both 𝜖 > 0, 𝛿 > 0, which proves (ii). finally, we prove that 𝜖+𝛿√𝐷 𝑘 is the smallest solution of (1.3) for the fixed classes 𝑖, 𝑗. on the contrary suppose 𝜖+𝛿√𝐷 𝑘 is positive solution of (1.3) but not the smallest solution of any fixed classes 𝑖, 𝑗. in this case 𝜖+𝛿√𝐷 𝑘 (𝑥1 − 𝑦1√𝐷) will be the positive solution of (1.3). then by (2.7) we have 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ≤ 𝜖+𝛿√𝐷 𝑘 < ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) (𝑥1 + 𝑦1√𝐷), which gives ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) (𝑥1 − 𝑦1√𝐷) ≤ 𝜖+𝛿√𝐷 𝑘 (𝑥1 − 𝑦1√𝐷) < 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 . on the solutions of pellian equation 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 215 here 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 runs through every fundamental solution of (1.3). thus, we can now say that 𝜖+𝛿√𝐷 𝑘 (𝑥1 − 𝑦1√𝐷) is a positive solution of (1.3) smaller than all the fundamental solutions 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 of (1.3), which is a contradiction. this contradiction finally assures that there cannot exist any solution of (1.3) which is not covered by (2.5), as required. we illustrate this theorem by the following example: illustration. once again we consider the pellian equation 𝑈2 − 7𝑉2 = 18 whose all the fundamental solutions are given by 𝐴11+𝐵11√2 3 = 19+7√7 3 ; 𝐴12+𝐵12√2 3 = 5+√7 3 and 𝐴13+𝐵13√2 3 = 9+3√7 3 and 𝑥1 + 𝑦1√𝐷 = 8 + 3√7 is the fundamental solution of 𝑥2 − 7𝑦2 = 1. then the above theorem asserts that all the integer solutions 𝑢1𝑗,𝑛+𝑣1𝑗,𝑛√7 3 of pellian equation 𝑈2 − 7𝑉2 = 18 are covered by 𝑢1𝑗,𝑛+𝑣1𝑗,𝑛√7 3 = ( 𝐴1𝑗+𝐵1𝑗√7 3 ) (𝑥1 + 𝑦1√𝐷) 𝑛 ; 𝑗 = 1,2,3 ; 𝑛 ≥ 0. this gives 𝑢11,𝑛+𝑣11,𝑛√7 3 = ( 19+7√7 3 ) (8 + 3√7) 𝑛 ; 𝑢12,𝑛+𝑣12,𝑛√7 3 = ( 5+√7 3 ) (8 + 3√7) 𝑛 and 𝑢13,𝑛+𝑣13,𝑛√7 3 = ( 9+3√7 3 ) (8 + 3√7) 𝑛 . we now prove some recurrence relations for the values of 𝑢𝑖𝑗,𝑛 and 𝑣𝑖𝑗,𝑛. we assume that 𝑥1 + 𝑦1√𝐷 is the fundamental solution of (1.2), 𝑢𝑖 + 𝑣𝑖 √𝐷 runs through all the fundamental solutions of (1.1), 𝑟𝑗+𝑠𝑗√𝐷 𝑘 runs through all the fundamental solutions of 𝑅2 − 𝐷𝑆2 = 𝑘2, 𝑟 ≥ 1 is a fixed integer and 𝑛 ≥ 1. theorem 2.5. a) 𝑢𝑖𝑗,𝑛+𝑟 = 𝑥𝑟𝑢𝑖𝑗,𝑛+𝐷𝑦𝑟𝑣𝑖𝑗,𝑛 𝑘 , 𝑣𝑖𝑗,𝑛+𝑟 = 𝑦𝑟𝑢𝑖𝑗,𝑛+𝑥𝑟𝑣𝑖𝑗,𝑛 𝑘 . b) 𝑢𝑖𝑗,𝑛+𝑟 = 𝑥1𝑢𝑖𝑗,𝑛+𝑟−1+𝐷𝑦1𝑣𝑖𝑗,𝑛+𝑟−1 𝑘 , 𝑣𝑖𝑗,𝑛+𝑟 = 𝑦1𝑢𝑖𝑗,𝑛+𝑟−1+𝑥1𝑣𝑖𝑗,𝑛+𝑟−1 𝑘 . c) 𝑢𝑖𝑗,𝑛+2𝑟 = 2𝑘𝑥𝑟𝑢𝑖𝑗,𝑛+𝑟−𝑢𝑖𝑗,𝑛 𝑘2 , 𝑣𝑖𝑗,𝑛+2𝑟 = 2𝑘𝑥𝑟𝑣𝑖𝑗,𝑛+𝑟−𝑣𝑖𝑗,𝑛 𝑘2 . proof. (a) by (2.5), we have 𝑢𝑖𝑗,𝑛+𝑟+𝑣𝑖𝑗,𝑛+𝑟√𝐷 𝑘 = ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) (𝑥1 + 𝑦1√𝐷) 𝑛+𝑟 = ( 𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 ) (𝑥𝑟 + 𝑦𝑟 √𝐷) = (𝑥𝑟𝑢𝑖𝑗,𝑛+𝐷𝑦𝑟𝑣𝑖𝑗,𝑛)+(𝑦𝑟𝑢𝑖𝑗,𝑛+𝑥𝑟𝑣𝑖𝑗,𝑛)√𝐷 𝑘 . hence 𝑢𝑖𝑗,𝑛+𝑟 = 𝑥𝑟𝑢𝑖𝑗,𝑛+𝐷𝑦𝑟𝑣𝑖𝑗,𝑛 𝑘 , 𝑣𝑖𝑗,𝑛+𝑟 = 𝑦𝑟𝑢𝑖𝑗,𝑛+𝑥𝑟𝑣𝑖𝑗,𝑛 𝑘 . (b) to prove the second result, we first consider 𝑟 = 1 and replace 𝑛 by 𝑛 − 1 in the above result. we thus get bilkis m. madni, devbhadra v. shah 216 𝑢𝑖𝑗,𝑛 = 𝑥1𝑢𝑖𝑗,𝑛−1+𝐷𝑦1𝑣𝑖𝑗,𝑛−1 𝑘 , 𝑣𝑖𝑗,𝑛 = 𝑦1𝑢𝑖𝑗,𝑛−1+𝑥1𝑣𝑖𝑗,𝑛−1 𝑘 . then 𝑢𝑖𝑗,𝑛+𝑟 = 𝑥𝑟𝑢𝑖𝑗,𝑛+𝐷𝑦𝑟𝑣𝑖𝑗,𝑛 𝑘 = 1 𝑘 {𝑥𝑟 ( 𝑥1𝑢𝑖𝑗,𝑛−1+𝐷𝑦1𝑣𝑖𝑗,𝑛−1 𝑘 ) + 𝐷𝑦𝑟 ( 𝑦1𝑢𝑖𝑗,𝑛−1+𝑥1𝑣𝑖𝑗,𝑛−1 𝑘 )} = 1 𝑘 {𝑥1 ( 𝑥𝑟𝑢𝑖𝑗,𝑛−1+𝐷𝑦𝑟𝑣𝑖𝑗,𝑛−1 𝑘 ) + 𝐷𝑦1 ( 𝑥𝑟𝑣𝑖𝑗,𝑛−1+𝑦𝑟𝑢𝑖𝑗,𝑛−1 𝑘 )} = 𝑥1𝑢𝑖𝑗,𝑛+𝑟−1+𝐷𝑦1𝑣𝑖𝑗,𝑛+𝑟−1 𝑘 value of 𝑣𝑖𝑗,𝑛+𝑟 can be obtained accordingly. (c) to prove the final part, we replace 𝑛 by 𝑛 + 𝑟 in the first result and using that in (a) above, we obtain 𝑢𝑖𝑗,𝑛+2𝑟 = 1 𝑘 {𝑥𝑟 𝑢𝑖𝑗,𝑛+𝑟 + 𝐷𝑦𝑟 ( 𝑦𝑟𝑢𝑖𝑗,𝑛+𝑥𝑟𝑣𝑖𝑗,𝑛 𝑘 )} = 1 𝑘 {𝑥𝑟 𝑢𝑖𝑗,𝑛+𝑟 + 𝐷𝑦𝑟 2𝑢𝑖𝑗,𝑛 𝑘 + 𝑥𝑟 ( 𝐷𝑦𝑟𝑣𝑖𝑗,𝑛 𝑘 )} = 1 𝑘 {𝑥𝑟 𝑢𝑖𝑗,𝑛+𝑟 + 𝐷𝑦𝑟 2𝑢𝑖𝑗,𝑛 𝑘 + 𝑥𝑟 (𝑢𝑖𝑗,𝑛+𝑟 − 𝑥𝑟𝑢𝑖𝑗,𝑛 𝑘 )} = 1 𝑘 {2𝑥𝑟 𝑢𝑖𝑗,𝑛+𝑟 − 𝑢𝑖𝑗,𝑛 𝑘 (𝑥𝑟 2 − 𝐷𝑦𝑟 2)} since 𝑥𝑟 + 𝑦𝑟 √𝐷 is a solution of (1.2), we have 𝑥𝑟 2 − 𝐷𝑦𝑟 2 = 1. thus, we obtain 𝑢𝑖𝑗,𝑛+2𝑟 = 2𝑘𝑥𝑟𝑢𝑖𝑗,𝑛+𝑟−𝑢𝑖𝑗,𝑛 𝑘2 . value of 𝑣𝑖𝑗,𝑛+2𝑟 can also be obtained accordingly. we further derive some more interesting properties related with the value of 𝑢𝑖𝑗,𝑛 and 𝑣𝑖𝑗,𝑛. the following interesting recursive formula connects three 𝑢𝑖𝑗,𝑛’s as well as 𝑣𝑖𝑗,𝑛’s when the suffixes are in arithmetic progression. corollary 2.6. (a) 𝑢𝑖𝑗,𝑛𝑢𝑖𝑗,𝑛+2𝑟 − 𝑢𝑖𝑗,𝑛+𝑟 2 = 𝑘2𝐷𝑁𝑦𝑟 2. (b) 𝑣𝑖𝑗,𝑛𝑣𝑖𝑗,𝑛+2𝑟 − 𝑣𝑖𝑗,𝑛+𝑟 2 = −𝑘2𝑁𝑦𝑟 2. proof. by (2.5) we have 𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 = ( 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 ) (𝑥1 + 𝑦1√𝐷) 𝑛 . taking its surd conjugate, we get 𝑢𝑖𝑗,𝑛−𝑣𝑖𝑗,𝑛√𝐷 𝑘 = ( 𝐴𝑖𝑗−𝐵𝑖𝑗√𝐷 𝑘 ) (𝑥1 − 𝑦1√𝐷) 𝑛 . for convenience we write 𝛾 = 𝑥1 + 𝑦1√𝐷, �̅� = 𝑥1 − 𝑦1√𝐷 and 𝜇𝑖𝑗 = 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 , 𝜇𝑖𝑗̅̅̅̅ = 𝐴𝑖𝑗−𝐵𝑖𝑗√𝐷 𝑘 . then we have 𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 = 𝜇𝑖𝑗 𝛾 𝑛 and 𝑢𝑖𝑗,𝑛−𝑣𝑖𝑗,𝑛√𝐷 𝑘 = 𝜇𝑖𝑗̅̅̅̅ �̅� 𝑛. adding and subtracting these two relations, we get 𝑢𝑖𝑗,𝑛 = 𝑘 2 {𝜇𝑖𝑗 𝛾 𝑛 + 𝜇𝑖𝑗̅̅̅̅ �̅� 𝑛} and 𝑣𝑖𝑗,𝑛 = 𝑘 2√𝐷 {𝜇𝑖𝑗 𝛾 𝑛 − 𝜇𝑖𝑗̅̅̅̅ �̅� 𝑛}. on the solutions of pellian equation 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 217 it can be easily observed that 𝛾�̅� = 𝑥1 2 − 𝐷𝑦1 2 = 1 and 𝜇𝑖𝑗 𝜇𝑖𝑗̅̅̅̅ = 𝐴𝑖𝑗 2 −𝐷𝐵𝑖𝑗 2 𝑘2 = 𝑁. then 𝑢𝑖𝑗,𝑛𝑢𝑖𝑗,𝑛+2𝑟 − 𝑢𝑖𝑗,𝑛+𝑟 2 = 𝑘2 4 { (𝜇𝑖𝑗 𝛾 𝑛 + 𝜇𝑖𝑗̅̅̅̅ �̅� 𝑛)(𝜇𝑖𝑗 𝛾 𝑛+2𝑟 + 𝜇𝑖𝑗̅̅̅̅ �̅� (𝑛+2𝑟)) −(𝜇𝑖𝑗 𝛾 𝑛+𝑟 + 𝜇𝑖𝑗̅̅̅̅ �̅� (𝑛+𝑟)) 2 } = 𝑘2 4 { 𝜇𝑖𝑗 𝜇𝑖𝑗̅̅̅̅ (𝛾 𝑛+2𝑟 �̅� 𝑛 + 𝛾 𝑛�̅� (𝑛+2𝑟)) −2𝜇𝑖𝑗 𝜇𝑖𝑗̅̅̅̅ 𝛾 𝑛+𝑟 �̅� (𝑛+𝑟) } = 𝑘2 4 {𝑁(𝛾 2𝑟 + �̅� 2𝑟) − 2𝑁} = 𝑘2𝑁 4 (𝛾 𝑟 − �̅� 𝑟 )2. since 𝑥𝑟 + 𝑦𝑟 √𝐷 = (𝑥1 + 𝑦1√𝐷) 𝑟 = 𝛾 𝑟 and 𝑥𝑟 − 𝑦𝑟 √𝐷 = (𝑥1 − 𝑦1√𝐷) 𝑟 = �̅� 𝑟, we get 𝑢𝑖𝑗,𝑛𝑢𝑖𝑗,𝑛+2𝑟 − 𝑢𝑖𝑗,𝑛+𝑟 2 = 𝑘2𝐷𝑁𝑦𝑟 2. second result can be proved accordingly. following result gives some more recurrence relations in the form of a determinant. theorem 2.7. a) | 𝑢𝑖𝑗,𝑛 𝑢𝑖𝑗,𝑛+𝑟 𝑣𝑖𝑗,𝑛 𝑣𝑖𝑗,𝑛+𝑟 | = 𝑘𝑦𝑟 𝑁 b) | 𝑢𝑖𝑗,𝑛+𝑟−1 𝑢𝑖𝑗,𝑛+𝑟 𝑣𝑖𝑗,𝑛+𝑟−1 𝑣𝑖𝑗,𝑛+𝑟 | = 𝑘𝑦1𝑁 c) | 1 1 1 𝑢𝑖𝑗,𝑛−𝑟 𝑢𝑖𝑗,𝑛 𝑢𝑖𝑗,𝑛+𝑟 𝑣𝑖𝑗,𝑛−𝑟 𝑣𝑖𝑗,𝑛 𝑣𝑖𝑗,𝑛+𝑟 | = −2𝑘𝑁𝑦𝑟 (𝑥𝑟 − 1). proof. we only prove (c), since first two results follow easily through theorem 2.5. now | 1 1 1 𝑢𝑖𝑗,𝑛−𝑟 𝑢𝑖𝑗,𝑛 𝑢𝑖𝑗,𝑛+𝑟 𝑣𝑖𝑗,𝑛−𝑟 𝑣𝑖𝑗,𝑛 𝑣𝑖𝑗,𝑛+𝑟 | = | 𝑢𝑖𝑗,𝑛 𝑢𝑖𝑗,𝑛+𝑟 𝑣𝑖𝑗,𝑛 𝑣𝑖𝑗,𝑛+𝑟 | − | 𝑢𝑖𝑗,𝑛−𝑟 𝑢𝑖𝑗,𝑛+𝑟 𝑣𝑖𝑗,𝑛−𝑟 𝑣𝑖𝑗,𝑛+𝑟 | + | 𝑢𝑖𝑗,𝑛−𝑟 𝑢𝑖𝑗,𝑛 𝑣𝑖𝑗,𝑛−𝑟 𝑣𝑖𝑗,𝑛 | = 𝑘𝑦𝑟 𝑁 − 𝑘𝑦2𝑟 𝑁 + 𝑘𝑦𝑟 𝑁 = 𝑘𝑁(2𝑦𝑟 − 𝑦2𝑟 ). now (𝑥1 + 𝑦1√𝐷) 2𝑟 = 𝑥2𝑟 + 𝑦2𝑟 √𝐷 = (𝑥𝑟 + 𝑦𝑟 √𝐷) 2 . this gives 𝑦2𝑟 = 2𝑥𝑟 𝑦𝑟 . thus | 1 1 1 𝑢𝑖𝑗,𝑛−𝑟 𝑢𝑖𝑗,𝑛 𝑢𝑖𝑗,𝑛+𝑟 𝑣𝑖𝑗,𝑛−𝑟 𝑣𝑖𝑗,𝑛 𝑣𝑖𝑗,𝑛+𝑟 | = 𝑘𝑁(2𝑦𝑟 − 2𝑥𝑟 𝑦𝑟 ) = −2𝑘𝑁𝑦𝑟 (𝑥𝑟 − 1). bilkis m. madni, devbhadra v. shah 218 3. number of solutions up to a desired limit in this final section, we define and obtain the values of the sums 𝑅(𝑍) = ∑ 1𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 ≤ 𝑍 𝑈2−𝐷𝑉2=𝑘2𝑁 , 𝑆(𝑍) = ∑ 1𝑢𝑖𝑗,𝑛 ≤ 𝑍 𝑈2−𝐷𝑉2=𝑘2𝑁 and 𝑇(𝑍) = ∑ 1𝑣𝑖𝑗,𝑛≤ 𝑍 𝑈2−𝐷𝑉2=𝑘2𝑁 , the total number of positive solutions 𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 , 𝑢𝑖𝑗,𝑛 and 𝑣𝑖𝑗,𝑛 respectively of (1.3) that do not exceed any given large positive real number 𝑍. for convenience, we denote 𝛿 = 1 log 𝛾 , 𝛾 = 𝑥1 + 𝑦1√𝐷 and let 𝒜𝑖𝑗 = 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 runs through all the fundamental solutions of (1.3) for any fixed class 𝑖. we also assume that (1.1) has 𝛽 fundamental solutions and the equation 𝑅2 − 𝐷𝑆2 = 𝑘2 has 𝜂 fundamental solutions. thus, throughout we have 1 ≤ 𝑖 ≤ 𝛽 and 1 ≤ 𝑗 ≤ 𝜂. we first obtain the value of 𝑅(𝑍) which gives the number of all the solutions of (1.3) not exceeding any fixed given positive real number 𝑍. theorem 3.1. 𝑅(𝑍) = 𝛿 {𝛽𝜂 log(𝑍) − log(∏ ∏ 𝒜𝑖𝑗 𝜂 𝑗=1 𝛽 𝑖=1 )} + 𝐶, where 𝐶 is the effective constant such that 0 ≤ 𝐶 < 𝛽𝜂. proof. to find the value of 𝑅(𝑍) = ∑ 1𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 ≤ 𝑍 𝑈2−𝐷𝑉2=𝑘2𝑁 , we first find the number of positive solutions 𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 of (1.3) that do not exceed 𝑍 for some fixed class 𝑖 = 𝛼 (1 ≤ 𝑗 ≤ 𝜂). since 𝛾 and 𝒜𝑖𝑗 are solutions of (1.2) and (1.3) respectively, (2.5) can be written as 𝑢𝛼𝑗,𝑛+𝑣𝛼𝑗𝑛√𝐷 𝑘 = 𝒜𝛼𝑗 𝛾 𝑛. now, for any given 𝑍, it is clear that for some fixed class 𝑖 = 𝛼, there exists some n such that 𝑢𝛼𝑗,𝑛+𝑣𝛼𝑗,𝑛√𝐷 𝑘 ≤ 𝑍 < 𝑢𝛼𝑗,𝑛+1+𝑣𝛼𝑗,𝑛+1√𝐷 𝑘 . then we get 𝒜𝛼𝑗 𝛾 𝑛 ≤ 𝑍 < 𝒜𝛼𝑗 𝛾 𝑛+1. this implies 𝑛 < log 𝑍−log 𝒜𝛼𝑗 log 𝛾 < 𝑛 + 1. since n is an integer, we get 𝑛 = [ log 𝑍−log 𝒜𝛼𝑗 log 𝛾 ], where [𝑥] denotes the integer part of 𝑥. now since [𝑥] = 𝑥 − {𝑥}, where {𝑥} is the fractional part of 𝑥 and as 0 ≤ {𝑥} < 1, we have 𝑅(𝑍) = ∑ ∑ [ log 𝑍−log 𝒜𝑖𝑗 log 𝛾 ] 𝜂 𝑗=1 𝛽 𝑖=1 = ∑ ∑ ( log 𝑍−log 𝒜𝑖𝑗 log 𝛾 + 𝑐′) 𝜂 𝑗=1 𝛽 𝑖=1 , where 0 ≤ c′ < 1. thus 𝑅(𝑍) = 1 𝑙𝑜𝑔 𝛾 ∑ ∑ (log 𝑍 − log 𝒜𝑖𝑗 ) 𝜂 𝑗=1 𝛽 𝑖=1 + 𝛽𝜂𝑐 ′. if we write 𝐶 = 𝛽𝜂𝑐′, then we get 0 ≤ 𝐶 < 𝛽𝜂 and 𝑅(𝑍) = 𝛿 {𝛽𝜂 log(𝑍) − log(∏ ∏ 𝒜𝑖𝑗 𝜂 𝑗=1 𝛽 𝑖=1 )} + 𝐶. on the solutions of pellian equation 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 219 we next find the value of 𝑆(𝑍). theorem 3.2. 𝑆(𝑍) = 𝛿 {𝛽𝜂 log(2𝑍/𝑘) − log(∏ ∏ 𝒜𝑖𝑗 𝜂 𝑗=1 𝛽 𝑖=1 )} + 𝐶, where 𝐶 is the effective constant such that −𝛽𝜂 ≤ 𝐶 < 𝛽𝜂. proof. to find the value of 𝑆(𝑍), we first find the number of positive solutions of (1.3) where the values 𝑢𝑖𝑗,𝑛 of 𝑈 do not exceed 𝑍 for some fixed class 𝑖 = 𝛼 (1 ≤ 𝑗 ≤ 𝜂). now (2.5) can be written as 𝑢𝛼𝑗,𝑛+𝑣𝛼𝑗𝑛√𝐷 𝑘 = 𝒜𝛼𝑗 𝛾 𝑛. then for any given 𝑍, it is clear that for some fixed class 𝑖 = 𝛼, there exists some n such that 𝑢𝛼𝑗,𝑛 ≤ 𝑍 < 𝑢𝛼𝑗,𝑛+1. since 𝒜𝑖𝑗 = 𝐴𝑖𝑗+𝐵𝑖𝑗√𝐷 𝑘 , we write 𝒜𝑖𝑗̅̅ ̅̅ ̅ = 𝐴𝑖𝑗−𝐵𝑖𝑗√𝐷 𝑘 . we also have 𝛾 −1 = 𝑥1 − 𝑦1√𝐷. since 𝛾 and 𝒜𝑖𝑗 , 𝒜𝑖𝑗̅̅ ̅̅ ̅ are the solutions of (1.2) and (1.3) respectively, we have 𝛾𝛾 −1 = 1 and 𝒜𝛼𝑗 𝒜𝛼𝑗̅̅ ̅̅ ̅ = 𝑁. then (2.5) can be written as 𝑢𝛼𝑗,𝑛+𝑣𝛼𝑗,𝑛√𝐷 𝑘 = 𝒜𝛼𝑗 𝛾 𝑛. (3.1) now taking surd-conjugate of (3.1) we get 𝑢𝛼𝑗,𝑛−𝑣𝛼𝑗,𝑛√𝐷 𝑘 = 𝒜𝛼𝑗̅̅ ̅̅ ̅𝛾 −𝑛. adding this with (3.1) we now have 𝑢𝛼𝑗,𝑛 = 𝑘 2 {𝒜𝛼𝑗 𝛾 𝑛 + 𝒜𝛼𝑗̅̅ ̅̅ ̅𝛾 −𝑛} = 𝑘 2 {𝒜𝛼𝑗 𝛾 𝑛 + 𝑁 𝒜𝛼𝑗 𝛾 −𝑛}. since 𝑢𝛼𝑗,𝑛 ≤ 𝑍 < 𝑢𝛼𝑗,𝑛+1, for some n, we get 𝒜𝛼𝑗 𝛾 𝑛 + 𝑁 𝒜𝛼𝑗 𝛾 −𝑛 ≤ 2𝑍 𝑘 < 𝒜𝛼𝑗 𝛾 𝑛+1 + 𝑁 𝒜𝛼𝑗 𝛾 −𝑛−1. also, since 𝑁 𝒜𝛼𝑗 𝛾 −𝑛 > 0, 𝒜𝛼𝑗̅̅ ̅̅ ̅ = 𝑁 𝒜𝛼𝑗 < 𝒜𝛼𝑗 and 𝛾 −1 < 𝛾, we have 𝒜𝛼𝑗 𝛾 𝑛 < 2𝑍 𝑘 < 2𝒜𝛼𝑗 𝛾 𝑛+1. now 𝛾 = 𝑥1 + 𝑦1√𝐷 > 2. then we have 𝒜𝛼𝑗 𝛾 𝑛 < 2𝑍 𝑘 < 𝒜𝛼𝑗 𝛾 𝑛+2. this implies 𝑛 < log 2𝑍−log 𝒜𝛼𝑗−log 𝑘 log 𝛾 < 𝑛 + 2. since n is an integer, we get 𝑛 ≤ [ log 2𝑍−log 𝒜𝛼𝑗−log 𝑘 log 𝛾 ] ≤ 𝑛 + 1. this implies 𝑛 = [ log 2𝑍−log 𝒜𝛼𝑗 −log 𝑘 log 𝛾 ] or 𝑛 = [ log 2𝑍−log 𝒜𝛼𝑗 −log 𝑘 log 𝛾 ] − 1. thus 𝑆(𝑍) = ∑ ∑ [ log 2𝑍−log 𝒜𝛼𝑗−log 𝑘 log 𝛾 ] 𝜂 𝑗=1 𝛽 𝑖=1 + 𝑐, where 𝑐 = 0 or −1. then 𝑆(𝑍) = 1 𝑙𝑜𝑔 𝛾 ∑ ∑ (log 2𝑍 − log 𝒜𝑖𝑗 − log 𝑘 + 𝑐 + 𝑐 ′) 𝜂 𝑗=1 𝛽 𝑖=1 , where 0 ≤ c′ < 1. now considering 𝑐 + 𝑐′ = 𝐶′, we have −1 ≤ 𝐶′ < 1. we can now write 𝑆(𝑍) = 𝛽𝜂 𝑙𝑜𝑔 𝛾 log(2𝑍/𝑘) − 1 𝑙𝑜𝑔 𝛾 ∑ ∑ log 𝒜𝑖𝑗 𝜂 𝑗=1 𝛽 𝑖=1 + 𝛽𝜂𝐶 ′. if we write 𝐶 = 𝛽𝜂𝐶′, then we get −𝛽𝜂 ≤ 𝐶 < 𝛽𝜂 and 𝑆(𝑍) = 𝛿 {𝛽𝜂 log(2𝑍/𝑘) − log(∏ ∏ 𝒜𝑖𝑗 𝜂 𝑗=1 𝛽 𝑖=1 )} + 𝐶. finally, we find the value of 𝑇(𝑍). bilkis m. madni, devbhadra v. shah 220 theorem 3.3. 𝑇(𝑍) = 𝛿 {𝛽𝜂 log(2√𝐷𝑍/𝑘) − log(∏ ∏ 𝒜𝑖𝑗 𝜂 𝑗=1 𝛽 𝑖=1 )} + 𝐶, where 𝐶 is the effective constant such that 0 ≤ 𝐶 < 2𝛽𝜂. proof. to find the value of 𝑇(𝑍), we first find the number of positive solutions of (1.3) where the values 𝑣𝑖𝑗,𝑛 of 𝑉 do not exceed 𝑍 for some fixed class 𝑖 = 𝛼 (1 ≤ 𝑗 ≤ 𝜂). now by (2.5) since 𝑢𝛼𝑗,𝑛+𝑣𝛼𝑗,𝑛√𝐷 𝑘 = 𝒜𝛼𝑗 𝛾 𝑛 and 𝑢𝛼𝑗,𝑛−𝑣𝛼𝑗,𝑛√𝐷 𝑘 = 𝒜𝛼𝑗̅̅ ̅̅ ̅𝛾 −𝑛, on subtraction, we get 𝑣𝛼𝑗,𝑛 = 𝑘 2√𝐷 {𝒜𝛼𝑗 𝛾 𝑛 − 𝒜𝛼𝑗̅̅ ̅̅ ̅𝛾 −𝑛} = 𝑘 2√𝐷 {𝒜𝛼𝑗 𝛾 𝑛 − 𝑁 𝒜𝛼𝑗 𝛾 −𝑛}. since 𝑣𝛼𝑗,𝑛 ≤ 𝑍 < 𝑣𝛼𝑗,𝑛+1, for some n, we have 𝒜𝛼𝑗 𝛾 𝑛 − 𝑁 𝒜𝛼𝑗 𝛾 −𝑛 ≤ 2√𝐷𝑍 𝑘 < 𝒜𝛼𝑗 𝛾 𝑛+1 − 𝑁 𝒜𝛼𝑗 𝛾 −𝑛−1. thus, we have 𝒜𝛼𝑗 𝛾 𝑛−1 ≤ 2√𝐷𝑍 𝑘 < 𝒜𝛼𝑗 𝛾 𝑛+1. this implies 𝑛 − 1 < log 2√𝐷𝑍−log 𝒜𝛼𝑗 −log 𝑘 log 𝛾 < 𝑛 + 1. since n is an integer, we have 𝑛 = [ log 2√𝐷𝑍−log 𝒜𝛼𝑗− log 𝑘 log 𝛾 ] or𝑛 = [ log 2√𝐷𝑍−log 𝒜𝛼𝑗−log 𝑘 log 𝛾 ] + 1. thus 𝑇(𝑍) = ∑ ∑ [ log 2√𝐷𝑍−log 𝒜𝛼𝑗−log 𝑘 log 𝛾 ] 𝜂 𝑗=1 𝛽 𝑖=1 + 𝑐, where 𝑐 = 0 or 1. then 𝑇(𝑍) = 1 𝑙𝑜𝑔 𝛾 ∑ ∑ (log 2√𝐷𝑍 − log 𝒜𝑖𝑗 − log 𝑘 + 𝑐 + 𝑐 ′) 𝜂 𝑗=1 𝛽 𝑖=1 , where 0 ≤ c′ < 1. considering 𝑐 + 𝑐′ = 𝐶′, we have 0 ≤ 𝐶′ < 2. thus, we now write 𝑇(𝑍) = 𝛽𝜂 𝑙𝑜𝑔 𝛾 log(2√𝐷𝑍/𝑘) − 1 𝑙𝑜𝑔 𝛾 ∑ ∑ log 𝒜𝑖𝑗 𝜂 𝑗=1 𝛽 𝑖=1 + 𝛽𝜂𝐶 ′. if we write 𝐶 = 𝛽𝜂𝐶′, then we get 0 ≤ 𝐶 < 2𝛽𝜂 and 𝑇(𝑍) = 𝛿 {𝛽𝜂 log(2√𝐷𝑍/𝑘) − log(∏ ∏ 𝒜𝑖𝑗 𝜂 𝑗=1 𝛽 𝑖=1 )} + 𝐶. the following interesting conclusions are now an easy consequence from these theorems. corollary 3.4. 𝑇(𝑍) − 𝑆(𝑍) ≈ 𝛿𝛽𝜂 log √𝐷. corollary 3.5. if the solutions 𝑢𝑖𝑗,𝑛+𝑣𝑖𝑗,𝑛√𝐷 𝑘 of 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 are considered as lattice points within the square [0, 𝑍] × [0, 𝑍], then density of these lattice points is zero. this follows from the fact that lim 𝑛→∞ log 𝑍 𝑍 = 0. on the solutions of pellian equation 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 221 4. conclusions in this paper, we derived the necessary and sufficient condition for any two solutions of 𝑈2 − 𝐷𝑉2 = 𝑘2𝑁 to belong to the same class and the bounds for the values of 𝑢, 𝑣 occurring in the fundamental solution. we also derived an explicit formula which gives all its positive solutions. we further obtained some interesting recurrence relations connecting the values of 𝑢, 𝑣. finally, we obtained the results for total number of its positive solutions not exceeding any given positive real number 𝑍. references [1] andreescu titu, andrica dorin, cucurezeanu ion. an introduction to diophantine equations. birkhäuser boston inc, secaucus, united states. 2010. [2] burton d. m. elementary number theory. tata mc graw–hill pub. co. ltd., new delhi. 2007. [3] kaplan p., williams k. s. pell’s equation 𝑥2 − 𝐷𝑦2 = −1, −4 and continued fractions. journal of number theory. 23, 169 – 182, 1986. [4] leveque william j. topics in number theory, vol. i. addition-wesley pub. co., inc. 137 – 158, 1956. [5] madni bilkis m., shah devbhadra v. alternate proofs for the infinite number of solutions of pell’s equation. international journal of engineering, science and mathematics. 7 (4), 255 – 259, april 2018. [6] matthews k. the diophantine equation 𝑥2 − 𝐷𝑦2 = 𝑁, 𝐷 > 0. expositiones mathematicae.18, 323 – 331, 2000. [7] mollin r. a., poorten a. j., williams h. c. halfway to a solution of 𝑥2 − 𝐷𝑦2 = −3. journal de theorie des nombres bordeaux. 6, 421 – 457, 1994. [8] shah devbhadra v. on the solutions of diophantine equations 𝑥2 − 𝐷𝑦2 = ±4𝑁. journal of v.n.s.g. university. 4 – b, 56 – 65, 2007. [9] steuding jörn. diophantine analysis. chapman & hall/crc, taylor & francis group, boca raton. 2005. [10] stevenhagen p. a density conjecture for the negative pell equation. computational algebra and number theory, mathematics and its applications. 325, 1995. [11] stolt bengt. on the diophantine equation 𝑢2 − 𝐷𝑣2 = ±4𝑁, part i. arkiv för matematik. 2 (1), 1 – 23, 1951. [12] stolt bengt. on the diophantine equation 𝑢2 − 𝐷𝑣2 = ±4𝑁, part ii. arkiv för matematik. 2 (10), 251 – 268, 1951. bilkis m. madni, devbhadra v. shah 222 [13] tekcan a. the pell equation 𝑥2 − 𝐷𝑦2 = ±4. applied mathematical sciences. 1 (8), 363 – 369, 2007. [14] telang s. g. number theory. tata mc graw–hill pub. co. ltd., new delhi. 2004. ratio mathematica volume 46, 2023 domination in m− polar soft fuzzy graphs s. ramkumar * r. sridevi † abstract in this article, we have introduced dominating set, minimal dominating set, independent dominating set, maximal independent dominating set in m polar soft fuzzy graphs. we proved theorems and also some properties of dominating set in m polar soft fuzzy graphs keywords: dominating set, independent dominating set, maximal independent dominating set in m − polar soft fuzzy graphs. 2020 ams subject classifications: 03e72, 05c72. 1 *pg and research department of mathematics, sri s. ramasamy naidu memorial college, sattur-626 203, tamil nadu, india. affiliated to madurai kamaraj university, madurai-625 021, tamil nadu, india; ramkumarmat2015@gmail.com. †pg and research department of mathematics, sri s. ramasamy naidu memorial college, sattur-626 203, tamil nadu, india. affiliated to madurai kamaraj university, madurai-625 021, tamil nadu, india;danushsairam@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1070. issn: 1592-7415. eissn: 2282-8214. ©s.ramkumar et al. this paper is published under the cc-by licence agreement. 138 s.ramkumar and r.sridevi 1 introduction the work of zadeh [13] in 1975, which interacted with ambiguity and imprecision between absolute true and absolute false, is credited with giving rise to fuzzy set theory. a fuzzy set’s possible values fall between [0,1]. fuzzy set theory’s astonishing discovery opened up new possibilities for handling uncertainty in a variety of scientific and technological fields. due to its use in engineering, communication networks, computer sciences, and artificial intelligence, graph theory is swiftly becoming a mainstream topic in mathematics.graphs and fuzzy graphs are investigated in [10, 11].the idea of domination in fuzzy graphs was propounded by a. somasundaram and s. somasundaram [14] in 1998.soft set and hybrid models are used to deal with uncertainty based on parametrization tool. soft set, fuzzy soft set and m-polar fuzzy sets are studied in [1, 4, 6, 7]. the possibilityof domination in m− polar fuzzy graphs was introduced by m. akram et.al [2] in 2017. domination in graphs and domination in fuzzy graphs are analysedin [9, 5]. mohintasumit and samanta.t.k. [8] proposed the thought of fuzzy soft graph. s. ramkumar and r. sridevi [12] presented their perception in m−polar soft fuzzy graphs. these concepts motivate us to introduce domination inm− polar soft fuzzy graphs. 2 dominating set, minimal dominating set, maximal independent set in m− polar soft fuzzy graphs in this paper m-psf-graph denote m− polar soft fuzzy graph. definition 2.1. an edge in an m-psf-graph g̃p,v = (g∗, ρ̃, µ̃,p)is defined as an effective edge if µ̃ex1(uv) = (ρ̃ex1(u) ∧ ρ̃ex1(v)) µ̃ex2(uv) = (ρ̃ex2(u) ∧ ρ̃ex2(v)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µ̃exm(uv) = (ρ̃exm(u) ∧ ρ̃exm(v)) in nhd(u) = {v ∈ v/v is a neighbor of u} is called the neighbourhood of u. definition 2.2. a vertex u ∈ v in an m-psf-graph g̃p,v = (g∗, ρ̃, µ̃,p) is said to be an isolated vertex if 139 domination in m− polar soft fuzzy graphs µ̃ex1(uv) < (ρ̃ex1(u) ∧ ρ̃ex1(v)) µ̃ex2(uv) < (ρ̃ex2(u) ∧ ρ̃ex2(v)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µ̃exm(uv) < (ρ̃exm(u) ∧ ρ̃exm(v))∀v ∈ v \{u} in h̃p,v (e) ∀ e ∈ p. so that nhd(u) = φ. example 2.1. consider a 3 −psf −graphg̃p,v . t t tt t t t t h̃p,v (e1) h̃p,v (e2) a1(0.8, 0.7, 0.9) (0.2, 0.3, 0.5) a2(0.3, 0.5, 0.6) (0.3, 0.4, 0.6) a3(0.6, 0.4, 0.9) (0.2, 0.4, 0.5) a4(0.2, 0.6, 0.5) (0.1, 0.5, 0.4) a1(0.7, 0.5, 0.8) (0.4, 0.2, 0.6) a2(0.5, 0.9, 0.7) (0.5, 0.5, 0.7) a3(0.7, 0.5, 0.8) (0.3, 0.5, 0.4) a4(0.3, 0.9, 0.4) (0.2, 0.1, 0.3) figure.1. 3− psf-graph g̃p,v = {h̃p,v (e1),h̃p,v (e2)}. in this example, a2a3 and a3a4 are effective edges. also nhd(a1) = {φ},nhd(a2) = {a3},nhd(a3) = {a4a2},nhd(a4) = {a3}. here a1 is an isolated vertex. definition 2.3. let g̃p,v = (g∗, ρ̃, µ̃,p) be an m-psf-graph. for any two vertices u,v ∈ v, we call u dominates v in g̃p,v if µ̃ex1(uv) = (ρ̃ex1(u) ∧ ρ̃ex1(v)) µ̃ex2(uv) = (ρ̃ex2(u) ∧ ρ̃ex2(v)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µ̃exm(uv) = (ρ̃exm(u) ∧ ρ̃exm(v)) in h̃p,v (e) ∀e ∈ p and ∀u,v ∈ v . definition 2.4. a subset s̃ of v be an m− psf-graph g̃p,v . then cardinality of s̃ is defined as, | s̃ |= ∑ e∈p ( ∑ u∈s̃ ρ̃e(u)) in h̃p,v (e)p. 140 s.ramkumar and r.sridevi definition 2.5. a subset s̃ of v is called a dominating set of an m − psf − graphg̃p,v if for every vertex u ∈ v \ s̃ then ∃ v ∈ s̃ such that u dominates v in h̃p,v (e) ∀e ∈ p.the domination number γ(g̃p,v ) means the infimum cardinality of all dominating set in g̃p,v and γ(g̃p,v ) = mins̃∈v ∑ e∈p ( ∑ u∈s̃ ρ̃e(u)). definition 2.6. a dominating set s̃ is called a minimal dominating set of m − psf − graphg̃p,v = (g∗, ρ̃, µ̃,p) if for any a ∈ s̃, s̃ \ {a} is not a dominating set in h̃p,v (e) ∀ e ∈ p. definition 2.7. lower and upper domination number of an m−psf-graph g̃p,v = (g∗, ρ̃, µ̃,p) is denoted by γ(g̃p,v ) and γ(g̃p,v ), respectively, and defined by infimum cardinality and supremum cardinality of all minimal dominating set of that m−psf-graph, respectively. example 2.2. consider a 3−psf-graph g̃p,v . @ @ @ @� � � � q q q q �� � �@ @ @ @ q q qr � � � �� q q q q a1(0.7, 0.8, 0.9) (0.3, 0.2, 0.6) (0.3, 0.2, 0.6) (0.4, 0.5, 0.6) a2(0.4, 0.5, 0.6) a3(0.3, 0.2, 0.6) a4(0.8, 0.6, 0.4) (0.3, 0.2, 0.4) h̃p,v (e1) a1(0.8, 0.6, 0.3) (0.3, 0.6, 0.1) a3(0.3, 0.7, , 0.1)a2(0.2, 0.4, 0.8) (0.2, 0.4, 0.3) a4(0.4, 0.6, 0.3) (0.3, 0.6, 0.1)(0.2, 0.4, 0.1) h̃p,v (e2) a1(0.6, 0.8, 0.4) (0.6, 0.8, 0.4) a3(0.8, 0.9, 0.4) a4(0.6, 0.7, 0.3) (0.6, 0.7, 0.3) (0.3, 0.6, 0.4) a2(0.3, 0.6, 0.7) (0.3, 0.6, 0.4) h̃p,v (e3) figure.2. minimal dominating set of 3-psf-graph in figure.2. here, the minimal dominating sets in each parameterized graph is s̃1 = {a1,a4}, s̃2 = {a2,a4}, s̃3 = {a3}. theorem 2.1. a dominating set s̃ is minimal if and only if one of the below mentioned criteria holds. 141 domination in m− polar soft fuzzy graphs 1. nhd(a) ∩s̃ = φ. 2. there is a vertex b ∈ v \ s̃ such that nhd(b) ∩s̃ = {a}, for each a ∈ s̃. proof. for a minimal dominating set s̃ of a 3−psf-graph g̃p,v , for every vertex a ∈ s̃, s̃ \{a} is not dominating set and so then ∃ b ∈ v \(s̃ \{a}) which is not dominated by any vertex in s̃ \ {a} of h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. if a = b then nhd(a) ⊆ v \s̃ ⇒ nhd(a)∩s̃ = φ in h̃p,v (ei) for i = 1, 2, . . . ,n . if a 6= b, then b is not dominated by s̃\{a} but is dominated by s̃, i.e., b is dominated only by a in s̃ in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. ∴ nhd(b) ∩s̃ = {a} in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. hence nhd(b) ∩s̃ = {a} in 3−psf-graph g̃p,v . conversely, let s̃ holds one of the two given criterias. if s̃ is not minimal dominating set in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. then ∃ a vertex a ∈ s̃ such that s̃ \ {a} is a dominating set in h̃p,v (ei) for i = 1, 2, . . . ,n. thus a is dominated by atleast one vertex in s̃ \ {a} in h̃p,v (ei) for i = 1, 2, . . . ,n. then nhd(a) * s̃ \ {a} in h̃p,v (ei) for i = 1, 2, . . . ,n. hence condition (1) does not hold. also, if s̃ \ {a} is a dominating set in h̃p,v (ei) for i = 1, 2, . . . ,n. then every vertex b in v \ (s̃ \ {a}) is dominated by at least one vertex in s̃ \ {a} in h̃p,v (ei) for i = 1, 2, . . . ,n. so nhd(b) ∩ s̃ 6= {a} in h̃p,v (ei) for i = 1, 2, . . . ,n. hence condition (2) does not hold. this leads to a ⇒⇐ . ∴ s̃ must be a minimal dominating set in h̃p,v (ei) for i = 1, 2, . . . ,n. hence s̃ must be a minimal dominating set in 3− psf-graph g̃p,v . theorem 2.2. let g̃p,v = ((p,ρ̃), (p,µ̃)) be a 3−psf-graph without isolated vertices. if s̃ is a minimal dominating set of g̃p,v then v \ s̃ is a dominating set of g̃p,v . proof. let s̃ be a minimal dominating set and a ∈ s̃. since g̃p,v has no isolated vertices ∃ a vertex b ∈ nhd(a) in h̃p,v (ei) ∀ei ∈ p for i = 1, 2, . . . ,n. utilization of the same content similar to the proof given for theorem 2.1, we get that b in v \s̃. thus every element of s̃ is dominated by some element of v \s̃ in h̃p,v (ei) ∀ei ∈ p for i = 1, 2, . . . ,n. consequently v \s̃ is a dominating set of g̃p,v . theorem 2.3. superset of a dominating set in g̃p,v is a dominating set. proof. proof is obvious. remark 2.1. subset of a dominating set in g̃p,v need not to be dominating set. example 2.3. consider a 3−psf-graph. 142 s.ramkumar and r.sridevi @ @ @ @ @ r r r s r r r r r rr r r ra1(0.7, 0.9, 0.9) a4(0.6, 0.7, 0.8) a5(0.6, 0.9, 0.8) a6(0.7, 0.8, 0.9) a7(0.6, 0.7, 0.9) a2(0.4, 0.3, 0.2) (0.6, 0.3, 0.4) a3(0.6, 0.3, 0.4) (0.6, 0.7, 0.8) (0.6, 0.8, 0.8)(0.6, 0.3, 0.4) (0.6, 0.3, 0.4) (0.4, 0.3, 0.2) (0.4, 0.3, 0.2) (0.6, 0.3, 0.4) h̃p,v (e1) a1(0.8, 0.9, 0.6) (0.6, 0.7, 0.6) (0.4, 0.3, 0.5) a2(0.4, 0.3, 0.5) a7(0.7, 0.6, 0.8) a6(0.9, 0.6, 0.8) a5(0.6, 0.7, 0.8) a4(0.3, 0.4, 0.6) a3(0.6, 0.7, 0.8) (0.3, 0.4, 0.6) (0.3, 0.4, 0.6) (0.6, 0.6, 0.8) (0.6, 0.6, 0.8) (0.4, 0.3, 0.5) (0.6, 0.6, 0.8) h̃p,v (e2) figure.3. 3-psf-graphs. here each parameterized graph is s̃ = {a2,a3,a5} and s̃∪{b} = {a1,a2,a3,a5} s̃ \{b} = {a2,a5}. here {a1,a2,a3,a5} is a dominating set. but {a2,a5} is not a dominating set. definition 2.8. a set s̃ ⊆ v of an m−psf-graph g̃p,v is called an independent set if µ̃ex1(uv) < (ρ̃ex1(u) ∧ ρ̃ex1(v)) µ̃ex2(uv) < (ρ̃ex2(u) ∧ ρ̃ex2(v)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µ̃exm(uv) < (ρ̃exm(u) ∧ ρ̃exm(v)) in h̃p,v (e) ∀ e ∈ p and for all u,v ∈ s̃. definition 2.9. an independent set s̃ of an m− psf-graph g̃p,v = (g∗, ρ̃, µ̃,p)is said to be maximal independent set if for every vertex u ∈ v \ s̃, the set s̃ ∪{u} is not independent in h̃p,v (e) ∀ e ∈ p. 143 domination in m− polar soft fuzzy graphs definition 2.10. lower and upper independence number of an m−psf −graph g̃p,v = (g ∗, ρ̃, µ̃,p) is denoted by i(g̃p,v ) and i(g̃p,v ), respectively, and defined by infinimum cardinality and supremum cardinality among all the maximum independent set of that m−psf-graph, respectively. example 2.4. consider a 3−psf-graph g̃p,v . r r rr r r r r r ra1(0.6, 0.7, 0.8) (0.6, 0.7, 0.8) a2(0.7, 0.9, 0.6) (0.3, 0.4, 0.5) a3(0.3, 0.4, 0.5) (0.3, 0.4, 0.5) (0.6, 0.7, 0.7) a5(0.4, 0.6, 0.5) (0.4, 0.6, 0.5) a4(0.6, 0.8, 0.7) h̃p,v (e1) h̃p,v (e2) (0.6, 0.4, 0.2) a3(0.8, 0.9, 0.2) (03, 0.4, 0.2) (0.3, 0.4, 0.5) a4(0.3, 0.4, 0.5) a2(0.6, 0.4, 0.2) a1(0.4, 0.6, 0.8) a5(0.7, 0.9, 1.0) (0.3, 0.4, 0.5) (0.4, 0.4, 0.2) r r r r ra1(0.6, 0.7, 0.8) a5(0.7, 0.8, 0.9) (0.6, 0.7, 0.6) (0.7, 0.8, 0.6) a4(0.8, 0.9, 0.6) (0.4, 0.6, 0.6) a3(0.4, 0.6, 0.8) (0.3, 0.5, 0.4) a2(0.3, 0.5, 0.4) (0.3, 0.5, 0.4) h̃p,v (e3) figure.4. independent set of a 3-psf-graphs. in this example, the dominating set in each parameterized graph is {{a3,a4}, {a1,a3,a5},{a2,a4}}. in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, 3. the maximal independent set is {{a2,a4},{a1,a3,a5}}. here {a2,a4} is a maximal independent set of g̃p,v with infimum cardinality and i(g̃p,v ) = (3.3, 3.9, 3.0). also here {a1,a3,a5} is maximal independent set of g̃p,v with supremum cardinality and i(g̃p,v ) = (4.9, 6.2, 6.7). 144 s.ramkumar and r.sridevi proposition 2.1. for any 3 − psf − graph g̃p,v = (g∗, ρ̃, µ̃,p),γ(g̃p,v ) ≤ i(g̃p,v ). example 2.5. consider a 3− psf-graph. q q q q q q q q e e e ee qq h̃p,v (e1) h̃p,v (e2) a1(0.9, 0.7, 0.3) a5(0.7, 0.8, 0.9) (0.4, 0.3, 0.6) a4(0.4, 0.3, 0.6) (0.4, 0.3, 0.6) a3(0.8, 0.4, 0.7) (0.5, 0.4, 0.7) a2(0.5, 0.6, 0.9) (0.5, 0.6, 0.3) a1(0.6, 0.9, 0.7) a5(0.3, 0.2, 0.1) (0.3, 0.2, 0.1) a4(0.4, 0.3, 0.8) (0.4, 0.3, 0.6) a3(0.7, 0.9, 0.6) (0.4, 0.6, 0.6) a2(0.4, 0.6, 0.8) (0.4, 0.6, 0.7) figure.5. independent set of a 3-psf-graphs in fig.5. that minimum dominating set of a 3−psf-graph g̃p,v is {a2,a4} and the maximal independent set of g̃p,v is {a1,a3,a5} in h̃p,v (ei) ∀ e ∈ p for i = 1, 2. also here γ(g̃p,v ) = (1.7, 1.8, 3.1) and i(g̃p,v ) = (4.0, 3.9, 3.3) in h̃p,v (ei) ∀ e ∈ p for i = 1, 2. clearly, γ(g̃p,v ) ≤ i(g̃p,v ). theorem 2.4. a set s̃ ⊆ v is a maximal independent set of a 3 −psf − graph g̃p,v if and only if it is independent and dominating set. proof. assume that s̃ is a maximal independent set of g̃p,v . then for each vertex a ∈ v \s̃, the set s̃∪{a} is not independent set in h̃p,v (ei) for i = 1, 2, . . . ,n. in this manner for every vertex a ∈ v \s̃,then ∃ vertex b ∈ s̃ such that b dominates a in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. hence s̃ is a dominating set in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. therefore s̃ is both independent and dominating set in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. hence s̃ is both independent and dominating set in g̃p,v conversely if we suppose that s̃ is not maximal independent set in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. then ∃ a vertex a ∈ v \ s̃, such that s̃ ∪{a} is independent set. thus there @ any vertex b in s̃ which dominates a in h̃p,v (ei) for i = 1, 2, . . . ,n. so s̃ is not a dominating set in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. which ⇒⇐ to the choice of s̃. accordingly s̃ is a maximal independent set in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. hence s̃ is a maximal independent set of g̃p,v 145 domination in m− polar soft fuzzy graphs theorem 2.5. in a 3 − psf − graph g̃p,v = ((p,µ̃), (p,ρ̃)), every maximal independent set is a minimal dominating set proof. for a maximal independent s̃ of g̃p,v . by theorem 2.4, s̃ is dominating set in g̃p,v . if we consider s̃ being not minimal dominating set in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. then ∃ atleast one vertex a ∈ s̃ so that s̃ \ {a} is a dominating set that means s̃ \ {a} dominates v \ (s̃ \ {a}). thus, there exists atleast one vertex in s̃ which dominates a. this contradict our assumption. therefore s̃ is a minimal dominating set in h̃p,v (ei) ∀ ei ∈ p for i = 1, 2, . . . ,n. hence s̃ is a minimal dominating set in gp,v . 3 conclusions due to the large range of applications and domination characteristics that can be defined, domination theory research is interesting. this work introduces the idea of dominating sets, independent sets, domination number etc. for m-polar soft fuzzy graphs and shows some intriguing findings. in a similar situation, future studies can define and examine additional domination parameters. references m. n. w. akram and b. davvaz. certain types of domination in m-polar fuzzy graphs. journal of multiple-valued logic and soft computing, 29(6), 2017. s. ramkumar and r. sridevi. proper m polar soft fuzzy graphs. adv.math.,sci.j., 10(4):1845–1856, 2021. a. somasundaram and s. somasundaram. domination in fuzzy graph-i. pattern recognition letter, 19(9):77–95, 1998. m. sumit and t. samanta. an introduction to fuzzy soft graph. mathematica moravica, 19(3):35–48, 2015. 146 ratio mathematica issn: 1592-7415 vol. 31, 2016, pp. 93--110 eissn: 2282-8214 93 groups of transformations with a finite number of isometries: the cases of tetrahedron and cube ferdinando casolaro1, luca cirillo2, raffaele prosperi3 1department di architecture university “federico ii” of naples, italy ferdinando.casolaro@unina.it 2 university of sannio in benevento, italy luca.cirillo@unisannio.it 3 disuff university of salerno, italy rprosperi@unisa.it received on: 3-11-2016. accepted on: 11-11-2017. published on: 28-02-2017 doi: 10.23755/rm.v31i0.322 © casolaro et al. abstract this paper deals with groups of transformations with finite number of isometries and extends previous studies (casolaro, f. l. cirillo and r. prosperi 2015) which are related to endless groups of transformations with isometrics. in particular, isometries of the tetrahedron and cube, which turn these figures in itself, are presented. keywords: geometric transformations, isometries, symmetry. 2010 ams subject classification: 97g50; 51n25. ferdinando casolaro, luca cirillo, raffaele prosperi 94 1. introduction compared with the operation of product of isometries, in previous studies, we presented some examples of infinite groups of transformations, whose we highlighted the following properties: the isometries of the space form a group. the direct isometries of the space form a group, subgroup of the previous group. the translations of the space form a group, subgroup of the group of direct isometries. rotations around a straight form a group, subgroup of direct isometries. the helical movements all having the same axis form a group, subgroup of the group of direct isometries. in this case, since the helical movements turn out to be products of rotations for translations having the direction of the axis of rotation, also translations (the rotation is reduced to the identity) and rotations (the translation is reduced to the identity) may be considered helical movements. it is also possible to obtain groups of transformation with a finite number of isometries. in particular: about the tetrahedron, we show the axial symmetry μ having as an axis line r, rotations ρ of 120 ° and 240 ° around the height of the tetrahedron outgoing from a fixed vertex, planar symmetry σ relative to the plan π passing through two vertices of the tetrahedron and through the midpoint of the edge that joins the other two vertices; about the cube, rotations ρ around a line r connecting the centers of two opposite faces, rotations ρ around the line r joining the midpoints of two opposite edges, planar symmetry σ relative to the plan π passing through two vertices of the tetrahedron and through the midpoint of the edge that joins the other two vertices, planar symmetry σ relative to the pane π parallel to two faces passing through the midpoints of the four edges perpendicular to these two faces, planar symmetries σ relative to the pane π passing through two opposite edges that do not have face in common and a vertex in common. groups of transformations with a finite number of isometries: the cases of tetrahedron and cube 95 figure 1 consider three straight lines x, y, z, passing through the same point o and perpendicular to each other two by two. the three planes α, β, γ, respectively determined by the straight lines x and y, x and z, and y and z, are also perpendicular to each other two by two (figure 1). let be: i the identity, sx the axial symmetry having as an axis the line x, sy the axial symmetry having as an axis the line y, sz the axial symmetry having as an axis the line z, sα the planar symmetry relative to the plane α, sβ the planar symmetry relative to the plane β, sγ the planar symmetry relative to the plane γ, so the symmetry with center o, it occurs that these eight isometries form a group. for this purpose, it is sufficient to prove that the product of any two of them is still one of the eight indicated isometries. 2. tetrahedron’s isometries other examples of finite groups of isometries can be obtained considering all the isometries which leave fixed a given figure f, that is, such that in each of them f is united (f is transformed into itself). abcd and a'b'c'd 'are two congruent tetrahedra. then there exists one and only one isometry that transforms the vertices a, b, c, d neatly in the vertices a ', b', c ', d' (figure 2). this isometry is direct or reverse depending on whether or not the two tetrahedra are equally oriented. ferdinando casolaro, luca cirillo, raffaele prosperi 96 figure 2 isometries that turn a tetrahedron t into itself are 24 (twenty-four). they form a group st , obviously isomorphic to the group s4 of the 24 permutations on four letters a, b, c, d. among the isometries ϕ that transform the tetrahedron t into itself, we present the following: a) the axial symmetry μ having as an axis the straight line r, joining the midpoints of two opposite sides (bimedian), is a rotation of 180 ° around the straight line r. the symmetries of this type present in the group are 3 (as many as the pairs of opposite sides of the tetrahedron); they have evidently period 2. therefore there are 3 axial symmetries that leave t globally invariant, as many as the pairs of opposite sides. a substitution is associated with each of these symmetries (m. impedovo 1998). with symmetry μ1 about the line r1 joining the midpoints of the sides ab and cd, the following substitution is associated: with symmetry μ2 about the line r2 joining the midpoints of the sides ac e bd the following substitution is associated: with symmetry μ3 about the line r3 joining the midpoints of the sides ad e bc the following substitution is associated: groups of transformations with a finite number of isometries: the cases of tetrahedron and cube 97 b) the rotations ρ of 120 ° and 240 ° around the height of the tetrahedron outgoing from a fixed vertex. for each height of the tetrahedron, you have two rotations of period 3 which hold the summit fixed. since the tetrahedron heights are 4, these rotations are 8; therefore, there are 8 rotations of this type which transform t into itself, two for each height of the tetrahedron. a substitution is associated with each of these rotations. with rotation ρ1 about the height outgoing from a the following substitution is associated: relative to the amplitude of 120° relative to the amplitude of 240° with rotation ρ3 about the height outgoing from b the following substitution is associated: relative to the amplitude of 120° relative to the amplitude of 240° with rotation ρ5 about the height outgoing from c the following substitution is associated: relative to the amplitude of 120° relative to the amplitude of 240° with rotation ρ7 about the height outgoing from d the following substitution is associated: ferdinando casolaro, luca cirillo, raffaele prosperi 98 relative to the amplitude of 120° relative to the amplitude of 240° c) the planar symmetry σ relative to the plan π passing through the two vertices of the tetrahedron and the midpoint of the edge that joins the other two vertices. the σ symmetry σ is uniquely determined by the initial vertex. the symmetries of this type are 6 (as many as the pairs of vertices of the tetrahedron), and have period 2. a substitution is associated with each of these symmetries with symmetry about the plane abm1, with m1 medium point of cd, the following substitution is associated: with symmetry about the plane acm2, with m2 medium point of bd, the following substitution is associated: with symmetry about the plane adm3, with m3 medium point of bc, the following substitution is associated: with symmetry about the plane bcm4, with m4 medium point of ad, the following substitution is associated: groups of transformations with a finite number of isometries: the cases of tetrahedron and cube 99 with symmetry about the plane bdm5, with m5 medium point of ac, the following substitution is associated: with symmetry about the plane cdm6, with m6 medium point of ab, the following substitution is associated: it is observed that the two sets of isometries described in points a) and b) each supplemented with the identity are closed about to the product. the first set is a g1 group of order 4 of involutorie transformations. the second set is a g2 group of order 9 of periodic transformations of order 3. the union of the two groups is a g3 group of order 12, which is the group of direct isometries of t. we will now examine the product of three symmetries, or we will fix an isometry σk of type c) (planar symmetry), and we will consider an isometry αt (t = 1, 2, … , 12) variable in the g3 group. the product σk º αt is still an isometry that changes the tetrahedron t into itself. they are in number of 12; in fact, if we fix, for example, the isometry multiplying each isometry of the g3 group for σ1, we will get 12 reverse isometries reverse, which can be summarized as: ferdinando casolaro, luca cirillo, raffaele prosperi 100 it is easily seen that it results: ϕ 12 = σ 1 , ϕ 5 = σ 2 , ϕ 4 = σ 3 , ϕ 7 = σ 4 , ϕ 6 = σ 5 , ϕ 1 = σ 6 that is the 12 isometries σk º αt are given by the 6 planar symmetries σk of the type c) and by the 6 antirotations ϕ k, with period 4. the isometries ϕ k do not take firm no vertex and no edge of the tetrahedron. in summary, we can say that the three axial symmetries of the g1 group, the 8 rotations of the g2 group, the 6 planar symmetries, the 6 latest found isometries, along with the identity, are the 24 isometries that leave the tetrahedron t globally invariant; their set is the st group of isometries of t. st is the group of isometries that change the tetrahedron t in itself. groups of transformations with a finite number of isometries: the cases of tetrahedron and cube 101 3. isometries of cube some examples of finite groups of isometries can be had considering all isometries leaving globally invariant a cube (a. morelli, 1989). abcdefgh e a’b’c’d’e’f’g’h’ are two equal cubes. then there exists one and only one isometry that transforms the vertices a, b, c, d, e, f, g, h, neatly in the vertices a’, b’, c’, d’, e’, f’, g’, h’ (figure 3). this isometry is direct or reverse depending on whether or not the two cubes are equally oriented. figure 3 isometries that transform a c cube to itself are forty eight. they forming a sc group evidently isomorphic to s8 group of forty eight permutations on eight letters a, b, c, d, e, f, g, h. among the isometries that transform the c cube itself there are obviously the following: a) the rotations around a straight line r which joins the centers of two opposite faces. since the faces of the cube are six, these lines are three; for each of these straight lines the cube is transformed into itself by the amplitude rotations, respectively, 90°, 180°, 270°. therefore you have nine rotations of this type which transform c itself. for each of these rotations it is associated a substitution. to 1 rotation around the straight through m1m2, with m1 the center of the abcd face and m2 the center of the efgh face, is associated the substitution:       ehgfcbad hgfedcba : 1  relative to the amplitude of 90° ferdinando casolaro, luca cirillo, raffaele prosperi 102       fehgbadc hgfedcba : 2  relative to the amplitude of 180°       gfehadcb hgfedcba : 3  relative to the amplitude of 270° to 4 rotation around the straight through m3m4, with m3 the center of the abfe face and m4 the center of the dcgh face, is associated the substitution:       ahedcfgb hgfedcba : 4  relative to the amplitude of 90°       badcfehg hgfedcba : 5  relative to the amplitude of 180°       gbcfedah hgfedcba : 6  relative to the amplitude of 270° to 7 rotation around the straight through m5m6, with m5 the center of the aehd face and m6 the center of the bfgc face, is associated the substitution:       efcdabgh hgfedcba : 7  relative to the amplitude of 90°       dcbahgfe hgfedcba : 8  relative to the amplitude of 180°       abghefcd hgfedcba : 9  relative to the amplitude of 270° b) the rotations  around the straight line r that connects the midpoints of two opposite edges. since the edges of the cube are twelve, these lines are six; for groups of transformations with a finite number of isometries: the cases of tetrahedron and cube 103 each of these straight lines the cube is transformed into itself by rotations of 180 ° amplitude. for each of these rotations it is associated a substitution. to rotation 10 around the straight line joining the midpoints of ab and ef edges, is associated the substitution:       dcfehgba hgfedcba : 1 0  to rotation 11 around the straight line joining the midpoints of cd and hg edges, is associated the substitution:       hgbadcfe hgfedcba : 1 1  to rotation 12 around the straight line joining the midpoints of bc and he edges, is associated the substitution:       hadefcbg hgfedcba : 1 2  to rotation 13 around the straight line joining the midpoints of ad and fg edges, is associated the substitution:       bgfcdeha hgfedcba : 1 3  to rotation 14 around the straight line joining the midpoints of bc and he edges, is associated the substitution:       dadefcbg hgfedcba : 1 4  to rotation 15 around the straight line joining the midpoints of ad and fg edges, is associated the substitution: ferdinando casolaro, luca cirillo, raffaele prosperi 104         bgfcdeha hgfedcba :1 5 c) the rotations  around the straight line r that contains a diagonal. the number of se lines is four; for each of these straight lines the cube is transformed into itself by the amplitude rotations respectively 120° and 240°. therefore there are eight rotations of this type which transform c to itself. for each of these rotations it is associated a substitution. to rotation 16 around the diagonal af, it is associated the substitution:       fcdehabg hgfedcba : 1 8  relative to the amplitude of 120°       dahefgbc hgfedcba : 1 9  relative to the amplitude of 240° to rotation 18 around the diagonal be, it is associated the substitution:       fcdehabg hgfedcba : 1 8  relative to the amplitude of 120°       dahefgbc hgfedcba : 1 9  relative to the amplitude of 240° to rotation 20 around the diagonal ch, it is associated the substitution:       hedabcfg hgfedcba : 2 0  relative to the amplitude of 120°       habgfcde hgfedcba : 2 1  relative to the amplitude of 240° to rotation 21 around the diagonal dg, it is associated the substitution: groups of transformations with a finite number of isometries: the cases of tetrahedron and cube 105       bghadefc hgfedcba : 2 2  relative to the amplitude of 120°       fgbcdahe hgfedcba : 2 3  relative to the amplitude of 240° d) the planar symmetry with respect to  plane parallel to two faces through the midpoints of the four edges perpendicular to these two faces. the symmetries of the type indicated are three. for each of these symmetries it is associated a substitution. at the planar symmetry 1 with respect to the plane 1 parallel to abgh and efcd faces, is associated the substitution:       efghabcd hgfedcba : 1  at the planar symmetry 2 with respect to the plane 2 parallel to abdc and hgef faces, is associated the substitution:         abcdefgh hgfedcba :2 at the planar symmetry  with respect to the plane 3 parallel to bcgh and adhe faces, is associated the substitution:         abcdefgh hgfedcba :3 e) the symmetries with respect to the  plan through two opposite edges that do not have common face and vertex. the symmetries of the type indicated are six. for each of these symmetries it is associated a substitution. at the planar symmetry 4 respect to the 4 plan through the edges ad and gf is associated with the substitution: ferdinando casolaro, luca cirillo, raffaele prosperi 106       bgfcdeha hgfedcba : 4  at the planar symmetry 5 respect to the 5 plan through the edges bc and he is associated with the substitution:       hadefcbg hgfedcba : 5  at the planar symmetry 6 respect to the 6 plan through the edges ab and ef is associated with the substitution:       dcfehgba hgfedcba : 6  at the planar symmetry 7 with respect to the 7 plan through the edges cd and hg is associated with the substitution:       hgbadcfe hgfedcba : 7  at the planar symmetry 8 with respect to the 8 plan through the edges ah and cf is associated with the substitution:       hefgbcda hgfedcba : 8  at the planar symmetry 9 with respect to the 9 plan through the edges bg and de is associated with the substitution:       fghedabc hgfedcba : 9  note that the two sets of isometry described in points a), b) and c), each supplemented with the identity: groups of transformations with a finite number of isometries: the cases of tetrahedron and cube 107       hgfedcba hgfedcba i : , are closed with respect to the product. the first set g1 is a group of order ten, the second set is a group g2 of order seven, the third set is a group g3 of order nine. the union of these three groups is a g4 group of order twenty four which constitutes the group of direct isometries of c. let us now examine the product of three symmetries, that is fix an type d) isometry k (planar symmetry), and consider an isometry t (t = 1, 2, ..., 24) variable in the g4 group. the product k t is still an isometry which changes the c cube to itself. the number of these product is twenty four; in fact, it fixed eg. the isometry       efghabcd hgfedcba : 1  , multiplying each isometry of the g4 group , you get twentyfour reverse isometries, which can be summarized as:        fghedabc hgfedcba : 111   ,        ghefcdab hgfedcba : 221   ,        hefgbcda hgfedcba : 331   ,        dehabgfc hgfedcba : 441   ,        cdabghef hgfedcba : 551   ,        fcbghade hgfedcba : 661   , ferdinando casolaro, luca cirillo, raffaele prosperi 108        dcfehgba hgfedcba : 771   ,        abcdefgh hgfedcba : 881   ,        hgbadcfe hgfedcba : 991   ,        efcdabgh hgfedcba : 1 01 01   ,        ebghefcd hgfedcba : 1 11 11   ,        edahgbcf hgfedcba : 1 21 21   ,        cfgbahed hgfedcba : 1 31 31   ,        edahgbcf hgfedcba : 1 41 41   ,        cfgbahed hgfedcba : 1 51 51   ,        gfcbadeh hgfedcba : 1 61 61   ,        cfedahgb hgfedcba : 1 71 71   ,        edcfgbah hgfedcba : 1 81 81   , groups of transformations with a finite number of isometries: the cases of tetrahedron and cube 109        ehadcbgf hgfedcba : 1 91 91   ,        adehgfcb hgfedcba : 2 02 01   ,        gbahedcf hgfedcba : 2 12 11   ,        ahgbcfed hgfedcba : 2 22 21   ,        cbgfehad hgfedcba : 2 32 31   ,        efghabcd hgfedcba i : 2 41   . it is easily seen that results:  24 =  1 ,  8 =  2 ,  2 =  3 ,  7 =  6 ,  9 = 7 ,  3 =  8 ,  1 =  9 that is, the twentyfour isometries k t are given from nine symmetries k planar type d), e), and fifteen anti rotations k. in summary therefore it can be said that the twenty three rotations of the g4 group, the nine planar symmetries and the latest isometries found, along with the identity, are the forty eight isometries which leave the cube c globally invariant; their set is the sc group of isometries of the cube c. sc is the group of the isometries that change c cube to itself. conclusions as already shown in a previous work (casolaro, f., cirillo, l. and prosperi, r. 2015), the geometric universe is three-dimensional, so the transformations taking place in it are generated in space. then, we believe, for a correct analysis of the physical phenomena that occur in the universe, that it is essential to the knowledge of the real transformations that take place in it. recent results of other branches of mathematics, in particular the modern algebra, have ferdinando casolaro, luca cirillo, raffaele prosperi 110 highlighted the interrelationships between movements in the plane and in space with some properties of the theory of groups (casolaro, f. 1992), for which we consider essential to the deepening of these issues both in education and in the field of pure research (casolaro, f. and eugeni, f. 1996). unfortunately, teaching (casolaro f. 2014) in both the secondary school that the university has been anchored to old programs that do not take into account the development of mathematics in the last 150 years, so we hope that this work will stimulate teachers and researchers to expand their views. references casolaro, f. l. cirillo and r. prosperi (2015). “le trasformazioni geometriche nello spazio: isometrie”. science &philosophy n. 1, 2015. casolaro, f. and eugeni, f. (1996). “trasformazioni geometriche che conservano la norma nelle algebra reali doppie”. ratio matematica n. 1, 1996. casolaro, f. and prosperi, r. (2011). “la matematica per la scuola secondaria di secondo grado: un contributo per il docente di matematica". atti della “scuola estiva mathesis” 26-30 luglio 2011. terni: editore 2c contact. casolaro, f. and paladino l. (2012). “evolution of the geometry through the arts”. 11th international conference aplimat 2012 in the faculty of mechanical engineering slovak university of tecnology in bratislava, febbraio 2012. casolaro, f. (2014). “l'evoluzione della geometria negli ultimi 150 anni ha modificato la nostra cultura. lo sa la scuola?”. “science&philosophy journal of epistemology", volume 2, numero 1, 2014. cundari, c. (1992). “disegno e matematica per una didattica finalizzata alle nuove tecnologie”. progetto del m.p.i. e del dipartimento di progettazione e rilievo dell’università “la sapienza” di roma, 11-15 dicembre 1990; 6-10 maggio 1991; 8-12 dicembre 1991. casolaro, f. (1992). “gruppo delle affinità, gruppo delle similitudini, gruppo delle isometrie”. da c. cundari, progetto del m.p.i. e del dipartimento di progettazione e rilievo dell’università “la sapienza” di roma (1992). morelli, a. (1989). geometria per il biennio delle scuole medie superiori. napoli: edizione loffredo. impedovo, m. (1998). “matrici e isometrie nello spazio”. un utilizzo didattico di maple. l’insegnamento della matematica e delle scienze integrate, vol. 21 b, n° 1, febbraio 1998. ratio mathematica volume 42, 2022 on the approximation of conjugate of functions belonging to the generalized lipschitz class by euler-matrix product summability method of conjugate series of fourier series jitendra kumar kushwaha* krishna kumar † abstract in this paper, a new theorem on the approximation of conjugate of functions belonging to the generalized lipschitz class by euler-matrix product summability method of conjugate series of fourier series has been obtained. sometimes a series is not summable by any individual summability method. but it becomes summable by taking product summability means of given series. so working in this direction we have used euler-matrix product summability method. since lipα lip(α,p) classes are the particular cases of generalized lipschitz class. therefore, many of the known results may become particular cases of our result. on the bases of above facts we can say that our result may be useful for the coming researchers in future. keywords: generalized lipschitz class, conjugate series of fourier series, product summability method, euler mean, matrix mean. 2022 ams subject classifications: 42b05, 42b08. 1 *department of math and stat, ddu gorakhpur university; kjitendrakumar@yahoo.com. †department of math and stat, ddu gorakhpur university; kkmaths1986@gmail.com. 1received on may 17th, 2022. accepted on june 27th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v41i0.788. issn: 1592-7415. eissn: 2282-8214. ©jitendra kumar kushwaha and krishna kumar. 271 j.k.kushwaha, k.kumar 1 introduction the degree of approximation of functions belonging to various classes by using cesáro, nörlund and generalized nörlund summability methods has been obtained by a number of researchers like chandra [1], holland [3], lal et al ([5],[6]) and kushwaha [4], qureshi [7]. later on tiwary et al [9] has discussed the degree of approximation of functions by using (e,q)a product summability means of fourier series. no work seems to have been done so for to find the degree of approximation of conjugate of functions belonging to generalized lipschitz class by using euler-matrix product summability means. now, in this paper, we are presenting a new theorem on the degree of approximation of conjugate functions belonging to the generalized lipschitz class by euler-matrix product summability method. this new result may become the generalization of many of the known results. 2 definitions in this section,we have given following definitions: definition 2.1. a function f ∈ lipα if |f(x + t) − f(x − t)| = o(|t|α) for 0 ≤ α < 1. definition 2.2. a function f ∈ lip(α,p) if( 2π∫ 0 |f(x + t) − f(x − t)|p dx )1/p = o(|tα|),0 ≤ α < 1,p ≥ 1. given a positive increasing function ξ(t) and integer p ≥ 1, f ∈ lip(ξ(t),p) if  2π∫ 0 |f(x + t) − f(x − t)|p dx  1/p = o (ξ(t)) . if ξ(t) = tα, then lip(ξ(t),p) coincides to lip(α,p). 272 on the approximation conjugate of functions belonging to....... definition 2.3. l∞-norm of a function f : r → r is defined by ∥f∥∞ = sup{|f(x)|/f : r → r} . lp-norm is defined by ∥f∥p =   2π∫ 0 |f(x)|p  1/p ,p ≥ 1. the degree of approximation of a function f : r → r by a trigonometric polynomial tn(zygmund [11]) is defined by ∥tn − f∥∞ = sup{|tn − f| : x ∈ r}or∥tn − f∥p = min∥tn − f∥. let f be 2π periodic and integrable over (−π,π) in lebesgue sense and f ∈ lip(ξ(t),p) . let its fourier series be given by f(t) = 1 2 a0 + ∞∑ n=1 (an cosnx + bn sinnx) = 1 2 a0 + ∞∑ n=1 an(x). (1) the conjugate series of the fourier series (1) is given by ∞∑ n=1 (an sinnx − bn cosnx) = − ∞∑ n=1 bn(x). (2) if f is lebesgue integrable, then f(x) = − 1 2π π∫ 0 ψ(t) cot(t/2)dt = − 1 2π lim ϵ→0 π∫ 0 ψ(t) cot(t/2)dt exists for almost all x (zygmund [11]). 273 j.k.kushwaha, k.kumar let t = (an,k) be an infinite lower triangular matrix satisfying töeplitz (p. 131) condition of regularity i.e. an,k → 1 as n → ∞, an,k = 0 for k > n and n∑ k=0 |an,k| ≤ m a finite constant. let ∞∑ n=0 un be an infinite series whose kth partial sums is sk = k∑ n=0 un. the sequence to sequence transformation tn = n∑ k=0 aa,ksk defines the sequence {tn} of lower triangular matrix summability means of sequence {sn} generated by the sequence of coefficients (an,k). the series ∞∑ n=0 un is said to summable to sum s by lower triangular matrix method if lim n→∞ tn exists and is equal to s (zygmund; p.74) and we write tn → s(t), as n → ∞. the (e,q) means of {sn} is defined by wn = 1 (1 + q)n n∑ k=0 ( n k ) qn−ksk. the (e,q) transform of matrix transform a of sn is defined by ηn = 1 (1 + q)n n∑ k=0 ( n k ) qn−ktk = 1 (1 + q)n n∑ k=0 ( n k ) qn−k { k∑ ν=0 aν,ksk } . if ηn → ∞ as n → ∞, then the series ∞∑ n=0 un is said to be (e,q)a-summable to sum s. we use the following notations : (i) ψ(x,t) = f(x + t) − f(x − t). (ii) mn(t) = 1 2π(1+q)n n∑ k=0 ( n k ) qn−k { k∑ ν=0 aν,k cos(ν+1/2)t sin(t/2) } . 274 on the approximation conjugate of functions belonging to....... 3 lemmas:for the proof of our theorem, we have required following lemmas: lemma 3.1. mn(t) = o ( 1 t ) for 0 ≤ t ≤ (n + 1)−1. proof. for 0 ≤ t ≤ (n + 1)−1, we have ∣∣mn(t)∣∣ = 1 2π(1 + q)n ∣∣∣∣∣ [ n∑ k=0 ( n k ) qn−k { k∑ ν=0 aν,k cos(ν + 1/2)t sin(t/2) }]∣∣∣∣∣ ≤ 1 2π(1 + q)n ∣∣∣∣∣ n∑ k=0 ( n k ) qn−k { k∑ ν=0 aν,k | cos(ν + 1/2)t| (t/π) }] ≤ 1 2π(1 + q)n ∣∣∣∣∣ n∑ k=0 ( n k ) qn−k { k∑ ν=0 aν,k }] = o ( 1 t ) . lemma 3.2. mn(t) = o ( an,τ t ) for (n + 1)−1 ≤ t ≤ π. proof. for (n + 1)−1 ≤ t ≤ π, we have by jordan’s lemma, sin(t/2) ≥ (t/π), then ∣∣mn(t)∣∣ = 1 2π(1 + q)n ∣∣∣∣∣ [ n∑ k=0 ( n k ) qn−k { k∑ ν=0 aν,k cos(ν + 1/2)t sin(t/2) }]∣∣∣∣∣ ≤ 1 2π(1 + q)n ∣∣∣∣∣ [ n∑ k=0 ( n k ) qn−k { k∑ ν=0 aν,k cos(ν + 1/2)t (t/π) }]∣∣∣∣∣ = 1 2t(1 + q)n ∣∣∣∣∣ [ n∑ k=0 ( n k ) qn−k { k∑ ν=0 aν,k cos(ν + 1/2)t }]∣∣∣∣∣ = 1 2t(1 + q)n ∣∣∣∣∣ [ n∑ k=0 ( n k ) qn−k { o (ak,k−τ−1 t ) + ak,τ }]∣∣∣∣∣ = o ( an,τ t ) . 275 j.k.kushwaha, k.kumar 4 theorem in this section, we have proved the theorem: theorem 4.1. let f be a 2π-periodic, lebesgue integrable function belonging to lip(ξ(t),p) class and t = (am,n) be an infinite lower triangular matrix. then the degree of approximation of conjugate function by (e,q)a-summability means of its conjugate series of fourier series is given by ∥η − f(x)∥p = o ( (n + 1)1/pξ ( 1 n + 1 )) , provided ξ(t) satisfies following conditions:   1/(n+1)∫ 0 ( t|ψ(t)| ξ(t) )p  1/p = o ( 1 n + 1 ) , (3) and   π∫ 1/(n+1) ( t−δ|ψ(t)| ξ(t) )q  1/q = o ( (n + 1)δ ) , (4) where δ ia an arbitrary number such that q(1 − δ) − 1 > 0, p−1 + q−1 = 1 such that 1 ≤ p < ∞, conditions (3) and (4) holds uniformly in x. proof. the kth partial sums of conjugate series of fourier series (2) is given by sk(f;x) = − 1 2π π∫ 0 cot(t/2)ψ(t)dt + 1 2π 2π∫ 0 cos(k + 1/2)t sin(t/2) ψ(t)dt. sk(f;x) = − 1 2π π∫ 0 cot(t/2)ψ(t)dt = 1 2π 2π∫ 0 cos(k + 1/2)t sin(t/2) ψ(t)dt. therefore making (a-transform) of sk(f;x), we get tn − f(x) = 1 2π π∫ 0 ψ(t) n∑ k=0 an,k cos(ν + 1/2)t sin(t/2) dt. 276 on the approximation conjugate of functions belonging to....... now, making (e,q)a-transform of sk(f;x), we get ηn − f(x) = 1 2π(1 + q)n × π∫ 0 ψ(t) n∑ k=0 ( n k ) qn−k { k∑ ν=0 an,k cos(ν + 1/2)t sin(t/2) } dt = π∫ 0 ψ(t)mn(t)dt =   1/(n+1)∫ 0 + π∫ 1/(n+1)  ψ(t)mn(t)dt = i1 + i2. (5) clearly, |ψ(x + t) − ψ(x − t)| ≤ |f(u + x + t) − f(x + t)| + |f(u − x − t) − f(x − t)|. now,let ψ(x,t) = ψ(x + t) − ψ(x − t) ψ1(u,x,t) = f(u + x + t) − f(x + t) ψ2(u,x,t) = f(u − x − t) − f(x − t) hence, by minkowski’s inequality  2π∫ 0 |ψ(x,t)|pdx   1/p ≤   2π∫ 0 |ψ1(u,x,t)|pdx   1/p +   2π∫ 0 |ψ2(u,x,t)|pdx   1/p = o (ξ(t)) . (6) 277 j.k.kushwaha, k.kumar then f ∈ lip(ξ(t),p) ⇒ ψ ∈ lip(ξ(t),p) . using the hölder’s inequality, ψ(t) ∈ lip(ξ(t),p), condition (3), sint ≥ (2π/t), lemma 1, and second mean value theorem for integrals, we have |i1| ≤   1/(n+1)∫ 0 ( tψ(t) ξ(t) )p dt   1/p   1/(n+1)∫ 0 ( ξ(t)|mn(t)| t )q dt   1/q = o ( (n + 1)−1 )  1/(n+1)∫ 0 ( ξ(t)|mn(t)| t )q dt   1/q = o ( (n + 1)−1 )  1/(n+1)∫ 0 ( ξ(t) t2 )q dt   1/q = o ( (n + 1)1/pξ ( 1 n + 1 )) ,p−1 + q−1 = 1. (7) using the hölder’s inequality, lemma 2, | sint| ≤ 1, sint ≥ (2π/t), and condition (4), we have |i2| ≤   π∫ 1/(n+1) ( t−δψ(t) ξ(t) )p dt   1/p  π∫ 1/(n+1) ( ξ(t)|mn(t)| t−δ )q dt   1/q ≤ o ( (n + 1)δ )  π∫ 1/(n+1) ( ξ(t)|mn(t)| t−δ )q dt   1/q ≤ o ( (n + 1)δ )  π∫ 1/(n+1) ( ξ(t) t−δ o ( an,τ t ))q dt   1/q 278 on the approximation conjugate of functions belonging to....... = o ( (n + 1)δ )  π∫ 1/(n+1) (( ξ(t) t1−δ an,τ ))q dt   1/q = o ( (n + 1)δ )  (n+1)∫ 1/π ( ξ(1/y) yδ−1 an,[y] )q dy y2   1/q = o ( (n + 1)δξ ( 1 n + 1 ))  (n+1)∫ 1/π dy yδq−q+2   1/q = { (n + 1)δξ ( 1 n + 1 ) ( o(n + 1)−q(δ−1)−1 )1/q} = o ( (n + 1)1/pξ ( 1 n + 1 )) ,p−1 + q−1 = 1. (8) collecting equations from (5) to (8), we get ∥η − f(x)∥p = o ( (n + 1)1/pξ ( 1 n + 1 )) ,1 ≤ p < ∞. 5 corollaries:corolary 5.1. if ξ(t) = tα then the degree of approximation of a function f(x), conjugate of f ∈ lip(α,p), 1 p < α < 1 by (e,q)a means is given by ∥η − f(x)∥p = o ( (n + 1)−α+1/p ) ,1 ≤ p < ∞. corolary 5.2. if p → ∞ in case 1, then for 0 < α < 1, the degree of approximation of a function f(x), conjugate of f ∈ lipα by (e,q)a means is given by ∥η − f(x)∥p = o ((n + 1)−α) . 6 acknowledgement auther is highley thankful to professeor shyam lal, department of mathematics, institute of science, banaras hindu university, varanasi, india for his encouragement and support to this work. 279 j.k.kushwaha, k.kumar references 1. p. chandra, trigonometric approximation of functions in lp-norm,j. math. anal. appl., 275, no 1 (2002), 13-26. 2. g. h. hardy, divergent series, american mathematical society (2000). 3. a. s. b. holland and b. n. sahney, on the degree of approximation by (e,q) means, studia sci. math. hunger, 11, (1976), 431-435. 4. j. k. kushwaha, on the approximation of conjugate function by almost triangular matrix summability means, int. j. of management tech. and engi., 9, no 3 (2019), 4382-4389; doi: 16.10089.ijmte.2019.v913.19.27979. 5. s. lal and j. k. kushwaha, degree of approximation of lipschitz function by product summability methods, international mathematical forum, 4, no 43, (2009), 2101-2107. 6. s. lal and j. k. kushwaha, approximation of conjugate of functions belonging to generalized lipschitz class by lower triangular matrix means, int. journal of math. analysis, 3, no 21, (2009), 1031-1041. 7. k. qureshi, on the degree of approximation of a function belonging to the weighted class, indian jour. of pure and appl. math, 4, no 13, (1982), 471-475. 8. e. c. titchmarsh, the theory of functions, oxford university press (1939). 9. s. k. tiwary and u. upadhyay, degree of approximation of functions belonging to the generalized lipschitz class by product means of its fourier series, ultra scientist, 3, no 25, (2013), 411-416. 10. o. töeplitz, über die lineare mittelbi-dungen prace, mat. fiz., no 22, (1911), 113-119. 11. a. zygmund, trigonometric series, cambridge university press, cambridge (1959). 12. c.k.chui. an introduction to wavelets (wavelets analysis and it’s application), vol. 1,academic press, usa,1992. 13. jitendra kumar kushwaha, on the approximation of generalized lipschitz function by euler means of conjugate series of fourier series.the scientific world journal. vol.2013,article id 508026. 280 on the approximation conjugate of functions belonging to....... 14. sandeep kumar tiwari and uttam upadhyay, degree of approximation of function belonging to the w(lr,ξ(t)) class by (e,q) a-product means of its fourier series.ijm archive4(8),2013,266-272. 15. xhevat z. krasniqi, on the degree of approximation of functions belonging to the lipschitz class by (e,q)(c,α,β) means.khayyam j.math. 1(2015),no.2 243-252. 16. jitendra kumar kushwaha,approximation of functions by (c,2)(e,1)product summability method of fourier series. ratio mathematica.vol 38,2020,pp. 341-348. 281 ratio mathematica volume 47, 2023 on real roots of complement degree polynomial of graphs safeera k.* anil kumar v.† abstract many studies have been carried out on the roots of graph polynomials such as the matching polynomials, the characteristic polynomial, the chromatic polynomial, and many others. in this paper, we study the real roots of the complement degree polynomials of some graphs. moreover, we investigate the location of the roots of the complement degree polynomials of some graphs. keywords: complement degree polynomial, cd-roots. 2020 ams subject classifications: 05c31,30c15 1 *department of mathematics, university of calicut, malappuram, kerala, india 673 635; safeerakoralatil@gmail.com. †department of mathematics, university of calicut, malappuram, kerala, india 673 635; anil@uoc.ac.in. 1received on june 13, 2022. accepted on february 6, 2023. published on march 10, 2023. doi: 10.23755/rm.v41i0.795. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 79 1 introduction in mathematics, a graph polynomial is a graph invariant whose values are polynomials. in algebraic graph theory, invariants of this kind are explored [shi et al., 2016]. these are some crucial graph polynomials: characteristic polynomial, chromatic polynomial, dichromatic polynomial, flow polynomial, ihara zeta function, martin polynomial, matching polynomial, reliability polynomial, tutte polynomial. there is a lot of research on the roots of graph polynomials, including the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others. the location and nature of the roots have been important research areas for several graph polynomials. recently, the present authors[safeera and kumar, a] introduced the complement degree polynomial of a graph. definition 1.1. let g = (v, e) be a finite simple graph of order n and let cd(g, i) be the set of vertices of degree i in complement graph g and let cdi(g) = |cd(g, i)|. then complement degree polynomial of g is the polynomial: cd[g, x] = ∆(g)∑ i=δ(g) cdi(g)x i, (1) where δ(g) and ∆(g) respectively denote the minimum degree and maximum degree of the complement graph g [safeera and kumar, a]. the authors also derived the complement degree polynomial of some wellknown graphs and some graph operations [safeera and kumar, a,b]. in this paper, we study the real roots of the complement degree polynomial of some graphs obtained in [safeera and kumar, a]. in particular, we investigate the location of the roots of the polynomials so obtained. 2 main results the roots of several graph polynomials have drawn a lot of interest, both for what they represent and what their nature and location indicated. in this section, we investigate the roots of the complementary degree polynomial investigated in [safeera and kumar, a]. definition 2.1. the roots of polynomial defined in equation (1) are called cd-roots of g. the number of real cd-roots of a graph g where the multiplicities counted, is denoted by cd(g). 80 theorem 2.1. zero is a cd-root of a complement degree polynomial of a graph g with n vertices if and only if ∆(g) ≤ n − 2. proof. let g be a graph of order n and zero is a cd-root of the polynomial cd[g, x]. if g has a vertex, say v which is adjacent to all other vertices, then v is an isolated vertex in g. this implies that cd[g, x] has a constant term. this is a contradiction because zero is a cd-root of cd[g, x]. therefore, g has no vertices adjacent to all other vertices. conversely, assume that ∆(g) ≤ n − 2. then δ(g) ≥ 1. equivalently, cd0(g) = 0. this tells us that the constant term of cd[g, x] is zero, and hence the result follows.2 corolary 2.1. if g has no isolated vertices, then zero is a root of cd[g, x] with multiplicity δ(g). theorem 2.2. if g is the non complete graph of order n, then zero is the only cd-root of cd[g, x] if and only if g is a regular graph. proof. first, assume that zero is the only cd-root of a graph g with n vertices. then it follows that the complement degree polynomial of g is cd[g, x] = nxr. this implies that the degree of every vertex in g is the same. equivalently, g is regular. conversely, assume that g is a r-regular graph. then we have cd[g, x] = nxn−r−1. it follows that zero is only cd-root of cd[g, x]. 2 corolary 2.2. if g is the r-regular graph with n vertices, then cd(g) = n−r−1. theorem 2.3. let g be a graph with n vertices. then (1) cd[g, x] is a strictly increasing function in [0, ∞). (2) let g be a graph and h be any spanning subgraph of g. then the degree of cd[g, x]) is less than or equal to the degree of cd[h, x]. (3) let g be a graph and h be any induced subgraph of g. then the degree of cd[g, x]) greater than equal to the degree of cd[h, x]. (4) let g be a graph of order n with t isolated vertices in g and r isolated vertices in g. then cd0(g) = r and cdn−1(g) = t. proof proof of the above result follows from the definition of complement degree polynomial of a graph. theorem 2.4. for a cosplitting graph cs(g) of r-regular graph g with n vertices, cd(cs(g)) = n. 81 proof. observe that cd[cs(g), x] = nxn−1(1 + xr) [safeera and kumar, a]. it is clear that x = 0 is a cd-root of g with multiplicity n − 1. note that the polynomial 1 + xr has no real roots if r is even and one real root if r is odd. thus we have, cd(cs(g)) = { n − 1 if r is even, n if r is odd . this completes the proof.2 theorem 2.5. for a path graph pn, cd(pn) = { 0, n = 2 n − 2, n ≥ 3. proof. for a path graph pn, we have [safeera and kumar, a]: cd[pn, x] = { 2xn−2, n = 2 (n − 2)xn−3 + 2xn−2, n ≥ 3 . here we consider two cases: if n = 2, then cd[p2, x] = 2, which has no zeros. if n > 2, then we have cd[pn, x] = x n−3(2x + n − 2). obviously, x = 0 is the cd-root of cd[pn, x] with multiplicity n − 3 and x = −(n − 2)/2 is the another cd-root of cd[pn, x]. thus cd(pn) = { 0, n = 2 n − 2, n ≥ 3. this completes the proof.2 theorem 2.6. let g be a graph with order n and g=g ∪ g ∪ . . . ∪ g (m times). then cd(g) = n(m − 1) + cd(g). proof. let g be a graph of order n and g=g ∪ g ∪ . . . ∪ g (m times). then, cd[g, x] = mx(m−1)ncd[g, x]. observe that x = 0 is a zero of cd[g, x] of multiplicity n(m−1). consequently, cd(g) = n(m−1)+cd(g). this completes the proof.2 theorem 2.7. for a ladder graph ln, cd(ln) = 2n − 3 for n ≥ 2. proof. obviously, cd-roots of cd[ln, x] are x = 0 with multiplicity 2n − 4 and x = −n − 2/2 with multiplicity one. hence the result follows.2 theorem 2.8. for a cocktail party graph cpn, cd(cpn) = 3 for n ≥ 2. 82 proof. in [safeera and kumar, a], the authors proved that cd[cpn, x] = 2x2(2+(n−2)x). it follows that cd[cpn, x] has cd-roots x = 0 with multiplicity 2 and x = −2/(n − 2) with multiplicity one. thus cd(cpn) = 3. hence the proof follows.2 theorem 2.9. if s(g) is a splitting graph of a graph g with n vertices, then cd(s(g)) ≥ 1. proof. observe that cd[s(g), x] do not have a constant term(see [safeera and kumar, a]). hence the result2. theorem 2.10. for a bull graph bl, cd(bl) = 1. proof. note that cd[bl, x] = x(2x2 + x + 2) [safeera and kumar, a]. the roots of this polynomial are x = 0, −1±i √ 15 4 . obviously, x = 0 is the only real root of cd[bl, x]. hence the result follows.2 theorem 2.11. for a sunlet graph sln, cd(sln) = 2n − 4 for n ≥ 3 . proof. note that cd[sln, x] = nx2n−4(1 + x2) [safeera and kumar, a]. then the cd-roots are x = 0 with multiplicity 2n − 4 and x = ±i. thus cd(sln) = 2n − 4.2 theorem 2.12. for a tadpole graph tm,n, cd(tm,n) = m + n − 2 for m ≥ 3 and n ≥ 1. proof. note that cd[tm,n, x] = xm+n−4(x2 + (m + n − 2)x + 1) [safeera and kumar, a]. since the discriminant of the polynomial x2 +(m+n−2)x+1 is always greater than or equal to zero, it follows that cd(tm,n) = m + n − 2. this completes the proof.2 theorem 2.13. for a bistar graph bn,n (n ≥ 1), cd(bn,n) = { n if n is even n + 1 if n is odd. proof. note that cd[bn,n, x] = 2xn(nxn + 1). if n is even, then nxn + 1 has only complex roots. if n is odd, then nxn + 1 has only one real root and n − 1 complex roots. hence , cd(bn,n) = { n if n is even n + 1 if n is odd. this completes the proof.2 83 theorem 2.14. for a web graph wbn, cd(wbn) = 3n − 4 for n ≥ 3. proof. note that cd[wbn, x] = nx3n−5(x3 + x + 1). obviously, the cd-roots of cd[wbn, x] are 0 with multiplicity 3n − 5, −0.68233, 0.34116 ± 1.16154i. hence cd(wbn) = 3n − 4.2 theorem 2.15. for a armed crown graph cn ⊙ pm, cd(cn ⊙ pm) = { n(m + 1) − 4, if m=1,2 n(m + 1) − 2, if m≥ 3. proof. note that cd[cn ⊙ pm, x] = x 2 + (m − 1)x + 1. for m = 1, 2, the zeros of x2 + (m − 1)x + 1 are complex numbers. if m > 2, the zeros of x2 + (m − 1)x + 1 are real numbers. thus cd(cn ⊙ pm) = { n(m + 1) − 4 if m=1,2 n(m + 1) − 2 if m≥ 3. this completes the proof.2 theorem 2.16. for a sungraph sn, n ≥ 3, cd(sn) = { n − 2, if n is odd n − 1, if n is even. proof. note that cd[sn, x] = nxn−2(xn−1 +1). since xn−1 +1 has real roots if and only if n is even. this tells us that the real cd-roots of cd[sn, x] are 0 and −1 if n is even. if n is odd x = 0 is the only real root of cd[sn, x]. therefore, we have cd(sn) = { n − 2, if n is odd n − 1, if n is even. this completes the proof.2 theorem 2.17. for a bipartite cocktaill party graph bn(n ≥ 2), we have cd(bn) = n. proof. the result follows from the fact that cd[bn, x] = 2nxn.2 84 3 location of the cd-roots of the some graphs in this section, we investigate the location of the roots of some complement degree polynomials. here need the following result [prasolov, 2009]. theorem 3.1. let f(z) = zn + a1zn−1 + . . . + an, where ai ∈ c. then, inside the circle |z| = 1 + max|ai|, there are exactly n roots of f, multiplicities counted. theorem 3.2. all the cd-roots of the gear graph gn lie inside the circle with center (0, 0) and radius n + 1. proof. observe that cd[gn, x] = xn + nx2n−3 + nx2n−2. in this case max|ai| = n, where a′is are the coefficients of cd[gn, x] for i = 1, 2, . . . , 2n−2. then by theorem 3.1, the result follows.2 theorem 3.3. all the cd-roots of the wheel graph wn lie inside the circle with center (0, 0) and radius n. proof. it follows from the fact that cd[wn, x] = (n − 1)xn−4 + 1.2 theorem 3.4. all the cd-roots of the bull graph bl lie on the unit circle centered at the origin. proof. note that the cd-roots of bl are x = 0, −1±i √ 15 4 . these three roots lie on the unit circle centered at the origin.2 theorem 3.5. all the cd-roots of the sunlet graph sln lies in the disk |z| ≤ 1. proof. the cd-roots of the sunlet graph cd[sln, x] are x = 0 and x = ±i. hence the result follows. 2 theorem 3.6. all the cd-roots of the sun graph sn lies in the disk |z| ≤ 1. proof. note that cd[sn, x] = nx2n−3 + nxn−2 = nxn−2(xn−1 + 1). obviously, roots of xn−1 + 1 lie on the unit circle. hence the result.2 4 conclusions in this paper, we introduced cd-roots of the complement degree polynomial of some graphs. moreover, we investigated the location of the cdroots of some complementary degree polynomials. 85 acknowledgements the authors are grateful to the referee for valuable suggestions. the first author acknowledges council of scientific and industrial research(csir) for financial support. references a. anto and p. p. hawkins. vertex polynomial of graphs with new results. global journal of pure and applied mathematics, 15:469–475, 2019. f. harary. graph theory. springer, new york, 1969. v. v. prasolov. polynomials. springer, new york, 2009. k. safeera and v. a. kumar. complement degree polynomial of graphs. gulf journal of mathematics (communicated), a. k. safeera and v. a. kumar. complement degree polynomials of some graph operations. palestine journal of mathematics (communicated), b. k. safeera and v. a. kumar. stability of complement degree polynomial of graphs. south east asian journal of mathematics and mathematical sciences( communicated), c. y. shi, m. dehmer, x. li, and i. gutman. graph polynomials. crc press, 2016. m. shikhi. a study on common neighbor polynomial of graphs. phd thesis, 2019. s. s. r. v. jeba rani and t. s. i. mary. vertex polynomial of ladder graphs. infokara research, 8:169–179, 2019. 86 ratio mathematica volume 42, 2022 common neighbor polynomial of some special trees shikhi mandattil* anil kumar vasu† abstract let g(v, e) be a simple graph of order n with vertex set v and edge set e. let (u, v) denotes an unordered vertex pair of distinct vertices of g and let n(u) denote the open neighborhood of the vertex u in g. the i-common neighbor set of g is defined as n(g, i) := {(u, v) : u, v ∈ v, u ̸= v and |n(u) ∩ n(v)| = i}, for 0 ≤ i ≤ n − 2. the polynomial n[g; x] = ∑(n−2) i=0 |n(g, i)|x i is defined as the common neighbor polynomial of g. in this paper, we study the common neighbor polynomial of some special type of trees such as complete m-ary tree, caterpillar tree, star like tree and fire cracker graph. keywords: common neighbor, common neighbor polynomial. 2020 ams subject classifications: 05c31, 05c39.1 *shikhi mandattil, department of mathematics, govt. arts and science college, kozhikode, kerala, india; shikhianil@gmail.com. †anil kumar vasu, department of mathematics, university of calicut, malappuram, kerala, india; anil@uoc.ac.in. 1received on april 9th, 2022. accepted on june 29th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v41i0.755. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 195 shikhi m, anil kumar v. 1 introduction vertex similarity of nodes is a well studied concept in graph theory as it is highly significant in various fields such as in the study of molecular structure of chemical graphs, measuring consensus rate of different individuals/organizations in network analysis etc. the number of common neighbors shared by two nodes in a network system can be treated as a measure of consensus among the corresponding individuals. this motivates the authors to define the i-common neighbor set and the common neighbor polynomial of a graph. let g(v, e) be a graph (simple graph) of order n with vertex set v and edge set e. let (u, v) denotes an unordered vertex pair of distinct vertices of g and let n(u) denote the open neighborhood of the vertex u in g. the i-common neighbor set of g is defined as n(g, i) := {(u, v) : u, v ∈ v, u ̸= v and |n(u) ∩ n(v)| = i}, for 0 ≤ i ≤ n − 2. the polynomial n[g; x] = ∑(n−2) i=0 |n(g, i)|x i is defined as the common neighbor polynomial of g. this polynomial was introduced by the present authors in (3). trees are commonly used to represent hierarchical data structures involved in network file system, possibility spaces, algorithmic routing etc. it is easy to dissect a tree data structure to access the information about a particular part of a huge data. due to this reason, many researches were conducted to explore the properties of trees. in this paper, we derive the common neighbor polynomial of some special trees like rooted trees, caterpillars etc. 2 main results a rooted tree is a tree in which one of the vertices is distinguished as the root. according to the distance of other vertices from the root vertex, there is a hierarchy on the vertices of a rooted tree. the distance of a vertex v from the root is called the depth or level of the vertex. the height of a rooted tree is the greatest depth of a vertex of the tree. considering a path from the root to a vertex w, if a vertex v immediately precedes w , then v is called the parent of w and w is called the child of v. vertices having same parent are called siblings. an m-ary tree (m ≥ 2) is a rooted tree in which every vertex has m or fewer number of children. a complete m-ary tree is an m-ary tree in which every internal vertices has exactly m children and all leaves are of same distance from the root. theorem 1. let t be a tree on n vertices. let v be a vertex of t with degree k. if t ′ is a tree obtained from t by attaching p pendent edges at the vertex v, we have the following: n[t ′; x] = n[t ; x] + p 2 (2k + p − 1)x + p(n − k). 196 common neighbor polynomial of some special trees proof. let {v1, v2, . . . , vk} be the neighbors of v in t . when we attach a pendent edge vw to v, n number of new pairs of vertices are introduced in which the pairs (vi, w) where i ∈ {1, 2, . . . , k} have one common neighbor v and the remaining n − k new pairs have no common neighbors. there will be no change in the number of common neighbors of pairs of vertices of t by the introduction of the pendent edge vw. hence the common neighbor polynomial becomes n[t; x] + kx+(n−k). repeating the process p times, after attaching the p-th pendent edge to v, the common neighbor polynomial of resulting graph becomes, n[t ′; x] = n[t; x] + kx + (n − k) + (k + 1)x + (n − k) + . . . + (k + p − 1)x + (n − k) = n[t; x] + [k + (k + 1) + (k + 2) + . . . + (k + p − 1)]x + p(n − k) = n[t; x] + p 2 (2k + p − 1)x + p(n − k). this completes the proof. theorem 2. let t be a complete m-ary tree with p levels where the root vertex is considered to be in the 0-th level. then we have the following: n[t; x] = m2(mp−1 − 1) m − 1 x + ( m 2 ) mp − 1 m − 1 x + m[m2p−2 − mp + m − 1] m − 1 + m2p−1 + p−3∑ i=0 [m2i+3 − mp+i+1 1 − m ] + m2[m2p − mp+1 − mp + m] 2(m2 − 1) . level 1 level 2 level 3 figure 1: complete binary tree with 3 levels of vertices proof. let (u, v) be any pair of vertices of t . here we consider 4 different cases according to the levels in which the vertices u and v lie in t . case(i) for i ∈ {0, 1, 2, . . . , p−2} let u be a vertex in the i-th level and v a vertex in the (i + 1)th or (i + 2)th level. if v is an (i+1)th level vertex, the vertex pair (u, v) have no common neighbors and there are mimi+1 such pairs of vertices in t . if v is in the (i + 2)th 197 shikhi m, anil kumar v. level, then there are mimi+2 pairs of vertices (u, v) in which mi[mi+2 −m2] pairs of vertices have no common neighbors and mim2 pairs have exactly one common neighbor. case(ii) let u be a vertex in the (p − 1)-th level and v a vertex in the p-th level. in this case the pairs of vertices (u, v) have no common neighbors and there are mp−1mp such pairs of vertices. case(iii) for i ∈ {0, 1, 2, . . . , p − 3} let u be a vertex in the i-th level and v a vertex in the j-th level where j = i + 3, i + 4, . . . , p. all the pairs of vertices under this case have no common neighbors and there are mi[mi+3 + mi+4 + . . . + mp] such pairs of vertices. case(iv) for i ∈ {1, 2, . . . , p} let u and v be vertices of same level. in this case 1 2 mi[mi−m] distinct pairs of vertices which are not siblings have no common neighbors and ( m 2 ) mi−1 pairs of vertices which are siblings have exactly one common neighbor. from the above cases, it follows that n[t; x] = p−2∑ i=0 mim2x + ( m 2 ) p∑ i=1 mi−1x + p−2∑ i=0 mi[mi+1 + mi+2 − m2] + m2p−1 + p−3∑ i=0 mi[mi+3 + mi+4 + . . . + mp] + p∑ i=1 1 2 mi(mi − m). = m2(mp−1 − 1) m − 1 x + ( m 2 ) mp − 1 m − 1 x + m[m2p−2 − mp + m − 1] m − 1 + m2p−1 + p−3∑ i=0 [mp+i+1 − m2i+3 m − 1 ] + m2[m2p − mp+1 − mp + m] 2(m2 − 1) . this completes the proof. a derivative of a graph g is a graph obtained from g by deleting all the pendent vertices of g. a caterpillar is a tree whose derivative is a path graph (5). consequently, a caterpillar pn(m1, m2, . . . , mn) is obtained by attaching mi pendent edges to the vertex vi of a path pn where i ∈ {1, 2, . . . , n}. theorem 3. the common neighbor polynomial of a caterpillar tree pn(m1, m2, . . . , mn) is given by the following: 198 common neighbor polynomial of some special trees v (1) 1 v (1) 2 v1 v2 v3 v4 v (2) 1 v (2) 2 v (2) 3 v (3) 1 v (4) 1 v (4) 2 v (4) 3 figure 2: the caterpillar p4(2, 3, 1, 3) n[pn(m1, m2, . . . , mn); x] = n[pn; x] + n∑ j=1 n[k1,mj; x] + ∑ l,k∈{1,2,...,n} l ̸=k mlmk + [ m1 + mn + 2 n−1∑ j=2 mj ] x + (n − 2)(m1 + mn) + (n − 3) n−1∑ j=2 mj. proof. let pn(m1, m2, . . . , mn) be a caterpillar and let v1, v2, . . . , vn be the vertices of its derived graph which is a path. also let v(j)1 , v (j) 2 , . . . , v (j) mj be the pendent vertices of the caterpillar tree attached to the vertex vj where j ∈ {1, 2, . . . , n}. let (u, v) be any pair of vertices of the caterpillar. we consider the following cases to build up its common neighbor polynomial. case(i) let u, v ∈ {v1, v2, . . . , vn}. here the number of pairs of vertices (u, v) with i common neighbors equals |n(pn, i)|. so the pairs of vertices under this case contribute the term n[pn; x] to the common neighbor polynomial of the caterpillar. case(ii) let u, v ∈ {vj, v (j) 1 , v (j) 2 , . . . , v (j) mj}. here the vertices of the set under consideration spans a star graph k1,mj and hence the number of pairs of vertices (u, v) with i common neighbors equals |n(k1,mj, i)|. case(iii) let u ∈ {v(l)1 , v (l) 2 , . . . , v (l) ml} and v ∈ {v (k) 1 , v (k) 2 , . . . , v (k) mk} where l ̸= k and l, k ∈ {1, 2, . . . , n}. here u and v are the pendent vertices attached to the vertices vl and vk respectively where l ̸= k. no pair of vertices under this case have common neighbors and there are ∑ l,k l ̸=k mlmk such pairs. case(iv) let u ∈ {v1, vn}, a pendent vertex of the derived graph pn and let v be any vertex attached to the vertices of pn such that uv is not an edge of the caterpillar. 199 shikhi m, anil kumar v. in this case pairs of vertices of the form (v1, v (2) l ) and (vn, v (n−1) k ) where l ∈ {1, 2, . . . , m2} and k ∈ {1, 2, . . . , mn−1} have exactly one common neighbor each and there are m2+mn−1 such pairs of vertices. there remains m1 +m2 +mn−1 +mn +2 ∑n−2 j=3 mj pairs of vertices under this case which have no common neighbors. case(v) let u ∈ {v2, v3 . . . , vn−1} and let v be any vertex selected in a way same as in case(iv). for i ∈ {2, 3, . . . , n − 1}, the pairs of vertices of the form (vi, v (i−1) l ) where l ∈ {1, 2, . . . , mi−1} have exactly one common neighbor vi−1 and pairs of vertices of the form (vi, v (i+1) k ) where k ∈ {1, 2, . . . , mi+1} have exactly one common neighbor vi+1. there are m1 +m2 +mn−1 +mn +2 ∑n−2 j=3 mj such pairs of vertices. the remaining pairs of vertices under this case have no common neighbors and there are (n − 3)(m1 + mn) + (n − 4)(m2 + mn−1) + (n − 5) ∑n−2 j=3 mj such pairs of vertices. from the above cases, it follows that n[pn(m1, m2, . . . , mn); x] = n[pn; x] + n∑ j=1 n[k1,mj; x] + ∑ l,k l ̸=k mlmk + [m2 + mn−1]x + m1 + m2 + mn−1 + mn + 2 n−2∑ j=3 mj + [ m1 + m2 + mn−1 + mn + 2 n−2∑ j=3 mj ] x + (n − 3)(m1 + mn) + (n − 4)(m2 + mn−1) + (n − 5) n−2∑ j=3 mj. now the result follows after some rearrangement of the terms. corollary 4. for a caterpillar pn(m, m, . . . , m) where m pendent edges are attached to each vertex of the path pn, we have n[pn(m, m, . . . , m); x] = n[pn; x] + nn[k1,m; x] + ( n 2 ) m2 + 2m(n − 1)x + m(n − 1)(n − 2). a star like tree(2) s(n1, n2, . . . , nk) is a graph having only one vertex w of degree greater than 2 such that deletion of w results in a disjoint union of the path 200 common neighbor polynomial of some special trees graphs pn1, pn2, . . . , pnk . the star like tree graphs are used to represent proteins which will have generally 20 branches where each branch indicates the presence of one of the 20 natural amino acids. for example, the strand a of human insulin has 21 amino acids of 11 kinds which can be modelled by a star like tree graph with 11 branches as shown in figure 3. figure 3: strand a of human insulin theorem 5. the common neighbor polynomial of a star like tree s(n1, n2, . . . , nk) with n + 1 vertices is given by n[s(n1, n2, . . . , nk); x] = k∑ r=1 n[pnr+1; x] + ( k 2 ) x + ( n 2 ) − k∑ r=1 ( nr 2 ) − ( k 2 ) where n = n1 + n2 + . . . + nk. proof. let s(n1, n2, . . . , nk) be a star like tree with a vertex w such that s(n1, n2, . . . , nk)−w = pn1 ∪pn2 ∪. . .∪pnk . any pair of vertices (u, v) ∈ pnr ∪ {w} where r ∈ {1, 2, . . . , k}, has as many common neighbors in s(n1, n2, . . . , nk) as it has in pnr+1. let (u, v) be a pair of vertices in s(n1, n2, . . . , nk) such that u ∈ pnr and v ∈ pns where r ̸= s, r, s ∈ {1, 2, . . . , k}. then the vertex pair (u, v) has a single common neighbor w if both u and v are adjacent to w and there are no common neighbors otherwise. hence there are ( k 2 ) pairs of vertices with one common neighbor and ( n 2 ) − ∑k r=1 ( nr 2 ) − ( k 2 ) pairs of vertices with no common neighbors. hence the result follows. corollary 6. the common neighbor polynomial of the graphical representation of strand a of human insulin is given by n[g; x] = 65x + 166. proof. the proof follows from the fact that strand a of human insulin can be graphically represented as a star like tree graph s(1, 2, 4, 2, 2, 1, 2, 2, 1, 2, 2). 201 shikhi m, anil kumar v. the (n, k) firecracker graph(1) is obtained by identifying each vertex of a path pn with one of the pendent vertices of the star graph k1,k. in particular, the (n, 2) firecracker graph is known as the centipede graph. u2 v2 v (2) 2 v (2) 3v (2) 1 figure 4: the (4, 4)firecracker graph lemma 7. (see (3)) for a complete bipartite graph km,n, we have n[km,n; x] = ( m 2 ) xn + ( n 2 ) xm + mn. lemma 8. (see (3)) for a path pn with n ≥ 2 vertices, we have n[pn; x] = (n − 2)x + ( n − 1 2 ) + 1. theorem 9. the common neighbor polynomial of (n, k) firecracker graph g is given by the following: n[g; x] = [ n ( k 2 ) + 3n − 4 ] x + nk + ( n−1 2 ) + 1 + ( n 2 ) k2 + (n − 1)(n − 2) + n(n − 1)(k − 1). proof. for j ∈ {1, 2, . . . , n}, let vj1, v j 2, . . . , v j k be the pendent vertices of the j th star where the vertex vjk is identified with the vertex uj of the path pn. let vj be the center vertex of the jth star attached to the vertex uj of pn. let (u, v) be any pair of vertices of g. here we consider 5 cases: case(i) let u, v ∈ {vj, v j 1, v j 2, . . . , v j k} where j ∈ {1, 2, . . . , n}. in this case, the pair (u, v) has as many common neighbors in g as in k1,k. case(ii) let u, v ∈ {u1, u2, . . . , un}. here the vertex pair (u, v) has as many common neighbors in g as in pn. case(iii) let u = vj and v ∈ {u1, u2, . . . , uj−1, uj+1, . . . , un} where j ∈ {1, 2, . . . , n} . in this case, for each j, the pairs (vj, uj−1) and (vj, uj+1) has exactly one 202 common neighbor polynomial of some special trees common neighbor. also the pairs (v1, u2) and (vn, un−1) have one common neighbor each. hence there are 2(n − 1) pairs of vertices (u, v) with one common neighbor and all other (n − 1)(n − 2) pairs of vertices have no common neighbors. case(iv) let u ∈ {vr, vr1, vr2, . . . , vrk−1} and v ∈ {vs, v s 1, v s 2, . . . , v s k−1} where r, s ∈ {1, 2, . . . , n}. in this case, there are ( n 2 ) k2 pairs of vertices which have no common neighbors. case(v) let u ∈ {vj1, v j 2, . . . , v j k−1} and v ∈ {u1, u2, . . . , uj−1, uj+1, . . . , un} where j ∈ {1, 2, . . . , n}, in this case, there are n(n − 1)(k − 1) pairs of vertices which have no common neighbors. using lemmas 7 and 8, it follows that, n[g; x] = nn[k1,k; x] + n[pn; x] + 2(n − 1)x + (n − 1)(n − 2) + ( n 2 ) k2 + n(n − 1)(k − 1) = n [(k 2 ) x + k ] + (n − 2)x + ( n − 1 2 ) + 1 + 2(n − 1)x + ( n 2 ) k2 + (n − 1)(n − 2) + n(n − 1)(k − 1) = [ n ( k 2 ) + 3n − 4 ] x + nk + ( n − 1 2 ) + 1 + ( n 2 ) k2 + (n − 1)(n − 2) + n(n − 1)(k − 1). this completes the proof. corollary 10. the common neighbor polynomial of the centipede graph g is given by n[g; x] = 4(n − 1)x + 2n + 3(n − 1)(3n − 2) 2 + 1. proof. the proof follows from the fact that the centipede graph is a special case of (n, k)firecracker graph when k = 2. 203 shikhi m, anil kumar v. references [1] eric w. weisstein, firecracker graph, http://mathworld.wolfram.com /firecrackergraph.html [2] m. lepovic, i. gutman, some spectral properties of starlike trees, bulletin t.cxxii de l academie serbe des sciences et des arts 2001. [3] m. shikhi and v. anil kumar, common neighbor polynomial of graphs, far east journal of mathematical sciences,volume 102, number 6, 2017, pages 1201-1221. [4] m. shikhi and v. anil kumar, common neighbor polynomial of graph operations, far east journal of mathematical sciences, volume 102, issue 11, 2017,pages 2629 2641. [5] sherif elbasil, applications of caterpillar trees in chemistry and physics, journal of mathematical chemistry,1987,pages 153-174. 204 introduction main results combination of discounting functions ratio mathematica, 21, 2011, pp. 59-74 59 combination of survival probabilities of the components in a system. an application to longterm financial valuation 1 salvador cruz rambaud 2 abstract. the net present value (npv) is a well-known method to value an investment project. nevertheless, this methodology exhibits a serious problem when the used discounting function decreases very rapidly, especially in (very) long-term projects, because the future cash-flows are not significant in the expression of the npv. for this reason, this paper introduces a methodology to correct the discounting function used for valuing. to do this, a new operation between discounting functions is defined by reducing the (cumulative) instantaneous discount rate corresponding of the valuing discounting function with another appropriate discounting function. the result is a new discounting function which can be more adequate to value this class of investment projects. keywords. combination, discounting function, (cumulative) instantaneous discount rate, net present value, investment project. 1. introduction 1 this paper has been partially supported by the project “valoración de proyectos gubernamentales a largo plazo: obtención de la tasa social de descuento”, reference: p09-sej-05404, proyectos de excelencia de la junta de andalucía and fondos feder. 2 departamento de dirección y gestión de empresas. universidad de almería (spain), e-mail: scruz@ual.es. mailto:scruz@ual.es 60 it is well-known that traditionally the exponential discounting has been used in the valuation of investment projects, as discounting function. but the main problem that exhibits this type of discount is the geometrical diminishing of its corresponding factors. in effect, the expression t i   )1( decreases exponentially whereby future cashflows, being very important, are not significant in the expression of the net present value (npv). this is the reason whereby our aim is to considerer a diminishing discount rate which would imply, at least, a decay of the corresponding (cumulative) instantaneous discount rate. on the other hand, in a previous work, cruz and muñoz (2005 and 2007) introduced a new point of view of determining the social rate of discount and, more concretely, the discount function to be applied in the valuation of (very) long-term environmental and governmental projects. to do this, they started from the hazard rate of the system to which the project we are trying to value is addressed. in this way, if we are trying to value the construction of a public good (for example, a highway), the hazard rate corresponding to this construction along his useful life will supply us its survival probability (defined as the complement to the unit of the corresponding distribution function) which we will identify with the discounting function to be used in the valuation. thus, the instantaneous hazard rate of an investment is identified with the instantaneous discount rate corresponding to the discounting function necessary to value the project. as a consequence, the discounting function will be the survival probability of the system. more widely, the following table establishes the correspondence between several concepts from finance (see, for example, gil, 1993) and from reliability theory (see, for example, barlow and proschan, 1996). reliability theory finance survival probability (distribution tail) )(1)( tfts  discounting function )(ta instantaneous hazard rate )( d/)(d )( ts tts th  instantaneous discount rate )( d/)(d )( ta tta t  61 density function t ts tf d )(d )(  cumulative inst. discount rate t ta t d )(d )(  conditional probability )(1 )(1 tf stf   discounting factor )( )( ta sta  table 1. correspondence of concepts from reliability theory and finance. starting from this methodology, we can obtain a (cumulative) instantaneous discount rate with two important advantages. the first one is that this magnitude is variable and the second one is that it can be diminishing. in this way, we agree with harvey’s (1986) position who proposes the hyperbolic or hyperbola-like discounting function and later (1994) defends variable discount rates. harvey (1994) examines “the reasonableness for public policy analysis of non-constant discounting method that, unlike constant discounting, can accord considerable importance to outcomes in the distant future”. in his work, he proposes a method with positive discount rates that decrease and converge to zero as time converges to infinity. the organization of this paper is as follows. in section 2, we introduce a new algebraic operation between the survival probabilities of two components in a system. taking into account table 1, this is the same as define an algebraic operation between two discounting functions. section 2 introduces a novel classification of discounting functions in singular and regular ones. later, section 4 presents a noteworthy application of section 2 for the valuation of (very) longterm investment projects, avoiding the problems exhibited by a rapidly decreasing discounting function. finally, section 5 summarizes and concludes. 2. combination of survival probabilities of the components in a system. combination of discounting functions let us consider a structure composed by two independent components whose i-th component (i = 1, 2) has probability 62 )(1)( tftp ii  of still being operative at time t. if )(ph is the structure reliability function, where ),( 21 ppp , and )(1)( tfts  is the probability of the structure survival past time t, then since ))(()( thts p , we obtain:                    t p p h t p p h t s d d d d d d 2 2 1 1 . (1) it is well-known that )(: d d tf t s  is the density function of the variable t describing the useful life of the system, and )(: d d 1 1 tf t p  and )(: d d 2 2 tf t p  are the density functions of variables 1 t and 2 t describing the useful life of components 1 and 2, respectively. a noteworthy case is that in which the density function of the structure survival is the density function of component 1 but reduced by the effect of the survival probability of component 2, that is to say: )( d )(d )()()( 2 1 21 tp t tp tptftf  . (2) in this case,   .d)()(1)(d)(1 d)( d )(d 1 d d d 1))((:)( 0 0 2112 0 2 1 0 21        t t t t xxpxfxpxp xxp x xp x x s tppts (3) 63 that is, )(d 1 tp , as an approximation of )()( 111 tphtpp  , is reduced by the survival probability of component 2. this new function 21 pp  will be called the combination of survival probabilities 1 p and 2 p . taking into account table 1, we can export this concept to finance. thus, the combination 21 aa  of two discounting functions 1 a and 2 a :         tt t t xxaxxaxa xxa x xa x x a taata 0 21 0 12 0 2 1 0 21 d)()(1)(d)(1 d)( d )(d 1 d d d 1))((:)(  (4) can be interpreted as a methodology to reduce the time perception of an objective discounting function ( 1 a ) by the effect of another discounting function ( 2 a ). this is because the more aged the individuals, the less time perception. in effect, a year of future time is not the same for a person r years old than a person s years old, being sr  . in this case, the time perception is greater for the second one. in what follows and taking into account the aim of this paper, we will only refer to the combination of two discounting functions. example 1 let us consider the combination of two simple discounting functions of parameters d and d:        2 11)( t dtdta . the following proposition supplies a preliminary basic inequality. proposition 1   )()(11)()( 211 tatatata  . 64 proof. in effect, for the first inequality, as 1)(0 2  xa , it is verified that:   tt xxxxax 0 1 0 21 d)(d)()(  and so )(d)(1d)()(1)( 1 0 1 0 21 taxxxxaxta tt    . for the second inequality, as function 2 a is strictly decreasing, take into account that:   )()(1d)()(d)()( 21 0 12 0 21 tataxxtaxxax tt    . thus,   )()(11d)()(1)( 21 0 21 tataxxaxta t   .  a graphic representation of the discounting function obtained in example 1 for 05.0d and 06.0d , and a confirmation of the result deduced in proposition 1, can be seen in figure 1. figure 1. confirming proposition 1 0 0,2 0,4 0,6 0,8 1 1,2 0 2 4 6 8 10 time d is c o u n ti n g f u n c ti o n s a_1(t) a(t) 1-[1-a_1(t)]*a_2(t) 65 with respect to the temporal domain, )(td , of the new discounting function, two cases can occur ( )( 1 td and )( 2 td are the time discounting domains of the discounting functions 1 a and 2 a , respectively):  if )()( 12 tdtd  , then )()( 2 tdtd  .  if )()( 21 tdtd  , then )()()( 21 tdtdtd  , because, if   11 ,0)( ttd  , there can exist a non-empty interval   )()(, 1221 tdtdtt  where 0 1 a and 0 d d 1  t a . in this case,   211 ,)()( tttdtd  . observe that eventually )(td can coincide with )( 2 td . in example 1,        d td 1 ,0)( 1 and         d td 1 ,0)( 2 . consequently, two cases can occur:  if dd  , dd 11   and so )()( 12 tdtd  . thus,         d tdtd 1 ,0)()( 2 .  if dd  , dd   11 and so )()( 21 tdtd  . as 1 a is decreasing and 0)( 2 ta in       dd 1 , 1 : 1 d 1 d  1 66 then there exists a 2 t such that        21 , 1 )()( t d tdtd . to calculate 2 t , we have to solve the equation: 01 2 2  dt t dd , which only has a solution if and only if 2 d d  . in this case the obtained solution is d d d dd dddd t        211 2 2 , from where d d d t     211 2 , which obviously lesser than d  1 . on the other hand, writing the solution as dd dddd t    22 2 )( , we can show that ddd ddd t 1)( 2     . definition 1 let 1 a and 2 a be two discounting functions. the ordinary product of both functions, denoted by 21 aa  , is defined in the following way: )()())(( 2121 tatataa  . observe that this algebraic operation reflects the “multiplicative” superposition of the effects due to both discounting functions over a certain temporal interval. definition 2 let 1 a and 2 a be two discounting functions. the reduced sum of both functions, denoted by 21 aa  , is defined in the following way: 67 1)()())(( 2121  tatataa . once defined these algebraic operations, we can enunciate the following proposition 2 212121 )()( aaaaaa  . proof. in effect, by calculating the integral  t xxax 0 21 d)()( by parts, we have:   tt xxxatataxxax 0 2121 0 21 d)()(1)()(d)()(  , from where we can easily deduce the required equality.  the following theorem relates the convexity of discounting functions a and 1 a . theorem 1 if 1 a is convex, then 21 aaa  is also convex, independently of the convexity or concavity of 2 a . proof. from equation (4), )( d )(d d )(d 2 1 ta t ta t ts  . differentiating again with respect to x: t ta t ta ta t ta t ts d )(d d )(d )( d )(d d )(d 21 22 1 2 2 2  . as 1 p is convex, 0 d )(d 2 1 2  t ta . moreover, as t ta d )(d 1 and t ta d )(d 2 are negative, and obviously 0)( 2 ta , then s is convex.  68 example 2 let us consider the combination of the simple discounting function of parameters d and the hyperbolic discounting of parameter i: )1ln(1)( tidta  . a graphic representation of the discounting function obtained in example 2 for 05.0d and 06.0i , and a confirmation of the result deduced in theorem 1, can be seen in figure 2. figure 2. confirming theorem 1 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 0 2 4 6 8 10 time d is c o u n ti n g f u n c ti o n s a_1(t) a_2(t) a(t) obviously, the operation  does not verify the commutative property, but we can easily show the following proposition. observe that there exists an “exchange by quotient” between the cumulative instantaneous discount rates of the two combinations of discounting functions towards the instantaneous discount rates corresponding to components 1 and 2. proposition 2 the following equality holds: )( )( ))((d ))((d 2 1 12 21 t t taa taa      . 69 proof. it is obvious taking into account that )( d )(d d ))((d 2 121 ta t ta t taa   , )( d )(d d ))((d 1 212 ta t ta t taa   , )( d/)(d )( 1 1 1 ta tta t  and )( d/)(d )( 2 2 2 ta tta t  .  we can check the result obtained in proposition 2 with the following example. example 3 combination of two exponential discounting functions of parameters i and i ( ii  ):   tiii t ta   )1)(1()1ln( d )(d and   tiii t ta    )1)(1()1ln( d )(d . therefore, we can obviously check that: )( )( )1ln( )1ln( ))((d ))((d 2 1 12 21 t t i i taa taa         . proposition 3 the operation  does not verify the commutative property except for equal elements. moreover, it is cancellative on the left and on the right. proof. in effect, if ))(())(( 1221 taataa  , then )( d )(d )( d )(d 1 2 2 1 ta t ta ta t ta  and, consequently, )( )( 1 d )(d )( 1 d )(d )( 2 2 2 1 1 1 t tat ta tat ta t   . thus )()( 21 tata  . on the other hand, if ))(())(( 3121 taataa  , by definition, )( d )(d )( d )(d 3 1 2 1 ta t ta ta t ta  and then )()( 32 tata  . finally, if 70 ))(())(( 3231 tpptpp  , by definition, )( d )(d )( d )(d 3 2 3 1 ta t ta ta t ta  and then )()( 21 tata  .  3. singular and regular discounting functions definition 3 (maravall, 1970) a discounting function )(ta is said to be singular if 0)(lim   ta t or there exists a real number 0 t such that 0)( 0 ta . otherwise, )(ta is said to be regular. example 4 the discounting function is tj ti ta    1 1 )( , where ji  , is singular because, obviously, j i ta t   )(lim . obviously, hyperbolic discounting is regular. a singular discounting function is a peculiar discounting function which has a horizontal asymptote at ly  , where l can be interpreted as the mass of probability at infinity of the corresponding distribution function. representing this function in the extended real numbers: 1 l 0  and so its corresponding distribution function: 71 1 – l 0  definition 3 provides a classification of discounting functions: 1. singular discounting functions with bounded domain:   0 ,0)( ttd  and 0)( 0 ta . more specifically, 1)(0 0  ta 2. regular discounting functions with bounded domain:   0 ,0)( ttd  and 0)( 0 ta . 3. singular discounting functions:   ,0)(td and 0)(lim   ta t . more specifically, 1)(lim0   ta t . 4. regular discounting functions:   ,0)(td and 0)(lim   ta t . it is possible to provide some results on all possible combinations of different class of discounting functions. for instance, it can be shown that the combination of two singular discounting functions with bounded domain is also singular with bounded domain (see example 1) and that the combination of a singular and a regular discounting function with bounded domain is singular with bounded domain. finally, next examples show the result of combining of some wellknown discounting function. example 5 combination of a hyperbolic discounting function of parameter i and a simple discounting of parameter d: 72                it it i d iti ta 1 1 )1ln(1 1 1 1 1 1)( 2 . example 6 combination of two hyperbolic discounting functions of parameters i and j:               itji i jt it ji j ta j i 1 1 1 )1( )1( ln )( 1)( 2 (singular). example 7 combination of a simple discounting function of parameter d and an exponential discounting function of parameter k:  kte k d ta   11)( (singular). example 8 combination of an exponential discounting function of parameter k and a simple discounting function of parameter d:  ktkt e k d edtta   1)1()( (singular). 4. the combination of discounting functions in the valuation of governmental projects consider the case in which a government must decide if a (very) long-term investment project is feasible. it is well-known that, to valuate this project, the most important discounting function to be used in the net present value (npv) formula is the exponential one t ita   )1()( 1 , being i the technical interest rate:    n k k kacfanpv 1 1 )( , 73 where:  npv is the net present value of the project;  a is the initial payment of the project;  n is the useful life of the project;  k cf is the k-th cash-flow corresponding to the project. assume that the survival probability of the system or the perception time of the population is described by the discounting function )( 2 ta . in this case, it could be convenient to reinforce the first discounting function with the aim of preserve the future cashflows. thus, the formula to be employed would be:    n k k kaacfanpv 1 21 ))(( , leading to smaller discount rates, more appropriate to value the aforementioned governmental projects. 5. conclusion in (very) long-term project appraisal (for example, governmental and environmental projects), the exponential discounting function has been traditionally used to update the future cash-flows at the present moment. despite its generalized use, exponential discounting presents an obvious problem: the geometric diminishing of the actualization factors “almost annihilates” the most distant cash-flows. therefore, it is necessary to increase the discounting function with the aim of reaching a higher presence of the further cash-flows. to do this, there are several procedures. the methodology used in this paper is based on the idea of a diminishing perception of future time. indeed, empirical researches show that for most people the 74 larger the age of the person, the shorter the time periods. thus, the “perceived” discounted amounts must be lesser and this fact must be reflected in the mathematical expression of the “true” discounting function. in this work, the reduction in the discounted values can be reached with another discounting function through the so-called combination of discounting functions. bibliography [1] barlow r.e. and proschan f. (1996) mathematical theory of reliability, siam, new york [2] cruz rambaud s. and muñoz torrecillas m.j. (2005) some considerations on the social discount rate, environmental science and policy, 8(4), 343-355 [3] cruz rambaud s. and muñoz torrecillas m.j. (2007) obtención de la tasa social de descuento a partir de la tasa de fallo de una distribución estadística: aplicación empírica, estudios de economía aplicada, 25(1), 49-82 [4] gil peláez l. (1993) matemática de las operaciones financieras, ed. ac, madrid [5] harvey c.m. (1986) value functions for infinite-period planning, management science, 32, 1123-1139 [6] harvey c.m. (1994) the reasonableness of non-constant discounting, journal of public economics, 53, 31-51 [7] maravall d. (1970) matemática financiera, ed. dossat, madrid ratio mathematica volume 41, 2021, pp. 7-18 on the stability of a multiplicative type sum form functional equation surbhi madan* shveta grover† dhiraj kumar singh‡§ abstract in this paper we intend to discuss the stability of a sum form functional equation n∑ i=1 m∑ j=1 f (piqj) = n∑ i=1 k (pi) m∑ j=1 q β j where f,k are real valued mappings each having the domain i; (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm; n ≥ 3, m ≥ 3 are fixed integers and β is a fixed positive real power different from 1 satisfying the conventions 0β := 0 and 1β := 1. keywords: additive mapping; bounded mapping; functional equation; stability of a sum form functional equation. 2020 ams subject classifications: 39b52, 39b82. 1 *department of mathematics, shivaji college (university of delhi), raja garden, ring road, new delhi-110027, india; surbhimadan@gmail.com, surbhi@shivaji.du.ac.in. †department of mathematics, university of delhi, delhi 110007, india; srkgrover9@gmail.com. ‡department of mathematics, zakir husain delhi college (university of delhi), jawaharlal nehru marg, delhi 110002, india; dhiraj426@rediffmail.com, dksingh@zh.du.ac.in. §corresponding author 1received on november 23, 2021. accepted on december 27, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.690. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 7 s. madan, s. grover, d.k. singh 1 introduction throughout this paper, n denotes the set of natural numbers; r denotes the set of real numbers and i denotes the closed unit interval [0, 1]. for n ∈ n, γn = { (p1, . . . ,pn); pi ≥ 0, i = 1, . . . ,n; n∑ i=1 pi = 1 } denotes the set of all finite n-component complete discrete probability distributions. a mapping a : i → r is said to be additive on i or on the unit triangle ∆ = {(x,y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ x + y ≤ 1} if it satisfies the equation a(x + y) = a(x) + a(y) for all (x,y) ∈ ∆. further, a mapping a : r → r is said to be additive on r if it satisfies the equation a(x + y) = a(x) + a(y) for all x ∈ r, y ∈ r. an interesting relation between these two additive mappings was established by daróczy and losonczi (2). they proved that additive mapping a : i → r can be uniquely extended to the set of real numbers. the stability of functional equations has intrigued mathematicians for more than eight decades now. the problem primarily aims to study the functional inequality corresponding to the given functional equation. thereafter, it focuses on examining the proximity of their solutions. it needs to be remarked that without knowing the general solution of an equation we can not discuss its stability. the seminal problem entered the corpus of sum form functional equations with the paper of maksa (9). one of the interesting aspect explored in this field is to obtain general solution and discuss the stability of those sum form functional equations which are useful in characterizing entropies. working in this direction, nath and singh (15; 16; 18); singh and grover (25) have recently addressed few sum form functional equations. these equations were useful in characterizing the shannon entropies (19) defined as: hn (p1, . . . ,pn) = − n∑ i=1 pilog2pi (with 0 log2 0 := 0) (1.1) where hn : γn → r, n ∈ n and (p1, . . . ,pn) ∈ γn. with the aim to further delve in the area of entropies, havrda and charvát (4) generalized the shannon entropies (1.1) by introducing entropies of degree β defined as: hβn (p1, . . . ,pn) = (1 − 2 1−β)−1 [ 1 − n∑ i=1 p β i ] (1.2) 8 on the stability of a multiplicative . . . where hβn : γn → r, n ∈ n; (p1, . . . ,pn) ∈ γn and β is a fixed positive real power different from 1, such that 0β := 0 and 1β := 1. losonczi and maksa (8) were first to address the sum form functional equation that characterized the entropies (1.2) by considering the multiplicative type functional equation n∑ i=1 m∑ j=1 f (piqj) = n∑ i=1 f (pi) m∑ j=1 f (qj) (1.3) where f : i → r; (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm and n ≥ 3,m ≥ 3 are fixed integers. thereafter, nath and singh have analysed pexiderized forms of (1.3), containing two and three unknown mappings for n ≥ 3,m ≥ 3 being fixed integers in (11) and (12). it needs to be highlighted that many research papers in reference to the sum form functional equations characterizing several entropies have been written. in brief, these papers reflected upon: some generalizations; pexiderized forms; importance and applications. some significant contributions are: nath and singh (10; 17); singh and dass (20); singh and grover (21; 22; 23). the primary focus of these authors had been to obtain the general solution (or solutions) of the sum form functional equations for fixed integers n ≥ 3, m ≥ 3 or n ≥ 3, m ≥ 2. the stability problem for some of these sum form functional equations has been discussed by the authors but for most of them it remains unaddressed. one of the interesting equation which motivated us is the functional equation n∑ i=1 m∑ j=1 f (piqj) = n∑ i=1 f (pi) m∑ j=1 q β j (1.4) where f : i → r; (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm and β is a fixed positive real power different from 1 satisfying 0β := 0 and 1β := 1. in the recent past, nath and singh (13) have studied the equation (1.4) and obtained its general solutions for n ≥ 3,m ≥ 3 being fixed integers. the authors have further explained the relation of these solutions with entropies (1.2). the stability of (1.4) is established by singh and grover (24) for n ≥ 3,m ≥ 3 being fixed integers. indeed, with the goal of getting a deeper insight of the equation (1.4), nath and singh (14); garg, grover and singh (3) have recently studied a pexiderized form (1.4), that is n∑ i=1 m∑ j=1 f (piqj) = n∑ i=1 k (pi) m∑ j=1 q β j (1.5) where f : i → r, k : i → r; (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm and β is a fixed positive real power different from 1 satisfying 0β := 0 and 1β := 1. in (14) 9 s. madan, s. grover, d.k. singh and (3), the authors have obtained the general solutions of (1.5) for n ≥ 3,m ≥ 3 and n ≥ 3,m ≥ 2. the authors have further reflected upon the significance of general solutions in the light of entropies of degree β and diversity index. so far the stability problem remains unaddressed. the objective of this paper is to discuss the stability of functional equation (1.5). for the problem of stability concerning functional equations, we refer to the survey paper of hyers and rassias (5) and hyers, isac and rassias (6). the problem of stability of the functional equation (1.5) is given along the following lines: let n ≥ 3, m ≥ 3 be fixed integers; 0 ≤ ε ∈ r be fixed. find all the mappings f : i → r, k : i → r satisfying the functional inequality∣∣∣∣∣ n∑ i=1 m∑ j=1 f(piqj) − n∑ i=1 k(pi) m∑ j=1 q β j ∣∣∣∣∣ ≤ ε (1.6) for all (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm. below we will provide some known results. lemma 1.1 ((8)). suppose a mapping φ : i → r satisfies the functional equation n∑ i=1 φ(pi) = c1 for all (p1, . . . ,pn) ∈ γn, n ≥ 3 a fixed integer and c1 a real constant. then there exists an additive mapping a : r → r such that φ(p) = a(p) − 1 n a(1) + c1 n for all p ∈ i. lemma 1.2 ((9)). let 0 ≤ ε ∈ r, n ≥ 3 be fixed integer and ψ : i → r be a mapping which satisfies the functional inequality ∣∣∣∣ n∑ i=1 ψ(pi) ∣∣∣∣ ≤ ε for all (p1, . . . ,pn) ∈ γn. then there exist an additive mapping a1 : r → r and a mapping b1 : r → r such that |b1(p)| ≤ 18ε for all p ∈ r, b(0) = 0 and ψ(p) −ψ(0) = a1(p) + b1(p) for all p ∈ i. lemma 1.3 ((7)). let a2 : r → r be an additive mapping, m : i → r a multiplicative mapping, b2 : r → r a bounded mapping and c2 ∈ r. if a2(p) = m(p)+c2 for all p ∈ i, then a2(p) = dp, p ∈ r for some d ∈ r and m(p) = 0 or m(p) = p, p ∈ i. also if a2(p) = m(p) + b2(p) for all p ∈ i, then a2(p) = dp, p ∈ r for some d ∈ r and m(p) = 0 or m(p) = pα, p ∈ i for some 0 ≤ α ∈ r. lemma 1.4 ((26)). if f is a solution of the functional equation f(x+y) = f(x) + f(y) which is bounded over an interval [a,b], then it is of the form f(x) = c3x for some real number c3. lemma 1.5 ((3)). let n ≥ 3, m ≥ 2 be fixed integers; β be fixed positive real power different from 1 satisfying the conventions 0β := 0, 1β := 1 and f : i → r, 10 on the stability of a multiplicative . . . k : i → r. the pair (f,k) satisfies (1.5) if and only if there exist the additive mappings a1,a2 : r → r and c ∈ r such that (i) f(p) = cpβ + a1(p) − 1nma1(1), (ii) k(p) = cpβ + a2(p) − 1na2(1). } 2 the stability of the functional equation (1.5) in this section our primary aim is to find the solutions of inequality (1.6). thereafter, we need to observe: what is the difference between these solutions and the solutions (given by lemma 1.5) of equation (1.5)? in the sense of hyers, isac and rassias (6), if the difference is only a bounded mapping, we would say that functional equation (1.5) is stable. following this we establish the the main result as follows: theorem 2.1. let n ≥ 3, m ≥ 3 be fixed integers; β be fixed positive real power different from 1 satisfying the conventions 0β := 0 and 1β := 1; ε be a nonnegative real constant and let f : i → r, k : i → r be real valued mappings. suppose the pair (f,k) satisfies (1.6), then there exist the additive mappings a1,a2 : r → r, the bounded mappings b1,b2 : r → r and 0 6= c,c ∈ r such that (i) f(p) −f(0) = cpβ + a1(p) + b1(p), (ii) k(p) −k(0) = cpβ + a2(p) + b2(p) } (2.1) with (i) |b1(p)| ≤ 1296εc (2m + 1), b1(0) = 0, (ii) |b2(p)| ≤ 1296εc (2m + 1) + 18ε, b2(0) = 0. } (2.2) proof. let us put q1 = 1, q2 = . . . = qm = 0 in (1.6). we get∣∣∣∣∣ n∑ i=1 [ f(pi) + (m− 1)f(0) −k(pi) ]∣∣∣∣∣ ≤ ε for all (p1, . . . ,pn) ∈ γn. by lemma 1.2, there exists an additive mapping a1 : r → r and a mapping b∗1 : r → r with |b∗1 (p)| ≤ 18ε, b∗1 (0) = 0, such that f(p) −k(p) −f(0) + k(0) = a1(p) + b∗1 (p) for all p ∈ i. from this, we obtain the expression k(p) = f(p) −a1(p) −b1(p) (2.3) 11 s. madan, s. grover, d.k. singh where b1 : r → r, defined as b1(x) = b∗1 (x) + f(0) − k(0) is a bounded mapping. with the aid of (2.3), inequality (1.6) can be written as∣∣∣∣∣ n∑ i=1 [ m∑ j=1 f(piqj) − [f(pi) −a1(1)pi −b1(pi)] m∑ j=1 q β j ]∣∣∣∣∣ ≤ ε for all (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm; n ≥ 3, m ≥ 3 being fixed integers. by lemma 1.2, there exists a mapping a2 : r × γm → r, additive in the first variable and a mapping b2 : r × γm → r, bounded in the first variable by 18ε with b2(0; q1, . . . ,qm) = 0, such that m∑ j=1 f(pqj) − [f(p) −a1(1)p−b1(p) −f(0) + b1(0)] m∑ j=1 q β j −mf(0) = a2(p; q1, . . . ,qm) + b2(p; q1, . . . ,qm). (2.4) let x ∈ i and (r1, . . . ,rm) ∈ γm be an arbitrary probability distribution. now, replacing p successively by xrt, t = 1, . . . ,m in (2.4); summing the resulting m equations so obtained and using the additivity of the mapping a2 : r × γm → r in the first variable, we have m∑ t=1 m∑ j=1 f(xrtqj)− [ m∑ t=1 f(xrt)−a1(1)x− m∑ t=1 b1(xrt)−mf(0)+mb1(0) ] m∑ j=1 q β j −m2f(0) = a2(x; q1, . . . ,qm) + m∑ t=1 b2(xrt; q1, . . . ,qm) (2.5) for all x ∈ i, (q1, . . . ,qm) ∈ γm and (r1, . . . ,rm) ∈ γm. now for p = x and q1 = r1, . . . ,qm = rm, functional equation (2.4) gives m∑ t=1 f(xrt) = [f(x) −a1(1)x−b1(x) −f(0) + b1(0)] m∑ t=1 r β t + mf(0) + a2(x; r1, . . . ,rm) + b2(x; r1, . . . ,rm). (2.6) from (2.5) and (2.6), we get m∑ t=1 m∑ j=1 f(xrtqj) − [f(x) −a1(1)x−b1(x) −f(0) + b1(0)] m∑ t=1 r β t m∑ j=1 q β j −m2f(0) = [ a2(x; r1, . . . ,rm)+b2(x; r1, . . . ,rm)−a1(1)x− m∑ t=1 b1(xrt) + mb1(0) ] m∑ j=1 q β j + a2(x; q1, . . . ,qm) + m∑ t=1 b2(xrt; q1, . . . ,qm). (2.7) 12 on the stability of a multiplicative . . . we see that, the left hand side of (2.7) is commutative in rt and qj, t = 1, . . . ,m; j = 1, . . . ,m (acźel (1)). so, the commutativity on the right hand side implies a2(x; r1, . . . ,rm) [ 1 − m∑ j=1 q β j ] −a2(x; q1, . . . ,qm) [ 1 − m∑ t=1 r β t ] = m∑ t=1 b2(xrt; q1, . . . ,qm) − m∑ j=1 b2(xqj; r1, . . . ,rm) + [ b2(x; r1, . . . ,rm) −a1(1)x− m∑ t=1 b1(xrt) + mb1(0) ] m∑ j=1 q β j − [ b2(x; q1, . . . ,qm) −a1(1)x− m∑ j=1 b1(xqj) + mb1(0) ] m∑ t=1 r β t . for fixed (q1, . . . ,qm) ∈ γm and (r1, . . . ,rm) ∈ γm, the right hand side of the above equation is bounded on i while the left hand side is additive in x ∈ i. so, by lemma 1.4, it follows that [a2(x; r1, . . . ,rm) −xa2(1; r1, . . . ,rm)] [ 1 − m∑ j=1 q β j ] = [a2(x; q1, . . . ,qm) −xa2(1; q1, . . . ,qm)] [ 1 − m∑ t=1 r β t ] . (2.8) now, we assert that for m ≥ 3 and fixed positive real power β 6= 1, 1− m∑ t=1 r β t does not vanish identically on γm. to the contrary, suppose 1− m∑ t=1 r β t vanishes identically on γm. then, 1 = m∑ t=1 r β t for all (r1, . . . ,rm) ∈ γm. by lemma 1.1, there exists an additive mapping a : r → r such that rβ = a(r) with a(1) = 1. by lemma 1.3, rβ = 0 or rβ = r for all r ∈ i. this gives a contradiction as for the former case, our supposition 1 = m∑ t=1 r β t is contradicted while for the latter case, our assumption β 6= 1 is contradicted. this proves our assertion and so there exists a probability distribution (r∗1, . . . ,r ∗ m) ∈ γm such that 1− m∑ t=1 r ∗β t 6= 0. therefore with the substitution rt = r∗t , t = 1, . . . ,m, functional equation (2.8) reduces to a2(x; q1, . . . ,qm) = a0(x) [ 1 − m∑ j=1 q β j ] + x a2(1; q1, . . . ,qm) (2.9) 13 s. madan, s. grover, d.k. singh where a0 : r → r defined as a0(x) = [ 1 − m∑ t=1 r∗ β t ]−1 [a2(x; r ∗ 1, . . . ,r ∗ m) −x a2(1; r ∗ 1, . . . ,r ∗ m)] is an additive mapping with a0(1) = 0. further from (2.4), we have a2 (1; q1, . . . ,qm) = m∑ j=1 f(qj)−[f(1)−a1(1)−b1(1)−f(0)+b1(0)] m∑ j=1 q β j −mf(0) −b2 (1; q1, . . . ,qm) . (2.10) from (2.4), (2.9), (2.10) with a0(1) = 0, we gather that m∑ j=1 [f(pqj)−f(0)−a0(pqj)− (f(1)−f(0))pqj]−[f(p)−f(0)−a0(p) −p(f(1)−f(0))] m∑ j=1 q β j −p m∑ j=1 [f(qj)−f(0)−a0(qj)−(f(1)−f(0))qj] = [b1(0)−b1(p)+pb1(1)−pb1(0)] m∑ j=1 q β j −pb2 (1; q1, . . . ,qm) (2.11) for all p ∈ i and (q1, . . . ,qm) ∈ γm. now, define a mapping f : i → r as f(x) = f(x) −f(0) −a0(x) − (f(1) −f(0))x (2.12) for all x ∈ i. with the aid of (2.12), equation (2.11) can be written as m∑ j=1 f(pqj) −f(p) m∑ j=1 q β j −p m∑ j=1 f(qj) = [b1(0)−b1(p)+pb1(1)−pb1(0)] m∑ j=1 q β j −pb2 (1; q1, . . . ,qm) . (2.13) it clearly follows from (2.12), that f(0) = 0. also we observe, the right hand side of (2.13) is bounded by 18ε(2m + 1), consequently by lemma 1.2, along with f(0) = 0, there exists a mapping a3 : i×r → r, additive in the second variable and a mapping b3 : i×r → r, bounded in the second variable by 324ε(2m + 1) with b3(p; 0) = 0, such that f(pq) − qβ f(p) −p f(q) = a3(p; q) + b3(p; q). (2.14) 14 on the stability of a multiplicative . . . define a mapping k : i × i → r as k(p; q) = f(pq) − qβ f(p) −p f(q) (2.15) for all p ∈ i, q ∈ i. with the help of (2.15), it can be verified easily that f(pqr) −pq f(r) − qβrβ f(p) −prβ f(q) = k (pq; r) + rβk (p; q) = k (p; qr) + pk(q; r) (2.16) for all p ∈ i, q ∈ i and r ∈ i. from (2.14), (2.15) and (2.16), it follows that a3 (p; qr) + pa3 (q; r) −a3 (pq; r) = b3 (pq; r) + r βa3 (p; q) + r βb3 (p; q) −b3 (p; qr) −pb3 (q; r) . (2.17) apparently, the left hand side of (2.17) is additive in r ∈ i, while its right hand side is bounded on i. consequently by lemma 1.4, its left hand side must be linear therefore, we get a3 (p; qr)+pa3 (q; r)−a3 (pq; r) =r [a3 (p; q) + pa3 (q; 1)−a3 (pq; 1)] . (2.18) now, for the substitution r = 1, equation (2.17) gives pa3 (q; 1) −a3 (pq; 1) = b3 (pq; 1) −pb3(q; 1) . (2.19) from (2.17), (2.18) and (2.19), we obtain( rβ − r ) a3 (p; q) = rb3 (pq; 1) − rpb3(q; 1) −b3 (pq; r) − rβb3 (p; q) + b3 (p; qr) + pb3 (q; r) . (2.20) since for fixed positive real number β 6= 1, we have ‘rβ − r’ does not vanish identically on i, there exists some r∗ ∈ i, such that r∗β − r∗ 6≡ 0. (2.21) using this in (2.20), it follows that a3 (p; q) = ( r∗β − r∗ )−1{ r∗b3 (pq; 1) − r∗pb3(q; 1) −b3 (pq; r∗) − r∗βb3 (p; q) + b3 (p; qr∗) + pb3 (q; r∗) } for all p ∈ i, q ∈ i. this yield that additive mapping a3(p; q) is bounded in the second variable also. hence by lemma 1.4, it must be linear therein. so, a3 (p; q) = qa3(p; 1) (2.22) 15 s. madan, s. grover, d.k. singh for all p ∈ i, q ∈ i. from (2.19), with the substitution q = 1 it follows that a3 (p; 1) = pa3 (1; 1) −b3 (p; 1) + pb3(1; 1) (2.23) for all p ∈ i. from (2.22) and (2.23), it can be concluded that mapping a3(p; q) is bounded. moreover, from (2.14) and (2.23) with f(1) = 0 we obtain, a3(p; 1) = −b3(p; 1). consequently we get |a3(p; q)| ≤ 324ε(2m + 1). hence, using this in (2.14), it follows that |f(pq) − qβ f(p) −p f(q)| ≤ 648ε(2m + 1) (2.24) for all p ∈ i, q ∈ i. now on interchanging the places of p and q in the functional inequality (2.24), we have |f(pq) −pβ f(q) − q f(p)| ≤ 648ε(2m + 1). (2.25) applying triangle inequality to functional inequalities (2.24) and (2.25), we obtain |(qβ − q) f(p) − (pβ −p) f(q)| ≤ 1296ε(2m + 1) (2.26) where p ∈ i, q ∈ i and β is a fixed positive real power different from 1. with the aid of (2.21) we get, q∗β − q∗ 6≡ 0 for some q∗ ∈ i. on taking c := (q∗β − q∗)−1 ∈ r; c := f(q∗)(q∗β − q∗)−1 ∈ r in (2.26), it follows that there exist a mapping b1 : r → r such that (2.2)(i) holds for all p ∈ r and f(p) = cpβ − cp + b1(p) (2.27) for all p ∈ i. thus, the solution (2.1)(i) follows from (2.12) and (2.27) by defining additive mapping a1 : r → r as a1(x) = a0(x)+(f(1)−f(0))x−cx. further, the solution (2.1)(ii) with (2.2)(ii) follows from (2.1)(i), (2.2)(i) and (2.3) by defining additive mapping a2 : r → r as a2(x) = a1(x) − a1(x) and bounded mapping b2 : r → r as b2(x) = b1(x) −b∗1 (x). this completes the proof. 3 acknowledgements the third author is grateful for the support from the serb-matrics scheme (mtr/2020/000508) of the department of science and technology, government of india. references [1] aczel 1966 j. acźel. lectures on functional equations and their applications, academic press, new york and london, 1966. 16 on the stability of a multiplicative . . . 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[26] g.s. young. the linear functional equation, the american mathematical monthly, 65(1) (1958), 37 – 38. 18 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica 28 (2015) 65-83 issn: 1592-7415 65 beneish m-score and detection of earnings management in italian smes christian corsi a , daniela di berardino b , tiziana di cimbrini c a faculty of communication sciences, university of teramo, italy, ccorsi@unite.it b department of management and business administration, university of chieti-pescara, italy, daniela.diberardino@unich.it c faculty of political sciences, university of teramo, italy, tdicimbrini@unite.it abstract accounting literature on the reliability of financial information presents several mathematical models whose purpose is to identify the existence of values manipulations. the phenomenon is described as earnings management and presents a broad discussion concerning the search for suitable models to measure the distortions in values. in this respect, the present paper aims to compare the ability of two versions of the same mathematical model of classify the risk of earnings manipulation in a discriminant way. keywords: beneish m-score; accounting ratios; regression analysis; earnings management 2010 ams subject classification: 03h10; 62p20; 91b02; 91g70. doi: 10.23755/rm.v28i1.28 1 introduction the paper aims to compare the ability of two beneish models, the m-score5 and the modified m-scoreit, to detect earnings management. these methods have not been evaluated by prior research, and it is unclear which type of model dominates, as each models relies on the same assumptions and only empirically we can verify which of them is more descriptively valid. davidson, stickney and weil (1987) [1] define the earnings management as the process by which managers, staying within accepted accounting principles, try to get at a certain desired level of profit to be marked on the http://dx.doi.org/10.23755/rm.v28i1.28 corsi, di berardino, di cimbrini 66 outside. healy and wahlen (1999) [2] state that earnings management occurs when “managers use their own judgment in reporting the financial data and in structuring transactions in order to alter financial reports to deceive stakeholders on the fundamental economic performance the company or to influence the consequences of contracts that depend on accounting data reported”. this perspective focuses on the matter to the judgment of the managers in the definition of financial data. technically, earnings management activities include a spectrum of activities ranging from conservative accounting fraud through aggressive accounting and the neutral, through a wide range of accounting choices [3]. there are several ways in which managers can apply judgment to influence the financial reports. for example by means of the estimates that relate to the final value and duration of a certain good, or about possible future expenses are not yet done. for this reason this phenomenon is linked to the discretionaryaccruals components present in the financial statement. literature on earnings management has extensively examined a set of models to estimate discretionary accruals. these models range from the simple mathematical equation, in which total accruals are used as a measure of discretionary accruals, to sophisticated regression models, which decompose accruals into discretionary and nondiscretionary components and aim to forecast the presence of fraud and financial distress. conversely, other models consider only a set of interrelated accounting ratios, comparing the values among several years in order to find some abnormalities. attention to earnings management policies comes from the social and financial consequences which produces the distortion of information on the financial results of the company. famous scandals of major companies are proof, for that reason more than thirty years research on mathematical methods able to adequately identify the phenomenon showed continued growth. prior studies concluded that managers use discretionary accruals to convey their private information to investors, examining the time-series of discretionary-accruals (hansen, 1996) [4] or the association between stock returns, discretionary accruals and nondiscretionary earnings [5]. several studies are focused on listed firms or on financial statements based on us gaap. in this study we observe a sample of 99 italian academic spin offs, with homogeneous activity and omogeneous accounting rule system. these firms are mainly small and medium and not listed, for these reasons all the statistic models linked to market price of equity, stock volatility and us gaap principles may be inadequate in detecting earnings management practices. in the next section we describe the main attribute of academic spin offs, followed by the concept and the consequences of earnings management and from the properties of beneish model. descriptive statistics, the comparative calculation and the regression analysis will be presented in section 5. beneish m-score and detection of earnings management in italian smes 67 finally we drawn some conclusion about the ability of two versions of beneish model in detecting earnings management within italian sme. 2 academic spin-offs: an overview the current improvement of the spin-off phenomenon in europe has provided a treasured approach to spread new technologies and knowledge [6], driving up the business prospects for the academics and other players involved in projects directed to increase the outcome of the university scientific research [7;8]. simultaneously, the spin-off process from a parent organization, especially from universities, has recently received growing attention both from the academic literature [9;10;11;12] and in the practice [13]. furthermore, thanks to their capability in generating wealth and inspiring the development of scientific knowledge, policy-makers have showed an emergent interest in the academic spin-offs, considering them an active tool to encourage the development of knowledge-based economies in different institutional settings [14;15] so that their creation has become a crucial matter for policy-makers all around the world [16]. this is also due to the fact that either academicians, policy-makers either practitioners agree about the role played by universities as one of the main sources of innovations and their successful diffusion in the society [17;18]. indeed, several scholars [19;20;21; 22] underline that the formation of a firm by a research institution is an outstanding method to commercialize the outcomes of the public research, as well as in contributing to the economic and social welfare and to the regional development. scholars usually highlight the eminence of the foundation and diffusion of knowledge by universities as a noteworthy driving force for technological innovation in an economy, both at local and at national level [23]. the existing literature remarks that the new model of "open innovation", embraced by numerous organizations with the aim to contribute to the dissemination of knowledge, [24;25] has become a critically method in cooperating either with new technology-based firms (ntbfs) either with scientific foundations, such as academics spin-offs, which provide new research settings and a multidisciplinary approach for the development of innovation processes [26;27]. academic spin-offs (asos) are firms generated in order to exploit knowledge originated within universities. more specifically, the current literature defines academic spin-offs as “those companies that germinate from a university, where a group of researchers composes the entrepreneurial unit aiming at the exploitation of skills and results from the research developed within the university” [28] or “company composed by individuals who were former employees of the parent organization, and where the technology and the academic inventors may spin-off both from the institution, or where the technology spins out from the institution but the academic inventor is employed in the university, or, lastly, where only the technology spins out, while the corsi, di berardino, di cimbrini 68 academic inventor does not maintain relationships with the new firm but may have equity” [29]. the establishment of the knowledge/technology employed by an academic spin off is a multi-stage process. generally, literature identified three main models of academic spin-off creation and development. in the first model, ndonzuau, pirnay, and surlemont [30] recognized four central stages in the growth of academic spin-offs: i) creating a sustainable business idea, ii) converting the idea into a business process, iii) building a firm and iv) contributing value to customers, employees, investors and all other stakeholders. these four stages are reciprocally dependent, since choices made in the earlier stages may effective influence the later stages. the second model, developed by shane [7], embraces five stages in typifying a distinctive process to build an academic spin off. the first state is merely academic but the model also allows for tangential technologies that have the prospective to easily enable the development of new products. in cases where the researcher considers that their new technology is an invention which can be commercialized, then, they reveal it to the technology transfer offices (tto). next, in the third stage, the prospective for intellectual property protection is estimated and a patent application may be made. based on the limited monopoly via the patent, the tto can either license the technology to a foundation firm or the researcher may start an academic start-up. moving from the models by ndonzuau et al. and shane, vohora, wright and lockett [31] provided a new perspective on the expansion of academic spin offs. their model also has five stages, but it stresses four pivotal junctures that must be overlapped before transitioning to the next stage: i) research (opportunity recognition), ii) opportunity framing (entrepreneurial commitment), iii) preorganization (threshold of credibility), iiii) re-orientation, iiiii) threshold of sustainability (sustainable returns). considering the above arguments, it is worthwhile to observe that the awareness demonstrated by literature in the success factors and supporting mechanisms of university entrepreneurship, through asos, has increased in the last years [15]. indeed, several scholars [32;33] deal with the elements fostering their creation and growth, which are classified into different categories. a first category refers to the institutional supporting measures [26] such as government laws, financial and non-financial incentives. a second type is associated to university policies [34] such as business plan competitions, spin-off regulations, university business incubators. the third, instead, refers to the external critical factor of the spin off activity [35] such as, for example, entrepreneurial support mechanisms, venture capital, science parks, proximity to parent organization and prospects available from industry. finally, a fourth type is related to the technology features [36], e.g. the prospective of commercialization, the appropriability and the value to customers. often scholars [37] associate features affecting the growth dynamics of academic spin-offs with three different levels of analysis, employed to beneish m-score and detection of earnings management in italian smes 69 investigate the phenomenon with a more comprehensive approach: micro, meso and macro levels. regarding the first level of analysis, the macro one, the focus is on the national systems of innovation and, above all, on the role that policymakers may have in the foundation of academic spin-offs [26;32]. hence, the studies on the creation of academic spin-offs focus on the occurrence of venture capitalists, legal protection of innovations, regional infrastructures and on the business environment in which the universities are regulated but, at the same time, they are less interested in what is happening within the parent organization, which is the university. that’s the reason why the theoretical framework that helps to explain the effects generated at the macro level of analysis – particularly as regards the ntbfs, of which the asos are a specific typology – is the knowledge spillover theory of entrepreneurship applied to the regional context [38]. as for the meso level of analysis, this is focused on the study of university and the tto and tries to identify the fostering mechanisms or factors by which universities promote the effective creation/development of academic start-ups [39;40], as well as, it tries to explore the success of spinning out processes such as a university technology transfer mechanism. frequently, the theoretical framework used to carry out this type of analysis is the resource-based view, according to which internal factors define or influence the formation of academic spin-offs. lastly, the last level of analysis, the micro one, concerns the role played by the individual characteristics of the entrepreneurs or the managerial team, jointly with their social ties, in encouraging the spinout foundation process. in this case, the theoretical framework of reference involves the field of entrepreneurial theories [41;42], which studies the individual characteristics, in conjunctions with the resource-based view which explores the personal assets influencing and affecting the foundation of the academic spin-offs [43]. following a theoretical approximation, the first two levels of analysis can be attributed to those ones which the literature defines university fostering mechanisms of academic entrepreneurship [26;32;35], while the second may be included in those contextual elements which form the local context factors in the development of the academic spin-offs [15]. 3 earnings management in smes earnings management can be loosely defined as a strategy of generating accounting earnings, which “is accomplished through managerial discretion over accounting choices and operating cash flows” [44]. it occurs when managers use judgment in financial reporting and in structuring transactions to alter financial reports to either mislead some stakeholders about the underlying economic corsi, di berardino, di cimbrini 70 performance of the company or to influence contractual outcomes that depend on reported accounting numbers [45]. earnings management is an umbrella for acts that affect the reported accounting earnings or their interpretation, starting from production and investment decisions that partly determine the underlying economic earnings, going through the choice of accounting treatment and the size of accruals when preparing the periodic reports, and ending in actions that affect the interpretation of the reported earnings. not all earnings management is misleading. investors, for example, prefer to separate persistent earnings from one-time shocks. firms that manage earnings in order to allow investors to better distinguish between the two components do not distort earnings. on the contrary, they enhance the informational value of their reported earnings. thus, depending on the will to signal of hide the short or long term performance, it can be beneficial, pernicious or neutral [46]. the studies usually relate the level and type of the earning management adopted by firms with the interests of the key players on the financial accounting scene, which can be grouped into three main categories: management, users and gatekeepers or monitors [46]. management reports earnings, users use earnings as an input to their decision making, and gatekeepers provide valuable signals to other users regarding the credibility and the informational value of the reported earnings [47]. the literature about earnings management has mainly explored the effect of these key players in large firms because financial information published by these firms is easily accessible. large companies are generally listed companies with publicly available financial information while smes are subject to less demand for financial information. in the last years. the literature is focusing on the level and type of earnings management in smes as a result of intuition that firm size affects the incentives to this practice. there should be a little interest of management in managing earnings in smes for its own advantages because small companies are less subject to agency problems, especially when shareholders and managers are the same people, like in family firms. however, incentives to manage earnings also exist in smes when the company needs external financing, for example from banks. also tax purposes are often advanced to explain accounting choices in small firms, especially when alignment between financial and tax reporting is high [48]. on one hand, several studies tried to explain the objectives of financial reporting in smes. lavigne [49] shows that, according to the managers of canadian smes, financial reporting respond to both internal management and tax purposes. he shows that structural factors, such as firm size, ownership structure and debt also influence accounting policies. in the same context of canadian firms, maingot and zeghal [50] find that the objectives of financial reporting are linked to taxes and debt. the performance of the firm can also influence financial reporting. saboly [51] shows that managers of small distressed firms can manage earnings to influence stakeholders. in australia, mcmahon [52] finds that financial reporting quality beneish m-score and detection of earnings management in italian smes 71 in smes is associated with firm size, but not with performance and growth. on the other hand, literature has also focused on the issue of earnings management’s intensity and typology in comparative terms between sme’s and large firms. moses [53] finds evidence that large firms have a bigger incentive to smooth earnings than small firms and michaelson, james, and charles [54] also find consistent evidence. differently, albrecht and richardson [55] find evidence that large firms have less incentive to smooth earnings than small firms. burgstahler and dichev [56] analyze the impact of earnings management on the company's losses, in a sample of 300 companies and the results show that large firms and small ones manage their earnings in order to avoid small losses or small profits decline. rangan [57] finds a significant relationship between earnings management and performance of experienced equity offerings. he suggests that older and largest firms were maneuvering the current accruals to exaggerate the earnings of the experienced equity offerings. degeorge, patel, and zeckhauser [58] indicate that large companies manipulate the earnings of the company to avoid the negative earnings. lee and choi [59] also find that firm size is a variable that could influence a firm's tendency to manage earnings: smaller firms are more likely to manage earnings to avoid reporting losses than larger firms. barton and simko [60] show that big companies face more influence to get the analysts’ demands to manage earnings more effectively. nelson, elliott, and tarpley [61] showed that sometimes auditors might ignore the earnings management of large sized firms. he argues that, since audit fees increase with client size, the probability of adjustments in the financial statements by the auditor becomes lower when increasing the client size. ching, firth, and rui, [62] examine that whether unrestricted current accruals forecasted the returns and earnings performance and resulted that larger firms manipulate current accruals to overstate earnings than the small sized firms. siregar and utam [63] find inconsistent evidence with regard to the impact of firm size on type of earnings management while persons [64] analysis of frauds reveals evidence of more fraudulent activity in smaller firms. the contributions above outlined testify that literature do not converge towards a homogeneous scenario and demonstrate that there is still much to say about sme’s propensity to earnings management. 4 beneish manipulation-score for italian asos literature on earnings management examines the amount of discretionary and non discretionary accruals within the financial statement, considering these values the main sources of manipulation. the pioneering healy [65] contribution assumes that profits derive from a cash part and accruals, the corsi, di berardino, di cimbrini 72 increase of which denotes the presence of a not really cashed income and hence more maneuverable. accruals include revenue and expenditure that have taken place in a certain period, but that did not generate a cash flow during the same period. discretionary accruals are measured as the accruals that cannot be explained by a change in sales and the level of fixed assets, thus, their measure will capture changes in any number of expenses, some revenues, and changes in various working capital accounts. marquardt and wiedman [66] demonstrate that firms issuing equity manage accruals by increasing revenue and decreasing depreciation expense. in other researches [67] emerge that changes to pension assumptions, inventory method, depreciation method and estimates, as well as lifo liquidations are used to manage earnings. other researches associate manipulation of results sudden adoption of more favourable credit terms, the increase in product inventories, the increase in discretionary spending such as research and development, advertising and maintenance [68]. as a result of the earnings management research the analyst will understand that some firms manipulate accounting numbers to manage earnings and that the vehicles chosen for manipulation vary in predictable ways. other than the earnings number, however, it is not known in any given context which numbers are likely to have been manipulated. deangelo et al. [69] state that abnormal changes in accruals between one year and the other are associated with intentional distortion of income, related to the managers’ desire to increase their profit margins in order to achieve their goals. there are different models that estimate accruals, based on statistic index or accounting ratios. the most popular models are the deangelo model (1986), healy model (1985), jones model (1991) and the modified jones model (dechow, sloan, and sweeney 1995), the industry model [45], the cross-sectional jones model [70] and the beneish m-score [71]. the first seven models attempt to measure the earning manipulation through the ratio between the discretionary and non discretionary accruals and three of them, the industry model, the healy model and the jones model, are estimated over an eight-year period ending just prior to the event year. in this analysis we use the beneish model adapted to italian smes by giunta, bini and dainelli [72], which consider the disparate effects on accruals played by the italian accounting principles. beneish m-score is a mathematical model that adopts some financial metrics to identify the extent of a company’s earnings. this model observes the value alteration phenomena in non-listed companies, where value emerges mainly from the financial statements. the original manipulation score (mscore) includes an intercept and eight variables that capture the financial statement distortions that can result from earnings manipulation or that indicate a predisposition to engage in earnings manipulation [73]. one advantage of the m-score is that the treatment sample consists of firm that have indeed managed earnings and that determination is independent of abnormal accrual models [71]. the formula is as follows: beneish m-score and detection of earnings management in italian smes 73 1) m-score8= -4,840 + 0.920dsri + 0.528gmi + 0.0404aqi + 0.892sgi + 0.115depi – 0.172sgai 0.327lvgi + 4.679tata days sales in receivables index (dsri) measures the ratio of days that sales are in accounts receivable in a year compared to that of a prior year and an index higher than 1 describes the increased percentage of non cash sales compared to the prior year. a disproportionate increase in accounts receivable may be indicative of inflated revenues. gross margin index (gmi) measures the variation of gross operating margin and when it’s greater than 1 shows that the profit has worsened in the period under review with the consequence that the firm is likely to manipulate its revenues. asset quality index (aqi) is the ratio of current (ca) and non current asset (property, plants and equipments-ppe) to total assets in one year to a prior year. an increase in aqi index may represent additional expenses that are being capitalized to preserve profitability [71]. indeed, an index greater than 1indicates that the firm has potentially increased its cost deferral or increased its intangible assets, implementing a potential earnings manipulation. sales growth index (sgi) is a measure of growth in revenue and if it’s greater than 1 there is a positive growth in the year under review. callen et al. [74] show that the likelihood of revenue manipulation is increasing with the credit loss ratio, leverage and with the volatility of equity returns and with the ratio of accounts receivable to sales. depreciation index (depi) is the ratio of depreciation expense and gross value of ppe in one year over a prior year. an index above 1 could be a reflection of an upward adjustment of the useful life of ppe. leverage index (lvgi) measures the ratio of total debt to total assets, describing the long-term risks of a company. an index of greater than 1 is interpreted as an increase in the gearing of the company and for that matter exposed to manipulation. total accruals to total assets index (tata) measures the quality of cash flows of the firms. the total accruals metric is computed as change in current assets (except cash and equivalent) less depreciation and the current portion of debts. an increasing degree of accruals as part of total assets would indicate a higher chance of manipulation. another version of the index was empirically derived from the university of lille with another european companies samples [72]. in this case only 5 variables were significant for the purpose of earning manipulation. the formula assumes the following definition: 2) m-score5 = -6,065 + 0.823dsri + 0.906gmi + 0.593aqi + 0.717sgi + 0.107depi empirically, when the m-score5 is greater than -2.22 is high the probability of earning manipulation. some of these variables (dsri, gmi and tata) describe the firms’ ability to generate cash and profits from their business corsi, di berardino, di cimbrini 74 operations. two of them (sgi and lvgi) try to capture the company’s skills and motivations that could lead to the manipulation of accounting rules. finally, the others (aqi, depi and sgai) evaluate investments in assets of the firm and the ability to control costs. the application of models based on the estimation of accounting parameters affected by accounting principles applicable in italy requires a revision in the calculation of the indicators and in their selection. applying the initial formula to a sample of italian listed companies, giunta et al. [73] found a large number of false positives and a predictive power of less than 47%. for this reason the model has been adapted to the italian system, dominated by smes who base their financial statements on the civil code rules, on national accounting principles (oic) based on the principle of prudent estimates of costs and provisions. readjustment affected the structure and the number of variables and related weights. sgi and tata indicators were removed considering their low significance in the sample for the earning manipulation event. therefore, the formula that we could consider for italian smes is the following [72]: 3) m-scoreit = 6,2273 + 0.448dsri + 0.1871gmi + 0.2001aqi + 0.2819depi + 0.6288lvgi the variable weights were estimated using the maximum likelihood analysis, starting from a sample of manipulative society compared with a control group of non-manipulative firms. in this case, the cut-off value for m-score is -4.14. giunta et al. (2014) shows that with this value the model reduces the errors for false positive at level 7.14% and correctly identifies the 92% of manipulations. table 1 describes the formula for each variable considered for m-score; in this analysis we compare m-score5 and m-scoreit. table 1 – variables description code name formula dsri days sales in receivables index (accounts receivablest/salest) / (accounts receivablest-1/salest1) gmi gross margin index [(revenues t-1 – costs of goods sold t-1)/ revenues t-1 ] / [(revenues t – costs of goods sold t)/ revenues t ] aqi asset quality index {1 – [(ca t +ppe t)/total assets t] } / {1 – [(ca t-1 +ppe t1)/total assets t-1]} where ca = current assets ppe = property, plant and equipment depi depreciation index [depreciation and amortizationt-1 / (depreciation and amortizationt-1 + ppe t-1)] / [depreciation and amortizationt / (depreciation and amortizationt + ppe t)] lvgi leverage index (total debts t / total assets t) / (total debts t-1 / total assets t-1) sgi sales growth index revenuest / revenuest-1 beneish m-score and detection of earnings management in italian smes 75 5 research model and results in order to analyse the effects of beneish model in signalling the manipulative firms, the research observes a sample of italian academic spin offs born in 2004 and 2005 and existing until 2015, taken from the database of national network of italian academic spin offs and patents (netval). this analysis considers the performances during the period 2009-2010, just after the beginning of the financial crisis, that is considered a pivotal event for earning managements. data were collected through infocamere database (the national register of italian companies), aidabvdep system and from company websites. we excluded the inactive firms, those with no financial statements after the 2010, distressed firms and others in liquidations. the final sample includes 99 firms, around the 12% of those academic spin offs existing on netval database in 2010 and 66% of those born in 2004-2005. the variation index of the net income in the period t-t-1 is the proxy used to estimate the manipulation risk. descriptive statistics in table 2 show the higher volatility of gmi and sgi indicators that affect the value of m-score5. the mean value of in table 2 we compare the m-score5 model with m-scoreit. always for sgi index, the median value exceeds the unit, showing for it a high associated risk of earnings manipulation related to the revenues management. table 2 – descriptive statistics no. min max mean stddev median dsri 99 0 16.61 0.744572 1.957968 0 gmi 99 -83.2 4268.56 42.71517 429.198 0.743818 aqi 99 0 18.45 1.643234 2.580883 0.991328 sgi 99 0 4913.6 50.7929 493.7183 1.079366 depi 99 0 3.11 1.043452 0.53463 0.927252 lvgi 99 0 5.9 1.086904 0.711245 0.984005 m-score5 99 -76.7 7385.16 70.75233 742.7017 -3.27986 m-scoreit 99 -18.41 793.47 3.405278 80.24861 -4.73452 varprofit t/t-1 99 -8979.53 26.95 -0.03542 0.396172 0.450448 corsi, di berardino, di cimbrini 76 table 3 – comparative analysis for m-score high risk low risk m-score5 33% 66% m-scoreit 31% 69% according to model based on 5 variables, the 33% of the sample presents a high risk of earnings manipulation, while the m-scoreit identifies a lower number of potentially manipulative firms, despite it assumes a lower threshold value. thinking about possible sources of bias, we may assume that the variable with the greatest impact on the difference of the two scores is associated with fluctuations in sales revenue (sgi), considering the high standard deviation that takes in the sample. the reasons can be adduced both to the fact that sgi is not scaled by total assets, as happens for the other and also for the nature of academic spin offs. in fact, the instability in sales is quite common and frequent in these firms, whereas many of them have to wait long periods before concluding the development of research and bring to market the goods obtained. however, the gap between the two indices is rather small, is to be concluded that the classification to which they lead is quite similar, therefore emerges not a significant contribution from the m-scoreit model in discriminating manipulative companies compared to the m-score5 based on accruals. considering that the beneish m-score is a probabilistic model, its limit is that the ability to detect potential fraud is not with 100% accurancy. for this reasons in this analysis we consider only the risk of profit manipulation, linking the variation of net income to the m-score variables, examining the linear regression as follows: 4) varprofit = β0 + β1dsri+ β2gmi + β3aqi+ β4depi + β5lvgi + β6sgi + εi the stepwise procedure (table 4) shows that only the aqi is significant to explain the variation of net income in the period observed. aqi in the sample assumes a mean value greater than 1 and a median value close to 1 that could indicate that the academic spin offs have potentially increased the deferred cost. the negative coefficient in the regression analysis shows that when the firms increase the capitalization of cost related to intangible assets, such as r&d costs, the variability of profit decreases between one period and another, leaving to hypothesize that the budgeting of costs related to r&d could ensure a certain stability in the level of profit. therefore, the systematic capitalization of these deferred costs would allow to homogenize the income levels over time, leaving to assume the existence of an earnings management policy. beneish m-score and detection of earnings management in italian smes 77 table 4 – regression analysis model non-standardized coefficients standardized coefficients t sig. b std.error beta 1 costant 64,384 102,790 ,626 ,533 aqi -92,196 33,884 -,264 -2,721 ,008 stepwise selection: prob f in <=0,050; prob f out>=0,100 model r r-square r-square adj std. error 1 ,264(a) ,070 ,060 866,29650 a. (constant), aqi 6 conclusion literature on earning management has largely focused on methods able to detect manipulative companies, minimizing classification errors, considering that the inadequacy of the calculation method can lead to important social and economic consequences. if on one hand the statistic accrual prediction models neglect some operational dynamics of the company and don’t describe in a significant way the phenomena when the samples are small, on the other hand, the accounting models are less stringent, and built on the basis of accounting standards adopted in selected countries. this paper assumes that in earnings management analysis is important consider the contingent features of the business, of corporate governance, the economic situation and the specific accounting rules of each country. these items affect the business trend of the firms, influencing the accounting policies and favouring opportunistic behaviour. applying the beneish m-score model to a sample of italian sme, in order to detect earnings management, rather than forecasting fraud and financial distress, we didn’t found deep differences between the adjusted version of mscoreit and the simplified model for the european firms. regression analysis also confirmed that the typicality of economic activity, from which descend the investment decisions, is the most effective on the variability of profit margins so for the purpose of detection of earnings management should be considered also expressive variables of this situation. an appropriate weighting system could adequately quantify the impact of sectoral differences, as well as the company size, then the complexity of corporate governance. corsi, di berardino, di cimbrini 78 bibliography [1] davidson s., stickney, c.p. and weil, r.l. 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(2008). revenue manipulations and restatements by loss firms. auditing: a journal of practice and theory 27(2), 129. approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 41, 2021, pp. 214-226 214 properties of an anti-vague filter in blalgebras s. yahya mohamed* p. umamaheswari† abstract in this paper, the concept of an anti-vague filter of a bl-algebra is introduced with suitable illustration, and also obtained some related properties. further, we have investigated some more equivalent conditions of anti-vague filter. keywords: bl-algebra; filter; implicative filter; vague set; vague filter; anti-vague filter 2010 ams subject classification‡: 03b50; 03b52; 03e72; 06d35. *assistant professors, pg and research department of mathematics, government arts college, tiruchirappalli-620 022. affiliated to bharathidasan university, trichirappalli, tamilnadu, india; yahya_md@yahoo.com †assistant professors, pg and research department of mathematics, government arts college, tiruchirappalli-620 022. affiliated to bharathidasan university, trichirappalli, tamilnadu, india; umagactrichy@gmail.com ‡ received on august 28, 2021. accepted on november 13, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.650. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. properties of an anti-vague filter in blalgebras 215 1. introduction hảjek [5] introduced the idea of bl-algebras as the algebraic structure for his basic logic. the interval [0, 1] endowed with the structure induced by a continuous tnorm is a well-known example of blalgebra. the mvalgebras, on the other hand, are one of the most well-known groups of blalgebras, having been introduced by chang [2] in 1958. in 1965, zadeh [12] introduced the concept of a fuzzy set. the flaw in fuzzy sets is that they only have one feature, which means they cannot convey supporting and opposing data. gau and buehrer [4] introduced the principle of vague set in 1993 as a result of this. the authors [7, 8, 9, 10] discussed the vague filter, implicative filter, prime, and boolean implicative filters of blalgebras, as well as some of their properties. the frame work of this study is constructed as follow: some basic observations connected to anti-vague filter are provided in “preliminaries”. “anti-vague filter” presents the new notions of anti-vague filter in bl-algebra and investigated some related properties, also derived some equivalent conditions for an anti-vague filter to be a vague filter. finally, the conclusion is presented in “conclusion”. 2. preliminaries in this section, we will go through some basic bl-algebra, filter, and vague set concepts, as well as their properties, which will help in the development of the main results. definition 2.1[5] a bl-algebra is an algebra (𝐴, ∨, ∧, ∗, →, 0, 1) of type (2, 2, 2, 2, 0, 0) such that (i) (𝐴, ∨, ∧, 0, 1) is a bounded lattice, (ii) (𝐴, ∗, 1)is a commutative monoid, (iii) ∗ and → form an adjoint pair, that is, 𝑧 ≤ 𝑥 → 𝑦 if and only if 𝑥 ∗ 𝑧 ≤ 𝑦 for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, (iv) 𝑥 ∧ 𝑦 = 𝑥 ∗ (𝑥 → 𝑦), (v) (𝑥 → 𝑦) ∨ (𝑦 → 𝑥) = 1. proposition 2.2[6] in a blalgebra a, the following properties are hold for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, (i) 𝑦 → (𝑥 → 𝑧) = 𝑥 → (𝑦 → 𝑧) = (𝑥 ∗ 𝑦) → 𝑧, (ii) 1 → 𝑥 = 𝑥, (iii) 𝑥 ≤ 𝑦 if and only if 𝑥 → 𝑦 = 1, (iv) 𝑥 ∨ y = ((𝑥 → 𝑦) → 𝑦) ∧ ((𝑦 → 𝑥) → 𝑥), (v) 𝑥 ≤ 𝑦 implies 𝑦 → 𝑧 ≤ 𝑥 → 𝑧, s. yahya mohamed and p. umamaheswari 216 (vi) 𝑥 ≤ 𝑦 implies 𝑧 → 𝑥 ≤ 𝑧 → 𝑦, (vii) 𝑥 → 𝑦 ≤ (𝑧 → 𝑥) → (𝑧 → 𝑦), (viii) 𝑥 → 𝑦 ≤ (𝑦 → 𝑧) → (𝑥 → 𝑧), (ix) 𝑥 ≤ (𝑥 → 𝑦) → 𝑦, (x) 𝑥 ∗ (𝑥 → 𝑦) = 𝑥 ∧ y, (xi) 𝑥 ∗ 𝑦 ≤ 𝑥 ∧ 𝑦 (xii) 𝑥 → 𝑦 ≤ (𝑥 ∗ 𝑧) → (𝑦 ∗ 𝑧), (xiii) 𝑥 ∗ (𝑦 → 𝑧) ≤ 𝑦 → (𝑥 ∗ 𝑧), (xiv) (𝑥 → 𝑦) ∗ (𝑦 → 𝑧) ≤ 𝑥 → 𝑧, (xv) (𝑥 ∗ 𝑥−) = 0. note. in the sequel, we shall use 𝐴 to denote as blalgebras and the operation ∨, ∧, ∗ have priority towards the operations " → ". note. in a blalgebra 𝐴, we can define 𝑥 − = 𝑥 → 0 for all 𝑥 ∈ 𝐴. definition 2.3[13] a filter of a blalgebra 𝐴 is a non-empty subset f of 𝐴 such that for all 𝑥, 𝑦 ∈ 𝐴, (i) if 𝑥, 𝑦 ∈ 𝐹, then 𝑥 ∗ 𝑦 ∈ 𝐹, (ii) if 𝑥 ∈ 𝐹 and 𝑥 ≤ 𝑦, then 𝑦 ∈ 𝐹. proposition 2.4[13] let 𝐹 be a non-empty subset of a blalgebra a. then, 𝐹 is a filter of 𝐴 if and only if the following conditions are hold (i) 1 ∈ 𝐹, (ii) 𝑥, 𝑥 → 𝑦 ∈ 𝐹 implies 𝑦 ∈ 𝐹. a filter f of a bl-algebra a is proper if 𝐹 ≠ 𝐴. definition 2.5[1, 3, 4] a vague set 𝑆 in the universe of discourse 𝑋 is characterized by two membership functions given by (i) a truth membership function 𝑡𝑆 : 𝑋 → [0, 1], (ii) a false membership function 𝑓𝑆 : 𝑋 → [0, 1]. where 𝑡𝑆 (𝑥) is lower bound of the grade of membership of x derived from the ‘evidence for x’, and 𝑓𝑆 (𝑥) is a lower bound of the negation of x derived from the ‘evidence against x’ and 𝑡𝑆 (𝑥)+𝑓𝑆 (𝑥) ≤ 1. thus the grade of membership of x in the vague set s is bounded by a subinterval [𝑡𝑆 (𝑥), 1 − 𝑓𝑆 (𝑥)] of [0, 1]. the vague set 𝑆 is written as 𝑆 = {(𝑥, [ 𝑡𝑆 (𝑥), 𝑓𝑆 (𝑥)])/𝑥 ∈ 𝑋}, where the interval [𝑡𝑆 (𝑥), 1 − 𝑓𝑆 (𝑥)] is called the value of x in the vague set 𝑆 and denoted by 𝑉𝑆(𝑥). definition 2.6[4] a vague set 𝑆 of a set 𝑋 is called (i) the zero vague set of 𝑋 if 𝑡𝑆 (𝑥) = 0 and 𝑓𝑆 (𝑥) = 1 for all 𝑥 ∈ 𝑋, (ii) the unit vague set of 𝑋 if 𝑡𝑆 (𝑥) = 1 and 𝑓𝑆 (𝑥) = 0 for all 𝑥 ∈ 𝑋, properties of an anti-vague filter in blalgebras 217 (iii) the 𝛼vague set of 𝑋 if 𝑡𝑆 (𝑥) = 𝛼 and 𝑓𝑆 (𝑥) = 1 − 𝛼 for all 𝑥 ∈ 𝑋, where 𝛼 ∈ (0, 1). definition 2.7[4] let 𝑆 be a vague set of 𝑋 with truth membership function 𝑡𝑆 and the false membership function 𝑓𝑆. for all 𝛼, 𝛽 ∈ [0, 1], the (𝛼, 𝛽)-cut of the vague set 𝑋 is crisp subset 𝑆(𝛼,𝛽)of the set 𝑋 by 𝑆(𝛼,𝛽) = {𝑉(𝑥) ≥ [𝛼, 𝛽]/𝑥 ∈ 𝑋}. obviously, 𝑆(0,0) = 𝑋. definition 2.8[4] let 𝐷[0, 1] denote the family of all closed subintervals of [0, 1]. now, we define refined maximum (rmax) and “≥ " on elements 𝐷1 = [𝑎1, 𝑏1] and 𝐷2[𝑎2, 𝑏2] of 𝐷[0, 1] as 𝑟𝑚𝑎𝑥 (𝐷1, 𝐷2) = [max{𝑎1, 𝑎2} , max{𝑏1, 𝑏2}]. similarly, we can define ≤, = and rmin. 3. anti-vague filter in this section, we introduce the notion of an anti-vague filter of bl algebra with illustration. moreover, we discuss some related properties. definition 3.1 let 𝑆 be vague set of a bl-algebra 𝐴 is called an anti vague filter of 𝐴 if it satisfies the following axioms (i) 𝑉𝑆(1) ≤ 𝑉𝑆(𝑥), (ii) 𝑉𝑆(𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆(𝑥)} for all 𝑥, 𝑦 ∈ 𝐴. proposition 3.2 let 𝑆 be vague set of bl-algebra 𝐴. 𝑆 is an anti vague filter of 𝐴 if and only if the following hold if for all 𝑥, 𝑦 ∈ 𝐴, (i) 𝑡𝑆 (1) ≤ 𝑡𝑆 (𝑥) and 1 − 𝑓𝑆 (1) ≤ 1 − 𝑓𝑆 (𝑥), (ii) 𝑡𝑆 (𝑦) ≤ max{𝑡𝑆 (𝑥 → 𝑦), 𝑡𝑆 (𝑥)} and 1 − 𝑓𝑆 (𝑦) ≤ max {1 − 𝑓𝑆 (𝑥 → 𝑦), 1 − 𝑓𝑆 (𝑦)}. proof: let 𝑆 be an anti-vague filter of 𝐴. then from (i) of definition 3.1 and the definition of 𝑉𝑆, we have (i) straight forward. from (ii) of definition 3.1and the definition of 𝑉𝑆, (ii) is obvious.∎ the following is the example of definition 3.1 and proposition 3.2. example 3.3 let 𝐴 = {0, 𝑎, 𝑏, 1}. the binary operations ′ ∗ ′ and ′ → ′ give by the following tables 3.1 and 3.2: s. yahya mohamed and p. umamaheswari 218 table3.1: ‘ ∗ ′operator table 3.2: ‘→’ operator then (𝐴, ∨, ∧, ∗, →, 0, 1) is a blalgebra. define a vague set 𝑆 of 𝐴 as follows: 𝑆 = {(1, [0.2, 0.7]), (𝑎, [0.3, 0.5]), (𝑏, [0.3, 0.5]), (0, [0.2, 0.7])}. it is easily verified that 𝑆 is an anti-vague filter of 𝐴 and satisfy the conditions (i) and (ii) of proposition 3.2. proposition 3.4 every anti-vague filter 𝑆 of blalgebra 𝐴 is order preserving. proof: let 𝑆 be an anti-vague filter of bl-algebra 𝐴. then, we prove that if 𝑥 ≤ 𝑦, then 𝑉𝑆(𝑥) ≥ 𝑉𝑆(𝑦) for all 𝑥, 𝑦 ∈ 𝐴. from (ii) of the proposition 3.2, we have, 𝑡𝑆 (𝑦) ≤ max{𝑡𝑆 (𝑥 → 𝑦), 𝑡𝑆 (𝑥)} ∗ 0 a b 1 0 0 0 0 0 a 0 0 a b b 0 a b b 1 0 a b 1 → 0 a b 1 0 1 1 1 1 a a 1 1 1 b 0 a 1 1 1 0 a b 1 properties of an anti-vague filter in blalgebras 219 = max {𝑡𝑆 (1), 𝑡𝑆 (𝑥)}, [from (iii) of proposition 2.2] also, we have 1 − 𝑓𝑆 (𝑦) ≤ max{1 − 𝑓𝑆 (𝑥 → 𝑦), 1 − 𝑓𝑆 (𝑦)}. from (i) of the proposition 3.2, we have 𝑡𝑆 (1) ≤ 𝑡𝑆 (𝑥) and 1 − 𝑓𝑆 (1) ≤ 1 − 𝑓𝑆 (𝑥). thus, 𝑡𝑆 (𝑦) ≤ max{𝑡𝑆(𝑥), 1 − 𝑓𝑆 (𝑦)} ≤ 1 − 𝑓𝑆 (𝑦), and so 𝑉𝑆 (𝑦) = [𝑡𝑆(𝑦), 1 − 𝑓𝑆 (𝑦)] ≤ [𝑡𝑆 (𝑥), 1 − 𝑓𝑆 (𝑥)] = 𝑉𝑆(𝑥). hence 𝑉𝑆(𝑥) ≥ 𝑉𝑆(𝑦). ∎ proposition 3.5 let 𝑆 be a vague set of blalgebra 𝐴, 𝑆 be an anti-vague filter of a if and only if 𝑥 → (𝑦 → 𝑧) = 1 implies 𝑉𝑆(𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴. proof: let 𝑆 be an anti-vague filter of bl-algebra 𝐴. then, from (ii) of the definition 3.1, we have 𝑉𝑆(𝑧) ≥ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑧 → 𝑦), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴. now, 𝑉𝑆(𝑧 → 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → (𝑦 → 𝑧), 𝑉𝑆(𝑥)}. if 𝑥 → (𝑦 → 𝑧) = 1, then we have 𝑉𝑆 (𝑧 → 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(1), 𝑉𝑆(𝑥)} = 𝑉𝑆 (𝑥). so, 𝑉𝑆(𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆 (𝑦)}. conversely, since 𝑥 → (𝑥 → 1) = 1 for all 𝑥 ∈ 𝐴. then 𝑉𝑆 (1) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑥)} = 𝑉𝑆(𝑥). on the other hand, from (𝑥 → 𝑦) → (𝑥 → 𝑦) = 1. it follows that 𝑉𝑆 (𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆 (𝑥)}. from the definition 3. 1, 𝑆 is the anti vague filter of 𝐴.∎ from (i) of the proposition 2.2, and the proposition 3.5, we have the following. s. yahya mohamed and p. umamaheswari 220 corollary 3.6 let 𝑆 be vague set of blalgebra 𝐴, 𝑆 be an anti vague filter of 𝐴 if and only if 𝑥 ∗ 𝑦 ≤ 𝑧 or 𝑦 ∗ 𝑥 ≤ 𝑧 implies 𝑉𝑆(𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦, 𝑧 ∈ 𝐴. proposition 3.7 let 𝑆 be a vague set of blalgebra 𝐴, 𝑆 be an anti vague filter of 𝐴 if and only if (i) 𝑥 ≤ 𝑦, then 𝑉𝑆(𝑥) ≥ 𝑉𝑆 (𝑦), (ii) 𝑉𝑆(𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦, ∈ 𝐴. proof: let 𝑆 be an anti vague filter of blalgebra 𝐴. then, from the proposition 3.4, we have 𝑥 ≤ 𝑦, 𝑉𝑆(𝑥) ≥ 𝑉𝑆(𝑦). since 𝑥 ∗ 𝑦 ≤ 𝑥 ∗ 𝑦 and corollary 3.6, we have 𝑉𝑆(𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. conversely, let 𝑆 be a vague set and satisfies (i) and (ii). for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, if 𝑥 ∗ 𝑦 ≤ 𝑧, then from (i) and (ii), we get 𝑉𝑆(𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆 (𝑦)}. from corollary 3.6, we have 𝑆 is an anti vague filter.∎ proposition 3.8 let 𝑆 be a vague set of blalgebra𝐴. let 𝑆 be an anti vague filter of 𝐴. the following holds for all 𝑥, 𝑦, 𝑧 ∈ 𝐴, (i) if 𝑉𝑆(𝑥 → 𝑦) = 𝑉𝑆 (1), then 𝑉𝑆(𝑥) ≥ 𝑉𝑆 (𝑦), (ii) 𝑉𝑆(𝑥 ∨ 𝑦) = 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}, (iii) 𝑉𝑆(𝑥 ∗ 𝑦) = 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}, (iv) 𝑉𝑆(1) = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑥 −)}, (v) 𝑉𝑆(𝑥 → 𝑧) ≤ 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆(𝑦 → 𝑧)}, (vi) 𝑉𝑆(𝑥 → 𝑦) ≥ 𝑉𝑆(𝑥 ∗ 𝑧 → 𝑦 ∗ 𝑧), (vii) 𝑉𝑆(𝑥 → 𝑦) ≥ 𝑉𝑆((𝑦 → 𝑧) → (𝑥 → 𝑧)), (viii) 𝑉𝑆(𝑥 → 𝑦) ≥ 𝑉𝑆((𝑧 → 𝑥) → (𝑧 → 𝑦)). proof: (i) let 𝑆 be an anti vague filter of blalgebra 𝐴. then, from the definition 3.1, and since 𝑉𝑆(𝑥 → 𝑦) = 𝑉𝑆 (1). we have 𝑉𝑆(𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑥 → 𝑦)} properties of an anti-vague filter in blalgebras 221 = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(1)} = 𝑉𝑆(𝑥). thus, 𝑉𝑆 (𝑥) ≥ 𝑉𝑆 (𝑦). (ii) since 𝑥 ∨ 𝑦 ≥ 𝑥 and 𝑥 ∨ 𝑦 ≥ 𝑦. from the proposition 3.4, we get 𝑉𝑆(𝑥 ∨ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. from the definition 3.1 we have 𝑉𝑆 (𝑥 ∨ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → (𝑥 ∨ 𝑦)), 𝑉𝑆(𝑥)} = 𝑟𝑚𝑎𝑥{𝑉𝑆((𝑥 → 𝑥) ∨ (𝑥 → 𝑦)), 𝑉𝑆(𝑥)} = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆(𝑥)} ≤ 𝑟𝑚𝑎𝑥{𝑟𝑚𝑎𝑥{𝑉𝑆(𝑦 → (𝑥 → 𝑦)), 𝑉𝑆(𝑦)}, 𝑉𝑆 (𝑥)} = 𝑟𝑚𝑎𝑥{𝑟𝑚𝑎𝑥{𝑉𝑆(1), 𝑉𝑆(𝑦)}, 𝑉𝑆(𝑥)} = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑦), 𝑉𝑆(𝑥)} = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} hence, 𝑉𝑆(𝑥 ∨ 𝑦) = 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆 (𝑦)}. (iii) from (ii) of proposition 3.7, we have 𝑉𝑆 (𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆 (𝑦)}. since 𝑥 ∗ 𝑦 ≥ 𝑥 ∨ 𝑦, proposition 3.4, and (ii), we have 𝑉𝑆(𝑥 ∗ 𝑦) ≥ 𝑉𝑆(𝑥 ∨ 𝑦) = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. thus, 𝑉𝑆(𝑥 ∗ 𝑦) = 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. (iv) from (iii), we have 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆(𝑥 −)} = 𝑉𝑆(𝑥 ∗ 𝑥 −) = 𝑉𝑆(1). therefore, 𝑉𝑆(1) = 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥), 𝑉𝑆(𝑥 −)}. (v) from (iii) and proposition 3.4, since (𝑥 → 𝑦) ∗ (𝑦 → 𝑧) ≤ 𝑥 → 𝑧, we get 𝑉𝑆((𝑥 → 𝑦) ∗ (𝑦 → 𝑧)) ≥ 𝑉𝑆((𝑥 → 𝑧), 𝑟𝑚𝑎𝑥{𝑉𝑆((𝑥 → 𝑦), 𝑉𝑆(𝑦 → 𝑧))} ≥ 𝑉𝑆 ((𝑥 → 𝑧). therefore, we have 𝑉𝑆(𝑥 → 𝑧) ≤ 𝑟𝑚𝑎𝑥 {𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆(𝑦 → 𝑧)}. from the proposition 2.2 and (i) of proposition 3.7 we can prove (vi), (vii) and (viii) easily.∎ s. yahya mohamed and p. umamaheswari 222 proposition 3.9 let 𝑆 be a vague set of blalgebra a, 𝑆 be an anti vague filter of a if and only if (i) 𝑉𝑆(1) ≤ 𝑉𝑆(𝑥), (ii) 𝑉𝑆 ((𝑥 → (𝑦 → 𝑧)) → 𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦, ∈ 𝐴. proof: let s be an anti vague filter of a. by the definition 3.1, (i) is straight forward. since, 𝑉𝑆 ((𝑥 → (𝑦 → 𝑧)) → 𝑧) ≤ 𝑟𝑚𝑎𝑥 {𝑉𝑆 ((𝑥 → (𝑦 → 𝑧)) → (𝑦 → 𝑧)) , 𝑉𝑆 (𝑦)}. (3.1) now, we have (𝑥 → (𝑦 → 𝑧)) → (𝑦 → 𝑧) = 𝑥 ∨ (y → z) ≥ 𝑥. 𝑉𝑆((𝑥 → (𝑦 → 𝑧)) → (𝑦 → 𝑧)) ≤ vs(𝑥). [from the proposition 3.4] (3.2) using (3.2) in (3.1), we have 𝑉𝑆((𝑥 → (𝑦 → 𝑧)) → 𝑧) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. conversely, suppose (i) and (ii) hold. since 𝑉𝑆(𝑦) = 𝑉𝑆(1 → 𝑦) = 𝑉𝑆(((𝑥 → 𝑦) → (𝑥 → 𝑦) → 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥 → 𝑦), 𝑉𝑆(𝑦). from (i), s is an anti vague filter of 𝐴. ∎ proposition 3.10 intersection of two anti vague filters of 𝐴 is also an anti vague filter of 𝐴. proof: let 𝑈 and 𝑊 be two anti vague filters of 𝐴. to prove: 𝑈 ∩ 𝑊 is an anti vague filter of 𝐴. for all 𝑥, 𝑦, 𝑧 ∈ 𝐴 such that 𝑧 ≤ 𝑥 → 𝑦, then 𝑧 → (𝑥 → 𝑦) = 1. since, 𝑈, 𝑊 are two anti vague filters 𝐴, we have 𝑉𝑈 (𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑈 (𝑧), 𝑉𝑈 (𝑥)} and 𝑉𝑊(𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑊(𝑧), 𝑉𝑊(𝑥)}. that is,𝑡𝑈 (𝑦) ≤ max {𝑡𝑈 (𝑧), 𝑡𝑈 (𝑥)} and 1 − 𝑓𝑈 (𝑦) ≤ max{1 − 𝑓𝑈 (𝑧), 1 − 𝑓𝑈 (𝑥)} , 𝑡𝑊(𝑦) ≤ max{𝑡𝑊(𝑧), 𝑡𝑊(𝑥)} and 1 − 𝑓𝑊(𝑦) ≤ max{1 − 𝑓𝑊(𝑧), 1 − 𝑓𝑊(𝑥)}. since, 𝑡𝑈∩𝑊(𝑦) = min{𝑡𝑈 (𝑦), 𝑡𝑊(𝑦)} properties of an anti-vague filter in blalgebras 223 ≤ max {max{𝑡𝑈 (𝑧), 𝑡𝑈 (𝑥)} , max{𝑡𝑊(𝑧), 𝑡𝑊(𝑥)} } = max {max{𝑡𝑈 (𝑧), 𝑡𝑊(𝑧)} , max{𝑡𝑈 (𝑥), 𝑡𝑊(𝑥)} } = max {max{𝑡𝑈∩𝑊(𝑧), 𝑡𝑈∩𝑊(𝑥)} } and 1 − 𝑓𝑈∩𝑊 (𝑦) = max{1 − 𝑓𝑈 (𝑦), 1 − 𝑓𝑊(𝑦)} ≤ max {max{1 − 𝑓𝑈 (𝑧), 1 − 𝑓𝑈 (𝑥)} , max{1 − 𝑓𝑊(𝑧), 1 − 𝑓𝑊 (𝑥)} } = max {max{1 − 𝑓𝑈 (𝑧), 1 − 𝑓𝑊(𝑧)} , max{1 − 𝑓𝑈 (𝑥), 1 − 𝑓𝑊(𝑥)} } = max{max{1 − 𝑓𝑈∩𝑊(𝑧), 1 − 𝑓𝑈∩𝑊(𝑥)} }. hence, 𝑉𝑈∩𝑊 (𝑦) = [𝑡𝑈∩𝑊(𝑦), 1 − 𝑓𝑈∩𝑊(𝑦)] ≤ 𝑟𝑚𝑎𝑥{𝑉𝑈∩𝑊(𝑧), 𝑉𝑈∩𝑊(𝑥)}. thus 𝑈 ∩ 𝑊 is an anti vague filter of 𝐴. ∎ corollary 3.11 let 𝑅𝑗 be a family of anti vague filters of 𝐴, where 𝑗 ∈ 𝐼, 𝐼 is a index set, then ⋂ 𝑅𝑗𝑗∈𝐼 is an anti vague filter of 𝐴. note: union two anti vague filters of blalgebra 𝐴 need not be an anti vague filter of 𝐴. proposition 3.12 a 𝜌vague set and zero vague set of a bl-algebra 𝐴 are anti vague filters of 𝐴. proof: let 𝑆 be a 𝜌-vague set of bl-algebra 𝐴, and 𝑆 be an anti vague filter of 𝐴. then, from the proposition 3.4, we have if 𝑥 ≤ 𝑦, then 𝑉𝑆(𝑥) ≥ 𝑉𝑆(𝑦) for all 𝑥, 𝑦, ∈ 𝐴. to prove: 𝑉𝑆 (𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆 (𝑦)} for all 𝑥, 𝑦, ∈ 𝐴. now,𝑡𝑆 (𝑥 ∗ 𝑦) = 𝜌 = max{𝜌, 𝜌} = max { 𝑡𝑆 (𝑥), 𝑡𝑆 (𝑦)} (3.3) and 1 − 𝑓𝑆 (𝑥 ∗ 𝑦) = 𝜌 = max{𝜌, 𝜌} = max {1 − 𝑓𝑆 (𝑥), 1 − 𝑓𝑆 (𝑦)} for all 𝑥, 𝑦, ∈ 𝐴 (3.4) from (3.3) and (3.4), we have 𝑉𝑆(𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)}. thus, 𝜌vague set is an anti vague filter of 𝐴. similarly, we prove zero vague set is an anti vague of 𝐴. ∎ s. yahya mohamed and p. umamaheswari 224 theorem 3.13 let 𝑆 be a vague set of bl-algebra 𝐴, 𝑆 be an anti vague filter of 𝐴 if and only if the set 𝑆(𝜌,𝜎) is either empty or a filter of 𝐴 for all 𝜌, 𝜎 ∈ [0, 1], where 𝜌 ≤ 𝜎. proof: let 𝑆 be an anti vague filter of bl-algebra 𝐴 and 𝑆(𝜌,𝜎) ≠ ∅ for all 𝜌, 𝜎 ∈ [0, 1]. to prove: 𝑆(𝜌,𝜎) is a filter of 𝐴. if 𝑥 ≤ 𝑦 and 𝑥 ∈ 𝑆(𝜌,𝜎). from the proposition 3.12, we have 𝑉𝑆(𝑦) ≤ 𝑉𝑆(𝑥) ≤ [𝜌, 𝜎] for all 𝑥, 𝑦 ∈ 𝐴. thus, 𝑦 ∈ 𝑆(𝜌,𝜎). if 𝑥, 𝑦 ∈ 𝑆(𝜌,𝜎), then 𝑉𝑆 (𝑥) and 𝑉𝑆(𝑦) ≤ [𝜌, 𝜎]. from (ii) of the proposition 3.7, we have 𝑉𝑆(𝑥 ∗ 𝑦) ≤ 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} ≤ [𝜌, 𝜎]. thus, 𝑥 ∗ 𝑦 ∈ 𝑆(𝜌,𝜎). hence 𝑆(𝜌,𝜎) is a filter of 𝐴. conversely, if for all 𝜌, 𝜎 ∈ [0, 1], the set 𝑆(𝜌,𝜎) is either empty or a filter of 𝐴. let 𝑡𝑆 (𝑥) = 𝜌1, 𝑡𝑆 (𝑦) = 𝜌2, 1 − 𝑓𝑆 (𝑥) = 𝜎1 and 1 − 𝑓𝑆 (𝑦) = 𝜎2. put 𝜌 = max{𝜌1, 𝜌2} and 𝜎 = max{1 − 𝜎1, 1 − 𝜎2}. then, 𝑡𝑆 (𝑥) , 𝑡𝑆 (𝑦) ≤ 𝜌 and 1 − 𝑓𝑆 (𝑥), 1 − 𝑓𝑆 (𝑦) ≤ 𝜎. thus, 𝑉𝑆(𝑥) and 𝑉𝑆(𝑦) ≤ [𝜌, 𝜎], that is 𝑥, 𝑦 ∈ 𝑆(𝜌,𝜎). thus, 𝑆(𝜌,𝜎) ≠ ∅. hence, by the assumption 𝑆(𝜌,𝜎) is a filter of 𝐴. to prove: 𝑆 is an anti vague filter of 𝐴. if 𝑥 ≤ 𝑦, 𝑡𝑆 (𝑥) = 𝜌 and 1 − 𝑓𝑆 (𝑥) = 𝜎. then 𝑥 ∈ 𝑆(𝜌,𝜎). since, 𝑆(𝜌,𝜎) is a filter, 𝑦 ∈ 𝑆(𝜌,𝜎), that is, 𝑉𝑆(𝑦) ≤ [𝜌, 𝜎]. (3.5) since, 𝑆(𝜌,𝜎) is filter of 𝐴, 𝑥 ∗ 𝑦 ∈ 𝑆(𝜌,𝜎). that is, 𝜗𝑆 (𝑥 ∗ 𝑦) ≤ [𝜌, 𝜎] for all 𝑥, 𝑦 ∈ 𝐴 properties of an anti-vague filter in blalgebras 225 = [max{𝜌1, 𝜌2}, max{1 − 𝜎1, 1 − 𝜎2}] = 𝑟𝑚𝑎𝑥{[𝑡𝑆 (𝑥), 1 − 𝑓𝑆 (𝑥)], [𝑡𝑆 (𝑦), 1 − 𝑓𝑆 (𝑦)] = 𝑟𝑚𝑎𝑥{𝑉𝑆(𝑥), 𝑉𝑆(𝑦)} for all 𝑥, 𝑦 ∈ 𝐴. (3.6) from (3.5) and (3.6), 𝑆 is an anti vague filter of 𝐴. ∎ note. the filter 𝑆(𝜌,𝜎) is called a vague-cut filter of blalgebra 𝐴. proposition 3.14 let 𝑆 be an anti vague filter of bl-algebra 𝐴. then 𝑆𝜌 is either empty or a filter of 𝐴 for all 𝜌 ∈ [0, 1]. proof: let 𝑆 be an anti vague filter of bl-algebra ℬ. then from the theorem 3.13, the proof is obvious. ∎ 4. conclusion in the present paper the notion of an anti-vague filter in blalgebra with suitable examples are studied. also investigated some related properties with the help of more implication of an anti-vague filter of bl-algebra. references [1]. r.biswas, vague groups, international journal of computational cognition, vol. 4(2) (2006), 20-23. [2]. c. c. chang, algebraic analysis of many valued logics, trans. amer. math. soc. 88 (1958), 467490. [3]. t. eswarlal, vague ideals and normal vague ideals in semirings, international journal of computational cognition, vol. 6, (2008), 6065. [4]. w. l. gau, d. j. buehrer, vague sets, ieee transactions on systems, man and cybernetics, vol. 23 (2), (1993), 610-614. [5]. p. hảjek, metamathematics of fuzzy logic, klower academic publishers, dordrecht, 1999. [6]. l. z. liu, k. t. li, fuzzy filters of bl-algebras, information sciences, 173 (2005), 141-154. s. yahya mohamed and p. umamaheswari 226 [7]. s. yahya mohamed and p. umamaheswari, vague filter of blalgebras, journal of computer and mathematical sciences, 9(8), (2018), 914-920. [8]. s. yahya mohamed and p. umamaheswari, vague prime and boolean filters of blalgebras, journal of applied science and computations, 5(11),(2018), 470-474. 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[13]. x. h. zhang, fuzzy logic and its algebraic analysis, science press, beijing (2008). ratio mathematica volume 45, 2023 strong forms of b-continuous multifunctions suresh r* pasunkilipandian s† hari siva annam g‡ selva banu priya t § abstract in this paper we have introduced strong forms of b-continuous multifunctions namely b#-multicontinuity and *b-multicontinuity and studied their properties and characterizations. also investigate the relationship with other type of functions with suitable examples. keywords: b-open, multi-function, u.b#-c, u.*b-c 2010 ams subject classification: 54c05, 54c08, 54c60**. *research scholar (19112102091007), manonmaniam sundaranar university, tirunelveli-12, india. rsuresh211089@gmail.com. †dept. of mathematics, aditanar college of arts and science, tiruchendur, affiliated to manonmaniam sundaranar university, tirunelveli-12, india. pasunkilipandian@gmail.com. ‡dept. of mathematics, kamaraj college, tuticorin, affiliated to manonmaniam sundaranar university, tirunelveli-12, tamilnadu, india. hsannam@yahoo.com. §department of artificial intelligence and data science, panimalar engineering college, chennai 600123, india. priya8517@gmail.com. ** received on july 10, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.1027. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 280 mailto:priya8517@gmail.com suresh, pasunkilipandian, hari siva annam, selva banu priya 1. introduction recently topologists concentrate their research in several types of continuous multi functions. a weak form of b-continuous multifunctions was studied in [4]. the variations of multi continuity were discussed in [5]. the weak and strong forms of continuity of multi functions were introduced in [6]. certain properties of topological spaces preserved under multivalued continuous mappings were investigated in [7]. certain strong forms of mixed continuous multi functions were characterized in [8] and the upper and lower -continuous multi functions were studied in [11]. the notions of b#-continuity and *b-continuity were respectively discussed and studied in [9] and [3]. in this paper we have introduced strong forms of b-continuous multifunctions namely b#-multicontinuity and *b-multicontinuity and also studied their properties and characterizations with suitable examples. 2. preliminaries throughout this paper it is assumed that x and y are non-empty sets and  and  are topologies on x and y respectively and  and  denote the collections of closed sets in x and y respectively. the notation p: x⇉y is used for a multivalued function. for the notations in multifunction theory, the reader may consult (thangavelu, premakumari, 2015). we use the following abbreviations and notations. “continuous” =”c”, “upper continuous” = “u.c” and “lower continuous” = “l.c”. further v (, x, p(x),)  v , xx and p(x)v. u[, x, p, v,]  u , xu and p(u)v. v (, x, p(x), )  v , xx and p(x)v. u [, x, p, v, ]  u, xu and p(u)v uu. {b#, *b}. definition 2.1. the set a is called (resp. b, *b)-open[1] (resp.[2], resp.[3]) if a cl(int(cl(a))) (resp. cl(int(a))int(cl(a)), cl(int(a))int(cl(a)) and b#-open [9,10] if a=cl(int(a))int(cl(a)). the complements of (resp. b,*b, b#)-open sets are (resp. b,*b, b#)-closed sets. lemma 2.2. the set b is (i) -open  b-open (ii) open  b-open (iii) b-open-open definition 2.3. the multifunction p is u.c [5,6,7] if  v(, x, p(x), ),  u[, x, p,v, ] and is l.c if  v(, x, p(x), ),  u[, x, p, v, ]. 281 strong forms of b-continuous multi functions analogously u.b-c [4] and u.-c [11] may be defined by replacing “” in [, x, p, v,] respectively by “bo(x,)” and “o(x,)”. also l.b-c [4] and l.-c [11] may be defined by replacing “” in [, x, p, v, ] by “bo(x,)” and “o(x,)” respectively. definition 2.4. the multifunction p is c if p is u.c and l.c. the notions b-c and -c can be similarly defined. 3.-multi continuity where {*b, b#} definition 3.1. the multivalued function p is u.b#-c (resp. u.*b-c ) if p+(v) is b#-open (resp. *b-open)  v. proposition 3.2. consider the following statements. (i)p is u.-c. (ii)p −(b) is -closed  b. (iii)p −(cl (b)) is -closed  by. (iv)p+ (int (b)) is -open  by. the implications (i)  (ii)  (iii)  (iv) always hold. proof: suppose (i) holds. let b that implies, p+(y\b) is -open so that x\ p−(b) = p+(y\b) is -open that further shows that p−(b) is -closed. this proves (i) (ii). now we assume (ii). let v that implies by (ii), p−(y\v) is -closed so that x\ p+(v) is -closed that further shows that p+(v) is -open. this proves (ii) (i). other implications follow easily. proposition 3.3. if p is u.-c then  v(, x, p(x), ), u[ o(x,), x, p, v, ]. proof: let p be u.-c and v(, x, p(x), ). since p(x)v, xp+(v). since p+(v) is -open  a -open set u with xu p+(v). clearly u[ o(x,), x, p, v, ]. proposition 3.4. p is u.-c  it is u.b-c and u.-c. definition 3.5. the multifunction p is l.b#-c(resp. l.*b-c) if p−(v) is b#-(resp.*b)-open  v. proposition 3.6. consider the following statements. (i)p is l.-c. (ii) p+ (b) is -closed  b. (iii) p+ (cl (b)) is -closed  b y. (iv) p− (int (b)) is -open  b y. the implications (i)  (ii)  (iii)  (iv) always hold. 282 suresh, pasunkilipandian, hari siva annam, selva banu priya proof: suppose (i) holds. let b  that implies p−(y\b) is -open so that x\ p+(b) is -open that further shows that p+(b) is -closed. this proves (i) (ii). now we assume (ii). let v that implies by (ii)), p+(y\v) is -closed so that x\ p−(v) is -closed that further shows that p−(v) is -open. this proves (ii) (i). the rest follows easily. proposition 3.7. if p is l.-c then v (, x, f(x), ),  u[o(x,), x, p, v, ]. proof: analogous to proposition 3.3. proposition 3.8. p is l.-c it is l.b-c and l.-c definition 3.9. p is b#-c (resp.*b-c) if it is u.b#-c (resp.u.*b-c) and l.b#-c(resp. l.*b-c). the next proposition follows from previous definition, proposition 3.2 and proposition 3.6. proposition 3.10. consider the following statements. (i) p is -continuous. (ii) p+ (v) and p− (v) are -open  v. (iii) p+ (b) and p− (b) are -closed  b. (iv) p+ (int (b)) and p− (int (b)) are -open b  y. (v) p+ (cl (b)) and p− (cl (b)) are -closed b  y. the implications (i)  (ii)  (iii)  (iv)  (v) always hold. the following diagrams always hold. diagram 3.11. let t=u or l. (i) t.b#-c  t.b-c  t.*b-c. (ii) t.c  t.b-c  t.-c. examples 3.12. in this section some examples are given to illustrate certain results in the third section. let x = {p, q r, s}, y = {1, 2, 3},  = {, {1}, y},  = {, {r}, {q}, {q, r}, {p, q}, {p, q, r}, {q, r, s}, x}. (i) f1(p) = {1, 2}, f1(q)={1, 3} f1(r) = {1} and f1(s) = {1} then f1 +({1})={r, s} is b#open so that f1 is u.b #-c. (ii) if f2(p) = {1, 2}, f2(q)={1}, f2(r) = {1, 3} and f2(s) = {3} then f2 +({1})={r} is *b-open that implies f2 is u.*b-c. 283 strong forms of b-continuous multi functions (iii) if f3(p) = {1}, f3(q)={1}, f3(r) ={1,2} and f3(s) = {1} then f3 +({1})={p, q, s} is bopen and -open and hence f3 is u.b-c and u.-c. however f3 is not u.-c as f3 +({1})= {p, q, s} is not -open. (iv) if f4(p) ={2}, f4(q)={3}, f4(r) ={1, 2} and f4(s) = {1, 3} then f4 −({1})= {r, s} is b#-open that implies f4 is l.b #-c. (v) if f5(p) = {1, 2}, f5(q)={1, 3}, f5(r) = {2} and f5(s) = {3} then f5 −({1})={p, q} is *b-open and hence f5 is l.*b-c. (vi) if f6(p) ={2}, f6(q)={1, 2}, f6(r) = {3} and f6(s) = {1, 3} then f6 −({1})={q, s}is b-open and -open so that f6 is l.b-c and l.-c. however f6 is not l.-c as f6 −({1})= {q, s} is not -open. (vii) if f7(p) = y, f7(q)={1, 3}, f7(r) = {1} and f7(s)={1} then f7 +({1})={r, s} and f7 −({1})= x are b#-open we see that f7 is u.b #-c and l.b#-c and hence b#-c. (viii) if g1(p) = {2}, g1(q)={1}, g1(r) = {1, 2} and g1(s) = {3} then g1 +({1})= {q} and g1 −({1})= {q, r} are *b-open so that g1 is u.*b-c and l.*b-c and hence *b-c. (ix) if g2(p) = {1, 3}, g2(q)={1}, g2(r) = {2} and g2(s) = {1} then g2 +({1})= {q, s} and g2 −({1})= {p, q, s} are b-open and -open we see that g2 is b-c and -c. however g2 is not -c as g2 +({1})={q, s} and g2 −({1})={p, q, s} are not -open . (x) if g3(p) = {2, 3}, g3(q)={1}, g3(r) ={1} and g3(s) = {1} then g3({1})= {q, r, s}is open that implies g3 is u.c. however g3 is not u.b #-c as g3 +({1})= {q, r, s} is not b#open. if g4(p)={2, 3}, g4(q)={1, 2}, g4(r)={1} and g4(s) ={1} then g4 +({1})={r, s} is b#-open so that g4 is u.b #.c. however g4 is not u.c as g4 +({1})={r, s} is not open. (xi) if g5(p) = {2, 3}, g5(q)={1, 2}, g5(r) = {1, 3} and g5(s) = {1} then g5 −({1})= {q, r, s} is open we see that g5 is l.c . however g5 is not l.b #-c as g5 + ({1}) = {q, r, s} is not b#-open. if g6 (p) = {3}, g6 (q) = {2}, g6(r) = y and g6(s) = {1, 3} then g6 −({1})={r, s} is b#-open we see that g6 is l.b #-c. however g6 is not l.c as g6 − ({1}) ={r, s} is not open. (xii) if f(p) = {2, 3}, f(q)={1}, f(r) = {1, 3} and f(s) = {1, 2} then f+({1})={q} and f−({1})={q, r, s} are open we see that f is u.c and l.c so that it is c. however f is neither u.b#-c nor l.b#-c as f+ ({1}) = {q} and f− ({1}) = {q, r, s} are not b#-open. if g(p) = {3}, g(q)={2, 3}, g(r) ={1} and g(s) = {1}then g+({1})={r, s}= g−({1}) is b#open we see that g is u.b#-c and l.b#-c so that it is b#-c. however g is not c as g+ ({1}) = {r, s} = g− ({1}) is not open. 284 suresh, pasunkilipandian, hari siva annam, selva banu priya 4. conclusions the concepts of strong forms of b-continuous multifunctions namely b#multicontinuous and *b-multicontinuous functions are suitable for future extension research. acknowledgements the authors are grateful and thankful to the reviewers who gave some constructive comments to improve the paper and also to the editorial board for the processing the paper. references [1] m.e. abd el-monsef, s.n.el-deeb, r.a.mahmoud, -open sets and -continuous mappings, bull. fac. sci. assiut univ., 12(1), 77-90, 1983. 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[11] valeriu popa, takashi noiri, on upper and lower -continuous multifunctions, real analysis exchange, 22(1), 362-376, 1996.   285 ratio mathematica volume 44, 2022 totally magic d-lucky number of graphs n. mohamed rilwan * a. nilofer † abstract in this paper we introduce a new labeling named as, totally magic d-lucky labeling, find the totally magic d-lucky number of some standard graphs like wheel, cycle, bigraph etc. and find the totally magic d-lucky number of some zero divisor graphs. a totally magic d-lucky labeling of a graph g = (v, e) is a labeling of vertices and label the graph's edges using the total label of its incident vertices in such a way that for any two different incident vertices u and v, their colors dgu, dtv= nvtu+ dg v are distinct and for any different edges in a graph, their weights are same where represents the degree of u in a graph and n represents the open neighbourhood of u in a graph. keywords: totally magic d-lucky labeling, totally magic d-lucky number, zero divisor graphs. 2010 ams subject classification: 05c78 ‡ * assistant professor, department of mathematics, sadakathullah appa college (autonomus), rahmath nagar, tirnelveli-627011, india; e-mailrilwan2020@gmail.com. † research scholar, register number 18221192092017, research center, department of mathematics, sadakathullah appa college (autonomus), rahmath nagar, tirunelveli 627011, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627012, india; e-mail lourdhunilofer@gmail.com ‡ received on june 9 th , 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.920. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 316 n. mohamed rilwan, a. nilofer 1. introduction in [2], the idea of lucky labeling was first proposed by czerwinski, grytczuk, and zelezny. in [1], the idea of "d-lucky labeling" was developed by indira rajasingh, d. ahima emilet, and d. azhubha jemilet. [1] let l: v(g) → n is a vertex labeling. if for each pair of incident vertices of u and v, c(u) ≠ c(v) holds where c(u) = , c(v) = , represents the degree of u and n(u) represents the open neighbourhood of the vertex u in a graph, then the labeling l is a dlucky labeling. a graph's d-lucky number is the smallest value of labeling required to label the graph. motivated by this labeling, we introduce totally magic d-lucky labeling. a graph's total labeling is a mapping from the union of the vertex set and the edge set to positive integers. if the sum of the edge label and the label of the edge's end points has the same constant, the total labeling is said to be totally magic labeling. in [5,] we learned about the totally magic labeling. a totally magic d-lucky labeling of a graph g = (v, e) is a labeling of vertices and label the edges of the graph by the sum of the labels of its incident vertices in such a way that for any two different incident vertices u and v, their colors , are distinct and for any different edges in a graph, their weights are same where represents the degree of u in a graph and n represents the open neighborhood of u in a graph. 2. totally magic d-lucky labeling in this section we introduce a new labeling named as the totally magic d-lucky labeling and apply it on the cycle, path, complete graph, bigraph, and wheel. definition 2.1 define t: v(g) → {1, 2, …, p}and label the edges of e(g) as the label of the edge's incident vertices added together. the labeling is said to be totally magic dlucky labeling if and t(u) + t(v) + t(uv) 0 (mod 2) where u, v v(g) and . the totally magic d-lucky number of g, tdln(g) is defined as the lowest value of p for which the graph g has totally magic d-lucky labeling. theorem 2.2. for a cycle graph cn, tdln (cn) = proof. let g be the cycle graph. let v(g) = {vi: 1≤ i ≤ n} and e(g) = {vi vi+1: 1≤ i ≤ n-1} ᴜ {vnv1} case. (i). when . let t: v(cn)→{1,2,...,p}defined by for 1≤i≤n-1 t(vi)=i for 1≤ i ≤ n then the induced edge labelling is, t(vivi+1) =2i+1 for 1≤ i ≤ n-1 317 totally magic d-lucky number of graphs t(vnv1) = n+1 we observe that, dt(v1) = n+4 dt(vi) = 2i+2, for 2≤i≤n-1 dt(vn) = n+2 dt(vi) ≠ dt(vi+1) and dt(v1) ≠ dt(vn) t(vi)+ t(vi+1) + t(vivi+1)=4i+2 ≡ 0 (mod 2) for 2≤ i≤ n-1 and t(v1)+ t(vn)+t(v1vn) = 2n+2≡ 0(mod2) case (ii). when n 0 (mod 2) define t: v(cn) → {1,2, …, p}as follows, for t(vi) = then the induced edge labelling is, t(vivi+1) = 3 for 2 ≤ i ≤ n-1 t(vnv1) = 3 we observe that, dt(vi) = 6 if i is odd dt(vi) = 4 if i is even dt(vi) ≠ dt(vi+1) and t(vi)+ t(vi+1) + t(vivi+1)= 6 0 (mod 2) it can be easily verified that weights of the incident vertices are pair wise distinct and have the common totally magic d-lucky constant for its edges. thus, the totally magic d-lucky number of cycle graph is 2. ■ theorem 2.3 every path pn has tdln(pn)=2 proof let pn be the path graph, v(pn)={vi: } and e(pn) ={vivi+1: for } define t: v(pn) → {1,2,…,p}as follows: t(vi) = then the induced edge labelling is, t(vi vi+1) =3 for all edges in pn we observe that, when n is even, dt(v1) = 3 dt(vi) = 4 if i ≡ 0(mod 2) dt(vi) = 6 if i ≡ 1(mod 2) dt(vn) = 2 dt(vi) ≠ dt(vi+1) for all i and t(vi) + t(vi+1) + t(vivi+1) = 6 ≡ 0(mod 2). when n is odd dt(v1) = 3 = dt(vn) 318 n. mohamed rilwan, a. nilofer dt(vi) = 4 if i ≡ 0(mod 2) dt(vi) = 6 if i ≡ 1 (mod 2) dt(vi) ≠ dt(vi+1) for all i and t(vi) + t(vi+1) + t(vivi+1) = 6 ≡ 0 (mod 2). hence tdln (pn) = 2. ■ theorem 2.4 for a complete graph kn, tdln (kn) = n proof in complete graph kn, each and every pair of vertices are close together. define t:v(kn)→{1,2,…,p}as follows: t(vi) = i : then the induced edge labelling is, t(vivj) = i+j for all edges in pn we observe that, for 1 ≤ i ≤ n dt(vi) = dt(vi) ≠ dt(vj) and t(vi) + t(vj) + t(vivj) = 2(i+j) ≡ 0(mod 2). tdln (kn) = n. it is simple to confirm that the colors of the pair wise incident vertices are distinct and that the sum of the labels for each edge and the incident vertices of its edges is even. ■ theorem 2.5 for a bigraph km, n, tdln (km, n) = 1. proof a bigraph's vertices can be divided into two separate subsets, v1 and v2, and each edge of the bigraph connects a point on each subset. km, n indicates a bigraph. let v (km, n) = v1 v2 where v1 = { } and v2 = and e (km, n) = { }. define t: v (km, n) → {1, 2, …, p} as follows: t(ui) = 1, t(vj) =1 then the induced edge labeling is t(uivj) = 2 for all edges in km, n we observe that, , , t(ui) + t(vj) + t(uivj) =2(i+j) ≡ 0 (mod 2). it is obvious that all incident vertices have pair wise different colors and that all of the edges in the km, n graph have the same totally magic d-lucky constant. hence tdln (km, n) =1. ■ theorem 2.6 for a wheel graph wn, tdln (wn) = . proof a wheel graph is obtained by joining a vertex to all the vertices of a cycle graph. it is denoted by wn for n>3, where n is the number of vertices in the graph. let v(wn) = { } and e(wn) = { } case(i) when n 1(mod2) 319 totally magic d-lucky number of graphs define a labeling t:v(wn) → {1,2,…,p}as follows: t( ) = 1, t( ) = i-1 for then the induced edge labelling is t( ) = i for we observe that, dt(u1) = dt( ) = n+5 dt( ) = 2i+2 for 3≤ i ≤n-1 dt(un) = n+3 dt(u1) ≠ dt(ui) for 2 dt(ui) ≠ dt(ui+1) for dt(un) ≠ dt(u1) t(u1)+t(ui)+t(u1ui) = 2i ≡ 0 (mod 2); t(ui)+t(ui+1)+t(uiui+1) = 4i-2 ≡ 0 (mod 2), for 2≤ i ≤ n-1; t(u1)+t(un)+t(un) = 2n ≡ 0(mod 2) hence tdln (wn) = n-1 case (ii) when n ≡ 0 (mod 2) define t:v(wn)→{1,2,…,n} as follows: t(ui) = i for then the induced edge labelling is ; = 2i+1 -1 t(unu2)=n+2 we observe that, dt( ) = dt( ) = n+7 dt(ui) = 2i+4 for 3≤ i ≤ n-1 dt(un) = n+5 dt ≠ dt(ui) for 2 dt(ui) ≠ dt(ui+1) for dt(un) ≠ dt( ) for t( ) + t(ui+1) + t( ui+1 ) = 2i+2 ≡ 0 (mod 2), t(ui) + t(ui+1) + t(uiui+1) = 4i+2 ≡ 0 (mod 2), t( ) + t(un) + t( un) = 2n+4 ≡ 0 (mod 2). hence tdln (wn)= n it is simple to confirm that all incident vertices' colors are pairwise different and preserve the totally magic d-lucky constant for all of the graph's edges wn. ■ 320 n. mohamed rilwan, a. nilofer 3. totally magic d-lucky number of some zero divisor graphs in this part, the totally magic d-lucky number of some zero divisor graphs is examined. theorem. 3.1 for r = zk , k = mn, m=2,3 and n>3 be a prime number, tdln(γ(r)) = 1 proof consider g0=γ(r) where r=zk, k=mn case(i) when m=2,n >3be a prime. by the definition of zero divisor graph, assume v(g0) = {2, 4... 2 (n-1), n} = {vi: 1 ≤ i≤ n}, e(g0) = {vi vn: vi v (γ (r))-{vn}}. we have dg(vi) = m-1, dg(vn) = n-1, 1 ≤ i ≤ 2(n-1) define t:v(g0)→{1,2,…,p} as follows: t(vi) = 1for 1≤i≤n then the induced edge labeling is, t(e) = 2 for all edges e in g0 we observe that, dt(vi) = 2m-2, dt(vn) = 2n-2 dt(vi)≠ dt(vn) for and t(vi)+ t(vn) + t(vivn) = 4 ≡ 0 (mod 2) hence tdln(g0) = 1 case(ii) when m=3, n>3 be a prime. in this graph, we have v(g0) = v1(g0) v2(g0) where v1(g0) = {n,2n}, v2(g0) = {3i: 1≤i≤n-1} and e(g0) = {uv: u v1(g0), v v2(g0)}. hence dg(u) = n-1 for all u v1(g0), dg(v) = m-1, for all v v2(g0) and |e(g0)| = 2n-2 define a labeling t: v(g0)→{1,2,…,p} as follows: t(u) = 1 for all u v1(g0) t(v) = 1 for all v v2(g0) then the induced edge labeling is, t(uv) = 2 for all uv e(g0) we observe that, dt(u) = 2n-2, dt(v) = 2m-2 dt(u) dt(v) for all u v1 , for all v v2 and t(u) + t(v) + t(uvi) = 4 ≡ 0(mod 2) for all uv e(g0) hence tdln(g0) = 1. ■ theorem 3.2 for r = zk, k =m 2 n, n >3 be a prime number, tdln(γ(r)) = proof assume g0 = γ(r) 321 totally magic d-lucky number of graphs case(i) when m=2, in this case(g0) has partitioned into two sets v1(g0),v2(g0). v1(g0) contains the multiples of n in zk, v2(g0) contains the multiples of m excluding 2n in zk. let v1(g0) = { } and v2(g0) = { } |v(g0)| =2n+1 e(g0) = |e(g0)| = 4n-4 hence dg(ri) = n-1, for i {1,3}; dg( ) = 2n-2; dg(sj) = m+1, sj ; dg(sj) = m-1, sj v2(g0) – {4, 8, …, 4n-4} define a labeling t: v(g0) → {1, 2, …, p} as follows: t(ri) = 1 for 1 ≤ i ≤ 3; t(sj) = 1 for 1 ≤ j ≤ 2n-2; then the induced edge labelling is, t(risj) = 2 for all risj e(g0) we observe that, dt(ri) = 2n-2, i {1,3}, dt( ) = 4n-4, dt(sj) = 2m-2, sj v(γ(r))-{4,8,…,4n-4}, dt(sj) = 2m+2, sj {4,8,…,4n-4} dt( ) ≠ dt(sj) for all v1(g0) , sj` v2(g0) and t(ri) + t(sj) + t (risj) = 4 ≡ 0 (mod 2) for all edges in g0 hence tdln(g0)=1. case(ii) when m = 3, in this case, the vertex set of g0 partitioned into two sets v1 and v2. wherev1 = {n, 2n, 3n, 4n, 5n, 6n, 7n, 8n} = {u1, u2, u3, u4, u5, u6, u7, u8}and v2 = {3, 6, 9, …, 9n-3} {n, 2n} = {v1, v2, v3, …, v3n-1}. e(g0) = {uivi: for all ui v1 , vi {9,18,27,…,9(n-1)} {uivi : ui {3n,6n} vi v2 } {u3,u6}. hence dg(ui) =n-1 for all ui v1-{3n,6n}; dg(ui) = 3n-2, i={3,6}; dg(vi) = 8 for all vi {9,18,…,9(n-1)}; dg(vi) = 2, vi v2-{9,18,…,9(n-1)} . define the labelling t v(g0)→ {1,2,…,p} as follows: t(ui) = 1, for ,ui v1; t(u6) = 2, u6 v1; t(vi) = 1, for all vi v2. then the induced edge labellings are, t(uivi) = 2 for all uivi e(g0) t(u3u6) = 3 t(u6vi) = 3 for all vi v2 we observe that, dt(ui) = 2n-2, 322 n. mohamed rilwan, a. nilofer dt( ) = 6n-3, dt( ) = 6n-4, dt(vi) = 5 for all vi v2(g0)-{9,8,…,9(n-1)} dt(vi) = 17, vi {9,8,…,9(n-1)} dt(ui) dt(vi), dt( ) dt( ) and t(ui)+t(vi)+t(uivi) = 4 ≡ 0 (mod 2) for all edges in g0 t(u3) + t(u6) + t(u3u6) = 6 ≡ 0 (mod 2) t(u6) + t(vi) + t(u6vi) = 3 for all vi v2 it can be easily verified that weights of all the incident vertices are distinct and all the edges of the graph have common totally magic d-lucky constant. hence tdln (g0) = 2. ■ theorem 3.3 let r = be a commutative ring with unity. for the zero-divisor graph γ(r), tdln(γ(r)) = m-1 where m = (m1, m2, m3, …, mk), mi’s are distinct prime numbers, ni’s are positive integers. proof consider g0 = γ(r) be a zero-divisor graph of commutative ring r = where mi’s are prime numbers and ni’s are positive integers. the vertex set of g0 consists of different blocks, v(g0) = where ( , ) (0, 0…, 0) and ( , ) ( ). = {( , ): =0 if = and | and ∤ if {0,1,2,…,ni-1}} all the vertices in are adjacent to all the vertices in if for all i= 1,2,…,k. the vertices in form a clique in g0 if for all i= 1,2,..,k hence we have, for each u , dg(u) = if is clique ; dg(u) = -1 + if is not a clique. define a labeling t: v(g0) {1, 2, …, p}as follows, label the vertices of the block as 1 if the block is not form a clique. if the block is form a clique, label the vertices of clique uj as , where , then the induced edge labellings are, if the block is not form a clique, t(e) = 2 for all edge e in this block if the block is form a clique, t(ujuj+1) = 2j+1 for all , t(uqu1) = q+1 323 totally magic d-lucky number of graphs let t = max ( ) if form a clique we observe that, for each u , we have, , dt(u) = t(u)+t(v)+t(uv) ≡ 0 (mod 2) for all uv e(g0) hence tdln(g0) = t= m-1 where m=max( ). it can be easily verified that colors of all the incident vertices are pairwise distinct and have the common constant for all the edges of the our given graph. ■ 4. conclusions in this paper, we introduced a new labeling, totally magic d-lucky labeling, found the totally magic d-lucky number of some standard graphs and some zero divisor graphs. in future, we use this labeling in some other graphs. acknowledgement the author is very grateful to the referees for their insightful comments on the revised version of the manuscript. references [1] mirka miller, indira rajasingh, d.ahima emilet, d.azubha jemilet, d-lucky labeling of graphs. precedia computer science 57(2015) 766-771. [2] s. czerwinski, j. grytczuk, v. zelazny, lucky labeling of graphs. information processing letters,109 (2009) 1078-1081. [3] a. d. garciano, rm marcelo, mjp ruiz and mac tolentino, on the sigma chromatic number of the zero divisor graphs of the ring of integers modulo. journal of physics, conf. series 1836(2021) 012013, iop publishing. [4] anderson df and livingston ps 1999, journal of algebra, 217 (434-447). [5] g. exoo, a. ling, j. mcsoriey, n. philips, and w. wallis, totally magic graphs. discrete mathematics, 254 (2002) 103-113. [6] sarifa khatun, graceful labeling of some zero divisor graphs. electronic notes in discrete mathematics 63(2017), 189-196. [7] i. cahit, some totally modular cordial graphs. discussiones mathematicae graph theory 22 (2002), 247-258. 324 ratio mathematica volume 42, 2022 insertion of terms satisfying the recurrence relations of horadam sequence and bifurcating fibonacci sequences khushbu das * devbhadra shah † abstract in this article, we consider the problem of finding general formula for the terms introduced between two given positive integers ‘𝑎’ and ‘𝑏’ in such a way that the terms of newly formed finite sequence satisfy the recurrence relations of horadam sequence and some bifurcating fibonacci sequences. keywords: generalized fibonacci sequences, horadam sequence, bifurcating sequence, inserted terms, missing terms. 2010 ams subject classification: 11b37, 11b39, 11b99. ‡ * khushbu das (department of mathematics, veer narmad south gujarat university, surat, india); khushbudas14@gmail.com. † devbhadra shah (department of mathematics, veer narmad south gujarat university, surat, india); drdvshah@yahoo.com. ‡ received on september 15th, 2021. accepted on april 17th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v39i0.639. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 7 khushbu j. das and devbhadra v. shah 1 introduction consider any two fixed given positive integers 𝑎 and 𝑏 such that 𝑎 < 𝑏. we insert 𝑘 number of terms between 𝑎 and 𝑏 so that every term inserted between follows the given recurrence relation of any generalized fibonacci sequence. then we say 𝑥𝑛 (1 ≤ 𝑛 ≤ 𝑘) to be the inserted term (or missing term, as defined by some of the authors) in the finite (generalized fibonacci-like) sequence 𝑎,𝑥1,𝑥2,𝑥3,…. ,𝑥𝑘,𝑏. the problem of insertion of terms in arithmetic, harmonic, geometric sequence is considered to be elementary which occurs in the high school mathematics. however, not much work is done regarding the general formula for the inserted terms in a fibonacci-like sequences. howell [8] presented a proof for finding the 𝑛th term of fibonacci sequence using vectors and eigenvalues. but he said nothing for the fibonacci-like sequences. agnes, et al. [1] provided a formula for inclusion of three consecutive missing terms in fibonacci-like sequence. horadam [7] defined a linear recurrence sequence of second order 𝐹𝑛(𝑝,𝑞), acknowledged as horadam sequence, by the recurrence relation 𝐹𝑛 (𝑝,𝑞) = 𝑝𝐹𝑛−1 (𝑝,𝑞) + 𝑞𝐹𝑛−2 (𝑝,𝑞) with the initial conditions 𝐹0 (𝑝,𝑞) = 𝑎,𝐹1 (𝑝,𝑞) = 𝑐, where 𝑎,𝑐 and 𝑝,𝑞 are arbitrary positive integers. first few terms of this sequence are shown in the table 1. n 𝑭𝒏 (𝒑,𝒒) 0 𝑎 1 𝑐 2 𝑝𝑐 + 𝑞𝑎 3 𝑝2𝑐 + 𝑝𝑞𝑎 + 𝑞𝑐 4 𝑝3𝑐 + 𝑝2𝑞𝑎 + 2𝑝𝑞𝑐 + 𝑞2𝑎 5 𝑝4𝑐 + 𝑝3𝑞𝑎 + 3𝑝2𝑞𝑐 + 2𝑝𝑞2𝑎 + 𝑞2𝑐 6 𝑝5𝑐 + 𝑝4𝑞𝑎 + 4𝑝3𝑞𝑐 + 3𝑝2𝑞2𝑎 + 3𝑝𝑞2𝑐 + 𝑞3𝑎 table 1 8 insertion of terms satisfying the recurrence relations of horadam sequence and bifurcating fibonacci sequences the binet-type explicit formula for the terms of this sequence is given by 𝐹𝑘 (𝑝,𝑞) = (𝑐−𝑎𝛽)𝛼𝑘−(𝑐−𝑎𝛼)𝛽𝑘 𝛼−𝛽 , where 𝛼 = 𝑝+√𝑝2+4𝑞 2 ,𝛽 = 𝑝−√𝑝2+4𝑞 2 . here we note that 𝛼 − 𝛽 = √𝑝2 + 4𝑞, 𝛼 + 𝛽 = 𝑝 and 𝛼𝛽 = −𝑞. diwan, shah [4,5] considered the sequence {𝐹𝑛 (𝑝,𝑞,𝑟,𝑠) } defined by the recurrence relation 𝐹𝑛 (𝑝,𝑞,𝑟,𝑠) = 𝑝𝜒(𝑛)𝑞1−𝜒(𝑛)𝐹𝑛−1 (𝑝,𝑞,𝑟,𝑠) + 𝑟𝜒(𝑛)𝑠1−𝜒(𝑛)𝐹𝑛−2 (𝑝,𝑞,𝑟,𝑠) , (1.1) where 𝑝,𝑞,𝑟,𝑠 are any fixed positive integers and 𝜒(𝑛) = { 1;if 𝑛 is odd 0;if 𝑛 is even . they studied this sequence extensively for some explicit values of 𝑝,𝑞,𝑟,𝑠. verma and bala, bilgici, yayenie [2,9,10] also studied this sequence for some specific values of 𝑝,𝑞,𝑟,𝑠. it is easy to observe that this sequence is bifurcating sequence depending on the parity of 𝑛. in this paper we derive the general formula which gives the value of any inserted term 𝑥𝑛 (1 ≤ 𝑛 ≤ 𝑘) for the sequence {𝐹𝑛(𝑝,𝑞)} and various bifurcating subsequence of {𝐹𝑛 (𝑝,𝑞,𝑟,𝑠) } by considering some fixed values of 𝑝,𝑞,𝑟,𝑠. throughout we assume that the positive integers 𝑎,𝑏 are the given first and last term respectively in the sequence to be considered. 2 insertion of terms satisfying the recurrence relation of horadam sequence in this section, we find the formula for the inserted terms between the given fixed positive integers 𝑎,𝑏 so that terms of the sequence 𝑎,𝑥1,𝑥2,𝑥3,…,𝑥𝑘,𝑏 satisfies the recurrence relation of 𝐹𝑛(𝑝,𝑞). before finding the general formula, we find the formula for the first term 𝑥1 in the finite sequence 𝑎,𝑥1,𝑥2,𝑥3,…. ,𝑥𝑘,𝑏. this value of 𝑥1 will be further used to find the general formula for any 𝑥𝑛 (1 ≤ 𝑛 ≤ 𝑘). in [7], horadam obtained only the formula for the first missing term (inserted term) 𝑥1 when 𝑘 terms are inserted between given two positive 9 khushbu j. das and devbhadra v. shah integers 𝑎 and 𝑏, so that all 𝑥𝑖’s satisfy the recurrence relation of 𝐹𝑛 (𝑝,𝑞) . in fact, he proved that 𝑥1 = 𝑏−𝑎𝑞𝐹𝑘 (𝑝,𝑞) 𝐹 𝑘+1 (𝑝,𝑞) . (2.1) in the following theorem, we obtain the general formula for any arbitrary term 𝑥𝑛 (1 ≤ 𝑛 ≤ 𝑘) when 𝑘 terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between positive integers 𝑎 and 𝑏, so that all 𝑥𝑖’s satisfy the recurrence relation of 𝐹𝑛 (𝑝,𝑞) . theorem 2.1. when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given integers 𝑎 and 𝑏 in such a way that the finite sequence 𝑎,𝑥1,𝑥2,𝑥3,…,𝑥𝑘,𝑏 satisfy the recurrence relation of 𝐹𝑛 (𝑝,𝑞) , then the general formula for any arbitrary term 𝑥𝑛 (1 ≤ 𝑛 ≤ 𝑘) is given by 𝑥𝑛 = 𝐹𝑛 (𝑝,𝑞) 𝑏(𝛼−𝛽)−𝑎𝑞𝐹𝑛 (𝑝,𝑞) {(𝑐−𝑎𝛽)𝛼𝑘−(𝑐−𝑎𝛼)𝛽 𝑘 }+𝑎𝑞𝐹𝑛−1 (𝑝,𝑞) {(𝑐−𝑎𝛽)𝛼𝑘+1−(𝑐−𝑎𝛼)𝛽 𝑘+1 } (𝑐−𝑎𝛽)𝛼𝑘+1−(𝑐−𝑎𝛼)𝛽 𝑘+1 . (2.2) proof. we prove the result by the principle of mathematical induction. for we 𝑘 = 1, we get 𝑛 = 1 and 𝑥1 = 𝐹1 (𝑝,𝑞) 𝑏(𝛼−𝛽)−𝑎𝑞𝐹1 (𝑝,𝑞) {(𝑐−𝑎𝛽)𝛼−(𝑐−𝑎𝛼)𝛽}+𝑎𝑞𝐹0 (𝑝,𝑞) {(𝑐−𝑎𝛽)𝛼2−(𝑐−𝑎𝛼)𝛽 2 } (𝑐−𝑎𝛽)𝛼2−(𝑐−𝑎𝛼)𝛽 2 = 𝑏−𝑎𝑞𝑐 𝑐𝑝+𝑎𝑞 , which is same as (2.1). next, we assume that (2.2) holds for some positive integer not exceeding 𝑛. then both the following results hold: 𝑥𝑛−1 = 𝐹𝑛−1 (𝑝,𝑞) 𝑏(𝛼−𝛽)−𝑎𝑞𝐹𝑛−1 (𝑝,𝑞) {(𝑐−𝑎𝛽)𝛼𝑘−(𝑐−𝑎𝛼)𝛽 𝑘 }+𝑎𝑞𝐹𝑛−2 (𝑝,𝑞) {(𝑐−𝑎𝛽)𝛼𝑘+1−(𝑐−𝑎𝛼)𝛽 𝑘+1 } (𝑐−𝑎𝛽)𝛼𝑘+1−(𝑐−𝑎𝛼)𝛽 𝑘+1 and 𝑥𝑛 = 𝐹𝑛 (𝑝,𝑞) 𝑏(𝛼−𝛽)−𝑎𝑞𝐹𝑛 (𝑝,𝑞) {(𝑐−𝑎𝛽)𝛼𝑘−(𝑐−𝑎𝛼)𝛽 𝑘 }+𝑎𝑞𝐹𝑛−1 (𝑝,𝑞) {(𝑐−𝑎𝛽)𝛼𝑘+1−(𝑐−𝑎𝛼)𝛽 𝑘+1 } (𝑐−𝑎𝛽)𝛼𝑘+1−(𝑐−𝑎𝛼)𝛽 𝑘+1 now consider 𝑝𝑥𝑛 + 𝑞𝑥𝑛−1 = (𝑝𝐹𝑛 (𝑝,𝑞) +𝑞𝐹𝑛−1 (𝑝,𝑞) )𝑏(𝛼−𝛽)−𝑎𝑞(𝑝𝐹𝑛 (𝑝,𝑞) +𝑞𝐹𝑛−1 (𝑝,𝑞) ){(𝑐−𝑎𝛽)𝛼𝑘−(𝑐−𝑎𝛼)𝛽𝑘} +𝑎𝑞(𝑝𝐹𝑛−1 (𝑝,𝑞) +𝑞𝐹𝑛−2 (𝑝,𝑞) ){(𝑐−𝑎𝛽)𝛼𝑘+1−(𝑐−𝑎𝛼)𝛽𝑘+1} (𝑐−𝑎𝛽)𝛼𝑘+1−(𝑐−𝑎𝛼)𝛽𝑘+1 = 𝐹𝑛+1 (𝑝,𝑞) 𝑏(𝛼−𝛽)−𝑎𝑞𝐹𝑛+1 (𝑝,𝑞) {(𝑐−𝑎𝛽)𝛼𝑘−(𝑐−𝑎𝛼)𝛽𝑘} +𝑎𝑞𝐹𝑛 (𝑝,𝑞) {(𝑐−𝑎𝛽)𝛼𝑘+1−(𝑐−𝑎𝛼)𝛽𝑘+1} (𝑐−𝑎𝛽)𝛼𝑘+1−(𝑐−𝑎𝛼)𝛽𝑘+1 . it can be observed that the right side of this result simplifies to 𝑥𝑛+1. thus, by (2.2) is true for every positive integer 𝑛. 10 insertion of terms satisfying the recurrence relations of horadam sequence and bifurcating fibonacci sequences 3 insertion of terms satisfying the recurrence relation of bifurcating sequence 𝑭𝒏 (𝒑,𝒒,𝟏,𝟏) in this section, we find the formula for the inserted terms between the numbers 𝑎 and 𝑏 so that terms of the sequence 𝑎,𝑥1,𝑥2,𝑥3,…,𝑥𝑘,𝑏 satisfies the recurrence relation (1.2). if we let 𝑝 = 𝑞 = 𝑟 = 𝑠 = 1, the sequence {𝐹𝑛 (𝑝,𝑞,𝑟,𝑠) } is the sequence of usual fibonacci numbers. if we define 𝐹0 (𝑝,𝑞,𝑟,𝑠) = 0,𝐹1 (𝑝,𝑞,𝑟,𝑠) = 1, then first few terms of this sequence are shown in table 2. n 𝑭𝒏 (𝒑,𝒒,𝒓,𝒔) 0 0 1 1 2 𝑞 3 𝑝𝑞 + 𝑟 4 𝑝𝑞2 + 𝑞𝑟 + 𝑞𝑠 5 𝑝2𝑞2 + 2𝑝𝑞𝑟 + 𝑝𝑞𝑠 + 𝑟2 6 𝑝2𝑞3 + 2𝑝𝑞2𝑟 + 2𝑝𝑞2𝑠 + 𝑟2𝑞 + 𝑠𝑞𝑟 + 𝑞𝑠2 table 2 if we consider 𝑟 = 𝑠 = 1 in (1.1), we get the sequence {𝐹𝑛 (𝑝,𝑞,1,1) } whose terms are defined by the recurrence relation 𝐹𝑛 (𝑝,𝑞,1,1) = 𝑝𝜒(𝑛)𝑞1−𝜒(𝑛)𝐹𝑛−1 (𝑝,𝑞,1,1) + 𝐹𝑛−2 (𝑝,𝑞,1,1) , where 𝐹0 (𝑝,𝑞,1,1) = 0,𝐹1 (𝑝,𝑞,1,1) = 1. this can be written in the form 𝐹𝑛 (𝑝,𝑞,1,1) = { 𝑝𝐹𝑛−1 (𝑝,𝑞,1,1) + 𝐹𝑛−2 (𝑝,𝑞,1,1) ;when 𝑛 is odd 𝑞𝐹𝑛−1 (𝑝,𝑞,1,1) + 𝐹𝑛−2 (𝑝,𝑞,1,1) ;when 𝑛 is even (𝑛 ≥ 2) (3.1) with the initial conditions 𝐹0 (𝑝,𝑞,1,1) = 0,𝐹1 (𝑝,𝑞,1,1) = 1. first few terms of this sequence are shown in table 3. this sequence was studied in detail by diwan, shah [4] as well as edson, yayenie [6]. they derived the binet-type explicit formula for the terms of this sequence as 𝐹𝑘 (𝑝,𝑞,1,1) = 𝑞1−𝜒(𝑘) (𝑝𝑞) [ 𝑘 2 ] ( 𝛼𝑘−𝛽𝑘 𝛼−𝛽 ), where 𝛼 = 𝑝𝑞+√𝑝2𝑞2+4𝑝𝑞 2 ,𝛽 = 𝑝𝑞−√𝑝2𝑞2+4𝑝𝑞 2 . here we note that 𝛼 − 𝛽 = √𝑝2𝑞2 + 4𝑝𝑞, 𝛼 + 𝛽 = 𝑝𝑞 and 𝛼𝛽 = −𝑝𝑞. 11 khushbu j. das and devbhadra v. shah n 𝑭𝒏 (𝒑,𝒒,𝟏,𝟏) 0 0 1 1 2 𝑞 3 𝑝𝑞 + 1 4 𝑝𝑞2 + 2𝑞 5 𝑝2𝑞2 + 3𝑝𝑞 + 1 6 𝑝2𝑞3 + 4𝑝𝑞2 + 3𝑞 table 3 when we consider the sequence 𝑎,𝑥1,𝑏, where 𝑥1 is the only inserted term between 𝑎 and 𝑏, then using 𝐹2 (𝑝,𝑞,1,1) = 𝑞𝐹1 (𝑝,𝑞,1,1) + 𝐹0 (𝑝,𝑞,1,1) , we get 𝑏 = 𝑞𝑥1 + 𝑎. thus 𝑥1 = 𝑏−𝑎 𝑞 . when we consider the finite sequence 𝑎,𝑥1,𝑥2,𝑏, so that it satisfies (3.1), we observe that 𝑥1 = 𝑥2−𝑎 𝑞 and 𝑥2 = 𝑏−𝑥1 𝑝 . this gives 𝑥2 = 𝑏𝑞+𝑎 𝑝𝑞+1 and 𝑥1 = 𝑏−𝑎𝑝 𝑝𝑞+1 . further, considering the sequence 𝑎,𝑥1,𝑥2,𝑥3,𝑏, it is now easy to observe that 𝑥1 = 𝑥2−𝑎 𝑞 ,𝑥2 = 𝑥3−𝑥1 𝑝 and 𝑥3 = b−𝑥2 𝑞 . solving these three equations in three variables 𝑥1,𝑥2,𝑥3 we get 𝑥1 = 𝑏−𝑎(𝑝𝑞+1) 𝑝𝑞2+2𝑞 ,𝑥2 = 𝑏𝑞+𝑎𝑞 𝑝𝑞2+2𝑞 ,𝑥3 = 𝑏−𝑎+𝑏𝑝𝑞 𝑝𝑞2+2𝑞 . if we continue extending the above finite sequence one more time, then we get the sequence 𝑎,𝑥1,𝑥2,𝑥3,𝑥4,𝑏. using the similar approach as above, we obtain 𝑥1 = 𝑏−𝑎(𝑝2𝑞+2𝑝) 𝑝2𝑞2+3𝑝𝑞+1 ,𝑥2 = 𝑏𝑞+𝑎𝑝𝑞+𝑎 𝑝2𝑞2+3𝑝𝑞+1 ,𝑥3 = 𝑏−𝑎𝑝+𝑏𝑝𝑞 𝑝2𝑞2+3𝑝𝑞+1 and 𝑥4 = 𝑎+2𝑏𝑞+𝑏𝑝𝑞2 𝑝2𝑞2+3𝑝𝑞+1 . we mention these results in the table 4. number of inserted terms formula x1 x2 x3 x4 1 𝑏 − 𝑎 𝑞 ------ 2 𝑏 − 𝑎𝑝 𝑝𝑞 + 1 𝑏𝑞 + 𝑎 𝑝𝑞 + 1 ---- 3 𝑏 − 𝑎(𝑝𝑞 + 1) 𝑝𝑞2 + 2𝑞 𝑏𝑞 + 𝑎𝑞 𝑝𝑞2 + 2𝑞 𝑏 − 𝑎 + 𝑏𝑝𝑞 𝑝𝑞2 + 2𝑞 -- 4 𝑏 − 𝑎(𝑝2𝑞 + 2𝑝) 𝑝2𝑞2 + 3𝑝𝑞 + 1 𝑏𝑞 + 𝑎𝑝𝑞 + 𝑎 𝑝2𝑞2 + 3𝑝𝑞 + 1 𝑏 − 𝑎𝑝 + 𝑏𝑝𝑞 𝑝2𝑞2 + 3𝑝𝑞 + 1 𝑎 + 2𝑏𝑞 + 𝑏𝑝𝑞2 𝑝2𝑞2 + 3𝑝𝑞 + 1 table 4 12 insertion of terms satisfying the recurrence relations of horadam sequence and bifurcating fibonacci sequences from this table, we observe that there is a similar pattern for the first inserted term in case of any number of inserted terms. we thus generalize it for the case of any number of inserted terms and when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given two positive integers 𝑎 and 𝑏, we can now write the general formula for 𝑥1 as 𝑥1 = 𝑏−𝑎𝑝1−𝜒(𝑘)𝑞𝜒(𝑘)−1𝐹 𝑘 (𝑝,𝑞,1,1) 𝐹 𝑘+1 (𝑝,𝑞,1,1) . (3.2) next, when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given two positive integers 𝑎 and 𝑏, we obtain the general formula for 𝑛th inserted term (1 ≤ 𝑛 ≤ 𝑘) using the recurrence relation (3.1). theorem 3.1. when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given integers 𝑎 and 𝑏, so that all 𝑥𝑖’s satisfy the recurrence relation of 𝐹𝑛 (𝑝,𝑞,1,1) , the general formula for any arbitrary term 𝑥𝑛 (1 ≤ 𝑛 ≤ 𝑘) is given by 𝑥𝑛 = 𝐹𝑛 (𝑝,𝑞,1,1) 𝑏−𝑎𝐹𝑛 (𝑝,𝑞,1,1) 𝑝1−𝜒(𝑘)𝑞𝜒(𝑘)−1𝐹𝑘 (𝑝,𝑞,1,1) +𝑎𝑝𝜒(𝑛)𝑞−𝜒(𝑛)𝐹𝑛−1 (𝑝,𝑞,1,1) 𝐹𝑘+1 (𝑝,𝑞,1,1) 𝐹 𝑘+1 (𝑝,𝑞,1,1) . proof. we use the principle of mathematical induction to prove the result. since by (3.2), we have 𝑥1 = 𝑏−𝑎𝑝1−𝜒(𝑘)𝑞𝜒(𝑘)−1𝐹 𝑘 (𝑝,𝑞,1,1) 𝐹 𝑘+1 (𝑝,𝑞,1,1) , which proves the result for 𝑛 = 1. we next assume that it is true for all positive integers not exceeding 𝑛. then the following holds: 𝑥𝑛−1 = 𝐹𝑛−1 (𝑝,𝑞,1,1) 𝑏−𝑎𝐹𝑛−1 (𝑝,𝑞,1,1) 𝑝1−𝜒(𝑘)𝑞𝜒(𝑘)−1𝐹𝑘 (𝑝,𝑞,1,1) +𝑎𝑝𝜒(𝑛−1)𝑞−𝜒(𝑛−1)𝐹𝑛−2 (𝑝,𝑞,1,1) 𝐹𝑘+1 (𝑝,𝑞,1,1) 𝐹 𝑘+1 (𝑝,𝑞,1,1) 𝑥𝑛−2 = 𝐹𝑛−2 (𝑝,𝑞,1,1) 𝑏−𝑎𝐹𝑛−2 (𝑝,𝑞,1,1) 𝑝1−𝜒(𝑘)𝑞𝜒(𝑘)−1𝐹𝑘 (𝑝,𝑞,1,1) +𝑎𝑝𝜒(𝑛−2)𝑞−𝜒(𝑛−2)𝐹𝑛−3 (𝑝,𝑞,1,1) 𝐹𝑘+1 (𝑝,𝑞,1,1) 𝐹 𝑘+1 (𝑝,𝑞,1,1) . now, 𝑝𝜒(𝑛)𝑞1−𝜒(𝑛)𝑥𝑛−1 + 𝑥𝑛−2 = { (𝑝 𝜒(𝑛)𝑞1−𝜒(𝑛)𝐹𝑛−1 (𝑝,𝑞,1,1) +𝐹𝑛−2 (𝑝,𝑞,1,1) )𝑏−𝑎(𝑝𝜒(𝑛)𝑞1−𝜒(𝑛)𝐹𝑛−1 (𝑝,𝑞,1,1) +𝐹𝑛−2 (𝑝,𝑞,1,1) ) ×𝑝1−𝜒(𝑘)𝑞𝜒(𝑘)−1𝐹𝑘 (𝑝,𝑞,1,1) +𝑎(𝑝𝜒(𝑛)𝑞1−𝜒(𝑛)𝑝𝜒(𝑛−1)𝑞−𝜒(𝑛−1)𝐹𝑛−2 (𝑝,𝑞,1,1) +𝑝𝜒(𝑛−2)𝑞−𝜒(𝑛−2)𝐹𝑛−3 (𝑝,𝑞,1,1) )𝐹𝑘+1 (𝑝,𝑞,1,1) } 𝐹 𝑘+1 (𝑝,𝑞,1,1) = { 𝐹𝑛 (𝑝,𝑞,1,1) 𝑏−𝑎𝐹𝑛 (𝑝,𝑞,1,1) 𝑝1−𝜒(𝑘)𝑞𝜒(𝑘)−1𝐹𝑘 (𝑝,𝑞,1,1) +𝑎(𝑝𝜒(𝑛)+𝜒(𝑛−1)𝑞1−(𝜒(𝑛)+𝜒(𝑛−1))𝐹𝑛−2 (𝑝,𝑞,1,1) +𝑝𝜒(𝑛)𝑞−𝜒(𝑛)𝐹𝑛−3 (𝑝,𝑞,1,1) )𝐹𝑘+1 (𝑝,𝑞,1,1) } 𝐹 𝑘+1 (𝑝,𝑞,1,1) 13 khushbu j. das and devbhadra v. shah = { 𝐹𝑛 (𝑝,𝑞,1,1) 𝑏−𝑎𝐹𝑛 (𝑝,𝑞,1,1) 𝑝1−𝜒(𝑘)𝑞𝜒(𝑘)−1𝐹𝑘 (𝑝,𝑞,1,1) +𝑎(𝑝𝐹𝑛−2 (𝑝,𝑞,1,1) +𝑝𝜒(𝑛)𝑞−𝜒(𝑛)𝐹𝑛−3 (𝑝,𝑞,1,1) )𝐹𝑘+1 (𝑝,𝑞,1,1) } 𝐹 𝑘+1 (𝑝,𝑞,1,1) . now, when 𝑛 is odd, we get 𝑝𝐹𝑛−2 (𝑝,𝑞,1,1) + 𝑝𝜒(𝑛)𝑞−𝜒(𝑛)𝐹𝑛−3 (𝑝,𝑞,1,1) = 𝑝𝜒(𝑛)𝑞1−𝜒(𝑛)𝐹𝑛−2 (𝑝,𝑞,1,1) + 𝑝𝜒(𝑛)𝑞−𝜒(𝑛)𝐹𝑛−3 (𝑝,𝑞,1,1) = 𝑝𝜒(𝑛)𝑞−𝜒(𝑛) (𝑞𝐹𝑛−2 (𝑝,𝑞,1,1) + 𝐹𝑛−3 (𝑝,𝑞,1,1) ) = 𝑝𝜒(𝑛)𝑞−𝜒(𝑛) (𝑝𝜒(𝑛−1)𝑞1−𝜒(𝑛−1)𝐹𝑛−2 (𝑝,𝑞,1,1) + 𝐹𝑛−3 (𝑝,𝑞,1,1) ) = 𝑝𝜒(𝑛)𝑞−𝜒(𝑛)𝐹𝑛−1 (𝑝,𝑞,1,1) . also, when 𝑛 is even, we get 𝑝𝐹𝑛−2 (𝑝,𝑞,1,1) + 𝑝𝜒(𝑛)𝑞−𝜒(𝑛)𝐹𝑛−3 (𝑝,𝑞,1,1) = 𝑝1−𝜒(𝑛)𝑞𝜒(𝑛)𝐹𝑛−2 (𝑝,𝑞,1,1) + 𝑝𝜒(𝑛)𝑞−𝜒(𝑛)𝐹𝑛−3 (𝑝,𝑞,1,1) = 𝑝𝜒(𝑛)𝑞−𝜒(𝑛) (𝑝1−2𝜒(𝑛)𝑞2𝜒(𝑛)𝐹𝑛−2 (𝑝,𝑞,1,1) + 𝐹𝑛−3 (𝑝,𝑞,1,1) ) = 𝑝𝜒(𝑛)𝑞−𝜒(𝑛) (𝑝𝜒(𝑛−1)𝑞1−𝜒(𝑛−1)𝐹𝑛−2 (𝑝,𝑞,1,1) + 𝐹𝑛−3 (𝑝,𝑞,1,1) ) = 𝑝𝜒(𝑛)𝑞−𝜒(𝑛)𝐹𝑛−1 (𝑝,𝑞,1,1) therefore, we have 𝑥𝑛 = 𝑝 𝜒(𝑛)𝑞1−𝜒(𝑛)𝑥𝑛−1 + 𝑥𝑛−2 = { 𝐹𝑛 (𝑝,𝑞,1,1) 𝑏−𝑎𝐹𝑛 (𝑝,𝑞,1,1) 𝑝1−𝜒(𝑘)𝑞𝜒(𝑘)−1𝐹𝑘 (𝑝,𝑞,1,1) +𝑎(𝑝𝜒(𝑛)𝑞−𝜒(𝑛)𝐹𝑛−1 (𝑝,𝑞,1,1) )𝐹𝑘+1 (𝑝,𝑞,1,1) } 𝐹 𝑘+1 (𝑝,𝑞,1,1) , which proves the result for every positive integer 𝑛. 4 insertion of terms satisfying the recurrence relation of bifurcating sequence 𝑭𝒏 (𝒑,𝟏,𝟏,𝒔) if we consider 𝑞 = 𝑟 = 1 in (1.1), the sequence {𝐹𝑛 (𝑝,1,1,𝑠) } whose terms are defined by the recurrence relation 𝐹𝑛 (𝑝,1,1,𝑠) = 𝑝𝜒(𝑛)𝐹𝑛−1 (𝑝,1,1,𝑠) + 𝑠1−𝜒(𝑛)𝐹𝑛−2 (𝑝,1,1,𝑠) , where 𝐹0 (𝑝,1,1,𝑠) = 0,𝐹1 (𝑝,1,1,𝑠) = 1. this can be written in the form 𝐹𝑛 (𝑝,1,1,𝑠) = { 𝑝𝐹𝑛−1 (𝑝,1,1,𝑠) + 𝐹𝑛−2 (𝑝,1,1,𝑠) ;when 𝑛 is odd 𝐹𝑛−1 (𝑝,1,1,𝑠) + 𝑠𝐹𝑛−2 (𝑝,1,1,𝑠) ;when 𝑛 is even (𝑛 ≥ 2) (4.1) 14 insertion of terms satisfying the recurrence relations of horadam sequence and bifurcating fibonacci sequences with the initial conditions 𝐹0 (𝑝,1,1,𝑠) = 0,𝐹1 (𝑝,1,1,𝑠) = 1. first few terms of this sequence are shown in table 5. n 𝑭𝒏 (𝒑,𝟏,𝟏,𝒔) 0 0 1 1 2 1 3 1 + 𝑝 4 1 + 𝑝 + 𝑠 5 1 + 2𝑝 + 𝑝𝑠 + 𝑝2 6 1 + 𝑠 + 2𝑝 + 2𝑝𝑠 + 𝑝2 + 𝑠2 table 5 this sequence was studied by diwan, shah [4]. they obtained binet-type explicit formula for the terms of this sequence as 𝐹𝑛 (𝑝,1,1,𝑠) = (𝛼−𝑠)𝜒(𝑛)𝛼 [ 𝑛 2 ] −(𝛽−𝑠)𝜒(𝑛)𝛽 [ 𝑛 2 ] 𝛼−𝛽 , where 𝛼 = (𝑝+𝑠+1)+√(𝑝+𝑠+1)2−4𝑠 2 ,𝛽 = (𝑝+𝑠+1)−√(𝑝+𝑠+1)2−4𝑠 2 . this gives 𝛼 − 𝛽 = √(𝑝 + 𝑠 + 1)2 − 4𝑠,𝛼 + 𝛽 = 𝑝 + 𝑠 + 1 and 𝛼𝛽 = 𝑠. in this section, we find the formula for the inserted terms between the numbers 𝑎 and 𝑏 so that terms of the sequence 𝑎,𝑥1,𝑥2,𝑥3,…,𝑥𝑘,𝑏 satisfies the recurrence relation (4.1). when we consider the sequence 𝑎,𝑥1,𝑏, by using 𝐹2 (𝑝,1,1,𝑠) = 𝐹1 (𝑝,1,1,𝑠) + 𝑠𝐹0 (𝑝,1,1,𝑠) , we get 𝑏 = 𝑥1 + 𝑠𝑎. thus 𝑥1 = 𝑏 − 𝑠𝑎. this basic formula will be used to find the other terms. when we consider the finite sequence as 𝑎,𝑥1,𝑥2,𝑏, so that it satisfies (4.1), we observe that 𝑥1 = 𝑥2 − 𝑠 and 𝑥2 = 𝑏−𝑥1 𝑝 . this gives 𝑥2 = 𝑏+𝑠𝑎 1+𝑝 and 𝑥1 = 𝑏−𝑠𝑎𝑝 1+𝑝 . further, considering the sequence 𝑎,𝑥1,𝑥2,𝑥3,𝑏, it is now easy to observe that 𝑥1 = 𝑥2 − 𝑠𝑎,𝑥2 = 𝑥3−𝑥1 𝑝 and 𝑥3 = 𝑏 − 𝑠𝑥2. solving these three equations in three variables 𝑥1,𝑥2,𝑥3 we get 𝑥1 = 𝑏−𝑎𝑠(𝑝+𝑠) 1+𝑝+𝑠 ,𝑥2 = 𝑏+𝑠𝑎 1+𝑝+𝑠 ,𝑥3 = 𝑏+𝑏𝑝−𝑎𝑠2 1+𝑝+𝑠 . if we continue extending the above finite sequence one more time, then we get the sequence 𝑎,𝑥1,𝑥2,𝑥3,𝑥4,𝑏. using the similar approach as above, we 15 khushbu j. das and devbhadra v. shah obtain 𝑥1 = 𝑏−𝑠𝑎(𝑝2+𝑝+𝑝𝑠) 1+2𝑝+𝑝𝑠+𝑝2 ,𝑥2 = 𝑏+𝑠𝑎𝑝+𝑠𝑎 1+2𝑝+𝑝𝑠+𝑝2 ,𝑥3 = 𝑏+𝑝𝑏−𝑠2𝑝𝑎 1+2𝑝+𝑝𝑠+𝑝2 and 𝑥4 = 𝑏+𝑝𝑏+𝑠𝑏+𝑠2𝑎 1+2𝑝+𝑝𝑠+𝑝2 . we mention these results in the table 6. number of inserted terms formula 𝑥1 𝑥2 𝑥3 𝑥4 1 𝑏 − 𝑠𝑎 ------ 2 𝑏 − 𝑠𝑎𝑝 1 + 𝑝 𝑏 + 𝑠𝑎 1 + 𝑝 ---- 3 𝑏 − 𝑠𝑎(𝑝 + 𝑠) 1 + 𝑝 + 𝑠 𝑏 + 𝑠𝑎 1 + 𝑝 + 𝑠 𝑏 + 𝑏𝑝 − 𝑎𝑠2 1 + 𝑝 + 𝑠 -- 4 𝑏 − 𝑠𝑎(𝑝2 + 𝑝 + 𝑝𝑠) 1 + 2𝑝 + 𝑝𝑠 + 𝑝2 𝑏 + 𝑠𝑎𝑝 + 𝑠𝑎 1 + 2𝑝 + 𝑝𝑠 + 𝑝2 𝑏 + 𝑝𝑏 − 𝑠2𝑝𝑎 1 + 2𝑝 + 𝑝𝑠 + 𝑝2 𝑏 + 𝑝𝑏 + 𝑠𝑏 + 𝑠2𝑎 1 + 2𝑝 + 𝑝𝑠 + 𝑝2 table 6 from this table, we observe that there is a similar pattern for the first inserted term in case of any number of inserted terms. we thus generalize it for the case of 𝑘 number of inserted terms and we can now write the general formula for 𝑥1 as 𝑥1 = 𝑏−𝑠𝑎{𝐹 𝑘+1 (𝑝,1,1,𝑠) −𝐹 𝑘−1 (𝑝,1,1,𝑠) } 𝐹 𝑘+1 (𝑝,1,1,𝑠) . (4.2) next, when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given two positive integers 𝑎 and 𝑏, we obtain the general formula for 𝑛th inserted term (1 ≤ 𝑛 ≤ 𝑘) using the recurrence relation (4.1). theorem 4.1. when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given integers 𝑎 and 𝑏, so that all 𝑥𝑖’s satisfy the recurrence relation of 𝐹𝑛 (𝑝,1,1,𝑠) , the general formula for any arbitrary term 𝑥𝑛 (1 ≤ 𝑛 ≤ 𝑘) is given by 𝑥𝑛 = 𝐹𝑛 (𝑝,1,1,𝑠) 𝑏+𝑠𝑎𝐹𝑛 (𝑝,1,1,𝑠) 𝐹𝑘−1 (𝑝,1,1,𝑠) −𝑠𝑎𝐹𝑛−2 (𝑝,1,1,𝑠) 𝐹𝑘+1 (𝑝,1,1,𝑠) 𝐹 𝑘+1 (𝑝,1,1,𝑠) . proof. we use the principle of mathematical induction to prove the result. since by (4.2), we have 𝑥1 = 𝑏−𝑠𝑎{𝐹 𝑘+1 (𝑝,1,1,𝑠) −𝐹 𝑘−1 (𝑝,1,1,𝑠) } 𝐹 𝑘+1 (𝑝,1,1,𝑠) , which proves the result for 𝑛 = 1. we next assume that it is true for all positive integers not exceeding 𝑛. then the following holds: 16 insertion of terms satisfying the recurrence relations of horadam sequence and bifurcating fibonacci sequences 𝑥𝑛−1 = 𝐹𝑛−1 (𝑝,1,1,𝑠) 𝑏+𝑠𝑎𝐹𝑛−1 (𝑝,1,1,𝑠) 𝐹𝑘−1 (𝑝,1,1,𝑠) −𝑠𝑎𝐹𝑛−3 (𝑝,1,1,𝑠) 𝐹𝑘+1 (𝑝,1,1,𝑠) 𝐹 𝑘+1 (𝑝,1,1,𝑠) and 𝑥𝑛−2 = 𝐹𝑛−2 (𝑝,1,1,𝑠) 𝑏+𝑠𝑎𝐹𝑛−2 (𝑝,1,1,𝑠) 𝐹𝑘−1 (𝑝,1,1,𝑠) −𝑠𝑎𝐹𝑛−4 (𝑝,1,1,𝑠) 𝐹𝑘+1 (𝑝,1,1,𝑠) 𝐹 𝑘+1 (𝑝,1,1,𝑠) . now, 𝑝𝜒(𝑛)𝑥𝑛−1 + 𝑠 1−𝜒(𝑛)𝑥𝑛−2 = { (𝑝𝜒(𝑛)𝐹𝑛−1 (𝑝,1,1,𝑠) +𝑠1−𝜒(𝑛)𝐹𝑛−2 (𝑝,1,1,𝑠) )𝑏+𝑠𝑎(𝑝𝜒(𝑛)𝐹𝑛−1 (𝑝,1,1,𝑠) +𝑠1−𝜒(𝑛)𝐹𝑛−2 (𝑝,1,1,𝑠) )𝐹𝑘−1 (𝑝,1,1,𝑠) −𝑠𝑎(𝑝𝜒(𝑛)𝐹𝑛−3 (𝑝,1,1,𝑠) +𝑠1−𝜒(𝑛)𝐹𝑛−4 (𝑝,1,1,𝑠) )𝐹𝑘+1 (𝑝,1,1,𝑠) } 𝐹 𝑘+1 (𝑝,1,1,𝑠) = {𝐹𝑛 (𝑝,1,1,𝑠) 𝑏+𝑠𝑎𝐹𝑛 (𝑝,1,1,𝑠) 𝐹𝑘−1 (𝑝,1,1,𝑠) −𝑠𝑎( 𝑝𝜒(𝑛−2)𝐹𝑛−3 (𝑝,1,1,𝑠) +𝑠1−𝜒(𝑛−2)𝐹𝑛−4 (𝑝,1,1,𝑠) )𝐹𝑘+1 (𝑝,1,1,𝑠) } 𝐹 𝑘+1 (𝑝,1,1,𝑠) = 𝐹𝑛 (𝑝,1,1,𝑠) 𝑏+𝑠𝑎𝐹𝑛 (𝑝,1,1,𝑠) 𝐹𝑘−1 (𝑝,1,1,𝑠) −𝑠𝑎𝐹𝑛−2 (𝑝,1,1,𝑠) 𝐹𝑘+1 (𝑝,1,1,𝑠) 𝐹 𝑘+1 (𝑝,1,1,𝑠) = 𝑥𝑛, which proves the result for every positive integer 𝑛. 5 insertion of terms satisfying the recurrence relation of bifurcating sequence 𝑭𝒏 (𝟏,𝒒,𝒓,𝟏) if we consider 𝑝 = 𝑠 = 1 in (1.1), then we get the sequence {𝐹𝑛 (1,𝑞,𝑟,1) } whose terms are defined by the recurrence relation 𝐹𝑛 (1,𝑞,𝑟,1) = 𝑞𝜒(𝑛)𝐹𝑛−1 (1,𝑞,𝑟,1) + 𝑟1−𝜒(𝑛)𝐹𝑛−2 (1,𝑞,𝑟,1) , where 𝐹0 (1,𝑞,𝑟,1) = 0,𝐹1 (1,𝑞,𝑟,1) = 1.this can be written in the form 𝐹𝑛 (1,𝑞,𝑟,1) = { 𝐹𝑛−1 (1,𝑞,𝑟,1) + 𝑟𝐹𝑛−2 (1,𝑞,𝑟,1) ;when 𝑛 is odd 𝑞𝐹𝑛−1 (1,𝑞,𝑟,1) + 𝐹𝑛−2 (1,𝑞,𝑟,1) ;when 𝑛 is even (𝑛 ≥ 2) (5.1) with the initial conditions 𝐹0 (1,𝑞,𝑟,1) = 0,𝐹1 (1,𝑞,𝑟,1) = 1. first few terms of this sequence are shown in table 7. diwan, shah [4] studied this sequence and obtained the binet-type explicit formula for the terms of this sequence as 𝐹𝑛 (1,𝑞,𝑟,1) = 𝑞1−𝜒(𝑛) { (𝛼−1)𝜒(𝑛)𝛼 [ 𝑛 2 ] −(𝛽−1)𝜒(𝑛)𝛽 [ 𝑛 2 ] 𝛼−𝛽 }, where 𝛼 = (𝑟+𝑞+1)+√(𝑟+𝑞+1)2−4𝑞 2 ,𝛽 = (𝑟+𝑞+1)−√(𝑟+𝑞+1)2−4𝑞 2 with 𝛼 − 𝛽 = √(𝑟 + 𝑞 + 1)2 − 4𝑞,𝛼 + 𝛽 = 𝑟 + 𝑞 + 1,𝛼𝛽 = 𝑟. 17 khushbu j. das and devbhadra v. shah n 𝑭𝒏 (𝟏,𝒒,𝒓,𝟏) 0 0 1 1 2 𝑞 3 𝑟 + 𝑞 4 𝑞 + 𝑟𝑞 + 𝑞2 5 𝑞 + 2𝑞𝑟 + 𝑟2 + 𝑞2 6 𝑞 + 𝑟𝑞 + 2𝑟𝑞2 + 2𝑞2 + 𝑟2𝑞 + 𝑞3 table 7 in this section, we find the formula for the inserted terms between the numbers 𝑎 and 𝑏 so that terms of the sequence 𝑎,𝑥1,𝑥2,𝑥3,…,𝑥𝑘,𝑏 satisfies the recurrence relation (5.1). when we consider the sequence 𝑎,𝑥1,𝑏, using 𝐹2 (1,𝑞,𝑟,1) = 𝑞𝐹1 (1,𝑞,𝑟,1) + 𝐹0 (1,𝑞,𝑟,1) , we get 𝑏 = 𝑞𝑥1 + 𝑎. thus 𝑥1 = 𝑏−𝑎 𝑞 . when we consider the sequence as 𝑎,𝑥1,𝑥2,𝑏, so that it satisfies (5.1), we observe that 𝑥1 = 𝑥2−𝑎 𝑞 and 𝑥2 = 𝑏 − 𝑟𝑥1. this gives 𝑥2 = 𝑞𝑏+𝑟𝑎 𝑟+𝑞 and 𝑥1 = 𝑏−𝑎 𝑟+𝑞 . further, considering the sequence 𝑎,𝑥1,𝑥2,𝑥3,𝑏, it is now easy to observe that 𝑥1 = 𝑥2−𝑎 𝑞 , 𝑥2 = 𝑥3 − 𝑟𝑥1, 𝑥2 = 𝑥3 − 𝑟𝑥1 and 𝑥3 = 𝑏−𝑥2 𝑞 . solving these three equations in three variables 𝑥1,𝑥2,𝑥3 we get 𝑥1 = 𝑏−𝑎(𝑞+1) 𝑞2+𝑞+𝑟𝑞 , 𝑥2 = 𝑏𝑞+𝑟𝑞𝑎 𝑞2+𝑞+𝑟𝑞 ,𝑥3 = 𝑏𝑞+𝑏𝑟−𝑟𝑎 𝑞2+𝑞+𝑟𝑞 . if we continue extending the above finite sequence one more time, then we get the sequence 𝑎,𝑥1,𝑥2,𝑥3,𝑥4,𝑏. using the similar approach as above, we obtain 𝑥1 = 𝑏−𝑎(1+𝑟+𝑞) 𝑞2+𝑟2+2𝑟𝑞+𝑞 ,𝑥2 = 𝑏𝑞+𝑎𝑟2+𝑎𝑟𝑞 𝑞2+𝑟2+2𝑟𝑞+𝑞 ,𝑥3 = 𝑏𝑞+𝑟𝑏−𝑎𝑟 𝑞2+𝑟2+2𝑟𝑞+𝑞 ,𝑥4 = 𝑏𝑞2+𝑟𝑏𝑞+𝑞𝑏+𝑎𝑟2 𝑞2+𝑟2+2𝑟𝑞+𝑞 . we mention these results in the table 8. from this table, we observe that there is a similar pattern for the first inserted term in case of any number of inserted terms. we thus generalize it for the case of any number of inserted terms and when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given two positive integers 𝑎 and 𝑏, we can now write the general formula for 𝑥1 as 𝑥1 = 𝑏−𝑎{[( 1−𝑟 𝑞 )𝐹 𝑘−1 (1,𝑞,𝑟,1) +𝐹 𝑘 (1,𝑞,𝑟,1) ] 𝜒(𝑘) [ 1 𝑞 𝐹 𝑘 (1,𝑞,𝑟,1) ] 1−𝜒(𝑘) } 𝐹 𝑘+1 (1,𝑞,𝑟,1) . (5.2) 18 insertion of terms satisfying the recurrence relations of horadam sequence and bifurcating fibonacci sequences number of inserted term formula 𝑥1 𝑥2 𝑥3 𝑥4 1 𝑏 − 𝑎 𝑞 ------ 2 𝑏 − 𝑎 𝑟 + 𝑞 𝑞𝑏 + 𝑟𝑎 𝑟 + 𝑞 ---- 3 𝑏 − 𝑎(𝑞 + 1) 𝑞2 + 𝑞 + 𝑟𝑞 𝑏𝑞 + 𝑟𝑞𝑎 𝑞2 + 𝑞 + 𝑟𝑞 𝑏𝑞 + 𝑏𝑟 − 𝑟𝑎 𝑞2 + 𝑞 + 𝑟𝑞 -- 4 𝑏 − 𝑎(1 + 𝑟 + 𝑞) 𝑞2 + 𝑟2 + 2𝑟𝑞 + 𝑞 𝑏𝑞 + 𝑎𝑟2 + 𝑎𝑟𝑞 𝑞2 + 𝑟2 + 2𝑟𝑞 + 𝑞 𝑏𝑞 + 𝑟𝑏 − 𝑎𝑟 𝑞2 + 𝑟2 + 2𝑟𝑞 + 𝑞 𝑏𝑞2 + 𝑟𝑏𝑞 + 𝑞𝑏 + 𝑎𝑟2 𝑞2 + 𝑟2 + 2𝑟𝑞 + 𝑞 table 8 next, when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given two positive integers 𝑎 and 𝑏, we obtain the general formula for 𝑛th inserted term (1 ≤ 𝑛 ≤ 𝑘) using the recurrence relation (5.1). theorem 5.1 when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given integers 𝑎 and 𝑏, so that all 𝑥𝑖’s satisfy the recurrence relation of 𝐹𝑛 (1,𝑞,𝑟,1) , the general formula for any arbitrary term 𝑥𝑛 (1 ≤ 𝑛 ≤ 𝑘) is given by 𝑥𝑛 = { 𝐹𝑛 (1,𝑞,𝑟,1) 𝑏−𝐹𝑛 (1,𝑞,𝑟,1) 𝑎{[( 1−𝑟 𝑞 )𝐹 𝑘−1 (1,𝑞,𝑟,1) +𝐹 𝑘 (1,𝑞,𝑟,1) ] 𝜒(𝑘) [ 1 𝑞 𝐹 𝑘 (1,𝑞,𝑟,1) ] 1−𝜒(𝑘) } +𝑎{[( 1−𝑟 𝑞 )𝐹𝑛−2 (1,𝑞,𝑟,1) +𝐹𝑛−1 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛) [ 1 𝑞 𝐹𝑛−1 (1,𝑞,𝑟,1) ] 𝜒(𝑛) }𝐹 𝑘+1 (1,𝑞,𝑟,1) } 𝐹 𝑘+1 (1,𝑞,𝑟,1) . proof. we use the principle of mathematical induction to prove the result. by (5.2), we have 𝑥1 = 𝑏−𝑎{[( 1−𝑟 𝑞 )𝐹 𝑘−1 (1,𝑞,𝑟,1) +𝐹 𝑘 (1,𝑞,𝑟,1) ] 𝜒(𝑘) [ 1 𝑞 𝐹 𝑘 (1,𝑞,𝑟,1) ] 1−𝜒(𝑘) } 𝐹 𝑘+1 (1,𝑞,𝑟,1) , which proves the result for 𝑛 = 1. we next assume that it is true for all positive integers not exceeding 𝑛. then the following holds: 𝑥𝑛−1 = { 𝐹𝑛−1 (1,𝑞,𝑟,1) 𝑏−𝐹𝑛−1 (1,𝑞,𝑟,1) 𝑎{[( 1−𝑟 𝑞 )𝐹 𝑘−1 (1,𝑞,𝑟,1) +𝐹 𝑘 (1,𝑞,𝑟,1) ] 𝜒(𝑘) [ 1 𝑞 𝐹 𝑘 (1,𝑞,𝑟,1) ] 1−𝜒(𝑘) } +𝑎{[( 1−𝑟 𝑞 )𝐹𝑛−3 (1,𝑞,𝑟,1) +𝐹𝑛−2 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛−1) [ 1 𝑞 𝐹𝑛−2 (1,𝑞,𝑟,1) ] 𝜒(𝑛−1) }𝐹 𝑘+1 (1,𝑞,𝑟,1) } 𝐹 𝑘+1 (1,𝑞,𝑟,1) 𝑥𝑛−2 = { 𝐹𝑛−2 (1,𝑞,𝑟,1) 𝑏−𝐹𝑛−2 (1,𝑞,𝑟,1) 𝑎{[( 1−𝑟 𝑞 )𝐹 𝑘−1 (1,𝑞,𝑟,1) +𝐹 𝑘 (1,𝑞,𝑟,1) ] 𝜒(𝑘) [ 1 𝑞 𝐹 𝑘 (1,𝑞,𝑟,1) ] 1−𝜒(𝑘) } +𝑎{[( 1−𝑟 𝑞 )𝐹𝑛−4 (1,𝑞,𝑟,1) +𝐹𝑛−3 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛−2) [ 1 𝑞 𝐹𝑛−3 (1,𝑞,𝑟,1) ] 𝜒(𝑛−2) }𝐹 𝑘+1 (1,𝑞,𝑟,1) } 𝐹 𝑘+1 (1,𝑞,𝑟,1) 19 khushbu j. das and devbhadra v. shah now, 𝑞1−𝜒(𝑛)𝑥𝑛−1 + 𝑟 𝜒(𝑛)𝑥𝑛−2 = { (𝑞 1−𝜒(𝑛)𝐹𝑛−1 (1,𝑞,𝑟,1) +𝑟𝜒(𝑛)𝐹𝑛−2 (1,𝑞,𝑟,1) )𝑏−𝑎(𝑞1−𝜒(𝑛)𝐹𝑛−1 (1,𝑞,𝑟,1) +𝑟𝜒(𝑛)𝐹𝑛−2 (1,𝑞,𝑟,1) ) {[( 1−𝑟 𝑞 )𝐹 𝑘−1 (1,𝑞,𝑟,1) +𝐹 𝑘 (1,𝑞,𝑟,1) ] 𝜒(𝑘) [ 1 𝑞 𝐹 𝑘 (1,𝑞,𝑟,1) ] 1−𝜒(𝑘) } +𝑎 { 𝑞1−𝜒(𝑛)[( 1−𝑟 𝑞 )𝐹𝑛−3 (1,𝑞,𝑟,1) +𝐹𝑛−2 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛−1) [ 1 𝑞 𝐹𝑛−2 (1,𝑞,𝑟,1) ] 𝜒(𝑛−1) +𝑟𝜒(𝑛)[( 1−𝑟 𝑞 )𝐹𝑛−4 (1,𝑞,𝑟,1) +𝐹𝑛−3 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛−2) [ 1 𝑞 𝐹𝑛−3 (1,𝑞,𝑟,1) ] 𝜒(𝑛−2) } 𝐹 𝑘+1 (1,𝑞,𝑟,1) } 𝐹 𝑘+1 (1,𝑞,𝑟,1) = { 𝐹𝑛 (1,𝑞,𝑟,1) 𝑏−𝑎𝐹𝑛 (1,𝑞,𝑟,1) {[( 1−𝑟 𝑞 )𝐹𝑘−1 (1,𝑞,𝑟,1) +𝐹𝑘 (1,𝑞,𝑟,1) ] 𝜒(𝑘) [ 1 𝑞 𝐹𝑘 (1,𝑞,𝑟,1) ] 1−𝜒(𝑘) } +𝑎 { 𝑞1−𝜒(𝑛)[( 1−𝑟 𝑞 )𝐹𝑛−3 (1,𝑞,𝑟,1) +𝐹𝑛−2 (1,𝑞,𝑟,1) ] 𝜒(𝑛) [ 1 𝑞 𝐹𝑛−2 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛) +𝑟𝜒(𝑛)[( 1−𝑟 𝑞 )𝐹𝑛−4 (1,𝑞,𝑟,1) +𝐹𝑛−3 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛) [ 1 𝑞 𝐹𝑛−3 (1,𝑞,𝑟,1) ] 𝜒(𝑛) } 𝐹 𝑘+1 (1,𝑞,𝑟,1) } 𝐹 𝑘+1 (1,𝑞,𝑟,1) we calculate the value of third term of numerator separately. now, when 𝑛 is odd, we get 𝑞1−𝜒(𝑛) [( 1−𝑟 𝑞 )𝐹𝑛−3 (1,𝑞,𝑟,1) + 𝐹𝑛−2 (1,𝑞,𝑟,1) ] 𝜒(𝑛) [ 1 𝑞 𝐹𝑛−2 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛) +𝑟𝜒(𝑛) [( 1−𝑟 𝑞 )𝐹𝑛−4 (1,𝑞,𝑟,1) + 𝐹𝑛−3 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛) [ 1 𝑞 𝐹𝑛−3 (1,𝑞,𝑟,1) ] 𝜒(𝑛) = 𝑞[ 1 𝑞 𝐹𝑛−2 (1,𝑞,𝑟,1) ] + ( 1−𝑟 𝑞 )𝐹𝑛−4 (1,𝑞,𝑟,1) + 𝐹𝑛−3 (1,𝑞,𝑟,1) = 1 𝑞 [𝑞𝐹𝑛−2 (1,𝑞,𝑟,1) − 𝑟𝐹𝑛−4 (1,𝑞,𝑟,1) + (𝑞𝐹𝑛−3 (1,𝑞,𝑟,1) + 𝐹𝑛−4 (1,𝑞,𝑟,1) )] = 1 𝑞 [𝑞𝐹𝑛−2 (1,𝑞,𝑟,1) − 𝑟𝐹𝑛−4 (1,𝑞,𝑟,1) + 𝐹𝑛−2 (1,𝑞,𝑟,1) ] = 1 𝑞 [(1 + 𝑞)𝐹𝑛−2 (1,𝑞,𝑟,1) − 𝑟𝐹𝑛−4 (1,𝑞,𝑟,1) ] = 1 𝑞 [(1 − 𝑟)𝐹𝑛−2 (1,𝑞,𝑟,1) + 𝑞𝐹𝑛−2 (1,𝑞,𝑟,1) + 𝑟𝑞𝐹𝑛−3 (1,𝑞,𝑟,1) ] = ( 1−𝑟 𝑞 )𝐹𝑛−2 (1,𝑞,𝑟,1) + 𝐹𝑛−1 (1,𝑞,𝑟,1) = [( 1−𝑟 𝑞 )𝐹𝑛−2 (1,𝑞,𝑟,1) + 𝐹𝑛−1 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛) . also, when 𝑛 is even, we get 𝑞1−𝜒(𝑛) [( 1 − 𝑟 𝑞 )𝐹𝑛−3 (1,𝑞,𝑟,1) + 𝐹𝑛−2 (1,𝑞,𝑟,1) ] 𝜒(𝑛) [ 1 𝑞 𝐹𝑛−2 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛) +𝑟𝜒(𝑛) [( 1 − 𝑟 𝑞 )𝐹𝑛−4 (1,𝑞,𝑟,1) + 𝐹𝑛−3 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛) [ 1 𝑞 𝐹𝑛−3 (1,𝑞,𝑟,1) ] 𝜒(𝑛) = [( 1 − 𝑟 𝑞 )𝐹𝑛−3 (1,𝑞,𝑟,1) + 𝐹𝑛−2 (1,𝑞,𝑟,1) ] + 𝑟[ 1 𝑞 𝐹𝑛−3 (1,𝑞,𝑟,1) ] = 1 𝑞 [𝑞𝐹𝑛−2 (1,𝑞,𝑟,1) + 𝐹𝑛−3 (1,𝑞,𝑟,1) ] = 1 𝑞 [𝐹𝑛−1 (1,𝑞,𝑟,1) ] = [ 1 𝑞 𝐹𝑛−1 (1,𝑞,𝑟,1) ] 𝜒(𝑛) 20 insertion of terms satisfying the recurrence relations of horadam sequence and bifurcating fibonacci sequences therefore, we have 𝑥𝑛 = 𝑞 1−𝜒(𝑛)𝑥𝑛−1 + 𝑟 𝜒(𝑛)𝑥𝑛−2 = { 𝐹𝑛 (1,𝑞,𝑟,1) 𝑏−𝐹𝑛 (1,𝑞,𝑟,1) 𝑎{[( 1−𝑟 𝑞 )𝐹 𝑘−1 (1,𝑞,𝑟,1) +𝐹 𝑘 (1,𝑞,𝑟,1) ] 𝜒(𝑘) [ 1 𝑞 𝐹 𝑘 (1,𝑞,𝑟,1) ] 1−𝜒(𝑘) } +𝑎{[( 1−𝑟 𝑞 )𝐹𝑛−2 (1,𝑞,𝑟,1) +𝐹𝑛−1 (1,𝑞,𝑟,1) ] 1−𝜒(𝑛) [ 1 𝑞 𝐹𝑛−1 (1,𝑞,𝑟,1) ] 𝜒(𝑛) }𝐹 𝑘+1 (1,𝑞,𝑟,1) } 𝐹 𝑘+1 (1,𝑞,𝑟,1) , which proves the result for every positive integer 𝑛. 6 insertion of terms satisfying the recurrence relation of bifurcating sequence 𝑭𝒏 (𝟏,𝟏,𝒓,𝒔) if we consider 𝑝 = 𝑞 = 1 in (1.1), then we get the sequence {𝐹𝑛 (1,1,𝑟,𝑠) } whose terms are defined by the recurrence relation 𝐹𝑛 (1,1,𝑟,𝑠) = 𝐹𝑛−1 (1,1,𝑟,𝑠) + 𝑟𝜒(𝑛)𝑠1−𝜒(𝑛)𝐹𝑛−2 (1,1,𝑟,𝑠) , where 𝐹0 (1,1,𝑟,𝑠) = 0,𝐹1 (1,1,𝑟,𝑠) = 1.this can be written in the form 𝐹𝑛 (1,1,𝑟,𝑠) = { 𝐹𝑛−1 (1,1,𝑟,𝑠) + 𝑟𝐹𝑛−2 (1,1,𝑟,𝑠) ;when n is odd 𝐹𝑛−1 (1,1,𝑟,𝑠) + 𝑠𝐹𝑛−2 (1,1,𝑟,𝑠) ;when n is even (𝑛 ≥ 2) (6.1) with the initial conditions 𝐹0 (1,1,𝑟,𝑠) = 0,𝐹1 (1,1,𝑟,𝑠) = 1. first few terms of this sequence are shown in table 9. n 𝐹𝑛 (1,1,𝑟,𝑠) 0 0 1 1 2 1 3 1 + 𝑟 4 1 + 𝑟 + 𝑠 5 1 + 2𝑟 + 𝑠 + 𝑟2 6 1 + 𝑟𝑠 + 2𝑟 + 2𝑠 + 𝑟2 + 𝑠2 table 9 diwan, shah [3] derived the binet-type explicit formula for the terms of this sequence as 21 khushbu j. das and devbhadra v. shah 𝐹𝑛 (1,1,𝑟,𝑠) = (𝛼−𝑠)𝜒(𝑛)𝛼 [ 𝑛 2 ] −(𝛽−𝑠)𝜒(𝑛)𝛽 [ 𝑛 2 ] 𝛼−𝛽 , where 𝛼 = (𝑟+𝑠+1)+√(𝑟+𝑠+1)2−4𝑟𝑠 2 ,𝛽 = (𝑟+𝑠+1)−√(𝑟+𝑠+1)2−4𝑟𝑠 2 with 𝛼 − 𝛽 = √(𝑟 + 𝑠 + 1)2 − 4𝑟𝑠, 𝛼 + 𝛽 = 𝑟 + 𝑠 + 1 and 𝛼𝛽 = 𝑟𝑠. in this section, we find the formula for the inserted terms between the numbers 𝑎 and 𝑏 so that terms of the sequence 𝑎,𝑥1,𝑥2,𝑥3,…,𝑥𝑘,𝑏 satisfies the recurrence relation (6.1). when we consider the sequence 𝑎,𝑥1,𝑏, where 𝑥1 is the only inserted term between 𝑎 and 𝑏. then using 𝐹2 (1,1,𝑟,𝑠) = 𝐹1 (1,1,𝑟,𝑠) + 𝑠𝐹0 (1,1,𝑟,𝑠) , we get 𝑏 = 𝑥1 + 𝑠𝑎. thus 𝑥1 = 𝑏 − 𝑠𝑎. when we consider the finite sequence as 𝑎,𝑥1,𝑥2,𝑏, so that it satisfies (6.1), we observe that 𝑥1 = 𝑥2 − 𝑠𝑎 and 𝑥2 = 𝑏 − 𝑟𝑥1. this gives 𝑥2 = 𝑏+𝑟𝑠𝑎 1+𝑟 .this also gives 𝑥1 = 𝑏−𝑠𝑎 1+𝑟 . further, considering the sequence𝑎,𝑥1,𝑥2,𝑥3,𝑏, it is now easy to observe that 𝑥1 = 𝑥2 − 𝑠𝑎, 𝑥2 = 𝑥3 − 𝑟𝑥1, 𝑥3 = 𝑏 − 𝑠𝑥2. solving these three equations in three variables 𝑥1,𝑥2,𝑥3 we get 𝑥1 = 𝑏−𝑎𝑠(1+𝑠) 1+𝑟+𝑠 , 𝑥2 = 𝑏−𝑟𝑠𝑎 1+𝑟+𝑠 , 𝑥3 = 𝑏+𝑏𝑟+𝑎𝑟𝑠2 1+𝑟+𝑠 . if we continue extending the above finite sequence one more time, then we get the sequence 𝑎,𝑥1,𝑥2,𝑥3,𝑥4,𝑏. using the similar approach as above, we obtain 𝑥1 = 𝑏−𝑠𝑎(1+𝑟+𝑠) 1+2𝑟+𝑠+𝑟2 ,𝑥2 = 𝑏+𝑠𝑟𝑎+𝑠𝑎𝑟2 1+2𝑟+𝑠+𝑟2 ,𝑥3 = 𝑏+𝑏𝑟−𝑠2𝑟𝑎 1+2𝑟+𝑠+𝑟2 and 𝑥4 = 𝑏+𝑏𝑟+𝑏𝑠+𝑟2𝑠2𝑎 1+2𝑟+𝑠+𝑟2 . we mention these results in the table 10. number of inserted term formula 𝑥1 𝑥2 𝑥3 𝑥4 1 𝑏 − 𝑠𝑎 ------ 2 𝑏 − 𝑠𝑎 1 + 𝑟 𝑏 + 𝑟𝑠𝑎 1 + 𝑟 ---- 3 𝑏 − 𝑠𝑎(1 + 𝑠) 1 + 𝑟 + 𝑠 𝑏 − 𝑟𝑠𝑎 1 + 𝑟 + 𝑠 𝑏 + 𝑏𝑟 + 𝑎𝑟𝑠2 1 + 𝑟 + 𝑠 -- 4 𝑏 − 𝑠𝑎(1 + 𝑟 + 𝑠) 1 + 2𝑟 + 𝑠 + 𝑟2 𝑏 + 𝑠𝑟𝑎 + 𝑠𝑎𝑟2 1 + 2𝑟 + 𝑠 + 𝑟2 𝑏 + 𝑏𝑟 − 𝑠2𝑟𝑎 1 + 2𝑟 + 𝑠 + 𝑟2 𝑏 + 𝑏𝑟 + 𝑏𝑠 + 𝑟2𝑠2𝑎 1 + 2𝑟 + 𝑠 + 𝑟2 table 10 from this table, we observe that there is a similar pattern for the first inserted term in case of any number of inserted terms. we thus generalize it for 22 insertion of terms satisfying the recurrence relations of horadam sequence and bifurcating fibonacci sequences the case of any number of inserted terms and when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given two positive integers 𝑎 and 𝑏, we can now write the general formula for 𝑥1 as 𝑥1 = 𝑏−𝑠𝑎{(𝐹𝑘 (1,1,𝑟,𝑠) ) 1−𝜒(𝑘) (∑ 𝑠 [ 𝑘−1 2 ]−𝑚[ 𝑘−1 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑘−1 2 ] ) 𝜒(𝑘) } 𝐹 𝑘+1 (1,1,𝑟,𝑠) , (6.2) next, when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given two positive integers 𝑎 and 𝑏, we obtain the general formula for 𝑛th inserted term (1 ≤ 𝑛 ≤ 𝑘) using the recurrence relation (6.1). theorem 6.1 when the terms 𝑥1,𝑥2,𝑥3,…,𝑥𝑘 are inserted between given integers 𝑎 and 𝑏, so that all 𝑥𝑖’s satisfy the recurrence relation of 𝐹𝑛 (1,1,𝑟,𝑠) , the general formula for any arbitrary term 𝑥𝑛 (1 ≤ 𝑛 ≤ 𝑘) is given by 𝑥𝑛 = { 𝐹𝑛 (1,1,𝑟,𝑠) 𝑏−𝐹𝑛 (1,1,𝑟,𝑠) 𝑠𝑎{(𝐹𝑘 (1,1,𝑟,𝑠) ) 1−𝜒(𝑘) (∑ 𝑠 [ 𝑘−1 2 ]−𝑚[ 𝑘−1 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑘−1 2 ] ) 𝜒(𝑘) } +𝑠𝑎𝐹𝑘+1 (1,1,𝑟,𝑠) (𝐹𝑛−1 (1,1,𝑟,𝑠) ) 𝜒(𝑛) (∑ 𝑠 [ 𝑛−2 2 ]−𝑚[ 𝑛−2 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑛−2 2 ] ) 1−𝜒(𝑛) } 𝐹 𝑘+1 (1,1,𝑟,𝑠) . proof. we use the principle of mathematical induction to prove the result. since by (6.2), we have 𝑥1 = 𝑏−𝑠𝑎{(𝐹𝑘 (1,1,𝑟,𝑠) ) 1−𝜒(𝑘) (∑ 𝑠 [ 𝑘−1 2 ]−𝑚[ 𝑘−1 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑘−1 2 ] ) 𝜒(𝑘) } 𝐹 𝑘+1 (1,1,𝑟,𝑠) , which proves the result for 𝑛 = 1. we next assume that it is true for all positive integers not exceeding 𝑛. then the following holds: 𝑥𝑛−1 = { 𝐹𝑛−1 (1,1,𝑟,𝑠) 𝑏−𝐹𝑛−1 (1,1,𝑟,𝑠) 𝑠𝑎{(𝐹𝑘 (1,1,𝑟,𝑠) ) 1−𝜒(𝑘) (∑ 𝑠 [ 𝑘−1 2 ]−𝑚[ 𝑘−1 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑘−1 2 ] ) 𝜒(𝑘) } +𝑠𝑎𝐹𝑘+1 (1,1,𝑟,𝑠) (𝐹𝑛−2 (1,1,𝑟,𝑠) ) 𝜒(𝑛−1) (∑ 𝑠 [ 𝑛−3 2 ]−𝑚[ 𝑛−3 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑛−3 2 ] ) 1−𝜒(𝑛−1) } 𝐹 𝑘+1 (1,1,𝑟,𝑠) , 23 khushbu j. das and devbhadra v. shah 𝑥𝑛−2 = { 𝐹𝑛−2 (1,1,𝑟,𝑠) 𝑏−𝐹𝑛−2 (1,1,𝑟,𝑠) 𝑠𝑎{(𝐹𝑘 (1,1,𝑟,𝑠) ) 1−𝜒(𝑘) (∑ 𝑠 [ 𝑘−1 2 ]−𝑚[ 𝑘−1 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑘−1 2 ] ) 𝜒(𝑘) } +𝑠𝑎𝐹𝑘+1 (1,1,𝑟,𝑠) (𝐹𝑛−3 (1,1,𝑟,𝑠) ) 𝜒(𝑛−2) (∑ 𝑠 [ 𝑛−3 2 ]−𝑚[ 𝑛−3 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑛−3 2 ] ) 1−𝜒(𝑛−2) } 𝐹 𝑘+1 𝑅(1,1,𝑟,𝑠) . now, 𝑥𝑛−1 + 𝑟 𝜒(𝑛)𝑠1−𝜒(𝑛)𝑥𝑛−2 = { (𝐹𝑛−1 (1,1,𝑟,𝑠) +𝑟𝜒(𝑛)𝑠1−𝜒(𝑛)𝐹𝑛−2 (1,1,𝑟,𝑠) )𝑏−(𝐹𝑛−1 (1,1,𝑟,𝑠) +𝑟𝜒(𝑛)𝑠1−𝜒(𝑛)𝐹𝑛−2 (1,1,𝑟,𝑠) )𝑠𝑎 ×{(𝐹𝑘 (1,1,𝑟,𝑠) ) 1−𝜒(𝑘) (∑ 𝑠 [ 𝑘−1 2 ]−𝑚[ 𝑘−1 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑘−1 2 ] ) 𝜒(𝑘) } +𝑠𝑎𝐹𝑘+1 𝑅(1,1,𝑟,𝑠) { (𝐹𝑛−2 (1,1,𝑟,𝑠) ) 𝜒(𝑛−1) (∑ 𝑠 [ 𝑛−3 2 ]−𝑚[ 𝑛−3 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑛−3 2 ] ) 1−𝜒(𝑛−1) +𝑟𝜒(𝑛)𝑠1−𝜒(𝑛)(𝐹𝑛−3 (1,1,𝑟,𝑠) ) 𝜒(𝑛−2) ×(∑ 𝑠 [ 𝑛−4 2 ]−𝑚[ 𝑛−4 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑛−4 2 ] ) 1−𝜒(𝑛−2) } } 𝐹 𝑘+1 (1,1,𝑟,𝑠) = { 𝐹𝑛 (1,1,𝑟,𝑠) 𝑏−𝐹𝑛 (1,1,𝑟,𝑠) 𝑠𝑎×{(𝐹𝑘 (1,1,𝑟,𝑠) ) 1−𝜒(𝑘) (∑ 𝑠 [ 𝑘−1 2 ]−𝑚[ 𝑘−1 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑘−1 2 ] ) 𝜒(𝑘) } +𝑠𝑎𝐹𝑘+1 (1,1,𝑟,𝑠) { (𝐹𝑛−2 (1,1,𝑟,𝑠) ) 1−𝜒(𝑛) (∑ 𝑠 [ 𝑛−3 2 ]−𝑚[ 𝑛−3 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑛−3 2 ] ) 𝜒(𝑛) +𝑟𝜒(𝑛)𝑠1−𝜒(𝑛)(𝐹𝑛−3 (1,1,𝑟,𝑠) ) 𝜒(𝑛) (∑ 𝑠 [ 𝑛−4 2 ]−𝑚[ 𝑛−4 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑛−4 2 ] ) 1−𝜒(𝑛) } } 𝐹 𝑘+1 (1,1,𝑟,𝑠) we calculate the value of third term of numerator separately. now, when 𝑛 is odd, we get (𝐹𝑛−2 (1,1,𝑟,𝑠) ) 1−𝜒(𝑛) (∑ 𝑠 [ 𝑛−3 2 ]−𝑚[ 𝑛−3 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−3 2 ] ) 𝜒(𝑛) +𝑟𝜒(𝑛)𝑠1−𝜒(𝑛)(𝐹𝑛−3 (1,1,𝑟,𝑠) ) 𝜒(𝑛) (∑ 𝑠 [ 𝑛−4 2 ]−𝑚[ 𝑛−4 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−4 2 ] ) 1−𝜒(𝑛) = 𝐹𝑛−2 (1,1,𝑟,𝑠) + 𝑠(∑ 𝑠 [ 𝑛−4 2 ]−𝑚[ 𝑛−4 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−4 2 ] ) = 𝐹𝑛−2 (1,1,𝑟,𝑠) + 𝑠( 𝑠 [ 𝑛−4 2 ] 𝐹0 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−6 2 ] 𝐹1 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−8 2 ] 𝐹2 (1,1,𝑟,𝑠) +⋯+ 𝐹𝑛−4 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−4 2 ] ) 24 insertion of terms satisfying the recurrence relations of horadam sequence and bifurcating fibonacci sequences = 𝐹𝑛−2 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−2 2 ] 𝐹0 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−4 2 ] 𝐹1 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−6 2 ] 𝐹2 (1,1,𝑟,𝑠) +⋯+ 𝑠𝐹𝑛−4 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−2 2 ] = ∑ 𝑠 [ 𝑛−2 2 ]−𝑚[ 𝑛−2 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−2 2 ] = (∑ 𝑠 [ 𝑛−2 2 ]−𝑚[ 𝑛−2 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−2 2 ] ) 1−𝜒(𝑛) . also, when 𝑛 is even, we get (𝐹𝑛−2 (1,1,𝑟,𝑠) ) 1−𝜒(𝑛) (∑ 𝑠 [ 𝑛−3 2 ]−𝑚[ 𝑛−3 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−3 2 ] ) 𝜒(𝑛) +𝑟𝜒(𝑛)𝑠1−𝜒(𝑛)(𝐹𝑛−3 (1,1,𝑟,𝑠) ) 𝜒(𝑛) (∑ 𝑠 [ 𝑛−4 2 ]−𝑚[ 𝑛−4 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−4 2 ] ) 1−𝜒(𝑛) = ∑ 𝑠 [ 𝑛−3 2 ]−𝑚[ 𝑛−3 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−3 2 ] + 𝑟𝐹𝑛−3 (1,1,𝑟,𝑠) . to prove the main result, we need to prove that this value should be 𝐹𝑛−1 (1,1,𝑟,𝑠) . clearly this result holds for 𝑛 = 3, since 𝐹2 (1,1,𝑟,𝑠) = 1. assume that this result holds for all positive integers not exceeding 𝑛. now, 𝐹𝑛−2 (1,1,𝑟,𝑠) = ∑ 𝑠 [ 𝑛−4 2 ]−𝑚[ 𝑛−4 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−4 2 ] + 𝑟𝐹𝑛−4 (1,1,𝑟,𝑠) and 𝐹𝑛−3 (1,1,𝑟,𝑠) = ∑ 𝑠 [ 𝑛−5 2 ]−𝑚[ 𝑛−5 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−5 2 ] + 𝑟𝐹𝑛−5 (1,1,𝑟,𝑠) . now, 𝐹𝑛−2 (1,1,𝑟,𝑠) + 𝑠𝐹𝑛−3 (1,1,𝑟,𝑠) = ∑ 𝑠 [ 𝑛−4 2 ]−𝑚[ 𝑛−4 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−4 2 ] + 𝑟𝐹𝑛−4 (1,1,𝑟,𝑠) +𝑠(∑ 𝑠 [ 𝑛−5 2 ]−𝑚[ 𝑛−5 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−5 2 ] + 𝑟𝐹𝑛−5 (1,1,𝑟,𝑠) ) = ∑ 𝑠 [ 𝑛−4 2 ]−𝑚[ 𝑛−4 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−4 2 ] + 𝑟𝐹𝑛−4 (1,1,𝑟,𝑠) +∑ 𝑠 [ 𝑛−3 2 ]−𝑚[ 𝑛−5 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−3 2 ] + 𝑟𝑠𝐹𝑛−5 (1,1,𝑟,𝑠) = (∑ 𝑠 [ 𝑛−4 2 ]−𝑚[ 𝑛−4 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−4 2 ] + ∑ 𝑠 [ 𝑛−3 2 ]−𝑚[ 𝑛−5 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−3 2 ] ) +𝑟(𝐹𝑛−4 (1,1,𝑟,𝑠) + 𝑠𝐹𝑛−5 (1,1,𝑟,𝑠) ) = ∑ 𝑠 [ 𝑛−3 2 ]−𝑚[ 𝑛−3 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) + 𝑠 [ 𝑛−3 2 ] + 𝑟𝐹𝑛−3 (1,1,𝑟,𝑠) = 𝐹𝑛−1 (1,1,𝑟,𝑠) = (𝐹𝑛−1 (1,1,𝑟,𝑠) ) 𝜒(𝑛) . . therefore, we have 25 khushbu j. das and devbhadra v. shah 𝑥𝑛 = 𝑥𝑛−1 + 𝑟 𝜒(𝑛)𝑠1−𝜒(𝑛)𝑥𝑛−2 = { 𝐹𝑛 (1,1,𝑟,𝑠) 𝑏−𝐹𝑛 (1,1,𝑟,𝑠) 𝑠𝑎{(𝐹𝑘 (1,1,𝑟,𝑠) ) 1−𝜒(𝑘) (∑ 𝑠 [ 𝑘−1 2 ]−𝑚[ 𝑘−1 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑘−1 2 ] ) 𝜒(𝑘) } +𝑠𝑎𝐹𝑘+1 (1,1,𝑟,𝑠) (𝐹𝑛−1 (1,1,𝑟,𝑠) ) 𝜒(𝑛) (∑ 𝑠 [ 𝑛−2 2 ]−𝑚[ 𝑛−2 2 ] 𝑚=0 𝐹2𝑚 (1,1,𝑟,𝑠) +𝑠 [ 𝑛−2 2 ] ) 1−𝜒(𝑛) } 𝐹 𝑘+1 (1,1,𝑟,𝑠) , which proves the result for every positive integer 𝑛. 7 conclusion in this paper we derived the general formula which gives the value of any inserted term 𝑥𝑛 (1 ≤ 𝑛 ≤ 𝑘) between the given fixed positive integers 𝑎,𝑏 so that terms of the sequence 𝑎,𝑥1,𝑥2,𝑥3,…,𝑥𝑘,𝑏 satisfies the recurrence relation of 𝐹𝑛(𝑝,𝑞) and various bifurcating subsequence of {𝐹𝑛 (𝑝,𝑞,𝑟,𝑠) } by considering some fixed values of 𝑝,𝑞,𝑟,𝑠. references [1] agnes m., buenaventura n., labao j. j., soria c. k., limbaco k. a., and l. natividad, inclusion of inserted terms (fibonacci mean) in a fibonacci sequence, math investigatory project, central luzon state university, 2010. [2] bilgici g., new generalizations of fibonacci and lucas sequences, applied mathematical sciences, vol. 8, no. 29, 2014, 1429 – 1437. [3] diwan d. m, shah d. v., extended binet’s formula for the class of generalized fibonacci sequences, proceeding 19th annual cum 4th international conference of gwalior academy of mathematical sciences (gams), svnit, surat, oct 3-6, 2014, 109 – 113. [4] diwan d. m, shah d. v., extended binet’s formula for the class of generalized fibonacci sequences, vnsgu journal of science and technology, vol. 4, no. 1, 2015, 205 – 210. [5] diwan d. m, shah d. v., explicit and recursive formulae for the class of generalized fibonacci sequence, international journal of advanced research in engineering, science and management, vol. 1, issue 10, july 2015, 1 – 6. [6] edson m., yayenie o., a new generalization of fibonacci sequence and extended binet’s formula, integers, vol. 9, 2009, 639 – 654. [7] horadam a. f., basic properties of certain generalized sequence of numbers, fibonacci quarterly, 3, 1965, 161-176. 26 insertion of terms satisfying the recurrence relations of horadam sequence and bifurcating fibonacci sequences [8] howell p., nth term of the fibonacci sequence, from math proofs: interesting mathematical results and elegant solutions to various problems, http://mathproofs.blogspot.com/2005/04/nth-term-offibonacci-sequence. [9] verma, bala a., on properties of generalized bi-variate bi-periodic fibonacci polynomials, international journal of advanced science and technology,29(3), 2020, 8065–8072. [10] yayenie o., a note on generalized fibonacci sequence, applied mathematics and computation, 217, 2011, 5603–5611. 27 ratio mathematica volume 42, 2022 a deep learning model for classifying the hate and offensive language in social media text nidhi bhandari1 rachna navlakhe2 g.l. prajapati3 abstract social media is a big data source for analyzing content. it requires a significant amount of computational cost. recently introduced model for identifying and removing toxic content from twitter, using an information retrieval (ir) model is working accurately but with small amount of data. therefore, in this paper, a deep learning technique is used to process large-scale social media text data. first, it uses natural language processing (nlp) based feature extraction to create four different sets of training samples i.e. tf-idf-based features, pos tagged features, a reduced feature vector of pos and the combined vector of tf-idf and pos tagged features. the deep convolutional neural networks (cnn) is used to train the model and to classify hate and offensive language. the dataset has been obtained from kaggle. the performance in terms of training accuracy, validation accuracy, training loss and validation loss has been measured with the time complexity. in addition, the class-wise precision, recall, f1-score and mean accuracy have also been investigated. from experimental results, we found tf-idf and pos-based combined features provide 82% and 83% of accuracy, respectively. keywords: text mining, social media, semantic knowledge, sentiment analysis, deep learning, hate and offensive language.* 1department of applied mathematics, iet, davv, indore. nidhi.bhandari1@gmail.com. 2department of applied mathematics & compu. sci., sgsits, indore, sgsits.rachna@gmail.com. 3department. of comp. engi., iet, davv, indore, glprajapati1@gmail.com. *received on january 21st, 2022. accepted on june 20th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v39i0.705. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 73 nidhi bhandari, rachna navlakhe and g.l.prajapati 1. introduction information retrieval (ir) is a technique used to locate precise information and supports different data formats, i.e. text, image, video, etc. among these data formats, the text has a significant contribution. the text ir model uses text mining techniques. in ir techniques, text mining or data mining algorithms are employed to recover user query relevance information [1].the ir model contains three key components: (1) user query (2) query processing (3) generation of outcomes[2]. however, the deficiency of these components can impact the performance of ir model such as lack of specific user query keywords, inappropriate keyword selection, lack of similar data, ranking of results etc. [3]. our study is initiated with an ir model named soir (semantic query optimizationbased information retrieval). this ir model incorporates query optimization and an fcm clustering technique. the results indicate the performance of the soir is better than previous models [4]. however, there are various applications of the ir systems. beyond these applications, the ir model can also be applied for pattern recognition. in this context, the proposed work is extended in order to be used with the social media toxic content filtering. this paper is an extension of the soir model, which will be used for classifying the toxic contents from the social media posts. this model is a promising technique for handling various negative tweets from social media by using lexical as well as semantic pattern analysis. in this paper, we are going to evaluate the soir and dgm based toxic content classification system on a large dataset. additionally, we do an investigation of a deep learning technique for classifying the different nlp based text features. we organize the contents in the following manner: 1. background: this section discusses the previously introduced soir and directed graph model (dgm) for text ir and toxic content classification. 2. proposed work: this section provides the understanding of the proposed deep learning model for classifying hate speech, offensive language and normal text. 3. results analysis: this section contains the experimental analysis of the proposed model and the different parameters have been provided. 4. conclusion: in this section, finally the work is concluded and future extension of work has been suggested. 2. background this section provides insight into soir model and previously developed dgm classification model. a. soir model this model is aimed to improve the ir model for running time and relevancy. due to the large amount of data, a significant amount of time is required to locate the precise information. in the documents database, a number of documents are present which do not belong to the similar category and content. this nature of database increases search 74 a deep learning model for classifying the hate and offensive language in social media text space. therefore, some improvements have been made to enhance the user query representation and subjective data categorization. the query optimization involves a semantic model to recognize similar words to optimize the query. first, we pre-process the data to improve the quality of data and exclusion of noise by removing stop words and special characters. next, feature selection techniques are used to reduce data dimensions and speed up the search process. thus, the tf-idf is used [5], to compute the weight w for identifying important tokens. the fixed size of the feature i.e. 30 tokens is considered. further, fcm clustering is being used to categorize feature vectors [6]. the clustering results are organized as a list of features: 𝐹 =< 𝐹𝑛 , 𝑘1,2,…𝑛, 𝐶 > where, f is the feature set, 𝐹𝑛 is the file name or index, 𝑘1,2,…𝑛 is the list of keywords, and c is the class name or subject. the training feature vector f is stored in a database. the categorized features are helpful for efficient data retrieval. on the other hand, the user query is transformed into a vector q as a set of keywords: 𝑄 = {𝑞1, 𝑞2, … , 𝑞𝑘 } in order to optimize the query, we initialize a set of queries by using synonyms. additionally, a map is prepared that contains the keywords and the synonyms. in this algorithm, a single keyword is twisted multiple times to generate new queries using similar words. the different search query increases the chances of finding accurate data. the search process is developed on the basis of the k-nn (k-nearest neighbour) algorithm [7]. the distance between the query and data less than 0.25 is counted as the result. the developed soir model has been compared with the cosine similarity-based and k-nn-based ir models. the precision, recall and f-score are calculated and visualized in figure 1. figure 1. mean performance of soir model 75 nidhi bhandari, rachna navlakhe and g.l.prajapati in this diagram, x-axis shows the measured matrix, and the y-axis shows the precision, recall and f-score. according to the results, we found that the technique improves the precision, recall and f-score as we utilizing a large size of learning data. b. soir for toxic content classification in order to keep clean social media, we need an accurate model to identify malicious and toxic posts. we used an soir model for this task [4]. we summarized the negative emotions identified in table 1 below, which consists of emotion classes and the relevant flow of emotions [8]. there are five main classes, and each class consists of its own subclasses. thus, we need a multiclass classification system. so, we solve this problem by using the soir-based model. table 1. emotion classes emotions content distressed sad, disappointed, guilty, missed surprised surprised fearful panic, frightened, shy angry angry disgusted dissatisfied, annoyed, doubtful, hateful in order to train the model, we have collected more than 3329 tweets and categorized them. then we considered 2002 tweets for the experiment. the pre-processing has been applied for removing tags, special characters, and stop words. further, part of speech tagging (pos) is used to understand the lexical structure [9]. the pos tags and sentiment classes are used to prepare a new dataset. on the other hand, tweets are also processed using the tf-idf weights. the top 20 weighted tokens are picked and used. in the next process, we prepared thresholds using the tweets' pos tags. in the first step, we compute the mean feature of each feature group using: 𝑀𝐹 = 1 𝑁 ∑ 𝐹𝑖 𝑁 𝑖=1 where 𝑀𝐹 is the mean of the feature, n is the total number of samples in the groups. after measuring the mean value, we compute the distance from each point as the limit. 𝐿𝐹 = 1 𝑁 ∑ |𝐹𝑖 − 𝑀𝐹 | 𝑁 𝑖=1 thus the threshold of the feature can be defined as: 𝑇𝐹 𝐺𝑟𝑜𝑢𝑝 = 𝑀𝐹 ± 𝐿𝐹 76 a deep learning model for classifying the hate and offensive language in social media text using this equation, we calculate the feature map fm for all the pos features. after that, we append features of the tf-idf. here the fcm clustering is used for creating the dictionary learning. we are just using the membership function of fcm for preparing the dictionary. the membership between data instance 𝑖 and centroid j is measured using: 𝜇𝑖,𝑗 = 1 ∑ ( 𝑑𝑖,𝑗 𝑑𝑖,𝑘 ) 2 𝑚−1 𝑐 𝑘=1 according to this process, the emotion-labeled data is being used from each group. a tweet is tokenized and then inserted into the dictionary d with the tf-idf weight. if a token exists in the dictionary, then we update the weight of the token and if the token is not available, then we simply insert the token. in order to compute the updated weight, we used: 𝑁𝑒𝑤𝑊𝑒𝑖𝑔ℎ𝑡 = 0.5 ∗ 𝑂𝑙𝑑𝑤𝑒𝑖𝑔ℎ𝑡 + 0.5 ∗ 𝜇𝑖,𝑗 where, 𝜇𝑖,𝑗 is the membership between previous and new weight for the token. the dictionaries are preserved in a database. now we classify new tweets. for the classification, 50% of training samples and 50% of new tweets from tweeter have been used. then, we perform the pos tagging of test data. the tagged feature vector 𝑇𝑣 is used to compare with the thresholds. for both pattern matching and decision making here the mean value 𝑀 and limit 𝐿 are used for computing upper threshold 𝐿𝑚𝑎𝑥 and lower threshold 𝐿𝑚𝑖𝑛.the patterns between these two limits are used to compute the distance among queried tweets. pos tag feature and all the features threshold are used from feature map fm. finally the higher matched value based class label is predicted as decision. after lexical pattern based decision making, we use the semantics for categorizing a tweet as final class label. thus, the tweets are tokenized to regenerate it, as in soir. let us have a tweet such that: 𝑇𝑊 = {𝑘1, 𝑘2, … , 𝑘𝑛} after regeneration, we get: 𝑇𝑊𝑚,𝑛 = { 𝑘1,1, 𝑘1,𝑠, … … … … . , 𝑘1,𝑛 𝑘2,1, 𝑘2,2, … … … … , 𝑘2,𝑛 𝑘𝑚,1, 𝑘𝑚,2, … … … … , 𝑘𝑚,𝑛 after preparing the set of similar keywords, we train the model using keywords and relevant weights for all the classes. the trained model can be defined as: 𝐷𝑜,𝑝 = { 𝐷1 = {[𝑇1,1, 𝑊1,1], [𝑇1,2, 𝑊1,2] … . . , [𝑇1,𝑝, 𝑊1,𝑝]} 𝐷2 = {[𝑇2,1, 𝑊2,1], [𝑇2,2, 𝑊2,2] … . . , [𝑇2,𝑝, 𝑊2,𝑝]} 𝐷𝑜 = {[𝑇𝑜,1, 𝑊𝑜,1], [𝑇𝑜,2, 𝑊𝑜,2] … . . , [𝑇𝑜,𝑝, 𝑊𝑜,𝑝]} 77 nidhi bhandari, rachna navlakhe and g.l.prajapati this is a function that generates a virtual directed graph using a weight matrix for decision-making. the matrix in the form of a graph model describes the association of a tweet with the given sentiment dictionary. an algorithm is developed to get the class label of the tweet, as given in table 2. table 2. dgm based classification input: set of semantically similar twits 𝑇𝑊𝑚,𝑛, trained dictionary model 𝐷𝑜,𝑝 output: class label c process: 1. 𝑓𝑜𝑟(𝑖 = 1; 𝑖 < 𝑚; 𝑖 + +) a. 𝑓𝑜𝑟(𝑗 = 1; 𝑗 < 𝑛; 𝑗 + +) i. 𝑖𝑓 (𝐷𝑖 . 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑇𝑊𝑖,𝑗 )) 1. 𝑊𝑖 = 𝑊𝑖−1 + 𝑊𝑖,𝑗 ii. 𝑒𝑛𝑑 𝑖𝑓 b. 𝑒𝑛𝑑 𝑓𝑜𝑟 c. 𝐷𝑊𝑖 = 𝑊𝑖 𝑚 2. end for 3. 𝐶 = 𝑔𝑒𝑡𝑀𝑎𝑥𝑉𝑎𝑙(𝐷𝑊𝑖 ). 𝑐𝑙𝑎𝑠𝑠𝑙𝑎𝑏𝑒𝑙 4. return c the above algorithm searches each word in the dictionary and the relevant weights are aggregated. a greater value of weights is used as the final sentiment label. in order to make final decision, we use a function as: 𝑓(𝐵, 𝐶) = { 𝐵 = 𝐶 𝑡ℎ𝑒𝑛 𝐶 𝐵! = 𝐶 𝑖𝑠 𝐵𝑖−1 == 𝐶 𝑡ℎ𝑒𝑛 𝐶 finally, if we find the class label c belongs to shy, panic, sad and guilty then we label the tweet as nontoxic else c returns the toxic labels. this model is named as directional graph model. figure 2 below shows the precision, recall and f-score of different techniques. the results indicate the dgm shows better accuracy as compared to other models. however, the dgm performance is higher but the soir is consistent as compared to dgm. 78 a deep learning model for classifying the hate and offensive language in social media text figure 2. performance summaries of classification models secondly, we measured the required time for training and reported it in figure 3. according to the results, the cosine-based and k-nn based techniques are winners. the main advantage of this model is that we can preserve the previously trained model for future use. additionally, new data will also learn continuously. the model's performance replicates the efficient and accurate identification of toxic tweets. figure 3. time consumption 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 precision recall f-score p e rf o rm a n ce ( % ) parameters cosin knn soir dgm 79 nidhi bhandari, rachna navlakhe and g.l.prajapati 3. the proposed work emotion classification has become a common and classical domain of research and development but still there are many opportunities available for discovering something new. initially, we started our work with the study of text mining and information retrieval. but due to new kinds of problems associated with text mining attracted us to classify social media text for finding toxic content. in order to design an effective method, we have implemented a dgm approach which is accurate and efficient for modelling negative emotions. but during the investigation, we found the following issues we tried to resolve in this study. 1. social media content these days is full of offensive language even during formal communication, comments and others. therefore, there is a need of distinguishing between the content as normal and toxic. 2. the previous experiments have been done with a small size of samples, but social media involve big data problems. therefore, a previously defined model is reframed using the deep learning technique to deal with a large amount of data. in this context, we have obtained a dataset of twitter from kaggle. the dataset is hate speech and offensive language dataset. the dataset is distributed in three class labels hate speech, offensive and neither. the dataset involves 1430 instances as hate speech post of twitter, 19190 tweets as the offensive language and 4163 instances as the normal text. the dataset consists of 24783 instances of data. among them, we make a split of 18587 instances for training and 6196 for testing. however, this dataset contains more data than the dataset on which we trained our previously designed models. therefore, initially, we have employed our dgm model for extracting features. during this, we found a significant drawback of this method, which we tried to demonstrate in figure 4. figure 4 shows the time consumed during the experiment with only 50% of entire dataset for constructing the required features for dgm technique. however, the dgm based technique has accurate results but having huge time complexity for training with a large amount of data. figure 4. time consumed with dgm 0 500 1000 1500 2000 2000 3000 5000 7000 8000 9000 12000 t im e i n s e c dataset size time consumed 80 a deep learning model for classifying the hate and offensive language in social media text therefore we drop the idea of utilizing the traditional classification technique for processing such a huge amount of data. the main reason behind the time consumption is computation of tf-idf features and then computing the feature representation, which increases the time consumption significantly. additionally, the measured features that need to train with the machine learning algorithms consume a significant amount of time for training. therefore, the idea of working with the large amount of data with the classical and traditional feature construction method is becoming expensive. finally, we proposed to use the deep learning technique for classifying the hate speech and offensive text to deal with the large amount of data. however, the deep learning models are mainly developed for image classification, but these techniques can also be used for other relevant classification and prediction tasks. the proposed model consists of the following steps to conduct the experiments. data preparation in any machine learning model, data preparation is an essential step of data analysis. we have placed the obtained data from kaggle into our google drive and use the google colab to process data using the different techniques of text mining and machine learning. figure 5. dataset samples in the first step, we load the dataset, the initial dataset samples are demonstrated in figure 5 with their relevant attributes. however, the dataset contains various attributes, but we are just utilizing the class labels and tweets here for data analysis. data pre-processing in basic terminology, the data pre-processing is a technique by which we prepare the data to apply to the learning algorithm. therefore in this context, we studied various nlp based text pre-processing techniques. from them, some essential techniques are applied for pre-processing of data. here we applied the following pre-processing techniques: a. punctuation removal: in this phase, we removed all the punctuations from the input dataset instances by analyzing each character of the input strings. 81 nidhi bhandari, rachna navlakhe and g.l.prajapati b. tokenization: in this step, the aim is to divide the tweet strings into chunks of words. c. stop-word removal: in this step, the aim is to remove english stop words. the stop words are those words which frequently occur in the text but not essential for domain identification such as, ''this', ''that', ''is', '''am' so on. in this context, we apply the natural language toolkit (nltk) library for stop word removal. d. stemming and lemmatization: both are generated form of the same words. the difference is that stem might not be an actual word and lemma is an actual language word. stemming usage an algorithm to which makes word processing faster. on the other hand, in lemmatization, we used wordnet to produce lemma, which is slower than stemming. figure 6. pre-processed data features analysis in this phase, we transform the tweets into a structured manner. so, we have used the following different ways to prepare four different set of feature vector. a. pos tagging: in this phase, we utilized the nltk based pos tagger to process each string of the dataset. after that we prepared the vector for making it training and testing set. in this process we get 37 tags or features for training data. the transformed data is demonstrated in figure 7. figure 7. sample tagged data b. pos with reduced features: in this phase, the 37 features of nlp-pos tags are used for experimentation. in order to reduce the amount of features, we have used principle component analysis (pca) and reduced the features, finally we have getting only 20 features for applying the algorithms. 82 a deep learning model for classifying the hate and offensive language in social media text c. term frequency – inverted document frequency (tf-idf): in this feature vector, we utilized tf-idf vectorizer to extract weighed keywords from the data. the dimensions of data are kept maximum 5000 keywords. the extracted features are demonstrated in figure 8. figure 8. extracted tf-idf features d. combined features: in this phase, we just combined tf-idf features, and pos tagged features in a vertical manner to construct total of 5037 columns and 24783 rows of data. applying cnn here, we trained the high-dimensional data using convolutional neural network (cnn). therefore, we have developed a simple three-layer sequential model. first layer is configured with the total number of attributes in training features, and activation function '''relu' is used. the second hidden layer consists of 10 neurons and the activation function '''sigmoid' has been used. the final layer contains three output neurons and 'soft-'max' activation function. the model takes very less amount of time to train the cnn model. the training time of the input training vectors has given in figure 9. the model will be trained in a few times even the combined features, which consist of 5037 columns and 24783 data. therefore, we can use the large data for learning with deep learning techniques. figure 9. training time 0 50 100 150 200 250 tf-idf pos pos f20 comined t ra in in g t im e i n s e c training time 83 nidhi bhandari, rachna navlakhe and g.l.prajapati performance analysis in this phase, we evaluated our model for obtaining the performance matrix for learning as well as the validation of the model. therefore, the training accuracy, validation accuracy, training loss and validation loss of the model have been measured. these performance parameters are discussed in next section. in addition, the class-wise performance report has also been prepared. 4. results this section details the conducted experiments and the obtained performance results. in deep learning, the accuracy metric is used to measure the 'algorithm's performance in an interpretable manner. the accuracy is determined in the form of a percentage. it measures how accurate our model's prediction is with the true data. figure 10. (a) training accuracy (%) in this experiment, we have used four different set of feature vectors namely tf-idf based weighted feature vector, second the pos tagged features, third we have reduce the size of pos tagged data for classification and finally the combined features of pos and tf-idf has been employed with the cnn model. the performance of the model with the different sets of selected features in terms of accuracy (%) has been reported using figure 10(a). additionally, the accuracy for validation or test dataset has been demonstrated in 10(b). according to the obtained consequences in terms of these two parameters, we found that the tf-idf based and the combination of tf-idf and pos 84 a deep learning model for classifying the hate and offensive language in social media text produces a higher performance than the other two algorithms. during training, we found the pos tagged data shows approximately 74.5 to 75.2% performance. figure 10. (b) validation accuracy (%) on the other hand, when we used the reduced 20 features with the pos tagged data then we found a small improvement and we get an accuracy of 75.5%, which is higher than the pos tagged complete dataset, which has 36 features. figure 10. (c) training loss on the other hand, when we train the model with the same data but tf-idf features, model returns the total of 98.8% accurate results in consistent manner. and when we apply the combination of pos and tf-idf features, we get improved results up to 85 nidhi bhandari, rachna navlakhe and g.l.prajapati 99.8%. however, when working with the validation set, the performance of algorithm has dropped continuously. in figure 10(b), we can see the performance of all the feature based technique has been decreased significantly. figure 10. (d) validation loss in addition, we have also computed the loss for training and validation. the loss function is used to optimize a machine learning algorithm. the loss is calculated on the training set and validation set and its interpretation is based on how well the model is doing in these two sets. it is the sum of errors made for each sample in training or test sets. loss value implies how poorly a model behaves after each round of optimization. a loss is a number indicating how bad the 'model's prediction was on a sample. if the 'model's prediction is perfect, the loss is zero; otherwise, the loss is greater. the goal of a model is to find the set of weights and biases that have low loss. higher loss is worse for any model. a loss is not a percentage. it is a sum of the errors made for each sample in training or validation. figure 11. shows the loss measuring example 86 a deep learning model for classifying the hate and offensive language in social media text figure 11 shows two graphs demonstrating the losses of two different models, the left graph has a high loss and the right graph has a low loss. • the arrows represent a loss. • the blue lines represent predictions. the loss of the proposed deep learning model has demonstrated in figure 10(c) and 10(d). the figure 10(c) shows the training loss and 10(d) shows the validation loss of the model. in the diagram 10(c), we found that the loss of training reduced and become consistent. on the other hand, 10(d) demonstrates the loss during the validation of the model. the performance indicates when number of epoch has increases the loss of validation has also been increases in the case of tf-idf based features learning, as well as also for the combined features, but the for combined features the loss function is less incline. on the other hand for pos tagged data we can see the model produces similar or consistent performance in all the cases. finally, for more simple understanding we have also measured the classification report. the classification report for the implemented techniques has been given in table 3.the class-wise performance analysis is demonstrated in the table 3. that demonstrates the precision, recall and f1-score. additionally, shows the accuracy, marco avg, and weighted avg. according to these values we can say the combined and tf-idf based features are providing higher accurate results. the accuracy has also visualized in figure 12. according to the given results of the implemented techniques, we found that the accuracy of pos based both the techniques i.e. entire features and top 20 features based approach demonstrate the similar accuracy of 77%. on the other hand the tf-idf and combined features based technique provide the 82% and 83% of accuracy. thus we can say the combined features based classification approach provide higher accurate results. 5. conclusions social media is one of the user-centric and user-generated data source. the data may have a significant amount of knowledge, challenges and opportunity for diverse areas of table 3. performance summary of the models pos with 20 feature pos tagged tf-idf combined (tf-idf + pos) precisi on recal l f1sscor e precisio n recal l f1sscor e precisio n recal l f1sscor e precisio n recal l f1sscor e class 0 0. 0 0.0 0.0 0.0 0.0 0.0 0.25 0.31 0.28 0.23 0.29 0.26 class 1 0.78 0.99 0.87 0.77 0.99 0.87 0.90 0.89 0.89 0.90 0.89 0.90 class 2 0.51 0.06 0.51 0.42 0.05 0.09 0.70 0.70 0.70 0.74 0.72 0.73 accuracy 0.77 0.77 0.82 0.83 marco avg 0.43 0.35 0.43 0.40 0.35 0.32 0.62 0.63 0.62 0.62 0.63 0.63 weighted avg 0.69 0.77 0.69 0.67 0.77 0.68 0.83 0.82 0.83 0.84 0. 83 0.83 87 nidhi bhandari, rachna navlakhe and g.l.prajapati applications. however, there are a number of opportunities hidden in the social media data, but toxic intension based data contains can create the biggest issues in healthy social media surroundings. in this context, a dgm model is analysed, which is limited to working with a small amount of data. therefore, in this paper, a deep learning model is proposed for classifying hate speech, offensive language and the normal social media posts. the finding says the deep learning techniques are better for classifying large datasets. figure 12. comparison of mean accuracy the dgm model utilizes the classical techniques for accurate classification of text but a limitation of the model motivates us to work with deep learning models. the proposed model includes the text feature analysis techniques and cnn algorithm for providing an accurate text classification technique. this model aims to differentiate between hate speech, offensive text and normal text by only considering text in social media post. during the experiments we conclude the following observations: 1. dgm is accurate but is limited to work with the less dimensional data on the other hand the deep learning is effective for learning with large scale data 2. deep learning technique provides relevant or higher accuracy in less amount of time which is proved in the obtained performance. 3. in classification of offensive text and hate speech text, the key words are more essential then the sentence structures or sentence formation. 4. pos tagger is less effective for representing the text features to model difference between offensive and hate speech text. 5. the combination of keyword based features and pos tagged data can improve the performance significantly. 0,74 0,75 0,76 0,77 0,78 0,79 0,8 0,81 0,82 0,83 0,84 pos pos f20 tf-idf tf-idf+ pos a cc u ra cy (% ) methods accuracy 88 a deep learning model for classifying the hate and offensive language in social media text this study is helpful as a tutorial to demonstrate how the classical machine learning techniques are accurate but can be expensive to solve real-world problems. on the other hand, this study also tried to demonstrate how we can make use the deep learning techniques for processing the text data and get higher classification accuracy. in the near future, the following directions can be helpful for the extension of this study: 1. the deep learning techniques involve more complex data models, which will provide more accurate data analysis. among them the lstm has become much more popular. for extension, we can use them in future implementations. 2. the combination of two different kinds of features increases the dimension of the data to be trained. at the same time, in the selected tf-idf features, various non-meaningful keywords are available. therefore, the need is to design an enhanced semantic keyword selection technique for feature representation. references [1] h. wang, q. zhang, & j. yuan, "semantically enhanced medical information retrieval system: a tensor factorization based approach”, ieee, 2169-3536, 2017. [2] s. bergamaschi, e. domnor, f. guerra, m. orsini, r. t. lado, y. velegrakis, “keymantic: semantic keyword-based searching in data integration systems”, proceedings of the vldb endowment, acm, vol. 3, no. 2, 2010. [3] m. chahal, “information retrieval using jaccard similarity coefficient”, international journal of computer trends and technology, volume 36, number 3, june 2016. [4] n. bhandari, r. navlakhe, g. l. prajapati, “semantic query optimization based information retrieval technique”, the journal of oriental research madras, issn:0022-3301,vol. xcii-lxxvii, 2021 [5] p. bafna, d. pramod, a. vaidya, “document clustering: tf-idf approach”, international conference on electrical, electronics, and optimization techniques, ieee, 2016. [6] r. k. roul, j. k. sahoo, k. arora, “modified tf-idf term weighting strategies for text categorization”, ieee, 2017. [7] j. chen, x. tang, “ensemble of multiple k-nn classifiers for societal risk classification”, j syst sci syst eng, 26(4): 433-447, aug 2017. [8] h. xu, w. yang, j. wang, “hierarchical emotion classification and emotion component analysis on chinese micro-blog posts”, expert systems with applications, 42, 8745–8752, 2015. [9] c. pasquier, c. da costa pereira, a. g. b. tettamanzi, “extending a fuzzy polarity propagation method for multi-domain sentiment analysis with word embedding and pos tagging”, ecai, the ios press, 2020. 89 microsoft word articolo6.doc numeri q-perfetti e q-amicabili di seconda specie e altre generalizzazioni dei numeri perfetti di seconda specie eugeni franco università degli studi di teramo ippoliti gianluca dottorando dell'università degli studi di teramo nel presente lavoro si affronta, tra le curiosità matematiche, il problema dei numeri perfetti di seconda specie, riprendendo anche un lavoro del 1979 di franco eugeni e bruno rizzi, utilizzato come preambolo e spunto per le problematiche lasciate aperte su tali concetti, e le loro generalizzazioni tra le quali quella dei numeri q-perfetti di 2a specie. il presente file si compone infatti di: 1. su alcune generalizzazioni dei numeri perfetti, tratto dal periodico di matematiche serie v volume 56 del 1980 riguardanti i numeri 1-perfetti di 2a specie e alcune problematiche dei numeri q-perfetti di 2a specie. il testo è corredato di note scritte in questa occasione. 2. un lavoro che appare qui per la prima volta in cui è trattato, per quanto sia possibile, il caso q > 1 per la suddetta generalizzazione dei numeri perfetti di 2a specie e una diversa generalizzazione dei numeri perfetti di 2a specie insieme a risultati sul problema delle coppie di numeri amicabili di seconda specie, per entrambe le generalizzazioni. introduzione. il problema dei numeri perfetti ha origini antiche, essendo dovuto ad euclide (iv secolo a. c.). egli definì numeri perfetti quei naturali la cui somma dei divisori inferiori eguaglia il numero stesso. definita la funzione σ come σ(n) = ∑ nd d | è chiaro che il problema si può esprimere nei seguenti termini: n è perfetto se e solo se σ(n) – 2n = 0. euclide stesso, posto il problema, caratterizzò completamente i numeri perfetti pari, mentre ancora oggi non è stato risolto, in maniera generale, quello dei perfetti dispari. in particolare non si conosce nessun numero perfetto dispari e né se ne esistano. la cosa però veramente interessante è che intorno al problema dei numeri perfetti siano nati, nel corso dei secoli, altri concetti, generalizzazioni e problemi che riguardano tali generalizzazioni, problemi, alcuni di essi tuttora irrisolti, che hanno affascinato schiere di matematici professionisti e dilettanti, uomini di cultura, studiosi e un vero e proprio popolo di curiosi. in questo lavoro ci occupiamo, appunto, di una parte di questi problemi, con qualche risultato originale. una di queste problematiche riguarda le coppie di numeri amicabili che sono quelle coppie di numeri naturali per le quali la somma dei divisori inferiori dell’uno eguaglia il valore dell’altro (è facile provare che (m,n) è una coppia di numeri amicabili se e solo se σ(n) = σ(m) = m + n). nella prima parte riportiamo anastaticamente l’articolo summenzionato, corredato, però da note a piè di pagina scritte per l’occasione, dove incontreremo una prima variazione, la più nota, sul concetto di numero perfetto (ovvero quella di numero perfetto di seconda specie) e una sua generalizzazione (ovvero quella di numero q-perfetto di seconda specie). nella seconda parte riprendiamo alcuni concetti/problemi della prima parte e forniamo una ulteriore generalizzazione dei numeri perfetti (ovvero quella che noi abbiamo chiamato numeri perfetti di seconda specie di ordine k) e il legame tra le due diverse generalizzazioni e risolviamo completamente i problemi collegati alle definizioni generalizzate dei numeri amicabili di seconda specie. si sono usate, qui, le stesse notazioni e simboli della prima parte tranne per quello che riguarda la funzione prodotto dei divisori di un numero naturale, per la quale nella seconda parte si è preferito usare π(n) piuttosto che p(n), e per l’insieme dei numeri primi che, sempre nella seconda parte, si è denotato con ℙ. 1. 1 1 diamo qui una breve dimostrazione: l’identità di pellegrino è equivalente a ν(n)·f(n) = ∑ nd df | )(2 . indicando con d1 d2,…, dν(n) i divisori di n e con di ’ il divisore “complementare” di di ovvero tale che n = di·di ’, si ha, essendo f incondizionatamente additiva, f(n) = f(di)+ f(di ’), ∀ i = 1,…, ν(n). si ha, allora, che ∑ = )( 1 )( n i nf ν = ∑ = )( 1 n i ν f(di)+ f(di ’) cioè ν(n)·f(n) = ∑ nd df | )(2 . 2 2 naturalmente questo significa che ∀ n ∈ ℕ1 – ℙ, esiste un’opportuna scelta della coppia (m, q) in modo che l’identità (2.5) sia vera. precisamente se n non è un quadrato perfetto (dunque ν(n) è pari ed è comunque ν(n)·> 2 quindi 2 )(nν – 1 ∈ ℕ1): q = 2 )(nν – 1 e βi = αi ( dunque m = n); se n è un quadrato perfetto (dunque αi ∈ 2ℕ, ν(n)·è dispari ma comunque ν(n) > 2): βi = 2 iα (dunque m = n ) e q = ν(n) – 2. 3 3 refuso si stampa: α = 2q + 1 e α1 = 1, α2 = p – 1 = q quindi 12 1 +qp , qpp 21 con p1 e p2 primi arbitrari distinti sono tutti e soli i numeri q-perfetti di 2a specie quando q + 1 è primo. 2.1 le generalizzazioni dei numeri perfetti di seconda specie. le generalizzazioni dei numeri perfetti di seconda specie più note sono di due tipi. la prima l’abbiamo vista nella prima parte e riguarda i numeri q-perfetti di 2a specie che qui, brevemente, richiamiamo e sulla quale puntualizziamo i risultati noti. definizione 2.1.1. sia q∈ ℕ1, un numero n ∈ ℕ2 è detto q-perfetto di 2a specie se il prodotto dei suoi divisori inferiori ad n eguaglia la potenza di ordine q di n stesso. è chiaro che per q = 1 si ritrovano i numeri perfetti di 2a specie che qui chiameremo, dunque, 1-perfetti di 2a specie. utilizzando la funzione π, che associa ad un numero n il prodotto di tutti i suoi divisori: π(n) = d nd | π , un numero n è q-perfetto di 2a specie se n d nd π | = nq ovvero se d nd | π = nq+1. si ha così modo di sfruttare l'identità πκ(n) = ∏ nd kd | = 2 )( n k n ν , (2.4) della prima parte con k = 1, dove con ν(n) si è indicata la funzione numero di divisori. sarà appena il caso di ricordare che, se 11 αp ··· rrp α è la fattorizzazione in primi di n, l'espressione per ν(n) è data da ν(n) = ( )∏ = + r i i 1 1 α , come già visto nella prima parte. è questa una delle più note funzioni aritmetiche insieme alla funzione σ, somma dei divisori, già incontrata con i numeri perfetti di 1a specie, la quale è moltiplicativa4. ricordiamo inoltre che ν(0) non è definita, ν(1) = 15, ν(n) = 2 se e solo se n è primo, ν(n)=3 se e solo se n è il quadrato di un primo6, ν(n) ≥ 4 per tutti gli altri naturali, dove l'uguaglianza si ha solo per i numeri secondi (non quadrati) e per i cubi di un primo i quali, come abbiamo già visto nella prima parte, costituiscono tutti e soli i numeri 1-perfetti di 2a specie. infine ricordiamo (vedi nota 4 della prima parte) che ν(n) è pari se e solo se n non è un quadrato perfetto e, quindi, 2 )( n n ν è una potenza intera di n (o della sua radice quadrata, se n è un quadrato perfetto). 4 una funzione aritmetica f è moltiplicativa se f(m·n) = f(m)·f(n) ∀ m, n ∈ ℕ1 tali che (m, n) = 1, ovvero relativamente primi tra loro. 5 sebbene 1 soddisfi l’uguaglianza (2.1.1), per definizione esso non è q-perfetto qualunque sia il valore di q. 6 questi ultimi due casi contemplano tutti numeri che, in analogia con quanto accade con i numeri perfetti di prima specie, a partire dalla definizione 2.1.2 chiameremo q-mancanti di seconda specie, qualunque sia il valore di q ∈ ℕ1. a questo punto possiamo continuare dicendo che un numero n ∈ℕ2 è q-perfetto di 2a specie se e solo se (2.1.1) 2 )( n n ν = nq+1, ovvero, essendo n > 1, se e solo se 2 )(nν = q + 1 ovvero se e solo se, infine, ν(n)=2(q+1). dunque essere q-perfetto per n ∈ℕ2 dipende solo ed esclusivamente dal numero dei suoi divisori e dalla fattorizzazione di 2(q + 1). di seguito riportiamo una prima casistica non generale di numeri q-perfetti di seconda specie al variare di q, per i quali non riportiamo le dimostrazioni. valori di q valori di ν(n) caratterizzazione dei numeri q-perfetti di 2a specie q = 0 ν(n) = 2 numeri primi q = 1 ν(n) = 4 numeri perfetti di 2a seconda specie, ovvero numeri secondi non quadrati oppure cubi di un primo q = 2 ν(n) = 6 n = 51p oppure n = 2 2 1 pp con p1, p2 ∈ ℙ, diversi tra loro q = 3 ν(n) = 8 n = 71p oppure n = 2 3 1 pp oppure 321 ppp con p1, p2, p3 ∈ ℙ, diversi tra loro q = 4 ν(n) = 10 n = 91p oppure n = 2 4 1 pp con p1, p2 ∈ ℙ, diversi tra loro q = 5 ν(n) = 12 n = 111p oppure n = 251 pp oppure n = 2 2 3 1 pp oppure 32 2 1 ppp con p1, p2, p3 ∈ ℙ, diversi tra loro e se estendessimo la definizione di numeri q-perfetti a valori di q non interi? in questo caso si seguiterebbero ad avere numeri q-perfetti a condizione che 2q ∈ ℕ1, più precisamente q = 2 m per qualche m ∈ ℕ1 – 2ℕ1. in tal caso la condizione ν(n) = 2(q + 1) diventa ν(n) = m + 2 con m dispari e tutto dipende dalla fattorizzazione di m + 2. possiamo dunque affermare che i valori ammissibili per q, ovvero quei valori che rendono non vuota la definizione di numeri q-perfetti di 2a specie sono quelli dell’insieme q0 ≝ { 2 m | m ∈ ℕ0} dove si è usato l’indice 0 per distinguerlo dall’insieme q1 ≝ { 2 m | m ∈ ℕ1} che in seguito useremo. riportiamo, di seguito una tabella con i primi valori non interi di q, anche questa senza dimostrazioni. valori valori caratterizzazione dei numeri q-perfetti di 2a specie di q di ν(n) 2 1 3 n = 21p con p1 ∈ ℙ 2 3 5 n = 41p con p1 ∈ ℙ 2 5 7 n = 61p con p1 ∈ ℙ 2 7 9 n = 81p oppure n = 2 2 2 1 pp con p1, p2 ∈ ℙ, diversi tra loro 2 9 11 n = 10 1p con p1 ∈ ℙ 2 11 13 n = 121p con p1 ∈ ℙ consideriamo ora qualche caso generale enunciato sotto forma di teoremi: teorema 2.1.1. i numeri della forma n = 121 +qp e n = 21 pp q ⋅ con p1, p2 ∈ ℙ, diversi tra loro, sono q-perfetti, i primi per ogni q ∈ q0 e i secondi per ogni q ∈ ℕ0. dimostrazione sia n = 121 +qp per qualche primo p1 ∈ ℙ, per qualche q ∈ q0, allora (2q + 1) ∈ ℕ1 e n ∈ℕ2 e si ha, per la nota identità che esprime ν(n) come = ( )∏ = + r i i 1 1 α , essendo 1 1 αp ··· rrp α la fattorizzazione in primi di n, ν(n) = 1 + 2q + 1 = 2(q + 1), dunque n è q-perfetto di 2a specie. sia n = qp1 p2 per qualche coppia di primi distinti p1, p2 ∈ ℙ, per qualche q ∈ ℕ0, allora, sempre per l’identità summenzionata, si ha ν(n) = (1 + 1)(1 + q) = 2(q + 1) ovvero n è q-perfetto di 2a specie. teorema 2.1.2. sia r ∈ℕ1.e sia q = 2r – 1 − 1. i numeri q-perfetti sono tutti e soli quei numeri n del tipo n = 121 1 −rp ·…· 12 − kr kp con k, r1, …, rk ∈ ℕ1 tali che ∑ = k i ir 1 = r e p1, …, pk ∈ ℙ diversi tra loro. sia q = 2r – 1 − 1 per qualche r ∈ℕ1. n è q-perfetto di 2a specie se e solo se ν(n) = 2r. un sistema completo di divisori di 2r è del tipo 12r ,…, kr2 , per qualche r1, …, rk ∈ℕ1 tali che ∑ = k i ir 1 = r per qualche k ∈ ℕ1 tale che k ≤ r = ω(n). dunque n è qperfetto di 2a specie se e solo se ∃ k ∈ ℕ1 con k ≤ r tale che n = 121 1 −rp ·…· 12 − kr kp con r1, …, rk ∈ ℕ1 tali che ∑ = k i ir 1 = r e p1, …, pk ∈ ℙ diversi tra loro. al concetto di numeri q-perfetti di 2a specie è collegato, come accade con i numeri perfetti (numeri perfetti di 1a specie), quello dei numeri q-abbondanti e qmancanti di 2a specie, per i quali forniamo la seguente: definizione 2.1.2. sia n ∈ ℕ2. n è q-mancante (q-abbondante) di 2a specie se ∏ ≠nd nd d | < nq ( ∏ ≠nd nd d | > nq). naturalmente, come accade per i numeri perfetti di 1a specie, anche qui l’insieme dei numeri q-perfetti, q-mancanti e q-abbondanti costituisce una partizione di ℕ2. tale definizione ci conduce, inoltre, alla dimostrazione di due lemmi, a noi non noti in letteratura, che risulteranno utili in seguito. lemma 2.1.1. sia n q-perfetto di 2a specie o q-mancante di 2a specie allora ogni sottomultiplo proprio di n è q-mancante. sia n = ∏ = r i i ip 1 α , se n è q-perfetto o q-mancante allora ν(n) ≤ 2·(q + 1) ma ν(n) = ( )∏ = + r i iα 1 1 . sia m un sottomultiplo proprio di n, allora m = ∏ = r i i ip 1 β con 0 ≤ βi ≤ αi ∀ i = 1,…,r e, inoltre, esiste almeno un indice k ≤ r tale che βk < αk. dunque ν(m) = ( )∏ = + r i i 1 1 β < ( )∏ = + r i i 1 1 α = ν(n) ≤ 2·(q + 1) essendo (1+ βi) ≤ (1+ αi) e (1+ βk) < (1+ αk) ovvero m è q-mancante di 2a specie. c.v.d. lemma 2.1.2. (duale). sia n q-perfetto o q-abbondante di 2a specie allora ogni multiplo proprio di n è q-abbondante. supponiamo che esista un numero n q-perfetto o q-abbondante con un multiplo proprio m q-perfetto o q-mancante allora tale multiplo m avrebbe n come sottomultiplo proprio ma n è q-perfetto di 2a specie o q-abbondante di 2a specie e ciò è assurdo per il lemma precedente. c.v.d. la seconda generalizzazione dei numeri perfetti di 2a specie è quella collegata al già citato problema di halcke, ed è illustrata dalla seguente: definizione 2.1.3. sia n ∈ ℕ1, n è perfetto di 2a specie di ordine k ∈ ℕ1 se ∏ ≠ = nd nd k nd | . è immediato provare che tale definizione è equivalente a πk(n) = nk+1 dove si è utilizzata la funzione πk(n) (prodotto delle potenze k-esime di tutti i divisori di n) per la quale abbiamo già visto nella prima parte essere vera l’identità πk(n) = ( ) 2 n k n ν . dunque, se si eccettua 1 che risulta, analogamente al caso precedente, perfetto di 2a specie di ordine k per ogni k∈ ℕ1, i numeri n perfetti di 2a specie di ordine k risultano completamente caratterizzati dalla seguente identità ( ) 2 n k ν – (k+1) = 0 cioè k = ( ) 2 2 −nν . dunque esistono perfetti di 2a specie di ordine k ∈ ℕ1 solo per k = 1 o k = 2. precisamente n è perfetto di 2a specie di ordine 1 se e solo se ν(n) = 4 ovvero se e solo se è 1-perfetto di 2a specie, le cui caratterizzazioni abbiamo già visto (infatti in tal caso le due diverse generalizzazioni coincidono entrambe con i numeri perfetti di 2a specie). nell’altro caso n è perfetto di seconda specie di ordine 2 se e solo se ν(n) = 3 ovvero se e solo se è 1-perfetto di 2a specie. ricordando che q ≝ { 2 m | m ∈ ℕ1}, riportiamo ora un teorema che non ci risulta noto in letteratura e che rappresenta una risultato che lega tra loro le due generalizzazioni dei numeri perfetti di 2a specie e: teorema 2.1.4. sia k l’insieme costituito dagli inversi degli elementi dell’insieme q. n è perfetto di 2a specie di ordine k ∈ k se e solo se n è q-perfetto di 2a specie con q = k 1 . dimostrazione sia n perfetto di 2a specie di ordine k allora k = ( ) 2 2 −nν ovvero ( )nν = ( ) k k+12 = 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + 1 1 k = = 2(q+1) ponendo q = k 1 dunque n è q-perfetto di 2a specie con q = k 1 . analogamente per il viceversa. c.v.d. di conseguenza restano caratterizzati tutti i numeri di 2a specie di ordine k come i numeri k 1 -perfetti e tutti i risultati validi per i secondi si possono vedere come risultati sui primi a patto di invertire l’indice. definizione 2.1.4. sia n ∈ ℕ2, n è mancante (abbondante) di 2a specie di ordine k se ∏ ≠nd nd kd | < n ( ∏ ≠nd nd kd | > n). 2.2 le generalizzazioni delle coppie di numeri amicabili di seconda specie. consideriamo ora le generalizzazioni delle coppie di numeri amicabili di seconda specie, illustrate con le seguenti definizioni che dimostreremo essere entrambe vuote. definizione 2.2.1. sia q ∈ q 7e siano m, n ∈ ℕ2 distinti. m, n sono q-amicabili di 2a specie se: ( ) ( )⎪ ⎩ ⎪ ⎨ ⎧ = = q q n m m m n n π π . definizione 2.2.2. sia k ∈ k e siano m, n ∈ ℕ2 distinti. m, n sono amicabili di 2a specie di ordine k se: ( ) ( )⎪ ⎩ ⎪ ⎨ ⎧ = = n m m m n n k k k k π π . teorema 2.2.1. non esistono q-amicabili di 2a specie per ogni q ∈ q. dimostrazione sia q ∈ q e siano m, n ∈ ℕ2 tali che ( ) ( )⎪ ⎩ ⎪ ⎨ ⎧ = = q q n m m m n n π π (1) o, equivalentemente 7 non consideriamo il caso q = 0, poco significativo e che, come è facile verificare, condurrebbe a definire ogni coppia di primi distinti una coppia di numeri amicabili. ( ) ( ) ⎪⎩ ⎪ ⎨ ⎧ = = − − q m q n nm mn 1 2 1 2 ν ν 8 (2) allora m ed n hanno gli stessi divisori primi. siano dunque n = ∏ = r i i ip 1 α ed m = ∏ = r i i ip 1 β , con αi, βi ∈ℕ1, le fattorizzazioni in primi di n ed m. le (1) sono equivalenti a (3) ( ) ( ) ⎪ ⎪ ⎩ ⎪⎪ ⎨ ⎧ =⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − =⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ii ii q m q n α ν β β ν α 1 2 1 2 ∀ i = 1,…,r. dalla (1) segue, moltiplicando membro a membro le due equazioni, che ( ) ( ) 11 ++ ⋅ qq m m n n ππ = 1. possiamo considerare due casi: primo caso: ( ) ( ) 11 1 ++ == qq m m n n ππ ovvero m ed n sono q-perfetti di 2a specie; ma dalla (2) segue che nq = mq ovvero m = n; secondo caso: ( ) 1+qn nπ < 1 e ( ) 1+qm mπ > 1 (o viceversa) ovvero n è q-mancante ed m è q-abbondante di 2a specie. in questo caso la (1) implica ⎩ ⎨ ⎧ > < + + 1 1 qq qq mmn nnm ovvero m < n; avendo m ed n hanno gli stessi fattori primi questo implica che m è un sottomultiplo proprio di n, ma, come abbiamo già dimostrato nel lemma 2.1.1 ciò implica m è q-mancante contro la posizione iniziale. c.v.d. teorema 2.2.1. non esistono coppie di numeri amicabili di seconda specie di ordine k per ogni k∈ℕ1. 8 al solito si è utilizzata l’identità (2.4) della prima parte con k = 1: ( ) 2)( n n ν π ∀ n ∈ ℕ1. dimostrazione ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = = ∏ ∏ ≠ ≠ md md k nd nd k nd md | | ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ⋅= ⋅= ∏ ∏ ≠ ≠ md md kk nd nd kk mnd nmd | | m ≠ n, m,n ∈ℕ2, k∈ ℕ2. πk(n) · πk(m) = (m · n)k + 1 e p|m ⇒ p|n e viceversa. ( ) ( ) 22 m k n k mn νν ⋅ = mk+1 · nk+1 ( ) 2 1 m kk m ν −+ · ( ) 2 1 n kk n ν −+ = 1 ν(m) = k k )1(2 + = 2·(1+ k 1 ) ν(n) = k k )1(2 + = 2·(1+ k 1 ) se k = 2 ⇒ ν(m) =3 e ν(n) =3 ⇒ m = p2 e n = p2. bibliografia (0) emilio ambirsi, numeri amici, mirabili e perfetti, periodico di matematiche, serie vi, volume 67, 1991. (1) p. hoffman, la vendetta di archimede, bompiani, 1990, p. 31. (2) h. steinhaus, cento problemi di matematica elementare, boringhieri. 1987. (3) d. r. hofstadter: godel, escher, bach: un’eterna ghirlanda brillante, adelphi 1984, pag. 434. (4) m. gardner, enigmi e giochi matematici, vol. 5, sansoni, 1976, pag. 106. ratio mathematica vol. 32, 2017, pp. 21–35 issn: 1592-7415 eissn: 2282-8214 on a functional equation related to information theory p. nath1, d.k. singh2∗ 1department of mathematics, university of delhi, delhi 110007, india pnathmaths@gmail.com 2department of mathematics, zakir husain delhi college (university of delhi) jawaharlal nehru marg, delhi 110002, india dhiraj426@rediffmail.com, dksingh@zh.du.ac.in received on: 28-05-2017. accepted on: 15-06-2017. published on: 30-06-2017 doi: 10.23755/rm.v32i0.332 c©p. nath and d.k. singh abstract the main aim of this paper is to obtain the general solutions of the functional equation (1.3) without imposing any regularity condition on the mappings appearing in it. to do so, the general solutions of the functional equation (1.5), without imposing any regularity condition on the mappings appearing in it are needed. to meet this need, the general solutions of the functional equation (1.6) without imposing any regularity condition on a mapping appearing have to be investigated. one solution of (1.3) is useful in information theory. thus, indeed, is the reason to consider (1.3). keywords: functional equation; additive mapping; multiplicative mapping. 2010 ams subject classifications: 39b22, 39b52, 94a15, 94a17. ∗corresponding author 21 p. nath and d.k. singh 1 introduction for n = 1, 2, . . ., let γn = {(p1, . . . ,pn) : 0 6 pi 6 1, i = 1, . . . ,n; n∑ i=1 pi = 1}, denote the set of all discrete n-component complete probability distributions with non-negative elements. let i = {x ∈ r : 0 6 x 6 1} = [0, 1], r denoting the set of all real numbers. the axiomatic characterization of the non-additive entropy of degree α (see [2]) defined as hαn (p1, . . . ,pn) = (2 1−α − 1)−1 ( n∑ i=1 pαi − 1 ) , α 6= 1 leads to the study of the functional equation n∑ i=1 m∑ j=1 f(piqj) = n∑ i=1 f(pi) + m∑ j=1 f(qj) + λ n∑ i=1 f(pi) m∑ j=1 f(qj) (1.1) in which f : i → r is an unknown mapping, (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm, λ 6= 0, λ ∈ r and n, m being positive integers. by a general solution of a functional equation, we mean a solution obtained without imposing any condition such as differentiability, continuity, continuity at a point, measurability, boundedness, monotonicity etc on a(the) mapping(s) appearing in the functional equation under consideration. the general solutions of (1.1), for fixed integers n > 3, m > 3 and (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm have been obtained in [5]. losonczi [4] considered the functional equation n∑ i=1 m∑ j=1 fij(piqj) = n∑ i=1 hi(pi) + m∑ j=1 kj(qj) + λ n∑ i=1 hi(pi) m∑ j=1 kj(qj) (1.2) with (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm, λ 6= 0, λ ∈ r, fij : i → r, hi : i → r, kj : i → r, i = 1, . . . ,n; j = 1, . . . ,m, as unknown mappings. he found the measurable (in the sense of lebesgue) solutions of (1.2) for all (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm by taking n > 3, m > 3 as fixed integers, in theorem 6 on p-69 in [4]. for the last more than two decades, the general solutions of (1.2) for all (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm, n > 3, m > 3 being fixed integers, are still not known so far. the main aim of this paper is to obtain the general solutions of the functional equation n∑ i=1 m∑ j=1 h(piqj) = n∑ i=1 h(pi) + m∑ j=1 kj(qj) + λ n∑ i=1 h(pi) m∑ j=1 kj(qj) (1.3) 22 on a functional equation related to information theory which contains m + 1 unknown real-valued mappings h and kj (j = 1, . . . ,m), each defined on i = [0, 1]; λ ∈ r, λ 6= 0 and n > 3, m > 3 being fixed integers. these general solutions have been obtained without making use of the difference operator dri suggested on p-58 by losonczi [4]. this paper is an improved version of the manuscript [9]. nath and singh [8] have also obtained the general solutions of n∑ i=1 m∑ j=1 f(piqj) = n∑ i=1 g(pi) + m∑ j=1 hj(qj) + λ n∑ i=1 g(pi) m∑ j=1 hj(qj) with f : i → r, g : i → r, hj : i → r, j = 1, . . . ,m; λ 6= 0, (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm, n > 3, m > 3 being fixed integers. the functional equation (1.3) is a special case of (1.2). a particular case of (1.3) is the following n∑ i=1 m∑ j=1 h(piqj) = n∑ i=1 h(pi) + m∑ j=1 k(qj) + λ n∑ i=1 h(pi) m∑ j=1 k(qj) in which h : i → r, k : i → r and (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm. nath and singh [7] have obtained its general solution(s) for fixed integers n > 3, m > 3. let us define the mappings f : i → r and gj : i → r, j = 1, . . . ,m as f(x) = x + λh(x); gj(x) = x + λkj(x) (1.4) for all x ∈ i. then (1.3) reduces to the pexider type functional equation n∑ i=1 m∑ j=1 f(piqj) = n∑ i=1 f(pi) m∑ j=1 gj(qj) . (1.5) we would like to mention that kannappan and sahoo [3] have obtained the general solutions of (1.3) and (1.5) on an open domain. in our case, the process of finding the general solutions of (1.5), for fixed integers n > 3, m > 3, needs determining the general solutions of the functional equation n∑ i=1 m∑ j=1 ϕ(piqj) = n∑ i=1 ϕ(pi) m∑ j=1 ϕ(qj) + m(n− 1) ϕ(0) n∑ i=1 ϕ(pi) (1.6) where ϕ : i → r and n > 3, m > 3 are fixed integers. this task has been accomplished in section 3. the corresponding general solutions of (1.5) and (1.3) have been investigated in sections 4 and 5 respectively. at the end of section 5, we have analysed the importance of the solutions of functional equation (1.3) from information-theoretic point of view. section 2 contains some known definitions and results needed for the subsequent development of this paper. 23 p. nath and d.k. singh 2 some preliminary results in this section, we mention some known definitions and results. a mapping a : i → r is said to be additive on i or on the unit triangle ∆ = {(x,y) : 0 6 x 6 1, 0 6 y 6 1, 0 6 x + y 6 1} if it satisfies the equation a(x + y) = a(x) + a(y) for all (x,y) ∈ ∆. a mapping a : r → r is said to be additive on r if it satisfies the equation a(x + y) = a(x) + a(y) for all x ∈ r, y ∈ r. it is known [1] that if a mapping a : i → r is additive on i, then it has a unique additive extension a : r → r in the sense that a is additive on r and a(x) = a(x) for all x ∈ i. a mapping m : i → r is said to be multiplicative if m(pq) = m(p) m(q) holds for all p ∈ i, q ∈ i. result 2.1. [5] let n > 3 be a fixed integer and c be a given constant. suppose that a mapping ψ : i → r satisfies the functional equation n∑ i=1 ψ(pi) = c for all (p1, . . . ,pn) ∈ γn. then there exists an additive mapping b : r → r such that ψ(p) = b(p) − 1 n b(1) + c n for all p ∈ i. result 2.2. [4] let d be a given real constant and ψj : i → r, j = 1, . . . ,m, be mappings which satisfy the functional equation m∑ j=1 ψj(qj) = d for all (q1, . . . ,qm) ∈ γm, m > 3 being a fixed integer. then there exists an additive mapping a : r → r and real constants cj (j = 1, . . . ,m) such that ψj(p) = a(p) + cj for all p ∈ i with a(1) + m∑ j=1 cj = d. 3 the functional equation (1.6) in this section, we prove: theorem 3.1. let n > 3, m > 3 be fixed integers and ϕ : i → r be a mapping which satisfies the functional equation (1.6) for all (p1, . . . ,pn) ∈ γn and (q1, . . . ,qm) ∈ γm. then ϕ is of the form ϕ(p) = a(p) + ϕ(0) (3.1) 24 on a functional equation related to information theory where a : r → r is an additive mapping with   (i) a(1) = −nmϕ(0) if ϕ(1) + (n− 1) ϕ(0) 6= 1 or (ii) a(1) = 1 −nϕ(0) if ϕ(1) + (n− 1) ϕ(0) = 1 (3.2) or ϕ(p) = m(p) −b(p) (3.3) where b : r → r is an additive mapping with b(1) = 0 and m : i → r is a multiplicative mapping which is not additive and m(0) = 0, m(1) = 1. proof. let us put p1 = 1, p2 = . . . = pn = 0 in (1.6). we obtain [ϕ(1) + (n− 1) ϕ(0) − 1] [ m∑ j=1 ϕ(qj) + m(n− 1) ϕ(0) ] = 0 (3.4) for all (q1, . . . ,qm) ∈ γm. we divide our discussion into two cases. case 1. ϕ(1) + (n− 1) ϕ(0) 6= 1. in this case, (3.4) reduces to m∑ j=1 ϕ(qj) = −m(n−1) ϕ(0) for all (q1, . . . ,qm) ∈ γm. by result 2.1, there exists an additive mapping a : r → r such that ϕ is of the form (3.1) with a(1) as in (3.2)(i). thus, we have obtained the solution (3.1) satisfying (i) in (3.2). case 2. ϕ(1) + (n− 1) ϕ(0) − 1 = 0. let us write (1.6) in the form m∑ j=1 { n∑ i=1 ϕ(piqj) −ϕ(qj) n∑ i=1 ϕ(pi) −m(n− 1)ϕ(0)qj n∑ i=1 ϕ(pi) } = 0 . 25 p. nath and d.k. singh by result 2.1, there exists a mapping a1 : γn × r → r, additive in the second variable, such that n∑ i=1 ϕ(piq) −ϕ(q) n∑ i=1 ϕ(pi) −m(n− 1) ϕ(0) q n∑ i=1 ϕ(pi) (3.5) = a1(p1, . . . ,pn; q) −ϕ(0) n∑ i=1 ϕ(pi) + nϕ(0) valid for all (p1, . . . ,pn) ∈ γn and q ∈ i with a1(p1, . . . ,pn; 1) = mϕ(0) [ n∑ i=1 ϕ(pi) −n ] . (3.6) let x ∈ i and (r1, . . . ,rn) ∈ γn. putting successively q = xrt, t = 1, . . . ,n in (3.5), adding the resulting n equations so obtained and then substituting the value of n∑ t=1 ϕ(xrt) calculated from (3.5), we get the equation n∑ i=1 n∑ t=1 ϕ(xpirt) − [ϕ(x) + m(n− 1) ϕ(0) x−ϕ(0)] (3.7) × n∑ i=1 ϕ(pi) n∑ t=1 ϕ(rt) −n2 ϕ(0) = a1(p1, . . . ,pn; x) + m(n− 1) ϕ(0) x n∑ i=1 ϕ(pi) + a1(r1, . . . ,rn; x) n∑ i=1 ϕ(pi) . the symmetry of the left hand side of (3.7), in pi and rt, i = 1, . . . ,n; t = 1, . . . ,n gives rise to the equation [a1(p1, . . . ,pn; x) + m(n− 1) ϕ(0) x] [ n∑ t=1 ϕ(rt) − 1 ] (3.8) = [a1(r1, . . . ,rn; x) + m(n− 1) ϕ(0) x] [ n∑ i=1 ϕ(pi) − 1 ] . case 2.1. n∑ t=1 ϕ(rt) − 1 vanishes identically on γn. 26 on a functional equation related to information theory in this case, by result 2.1, there exists an additive mapping a : r → r such that ϕ is of the form (3.1) but now a(1) is as in (3.2)(ii). case 2.2. n∑ t=1 ϕ(rt) − 1 does not vanish identically on γn. then, there exists a probability distribution (r∗1, . . . ,r ∗ n) ∈ γn such that[ n∑ t=1 ϕ(r∗t ) − 1 ] 6= 0 . (3.9) setting rt = r∗t , t = 1, . . . ,n in (3.8) and using (3.9), we obtain the equation a1(p1, . . . ,pn; x) = b(x) [ n∑ i=1 ϕ(pi) − 1 ] −m(n− 1) ϕ(0) x (3.10) where b : r → r is defined as b(x) = [ n∑ t=1 ϕ(r∗t ) − 1 ]−1 [a1(r ∗ 1, . . . ,r ∗ n; x) + m(n − 1) ϕ(0) x] for all x ∈ r. it can be easily verified that b : r → r is an additive mapping with b(1) = mϕ(0). from (3.5), (3.10), b(1) = mϕ(0) and the additivity of b : r → r, it follows that n∑ i=1 [m(piq) −m(q)m(pi) + n(m− 1) ϕ(0) m(q) pi] = 0 (3.11) where m : i → r is defined as m(x) = ϕ(x) + b(x) + m(n− 1)ϕ(0)x−ϕ(0) (3.12) for all x ∈ i. from (3.12), it is easy to see that m(0) = 0 as b(0) = 0. applying result 2.1 on (3.11), there exists a mapping e : r × i → r, additive in the first variable such that m(pq) −m(p)m(q) + n(m− 1) ϕ(0) m(q) p = e(p,q) − 1 n e(1,q) (3.13) for all p ∈ i, q ∈ i. the substitution p = 0 in (3.13) and the use of m(0) = 0 gives e(1,q) = 0 for all q ∈ i. consequently, m(pq) −m(p)m(q) + n(m− 1) ϕ(0) m(q) p = e(p,q) (3.14) 27 p. nath and d.k. singh for all p ∈ i, q ∈ i. since e(1,q) = 0, therefore e(1, 1) = 0. now, putting p = q = 1 in equation (3.14), we obtain m(1)[1 −m(1) + n(m− 1)ϕ(0)] = 0 . (3.14a) we prove that m(1) 6= 0. to the contrary, suppose that m(1) = 0. putting q = 1 in (3.14) and using m(1) = 0, we get m(p) = e(p, 1) for all p ∈ i. so, m is additive on i. also, if we put x = 1 in (3.12), use m(1) = 0 and ϕ(1) + (n− 1)ϕ(0) = 1, we obtain n(m− 1)ϕ(0) = −1. now from (3.9), (3.12) and the additivity of m on i, we have 1 6= n∑ t=1 ϕ(r∗t ) = 1 a contradiction. hence m(1) 6= 0. now, from (3.14a), it follows that m(1) − 1 = n(m− 1)ϕ(0) . (3.15) our next task is to prove that m : i → r, defined by (3.12), is not additive. to the contrary, suppose that m is additive. now from (3.9), (3.12), the additivity of m on i and (3.15), we have 1 6= n∑ t=1 ϕ(r∗t ) = m(1) −n(m− 1)ϕ(0) = 1 a contradiction. hence m : i → r is not additive. now we prove that, indeed, m(1)−1 = 0. if possible, suppose [m(1)−1] 6= 0. putting q = 1 in (3.14) and using (3.15), we obtain [m(1)p−m(p)] = [m(1) − 1]−1e(p, 1) for all p ∈ i. since p 7−→ e(p, 1) is additive on i, it follows that p 7−→ m(1)p−m(p) must also be additive on i. but p 7−→ m(1)p is additive on i. hence m is additive on i contradicting the fact that m is not additive. hence m(1) − 1 = 0, that is, m(1) = 1. 28 on a functional equation related to information theory now, from (3.15), it follows that ϕ(0) = 0. consequently, equation (3.14) reduces to the equation m(pq) −m(p) m(q) = e(p,q) (3.16) for all p ∈ i, q ∈ i and (3.12) reduces to (3.3) for all p ∈ i with b(1) = 0. the left hand side of (3.16) is symmetric in p and q. hence e(p,q) = e(q,p) for all p ∈ i, q ∈ i. consequently, e is also additive on i in the second variable. we may assume that e(p, ·) has been extended additively to the whole of r. let p ∈ i, q ∈ i, r ∈ i. from (3.16), we have e(pq,r) + m(r) e(p,q) = m(pqr) −m(p) m(q) m(r) (3.17) = e(qr,p) + m(p) e(q,r) . we prove that e(p,q) = 0 for all p ∈ i, q ∈ i. if possible, suppose there exists a p∗ ∈ i and a q∗ ∈ i such that e(p∗,q∗) 6= 0. then, (3.17) gives m(r) = [e(p∗,q∗)]−1{e(q∗r,p∗) + m(p∗)e(q∗,r) −e(p∗q∗,r)} from which it follows that m is additive on i contradicting the fact that m is not additive. hence e(p,q) = 0 for all p ∈ i, q ∈ i. now, from (3.16), it follows that m(pq) = m(p) m(q) for all p ∈ i, q ∈ i. thus, m : i → r is a multiplicative mapping which is not additive and m(0) = 0, m(1) = 1. 4 the functional equation (1.5) in this section, we prove: theorem 4.1. let n > 3, m > 3 be fixed integers and f : i → r, gj : i → r, j = 1, . . . ,m be mappings which satisfy the functional equation (1.5) for all 29 p. nath and d.k. singh (p1, . . . ,pn) ∈ γn and (q1, . . . ,qm) ∈ γm. then, any general solution of (1.5), for all p ∈ i, is of the form{ f(p) = b(p) gj any arbitrary real-valued mapping (4.1) or   f(p) = [f(1) + (n− 1) f(0)] a(p) + f(0), [f(1) + (n− 1) f(0)] 6= 0 gj(p) = a(p) + a ∗(p) + gj(0) (4.2) or { f(p) = f(1)[m(p) −b(p)] , f(1) 6= 0 gj(p) = m(p) −b(p) + a∗(p) + gj(0) (4.3) where b : r → r, a : r → r, a∗ : r → r, b : r → r are additive mappings with   (i) b(1) = 0 (ii) b(1) = 0 (iii) a(1) = 1 −nf(0)[f(1) + (n− 1)f(0)]−1 (iv) a∗(1) = − m∑ j=1 gj(0) + nmf(0)[f(1) + (n− 1)f(0)]−1 (4.4) and m : i → r is a multiplicative mapping which is not additive and m(0) = 0, m(1) = 1. proof. put p1 = 1, p2 = . . . = pn = 0 in (1.5). we obtain m∑ j=1 [f(qj) + (n− 1)f(0)] = [f(1) + (n− 1) f(0)] m∑ j=1 gj(qj) (4.5) for all (q1, . . . ,qm) ∈ γm. case 1. f(1) + (n− 1) f(0) = 0 . then, (4.5) reduces to the equation m∑ j=1 f(qj) = −m(n−1) f(0). put q1 = 1, q2 = . . . = qm = 0 in this equation and using the fact f(1) + (n−1) f(0) = 0, we have 30 on a functional equation related to information theory f(0) = 0 = f(1). hence m∑ j=1 f(qj) = 0. by result 2.1, there exists an additive mapping b : r → r such that f(p) = b(p) with b(1) = 0. consequently, for all (p1, . . . ,pn) ∈ γn, (q1, . . . ,qm) ∈ γm, it is easy to verify that n∑ i=1 m∑ j=1 f(piqj) = n∑ i=1 f(pi) = b(1) = 0. now, from (1.5), it follows that gj (j = 1, . . . ,m) are, indeed, arbitrary real-valued mappings. thus, we have obtained the solution (4.1) of (1.5) where b(1) is given by (4.4)(i). case 2. f(1) + (n− 1) f(0) 6= 0. in this case, (4.5) can be written in the form m∑ j=1 { gj(qj) − [f(1) + (n− 1) f(0)]−1[f(qj) + (n− 1) f(0)] } = 0 . (4.6) by result 2.2, there exists an additive mapping a∗ : r → r such that gj(p) = [f(1) + (n− 1) f(0)]−1[f(p) −f(0)] + a∗(p) + gj(0) (4.7) for j = 1, . . . ,m with a∗(1) given by (iv) in (4.4). the elimination of m∑ j=1 gj(qj) from equations (1.5) and (4.6) gives the equation n∑ i=1 m∑ j=1 f(piqj) = [f(1) + (n− 1) f(0)]−1 n∑ i=1 f(pi) m∑ j=1 f(qj) (4.8) + [f(1) + (n− 1) f(0)]−1m(n− 1) f(0) n∑ i=1 f(pi) valid for all (p1, . . . ,pn) ∈ γn and (q1, . . . ,qm) ∈ γm. define a mapping ϕ : i → r as ϕ(x) = [f(1) + (n− 1) f(0)]−1 f(x) (4.9) for all x ∈ i. then (4.8) reduces to the functional equation (1.6) with ϕ(1) + (n− 1) ϕ(0) = 1. so, we need to consider only those solutions of (1.6) which satisfy the requirement ϕ(1) + (n− 1) ϕ(0) = 1. 31 p. nath and d.k. singh the solutions (3.1) (with (3.2)(ii)) and (3.3) of (1.6) satisfy the condition ϕ(1) + (n − 1) ϕ(0) = 1. making use of (4.9), (4.7), (3.1) (with (3.2)(ii)) and (3.3), the solutions (4.2) and (4.3) can be obtained in which b(1), a(1) and a∗(1) are given respectively by (ii), (iii) and (iv) in (4.4). 5 the functional equation (1.3) in this section, we prove: theorem 5.1. let n > 3, m > 3 be fixed integers and h : i → r, kj : i → r, j = 1, . . . ,m be mappings which satisfy the functional equation (1.3) for all (p1, . . . ,pn) ∈ γn and (q1, . . . ,qm) ∈ γm and λ 6= 0. then, any general solution of (1.3), for all p ∈ i, is of the form   h(p) = 1 λ [b(p) −p] kj any arbitrary real-valued mapping (5.1) or   h(p) = 1 λ { [λ(h(1) + (n− 1) h(0)) + 1] a(p) −p } + h(0), [λ(h(1) + (n− 1) h(0)) + 1] 6= 0 kj(p) = 1 λ { a(p) + a∗(p) −p } + kj(0) (5.2) or   h(p) = 1 λ { [λh(1) + 1][m(p) −b(p)] −p } , [λh(1) + 1] 6= 0 kj(p) = 1 λ { m(p) −b(p) + a∗(p) −p } + kj(0) (5.3) where b : r → r, a : r → r, a∗ : r → r, b : r → r are additive mappings 32 on a functional equation related to information theory with  (i) b(1) = 0 (ii) b(1) = 0 (iii) a(1) = 1 −λnh(0)[λ(h(1) + (n− 1)h(0)) + 1]−1 (iv) a∗(1) = −λ m∑ j=1 kj(0)+λnmh(0)[λ(h(1) +(n− 1)h(0))+ 1]−1 (5.4) and m : i → r is a multiplicative mapping which is not additive and m(0) = 0, m(1) = 1. proof. from (1.4) and the solutions of the functional equation (1.5) i.e., (4.1), (4.2), (4.3) with (4.4); we obtain respectively the solutions (5.1), (5.2), (5.3) with (5.4); of the functional equation (1.3). the details are omitted. remarks. the object of this remark is to point out the importance of various solutions of theorem 5.1 from information-theoretic point of view. 1. the summand n∑ i=1 h(pi) of the mapping h appearing in (5.1) is independent of the probabilities p1, . . . ,pn. the solution (5.1) may be of some importance in information theory provided kj is chosen as a suitable mapping of probability p, p ∈ i. 2. in solution (5.2), the summands n∑ i=1 h(pi) and m∑ j=1 kj(qj) are independent of the probabilities p1, . . . ,pn and q1, . . . ,qm respectively. so, this solution does not seem to be of any relevance in information theory. 3. in solution (5.3) n∑ i=1 h(pi) = 1 λ { β1 n∑ i=1 m(pi) − 1 } 33 p. nath and d.k. singh and m∑ j=1 kj(qj) = 1 λ { m∑ j=1 m(qj) − 1 } + β2 where β1 = λh(1) + 1 β2 = nmh(0)[λ(h(1) + (n− 1)h(0)) + 1]−1 . if β1 = 1 and β2 = 0, then n∑ i=1 h(pi) = l λ n(p1, . . . ,pn) and m∑ j=1 kj(qj) = l λ m(q1, . . . ,qm) where (see nath and singh [6]) lλt (x1, . . . ,xt) = 1 λ [ t∑ i=1 m(xi) − 1 ] . (5.5) the non-additive measure of entropy hαt (x1, . . . ,xt) = (2 1−α−1)−1( t∑ i=1 xαi −1), α 6= 1, is a particular case of (5.5) when λ = 21−α − 1, α > 0, α 6= 1 and m : i → r is of the form m(p) = pα, p ∈ i, α 6= 1, α > 0, 0α := 0, 1α := 1. references [1] z. daróczy and l. losonczi, über die erweiterung der auf einer punktmenge additiven funktionen, publ. math. (debrecen) 14 (1967), 239–245. [2] j. havrda and f. charvát, quantification method of classification process, concept of structural α-entropy, kybernetika (prague) 3 (1967), 30–35. [3] pl. kannappan and p.k. sahoo, on the general solution of a functional equation connected to sum form information measures on open domain-vi, radovi matematicki 8 (1992), 231–239. [4] l. losonczi, functional equations of sum form, publ. math. (debrecen) 32 (1985), 57–71. 34 on a functional equation related to information theory [5] l. losonczi and gy. maksa, on some functional equations of the information theory, acta math. acad. sci. hung. 39 (1982), 73–82. [6] p. nath and d.k. singh, on a multiplicative type sum form functional equation and its role in information theory, applications of mathematics 51(5) (2006), 495–516. [7] p. nath and d.k. singh, on a sum form functional equation and its role in information theory, in proc.: 8th national conf. isita on “information technology: setting trends in modern era”, (2008), 88–94. [8] p. nath and d.k. singh, on a functional equation containing an indexed family of unknown mappings, functional equations in mathematical analysis (edited by themistocles rassias and janusz brzdek), springer 52 (2011), 671–687. [9] p. nath and d.k. singh, a sum form functional equation on a closed domain and its role in information theory, arxiv:1508.05910v1 (2015), 1–18. 35 ratio mathematica issue n. 30 (2016) pp. 3-21 issn (print): 1592-7415 issn (online): 2282-8214 a geometric view on pearson’s correlation coefficient and a generalization of it to non-linear dependencies priyantha wijayatunga department of statistics, umeå school of business and economics, umeå university, umeå 901 87, sweden priyantha.wijayatunga@umu.se abstract measuring strength or degree of statistical dependence between two random variables is a common problem in many domains. pearson’s correlation coefficient ρ is an accurate measure of linear dependence. we show that ρ is a normalized, euclidean type distance between joint probability distribution of the two random variables and that when their independence is assumed while keeping their marginal distributions. and the normalizing constant is the geometric mean of two maximal distances; each between the joint probability distribution when the full linear dependence is assumed while preserving respective marginal distribution and that when the independence is assumed. usage of it is restricted to linear dependence because it is based on euclidean type distances that are generally not metrics and considered full dependence is linear. therefore, we argue that if a suitable distance metric is used while considering all possible maximal dependences then it can measure any non-linear dependence. but then, one must define all the full dependences. hellinger distance that is a metric can be used as the distance measure between probability distributions and obtain a generalization of ρ for the discrete case. keywords: metric/distance; probability simplex; normalization. 2010 ams subject classifications: 62h20 doi: 10.23755/rm.v30i1.5 3 priyantha wijayatunga 1 introduction measuring association between two random quantities is of interest in many types statistical analyses and applications in various disciplines. pearson’s product moment correlation coefficient is the standard in statistical textbooks and applications for measuring linear association. and spearman’s rank correlation coefficient is capable of measuring any monotonic dependence between two random variables. for two ordinal variables cramér’s v-statistic is widely used whereas tchuprow’s t-statistic is less-known and therefore less often used (see [14] and references therein). furthermore, there are many other kinds of dependence measures used in statistical literature, especially in applied statistical analyses. in statistical genetics for evaluation of linkage disequilibrium between genetic markers, authors of [2] use volume tests that are discussed in [10] as a measures of dependence between ordinal variables with fixed margins. for massive datasets in [8] it is used mutual information dimension that is defined in terms of information dimension descried in [1]. in [9] it is said that “although it is customary in bivariate data analysis to compute a correlation measure of some sort, one number (or index) alone can never fully reveal the nature of dependence; hence a variety of measures are needed”. it is also stated therein that “if (two quantities are) not totally dependent, then it may be helpful to find some quantities that can measure the strength or degree of dependence between them”. in this article we try to develop a measure that can indicate ‘the’ degree or strength of association between two discrete variables. our measure can be seen as a generalization of the pearson’s correlation coefficient ρ using a suitable distance metric between joint probability distributions, instead of simple euclidean type distances that are used in ρ (see below). given the joint probability distribution (jpd) of two discrete variables, say, x and y , the degree of dependence (also called association) between them is expressed as the normalized distance between the jpd of them and that of when the independence of them is assumed. the associated normalizing constant is geometric mean of distances between the latter and all possible jpds where full dependence between x and y is assumed while retaining each marginal distribution at a time. these latter distances are in fact the maximal distances since we obtain them by assuming full dependence. in the following we show that the pearson’s correlation coefficient is measure of this nature based on some euclidean type distances. that is, it is the ratio of the distance between dependence and independence, and the geometric mean of the distances that are between full linear dependences and independence. therefore, our measure can be regarded as a generalization of ρ using a suitable distance between probability distributions and considering non-linear dependencies. one thing that ρ shows us is that if we need to define a strength of a dependence then we must find or hypothesize the full dependence(s) corre4 a geometric view on pearson’s correlation coefficient and a generalization of it to non-linear dependencies sponding to the given dependence. this aspect can make numerical evaluation of the measure algorithmic or computational since sometimes it may not be possible to obtain the full dependences easily. however, here we do not deal with such computational issues but our consideration is on defining a measure following the structure of ρ. for a given dependence (in terms of a jpd) finding efficiently related jpds representing the full dependences that preserve either of the marginal is an open problem. first we show that, in the simple case of binary x and y , the ρ measures the degree of dependence with a certain type of euclidean distance, but for multinary case (and also for continuous variables) a distance in terms another type of euclidean area is used. but these euclidean type distances are appropriate for measuring only linear dependences. since we are interested in measuring any non-linear dependence we propose to use hellinger distance between joint probability distributions, that is called as matsusita distance in the discrete (see [6]). the hellinger distance is a metric and it possesses the so-called linear invariance properties, so it is more suitable for measuring distances between the probability distributions. therefore, it can be used to measure any type of dependence. 2 pearson’s correlation coefficient ρ for random variables x and y, the pearson’s correlation coefficient ρ(x,y ) is such that |ρ(x,y )| ≤ 1. the equality holds if and only if x and y are fully linearly dependent and ρ(x,y ) = 0 if they are linearly independent. and the converse of the latter is not always true unless x and y are binary. note that the full dependence is linear in the binary (also called 2 × 2) case where then the ρ(x,y ) is often called φ-coefficient. 2.1 2 × 2 case: φ-coefficient let x and y be two binary variables with a common state space {0, 1} where their jpds and marginal probability distributions are written as pxy = p(x = x,y = y), px = p(x = x) and qy = p(y = y) for x,y = 0, 1. let p =( p00 p01 p10 p11 ) for short. as shown in [12], any such p can be represneted as a point in the probability simplex shown in the figure 1. the jpd of x and y under the assumption that they are independent while keeping the marginal distributions fixed is pi = ( p0q0 p0q1 p1q0 p1q1 ) and the set of such probability distributions for all p makes a surface (shown by lines) in the probability simplex. the φ-coefficient 5 priyantha wijayatunga of x and y is defined by φ = p11 −p1q1√ p1(1 −p1)q1(1 − q1) , which is a measure of degree of association between x and y . now let x and y be positively correlated, then there are two jpds under the assumption that the two variables are fully dependent. they are px = ( p0 0 0 p1 ) and py = ( q0 0 0 q1 ) , where px is when the marginal distribution of x is preserved and py is when the marginal distribution of y is preserved. note that each full dependence is obtained from p while preserving respective marginal distribution, then the marginal distribution of the other variable should be assumed by it. therefore in these cases, the full dependence is essentially linear. for a generalization of ρ to measure ‘any’ type of dependence we need to look at its structure and construction. first we consider the case of two binary variables by examining the φ-coefficient. let dp i,p be p11 −p1q1 that is the (2, 2)th component euclidean distance between the two probability distributions pi and p . it is a measure of how far the dependence (under p ) from the independence (under pi ) when marginals of x and y are fixed. note that in the 2 × 2 case it is sufficient to consider a single component difference (between the two probability matrices) since all the components have same absolute difference. similarly, we have dp i,p x = p1(1 − q1) and dp i,p y = q1(1 − p1). since px and py are the two full dependences that we can obtain from p while preserving respective marginal in each case, we have that dp i,p ≤ dp i,p x and dp i,p ≤ dp i,p y . in fact dp i,p = p11−p1q1 = p1(p11/p1−q1) ≤ p1(1−q1) = dp i,p x since p1 ≥ p11 and similarly the other inequality. it is easy to see that the denominator of the φ-coefficient is the geometric mean of dp i,p x and dp i,p y (the two maximal distances) and the numerator is dp i,p . therefore, the φ-coefficient can be thought of as the normalized distance between p and pi where the normalizing constant is the geometric mean of the two maximal distances. hence the φ-coefficient is 1 if and only if p = px = py (full dependence) and it is 0 if and only if pi = p (independence). 2.2 n×m case let x and y be two multinary random variables where their state spaces are {0, 1, ..,n − 1} and {0, 1, ..,m − 1} respectively for n,m > 2. for any given jpd of x and y, p = (p00, ...,p0(m−1); p10, ...,p1(m−1); ...; p(n−1)1, ...,p(n−1)(m−1)) where pij = p(x = i,y = j) for i = 0, ..,n − 1 and j = 1, ...m − 1, we define the probability simplex, ∆ = {p = (pij)n×m : ∑ ij pij = 1,pij ≥ 0; i = 6 a geometric view on pearson’s correlation coefficient and a generalization of it to non-linear dependencies (0, 1, 0, 0) (1, 0, 0, 0) (0, 0, 1, 0) (0, 0, 0, 1) figure 1: probability simplex for binary x and y where their jpd p = (p00,p10,p01,p11) is a point in it. any jpd on surface shown by lines represents independence of x and y. 0, 1, ..,n − 1; j = 0, 1, ...,m − 1} similar to the case of two binary random variables. but here visualization of it is more difficult. recall that ρ(x,y ) = cov(x,y )/ √ var(x)v ar(y ), where cov(x,y ) = ∑ x,y xyp(x,y) − ∑ x xp(x) ∑ y yp(y) and var(x) = ∑ x x2p(x) −{ ∑ x xp(x)}2. in the following we try to visualize the ρ and its structure for understanding how it measures the dependence. let us take the case where n = m, thus allowing us to have perfect (one-toone) dependence between x and y, linear or non-linear. it can be seen that when x and y are assigned to two perpendicular axes, cov(x,y ) is area difference between two rectangular euclidean areas, that is shown as the dark area in the figure 2. the first area (i.e., ∑ x,y xyp(x,y)) is the weighted average area created by the values of x and y, where, for each component area that is being weighted is with side lengths x = x and y = y and its weight is the respective joint probability of x = x and y = y, i.e., p(x = x,y = y). this area represents the 7 priyantha wijayatunga x y x1 x2 xn y1 y2 yn e{xy} e{x} e{y} cov(x,y ) figure 2: covariance of x and y is the weighted averaged euclidean area difference. dependence between x and y . and the second area (i.e., ∑ x xp(x)× ∑ y yp(y)) is the area created by the side lengths that are the weighted average of values of x (i.e., e{x}) and that of y (i.e., e{y}) where the weights are the respective marginal probabilities. since the lengths or values e{x} and e{y} are also on same axes as x and y are, respectively, we can see the difference of the two areas. note that it can be seen that the second area (i.e., ∑ x,y xyp(x)p(y)) is also calculated in the similar way as the first, but assuming the independence of x and y , i.e., it is the weighted average area created by the values of x and y , where for each component area that is being weighted is with side lengths x = x and y = y and the weight associated with it is the respective joint probability of x = x and y = y assuming independence p(x = x,y = y) = p(x = x)p(y = y). so the second area represents the scenario of the independence of x and y . therefore one can view that the two areas refer to those when a dependence between x and y is assumed and when their independence is assumed while keeping the marginal distributions fixed, therefore cov(x,y ) is a ‘distance’ in terms of a euclidean area difference between dependence and independence of the two variables. moreover var(x) can be interpreted in the same way. now x is assumed to be on both axes meaning that y is replaced by x (taken as if y were x). this is a context of assuming a full dependence of x and y when the marginal of x is 8 a geometric view on pearson’s correlation coefficient and a generalization of it to non-linear dependencies preserved. assuming one variable by the other is ‘a way’ to consider a case of full dependence between the two variables. then we are assuming the marginal of y by that of x. this assumption is easily seen when both variables have same sizes in their state spaces but it is hard to see when they are different. so the e{x2} is indicated by the weighted average area that we obtain when y is x where weight for each component area x2 is p(x,y) = p(x), i.e., when the marginal of x is preserved. this is a sensible area under full dependence. and e{x}2 is indicated by the area when the respective weight is p(x)p(y) = p(x)2 where x = y. this is a hypothetical case where it is taken as if y were x, yet their joint probability is taken as if they were independent. so, var(x) is deviation of the full dependence from independence if y were x. and the same interpretation applies for var(y ). thus, ρ(x,y ) is the normalized area difference referring to cov(x,y ) with the normalizing constant being the geometric mean of the two maximal area differences referring to cov(x,y ) where they are such that, one is when y is assumed to be x (i.e., var(x)) and the other is when x is assumed to be y (i.e., var(y )). that is, the normalizing constant is obtained by assuming the full dependence between x and y. however the full dependence quantified in this way is appropriate only for doing so for linear dependences. since there are two such cases of full linear dependence the geometric mean of these two maximal area differences is taken. note that the above interpretation is valid for the case of x and y have continuous state spaces. one thing that we need to show is that cov(x,y ) is maximal (or minimal) when x and y are strictly monotonically related, for example, linearly related positively (negatively), among all cases of full ono-to-one dependencies between x and y for fixed maginals of x and y . this indicates that ρ is not able to identify non-monotonic relations since their covariance values can not be ordered. to see that cov(x,y ) is maximal when y is strictly increasing with x, let x = {a1 < ... < an} be the state space of x and y = {b1 < ... < bn} be that of y . then considering inequalities (ai−aj)(bi−bj) > 0 for i,j = 1, ...,n (i.e., we have aibi + ajbj > aibj + ajbi) it can be shown that ∑ i aibi > ∑ i,j:j=f(i) aibj where f is any one-to-one function from x to y such that f(i) 6= i for at least two distinct values of i (i.e., f is not a strictly increasing function of i). now if the marginals of x and that of y are (p1, ...,pn) and (q1, ...,qn), where pi = qi for all i = 1, ...,n when y is monotonically increasing with x and otherwise pi = qj for some appropriate i 6= j for i,j = 1, ....,n, then ∑ i aibipi > ∑ i,j:j=f(i) aibjpi meaning that e{xym} ≥ e{xy} where ym is y when it is strictly increasing with x. this implies that cov(x,ym ) ≥ cov(x,y ) for fixed marginals of x and y . therefore, for discrete x and y , ρ(x,y ) is maximal when y is strictly increasing in x, among all one-to-one relationships between them. so, if this is the case ρ(x,y ) = 1 (maximal) since cov(x,y ) ≤ var(x) and cov(x,y ) ≤ var(y ). 9 priyantha wijayatunga 3 some other popular measures of dependence there are a few popular measures of dependence that have similar structure in their definition. we review them briefly by giving some interpretations that support our definition of dependence measure. 3.1 spearman’s rank correlation coefficient ρs in many statistical analyses, especially for non-normal data a popular measure of dependence between two random variables, say, x and y , is the spearman’s rank correlation coefficient. ρs = 1 − 6 ∑n i=1 d 2 i n(n2 − 1) where di = x(i) − y(i) and x(i) is the ith smallest value in the data sample of x and similarly for y(i). it is obvious that ρs = 1 if and only if two components of data pair (xi,yi) has the same ranking, for all data pairs since then di = 0 for all i. and one can see that for a perfect negative dependence ∑n i=1 d 2 i should be its maximal value that is n(n2 − 1)/3 in order to get ρsx,y = −1. therefore the normalizing constant is taken as n(n2 − 1)/6 but due to the structure of the definition of the coefficient it is applied to the term ∑n i=1 d 2 i . therefore the ρ s is an accurate measure any monotonic dependence between the two variables. however, when the two variables are not having a strictly monotonic relationship the measure can not give a correct picture of the dependence. 3.2 information theoretic measures another popular measure of dependence, especially in machine learning literature and applied statistics is so-called mutual information (see, for example, [11]). for discrete random variables x and y , it is defined as i(x,y ) = ∑ x,y p(x,y)log p(x,y) p(x)p(y) and furthermore, conditional mutual information between x and y given another variable z is defined as ci(x,y,z) = ∑ x,y,z p(x,y,z)log p(x,y|z) p(x|z)p(y|z) (1) if x and y are independent then the i(x,y ) = 0 and if x and y are conditionally independent given z then the ci(x,y,z) = 0. in fact, these dependence measures are also based on so-called kullback-leibler (kl) distance or 10 a geometric view on pearson’s correlation coefficient and a generalization of it to non-linear dependencies rather divergance, [13]. it is easy to see that i(x,y ) is the kl divergence between the joint probability distribution of x and y , and that when independence is assumed, therefore it measures the dependence in terms of ‘departure’ from independence. in fact, i(x,y ) is the weighted average of euclidean distance between logarithmic of the joint probability p(x,y) and that when independence is assumed, where weights are the respective joint probabilities. that is, it is the expectation, under the joint probability, of the difference between the logarithmic of the joint probability p(x,y) and that when independence is assumed. note that though 0 ≤ i(., .) ≤ 1, there is no normalization (with respect to any maximal dependence) is involved. though these information measures are used to identify respective dependences they are not metrics since kl-divergance is not a true distance (metric), therefore they can not be used to measure the degree of dependence between variables. for example, as shown in [7] let p(x,y) and q(x,y) define two dependencies between x and y where p(x,y) = ( 3/8 1/8 1/8 3/8 ) and q(x,y) =( 1/2 0 1/8 3/8 ) . obviously probability distribution q shows a higher dependency than that of p but its mutual information is lower than that of p, (mip(x,y ) > miq(x,y )). note that q is obtained from p without preserving the marginal distributions of x and y . now let r(u,v) and s(u,v) define two dependencies between random variables u and v where r(u,v) =   0 1/7 1/71/7 1/7 1/7 1/7 1/7 0   and s(u,v) =   0 0 2/71/7 2/7 0 1/7 1/7 0   . then we have that mir(u,v ) < mis(u,v ). note that s shows a higher dependency than that of r and it is obtained from r by preserving the marginal distributions of u and v . furthermore, all zeros in r are also in s. if this is the case then higher dependency implies higher mutual information. so mutual information is restricted measure of degree of dependence. 3.3 chi squared test statistic χ2 we can see that well-known chi squared test statistic χ2 that is used for testing independence of two discrete random variables uses a certain dependence measure in it for performing the test. let x and y take values i = 1, ...,α and j = 1, ...,β, respectively and let us write the joint probability of x = i and y = j as pij, marginal probability of x = i as pi. and that of y = i as p.j. so, the conditional probability of x = i given y = j is pi|j = pij/p.j and similarly pj|i is defined. 11 priyantha wijayatunga then, χ2 = ∑ i,j n (pij −pi.p.j)2 pi.p.j = n {∑ i,j p2ij pi.p.j − 1 } = n {∑ i,j pij pij −pi.p.j pi.p.j } = n {∑ i,j pij pi|j −pi. pi. } = n {∑ i,j pij pj|i −p.j p.j } = ne{a} where a is a random variable taking the value pi|j−pi. pi. = pj|i−p.j p.j with probability pij, for i = 1, ...,α and j = 1, ...,β, and e denotes the expectation. that is, χ2 is n-multiple of the expectation of a random variable whose (i,j)th value is a ‘normalized’ distance between the probability value pi|j and pi. where the normalizing constant is pi., for all i,j, and vice versa. note that pi|j−pi. pi. may be referred to as the ‘degree’ of dependence between the two events x = i and y = j. in fact, it is the certainty factor for the case pi|j < pi., as described in [4] for measuring the dependency between the two events and it is a symmetric measure. however, here it is used without the condition. so, e{a} is the expectation of a degree of dependence between the events x = x and y = y for all x,y. therefore, e{a} can be thought of as measure of degree of dependence between x and y. and the term n in χ2 makes it a statistic. that is, a statistic for testing dependence between two variables can be seen as a product of two factors; one is a quantity related the degree of dependence between two variables and the other is that of total number of data cases that are used to estimate the probabilities related to them (i.e., sample information). 3.4 test of two proportions sometimes one may be interested in testing equality of two proportions to see if given two variables are independent, for example, when the outcome (y ) of interest is binary, such as voting, denoted by y = 1 (or not, denoted by y = 0), for a political candidate in an election for two groups/populations (x) such as men, denoted by x = 1, and women, denoted by x = 0. then one can test if two proportions are equal, i.e., p(y = 1|x = 1) = p(y = 1|x = 0) (let us write it as p1 = q1) by the z statistics z = 1√ 1/a + 1/b p1 − q1√ p(1 −p) where a and b are the sizes of the two samples of y when x = 1 and x = 0, respectively, and p = p(y = 1). now we can interpret that the factor p1−q1√ p(1−p) as a measure of degree of dependence between the two variables due to the term 12 a geometric view on pearson’s correlation coefficient and a generalization of it to non-linear dependencies (p1−q1) in it, where the term √ p(1 −p) should be taken as the normalizing constant. note that the latter is constructed assuming full dependence between the two variables where, then their joint probability distribution is p = ( 1 −p 0 0 p ) or similar. instead of just using p which is the pooled proportion, the geometric mean of p and (1 − p) should be used as the normalizing constant. this is necessary to yield the same test statistic value for testing the same hypothesis with complementary probabilities i.e., p(y = 0|x = 1) and p(y = 0|x = 0). and the term 1√ 1/a+1/b which is a function of sample sizes (sample information) makes z a statistics. so, similar to χ2 statistic, z has a measure of degree of dependence between the two variables in it, in addition to information on the sample sizes. 4 axioms of an ideal measure of dependence before we define our measure of strength/degree of dependence (or rather a generalization of ρ) it is appropriate to mention axioms that an ideal measure should possess as shown in [3]. however, it is hard to find dependence measures satisfying all these axioms. our generalization of ρ seems to have a bigger potential in satisfying them, but we omit the discussion here. following are the axioms; 1. it is well-defined for both continuous and discrete case 2. it is normalized such that its value 0 implies the independence and value 1 implies the full dependence (one variable is a deterministic function of the other), where all intermediate degrees of dependencies lie between 0 and 1 3. it is equal or has a simple relationship with the pearson’s correlation coefficient in the case of a bivariate normal distribution 4. it is a metric, i.e., it is a true measure of distance (between the independence and dependence of interest) not just a divergence 5. it is invariant under continuous and strictly increasing transformations. these axioms are straightforward and require no further explanation. in the following we define our measure following the structure and the construction of ρ but using a true distance metric. we propose to use so-called hellinger distance but one may use another suitable distance metric. since we are keeping the structure of the ρ the same but replacing its distance measure with a better one (a metric) when defining our dependence measure, we call it as a generalization of the ρ. this means that for any given dependence we should be 13 priyantha wijayatunga able to define the corresponding all possible full dependences, since the measure should be a ratio between a distance from independence to the given dependence and geometric average of distances from independence to the full dependences. 5 defining a measure of degree of dependence as we have seen earlier, in the two binary variables (2 × 2) case where only the linear dependence exists the dependence can be measured by using a single component euclidean distance between joint probability distributions. however, in the case of two multinary variables (n × n, where n > 2) we can have many types of dependences, and therefore distances among probability distributions can not be defined through only a single component or a weighted average area difference, that are euclidean type distances and capable of measuring only linear dependences. therefore we need to use some other suitable distance to measure any non-linear dependences. in the following we discuss a possible distance that is a true metric. 5.1 a metric distance between two probability distributions we propose to use hellinger distance between probability distributions (also called matsushita distance for the discrete case) which is a metric in the probability simplex for our task of measuring dependence. recall that our dependence measure should be the normalized distance between the given joint probability distribution of the two variables and that when their independence is assumed while preserving the marginals, where the normalizing constant is obtained by considering similar distances related to the all possible maximal dependences but preserving only one of the marginals at each time. let φ and ψ be two discrete distribution functions (φ and ψ are probability distributions or mass functions) then the hellinger distance between φ and ψ is defined as m(φ, ψ) = { 1 2 ∑ x {√ φ(x) − √ ψ(x) }2}1/2 in addition to satisfying properties of a metric m(., .) also satisfies the following properties: (1) 0 ≤ m(φ, ψ) ≤ 1, (2) m(φ(t), ψ(t)) = m(φ(t + a), ψ(t + a)) for any constant a, and (3) m(φ(t), ψ(t)) = m(φ(ct), ψ(ct)) for any constant c 6= 0 where the last two are called the linear invariance properties of the probability metric. note that ( m(., .) )2 is not a metric. first we should have an idea about the furtherest jpd(s) for a given jpd that may represent independence. in fact we can see that the furtherest probability 14 a geometric view on pearson’s correlation coefficient and a generalization of it to non-linear dependencies distribution to a distribution that represent independence is not useful but those with fixed marginals, each at a time. for a given distribution function, say, φ let us find the maximally hellinger-distanced distribution function ψ. the following proposition shows how to find it. proposition 5.1. for positive probability distribution φ maximally hellinger-distanced probability distribution ψ is given by ψ(t) = { 1, if t = argminu φ(u) 0, otherwise. and then, m(φ, ψ) = { 1 − √ min{φ(t) : t ∈t} }1/2 < 1. proof. let |t | = n, φ(ti) = φi and ψ(ti) = ψi for i = 1, ...,n. let re-index all φi’s such that φ(1) ≥ φ(2) ≥ .... ≥ φ(n) and possibly some of the ψi’s can be zeros. m(φ, ψ) is maximal when ∑ t∈t √ φ(t)ψ(t) is minimal. n∑ i=1 √ φiψi = ( √ ψ1 + ... + √ ψn) √ φ(n) +( √ ψ1 + ... + √ ψn−1)( √ φ(n−1) − √ φ(n)) ... + √ ψ1( √ φ(1) − √ φ(2)) ≥ √ φ(n) that is, ∑n i=1 √ φiψi is minimal when ψ1 = ... = ψn−1 = 0 and ψn = 1. so we obtain the maximally hellinger-distanced distribution function ψ and therefore m(φ, ψ).2 but then t is deterministic variable with respect to ψ! this theorem says that for any given probability distribution, bivariate discrete in our case, the maximally hellinger-distanced probability distribution is represented by a vertex of the probability simplex. all its component are zeros except for one place that has 1 that is corresponding to the smallest probability value of the reference probability distribution. this is a degenerate case as far as dependence of the two variables are concerned since it represents that both variables are deterministic and having full dependence. therefore, such a full dependence can not be used for the normalization since it does not generally preserve the marginals. for a given jpd p of x and y, the dependence of them that it represents should be measured with a suitable normalized distance between p and pi . it is clear from above that the normalizing constant should be the geometric mean of distances from independence to all possible full dependences where each such 15 priyantha wijayatunga full dependence should be preserving either of marginals. this rule is to follow the correlation coefficient definition. therefore, an essential step is to find the two types of probability distributions px (jpd(s) representing full dependence when marginal of x is fixed) and py (jpd(s) representing full dependence when marginal of y is fixed) in order to find the normalizing constant. as you will see in some cases there may be multiple candidates for each of them. therefore we have the following definition. note that there are some instances such as in [3] and [5] where hellinger distance between the jpd and that of when independence is assumed is used for measuring the dependence, but in such work no normalization is done. however, the above proposition implies that distance between any non-deterministic jpd representing independence and that representing a full dependence can be strictly less than 1 for two discrete random variables, therefore normalization is necessary if one wants to have a measure that shows strength of dependence. definition 5.1. when m is a metric in the probability simplex of two discrete random variables x and y, m-based measure of degree of dependence between x and y represented by their joint distribution function p is defined as ρm (x,y ) = m(pi,p)∏ p x∈pxmax ∏ p y ∈pymax { m(pi,px )m(pi,py ) }1/|pymax+pxmax| where pi is the joint distribution function of x and y when their independence is assumed, pxmax denotes the set of all joint distribution functions, each representing a maximal dependence while preserving the marginal distribution of x and similarly for pymax, |a| is the cardinality of the set a, and m(p,q) is the distance metric between two probability distributions p and q. note that the denominator is the geometric mean of the maximal distances between full dependences and the independence. and we use hellinger distance as the distance measure. since ρm is defined following the structure of the pearson’s correlation coefficient it can be regarded as a generalization of it for the case of discrete variables. for linear relationships measuring the dependence is relatively easy since both px and py represent perfect linear dependence. this is when they have all their entries zero except for those, but may not be all, in each diagonal in respective case. for example, for a positive linear relation, px is obtained by assigning each main diagonal entry with the sum of all entries in the respective row. this assures that the marginal probability of x is preserved when obtaining full dependence, and similarly for py . note that positive linear relationship is selected if main diagonal entries are generally larger than the other entries in the joint probability value matrix p . but when we allow non-linear relationships between x and y 16 a geometric view on pearson’s correlation coefficient and a generalization of it to non-linear dependencies there are no pre-specified px and py , therefore multiple candidates may exist for each of them. we argue that they should be induced from the jpd in a similar way to the case of linear dependence. so we propose following simple rule for obtaining px and py . definition 5.2. for each x, when there exists a single value y′ such that y′ = argmaxy p(x = x,y = y), then let p x (x = x,y = y′) = p(x = x) and px (x = x,y 6= y′) = 0 to obtain px . if there are multiple such y′ values then obtain multiple px , each refering to one of those y′ values, assuming that it is the only value where maxima exists. and similarly py is defined. by this way, we get one or more jpds each representing a maximal dependence that preserves respective marginal. 6 examples of n×n case where n ≥ 2 now we consider some different cases of p and demonstrate how we can calculate our measure and compare its value to those of some trational measures. case 1 suppose a simple case of each row and column of p having a single maximal entry that is common to both its row and column. then the other entries in the row are summed onto the maximal entry in the row for each row to yield px and similarly py is obtained. therefore, px and py are on the boundary of ∆, so they are the furtherest probability distributions from pi while preserving respective marginals. then the degree of dependence between x and y is defined as (since |pxmax| = |pymax| = 1 ) ρm (x,y ) = m(pi,p)√ m(pi,px )m(pi,py ) example 6.1. for binary x and y with p = ( 0.3 0.2 0.1 0.4 ) , φ = 0.4082 and ρm = 0.2783 (cramer’s v and tschuprow’s t are 0.4082). and interchanging off-main diagonal entries but keeping the main diagonal entries as they were, i.e., having p = ( 0.3 0.1 0.2 0.4 ) , gives the same results for all measures. example 6.2. let state spaces of x and y be {1, 2, 3} and their joint probability p =   0.05 0.03 0.200.30 0.07 0.05 0.04 0.20 0.06   that is a non-linear dependence and then 17 priyantha wijayatunga pi =   0.1092 0.084 0.08680.1638 0.126 0.1302 0.1170 0.090 0.0930  , px =   0.00 0.00 0.280.42 0.00 0.00 0.00 0.30 0.00   and py =   0.00 0.00 0.310.39 0.00 0.00 0.00 0.30 0.00  . and then ρ = −0.2025 but ρm = 0.4113 (cramer’s v and tschuprow’s t are 0.5472). but had that p ==   0.05 0.03 0.200.04 0.20 0.05 0.30 0.07 0.06   which is a linear dependence then ρ = −0.5474 and ρm = 0.4075 (cramer’s v and tschuprow’s t are 0.5467). note the change in the degree of dependence is small since linear dependence is obtained from nonlinear case by just interchanging probability values in p . case 2 when each row and column of p has a single maximal entry that may not be common to both its row and column we still can obtain a single px and a single py . therefore, we can apply the above definition. example 6.3. when p =   0.30 0.03 0.200.05 0.07 0.05 0.04 0.20 0.06   we have ρ = 0.1383 and ρm = 0.450011. note that here we have that cramer’s v and tschuprow’s t are 0.4257843 that are lesser than our measure. case 3 when there are more than one maximal entry in a row or a column we have multiple px ’s and multiple py ’s. note that here we try to obtain a similar situation in the above two cases. that is, each row of px has only one non-zero element (it is obtained by summing up all entries in the corresponding row of p , thereby preserving the marginal probability distribution of x). assume that we get a number of px ’s, say, px1, ...,pxa and b number of py , say, py1, ...,pyb . let us consider the following example. example 6.4. when p =   0.11 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.25 0.01 0.10 0.10 0.01 0.01 0.01 0.01 0.01 0.15 0.01 0.01 0.10 0.01 0.01 0.01   then we make two px ’s; 18 a geometric view on pearson’s correlation coefficient and a generalization of it to non-linear dependencies px1 =   0.15 0.000 0.000 0.00 0.00 0.00 0.000 0.000 0.00 0.29 0.00 0.230 0.000 0.00 0.00 0.00 0.000 0.000 0.19 0.00 0.00 0.140 0.000 0.00 0.00   and px2 =   0.15 0.000 0.000 0.00 0.00 0.00 0.000 0.000 0.00 0.29 0.00 0.000 0.230 0.00 0.00 0.00 0.000 0.000 0.19 0.00 0.00 0.140 0.000 0.00 0.00  . therefore we have two maximal distances to these two full dependences. they are m(pi,px1 ) and m(pi,px2 ) and similarly we obtain another two full dependences when marginal of y is preserved. therefore, ρm (x,y ) = m(p,pi )∏2 i=1 ∏2 j=1 { m(pxi,pi )m(pyj,pi ) }1 4 then ρ = −0.0491 and ρm = 0.5731. note that here we have that cramer’s v and tschuprow’s t are 0.6652. 7 conclusion we have looked at the structure and the construction of the pearson’s correlation coefficient ρ in order to have a generalization of it for measuring any non-linear dependence between two random variables. we have shown that it is simple do it geometrically for discrete variables. it can be shown that ρ is a normalized ‘euclidean’ type distance between the joint probability distribution of the two random variables and that when their independence is assumed in the probability simplex of the two variables where normalizing constant is the geometric mean of two maximal such distances; each between full linear dependence of the two variables and their independence while preserving the marginal distribution of respective variable. so, we have shown that if we consider all possible full dependences and use an appropriate distance such as hellinger then we can have a genaralization of ρ. but generally it is not easy to find all possible maximal distances, which is an open problem that may need algorithmic or computational solutions. however we have shown some examples after having defined a generalization. acknowledgments: financial support for this research is from swedish research council for health, working life and welfare (forte) and swedish initiative for microdata research in the medical and social sciences (simsam). 19 priyantha wijayatunga references [1] a. rényi, probability theory north-holland publishing company and akadémiai kiadó, publishing house of the hungarian academy of sciences. republished dover usa, 2007. [2] c. sabatti, measuring dependency with volume tests, the american statistician 56 3 (2002), 191-195. doi: 10.1198/000313002128. [3] c. w. granger, e. maasoumi and j. racine, a dependence metric for possibly nonlinear processes, the journal of time series analysis 25 5 (2004), 649-669. [4] f. berzal, i. blanco, d. sanchez and m. -a. vila, measuring the accuracy and interest of association rules: a new framework, intelligent data analysis 6 3 (2002), 221-235. [5] h. skaug and d. tjostheim, testing for serial independence using measures of distance between densities, p. m. robinson and m. rosenblatt (eds): athens conference on applied probability and time series, volume ii: time series analysis in memory of e.j. hannan, springer lecture notes in statistics 115 (1996), 363-377. [6] k. matsusita, decision rules, based on distance, for problems of fit, two samples, and estimation, annals of mathematical statistics 26 4 (1955), 631640. [7] m. studeny and j. vejnarova, the multiinformation function as a tool for measuring stochastic dependence, m. i. jordan (eds): learning in graphical models, kluwer academic publishers (1998), 261-297. [8] m. sugiyama and k. m. borgwardt, measuring statistical dependence via the mutual information dimension, proceedings of the twenty-third international joint conference on artificial intelligence (ijcai’13) aaai press (2013), 1692-1698. [9] n. balakrishnan and c. -d. lai, continuous bivariate distributions, springer, 2009. [10] p. diaconis and b. efron, testing for independence in a two-way table: new interpretations of chi-square statistics, the annals of statistics 13 (1985), 845-874. 20 a geometric view on pearson’s correlation coefficient and a generalization of it to non-linear dependencies [11] p. wijayatunga, s. mase and m. nakamura, appraisal of companies with bayesian networks, international journal of business intelligence and data mining 1 3 (2006), 326-346. [12] s. e. fienberg and j. p. gilbert, the geometry of a two by two contingency table, journal of the american statistical association 65 (1970), 694-701 [13] s. kullback and r. a. leibler, on information and sufficiency, the annals of mathematical statistics 22 1 (1951), 79-86 [14] w. bergsma, a bias-correction for cramér’s v and tschuprow’s t, journal of the korean statistical society 42 3 (2013), 323-328. http://dx.doi.org/10.1016/j.jkss.2012.10.002. 21 ratio mathematica volume 44, 2022 independent restrained k rainbow dominating function m. esakki dharani* a. nagarajan† k. palani‡ abstract let g be a graph and let f be a function that assigns to each vertex a set of colors chosen from the set {1, 2…, k} that is f: v(g) → p [1, 2, …, k]. if for each vertex v ∈ v(g) such that f(v) = ∅ .we have ⋃ 𝑓(𝑢) = {1, 2, . . , 𝑘} 𝑢∈𝑁[𝑣] then f is called the k – rainbow dominating function (krdf) of g. a k – rainbow dominating function is said to be independent restrained k rainbow dominating function if (i)the set of vertices assigned with non – empty set is independent. (ii)the induced subgraph of g, by the vertices with label ∅ has no isolated vertices. the weight w(f) of a function f is defined as w(f) = ∑ |𝑓(𝑣)|𝑣 ∈ 𝑉(𝐺) .the independent restrained k – rainbow domination number is the minimum weight of g. in this paper we introduce independent restrained k – rainbow domination and find for some graphs keywords: independent, restrained, rainbow domination number, weight. 2010 ams subject classification: 05c69§ *research scholar (reg.no.21212232092011), pg & research department of mathematics, v.o. chidambaram college, thoothukudi-628008, tamil nadu, india. affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india.dhanushiya1411@gmail.com †associate professor (retd.), v.o. chidambaram college, thoothukudi-628008, tamil nadu, india affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india.nagarajan.voc@gmail.com ‡associate professor, a.p.c. mahalaxmi college for women, thoothukudi-628008, tamil nadu, india. affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india.palani@apcmcollege.ac.in § received on june 10th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.924. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 349 m. esakki dharani, a. nagarajan, and k. palani 1. introduction domination in graphs originates from location problems in operations research. as a variation of domination in graphs, rainbow domination was introduced by bresar et al. [2]. shao et al. [7] gave bounds for the k – rainbow domination number on an arbitrary graph. hao et al. [4] studied the k – rainbow domination number of directed graphs. independent rainbow domination was introduced by zehui shao et al. [8]. amjadi et al. [1] was investigated the rainbow restrained domination number. in this paper we introduce independent restrained k – rainbow domination and find for some graphs. 2. preliminaries a graph g consists of pair(v(g), e(g)) where v(g) is a non-empty finite set whose elements are called points or vertices and e(g) is a set of unordered pair of distinct elements of v(g). the elements of e(g) are called lines or edges of the graph g. for any vertex u in g, the open neighbourhood of u, is denoted by n(u) is the set of vertices adjacent to u and the closed neighbourhood of u, is denoted by n[u] = n(u) ∪ {u}. a set of vertices in a graph is said to be an independent set of vertices or simply an independent if no two vertices in the set are adjacent. the splitting graph of g is denoted by s[g]for each vertex v of a graph g, take a new vertex 𝑣′ and join 𝑣′ to all vertices adjacent to v in g.the corona product of two graphs g and h is defined as the graph obtained by taking one copy of g and |𝑉(𝐺)| copies of h and joining the 𝑖𝑡ℎ vertex of g to every vertex in the 𝑖𝑡ℎ copy of h. it is denoted as 𝐺 ∘ 𝐻. let g be a graph and let f be a function that assigns to each vertex a set of colors chosen from the set {1, 2, …, k} that is f : v(g) → p [1,2,…,k]. if for each vertex v∈ v(g) such that f(v) = ∅ .we have ⋃ 𝑓(𝑢) = {1,2, . . , 𝑘} 𝑢∈𝑁[𝑣] then f is called the k – rainbow dominating function (krdf) of g. a k – rainbow dominating function is called independent k – rainbow domination if vertices assigned with non – empty sets are pairwise nonadjacent. a k – rainbow dominating function is called rainbow restrained domination function if vertices assigned with empty sets has no isolated vertex. 3. main results definition 3.1. let g be a graph and let f be a function that assigns to each vertex a set of colors chosen from the set {1, 2, …, k} that is f : v(g) → p [1,2,…,k]. if for each vertex v∈ v(g) such that f(v) = ∅ . we have ⋃ 𝑓(𝑢) = {1,2, . . , 𝑘} 𝑢∈𝑁[𝑣] then f is called the k – rainbow dominating function (krdf) of g. a k – rainbow dominating function is said to be independent restrained krainbow dominating function if i. the set of vertices assigned with non – empty set is independent. ii. the induced subgraph of g, by the vertices with label ∅ has no isolated vertices. 350 independent restrained k rainbow dominating function the weight w(f) of a function f is defined as w(f) = ∑ |𝑓(𝑣)|𝑣∈𝑉(𝐺) . the independent restained k – rainbow domination number is the minimum weight of g. observation 3.2. 1. let u be a pendant vertex to v. if one of the vertex u and v is assigned {1, 2, … k} then the other may be assigned ∅. (i.e) if the end vertex is assigned {1, 2, … k} then the support vertex may be assigned ∅ 2. always 𝑘 ≤ 𝛾𝑖𝑟𝑘𝑟 ≤ 𝑛𝑘 3. if a graph g has a clique as a subgraph, then one of the vertices of the clique should be labelled {1, 2, … k} and all the remaining vertices are labelled ∅. theorem 3.3. let g be a graph and let v be a full degree vertex in g. suppose g – {v} has no isolated vertex then 𝛾𝑖𝑟𝑘𝑟(𝐺) = 𝑘 proof: define f: v(g) → p [1,2,…,k] by f(x) = { { 1,2, … , 𝑘}𝑖𝑓𝑥 = 𝑣 ∅ otherwise } then clearly f is an independent restrained rainbow dominating function. further w(f) = k as 𝛾𝑖𝑟𝑘𝑟(𝐺) ≥ 𝑘, f is a minimum independent restrained k rainbow dominating function. therefore, 𝛾𝑖𝑟𝑘𝑟(𝐺) = 𝑘 corollary 3.4: complete graph, wheel graph, fan graph, flower graph all have independent restrained k rainbow domination number is equal to k. theorem 3.5: the independent restrained krainbow dominating function exists for path graph 𝑃𝑛 if and only if 𝑛 ≡ 1 𝑚𝑜𝑑 3 and then 𝛾𝑖𝑟𝑘𝑟 (𝑃𝑛) = ⌈ 𝑛 3 ⌉ k, where 𝑛 ≡ 1 𝑚𝑜𝑑 3 proof: when 𝑛 ≡ 0 𝑜𝑟 2 𝑚𝑜𝑑 3, the graph g does not admit any independent restrained k rainbow dominating function. since any k rainbow dominating function f fails to satisfy either independent or restrained condition. let 𝑣1, 𝑣2, 𝑣3, … 𝑣𝑛 be the vertices of the path graph. define f: v(g) → p [1, 2, …, k] by f(x) = { { 1,2, … , 𝑘}𝑖𝑓𝑥 = 𝑣𝑖 𝑤ℎ𝑒𝑟𝑒𝑖 ≡ 1 𝑚𝑜𝑑 3 𝑎𝑛𝑑 0 ≤ 𝑖 ≤ 𝑛 ∅ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 } then clearly f is an independent restrained k rainbow dominating function. therefore, w(f) = ∑|𝑓(𝑣𝑖 )| = |𝑓(𝑣1)| + |𝑓(𝑣4)| + |𝑓(𝑣7)| + ⋯ + |𝑓(𝑣𝑛)| therefore, 𝛾𝑖𝑟𝑘𝑟(𝑃𝑛) = ⌈ 𝑛 3 ⌉ k 351 m. esakki dharani, a. nagarajan, and k. palani theorem 3.6: the independent restrained krainbow dominating function exists for cycle graph 𝐶𝑛 if and only if 𝑛 ≡ 0 𝑚𝑜𝑑 3 and then 𝛾𝑖𝑟𝑘𝑟 (𝐶𝑛) = 𝑛𝑘 3 , 𝑛 ≡ 0 𝑚𝑜𝑑 3 proof: when 𝑛 ≡ 1 𝑜𝑟 2 𝑚𝑜𝑑 3 , the graph g does not admit any independent restrained k rainbow dominating function. since any k rainbow dominating function f fails to satisfy either independent or restrained condition. let 𝑣1, 𝑣2, 𝑣3, … 𝑣𝑛 be the vertices of the cycle graph. define f: v(g) → p [1, 2, …, k] by f(x) = { { 1,2, … , 𝑘}𝑖𝑓𝑥 = 𝑣𝑖 𝑤ℎ𝑒𝑟𝑒𝑖 ≡ 0 𝑚𝑜𝑑 3 𝑎𝑛𝑑 0 ≤ 𝑖 ≤ 𝑛 ∅ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 } then clearly f is an independent restrained k rainbow dominating function. therefore, w(f) = ∑|𝑓(𝑣𝑖 )| = |𝑓(𝑣3)| + |𝑓(𝑣6)| + |𝑓(𝑣9)| + ⋯ + |𝑓(𝑣𝑛)| therefore, 𝛾𝑖𝑟𝑘𝑟(𝐶𝑛) = ⌈ 𝑛 3 ⌉ k theorem 3.7: let 𝐻𝑛 be the helm graph obtained from the wheel by attaching a pendant vertex to each rim vertex. then 𝛾𝑖𝑟𝑘𝑟(𝐻𝑛) = k + n 2 for n ≥ 4 proof: let 𝑣1, 𝑣2, 𝑣3, … 𝑣𝑛−1 be the rim vertex. let u be the apex vertex. let 𝑤1, 𝑤2, … 𝑤𝑛−1such that 𝑤𝑖 is adjacent to 𝑣𝑖 for 1 ≤ 𝑖 ≤ 𝑛 − 1 . define f: v(g) → p [1, 2, …, k] by f(x) = { { 1}𝑖𝑓𝑥 = 𝑤𝑖 where 1 ≤ 𝑖 ≤ 𝑛 − 1 { 2,3, … , 𝑘 }𝑖𝑓𝑥 = 𝑢 ∅ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 } obviously, every vertex v with labelled ∅ satisfies the condition ⋃ 𝑓(𝑢) =𝑢∈𝑁[𝑣] {1,2, . . , 𝑘} ∴f is a k rainbow dominating function. let s be the set of vertices assigned ∅ labelled then s = {𝑣1, 𝑣2, 𝑣3, … 𝑣𝑛−1}. here <s> has no isolated vertices. let s’ be the set of vertices assigned non empty labelled. then s’ = {𝑢, 𝑤1, 𝑤2, … 𝑤𝑛−1} is independent. then clearly f is an independent restrained k rainbow dominating function. ∴ w(f) = ∑ (|𝑓(𝑢)| + |𝑓(𝑤𝑖 )| ) where i = 1 to n – 1 therefore, 𝛾𝑖𝑟𝑘𝑟(𝐻𝑛) = k + n – 2 theorem 3.8: let (𝐶𝑛 ∘ 𝐾1)be the crown graph obtained by joining a pendant edge to each vertex of 𝐶𝑛. then for n ≥ 1, 𝛾𝑖𝑟𝑘𝑟 (𝐶𝑛 ∘ 𝑘1) = nk proof: let 𝑣1, 𝑣2, 𝑣3, … 𝑣𝑛 be the vertices of the cycle. let 𝑤1, 𝑤2, 𝑤3, … 𝑤𝑛 be the set of end vertices of the crown graph, where 1 ≤ 𝑖 ≤ 𝑛 . define f: v(g) → p [1, 2, …, k] by f(x) = { { 1,2, … , 𝑘}𝑖𝑓𝑥 = 𝑤𝑖 ; 1 ≤ 𝑖 ≤ 𝑛 ∅ otherwise } 352 independent restrained k rainbow dominating function obviously, every vertex v with labelled ∅ satisfies the condition ⋃ 𝑓(𝑤) =𝑤∈𝑁[𝑣] {1,2, . . , 𝑘} ∴f is a k rainbow dominating function. by observation 3.2(1), we assigned { 1,2, … , 𝑘} to the end vertices, which is independent and we assigned∅ to all the support vertices. then the induced subgraph of empty set is also connected. hence it satisfies both independent and restrained condition. then clearly f is an independent restrained k rainbow dominating function. ∴ w(f) = ∑ (|𝑓(𝑤𝑖 )| ) where i = 1 to n = |𝑓(𝑤1)| + |𝑓(𝑤2)| + |𝑓(𝑤3)| + ⋯ + |𝑓(𝑤𝑛)| = nk therefore, 𝛾𝑖𝑟𝑘𝑟 (𝐶𝑛 ∘ 𝑘1) = nk remark 3.9: (i) in observation 2.2 (2), equality holds. since 𝛾𝑖𝑟𝑘𝑟 (𝐾𝑛) = nk and 𝛾𝑖𝑟𝑘𝑟 (𝐶𝑛 ∘ 𝑘1) = nk. (ii) the inequality is also strict. since 𝛾𝑖𝑟𝑘𝑟(𝐻𝑛) = k + n – 2 references [1] j. amjadi s.m. sheikholeslami. l. volkmann. rainbow restrained domination numbers in graphs, in ars combinatoria [2] b. bresar, m. henning, d. rall, rainbow domination in graphs, taiwanese j. math. 12 (2008), 213–225. [3] fujita. s, furuyana. m, magnant c general bounds on rainbow domination numbers. graphs combin 2015,31, 601-613. [4] hao, g.l, quian, j. g. on the rainbow domination number of digraphs. graphs combin 2016, 32,1903 -1913. [5] t.w. haynes, s. t. hedetniemi and p.j. slater, fundamentals of domination in graphs, marcel dekker, new york, 1998. [6] hong gao, changqing xi, yuansheng yang. the 3-rainbow domination number of the cartesian product of the cycles. [7] shao z h, liang m l, yin c, pavlic p, zerovnik j. on rainbow domination number of graphs. inform. sci. 2014, 254,225-234. [8] zehui shao, zepeng li, aljosa peperko, jiafu wan. independent rainbow domination of graphs. 353 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 41, 2021, pp. 255-282 255 a dynamic model of typhoid fever with optimal control analysis chinwendu emilian madubueze* reuben iortyer gweryina† kazeem austin tijani‡ abstract in this study, a deterministic mathematical model of typhoid fever dynamics with control strategies; vaccination, hygiene practice, sterilization and screening is studied. the model is first analyzed for stability in terms of the control reproduction number, rc with constant controls. the disease-free equilibrium and endemic equilibrium of the model exist and is shown to be stable whenever rc < 1 and rc > 1, respectively. the model by investigation shows a forward bifurcation and the sensitivity analysis conducted revealed the most biological parameters to be targeted by policy health makers for curtailing the spread of the disease. the optimal control problem is obtained through application of pontryagin maximum principle with respect to the above-mentioned control strategies. simulations of the optimal control system and sensitivity of the constant control system confirms that hygiene practice with sterilization could be the best strategy in controlling the disease. keywords: typhoid fever; global stability; sensitivity analysis; optimal control; bifurcation. 2010 ams subject classification§: 92b05, 34d23, 49k15, 37n25, 34d05. * department of mathematics, joseph sarwuan tarka university, makurdi, nigeria; ce.madubueze@gmail.com. † department of mathematics, joseph sarwuan tarka university, makurdi, nigeria; gweryina.reuben@uam.edu.ng. ‡ department of mathematics, joseph sarwuan tarka university, makurdi, nigeria; kazeemtijani1987@yahoo.com. § received on september 11, 2021. accepted on december 21, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.657. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. c. e. madubueze, r.i. gweryina, and k. a. tijani 256 1. introduction typhoid fever is a life-threatening infection that is usually caused by salmonella enteric serovar typhi (s. typhi) and salmonella enteric serovar paratyphi (s. paratyphi, that is paratyphi a, b, and, uncommonly is s. paratyphi c) [1]. typhoid fever has been a public health challenge globally. however, the disease is endemic in most developing countries in africa and south-east asia where potable clean water, sanitation and hygiene are either grossly inadequate or non-existent. the transmission of s. typhi and s. paratyphi occur through the consumption of contaminated food or water resulting from inadequate environmental sanitation and hygiene practices [2]. people that are clinically ill from typhoid fever and those who have recovered from it pass out the bacteria in their stools (carriers) and urine [3]. a chronic carrier sheds salmonella typhoid more than 12 months after onset of illness. human beings are the only known reservoir of typhoid and the mode of transmission happens through food and water contaminated by acutely ill or chronic carriers of the bacteria [4]. vaccine can be taken to prevent typhoid fever but does not provide longterm immunity [5]. on the other hand, educating travelers moving to typhoid endemic regions on the importance of sanitation and hygiene precautions as well as vaccination will help immensely to preventing the rapid spread of typhoid disease [4]. mathematical models of infectious diseases are used to test and compare various intervention strategies especially when there are limited resources [6]. in controlling typhoid fever, several mathematical models have been formulated. for instance, mushayabas [7] considered the impact of education campaigns and treatment on the dynamics of typhoid fever and abboubakar and racke [8] carried out a human and bacteria model without considering hygiene practice and individuals protected through vaccination in the population, while karunditu et al. [9], peter et al. [10], nyerere et al. [11], peter et al. [12], edward and nyerere [13], kgosimore and kelatlehegile [14] and aji et al. [15] considered only human population without factoring in the bacteria concentration in the contaminated food and or water. tilalum et al. [16], okolo and abu [17], peter et al. [18], abboubakar and racke [19] and awoke [20] studied the optimal control of typhoid transmission with control measures. none of the aforementioned works studied the combined control measures such as vaccination, hygiene practice, screening of carriers and sterilization of the bacteria in the environment as autonomous or non-autonomous system of equations. this study will bridge these gaps and form a novel contribution to the existing body of knowledge on the subject matter. peter et al. [10] forms the motivation of this work. they considered protected, susceptible, infected, treated and recovery model without the bacteria concentration and the effect of screening of infected carriers and a dynamic model of typhoid fever with optimal control analysis 257 hygiene practice on transmission dynamics of typhoid fever. they assumed that the protected class belongs to only individuals that have been vaccinated before entrance into typhoid endemic population and also optimal control and numerical simulation were not considered in their work. modifying the work of peter et al. [10], we consider protected, susceptible, infected individuals, carriers, recovery and bacteria concentration model in which some susceptible individuals are protected through vaccination and population practices hygiene which reduces the transmission rate. hygiene practices which include safe water, sanitation and personal hygiene are crucial in preventing and controlling the spread of typhoid. in addition, the screening and treating of carriers who are silent spreaders of the disease due to their asymptomatic nature and the sterilization of bacteria concentration in the immediate environment are also important in elimination of typhoid fever in the population. this work will be the first to consider sterilization of the bacteria concentration as a control measure for typhoid fever. also, the sensitivity analysis for the prediction of appropriate intervention strategies for the control of typhoid fever spread and the optimal control analysis are carried out in this work. therefore, a modified version of the work of peter et al. [10] is formulated in section 2 and a comprehensive mathematical analysis of the model in section 3. the sensitivity analysis and optimal control strategies of the typhoid fever dynamics are considered in section 4 while the numerical simulations and discussion are given in section 5. section 6 is the conclusion of the work. 2. model description and formulation in this section, the work of peter et al. [10] is modified by considering human population (infected carriers) as well as bacteria concentration. the human population at any time, 𝑡 is subdivided into five subpopulations namely; protected population, 𝑃(𝑡), susceptible population, 𝑆(𝑡), infected population, 𝐼(𝑡), carrier population, 𝐼𝑐(𝑡) and recovered individuals, 𝑅(𝑡). the bacteria concentration is represented by 𝐵𝑐(𝑡). in this study, the protected population, 𝑃(𝑡), are susceptible individuals that are vaccinated and individuals coming in from the population that is not at risk of typhoid fever into typhoid fever endemic population. infected population, 𝐼(𝑡), are infected individuals that are showing symptoms of the disease and are capable of spreading the bacteria in the environment while carrier population, 𝐼𝑐(𝑡), represents asymptomatic infected individuals that are treated but still carrying the salmonella typhi. recovered individuals, 𝑅(𝑡), are individuals who have recovered from the disease by treatment or natural immunity. the protected population, 𝑃(𝑡), of the proportion, 𝛼 ∈ (0,1) is increased by birth or immigration at a rate, 𝛬, and also from susceptible individuals that are protected through vaccination at a rate, ƞ. the protected population loses c. e. madubueze, r.i. gweryina, and k. a. tijani 258 immunity when the vaccine wanes at a rate, 𝛾. the susceptible population is increased at a rate, (1 − 𝛼)𝛬, of the unprotected population through birth or immigration and also from recovered population,𝑅(𝑡), after losing their temporary immunity at a rate, ɸ. susceptible population contract typhoid disease through food, water or environment contaminated by salmonella bacteria as a result of inadequate hygiene practice measure at a rate, (1 − 𝑝)𝜆 and progress to infected population. here, λ = 𝛽𝐵𝑐 𝐾+𝐵𝑐 is the force of infection, 𝛽 is the ingestion or consumption rate of the contaminated food, water or environment, 𝐾 is the carrying capacity of the bacteria in food, water or environment and 𝑝 ∈ (0,1) is the hygiene practice control measure. infected individuals progress to carrier class at a rate, 𝜎 while some infected individuals recovered fully by treatment at rate 𝜏1 or they die of the disease (bacteria) at a rate, d. carrier class, 𝐼𝑐(𝑡), recovered by natural immunity at a rate, 𝜏2 or by early treatment when they are screened at a rate, ψ with 𝜃 as the treatment rate. the natural death rate, 𝜇 is assumed for all the human population. for the bacteria concentration, 𝐵𝑐(𝑡), in the environment, they increased through the shedding from carriers and symptomatic population, 𝐼𝑐(𝑡) and 𝐼(𝑡) at the rates, 𝜋1 and 𝜋2 respectively. the shedding rates, 𝜋1 and 𝜋2 are reduced by 𝑝, the level of hygiene practice the infected populations, 𝐼𝑐(𝑡) and 𝐼(𝑡), observed. the bacteria decays in the environment at a rate, 𝜇bc. we assume that there is no human to human transmission but rather human aids in shedding the bacteria in the environment or contaminating the environment; neither there is immigration of infectious humans. also, disease induced death does not occur in carrier class since they are asymptomatic, that is before the bacteria can cause death, it must have progressed to symptomatic stage. the systematic diagram of model is given in figure 1. a dynamic model of typhoid fever with optimal control analysis 259 figure 1. the systematic diagram for typhoid fever model. the system of differential equation is derived using the figure 1 as follows. 𝑑𝑃 𝑑𝑡 = αʌ + ƞs – (γ + µ)p 𝑑𝑆 𝑑𝑡 = (1 − α)ʌ + γp + ɸr – (ƞ + µ + (1 − p)λ)s 𝑑𝐼 𝑑𝑡 = (1 − p)λs – (σ + 𝜏1 + µ + 𝑑)i 𝑑𝐼𝑐 𝑑𝑡 = σ𝐼 – (𝜏2 + ѱ𝜃 + 𝜇)𝐼𝐶 𝑑𝑅 𝑑𝑡 = 𝜏1i + (𝜏2 + ѱ𝜃)𝐼𝐶 – (µ + ɸ)r 𝑑𝐵𝑐 𝑑𝑡 = 𝜋2(1 − 𝑝)i + 𝜋1(1 − 𝑝)𝐼𝐶 − µ𝐵𝐵𝑐 } (1) with initial conditions, 𝑃(0) > 0, 𝑆(0) > 0, 𝐼𝐶(0) ≥ 0,i(0) ≥ 0, 𝑅(0) ≥ 0, 𝐵𝐶(0) ≥ 0, where λ= 𝛽𝐵𝑐 (𝐾+𝐵𝑐) and the model parameters are assumed to be nonnegative. c. e. madubueze, r.i. gweryina, and k. a. tijani 260 3. mathematical analysis of the model 3.1 invariant region invariant region is a region where the model solutions are uniformly bounded. theorem 1. all feasible solutions of the model are uniformly bounded in a proper subset 𝐷 = 𝐷𝐻 𝑋 𝐷𝐵𝑐, where dh = {(p,s, i, 𝐼𝐶,r) ∈ ɍ+ 5 : n(t) ≤ ʌ µ } is a subset for human population and 𝐷𝐵𝑐 = {𝐵𝑐 ∈ ℝ+: 𝐵𝑐 ≤ [(𝜋2 + 𝜋1)(1−𝑝)]ʌ µµ𝐵 } is a subset for bacteria concentration in environment. proof. the total human population, 𝑁(𝑡) is given by 𝑁= 𝑃 + 𝑆 + 𝐼 + 𝐼𝑐 + 𝑅 with initial conditions 𝑁(0) = 𝑁0 and 𝐵𝑐(0) = 𝐵𝑐0 for the bacteria in the environment. this implies that from equation (1) that 𝑑𝑁 𝑑𝑡 = 𝛬 − 𝜇𝑁 − 𝑑𝐼. in the absence of disease-induced death rate, that is, 𝑑 = 0, we have 𝑑𝑁𝐻 𝑑𝑡 ≤ 𝛬 − 𝜇𝑁 which by method of integrating factor and the initial condition, 𝑁(0) = 𝑁0 gives 𝑁(𝑡) ≤ ʌ µ + (𝑁0 − ʌ µ )𝑒−µ𝑡. (2) as 𝑡 → ∞ in equation (2), we have 𝑁(𝑡) ≤ ʌ µ . this means that the feasible solutions of the model for the human population are in the region, dh = {(p,s, i, 𝐼𝐶,r) ∈ ɍ+ 5 : n(t) ≤ ʌ µ }. for bacteria concentration since 𝑁(𝑡) ≤ ʌ µ , it means that 𝐼 ≤ ʌ µ and 𝐼𝐶 ≤ ʌ µ , we have from the last equation of (1) that 𝑑𝐵𝑐 𝑑𝑡 = 𝜋2(1 − 𝑝)𝑁 + 𝜋1(1 − 𝑝)𝑁 − µ𝐵𝐵𝑐 ≤ 𝜋2(1 − 𝑝) ʌ µ + 𝜋1(1 − 𝑝) ʌ µ − µ𝐵𝐵𝑐 . (3) solving equation (3) with 𝐵𝑐(0) = 𝐵𝑐0 as the initial condition yields 𝐵𝑐 ≤ (𝜋2 +𝜋1)(1−𝑝)ʌ µµ𝐵 + (𝐵𝑐0 − (𝜋2 +𝜋1)(1−𝑝)ʌ µµ𝐵 )𝑒−µ𝐵𝑡. (4) a dynamic model of typhoid fever with optimal control analysis 261 as t → ∞ in equation (4), we have 𝐵𝑐 ≤ (𝜋2 +𝜋1)(1−𝑝)ʌ µµ𝐵 . therefore, the feasible solution of the bacterial population enters the region 𝐷𝐵𝑐 = {𝐵𝑐 ∈ ℝ+: 𝐵𝑐 ≤ [(𝜋2 + 𝜋1)(1−𝑝)]ʌ µµ𝐵 }. this completes the proof. theorem 1 implies that the model is well posed mathematically and epidemiologically. therefore, it is sufficient enough to study the dynamics of the model (1) in the region 𝐷 = 𝐷𝐻 × 𝐷𝐵𝑐. 3.2 positivity of the solutions theorem 2. let d = {𝑃,𝑆,𝐼, 𝐼𝐶,𝑅, 𝐵𝑐} ∈ ℝ+ 6 be solution set such that 𝑃(0) = 𝑃0, 𝑆(0) = 𝑆0, 𝐼𝐶(0) = 𝐼𝐶0, i(0) = 𝐼0, 𝑅(0) = 𝑅0 and 𝐵𝐶(0) = 𝐵𝐶0 are positive, then the elements of the solution set 𝐷 are all positive for 𝑡 ≥ 0. proof. from the first equation of the model equations (1), we have 𝑑𝑃 𝑑𝑡 = 𝛼λ + 𝜂𝑆 − (𝛾 + µ)p ≥ −(𝛾 + µ)p. (5) integrating equation (5) with initial conditions 𝑃(0) = 𝑃0 yields 𝑃(𝑡) ≥ 𝑃0𝑒 – (γ + µ)t ≥ 0 . in a similar way, the rest of the equations of the model equation (1) with initial conditions, 𝑆(0) = 𝑆0, 𝐼(0) = 𝐼0𝐼𝐶 (0) = 𝐼𝑐0,𝑅(0) = 𝑅0 and 𝐵𝑐(0) = 𝐵𝑐0 give 𝑆(𝑡) ≥ 𝑆0exp(∫ – (ƞ + µ + (1 − 𝑝)λ) 𝑡 0 )𝑑𝑢 ≥ 0, 𝐼(𝑡) ≥ 𝐼0exp{−(σ + 𝜏1 + µ + 𝑑)t} ≥ 0, 𝐼𝐶 (𝑡) ≥ 𝐼𝑐0exp{− (𝜏2 + ѱ𝜃 + 𝜇)𝑡} ≥ 0, 𝑅(𝑡) ≥ 𝑅0exp{–(µ + ɸ)t} ≥0, 𝐵𝑐(𝑡) ≥ 𝐵𝑐0exp(− µ𝐵𝑡) ≥ 0. therefore, the solution set {𝑃(𝑡), 𝑆(𝑡), 𝐼(𝑡),𝐼𝑐(𝑡), 𝑅(𝑡), 𝐵𝑐(𝑡)}, of the system (1) is positive for all 𝑡 ≥ 0 since exponential functions and their initial conditions are positive. 3.3 disease-free equilibrium point and control reproduction number we compute the control reproduction number,𝑅𝑐, which is define as the average number of secondary cases reproduced when an infected person is introduced into a population where control measures like vaccination, screening, sanitation and hygiene are in place. c. e. madubueze, r.i. gweryina, and k. a. tijani 262 in obtaining this, we apply the next-generation matrix approach [21] at the disease-free disease (dfe). the disease-free equilibrium (dfe) is obtained by equating the right hand side of the equation (1) to zero and solve simultaneously for the disease-free equilibrium, 𝐸0 = (𝑃 0,𝑆0, 𝐼0, 𝐼𝐶 0,𝑅0,𝐵𝐶 0). we have dfe, 𝐸0 = ( ʌ(𝛼µ+ƞ) µ(𝛾+ƞ+µ) , ʌ(𝛾+𝜇(1−𝛼)) µ(𝛾+ƞ+µ) ,0,0,0,0). by the principle of next-generation matrix approach, we have 𝐹 = ( 0 0 a𝛽𝑆0 𝐾 0 0 0 0 0 0 ), 𝑉 = ( k3 0 0 −𝜎 k4 0 − 𝜋2(1 − 𝑝) −𝜋1(1 − 𝑝) µ𝐵 ), (6) where 𝑎 = (1 − p), 𝑘1 = (γ + µ),𝑘2 = (ƞ + µ), 𝑘3 = (σ + 𝜏1 + µ + 𝑑), 𝑘4 = (𝜏2 + ѱ𝜃 + 𝜇), 𝑘5 = (µ + ɸ). (7) solving for the maximum eigenvalue of the matrix, 𝐹𝑉−1, we have 𝑅𝑐 = 𝑎𝛽s0[(𝜎𝜋1+𝜋2𝐾4)(1−𝑝)] 𝐾µ𝐵𝐾3𝐾4 . (8) with the definition of equation (7), we have 𝑅𝑐 = 𝛽𝛬(1−p)(𝛾+(1−𝛼)𝜇)[(𝜎𝜋1+𝜋2(𝜏2+ ѱ𝜃+𝜇))(1−𝑝)] 𝜇𝐾𝜇𝐵(𝛾+ƞ+µ)(σ+𝜏1+ µ+𝑑)(𝜏2+ ѱ𝜃+𝜇) . (9) the control reproduction number, 𝑅𝑐, can be written as 𝑅𝑐=𝑅𝐼 + 𝑅𝐼𝐶 , (10) where 𝑅𝐼 = 𝛽(1−p)2𝜋2𝑆0 𝐾µ𝐵(σ+𝜏1+ µ+𝑑) , 𝑅𝐼𝑐 = 𝛽(1−p)2𝜎𝜋1𝑆0 𝐾µ𝐵(𝜏2 + ѱ𝜃+𝜇)(σ+𝜏1+ µ+𝑑) (11) denote the reproduction numbers which the infected population and carrier population contributed respectively through their shedding in the environment. 3.4 local stability of the disease-free equilibrium, 𝑬𝟎 a dynamic model of typhoid fever with optimal control analysis 263 theorem 3. if 𝐸0 is the dfe of the model, then 𝐸0 is locally asymptomatically stable if 𝑅𝑐 < 1, otherwise it is unstable if 𝑅𝑐 > 1. proof. in proving this theorem, the jacobian matrix of equation (1) at the disease-free equilibrium, 𝐸0 is given as 𝐽(𝐸0) = ( –k1 ƞ 0 0 0 0 γ −k2 0 0 ɸ − a𝛽𝑆0 𝐾 0 0 – k3 0 0 a𝛽𝑆0 𝐾 0 0 σ – k4 0 0 0 0 𝜏1 (𝜏2 + ѱ𝜃) – k5 0 0 0 𝜋2(1 − 𝑝) 𝜋1(1 − 𝑝) 0 −µ𝐵 ) . (12) the eigenvalues of the jacobian matrix (12) are −k5 and the solutions of the polynomial 𝜆5 + 𝐴𝜆4 + 𝐵𝜆3 + 𝐶𝜆2 + 𝐷𝜆 + 𝐸 = 0 (13) where a = k1 + k2 + k3 + k4 + μb, 𝐵 = (k1 + k2 )(k3 + k4 + μb) + k4(k3 + μb) + 𝜇(k2 + γ) + k3μb(1 − 𝑅𝐼), 𝐶 = 𝜇(k4 + μb)(k2 + γ) + k1k2k3 + k3μb(k1 + k2 )(1 − 𝑅𝐼) + k3k4μb(1 − 𝑅𝐶) + k4(k1 + k2 )(k3 + μb), 𝐷 = 𝜇k3μb(k2 + γ)(1 − 𝑅𝐼) + 𝜇k4(k3 + μb)(k2 + γ) + k3k4μb(k1 + k2 )(1 − 𝑅𝐶), 𝐸 = 𝜇(k2 + γ)k3k4μb(1 − 𝑅𝐶). using the theorem in heffernan et al. [22], the roots of the polynomial (13) have negative real part if 𝐴,𝐵,𝐶,𝐷,𝐸 > 0. with the definition of 𝑅𝑐 in equation (10), we have 𝐴,𝐵,𝐶,𝐷,𝐸 > 0 if 𝑅𝑐 < 1. therefore, the jacobian matrix (12) has negative real eigenvalues if 𝑅𝐶 < 1. hence, the disease-free equilibrium, 𝐸0, is locally asymptotically stable if 𝑅𝐶 < 1. this ends the proof. 3.5 global stability of disease-free equilibrium theorem 4. the disease-free equilibrium, 𝐸0, is globally asymptotically stable if 𝑅𝑐 < 1. proof. we construct a lyapunov function using the infected classes only and this is given by c. e. madubueze, r.i. gweryina, and k. a. tijani 264 𝐿 = [𝜎𝜋1(1−𝑝)+𝜋2(1−𝑝)k4] µ𝐵k4k3 𝐼 + 𝜋1(1−𝑝) µ𝐵k4 𝐼𝐶 + 1 µ𝐵 𝐵𝑐 . (14) differentiating (14) with respect to time, 𝑡, along the solutions of the model (1) gives 𝑑𝐿 𝑑𝑡 = ( 𝜎𝜋1(1−𝑝)+𝜋2(1−𝑝)k4 µ𝐵k4k3 )( 𝑎𝛽𝐵𝑐 (𝐾+𝐵𝑐) s – k3i) + 𝜋1(1−𝑝) µ𝐵k4 (σ𝐼 – k4𝐼𝐶) + 1 µ𝐵 (𝜋2(1 − 𝑝)i + 𝜋1(1 − 𝑝)𝐼𝐶 − µ𝐵𝐵𝑐). (15) expanding and simplifying (15) yields 𝑑𝐿 𝑑𝑡 = ( 𝑎𝛽𝐵𝑐 (𝐾+𝐵𝑐) ( 𝜎𝜋1(1−𝑝)+𝜋2(1−𝑝)k4 µ𝐵k4k3 )𝑆 − 1)𝐵𝑐 = ( 𝑅𝑐𝐾𝑆 (𝐾+𝐵𝑐) 𝑆0 − 1)𝐵𝑐 . since 𝑆 ≤ 𝑆0 and 𝐾 (𝐾+𝐵𝑐) ≤ 1, we have 𝑑𝐿 𝑑𝑡 ≤ 𝐵𝑐(𝑅𝑐 − 1). clearly, 𝑑𝐿 𝑑𝑡 ≤ 0 if 𝑅𝑐 ≤ 1. if 𝐵𝑐 = 0, 𝑑𝐿 𝑑𝑡 = 0. by virtue of lasalle’s invariance principle, the disease-free equilibrium,𝐸0, is globally asymptotically stable (gas) whenever 𝑅𝑐 < 1. 3.6 endemic equilibrium state the endemic equilibrium state 𝐸∗ is a state where the disease is present in the population. at 𝑑𝑃 𝑑𝑡 = 𝑑𝑆 𝑑𝑡 = 𝑑𝐼 𝑑𝑡 = 𝑑𝐼𝑐 𝑑𝑡 = 𝑑𝑅 𝑑𝑡 = 𝑑𝐵𝑐 𝑑𝑡 = 0 , we obtain after solving simultaneously that 𝐼∗ = k4𝑘5(rc−1) 𝐵 , 𝐼𝑐 ∗ = 𝜎𝑘5(rc−1) b , 𝑅∗ = (𝜏1k4+𝜎 (𝜏2 +ѱ𝜃))(rc−1) b , 𝐵𝑐 ∗ = [𝜋2(1−𝑝)k4+𝜎𝜋1(1−𝑝)]𝑘5(rc−1) b𝜇𝐵 , 𝑆∗ = 𝛬(k1−αμ)𝐵+k1[𝜎(𝜏2 +ѱ𝜃)ɸ+ɸk4τ1−k3k4k5](rc−1) 𝐵(k1k2−ƞ𝛄) , 𝑃∗ = 𝛼𝛬(b(k1k2−ƞ𝛄))+ƞ[𝛬(k1−αμ)𝐵+k1[𝜎(𝜏2 +ѱ𝜃)ɸ+ɸk4τ1−k3k4k5](rc−1)] k1b(k1k2−ƞ𝛄) . then, 𝑃∗,𝑆∗, 𝐼∗, 𝐼𝑐 ∗,𝑅∗, 𝐵𝑐 ∗ are all positive if and only if rc > 1, which established that the endemic equilibrium state, 𝐸∗ = (𝑃∗,𝑆∗, 𝐼∗, 𝐼𝑐 ∗,𝑅∗, 𝐵𝑐 ∗) exists for rc > 1. a dynamic model of typhoid fever with optimal control analysis 265 3.7 global stability of endemic equilibrium the global stability of endemic equilibrium, 𝐸∗, is established in the absence of disease induced death. theorem 4.the endemic equilibrium, 𝐸∗, is globally asymptotically stable if rc > 1 and 𝑑 = 0. proof. we construct the lyapunov function given by 𝐿 = 1 2 [(𝑃 − 𝑃∗ ) + (𝑆 − 𝑆∗ ) + (𝐼 − 𝐼∗ ) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗ )]2 + (𝐵𝑐 − 𝐵𝑐 ∗ − 𝐵𝑐 ∗ ln 𝐵𝑐 𝐵𝑐 ∗). taking the derivative of 𝐿 along the solutions of equation (1) yields 𝐿′ = [(𝑃 − 𝑃∗) + (𝑆 − 𝑆∗) + (𝐼 − 𝐼∗) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗)] 𝑑 𝑑𝑡 (𝑃 + 𝑆 + 𝐼 + 𝐼𝑐 + 𝑅) +(1 − 𝐵𝑐 ∗ 𝐵𝑐 ) 𝑑𝐵𝑐 𝑑𝑡 ̇ which upon substitution gives 𝐿′ = [(𝑃 − 𝑃∗) + (𝑆 − 𝑆∗) + (𝐼 − 𝐼∗) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗)](𝛬 −𝜇(𝑃 + 𝑆 + 𝐼 + 𝐼𝑐 + 𝑅) − 𝑑𝐼) +(1 − 𝐵𝑐 ∗ 𝐵𝑐 )(𝜋2(1 − 𝑝)i + 𝜋1(1 − 𝑝)𝐼𝐶 − µ𝐵𝐵𝑐). ̇ (16) substituting at endemic equilibrium, 𝛬 = 𝜇(𝑃∗ + 𝑆∗ + 𝐼∗ + 𝐼𝑐 ∗ + 𝑅∗) + 𝑑𝐼∗, µ 𝐵 = 𝜋2(1−𝑝)𝐼 ∗ 𝐵𝑐 ∗ + 𝜋1(1−𝑝)𝐼𝑐 ∗ 𝐵𝑐 ∗ in equation (16) and simplify, we have 𝐿′ = −𝜇[(𝑃 − 𝑃∗) + (𝑆 − 𝑆∗) + (𝐼 − 𝐼∗) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗)]2 + 𝜋2(1 − 𝑝)𝐼∗ [1 + 𝐼 𝐼∗ − 𝐵𝑐 𝐵𝑐 ∗ − 𝐵𝑐 ∗𝐼 𝐼∗𝐵𝑐 ] + 𝜋1(1 − 𝑝)𝐼𝑐 ∗ [1 + 𝐼𝑐 𝐼𝑐 ∗ − 𝐵𝑐 𝐵𝑐 ∗ − 𝐵𝑐 ∗𝐼 𝐼∗𝐵𝑐 ] − 𝑑(𝐼 − 𝐼∗)[(𝑃 − 𝑃 ∗) + (𝑆 − 𝑆∗) + (𝐼 − 𝐼∗) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗)]. using the hypothesis that 𝑑 = 0, we have 𝐿′ = −𝜇[(𝑃 − 𝑃∗) + (𝑆 − 𝑆∗) + (𝐼 − 𝐼∗) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗)]2 + (𝜋2(1 − 𝑝)𝐼 ∗ + 𝜋1(1 − 𝑝)𝐼𝑐 ∗)[2 − 𝐵𝑐 𝐵𝑐 ∗ − 𝐵𝑐 ∗ 𝐵𝑐 ] with 𝐼𝑐 𝐼𝑐 ∗ ≤ 1, 𝐼 𝐼∗ ≤ 1. this implies that 𝐿′ ≤ 0 since 2 − 𝐵𝑐 𝐵𝑐 ∗ − 𝐵𝑐 ∗ 𝐵𝑐 ≤ 0, by arithmetic and geometric theorem and 𝐿 = 0 if 𝑃 = 𝑃∗ , 𝑆 = 𝑆∗ , 𝐼 = 𝐼∗ , 𝐼𝑐 = c. e. madubueze, r.i. gweryina, and k. a. tijani 266 𝐼𝑐 ∗ , 𝑅 = 𝑅∗ and 𝐵𝑐 = 𝐵𝑐 ∗. this means that the endemic equilibrium, 𝐸∗, is globally asymptotically stable (gas) whenever 𝑅𝑐 > 1 and 𝑑 = 0 according to lasalle’s invariance principle. 3.8 local stability of endemic equilibrium state due to the mathematical complexity of the stability of endemic equilibrium, the centre manifold theory approach is used to establish the local stability of endemic equilibrium by proving the existence of a forward bifurcation of the system. a forward bifurcation means that the endemic equilibrium is locally asymptotically stable if 𝑅𝑐 > 1 but near unity. theorem 5. the endemic equilibrium is locally asymptotically stable whenever if 𝑅𝑐 > 1 but near unity. proof. using the approach of centre manifold theory by castillo-chavez and song [23], let 𝛽 = 𝛽∗ be a bifurcation parameter at 𝑅𝑐 = 1 so that 𝛽 = 𝛽∗ = 𝐾µ𝐵𝑘3𝑘4 𝑎s0[(𝜎𝜋1+𝜋2𝑘4)(1−𝑝)] . this implies that the jacobian matrix of equation (12) has negative eigenvalues and a simple zero eigenvalue. the left and right eigenvectors associated with the jacobian matrix (12) are 𝑤 = (𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝑤5, 𝑤6) and 𝑣 = (𝑣1, 𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣6) respectively where 𝑤1 = ƞ(k3k4k5−ɸ𝜏1k4−𝜎ɸ(𝜏2 + ѱ𝜃))𝑤3 k4k5(ƞ𝛾−𝑘1k2) ,𝑤2 = 𝑘1(k3k4k5−ɸ𝜏1k4−𝜎ɸ(𝜏2 + ѱ𝜃))𝑤3 k4k5(ƞ𝛾−𝑘1k2) ,𝑤4 = 𝜎𝑤3 k4 ,𝑤5 = (𝜏1k4+σ(𝜏2 + ѱ𝜃) )𝑤3 k4k5 ,𝑤6 = k3𝑤3 𝑐 ,𝑣1 = 𝑣2 = 𝑣5 = 0,𝑣4 = 𝜋1(1−𝑝)c𝑣3 k4µ𝐵 ,𝑣6 = 𝑐𝑣3 µ𝐵 ,𝑤3,𝑣3 > 0,𝑐 = a𝛽𝑆0 𝐾 . representing the state variables 𝑃 = 𝑥1, 𝑆 = 𝑥2, 𝐼 = 𝑥3, 𝐼𝑐 = 𝑥4, 𝑅 = 𝑥5, 𝐵𝑐 = 𝑥6 so that the system (1) becomes 𝑑𝑋 𝑑𝑡 = 𝐹 = (𝑓1,𝑓2,𝑓3,𝑓4,𝑓5,𝑓6) 𝑇 with 𝑓𝑖 = 𝑓𝑖(𝑥1,𝑥2,𝑥3,𝑥4,𝑥5,𝑥6), we have the non-zero second order partial derivatives at 𝐸0 given as 𝜕2𝑓3(𝐸0) 𝜕𝑥2𝜕𝑥6 = (1−𝑝)𝛽∗ 𝑘 , 𝜕2𝑓3(𝐸0) 𝜕𝑥6 2 = − 2(1−𝑝)𝛽∗𝑆0 𝑘2 , 𝜕2𝑓6(𝐸0) 𝜕𝑥6𝜕𝛽 ∗ = (1−𝑝)𝑆0 𝑘 . we compute the coefficients, 𝑚 and 𝑛 as follows 𝑚 = 𝑣3 (𝑤2𝑤6 𝜕2𝑓3 𝜕𝑥2𝜕𝑥6 + 𝑤6 2 𝜕 2𝑓3 𝜕𝑥6 2) and 𝑛 = 𝑤3𝑣3 𝜕2𝑓3 𝜕𝑥3𝜕𝛽 ∗ > 0. upon substituting, we have 𝑚 = − k3(1−𝑝)𝛽 ∗𝑣3𝑤3 2 𝑐𝑘 ( 𝑘1(k3k4k5−ɸ𝜏1k4−𝜎ɸ(𝜏2 + ѱ𝜃)) k4k5(𝑘1k2−ƞ𝛾) + 2k3𝑆0 𝑐𝑘 ) < 0, a dynamic model of typhoid fever with optimal control analysis 267 𝑛 = 𝑤3k3𝑣3(1−𝑝)𝑆0 𝑐𝑘 > 0 , which implies that a forward bifurcation exists. thus, the endemic equilibrium is locally asymptotically stable if 𝑅𝑐 > 1 but near unity. this is shown graphically in figure 2. figure 2. forward bifurcation for typhoid model. 4. sensitivity analysis and optimal control analysis 4.1 sensitivity analysis of the model parameters sensitivity analysis is used to examine the connection between uncertain parameters of a mathematical model and a property of the observable output [24]. it is used to determine the model parameters that have a great impact on reproduction number, 𝑅𝑐 for the purpose of targeting such by intervention strategies [25]. in carrying out the sensitivity analysis, we adopted normalized forward index method as used by rodrigues et al. [25] and this is given by 𝑆𝑌 𝑅𝑐 = 𝜕𝑅𝑐 𝜕𝑌 × 𝑌 𝑅𝑐 , where 𝑌 is the parameters reflecting in the control reproduction number, 𝑅𝑐. the sensitivity indices of 𝑅𝑐 are given in table 2 using the parameter values in table 1. 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 2 4 6 8 10 12 14 16 18 20 r c in fe c te d i n d iv id u a ls , i* ( t) stable dfe stable ee unstable dfe c. e. madubueze, r.i. gweryina, and k. a. tijani 268 table 1. parameter values of the model with their sources table 2. sensitivity index of the parameters values parameter index sign index sign sensitivity index values parameter index sign sensitivity index values 𝛾 + 0.66520217 𝑑 − 0.53789731 𝛼 − 0.03617419 π1 + 0.161676646 𝜎 − 0.08282213 π2 + 0.838323354 𝜏1 − 0.01629918 μb − 0.999999999 𝜏2 − 0.00027716 ѱ − 0.138579983 𝑝 − 1.99999999 𝜃 − 0.138579983 ƞ − 0.67891735 𝛽 + 1.00000000 from table 2, the parameters with positive indices (𝜋1,𝜋2,𝛾 ) indicate that they have impact on expanding the disease in the population if their values are increasing because the control reproduction number increases as their values increase. also, the parameters in which their sensitivity indices are negative have influence in reducing the burden of the bacteria in the population as their values increase because the control reproduction number decreases as their values increase, which will lead to reducing the endemicity of the bacteria in the population. parameter value source parameter value source λ 100 [16] 𝜇 0.0247 [16] 𝛾 0.33 [26] 𝑑 0.066 [30] 𝛼 0.5 [27] π1 0.9 [30] ɸ 0.000904 [28] π2 0.8 [16] 𝜎 0.03-0.05 [28] μb 0.0345 [31] 𝜏1 0.002 [16] ѱ 0.75 [11] 𝜏2 0.0003 [16] 𝜃 0.2 [16] p ƞ 0.3 0.75 assumed [11] 𝛽 𝐾 0.9 500000 [16] [29] a dynamic model of typhoid fever with optimal control analysis 269 (a) (b) (c) (d) figure 3. simulations for the impact of model parameters on control reproduction number. according to the phase plane (figure 3a), the value of 𝑅𝑐 decreases drastically as 𝜌 and 𝜇𝐵 increases. also, the value of 𝑅𝑐 decreases sharply in figure 3b as 𝜌 increases, but the change of 𝜂 has a significantly lower impact on 𝑅𝑐. the phase planes in figures 3c and 3d illustrate similar results, which shows that 𝑅𝑐 is much sensitive to 𝜇𝐵 than to 𝜂 and 𝜓, respectively. therefore, from all cases, 𝜇𝐵 has shown to be a superior force in reducing the burden of typhoid fever. however, the combination of 𝜌 and 𝜇𝐵 has proven to be the best control strategy as compared to the rest. 4.2 optimal control analysis the optimal control model is formulated from system (1) when the constant parameters, ƞ,p,ψ,µ𝐵 are time dependent that is ƞ(t), p(t), ѱ(t) and µ𝐵(t) c. e. madubueze, r.i. gweryina, and k. a. tijani 270 where ƞ(t) is vaccination control, p(t) is hygiene practice control, ѱ(t) is the screening control and µ𝐵(t) is the sterilization control. the objective function to be minimized is given as 𝐽(ƞ(t), 𝑝(t), ѱ(t), µ𝐵(t)) = ∫ (𝐼 + 𝐼𝐶 + bc + 𝑚1ƞ 2(t) 2 + 𝑚2𝑝 2(t) 2 + 𝑡𝑓 0 𝑚3ѱ 2(𝑡) 2 + 𝑚4𝜇𝐵 2(t) 2 ) (18) subject to equation (1) with ƞ = ƞ(𝑡),p = p(t),ψ = ψ(t),µ𝐵 = µ𝐵(𝑡). (19) the coefficients, 𝑎,𝑏,𝑐,are the weight constants for the infected, carriers and the bacteria concentration respectively whereas 𝑚𝑖, i = 1,2,3,4 are cost of implementing these control measures. we assume a quadratic expression for the costs based on literature. the control measures, ƞ(t), 𝑝(t), ѱ(t), µ𝐵(t) are lebesgue measurable with 0 ≤ ƞ(t) < 0.9,0 ≤ 𝑝(t) < 1,0 ≤ ѱ(t) < 1, 0 ≤ µ𝐵(t) < 1 for 0 ≤ 𝑡 ≤ 𝑡𝑓. we aimed to minimize the number of infectives, carriers, bacteria concentration and their costs of implementations, that is, 𝐽(ƞ(t)∗, 𝑝(t)∗,ѱ(t)∗,µ𝐵(t) ∗) = 𝑚𝑖𝑛 𝐽(ƞ(t), 𝑝(t), ѱ(t), µ𝐵(t)). the optimal control pair is obtained using pontryagin maximum principle [32]. this principle converts equations (18) and (1) with (19) into a problem of minimizing pointwise a hamiltonian h with respect to ƞ(t), 𝑝(t), ѱ(t), µ𝐵(t) such that; 𝐻(𝑃,𝑆,𝐼, 𝐼𝐶,𝑅,𝐵𝐶) = 𝑑𝐽 𝑑𝑡 + 𝜆1 𝑑𝑃 𝑑𝑡 + 𝜆2 𝑑𝑆 𝑑𝑡 + 𝜆3 𝑑𝐼 𝑑𝑡 + 𝜆4 𝑑𝐼𝑐 𝑑𝑡 + 𝜆5 𝑑𝑅 𝑑𝑡 + 𝜆6 𝑑𝐵𝐶 𝑑𝑡 . thus, 𝐻(𝑃,𝑆,𝐼, 𝐼𝐶,𝑅,𝐵𝐶) = (𝑎𝐼 + 𝑏𝐼𝐶 + 𝑚1ƞ(t) 2 2 + 𝑚2𝑝(t) 2 2 + 𝑚3ѱ(t) 2 2 + 𝑚4µ𝐵(t) 2 2 ) + 𝜆1 𝑑𝑃 𝑑𝑡 + 𝜆2 𝑑𝑠 𝑑𝑡 + 𝜆3 𝑑𝐼 𝑑𝑡 + 𝜆4 𝑑𝐼𝐶 𝑑𝑡 + 𝜆5 𝑑𝑅 𝑑𝑡 + 𝜆6 𝑑𝐵𝐶 𝑑𝑡 , where 𝜆1,𝜆2,𝜆3,𝜆4,𝜆5 and𝜆6 are the adjoint variable functions. so, 𝐻 = (𝑎𝐼 + 𝑏𝐼𝑐 + 𝑐𝐵𝑐 + 𝑚1ƞ(t) 2 2 + 𝑚2𝑝(t) 2 2 + 𝑚3ѱ(t) 2 2 + 𝑚4µ𝐵(t) 2 2 ) + 𝜆1(αʌ + ƞ(t)s – (γ + µ)p) + 𝜆2 ((1 − α)ʌ + γp + ɸr – (ƞ(t) + µ + (1 − p(t)) 𝛽𝐵𝑐 (𝐾+𝐵𝑐) )s) + 𝜆3 ((1 − p(t)) 𝛽𝐵𝑐 (𝐾+𝐵𝑐) s – (σ + 𝜏1 + µ + 𝑑)i ) + a dynamic model of typhoid fever with optimal control analysis 271 𝜆4(σ𝐼 – (𝜏2 + ѱ(t)𝜃 + 𝜇)𝐼𝑐) + 𝜆5(𝜏1i + (𝜏2 + ѱ(t)𝜃)𝐼𝑐 – (µ + ɸ)r ) + 𝜆6(𝜋2(1 − 𝑝(t))i + 𝜋1(1 − 𝑝(t))𝐼𝑐 − µ𝐵(t)𝐵𝑐). (20) theorem 5. given an optimal control ƞ(t)∗, 𝑝(t)∗,ѱ(t)∗,µ𝐵(t) ∗ and corresponding state variables 𝑃,𝑆,𝐼,𝐼𝐶,𝑅,𝑎𝑛𝑑 𝐵𝐶 that minimize the objective function 𝐽(ƞ(t), 𝑝(t), ѱ(t), µ𝐵(t)) over u, there exist adjoint functions 𝜆1,…𝜆6 satisfying; 𝑑𝜆1 𝑑𝑡 = (𝜆1 − 𝜆2)𝛾 + μ𝜆2, 𝑑𝜆2 𝑑𝑡 = (𝜆2 − 𝜆1)ƞ(t)+ μ𝜆2 + (𝜆2 − 𝜆3)(1 − p(𝑡)) 𝛽𝐵𝑐 (𝐾+𝐵𝑐) , 𝑑𝜆3 𝑑𝑡 = −𝑎 + (𝜆3 − 𝜆4)σ + (𝜆3 − 𝜆5)𝜏1 + 𝜆3(µ + 𝑑) − 𝜋2(1 − 𝑝(t))𝜆6 , 𝑑𝜆4 𝑑𝑡 = −𝑏 + (𝜆4 − 𝜆5)(𝜏2 + ѱ(t)𝜃) + 𝜆4𝜇 − 𝜋1(1 − 𝑝(t))𝜆6, 𝑑𝜆5 𝑑𝑡 = (𝜆5 − 𝜆4)ɸ + 𝜆5𝜇 , 𝑑𝜆6 𝑑𝑡 = −𝑐 + (𝜆2 − 𝜆3)(1 − p(t)) 𝛽 (𝐾+𝐵𝑐) (1 − 𝐵𝑐 (𝐾+𝐵𝑐) ) + 𝜆6µ𝐵(t). } (21) with the transversality condition, 𝜆𝑖(𝑡𝑓) = 0,for 𝑖 = 1(1)6 and the controls ƞ∗(t),𝑝∗(t),ѱ∗(t)and 𝜇𝐵 ∗(t) satisfying the optimality condition; ƞ∗ = 𝑚𝑎𝑥{0,𝑚𝑖𝑛 (1, (𝜆2−𝜆1)𝑆 𝑚1 )} , p∗ = {0,𝑚𝑖𝑛 (1, (𝜆3−𝜆2)𝛽𝐵𝑐𝑆+𝜆6𝜋2𝐼(𝐾+𝐵𝑐)+𝜆6𝜋1𝐼𝑐(𝐾+𝐵𝑐) 𝑚2(𝐾+𝐵𝑐) )} , ѱ∗ = 𝑚𝑎𝑥{0,𝑚𝑖𝑛 (1, (𝜆4−𝜆5)𝜃𝐼𝑐 𝑚3 )} , µ𝐵 ∗ = 𝑚𝑎𝑥{0,𝑚𝑖𝑛 (1, 𝜆6𝐵𝑐 𝑚4 )} . } (22) proof. using pontryagin maximum principle, we obtained the adjoint equation and tranversality conditions by differentiating the hamiltonian function with respect to state variables 𝑃,𝑆,𝐼,𝐼𝐶,𝑅 and 𝐵𝐶 respectively which is evaluated at the optimal control functions ƞ(t), 𝑝(t), ѱ(t), µ𝐵(t). so, the adjoint system (21) is obtained using the following derivatives 𝑑𝜆1 𝑑𝑡 = − 𝜕𝐻 𝜕𝑃 , 𝑑𝜆2 𝑑𝑡 = − 𝜕𝐻 𝜕𝑆 , 𝑑𝜆3 𝑑𝑡 = − 𝜕𝐻 𝜕𝐼 , 𝑑𝜆4 𝑑𝑡 = − 𝜕𝐻 𝜕𝐼𝐶 , 𝑑𝜆5 𝑑𝑡 = − 𝜕𝐻 𝜕𝑅 , 𝑑𝜆6 𝑑𝑡 = − 𝜕𝐻 𝜕𝐵𝑐 while the interior of the control set of equation (22) is obtained by solving for ƞ(t), 𝑝(t), ѱ(t), µ𝐵(t) in the respective equations c. e. madubueze, r.i. gweryina, and k. a. tijani 272 𝜕𝐻 𝜕ƞ(t) = 0, 𝜕𝐻 𝜕p(t) = 0, 𝜕𝐻 𝜕ѱ(t) = 0 , 𝜕𝐻 𝜕µ𝐵(t) = 0. this completes the proof. the optimality system involves equation (1) with (19), equations (21) and (22). 5. numerical simulations and discussion the numerical simulations of the optimality system involving equations (1) with (19), (21) and (22) are implemented using runge-kutta method with the aid of matlab r2007b. the simulations are carried out to examine the impact of the control measures on typhoid fever. the parameter values used for the simulations are in table 2 while the initial c onditions are from mushanyu et al. (2018) as follows; s(0) = 10000,i(0) = 10,ic(0) = 10,r(0) = 0,bc(0) = 100000. p(0) = 100 is assumed. the weight constants for simulation are given as m1 = 9 × 10 −1,m2 = 5 × 105,m3 = 7 × 10 2 and m4 = 4 × 10 6. (a) optimal and constant control. the importance of time-dependent control measures is considered in figure 3. with optimal control, a typhoid-free population is attained within 200 days compared with constant control which shows the endemicity of the typhoid in the population. this is achieved when 𝑢1 is at the upper bound for 150 days and 𝑢2, 𝑢3 and 𝑢4 are below a bound of 0.3 for 175 days before they decline to their final time. this implies that control measures should be implemented in time to achieve a typhoid free population. figure 4. solutions of typhoid model for the infected state variables with and without control measures with control profile. 0 50 100 150 200 0 500 1000 1500 time (days) in fe c te d i n d iv id u a ls , i( t) a optimal constant 0 50 100 150 200 0 200 400 600 800 time (days) in fe c te d c a rr ie rs , i c (t ) b optimal constant 0 50 100 150 200 0 5 10 x 10 4 time (days) b a c te ri a c o n c e n tr a ti o n , b (t ) c optimal constant 0 50 100 150 200 0 0.5 1 time (days) c o n tr o l p ro fi le d 1 2 3 4 a dynamic model of typhoid fever with optimal control analysis 273 (b) vaccination and hygiene practices. we minimize the objective function for vaccination and hygiene practices (𝑢1,𝑢2 ≠ 0,𝑢3 = 𝑢4 = 0) to assess their effect on the disease. the number of infected individuals and bacteria concentration are reduced when compared to without control (see figure 5). this is obtained when 𝑢1 is at its upper bound for all the time 200 days and 𝑢2 attains a bound of 0.9 and decline after 5 days (figure 5d). however, typhoid disease still remains in the population. figure 5. solutions of typhoid model for the infected state variables without and with vaccination (𝑢1) and hygiene practices (𝑢2) control measures only. w/o means without. (c) vaccination and screening. we minimize the objective function for vaccination and screening (𝑢1,𝑢3 ≠ 0,𝑢2 = 𝑢4 = 0). they reduced the number of infected persons and bacteria concentration but not as in case (b) (see figures 5 and 6) as the number of carriers reduces in figure 6b than figure 5b. this may be as a result of screening in the combined control measures. this is achieved when the control, 𝑢1, is maintain at the upper bound for all time (200 days) while 𝑢3 decline after attaining upper bound for 110 days (figure 6d). 0 50 100 150 200 0 2000 4000 6000 time (days) in fe ct ed i nd iv id ua ls , i( t) a u 1 , u 2 w/o control 0 50 100 150 200 0 500 1000 1500 2000 time (days) in fe ct ed c ar rie rs , i c( t) b u 1 , u 2 w/o control 0 50 100 150 200 1 2 3 4 5 x 10 5 time (days) b ac te ria c on ce nt ra tio n, b (t ) c u 1 , u 2 w/o control 0 50 100 150 200 0 0.5 1 time (days) c on tr ol p ro fil e d u 1 u 2 c. e. madubueze, r.i. gweryina, and k. a. tijani 274 figure 6. solutions of typhoid model for the infected state variables without and with vaccination (𝑢1) and screening (𝑢3) control measures only. w/o means without. (d) vaccination and sterilization. we minimize the objective function for vaccination and sterilization (𝑢1,𝑢4 ≠ 0,𝑢2 = 𝑢3 = 0). the simultaneous implementation of 𝑢1 and 𝑢4 reduced the number of infected persons and bacteria concentration to zero after 70 days and 30 days respectively while the number of carriers in the population is almost zero as at 200 days. the control, 𝑢1, maintains an upper bound for 200 days while 𝑢4 attains a bound of 0.2 for 190 days before decline to its final time. 0 50 100 150 200 0 2000 4000 6000 time (days) in fe ct ed i nd iv id ua ls , i( t) a u 1 , u 3 w/o control 0 50 100 150 200 0 500 1000 1500 2000 time (days) in fe ct ed c ar rie rs , i c( t) b u 1 , u 3 w/o control 0 50 100 150 200 1 2 3 4 5 x 10 5 time (days) b ac te ria c on ce nt ra tio n, b (t ) c 0 50 100 150 200 0 0.5 1 time (days) c on tr ol p ro fil e d u 1 u 3 u 1 , u 3 w/o control a dynamic model of typhoid fever with optimal control analysis 275 figure 7. solutions of typhoid model for the infected state variables without and with vaccination (𝑢1) and sterilization (𝑢4) control measures only. here, w/o means without. (e) hygiene practices and screening. we minimize the objective function for hygiene practices and screening (𝑢2,𝑢3 ≠ 0,𝑢1 = 𝑢4 = 0). the observed effect is similar to case (c) except that 𝑢3 attains an upper bound and declines after 70 days while 𝑢2 of a bound of 0.55 and declines immediately to final time. the disease still remains endemic in the population. 0 50 100 150 200 0 2000 4000 6000 time (days) in fe ct ed i nd iv id ua ls , i( t) a u 1 , u 4 w/o control 0 50 100 150 200 0 500 1000 1500 2000 time (days) in fe ct ed c ar rie rs , i c( t) b u 1 , u 4 w/o control 0 50 100 150 200 0 2 4 6 x 10 5 time (days) b ac te ria c on ce nt ra tio n, b (t ) c u 1 , u 4 w/o control 0 50 100 150 200 0 0.5 1 time (days) c on tr ol p ro fil e d u 1 u 4 c. e. madubueze, r.i. gweryina, and k. a. tijani 276 figure 8. solutions of typhoid model for the infected state variables without and with hygiene practices (𝑢2) and screening (𝑢3) control measures only. here, w/o means without. (f) hygiene practices and sterilization. we minimize the objective function for hygiene practices and sterilization as control measures (𝑢2,𝑢4 ≠ 0,𝑢1 = 𝑢3 = 0). the combined implementation of 𝑢2 and 𝑢4 reduces the number of infected persons and bacteria concentration to zero after 110 days and 50 days respectively while there is still some infected carriers in the population after 200 days. the hygiene practice 𝑢2, initially increases from 0.18 to 0.28 bound within 8 days and declines after 120 days while 𝑢4 attains a bound of 0.2 for 195 days before declining to its final time. 0 50 100 150 200 0 2000 4000 6000 time (days) in fe ct ed i nd iv id ua ls , i( t) a u 2 , u 3 w/o control 0 50 100 150 200 0 500 1000 1500 2000 time (days) in fe ct ed c ar rie rs , i c( t) b u 2 , u 3 w/o control 0 50 100 150 200 1 2 3 4 5 x 10 5 time (days) b ac te ria c on ce nt ra tio n, b (t ) c u 2 , u 3 w/o control 0 50 100 150 200 0 0.5 1 time (days) c on tr ol p ro fil e d u 2 u 3 a dynamic model of typhoid fever with optimal control analysis 277 figure 9. solutions of typhoid model for the infected state variables without and with hygiene practices (𝑢2) and sterilization (𝑢4) control measures only. here, w/o means without. (g) screening and sterilization. we minimize the objective function for screening and sterilization as control measures (𝑢3,𝑢4 ≠ 0,𝑢1 = 𝑢2 = 0). the simultaneous implementation of 𝑢3 and 𝑢4 behaves similar as cases (e) and (f). here, the number of infected persons, carriers and bacteria concentration reduce to zero after 75 days, 100 days and 45 days respectively. this is achieved when 𝑢3 and 𝑢4 are at bound 0.28 for 170 days and 0.19 for 190 days respectively before declining to their final time. 0 50 100 150 200 0 2000 4000 6000 time (days) in fe ct ed i nd iv id ua ls , i( t) a u 2 , u 4 w/o control 0 50 100 150 200 0 500 1000 1500 2000 time (days) in fe ct ed c ar rie rs , i c( t) b u 2 , u 4 w/o control 0 50 100 150 200 0 2 4 6 x 10 5 time (days) b ac te ria c on ce nt ra tio n, b (t ) c u 2 , u 4 w/o control 0 50 100 150 200 0 0.1 0.2 0.3 0.4 time (days) c on tr ol p ro fil e d u 2 u 4 c. e. madubueze, r.i. gweryina, and k. a. tijani 278 figure 10. solutions of typhoid model for the infected state variables without and with screening (𝑢3) and sterilization (𝑢4) control measures only. here, w/o means without. (h) three combine control measures we minimize the objective function for three control measures that is 𝑢1,𝑢2,𝑢3,≠ 0,𝑢4 = 0 (123), 𝑢1,𝑢2,𝑢4,≠ 0,𝑢3 = 0 (124), 𝑢1,𝑢3,𝑢4,≠ 0,𝑢2 = 0 (134) and 𝑢2,𝑢3,𝑢4,≠ 0,𝑢1 = 0 (234). we notice from figure (10) that bacteria clearance reduces the number of infected populations (𝐼(𝑡),𝐼𝑐(𝑡)) and bacteria concentration. however, the combine implementation of vaccination, screening and sterilization gives a better result compared to 𝑢1,𝑢2,𝑢4, and 𝑢1,𝑢3,𝑢4, as it achieves a typhoid-free population in shortest period of time than others. 0 50 100 150 200 0 2000 4000 6000 time (days) in fe ct ed i nd iv id ua ls , i( t) a u 3 , u 4 w/o control 0 50 100 150 200 0 500 1000 1500 2000 time (days) in fe ct ed c ar rie rs , i c( t) b u 3 , u 4 w/o control 0 50 100 150 200 0 2 4 6 x 10 5 time (days) b ac te ria c on ce nt ra tio n, b (t ) c u 3 , u 4 w/o control 0 50 100 150 200 0 0.1 0.2 0.3 0.4 time (days) c on tr ol p ro fil e d u 3 u 4 a dynamic model of typhoid fever with optimal control analysis 279 figure 11. solutions of typhoid model for the infected state variables with optimal control. here, 123 means 𝑢1,𝑢2,𝑢3 combine, 124 means 𝑢1,𝑢2,𝑢4 combine, 134 means 𝑢1,𝑢3,𝑢4 combine, 234 means 𝑢2,𝑢3,𝑢4 combine. 7. conclusion in this study, the mathematical model of typhoid fever dynamics with protected human population and bacteria concentration is examined. the control measures such as vaccination, hygiene practice and screening are taken into consideration. the disease-free and endemic equilibrium states are both locally and globally stable whenever 𝑅𝑐 < 1 and 𝑅𝑐 > 1 respectively. the local stability of endemic equilibrium state is established using centre manifold theorem in order to show existence of forward bifurcation while the global stability is done when disease-related death rate is neglected. the sensitivity analysis of the control reproduction number is carried out and the result indicates that the typhoid fever disease will be controlled in the population if susceptible people are vaccinated with high practice of personal hygiene as well as screening of the carriers are screened and also the bacteria in the environment is disinfect or sterilization. the optimal control analysis is carried out for time-dependent control functions to form non-autonomous system. the pontryagin maximum principle is used to establish the optimality conditions for the system. this is solved numerically to establish that optimal control implementation achieved infectionfree population on time compare to constant control. considering when there is limited resources to implement all the controls together, screening and bacteria sterilization should be adopted for two combined controls, while vaccination, 0 50 100 150 200 0 1000 2000 3000 time (days) in fe ct ed in di vi du al s, i( t) a 123 124 134 234 0 50 100 150 200 0 200 400 600 time (days) in fe ct ed c ar rie rs , i c( t) b 123 124 134 234 0 50 100 150 200 0 5 10 15 x 10 4 time (days) b ac te ria c on ce nt ra tio n, b (t) c 123 124 134 234 c. e. madubueze, r.i. gweryina, and k. a. tijani 280 screening and bacteria sterilization should be implemented together for three combine controls. however, the combined implementation of all controls is more effective in eradicating the disease from the environment. it is therefore recommended that these preventive measures (vaccination, hygiene practice, screening and sterilization) should be adopted by the policy makers to eliminate the typhoid bacteria from the population. references [1] who. typhoid. www.whoints/news-room/fact-sheets/detail/typhoid, 2020. accessed on 10th may, 2021. 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[32] l. s. pontryagin, v. g. boltyanskii, r. v. gamkrelidze and e. f. mishchenko. the mathematical theory of optimal processes, john wiley &sons, lonon, uk, 1962. ratio mathematica volume 44, 2022 nano semi* -open sets reena c * kanaga m † abstract in this paper, we introduce a new class of sets called nano semi* -open sets and discuss some of its properties in nano topological space. we also, present nano semi* -interior, nano semi* -closure and study some of its fundamental properties. keywords: nano semi* -open, nano semi* -closed, nano semi* -interior, nano semi* –closure. ams subject classification: 54a05 ‡ * assistant professor, department of mathematics, st. mary’s college (autonomous), (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli), thoothukudi-1, tamilnadu, india; reenastephany@gmail.com. † sec research scholar, reg.no. 21122212092007, department of mathematics, st. mary’s college (autonomous), (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli), thoothukudi-1, tamilnadu, india; kanahaspm@gmail.com. ‡ received on june 12th, 2022. accepted on sep 1 st , 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.917. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 288 mailto:reenastephany@gmail.com c. reena and m. kanaga 1. introduction the notion of nano topology was introduced by lellis thivagar [8] which was defined in terms of approximations and boundary region of a subset of a universe using an equivalence relation on it and also defined nano closed sets, nano-interior and nano-closure. he also introduced the weak forms of nano open sets namely nano open sets, nano semi-open sets and nano pre-open sets. in 2017, the concept of nano semi open sets was introduced by qays hatem imran [3]. in 2014, a. robert and s. pious missier [7] have introduced and studied semi* -open sets in general topology. in this paper we introduce nano semi* -open sets and nano semi * -closed sets in nano topological spaces. we investigate its fundamental properties and find its relation with other nano sets and study some of its properties. 2. preliminaries throughout this chapter (u, (x)) is a nano topological space with respect to x where x ⊆ u, r is an equivalence relation on u, u/r denotes the family of equivalence classes of u by r. definition 2.1 [8]: let u be a non-empty finite set of objects called the universe and r be an equivalence relation on u named as the indiscernibility relation. then u is divided into disjoint equivalence classes. elements belonging to the same equivalence class are said to be discernible with one another. the pair (u, r) is said to be the approximation space. let x⊆u 1. the lower approximation of x with respect to r is the set of all objects which can be for certain classified as x with respect to r and it is denoted by . that is (x) = ⊆ where r(x) denotes the equivalence class determined by x. 2. the upper approximation of x with respect to r is the set of all objects which can be possibly defined as x with respect to r and it is denoted by ur(x). that is ur(x)= 3. the boundary region of x with respect to r is the set of all objects which can be classified neither as x nor as not x with respect to r and is denoted by br(x). that is br(x) = ur(x) – lr(x). proposition 2.2 [8] if (u, r) is an approximation space and x, y ⊆ u, then 1. lr(x) ⊆ x ⊆ ur(x) 2. lr ( ) = ur ( ) = and lr (u) = ur(u) = u 3. ur (x ∪y) = ur(x) ∪ur(y) 4. ur (x ∩ y) ⊆ ur(x) ∩ ur(y) 5. lr (x ∪y) ⊇ lr(x) ∪ lr(y) 6. lr (x ∩ y) = lr(x) ∩ lr(y) 7. lr(x) ⊆lr(y) and ur(x) ⊆ur(y) whenever x ⊆y 8. ur(x c ) = [lr(x)] c and lr(x c ) = [ur(x)] c 9. urur(x) = lr ur(x) = ur(x) 10. lrlr(x) = ur lr(x) = (x) 289 nano semi* -open sets definition 2.3 [8]: let u be the universe, r be an equivalence relation on u and (x) = {u, , lr(x), ur(x), br(x)} where x ⊆ u. then by the proposition 2.2, r(x) satisfies the following axioms: 1. u and (x) 2. the union of the elements of any subcollection of (x) is in (x). 3.the intersection of the elements of any finite subcollection of (x) is in (x). that is (x) is a topology on u called the nano topology on u with respect to x. we call ( , (x)) as the nano topological space. the elements of (x) are called as nano-open sets. definition 2.4 [8]: if (u, (x)) is a nano topological space with respect to x and if a ⊆ u, then (i) nano interior of a is defined as the union of all nano-open sets contained in a and is denoted by nint (a). that is, nint(a) is the largest nano-open subset of a. (ii) nano closure of a is defined as the intersection of all nano-closed sets containing a and it is denoted by ncl(a). that is, ncl(a) is the smallest nano-closed set containing a. definition 2.5 [2]: let (u, (x)) be a nano topological space. a subset a of (u, (x)) is called nano generalized-closed (briefly ngclosed) if ncl(a) ⊆v where a ⊆v and v is nano-open. the complement of nano generalized -closed set is called as nano generalized-open. definition 2.6 [2]: for every set a ⊆u, the nano generalized closure of a is defined as the intersection of all ngclosed sets containing a and is denoted by ncl*(a). definition 2.7 [2]: for every set a ⊆u, the nano generalized interior of a is defined as the union of all ngopen sets contained in a and is denoted by nint*(a). definition 2.8: let (u, (x)) be a nano topological space and a ⊆u. then a is said to be (i) nano -open [8] if a ⊆ n int (ncl (nint (a))) (ii) nano semi*-open [1] if a ⊆ ncl*(nint(a)) (iii) nano semi -open [3] if a ⊆ ncl (nint (ncl (nint a))) (iv) nano semi pre-open [6] if a ⊆ ncl (nint (ncl (a))) (v) nano regular-open [8] if a = nint (ncl (a)) (vi) nano regular *-open [5] if (vii) nano pre *-open [4] if ⊆ (viii) nano pre-open [8] if a ⊆ nint (ncl (a)) (ix) nano -open [9], if for each x a, there exists a nano open set g such that x g ⊆ ncl (a) ⊆ a. the complements of the above-mentioned sets are called their respective nano-closed sets. 290 c. reena and m. kanaga 3. nano semi* -open sets definition 3.1: a subset a of a nano topological space is called nano semi* open if there is a nano -open set g in u such that ⊆ ⊆ . the collection of all nano semi* -open sets is denoted by . example 3.2: let with u/r= {{ },{b,c,d}} let then the nano-closed sets are the nano generalizedclosed sets are . the nano generalized-open sets are . theorem 3.3: for a subset a of a nano topological space the following statements are equivalent: (i)a is nano -open. (ii) ⊆ (iii) proof: (i) (ii) if a is a nano semi * -open, then there is a nano -open set g in u such that ⊆ ⊆ now ⊆ g= ⊆ a ⊆ ⊆ . (ii) (iii)by assumption, a⊆ . we have (a)⊆ = now ⊆ implies that ⊆ therefore (iii) (i) take g= then g is a nano -open set in u such that g⊆ ⊆ therefore by definition 3.1, a is nano semi * open. theorem 3.4: arbitrary union of nano semi * -open set is nano semi * -open. proof: let { } be a collection of nano semi * -open sets in nano topological space u. then there exists a nano -open set such that ⊆ ⊆ for each hence ∪ ⊆ ⊆ ⊆ ∪ .since ∪ is nano -open , by definition 3.1 ∪ is nano semi * -open. remark 3.5: the intersection of two nano semi * open sets need not be a nano semi * -open as seen from the following example. example 3.6: let , .let .then , and . here the subsets } are nano semi * open sets, but a is not nano semi * open. 291 nano semi* -open sets theorem 3.7: if a is nano semi * -open in u and b is nano open in x ,then is nano semi * -open in u. proof: since a is nano semi * -open in u,there is an nano -open sets g such that ⊆ ⊆ .since b is nano open , ⊆ ⊆ ⊆ since is nano -open,by definition 3.1, is nano semi * -open. theorem 3.8: every nano -open set is nano semi * -open. proof: let a be a nano -open set in u. then and hence ⊆ hence a is nano semi * -open. remark 3.9: the converse of the above theorem is not true as shown in the following example. example 3.10: let , .let . then and . clearly the sets{ and are nano semi * -open but not nano open. theorem 3.11: every nano open set is nano semi * -open. proof: let a be any nano open set. since every nano open set is nano -open and hence by theorem 3.8, a is nano semi * -open. remark 3.12: the converse of the above theorem is not true as shown in the following example. example 3.13: let , .let . then and , .clearly the sets and is nano semi * open but not nano open. theorem 3.14: every nano semi * -open set is nano semi * -open. proof: let a be any nano semi * open set. then there is a nano open set g in u such that ⊆ ⊆ .since every nano open set is nano -open ,a is nano semi * -open. remark 3.15: the converse of the above theorem is not true as shown in the following example. example 3.16: let let x = }.then , . . clearly the sets , are nano semi * -open but not nano semi * -open. theorem 3.17: every nano semi * open set is nano semi -open. 292 c. reena and m. kanaga proof: let a be any nano semi * open set. then there is a nano -open set g in u such that ⊆ ⊆ .since ⊆ ,we have ⊆ ⊆ hence a is nano semi -open. remark 3.18: the converse of the above theorem is not true as shown in the following example. example3.19: let . let . }.clearly the subset and are semi nano -open but not nano semi * -open. theorem 3.20: every nano semi * open set is nano semi pre-open. proof: let a be any nano semi * -open set. then there is a nano -open set g such that ⊆ ⊆ .since every nano -open set is nano pre-open and ⊆ , a is nano semi pre-open. remark 3.21: the converse of the above theorem is not true as shown in the following example. example 3.22: let u={ }, u/r= { }.let x= . . and ={ , , , . clearly the subsets , , are nano semi pre-open but not nano semi * -open. theorem 3.23: every nano regular open set is nano semi * -open. proof: let a be any nano regular open set. since every nano regular open set is nano open and by theorem 3.11, we have a is nano semi * -open. remark 3.24: the converse of the above theorem is not true as shown in the following example. example 3.25: let . let . . . = { .clearly the subset is nano semi * -open but not nano regular open. theorem 3.26: every nano regular*-open set is nano semi * -open. proof: let a be any nano regular*-open set. since every nano regular*-open set is nano open and by theorem 3.11, we have a is nano semi * -open. 293 nano semi* -open sets remark 3.27: the converse of the above theorem is not true as shown in the following example. example 3.28: let let x = }.then }, , , . * .clearly the subsets , , , are nano semi * -open but not nano regular*-open. theorem 3.29: every nano semi * -open set is nano pre *-open. proof: let a be any nano semi * -open set. then there is a nano -open set g in u such that ⊆ ⊆ .since every nano -open set is nano pre * -open, we have a is nano pre*open. remark 3.30: the converse of the above theorem is not true as shown in the following example. example3.31: let let x = }.then { , } and * , { , clearly the subsets , { , are nano pre * -open but not nano semi * -open. remark 3.32: the concept of nano semi * -open sets and nano pre-open sets are independent as shown in the following example. example 3.33: let let x = .then , , and ={ , , , , clearly the subset is nano semi * -open but not nano pre open and the subsets , , are nano pre-open but not nano semi * -open. remark 3.34: the concept nano semi * -open and nano -open sets are independent as shown in the following example. example 3.35: let . let . . and .clearly the subsets are nano semi * -open but not nano -open and the subsets are nano -open but not semi * -open. diagram 3.36: from the above discussions we have the following diagram. 294 c. reena and m. kanaga 4. nano semi * closed sets definition 4.1: the complement of nano semi * -open set is called as nano semi * closed. the collection of all nano semi * -open sets is denoted by . example 4.2: let with let . then the nano-closed sets are {u, }. the nano generalized – closed sets are { }. the nano generalized open sets are { . . theorem 4.3: arbitrary intersection of nano semi * -closed sets is nano semi * closed. proof: let be a collection of nano semi * -closed sets in u. since each is nano semi * closed, is a nano semi * -open. since ∪ and hence by thm 3.4, is nano semi * -open. hence is nano semi * -closed. remark 4.4: union of two nano semi * -closed sets need not be nano semi * -closed as shown in the following example. example 4.5: consider , u/r = { }. . then , , and . the sets and are nano semi * -closed but their union ∪ is not nano semi * -closed. theorem 4.6: in any nano topological space. (i)every nano -closed set is nano semi * -closed. (ii) every nano-closed set is nano semi * -closed. (iii) every nano semi * -closed set is nano semi * -closed. (iv) every nano semi * -closed set is nano semi -closed. (v) every nano semi * closed set is nano semi pre-closed. (vi) every nano regular closed set is nano semi * -closed. 295 nano semi* -open sets (vii) every nano regular*closed set is nano semi * -closed. (viii) every nano semi * closed set is nano pre *-closed. proof: (i)let a be any nano -closed set in u, then u\a is nano -open. by theorem 3.8, u\a is nano semi * -open. hence a is nano semi * -closed. (ii) let a be any nanoclosed set in u. then u\a is nano open. by theorem 3.11, u\a is nano semi * -open. hence a is nano semi * -closed. (iii)let a be any nano semi * -closed set in u, then u\a is nano semi * -open. by theorem 3.14, u\a is nano semi * -open. hence a is nano semi * -closed.(iv) let a be a nano semi * -closed set in u. then u\a is nano semi * open. by theorem 3.16, u\a is nano semi -open. hence a is nano semi -closed. (v) let a be a nano semi * -closed set in u, then u\a is nano semi * -open. by theorem 3.20, u\a is nano semi pre-open. hence a is nano semi pre-closed. (vi) let a be a nano regular closed set in u. then u\a is nano regular open. by theorem 3.23, u\a is nano semi * -open. hence a is nano semi * -closed. (vii) let a be a nano regular*-closed set in u. then u\a is nano regular*-open. by theorem 3.26, u\a is nano semi * open .hence a is nano semi * -closed .(viii) let a be a nano semi * -closed set in u. then u\a is nano semi * -open set. by theorem 3.28, u\a is nano pre * -open. hence a is nano pre * -closed. remark 4.7: the converse of each of the statements in theorem 4.6 is not true as shown in the following examples. example 4.8: let with u/r={ .let .then and , and .clearly the subsets and are nano semi * -closed but not nano closed . example 4.9: let with u/r = { }. let x = { .then and .clearly the subsets nano semi * -closed but not nano-closed. example 4.10: let with u/r = { }. let x = . then and . ={ }. clearly the subsets are nano semi * -closed but not nano semi * -closed. example 4.11:let , .then and .clearly the subsets are nano semi closed but not nano semi * -closed . example 4.12: let with let . then and . 296 c. reena and m. kanaga . clearly the subsets {b}, {c} are nano semi preclosed but not nano semi * -closed. example 4.13: let with let . then , }. . clearly the subset is nano semi * -closed but not nano regular-closed. example 4.14: let withu/r let . then , . clearly the subset is nano semi * closed but not nano regular*closed. example 4.15: let with let . then , . , . clearly the subsets are nano pre * closed but not nano semi * -closed. remark 4.16: the concept of nano semi * -closed sets and nano pre-closed sets are independent as shown in the following example. example 4.17: let let x = . then ={ , . ={ , . , ,{c,d}, , , . clearly the subset is nano semi * -closed but not nano pre closed and the subsets , ,{c,d}, , , , , are nano preclosed but not nano semi * -closed. remark 4.18: the concept of nano semi * -closed sets and nano closed sets are independent as shown in the following example. example 4.19: let u= ,u/r = { }. let x= { }. = {u, {d}, }. then = , .clearly the subsets are nano semi * -closed but not nano closed and the subsets are nano -closed but not semi * -closed. 5.nano semi* interior and nano semi* -closure definition 5.1: the nano semi* interior of a is defined as the union of all nano semi* open sets contained in a. it is denoted by s* int(a). definition 5.2: let a be a subset of u. a point u in u is called a nano semi* -interior point of a if a contains a nano semi* -open set containing u. 297 nano semi* -open sets theorem 5.3: if a is any subset of a nano topological space ,then (i) s* int(a) is the largest nano semi* -open set contained in a. (ii)a is nano semi* -open if and only if s* int(a)=a. proof: (i) being the union of all nano semi* -open subsets of a, by theorem 3.4, s* int(a) is nano semi* -open and contains every nano semi* -open subsets of a. (ii) a is nano semi* -open implies s* int(a)=a is obvious from definition 5.1. on the other hand, suppose s* int(a)=a. hence by (i) s* int(a) is nano semi* open and hence a is nano semi* -open. theorem 5.4: in any nano topological space (u, (x), if a and b are subsets of u, then the following results hold: i) s* int ( )= ii) s* int(u)=u iii) s* int(a)⊆a iv) a⊆b s* int(a)⊆s* int(b) v) s* int(s* int(a)) = s* int(a). vi) int(a)⊆s* int(a)⊆s int(a)⊆a vii) s* int(a∪ )⊇s* int(a)∪s* int(b) viii) s* int(a b) ⊆s* int(a) s* int(b) proof: (i), (ii), (iii) and (iv) follows from definition 5.1. by theorem 5.3(i), s* int(a) is nano semi* -open and by theorem 5.3(ii), s* int(s* int(a)) = s* int(a). thus (v) proved. (vi) follows from theorem 3.11 and 3.17. (vii)since a⊆a∪b, from statement (iv) we have s* int(a) ⊆ s* int(a∪b). similarly, s* int(b) ⊆ s* int(a∪b). then s* int(a∪ ) ⊇ s* int(a) ∪ s* int(b). (viii) since ⊆ ,from statement (iv) we have s* int( ) ⊆ s* int(a).similarly s* int( ) ⊆ s* int(b). therefore s* int( ) ⊆ s* int(a) int(b). remark 5.5: in theorem 5.4(vi), each of the inclusions may be strict and equality may also hold. this can be seen from the following examples: example 5.6: let , , .then and , let .then . let then int (b)= , , . here let then , , . let ,then , . here . let .then , , . here 298 c. reena and m. kanaga remark 5.7: in theorem 5.5(vii) and (viii), each of the inclusions may be strict and equality may also hold. this can be seen from the following examples: example 5.8: let , ,x= . then and = { , { . let b = { .then ∪ and , , ∪ . therefore ∪ ∪ letc , . then ∪ and , , ∪ therefore ∪ ∪ let , .then and , , . therefore let g , then and , , . therefore definition 5.9: if a is a subset of a nano topological space u, the nano semi * -closure of a is defined as the intersection of all nano semi * -closed sets in u containing a. it is denoted by . theorem 5.10: if a is any subset of a nano topological space (u, ,then (i) is the smallest nano semi* -closed set in u containing a. (ii)a is nano semi* -closed if and only if proof: (i) since is the intersection of all nano semi* -closed subsets of u containing a, by theorem 4.3, it is nano semi* -closed and it is contained in every nano semi* -closed set containing a and hence it is the smallest nano semi* -closed set in u containing a. (ii)if a is nano semi* -closed, then is obvious. conversely, let , by (i) is nano semi* -closed and hence a is nano semi* -closed. theorem 5.11: in any nano topological space (u, (x), if a and b are subsets of u, then the following results hold: (i)s* cl( )= (ii)s* cl(u)=u (iii)a⊆s* cl(a) (iv)a⊆b s* cl(a)⊆s* cl(b) (v)s* (s* (a))= s* (a). (vi)a⊆s cl(a)⊆s* cl(a)⊆ cl(a) (vii)s* cl(a∪ )⊇s* cl(a) ∪ s* cl(b) (viii)s* cl(a b)⊆s* cl(a) s* cl(b) 299 nano semi* -open sets proof: (i), (ii), (iii) and (iv) follows from definition 5.7. from theorem 5.10(i) s* cl(a) is the nano semi* -closed and from theorem5.10(ii) s* (s* (a))= s* (a).this proves (v). (vi) follows from theorem 4.6 and 4.9. (vii) since a⊆a∪b, from statement (iv) we have s*ncl(a) ⊆ s* ncl(a∪b). similarly, s* cl(b) ⊆ s* cl(a∪ b). then s* cl(a∪b) ⊇s* cl(a) ∪ s* cl(b) (viii) since ⊆ ,from statement (iv) we have s* cl( ) ⊆ s* cl(a).similarly s* cl( ) ⊆ s* cl(b). therefore s* cl( ) ⊆ s* int(a) s* cl(b). remark 5.12: in theorem 5.11(vi), each of the inclusions may be strict and equality may also hold. this can be seen from the following examples: example 5.13: let , , , then , and , { . let .then s* cl(a)= s cl(a)= cl(a)=u. let .then s* cl(b) , s cl(b) , here cl(a). let .thens* cl(c) , , here cl(c). let }. then s* cl(d) , s cl( , cl(a)= here cl(d). let e .then s* cl(e) , s cl(e , cl(e)= here cl(e). 6. conclusions in this article, we have introduced nano semi* -open sets and nano semi * -closed sets in nano topological spaces and studied their characterizations with other nano open sets. a diagramatic explanation gives a clear explanation of this article. references [1] j. arul jesti, k. heartlin, a new class of nearly open sets in nano topological spaces, international journal of advanced science and technology, vol.29, no.9s, pp.4784-4793, 2020. 300 c. reena and m. kanaga [2] k. bhuvaneswari, k. mythilignanapriya, on nano generalized closed sets in nano topological space. international journal of scientific and research publications, volume 4, issues 5, issn 22503153, may 2014. [3] qays hatem imran, on nano semi alpha open sets, journal of science and arts, year 17, no. 2(39), pp. 235-244, 2017. [4] c. reena, r. raxi and s. buvaneswari, nano pre*open sets, proceeding of the international conference on analysis and applied mathematics-2022, organized by ayya nadar janaki ammal college (autonomous), sivakasi. [5] c. reena, b. santhalakshmi and s.m janu priyadharshini , nano regular*open sets, proceeding of the international conference on analysis and applied mathematics-2022, organized by ayya nadar janaki ammal college (autonomous), sivakasi. [6] a. revathy, g. ilango, on nano open sets. int.j.eng. contemp. math sci 1(2),16(2015). [7] a. robert and s. pious missier, a new class of sets weaker than -open sets, international journal of mathematics and soft computing, vol.4, no.2., 197 206. 2014. [8] m.l. thivagar, c. richard, on nano forms of weekly open sets. int. j. math. stat: inven. 1(1), 31 – 37(2013). [9] a. vadivel, p. dhanasekaran, g. saravanakumar, m. angayarkanni, generalizations of nano closed sets in nano topological spaces. 301 ratio mathematica volume 45, 2023 monophonic distance energy for join of some graphs sinju manohar v s1 binu selin t2 abstract let 𝐺 be a connected graph with 𝑛 vertices and 𝑚 edges. let µ1, µ2 ,..., µn be the eigen values of distance matrix of 𝐺. the distance energy of a graph ed (g) = ∑ |𝜇𝑖 | 𝑝 𝑖=1 ,was already studied. we now define and investigate the monophonic distance energy as em (g) = ∑ |𝜇𝑖 𝑀|𝑛𝑖=1 , where µ1 m, µ2 m..., µn m are the eigen values of monophonic distance matrix of graphs. in this paper we find the monophonic distance energy for join of some graphs. keywords: join of graphs; monophonic distance matrix; monophonic distance energy. 2010 mathematics subject classification: 05c12, 05b203. 1 research scholar, reg no: 20113162092016, department of mathematics, scott christian college (autonomous), nagercoil 629 003, india. (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627 012, tamil nadu, india.). email: sinjusinju2124@gmail.com 2 assistant professor, department of mathematics, scott christian college (autonomous), nagercoil 629 003, india. (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627 012, tamil nadu, india.). email: binuselin@gmail.com 3 received on july 19, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.979. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 60 sinju manohar v s, binu selin t 1. introduction in this paper we considered simple, connected and undirected graphs. the concept of energy of a graph was introduced by i. gutman [2] in the year 1978. let 𝐺 be a connected graph with 𝑛 vertices and 𝑚 edges. let 𝐴 = (𝑎𝑖𝑗 ) be the adjacency matrix of the graph. the eigenvalues λ1, λ2, ..., λn of a, assumed in non-increasing order, are the eigen values of the graph 𝐺. the energy 𝐸(𝐺) of 𝐺 is defined to be the sum of the absolute values of the eigen values of 𝐺. ie., 𝐸(𝐺) = ∑ |𝜆𝑖 |𝑛𝑖=1 . also, distance energy of a graph was introduced by i. gutman and others [4] in the year 2008. a. p. santhakumaran and others introduced the monophonic number of a graph in 2014 [8]. for any two vertices 𝑢 and 𝑣 in a connected graph𝐺, a 𝑢 − 𝑣 path is a monophonic path if it contains no chords, and the monophonic distance 𝑑m(𝑢, 𝑣) is the length of a longest 𝑢 − 𝑣 monophonic path in 𝐺. based on these we introduce a new concept monophonic distance energy of a graph. based on these we introduce a new concept monophonic distance energy of a graph. in this paper we investigate the monophonic distance energy of 𝐾1,𝑛 + 𝐾1,𝑛, 𝐾𝑛,𝑛 + 𝐾𝑛, 𝐾𝑛,𝑛 + 𝐾𝑛,𝑛 and 𝐾𝑛 + 𝐾1,𝑛. 2. definitions definition 2.1. let 𝐺 be a connected graph with vertex set 𝑣1, 𝑣2, . . . , 𝑣n. the monophonic distance matrix of 𝐺 is defined as 𝑀 = 𝑀 [𝐺] = (𝑑𝑚𝑖𝑗 ) 𝑛×𝑛 where 𝑑𝑚𝑖𝑗 = { 𝑑𝑚 (𝑣𝑖 , 𝑣𝑗 ), 𝑖𝑓 𝑖 ≠ j 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 . here 𝑑m(𝑣i, 𝑣j) is the monophonic distance of 𝑣i to 𝑣j . the eigen values of monophonic distance matrix 𝑀(𝐺) are denoted by µ1 m, µ2 m ..., µn m and are said to be 𝑀-eigen values of 𝐺 and to form the 𝑀-spectrum of 𝐺, denoted by 𝑠𝑝𝑒𝑐m (𝐺). we note that since the monophonic distance matrix is symmetric, its eigen values are real and can be ordered as µ1 m ≤ µ2 m ≤ ... ≤ µn m. we can define the monophonic distance energy of a graph as 𝐸m(𝐺) = ∑ |𝜇𝑖 𝑀|𝑛𝑖=1 . 3. main results definition 3.1. [3] the join 𝐺 = 𝐺1 + 𝐺2 of graphs 𝐺1 and 𝐺2 with disjoint point sets 𝑉1 and 𝑉2 and edge sets 𝑋1 and 𝑋2 is the graph union 𝐺1 ∪ 𝐺2 together with all the edges joining 𝑉1 and 𝑉2. theorem 3.2. for the star graph 𝐾1,𝑛, 𝐸𝑀(𝐾1,𝑛 + 𝐾1,𝑛) = 5(𝑛 − 1) + √9𝑛 2 − 2𝑛 + 9. proof. from the definition of join, the monophonic distance matrix of 𝐾1,𝑛 + 𝐾1,𝑛 can be written as 61 monophonic distance energy for join of some graphs 𝑀(𝐾1,𝑛 + 𝐾1,𝑛) = [ 𝑀(𝐾1,𝑛) 𝐽𝑛 𝐽𝑛 𝑀(𝐾1,𝑛) ] where 𝑀(𝐾1,𝑛) be a monophonic distance matrix of 𝐾1,𝑛. we have 𝑀(𝐾1,𝑛 + 𝐾1,𝑛) = (𝑑𝑚𝑖𝑗 )2(𝑛+1)×2(𝑛+1). the monophonic distance spectrum 𝑆𝑝𝑒𝑐m (𝐾1,𝑛 + 𝐾1,𝑛) is ( −2 −1 (3𝑛 − 1) − √9𝑛2 − 2𝑛 + 9 2 (𝑛 − 1) 1 1 (𝑛 − 2) (3𝑛 − 1) + √9𝑛2 − 2𝑛 + 9 2 1 1 ). thus the monophonic distance energy of 𝐾1,𝑛 + 𝐾1,𝑛 is 𝐸m(𝐾1,𝑛 + 𝐾1,𝑛) = ∑ |𝜇𝑖 𝑀| 2(𝑛+1) 𝑖=1 ,where 𝜇1 𝑀, 𝜇2 𝑀 , … , 𝜇2(𝑛+1) 𝑀 are the eigen values of monophonic distance matrix 𝑀(𝐾1,𝑛 + 𝐾1,𝑛). for 𝑛 > 1, 𝐸m(𝐾1,𝑛 + 𝐾1,𝑛) = |−2| + |−2| + ⋯ + |−2| + |−1| + | (3𝑛 − 1) − √9𝑛2 − 2𝑛 + 9 2 | + |(𝑛 − 2)| + | (3𝑛 − 1) + √9𝑛2 − 2𝑛 + 9 2 | = 2 (2(𝑛 − 1) + 1 − (3𝑛 − 1) + √9𝑛2 − 2𝑛 + 9 2 + (𝑛 − 2) + (3𝑛 − 1) + √9𝑛2 − 2𝑛 + 9 2 = 5(𝑛 − 1) + √9𝑛2 − 2𝑛 + 9. theorem 3.3. for the complete bipartite graph 𝐾𝑛,𝑛 and complete graph 𝐾𝑛, 𝐸𝑀(𝐾𝑛,𝑛 + 𝐾𝑛) = { (6𝑛 − 7) + √12𝑛2 − 4𝑛 + 1 𝑖𝑓 1 ≤ 𝑛 ≤ 4 10(𝑛 − 1) 𝑖𝑓 𝑛 ≥ 5 . proof. from the definition of join, the monophonic distance matrix of 𝐾𝑛,𝑛 + 𝐾𝑛 can be written as 𝑀(𝐾𝑛,𝑛 + 𝐾𝑛) = [ 𝑀(𝐾𝑛,𝑛) 𝐽2𝑛,𝑛 𝐽𝑛,2𝑛 𝑀(𝐾𝑛) ] where 𝑀(𝐾𝑛,𝑛) be a monophonic distance matrix of 𝐾𝑛,𝑛 and m(𝐾𝑛) be a monophonic distance matrix of 𝐾𝑛. we have 𝑀(𝐾𝑛,𝑛 + 𝐾𝑛) = (𝑑𝑚𝑖𝑗 )3𝑛×3𝑛. the monophonic distance spectrum 𝑆𝑝𝑒𝑐m (𝐾𝑛,𝑛 + 𝐾𝑛) is ( −2 −1 (4𝑛 − 3) − √12𝑛2 − 4𝑛 + 1 2 2(𝑛 − 1) (𝑛 − 1) 1 (4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1 2 𝑛 − 2 1 1 ). thus the monophonic distance energy of 𝐾𝑛,𝑛 + 𝐾𝑛 is 62 sinju manohar v s, binu selin t 𝐸m(𝐾𝑛,𝑛 + 𝐾,𝑛) = ∑ |𝜇𝑖 𝑀|3𝑛𝑖=1 ,where 𝜇1 𝑀, 𝜇2 𝑀 , … , 𝜇3𝑛 𝑀 are the eigen values of monophonic distance matrix 𝑀(𝐾𝑛,𝑛 + 𝐾𝑛). for 1 ≤ 𝑛 ≤ 4, 𝐸m(𝐾𝑛,𝑛 + 𝐾𝑛) = |−2| + |−2| + ⋯ + |−2| + |−1| + |−1| + ⋯ + |−1| + | (4𝑛 − 3) − √12𝑛2 − 4𝑛 + 1 2 | + | (4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1 2 | + (𝑛 − 2) = 2 (2(𝑛 − 1) + (𝑛 − 1) − (4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1 2 + (4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1 2 + (𝑛 − 2) = (6𝑛 − 7) + √12𝑛2 − 4𝑛 + 1 for 𝑛 ≥ 5, 𝐸m(𝐾𝑛,𝑛 + 𝐾𝑛) = |−2| + |−2| + ⋯ + |−2| + |−1| + |−1| + ⋯ + |−1| + | (4𝑛 − 3) − √12𝑛2 − 4𝑛 + 1 2 | + | (4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1 2 | + (𝑛 − 2) = 2 (2(𝑛 − 1) + (𝑛 − 1) + (4𝑛 − 3) − √12𝑛2 − 4𝑛 + 1 2 + (4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1 2 + (n − 2) = 10(𝑛 − 1). theorem 3.4. for the complete bipartite graph 𝐾𝑛,𝑛 , 𝐸𝑀(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) = { 2(5𝑛 − 2) 𝑖𝑓 𝑛 = 1 16(𝑛 − 1) 𝑖𝑓 𝑛 ≥ 2 . proof. from the definition of join, the monophonic distance matrix of 𝐾𝑛,𝑛 + 𝐾𝑛,𝑛 can be written as 𝑀(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) = [ 𝑀(𝐾𝑛,𝑛) 𝐽2𝑛 𝐽2𝑛 𝑀(𝐾𝑛,𝑛) ] where 𝑀(𝐾𝑛,𝑛) be a monophonic distance matrix of 𝐾𝑛,𝑛 we have 𝑀(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) = (𝑑𝑚𝑖𝑗 )4𝑛×4𝑛. the monophonic distance spectrum 𝑆𝑝𝑒𝑐m(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) is ( −2 (𝑛 − 2) (5𝑛 − 2) 4(𝑛 − 1) 3 1 ). thus the monophonic distance energy of 𝐾𝑛,𝑛 + 𝐾𝑛,𝑛 is 𝐸m(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) = ∑ |𝜇𝑖 𝑀|4𝑛𝑖=1 ,where 𝜇1 𝑀, 𝜇2 𝑀, … , 𝜇4𝑛 𝑀 are the eigen values of monophonic distance matrix 𝑀(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛). for 𝑛 = 1, 63 monophonic distance energy for join of some graphs 𝐸m(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) = |−2| + |−2| + ⋯ + |−2| + |𝑛 − 2| + ⋯ + |𝑛 − 2| + |5𝑛 − 2| = 2 (4(𝑛 − 1) + 3(−𝑛 + 2) + 5𝑛 − 2 = 8𝑛 − 8 − 3𝑛 + 6 + 5𝑛 − 2 = 10𝑛 − 4 = 2(5𝑛 − 2) for 𝑛 ≥ 2, 𝐸m(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) = |−2| + |−2| + ⋯ + |−2| + |𝑛 − 2| + ⋯ + |𝑛 − 2| + |5𝑛 − 2| = 2 (4(𝑛 − 1) + 3(𝑛 − 2) + 5𝑛 − 2 = 8𝑛 − 8 + 3𝑛 − 6 + 5𝑛 − 2 = 16𝑛 − 16 = 16(𝑛 − 1). theorem 3.5. for the star graph 𝐾1,𝑛 and complete graph 𝐾𝑛, 𝐸𝑀(𝐾𝑛 + 𝐾1,𝑛) = { (3𝑛 − 2) + √5𝑛2 + 4 𝑖𝑓 𝑛 ≤ 2 2(3𝑛 − 2) 𝑖𝑓 𝑛 ≥ 3 . proof. . from the definition of join, the monophonic distance matrix of 𝐾𝑛 + 𝐾1,𝑛 can be written as 𝑀(𝐾𝑛 + 𝐾1,𝑛) = [ 𝑀(𝐾𝑛) 𝐽𝑛,𝑛+1 𝐽𝑛+1,𝑛 𝑀(𝐾1,𝑛) ] where 𝑀(𝐾1,𝑛) be a monophonic distance matrix of 𝐾1,𝑛 and m(𝐾𝑛) be a monophonic distance matrix of 𝐾𝑛. we have 𝑀(𝐾𝑛 + 𝐾1,𝑛) = (𝑑𝑚𝑖𝑗 )(2𝑛+1)×(2𝑛+1). the monophonic distance spectrum 𝑆𝑝𝑒𝑐m (𝐾𝑛,𝑛 + 𝐾𝑛) is ( −2 −1 (3𝑛 − 2) − √5𝑛2 + 4 2 (𝑛 − 1) 𝑛 1 (3𝑛 − 2) + √5𝑛2 + 4 2 1 ). thus the monophonic distance energy of 𝐾𝑛 + 𝐾1,𝑛 is 𝐸m(𝐾𝑛 + 𝐾1,𝑛) = ∑ |𝜇𝑖 𝑀|2𝑛+1𝑖=1 ,where 𝜇1 𝑀, 𝜇2 𝑀 , … , 𝜇2𝑛+1 𝑀 are the eigen values of monophonic distance matrix 𝑀(𝐾𝑛 + 𝐾1,𝑛). for 𝑛 ≤ 2, 𝐸m(𝐾𝑛 + 𝐾1,𝑛) = |−2| + |−2| + ⋯ + |−2| + |−1| + |−1| + ⋯ + |−1| + | (3𝑛 − 2) − √5𝑛2 + 4 2 | + | (3𝑛 − 2) + √5𝑛2 + 4 2 | = 2 (𝑛 − 1) + 𝑛 − (3𝑛 − 2) + √5𝑛2 + 4 2 + (3𝑛 − 2) + √5𝑛2 + 4 2 = 2𝑛 − 2 + 𝑛 + √5𝑛2 + 4 = (3𝑛 − 2) + √5𝑛2 + 4. for 𝑛 ≥ 5, 𝐸m(𝐾𝑛 + 𝐾1,𝑛) = |−2| + |−2| + ⋯ + |−2| + |−1| + |−1| + ⋯ + |−1| 64 sinju manohar v s, binu selin t + | (3𝑛 − 2) − √5𝑛2 + 4 2 | + | (3𝑛 − 2) + √5𝑛2 + 4 2 | = 2 (𝑛 − 1) + 𝑛 + (3𝑛 − 2) − √5𝑛2 + 4 2 + (3𝑛 − 2) + √5𝑛2 + 4 2 = 2𝑛 − 2 + 𝑛 + 3𝑛 − 2 = 6𝑛 − 4 = 2(3𝑛 − 2). 4. conclusions in this paper we discussed the concept of monophonic distance energy and obtained some results on monophonic distance energy for certain join graphs. in future we planned to extend our research for various graph operations. references [1] davis. p.j, circulant matrices, wiley, new york, 1979. [2] gutman. i, the energy of a graph, ber. math-statist. sekt. forschungsz. graz, 103, 122, 1978. [3] harary. f, graph theory, addison-wesley, boston, 1969. [4] indulal. g, gutman. i and vijayakumar. a, on distance energy of a graph, match commun. math. comput. chem (2008), 461-472. [5] indulal. g and vijayakumar. a, on a pair of equienergetic graphs, match commun. math. comput. chem (2008), 83-90. [6] indulal. g and gutman. i, on the distance spectra of some graphs, mathematical communications 13(2008), 123-131. [7] samir k. vaidya and kalpesh m. popat, some new results on energy of graphs, match commun. math. comput. chem (2017), 589-594. [8] santhakumaran. a, p, titus. p and ganesamoorthy. k, on the monophonic number of a graph, j. appl. math. and informatics vol. 32 (2014), no.1-2, pp.255-266. 65 ratio mathematica volume 44, 2022 conjunction weighted average method with fuzzy expert system for weather event forecasting – a monthly outlook u. ramya devi1 k. uma2 abstract fuzzy logic as a limiting case of approximate reasoning is viewed in exact reasoning, consider everything in a matter of degree. a collection of elastic or equivalently interpreted to knowledge, a collection of variables in fuzzy constraint. inference is process as a propagation of elastic constraints. every logical system is fuzzified in fuzzy logic. fuzzy logic is fascinating area of research, it trading off between significance and precision. it is convenient way to map space of input to a space of output. fuzzy logic as so far as the laws of mathematics refers to reality, they are not certain and so far, as they are certain as complexity rises, precise statements lose meaning and meaningful statements lose precision. most meteorological infrastructure is surprisingly versatile. for example, the same radar system that can detect oncoming storms will also be useful for gathering general rainfall data for the farming sector. being able to predict and forecast the weather also allows for data to be gathered to build up a more detailed picture of a nation’s climate, and trends within it. keywords: fuzzy logic, rainfall and weather forecasting mathematical classification: 94d053 1 pg and research department of mathematics, poompuhar college, (autonomous), melaiyur nagapattinam affiliated by bharathidasan university-trained, india, e-mail: ramyadeviu@yahoo.com email: umak197630@gmail.com 2 pg and research department of mathematics, poompuhar college, (autonomous),melaiyur nagapattinam affiliated by bharathidasan university tamil nadu, india, e-mail: ramyadeviu@yahoo.com e-mail:umak197630@gmail.com 3 received on june 6th, 2022. accepted on sep 9st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.928. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 379 u. ramya devi & k. uma 1. introduction fuzzy logic models have been developed as a solution forecasting method, because the weather parameters can be easily classified unlike the other techniques mentioned above [4, 6]. also, it does not require a computational mapping of inputs to outputs or no need for precise inputs. fuzzy logic is simply a means of representing human reasoning. the main components of fuzzy logic are fuzzy set, membership function and fuzzy if-then rule base. if – then rule base is used to convert the fuzzy input into the fuzzy output [8]. in this research, the temperature, pressure, wind speed, humidity and the precipitation will serve as the input parameters for the month forecasting. the aim of this work is to develop fuzzy logic methodology for weather forecasting with the following objectives: to formulate fuzzy logic membership function that will facilitate the monthly weather forecasting. also, proposed monthly weather forecasting using fuzzy logic based on weather historical data (temperature, humidity, wind speed, pressure and precipitation) for the year 2021. weather data used is from state tamil nadu, india. the work focused on month wise for the year 2021. rainfall forecasts have significant value for resources planning and management e.g., reservoir operations, agricultural practices and flood emergency responses. to mitigate this, effective planning and management of water resources is necessary. in the short term, this requires a good idea of the upcoming season. in the long term, it needs realistic projections of scenarios of future variability and change [3]. 2. monthly weather forecasting monthly forecasting has the potential to play a significant role in enhancing end users’ resilience to the impacts of climate change and variability. smallholder farmers, for example, are vital to the economies and food security within the majority of the developing world, yet they are confronted with increasingly scarce resources, changing weather patterns, and extreme events that pose significant threats to the stability of both production and income. monthly forecasts can provide this key stakeholder group with information to support their decision making regarding which crops they should plant, when to plant and harvest, and when to apply fertilizer and other inputs to maximize their yields and mitigate their losses. 2.1 data and area of study the study area is tamil nadu, which is one of the states in india where the climate is influenced mainly by the rain-bearing southwest monsoon winds from the ocean and the dry northwest winds from the sahara desert. when using monthly forecasting as part of a forecast-based action system, it is important to comprehend how the prediction relates to the harm you are trying to mitigate. for instance, even a "above normal" rainfall season may have less flooding than a "normal" rainfall season depending on the type of rainfall (i.e., whether it falls gradually over several days or weeks versus all at once during a period of several hours). however, if there is a 380 conjunction weighted average method with fuzzy expert system for weather event forecasting – a monthly outlook forecast for below-average rainfall and a humanitarian organization wants to take proactive measures to address a drought, they may do so by using the monthly outlook. table 1: seasons of tamil nadu 3. architecture of proposed model 4. methodology a fuzzy set is a function that transfers the universe object y on to the interval [0, 1]. a set b's fuzzy membership function represented mathematically, where the functional mapping is provided. 𝜇�̃� (y)𝜖 [ 0, 1] similarly, the symbol 𝜇�̃� (y) in the degree of membership element y in the fuzzy set �̃�. a membership function that maps a component of a domain, space, or universe of discourse to the unit interval [0, 1] defines a fuzzy set. a fuzzy set �̃� in a universe of discourse y is defined as following set of pairs �̃� = { 𝑦,𝜇�̃� (y); y ∈ 𝑌 } 381 u. ramya devi & k. uma here, 𝜇�̃�: y→ [0,1] is a mapping called degree of membership function of fuzzy set �̃� and 𝜇�̃� (y) is called the membership value of y ∈ 𝑌 in the fuzzy set �̃�. these membership levels are frequently expressed as real numbers between [0, 1]. 4.1 algorithm step 1: construct a set∑ 𝑊�̃� 12 𝑖=1 = ∑ ∑ �̃�𝑖 5 𝑗=1 12 𝑖=1 �̃�𝑗 step 2: create a triangular fuzzy membership function with respect to the decision makers parameters of their own choice. step 3: construct the membership function for pressure, temperature, humidity, wind speed, rainfall. membership function of wind speed membership function of temperature step 4: determine the product fuzzy conjunction 𝜇(�̃�) = 𝜇(�̃�1�̃�1)⋀𝜇(�̃�1�̃�2)⋀𝜇(�̃�1�̃�3)⋀𝜇(�̃�1�̃�4)⋀ 𝜇(�̃�1�̃�5) step 5: 382 conjunction weighted average method with fuzzy expert system for weather event forecasting – a monthly outlook calculate the product fuzzy weighted average defuzzification method �̃� = ∑ ∑ �̃�𝑖∙𝜇 ( 𝑊�̃�) 𝜇 ( 𝑊�̃�) 5 𝑗=1 12 𝑖=1 5. case study consider the set �̃� = { �̃�1,�̃�2, �̃�3 ………….�̃�12 } as a universal sets where �̃�1,�̃�2, �̃�3 ………….�̃�12 represent the month from january to december for the year 2021 and let the set �̃� = { �̃�1, �̃�2, �̃�3, �̃�4, �̃�5} where �̃�1 − pressure �̃�2 − temperature �̃�3 − humidity �̃�4 − wind speed �̃�5 − rainfall the set �̃�represent the parameter environmental factors exposure to monthly weather forecasting outlook. it gives the relationship �̃� called the set month and parameter data. here, following steps made for weather forecasting in january 2021 in tamil nadu. step 1& 2: 𝑊1̃ = { (�̃�1�̃�1 ),(�̃�1�̃�2),( �̃�1�̃�3),(�̃�1�̃�4),(�̃�1�̃�5) } = {(97.86), (22.85), (86.88), (2.27), (100.2)} step 3: when �̃�1�̃�1 = 97.86 𝜇𝐿𝑜𝑤(�̃�1�̃�1 ) = { 0 �̃�1�̃�1 ≥ 98.10 1 �̃�1�̃�1 = 96.01 (98.10 − �̃�1�̃�1 ) (98.10 − 96.01) 96.01 < �̃�1�̃�1 < 98.10 = 0.11 𝜇𝑀𝑜𝑑𝑒𝑟𝑎𝑡𝑒(�̃�1�̃�1 ) = { 0 �̃�1�̃�1 ≤ 98.11 𝑜𝑟�̃�1�̃�1 ≥ 101.5 (�̃�1�̃�1 − 98.11) (99.13 − 101.5) 98.11 < �̃�1�̃�1 < 99.13 1 �̃�1�̃�1 = 101.5 (101.5 − �̃�1�̃�1 ) (101.5 − 99. .13) 101.5 < �̃�1�̃�1 < 120.0 𝜇𝐻𝑖𝑔ℎ(�̃�1�̃�1 ) = { 0 �̃�1�̃�1 ≤ 101.6 1 �̃�1�̃�1 = 120.0 (�̃�1�̃�1 − 101.6) (120.0 − 101.6) 101.6 < �̃�1�̃�1 < 120.0 step 4: 383 u. ramya devi & k. uma = 𝜇𝐿𝑜𝑤(�̃�1�̃�1 )⋀𝜇𝐿𝑜𝑤(�̃�1�̃�2)⋀𝜇𝐻𝑖𝑔ℎ(�̃�1�̃�3)⋀𝜇𝐿𝑜𝑤(�̃�1�̃�4)⋀𝜇𝐿𝑜𝑤(�̃�1�̃�5) = 0.11 ⋀0.086 ⋀0.43 ⋀0.9294 ⋀0.0457 step 5: = 0.11 𝑋 97.86+0.086 𝑋 22.85+0.43 𝑋 86.88+0.9294 𝑋 2.27+0.0457 𝑋 100.2 0.11+0.086+0.43+0.9294+0.0451 = 56.7769 1.6011 = 35.46 % (i.e) january month got 35.46 % that it is to be chance of possibility of to get likely weather event. 6. result and justification in this effort, pressure, temperature, humidity, wind speed and rainfall are taken as a important parameter of weather forecasting for monthly outlook based on fuzzy expert system with product conjunction. the result appear for the month of february is 15.09% (i.e.,) very less rainfall and high temperature so it is considered to be unlikely weather event, meanwhile july 92.24% and november 97.67% (i.e.) more rainfall and less temperature so it is considered to be a most likely weather event month. for, remaining month weather event are given in table 2. this method is used to determination of monthly outlook weather forecasting with high accuracy. table 2: weather event prediction – a monthly outlook. terminologies of likely(l), unlikely (ul), most likely (ml) are justified with standard operation procedure – weather forecasting and warning services given by indian metrological department also discussed with expert metrologiest. 7.conclusion the effect of the shape of membership functions upon the solution is very important. broader input membership functions, those with an extended domain with membership of 100%, have a larger weighting during the rule evaluation. this will be reflected in the final solution. in this work, fuzzy methodology for one-year 2021of tamil nadu weather event forecasting is discussed. in this model, that is month wise weather event prediction model, we applied the notion of fuzzy triangular membership function in fuzzy expert system. the benefit of this 384 conjunction weighted average method with fuzzy expert system for weather event forecasting – a monthly outlook model is that if the weather parameter and triangular fuzzy number are known, it is possible to find out monthly outlook of weather event as forecasting. references [1] zadeh, l.a., 1965. fuzzy sets information and control, pp: 338 – 353. [2] kosko, b., 1992. neural networks and fuzzy systems. prentice hall. englewood cliffs, n.j. [3] abraham, a., philip n. and joseph b. (2001); “soft computing models for long term rainfall forecasting”: in: 15th european simulation multi conference (esm, august/september 2001), modeling and simulation 2000, kerckhoffs, e.j.h. and m. snorek (eds.). czech republic, prague, pp: 1044 – 1048. [4] hari s. and saravanan, r. “short term electric load prediction using fuzzy bp”, journal of computing and information technology, vol. 3, 2007, pp.1 – 15 [5] bardossy, a., duckstein l. and bogardi i. (1995); “fuzzy rule-based classification of atmospheric circulation patterns”: int. j. climatol., 15: 1087 – 1097. [6] hong t, “shortterm electric load forecasting”, phd thesis, graduate faculty of north carolina state university, october, 2010, pp. 1 – 175. [7] edvin and yudha (2008); “application of multivariate anfis for daily rainfall prediction: influences of training data size”: makara, sains, volume 12, no. 1, april 2008: 7 – 14 [8] swaroop, r. and. hussein, a. a. “load forecasting for power system planning using fuzzy-neural network”, proceeding of the world congress on engineering and computer science, san fransico, usa, vol. 1, october 24-26, 2012, pp. 1-5. [9] wong, k.w., wong p.m., gedeon t.d. and fung c.c. (2003); “rainfall prediction model using soft computing technique”: soft comput. fusion foundat. methodol. appli.7:434 – 438. [10] özelkan, e.c., ni f. and duckstein l. (1996); “relationship between monthly atmospheric circulation patterns and precipitation: fuzzy logic and regression approaches”: water resour. res., 32: 2097 – 2103. 385 ratio mathematica volume 41, 2021, pp. 53-63 an image compression method based on ramanujan sums and measures of central dispersion sadanandan sajikumar * john dasan † vasudevan hema ‡ abstract this paper introduces a simple lossy image compression method based on ramanujan sums cq(n) and the statistical measures of numerical data such as mean and standard deviation. the ramanujan sum cq(n) has been used in digital signal processing for a variety of applications nowadays. some of them include the recently developed image kernels for edge detection, extraction of periodicity from signals, etc. the presented compression algorithm is an extension of the edge detection algorithm using an integer image kernel based on ramanujan sums. we propose a block-based compression algorithm that detects edges in the images using this image kernel and then compresses the image by storing kernel operation values, the mean and standard deviation for each block instead of pixel values. the proposed method has the advantage of low computational complexity and shows its ability in fast reconstruction and high compression that can be achieved for different block sizes. keywords: ramanujan sum cq(n); lossy image compression; mean; standard deviation. 2020 ams subject classifications: 11z05; 97m10. 1 *first author’s affiliation (college of engineering trivandrum, thiruvananthapuram, kerala, india); sajikumar.s@cet.ac.in. †second author’s affiliation (college of engineering trivandrum, thiruvananthapuram, kerala, india); dasanj@cet.ac.in. ‡third author’s affiliation (college of engineering trivandrum, thiruvananthapuram, kerala, india); hema@cet.ac.in. 1received on october 28, 2021. accepted on december 20, 2021. published on december 31, 2021. doi: doi:10.23755/rm.v41i0.683. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. s. sajikumar, j. dasan, v. hema 1 introduction in the rapid popularization of the internet and social media, the role of a digital image is indispensable. as we have to transmit a lot of information over communication networks, bandwidth reduction is necessary. to achieve this, audio and video signals need to be compressed. image or video signals in compressed form are convenient for editing, storing, utilizing, and transmitting. image compression techniques can be classified into two categories. if the information retained after decompression is 100%, it is called lossless compression otherwise lossy. if we take a pixel in an image at random there is a good chance that its neighbours will have the same intensity or very similar intensity. typically hence, image compression is based on the fact that the neighbouring pixels are highly correlated ([salomon, 2007], [sayood, 2012]). most image compression methods exploit this feature to obtain efficient compression. lossless compression can be achieved with the techniques like run length encoding (rle), huffman coding, arithmetic coding etc.([gallager, 1978], [jain, 1989], [taubman and marcellin, 2012], [witten et al., 1987]). lossy techniques include transform coding methods such as discrete cosine transform (dct), jpeg, jpeg2000 etc.([pennebaker and mitchell, 1992], [gonzalez and woods, 2008], [goyal, 2001]). polynomial-based compression is another lossy compression method ([sadeh, 1996],[eden et al., 1986]). sajikumar s et al., [sajikumar and anilkumar, 2017] introduced a compression scheme using chebyshev polynomials. the proposed compression algorithm differs from the standard compression algorithms in its low computational complexity and fast reconstruction. lossy compression techniques are tested for their performance based on three commonly used measures, the root mean square error (rmse), peak signal to noise ratio (psnr) and the compression ratio (cr). the rmse between original image f(x,y) and reconstructed image f̂(x,y) of size m ×n is defined by [joshi, 2018]: rmse = √√√√ 1 mn m−1∑ x=0 n−1∑ y=0 [ f(x,y)− f̂(x,y) ]2 (1) for an 8bit gray level image, psnr = 10 log10 ( 2552 mse ) (db) (2) cr = compressed image size uncompressed image size % (3) an image kernel is a matrix used to obtain effects like blurring, sharpening, outlining, etc. computer vision applications of image kernel mainly include feature extraction and edge detection. a geometric perspective of kernel methods 54 an image compression method based on ramanujan sums and measures of central dispersion can be seen in [lampert, 2009]. zhang et al., studies various non-local kernel regression for image and video restoration tasks [zhang et al., 2010]. odone et al., describes methods for building kernels from binary strings for image matching [odone et al., 2005]. in 2016, krishnaprasad p et al., [krishnaprasad and ramanujan, 2016] presented an image kernel based on ramanujan sums to detect edges. for each 3 × 3 block of pixels in the image, they multiplied each pixel by the corresponding entry of the 3×3 kernel matrix constructed from c3(n) and then takes the sum. this sum is considered as a new pixel in the image. ramanujan sums cq(n) is a family of trigonometric sums defined by srinivasa ramanujan in 1918 [ramanujan, 1918]. in the last fifteen years, ramanujan sums have aroused some interest in signal processing. cohen initially introduced ramanujan sums to the signal processing community ([cohen, 1955], [cohen, 1958]). in 1950, he observed that the dft coefficients of even symmetric signals can be computed by integer-valued weighting coefficients. later it was proved that these integer-valued coefficients are nothing but the well-known ramanujan sums. the rest of the paper is organized as follows. a brief description of the ramanujan sum is given in section 2. image kernel construction and the proposed compression algorithm are given in sections 3 and 4 respectively. results and discussion are included in section 5 and section 6 concludes the paper. 2 review of ramanujan sums the ramanujan sum cq(n) has been used by mathematicians to derive many important infinite series expansions for arithmetic functions in number theory [apostol, 1976]. interestingly, this sum has many properties which are attractive from the point of view of digital signal processing. srinivasa ramanujan defined the qth ramanujan sum by cq(n) = q∑ k=1 (k,q)=1 wknq = q∑ k=1 (k,q)=1 w−knq (4) where wq = e−i2π/q, i = √ −1 and (k,q) denotes the gcd of k and q. here the sum runs over those k satisfying (k,q) = 1 means that we are considering all the integers which are coprime to q in the summation. for example, if q = 8 then k ∈{1,3,5,7} so that c8(n) = e i2nπ/8 + ei6nπ/8 + ei10nπ/8 + ei14nπ/8 55 s. sajikumar, j. dasan, v. hema in number theory, the number of integers less than or equal to q and coprime to q is called the euler’s totient function φ(q) apostol [1976]. since 1,3,5,7 are coprime to 8, φ(8) = 4. so the sum given in equation (4) has precisely φ(q) terms and it is clear that cq(0) = φ(q). also cq(n + q) = q∑ k=1 (k,q)=1 ei2nπk/q.ei2πk = q∑ k=1 (k,q)=1 ei2nπk/q = cq(n) that is cq(n) is periodic with period q. if (k,q) = 1, we have (q −k,q) = 1. therefore, (wkq ) ∗ = w−kq = w −(q−k) q = w k q where ∗ is the complex conjugate. this implies that the summation (4) is real valued and it can also be written as : cq(n + q) = q∑ k=1 (k,q)=1 cos 2nπk q (5) from (5), cq(n) = cq(−n) shows that cq(n) is symmetric. thus cq(n) is a real, symmetric, and periodic sequence in n. for 0 ≤ n ≤ q −1, first few ramanujan sequences are c1(n) = 1 c2(n) = 1,−1 c3(n) = 2,−1,−1 c4(n) = 2,0,−2,0 c5(n) = 4,−1,−1,−1,−1 note that cq(n) is integer-valued and further properties can be seen in [vaidyanathan, 2014]. 3 image kernel krishnaprasad et al., [krishnaprasad and ramanujan, 2016], has introduced a kernel matrix mq of size q×q constructed from cq(n) by considering the circular 56 an image compression method based on ramanujan sums and measures of central dispersion shifts of the q elements cq(0),cq(1), · · · ,cq(q−1) in each row. the first row of the kernel matrix contains the q elements in the order cq(0),cq(1), · · · ,cq(q −1) where cq(r) = q∑ k=1 (k,q)=1 ei2πkr/q for 0 ≤ r ≤ q −1. the second-row elements are cq(q−1),cq(0),cq(1), · · · ,cq(q−2) and so on. thus mq =   cq(0) cq(1) cq(2) .... cq(q −1) cq(q −1) cq(0) cq(1) .... cq(q −2) ... ... ... ... ... cq(1) cq(2) cq(3) .... cq(0)   4 proposed method partition the input image into non-overlapping blocks of size q × q. test images of size 256 × 256 with 8bit gray levels between 0 and 255 are considered. multiply each pixel in the q × q block with the corresponding elements of the kernel mq and take their sum. this sum is stored for edge detection. represent the entire block of pixel values with this sum obtained. after the edge detection process, we move on to the compression part. in this step, we are computing the mean and standard deviation of each block of pixels to obtain the texture at decompression. two different ways of compressing an image with the statistical measures of pixel values are proposed. a. method 1 for each q×q block, a block value is computed by adding the kernel multiplication sum which is obtained at the edge detection stage, the mean and standard deviation of each block of pixels. represent the entire block with this sum at the reconstruction stage. by varying the block size we can compress the image at different compression levels. b. method 2 instead of taking the sum of three quantities to represent each block, consider the kernel sum and the mean value only. thus we need to store only two values in place of q2 pixels and hence high compression is achieved. 57 s. sajikumar, j. dasan, v. hema experimental results with the test images are given in tables 1-3 and figures 1-2. in both methods, no quantization or postprocessing is done at the reconstruction step. algorithm step 1 load an input gray image. step 2 partition the image matrix into non-overlapping blocks of size q × q. step 3 for each block compute the elementwise product sum with the kernel matrix and the pixel values of the q×q block. also find the mean and standard deviation of the q2 pixels. store these values for the reconstruction of each block. step 4 replace the q2 gray values by the elementwise product sum computed in step 3 to detect edges. step 5 replace the q2 gray values by the sum of the three quantities stored in step 3 to compress the image. or replace the q2 gray values by the sum of the elementwise product sum and the mean to achieve high compression. 5 results and discussion edge detection results for the test images of different block sizes 2×2, 3×3, 4×4 are given in figure 1. edge detection with 2×2 blocks shows better results as compared to others. in the first method , we replace q2 pixel values with three quantities. hence in 2 × 2, 3 × 3, and 4 × 4 blocks compression ratios are 75%, 33.33%, and 18.75% respectively. but in the second method, we need to store only two quantities instead of q2 values in each block. thus the compression ratios are 50% for 2 × 2 blocks, 22.22% for 3×3 blocks, and 12.5% for 4×4 blocks. from figure 2, and tables 1-3 we can conclude that method 2 shows better cr with reasonable reconstructed image quality measures psnr and rmse. the test image rice achieves an appreciable psnr 30.3117(db) with rmse 7.7796 in the case of 2×2 blocks. as the quality measures psnr decreases and rmse increases with an increase of the block size, the algorithm works better with smaller blocks. 58 an image compression method based on ramanujan sums and measures of central dispersion experrimental results shows that psnr values are greater than 22(db) and rmse is less than 20 for different test images when we apply the second method of compression. in practical applications this is an acceptable range at the cr 50%. also, as the kernel opertion doesn’t involve usual convolution product at the edge detection stage the number of additions and multiplications required is reduced and hence saves a lot of computation time. test image bolck size block value psnr(db) rmse lena 2×2 method 1 22.8412 18.3857 2×2 method 2 24.8995 14.5066 cameraman 2×2 method 1 20.7047 23.5127 2×2 method 2 22.6443 18.8073 aerial 2×2 method 1 19.9407 25.6747 2×2 method 2 22.3203 19.5221 rice 2×2 method 1 27.7290 10.4734 2×2 method 2 30.3117 7.7796 table 1: compression quality measures with 2×2 blocks using methods 1& 2 test image bolck size block value psnr(db) rmse lena 3×3 method 1 16.9428 36.2579 3×3 method 2 18.0869 31.7830 cameraman 3×3 method 1 14.9823 45.4389 3×3 method 2 15.8706 41.0294 aerial 3×3 method 1 14.3263 49.0033 3×3 method 2 15.1876 44.3774 rice 3×3 method 1 21.5605 21.3067 3×3 method 2 23.6949 16.7801 table 2: compression quality measures with 3×3 blocks using methods 1& 2 59 s. sajikumar, j. dasan, v. hema test image bolck size block value psnr(db) rmse lena 4×4 method 1 12.5169 60.3522 4×4 method 2 12.8137 58.3253 cameraman 4×4 method 1 10.5412 75.7666 4×4 method 2 10.9338 72.4182 aerial 4×4 method 1 8.7710 92.8943 4×4 method 2 9.0363 90.0997 rice 4×4 method 1 16.8793 36.5237 4×4 method 2 17.5469 33.8217 table 3: compression quality measures with 4×4 blocks using methods 1& 2 figure 1: the first column: orinal images; the second column: edges by m2; the third column: edges by m3; the fourth column: edges by m4. 60 an image compression method based on ramanujan sums and measures of central dispersion figure 2: the first column: orinal images; the second column: reconstructed image using method 2 at the cr 50%; the third column: reconstructed image using method 2 at the cr 22.22%; the fourth column: reconstructed image using method 2 at the cr 12.5%. 6 conclusions in this paper, we presented an edge detection and compression algorithm based on ramanujan sums and measures of central tendency and dispersion such as mean and standard deviation. the edge detection algorithm using kernels constructed from ramanujan sums has been extended to a compression algorithm. here the compression is achieved by replacing each block of pixels with a single 61 s. sajikumar, j. dasan, v. hema value obtained by adding edges with texture. the advantage of this method is its low computational complexity and fast reconstruction. references tom m apostol. analytic number theory. 1976. eckford cohen. a class of arithmetical functions. proceedings of the national academy of sciences of the united states of america, 41(11):939, 1955. eckford cohen. representations of even functions (mod r), arithmetical identities. duke mathematical journal, 25(3):401–421, 1958. murray eden, michael unser, and riccardo leonardi. polynomial representation of pictures. signal processing, 10(4):385–393, 1986. robert gallager. variations on a theme by huffman. ieee transactions on 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classification: 05c72, 05c99.‡ 1. introduction graph theory is used to model various types of relations that exist in different fields of physics, chemistry, medicine, electrical network, computer science such as networking, image processing etc. whenever the information provided is imprecise, uncertainty exists. molodtsov [1] initiated the concept of soft sets to deal with uncertainty. a. rosenfeld [2] developed the theory of fuzzy graphs in 1975 based on fuzzy sets which were initiated by zadeh [3] in 1965. maji et al. [5,7] presented the definition of fuzzy soft sets and applied it in decision making problems. later many researchers progressively worked on these concepts and developed it. operations on fuzzy graphs were demonstrated by j. n. mordeson and c. s. peng [6]. akram and saira nawaz [8,11] introduced fuzzy soft graphs, studied some of its properties and applied these concepts in social network and road network. shashikala s and anil p n [12,15] discussed connectivity in fuzzy soft graphs and studied hamiltonian fuzzy soft cycles. a. pouhassani and h. doostie [13] studied degree, total degree, regularity and total regularity of fuzzy soft graph and its properties. * shashikala s (global academy of technology, bangalore, india); shashikala.s@gat.ac.in † anil p n (global academy of technology, bangalore, india); anilpn@gat.ac.in ‡received on august 19, 2021. accepted on december 1, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.644. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. shashikala s and anil p n 228 regular fuzzy soft graphs and its related properties are studied by b akhilandeswari [16]. shovan dogra [10] studied some types of fuzzy graph products such as modular product, homomorphic product and determined its degree of vertices. union and intersection of fuzzy soft graphs and some of its properties are studied by mohinta and samanta [9]. fuzzy soft theory provides a clear picture of the problems that allows parameterization finds applications in many areas. recently, it is used to represent the oligopolistic competition among the wireless internet connection providers in malaysia by akram and saba nawaz [17]. in this paper, some products of fuzzy soft graphs namely alpha, beta and gamma products are defined and degree of a vertex in these products are determined and its regular properties are studied. 2. preliminaries definition 2.1: [11] a fuzzy soft graph g ~ over a graph ),(: * evg is a triple ), ~ , ~ ( akf where: a) a is a nonempty set of parameters b) ), ~ ( af is a fuzzy soft set over v c) ), ~ ( ak is a fuzzy soft set over e d) ( ))(~),(~ ii ekef is a fuzzy graph on aeg i  * i.e. )})(( ~ ),)(( ~ min{))(( ~ yefxefxyek iii  for all aei  and ., vyx  definition 2.2: [14] the underlying crisp graph of a fuzzy soft graph g ~ is denoted by ( )*** , kfg = where  aesomeforxefvxf ii = 0))(( ~ : * ,  aesomeforyxekvvyxk ii = 0),)(( ~ :),( * . definition 2.3: [8] let g ~ be a fuzzy soft graph on *g . the degree of a vertex x is defined as     = ae xyvy ig i xyekx , ~ ))(( ~ )(deg . definition 2.4: [8] ] g ~ is said to be a regular fuzzy soft graph if ( ))(~),(~ ii ekef is regular fuzzy graph for all ae i  . if ( ))(~),(~ ii ekef is a regular fuzzy graph of degree k for all ae i  then g ~ is a k-regular fuzzy soft graph. definition 2.5: [4] the degree )(* xd g of a vertex v in *g is the number of edges incident with x . some studies on products of fuzzy soft graphs 229 in this paper, we assume that ),(: * evg of any fuzzy soft graph g ~ is finite and simple. notation: let ), ~ , ~ (: ~ 1111 akfg and ), ~ , ~ (: ~ 2222 akfg be two fuzzy soft graphs. the relation )( ~ )( ~ 21 ji ekef  for all 1 ae i  , 2ae j  means that ))(( ~ ))(( ~ 21 eekxef ji  21 , eevx  where 1 ~ f is a fuzzy soft subset of 1 v and 2 ~ k is a fuzzy soft subset of 2 e . 3. alpha product )( product− , beta product )( product− and gamma product )( product− of fuzzy soft graph definition 3.1: let ), ~ , ~ (: ~ 1111 akfg and ), ~ , ~ (: ~ 2222 akfg be two fuzzy soft graphs on * 1 g and * 2 g respectively. the product− ), ~~ , ~~ (: ~~ 21212121 aakkffgg   is defined as follows: )(:) ~~ ( 212121 vvfsaaff →  by 21212121 ,,)()( ~ )()( ~ )(),() ~~ ( vvyxaeaeyefxefyxeeff lkjiljkilkji =  and )(:) ~~ ( 212121 eefsaakk →  by               = = = 2 1211 2 1122 112 221 21 ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()(),() ~~ ( eyy exxifyyekxefxef eyy exxifxxekyefyef exxyyifxxekyef eyyxxifyyekxef yxyxeekk nl mknljmiki nl mkmkinjlj mknlmkilj nlmknljki nmlkji  definition 3.2: let ), ~ , ~ (: ~ 1111 akfg and ), ~ , ~ (: ~ 2222 akfg be two fuzzy soft graphs on * 1 g and * 2 g respectively. the product− ), ~~ , ~~ (: ~~ 21212121 aakkffgg   is defined as follows: )(:) ~~ ( 212121 vvfsaaff →  by 21212121 ,,)()( ~ )()( ~ )(),() ~~ ( vvyxaeaeyefxefyxeeff lkjiljkilkji =  and )(:) ~~ ( 212121 eefsaakk →  by shashikala s and anil p n 230                = 2121 1 122 2 211 21 ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ )()(),() ~~ ( eyyexxifyyekxxek exx yyifxxekyefyef eyy xxifyyekxefxef yxyxeekk nlmknljmki mk nlmkinjlj nl mknljmiki nmlkji  definition 3.3: let ), ~ , ~ (: ~ 1111 akfg and ), ~ , ~ (: ~ 2222 akfg be two fuzzy soft graphs on * 1 g and * 2 g respectively. the product− ), ~~ , ~~ (: ~~ 21212121 aakkffgg   is defined as follows: )(:) ~~ ( 212121 vvfsaaff →  by 21212121 ,,)()( ~ )()( ~ )(),() ~~ ( vvyxaeaeyefxefyxeeff lkjiljkilkji =  and )(:) ~~ ( 212121 eefsaakk →  by                 = = = 2121 1 122 2 211 112 221 21 ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()(),() ~~ ( eyyexxifyyekxxek exx yyifxxekyefyef eyy xxifyyekxefxef exxyyifxxekyef eyyxxifyyekxef yxyxeekk nlmknljmki mk nlmkinjlj nl mknljmiki mknlmkilj nlmknljki nmlkji  example 3.4: consider two fuzzy soft graphs ), ~ , ~ (: ~ 1111 akfg and ), ~ , ~ (: ~ 2222 akfg on ),(: 11 * 1 evg and ),(: 22 * 2 evg respectively such that },{ 211 xxv = , }{ 211 xxe = , },,{ 3212 yyyv = , },{ 32212 yyyye = , }{ 1 i ea = where i=1,2 and }{2 jea = where j=3,4. let ), ~ ( 11 af , ), ~ ( 22 af , ), ~ ( 11 ak and ), ~ ( 22 ak be represented by the following table 1. 1 ~ f 1 x 2 x 2 ~ f 1 y 2y 3 y 1 e 0.4 0.6 3e 0.3 0.5 0.8 2 e 0.7 0.5 4 e 0.5 0.6 0.7 1 ~ k 21 xx 2 ~ k 21 yy 32 yy 1 e 0.1 3e 0.3 0.4 2 e 0.3 4 e 0.4 0.5 table 1 : tabular representation of two fuzzy soft graphs some studies on products of fuzzy soft graphs 231 1 ~ .1. gfig 2 ~ .2. gfig 21 ~~ .3. ggfig   21 ~~ .4. ggfig   shashikala s and anil p n 232 21 ~~ .5. ggfig   4. degree of a vertex in alpha product )( product− of two fuzzy soft graphs and its regular properties theorem 4.1: let ), ~ , ~ (: ~ 1111 akfg and ), ~ , ~ (: ~ 2222 akfg be two fuzzy soft graphs on ),(: 11 * 1 evg and ),(: 22 * 2 evg respectively. if )( ~ )( ~ 21 ji ekef  and )( ~ )( ~ 12 ij ekef  then )()](1[)()](1[),(deg 1 ~ 11 ~ 111 ~~ 1 * 2 2 * 1 21 xdeydydexdyx gjgg i ggg cc +++=  proof:      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2111221211 ae ae exx yy ij ae ae eyy xx ji j i j i xxekyefyyekxef       + 2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2112212 ae ae eyy exx ijj j i xxekyefyef        2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2122111 ae ae eyy exx jii j i yyekxefxef ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ 211212 ae ae exx yy i ae ae eyy xx j j i j i xxekyyek       + 2 1 221 121 ))(( ~ 211 ae ae eyy exx i j i xxek some studies on products of fuzzy soft graphs 233       2 1 221 121 ))(( ~ 212 ae ae eyy exx j j i yyek )()](1[)()](1[ )()()()()()( 1 ~ 11 ~ 1 1 ~ 11 ~ 11 ~ 1 ~ 1 * 2 2 * 1 2 * 1 1 * 2 12 xdeydydexd ydxdexdydexdeyde gjgg i g gg igg jgjgi cc cc +++= +++= this is true for any vertex ),( 11 yx in 21 ~~ gg   with 21 ~~ kf  and 12 ~~ kf  . theorem 4.2 : let 1 ~ g and 2 ~ g be two fuzzy soft graphs on ),(: 11 * 1 evg and ),(: 22 * 2 evg respectively. if )( ~ )( ~ 21 ji ekef  and )( ~ 1 i ef is a constant function with 11 ))(( ~ vxcxef kki = then )](1)[()](1)[(),(deg 11 ~ 1111 ~~ * 2 1 * 1 * 221 ydxdexdydeecyx cc gg j gg jigg +++=   proof: given )( ~ )( ~ 21 ji ekef  then )( ~ )( ~ 12 ij ekef  . 11 ))(( ~ vxcxef kki = , for any 21 ),( vvyx lk       = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2111221211 ae ae exx yy ij ae ae eyy xx ji j i j i xxekyefyyekxef       + 2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2112212 ae ae eyy exx ijj j i xxekyefyef        2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2122111 ae ae eyy exx jii j i yyekxefxef ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ 21111 ae ae exx yy i ae ae eyy xx i j i j i xxekxef       + 2 1 221 121 ))(( ~ 211 ae ae eyy exx i j i xxek       2 1 221 121 ))(( ~ 11 ae ae eyy exx i j i xef +++=   )()()())(( ~ )( 1 ~ 11 ~ 111 1 * 2 1 1 * 2 xdydexdexefyde gg jgj ae igj c i )())(( ~ )( 1111 * 2 1 * 1 ydxefxde g ae i g j i c   shashikala s and anil p n 234 )()()()()()( 111 ~ 11 ~ 1 * 2 * 1 1 * 2 1 * 2 ydecxdexdydexdeecyde gig jgg jgjigj cc +++= )](1)[()](1)[( 11 ~ 11 * 2 1 * 1 * 2 ydxdexdydeec cc gg j gg ji +++= theorem 4.3 : let 1 ~ g and 2 ~ g be two fuzzy soft graphs on ),(: 11 * 1 evg and ),(: 22 * 2 evg respectively. if )( ~ )( ~ 12 ij ekef  and )( ~ 2 j ef is a constant function with 22 ))(( ~ vymyef lli = then )](1)[()](1)[(),(deg 11 ~ 1111 ~~ * 1 2 * 2 * 121 xdydeydxdeemyx cc gg i gg jigg +++=   proof: given )( ~ )( ~ 12 ij ekef  then )( ~ )( ~ 21 ji ekef       = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2111221211 ae ae exx yy ij ae ae eyy xx ji j i j i xxekyefyyekxef       + 2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2112212 ae ae eyy exx ijj j i xxekyefyef        2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2122111 ae ae eyy exx jii j i yyekxefxef ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ 12212 ae ae exx yy j ae ae eyy xx j j i j i yefyyek       + 2 1 221 121 ))(( ~ 12 ae ae eyy exx j j i yef       2 1 221 121 ))(( ~ 212 ae ae eyy exx j j i yyek +++=   )())(( ~ )())(( ~ )()( 11211211 ~ * 1 2 * 2 2 * 12 xdyefydeyefxdeyde g ae j g i ae jgigi j c j )()( 1 ~ 1 2 * 1 ydxde gg i c )](1)[()](1)[( 11 ~ 11 * 1 2 * 2 * 1 xdydeydxdeem cc gg i gg ji +++= theorem 4.4 : let 1 ~ g and 2 ~ g be two fuzzy soft graphs on complete graphs ),(: 11 * 1 evg and ),(: 22 * 2 evg respectively. if )( ~ )( ~ 21 ji ekef  and )( ~ )( ~ 12 ij ekef  then )()(),(deg 1 ~ 1 ~ 11 ~~ 1221 xdeydeyx gjgigg +=   proof:      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2111221211 ae ae exx yy ij ae ae eyy xx ji j i j i xxekyefyyekxef some studies on products of fuzzy soft graphs 235       + 2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2112212 ae ae eyy exx ijj j i xxekyefyef        2 1 221 121 ))(( ~ ))(( ~ ))(( ~ 2122111 ae ae eyy exx jii j i yyekxefxef ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ 211212 ae ae exx yy i ae ae eyy xx j j i j i xxekyyek       + 2 1 221 121 ))(( ~ 211 ae ae eyy exx i j i xxek       2 1 221 121 ))(( ~ 212 ae ae eyy exx j j i yyek )()](1[)()](1[ 1 ~ 11 ~ 1 1 * 2 2 * 1 xdeydydexd gjgg i g cc +++= )()( 1 ~ 1 ~ 12 xdeyde gjgi += (since * 1 g and * 2 g are complete graphs) theorem 4.5 : let ), ~ , ~ (: ~ 1111 akfg and ), ~ , ~ (: ~ 2222 akfg be two fuzzy soft graphs on regular graphs ),(: 11 * 1 evg and ),(: 22 * 2 evg respectively. if )( ~ )( ~ 21 ji ekef  and )( ~ )( ~ 12 ij ekef  then 21 ~~ gg   is a regular fuzzy soft graph if and only if 1 ~ g and 2 ~ g are regular fuzzy soft graphs. proof: let 1 ~ g and 2 ~ g be regular fuzzy soft graphs of degree 1k and 2 k respectively. for any vertex 2111 )( vvyx  , )()](1[)()](1[),(deg 1 ~ 11 ~ 111 ~~ 1 * 2 2 * 1 21 xdeydydexdyx gjgg i ggg cc +++=  (from theorem 4.1) 122111 ~~ ]1[]1[),(deg 21 keekeeyx j c i c gg +++=   this is true 2111 )( vvyx  hence, 21 ~~ gg   is a regular fuzzy soft graph. conversely, let 21 ~~ gg   be a regular fuzzy soft graph. for any two vertices )( 11 yx and )( 22 yx in 21 vv  , =  ),(deg 11 ~~ 21 yx gg  ),(deg 22 ~~ 21 yx gg   )()](1[)()](1[ 1 ~ 11 ~ 1 1 * 2 2 * 1 xdeydydexd gjgg i g cc +++ )()](1[)()](1[ 2 ~ 22 ~ 2 1 * 2 2 * 1 xdeydydexd gjgg i g cc +++= (from theorem 4.1) fix 1 vx , consider )( 11 yx and )( 22 yx in 21 vv  , shashikala s and anil p n 236 )()](1[)()](1[ 1 * 2 2 * 1 ~ 11 ~ xdeydydexd gjgg i g cc +++ )()](1[)()](1[ 1 * 2 2 * 1 ~ 22 ~ xdeydydexd gjgg i g cc +++= )()](1[)()](1[ 2 ~ 1 ~ 2 * 1 2 * 1 ydexdydexd gigg i g cc +=+ )()( 2 ~ 1 ~ 22 ydyd gg = this is true for all vertices of 2 v . 2 ~ g is a regular fuzzy soft graph. fix 2 vy , consider )( 11 yx and )( 22 yx in 21 vv  , )()](1[)()](1[ 1 ~~ 1 1 * 2 2 * 1 xdeydydexd gjgg i g cc +++ )()](1[)()](1[ 2 ~~ 2 1 * 2 2 * 1 xdeydydexd gjgg i g cc +++= )()](1[)()](1[ 2 ~ 1 ~ 1 * 2 1 * 2 xdeydxdeyd gjgg j g cc +=+ )()( 2 ~ 1 ~ 11 xdxd gg = this is true for all vertices of 1 v . 1 ~ g is a regular fuzzy soft graph. theorem 4.6 : let 1 ~ g and 2 ~ g be two fuzzy soft graphs on complete graphs ),(: 11 * 1 evg and ),(: 22 * 2 evg respectively. if )( ~ )( ~ 21 ji ekef  and )( ~ )( ~ 12 ij ekef  then 21 ~~ gg   is regular if and only if 1 ~ g and 2 ~ g are regular fuzzy soft graphs. proof: let 1 ~ g and 2 ~ g be regular fuzzy soft graphs of degree 1k and 2 k respectively. let * 1 g and * 2 g are complete graphs. )()(),(deg 1 ~ 1 ~ 11 ~~ 1221 xdeydeyx gjgigg +=   (from theorem 4.4) this is true 2111 )( vvyx  hence, 21 ~~ gg   is a regular fuzzy soft graph. conversely, let 21 ~~ gg   be a regular fuzzy soft graph. for any two vertices )( 11 yx and )( 22 yx in 21 vv  , =  ),(deg 11 ~~ 21 yx gg  ),(deg 22 ~~ 21 yx gg   )()( 1 ~ 1 ~ 12 xdeyde gjgi + )()( 2~2~ 12 xdeyde gjgi += fix 1 vx , consider )( 11 yx and )( 22 yx in 21 vv  , )()( 12 ~ 1 ~ xdeyde gjgi + )()( 12 ~ 2 ~ xdeyde gjgi += )()( 2 ~ 1 ~ 22 ydyd gg = some studies on products of fuzzy soft graphs 237 this is true for all vertices of 2 v . 2 ~ g is a regular fuzzy soft graph. fix 2 vy , consider )( 11 yx and )( 22 yx in 21 vv  , )()( 1 ~~ 12 xdeyde gjgi + )()( 2 ~~ 12 xdeyde gjgi += )()( 2 ~ 1 ~ 11 xdxd gg = this is true for all vertices of 1 v . 1 ~ g is a regular fuzzy soft graph. theorem 4.7 : let 1 ~ g and 2 ~ g be two fuzzy soft graphs on complete graphs * 1 g and * 2 g respectively then 21 ~~ gg   becomes a cartesian product of fuzzy soft graphs. proof: by the definition of alpha product of fuzzy soft graphs, for any two vertices )( 11 yx and )( 22 yx in 21 vv  ,               = = = 221 1212122111 221 1212112212 1212121112 2212121211 221121 ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ )()( ~ ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()(),() ~~ ( eyy exxifyyekxefxef eyy exxifxxekyefyef exxyyifxxekyef eyyxxifyyekxef yxyxeekk jii ijj ij ji ji  since * 1 g and * 2 g are complete graphs,     = = = 1212121112 2212121211 221121 ,)()( ~ )()( ~ ,)()( ~ )()( ~ )()(),() ~~ ( exxyyifxxekyef eyyxxifyyekxef yxyxeekk ij ji ji  )()(),() ~~ ()()(),() ~~ ( 221121221121 yxyxeekkyxyxeekk jiji =  is a cartesian product of 1 ~ g and 2 ~ g . theorem 4.8 : let 1 ~ g and 2 ~ g be two fuzzy soft graphs such that )( ~ )( ~ 21 ji ekef  . let )( ~ 1 i ef be a constant and * 1 g is a complete graph then 21 ~~ gg   is regular fuzzy soft graph if and only if 1 ~ g is a regular fuzzy soft graph and * 2 g is a regular graph. proof: let 11 ))(( ~ vxcxef kki = given )( ~ )( ~ 21 ji ekef  then )( ~ )( ~ 12 ij ekef  . let * 1 g be a complete graph. from theorem 4.2, )](1)[()](1)[(),(deg 11 ~ 1111 ~~ * 2 1 * 1 * 221 ydxdexdydeecyx cc gg j gg jigg +++=   )()()()( 1 ~ 11 ~ 1 1 * 2 1 * 2 xdydexdeydeec gg jgjgji c++= shashikala s and anil p n 238 let 1 ~ g be a regular fuzzy soft graph with degree m and * 2 g is a regular graph of degree n. mydemeneecyx c g jjjigg )(),(deg 111 ~~ * 2 21 ++=   ]1[ 2 c jji emecnee ++= hence, 21 ~~ gg   is regular fuzzy soft graph. conversely, assume that 21 ~~ gg   is regular fuzzy soft graph. ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx gggg   = =+++ )()()()()()( 111 ~ 11 ~ 1 * 2 * 1 1 * 2 1 * 2 ydecxdexdydexdeydeec gig jgg jgjgji cc )()()()()()( 222 ~ 22 ~ 2 * 2 * 1 1 * 2 1 * 2 ydecxdexdydexdeydeec gig jgg jgjgji cc +++ fix 1 vx , consider )( 1 xy and )( 2 xy in 21 vv  , )](1)[()()](1)[()( 2 ~ 21 ~ 1 * 2 1 * 2 * 2 1 * 2 ydxdeydeecydxdeydeec cc gg jgjigg jgji ++=++ this is true when degree of all vertices in * 2 g as well as in its complement are equal. hence, * 2 g is regular. fix 2 vy , )](1)[()()](1)[()( * 2 1 * 2 * 2 1 * 2 2 ~ 1 ~ ydxdeydeecydxdeydeec cc gg jgjigg jgji ++=++ )()( 2 ~ 1 ~ 11 xdxd gg = hence, 1 ~ g is a regular fuzzy soft graph. theorem 4.9 : let 1 ~ g and 2 ~ g be two fuzzy soft graphs such that )( ~ )( ~ 21 ji ekef  . if 21 ~~ gg   , 1 ~ g are regular fuzzy soft graphs, * 1 g and * 2 g are regular graphs then   1 ))(( ~ 1 ae ki i xef is same for all k=1,2,3..... proof: let 1 ~ g be a regular fuzzy soft graph of degree m, * 1 g and * 2 g are regular graphs of degree s and n respectively. given )( ~ )( ~ 21 ji ekef  then )( ~ )( ~ 12 ij ekef  . using the definition of 21 ~~ gg   ,      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   some studies on products of fuzzy soft graphs 239 ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ 21111 ae ae exx yy i ae ae eyy xx i j i j i xxekxef       + 2 1 221 121 ))(( ~ 211 ae ae eyy exx i j i xxek        2 1 221 121 ))(( ~ ))(( ~ 2111 ae ae eyy exx ii j i xefxef   +++= 1 * 2 * 1 1 * 2 1 1 * 2 ))(( ~ )()()()()())(( ~ )( 1111 ~ 11 ~ 111 ae kigg jgg jgj ae igj i cc i xefydxdexdydexdexefyde )](1)[(])(1[))(( ~ )( 11 ~ 1111 * 2 1 * 1 1 * 2 ydxdexdxefyde cc i gg j g ae igj +++=   since 21 ~~ gg   is a regular fuzzy soft graph, ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx gggg   = =+++  )](1)[(])(1[))(( ~ )( 11 ~ 1111 * 2 1 * 1 1 * 2 ydxdexdxefyde cc i gg j g ae igj )](1)[(])(1[))(( ~ )( 22 ~ 2212 * 2 1 * 1 1 * 2 ydxdexdxefyde cc i gg j g ae igj +++  =+++  )](1[])(1[))(( ~ 1111 * 2 * 1 1 ydmexdxefne cc i g j g ae ij )](1[])(1[))(( ~ 2221 * 2 * 1 1 ydmexdxefne cc i g j g ae ij +++  ])(1[))(( ~ ])(1[))(( ~ 221111 * 1 1 * 1 1 xdxefnexdxefne c i c i g ae ij g ae ij +=+   (since c g * 1 is regular)   = 11 ))(( ~ ))(( ~ 2111 ae i ae i ii xefxef theorem 4.10 : let 1 ~ g and 2 ~ g be two fuzzy soft graphs such that )( ~ )( ~ 12 ij ekef  . if 21 ~~ gg   , 2 ~ g are regular fuzzy soft graphs, * 1 g and * 2 g are regular graphs then   21 ))(( ~ 1 ae lj j yef is same for all ...3,2,1=l proof: proof is similar to the proof of theorem 4.9. theorem 4.11 : let 1 ~ g and 2 ~ g be two fuzzy soft graphs with )( ~ )( ~ 12 ij ekef  and cef j =)( ~ 2 then 21 ~~ gg   is a regular fsg if and only if * 1 g and * 2 g is a regular graph and 2 ~ g is a regular fsg. proof: given cef j =)( ~ 2 , since )( ~ )( ~ 12 ij ekef  , )( ~ )( ~ 21 ji ekef  . let 2 ~ g be a regular fsg of degree m, * 1 g and * 2 g are regular graphs of degree n and p respectively. shashikala s and anil p n 240 from theorem 4.3, )](1)[()](1)[(),(deg 11 ~ 1111 ~~ * 1 2 * 2 * 121 xdydeydxdeecyx cc gg i gg jigg +++=   )](1[)](1[ 11 * 1 * 2 xdemydeecn cc g i g ji +++= 21 ~~ gg   is a regular fsg. conversely, let 21 ~~ gg   be a regular fsg. ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx gggg   = =+++ )](1)[()](1)[( 11 ~ 11 * 1 2 * 2 * 1 xdydeydxdeec cc gg i gg ji )](1)[()](1)[( 22 ~ 22 * 1 2 * 2 * 1 xdydeydxdeec cc gg i gg ji +++ fix 1 vx , =+++ )](1)[()](1)[( * 1 2 * 2 * 1 1 ~ 1 xdydeydxdeec cc gg i gg ji )](1)[()](1)[( * 1 2 * 2 * 1 2 ~ 2 xdydeydxdeec cc gg i gg ji +++ )()( 2 ~ 1 ~ 22 ydyd gg = and )()( 21 * 2 * 2 ydyd cc gg = 2 ~ g is a regular fsg and * 2 g is regular. fix 2 vy , =+++ )](1)[()](1)[( 1 ~ 1 * 1 2 * 2 * 1 xdydeydxdeec cc gg i gg ji )](1)[()](1)[( 2 ~ 2 * 1 2 * 2 * 1 xdydeydxdeec cc gg i gg ji +++ )()( 21 * 1 * 1 xdxd gg = and )()( 21 * 1 * 1 xdxd cc gg = this is true when * 1 g is regular. theorem 4.12 : let 1 ~ g and 2 ~ g be two fuzzy soft graphs where * 1 g is a complete graph. if )( ~ 1 i ef and )( ~ 1 i ek are constant and )( ~ )( ~ 21 ji ekef  then 21 ~~ gg   is a regular fsg if and only if 1 ~ g is regular and * 2 g is a regular graph. proof: let cekef ii == )( ~ )( ~ 11 and 21 ~~ gg   be regular. given )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  . ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx gggg   = using the result of theorem 4.9, =+++  )](1)[(])(1)[)(( ~ )( 11 ~ 1111 * 2 1 * 1 1 * 2 ydxdexdxefyde cc i gg j g ae igj some studies on products of fuzzy soft graphs 241 )](1)[(])(1)[)(( ~ )( 22 ~ 2212 * 2 1 * 1 1 * 2 ydxdexdxefyde cc i gg j g ae igj +++  =++  )](1)[())(( ~ )( 11 ~ 111 * 2 1 1 * 2 ydxdexefyde c i gg j ae igj )](1)[())(( ~ )( 22 ~ 212 * 2 1 1 * 2 ydxdexefyde c i gg j ae igj ++  (since * 1 g is a complete graph) given cef i =)( ~ 1 )](1)[()()](1)[()( 22 ~ 211 ~ 1 * 2 1 * 2 * 2 1 * 2 ydxdydecydxdydec cc ggg i ggg i ++=++ fix 1 vx , )](1)[()()](1)[()( 2 ~ 21 ~ 1 * 2 1 * 2 * 2 1 * 2 ydxdydecydxdydec cc ggg i ggg i ++=++ )()( 21 * 2 * 2 ydyd gg = and )()( 21 * 2 * 2 ydyd cc gg = this is true only when * 2 g is a regular graph. similarly, fix 2 vy , )()( 2 ~ 1 ~ 11 xdxd gg = 1 ~ g is regular fsg. conversely, let 1 ~ g and * 2 g be regular with degree m and n respectively. )](1)[(])(1)[)(( ~ )(),(deg 11 ~ 111111 ~~ * 2 1 * 1 1 * 221 ydxdexdxefydeyx cc i gg j g ae igjgg +++=     )1( pmecene jij ++= therefore, 21 ~~ gg   is a regular fsg. 5. degree of a vertex in beta product )( product− of two fuzzy soft graphs and its regular properties theorem 5.1: let 1 ~ g and 2 ~ g be two fsgs on complete graphs * 1 g and * 2 g respectively. i) if )( ~ )( ~ 21 ji ekek  then )()(),(deg 11 ~ 11 ~~ * 2121 ydxdeyx ggjgg =   ii) if )( ~ )( ~ 12 ij ekek  then )()(),(deg 11 ~ 11 ~~ * 1221 xdydeyx ggigg =   proof:      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   +=       2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 2122111 ae ae eyy xx jii j i yyekxefxef shashikala s and anil p n 242 +      2 1 121 21 ))(( ~ ))(( ~ ))(( ~ 2112212 ae ae exx yy ijj j i xxekyefyef        2 1 221 121 ))(( ~ ))(( ~ 212211 ae ae eyy exx ji j i yyekxxek i) for any vertex )( 11 yx 21 vv  since * 1 g and * 2 g are complete graphs, we have        = 2 1 221 121 21 ))(( ~ ))(( ~ ),(deg 21221111 ~~ ae ae eyy exx jigg j i yyekxxekyx  given )( ~ )( ~ 21 ji ekek         = 2 1 221 121 21 ))(( ~ ),(deg 21111 ~~ ae ae eyy exx igg j i xxekyx  )()( 11 ~ * 21 ydxde ggj = ii)        = 2 1 221 121 21 ))(( ~ ))(( ~ ),(deg 21221111 ~~ ae ae eyy exx jigg j i yyekxxekyx  since )( ~ )( ~ 12 ij ekek         = 2 1 221 121 21 ))(( ~ ),(deg 21211 ~~ ae ae eyy exx jgg j i yyekyx  )()( 11 ~ * 12 xdyde ggi = theorem 5.2: let ), ~ , ~ (: ~ 1111 akfg and ), ~ , ~ (: ~ 2222 akfg be two fuzzy soft graphs on ),(: 11 * 1 evg and ),(: 22 * 2 evg respectively. if )( ~ )( ~ 21 ji ekef  and )( ~ )( ~ 12 ij ekef  then i) if )( ~ )( ~ 21 ji ekek  then )()()]()()[(),(deg 1 ~ 1111 ~ 11 ~~ 1 * 1 * 2 * 2 121 ydxdeydydxdeyx gg iggg jgg cc ++=   ii) if )( ~ )( ~ 12 ij ekek  then )()()]()()[(),(deg 1 ~ 1111 ~ 11 ~~ 1 * 2 * 1 * 1 221 xdydexdxdydeyx gg jggg igg cc ++=   proof:      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   +=       2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 2122111 ae ae eyy xx jii j i yyekxefxef some studies on products of fuzzy soft graphs 243 +      2 1 121 21 ))(( ~ ))(( ~ ))(( ~ 2112212 ae ae exx yy ijj j i xxekyefyef        2 1 221 121 ))(( ~ ))(( ~ 212211 ae ae eyy exx ji j i yyekxxek i) given )( ~ )( ~ 21 ji ekef  and )( ~ )( ~ 12 ij ekef  . also )( ~ )( ~ 21 ji ekek              += 2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ ae ae exx yy i ae ae eyy xx jgg j i j i xxekyyekyx        + 2 1 221 121 ))(( ~ 211 ae ae eyy exx i j i xxek )()()()([)()( 11 ~ 1 ~ 11 ~ 1 * 211 * 2 2 * 1 ydxdexdydeydxde ggjgg jgg i cc ++= )()()]()()[( 1 ~ 1111 ~ 1 * 1 * 2 * 2 1 ydxdeydydxde gg iggg j cc ++= ii) given )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  & )( ~ )( ~ 12 ij ekek              += 2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ ae ae exx yy i ae ae eyy xx jgg j i j i xxekyyekyx        + 2 1 221 121 ))(( ~ 212 ae ae eyy exx j j i yyek )()()()([)()( 11 ~ 1 ~ 11 ~ 1 * 121 * 2 2 * 1 xdydexdydeydxde ggigg jgg i cc ++= )()()]()()[( 1 ~ 1111 ~ 1 * 2 * 1 * 1 2 xdydexdxdyde gg jggg i cc ++= theorem 5.3: let ), ~ , ~ (: ~ 1111 akfg and ), ~ , ~ (: ~ 2222 akfg be two fuzzy soft graphs on ),(: 11 * 1 evg and ),(: 22 * 2 evg respectively. if )( ~ )( ~ 21 ji ekef  and )( ~ 1 i ef is a constant function with cxef ki =))(( ~ 1 1vxk  then )()()]()()[(),(deg 11111 ~ 11 ~~ * 2 * 1 * 2 * 2 121 ydxdeecydydxdeyx gg jiggg jgg cc ++=   proof:      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   using the given conditions,                 ++= 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 21121111 ae ae eyy exx i ae ae exx yy i ae ae eyy xx i j i j i j i xxekxxekxef shashikala s and anil p n 244 )()()()()()(),(deg 11 ~ 11 ~ 1111 ~~ * 21 * 2 1 * 2 * 1 21 ydxdeydxdeeydxdceyx ggjgg jjgg igg cc ++=   )()()]()()[( 11111 ~ * 2 * 1 * 2 * 2 1 ydxdeecydydxde gg jiggg j cc ++= theorem 5.4: let 1 ~ g and 2 ~ g be two fsgs on ),(: 11 * 1 evg and ),(: 22 * 2 evg respectively. if )( ~ )( ~ 12 ij ekef  and )( ~ 2 j ef is a constant function with myef lj =))(( ~ 2 2vyl  then )()()]()()[(),(deg 11111 ~ 11 ~~ * 1 * 2 * 1 * 1 221 xdymdeexdxdydeyx gg jiggg igg cc ++=   proof:      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   using the given conditions,                 ++= 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 21212212 ae ae eyy exx j ae ae exx yy j ae ae eyy xx j j i j i j i yyekyefyyek )()()()()()( 11 ~ 1111 ~ * 12 * 1 * 2 * 1 2 xdydeexdymdexdyde ggiigg j gg i cc ++= )()()]()()[( 11111 ~ * 1 * 2 * 1 * 1 2 xdymdeexdxdyde gg jiggg i cc ++= theorem 5.5: let 1 ~ g and 2 ~ g be two fsgs on complete graphs * 1 g and * 2 g respectively. 21 ~~ gg   is a regular fsg if and only if 1 ~ g and 2 ~ g are regular. proof: let 1 ~ g and 2 ~ g be regular fsgs with degrees m and n respectively. let * 1 g and * 2 g be complete graphs with degrees p and q respectively.      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg         = 2 1 221 121 ))(( ~ ))(( ~ 212211 ae ae eyy exx ji j i yyekxxek case (i): if )( ~ )( ~ 21 ji ekek  then        = 2 1 221 121 21 ))(( ~ ),(deg 21111 ~~ ae ae eyy exx igg j i xxekyx  )()( 11~ * 21 ydxde ggj = qme j = 21 ~~ gg   is a regular fsg. some studies on products of fuzzy soft graphs 245 case (i): if )( ~ )( ~ 12 ij ekek  then =  ),(deg 11 ~~ 21 yx gg        2 1 221 121 ))(( ~ 212 ae ae eyy exx j j i yyek )()( 11 ~ * 12 xdyde ggi = pnei= 21 ~~ gg   is a regular fsg. conversely, let 21 ~~ gg   be a regular fsg. ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx gggg   = case (i): if )( ~ )( ~ 21 ji ekek  then )()( 11 ~ * 21 ydxde ggj )()( 22 ~ * 21 ydxde ggj = fix 1 vx , )()()()( 2 ~ 1 ~ * 21 * 21 ydxdydxd gggg = )()( 21 * 2 * 2 ydyd gg = * 2 g is a regular graph. fix 2 vy , )()()()( * 21 * 21 2 ~ 1 ~ ydxdydxd gggg = )()( 2 ~ 1 ~ 11 xdxd gg = 1 ~ g is regular. case (ii): similarly for )( ~ )( ~ 12 ij ekek  , we get * 1 g and 2 ~ g as regular. theorem 5.6: let 1 ~ g and 2 ~ g be two fsgs and * 1 g is a complete graph and * 2 g is a regular graph. if )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  and )( ~ )( ~ 21 ji ekek = then 21 ~~ gg   is a regular fsg if and only if 1 ~ g is a regular fsg. proof: let * 2 g be a regular graph of degree p and * 1 g is a complete graph. let cekek ji == )( ~ )( ~ 21 , )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  . let us assume that 1 ~ g is a regular fsg of degree m. using the definition of degree of a vertex in beta product of fsgs and the above conditions, we get shashikala s and anil p n 246                  ++= 2 1 221 121 2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ))(( ~ ),(deg 21121121211 ~~ ae ae eyy exx i ae ae exx yy ij ae ae eyy xx gg j i j i j i xxekxxekyyekyx             += 2 1 221 121 2 1 121 21 ))(( ~ ))(( ~ 211211 ae ae eyy exx i ae ae exx yy i j i j i xxekxxek (since * 1 g is a complete graph) )()()()( 11 ~ 1 ~ 1 * 211 * 2 ydxdexdyde ggjgg j c += )]()([)( 111 ~ * 2 * 21 ydydxde c ggg j += ][ spme j += 21 ~~ gg   is a regular fsg. conversely, let 21 ~~ gg   be a regular fsg, * 2 g is a regular graph of degree p and * 1 g is a complete graph. ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx gggg   = )]()([)()]()([)( 222 ~ 111 ~ * 2 * 21 * 2 * 21 ydydxdeydydxde cc ggg j ggg j +=+ ][)(][)( 2 ~ 1 ~ 11 tpxdetpxde gjgj +=+ where tyd c g =)( * 2 )()( 2 ~ 1 ~ 11 xdxd gg = 1 ~ g is regular. theorem 5.7: let 1 ~ g and 2 ~ g be two fsgs and * 2 g is a complete graph and * 1 g is a regular graph. if )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  and )( ~ )( ~ 21 ji ekek = then 21 ~~ gg   is a regular fsg if and only if 1 ~ g is a regular fsg. proof: proof is similar to the proof of theorem 5.6. theorem 5.8: if 1 ~ g and 2 ~ g are two regular fsgs with )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  , * 1 g and * 2 g are regular but not complete graphs, then beta product of two fsgs 1 ~ g and 2 ~ g is a regular fsg. proof: let 1 ~ g and 2 ~ g be regular fsgs with degrees m and n respectively. let * 1 g and * 2 g be regular graphs of degrees p and q respectively and suppose that * 1 g and * 2 g are not complete graphs. let )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  . some studies on products of fuzzy soft graphs 247      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   case (i) : assume that * 1 g and * 2 g are isomorphic graphs. let cekek ji == )( ~ )( ~ 21 +=        2 1 221 21 21 ))(( ~ ))(( ~ ))(( ~ ),(deg 212211111 ~~ ae ae eyy xx jiigg j i yyekxefxefyx  +      2 1 121 21 ))(( ~ ))(( ~ ))(( ~ 2112212 ae ae exx yy ijj j i xxekyefyef        2 1 221 121 ))(( ~ ))(( ~ 212211 ae ae eyy exx ji j i yyekxxek                 ++= 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 211211212 ae ae eyy exx i ae ae exx yy i ae ae eyy xx j j i j i j i xxekxxekyyek )()()()()()( 11 ~ 1 ~ 11 ~ 1 * 211 * 2 2 * 1 ydxdexdydeydxde ggjgg jgg i cc ++= nsesqme ij ++= )( (since * 1 g and * 2 g are regular graphs of degree p and q and are isomorphic, sydxd cc gg == )()( 11 * 2 * 1 ) 21 ~~ gg   is regular. case (ii) : assume that * 1 g and * 2 g are not isomorphic but are regular graphs of degrees p and q . by the definition, ++=             2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ ae ae exx yy i ae ae eyy xx jgg j i j i xxekyyekyx         2 1 221 121 ))(( ~ ))(( ~ 212211 ae ae eyy exx ji j i yyekxxek if )( ~ )( ~ 21 ji ekek  , ++=             2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ ae ae exx yy i ae ae eyy xx jgg j i j i xxekyyekyx        2 1 221 121 ))(( ~ 211 ae ae eyy exx i j i xxek )()()]()()[( 11 ~ 111 ~ * 1 2 * 2 * 21 xdydeydydxde cc gg i ggg j ++= snetqme ij ++= ][ shashikala s and anil p n 248 21 ~~ gg   is regular. if )( ~ )( ~ 12 ij ekek  , ++=             2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ ae ae exx yy i ae ae eyy xx jgg j i j i xxekyyekyx        2 1 221 121 ))(( ~ 212 ae ae eyy exx j j i yyek )()()]()()[( 11 ~ 111 ~ * 2 1 * 1 * 12 ydxdexdxdyde cc gg j ggg i ++= tmespne ji ++= ][ 21 ~~ gg   is regular. 6. degree of a vertex in gamma product )( product− of two fuzzy soft graphs and its regular properties theorem 6.1: let 1 ~ g and 2 ~ g be two fsgs on complete graphs * 1 g and * 2 g respectively and )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  i) if )( ~ )( ~ 21 ji ekek  then )()](1)[(),(deg 1 ~ 11 ~ 11 ~~ 2 * 2121 ydeydxdeyx giggjgg ++=   ii) if )( ~ )( ~ 12 ij ekek  then )()](1)[(),(deg 1 ~ 11 ~ 11 ~~ 1 * 1221 xdexdydeyx gjggigg ++=   proof: given )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  .for any vertex 2111 )( vvyx  ,      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ++=       =    = 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2111221211 ae ae exx yy ij ae ae eyy xx ji j i j i xxekyefyyekxef +      2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 2122111 ae ae eyy xx jij j i yyekxefxef        2 1 121 21 ))(( ~ ))(( ~ ))(( ~ 2112212 ae ae exx yy ijj j i xxekyefyef some studies on products of fuzzy soft graphs 249 ))(( ~ ))(( ~ 212211 2 1 221 121 yyekxxek j ae ae eyy exx i j i        i) given )( ~ )( ~ 21 ji ekek  , ++=       =    =  2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 21121211 ~~ ae ae exx yy i ae ae eyy xx jgg j i j i xxekyyekyx                  ++ 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 211211212 ae ae eyy exx i ae ae exx yy i ae ae eyy xx j j i j i j i xxekxxekyyek since * 1 g and * 2 g are complete graphs,            =    = ++= 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 211211212 ae ae eyy exx i ae ae exx yy i ae ae eyy xx j j i j i j i xxekxxekyyek )()()()(),(deg 11 ~ 1 ~ 1 ~ 11 ~~ * 212121 ydxdeydexdeyx ggjgigjgg ++=   )()](1)[( 1~11~ 2 * 21 ydeydxde giggj ++= ii) given )( ~ )( ~ 12 ij ekek       = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   ))(( ~ ))(( ~ ))(( ~ 212212211 2 1 221 121 2 1 221 21 2 1 121 21 yyekyyekxxek j ae ae eyy exx ae ae eyy xx j ae ae exx yy i j i j i j i            =    = ++= )()()()(),(deg 11 ~ 1 ~ 1 ~ 11 ~~ * 122121 xdydeydexdeyx ggigigjgg ++=   )()](1)[( 1~11~ 1 * 12 xdexdyde gjggi ++= theorem 6.2: let 1 ~ g and 2 ~ g be two fsgs on crisp graphs * 1 g and * 2 g respectively and )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  i) if )( ~ )( ~ 21 ji ekek  then )](1)[()]()(1)[(),(deg 11 ~ 111 ~ 11 ~~ * 1 2 * 2 * 2 121 xdydeydydxdeyx cc gg iggg jgg ++++=   ii) if )( ~ )( ~ 12 ij ekek  then )](1)[()]()(1)[(),(deg 11 ~ 111 ~ 11 ~~ * 2 1 * 1 * 1 221 ydxdexdxdydeyx cc gg jggg igg ++++=   proof: given )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  i) if )( ~ )( ~ 21 ji ekek  then shashikala s and anil p n 250 using the definition of degree of 21 ~~ gg        = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   +++=            =    = 2 1 221 21 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 212211212 ae ae eyy xx j ae ae exx yy i ae ae eyy xx j j i j i j i yyekxxekyyek            + 2 1 221 121 2 1 121 21 ))(( ~ ))(( ~ 211211 ae ae eyy exx i ae ae exx yy i j i j i xxekxxek )](1)[()]()(1)[(),(deg 11 ~ 111 ~ 11 ~~ * 1 2 * 2 * 2 121 xdydeydydxdeyx cc gg iggg jgg ++++=   ii) if )( ~ )( ~ 12 ij ekek  then      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   +++=            =    = 2 1 221 21 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ 212211212 ae ae eyy xx j ae ae exx yy i ae ae eyy xx j j i j i j i yyekxxekyyek             + 2 1 121 21 2 1 221 121 ))(( ~ ))(( ~ 212211 ae ae exx yy j ae ae eyy exx i j i j i yyekxxek )](1)[()]()(1)[(),(deg 11 ~ 111 ~ 11 ~~ * 2 1 * 1 * 1 221 ydxdexdxdydeyx cc gg jggg igg ++++=   theorem 6.3: let ), ~ , ~ (: ~ 1111 akfg and ), ~ , ~ (: ~ 2222 akfg be two fuzzy soft graphs on * 1 g and * 2 g respectively. if )( ~ )( ~ 21 ji ekef  and )( ~ 1 i ef is a constant function with cxef ki =))(( ~ 1 1vxk  , then )](1)[()]()(1)[(),(deg 11111 ~ 11 ~~ * 1 * 2 * 2 * 2 121 xdydeecydydxdeyx cc gg jiggg jgg ++++=   proof: given )( ~ )( ~ 21 ji ekef  then )( ~ )( ~ 12 ij ekef  , )( ~ )( ~ 21 ji ekek  and cxef i =))(( ~ 11 by the definition, ++=       =    =  2 1 121 21 2 1 221 21 21 ))(( ~ ))(( ~ ),(deg 2111111 ~~ ae ae exx yy i ae ae eyy xx igg j i j i xxekxefyx                  ++ 2 1 221 121 2 1 121 21 2 1 221 21 ))(( ~ ))(( ~ ))(( ~ ))(( ~ 2112112111 ae ae eyy exx i ae ae exx yy i ae ae eyy xx ij j i j i j i xxekxxekxefxef some studies on products of fuzzy soft graphs 251 +++=    jgg igj ae igjgg eydxcdexdexefydeyx c i )()()())(( ~ )(),(deg 111 ~ 11111 ~~ * 2 * 1 1 1 * 221  )()()()( 11 ~ 11 ~ * 21 * 2 1 ydxdeydxde ggjgg j c + )](1)[()]()(1)[( 11111 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc gg jiggg j ++++= theorem 6.4: let ), ~ , ~ (: ~ 1111 akfg and ), ~ , ~ (: ~ 2222 akfg be two fuzzy soft graphs on * 1 g and * 2 g respectively. if )( ~ )( ~ 12 ij ekef  and )( ~ 2 j ef is a constant function with cyef lj =))(( ~ 2 2vyl  , then )](1)[()]()(1)[(),(deg 11111 ~ 11 ~~ * 2 * 1 * 1 * 1 221 ydxdeecxdxdydeyx cc gg jiggg igg ++++=   proof: proof is analogues to the proof of theorem 6.3. theorem 6.5 : let 1 ~ g and 2 ~ g be two fuzzy soft graphs with )( ~ )( ~ 21 ji ekef  and 11 ))(( ~ vxcxef kki = then 21 ~~ gg   is a regular fsg if and only if 1 ~ g is regular, * 1 g and * 2 g are regular graphs. proof: given cxef ki =))(( ~ 1 , )( ~ )( ~ 21 ji ekef  , )( ~ )( ~ 12 ij ekef  then )( ~ )( ~ 21 ji ekek  . let 1 ~ g be a regular fsg of degree m, * 1 g and * 2 g are regular graphs of degree p and q respectively.      = aee eyxyx jigg ji yxyxeekkyx ),( )()( 22112111 ~~ 2211 21 )())(,() ~~ (),(deg   using theorem 6.3, )](1)[()]()(1)[( 11111 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc gg jiggg j ++++= ]1[]1[ 122 c ji c j epeeceeme ++++= 21 ~~ gg   is a regular fsg. conversely, let 21 ~~ gg   be a regular fsg. ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx gggg   = =++++ )](1)[()]()(1)[( 11111 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc gg jiggg j )](1)[()]()(1)[( 22222 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc gg jiggg j ++++= fix 2 vy , shashikala s and anil p n 252 =++++ )](1)[()]()(1)[( 11 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc gg jiggg j )](1)[()]()(1)[( 22 ~ * 1 * 2 * 2 * 2 1 xdydeecydydxde cc gg jiggg j ++++= )()( 2 ~ 1 ~ 11 xdxd gg = and )()( 21 * 1 * 1 xdxd cc gg = i.e. 1 ~ g and * 1 g are regular. fix 1 vx , =++++ )](1)[()]()(1)[( * 1 * 2 * 2 * 2 1 111 ~ xdydeecydydxde cc gg jiggg j )](1)[()]()(1)[( * 1 * 2 * 2 * 2 1 222 ~ xdydeecydydxde cc gg jiggg j ++++= this holds good when )()( 21 * 2 * 2 ydyd gg = and )()( 21 * 2 * 2 ydyd cc gg = * 2 g is regular. theorem 6.6: let 1 ~ g and 2 ~ g be two fsgs on complete graphs * 1 g and * 2 g respectively. if )( ~ )( ~ 21 ji ekef  and )( ~ )( ~ 12 ij ekef  then 21 ~~ gg   is a regular fsg if and only if 1 ~ g and 1 ~ g are regular fsgs. proof: let 1 ~ g and 2 ~ g be regular fsgs of degree m and n respectively. given )( ~ )( ~ 21 ji ekef  and )( ~ )( ~ 12 ij ekef  , * 1 g and * 2 g are complete graphs of degree p and q respectively. from theorem 6.1, i) if )( ~ )( ~ 21 ji ekek  then )()](1)[(),(deg 1 ~ 11 ~ 11 ~~ 2 * 2121 ydeydxdeyx giggjgg ++=   neqme ij ++= ]1[ 21 ~~ gg   is a regular fsg. ii) if )( ~ )( ~ 12 ij ekek  then )()](1)[(),(deg 1 ~ 11 ~ 11 ~~ 1 * 1221 xdexdydeyx gjggigg ++=   mepne ji ++= ]1[ 21 ~~ gg   is a regular fsg. conversely, let 21 ~~ gg   be a regular fsg. some studies on products of fuzzy soft graphs 253 ),(deg),(deg 22 ~~ 11 ~~ 2121 yxyx gggg   = for )(~)(~ 21 ji ekek  , )()](1)[()()](1)[( 2 ~ 22 ~ 1 ~ 11 ~ 2 * 212 * 21 ydeydxdeydeydxde giggjgiggj ++=++ fix 1 vx , )()](1)[()()](1)[( 2 ~ 2 ~ 1 ~ 1 ~ 2 * 212 * 21 ydeydxdeydeydxde giggjgiggj ++=++ since * 1 g and * 2 g are complete graphs, we get )()( 2~1~ 22 ydyd gg = 2 ~ g is a regular fsg. fix 2 vy , )()](1)[()()](1)[( 2 * 212 * 21 ~ 2 ~~ 1 ~ ydeydxdeydeydxde giggjgiggj ++=++ )()( 2 ~ 1 ~ 11 xdxd gg = 1 ~ g is a regular fsg. ii) for )( ~ )( ~ 12 ij ekek  , similar process is followed and we get 1 ~ g and 2 ~ g as regular fsgs. theorem 6.7 : let 1 ~ g and 2 ~ g be two fsgs with )( ~ )( ~ 12 ij ekef  and 22 ))(( ~ vycyef llj = then 21 ~~ gg   is a regular fsg if and only if 2 ~ g is regular, * 1 g and * 2 g are regular graphs. proof: proof is similar to the proof of theorem 6.5. 7 conclusions in this paper, different products of fuzzy soft graphs are defined and demonstrated with some examples. the degree of a vertex in these product fsgs under some conditions and the regular properties associated with it are analyzed. when fuzzy soft graphs are very large these formulas will play a major role which helps in studying some of its properties. references [1] d. molodtsov, soft set theory-first results, computers and mathematics with applications, 37 (1999) 19-31. [2] a. rosenfeld, fuzzy graphs, in: l a zadeh, k s fu, m shimura (eds), fuzzy sets and their applications (new york: academic press 1975), 77-95. [3] lotfi a zadeh, fuzzy sets, information and control, 8 (3) (1965) 338-353. shashikala s and anil p n 254 [4] frank harary, graph theory, narosa/addison wesley, indian student edition, (1988). [5] p.k. maji, a. r. roy, r. biswas, fuzzy soft sets, the journal of fuzzy mathematics 9(3) (2001) 589-602. [6] j. n. mordeson and c. s. peng, operations on fuzzy graphs, information science, 79 (1994) 159-170. [7] a. r. roy and p. k. maji, a fuzzy soft set theoretic approach to decision making problems, journal of computational and applied mathematics, 28 (3) (2007) 412-418. [8] m. akram and s. nawaz, on fuzzy soft graphs, italian journal of pure and applied mathematics, 34 (2015) 463-480. [9] sumit mohinta and t. k. samanta, an introduction to fuzzy soft graph, mathematica moravica , 19( 2) (2015) 35-48. [10] shovan dogra, different types of product of fuzzy graphs, progress in nonlinear dynamics and chaos, 3(1) (2015) 41-56. [11] m. akram and s. nawaz, fuzzy soft graphs with applications, journal of intelligent & fuzzy system, 30 (2016) 3619-3632. [12] shashikala s and anil p n, connectivity in fuzzy soft graph and its complement, iosr journal of mathematics, 12 (5) (2016) 95-99. [13] a. pouhassani and h. doostie, on the properties of fuzzy soft graphs, journal of information and optimization sciences, 38: 3-4 (2017) 541-557. [14] anas al-masarwah, majdoleen abuqamar, certain types of fuzzy soft graphs, new mathematics and natural computation, (2017) 541-557. [15] shashikala s and anil p n, fuzzy soft cycles in fuzzy soft graphs, journal of new results in science, 8(1) (2019) 26-35. [16] b akhilandeswari, on regular fuzzy soft graphs, international journal of scientific research and engineering trends, 5(3) (2019) 831-834. [17] hafiza saba nawaz, muhammad akram, oligopolistic competition among the wireless internet service providers of malaysia using fuzzy soft graphs, journal of applied mathematics and computing, (2021) 1-36. gamma modules r. ameri, r. sadeghi department of mathematics, faculty of basic science university of mazandaran, babolsar, iran e-mail: ameri@umz.ac.ir abstract let r be a γ-ring. we introduce the notion of gamma modules over r and study important properties of such modules. in this regards we study submodules and homomorphism of gamma modules and give related basic results of gamma modules. keywords: γ-ring, rγ-module, submodule, homomorphism. 1 introduction the notion of a γ-ring was introduced by n. nobusawa in [6]. recently, w.e. barnes [2], j. luh [5], w.e. coppage studied the structure of γ-rings and obtained various generalization analogous of corresponding parts in ring theory. in this paper we extend the concepts of module from the category of rings to the category of rγ-modules over γ-rings. indeed we show that the notion of a gamma module is a generalization of a γ-ring as well as a module over a ring, in fact we show that many, but not all, of the results in the ratio mathematica 20, 2010 127 theory of modules are also valid for rγ-modules. in section 2, some definitions and results of γ − ring which will be used in the sequel are given. in section 3, the notion of a γ-module m over a γ − ring r is given and by many example it is shown that the class of γ-modules is very wide, in fact it is shown that the notion of a γ-module is a generalization of an ordinary module and a γ − ring. in section 3, we study the submodules of a given γ-module. in particular, we that l(m ), the set of all submodules of a γ-module m constitute a complete lattice. in section 3, homomorphisms of γ-modules are studied and the well known homomorphisms (isomorphisms) theorems of modules extended for γ-modules. also, the behavior of γ-submodules under homomorphisms are investigated. 2 preliminaries recall that for additive abelian groups r and γ we say that r is a γ − ring if there exists a mapping · : r × γ × r −→ r (r, γ, r′) 7−→ rγr′ such that for every a, b, c ∈ r and α, β ∈ γ, the following hold: (i) (a + b)αc = aαc + bαc; a(α + β)c = aαc + aβc; aα(b + c) = aαb + aαc; (ii) (aαb)βc = aα(bβc). a subset a of a γ-ring r is said to be a right ideal of r if a is an additive subgroup of r and aγr ⊆ a, where aγr = {aαc| a ∈ a, α ∈ γ, r ∈ r}. a left ideal of r is defined in a similar way. if a is both right and left ideal, we say that a is an ideal of r. ratio mathematica 20, 2010 128 if r and s are γ-rings. a pair (θ, ϕ) of maps from r into s such that i) θ(x + y) = θ(x) + θ(y); ii) ϕ is an isomorphism on γ; iii) θ(xγy) = θ(x)ϕ(γ)θ(y). is called a homomorphism from r into s. 3 rγ -modules in this section we introduce and study the notion of modules over a fixed γ-ring. definition 3.1. let r be a γ-ring. a (left) rγ-module is an additive abelian group m together with a mapping . : r × γ × m −→ m ( the image of (r, γ, m) being denoted by rγm), such that for all m, m1, m2 ∈ m and γ, γ1, γ2 ∈ γ, r, r1, r2 ∈ r the following hold: (m1) rγ(m1 + m2) = rγm1 + rγm2; (m2) (r1 + r2)γm = r1γm + r2γm; (m3) r(γ1 + γ2)m = rγ1m + rγ2m; (m4) r1γ1(r2γ2m) = (r1γ1r2)γ2m. a right rγ − module is defined in analogous manner. definition 3.2. a (left) rγ-module m is unitary if there exist elements, say 1 in r and γ0 ∈ γ, such that, 1γ0m = m for every m ∈ m . we denote 1γ0 by 1γ0 , so 1γ0 m = m for all m ∈ m . remark 3.3. if m is a left rγ-module then it is easy to verify that 0γm = r0m = rγ0 = 0m . if r and s are γ-rings then an (r, s)γ-bimodule m is both a left rγ-module and right sγ-module and simultaneously such that (rαm)βs = rα(mβs) ∀m ∈ m, ∀r ∈ r, ∀s ∈ s and α, β ∈ γ. ratio mathematica 20, 2010 129 in the following by many examples we illustrate the notion of gamma modules and show that the class of gamma module is very wide. example 3.4. if r is a γ-ring, then every abelian group m can be made into an rγmodule with trivial module structure by defining rγm = 0 ∀r ∈ r, ∀γ ∈ γ, ∀m ∈ m . example 3.5. every γ-ring r, is an rγ-module with rγ(r, s ∈ r, γ ∈ γ) being the γ-ring structure in r, i.e. the mapping . : r × γ × r −→ r. (r, γ, s) 7−→ r.γ.s example 3.6. let m be a module over a ring a. define . : a × r × m −→ m , by (a, s, m) = (as)m, being the r-module structure of m . then m is an aa-module. example 3.7. let m be an arbitrary abelian group and s be an arbitrary subring of z, the ring of integers. then m is a zs-module under the mapping . : z × s × m −→ m (n, n′, x) 7−→ nn′x example 3.8. if r is a γ-ring and i is a left ideal of r .then i is an rγ-module under the mapping . : r × γ × i −→ i such that (r, γ, a) 7−→ rγa . example 3.9. let r be an arbitrary commutative γ-ring with identity. a polynomial in one indeterminate with coefficients in r is to be an expression p (x) = anx n + + a2x 2 + a1x + a0 in which x is a symbol, not a variable and the set r[x] of all polynomials is then an abelian group. now r[x] becomes to an rγ-module, under the mapping . : r × γ × r[x] −→ r[x] (r, γ, f (x)) 7−→ r.γ.f (x) = ∑n i=1(rγai)x i. ratio mathematica 20, 2010 130 example 3.10. if r is a γ-ring and m is an rγ-module. set m [x] = { ∑n i=0 aix i | ai ∈ m}. for f (x) = ∑n j=0 bjx j and g(x) = ∑m i=0 aix i, define the mapping . : r[x] × γ × m [x] −→ m [x] (g(x), γ, f (x)) 7−→ g(x)γf (x) = ∑m+n k=1 (ak.γ.bk)x k. it is easy to verify that m [x] is an r[x]γ-module. example 3.11. let i be an ideal of a γ-ring r. then r/i is an rγ-module, where the mapping . : r × γ × r/i −→ r/i is defined by (r, γ, r′ + i) 7−→ (rγr′) + i. example 3.12. let m be an rγ-module, m ∈ m . letting t (m) = {t ∈ r | tγm = 0 ∀γ ∈ γ}. then t (m) is an rγ-module. proposition 3.12. let r be a γ-ring and (m, +, .) be an rγ-module. set sub(m ) = {x| x ⊆ m}, then sub(m ) is an rγ-module. proof. define ⊕ : (a, b) 7−→ a ⊕ b by a ⊕ b = (a\b) ∪ (b\a) for a, b ∈ sub(m ). then (sub(m ), ⊕) is an additive group with identity element ∅ and the inverse of each element a is itself. consider the mapping: ◦ : r × γ × sub(m ) −→ sub(m ) (r, γ, x) 7−→ r ◦ γ ◦ x = rγx, where rγx = {rγx | x ∈ x}. then we have (i) r ◦ γ ◦ (x1 ⊕ x2) = r · γ · (x1 ⊕ x2) = r · γ · ((x1\x2) ∪ (x2\x1)) = r · γ · ({a | a ∈ (x1\x2) ∪ (x2\x1)} = {r · γ · a | a ∈ (x1\x2) ∪ (x2\x1)}. and r ◦ γ ◦ x1 ⊕ r ◦ γ ◦ x2 = r · γ · x1 ⊕ r · γ · x2 = (r · γ · x1\r · γ · x2) ∪ (r · γ · x2\r · γ · x1) ratio mathematica 20, 2010 131 = {r · γ · x | x ∈ (x1\x2)} ∪ {r · γ · x | x ∈ (x2\x1)}. = {r · γ · x | x ∈ (x1\x2) ∪ (x2\x1)}. (ii) (r1 + r2) ◦ γ ◦ x = (r1 + r2) · γ · x = {(r1 + r2) · γ · x | x ∈ x} = {r1 · γ · x + r2 · γ · x | x ∈ x} = r1 · γ · x + r2 · γ · x = r1 ◦ γ ◦ x + r2 ◦ γ ◦ x. (iii) r ◦ (γ1 + γ2) ◦ x = r · (γ1 + γ2) · x = {r · (γ1 + γ2) · x | x ∈ x} = {r · γ1 · x + r · γ2 · x | x ∈ x} = r · γ1 · x + r · γ2 · x = r ◦ γ1 ◦ x + r ◦ γ2 ◦ x. (iv) r1 ◦ γ1 ◦ (r2 ◦ γ2 ◦ x) = r1 · γ1 · (r2 ◦ γ2 ◦ x) = {r1.γ1.(r2 ◦ γ2 ◦ x)|x ∈ x} = {r1.γ1.(r2.γ2.x) | x ∈ x} = {(r1.γ1.r2).γ2.x | x ∈ x} = (r1.γ1.r2).γ2.x. corollary 3.13. if in proposition 3.12, we define ⊕ by a ⊕ b = {a + b|a ∈ a, b ∈ b}. then (sub(m ), ⊕, ◦) is an rγ-module. proposition 3.14. let (r, ◦) and (s, •) be γ-rings. let (m, .) be a left rγ-module and right sγ-module. then a = {   r m 0 s   | r ∈ r, s ∈ s, m ∈ m} is a γ-ring and aγ-module under the mappings ? : a × γ × a −→ a (   r m 0 s   , γ,   r1 m1 0 s1  ) 7−→   r ◦ γ ◦ r1 r.γ.m1 + m.γ.s1 0 s • γ • s1   . � proof. straightforward. example 3.15. let (r, ◦) be a γ-ring . then r ⊕ z = {(r, s) | r ∈ r, s ∈ z} is an left rγ-module, where ⊕ addition operation is defined (r, n) ⊕ (r′, n′) = (r +r r′, n +z n′) and the product · : r × γ × (r ⊕ z) −→ r ⊕ z is defined r′ · γ · (r, n) −→ (r′ ◦ γ ◦ r, n). ratio mathematica 20, 2010 132 example 3.16. let r be the set of all digraphs (a digraph is a pair (v, e) consisting of a finite set v of vertices and a subset e of v × v of edges) and define addition on r by setting (v1, e1) + (v2, e2) = (v1 ∪ v2, e1 ∪ e2). obviously r is a commutative group since (∅, ∅) is the identity element and the inverse of every element is itself. for γ ⊆ r consider the mapping · : r × γ × r −→ r (v1, e1) · (v2, e2) · (v3, e3) = (v1 ∪ v2 ∪ v3, e1 ∪ e2 ∪ e3 ∪ {v1 × v2 × v3}), under condition (∅, ∅) = (∅, ∅) · (v1, e1) · (v2, e2)(v1, e1) · (∅, ∅) · (v2, e2) = (v1, e1) · (∅, ∅) · (v2, e2) = (v1, e1) · (v2, e2) · (∅, ∅). it is easy to verify that r is an rγ-module . example 3.17. suppose that m is an abelian group. set r = mmn and γ = mnm, so by definition of multiplication matrix subset r (t) mn = {(xij) | xtj = 0 ∀ j = 1, ...m} is a right rγ-module. also, c (k) mn = {xij) | xik = 0 ∀i = 1, ..., n} is a left rγ-module. example 3.18. let (m, •) be an rγ-module over γ-ring (r, .) and s = {(a, 0)|a ∈ r}. then r × m = {(a, m)|a ∈ r, m ∈ m} is an sγ-module, where addition operation is defined by (a, m) ⊕ (b, m1) = (a +r b, m +m m1). obviously, (r × m, ⊕) is an additive group. now consider the mapping ◦ : s × γ × (r × m ) −→ r × m ((a, 0), γ, (b, m)) 7−→ (a, 0) ◦ γ ◦ (b, m) = (a.γ.b, a • γ • m). then it is easy to verify that r × m is an sγ-module. example 3.19 let r be a γ-ring and (m, .) be an rγ-module. consider the mapping α : m −→ r. then m is an mγ-module, under the mapping ratio mathematica 20, 2010 133 ◦ : m × γ × m −→ m (m, γ, n) 7−→ m ◦ γ ◦ n = (α(m)).γ.n. example 3.20. let (r, ·) and (s, ◦) be γrings. then (i) the product r × s is a γring, under the mapping ((r1, s1), γ, (r2, s2)) 7−→ (r1 · γ · r2, s1 ◦ γ ◦ s2). (ii) for a = {   r 0 0 s   | r ∈ r, s ∈ s} there exists a mapping r × s −→ a, such that (r, s) −→   r 0 0 s   and a is a γring. moreover, a is an (r × s)γmodule under the mapping (r × s) × γ × a −→ a ((r1, s1), γ,   r2 0 0 s2  ) −→   r1 · γ · r2 0 0 s1 ◦ γ ◦ s2   . example 3.21. let (r, ·) be a γ-ring. then r×r is an rγ-module and (r×r)γmodule. consider addition operation (a, b)+(c, d) = (a+r c, b+r d). then (r×r, +) is an additive group. now define the mapping r×γ×(r×r) 7−→ r×r by (r, γ, (a, b)) 7−→ (r·γ·a, r·γ·b) and (r×r)×γ×(r×r) −→ r×r by ((a, b), γ, (c, d)) 7−→ (a·γ·c+b·γ·d, a·γ·d+b·γ·c). then r × r is an (r × r)γmodule. 4 submodules of gamma modules in this section we study submodules of gamma modules and investigate their properties. in the sequel r denotes a γ-ring and all gamma modules are rγ-modules definition 4.1. let (m, +) be an rγ-module. a nonempty subset n of (m, +) is said to be a (left) rγ-submodule of m if n is a subgroup of m and rγn ⊆ n , where ratio mathematica 20, 2010 134 rγn = {rγn|γ ∈ γ, r ∈ r, n ∈ n}, that is for all n, n′ ∈ n and for all γ ∈ γ, r ∈ r; n − n′ ∈ n and rγn ∈ n . in this case we write n ≤ m . remark 4.2. (i) clearly {0} and m are two trivial rγ-submodules of rγ-module m , which is called trivial rγ-submodules. (ii) consider r as rγ-module. clearly, every ideal of γ-ring r is submodule, of r as rγ-module. theorem 4.3. let m be an rγ-module. if n is a subgroup of m , then the factor group m/n is an rγ-module under the mapping . : r × γ × m/n −→ m/n is defined (r, γ, m + n ) 7−→ (r.γ.m) + n. proof. straight forward. theorem 4.4. let n be an rγ-submodules of m . then every rγ-submodule of m/n is of the form k/n , where k is an rγ-submodule of m containing n . proof. for all x, y ∈ k, x + n, y + n ∈ k/n ; (x + n ) − (y + n ) = (x − y) + n ∈ k/n , we have x − y ∈ k, and ∀r ∈ r ∀γ ∈ γ, ∀x ∈ k, we have rγ(x + n ) = rγx + n ∈ k/n ⇒ rγx ∈ k. then k is a rγ-submodule m . conversely,it is easy to verify that n ⊆ k ≤ m then k/n is rγ-submodule of m/n . this complete the proof. � proposition 4.5. let m be an rγ-module and i be an ideal of r. let x be a nonempty subset of m . then iγx = { ∑n i=1aiγixi | ai ∈ irγ i ∈ γ, xi ∈ x, n ∈ n} is an rγ-submodule of m . proof. (i) for elements x = ∑n i=1aiαixi and y = ∑m j=1xa′j βj yj of iγx, we have x − y = ∑m+n k=1 bkγkzk ∈ iγx. now we consider the following cases: case (1): if 1 ≤ k ≤ n, then bk = ak, γk = αk, zk = xk. ratio mathematica 20, 2010 135 case(2): if n + 1 ≤ k ≤ m + n, then bk = −a′k−n, γk = βk−n, zk = yk−n. also (ii) ∀r ∈ r, ∀γ ∈ γ, ∀a = ∑n i=1 aiγixi ∈ iγx, we have rγx = ∑n i=1 rγ(aiγixi) =∑n i=1(rγai)γixi. thus iγx is an rγ-submodule of m. � corollary 4.6. if m is an rγ-module and s is a submodule of m . then rγs is an rγ-submodule of m . let n ≤ m . define n : m = {r ∈ r|rγm ∀γ ∈ γ ∀m ∈ m}. it is easy to see that n : m is an ideal of γ ring r. theorem 4.7. let m be an rγ-module and i be an ideal of r. if i ⊆ (0 : m ), then m is an (r/i)γ-module. proof. since r/i is γ-ring, def inethemapping• : (r/i) × γ × m −→ m by (r + i, γ, m) 7−→ rγm.. the mapping • is well-defined since i ⊆ (0 : m ). now it is straight forward to see that m is an (r/i)γ-module. � proposition 4.8. let r be a γ-ring, i be an ideal of r, and (m, .) be a rγ-module. then m/(iγm ) is an (r/i)γmodule. proof. first note that m/(iγm ) is an additive subgroup of m . consider the mapping γ • (m + iγm ) = r.γ.m + iγm )n owitisstraightf orwardtoseethatmisan(r/i)γ-module. � proposition 4.9. let m be an rγ-module and n ≤ m , m ∈ m . then (n : m) = {a ∈ r | aγm ∈ n ∀γ ∈ γ} is a left ideal of r. proof. obvious. proposition 4.10. if n and k are rγ-submodules of a rγ-module m and if a, b are nonempty subsets of m then: (i) a ⊆ b implies that (n : b) ⊆ (n : a); (ii) (n ∩ k : a) = (n : a) ∩ (k : a); (iii) (n : a) ∩ (n : b) ⊆ (n : a + b), moreover the equality hold if 0m ∈ a ∩ b. ratio mathematica 20, 2010 136 proof. (i) easy. (ii) by definition, if r ∈ r, then r ∈ (n ∩ k : a) ⇐⇒ ∀a ∈ ar ∈ (n ∩ k : a) ⇐⇒ ∀γ ∈ γ; rγa ∈ n ∩ k ⇐⇒ r ∈ (n : a) ∩ k : a). (iii) if r ∈ (n : a) ∩ (n : b). then ∀γ ∈ γ, ∀a ∈ a, ∀b ∈ b, rγ(a + b) ∈ n and r ∈ (n : a + b). conversely, 0m ∈ a + b =⇒ a ∪ b ⊆ a + b =⇒ (n : a + b) ⊆ (n : a ∪ b) by(i). again by using a, b ⊆ a ∪ b we have (n : a ∪ b) ⊆ (n : a) ∩ (n : b). � definition 4.11. let m be an rγ-module and ∅ 6= x ⊆ m . then the generated rγ-submodule of m , denoted by < x > is the smallest rγ-submodule of m containing x, i.e. < x >= ∩{n|n ≤ m}, x is called the generator of < x >; and < x > is finitely generated if |x| < ∞. if x = {x1, ...xn} we write < x1, ..., xn > instead < {x1, ..., xn} >. in particular, if x = {x} then < x > is called the cyclic submodule of m , generated by x. lemma 4.12. suppose that m is an rγ-module. then (i) let {mi}i∈i be a family of rγ-submodules m . then ∩mi is the largest rγ-submodule of m , such that contained in mi, for all i ∈ i. (ii) if x is a subset of m and |x| < ∞. then < x >= { ∑m i=1 nixi + ∑k j=1 rjγjxj|k, m ∈ n, ni ∈ z, γj ∈ γ, rj ∈ r, xi, xj ∈ x} . proof. (i) it is easy to verify that ∩i∈i mi ⊆ mi is a rγ-submodule of m . now suppose that n ≤ m and ∀ i ∈ i, n ⊆ mi, then n ⊆ ∩mi. (ii) suppose that the right hand in (b) is equal to d. first, we show that d is an rγ-submodule containing x. x ⊆ d and difference of two elements of d is belong to d and ∀r ∈ r ∀γ ∈ γ, ∀a ∈ d we have rγa = rγ( ∑m i=1 nixi + ∑k j=1 rjγjxj) = ∑m i=1 ni(rγxi) + ∑k j=1(rγrj)γjxj ∈ d. also, every submodule of m containing x, clearly contains d. thus d is the smallest ratio mathematica 20, 2010 137 rγ-submodules of m , containing x. therefore < x >= d. � for n, k ≤ m , set n + k = {n + k|n ∈ n, k ∈ k}. then it is easy to see that m + n is an rγ-submodules of m , containing both n and k. then the next result immediately follows. lemma 4.13. suppose that m is an rγ-module and n, k ≤ m . then n + k is the smallest submodule of m containing n and k. set l(m ) = {n|n ≤ m}. define the binary operations ∨ and ∧ on l(m ) by n ∨ k = n + k andn ∧ k = n ∩ k. in fact (l(m ), ∨, ∧) is a lattice. then the next result immediately follows from lemmas 4.12. 4.13. theorem 4.13. l(m ) is a complete lattice. 5 homomorphisms gamma modules in this section we study the homomorphisms of gamma modules. in particular we investigate the behavior of submodules od gamma modules under homomorphisms. definition 5.1. let m and n be arbitrary rγ-modules. a mapping f : m −→ n is a homomorphism of rγ-modules ( or an rγ-homomorphisms) if for all x, y ∈ m and ∀r ∈ r, ∀γ ∈ γ we have (i) f (x + y) = f (x) + f (y); (ii) f (rγx) = rγf (x). a homomorphism f is monomorphism if f is one-to-one and f is epimorphism if f is onto. f is called isomorphism if f is both monomorphism and epimorphism. we denote the set of all rγ-homomorphisms from m into n by homrγ (m, n ) or shortly by ratio mathematica 20, 2010 138 homrγ (m, n ). in particular if m = n we denote hom(m, m ) by end(m ). remark 5.2. if f : m −→ n is an rγ-homomorphism, then kerf = {x ∈ m|f (x) = 0}, imf = {y ∈ n|∃x ∈ m ; y = f (x)} are rγ-submodules of m . example 5.3. for all rγ-modules a, b, the zero map 0 : a −→ b is an rγ-homomorphism. example 5.4. let r be a γ-ring. fix r0 ∈ γ and consider the mapping φ : r[x] −→ r[x] by f 7−→ f γ0x. then φ is an rγ-module homomorphism, because ∀r ∈ r, ∀γ ∈ γ and ∀f, g ∈ r[x] : φ(f + g) = (f + g)γ0x = f γ0x + gγ0x = φ(f ) + φ(g) and φ(rγf ) = rγf γ0x = rγφ(f ). example 5.5. if n ≤ m , then the natural map π : m −→ m/n with π(x) = x + n is an rγ-module epimorphism with kerπ = n . proposition 5.6. if m is unitary rγ-module and end(m ) = {f : m −→ m|f is rγ − homomorphism}. then m is an end(m )γ-module. proof. it is well known that end(m ) is an abelian group with usual addition of functions. define the mapping . : end(m ) × γ × m −→ m (f, γ, m) 7−→ f (1.γ.m) = 1γf (m), where 1 is the identity map. now it is routine to verify that m is an end(m )γ-module.� lemma 5.7. let f : m −→ n be an rγ-homomorphism. if m1 ≤ m and n1 ≤. then (i) kerf ≤ m , imf ≤ n ; (ii) f (m1) ≤ imf ; (iii) kerf−1(n1) ≤ m . ratio mathematica 20, 2010 139 example 5.8. consider l(m ) the lattice of rγ-submodules of m . we know that (l(m ), +) is a monoid with the sum of submodules. then l(m ) is rγ-semimodule under the mapping . : r × γ × t −→ t , such that (r, γ, n ) 7−→ r.γ.n = rγn = {rγn|n ∈ n}. example 5.9. let θ : r −→ s be a homomorphism of γ-rings and m be an sγ-module. then m is an rγ-module under the mapping • : r × γ × m −→ m by (r, γ, m) 7−→ r • γ • m = θ(r). moreover if m is an sγ-module then m is a rγ-module for r ⊆ s. example 5.10. let (m, .) be an rγ-module and a ⊆ m . letting m a = {f|f : a −→ m is a map}. then m a is an rγ-module under the mapping ◦ : r × γ × m a −→ m a defined by (r, γ, f ) 7−→ r ◦ γ ◦ f = rγf (a), since m a is an additive group with usual addition of maps. example 5.11. let(m, .) and (n, •) be rγ-modules. then hom(m, n ) is a rγ-module, under the mapping ◦ : r × γ × hom(m, n ) −→ hom(m, n ) (r, γ, α) 7−→ r ◦ γ ◦ α, where (r • γ • α)(m) = rγα)(m). example 5.12. let m be a left rγ-module and right sγ-module. if n be an rγ-module, then (i) hom(m, n ) is a left sγ-module. indeed ◦ : s × γ × hom(m, n ) −→ hom(m, n ) (s, γ, α) −→ s ◦ γ ◦ α : m −→ n m 7−→ α(mγs) (ii) hom(n, m ) is right sγ-module under the mapping ratio mathematica 20, 2010 140 ◦ : hom(n, m ) × γ × s −→ hom(n, m ) (α, γ, s) 7−→ α ◦ γ ◦ s : n −→ m n 7−→ α(n).γ.s example 5.13. let m be a left rγ-module and right sγ-module and α ∈ end(m ) then α induces a right s[t]γ-module structure on m with the mapping ◦ : m × γ × s[t] −→ m (m, γ, ∑n i=0 sit i) 7−→ m ◦ γ ◦ ( ∑n i=0 sit i) = ∑n i=0(mγsi)α i proposition 5.14. let m be a rγ-module and s ⊆ m . then sγm = { ∑ siγiai | si ∈ s, ai ∈ m, γi ∈ γ} is an rγ-submodule of m . proof. consider the mapping ◦ : r × γ × (sγm ) −→ sγm (r, γ, ∑n i=1 siγiai) 7−→ ∑n i=1 siγi(rγai). now it is easy to check that sγm is a rγ-submodule of m . example 5.16. let (r, .) be a γ-ring. let z2, the cyclic group of order 2. for a nonempty subset a, set hom(r, ba) = {f : r −→ ba}. clearly (hom(r, ba), +) is an abelian group. consider the mapping ◦ : r × γ × hom(r, ba) −→ hom(r, ba) that is defined (r, γ, f ) 7−→ r ◦ γ ◦ f, where (r ◦ γ ◦ f )(s) : a −→ b is defied by (r ◦ γ ◦ f (s))(a) = f (sγr)(a). now it is easy to check that hom(r, ba) is an γ-ring. example 5.17. let r and s be γ-rings and ϕ : r −→ s be a γ-rings homomorphism. then every sγ-module m can be made into an rγ-module by defining ratio mathematica 20, 2010 141 rγx (r ∈ r, γ ∈ γ, x ∈ m ) to be ϕ(r)γx. we says that the rγ-module structure m is given by pullback along ϕ. example 5.18. let ϕ : r −→ s be a homomorphism of γ-rings then (s, .) is an rγ-module. indeed ◦ : r × γ × s −→ s (r, γ, s) 7−→ r ◦ γ ◦ s = ϕ(r).γ.s example 5.19. let (m, +) be an rγ-module. define the operation ◦ on m by a ⊕ b = b.a. then (m, ⊕) is an rγ-module. proposition 5.20. let r be a γ-ring. if f : m −→ n is an rγ-homomorphism and c ≤ kerf , then there exists an unique rγ-homomorphism f̄ : m/c −→ n , such that for every x ∈ m ; kerf̄ = kerf /c and imf̄ = imf and f̄ (x + c) = f (x), also f̄ is an rγ-isomorphism if and only if f is an rγ-epimorphism and c = kerf . in particular m/kerf ∼= imf . proof. let b ∈ x + c then b = x + c for some c ∈ c, also f (b) = f (x + c). we know f is rγ-homomorphism therefore f (b) = f (x + c) = f (x) + f (c) = f (x) + 0 = f (x) (since c ≤ kerf ) then f̄ : m/c −→ n is well defined function. also ∀ x + c, y + c ∈ m/c and ∀ r ∈ r, γ ∈ γ we have (i) f̄ ((x + c) + (y + c)) = f̄ ((x + y) + c) = f (x + y) = f (x) + f (y) = f̄ (x + c) + f̄ (y + c). (ii) f̄ (rγ(x + c)) = f̄ (rγx + c) = f (rγx) = rγf (x) = rγf̄ (x + c). then f̄ is a homomorphism of rγ-modules, also it is clear imf̄ = imf and ∀(x + c) ∈ kerf̄ ; x + c ∈ kerf̄ ⇔ f̄ (x + c) = 0 ⇔ f (x) = 0 ⇔ x ∈ kerf then kerf̄ = kerf /c. then definition f̄ depends only f , then f̄ is unique. f̄ is epimorphism if and only if f is epimorphism. f̄ is monomorphism if and only if kerf̄ be trivial rγ-submodule of m/c. in actually if and only if kerf = c then m/kerf ∼= imf .� ratio mathematica 20, 2010 142 corollary 5.21. if r is a γ-ring and m1 is an rγ-submodule of m and n1 is rγ-submodule of n , f : m −→ n is a rγ-homomorphism such that f (m1) ⊆ n1 then f make a rγ-homomorphism f̄ : m/m1 −→ n/n1 with operation m + m1 7−→ f (m) + n1. f̄ is rγ-isomorphism if and only if imf + n1 = n, f −1(n1) ⊆ m1. in particular, if f is epimorphism such that f (m1) = n1, kerf ⊆ m1 then f is a rγ-isomorphism. proof. we consider the mapping m −→f n −→π n/n1. in this case; m1 ⊆ f−1(n1) = kerπf (∀m1 ∈ m1, f (m1) ∈ n1 ⇒ πf (m1) = 0 ⇒ m1 ∈ kerπf ). now we use proposition 5.20 for map πf : m −→ n/n1 with function m 7−→ f (m) + n1 and submodule m1 of m . therefore, map f̄ : m/m1 −→ n/n1 that is defined m + m 7−→ f (m) + n1 is a rγ-homomorphism. it is isomorphism if and only if πf is epimorphism, m1 = kerπf . but condition will satisfy if and only if imf + n1 = n , f −1(n1) ⊆ m1. if f is epimorphism then n = imf = imf + n1 and if f (m1) = n1 and kerf ⊆ m1 then f−1(n1) ⊆ m1 so f̄ is isomorphism.� proposition 5.22. let b, c be rγ-submodules of m . (i) there exists a rγ-isomorphism b/(b ∩ c) ∼= (b + c)/c. (ii) if c ⊆ b, then b/c is an rγ-submodule of m/c and there is an rγ-isomorphism (m/c)/(b/c) ∼= m/b . proof. (i) combination b −→j b + c −→π (b + c)/c is an rγ-homomorphism with kernel= b ∩ c, because kerπj = {b ∈ b|πj(b) = 0(b+c)/c} = {b ∈ b|π(b) = c} = {b ∈ b|b + c = c} = {b ∈ b|b ∈ c} = b ∩ c therefore, in order to proposition 5.20., b/(b ∩ c) ∼= im(πj)(?), every element of (b + c)/c is to form (b + c) + c, thus (b + c) + c = b + c = πj(b) then πj is epimorphism and imπj = (b + c)/c in attention (?), b/(b ∩ c) ∼= (b + c)/c. (ii) we consider the identity map i : m −→ m , we have i(c) ⊆ b, then in order to ratio mathematica 20, 2010 143 apply proposition 5.21. we have rγ-epimorphism ī : m/c −→ m/b with ī(m + c) = m + b by using (i). but we know b = ī(m + c) if and only if m ∈ b thus ker ī = {m + c ∈ m/c|m ∈ b} = b/c then kerī = b/c ≤ m/c and we have m/b = imī ∼= (m/c)/(b/c).� let m be a rγ-module and {ni|i ∈ ω} be a family of rγ-submodule of m . then ∩i∈ωni is a rγ-submodule of m which, indeed, is the largest rγ-submodule m contained in each of the ni. in particular, if a is a subset of a left rγ-modulem then intersection of all submodules of m containing a is a rγ-submodule of m , called the submodule generated by a. if a generates all of the rγ-module, then a is a set ofgenerators for m . a left rγ-module having a finite set of generators is finitely generated. an element m of the rγ-submodule generated by a subset a of a rγ-module m is a linear combination of the elements of a. if m is a left rγ-module then the set ∑ i∈ω ni of all finite sums of elements of ni is an rγ-submodule of m generated by ∪i∈ωni. rγ-submodule generated by x = ∪i∈ωni is d = { ∑s i=1 riγiai + ∑t j=1 njbj|ai, bj ∈ x, ri ∈ r, nj ∈ z, γi ∈ γ} if m is a unitary rγ-module then d = rγx = { ∑s i=1 riγiai|ri ∈ r, γi ∈ γ, ai ∈ x}. example 5.23. let m, n be rγ-modules and f, g : m −→ n be rγ-module homomorphisms. then k = {m ∈ m | f (m) = g(m)} is rγ-submodule of m . example 5.24. let m be a rγ-module and let n, n ′ be rγsubmodules of m . set a = {m ∈ m | m + n ∈ n ′ f or some n ∈ n} is an rγ-module of m containing n ′. proposition 5.25. let (m, ·) be an rγmodule and m generated by a. then there exists an rγ-homomorphism r (a) −→ m , such that f 7−→ ∑ a∈a,a∈supp(f ) f (a) · γ · a. remark 5.26. let r be a γring and let {(mi, oi)|i ∈ ω} be a family of left rγmodules. then ×i∈ωmi, the cartesian product of mi’s also has the structure of a left rγ-module under componentwise addition and mapping ratio mathematica 20, 2010 144 · : r × γ × (×mi) −→ ×mi (r, γ, {mi}) −→ r · γ · {mi} = {roiγoimi}ω. we denote this left rγ-module by ∏ i∈ω mi. similarly,∑ i∈ω mi = {{mi} ∈ ∏ mi|mi = 0 for all but finitely many indices i} is a rγ-submodule of ∏ i∈ω mi. for each h in ω we have canonical rγhomomorphisms πh : ∏ mi −→ mh and λh : mh −→ ∑ mi is defined respectively by πh :< mi >7−→ mh and λ(mh) =< ui >, where ui =   0 i 6= h mh i = h the rγ-module ∏ mi is called the ( external)direct product of the rγmodules mi and the rγmodule ∑ mi is called the (external) direct sum of mi. it is easy to verify that if m is a left rγ-module and if {mi|i ∈ ω} is a family of left rγ-modules such that, for each i ∈ ω, we are given an rγ-homomorphism αi : m −→ mi then there exists a unique rγhomomorphism α : m −→ ∏ i∈ω mi such that αi = απi for each i ∈ ω. similarly, if we are given an rγ-homomorphism βi : mi −→ m for each i ∈ ω then there exists an unique rγhomomorphism β : ∑ i∈ω mi −→ m such that βi = λiβ for each i ∈ ω . remark 5.27. let m be a left rγ-module. then m is a right r op γ -module under the mapping ∗ : m × γ × rop −→ m (m, γ, r) 7−→ m ∗ γ ∗ r = rγm. definition 5.28. a nonempty subset n of a left rγ-module m is subtractive if and only if m + m′ ∈ n and m ∈ n imply that m′ ∈ n for all m, m′ ∈ m . similarly, n is strong subtractive if and only if m + m′ ∈ n implies that m, m′ ∈ n for all m, m′ ∈ m . ratio mathematica 20, 2010 145 remark 5.29. (i) clearly, every submodule of a left rγ-module is subtractive. indeed, if n is a rγ-submodule of a rγ-module m and m ∈ m, n ∈ n are elements satisfying m + n ∈ n then m = (m + n) + (−n) ∈ n . (ii) if n, n ′ ⊆ n are rγ-submodules of an rγ-module m , such that n ′ is a subtractive rγ-submodule of n and n is a subtractive rγ-submodule of m then n ′ is a subtractive rγ-module of m . note. if {mi|i ∈ ω} is a family of (resp. strong) subtractive rγ-submodule of a left rγ-module m then ∩i∈ωmi is again (resp. strong) subtractive. thus every rγ -submodule of a left rγ-module m is contained in a smallest (resp. strong) subtractive rγ-submodule of m , called its (resp. strong) subtractive closure in m . proposition 5.30 let r be a γ-ring and let m be a left rγ -module. if n, n ′ and n ′ ′ ≤ m are submodules of m satisfying the conditions that n is subtractive and n ′ ⊆ n , then n ∩ (n ′ + n ′′) = n ′ + (n ∩ n ′′). proof. let x ∈ n ∩ (n ′ + n ′′). then we can write x = y + z, where y ∈ n ′ and z ∈ n ′′. by n ′ ⊆ n , we have y ∈ n and so, z ∈ n , since n is subtractive. thus x ∈ n ′ + (n ∩ n ′′), proving that n ∩ (n ′ + n ′′) ⊆ n ′ + (n ∩ n ′′). the reverse containment is immediate.� proposition 5.31. if n is a subtractive rγ-submodule of a left rγ-module m and if a is a nonempty subset of m then (n : a) is a subtractive left ideal of r. proof. since the intersection of an arbitrary family of subtractive left ideals of r is again subtractive, it suffices to show that (n : m) is subtractive for each element m. let a ∈ r and b ∈ (n : m ) (for γ ∈ γ ) satisfy the condition that a + b ∈ (n : m ). then aγm + bγm ∈ n and bγm ∈ n so aγm ∈ n , since n is subtractive. thus a ∈ (n : m ).�. proposition 5.32. if i is an ideal of a γ-ring r and m is a left rγ-module. then ratio mathematica 20, 2010 146 n = {m ∈ m | iγm = {0}} is a subtractive rγ-submodule of m. proof. clearly, n is an rγ-submodule of m . if m, m ′ ∈ m satisfy the condition that m and m + m′ belong to n then for each r ∈ i and for each γ ∈ γ we have 0 = rγ(m + m′) = rγm + rγm′m′ = rγm′, and hence m′ ∈ n . thus n is subtractive. � proposition 5.33. let (r, +, ·) be a γ-ring and let m be an rγ-module and there exists bijection function δ : m −→ r. then m is a γ-ring and mγ-module. proof. define ◦ : m × γ × m −→ m by (x, γ, y) 7−→ x ◦ γ ◦ y = δ−1(δ(x) · γδ(y)). it is easy to verify that r is a γring. if m is a set together with a bijection function δ : x −→ r then the γ-ring structure on r induces a γ-ring structure (m, ⊕, �) on x with the operations defined by x ⊕ y = δ−1(δ(x) + δ(y)) and x � γ � y = δ−1(δ(x) · γ · δ(y)).� acknowledgements the first author is partially supported by the ”research center on algebraic hyperstructures and fuzzy mathematics, university of mazandaran, babolsar, iran”. references [1] f.w.anderson ,k.r.fuller , rings and categories of modules, springer verlag ,new york ,1992. [2] w.e.barens,on the γ-ring of nobusawa, pacific j.math.,18(1966),411-422. [3] j.s.golan,semirinngs and their applications.[4] t.w.hungerford ,algebra.[5] j.luh,on the theory of simple gamma rings, michigan math .j.,16(1969),65-75.[6] n.nobusawa ,on a generalization of the ring theory ,osaka j.math. 1(1964),81-89. ratio mathematica 20, 2010 147 enlarging hv-structures a hyperoperation defined on a groupoid equipped with a map thomas vougiouklis democritus university of thrace, school of education 681 00 alexandroupolis, greece e-mail: tvougiou@eled.duth.gr abstract the hv-structures are hyperstructures where the equality is replaced by the non-empty intersection. the fact that this class of the hyperstructures is very large, one can use it in order to define several objects that they are not possible to be defined in the classical hypergroup theory. in the present paper we introduce a kind of hyperoperations which are defined on a set equipped with an operation or a hyperoperation and a map on itself. ams subject classification: 20n20 key words: hyperstructures, hv-structures. 1. introduction the object of this paper is the hyperstructures called hvstructures introduced in 1990 [5], which satisfy the weak axioms where the non-empty intersection replaces the equality. recall some basic definitions: definitions 1. in a set h equipped with a hyperoperation ⋅:h×h→p(h), we abbreviate by wass the weak associativity: (xy)z∩x(yz)≠∅, ∀x,y,z∈h and by cow the weak commutativity: xy∩yx≠∅, ∀x,y∈h. the hyperstructure (h,⋅) is called hv-semigroup if it is wass, is called hv-group if it is reproductive hv-semigroup. the hyperstructure (r,+,⋅) is called hv-ring if (+) and (⋅) are wass, the reproduction axiom is valid for (+) and (⋅) is weak distributive with respect to (+): x(y+z)∩(xy+xz)≠∅, (x+y)z∩(xz+yz)≠∅, 25 ∀x,y,z∈r. hv-modulus and hv-vector spaces are also defined in a similar way. for more definitions, results and applications on hv-structures, see books [6,2] and on some papers such as [3-11]. a special class [6]: an hv-structure is called very thin iff all its hyperoperations are operations except one, which all hyperproducts are singletons except only one, which has cardinality more than one. the fundamental relations β*, γ* and ε* are defined, in hvgroups, hv-rings and hv-vector spaces, respectively, as the smallest equivalences so that the quotient would be group, ring and vector space, respectively (see [1,6]). the way to find the fundamental classes is given by analogous theorems to the following [5,6,7]: theorem. let (h,⋅) be hv-group and let us denote by u the set of all finite products of elements of h. we define the relation β in h as follows: xβy iff {x,y}⊂u where u∈u. then the fundamental relation β* is the transitive closure of β. proof. the main point is: take x,y such that {x,y}⊂u∈u and any hyperproduct where one of the elements x,y, is used. then, if this element is replaced by the other, the new hyperproduct is inside the same fundamental class where the first hyperproduct is. therefore, if the hyperproducts of the above β-classes are products, then, they are fundamental classes. analogous remarks for the relations γ*, in hvrings, and ε*, in hv-vector spaces, are also applied. an element is called single if its fundamental class is singleton. the fundamental relations are used for general definitions. thus, to define the hv-field the γ* is used: an hv-ring (r,+,⋅) is called hv-field if r/γ* is a field [5], and in the sequence the general hvvector space is defined. let (h,⋅), (h,*) be hv-semigroups defined on the same set h. (⋅) is called smaller than (*), and (*) greater than (⋅), iff there exists an automorphism f∈aut(h,*) such that xy⊂f(x*y), ∀x,y∈h. then we 26 write ⋅≤* and we say that (h,*) contains (h,⋅). if (h,⋅) is a structure then it is called basic structure and (h,*) is called hb-structure. theorem. greater hyperoperations of the ones which are wass or cow, are also wass or cow, respectively. remark 2. the weak axioms lead to a great number of hyperoperations and these hyperoperations define hyperstructures which can be now studied in detail and, in any case, they have a substance; hence they can be considered as hyperstructures with interesting properties. these are many hyperoperations which, in the past, were unlikely to be considered because not even one property was valid in them. we can see that the hyperoperations introduced here are associative only in very special cases and before 1990 such hyperoperations could hardly be considered, even though they appeared in the research. nevertheless, the created theory can now give results and discover new properties of the obtained hyperstructure. thus, algebraic domains reveal constructions which seem to be chaotic. even more so, in certain cases, some of these hyperstructures contain well known structures or hyperstructures, see also [11,12]. this remark follows that constructions and hyper-constructions are needed to be enlarged or to become smaller and we can do this: definitions 3. let (h,⋅) be a hypergroupoid. we say that remove h∈h, if we consider the restriction of (⋅) in h-{h}. we say that h∈h absorbs h∈h if we replace h by h. we say that h∈h merges with h∈h, if we take as product of any x∈h by h, the union of the results of x with both h, h, and consider h and h as one class, with representative h. most of these constructions are needed in the representation theory. representations of hv-groups can be considered either by generalized permutations or by hv-matrices [6]. the representation problem by hv-matrices is the following: hv-matrix is a matrix with entries of an hv-ring. the hyperproduct of hv-matrices a=(aij) and b=(bij), of type m×n and n×r, respectively, is a set of m×r hv-matrices: 27 a⋅b = (aij)⋅(bij) = { c = (cij)  cij ∈ ⊕σaik⋅bkj }, where ⊕ denotes the n-ary circle hyperoperation on the hyperaddition. definition 4. let (h,⋅) be hv-group, take a hv-ring (r,+,⋅) and a set mr ={ (aij)aij∈r }, then any map t:h→mr: h→t(h) with t(h1h2)∩t(h1)t(h2)≠∅, ∀h1,h2∈h, is a hv-matrix representation. if t(h1h2)⊂t(h1)t(h2), then t is an inclusion, if t(h1h2)=t(h1)t(h2), then t is a good and an induced representation for the hypergroup algebra is obtained. in the same attitude recently we defined, using hyperstructure theory, hyperoperations on any type of matrices: definition 5 [12]. let a=(aij)∈mm×n be matrix and s,t∈n with 1≤s≤m, 1≤t≤n. then helix-projection is a map st:mm×n→ms×t:a→ast=(aij), where ast has entries aij = { ai+κs,j+λt 1≤i≤s, 1≤j≤t and κ,λ∈n, i+κs≤m, j+λt≤n } let a=(aij)∈mm×n, b=(bij)∈mu×v be matrices, s=min(m,u), t=min(n,v). we define a hyper-addition, called helix-addition, as follows ⊕:mm×n×mu×v→p(ms×t):(a,b)→a⊕b=ast+bst=(aij)+(bij)⊂ms×t where (aij)+(bij)= {(cij)=(aij+bij) aij∈aij and bij∈bij)}. let a=(aij)∈mm×n and b=(bij)∈mu×v be matrices and s=min(n,u). we define a hyper-multiplication, called helix-multiplication, as follows ⊗:mm×n×mu×v→p(mm×v):(a,b)→a⊗b=ams⋅bsv=(aij)⋅(bij)⊂mm×v where (aij)⋅(bij)= {(cij)=(∑aitbtj) aij∈aij and bij∈bij)}. the helix-addition is commutative, is wass, not associative. the helix-multiplication is wass, not associative and it is not distributive, not even weak, to the helix-addition. if all used matrices are of the same type, then the inclusion distributivity, is valid. from the definition of representations by hv-matrices, we have two difficulties. the first one is to find an appropriate hv-ring and the 28 second one is to find an appropriate set of hv-matrices. however, with the above hyper-multiplication we can use subsets of matrices of type mm×n with m≠n. thus, the representation problem is reduced, as in the classical theory, in searching appropriate sets from usual matrices. this is so, because we have now a hyperalgebra over non-square matrices. 2. new hyperoperations we will define a hyperoperation in a groupoid equipped with a map f on it. the map plays crucial role so the hyperoperation is called map and it is denoted by ∂f, because the motivation to obtain this is the property which the ‘derivative’ has on the product of functions. however, since there is no confusion, we will write simply theta ∂. definition 6. let (g,⋅) be a groupoid (respectively, hypergroupoid) and f:g→g be any map. we define a hyperoperation (∂), we call it theta-operation, on g as follows x∂y = {f(x)⋅y, x⋅f(y) } (respectively, x∂y = (f(x)⋅y)∪(x⋅f(y)) if (⋅) is commutative then (∂) is also commutative. if (⋅) is a cow hyperoperation, then (∂) is also cow hyperoperation. remark. one can use instead of single valued map f, a multivalued map as well. we will not consider this problem here. remark. motivation for this definition was the map ‘derivative’ where only the multiplication of functions can be used. in other words, if we ‘do not know’ the addition of functions. therefore, for any functions s(x), t(x), we have s∂t={s′t, st′} where (′) denotes the derivative. properties 7. if (g,⋅) is a semigroup then: (a) for every f, the hyperoperation (∂) is wass. (b) if f is homomorphism then, again, (∂) is wass. (c) if f is homomorphism and projection, or idempotent, i.e. f2 =f, then (∂) is associative. proof. 29 (a) for all x,y,z in g we have (x∂y)∂z = {f(x)⋅y, x⋅f(y)}∂z = = { f(f(x)⋅y)⋅z, (f(x)⋅y)⋅f(z), f(x⋅f(y))⋅z, (x⋅f(y))⋅f(z) } = = { f(f(x)⋅y)⋅z, f(x)⋅y⋅f(z), f(x⋅f(y))⋅z, x⋅f(y)⋅f(z) } x∂(y∂z) = x∂{f(y)⋅z, y⋅f(z)} = = { f(x)⋅(f(y)⋅z), x⋅f(f(y)⋅z), f(x)⋅(y⋅f(z)), x⋅f(y⋅f(z)) } = = { f(x)⋅f(y)⋅z, x⋅f(f(y)⋅z), f(x)⋅y⋅f(z), x⋅f(y⋅f(z)) } therefore (x∂y)∂z∩x∂(y∂z) = { f(x)⋅y⋅f(z) }≠∅, so (∂) is wass. (b) if f is homomorphism then we obtain (x∂y)∂z = { f(f(x))⋅f(y)⋅z, f(x)⋅y⋅f(z), f(x)⋅f(f(y))⋅z, x⋅f(y)⋅f(z) } x∂(y∂z) = { f(x)⋅f(y)⋅z, x⋅f(f(y))⋅f(z), f(x)⋅y⋅f(z), x⋅f(y)⋅f(f(z)) } so, again (x∂y)∂z∩x∂(y∂z)={f(x)⋅y⋅f(z)}≠∅ and (∂) is wass. (c) if f is homomorphism and projection then we have (x∂y)∂z = { f(x)⋅f(y)⋅z, f(x)⋅y⋅f(z), x⋅f(y)⋅f(z) }=x∂(y∂z). therefore, (∂) is an associative hyperoperation. notice that only projection without homomorpthism does not give the associativity. commutativity does not improve the results. 3. properties and characteristic elements. we will discuss now some properties in the general case where (g,⋅) be a groupoid and f:g→g be a map. properties 8. reproductivity. for the reproductivity we must have x∂g =∪g∈g{f(x)⋅g, x⋅f(g)}=g and g∂x =∪g∈g{f(g)⋅x, g⋅f(x)}=g. thus, if (⋅) is reproductive then (∂) is also reproductive, because ∪g∈g{f(x)⋅g}=g and ∪g∈g{g⋅f(x)}=g. 30 commutativity. if (⋅) is commutative then x∂y={f(x)⋅y, x⋅f(y)}=y∂x, so (∂) is commutative. if f is into zg={z∈g z⋅g=g⋅z,∀g∈g}, the centre of g, then (∂) is a commutative hyperoperation. if (g,⋅) is a cow hypergroupoid then, obviously (∂) is a cow hypergroupoid. unit elements. in order to have a right unit element u we must have x∂u={f(x)⋅u, x⋅f(u)}∋x. but, the unit must not depend on the f(x), so we must have f(u)=e, where e be unit in (g,⋅) which must be a monoid. the same it is obtained for the left units. therefore, the elements of the kernel of f, i.e. u for which f(u)=e, are the units of (g,∂). inverse elements. let u be a unit in (g,∂), then (g,⋅) is a monoid with unit e and f(u)=e. for given x in order to have an inverse element x′ with respect to u, we must have x∂x′= {f(x)⋅x′, x⋅f(x′)}∋u and x′∂x={f(x′)⋅x, x′⋅f(x)}∋u. so the only cases, which do not depend on the image f(x′), are x′ = (f(x))-1u and x′ = u(f(x))-1 the right and left inverses, respectively. we have two-sided inverses iff f(x)u = uf(x). for example, if u belongs to the centre of g. in some cases, some elements may have a second inverse. proposition 9. let (g,⋅) be a group and f(x)=a, a constant map on g. then (g,∂)/β* is a singleton. proof. for all x in g we can take the hyperproduct of the elements, a1 and a-1x a-1∂(a-1⋅x) = {f(a-1)⋅a-1⋅x, a-1⋅f(a-1⋅x)} = {x,a}. thus xβa, ∀x∈g, so β*(x)=β*(a) and (g,∂)/β* is singleton. q.e.d. remark. if (g,⋅) be a group and f(x)=e, then we obtain x∂y={x,y} which is the smallest incidence hyperoperation. remark. every f:g→g defines a partition of g by setting two elements x,y in the same class iff f(x)=f(y), we shall call this partition f-partition and we will denote the class of x by f[x]. so, in the above proposition, we have f[x] = g = β*(x) for all x in g. 31 proposition 10. let (g,⋅) be a group, e the unit, and f homomorphism, then for (g,∂), we have xβf(x). proof. indeed e∂x = {f(e)⋅x, e⋅f(x)} = {x, f(x)}. q.e.d. obviously we have x β f(x) β f(f(x)) β... theorem 11. let (g,⋅) be a group and f be an homomorphism, then f[x] ⊂ β*(x) for all x in g. proof. let y∈f[x], then f(y)=f(x) but from proposition10, we have xβf(x) = f(y)βy, so xβ*y. q.e.d. 4. special cases and applications in this paragraph we present some applications and we give some examples in order to see that a large field of research is open. application 12. taking the application on the derivative, consider all polynomials of first degree gi(x) = aix+bi. we have g1∂g2 = {a1a2x+ a1b1, a1a2x+b1b2}, so it is a hyperoperation inside the set of first degree polynomials. moreover all polynomials x+c, where c be a constant, are units. application 13. if r+ be the set of positive reals and a∈r+, then we take the exponential map x→xa. the theta-operation takes the form x∂y = { xay, xya} for all x,y in r+. the only one unit is the 1. in order to find the inverses x′, of the element x∈r+, we must have x∂x′={xax′, x(x′)a}∋1. from which we obtain that for every element x, there are two inverses, the x-a and x-1/a. example 14. in the group (z5-{0},⋅) we consider the map f: 1→1, 2→2, 3→3, 4→2. then we obtain the multiplicative table ∂ 1 2 3 4 1 1 2 3 {4, 2} 32 2 2 4 1 {3, 4} 3 3 1 4 {2, 1} 4 {4, 2} {3, 4} {2, 1} 3 we remark that there exists only one fundamental class. the maphyperoperation is not associative but it is wass, because, for example, 2∂(4∂4) = {1} and (2∂4)∂4 = {1, 2, 3}. example 15. consider the group (z6,+) and the map f:z6→z6:x→x-1. then the map-operation is given from the table ∂ 0 1 2 3 4 5 0 0 {1, 5} {2, 4} 3 {2, 4} {1, 5} 1 {1, 5} 0 {1, 5} {2, 4} 3 {2, 4} 2 {2, 4} {1, 5} 0 {1, 5} {2, 4} 3 3 3 {2, 4} {1, 5} 0 {1, 5} {2, 4} 4 {2, 4} 3 {2, 4} {1, 5} 0 {1, 5} 5 {1, 5} {2, 4} 3 {2, 4} {1, 5} 0 this is a commutative hyperoperation, it is wass, because, for example, 1∂(1∂2) = {2, 4} and (1∂1)∂2 = {0, 2, 4}, so (z6,∂) is a commutative hv-group. one can obtain that (z6,∂)/β* = {{0, 2, 4},{1, 3, 5}} ≅ z2. this is not cyclic since x∂x = {0} for all x in z6, i.e. every element has itself as the only one inverse element. example 16. consider the group (z6,+) and the map f: 0→0, 1→1, 2→2, 3→3, 4→4, 5→2. then the map-operation is given from the table 33 ∂ 0 1 2 3 4 5 0 0 1 2 3 4 {2, 5} 1 1 2 3 4 5 {0, 3} 2 2 3 4 5 0 {1, 4} 3 3 4 5 0 1 {2, 5} 4 4 5 0 1 2 {0, 3} 5 {2, 5} {0, 3} {1, 4} {2, 5} {0, 3} 1 one can obtain that (z6,∂)/β* = {{0, 3},{1, 4},{2, 5}} ≅ z3. (z6,∂) is a cyclic hv-group where 1 and 5 are generators of period 5. example 17. consider the group (z6,+) and the map f: 0→0, 1→1, 2→2, 3→3, 4→2, 5→5. then the map-operation is given from the table ∂ 0 1 2 3 4 5 0 0 1 2 3 {2, 4} 5 1 1 2 3 4 {3, 5} 0 2 2 3 4 5 {0, 4} 1 3 3 4 5 0 {1, 5} 2 4 {2, 4} {3, 5} {0, 4} {1, 5} 0 {1, 3} 5 5 0 1 2 {1, 3} 4 one obtains that (z6,∂)/β* = {{0, 2, 4},{1, 3, 5}} ≅ z2. 34 for the reproductivity, the element 4+4 which does not appeared in the normal position in the result it appears, in the general case, as follows: x+x∈ x∂(x+x-f(x))= {f(x)+x+x-f(x), x+f(x+x-f(x))}, ∀x∈z6, so the reproductivity is clear. we conclude with a theorem on this field. theorem 18. consider the commutative group of integers (z,+) and let n≠0 be a natural number. take the map f such that f(n)=0 and f(x)=x for all x in z-{n}. then (z,∂)/β* ≅ zn. proof. first, for all x,y in z-{n} we have, for the theta-operation, x∂y = {f(x)+y, x+f(y)} = {x+y}, so the hypersum is a singleton and coincides with the usual sum in z. for all x in z-{n} we have x∂n = n∂x = {f(x)+n, x+f(n)} = {x+n, x}. finally n∂n = {f(n)+n, n+f(n)} = {n}. therefore xβ(x+n). moreover, from the above, we obtain that for all x,y in z, the hypersum {x, x+n}∂{y, y+n} belongs to the same class modn. thus, the fundamental classes are the classes modn. therefore (z,∂)/β* ≅ zn. q.e.d. remark that this construction is an analogous case to the case of the uniting the elements 0 and n, see [6]. references 35 [1] corsini, p., prolegomena of hypergroup theory, aviani editore, 1993. [2] corsini, p., leoreanu, v., applications of hypergroup theory, kluwer academic publishers, 2003. [3] davvaz, b., on hv-rings and fuzzy hv-ideals, j.fuzzy math. v.6, n.1(1998),33-42 [4] ___, abrief survey of the theory of hv-structures, 8th aha congress, spanides press (2003),39-70. [5] vougiouklis, t., the fundamental relation in hyperrings. the general hyperfield, proc. 4thaha, world scientific (1991),203211. [6] ___, hyperstructures and their representations, monographs in mathematics, hadronic press, 1994. [7] ___, some remarks on hyperstructures, contemporary mathematics, amer. math. society, 184,(1995), 427-431. [8] ___, constructions of hv-structures with desired fundamental structures, new frontiers in hyperstructures, hadronic press (1996), 177-188. [9] ___, enlarging hv-structures, algebras and combinatorics, icac’97, hong kong, springer verlag (1999),455-463. [10] ___, on hv-rings and hv-representations, discrete mathematics, elsevier, 208/209 (1999), 615-620. [11] ___, finite hv-structures and their representations, rendiconti seminario matematico di messina s.ii, v.9 (2003),245-265. [12] vougiouklis, t., vougiouklis, s., the helix hyperoperations, to appear in italian journal of pure and applied mathematics. 36 on properties of fuzzy subspaces of vectorspaces h. hedayati department of mathematics, faculty of basic science, babol university of technology, babol, iran e-mail: h.hedayati@nit.ac.ir, hedayati143@yahoo.com abstract in this paper, we introduce the notion of normal fuzzy subspace of vector spaces. by using it, we construct new fuzzy subspaces. we also show that, under certain conditions, a fuzzy subspace of a vector space is two-valued and takes 0 and 1. mathematics subject classification: 08a72 keywords: vector space, fuzzy subspace, normal fuzzy subspace 1 introduction zadeh in [17] introduced the notion of fuzzy set and started a generalized logic. after that reconsideration of mathematics concepts begun. also there have been a number of generalizations of this fundamental concept. fuzzy algebraic ratio mathematica, 19, pp. 1-10 1 structures play a prominent role in mathematics with wide applications in many other branches such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, logic, set theory, group theory, groupoids, real analysis, measure theory etc (see [4], [5], [10] and [12]). in 1977, katsaras and liu [7] formulated and studied the concept of a fuzzy subspace of a vector space. since then, a host of mathematicians are involved in extending the basic concepts and results from the theory of crisp vector spaces to the broader framework of the fuzzy setting. however, not all the results can be fuzzified. in [8], among other concepts and results, the fuzzy coset of a fuzzy subspace is defined and the algebraic nature of fuzzy subspaces under homomorphism is studied. in [9], the fuzzy basis and dimension of a fuzzy subspace are defined and studied. in [1] and [3], the fuzzy subspaces over fuzzy fields are discussed. in this paper, some properties of fuzzy subspaces of vector spaces are investigated. specially, some ways are created to construct new fuzzy subspaces from the old. also the notion of normal fuzzy subspace of vector spaces is introduced. we see some normal fuzzy subspaces can be constructed by a fuzzy subspace. finally, we show that, when a non-constant normal fuzzy subspace be a maximal in the partial ordered set of normal fuzzy subspaces of a vector space, then this fuzzy subspace is two-valued and takes the values 0 and 1. 2 preliminaries an abelian group (v, +) on a field f is called a vector space on f if there exists a map . : v × f −→ v such that for all x, y ∈ v and a, b ∈ f the following ratio mathematica, 19, pp. 1-10 2 conditions hold: (i) 1x = x, (ii) (ab)x = a(bx), (iii) a(x + y) = ax + ay, (v) (a + b)x = ax + by. also a non-empty subset w of a vector space v is called a subspace, if w is a vector space on f . let x be an ordinary set. by a fuzzy set µ in x, we mean a function µ : x −→ [0, 1] with the grade of membership µ(x) for x ∈ x. if t ∈ [0, 1), then µt = {x ∈ x| µ(x) ≥ t} is called a level subset of µ. 3 fuzzy normal subspaces in what follows, v is a vector space on a field f , unless otherwise specified. definition 3.1. ([7], [11]) a fuzzy set µ of v is called a fuzzy subspace of v, if for all x, y ∈ v and a ∈ f the following conditions hold: (i) µ(x + y) ≥ µ(x) ∧ µ(y), (ii) µ(−x) ≥ µ(x), (iii) µ(ax) ≥ µ(x). clearly, if µ is a fuzzy subspace of v, then µ(0) ≥ µ(x) for all x ∈ v. also, µ is a fuzzy subspace of v if and only if µt is a subspace of v for all t ∈ [0, 1). lemma 3.2. if µ is a fuzzy subspace of v, then the set vµ = {x ∈ v | µ(x) = µ(0)} is a subspace of v. proof. let x, y ∈ vµ. then µ(x) = µ(y) = µ(0). since µ is a fuzzy ratio mathematica, 19, pp. 1-10 3 subspace, it follows that µ(x − y) ≥ µ(x) ∧ µ(y) = µ(0) ∧ µ(0) = µ(0). on the other hand µ(x − y) ≤ µ(0). hence we have µ(x − y) = µ(0) and so x − y ∈ vµ. also for any x ∈ vµ and a ∈ f, we get µ(ax) ≥ µ(x) = µ(0). on the other hand µ(ax) ≤ µ(0). hence, we obtain µ(ax) = µ(0), which shows that ax ∈ vµ. consequently, the set vµ is a subspace of v. � definition 3.3. a fuzzy subspace of v is said to be normal if there exists x ∈ v such that µ(x) = 1. note that if a fuzzy subspace of v is normal, then µ(0) = 1. hence µ is a normal fuzzy subspace if and only if µ(0) = 1. theorem 3.4. let µ be a fuzzy subspace of v and let µ̃ be a fuzzy set in v defined by µ̃(x) = µ(x) + 1 − µ(0) for all x ∈ v. then µ̃ is a normal fuzzy subspace of v containing µ. proof. let x, y ∈ v and a ∈ f. then µ̃(x − y) = µ(x − y) + 1 − µ(0) ≥ (µ(x) ∧ µ(y)) + 1 − µ(0) = (µ(x) + 1 − µ(0)) ∧ (µ(y) + 1 − µ(0)) = µ̃(x) ∧ µ̃(y). also we have µ̃(ax) = µ(ax) + 1 − µ(0) ≥ µ(x) + 1 − µ(0) = µ̃(x). clearly, µ̃(0) = 1 and µ ⊆ µ̃. this completes the proof. � corollary 3.5. if µ is a fuzzy subspace of v satisfying µ̃(x) = 0 for some x ∈ v, then µ(x) = 0. ratio mathematica, 19, pp. 1-10 4 lemma 3.6. let χw be the characteristic function of a subset w ⊆ v. then w is a subspace of v if and only if χw is a fuzzy subspace of v. proof. it is directly followed from discussion after definition 3.1. � theorem 3.7. for any subspace w of v, the characteristic function χw is a normal fuzzy subspace of v and vχw = w. proof. straightforward. � theorem 3.8. a fuzzy subspace µ of v is normal if and only if µ̃ = µ. proof. if µ̃ = µ, then it is obvious that µ is a normal fuzzy subspace of v. assume that µ is a normal fuzzy subspace of v and let x ∈ v. then µ̃(x) = µ(x) + 1 − µ(0) = µ(x), and hence µ̃ = µ. � theorem 3.9. if µ is a fuzzy subspace of v, then (̃µ̃) = µ̃. proof. straightforward. � theorem 3.10. let µ be a fuzzy subspace of v. if there exists a fuzzy subspace ν of v satisfying ν̃ ⊆ µ, then µ is a normal fuzzy subspace of v. proof. suppose there exists a fuzzy subspace ν of v such that ν̃ ⊆ µ. then 1 = ν̃(0) ≤ µ(0), and therefore µ(0) = 1. � corollary 3.11. let µ be a fuzzy subspace of v. if there exists a fuzzy subspace ν of v satisfying ν̃ ⊆ µ, then µ̃ = µ. proof. it is immediately obtained from theorem 3.10 and definition of µ̃. � theorem 3.12. let µ be a fuzzy subspace of v and f : [0, µ(0)] −→ [0, 1] ratio mathematica, 19, pp. 1-10 5 be an increasing map. define a fuzzy set µf : v −→ [0, 1] by µf (x) = f (µ(x)) for all x ∈ v. then µf is a fuzzy subspace of v. in particular, if f (t) ≥ t for all t ∈ [0, µ(0)] then µ ⊆ µf . proof. let x, y ∈ v. then µf (x − y) = f (µ(x − y)) ≥ f (µ(x) ∧ µ(y)) = f (µ(x)) ∧ f (µ(y)) = µf (x) ∧ µf (y). also if a ∈ f and x ∈ v, then µf (ax) = f (µ(ax)) ≥ f (µ(x)) = µf (x). hence µf is a fuzzy subspace of v. assume that f (t) ≥ t for all t ∈ [0, µ(0)]. then µf (x) = f (µ(x)) ≥ µ(x) for all x ∈ v, which means µ ⊆ µf . � theorem 3.13. let µ be a non-constant normal fuzzy subspace of v, which is maximal in the partial ordered set of normal fuzzy subspaces of v under fuzzy sets inclusion. then µ is a two-valued fuzzy subspace and takes the values 0 and 1. proof. we know µ(0) = 1. let x ∈ v be such that µ(x) 6= 1. it is enough to show that µ(x) = 0. assume that there exists x′ ∈ v such that 0 < µ(x′) < 1. define a fuzzy set ν : v −→ [0, 1] by ν(x) = 1/2(µ(x) + µ(x′)) for all x ∈ v. then clearly ν is well-defined. let x, y ∈ v. then ν(x − y) = 1/2(µ(x − y) + µ(x′)) ≥ 1/2((µ(x) ∧ µ(y)) + µ(x′)) = (1/2(µ(x) + µ(x′))) ∧ (1/2(µ(y) + µ(x′))) = ν(x) ∧ ν(y). also if a ∈ f and x ∈ v, then ratio mathematica, 19, pp. 1-10 6 ν(ax) = 1/2(µ(ax) + µ(x′)) ≥ 1/2(µ(x) + µ(x′)) = ν(x). hence ν is a fuzzy subspace of v. now we have ν̃(x) = ν(x) + 1 − ν(0) = 1/2(µ(x) + µ(x′)) + 1 − 1/2(µ(0) + µ(x′)) = 1/2(µ(x) + 1). so ν̃(0) = 1/2(µ(0) + 1) = 1. thus ν̃ is a normal fuzzy subspace of v. also ν̃(0) = 1 > ν̃(x′) = 1/2(µ(x′) + 1) > µ(x′). we know that ν̃ is non-constant. so by ν̃(x′) > µ(x′), it follows that µ is not maximal, which is a contradiction. therefore µ takes only the values 0 and 1. � theorem 3.14. let µ be a fuzzy subspace of v and let µ be a fuzzy set in v defined by µ(x) = µ(x)/µ(0) for all x ∈ v. then µ is a normal fuzzy subspace of v containing µ. proof. for any x, y ∈ v, we have µ(x − y) = µ(x − y)/µ(0) ≥ (1/µ(0))(µ(x) ∧ µ(y)) = (µ(x)/µ(0)) ∧ (µ(y)/µ(0)) = µ(x) ∧ µ(y). also if a ∈ f and x ∈ v we get µ(ax) = (µ(ax)/µ(0)) ≥ (µ(x)/µ(0)) = µ(x). hence µ is a fuzzy subspace of v. clearly µ(0) = 1 and µ ⊆ µ. � corollary 3.15. if µ is a fuzzy subspace of v satisfying µ(x) = 0 for some x ∈ v, then µ(x) = 0. ratio mathematica, 19, pp. 1-10 7 proof. obvious. � theorem 3.16. let µ be a non-constant fuzzy subspace of v such that µ̃ is a maximal in the partial ordered set of normal fuzzy subspace of v under fuzzy sets inclusion. then (1) µ is normal. (2) µ takes only the values 0 and 1. (3) χvµ = µ. (4) vµ is a maximal subspace of v. proof. since µ is non-constant, so µ̃ is non-constant maximal. also µ̃ is normal, which implies µ̃ takes only values 0 and 1 by theorem 3.13. . if µ̃(x) = 1, then µ(x) = µ(0) and if µ̃(x) = 0, then µ(x) = µ(0)−1. by corollary 3.5, we have µ(x) = 0 which implies µ(0) = 1. therefore µ is normal, and also µ̃ = µ by theorem 3.8, which proves (1) and (2). (3) clearly χvµ ⊆ µ and χvµ takes only the values 0 and 1. let x ∈ v and µ(x) = 0, then µ ⊆ χvµ . if µ(x) = 1 then x ∈ vµ and so χvµ (x) = 1. in any case µ ⊆ χvµ . (4) since µ is non-constant, vµ is a proper subspace of v. let w be a subspace of v such that vµ ⊆ w. then we obtain µ = χvµ ⊆ χw. since µ and χw are normal and µ = µ̃ is maximal in the partial ordered set of normal fuzzy subspaces under fuzzy sets inclusion, we have µ = χw or χw(x) = 1 for all x ∈ v, so w = v. if µ = χw then vµ = vχw = w by theorem 3.7 . therefore vµ is a maximal subspace of v. � ratio mathematica, 19, pp. 1-10 8 references [1] m. t. abu osman, ”on t−fuzzy subfield and t−fuzzy vector subspace”, fuzzy sets syst 33 (1989), 111-117. [2] m. t. abu osman, ”on fuzzy vector spaces via triangular norms”, bull. malaysian math. soc. 9(1) (1986), 33-42. [3] r. biswas, ”fuzzy fields and fuzzy linear spaces redefiened ”, fuzzy sets syst 33 (1989), 257-259. [4] c. l. chang, ”fuzzy topological spaces”, j. math. anal. appl. 24 (1968), 182-190. [5] p. das, ”fuzzy vector spaces under triangular norms”, fuzzy sets syst 25 (1988), 73-85. 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[17] l. a. zadeh, ”fuzzy sets” inform. control, 8(1965), 338-353. ratio mathematica, 19, pp. 1-10 10 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 41, 2021, pp. 119-136 119 a unified shape model for sunspot number cycles beena girija puthumana* sabarinath amaranathan† anilkumar ajimandiram krishnankuttynair‡ abstract we proposed a model which can unify many of the shape models existing in the literature and show that the shape of the sunspot number cycle can be described as a product of a polynomial and a negative exponential function. the proposed model has certain free parameters, which need to be estimated from the observed sunspot number data. since all the models reviewed in this paper are a product of a polynomial and a negative exponential along with a number of parameters, we have seen that all these models can be derived from a modified generalized gamma probability density function by transforming certain parameters and fixing certain parameters. in this paper, we estimate the parameters of the model from the revised monthly averaged sunspot numbers available in the sidc website. finally, a preliminary level prediction has also been attempted to forecast the characteristics of sunspot number cycle 25. keywords: sunspot numbers; gamma distribution; free parameter; etc. 2010 ams subject classification: 70f15; 97m10.§ *assistant professor, st. gregorios college, kottarakkara, kerala, india; beenamabhi@gmail.com. †scientist, vikram sarabhai space centre, thiruvananthapuram, kerala, india; a_sabarinath@yahoo.co.in. ‡ director, dssam, isro head quarters, bangalore, karnataka, india; ak_anilkumar@isro.gov.in. § received on xxx, xxx. accepted on xxx, xxx. published on xxx, xxx. doi: xxx. issn: xxx. eissn: doi. 10.23755/rm.v41i0.671.. ©the authors. this paper is published under the cc-by licence agreement. beena g p, sabarinath a, anilkumar a k 120 1. introduction solar activity is the key factor which drives the space weather. refined modelling and accurate prediction of the solar activity intensity has been an important activity of space faring agencies across the world due to the impact of solar activity on satellites as well as on weather (haigh, 2007, hathaway et.al., 2004). solar flux causes the upper atmosphere density variation and in turn it affects directly the lifetime of the earth-orbiting satellites especially in the low earth orbit. solar activity intensity has been measured as the number of dark spots, called sunspot numbers, appears in the visible solar disc through direct observation since 1749 onwards. irrespective of the measurement interval (daily, monthly, and yearly), a definite pattern is existing in the sunspot number time series. accurate predictions of the intensity of solar activity are increasingly important as we become more reliant upon satellites in low-earth orbits, which provide crucial contribution in communication, national defence and earth mapping. also, such satellites provide an abundance of scientific data. during higher solar activity period, the increased ultraviolet emission from sun heats up the earth’s upper atmosphere and this causes the atmosphere to expand and results in the increased drag on low earth orbits satellites, thereby leading to early decay into the earth’s atmosphere. therefore, better predictions of solar activity are essential to help mission planning and design of satellites (vallado et.al., 2014). sunspot number cycle time series is one of the longest time series which was studied by many experts. first of all, this time series is non stationary, cyclic and highly nonlinear in the time domain. the more interesting and difficult part to deal with is the high dispersion between consecutive observations (withbroe,1989) which in fact makes the prediction of sunspot numbers tedious. many attempts to model and predict the future behaviour of the solar activity are well documented in the literature. depending on the nature of the prediction methods, we can classify the methodology in to five classes as: 1) curve fitting 2) precursor 3) spectral 4) neural networks and 5) climatology. the first attempt using the curve fitting methodology was by the mcnishlincolon curve fitting (hathaway, d. h, 2015). subsequently, several authors have studied the highly nonlinear behaviour of sunspot number cycle and proposed various models to handle the studies related to the prediction. many mathematical functions have been appeared in the literature as model of the shape of the sunspot number cycle. due to the exponential rise and decay of sunspot number cycle, a model involving exponential function was proposed by nordemann (1992). the bell shape and the asymmetry along the peak amplitude of most of the sunspot number cycle were explored and there by a suitable mathematical function was introduced by hathaway et.al. (1994). few a unified shape model for sunspot cycles 121 statistical probability distribution functions were also proposed by various authors (sabarinath et.al.,2008, du et.al., 2011, li et.al., 2017, sabarinath et.al., 2020) for modeling the shape of sunspot number cycles. de meyer (1981) proposed a model using periodic functions. as far as prediction of a future sunspot number cycle is concerned, statistical averaged models are used as an initial estimate of the future sunspot number cycle. 2. existing models several authors developed different mathematical functions to describe the shape of the sunspot number cycle. in particular stewart and panofsky (sp) (1938) proposed a function for the shape of a sunspot number cycle with the form, 𝑅(𝑡) = 𝑎(𝑡 − 𝑡0) 𝑏 𝑒−𝑐(𝑡−𝑡0) (1) where 𝑎, 𝑏, 𝑐 and 𝑡0 are parameters that vary from cycle to cycle and 𝑡 is the independent variable represents time. the important thing to be noticed is that, this model resembles a power law for the rising phase of a sunspot number cycle and an exponential for the declining phases of a cycle. nordemann (n) (1992) proposed another fit. he used the solution of the differential equation 𝑑𝑁 𝑑𝑡 = 𝐾𝑁, in analogy with the nuclear decay process. the declining phase (maximum to minimum) of a sunspot number cycle is represented by 𝑁 = 𝑁0𝑒 𝐾𝑡 , 𝐾 < 0 (2) and the solution of 𝑑𝑁 𝑑𝑡 = 𝐴 + 𝐾𝑁, is used to represent the first phase or ascent phase (minimum to maximum) of a sunspot number cycle. thus, the model for the ascent phase is: 𝑁 = 𝐴 𝐾 (1 − 𝑒𝐾𝑡 ), 𝐾 < 0 (3) where 𝑁 represents sunspot numbers, 𝐾 is the decay constant and 𝐴 is a production parameter. hathaway et.al. (h) (1994) established a model with four parameters along with a measure for the goodness of fit. the functional form is: 𝑓(𝑡) = 𝑎(𝑡 − 𝑡0) 3 𝑒 [ (𝑡−𝑡0) 2 𝑏2 ] − 𝑐 (4) where, 𝑎 represents the amplitude; 𝑏 represents the time in months and 𝑡0 denotes the starting time; 𝑐 gives the asymmetry of the sunspot number cycle. this function is derived from stewart and panofsky model, but requires two beena g p, sabarinath a, anilkumar a k 122 parameters for each sunspot number cycle. where 𝑐 = 0.71, and 𝑏 is a dependent parameter given by, 𝑏 = 27.12 + 25.15 [𝑎 × 103] 1 4 (5) sabarinath et.al. (s) (2008) used a binary mixture of a modified laplace distribution. laplace distribution is a function 𝑓 of two parameters 𝑚 and 𝑠 is given by, 𝑓(𝑥) = 1 2𝑠 𝑒 −|𝑥−𝑚| 𝑠 (6) where, 𝑚 is the location parameter and 𝑠 is the scale parameter. they modified this form and used the binary mixture of this distribution and reducing the number of parameters into six (later reduced to two floating parameters) to fit the predominant double peaks during the high solar activity regime of a sunspot number cycle. modified final model is: volobuev (v) (2009) found a function similar to that used by stewart and panofsky to fit the shape of sunspot number cycle that requires only one parameter for each cycle. the empirical model used is: 𝑅 = ( 𝑡 − 𝑡0 𝑇𝑠 ) 2 𝑒 −( 𝑡−𝑡0 𝑇𝑑 ) 2 (8) we can see that this model is also similar to that of stewart and panofsky (1938) by putting 𝑏 = 2 and modifying the growth multiplier and decay multiplier properly by introducing the new parameters 𝑇𝑠 and 𝑇𝑑. du (2011) suggested modified gaussian function with four parameters viz. peak size 𝐴, peak timing 𝑡𝑚, width 𝐵, and asymmetry 𝛼, in the form: 𝑅(𝑡) = 𝐴 𝑒𝑥𝑝 ( −(𝑡 − 𝑡𝑚) 2 2𝐵2[1 + 𝛼(𝑡 − 𝑡𝑚)] 2 ) (9) li et al (l) (2017) used a binary mixture of gaussian function (suggested by du), 𝑓(𝑥) = 𝐴1 𝑒𝑥𝑝 (− (𝑥 − 𝑚1) 2 𝑠1 ) + 𝐴2 𝑒𝑥𝑝 (− (𝑥 − 𝑚2) 2 𝑠2 ) (10) this model has six parameters. peak sizes are denoted by and 𝐴1, 𝐴2, 𝑠1 and 𝑠2 represents gradients and peak time is represented by 𝑚1 and 𝑚2. sabarinath et. al. (sb) (2020) fit the full sunspot number cycle perfectly with modified maxwell boltzmann probability distribution function. the final modified model is: 𝑓(𝑡) = 𝐴1 33.2 𝑒𝑥𝑝 ( −|𝑡 − 𝑡0 − 41.7| 16.6 ) + 𝐴2 46 𝑒𝑥𝑝 ( −|𝑡 − 𝑡0 − 67.3| 23 ) (7) a unified shape model for sunspot cycles 123 𝑓(𝑥; 𝛼; 𝐴) = 𝐴 𝛼3 √ 2 𝜋 𝑥2𝑒 − 𝑥2 2𝛼2 (11) where 𝐴 is the area parameter, 𝛼 > 0. all these models discussed so far has a common form that is these models are a product of a polynomial and a negative exponential function. since the solar activity like process can be modelled by a bell-shaped curve viz., gamma family of probability density function, we propose that the sunspot number cycles can be effectively modelled as a generalized gamma distribution with certain free parameters. these free parameters can be estimated in the maximum likelihood sense from the sunspot data. also, all these models discussed so far can be derived from this generalized gamma distribution model as special cases. in the next section we brief on the derivation of the proposed model. 3. methodology new model-generalized gamma distribution the gamma distribution is often used to describe variables bounded on one side. a version of this distribution is obtained by adding a third parameter and gets the generalized gamma distribution (walck, 2001). probability density function is, 𝑓 (𝑥; 𝑝, 𝑞, 𝑟) = 𝑝𝑟 𝛤(𝑞) (𝑝𝑥)𝑞𝑟−1𝑒−(𝑝𝑥) 𝑟 (12) where 𝑝 (a scale parameter) and 𝑞 are the real positive parameters and a third parameter 𝑟 has been added (𝑟 = 1 for the ordinary gamma distribution) to generalize the gamma distribution. this new parameter takes any real value but normally we consider the case where 𝑐 > 0 put the following substitutions, 𝑐 = 𝑝𝑟 𝛤(𝑞) 𝑝𝑞𝑟−1  = 𝑞𝑟 𝐾 = 𝑝𝑟 𝛿 = 𝑟 (13) then, equation (12) becomes 𝑓 (𝑥; 𝑐, 𝑘, 𝛼, 𝛿) = 𝑐𝑥𝛼−1𝑒−𝑘𝑥 𝛿 (14) where 𝑘 is the scale parameter, 𝛼 is the shape parameter and 𝛿 is the location parameter. depending on the values of the parameters we can arrive all the beena g p, sabarinath a, anilkumar a k 124 existing models. table1 list the existing models and their derivation into equation (14). once we fix a model, the next step is to evaluate the best estimates of the model parameters in statistical parameter estimation sense. here our measurement data is the monthly averaged sunspot numbers. the model we indent to fit over this data is given in equation (14). we estimate the parameters by least square method. using simple random search technique, we estimate the parameters. the mathematical algorithms used are given in detail in the next part. data in the present study, we use the monthly averaged sunspot numbers, and these sunspot numbers for all the 24 sunspot number cycles were used. on july 1st, 2015, the sunspot number series has been replaced by a new improved version called version 2.0 data, that includes several corrections of past in homogeneities in the sunspot number time series. in the new version 2.0 data, the conventional 0.6 zürich scale factor, has been replaced by a factor of 1/0.6. this scale change, when combined with the recalibration, leads to a net increase of about 45% (correction variable with time) of the most recent part of the series, after 1947. this data can be obtained from https://www.bis.sidc.be/silso/data/sn_m_tot_v2.0.txt. estimation techniques the function in which parameters to be estimated is, 𝑓 (𝑥; 𝑐, 𝑘, 𝛼, 𝛿) = c𝑥𝛼−1𝑒−𝑘𝑥 𝛿 (15) the maximum likelihood estimates of the parameters 𝛼, 𝛿 and 𝑘 are considered to be the best unbiased, consistent and sufficient estimate of the parameters (sorenson, 1980). practically, the least square estimate is considered to be the maximum likelihood estimate. the simple mathematical procedure to estimate the parameters is to minimize the sum of squared error function 𝐽. 𝐽 = ∑ 𝑒𝑟 2 𝑟 (16) where 𝑒𝑟 is the error. the minimum of 𝐽 can be found by differentiating 𝐽 with respect to the parameters 𝛼, 𝛿 and 𝑘. in the present study, if we consider without loss of generality, a sunspot number cycle having a length of 132 months (11 year), and if we assume {𝑠𝑛: 𝑛 = 1,2, … ,132} as the realised sunspot number values, then the function 𝐽 can be written as, https://www.bis.sidc.be/silso/data/sn_m_tot_v2.0.txt a unified shape model for sunspot cycles 125 𝐽 = ∑[𝑠𝑛 − 𝑓(𝑥𝑛, 𝛼, , 𝑘)] 2 132 𝑖=1 (17) where, 𝑥𝑛 = 1,2, … ,132 represents the months for each 𝑛 = 1,2, … . ,132. then our objective is to compute and solve 𝛼, 𝛿 and 𝑘 from 𝜕𝐽 𝜕𝛼 = 0 (18) 𝜕𝐽 𝜕 = 0 (19) 𝜕𝐽 𝜕𝑘 = 0 (20) analytically solving the equations (18) to (20) for 𝛼, 𝛿 and 𝑘 is cumbersome. hence, we go with numerical procedures for estimating the parameters. monte carlo based simple random search-based procedure is considered here to estimate the parameters. this procedure is described below as an algorithm. step-1. start with a search region 𝛼, 𝛿 and 𝑘. let 𝑆𝛼 , 𝑆𝛿 and 𝑆𝑘 are the bounded search regions of 𝛼, 𝛿 and 𝑘 respectively. our objective is to find an 𝛼0 ∈ 𝑆𝛼 , 𝛿0 ∈ 𝑆𝛿 and 𝑘0 ∈ 𝑆𝑘 such that, 𝐽𝛼0,𝐴0 = ∑[𝑠𝑛 − 𝑓(𝑥𝑛, 𝛼0, 𝛿0, 𝑘0)] 2 132 𝑖=1 (21) is minimum. that is, 𝐽𝛼0,𝛿0,𝑘0 ≤ 𝐽𝛼,,𝑘 (22) for any 𝛼 ∈ 𝑆𝛼, 𝛿 ∈ 𝑆𝛿 and 𝑘 ∈ 𝑆𝑘 step-2. start with a random initial value of 𝛼 in 𝑆𝛼, 𝛿 in 𝑆𝛿 and 𝑘 in 𝑆𝑘. compute 𝐽 and in each iteration keep the minimum value of 𝐽, 𝛼, 𝛿 and 𝑘. after a very large number of iterations take the value of 𝛼, 𝛿 and 𝑘 corresponds to the global minimum value of 𝐽. 4. results estimates for the parameters table 2 shows typical converged values of the four model parameters 𝑐, 𝛼, 𝑘 and 𝛿. if we do a monte carlo based estimation of these parameters, due to the initial random number variation the optimum value differ numerically due to different starting points. but the variation is insignificant. this was proved in many monte carlo based optimizations (ji et.al., 2006). hence without loss of generality we consider a typical monte carlo run and a converged value of the parameters for further analysis. table 2 gives one such value. the range of beena g p, sabarinath a, anilkumar a k 126 search or feasible region was found by trial-and-error method. the range considered for the simulation is given below. 0.0001 ≤ 𝑐 ≤ 0.0030, 3 ≤ 𝛼 ≤ 7, 0 ≤ 𝑘 ≤ 0.9, and 0 ≤ 𝛿 ≤ 3.1. model name and year in which it is proposed functional form parameters in equation (14) generalized form 𝑐 𝛼 𝛿 𝑘 sp (1938) a(𝑡 − 𝑡0) 𝑏 𝑒𝑥𝑝(−𝑐(𝑡 − 𝑡0)) free free 1 free a𝑥 𝛼 𝑒𝑥𝑝 (−𝑐𝑥) n (1992) 𝑁 = 𝑁0𝑒𝑥𝑝(𝑘𝑡) 𝐴 𝐾 (1 − 𝑒𝑥𝑝(−𝑘𝑡)) free 1 1 free a𝑒𝑥𝑝 (−𝑐𝑥) h (1994) 𝑎(𝑡 − 𝑡0) 3 𝑒𝑥𝑝 ( (𝑡 − 𝑡0) 2 𝑏2 ) − 𝑐 free 4 2 free 𝑎𝑥3𝑒𝑥𝑝(−𝑘𝑥2) s (2008) 𝐴1 33.2 exp ( −|𝑡 − 41.7| 16.6 ) + 𝐴2 46 𝑒𝑥𝑝 ( −|𝑡−67.3| 23 ) free 1 1 fixed 𝑐1𝑒𝑥𝑝(−𝑘1x) +𝑐2𝑒𝑥𝑝(−𝑘2x) v (2009) (𝑡 − 𝑡0) 2 𝑇𝑠 2 𝑒𝑥𝑝 ( −(𝑡 − 𝑡0) 2 𝑇𝑑 2 ) free 3 2 free 𝑐𝑥2𝑒𝑥𝑝(−𝑘𝑥2) du (2011) 𝐴 𝑒𝑥𝑝 (− (𝑡 − 𝑡𝑚) 2 2𝐵2(1 + 𝛼(𝑡 − 𝑡𝑚) 2 ) free 1 2 free 𝑐𝑒𝑥𝑝(−𝑘𝑥2) l (2017) 𝐴1𝑒𝑥𝑝 ( −(𝑡−𝑚1) 2 𝑠1 )+𝐴2𝑒𝑥𝑝 ( −(𝑡−𝑚2) 2 𝑠2 ) free 1 2 free 𝑐1𝑒𝑥𝑝(−𝑘1𝑥 2) +𝑐2𝑒𝑥𝑝(−𝑘2𝑥 2) sb (2020) 𝐴 𝛼3 √2 𝜋⁄ 𝑡 2 𝑒𝑥𝑝 ( −𝑡2 2𝛼2 ) free 3 2 free 𝑐𝑥2𝑒𝑥𝑝(−𝑘𝑥2) table 1. different models, their parameters and its values. figure 1, 2, and 3 shows the model fit of the model on sunspot number cycles 13, 23 and 24, respectively. a unified shape model for sunspot cycles 127 figure 1. generalized gamma distribution fit on the monthly averaged sunspot number cycle 13. figure 2. generalized gamma distribution fit on the monthly averaged sunspot number cycle 23. beena g p, sabarinath a, anilkumar a k 128 figure 3. generalized gamma distribution fit on the monthly averaged sunspot number cycle 24. analysis of the parameters the parameters are estimated on each of the sunspot number cycle independently. figures 4 and 5 show these estimated model parameters for sunspot number cycle 1 to 24. the parameters are estimated in the maximum likelihood (ml) sense using the random search method. in table 2 column number 2 to 5 gives the ml estimate of the parameters 𝑐, 𝛼, 𝑘 and 𝛿. column 6 gives the coefficient of determination 𝑅2 value. from these values one can see that the goodness of fit of the model is fair and on all modern sunspot number cycles are of high degree, since the coefficient of determination is greater than 0.8. the important thing to be noted is that, since the model is derived from the gamma distribution probability density function as given in equation (12), there must be a correlation among the model parameters. as evident through the set of equations (13) the correlation between the model parameters is given in table 3. 𝛼 has a very high correlation with 𝑘 and 𝛿. figure 6 and 7 shows this correlation along with the linear regression model derived out of this correlation. of course, 𝛼 and 𝑘 has a positive correlation and between 𝛼 and 𝛿 a negative correlation. the corresponding linear regression fits are given in equation (23) and (24). a unified shape model for sunspot cycles 129 𝑘 = 0.29𝛼 − 1.3 (23) and 𝛿 = −0.32𝛼 + 2.7 (24) now, substitute equation (23) and (24) in equation (14), we can reduce the proposed model into a two-parameter model. again, if we re-estimate the two parameters in a maximum likelihood sense, we can again obtain a correlation between the parameters, there by the model reduce to a one parameter model as proposed in the shape of the sunspot number cycle-a one parameter fit by volobuev (2009). in fact, the reduction of model parameters into a single parameter does not add any predictive power in the characterisation of a sunspot number cycle via the prediction of the peak amplitude, location of the sunspot maximum, cycle length etc. hence, an indicative parameter for these characters of a sunspot number cycle is essential in a sunspot model. so minimum two parameters, a location parameter and a scale parameter must be there in a sunspot model. if there is a shape parameter, it can characterize the degree of asymmetry present in a cycle. figure 4. variation of the ml estimate of the parameter 𝑐 over different sunspot number-cycles coefficient of determination (𝑅2) is considered as one of the measures of goodness of fit for a regression fit. for each cycle the coefficient of determination for the best fit are computed and is given in table 2, column 6 and the pictorial version is given in figure 8. beena g p, sabarinath a, anilkumar a k 130 sunspot number cycle number 𝑐 α 𝑘 δ coefficient of determination 1 0.00095 4.62004 0.03367 1.10253 0.7 2 0.00259 5.86033 0.52436 0.69427 0.7 3 0.00104 6.79747 0.80867 0.65381 0.9 4 0.00278 5.83371 0.56733 0.66284 0.9 5 0.00213 4.21489 0.01729 1.22798 0.8 6 0.00144 3.97518 0.00191 1.62772 0.6 7 0.00114 4.23167 0.00323 1.53445 0.7 8 0.00014 6.71511 0.47050 0.72610 0.8 9 0.00044 5.30111 0.10092 0.93291 0.8 10 0.00033 6.01638 0.37419 0.72687 0.9 11 0.00153 5.29585 0.14993 0.89630 0.9 12 0.00119 4.53186 0.01235 1.33943 0.8 13 0.00226 6.11178 0.74136 0.63797 0.9 14 0.00277 5.06902 0.26096 0.77246 0.7 15 0.00278 4.34677 0.00650 1.48307 0.8 16 0.00283 4.81611 0.08682 0.98842 0.8 17 0.00096 4.98223 0.04514 1.10391 0.9 18 0.00238 4.96288 0.06138 1.07036 0.9 19 0.00298 5.12209 0.08539 1.02239 0.9 20 0.00242 4.74031 0.08620 0.95019 0.9 21 0.00236 5.06018 0.08613 1.00536 0.9 22 0.00205 5.00316 0.06184 1.07074 0.9 23 0.00288 4.78601 0.07280 1.00605 0.9 24 0.00280 4.84851 0.09596 0.98271 0.8 table 2. the estimated model parameters for sunspot number cycles 1 to 24. it may be seen that the 𝑅2 value for most of the cycles are greater than 0.8. especially, for modern cycles (cycle 15 to 24) the 𝑅2 values are greater than 0.8. this shows that the generalized gamma model is best for modelling the shape of a sunspot number cycle. 𝑐 𝛼 𝑘 𝛿 𝑐 1 -0.27 -0.09 -0.02 𝛼 1 0.89 -0.86 𝑘 1 -0.78 𝛿 1 table 3. the correlation among the estimated model parameters. a unified shape model for sunspot cycles 131 figure 5. variation of the ml estimate of the parameter 𝛼, 𝑘, and 𝛿 over different sunspot number cycles. figure 6. linear correlation between the model parameters 𝛼 and 𝑘. beena g p, sabarinath a, anilkumar a k 132 figure 7. linear correlation between the model parameters 𝛼 and 𝛿. figure 8. coefficient of determination (𝑅2) of the ml fit on different cycles. a unified shape model for sunspot cycles 133 prediction of sunspot number cycle 25 as an attempt has been made to predict the shape of the sunspot number cycle 25. before attempting the predict cycle 25 from the proposed model, a direct comparison of the model of all the sunspot number cycles from 1 to 24 has been done. figure 9 shows this model comparison. it may be noted that, the length of the cycle is not being considered here. without loss of generality one can assume the length as 132 months. the variation in amplitude and the location of the peak amplitude varies between cycles. the range of variation of peak amplitude is from 70 to 300 units of sunspot numbers. since the range of variation of the peak amplitude is quite large, the problem of prediction of cycle 25 is cumbersome from the previous cycle’s model parameters alone. hence, we make two kinds of predictions which are statistically more probable forecasts. they are (1) since the parameters 𝛼, 𝑘 and 𝛿 has lesser variation if we consider cycle 16-24, we fix these parameters as their average over cycle 16 to 24 and re-estimated the other parameter 𝑐 alone for these cycles. then for cycle 25, the parameter is 𝑐 taken as the average of the re-estimated values of 𝑐 for cycle 16 to 24. this value is 0.002. the remaining parameters are, 𝛼 = 4.9246, 𝑘 = 0.0757, 𝛿 = 1.02. the prediction of cycle 25 in this direction is plotted in figure 10. figure 9. model of all the 24 sunspot number cycles. this prediction shows the peak amplitude as 157 units occurring at 47 months from the beginning of sunspot number cycle 25. the second methods are (2) keeping the parameters 𝛼, 𝑘, and 𝛿 similar to cycle 24 and keeping the beena g p, sabarinath a, anilkumar a k 134 parameter 𝑐 as taken in method (1). in this way the peak amplitude of cycle 25 is 81 units of sunspot numbers occurring at 44 months from the beginning of the cycle 25. these two predictions can be considered as a band, inside which the actual cycle 25 may occur. figure 10. prediction of sunspot number cycle 25 maximum peak occurring at 44 months with a sunspot number value of 81 units and 47 months with a sunspot value of 157 units. 5 conclusion proposed a model which can unify many of the shape models existing in the literature. also, it is shown that the shape model of sunspot number cycle can be described by a product of a polynomial and a negative exponential function. since all the models reviewed in this paper are a product of a polynomial and a negative exponential, we proposed that all these models can be derived from the generalized gamma probability density function by giving suitable parameter values. in this paper, we derived the existing models from the proposed generalized gamma model and estimated the parameters of the proposed model from the revised version-2 monthly averaged sunspot numbers available in the sidc’s website. prediction of sunspot number cycle 25 shows that the peak amplitude of cycle 25 can vary between 81 to 157 units of sunspot numbers and this peak amplitude may occur between 44 to 47 months from the beginning of cycle 25. in actual date, this shows cycle 25 may peak during august 2023 to november 2023. a unified shape model for sunspot cycles 135 references [1] boslaugh, s. watters, p.a. statistics in a nutshell. sebastopol, ca: oreilly media. 2008. 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[15] vallado, d. a. finkleman, d. a critical assessment of satellite drag and atmospheric density modeling. acta astronautica, 95: 141-165, 2014. beena g p, sabarinath a, anilkumar a k 136 [16] volobuev, d. m. the shape of the sunspot cycle: a one-parameter fit. solar physics, 258(2): 319-330, 2019. [17] withbroe,g.l. solar activity cycle-history and predictions. journal of spacecraft and rockets, 26(6): 394-402. 22. 1989. [18] walck, c. hand-book on statistical distributions for experimentalists. internal report suf–pfy/96–01, particle physics group, university of stockholm. 2001. ratio mathematica volume 43, 2022 the extension of generalized intuitionistic topological spaces mathan kumar gk* g. hari siva annam† abstract in this paper, irresolute functions in generalized intuitionistic topological spaces were introduced. regarding these functions, we attempted to unveil the notions of some minimal and maximal irresolute functions. in addition, the generalized intuitionistic topological spaces were extended by using their open sets which are finer than of it and their basic characterizations were investigated. some continuous functions in the extension of generalized intuitionistic topological spaces are also been discussed in this paper. keywords: mn-µi -ops, mx-µi -ops, pµi -ops, sµi -ops, mn-µi -cts, mx-µi -cts, mn-µi -irresolute, mx-µi irresolute. 2020 ams subject classifications: 54a05, 54c08, 54c10. 1 *research scholar [19212102091012], pg and research department of mathematics, kamaraj college, thoothukudi-628003, tamil nadu, india. mathangk96@gmail.com. affiliated to manonmaniam sundaranar university, tirunelveli-627012, tamil nadu, india. †assistant professor, pg and research department of mathematics, kamaraj college, thoothukudi-628003, tamil nadu, india. hsannam84@gmail.com. affiliated to manonmaniam sundaranar university, tirunelveli-627012, tamil nadu, india. 1received on november 1st, 2022. accepted on december 29th, 2022. published on december 30th, 2022. doi: 10.23755/rm.v41i0.949. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. mathan kumar gk, g. hari siva annam 1 introduction the concept of an intuitionistic set which is a generalization of an ordinary set and the specialization of an intuitionistic fuzzy set was given by coker[2]. after that time, intuitionistic topological spaces were introduced [3]. a.csaszar[1] introduced many closed sets in generalized topological spaces based on their basics. in 2019 [9], some new generalized closed sets in ideal nano topological spaces were developed. in 2022 [6], we have introduced a new type of topology called generalized intuitionistic topological spaces with the help of intuitionistic closed sets. after that time we introduced and studied µi -maps in generalized intuitionistic topological spaces. in addition we have introduced and defined a new structure of minimal and maximal µi -open sets in generalized intuitionistic topological spaces. in 2011 [10], the subject like minimal and maximal continuous, minimal and maximal irresolute, t-min space etc. were investigated on basic topological spaces. in 2022 [7], the characterizations of niαg-closed sets are proved. in that paper authors has been used kuratowski’s closure operator. taking it as an inspiration we introduce µi -irresolute functions in generalized intuitionistic topological spaces throughout this paper. also, some minimal and maximal µi -irresolute functions were introduced and studied in detail. the aim of this paper is, to introduce the µi (a)-topology which is finer than µi topology by using the formula u ∪ (v ∩ a), where u and v are µi -open. in addition, some important and interesting results were discussed by using µi continuous maps on the extension of µi -topology. also, some counterexamples are given to support this work. 2 preliminaries definition 2.1 (6). a µi topology on a non-empty set x is a family of intuitionistic subsets of x satisfying the following axioms: 1. ∅ ∈ µi 2. arbitrary union of elements of µi belongs to µi . for a gits (x,µi ), the elements of µi are called µi -open sets(briefly µi -ops) and the complement of µi -open sets are called µi -closed sets(briefly µi -cds). note:[6] cµi (∅) ̸= ∅, cµi (x) = x, iµi (∅) = ∅ and iµi (x) ̸= x. the extension of generalized intuitionistic topological spaces definition 2.2 (6). let (x,µi ) be a gits. 1. a proper non-null µi -ops g of (x, µi ) is said to be a mn-µi -ops if any µi ops which is contained in g is ∅ or g. 2. a proper non-null µi -ops g( ̸= mµi ) of (x,µi ) is said to be a mx-µi -ops set if any µi -ops which contains g is mµi or g. definition 2.3 (6). let (x,µi ) and (y,σi ) be the topological spaces. a map f: (x,µi ) → (y,σi ) is called, 1. mn-µi -cts if f−1(g) is a µi -ops in (x, µi ) for every mn-µi -ops g in (y,σi ). 2. mx-µi -cts if f−1(g) is a µi -ops in (x, µi ) for every mx-µi -ops set g in (y,σi ). results: [6] 1. every µi -cts map is mn-µi -cts. 2. every µi -cts map is mx-µi -cts. 3. mn-µi -cts and mx-µi -cts maps are independent of each other. 4. if f: (x,µi ) → (y,σi ) is µi -cts and g: (y,σi ) → (z,ρi ) is mn-µi -cts then g◦f: (x,µi ) → (z,ρi ) is mn-µi -cts. 5. f: (x,µi ) → (y,σi ) is µi -cts and g: (y,σi ) → (z,ρi ) is mx-µi -cts then g◦f: (x,µi ) → (z,ρi ) is mx-µi -ops. definition 2.4 (4). let x be a µi -topological spaces. a subset a of x is said to be µi -dense if cµi (a) = x. clearly, x is the only µi -closed set dense in (x,µi ). theorem 2.1. let (x,µi ) be a gits with closed under intersection property. then cµi (a ∪ b) = cµi (a) ∪ cµi (b). proof: since a ⊂ a ∪ b and b ⊂ a ∪ b, cµi (a) ⊂ cµi (a ∪ b) and cµi (b) ⊂ cµi (a ∪ b). now we have to prove the second part, since a ⊆ cµi (a) and b ⊆ cµi (b), a∪b ⊆ cµi (a)∪cµi (b) which is µi -closed. then cµi (a∪b) ⊆ cµi (a) ∪ cµi (b). hence the theorem. mathan kumar gk, g. hari siva annam 3 µi-irresolute in gits definition 3.1. a mapping k: (x,µi ) → (y,σi ) is said to be a 1. semi µi -irresolute function(briefly sµi -irresolute) if the inverse image of semi µi -open sets(briefly sµi -ops) in (y,σi ) is sµi -op in (x,µi ). 2. pre µi -irresolute function(briefly pµi -irresolute) if the inverse image of pre µi -open sets(briefly pµi -ops) in (y,σi ) is pµi -op in (x,µi ). 3. αµi -irresolute function if the inverse image of αµi -ops in (y,σi ) is αµi open in (x, µi ). 4. βµi -irresolute function if the inverse image of βµi -ops in (y,σi ) is βµi -open in (x,µi ). theorem 3.1. let k: (x,µi ) → (y,σi ) be a semi µi -irresolute function if and only if the inverse image of semi µi -cds in (y,σi ) is semi µi -closed in (x,µi ). proof: necessary part: let k: (x,µi ) → (y,σi ) be a semi µi -irresolute function and a be a semi µi -cds in (y,σi ). since f is sµi -irresolute, k−1(y − a) = x − k−1(a) is sµi -open in (x,µi ). hence k−1(a) is sµi -closed in (x,µi ). sufficient part: assume that k−1(a) is sµi -closed in (x,µi ) for each sµi -closed set in (y,σi ). let v be a sµi -ops in (y,σi ) which yields that y − v is sµi -cds in (y,σi ). then we get k(−1)(y − v ) = x − k(−1)(v ) is sµi -closed in (x,µi ) this implies k−1(v ) is sµi -open in (x,µi ). hence k is sµi -irresolute. theorem 3.2. if k is sµi -irresolute then k is sµi -cts. proof: suppose k is sµi -irresolute. let a be any sµi -ops in (y,σi ). since every µi -ops is sµi -open and since a is sµi -open, k−1(a) is sµi -open in (x,µi ). hence k is sµi -cts. remark 3.1. since every sµi -ops need not be µi -open, we cannot deduce the reversal concept of the above statement. theorem 3.3. let (x,µi ), (y,σi ) and (z,ρi ) be three µi -topological spaces. for any sµi -irresolute map k: (x,µi ) → (y,σi ) and any sµi -cts ℏ: (y,σi ) → (z,ρi ) the composition ℏ ◦ k: (x,µi ) → (z,ρi ) is sµi -cts. proof: let a be any µi -ops in (z,ρi ). since ℏ is sµi -cts, ℏ−1(a) is sµi -open in (y,σi ). by using k is semi µi -irresolute, we get k−1[ℏ−1(a)] is sµi -open in (x,µi ). the extension of generalized intuitionistic topological spaces but k−1[ℏ−1(a)] = (ℏ ◦ k)−1(a). therefore, inverse image of µi -ops in (z,ρi ) is sµi -open in (x,µi ). hence ℏ ◦ k: (x,µi ) → (z,ρi ) is sµi -cts. theorem 3.4. if k: (x,µi ) → (y,σi ) and ℏ: (y,σi ) → (z,ρi ) are both sµi -irresolute then ℏ ◦ k: (x,µi ) → (z,ρi ) is also sµi -irresolute. proof: let a be any sµi -ops in (z,ρi ). since k and ℏ are sµi -irresolute, ℏ−1(a) is sµi -open in (y,σi ) and k−1[ℏ−1(a)] is sµi -open in (x,µi ). hence (ℏ ◦ k)−1(a) = k−1[ℏ−1(a)] is sµi -open and so ℏ ◦ k: (x,µi ) → (z,ρi ) is sµi -irresolute. theorem 3.5. let k: (x,µi ) → (y,σi ) be a pµi -irresolute(resp. αµi -irresolute and βµi -irresolute) function if and only if the inverse image of pµi -closed(resp. αµi -closed and βµi -closed) sets in (y,σi ) is pµi -closed(resp. αµi -closed and βµi -closed) in (x,µi ). proof: we can prove this theorem as we have done in the theorem 3.2. theorem 3.6. if f is pµi -irresolute(resp. αµi -irresolute and βµi -irresolute) then f is pµi -continuous(resp. αµi -cts and βµi -cts). proof: we can prove this theorem as we have done in the theorem 3.3. remark 3.2. since every pµi -open(resp. αµi -open and βµi -open) set need not be µi -open, we cannot deduce the reversal concept of the above statement. theorem 3.7. let (x,µi ), (y,σi ) and (z,ρi ) be three µi -topological spaces. for any pµi -irresolute(resp. αµi -irresolute and βµi -irresolute) map k: (x,µi ) → (y,σi ) and any pµi -cts(resp. αµi -cts and βµi cts) ℏ: (y,σi ) → (z,ρi ) the composition ℏ ◦ k: (x,µi ) → (z,ρi ) is pµi -cts(resp. αµi -cts and βµi -cts). proof: we can prove this theorem as we have done in the theorem 3.5. theorem 3.8. if k: (x,µi ) → (y,σi ) and ℏ: (y,σi ) → (z,ρi ) are both pµi irresolute(resp. αµi -irresolute and βµi -irresolute) then ℏ ◦ k: (x,µi ) → (z,ρi ) is also pµi -irresolute(resp. αµi -irresolute and βµi -irresolute). proof: we can prove this theorem as we have done in the theorem 3.6 4 minimal and maximal µi-irresolute definition 4.1. let (x,µi ) and (y,σi ) be the topological spaces. a map k: (x,µi ) → (y,σi ) is called, 1. mn-µi -irresolute if the inverse image of every mn-µi -ops in (y,σi ) is mn-µi open in (x,µi ). mathan kumar gk, g. hari siva annam 2. mx-µi -irresolute if the inverse image of every mx-µi -ops in (y,σi ) is mx-µi open in (x,µi ). example 4.1. let x = {a, b, c, d} and y = {t, u, v, w} with µi = {∅ , < x, ∅, {b} >, < x, ∅, {d} >, < x, {a, d}, ∅ >, < x, {a}, ∅ >, < x, ∅, ∅ >, < x, ∅, {c, d} >, < x, ∅, {c} >, < x, {d}, ∅ >, < x, {d}, {b} >} and σi = {∅, < x, ∅, {v} >, < x, ∅, {w} >, < x, ∅, {u, v} >, < x, ∅, ∅ >, < x, {v}, ∅ >, < x, {v}, {w} >}. define k: (x,µi ) → (y,σi ) by k(a) = t, k(b) = w, k(c) = u and k(d) = v. hence k is a mn-µi -irresolute map. theorem 4.1. every mn-µi -irresolute map is mn-µi -cts. proof: let k: (x,µi ) → (y,σi ) be a mn-µi -irresolute map. let g be any mn-µi ops in (y,σi ). since k is mn-µi -irresolute, k−1(a) is a mn-µi -ops in (x,µi ). that is k−1(a) is a µi -ops in (x,µi ) hence k is mn-µi -cts. remark 4.1. the reversal statement of the above theorem is not necessarily true. in example 4.3, k is mn-µi -cts but not mn-µi -irresolute. since k−1(¡x,w,∅¿) = ¡x,b,∅¿ which is not minimal µi -open in (x,µi ). theorem 4.2. every mx-µi -irresolute map is mx-µi -cts. proof: we can prove this theorem as we have done in the theorem 4.4. remark 4.2. the reversal statement of the above theorem is not necessarily true. in example 4.2, k is mx-µi -cts but not mx-µi -irresolute. since k−1(¡x,v,w¿ = ¡x,d,b¿ which is not mx-µi -open in (x,µi ). remark 4.3. in example 4.2, k is a mn-µi -irresolute map but not mx-µi -irresolute. in example 4.3, k is a mx-µi -irresolute map but not mn-µi -irresolute. that is mnµi -irresolute maps and mx-µi -irresolute maps are independent of each other. remark 4.4. since mn-µi -ops and mx-µi -ops are independent of each other, 1. mn-µi -irresolute and mx-µi -cts are independent of each other. 2. mx-µi -irresolute and mn-µi -cts are independent of each other. theorem 4.3. let k: (x,µi ) → (y,σi ) be a mn-µi -irresolute map if and only if the inverse image of each mx-µi -closed in (y,σi ) is a mx-µi -closed in (x,µi ). proof: we can prove this theorem by using the result, if g is a mn-µi -ops if and only if gc is a mx-µi -closed set. theorem 4.4. if k: (x,µi ) → (y,σi ) and ℏ: (y,σi ) → (z,ρi ) are mn-µi -irresolute then ℏ ◦ k: (x,µi ) → (z,ρi ) is a mn-µi -irresolute map. proof: let g be any mn-µi -ops in (z,ρi ). since ℏ is mn-µi -irresolute, ℏ−1(g) is a mn-µi -ops in (y,σi ). also since k is mn-µi -irresolute, k−1[ℏ−1(g)] = (ℏ◦k)−1(g) is a mn-µi -ops in (x,µi ). hence ℏ ◦ k is mn-µi -irresolute. the extension of generalized intuitionistic topological spaces theorem 4.5. let k: (x,µi ) → (y,σi ) be a mx-µi -irresolute map if and only if the inverse image of each mn-µi -closed in (y,σi ) is a mn-µi -closed in (x,µi ). proof: we can prove this theorem by using the result, if g is a mx-µi -ops if and only if gc is a mn-µi -cds. theorem 4.6. if k: (x,µi ) → (y,σi ) and ℏ: (y,σi ) → (z,ρi ) are mx-µi -irresolute then ℏ ◦ k: (x,µi ) → (z,ρi ) is a mx-µi -irresolute map. proof: similar to that of theorem 4.11. 5 the simple extension of µi-topology over a µi-set in (x,µi ) a subset a of x, we denote by µi(a) the simple extension of µi over a, that is the collection of sets u∪(v∩a), where u,v ∈ µi . note that µi(a) is finer than µi . theorem 5.1. if a is µi -dense subset of the space (x,µi ), then a is also µi -dense in (x,µi(a)). proof: since µi(a) is finer than µi , µi ⊂ µi(a). this gives cµi(a)(a) ⊂ cµi (a). to prove cµi (a) ⊂ cµi(a)(a). let x ∈ cµi (a) and let g be a µi -ops of x in µi(a). then x∈g = h∪(j∩a) where h,j ∈ µi . if x∈h then h∩a ̸= ∅ and g∩a ̸= ∅. if x∈j∩a then j∩a ̸= ∅ and g∩a ̸= ∅. hence x ∈ cµi(a)(a). therefore cµi(a)(a) = cµi (a). theorem 5.2. let (x,µi ) be a µi -topological space with closed under intersection property. let a be a µi -dense subset of the space (x,µi ). then for every µi -open subset g of the space (x,µi(a)) we have cµi (g) = cµi(a)(g) and for every µi closed subset f of the space (x,µi(a)) we have iµi (f) = iµi(a)(f). proof: let v ∈ µi . since µi(a) is finer than µi , cµi(a)(v ) ⊂ cµi (v ). now to prove, cµi (v ) ⊂ cµi(a)(v ). let x ∈ cµi (v ) and let g be a µi -open neighborhood of x in (x,µi(a)). then x∈g = h∪(j∩a) where h,j ∈ µi . if x∈h then h∩v ̸= ∅. again if x ∈ j∩a⊂j then j∩v ̸= ∅ and hence j∩v∩a ̸= ∅, since j∩v ∈ µi and since a is µi -dense. thus also in this case g∩v ̸= ∅ and hence x ∈ cµi(a)(v ). this implies cµi (v ) ⊂ cµi(a)(v ). henceforth cµi (v ) = cµi(a)(v ) for each v ∈ µi . let g ∈ µi(a) then g = h∪(j∩a) where h,j ∈ µi . clearly cµi (h) = cµi(a)(h). since j ∈ µi(a) and since a is a µi -dense subset of (x,µi(a)), cµi(a)(j∩a) = cµi(a)(j) = cµi (j) = cµi (j∩a). thus cµi(a)(g) = cµi (h) ∪ cµi (j∩a) =cµi (h∪(j∩a)) = cµi (g). proceeding like this we can prove iµi (f) = iµi(a)(f). corolary 5.1. let (x,µi ) be a gits with closed under intersection property. if a is a µi -dense subset of the space (x,µi ). then for every v ∈ µi(a) we have iµi (cµi (v )) = iµi(a)(cµi(a)(v )). hence the set v is a regular µi -open subset of mathan kumar gk, g. hari siva annam (x,µi ) if and only if it is regular µi -open in (x,µi(a)). proof: from the previous theorem we have iµi (cµi (v )) = iµi (cµi(a)(v )) = iµi(a)(cµi(a)(v )). 6 the characterization of extension on µi-topology remark 6.1. if k: (x,µi(a)) → (y,σi ) is µi -cts. then the restriction of k on (x,µi ) [shortly, k|(x,µi )] need not be µi -cts. example 6.1. let x = {a, b, c} and y = {u, v, w} with µi = {∅, < x, ∅, {a} >, < x, ∅, {b} >, < x, ∅, ∅ >, < x, ∅, {a, b} >, < x, {a, b}, ∅ >}, µi(a) = { ∅, < x, ∅, {a} >, < x, ∅, {b} >, < x, ∅, ∅ >, < x, ∅, {a, b} >, < x, {a, b}, ∅ >, < x, {b}, ∅ >} and σi = {∅, < x, ∅, {u} >, < x, ∅, {v} >, < x, ∅, ∅ >, < x, {v}, ∅ >}. define k: (x,µi(a)) → (y,σi ) by k(a) = u, k(b) = v and k(c) = w. hence k is µi(a)-cts. but k|(x,µi(a)) is not µi -cts, since k−1(< x, {v}, ∅ >) = < x, {b}, ∅ > /∈ µi . remark 6.2. since µi(a) is finer than µi , some elements of µi(a) does not belongs to µi and the elements of µi(a) which is not in µi need not be mn-µi -open in (x,µi ). for, u ⊂ u∪(v∩a) /∈ µi and u ∈ µi(a), u∪(v∩a) should not be mn-µi -open in (x,µi(a)). by the previous example, we may conclude that every mx-µi -ops in (x,µi(a)) need not be µi -open in (x,µi ). remark 6.3. a function k is mn-µi(a)-cts in (x,µi(a)) then k|(x,µi ) is mn-µi cts. in example 6.2, a function f is mx-µi(a)-cts in (x,µi(a)) then f|(x,µi ) need not be mx-µi -cts. 7 conclusions in example 4.2, k is a mn-µi -irresolute map but not mx-µi -irresolute and in example 4.3, k is a mx-µi -irresolute map but not mn-µi -irresolute. this examples evinces mn-µi -irresolute maps and mx-µi -irresolute maps are independent of each other. remark 6.1 propounded the restriction of the function k on (x,µi ) need not be a µi -continuous function. in remark 6.3, we discussed the connections between minimal µi -open sets in (x,µi ) and in (x,µi(a)). we hope that we improved some results concerning µi(a)-topological spaces. we will extend our research in kernel and contra continuous of µi -topological spaces. the extension of generalized intuitionistic topological spaces acknowledgements my completion of this paper could not have been accomplished without the support of my guide and i cannot express enough thanks to my guide for the continued support and encouragement references [1] a.csaszar, generalized topology, generalized continuity, acta mathematics, hungar, 96(2002). [2] dogan coker, a note on intuitionistic sets and intuitionistic points, tr.j. of mathematics, 20(1996), 343-351. [3] j.h.kim, p.k.lim, j.g.lee, k.hur, intuitionistic topological spaces, annals of fuzzy mathematics and informations, 14 december 2017. [4] julian dontchev, on submaximal spaces, tamking journal of mathematics, volume 26, number 3, autumn 1995. [5] karthika m, parimala m, jafari s, smarandache f, alshumrani m, ozel c, and udhayakumar r (2019), ”neutrosophic complex ?? connectedness in neutrosophic complex topological spaces”, neutrosophic sets and systems, 29, 158-164. [6] mathan kumar gk and g.hari siva annam, minimal and maximal µi -open sets in gits, advances and applications in mathematical sciences, mili publications, volume 21, issue 7, may 2022, pages 4097-4109. [7] m.parimala, d.arivuoli and r. udhayakumar, niαg-closed sets and normality via niαg-closed sets in nano ideal topological spaces, punjab university journal of mathematics, vol. 52(4)(2020) pp. 41-51. [8] mani, p, muthusamy k, jafari s, smarandache f and ramalingam u. decision-making via neutrosophic support soft topological spaces. symmetry 2018, 10, 217. https://doi.org/10.3390/sym10060217. [9] raghavan asokan, ochanan nethaji and ilangovan rajasekaran, new generalized closed sets in ideal nano topological spaces, bulletin of the international mathematical virtual institute, vol. 9(2019), 535-542, www.imvibl.org /journals / bulletin, http://dx.doi.org/10.7251/bimvi1903535a mathan kumar gk, g. hari siva annam [10] s.s.benchalli, basavaraj m. ittanagi and r.s.wali, on minimal open sets and maps in topological spaces, j. comp. and math. sci. vol.2 (2), 208-220 (2011). ratio mathematica volume 47, 2023 p-clean properties in amalgamated rings selvaganesh thangaraj* selvaraj chelliah† abstract let a be a ring. then a is called p-clean ring if each element in a express as the sum of an idempotent and pure element. let f : a → b be a ring homomorphism and j be an ideal of b. the amalgamation of a with b along j with respct to f is a new ring structure introduced and studied by anna et al. in 2009. this construction is a generalization of the amalgamated duplication of a ring along an ideal and other classical constructions such as the a + xb[x] and a + xb[[x]] constuctions. in this paper, the transfer of the notion of p-clean rings to the amalgamation of rings along ideal is studied. in particular, the necessary and sufficient conditions for amalgamation to be a p-clean ring are studied. keywords: pure element, amalgamation ring, p-clean ring 2020 ams subject classifications: 13a15, 13d05. 1 1 introduction throughout this paper all rings are commutative with identity. let a and b be two rings with unity, let j be an ideal of b and let f : a → b be a ring homomorphism. anna et al. [2009] introduced and studied the new ring structure of the following subring of a × b: a ▷◁f j := {(a, f(a) + j) | a ∈ a, j ∈ j} *department of mathematics, ramco institute of technology, rajapalayam 626 117, virudhunagar, tamilnadu, india, selvaganeshmaths@gmail.com †department of mathematics, periyar university, salem 636 011, tamilnadu, india, selvavlr@yahoo.com 1received on november 19, 2022. accepted on june 30, 2023. published on june 30, 2023. doi: 10.23755/rm.v39i0.958. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 375 selvaganesh thangaraj and selvaraj chelliah called the amalgamation of a with b along j with respect to f. this new ring structure construction is a generalization of the amalgamated duplication of a ring along an ideal. the amalgamated duplication of a ring along an ideal was introduced and studied in [anna [2006], anna and fontana [2007]]. in [anna et al. [2009], section 4], the authors studied the amalgamation can be in the frame of pullback constructions and also the basic properties of this construction [e.g., characterizations for a ▷◁f k to be a noetherian ring, an integral domain, a reduced ring] and they characterized those distinguished pullbacks that can be expressed as an amalgamation. aruldoss et al. [2022] and vijayanand and selvaraj [2023] studied some rings and module characterized via amalgamation construction. the notion of regular element was first introduced by von neumann [1936], where an element a ∈ a is called regular if there exists b ∈ a such that a = aba. a ring a is called regular ring if each element in a is regular. many other authors interested in studying regular rings, for example see vas [2010] and wardayani et al. [2020]. let the set of regular elements in a be denoted by reg(a) that is reg(a) = {r ∈ a; r = rsr, for some s ∈ a}. the concept of clean ring first introduced by nicholson [1977], where the ring a is called clean ring if for each a ∈ a there exist e ∈ id(a) and u ∈ u(a) such that a = e + u. many other authors interested in studying clean rings, for example see chen and cui [2008] and ashrafi and nasibi [2013]. ashrafi and nasibi [2013], introduced the concept of r−clean ring , where the ring a is called r-clean ring if for each a ∈ a there exist e ∈ id(a) and r ∈ reg(a) such that a = e + r. many authors have studied r-clean rings such as andari [2018] and sharma and singh [2018]. let a be a ring, then an element p ∈ a is called pure element if there exists q ∈ a such that p = pq [majidinya et al. [2016]] and the set of pure elements in a write pu(a) = {p ∈ a : p = pq, for some q ∈ a}. the concept of von neumann local ring was studied by anderson and badawi [2012], where a ring a is called von neumann local ring if for each a ∈ a we have either a ∈ reg(a) or 1 − a ∈ reg(a). mohammed et al. [2021] studied the class of rings namely rings in which each element express as the sum of an idempotent and pure. this paper aims at studying the transfer of the notion of p-clean rings to the amalgamation of rings along ideal. in particular, we study the necessary and sufficient condition for amalgamation to be a p-clean ring. we denote by u(a), id(a), nilp(r) and pu(a), the set of unit elements, the set of idempotents, the set of nilpotent elements and set of all pure elements of a, respectively. 2 p-clean ring we start with the following definition. 376 p-clean properties in amalgamated rings definition 2.1 (mohammed et al. [2021]). an element a ∈ a is called p-clean if there exist e ∈ id(a) and p ∈ pu(a) such that a = e + p. a ring a is called p-clean ring if each element in a is p-clean element. proposition 2.1 (mohammed et al. [2021]). (i) the class of p-clean is closed under homomorphic images. (ii) if i is an ideal of a p-clean ring a, then a/i is a p-clean ring. proposition 2.2. the ring a is p-clean if and only if the ring a[[x]] of formal series over a is p-clean. proof. if a[[x]] is p-clean, then it follows by the isomorphism a ∼= a[[x]]/(x) and by proposition 2.1 (i) that a is p-clean. conversely, suppose that a is p-clean. let f(x) = ∑∞ i=0 aix i ∈ a[[x]], then f(x) = a0 + a1x + a2x 2 + · · · . since a is p-clean, then a0 = p + e, where p ∈ pu(a) and e ∈ id(a). so f(x) = p + e + a1x + a2x 2 + · · · ∈ id(a[[x]]), with p ∈ pu(a) ⊆ pu(a[[x]]) and e + a1x + a2x2 + · · · ∈ id(a[[x]]). thus, a[[x]] is a p-clean ring. proposition 2.3. if a is a p-clean ring, then a[[{xα}]] is a p-clean ring. proof. let f(x) ∈ a[[{xα}]]. then f = f0+f ′ , where f0 ∈ a and f ′ ∈ ({xα}). since f0 ∈ a, we can write f0 = p + e, where p ∈ pu(a) and e ∈ id(a). now f = p + e + f ′ = (p + f ′ ) + e, where p + f ′ ∈ pu(a[[{xα}]]) and e ∈ id(a) ⊆ id(a[[{xα}]]). hence, a[[{xα}]] is p-clean. let a be a ring and m an a-module. the trivial extension of a by m is the ring a ∝ m = {(a, m) : a ∈ a, m ∈ m} under coordinatewise addition and an adjusted multiplication defined by (a, m)(a ′ , m ′ ) = (aa ′ , am ′ + a ′ m, ) for all a, a ′ ∈ a, m, m′ ∈ m. theorem 2.1. let a be a ring and m an a-module. then a ∝ m is p-clean if and only if a is p-clean. proof. note that a ∼= (a ∝ m)/({0} × m) is a homomorphic image of a ∝ m. hence if a ∝ m is p-clean, so by proposition 2.1 (i), a is p-clean. conversely, suppose that a is p-clean. recall that 1a∝m = (1, 0) and observe that if p ∈ pu(a), then (p, m) ∈ pu(a ∝ m) for each m ∈ m and if e ∈ id(a), then (e, 0)2 = (e2, 0) = (e, 0) in a ∝ m. hence, if a ∈ a with a = p + e, where p ∈ pu(a) and e ∈ id(a), then for m ∈ m, (a, m) = (p+e, m) = (p, m)+(e, 0), where (p, m) ∈ pu(a ∝ m) and (e, 0) ∈ id(a ∝ m). hence, a ∝ m is pclean. 377 selvaganesh thangaraj and selvaraj chelliah lemma 2.1. let f = ∑n i=0 aix i ∈ a[x] be a pure element. then a0 is pure and ai is nilpotent for each i > 0. proof. since f is a pure element, there exists g = ∑n i=0 bix i ∈ a[x] such that f = fg. therefore, a0 = a0b0 and so a0 is pure. now let p be a prime ideal of a. then (a/p)[x] is an integral domain. define ϕ : a[x] → (a/p)[x] by ϕ( ∑k i=0 aix i) = ∑k i=0(ai + p)x i. clearly, ϕ is an epimorphism. then we have ϕ(f)ϕ(g) = ϕ(fg) = ϕ(f). so deg(ϕ(f)ϕ(g)) = deg(ϕ(f)). thus, deg(ϕ(f)) + deg(ϕ(g)) = deg(ϕ(f)). therefore deg(ϕ(f)) = 0. hence, ai ∈ p for i = 1, . . . , n. since p is arbitrary, ai is nilpotent for each i > 0. theorem 2.2. a[x] is not p-clean. proof. let f = ∑n i=0 aix i ∈ a[x]. suppose f is p-clean. then f = p + e, where p ∈ pu(a[x]) and e ∈ id(a[x]). since idempotents of a are exactly that of a[x], p = f −e is pure. hence by lemma 2.1, the element 1 should be nilpotent, which is a contradiction. theorem 2.3. for every ring a, we have the following statements: (i) if e is a central idempotent element of a and eae and (1−e)a(1−e) are both p-clean, then so is a; (ii) let e1, e2, · · · , en be orthogonal central idempotents with e1+e2+· · ·+en = 1. then eirei is p-clean for each i, if and only if so is a. (iii) if a is p-clean, then so is the matrix ring mn(a) for any n > 1. proof. (i) for convenience, write e = 1−e for each e ∈ id(a). we use the pierce decomposition of a: we have a = eae ⊕ eae ⊕ eae + eae. since e, e are central, we have a = eae ⊕ eae ∼= ( eae 0 0 eae ) . so each matrix b ∈ a is of the form ( a 0 0 b ) , where a ∈ ere, b ∈ eae. by hypothesis, a and b are p-clean. then a = p1 + e1, b = p2 + e2, where p1 ∈ pu(eae) ⊆ pu(a), p2 ∈ pu(eae) ⊆ pu(a), e1 ∈ id(eae) ⊆ id(a), e2 ∈ id(eae) ⊆ id(a). so b = ( a 0 0 b ) = ( p1 + e1 0 0 p2 + e2 ) = ( p1 0 0 p2 ) + ( e1 0 0 e2 ) . 378 p-clean properties in amalgamated rings since p1 and p2 are pure elements of a, there exist q1, q2 in a such that p1 = p1q1 and p2 = p2q2. therefore, we have( p1 0 0 p2 ) ( q1 0 0 q2 ) = ( p1q1 0 0 p2q2 ) = ( p1 0 0 p2 ) . so ( p1 0 0 p2 ) is pure. since ( e1 0 0 e2 ) is an idempotent, we have a is p-clean. (ii) one direction of (ii) follows from (i) by induction. the other direction follows from proposition 2.1(ii). (iii) follows from (ii). 3 p-clean properties in amalgamated ring proposition 3.1. let f : a → b be a ring homomorphism and j an ideal of b. if a ▷◁f j is p-clean ring, then a and f(a) + j are p-clean rings. proof. define pa : a ▷◁f j → a by pa(a, f(a) + k) = a and pb : a ▷◁f j → b by pb(a, f(a) + k) = f(a) + k. then a ▷◁f j/({0} × j) ∼= a and a ▷◁f j/(f(−1)(j) × {0}) ∼= f(a) + j. since every homomorphic image of p-clean ring is p-clean, a and f(a) + j are p-clean rings. the converse of the above proposition is not true. proposition 3.2. let f : a → b be a ring homomorphism and j an ideal of b. assume that (f(a)+j)/j is uniquely p-clean and b is an integral domain. then a ▷◁f j is p-clean ring if and only if a and f(a) + j are p-clean rings. proof. by proposition 3.1, a ▷◁f j is p-clean ring implies a and f(a) + j are p-clean rings. conversely, assume that a and f(a) + j are p-clean rings. since a is p-clean, we can write a = e + p with e ∈ id(a), p ∈ pu(a). similarly, since f(a) + j is p-clean, we can write f(a) + j = f(e1) + j1 + f(p1) + j2 with f(e1) + j1 is an idempotent element and f(p1) + j2 is a pure element. clearly, f(e1) = f(e1) + j1 (resp., f(e)) and f(p1) = f(p1) + j2 (resp.,f(p)) are respectively idempotent and pure element of (f(a) + j)/j. then we have f(a) = f(e) + f(p) = f(e1) + f(p1). since (f(a) + j)/j is uniquely p-clean, f(e) = f(e1) and f(p) = f(p1). consider j ′ 1, j ′ 2 ∈ j such that f(e1) = f(e) + j ′ 1 and f(p1) = f(p) + j ′ 2. then (a, f(a) + j) = (e + p, f(e1) + j1 + f(p1) + j2) = (e, f(e) + j ′ 1 + j1) + (p, f(p) + j ′ 2 + j2). clearly, (e, f(e) + j ′ 1 + j1) is an idempotent element of a ▷◁f j. since f(p) + j ′ 2 + j2 is pure in f(a) + j, there exists 379 selvaganesh thangaraj and selvaraj chelliah an element f(α0) + j0 such that f(p) + j ′ 2 + j2 = (f(p) + j ′ 2 + j2)(f(α0) + j0). since p = pq for some q ∈ a, we have f(p)f(q) = f(p) = f(p)f(α0). since b is an integral domain, f(q) = f(α0). this implies f(α0) = f(q) + j ′ 0 and hence f(α0) + j0 = f(q) + j ′ 0 + j0. therefore, f(p) + j ′ 2 + j2 = (f(p) + j ′ 2 + j2)(f(q) + j ′ 0 + j0). hence, (p, f(p) + j ′ 2 + j2) = (pq, (f(p) + j ′ 2 + j2)(f(q) + j ′ 0 + j0) = (p, (f(p) + j ′ 2 + j2))(q, (f(q) + j ′ 0 + j0)). therefore, (p, f(p) + j ′ 2 + j2) is a pure element in a ▷◁f j. hence, a ▷◁f j is p-clean. remark 3.1. let f : a → b be a ring homomorphism and j an ideal of b. 1. if b = j, then a ▷◁f b is p-clean if and only if a and b are p-clean since a ▷◁f j = a × b 2. if f−1(j) = 0, then by [anna et al. [2009], proposition 5.1(3)], a ▷◁f j is p-clean if and only if f(a) + j is p-clean . corolary 3.1. let a be a ring and i an ideal such that a/i is uniquely p-clean. then a ▷◁f i is p-clean if and only if a is p-clean. theorem 3.1. let f : a → b be a ring homomorphism and j an ideal of b such that f(p) + j is pure (in b) for each p ∈ pu(a) and j ∈ j. then a ▷◁f j is p-clean if and only if a is p-clean. proof. a ▷◁f j is p-clean implies a is p-clean by proposition 3.1. conversely, assume that a is p-clean and f(p) + j is pure (in b) for each p ∈ pu(a) and j ∈ j. since a is p-clean, a = e + p, where e and p are idempotent and pure elements, respectively in a. since p is pure in a, then there exists q ∈ b such that p = pq. therefore (p, f(p) + j)(q, f(q) + j) = (pq, (f(p) + j)(f(q) + j)) = (pq, f(p)f(q)+j) = (p, f(p)+j). thus, (p, f(p)+j) is pure in a ▷◁f j. hence, (a, f(p)+j) = (e, f(e))+(p, f(p)+j) is a sum of idempotent and pure elements in a ▷◁f j. therefore, a ▷◁f j is p-clean. theorem 3.2. let f : a → b be a ring homomorphism and j an ideal of b. set a = a/nilp(a), b = a/nilp(b). π : b → b, the canonical projection and j = π(j). consider a ring homomorphism f : a → b defined by f(a) = f(a). then a ▷◁f j is p-clean (resp., uniquely p-clean) if and only if a ▷◁f j is p-clean (resp., uniquely p-clean) proof. clearly f is well defined and ring homomorphism. consider the map χ : (a ▷◁f j)/nilp(a ▷◁f j) → a ▷◁f j defined by χ((a, f(a + j)) = (a, f(a + j). if (a, f(a + j) = (b, f(b + j′), then (a − b, f(a − b) + j − j′) ∈ nilp(a ▷◁f j). therefore, a − b ∈ nilp(a) and j − j′ ∈ nilp(b). then a = b and j = j′. hence χ is well defined. we can easily check that χ is a ring homomorphism. 380 p-clean properties in amalgamated rings moreover,(a, f(a + j) = (0, 0) implies that a ∈ nilp(a) and j ∈ nilp(b). consequently, (a, f(a) + j) ∈ nilp(a ▷◁f j). hence, (a, f(a + j) = (0, 0). this implies that χ is injective. clearly, by the construction, χ is surjective. hence, χ is an isomorphism. proposition 3.3. let f : a → b be a ring homomorphism and let (e) be an ideal of b generated by the idempotent element e of b. then a ▷◁f (e) is p-clean if and only if a and f(a) + (e) are p-clean. in particular, if e is an element of a, then a ▷◁f (e) is p-clean if and only if a is p-clean. proof. by theorem 3.1, a ▷◁f (e) is p-clean implies a and f(a) + (e) are pclean. conversely, assume that a and f(a) + (e) are p-clean. let (a, f(a) + re) be an element of a ▷◁f (e) with a ∈ a and r ∈ b. since a is p-clean. there exist an idempotent element v and pure element p in a such that a = v + p. also, since f(a)+(e) is p-clean, there exist an idempotent element v′ and pure element p′ in f(a) + (e) such that f(a) + re = v′ + p′. we have (a, f(a) + re) = (v, f(v) + (v′ − f(v))e) + (p, f(p) + (p′ − f(p))e). on the other hand, [f(v) + (v′ − f(v))e]2 = [f(v)(1 − e) + v′e]2 = f(v)(1 − e) + v′e = f(v) + (v′ − f(v))e and [f(p) + (p′ − f(p))e][f(q) + (q′ − f(q))e] = [f(p)(1 − e) + p′e][f(q)(1 − e)+q′e] = f(pq)(1−e)+p′q′e = f(p)(1−e)+p′e = f(p)+(p′ −f(p))e. then (v, f(v) + (v′ − f(v))e) and (p, f(p) + (p′ − f(p))e) are respectively idempotent and pure element in a ▷◁f (e). hence a ▷◁f (e) is p-clean. finally, if a = b and f = ida, then a ▷◁f (e) = a ▷◁ (e) and f(a) + (e) = a. then a is p-clean. theorem 3.3. let f : a → b be a ring homomorphism and j an ideal of b. f(p) + j is pure (in b) for each p ∈ pu(a) and j ∈ j. then a ▷◁f j is a von neumann local ring if and only if a is a von neumann local ring. proof. note in first that a commutative ring is von neumann local ring if and only if it is an indecomposable p-clean ring (that is a p-clean ring where {0, 1} is the set of all idempotent elements) by [mohammed et al. [2021], theorem 1.8]. assume that a ▷◁f j is a von neumann local ring. then a ▷◁f j is an indecomposable p-clean ring. then a must be p-clean ring. also if e ∈ id(a), then (e, f(e)) ∈ id(a ▷◁f j) = {(0, 0), (1, 1)}. then id(a) = {0, 1}. this implies that a is an indecomposable p-clean ring, and so a is von neumann local ring. conversely, assume that a is a von neumann local ring. again by [mohammed et al. [2021], theorem 1.8 ], a is p-clean ring. also by theorem 3.1, a ▷◁f j is p-clean. on the other hand, by [chhiti et al. [2015], lemma 2.5], id(a ▷◁f j) = {(e, f(e))|e ∈ id(a)} = {(0, 0), (1, 1)}. thus a ▷◁f j is an indecomposable p-clean ring. this implies that a ▷◁f j is a von neumann local ring. 381 selvaganesh thangaraj and selvaraj chelliah corolary 3.2. let f : a → b be a ring homomorphism and j an ideal of b. if f(p) + j is pure (in b) for each p ∈ pu(a) and j ∈ j. then the following are equivalent: (i) a ▷◁f j is a von neumann local and uniquely p-clean ring. (ii) a is a von neumann local and uniquely p-clean ring. in particular, if a is a ring and j an ideal of b, then a ▷◁ j is a von neumann local and uniquely p-clean ring if and only if a is a von neumann local and uniquely p-clean ring. proof. from proposition 3.1 and theorem 3.3, a ▷◁f j is a von neumann local and uniquely p-clean ring implies that a is a von neumann local and uniquely p-clean ring. the converse is immediate. proposition 3.4. let f : a → b be a ring homomorphism and let (e) be an ideal of b generated by the idempotent element e. then a ▷◁f (e) is p-clean ring if and only if a and f(a) = (e) are p-clean ring. in particular, if e is an idempotent element of a, then a ▷◁ (e) is p-clean ring if and only if a is p-clean ring. proof. let a and f(a) + (e) are p-clean ring. we show that a ▷◁f (e) is p-clean ring. let (a, f(a) + re) be an element of a ▷◁f (e) with a ∈ a and r ∈ b. since a and f(a) + (e) are p-clean, there exist p and v (resp., p′ and v′) in a (resp., f(a) + (e)) which are respectively pure and idempotent element such that a = p + v and f(a) = re = p′ + v′. we have (a, f(a) + re) = (p, f(p) + (p′ − f(p)e)+(v, f(v)+(v′ −f(v)e)). on the other hand, [f(p)+(p′ −f(p)e)][f(q)+ (q′ − f(q)e)] = [f(p)(1 − e) + p′e][f(q)(1 − e) + q′e] = f(pq)(1 − e) + p′q′e = f(p)(1−e)+p′e = f(p)+(p′ −f(p)e). also [f(v)+(v′ −f(v)e)]2 = [f(v)(1− e)+v′e]2 = f(v)(1−e)+v′e = f(v)+(v′ −f(v)e). then (p, f(p)+(p′ −f(p)e) and (v, f(v) + (v′ − f(v)e)) are respectively pure and idempotent in a ▷◁f (e). consequently, a ▷◁f (e) is p-clean as desired. finally, if a = b and f = ida, then a ▷◁f (e) = a ▷◁ (e) and f(a) + (e) = a. thus, the particular case is obvious. 4 conclusion this paper studies the transfer of the notion of p-clean rings to the amalgamation of rings along ideal. in particular, the necessary and sufficient conditions for amalgamation to be a p-clean ring are studied. this study will further help in studying the properties of other ring structure such as the a + xb[x] and a + xb[[x]] constructions. furthermore, this work provides an answer to the question of when a ▷◁f j is a von neumann local ring. in future, there is a scope to study the generalization of amalgamated ring namely bi-amalgamation ring with p-clean properties. 382 p-clean properties in amalgamated rings acknowledgements the first author would like to thank ramco institute of technology, rajapalayam for providing the excellent facilities and constant support extended to carry out the research work. the second author was supported by dst fist (letter no: sr/fst/msi-115/2016 dated 10th november 2017). references a. andari. the relationships between clean rings, r-clean rings, and f-clean rings. aip con. proc., 2021 (1):1–4, 2018. d. anderson and a. badawi. von neumann regular and related elements in commutative rings. algebra colloq., 19 (01):1017–1040, 2012. m. anna. a construction of gorenstein rings. j. algebra, 306 (2):507–519, 2006. m. anna and m. fontana. an amalgamated duplication of a ring along an ideal: the basic properties. j. algebra appl., 6 (3):443–459, 2007. m. anna, c. finocchiaro, and m. fontana. amalgamated algebras along an ideal. de gruyter, new york, 2009. a. aruldoss, c. selvaraj, and b. davvaz. coherence properties in bi-amalgamated modules. gulf journal of mathematics, 14 (1):13–24, 2022. n. ashrafi and e. nasibi. rings in which elements are the sum of an idempotent and a regular element. bull. iran. math. soc., 39 (3):579–588, 2013. w. chen and s. cui. on clean rings and clean elements. southeast asian bull. math., 32 (5):855–861, 2008. m. chhiti, n. mahdou, and m. tamekkante. clean property in amalgamated algebras along an ideal. hacettepe j. math. stat., 44 (1):41–49, 2015. a. majidinya, a. moussavi, and k. paykan. rings in which the annihilator of an ideal is pure. algebra colloq., 22 (01):947–968, 2016. a. s. mohammed, i. s. ahmed, and s. h. asaad. study of the rings in which each element express as the sum of an idempotent and pure. int. j. nonlinear anal. appl., 12 (2):1719–1724, 2021. w. nicholson. lifting idempotents and exchange rings. trans. amer. math. soc., 229:269–278, 1977. 383 selvaganesh thangaraj and selvaraj chelliah g. sharma and a. singh. strongly r-clean rings introduction. int. j. math. comput. sci., 13 (2):207–214, 2018. l. vas. *-clean rings; some clean and almost clean baer *-rings and von neumann algebras. j. algebra, 24 (12):3388–3400, 2010. v. vijayanand and c. selvaraj. amalgamated rings with semi nil-clean properties. gulf journal of mathematics, 14 (1):173–181, 2023. j. von neumann. on regular rings. proceedings of the national academy of sciences of the united states of america, 22 (12):707–713, 1936. a. wardayani, i. kharismawati, and i. sihwaningrum. regular rings and their properties. j. phys: conf. ser., 1494 (1):1–4, 2020. 384 ratio mathematica volume 44, 2022 the outer connected detour monophonic number of a graph n.e. johnwin beaula1 s. joseph robin2 abstract for a connected graph 𝐺 = (𝑉, 𝐸) of order 𝑛 ≥ 2, a set 𝑀 ⊆ 𝑉 is called a monophonic set of 𝐺if every vertex of 𝐺is contained in a monophonic path joining some pair of vertices in 𝑀. the monophonic number 𝑚(𝐺) of 𝐺 is the minimum cardinality of its monophonic sets. if 𝑀 = 𝑉 or the subgraph 𝐺[𝑉 – 𝑀]is connected, then a detour monophonic set 𝑀 of a connected graph 𝐺 is said to be an outer connected detour monophonic setof 𝐺.the outer connecteddetourmonophonic number of 𝐺, indicated by the symbol 𝑜𝑐𝑑𝑚(𝐺), is the minimum cardinality of an outer connected detour monophonic set of 𝐺. the outer connected detour monophonic number of some standard graphs are determined. it is shown that for positive integers 𝑟𝑚, 𝑑𝑚and 𝑙 ≥ 2 with 𝑟𝑚 < 𝑑𝑚 ≤ 2𝑟𝑚,there exists a connected graph 𝐺with𝑟𝑎𝑑𝑚𝐺 = 𝑟𝑚 , 𝑑𝑖𝑎m𝑚𝐺 = 𝑑𝑚and 𝑜𝑐𝑑𝑚(𝐺)= 𝑙. also, it is shown that for every pair of integers 𝑎and b with 2 ≤ 𝑎 ≤ 𝑏, there exists a connected graph 𝐺with𝑑𝑚(𝐺) = 𝑎 and𝑜𝑐𝑑𝑚(𝐺) = 𝑏. keywords: chord, monophonic path, monophonic number, detour monophonic path, detour monophonic number, outer connected detour monophonic number. ams subject classification: 05c383 1 register number.20123162092018, research scholar. scott christian college (autonomous), nagercoil – 629003, india. beaulajohnwin@gmail.com 2 department of mathematics, scott christian college (autonomous), nagercoil – 629003, india prof.robinscc@gmail.com affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamil nadu, india 3 received on june15th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.921. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement 325 mailto:beaulajohnwin@gmail.com mailto:prof.robinscc@gmail.com n.e. johnwin beaula and s. joseph robin 1. introduction a finite, undirected connected graph with no loops or many edges is referred to as a graph 𝐺 = (𝑉, 𝐸). by 𝑛 and 𝑚, respectively, we indicate the order and size of 𝐺. we refer to [1] for the fundamental terms used in graph theory. if 𝑢𝑣 is an edge of g, then two vertices 𝑢 and 𝑣 are said to be adjacent. if two edges of 𝐺 share a vertex, they are said to to be adjacent. let 𝑆 ⊂ 𝑉 be any subset of vertices of 𝐺. then the graph with 𝑆 as its vertex set and all of its edges in 𝐸 having both of their end points in 𝑆 is the induced subgraph 𝐺[𝑆]. a vertex 𝑣 is an extreme vertex of a graph g if the subgraph induced by its neighbors is complete. the length of the shortest path in a connected graph g is equal to the distance d(u, v) between two vertices u and v. an u − vgeodesic is a u − v path with length d(u, v). an edge that connects two non-adjacent vertices of a path p is called the chord of p . a chordlessu − v path is referred to as a monophonic path. the monophonic distance 𝑑𝑚(𝑢, 𝑣)for two vertices 𝑢 and 𝑣 in a connected graph 𝐺 is the length of a longest 𝑢 − 𝑣 monophonic path in 𝐺. an u − vdetour monophonic path is one that has a length of dm(u, v).the monophonic eccentricity of a vertex v, denoted by em(v) is the monophonic distance between v and a vertex farthest from 𝑣. the monophonic radius, 𝑟𝑎𝑑𝑚(𝐺), and the monophonic diameter, 𝑑𝑖𝑎𝑚𝑚(𝐺) are the vertices respective minimum and maximum monophonic eccentricities. the closed interval 𝐽𝑑𝑚[𝑢, 𝑣] for two vertices 𝑢 and 𝑣 is consists of all the vertices along an 𝑢 − 𝑣 detour monophonic path, including the vertices 𝑢 and 𝑣. if 𝑣 ∈ 𝐸 , then𝐽𝑑𝑚[𝑢, 𝑣] = {𝑢, 𝑣}. for a set 𝑀 of vertices, let 𝐽𝑑𝑚[𝑀] = ∪𝑢,𝑣∈𝑀 𝐽[𝑢, 𝑣]. then certainly 𝑀 ⊆ 𝐽𝑑𝑚[𝑀]. if 𝐽𝑑𝑚[𝑀] = 𝑉,a set 𝑀 ⊆ 𝑉(𝐺) is referred to as a detour monophonic set of 𝐺.the detour monophonic number 𝑑𝑚(𝐺) of 𝐺 is the minimum order of its detour monophonic sets.any detour monophonic set of order 𝑑𝑚(𝐺) is referred to as an 𝑑𝑚-set of 𝐺. in [2-4], these concepts were investigated. the following theorem is used in sequel. theorem 1.1. [4] each extreme vertex of a connected graph 𝐺 belongs to everydetour monophonic set of 𝐺. 2. the outer connected detour monophonic number of a graph definition 2.1. if 𝑀 = 𝑉 or the subgraph 𝐺[𝑉 – 𝑀]is connected, then a detour monophonic set 𝑀 of a connected graph 𝐺 is said to be an outer connected detour monophonic setof 𝐺.the outer connecteddetourmonophonic number of 𝐺, indicated by the symbol 𝑜𝑐𝑑𝑚(𝐺), is the minimum cardinality of an outer connected detour monophonic set of 𝐺. the 𝑜𝑐𝑑𝑚-set of 𝐺 is a minimum cardinality of an outer connected detour monophonic setof 𝐺. 326 the outer connected detour monophonic number of a graph example 2.2. 𝑀 = {𝑣2, 𝑣4} is a 𝑑𝑚-set of the graph 𝐺 in figure 2.1 such that 𝑑𝑚(𝐺) = 2. 𝑀 is not an outer connected detour monophonic set of 𝐺 because 𝐺[𝑉 − 𝑀 ] is not connected, and as a result, 𝑜𝑐𝑑𝑚(𝐺) ≥ 3. now since 𝑀1= {𝑣2, 𝑣3, 𝑣4}, is a 𝑜𝑐𝑑𝑚-set of 𝐺, and 𝑜𝑐𝑑𝑚(𝐺) = 3 as a result. observation 2.3. (i) each extreme vertex of a connected graph 𝐺 belongs to every outer connected detour monophonic set of 𝐺. (ii) no cut vertex of 𝐺 belongs to any 𝑜𝑐𝑑𝑚-set of 𝐺. (iii)for any connected graph g of order𝑛, 2 ≤ 𝑑𝑚(𝐺) ≤ 𝑜𝑐𝑑𝑚(𝐺) ≤ 𝑛. theorem 2.4. 𝑜𝑐𝑑𝑚 (𝐺) = 𝑛, for the complete graph 𝐺 = 𝐾𝑛 (n ≥ 2). proof. the vertex set of 𝐾𝑛is the unique outer connected detour monophonic set of 𝐾𝑛 since every vertex of the complete graph 𝐾𝑛(𝑛 ≥ 2)is the extreme vertex. therefore, 𝑜𝑐𝑑𝑚(𝐺) = 𝑛. ∎ theorem 2.5. 𝑜𝑐𝑑𝑚(𝑇) = 𝑘, for any tree t with k end vertices. proof. let 𝑀 represent the collection of 𝑇′𝑠 end vertices. according to observation 2.3(i) and (ii), 𝑜𝑐𝑑𝑚 (𝑇) ≥ |𝑀|. 𝑀 is the unique outer connected detour monophonic set of 𝑇,since the subgraph 𝐺[𝑉 − 𝑀] is connected. consequently, 𝑜𝑐𝑑𝑚(𝑇) = |𝑀| = 𝑘.∎ corollary2.6. 𝑜𝑐𝑑𝑚(𝑃𝑛) = 2 for the non-trivial path 𝑃𝑛 (𝑛 ≥ 3). corollary2.7. 𝑜𝑐𝑑𝑚(𝐾1,𝑛−1) = 𝑛 − 1 for star 𝐾1,𝑛−1(𝑛 ≥ 3). theorem 2.8. 𝑜𝑐𝑑𝑚(𝐺) = 3, for the cycle 𝐺 = 𝐶𝑛(𝑛 ≥ 4). 𝑣1 𝐺 figure 2.1 𝑣4 𝑣3 𝑣5 𝑣2 327 n.e. johnwin beaula and s. joseph robin proof. set the cycle 𝐶𝑛 to be 𝑣1, 𝑣2, … , 𝑣𝑛 , 𝑣1. then, 𝑀 = {𝑣1, 𝑣2, 𝑣3} is a 𝐺 ′𝑠 outer connected detour monophonic set, resulting in 𝑜𝑐𝑑𝑚(𝐺) ≤ 3. we establish𝑜𝑐𝑑𝑚(𝐺) = 3. assume that 𝑜𝑐𝑑𝑚(𝐺)= 2. then 𝐺′𝑠𝑜𝑐𝑑𝑚-set is 𝑀1 = {𝑥, 𝑦}. it is obvious that 𝑥 and 𝑦 are not adjacent. a contradiction results since 𝐺[𝑉 − 𝑀1]is not connected and 𝑀1 is not a 𝐺 ′𝑠 𝑜𝑐𝑑𝑚 − set. ,consequently, 𝑜𝑐𝑑𝑚(𝐺) = 3. ∎ theorem 2.9. 𝑜𝑐𝑑𝑚 (𝐺) = { 𝑠, 𝑖𝑓 𝑟 = 1, 𝑠 ≥ 2 3, 𝑖𝑓 𝑟 = 𝑠 = 2 4, 𝑖𝑓 2 < 𝑟 ≤ 𝑠 for the complete bipartite graph 𝐺= 𝐾𝑟,𝑠 proof. 𝐺 = 𝐾𝑟,𝑠 is a tree with 𝑠 end vertices when 𝑟 = 1 and 𝑠 ≥ 2.therefore,𝑜𝑐𝑑𝑚 (𝐾1,𝑠) = 𝑠 as per corollary2.7.𝐺 = 𝐾2,2is the cyclec4 when 𝑟 = 𝑠 = 2 , thus, according to theorem 2.8, 𝑜𝑐𝑑𝑚(𝐾2,2) = 3. let 2 < 𝑟 ≤ 𝑠.let 𝑋 = {𝑥1, 𝑥2, … , 𝑥𝑚},𝑌 = {𝑦1, 𝑦2, … , 𝑦𝑛} be the bipartitions of 𝐺. let 𝑀 = {𝑥𝑖 , 𝑥𝑗 , 𝑦𝑘 , 𝑦𝑙 }, where𝑖 ≠ 𝑗, 𝑘 ≠ 𝑙.then m is adetour monophonic set of 𝐺. 𝑀 is anouter connected detour monophonic set of 𝐺 because the subgraph𝐺[𝑉 − 𝑀]is connected, and as a result, 𝑜𝑐𝑑𝑚(𝐺) ≤ 4. we demonstrate that 𝑜𝑐𝑑𝑚(𝐺) = 4. let's assume that 𝑜𝑐𝑑𝑚(𝐺) ≤ 3.then |𝑀| ≤ 3 and there exists a𝑜𝑐𝑑𝑚(𝐺)-set 𝑀. if 𝑀 ⊆ 𝑋 or 𝑀 ⊆ 𝑌 then 𝐺[𝑉 − 𝑀] is not connected. consequently, 𝑀 ⊂ 𝑋𝑈𝑌.which suggests 𝑀 is not aouter connected detour monophonic set of g, which is in contrast with the statement made earlier. thus𝑜𝑐𝑑𝑚(𝐺) = 4. ∎ theorem 2.10 𝑜𝑐𝑑𝑚 (𝐺) = { 2 𝑖𝑓 𝑛 = 5 3 𝑖𝑓 𝑛 ≥ 6 for the wheel 𝐺 = 𝐾1 + 𝐶𝑛−1 (𝑛 ≥ 5). proof. let's say that 𝑉(𝐾1) = 𝑥 and 𝑉(𝐶𝑛−1) = {𝑣1, 𝑣2, … … … , 𝑣𝑛−1 } . 𝑀 = {𝑣1, 𝑣3,} is an 𝑜𝑐𝑑𝑚 − 𝑠𝑒𝑡 of 𝐺 for 𝑛 = 5. therefore, 𝑜𝑐𝑑𝑚(𝐺) = 2. so, let 𝑛 ≥ 6. hence it follows that 𝑜𝑐𝑑𝑚(𝐺) ≥ 3 let 𝑀1 = {𝑣1, 𝑣2,𝑣3}. then 𝑀1 is an outer connected detour monophonic set of 𝐺. consequently, 𝑜𝑐𝑑𝑚(𝐺) = 3. ∎ theorem 2.11 𝑜𝑐𝑑𝑚 (𝐺) = { 2 𝑖𝑓 𝑛 = 4 3 𝑖𝑓 𝑛 > 4 ,for the graph 𝐺 = 𝐾1 + 𝑃𝑛−1 . proof. let's say that 𝑉(𝐾1) = 𝑥 and 𝑉(𝑃𝑛−1) = {𝑣1, 𝑣2, … … … , 𝑣𝑛−1 }. 𝑀1 = {𝑣1, 𝑣3 } is a𝑜𝑐𝑑𝑚 − 𝑠𝑒𝑡 𝑜𝑓 𝐺 for n = 4 , and 𝑜𝑐𝑑𝑚(𝐺) = 2. so, let 𝑛 ≥ 5. let 𝑀 = {𝑣1, 𝑣𝑛−1} be the extreme vertices of 𝐺.by observation 2.3(i)𝑀 is a subset of every 𝑜𝑐𝑑𝑚 − 𝑠𝑒𝑡 𝑜𝑓 𝐺. since m is not a outer connected detour monophonic set of g, 𝑜𝑐𝑑𝑚(𝐺) ≥ 3. now 𝑀2 = 𝑀 ∪ {𝑥} is a 𝑜𝑐𝑑𝑚 − 𝑠𝑒𝑡 𝑜𝑓 𝐺. 𝑠𝑜 𝑡ℎ𝑎𝑡 𝑜𝑐𝑑𝑚(𝐺) = 3. ∎ theorem 2.12. consider the connected graph 𝐺, where 𝑑𝑚(𝐺) = 2. if deg (𝑥) ≥ 3 for every 𝑥∈𝑉, then 𝑜𝑐𝑑𝑚(𝐺) = 2. proof. let the detour monophonic set of 𝐺 be 𝑀{𝑢, 𝑣},𝑑𝑒g(𝑥) ≥ 3 for𝑥 ∈ 𝑉, so 𝐺[𝑉 − 𝑀]is connected. as a result, 𝑀 is an outer connected detour monophonic set of 𝐺 so that 328 the outer connected detour monophonic number of a graph 𝑜𝑐𝑑𝑚(𝐺) = 2. ∎ theorem 2.13. suppose 𝐺 is a connected graph with d𝑚(𝐺) = 2. if 𝐺 has 2 possible outermost vertices 𝑢, 𝑣 ∈ 𝑉 that are not relatedand ∆[〈𝑉 − {𝑢, 𝑣}〉] = 𝑛 − 3.then𝑜𝑐𝑑𝑚(𝐺)= 2. proof. let 𝑢 and 𝑣 represent the outermost vertices of 𝐺. let 𝑀 = {𝑢, 𝑣}. 𝑀 is therefore a 𝐺 detour monophonic set. given that ∆ [〈𝑉 − {𝑢, 𝑣}〉] = 𝑛 − 3, and that 𝑢 and 𝑣 are not adjacent outermost vertices of 𝐺, 𝐺[𝑉 – 𝑀]is connected. as a result, 𝑀 is a outer connected detour monophonic set of 𝐺, and𝑜𝑐𝑑𝑚(𝐺) = 2. ∎ theorem 2.14. let 𝐺 be a connected graph of order 𝑛 that has precisely one vertex that is not a cut vertex and has a degree of n 1. then𝑜𝑐𝑑𝑚(𝐺) ≤ 𝑛 − 3. proof. let 𝑥 represent the non-cut vertex of g at the vertex of degreen– 1.since 𝑁(𝑥)is not complete, there exist at least two non-adjacent vertices, say 𝑦 and 𝑧 that are both members of 𝑁(𝑥) .there are at least two vertices say 𝑥1and 𝑥2 because 𝑁(𝑥)is the unique vertex of degree 𝑛 − 1, and they are both located on the 𝑦 − 𝑧detour monophonic path such that 𝑥1 ≠ 𝑥, 𝑥2 ≠ 𝑥. 𝑀 = 𝑉(𝐺) −{𝑥, 𝑥1,𝑥2} is a detour monophonic set of 𝐺. 𝑀 is an outer connected detour monophonic set of 𝐺 since the subgraph g[𝑉–𝑀]is connected, which causes𝑜𝑐𝑑𝑚(𝐺) ≤ 𝑛 − 3. ∎ theorem 2.15. let 𝐺 be an order 𝑛 ≥ 3. 𝑜𝑐𝑑𝑚(𝐺) ≤ 𝑛 – 1 if 𝐺 contains a cut vertex of degree n – 1. proof. let 𝑀 be a minimum outer connected detour monophonic set and 𝑣 be the cut vertex of degree 𝑛 − 1 in 𝐺. observation 2.3(ii) says that 𝑣 ∉ 𝑀. it is obvious that𝑜𝑐𝑑𝑚(𝐺) ≤ 𝑛 – 1, 𝑀 = 𝑉(𝐺) − {𝑣} is a detour monophonic set of 𝐺. 𝑀 is an outer connected detour monophonic set of 𝐺 since 𝑣 is a universal cut vertex of 𝐺, the subgraph g〈𝑉 − 𝑀〉is connected. as a result, 𝑀 is an outer connected detour monophonic set of 𝐺. which causes 𝑜𝑐𝑑𝑚(𝐺) ≤ 𝑛 − 1. ∎ theorem 2.16 there exists a connected graph 𝐺 with 𝑟𝑎𝑑𝑚𝐺= 𝑟𝑚, 𝑑𝑖𝑎𝑚𝐺 = 𝑑𝑚and 𝑜𝑐𝑑𝑚(𝐺) = 𝑙 for positive integers 𝑟𝑚, 𝑑𝑚and 𝑙 ≥ 2 with 𝑟𝑚 < 𝑑𝑚 ≤ 2𝑟𝑚. proof. we make the convenient assumptions that 𝑟𝑚= 𝑟 and 𝑑𝑚= 𝑑. let 𝐺 = 𝐾1,𝑙 when 𝑟 = 1. theorem 2.5 states that 𝑜𝑐𝑑𝑚(𝐺) = 𝑙 . let 𝑟𝑚 ≥ 2. let 𝐶𝑟+2: 𝑣1, 𝑣2, … 𝑣𝑟+2be a cycle of length 𝑟 + 2 and let 𝑃𝑑𝑚−𝑟𝑚+1: 𝑢0, 𝑢1, 𝑢2, … , 𝑢𝑑𝑚−𝑟𝑚 be that cycle. by locating 𝑣1 in 𝐶𝑟+2and 𝑢0in 𝑝𝑑𝑚−𝑟𝑚+1 , we may construct the graph 𝐻.the graph shown in figure 2.2 is then created by joining each of the 𝑤𝑖vertices (1 ≤ 𝑖 ≤ 𝑙 − 3) to the vertex 𝑢𝑑𝑚−𝑟𝑚−1 and adding new vertices𝑤1, 𝑤2, … , 𝑤𝑙−3to 𝑢𝑑𝑚−𝑟𝑚−1 so, 𝑟𝑎𝑑𝑚𝐺 = 𝑟𝑚, 𝑑𝑖𝑎𝑚𝐺 = 𝑑𝑚. the set of all 𝐺′𝑠 end vertices, 𝑊 ={𝑤1, 𝑤2,…,𝑤𝑙−3,𝑢𝑑𝑚−𝑟𝑚 }shall be defined. 𝑊 is then contained in every detour monophonic detour set of 𝐺 according to observation 2.3(i). since𝐽𝑑𝑚[𝑀] ≠ 𝑉, 𝑊 is not an outer connected detour monophonic set of 𝐺 and so 𝑜𝑐𝑑𝑚(𝐺) ≥ 𝑙 − 1. 𝑜𝑐𝑑𝑚(𝐺) ≥ 𝑙 because it is obvious that 𝑀 is not an outer connected detour 329 n.e. johnwin beaula and s. joseph robin monophonic set of 𝐺, where𝑀 = 𝑊 ∪{𝑢0} and𝑢0 ∉ 𝑀. 𝑜𝑐𝑑𝑚(𝐺) = 𝑙 because it is obvious that 𝑀 is an outer connected detour monophonic set of 𝐺, where 𝑀 = 𝑊 ∪{𝑣2, 𝑣3}. theorem 2.17. there is a connected graph g with 𝑑𝑚(𝐺) = 𝑎 and 𝑜𝑐𝑑𝑚(𝐺) = 𝑏 for every pair of positive integers a and b such that 2 ≤ 𝑎 ≤ 𝑏. proof. let 𝑉(𝐾2̅̅ ̅) = {𝑥, 𝑦}. let a graph be created by adding additional vertices to (𝐾2̅̅ ̅) as follows: 𝑧1, 𝑧2, … , 𝑧𝑎−1, 𝑣1, 𝑣2, … , 𝑣𝑏−𝑎, and connecting each 𝑧𝑖 (1 ≤ 𝑖 ≤ 𝑎 − 1) with 𝑥 and 𝑦. graph 𝐺 is displayed in figure 2.3. first, we demonstrate that 𝑑𝑚(𝐺) = 𝑎. assume that 𝑍 = {𝑧1, 𝑧2, … , 𝑧𝑎−1} is the collection of all end vertices of 𝐺. according to theorem1.1, every detour monophonic set of 𝐺 has 𝑍 as a subset. since it is obvious that 𝑍 is not a monophonic detour set of 𝐺, 𝑑𝑚(𝐺) ≥ 𝑎. now that 𝑍∪{𝑦} is a monophonic set, 𝑑𝑚(𝐺) = 𝑎. we then demonstrate that 𝑜𝑐𝑑𝑚(𝐺) = 𝑏. 𝑍 ∪ {𝑦}is not an outer connected detour monophonic set of 𝐺 because 𝐺[𝑉 − ( 𝑍 ∪ {𝑦})] is not connected. according to observation 2.3(i), each outer connected detour monophonic set of 𝐺 has the vertex 𝑧𝑖 (1 ≤ 𝑖 ≤ 𝑎 − 1). additionally, it is simple to see that any outer connected detour monophonic set of g contains each 𝑣𝑖 (1 ≤ 𝑖 ≤ 𝑏 − 𝑎),which means that 𝑜𝑐𝑑𝑚(𝐺) ≥𝑎 − 1 + 𝑏 − 𝑎 = 𝑏 – 1. let 𝑀 = 𝑍 ∪ {𝑣1, 𝑣2, … , 𝑣𝑏−𝑎}. since m is not an outer connected detour monophonic set of 𝐺, then 𝑜𝑐𝑑𝑚(𝐺) ≥ 𝑏. now that 𝑀 ∪ {𝑥} is an outer connected detour monophonic set of 𝐺 𝑜𝑐𝑑𝑚(𝐺) = 𝑏. 𝐺 figure 2.2 𝑢𝑑𝑚−𝑟𝑚 𝑢𝑑𝑚−𝑟𝑚−2 𝑢𝑑𝑚−𝑟𝑚−1 𝑣3 𝑣2 𝑣1 = u0 𝑢1 𝑢2 𝑤1 𝑤2 𝑤𝑙−3 𝑣𝑟+2 330 the outer connected detour monophonic number of a graph 3. conclusion this article established a novel detour monophonic distance parameter called the outer connected detour monophonic number of graphs. we will develop this concept to incorporate more distance considerations in a subsequent investigation. acknowledgements we are thankful to the referees for their constructive and detailed comments and suggestions which improved the paper overall. references [1] t. w. haynes, s. t. hedetniemi and p. j, slater, fundamentals of domination in graphs, marcel dekker, new york, (1998). [2] j. john, the forcing monophonic and the forcing geodetic numbers of a graph, indonesian journal of combinatorics 4(2), (2020), 114-125. [3] j. john and s. panchali 2, the upper monophonic number of a graph, int. j. math.combin. 4, (2010),46 – 52. [4] p. titus, k. ganesamoorthy and p. balakrishnan, the detour monophonic number of a graph. j. combin. math. combin. comput., (84), (2013),179-188. 𝑣1 𝑣2 𝑣3 𝑣𝑏−𝑎 𝑦 𝑧2 𝑧𝑎−1 𝑧1 𝑥 𝐺 figure 2.3 331 ratio mathematica issn: 1592-7415 vol. 31, 2016, pp. 37--64 eissn: 2282-8214 37 max-min fuzzy relation equations for a problem of spatial analysis 1ferdinando di martino, 2salvatore sessa 1università degli studi di napoli federico ii, dipartimento di architettura via toledo 402, 80134 napoli, italy fdimarti@unina.it 2università degli studi di napoli federico ii, dipartimento di architettura via toledo 402, 80134 napoli, italy salvatore.sessa@unina.it received on: 22-12-2016. accepted on: 25-01-2017. published on: 28-02-2017 doi: 10.23755/rm.v31i0.318 © ferdinando di martino and salvatore sessa abstract we implement an algorithm that uses a system of max-min fuzzy relation equations (sfre) for solving a problem of spatial analysis. we integrate this algorithm in a geographical information systems (gis) tool. we apply our process to determine the symptoms after that an expert sets the sfre with the values of the impact coefficients related to some parameters of a geographic zone under study. we also define an index of evaluation about the reliability of the results. keywords: fuzzy relation equation, max-min composition, gis, triangular fuzzy number 2010 ams subject classification: 03e72, 94d05. ferdinando di martino, salvatore sessa 38 1. introduction a geographical information system (gis) is used as a support decision system for problems in a spatial domain. we use a gis to analyse spatial distribution of data, the impact of event data on spatial areas: this analysis implies the creation of geographic thematic maps. several authors (cfr., e. g., [3], [4], [7], [8], [25]) solve spatial problems using fuzzy relational calculus. in this paper, we propose an inferential method to solve such problems based on an algorithm for the resolution of a system of fuzzy relation equations (shortly, sfre) given in [20] (cfr. also [21], [22]) and applied in [10] to solve industrial application problems. here we integrate this algorithm in the context of a gis architecture. usually a sfre with max-min composition is read as           mnmnm nn nn bxaxa bxaxa bxaxa )(...)( ... )(...)( )(...)( 11 2212 1 1111 1 (1) the system (1) is said consistent if it has solutions. sanchez [23] determines its greatest solution, moreover many researchers have found algorithms which determine minimal solutions of (1) (cfr., e. g., [1], [2], [5], [6], [9], [11]÷[24], [26]). in [20] and [21] a method is described for the consistence of the system (1). this method has been applied in this paper to real spatial problem in which the input data vary for each subzone of the geographical area. the expert starts from a valuation of input data and he uses linguistic labels for the determination of the output results for each subzone. the input data are the facts or symptoms, the parameters to be determined are the causes. for example, let us consider a planning problem. a city planner needs to determine in each subzone the mean state of buildings (x1) and the mean soil permeability (x2), knowing the number of collapsed building in the last year (b1) and the number of flooding in the last year (b2). the expert creates the sfre (1) for each subzone by setting the impact matrix a, whose entries aij (i=1,…,n and j=1,…,m) represent the impact of the j-th cause xj to the production of the i-th symptom bi, where the value of bi is the membership degree in the corresponding fuzzy set and let b=[b1,…,bm]. in another subzone, the input data vector b and the matrix a can vary. max-min fuzzy relation equations for a problem of spatial analysis 39 input extraction i1 0 1 ik input fuzzification b1 b2 b3 b4 b5 b6 b7 b8 b9 i2 bm-3 bm-2 bm-1 bm results fuzzification 0 1 x1 x2 x3 x4 x5 x6 x7 x8 x9 xn-2 xn-1 xn 0 1 output extraction o1 o2 oh sfre a x = b 10 10 10 0 1 0 1 0 1 sfre solving fig. 1. resolution process of a sfre the process of the resolution of the system (1) is schematized in fig. 1. we can determine the maximal interval solutions of (1). each maximal interval solution is an interval whose extremes are the values taken from a lower solution and from the greatest solution. every value xi belongs to this interval. if the sfre (1) is inconsistent, it is possible to determine the rows for which no solution is permitted. if the expert decides to exclude the row for which no solution is permitted, he considers that the symptom bi (for that row) is not relevant to its analysis and it is not taken into account. otherwise, the expert can modify the setting of the coefficients of the matrix a to verify if the new system has some solution. in general, the sfre (1) has t maximal interval solutions xmax(1),…,xmax(t). in order to describe the extraction process of the solutions, let xmax(t), t{1,…,t}, be a maximal interval solution given below, where x low is a lower solution and xgr is the greatest solution. our aim is to assign the linguistic label of the most appropriate fuzzy sets, usually triangular fuzzy numbers (briefly, tfn), corresponding to the unknown { sjjj xxx ,...,, 11 } related to an output variable os, s = 1,…,k. for example, assuming that inf(j), mean(j), sup(j) are the three fundamental values of the generic tfn xj , j=j1, …, js, respectively, we can write their membership functions hjjj  ,...,, 21 as follows: ferdinando di martino, salvatore sessa 40             otherwise 0 )(x )( if )()( )( )(x )inf(j if 1 11 11 1 11 j1 jsupjmean jmeanjsup xjsup jmean  (2) }j,...,{jj and otherwise 0 )(sx m ean(j) if )()( )( )(mx inf(j) if )()( )( 1-s2j                  jup jmeanjsup xjsup jean jinfjmean jinfx  (3)              otherwise 0 )(x )(ean if 1 )(x )( if )()( )( js ss ss ss s jsupjm jmeanjinf jinfjmean jinfx  (4) if xmint(j) (resp. xmaxt(j)) is the min (resp., max) value of every interval corresponding to the unknown xj, we can calculate the arithmetical mean value xmeant(j) of the j-th component of the above maximal interval solution xmax(t) as 2 )()( )( jxmaxjxmin jxmean tt t   (5) and we get the vector column xmeant = [xmeant(1),…, xmeant(n)] -1. the value given from max{xmeant(j1),…,xmeant(js)} obtained for the unknowns sj x,...,x 1j corresponding to the output variable os, is the linguistic label of the fuzzy set assigned to os and it is denoted by scoret(os), defined also as reliability of os in the interval solution t. for the output vector o = [o1,…,ok], we define the following reliability index in the interval solution t as max-min fuzzy relation equations for a problem of spatial analysis 41       k s stt oscore k ol 1 1 re (6) and then as final reliability index of o, the number rel(o)=max{relt(o):t=1,…,t}. the reliability of our solution is higher, the more the final reliability index rel(o) close to 1 is. in section 2 we give an overview of how finding the whole set of the solutions of a sfre. in section 3 we show how the proposed algorithm is applied in spatial analysis. section 4 contains the results of our simulation and it is divided in five subsections. 2. sfre: an overview the sfre (1) is abbreviated in the following known form: a ○ x = b where a = (aij), is the matrix of coefficients, x = (x1, x2,…, xn) -1 is the column vector of the unknowns and b = (b1,b2,…,bm) -1 is the column vector of the known terms, being aij, xj, bi  [0,1] for each i = 1,…,m and j = 1,…,n. we have the following definitions and terminologies: the whole set of all solutions x of the sfre (1) is denoted by  . a solution x̂  is called a minimal solution if x ≤ x̂ for some x  implies x= x̂ , where “≤” is the partial order induced in  from the natural order of [0, 1]. we also recall that the system (1) has the unique greatest (or maximum) solution 1 21 ),...,,(   g r n g rg rg r xxxx if  ≠ø [23]. a matrix interval xinterval of the following type: ],[ [...,...] ],[ ],[ 22 11 in t                nn erva l ba ba ba x where [aj,bj]  [0,1] for each j=1,…,n, is called an interval solution of the sfre (1) if every x=(x1,x2,…,xn) -1 such that ],[ jjj bax  for each j = 1,…,n, belongs to  . if aj is a membership value of a minimal solution and bj is a membership value of xgr for each j = 1,…,n, then xinterval is called a maximal interval solution of the sfre (1) and it is denoted by xmax(t) , where t varies from 1 till to the ferdinando di martino, salvatore sessa 42 number of minimal solutions. the sfre (1) is said to be in normal form if b1≥b2≥…≥bm. the time computational complexity to reduce a sfre in a normal form is polynomial [20, 22]. now we consider the matrix )(   ij aa so defined:          i ii i ij b bb b a ij ij ij * a if 1 a if a if 0 where i = 1,…,m and j = 1,…,n, that is  ij a is s—type coefficient (smaller) if aij<bi, e—type coefficient (equal) if aij=bi and g—type coefficient (greater) if aij>bi.  a is called augmented matrix and the system bxa    is said associated to the sfre (1). without loss of generality, from now on we suppose that the system (1) is in normal form. we also the following definitions and results from [16, 17, 20, 22]. definition 1. let sfre (1) be consistent and },...,{ 1   mjjj aaa . if  j a contains g-type coefficients and k{1,…,m} is the greatest index of row such that 1  kj a , then the following coefficients in  j a are called selected:  ij a for i{1,…,k} with kiij bba   ,  ij a for i{k+1,…,m} with iij ba   . definition 2. if  j a not contains g-type coefficients, but it contain e-type coefficients and r {1,…,m} is the smallest index of row such that rrj ba   , then any iij ba   in  j a for i{r,…,m} is called selected. theorem 1. let us consider a sfre (1). then the sfre (1) is consistent if and only if there exist at least one selected coefficient for each i-th equation, i=1,…,m. the complexity time function for determining the consistency of the sfre (1) is o(m∙n). consequently, when a sfre (1) is inconsistent, the equations for which no element is a selected coefficient, could not be satisfied simultaneously with the other equations having at least one selected coefficient. furthermore a vector ind=(ind(1),…,ind(m)) is defined by setting ind(i) equal to the number of selected coefficients in the ith equation for each i = l,...,m. if ind(i) = 0, then max-min fuzzy relation equations for a problem of spatial analysis 43 all the coefficients in the ith equation are not selected and the system is inconsistent. the system is consistent if ind(i) ≠ 0 if for each i = l,...,m and the product    m i iindpn 1 )(2 gives the upper bound of the number of the eventual minimal solutions. theorem 2. let sfre (1) be consistent. then the sfre has an unique greatest solution xgr with component k g r j bx  if the jth column  j a contains selected g-type coefficients kja and 1 g r j x otherwise. the complexity time function for computing xgr is o(m∙n). a help matrix h=[hij], i = 1,…,m and j = 1,…,n, is defined as follows:       otherwise 0 selected is a if iji ij b h let |hi| be the number of coefficients hij in the ith equation of the sfre (1). then the number of potential minimal solutions cannot exceed the value    m i i hpn 1 1 and one has 12 pnpn  . definition 3. let ),...,,( 21 iniii hhhh  and ),...,,( 21 knkkk hhhh  be the ith and the kth rows of the matrix h. if for each j=1,…n, 0ijh implies both 0kjh and ijkj hh  , then the ith row (resp. equation) is said dominant over the kth row in h (resp. equation) or that the kth row (resp. equation) is said dominated by the ith row (resp. equation). if the ith equation is dominant over the kth equation in (1), then the kth equation is a redundant equation of the system. by using definition 3, we can build a matrix of dimension m×n, called dominance matrix h*, having components: otherwise equationanother by dominated is equation ith theif 0 *     ij ij h h ferdinando di martino, salvatore sessa 44 for each i= 1, ...,m, now we set |  i h | as the number of coefficients 0  iij bh in the ith row of the dominance matrix h*. when this value is 0, we set |  i h | = 1. then the number of potential minimal solutions of the sfre cannot exceed the value    m i i hpn 1 *3 being 123 pnpnpn  [17, 20 ,22]. there the authors use the symbol j b i to indicate the coefficients 0  iij bh . we have ijij bxh   if ]1,[ ij bx  and ij bx  is the jth component of a minimal solution. a solution of the ith equation can be written as    n j i i j b h 1 in [20,22] the concept of concatenation w is introduced to determine all the components of the minimal solutions and it is given by              m i n j i m i i j b hw 1 11 we can determine the minimal solutions 1)()( 2 )( 1 )( ),...,,(   tlo w n tlo wtlo wtlo w xxxx , t )}3(,...,1{ pn , with components otherwise 0 0b if b tt ii)(      tlo w j x in order to determine if a sfre is consistent, hence its greatest solution and minimal solutions, we have used the universal algorithm of [20,22] based on the above concepts. for brevity of presentation, here we do not give this algorithm which has been implemented and tested under c++ language. the c++ library has been integrated in the esri arcobject library of the tool arcgis 9.3 for a problem of spatial analysis illustrated in the next section 3. 3. sfre in spatial analysis we consider a specific area of study on the geographical map on which we have a spatial data set of “causes” and we want to analyse the possible “symptoms”. max-min fuzzy relation equations for a problem of spatial analysis 45 we divide this area in p subzones where a subzone is an area in which the same symptoms are derived by input data or facts, and the impact of a symptom on a cause is the same one as well. it is important to note that even if two subzones have the same input data, they can have different impact degrees of symptoms on the causes. for example, the cause that measures the occurrence of floods may be due with different degree of importance to the presence of low porous soils or to areas subjected to continuous rains. afterwards the area of study is divided in homogeneous subzones, hence the expert creates a fuzzy partition for the domain of each input variable and he determines the values of the symptoms bi, as the membership degrees of the corresponding fuzzy sets (cfr., input fuzzification process of fig. 1) for each subzone on which the expert sets the most significant equations and the values aij of impact of the j-th cause to the ith symptom. after the determination of the set of maximal interval solutions, the expert for each interval solution calculates, for each unknown xj, the mean interval solution xmean(t) with (5). the linguistic label relt(os) is assigned to the output variable os . then he calculates the reliability index relt(o), given from formula (6), associated to this maximal interval solution t. after the iteration of this step, the expert determines the reliability index (6) for each maximal interval solution, by choosing the output vector o for which rel(o) assumes the maximum value. iterating the process for all the subzones (cfr., fig. 2), the expert can show the thematic map of each output variable. if the sfre related to a specific subzone is inconsistent, the expert can decide whether or not eliminate rows to find solutions: in the first case, he decides that the symptoms associated to the rows that make the system inconsistent are not considered and eliminates them, so reducing the number of the equations. in the second case, he decides that the corresponding output variable for this subzone remain unknown and it is classified as unknown on the map. 4. simulation results here we show the results of an experiment in which we apply our method to census statistical data agglomerated on four districts of the east zone of naples (italy). we use the year 2000 census data provided by the istat (istituto nazionale di statistica). these data contain informations on population, buildings, housing, family, employment work for each census zone of naples. every district is considered as a subzone with homogeneous input data given in table 2. in this experiment, we consider the following four output variables: “o1 = economic prosperity” (wealth and prosperity of citizens), “o2 = transition into the job” (ease of finding work), “o3 = social environment” (cultural levels of ferdinando di martino, salvatore sessa 46 citizens) and “o4 = housing development” (presence of building and residential dwellings of new construction). for each variable, we create a fuzzy partition composed by three tfns called “low”, “mean” and “high” presented in table 1. moreover, we consider the following seven input parameters: i1=percentage of people employed=number of people employed/total work force, i2=percentage of women employed=number of women employed/number of people employed, fig. 2. area of study: four districts at east of naples (italy) table 1. values of the tfns low, mean, high output low mean high inf mean sup inf mean sup inf mean sup o1 0.0 0.3 0.5 0.3 0.5 0.8 0.5 0.8 1.0 o2 0.0 0.3 0.5 0.3 0.5 0.8 0.5 0.8 1.0 o3 0.0 0.3 0.5 0.3 0.5 0.8 0.5 0.8 1.0 o4 0.0 0.3 0.5 0.3 0.5 0.8 0.5 0.8 1.0 i3=percentage of entrepreneurs and professionals = number of entrepreneurs and professionals/number of people employed, i4 = percentage of residents graduated=numbers of residents graduated/number of residents with age > 6 years, i5=percentage of new residential buildings=number of residential max-min fuzzy relation equations for a problem of spatial analysis 47 buildings built since 1982/total number of residential buildings, i6 = percentage of residential dwellings owned=number of residential dwellings owned/ total number of residential dwellings, i7 = percentage of residential dwellings with central heating system = number of residential dwellings with central heating system/total number of residential dwellings. in table 4 we show these input data for the four subzones. table 2. input data given for the four subzones district i1 i2 i3 i4 i5 i6 i7 barra 0.604 0.227 0.039 0.032 0.111 0.424 0.067 poggioreale 0.664 0.297 0.060 0.051 0.086 0.338 0.149 ponticelli 0.609 0.253 0.039 0.042 0.156 0.372 0.159 s. giovanni 0.576 0.244 0.041 0.031 0.054 0.353 0.097 table 3. tfns values for the input domains input var low mean high inf mean sup inf mean sup inf mean sup i1 0.00 0.40 0.60 0.40 0.60 0.80 0.60 0.80 1.00 i2 0.00 0.10 0.30 0.10 0.30 0.40 0.30 0.50 1.00 i3 0.00 0.04 0.06 0.04 0.06 0.10 0.07 0.20 1.00 i4 0.00 0.02 0.04 0.02 0.04 0.07 0.04 0.07 1.00 i5 0.00 0.05 0.08 0.05 0.08 0.10 0.08 0.10 1.00 i6 0.00 0.10 0.30 0.10 0.30 0.60 0.30 0.60 1.00 i7 0.00 0.10 0.30 0.10 0.30 0.50 0.30 0.50 1.00 ferdinando di martino, salvatore sessa 48 table 4: tfns for the symptoms b1 ÷ b12 subzone b1: i1 = low b2: i1 = me an b3: i1 = hi-gh b4: i2 = low b5: i2= me an b6: i2 = high b7: i3 = low b8: i3 = me an b9: i3 = high b10: i4 = low b11: i4 = mean b12: i4 = high barra 0.00 0.98 0.02 0.36 0.63 0.00 1.00 0.00 0.00 0.40 0.60 0.00 poggioreale 0.00 0.93 0.07 0.01 0.99 0.00 0.00 1.00 0.00 0.00 0.63 0.37 ponticelli 0.00 0.91 0.05 0.23 0.76 0.00 1.00 0.00 0.00 0.00 0.93 0.07 s. giovanni 0.12 0.88 0.00 0.28 0.72 0.00 0.95 0.05 0.00 0.45 0.55 0.00 the expert indicates a fuzzy partition for each input domain formed from three tfns labeled “low”, “mean” and “high”, whose values are reported in table 3. in tables 4 and 5 we show the values of tfns for the 21 symptoms b1,...,b21. in order to form the sfre (1) in each subzone, the expert defines the most significant symptoms. table 5: tfns for the symptoms b13 ÷ b21 subzone b13: i5 = low b14: i5 = mean b15: i5 = high b16: i6 = low b17: i6 = mean b18: i6 = high b19: i7 = low b20: i7 = mean b21: i7 = high barra 0.00 0.00 0.10 0.00 0.59 0.41 1.00 0.00 0.00 poggioreale 0.00 0.70 0.30 0.00 0.87 0.13 0.75 0.25 0.00 ponticelli 0.00 0.00 1.00 0.00 0.76 0.24 0.70 0.30 0.00 s. giovanni 0.87 0.13 0.00 0.00 0.82 0.18 1.00 0.00 0.00 max-min fuzzy relation equations for a problem of spatial analysis 49 4.1 subzone “barra” the expert chooses the significant symptoms b2, b4, b5, b7, b10, b11, b15, b17, b18, b19, by obtaining a sfre (1) with m = 10 equations and n = 12 unknowns. the matrix a of the impact values aij has dimensions 10×12 and the vector b of the symptoms bi has dimension 10×1 and both are given below. the sfre (1) is inconsistent and eliminating the rows for which the value ind(j) = 0, we obtain four maximal interval solutions xmax(t) (t=1,…,4) and we calculate the vector column xmeant on each maximal interval solution. hence we associate to the output variable os (s = 1,…,4), the linguistic label of the fuzzy set with the higher value calculated with formula (5) obtained for the corresponding unknowns sj x,...,x 1j and given in table 6. for determining the reliability of our solutions, we use the index given by formula (6). we obtain that relt(o1) = relt(o2) = relt(o3) = relt(o4) = 0.6025 for t=1,…,4 and hence rel(o)=max{relt(o): t=1,…,4}=0.6025 where o={o1,…o4}. we note that the same final set of linguistic labels associated to the output variables o1 = “high”, o2 = “mean”, o3 = “low”, o4 = “low” is obtained as well. the relevant quantities are given below.                                                                   00.1 41.0 59.0 10.0 60.0 40.0 00.1 63.0 36.0 98.0 b 0.01.00.10.03.04.00.03.04.00.02.05.0 5.04.02.05.05.01.04.04.01.04.04.01.0 3.07.03.01.05.02.01.04.01.02.05.02.0 3.03.01.01.01.02.01.02.01.01.01.01.0 1.02.01.03.07.02.03.07.03.03.07.03.0 0.00.01.01.04.06.01.04.06.01.03.05.0 0.00.03.02.02.08.01.03.08.00.02.00.1 0.00.00.02.07.02.02.07.02.02.07.02.0 0.00.00.02.06.03.04.05.04.02.05.03.0 2.03.01.03.07.02.02.00.10.40.01.00.5 a                                                                                                                                                             ]10.0,00.0[ ]10.0,00.0[ ]00.1,00.1[ ]41.0,41.0[ ]36.0,36.0[ ]00.1,00.0[ ]36.0,00.0[ ]00.1,00.0[ ]36.0,36.0[ ]00.1,00.0[ ]36.0,00.0[ ]40.0,40.0[ ]10.0,00.0[ ]10.0,00.0[ ]00.1,00.1[ ]41.0,41.0[ ]36.0,00.0[ ]00.1,00.0[ ]36.0,36.0[ ]00.1,00.0[ ]36.0,00.0[ ]00.1,00.0[ ]36.0,00.0[ ]40.0,40.0[ ]10.0,00.0[ ]10.0,00.0[ 00.1,00.1[ ]41.0,41.0[ ]36.0,00.0[ ]00.1,00.0[ ]36.0,00.0[ ]00.1,00.0[ ]36.0,36.0[ ]00.1,00.0[ ]36.0,00.0[ ]40.0,40.0[ ]10.0,00.0[ ]10.0,00.0[ ]00.1,00.1[ ]41.0,41.0[ ]36.0,00.0[ ]00.1,00.0[ ]36.0,00.0[ ]00.1,00.0[ ]36.0,00.0[ ]00.1,00.0[ ]36.0,36.0[ ]40.0,40.0[ )4max ()3max ()2max ()1max ( xxxx ferdinando di martino, salvatore sessa 50                                                                                                                                                             05.0 05.0 00.1 41.0 36.0 50.0 18.0 50.0 36.0 05.0 18.0 40.0 05.0 05.0 00.1 18.0 18.0 50.0 36.0 50.0 18.0 50.0 18.0 40.0 05.0 05.0 00.1 41.0 18.0 50.0 18.0 50.0 36.0 50.0 18.0 40.0 05.0 05.0 00.1 41.0 18.0 50.0 18.0 50.0 18.0 50.0 36.0 40.0 4321 xmeanxmeanxmeanxmean table 6. final linguistic labels for the output variables in the district barra output variable score1(os) score2(os) score3(os) score4(os) o1 high high high high o2 mean mean mean mean o3 low low low low o4 low low low low for determining the reliability of our solutions, we use the index given by formula (6). we obtain rel(ok) = 0.4675 for k = 1,..,12. then we obtain two final sets of linguistic labels associated to the output variables: o1 = “low”, o2 = “low”, o3 = “low”, o4 = “low”, and o1 = “low”, o2 = “low”, o3 = “low”, o4 = “mean”, with a same reliability index value 0.4675. the expert prefers to choose the second solution: o1 = “low”, o2 = “low”, o3 = “low”, o4 = “mean” because he considers that in the last two years in this district the presence of building and residential dwellings of new construction has increased although marginally. 4.2 subzone “poggioreale” the expert choices the significant symptoms b2, b5, b8, b11, b12, b14, b15, b17, b18, b19, b20, by obtaining a sfre (1) with m = 11 equations and n = 12 unknowns. the matrix a of the impact values aij has sizes dimension 11×12 and the column max-min fuzzy relation equations for a problem of spatial analysis 51 vector b of the symptoms bi has sizes 11×1 are given below. the sfre (7) is inconsistent and eliminating the rows for which the value ind(j) = 0, we obtain 12 maximal interval solutions xmax(t) (t=1,…,12) and we calculate the vector column xmeant on each maximal interval solution. table 7 contains the output variables and the relevant quantities are given below.                                                                           25.0 75.0 13.0 87.0 30.0 70.0 37.0 63.0 00.1 99.0 93.0 b 2.06.03.01.02.01.01.02.01.01.02.01.0 0.03.07.01.03.05.03.05.08.00.01.04.0 4.01.00.05.02.01.05.02.01.05.01.00.0 2.08.02.02.08.02.01.09.01.01.09.01.0 2.01.00.06.04.02.06.04.03.06.04.02.0 1.02.01.03.07.02.03.07.03.03.07.03.0 1.00.00.06.05.03.06.05.03.06.05.04.0 2.02.01.03.07.02.03.07.03.03.07.03.0 0.00.00.02.00.12.02.00.12.02.00.12.0 0.00.00.02.09.02.02.00.12.02.00.12.0 2.03.01.03.07.02.02.00.10.40.01.00.5 a                                                                                                                                                             ]13.0,00.0[ ]25.0,25.0[ ]25.0,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]00.1,00.0[ ]13.0,00.0[ ]13.0,13.0[ ]75.0,75.0[ ]13.0,00.0[ ]30.0,00.0[ ]37.0,37.0[ ]13.0,00.0[ ]25.0,00.0[ ]25.0,25.0[ ]13.0,00.0[ ]13.0,00.0[ ]00.1,00.0[ ]13.0,00.0[ ]13.0,13.0[ ]75.0,75.0[ ]13.0,00.0[ ]30.0,00.0[ ]37.0,37.0[ ]13.0,00.0[ ]25.0,25.0[ ]25.0,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]00.1,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]75.0,75.0[ ]13.0,13.0[ ]30.0,00.0[ ]37.0,37.0[ ]13.0,00.0[ ]25.0,00.0[ ]25.0,25.0[ ]13.0,00.0[ ]13.0,00.0[ ]00.1,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]75.0,75.0[ ]13.0,13.0[ ]30.0,00.0[ ]37.0,37.0[ )4max ()3max ()2max ()1max ( xxxx ferdinando di martino, salvatore sessa 52                                                                                                                                                             ]13.0,00.0[ ]25.0,25.0[ ]25.0,00.0[ ]13.0,00.0[ ]13.0,13.0[ ]00.1,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]75.0,75.0[ ]13.0,00.0[ ]30.0,00.0[ ]37.0,37.0[ ]13.0,00.0[ ]25.0,00.0[ ]25.0,25.0[ ]13.0,00.0[ ]13.0,13.0[ ]0.1,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]75.0,75.0[ ]13.0,00.0[ ]30.0,00.0[ ]37.0,37.0[ ]13.0,00.0[ ]25.0,25.0[ ]25.0,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]00.1,00.0[ ]13.0,00.0[ ]13.0,13.0[ ]75.0,75.0[ ]13.0,00.0[ ]30.0,00.0[ ]37.0,37.0[ ]13.0,00.0[ ]25.0,00.0[ ]25.0,25.0[ ]13.0,00.0[ ]13.0,00.0[ ]00.1,00.0[ ]13.0,00.0[ ]13.0,13.0[ ]75.0,75.0[ ]13.0,00.0[ ]30.0,00.0[ ]37.0,37.0[ )8max ()7max ()6max ()5max ( xxxx                                                                                                                                                             ]13.0,13.0[ ]25.0,25.0[ ]25.0,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]00.1,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]75.0,75.0[ ]13.0,00.0[ ]30.0,00.0[ ]37.0,37.0[ ]13.0,13.0[ ]25.0,00.0[ ]25.0,25.0[ ]13.0,00.0[ ]13.0,00.0[ ]00.1,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]75.0,75.0[ ]13.0,00.0[ ]30.0,00.0[ ]37.0,37.0[ ]13.0,00.0[ ]25.0,25.0[ ]25.0,00.0[ ]13.0,13.0[ ]13.0,00.0[ ]00.1,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]75.0,75.0[ ]13.0,00.0[ ]30.0,00.0[ ]37.0,37.0[ ]13.0,00.0[ ]25.0,00.0[ ]25.0,25.0[ ]13.0,13.0[ ]13.0,00.0[ ]00.1,00.0[ ]13.0,00.0[ ]13.0,00.0[ ]75.0,75.0[ ]13.0,00.0[ ]30.0,00.0[ ]37.0,37.0[ )1 2max ()1 1max ()1 0max ()9max ( xxxx                                                                                                                                                             065.0 250.0 125.0 065.0 065.0 500.0 065.0 130.0 750.0 065.0 150.0 370.0 065.0 125.0 250.0 065.0 065.0 500.0 065.0 130.0 750.0 065.0 150.0 370.0 065.0 250.0 125.0 065.0 065.0 500.0 065.0 065.0 750.0 130.0 150.0 370.0 050.0 125.0 250.0 065.0 065.0 500.0 065.0 065.0 750.0 130.0 150.0 370.0 4321 xmeanxmeanxmeanxmean max-min fuzzy relation equations for a problem of spatial analysis 53                                                                                                                                                             065.0 250.0 125.0 065.0 130.0 500.0 065.0 065.0 750.0 065.0 150.0 370.0 065.0 125.0 250.0 065.0 130.0 500.0 065.0 065.0 750.0 065.0 150.0 370.0 050.0 250.0 125.0 065.0 065.0 500.0 130.0 065.0 750.0 065.0 150.0 370.0 05.0 125.0 250.0 065.0 065.0 500.0 130.0 065.0 750.0 065.0 150.0 370.0 8765 xmeanxmeanxmeanxmean                                                                                                                                                             130.0 250.0 125.0 065.0 065.0 500.0 065.0 065.0 750.0 065.0 150.0 370.0 130.0 125.0 250.0 065.0 065.0 500.0 065.0 065.0 750.0 065.0 150.0 370.0 050.0 250.0 125.0 130.0 065.0 500.0 065.0 065.0 750.0 065.0 150.0 370.0 050.0 125.0 250.0 130.0 065.0 500.0 065.0 065.0 750.0 065.0 150.0 370.0 1 21 11 09 xmeanxmeanxmeanxmean for determining the reliability of our solutions, we use the index given by formula (6). we obtain rel(ok) = 0.4675 for k = 1,..,12. then we obtain two final sets of linguistic labels associated to the output variables: o1 = “low”, o2 = “low”, o3 = “low”, o4 = “low”, and o1 = “low”, o2 = “low”, o3 = “low”, o4 = “mean”, with a same reliability index value 0.4675. the expert prefers to choose the second solution: o1 = “low”, o2 = “low”, o3 = “low”, o4 = “mean” because he considers that in the last two years in this district the presence of building and residential dwellings of new construction has increased although marginally. ferdinando di martino, salvatore sessa 54 table 7. final linguistic labels for the output variables in the district “poggioreale” l i n g u i s t i c l a b e l s a s s o c i a t e d t o o u tp u t v a ri a b le x m e a n 1 x m e a n 2 x m e a n 3 x m e a n 4 x m e a n 5 x m e a n 6 x m e a n 7 x m e a n 8 x m e a n 9 x m e a n 1 0 x m e a n 1 1 x m e a n 1 2 o1 low low low high low low low high low low low high o2 low low low mea n low low low mea n low low low mea n o3 low low low low low low low low low low low low o4 low m e a n low m e a n low m e a n low m e a n low m e a n low m e a n 4.3 subzone: district ponticelli the expert choices the significant symptoms b2, b4, b5, b7, b11, b15, b17, b18, b19, b20, obtaining a sfre (7) with m = 10 equations and n = 12 variables: the matrix a of sizes 10×12 and the column vector b of dimension 10×1 are given by:                                                                   0.30 0.70 0.24 0.76 1.00 0.93 1.00 0.76 0.23 91.0 b 1.05.03.00.02.01.00.02.01.00.02.01.0 0.02.07.01.02.04.01.02.04.00.01.02.0 2.01.00.02.01.00.02.01.00.02.01.00.0 3.07.03.02.08.02.02.08.02.03.07.03.0 0.11.00.07.03.01.07.03.01.00.11.00.0 0.03.01.01.08.02.01.09.03.01.08.04.0 0.01.03.02.02.08.00.01.00.10.02.00.1 0.00.00.02.08.02.02.08.02.02.08.02.0 0.00.00.00.01.02.00.01.02.00.01.02.0 2.03.01.03.07.02.02.00.10.40.01.00.5 a max-min fuzzy relation equations for a problem of spatial analysis 55 the sfre (7) is inconsistent and eliminating the rows for which the value ind(j) = 0, we obtain 8 maximal interval solutions xmax(t) (t=1,…,8) and we calculate the vector column xmeant on each maximal interval solution. table 10 contains the output variables and the relevant quantities are given below.                                                                                                                                                             ]00.1,00.1[ ]30.0,00.0[ ]00.1,70.0[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.0[ ]00.1,00.0[ ]76.0,76.0[ ]00.1,00.1[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.0[ ]00.1,00.1[ ]30.0,00.0[ ]00.1,70.0[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.0[ ]00.1,00.0[ ]76.0,76.0[ ]00.1,00.0[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.1[ ]00.1,00.0[ ]30.0,00.0[ ]00.1,70.0[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.0[ ]00.1,00.0[ ]76.0,76.0[ ]00.1,00.1[ ]00.1,00.1[ ]76.0,00.0[ ]00.1,00.0[ ]00.1,00.0[ ]30.0,00.0[ ]00.1,70.0[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.0[ ]00.1,00.0[ ]76.0,76.0[ ]00.1,00.0[ ]00.1,00.1[ ]76.0,00.0[ ]00.1,00.1[ )4max ()3max ()2max ()1max ( xxxx                                                                                                                                                             ]00.1,00.1[ ]30.0,00.0[ ]00.1,70.0[ ]00.1,00.0[ ]76.0,76.0[ ]00.1,00.0[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.1[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.0[ ]00.1,00.1[ ]30.0,00.0[ ]00.1,70.0[ ]00.1,00.0[ ]76.0,76.0[ ]00.1,00.0[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.0[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.1[ ]00.1,00.0[ ]30.0,00.0[ ]00.1,70.0[ ]00.1,00.0[ ]76.0,76.0[ ]00.1,00.0[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.1[ ]00.1,00.1[ ]76.0,00.0[ ]00.1,00.0[ ]00.1,00.0[ ]30.0,00.0[ ]00.1,70.0[ ]00.1,00.0[ ]76.0,76.0[ ]00.1,00.0[ ]00.1,00.0[ ]76.0,00.0[ ]00.1,00.0[ ]00.1,00.1[ ]76.0,00.0[ ]00.1,00.1[ )8max ()7max ()6max ()5max ( xxxx                                                                                                                                                             00.1 15.0 85.0 50.0 38.0 50.0 50.0 76.0 00.1 50.0 38.0 50.0 00.1 15.0 85.0 50.0 38.0 50.0 50.0 76.0 50.0 50.0 38.0 00.1 50.0 15.0 85.0 50.0 38.0 50.0 50.0 76.0 00.1 00.1 38.0 5.0 50.0 15.0 85.0 50.0 38.0 50.0 50.0 76.0 50.0 00.1 38.0 00.1 4321 xmeanxmeanxmeanxmean ferdinando di martino, salvatore sessa 56                                                                                                                                                             00.1 15.0 85.0 50.0 76.0 50.0 50.0 38.0 00.1 50.0 38.0 50.0 00.1 15.0 85.0 50.0 76.0 50.0 50.0 38.0 50.0 50.0 38.0 50.0 50.0 15.0 85.0 50.0 76.0 50.0 50.0 38.0 00.1 00.1 38.0 50.0 50.0 15.0 85.0 50.0 76.0 50.0 50.0 38.0 50.0 00.1 38.0 00.1 8765 xmeanxmeanxmeanxmean now we associate to the output variables os k = 1,…,4, the linguistic label of the fuzzy set with the higher xmeanj obtained for the corresponding unknowns 1j x ,…, sj x obtaining: table 8. final linguistic labels for the output variables in the district “ponticelli” l i n g u i s t i c l a b e l s a s s o c i a t e d t o o u tp u t v a ri a b le x m e a n 1 x m e a n 2 x m e a n 3 x m e a n 4 x m e a n 5 x m e a n 6 x m e a n 7 x m e a n 8 o1 low-high high low low -high low -high high low low -high o2 mean low mea n low low -high low low -high low o3 low-high low-high low -high low -high mea n mea n mea n mea n o4 low low low low low low low low here “low-high” indicates that the membership degree of both the fuzzy sets with linguistic labels “low” and “high” have the maximal value for that output variable. we obtain for each solution rel(o1) =0.565, rel(o2) = 0.625, rel(o3) max-min fuzzy relation equations for a problem of spatial analysis 57 = 0.565 rel(o4) = 0.5, rel(o5) =0.565, rel(o6) = 0.69, rel(o7) = 0.565 rel(o8) = 0.565. thus we choice the solution o6 which have the greatest reliability rel(o6) = 0.69. our solution for this subzone is: o1 = “high”, o2 = “low”, o3 = “mean”, o4 = “low”. 4.4 subzone: district s. giovanni the expert choices the significant symptoms b2, b4, b5, b7, b11, b15, b17, b18, b19, b20, obtaining a sfre (1) with m = 12 equations and n = 12 variables: the matrix a of sizes 12×12 and the column vector b of sizes 12×1 are given by:                                                                               0.1 18.0 0.82 0.13 0.87 0.55 0.45 0.95 0.72 0.28 0.88 12.0 b 0.00.00.10.01.04.00.01.04.01.02.05.0 5.01.00.01.00.00.01.00.00.01.00.00.0 1.07.03.03.06.03.03.06.03.03.06.03.0 1.04.01.00.01.00.00.01.00.00.01.00.0 0.02.08.01.02.05.01.02.05.01.03.06.0 0.02.00.02.08.02.02.05.02.02.06.03..0 0.01.02.01.03.06.01.03.05.01.03.05.0 0.01.03.00.01.09.00.01.00.10.02.00.1 0.02.00.02.08.02.02.08.02.02.08.02.0 0.00.02.00.01.04.00.01.04.00.01.04.0 0.03.00.01.09.01.01.09.01.01.09.01.0 0.00.01.00.01.03.00.01.00.30.00.10.3 a the sfre (1) is inconsistent and eliminating the rows for which the value ind(j) = 0, we obtain 6 maximal interval solutions xmax(t) (t=1,…,6) and we calculate the vector column xmeant on each maximal interval solution. table 11 contains the output variables and the relevant quantities are given below. ferdinando di martino, salvatore sessa 58 ]18.0,18.0[ ]13.0,13.0[ ]00.1,00.1[ ]00.1,00.0[ ]55.0,00.0[ ]12.0,00.0[ ]00.1,00.0[ ]72.0,72.0[ ]12.0,12.0[ ]00.1,00.0[ ]55.0,55.0[ ]12.0,00.0[ ]18.0,18.0[ ]13.0,13.0[ ]00.1,00.1[ ]00.1,00.0[ ]55.0,55.0[ ]12.0,00.0[ ]00.1,00.0[ ]72.0,72.0[ ]12.0,00.0[ ]00.1,00.0[ ]55.0,00.0[ ]12.0,12.0[ ]18.0,18.0[ ]13.0,13.0[ ]00.1,00.1[ ]00.1,00.0[ ]55.0,00.0[ ]12.0,00.0[ ]00.1,00.0[ ]72.0,72.0[ ]12.0,00.0[ ]00.1,00.0[ ]55.0,55.0[ ]12.0,12.0[ )3(max ,)2(max ,)1max (                                                                                                                      xxx ]18.0,18.0[ ]13.0,13.0[ ]00.1,00.1[ ]00.1,00.0[ ]55.0,55.0[ ]12.0,12.0[ ]00.1,00.0[ ]72.0,72.0[ ]12.0,00.0[ ]00.1,00.0[ ]55.0,00.0[ ]12.0,00.0[ ]18.0,18.0[ ]13.0,13.0[ ]00.1,00.1[ ]00.1,00.0[ ]55.0,00.0[ ]12.0,12.0[ ]00.1,00.0[ ]72.0,72.0[ ]12.0,00.0[ ]00.1,00.0[ ]55.0,55.0[ ]12.0,00.0[ ]18.0,18.0[ ]13.0,13.0[ ]00.1,00.1[ ]00.1,00.0[ ]55.0,55.0[ ]12.0,00.0[ ]00.1,00.0[ ]72.0,72.0[ ]12.0,12.0[ ]00.1,00.0[ ]55.0,00.0[ ]12.0,00.0[ )6max ()5max ()4max (                                                                                                                      xxx 18.0 13.0 00.1 50.0 275.0 06.0 50.0 72.0 12.0 50.0 55.0 06.0 18.0 13.0 00.1 50.0 55.0 06.0 50.0 72.0 06.0 50.0 275.0 12.0 18.0 13.0 00.1 50.0 275.0 06.0 50.0 72.0 06.0 50.0 55.0 12.0 321                                                                                                                      xmeanxmeanxmean max-min fuzzy relation equations for a problem of spatial analysis 59 180.0 130.0 000.1 500.0 550.0 120.0 500.0 720.0 060.0 500.0 275.0 060.0 18.0 13.0 00.1 50.0 275.0 06.0 50.0 72.0 06.0 50.0 55.0 06.0 18.0 13.0 00.1 50.0 55.0 06.0 50.0 72.0 12.0 50.0 275.0 06.0 6514                                                                                                                      xmeanxmeanxmean table 9. final linguistic labels for the output variables in the district “san giovanni” output variabl e linguistic label associate d to xmean1 linguistic label associate d to xmean2 linguistic label associate d to xmean3 linguistic label associate d to xmean4 linguistic label associate d to xmean5 linguistic label associate d to xmean6 o1 mean high mean high mean high o2 mean mean mean mean mean mean o3 high mean high mean high mean o4 low low low low low low we obtain rel(ok) = 0.6925 for k = 1,…,6. thus we obtain two final sets of linguistic labels associated to the output variables: o1 = “mean”, o2 = “mean”, o3 = “high”, o4 = “low”, and o1 = “high”, o2 = “mean”, o3 = “mean”, o4 = “low” with the same reliability index value 0.6925. the expert prefers to choose the first solution: o1 = “mean”, o2 = “mean”, o3 = “high”, o4 = “low”, because he considers in this district that in the two years the presence of residents was graduated and consequently, the cultural level of citizens has increased, whereas the average pro capite wealth of citizens has decreased. ferdinando di martino, salvatore sessa 60 4.5 thematic maps and conclusions finally, we obtain four final thematic maps shown in figs. 3, 4, 5, 6 for the output variable o1, o2, o3, o4, respectively. fig. 3. thematic map for output variable o1 (economic prosperity) fig. 4. thematic map of the output variable o2 (transition into the job) fig. 5. thematic map for the output variable o3 (social environment) max-min fuzzy relation equations for a problem of spatial analysis 61 fig. 6. thematic map for the output variable o4 (housing development) the results show that there was no housing development in the four districts in the last 10 years and there is difficulty in finding job positions. in fig. 7 we show the histogram of the reliability index rel(o) for each subzone, where o=[o1,o2,o3,o4]. fig. 7. histogram of the reliability index rel(o) for the four subzones. this paper is a new reformulation of our work titled “spatial analysis and fuzzy relation equations” published in advances in fuzzy systems, volume 2011 (2011), article id 429498, 14 pages (http://dx.doi.org/10.1155/2011/429498) (under common license) where an extended version of the first three sections can be found, indeed an extended version of section 4 is here more complete with respect to section 4 presented there. ferdinando di martino, salvatore sessa 62 references 1. chen, l., wang, p.: fuzzy relational equations (i): the general and specialized solving algorithms. soft computing 6, 428—435 (2002) 2. de baets, b.: analytical solution methods for fuzzy relational equations. in: dubois, d., prade, h., (eds.) fundamentals of fuzzy sets, the handbooks of fuzzy sets series, vol. 1, pp. 291—340. kluwer academic publishers, dordrecht (2000) 3. di martino, f., loia, v., sessa, s.: a fuzzy-based tool for modelization and analysis of the vulnerability of aquifers: a case study. international journal of approximate reasoning 38, 98—111 (2005) 4. di martino, f., loia, v., sessa, s., giordano, m.: an evaluation of the reliability of a gis based on the fuzzy 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for land suitability classification. agricultural systems 83, 49—75 (2005) 26. wu, y.k., guu, s.m.: an efficient procedure for solving a fuzzy relational equation with max-archimedean t-norm composition. ieee transactions on fuzzy systems 16 (1), 73—84 (2008) ratio mathematica volume 47, 2023 mathematical modelling and application of reduced differential transform method for river pollution manan a. maisuria* priti v. tandel† abstract this paper presents the mathematical model of pollutant transport in a river. to effectively find the analytical solution of the advectiondiffusion equation under various forms of suitable initial conditions, the reduced differential transform method (rdtm) is used. three different initial concentration function cases, including rational, exponential, and power, are analyzed for the present model. a 2d and 3d visual comparison of the solutions obtained for each case is also shown. this article discusses the sufficient condition for convergence of the reduced differential transform approach to solving non-linear differential equations.the convergence results for the concentration functions in each case are briefly described. the present method is highly effective and more efficient in solving real-world problems. for all cases, the amount of phosphate pollutant concentration at various distances and time levels has been examined using numerical and graphical representations. while analyzing actual world problems, the current study demonstrates its effectiveness. keywords: pollutant transport equation; reduced differential transform method; convergence; river pollution 2020 ams subject classifications: 35a22 ,35c10, 35g05 1 *veer narmad south gujarat university, surat, gujarat, india; mananmaisuria.maths21@vnsgu.ac.in. †veer narmad south gujarat university, surat, gujarat, india; pvtandel@vnsgu.ac.in. 1received on november 17, 2022. accepted on march 21, 2023. published online on april 10, 2023. doi: 10.23755/rm.v41i0.955. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 342 manan a. maisuria, priti v. tandel 1 introduction accidents involving environmental contamination are common in the process plant, particularly in sectors like the chemical sector, manufacture of agricultural chemicals, natural gas extraction, etc [li et al., 2009]. organic materials and heavy metals are frequent and harmful contaminants in water pollution incidents, and examples of organic materials include benzene, naphthalene, phenol, anthracene, alcohol, etc. as a typical hydrocarbon, benzene is poisonous, and cancer-causing [nomura et al., 2019] it may be inhaled or absorbed via the skin. because of its low vapor pressure, benzene may be easily spread through the air. exposure to benzene in the past is a common factor in developing leukemia [jiang et al., 2018, tsuji et al., 2018, meszaros et al., 2017]. in many places, industrial or household human activity-related water contamination is a serious issue [tchobanoglous et al., 1991]. the contamination of water sources is responsible for the deaths of over 25 million people annually. models to manage and forecast water quality are vital. when analyzing river water quality, numerous aspects must be addressed, including dissolved oxygen, nitrates, chlorides, phosphates, suspended particles, environmental hormones, and chemical oxygen demand, such as heavy metals and bacteria. agricultural pollution may degrade surface, and groundwater [knight et al., 2000]. to satisfy its many needs, society relies heavily on river water, one of the few abundant sources of freshwater [shi et al., 2019]. to ensure an undisturbed freshwater supply, specific water quality requirements along the rivers must be maintained [chen et al., 2016]. agricultural non-point source pollution (anpsp), caused by the use of agrochemicals in farming, significantly impacts water quality and aquatic ecosystems [bryan and kandulu, 2011, borges et al., 2017]. in 1925, the well-known model of streeter and phelps characterized the equilibrium of dissolved oxygen in rivers, marking the beginning of the era of mathematical water quality models. since then, there have been many updates to this model [streeter, 1925, james, 1978]. weighted discretizations and the two-dimensional modified equation method solved the linear, constant coefficient advection-diffusion equation. the modified equivalent equation determines oneand multi-dimensional finite difference method accuracy [noye and tan, 1989]. the eulerian-lagrangian localized adjoint method (ellam) solves the nonlinear buckley-leverea equation, which has degenerate diffusion and sharpening near-shock solutions [dahle et al., 1995]. it is thought that a suitable strategy for identifying and evaluating the production of nutrients produced by management scenarios, which may aid in project prioritization and improve water quality, is extensive modeling of the surface water using tried-and-true techniques. they should be simulated as management scenarios before implementing plans to evaluate their effectiveness 343 mathematical modelling and application of reduced differential transform method for river pollution [fakouri et al., 2019]. numerous research on the impact of various water management strategies on water quality and quantity have been carried out using multiple models and experimental techniques in diverse places of different sizes with varying objectives. they used a mike11 pattern on the pasikhan river and simulated nitrate and phosphate contaminant concentration. the effects of dumping waste water and draining water into rivers are significant and impact the river’s water quality. in addition, kerich assessed the chemicals used in the ahero irrigation scheme and provided many recommendations to enhance the quality of water retrieved from the drainage canals. for this reason, the most efficient means of purifying water for human consumption in the area were biodegradable chemicals for pest and herbicide management and bio-sand filters [kerich, 2020]. groundwater quality in the blinaja river basin was also investigated by çadraku using irrigation water quality criteria. according to the findings, the groundwater in the research region is of sufficient quality for irrigating the crops. as well as addressing surface water issues, specific recommendations were made for preserving groundwater quality [çadraku, 2021]. to solve the advection-dispersion equation (ade) in rivers backward in time and a one-dimensional domain for different pollution loading patterns, an unique analytical approach was devised using the quasi-reversibility (qr) technique and the fourier transform tool. to avoid the issue being ill-posed during the inverse solution process, a stability factor is added to the initial transport equation in this approach [permanoon et al., 2022]. mass movement is regulated by the molecular diffusion of solutes between mobile and static water in aquifers like the chalk, which have long diffusion path lengths [bibby, 1981]. in this paper, we formulate a one-dimensional mathematical model of pollutant transport. the governing equation is a 1d advection-diffusion equation solved by the reduced differential transform method (rdtm). this method requires an initial condition. to generate the initial condition, we have used the concentration of the river khobistskali for po4 pollutant component [tsuji et al., 2018]. also, we have discussed the convergence of analytic solutions obtained by rdtm. section 2 covers the mathematical formulation of this problem. section 3 contains the fundamental ideas behind the reduced differential transform method. in section 4, the process for achieving the convergence of the analytic series solution given by rdtm has been discussed. section 5 includes the numerical outcomes and the convergence of the method for its effectiveness. 2d and 3d plots show a visual representation of the obtained solutions. section 6 provides a summary of the conclusion. 344 manan a. maisuria, priti v. tandel 2 mathematical formulation of the problem reaction, diffusion, advection, absorption, and sedimentation all have a role in transporting the pollutant material farther downstream. variables such as the kind of pollution, its physicochemical characteristics, flow characteristics, and the surrounding environment all have a role. thus, parameters linked to the flow of pollution are prioritized above those relating to the pollutant’s nature [kim and chapra, 1997]. a linear partial differential equation, the mass transfer equation, is often utilized in research on water, soil, petroleum, the living environment, and several engineering subjects. a linear parabolic partial differential equation, the aforementioned equation is of the first and second orders in terms of time and space, respectively. in the one-dimensional domain (along the river length), the general form of this equation under unstable and non-uniform flow regimes is as follows [amiri et al., 2021]. a ∂c ∂t = ∂ ∂x ( adx ∂c ∂x ) − av ∂c ∂x − akc + af (1) where c =pollutant concentration, dx =diffusion coefficient along the xdirection, v = mean flow velocity, k = coefficient of non-conservation, a = flow area, f = source term, x = distance from starting point of domain, t = time dimension. for the entire study, the area of the cross-section of the river is considered a constant. to analyze this problem, we have used the concentration of pollutant substance po4 of river khobistskali. here khobistskali river’s length is 44800 m. using past collected data, at time t = 0 (any fixed time), the concentration is assumed as a rational, exponential, and power form, and the following values of parameters are used to obtain the solutions. dx = 0.55 m2 sec , v = 0.534930 m sec , k = 0 1 sec and f = 0 [kachiashvili et al., 2007]. goodness of fit curve fitting sse r-square adjusted r-square rmse rational 3.89e-06 0.9884 0.9876 0.0003662 exponential 1.62e-05 0.9515 0.9499 0.0007354 power 5.55e-06 0.9834 0.9823 0.0004376 table 1: statistical indices of initial function graphical representations of curve fitting for the initial condition are shown in figures 1, 2, and 3. table 1 lists the goodness of fit values for various initial conditions. hence this problem is studied for three following different cases: 345 mathematical modelling and application of reduced differential transform method for river pollution 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x(m) distance 104 0.004 0.006 0.008 0.01 0.012 0.014 0.016 c (m g /l ) -c o n c e n tr a ti o n rational curve fitting of c(x,0) c vs. x untitled fit 1 goodness of fit: sse: 3.889e-06 r-square: 0.9884 adjusted r-square: 0.9876 rmse: 0.0003662 figure 1: rational curve fitting of c(x, 0). 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x(m) distance 104 0.004 0.006 0.008 0.01 0.012 0.014 0.016 c (m g /l ) c o n c e n tr a ti o n exponential curve fitting of c(x,0) c vs. x untitled fit 1 goodness of fit: sse: 1.623e-05 r-square: 0.9515 adjusted r-square: 0.9499 rmse:0.0007354 figure 2: exponential curve fitting of c(x, 0). 346 manan a. maisuria, priti v. tandel 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x(m) distance 104 0.004 0.006 0.008 0.01 0.012 0.014 0.016 c (m g /l ) c o n c e n tr a ti o n power curve fitting of c(x,0) c vs. x untitled fit 1 goodness of fit: sse: 5.552e-06 r-square: 0.9834 adjusted r-square: 0.9823 rmse: 0.0004376 figure 3: power curve fitting of c(x, 0). case-1 rational initial function for this case, c(x, 0) = 0.0004008x + 251.4 x + 1.39e + 04 (2) with c(44800, t) = 5.846e − 13t2 + 3.463e − 08t + 0.004597 (3) case-2 exponential initial function for this case, c(x, 0) = 0.01533e−3e−05x − 0.0001573 (4) with c(44800, t) = 6.887e − 13t2 + 6.167e − 08t + 0.003846 (5) case-3 power initial function for this case, c(x, 0) = −0.007562x0.1384 + 0.03762 (6) with c(44800, t) = 6.846e − 08t + 0.004247 (7) 347 mathematical modelling and application of reduced differential transform method for river pollution 3 reduced differential transform method let b(ω, τ) be a two-variable function. suppose b(ω, τ) is written as b(ω, τ) = f(ω)g(τ).b(ω, τ) can be represented as the following using the features of the differential transform: b(ω, τ) = ∞∑ i=0 f(i)ω i ∞∑ j=0 g(j)τ j = ∞∑ k=0 bk(ω)τ k (8) where bk(ω) is referred to as the t-dimensional spectrum function of b(ω, τ). bm(ω) = 1 m! [ ∂m ∂τm b(ω, τ) ] τ=0 (9) the original function is denoted by the lowercase [b(ω, τ)], whereas the altered function is denoted by the capital [b(ω, τ)]. the way to define the differential inverse transform of bk(ω) is as follows: b(ω, τ) = ∞∑ m=0 bm(ω)τ m (10) from equations (9) and (10), we get b(ω, τ) = ∞∑ m=0 1 m! [ ∂m ∂τm b(ω, τ) ] τ=0 τm (11) let us consider the following nonlinear pde, to understand the basic concept of rdtm. tb(ω, τ) + pb(ω, τ) + ob(ω, τ) = f(ω, τ) (12) with initial condition b(ω, 0) = η(ω) , where t = ∂ ∂τ , pb(ω, τ) is a linear term that has partial derivatives, while ob(ω, τ) is a non-linear term, and f(ω, τ) is a source term [al-amr, 2014]. by applying the transform on equation (12), we get (m + 1)bm+1(ω) = fm(ω) − pbm(ω) − obm(ω) (13) where bm(ω),fm(ω),pbm(ω) and obm(ω) are transform of b(ω, τ),f(ω, τ),pb(ω, τ) 348 manan a. maisuria, priti v. tandel and ob(ω, τ) respectively. we are able to write this down based on the initial condition. b0(ω) = η(ω) (14) from equations (13) and (14), we get the values of bm(ω). after that, an approximation solution is produced by carrying out an inverse transformation on the set of values {bm(ω)} n m=0. this transformation yields an approximation solution as b̃n(ω, τ) = n∑ m=0 bm(ω)τ m (15) where n is the order of approximation answer. consequently, the exact solution is given by [al-amr, 2014], b(ω, τ) = lim n→∞ b̃n(ω, τ) (16) function transformation b(ω, τ) bm(ω) = 1 m! [ ∂m ∂τm b(ω, τ) ] τ=0 αf(ω, τ) ± βg(ω, τ) αfm(ω) + βgm(ω) ωkτn ωkδ(m − n) ωkτnb(ω, τ) ωkbm−n(ω) l(ω, τ) = f(ω, τ)g(ω, τ) lm(ω) = ∑m r=0 fr(ω)gm−r(ω) ∂r ∂τr b(ω, τ) (m+r)! m! bm+r(ω) ∂ ∂ω b(ω, τ) ∂ ∂ω bm(ω) table 2: transform table[al-amr, 2014, keskin and oturanc, 2010, srivastava et al., 2014] 4 convergence of rdtm to understand the convergence, let us consider the solution of equation (13) in power series form as follow b(ω, τ) = ∞∑ n=0 bn(ω)τ n = ∞∑ n=0 bnτ n (17) 349 mathematical modelling and application of reduced differential transform method for river pollution which is obtained by equation (16) [moosavi noori and taghizadeh, 2021, saeed and mustafa, 2017]. theorem 4.1. if ∑∞ n=0 bnτ n is given series, [1] ∃ 0 < β < 1 ∋ ∥bn+1∥∥bn∥ ≤ β ⇒ series is convergent. [2] ∃ β > 1 ∋ ∥bn+1∥∥bn∥ ≥ β ⇒ series is divergent. proof. let (c[l], ∥.∥) represent the banach space that contains all continuous functions on l that satisfy the norm ∥.∥. also, let’s suppose that ∥b0(ω)∥ ≤ m, where m is an integer in the positive range. let {δn}∞n=0 be a partial sum δn = b0 + b1 + b2 + ... + bn if we can prove that {δn}∞n=0 is a cauchy sequence in banach space, then we can conclude that the sequence of partial sum is convergent in banach space. as a result, at this point, we shall demonstrate that the series of partial sums follows the cauchy sequence. we take ∥δn+1 − δn∥ = ∥bn+1∥ ≤ β∥bn∥ ≤ ... ≤ βn+1∥b0∥ ≤ βn+1m therefore,∀n, m ∈ n, n ≥ m ,we have ∥δn − δm∥ = ∥(δn − δn−1) + (δn−1 − δn−2) + ... + (δm+1 − δm)∥ ≤ ∥δn − δn−1∥ + ∥δn−1 − δn−2∥ + ... + ∥δm+1 − δm∥ ≤ 1−β n−m 1−β β m+1∥b0∥ now here 0 < β < 1, we obtain lim n,m→∞ ∥δn − δm∥ = 0 hence {δn}∞n=0 is cauchy sequence in banach space. therefore, given series is convergent. 350 manan a. maisuria, priti v. tandel 5 result and discussion we take three cases of initial function in rational, exponential and power form and obtain three solutions using reduced differential transform method. case-1 rational initial function solving equation (1) by rdtm with initial condition (2), we get c(x, t) = m0 + m1t + m2t 2 + m3t 3 + m4t 4 + ... where m0 = 0.0004008x+251.4 x+1.39e+04 m1 = 0.0270 ( 4863 x + 67605700 ) (x+13900)3 m2 = 8.9236e−06   x2 + 219193889400 x +1523735609830000   (x+13900)5 m3 = 9.8159e−10   x3 + 1599028070073300 x2 +22233066963300870000 x +103043693991250631000000   (x+13900)7 m4 = 1.0798e−13   186421425071787 x4 +10368864699446257200 x3 +216270792211342627620000 x2 +2004850662624966111612000000 x +6969433288195869001126700000000   (x+13900)9 here, ∥m1∥ ∥m0∥ = 3.4116405e − 05 < 1, ∥m2∥ ∥m1∥ = 3.4972143e − 05 < 1, ∥m3∥ ∥m2∥ = 3.4976844e − 05 < 1, ∥m4∥ ∥m3∥ = 3.4981545e − 05 < 1 therefore, the solution function c(x, t) is convergent. 351 mathematical modelling and application of reduced differential transform method for river pollution 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 figure 4: c(x, t) at t = 1000 sec 352 manan a. maisuria, priti v. tandel 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 figure 5: c(x, t) at x = 7000 m 353 mathematical modelling and application of reduced differential transform method for river pollution two-dimensional representations of the concentration function for case-1 are provided in figures 4, and 5, respectively, for the fixed values of (t = 1000sec),and (x = 7000m).as shown in figure 4, we can see that the value of concentration decreases as the length (x) variable increases. in figure 5, we can see that the value of concentration rises as the passage of time (t) increases. x(m)\t(sec) 4500 9000 13500 18000 22500 27000 31500 36000 1400 0.019467 0.023775 0.030123 0.039479 0.053046 0.072264 0.098812 0.134601 5740 0.014666 0.016968 0.020065 0.024264 0.029941 0.03754 0.047573 0.060618 10080 0.011796 0.013224 0.015034 0.017353 0.020329 0.024138 0.028979 0.035078 14420 0.009888 0.010858 0.01204 0.013494 0.015289 0.017506 0.020238 0.023587 18760 0.008527 0.009229 0.01006 0.011051 0.01224 0.013668 0.015383 0.017437 23100 0.007507 0.008038 0.008654 0.009372 0.010213 0.011201 0.012363 0.013728 27440 0.006715 0.007131 0.007605 0.008148 0.008772 0.009493 0.010326 0.01129 31780 0.006082 0.006416 0.006792 0.007217 0.007698 0.008246 0.00887 0.009581 36120 0.005564 0.005839 0.006144 0.006485 0.006868 0.007297 0.00778 0.008325 40460 0.005133 0.005362 0.005616 0.005896 0.006206 0.006551 0.006936 0.007365 44800 0.004768 0.004963 0.005176 0.00541 0.005667 0.00595 0.006264 0.00661 table 3: c(x, t) for case-1 the values of the concentration function for case-1 are shown in table 3. case-2 exponential initial function solving equation (1) by rdtm with initial condition (4), we get c(x, t) = a0 + a1t + a2t 2 + a3t 3 + a4t 4 + ... where a0 = 0.01533e −3e−05x − 0.0001573 a1 = 2.4602e − 07 e− 3 x 100000 a2 = 1.9741e − 12 e− 3 x 100000 a3 = 1.0561e − 17 e− 3 x 100000 a4 = 4.2370e − 23 e− 3 x 100000 here, ∥a1∥ ∥a0∥ = 1.6221987e − 05 < 1, ∥a2∥ ∥a1∥ = 8.0241975e − 06 < 1, 354 manan a. maisuria, priti v. tandel ∥a3∥ ∥a2∥ = 5.3494650e − 06 < 1, ∥a4∥ ∥a3∥ = 4.0120987e − 06 < 1 therefore, the solution function c(x, t) is convergent. figures 6 and 7 illustrate, 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 2 4 6 8 10 12 14 16 10 -3 figure 6: c(x, t) at t = 1000 sec respectively, two-dimensional representations of the concentration function for case-2 with the fixed values of (t = 1000 sec) and (x = 7000 m). as shown in figure 6, the concentration value falls as the distance (x) variable rises. it is clear from graph 7 that when time (t) grows, so does the value of concentration. the values of a concentration function for case-2 are presented in table 4. case-3 power initial function solving equation (1) by rdtm with initial condition (6), we get c(x, t) = p0 + p1t + p2t 2 + p3t 3 + p4t 4 + ... 355 mathematical modelling and application of reduced differential transform method for river pollution 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.02 0.021 0.022 figure 7: c(x, t) at x = 7000 m x(m)\t(sec) 4500 9000 13500 18000 22500 27000 31500 36000 1400 0.015643 0.016826 0.018098 0.019465 0.020934 0.022512 0.024208 0.026029 5740 0.013714 0.014753 0.015869 0.01707 0.018359 0.019745 0.021233 0.022832 10080 0.012021 0.012933 0.013913 0.014966 0.016099 0.017315 0.018622 0.020025 14420 0.010534 0.011335 0.012195 0.01312 0.014114 0.015182 0.016329 0.017561 18760 0.009229 0.009932 0.010687 0.011499 0.012372 0.01331 0.014317 0.015398 23100 0.008083 0.0087 0.009363 0.010076 0.010842 0.011665 0.01255 0.013499 27440 0.007077 0.007619 0.008201 0.008827 0.0095 0.010222 0.010998 0.011832 31780 0.006194 0.006669 0.007181 0.00773 0.008321 0.008955 0.009637 0.010368 36120 0.005418 0.005836 0.006285 0.006767 0.007286 0.007843 0.008441 0.009083 40460 0.004738 0.005104 0.005498 0.005922 0.006377 0.006866 0.007391 0.007955 44800 0.00414 0.004462 0.004808 0.00518 0.005579 0.006009 0.00647 0.006965 table 4: c(x, t) for case-2 356 manan a. maisuria, priti v. tandel where p0 = −0.007562x0.1384 + 0.03762 p1 = 3.4537e−07 (1621 x+1436) x2327/1250 p2 = 1.2902e−04 x2+4.9388e−04 x+7.2656e−04 x4827/1250 p3 = 4.2826e−05 x3+3.7801e−04 x2+0.0015 x+0.0025 x7327/1250 p4 = 1.6389e−05 x4+2.6028e−04 x3+0.0020 x2+0.0078 x+0.0138 x9827/1250 here, ∥p1∥ ∥p0∥ = 6.4108463e − 05 < 1, ∥p2∥ ∥p1∥ = 0.0001649 < 1, ∥p3∥ ∥p2∥ = 0.0002379 < 1, ∥p4∥ ∥p3∥ = 0.0002747 < 1 therefore, the solution function c(x, t) is convergent. figures 8 and 9 are twodimensional representations of the concentration function for case-3 with fixed parameters ( t = 1000sec) and (x = 7000m), respectively. as shown in figure 8, as the distance (x) variable increases, the concentration value decreases. graph 9 demonstrates that as time (t) increases, so does the value of concentration. the x(m)\t(sec) 4500 9000 13500 18000 22500 27000 31500 36000 1400 0.034284 0.149754 0.560422 1.579016 3.633997 7.269557 13.14562 22.03783 5740 0.014373 0.017404 0.02281 0.032238 0.047824 0.072202 0.108497 0.160329 10080 0.011539 0.012841 0.014609 0.017063 0.020482 0.025201 0.031609 0.040152 14420 0.00987 0.010713 0.011738 0.013011 0.014609 0.016626 0.019168 0.022356 18760 0.00866 0.009291 0.010017 0.010867 0.011871 0.013066 0.014494 0.016203 23100 0.007701 0.008209 0.008775 0.009415 0.010143 0.010977 0.011937 0.013046 27440 0.006902 0.007329 0.007796 0.008311 0.008883 0.009521 0.010237 0.011043 31780 0.006215 0.006584 0.006983 0.007416 0.007888 0.008405 0.008974 0.009603 36120 0.00561 0.005937 0.006285 0.006659 0.007062 0.007498 0.007971 0.008486 40460 0.005069 0.005362 0.005673 0.006003 0.006355 0.006732 0.007137 0.007573 44800 0.004579 0.004845 0.005126 0.005422 0.005735 0.006068 0.006423 0.006801 table 5: c(x, t) for case-3 values of a concentration function for case-3 are shown in table 5. figure 10 give the 3d graphical comparison of solution obtained for all cases. also figures 11 and 12 give the 2d graphical comparison of solution obtained for all cases for t = 3000 sec and x = 40000 m respectively. 357 mathematical modelling and application of reduced differential transform method for river pollution 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 figure 8: c(x, t) at t = 1000 sec 358 manan a. maisuria, priti v. tandel 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 figure 9: c(x, t) at x = 7000 m 359 mathematical modelling and application of reduced differential transform method for river pollution figure 10: 3d comparison of c(x, t) 360 manan a. maisuria, priti v. tandel 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 4 0 0.005 0.01 0.015 0.02 0.025 figure 11: 2d comparison of c(x, 3000) 361 mathematical modelling and application of reduced differential transform method for river pollution 0 0.5 1 1.5 2 2.5 3 3.5 4 10 4 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 10 -3 figure 12: 2d comparison of c(40000, t) 362 manan a. maisuria, priti v. tandel 6 conclusion in this paper, we outlined the main components of a mathematical model that various ways to predict chemical concentrations in rivers due to pollutant discharges. we get initial condition from old collected data in different form like rational, exponential and power. we attain three different solutions from different form of initial conditions. we conclude that concentration rises as time(t) rises. concentration decreases as length(x) increases. the fundamental benefit of the rdtm is that it offers the user a quick converging power series form with neatly calculated terms that contains an analytical approximation, and in many situations, an exact solution. there is no discretization or unavoidable presumptions while using rdtm. sometimes rdtm is superior to other 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for a contractive mapping via (α, β, ψ)-admissibility in b-metric space jahir hussain rasheed* maheshwaran kanthasamy† dhamodharan durairaj‡ abstract in this paper, we establish the concept of a common fixed point theorem for new type of generalized contractive mappings. furthermore, we employ our main result to shows a common fixed point theorem for a pair of self-mappings (r,s) in b-metric space via (α,β,ψ)admissibility type contractive condition. an example is also given to verify the main result. keywords: α-admissible mapping, common fixed point, b-metric spaces. 2020 ams subject classifications: 47h10, 54h25, 54m20. 1 *department of mathematics, jamal mohamed college (autonomous) (affiliated to bharathidasan university), tiruchirappalli-620020, tamilnadu, india; hssn jhr@yahoo.com. †department of mathematics, jamal mohamed college (autonomous) (affiliated to bharathidasan university), tiruchirappalli-620020, tamilnadu, india; mahesksamy@gmail.com. ‡department of mathematics, jamal mohamed college (autonomous) (affiliated to bharathidasan university), tiruchirappalli-620020, tamilnadu, india; dharan raj28@yahoo.co.in. 1received on may 1st, 2022. accepted on june 28th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v41i0.781. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 245 r. jahir hussain, k. maheshwaran, d. dhamodharan 1 introduction in the last fifty years, fixed point theories lie in finding and proving the uniqueness of solutions for many questions of applied sciences such as physics, chemistry, economics, and engineering. in 1922, stefan banach [s.banach [1922]] proved a fixed point theorem for contractive mappings in complete metric spaces. in 1969, nadler [nadler [1969]] introduced the concept of multi-value function. later, czerwik [czerwik [1993]] and bakhtin [bakhtin [1989]] initiate the concept of b-metrics metric space. khan [khan et al. [1984]] introduced the altering distance mapping to formulate a new contractive condition in fixed point theory in order to extend the banach fixed point theorem to new forms. for some extension to the banach contraction theorem. recently, abodayeh et al. [abodayeh et al. [2017]] introduced a new notion, named almost perfect function, to formulate new contractive conditions to modify and extend some fixed point theorems known in the literature. now, we mention the notions of altering distance function and almost perfect function. 2 preliminaries definition 2.1 (khan et al. [1984] ). a self-function ψ on r+ ∪ {0} is called an altering distance function if ψ satisfies the following conditions: (1) ψ(s) = 0 ⇐⇒ s = 0. (2) ψ is a nondecreasing and continuous function. definition 2.2 (abodayeh et al. [2017]). a nondecreasing self-function ψ on r+∪ {0} is called an almost perfect function if ψ satisfies the following conditions: (1) ψ(s) = 0 ⇐⇒ s = 0. (2) if for all sequence (sn) in r+ ∪ {0} with ψ(sn) → 0 it holds sn → 0. definition 2.3 (samet et al. [2012]). let r be a self-mapping on x and α : x × x → r+ ∪ {0} be a function. then, r is called α-admissible if for all v, w ∈ x with α(v, w) ≥ 1 it holds α(rv, rw) ≥ 1. the definition of triangular α-admissibility for a single mapping definition 2.4 (karupinar et al. [2013]). let r be a self-mapping on x and α : x × x → r+ ∪ {0}. then, we call r triangular α-admissible if (1) r is α-admissible; and 246 common fixed point for a contractive mapping via (α, β, ψ)-admissibility in b-metric space (2) for all v, w, u ∈ x with α(v, w) ≥ 1 and α(w, u) ≥ 1 it holds α(v, w) ≥ 1. definition 2.5 (abdeljawad [2013]). let r and s be two self mappings on x and α : x × x → r+ ∪ {0} be a function. then, the pair (r, s) is called α-admissible if z, w ∈ x and α(z, w) ≥ 1 imply α(rz, sw) ≥ 1 and α(sz, rw) ≥ 1. definition 2.6 (hussain et al. [2014]). let db be a metric on a set x and α, β : x × x → r+ ∪ {0} be functions. then, x is called α, β-complete if and only if {xn} is a cauchy sequence in x and α(xn, xn+1) ≥ β(xn, xn+1) for all n ∈ n imply (xn) converges to some x ∈ x. definition 2.7 (hussain et al. [2014]). let db be a metric on a set x and α, η : x × x → r+ ∪ {0} be functions. a self-mapping s on x is called α, βcontinuous if {xn} is a sequence in x, xn → x as n → ∞ and α(xn, xn+1) ≥ β(xn, xn+1) for all n ∈ n imply sxn → sx as n → ∞. definition 2.8 (mehemet and kiziltunc [2013]). let x be a non-empty set and let s ≥ 1 be a given real number. a function db : x × x → r+ ∪ {0} is called a b-metric provide that, for all x, y, z ∈ x, (1) db(x, y) = 0 if and only if x = y (non-negative axiom) (2) db(x, y) = db(y, x) (symmetric axiom) (3) db(x,z) ≤ s[db(x, y) + db(y, z)]. (s-triangular inequality). a pair (x, db) is called a b-metric space. definition 2.9. let r,s, be two self-mappings on the set x and α, β : x×x → r+ ∪ {0} be functions. we say that (r, s) is a pair of (α, β)-admissibility if z, w ∈ x and α(z, w) ≥ β(z, w) imply α(rz, sw) ≥ β(rz, sw) and α(sz, rw) ≥ β(sz, rw). example 2.1. define self-mappings r and s on a set of real numbers by ru = u2 and su = { −u2, if u < 0; u2, if u ≥ 0. additionally, define α,β : x × x → r+ ∪ {0} via α(u,v) = eu+v and β(u,v) = eu. then, (r,s) is a pair of (α,β)-admissibility. 247 r. jahir hussain, k. maheshwaran, d. dhamodharan 3 main results definition 3.1. let ψ be a nondecreasing function on r+ ∪ {0} . we call ψ a perfect control function if the following conditions hold: (i) ψ(t) = 0 ⇐⇒ t = 0. (ii) if (tn) is a sequence in r+ ∪ {0} and ψ(tn) → 0 as n → +∞ implies tn → 0 as n → +∞. (iii) ψ(u + v) ≤ ψ(u) + ψ(v) for all u, v ∈ r+ ∪ {0} . (iv) ψn(λx) = λnψ(x). definition 3.2. let (x,db) be a b-metric space with constant s ≥ 1. let r, s be two self-mappings on x, ψ be a perfect self-mapping on r+ ∪ {0}, α, β : x × x → r+ ∪ {0} be functions. we say that the pair (r, s) is an (α, β, ψ)admissibility type contraction if there exists λ ∈ [0, 1) such that z, w ∈ x and α(z, w) ≥ β(z, w) imply ψ(db(rz,sw)) ≤ λψ(db(z,w)) + λψ(db(z,rz)) + λψ(db(w,sw)) +λψ(db(w,rz)) + λψ(db(z,sw)) (1) and ψ(db(sz,rw)) ≤ λψ(db(z,w)) + λψ(db(z,sz)) + λψ(db(w,rw)) +λψ(db(w,sz)) + λψ(db(z,rw)) (2) theorem 3.1. let (x,db) be a b-metric space with constant s ≥ 1. let α, β : x × x → r+ ∪ {0} be function and (r,s) be a self-mappings on x. assume following conditions: (i) (x,db) is an α,β-complete b-metric space. (ii) r and s are α,β-continuous. (iii) (r,s) is pair of (α,β)-admissibility. (iv) if v,w,z are in x, with α(v,w) ≥ β(v,w) and α(w,z) ≥ β(w,z), then α(v,z) ≥ β(v,z). (v) there exists x0 ∈ x such that α(rx0,srx0) ≥ β(rx0,srx0) and α(srx0,rx0) ≥ β(srx0,rx0). then r and s have a common fixed point. 248 common fixed point for a contractive mapping via (α, β, ψ)-admissibility in b-metric space proof. in view of condition (v) we start with x0 ∈ x in such away that α(rx0,srx0) ≥ β(rx0,srx0) and α(srx0,rx0) ≥ β(srx0,rx0). now, let x1 = rx0 and x2 = sx1. then α(x0,x1) ≥ β(x0,x1) and α(x1,x0) ≥ β(x1,x0). in view of condition (iii), we have α(x1,x2) = α(rx0,sx1) ≥ β(rx0,sx1) = β(x1,x2) and α(x2,x1) = α(sx1,rx0) ≥ β(sx1,rx0) = β(x2,x1) again we put x3 = sx2. then condition (iii) implies that α(x2,x3) = α(sx1,rx2) ≥ β(sx1,rx2) = β(x2,x3) and α(x3,x2) = α(rx2,sx1) ≥ β(rx2,sx1) = β(x3,x2) putting x4 = sx3 and referring to condition (iii) we conclude α(x3,x4) = α(rx2,sx3) ≥ β(rx2,sx3) = β(x3,x4) and α(x4,x3) = α(sx3,rx2) ≥ β(sx3,rx2) = β(x4,x3) continuing in the same manner, we contract a sequence (xn) in x with x2n+1 = rx2n and x2n+2 = sx2n+1 such that α(xn,xn+1) ≥ β(xn,xn+1) ∀ n ∈ n and α(xn+1,xn) ≥ β(xn+1,xn) ∀ n ∈ n from condition (iv) we see that α(xn,xm) ≥ β(xn,xm) ∀ n,m ∈ n if there exists q ∈ n such that x2q = x2q+1, then x2q = rx2q and hence r has a fixed point. from contractive condition (1), we have ψ(db(x2q+1,x2q+2)) = ψ(db(rx2q,sx2q+1)) ≤ ψ  λ(db(x2q,x2q+1)) + λ(db(x2q,rx2q)) + λ(db(x2q+1,sx2q+1)) +λ(db(x2q+1,rx2q)) + λ(db(x2q,sx2q+1))   ≤ ψ ( 2λ(db(x2q,x2q+1)) + λ(db(x2q+1,x2q+2)) +λs[(db(x2q,x2q+1) + db(x2q+1,x2q+2))] ) ≤ ψ ( λ(2 + s)(db(x2q,x2q+1)) +λ(1 + s)(db(x2q+1,x2q+2)) ) 249 r. jahir hussain, k. maheshwaran, d. dhamodharan ≤ ψ ( λ(2+s) 1−λ(1+s)(db(x2q,x2q+1)) ) (3) the last inequality is correct only if ψ( λ(2+s) 1−λ(1+s)(db(x2q,x2q+1))) = 0. the properties of ψ and db imply that x2q+1 = x2q+2. hence, x2q = rx2q = sx2q. thus, r and s have a common fixed point of r and s. if there exists q ∈ n such that x2q+1 = x2q+2 then x2q+1 = tx2q+1 and hence s has a fixed point. from contractive condition (2), we have ψ(db(x2q+2,x2q+3)) = ψ(db(sx2q+1,rx2q+2)) ≤ ψ ( λ(db(x2q+1,x2q+2)) + λ(db(x2q+1,sx2q+1)) + λ(db(x2q+2,rx2q+2)) +λ(db(x2q+2,sx2q+1)) + λ(db(x2q+1,rx2q+2)) ) ≤ ψ ( 2λ(db(x2q+1,x2q+2)) + λ(db(x2q+2,x2q+3)) +λs[(db(x2q+1,x2q+2) + db(x2q+2,x2q+3))] ) ≤ ψ ( λ(2 + s)(db(x2q+1,x2q+2)) +λ(1 + s)(db(x2q+2,x2q+3)) ) ≤ ψ ( λ(2+s) 1−λ(1+s)(db(x2q+1,x2q+2)) ) (4) the last inequality is correct only if ψ( λ(2+s) 1−λ(1+s)(db(x2q+1,x2q+2))) = 0. the properties of ψ and db imply that x2q+2 = x2q+3. hence, x2q+1 = rx2q+1 = sx2q+1. thus, r and s have a common fixed point of r and s. now, assume that xn ̸= xn+1 ∀ n ∈ n. for n ∈ n ∪ {0}, we get ψ(db(x2n+1,x2n+2)) = ψ(db(rx2n,sx2n+1)) ≤ ψ ( λ(db(x2n,x2n+1)) + λ(db(x2n,rx2n)) + λ(db(x2n+1,sx2n+1)) +λ(db(x2n+1,rx2n)) + λ(db(x2n,sx2n+1)) ) ≤ ψ ( 2λ(db(x2n,x2n+1)) + λ(db(x2n+1,x2n+2)) +λs[(db(x2n,x2n+1) + db(x2n+1,x2n+2))] ) ≤ ψ ( λ(2 + s)(db(x2n,x2n+1)) +λ(1 + s)(db(x2n+1,x2n+2)) ) ≤ ψ ( λ(2+s) 1−λ(1+s)(db(x2n,x2n+1)) ) (5) let [δ = λ(2+s) 1−λ(1+s)]. hence ψ(db(x2n+1,x2n+2)) ≤ ψ(δ(db(x2n,x2n+1))) 250 common fixed point for a contractive mapping via (α, β, ψ)-admissibility in b-metric space using argument similar to the above, we may show that ψ(db(x2n,x2n+1)) = ψ(db(sx2n−1,rx2n)) ≤ ψ ( λ(db(x2n−1,x2n)) + λ(db(x2n−1,sx2n−1)) + λ(db(x2n,rx2n)) +λ(db(x2n,sx2n−1)) + λ(db(x2n−1,rx2n)) ) ≤ ψ ( 2λ(db(x2n−1,x2n)) + λ(db(x2n,x2n+1)) +λs[(db(x2n−1,x2n) + db(x2n,x2n+1))] ) ≤ ψ ( λ(2 + s)(db(x2n−1,x2n)) +λ(1 + s)(db(x2n,x2n+1)) ) ≤ ψ ( λ(2+s) 1−λ(1+s)(db(x2n−1,x2n))) ) ≤ ψ ( (δ(db(x2n−1,x2n))) ) (6) combining equation (5) and (6) together, we reach ψ(db(xn,xn+1)) = ψ(db(sxn−1,rxn)) ≤ ψ ( λ(db(xn−1,xn)) + λ(db(xn−1,sxn−1)) + λ(db(xn,rxn)) +λ(db(xn,sxn−1)) + λ(db(xn−1,rxn)) ) ≤ ψ ( 2λ(db(xn−1,xn)) + λ(db(xn,xn+1)) +λs[(db(xn−1,xn) + db(xn,xn+1))] ) ≤ ψ ( λ(2 + s)(db(xn−1,xn)) +λ(1 + s)(db(xn,xn+1)) ) ≤ ψ ( λ(2+s) 1−λ(1+s)(db(xn−1,xn)) ) ≤ ψ ( δ(db(xn−1,xn)) ) (7) by recurring equation (7) n-times, we deduce ψ(db(xn,xn+1)) ≤ ψ(δ(db(xn−1,xn))) ≤ δψ(db(xn−2,xn−1)) ≤ δ(δψ(db(xn−2,xn−1))) = δ2ψ(db(xn−2,xn−1)) ... 251 r. jahir hussain, k. maheshwaran, d. dhamodharan ≤ δnψ(db(x0,x1)). (8) on allowing n → ∞ in equation (8), we get lim n→+∞ ψ(db(xn,xn+1)) = 0 (9) the properties of ψ implies that lim n→+∞ db(xn,xn+1) = 0 (10) we intend to prove that (xn) is cauchy sequence in x, take n,m ∈ n with m > n. we divide the proof into four cases: case 1: n is an odd integer and m is an even integer. therefore, there exists t ∈ n and an odd integer h such that n = 2t + 1 and m = 2t + 1 + h. since α(xn,xm) ≥ β(xn,xm), we have ψ(db(xn,xm)) = ψ(db(x2t+1,x2t+1+h)) = ψ(db(rx2t,sx(2t+h))) ≤ ψ ( λ(db(x2t,x2t+h)) + λ(db(x2t,rx2t)) + λ(db(x2t+h,sx2t+h)) +λ(db(x2t+h,rx2t)) + λ(db(x2t,sx2t+h)) ) = ψ ( λ(db(x2t,x2t+h)) + λ(db(x2t,x2t+1)) + λ(db(x2t+h,x2t+1+h)) +λ(db(x2t+h,x2t+1)) + λ(db(x2t,x2t+1+h)) ) ≤ ψ ( λ ∑2t+h−1 i=2t (db(xi,xi+1)) + λ(db(x2t,x2t+1)) + λ(db(x2t+h,x2t+1+h)) +λ ∑2t+h−1 i=2t+1 (db(xi,xi+1)) + λ(db(x2t,x2t+1+h)) ) ≤ ψ ( λ(2 + s) ∑∞ i=2t(db(xi,xi+1)) + λ(db(x2t,x2t+1)) +λ(db(x2t+h,x2t+1+h)) + λs(db(x2t,x2t+1)) ) where,k = λ(2 + s) ≤ ψ ( k ∑∞ i=2t(db(xi,xi+1)) + λ(db(x2t,x2t+1)) +λ(db(x2t+h,x2t+1+h)) + λs(db(x2t,x2t+1)) ) ≤ ψ ( k2t+1 1−k (db(x0,x1)) + λ(db(x2t,x2t+1)) +λ(db(x2t+h,x2t+1+h)) + λs(db(x2t,x2t+1)) ) by permitting n,m → ∞ in above inequalities and considering equation (9) lim n→+∞ ψ(db(xn,xm)) = 0 252 common fixed point for a contractive mapping via (α, β, ψ)-admissibility in b-metric space the properties of ψ implies that lim n→+∞ db(xn,xm) = 0 (11) case 2:n and m are both even integers. applying the triangular inequality of the b-metric db, we have db(xn,xm) ≤ s[db(xn,xn+1) + db(xn+1,xm)], for m ≥ n letting n → ∞ and in view of equation (10) and (11),we get lim n→+∞ ψ(db(xn,xm)) = 0. case 3: n is an even integer and m is an odd integer. applying the triangular inequality of the b-metric db, we have db(xn,xm) ≤ s[db(xn,xn+1) + db(xn+1,xm)] db(xn,xm) ≤ s[db(xn,xn+1)+s[db(xn+1,xm−1)+db(xm−1,xm)]] , for m ≥ n on permitting m,n → ∞ and considering equation (10) and (11), we get lim n→+∞ ψ(db(xn,xm)) = 0. case 4: n and m are both odd integers. applying the triangular inequality of the b-metric db, we have db(xn,xm) ≤ s[db(xn,xm−1) + db(xm−1,xm)], for m ≥ n on permitting n → ∞ and in view of equation (10) and (11), we get lim n→+∞ ψ(db(xn,xm)) = 0. combining all cases with each other, we conclude that lim n→+∞ ψ(db(xn,xm)) = 0. thus, we conclude that (xn) is a cauchy sequence in x. the α, β-completeness of the b-metric space (x,db) ensures that there is x ∈ x such that xn → x. using the α, β-continuity of the mappings r and s, we deduce that x2n+1 = rx2n → rx and x2n+2 = sx2n+1 → sx . by uniqueness of limit, we obtain rx = sx = x. thus, x is a fixed point of r.2 253 r. jahir hussain, k. maheshwaran, d. dhamodharan example 3.1. define db : r+0 ×r + 0 → r + 0 by db(z,w) = |z−w| and let r,s be two self-mappings on r+0 define by rz = z 2 and sw = w 4 . in addition, define the function ψ : r+0 → r + 0 by ψ nλx = λnψ(x), where ψ(x) = x 1+x . furthermore, the functions α,β : x × x → r+0 define by α(p,q) = { ep+q, if p,q ∈ [0,1]; 0, if p > 1 or q > 1. . and β(p,q) = { ep, if p,q ∈ [0,1]; 1, if p > 1 or q > 1. . then: 1. ψ is a perfect control function. 2. there exists u0 ∈ x such that α(ru0,ru1) ≥ β(ru0,ru1) and α(ru1,ru0)] ≥ β(ru1,ru0). 3. (r,s) is a pair of (α,β)-admissibility. 4. r and s are α, β-continuous. 5. (x,db) is an α, β-complete b-metric space. 6. (r,s) is an (α,β,ψ)-contraction. proof. it is an easy matter to see equations (1) to (3). to prove (4), let (un) be any sequence in r+0 whenever un → u ∈ r + 0 and α(un,un+1) ≥ β(un,un+1) ∀ n ∈ n. case 1: if un = u for all n, where un ∈ [0,1] ∀ n ∈ n. we conclude that run → ru as n → ∞. case 2: if un ̸= u, for all n, we notice that u = 0. hence, un → 0 in ([0,1], |.|). therefore, |u 2 ,0| → 0 = ru in (r+0 ,db); that is r is α,β-continuous. to prove (5), let (un) be a cauchy sequence in (r + 0 ,db) such that α(un,un+1) ≥ β(un,un+1). then, un ∈ [0,1] ∀n ∈ n. if there exists u ∈ [0,1] such that un = u for all n, then, un → u as n → +∞. now, suppose the elements of (un) are distinct. give ϵ > 0, since (un) is a cauchy sequence in (r + 0 ,db), then there exists n0 ∈ n such that |un,um| < ϵ ∀ m > n ≥ n0. therefore, |un,0| < 0 ∀ n ≥ n0. so, un → 0 in (r+0 ,db). thus, (r + 0 ,db) is an α,β-complete b-metric space. to prove (6), let z,w ∈ x be such that α(z,w) ≥ β(z,w). then, z,w ∈ [0,1]. 254 common fixed point for a contractive mapping via (α, β, ψ)-admissibility in b-metric space so, ψ(db(rz,sw)) = ψ(db( z 2 , w 4 )) = ψ(|z 2 , w 4 |) = | z 2 , w 4 | 1+| z 2 , w 4 | = 1 2 |z, w 2 | 1+ 1 2 |z, w 2 | = |z, w 2 | 2+|z, w 2 | ≤ 1 6 ( |z,w| 1+|z,w|) ≤ 1 6 ψ(db(z,w)) ψ(db(rz,sw)) ≤ 16ψ(db(z,w)) + 1 6 ψ(db(z,rz)) + 1 6 ψ(db(w,sw))+ 1 6 ψ(db(w,rz)) + 1 6 ψ(db(z,sw)), (∵ λ = 1 6 ) similarly, we can show that ψ(db(sz,rw)) ≤ 16ψ(db(z,w)) + 1 6 ψ(db(z,sz)) + 1 6 ψ(db(w,rw))+ 1 6 ψ(db(w,sz)) + 1 6 ψ(db(z,rw)), (∵ λ = 1 6 ) hence, r and s satisfy definition 3.2. therefore, r and s satisfy all the condition of theorem. therefore, r and s have a common fixed point. 4 conclusions in theorem 3.1 we have formulated a new contractive conditions to modify and extend some common fixed point theorem for a pair of self-mappings (r,s) in b-metric space via (α,β,ψ)-admissibility type. the existence and uniqueness of the result is presented in this article. we have also given some example which satisfies the condition of our main result. our result may be the vision for other authors to extend and improve several results in such spaces and applications to other related areas. 255 r. jahir hussain, k. maheshwaran, d. dhamodharan 5 acknowledgements the authors thanks the management, ratio mathematica for their constant support towards the successful completion of this work. we wish to thank the anonymous reviewers for a careful reading of manuscript and for very useful comments and suggestions. references t. abdeljawad. meir-keeler α-contractive fixed and common fixed point theorems. fixed point theory appl, 19:1–10, 2013. k. abodayeh, a. bataihah, a. ansari, and w. shatanawi. some fixed point and common fixed point results throughw-distance under nonlinear contractions. gazi university journal of science, 30:293–302, 2017. i. bakhtin. the contraction mapping principle in quasimetric spaces. functional analysis, 30:26–37, 1989. s. czerwik. contraction mappings in b-metric spaces. acta math inform univ ostraviensis, 1:5–11, 1993. n. hussain, m. kutbi, and p. salimi. fixed point theory in α-complete metric spaces with applications. abstract and appled analysis, 2014:1–11, 2014. e. karupinar, p. kumam, and p. salimi. on α − ψ-meir-keeler contractive mappings. fixed point theory and applications, 94:1–13, 2013. m. khan, m. swaleh, and s. sessa. fixed point theorems by altering distances between the points. bulletin of the australian mathematical society, 30:1–9, 1984. k. mehemet and h. kiziltunc. on some well-known fixed point theorems in b-metrics spaces. turkish journal of analysis and number theory, 1:13–16, 2013. s. nadler. multi-valued contraction mappings. pacific journal of mathematics, 30:475–488, 1969. b. samet, c. vetro, and p. vetro. fixed point theorems for a α − ψ-contractive mappings. nonlinerar analysis, 75:2154–2165, 2012. s.banach. sur les operations dans les ensembles et leur application aux equation sitegrales. fundamenta mathematicae, 3:133–181, 1922. 256 ratio mathematica volume 48, 2023 cactus graphs with cycle blocks and square product labeling keerthi g. mirajkar* priyanka g. sthavarmath† abstract in many real world problems, cactus graphs were considered as models from both algorithmic and theoretical point of view and this graph is a subclass of planar graph and superclass of a tree. in this article, the study has been carried out on some cactus graphs with cycle blocks for obtaining results on square product labeling. keywords: square product labeling (spl) and cactus graphs. 2020 ams subject classifications: 05c05, 05c38, 05c76, 05c78. 1 *associate professor, department of mathematics, karnatak university’s karnatak arts college, dharwad 580 001, karnataka, india; email:keerthi.mirajkar@gmail.com †research scholar, department of mathematics, karnatak university’s karnatak arts college, dharwad 580 001, karnataka, india; email:priyankasthavarmath1990@gmail.com 1received on february 11, 2023. accepted on july 15, 2023. published on august 1, 2023. doi: 10.23755/rm.v39i0.1131. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. k. g. mirajkar and p. g. sthavarmath 1 introduction graph labeling is one of the field in graph theory and it is the labeling of integers to the vertices or edges, or both under particular conditions which was introduced by a. rosa [1967] in 1967. graph labeling has a wide range of applications such as in x-ray crystallography, coding theory, radar, astronomy, circuit design, network theory, communication networks and database management. nowadays, research in labeling of graph is increasingly expanding by studing more than 300 kinds of labelings. one such labeling is square sum labeling introduced by v. ajitha et al. [2009] and they explored some results on it. j. b. babujee and babitha [2012] also worked on square sum labeling and obtained the results on it. further, j.shiama [2012] has worked on square difference labeling and proved some results. k. g. mirajkar and sthavarmath [2022] initiated square product labeling and obtained results for some class of graphs, cycle related graphs, cartesian product of graphs and silicate and oxide networks. khan et al. [2010a] have studied on cactus graphs for proving the results for (2, 1)− total labeling and for l(2, 1)− labeling of cactus graphs in khan et al. [2010b]. k. kalaiarasi and mahalakshmi [2022] shown the applications for cactus fuzzy labeling graphs. further, s. philomena, m. pal, and k. thirusanga philomena et al. [2014] investigated some results for square and cube difference labeling on cactus graphs. in this article, the results are obtained for cactus graphs with cycle blocks on square product labeling and applied number theory concepts to establish the results. all examined graphs here are finite, undirected, simple and connected. for undefined expressions and symbols refer f.harary [1969], for number theory concepts refer burton [2006] and for different labeling concepts we refer gallian [2020]. 2 preliminaries definition 2.1. a graph g is said to be a square product labeling (spl), if there exists function f : v (g) → {1, 2, 3, . . . , p} which is bijective, here p is counting of vertices inducing f∗ : e(g) → n which is injective, where f∗(uv) = f(u)2f(v)2 and the resulting edges are distinctly labeled. definition 2.2. cactus graph khan et al. [2010a] is a connected graph, in which every block is a cycle or an edge, in other words, no edge belongs to more than one cycle. cactus graphs with cycle blocks and square product labeling 3 main results theorem 3.1. for two cycles cn and cm of any lengths n and m with a common cutvertex v admits square product labeling. proof. consider the cycles cn and cm of any lengths n and m having a common cutvertex v labeled by 1 with v (cn) = {v1, v2, v3, ..., vn} and v (cm) = {v′1, v ′ 2, v ′ 3, ..., v ′ m}, hence the number of vertices are (n + m) − 1 and edges are 2n. we consider six cases to prove the result with f : v (g) → {1, 2, 3, . . . , (n + m) − 1} case 1: for n < m f(vi) = 2i for 1 ≤ i ≤ n − 1, f(v ′ i) = f(vn−1) + 2i for 1 ≤ i ≤ m − n f(v ′ i+1) = 2i + 1 for m − n ≤ i ≤ m − 2 labeling of edges acquired here are distinct, as it forms quadratic sequences of the forms (4p(p +1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n−2} and (4p 2 −1)2, p ⩾ 1 for {f∗(v′m−n+1+(i−1)v ′ m−n+2+(i−1)) : m−n ≤ i ≤ m−2}. the remaining edges are labeled as follows, f∗(vvn) = (f(vn)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1)) 2 f∗(vv ′ n) = (f(v ′ n)) 2, f∗(v ′ iv ′ i+1) = (f(v ′ n)) 2(f(v ′ n)) 2 for 1 ≤ i ≤ m − n case 2: for n > m f(vi) = 2i for 1 ≤ i ≤ n − 2, f(vn−1) = 3 f(v ′ i) = f(vn−1) + 2i for 1 ≤ i ≤ m − 1 labeling of edges acquired here are distinct, as it forms quadratic sequences of the forms (4p(p + 1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n − 3} and (4p 2 + 16p + 15)2, p ⩾ 1 for {f∗(v′iv ′ i+1) : m − n ≤ i ≤ m − 2}. the remaining edges are labeled as follows, f∗(vvn) = (f(vn)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1)) 2 f∗(vn−1vn) = (f(vn−1)) 2 (f(vn)) 2 case 3: for n = m f(vi) = 2i, 1 ≤ i ≤ (n − 1), f(v ′ i) = 2i − 1, 1 ≤ i ≤ m − 1 f(v ′ i) = f(vn−1) + 2i, 1 ≤ i ≤ m − 1 labeling of edges acquired here are distinct, as it forms quadratic sequences of the forms (4p(p +1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n−2} and (4p 2 −1)2, k. g. mirajkar and p. g. sthavarmath p ⩾ 1 for {f∗(v′iv ′ i+1) : m − n ≤ i ≤ m − 2}. the remaining edges are labeled as follows, f∗(vvn) = (f(vn)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1)) 2 f∗(v ′ mv) = (f(v ′ ))2 case 4: for n = 2p(p + 1), p ⩾ 1 and n < m for n = 2p(p + 1), p = 1 f(v1) = 2, f(v2) = 6, f(v3) = 4, f(v ′ 1) = f(v2) + 2 f(v ′ m−1 − (i − 1)) = 2i + 1 for 1 ≤ i ≤ m − n if n ⩾ 4, n = 2p(p + 1), p > 1 f(vi) = 2i for 1 ≤ i ≤ n − 1, f(v ′ 1) = 3 f(v ′ i+1) = 2i + 1 for 2 ≤ i ≤ m − 3 f(v ′ i+1) = f(vn−1) + 2 + 2(i − 1) for 1 ≤ i ≤ m − n labeling of edges acquired here are distinct, as it forms quadratic sequences of the forms (4p(p +1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n−3} and (4p 2 −1)2, p ⩾ 1 for {f∗(v′(m−1)−(i−1)v ′ (m−2)−(i−1)) : 1 ≤ i ≤ m−3}. the remaining edges are labeled as follows, f∗(vvn) = (f(vn)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1)) 2, f∗(v ′ mv) = (f(v ′ ))2 f∗(vn−1v) = (f(vn−1)) 2, f∗(v1v2) = f(v1) 2 f(v2) 2 f∗(v2v3) = f(v2) 2 f(v3) 2, f∗(v ′ 1v ′ 2) = (f(v ′ 1)) 2 (f(v ′ 2)) 2 case 5: for n = 2p 2 + 2p + 1 m ⩾ (n − 1) f(vi) = 2i for 1 ≤ i ≤ n − 3, f(vn−2) = (2n − 2) f(vn−1) = (2n − 4) f(v ′ 1) = 3, f(v ′ i+1) = f(vn−2) + 2i for 1 ≤ i ≤ m − n f(v ′ (m−n+1)+(i−1)) = 2i + 1 for m − n ≤ i ≤ m − 2 labeling of edges acquired here are distinct, as it forms quadratic sequences of the forms (4p(p + 1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n − 3} and (4p 2 + 16p + 15)2, p ⩾ 1 for {f∗(v′iv ′ i+1) : m − n ≤ i ≤ m − 2}. the remaining edges are labeled as follows, f∗(vvn) = (f(vn)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1)) 2 f∗(vn−1vn) = (f(vn−1)) 2 (f(vn)) 2, f∗(v ′ 1v ′ 2) = 9 (f(vn−1) + 2)) 2 if n = 5, f∗(v ′ 2v ′ 3) = (f(vn−1) + 2)) 2 (f(v ′ 3)) 2 f∗(v ′ i+2v ′ i+3) = (4p 2 + 16p + 15)2 for 1 ≤ i ≤ m − 3 cactus graphs with cycle blocks and square product labeling case 6: for n = 4p 2 +8p +3 and n = 4(p 2 +2p +1) number of vertices where p ≥ 1 v ′ n−1 = n+m−2 and v ′ n−2 = n+m−1 and if f(v ′ i) is labeled with even numbers before the label 3 for n = 4(p 2 + 2p + 1) then change the label with preceding vertex label. the remaining labels of both the edges and vertices are same as in the above cases. the labels of both vertices and edges are distinct in all cases, hence the result. 2. example 3.1: the square product labeling of two cycles of any lengths n and m with a common cut vertex v is as shown in below figure. figure 1: square product labeling of two cycles of any lengths n and m with a common cutvertex v theorem 3.2. for three cycles cn, cm, and cl of any lengths n, m, and l with a common cut vertex v admits spl. proof. consider three cycles cn, cm, and cl having a common cut vertex v with fixed label as 1 and v (cn) = {v1, v2, v3, ..., vn}, v (cm) = {v ′ 1, v ′ 2, v ′ 3, ..., v ′ m}, and v (cl) = {v ′′ 1 , v ′′ 2 , v ′′ 3 , ..., v ′′ l }, hence the number of vertices are (n+m+l)−2 and edges are 3n. we consider nine cases to prove the result and in each case k. g. mirajkar and p. g. sthavarmath the labels of vertices of cactus graph starts from even numbers among the total number of vertices of that graph later the graph g is given odd numbers. let f : v (g) → {1, 2, 3, . . . , (n + m + l) − 2}, thus both the vertices and edges for all nine cases are labeled as follows, case 1: for n = 2p(p + 1), p ≥ 1 and n < m if n = 2p(p + 1), p = 1 f(v1) = 2, f(v2) = 6, f(v3) = 4, f(v ′ i) = f(v2) + 2 i for 1 ≤ i ≤ m + n − l f(v ′ m+n−l + i) = 2i + 1 for 1 ≤ i ≤ l − m, if l > m if n = 2p(p + 1), p > 1 f(vi) = 2i, 1 ≤ i ≤ (n − 2), f(v ′ i) = f(vn−2) + 2i for 1 ≤ i ≤ m + n − l f(v ′′ i ) = f(v ′ n−1) + 2i for 1 ≤ i ≤ n + l − m + 1 f(vn−2) = 2n, f(vn−1) = 2n − 2 the edge labels acquired here are distinct, as it forms a quadratic sequence of the form (4p(p + 1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n − 3}. the remaining edges are labeled as follows, f∗(vvn) = (f(vn)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1)) 2 f∗(v1v2) = (f(v1)) 2 (f(v2)) 2, f∗(v2v3) = (f(v2)) 2 (f(v3)) 2 f∗(vv ′ n−1) = (f(v ′ n−1)) 2, f∗(v ′ iv ′ i+1) = (f(vn−2) + 2i)) 2(f(v ′ i) + 2i)) 2, for 1 ≤ i ≤ m − 3 f∗(v ′ n−2v ′ n−1) = (f(v ′ n−2)) 2(f(v ′ n−1)) 2 f∗(v ′′ i v ′′ i+1) = (f(v ′ n−2) + 2i)) 2(f(v ′′ i ) + 2i)) 2 for 1 ≤ i ≤ l + n − m case 2: for n = 2p 2 + p + 1, p ≥ 1 f(vi) = 2i for 1 ≤ i ≤ n − 1 2 , f(vn−2) = (2n − 2), f(vn−1) = 2n − 4 f(v ′ i) = f(vn−2) + 2i for 1 ≤ i ≤ m + n − l f(v ′′ i ) = f(v ′ n−1) + 2i for 1 ≤ i ≤ (n + l − m) + 1 the edge labels acquired here are distinct, as it forms quadratic sequences of the form (4p(p + 1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n − 3}, (4p 2 + 24p + 35))2, p ⩾ 1 for {f∗(v′′i v ′′ i+1) : 1 ≤ i ≤ l − 2} and (4p 2 − 1)2, p ⩾ 1 for cactus graphs with cycle blocks and square product labeling {f∗(v′(n+l−m)−1+iv ′ (n+l−m)+1+(i−1)) : 1 ≤ i ≤ m − l} if l < m. the remaining edges are labeled as follows, f∗(vvn−1) = (f(vn−1)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1)) 2 f∗(v ′ n−2v ′ n−1) = (f(v ′ n−2)) 2 (f(v ′ n−1)) 2 f∗(v ′ n−3v ′ n−2) = (f(v ′ n−3)) 2 (f(v ′ n−2)) 2 f∗(vv ′ n−1) = (f(v ′ n−1)) 2, f∗(vv ′′ n−1) = (f(v ′′ n−1)) 2 f∗(v ′ iv ′ i+1) = (f(vn−2) + 2i)) 2(f(v ′ i) + 2i)) 2 for 1 ≤ i ≤ (l + n − m) − 2 f∗(v ′ (n+l−m)−1v ′ (n+l−m)+1)) = (f(v ′ (n+l−m)−1))) 2(f(v ′ (n+l−m)+1)) 2 case 3: for n = m = l f(vi) = 2i for 1 ≤ i ≤ n − 1, f(v ′ i) = f(vn−1) + 2i for 1 ≤ i ≤ m − 4 f(v ′ (m−4)+i) = 2i + 1 for 1 ≤ i ≤ m − 4, f(v ′′ i ) = f(v ′ n−1) + 2i for 1 ≤ i ≤ l − 1 the edge labels acquired here are distinct, as it forms quadratic sequences of the form (4p(p + 1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n − 2} and (4p 2 − 1)2, p ⩾ 1 for {f∗(v(m−3)+(i−1)v(m−2)+(i−1)) : 1 ≤ i ≤ m−5}. the remaining edges are labeled as follows, f∗(vvn−1) = (f(vn−1)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1)) 2 f∗(vv ′ n−1) = (f(v ′ n−1)) 2, f∗(vv ′′ n−1) = (f(v ′′ n−1)) 2 f∗(v ′ iv ′ i+1) = (f(vn−1) + 2i)) 2(f(v ′ i) + 2i)) 2 for 1 ≤ i ≤ m − 5 f∗(vm−4vm−3) = (f(vm−4)) 2 (f(vm−3)) 2 f∗(v ′′ i v ′′ i+1) = (f(v ′ m−1) + 2i)) 2(f(v ′ i) + 2i)) 2 for 1 ≤ i ≤ l − 2 case 4: for v (g) = 4p 2 + 8p + 3, p ≥ 1, n < m and n ⩽ l f(vi) = 2i for 1 ≤ i ≤ n − 1, f(v ′′ n−2) = (n + m + l − 2) f(v ′′ n−1) = (n + m + l − 4) f(v ′ i) = f(vn−1) + 2i for 1 ≤ i ≤ m − l − 1 f(v ′ m−(l+1)+i) = 2i + 1 for 1 ≤ i ≤ m − 2 f(v ′′ i ) = f(v ′ n−1) + 2i for 1 ≤ i ≤ l − 3 the edge labels acquired here are distinct, as it forms a quadratic sequence of the form (4p(p + 1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n − 2}. the remaining k. g. mirajkar and p. g. sthavarmath edges are labeled as follows, f∗(vvn−1) = (f(vn−1)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1)) 2 f∗(v ′′ n−2v ′′ n−1) = (f(v ′′ n−2)) 2, f∗(v ′ 1v ′ 2) = (f(v ′ 1)) 2 (f(v ′ 2)) 2 f∗(vv ′ n−1) = (f(v ′ n−1)) 2, f∗(vv ′′ n−1) = (f(v ′′ n−1)) 2, f∗(vv ′′ 1 ) = (f(v ′′ 1 )) 2 f∗(v ′ iv ′ i+1) = (f(vn−2) + 2i)) 2(f(v ′ i) + 2i)) 2 for 2 ≤ i ≤ m − 2 f∗(v ′′ i v ′′ (i+1)) = (f(v ′ (n−1) + 2i)) 2(f(v ′ i) + 2i) 2 for 1 ≤ i ≤ l − 3 case 5: for v (g) = 4(p 2 + 2p + 1), p ≥ 1 f(vi) = 2i for 1 ≤ i ≤ (n − 1), f(v ′ i) = f(vn−1) + 2i for 1 ≤ i ≤ m − l f(v ′ (m−4)+i) = 2i + 1 for 1 ≤ i ≤ m − 4, f(v ′ (m−l)+1) = f(v ′ (m−l)+4) f(v ′ (m−l)+2) = f(v ′ (m−l)+2), f(v ′′ i ) = f(v ′ n−1) + 2i for 1 ≤ i ≤ l − 3 f(v ′′ n−2) = f(v ′ (m−l)) + 3, f(v ′′ n−1) = f(v ′ (n−1)) − 2 the edge labels acquired here are distinct, as it forms quadratic sequences of the form (4p(p + 1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n − 2} and (4p 2 − 1)2, p ⩾ 1 for {f∗(v′(m−3)+(i−1)v ′ (m−2)+(i−1)) : 1 ≤ i ≤ m−5}. the remaining edges are labeled as follows, f∗(vvn−1) = (f(vn−1)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1)) 2 f∗(v ′ (m−l)+1v ′ (m−l)+2) = (f(v ′ (m−l) + 4)) 2 (f(v ′ (m−l) + 2)) 2, f∗(v ′ m−4v ′ m−3) = (f(v ′ m−4)) 2 (f(v ′ m−3)) 2 f∗(vv ′ n−1) = (f(v ′ n−1)) 2, f∗(vv ′′ n−1) = (f(v ′′ n−1)) 2 f∗(v ′ iv ′ i+1) = (f(vn−1) + 2i)) 2(f(v ′ i) + 2i)) 2 for 1 ≤ i ≤ m − l f∗(v ′′ i v ′′ (i+1)) = (f(v ′ (n−1)) + 2i) 2 (f(v ′ i) + 2i)) 2 for 1 ≤ i ≤ l − 2 case 6: for v (v ) = 12p 2 + 4p + 1 and 12p 2 + 4p + 1, p ≥ 1, where n < m f(vi) = 2i for 1 ≤ i ≤ n − 1 f(v ′ i) = f(vn−1) + 2i for 1 ≤ i ≤ m − n f(v ′ (m−n) + 1) = (n + m + l − 2), f(v ′ (m−n)+2) = (n + m + l − 4) f(v ′ (m−n)+2 + i) = 2i + 1 for 1 ≤ i ≤ (m − n) + 2 f(v ′′ n−2) = f(v ′ (m−l)) + 3, f(v ′′ i ) = f(v ′ (n−1)) + 2i for 1 ≤ i ≤ l − 2 the edge labels acquired here are distinct, as it forms quadratic sequences of the forms (4p(p + 1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n − 2} and (4p 2 − 1)2, cactus graphs with cycle blocks and square product labeling p ⩾ 1 for {f∗(v′(m−n)+2+iv ′ (m−n)+4+i) : 1 ≤ i ≤ (m − n) + 1}. the remaining edges are labeled as follows, f∗(vvn−1) = (f(vn−1)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(vn−1) + 2) 2 f∗(vv ′′ 1 ) = (f(v ′ n−1) + 2) 2 f∗(vv ′ n−1) = (f(v ′ n−1)) 2, f∗(vv ′′ n−1) = (f(v ′′ n−1)) 2 f∗(v ′ iv ′ i+1) = (f(vn−1) + 2i)) 2(f(v ′ i) + 2i)) 2 for 1 ≤ i ≤ m − n f∗(v ′′ i v ′′ (i+1)) = (f(v ′ (n−1)) + 2i) 2 (f(v ′′ i ) + 2i)) 2 for 1 ≤ i ≤ l − 2 f(v ′ (m−n)+1v ′ (m−n)+2) = (n + m + l − 2) 2 (n + m + l − 4)2 case 7: for n + m = 2p 2 + 6p + 5 p ≥ 1 and l = 2p 2 + 6p + 4, p ≥ 1 f(vi) = 2i for 1 ≤ i ≤ (n − 2), f(v ′ i) = f(vn−1) + 2i, 1 ≤ i ≤ (m − 3) f(v ′ (n−2)) = 2(n + m) − 2, f(v ′ (n−1)+2) = 2(n + m) − 4, f(v ′′ i ) = 2i + 1 for 1 ≤ i ≤ l − 1 the edge labels acquired here are distinct, as it forms quadratic sequences of the form (4p(p + 1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n − 2} and (4p 2 − 1)2, p ⩾ 1 for {f∗(v′′i v ′′ i+1) : 1 ≤ i ≤ l − 2}. the remaining edges are labeled as follows, f∗(vvn−1) = (f(vn−1)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1) 2, f∗(vv ′′ 1 ) = (f(v ′′ 1 ) 2 f∗(vv ′ n−1) = (f(v ′ n−1)) 2, f∗(vv ′′ n−1) = (f(v ′′ n−1)) 2 f∗(v ′ iv ′ i+1) = (f(vn−1) + 2i)) 2(f(v ′ i) + 2i)) 2 for 1 ≤ i ≤ m − 4 f∗(v ′ m−3v ′ (m−2)) = (f(v ′ (m−3))) 2 (f(v ′ m−2)) 2 f(v ′ (m−2)v ′ (m−1)) = (2(n + m) − 2) 2 (2(n + m) − 4)2 case 8: for n + m = (2p 2 + 6p + 4), p ≥ 1 and l = (2p 2 + 6p + 5), p ≥ 1 f(vi) = 2i for 1 ≤ i ≤ (n − 2), f(v ′ i) = f(vn−1) + 2i for 1 ≤ i ≤ m − 2 f(v ′′ 2 ) = 2(l − 1), f(v ′′ 1 ) = 3, f(v ′′ 1 ) = 9 f(v ′′ 1 ) = 5, f(v ′′ 1 ) = 7, f(v ′′ 5+i) = 2i + 9 for 1 ≤ i ≤ l − 6 the edge labels acquired here are distinct, as it forms quadratic sequences of the form (4p(p + 1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n − 2} and (2p + 9)2, k. g. mirajkar and p. g. sthavarmath p ⩾ 1 for {f∗(v′′5+iv ′′ 6+i) : 1 ≤ i ≤ l − 7}. the remaining edges are labeled as follows, f∗(vvn−1) = (f(vn−1)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v(n−1)+2) 2 f∗(vv ′′ 1 ) = (f(v ′′ 1 ) 2 f∗(vv ′ n−1) = (f(v ′ n−1)) 2, f∗(vv ′′ n−1) = (f(v ′′ n−1)) 2 f∗(v ′ iv ′ i+1) = (f(vn−1) + 2i)) 2(f(v ′ i) + 2i)) 2 for 1 ≤ i ≤ m − 2 f∗(v ′′ 5 v ′′ 6 ) = (f(v ′′ 5 )) 2 (f(v ′′ 6 )) 2, f∗(v ′′ 1 v ′′ 2 ) = (f(v ′′ 1 )) 2, (f(v ′′ 2 )) 2 f∗(v ′′ 3 v ′′ 4 ) = (f(v ′′ 3 )) 2 (f(v ′′ 4 )) 2, f∗(v ′′ 4 v ′′ 5 ) = (f(v ′′ 4 )) 2 (f(v ′′ 5 )) 2 case 9: for all n, m, and l except the above cases f(vi) = 2i for 1 ≤ i ≤ (n − 1), f(v ′ i) = f(vn−1) + 2i for 1 ≤ i ≤ n − m, if n > m and 1 ≤ i ≤ m − n if n < m f(v ′ (n−m)+i) = 2i + 1 for 1 ≤ i ≤ m − 2, f(v ′′ i ) = f(v ′ n−1) + 2i for 1 ≤ i ≤ l − 1 f(v ′′ i ) = 2i + 1 for 1 ≤ i ≤ l − 1. the edge labels acquired here are distinct, as it forms quadratic sequence of the forms (4p(p + 1))2, p ⩾ 1 for {f∗(vivi+1) : 1 ≤ i ≤ n − 2} and (4p 2 − 1)2, p ⩾ 1 for {f∗(v′(n−m)+(i−1))v ′ (n−m+1)+(i−1)) : 1 ≤ i ≤ m − 3}. the remaining edges are labeled as follows, f∗(vvn−1) = (f(vn−1)) 2, f∗(vv1) = 4, f ∗(vv ′ 1) = (f(v ′ 1) 2, f∗(vv ′′ 1 ) = (f(v ′′ 1 ) 2 f∗(vv ′ n−1) = (f(v ′ n−1)) 2, f∗(vv ′′ n−1) = (f(v ′′ n−1)) 2 f∗(v ′ iv ′ i+1) = (f(vn−1) + 2i)) 2(f(v ′ i) + 2i)) 2 for 1 ≤ i ≤ n − m if n > m f∗(v ′′ i v ′′ (i+1)) = (f(v ′ (n−1) + 2i)) 2 (f(v ′′ i + 2i)) 2 for 1 ≤ i ≤ l − 2 in all the cases, the labeling pattern of edges and vertices are distinct so, for three cycles cn, cm, and cl of any lengths n, m, and l with a common cut vertex v admits spl. 2 example 3.2: the square product labeling of three cycles of any lengths n, m, and l with a common cut vertex v is shown in the below figure. cactus graphs with cycle blocks and square product labeling figure 2: three cycles of any lengths n, m, and l with a common cutvertex v theorem 3.3. a graph g having r number of cycle of length n, with a common cutvertex v, except for the graph g with r number of cycle of length 3 having a common cutvertex v admits spl. k. g. mirajkar and p. g. sthavarmath proof. consider the graph g having r number of cycles of length n with a common cutvertex v with nr − (r − 1) vertices and nr edges, here we consider two cases for prove the result with f : v (g) → {1, 2, 3, . . . , (nr − (r − 1))}. case 1: suppose g contains even copies of cycles of length n in this case, the vertices of first half copies r 2 of cycle of length n are labeled by even numbers then next half copies by odd numbers. in first r 2 copies of cycle, the vertices are labeled as, f(v11) = 2, f(v i 2) = f(v i 1) + 2 for 1 ≤ i ≤ r 2 f(vin−1) = f(v i 1) + 4 for 1 ≤ i ≤ r 2 f(vi3) = f(v i n−1) + 2 for 1 ≤ i ≤ r 2 , f(vi3+i) = f(v i 3) + 2i for 1 ≤ i ≤ n − 5 f(v(i + 1)1) = f(v i n−2) + 2 for 1 ≤ i ≤ r − 2 2 in second r 2 copies of cycle, the vertices are labeled by, f ( v r+2 2 1 ) = 3, f ( v r+2 2 2 ) = f ( v r+2 2 1 ) + 2 for 1 ≤ i ≤ r 2 f ( v r+2 2 n−1 ) = f ( v r+2 2 1 ) + 2 for 1 ≤ i ≤ r 2 f ( v r 2 +(i−1) 1 ) = f ( v r 2 +i n−2 ) + 2 for 1 ≤ i ≤ r − 2 2 f ( v r 2 +i 2 ) = f ( v r 2 +i n−1 ) + 2 for 1 ≤ i ≤ r 2 f ( v r 2 +i 2+i ) = f ( v r 2 +i 2 ) + 2 for 1 ≤ i ≤ n − 4 the labels of vertices of f(vi1) and f(v i n−1) with 1 yields distinct edge labels where 1 ≤ i ≤ r, the remaining edge labels are as below f∗(vi1v i 2) = (f(v i 1)) 2 (f(vi2)) 2 for 1 ≤ i ≤ r 2 , f∗(vi2v i 3) = (f(v i 1) + 2) 2 (f(vin−1) + 2) 2 for 1 ≤ i ≤ r 2 f∗(vin−2v i n−1) = (f(v i n−2)) 2 (f(vi1) + 4) 2 for 1 ≤ i ≤ n − 4 f∗(vi3v i 3+i) = (f(v i n−1 + 2)) 2 (f(vi3 + 2i)) 2 for 1 ≤ i ≤ n − 5 the labels of edges of remaining r 2 copies of g are, f∗ ( v r 2 +i 1 v r 2 +i 2 ) = ( f(v r 2 +i 1 ) )2 (f(v r 2 +i n−1 + 2) )2 for 1 ≤ i ≤ r − 2 2 f∗ ( v r 2 +i n−2v r 2 +i n−1 ) = ( f(v r 2 +i n−2) )2 ( f(v r 2 +i 1 + 2) )2 for 1 ≤ i ≤ r − 2 2 f∗ ( v r 2 +i 2 v r 2 +i 2+i ) = ( f(v r 2 +i n−1 + 2) )2 ( f(v r 2 +i 2 + 2i) )2 for 1 ≤ i ≤ n − 4 cactus graphs with cycle blocks and square product labeling case 2: suppose g contains odd copies of cycles of length n in this case, the vertices of r copies of cycle of length n are first labeled by even numbers then by odd numbers as follows, f(v11) = 2, f(v i 2) = f(v i 1) + 2 for 1 ≤ i ≤ r − 1 2 f(vi+11 ) = f(v i n−2) + 2 for 1 ≤ i ≤ r − 3 2 f(vin−1) = f(v i 1) + 4 for 1 ≤ i ≤ r − 1 2 f(vi2) = f(v i 1) + 2i for 1 ≤ i ≤ r − 1 2 f(v(i)3) = f(v i n−1) + 2 for 1 ≤ i ≤ r − 1 2 f(v(i)3+i) = f(v i 3) + 2i for 1 ≤ i ≤ n − 5 f ( v r+1 2 1 ) = f(v r−1 2 n−2 ) + 2, f ( v r+1 2 n−1 ) = f ( v r+1 2 1 ) + 4 f ( v r−1 2 +i 2 ) = f ( v r−1 2 +i 1 ) + 2 for 1 ≤ i ≤ n − 6 f ( v r−1 2 +i n−4 ) = 2i + 1 for 1 ≤ i ≤ n − 4 f ( v r+1 2 +i 1 ) = f ( v r+1 2 +i n−2 ) + 2 for 1 ≤ i ≤ r − 1 2 f ( v r+1 2 +i n−1 ) = f ( v r+1 2 +i 1 ) + 2 for 1 ≤ i ≤ r − 1 2 f ( v r+1 2 +j i+1 ) = f ( v r+1 2 +j 1 ) + 4 + 2(i − 1) for 1 ≤ i ≤ n − 3 and 1 ≤ j ≤ r − 1 2 the labels of vertices of f(vi1) and f(v i n−1) with cutvertex label 1 yields distinct edge labels where 1 ≤ i ≤ r, the remaining edge labels are as below. f∗(vi1v i 2) = (f(v i 1)) 2 (f(vi1 + 2)) 2 for 1 ≤ i ≤ r − 1 2 , f∗(vi2v i 3) = (f(v i 1 + 2)) 2 (f(vin−1 + 2)) 2 for 1 ≤ i ≤ r − 1 2 , f∗(vin−2v i n−1) = (f(v i n−2)) 2 (f(vi1 + 4)) 2 for 1 ≤ i ≤ r − 1 2 f∗(vi3v i 3+i) = (f(v i n−1 + 2)) 2 (f(vi3 + 2i)) 2 for 1 ≤ i ≤ n − 4 f∗ ( v r−1 2 +i n−2 v r−1 2 +i n−1 ) = ( f(v r−1 2 +2 1 ) )2 ( f(v r−1 2 +i 1 + 4) )2 for 1 ≤ i ≤ r − 3 2 f∗ ( v r+1 2 +i 1 v r+1 2 +i 2 ) = ( f(v r−1 2 +i n−2 + 2) )2 ( f(v r−1 2 +i n−1 + 2) )2 for 1 ≤ i ≤ r − 1 2 f∗ ( v r+1 2 +i i v r+1 2 +i 2+i ) = ( f(v r+1 2 +i n−1 + 2) )2 ( f(v r+1 2 +i 2 + 2i) )2 for 1 ≤ i ≤ n − 3 k. g. mirajkar and p. g. sthavarmath subcase 1: for cycle of length 4 with even cpoies f(v11) = 2, f(v i n−1) = f(v i 1) + 2 for 1 ≤ i ≤ r 2 f(vi+11 ) = f(v i 2) + 2 for 1 ≤ i ≤ r − 2 2 f(vi2) = f(v i n−1) + 2 for 1 ≤ i ≤ r 2 , f ( v r+2 2 1 ) = 3 f ( v r+2 2 n−1 ) = f ( v r 2 +i 1 ) + 2 for 1 ≤ i ≤ r 2 , f ( v r+2 2 +i 1 ) = f ( v r 2 2 ) + 2 for 1 ≤ i ≤ r − 1 2 f ( v r 2 2 ) = f ( v r 2 +i n−1 ) + 2 for 1 ≤ i ≤ r 2 the labels of vertices of f(vi1) and f(v i n−1) with 1 yields distinct edge labels where 1 ≤ i ≤ r, the remaining edge labels are as below f∗(vi1v i 2) = (f(v i 1)) 2 (f(vin−1 + 2)) 2 for 1 ≤ i ≤ r 2 f∗(vi2v i n−1) = (f(v i n−1 + 2)) 2 (f(vi1 + 2)) 2 for 1 ≤ i ≤ r 2 f∗ ( v r 2 +i 1 v i 2 ) = ( f(v r 2 +i 1 ) )2 ( f(v r 2 +i n−1 + 2) )2 for 1 ≤ i ≤ r 2 f∗ ( v r+2 2 +i 2 v r 2 +i n−1 ) = ( f(v r 2 +i n−1 + 2) )2 ( f(v r 2 +i 1 + 2) )2 for 1 ≤ i ≤ r 2 subcase 2: for cycle of length 5 with odd cpoies f(v11) = 2, f(v i 2) = f(v i 1) + 2 for 1 ≤ i ≤ r − 1 2 f(vi+11 ) = f(v i n−2) + 2 for 1 ≤ i ≤ r − 2 2 f(vin−1) = f(v i 1) + 4 for 1 ≤ i ≤ r − 1 2 , f(vi2) = f(v i 1) + 2 for 1 ≤ i ≤ r − 1 2 f(vi3) = f(v i n−1) + 2 for 1 ≤ i ≤ r − 1 2 f ( v r+1 2 1 ) = f ( v r−1 2 3 ) + 2, f ( v r+1 2 3 ) = 3, f ( v r+1 2 n−1 ) = 7 f ( v r+1 2 +i 1 ) = f ( v r−1 2 +i n−1 ) + 2 for 1 ≤ i ≤ r − 1 2 f ( v r+3 2 2 ) = 9, f ( v r+1 2 +i 3 ) = f ( v r+3 2 +i n−1 ) + 2i for 1 ≤ i ≤ r − 1 2 f ( v r+1 2 +j 2 ) = f ( v r+3 2 +j 1 ) + 2i for 1 ≤ i ≤ n − 2 and 1 ≤ j ≤ r − 3 2 cactus graphs with cycle blocks and square product labeling the labels of vertices of f(vi1) and f(v i n−1) with 1 yields distinct edge labels where 1 ≤ i ≤ r, the remaining edge labels are as below f∗(vi1v i 2) = (f(v i 1)) 2 (f(vin−1 + 2)) 2 for 1 ≤ i ≤ r − 1 2 f∗(vi2v i 3) = (f(v i 1 + 2)) 2 (f(vin−1 + 2)) 2 for 1 ≤ i ≤ r − 1 2 f∗(vin−2v i n−1) = (f(v i n−2)) 2 (f(vi1 + 4)) 2 for 1 ≤ i ≤ r − 1 2 f∗(vi3v i 3+i) = (f(v i n−1 + 2)) 2 (f(vi3 + 2i)) 2 for 1 ≤ i ≤ n − 4 f∗ ( v r+1 2 1 v r+1 2 2 ) = ( f(v r−1 2 n−2 + 2) )2 ( f(v r+1 2 1 + 2) )2 f∗ ( v r+1 2 2 v r+1 2 3 ) = 9 ( f(v r+1 2 1 + 2) )2 f∗ ( v r+1 3 2 v r+1 2 n−1 ) = 9 ( f(v r+1 2 n−1) )2 f∗ ( v r+1 2 1 v r+1 2 2 ) = 25 ( f(v r+1 2 n−1 + 2) )2 f∗ ( v r+3 2 2 v r+3 2 3 ) = ( f(v r+1 2 n−1 + 2) )2 ( f(v r+3 2 n−1 + 2) )2 f∗ ( v r+3 3 2 v r+3 2 n−1 ) = ( f(v r+3 2 n−1 + 2) )2 ( f(v r+3 2 n−2 + 2) )2 f∗ ( v r+3 3 +i 2 v r+3 2 +i 2 ) = ( f(v r+1 2 +i n−1 + 2) )2 ( f(v r+3 2 +i n−2 + 2i) )2 for 1 ≤ i ≤ r − 3 2 for r copies of cycle of length 3 with common cut vertex v, | v (g) |≥ (v11)2 (v12)2 are not square product graphs. consider a graph g containing r cycles of length 3 with fixed cut vertex labeled as 1, labeling of vertices carries distinct non negative integers and v (g) is bijective. the cut vertex v is adjacent with all other vertices since the graph is c3 with r copies. hence, while labeling the edges the product of labels of vertices with the fixed cut vertex 1 is not injective for | v (g) |≥ (v1)2 (v2)2. in the below figure, f∗(v1v2) = 36 and f ∗(vv21) = 36 which is not injective. 2 k. g. mirajkar and p. g. sthavarmath example 3.3: the square product labeling of even copies of c6, odd copies of c6, even copies of c4, odd copies of c5 and r copies of c3 are shown in the below figures. figure 3: square product labeling of even copies of c6 and c4 and odd copies of c6 and c5 cactus graphs with cycle blocks and square product labeling figure 4: square product labeling of r copies of c3 4 conclusion in this article, results on square product labeling for cactus graphs with cycle blocks are established. here the limitation is results can be established only for r copies of cycle of any length with a common cutvertex except for r copies of cycles of length 3 which are not square product graphs for | v (g) |≥ (v1)2 (v2)2. in this article, the results are established only for cactus graphs with cycle blocks. the results on square product labeling can also be extended to trees. acknowledgements authors are thankful to karnatak university, dharwad, karnataka, india for the support through university research studentship (urs), no. ku/sch/urs/2021/623, dated: 26/10/2021. k. g. mirajkar and p. g. sthavarmath references v. ajitha, s. arumugam, and k. germina. on square sum graphs. akce international journal of graphs and combinatorics, 6(1):1–10, 2009. j. babujee and s. babitha. on square sum labeling in graphs. int.rev.fuzzy math., 7(2):81–87, 2012. d. burton. elementary number theory. tata magraw hill, 2006. f.harary. graph theory. addison-wesleyl, 1969. j. a. gallian. a dynamic survey of graph labeling. electronic journal of combinatorics, (dynamicsurveys), 2020. k. kalaiarasi and l. mahalakshmi. application of cactus fuzzy labeling graphs. adv. appl. math. sci., 21(5):2841–2492, 2022. n. khan, m. pal, and a. pal. (2, 1)-total labeling of cactus graphs. j. comput. inf. sci. eng., 5(4):243–260, 2010a. n. khan, m. pal, and a. pal. mapana j. sci., 11(4):15–42, 2010b. k. mirajkar and p. sthavarmath. on square product labeling. accepted for publication in south east asian j. math. math. sci., 2022. s. philomena, m. pal, and k. thirusanga. square and cube difference labeling of cycle cactus, special trees and a new key graphs. ann. pure appl. math., 8(2): 115–121, 2014. a. rosa. on certain valuations of the vertices of a graph, theory of graphs (internat. symposium, rome, july 1966), 1967. j. shiama. square difference labeling for some graphs. international journal of computer applications, 44(4):30–33, 2012. approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 41, 2021, pp. 28-44 28 a result on b-metric space using − compatible mappings thirupathi thota* srinivas veladi† abstract the objective of this paper is to generate a common fixed point theorem in b-metric space using  -compatible and -continuous mappings. this result generalizes the theorem proved by j.r. roshan and others. further our findings are supported by discussing some valid examples. keywords: b-metric space; fixed point;  -fixed point;  -compatible;  -continuous mappings. 2010 ams subject classification: 54h25, 47h10. *mathematics department, sreenidhi institute of science & technology, ghatkesar, hyderabad, telangana, india-501301. e-mail: thotathirupathi1986@gmail.com. †mathematics departments, university college of science, saifabad, osmania university, hyderabad, india. srinivasmaths4141@gmail.com. *received on september 26, 2021. accepted on december 5, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.666. issn: 1592-7415. eissn: 2282-8214. a result on b-metric space using − compatible mappings 29 1. introduction fixed point theory plays an important role in mathematics and it is fast growing in the fields of analysis because of its applications in mathematics and allied subjects. several authors [1, 2, 3, 4] established many results in fixed point theory using various weaker conditions. in the recent past, b-metric space was emerged as one of the generalizations of metric space. during this period czerwik [5] introduced the concept of −b metric space. in recent years, number of well-known fixed point theorems have been established in b-metric space such as [6, 7, 8, 9]. the concept of alpha compatible and alpha continuous mappings were introduced in metric space [10] and some results were established in the recent past under certain weaker conditions. j.r. roshan, n. shobkolaei, s.sedghi and m.abbas [11] proved a common fixed point theorem using compatible and continuous mappings in b-metric space. in this paper, we use the concept of  -compatible and  -continuous maps and generate a fixed point theorem in b-metric space. 2.2. preliminaries definition 2.1. a function + → rxxd : where x is a nonempty set and 1m is a b-metric space if and only if for each x ,, (i)  == 0),(d (ii) ),(),(  dd = (iii)   .),(),(),(  ddmd + definition 2.2. two self maps m, n of a b-metric space x is said to be compatible if d ( ) 0, =kk nmmn  whenever sequence in x such that  == kk nm for some as definition 2.3. a point x is said to be an  fixed point of map xxm →: if ( )  =m . remark 2.4. a fixed point is not necessarily  fixed point and  fixed point is not necessarily a fixed point. if  =1, the identity map then they coincide. example2.5. let m,  : rr → be defined by m(u) = 1 2 −u and .1)( 3 += uu   k  x .→k thirupathi thota and srinivas veladi 30 then 0)1()0)(( =−= m and 1)0()1)(( == m . therefore 0 and 1 are fixed points but not fixed points. example 2.6. let m,  : rr → be defined by m(u) = 3 2 u and .)( 3 uu = here 0)0()0)(( == m and m(0) = 0. therefore 0 is fixed point of m which is also fixed point of m. definition 2.7. a pair of self maps m and n of a b-metric space x is called − commuting if ( ) ( ) )()()()( mnnm = for all x . the preceding example show the relation between commuting and − commuting mappings. example 2.8. let m, n,  : rr → be defined by 4 )( uum = , uun =)( and uu 3)( = for all ru  . 24 )()()( uuumumnthen === and 24 )()( uununm == . therefore )()( unmumn = . hence m and n are commuting mappings. `also for ru  , 44 3)())(( uuum ==  , .3)())(( uuun == 2524 3)3()3)(())(()( uuumunm ===  .33)3()3)(())(()( 244 uuunumnand ===  therefore ).)(()())(()( unmumn   hence m and n are not − commuting mappings. example 2.9. suppose    00:,, −→− rrnm  given by 4 )( uum = , 5 )( uun = and u u 1 )( = for all .ru  2 05 )()( uumumnthen == and 2 04 )()( uununm == . therefore )()( unmumn = . hence m and n are commuting mappings. also for  0− ru , 4 4 1 )())(( u uum == , 5 5 1 )())(( u uun == 2 0 2 055 111 )())(()( u uu t u munm =      =            =      =  . a result on b-metric space using − compatible mappings 31 2 0 2 044 111 )())(()( u uu s u numnand =      =            =      =  . therefore ))(()())(()( umnunm  = hence m and n are − commuting mappings. example 2.10. let m, n,  : rr → be defined by 4 )( uum = , uun 4)( = and 4 )( u u = for all ru  . 444 4)4()4()( uuumumn === , 44 4)()( uununm == . therefore )()( unmumn  . hence m and n are not commuting mappings. also for ru  , 4 )())(( 4 4 u uum == , uuun == )4())((  4 )())(())(()( 4 4 u uumunm ===  4 )( 4 )())(()( 4 4 4 u u u numn ==        =  . therefore, ).)(()())(()( umnunm  = hence m and n are − commuting mappings. example 2.11. suppose    00:,, −→− rrnm  given by 3 )( uum = , 2 3)( uun = and 2 1 )( u u = for all ru  . 632 3)3()( uumumn == , 63 3)()( uununm == . therefore )()( unmumn  . hence m and n are not commuting mappings. also for  0− ru , 6 3 1 )())(( u uum == , 42 2 3 1 )3())(( u uun == 2 46 1 2344 9 9 1 9 1 9 1 )())(()( u uu m u munm =      =            =      =  . 9 1 3 11 )())(()( 2 4 1 266 u uu s u numn =            =            =      =  . therefore, ).)(()())(()( umnunm   thirupathi thota and srinivas veladi 32 hence, m and n are not − commuting mappings. definition 2.12. a pair of self maps of a b-metric space (x, d) is called weakly − commuting mappings if )( m and )( n are weakly commuting maps. i.e ( ) ( )( ) ( )))((),)(()()()(),()()( unumdumnunmd   for all .xu  definition 2.13. the self maps m,nofa b-metric space x are called  compatible maps if ( )m and ( )n are compatible if whenever  nu is a sequence in x such that ( )( ) thenxunumd nn ,)(),)(( →  ( ) ( )( ) .0)()()(),()()( →→ nasumnunmd nn  definition 2.14. two self maps m and n are called weakly − compatible if )( m and )( n are weakly compatible,i.e )( m and )( n commute at their coincidence points. remark 2.15. it may be observed that − commuting maps are weakly − commuting maps,weakly − commuting maps are − compatible maps and − compatible mappings are weakly − compatible maps.but converse is not true in each case. these facts are presented in the following example. example2.16. let m, n,  :    00 −→− rr given by 5)( uum = , 4 )( uun = and u u 1 )( = for all ru  . here 5 5 1 )())(( u uum == , 4 4 1 )())(( u uun == 20 204 11 )())(()( u uu munm =      =      =  . 20 205 11 )())(()( u uu numnand =      =      =  . therefore, ))(()())(()( umnunm  = hence, n and m are − commuting mappings. also for  0− ru ( ) ( )( ) 0)()()(),()()( 2 2020 =−= uuumnunmd  ( ) 2 5 2 45 111 )(,)( u u uu unumd − =−= a result on b-metric space using − compatible mappings 33 ( ) ( )( ) 0)()()(),()()( =umnunmd  ( ).)(,)( 1 2 5 unumd u u = −  therefore m and n are weakly − commuting maps. now ( ) 5 1 )( n n u um = , ( ) 4 1 )( n n u un = ( ) .10 11 ))((),)(( 2 45 →→−= n nn nn uas uu unumd  hence m and n are − compatible mappings. definition 2.16. the self mapping m of a b-metric space (x,d) is said to be − continuous if ( )m is continuous .in other words for every ,0 for all 0 such that ( ) ( ) .)(,)(,   vmumdyxd the following theorem was proved in [11]. theorem 2.17[10]. let f, g, s and t be four self mappings defined on a complete b-metric space(x, d) with the following conditions: (c1) ( ) ( )xtxf  and ( ) ( )xsxg  (c2) ( ) ( ) ( ) ( ) ( ) ( )( )       + tvfudgvsudtvgvdsdufudtvsud k q gvfud ,, 2 1 ,,,,,,max, 4 holds for every xvu , with 10  q . (c3) the self mappings t and s are both continuous (c4) two pairs ( )sf , and ( )tg , are compatible. then the above four maps will be having a unique fixed point which is common. now we prove the generalization of theorem (2.17) in the preceding theorem under some modified conditions. to do so, we'll need to recall the following lemmas. thirupathi thota and srinivas veladi 34 lemma 2.18[10]. let ( )dx , be a −b metric space with 1k and two sequences }{ nu and }{ nv are b-convergent to u and v respectively. then we have ( ) ).,(),(suplim,inflim),( 1 2 2 vudkvudvudvud k nn n nn n  →→ lemma 2.19[9]. let m and n be − compatible mappings from a b-metric .,)(lim)(limsuchthat itselfintod)(x,space xsomeforunum n j n j == →→  then ( )  == → )()()(lim munm n j , if m is − continuous. 3. main result theorem 3.1. let m, n, p and q be four self maps and  as defined on a bmetric space (x,d) which is complete with the given conditions: (b1) ( )( ) ( )( )xqxm   and ( )( ) ( )( )xpxn   (b2) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) , , , 2 1 ,, ,, ,, max, 4                         +  vqumd vnupd vqvnd upumd vqupd k q vnumd       holds for every xvu , with ( ).1,0q (b3) the mappings q and p are − continuous (b4) the pair of maps ( )pm , and ( )qn , are − compatible. then the above four maps will be having a unique fixed point which is common. proof: using the condition (b1) for the point xu 0  xu 1 such that ( ) ( ) . 10 uqum  = for this point 1u we can select a point xu 2 such that ( ) ( ) 21 upun  = and so on. continuing this process it is possible to construct a sequence }{ jv such that ( ) ( ) 1222 +== jjj uqumv  and ( ) ( ) 221212 +++ == jjj upunv   .0j we now demonstrate that }{ j v is a cauchy sequence. take ( ) ( ) ( )( ) 122122 ,, ++ = jjjj unumdvvd  a result on b-metric space using − compatible mappings 35 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ))},(),(( 2 1 ),,(),,(),,(max{ 122122 1212221224 ++ +++ +  jjjj jjjjjj uqumdunupd uqundupumduqupd k q   ))}.,(),(( 2 1 ),,(),,(),,(max{ 221212 2121222124 jjjj jjjjjj vvdvvd vvdvvdvvd k q + = +− +−− } 2 ),( ),,(),,(max{ 1212 1222124 +− +− = jj jjjj vvd vvdvvd k q ))}.,(),(( 2 ),,(),,(max{ 1222121222124 +−+− + jjjjjjjj vvdvvd k vvdvvd k q if ( ) ( ) jjjj vvdvvd 212122 ,, −+  for some j, then the above inequality gives ( ) ( ) 1223122 ,, ++  jjjj vvd k q vvd a contradiction. hence ( ) ( ) jjjj vvdvvd 212122 ,, −+  for all nj  . now the above inequality gives ( ) ( ).,, 2123122 jjjj vvd k q vvd −+  -------------------(1) similarly ( ) ( ).,, 12223212 −−−  jjjj vvd k q vvd ------------(2) from (1) and (2) we have ( ) ( ),,, 211 −−−  jjjj vvdvvd  where 1 3 = k q  and 2j . hence for all 2j ,we obtain ( ) ( ) ),(..........,, 01 1 211 vvdvvdvvd j jjjj − −−−   .----------(3) so for all lj  , we have ( ) ( ) ),(..........),(,, 1 1 21 2 1 jj lj lllllj yydkyydkyykdyyd − −− +++ +++ . now from (3),we have ( ) ),()..............(, 01 1112 vvdkkkvvd jljll lj −−−+ +++  ),(....)..........1( 01 22 vvdkkk l +++  ).,( 1 01 vvd k k l   −  taking limits as →jl, ,we have 0),( → lj vvd as k is less than one. thirupathi thota and srinivas veladi 36 therefore }{ j v is a cauchy sequence in x and by completeness of x, it converges to some point  in x such that ( ) ( ) ( ) ( )  ==== + → + → + →→ 2212122 limlimlimlim j j j j j j j j upunuqum . since p is − continuous, therefore ( ) ( ) ( ) pupp j j = + → 22 lim and ( ) ( ) ( ) .lim 2  pump j j = → by (b4) we have (m,p) is − compatible, ( ) ( ) ( ) ( )( ) 0,lim 22 = → jj j umpupmd  so by lemma (2.19), we have ( ) ( ) ( ) pupm j j = → 2 lim . now putting j upu 2 )(= and 12 + = j uv in (b2), we get take sup limit as →j on both the sides and by lemma (2.18),we get ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) , ,)( ,)( 2 1 ,, ,)(,)( ,,)( max,)( 122 122 1212 22 122 4122                           +  + + ++ + + jj jj jj jj jj jj uqupmd unuppd uqund uppupmd uquppd k q unupmd       ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) , ,)(suplim ,)(suplim 2 1 ,,suplim ,)(,)(suplim ,,)(suplim max ,)(lim ,)( 122 122 1212 22 122 4 1222                                   +   + → + → ++ → → + → + → jj j jj j jj j jj j jj j jj j uqupmd unuppd uqund uppupmd uquppd k q unupmd k pd        ( ) ( ) ( ) ( ) ( )( )           +    ,)(,)( 2 1 ,,,)(,)(,,)( max 2 4 pdpd dppdpdk k q ( ) ( )          ,)( ,0,0,,)( max 2 4 pd pdk k q ( ) ( )          ,)( ,0,0,,)( max 2 4 pd pdk k q = ( ) ,)(2 4 pdk k q a result on b-metric space using − compatible mappings 37 = ( ).,)( 2 pd k q therefore ( ) ( ) ,)(,)( pqdpd  . as ,10  q so .)(  =p since q is − continuous, therefore ( ) ( ) ( ) tutt j j = + → 22 lim and ( ) ( ) ( ) .lim 2  qunq j j = → since the pair (n,q) is − compatible, we have ( ) ( ) ( ) ( )( ) .0,lim 22 = → jj j unquqnd  so by lemma (2.18) we have ( ) ( ) ( ) quqn j j = → 2 lim . now putting j uu 2 = and ( ) 12 + = j uqv  in (b2), we get ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )                           +  + + ++ + + 122 122 1212 2 122 4122 , , 2 1 ,, ,, ,, max, jj jj jj j jj jj uqqumd uqnupd uqquqnd upumd uqqupd k q uqnumd       take sup limit as →j on both the sides and by lemma (2.18),we get ( )( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )                +  +     qd qdk qdk k q uqnumdqd jj , , 2 ,0,0,, max,, 2 2 4122 ( )( ) ( ) )(,, 2 qd k q qd  which implies that .)(  q= therefore .)()(  == qp ---------(4) again putting =u and 12 + = j uv in (b2) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) , , , 2 1 ,, ,, ,, max, 12 12 1212 12 412                           +  + + ++ + + j j jj j j uqmd unpd uqund pmd uqpd k q unmd       take sup limit as →j on both the sides and by lemma (2.18) we get thirupathi thota and srinivas veladi 38 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) , , , 2 ,, ,,,, max, 2 2 22 4                         +       md pdk dk pmdkpdk k q md = ( )( )., 2 md k q this implies that ( )( ) .0, =md that gives ( )  =m as .10  q again putting =u and =v in (b2), we get ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) , , , 2 1 ,, ,, ,, max, 4                         +        qmd npd qnd pmd qpd k q nmd ( )( ) ( ) ( ) ( )( ) ( )( ) ( )( ) , ,, 2 1 ,,,,,, max, 4           +     dnd nddd k q nd ( )( ) ( )( ) ( )( )( ) , 0, 2 1 ,,,0,0 max, 4           +     nd nd k q nd ( )( ) ( )( ) ( )( )( ) , 0, 2 1 ,,,0,0 max, 4           +     nd nd k q nd ( )( ) , 2 nd k q = ( )( ) ,nqd which implies that ( )( ) 0, = nd ( ) n= . therefore ( ) ( ) . == nm ----------(5) hence from (4) and (5) we obtain  ==== )()()()( nmqp . therefore  is a common − fixed point of m, n, p and q. a result on b-metric space using − compatible mappings 39 uniqueness: assume that )(   is another common fixed point of the four mappings m, n, p, and q. put =u and =v in (b2) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )                         +        qmd npd qnd pmd qpd k q nmd , , 2 1 ,, ,, ,, max, 4 ( ) ( ) ( ) ( ) ( )( )           +    ,, 2 ,,,,,, max 2 222 4 dd k dkdkdk k q ( ) ),(,0,0,,max 22 4  dkdk k q = ),( ),( 2 42   dk k q k d = ).,(),(  qdd  10  qas ,so . = hence the four maps m, n, p and q will be having a unique common fixed point. now we give an illustration to support our result. example 3.2: suppose  1,0=x is a b-metric space ( ) 2 , vuvud −= where xvu , . define the four self maps m, n, p , q and  as follows ( )           =  + === 1 2 1 , 6 1 2 1 , 2 1 2 1 0, 6 2 )(;)( u u u u unumuu ; thirupathi thota and srinivas veladi 40 ( )      −  + == 1 2 1 ,1 2 1 0, 6 34 )( uu u upuq  ; ( ) (        == 2 1 33.0,16.0)( xnxm , ( ) ( )       == 2 1 ,0xqxp , clearly the condition (b1) is satisfied. take a sequence as j u j 1 2 1 −= for . now 6 5 6 1 6 5 6 2 1 2 1 lim 1 2 1 limlim =−=       +− =      −= →→→ n j j mmu jj j j and 6 5 6 3 1 2 1 4 lim 1 2 1 limlim =         +      − =      −= →→→ j j qpu jj j j that is a sequence }{ ju in x such that 6 5 limlim == →→ j j j j pumu . similarly . 6 5 limlim == →→ j j j j qunu also andkas j m j m j mpmpu j →=      −=               +      − =      −= 2 1 3 2 6 5 6 3 1 2 1 4 1 2 1 . 6 1 6 1 6 5 1 6 1 6 5 6 2 1 2 1 1 2 1 →=+−=      −=               +      − =      −= jas nn p j p j pmpmu j 0j  a result on b-metric space using − compatible mappings 41 ( ) .0 36 4 6 5 2 1 6 5 , 2 1 ,lim 2 =−=      = → dpmumpudthatso jj j ( ) .0,lim  → jj j qnunqudsimilarly showing that the pairs ( )pm , and ( )qn, are not compatible mappings. ( ) ( ) →=      −=               +      − =      −= jas j j j mumagain j 6 5 6 1 6 5 6 2 1 2 1 1 2 1 )(  ( ) ( ) . 6 5 3 2 6 5 6 3 1 2 1 4 1 2 1 )( →=      −=               +      − =      −= jas j j j pupalsoand j  ( ) ( ) ( ) andjas j m j mupm j →=      =            −=      −= 6 1 6 1 3 2 6 5 3 2 6 5  ( ) ( ) ( ) . 6 1 6 1 6 1 6 1 6 5 1 6 1 6 5 →=      +=       +−=      −= jas nnj qump j  ( ) ( ) ( ) ( )( ) .0 6 1 6 1 6 1 , 6 1 ,lim 2 =−=      = → dumpupmdthatso jj j  ( ) ( ) ( ) ( )( ) .0,lim = → jj j uqnunqdsimilarly  showing that the pairs ( )pm  , and ( )qn  , are  -compatible mappings. now we fulfill the requirement that the mappings m, n, p and q satisfy the condition (b2). thirupathi thota and srinivas veladi 42 we have ( ) 0)( =um , ( ) 16 )( v vn = ( ) . 2561616 ,0)(,)( 22 vvv dvnumd ==      = also ( ) uup =)( and ( ) 4 )( v vq = , ( ) , 44 ,)(,)( 2 v u v udvqupd −=      = ( ) ( ) ,,0)(,)( 2uudupumd == ( ) , 256 9 4164 , 16 )(,)( 22 vvvvv dvtvgd =−=      = ( ) , 1616 ,)(,)( 2 v u v udvgusd −=      = ( ) . 164 ,0)(,)( 2 vv dvqumd =      = substituting all these in the inequality (b2), we obtain                 +      −      − 16162 , 256 9 ,, 4 max 256 2222 22 2 2 4 2 vv u kv kuk v uk k qv if we choose 5.0=u , 9.0=v and k = 2 we obtain  4948.0,112.0,1,3024.0max 16 00316.0 2 k q  )1( 16 00316.0 q  ).1,0(05.0)1( 16 00316.0 = q q hence the condition (b2) is satisfied. 4. conclusion this work is focused to generate the existence of common fixed point theorem proved by j.r.roshan and others mentioned in theorem (2.17) by employing a result on b-metric space using − compatible mappings 43 some weaker conditions − compatible and − continuous mappings instead of compatible and continuous mappings. at the end of the theorem our result is justified with a suitable example. references [1] v.srinivas and k.mallaiah. a resulton multiplicative metric space. journal of mathematical and computational science, 10(5),1384-1394, 2020. [2] v.nagaraju,bathini raju and p.thirupathi. common fixed point theorem for four self maps satisfying common limit point property. journal of mathematical and computational science, 10(4), 1228-1238, 2020. [3] b.vijayabaskar reddy and v.srinivas. fixed point results on multiplicative semi metric space. journal of scientific research, 12(3), 341-348, 2020. [4] v.srinivas,t.thirupathi and k.mallaiah. a fixed point theorem using e.a property on multiplicative metric space. journal of mathematical and computational science, 10 (5), 1788-1800, 2020. [5] czerwik,s.nonlinear set valued contraction mappings in b-metric spaces. atti.sem.mat.fis.uni.sem.mat.fis.univ.modena, 46(2), 263-276, 1998. [6] boriceanu, monica, marius bota and adrian petruşel. maltivalued fractals in b-metic spaces. central european journal of mathematics, 8(2), 367-377, 2010. [7] t.thirupathi and v. srinivas. some outcomes on b-metric space. journal of mathematical and computational science, 10(6), 3012-3025, 2020. [8] a. aghajani, m.abbas and j. roshan. common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces. mathematica slovaca ,64 (4), 941-960, 2014. [9] t.thirupathi and v.srinivas. certain results in b-metric space using subcompatible, faintly compatible mappings. journal of mathematical and computational science, 11(6), 8382-8399, 2021. https://link.springer.com/journal/11533 thirupathi thota and srinivas veladi 44 [10] shivram sharma and praveen kumar sharma. on common fixed point theorems. journal of mathematical and computational science,11(1), 87-108, 2021. [11] j.r.roshan,n.sobkolaei,s.sedghi and m.abbas. common fixed point of four maps in b-metric spaces. hacettepe journal of mathematics and statistics, 43(4), 613-624, 2014. − ratio mathematica volume 41, 2021, pp. 283-290 contra nα-i-continuity over nano ideals s. vijaya* p. santhi† a. yuvarani‡ abstract the conceptualization of nα-i-open sets and nα-i-continuous functions in nano ideal topology are used to study contra nα-i-continuity. also the characteristics and behaviours of contra nα-i-continuity based on nano urysohn space and nano ultra hausdorff space are discussed. keywords: cnα-cts function, cnα-i-cts function, nα-i-t2 space, nα-i-connected. 2020 ams subject classifications: 54a05, 54b05. 1 *thiagarajar college, madurai, tamil nadu, india; viviphd.11@gmail.com. †the standard fireworks rajaratnam college for women, sivakasi, tamil nadu, india; saayphd.11@gmail.com. ‡the american college, madurai, tamil nadu, india; yuvamaths2003@gmail.com. 1received on august 7, 2021. accepted on november 30, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.651. issn: 1592-7415. eissn: 2282-8214. ©vijaya et al. this paper is published under the cc-by licence agreement. 283 s. vijaya, p. santhi, a. yuvarani 1 introduction the ideal concept in topology was developed by kuratowski [kuratowski, 1966].the notion of α-i-continuity was introduced in 2004 [a. acikgoz and yuksel, 2004].the conception of nano topology was initated by l.thivagar [thivagar and richard, 2013a].in addition to that the concept of continuity, α-continuity, kernal and clopen in nano topology was introduced by [karthiksankari and subbulakshmi, 2019] [thivagar and richard, 2013b] and [m. lellis thivagar and suthadevi, 2017].parimala and jafari [parimala and jafari, 2018] had worked on nano ideals.this work aims the introduction of contra nα-i-continuous functions by applying the concept of nα-i-open and nα-i-continuity in nano ideal topology.also this contra nα-i-continuity are compared with some existing functions.moreover, new class of functions are obtained. at every places the new notions have been substantiated with suitable examples.throughout this article we use the notation nts, nits, n-regular, n-open, nα-open, n-clopen, nα-cts for nano topological spaces, nano ideal topological spaces, nano regular space, nano open, nano α-open, nano clopen, nano α-continuous respectively.similar notations are used for their respective closed sets. 2 preliminaries definition 2.1. [m. lellis thivagar and suthadevi, 2017] let (u, τr(x)) be a nts and s is a subset of u.the nano kernel of s is defined as nker(s)=∩{u : s is a subset of u, u ∈ τr(x)}. theorem 2.1. [m. lellis thivagar and suthadevi, 2017] let (u, τr(x)) be a nts and a1, a2 ⊆ u.we have 1. x ∈ nker(a1) iff for any n-closed set f containing x, a1 and f are disjoint, 2. if a1 ⊆ nker(a1) and then a1= ker(a1) if a1 is n-open in u, 3. if a1 ⊆ a2, then nker(a1) ⊆ nker(a2). definition 2.2. [thivagar and suthadevi, 2016] a nts (u, τr(x)) along with an ideal i defined on u is called as a nits and is denoted by (u, τr(x),i).throughout this paper u represents a nts (u, τr(x)) and ui represents a nits (u, τr(x),i). definition 2.3. [rajasekaran and nethaji, 2018] let (u, τr(x),i) be a nano ideal topological space and a ⊆ u.then a is said to be nα-i-open if a ⊆ nint(ncl∗(nint (a))).the complements of nα-i-open is nα-i-closed set. 284 contra nα-i-continuity over nano ideals theorem 2.2. [v. inthumathi and krishnaprakash, 2020] let (u1, τr(x1),i) be a nits and (u2, τr(x1)) be a nts.then h : u1 → u2 is called nα-i-cts on u1 if h−1(s) is nα-i-open in u1 for any n-open set s in u2. definition 2.4. bhuvaneswari and nagaveni [2018] a nts (u, τr(x)) is called n-regular space, if for each n-closed set t and each point x 6∈ t, ∃ disjoint n-open sets g and h such that x ∈ g and t ⊂ g. 3 contra nα-i-continuity the notations used are cnα-open, cn-cts function, cnα-cts function, cnαi-cts function for contra nano α-open, contra nano continuous, contra nαcontinuous, contra nα-i-continuous function resp. definition 3.1. let (u1, τr(x1)) and (u2, τr′ (x2)) be nts.then h : u1 → u2 is cnα-cts if h−1(s) is nα-closed in u1 whenever s is n-open set in u2. definition 3.2. let h : (u1, τr(x1),i) → (u2, τr′ (x2)) is cnα-i-cts if h−1(s) is nα-i-closed in u1 whenever s is n-open set in u2. example 3.1. let u1={i,j,k,l}, u1/r={{i},{j},{k},{l}} and x1={i}.then τr(x1) ={u1,φ,{i}}.let i={φ}.here the n α-i-open sets are{u1,φ,{i},{i,j},{i,k},{i,l},{i, j,k},{i,j,l},{i,k,l}}. let u2={m,n,o,p} with u2/r′={{m},{n},{o,p}} and x2={n, o}.then τr′ (x2)={ u2,φ,{n},{o,p},{n,o,p}}.we define h:(u1,τr(x1),i)→( u2, τr′ (x2)) as f(i)=m, f(j)=n, f(k)=o and f(l)=p.then h−1(s) is nα-i-closed in u1 whenever s is n-open in u2.therefore h is cnα-i-cts. proposition 3.1. 1. any cnα-i-cts function is cnα-cts. 2. any cn-cts function is cnα-i-cts. proof. (i) let h : (u1, τr(x1),i) → (u2, τr′ (x2)) be a cnα-i-cts function.let s be a n-open in u2.since h is cnα-i-cts, h−1(s) is nα-i-closed in u1.we know that each nα-i-closed set is nα-closed.hence h−1(s) is nα-closed in u1.hence h is cnα-cts. (ii) let h : (u1, τr(x1),i) → (u2, τr′ (x2)) be a cn-cts function.let s be a nopen set in u2.since h is cn-cts, h−1(s) is n-closed in u1.it is obvious that every n-closed set is nα-i-closed.thus h−1(s) is nα-i-closed in u1.which implies h is cnα-i-cts function.2 example 3.2. cnα-cts ; cnα-i-cts let u1={i,j,k,l} with u1/r={{i},{j,k},{l}} and x1={l}.then τr(x1)={u1,φ,{l}}. let i={φ,{l}}.here the nα-open sets are {u1,φ,{l},{i,l},{j,l},{k,l},{i,j,l},{i,k,l}, 285 s. vijaya, p. santhi, a. yuvarani {j,k,l}} and nα-i-open sets are {u1,φ,{l}}.let u2={m,n,o,p} with u2/r′={{m}, {n,o},{p}} and x2={m,n}.then τr′ (x2) = {u2,φ,{m},{n,o},{m,n,o}}. we define h : (u1, τr(x1),i) → (u2, τr′ (x2)) as h(i)=m, h(j)=n, h(k)=o and h(l)=p.then h−1(s) is nα-closed in u1 but not nα-i-closed whenever s is n-open set in u2. hence h is cnα-cts but not cnα-i-cts function. example 3.3. cnα-i-cts ; cn-cts let u1={i,j,k,l} with u1/r= {{i},{j},{k},{l}} and x1={i}.then τr(x1)={u1,φ,{ i}}.let i = {φ}.here the nα-i-open sets are {u1,φ,{i},{i,j},{i,k},{i,l},{i,j,k},{i,j, l},{i,k,l}}.let u2={m,n,o,p} with u2/r′={{m},{o,p},{n}} and x2={n,o}.then τr′ (x2) = {u2,φ,{n},{o,p},{n,o,p}}.we define h : (u1, τr(x1),i) → (u2, τr′ (x2)) as h(i) = m, h(j) = n, h(k) = o and h(l) = p.then h−1(s) is nα-i-closed in u1 but not n-closed whenever s is n-open set in u2. hence h is cnα-i-cts but not cn-cts function. theorem 3.1. let h : (u1, τr(x1),i) → (u2, τr′ (x2)), then the following statements are equivalent: 1. h is cnα-i-cts, 2. for each n-closed subset t of u2, h−1(t) ∈ nαio(u1), 3. for each x ∈ u1 and each n-closed set t of u2 containing h(x), ∃ u ∈ nαio(u1) such that h(u) ⊂ t, 4. h(nαi-cl(v)) ⊂ nker(h(v)) for each v ⊆ u1, 5. nαi-cl(h−1(w)) ⊂ h−1(nker(w)) for each w ⊆ u2. proof. (i) ⇒ (ii) and (ii) ⇒ (iii) are obvious. (iii) ⇒ (ii) let t be any n-closed set of u2 and x ∈ h−1(t).then h(x) ∈ t and ∃ ux ∈ nαio(u1) such that h(ux) ⊂ t.therefore, we obtain h−1(t) = ∪{ ux : x ∈ h−1(t)} and hence h−1(t) ∈ nαio(u1). (ii) ⇒ (iv)let v ⊆ u1.if y 6∈ nker(h(v)), then by thm 2.1, ∃ a n-closed set t of u2 containing y such that h(v) ∩ t=φ.therefore v ∩ h−1(t) = φ and nαi-cl(v) ∩ h−1(t)=φ.hence h(nαi-cl(v)) ∩ t=φ and y 6∈ h(nαi-cl(v)).thus h(nαi-cl(v)) ⊂ nker(h(v)). (iv) ⇒ (v) let w ⊆ u2.by the hypothesis and thm 2.1, h(nαi-cl(h−1(w))) ⊂ nker(h(h−1(w))) ⊂ nker(w) and nαi-cl(h−1(w)) ⊂ h−1(nker(w)). (v) ⇒ (i) let w be a n-open set of u2.by thm 2.1, nαi-cl(h−1(w)) ⊂ h−1(nker(w)) = h−1(w) and nαi-cl(h−1(w)) = h−1(w).therefore h−1(w) is nα-i-closed in (u1, τr(x),i).2 286 contra nα-i-continuity over nano ideals theorem 3.2. if a function h : (u1, τr(x1),i) → (u2, τr′ (x2)) is cnα-i-cts and v is n-regular, then h is nα-i-cts. proof. let x ∈ u1 and y a n-open set of u2 containing h(x).since u2 is n-regular, ∃ a n-open set z in u2 containing h(x) such that ncl(z) ⊂ y.since h is cnα-i-cts, by the above theorem, ∃ x ∈ nαio(u1) such that h(x) ⊂ ncl(z).therefore h(x) ⊂ ncl(z) ⊂ y.hence h is nα-i-cts.2 definition 3.3. a function h : (u1, τr(x1),i) → (u2, τr′ (x2)) satisfy the nα-iinteriority rule if nαi-int(h−1(ncl(w))) ⊂ h−1(w) whenever w is n-open set of (u2, τr′ (x2)). theorem 3.3. if a function h : (u1, τr(x1),i) and (u2, τr′ (x2)) is cnα-i-cts and satisfies nα-i-interiority rule, then h is nα-i-cts. proof. let y be any n-open set of u2. since h is cnα-i-cts and ncl(y) is n-closed, by thm 3.1, h−1(ncl(y)) is nα-i-open in (u1, τr(x),i).by hypothesis of h, h−1(y) ⊂ h−1(ncl(y)) ⊂ nαi-int(h−1(ncl(y))) ⊂ nαi-int(h−1(y)) ⊂ h−1(y).thus, we obtain h−1(y)=nαi-int(h−1(y)) and consequently h−1(y) ∈ nαio(u).therefore h is nα-i-cts.2 theorem 3.4. let (u1, τr(x1),i) be any nits and h : (u1, τr(x1),i) → (u2, τr′ (x2)) be a function and g : u1 → u1 × u2 be the graph function, given by g(x) = (x, h(x)) for every x ∈ u1.then f is cnα-i-cts if and only if g is nα-i-cts. proof. let x ∈ u1 and let t be a n-closed subset of u1 × u2 containing g(x).then t ∩ ({x}× u2) is n-closed in {x}× u2 containing g(x).also {x}× u2 is homeomorphic to u2. hence {y ∈ u2 : (x, y) ∈ t} is a n-closed subset of u2.since h is cnα-i-cts, ∪ { h−1(y) ∈ u2 : (x, y) ∈ t } is a nα-i-open subset of (u1, τr(x1),i).further, x ∈∪ { h−1(y) ∈ u2 : (x, y) ∈ t }⊂ g−1(t).hence g−1(t) is nα-i-open.then g is cnα-i-cts. conversely, let f be a n-closed subset of u2.then u1 × f is a n-closed subset of u1 × u2.since g is cnα-i-cts, g−1(u1 × f) is a nα-i-open subset of u1. also, g−1(u1 × f)=h−1(f).hence h is cnα-i-cts.2 definition 3.4. a nits (u1, τr(x1),i) is called nα-i-t2 if for any distinct two points x, y ∈ u1, ∃ x, y ∈ nαio(u1) containing x and y, resp., such that x ∩ y=φ. definition 3.5. 1. a nts (u1, τr(x1)) is termed as a n-urysohn space if for any two distinct points x, y ∈ u1, ∃ disjoint n-open subsets x ∈ a, y ∈ b such that the n-closures a and b are disjoint n-closed subsets of u1. 2. a nts (u1, τr(x1)) is called n-ultra hausdorff if any two distinct points of u1 can be separated by disjoint n-clopen sets. theorem 3.5. if (u1, τr(x1),i) is an nits and for any two distinct points x1 , x2 ∈ u1, ∃ a function h into a n-urysohn space (u2, τr′ (x2)) such that h(x1) 6= h(x2) and h is cnα-i-cts at x1 , x2, then (u1, τr(x1),i) is nα-i-t2. 287 s. vijaya, p. santhi, a. yuvarani proof. let x1 , x2 be any two distinct points of u1.then by hypothesis there is a nurysohn space (u2, τr′ (x2)) and a function h : (u1, τr(x1),i) and (u2, τr′ (x2)), which satisfies the required condition.let yi= h(xi) for i=1,2.then y1 6= y2. since (u2, τr′ (x2)) is n-urysohn, ∃ n-open neighbourhoods xy1 and xy2 of y1, y2 respectively in u2 such that ncl(xy1 ) ∩ ncl(xy2 )=φ.since h is cnα-i-cts at xi, ∃ nα-i-open neighbourhoods wxi of xi in u1 such that h(wxi ) ⊂ ncl(xyi ) for i=1,2. hence we get wx1 ∩ wx2 =φ because ncl(xy1 ) ∩ ncl(xy2 )=φ.therefore (u1, τr(x1),i) is nα-i-t2.2 corolary 3.1. if h is a cnα-i-cts injective function of a nits (u1, τr(x1),i) into a n-urysohn space (u2, τr′ (x2)), then (u1, τr(x1),i) is a nα-i-t2 space. proof. for any to two distinct points x1 , x2 in u1, h is cnα-i-cts function of u1 into a n-urysohn space (u2, τr′ (x2)) such that h(x1) 6= h(x2) because h is injective.by thm 3.5, the space (u1, τr(x1),i) is nα-i-t2.2 theorem 3.6. if h is a cnα-i-cts injective function of a nts (u1, τr(x1),i) into n-ultra hausdorff space (u2, τr′ (x2)), then (u1, τr(x1),i) is a nα-i-t2 space. proof. let the pair of distinct points of u1 be x1 , x2.since f is injective, u2 is n-ultra hausdorff h(x1) 6= h(x2) ∃ n-clopen sets z1, z2 such that h(x1) ∈ z1, h(x2) ∈ z2 and z1 ∩ z2=φ.then xi ∈ h−1(zi) ∈ nαio(u1) for i=1,2 and h−1(z1) ∩ h−1(z2)=φ.therefore (u1, τr(x),i) is a nα-i-t2 space.2 definition 3.6. let h : (u1, τr(x1),i) → (u2, τr′ (x2)).the graph g(h) of the function h is called be cnα-i-closed in u1 × u2 if for any (x1, x2) ∈ (u1 × u2)\g(h), ∃ a ∈ nαio(u1) and a n-closed set t of u2 containing x2 such that (u1 × u2) ∩ g(h)=φ. lemma 3.1. let h : (u1, τr(x1),i) → (u2, τr′ (x2)).the graph g(h) of the function h is cnα-i-closed in u1 × u2 if and only if for each (x1, x2) ∈ (u1 × u2)\g(h), ∃ a ∈ nαio(u1, x1) such that h(a) ∩ ncl(t)=φ where t is a n-closed subset of u1 × u2 containing g(x1). theorem 3.7. if h : (u1, τr(x1),i) → (u2, τr′ (x2)) is a cnα-i-cts function and u2 is a n-urysohn space, then g(h) is cnα-i-closed in u1 × u2. proof. let (x1, x2) ∈ (u1 × u2)\g(h).then x2 6= h(x1) and ∃ n-open set a, b of u2 such that h(x1) ∈ a, x2 ∈ b and ncl(a) ∩ ncl(b)=φ.since h is cnα-i-continuous, ∃ u ∈ nαio(u1,x1) such that h(u) ⊂ ncl(a).therefore h(u) ∩ ncl(b) =φ.hence g(h) is cnα-i-closed.2 theorem 3.8. if h : (u1, τr(x1),i) → (u2, τr′ (x2)) is a cnα-i-cts function and (u2, τr′ (x2)) is t2, then g(h) is cnα-i-closed. proof. let (x1, x2) ∈ (u1 × u2)\g(h).then x2 6= h(x1) and ∃ n-open set b of u2 such that h(x1) ∈ b, x2 6= b. since h is cnα-i-cts, ∃ u ∈ nαio(u1,x1) such that 288 contra nα-i-continuity over nano ideals h(u) ⊂ ncl(b).therefore h(u) ∩ (u2 b)=φ and u2-b is a n-closed set of u2 containing x2.hence g(h) is cnα-i-closed.2 definition 3.7. a nits (u, τr(x),i) is called nα-i-connected if there exists nα-iopen sets a and b which form a separation of x. proposition 3.2. a cnα-i-cts image of a nα-i-connected space is connected. definition 3.8. a nits (u, τr(x),i) is called nα-i-normal if given any non-empty disjoint n-closed sets t and f such that ∃ nα-i-open sets a of t and b of f such that a ∩ b=φ. definition 3.9. a nts (u, τr(x)) is called n-ultra normal if given any non-empty disjoint n-closed sets t and f such that ∃ n-clopen sets a of t and b of f such that a ∩ b=φ. theorem 3.9. if h : (u1, τr(x1),i) → (u2, τr′ (x2)) is a cnα-i-cts closed injective function and (u2, τr′ (x2)) is n-ultra-normal space, then (u1, τr(x1),i) is a nαi-normal space. proof. let the two disjoint n-closed subsets of u1 be f1 and f2. since h is nclosed and injective, h(f1) ∩ h(f2)=φ where h(f1) and h(f2) are n-closed subsets of u2.since u2 is n-ultra normal, ∃ n-clopen sets y1 of h(f1) and y2 of h(f2) in u2 such that y1 ∩ y2=φ.hence fi ⊂ f−1(yi), f−1(yi) ∈ nαio(u) for i=1,2 and f−1(y1) ∩ f−1(y2)=φ.therefore (u1, τr(x),i) is a nα-i-normal.2 theorem 3.10. for the functions h : (u1, τr(x1),i) → (u2, τr′ (x2)) and g : (u2, τr′ (x2),i’) → (u3, τr′′ (x3)), we have 1. g ◦ h is nα-i-cts, if h is cnα-i-cts and g is cn-cts. 2. g ◦ h is cnα-i-cts, if h is cnα-i-cts and g is n-cts. remark 3.1. in general, g ◦ h is not cnα-i-cts functions if g and f are cnα-i-cts functions.the below example illustrate this result. example 3.4. let u1={i,j,k,l} with u1/r={{i,k},{j},{l}}, and x1={i,l}.then τr(x1)={u1,φ,{l},{i,k},{i,k,l}}.let i1={φ,j}.let u2={m,n,o,p} with u2/r′={{m, n},{o,p}} and y={o,p}.then τr′ (x2)={u2,φ,{p},{m,o},{m,o,p}}.let i2={φ,m}. let w={t,u,v,w} with w/r′′={{t},{u,v},{w}} and z={w}.then τr′′ (z)={w,φ,{w} }.define h : (u1, τr(x),i1) → (u2, τr′ (x2)) by h(i)=n, h(j)=p, h(k)=m, h(l)=o and g : (u2, τr′ (y),i2) → (u3, τr′′ (z)) by g(m)=w, g(n)=t, g(o)=u, g(p)=v.then h and g are cnα-i-cts functions but (g ◦ h)−1(w)=k which does not belongs to nα-iclosed in (u1, τr(x1),i). 289 s. vijaya, p. santhi, a. yuvarani 4 conclusion through the above discussions we have summarized the conceptulation of contra nα-i-continuity and its characteristics based on nano urysohn space and nano ultra hausdorff space.also, we compared contra nα-i-continuity with some existing functions using suitable examples.further, this concept may be extended to frechet urysohn space and completely hausdorff space in nano ideal topology. references t. noiri a. acikgoz and s. yuksel. on α-i-continuous and α-i-open functions. acta mathematica hungarica, 105:27–37, 2004. m. bhuvaneswari and n. nagaveni. on nwg-normal and nwg-regular spaces. international journal of mathematics trends and technology, 2018. p. karthiksankari and p. subbulakshmi. on nano totally semi continuous functions in nano topological space. international journal of mathematical archive, 10: 14–18, 2019. k. kuratowski. topology. academic press, new york, i, 1966. saeid jafari m. lellis thivagar and v. suthadevi. on new class of contra continuity in nano topology. italian journal of pure and applied mathematics, 43: 1–11, 2017. m. parimala and s. jafari. on some new notions in nano ideal topological space. eurasian bulletin of mathematics, 1:51–64, 2018. i. rajasekaran and o. nethaji. simple forms of nano open sets in an ideal nano topological spaces. journal of new theory, 24:35–43, 2018. m. lellis thivagar and carmel richard. on nano forms of weakly open sets. international journal of mathematics and statistics invention, i:31–37, 2013a. m. lellis thivagar and carmel richard. on nano continuity. mathematical theory and modelling, 3:32–37, 2013b. m. lellis thivagar and v. suthadevi. new sort of operators in nano ideal topology. journal of ultra scientist of physical sciences, 28:51–64, 2016. s. narmatha v. inthumathi and s. krishnaprakash. decompositions of nano-ri-continuity. international journal of advanced science and technology, 29: 2578–2582, 2020. 290 ratio mathematica volume 44, 2022 the edge-to-vertex triangle free detor distance in graphs s. lourdu elqueen1 g. priscilla pacifica2 abstract for every connected graph g, the triangle free detour distance d∆f (u, v) is the length of a longest uv triangle free path in g, where u, v are the vertices of g. a u-v triangle free path of length d∆f (u, v) is called the u-v triangle free detour. in this article, the edge-to-vertex triangle free detour distance is introduced. it is found that the edge -tovertex triangle free detour distance differs from the edge -to-vertex distance and edgeto-vertex detour distance. the edge-to-vertex triangle free detour distance is found for some standard graphs. their bounds are determined and their sharpness is checked. certain general properties satisfied by them are studied. keywords: connected graph, edge -to-vertex distance and edge-to-vertex detour distance 2010 ams subject classification: 05c12, 05c693 1reg no: 19212212092009, ph. d research scholar (full time) of mathematics. mary’s college (autonomous) thoothukudi affiliated under manonmaniam sundaranar university, abishekapatti, tirunelveli, tamil nadu, south india. sahayamelqueen@gmail.com. 2department of mathematics, st. mary’s college (autonomous), thoothukudi, india, priscillamelwyn@gmail.com. 3 received on june28th, 2022. accepted on sep 1st, 2022.published on nov 30th, 2022.doi: 10.23755/rm.v44i0.894. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 86 s. lourdu elqueen & g. priscilla pacifica 1. introduction the facility location problem was introduced as edge-to-vertex distance by santhakumaran [9], in 2010. for an edge e and a vertex v in a connected graph, the edge-to-vertex distance is defined by 𝑑(𝑒, 𝑣) = 𝑚𝑖𝑛{𝑑(𝑢, 𝑣) ∶ 𝑢 ∈ 𝑒}. the edge-tovertex eccentricity of e is defined by 𝑒2(𝑒) = 𝑚𝑎𝑥{𝑑(𝑒, 𝑣) ∶ 𝑣 ∈ 𝑉}. a vertex v of g such that 𝑒2 (𝑒) = 𝑑(𝑒, 𝑣) is called an edge-to-vertex eccentric vertex of v. the edgeto-vertex radius 𝑟2 of g is defined by 𝑟2 = 𝑚𝑖𝑛{𝑒2(𝑒) ∶ 𝑒 ∈ 𝐸} and the edge-to-vertex diameter 𝑑2 of g is defined by𝑑2 = 𝑚𝑎𝑥{𝑒2 (𝑒) ∶ 𝑒 ∈ 𝐸}. an edge e for which 𝑒2(𝑒) is minimum is called an edge-to-vertex central edge of 𝐺and the set of all edge-tovertex central edges of 𝐺 is the edge-to-vertex center 𝐶2(𝐺) of𝐺. an edge 𝑒 for which 𝑒2(𝑒) is maximum is called an edge-to-vertexperipheral edge of 𝐺 and the set of all edge-tovertex peripheraledges of 𝐺 is the edge-to-vertex periphery 𝑃2 (𝐺) of 𝐺. if every edgeof 𝐺is an edge-to-vertex central edge then 𝐺 is called the edge-to-vertex self-centered graph. this concept is useful in channel assignment problem in radio technology and security-based communication network design. the concept of edge-to-vertex detour distance was introduced by i. keerthi asir [6], let 𝑒 be an edge and 𝑣 a vertex in a connected graph 𝐺. an edge-to-vertex 𝑒 − 𝑣 path 𝑃 is a 𝑢 − 𝑣 path, where𝑢 is a vertex in 𝑒 such that 𝑃 contains no vertices of e other than 𝑢. the edge-to-vertex detour distance 𝐷(𝑒, 𝑣) is the length of a longest 𝑒 − 𝑣path in 𝐺. an𝑒 − 𝑣 path of length 𝐷(𝑒, 𝑣) is called an edge-to-vertex 𝑒 − 𝑣 detour or simply 𝑒 − 𝑣 detour. forour convenience an𝑒 − 𝑣 path of length 𝑑(𝑒, 𝑣) is called an edge-to-vertex 𝑒 − 𝑣 geodesic or simply 𝑒– 𝑣 geodesic. the following theorems are used in the article. theorem: 1.1.[6] for any edge 𝑒 and a vertex 𝑣in a non-trivial connected graph of order𝑛, 0 ≤ 𝑑(𝑒, 𝑣) ≤ 𝐷(𝑒, 𝑣) ≤ 𝑛 − 2 . theorem: 1.2.[6] let𝐾𝑛,𝑚 (𝑛 < 𝑚) be a complete bipartite graph with partition𝑉1, 𝑉2 of 𝑉(𝐾𝑛,𝑚) such that |𝑉1| = 𝑛 and |𝑉2| = 𝑚. let 𝑒 be an edge and 𝑣 a vertex such that 𝑣 ∉ 𝑒 in 𝐾𝑛,𝑚, then 𝐷(𝑒, 𝑣) = { 2𝑛 − 2 𝑖𝑓𝑣 ∈ 𝑉1 2𝑛 − 1 𝑖𝑓𝑣 ∈ 𝑉2 2. edge-to-vertex triangle free detour distance definition. 2.1 let 𝐺be a connected graph. let 𝑒be an edge and 𝑢 a vertex in𝐺. an edge-to-vertex 𝑒 − 𝑢triangle free path 𝑃is a 𝑢 − 𝑣 triangle free path, where𝑣is a vertex in 𝑒such that 𝑃contains no vertices of 𝑒 other than 𝑣. the edge-to-vertex triangle free detour distance is the length of the longest 𝑒 − 𝑢 triangle free path in 𝐺.it is denoted by 𝐷∆f (𝑒, 𝑣). an𝑒 − 𝑢 triangle free path of length𝐷∆f (𝑒, 𝑣)is called an edge-to-vertex 𝑒 − 𝑢triangle free detour. 87 the edge-to-vertex triangle free detor distance in graphs example: 2.1 consider the graph 𝐺given in the figure: 2.1. let 𝑒 = {𝑢6, 𝑢7}and 𝑣 = 𝑢4. the paths between 𝑒and 𝑣 are 𝑃1: 𝑢6, 𝑢5, 𝑢4;𝑃2: 𝑢7, 𝑢2, 𝑢4;𝑃3: 𝑢7, 𝑢2, 𝑢3, 𝑢4 ;𝑃4: 𝑢7, 𝑢8, 𝑢9, 𝑢1, 𝑢2, 𝑢4; and 𝑃5: 𝑢7, 𝑢8, 𝑢9, 𝑢1, 𝑢2, 𝑢3, 𝑢4 ;the paths 𝑃1, 𝑃2, 𝑃4are triangle free 𝑒 − 𝑣 paths and 𝑃3 and 𝑃5 are not triangle free 𝑒 − 𝑣 paths. thus edge-to-vertex distance 𝑑(𝑒, 𝑣) = 2, edge-to-vertex triangle free detour distance 𝐷∆f (𝑒, 𝑣) = 5 and edge-to-vertex detour distance d(e, v) = 6. figure: 2.1 g thus edge-to-vertex triangle free detour distance differs from the edge-to-vertex distance and edge-to-vertex detour distance. theorem. 2.1 let 𝐺 be a connected graph of order 𝑛. let 𝑒 be an edge and 𝑢a vertex of𝐺, then 0 ≤ 𝑑(𝑒, 𝑣) ≤ 𝐷∆𝑓 (𝑒, 𝑣) ≤ 𝐷(𝑒, 𝑣) ≤ 𝑛 − 2 . proof. by theorem 1.1 , we can conclude that 0 ≤ 𝑑(𝑒, 𝑣) ≤ 𝐷(𝑒, 𝑣) ≤ 𝑛 − 2 . it is enough to prove that (i)𝑑(𝑒, 𝑣) ≤ 𝐷∆𝑓 (𝑒, 𝑣) and (ii) 𝐷∆𝑓 (𝑒, 𝑣) ≤ 𝐷(𝑒, 𝑣). thus (i) is true by the definition of edge-to-vertex distance and edge-to-vertex triangle free detour distance. to prove :(ii) case(i): if the detour path does not induce a triangle in g, then𝐷∆𝑓 (𝑒, 𝑣) = 𝐷(𝑒, 𝑣) . case(ii): if the detour path induces a triangle in g, then 𝐷∆𝑓 (𝑒, 𝑣) < 𝐷(𝑒, 𝑣) remark 2.1. the bounds in the theorem 2.1 are sharp. let 𝐺be a graph and 𝑒 be an edge, 𝑑(𝑒, 𝑢) = 𝐷∆𝑓 (𝑒, 𝑢) = 𝐷(𝑒, 𝑢) = 0iff𝑢 ∈ 𝑒.let 𝐺be a path with vertices {𝑣1 , 𝑣2, … . 𝑣𝑛 }. then𝑑(𝑒, 𝑢) = 𝐷∆𝑓 (𝑒, 𝑢) = 𝐷(𝑒, 𝑢) = 𝑛 − 2, where 𝑒 = 88 s. lourdu elqueen & g. priscilla pacifica {𝑣𝑛−1, 𝑣𝑛 }and 𝑢 = 𝑣1. let 𝐺be a tree, 𝑑(𝑒, 𝑢) = 𝐷∆𝑓 (𝑒, 𝑢) = 𝐷(𝑒, 𝑢) for every edge 𝑒and vertex 𝑢of 𝐺. for the graph 𝐺 given in the figure:2.1, 𝑒 = {𝑢6, 𝑢7} and 𝑣 = 𝑢4. the paths between 𝑒and 𝑣 are 𝑃1: 𝑢6, 𝑢5, 𝑢4; 𝑃2: 𝑢7, 𝑢2, 𝑢4; 𝑃3: 𝑢7, 𝑢2, 𝑢3, 𝑢4 ; 𝑃4: 𝑢7, 𝑢8, 𝑢9, 𝑢1, 𝑢2, 𝑢4; and 𝑃5: 𝑢7, 𝑢8, 𝑢9, 𝑢1, 𝑢2, 𝑢3, 𝑢4 ;the paths 𝑃1, 𝑃2, 𝑃4are triangle free 𝑒 − 𝑣 paths and 𝑃3 and 𝑃5 are not triangle free 𝑒 − 𝑣 paths. thus edge-tovertex distance 𝑑(𝑒, 𝑣) = 2, edge-to-vertex triangle free detour distance 𝐷∆f (𝑒, 𝑣) = 5 and edge-to-vertex detour distanced(e, v) = 6. thus 0 < 𝑑(𝑒, 𝑣) < 𝐷∆𝑓 (𝑒, 𝑣) < 𝐷(𝑒, 𝑣) < 𝑛 − 2 . theorem. 2.2 for a complete bipartite graph 𝐺with partitions 𝑉1and 𝑉2such that |𝑉1| = 𝑛and |𝑉2| = 𝑚(𝑛 < 𝑚).let 𝑒 be an edge of 𝐺 and 𝑢a vertex such that 𝑢 ∉ 𝑒 in g. then, 𝐷∆𝑓 (𝑒, 𝑢) = { 2𝑛 − 2 𝑖𝑓𝑢 ∈ 𝑉1 2𝑛 − 1 𝑖𝑓𝑢 ∈ 𝑉2 proof. since any vertex subset of 𝐺 do not induce a cycle 𝐶3in 𝐺. thus edge-to-vertex triangle free detour distance is equal to edge-to-vertex detour distance. by theorem: 1.2, 𝐷∆𝑓 (𝑒, 𝑢) = { 2𝑛 − 2 𝑖𝑓𝑢 ∈ 𝑉1 2𝑛 − 1 𝑖𝑓𝑢 ∈ 𝑉2 corollary:2.1 let 𝐺 be a complete bipartite graph 𝐾𝑛,𝑛 with partitions 𝑉1and 𝑉2 .let 𝑒 be an edge and 𝑢 be a vertex such that 𝑢 ∉ 𝑒in𝐺. then𝐷∆𝑓 (𝑒, 𝑢) = 2𝑛 − 2. theorem: 2.3 let 𝐺 be a tree, then for every edge 𝑒 and a vertex 𝑣in 𝐺, 𝑑(𝑒, 𝑣) = 𝐷∆𝑓 (𝑒, 𝑣) = 𝐷(𝑒, 𝑣). remark: 2.2 the converse of the theorem:2.3 need not be true. consider the graph,𝐺 = 𝐶4, where 𝑑(𝑒, 𝑣) = 𝐷∆𝑓 (𝑒, 𝑣) = 𝐷(𝑒, 𝑣) = 1 if 𝑣 ∉ 𝑒 and 𝑑(𝑒, 𝑣) = 𝐷∆𝑓 (𝑒, 𝑣) = 𝐷(𝑒, 𝑣) = 0 if 𝑣 ∈ 𝑒. definition: 2.2 the edge-to-vertex triangle free detour eccentricity 𝑒∆𝑓2 (𝑒) of an edge 𝑒 in a connected graph 𝐺 is defined as𝑒∆𝑓2(𝑒) = 𝑚𝑎𝑥{𝐷∆𝑓 (𝑒, 𝑣) ∶ 𝑣 ∈ 𝑉}. a vertex 𝑣 for which 𝑒∆𝑓2(𝑒) = 𝐷∆𝑓 (𝑒, 𝑣) is called an edge-to-vertex triangle free detour eccentric vertex of 𝑒. the edge-to-vertex triangle free detour radius of g is defined as 𝑅∆𝑓2 = 𝑟𝑎𝑑∆𝑓2(𝐺) = 𝑚𝑖𝑛{𝑒∆𝑓2(𝑒): 𝑒 ∈ 𝐸}. the edge-to-vertex triangle free detour diameter of g is defined as 𝐷∆𝑓2 = 𝑑𝑖𝑎𝑚∆𝑓2(𝐺) = 𝑚𝑎𝑥{𝑒∆𝑓2(𝑒): 𝑒 ∈ 𝐸}. definition: 2.3 an edge 𝑒 is called an edge-to-vertex triangle free detour central edge if 𝑒∆𝑓2(𝑒) = 𝑅∆𝑓2. the edge-to-vertex triangle free detour center of 𝐺is defined as 𝐶∆𝑓2(𝐺) = 𝐶𝑒𝑛∆𝑓2(𝐺) = {𝑒 ∈ 𝐸: 𝑒∆𝑓2(𝑒) = 𝑅∆𝑓2}. definition: 2.4 an edge 𝑒 is called an edge-to-vertex triangle free detour peripheral edge if 𝑒∆𝑓2(𝑒) = 𝐷∆𝑓2. the edge-to-vertex triangle free detour periphery of 𝐺 is defined as 𝑃∆𝑓2(𝐺) = 𝑃𝑒𝑟∆𝑓2(𝐺) = {𝑒 ∈ 𝐸: 𝑒∆𝑓2(𝑒) = 𝐷∆𝑓2}. 89 the edge-to-vertex triangle free detor distance in graphs definition. 2.5 if every edge of a graph 𝐺 is a edge-to-vertex triangle free detour central edge, then 𝐺 is called edge-to-vertex triangle free detour self centered graph. definition. 2.6 if 𝐺 is the edge-to-vertex triangle free detour self centered graph, then𝐺 is called edge-to-vertex triangle free detour periphery. example. 2.2 for the graph 𝐺given in the figure: 2.2, 𝑒1 = {𝑢1, 𝑢2}, 𝑒2 = {𝑢2, 𝑢3}, 𝑒3 = {𝑢3, 𝑢4}, 𝑒4 = {𝑢4, 𝑢5}, 𝑒5 = {𝑢5, 𝑢6}, 𝑒6 = {𝑢6, 𝑢7}, 𝑒7 = {𝑢7, 𝑢8}, 𝑒8 = {𝑢1, 𝑢8}, 𝑒9 = {𝑢8, 𝑢2}, 𝑒10 = {𝑢7, 𝑢5}, 𝑒11 = {𝑢5, 𝑢2}, 𝑒12 = {𝑢3, 𝑢5} are the edges of 𝐺. figure:2.2 𝐺 the edge-tovertex triangle free detour distances of the graph 𝐺, are provided in the following table. 𝒖𝟏 𝒖𝟐 𝒖𝟑 𝒖𝟒 𝒖𝟓 𝒖𝟔 𝒖𝟕 𝒖𝟖 𝒆∆𝒇𝟐 𝒆𝟏 0 0 4 4 3 3 2 3 4 𝒆𝟐 1 0 0 4 3 3 2 3 4 𝒆𝟑 4 4 0 0 1 4 3 3 4 𝒆𝟒 3 3 1 0 0 5 4 3 5 𝒆𝟓 3 3 4 5 0 0 3 2 5 𝒆𝟔 3 2 3 4 3 0 0 3 4 𝒆𝟕 3 2 2 3 2 3 0 0 3 𝒆𝟖 0 3 3 3 2 3 3 0 3 𝒆𝟗 1 0 3 3 2 2 2 0 3 𝒆𝟏𝟎 2 2 3 4 0 1 0 2 4 𝒆𝟏𝟏 3 0 2 2 0 3 2 2 3 𝒆𝟏𝟐 3 3 0 1 0 4 3 2 4 table:2.1 the following table provides the edge-tovertex distances, edge-to-vertex triangle free detour distances and edge-tovertex detour distances of the graph 𝐺in figure:2.2 90 s. lourdu elqueen & g. priscilla pacifica 𝒆𝟏 𝒆𝟐 𝒆𝟑 𝒆𝟒 𝒆𝟓 𝒆𝟔 𝒆𝟕 𝒆𝟖 𝒆𝟗 𝒆𝟏𝟎 𝒆𝟏𝟏 𝒆𝟏𝟐 𝒆𝟐 2 2 2 2 2 2 2 3 2 2 1 2 𝒆∆𝒇𝟐 4 4 4 5 5 4 3 3 3 4 3 4 𝒆𝑫𝟐 6 6 6 6 6 6 6 6 5 5 4 5 table: 2.2 the edge-to-vertex radius𝑟2 = 1, the edge-to-vertex triangle free detour radius 𝑅∆𝑓2 = 3 , the edge-tovertex detour radius 𝑅2 = 4. thus, the edge-to-vertex triangle free detour radius is different from the edge-tovertex radius and the edge-tovertex detour radius. the edge-to-vertex diameter 𝑑2 = 3, the edge-tovertex triangle free detour diameter 𝐷∆𝑓2 = 6 , the edge-tovertex detour diameter 𝐷2 = 6. thus, the edgetovertex triangle free detour diameter is different from the edge-tovertex diameter and the edge-tovertex detour diameter. the edge-to-vertex center 𝐶2(𝐺) = {𝑒11}, the edge-to-vertex triangle free detour center 𝐶∆𝑓2(𝐺) = {𝑒7, 𝑒8, 𝑒9, 𝑒11}, the edge-to-vertex detour center 𝐶𝐷2(𝐺) = {𝑒9, 𝑒10, 𝑒11}thus the edge-tovertex triangle free detour center is different from the edge-tovertex center and the edge-tovertex detour center. the edge-to-vertex periphery 𝑃2(𝐺) = {𝑒8}, the edge-to-vertex triangle free detour periphery 𝑃∆𝑓2(𝐺) = {𝑒4, 𝑒5}, the edge-to-vertex detour periphery 𝑃𝐷2(𝐺) = {𝑒1, 𝑒2 , 𝑒3 , 𝑒4, 𝑒5, 𝑒6, 𝑒7, 𝑒8}. thus, the edge-tovertex triangle free detour periphery is different from the edge-to vertex periphery and the edge-tovertex detour periphery. the edge-to-vertex triangle free detour radius 𝑅∆𝑓2 and the edge-to-vertex triangle free detour diameter 𝐷∆𝑓2 of some standard graphs are provided in the table:2.3 𝑮 𝑲𝒏 𝑷𝒏 𝑪𝒏 (𝒏 ≥ 𝟒) 𝑾𝒏(𝒏 ≥ 𝟓) 𝑲𝒏,𝒎(𝒏 ≥ 𝒎) 𝑹∆𝒇𝟐 1 ⌊ 𝑛 − 2 𝑛 ⌋ 𝑛 − 2 𝑛 − 2 { 2(𝑛 − 1), 𝑖𝑓𝑛 = 𝑚 2𝑛 − 1, 𝑖𝑓𝑛 > 𝑚 𝑫∆𝒇𝟐 1 𝑛 − 2 𝑛 − 2 𝑛 − 2 { 2(𝑛 − 1), 𝑖𝑓𝑛 = 𝑚 2𝑛 − 1, 𝑖𝑓𝑛 > 𝑚 example: 2.3 the complete graph 𝐾𝑛, the cycle graph 𝐶𝑛 (𝑛 ≥ 4) and the wheel graph 𝑊𝑛(𝑛 ≥ 5) are the edge-to-vertex triangle free detour self centered graph. theorem:2.4 for a connected graph 𝐺 of order 𝑛. then (i)0 ≤ 𝑒2(𝑒) ≤ 𝑒∆𝑓2(𝑒) ≤ 𝑒𝐷2(𝑒) ≤ 𝑛 − 2, for every edge 𝑒of 𝐺. (ii)0 ≤ 𝑟2 ≤ 𝑅∆𝑓2 ≤ 𝑅2 ≤ 𝑛 − 2. (iii) 0 ≤ 𝑑2 ≤ 𝐷∆𝑓2 ≤ 𝐷2 ≤ 𝑛 − 2. remark: 2.3 the bounds in the theorem:2.4are sharp. if 𝐺 = 𝑃2, then 𝑒2(𝑒) = 𝑒∆𝑓2(𝑒) = 𝑒𝐷2(𝑒) = 0. if 𝐺 = 𝐶𝑛 (𝑛 ≥ 4), then 𝑒2(𝑒) = 𝑒∆𝑓2(𝑒) = 𝑒𝐷2(𝑒) = 𝑛 − 2. for the graph 𝐺 given in the figure:2.2, 0 < 𝑒2(𝑒) < 𝑒∆𝑓2(𝑒) < 𝑒𝐷2(𝑒) < 𝑛 − 2, for the edges 𝑒 = 𝑒9, 𝑒10, 𝑒11, 𝑒12. 91 the edge-to-vertex triangle free detor distance in graphs references [1] h. bielak and m. m. syslo, peripheral vertices in graphs, studies. math. ungar., 18 (1983), 269-275. [2] g. chartrand and h. escuadro and p. zhang, detour distance in graphs, j. combin. math. combin. comput., 53 (2005), 75-94. [3] g. chartrand and p. zhang, distance in graphs taking the long view, akcej. graphs. combin., 1 (2004), 1–13. [4] g. chartrand and p. zhang, introduction to graph theory, tata mcgraw-hill new delhi, 2006. [5] i. keerthi asir and s. athisayanathan, triangle free detour distance in graphs, j. combin. math. combin. comput.,105(2016). [6] i. keerthi asir and s. athisayanathan, edge-to-vertex detour distance in graphs, scienciaactaxaverianaan international science journal, volume 8 no. 1, 115-133 [7] p.a. ostrand, graphs and specified radius and diameter, discrete math., 4(1973),7175. [8] a. p. santhakumaran and p. titus, monophonic distance in graphs, discrete math. algorithms appl., 3 (2011), 159–169. [9] a. p. santhakumaran, center of a graph with respect to edges, scientia series a: mathematical sciences, 19 (2010), 13-23. [10] sr little femilin jana. d., jaya. r., arokia ranjithkumar, m., krishnakumar. s., “resolving sets and dimension in special graphs”, advances and applications in mathematical sciences 21 (7) (2022), 3709 – 3717. 92 https://scholar.google.com/citations?view_op=view_citation&hl=en&user=xwcp70yaaaaj&sortby=pubdate&authuser=1&citation_for_view=xwcp70yaaaaj:ijcspb-oge4c e:\uziv\sarka\clanky\rm_23\spart\gah2.dvi ratio mathematica 23 (2012), 51–64 issn:1592-7415 directed graphs representing isomorphism classes of c-hypergroupoids antonios kalampakas, stefanos spartalis, kassiani skoulariki department of production engineering and management laboratory of computational mathematics school of engineering, democritus university of thrace v.sofias 12, prokat, building a1, 67100xanthi, greece akalampakas@gmail.com, sspart@pme.duth.gr abstract we investigate the relation of directed graphs and hyperstructures by virtue of the graph hyperoperation. a new class of graphs arises in this way representing isomorphism classes of c-hypergroupoids and we present the 17 such graphs that correspond to the 73 chypergroupoids associated with binary relations on three element sets. as it is shown they constitute an upper semilattice with respect to graph inclusion. key words: hyperoperations, hypergroupoids, directed graphs. mcs2010: 20n20, 68r10, 97k30. 1 introduction the correlation between hyperstructures and binary relations has been intensively investigated in the last 20 years by several researchers ([10], [11], [12], [14], [18], [6], [7], [8]) while of particular importance are the hypergroupoids that derive from binary relations known as c-hypergroupoids which were introduced by corsini in [9] (see also [19], [20], [22], [21]). the purpose of the present paper is to further expand the ongoing research on hypergroupoids by employing concepts from graph theory. while 51 a. kalampakas, s. spartalis, k. skoulariki the foundations of graph theory can be traced back to l. euler and his “königsberg bridge problem” [1] (see also [13, 2]), its growth in recent years has been explosive covering a large number of disciplines ranging from mathematical foundations of computer science [3, 4] to physical chemistry [5] and natural language processing [17]. in section 2, basic concepts and results on directed graphs and hypergroupoids are presented. in section 3 we define the graph hyperoperation which is actually corsini’s hyperoperation applied on graphs. a particular class of directed graphs arises in this way, representing isomorphism classes of c-hypergroupoids, namely corsini’s graphs. as it is evident from their construction, corsini’s graphs constitute a useful apparatus in order to represent and arrange the hypergroupoid classes they represent which thus results in a hierarchy inside the class of c-hypergroupoids. we identify and present the 17 corsini’s graphs with 3 nodes and we find that they constitute an upper semilattice with graph inclusion as the partial order. 2 preliminaries on hyperstructures and graph theory a partial hypergroupoid is a pair (h, ∗), where h is a non-empty set, and ∗ is a hyperoperation i.e. ∗ : h × h → p(h), (x, y) 7→ x ∗ y. if a, b ∈ p(h)-{∅}, then a ∗ b = ⋃ a∈a,b∈b a ∗ b. we denote by a ∗ b (respectively, a ∗ b) the hyperproduct a ∗ b in the case that the set a (respectively, the set b) is the singleton {a} (respectively, {b}). moreover, (h, ∗) is called hypergroupoid if x ∗ y 6= ∅, for all x, y ∈ h and it is called a degenerative (respectively, total) hypergroupoid in the case that for all x, y ∈ h, x ∗ y = ∅ (respectively, x ∗ y = h). given a binary relation r ⊆ h × h the corsini’s hyperoperation (cf. [9]) ∗r : h × h → p(h) is defined in the following way: (x, y) 7→ x ∗r y = {z ∈ h | (x, z) ∈ r and (z, y) ∈ r}. 52 directed graphs representing isomorphism classes of c-hypergroupoids the hyperstructure (h, ∗r) is called corsini’s partial hypergroupoid associated with the binary relation r or simply partial c-hypergroupoid and is denoted hr (cf. [19], [20]). in the case that x ∗r y 6= ∅, for all x, y ∈ h, then (h, ∗r) is called c-hypergroupoid. it can be easily seen that a partial chypergroupoid hr is a c-hypergroupoid if and only if it holds r◦r = h×h, where ◦ is the usual relation composition. let r ⊆ h×h be a binary relation on the set h = {x1, x2, . . . , xn} then the n × n matrix mr = [mi,j ]n×n, with mi,j = 1 if (xi, xj ) ∈ r and mi,j = 0, else is called the boolean matrix of r. formally a concrete directed graph g is a pair (vg, eg) where: vg is a finite set, the elements of which we call vertices and eg ⊆ vg × vg is a set of ordered pairs of vg the elements of which we call edges. a vertex is simply drawn as a node and an edge as an arrow connecting two vertices the head and the tail of the edge. a graph g′ = (vg′, eg′ ) is a subgraph of the graph g = (vg, eg) if it holds v ′ g ⊆ vg and e ′ g ⊆ eg. in the other direction, a supergraph of a graph g is a graph that has g as a subgraph. we say that the graph h is included in the graph g (h ≤ g) if g has a subgraph that is equal or isomorphic to h. the relation ≤, which is called graph inclusion, is a graph invariant. given a graph g = (vg, eg) and v ∈ vg the number of edges that “leave” the vertex v is called the out degree of v and the number of edges that “enter” the vertex is called the in degree of v. moreover we denote by degreeout(g) the set of all the out degrees of g’s vertices and similarly for degreein(g). the order of a graph is the number of its vertices, i.e. |vg|, and the size of a graph is the number of its edges, i.e. |eg|. a loop is an edge whose head and tail is the same vertex. an edge is multiple if there is another edge with the same head and the same tail. a graph is called simple if it has no multiple edges. a vertex is called isolated if there is no edge connected to it. two graphs g and h are said to be isomorphic if there exists an isomorphism f between the vertices of the two graphs that respects the edges, i.e. it holds (x, y) ∈ eg if and only if (f (x), f (y)) ∈ eh . since the specific sets vg, eg chosen to define a concrete directed graph g are actually irrelevant we don’t distinguish between two isomorphic graphs. hence the following definition of an abstract graph. the equivalence class of a concrete directed graph with respect to isomorphism is called an abstract directed graph or simply graph. a graph property is called invariant if it is invariant under graph isomorphisms. examples of graph invariants are 53 a. kalampakas, s. spartalis, k. skoulariki order, size and diameter (the longest of the shortest path lengths between pairs of vertices) vertex (edge) connectivity, the smallest number of vertices (edges) whose removal disconnects the graph vertex (edge) chromatic number, the minimum number of colors needed to color all vertices (edges) so that adjacent vertices (edges) have a different color vertex (edge) covering number, the minimal number of vertices (edges) needed to cover all edges (vertices) in what follows we consider simple graphs without isolated vertices. 3 the graph hyperoperation given a concrete directed graph g = (vg, eg), we introduce the graph hyperoperation ◦g defined on the nodes of the graph g: ◦g : vg × vg → p(vg), (x, y) 7→ x ◦g y = {z ∈ vg | (x, z), (z, y) ∈ eg}. it can be easily seen that it holds ◦g = ∗eg , where ∗eg is the corsini’s hyperoperation derived by the binary relation eg. hence ◦g structures the set vg into a (partial) c-hypergroupoid called the corsini’s (partial) hypergroupoid associated with g. if the hyperoperation ◦g structures vg into a c-hypergroupoid then the same holds for every graph g′ isomorphic with g, hence this property is a graph invariant. an (abstract) graph is called corsini’s graph if the hyperoperation ◦g structures vg into a chypergroupoid. remark 3.1 a hyperoperation for undirected graphs has been presented in [15] by virtue of spanning trees. proposition 3.1 the graph hyperoperation associated with g = (vg, eg) structures vg into a c-hypergroupoid if and only if, there exists a path of length 2 between every pair of nodes of g. proof. the corsini’s hyperoperation associated with g is a hypergroupoid if and only if x ◦g y 6= ∅ for all x, y ∈ vg if and only if for every pair of nodes x, y ∈ vg there exists a node z ∈ vg such that (x, z), (z, y) ∈ eg, if and only if there exists a path of length 2 between x and y. 2 it is easy to prove that there are two corsini’s graphs with two nodes, depicted in the following figure. 54 directed graphs representing isomorphism classes of c-hypergroupoids we note that the second graph is actually a subgraph of the first. remark 3.2 the adjacency matrix ag of the graph g is identical with the boolean matrix meg of the relation eg. from this remark we obtain an alternative proof of proposition 3.1. indeed as it is stated in [9] a hyperoperation derived from the binary relation r ⊆ h × h structures h into a hypergroupoid if and only if the boolean matrix mr of the relation has the property m 2 r = 1, where 1 is the matrix that has the unit at every entry. since in the case of a hyperoperation associated with a graph g this matrix is equal with the adjacency matrix ag of g it follows that a2g = 1. hence there exists a path of length 2 between any given pair of vertices. proposition 3.2 the number of corsini’s graphs with order n is always smaller than the number of different corsini’s hypergroupoids derived from all binary relations r ⊆ h × h with card|h| = n. proof. indeed, although there exist isomorphic graphs with different adjacency matrices, different binary relations r ⊆ h × h always correspond to different boolean matrices. hence there exist different corsini’s hypergroupoids corresponding to two distinct boolean matrices that represent the same graph up to isomorphism. 2 as an example for the above proposition we recall that there are three different corsini’s hypergroupoids deriving from binary relations with 2 elements (cf. [9]) but only two corsini’s graphs as we noted before. proposition 3.3 if g is a corsini’s graph then every graph g′ with g ≤ g′ is also corsini’s. proof. this is obtained by applying proposition 3.1 since if there exists a path of length 2 between two nodes in a graph g then there exists such a path in every supergraph of g. 2 now let h = {1, 2, 3}, r ⊆ h × h and mr the 3 × 3 boolean matrix of r. we say that mr has the form (p1, p2, p3) if |{j ∈ h | (k, j) ∈ r}| = pk for all k ∈ h 55 a. kalampakas, s. spartalis, k. skoulariki or equivalently if, for all k ∈ h, the sum of the elements of the kth line of mr is pk. we call the matrix mr good if the corsini hyperoperation ∗r structures (h, ∗r) into a c-hypergroupoid. as it is shown in [9] there are 30 good matrices with form (p1, p2, p3) such that p1 +p2 +p3 = 6. more precisely there are 12 matrices such that pi = 2, where i ∈ h, and 18 matrices such that pi = 1, pj = 2, pk = 3, where i, j, k ∈ h. proposition 3.4 the 30 good boolean matrices with form (p1, p2, p3), where p1 + p2 + p3 = 6, correspond to 7 different non-isomorphic corsini’s graphs. proof. first we examine the 12 good matrices with form (2, 2, 2). the matrices   1 1 0 1 0 1 1 0 1  ,   1 0 1 1 1 0 1 1 0  ,   1 0 1 1 0 1 0 1 1  ,   1 1 0 0 1 1 1 1 0  ,   0 1 1 1 1 0 0 1 1  ,   0 1 1 0 1 1 1 0 1  , which are respectively the matrices (1), (3), (5), (8), (10) and (12) of [9], all represent corsini’s graphs isomorphic with the following graph. (g1) the matrices   1 1 0 1 0 1 0 1 1  ,   1 0 1 0 1 1 1 1 0  ,   0 1 1 1 1 0 1 0 1  , which are respectively the matrices (2), (6) and (9) of [9], correspond to graphs isomorphic with the graph below. (g2) 56 directed graphs representing isomorphism classes of c-hypergroupoids the matrices   1 0 1 1 1 0 0 1 1  ,   1 1 0 0 1 1 1 0 1  , which are respectively the matrices (4) and (7) of [9], both represent corsini’s graphs isomorphic with the next graph. (g3) the matrix   0 1 1 1 0 1 1 1 0  , is the matrix (11) of [9] and represents the following corsini’s graph. (g4) now we examine the rest 18 good matrices with the desired form. the matrices   0 0 1 1 1 0 1 1 1  ,   1 1 0 0 0 1 1 1 1  ,   0 1 0 1 1 1 1 0 1  ,   1 0 1 1 1 1 0 1 0  ,   1 1 1 1 0 0 0 1 1  ,   1 1 1 0 1 1 1 0 0  , which are respectively the matrices (13), (16), (20), (23), (27) and (30) of [9], represent corsini’s graphs isomorphic with the following graph. 57 a. kalampakas, s. spartalis, k. skoulariki (g5) the matrices   0 0 1 1 0 1 1 1 1  ,   0 1 1 0 0 1 1 1 1  ,   0 1 0 1 1 1 1 1 0  ,   0 1 1 1 1 1 0 1 0  ,   1 1 1 1 0 0 1 1 0  ,   1 1 1 1 0 1 1 0 0  , which are respectively the matrices (14), (18), (19), (24), (25) and (29) of [9], represent corsini’s graphs isomorphic with the following graph. (g6) the matrices   0 0 1 0 1 1 1 1 1  ,   1 0 1 0 0 1 1 1 1  ,   0 1 0 1 1 1 0 1 1  ,   1 1 0 1 1 1 0 1 0  ,   1 1 1 1 0 0 1 0 1  ,   1 1 1 1 1 0 1 0 0  , which are respectively the matrices (15), (17), (19), (22), (25) and (28) of [9], represent corsini’s graphs isomorphic with the next graph. (g7) 58 directed graphs representing isomorphism classes of c-hypergroupoids 2 remark 3.3 a boolean matrix with form (p1, p2, p3) corresponds to a graph g with a degreeout(g) = {p1, p2, p3}. hence graphs g1-g4 have degreeout(g) = {2, 2, 2} and graphs g5-g7 have degreeout(g) = {1, 2, 3}. the number of corsini’s graphs with order n is equal with the number of boolean n × n matrices forming non-isomorphic hypergroupoids. although massouros and tsitouras [16] have calculated this number for n = 3 there are 17 such matrices no actual representation of these 17 hypergroupoid classes has been demonstrated in [16]. such a representation is of much greater importance than merely computing their number since it will allow us to compare and correlate these isomorphism classes. in what follows we present the remaining corsini’s graphs and moreover we discover that they constitute an upper semilattice with respect to graph inclusion. this hierarchy actually determines a hierarchy inside the set of 73 c-hypergroupoids (cf. [22]) deriving from binary relations on 3 elements. since in the previous proposition we found 7 corsini’s graphs with 3 vertices it follows that there are 10 more. from these 1 has degreeout(g) = {1, 1, 3} (g8) 5 have degreeout(g) = {2, 2, 3} (g9) (g10) (g11) (g12) (g13) 1 has degreeout(g) = {1, 3, 3} 59 a. kalampakas, s. spartalis, k. skoulariki (g14) 2 have degreeout(g) = {2, 3, 3} (g15) (g16) and 1 (the complete graph with 3 nodes) has degreeout(g) = {3, 3, 3}. (g17) as it is implied by proposition 3.3, the graphs g1-g17 form a partially ordered set with respect to graph inclusion. more precisely they form the following upper semilattice as it can be verified by merely inspecting the drawings of graphs g1-g17. 60 directed graphs representing isomorphism classes of c-hypergroupoids g8 g1 g2 g3 g4 g5 g6 g7 g9 g10 g11 g12 g13 g14 g15 g16 g17 references [1] n. biggs, e. lloyd and r. wilson, graph theory, 1736-1936, oxford university press, 1986. 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[22] s. spartalis and c. mamaloukas, on hyperstructures associated with binary relations, computers and mathematics with applications 51, (2006), 41-50. 63 a. kalampakas, s. spartalis, k. skoulariki 64 ratio mathematica 23 (2012), 81–86 issn: 1592-7415 closed, reflexive, invertible, and normal subhypergroups of special hypergroups pavĺına račková university of defence, brno, czech republic pavlina.rackova@unob.cz abstract in [5] j. jantosciak introduced several special types of subhypergroups (invertible, closed, normal, reflexive) of a general hypergroup and studied their relationship. in this article, the full description of such subhypergroups in hypergroups induced by quasiordered groups is given. further, it is shown that there are no such non-trivial subhypergroups in quasiorder hypergroups. key words: hypergroup, transposition hypergroup, join space, closed, reflexive, invertible, and normal subhypergroup, quasiordered group, quasiorder hypergroup. msc 2010: 20n20. 1 introduction we will sum up basic concepts from hypergroup theory and results which will be needed in the following text. • let h 6= ∅. a mapping ∗: h × h → p∗ is called binary hyperoperation on h. the pair (h,∗) is hypergroupoid. • a hypergroupoid (h,∗) is extensive if {a, b}⊆ a ∗ b for any a, b ∈ h. • a hypergroupoid (h,∗) is called hypergroup if the hyperoperation ∗ is associative, i.e (x ∗ y) ∗ z = x ∗ (y ∗ z) for any x, y, z ∈ h, pavĺına račková and the reproduction axiom a ∗ h = h = h ∗ a for any a ∈ h is satisfied. • let (h,∗) be a hypergroup and s ⊆ h. assume that a ∗ b ⊆ s for any a, b ∈ s. thus (s,∗) is an associative hypergroupoid (so called semihypergroup). if, moreover, it satisfies the reproduction axiom, that is, it is a hypergroup, we say that (s,∗) is subhypergroup of (h,∗). • a hypergroup (h,∗) is called transposition hypergroup if the transposition axiom is satisfied, that is, for any quadruple of elements a, b, c, d ∈ h the implication: if b\a ≈ c/d, then a ∗ d ≈ b ∗ c holds, where b\a = {x ∈ h : a ∈ b ∗ x}, c/d = {x ∈ h : c ∈ x ∗ d} are left and right extensions, respectively. (for two set a, b, the symbol a ≈ b means that a and b are incident, i.e. a ∩ b 6= ∅.) • a commutative transposition hypergroup (h,∗) is called join space. the following concepts play the key role in the formulation of the main results. definition 1.1 a subhypergroup (s,∗) of a hypergroup (h,∗) is called • closed if a/b ⊆ s and b\a ⊆ s for any a, b ∈ s, • invertible if a/b ≈ s implies b/a ≈ s and b\a ≈ s implies a\b ≈ s for any a, b ∈ h, • reflexive if a\s = s/a for any a ∈ h, • normal if a ∗ s = s ∗ a for any a ∈ h. j. jantosciak (see [5]) proved that • invertible subhypergroup of any hypergroup is closed, • closed and normal subhypergroup of a transposition hypergroup is reflexive, • invertible and normal subhypergroup of a transposition hypergroup is closed and reflexive. 82 closed, reflexive, invertible, and . . . 2 hypergroups induced by quasiordered groups first the relationship among closed, normal, invertible, and reflexive subhypergroups of hypergroups induced by quasiordering will be tackled. definition 2.1 an ordered (quasiordered) group is a triple (h, ·,≤), where (h, ·) is a group and “≤” is an ordering (quasiordering) on h having the substitution property on (h, ·), that is for any quadruple a, b, c, d ∈ h such that a ≤ b, c ≤ d, there is a · c ≤ b · d. a hyperoperation is naturally induced on each (quasi)ordered group (h, ·,≤). for x ∈ h, let us denote [x)≤ the principal end generated by x, that is [x)≤ = {y ∈ h : x ≤ y}. analogously the principal beginning (x]≤ is defined. for x, y ∈ h, let us denote x ∗ y = [x · y)≤. then (h,∗) is a hypergroupoid associated with (h, ·,≤). the following result can be proven (see [3, 6, 7]): theorem 2.1 let (h, ·,≤) be a quasiordered group and (h,∗) the hypergroupoid associated with it. then (h,∗) is the transposition hypergroup. let (g, ·,≤) be a quasiordered group, (g,∗), where a∗b = [a ·b)≤, be the induced (transposition) hypergroup. let (s,∗) be its subhypergroup, that is, [x · y)≤ ⊂ s holds for x, y ∈ s, namely x · y ∈ s. for the right and left extensions we have: a/b = {x ∈ g : a ∈ x ∗ b} = {x ∈ g : x · b ≤ a} = = {x ∈ g : x ≤ a · b−1}, b\a = {x ∈ g : a ∈ b ∗ x} = {x ∈ g : b · x ≤ a} = = {x ∈ g : x ≤ b−1 · a}. theorem 2.2 a subhypergroup (s,∗) is closed iff (s, ·) is a subgroup of the group (g, ·) and (a]≤ ∪ [a)≤ ⊂ s for any a ∈ s. proof. necessity: for a ∈ s there is a/a = (e]≤, hence e ∈ s (e is the neutral element). if a ∈ s, then e/a = (a−1]≤, hence a−1 ∈ s. if a ∈ s, then a/e = (a]≤, hence (a]≤ ⊂ s. because (s,∗) is a subhypergroup we also get a ∗ e = [a)≤ ⊂ s. sufficiency: let a, b ∈ s. then a · b−1 ∈ s, so (a · b−1]≤ = a/b ⊂ s. analogously for b\a. 2 83 pavĺına račková theorem 2.3 a subhypergroup (s,∗) is closed iff it is invertible. proof. each invertible subhypergroup is closed. now, let us assume that (s,∗) is closed. if a/b ≈ s, there exists x ∈ s such that x ≤ a · b−1. due to the previous theorem a · b−1 ∈ s and also (a·b−1)−1 = b·a−1 ∈ s, therefore b/a ⊂ s. especially, b/a ≈ s. analogously the statement for b\a can be proven. 2 theorem 2.4 a subhypergroup (s,∗) is reflexive iff the following property holds: if for x, y ∈ g the element x · y is covered by an element of s, then y · x is also covered by an element of s. proof. the following equalities hold: a\s = ⋃ b∈s a\b = ⋃ b∈s (a−1 · b]≤ , s/a = ⋃ b∈s b/a = ⋃ b∈s (b · a−1]≤ . hence, x ∈ a\s iff b ∈ s exists such that x ≤ a−1 · b, that is, a · x ≤ b. analogously x ∈ s/a iff c ∈ s exists such that x · a ≤ c. this implies the statement. 2 if (s,∗) is a closed subhypergroup, according to the previous result, the condition “to be comparable with an element of s” is equivalent with the condition “to be in s”. therefore, we get: corollary 2.1 a closed subhypergroup (s,∗) is reflexive iff the following property holds: if for x, y ∈ g the element x · y ∈ s, then also y · x ∈ s. theorem 2.5 a subhypergroup (s,∗) is normal iff the following property holds: if x · y, where x, y ∈ g, covers an element of s, then y · x also covers an element of s. proof. the following equalities hold: a ∗ s = ⋃ b∈s a ∗ b = ⋃ b∈s [a · b)≤ , s ∗ a = ⋃ b∈s b ∗ a = ⋃ b∈s [b · a)≤ . hence, x ∈ a ∗ s iff b ∈ s exists such that a · b ≤ x, that is, b ≤ a−1 · x. analogously x ∈ s ∗a iff c ∈ s exists such that c ≤ x ·a−1. this implies the statement. 2 84 closed, reflexive, invertible, and . . . in case (s,∗) is a closed subhypergroup, analogously to reflexive subhypergroups we have: corollary 2.2 a closed subhypergroup (s,∗) is normal iff the following property holds: if for x, y ∈ g the element x · y ∈ s, than also y · x ∈ s. joining the previous two results we get: corollary 2.3 let (s,∗) be a closed subhypergroup. then (s,∗) is normal iff it is reflexive. 3 quasiorder hypergroups now the relationship among closed, normal, invertible, and reflexive subhypergroups of quasiorder hypergroups will be tackled. if r is a quasiordering on h we denote r(a) = {x ∈ h : a r x}. further, for a ⊆ h we set r(a) = ⋃ a∈a r(a). theorem 3.1 let (h, r) be a quasiordered set. for any pair a, b ∈ h we set a ∗r b = r(a) ∪ r(b) = r({a, b}). then (h,∗r) is commutative extensive hypergroup. for the proof see [3, p. 150, th. 2.1]. definition 3.1 a hypergroup (h,∗) is said quasiorder if the following conditions are satisfied: • a ∈ a3 ⊆ a2, • a ∗ b = a2 ∪ b2 for any a, b ∈ h. the following theorem characterizes all quasiorder hypergroups. theorem 3.2 a hypergroup (h,∗) is quasiorder iff a quasiordering r on h exists such that for any a, b ∈ h there is a ∗ b = r(a) ∪ r(b) = r({a, b}). for the proof see [1, p. 96]. 85 pavĺına račková theorem 3.3 let (g,∗) be a quasiorder hypergroup. then any subhypergroup of (g,∗) is reflexive and normal. moreover, (g,∗) contains no proper closed or invertible subhypergroup. proof. suppose that (g,∗) is a quasiorder hypergroup and r is a quasiordering from the previous theorem. the hyperoperation ∗ is commutative, hence any subhypergroup (s,∗) is reflexive and normal. further, a/b = {x ∈ g : a ∈ b ∗ x} = {x ∈ g : a ∈ r(b) ∪ r(x)}. especially a/a = g. if (s,∗) is closed, then necessarily s = g. therefore, no proper closed subhypergroup exists. analogously, any invertible subhypergroup is closed, therefore, no proper invertible subhypergroup exists. 2 references [1] corsini, p., leoreanu, v., applications of hyperstructure theory. kluwer ap, dordrecht, boston, london 2003, 322 pp. [2] hošková, š., representation of quasi-order hypergroups. global journal of pure and applied mathematics, vol. 1, number 2, india (2005), pp. 173–176. [3] chvalina, j., funkcionálńı grafy, kvaziuspořádané množiny a komutativńı hypergrupy. brno, mu 1995, 206 pp. (in czech) [4] jantosciak, j., transposition in hypergroups. internat. congress on aha 6 (prague 1996), democtritus univ. of thrace press, alexandropolis, 1997, pp. 77–84. [5] jantosciak, j., transposition hypergroups: noncommutative join spaces. j. algebra 187 (1997), pp. 97–119. [6] račková, p., akce polohypergrup a modelováńı hypergrup integrálńımi operátory. phd. thesis, faculty of science, palacký university, olomouc 2008, czech rep., 73 pp. (in czech) [7] račková, p., hypergroups of symmetric matrices. 10th international congress on aha (brno 2008), proceedings, pp. 267–271, isbn 97880-7231-688-5. 86 homomorphism and quotient of fuzzy k-hyperideals r. ameria adepartment of mathematics, university of mazandaran, babolsar, iran e-mail: ameri@umz.ac.ir h. hedayatib bdepartment of mathematics, faculty of basic science, babol university of technology, babol, iran e-mail: h.hedayati@nit.ac.ir, hedayati143@yahoo.com abstract in [15], we introduced the notion of weak (resp. strong) fuzzy khyperideal. in this note we investigate the behavior of them under homomorphisms of semihyperrings. also we define the quotient of fuzzy weak (resp. strong) k-hyperideals by a regular relation of semihyperring and obtain some results. mathematics subject classification: 20n20 keywords: (semi-) hyperring, homomorphism, fuzzy weak (strong) khyperideals, regular relation, (fuzzy) quotient of k-hyperideals ratio mathematica 20, 2010 148 1 introduction following the introduction of fuzzy set by l. a. zadeh in 1965 ([26]), the fuzzy set theory developed by zadeh himself and can be found in mathematics and many applied areas. the concept of a fuzzy group was introduced by a. rosenfeld in [24]. the notion of fuzzy ideals in a ring was introduced and studied by w. j. liu [20]. t.k. dutta and b. k. biswas studied fuzzy ideals, fuzzy prime ideals of semirings in [14, 16] and they defined fuzzy ideals of semirings and fuzzy prime ideals of semirings and characterized fuzzy prime ideals of non-negative prime integers and determined all it’s prime ideals. recently, y. b. jun, j. neggeres and h. s. kim ([16]) extended the concept of a l-fuzzy (characteristic) ideal left(resp. right) ideal of a ring to a semiring. s. i. baik and h. s. kim introduced the notion of fuzzy k-ideals in semirings [6]. also a hypergroup was introduced by f. marty ([23]), today the literature on hypergroups and related structures counts 400 odd items [8, 9, 25]. among the several contexts which they aries is hyperrings. first m. krasner studied hyperrings, which is a triple (r, +, .), where (r, +) is a canonical hypergroup and (r, .) is a semigroup, such that for all a, b, c ∈ r, a(b + c) = ab + ac, (b + c)a = ba + ca ([18]). zahedi and others introduced and studied the notion of fuzzy hyperalgebraic structures [3, 4, 5, 11, 12, 19, 27]. in [15] we introduced the notion of fuzzy weak (strong) k-hyperideal and then we obtained some related basic results. in this note we investigate the behavior of them under homomorphisms of semihyperrings. also we define the quotient of fuzzy weak (strong) k-hyperideals by a regular relation of semihyperring and obtain some results. ratio mathematica 20, 2010 149 2 preliminaries in this section we gather all definitions and simple properties we require of semihyperrings and fuzzy subsets and set the notions. a map ◦ : h × h −→ p∗(h) is called hyperoperation or join operation. a hypergroupoid is a set h with together a (binary) hyperoperation ◦. a hypergroupoid (h, ◦), which is associative, that is x ◦ (y ◦ z) = (x ◦ y) ◦ z, ∀x, y, z ∈ h is called a semi-hypergroup . a hypergroup is a semihypergroup such that ∀x ∈ h we have x ◦ h = h = h ◦ x, which is called reproduction axiom. let h be a hypergroup and k a nonempty subset of h. then k is a subhypergroup of h if itself is a hypergroup under hyperoperation restricted to k. hence it is clear that a subset k of h is a subhypergroup if and only if ak = ka = k, under the hyperoperation on h. a set h together a hyperoperation ◦ is called a polygroup if the following conditions are satisfied: (1) (x ◦ y) ◦ z = x ◦ (y ◦ z) ∀x, y, z ∈ h; (2) ∃e ∈ h as unique element such that e ◦ x = x = x ◦ e ∀x ∈ h; (3) ∀x ∈ h there exists an unique element, say x′ ∈ h such that e ∈ x ◦ x′ ∩ x′ ◦ x ( we denote x′ by x−1). (4) ∀x, y, z ∈ h, z ∈ x ◦ y =⇒ x ∈ z ◦ y−1 =⇒ y ∈ x−1 ◦ z. a non-empty subset k of a polygroup (h, ◦) is called a subpolygroup if (k, ◦) is itself a polygroup. in this case we write k <p h. a commutative polygroup is called canonical hypergroup. definition 2.1. a hyperalgebra (r, +, .) is called a semihyperring if and only ratio mathematica 20, 2010 150 if (i) (r, +) is a semihypergroup and (r, .) is a semigroup; (ii) a.(a + b) = a.b + a.c and (b + c).a = b.c + c.a ∀a, b, c ∈ r. a semihyperring is called with zero, if there exists an element, say 0 ∈ r such that 0.x = 0 = x.0 and 0 + x = x = x + 0 ∀x ∈ r. also a semihyperring (r, +, .) is called a hyperring provided (r, +) is a canonical hypergroup. a hyperring (r, +, .) is called (i) commutative if and only if a.b = b.a ∀a, b ∈ r; (ii)with identity, if there exists an element, say 1 ∈ r such that 1.x = x.1 = x ∀x ∈ r. let (r, +, .) be a hyperring, a nonempty subset s of r is called a subhyperring of r if (s, +, .) is itself a hyperring. definition 2.2. a subhyperring i of a hyperring r is a (resp. left) right hyperideal of r provided that ( resp. x.r ∈ i) r.x ∈ i ∀r ∈ r, ∀x ∈ i. i is called a hyperideal if i is both left and right hyperideal. we use i = [0, 1], the real unit interval as a chain with the usual ordering, in which ∧ stands for infimum (inf ) (or intersection ) and ∨ stands for supremum ( sup ) (or union), for the degree of membership. a fuzzy subset of a given set x is a mapping µ : x −→ i. we denote the set of all fuzzy subsets of x by f s(x). for µ ∈ f s(x), the level subset of µ is defined by µt = {x ∈ x| µ(x) ≥ t} ∀t ∈ i. for a fuzzy subset µ of x we denote by im(µ) the image of µ. let {µi | ratio mathematica 20, 2010 151 i ∈ i} be a family of fuzzy subsets, intersection of µi’s is defined by ( ⋂ i∈i µi)(x) = ∧ i∈i µi(x). definition 2.3. let (g, .) be a group and µ ∈ f s(g). then µ is said to be a fuzzy subgroup of g if ∀x, y ∈ g we have : (i) µ(xy) ≥ µ(x) ∧ µ(y); (ii) µ(x−1) ≥ µ(x). definition 2.4. if f : x −→ y be a function and µ ∈ f s(x) , then we say µ is f−invariant if and only if f (a) = f (b) =⇒ µ(a) = µ(b). in the sequel by r we mean a semihyperring. definition 2.5.[1] a nonempty subset i of r is called (i) a left (resp. right) hyperideal of r if and only if (1) (i, +) is a semihypergroup of (r, +); (2) rx ∈ i (resp. xr ∈ i), for all r ∈ r and for all x ∈ i. (ii) a hyperideal of r if it is both a left and a right hyperideal of r. by i <h r, we mean hyperideal of r. (iii) a left hyperideal i of r is called weak left k-hyperideal of r if for a ∈ i and x ∈ r we have a + x ⊆ i or x + a ⊆ i =⇒ x ∈ i. a left hyperideal i of r is called strong left k-hyperideal of r if for a ∈ i and x ∈ r we have a + x ≈ i or x + a ≈ i =⇒ x ∈ i, ratio mathematica 20, 2010 152 where by a ≈ b, we mean a ∩ b 6= ∅, for all nonempty subsets a and b of r. a right(resp. strong) weak k-hyperideal is defined dually. a two sided (resp. strong) weak k-hyperileal or simply a (resp. strong) weak k-hyperideal is both left and right (resp. strong) weak k-hyperideal. we denote i <w.k.h r (resp. i <s.k.h r) for weak (resp. strong) k-hyperideal of r. clearly, every (strong) weak k− hyperideal is a hyperideal, but the converse is not true. example. consider z, the set of integer numbers. define new hyperoperations ⊕ and ◦ on z as follow m ⊕ n = {m, n} and m ◦ n = mn ∀m, n ∈ z. clearly (z, ⊕, ◦) is a semihyperring. now it is easy to verify that i =< 2 >= {2k | k ∈ z}, is a hyperideal of z, but it isn’t strong k−hyperideal, since 3 ⊕ 2 = {3, 2} ≈ i and 2 ∈ i but 3 6∈ i. definition 2.6 .[7] let r and s be semihyperrings. a mapping f : r −→ s is said to be (i) homomorphism if and only if f (x + y) ⊆ f (x) + f (y) and f (x.y) = f (x).f (y) ∀x, y ∈ r. (ii) good homomorphism if and only if f (x + y) = f (x) + f (y) and f (x.y) = f (x).f (y) ∀x, y ∈ r. definition 2.7 .[15] a fuzzy subset µ of a semihyperring r is called a fuzzy ratio mathematica 20, 2010 153 left hyperideal of r if and only if (i) ∧ z∈x+y µ(z) ≥ µ(x) ∧ µ(y) ∀x, y ∈ r; (ii) µ(xy) ≥ µ(y) ∀x, y ∈ r. a fuzzy right hyperideal is defined dually. a fuzzy left and right hyperideal is called a fuzzy hyperideal. we denote µ <f.h r for fuzzy hyperideal of r. definition 2.8.[15] a fuzzy hyperideal µ of r is called (i) a weak fuzzy k-hyperideal of r if and only if µ(x) ≥ [( ∧ u∈x+y µ(u)) ∨ ( ∧ v∈y+x µ(v))] ∧ µ(y) ∀x, y ∈ r. (ii) a strong fuzzy k-hyperideal of r if and only if µ(x) ≥ (µ(z) ∨ µ(z′)) ∧ µ(y) ∀z ∈ x + y, ∀z′ ∈ y + x. note that if (r, +) is a commutative semihyperring, then the above conditions reduce to the following conditions: µ(x) ≥ ( ∧ u∈x+y µ(u)) ∧ µ(y) ∀x, y ∈ r. and µ(x) ≥ µ(z) ∧ µ(y) ∀z ∈ x + y. we denote by µ <w.f.k.h r (resp. µ <s.f.k.h r), for a weak fuzzy k−hyperideal (resp. strong fuzzy k−hyperideal) of r. proposition 2.9.[15] let r be a semihyperring and µ ∈ f s(r). then (i) µ is a fuzzy hyperideal of r if and only if every nonempty level subset, µt is a hyperideal of r. ratio mathematica 20, 2010 154 (ii) µ is a weak fuzzy k-hyperideal of r if and only if every nonempty level subset, µt is a weak k-hyperideal of r. (iii) µ is a strong fuzzy k-hyperideal of r if and only if every nonempty level subset, µt is a strong k-hyperideal of r. lemma 2.10. let r be a semihyperring with zero and µ be a fuzzy hyperideal of r. then µ(x) ≤ µ(0) for all x ∈ r. 3 homomorphisms of fuzzy k-hyperideals in this section we investigate the behavior of fuzzy weak (strong) k-hyperideals under homomorphisms of semihyperrings. proposition 3.1. let f : r −→ r′ be a homomorphism of semihyperrings. if ν <s.f.k.h r ′, then f−1(ν) <s.f.k.h r. proof. we know that f−1(ν)(x) = ν(f (x)). let x, y ∈ r and z ∈ x + y, then we have f (z) ∈ f (x + y) ⊆ f (x) + f (y), and since ν <f.h r′, it concluded that ν(f (z)) ≥ ν(f (x)) ∧ ν(f (y)). also ν(f (xy)) = ν(f (x)f (y)) ≥ ν(f (x)) ∨ ν(f (y)). therefore f−1(ν) <f.h r. now let z ∈ x + y and z′ ∈ y + x, thus f (z) ∈ f (x) + f (y) and f (z′) ∈ f (y) + f (x), then ν <s.f.k.h r ′ implies that ν(f (x)) ≥ [ν(f (z)) ∨ ν(f (z′))] ∧ ν(f (y)) as required. ratio mathematica 20, 2010 155 proposition 3.2. let f : r −→ r′ be a good homomorphism of semihyperrings. if ν <w.f.k.h r ′ , then f−1(ν) <w.f.k.h r. proof. we know that f−1(ν)(x) = ν(f (x)). first we prove that f−1(ν) is a fuzzy hyperideal of r. let x, y ∈ r and z ∈ x + y, we should prove that ν(f (z)) ≥ ν(f (x)) ∧ ν(f (y)) (1) (1) is valid because ν is a fuzzy hyperideal and f is a good homomorphism, then for z ∈ x + y we have f (z) ∈ f (x + y) = f (x) + f (y). also similar previous proposition ν(f (xy)) ≥ ν(f (x)) ∨ ν(f (y)). therefore f−1(ν) <f.h r. now we prove that f−1(ν) <w.f.k.h r, that is f−1(ν)(x) ≥ {( ∧ t∈x+y f−1(ν)(t)) ∨ ( ∧ t ′∈y+x f−1(ν)(t ′ ))} ∧ f−1(ν)(y) (2). note that since f is a good homomorphism, then t ∈ x + y if and only if f (t) ∈ f (x) + f (y), and also ν <w.f.k.h r′, we have ν(f (x)) ≥ {( ∧ f (t)∈f (x)+f (y) ν(f (t))) ∨ ( ∧ f (t ′ )∈f (y)+f (x) ν(f (t ′ )))} ∧ ν(f (y)). the last relation implies (2), and this complete the proof. proposition 3.3. let f : r −→ r′ be a good epimorphism of semihyperrings. if µ <w.f.k.h r (resp. µ <s.f.k.h r) and µ be f−invariant, then f (µ) <w.f.k.h r′ (resp. f (µ) <s.f.k.h r). proof. first we show that f (µ) <f.h r ′ . ratio mathematica 20, 2010 156 let a, b ∈ r′ and c ∈ a + b, we should prove that f (µ(c)) ≥ f (µ(a)) ∧ f (µ(b)). we have f (µ(c)) = ∨ z∈f−1(c) µ(z), f (µ(a)) = ∨ x∈f−1(a) µ(x), f (µ(b)) = ∨ y∈f−1(b) µ(y). since µ is f−invariant, then ∃z0 ∈ f−1(c), f (µ(c)) = µ(z0), ∃x0 ∈ f−1(a), f (µ(a)) = µ(x0), ∃y0 ∈ f−1(b), f (µ(b)) = µ(y0), therefore f (z0) = c, f (x0) = a, f (y0) = b =⇒ f (z0) ∈ f (x0) + f (y0) =⇒ z0 ∈ x0 + y0 ( f is a good homomorphism ) =⇒ µ(z0) ≥ µ(x0) ∧ µ(y0) (µ <f.h r) =⇒ f (µ(c)) ≥ f (µ(a)) ∧ f (µ(b)). for proving the second condition of a fuzzy hyperideal, we should prove that f (µ)(r ′ x ′ ) ≥ f (µ)(x′) ∨ f (µ)(r′) ∀ r′, x′ ∈ r′ ratio mathematica 20, 2010 157 since f is onto, then r′ = f (r) and x′ = f (x) for some r and x in r, thus f (µ)(r ′ x ′ ) = ∨ rx∈f−1(r′x′) µ(rx) = µ(r0x0) ∃r0 ∈ f−1(r′), x0 ∈ f−1(x′) (µ is f -invariant) ≥ µ(x0) ∨ µ(r0) ( µ <f.h r) = f (µ)(x ′ ) ∨ f (µ)(r′) (µ is f -invariant). therefore f (µ)(r′x′) ≥ f (µ)(r′) ∨ f (µ)(x′). now we prove that f (µ) <w.f.k.h r ′ . let a, b ∈ r, we show that f (µ)(a) ≥ [( ∧ t∈a+b f (µ)(t)) ∨ ( ∧ t ′∈b+a f (µ)(t ′ ))] ∧ f (µ)(b) (1) since f is onto and µ is f−invariant, then f (µ)(a) = µ(x0), f (µ)(t) = µ(z0), f (µ)(t ′ ) = µ(z ′ 0), f (µ)(b) = µ(y0), where x0 ∈ f−1(a), y0 ∈ f−1(b), z0 ∈ f−1(t), z ′ ∈ f−1(t′). hence (1) reduced to the form µ(x0) ≥ [( ∧ t∈a+b µ(z0)) ∨ ( ∧ t ′∈b+a µ(z ′ 0))] ∧ µ(y0) (2) on the other hand from above discussion and since f is a good homomorphism t ∈ a + b if and only if f (z0) ∈ f (x0) + f (y0) if and only if z0 ∈ x0 + y0. similarly, t′ ∈ b + a if and only if z′0 ∈ y0 + x0. ratio mathematica 20, 2010 158 therefore by (2), it is enough that we prove that µ(x0) ≥ [( ∧ z0∈x0+y0 µ(z0)) ∨ ( ∧ z ′ 0∈y0+x0 µ(z ′ 0))] ∧ µ(y0), but clearly the last statement is true, since µ <w.f.k.h r. this complete the proof. in this part we define the quotient of fuzzy weak (strong) k-hyperideals by a regular relation of semihyperring let r be a semihyperring and θ be an equivalence relation on r. naturally we can extend θ to θ to the subsets of r as follow: let a, b be nonempty subsets of r. define aθb ⇐⇒ ∀a ∈ a ∃b ∈ b : aθb, ∀b ∈ b ∃ a ∈ a : bθa. an equivalence relation θ on r is said to be regular if for all a, b, x ∈ r we have (i) aθb =⇒ (a + x)θ(b + x) and (x + a)θ(x + b), (ii) aθb =⇒ (ax)θ(bx) and (xa)θ(bx). by r : θ we mean the set of all equivalence classes with respect to θ, that is r : θ = {rθ|r ∈ r}. remark 3.4. we know that if r is a semihyperring and θ is a regular equivalence relation on r , then r : θ by hyperoperations ⊕ and ¯ is defined as follow xθ ⊕ yθ = {xθ|z ∈ x + y}, xθ ¯ yθ = (xy)θ. ratio mathematica 20, 2010 159 is a semihyperring. for µ ∈ f s(r), define (µ : θ)(xθ) = ∨ y∈xθ µ(y). also we know that the mapping ϕ : r −→ r : θ defined by ϕ(a) = aθ is a good epimorphism. now if µ <w.f.k.h r and µ be ϕ−invariant then by proposition 3.3 it concludes that ϕ(µ) = µ : θ <w.f.k.h r : θ . proposition 3.5. if µ <w.f.k.h r and r has zero, then µ∗ = {x ∈ r | µ(x) = µ(0)} is a weak k−hyperideal of r. proof. first we prove that µ∗ <h r. for x, y ∈ µ∗ and z ∈ x + y, then µ(z) ≥ µ(x) ∧ µ(y) = µ(0), hence by lemma 2.10 µ(z) = µ(0), therefore z ∈ µ∗. let r ∈ r and x ∈ µ∗, then we have µ(rx) ≥ µ(r) ∨ µ(x) = µ(r) ∨ µ(0) ( x ∈ µ∗ ) = µ(0) ( by lemma 2.10 ) =⇒ µ(rx) = µ(0) ( by lemma 2.10 ) =⇒ rx ∈ µ∗. now suppose r + x ⊆ µ∗ or x + r ⊆ µ∗ and x ∈ µ∗, we show that r ∈ µ∗. from µ <w.f.k.h r then we have : µ(r) ≥ [( ∧ z∈r+x µ(z)) ∨ ( ∧ z′∈x+r µ(z′))] ∧ µ(x). since µ(x) = µ(0) and ∧ z∈r+x µ(z) = µ(0) and ∧ z′∈x+r µ(z′) = µ(0), then µ(r) ≥ µ(0), and then by lemma 2.10, µ(r) = µ(0). therefore µ∗ <w.k.h r. proposition 3.6. if µ <s.f.k.h r, then µ ∗ = {x ∈ r | µ(x) > 0} is a strong k-hyperideal of r. ratio mathematica 20, 2010 160 proof. let x, y ∈ µ∗ and z ∈ x + y, then by hypothesis yields µ(z) ≥ µ(x) ∧ µ(y) > 0, thus z ∈ µ∗. if r ∈ r and x ∈ µ∗, then we have µ(rx) ≥ µ(r) ∨ µ(x) ≥ µ(x) > 0, therefore rx ∈ µ∗. similarly xr ∈ µ∗. thus µ∗ <h r. now if r + x ≈ µ∗ or x + r ≈ µ∗ and x ∈ µ∗. by hypothesis we have µ(r) ≥ (µ(z) ∨ µ(z′)) ∧ µ(x) > 0 ∀z ∈ r + x ≈ µ∗, ∀z′ ∈ x + r ≈ µ∗, that is r ∈ µ∗, and hence µ∗ <s.k.h r. proposition 3.7. let r be a semihyperring with zero and x, y ∈ r: (i) if µ <w.f.k.h r and µ(t) = µ(0) = µ(t ′) for all t ∈ x+y and t′ ∈ y +x, then µ(x) = µ(y). (ii) if µ <s.f.k.h r and µ(u) = µ(0) = µ(v) for some u ∈ x + y and v ∈ y + x, then µ(x) = µ(y). proof. (i) since µ <w.f.k.h r and µ(t) = µ(0) = µ(t ′) for all t ∈ x + y and t′ ∈ y + x, then ∧ t∈x+y µ(t) = µ(0) = ∧ t′∈y+x µ(t′), thus µ(x) ≥ [( ∧ t∈x+y µ(t)) ∨ ( ∧ t′∈y+x µ(t′))] ∧ µ(y) = µ(0) ∧ µ(y) = µ(y) (by lemma 2.10) =⇒ µ(x) ≥ µ(y). ratio mathematica 20, 2010 161 similarly we conclude that µ(y) ≥ µ(x). therefore µ(x) = µ(y). (ii) suppose u ∈ x + y and v ∈ y + x such that µ(u) = µ(0) = µ(v), since µ <s.f.k.h r, then µ(y) ≥ (µ(u) ∨ µ(v)) ∧ µ(x) = µ(0) ∧ µ(x) ( by hypothesis ) = µ(x) (by lemma 2.10) =⇒ µ(y) ≥ µ(x). similarly we obtain µ(x) ≥ µ(y). therefore µ(x) = µ(y). refrences [1] r. ameri and m.m. zahedi ”hyperalgebraic system”, italian journal of pure and applied mathematics, 6: (1999) 21-32. [2] r. ameri, ”fuzzy transposition hypergroups”, italian journal pure and applid mathematics, no. 18 (2005), 167-174. 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[26] l. a. zadeh , ”fuzzy sets”, inform. and control, vol. 8 (1965) 338-353. ratio mathematica 20, 2010 164 [27] m.m. zahedi , m. bolurian, a. hasankhani, ”on polygroups and fuzzy subpolygroups”, j. of fuzzy mathematics, no.1, (1995) 1-15. ratio mathematica 20, 2010 165 ratio mathematica volume 45, 2023 micro sp-open sets in micro topological spaces m. maheswari1 s. dhanalakshmi2 n. durgadevi3 abstract in this paper, a new class of open sets called micro spopen sets in micro topological spaces are introduced and its fundamental properties are analyzed. also, some operations on micro sp-open sets are investigated. keywords: micro-open, micro-semi open, micro-pre-open, micro-pre closed and micro sp-open, micro sp-closed. 2010 mathematics classification:46s40, 34a07, 03e724 1 research scholar, department of mathematics, sri parasakthi college for women, courtallam. (affiliated to manonmaniam sundaranar university, tirunelveli, tamilnadu). 2 research scholar, department of mathematics, sri parasakthi college for women, courtallam. (affiliated to manonmaniam sundaranar university, tirunelveli, tamilnadu). 3 assistant professor, department of mathematics, sri parasakthi college for women, courtallam (affiliated to manonmaniam sundaranar university, tirunelveli, tamilnadu). email: durgadevin@sriparasakthicollege.edu.in 4received on july 20, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm. v45i0.985. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 90 mailto:durgadevin@sriparasakthicollege.edu.in m. maheswari, s. dhanalakshmi and n. durgadevi 1. introduction the concept of nano topology was introduced by lellis thivagar [3] which is in terms of the lower and upper approximations and the boundary region of a subset of an universe. the notion of approximations and boundary region of a set was originally proposed by pawlak [4] in order to introduce the concept of rough set theory. chandrasekar [5] introduced the concept of micro topology which is a simple extension of nano topology and he also studied the concepts of micro pre-open and micro semiopen sets. in 2010, shareef introduced the class of semi open sets called sp-open sets. chandrasekar and swathi [1] introduced micro -open in micro topological space. in this paper a new class of sets in micro topological spaces called micro sp-open set is introduced and some of its properties are derived. 2. preliminaries definition 2.1. [4] let u be a non-empty finite set of objects called the universe and r be an equivalence relation on u named as the indiscernibility relation. elements belonging to the same equivalence class are said to be indiscernible with one another. the pair (u, r) is said to be the approximation space, let x  u, then (i) the lower approximation of x with respect to r is the set of all objects which can be certain classified as x with respect to r and is denoted by lr(x). that is lr(x) =  ux xr  )({ : r(x)  x} where r(x) denotes the equivalence class determined by x u. (ii) the upper approximation of x with respect to r is the set of all objects, which can be possibly classified as x with respect to r and it is denoted by ur(x). that is ur(x) = u ux xr  )({ : r(x) x}. (iii) the boundary region of x with respect to r is the set of all objects, which can be classified as neither as x nor as not-x with respect to r and is denoted by br(x). that is br(x) = ur(x) – lr(x). definition2.2. [3] let r be an equivalence relation on the universe u and r(x) = {u,, lr(x), ur(x), br(x)} and xu. then r(x)satisfies the following axioms. (i) u and r(x). (ii) the union of the elements of any sub collection of r(x) is in r(x). (iii) the intersection of the elements of any finite subcollection of r(x) is in r(x). that is,r(x) is a topology on u called the nano topology on u with respect to x. thus (u, r(x)) as called as nano topological space. the elements of r(x) are called as nano open sets. a subset f of u is nano closed if its complement is nano open. definition 2.3. [5] let (u,r(x)) be a nano topological space. then r(x) = {n ∪ (n):n, nr(x)} and r(x)) is called the micro topology in u with respect to x. the triplet (u,r(x), r(x)) is called micro topological space and then elements of 91 micro sp-open sets in micro topological spaces r(x) are called micro open sets and the complement of a micro open set is called a micro closed set. definition 2.4. [1] let (u,r(x), r(x))be a micro topological space and au. then a is called (i) micro -open if a  mic-int (mic-cl (mic-int (a))) (ii) micro pre-open if a  mic-int (mic-cl (a)). (iii) micro semi-open if a  mic-cl (mic-int (a)). definition 2.5. [5] let (u, r(x), r(x)) be a micro topological space. let a and b be any two subsets of u. then (i) a is micro open set if and only if mic-int(a)=a. (ii) a is micro closed set if an only if mic-cl(a) = a. (iii) mic-int(u\a) = u\mic-cl(a). (iv) mic-cl(u\a) =u\mic-int(a). 3. micro sp-open sets definition 3.1. let (u, r(x), r(x)) be a micro topological space and a  u. then a is said to be micro sp-open (briefly mic sp-open) if for each xamic-so (u, x), there exists a micro pre-closed set f such that xf  a. the set of all micro sp-open sets is denoted by mic sp-o (u, x). definition 3.2. let (u, r(x), r(x)) be a micro topological space. a subset b of u is called micro sp-closed (briefly mic sp-closed) if and only if its complement is micro sp-open and mic sp-cl (u, x) denotes the collection of all micro sp-closed sets. example 3.3. let u = {a, b, c, d} with u|r = {{a}, {c}, {b, d}}, x = {b, d}  u then r(x) = {u, , {b, d}}. if  = {a}. then r(x) = {u,, {a}, {b, d}, {a, b, d}} and micro sp-open sets are {u, , {a, c}, {b, d}, {b, c, d}} remark 3.4. every micro sp-open set is a micro semi open set but the converse need not always be true as shown from the following example. example 3.5. in example 3.3, mic-so(u, x) = {u, , {a},{a, c}, {b, d}, {a, b, d}, {b, c, d}}.mic sp-o (u, x) = {u,, {a, c}, {b, d}, {b, c, d}} the set {a, b, d} is micro semiopen but not micro sp-open. theorem 3.6. an arbitrary union of any family of micro sp-open sets is micro sp-open. proof: let {ai: i} be a family of mic sp-open sets. if xa, then for each x i i a  mic-so (u, x) there exists a micro pre-closed set f such that x  f  ai u i i a which 92 m. maheswari, s. dhanalakshmi and n. durgadevi implies x  f  u i i a . therefore u i i a is micro sp-open. remark 3.7. from the above theorem 3.6, arbitrary intersection of micro sp-closed sets of a micro topological space is micro sp-closed as shown by the following example. example 3.8. in example 3.3, mic sp-o (u, x) = {u,, {a, c}, {b, d}, {b, c, d}} and mic sp-cl (u, x) = {u, , {b, d}, {a, c}, {a}}. here {b, d}  {b, c, d} = {b, d} which is a micro sp-closed set. remark 3.9.the intersection of any two mic sp-open sets need not be a mic sp-open set. from example 3.3, {a, c}, {b, c, d} are mic sp-open sets but {a, c}  {b, c, d} = {c} is not a mic sp-open set. proposition 3.10. if a subset a of a micro topological space (u, r(x), r(x)) is mic spopen. then a is a micro semi-open set and a is a union of micro pre-closed sets. proof: let a mic sp-o (u, x) and a  u. then for each x  a  mic-so (u, x), there exists a micro pre-closed set f containing x such that x  f  a. thus a is a micro semi-open set and also f  a which implies a is the union of micro pre-closed sets. remarks 3.11. the converse of the above proposition 3.10 need not be true as shown in the following example. example 3.12. in example 3.3, mic-so (u, x) = {u, , {a}, {a, c}, {b, d}, {a, b, d}, {b, c, d}} and mic-pcl (u, x) = {u, , {b}, {c}, {d}, {a, c}, {b, c}, {c, d}, {a, b, c}, {a, c, d}, {b, c, d}}. then {a, b, d}  mic-so (u, x) and {a, b, d} is not in the union of micro pre-closed sets, also {a, b, d}  mic sp-o (u, x). 4. operations on micro sp-open sets definition 4.1. a point xu is said to be a micro sp-interior point of a if there exists a micro sp-open set v containing x such that va. the set of all mic sp-interior points of a is said to be micro sp-interior of a and is denoted by mic sp-int(a). definition 4.2. let a be any subset of a micro topological space (u, r(x), r(x)). then a point xu is in the micro sp-closure of a if and only if a  h  for every h mic sp-o (u, x) containing x. the intersection of all micro sp-closed sets containing h is called the micro sp-closure of f and is denoted by mic sp-cl (a). 93 micro sp-open sets in micro topological spaces theorem 4.3. let a be any subset of a micro topological space (u, r(x), r(x)). if a point x mic sp-int (a), then there exists f mic-pcl (u, x) containing x such that f  a. proof: suppose that x  mic sp-int (a). then v mic sp-o (u, x) containing x such v  a. since v mic sp-o (u, x), then there exists f mic-pcl (u, x) containing x such that f  u  a. hence x  f  a. theorem 4.4. let a be a subset of a micro topological space (u,r(x), r(x)). if a  f   for every fmic-pcl (u, x) containing x, then x mic sp-cl (a). proof: let v mic sp-o (u, x) containing x, then there exists a micro pre-closed set f containing x such that fv. since a  f , x  mic sp-cl (a). theorem 4.5. for any two subsets a and b of a micro topological space (u, r(x), r(x)), the following properties are true. (i) mic sp-int (mic sp-int (a)) = mic sp-int (a). (ii) mic sp-int (a) = u – mic sp-cl (u – a). (iii) if a  b, then mic sp-int(a)  mic sp-int (b) (iv) mic sp-int (a) mic sp-int (b)  mic sp-int (a  b). (v) mic sp-int (a  b)  mic sp-int (a)  mic sp-int (b). proof: obvious. the converse of (iii), (iv) and (v) of theorem 4.5 need not be true as shown in the following example. example 4.6.consider u = {p, q, r, s, t} with u|r = {{p, q, r}, {s}, {t}}, x = {p, q}  u, then r(x) = {u,, {p, q, r}}. if  = {t}, then r(x) = {u,, {t}, {p, q, r}, {p, q, r, t}} and mic sp-o(u, x) = {u, , {s, t}, {p, q, r}, {p, q, r, s}}. (iii) if a = {s} and b = {q, r, t}, then mic sp-int ({s}) =  = mic sp-int ({q, r, t}) but a  b. (iv) let a = {p, q} and b = {r, s}, then mic sp-int ({p, q})  mic sp-int ({r, s}) =  =. but mic sp-int ({p, q}  {r, s}) = mic sp-int ({p, q, r, s}) = {p, q, r, s} which implies that mic sp-int (a  b)  mic sp-int (a)  mic sp-int (b). (v) consider a = {p, q, r, s} and b = {p, s, t}, then mic sp-int ({p, q, r, s})  mic sp-int ({p, s, t}) = {p, q, r, s}  {s, t} = {s}. but mic sp-int ({p, s, t}  {p, q, r, s}) = mic sp-int ({p, s}) =. therefore, mic sp-int (a)  mic sp-int(b)  mic spint(ab). theorem 4.7. for any two subsets a and b of a micro topological space (u, r(x), r(x)). the following properties are true. (i) mic sp-cl (mic sp-cl (a)) = mic sp-cl (a). (ii) mic sp-cl (a) = u – mic sp-int (u – a). (iii) if a  b, then mic sp-cl (a)  mic sp-cl (b) 94 m. maheswari, s. dhanalakshmi and n. durgadevi (iv) mic sp-cl(a)  mic sp-cl(b)  mic sp-cl(ab). (v) mic sp-cl(ab)  mic sp-cl(a)  mic sp-cl(b). proof: obvious, the converse of (iii), (iv) and (v) of theorem 4.7 need not be true as shown in the following example. example 4.8. consider u= {p, q, r, s, t} with u|r = {{p, q, r}, {s}, {t}}, x = {p, q}  u, then r(x) = {u,, {p, q, r}}. if  = {t}, then r(x) = {u, , {t}, {p, q, r}, {p, q, r, t}} and mic sp-o(u, x) = {u,, {s, t}, {p, q, r}, {p, q, r, s}}, mic sp-cl (u, x) = {u, , {p, q, r}, {s, t}, {t}}. (iii) let a = {p, t} and b = {q, s, t}, then mic sp-cl ({p, t}) = u and mic sp-cl ({q, s, t}) = u but a  b. (iv) let a = {p, q, r} and b = {t}, then mic sp-cl ({p, q, r})  mic sp-cl ({t}) = {p, q, r}  {t} = {p, q, r, t}. but mic sp-cl ({p, q, r}  {t}) = mic sp-cl ({p, q, r, t}) = u. therefore mic sp-cl (ab)  mic sp-cl (a)  mic sp-cl (b). (v)in general, for any closure operator cl (f)  cl (e) = cl (f  e) and for most of the closure operators cl (f)  cl (e)  cl (fe). in the case of mic sp-closure operator, the equality sign need not hold for both the cases and it is justified by the following example. this obviously leads to the conclusion that mic sp-closure operator is not a kuratowski’s operator. for, let a = {r} and b = {p, t} then mic sp-cl ({r}) mic sp-cl ({p, t}) = {p, q, r}  u = {p, q, r}. but mic sp-cl ({r} {p, t}) = mic spcl()= . hence mic sp-cl(a)  mic sp-cl(b)  mic sp-cl(ab). 5.conclusion in this paper mic sp -open sets and mic sp -closed sets are defined and some of their properties are discussed. this shall be extended in the future research with some applications. acknowledgement it is our pleasant duty to thank referees for their useful suggestions which helped us to improve our manuscript. references [1] chandrasekar. s and swathi. g: micro -open sets in micro topological spaces, international journal of research in advent technology, vol.6, no. 10, october 2018. [2] hariwan. z. ibrahim: micro -open sets in micro topology, gen. lett., 8(1) (2020),8-15. 95 micro sp-open sets in micro topological spaces [3] lellis thivagar. m and richard. c: on nano forms of weakly open sets, international journal of mathematics and statistics invention, 1/1 (2013), 31-37. [4] pawlak. z: rough sets., international journal of information and computer sciences, 11(1982), 341-356. [5] sakkraiveeranan chandrasekar: on micro topological spaces, journal of new theory 26(2019), 23-31. [6] saravanakumar. d, sathiyanandham. t and shalini. v. c: nsp-open sets and nspclosed sets in nano topological spaces, international journal of pure and applied mathematics, volume 113, no.12, 2017, 98-106. [7] shareef. a.h: sp-open sets, sp-continuity and sp-continuity and sp-compactness in topological spaces, m.sc. thesis, college of science, sulaimani univ., 2007. [8] quays hatem imran: on nano semi alpha open sets, journal of science and arts, year 17, no.2(39), pp 235-244, 2017. 96 ratio mathematica volume 44, 2022 the detour monophonic convexity number of a graph m. sivabalan* s. sundar raj† v. nagarajan‡ abstract a set 𝑆 is detour monophonic convexif𝐽𝑑𝑚 [𝑆} = 𝑆. the detour monophonic convexity number is denoted by 𝐶𝑑𝑚(𝐺), is the cardinality of a maximum proper detour monophonic convex subset of 𝑉.some general properties satisfied by this concept are studied. the detour monophonic convexity number of certain classes of graphs are determined. it is shown that for every pair of integers 𝑎 and 𝑏 with 3 ≤ 𝑎 < 𝑏, there exists a connected graph 𝐺 such that 𝐶𝑚(𝐺) = 𝑎 and 𝐶𝑑𝑚(𝐺) = 2(𝑏 + 1), where𝐶𝑚(𝐺) is the monophonic convexity number of 𝐺. keywords: convex, detour, chord, detour monophonic path, monophonic convexity number, detour monophonic, convexity number. ams subject classification: 05c12,05c38§. *register number.12567, research scholar, department of mathematics, s.t. hindu college, (nagercoil – 629002, india); e-mail: sivabalanvkc@gmail.com †department of mathematics, vivekananda college, (agasteeswaram – 629701, india); e-mail: sundarrajvc@gmail.com ‡department of mathematics, s.t. hindu college, (nagercoil – 629002, india); e-mail: sthcrajan@gmail.com affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamil nadu, india § received on june 12 th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.918. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 302 mailto:sivabalanvkc@gmail.com mailto:sthcrajan@gmail.com m. sivabalan, s. sundar raj and v. nagarajan, 1. introduction by a graph 𝐺 = (𝑉, 𝐸), we mean a finite undirected connected graph without loops or multiple edges. the order and size of 𝐺 are denoted by 𝑝 and 𝑞 respectively. for basic graph theoretic terminology, we refer to [1]. a vertex v is adjacent to another vertex 𝑢 if and only if there exists an edge 𝑒 = 𝑢𝑣 ∈ 𝐸(𝐺). if 𝑢𝑣 ∈ 𝐸(𝐺), we say that 𝑢 is a neighbor of 𝑣 and denote by 𝑁𝐺 (𝑣), the set of neighbors of 𝑣. a vertex 𝑣 is said to be universal vertex if deg𝐺 (𝑣) = 𝑝 − 1. a vertex 𝑣 is called an extreme vertex if the subgraph induced by 𝑣 iscomplete. the length of a path is the number of its edges. let u and v be vertices of a connected graph g. a shortest u-v path is also called a u-vgeodesic. the (shortest path) distance is defined as the length of a 𝑢-𝑣 geodesic in 𝐺 and is denoted by𝑑𝐺 (𝑢, 𝑣) or 𝑑(𝑢, 𝑣) for short if the graph is clear from the context. for a set 𝑆 of vertices, let 𝐼[𝑆] = ⋃ 𝐼[𝑥, 𝑦].𝑥,𝑦∈𝑆 a set 𝑆 ⊂ 𝑉 is called a convex set of 𝐺 if 𝐼[𝑆] = 𝑆. these concepts were studied in [1, 3] a chord of a path p is an edge which connects two non-adjacent vertices of p. a uv path is called a monophonic path if it is a chordless path. for two vertices u and v, the closed interval j[u, v] consists of all the vertices lying in a u − v monophonic path including the vertices u and v. if u and v are adjacent, then j[u, v] = {u, v}. for a set m of vertices, let j[m] = ∪u,v∈m j[u, v]. then certainly m ⊆ j[m]. a set m ⊆ v(g) is called a monophonic set of 𝐺 if 𝐽[𝑀] = 𝑉. the monophonic number 𝑚(𝐺) of 𝐺 is the minimum order of its monophonic sets and any monophonic set of order 𝑚(𝐺) is called a 𝑚-set of 𝐺. a set 𝑀 ⊆ 𝑉(𝐺) is called a monophonic convex set of g if 𝐽(𝑀) = 𝑀. the monophonic convexity number 𝐶𝑚(𝐺) of g is the cardinality of a maximum proper monophonic convex subset of v. these concepts were studied in [5-10]. the detour distance 𝐷(𝑢, 𝑣) between two vertices 𝑢 and 𝑣 in a connected graph 𝐺 from 𝑢 to 𝑣 is defined as the length of a longest 𝑢-𝑣 path in 𝐺. an 𝑢-𝑣 path of length 𝐷(𝑢, 𝑣) is called an 𝑢-𝑣 detour. the detour monophonic distance 𝑑𝑚(u,v) between two vertices u and v is the length of a longest u-v monophonic path in g, any monophonic path of length dm(u,v) is called u-v detour monophonic path. for two vertices 𝑢, 𝑣 ∈ 𝑉, let 𝐽𝑑𝑚[𝑢, 𝑣] denotes the set of all vertices that lies in 𝑢-𝑣detour monophonic path including 𝑢 and 𝑣, and 𝐽𝑑𝑚(𝑢, 𝑣)denotes the set of all internal vertices that lies in 𝑢 − 𝑣detour monophonic path . for 𝑀 ⊆ 𝑉, let 𝐽𝑑𝑚 [𝑀] = ⋃ 𝐽𝑑𝑚[𝑢, 𝑣]𝑢,𝑣∈𝑀 .a set 𝑀 ⊆ 𝑉 is a detour monophonic set if 𝐽𝑑𝑚[𝑀] = 𝑉.the minimum cardinality of a detour monophonic set of g is the detour monophonic number of g and is denoted by 𝑑𝑚(𝐺). the detour monophonic set of cardinality 𝑑𝑚(𝐺) is called dm-set. these concepts were studied in [2, 4, 11]. 2.the detour monophonic convexity number of a graph definition 2.1. a set 𝑆 is detour monophonic convex if𝐽𝑑𝑚[𝑆} = 𝑆. clearly 𝑆 = {𝑣} or 𝑆 = 𝑉 then 𝑆 is detour monophonic convex. the detour monophonic convexity number 303 the detour monophonic convexity number of a graph is denoted by 𝐶𝑑𝑚(𝐺), is the cardinality of a maximum proper detour mono-phonic convex subset of 𝑉. example 2.2. for the graph 𝐺in figure 2.1, 𝑀1 = {𝑣1, 𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣6, 𝑣7, 𝑣8}is a 𝐶𝑑𝑚set of 𝐺 so that 𝐶𝑑𝑚(𝐺) = 8. also 𝑀2 = {𝑣1, 𝑣2}is a 𝐶𝑚-set of 𝐺 so that 𝐶𝑚(𝐺) = 2. observation 2.3. let 𝐺 be a connected graph of order 𝑝 ≥ 3. then 2 ≤ 𝐶𝑑𝑚(𝐺) ≤ 𝑝 − 1. theorem 2.4. let 𝐺 be a connected graph of order 𝑝 and 𝐺 contains an extreme vertex. then 𝐶𝑑𝑚(𝐺) = 𝑝 − 1. proof. let 𝐺 contain an extreme vertex, say 𝑣. then 𝑆 = 𝑉(𝐺) − {𝑣} is a 𝑑𝑚-convex set of 𝐺 so that 𝐶𝑑𝑚(𝐺) = 𝑝 − 1. theorem 2.5. let 𝐺 be a connected graph of order 𝑝 ≥ 3. then2 ≤ 𝜔(𝐺) ≤ 𝐶𝑑𝑚(𝐺) ≤ 𝑝 − 1, where 𝜔(𝐺) is the clique number of 𝐺. proof. since 𝐺 is a connected graph of order 𝑝 ≥ 3, 𝜔(𝐺) ≥ 2. let 𝐻 be a subgraph of 𝐺 such that 〈𝑉(𝐻)〉 is a maximal complete subgraph of 𝐺 so that 𝐶𝑑𝑚(𝐺) ≥ |𝑣(𝐻)| = 𝜔(𝐺). let 𝑆 be a 𝑑𝑚-convex set of 𝐺. then 𝑆 is a convex set of 𝐺 so that 𝐶𝑑𝑚 (𝐺) ≤ 𝐶(𝐺). since every convex set of 𝐺 is a proper subset of 𝐺, 𝐶(𝐺) ≤ 𝑝 − 1. therefore 2 ≤ 𝜔(𝐺) ≤ 𝐶𝑑𝑚 (𝐺) ≤ 𝑝 − 1.. corollary 2.6. (i) for the complete graph 𝐺 = 𝐾𝑝 (𝑝 ≥ 3), 𝐶𝑑𝑚(𝐺) = 𝑝 − 1. (ii) for a trivial tree g of order 𝑝 ≥ 3, 𝐶𝑑𝑚(𝐺) = 𝑝 − 1. (iii) for the fan graph 𝐺 = 𝐾1 + 𝑃𝑝−1 (𝑝 ≥ 4), 𝐶𝑑𝑚(𝐺) = 𝑝 − 1. theorem 2.7. for the cycle 𝐺 = 𝐶𝑝, (𝑝 ≥ 3), 𝐶𝑑𝑚(𝐺) = 2. 304 m. sivabalan, s. sundar raj and v. nagarajan, proof. let𝑆 = {𝑥, 𝑦}be a set of two adjacent vertices of 𝐺. then 𝐽𝑑𝑚[𝑆] = 𝑆, it follows that s is a 𝑑𝑚-convex set of 𝐺 so that 𝐶𝑑𝑚 (𝐺) ≥ 2. we prove that 𝐶𝑑𝑚(𝐺) = 2. suppose that 𝐶𝑑𝑚(𝐺) ≥ 3. then there exists a dm-convex set 𝑆1 such that |𝑆1| ≥ 3. hence it follows that 𝑆1 contains two independent vertices of 𝐺. then 𝐽𝑑𝑚[𝑆1] ≠ 𝑆1. therefore 𝐶𝑑𝑚(𝐺) = 2.∎ theorem 2.8. for the complete bipartite graph 𝐺 = 𝐾𝑚,𝑛, 𝐶𝑑𝑚(𝐺) = 2. proof: let (𝑉1, 𝑉2) be a partition of 𝐺. since 𝜔(𝐺) = 2, 𝐶𝑑𝑚(𝐺) ≥ 2. we prove that 𝐶𝑑𝑚(𝐺) = 2. suppose that 𝐶𝑑𝑚(𝐺) ≥ 3. then there exists two vertices 𝑥 and 𝑦 belong to the same partite 𝑉1 (or 𝑉2). since 𝑑(𝑥, 𝑦) = 2 in 𝐺, every vertex in 𝑉1 (or 𝑉2) lie on 𝑥 − 𝑦 detour monophonic. hence it follows that 𝐽𝑑𝑚 [𝑆] ≠ 𝑆. therefore 𝐶𝑑𝑚(𝐺) = 2. ∎ theorem 2.9. for the wheel graph g= 𝑊𝑝 = 𝐾1 + 𝐶𝑝−1(𝑝 ≥ 4), 𝐶𝑑𝑚(𝐺) = 3. proof. let𝑉(𝐾1) = 𝑥and𝑉(𝐶𝑝−1) = {𝑣1, 𝑣2, … 𝑣𝑝−1}. then 𝑆 = {𝑥, 𝑣1, 𝑣2} is a det our monophonic convex set of 𝐺 so that 𝐶𝑑𝑚(𝐺) ≥ 3.we prove that 𝐶𝑑𝑚(𝐺) = 3. suppose that 𝐶𝑑𝑚(𝐺) ≥ 4. then there exists 𝑑𝑚-convex set 𝑆1 such that |𝑆1| ≥ 4. hence it follows that 𝑆1 contains two independent vertices of 𝐺. then 𝐽𝑑𝑚[𝑆1] ≠ 𝑆1. therefore𝐶𝑑𝑚(𝐺) = 3. ∎ theorem 2.10. for any two positive integers such that 2 ≤ 𝑎 ≤ 𝑏, there exists a connected graph 𝐺 such that 𝜔(𝐺) = 𝑎 and 𝐶𝑑𝑚(𝐺) = 𝑏. proof. for 𝑎 = 𝑏,let 𝐺 = 𝐾𝑎+1 − {𝑒}. then 𝜔(𝐺) = 𝐶𝑑𝑚(𝐺) = 𝑎.for 𝑎 < 𝑏, let 𝐾𝑎 be the complete graph with vertices 𝑣1,𝑣2, … , 𝑣𝑎 . let 𝑃: 𝑢1,𝑢2, … , 𝑢𝑏−𝑎 , 𝑢𝑏−𝑎+1,… , 𝑢𝑐 where 𝑐 > 𝑏 − 𝑎 a path on 𝑐 vertices. let 𝐺 be the graph obtained from 𝐾𝑎 and 𝑃 by joining 𝑢1 with 𝑣𝑎−1and 𝑣𝑎 each𝑢𝑖 (2 ≤ 𝑖 ≤ 𝑏 − 𝑎) with 𝑣𝑎−1 and 𝑢𝑐 with 𝑣𝑎−1. the graph 𝐺 is shown in figure 2.2. first, we prove that 𝜔(𝐺) = 𝑎. let 𝑆 = {𝑣1,𝑣2, … , 𝑣𝑎 }. it is clear that 𝑆 is a maximal complete subgraph of 𝐺 such that 𝜔(𝐺) = 𝑎. next, we prove that 𝐶𝑑𝑚(𝐺) = 𝑏. let 𝑊 = {𝑣1,𝑣2, … , 𝑣𝑎, 𝑢1,𝑢2, … , 𝑢𝑏−𝑎}. it is clear that 𝑊 is a 𝑑𝑚-convex set of 𝐺 so that𝐶𝑑𝑚 (𝐺) ≥ 𝐺 . we prove that 𝐶𝑑𝑚(𝐺) = 𝑏. suppose that 𝐶𝑑𝑚(𝐺) > 𝑏. let 𝑆1 be a dm-convex set with |𝑆1| ≥ 𝑏 + 1.then there exists a vertex𝑢𝑖 (𝑏 − 𝑎 + 1 ≤ 𝑖 ≤ 𝑐)such that 𝑢𝑖 ∈ 𝑆1. then 𝐽𝑑𝑚[𝑆1] ≠ 𝑆1.therefore𝐶𝑑𝑚(𝐺) = 𝑏. ∎ 305 the detour monophonic convexity number of a graph theorem 2.11. for every pair of integers 𝑎 and 𝑏 with 3 ≤ 𝑎 < 𝑏, there exists a connected graph 𝐺 such that 𝐶𝑚(𝐺) = 𝑎 and 𝐶𝑑𝑚(𝐺) = 2(𝑏 + 1). proof. let 𝑉(�̅�2) = {𝑥, 𝑦}. let 𝑃𝑖 : 𝑢𝑖,𝑣𝑖 (1 ≤ 𝑖 ≤ 𝑏)be a copy of path of order two. let 𝐺 be the graph obtained from �̅�2, 𝑃𝑖 (1 ≤ 𝑖 ≤ 𝑏) and 𝐾𝑎−1 by joining 𝑥 with each 𝑢𝑖 (1 ≤ 𝑖 ≤ 𝑏) and y with each 𝑣𝑖 (1 ≤ 𝑖 ≤ 𝑏)and 𝑥 and 𝑦 with each vertex of 𝐾𝑎. the graph 𝐺 is shown in figure 2.3. first, we prove that 𝐶𝑚(𝐺) = 𝑎. let 𝑀 = 𝑉(𝐾𝑎) ∪ {𝑥}. then 𝑀 is a monophonic convex set of 𝐺 and so 𝐶𝑚(𝐺) ≥ 𝑎. we prove that 𝐶𝑚(𝐺) = 𝑎. suppose that 𝐶𝑚(𝐺) ≥ 𝑎 + 1. let 𝑀1 be 𝑚-convex set with |𝑆| ≥ 𝑎 + 1. then there exists at least one vertex, say 𝑥 such that 𝑥 ∈ 𝑀1 and 𝑥 ∉ 𝑀. hence it follows that 𝑥 = 𝑢𝑖 or 𝑣𝑖 or 𝑦 for some 𝑖 (1 ≤ 𝑖 ≤ 𝑏). then 𝐽𝑚[𝑀1] ≠ 𝑀1, which is a contradiction. therefore 𝐶𝑚(𝐺) = 𝑎. next we prove that 𝐶𝑑𝑚(𝐺) = 2(𝑏 + 1). let 𝑆 = 𝑉(𝐺) − 𝑉(𝐾𝑎). then 𝑆 is a detour monophonic convex set of 𝐺 and so 𝐶𝑑𝑚(𝐺) ≥ 2(𝑏 + 1). we prove that 𝐶𝑑𝑚(𝐺) = 2(𝑏 + 1). on the contrary 𝐶𝑑𝑚(𝐺) > 2(𝑏 + 1). let 𝑆1 be a dm-convex set with |𝑆1| ≥ 2(𝑏 + 1) + 1. then there exists a vertex 𝑥 ∈ 𝑆1 such that 𝑥 ∉ 𝑆. hence it follows that 𝑥 ∈ 𝐾𝑎. then 𝐽𝑑𝑚[𝑆1] ≠ 𝑆1. therefore 𝐶𝑑𝑚(𝐺) = 2(𝑏 + 1).∎ 306 m. sivabalan, s. sundar raj and v. nagarajan, 3. conclusions in this paper, we investigated the detour monophonic convexity number of some standard graphs. also, we proved for every pair of integers 𝑎 and 𝑏 with 3 ≤ 𝑎 < 𝑏, there exists a connected graph 𝐺 such that 𝐶𝑚(𝐺) = 𝑎 and 𝐶𝑑𝑚(𝐺) = 2(𝑏 + 1). acknowledgements the author would like to express her gratitude to the referees for their valuable comments and suggestions. references [1] f. buckley and f. harary, distance in graphs, addition-wesley, redwood city, ca, 1990. 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[11] p. titus, k. ganesamoorthy and p. balakrishnan, the detour monophonic number of a graph, ars combinatoria, 84, 179-188,2013. 308 veblen and bussey [ ], [ ] have defined a finite projective geometry, which is said to be a geometry of a k-dimensional space, in the following way ratio mathematica, 21, 2011, pp. 43-58 43 on geometrical hyperstructures of finite order achilles dramalidis school of sciences of education, democritus university of thrace, 681 00 alexandroupolis, greece adramali@psed.duth.gr abstract it is known that a concrete representation of a finite k-dimensional projective geometry can be given by means of marks of a galois field gf [p n ], denoted by pg(k, p n ). in this geometry, we define hyperoperations, which create hyperstructures of finite order and we present results, propositions and examples on this topic. additionally, we connect these hyperstructures to join spaces. ams classification : 20n20, 16y99, 51a45 keywords : hypergroups, hv-groups, hv-rings 1. introduction the algebraic hyperstructures, which constitute a generalization of the ordinary algebraic structures, were introduced by marty in 1934 [5]. since then, many researchers worked on hyperstructures. the results of this work, as well as, applications of the hyperstructures theory can be found in the books [2] and [3]. vougiouklis in 1991 introduced a larger class than the known hyperstructures, so called hvstructures [8] and all about them can be found in his book [9]. let us give some basic definitions, appearing in [3], [9]: mailto:adramali@psed.duth.gr 44 let h be a set, p(h) the family of nonempty subsets of h and () a hyperoperation in h, that is  : hh  p(h) if (x,y)hh, its image under () is denoted by xy or xy. if a , b  h then ab or ab is given by ab =  {xy / xa , yb}. xa is used for {x}a and ax for a{x}. generally, the singleton {x} is identified with its member x. the hyperoperation () is called associative in h if (xy)z = x(yz) for all x,y,zh the hyperoperation () is called commutative in h if xy = yx for all x,yh a hypergroupoid (h,) that satisfies reproducibility, xh = hx = h for all xh, and associativity, is called hypergroup. a join operation () [6] in a set j is a mapping of jj into the family of subsets of j. a join space is defined as a system (j,), where () is a join operation in the arbitrary set j, which satisfies the postulates: i) ab≠ ii) ab = ba iii) (ab)c=a(bc) iv) a/b  c/d≠  ad  bc≠ v) a/b = {xj / abx}≠ . the hv-structures are hyperstructures satisfying the weak axioms, where the non-empty intersection replaces the equality. let h≠ be a set equipped with the hyperoperations (+), (), then the weak associativity in () is given by the relation (xy) z  x(yz)  , x,y,zh. the (), is called weak commutative if xy  yx  , x,yh. the hyperstructure (h,) is called hv-semigroup if () is weak associative and it is called hv-quasigroup if the reproduction axiom is valid, i.e. xh=hx=h, xh. the hyperstructure (h,) is called hv-group if it is an hv-quasigroup and an hv-semigroup. it is called hv-commutative group if it is an hvgroup and the weak commutativity is valid. 45 the weak distributivity of () with respect to (+) is given for all x,y,zh, by x(y+z)  (xy+xz)   , (x+y)z  (xz+yz)  . using these axioms, the hv-ring, which is the largest class of algebraic systems that satisfy ring-like axioms, is defined to be the triple (h,+,), where in both (+) and () the weak associativity is valid, the weak distributivity is also valid and (+) is reproductive, i.e x+h = h+x = h, xh. an hv-ring (r,+,) is called dual hv-ring if the hyperstructure (r,,+) is also an hv-ring [4]. let (h,) be a hypergroup or an hv-group. the β* relation is defined as the smallest equivalence relation, one can say also congruence, such that, the quotient h/β* is a group. the β* is called fundamental equivalence relation. 2. representation of the geometry of a k-dimensional space by means of galois fields veblen and bussey [7] have defined a finite projective geometry, which is said to be a geometry of a k-dimensional space, in the following way. it consists of a set of elements, called points for suggestiveness, which are subjected to the following five conditions or postulates: i. the set contains a finite number of points. it contains one or more subsets called lines, each of which contains at least three points. ii. if a and b are distinct points, there is one and only one line that contains both a and b. iii. if a, b, c are noncollinear points and if a line l contains a point d of the line ab and a point e of the line bc but does not contain a or b or c, then the line l contains a point f of the line ca. iv. if m is an integer less than k, not all the points considered are in the same m-space. 46 v. if (iv) is satisfied, there exists in the set of points considered no (k+1)-space. furthermore, a point is called 0-space, a line is called 1-space and a plane is called 2-space. by means of marks of a galois field, we shall now give a concrete representation of a finite k-dimensional projective geometry. we denote a point of the geometry by the ordered set of coordinates (μ0, μ1, μ2,…., μk), where μ0, μ1, μ2,…., μk are marks of the gf[p n ], at least one of which is different from zero. it is understood that the foregoing symbol (μ0, μ1, μ2,…., μk) denotes the same point as the symbol (μμ0, μμ1, μμ2,…., μμk), where μ is one of the p n -1 nonzero marks of the field. the ordered set of marks μ0, μ1, μ2,…., μk may be chosen in (p n ) κ+1 ways, but since the symbol (0, 0, 0,….,0) is excepted, then it may be chosen in (p n ) κ+1 -1 ways. so, there exists (p n ) κ+1 -1 points. in this totality, each point is represented in p n -1 ways (there are p n -1 nonzero marks in the field) and thus, it follows that the number of points defined is knn n kn pp p p     .....1 1 1)( 1 this representation of the finite κ-dimensional projective geometry by means of the marks of the gf[p n ] constitute the projective geometry pg(κ, p n ) [1]. now, for the line containing the two distinct points (μ0, μ1, μ2,…., μk) and (ν0, ν1, ν2,…., νk) we consider the set of points : ( μμ0 + νν0 , μμ1 + νν1 , μμ2 + νν2 , ……, μμk + ννk ) where μ and ν run independently over the marks of the gf[p n ], subjected to the condition that μ and ν shall not be simultaneously zero. then the number of possible combinations of μ and ν is (p n ) 2 – 1 and for each of these the corresponding symbol denotes a point, since not all the k+1 coordinates are zero. but the same point is 47 represented p n -1 times, due to the factor of proportionality involved in the definition of points. therefore, a line so defined contains 1 1 1)( 2    n n n p p p points. it is obvious that any two points on the line may be used in this way to define the same line. the five postulates given above for the k-dimensional space are satisfied by the concrete elements thus introduced [1]. 3. on a hypergroup of finite order let us denote by v the set of the elements of the pg(κ, p n ) and for x,yv let us denote by lxy the line which is defined by the points x and y. by lxis denoted the line which is defined by the point x and any other point of v. we define the hyperoperation (∙) on v, as follows : definition 1. for every x,yv , ∙ : vv  p(v) , such that x∙y =      yxifl yxifx xy obviously, the above hyperoperation is a commutative one, since x∙y = lxy = lyx = y∙x for every x,yv and x ≠ y. one can compare the above defined hyperoperation with the join operation [6], when euclidean geometry is converted into join spaces by defining ab with a≠b, to be the open segment, whose endpoints are a and b. moreover, aa is defined to be a. proposition 2. for every noncollinear x,y,zv, ∣x∙(y∙z)∣=∣(x∙y)∙z∣ = p 2n +p n +1. 48 proof. all the lines defined in v are having one point in common, at most. first, let us calculate the points of the set x∙(y∙z). for y ≠ z, the set y∙z = lyz consists of p n +1 points, including y and z. on the other hand, the point x (x ≠ y,z), with each of the p n +1 points of the line lyz, creates p n +1 lines of the type lxwhich are having the point x in common. this means that the p n +1 lines of the type lx are having no other point in common. so, each line lx is having p n different points from the others. then it follows that the set x∙(y∙z) consists of (p n +1)∙p n +1= p 2n +p n +1 different points. similarly, it arises that ∣(x∙y)∙z∣ = p 2n +p n +1.■ proposition 3. the hyperstructure (v, ∙) is a hypergroup. proof. easily follows, that for every xv x∙v =   )()( xvvx vvvv  v∙x = v now, for every x, y, zv if x = y = z then x∙(y∙z) = (x∙y)∙z = x if x = y ≠ z then x∙(y∙z) = (x∙y)∙z = lxz if x = z ≠ y then x∙(y∙z) = (x∙y)∙z = lxy if y = z ≠ x then x∙(y∙z) = (x∙y)∙z = lxy if x ≠ y ≠ z and x, y, z collinear, then x∙(y∙z) = (x∙y)∙z = lxy if x, y, z noncollinear then for the line lyz containing the two distinct points y(y0, y1,..,yk) and z(z0, z1,…, zk) we take the set of points : ( μy0 + νz0 , μy1 + νz1 ,……, μyk + νzk ) where μ and ν run independently over the marks of the gf[p n ] subjected to the condition that μ and ν shall not be simultaneously zero. let x(x0, x1,...,xk). then for the set x∙(y∙z) we take the set of points : (ρx0 + λ(μy0 + νz0) , ρx1 + λ(μy1 + νz1) ,……, ρxk + λ(μyk + νzk)) (1) 49 where ρ and λ run independently over the marks of the gf[p n ] subjected to the condition that ρ and λ shall not be simultaneously zero. let w x∙(y∙z) , then the coordinates of the point w is of the form of (1). for some i0,1,…..,k we have ρxi+λ(μyi+νzi) = ρxi+λμyi+λνzi = ρxi+λμyi+νzi, where νgf[p n ] if ρ=0 (2) then ρ=λ0 for every λgf[p n ] and then ρxi + λμyi + νzi = λ0xi + λμyi + νzi = λ(0xi + μyi ) + νzi if ρ≠0 (3) then ρ=λμ for every λ, μgf[p n ]-{0} and then ρxi + λμyi + νzi = λμxi + λμyi + νzi = λ(μxi + μyi ) + νzi the coordinates of the points of the set (x∙y)∙z are of the form κ(τxi + τyi ) + κzi where τ, τ, κ, κ run independently over the marks of the gf[p n ] subjected to the condition that τ, τ and κ, κ shall not be simultaneously zero. due to the conditions (2) and (3) we get that w x∙(y∙z)  w (x∙y)∙z which means that x∙(y∙z)  (x∙y)∙z . in a similar way, it can be proven that for w(x∙y)∙z  wx∙(y∙z), which means that (x∙y)∙z  x∙(y∙z). so, x∙(y∙z) = (x∙y)∙z for every x,y,zv.■ remark 4. for the hypergroup (v,∙), since {x,y}  x∙y x,yv, the vβ* is a singleton. proposition 5. the hypergroup (v, ∙) is a join space. proof. since the hyperoperation (∙) is commutative, the hyperstructure (v, ∙) is a commutative hypergroup. 50 moreover, let a/b  c/d ≠  , a,b,c,dv . then, there exists wv such that wa/b which implies that a lbw and wc/d which implies that c ldw. since the lines of v are having one point in common at most, the lines lbw and ldw intersect at w. let the ordered set of coordinates of the points w, a, d be (w0, w1,...,wk) , (a0, a1,...,ak), (d0, d1,...,dk) respectively. then, the coordinates of the point b will be of the form ( λa0 + μw0 , λa1 + μw1 ,……, λak + μwk ), where λ,μ gf[p n ] and the coordinates of the point c will be of the form ( κd0 + ρw0 , κd1 + ρw1,… ……, κak + ρwk ), where κ,ρ gf[p n ]. since the points w, a, d do not belong to the line lbc , the marks λ,μ,κ,ρ of the gf[p n ] are not zero. now, lbc consists of the points of the form (νλa0+νμw0+ τκd0+τρw0 , νλa1+νμw1+ τκd1+τρw1 ,……, νλak+νμwk+ τκdk+τρwk ), where ν and τ run independently over the marks of the gf[p n ] and they are not simultaneously zero. it is known that for the nonzero marks μ and ρ, there exist nonzero marks ν and τ such that: νμ+τρ = 0 . then, we get νμw0 + τρw0 = (νμ+τρ)w0 = 0, νμw1 + τρw1 = (νμ+τρ)w1 = 0 ,…….., νμwk + τρwk = (νμ+τρ)wk = 0 . in that case, the point (νλa0+τκd0 , νλa1+τκd1 ,……., νλak+τκdk) of the line lbc is additionally a point of the line lad. so, the lines lbc, lad intersect and then: a∙d  b∙c ≠  for all a,b,c,dv . ■ 4. on a hv-group of finite order now, we define a new hyperoperation (◦) on v as follows : definition 6. for every x,yv , ◦ : vv  p(v) , such that x◦y =      yxifxl yxifx xy }{ 51 every line of the set v contains pn+1 points. the hyperoperation (◦) is weak commutative, since the lines lxy-{x} and lyx-{y} are having p n +1-2 = p n -1 points in common, so (x◦y)  (y◦x) ≠  for every x,yv . proposition 7. for every noncolliner x,y,zv, ∣x◦(y◦z)∣= p 2n > ∣(x◦y)◦z∣ = p 2n -p n +1 proof. since the line y◦z does not contain the point y, we get that ∣y◦z∣ = p n . the point x creates p n points with each of the p n points of the line y◦z (the point x is not participating according to the hyperoperation (◦)). so, ∣x◦(y◦z)∣= p n ‧p n = p 2n . on the other hand, the line x◦y consists of p n points. each of these points creates p n points each time together with the point z, but since the point z appears p n times, we get ∣(x◦y)◦z∣ = p n ‧p n p n +1 = p 2n -p n +1. since p is prime and nin, easily follows that ∣x◦(y◦z)∣ > ∣(x◦y)◦z∣.■ proposition 8. the hyperstructure (v, ◦) is an hv-group. proof. indeed, for every xv x◦v = )( vx vv   = (x◦x)         )( }{ vx xvv  = x         }){( }{ xl xv xvv  = v since every line lxv always contains the point vv. on the other hand, for every xv v◦x = )( xv vv   = (x◦x)         )( }{ xv xvv  = x         }){( }{ vl vx xvv  = v indeed, having the fact that every line of v contains at least 3 points, for every line lvx–{v}=v◦x there exists at least one point vlvx –{v}, that v lvx– {v} = v◦x. so, x◦v = v◦x = v for every xv. 52 the hyperoperation (◦) is weak associative, since for every x,y,zv x◦(y◦z)  x◦z  z and (x◦y)◦z  y◦z  z but, we shall go further on, proving that the inclusion on the right parenthesis is valid, i.e (x◦y)◦z  x◦(y◦z). from the proposition 7 we get that ∣x◦(y◦z)∣= p 2n , since the p n +1 points of the line lxy are not contained into the set x◦(y◦z). similarly, the set (x◦y)◦z does not contain the points of the line lxy, since (x◦y)◦z = (lxy – {x})◦z = (x1◦z ) (x2◦z ) …… ( 1 n p x ◦z ) (y◦z ), where x1, x2,……., 1 n p x lxy. also, the set (x◦y)◦z does not contain the points of the line lxz, since x◦y=lxy– {x}. as the lines lxy and lxz intersect at the point x, they don’t have any other points in common. that means that the p 2n -p n +1 points of the set (x◦y)◦z (proposition 7), are also points of the set x◦(y◦z). so, we proved that (x◦y)◦z  x◦(y◦z), which, generally, means that (x◦y)◦z  x◦(y◦z) ≠  for every x,y,zv. ■ remark 9. for the hv-group (v,◦), since yx◦y x,yv, the vβ* is a singleton. since, yx◦y for every x,yv, we get the following proposition: proposition 10. every element of the hv-commutative group (v,◦) is simultaneously a right zero and a left unit element. example 11. by means of the marks of a galois field, we shall now give a concrete representation of a finite 2-dimensional projective geometry. we denote a point of the geometry by the ordered set of coordinates (μ0, μ1, μ2). the μ0, μ1, μ2 are marks of the gf[2 2 ] defined by means of the function x 2 +x+1. at least one of μ0, μ1, μ2 is different from 53 zero. let us denote the marks of the gf[2 2 ] by 0, 1, a, b, then we have the following tables : + 0 1 a b ∙ 0 1 a b 0 0 1 a b 0 0 0 0 0 1 1 0 b a 1 0 1 a b a a b 0 1 a 0 a b 1 b b a 1 0 b 0 b 1 a the ordered set of marks μ0, μ1, μ2 may be chosen by (2 2 ) 2+1 = 64 ways, but since the symbol (0, 0, 0) is excepted, it may be chosen by 64-1=63 ways. so, there exist 63 points. in this totality, each point is represented by 3 ways (3 sets of symbols, since there are 3 nonzero marks in the field). then the number of points defined is 63 ÷ 3 = 21. this representation of the finite 2-dimensional projective geometry by means of the marks of the gf[2 2 ], constitute the projective geometry pg(2, 2 2 ). the 21 points of pg(2, 2 2 ) = v will be denoted by letters in accordance with the following scheme : a(001) b(010) c(011) d(01a) e(01b) f(100) g(101) h(10a) i(10b) j(110) k(111) l(11a) m(11b) n(1a0) o(1a1) p(1aa) q(1ab) r(1b0) s(1b1) t(1ba) u(1bb) now, for the line containing the two distinct points a(001) and b(010) we take the set of points : ( μ0 + ν0 , μ0 + ν1 , μ1 + ν0 ) where μ and ν run independently over the marks of the gf[2 2 ] subjected to the condition that μ and ν shall not be simultaneously zero. then the number of possible combinations of the μ and ν is (2 2 ) 2 – 1 = 15. for each of these combinations, the corresponding symbol 54 denotes a point. but the same point is represented by 3 of these combinations of μ and ν, due to the factor of proportionality involved in the definition of points. so, we get the following scheme: a : (001) , (00a) , (00b) b : (010) , (0a0) , (0b0) c : (011) , (0aa) , (0bb) d : (01a) , (0b1) , (0ab) e : (01b) , (0ba) , (0a1) therefore, a line so defined, contains the 15 ÷ 3 = 5 points a,b,c,d,e. it is obvious that any two points on the line may be used in this way to define the same line. the 21 lines are those given in the following scheme and the letters in a given column denoting a line: a a a a a b b b b c c c c d d d d e e e e b f j n r f g h i f g h i f g h i f g h i c g k o s j k l m k j m l l m j k m l k j d h l p t n o p q p q n o q p o n o n q p e i m q u r s t u u t s r s r u t t u r s let us take the noncolliner points a, b, f, then a◦(b◦f) = a◦{f, j, n, r} = ={f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u}. (a◦b)◦f = {b, c, d, e}◦f = {f, j, n, r, k, p, u, l, q, s, m, o, t}. then, it follows that (a◦b)◦f  a◦(b◦f). also, for the hyperoperation (∙) we proved that ∣x∙(y∙z)∣= ∣(x∙y)∙z∣ = p 2n +p n +1 and since the set v = pg(2, 22) consists of 21 points, it follows that x∙(y∙z)= (x∙y)∙z = v for every x,y,zv. 5. on a dual hv-ring of finite order working on dual hv-rings (h,+,), one needs to prove not only the weak distributivity of () with respect to (+) but also the weak distributivity of (+) with respect to (). 55 since the set v is now equipped with the hyperoperations () and (◦) mentioned above, the next propositions 12 to 19 serve the above purpose. similarly, as in proposition 2, we can prove that: proposition 12. for every noncollinear x,y,zv ∣(x∙y)∙(x∙z)∣ = ∣(x∙z)∙(y∙z)∣= p 2n +p n +1 following a similar procedure, as in proposition 3 and according to the propositions 2 and 12, we get the next proposition: proposition 13. for every x,y,zv x∙(y∙z) = (x∙y)∙(x∙z) and (x∙y)∙z = (x∙z)∙(y∙z) proposition 14. for every noncolliner x,y,zv, ∣x∙(y◦z)∣= p 2n +1 and ∣(x∙y)◦(x∙z)∣= p 2n +p n +1 proof. first we consider the set x∙(y◦z). since the point y does not belong to the line y◦z, we get that ∣y◦z∣ = p n . also, for every wy◦z we get that ∣x∙w∣=p n +1. since the point x appears p n times in the set x∙(y◦z), we have ∣x∙(y◦z)∣=(p n +1) p n p n +1 = p 2n +1. consider now the set (x∙y)◦(x∙z). each of the lines x∙y and x∙z contains p n +1 points, having in common only the point x, since the points x,y,z are noncollinear. then, we get the following: i) x◦x = x, by definition. ii) due to the hyperoperation (◦), the point xx∙y together with the rest p n points of the line x∙z create the p n points of the line x∙z. iii) due to the hyperoperation (◦), the point yx∙y together with the p n +1 points of the line x∙z, create (p n +1)-2 = p n -1 points each time. indeed, the point y does not participate (by definition) and the point wx∙z (which appears due to the hyperoperation y◦w), already exists due to the hyperoperation x◦w of the case (ii). 56 iv) also, the point y(x∙y)◦(x∙z). indeed, since there exists wx∙y such that yw◦x (where xx∙z). so, from the above 4 cases we get that: ∣(x∙y)◦(x∙z)∣= 1+ p n +( p n -1)( p n +1)+1 = p 2n +p n +1.■ in a similar way, we get the following proposition: proposition 15. for every noncolliner x,y,zv ∣(x◦y)∙z∣= p 2n +1 and ∣(x∙z)◦(y∙z)∣= p 2n +p n +1 proposition 16. for every noncolliner x,y,zv, ∣x◦(y∙z)∣=∣(x◦y)∙(x◦z)∣= p 2n +p n proof. first, consider the set x◦(y∙z). the line y∙z consists of p n +1 points. the point x (due to the hyperoperation (◦)) together with the points of the line y∙z creates each time (p n +1) p n = p 2n +p n points, since x is not participating. consider now, the set (x◦y)∙(x◦z). each of the lines x◦y and x◦z contains p n different points, since the point x is not participating. then, we get the following: i) the point yx◦y (due to the hyperoperation (∙)), together with every wx◦z creates p n +1 points each time, but since the point y appears p n times we get that the number of points in this case is (p n +1) p n p n +1 = p 2n +1. ii) the point w(x◦y)-{y} (due to the hyperoperation (∙)), together with the point wx◦z creates each time the point w, since the point w already exists from case (i). the number of those w’s is p n -1. then, the set (x◦y)∙(x◦z) consists of (p 2n +1)+(p n -1) = p 2n +p n points. as we mentioned, x is the only point which is not participating.■ similarly, we get the following two propositions: proposition 17. for every noncolliner x,y,zv 57 ∣(x∙y)◦z∣= p 2n and ∣(x◦z)∙(y◦z)∣= p 2n +p n +1 proposition 18. for every noncollinear x,y,zv ∣(x◦y)◦(x◦z)∣ = p 2n and ∣(x◦z)◦(y◦z)∣= p 2n +p n +1 following a similar procedure as above and according to the proposition 18 we get the next proposition: proposition 19. for every x,y,zv x◦(y◦z) = (x◦y)◦(x◦z) and (x◦y)◦z  (x◦z)◦(y◦z) proposition 20. the hyperstructure (v,▫,▪), where ▫,▪{∙,◦}, is a dual hv-ring. proof. there are four hyperstructures: (v,∙,∙) , (v,◦,◦) , (v,∙,◦) , (v,◦,∙). the hyperoperations (∙), (◦) (by propositions 3 and 8 respectively) are satisfying the reproduction axiom. the hyperoperation (∙) (by proposition 3) is associative and the hyperoperation (◦) (by proposition 8) is weak associative. now, for the distributivity or the weak distributivity of (▪) with respect to (▫) we have the following cases: by proposition 13: x∙(y∙z) = (x∙y)∙(x∙z) and (x∙y)∙z = (x∙z)∙(y∙z) for every x,y,zv. by proposition 19: x◦(y◦z) = (x◦y)◦(x◦z) and (x◦y)◦z  (x◦z)◦(y◦z)≠  x,y,zv. following a similar procedure as for the distributivity of the above hyperoperations and taking into account the propositions 16, 17, 18, 19 we get that: x◦(y∙z)=(x◦y)∙(x◦z) for every x,y,zv. on the right side, (x∙y)◦z  (x◦z)∙(y◦z) is valid, which means that (x∙y)◦z  (x◦z)∙(y◦z)≠ for every x,y,zv. also, x∙(y◦z)  (x∙y)◦(x∙z), which means that 58 x∙(y◦z)  (x∙y)◦(x∙z) ≠ for every x,y,zv. finally, on the right hand side (x◦y)∙z  (x∙z)◦(y∙z), which means that (x◦y)∙z  (x∙z)◦(y∙z) ≠ for every x,y,zv. so, the hyperstructure (v,▫,▪), where ▫,▪{∙,◦}, is a dual hv-ring.■ references [1] carmichael r.: introduction to the theory of groups of finite order, dover publications, inc., new york. [2] corsini p. : prolegomena of hypergroup theory, second edition, aviani (1993). [3] corsini p. – leoreanu v. : applications of hyperstructures theory, kluwer academic publishers, boston / dordrecht / london. [4] dramalidis, a., dual hv-rings, rivista di mathematica pura ed applicata, vol. 17, 55-62 (1996). [5] marty f. : sur une generalization de la notion de group, 8 iem congres math. scandinaves, stockholm (1934), 45-49. [6] prenowitz w. – jantosciak j.: geometries and join spaces, journal mathematic, v. 257, 101-128 (1972). [7] veblen o. – bussey w.h.: finite projective geometries, trans. amer. math. soc., v. 7, 241-259 (1906). [8] vougiouklis t. : the fundamental relation in hyperrings. the general hyperfield, proc. of the 4 th aha, xanthi (1990), world scientific, 1991, 203 – 211. [9] vougiouklis t. : hyperstructures and their representations, monographs, hadronic press 1994, usa. microsoft word 2005_labirinti_ratiomath.doc 23 ratio mathematica nr. 16 (2005) la filosofia dei labirinti: dal mito all’intelligenza artificiale raffaele mascella dipartimento di scienze della comunicazione, università degli studi di teramo rmascella@unite.it 1. dal mito alla matematica ricreativa in una prima grossolana indicazione possiamo affermare che un labirinto è una struttura, per solito di vaste dimensioni, costruita con un ingresso, una uscita, una serie intricata di vie in modo tale che una volta entrati ci sia difficile trovare l’uscita. nel linguaggio comune è anche un sinonimo di rompicapo. il labirinto più famoso e leggendario appare nelle opere mitologiche come labirinto di cnosso, giunto a noi sul verso di monete cretesi di epoca minoica. di questo labirinto si disse che una struttura così ingannatrice così ingegnosamente concepita non fu mai vista al mondo allora ed in epoche successive.* labirinti architettonici di questo tipo non erano rari nel mondo antico. in alcune iscrizioni rupestri della val camonica, circa 1500 a.c., fra immancabili immagini rituali di dischi solari e segni geometrici, guerrieri e scene di danze, appare spesso la raffigurazione schematica e simbolica del labirinto. spesso rappresentato in forma circolare, il labirinto camuno coincide fin troppo con l'immagine simbolica del labirinto di cnosso. e la stessa immagine, unita spesso all'onnipresente spirale cosmica, simboleggiante l'eternità, si ritrova non solo nell'area mediterranea e microasiatica ma anche nel nordeuropa, nelle tracce lasciate dalle civiltà megalitiche della fascia atlantica e dalle popolazioni celtiche in irlanda e scandinavia meridionale. anche in egitto lo schema labirintico non è sconosciuto. erodoto parla di un labirinto, in parte da lui stesso visitato, con circa 3000 camere, sviluppato su due piani con edifici collegati ed un muro esterno che li racchiude. * si narra che esso venne fatto costruire dal re minosse nell'isola di creta per rinchiudervi il mostruoso minotauro, nato da una irreale unione tra la moglie del re e un toro. questo labirinto era un complesso intrico di strade contornate da alti muri, stanze con molte porte, cunicoli e gallerie. il mitico architetto ideatore fu un tale dedalo, che a costruzione ultimata venne fatto rinchiudere, dal re minosse, assieme al figlio icaro, perché non potessero rivelarne i segreti costruttivi. il geniale dedalo costruì delle ali, che applicò con la cera alle spalle swue e di suo figlio. entrambi allora uscirono volando, unico modo di uscire nell’immaginario collettivo di allora, ma il figlio icaro colpito da orgoglio volò troppo vicino al sole, la cera che teneva le ali si sciolse ed icaro precipitò miseramente verso il basso. minosse impose che ogni anno sette giovani e sette fanciulle di atene, fossero date in pasto al minotauro. fu l’eroe teseo che aiutato da un lungo filo, donatogli da una tale arianna, filo che aveva lasciato scorrere lungo il percorso, riuscì ad orientarsi e a muoversi nel labirinto, trovare il mostro e ucciderlo. 24 r. mascella nel cristianesimo i fedeli si ritrovano spesso a dover costruire e frequentare veri e propri labirinti sotterranei dal disegno intricato e dagli accessi oscuri e pericolosi: le catacombe, rifugio e nascondiglio oltre che luogo di culto e di sepoltura dei defunti. ma fin dai primi secoli d.c. si realizza una interessante fusione del pensiero classico con quello cristiano, ovvero, il trascorrere tortuoso e faticoso della vita, dalla nascita fino alla conclusione, che da una parte era semplicemente nel regno dei morti, dall’altra era nel paradiso celeste. la cultura cristiana, infatti, già nel iv secolo affianca gli eroi classici con il simbolismo cristiano; e così la chiesa, così come arianna, indica la strada per raggiungere la meta finale al centro della schema, ovvero la gerusalemme celeste, il regno dei cieli. molte cattedrali cristiane edificate in tempo medievale avevano così schemi labirintici incastonati nei loro pavimenti, in realtà nell'iconografia tradizionale semplici schemi unicursali, ovvero “falsi labirinti” (una sola strada tortuosamente aggrovigliata su se stessa ma del tutto priva di biforcazioni od incroci), successivamente quasi tutti distrutti da canonici scandalizzati, tra il xvii e il xviii secolo, perché sembra che le persone, invece di seguire le funzioni, iniziavano a camminarci sopra, seguendone le giravolte e cercandone la soluzione. e questo pareva inaccettabile. ma il loro significato simbolico, prima di essere smarrito, era comunque profondo, ad indicare il percorso dei fedeli, impegnati in un cammino terreno ascetico e mistico. per altri versi il labirinto ha rappresentato anche il lungo e tortuoso cammino iniziatico che l'eroe affronta per poter incontrare il mostro, cammino che può essere abbreviato e semplificato solo dal consiglio e dalla saggezza di chi è già iniziato. così come dedalo, artefice e creatore del labirinto nonché co-responsabile della nascita del minotauro, che suggerisce ad arianna il famoso stratagemma del filo. dunque il labirinto nella storia rappresenta e concentra in sé una serie di miti e simboli, da quelli profondamente radicati nella nostra coscienza a quelli ottenuti con rielaborazioni successive sia dal punto di vista architettonico che simbolico, avvenute in un ampio intervallo di tempo e di spazio. perciò si passa dal “viaggio iniziatico” dei popoli primitivi al “cammino della salvezza” dei cristiani, dalla “discesa degli inferi” alla “peregrinazione impedita” ed alla “ricerca della conoscenza”. e anche la letteratura ne ha tenuto conto. lo scrittore inglese joseph addison scrisse un’interessante opera su rosamunda dalla quale algernon charles swiburne trasse il poema drammatico “rosamunda” il cui tema centrale è un labirinto.† lo stesso shakespeare cita questi curiosi labirinti fra il selvatico e il verde che ornavano i prati davanti le chiese. e poi esempi più diretti, come quello di james joyce, la cui intera opera, non a caso, è un gioco labirintico con il lettore. o umberto eco che incarna in modo spettacolare il concetto labirintico della conoscenza, attraverso la biblioteca-labirinto de il nome della rosa, anche se questa immagine, in realtà, è presa a prestito dalla labirintica biblioteca di babele di borges. anche nell'arte barocca vi è interesse per il labirinto come ornamento o passatempo, anche se scevro di qualsiasi connotazione mistica o religiosa. i grandi palazzi patrizi vedono il sorgere di giardini ornati da siepi che formano percorsi labirintici, ad uso dei giochi di società dei loro nobili ed annoiati signori. famoso il labirinto tardo-rinascimentale † nel xii secolo il re d’inghilterra enrico ii fece costruire, in un parco a woodstock, il cosidetto rosamunda’s bower (rifugio di rosamunda), ai fini di nascondere la sua amante rosamunda alla moglie eleonora d’aquitania. il rifugio era al centro di un intricato labirinto. tuttavia la moglie eleonora ricorrendo alla tecnica del filo di arianna arrivò al centro del labirinto e costrinse la bella rivale a bere un potente veleno e quindi a tornare con il suo filo nei suoi regali appartamenti. la filosofia dei labirinti: dal mito all’intelligenza artificiale 25 creato nel 1690 per il palazzo di hampton court, residenza di guglielmo d’orange. nei primi anni del ‘900 nell’indiana (usa) precisamente ad harmony una setta di emigrati tedeschi edificò un labirinto di siepi che fu preso come simbolo della tortuosità del peccato e sulla difficoltà a trovare la retta via. il labirinto fu distrutto e poi riedificato nel 1942 su disegni forse non originali. il labirinto viene studiato dal punto di vista geometrico e matematico, perfino da leonardo. da allora in poi, e fino ad oggi, esso si insinua spesso inconsapevolmente nell'arte, nella musica, nella pittura in mille e mille modi che è difficile descriverli tutti. ma apparentemente oggi cosa ci rimane dei labirinti? ad una prima occhiata profana la loro conoscenza sembra legata esclusivamente al gioco, come capitolo della matematica ricreativa. ed infatti i “matematici” si sono occupati dei labirinti, in particolare di quelli classici, cioé senza incroci. con un po’ di formalismo, un (e,u)-labirinto può essere definito come un grafo nel quale sono dati un certo numero e di vertici detti entrate ed un certo numeri u di vertici detti uscite con le condizioni seguenti: 1. esistono nel grafo cammini che da ogni entrata conducano ad almeno una uscita, 2. il numero v dei vertici è molto grande rispetto ad e ed u, 3. il numero dei cammini entrata –uscita è molto piccolo rispetto a v, 4. il numero di archi (link) è elevato. limitandoci agli (1,1)-labirinti, detti semplicemente labirinti, ci si può chiedere: quanti ne esistono? per meglio dire: è possibile classificare tutti i labirinti? numerando in modo progressivo dall'esterno verso l'interno del labirinto le circonvoluzioni che lo compongono, ad ogni labirinto si può associare la sequenza numerica con cui vengono percorse le circonvoluzioni; chiamando “livello del labirinto” il numero massimo di circonvoluzioni che lo compongono (con il cerchio esterno indicato convenzionalmente dal numero zero), phillips ha classificato i labirinti fino al livello 22. e fino a questo livello, il loro numero risulta essere di 73.424.650. i matematici (e gli informatici) hanno cercato anche delle regole che permettessero di trovare in modo efficiente la via da percorrere. se di un labirinto si possiede la pianta si può procedere con un metodo tipicamente esaustivo, cioè percorrendo tutti i cammini, magari in maniera ordinata (tanto per semplificarsi la vita…) annerendo tutti i cammini che ad un certo punto si chiudono. alla fine si trova necessariamente il cammino che porta all’uscita. diverso è il problema dell’individuo che percorre un labirinto di cui non ha la pianta. potremmo pensare, con un piccolo sforzo di immaginazione, che sarebbe come trovarsi nel fiabesco mondo piatto di abbott, in cui le figure geometriche piane si muovono in condizioni visuali ridotte, ovvero senza l’uso della tridimensionalità. e qui, senza voler approfondire troppo la questione, qualunque tipo di ricerca di una via, un monumento, un edificio, è approssimabile, stavolta non in senso matematico, alla ricerca di una soluzione labirintica. una regola pragmatica, in questo caso, è quella della mano destra. se percorro un (1,1)-labirinto in cui tutti muri sono tra loro collegati (cioè è “semplicemente connesso”) tenendo la mano destra sempre a contatto del muro prima o troverò l'uscita. ma se si tratta di un labirinto “molteplicemente connesso” ritornerò all'entrata. e lo stesso problema nasce quando vi è invece un centro da raggiungere poiché eventuali cammini che girano attorno al centro (pensate ad un cerchio con un buco) portano al punto di partenza. la mancanza di tali cammini circolari chiusi corrisponde alla semplice connessione delle figure piane. problemi analoghi nascono nel caso di labirinti con un numero maggiore di entrate. dunque abbiamo a che fare con una questione matematico-informatica, modellizzabile come problema di visite in un grafo, in cui la topologia dello schema non è secondaria rispetto alla ricerca di soluzioni “semplici”. 26 r. mascella ma esiste un algoritmo che possa risolvere un labirinto? la risposta è positiva e un algoritmo può essere dedotto da alcune regole dovute ai matematici francesi g. tarry e m. trémaux, alla fine del xix secolo. esse sono riportate nel libro di edouard lucas, récréations mathématiques (vol. i, 1882), e sarà nostra cura preparare la traduzione integrale commentata di quest’opera. di questo algoritmo, conosciuto come algoritmo di tremaux, o meglio di una sua versione semplificata, si avvalse anche il padre della teoria dell'informazione c. shannon, e qui siamo infine giunti alla connessione con il nostro discorso che si diramerà nei prossimi paragrafi, ovvero alla connessione tra labirinti e calcolatori. all’ottava “cybernetics conference” del 1952, shannon presentò un piccolo robot semovente, una specie di topolino denominato “teseo”, in grado di imparare ad uscire da un labirinto sconosciuto, proprio come un topo di laboratorio. il labirinto era articolato su un quadrato 5x5, che poteva essere modificato disponendo diversamente i muretti divisori, e anche l’obiettivo da raggiungere poteva essere così cambiato a piacimento. nella versione semplificata, l'algoritmo eliminava dal percorso trovato i rami inutili ed i giri viziosi per potere successivamente ripercorrere lo stesso labirinto lungo un tragitto più breve (anche se non precisamente ottimale). tutto ciò, grazie ad una duplice strategia. da un lato, quella di tipo esplorativo, per muoversi nel labirinto, dall’altro quella relativa all’obiettivo da raggiungere. molte furono già le prime evoluzioni del topo “teseo-trémaux”. il robot solutore di jariosh deutsch di oxford utilizza scorciatoie e ripropone le strategie anche su un labirinto topologicamente equivalente ad uno dato e da lui conosciuto. questi furono i rudimenti iniziali di questi robot allo stato embrionale. nel 1951, quando era ancora studente, lo stesso m. minsky nella costruzione dello snarc (stochastic neural-analog reinforcement computer), la prima rete neurale artificiale del mondo, modellò il processo di apprendimento di cui un topo si avvale quando cerca di uscire da un labirinto. tutto ciò avveniva alle soglie di un nuovo movimento culturale prima che informaticoteorico inteso a studiare il comportamento intelligente degli uomini e degli animali e a riprodurlo sulle macchine. in altre parole, l'intelligenza artificiale. 2. i labirinti degli ipertesti da tempo sono comparsi nel mondo letterario i cosiddetti “ipertesti”, che sono di fatto nuovi oggetti testuali. essi sono caratterizzati dal fatto di essere registrati su di una memoria magnetica/ottica/elettronica invece che su carta, ovvero per dirla in altri termini, basati su dispositivi di memoria ad accesso casuale, che permettono di raggiungere direttamente ogni singola unità, e ciò diversamente da quanto accade nei dispositivi ad accesso seriale, nei quali si devono scorrere tutti i dati prima di giungere a quello desiderato. questa loro caratteristica ne consente modalità di lettura e principalmente d’uso ben diverse da quelle del tradizionale libro a stampa. si aprono così una serie di interrogativi che toccano da vicino i mutui rapporti tra i concetti di autore, opera, lettore, sequenzialità. per definire un ipertesto, o se vogliamo di ipermedia, accettando anche un compromesso attualizzante con la multimedialità, possiamo asserire che esso è un grafo i cui vertici o nodi sono oggetti di varia natura (file, immagini, filmati, musica, informazioni codificate) e i cui lati o archi sono i link tra i nodi colleganti le varie informazioni. forse vi sono molti ingressi, teoricamente anche tutti i vertici sono ingressi, e molte uscite, forse tutti i vertici la filosofia dei labirinti: dal mito all’intelligenza artificiale 27 ancora. i numeri non sono sempre quelli di pochi cammini e molti vertici, gli obiettivi appaiono pure differenti: nel procedere in un cammino vogliamo impadronirci di molti vertici-informazioni, ma le strutture matematiche sottese sono le medesime ed anche le metodologie per andare da un ingresso, quale che sia, ad una uscita, quale che sia. gli ipertesti più semplici sono, in prima analisi, files di testo registrati su memoria magnetica in cui le singole sotto-unità (che possono essere indifferentemente pagine, capitoli, paragrafi, frasi, brani, ecc.) non sono disposte, e quindi leggibili, secondo un ordine sequenziale (come le pagine, o i paragrafi, o i capitoli, all'interno di un libro), bensì secondo un grafo ovvero secondo un ordinamento reticolare. ne consegue che da ogni sotto-unità di un ipertesto, che è di per se un nodo (vertice del grafo) si può accedere direttamente a qualsiasi altra sotto-unità/nodo ad essa collegata. i collegamenti tra le sottounità sono chiamati links, e sono legami arbitrari che l'amministratore dell’ipertesto crea liberamente, e che può modificare secondo le successive esigenze. l’ipertesto dunque per sua iniziale costruzione non è mai definitivo. oggi nessuno si meraviglia più del fatto che è possibile parlare con chiunque in qualunque parte del mondo, usando il telefono; allo stesso modo è ormai quasi-possibile sempre via telefono leggere e scrivere testi in qualunque angolo del mondo. si profila nell’ipertesto una dimensione molto più ampia, che non era così ovvia quando fin dal 1990 cominciarono a girare le affermazioni, definite utopiche e visionarie di vannevar bush e ted nelson, che concepirono l'idea di un macrosistema di relazioni tra nodi testuali, che potesse permettere all’utente di andare (navigare) fra testi sparsi ai quattro angoli dell'universo, e non solo tra testi, purché collegati per via telematica. l'ipertesto, in questo senso, non sarebbe solamente un testo più complesso, denso di collegamenti interni; sarebbe l'insieme di tutti i testi esistenti, e di tutte le loro relazioni. a titolo di esempio vediamo cosa succede leggendo un ipertesto. sullo schermo del computer appare una schermata iniziale con varie indicazioni ( frecce, rimandi, parole azzurre sottolineate, o altro), che stanno ad indicare come da quel punto con un semplice click su una parola si può “saltare” verso un altro nodo dell'ipertesto, e cioè verso un'altra unità testuale. scegliendo una di queste opzioni con il sistema di puntamento, mouse o frecce di direzione, ci si trova in automatico in un altro elemento testuale, che a sua volta presenta un'altra serie di links verso altri nodi, tra i quali è possibile scegliere di nuovo; e così via. l’atteggiamento è identico a quello di teseo che si muove nel labirinto. ad ogni passo vanno fatte delle scelte. il leggere su uno schermo e non su carta, almeno dal punto di vista teorico dovrebbe essere irrilevante, ma chiaramente non è così se non ci si abitua e appare invece chiaro come molte siano sono le differenze rispetto ad un testo “normale”. in più, proprio perché abbiamo consapevolezza della difficoltà di orientamento nella navigazione tra i meandri testuali, l'ausilio tecnologico, il browser, ne conserva memoria, passaggio dopo passaggio, liberandoci da uno sforzo cognitivo paragonabile a quello effettivamente utilizzato nei concetti testualizzati. vediamo un rapido elenco degli aspetti principali, presentandoli così come compaiono all'osservazione empirica: 1) mancanza di sequenzialità. è la caratteristica che più colpisce il lettore di un ipertesto, già la prima volta che vi si accosta. il lettore “naviga” da un nodo testuale ad un altro in totale libertà, senza dover rispettare nessun ordine e scegliendo liberamente tra i links disponibili. il limite a questa libertà è dato esclusivamente da quanti nodi l'autore ha inserito nell'ipertesto, e dai collegamenti che ha istituito tra di essi. 28 r. mascella 2) possibilità di letture multiple. il lettore si accorge facilmente che in un ipertesto non c'è una lettura unica ma la lettura o l’ascolto o la visione dipendono dalla scelta della sequenza dei link determinata esclusivamente dalle scelte che si compiono volta per volta, durante il percorso di lettura, non prima. del resto per “lettura” intendiamo l'ordine con cui i singoli elementi costituenti i nodi scelti appaiono alla nostra attenzione. questo ordine, nel caso del libro, dipende dalla successione delle pagine così come sono state rilegate, secondo la volontà dell'autore. 3) multimedialità. da un testo è possibile aprire collegamenti ad altri files, di genere non testuale, quindi è possibile incorporare brani musicali, immagini fisse e filmati. il concetto di nodo è quindi estremamente largo e può comprendere oggetti testuali di varia natura, aprendo notevoli possibilità in campo didattico e saggistico si potrebbe dire che tutto questo si può fare anche mettendo insieme un libro, un registratore audio e un registratore video; ma così facendo andrebbe indubbiamente persa quell'unità testuale che invece nell'ipertesto multimediale è racchiusa in un solo apparecchio che nel caso di un computer portatile non è ormai più grande di un libro. 4) organizzazione reticolare delle unità testuali. l’utente di un ipertesto si rende conto facilmente che ogni nodo è connesso contemporaneamente con molti altri. non hanno senso i concetti assoluti di “pagina precedente”, “pagina seguente” ed ogni nodo può essere il successivo di molti altri, ed essere a sua volta il punto di partenza per diversi altri. 5) espandibilità del testo e non distinzione autore/lettore. il testo su memoria magnetica non è mai definitivo. si incrina il concetto di testo come qualcosa di definito, di delimitato. in parte perché è impossibile controllare i bordi dell'ipertesto. ad un ipertesto l'autore/lettore può aggiungere in qualsiasi momento nuovi elementi, con i soli limiti del sistema hardware di cui dispone: può aggiungere la traduzione del testo, o diverse traduzioni in più lingue; può aggiungere commenti, note, annotazioni; può aggiornare la bibliografia, può aprire nuovi collegamenti, ecc. l'utente può inoltre passare da “autore” a “lettore” essendo queste in un testo interattivo di due “modalità” che si possono scambiare a piacimento. 6) cooperazione. un ipertesto può essere costruito da autori fisicamente lontani ma intellettualmente vicini essendo le distanza annullate dal potersi muovere nel cosiddetto cyberspazio. ampliamo l’idea di labirinto verso altri aspetti. la moltiplicazione delle possibilità di scelta, l'ansia dello smarrimento, il gusto dell'esplorazione, la convinzione di trovare un significato nascosto sotto il groviglio dei percorsi, sono tutti elementi che abbiamo visto caratterizzare gli ipertesti, e che d'altra parte rimandano direttamente anche alla simbologia del labirinto e alle varie forme con cui questa figura archetipica riaffiora costantemente nelle arti e nella letteratura. il più grande ipertesto conosciuto è internet. in esso ogni vertice è ingresso ed uscita. la ricerca di un filo d’arianna è ugualmente importante anche se per scopi differenti. il sapere sostituisce il trovare l’uscita, ma forse l’uscita è un uscita da un problema intricato, è un procurarsi informazioni corrette, e così via. l'accostamento ipertesto-labirinto è estremamente ovvio, ed è tra le prime cose che vengono in mente usando un ipertesto; d'altra parte evocare il labirinto semplicemente come generico richiamo figurale lascia in la filosofia dei labirinti: dal mito all’intelligenza artificiale 29 ombra l'interesse epistemologico che, in vari campi del sapere, si è sollevato intorno ai concetti connessi con il labirinto di rete, di molteplicità, di complessità. circa la complessità, secondo noi operativa, questa sembrerebbe essere data dalla qualità. internet secondo molti è un enorme immondezzaio all’interno del quale poche e rare perle del sapere potranno essere un domani evidenziate da brillanti algoritmi intelligenti, creati dall’uomo ma capaci di operare in proprio. e proprio la figura del labirinto è stata sottoposta ad un intenso lavoro di rielaborazione teorica e di classificazione tipologica, secondo la quale per rosenstiehl anche i labirinti si possono suddividere in tre grandi categorie: unicursali, arborescenti e ciclomatici. a) il labirinto unicursale lo si può immaginare come un serpente arrotolato, oppure una fune avvolta su se stessa. l'impressione di groviglio inestricabile è in realtà un'illusione, in quanto chi segue il corpo del serpente non corre il rischio di sbagliare, lo percorre tutto dall'inizio alla fine, curva dopo curva. il labirinto unicursale dunque è un labirinto in cui non ci si perde, ma che anzi, attraverso una notevole complessità figurale, riesce a dare alla fine un senso di sollievo e di soddisfazione per la prova superata. b) il labirinto arborescente, è quella avente come grafo di base un albero: quindi una struttura ramificata, dicotomica, che si manifesta in una successione di bivi. a differenza dei labirinti del primo tipo, dove l'esploratore va sempre avanti qui ci troviamo in uno schema di percorso che consente di andare avanti fino alla fine di un ramo; poi di nuovo indietro fino al bivio; poi di nuovo avanti fino alla fine di un altro ramo, e così via. c) il labirinto ciclomatico è di fatto una struttura del tutto generale con la presenza di passaggi trasversali da un ramo all'altro. in questa struttura sono presenti isole (o zone centrali o buchi) attorno alle quali si possono creare dei loop (cicli) di nodilink che ritornano al punto di partenza. in questo tipo di labirinto è possibile perdersi, e l'esploratore rischia di restare intrappolato; a meno che non riesca a tenere sotto controllo il proprio itinerario. come si può facilmente notare, la classificazione dei labirinti (unicursale, arborescente, ciclomatico) ricalca perfettamente quella delle strutture di dati (lineare, ad albero, a rete): e forse a questo punto abbiamo la possibilità di saldare completamente i due oggetti ipertesto e labirinto. rimane da vedere se le caratteristiche tecniche della memoria dei computer, la suggestione archetipica dei labirinti e se si vuole le antiche tecniche dell’arte della memoria possano convergere verso una teoria generale e creare sinergie tra gli ambienti. 3. i labirinti dell’intelligenza artificiale l'idea della macchina pensante e dell'uomo artificiale (non solo meccanico) è molto antica. a questa idea si aggancia la parallela idea dell’uomo che, nelle sue splendide autopoiesi, crea e progetta mutazioni di se stesso anche e specialmente mediante protesi di vario genere e varia natura. si tratta di antiche aspirazioni umane, cantate al tempo della comunicazione attraverso una oralità mimetico poetica, scritta al tempo della carta imperante e diffusa per mezzo della letteratura anche popolare che ricorrendo ad immaginarie e possibili fantasie pseudoscientifiche ha proceduto e procede all'invenzione e fabbricazione dei miti moderni, eredi della sana mitologia. la personificazione di questa vecchia idea è il mostro, idea 30 r. mascella difficile da definire, parola che al di là della sua radice latina, personifica un insieme di forze che provengono dal nostro immaginario collettivo ma si intrecciano con forze sociali e spirituali, che non sono nette esprimendo in realtà solo tendenze e che comunque non appaiono controllate all'interno di un ordinamento razionale e ancor meno in un nuovo chiaro paradigma. il mostro è dunque l’incognita del futuro, attualmente è il nostro io proiettato in una accoppiata con una macchina, è il nostro io modificato geneticamente, è un nostro possibile clone come la pecora dolly, e il nostro discendente che forse vive fino a 150 anni e sostituisce con protesi intelligenti gran parte dei suoi organi danneggiati. ieri il mostro era altro ma sempre un quid che irrompe dall'esterno, che produce una alterità nell'ordine umano, e che si presenta al mondo con fattezze ultraumane o forse post-umane. il mostro è comunque vissuto come una minaccia, un sovvertimento come sostanzialmente il minotauro che divora i giovani in un luogo non sicuro, celato da forze arcane come i percorsi non comprensibili del labirinto. il mostro opera in uno spazio simbolico che possiede una dimensione interiore che è incontrollabile, speculare in un mondo rovesciato, nel quale il terrore convive con una sorta di attrazione onirica prodotta dal vuoto del mostro. e il mostro rappresenta anche l'invasione di quello spazio intermedio che separa l'umano dal divino per i credenti, dall'ultraterreno per gli occultisti, dall’irrazionalità per lo scienziato: il mostro è il superamento, l’accostamento di questi due confini. occorre un eroe, un teseo e un metodo come il filo d’arianna ovvero la forza, la scienza, la fede, la ragione, la conoscenza, l’intuito per abbatterlo e ristabilire la supremazia dell’uomo. nella tradizione orale, letteraria ed oggi anche filmica, non sempre il mostro è connotato negativamente ma in ogni caso rappresenta sempre una rottura dell'ordinamento supposto naturale, quello stabilmente dato e perciò in qualche modo conoscibile e controllabile. sono le anime innocenti e forse incoscienti ad accettare frankenstein, dracula ed e.t.‡. l’uomo comune davanti al mostro prova angoscia, terrore e ribrezzo giusta reazione alla rottura di equilibrio, assieme tuttavia all'attrazione per il mostro e persino al subire o cercare una forte seduzione estetica per il diverso. ciò che unifica la serie delle mostruosità immaginate e sedimentate nella nostra fantasia nel corso dei secoli da queste proiezioni psicologiche è insomma il mondo dell'inconscio che reagisce all'eterna paura dell'altro, alla paura del non conosciuto. tradizionalmente il mostruoso riguarda l'informe, il deforme, l'ibrido e la chimera. a partire dagli anni cinquanta, l'iniziativa di costruire forme artificiali di intelligenza ha preso piede in modo deciso. ed il primo, naturale obiettivo, tuttora ancora il cardine di tutto il progetto, è rappresentato dalla comprensione e dalla riproduzione dei nostri meccanismi cerebrali. il nostro cervello, proprio per la struttura così come oggi la conosciamo, fatta di neuroni, i nodi, e di connessioni sinaptiche, gli archi, è la rete più complessa su cui oggi esiste un obiettivo preciso di schematizzazione. in altre parole esisteranno anche reti più complesse, ma queste sono aldilà del muro, inaccessibili o poco interessanti in rapporto all'obiettivo-cervello. una delle prime visioni ed il primo serio approccio, per molti da ricollegare al pensiero hobbesiano, hanno considerato che la nostra struttura cerebrale non sia né più né meno che ‡ in un vecchio film della walt disney dal titolo “il pianeta proibito” interpretato da un magistrale walter pidgeon (nella parte del prof. morbius) degli ingenui astronauti “stile anni ’50” raggiungono un pianeta ove il suddetto professore ha scoperto i resti di una razza antica che aveva raggiunto vette tecnologiche elevatissime ma che non aveva tenuto in conto i “mostri dell’id”, vere proiezioni esterne del loro io interno e della personificazione di quanto proviamo anche nei confronti dell’altro, pensato come il mostro che si oppone a noi la filosofia dei labirinti: dal mito all’intelligenza artificiale 31 una sorta di calcolatore e che tutto il nostro pensiero non sia altro che fare calcoli, più complessi di somma e sottrazione, ma pur sempre semplicemente calcoli. anni di ricerca computazionalistica hanno però portato dapprima a far evolvere l'idea, a favore della visione connessionistica, in cui il calcolo parallelo gioca un ruolo di primo piano e poi, a superarlo del tutto. in molti, oggi, sono convinti che la metafora del calcolatore non sia più appropriata, non risponde efficacemente alle istanze strutturali dello stato di coscienza e delle emozioni. ed in fondo il problema della demarcazione tra mente e corpo, monistica o dualistica, è ancora aperto è tale è destinato a restare a lungo. ma se per un attimo, utilizzando ancora la metafora del calcolatore solo in termini di approccio schematico esterno, e volendo pensare al nostro apparato del pensiero come una macchina molto complessa, che reagisce in risposta ad un input ambientale e fornisce un output comportamentale, è abbastanza immediato identificare la scatola al centro, in cui i segnali elettrici compiono percorsi di attivazione dei circuiti neuronali, ovvero di cammini paralleli del pensiero, come una struttura labirintica. e' la stessa struttura, dunque, ad avere la caratteristica che una volta entrati sia difficile uscire. molto di più, perlomeno apparentemente, dei labirinti di pensiero in cui ci perdiamo. così come diceva leibniz, infatti, il nostro pensiero ha labirinti nei quali non riesce a cavarsela, ed i più famosi, per lui, erano: “l'uno riguarda la questione del libero e del necessario, soprattutto nella produzione e nell'origine del male; l'altro consiste nella discussione circa la continuità e gli indivisibili, che risultano esserne gli elementi, e in cui deve entrare la considerazione dell'infinito”. ed anche bacone, pensando a tutto il nostro universo conoscitivo, osservava: […] l'edificio di questo universo appare nella sua struttura come un labirinto all'intelletto umano che lo contempla; e sembra tutto occupato da vie ambigue, da somiglianze ingannevoli di segni e di cose, dai giri contorti e dai nodi intricati delle nature. il cammino poi deve esser percorso sempre sotto l'incerta luce del senso, ora accecante ora opaca, e bisogna aprirsi continuamente la strada attraverso le selve dell'esperienza e dei fatti particolari. anche coloro che si offrono (come si è detto) come guide nel cammino, vi sono essi stessi implicati e accrescono con simile guida il numero degli errori e degli erranti.” e conclude: “in mezzo a tante difficoltà, bisogna per forza dubitare della esattezza del giudizio umano, sia quanto alla sua propria forza, sia quanto a un successo fortuito: non c'è eccellenza d'ingegno, per quanto grande essa sia, né probabilità di esperimento, per quanto spesso ripetuto, che possa vincere quelle difficoltà. ci occorre un filo conduttore per guidare i nostri passi, e tracciare la via fin dalle prime percezioni dei sensi. sempre a questo proposito, lo psicologo g. mantovani osserva che il cervello, così come l'ipertesto, è organizzato per strutture labirintiche: “gli ipertesti e gli ipermedia, in quanto adottano il paradigma non sequenziale, si propongono come più aderenti e fedeli alle caratteristiche dei processi del pensiero”, pertanto essi consentono “di superare la frattura, propria del funzionamento cognitivo umano, tra processi di pensiero, non sequenziali, e modalità di trasmissione dell'informazione, sequenziali e vincolate da un ordine”. e' proprio la molteplicità delle connessioni, o la potenzialità di connessione, che generano uno stato di smarrimento, tanto nella comprensione del cervello, quanto nella navigazione ipertestuale. ci si muove infatti, infatti, in una giungla neuronale e sinaptica di una complessità assolutamente strabiliante. ci sono 100 miliardi di neuroni nel cervello e ogni neurone è capace di 10.000 contatti con altre cellule nervose per un saldo di 500 milioni di contatti a 32 r. mascella millimetro cubo. ed in questo microuniverso sconfinato le neuroscienze avanzano in mille modi differenti, procedendo dal semplice al complesso ma anche viceversa e separano, distinguono, sfaldano funzioni psicologiche articolate così da stabilire, tra neuronale e psicologico, corrispondenze verosimili. uno dei problemi centrali è comunque quello delle rappresentazioni, tanto per lo studio del comportamento animale, tanto per la progettazione di robot “autonomi”. ad esempio la codifica di configurazioni spaziali è molto importante nel filone della nuova robotica. qui vi è interesse ai compiti che si svolgono nell’ambiente reale. da questa angolazione sono state mosse una serie di critiche alla robotica classica, che si riassumono nel rifiuto del cosiddetto paradigma simbolico, per il quale la conoscenza del mondo non è possibile se non attraverso la mediazione di rappresentazioni, intese come strutture linguistiche manipolate attraverso una serie di regole esplicite. questo paradigma viene solitamente riferito all’ipotesi del sistema fisico dei simboli, originariamente formulata da newell e simon nel 1976. da un lato c’è l’approccio cosiddetto “dall’alto”, che prende in esame i problemi più complessi, ad esempio come noi facciamo a fare un ragionamento astratto; dall’altro, un approccio che potremmo definire “dal basso”, vuole essere l’opposto, ovvero mettere al centro dell’attenzione i problemi di interazione senso-motoria dell’agente con l’ambiente. in altre parole più che pensare agli esseri umani ed ai compiti complessi, la fonte di ispirazione è il comportamento degli animali, anche semplici, comunque ritenuti intelligenti, perché mostrano adattamento all’ambiente ed apprendimento in ambienti reali. un’efficace battuta è quella di r. brooks che a questo proposito disse: “gli elefanti non giocano a scacchi”. l’architettura proposta da brooks si chiama “della sussuzione” in cui l’agente è composto da più livelli funzionalmente distinti, che nell’interazione con l’ambiente agiscono in relativa indipendenza l’uno dall’altro, senza la supervisione di un unico sistema centrale di controllo. questo approccio ha permesso di costruire robot mobili, per esempio testati nella ricerca di un obiettivo distante, ma che nello stesso tempo continuano ad evitare gli ostacoli che incontrano lungo la via. dunque alcuni risultati si sono ottenuti rispetto all’approccio classico, tanto che alcuni psicologi e filosofi hanno prontamente concluso che questa nuova robotica, di tipo behaviour-based, ha accantonato la disembodied mind dell’ai classica, in cui mente e corpo, così come sosteneva ad esempio cartesio, sono due cose ben distinte. dopo questa fase euforica, intorno agli anni ‘70, il panorama delle ricerche si è evoluto, ed ormai esistono robot “ibridi”, che usano buone caratteristiche di robustezza e azione in tempo reale e, nello stesso tempo, usano classici sistemi di pianificazione e rappresentazione della conoscenza. però questo approccio, integrato con quello dall’alto, è stato e continua ad essere decisivo. la connessione con i labirinti è dunque evidente. in questi studi si pone l’accento sulla capacità di svolgere un compito, e l’esempio classico è la ricerca di un cammino, caratterizzato da una sequenza stimolo-azione che dovrebbe permettere di decidere in tempi brevi, e da una certa capacità di memorizzazione come forma specializzata di apprendimento. e una forma tra le più complesse, si è visto, è quella che richiede una navigazione verso un punto che è al di fuori della scena visiva, o comunque non a portata dell’apparato sensoriale. in questo caso vi è una rappresentazione topologica che ad un livello basso non permette di capire, ad esempio quando un cammino viene seguito più volte e ci diviene familiare, la presenza di eventuali scorciatoie. per fare questo abbiamo bisogno di un tipo di informazione in più che consiste in quella metrica, ovvero la filosofia dei labirinti: dal mito all’intelligenza artificiale 33 l’informazione sulle distanze percorse. le relazioni metriche spaziali sono fondamentali, sia per noi che per riprodurre artificialmente intelligenza. dunque ci sono due quadri complessivi di rappresentazione, una in cui l’agente è in interazione diretta con l’ambiente, ed un’altra che permette di creare un quadro di riferimento più globale.e qui, a questo secondo livello, oggi abbiamo ancora qualche problema. mentre modelli che manifestano capacità accettabili al primo livello ormai abbondano, modelli che consentono di rispondere efficacemente al secondo livello, ovvero con delle prestazioni anche solo modestamente cognitive, scarseggiano. per inciso, la loro prestazione varia a seconda di cosa devono fare e del tipo di rappresentazione che implementano, e quindi si creano problemi ancora difficili da risolvere per i progettisti. questa ricerca è fondamentale per comprendere il comportamento degli animali. basti pensare agli uccelli che tornano al nido, le api che tornano all’alveare e le formiche che tornano al formicaio, pur essendosi allontanate di decine o centinaia di metri. uno degli esseri viventi che più è stato studiato è la formica del deserto sahariano, cataglyphis. gli studi dicono che in questo caso la questione metrica sia risolta da una specie di bussola solare. inoltre, sembra che questa formica abbia in aiuto anche una capacità di memorizzazione, non di tutto l’ambiente ma solo di scenari, di snapshot. di fronte ad una scena visiva, insomma, cataglyphis fa dei confronti con scenari che ha già in memoria, e questo gli consente di orientarsi. alcuni agenti-robot sono stati costruiti a questo proposito, per comprendere e verificare le ipotesi via via sul tappeto. tra questi, uno dei più noti è il robot di wieth e browning, che consiste in un topo robot solutore di labirinti, che permette una rappresentazione non solo di basso livello, ma anche di alto livello. in questo caso il labirinto ha dimensioni 16x16, variamente configurabile. wieth e browning prevedevano tre diversi schemi o livelli di importanza: 1. il primo relativo alla reazione con l’ambiente, quindi per evitare ostacoli, girare, e così via, ma senza accedere alla memoria. dunque un vero e proprio schema percettivo; 2. il secondo, invece, fa uso della mappa del labirinto per pianificare l’azione; quindi sulla base degli schemi del livello inferiore, decide l’azione da compiere. dunque uno schema per la rappresentazione del mondo; 3. il terzo, che è un livello motivazionale, genera le mete e dispone di una serie di valori da usare per valutare le strategie più appropriate. oggi, da questa e da altre esperienze, si pensa che possano coesistere forme di rappresentazione dello spazio diverse, alla base di diverse strategie di esplorazione dell’ambiente, e quindi una concezione di modelli multipli, così come indicato da prescott. proprio come fa cataglyphis. 4. conclusioni vi sono campi scientifici nei quali l’interesse per i labirinti è elevato. nel campo della psicologia i labirinti forniscono ottimi test per il comportamento di uomini ed animali nei confronti dell’apprendimento. così anche un lombrico o una formica può percorrere un labirinto ed “imparare” a muoversi. i programmi, anche loro, sono qualcosa di analogo a forme di vita primordiale e anche essi possono imparare a muoversi nei labirinti. 34 r. mascella interessante l’analisi di bolter (1984) riguardo al cosiddetto “uomo di turing”, l'uomo abituato a convivere con lo spazio del computer. uno spazio limitato, caratterizzato da una infinita riscrivibilità, entro cui si perde il senso di cosa significa una scelta definitiva: “all'uomo di turing manca l'intensità emotiva dei suoi predecessori. egli, nei suoi giochi, non investe tutto se stesso precisamente perché i giochi che fa non sono irrevocabili.” e’ il carattere finito dello spazio del computer ad innescare questa dimensione ludica, questo senso di chiusura che deriva anche dalla pratica della programmazione, intesa come sequenza da sperimentare, modificare e rielaborare all'infinito. e ritornando al topolino “teseo-tremaux” di shannon, quando questo giunge ad un incrocio e deve fare una scelta, non la fa a caso, come faremmo noi, ma la fa usando un criterio preciso esistente nella sua programmazione. l’accoppiata di un algoritmo e di un piccolo robot che lo usa è molto più efficiente di quella umana ovvero animale. a riguardo ebbe a dire shannon: “e’ piuttosto difficile trovare guasti in una macchina che opera su elementi casuali. e’ difficile pure dire se una macchina del genere sta funzionando male quando in effetti non è possibile prevedere ciò che deve fare”. il topo “teseo-trémaux” una volta trovato un cammino per uscire su un labirinto lo ripercorre a gran velocità una seconda volta avendo memorizzato tutte le scelte. un topo reale invece dovrebbe fare scelte veramente casuali impiegando più tempo. e allora, possiamo concludere con un’immagine storico-evolutiva. i labirinti, da strumento utilizzato per mettere in difficoltà l’essere umano, salvo poi ricompensarlo a compito eseguito, è infine divenuto strumento per mettere in difficoltà le macchine. e la ricompensa, stavolta, assegnando in un impeto una valenza emotiva alle macchine che costruiamo, è l’avvicinamento all’uomo. bibliografia barthes r., la mort de l'auteur, in le bruissement de la langue, seuil, paris, 1984. trad. it. la morte dell'autore, in id., il brusio della lingua, 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(a cura di), futuri immaginari, logica university press, roma, 1998. newell a., “physical symbol systems”, in cognitive science, n. 4, 1980, pp. 135-183. newell a. e simon h.a., “computer science as empirical enquiry: symbolsand search”, in communications of the acm, n. 19, 1976, pp. 113-126. odifreddi p., jorge luis borges. i labirinti dello spirito, reperibile su www.vialattea.net, 1992. prescott t.j., from animals to animats 3, mit press, cambridge (ma), 1994. queneau r., conte à votre façon, 1967. trad. it. un racconto a modo vostro, in id., segni, cifre e lettere, einaudi, torino, 1981. rosenstiehl p., “labirinto”, in enciclopedia, s.v., einaudi, torino, 1979. vanlehn k., architecture for intelligence, erlbaum, hillsdale (nj), 1991. vera a.h. e simon h.a., “situated action: a symbolic interpretation”, in cognitive science, n. 17, 1993, pp.7-48. ratio mathematica volume 42, 2022 a new class of almost continuity in topological spaces jagadeesh b.toranagatti* abstract in this paper, we apply the notion of δgβ-open sets due to benchalli et al.[benchalli et al., 2017] to present a new class of functions called almost δgβ-continuous functions along with its several properties, characterizations and mutual relationships. keywords: almost continuity,almost β-continuity, δgβ-continuity,almost δgβ-continuity. 2020 ams subject classifications: 54a05,54c08. 1 *department of mathematics, karnatak university’s karnatak college, dharwad 580001, karnataka, india; jagadeeshbt2000@gmail.com 1received on january 28th, 2022. accepted on may 22nd, 2022. published on june 30th, 2022. doi: 10.23755/rm.v39i0.708. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 91 jagadeesh b.toranagatti 1 introduction the notion of continuity is an important concept in general topology as well as all branches of mathematics. of course it′s weak forms and strong forms are important, too. singal and singal[singal et al., 1968], in 1968, defined almost continuous functions as a generalization of continuity.noiri and popa[noiri and popa, 1998], in 1998, defined almost β-continuous functions as a generalization of almost continuity.recently, benchalli et al.[benchalli et al., 2017] introduced the notion of δgβ-continuous functions in topological spaces. in this article, using the notion of δgβ-open sets given in [benchalli et al., 2017], we introduce and study a new class of functions called almost δgβ-continuous functions. we investigate several properties of this class. the class of almost δgβcontinuity is a generalization of almost β-continuity and δgβ-continuity. 2 preliminaries throughout this paper (x,τ),(y,σ) and (z,η)(or simply x,y and z ) represent nonempty topological spaces on which no separation axioms are assumed unless otherwise stated. let m be a subset of x. the closure of m and the interior of m are denoted by cl(m) and int(m), respectively. definition 2.1. a set m ⊆ x is called β-closed[abd, 1983](=semi preclosed[andrijević, 1986] (resp.,pre-closed[mashhour, 1982], regular closed[stone, 1937], semi-closed [levine, 1963] if int(cl(int(m)) ⊆ m (resp.,cl(int(m)) ⊆ m, m = cl(int(m),int(cl(m)) ⊆ m) . definition 2.2. a set m ⊆ x is called δ-closed [velicko, 1968] if m = clδ(m) where clδ(m) = { p ∈ x :int(cl(n)) ∩ m ̸= ϕ, n ∈ τ and p ∈ n }. definition 2.3. a set m ⊆ x is called gβ-closed [tahiliani, 2006](resp., gsprclosed [navalagi et al.] and δgβ-closed[benchalli et al., 2017] if βcl(m) ⊆ g whenever m ⊆ g and g is open(resp. regular open and δ-open) in x. the complements of the above mentioned closed sets are their respective open sets. the class of δgβ-open (resp., δgβ-closed, open, closed, regular open, regular closed, preopen, semiopen and β-open) sets of (x,τ) containing a point p ∈ x is denoted by δgβo(x,p)(resp., δgβc(x,p),o(x,p), c(x,p), ro(x,p), rc(x,p), po(x,p), so(x,p)and βo(x,p)). 92 a new class of almost continuity in topological spaces definition 2.4. a function f:x → y is called almost continuous[singal et al., 1968](resp., almost β-continuous[noiri and popa, 1998] and almost gspr-continuous) if the inverse image of every regular open set g of y is open (resp., β-open and gspr-open) in x. definition 2.5. [benchalli et al., 2017] a function f:x → y is called δgβ-continuous (resp.,δgβ-irresolute) if the inverse image of every open(resp.,δgβ-open) set g of y is δgβ-open in x. definition 2.6. a function f:x → y is called almost contra continuous [baker, 2006](resp. almost contra super-continuous[ekici, 2004] and contr r-map[ekici, 2006] if the inverse image of every regular closed set g of y is open(resp. δ-open and regular open) in x. definition 2.7. a space x is said to be: (i) nearly compact [singal and mathur, 1969] if every regular open cover of x has a finite subcover, (ii) r-t1-space[ekici, 2005] if for each pair of distinct points x and y of x, there exist regular open sets u and v such that x ∈ u, y /∈u and x /∈ v, y ∈ v, (iii) r-t2-space [ekici, 2005] if for each pair of distinct points x and y of x, there exist regular open sets u and v such that x ∈ u, y ∈ v and u∩v =ϕ, (iv) δgβ-t1 space if for any pair of distinct points p and q, there exist g,h ∈δgβo(x) such that p ∈ g, q /∈ g and q ∈ h, p /∈h, (v) δgβ-t2 space[benchalli et al., 2017] if for each pair of distinct points x and y of x, there exist g,h ∈δgβo(x) such that x ∈ g, y ∈ h and g∩h =ϕ. definition 2.8. [benchalli et al., 2017] a space x is said to be tδgβ(resp.,δgβt 1 2 )space if δgβo(x)=o(x) (rep.,δgβo(x) = βo(x)). definition 2.9. [carnahan, 1972] a subset m of a space x is said to be n-closed relative to x if every cover of m by regular open sets of x has a finite subcover. theorem 2.1. [benchalli et al., 2017] if a and b are δgβ-open subsets of a extremely disconnected and submaximal space x, then a∩b is δgβ-open in x. definition 2.10. [jankovic, 1983] a space x is called locally indiscrete if o(x)=ro(x). lemma 2.1. [noiri, 1989] let (x,τ) be a space and let m be a subset of x. m ∈ po(x) if and only if scl(m) = int(cl(m)). 93 jagadeesh b.toranagatti 3 almost δgβ-continuous functions definition 3.1. a function f: x → y is said to be almost δgβ-continuous at p ∈ x if for each n ∈ δo(y,f(p)), there exists m ∈ δgβo(x,p) such that f(m) ⊆ n. if f is almost δgβ-continuous at every point of x, then it is called almost δgβ-continuous. remark 3.1. we have the following implications almost β-continuity−→ almost δgβ-continuity−→almost gspr-continuity. ↑ δgβ-continuity. none of these implications is reversible. example 3.1. let x = {p,q,r,s}, τ = {x, ϕ, {p}, {q}, {p,q}, {p,q,r}} and σ = {y, ϕ, {p}, {q}, {p,q}, {p,r}, {p,q,r}}. define f: (x,τ) → (x,σ) by f(p) = f(r) = q , f(q) = p and f(s) = r. clearly f is almost δgβ-continuous but for {q}∈ ro((x,σ), f−1({q}) = {p,r} /∈ gβo(x,τ). therefore f is not almost gβ-continuous. define g: (x,τ) → (x,σ) by g(p) = p, g(q) = s, g(r) = r and g(s) = q.then g is almost δgβ-continuous but for {p} ∈ o(x,σ), g−1({p}) = {p} /∈ δgβo(x,τ).therefore g is not δgβ-continuous. define h: (x,τ) → (x,σ) by h(p) = h(q) = q, h(r) = p and h(s) = r.then h is almost gspr-continuous but for {q}∈ ro(x,σ), h−1({q}) = {p,q} /∈ δgβo(x,τ). therefore h is not almost δgβ-continuous theorem 3.1. if f:x→y is almost δgβ-continuous and y is locally indiscrete space,then f is δgβ-continuous. proof: it follows from the definition 2.10 theorem 3.2. let x be a locally indiscrete space and m⊆x,then the following properties are equivalent: (i) m is gspr-closed; (ii) m is δgβ-closed; (iii) m is gβ-closed. as a consequence of above theorem,we have the following result; theorem 3.3. let x be a locally indiscrete space,then the following properties are equivalent: (i) f:x→y is almost gspr-continuous; (ii) f:x→y is almost δgβ-continuous; (iii) f:x→y is almost gβ-continuous. 94 a new class of almost continuity in topological spaces theorem 3.4. let x be a δgβt 1 2 -space. then the following are equivalent: (i) f: x → y is almost β-continuous; (ii) f: x → y is almost gβ-continuous; (iii) f: x → y is almost δgβ-continuous. theorem 3.5. let x be a tδgβ-space. then the following are equivalent: (i) f: x → y is almost continuous; (ii) f: x → y is almost β-continuous; (iii) f: x → y is almost gβ-continuous; (iv) f: x → y is almost δgβ-continuous; (v) f: x → y is almost gspr-continuous. lemma 3.1. [benchalli et al., 2017] for a subset m of a space x ,the following are equivalent: (i) m is δ-open and δgβ-closed; (ii) m is regular open; (iii) m is open and β-closed. theorem 3.6. the following statements are equivalent for a f: x → y: (i) f is almost contra super-continuous and almost δgβ-continuous; (ii) f is contra r-map; (iii) f is almost contra continuous and almost b-continuous. theorem 3.7. the following statements are equivalent for a f: x → y: (i) f is almost δgβ-continuous; (ii) for each point p∈x and each g∈δc(y) with f(p) /∈g,there exists a h ∈δgβc(x) and p /∈h such that f−1(g)⊆h; (iii) for each point p∈x and each n∈ro(y,f(p)),there exists an m ∈ δgβo(x,p) such that f(m)⊆n; (iv) for each point p∈x and each g∈rc(y) with f(p) /∈g,there exists a h ∈ δgβc(x) and p /∈h such that f−1(g)⊆h; 95 jagadeesh b.toranagatti (v) for each p ∈ x and each n∈o(y,f(p)),there exists m ∈ δgβo(x,p) such that f(m) ⊂ int(cl(n)); (vi) for each p ∈ x and each n∈o(y,f(p)), there exists m ∈ δgβo(x,p) such that f(m) ⊂ scl(n). proof: (i)←→(ii)−→(iv)←→(iii)←→(v)←→(vi): obvious. (iii)−→(i): let n ∈ δo(y) such that f(p) ∈ n, then there exists g ∈ ro(y) such that f(p) ∈ g ⊆ n. by (iii), there exists an m ∈ δgβo(x,p) such that f(m)⊆g ⊆ n. definition 3.2. a space x is said to be δgβ-additive if δgβo(x) is closed under arbitrary union. theorem 3.8. let x be a δgβ-additive space.then m ⊆ x is δgβ-closed(resp., δgβ-open) if and only if δgβcl(m) = m (resp., δgβint(m) = m ). theorem 3.9. the following statements are equivalent for a f: x → y where x is δgβ-additive: (i) f is almost δgβ-continuous; (ii) f(δgβcl(m)) ⊆ clδ(f(m)) for each m ⊆ x; (iii) δgβcl(f−1(n)) ⊆ f−1(clδ(n)) for each n ⊆ y; (iv) f−1(g)∈δgβc(x) for each δ-closed set g of y; (v) f−1(h)∈δgβo(x) for each δ-open set h of y; (vi) f−1(g)∈δgβc(x) for each regular closed set g of y; (vii) f−1(h)∈δgβo(x) for each regular open set h of y. proof: (i)−→(ii) let n ∈ δc(y) such that f(m) ⊆ n. observe that n = clδ(n) =⋂ {f:n ⊆ f and f ∈ rc(y)} and so f−1(n) = ⋂ {f−1(f):n ⊆ f and f ∈ rc(y)}. by (i) and definition 3.2, we have f−1(n) ∈ δgβc(x) and m ⊆ f−1(n). hence δgβcl(m) ⊆f−1(n), and it follows that f(δgβcl(m)) ⊆ n. since this is true for any δ-closed set n containing f(m), we have f(δgβcl(m)) ⊆ clδ(f(m)). (ii)−→(iii) let d ⊆ y, then f−1(d) ⊆ x. by (ii), f(δgβcl(f−1(d))) ⊆ clδ(f(f−1(d))) ⊆δgβcl(d). so that δgβcl(f−1(d)) ⊆f−1(clδ(d)). (iii)−→(iv) let g be a δ-closed subset of y.then by (iii), δgβcl(f−1(g)) ⊆f−1(clδ(g)) = f−1(g).in consequence, δgβcl(f−1(g)) = f−1(g) and hence by theorem 3.8, f−1(g) ∈ δgβc(x). (iv)−→(v):clear. (v)−→(i): let n ∈ ro(y).then n is δ-open in y. by (v), f−1(n) ∈ δgβo(x). hence f is almost δgβ-continuous 96 a new class of almost continuity in topological spaces theorem 3.10. the following statements are equivalent for a f: x → y where x is δgβ-additive: (i) f is almost δgβ-continuous; (ii) for every open subset k of y,f−1(int(cl(k)∈δgβo(x); (iii) for every closed subset m of y,f−1(cl(int(m)∈δgβc(x); (iv) for every β-open subset k of y,δgβcl(f−1(k)) ⊆ f−1(cl(k)); (v) for every β-closed subset m of y,f−1(int(m)) ⊆ δgβint(f−1(m)); (vi) for every semi-closed subset m of y,f−1(int(m)) ⊆ δgβint(f−1(m)); (vii) for every semi-open subset k of y,δgβcl(f−1(k)) ⊆ f−1(cl(k)); (viii) for every pre-open subset m of y,f−1(m) ⊆ δgβint(f−1(int(cl(m)). proof: (i)←→(ii): let k ⊆ y. since int(cl(n)) is regular open in y. then by (i), f−1(int(cl(n)) ∈ δgβo(x). the converse is similar. (i)←→(iii)it is similar to (i)←→(ii). (i)−→ (iv): let k ∈ βo(y),then cl(k) is regular closed in y. so by(i),f−1(cl(k)) ∈ δgβc(x). since f−1(n) ⊆ f−1(cl(n)),then δgβcl(f−1(n)) ⊆ f−1(cl(n)). (iv)−→ (v) and (vi)−→ (vii):obvious (v)−→ (vi):it follows from the fact that every semiclosed set is β-closed. (vii)−→ (i):it follows from the fact that every regular closed set is semi-open. (i)←→ (viii): let m ∈ po(y). since int(cl(n)) is regular open in y,then by (i), f−1(int(cl(n))) ∈δgβo(x) and hence f−1(n) ⊆f−1(int(cl(n))) = δgβint(f−1(int(cl(n)))). conversely,let h ∈ ro(y). since h is preopen in y, f−1(h) ⊆δgβint(f−1(int(cl(n)))) = δgβint(f−1(n)),in consequence, δgβint(f−1(h))=f−1(h) and by theorem 3.8, f−1(n) ∈ δgβo(x). theorem 3.11. the following statements are equivalent for a f: x → y where x is δgβ-additive: (i) f is almost δgβ-continuous; (ii) for every e∗-open set k of y,f−1(clδ(k)) is δgβ-closed in x; (iii) for every δ-semiopen subset k of y,f−1(clδ(k)) is δgβ-closed set in x; (iv) for every δ-preopen subset k of y,f−1(int(clδ(k))) is δgβ-open set in x; (v) for every open subset k of y,f−1(int(clδ(k))) is δgβ-open set in x; 97 jagadeesh b.toranagatti (vi) for every closed subset k of y,f−1(cl(intδ(k))) is δgβ-closed set in x. proof: (i)→(ii):let k be a e∗-open subset of y. then by lemma 2.7 of [ayhan and özkoç, 2018], clδ(k) ∈ rc(y). by (i),f−1(clδ(k)) ∈ δgβc(x). (ii)→(iii):obvious since every δ-semiopen set is e∗-open. (iii)→(iv):let k be a δ-preopen subset of y,then intδ(y\k) ∈ δso(y). by (iii), f−1(clδ(intδ(y\k)) ∈ δgβc(x) which implies f−1(int(clδ(k)) ∈ δgβo(x). (iv)→(v):obvious since every open set is δ-preopen. (v)→(vi):clear (vi)→(i):let k ∈ ro(y). then k = int(clδ(k)) and hence (y\k) is closed in x. by (vi), f−1(y\k) = x\f−1(int(clδ(k))) = f−1(cl(intδ(y\k)) ∈ δgβc(x). thus f−1(k) is δgβ-open in x. theorem 3.12. the following are equivalent for a function f: x → y where x is δgβ-additive: (i) f is almost δgβ-continuous; (ii) for every e∗-open subset g of y,f−1(a-cl(g)) is δgβ-closed set in x; (iii) for every δ-semiopen subset g of y,f−1(δ-pcl(g)) is δgβ-closed set in x; (iv) for every δ-preopen subset g of y,f−1(δ-scl(g))) is δgβ-open set in x. proof:follows from the lemma 3.1 of [ayhan and özkoç, 2018] theorem 3.13. if an injective function f:x → y is almost δgβ-continuous and y is r-t1, then x is δgβ-t1. proof: let (y,σ) be r-t1 and p1,p2 ∈ x with p1 ̸= p1. then there exist regular open subsets g, h in y such that f(p1) ∈ g, f(p2) /∈ g, f(p1) /∈ h and f(p2) ∈ h. since f is almost δgβ-continuous, f−1(g) and f−1(h) ∈ δgβo(x) such that p1 ∈f−1(g), p2 /∈ f−1(g), p1 /∈ f−1(h) and p2 ∈ f−1(h). hence x is δgβ-t1 . theorem 3.14. if f:x → y is an almost δgβ-continuous injective function and y is r-t2, then x is δgβ-t2. proof: similar to the proof of theorem 3.13 theorem 3.15. if f,g:x → y are almost δgβ-continuous where x is submaximal, extremely disconnected and δgβ-additive and y is hausdorff, then the set {x ∈ x : f(x) = g(x)} is δgβ-closed in x. proof: let d = {x ∈ x : f(x) = g(x)} and x /∈ (x\d). then f(x) ̸= g(x). since y is hausdorff, there exist open sets v and w of y such that f(x) ∈ v, g(x)∈ w and v ∩ w = ϕ, hence int(cl(v)) ∩ int(cl(w)) = ϕ. since f and g are almost δgb-continuous, there exist g,h ∈ δgbo(x,x)) such that f(g) ⊆ int(cl(v )) and 98 a new class of almost continuity in topological spaces g(h) ⊆ int(cl(w)). now, put u = g ∩ h, then u ∈ δgbo(x,x)) and f(u) ∩ g(u) ⊆ int(cl(v)) ∩ int(cl(w)) =ϕ. therefore, we obtain u ∩ d = ϕ and hence x /∈ δgbcl(d) then d = δgbcl(d). since x is δgb-additive, d is δgb-closed in x. definition 3.3. a space x is called δgβ-compact if every cover of x by δgβ-open sets has a finite subcover. definition 3.4. a subset m of a space x is said to be δgβ-compact relative to x if every cover of m by δgβ-open sets of x has a finite subcover. theorem 3.16. if f:x → y is almost δgβ-continuous and k is δgβ-compact relative to x, then f(k) is n-closed relative to y. proof: let { gα: α ∈ ω } be any cover of f(k) by regular open sets of y . then {f−1(gα):α∈ω} is a cover of k by δgβ-open sets of x. hence there exists a finite subset ωo of ω such that k ⊂∪{f−1(gα):α∈ωo }. therefore, we obtain f(k) ⊂ {gα: α∈ωo}. this shows that f(k) is n-closed relative to y . corolary 3.1. if a surjective function f:x → y is almost δgβ-continuous and x is both δgβ-compact and δgβ-additive, then y is nearly compact. lemma 3.2. let x be a δgβ-compact , submaximal and extremely disconnected and n⊂x.then n is δgβ-compact relative to x if n is δgβ-closed. proof: let { bα: α ∈ ω } be a cover of n by δgβ-open sets of x. note that (x-n) is δgβ-open and that the set (x-n) ∪{ bα: α ∈ ω } is a cover of x by δgβ-open sets. since x is δgβ-compact, the exists a finite subset ωo of ω such that the set (x-n) ∪{ bα: α ∈ ωo } is a cover of x by δgβ-open sets in x. hence { bα: α ∈ ωo } is a finite cover of n by δgβ-open sets in x. theorem 3.17. if the graph function g: x → x×y of f: x → y,defined by g(x)=(x,f(x)) for each x∈x is almost δgβ-continuous. then f is almost δgβ-continuous. proof:let n∈ro(y), then x×v ∈ ro(x×y). as g is almost δgβ-continuous, f−1(n) = g−1(x×n) ∈ δgβo(x). theorem 3.18. if the graph function g: x → x×y of f: x → y, defined by g(x)=(x,f(x)) for each x ∈ x. if x is a submaximal and extremely disconnected space and δgβ-additive, then g is almost δgβ-continuous if and only if f is almost δgβ-continuous. proof: we only prove the sufficiency. let x ∈ x and w ∈ro(x×y). then there exist regular open sets u1 and v in x and y, respectively such that u1×v ⊂ w. if f is almost δgβ-continuous, then there exists a δgβ-open set u2 in x such that x ∈ u2 and f(u2)⊂v . put u = (u2∩u2).then u is δgβ-open and g(u) ⊂ u1×v ⊂ w. thus g is almost δgβ-continuous. recall that for a f:x → y, the subset gf = {(x,f(x)):x ∈x}⊂ x×y is said to be graph of f. 99 jagadeesh b.toranagatti definition 3.5. a graph gf of a function f:x → y is said to be strongly δgβclosed if for each (p,q) /∈ gf , there exist v∈δgβo(x,p) and w∈ro(y,q) such that (v×w)∩ gf = ϕ. lemma 3.3. for a graph gf of a function f: x → y, the following properties are equivalent: (i) gf is strongly δgβ-closed in x×y; (ii) for each (p,q) /∈gf , there exist u∈δgβo(x,p) and v∈ro(y,q) such that f(u)∩v = ϕ. theorem 3.19. let f: x → y have a strongly δgβ-closed graph gf . if f is injective, then x is δgβ-t1. proof:let x1,x2∈x with x1 ̸=x2.then f(x1)̸=f(x2) as f is injective so that (x1,f(x2)) /∈gf .thus there exist u∈δgβo(x,x1) and v∈ro(y,f(x2)) such that f(u)∩v = ϕ.then f(x2) /∈f(u) implies x2 /∈u and it follows that x is δgβ-t1. theorem 3.20. let f: x → y and g: y → z be any two functions. (i) if f is δgβ-continuous and g is almost continuous, then (g◦f) is almost δgβcontinuous. (ii) if f is δgβ-irresolute and g is almost δgβ-continuous,then (g◦f) is almost δgβcontinuous. (iii) if f is almost δgβ-continuous and g is r-map, then (g◦f) is almost δgβcontinuous. proof:(i) let n ∈ ro(z). then g−1(n) is open in y since g is almost continuous. the δgβ-continuity of f implies f−1[g−1(n))] = (g◦f)−1((n)) ∈ δgβo(x). hence g◦f is almost δgβ-continuous. the proofs of (ii) and (iii) are similar to (i). definition 3.6. a function f: x → y is said to be δgβ∗continuous if for each p ∈ x and each n∈o(y,f(p)), there exists m ∈ δgβo(x,p) such that f(m) ⊂ cl(n). theorem 3.21. if f: x → y is δgβ∗-continuous and k is δgβ-compact relative to x, then f(k) is h-closed relative to y. proof: similar to the proof of theorem 3.16 theorem 3.22. if for each pair of distinct points p1 and p2 in a space x, there exists a function f of x into a hausdorff space y such that (i) f(p1) ̸= f(p2), 100 a new class of almost continuity in topological spaces (ii) f is δgβ∗-continuous at p1 and (iii) almost δgβ-continuous at p2,then x is δgβ-t2. proof:as y is hausdorff, there exist disjoint open sets w1 and w2 of y such that f(p)∈ w1, f(q) ∈ w2. hence cl(w1) ∩ int(cl(w2)) = ϕ. since f is δgβ∗-continuous at p1, there exists u1 ∈ δgβo(x,p1) such that f(u1) ⊂ cl(w1). since f is almost δgβ-continuous at p2, there exists u2 ∈δgβo(x,p2) such that f(u2) ⊂ int(cl(w2)). therefore, we obtain u1∩ u2 = ϕ, x is δgβ-t2. 4 conclusion the notions of closed sets and continuous functions have been found to be useful in computer science and digital topology[[khalimsky et al., 1990],[kong et al., 1991]]. professor el-naschie[el naschie, 2000] showed that the notion of fuzzy topology may be related to quantum physics in connection with string theory and ϵ∞ theory. therefore, the fuzzy topological version of the notions and results given in this paper will turn out to be useful in quantum physics. 5 acknowledgment the author is thankful to the karnatak university dharwad for financial support to this research work under karnatak university research seed grant policy (grant no.: 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velicko. h-closed topological spaces. amer. math. soc. transl., 78:103–118, 1968. 103 ratio mathematica 23 (2012), 65–80 issn: 1592-7415 el–hyperstructures: an overview michal novák faculty of electrical engineering and communication, brno university of technology, brno, czech republic novakm@feec.vutbr.cz abstract this paper gives a current overview of theoretical background of a special class of hyperstructures constructed from quasi / partially ordered (semi) groups using a construction known as the ”ends lemma”. the paper is a collection of both older and new results presented at aha 2011. key words: group, hyperstructure, partial ordering, quasi ordering, semigroup. msc2010: 20n20. 1 introduction this paper is a written version of a lecture given at the 11th international conference on algebraic hyperstructures and applications in october 2011. together with results new at that time i presented some older results published in journals or proceedings of rather local impact thus unknown to the international hyperstructure community. the new results presented at the conference were meanwhile published as [20]. therefore, this article instead of presenting new unpublished results gives an overview of what has so far been achieved in the area of el–hyperstructures. for proofs of respective theorems as well as for examples explaining their meaning cf either [20] or works indicated throughout the paper. where not stated otherwise all definitions of hyperstructure concepts or properties of hyperstructures are used in the sense of the standard book [3]. michal novák 2 the ”ends lemma” the el–hyperstructures are hyperstructures constructed from quasi / partially ordered (semi)groups using the ”ends lemma”, which has the form of the following theorems from [4]. the notation [a)≤ used below stands for the set {x ∈ h; a ≤ x}, where the properties of h are specified in the respective theorems (typically (h, ·,≤) is a quasi-ordered / partially ordered set / semigroup / group). lemma 2.1 ([4], theorem 1.3, p. 146) let (s, ·,≤) be a partially ordered semigroup. binary hyperoperation ∗ : s ×s →p′(s) defined by a∗ b = [a · b)≤ (1) is associative. the semi-hypergroup (s,∗) is commutative if and only if the semigroup (s, ·) is commutative. in accordance with other papers regarding this topic, the hyperstructure (s,∗) constructed in this way will further on be called the associated hyperstructure to the single-valued structure (s, ·) or an ”ends lemma”–based hyperstructure, or an el–hyperstructure for short. instead of s the carrier set will from lemma 2.3 onward be denoted by h. lemma 2.2 ([4], theorem 1.4, p. 147) let (s, ·,≤) be a partially ordered semigroup. the following conditions are equivalent: 10 for any pair (a,b) ∈ s2 there exists a pair (c,c′) ∈ s2 such that b · c ≤ a and c′ · b ≤ a 20 the associated semi-hypergroup (s,∗) is a hypergroup. remark 2.1 if (s, ·,≤) is a partially ordered group, then if we take c = b−1 ·a and c′ = a · b−1, then condition 10 is valid. therefore, if (s, ·,≤) is a partially ordered group, then its associated hyperstructure is a hypergroup. remark 2.2 the wording of the above lemmas is the exact translation of theorems from [4]. the respective proofs, however, do not change in any way, if we regard quasi-ordered structures instead of partially ordered ones as the anti-symmetry of the relation ≤ is not needed (with the exception of the ⇐ implication of the part on commutativity, which does not hold in this case). the often quoted version of the ”ends lemma” is therefore the version assuming quasi–ordered structures. 66 el–hyperstructures: an overview the ”ends lemma” was later extended (cf e.g. [23]). notice that if (h, ·) is commutative, then (h,∗) is a join space. also notice that unlike in the original ”ends lemma” the underlying single-valued structure in the following theorem is a group (not a semigroup). lemma 2.3 ([23], theorem 4) let (h, ·,≤) be a quasi-ordered group and (h,∗) be the associated hypergroupoid. then (h,∗) is the transposition hypergroup. initially, the typical use of the ”ends lemma” was creating hyperstructures and proving or deriving their properties at random without any (or with a very limited) theoretical background. this model is used in e.g. [6, 8, 13, 21, 23]. in order to overcome this inconvenience, theoretical background of the ”ends lemma” is being developed. 3 extending the lemma, identities and inverses after the ”ends lemma” extension, i.e. lemma 2.3, was proved, there arose a question of whether one can go any further to stronger hyperstructures such as canonical hypergroups, strongly canonical hypergroups, etc. a positive answer to this question would mean that numerous ring-like analogies of el–hyperstructures could be studied extensively. unfortunately, the answer – obtained in [18] – turned out to be negative. theorem 3.1 let (h, ·,≤) be a non-trivial quasi-ordered group, where the relation ≤ is not the identity relation, and let (h,∗) be its associated transposition hypergroup. then: 1. (h,∗) does not have a scalar identity. 2. regardless of commutativity, (h,∗) cannot be a canonical hypergoup. naturally, if we regard the definition of a canonical hypergroup, 2 immediately follows from 1. once it was established that looking for scalar identities in el–hyperstructures based on groups is of no point, at least the issue of identities was explored. in [18] the following simple results concerning identities were obtained. theorem 3.2 let (h,∗) be the semi-hypergroup associated to a quasi-ordered semigroup (h, ·,≤) with the identity u. 67 michal novák 1. an element e ∈ h is an identity of (h,∗) if and only if e ≤ u. 2. if (h, ·) is a group, then the identity of (h, ·) is an identity of (h,∗). again, 2 is naturally an immediate corollary of 1 yet we set it aside due to uniqueness of the single-valued group identity. lemma 3.1 let (h,∗) be the join space associated to a quasi-ordered commutative group (h, ·,≤). if an element e ∈ h is an identity of (h,∗), then e ≤ e−1. since the concept of an inverse in a hyperstructure is defined using the concept of an identity, the issue of inverses was touched upon in the same paper and the following result was obtained for the set i(a) of inverses of an arbitrary element a ∈ h in (h,∗). theorem 3.3 let (h,∗) be the transposition hypergroup associated to a quasi-ordered group (h, ·,≤). then for an arbitrary a ∈ h there holds i(a) = {a′ ∈ h; a′ ≤ a−1} = (a−1]≤, where a−1 is the inverse of a in (h, ·). corollary 3.1 let (h,∗) be the transposition hypergroup associated to a quasi-ordered group (h, ·,≤). then (h,∗) is regular. 4 ring-like hyperstructures since it turned out that once we start with groups the ”ends lemma” cannot effectively be used to construct canonical hypergroups, the scope for use of this idea in the area of ring-like hyperstructures narrowed. recall that there is a great variety of definitions of ring-like hyperstructures, yet many of them including the most often used one – that of krasner hyperring – is built on canonical hypergroups. however, the idea of limits of the ”ends lemma” in the area of hyperstructures with two (hyper)operations is still worth exploring. in [19] (published before the author was able to get [12]) three possible extensions are suggested and explored: 1. let (h, +) and (h, ·) be two single-valued structures. we can define a hyperoperation using one of the operations + or · by e.g. a∗b = [a+b)≤ – thus we get an el–hyperstructure (h,∗). the hyperstructure will then be a triplet (h,∗, ·) where ∗ is a hyperoperation based on the single-valued operation +. 68 el–hyperstructures: an overview 2. let (h, +) and (h, ·) be two single-valued structures. we can define two hyperoperations, each based on one single-valued operation, i.e. for an arbitrary pair (a,b) ∈ h2 we can define a ∗ b = [a + b)≤ and a ◦ b = [a · b)≤. thus we get a triplet (h,∗,◦), where ∗ and ◦ are hyperoperations. 3. however, we can also start with a single single-valued structure (h, ·) and using it define a hyperoperation ∗ by a ∗ b = [a · b)≤. the hyperstructure will then be a triplet (h,∗, ·) where ∗ is a hyperoperation based on the single-valued operation ·. if we have a triplet (h, +, ·), where symbols + and · may stand for both single-valued and multivalued operation, then for an arbitrary triplet (a,b,c) ∈ h3 we may either ask that a ·(b + c) = a ·b + a ·c and (a + b) ·c = a·c+b·c or we may ask that inclusions holds instead of equalities.1 each of the three ways to create ring-like hyperstructures was explored with respect to both of these types of distributivity and the following results were obtained in [19]. notice the variety of conditions imposed on the respective structures (group / semigroup, quasi-ordered / partially ordered, single-valued / multivalued). definition 4.1 [cf [28], p. 21, included as plain text] (r, +, .) is a hyperring in the general sense if (r, +) is a hypergroup2, (·) is associative hyperoperation and the distributive law3 x(y + z) ⊆ xy + xz, (x + y)z ⊆ xz + yz is satisfied for every x,y,z of r. additive hyperring is the one of which only (+) is a hyperoperation, multiplicative hyperring is the one of which only (·) is a hyperoperation. [. . .] (r, +, ·) will be called semihyperring if (+), (·) are associative hyperoperations, where (·) is distributive with respect to (+). the rest of definitions are analogous. if the equality in the distributive law is valid, then the hyperring is called strong or good. theorem 4.1 let (h, +, ·) be a ring such that (h, +) is a group, (h, ·) a semigroup and ≤ quasi-ordering on h such that (h, +,≤) is a quasi-ordered group and (h, ·,≤) is a quasi-ordered semigroup. further, for an arbitrary pair of elements (a,b) ∈ h2 define a∗b = [a + b)≤ and a◦b = [a ·b)≤. then (h,∗,◦) is a hyperring in the general sense. 1the former distributivity is sometimes called good distributivity while the latter is often called weak distributivity. 2vougiouklis uses the term hypergroup of marty. 3vougiouklis uses the sign ⊂ in the sense of ⊆. 69 michal novák theorem 4.2 let (h, +) be a semigroup, (h, ·) a group and ≤ quasi-ordering on h such that (h, +,≤) is a quasi-ordered semigroup and (h, ·,≤) is a quasi-ordered group. further, for an arbitrary pair of elements (a,b) ∈ h2 define a ∗ b = [a + b)≤ and a ◦ b = [a · b)≤. finally, let · distribute over + from both left and right. then (h,∗,◦) is a good semihyperring in the sense of definition 4.1. theorem 4.3 let (h, +, ·) be a ring such that (h, +) is a group with neutral element 0, (h\{0}, ·) a group and ≤ quasi-ordering on h such that (h, +,≤ ) and (h, ·,≤) are quasi-ordered groups. further, for an arbitrary pair of elements (a,b) ∈ h2 define a∗b = [a+b)≤ and a◦b = [a·b)≤. then (h,∗,◦) is a good hyperring in the general sense. theorem 4.4 let (h, +) be a group and (h, ·,≤) a quasi-ordered semigroup and for an arbitrary pair of elements (a,b) ∈ h2 define a◦b = [a·b)≤. further, let (h, +, ·) be such that the operation · distributes over the operation + from both left and right. then (h, +,◦) is a good multiplicative hyperring. definition 4.2 [cf [14], definition 2.1 and remark] a hyperalgebra (r, +, ·) is called a semihyperring if and only if (i) (r, +) is a semihypergroup; (ii) (r, ·) is a semigroup; (iii) ∀a,b,c ∈ r, a · (b + c) = a · b + a · c and (b + c) ·a = b ·a + c ·a if we replace (iii) by ∀a,b,c ∈ r,a · (b + c) ⊆ a · b + a · c and (b + c) ·a ⊆ b ·a + c ·a we say that r is a weak distributive semihyperring. a semihyperring is called with zero element, if there exists a unique element 0 ∈ r such that 0 + x = x = x + 0 and 0 ·x = 0 = x · 0 for all x ∈ r. [. . .] a semihyperring is called a hyperring provided (r, +) is a canonical hypergroup. theorem 4.5 let (h, ·,≤) be a quasi-ordered semigroup such that · is a commutative idempotent operation. further, for an arbitrary pair of elements (a,b) ∈ h2 define a ◦ b = [a · b)≤. then (h,◦, ·) is a weak distributive semihyperring. unfortunately, from theorem 3.1 there follows that krasner hyperrings cannot be constructed using the ”ends lemma” if the underlying singlevalued structure (h, +) is a group. however, there are weaker structures 70 el–hyperstructures: an overview such as e.g. hyperringoids, which are defined as semihyperrings in the sense of definition 4.2 where (r, +) is a join space4, for the construction of which the ”ends lemma” might still be used. the assumptions of the following theorem seem rather complicated. the reason is simple: the requirement ”(h, +) is a group” results in trivialities. notice that condition 1 is the condition used in lemma 2.1 – the one which secures that (h,∗) is a hypergroup. theorem 4.6 let (h, +) be a commutative semigroup, (h, ·) a group and ≤ quasi-ordering on h such that 1. to every pair of elements (a,b) ∈ h2 such that a ≤ b there exists a pair of elements (c,c′) ∈ h2 such that b + c ≤ a, c′ + b ≤ a, 2. (h, +,≤) is a quasi-ordered semigroup and 3. (h, ·,≤) is a quasi-ordered group. moreover, for an arbitrary pair of elements (a,b) ∈ h2 define a∗b = [a+b)≤. finally, suppose that · distributes over + from both left and right. then if (h,∗) satisfies the transposition axiom, then (h,∗, ·) is a hyperringoid. corollary 4.1 if in theorem 4.6 we suppose that (h, +,≤) is a quasiordered semigroup without any further assumptions, then (h,∗, ·) is a semihyperring in the sense of definition 4.2. theorem 4.7 let (h, +,≤) be a non-trivial quasi-ordered group with neutral element 0 such that ≤ is not the identity relation, (h \{0}, ·) a group. moreover, for an arbitrary pair of elements (a,b) ∈ h2 define a∗b = [a+b)≤. finally, suppose that · distributes over + from both left and right. then (h,∗, ·) is a weak distributive hyperringoid. 5 the issue of a subhyperstructure in order to proceed to the study of properties of el–hyperstructures, one must clarify the concept of a subhyperstructure of an el–hyperstructure 4this definition is used in [3], chapter 6. on contrary massouros brothers in [17] call this hyperstructure a join hyperringoid, while they call hyperringoid a hyperstructure such that (r, +) is a hypergroup only. further on in theorem 4.6 the definition used in [3] is regarded. 71 michal novák since there are two possible approaches to it. if we regard a quasi-ordered semigroup (h, ·,≤) and define a hyperoperation ∗ on h by a∗ b = [a · b)≤ = {x ∈ h; a · b ≤ x} (2) for an arbitrary pair of elements (a,b) ∈ h2, we may in a subsemigroup (g, ·) of the semigroup (h, ·) set either a∗g b = [a · b)≤g = {x ∈ g; a · b ≤ x} (3) or a∗h b = [a · b)≤h = {x ∈ h; a · b ≤ x} (4) and thus create either (g,∗g) or (g,∗h ). none of these concepts is obviously the only possible and ”correct” one since the definition of both can be justified. in a way, the idea of ∗h conforms to the idea of the ”ends lemma” better. thus in [22], which discussed the issue of subhyperstructures of el– hyperstructures, it was this concept that was favoured. the concept of the upper set was introduced.5 definition 5.1 let (h, ·,≤) be a partially ordered semigroup and let g be a nonempty subset of h. 1. if for an arbitrary element g ∈ g there holds [g)≤ ⊆ g, we call g an upper end of h. 2. if there exists an element g ∈ g such that there exists an element x ∈ h \ g such that g ≤ x (i.e. x ∈ [g)≤)6, we say that g is not an upper end of h because of the element x. among other results concerning hyperoperation ∗h (4) the following was proved. theorem 5.1 let (h,∗) be the semihypergroup associated to a quasi-ordered semigroup (h, ·,≤). suppose that g is an upper end of h. if (g, ·) is a subgroup of (h, ·), then (g,∗) is a subhypergroup of (h,∗). 5the concept itself is naturally not a new invention; the definition was only tailored for use in the ”ends lemma” context. 6we could – probably more properly since x 6∈ g – write g < x and x ∈ [g)≤ \{g} yet in the definition we keep the ≤ notation of the ends lemma for consistency reasons. 72 el–hyperstructures: an overview proposition 5.1 let (h,∗) be the semihypergroup associated to a partially ordered semigroup (h, ·,≤) and g ⊆ h nonempty. if (g,∗) is a subhypergroup of (h,∗), then (g, ·) is a subsemigroup of (h, ·) and g is an upper end of h such that for any pair (a,b) ∈ g2 there exists a pair (c,c′) ∈ g2 such that b · c ≤ a and c′ · b ≤ a. theorem 5.2 let (h,∗) be the semihypergroup associated to a partially ordered semigroup (h, ·,≤). further, let g ⊆ h be non-empty and such that (g, ·) is a subgroupoid of (h, ·), and let the relation ≤g be a restriction of ≤ on g, i.e. ≤g=≤ ∩(g × g).7 finally – if it exists – denote u the identity of (h, ·) and define a new hyperoperation ∗g : g×g → p∗(g) for arbitrary elements a,b ∈ g by (3), i.e. by a∗g b = [a · b)≤g = {x ∈ g; a · b ≤g x}. then 1. (g, ·) is a semigroup if and only if (g,∗g) is a semihypergroup. 2. (g, ·) is a monoid if and only if (g,∗g) is a semihypergroup and u ∈ g. 3. if (g, ·) is a group, then (g,∗) is a transposition hypergroup. 4. if (g,∗) is a hypergroup, then (g, ·) is a semigroup such that for any pair (a,b) ∈ g2 there exists a pair (c,c′) ∈ g2 such that b · c ≤ a and c′ · b ≤ a. 6 properties of el–hyperstructures and their subhyperstructures results in this section were presented at aha 2011 and later included in [20]. theorem 6.1 let (h,∗) be the hypergroup associated to a quasi-ordered group (h, ·,≤) and (g, ·) its subgroup such that g is an upper end of h. then (g,∗), where ∗ is defined for an arbitrary pair (a,b) ∈ h2 as a∗b = [a·b)≤h , is invertible and closed in h. 7the exact quote from [22] at this place reads ”for arbitrary elements a, b ∈ g let a ≤ b ⇒ a ≤g b”. 73 michal novák theorem 6.2 let (h,∗) be the hypergroup associated to a quasi-ordered group (h, ·,≤) and (g,∗) its arbitrary subhypergroup associated to a subgroup (g, ·) of (h, ·), where g is an upper end of h (i.e. as defined in theorem 5.1 using hyperoperation ∗h ). denote u the identity of (h, ·). then 1. g is ultraclosed if and only if for any h ∈ h such that h ≤ u it follows that h ∈ g. 2. if g 6= h and if (h, ·,≤) has the smallest element, then (g,∗) is not ultraclosed. 3. if (h, ·) or (h,∗) is commutative, then g is a complete part of h if and only if for every h ∈ h such that h ≤ u there is h ∈ g. theorem 6.3 let (h,∗) be the associated hypergroup of a quasi-ordered group (h, ·,≤) and (g,∗) its arbitrary subsemihypergroup associated to a subsemigroup (g, ·) of (h, ·).8 if for arbitrary x ∈ h and g ∈ g there holds x ·g ·x−1 ∈ g, then (g,∗) is normal. corollary 6.1 let (h,∗) be the hypergroup associated to a quasi-ordered group (h, ·,≤) and (g, ·) its normal subgroup such that g is an upper end of h. then (g,∗), where ∗ is defined as a∗ b = [a · b)≤h , is reflexive. theorem 6.4 let (h,∗) be the hypergroup associated to a quasi-ordered group (h, ·,≤). then (h,∗) is reversible. as far as regularity of el–hyperstructures is concerned, cf theorem 3.3 and its corollary. in the following theorem notice that by a subhypergroup we mean a subhyperstructure defined by hyperoperation ∗h (4). this is important to consider since the definition of inner irreducibility relies on subhyperstructures as a commutative hypergroup (h,∗) is called inner irreducible if for any pair of its subhypergroups g1,g2 such that g1 ∗g2 = h there holds g1 ∩g2 6= ∅. theorem 6.5 let (h,∗) be the associated hypergroup of a partially ordered commutative group (h, ·,≤). 1. if for every x ∈ h such that x,x−1 are incomparable with respect to ≤ there is either [x)≤ ∩ [x−1)≤ 6= ∅ or (x]≤ ∩ (x−1]≤ 6= ∅, then (h,∗) is inner irreducible. 8in this context the fact whether we define a subhyperstructure by means of ∗h (4) or ∗g (3) is not important. 74 el–hyperstructures: an overview 2. if (h,≤) is a linear ordered set or if (h,≤) has the smallest or the greatest element, then (h,∗) is inner irreducible. naturally, 2 is a corollary of 1 since in linear ordered sets all elements are comparable. 7 the issue of origins of a hypergroup the ”ends lemma” describes a way to construct semihypergroups from quasi-ordered semigroups and hypergroups from semigroups with a special property. we know that groups are such structures that this property holds trivially. this means that we know that using the ”ends lemma” we may create a hypergroup from a group. thus one can ask: if have a hypergroup (itself or as a subhypergroup of a larger structure) created in the ”ends lemma” fashion, is there a way to determine whether its underlying single-valued structure is a group or a semigroup? answering this question is not academic only as proofs of some of the above theorems have to answer this question in a rather complicated way. this issue is also connected to the issue of the converse of the ”ends lemma”, which was already necessary to complete some proofs of theorems on subhyperstructures. the proof of the following theorem can be found in [22]. theorem 7.1 let (h, ·) be a non-trivial groupoid and ≤ a binary partial ordering on h such that for an arbitrary pair of elements (a,b) ∈ h2, a ≤ b, and for arbitrary c ∈ h there holds c·a ≤ c·b and a·c ≤ b·c. further define a hyperoperation ∗ : h×h →p∗(h) for an arbitrary pair of elements (a,b) ∈ h2 by a∗ b = [a · b)≤ = {x ∈ h; a · b ≤ x}. then if the hyperoperation ∗ is associative, then the single-valued operation · is associative too. furthermore, if there exists an element e ∈ h such that for every a ∈ h there holds a ∗ e = e ∗ a = [a)≤, then this element e is the identity of the semigroup (h, ·). notice that if the relation ≤ is not antisymmetric, the above theorem is not true. this is caused by the fact that only for antisymmetric relations ≤ there holds that [a)≤ = [b)≤ implies that a = b. indeed, suppose a simple two element set m = {a,b} where the relation ≤ is defined as a ≤ a,a ≤ b,b ≤ a,b ≤ b. this reflexive and transitive relation ≤ is obviously not antisymmetric and there holds [a)≤ = [b)≤ yet a 6= b. a simple way of distinguishing between semigroups and groups is the study of idempotent elements, i.e. elements which (if we ignore the identity) 75 michal novák exist in semigroups only. in [20] a few basic results concerning idempotent elements are included. theorem 7.2 let (h,∗) be the semihypergroup associated to a quasi–ordered semigroup (h, ·,≤). for an arbitrary element a ∈ h there holds a∗a = {a}⇔ a is an idempotent and simultaneously a maximal element of (h, ·,≤). corollary 7.1 let (h,∗) be the associated hypergroup of such a quasi-ordered semigroup (h, ·,≤) that at least two distinct elements a,b ∈ h are in relation ≤ (i.e. ≤ is not trivial). if there exists an element a ∈ h such that a∗a = {a}, then (h, ·) is not a group. corollary 7.2 let (h,∗) be the hypergroup associated to a quasi-ordered semigroup (h, ·,≤) and (g,∗) a subhypergroup of (h,∗).9 1. denote u the identity of (h, ·). if u is the maximal element of (g,≤) and at least two distinct elements a,b ∈ g are in relation ≤, then (g, ·) is a subsemigroup of (h, ·) yet not a subgroup of (h, ·). 2. if for two distinct elements a,b ∈ h there holds a∗a = {a}, b∗b = {b}, then h does not have the greatest element. also – obviously – (h, ·) is not a group. 8 ”ends lemma” in a broader context naturally, the ”ends lemma” is not a revolutionary stand alone concept. the study of relation of hyperstructures and ordered sets or binary relations is included in [3] as chapter 3. this part of the ”canonical” book on hyperstructures was inspired by works of chvalina (especially [5]) and rosenberg (especially [24]). results concerning the relation of hyperstructures and ordered sets have also been included in [12], another ”canonical” book on hyperstructure theory. among older works concerning relation of ordered sets and hyperstructures there is e.g. [2], which studies the relation in general giving a number of possible ways to create hyperstructures from ordered sets, and [29], in which a concept in a way similar to the ”ends lemma” may be found. recent works related to the concept discussed in this article include e.g. works of cristea and ştefănescu which deal with n–ary relations on hypergroups (e.g. [9, 10]) or study of fundamental relations on hypergroupoids associated with binary relations (such as [11]), or works of e.g. spartalis or massouros (such as [16, 25, 26]). 9the statement is valid for subhypergroups based on both ∗h (4) or ∗g(3). 76 el–hyperstructures: an overview 9 open issues there are many loose ends that wait to be tied. most importantly, the full potential of the property included in lemma 2.1 must be explored. this is closely connected to the problem of reversing the ”ends lemma”, which has been partly answered by theorem 7.1, and to the problem of telling the origins of the hypergroup for which the idea of idempotent elements is only a first and insufficient attempt. clarifying this issue would also help in the study of ring-like el–hyperstructures and in the study of such properties of el–hyperstructures that rely on the concept of a subhypergroup. naturally, it might be very useful to set the issue of el–hyperstructures in a broader perspective of hyperstructures constructed from binary operations of single-valued (semi)groups. references [1] p. corsini, prolegomena of hypergroup theory, aviani editore, tricesimo, 1993. [2] p. corsini, hyperstructures associated with ordered sets, bul. of the greek math. soc. 48 (2003), 7–18. [3] p. corsini, v. leoreanu, applications of hyperstructure theory, kluwer academic publishers, dodrecht – boston – london, 2003. [4] j. chvalina, functional graphs, quasi-ordered sets and commutative hypergroups, masaryk university, brno, 1995 (in czech). [5] j. chvalina, relational product of join spaces determined by quasiorders, in: proceedings of the 6th international congress on aha and appl., democritus university of thrace press, xanthi, 1996. [6] j. chvalina, l. chvalinová, join spaces of linear ordinary differential operators of the second order, folia fsn universitatis masarykianae brunensis, mathematica 13, cdde – proc. colloquium on differential and difference equations, brno, (2002), 77–86. [7] j. chvalina, l. chvalinová, state hypergroups of automata, acta math. et inform. univ. ostraviensis 4(1) (1996), 105–120. [8] j. chvalina, m. novák, hyperstructures of preference relations, in: 10th international congress of algebraic hyperstructures and applications, 77 michal novák proceedings of aha 2008, university of defence, brno, 2009, pp. 131– 140. [9] i. cristea, several aspects on the hypergroups associated with n-ary relations, an. şt. univ. ovidius constanta, 17(3) (2009), 99-110. [10] i. cristea, m. ştefănescu, hypergroups and n-ary relations, european j. combin., 31(2010), 780-789, doi:10.1016/j.ejc.2009.07.005. [11] i. cristea, m. ştefănescu, c. angheluţă, about the fundamental relations defined on the hypergroupoids associated with binary relations, european j. combin., 32(2011), 72–81, doi¿10.1016/j.ejc.2010.07.013. [12] b. davvaz, v. leoreanu fotea, applications of hyperring theory, international academic press, palm harbor, 2007. 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[24] i. g. rosenberg, hypergroups and join spaces determined by relations, ital. j. pure appl. math., 4 (1998), 93–101. [25] s. spartalis, m. konstantinidou-serafimidou, a. taouktsoglou, chypergroupoids obtained by special binary relations, comput. math. appl. 59 (8) (2010), 2628–2635, doi: 10.1016/j.camwa.2010.01.031. [26] s. spartalis, c. mamaloukas, hyperstructures associated with binary relations, comput. math. appl., 51 (2006), 41–50, doi:10.1016/j.camwa.2005.07.011. [27] t. vougiouklis, generalization of p –hypergroups, rend. circ. mat. palermo, 36(ii) (1987), 114–121. [28] th. vougiouklis, on some representations of hypergroups, annales scientifiques de l’université de clermont-ferrand 2, tome 95, série mathematiques, 26 (1990), 21–29. [29] t. vougiouklis, representations of hypergroups by generalized permutations, algebr. universalis 29 (1992), 172–183. 79 michal novák 80 ratio mathematica volume 44, 2022 semi generalization of δi*-closed sets in ideal topological space dr k palani 1 m karthigai jothi2 abstract in this paper we introduce the notion of semi generalized i*-closed sets or gsi*closed sets using semi open sets and investigate its basic properties and characterizations in an ideal topological space. this class of sets is properly lies between the class of i*-closed sets and the class of g-closed sets. also, study the relationship with various existing closed sets in ideal topological spaces. moreover, we introduce and study the concept of maximal gsi*-closed sets. keywords: ideal topological space, i*-closed sets, gsi*-closed sets. 2010ams subject classification: 05c693 1associate professor and head, pg & research department of mathematics, a.p.c mahalaxmi college for women, thoothukudi-2. tamilnadu, india.e-mail: palani@apcmcollege.ac.in 2research scholar, reg. no: 21212012092005, pg & research department of mathematics, a.p.c mahalaxmi college for women, thoothukudi-2. manonmaniam sundaranar university, abishekapatti, tirunelveli -12. tamilnadu, india.e-mail: jothiperiyasamy05@gmail.com. 3received on june 10 th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.925. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 354 mailto:palani@apcmcollege.ac.in mailto:jothiperiyasamy05@gmail.com dr k palani m karthigai jothi 1. introduction and preliminaries an ideal i is a non-empty collection of subsets of x which satisfies: (i) a∈i and b ⊆ a implies b ∈i, and (ii) a∈i and b ∈i implies a ∪ b ∈i. given a topological space (x, τ) with an ideal i on x called ideal topological space denoted by (x, τ, i). kuratowski [5] and vaidhyanathaswamy [18] was studied the notion of ideal topological spaces, j. dontchev, m. ganster [3], navaneethakrishnan, p. paulraj joseph [13], d. jankovic, t. r. hamlett [4], m. n. mukherjee, r. bishwambhar, r. sen [10], a. a. nasef, r. a. mahmond [12] etc., were investigated applications to various fields of ideal topology. if p(x) is the collection of all subsets of x a set operator (.)*: p(x) → p(x) called a local function [5] for any subset a of x with respect to i and τ is defined as, a*(i, τ) = {x ∈x: u ∩ a ∉ i for every u ∈ τ(x)}, where τ(x) = {u ∈ τ / x ∈ u}. a kuratowski closure operator cl*(a) for a topology τ*(i, τ) called *topology finer than τ is defined by cl*(a) = a ∪a*(i, τ). a subset a of x is said to be δ-closed [19] set if clδ(a) = a, where clδ(a) = {x x: int(cl(u)) ∩ a , for every u (x)}. the complement of δ-closed set is δ-open set. a subset a subset a of a space (x, ) is an -open [14] (resp. semi open [7]) set if a int(cl(int(a))) (resp. a  cl(int(a))). the complement of a semi open (resp.-open) set is called a semi closed (resp.-open). definition 1.1. let (x,) be a topological space. a subset a of x is said to be (i) a generalized closed (briefly, g-closed) set [6] if cl(a)  u whenever a  u and u is open in (x, ). (ii) a generalized semi closed (briefly, gs-closed) set [1] if scl(a)  u whenever a  u and u is open set in (x, ). (iii) a semi-generalized closed (briefly, sg-closed) set [2] if scl(a)  u whenever a  u and u is semi open set in (x, ). (iv)an-generalized closed (briefly,g-closed) se [8]t if cl(a)  u whenever a  u and u is open in (x, ). (v) a generalized -closed (briefly, g-closed) set [9] if cl(a)  u whenever a  u and u is -open in (x, ). (vi) a ĝ (or) w-closed set [20] if cl(a)  u whenever a  u and u is semi open set in (x, ). definition 1.2. [21] let (x, , i) be an ideal topological space. a subset a of x is said to be an ig-closed set if a*  u whenever a  u and u is open in x. definition 1.3. [21] let (x, , i) be an ideal topological space, a a subset of x and x is a point of x. then 355 semi generalization of δi*-closed sets in ideal topological space (1) x is called a -i-cluster point of a if a ∩int(cl*(u)) ≠ , for each open neighborhood u of x. (2) the family of all -i-cluster points of a is called the -i-closure of a and is denoted by [a]  -i. (3) a subset a is said to be -i-closed if [a]-i = a. the complement of a -i-closed set of x is said to be -i-open. lemma 1.4. [21] let a and b be subsets of an ideal topological space (x, , i). then, the following properties hold. (1) a  [a]-i. (2) if a  b, then [a]-i [b]-i. (3) [a]-i = ∩ {f  x / a  f and f is -i-closed}. (4) if a is -i-closed set of xs for each , then ∩ {a / } is -i-closed. (5) [a]-i is -i-closed. lemma 1.5. [21] let (x, τ, i) be an ideal topological space and τ-i = {a  x / a is -iopen subset of (x, τ, i)}. then τ-i is a topology such that τsτ-i τ, where τs is the collection of -open sets. definition 1.6. [16] let (x, τ, i) be an ideal topological space and a a subset of x. then [a]*(i, τ) = {x ∈x: int[u]δ-i∩ a  for every u ∈τ(x)} is called local δi-closure function of a with respect to the ideal i and topology τ, where τ(x) = {u ∈ τ / x ∈ u}. a subset a is said to be δi-closed if [a]* = a. the complement of δi-closed set is called δi-open set. remark 1.7.[16] always, (i) [a]* is closed, (ii) []* =  and [x]* = x, (iii) a ⊆ [a]*. lemma.1.8. [16] let (x, τ, i) be an ideal topological space and a, b subsets of x. then for local δi-closure functions the following properties hold. (i) if a ⊆b then [a]*⊆[b]*. (ii) [a ∪b] *= [a]* ∪ [b]*. (iii) [a ∩ b] * ⊆[a]* ∩ [b]*. (iv) [[a]*] * = [a]*. lemma 1.9.[16] (i) cl(a) ⊆ [a]*, (ii) a* ⊆ [a]*, (iii) clδ(a) ⊆ [a]*, (iv) [a]δ-i⊆ [a]*. 356 dr k palani m karthigai jothi definition 1.10. [17] a subset a of an ideal space (x, , i) is called gδi*-closed if [a]* ⊆ u whenever a ⊆ u and u is open in (x, , i). the complement of a gδi*-closed set in (x, , i) is called gδi*open set in (x, , i). 2. gsi*closed sets in this section we introduce gsi*-closed sets and discuss the relationship with some existing sets. definition 2.1. a subset a of an ideal topological space (x, , i) is called gsi*-closed if [a]*⊆ u whenever a ⊆ u and u is semi open set in (x, , i). the complement of gsi*-closed set in (x, , i), is called gsi*-open set in (x, , i). theorem 2.2. everyi*-closed set is gsi*-closed. proof. let a be any i*-closed set and u be any semi open set containing a. since a is i*-closed, [a]* = a. therefore, a is gsi*-closed set in (x, , i). remark 2.3. the converse of the above theorem 2.2 is need not be true as shown in the following example 2.4. example 2.4. let x = {a, b, c},  = {x, , {b}, {c, d}, {b, c}, {b, c, d}}, i = {, {d}}. let a = {a, b, c}. then, a is gsδi*-closed but not δi*-closed. theorem 2.5. in an ideal topological space (x, , i), every gsi*-closed set is (i) ĝ -closed set in (x, ). (ii) g-closed (resp. gα, αg, sg, gs) -closed set in (x, ). (iii) ig -closed set in (x, , i). proof. (i) let a be a gsi*-closed set and u be any semi open set in (x, , i) containing a. since a is gsi*-closed, [a]*⊆ u. then cl(a) ⊆ u and hence a is ĝ -closed in (x, , i), by lemma 1.9. (ii) by [20], every ĝ-closed set is g-closed (resp. gα-closed, αg-closed, sg-closed, gsclosed) set in (x, , i). therefore, it holds. (iii) since every g-closed set is ig-closed, it holds. remark 2.6. the following example 2.7 shows that, the converse of the above theorem 2.5 (i) is not always true. example 2.7. let x = {a, b, c, d},  = {x, , {b}, {a, b}, {b, c}, {a, b, c}, {a, b, d}} and i = {, {b}}. let a = {c, d}. then a is ĝ-closed set but not gsi*-closed. remark 2.8. the following examples shows that, the converse of theorem 2.5 (ii) is not true. 357 semi generalization of δi*-closed sets in ideal topological space example 2.9. let x = {a, b, c, d},  = {x, , {b}, {c}, {b, c,}} and i = {, {d}}. let a = {d}. then a is g-closed, g-closed, g-closed but not gsi*-closed. example 2.10. let x = {a, b, c, d},  = {x, , {a}, {c, d}, {a, c, d}, {b, c, d}} and i = {, {a}}. let a = {a, b}. then a is gs-closed and sg-closed but not gsi* closed. remark 2.11. the following example 2.12 shows that, the converse of theorem 2.5 (iii) is not always true. example 2.12. let x = {a, b, c, d},  = {x, , {b}, {a, b}, {b, c}, {a, b, c}, {a, b, d}} and i = {, {b}}. let a = {b}. then a is ig -closed but not gsi*-closed. 3. characterizations in this section we study some of the basic properties and characterizations of gsi*closed sets. theorem 3.1. let (x, , i) be an ideal space and a a subset of x. then [a]* is semi closed. proof. by remark 1.7, [a]* is closed and hence it is semi closed. theorem 3.2. let (x, , i) be an ideal space and a ⊆ x. if a ⊆ b ⊆[a]*, then [a]* = [b]*. proof. since a ⊆ b, [a]*⊆[b]* and since b ⊆[a]*, [b]*⊆[[a]*] * = [a]*, by lemma 1.8 and lemma 1.9. therefore, [a]* = [b]*. theorem 3.3. let (x, , i) be an ideal space. then [a]* is always gsi*-closed for every subset a of x. proof. let [a]*⊆ u, where u is semi open. always, [[a]*] * = [a]*. hence [a]* is gsi*-closed. theorem 3.4. let (x, , i) be an ideal space and a ⊆ x. if sker(a) is gsi*-closed, then a is also gsi*-closed. proof. suppose that, sker(a) is a gsi*-closed set. if a ⊆ u and u is semi open, then sker(a) ⊆ u. since sker(a) is gsi*-closed, [sker(a)]*⊆ u. always, [a]*⊆[sker(a)]*. thus, a is gsi*-closed. the following example 3.5 shows that, the converse of the above theorem 3.4 is not always hold. example 3.5. in example 2.12, let a = {a, b}. then a is gsi*-closed. but, sker(a) = {a ,b, c} is not gsi*-closed. 358 dr k palani m karthigai jothi theorem 3.6. if a is gsi*-closed subset in (x, , i), then [a]* – a does not contain any nonempty closed set in (x, , i). proof. let f be any closed set in (x, , i) such that f ⊆[a]* – a then a ⊆ x – f and x – f is open and hence semiopen in (x, , i). since a is gsi*-closed, [a]*⊆x – f. hence, f ⊆ x – [a]*. therefore, f ⊆ ([a]* – a)  (x – [a]*) = . remark 3.7. the converse of the above theorem 3.6 is not always true as shown in the following example 3.8. example 3.8. let x = a, b, c,  = x,, a, {b}, {a, b} and i = , {c}, {d}, {c, d}. let a = a, b, c. then [a]* – a = x – a, b, c = d does not contain any nonempty closed set. but a is not a gsi*-closed subset of (x, , i). theorem 3.9. for a subset a of an ideal space (x, , i), cl(a) – a is gsi*-closed if and only if a  (x – cl(a)) is gsi*-open. proof. necessity let f = cl(a) – a. by hypothesis, f is gsi*-closed and x – f = x  (x – f) = x  (x – (cl(a) – a)) = a  (x – cl(a)). since x – f is gsi*-open, a (x– cl(a)) is gsi*-open. sufficiency-let u = a  (x – cl(a)). by hypothesis, u is gsi*-open. then x – u is gsi*-closed and x – u = x – (a  (x – cl(a))) = cl(a)  (x – a) = cl(a) – a. hence proved. theorem 3.10. let (x, , i) be an ideal space. then every subset of x is gsi*-closed if and only if every semiopen subset of x is i*-closed. proof. necessity suppose every subset of x is gsi*-closed. if u is a semiopen subset of x, then u is gsi*-closed and so [u]* = u. hence, u is i*-closed. sufficiency suppose a⊆ u and u is semiopen. by hypothesis, u is i*-closed. therefore, [a]*⊆[u]* = u and hence a is gsi*-closed. theorem 3.11. let (x, , i) be an ideal space. if every subset of x is gsi*-closed, then every open subset of x is i*-closed. proof. suppose every subset of x is gsi*-closed. if u is an open subset of x, then u is gsi*-closed and so [u]*⊆ u, since every open set is semiopen. hence, u is i*-closed. theorem 3.12. intersection of a gsi*-closed set and ai*-closed set is always gsi*closed. proof. let a be a gsi*-closed set and g be any i*-closed set of an ideal space (x, , i). suppose a  g ⊆ u and u is semiopen set in x. then, a⊆ u  (x – g). now, x – g is i*-open and hence open and so semiopen set. therefore, u  (x – g) is a semiopen set containing a. but a is gsi*-closed and therefore, [a]*⊆ u (x – g). 359 semi generalization of δi*-closed sets in ideal topological space therefore, [a]* g ⊆ u which implies that, [a  g] *⊆ u. hence, a  g is gsi*closed. theorem3.13. in an ideal space (x, , i), for each x x, either x is semiclosed or xc is gsi*-closed. proof. suppose that x is not a semiclosed set, then xc is not a semiopen set and hence x is the only semiopen set containing xc. therefore, [xc] *⊆ x and hence xc is gsi*-closed in (x, , i). theorem 3.14. every gsi*-closed, semiopen set is i*-closed. proof. let a be a gsi*-closed, semiopen set in (x, , i). since a is semiopen such that a ⊆ a, by hypothesis, [a]*⊆ a. thus, a is i*-closed. corollary 3.15. every gsi*-closed; open set is i*-closed set. theorem 3.16. if a and b are gsi*-closed sets in an ideal topological space (x, , i), then a  b is a gsi*-closed set in (x, , i). proof. suppose that a  b ⊆ u, where u is semi open set in (x, , i). then a⊆ u and b ⊆ u. since a and b are gsi*-closed sets in (x, , i), [a]*⊆ u and [b]*⊆ u. always, [a  b] * = [a]*[b]*. therefore, [a  b] *⊆ u, whenever u is semi open. hence, a  b is gsi*-closed set in (x, , i). theorem 3.17. let (x, , i) be an ideal space. if a is a gsi*-closed subset of x and a ⊆ b ⊆[a]*, then b is also gsi*-closed. proof. the proof is clear. theorem 3.18. a subset a of an ideal space (x, , i) is gsi*-closed if and only if [a]*⊆ sker(a). proof. necessity suppose a is gsi*-closed and x [a]*. if x  sker(a), then there exist a semiopen set u such that a ⊆ u but x  u. since a is gsi*-closed, [a]*⊆ u and so x [a]*, a contradiction. therefore, [a]* sker(a). sufficiency suppose that [a]*⊆ sker(a). if a ⊆ u and u is semiopen then sker(a) ⊆ u and so [a]*⊆ u. therefore, a is gsi*-closed. theorem3.19. let a be a semi set of an ideal space (x, , i). then a is gsi*-closed if and only if a is i*-closed. proof. necessity suppose a is gsi*-closed. then by theorem 3.18, [a]*⊆sker(a) = a, since a is semi set. therefore, a is i*-closed. sufficiency the proof is follows from the theorem 2.2. 360 dr k palani m karthigai jothi definition 3.20. a proper nonempty gsi*-closed subset a of an ideal space (x, , i) is said to be maximal gsi*-closed if any gsi*-closed set containing a is either x or a. example 3.21. let x = {a, b, c, d},  = {x, , {b}, {c, d}, {b, c, d}} and i = {, {d}}. then {a, b, c} is a maximal gsi*-closed set. theorem 3.22. in an ideal space (x, , i), the following are true. (i) let f be a maximal gsi*-closed set and g be a gsi*-closed set. then f  g = x or g ⊆f. (ii) let f and g be maximal gsi*-closed sets. then f  g = x or f = g. proof. (i)let f be a maximal gsi*-closed set and g be a gsi*-closed set. if f  g = x, then there is nothing to prove. assume that, f  g ≠ x. now, f ⊆ f  g. by theorem 3.16, f  g is a gsi*-closed set. since f is maximal gsi*-closed, we have f  g = x or f  g = f. hence, f  g = f and so g ⊆ f. (ii) let f and g be maximal gsi*-closed sets. if f  g = x, then there is nothing to prove. assume that, f  g ≠ x. then by (i), f ⊆ g and g ⊆ f, which implies that, f = g. theorem 3.23. a subset a of an ideal space (x, , i) is gsi*-open if and only if f ⊆[a]int* whenever f is semiclosed andf ⊆ a. proof. necessity suppose a is gsi*-open and f be a semiclosed set contained in a. then x – a ⊆ x – f and hence [x – a]*⊆ x – f. thus, f ⊆ x – [x – a]* = [a]int*. sufficiency suppose x – a ⊆ u, where u is semiopen. then x – u ⊆ a and x – u is semiclosed. then x – u ⊆[a]int*, which implies [x – a]*⊆u. therefore, x – a is gsi*-closed and hence a is gsi*-open. theorem 3.24. if a is a gsi*-open subset of an ideal space (x, , i) and [a]int*⊆ b ⊆a. then b is also a gsi*-open subset of (x, , i). proof. suppose f ⊆ b, where f is semiclosed set. then, f ⊆ a. since a is gsi*-open, f ⊆[a]int*. since [a]int*⊆[b]int*, we have f ⊆[b]int*. by the above theorem 3.23, b is gsi*-open. references [1] s. p. arya, t. nour, characterizations of s-normal spaces, indian j. pure appl. math., 21 (8), 717 719. 1990. 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[21] s. yuksel, a. acikgoz, t. noiri, on δ-i-continuous functions, turk j math, 29, pp.39-51, 2005. 362 ratio mathematica volume 42, 2022 an efficient block-based image compression and quality-wise decompression algorithm sadanandan sajikumar * john dasan † vasudevan hema ‡ abstract in this paper, we propose a block-based lossy image compression algorithm that makes use of spatial redundancies of neighboring pixels in image data. compression is achieved by replacing a block of pixels with their statistical mean. the algorithm helps in decompressing the image at different quality levels. quality matrices constructed from the quantization table of the jpeg baseline algorithm are used to achieve different qualities of the reconstructed data. experimental results show that the proposed method outperforms existing polynomial-based algorithms both in computation time and complexity. keywords: lossy compression;jpeg compression; polynomial-based compression. 2020 ams subject classifications: 97m10 1 *college of engineering trivandrum, thiruvananthapuram, kerala, india; sajikumar.s@cet.ac.in. †college of engineering trivandrum, thiruvananthapuram, kerala, india; dasanj@cet.ac.in. ‡college of engineering trivandrum, thiruvananthapuram, kerala, india; hema@cet.ac.in. 1received on april 23rd, 2022. accepted on june 24th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v41i0.771. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 205 s. sajikumar, j. dasan, v. hema 1 introduction image compression is a process that reduces the size of image data files while keeping necessary information. we can classify image compression schemes into four groups according to the process element as pixel-based, block-based, subband-based, and region-based. image compression has been a hot topic of research for many years and several image compression standards have been developed ([sonal, 2007], [memon and sayood, 1995]). among these, the blockbased compression scheme jpeg that uses discrete cosine transform (dct) and the subband-based scheme jpeg 2000 has got much attention and popularity ([wallace, 1992], [rabbani, 2002]). since these two methods involve transforms such as dct and wavelet transform, their computational complexity is very high. researches are still going on in developing simple and fast compression algorithms that can show better performance than the existing one. as an alternative to transform-based techniques, polynomial-based compression and statistical approach in compression are also developed([shukla et al., 2005], [ameer, 2009], [sajikumar and anilkumar, 2017], [sajikumar et al., 2021]). even though many algorithms have been reported in this field, research is still needed to cope with the continuous demand for efficient transmission or storage of image data. if the information retained after decompression is 100%, the compression method is called lossless otherwise it is lossy. if we take a pixel in an image at random there is a good chance that its neighbours will have the same intensity or very similar intensity. typically hence, image compression is based on the fact that the neighbouring pixels are highly correlated ([salomon, 2007], [sayood, 2012]). most image compression methods exploit this feature to obtain efficient compression. lossless compression can be achieved with the techniques like run length encoding (rle), huffman coding, arithmetic coding, etc.([gallager, 1978], [jain, 1989], [taubman and marcellin, 2012], [witten et al., 1987]). lossy techniques include transform coding methods such dct/jpeg, jpeg2000, etc. ([pennebaker and mitchell, 1992], [gonzalez and woods, 2008], goyal [2001]). polynomial-based compression is another type of lossy compression method ([sadeh, 1996], [eden et al., 1986]). s. sajikumar and a. k. anilkumar [sajikumar and anilkumar, 2017] introduced a compression scheme using chebyshev polynomials. lossy compression techniques tested for their performance based on three commonly used measures, the root mean square error (rmse), peak signal to noise ratio (psnr) and the compression ratio (cr). the rmse between original image f(x, y) and reconstructed image f̂(x, y) of size m × n is defined by [joshi, 2018]: rmse = [ 1 mn m−1∑ x=0 n−1∑ y=0 [ f(x, y) − f̂(x, y) ]2]12 (1) 206 an efficient block-based image compression and quality-wise decompression algorithm for an 8bit gray level image, psnr = 10 log10 ( 2552 mse ) (db) (2) cr = compressed image size uncompressed image size % (3) in digital image compression, the basic data redundancies are due to coding redundancy, inter-pixel redundancy, and psycho-visual redundancy. statistical approaches in image compression are the compression techniques that try to decorrelate this inter-pixel redundancy. dimensionality reduction is another aspect of image compression. the principal component analysis (pca) is the significant one in this area ([du and fowler, 2007], [sonal, 2007]). the pca approach is implemented via the statistical approach and the neural network approach [dony and haykin, 1995]. to reduce the storage space required, we can make use of statistical measures of central dispersion such as mean and variance of pixel values in an image [sajikumar et al., 2021]. this paper presents a simple and efficient lossy image compression method using the mean value of a block of pixels. the input image is partitioned into non-overlapping blocks and the mean pixel value for each block is used to represent the entire block of pixels followed by a quality gradation at the decompression stage. we compare the proposed method with polynomial-based compression techniques such as plane fitting model and chebyshev polynomial surface fit method ([ameer and basir, 2006], [sajikumar and anilkumar, 2017]). in comparison, it is found that the proposed method outperforms these algorithms. 2 proposed method divide the input image matrix into non-overlapping blocks of size n×n. subtract 128 from each pixel in the image matrix to change the gray levels from [0, 255] to values centered about zero. thus the modified range becomes [−128, 127]. compute the mean value of each block of pixels and store this mean for each block as the reconstruction parameter. at the reconstruction stage, replace all pixels in each block by the respective mean values. that is, n2 pixel values in each block are replaced by a single parameter and hence high compression can be achieved as the block size increases. to reduce the loss of information at the decompression stage, a quality matrix of dimension n×n is introduced. this matrix allows us to decompress output images at different quality levels. the quality determination process outputs images at different bit-rates of lower to a higher order. we have adopted this matrix from 207 s. sajikumar, j. dasan, v. hema the jpeg’s baseline compression algorithm where it is used as the quantization matrix ([wallace, 1992], [ahumada jr and peterson, 1992], [watson, 1993]). q50 =   16 11 10 16 24 40 51 61 12 12 14 19 26 58 60 55 14 13 16 24 40 57 69 56 14 17 22 29 51 87 80 62 18 22 37 56 68 109 103 77 24 35 55 64 81 104 113 92 49 64 78 87 103 121 120 101 72 92 95 98 112 100 103 99   (4) if n is the visual quality level of the decompressed image, then we can obtain different quality matrices qn using the following equation [khedr and abdelrazek, 2016]: qn =   ( 100 − n 50 ) q50, n > 50 (50 n ) q50, n < 50 (5) we have considered submatrices of size n × n with elements taken in order from the top left corner of the matrix q50. the quality matrix q50 for different block sizes 2 × 2 , 3 × 3, 4 × 4 are: ( 16 11 12 12 ) ,  16 11 1012 12 14 14 13 16  ,   16 11 10 16 12 12 14 19 14 13 16 24 14 17 22 29   3 experimental results test images of size 256 × 256 with gray levels in the range [0, 256] are considered. experimental results with block sizes 2 × 2, 3 × 3, 4 × 4 are given in tables 1-3 and figures 1-5. compression qualities are analyzed at different levels 5, 10, 50, 90, and 95. decompressed images at these levels are given in figures 2-5. in 2 × 2 blocks, four gray values are replaced by the mean and hence save 75% storage space with cr 25%. at this cr, all the test images show reasonable reconstructed image quality with low rmse and exhibit superior performance to polynomial-based compression schemes. with the 25% cr and quality index 95, rice image shows psnr value 31.5585 (db), lena 27.5988 (db), and cameraman 208 an efficient block-based image compression and quality-wise decompression algorithm 25.4852(db). these results are promising in comparison with the polynomialbased algorithms. detailed comparison results with different block sizes are given in the next section. for 3×3 blocks, cr is 11.11% and it is 6.25% for 4×4 blocks. as the block size increases reconstruction quality becomes poor with a marginal increase in rmse. test image cr % performance quality index 95 90 50 10 5 rice 25 psnr 31.5585 31.5340 30.4099 21.0875 16.0389 rmse 6.7393 6.7583 7.6921 22.4491 40.2340 lena 25 psnr 27.5988 27.5823 27.0974 21.7713 16.3263 rmse 10.6317 10.6518 11.2634 220.7958 38.9249 cameraman 25 psnr 25.4852 25.4779 25.1629 20.6225 17.0641 rmse 13.5604 13.5720 14.0733 23.7365 35.7547 table 1: compression performance at different reconstruction qualities in the case of 2 × 2 blocks. test image cr % performance quality index 95 90 50 10 5 rice 11.11 psnr 26.0410 26.0311 25.6843 20.7885 15.2735 rmse 12.7202 12.7346 13.2534 23.2871 43.9405 lena 11.11 psnr 23.7492 23.7424 23.5305 19.9055 15.5763 rmse 16.5607 16.5738 16.9831 25.77901 42.4353 cameraman 11.11 psnr 22.1119 22.1079 21.9610 19.4491 16.6085 rmse 19.9961 20.0053 20.3466 21.1698 37.6804 table 2: compression performance at different reconstruction qualities in the case of 3 × 3 blocks. 209 s. sajikumar, j. dasan, v. hema test image cr % performance quality index 95 90 50 10 5 rice 6.25 psnr 25.3334 25.3145 24.8320 18.4780 15.4024 rmse 13.7997 13.8298 14.6197 30.3836 43.2934 lena 6.25 psnr 23.3583 23.3464 23.0116 18.6289 14.3644 rmse 17.3231 17.3468 18.0285 29.8604 48.7890 cameraman 6.25 psnr 21.7695 21.7629 21.5635 17.9073 14.8582 rmse 20.8001 20.8160 21.2994 32.4471 46.0924 table 3: compression performance at different reconstruction qualities in the case of 4 × 4 blocks. figure 1: the first column: orinal images; the second column: reconstructed images using 2×2 blocks at quality index 95; the third column: reconstructed images using 3 × 3 blocks at quality index 95; the fourth column: reconstructed images using 4 × 4 blocks at quality index 95. 210 an efficient block-based image compression and quality-wise decompression algorithm figure 2: top left corner: orinal image; left to right: reconstructed lena image using 2 × 2 blocks at quality indices 5, 10, 50, 90, 95. figure 3: top left corner: orinal image; left to right: reconstructed aerial image using 2 × 2 blocks at quality indices 5, 10, 50, 90, 95. 211 s. sajikumar, j. dasan, v. hema figure 4: top left corner: orinal image; left to right: reconstructed rice image using 2 × 2 blocks at quality indices 5, 10, 50, 90, 95. figure 5: top left corner: orinal image; left to right: reconstructed cameraman image using 2 × 2 blocks at quality indices 5, 10, 50, 90, 95. 212 an efficient block-based image compression and quality-wise decompression algorithm 4 comparison with polynomial models experimental results are compared with the plane fitting model proposed by s. ameer and o. basir [ameer and basir, 2006] and the chebyshev polynomial surface fit model proposed by s. sajikumar and a. k. anilkumar [sajikumar and anilkumar, 2017]. both these methods are block-based algorithms and the proposed method outperforms these two for any block size. comparison results for 2 × 2 blocks and 4 × 4 blocks are given in tables 4-5. the plane fitting model and chebyshev polynomial model have cr’s 75% with 2 × 2 blocks and 18.75% with 4 × 4 blocks respectively. but the proposed method has a cr of only 25% with 2 × 2 blocks and it decreases as the block size increases. even at 25% cr, the proposed method can give an improved result. test image cr % performance plane model chebyshev poly. fit proposed method rice 25 psnr 28.9417 28.9417 31.5585 rmse 9.1104 9.1104 6.7393 lena 25 psnr 26.1790 26.1790 27.5988 rmse 12.5299 12.5299 10.6317 cameraman 25 psnr 23.8796 23.8796 25.4852 rmse 16.3095 16.3095 13.5607 table 4: performance comparison with polynomial fitting model and chebyshev polynomial surface fit model in the case of 2 × 2 blocks. test image cr % performance plane model chebyshev poly. fit proposed method rice 6.25 psnr 24.4904 24.9876 25.3334 rmse 15.1987 14.3527 13.7997 lena 6.25 psnr 23.2617 23.2985 23.3583 rmse 17.5214 17.4356 17.3231 cameraman 6.25 psnr 21.3521 21.3374 21.7695 rmse 21.8174 21.8632 20.8001 table 5: performance comparison with polynomial fitting model and chebyshev polynomial surface fit model in the case of 4 × 4 blocks. 213 s. sajikumar, j. dasan, v. hema 5 conclusions this paper presents a simple and efficient lossy image compression algorithm based on mean values of non-overlapping blocks of pixels. this mean value is taken as the parameter for reconstruction. a method for obtaining decompressed images at desired quality is also implemented. using the quality matrix for different block sizes, the end-user or application has a choice for getting decompressed images according to the use. the proposed method outperforms existing polynomial-based methods in its speed of execution and computational complexity. references albert j ahumada jr and heidi a peterson. luminance-model-based dct quantization for color image compression. in human vision, visual processing, and digital display iii, volume 1666, pages 365–374. international society for optics and photonics, 1992. salah ameer. investigating polynomial fitting schemes for image compression. 2009. salah ameer and otman a basir. a simple three-parameter surface fitting scheme for image compression. in visapp (1), pages 101–106, 2006. robert d dony and simon haykin. neural network approaches to image compression. proceedings of the ieee, 83(2):288–303, 1995. qian du and james e fowler. hyperspectral image compression using jpeg2000 and principal component analysis. ieee geoscience and remote sensing letters, 4(2):201–205, 2007. murray eden, michael unser, and riccardo leonardi. polynomial representation of pictures. signal processing, 10(4):385–393, 1986. robert gallager. variations on a theme by huffman. ieee transactions on information theory, 24(6):668–674, 1978. rafael c gonzalez and richard e woods. digital image processing. nueva jersey, 2008. vivek k goyal. theoretical foundations of transform coding. ieee signal processing magazine, 18(5):9–21, 2001. 214 an efficient block-based image compression and quality-wise decompression algorithm anil k jain. fundamentals of digital image processing. prentice-hall, inc., 1989. madhuri a joshi. digital image processing: an algorithmic approach. phi learning pvt. ltd., 2018. wael m khedr and mohammed abdelrazek. image compression using dct upon various quantization. international journal of computer applications, 137(1): 11–13, 2016. nasir d memon and khalid sayood. lossless image compression: a comparative study. in still-image compression, volume 2418, pages 8–20. international society for optics and photonics, 1995. william b pennebaker and joan l mitchell. jpeg: still image data compression standard. springer science & business media, 1992. majid rabbani. jpeg2000: image compression fundamentals, standards and practice. journal of electronic imaging, 11(2):286, 2002. i sadeh. polynomial approximation of images. computers & mathematics with applications, 32(5):99–115, 1996. s sajikumar and ak anilkumar. image compression using chebyshev polynomial surface fit. int. j. pure appl. math. sci, 10:15–27, 2017. sadanandan sajikumar, john dasan, and vasudevan hema. an image compression method based on ramanujan sums and measures of central dispersion. ratio mathematica, 41:53, 2021. david salomon. a concise introduction to data compression. springer science & business media, 2007. khalid sayood. introduction to data compression. newnes, 2012. rahul shukla, pier luigi dragotti, minh n do, and martin vetterli. ratedistortion optimized tree-structured compression algorithms for piecewise polynomial images. ieee transactions on image processing, 14(3):343–359, 2005. dinesh kumar sonal. a study of various image compression techniques. coit, rimt-iet. hisar, 8:97–102, 2007. david taubman and michael marcellin. jpeg2000 image compression fundamentals, standards and practice: image compression fundamentals, standards and practice, volume 642. springer science & business media, 2012. 215 s. sajikumar, j. dasan, v. hema gregory k wallace. the jpeg still picture compression standard. ieee transactions on consumer electronics, 38(1):xviii–xxxiv, 1992. andrew b watson. dct quantization matrices visually optimized for individual images. in human vision, visual processing, and digital display iv, volume 1913, pages 202–216. international society for optics and photonics, 1993. ian h witten, radford m neal, and john g cleary. arithmetic coding for data compression. communications of the acm, 30(6):520–540, 1987. 216 ratio mathematica volume 41, 2021, pp. 197-205 on cesàro’s means of first order of wavelet packet series manoj kumar* shyam lal† abstract wavelet packets have the capability of partitioning the higher-frequency octaves to yield better frequency localisation. ahmad and kumar [2000] have obtained the pointwise convergence of the wavelet packet series. but till now no work seems to have been done to obtain cesàro summability of order 1 of wavelet packet series. in an attempt to make an advanced study in this direction, a novel theory on (c, 1), cesàro summability of order 1 of wavelet packet series is obtained in this study. keywords: multiresolution analysis, (c, 1) summability, wavelet packets, periodic wavelet packets, wavelet packet expansions. 2020 ams subject classifications: 40a30, 42c15. 1 1 introduction several researchers, including s. e. kelly and raphael [1994a], s. e. kelly and raphael [1994b], kumar and lal [2013], meyer [1992], walter [1992], walter [1995], wickerhauser [1994], have investigated the problem of wavelet packet series convergence and demonstrated that wavelet packets are a basic yet effective wavelet and multiresolution analysis extension. wavelet packet functions are a collection of functions that can be used to create other functions. wavelet packet *applied sciences and humanities department, institute of engineering and technology, lucknow-226021, india; manojkumar@ietlucknow.ac.in. †department of mathematics, institute of science, banaras hindu university, varanasi221005, india; shyam lal@rediffmail.com. 1received on november 18, 2021. accepted on december 12, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.687. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 197 manoj kumar, shyam lal functions are still time-localized, but they have more versatility in describing diverse types of signals than wavelets. wavelet packets, in particular, are better at encoding signals with periodic behaviour. wavelet packets can partition higherfrequency octaves, resulting in more accurate frequency localization. working in slight different directions, ahmad and kumar [2000] have obtained the pointwise convergence of the wavelet packet series. but till now no work seems to have been done to obtain cesàro summability of order 1 of wavelet packet series. it is important to note that cesàro summability is a strong tool to obtain the convergence than that of ordinary convergence. this work establishes a new theory on cesàro summability of order 1 of wavelet packet series in an attempt to make a more advanced study in this field. 2 definitions and preliminaries let l2(r) be the space of measurable and square integrable functions over set of real numbers r . if a function φ ∈ l2(r) generates nested sequences of closed subspaces, it is said to produce an mra (multiresolution analysis), qı = span{φı, : ı, ∈ z}, where φı,(t) = 2ı/2φ(2ıt− ) and z is the set of integers , satisfying the following conditions (i) ... ⊂ q−2 ⊂ q−1 ⊂ q0 ⊂ q1 ⊂ q2 ⊂ ..., i.e. qı ⊂ qı+1, ı ∈ z; (ii) (∪ı∈zqı) = l2(r); (iii) ∩ı∈zqı = {0}; (iv) λ(t) ∈ qı ⇔ λ(2t) ∈ qı+1, ı ∈ z such that φ0, form a riesz basis of {q0}. a function φ which generates a multiresolution analysis, is called a scaling function . wavelet packets can be constructed with the help of multiresolution analysis. we know that if h is a hilbert space with onb (orthonormal basis) {�}∈z then, λ2k = √ 2 ∑ ∈z α2k−�, λ2k+1 = √ 2 ∑ ∈z β2k−�, where {αk}k∈z and {βk}k∈z are in l2(z), are orthonormal bases of two orthogonal closed subspaces h1 and h0 respectively, such that h = h1 ⊕h0. using the foregoing decomposition strategy, we now build the fundamental wavelet packets connected with the scaling function φ ∈ l2(r) which is already defined in multiresolution analysis. 198 on cesàro’s means of first order of wavelet packet series let {ξk, k = 0, 1, 2, ...,} denote a wavelet packet family that corresponds to the scaling function φ which is orthonormal. consider ξ0 = φ. recursively, the wavelet packets ξk, k = 0, 1, 2, ..., are defined by  ξ2k(t) = √ 2 ∑ ∈z hξk(2t− ) ξ2k+1(t) = √ 2 ∑ ∈z gξk(2t− ). (1) as a result, the {ξk} family is a generalisation of the orthonormal wavelet ξ1 = ψ, often known as the mother wavelet. for the hilbert space l2(r), the set {ξk(t−) : k = 0, 1, 2, ...,  ∈ z} form an onb. consider the family of subspaces of l2(r) as pkı = span{2 ıξk(2 ıt− ) :  ∈ z}, ı ∈ z, (2) formed by the family of wavelet packets {ξk} for each k = 0, 1, 2, .... observe that p0ı = qı and p 1 ı = wı, where {qı} is the multiresolution analysis of l2(r) produced by ξ0 = φ and {wı} is the sequence of orthogonal complimentary subspaces generated by the wavelet ξ1 = ψ. the orthogonal decomposition qı+1 = qı ⊕wı, ı ∈ z can then be expressed as p0ı+1 = p 0 ı ⊕p 1 ı . (3) as follows, this orthogonal decomposition can be extended from k = 0 to any k = 1, 2, 3, ... in the form of pkı+1 = p 2k ı ⊕p 2k+1 ı , ı ∈ z. (4) now we’ll state a result that will be employed in the theorem’s proof. the decomposition trick (4) produces wı = p 1 ı = p 2 ı−1 ⊕p 3 ı−1 = p4ı−2 ⊕p 5 ı−2 ⊕p 6 ı−2 ⊕p 7 ı−2 ... = p2  ı− ⊕p 2+1 ı− ⊕ ...⊕p 2+1−1 ı− ... = p2 ı 0 ⊕p 2ı+1 0 ⊕ ...⊕p 2ı+1−1 0 , (5) for each ı = 1, 2, ..., where (2) declares pkı . furthermore, the family { 2 ı− 2 ξr(2 ı−t− l) : l ∈ z } is an onb of prı−, where r = 2  + µ for each µ = 0, 1, 2, ..., 2 − 1,  = 1, 2, ...ı; 199 manoj kumar, shyam lal and ı = 1, 2, .... all of the elements of this base, however, have the same basic shape: ξı,k,(t) = 2 ı/2ξk(2 ıt− ). (6) let λ ∈ l2(r), then the function λ can be approximated by a wavelet packet series as follows: λ(t) ∼ ∑ ı∈z 2r+1−1∑ k=2r ∑ ∈z cl,k,ξl,k,(t), (7) where l = ı− r, r = 0 if ı < 0 and r = 0, 1, 2, ..., ı if ı ≥ 0; and the coefficients cl,k, defined by cl,k, = 〈λ,ξl,k,〉 , (8) are called the wavelet packet coefficients. wavelet packets are a scalable time signal analysis method that combines the advantages of windowed harmonic and wavelet processing. wavelet bundles, which are periodic as well, offer a fascinating supplement to fourier series. using the periodization techniques for period 1 on the basis functions, an mra for l2(r) can be transformed into an mra for l2(0, 1). let {ξk : k ∈ z} denote the family of wavelet packets presented previously which is nonstationary in nature. define general periodic wavelet packets ξperk,ı, by ξ per k,ı, = ∑ l∈z 2ı/2ξk(2 ı(t + l) − ) for 0 ≤  < 2ı and k,ı = 1, 2, 3, · · · . with ξperk , we now define an operator sνλ as follows: (sνλ)(t) = 2r+1−1∑ k=2r ν∑ =0 〈 λ,ξ per l,k, 〉 ξ per l,k,(t). (9) let sk = k∑ ν=0 aν be the kth partial sum of an infinite series ∞∑ k=0 ak. if σk = 1 k+1 k∑ ν=0 sν → s as k →∞ then the series ∞∑ k=0 ak is called summable to s by (c, 1) i.e. cesàro means of order 1 (titchmarsh titchmarsh [1939]). let dµ(µ = 1, 2, 3, · · ·) be the collection of constant dyadic step functions on the intervals [2−µ, (+ 1)2−µ); 0 <  ≤ 2µ. let d = ∪∞µ=1dµ. let b be a banach space and σζ be a bounded linear functional on b which must be generated by any function ζ ∈ d as σζλ = ∫ 1 0 λζ for λ ∈ b. 200 on cesàro’s means of first order of wavelet packet series we have |σζλ| ≤ ‖ζ‖∞‖λ‖b . now if we take b = lq and define ‖ζ‖r = ‖σζ‖ = sup ‖λ‖q≤1 ∫ 1 0 λζ for any ζ ∈ d. (10) then clearly ∣∣∣∣ ∫ 1 0 λζ ∣∣∣∣ ≤‖λ‖q ‖ζ‖r ,λ ∈ lq,ζ ∈ d. (11) let us write πıλ(t) = 2ı−1∑ µ=0 ( 1 µ + 1 µ∑ ν=0 (sνλ)(t) ) δ[µ2−ı,(µ+1)2−ı) = 2ı−1∑ µ=0 σµλ(t)δ[µ2−ı,(µ+1)2−ı) and aı = 2ı−1∑ µ=0 c per l,k,δ[µ2−ı,(µ+1)2−ı), where (9) defines sνλ and δi is the characteristic function on i ⊂ r. we’re going to define an operator now tı(t,x) = 2 −ı 2ı−1∑ =0 c per l,o,φ per ı, (t)φ per ı, (x) = 2−ı 2r+1−1∑ k=2r ∑ µ<ı 2ı−1∑ =0 ξ per l,k,(t)ξ per l,k,(x), where l = µ− r, r = 0 if µ < 0 and r = 0, 1, 2, ...,µ if 0 ≤ µ < ı. in this paper, an estimate for the cesàro summability of wavelet packet series has been determined in the following form: theorem 2.1. let λ be 1-periodic continuous function. then∥∥∥∥∥∥ ( 2−ı 2ı−1∑ µ=0 ∣∣∣∣∣ 1µ + 1 µ∑ ν=0 sνλ ∣∣∣∣∣ r)1/r∥∥∥∥∥∥ ∞ ≤ c‖λ‖∞ (12) if and only if ‖tı‖1 ≤ c‖aı‖q , (13) 201 manoj kumar, shyam lal where c > 0, a constant and 1 < r < ∞. furthermore, lim ı→∞ ‖πıλ(t) −λ(t)‖r = 0 uniformly in [0, 1]. proof. by equation 12 we have ( 2−ı 2ı−1∑ µ=0 ∣∣∣∣∣ 1µ + 1 µ∑ ν=0 sνλ ∣∣∣∣∣ r)1r = ‖πıλ (t)‖r = sup ‖aı‖q≤1 2−ı 2ı−1∑ µ=0 c per l,k,σµλ (t) = sup ‖aı‖q≤1 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 c per l,k,sνλ (t) = sup ‖aı‖q≤1 ∫ 1 0 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 c per l,k,kν (t,x) λ(x)dx ≤ ‖λ‖∞ sup ‖aı‖q≤1 1 µ + 1 µ∑ ν=0 ‖tν(t,x)‖1 ≤ ‖λ‖∞ sup ‖aı‖q≤1 1 µ + 1 µ∑ ν=0 ( c‖aı‖q ) , by (13) = ‖λ‖∞ sup ‖aı‖q≤1 c‖aı‖q ≤ c‖λ‖∞ , where kı (t,x) = 2ı−1∑ =0 φperı, (t)φ per ı, (x) = 2r+1−1∑ k=2r ∑ µ<ı 2ı−1∑ =0 ξ per l,k,(t)ξ per l,k,(x). if, on the other hand, (12) is true, we have 202 on cesàro’s means of first order of wavelet packet series ‖tı(t,x)‖1 = sup ‖λ‖∞≤1 ∫ 1 0 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 c per l,k,tν(0,x)λ(x)dx = sup ‖λ‖∞≤1 ∫ 1 0 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 c per l,k,(2 −ν 2r+1−1∑ k=2r 2ν−1∑ =0 ξ per l,k,(0)ξ per l,k,(x))λ(x)dx = sup ‖λ‖∞≤1 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 c per l,k,(2 −ν 2r+1−1∑ k=2r 2ν−1∑ =0 ξ per l,k,(0)) ∫ 1 0 λ(x)ξ per l,k,(x)dx = sup ‖λ‖∞≤1 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 c per l,k,(2 −ν 2r+1−1∑ k=2r 2ν−1∑ =0 〈 λ,ξ per l,k, 〉 ξ per l,k,(0)) = sup ‖λ‖∞≤1 2−ı 2ı−1∑ µ=0 1 µ + 1 µ∑ ν=0 c per l,k,(sνλ)(0) = sup ‖λ‖∞≤1 2−ı 2ı−1∑ µ=0 c per l,k,(σµλ)(0) = sup ‖λ‖∞≤1 ∫ 1 0 πıλ(0)aı ≤ sup ‖λ‖∞≤1 ‖aı‖q ‖πıλ(0)‖r ≤ ‖aı‖q sup ‖λ‖∞≤1 ∥∥∥∥∥∥ ( 2−ı 2ı−1∑ µ=0 ∣∣∣∣∣ 1µ + 1 µ∑ ν=0 (sνλ)(0) ∣∣∣∣∣ r)1/r∥∥∥∥∥∥ ∞ = ‖aı‖q sup ‖λ‖∞≤1 ∥∥∥∥∥∥ ( 2−ı 2ı−1∑ µ=0 |(σµλ)(0)| r )1/r∥∥∥∥∥∥ ∞ , by (12) ≤ ‖aı‖q sup ‖λ‖∞≤1 c‖λ‖∞ ≤ c‖aı‖q . now πlλ(t) −λ(t) = m∑ µ=0 ((σµλ)(t) −λ(t)) δ[µ2−l,(µ+1)2−l) = m∑ µ=0 1 µ + 1 µ∑ ν=0 ((sνλ)(t) −λ(t)) δ[µ2−l,(µ+1)2−l) 203 manoj kumar, shyam lal for any l ≥ m ≥ 2ı. as a result, ‖πlλ(t) −λ(t)‖ ≤ m∑ µ=0 ∥∥∥∥∥ 1µ + 1 µ∑ ν=0 ((sνλ)(t) −λ(t)) ∥∥∥∥∥ ∞ ∥∥∥δ[0,2−l)∥∥∥r ≤ m∑ µ=0 1 µ + 1 µ∑ ν=0 ‖sνλ−λ‖∞ ∥∥∥δ[0,2−l)∥∥∥r , since the limit of the characteristic function of [0, 2−ı) in all lr-space (1 < r < ∞) is 0 and thus the ultimate result is fallowed. the theorem’s proof is now complete.2 3 conclusions the estimate for the cesàro summability of order 1 of wavelet packet series has been determined in the form of lim ı→∞ ‖πıλ(t) −λ(t)‖r = 0 uniformly in [0, 1]. acknowledgements. one of the authors, shyam lal, is grateful to dst-cims for supporting his work. manoj kumar is thankful to the director, institute of engineering and technology, luchnow for promoting this research activity. references k. ahmad and r. kumar. pointwise convergence of wavelet packet series. atti sem. fis. univ. modena, 48:107–120, 2000. m. kumar and s. lal. on generalized carleson operators of periodic wavelet packet expansions. the scientific world journal, 2013:1–10, 2013. y. meyer. wavelets and operators. cambridge university press, cambridge, 1992. m. a. kon s. e. kelly and l. a. raphael. pointwise convergence of wavelet expansions. bull. amer. math. soc., 30:87–94, 1994a. m. a. kon s. e. kelly and l. a. raphael. local convergence for wavelet expansions. j. funct. anal., 126:102–138, 1994b. 204 on cesàro’s means of first order of wavelet packet series e. c. titchmarsh. the theory of functions. oxford university press, 1939. g. g. walter. approximation of delta function by wavelets. j. approx. theory, 71:329–343, 1992. g. g. walter. pointwise convergence for wavelet expansions. j. approx. theory, 80:108–118, 1995. m. v. wickerhauser. adapted wavelet analysis from theory to software. a.k. peters, ltd., wellesley, 1994. 205 ratio mathematica volume 44, 2022 soft -generalized continuous functions in soft topological spaces c. reena 1 m. karthika 2 abstract the aim of this paper is to define a new class of generalized continuous functions called soft -generalized continuous functions and soft -generalized irresolute functions in soft topological spaces. we discuss several characterizations of soft generalized continuous and irresolute functions and also investigate their relationship with other soft continuous functions. keywords: soft -generalized continuous functions and soft -generalized irresolute functions. 2010 ams subject classification: 54a05, 54a10 3 1 assistant professor, department of mathematics, st. mary’s college (autonomous), (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli), thoothukudi-1, tamilnadu, india. mail id: reenastephany@gmail.com 2 research scholar, reg.no. 21112212092004, department of mathematics, st. mary’s college (autonomous), (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli), thoothukudi-1, tamilnadu, india. mail id: karthikamarimuthu97@gmail.com 3 received on june 21st, 2022. accepted on aug 10 th , 2022. published on nov30th, 2022. doi: 10.23755/rm.v44i0.901. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by license agreement. 145 mailto:reenastephany@gmail.com mailto:karthikamarimuthu97@gmail.com c. reena and m. karthika 1. introduction in 1999 molostsov [6] initiated the study of soft set theory as a new mathematical tool to deal with uncertainties. muhammad shabir [7] and munazza naz (2011) introduced soft topological spaces which are defined over an initial universe with a fixed set of parameters. athar kharal [2] and ahmad introduced the concept of soft mapping. aras [1] and sonmez discussed the properties of soft continuous mappings. akdag m [5] and ozkan introduced soft pre-continuity in soft topological space. the authors [8] of this paper introduced a new glass of generalized closed set called soft -generalized closed sets in soft topological spaces. in this paper, we introduce soft -generalized continuous and soft -generalized irresolute functions in soft topological spaces. we investigate its fundamental properties and find its relation with other soft continuous functions. 2. preliminaries throughout this paper, , and are soft topological spaces. let be a subset of a soft topological space. then , , and denote the soft closure, soft interior, soft generalized closure and soft generalized interior respectively. definition 2.1: [6] let be an initial universe, e be a set of parameters, denote the power set of and be a non-empty set of . a pair is called soft set over , where is a mapping given by . definition 2.2: [7] let be a collection of soft sets over . then is called a soft topology on if i. , belons to . ii. the union of any number of soft sets in belongs to . iii. the intersection of any two soft sets in belongs to . the triplet is called soft topological space over . the members of are called soft open and their complements are called soft closed. definition 2.3. a function is said to be soft continuous [1] (respectively soft semi continuous [4], soft continuous, soft continuous [5], soft regular continuous [3], soft generalized continuous [9], soft generalized continuous and soft generalized pre continuous) if inverse image of every soft closed set in is soft closed (respectively soft semi-closed, soft -closed, soft closed, soft regular closed, soft generalized closed, soft generalized closed and soft generalized pre closed) in . definition 2.4. [8] a subset of a soft topological space is said to be soft generalized closed if whenever and 146 soft -generalized continuous functions in soft topological spaces is soft -open. the complement of soft -generalized closed is called soft generalized open. theorem 2.5. [8] in any topological space , i. every soft closed set is soft -generalized closed. ii. every soft regular-closed set is soft -generalized closed. iii. every soft -closed set is soft -generalized closed. iv. every soft generalized -closed set is soft -generalized closed. v. every soft generalized pre-closed set is soft -generalized closed. remark 2.6: the above theorem is true for soft -generalized open. 3. soft -generalized continuous functions definition 3.1. let and be soft topological spaces. let and be mappings. the function is said tobe soft generalized continuous function if the inverse image of every soft closed set in is soft -generalized closed in . the following soft sets are used in all the examples: let and . then the soft sets are similarly, let and then the soft sets are obtained by replacing , , and by , , and respectively in the above sets. example 3.2. let , , and . define and as , , p ( and . consider the soft topologies and . let be a soft mapping. since and and are soft generalized closed, is soft -generalized continuous. theorem 3.3. let be a soft continuous function. then is soft -generalized continuous. proof: let be a soft closed set in . since is soft continuous, is soft closed. then by theorem 2.5(i), is soft -generalized closed. hence is soft -generalized continuous. 147 c. reena and m. karthika remark 3.4. the converse of the above theorem need not be true as shown in the following example. example 3.5. let , , and . define and as , , p ( and . consider the soft topologies and . let be a soft mapping. since , and are soft -generalized closed but not soft closed, is soft generalized continuous but not soft continuous. theorem 3.6. let be a soft function. i. if is soft regular continuous, then is soft -generalized continuous. ii. if is soft continuous, then is soft -generalized continuous. iii. if is soft generalized continuous, then is soft -generalized continuous. iv. if is soft generalized pre continuous, then is soft -generalized continuous. proof: the proofs are similar to theorem 3.3. remark 3.7. the converse of each of the statements in above theorem need not be true. example 3. 8. let , , and . define and as , , and . consider the soft topologies and . let be a soft mapping. since , and are soft -generalized closed but not soft regular closed, is soft -generalized continuous but not regular continuous. example 3.9. let , , and . define and as , , and . consider the soft topologies and . let be a soft mapping. since and are soft -generalized closed but not soft -closed, is soft -generalized continuous but not soft continuous. example 3.10. let , , and . define and as , , and . consider the soft topologies and . let be a soft mapping. since , and are soft -generalized closed but not soft generalized -closed, is soft -generalized continuous but not soft generalized continuous. 148 soft -generalized continuous functions in soft topological spaces example 3.11. let , , and . define and as , , and . consider the soft topologies and . let be a soft mapping. since is soft -generalized closed but not soft generalized pre-closed, is soft -generalized continuous but not soft generalized pre continuous. remark 3.12. the concept of soft -generalized continuity and soft generalized continuity are independent as shown in the following example. example 3.13. let , , and . define and as , , , . consider the soft topologies and . let be a soft mapping. since and are soft generalized closed but not soft generalized closed, is soft -generalized continuous but not soft generalized continuous. example 3.14. let , , and . define and as , , , . consider the soft topologies and . let be a soft mapping. since is soft generalized closed but not soft -generalized closed, is soft generalized continuous but not soft -generalized continuous. remark 3.15. the concept of soft -generalized continuity and soft continuity are independent as shown in the following example. example 3.16. let , , and . define and as , , , . consider the soft topologies and .let be a soft mapping. since and are soft -generalized closed but not soft -closed, is soft -generalized continuous but not soft -continuous. example 3.17. let , , and . define and as , , , . consider the soft topologies and . let be a soft mapping. since is soft generalized closed but not soft -generalizedclosed, is soft -generalized continuous but not soft -generalized continuous. remark 3.18. the concept of soft -generalized continuity and soft semi continuity are independent as shown in the following example. 149 c. reena and m. karthika example 3.19. let , , and . define and as , , , . consider the soft topologies and . let be a soft mapping. since and are soft -generalized closed but not soft semi closed, is soft -generalized continuous but not soft semi continuous. example 3.20. let , , and . define and as , , , . consider the soft topologies and . let be a soft mapping. since is semi-closed but not soft -generalized closed, is soft semi continuous but not soft generalized continuous. from the above discussion we have the following diagram: theorem 3.21. let be a function. then the following are equivalent. i. is soft -generalized continuous. ii. the inverse image of every soft open set in is soft -generalized closed in . iii. for every subset of , . iv. for every subset of , . proof: (i) (ii): let be soft -generalized continuous and be a soft open set in y. then is soft closed in . since is soft soft generalized pre-continuous soft regular-continuous soft continuous soft generalized -continuous soft -continuous soft generalized-continuous soft semi-continuous soft semi*-continuous soft pre* generalized continuous 150 soft -generalized continuous functions in soft topological spaces generalized continuous is soft -generalized closed in . but . hence is soft -generalized open in x. (ii) (i): suppose the inverse image of every soft open set in is soft -generalized open in . let be soft closed in . then is open in . by assumption is soft -generalized open. . therefore is soft -generalized closed in . hence is soft generalized continuous. (i) (iii): let be a subset of . since is soft -generalized continuous, is soft -generalized closed in . then . now . this proves (ii). (iii) (iv): let be a subset of . then is a subset of . by our assumption, .but . thus . hence . this proves (iii). (iv) (i): let be soft subset of . then . by (iii) . that implies .but . thus and so is soft -generalized closed. hence is soft -generalized continuous. remark 3.22. the composition of two soft -generalized continuous functions need not be soft -generalized continuous as shown in the following example. example 3.23. let , , , , and where . define and as , , , . then the soft mapping is soft -generalized continuous. also, define and as , , , . then the soft mapping is soft -generalized continuous. now let be the composition of two soft generalized continuous functions. since is not soft -generalized closed, is not soft -generalized continuous. 151 c. reena and m. karthika theorem 3.24. if is soft -generalized continuous and is soft continuous then their composition is also soft -generalized continuous. proof: let be soft closed set in . since is soft continuous, is closed in and since is soft -generalized continuous, is soft -generalized closed in . this implies is soft -generalized closed in . thus is soft -generalized closed in for every soft closed subset of . hence is soft -generalized continuous. 4. soft -generalized irresolute functions definition 4.1. a function is said to be soft -generalized irresolute if is soft -generalized closed in for every soft generalized closed set in . example 4.2. let , , and . define and as , , p ( and . consider the soft topologies and . let be a soft mapping. since and and are soft generalized closed, is soft -generalized irresolute. theorem 4.3. if is a soft -generalized irresolute function then is soft -generalized continuous. proof: let be soft closed in . by theorem 2.10(i), is soft generalized closed. since is soft -generalized irresolute function, is soft -generalized closed in .hence is soft -generalized continuous. theorem 4.4. if and are soft generalized irresolute functions then is soft -generalized irresolute. proof: let be soft -generalized closed in . since is soft -generalized irresolute, is soft -generalized closed in . also, since is soft generalized irresolute, is soft -generalized closed in . hence is soft -generalized irresolute. theorem 4.5. let is soft -generalized irresolute and is soft -generalized continuous. then is soft -generalized continuous. proof: let be soft closed set in . since is soft -generalized continuous, is soft -generalized closed in . also, since is soft -generalized irresolute, is soft -generalized closed in .hence is soft -generalized continuous. 152 soft -generalized continuous functions in soft topological spaces theorem 4.6. let be a function. then the following are equivalent. i. is soft -generalized irresolute. ii. the inverse image of every soft -generalized open set in is soft generalized open in . iii. for every subset of . iv. for every subset of . proof: the proof is similar to theorem 3.21. theorem 4.7. a function is soft -generalized iresolute if and only if for every subset of . proof: let be soft -generalized irresolute. let be a subset of . then is soft -generalized open in . since is soft -generalized irresolute, is soft -generalized open in . then . thus . conversely, let be soft -generalized open in . then by (iv), .but . therefore and so is soft generalized open. hence is soft -generalized irresolute. references [1] aras c.g and sonmez a, on soft mappings, arxiv: 1305.4545, (2013). [2] athar kharal and ahmad. b, mappings on soft classes, new math. nat. comput. 7(3), 471-481 (2011) [3] janaki c and jeyanthi v, on soft -continuous in soft topological spaces, international journal of engineering research and technology, issn:2278-0181, vol 3, (2014). [4] mahanta j and das p.k., on soft topological space via semiopen and semiclosed soft sets, arxiv:1203.4133v1 (math.gn), (2012). [5] metin akdag and alkan ozkan, soft alpha-open sets and soft alpha-continuous functions, hindawi publishing corporation, abstract and applied analysis, 2014, 7 pages. 153 c. reena and m. karthika [6] molodtsov d.a., soft set theory-first results, computers and mathematics with applications37 (1999) 19-31. [7] muhammad shabir and munazza naz, on soft topological spaces, computers and mathematics with applications volume 61, issue 7, april 2011, 1786-1799. [8] reena c and karthika m, a new class of generalized closed sets in soft topological spaces, paper presented in the international virtual conference on current scenario in modern mathematics (iccsmm-2022). [9] sattar hameed hamzah and samer adnan jubair, on soft generalized continuous mappings, journal of al-qadisiyah for computer science and mathematics, 8(1) 83-92, (2016). 154 microsoft word articolo2.doc una presentazione dei quaternioni franco eugeni, daniela tondini, annamaria viceconte department of communication science,univeristy of teramo. e-mail : {eugeni, dtondini}@unite.it sia r il campo ordinato dei numeri reali e v uno spazio vettoriale 3−dimensionale reale. denotiamo con q : = r + v = r × v l’insieme delle “somme finali” di un numero reale con un vettore, ovvero un’espressione formale del tipo ua ρ + con a ∈ r, u r ∈ v ovvero una “coppia ordinata” ( ), a ur del prodotto cartesiano r × v. se, per ogni coppia ordinata ua ρ + , vb ρ + ∈ q, definiamo un’operazione (+) in q ponendo: ( ) ( ) ( ) ( ):a u b v a b u v+ + + = + + +r r r r la struttura algebrica (q, +) risulta essere un gruppo abeliano. una seconda operazione (∗) può essere definita ∀ ua ρ + , vb ρ + ∈ q ponendo: ( ) ( ) ( ):a u b v ab bu av u v u v+ ∗ + = + + + ∧ − ⋅r r r r r r r r si prova con facilità che la struttura (q, +, ∗) è un corpo, non valendo la proprietà commutativa della seconda operazione. gli elementi di q, nella sopraindicata struttura algebrica, si dicono quaternioni ed il corpo costruito si dice corpo dei quaternioni. se introduciamo ancora l’operazione “esterna” definita ponendo: ( ) :k a u k a k u ka ku⋅ + = ⋅ + ⋅ = +r r r ∀ k ∈ r, ∀ ua ρ+ ∈ q la struttura algebrica (q, +, ∗, ⋅) prende il nome di algebra dei quaternioni. è importante introdurre ora la nozione di norma. quale che sia ua ρ + ∈ q, si chiama norma l’applicazione || || : q → r definita ponendo: 2 2 2a u a u+ = + r essendo u la norma di u ρ in v. si chiama inoltre quaternione coniugato di ua ρ + il quaternione :a u a u+ = − r r è immediato provare che: (1) ( ) ( ) ( ) ( ) 2 2a u a u a u a u a u+ ∗ − = − ∗ + = +r r r r (2) ( )ua ua ua ρρ − + =+ − 22 1 1)( spesso è utile scrivere a u a br+ = + r r essendo r ρ un versore parallelo ad u ρ , ed anche: x y za u a u i u j u k+ = + + + rr rr essendo { }, , i j k rr r una base di versori di v. si prova facilmente che: (3) 12222 −==== rkji (4) ij ji k= − = ; jk kj i= − = ; jikki =−= si prova inoltre il seguente ovvio teorema 1. sia r r ∈ � un fissato versore. allora risulta { }( ), , a br+ + ∗ ≅r £ . teorema 2. sia ( ) q a br cos r senρ ϑ ϑ= + = +r r un quaternione. la trasformazione 1y q x q−= ∗ ∗ r r è una rotazione di asse r ρ ed ampiezza ϑ2 . dimostrazione. si ha ( ) ( )1 22 2 2y q x q cos x sen x r sen x r rϑ ϑ ϑ−= ∗ ∗ = − ∧ + ⋅r r r r r r r r la trasformazione è lineare; infatti ( ) ( ) ( )1 1 11 2 1 2q x x q q x q q x qλ µ λ µ− − −∗ + ∗ = ∗ ∗ + ∗ ∗ r r r r ed è tale che 11y q x q q x q x −−= ∗ ∗ = = r r r r per rx ρρ = risulta ry ρρ = cioè r ρ è un vettore unitario. sia a br cos rsinϑ ϑ+ = + r r un quaternione unitario. consideriamo: ( ) ( ) ( ) ( )a br x a br ax b r x r x a br+ ∗ ∗ − = + ∧ − ⋅ ∗ − =⎡ ⎤⎣ ⎦ r r r r r r r r r ( ) ( ){ } ( )b r x ax b r x a br= − ⋅ + + ∧ ∗ − =⎡ ⎤⎣ ⎦ r r r r r r ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2ab x r b x r r a x ab r x ax b r x br ax b r x br= − ⋅ + ⋅ + + ∧ + + ∧ ∧ − − + ∧ ⋅ − =⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ r r r r r r r r r r r r r r r r ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2ab x r b x r r a x ab r x ab x r b r x r ab x r b r x r= − ⋅ + ⋅ + + ∧ − ∧ − ∧ ∧ + ⋅ + ∧ ⋅ =r r r r r r r r r r r r r r r r r r ( ) ( ) ( )2 2 22b x r r a x ab x r b x x r r= ⋅ + − ∧ − − ⋅ =⎡ ⎤⎣ ⎦ r r r r r r r r r r ( ) ( ) ( )2 2 22 2a b x ab x r b x r r y= − − ∧ + ⋅ =r r r r r r r essendo, in generale: ( ) ( ) ( )a b c ac b bc a∧ ∧ = − supponiamo ora che 1x r sia ortogonale ad r r , cioè che sia: 1 0x r⋅ = r r . risulta allora: ( )1 1 12 2y cos x sin x rϑ ϑ= + ∧ r r r r e quindi: ( )1 12 0y r cos x rϑ⋅ = ⋅ = r r r r segue allora che l’angolo dei due piani ( ), x rr r ed ( ), y rr r è dato dall’angolo dei due vettori 1x r ed 1y r . si ha: 2 1 1 12y x cos xϑ= r r r e quindi · 1 1 2x y ϑ= r r poiché la trasformazione 1y q x q−= ∗ ∗ r r è lineare, deve esistere una matrice a = a(q) tale che: 1y q x q ax−= ∗ ∗ = r r r essendo, inoltre: ( ) ( )1 22 2 2y q x q cos x sen x r sen x r rϑ ϑ ϑ−= ∗ ∗ = − ∧ + ⋅r r r r r r r r passando alle componenti, risulta: 1 2 3y i y j y k+ + = rr r r r x r y r 1x r 1y r ( ) ( ) ( )21 2 3 1 2 3 1 1 2 2 3 3 1 2 3 1 2 3 2 2 2 i j k = cos x i x j x k sin x x x sin x r x r x r r i r j r k r r r θ ϑ θ+ + − + + + ⋅ + + rr r r rr r r r da cui: ( ) ( )21 1 2 3 2 3 1 1 1 2 2 3 32 2 2y cos x sin x r r x sin r x r x r x rθ θ θ= − − + + + ( ) ( )22 2 1 3 1 3 2 1 1 2 2 3 32 2 2y cos x sin x r r x sin r x r x r x rθ θ θ= + − + + + ( ) ( )23 3 1 2 1 2 3 1 1 2 2 3 32 2 2y cos x sin x r r x sin r x r x r x rθ θ θ= − − + + + ed ordinando: ( ) ( ) ( )2 2 2 21 1 1 3 1 2 2 2 1 3 32 2 2 2 2 2y cos r sin x r sin r r sin x r sin r r sin xθ θ θ θ θ θ= + + − + + + ( ) ( ) ( )2 2 2 22 3 1 2 1 2 2 1 2 3 32 2 2 2 2 2y r sin r r sin x + cos r sin x r sin r r sin xθ θ θ θ θ θ= + + + − + ( ) ( ) ( )2 2 2 23 2 1 3 1 1 2 3 2 3 32 2 2 2 2 2y r sin r r sin x + r sin r r sin x + cos r sin xθ θ θ θ θ θ= − + + + se ora poniamo: a cosϑ= b sinϑ= in maniera analoga otteniamo: ( )( ) ( ) ( )2 2 21 2 3 1 2 3 1 2 3 1 1 2 2 3 3 1 2 3 1 2 3 2 2 i j k y i y j y k a b x i x j x k ab x x x b x r x r x r r i r j r k r r r + + = − + + − + + + ⋅ + + rr r r r rr r r r r r da cui, passando alle componenti: ( ) ( ) ( )2 2 21 1 2 3 2 3 1 1 1 2 2 3 32 2y a b x ab x r r x b r x r x r x r= − − − + + + ( ) ( ) ( )2 2 22 2 1 3 1 3 2 1 1 2 2 3 32 2y a b x ab x r r x b r x r x r x r= − + − + + + ( ) ( ) ( )2 2 23 3 1 2 1 2 3 1 1 2 2 3 32 2y a b x ab x r r x b r x r x r x r= − − − + + + ovvero: ( ) ( ) ( )2 2 2 2 2 21 1 1 3 1 2 2 2 1 3 32 2 2 2 2y a b b r x abr b r r x abr b r r x= − + + − + + + ( ) ( ) ( )2 2 2 2 2 22 3 2 1 1 2 2 1 2 3 32 2 2 2 2y abr b r r x a b b r x abr b r r x= + + − + + − + ( ) ( ) ( )2 2 2 2 2 23 2 1 3 1 1 2 3 2 3 32 2 2 2 2y abr b r r x abr b r r x a b b r x= − + + + + − + se ora poniamo: 2 2 2 2 2 2 1 3 1 2 2 1 3 2 2 2 2 2 2 3 1 2 2 1 2 3 2 2 2 2 2 2 2 1 3 1 2 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 a b b r abr b r r abr b r r a abr b r r a b b r abr b r r abr b r r abr b r r a b b r ⎛ ⎞− + − + + ⎜ ⎟ = + − + − +⎜ ⎟ ⎜ ⎟− + + − +⎝ ⎠ risulta proprio y ax= r r . dobbiamo provare che questa matrice è ortogonale. si osservi che è funzione di 2ϑ (angolo di rotazione) e di r r . ( )( ) ( )( )2 2 2 2 2 2 2 2 2 21 3 1 2 3 1 2 22 2 2 2 2 2a b b r abr b r r abr b r r a b b r− + + + − + − + + ( )( ) ( )( )2 2 2 2 2 22 1 3 1 2 3 3 1 2 3 1 22 2 2 2 2 2 2 2abr b r r abr b r r a b abr b r r abr b r r+ + − + = − + − + = ( ) ( ) ( )2 2 2 2 2 2 2 2 21 2 1 3 1 2 2 3 1 24 2 2 2 2 2 2a b b r r b r abr b r r b r abr b r r= − + + + − + = ( ) ( )2 2 2 2 2 2 2 2 2 21 2 1 3 1 1 2 2 3 2 1 24 2 2 2 2 2a b b r r b r abr r b r r r abr r b r r= − + + − + ( ) 3 2 1 2 3 2 2 2 3 3 1 1 2 3 1 2 3 2 1 1 2 3 0 2 0 2 0 r r r r r a a b i ab r r b r r r r r r r r r r r −⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟= − + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ f.eugeni-d.tondini-a.viceconte www.eiris.it det = 0 det = 0 ratio mathematica volume 44, 2022 soft semi*𝜹-continuity in soft topological spaces reena c* yaamini k s† abstract in this paper, we introduce the concept of soft semi*𝛿-continuous functions and soft semi*𝛿-irresolute functions in soft topological spaces. also, we investigate its properties and study its relation with other soft continuous functions. keywords: soft semi*𝛿-open, soft semi*𝛿-closed, soft semi*𝛿-continuous, soft semi*𝛿-irresolute. 2010 ams subject classification: 54c05‡ *assistant professor, department of mathematics, st. mary’s college (autonomous), (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli), thoothukudi-1, tamil nadu, india; reenastephany@gmail.com †research scholar, reg. no. 21212212092002, department of mathematics, st. mary’s college (autonomous), (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli), thoothukudi-1, tamil nadu, india; ksyaamini@gmail.com ‡received on january 12th, 2022. accepted on may 12th, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.898. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 120 c. reena and k. s. yaamini 1. introduction the concept of soft set theory was first introduced by molotov [8] in 1999 to deal with uncertainty. according to him, a soft set over the universe is a parameterized family of subsets of the universe. in 2011, muhammad shabir and munazza naz [10] introduced soft topological spaces which are defined over an initial universe with a fixed set of parameters. meanwhile, in 2010, athar kharal and b. ahmad [4] defined the notion of soft mappings on soft classes. later, in 2013, aras and sonmez[2] introduced and studied soft continuous mappings. further, many authors defined and studied various forms of soft functions. recently, the authors[12] of this paper introduced a new class of soft sets namely soft semi*𝛿-open sets and soft semi*𝛿-closed sets. in this paper, we introduce the concept of soft semi*𝛿-continuous functions and soft semi*𝛿-irresolute functions in soft topological spaces. we also investigate its properties and study its relation with other soft continuous functions. 2. preliminaries throughout this work, (𝑋, �̃�,𝐸),(𝑌, �̃�,𝐾) and (𝑍,𝜇,𝐿) are soft topological spaces. 𝑆𝑡𝑐𝑙(𝐹,𝐴), 𝑆𝑡𝑖𝑛𝑡(𝐹,𝐴),𝑆𝑡𝑐𝑙 ∗(𝐹,𝐴) and 𝑆𝑡𝑖𝑛𝑡 ∗(𝐹,𝐴) denote soft closure, soft interior, soft generalized closure and soft generalized interior of (𝐹,𝐴) respectively. definition 2.1. [10] let 𝑋 be an initial universe and 𝐸 be a set of parameters. let 𝑃(𝑋) denote the power set of 𝑋 and 𝐴 be a non – empty subset of 𝐸. a pair (𝐹,𝐴) is called a soft set over 𝑋 where 𝐹 is a mapping given by 𝐹:𝐴 → 𝑃(𝑋). the collection of all soft sets over 𝑋 is called a soft class and denoted by 𝑆𝑡(𝑋,𝐸). definition 2.2. [10] let �̃� be the collection of soft set over 𝑋. then �̃� is said to be a soft topology on 𝑋 if 1) ϕ̃, �̃� belongs to �̃� 2) the union of any number of soft sets in �̃� belongs to �̃� 3) the intersection of any two soft sets in �̃� belongs to �̃�. the triplet (𝑋, �̃�,𝐸) is called a soft topological space. the members of �̃� are called soft open and its complements are called soft closed. definition 2.3. [4] let 𝑆𝑡(𝑋,𝐸) and 𝑆𝑡(𝑌,𝐾) be soft classes. let 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 be mappings. then a mapping 𝑓:𝑆𝑡(𝑋,𝐸) → 𝑆𝑡(𝑌,𝐾) is defined as: for a soft set (𝐹,𝐴) in 𝑆𝑡(𝑋,𝐸), (𝑓(𝐹,𝐴),𝐵),𝐵 = 𝑝(𝐴) ⊆ 𝐾 is a soft set in 𝑆𝑡(𝑌,𝐾) given by 𝑓(𝐹,𝐴)(𝛽) = { 𝑢( ⋃ 𝐹(𝛼) 𝛼∈𝑝−1(𝛽)∩𝐴 ), if 𝑝−1(𝛽) ∩ 𝐴 ≠ 𝜙 𝜙 otherwise 121 soft semi*𝛿-continuity in soft topological spaces for 𝛽 ∈ 𝐵 ⊆ 𝐾. (𝑓(𝐹,𝐴),𝐵) is called soft image of a soft set (𝐹,𝐴). if 𝐵 = 𝐾, then (𝑓(𝐹,𝐴),𝐾) is written as 𝑓(𝐹,𝐴). definition 2.4. [4] let 𝑓:𝑆𝑡(𝑋,𝐸) → 𝑆𝑡(𝑌,𝐾) be a mapping from a soft class 𝑆𝑡(𝑋,𝐸) to 𝑆𝑡(𝑌,𝐾) and (𝐺,𝐶) be a soft set in 𝑆𝑡(𝑌,𝐾) where 𝐶 ⊆ 𝐾. let 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 be mappings. then (𝑓−1(𝐺,𝐶),𝐷), 𝐷 = 𝑝−1(𝐶) is a soft set in 𝑆𝑡(𝑋,𝐸) defined as 𝑓−1(𝐺,𝐶)(𝛼) = { 𝑢−1(𝐺(𝑝(𝛼)), if 𝑝(𝛼) ∈ 𝐶 𝜙 otherwise for 𝛼 ∈ 𝐷 ⊆ 𝐸. (𝑓−1(𝐺,𝐶),𝐷) is called a soft inverse image of (𝐺,𝐶). we shall write (𝑓−1(𝐺,𝐶),𝐸) as 𝑓−1(𝐺,𝐶). definition 2.5. let (𝑋, �̃�,𝐸) and (𝑌, �̃�,𝐾) be soft topological spaces. a soft function 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) is soft continuous[2] (respectively soft semi-continuous[5], soft pre-continuous[9], soft α-continuous[6], soft β-continuous [14], soft b-continuous[7], soft regular continuous[3], soft δ-continuous [11], soft generalized continuous[13], soft semi*-continuous, soft pre*-continuous, soft β∗-continuous [1]) if 𝑓−1(𝐺,𝐵) is soft open (respectively soft semi-open, soft pre-open, soft α-open, soft 𝛽-open, soft b-open, soft regular open, soft 𝛿-open, soft generalized open, soft semi*-open, soft pre*-open, soft β∗-open) in (𝑋, �̃�,𝐸) for every soft open set (𝐺,𝐵) in (𝑌, �̃�,𝐾). definition 2.6.[12] a subset (𝐹,𝐴) of a soft topological space (𝑋, �̃�,𝐸) is called soft semi*𝜹-open set if there exists a soft 𝛿-open set (𝑂,𝐴) such that (𝑂,𝐴) ⊆̃ (𝐹,𝐴) ⊆̃ 𝑆𝑡𝑐𝑙 ∗(𝑂,𝐴). the complement of soft semi*𝛿-open set is called soft semi*𝛿-closed. the class of soft semi*𝛿-open sets in (𝑋, �̃�,𝐸) is denoted by 𝑆𝑡𝑆 ∗𝛿𝑂(𝑋, �̃�,𝐸). theorem 2.7.[12] in any soft topological space (𝑋, �̃�,𝐸), (i) every soft 𝛿-open set is soft semi*𝛿-open. (ii) every soft regular open set is soft semi*𝛿-open. (iii) every soft semi*𝛿-open set is soft semi-open. (iv) every soft semi*𝛿-open set is soft semi*-open. (v) every soft semi*𝛿-open set is soft 𝛽-open. (vi) every soft semi*𝛿-open set is soft 𝛽∗-open. (vii) every soft semi*𝛿-open set is soft b-open. remark 2.8:[12] the above theorem is also true for soft semi*𝛿-closed sets. 3. soft semi*𝜹-continuous functions definition 3.1. let (𝑋, �̃�,𝐸) and (𝑌, �̃�,𝐾) be soft topological spaces. let 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 be mappings. then the soft function 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) is said to be soft semi*𝛿122 c. reena and k. s. yaamini continuous if 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in (𝑋, �̃�,𝐸) for every soft open set (𝐺,𝐵) in (𝑌, �̃�,𝐾). the following soft sets are used in all examples let 𝑋 = {𝑎,𝑏} and 𝐸 = {𝑒1,𝑒2} . then the soft sets are 𝐹1 = {(𝑒1, {𝜙}),(𝑒2, {𝜙})} = ϕ̃ 𝐹9 = {(𝑒1, {𝑏}),(𝑒2, {𝜙})} 𝐹2 = {(𝑒1, {𝜙}),(𝑒2, {𝑎})} 𝐹10 = {(𝑒1, {𝑏}),(𝑒2, {𝑎})} 𝐹3 = {(𝑒1, {𝜙}),(𝑒2, {𝑏})} 𝐹11 = {(𝑒1, {𝑏}),(𝑒2, {𝑏})} 𝐹4 = {(𝑒1, {𝜙}),(𝑒2, {𝑎,𝑏})} 𝐹12 = {(𝑒1, {𝑏}),(𝑒2, {𝑎,𝑏})} 𝐹5 = {(𝑒1, {𝑎}),(𝑒2, {𝜙})} 𝐹13 = {(𝑒1, {𝑎,𝑏}),(𝑒2, {𝜙})} 𝐹6 = {(𝑒1, {𝑎}),(𝑒2, {𝑎})} 𝐹14 = {(𝑒1, {𝑎,𝑏}),(𝑒2, {𝑎})} 𝐹7 = {(𝑒1, {𝑎}),(𝑒2, {𝑏})} 𝐹15 = {(𝑒1, {𝑎,𝑏}),(𝑒2, {𝑏})} 𝐹8 = {(𝑒1, {𝑎}),(𝑒2, {𝑎,𝑏})} 𝐹16 = {(𝑒1, {𝑎,𝑏}),(𝑒2, {𝑎,𝑏})} = �̃� similarly, let 𝑌 = {𝑥,𝑦} and 𝐾 = {𝑘1,𝑘2}. then the soft sets 𝐺1,𝐺2,…,𝐺16 are obtained by replacing 𝑎,𝑏,𝑒1and 𝑒2 by 𝑥,𝑦,𝑘1and 𝑘2 respectively in the above sets. example 3.2. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹4,𝐹5,𝐹8} and �̃� = {�̃�,𝜙,̃𝐺2,𝐺13,𝐺14}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. then, 𝑓−1(𝐺2) = 𝐹5,𝑓 −1(𝐺13) = 𝐹4 and 𝑓 −1(𝐺14) = 𝐹8. here, 𝐹4,𝐹5,𝐹8 are soft semi*𝛿-open. hence 𝑓 is soft semi*𝛿-continuous. theorem 3.3. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft 𝛿-continuous function. then 𝑓 is soft semi*𝛿-continuous. proof: let (𝐺,𝐵) be a soft open set in 𝑌. since 𝑓 is soft 𝛿-continuous, 𝑓−1(𝐺,𝐵) is soft 𝛿-open in 𝑋. then by theorem 2.7(i), 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open. hence 𝑓 is soft semi*𝛿-continuous. remark 3.4. the converse of the above theorem need not be true. example 3.5. let𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹2,𝐹11,𝐹12} and �̃� = {�̃�,𝜙,̃𝐺6,𝐺11}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. since 𝑓−1(𝐺6) = 𝐹6 is soft semi*𝛿-open but not soft 𝛿-open, 𝑓 is soft semi*𝛿-continuous but not soft 𝛿-continuous. theorem 3.6. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft regular continuous function. then 𝑓 is soft semi*𝛿-continuous. proof. similar to theorem 3.3, the proof follows from theorem 2.7(ii). remark 3.7. the converse of the above theorem need not be true. example 3.8. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1. consider the soft 123 soft semi*𝛿-continuity in soft topological spaces topologies �̃� = {�̃�,𝜙,̃𝐹5,𝐹9,𝐹13} and �̃� = {�̃�,𝜙,̃𝐺2,𝐺3,𝐺4}.let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. since 𝑓−1(𝐺4) = 𝐹13 is soft semi*𝛿-open but not soft regular open, 𝑓 is soft semi*𝛿-continuous but not soft regular continuous. theorem 3.9. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft function. (i) if 𝑓 is soft semi*𝛿-continuous, then 𝑓 is soft semi-continuous. (ii) if 𝑓 is soft semi*𝛿-continuous, then 𝑓 is soft semi*-continuous. (iii) if 𝑓 is soft semi*𝛿-continuous, then 𝑓 is soft 𝛽-continuous. (iv) if 𝑓 is soft semi*𝛿-continuous, then 𝑓 is soft 𝛽∗-continuous. (v) if 𝑓 is soft semi*𝛿-continuous, then 𝑓 is soft b-continuous. proof. (i) let (𝐺,𝐵) be a soft open set in 𝑌. since 𝑓 is soft semi*𝛿-continuous, 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in 𝑋. then by theorem 2.7(iii), 𝑓−1(𝐺,𝐵) is soft semiopen. hence 𝑓 is soft semi-continuous. the other proofs are similar. remark 3.10. the converse of each of the statements in above theorem need not be true. example 3.11. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹5,𝐹9,𝐹13}and �̃� = {�̃�,𝜙,̃𝐺6,𝐺14}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. since 𝑓−1(𝐺6) = 𝐹6 and 𝑓 −1(𝐺14) = 𝐹8 are soft semi-open but not soft semi*𝛿-open, 𝑓 is soft semi-continuous but not soft semi*𝛿-continuous. example 3.12. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥,𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹6,𝐹14}and �̃� = {�̃�,𝜙,̃𝐺11,𝐺15}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. since 𝑓−1(𝐺11) = 𝐹6 and 𝑓 −1(𝐺15) = 𝐹14 are soft semi*-open but not soft semi*𝛿-open, 𝑓 is soft semi*-continuous but not soft semi*𝛿-continuous. example 3.13. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹4,𝐹9,𝐹12} and �̃� = {�̃�,𝜙,̃𝐺2,𝐺6,𝐺10,𝐺14}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. since 𝑓−1(𝐺2) = 𝐹2,𝑓 −1(𝐺6) = 𝐹6,𝑓 −1(𝐺10) = 𝐹10 and 𝑓−1(𝐺14) = 𝐹14 are both soft 𝛽-open and soft 𝛽 ∗-open but not soft semi*𝛿-open, 𝑓 is both soft 𝛽-continuous and soft 𝛽∗-continuous but not soft semi*𝛿-continuous. example 3.14. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥,𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹3,𝐹9,𝐹11} and �̃� = {�̃�,𝜙,̃𝐺2,𝐺4,𝐺6,𝐺8}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. since 𝑓−1(𝐺4) = 𝐹4 and 𝑓 −1(𝐺8) = 𝐹12 are soft b-open but not soft semi*𝛿-open, 𝑓 is soft b-continuous but not soft semi*𝛿-continuous. 124 c. reena and k. s. yaamini remark 3.15. the concept of soft semi*𝛿-continuity and soft continuity are independent as shown in the following example. example 3.16. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1.consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹4,𝐹5,𝐹8} and �̃� = {�̃�,𝜙,̃𝐺15}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. since 𝑓−1(𝐺15) = 𝐹12 is soft semi*𝛿-open but not soft open, 𝑓 is soft semi*𝛿-continuous but not soft continuous. now, consider the soft topology �̃� = {�̃�,𝜙,̃𝐹4,𝐹8,𝐹12} on 𝑋. here, since 𝑓 −1(𝐺15) = 𝐹12 is soft open but not soft semi*𝛿-open, 𝑓 is soft continuous but not soft semi*𝛿-continuous. remark 3.17. the concept of soft semi*𝛿-continuity and soft generalized continuity are independent as shown in the following example. example 3.18. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦, 𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹6,𝐹9,𝐹14} and �̃� = {�̃�,𝜙,̃𝐺6,𝐺11}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. since 𝑓−1(𝐺11) = 𝐹11 is soft semi*𝛿-open but not soft generalized open, 𝑓 is soft semi*𝛿-continuous but not soft generalized continuous. now, consider the soft topology �̃� = {�̃�,𝜙,̃𝐺13} on 𝑌. here, since 𝑓 −1(𝐺13) = 𝐹13 is soft generalized open but not soft semi*𝛿-open, 𝑓 is soft generalized continuous but not soft semi*𝛿continuous. remark 3.19. the concept of soft semi*𝛿-continuity and soft pre-continuity are independent as shown in the following example. example 3.20. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦, 𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹2,𝐹11,𝐹12}and �̃� = {�̃�,𝜙,̃𝐺5,𝐺6,𝐺15}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. since 𝑓−1(𝐺6) = 𝐹6 is soft semi*𝛿-open but not soft preopen, 𝑓 is soft semi*𝛿-continuous but not soft pre-continuous. now, consider the mapping 𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥, 𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1. here, since 𝑓 −1(𝐺5) = 𝐹3 and 𝑓−1(𝐺15) = 𝐹8 are soft pre-open but not soft semi*𝛿-open, 𝑓 is soft pre-continuous but not soft semi*𝛿-continuous. remark 3.21. the concept of soft semi*𝛿-continuity and soft pre*-continuity are independent as shown in the following example. example 3.22. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥, 𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹6,𝐹9,𝐹14} and �̃� = {�̃�,𝜙,̃𝐺11,𝐺12}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) 125 soft semi*𝛿-continuity in soft topological spaces be a soft mapping. since 𝑓−1(𝐺12) = 𝐹8 is soft semi*𝛿-open but not soft pre*-open, 𝑓 is soft semi*𝛿-continuous but not soft pre*-continuous. now, consider the soft topology �̃� = {�̃�,𝜙,̃𝐺3,𝐺9, 𝐺11}. here, since 𝑓 −1(𝐺3) = 𝐹2 and 𝑓 −1(𝐺9) = 𝐹5 are soft pre*-open but not soft semi*𝛿-open, 𝑓 is soft pre*-continuous but not soft semi*𝛿-continuous. remark 3.23. the concept of soft semi*𝛿-continuity and soft 𝛼-continuity are independent as shown in the following example. example 3.24. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥, 𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹4,𝐹9,𝐹12} and �̃� = {�̃�,𝜙,̃𝐺4,𝐺8,𝐺12}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌, �̃�,𝐾) be a soft mapping. since 𝑓−1(𝐺12) = 𝐹8 is soft semi*𝛿-open but not soft 𝛼-open, 𝑓 is soft semi*𝛿-continuous but not soft 𝛼-continuous. now, consider the soft topology �̃� = {�̃�,𝜙,̃𝐹4,𝐹8,𝐹12} on 𝑋. here, since 𝑓 −1(𝐺4) = 𝐹4, 𝑓 −1(𝐺8) = 𝐹12 and 𝑓 −1(𝐺12) = 𝐹8 are soft 𝛼-open but not soft semi*𝛿-open, 𝑓 is soft 𝛼-continuous but not soft semi*𝛿continuous. from the above discussions, we have the following diagram theorem 3.25. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft function. then the following statements are equivalent: (i) 𝑓 is soft semi*𝛿-continuous. (ii) the inverse image of every soft closed set in (𝑌, �̃�,𝐾) is soft semi*𝛿-closed in (𝑋, �̃�,𝐸). (iii) 𝑓(𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝐹,𝐴)) ⊆̃ 𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴)) for every soft set (𝐹,𝐴) over 𝑋. 𝑆𝑡𝑆-continuous 𝑆𝑡𝑆 ∗𝛿continuous 𝑆𝑡𝛿-continuous 𝑆𝑡𝑅-continuous 𝑆𝑡𝑏-continuous 𝑆𝑡𝛽-continuous 𝑆𝑡𝛽 ∗ -continuous 𝑆𝑡𝑆 ∗ -continuous 𝑆𝑡𝛼-continuous 𝑆𝑡𝑃-continuous 𝑆𝑡𝑃 ∗ -continuous 𝑆𝑡𝐺-continuous 𝑆𝑡-continuous figure 1 126 c. reena and k. s. yaamini (iv) 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)) ⊆̃ 𝑓−1(𝑆𝑡𝑐𝑙(𝐺,𝐵)) for every soft set (𝐺,𝐵) over 𝑌. proof. (i) ⟹(ii) let 𝑓 be a soft semi*𝛿-continuous function and (𝐻,𝐵) be a soft closed set in 𝑌. then (𝐻,𝐵)𝑐̃ is soft open in 𝑌. since 𝑓 is soft semi*𝛿-continuous, 𝑓−1((𝐻,𝐵)𝑐̃) is soft semi*𝛿-open in 𝑋. that is, (𝑓−1(𝐻,𝐵))𝑐̃ is soft semi*𝛿-open in (𝑋, �̃�,𝐸). hence 𝑓−1(𝐻,𝐵) is soft semi*𝛿-closed in (𝑋, �̃�,𝐸). (ii)⟹(i) let (𝐺,𝐵) be soft open in 𝑌. then (𝐺,𝐵)𝑐̃ be soft closed in 𝑌. by assumption, 𝑓−1((𝐺,𝐵)𝑐̃) is soft semi*𝛿-closed in 𝑋. that is, (𝑓−1(𝐺,𝐵))𝑐̃ is soft semi*𝛿-closed in 𝑋. hence, 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in 𝑋. therefore, 𝑓 is soft semi*𝛿-continuous. (ii)⟹(iii) let (𝐹,𝐴) be a soft set over 𝑋. now, (𝐹,𝐴) ⊆̃ 𝑓−1(𝑓(𝐹,𝐴)) implies (𝐹,𝐴) ⊆̃ 𝑓−1 (𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴))). since 𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴)) is soft closed in 𝑌, by assumption 𝑓−1 (𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴))) is a soft semi*𝛿-closed set containing (𝐹,𝐴). also, 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝐹,𝐴) is the smallest soft semi*𝛿-closed set containing (𝐹,𝐴). hence, 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝐹,𝐴) ⊆̃ 𝑓−1 (𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴))). therefore, 𝑓(𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝐹,𝐴)) ⊆̃ 𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴)). (iii)⟹(ii) let (𝐻,𝐵) be a soft closed set in 𝑌. then, by assumption, 𝑓(𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝑓−1(𝐻,𝐵))) ⊆̃ 𝑆𝑡𝑐𝑙(𝑓(𝑓 −1(𝐻,𝐵))) ⊆̃ 𝑆𝑡𝑐𝑙(𝐻,𝐵) = (𝐻,𝐵) implies 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝑓−1(𝐻,𝐵)) ⊆̃ 𝑓−1(𝐻,𝐵). also, 𝑓−1(𝐻,𝐵) ⊆̃ 𝑆𝑡𝑠 ∗𝛿𝑐𝑙 (𝑓−1(𝐻,𝐵)). hence, 𝑓−1(𝐻,𝐵) is soft semi*𝛿-closed in 𝑋. (iii)⟹(iv) let (𝐺,𝐵) be a soft set over 𝑌 and let (𝐹,𝐴) = 𝑓−1(𝐺,𝐵). by assumption, 𝑓(𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝐹,𝐴)) ⊆̃ 𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴)) = 𝑆𝑡𝑐𝑙(𝐺,𝐵). this implies 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)) ⊆̃ 𝑓−1(𝑆𝑡𝑐𝑙(𝐺,𝐵)). (iv)⟹(iii) let (𝐺,𝐵) = 𝑓(𝐹,𝐴). then, by assumption 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝐹,𝐴) = 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)) ⊆̃ 𝑓−1(𝑆𝑡𝑐𝑙(𝐺,𝐵)) = 𝑓 −1 (𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴))). this implies 𝑓(𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝐹,𝐴)) ⊆̃ 𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴)) (iv)⟹(i) let (𝐺,𝐵) be soft open in 𝑌. then, (𝐺,𝐵)𝑐̃ is soft closed in 𝑌. by assumption, 𝑓−1((𝐺,𝐵)𝑐̃) = 𝑓−1(𝑆𝑡𝑐𝑙(𝐺,𝐵) 𝑐̃) ⊇̃ 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)𝑐̃). also, we know that 𝑓−1((𝐺,𝐵)𝑐̃) ⊆̃ 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)𝑐̃). hence 𝑓−1((𝐺,𝐵)𝑐̃) = 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)𝑐̃). therefore, 𝑓−1((𝐺,𝐵)𝑐̃) is soft semi*𝛿-closed. that is, (𝑓−1(𝐺,𝐵))𝑐̃ is soft semi*𝛿closed in 𝑋. hence, 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in 𝑋. therefore, 𝑓 is soft semi*𝛿continuous. theorem 3.26. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft function. then 𝑓 is soft semi*𝛿continuous if and only if 𝑓−1(𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵)) ⊆̃ 𝑆𝑡𝑠 ∗𝛿𝑖𝑛𝑡(𝑓−1(𝐺,𝐵)) for every soft set (𝐺,𝐵) over 𝑌. proof. let 𝑓 be a soft semi*𝛿-continuous function and (𝐺,𝐵) be a soft set over 𝑌. then 𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵) is soft open set in 𝑌. by assumption, 𝑓 −1(𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵)) is soft 127 soft semi*𝛿-continuity in soft topological spaces semi*𝛿-open in 𝑋. now, 𝑓−1(𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵)) ⊆̃ 𝑓 −1(𝐺,𝐵) and 𝑆𝑡𝑠 ∗𝛿𝑖𝑛𝑡(𝑓−1(𝐺,𝐵)) is the largest soft semi*𝛿-open set contained in 𝑓−1(𝐺,𝐵). hence 𝑓−1(𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵)) ⊆̃ 𝑆𝑡𝑠 ∗𝛿𝑖𝑛𝑡(𝑓−1(𝐺,𝐵)). conversely, let (𝐺,𝐵) be soft open in 𝑌. then, 𝑓−1(𝐺,𝐵) = 𝑓−1(𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵)) ⊆̃ 𝑆𝑡𝑠 ∗𝛿𝑖𝑛𝑡(𝑓−1(𝐺,𝐵)). also, 𝑆𝑡𝑠 ∗𝛿𝑖𝑛𝑡(𝑓−1(𝐺,𝐵)) ⊆̃ 𝑓−1(𝐺,𝐵) . this implies 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in 𝑋. hence 𝑓 is soft semi*𝛿-continuous. remark 3.27. the composition of two soft semi*𝛿-continuous functions need not be soft semi*𝛿-continuous. example 3.28. let 𝑋 = {𝑎,𝑏,𝑐},𝑌 = {𝑥,𝑦},𝑍 = {𝑚,𝑛},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}, 𝐿 = {𝑙1, 𝑙2}. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹5,𝐹12,𝐹16}, �̃� = {�̃�,𝜙,̃𝐺2,𝐺7,𝐺8} and𝜇 = {�̃�,𝜙,̃𝐻10} where 𝐹5 = {(𝑒1, {𝜙}),(𝑒2, {𝑎,𝑏})}, 𝐹12 = {(𝑒1, {𝑎}),(𝑒2, {𝑐})}, 𝐹16 = {(𝑒1, {𝑎}),(𝑒2, {𝑎,𝑏,𝑐})}, 𝐺2 = {(𝑘1, {𝜙}),(𝑘2, {𝑥})}, 𝐺7 = {(𝑘1, {𝑥}),(𝑘2, {𝑦})}, 𝐺8 = {(𝑘1, {𝑥}),(𝑘2, {𝑥,𝑦})} and 𝐻10 = {(𝑙1, {𝑛}),(𝑙2, {𝑚})}. define 𝑢1:𝑋 ⟶ 𝑌 and 𝑝1:𝐸 ⟶ 𝐾 as 𝑢1(𝑎) = 𝑢1(𝑏) = 𝑥,𝑢1(𝑐) = 𝑦, 𝑝1(𝑒1) = 𝑘1,𝑝1(𝑒2) = 𝑘2. then the soft mapping 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑌,�̃�,𝐾) is soft semi*𝛿-continuous. also, define 𝑢2:𝑌 ⟶ 𝑍 and 𝑝2:𝐾 ⟶ 𝐿 as 𝑢2(𝑥) = 𝑚,𝑢2(𝑦) = 𝑛,𝑝2(𝑘1) = 𝑙1 and 𝑝2(𝑘2) = 𝑙2. then the soft mapping �̃�:(𝑌, �̃�,𝐾) ⟶ (𝑍,𝜇,𝐿) is soft semi*𝛿-continuous. now, let �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) be the composition of two soft semi*𝛿-continuous functions. then �̃� ∘ 𝑓 is not soft semi*𝛿-continuous since (�̃� ∘ 𝑓) −1 (𝐻10) = 𝑓 −1(�̃�−1(𝐻10)) = 𝑓−1(𝐺10) = {(𝑒1, {𝑐}),(𝑒2, {𝑎,𝑏})} is not soft semi*𝛿-open. theorem 3.29. let (𝑋, �̃�,𝐸), (𝑌, �̃�,𝐾) and (𝑍,𝜇,𝐿) be soft topological spaces and let (𝑌, �̃�,𝐾) be a space in which every soft semi*𝛿-open set is soft open. then the composition �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) of two soft semi*𝛿-continuous functions 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑌,�̃�,𝐾) and �̃�:(𝑌, �̃�,𝐾) ⟶ (𝑍,𝜇,𝐿) is soft semi*𝛿-continuous. proof. let (𝐻,𝐶) be any soft open set in 𝑍. since �̃� is soft semi*𝛿-continuous, �̃�−1(𝐻,𝐶) is soft semi*𝛿-open in 𝑌. then, by assumption �̃�−1(𝐻,𝐶) is soft open in 𝑌. also, since 𝑓 is soft semi*𝛿-continuous, 𝑓−1(�̃�−1(𝐻,𝐶)) = (�̃� ∘ 𝑓) −1 (𝐻,𝐶) is soft semi*𝛿-open in 𝑋. hence �̃� ∘ 𝑓 is soft semi*𝛿-continuous. theorem 3.30. let 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑌,�̃�,𝐾) be a soft semi*𝛿-continuous function and �̃�:(𝑌, �̃�,𝐾) ⟶ (𝑍,𝜇,𝐿) be a soft continuous function. then their composition �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) is soft semi*𝛿-continuous. 128 c. reena and k. s. yaamini proof. let (𝐻,𝐶) be any soft open set in 𝑍. since �̃� is soft continuous, �̃�−1(𝐻,𝐶) is soft open in 𝑌. also, since 𝑓 is soft semi*𝛿-continuous, 𝑓−1(�̃�−1(𝐻,𝐶)) = (�̃� ∘ 𝑓) −1 (𝐻,𝐶) is soft semi*𝛿-open in 𝑋. hence �̃� ∘ 𝑓 is soft semi*𝛿-continuous. 4. soft semi*𝜹-irresolute functions definition 4.1. let (𝑋, �̃�,𝐸) and (𝑌, �̃�,𝐾) be soft topological spaces. let 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 be mappings. then the soft function 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) is said to be soft semi*𝛿-irresolute if 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in (𝑋, �̃�,𝐸) for every soft semi*𝛿open set (𝐺,𝐵) in (𝑌, �̃�,𝐾). example 4.2. let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. define 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥, 𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹4,𝐹9,𝐹12} and �̃� = {�̃�,𝜙,̃𝐺4,𝐺5,𝐺8}.here, 𝑆𝑡𝑆 ∗𝛿𝑂(𝑋, �̃�,𝐸) = {�̃�, �̃�,𝐹4,𝐹8,𝐹9,𝐹12,𝐹13} and 𝑆𝑡𝑆 ∗𝛿𝑂(𝑌,�̃�,𝐾) = {�̃�, �̃�,𝐺4,𝐺5,𝐺8,𝐺12,𝐺13}. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. then, 𝑓−1(𝐺4) = 𝐹4, 𝑓 −1(𝐺5) = 𝐹9 𝑓−1(𝐺8) = 𝐹12, 𝑓 −1(𝐺12) = 𝐹8and 𝑓 −1(𝐺13) = 𝐹13. hence, 𝑓 is soft semi*𝛿irresolute. theorem 4.3. let (𝑋, �̃�,𝐸) and (𝑌, �̃�,𝐾) be soft topological spaces and let (𝑌, �̃�,𝐾) be a space in which every soft semi*𝛿-open set is soft open. if 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) is soft semi*𝛿-continuous, then 𝑓 is soft semi*𝛿-irresolute. proof. let (𝐺,𝐵) be soft semi*𝛿-open in 𝑌. then, by assumption (𝐺,𝐵) is soft open in 𝑌. since 𝑓 is soft semi*𝛿-continuous, 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in 𝑋. hence 𝑓 is soft semi*𝛿-irresolute. theorem 4.4. let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft function. then the following statements are equivalent: (i) 𝑓 is soft semi*𝛿-irresolute. (ii) the inverse image of every soft semi*𝛿-closed set in (𝑌, �̃�,𝐾) is soft semi*𝛿-closed in (𝑋, �̃�,𝐸). (iii) 𝑓(𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝐹,𝐴)) ⊆̃ 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝑓(𝐹,𝐴)) for every soft set (𝐹,𝐴) over 𝑋. (iv) 𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)) ⊆̃ 𝑓−1(𝑆𝑡𝑠 ∗𝛿𝑐𝑙(𝐺,𝐵)) for every soft set (𝐺,𝐵) over 𝑌. proof. the proof is similar to theorem 3.25 theorem 4.5. if 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) and �̃�:(𝑌, �̃�,𝐾) → (𝑍,𝜇,𝐿) are soft semi*𝛿irresolute functions, then their composition �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) is also soft semi*𝛿-irresolute. proof. let (𝐻,𝐶) be soft semi*𝛿-open in z. since �̃� is soft semi*𝛿-irresolute, �̃�−1(𝐻,𝐶) is soft semi*𝛿-open in 𝑌. again, since 𝑓 is soft semi*𝛿-irresolute, 129 soft semi*𝛿-continuity in soft topological spaces 𝑓−1(�̃�−1(𝐻,𝐶)) = (�̃� ∘ 𝑓) −1 (𝐻,𝐶) is soft semi*𝛿-open in 𝑋. hence �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) is soft semi*𝛿-irresolute. theorem 4.6. if 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) is soft semi*𝛿-irresolute and �̃�:(𝑌, �̃�,𝐾) → (𝑍,𝜇,𝐿) is soft semi*𝛿-continuous, then �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) is soft semi*𝛿continuous. proof. let (𝐻,𝐶) be a soft open set in 𝑍. since �̃� is soft semi*𝛿-continuous, �̃�−1(𝐻,𝐶) is soft semi*𝛿-open in 𝑌. now, since 𝑓 is soft semi*𝛿-irresolute, 𝑓−1(�̃�−1(𝐻,𝐶)) = (�̃� ∘ 𝑓) −1 (𝐻,𝐶) is soft semi*𝛿-open in 𝑋. hence �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) is soft semi*𝛿-continuous. 5. conclusions we have studied the concept of continuity in soft topological spaces by means of soft semi*𝛿-open sets. we have also introduced the concept of soft semi*𝛿-irresolute functions. further, we have compared it with other existing soft functions and we have also investigated the characterization of these functions. references [1] p. anbarasi rodrigo, s. maheshwari, functions related to soft 𝛽∗-closed sets in soft topological spaces, proceedings of international conference on recent advances in computational mathematics and engineering 2021. [2] c.g. aras, a. sonmez, h. cakalli, on soft mappings, arxiv: 1305.4545. 2013. [3] i. arockiarani, a. selvi, soft πg-continuous functions and irresolute functions, international journal of innovation and applied studies, volume 7(2), 440-446. 2014. [4] athar kharal and b. ahmad, mappings on soft classes, new mathematics and natural computation, volume 7, 471-481. 2010. [5] j. mahanta, p.k. das, on soft topological spaces via semi open and semi closed soft sets, kyungpook math. journal, 54, 221-236. 2014. [6] metin akdag, alkan ozkan, soft alpha – open sets and soft alpha – continuous functions, hindawi publishing corporation, abstract and applied mathematics, 2014. [7] metin akdag, alkan ozkan, soft b – open sets and soft b – continuous functions, math sci 8, 124. 2014. [8] d. molostsov, soft set theory – first results, computers and mathematics with applications, 37, 19 – 31. 1999. [9] mrudula ravindran, g. manju, some results on soft pre-continuity, malaya journal of mathematik, 10-17. 2015. 130 c. reena and k. s. yaamini [10] muhammud shabir, munazza naz, on soft topological spaces, computers and mathematics with applications, 61, 1786 – 1799. 2011. [11] ramadhan a. mohammed, o. r. sayed, a. eliow, some properties of soft delta – topology, academic journal of nawroz university, 2019. [12] c. reena, k.s. yaamini, a new class of soft sets weaker than soft δ-open sets, proceedings of international conference on smart technologies and applications (icsta 2022). [13] sattar hameed hamzah, samer adnan jubair, on soft generalized continuous mappings, journal of al-qadisiyah for computer science and mathematics, 8(1), 8392. 2016 [14] yenus yumak, aynur keskin kaymakci, soft β − open sets and their applications, journal of new theory, 80 – 89. 2015. 131 microsoft word migliorato-1.doc 37 euclid and the scientific thought in the third century b.c.1 renato migliorato giuseppe gentile2 department of mathematics, university of messina address: contrada papardo, 98121 messina (italy). e-mail: renato.migliorato@unime.it, gentile@dipmat.unime.it abstract the criticism on the texts of euclid, even assuming different positions, starts generally from the previous assumption that the author of the elements is totally inside the platonic-aristotelian tradition. the thesis affirmed in this paper is that many of the gaps and contradictions found by the criticism have their root in this assumption. the authors assert that euclid was a scientist that belonged in a full way to the new cultural climate of the hellenistic kingdoms, and particularly of the alexandria’s museum. in this climate, characterized by lively philosophical disputes, the scientists, and in particular euclid, tend to obtain coherent and stable results, voluntarily omitting to give their opinion on the real being of the scientific object and on the truth of the principles. 1. introduction even if important innovations in the critical studies on euclidean geometry don’t less in a more recent times, the period starting from the end of xix century until the beginning of the xx century is surely the more prolific one, that in which a critical order was constituted such to be considered until now almost definitive. from 1850 to 1928 the heiberg and menge's edition3 of the euclid's works was published; this is considered the more reliable text and the nearest to the original one. in the first half of the twentieth century there are many translations, with comments and remarks, founded on the heiberg's text, as that one of federigo enriques4 or that one of heath5, and many critical elaborations by the same and other authors6. among the more recent published works that propose some new interpretative hypotheses, it seems suitable to us to men 1 work supported by university of messina as local research project (p.r.a.). 2 this work, coordinated by r. migliorato, is the result of a collaboration that sow contemporarily engaged both the authors on all the treated aspects. also the searches that separately were effected, was discussed in all points before to decide which solution to adopt. the contribution of g. gentile, concretised however prevalently on the sections 2, 3 and 5, can be quantify as a third of the whole work. 3 euclides, 1883-1916. 4 enriques, 1912-1935. 5 heath, 1956. 6 see for instance tannery; vailati; veronese; amaldi; enriques, 1912. 38 tion two articles and a monograph of lucio russo7, a monograph of francesca incardona8 and a book of imre toth9. the articles of russo consider the first seven definitions in the first book of the elements, that he affirms to be posterior interpolations. with that, it would come to fall the greatest residual reasons in favour of a supposed platonism of euclid. we believe that such hypothesis, well argued and documented by the author, must be accepted, not only because it is historically reliable, but also because it seems to us most suitable to answer to difficult problems, remained unsolved, on the interpretation of the euclidean work. the book “la rivoluzione dimenticata” (the forgotten revolution), also by russo, would demand instead a more complex and articulated valuation that, in its entirety, is extraneous to the object and the purpose of the present paper. however the fundamental hypotheses that are the nucleus of the book, cannot remain excluded from our analysis. for our purposes, it seems meaningful to us the introductory text by which incardona accompanies her translation of euclid’s optic. it seems interesting to us, in particular, the reasonings that leads to interpret the work as a mathematical model of the phenomenon of the vision and tend to insert euclid in the context of a more wide evolution of the scientific and philosophical ideas at the beginning of third century. not well founded it appears instead the hypothesis, however only fleetingly pointed out by the author, that euclid could be absolutely assimilated to the area of the stoic philosophy. finally the book of imre toth10 is interesting for us because it calls our attention on some passages of aristotle that evidence the existence, already before euclid, of an open problem on the parallel straight lines. the purpose of the present paper is a re-examination of the euclidean text to the light of the last criticism’s history, with particular reference to the mentioned works, and of the most recent acquisitions of knowledge on the hellenistic society and culture of third century b.c. the conclusions to which we will reach, seem to answer to some questions that still now remained opened. 2. platonist or aristotelian? the date of the elements is set about 300 b.c. and a more precise dating is objectively difficult. but if we look at the indications of proclus11, still now the main source about this argument, we have to think that he written his works in the first decades of the third century. in effect, the contemporaneousness with the first ptolemy imposes only that euclid was present and active before 283 b.c., date of the death of ptolemy, but this doesn’t tell something of more precise about the period during which his work was elaborated; so, as principle, a lot of hypotheses would be possible. everything however allows us to suppose that the elements and the other principal known works of euclid were written within the museum and the library of alexandria. 7 russo, 1992, 1997, 1998. 8 incardona, 1996.. 9 toth, 1997. 10 it is difficult to summarize in a little space the content of an ample monograph that tends to a re-interpretation of the aristotelian thought to the light of the not-euclidean geometries. 11 “.....not much younger than these [hermotimus of colophon and philippus of medma] is euclid, who put together the elements. collecting many of eudoxus theorems, perfecting many of theaetetus, and also bringing to irrefragable proof the things which were only somewhat loosely proved by his predecessors. this man lived in the time of the first ptolemy. for archimedes, who lived immediately after the first (ptolemy), makes mention of euclid: and further, they say that ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry. he is then younger than the pupils of plato but older than eratosthenes and archimedes; for the latter were contemporary with one another as eratosthenes somewhere says. for his ideas euclid was platonic and had very families this philosophy, so much that the final purpose of the whole collection of elements was the construction of the so-called platonic figures.” (proclus, comm. eucl.., ii, 68). 39 for vastness and organization, but also for its intrinsic characters that we will see more, such work seems fully to reconcile with a great organized enterprise, what the school in alexandria was certainly, and with a historical phase in which the sciences were detached by the philosophy and this also has the tendency to abandon the great systems to follow more pragmatistic and empiristic ways12. regarding the supposed platonism of euclid, affirmed with so much safety by proclus, we observe that this is a partial and unreliable interpreter in order to this problem: his marked newplatonic positions can have induced him to suppose, and then also to find, evidences of platonic thought in a scientist that was certainly considered as the greatest mathematician of every time13. of course it was difficult to find contrary evidences, just for the lack of explicit affirmations and comments that could reveal his philosophical options. someone, in the xix century affirmed that “for proclus euclid had the great fortune not to be denied neither from the caldean oracles, neither from the speculations of the old and new pythagorean philosophers ”14. but as we will say forward, we can well hypothesize that the apparent aseptic style of the euclidean writings and other scientific works of the same century, are not determined by the fortune but by a precise choice; and it is possible that such hypothesis furnishes a key of reading of the alexandrine science, and at the same time an explanation of the survival of several scientific texts, while all the original texts of the same period that subtend a vision of the world are lost15. naturally what we said, doesn't confirm but doesn't even exclude an adhesion or a proximity of euclid to the platonism. and so the modern criticism, even if it doesn’t consider seriously the 12 for instance ludovico geymonat wrote “the museum of alexandria represents the triumph of the specialized culture: the so-called hellenistic culture. the field of the to knowledge is divided in well circumscribed departments. neither general philosophical systems nor vast syntheses are created, but rigorous researches are developed on particular problems facing them one for time. the whole type of teaching has the tendency just to form some researchers more rich in serious and sure doctrine. this tendency towards the detail explains the interest for the scientific investigation and, still more, the method of the specialization adopted by the hellenistic science. while the great philosophers treated, with equal boldness and competence, of physics and of mathematics, as plato did, for example, or of logic and natural sciences as aristotle and theophrastus, the scientists of the hellenistic age are not expert in philosophy, and inversely the philosophers neglect scientific investigation, to reduce himself to his own specific competences” (geymonat, p. 284). and enriques: “later, with the diffusion of the greek civilization that was subsequent to the macedonian conquests, other centres, splendid of culture, flourished, as rhodes, pergamos, and above all alexandria of egypt. in this hellenistic period the science is loose from the philosophy (that it seems by now exclusively dominated by moral interests) and it touches to a florid maturity: to which nevertheless the decadence follows soon, also if slowly”. (enriques 1925, 1, p. 15). 13 the affirmation that the purpose of the elements is constituted by the platonic bodies (the five regular polyhedrons), doesn't have however some bases, because although they constitute the last matter of the work, there are remarkable parts of this that haven’t any application to the regular polyhedrons. the same affirmation becomes laughable if we think that the principles and the theorems of the elements are used in following works, particularly in the optic and this, together with the notions of the elements, is applied to the phenomena. if we use the criterion of proclus then we should conclude instead that the elements have as finality to “save the phenomena”. (see more forward, the section 4). 14 “martin says rather neatly, «pour proclus , les éléments d'euclide ont l'heureuse chance de n'être contredits ni par les oracles chaldaïques, ni par les spéculations des pythagoriciens anciens et nouveaux... ».” heath, 1956, i, p. 30, note 2. 15 the authors prefer, for the time being, to suspend the judgment on the reasons and the dynamics that will bring, with the crisis and the following end of the hellenistic kingdoms, to a deep inversion of tendency in the vision of the world; but it is enough evident that in the immediately following centuries the philosophies of this period are strongly opposed. it is enough to think that all testimonies on the ancient stoa (from zeno of citium to chrysippus of soli) are in antagonistic key and often (see as ex. galen, but also cicero) denounce an impassioned aversion or quite acridity. but not less impassioned criticism appears towards all that texts (not only philosophical) of the third century by which a conception of the world or a theory of the knowledge could be deduced. 40 affirmation of proclus16, it doesn’t always exclude an ascendancy of platonic origin on the euclidean geometry, but rather it sometimes uses categories proper of the founder of the academy in the analysis of the elements. heat said on this argument: “he may himself have been a platonist, but this does not follow from the statements of proclus on the subject […] it is evident that it was only an idea of proclus own to infer that euclid was a platonist because his elements end with the investigation of the five regular solids, since a later passage shows him hard put to it to reconcile the view that the construction of the five regular solids was the end and aim of the elements with the obvious fact that they were intended to supply a foundation for the study of geometry in general”17. enriques observes instead that “plato (resp. 527) seems to disdain the use of postulates, where he criticizes the «too much ridiculous and miserable terminology» of the geometers, which, as it was treated of practical purpose, they speak always to square, to prolong, or to add, while the whole science is cultivated to the purpose to know” 18. effectively, from a platonist man we could expect at least a different formulation of the first three postulates that clearly introduce in the geometry a constructive character! neugebauer is instead very drastic; he denies in general whatever influence of plato on the mathematicians: “it seems evident to me that the role of plato has been enormously exaggerated. his direct contributions to the mathematical knowledge have openly been void. the fact that, for brief time, mathematicians of the level of eudoxus belong to his entourage, is not a proof of the influence of plato on the mathematical research. […] the doctrines of plato have, without doubt, practiced a great influence on the modern interpretation of the greek science. but if the modern studious had devoted to galen or to ptolemy as more attention as to plato and his followers, they would rather have reached different results and would not have invented the myth of the strong attitude of the so-called greek spirit to develop scientific theories without making petition to experiments or to empiric verifications”19. a point remains to examine, that in our opinion is very important, it is that concerning the first seven definitions or terms (o(/roi). these, according to someone, would be able to confirm the platonist thesis. but their uselessness is noticed by several commentators; in fact they are never used in the proofs of theorems. moreover, not always they are clear, as the terms 4 and 7, in contrast with the extreme linearity and transparency of the work in every other part20. the attempts to explain such definitions, and particularly the 4 and the 721, already mentioned, afar from to do clarity, create more problems than they solve. on this matter an answer can come from the already mentioned paper of russo22 that by an ample analysis of the texts and the sources, reaches the conclusion that the o(/roi 1-7 would probably be formulated by heron and, however, added in a later period, presumably for educational or explanatory motives. beyond the convincing historical reasonings, this hypothesis allows us to answer to questions that otherwise would be dark. so we are inclined for the acceptance of this thesis, but we have to say that the matter cannot be 16 nevertheless exposures don't miss, mostly of synthetic, encyclopaedic or educational character, that in absence of further precise statements, can confirm the thesis of proclo or an anterior dating to the 300 b.c. we mention only an example in electronic edition and therefore recent. to the item euclid (marinus taisbak, in encyclopædia britannica, on line edition, britannica.com inc., 1999-2001) we can read: “for his subject matter euclid doubtless drew upon all his predecessors, but it is clear that the whole design of his work was his own, culminating in the construction of the five regular solids, now known as the platonic solids.” 17 heath, 1956, vol. i, p. 2. 18 enriques, , 1912, i, p. 42. 19 neugebauer, pp. 183-184. 20 an exception is perhaps constituted by the following def. 8 and 9 and some others. but on this topic we will speak forward. 21 see for ex.. enriques, 1912, i, p. 30 ; heath, 1956, pp. 155-176. 22 russo, 1998. 41 considered as definitely closed; in fact, if euclid has left indefinite the fundamental geometric notions, why would had to define the concepts of unity and that one of number (o(/roi 1 and 2 of book vii)? this introduces a possibility of a more general discourse extended to these and perhaps other definitions. however there are strong signs of possible additions or changes during the time: for example the definition of the odd time odd numbers23, not only is useless for itself, but it doesn't appear in the teon’s edition24. we want only finally discuss the problem, already mentioned25, of the o(/roi 8 and 9, on which numerous questions have also risen26. in first position it is the reason for which euclid owes to give a general definition of angle, containing also that one of curved angles. it is true that before euclid such angles was considered, but in the elements the more general definition remains an aim for itself, because, as definitions 1-7, they are never used. we can hypothesize that also the o(/roi 8 and 9, at least in the form in which they appear in the text of heiberg, has been written in a posterior period. in this case however we can ask to us if the angle was originally treated as not defined term or if euclid fixed one only definition (and what) for the rectilinear angle. the actual definition, moreover, doesn't define anything because it is limited to set a relationship of synonymy between the words angle and inclination. it is not possible, in this paper, to give an answer to a question that would require a specific research but in every case it deserves to be deepened. at this point we can affirm, without further hesitations, that anything is found in the elements that can confirm a direct descent of the euclid’s thought from that one of plato, neither a such influence can be historically documented. there are contrarily a lot of reasons to think euclid enough away from a platonist conception of the science. more complex is the matter of the relationship between euclid and aristotle. while, in fact, many aspects of the aristotelian theory of the knowledge are present, in different form and measure, in the whole history of the science, from euclid to our days27, on the other hand it is difficult today to understand what euclid knew of the aristotelian doctrine that is also known to us28. it is permissible nevertheless to suppose that the aristotelian thought, anyway, had its diffusion and a role of primary importance in the formation of the school of alexandria, also for the 23 euclides, elem. vii, def. 10. 24 see. enriques, 1912-1935, ii, p. 170. 25 v. nota 20. 26 v. ad es. enriques, 1932, i, pp. 32-35 e heath, 1956, i, pp. 176-178. 27 it is true that the birth of the modern science is considered as the fruit of the galileo’s opposition to the aristotelian tradition, but it is also true that in every phase and in every passage of its development, the scientific thought had to confront itself with the great themes set by aristotle. among these themes we remember particularly the distinction between philosophy and particular sciences and the definition of scientific knowledge that ever since is indissolubly tied up to the concept of cause. for aristotle to know (in a not sophistic way) means in fact to know “the cause by which a thing is and can’t not to be” (post. anal. i, 2; metaph, i, 1). the simple description of things or facts or the presence of contingent relations doesn't re-enter therefore in the scientific knowledge. the search of the general causes for a class of phenomena was the base of all the development of the ancient (post-aristotelian) and modern sciences, even if the concept of cause lose, with hume, his metaphysic mean ( david hume: a treatise of human nature). when the final cause was expunged from the modern sciences, only the concept of efficient cause remained and produced the consequent deterministic vision of the nature. it can seem that the modern science, refusing the final cause, keeps, clearly and definitively distance from aristotle at least on this. but the birth of the cybernetics has forced the attention of the scientists on the “finalistic behaviours”, simulated until now by the mechanism of the feed back, and even if this is conceived in reality as a deterministic mechanism that simulates in the results only a finalistic behaviour, different conceptions there are such that can to stake the whole matter (on the subject see for instance h. von foerster: la verità è l’invenzione di un bugiardo, melteni, roma, 2001). 28 notoriously the so-called esoteric or achromatic papers of aristotle, and this means almost all the work that we today know, were found again to athens and subsequently, brought to rome by silla, they were ordered and published by andronicus of rhodes in the first century b.c. euclid therefore could not have read, of aristotle, the same works that we know and on which in the centuries the whole exegesis of the aristotelian thought was founded; however it is likely that he knew, by other ways, the aristotelian teaching. 42 prestige of which the philosopher of stagirus had to enjoy to the successors of alexander. so if it is true that to bring the aristotelian thought to account has been an unavoidable necessity of the whole scientific thought, then it is also true that we don’t have now to establish if euclid was aristotelian, but if his work introduced elements of novelty and originality in comparison to the past and particularly in comparison to the philosopher of stagirus. to such purpose first of all we will try to individuate what characters of the work of euclid are surely maintained in the line of the aristotelian science. these characters can be easily identified with the deductive structure, that even if it was somehow already delineated by precedents geometers, nevertheless aristotle was he who clearly theorized and systematized it. to this deductive structure the name elements (stoixei/a) surely alludes29 . for aristotle elements are the indivisible components of something to which they are immanent30, so in the case of the euclidean geometry, we can think that the “elements” to which the title alludes is (1) components of the geometric objects, that is solids, surfaces or lines, with exclusion of the points that cannot have components. or (2) components of proofs. the first interpretation is given by different commentators, according to which, points, lines, and surfaces would be the elementary components to which the title would point out. this interpretation however doesn't convince. already in the platonic conception, the mathematical beings, because they are ideas, are for itself indivisible. also in the aristotelian conception, all the objects of the geometry, except the point, are of course divisible as greatness, but they are not divisible as notion, as definition and as category31. so a surface divides a body that has the form of a geometric solid, but it is not a constituent element of this last, neither as notion, neither as definition or category (or as idea for the supporters of platonic thesis). equally a point is the limit or the division of a line but it is not an element of it32. besides to consider the points as constitutive elements of a geometric figure would involve the admission of the actual infinite. this is expressly excluded by aristotle and, thing more important for us, carefully avoided by euclid. moreover the fundamental objects as point, line, surface, are not the real object of the euclid’s elements; there the whole plain geometry, the solid one, the theory of numbers and the theory of proportions are developed, while on such fundamental objects too much little is said, or nothing if the first seven oro/i are considered apocryphal. the second interpretation (always following aristotle), would concern the first principles of the geometry, that is the postulates and, at the most, the common notions. these in fact are, for aristotle, the indivisible constituent parties of the proofs. 29 it isn’t important if this title was or not assigned by euclid himself; we are interested to observe that, as proclus said, before euclid others geometric works was already called elements, in particular that ones written by hyppocrates of chios (470-410). 30 aristotle articulates this definition in cases and sub-cases but all belonging to the same general concept that presupposes: (1) to be parts of something, (2) to be indivisible (see metaph., 1014 a, 26 –1014 b, 15.) . 31 see. metaph. 1016b. 32 aristotle defines the surface as limit or division of a solid, the line as limit or division of a surface and the point as limit or division of a line (for ex. “…when the bodies are set to contact or are divided, in the moment in which one touches another, one surface only is formed and, in the moment they are divided, they form two surfaces. accordingly, when the bodies are gathered, the two surfaces don't exist anymore and they result destroyed; when, instead, the bodies are separated, the two surfaces exist as first they didn't exist […]. in fact, all these things [lines and surfaces] are, in the same way or limits or divisions” metaph, 1002a, 39–102b, 10). these entities are not able however to exist, for aristotle, separated by the bodies, and only our mind is able to consider it separately and independently from them (for ex.: “it is shown therefore sufficiently that the mathematical being are not substances in taller degree of the bodies, and that, in comparison to the sensitive ones, they don't have a priority in the order of the notion and, finally, that are not able in some way to separately exist” metaph. 1077b, 12-15, and “so the mathematical sciences won't be sciences of sensitive things, but they won't be even sciences of other objects separated by the sensitive ones” ibid. 1078a, 3-5) 43 as we will see this conception finds some difficulties if we consider possible to assume as postulate indifferently one or another between two equivalent propositions. but as we will say forward, this position doesn't seem to be that one of aristotle because he supposes instead a fundamental asymmetry between what is proved (more complex) and what needs to prove it (more simple), until to arrive so to a base of not demonstrable propositions: only these would be therefore elements. in the euclidean geometry, instead, the fifth postulate, at least, doesn't seem to satisfy to this condition and this set some difficulties for the second interpretation also. but there is a third hypothesis: that euclid uses the word stoixei/a following not aristotle but a precedent tradition in which elements means of course the parts that constitute a proof but without the pretension that they are not demonstrable. in this way, the greatest part of theorems, besides the postulates, would be elements of other theorems in whose proofs they are used. this hypothesis could justify the choice of the term as title of a work that is characterized for a branched structure according to a well precise definable relationship of partial order as “a is element of b” and the “postulates” constitute a set of minimal elements. this hypothesis is better suitable if we also consider the fact that the proof of a theorem was conceived, in the greek tradition and by aristotle also, as a decomposition of a proposition in several more simple propositions that one uses in the proof. the fact that already before euclid some elements have been written, confirms this interpretation, because in the greek tradition the first proofs should concern the most complex and important propositions, as are those on the greatness and those concerning relationships (equalities, similarity, theorem of pythagoras, etc…) while the proofs were founded upon propositions more reliable about which there were not doubts. the gradual refinement of the critical analysis should have induced greater caution and smaller confidence towards the presumed truths that were previously admitted, until the constitution of an inductive chain, and so, because an endless chains is impossible, the search had beginning of the simplest “truths”. we can read on this subject the following passage by enriques: “…in the work of euclid…in fact appear to the first places theorems as those on the equality of the triangles that aren't able to belong to a primitive period of the geometric development because they haven’t meant for itself, but receive it only as begins or elements of a chain that conducts to really meaningful geometric properties: as the sum of the angles of the triangle and the relationship (pythagorean) among the squares of the sides of the rightangled triangle, the two fires to which the arrangement of the first book aims” 33. the deductive order is therefore a sure element of continuity with the aristotelian thought, but partly also with the tradition of the greek geometry, at least beginning from hyppocrates of chios34. and certainly it is not easy to distinguish the two aspects, also because aristotle himself broadly uses the logical structures already consolidated within the mathematical studies, as bricks to compose his philosophical system. this continuity with the preceding tradition is clearly implicit in the passage of proclus in which, coherently with our conclusions, the meaning of the word “elements” is explained as follows: “…besides, the term «element» can be used in two senses, as menaechmus says: …, what proves is element of what is proved, as in euclid the first proposition is element of the second one and fourth one of the fifth one…”35 it is interesting the reference to menaechmus because it proves that surely before euclid the term was really used with the meaning that we said. 33 enriques, l’evoluzione etc., 1912, p. 4. 34 we remember that proclus said that the use of the term elements, dates back at least to hyppocrates of chios (470 410 b.c). see proclus, comm. ii, iv, 66. 35 proclus, comm., ii, vii, 72. 44 3. the hellenistic science. the problem regarding the role that the particular sciences had in hellenistic age, and particularly from the third century to the second one b.c., is still away from to have a satisfactory and univocal solution. it is broadly shared common notion that after the peak of the alexandrine civilization, and precisely from the second half of the second century b.c., a phase beginnings of decadence, while the range and the extension of the scientific enterprise, besides the moment and the reasons for which this development would be interrupted, appear still problematic. the thesis is very accredited for which the hellenistic science would have reached the threshold of a scientific revolution, but without never crossing it, although there were (in everything or partly) the theoretical premises. the literature on this matter is very wide; it will be enough to quote some examples only. the first one is quoted by ludovico geymonat that wrote as follows: “in front of the first victorious affirmations of such method [application of the scientific principles to the technology], that becomes today the main base of the modern technical civilization, there is to wonder for what motive it has not had in the antiquity a greater development, and it remained instead conscripted to some isolated cases […]. it is a very complex problem, that in general way can be formulated as follows: way not even a sketch of mechanical civilization was not developed, in the ancient world, while undoubtedly there were the first theoretical premises, though in limited measure? […]. the cause […] probably can be found in the social structure of the greek-latin world, which didn't feel the need to invent new machines, sufficiently having already to own cheap disposition the «natural machines» of the slavery.we remember on such matter that marcus terentius varro, describing the tools by which the earth is worked, he textually reports that «somebody divide them in three categories: speaking tools, semi-speaking-tools and mute tools”36. the second example that we want to quote is by federigo enriques and giorgio de santillana which affirm: “but who wants to understand the motives for this superb flowering [the extraordinary cultural climate that was developed around the museum of alexandria] is induced to look, over the external environment, the intimate conditions of the work of the researchers: as we said the thought, forgotten the universalistic claims, is now circumscribed within specific fields of search and, on the base of simple postulates, it succeeds in answering to determined problems. this separation from the philosophy seems a liberation of the science that, renouncing to really know the nature of the things, acquires properties of its real object and tries to derive the most important positive results. at the same time the great means of study, the most frequent contacts of the reunited researchers in the museum and the practice of the teaching that disciplines together teachers and pupils, compete to form a school in the modern sense: not longer philosophical school, that receives impulse from the metaphysical idea of a chief, but scientific school where different intelligence unite their efforts, creating and preserving the tradition of the method. nevertheless it is easy to imagine that these reasons can’t be enough for long time to maintain the progress of the science, if the interest of the problems is not relighted by an always living philosophical vision, and the work of the narrow class of researchers doesn't feed from an underlying culture of the people. under such conditions cannot surprise that the flowers quickly budded of the scientific genius come soon to fade.” 37. 36 geymonat, 1973, p. 300. we has to observe as geymonat uses in this passage two criterions that we think debatable: a generic reference to a greek-latin world, without further precise space-temporal references and the appeal to a few pertinent source. although in fact terentius varro is not entirely out of the considered temporal arc, he purely belonged to a cultural latin and roman circle, that between ii and i centuries b.c. was still set in comparison to the hellenistic world in terms of conquest and therefore of difference. but on this theme we will return forward. 37 enriques, santillana, 1937, p. 148. 45 on the same theme, the following passage of ludwig edelstein seems to us particularly meaningful. “since the nineteenth century the great majority of scholars have held that ancient science and modern science are worlds apart. but if one reads through the texts collected in the source book38, he can not but agree with the editors that it is an error to date the rise of natural science in the seventeenth century and to consider the greeks «mere speculators». in mathematics, astronomy and mathematical geography, physics, chemistry and chemical technology, geology and meteorology, biology, medicine, physiological psychology, in all these branches of learning the greeks developed and followed methods that closely approximate, if they do not equal, the standards of modern science. to be sure, the material assembled in these chapters is mostly outdated. what is presented here is not yet modern science. nevertheless the link between the ancient investigations and those of modern times is obvious” 39. end after “that ancient science failed to lead to technological application is another one of those prejudices that die hard. yet. contrary to the assertions repeated over and over again and made the basis of far-reaching generalizations, like those of spengler, the greeks were not hostile to technology, plato, to be sure, blamed the «corrupters and destroyers of the pure excellence of geometry, which thus turned her back upon the incorporeal things of abstract thought and descended to the things of sense, making use, moreover, of objects which required such mean and manual labor». but plato is not all of antiquity. archytas, eudoxus, menaechmus constructed instruments and machines. aristotle admired mechanical toys. aristoxenus appreciated technical detail. although plutarch intimates that archimedes on account of his «lofty spirit», his « profound soul», that is, on account of his platonic leanings, did not write on his inventions it still remain true that this «geometrical briareus”, as the romans called him, did apply his knowledge to practical ends. the list of his inventions is impressive. geminus, among others, considered mechanics a branch of that part of mathematics which is "concerned with and applied to things perceived by the senses". …»”40. but edelstein himself warns: “modem science and ancient science, then, are not diametrically opposed. i hasten to add, however, that such a claim can be made good only so long as one is willing to do what the editors of the source book have done, namely to select as evidence that material «which would generally be regarded today as scientific in method, i.e., based, in principle, either on mathematics or on empirical verification». to put it differently, the impression that ancient science is modern in character is bought at the price of neglecting or omitting all the evidence to the contrary”41. admitting therefore that the things are exactly as described, even with different tones, two great problems are set. 1. why, even in presence of enough theoretical premises, a scientific-technological civilization, as that started in that age, would not have been developed? obviously this question doesn't subsist if we assume with russo42 that at least the start of such civilization would take place. 2. why the initial progress of the hellenistic civilization in brief time was stopped, declining with great rapidity? 38 the considered article was written by edelstein about the book of coen, m. r., 1948; 39 edelstein, p. 91. 40 ibid., pp. 96-97. 41 ibid., p. 93. 42 russo, 1997. (see also at sect 1. introduction) 46 the answers are obviously different. edelstein write: “the argument most commonly advanced, and advocated also in the now most widely read histories of ancient science, is that in a slave society labor is cheap; technical improvements therefore were unnecessary in antiquity. such an oversimplification seems no longer justifiable, for as the investigations of the past few decades have shown, ancient economy can hardly be called a pure slave economy. especially in the arts and crafts free labor continued to hold its place. in addition to slaves, metics and citizens were employed as artisans during the classical age; …. simple reference to ancient society as a slave economy, then, explains nothing. the exact numerical relation of the various components of the laboring class it is difficult to estimate. what is certain is that the number of slaves in antiquity was much smaller than was thought by historians of the nineteenth century , and this is true above all of the classical and roman ages. however, even assuming that the percentage of slaves was relatively high, slave labor was neither cheap, nor docile, as is evidenced by slave revolts and strikes.”43. after having criticized the positions founded in exclusive way upon economical evaluations, edelstein continues, in pressing opposition to benjamin farrington44, of which disapproves the two fundamental theses, that is that the missed scientific development in the antiquity would essentially be owed to two factors: 1) an ideological refusal of the technology; 2) the censorial intervention of the politics. the reasoning of edelstein is convincing, historically found upon valid data, and it doesn't seem to leave space to meaningful objections. but when he tries then to answer to the same questions, he thinks to found the causes that braked the scientifictechnological development in the individual and private character of the search and in the lack of an organized scientific enterprise. to show this he uses data and sources that originate or from the classical period or from the following greek-roman one, with exclusion of the alexandrine period: the only one, in the whole antiquity, in which the scientific search is surely organized and enjoys of conspicuous financings. the library and the museum of alexandria, other smaller institutions elsewhere existing, the realizations of archimedes, are certainly known and recognized, but they are considered few influential exceptions. it is evident that once more there is a refusal to hypothesize a clean cut and a sudden arrest that could had place in the following period, refusal that is present in the majority of the modern researchers. we think to see on such refusal a positivistic idea of progress, as a continuous and unstoppable to go forward. according to this idea, if some scientific and technological progress was started then it can’t was stopped without a very big exceptional and traumatic event. it is just in opposition to this, that the book of russo was inserted here. he sustains in fact that the scientific revolution of which he speaks, would be developed in the brief arc of time characterized by the economic, military, politic and cultural power of ptolemaic egypt and of the other hellenistic kingdoms, becoming unintelligible after the roman conquest. in such way most of the possible objections is overcome, because all the contrary results, that we have, concern preceding or following periods, or they are extraneous however to the cultural elaboration that is developed in the alexandrine area. 4. interpretation of the science and the swzein ta fainomena. the definition of such an ample problem, introduces not only great difficulties for the shortage of original documentary material45, but it risks to give disputes that are empty of their object. 43 edelstein., pp. 97-98. 44 see farrington, 1944. 45 almost all the philosophical texts and literary products of the third century b.c. are lost, while the scientific texts that remain, among which also some of the most important ones, are generally without comment and above all they don't declare the philosophical choices set to their base. 47 this because the different sustained theses are often reported to different conceptions of the science. already more above we quoted a passage of edelstein in which this difficulty comes to delineate in enough evident way (see quotation corresponding to note 43) if it is put in relationship with what edelstein himself says later: “the editors [of the book. see. note 38], like many other students of antiquity, seem inclined to classify «theories that are now known to be false or even ridiculous» as «magic, superstition, and religion.» they speak of «’pseudo science,' such as astrology and the like,» that «can be found in the writings of such sober greek scientists as aristotle and ptolemy»; they refer to «the intrusion of the occult» that is noticeable also in modern scientific writings from kepler to eddington. but astrology, the theory of humors, plato's mathematical scale of music are not «intrusions» in ancient science. theories like these, which do not pass the muster of modem criticism, constitute is in fact the greater part of the preserved material. to the greeks, they were, just as scientific as those other views which happen to seem acceptable to the modern scientist” 46. it is nevertheless very unlikely that this judgment can be adapted to the scientists of the alexandrine period as euclid or archimedes and apollonius, also because, as we already said (see note 45), in their scientific texts there are not beliefs and visions of the world. the reference to ptolemy, lived in the ii century b.c., doesn't add instead nothing to how much already said. nevertheless the quoted passage clarifies in effective way the problem to which we want to refer and that often makes not comparable among them the evaluations on the ancient sciences, just for the incommensurability of what is intended by the word “science” 47. one of the more live debates on the interpretation of the greek science, in general and not only of that one alexandrine, concerns the saying sw/zein ta\ faino/mena (to save the phenomena), used particularly by pierre duhem48 to reinterpret the most meaningful part of the ancient science in instrumentalist key49. we agree with a big part of the criticisms addressed to duhem when, forcing the sense of the words or distorting a translation, he attributes not demonstrable instrumental intents to the “most representative” of the ancient greek scientists and in particular way to ptolemy and proclus. nevertheless, geoffrey lloyd50 himself, showing many of the historical mistakes of duhem, warns against easy generalizations. he says in fact in the introduction of his "to save the phenomena": 46 edelstein, p. 93. 47 it is hardly the case to remember as a large part of the post-popperian criticism (among which t. kuhn, p. feyerabend, etc…) conducted to more and more vanished and mobile vision of the delimitations between science and ‘pseudo science’. in this perspective, also the conceptions, that are introduced in the history as soaked of elements of metaphysics or also of “occult” or “magic”, could not immediately and uncritically be liquidated as ‘superstition’ and ‘pseudo science’. 48 duhem, 1908. see also duhem 1956-73. 49 by the word instrumentalism we understand an attitude that sees the scientific theory as an artificially built tool to insert the observed data in a coherent system with the purpose to allow predictive deductions on the future phenomena to check experimentally. realism is the opposed attitude to the instrumentalism, that is a tendency to consider the scientific theories as real explanations of reality. on this matter, as it is known, the process to galileo a lot was played, because the cardinal bellarmino was well inclined to accept a purely mathematic description of the cosmological system in terms of a heliocentric model, to the condition that the conviction that “really the sun is immovable to the centre and the earth is moved” was repudiated, while galileo was well firm in his realistic position. duhem reinterprets the whole history of the physics and above all of the cosmology in terms of to save the phenomena, understood in the sense of a substantial and, often, radical instrumentalism, so he reach the conclusion that the positions of bellarmino were correct and wrong the realistic one of galileo. here however we are interested to the fact that the instrumentalist interpretation of duhem is extended to the greatest part of the scientific elaborations of the greek antiquity also extended to astronomers as ptolemy. 50 see for ex. lloyd, 1993. 48 “first of all, the pluralism of the ancient science must be remarked again. […] the purposes and the assumptions of the ancient scientists in the field of the astronomy, of the acoustics, of the optics, etc…, is different, and not only from discipline to discipline, but also inside every of them. so, inside the same astronomy there are different undertaken types of study or types of composed essays. nearby to the tradition represented by the construction of mathematical astronomic models by hipparchus and ptolemy, there is, on the one hand, a more descriptive work search and, on the other hand, a work of more mathematical character [...]. i cited the tightly geometric study of aristarchus «on the dimensions and the distances of the sun and the moon»: such tradition also includes the «spherical» of theodosius, the writing «on the mobile spheres» of autolycus and the «phenomena» of euclid” 51. and even if in the conclusion of the same essay, he affirms that: “there where in effect we have some documentations, […] they often contradict the interpretative line so emphatically sustained by duhem and then taken back by others. […] in the methodological declarations of geminus, theon and proclus and in the real scientific practice of ptolemy we find elements in support to the opposite point [of view]” 52, he premises nevertheless that: “…for a lot of the most important figures of the history of the greek astronomy we are not in condition to pronounce ourselves in a definitive way on their conceptions or on the status of the varied hypotheses from them used or on the more general matter of the nature of the astronomy and of his relationship with the physics” 53. now this last observation unfolds all its meaning if we reflect on the circumstance that really for euclid and for other scientists of the considered period, the lack of knowledge regarding their philosophical beliefs is not owed, as in other cases, to the loss of their works, but to the fact that they wontedly have omitted every pronouncement54. we will take back forward this problem that seems to us essential because we don't think that the apparent to be aseptic of the scientific writings of euclid and other coeval scientists can be due to the chance or to an absence of convictions; neither the fact can be casual that, contrarily to a few centuries from there, all those persons that will attend to mathematics and exact sciences, will remark the exigency to accompany the scientific text with ample comments and justifications of the chosen premises. it is at least the sign of a conceptual change, or, to saying it with kuhn, of a “gestalt reorientation” 55 towards the methods and the objects of the scientific ‘to know’. about the different conceptions of the science, we must say that russo, in his mentioned book56, specifies in precise way what he intends with the expression “exact sciences”, preciously: 1. “… [they are] constituted by theories, as thermodynamics, the euclidean geometry or the calculus of probability, with the followings main characteristic points: the scientific affirmations don't concern concrete objects but specific theoretical beings […]. 2. the theory has a rigorously deductive structure; it is constituted, that is, by few fundamental propositions («axioms», «postulates» or «principles») on his own characteristic beings and from an unitary and universally approved method to deduce a 51 ibid. p. 427. 52 ibid. p. 470. 53 ibid. p. 470. 54 nevertheless, as we will say forward, the “phenomena” of euclid are not entirely deprived of indications toward a mathematical model that «save the phenomena». 55 the gestalt psychology considers ambiguous figures that can be interpreted in different way in different moments. typical is for ex. the case of geometric figures that appear sometimes concave, sometimes convex, according to the mental disposition of the observer; disposition that can suddenly change without there is apparently a reason. this “mutability” of gestalt’s orientation is assumed by kuhn as metaphor of the change, not only of individual level but, for vast cultural areas, of a deeper and general vision of the world. (see for ex. the intervention of kuhn in “criticism and growth of the knowledge”, edited by imre lakatos and alan musgrave, italian translation, feltrinelli, milano, 1984 ). 56 russo, 1997. 49 beings and from an unitary and universally approved method to deduce a boundless number of consequences. in other words the theory furnishes general methods to resolve an indefinite number of problems. such problems, enunciable within the structure of the theory, are in reality «exercises»: problems, that is, on which there is a general accord among the experts on the methods that can be used to solve them and to check the correctness of the solution. the fundamental methods are the proofs and the calculus. 3. the «truth» of the «scientific» affirmations is therefore in this sense guaranteed. the applications to the real world are founded on the «rules of correspondence» between beings of the theory and «concrete objects»”57. we don't enter into a discussion on those we think to be the limits of a definition of the science which is expressed from russo58, also because to the purposes of the present paper, we don't believe that a complete answer is necessary to all the themes that we placed until now, neither we think necessary that a general characterization of the alexandrine-ptolemaic science is given. euclid, in fact, could participate only to the initial phase of the hellenistic age. instead it seems to us very important to consider some of the original characters of this period that we can’t anymore found in following phases of the hellenistic age, because the philosophies and the visions of the world that was the base of the scientific progress in this century will be subsequently refused. it is necessary however to don’t fall into temptation to look for an unitary and typically alexandrine vision of the world, rather it is just this lack of unitary visions and values that seems to be the real character of the century and it is just here that we must look for a key of reading of the character that the different sciences begin assuming. in the complex and variegated differentiation of the points of view that are faced, often in sour polemic among them, it is an obligation to make reference to those are considered the three main current of the philosophical post-aristotelian thought: the epicureanism, the scepticism and the stoicism even if we have to say that this is only a simplification for convenience. so, for example, we cannot ignore the contemporary existence of reality as the peripatetic school, but it doesn't seem that the immediate followers of aristotle has product something of meaningful for us. in the same way we have to consider the differentiations, also radicals, that divide philosophers that was too easily confused by the use of a same label as sceptics59, stoic, etc…. having therefore to make reference to the philosophical currents of the third century b.c., and specially to the relationships that can subsist with the mathematics and the exact sciences, we want here to contract the discourse to the stoic school founded from zeno of citium and to the sceptics of the academy beginning from arcesilaus of pitane. this choice is owed to the fact that the contrast of ideas between these two schools seems to us of particular interest; contrast 57 ibid, p. 34. 58 this is not the place to express our position on the epistemological criticism that in the xx century followed the logical positivism. we believe to be useful only to develop some particular considerations about this specific case. it seem to us that putting a too sever limitations to what is legitimated to call science (as russo do), while on the one hand can be useful, helping to separate confused things among them, on the other hand risks to artificially build a cut, so rigid to lose sight of meaningful parts of the complex historical circumstances. a sure merit of the book of russo with his definition of exact sciences, is to delimit in a precise way the problem of the validation of the historical sources. so, putting also a precise space-temporal delimitation, he starts using improper generalizations to which we have still now more times mentioned (see in particular the end of sect. 3). oppositely a rigid delimitation as we have seen, of what is right science, would too drastically exclude visions and systems as the aristotelian one that, if on the one hand, with his pretensions of metaphysical absoluteness, constituted an obstacle to the empirical investigation, on the other hand has been and it is a base that keeps permeating of itself also the more positivistic regions of the modern and contemporary science. (see note 27). 59 we remember as arcesilaus held a lot to specify a connection of descent with plato, to distinguish in clear way from pyrrhon and from the phirrhonian scepticism. 50 that during the dialectical opposition, first between arcesilaus and zeno, more forward between carneades and crysippus, produced certainly for both them a growth of depth of elaboration60. we don't think, as already said, that direct relationships of adhesion can be found, and this for a multiplicity of reasons both of chronological character, and for the already many times mentioned aseptic character of the scientific texts. it appears well funded, instead, the problem to verify, through the comparison with the different expressions of the coeval thought, what hypothesis on the greek science can be considered somehow compatible with other aspects of the culture of the same age. it is enough for us, to point out as the foundation of the hellenistic kingdoms constitutes an element of strong novelty because: 1. the cultural production loses its purely individual volunteer and private character, to become financed activity, organized and integrated in the government structure. 2. between science and technique there are interrelations that are not casual but surely required by the same government entity (the king) that finances the production and diffusion of the culture. it doesn't care, for the time being, how much wide was their economic importance and if the prevailing aim was of military type. 3. the hellenistic world is from now so complex and variegated that, in it, an unified vision of the world cannot be anymore thought. the science, intended as complex of particular sciences that can have some relationship with the te/xnh, can subsist then only under the condition to don’t express any opinion on the themes of the being as being and to leave the matter of the absolute and definitive truth of the scientific premises undecided61. if the first two points seem already enough clear and hardly disprovable, some precise statements can be useful on the point 3. if, for one hand, it is difficult to maintain a precise interest towards the particular sciences by the greatest representatives of the philosophical hellenistic schools62, we are not able nevertheless to deny that the new terms in which the philosophical debate is set, putting in discussion the same meaning of knowledge, can’t not have more direct consequences on the methodologies and on the bases of the scientific search. the simple distinction, for instance, within the stoic school, between reality and meaning of the discourse (lexto/n), sets in absolutely new general terms the problem of the knowledge and particularly to the scientific knowledge. scepticism and epicureanism, agree then in to assign to the phenomenical experience the only source of knowledge that we can have, even if only the first one between such two schools assigns a value to the deductive method. the most serious problem, however, from the point of view of the value that one wants to assign to the scientific knowledge, it is in the effective stability and objectivity of the possible knowledge. it is this one the point of real and apparently irremediable contrast among stoics and academicians. for the stoic ones the science (e/pisth/mh), differs from the opinion (do/ca) because the first one have reached such a stability that cannot be upside-down by reasonings. we can wonder “why they specify «by reasonings»?”. it is possible to think that they considered the possibility to change some previously admitted knowledge by a different way? for example if a new comprehensive representation (xatalhptixh\ fantasi/a) arrives as consequence to some new experiences? all the interpretations given by the ancient sources and by the modern exegetic researches, lead to a negative answer. but we have to consider also that the stoic phi 60 see in particular ioppolo, 1986. 61 on the hellenistic history and society, as well as on the organization of the scientific enterprise in ptolemaic egypt, see for ex. bergtson, 1989; bianchi bandinelli, 1977; fraser, 1972; gullini, 1998, flower. for the philosophical thought in the same period, and particularly for the dispute among stoics and academicians, see for ex.. geimonat, 1973; ioppolo, 1996; isnardi parente, 1994, 1999; lévi, 2002; canfora, 1995. 62 see for ex. the introductive essay in isnardi parente, 1999). 51 losophers asserted, of course, the possibility to become a wise man, but nobody says to be a wise man, so such possibility has to be considered as an ideal target. there is however a point that perhaps could produce a doubt, and however can show as the ancient stoic gnoseology was more complex that it seems. we find it among a fragment of galen, the author that, just for the passionate vehemence by which he expresses his aversion for crysippus, refers, to testify his objectivity, some integral passages of the stoic philosopher. we see in fact that according to galen63: “[crysippus] undertakes to show what is correct to believe on the ground of the opinion [doxa] of any testimonies of the common people and not according to the nature of the things” 64. obviously this is by itself few believable because in clear contradiction with the central nucleus of the stoic doctrine and particularly that one of crysippus. but galen continues: “i transcribe here his same expressions, that are about these: «about such things we will likewise make search, starting from the common opinion and from the discourses that according to this taken place» and with common opinion cryisippus wants point out what commonly appears to all people; then continuing he says: «by all of this, since the beginning of preference, they seem to be conducted to affirm that our directive part is in the heart». still treating more of this, he textually writes: «it seems to me that most men are generally inclined to affirm this, because in certain way they realized that, in concomitance with their psychic motions, it is verified something in their breast, and especially in the place where the heart is set…»”65 and so galen polemically describes the defence that crysippus is building to his thesis; but more than the matter on which they contend we are interested to the way by which the stoic philosopher attributes cognitive validity to the common opinions, to the tradition, to the myth, to the allegory (galen say sarcastically :“…after to have filled the whole book with verses of homerus, hesiodus, stesicorus, empedocles, orpheus,….”66). still in the testimony of galen, we find however in another point (on the matter if the heart was the source of the nerves) the admission by crysippus …of do not know really the matter because inexperienced of dissection. precisely galen says: “[…] nevertheless he makes tolerable his opinion by saying modestly that he doesn't suppose to say that the heart is the source of the nerves or that he knows really what is concerned to this matter, since it is pronounced inexperienced of the art of dissection.” 67 these last affirmations, that galen interprets as proof of a confessed incapability and incompetence of crysippus, can be interpreted instead as awareness that the cultural stratifications bequeathed by the language, the tradition and the myth, contain in itself an image of the world that possesses its own validity, surmountable68, as it seems, only by a reorganization of the knowledge founded on specific and organized experiences (in this case the dissection). and in fact, if crysippus declares to don’t know the reality on this matter, to what would be directed the long discussion on which galen referred, if not to the interpretation of the symbolic meanings of the myth and the semantic stratifications of the language, as another passage of crysippus, referred 63 the theme of the dispute concerns the location of the directive part of the soul that is, in the ancient language, the location of the our rational functions. the fact that crysippus defends the most retrograde thesis (that it is in the heart) has here a little importance: we are interesting only to some reasoning’s passages. 64 galeno, de hippocr. et ptat. plac., iii, 1, p. 254 müller = svf ii, 886 . cit in isnardi parente 1999, p. 400. 65 ibid. 66 ibid., iii, 4, p. 281, müller = svf ii, 907 . in isnardi parente, p. 413. 67 ibid., i, 5-10, p. 138-145, müller = svf ii, 897 . in isnardi parente, p. 407. 68 that the tradition, as the language and the myth can’t be, for crysippus, the final criterion of the truth, is clear for many others known fragments. in particular we remember that he wrote against the common opinions. 52 by galen, confirms?69 in particular way just the sentence of crysippus “besides we give birth in us to the products of the sciences” seems to oppose in radical way to the tightly realistic interpretations given by galen70. obviously this doesn't oppose with the fact that the stoic philosophers aimed to reach a definitive and stable knowledge. here are the fundamental terms of the opposition between dogmatists (stoics) and academicians (sceptics), controversy whose fundamental nucleus concerned the possibility to reach a stable and objective knowledge. such possibility, as we said, was admitted by the stoic philosophers, at least as principle, but as result of a long process of intellectual elaboration beginning from the data of the sensitive experience. it is here however essential the fact that the sensitive datum is not the knowledge by itself, because this passes through the creation of conceptual objects or categories of thought (prolh/yeij o koinai/ e)/nnoiai) that are expressed and perhaps partly identified with the ways of the language. so the logos becomes essential part of the rational knowledge which begin seems to be set about the seven year-old age71. the sceptical academic school, instead, that began when arcesilaus becomes chief of the academy, contested not so much the most complex procedures by which the knowledge should be acquired, but the idea in itself that a certainty on something could be however reached. they advocated therefore the necessity of the suspension of the judgment, but not for this reason they refused the practical forms of knowledge, intended as provisional acquisitions, always revisable and revolts to some finalities (te/xnh). it is prevailing, but not unanimous72, opinion that both the schools didn't look with interest at the particular sciences, but only to the ethical matters. what, however, is interesting for us and appears as a fact, is that the general frame was surely not incompatible with a more and more autonomous, and somehow not realistic, development of the particular sciences. rather we could say that the ideal frame for the development of scientific ideas was that one furnished by the philosophical disputes on the value of the knowledge, as those that took place especially between stoic and academic philosophers. a frame in which the science could earn authoritativeness assuming new logical dimensions and categories of thought, at least similar to those ones developed from the stoic school73, but also suspending the judgment, as the academicians wanted, on the reality of the things and on the truth of the principles. 69 "after this, crysippus [...] says: «such they are the things that are said of athena [the myth that wants athena having birth by the head of zeus after that this swallowed metis], and the allegory that results from them is another. for first thing metis is compared to the mind and the art of to live; for artwork of this we have to send down and to swallow the arts equally that we say to send down discourses of other peoples: it is consequently as to say that we almost have to gulp down them and to send down them in the abdomen. after this, it is reasonable that we give birth to this art that we swallowed, becoming with this similar to a mother that produces; besides we give birth in us to the products of the sciences...»" ( ibid., iii, 8, p. 321, müller = svf ii, 909 . in isnardi parente, p. 415). 70 however all the testimonies on crysippus, to a careful reading, bring to the representation of an extremely deep thought, beyond, for the most part, than the same witness not succeeded in intending. 71 “just that ability to reason in virtue of which we are said reasonable beings, they says that it is formed in us in base to the anticipations (prolh/yeij) and it reaches perfection around the seven year-old age”. (aetius, plac., ii, 1-4, dox., gr., pp. 400-401 =svf ii, 83, in isnardi parente, p. 700). we notice transiently only the almost coincidence of age, in the theory of jean piagét, with the passage from the pre-operating intelligence to that of the material operations. see for ex. flavell, j. h., the developmental psychology of jean piaget, princeton, n. j. , 1963) 72 see for ex. lévy. 73 as it regards the revaluation of the stoic logic, not all arrange on the value that it could have in relationship to the sciences. isnardi parenti thinks, for instance, that the logic of crysippus constitutes, in comparison to the aristotelian one, a return to the tradition of the fifth century. this particularly because the crysippus’s syllogism, unlike the aristotelian one, have as object individual and not universal things. we don't agree on this, and observe that in the geometric proofs the form of the aristotelian syllogism was never used. the reasoning in fact is always referred to singular objects, even they are supposed selected in a casual way. we say for instance: the point a.…, the point b, the straight line ab…., the angle abc…, etc…; never “all the points are…” or “all the straight lines are….”, etc…the generalization is obtained to a next action of thought in virtue of the arbitrariness with which the choices are intended effectued. syllogisms of the type “all the triangles are polygons, all the polygons are surfaces, therefore all the triangles are surfaces”, can help us to illustrate with examples the aristotelian syllogistic forms, but 53 5. common notions and postulates. a datum already meaningful for itself, is the fact that the authenticity of the common notions has been debated, in everything or partially. putting aside the recognized unreliability of some of them, we think interesting the debate to which an article of tannery given place, by which he sustained a not authenticity of all the common notions, that would been subsequently interpolated, and he formulated the hypothesis that apollonius would have add them. these conclusions have been however rejected by heath and by other authors. even if there are not certain documentary proofs, we are inclinable to accept the prevailing thesis, according to which at least the first three common notions (but probably more) would be authentic of euclid. any the factual truth would be, nevertheless, we consider very interesting the reasoning produced on both sides74. to such purpose we entirely report here what heath said on the subject: “the following are his main arguments. (1) if euclid had set about distinguishing between indemonstrable principles (a) common to all demonstrative sciences and (b) peculiar to geometry, he would, says tannery, certainly not have placed the common principles second and the special principles (the postulates) first. (2) if the common notions are euclid's, this designation of them must be his too; for he must have used some name to distinguish them from the postulates and, if he had used another name, such as axioms, it is impossible to imagine why that name was changed afterwards for a less suitable one. the word ©nnoia (notion), says tannery, never signified a notion in the sense of a proposition, but a notion of some object, nor is it found in any technical sense in plato and aristotle. (3) tannery's own view was that the formulation of the common notions dates from the time of apollonius, and that it was inspired by his work relating to the elements (we know from proclus that apollonius tried to prove the common notions). this idea, tannery thought, was confirmed by a «fortunate coincidence furnished by the occurrence of the word e(\nnoia (notion) in a quotation by proclus (p. 100, 6): "we shall agree with apollonius when he says that we have a notion (e(\nnoia) of a line when we order the lengths, only, of roads or walls to be measured.» in reply to argument (1) that it is an unnatural order to place the purely geometrical postulates first, and the common notions, which are not peculiar to geometry, last, it may be pointed out that it would surely have been a still more awkward arrangement to give the definitions first and then to separate from them, by the interposition of the common notions, the postulates, which are so closely connected with the definitions in that they proceed to postulate the existence of certain of the things defined, namely straight lines and circles. (2) though it is true that ©nnoia in plato and aristotle is generally a notion of an object, not of a fact or proposition, there are instances in aristotle where it does mean a notion of a fact: thus in the eth. nic. ix. 1171a32 he speaks of "the notion (or consciousness) that friends sympathise «(≤ ©nnoia toë sinalge›n toáw f¤louw) and again, b 14, of " they are too much trivial to be able to do indeed geometric proofs. if instead one had wanted to use however the universal quantification (as aristotle would like) in the real mathematical proofs, he would necessarily have had to bring the actual infinite to the account. this is so the problem that, with the development of the modern infinitesimal analysis, has not been more possible to elude and that has brought to the set theory. in terms of modern formal logic, this is gotten adding to the modus ponens the generalization rule, that affixing to a variable the universal quantificator (for all), generalize singular propositions to an actual infinity, whereas supposing only the arbitrariness of the choice (as in the case of euclid, but of cauchy also), the extension becomes only potential. in conclusion we think important, in the evolution of the mathematical thought, the aristotelian conception of the syllogistic reasoning as formal tool separated by the questions of truth and reality. if the real form of the syllogism is considered instead in the mathematical reasonings, then that of crysippus appears to be more qualified than the aristotelian one. 74 tannery, 1884. 54 the notion (or consciousness) that they are placed, at his good fortune.» it is true that plato and aristotle do not use the word in a technical sense; but neither was there apparently in aristotle's time any fixed technical term for what we call «axioms», since he speaks of them variously as «the so-called axioms in mathematics,» «the so-called common axioms,» «the common (things)» (tå koinñ), and even «the common opinions» (koina‹ dòjai). i see therefore no reason why euclid should not himself have given a technical sense to «common notions,» which is at least a distinct improvement upon «common opinions.» (3) the use of ©nnoia in proclus’ quotation from apollonius seems to me to be an unfortunate, rather than a fortunate, coincidence from tannery's point of view, for it is there used precisely in the old sense of the notion of an object (in that case a line)”75. we reported this long passage because some meaningful elements emerge from it for our analysis. but we want first to recall on this subject the judgment of enriques that affirms: “the distinction between the postulates and those that forward are designated as «common notions» (the «axioms» of the pythagoreans) is illustrated by aristotle and by the comment of proclus according to different points of view […]”76 and after: “but it is remarkable that aristotle never speaks of common notions, using the pythagorean term of axioms (a\ciw/mata = dignity); rather the word ©nnoia doesn't seem to be in technical meaning in plato or aristotle, but only later among the stoics. however, the deductions that someone (tannery) wanted to draw from this circumstance, doubting the authenticity of the euclidean notions, fall in front of the observation that the word ©nnoia is found in a fragment of democritus. and since among his lost works there is an essay of geometry that, for the disposition, is like to the euclid’s elements, it is permissible to deduce that the text of democritus could have this denomination of the axioms and that from it euclid has taken back” 77. the reasoning of heath can indeed frustrate those of tannery, and we agree with him when he says “i see therefore no reason why euclid should himself have given a technical sense to «common notions» “. of course! why should not he have does it? and we tell it beyond the use that aristotle does in different contexts of expressions as ta/ koina and koinai\ do/cai, without to notice the exigency to fix a technical term. what instead seems to us to unite the three authors is the fact that all are moved on an implicit common assumption that is present in almost all the critical literature on the elements and that can be resumed in this way: “the euclidean geometry is built within the platonist-aristotelian philosophy; therefore if something in it is not consistent with plato or aristotle, and only in this case, or it is a defect or it is a posterior addition”. this assumption his not proposable, we think, because it tends to prove the affiliation of euclid to a philosophic area by presuming such affiliation as premise. to great reason this holds 75 eath, 1956, p. 221. 76 enriques, f., 1912, p. 42. we remember that really aristotle distinguishes among the fundamental principles (not demonstrable), those that are to the base of single particular sciences from those that have instead a general character. he however doesn't seem to use for this a technical terminology, as it is noted above by heath. terminological distinction was instead done by aristotle not among the indemonstrable premises but among those that are assumed without proof even if they are demonstrable. such propositions are said by aristotle hypotheses if they are believed by the pupils, postulates in the other cases (see post. anal. i, 10). what seems to us important is that both of such two order of distinctions have a character instrumental and methodological but not gnoseological. the first distinction (between fundamental or particular principles), in fact, is referred to the generality degree, the second distinction (hypotheses or postulates) concern a precise rapport with well determined pupils (as it is well marked by aristotle) that can or not believe some premise. as we can see, all this do not regard any problem of being and of truth. to the first principles (indemonstrable) one riches, for aristotle, following an inductive process and a not better clarified intuition; therefore, to be such, they have to be as much clear, as evident and true. 77 ibid. pp. 47-48. 55 if we consider the complexity of the cultural, scientific and philosophical context of the third century b.c., during which, on the other hand, we don’t known what of the aristotelian works were known and circulating78. going back to the quoted passage of enriques, we note as, despite he had observed as the word ©nnoia was used by the stoics (and zeno of citium is contemporary of euclid), he preferred to resort to vague and captious hypotheses, also for the temporal distance that separates euclid from democritus. in the stoic school, not only the word ©nnoia is used, but we finds agai the whole expression koinai/ e)/nnoia, that for example isnardi parente thinks to be able to identify (at least from some citations of arrianus and plutarchus) with prolh/yeij (translatable as “idea” but also “scheme” or “mental image”; today we would perhaps speak of cognitive structure. see also section 4, note 71). we want immediately to observe as the stoic concept expressed by koinai/ e)/nnoiai, from the extensional point of view could partially correspond to what aristotle sometimes designates with axiomata, but on the conceptual ground differs deeply from it. from here, however, we don't think that we can draw hurried conclusions because, as we already told on introduction, we reject rigid relations of interdependence between euclidian geometry and stoic thought79. we think, however, that we will must consider with a lot of attention the hypothesis of repositioning the work of euclid from the narrow circle of the platonic-aristotelian philosophy to a more mature phase of evolution of the scientific and philosophical thought. if the knowledge’s procedure, in fact, was already theoretically delineated by aristotle as result of an inductive-deductive process, now it’s enriching itself on the one hand through the organized work of the scientists, on the other hand through a philosophical dispute that put in discussion the means of the acquisition and the same value of the knowledge. different elements seem to us to point out a coherence with our hypotheses. meanwhile we must observe that postulates and common notions, in the elements of euclid, differ not only for to be separately gathered, but also for the different way of enunciation. only for the postulates (aith/mata), in fact, the word h)ith/sqw80 is initially used; that word is translated by heath as “let the following be postulated” and by enriques more literally as “domanda” = ”ask” (as command) while in other cases it is ignored81. if we admit then that (at least some of) the common notions are authentic, in the sense that they have already been introduced in the original text, then the presence of the incipit h)ith/sqw in the postulates but not in the common notions, has of course to constitute a difference between these two classes of first principles. and this difference cannot be confined to the aristotelian distinction between general principles and principles of the specific sciences, but it has to refer rather to something that involves the state of truth 78 v nota 28. 79 the direct approach of euclid to some aspects of the stoic gnoseology could constitute a very strong and fascinating suggestion. beyond however of the difficulties that could emerge from a more careful analysis, there is a historical and chronological not eludible datum. in fact the most interesting aspects of the stoic thought in relationship to the theory of the knowledge seem to be those were delineating with crysippus, not in a casual way, but in the course and under the push of the controversy with the academy (see. joppolo). now, because crysppus borns in 281, it is not chronologically possible that he had some influences on the our alexandrine scientist, active certainly before he was born. we cannot be exclude instead that the new forms of thought that were affirming in the science could influence the philosophical debate: it would seem rather few believable to us that this had not happened. 80 h(i(te/sqw: imperative form of the verb ai(te/w (to ask), therefore literally «ask» as command. in the dialectical tradition of the greek philosophy the questions were put to the interlocutor before beginning a deductive reasoning and they were substantially a request of assent on some propositions, that after approved became the premises of a reasoning. the same aristotle sometimes uses the term asks with the meaning of premise of a syllogism. it is clear from this that the intent is to constitute a syllogistic premise, but it doesn't automatically follows from this, that such premise has a hypothetical value in the sense of suspension of the judgment on the truth. just for what the comparison with the common notions, can assume meaning in this sense. 81 see for ex. trudeau. 56 or any other criterion of acceptability. now, because the dialectical question is just a request of assent, the different way of setting the two classes of first principles can be formulated as follows: the assent of the reader was trivially admitted for the common notions, while for the postulates a convention, an agreement, or a hypothetical acceptance was required82. we want besides to do a last remark on this argument: as we said (see note 76), the term postulate had for aristotle a very different meaning, at least if we refer ourselves to the posterior analytics. in particular it is impossible to assimilate the euclidean distinction between postulates and common notions with the aristotelian distinction between postulates and hypotheses because: a) the aristotelian division is (as he explicitly remarks) relative to particular pupils and so it is possible only on the personal rapport of teaching and not in a write work. b) some of the euclidean common notions, is used by aristotle as example of indemonstrable general premise that he distinguishs from both of hypotheses and postulates. but beyond this indication, there are deeper motives to think that euclid, on the matters of truth and existence, had a different position than aristotle. we will consider in the next section the most meaningful topic under this aspect, that is that of the parallel straight lines. now we want instead to remember that euclid, besides the elements, also wrote other works and, among these ones, the optics in particular must be considered, just because they reflect the same hypotheticaldeductive structure of the elements. here also the enunciation of the fundamental principles (postulates) begins with a hypothetical form, although in this case the used term is different83. but what is here more meaningful is in the content itself of the postulates, in particular for the postulate 1, 3 and 4 that are formulated as follows: 1. let us to put therefore that the radiuses84 that depart from the eye are straight lines that have some distance among them. ………… 3. and that the things to which radiuses come are visible. 4. and that the things to which radiuses don't come are not visible. it is clear that the criterion of the truth and the self-evidence prescribed by aristotle to accept the prime principles, is here totally inapplicable, and if one wanted to adopt it, than it would bring only to the refusal of the theory; and really this historically happened85. in fact, also without giving a realistic interpretation to the radiuses (see note 84) and considering only them as geometric entity, it would be difficult to find a justification to the choice of a discreet model (the radiuses have distance among them) considering only the axioms in itself, neither the reasons because the 82 obviously all this would fall if one accepts, in accord with tannery, the non authenticity of all the common notions, and in this case a possible stoic origin of their denomination could be reconsidered. but the idea of tannery that the interpolation of common notions can be attributed to apollonius, beyond the reasons produced by heath, appears to us not proposable because, the custom of the comments and of the interpolations in other people's works seems begun only later, when the original scientific production become less meaningful and, above all, the scientist coincides often with a philosopher that tray to brought again the science inside some philosophical conception. securely this condition is realized starting from posidonius and geminus in the first century b.c. 83 u)poxei/sqw = let us to put. 84 in the translation, we used here the term radius and not ray, because such last one presuppose a realistic interpretation, that is rays of light; on the contrary we think, according to incardona, that it has to be interpreted as geometric straight lines conducted from the eye. 85 historically it was refused (for ex. by ptolemy and by leonardo of vinci) by virtue of a realistic interpretation of the “rays (or radiuses?) that go out of the eye” (see for ex. incardona). such interpretation kept on influencing the modern criticism also, despite the purely mathematical character of the euclidean text was clear. we read for example in boyer (that in turn mention cohen and drabkin): “the optics of euclid is notable for the exposure of an «emissive» theory of the vision according to which the eye utters rays that cross the space….”( boyer, 1976, p. 121). end immediately later: “note that the mathematical concepts of the perspective (otherwise from the physical description) are the same any it is the adopted theory” (ibid). nevertheless, considering that the euclidean work is only geometric, he don’t wonder from where we had to deduce a thought that the author doesn't express. but sometimes the strength of the tradition appears stronger than the observed evidences. 57 third and the fourth postulates would be included appear understandable if we don't analyse the consequences of those in conjunction with the first one. the things change instead if we assume as criterion of validation not the truth of the principles but the correspondence between the consequences of the postulates and the observable phenomena. in fact, this appears of an extraordinary clarity when one reads the theorems in whose proofs the aforesaid postulates are used. and in fact we find them in the proofs of theorem 2 (nearer objects are seen with greater resolution than more distant objects) and of theorem 3 (for every object there is a distance over which it is not seen anymore). as we can see, here not only the truth and the self-evidence fails as criterion for the acceptance of the postulates, but the same criterion of validation is moved from the premise to its consequence. neither the work phenomena must be neglected to this regard; in fact, even having a discursive style, it displays a notable interest by the point of view that we are examining, just because, as we already said, they are founded upon an initial hypothesis of mathematization that among all the hypothetical formulations of euclid, is the only one to have been expressly justified in a way that clearly reveals the intent to save the phenomena. 6. the question of the parallel straight lines. on the matter of the fifth postulate we quote by heath: “we know from aristotle that up to his time the theory of parallels had not been put on a scientific basis (anal. prior. ii, 16, 65a 4): there was apparently some petitio principii lurking in it. it seems therefore clear that euclid was the first to apply the bold remedy of laying down the indispensable principle of the theory in the form of an indemonstrable postulate”86. we will see the reason why, to our opinion, the theory of the parallel straight lines could not find a coherent organization inside the aristotelian conception of the indemonstrable principles, neither, all the more reason, in a platonist philosophy. we will see besides as from here can come down a reasonable key of reading for the two millennia long story of criticism to the fifth postulate. now however we have to analyze what was said on the subject by imre toth87 in the volume that we have already mentioned in introduction. to the beginning of his work toth mentions the same passage of aristotle that few above is cited by heath, and he says: “the interpretation of a known passage of the first analytic, ii 16, 65a 4-7, reach to the result that this prime principle can be only a proposition that is equivalent or also identical to the famous axiom of the parallel straight lines”88. this passage (of toth), requires itself some explanation. the quoted passage of aristotle of which toth doesn't give a translation, neither precise which is the interpretation to which alludes, is in fact the following: ú o(/per poiou=sin oi( ta\j parallh/loj oi))o/menoi gra/fein! lanqa/nousi ga\r a)utoi\ e)autu\j toiau=ta lamba/nontej a(\ ou)x oi\=o/n te a)podei=cai mh\ ou)sw=n tw=n parallh/lon. there are different discordant translations of it. we quote first the italian translation of g. colli that is the following: 86 heath, 1931, p. 358. 87 see. note 9. 88 toth, 2000, p.69. the first principle (arxh/))), on which toth spicks and as himself specifies in the same page, is a proposition from which the so-called theorem of the parallel straight lines derives, that is the twenty-ninth proposition of the elements. 58 “ed è proprio questo l'errore commesso da coloro che ritengono di tracciare delle rette parallele: essi infatti non si accorgono di assumere delle premesse tali, da non poter essere dimostrate, a meno che le rette non si presuppongano come parallele” 89. it is a redundant translation that tries to already furnish an interpretation of the text, but of which the meaning is not well understood in relationship to what we know on the problem of the parallels. we have to premise at this point the context in which the passage of aristotle is found: the philosopher had just illustrated the insubstantiality of the circular reasonings through which some people pretend to prove a through b when b has been in turn proved by c and this by a; the quoted passage is used as an exemplification of these false reasonings. it seems clear, at this point, that the expression “to draw the parallel lines” is referred to some geometric construction of the parallel line drawn by a point to a given straight line; this presuppose some proposition that assures that two straight lines, as are built, are indeed parallel. toth assumes, as it seems, that the proposition to which aristotle alludes was the theorem (or equivalent) that euclid sets as twenty-ninth proposition, the first one in the elements that is proved using the fifth postulate. if so, then indeed a first principle would need equivalent to the fifth postulate to prove such proposition. but what sense does have, in such case, to say that it is necessary to suppose already that the drawn straight lines are parallel? we know that to prove the parallelism between two straight lines drawn in such a way that corresponding angles are equal, needn’t the twenty-ninth proposition, but only the twenty-eighth one, which demonstration doesn't require the fifth postulate. a simple construction of the parallel line to the straight line r for a point p is got building first by p the perpendicular t to r (elem. prop. 12) and then still by p the perpendicular s to t (elem. prop. 11). by proposition 28 we have finally that r and s are parallel. the twenty-ninth proposition, instead, reversing the proposition 28, allows to prove the oneness (and not the existence) of the parallel line. in this perspective, the aristotle's passage, in the version of colli above quoted, doesn't seem to assume a reasonable sense. a rigorously literal translation of the aristotelian passage could be: “this is what those persons do that believe to trace (or to draw) parallel straight lines: in fact, without realizing it, they assume what is not possible to prove if the parallel straight lines don't exist.” in this version, that is comforted by other known translations90, the passage assumes a more coherent meaning; in fact the proposition 28, that allows us to recognize as parallel the built straight line, would not be demonstrable in a geometry in which the parallel straight lines don't exist. it is this in fact the case that corresponds to the hypothesis of the obtuse angle of gerolamo saccheri, in which these three facts, that are interdependent among them, contemporarily hold: a. the straight line is not infinitely extensible. b. the sum of the inside angles of a triangle is greater than two right angles. c. parallel straight lines don't exist. what aristotle seems therefore to affirm, according to what appears the most reasonable interpretation, is that to assure the validity of the construction of parallel straight lines (that is to 89 “and it is just this one the error committed by those that think to draw parallel straight lines: they in fact do not realize to assume some premises, that can’t prove unless the straight lines are not presupposed to be parallel”. aristotele, organon, edited by giorgio colli, adelphi, milano, 2003, p. 250. 90 for ex. the italian translation by marcello zanatta: “cosa che compiono quelli che credono di disegnare le parallele: costoro infatti non si avvedono di assumere cose tali che non è possibile se non esistono le parallele”90. (aristotele, organon, a cura di marcello zanatta, vol. i, utet, torino, 1996, p. 397) or the english one of a. j. jekinson: “this is what those persons do who suppose that they are constructing parallel straight lines: they don’t see that they are assuming facts which it is impossible to demonstrate unless the parallel straight lines exist” (aristotle: prior analytics, electronic edition of mit, massachusetts, on http://classics.mit.edu/aristotle/prior.html). 59 prove that two straight lines anyhow built, are parallel) it is necessary to suppose the existence of parallel straight lines or some equivalent proposition91, so if somebody thinks to prove the existence of the parallelism by building parallel straight lines, then he encounters the contradiction considered by aristotle92. from what we said it follows that the required proposition doesn't to be equivalent to the fifth postulate, rather, this cannot be that (or equivalent to that) wanted by aristotle: in fact the fifth postulate is trivially valid in the case in which the parallel straight lines don't exist. to prove the existence of the parallel straight lines, instead, it is enough a proposition that allows to deny what gerolamo saccheri sets as hypothesis of the obtuse angle. in the elements of euclid such proposition can be identified with the third postulate (unlimited prolongability of the straight line). all of this, to our opinion, doesn't invalidate the general discourse of toth, but it sets a demand of a more precise distinction between the two fundamental problems regarding the parallel straight lines: the problem of the existence, guaranteed when we declare the unlimited prolongability of the straight lines, and the problem of the singleness, that requires a proposition that must be equal or equivalent to the fifth postulate. this appears clearer, also from toth, in the introductory essay to the work of saccheri, written with elisabetta cattanei93. here she says in fact: “the hypothesis by which the sum of the angles of a given triangle is greater of two right angles is not by itself incoherent, and it possesses rather remarkable implications of geometric type, however it is incompatible with a fundamental property of the straight line, that is presupposed both by the hypothesis of the right angle, and by the hypothesis of the acute angle: the property of the straight line to be an open in both directions and endless line” 94. this is a distinction that, from what can be understood by the quoted passage of aristotle, didn't have to be still well clear before euclid, just because a first principle had not been found by which it is possible to derive the properties of the parallelism and that appeared, at the same time, simplest and self-evident according to the aristotelian demand. to confirm our interpretation of the aristotelian passage, there are all the other passages of aristotle that are mentioned by toth himself. in them in fact, more times and in varied contexts, not euclidean hypothesis is considered. every time however such hypothesis doesn't contemplate a geometry of hyperbolic type (sum of angles smaller then two right ones) but, on the opposite, a geometry of elliptic type (sum of angles greater then two right ones) in which cannot exist parallel straight lines. this results is well clear for instance in a passage of the prior analytics95, in which, aristotle speaks on the contradictions that can subsist after removing a false hypothesis 91 notoriously in the so-called elliptic geometry, what for ex. is that one on a spherical surface, however two distinct straight lines are given, they always have a common point. if we put ourselves on such geometry, it is possible to draw two lines by the same construction generally used to build parallel straight lines; that is we are always able from a point p to conduct the perpendicular s to the straight line r, and then still from p the perpendicular t to s. in this case however the straight lines r and t are not parallel because they have a point in common. it is correct therefore to affirm that who believes with the aforesaid construction “to have built (traced, drawn) the parallel straight lines” owes first to prove that the geometry is not elliptic, that is that in it the parallel straight lines exist. 92 otherwise we don’t see where is the circularity in the affirmation that to prove the parallelism between to straight lines, the existence of parallel must be before proved. 93 saccheri, 2001. 94 ibid., p. 24. 95 pr. anal. 66a 7-16: “consequently since the impossibility results whether the first assumption is suppressed or not, it would appear to be independent of that assumption. or perhaps we ought not to understand the statement that the false conclusion results independently of the assumption, in the sense that if something else were supposed the impossibility would result; but rather we mean that when the first assumption is eliminated, the same impossibility results through the remaining premises; since it is not perhaps absurd that the same false result should follow from several hypotheses, e.g. that parallels meet, both on the assumption that the interior angle is greater than the exterior and on the assumption that a triangle contains more than two right angles.“ 60 by another similarly false; here, to exemplification purpose, he considers two possible hypotheses: the first one denies the theorem of the external angle, the second one, equivalent to the first one, suppose that the sum of the angles of a triangle is greater then two right angles; both these hypotheses are such to lead to the same contradiction: the incidence of two straight lines already supposed parallel (“the parallel straight lines meet”). 7. conclusion. we can now wonder if the analysis until here taken allows us to answer, at least partly, to the questions that were set: that is to appraise how much of originality there is in euclid in comparison to the preceding tradition and if his work must be considered as the summa of a scientific thought developed in the precedent century, or as an opening of a new phase of the science. we don't say nothing new if we individuate the matter of the parallel lines as that by which the originality of the euclidean contribution emerges with evidence, but we want to understand if he gave only a technically correct answer to a preexistent well posed problem, or instead the solution itself determined a qualitative jump on the conception of this problem, of the geometry, of the sciences altogether. we will try to formulate an answer to different levels. a first level of novelty of the euclidean solution, can be deduced already by the comparison with the passages of aristotle in which a not euclidean hypothesis is formulated. these, as we have seen, don't consider any distinction between the two connected problems of the parallelism: that is the existence and the oneness of the parallel straight line. in the elements, instead, the first 28 propositions, rigorously proved without the use of the fifth postulate, seem to be placed just in that precise order to underline the impossibility, in virtue only of the first four postulates, of that we have for brevity more times pointed out as hypothesis of the obtuse angle. rather the twenty-eighth proposition could implicitly constitute an answer to aristotle's passage quoted by toth. this hypothesis however appears not well formulated if we remember that euclid doesn’t was able to read an aristotelian work that was published about two century later96, so we reformulate it in the following way: euclid had securely known the open problem that only shortly aristotle indicated, so to this problem must be addressed the answer of euclid that, as we said, strongly remarks the separation of the problem of the existence from that one of the oneness. only at this point, with the twenty-ninth proposition, the fifth postulate enters the scene to answer to the second question of the parallelism. this answer immediately calls a new question: why euclid, certainly conscious to have given an innovative answer to the science, doesn't take the opportunity to underline his own important contribution by a clear explanation? why he entrusts all his ideas to only the mathematical language? unfortunately the mathematical writings of the immediate predecessors of euclid are not arrived until us and we therefore don't know their style, but we know the style of immediately following scientists as archimedes and apollonius. in such cases there are some comments or explanations, but not to make clear some philosophical options put as base of their scientific works. to suppose that the lack, in euclid, of any explicit explanations is owed only to the chance or to a short attitude to write, is therefore totally not convincing. so much more than in the phenomena, the only case in which he put to the base an explanation, this escapes from every declaration on the being and on the truth to limit himself to save the phenomena. the initial page can be resumed in fact, for what it results essential to us, in the following way: 96 see note 28. 61 “since the stars are perceived [as things that are moved in a certain way, according to the proofs of the optics], then we have to set that [is moved… in that way] 97. the intent of “to save the phenomena”, giving a geometric explanation of it, could not be more evident. at this point we think that it haven’t any sense to ask if this was intended in a realistic way or in an instrumentalist one according to the interpretation of duhem. don't have sense simply because the author doesn't tell and had not perhaps any intension to declare it! as for the elements, any justification isn’t given, for instance, for the postulates of the optic98, and however here also it is not difficult to discover the intent to furnish the mathematical explanation of a phenomenon (in this case that of the vision); and the fact is not secondary that such mathematical explanation is recalled then in the phenomena. referring for the rest to what was said on the essay of incardona, we want only to underline a point that contributes to make transparent, on the optic, the character of mathematical model for a class of phenomena. that is, as we have already seen, the choice to use a discreet model and not a continuous one because such choice appears more suitable to explain (in descriptive sense) the loss of optic resolution, until to the complete disappearance of the object, when the distance increases. therefore we are able to affirm that works as optic and phenomena are set as their aim “to save the phenomena” without assume a position on the realistic character of the scientific knowledge. neither we see at this point some reason to believe otherwise for the elements, since the objects of the geometry seem to concern the spatial form of the bodies, as from us perceived and rationally organized. it is perhaps this one the aspect that appears the most innovative if compared to what we are able to know about the preceding epistemological status of the sciences. the fact is that the not pronouncing on the realistic character of the scientific knowledge, makes these equally valid beyond the different philosophical options in a historical age of great diversifications of the ideas. this doesn't mean however a generic indifference towards the problems of the philosophy, but rather a suspension of the judgment in comparison to the matters that, in the given historical period, divided the different schools of thought in an apparently incompatible way. such matters are just those that pertain to the status of the possible knowledge, and are expressed, in terms of theirs most radical opposition, by the dispute among academic (arcesilaus-carneades) and stoic (zeno-crysippus) schools. the fact that the sciences, as they appear in the work of euclid, are afar to ignore the philosophical problems, it is testified by the presence, to formal level, of almost all the characteristic elements of the aristotelian deductive structure. and nevertheless the innovative characters, in comparison to the conception of the science expressed by aristotle, appear at this point very clear: specially for what concerns the real and sure correspondence of the scientific knowledge with the reality for itself. euclid, in other words, misses any declaration on the being and on the truth beyond the phenomena, whereas the truth, with the simplicity and not derivability, was for aristotle the fundamental requisite of a proposition to be a scientific premise, just to guarantee the truth of the whole corpus of the scientific knowledge. but the formal differentiations also can’t be neglected, both in the terminology (as the expression koinai / e)/nnoiai)) and in what the change itself of terminology can subtend99. so today we can say that the setup given by euclid to the matter of the parallel straight lines doesn’t could precedently to be found neither easily approved, because also respecting the aristotelian formal deductive plant, it failed its substantial condition: the self-evidence of the premises, self-evidence that has would revealed through a 97 euclides, opera omnia, vol. 8, (phaenomena et scripta musica), 1916. 98 see incardona. 99 it is the case to remark the interpretation given by russo of another terminological innovation of euclid that, in fact, use the word shmei=o/n to denote the geometrical point in substitution of the old term stigmh/. russo riche the conclusion that such terminological change is finalized to abandon the old realistic meaning for another technical and not realistic one. 62 “not further decomposability” in more simple propositions and put substance in the oneness and absoluteness of the “truth”, essential condition of the being as being. if we accept this key of reading, it cannot surprise that the beginning of the criticism to the fifth postulate, coincides with a resumption of the aristotelian thought and it is always addressed to try to decompose (prove) the postulate in terms of more simple propositions (or absolutely simple in the sense of not subsequently demonstrable). on this matter it is still the case to mention from the eudemian ethics a passage in which the syllogistic derivation used in the mathematical proofs is assimilated to the relationship that intervenes between cause and effect100: a relationship in which the symmetry is excluded. therefore not all possible premises that allow us to prove the twenty-ninth proposition was for aristotle equivalent: it was necessary to look for the simplest and most evident. the history of the criticism on the fifth postulate begins therefore, for what we know, in the first century b.c., with posidonius of rhodes (135-50 b.c.), on the base of a demand that can perhaps be that one of aristotle, but that euclid evidently didn't consider necessary. that the period was the same in which the lost papers of aristotle was found and published can be casual, but it is not a coincidence that the criticism on the fifth postulate starts with posidonius and with his pupil, or follower, geminus. posidonius in fact was not a scientist in the sense of euclid or archimedes, but a philosopher that wanted to bring back the sciences inside the philosophy101 and, even if put himself in the stoic area, nevertheless he contrasted chrysippus for many aspects and tried to recover elements of the precedent platonic-aristotelian tradition. coming back to euclid, it seems therefore justified an interpretation that shows him as a figure of scientist (in a sense that is more modern than previously admitted) intent on produce coherent and stable results, in connection with the phenomenical reality and with the te/xnh, and under shelter by the philosophical disputes. that he, for himself, would have believed that the interrelations set among the phenomena had to reflect, or at least appreciate, the reality for itself, or that contrarily he assumed a instrumentalist position, is a question that transcends the scientific discourse to concern the individual personality of a man of which we don’t now nearly anything; it is something that we perhaps don’t will know never why he didn’t want to declare it, at least in the scientific works that arrived to us. today, with our knowledges, we can say that not only the suspension of the judgment on the ontological and metaphysical themes was the mean for a more general acceptance and a great efficiency of the scientific methods, but we also know that the same suspension of the judgment on the most controversial themes has also been the paid price that allowed important texts as those of euclid, of apollonius and of archimedes unlike those of zeno, or arcesilaus or crysippus, to cross the filter of the immediately following centuries and to survive after two thousand-three hundred years. 100.eud. eth. 1222 b 22-30: ”and since as in other matters the first principle is a cause of the things that exist or come into existence because of it, we must think as we do in the case of demonstrations. for example, if as the angles of a triangle are together equal to two right angles the angles of a quadrilateral are necessarily equal to four right angles, that the angles of a triangle are equal to two right angles is clearly the cause of that fact; and supposing a triangle were to change, a quadrilateral would necessarily change too for example if the angles of a triangle became equal to three right angles, the angles of a quadrilateral would become equal to six right angles, or if four, eight; also if a triangle does not change but is as described, a quadrilateral too must of necessity be as described”. it is true that, as specified some line above, the assimilation to the concept of cause for “the immovable things” as those of the geometry, is made by aristotle only “for analogy”, but a substantial asymmetry remains between the 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(received september 2003) ratio mathematica 22 (2012) 61-68 issn:1592-7415 codes on s-periodic errors pankaj kumar das* and vinod tyagi** *department of mathematics, shivaji college (university of delhi), raja garden, delhi-110 027, india **department of mathematics, shyam lal college (eve.)(university of delhi), shahdara, delhi-110032, india *pankaj4thapril@yahoo.co.in, **vinodtyagi@hotmail.com abstract in this paper, we study linear codes capable of detecting and correcting s-periodic errors. lower and upper bounds on the number of parity check digits required for codes detecting such errors are obtained. another bound on codes correcting such errors is also obtained. an example of a code detecting such errors is provided. key words: parity check matrix, syndromes, standard array, periodic error. 2000 ams subject classifications: 94b25, 94b60, 94-02. 1 introduction investigations in coding theory have been made in several directions but one of the most important directions has been the detection and correction of errors. it began with hamming codes[9] for single errors, golay codes[10, 11] for double and triple random errors and thereafter bch codes[12, 13, 14] were studied for multiple error correction. there is a long history towards the growth of the subject and many of the codes developed have found applications in numerous areas of practical interest. one of the areas of practical importance in which a parallel growth of the subject took place is that of burst error detecting and correcting codes. it has also been observed that in many communication channels, burst errors occur more frequently than random errors. a burst of length b may be defined as follows: 61 p. k. das and v. tyagi definition 1.1. a burst of length b is a vector whose only non-zero components are among some b consecutive components, the first and the last of which is non zero. extending the work of hamming[9], abramson[1] developed codes which dealt with the correction of single and double adjacent errors. the work due to fire[8] depicted a more general concept of burst errors. stone[19], and bridwell and wolf[4] considered multiple bursts. it was noted by chien and tang[5] that in several channels errors do occur in the form of a burst but not near the end of the vector. channels due to alexander, gryb and nast[2] fall in this category. in this light, chien and tang proposed a modification in the definition of a burst, now known as ct burst, according to which a ct burst of length b is defined as follows: definition 1.2. a ct burst of length b is a vector whose only non zero components are confined to some b consecutive positions, the first of which is non-zero. recently a new kind of error, known as repeated burst, has been observed by berardi, dass and verma[3]. for further work on this type of error, one may refer to [6, 7, 18] and references therein. it is very clear that the nature of error differ from channel to channel depending upon the behaviour of channels or the kind of errors which occur during the process of transmission. there is a need to deal with many types of error patterns and accordingly codes are to be constructed to combat such error patterns. though the errors are generally classified mainly in two categories random errors and burst errors, it has also been observed that the occurrence of errors may follow a pattern, different from random and burst. in certain communication channel like astrophotography[21], small mechanical error occurs periodically in the accuracy of the tracking in a motorized mount that results small movements of the target that can spoil long-exposure images, even if the mount is perfectly polar-aligned and appears to be tracking perfectly in short tests. it repeats at a regular interval the interval being the amount of time it takes the mount’s drive gear to complete one revolution. this type of error pattern is termed as periodic or alternate pattern. it was in this spirit that the codes correcting s-alternate errors were developed by tyagi and das [20]. an s-periodic error is defined as follows: 62 codes on s-periodic errors definition 1.3. an s-periodic error is an ntuple whose non zero components are located at a gap of s positions where s = 1, 2, 3,....,(n −1) and the number of its starting positions is among the first s + 1 components. for s=1, the 1-periodic error vectors are the ones where error may occur in 1st, 3rd, 5th...positions or 2nd, 4th, 6th,... positions. for example, in a vector of length 8, 1-periodic error vectors are of the type 10101000, 00101000, 0010101, 10101010, 10001010, 01010101, 01000101, 00000101, 00000001 etc. for s=2, the 2-periodic error vectors are those where error may occur in 1st, 4th, 7th,... positions or 2nd, 5th, 8th,...positions or 3rd, 6th, 9th,... positions. the 2-periodic error vectors may look like 10010010, 10000010, 00010010, 01000001, 01000000, 00001001, etc in a vector of length 8. for s=3, in a code length 8, the 3-periodic errors are 10001000, 01000100, 00100010, 00010001, 10000000, 01000000 etc. in what follows a linear code will be considered as a subspace of the space of all n-tuples over gf(q). the distance between two vectors shall be considered in the hamming sense. the rest of the paper is organized as follows: in section 2, we study the linear codes that detect any s-periodic error. we obtain lower and upper bounds on the parity check digits for codes detecting such errors. it is followed by an example of such a code. in second 3, we give a bound (based on reiger’s bound[16]) on codes correcting such errors . 2 codes detecting s-periodic errors we consider the linear codes that are capable of detecting any s-periodic error. clearly, the patterns to be detected should not be code words. in other words we consider codes that have no s-periodic error as a code word. firstly, we obtain a lower bound over the number of parity-check digits required for such a code. the proof is based on the technique used in theorem 4.13, peterson and weldon [15]. theorem 2.1. any (n, k) linear code over gf(q) that detects any s-periodic error must have at least ⌈ n s + 1 ⌉ parity-check digits. proof. the result will be proved on the basis that no detectable error vector can be a code word. 63 p. k. das and v. tyagi let v be an (n, k) linear code over gf(q). consider a set x of all those vectors such that the non-zero components are located at the first position and thereafter a gap of s positions. we claim that no two vectors of the set x can belong to the same coset of the standard array; else a code word shall be expressible as a sum or difference of two error vectors. assume on the contrary that there is a pair, say x1, x2 in x belonging to the same coset of the standard array. their difference viz. x1-x2 must be a code vector. but x1-x2 is a vector all of whose non-zero components are located at the 1st position or after a gap of s position and so is a member of x, i.e., x1-x2 is an s-periodic error, which is a contradiction. thus all the vectors in x must belong to distinct cosets of the standard array. the number of such vectors over gf(q) is clearly qp, where p = ⌈ n s + 1 ⌉ . the theorem follows since there must be at least this number of cosets. 2 in the following, an upper bound on the number of check digits required for the construction of a linear code discussed in theorem 2.1 is provided. this bound assures the existence of such a linear code and has been obtained by constructing a matrix under certain constraints. the proof is based on the well known technique used in varshomov-gilbert sacks bound (refer sacks[17], also theorem 4.7 peterson and weldon [15]). theorem 2.2. there exists an (n, k) linear code over gf(q) that has no s-periodic error as a code word provided that n − k ≥ ⌈ n s + 1 ⌉ . proof. the existence of such a code will be shown by constructing an appropriate (n − k) × n parity-check matrix h. the requisite parity-check matrix h shall be constructed as follows. select any non-zero (n − k)-tuples as the first j − 1 columns h1, h2,..., hj−1; the jth(j > s + 1) column hj is added provided that hj ̸= ∑p i=1 uihj−i(s+1) where ui ∈ gf(q) and p = ⌈ j s + 1 ⌉ − 1. this condition ensures that no s-periodic error will be a code word. the number of ways in which the coefficients ui can be selected is clearly q p. 64 codes on s-periodic errors at worst, all these linear combinations might yield a distinct sum. therefore a column hj can be added to h provided that qn−k > qp. or, n − k ≥ ⌈ j s + 1 ⌉ . for a code of length n, replacing j by n gives the result. 2 remark: the above two theorems can be combined as follows: for detecting s-periodic errors in a linear code of length n, ⌈ n s + 1 ⌉ parity check symbols are necessary and sufficient. example 2.1. consider a (7, 4) binary code with parity check matrix h =   1 1 1 0 0 1 00 1 0 1 1 1 0 0 0 1 0 1 1 1   this matrix has been constructed by the synthesis procedure, outlined in the proof of theorem 2.2, by taking s = 2 and n =7. it can be seen from table 1 that the syndromes of the different 2-periodic errors are nonzero, showing thereby that the code that is the null space of this matrix can detect all 2-periodic errors. table 1 error patterns syndromes 1000000 100 0001000 010 0000001 001 1001000 110 1000001 101 0001001 011 1001001 111 0100000 110 0000100 011 0100100 101 0010000 101 0000010 111 0010010 010 65 p. k. das and v. tyagi 3 codes correcting s-periodic errors the following theorem gives a bound on the number of parity-check digits for a linear code that corrects s-periodic errors. the proof is based on the technique used to establish reiger’s bound[16] (also refer theorem 4.15, peterson and weldon [15]) for correction of s-periodic errors. theorem 3.1. an (n, k) linear code over gf(q) that corrects all t-periodic errors, t = 2s + 1 must have at least ⌈ n s + 1 ⌉ parity-check digits. proof. any vector that has the form of an s-periodic error can be expressible as a sum or difference of two vectors, each of which is an t-periodic error. these component vectors must belong to different cosets of the standard array because both such errors are correctable errors. accordingly, such a vector viz. s-periodic error can not be a code vector. in view of theorem 2.1, such a code must have at least ⌈ n s + 1 ⌉ parity-check digits. acknowledgement the authors are very much thankful to prof. b. k. dass, department of mathematics, university of delhi for his valuable suggestions, revising the contents and bringing the paper to the current form. references [1] n. m. abramson, a class of systematic codes for non-independent errors, ire trans. on information theory, it-5, no. 4(1959), 150-157. [2] a. a. alexander, r. m. gryb and d. m. nast, capabilities of the telephone network for data transmission, bell system tech. j., vol. 39, no. 3(1960), 431-476. [3] l. berardi, b. k. dass and r. verma, on 2-repeated burst error detecting codes, journal of statistical theory and practice, vol. 3, no. 2(2009), 381-391. [4] j. d. bridwell and j. k. wolf, burst distance and multiple-burst correction, bell system tech. j., vol. 49(1970), 889-909. [5] r. t. chien and d. t. tang, on definitions of a burst, ibm journal of research and development, vol. 9, no. 4(1965), 292-293. 66 codes on s-periodic errors [6] b. k. dass and s. madan, repeated burst error locating linear codes, discrete mathematics, algorithms and applications, vol. 2, no. 2(2010), 181-188. [7] b. k. dass, and r. verma, repeated burst error detecting linear codes, ratio mathematica journal of applied mathematics, vol. 19(2009), 25-30. [8] p. fire, a class of multiple-error-correcting binary codes for nonindependent errors, sylvania report rsl-e-2, sylvania reconnaissance systems laboratory, mountain view, calif,(1959). [9] r. w. hamming, error-detecting and error-correcting codes, bell system technical journal, vol. 29(1950), 147-160. [10] m. j. e. golay, notes on digital coding, proc. ire, vol. 37(1949), 657. [11] m. j. e. golay, binary coding, ire trans., pgit-4(1954), 23-28. [12] r. c. golay and d. k. ray-chaudhuri, on a class of error correcting group codes, inf. and control, vol. 3(1960), 68-79. [13] r. c. golay and d. k. ray-chaudhuri, further results on error correcting binary group codes, inf. and control, vol. 3(1960), 279-290. [14] a. hocquenghem, codes corecteurs d’erreurs, chiffres, vol. 2(1959), 147-156. [15] w. w. peterson and e. j. weldon(jr.), error-correcting codes, 2nd edition, the mit press, mass, 1972. [16] s. h. reiger, codes for the correction of clustered errors, ire trans. inform. theory, it-6(1960), 16-21. [17] g. e. sacks, multiple error correction by means of parity-checks, ire trans. inform. theory it, 4(1958), 145 147. [18] b. d. sharma, b. rohtagi, some results on weights of vectors having 2-repeated bursts, cybernetics and information technologies, vol. 11, no. 1(2011), 36-44. [19] j. j. stone, multiple burst error correction, information and control, vol. 4(1961), 324-331. 67 p. k. das and v. tyagi [20] v. tyagi and p. k. das, s-alternate error correcting linear codes, j. of combinatorics, information & system sciences, vol. 35, no. 1-2(2010), 17-26. [21] www.themcdonalds.net/richard/index.php?title= astrophotography−mounts:−periodic−error−correction. 68 ratio mathematica volume 44, 2022 co-even geodetic number of a graph t. jebaraj * ayarlin kirupa.m † abstract let 𝐺 = (𝑉, 𝐸) be a graph with vertex set 𝑉 and edge set 𝐸. if 𝑆 is a set of vertices of 𝐺, then 𝐼[𝑆] is the union of all sets 𝐼[𝑢, 𝑣] for 𝑢, 𝑣 ∈ 𝑆. if 𝐼[𝑆] = 𝑉(𝐺), then 𝑆 is a geodetic set for 𝐺. the geodetic number 𝑔(𝐺) is the minimum cardinality of a geodetic set. a geodetic set 𝑆 is called coeven geodetic set if the degree of vertex 𝑣 is even number for all 𝑣 ∈ 𝑉 − 𝑆. the cardinality of a smallest coeven geodetic set of 𝐺, denoted by 𝑔𝑐𝑜𝑒(𝐺) is the coeven geodetic number of 𝐺. in this paper, we find the co even geodetic number of certain graphs and complement graphs. keywords: geodetic set, co-even geodetic set, co-even geodetic number 2010 ams subject classification: 05c12.‡ *assistant professor, research department of mathematics, malankara catholic college, mariagiri, kaliakkavilai, india; jebaraj.math@gmail.com †research scholar, reg. no.20113082092003, research department of mathematics, malankara catholic college, mariagiri, kaliakkavilai, india; ayarlin.kirupa19@gmail.com. ‡affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamilnadu, india; received on june 6 th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.922. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 332 t. jebaraj, ayarlin kirupa.m 1. introduction by a graph 𝐺 = (𝑉, 𝐸), we mean a finite undirected connected graph without loops or multiple edges. as usual 𝑛 = |𝑉| and 𝑚 = |𝐸| denote the number of vertices and edges of a graph 𝐺 respectively. the minimum and maximum degree 𝛿(𝐺) and δ(𝐺), respectively. in case where δ(𝐺) = 𝛿(𝐺), 𝐺 is called a regular graph. the distance 𝑑(𝑥, 𝑦) is the length of a shortest 𝑥 − 𝑦 path in 𝐺. it is known that the distance is a metric on the vertex set of g. an 𝑥 − 𝑦 path of length 𝑑(𝑥, 𝑦) is called an 𝑥 − 𝑦 geodesic. for any vertex 𝑢 of 𝐺,the eccentricity of 𝑢 is 𝑒(𝑢) = 𝑚𝑎𝑥{𝑑(𝑢, 𝑣) ∶ 𝑣 ∈ 𝑉}. a vertex 𝑣 is an eccentric vertex of 𝑢 if 𝑒(𝑢) = 𝑑(𝑢, 𝑣). the neighborhood of a vertex 𝑣 is the set 𝑁(𝑣) consisting of all vertices 𝑢 which are adjacent with 𝑣. a vertex 𝑣 is an extreme vertex of 𝐺 if the subgraph induced by its neighbors is complete. the closed interval 𝐼[𝑥, 𝑦] consists of all vertices lying on some 𝑥 − 𝑦 geodesic of 𝐺, while for 𝑆 ⊆ 𝑉 , [𝑆] = ⋃ 𝐼[𝑥, 𝑦]𝑥,𝑦∈𝑆 . a set s of vertices is a geodetic set if 𝐼[𝑆] = 𝑉 and the minimum cardinality of a geodetic set is the geodetic number 𝑔(𝐺). in this paper, we study the co-even geodetic number and is denoted by 𝑔𝑐𝑜𝑒(𝐺) also we discuss the co-even geodetic number of some standard graphs. 2. co-even geodetic number of a graph definition 2.1 a geodetic set 𝑺 is called co-even geodetic set if the degree of vertex 𝒗 is even number for all 𝒗 ∈ 𝑽 − 𝑺. the cardinality of a smallest co-even geodetic set of 𝑮, denoted by 𝒈𝒄𝒐𝒆(𝑮) is the co-even geodetic number of 𝑮. example 2.2 figure 2.1 in figure 2.1, 𝑆 = {𝑣1, 𝑣3, 𝑣4, 𝑣5} is a co-even geodetic set. here, the vertices 𝑣1 and 𝑣4 has odd degree. these two vertices do not make a geodetic set and no 3element subset of 𝐺 is a co-even geodetic set. then it is clear that 𝑔𝑐𝑜𝑒 (𝐺) = 4. 333 co-even geodetic number of a graph remark in figure 2.1, 𝑆 = {𝑣1, 𝑣3, 𝑣5} is the minimum geodetic set of 𝐺. ie) 𝑔(𝐺) = 3. thus, the geodetic number and co-even geodetic number of a graph 𝐺 can be different. proposition 2.3 let 𝐺 be a graph and 𝑆 is a co-even geodetic set . then, i) all vertices of odd degrees belong to every co-even geodetic set. ii) 𝑑𝑒𝑔(𝑣) ≥ 2 for all 𝑣 ∈ 𝑉 − 𝑆. proposition 2.4 if 𝐺 is 𝑝regular graph, then 𝑔𝑐𝑜𝑒 (𝐺) = { 𝑛 𝑖𝑓 𝑝 𝑖𝑠 𝑜𝑑𝑑 𝑔(𝐺) 𝑖𝑓 𝑝 𝑖𝑠 𝑒𝑣𝑒𝑛 theorem 2.5 if 𝐺 be a graph of order 𝑛, then 2 ≤ 𝑔(𝐺) ≤ 𝑔𝑐𝑜𝑒 (𝐺) ≤ 𝑛. proof: a geodetic set needs atleast two vertices. therefore, 𝑔(𝐺) ≥ 2. clearly, every co-even geodetic set is a geodetic set of 𝐺, 𝑔(𝐺) ≤ 𝑔𝑐𝑜𝑒(𝐺) . also, all the vertices of 𝐺 is the co-even geodetic set of 𝐺.ie) 𝑔𝑐𝑜𝑒 (𝐺) ≤ 𝑛. remark 2.6 the bounds of the theorem 2.5 are sharp. the co-even geodetic number of paths 𝑃𝑛 with 𝑛 vertices is 2. in this case, the smallest bounds is obtained. also, 𝐾𝑛 with 𝑛 vertices have the co-even geodetic number is 𝑛. then the upper bound is obtained. theorem 2.7 if 𝐺 is a non trivial connected graph with 𝑛 ≥ 2.if 𝑔𝑐𝑜𝑒 (𝐺) = 2 then 𝑔(𝐺) = 2. proof. it is follows from theorem 2.5. remark 2.8 the converse part of above theorem is need not be true for all graphs. in figure 2.2, the minimum geodetic number is 2 and the minimum co-even geodetic number is 3. figure 2.2 corollary 2.9 let 𝐺 be the non-trivial connected graph, 𝑔(𝐺) = 2 then 𝑔𝑐𝑜𝑒 (𝐺) = 2. proof. case (i) if 𝐺 = 𝐾2 it is easy to see 𝑔(𝐾2) = 2 then 𝑔𝑐𝑜𝑒(𝐾2) = 2. case (ii) all the vertices of 𝐺 should be even degree. consider the even cycle 𝐶2𝑛. all vertices have even degree for 𝐶2𝑛.we know that 𝑔(𝐶2𝑛) = 2. further more, 𝑔𝑐𝑜𝑒 (𝐶2𝑛) = 2. 334 t. jebaraj, ayarlin kirupa.m case (iii) a graph with exactly two odd degree vertices which only belongs to the minimum geodetic set. for example, in figure 2.3, the vertices 𝑣3 and 𝑣5 have odd degree and 𝑣1, 𝑣2, 𝑣4 have even degree. the minimum geodetic number of 𝐺 is 2. also, it is easily seen that 𝑔𝑐𝑜𝑒 (𝐺) = 2. remark all the graphs are not satisfied for the corollary 2.9 except the above three type graphs. observation. 2.10 𝑔𝑐𝑜𝑒(𝐶𝑛) = 𝑔(𝐶𝑛 ), where 𝐶𝑛 is a cycle of order 𝑛. proof. every cycle is the 2regular graph .by the proposition 2.4, we get 𝑔𝑐𝑜𝑒 (𝐶𝑛) = 𝑔(𝐶𝑛). theorem 2.11 for the wheel graph 𝑊𝑛 (𝑛 ≥ 4), then 𝑔𝑐𝑜𝑒 (𝑊𝑛) = { 𝑛 − 1 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 𝑛 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 proof. case (i) 𝑛 is odd let 𝑊𝑛 = 𝐾1 + 𝐶𝑛−1 and 𝑢 be the vertex of 𝐾1. it is easy to see that the 𝑛 − 1 vertices has odd degree except the vertex 𝑢. by the proposition 2.3, 𝑛 − 1 vertices belong to the co-even geodetic set 𝑆. also, the vertex 𝑢 ∈ 𝑉 − 𝑆, which has even degree. hence |𝑆| = 𝑛 − 1. case (ii) 𝑛 is even. every vertex of 𝑊𝑛 has odd degree. by the proposition 2.3, all the vertices of 𝑊𝑛 belongs to the co-even geodetic set. therefore, 𝑔𝑐𝑜𝑒 (𝑊𝑛) = 𝑛. corollary 2.12 for the wheel graph with 𝑛 ≥ 4 then 𝑔𝑐𝑜𝑒 (𝑊𝑛)= 2𝛼0(𝑊𝑛) − 2. proof. we prove this theorem by two cases. 335 co-even geodetic number of a graph case (i) 𝑛 is even we have 𝑔𝑐𝑜𝑒(𝑊𝑛) = 𝑛 if 𝑛 is even and 𝛼0(𝑊𝑛) = 𝑛+2 2 . we have 𝑔𝑐𝑜𝑒 (𝑊𝑛) = 𝑛 . 𝑔𝑐𝑜𝑒 (𝑊𝑛) + 2 = 𝑛 + 2. then 𝑔𝑐𝑜𝑒(𝑊𝑛)+2 2 = n+2 2 𝑔𝑐𝑜𝑒 (𝑊𝑛) 2 + 1 = 𝛼0(𝑊𝑛) 𝑔𝑐𝑜𝑒(𝑊𝑛) = 2𝛼0(𝑊𝑛) − 2 case (ii) 𝑛 is odd since 𝑔𝑐𝑜𝑒 (𝑊𝑛) = 𝑛 − 1 if n is odd and 𝛼0(𝑊𝑛) = 𝑛+1 2 we have 𝑔𝑐𝑜𝑒 (𝑊𝑛) = 𝑛 − 1 𝑔𝑐𝑜𝑒 (𝑊𝑛) + 1 2 = n − 1 + 1 2 𝑔𝑐𝑜𝑒 (𝑊𝑛) 2 = 𝑛 + 1 2 − 1 𝑔𝑐𝑜𝑒 (𝑊𝑛) = 2𝛼0(𝑊𝑛) − 2. theorem 2.13 if 𝐺 is the double fan graph 𝐹 = 𝑃𝑛 + 𝐾2̅̅ ̅ with 𝑛 ≥ 5, then 𝑔𝑐𝑜𝑒 (𝐺) = 4. proof figure 2.4 let 𝑝1, 𝑝2,…, 𝑝𝑛 be the vertices of path 𝑃𝑛 and let 𝑥 and 𝑦 be the two vertices of 𝐾2̅̅ ̅. all the vertices of path 𝑃𝑛 is adjacent to 𝑥 and 𝑦. now, the double fan graph 𝐹 = 𝑃𝑛 + 𝐾2̅̅ ̅ have the 𝑛 + 2 vertices. we prove this theorem by two cases. case (i) 𝑛 is odd if 𝑛 is odd then the end vertices of 𝑃𝑛 and the vertices of 𝐾2̅̅ ̅ have the odd degree. by the proposition 2.3, these four vertices 𝑝1, 𝑝𝑛 ,𝑥,𝑦 belongs to co-even geodetic set. also all the vertices of 𝐹 lies on any geodesic of the co-even geodetic set. thus 𝑔𝑐𝑜𝑒 (𝑃𝑛 + 𝐾2̅̅ ̅) = 4. 336 t. jebaraj, ayarlin kirupa.m case (ii) 𝑛 is even if 𝑛 is even then all the vertices of 𝐹 is even degree except the vertices 𝑝1 and 𝑝𝑛 belongs to co-even geodetic set. all the vertices of 𝐹 does not lies the 𝑝1 − 𝑝𝑛 geodesic. so we chosen the vertices 𝑥 and 𝑦 in the co-even geodetic set. now the set 𝑆 = {𝑝1, 𝑝𝑛 , 𝑥 , 𝑦} is the co-even geodetic set as well as all the vertices of 𝑉 − 𝑆 has even degree. therefore, 𝑔𝑐𝑜𝑒(𝑃𝑛 + 𝐾2̅̅ ̅) = 4. corollary 2.14 for the double fan graph 𝐹 = 𝑃𝑛 + 𝐾2̅̅ ̅ with 𝑛 ≥ 5 then, 𝑔𝑐𝑜𝑒 (𝑃𝑛 + 𝐾2̅̅ ̅) = { 2𝛼0(𝑃𝑛 + 𝐾2̅̅ ̅) − 𝑛 + 1 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 2𝛼0(𝑃𝑛 + 𝐾2̅̅ ̅) − 𝑛 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 theorem 2.15 for the ladder graph 𝐿𝑛 then, 𝑔𝑐𝑜𝑒 (𝐿𝑛) = 2𝑛 − 2. proof figure 2.5 the ladder graph 𝐿𝑛with 2𝑛 vertices. the geodetic number of 𝐿𝑛 is 2. 𝑆 = {𝑣1, 𝑣2𝑛 } or {𝑣𝑛, 𝑣𝑛+1} is the minimum geodetic set of 𝐿𝑛, which is not a co-even geodetic set. because some vertices of 𝑉 − 𝑆 has odd degree. therefore, the odd degree vertices {𝑣2, 𝑣3,…, 𝑣𝑛−1, 𝑣𝑛+2,…, 𝑣2𝑛+1} is belong to the co-even geodetic set of 𝐿𝑛. therefore, all the vertices of 𝐿𝑛 except two vertices make the co-even geodetic set. hence 𝑔𝑐𝑜𝑒 (𝐿𝑛) = 2𝑛 − 2. theorem2.16 for the cone graph 𝐶𝑚 + �̅�𝑛 then 𝑔𝑐𝑜𝑒(𝐶𝑚 + �̅�𝑛) = { 𝑛 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛, 𝑚 ≥ 5 𝑚 𝑖𝑓 𝑚 𝑖𝑠 𝑒𝑣𝑒𝑛 , 𝑛 𝑖𝑠 𝑜𝑑𝑑 𝑚 + 𝑛 𝑖𝑓 𝑚 𝑖𝑠 𝑜𝑑𝑑, 𝑛 𝑖𝑠 𝑜𝑑𝑑 proof. the cone graph 𝐶𝑚 + �̅�𝑛 is adding with cyclic graph 𝐶𝑚 and empty graph �̅�𝑛. the cone graph has 𝑚 + 𝑛 vertices. we prove this theorem by three cases. case (i) if 𝑛 is even in this case, we prove with two subcases. sub case (i) if 𝑛 is even, 𝑚 is odd for the cone graph 𝐶𝑚 + �̅�𝑛 , only 𝑛 vertices have odd degree. by the proposition 2.3, 𝑛vertices belongs to the co-even geodetic set. now, every vertex belongs to any geodesic of the co-even geodetic set. hence 𝑔𝑐𝑜𝑒 (𝐶𝑚 + �̅�𝑛) = 𝑛. sub case (ii) if 𝑛 is even, 𝑚 is even 337 co-even geodetic number of a graph both the vertices of 𝐶𝑚 + �̅�𝑛 has even degree. now, 𝑛vertices forms a co-even geodetic set of 𝐶𝑚 + �̅�𝑛. hence 𝑔𝑐𝑜𝑒(𝐶𝑚 + �̅�𝑛) = 𝑛. case (ii) if 𝑚 is even and 𝑛 is odd let 𝑚 is even number of vertices and 𝑛 is odd number of vertices. here, 𝐶𝑚 + �̅�𝑛 has 𝑚even vertices have odd degree and 𝑛-odd vertices have even degree. then it follows from the sub case (i) we get 𝑔𝑐𝑜𝑒(𝐶𝑚 + �̅�𝑛) = 𝑚. case (iii) if both 𝑚 and 𝑛 are odd for all the vertices of 𝐶𝑚 + �̅�𝑛 have odd degree. then it follows from the subcase (i). thus, we get, 𝑔𝑐𝑜𝑒 (𝐶𝑚 + �̅�𝑛) = 𝑚 + 𝑛. hence proved. 3.co-even geodetic number of complement of a graph theorem 3.1 if 𝑃𝑛 is a path graph with 𝑛 ≥ 5 ,then 𝑔𝑐𝑜𝑒 (𝑃�̅�) = { 4 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 𝑛 − 2 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 proof. let 𝑢 and 𝑣 be the end vertices of 𝑃𝑛. the vertices 𝑢 and 𝑣 are adjacent to 𝑛 − 2 vertices in 𝑃�̅�. the remaining vertices are adjacent to 𝑛 − 3 vertices in 𝑃�̅�. case (i) if 𝑛 is odd since 𝑢 and 𝑣 are adjacent to 𝑛 − 2 vertices in 𝑃�̅�. clearly, 𝑢 and 𝑣 are odd vertices. therefore {𝑢 , 𝑣} ∈ 𝑆. also, {𝑢 , 𝑣} is not a geodetic set. consider a vertex 𝑥, which is adjacent to 𝑣 and non adjacent to 𝑢. obviously, 𝑛 − 3 vertices lie on the 𝑥 − 𝑢 geodesic. choose a vertex 𝑦 there exist 𝑦 ∈ 𝑉(𝑃�̅�) such that 𝑦 ∉ 𝐼[𝑥 , 𝑢]. also no 3 element subset contains the co-even geodetic set. hence, 𝑆 = {𝑢 , 𝑣 , 𝑥 , 𝑦} is the minimum co-even geodetic set. case (ii) if 𝑛 is even for 𝑛 is even, clearly, 𝑢 and 𝑣 are even degree vertices. remaining 𝑛 − 2 vertices are adjacent to 𝑛 − 3 vertices. obviously, 𝑛 − 2 vertices is odd vertices. also, every vertex lies on the any geodesic of 𝑛 − 2 vertices. therefore, the minimum co-even geodetic number is 𝑛 − 2. ie) 𝑔𝑐𝑜𝑒 (𝑃�̅�) = 𝑛 − 2. theorem 3.2 for any gear graph 𝐺𝑛 with 𝑛 ≥ 3 then 𝑔𝑐𝑜𝑒 (𝐺𝑛̅̅ ̅) = 𝑛 + 1. proof. for the gear graph 𝐺𝑛 ,if 𝑛 is odd, then 𝐺𝑛̅̅ ̅ has 𝑛 + 1 odd vertices. by the proposition 2.3, 𝑛 + 1 vertices belong to co-even geodetic set. moreover, if 𝑛 is even, then the graph 𝐺𝑛̅̅ ̅ has 𝑛 vertices have odd degree. these 𝑛 vertices containing the co-even geodetic set. it is easy to see that all vertices do not lies any geodesic of coeven geodetic set. so we add one more vertex in co-even geodetic set. obviously, 𝑔𝑐𝑜𝑒 (𝐺𝑛̅̅ ̅) = 𝑛 + 1. theorem 3.3 for the complement of the cycle 𝐶𝑛̅̅ ̅ with 𝑛 ≥ 5, then 𝑔𝑐𝑜𝑒(𝐶𝑛̅̅ ̅) = { 3 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 𝑛 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 proof. this theorem follows from the theorem 3.1 338 t. jebaraj, ayarlin kirupa.m 4. conclusions in this paper, we obtained co-even geodetic number of some kind of graphs and complement of some graphs. also, we see the relation between vertex covering and co-even geodetic number of some graphs. references [1] f. buckley and f. harary, distance in graphs, addisonwesley, reading, ma (1990) [2] g. chartrand, f. harary and p. zhang, on the geodetic number of a graph, networks, 39(2002), 1-6. [3] manar m. shalaan and ahmed a. omran, co even domination in graphs, international journal of control and automation vol. 13. no. 3. (2020). pp. 330-334. [4] manar m. shalaan and ahmed a. omran, co even domination in some graphs, iop conf. series: materials science and engineering 928 (2020) 042015. [5] nima ghanbari, more on co even domination number, arxiv:2111.11817v2 [math.co] 19 jan 2022. 339 ratio mathematica volume 44, 2022 outer independent square free detour number of a graph k. christy rani * g. priscilla pacifica † abstract for a connected graph , a set of vertices is called an outer independent square free detour set if is a square free detour set of such that either or is an independent set. the minimum cardinality of an outer independent square free detour set of is called an outer independent square free detour number of and is denoted by we determine the outer independent square free detour number of some graphs. we characterize the graph which realizes the result that for any pair of integers and with there exists a connected graph of order with square free detour number and outer independent square free detour number keywords: square free detour set; outer independent square free detour set; outer independent square free detour number. 2010 subject classification: 05c12, 05c38 ‡ * research scholar, reg. no.: 20122212092002, pg and research department of mathematics, st. mary’s college (autonomous), thoothukudi-628 001, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627 012, india; christy.agnes@gmail.com. † assistant professor, department of mathematics, st. mary’s college (autonomous), thoothukudi628001, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, india; priscillamelwyn@gmail.com. ‡ received on june 14th, 2022. accepted on sep 1st, 2022. published on nov30th, 2022. doi: 10.23755/rm.v44i0.919. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 309 k. christy rani and g. priscilla pacifica 1. introduction in this article, a graph is considered to be a finite, undirected and connected graph of order with neither loops nor multiple edges. let be the longest path in and a path of is called detour. the parameters on detour concept were developed by chartrand [2]. the detour concept was extended to triangle free detour concept by s. athisayanathan et al. [1, 7]. the detour concept was applied in domination by number of authors. the detour domination number was studied and extended to outer independent detour domination by number of authors in [5, 6]. for any two vertices in a connected graph , the path p is called triangle free path if no three vertices of p induce a triangle. the triangle free detour distance ( ) is the length of a longest triangle free path in a path of length ( ) is called a triangle free detour. a set of is called a triangle free detour set of if every vertex of lies on a triangle free detour joining a pair of vertices of . the triangle free detour number of is the minimum order of its triangle free detour sets. this triangle free detour number was studied by s. athisayanathan and s. sethu ramalingam in [8]. this concept was extended to square free detour number by k. christy rani and g. priscilla pacifica [4]. a square free detour number of denoted by is defined as the minimum order of square free detour set consisting of every pair of vertices of all the square free detours in which every vertex of lies on. in this article, we introduce the outer independent square free detour number denoted by the outer independent square free detour number of some standard graphs and cycle related graphs are determined. for the basic terminologies we refer to chartrand [2]. 2. preliminaries the following theorems are used in the sequel. theorem 2.1 [3] for any connected graph theorem 2.2 [3] every end-vertex of a non-trivial connected graph belongs to every detour set of . theorem 2.3 [3] if t is a tree with end-vertices, then theorem 2.4 [4] if is the cycle ), then 310 outer independent square free detour number of a graph 3. outer independent square free detour number of a graph definition 3.1 let be a simple connected graph of order a set of vertices is called an outer independent square free detour set in if is a square free detour set such that either or is independent. the minimum cardinality of an outer independent square free detour number of is called outer connected square free detour number of and is denoted by example 3.2 for the graph shown in figure 1, the set is a minimum outer independent square free detour set and is a minimum square free detour set for and so and here we find that moreover, the sets and are also the minimum outer independent square free detour sets of hence there can be more than one minimum outer independent square free detour set for a graph figure 1: theorem 3.3 for any connected graph every end-vertex of belongs to every outer independent detour set of . proof. since every outer independent square free detour set is also a detour set of , the proof follows from theorem 2.2. theorem 3.4 for any connected graph proof. the result follows from theorems 2.1 and 3.3. remark 3.5 the bounds in theorem 3.4 are sharp. the set of two end-vertices of a path is its minimum outer independent square free detour set so that the bounds in theorem 3.4 are also strict. for the graph of order 11 given in figure 1, 311 k. christy rani and g. priscilla pacifica theorem 3.6 if is a path , then proof. let be a path of order and be a set of vertices. then we consider two cases. case 1. let be odd. let be the square free detour set such that is independent. hence is an outer independent square free detour set. thus case 2. let be even. let be a square free detour set such that is independent. thus the following corollary is immediate. corollary 3.7 for any connected graph if and only if theorem 3.8 if is a star , then proof. let be a star with end-vertices. then by theorem 3.3, where are the end-vertices of such that is independent. hence by theorem 2.3, is also a minimum detour set and so hence theorem 3.9 if is a complete bipartite graph , then proof. let be a complete bipartite graph of order with two partitions and where and let be a set of vertices of now, it is easy to verify that is independent. hence . remark 3.10 due to the connectivity of the complete graph it is not possible to find the outer independent square free detour number for . theorem 3.11 if is a cycle , then . proof. let be a cycle of order . let be any set of vertices of . we consider two cases. case 1. let be odd. let be a square free detour set such that is independent. hence . case 2. let be even. let be a square free detour set such that is independent. hence . 312 outer independent square free detour number of a graph from the above cases, we observe that . then by theorem 2.4, it follows that . theorem 3.12 if is a wheel, then . proof. let be a wheel of order . then is a square free detour set such that is independent. thus . theorem 3.13 if is a flower graph then proof. let be a flower graph of order n. let be the hub, be the vertices on the inner rim and be the vertices at square free detour distance 3 from the hub. then we have two cases. case 1. let be even. let is the square free detour set such that is independent. thus case 2. let be odd. then is a square free detour set such that is independent. hence theorem 3.14 if is a helm , then proof. let be a helm of order . let be any set in . then we have the following two cases. case 1. let be odd. then is a square free detour set of vertices where is the hub, are the vertices on the rim and are the pendent vertices of such that is independent. hence case 2. let be even. then is a square free detour set of vertices where is the hub, are the vertices on the rim and are the pendent vertices of such that is independent. hence theorem 3.15 if is a closed helm , then 313 k. christy rani and g. priscilla pacifica proof. let be a closed helm of order . let be the hub, and are the vertices of inner and outer rim of respectively. then we consider two cases. case 1. let be even. let is the square free detour set such that is independent. thus case 2. let be odd. then is a square free detour set such that is independent. hence theorem 3.16 for any pair of integers and with there exists a connected graph of order with square free detour number and outer independent square free detour number proof. we consider two cases. case 1. . any star with end-vertices has the desired property. case 2. . let be a graph obtained from by adding new vertices to let be the graph derived from by adding new vertices and identifying with let be the graph derived from by joining the remaining vertices to and of the resulting graph of order is shown in figure 2. by theorem 3.3, it is verified that and are the square free detour sets of and so figure 2: now, consider the set of all vertices finally added to to obtain it is easy to verify that is the outer independent square free detour set of and so 4. conclusion in this paper, we determined the outer independent square free detour number of some standard graphs and cycle related graphs. the relationship between the square free detour number and the outer independent square free detour number has been established. further investigation is 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[8] sethu ramalingam, s., and s. athisayanathan. upper triangle free detour number of a graph. discrete mathematics, algorithms and applications 14.01 (2022): 2150094. 315 ratio mathematica volume 41, 2021, pp. 101-118 some common fixed point theorems in complex valued fuzzy metric spaces md nazimul islam* abstract in this paper, we aim to prove some common fixed point theorems for pairs of any mappings, for pairs of occasionally weakly compatible mappings satisfying some conditions in complex valued fuzzy metric spaces. keywords: complex fuzzy set, complex valued continuous t-norm, complex valued fuzzy metric spaces, occasionally weakly compatible mappings, common fixed point. 2020 ams subject classifications: 47h10, 54h25. 1 *department of mathematics, naimouza high school, west bengal, india; n.islam000@gmail.com 1received on september 1, 2021. accepted on november 30, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.645 . issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 101 md n. islam 1 introduction the concept of fuzzy sets was initiated by l. a. zadeh [zadeh, 1965] in 1965. fuzzy set theory is a useful tool to describe situations in which the data are imprecise or vague. fuzzy sets handle such situation by attributing a degree to which a certain object belongs to a set. since then it has become a vigorous area of research in engineering, medical sciennce, social science, graph theory, metric space theory, complex analysis etc. deng [deng, 1982], erceg [ercez, 1979], kaleva and seikkala [kaleva and seikkala, 1984], kramosil and michalek [kramosil and michalek, 1975] have introduced the concepts of fuzzy metric spaces in different ways. george and veermani [george and veeramani, 1994] modified the notion of fuzzy metric spaces with the help of continuous t-norms. the concepts of compatible, weakly compatible, occasionally weakly compatible mappings in fuzzy metric space are present in the paper by c. t. aage and j. n. salunke [aage and salunke, 2009]. many researchers have obtained common fixed point theorems for mappings satisfying different types of commutativity conditions in fuzzy metric spaces. the concept of fuzzy complex numbers and fuzzy complex analysis were first introduced by buckley (see [buckley, 1987], [buckley, 1989], [buckley, 1991], [buckley, 1992]). motivated by the work of buckley some authers continued research in fuzzy complex numbers. in this series ramot et al. [d. ramot and kandel, 2002] extended fuzzy sets to complex fuzzy sets as a generalization. according to ramot et al. [d. ramot and kandel, 2002], the complex fuzzy set is characterized by a membership function, whose range is not limited to [0,1] but extended to the unit cirlce in the complex plane. azam et al. [a. azam and khan, 2011] introduced the notion of complex valued metric space which is a generalization of classical metric space and established sufficient conditions for the existence of common fixed points of a pair of mappings satisfying a contractive condition. the idea of complex valued metric spaces can be exploited to define complex valued normed spaces and complex valued hilbert spaces. additionally it offers numerous research activities in mathematical analysis. later, d. singh et al. [d. singh and kumam, 2016] defined the notion of complex valued fuzzy metric spaces with the help of complex valued continuous t -norm and also defined the notion of convergent sequence, cauchy sequence in complex valued fuzzy metric spaces. this paper presents some common fixed point theorems for pairs of any mappings and pairs of occasionally weakly compatible mappings satisfying some conditions in the complex valued fuzzy metric spaces. we also provide some examples which support the main results here. 102 some common fixed point theorems in complex valued fuzzy metric spaces 2 definitions and notations here and in the following, let r,r+0 ,c and n be the sets of real numbers, non negative real numbers, complex numbers and positive integers, respectively. definition 2.1. [j. choi and islam, 2017] define a partial order relation on c as follows: for z1, z2 ∈ c, z1 z2 if and only if <(z1) ≤<(z2) and =(z1) ≤=(z2). thus z1 z2 if any one of the following statements holds: (o1) <(z1) = <(z2) and =(z1) = =(z2) (o2) <(z1) < <(z2) and =(z1) = =(z2) (o3) <(z1) = <(z2) and =(z1) < =(z2) (o4) <(z1) < <(z2) and =(z1) < =(z2), where <(z) and =(z) indicates respectively the real and imaginary parts of the complex number z. we write z1 � z2 if z1 z2 and z1 6= z2, i.e., any one of (o2), (o3) and (o4) is satisfied and we write z1 ≺ z2 if only (o4) is satisfied. considering (o1)-(o4), the following properties for the partial order on c hold true: (i) 0 z1 z2 =⇒|z1| ≤ |z2| (ii) z1 z2 and z2 z3 =⇒ z1 z3 (iii) z1 z2 and λ > 0 (λ ∈ r) =⇒ λz1 λz2. note: z1 z2 and z2 % z1 have the same meaning. definition 2.2. [j. choi and islam, 2017] the max function for the partial order on c is defined as follows: (i) max{u,v} = v ⇔ u v (ii) u max{u,v}⇒ u v (iii) u max{v,w}⇒ u v or u w. for any 0 u, 0 v, we can easily prove that |max{u,v}| = max{|u| , |v|}, where |.| denotes the usual modulus of complex number. definition 2.3. [j. choi and islam, 2017] the min function for the partial order on c is defined as follows: (i) min{u,v} = u ⇔ u v (ii) min{u,v} v ⇒ u v (iii) min{u,w} v ⇒ u v or w v. for any 0 u, 0 v, we can easily prove that |min{u,v}| = min{|u| , |v|}. definition 2.4. [zadeh, 1965] a fuzzy set on a non empty set x is just a function µ : x → [0,1]. 103 md n. islam definition 2.5. [d. ramot and kandel, 2002] a complex fuzzy set s, defined on a universe discourse u, is characterized by a membership function µs(x) that assigns every element x ∈ u, a complex valued grade of membership in s. the values µs(x) lie within the unit circle in the complex plane and are thus of the form µs(x) = rs(x)e iws(x),(i = √ −1), where rs(x) and ws(x) both real valued, with rs(x) ∈ [0,1]. the complex fuzzy set s may be represented as the set of ordered pairs given by s = {(x,µs(x)) : x ∈ u}. definition 2.6. [d. singh and kumam, 2016] a binary operation ∗ : rseiθ × rse iθ → rseiθ, where rs ∈ [0,1] and a fix θ ∈ [0, π2 ], is called complex valued continuous t-norm if it satisfies the followings: (t1) ∗ is associative and commutative, (t2) ∗ is continuous, (t3) a∗eiθ = a,∀a ∈ rseiθ, (t4) a∗ b c∗d whenever a c and b d,∀ a,b,c,d ∈ rseiθ. example 2.1. [d. singh and kumam, 2016] the followings are examples for complex valued continuous t-norm: (i) a ∗ b = min(a,b), ∀a,b ∈ rseiθ and a fix θ ∈ [0, π 2 ] (ii) a∗ b = max(a + b−eiθ,0), ∀a,b ∈ rseiθ and a fix θ ∈ [0, π2 ]. definition 2.7. [d. singh and kumam, 2016, demir, 2021] the triplet (x,m,∗) is said to be complex valued fuzzy metric space if x is an arbitrary non empty set, ∗ is a complex valued continuous t-norm and m : x ×x × (0,∞) → rseiθ is a complex valued fuzzy set, where rs ∈ [0,1] and θ ∈ [0, π2 ], satisfying the following conditions: (cf1) 0 ≺ m(x,y,t), (cf2) m(x,y,t) = eiθ,∀t ∈ (0,∞) if and only if x = y, (cf3) m(x,y,t) = m(y,x,t), (cf4) m(x,y,t + s) % m(x,z,t)∗m(z,y,s), (cf5) m(x,y,t) : (0,∞) → rseiθ is continuous, for all x,y,z ∈ x ; t,s ∈ (0,∞); rs ∈ [0,1] and θ ∈ [0, π2 ]. the pair (m,∗) is called a complex valued fuzzy metric and m(x,y,t) denotes the degree of nearness between x and y with respect to t. it is noted that if we take θ = 0, then complex valued fuzzy metric simply goes to real valued fuzzy metric. note: it is clear that rseiθ eiθ and consequently, m(x,y,t) eiθ for all x,y ∈ x, t ∈ (0,∞), rs ∈ [0,1] and θ ∈ [0, π2 ]. example 2.2. [d. singh and kumam, 2016] let (x,d) be a real valued metric space. let a∗b = min{a,b}, for all a,b ∈ rseiθ, where rs ∈ [0,1] and θ ∈ [0, π2 ]. 104 some common fixed point theorems in complex valued fuzzy metric spaces for each t > 0, x,y ∈ x, we define m(x,y,t) = eiθ ktn ktn + md(x,y) , where k,m,n ∈ n. then (x,m,∗) is a complex valued fuzzy metric space. by choosing k = m = n = 1, we get m(x,y,t) = eiθ t t + d(x,y) . this complex valued fuzzy metric space induced by a metric d is referred to as a standard complex valued fuzzy metric space. example 2.3. [d. singh and kumam, 2016] let x = r. let a∗ b = min{a,b}, for all a,b ∈ rseiθ, where rs ∈ [0,1] and θ ∈ [0, π2 ]. for each t > 0, x,y ∈ x, we define m(x,y,t) = eiθe− |x−y| t . then (x,m,∗) is a complex valued fuzzy metric space. definition 2.8. [d. singh and kumam, 2016] let (x,m,∗) be a complex valued fuzzy metric space. we define an open ball b(x,r,t) with centre x ∈ x and radius r ∈ c with 0 ≺ r ≺ eiθ, t > 0 as: b(x,r,t) = {y ∈ x : m(x,y,t) � eiθ − r}, where θ ∈ [0, π 2 ]. a point x ∈ x is said to be interior point of a set a ⊂ x, whenever there exists r ∈ c with 0 ≺ r ≺ eiθ such that b(x,r,t) = {y ∈ x : m(x,y,t) � eiθ − r} ⊂ a, where θ ∈ [0, π 2 ]. a subset a of x is called open if every element of a is an interior point of a. if we define τ = {a ⊂ x : x ∈ a if and only if there exists t > 0 and r ∈ c, 0 ≺ r ≺ eiθ,θ ∈ [0, π 2 ] such that b(x,r,t) ⊂ a }. then one can easily check that τ is a topology on x. definition 2.9. [d. singh and kumam, 2016] let (x,m,∗) be complex valued fuzzy metric space and τ be the topology induced by complex valued fuzzy metric. let {xn} be any sequence in x. the sequence {xn} is said to converges to x ∈ x if and only if for any t > 0, m(xn,x,t) → eiθ as n →∞ or |m(xn,x,t)|→ 1 as n → ∞. a sequence {xn} in a complex valued fuzzy metric space (x,m,∗) is a cauchy sequence if and only if for any t > 0, m(xm,xn, t) → eiθ as m,n → ∞ or |m(xm,xn, t)| → 1 as m,n → ∞. a complex valued fuzzy metric space in which every cauchy sequece is convergent is called complex valued complete fuzzy metric space. for example (see [d. singh and kumam, 2016]), let x = r and a ∗ b = min{a,b}, for all a,b ∈ rseiθ, where rs ∈ [0,1] and θ ∈ [0, π2 ]. for each t > 0 and x,y ∈ x, we define m(x,y,t) = te iθ t+|x−y|. then (x,m,∗) is complex valued complete fuzzy metric space. 105 md n. islam definition 2.10. let (x,m,∗) be a complex valued fuzzy metric space and s, t : x → x be two mappings. a point x ∈ x is said to be a coincidence point of s and t if and only if sx = tx. we shall call w = sx = tx a point of coincidence of s and t . moreover, if sx = tx = x, then the point x ∈ x is called common fixed point of s and t . by study of the paper by c. t. aage and j. n. salunke (see [aage and salunke, 2009]), we find the definitions of compatible, weakly compatible, occasionally weakly compatible mappings in fuzzy metric space. in the similar way, we can define the same definitions in complex valued fuzzy metric space as follows: definition 2.11. let (x,m,∗) be a complex valued fuzzy metric space and s, t : x → x be two self mappings. the self maps s and t on x are said to be commuting if stx = tsx, for all x ∈ x. the self maps s and t are said to be compatible if lim n→∞ m(stxn,tsxn, t) = e iθ, t > 0 or lim n→∞ |m(stxn,tsxn, t)| = 1, t > 0, whenever {xn} is a sequence in x such that lim n→∞ sxn = lim n→∞ txn = x, for some x ∈ x. definition 2.12. let (x,m,∗) be a complex valued fuzzy metric space and s, t : x → x be two mappings. the self-maps s and t are said to be weakly compatible if stx = tsx whenever sx = tx, that is, they commute at their coincidence points. definition 2.13. let (x,m,∗) be a complex valued fuzzy metric space and s, t : x → x be two mappings. the self-maps s and t are said to be occasionally weakly compatible if and only if there is a coincidence point x (say) in x of s and t at which s and t commute, i.e., stx = tsx. it is needed to mention that every weakly compatible map is occasionally weakly compatible but converse is not always true. the supporting example is given below: example 2.4. let r be the set of real numbers with standard complex valued fuzzy metric space. define s,t : r → r by sx = x2 + x and tx = 2x for all x ∈ r. then sx = tx for x = 0,1 but st0 = ts0 and st1 6= ts1. therefore s and t are occasionally weakly compatible self maps but not weakly compatible. 106 some common fixed point theorems in complex valued fuzzy metric spaces 3 lemmas here we set lemmas which will be required in the sequel. lemma 3.1. [jungck and rhoades, 2006] let x be any set and s,t be occasionally weakly compatible self maps of x. if s and t have a unique point of coincidence w = sx = tx, then w is the unique common fixed point of s and t . d. singh et al. [d. singh and kumam, 2016] state and prove the following lemma for complex valued complete fuzzy metric space, but it is also true without the restriction of completeness. so we state the lemma without the restriction of completeness. lemma 3.2. let (x,m,∗) be a complex valued fuzzy metric space such that lim t→∞ m(x,y,t) = eiθ, for all x,y ∈ x. if m(x,y,kt) % m(x,y,t), for some 0 < k < 1, for all x,y ∈ x, t ∈ (0,∞), then x = y. lemma 3.3. [d. singh and kumam, 2016] let {xn} be a sequence in a complex valued fuzzy metric space (x,m,∗) with lim t→∞ m(x,y,t) = eiθ, for all x,y ∈ x. if there exists k ∈ (0,1) such that m(xn+1,xn+2,kt) % m(xn,xn+1, t), for all t > 0 and n = 0,1,2,3, .... then {xn} is a cauchy sequence in x. we can easily prove the following lemma. lemma 3.4. let (x,m,∗) be a complex valued fuzzy metric space. let {xn} be a sequence in x converging to x ∈ x. then any subsequence of {xn} converges to the same point x ∈ x. 4 main results here we present some fixed point theorems on complex valued fuzzy metric spaces. theorem 4.1. let (x,m,∗) be a complex valued complete fuzzy metric space with lim t→∞ m(x,y,t) = eiθ, for all x,y ∈ x,t > 0 and let s and t be selfmappings on x. if there exists k ∈ (0,1) such that m(sx,ty,kt) % m(x,y,t), (1) for all x,y ∈ x and for all t > 0, then s and t have a unique common fixed point in x. 107 md n. islam proof. let x0 ∈ x be an arbitrary point and we define the sequence {xn} by x2n+1 = sx2n and x2n+2 = tx2n+1;n = 0,1,2,3, ... now, for k ∈ (0,1) and for all t > 0, we have by the condition (1) m(x2n+1,x2n+2,kt) = m(sx2n ,tx2n+1,kt) % m(x2n ,x2n+1, t). and m(x2n,x2n+1,kt) = m(sx2n−1 ,tx2n,kt) % m(x2n−1 ,x2n, t). in general, we have m(xn+1,xn+2,kt) % m(xn,xn+1, t), for all t > 0 and k ∈ (0,1);n = 0,1,2,3, .... by lemma 3.3, {xn} is a cauchy sequence in x. since x is complete, then there exists u ∈ x such that xn → u as n → ∞. obviously {x2n} and {x2n+1} are subsequences of {xn} in x, by lemma 3.4, they are also converge to the same point u ∈ x, i.e., x2n → u and x2n+1 → u as n →∞. now, we claim that u is the common fixed point of s and t . for this, using condition (1), we have m(su,u,kt) = m(su,u, kt 2 + kt 2 ) % m(su,x2n+2, kt 2 )∗m(x2n+2,u, kt 2 ) = m(su,tx2n+1, kt 2 )∗m(x2n+2,u, kt 2 ) % m(u,x2n+1, t 2 )∗m(x2n+2,u, kt 2 ). taking limit as n →∞, we get, m(su,u,kt) % eiθ ∗eiθ = eiθ. so su = u. again, m(u,tu,kt) = m(u,tu, kt 2 + kt 2 ) % m(u,x2n+1, kt 2 )∗m(x2n+1,tu, kt 2 ) = m(u,x2n+1, kt 2 )∗m(sx2n,tu, kt 2 ) % m(u,x2n+1, kt 2 )∗m(x2n,u, t 2 ). taking limit as n →∞, we get m(u,tu,kt) % eiθ∗eiθ = eiθ. therefore tu = u. 108 some common fixed point theorems in complex valued fuzzy metric spaces thus su = tu = u and therefore u is a common fixed point of s and t . for uniqueness, let z be another fixed point of s and t . now, using condition (1), m(u,z,kt) = m(su,tz,kt). % m(u,z,t). by lemma 3.2, u = z. this completes the theorem. if we consider s = t in theorem 4.1, we get the following corollary. corollary 4.1. let (x,m,∗) be a complex valued complete fuzzy metric space with lim t→∞ m(x,y,t) = eiθ, for all x,y ∈ x, t > 0 and let s be self-mapping on x. if there exists k ∈ (0,1) such that m(sx,sy,kt) % m(x,y,t), for all x,y ∈ x and for all t > 0, then s has a unique fixed point in x. a supporting example to corollary 4.1 is given below. example 4.1. let x = r and a ∗ b = min{a,b}, for all a,b ∈ rseiθ, where rs ∈ [0,1] and θ ∈ [0, π2 ]. for each t > 0 and x,y ∈ x, we define m(x,y,t) = teiθ t + |x−y| . then certainly (x,m,∗) is complex valued complete fuzzy metric space with lim t→∞ m(x,y,t) = eiθ. we define the self map s on x by sx = x + 1 2 , for all x ∈ x. now, for any t > 0 and for k = 1 2 , m(sx,sy, t 2 ) = t 2 eiθ t 2 + |sx−sy| . = teiθ t + 2 ∣∣x+1 2 − y+1 2 ∣∣ = teiθt + |x−y| = m(x,y,t). thus all the conditions of corollary 4.1 are satisfied and x = 1 is the unique fixed point of s. 109 md n. islam theorem 4.2. let (x,m,∗) be a complex valued fuzzy metric space with lim t→∞ m(x,y,t) = eiθ, for all x,y ∈ x and let a,b,s and t be self-mappings on x. let the pairs {a,s} and {b,t} be occasionally weakly compatible. if there exists k ∈ (0,1) such that m(ax,by,kt) % min{m(sx,ty,t),m(sx,ax,t),m(by,ty,t), m(ax,ty,t),m(by,sx,t)}, (2) for all x,y ∈ x and for all t > 0, then a,b,s and t have a unique common fixed point in x. proof. since the pairs {a,s} and {b,t} are occasionally weakly compatible, so there are points x,y ∈ x such that ax = sx and by = ty. now, by the given condition (2), we get m(ax,by,kt) % min{m(sx,ty,t),m(sx,ax,t),m(by,ty,t), m(ax,ty,t),m(by,sx,t)} = min{m(ax,by,t),m(ax,ax,t),m(by,by,t), m(ax,by,t),m(by,ax,t)} = min{m(ax,by,t),eiθ,eiθ,m(ax,by,t),m(by,ax,t)} = m(ax,by,t). in view of lemma 3.2, we have ax = by and therefore ax = sx = by = ty. (3) suppose that the pair {a,s} have an another coincidence point z ∈ x, i.e., az = sz. now, m(az,by,kt) % min{m(sz,ty,t),m(sz,az,t),m(by,ty,t), m(az,ty,t),m(by,sz,t)} = min{m(az,by,t),m(az,az,t),m(by,by,t), m(az,by,t),m(by,az,t)} = min{m(az,by,t),eiθ,eiθ,m(az,by,t),m(by,az,t)} = m(az,by,t). again, in view of lemma 3.2, we have az = by. therefore az = sz = by = ty. (4) 110 some common fixed point theorems in complex valued fuzzy metric spaces from (3) and (4), ax = az and therefore the pair {a,s} have a unique point of coincidence w = ax = sx. thus by lemma 3.1, w is the unique common fixed point of {a,s}. similarly, we can show that the pair {b,t} have also a unique common fixed point. suppose this is u ∈ x. now, m(w,u,kt) = m(aw,bu,kt) % min{m(sw,tu,t),m(sw,aw,t),m(bu,tu,t), m(aw,tu,t),m(bu,sw,t)} = min{m(w,u,t),m(w,w,t),m(u,u,t), m(w,u,t),m(u,w,t)} = min{m(w,u,t),eiθ,eiθ,m(w,u,t),m(u,w,t)} = m(w,u,t). therefore using lemma 3.2, we have w = u and consequently, w is common fixed point of a,b,s and t . to show, this common fixed point is unique, let v is an another common fixed point of a,b,s and t . now, m(w,v,kt) = m(aw,bv,kt) % min{m(sw,tv,t),m(sw,aw,t),m(bv,tv,t), m(aw,tv,t),m(bv,sw,t)} = min{m(w,v,t),m(w,w,t),m(v,v,t), m(w,v,t),m(v,w,t)} = min{m(w,v,t),eiθ,eiθ,m(w,v,t),m(v,w,t)} = m(w,v,t). by lemma 3.2, we have w = v. hence a,b,s and t have a unique common fixed point. in the following an supporting example to theorem 4.2 is given. example 4.2. let x = r. consider the metric d(x,y) =| x | + | y |, for all x 6= y and d(x,y) = 0, for x = y on x. let a∗b = min{a,b}, for all a,b ∈ rseiθ, where rs ∈ [0,1] and θ ∈ [0, π2 ]. for each t > 0 and x,y ∈ x, we define m(x,y,t) = teiθ t + d(x,y) . 111 md n. islam then certainly (x,m,∗) is complex valued fuzzy metric space with lim t→∞ m(x,y,t) = eiθ. now, we define the self maps a,b,s and t on x by ax = x 3 ,bx = x 4 ,sx = x and tx = x 2 , for all x ∈ x. set k = 1 2 . now, for x 6= y, m(ax,by, t 2 ) = t 2 eiθ t 2 + | ax | + | by | = teiθ t + 2 |x| 3 + 2 |y| 4 % teiθ t + 2 |x| 2 + 2 |y| 4 = teiθ t + |x|+ |y| 2 = teiθ t + |sx|+ |ty| = m(sx,ty,t). for the case x = y, m(ax,by, t 2 ) = t 2 eiθ t 2 + 0 = eiθ = m(sx,ty,t). therefore, for any x,y ∈ x, m(ax,by, t 2 ) % m(sx,ty,t) = min{m(sx,ty,t),m(sx,ax,t),m(by,ty,t), m(ax,ty,t),m(by,sx,t)}. therefore, the maps a,b,s and t satisfies the condition (2) of theorem 4.2 for k = 1 2 . also the pairs {a,s} and {b,t} are obviously occasionally weakly compatible. thus all the conditions of theorem 4.2 are satisfied and x = 0 is the unique common fixed point of a,b,s and t in x. setting a = b and s = t in theorem 4.2, we get the following corollary. 112 some common fixed point theorems in complex valued fuzzy metric spaces corollary 4.2. let (x,m,∗) be a complex valued fuzzy metric space with lim t→∞ m(x,y,t) = eiθ, for all x,y ∈ x and let a,s be self-mappings on x. let the pair {a,s} be occasionally weakly compatible. if there exists k ∈ (0,1) such that m(ax,ay,kt) % min{m(ax,sy,t),m(ay,sx,t),m(sx,sy,t), m(sx,ax,t),m(sy,ay,t)}, for all x,y ∈ x and for all t > 0, then a and s have a unique common fixed point in x. theorem 4.3. let (x,m,∗) be a complex valued fuzzy metric space with lim t→∞ m(x,y,t) = eiθ, for all x,y ∈ x and let a,b,s and t be self-mappings on x. let the pairs {a,s} and {b,t} be occasionally weakly compatible. if there exists k ∈ (0,1) such that m(ax,by,kt) % g(min{m(sx,ty,t),m(sx,ax,t),m(by,ty,t), m(ax,ty,t),m(by,sx,t)}), for all x,y ∈ x and for all t > 0, where g : rseiθ → rseiθ with g(x) % x for all x ∈ rseiθ, where rs ∈ (0,1) and θ ∈ [0, π2 ]. then a,b,s and t have a unique common fixed point in x. proof. the proof follows from theorem 4.2. theorem 4.4. let (x,m,∗) be a complex valued fuzzy metric space with lim t→∞ m(x,y,t) = eiθ, for all x,y ∈ x, and let a,b,s and t be self-mappings on x. let the pairs {a,s} and {b,t} be occasionally weakly compatible. if there exists k ∈ (0,1) such that m(ax,by,kt) % m(sx,ty,t)∗m(ax,sx,t)∗m(by,ty,t)∗m(ax,ty,t), (5) for all x,y ∈ x and for all t > 0. then a,b,s and t have a unique common fixed point in x. proof. the pairs {a,s} and {b,t} are occasionally weakly compatible, so there are points x,y ∈ x such that ax = sx and by = ty. now, the condition (5) gives m(ax,by,kt) % m(sx,ty,t)∗m(ax,sx,t)∗m(by,ty,t) ∗m(ax,ty,t) = m(ax,by,t)∗m(ax,ax,t)∗m(by,by,t) ∗m(ax,by,t) = m(ax,by,t)∗eiθ ∗eiθ ∗m(ax,by,t) = m(ax,by,t). 113 md n. islam in view of lemma 3.2, we have ax = by and therefore ax = sx = by = ty. (6) suppose that the pair {a,s} have an another coincidence point z (say)∈ x, i.e., az = sz. now, m(az,by,kt) % m(sz,ty,t)∗m(az,sz,t)∗m(by,ty,t) ∗m(az,ty,t) = m(az,by,t)∗m(az,az,t)∗m(by,by,t) ∗m(az,by,t) = m(az,by,t)∗eiθ ∗eiθ ∗m(az,by,t) = m(az,by,t). using lemma 3.2, az = by and consequently az = sz = by = ty. (7) from (6) and (7) we get, ax = az and therefore the pair {a,s} have a unique point of coincidence w = ax = sx. thus by lemma 3.1, w is the unique common fixed point of {a,s}. similarly, we can show that there is a unique common fixed point u (say)∈ x of {b,t}. now, m(w,u,kt) = m(aw,bu,kt) % m(sw,tu,t)∗m(aw,sw,t)∗m(bu,tu,t) ∗m(aw,tu,t) = m(aw,bu,t)∗m(aw,aw,t)∗m(bu,bu,t) ∗m(aw,bu,t) = m(aw,bu,t)∗eiθ ∗eiθ ∗m(aw,bu,t) = m(aw,bu,t) = m(w,u,t). by lemma 3.2, we have w = u and consequently w is common fixed point of a,b,s and t . for uniqueness, let v is an another common fixed point of a,b,s and t. 114 some common fixed point theorems in complex valued fuzzy metric spaces therefore m(w,v,kt) = m(aw,bv,kt) % m(sw,tv,t)∗m(aw,sw,t)∗m(bv,tv,t) ∗m(aw,tv,t) = m(w,v,t)∗m(w,w,t)∗m(v,v,t)∗ m(w,v,t) = m(w,v,t)∗eiθ ∗eiθ ∗m(w,v,t). = m(w,v,t) and in view of lemma 3.2, we have w = v. hence a,b,s and t have a unique common fixed point. a supporting example to corollary 4.4 is given below. example 4.3. let x = r. define the metric (.x,y) =| x | + | y |, for all x 6= y and d(x,y) = 0, for x = y on x. let a∗b = min{a,b}, for all a,b ∈ rseiθ, where rs ∈ [0,1] and θ ∈ [0, π2 ]. for each t > 0 and x,y ∈ x, we define m(x,y,t) = eiθe− d(x,y) t . then certainly (x,m,∗) is complex valued fuzzy metric space with lim t→∞ m(x,y,t) = eiθ. now, we define the self maps a,b, s and t on x by ax = x 5 ,bx = x 15 ,sx = x and tx = x 4 , for all x ∈ x. let k = 1 3 . now for x 6= y, m(ax,by, t 3 ) = t 3 eiθ t 3 + | ax | + | by | = teiθ t + 3 |x| 5 + 3 |y| 15 % teiθ t + |x|+ |y| 4 = teiθ t + |sx|+ |ty| = m(sx,ty,t). 115 md n. islam for x = y, m(ax,by, t 3 ) = t 3 eiθ t 3 + 0 = eiθ = m(sx,ty,t). thus for any x,y ∈ x, m(ax,by, t 3 ) % m(sx,ty,t) = min{m(sx,ty,t),m(sx,ax,t),m(by,ty,t), m(ax,ty,t),m(by,sx,t)}. therefore the maps a,b,s and t satisfies the condition (5) of theorem 4.4 for k = 1 3 . also the pairs {a,s} and {b,t} are obviously occasionally weakly compatible. thus all the conditions of theorem 4.3 are satisfied and x = 0 is the unique common fixed point of a,b,s and t in x. setting a = b and s = t in theorem 4.4, we get the following corollary. corollary 4.3. let (x,m,∗) be a complex valued fuzzy metric space with lim t→∞ m(x,y,t) = eiθ, for all x,y ∈ x, and let a,s be self-mappings on x. let the pair {a,s} be occasionally weakly compatible. if there exists k ∈ (0,1) such that m(ax,ay,kt) % m(sx,sy,t)∗m(ax,sx,t)∗m(ay,sy,t)∗m(ax,sy,t), for all x,y ∈ x and for all t > 0, then a and s have a unique common fixed point in x. theorem 4.5. let (x,m,∗) be a complex valued fuzzy metric space with lim t→∞ m(x,y,t) = eiθ, for all x,y ∈ x. let the pair {a,s} be occasionally weakly compatible self maps on x. if there exists k ∈ (0,1) such that m(sx,sy,kt) % pm(ax,ay,t) + q min{m(ax,ay,t),m(sx,ax,t),m(sy,ay,t)}, (8) for all x,y ∈ x and for all t > 0, where p,q > 0 are reals with p + q ≥ 1. then a and s have a unique common fixed point in x. proof. the pair {a,s} are occasionally weakly compatible, then there exists x ∈ x such that ax = sx. suppose that there exists another y ∈ x for which ay = sy. 116 some common fixed point theorems in complex valued fuzzy metric spaces from the condition (8), m(sx,sy,kt) % pm(ax,ay,t) + q min{m(ax,ay,t),m(sx,ax,t),m(sy,ay,t)} = pm(sx,sy,t) + q min{m(sx,sy,t),m(sx,sx,t),m(sy,sy,t)} = pm(sx,sy,t) + q min{m(sx,sy,t),eiθ,eiθ} = pm(sx,sy,t) + qm(sx,sy,t) = (p + q)m(sx,sy,t). since p + q ≥ 1, m(sx,sy,kt) % m(sx,sy,t). in view of lemma 3.2, we have sx = sy and consequently ax = ay. therefore the pair {a,s} have a unique point of coincidence w = ax = sx. thus by lemma 3.1, a and s have a unique common fixed point in x. the following corollary is the direct consequence of theorem 4.5. corollary 4.4. let (x,m,∗) be a complex valued fuzzy metric space with lim t→∞ m(x,y,t) = eiθ, for all x,y ∈ x. let a be self map on x. if there exists k ∈ (0,1) such that m(ax,ay,kt) % αm(ax,ay,t), for all x,y ∈ x and for all t > 0, where α ≥ 1 real. then a has a unique fixed point in x. 5 conclusions in the theorem 4.1 we have shown a sufficient condition for existence of a unique common fixed point of pair of any mappings in complex valued fuzzy metric spaces. similar results are found in the remaining theorems 4.2, 4.3, 4.4 and 4.5 but for pairs of occasionally weakly compatible mappings. references f. brain a. azam and m. khan. common fixed point theorems in complex valued metric spaces. numer. funct. anal. optim., 32(3):243–253, 2011. c. t. aage and j. n. salunke. common fixed point theorems in fuzzymetric spaces. international journal of pure and applied mathematics, 56(2):155– 164, 2009. 117 md n. islam j. j. buckley. fuzzy complex numbers. proceedings of isfk,guangzhou, china, pages 597–700, 1987. j. j. buckley. fuzzy complex numbers. fuzzy sets and systems, 33(3):333–345, 1989. j. j. buckley. fuzzy complex analysis i: differentiation. fuzzy sets and systems, 41:269–284, 1991. j. j. buckley. fuzzy complex analysis ii: integration. 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point theorems for occasionally weakly compatible mappings. fixed point theory, 7(2):287–296, 2006. o. kaleva and s. seikkala. on fuzzy metric spaces. fuzzy sets and systems, 12: 215–229, 1984. i. kramosil and j. michalek. fuzzy metric and statistical metric spaces. kybernetika, 11:336–344, 1975. l.a. zadeh. fuzzy sets. information and control, 8:338–353, 1965. 118 ratio mathematica volume 44, 2022 more functions related to šα ∗ open set in soft topological spaces p. anbarasi rodrigo1 s. anitha ruth2 abstract in this paper, we introduce some soft functions like š strongly𝛼∗ continuous function, š perfectly 𝛼∗ continuous function, š totally 𝛼∗ continuous function. we study the connections of these function with other š function. also, we establish the relationships in between the above functions and also investigate various aspects of these functions. keywords: soft functions, continuous function 2010 ams subject classification: 54c053 1assistant professor, department of mathematics, st. mary’s college (autonomous), thoothukudi, affiliated by manonmaniam sundaranar university abishekapatti, tirunelveli, india email.id: anbu.n.u@gmail.com 2research scholar (part time), department of mathematics, st. mary’s college (autonomous), thoothukudi, register number: 19122212092001. affiliated by manonmaniam sundaranar university abishekapatti, tirunelveli, india email.id: anitharuthsubash@gmail.com *corresponding author: anitharuthsubash@gmail.com 3received on june 9th, 2022.accepted on sep 5st, 2022.published on nov 30th, 2022.doi: 10.23755/rm.v44i0.905. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 182 mailto:anbu.n.u@gmail.com mailto:anitharuthsubash@gmail.com mailto:anitharuthsubash@gmail.com p. anbarasi rodrigo and s. anitha ruth 1. introduction molodtsov introduced the concept of soft sets from which the difficulties of fuzzy sets, intuitionistic fuzzy sets, vague sets, interval mathematics and rough sets have been rectified. application of soft sets in decision making problems has been found by maji et al. whereas chen gave a parametrization reduction of soft sets and a comparison of it with attribute reduction in rough set theory. further soft sets are a class of special information. shabir and naz introduced soft topological spaces in 2011 and studied some basic properties of them. meanwhile generalized closed sets in topological spaces were introduced by levine in 1970 and recent survey of them is in which is extended to soft topological spaces in the year 2012. further kannan and rajalakshmi have introduced soft g – locally closed sets and soft semi star generalized closed sets. soft strongly g – closed sets have been studied by kannan, rajalakshmi and srikanth. chandrasekhara rao and palaiappan introduced generalized star closed sets in topological spaces and it is extended to the bitopological context by chandrasekhara rao and kannan. recently papers about soft sets and their applications in various fields have increased largely. modern topology depends strongly on the ideas of set theory. any research work should result in addition to the existing knowledge of a particular concept. such an effort not only widens the scope of the concept but also encourages others to explore new and newer ideas. therefore, in this work we introduce a new soft generalized set called šα∗ open set and its related properties. this may be another starting point for the new soft set mathematical concepts and structures that are based on soft set theoretic operations. 2. preliminaries in this section, this project x be an initial universe and ê be a set of parameters. let p (x) denote the power set of x and a be a non – empty subset of ξ. a pair (fš, a) denoted by fš a is called a soft set over x, where fš is a mapping given by f š: a → p (x). definition2.1.1 [8] for two soft sets (fš, a) and (g, b) over a common universe x, we say that (fš, a) is a soft subset of (g, b) denoted by (fš, a) s (g, b), if i. a s b and ii. fš (e) s g (e) for all e  ξ definition2.1.2 [8] the complement of a soft set (fš, a) denoted by (fš, a) c, is defined by ((fš, a)) c = ((fšc, a), where fšc : a → p (x) is a mapping given by fšc ( e ) = x fš (e), for all e  ξ. definition2.1.[8] a subset of a štopological space (x, τs, ξ) is said to be 1. a š semi-open set 2. if (fš, ê) s šcl (šint (f š, ê) and a š semi-closed set if š int (š cl (fš, ê) s (f š, ê). 3. a š pre-open set [1] if (fš, ê) s š int (š cl (f š, ê)) and a špre-closed set if š cl (š 183 more functions related to š𝛼∗ open set in soft topological spaces int (fš, ê) ⊆ s (f š, ê) a š α-open set [1] if (fš, ê) s š in (šcl (int(f š, ê) and a š α-closed set if šcl (šint (šcl (fš, ê) s (f š, ê)). 4. a š  -open set [1] if (fš, ê) s šcl (šint (š cl(f š, ê ))) and a š  -closed set if š int(šcl(int(fš, ê )) s (f š, ê)). 5. ašgeneralized closed set(briefly šgsclosed) if š cl(fš, ê ) s (g , ξ) whenever (fš, ê)⊆ s (g , ξ)and(g , ξ)is š open in (x, τs, ξ). the complement of a š gs-closed set is called a šgs-open set. 6. a š semi-generalized closed set (briefly š sg-closed) if š cl (fš, ê) s (g, ξ) whenever (fš, ê ) ⊆s (g , ξ)and(g , ξ)is šsemi open in(x, τs, ξ).the complement of ašsg-closed set is called a šsg-open set. 7. a generalized šsemi-closed set (briefly gs-closed) if šcl (fš, ê) s (g, ξ) whenever (fš, ê ) s (g , ξ) and(g , ξ)is šopenin(x, τs,ξ).thecomplementofa šgsclosed setiscalled ašgs-open set. 8. aš – closed [9] if š cl(f , ξ)⊆s (g , ξ) whenever (f , ξ)⊆ s (g , ξ) and (g , ξ) isšsemi openin (x, τs, ξ) 9. aš𝜔-closed [9] ifšcl(fš, ê ) ⊆s (g , ξ)whenever(f , ξ)⊆s (g , ξ)and(g , ξ) is š semi open 10. a š alpha-generalized closed set (briefly šαg-closed) if α š cl(fš, ê ) ⊆s (g , ξ) whenever (fš, ê ) ⊆s (g , ê) and(g, ê) is šαopen in (x, τs, ê).the complement of a šαg-closed set is called a šαg-open set. 11. a š generalized alpha closed set (briefly šgα-closed) ifα š cl(fš, ê ) ⊆s (g , ê) whenever (fš, ê) ⊆s (g , ê)and(g , ê) is š open in(x, τs, ê).the complement of aš gαclosed set is called a šgα-open set. 12. a šgeneralized pre closed set (briefly šgp-closed)[1] if p š cl(fš, ê ) ⊆s (g , ê) whenever a š gp-open set. 13. ašgeneralized pre regular closed set (briefly šgpr-closed) [5] if p šcl (fš, ê) ⊆s (g , ê) whenever (fš, ê ) ⊆s (g , ê) and (g , ê) isš regular open in (x, τs, ê). the complement ofa šgpr-closedset is called a š gpr – open set. 3.1 strongly š𝛂∗-continuous function definition 3.1.1: a š function f: (x, τs, ê) (ý, τs, k ) is said to be strongly šα∗-continuous function, if the inverse image of every š α*ô(ý)in (ý, τs, k ) is š ô(x) in (x, τs, ê). theorem 3.1.2: let f : (x, τs, ê) (ý, τs, k ) be strongly šα ∗-continuous function, then it isš-continuous function. proof: let (fš, ê) be š ô(x) in (ý, τs, k). since every š ô(x) is š α*ô(x), then (fš, ê) is š α*ô(x) in (ý, τs, k ). since, f is strongly šα ∗-continuous function, f -1(fš, ê) is š ô(x) in (x, τs, ê).therefore, f is š-continuous. 184 p. anbarasi rodrigo and s. anitha ruth remark 3.1.3: the converse of the above theorem need not be true. example 3.1.4: let x = ý = { x1, ,x2}, τs = {f š 1, f š 2, f š 3,f š 15,f š 16},and s = {f š 3, f š 11, fš12,f š 15,f š 16}, š α*ô(ý)= {fš1,f š 2, f š 3,f š 7,f š 8,f š 9,f š 10,f š 11,f š 12,f š 13,f š 14,f š 15,f š 16 }. let f : (x, τs, ê) (ý, τs, k )be defined by f(f š 1) = f š 3 , f(f š 2) = f š 11,f(f š 3) = f š 12,f(f š 4) = f š 1,f(f š 5) = fš2,f(f š 6) = f š 13 ,f(f š 7) = f š 4f(f š 8)= f š 14,f(f š 9) = f(f š 10) = f(f š 11)= f(f š 12)= f š 2,f(f š 13) = fš9 ,f(f š 14) = f š 6,f(f š 15) = f š 15,f(f š 16) = f š 16. clearly f is š– continuous but not strongly šα∗-continuous function, because f -1(fš1) = f š 4is not š ô(x) in (x, τs, ê). theorem 3.1.5: let f : (x, τs, ê) (ý, τs, k )be strongly šα ∗-continuous function iff the inverse image of every š α* ç(ý) in (ý, τs, k) is š ç(x) in (x, τs, ê) proof: assume that f is strongly šα∗-continuous function. let (fš, ξ ) be any š α* ç(x) in (ý, τs, k). then, (f š, ê)cis š α*ô(x) in (ý, τs, k). since f is strongly šα ∗-continuous function. f-1((fš, ê) c) is š ô(x) in (x, τs, ê). but f -1((fš, ξ ) c) = x f -1((fš, ξ ) is š ô(x) in (x, τs, ê) f -1((fš, ê) is š ç(x) in (x, τs, ê). conversely, assume that the inverse image of every š α*ç(x) in (ý, τs, ê) is š ç(x) in (x, τs, ê). let (f š, ê) be any š α*ô(x) in (ý, τs, k). then, (f š, ê)cis š α*ç(x) in (ý, τs, k). by assumption, f -1((fš, ê) c) is š ç(x) in (x, τs, ê). but f -1((fš, ê) c) = x f -1((fš, ê) is š ç(x) in (x, τs, ê) f -1((fš, ê) is š ô(x) in (x, τs, ê). hence f is strongly šα ∗-continuous function. theorem 3.1.6: let f : (x, τs, ê) (ý, τs, k )be strongly š -continuous function then it is strongly šα∗-continuous function. proof: let (fš, ê) be any š ô(x) in (ý, τs, k). since every š ô(x) is š α*ô(x), since f is strongly š continuous function, then f -1((fš, ê) is both š ô(x) and š ç(x) in (x, τs, ê). f-1((fš, ê) is š ô(x) in (x, τs, ê). hence f is strongly š α ∗continuous function. remark 3.1.7: the converse of the above theorem need not be true. example 3.1.8: let x = ý = { x1, ,x2}, τs = {f š 1,f š 2, f š 3, fš5,f š 7,f š 8,f š 9,f š 10,f š 12,f š 13,f š 14,f š 15,f š 16 }, τs c = {fš1,f š 2, f š 4, fš6,f š 7,f š 8,f š 9,f š 10,f š 13,f š 14,f š 15,f š 16 }, and s = {f š 1, f š 13f š 15,f š 16}, š α*ô(ý)= {fš1, f š 3,f š 7,f š 8,f š 11,f š 12,f š 13,f š 15,f š 16 }. let f : (x, τs, ê) (ý, τs, k )be defined by f(fš1) = f š 5 , f(f š 2) = f š 2, f(f š 3) = f š 3,f(f š 4) = f š 4f(f š 5) = f š 5, f(f š 6) = fš6f(f š 7) = f š 7,f(f š 8)= f š 8f(f š 9) =f š 9 ,f(f š 10) = f š 10,f(f š 11) =f š 11,f(f š 12) = f š 12,f(f š 13) = fš13,f(f š 14) = f š 4,f(f š 15) = f š 15,f(f š 16) = f š 16. clearly f is strongly šα ∗-continuous function but not strongly š-continuous function,sincef -1(fš1) = f š 5is š ô(x) but not š ç(x) theorem 3.1.13: let f : (x, τs, ê) (ý, τs, k )be strongly šα ∗-continuous function and g: (x, s, ê) (ž, s, k ) be šα ∗-continuous function, then 185 more functions related to š𝛼∗ open set in soft topological spaces go f : (x, τs, ê) (ž, s, k ) is šα ∗-continuous. proof: let(fš, ê) be any š ô(x) in (ž, s, k ). since g is šα ∗-continuous, then g-1((fš, ê)is š α*ô(x) in (ý, τs, k). since f is strongly šα ∗-continuous function, then f-1(g 1((fš, ξ )) is š ô(x) in (x, τs, ê)( go f) -1(fš, ê) is š ô(x) in (x, τs, ê). )( go f) 1(fš, ê) is šα∗ ô(x) in (x, τs, ê). hence gof isš α ∗-– continuous. theorem 3.1.14: let f : (x, τs, ê) (ý, s, k )be strongly š α ∗-continuous function and g: (ý, s, k) (ž, s, k ) be šα ∗-irresolute, then go f : (x, τs, ê) (ž, s, k ) is strongly šα ∗– continuous. proof: let (fš, ξ) be any šα∗ô(x) in (ž, s, k ). since g is šα ∗-irresolute, then g-1((fš, ê)is š α*ô(x) in (ý, s, k). since f is strongly šα ∗-continuous function, then f-1( g 1((fš, ê)) is š ô(x) in (x, τs, ê)( go f) -1((fš, ê) is š ô(x) in (x, τs, ê). hence gof is strongly šα∗– continuous. theorem 3.1.15: let f : (x, τs, ê) (ý, s, k )be šα ∗-continuous and g: (ý, s, k) (ž, s, k ) be strongly š α ∗-continuous function, then go f : (x, τs, ê) (ž, s, k ) is š α ∗–irresolute. proof: let (fš, ê) be any šα∗ô(x) in (ž, s, k ). since g is strongly šα ∗-continuous, then g -1((fš, ê)isš ô(x) in (ý, τs, k). since f is šα∗-continuous function, then f -1( g 1((fš, ê)) is šα∗ô(x) in (x, τs, ê)( go f) -1((fš, ê) is šα∗ô(x) in (x, τs, ê). hence gof is š α∗– irresolute. theorem 3.1.16: let f : (x, τs, ê) (ý, τs, k )be strongly š α ∗-continuous function and g: (x, s, ê) (ž, s, k ) be strongly šα ∗-continuous function, then go f : (x, τs, ê) (ž, s, k ) is strongly š α ∗– continuous. proof: let (fš, ê) be any šα∗ ô(x) in (ž, s, k ). since g is strongly šα ∗-continuous, then (g-1(fš, ê))is š ô(x) in (ý, τs, k). since f is strongly šα ∗-continuous function, then f-1( g -1((fš, ê)) is š ô(x) in (x, τs, ê)( go f) -1((fš, ê) is š ô(x) in (x, τs, ê). hence gof isstrongly š α∗– continuous. theorem 3.1.17: let f : (x, τs, ê) (ý, s, k )be š -continuous function and g: (ý, s, k) (ž, s, k ) be strongly šα ∗-continuous function, then go f : (x, τs, ê) (ž, s, k ) is strongly šα ∗– continuous. proof: let (fš, ê) be any šα∗ô(x) in (ž, s, k ). since g is strongly šα ∗-continuous, then g-1((fš, ê)is šô(x) in (ý, τs, k). since f is š-continuous function, then f -1( g -1((fš, ξ )) is š ô(x) in (x, τs, ê)( go f) -1((fš, ê ) is š ô(x) in (x, τs, ê). hence gof is strongly š α∗– continuous. 186 p. anbarasi rodrigo and s. anitha ruth references [1] p. anbarasi rodrigo & s. anitha ruth, "a new class of soft set in soft topological spaces", international conference on mathematics and its scientific applications, organized by sathyabama institute of science and technology. 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[9] m. shabir, and m. naz, "on soft topological spaces"., comput. math. appl., vol. 61, pp. 1786-1799, 2011. 187 ratio mathematica volume 44, 2022 extraction of aspects from online reviews using a convolution neural network vidya kamma1 sridevi gutta2 teja santosh dandibhotla3 abstract the quality of the product is measured based on the opinions gathered from product reviews expressed on a product. opinion mining deals with extracting the features or aspects from the reviews expressed by the users. specifically, this model uses a deep convolutional neural network with three channels of input: a semantic word embedding channel that encodes the semantic content of the word, a part of speech tagging channel for sequential labelling and domain embedding channel for domain specific embeddings which is pooled and processed with a softmax function. this model uses three input channels for aspect extraction. experiments are conducted on amazon review dataset. this model achieved better results. keywords: neural network, softmax function and extraction. ams classification: 68t074 1 research scholar, department of computer science and engineering, konerulakshmaiah education foundation, vaddeswaram, a.p., india. & assistant professor, department of computer science, neil gogte institute of technology. mail id: kammavidya@gmail.com 2 professor, konerulakshmaiah education foundation, vaddeswaram, a.p., india. mail id:sridevi.gutta2012@gmail.com 3 associate professor, sreenidhi institute of science and technology, yamnampet, ghatkesar, telangana, india. mail id:tejasantoshd@gmail.com 4received on june 16th, 2022. accepted on aug 10th , 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.909. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 213 mailto:kammavidya@gmail.com mailto:sridevi.gutta2012@gmail.com mailto:tejasantoshd@gmail.com vidya kamma, sridevi gutta and teja santosh dandibhotla 1. introduction customers are expressing views and opinions of products purchased. customers can leverage the information contained in such reviews to identify the best products. these reviews help organizations to identify customer needs. the rise of reviews is gaining greater attention in research community. when it comes to traditional sentiment analysis, talk about opinion on entire sentence. customer reviews often express opinion on different aspects of product rather than opinion on whole product schouten, k., & frasincar, f. (2015). this raised the problem of aspect extraction for sentiment analysis. many challenge to be addressed, identifying implicit aspects, multiple aspects in single statement. the major task of ae (aspect extraction) to extracting aspects of customer reviews cheung, d. 2009, et al. jakob, n., & gurevych, i. (2010), extracted aspects of a spatial dataset using conditional random fields, the limitation is it need to work of large datasets and is linear in nature. hu, m., & liu, b. (2004), used linguistic models for extracting aspects from reviews. the limitation is rules to be drafted manually. efficiency based on grammatical accuracy of sentences. neural network models are widely used for aspect extraction. existing cnn based models are lacking in extracting context-level features. ye, h. et al., (2017) and dong, f. et al., (2017), used cnn integrated with lstm for aspect extraction achieved better performance with increase the complexity of the model parameters. in this paper, to overcome above mentioned issues we contributed the following. to encode contextual information, three different input channels were used: i) word embeddings + lexicon ii) domain embeddings iii) pos tag embeddings. previous models used either pre-trained embeddings or review embeddings poria, s. et al., (2016). to increase performance of model we included lexicons in word embedding layer. the first two channels are used to capture semantic and syntactic information. third channel is used for sequential labelling of aspects. 2. related work among the many tasks involved in aspect level sentiment analysis, a major one is aspect extraction (liu b, 2012). current approaches to extracting features are based on deep learning. macháček, j. (2016) implemented traditional supervised machine learning method which uses biagram (bow) bag of words. li and lam, (2017) performed aspect extraction by considering annotated data and achieved better results. luo et al., (2018) integrated word embedding with crf&bilstm for better extracting the aspects. toh, z., &su, j. (2016) combined cnn with binary classifier and implemented anhybrid approach. approach was top rated in semeval 2016 task 5 competition. jihan, n. et al., (2017) used pre-processing pipeline for normalizing the data. they implemented multi-domain feature extraction and predicted aspect category using svm. khalil, t.et al (2016) initialized pre-trained word vectors for cnn and bag of words as features to ensemble classifier. to improve classification, they used secondary classifier. stéphan tulkens et al., (2020), implemented (cat) contrastive 214 extraction of aspects from online reviews using a convolution neural network attention mechanism an unsupervised approach based on rbf kernel for extraction aspects. ruidan he et al., (2017), extracted coherent aspects using neural approach. they used attention mechanism in the training phase to minimize irrelevant words. poria, s. et al., (2016), used deep cnn with 7 layers plus linguistics patterns for extracting aspects in product reviews. wang et al., (2016) proposed the method which combined both crf and dependency tree for better aspect extraction. wang et al., (2014), based on seeding words implemented the extraction of product aspects was done using two semi-supervised models. collobert et al., (2011) initialized cnn with word embeddings to solve semantic role labelling and named entity recognition problem. yinyang et al., (2017), proposed (cat-lda) a two-layer topic model to extract hierarchical aspects i.e parent and child category. lin wang (2015), used restricted boltzmann machines to extract aspects. xu et al., (2017) and li & lam (2018), used deep learning model for extracting aspect and opinion items. 3. model the proposed model has three input layers. three embedding layers word-embedding layer enriched word vectors using lexicons, domain embedding layer, pos embedding layer, cnn layers with relu-nonlinear activation function, fully connected layer, pooling layer for multiclass labels a softmax classifier is used for labelling y={b,i,o,e,s},with beginning, inside, outside, end and single of the aspect term. figure 1. shows the proposed cnn model figure 1. proposed convolution neural network model 215 vidya kamma, sridevi gutta and teja santosh dandibhotla pre-processing mainly pre-processing is performed to clean and convert the reviews for further processing. we used python with nlp libraries are used for removal of numbers, converting acronyms, white spaces, tabs, stop words, words with length of 1 and special characters. repeated letter words i.e “gooooood” or replaced with original form. all the review sentences are converted to lower case. word embeddings the three inputs google corpus, amazon reviews, pos tagging are mapped to vectors. padding is performed to maintain the sentences of equal length. let the input . i h w x + 1 ) .( , , n x x x=  the three embedding layers representations are gl w for google + lexicon embeddings, d w for domain embeddings and p w for pos tag embeddings are fed to cnn. convolution convolution layer is mainly used to extract features. filters of different sizes (3,4,5) are used on each matrix. the stride used in each layer if 1 to tag each word. the relu is the activation function is used with the convolution operation. the features produced will be of the form. 1 max(0, ( . )) n i i h i g w x b + = = + (1) where . i h w x + is weight and vector of h-gram in a sentence. max pooling layer pooling operation is applied on the feature maps. the purpose of this layer is to extract maximum efficient features so max pooling is applied. the feature map obtained is the concatenation of three feature maps. where n, m, o represents the filter sizes for google +lexicon features, domain features and pos features. 1 1 1 ........ ........ ........ n m o gl gl d d p p f f f f f f f=         (2) to choose hyperparameters we applied cross-validation strategy. overfitting is the major problem with deep neural networks if the parameters are high. dropout regularization is added. 50% is considered as the dropout rate to avoid over fitting. and features are added to fully connected layer 1 max(0, ( . )) n i i h i g w x b + = = + . ( * )k w f b= + (3) where w is the weight matrix, b is bias and  is relu activation function. softmax classifier this is the final layer used for classification of aspects into classes with highest probability. ˆ arg max ( / , , ) arg max( / 1 ) xwj aj k xwk aj j j k y p y j x w a e xe + + = = = = j w : weights of class j and j a : bias of class j .the above-mentioned process is repeated for the complete training samples. 216 extraction of aspects from online reviews using a convolution neural network 4. experiments these datasets are a collection of amazon reviews of four categories of products. product categories included in the analysis of reviews include laptops, smartphones, cameras and wristwatches. from the e-commerce applications, 100 products are considered for each product category. dataset details are presented in table 1. table.1: dataset details document attributes values documents reviewed 4,13,841 each review should include one sentence 1 max. sentences per review 43 customer review average 5.98 review count for the product on average 49.47 amazon reviews are used to train the domain embeddings. electronic product reviews are collected from amazon. in this work, word embeddings are obtained from the google news corpus word2vec embeddings. we remove all stop words and nonenglish words from the data during pre-processing. pos tags are applied to the obtained words. the table 2 below shows the experimental setup for the modified cnn model, which covers a variety of settings. table.2: experimental setup parameters value filter size (3,4,5) no. of feature maps 100 pooling function max regularization (dropout rate) 50% activation function relu number of epochs 10 by carefully adjusting the settings of the modified cnn, good accuracy was achieved. according to pedro domingos, "bigger data are better than smart algorithms" which is also the case for our modified cnn model which is clear from accuracy metrics when considering the size of the reviews collection. figure-2 and figure-3 shows that the model has different performances across the four different domains. 217 vidya kamma, sridevi gutta and teja santosh dandibhotla figure 2: comparison of the performance with in four domains figure 3: comparison of the performance with in four domains table.3: impact of the pos feature on the dataset. domain classifiers precision recall f-score camera we 80.74% 74.15% 74.76% camera we+pos+de 84.42% 75.21% 80.10% cell phone we 84.01% 80.32% 81.59% cell phone we+pos+de 89.05% 83.47% 84.68% wrist watches we 82.41% 73.26% 79.67% wrist watches we+pos+de 83.48% 77.25% 81.24% laptops we 86.35% 80.23% 82.17% laptops we+pos+de 86.40% 80.77% 83.49% pos features were used in conjunction with word embedding features to achieve the 65,00% 70,00% 75,00% 80,00% 85,00% 90,00% camera cellphone wrist watches laptops we precision recall f-score 70,00% 75,00% 80,00% 85,00% 90,00% 95,00% camera cellphone wrist watches laptops we+pos+de precision recall f-score 218 extraction of aspects from online reviews using a convolution neural network highest accuracy on this dataset. pos play an important role in extraction of aspects as well as word embedding, as demonstrated here. impact of the pos feature on the dataset is presented in table.3. 5. conclusion in this paper, three channel-based cnn for aspect extraction is introduced. experiments are conducted on five different domains smartphones, camera, laptop, wrist watches, books. python is used as acomputational environment. we determined the efficiency of using lexicon in word embedding layer. these carefully configured settings have helped the modified cnn architecture achieve good accuracy. references [1] cheung, d. w. l., song, i. l., chu, w., hu, x. and lin, j., “proceedings of the 18th acm conference on information and knowledge management”, association of computing machinery, 2009. 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[25] qiu, g., liu, b., bu, j. and chen, c., “opinion word expansion and target extraction through double propagation”, computational linguistics, 37(1), pp.9-27, 2011. 221 ratio mathematica volume 44, 2022 the forcing geodetic cototal domination number of a graph s. l. sumi1 v. mary gleeta2 j. befija minnie3 abstract let 𝑆 be a geodetic cototal domination set of 𝐺. a subset 𝑇 ⊆ 𝑆 is called a forcing subset for 𝑆 if 𝑆 is the unique minimum geodetic cototal domination set containing 𝑇. the minimum cardinality t is the forcing geodetic cototal domination number of s is denotedby 𝑓𝛾𝑔𝑐𝑡 (𝑆), is the cardinality of a minimum forcing subset of s. the forcing geodetic cototal domination number of 𝐺,denoted by 𝑓𝛾𝑔𝑐𝑡 (𝑆), is 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 𝑚𝑖𝑛{𝑓𝛾𝑔𝑐𝑡 (𝑆)}, where the minimum is takenover all 𝛾𝑔𝑐𝑡-sets 𝑆 in 𝐺. some general properties satisfied by this concept arestudied. it is shown that for every pair 𝑎, 𝑏 of integers with 0 ≤ 𝑎 < 𝑏, 𝑏 ≥ 2,there exists a connected graph 𝐺 such that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 𝑎 and 𝛾𝑔𝑐𝑡 (𝐺) = 𝑏. where𝛾𝑔𝑐𝑡 (𝐺) isthe geodetic cototal dominating number of 𝐺. keywords: geodetic set, cototal dominating set, geodetic cototal dominating set, geodetic cototal domination number, forcing geodetic cototal domination number. ams subject classification: 05c12, 05c694 1research scholar, register no.20123042092007, department of mathematics, holy cross college (autonomous), nagercoil 629004, affiliated by manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamil nadu, india. sumikrish123@gmail.com. 2assistant professor, department of mathematics, t.d.m.n.s college, t. kallikulam 627 113, affiliated by manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamil nadu, india. gleetass@gmail.com. 3assistant professor, department of mathematics, holy cross college (autonomous), nagercoil 629004, affiliated by manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamil nadu, india, mail id: befija@gmail.com 4received on june 11th, 2022.accepted on sep 9st, 2022.published on nov 30th, 2022.doi: 10.23755/rm.v44i0.895. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement 93 s. l. sumi, v. mary gleeta and j. befija minnie 1. introduction by a graph 𝐺 = (𝑉, 𝐸), we mean a finite, undirected connected graph without loops or multiple edges. the order and size of 𝐺 are denoted by 𝑚and 𝑛respectively. for basic definitions and terminologies, we refer to [1,2]. for vertices 𝑢 and 𝑣 in a connected graph 𝐺, the distance𝑑(𝑢, 𝑣) is the length of a shortest 𝑢– 𝑣 path in 𝐺. a 𝑢– 𝑣 path of length 𝑑(𝑢, 𝑣) is called a 𝑢– 𝑣geodesic. the eccentricity𝑒(𝑣) of a vertex 𝑣 in 𝐺 is the maximum distance from 𝑣 and a vertex of 𝐺. the minimum eccentricity among the vertices of 𝐺 is the radius, 𝑟𝑎𝑑𝐺 or𝑟(𝐺) and the maximum eccentricity is its diameter, 𝑑𝑖𝑎𝑚𝐺of 𝐺. let 𝑥, 𝑦 ∈ 𝑉and let𝐼[𝑥, 𝑦] be the set of all vertices that lies in 𝑥 − 𝑦 geodesic including 𝑥and 𝑦. let 𝑆 ⊆ 𝑉(𝐺)and 𝐼[𝑆] = ⋃ 𝐼[𝑥, 𝑦]𝑥,𝑦∈𝑆 . then 𝑆 is said to be a geodetic set of 𝐺, if 𝐼[𝑆] = 𝑉. the geodetic number𝑔(𝐺) of 𝐺is the minimum order of its geodetic sets and any geodetic set of order 𝑔(𝐺) is called a 𝑔setof 𝐺. a set 𝑆 ⊆ 𝑉 (𝐺) is called a dominating set if every vertex in 𝑉(𝐺) − 𝑆 is adjacent to at least one vertex of 𝑆. the domination number, 𝛾(𝐺), of a graph 𝐺 denotes the minimum cardinality of such dominating sets of 𝐺. a minimum dominating set of a graph 𝐺 is hence often called as a 𝛾-set of 𝐺. the domination concept was studied in [3]. a dominating set 𝑆 of 𝐺 is a cototal dominating set if every vertex 𝑣 ∈ 𝑉 ∖ 𝑆 is not an isolated vertex in the induced subgraph 〈𝑉 ∖ 𝑆〉. the cototal domination number 𝛾𝑐𝑡 (𝐺) of 𝐺 is the minimum cardinality of a cototal dominating set. the cototal domination number of a graph was studied in [4]. a set 𝑆 ⊆ 𝑉 is said to be a geodetic cototal dominating set of 𝐺, if𝑆 is both geodetic set and cototal dominating set of 𝐺. the geodetic cototal domination number of 𝐺 is the minimum cardinality among all geodetic cototal dominatingsets in 𝐺 and denoted by 𝛾𝑔𝑐𝑡 (𝐺). a geodetic cototal dominating set of minimumcardinality is called the 𝛾𝑔𝑐𝑡-set of 𝐺. the geodetic cototal domination number of agraph was studied in [6]. the following theorems are used in the sequel. theorem 1.1. [6] every end vertex of g belongs to every geodetic cototal dominating set of g. 2. the forcing geodetic cototal domination number of a graph even though every connected graph contains a minimum geodetic cototal dominating sets, some connected graph may contain several minimum geodetic cototal dominating sets. for each minimum geodetic cototal dominating set s in a connected graph there is always some subset t of s that uniquely determines s as the minimum geodetic cototal dominating set containing t such “forcing subsets” are considered in this section. the forcing concept was studied in [5] definition 2.1. let 𝑆 be a geodetic cototal domination set of 𝐺. a subset t ⊆ s is called a forcing subset for s if s is the unique minimum geodetic cototal domination set containing t. the minimum cardinality t is the forcing geodetic cototal domination 94 the forcing geodetic cototal domination number of a graph number of s is denoted by 𝑓𝛾𝑔𝑐𝑡 (𝑆), is the cardinality of a minimum forcing subset of s. the forcing geodetic cototal domination number of 𝐺, denoted by𝑓𝛾𝑔𝑐𝑡 (𝑆), is 𝑓𝛾𝑔𝑐𝑡 (𝐺) = min {𝑓𝛾𝑔𝑐𝑡 (𝑆)}, where the minimum is taken over all 𝛾𝑔𝑐𝑡-setss in g. example 2.2. for the graph 𝐺 of figure 2.1, 𝑆1 = {𝑣3, 𝑣6, 𝑣7} and 𝑆2 = {𝑣2, 𝑣5, 𝑣7}are the only two 𝛾𝑔𝑐𝑡-sets of 𝐺 so that 𝛾𝑔𝑐𝑡 (𝐺) = 3 and 𝑓𝛾𝑔𝑐𝑡(𝑆1) = 𝑓𝛾𝑔𝑐𝑡 (𝑆2) = 1 sothat 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 1. the following result follows immediately from the definitions of the geodetic cototal domination number and the forcing geodetic cototal domination number of a connected graph 𝐺. theorem 2.3. for every connected graph 𝐺, 0 ≤ 𝑓𝛾𝑔𝑐𝑡 (𝐺) ≤ 𝛾𝑔𝑐𝑡 (𝐺). remark 2.4. the bounds in theorem 2.3 are sharp. for the complete graph 𝐺 = 𝐾𝑛,𝑆 = 𝑉 is the unique 𝛾𝑔𝑐𝑡-set of 𝐺 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0. also, the bounds in theorem2.3 can be strict. for the graph 𝐺 given in figure 2.1, 𝛾𝑔𝑐𝑡 (𝐺)= 3 and 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 1. thus 0 < 𝑓𝛾𝑔𝑐𝑡 (𝐺) < 𝛾𝑔𝑐𝑡 (𝐺). theorem 2.5. let g be a connected graph. then (a) 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0 if and only if g has a unique minimum 𝛾𝑔𝑐𝑡-set. (b) 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 1 if and only if g has at least two minimum 𝛾𝑔𝑐𝑡-sets, one of which isa unique minimum 𝛾𝑔𝑐𝑡-set containing one of its elements and (c) 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 𝛾𝑔𝑐𝑡 (𝐺) if and only if no 𝛾𝑔𝑐𝑡-set of 𝐺 is the unique minimum 𝛾𝑔𝑐𝑡-set containing any of its proper subsets. 𝑣1 𝑣6 𝑣2 𝑣5 𝑣3 𝐺 figure 2.1 𝑣4 𝑣7 95 s. l. sumi, v. mary gleeta and j. befija minnie definition 2.6. a vertex 𝑣 of a connected graph 𝐺 is said to be a geodetic cototal dominating vertex of 𝐺 if 𝑣 belongs to every 𝛾𝑔𝑐𝑡-set of 𝐺. example 2.7. for the graph 𝐺 given in figure 2.2, 𝑆1 = {𝑣1, 𝑣3, 𝑣6} and 𝑆2 = {𝑣1, 𝑣3, 𝑣5} are the only two minimum 𝛾𝑔𝑐𝑡-sets of 𝐺 so that {𝑣1, 𝑣3} is the geodeticcototal dominating vertex of g.then 𝑓𝛾𝑔𝑐𝑡 (𝐺) ≤ 𝛾𝑔𝑐𝑡 (𝐺)– | 𝑊 |. remark 2.9. the bound in corollary 2.7 is sharp. for the graph 𝐺 of figure 2.2, 𝑆1 = {𝑣1, 𝑣3, 𝑣6} and 𝑆2 = {𝑣1, 𝑣3, 𝑣5} are the only two minimum 𝛾𝑔𝑐𝑡-sets of 𝐺 so that 𝑓𝛾𝑔𝑐𝑡 (𝑆1) = 𝑓𝛾𝑔𝑐𝑡 (𝑆2) = 1 so that 𝛾𝑔𝑐𝑡 (𝐺) = 3 and 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 1. also, 𝑊 = {𝑣1, 𝑣3} is theset of all geodetic cototal dominating vertices of 𝐺. now,𝛾𝑔𝑐𝑡 (𝐺) − |𝑊| = 3 − 2 = 1. thus 𝑓𝛾𝑔𝑐𝑡 (𝐺) < 𝛾𝑔𝑐𝑡 (𝐺) − |𝑊|. also, the bounds in theorem 2.7 can be strict. theorem 2.10. for the complete bipartite graph 𝐺 = 𝐾𝑟,𝑠 (1 ≤ 𝑟 ≤ 𝑠), 𝑓𝛾𝑔𝑐𝑡 (𝐺) = { 0, 𝑖𝑓 1 ≤ 𝑟 ≤ 3 4, 𝑖𝑓4 ≤ r ≤ s proof: let 𝑈 = {𝑢1, 𝑢2, . . . , 𝑢𝑟 } and 𝑊 = {𝑤1, 𝑤2, . . . , 𝑤𝑠} be the bipartite sets of 𝐺. for 1 ≤ 𝑟 ≤ 3. let 𝑆 = 𝑈 ∪ 𝑊 is the unique 𝛾𝑔𝑐𝑡-set of 𝐺 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0. let1 ≤ 𝑟 ≤ 3. if 𝑟 ≥ 4, then every 𝛾𝑔𝑐𝑡 (𝐺)-set is of the form 𝑆 = {𝑢𝑖1 , 𝑢𝑖2 , 𝑤𝑗1 , 𝑤𝑗2 }where1≤ 𝑖1 ≤ 𝑖2 ≤ 𝑟 and 1 ≤ 𝑗1 ≤ 𝑗2 ≤ 𝑠. since 𝑆 is not the unique geodetic cototaldominating set containing any of its proper subset, by theorem 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 4.∎ theorem 2.11. for the wheel 𝐺 = 𝐾𝑛 + 𝐶𝑛−1 (𝑛 ≥ 5), 𝑓𝛾𝑔𝑐𝑡 (𝐺) ={ 1, 𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛 2, 𝑖𝑓𝑛𝑖𝑠𝑜𝑑𝑑 𝑣1 𝑣2 𝑣5 𝑣6 𝑣3 𝑣4 𝐺 figure 2.2 figurfigfure 2.1 figure 2.1 e 1.1 96 the forcing geodetic cototal domination number of a graph proof: let 𝑥 be the central vertex of 𝐺 and 𝐶𝑛−1 be 𝑣1, 𝑣2, … , 𝑣𝑛−1, 𝑣𝑛 . case 1: 𝑛 is even. then 𝑆1 = {𝑣1, 𝑣3, 𝑣5, . . . , 𝑣𝑛−3, 𝑣𝑛−1}, 𝑆2 = {𝑣2, 𝑣4, 𝑣6, . . . , 𝑣𝑛−2, 𝑣𝑛 } are the only two𝛾𝑔𝑐𝑡-sets of 𝐺 such that 𝑓𝛾𝑔𝑐𝑡 (𝑆1) = 𝑓𝛾𝑔𝑐𝑡 (𝑆2) = 1 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 1. case 2: 𝑛 is odd. then 𝑆1 = {𝑣1, 𝑣3, 𝑣5, … , 𝑣𝑛 }, 𝑆2 = {𝑣2, 𝑣4, 𝑣6, … , 𝑣𝑛−1, 𝑣1}, … , 𝑆𝑛 2⁄ = {𝑣𝑛 2⁄ , 𝑣𝑛 2⁄ +1 , . . . , 𝑣1, 𝑣3, 𝑣𝑛 2⁄ −1 }are the 𝑛/2 𝛾𝑔𝑐𝑡-sets of 𝐺 such that 𝑓𝛾𝑔𝑐𝑡 (𝑆1) = 𝑓𝛾𝑔𝑐𝑡 ( 𝑆2) = . . . = 4𝑓𝛾𝑔𝑐𝑡 (𝑆𝑛 2⁄ ) = 2 sothat 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 2.∎ theorem 2.12. for the helm graph 𝐺 = 𝐻𝑟 , 𝐺 = 𝑇, 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0, for 𝑛 ≥ 6. proof: let 𝑆 be the set of end vertices and the cut vertices of 𝐺. then 𝑆 is theunique 𝛾𝑔𝑐𝑡-set of 𝐺 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0.∎ theorem 2.13. for the triangular snake graph 𝐺 = 𝑇𝑟, 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0. proof: let 𝑆 be the set of extreme vertices of 𝐺. then s is the unique 𝛾𝑔𝑐𝑡-set of 𝐺so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0.∎ theorem 2.14. for the fan graph 𝐹𝑛 = 𝐾1 + 𝑃𝑛−1, 𝑓𝛾𝑔𝑐𝑡 (𝐺) = { 0, 𝑖𝑓𝑛 − 1 𝑖𝑠𝑜𝑑𝑑 1, 𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛 proof: let 𝑉 (𝐾1) = {𝑥} and 𝑉(𝑃𝑛−1) = {𝑣1, 𝑣2, . . . , 𝑣𝑛−1}. let 𝑛 − 1 is odd. let 𝑛 − 1 = 2𝑘 + 1. then 𝑆 = {𝑣1, 𝑣3, 𝑣5, . . . , 𝑣2𝑘+1} is the unique𝛾𝑔𝑐𝑡-set of 𝐺 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0. let 𝑛 − 1 be even. let 𝑛 − 1 = 2𝑘. then 𝑆1 = {𝑣1, 𝑣3, 𝑣5, . . . , 𝑣2𝑘−1, 𝑣2𝑘 }, 𝑆2 = {𝑣1, 𝑣3, 𝑣5, . . . , 𝑣2𝑘−2 , 𝑣2𝑘 , 𝑣2}are the two 𝛾𝑔𝑐𝑡-sets of 𝐺 such that 𝑓𝛾𝑔𝑐𝑡 (𝑆1) = 𝑓𝛾𝑔𝑐𝑡 (𝑆2) = 1.so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 1. ∎ theorem 2.15. for the banana tree graph 𝐺 = 𝐵𝑟,𝑠, 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0. proof: let 𝑥 be the centre vertex of 𝐺 and the set of all end vertices of 𝐺. then𝑆 = 𝑍 ∪ {𝑥} is the unique 𝛾𝑔𝑐𝑡-set of 𝐺 so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0.∎ theorem 2.16. for the sunflower graph 𝐺 = 𝑆𝐹𝑛, 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0. proof: let 𝑆 be the set of extreme vertices of 𝐺. then 𝑆 is the unique 𝛾𝑔𝑐𝑡-set of 𝐺. so that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 0. ∎ theorem 2.17. for every pair𝑎, 𝑏of integers with 0 ≤ 𝑎 < 𝑏, 𝑏 ≥ 2, there exists a connected graph 𝐺 such that 𝑓𝛾𝑔𝑐𝑡 (𝐺) = 𝑎 and 𝛾𝑔𝑐𝑡 (𝐺) = 𝑏. proof: let 𝑃 ∶ 𝑢, 𝑣, 𝑧be a path of order three. let 𝑃𝑖 : 𝑢𝑖 , 𝑣𝑖 (1 ≤ 𝑖 ≤ 𝑎) be a copyof path on two vertices. let 𝐻be a graph obtained from 𝑃and 𝑃𝑖 (1 ≤ 𝑖 ≤ 𝑎) byjoining each 𝑢𝑖 (1 ≤ 𝑖 ≤ 𝑎) with 𝑣 and each 𝑣𝑖 (1 ≤ 𝑖 ≤ 𝑎) with 𝑧. let 𝐺 be thegraph obtained from 𝐻 by introducing new vertices𝑧1, 𝑧2, . . . , 𝑧𝑏−𝑎+1 joining each𝑧𝑖 (1 ≤ 𝑖 ≤ 𝑎) with 𝑧. the graph 𝐺 is given in figure 2.4. 97 s. l. sumi, v. mary gleeta and j. befija minnie first, we show that γgct(g) = b. let z = {u, z1, z2, . . . , zb−a+1} be the set of endvertices of g. by theorem 1.1, z is a subset of every geodetic cototal dominating set of g. let hi = {ui, vi}. then it is easily observed that every geodetic cototaldominating set containing at least one vertex from each hi(1 ≤ i ≤ a) and soγgct(g)≥ b– a + a = b. let s = z ∪ {u1, u2, . . . , ua}. then s is a minimum geodeticcototal dominating set of g so that γgct(g) = b. next, we prove that fγgct(g) = a. since every geodetic co-total dominating set ofgcontains z, it follows thatfγgct(g) ≤ γgct(g) − | z | = b − (b − a) = a.now, since γgct(g) = b and every γgct-set of g contains z, it is easily seen that everyγgct-set of g is of the form s = z ∪ {c1, c2, . . . , ca}, where ci ∈ hi (1 ≤ i ≤ a). let t beany proper subset of s with | t | < a. then there exists an edge ej (1 ≤ j ≤ a) suchthat ej ∉ t. let fj be an edge of hj distinct from ej. then w1 = (s − {ej} ∪ {fj}is a γgct-set properly containing t. thus w is not the unique γgct-set containingt. thus t is not a forcing subset of s. this is true for all minimum geodetic cototaldominating sets of g and so it follows that fγgct(g) = a. 3. conclusion in this paper we studied the concept of forcing geodetic cototal domination number of some standard graphs some general properties satisfied by this concept are studied. in future studies, the same concept is applied for the other graph operations. 𝑧𝑏−𝑎−1 𝑧2 𝑧1 𝑣2 𝑢1 𝑢 𝐺 figure 2.3 𝑣 𝑧 𝑣1 𝑢2 𝑢3 𝑢4 𝑢𝑎 𝑣4 𝑣3 𝑣𝑎 98 the forcing geodetic cototal domination number of a graph references [1] f. harary, graph theory, addison – wesley, (1969). [2] f. buckley and f. harary, distance in graphs, addition-wesley, redwood city, ca, (1990). [3] t. w. haynes, s. t. hedetniemi and p. j. slater, fundamentals of domination in graphs, marcel dekker, new york, 1998. [4] v.r. kulli, b. janakiram and radha rajamani iyer, the cototal domination number of a graph, journal of discrete mathematical sciences and cryptography, 2 (2), (1999), 179 – 184. [5] gary chartrand and p. zhang, the forcing geodetic number of a graph, discussiones mathematicae graph theory 19(1999), 45 – 58. [6] s. l. sumi, v. mary gleeta and j. befija minnie, the geodetic cototal domination number of a graph, icdm 2021, isbn:978-93-91077-53-2. 99 microsoft word nuovi criteri di divisibilità.doc 47 nuovi criteri di divisibilità bruno bizzarri, franco eugeni, daniela tondini1 1. – su tutti i testi scolastici di scuola media, nonostante siano riportati i criteri di divisibilità per i numeri 2, 3, 4, 5, 6, 8, 9, 10, 11, viene omesso, com’è immediato constatare, il criterio di divisibilità sia per 7 che per i successivi valori 12, 13, etc. ma allora esiste un criterio di divisibilità per 7? la risposta a tale domanda risulta essere affermativa anche se i vari criteri noti, sia per il 7 che per altri numeri, quali il 12 ed il 13, sono in generale difficili, non solo da applicare ma anche e soprattutto da ricordare. a) il più noto ed antico criterio consiste nel prendere un numero in rappresentazione decimale, scriverlo in ordine inverso come vettorecifre e moltiplicarlo scalarmente, ovvero effettuando la somma dei prodotti cifra per cifra, per i numeri della sequenza 1, 3, 2, 6, 4, 5 accorciata o ripetuta a seconda della lunghezza del vettore dato. esempio 1. il numero 219135 è divisibile per 7? se consideriamo il numero, cifra per cifra, scritto al rovescio (5, 3, 1, 9, 1, 2) e lo moltiplichiamo scalarmente per la sequenza (1, 3, 2, 6, 4, 5), otteniamo:  5×1 + 3× 3 + 1× 2 + 9 × 6 + 1× 4 + 2 × 5 = 84 = 70 +14 = 7 × 10 + 2 che è divisibile per 7! esempio 2. il numero 2191 è divisibile per 7? se consideriamo il numero, cifra per cifra, scritto al rovescio, ovvero (1, 9, 1, 2), e lo moltiplichiamo per (2, 6, 4, 5), otteniamo: 1× 2 + 9× 6 +1× 4 + 2 × 5 = 70 = 7 ×10 (divisibile per 7!) o ancora: 1×1 + 9 × 3 +1× 2 + 2 × 6 = 42 = 7 × 6 (divisibile per 7!) esempio 3. il numero 1753087 è divisibile per 7? se consideriamo il numero, cifra per cifra, scritto al rovescio, ovvero (7, 8, 0, 3, 5, 7, 1), e lo moltiplichiamolo per (1, 3, 2, 6, 4, 5, 1), otteniamo: 7 ×1 + 8× 3 + 0 × 2 + 3× 6 + 5× 4 + 7 × 5 +1×1 = 105 = 7 ×15 (divisibile per 7!) b) un ulteriore criterio consiste nel rovesciare sempre il numero, utilizzando, però, una sequenza più riduttiva, precisamente 1, 3, 2, 1, 3, 2. esempi. il numero 219135 è divisibile per 7 se e solo se lo è:    5, 3,1, 9,1, 2 1, 3, 2, 1, 3, 2 5 9 2 9 3 4 0 7 0             (divisibile per 7!) 1 dipartimento di scienze della comunicazione – università degli studi di teramo eugenif@tin.it, dtondini@unite.it 48 il numero 2191 è divisibile per 7 se e solo se lo è:    1, 9,1, 2 1, 3, 2, 1 1 27 2 2 28 7 4         (divisibile per 7!) il numero 1753087 è divisibile per 7 se e solo se lo è:    7,8, 0, 3, 5, 7,1 1, 3, 2, 1, 3, 2,1 7 24 0 3 15 14 1 0 7 0              (divisibile per 7!) c) un altro criterio, attribuito a david sence (the mathematical gazette, 1956) anche se, in realtà, era già stato scoperto dal russo andrej ziboski (cfr. e. dickson, history of theory of numbers), consiste nel sottrarre al numero originario, privato della sua ultima cifra, il doppio dell’ultima cifra stessa, iterando il ragionamento fino a quando non si ha la certezza di trovarsi di fronte ad un numero divisibile per 7. esempio. il numero 2191 è divisibile per 7 se e solo se lo è 219 2 1 217   che, a sua volta, è divisibile per 7 se e solo se lo è 21 2 7 7   (divisibile per 7!). scopo della presente nota è di illustrare, non solo un semplice criterio di divisibilità per 7, a nostro avviso sconosciuto e la cui dimostrazione risulta banale, ma anche una sua estensione, altrettanto facile, a casi più generali (cfr. paragrafo 3). possiamo enunciare suddetto criterio, applicabile ad un numero di almeno tre cifre, essendo il caso di due cifre ovvio, nel modo seguente: criterio di divisibilità per 7. un numero (in rappresentazione decimale) 1 2 1 0...n nn c c c c c è divisibile per 7 se e solo se lo è il numero 1 0 1 22 ...n nc c c c c  , ovvero la somma tra le ultime due cifre a destra ed il doppio della parte residua. esempio. il numero 2191 è divisibile per 7 se solo se lo è 91 2 21 133   che, a sua volta, è divisibile per 7 se solo se lo è 33 2 1 35 7 5     (divisibile per 7!). ancora il numero 219135 è divisibile per 7 se e solo se lo è 35 2 2191 4417   che, a sua volta, è divisibile per 7 se e solo se lo è 17 2 44 105   che, iterando il ragionamento, è divisibile per 7 se e solo se lo è 5 2 1 7   (divisibile per 7!). nel paragrafo 2, dopo aver richiamato i classici criteri di divisibilità, derivanti dal cosiddetto criterio generale di divisibilità, che denomineremo i criterio, ci soffermeremo sulle difficoltà che siamo costretti ad affrontare più frequentemente. nel paragrafo 3, invece, dopo aver dimostrato il criterio di divisibilità per 7, ci concentreremo sulle sue generalizzazioni, sì da giungere a quello che chiameremo ii criterio generale di divisibilità. nel paragrafo 4, infine, porremo in risalto le varie conseguenze che ne discendono, illustrando anche ulteriori criteri di divisibilità. 49 2. – com’è ben noto, se 0,1, 2,..., 9x  è la base per la numerazione decimale, allora ogni numero naturale n si esprime, in modo unico, nella forma seguente:   1 21 2 1 0 1 2 1 010... 10 10 ... 10 10 n n n n n nn c c c c c c c c c c          nel presente paragrafo richiameremo la teoria della divisibilità rispetto ad una base di numerazione così come essa si deduce dalla teoria delle congruenze e del gaussiano. allo scopo ricordiamo alcune definizioni. siano dati i numeri naturali m, a ed 'a . allora scrivere che  ' mod a a m (si legge a congruo ad 'a modulo m) significa dire che a ed 'a , divisi per m, hanno lo stesso resto. ciò premesso ricordiamo che si chiama gaussiano in base a di m il numero:       , , min t.c. 1 mod xg m a gauss m a x a m   sia data ora la successione: 2 31, , , ,..., ,...na a a a e sia 0 1 21, , ,..., ,...nr r r r la successione dei resti della divisione per m di tali numeri. se 1m è il più grande divisore di m costituito da fattori primi di a (in altre parole i fattori primi di a, comuni ad m, ma elevati all’esponente massimo con cui dividono m), allora: teorema 1. la successione dei resti sopra indicati è in generale periodico-mista con antiperiodo a e periodo g dati rispettivamente da:   1min t.c. 0 mod xa x a m  1 , m g gauss a m        n.b. se a ed m sono primi tra loro allora 1 1, 0,m a g g   ; la sequenza dei resti arriva fino all’indice 1g  ed inizia a ripetersi in corrispondenza dell’indice g, essendo 0g h hr r  . segue, in tal caso, che i resti che formano il periodo sono esattamente: 0 1 2 11, , ,..., gr r r r  osservazione. nel caso in cui sia 10a  (base decimale), allora  r è:  periodica semplice se m è primo con 2 ovvero con 5;  periodico composta se m è divisibile per 2 ovvero per 5. 50 vediamo due esempi significativi: a) sia 3m  ed 10a  . poiché  3;10 1 occorre calcolare    , 3,10g g m a g  ovvero   min t.c. 10 1 mod 3xx  . poiché 110 1 9  segue che è 1g  e che il periodo è costituito dal solo 0 1r  . b) sia 4m  ed 10a  . risulta 1 4m  , per cui dobbiamo calcolare:   min t.c. 10 0 mod 4xa x      1,10 min t.c. 10 1 mod 1xg gauss x   quindi: 2a x  e 1g x  . i resti sono: 1, 2, 0, 0, 0,... nel seguito conviene, per le prime potenze del 10, effettuare le divisioni per 2, 3, 4, 5, 6,...m  per ottenere la tabella che segue dalla quale si deducono vari criteri di divisibilità. accanto al resto r porremo anche r m nel caso in cui questi sia, in valore assoluto, inferiore ad r, essendo:   mod r m r m  in realtà il ricorso al teorema 1 occorre solo se l’analisi procede per molti numeri, specie nei casi di gaussiano di grandi dimensioni. tabella resti mod 2:    1, 0, 0, 0,...,1, 0r   antiperiodo = (1); periodo = (0) resti mod 3:    1,1,1,1,...,1, 1r   periodo = (1) resti mod 4:    1, 2, 0, 0,..., 0r   antiperiodo = (1, 2); periodo = (0) resti mod 5:    1, 0, 0, 0,...,1, 0r   antiperiodo = (1); periodo = (0) resti mod 6:    1, 4, 4, 4,..., 4, 4r   antiperiodo = (1); periodo = (4) = (2) resti mod 7:   1, 3, 2, 6, 4, 5,...r   periodo = (1, 3, 2, 6, 4, 5) ovvero   1, 3, 2, 1, 3, 2,...r     utilizzando anche resti negativi  m r resti mod 8:   1, 2, 4, 0, 0, 0,...r   antiperiodo = (1, 2, 4); periodo = (0) ovvero   1, 3, 2, 1, 3, 2,...r     utilizzando anche resti negativi  m r resti mod 9:    1,1,1,1,...,1, 1r   periodo = (1) resti mod 10:    1, 0, 0, 0,...,1, 0r   antiperiodo = (1); periodo = (0) resti mod 11:   1,10,1,10,1,10,...r   periodo = (1, 10) ovvero   1, 1,1, 1,1, 1,...r     utilizzando anche resti negativi  m r resti mod 12:   1,10, 4, 4, 4, 4,...r   antiperiodo = (1, 10); periodo = (4) ovvero   1, 2, 4, 4, 4, 4,...r   utilizzando anche resti negativi  m r resti mod 13:   1,10, 9,12, 3, 4,...r   periodo = (1, 10, 9, 12, 3, 4) ovvero   1, 3, 4, 1, 3, 4,...r     utilizzando anche resti negativi  m r 51 se ora vogliamo individuare dei criteri affinché un altro numero naturale m sia un divisore di:   1 21 2 1 0 1 2 1 010... 10 10 ... 10 10 n n n n n nn c c c c c c c c c c          dobbiamo, in generale, studiare il comportamento degli elementi della successione 2 11,10,10 ,...,10 ,10 ,...n n qualora ciascuno di essi venga diviso per m, ed in particolare, valutare la successione dei resti della divisione di tali elementi per l’intero m. a riguardo, denotata la successione dei resti e quella delle cifre rispettivamente con:   0 1 2 11, , ,..., , ,...n nr r r r r r  e quella delle cifre con:   0 1 2 1, , ,..., , ,...n nc c c c c c possiamo richiamare il seguente teorema, oramai ben noto in letteratura, la cui dimostrazione è banale. teorema 2 (primo criterio generale di divisibilità). dato il numero naturale: 1 2 1 2 1 0 1 2 1 0... 10 10 ... 10 10 n n n n n nn c c c c c c c c c c          risulta che n è divisibile per m se è solo se lo è l’intero     0 0 1 1 2 2 1 1... n n n nc r c r c r c r c r c r        . n.b. ogni singolo hr può banalmente sostituirsi con hr m . esempio 1. 1 2 1 0...n nn c c c c c è divisibile per 2 (ovvero per 5, ovvero per 10 ) se e solo se lo è: 0 1 1 2 2 1 1... n n n nc c r c r c r c r      con   1, 0, 0, 0,..., 0,...r  cioè se e solo se lo è 0c . esempio 2. 1 2 1 0...n nn c c c c c è divisibile per 3 (ovvero per 9) se e solo se lo è: 0 1 1 2 2 1 1... n n n nc c r c r c r c r      con   1,1,1,1,...,1,...r  , cioè se e solo se lo è la somma delle cifre. esempio 3. 1 2 1 0...n nn c c c c c è divisibile per 4 se e solo se lo è: 0 1 1 2 2 1 1... n n n nc c r c r c r c r      con   1, 2, 0, 0,..., 0,...r  , cioè se e solo è divisibile per 4 il complesso delle ultime due cifre a destra, che lo è se lo è il numero 0 12c c . esempio 4. 1 2 1 0...n nn c c c c c è divisibile per 6 se e solo se lo è: 0 1 1 2 2 1 1... n n n nc c r c r c r c r      con   1, 4, 4, 4,..., 4,...r  , cioè se e solo se lo è la somma tra l’ultima cifra a destra e quattro volte la somma delle rimanenti, ovvero  0 1 24 ... nc c c c    . esempio 5. 1 2 1 0...n nn c c c c c è divisibile per 7 se e solo se lo è: 0 1 1 2 2 1 1... n n n nc c r c r c r c r      con   1, 3, 2, 1, 3, 2,...r     52 esempio 6. 1 2 1 0...n nn c c c c c è divisibile per 8 se e solo se lo è: 0 1 1 2 2 1 1... n n n nc c r c r c r c r      con   1, 2, 4, 0, 0, 0,...r  , cioè se e solo è divisibile per 8 il complesso delle ultime tre cifre a destra, che lo è se lo è il numero 0 1 22 4c c c  . esempio 7. 1 2 1 0...n nn c c c c c è divisibile per 11 se e solo se lo è: 0 1 1 2 2 1 1... n n n nc c r c r c r c r      con   1, 1,1, 1,...r    , cioè se e solo se lo è la somma delle cifre pari meno quella delle cifre dispari. esempio 8. 1 2 1 0...n nn c c c c c è divisibile per 12 se e solo se lo è: 0 1 1 2 2 1 1... n n n nc c r c r c r c r      con   1, 2, 4, 4, 4, 4,...r   esempio 9. 1 2 1 0...n nn c c c c c è divisibile per 13 se e solo se lo è: 0 1 1 2 2 1 1... n n n nc c r c r c r c r      con   1, 3, 4, 1, 3, 4,...r     osserviamo che gli esempi 1, 2, ..., 7 rappresentano proprio i criteri citati in premessa, ivi compreso il criterio di divisibilità per 7; gli esempi 8 e 9, invece, esprimono i complicati criteri di divisibilità per 12 e per 13, derivanti dal primo criterio di divisibilità. 3. – nel presente paragrafo dimostreremo il criterio di divisibilità per 7, già enunciato nel paragrafo 1. in realtà, però, proveremo un po’ di più, precisamente il seguente: teorema (secondo criterio generale di divisibilità). sia dato un numero naturale n rappresentato, rispetto ad una base 0,1,..., 1x x  , da:   1 21 2 1 0 1 2 1 0... ...n nn n n nxn c c c c c c x c x c x c x c          allora n è divisibile per m se è solo se lo è l’intero:     1 1 1 2 1 0... ...r n n r r rx xm x km c c c c c c c c     dove k è un intero arbitrario, quindi, in particolare, il più grande intero tale: 0 r r xx km k m           dimostrazione. sia m un divisore di n. allora: 1 1 2 1 1 2 1 0... ... n n r r n n r rn hm c x c x c x c x c x c x c                11 1 2 1 0... ...r n r n rn n r r xx c x c x c c c c c                 1 1 2 1 0 1... ... ...n n r n n n n rx x xhm km c c c c c c c c km c c c           1 1 1... ... ...r n n r n n r n n rx x xx c c c c c c km c c c          1 1... ...r n n r n n rx xx km c c c c c c m     dunque m è un divisore di m. 53 supponiamo ora che m divida m. allora:     1 1... ...r n n r n n rx xm sm x km c c c c c c     da cui segue:        1 1 1 1... ... ... ...rn n r n n r n n r n n rx x x xm km c c c sm km c c c x c c c c c c n         dunque l’asserto è verificato. osserviamo, in ultima analisi che il precedente teorema vale, non solo per una base qualsiasi, e quindi in particolare per la base 10, ma anche per un indice qualsiasi 2r  e per un k qualsiasi, quindi anche per la suddetta parte intera. 4. – nuovi criteri a) criterio di divisibilità per 7. per 1 9x   , 7m  , 2r  e 14k  , ritroviamo il teorema enunciato nel primo paragrafo, essendo  210 7 2 0k   . b) criterio di divisibilità per 13. per 1 9x   , 13m  , 2r  e 7k  , otteniamo  210 13 9 0k   ; per 8k  , invece, abbiamo  210 13 4 0k    . possiamo, quindi, enunciare il criterio di divisibilità per 13 nel modo seguente: un numero (in rappresentazione decimale) 1 2 1 0...n nn c c c c c è divisibile per 13 se e solo se lo è il numero 1 0 1 29 ...n nc c c c c  [somma tra le ultime due cifre a destra e nove volte la parte residua], ovvero se e solo se lo è il numero 1 0 1 24 ...n nc c c c c  [differenza tra le ultime due cifre a destra e quattro volte la parte residua]. esempi. 1 9x   ; 7m  ; 2r  ; 2 14 13 x k m         (si parte allora dal successivo!!!) 1 9x   ; 8m  ; 2r  ; 2 12 x k m        ;  2 4x km  ; 1 0 1 24 ...n nc c c c c 1 9x   ; 9m  ; 2r  ; 2 11 x k m        ;  2 1x km  ; 1 0 1 2...n nc c c c c 1 9x   ; 10m  ; 2r  ; 10k  ;  2 0x km  ; 1 0c c 1 9x   ; 11m  ; 2r  ; 9k  ;  2 1x km  ; 1 0 1 2...n nc c c c c 1 9x   ; 12m  ; 2r  ; 8k  ;  2 4x km  ; 1 0 1 24 ...n nc c c c c 1 9x   ; 13m  ; 2r  ; 7k  ;  2 9 4x km    ; 1 0 1 2 1 0 1 29 ... 4 ...n n n nc c c c c c c c c c    1 9x   ; 14m  ; 2r  ; 7k  ;  2 2x km  ; 1 0 1 22 ...n nc c c c c 1 9x   ; 15m  ; 2r  ; 6k  ;  2 10 5x km    ; 1 0 1 25 ...n nc c c c c 1 9x   ; 16m  ; 2r  ; 6k  ;  2 4x km  ; 1 0 1 24 ...n nc c c c c 1 9x   ; 17m  ; 2r  ; 5k  ;  2 15 2x km    ; 1 0 1 22 ...n nc c c c c 1 9x   ; 18m  ; 2r  ; 5k  ;  2 10 8x km    ; 1 0 1 28 ...n nc c c c c 54 1 9x   ; 19m  ; 2r  ; 5k  ;  2 5x km  ; 1 0 1 25 ...n nc c c c c 1 9x   ; 20m  ; 2r  ; 5k  ;  2 0x km  ; 1 0c c 1 9x   ; 21m  ; 2r  ; 4k  ;  2 16x km  ; 1 0 1 25 ...n nc c c c c 1 9x   ; 22m  ; 2r  ; 4k  ;  2 12x km  ; 1 0 1 210 ...n nc c c c c 1 9x   ; 23m  ; 2r  ; 4k  ;  2 8x km  ; 1 0 1 28 ...n nc c c c c 1 9x   ; 24m  ; 2r  ; 4k  ;  2 4x km  ; 1 0 1 24 ...n nc c c c c 1 9x   ; 25m  ; 2r  ; 4k  ;  2 0x km  ; 1 0c c 1 9x   ; 26m  ; 2r  ; 3k  ;  2 22x km  ; 1 0 1 24 ...n nc c c c c 1 9x   ; 27m  ; 2r  ; 3k  ;  2 19x km  ; 1 0 1 28 ...n nc c c c c 1 9x   ; 28m  ; 2r  ; 3k  ;  2 16x km  ; 1 0 1 212 ...n nc c c c c 1 9x   ; 29m  ; 2r  ; 3k  ;  2 13x km  ; 1 0 1 213 ...n nc c c c c 1 9x   ; 30m  ; 2r  ; 3k  ;  2 10x km  ; 1 0 1 210 ...n nc c c c c 1 9x   ; 31m  ; 2r  ; 3k  ;  2 7x km  ; 1 0 1 27 ...n nc c c c c 1 9x   ; 32m  ; 2r  ; 3k  ;  2 4x km  ; 1 0 1 24 ...n nc c c c c 1 9x   ; 33m  ; 2r  ; 3k  ;  2 1x km  ; 1 0 1 2...n nc c c c c 1 9x   ; 34m  ; 2r  ; 2k  ;  2 32x km  ; 1 0 1 22 ...n nc c c c c 1 9x   ; 35m  ; 2r  ; 2k  ;  2 30x km  ; 1 0 1 25 ...n nc c c c c 1 9x   ; 36m  ; 2r  ; 2k  ;  2 28x km  ; 1 0 1 28 ...n nc c c c c 1 9x   ; 37m  ; 2r  ; 2k  ;  2 26x km  ; 1 0 1 211 ...n nc c c c c 1 9x   ; 38m  ; 2r  ; 2k  ;  2 24x km  ; 1 0 1 214 ...n nc c c c c 1 9x   ; 39m  ; 2r  ; 2k  ;  2 22x km  ; 1 0 1 217 ...n nc c c c c 1 9x   ; 40m  ; 2r  ; 2k  ;  2 20x km  ; 1 0 1 220 ...n nc c c c c 1 9x   ; 41m  ; 2r  ; 2k  ;  2 18x km  ; 1 0 1 218 ...n nc c c c c 1 9x   ; 42m  ; 2r  ; 2k  ;  2 16x km  ; 1 0 1 216 ...n nc c c c c 1 9x   ; 43m  ; 2r  ; 2k  ;  2 14x km  ; 1 0 1 214 ...n nc c c c c 1 9x   ; 44m  ; 2r  ; 2k  ;  2 12x km  ; 1 0 1 212 ...n nc c c c c 1 9x   ; 45m  ; 2r  ; 2k  ;  2 10x km  ; 1 0 1 210 ...n nc c c c c 1 9x   ; 46m  ; 2r  ; 2k  ;  2 8x km  ; 1 0 1 28 ...n nc c c c c 1 9x   ; 47m  ; 2r  ; 2k  ;  2 6x km  ; 1 0 1 26 ...n nc c c c c 1 9x   ; 48m  ; 2r  ; 2k  ;  2 4x km  ; 1 0 1 24 ...n nc c c c c 1 9x   ; 49m  ; 2r  ; 2k  ;  2 2x km  ; 1 0 1 22 ...n nc c c c c 1 9x   ; 50m  ; 2r  ; 2k  ;  2 0x km  ; 1 0c c 55 1 9x   ; 51m  ; 2r  ; 1k  ;  2 49x km  ; 1 0 1 22 ...n nc c c c c 1 9x   ; 52m  ; 2r  ; 1k  ;  2 48x km  ; 1 0 1 24 ...n nc c c c c 1 9x   ; 53m  ; 2r  ; 1k  ;  2 47x km  ; 1 0 1 26 ...n nc c c c c 1 9x   ; 54m  ; 2r  ; 1k  ;  2 46x km  ; 1 0 1 28 ...n nc c c c c 1 9x   ; 55m  ; 2r  ; 1k  ;  2 45x km  ; 1 0 1 210 ...n nc c c c c 1 9x   ; 56m  ; 2r  ; 1k  ;  2 44x km  ; 1 0 1 212 ...n nc c c c c 1 9x   ; 57m  ; 2r  ; 1k  ;  2 43x km  ; 1 0 1 214 ...n nc c c c c 1 9x   ; 58m  ; 2r  ; 1k  ;  2 42x km  ; 1 0 1 216 ...n nc c c c c 1 9x   ; 59m  ; 2r  ; 1k  ;  2 41x km  ; 1 0 1 218 ...n nc c c c c 1 9x   ; 60m  ; 2r  ; 1k  ;  2 40x km  ; 1 0 1 220 ...n nc c c c c 1 9x   ; 61m  ; 2r  ; 1k  ;  2 39x km  ; 1 0 1 222 ...n nc c c c c 1 9x   ; 62m  ; 2r  ; 1k  ;  2 38x km  ; 1 0 1 224 ...n nc c c c c 1 9x   ; 63m  ; 2r  ; 1k  ;  2 37x km  ; 1 0 1 226 ...n nc c c c c 1 9x   ; 64m  ; 2r  ; 1k  ;  2 36x km  ; 1 0 1 228 ...n nc c c c c 1 9x   ; 65m  ; 2r  ; 1k  ;  2 35x km  ; 1 0 1 230 ...n nc c c c c 1 9x   ; 66m  ; 2r  ; 1k  ;  2 34x km  ; 1 0 1 232 ...n nc c c c c 1 9x   ; 67m  ; 2r  ; 1k  ;  2 33x km  ; 1 0 1 233 ...n nc c c c c 1 9x   ; 68m  ; 2r  ; 1k  ;  2 32x km  ; 1 0 1 232 ...n nc c c c c 1 9x   ; 69m  ; 2r  ; 1k  ;  2 31x km  ; 1 0 1 231 ...n nc c c c c 1 9x   ; 70m  ; 2r  ; 1k  ;  2 30x km  ; 1 0 1 230 ...n nc c c c c 1 9x   ; 71m  ; 2r  ; 1k  ;  2 29x km  ; 1 0 1 229 ...n nc c c c c 1 9x   ; 72m  ; 2r  ; 1k  ;  2 28x km  ; 1 0 1 228 ...n nc c c c c 1 9x   ; 73m  ; 2r  ; 1k  ;  2 27x km  ; 1 0 1 227 ...n nc c c c c 1 9x   ; 74m  ; 2r  ; 1k  ;  2 26x km  ; 1 0 1 226 ...n nc c c c c 1 9x   ; 75m  ; 2r  ; 1k  ;  2 25x km  ; 1 0 1 225 ...n nc c c c c 1 9x   ; 76m  ; 2r  ; 1k  ;  2 24x km  ; 1 0 1 224 ...n nc c c c c 1 9x   ; 77m  ; 2r  ; 1k  ;  2 23x km  ; 1 0 1 223 ...n nc c c c c 1 9x   ; 78m  ; 2r  ; 1k  ;  2 22x km  ; 1 0 1 222 ...n nc c c c c 1 9x   ; 79m  ; 2r  ; 1k  ;  2 21x km  ; 1 0 1 221 ...n nc c c c c 1 9x   ; 80m  ; 2r  ; 1k  ;  2 20x km  ; 1 0 1 220 ...n nc c c c c 1 9x   ; 81m  ; 2r  ; 1k  ;  2 19x km  ; 1 0 1 219 ...n nc c c c c 1 9x   ; 82m  ; 2r  ; 1k  ;  2 18x km  ; 1 0 1 218 ...n nc c c c c 56 1 9x   ; 83m  ; 2r  ; 1k  ;  2 17x km  ; 1 0 1 217 ...n nc c c c c 1 9x   ; 84m  ; 2r  ; 1k  ;  2 16x km  ; 1 0 1 216 ...n nc c c c c 1 9x   ; 85m  ; 2r  ; 1k  ;  2 15x km  ; 1 0 1 215 ...n nc c c c c 1 9x   ; 86m  ; 2r  ; 1k  ;  2 14x km  ; 1 0 1 214 ...n nc c c c c 1 9x   ; 87m  ; 2r  ; 1k  ;  2 13x km  ; 1 0 1 213 ...n nc c c c c 1 9x   ; 88m  ; 2r  ; 1k  ;  2 12x km  ; 1 0 1 212 ...n nc c c c c 1 9x   ; 89m  ; 2r  ; 1k  ;  2 11x km  ; 1 0 1 211 ...n nc c c c c 1 9x   ; 90m  ; 2r  ; 1k  ;  2 10x km  ; 1 0 1 210 ...n nc c c c c 1 9x   ; 91m  ; 2r  ; 1k  ;  2 9x km  ; 1 0 1 29 ...n nc c c c c 1 9x   ; 92m  ; 2r  ; 1k  ;  2 8x km  ; 1 0 1 28 ...n nc c c c c 1 9x   ; 93m  ; 2r  ; 1k  ;  2 7x km  ; 1 0 1 27 ...n nc c c c c 1 9x   ; 94m  ; 2r  ; 1k  ;  2 6x km  ; 1 0 1 26 ...n nc c c c c 1 9x   ; 95m  ; 2r  ; 1k  ;  2 5x km  ; 1 0 1 25 ...n nc c c c c 1 9x   ; 96m  ; 2r  ; 1k  ;  2 4x km  ; 1 0 1 24 ...n nc c c c c 1 9x   ; 97m  ; 2r  ; 1k  ;  2 3x km  ; 1 0 1 23 ...n nc c c c c 1 9x   ; 98m  ; 2r  ; 1k  ;  2 2x km  ; 1 0 1 22 ...n nc c c c c 1 9x   ; 99m  ; 2r  ; 1k  ;  2 1x km  ; 1 0 1 2...n nc c c c c si noti che se un numero 0...nn c c non è divisibile per m allora il resto della divisione di n per m è dato dal risultato dell’ultima divisione effettuata. esempio. il numero 1123 non è divisibile per 7. operando con il criterio di divisibilità per 7, considerando che 2 11 23 45   e che 45 6 7 3   , risulta 1123 7 3   con 160  . 5. – divisibilità per 3 notiamo che un numero è divisibile per 3 se, comunque ripartito, il doppio della somma delle cifre della ripartizione di sinistra meno la somma delle cifre della ripartizione di destra è divisibile per 3. dimostrazione. sappiamo che un numero 0...nn c c è divisibile per 3 se e solo se lo è la somma delle sue cifre (decimali), cioè se e solo se lo è il numero 2 1 0...nm c c c c     . consideriamo l’identità:      2 1 0 1 1 02 ... 2 ... 2 ...n n i ic c c c c c c c c                 1 1 0 1 1 02 ... ... 3 ...n i i ic c c c c c c c            che chiaramente prova l’asserto. 57 esempio. il numero 845322 (con 8 4 5 3 2 2 24      ; 2 4 6  !) è divisibile per 3: ripartito, infatti, in 84 e 5322, risulta:    2 8 4 5 3 2 2 24 12 12        , che è divisibile per tre. bibliografia f. eugeni – m.a. garzia – d. tondini, la divisibilità nell’anello degli interi relativi, in: www.apav.it (vedi: comunicazione, scienze e società, voce numeri). a. chiellini – r. giannarelli, l’esame orale di matematica nei concorsi a cattedre di scuole, libreria eredi virgilio veschi, roma, 1962. ratio mathematica volume 47, 2023 equitable eccentric domination in graphs riyaz ur rehman a* a mohamed ismayil† abstract in this paper, we define equitable eccentric domination in graphs. an eccentric dominating set s ⊆ v (g) of a graph g(v, e) is called an equitable eccentric dominating set if for every v ∈ v − s there exist at least one vertex u ∈ v such that |d(v) − d(u)| ≤ 1 where vu ∈ e(g). we find equitable eccentric domination number γeqed(g) for most popular known graphs. theorems related to γeqed(g) have been stated and proved. keywords: eccentricity, equitable domination number, equitable eccentric domination number. 2020 ams subject classifications: 05c69. 1 *research scholar, pg & research department of mathematics, jamal mohamed college (affiliated to bharathidasan university), tiruchirappalli, india; mail id: fouzanriyaz@gmail.com. †associate professor, pg & research department of mathematics, jamal mohamed college (affiliated to bharathidasan university), tiruchirappalli, india; mail id: amismayil1973@yahoo.co.in. 1received on september 15, 2022. accepted on december 15, 2022. published online on january 10, 2023. doi: 10.23755/rm.v41i0.802. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 87 riyaz ur rehman a and a mohamed ismayil 1 introduction a graph is a representation of a pair of sets (v, e), where v is the set of vertices and e is the set of edges which are connecting the pair of vertices. graph theory has its application in many fields such as computation, social and natural science etc. any problems of mathematics, science and engineering can be represented in the form of a graph. the concept of graph theory was first introduced by leonard euler in the year 1736. he created the first graph as a solution to solve the problem of seven bridges of konigsberge built across the pregel river of prussia. graph theory has experienced tremendous growth, the main reason for this phenomena is applicability of graph theory in different disciplines. graph theory becomes interesting because graphs can be used to model situations that occur in real world problems. these problems can be studied with the aid of graphs. the concept of domination in graphs was studied by ore and berge. ore[11] introduced domination in graphs in his famous book ’theory of graphs’ in 1962. cockyane and hedetniemi[3] also contributed several results pertaining to domination. they unfolded different aspects, by swaying all available results bringing to light new ideas and emphasizing its applicable potential in a variety of scientific ideas in their paper ’towards a theory of domination in graphs’. t.w.haynes, s.hedetniemi and p.slater[6] have breifly discussed on various domination parameters in the book fundamentals of domination in graphs. t.n. janakiraman et al[9] introduced the concept of eccentric domination in graphs in 2010. kuppusamy markandan dharmalingam[4] introduced equitable graph of a graph. e. sampathkumar et al[1] introduced degree equitable sets in a graph. v swaminathan and k.m. dharmalingam[12] introduced degree equitable domination in graphs. basavanagoud et al[2] introduced equitable dominating graph. the concept of eccentricity by t.n. janakiraman et al has inspired researchers which has led to many invariants of eccentric dominations in graphs. some of the extended eccentric dominations are accurate eccentric domination[7] and equal eccentric domination[8]. the concept of geodesic distance is very important. the existing eccentric domination only highlighted the idea based on an eccentric vertex and its domination. the proposed equitable eccentric domination was mainly necessary because it highlights the properties of a vertex in a graph, it considers the connectivity between the vertices where the difference between their vertex degrees is less than or equal to one. equitable domination when incorporated with eccentric domination yeilds equitable eccentric domination which concentrates on the vertex degree, geodesic distance, eccentricity, eccentric vertex and domination. in this paper, we introduce equitable eccentric domination in graphs. we 88 equitable eccentric domination find equitable eccentric dominating set, equitable eccentric domination number γeqed(g), upper equitable eccentric dominating set and upper equitable eccentric domination number γeqed(g) of different standard graphs. for undefined graph terminologies refer the book ’graph theory’ by frank harary[5]. 2 preliminaries definition 2.1 (11). let g be a graph with the vertex set v . a subset d of v is a dominating set for g when every vertex not in d is the endpoint of some edge from a vertex in d. definition 2.2 ([10]). let γ(g) (called the domination number) and γ(g) (called the upper domination number) be the minimum cardinality and the maximum cardinality of a minimal dominating set of g, respectively. definition 2.3 ([6]). the degree deg(v) of v is the number of edges incident with v. definition 2.4 ([9]). the eccentricity e(v) of v is the distance to a vertex farthest from v. thus, e(v) = max{d(u, v) : u ∈ v }. for a vertex v, each vertex at a distance e(v) from v is an eccentric vertex. eccentric set of a vertex v is defined as e(v) = {u ∈ v (g)/d(u, v) = e(v)}. definition 2.5 ([9]). the radius r(g) is the minimum eccentricity of the vertices, whereas the diameter diam(g) is the maximum eccentricity. definition 2.6 ([9]). v is a central vertex if e(v) = r(g). the center c(g) is the set of all central vertices. v is a peripheral vertex if e(v) = diam(g). the periphery p(g) is the set of all peripheral vertices. definition 2.7 ([9]). a set d ⊆ v (g) is an eccentric dominating set if d is a dominating set of g and for every v ∈ v − d, there exists at least one eccentric point of v in d. if d is an eccentric dominating set, then every superset d′ ⊇ d is also an eccentric dominating set. but d′′ ⊆ d is not necessarily an eccentric dominating set. an eccentric dominating set d is a minimal eccentric dominating set if no proper subset d′′ ⊆ d is an eccentric dominating set. definition 2.8 ([9]). the eccentric domination number γed(g) of a graph g equals the minimum cardinality of an eccentric dominating set. that is, γed(g) = min|d|, where the minimum is taken over d in d, where d is the set of all minimal eccentric dominating sets of g. definition 2.9 ([4]). a subset d of v is called an equitable dominating set if for every v ∈ v − d there exists a vertex u ∈ d such that uv ∈ e(g) and |deg(u) − deg(v)| ≤ 1. the minimum cardinality of such a dominating set is denoted by γe and is called the equitable domination number of g. 89 riyaz ur rehman a and a mohamed ismayil 3 equitable eccentric domination in graphs in this section we introduce equitable eccentric domination, theorems related to equitable eccentric domination number of family of graphs are stated and proved. definition 3.1. an eccentric dominating set s ⊆ v (g) is called an equitable eccentric dominating set(eqed-set) if for every v ∈ v − s there exist at least one vertex u ∈ s such that vu ∈ e(g) and |d(v) − d(u)| ≤ 1. definition 3.2. an equitable eccentric dominating set s is called a minimal equitable eccentric dominating set if no proper subset of s is equitable eccentric dominating set. definition 3.3. the equitable eccentric domination number γeqed(g) of a graph g is the minimum cardinality among the minimal equitable eccentric dominating sets of g. definition 3.4. the upper equitable eccentric domination number γeqed(g) of a graph g is the maximum cardinality among the minimal equitable eccentric dominating sets of g. example 3.1. v4 v5 v2 v3 v6 v1 figure 2.1: graph g consider the graph g consists of 6 vertices given in figure 2.1. here the dominating set is s = {v1, v4} but not eccentric dominating set since e(v3) = {v2, v6} not in s. the eccentric dominating set is s = {v1, v6} but not equitable eccentric dominating set since |d(v4) − d(v6)| = 2. the equitable eccentric dominating set is s = {v1, v2, v6}. remark 3.1. for any path pn where n ≥ 3, 1. every minimum eqed-set contains the pendant vertices. 2. if d1, d2, d3 are minimum eqed-sets of paths pn−1, pn, pn+1 consecutively where n = 3k and k > 1. then |d1| = |d2| = |d3|. therefore for k = 2, γeqed(p5) = γeqed(p6) = γeqed(p7) = 3. theorem 3.1. for complete graph kn, γeqed(kn) = 1, ∀ n ≥ 2. 90 equitable eccentric domination proof. in a complete graph kn all the vertices are eccentric vertices to each other. if v ∈ v (kn) then the eccentric vertex e(v) = v (kn) − {v} and every singleton set forms a dominating set. for every vertex v ∈ d ∃ a vertex u ∈ v (kn) − d ∋ |deg(u) − deg(v)| ≤ 1 where uv ∈ e(kn). therefore every single vertex of kn is an eqed-set. hence γeqed(kn) = 1. theorem 3.2. for path graph pn where n > 1, γeqed(pn) = { 1, for n = 2 ⌊n+1 3 ⌋ + 1, ∀ n ≥ 3 proof. case(i): for a path p2, v (p2) = {v1, v2}. both the vertices are eccentric vertices to each other. therefore d = {v1} or {v2} and |deg(v2)−deg(v1)| = 0, where v1v2 ∈ e(p2). hence γeqed(p2) = 1. case(ii): for a path pn where n ≥ 3. the pendant or end vertices of the path form the eccentric vertices ie, if v (pn) = {v1, v2, v3, . . . vn}, e(v1) = {vn} and e(vn) = {v1}. e(vi) = {v1} or {vn} for any vi ∈ v (pn) where n is even. if n is odd then e(vi) = {v1} or {vn}. for pn where ′n′ is odd, the central vertex vi has two eccentric vertices ie, e(vi) = {v1, vn}. degree of end vertices is 1 and degree of all the intermediate vertices is 2. the eqed-set contains both the pendant vertices. both v1 and vn being pendant vertices dominate the vertices adjacent to them and the minimum dominating set among the intermediate vertices along with two pendant vertices forms an eqed-set. since |deg(u)−deg(v)| ≤ 1 where uv ∈ e(pn) for all u ∈ d and v ∈ v (pn) − d and ∃ an eccentric vertex u ∈ d for every v ∈ v (pn) − d. for pn where n = 3k and k > 2, number of vertices of p3k−1, p3k, p3k+1 are same. every minimum equitable eccentric domination set of d contains ⌊n+1 3 ⌋ + 1 number of vertices. theorem 3.3. for star graph sn, γeqed(sn) = { 2, if n = 3 0, if n ̸= 3 proof. case(i): if n = 3, then the star graph s3 is isometric to p3. from the theorem-3.2 γeqed(p3) = γeqed(s3) = 2. case(ii): if n ̸= 3 then sn is of the form s4, s5, s6, . . . for any graph sn where n ̸= 3, there can be many dominating sets and eccentric dominating sets but we cannot find a eqed-set because of the central vertex vi of the star graph has degree ≥ 3. the degree of every pendant vertex u of a star graph is 1, deg(u) = 1, u ∈ v (sn) − {vi}. the degree of central vertex vi of a star graph is given by deg(vi) = n − 1. since, central vertex vi ∈ v (sn) then either vi ∈ d or vi ∈ v (sn) − d. therefore |deg(vi) − deg(u)| > 1 always which doesnot satisfy the condition to be a eqed-set. hence γeqed(sn) = 0 where n ̸= 3. 91 riyaz ur rehman a and a mohamed ismayil theorem 3.4. for cycle graph cn where n ≥ 3, γeqed(cn) =   n 2 , if n is even ∀ n ≥ 4 ⌈n 3 ⌉, if n is odd & n = 3k ∀ k = 1, 3, 5, 7, . . . ⌈n 3 ⌉ + 1, otherwise proof. case(i): if ′n′ is even and n ≥ 4. let the cycles cn be of the form c4, c6, c8, c10, . . . c2n. in an even cycle if u ∈ v (cn) the eccentric vertex of u, e(u) = {v} is always placed at a distance of n 2 edges from it and every vertex has a unique eccentric vertex to form the first eccentric dominating set. the set d must contain n 2 vertices in such a way that for every v ∈ d then e(v) /∈ d or for some u ∈ v (cn) − d, e(u) /∈ v (cn) − d. then if the vertex u and e(u) ∈ d then we cannot construct a eccentric dominating set. further if we reduce the cardinality of d to less than n 2 we will have u and e(u) in v − d. therefore d must contain n 2 vertices with all the unique eccentric vertices in v − d. then for any u ∈ v (cn) − d ∃ a vertex v ∈ d such that |deg(u) − deg(v)| ≤ 1 where uv ∈ e(cn) for every vertex vi ∈ cn, deg(vi) = 2. therefore |deg(u) − deg(v)| = |2 − 2| = 0. therefore γeqed(cn) = n2 . case(ii): now we have the odd cycles of the form c3, c9, c15, c21, . . . c3k. every vertex u ∈ v (cn) has two eccentric vertices vi, vj such that e(u) = {vi, vj}. the eccentric vertices vi, vj will always be adjacent i.e., vi, vj ∈ e(cn). vi, vj are placed at a distance of n−1 2 edges from u. since every vertex u can dominate its adjacent vertices v, w. n 3 set of vertices form a dominating set of a cycle. the dominating set d = {vi, vj, vn} forms the eqed-set such that no eccentric vertices of vi ∈ d are in d. then ∀ v ∈ v (cn) − d ∃ a vertex u ∈ d ∋′ |deg(u) − deg(v)| = |2 − 2| = 0. therefore γeqed(cn) = ⌈n3 ⌉. case(iii): if n = 3k+1 where k is even. the cycles are of the form c7,c13,c19,. . . , c3k+1 and if n = 3k+1 where k is odd, the cycles are of the form c5,c11,c17,. . . , c3k+2. totally we have c5, c7, c11, c13, c17, c19, . . . c3k+1, c3k+2. similar to case(ii) every vertex vi ∈ v (cn) has two eccentric vertices vl, vm, e(vi) = {vl, vm} such that vl and vm are adjacent i.e., vl, vm ∈ e(cn). eccentric vertex vl and vm of vi are placed at a distance of n−1 2 from vi. if n = 3k we get 3, 9, 15, 21, . . . which are the multiples of 3 we get a whole number which forms the cardinality of a eqed-set as proved in case(ii). but when n = 3k + 1 or n = 3k +2 then n = 5, 7, 11, 13, 17, 19, . . . 3k +1, 3k +2 which are not multiples of 3 we get a fraction value and also we are left out with a vertex which is to be dominated. therefore the cardinality of the eqed-set of a cycles of the form c3k+1, c3k+2 increases by 1. hence γeqed(cn) = ⌈n3 ⌉ + 1. theorem 3.5. every eqed-set in a wheel graph wn, n ≥ 6 contains the central vertex. 92 equitable eccentric domination proof. let v1 be the central vertex of the wheel graph wn, n ≥ 6 then deg(v1) = n − 1 = ∆(wn). the degree of any non-central vertex u ∈ v (wn) is deg(u) = 3 = δ(wn). suppose the central vertex v1 ∈ v (wn) − d, u ∈ d and d is an minimal eccentric dominating set we need to check for the condition of equitable domination then for v1 ∈ v (wn) − d and u ∈ d, we have uv1 ∈ e(wn) |deg(v1) − deg(u)| = |∆(g) − δ(g)| |deg(v1) − deg(u)| = |(n − 1) − 3| |deg(v1) − deg(u)| = |n − 4| where n ≥ 6 |deg(v1) − deg(u)| > 1. which is a contradiction. therefore the central vertex v1 must belong to d, if the set d is a equitable eccentric dominating set of wn. theorem 3.6. let wn be a wheel graph where n ≥ 5 then eqed-set contains more than one vertex. proof. in any wheel graph wn where n ≥ 5. if the set d ⊆ v (wn) contains the central vertex v1 then d forms a dominating set as deg(v1) = n − 1 = ∆(g). but the eccentric vertices of a central vertex v1 is given by e(v1) = v − {v1} and the eccentric vertex of any non-central vertex u is given by e(u) = v − n[u]. therefore there is no eccentric dominating or equitable eccentric dominating set of cardinality 1 for wn where n ≥ 5. theorem 3.7. for wheel graph wn, where n ≥ 4 we have γeqed(wn) =   1, if n = 4 4, if n = 6 ⌊n 2 ⌋, if n is odd and n ≥ 5 ⌊n+1 3 ⌋ + 1, ∀ n ≥ 8 and n is even proof. case(i): if n = 4, w4 is isometric to k4, then by theorem-3.1 γeqed(w4) = γeqed(k4) = 1. case(ii): if n = 6, in a wheel graph w6, there are no eccentric dominating sets of cardinality 1 or 2. therefore we do not get an eqed-set of cardinality 1 or 2. there are sets of cardinality 3 which are eccentric dominating sets. but they do not form an eqed-set as the central vertex should not be present in v −d. since the degree of central vertex vi is deg(vi) = n − 1 = 5 and degree of any other non-central vertex is deg(vj) = 3. therefore |deg(vi) − deg(vj)| = 2 > 1 and in other cases if vi /∈ v − d then we find a combination of vertices of 3 cardinality which are eccentric dominating set but they dont form an ined-set since for some vertex v ∈ v − d there is no vertex u ∈ d such that u, v /∈ e(w6). but we find a eqed-set with cardinality 4 as we have the central vertex in d. then |deg(vi) − deg(vj)| ≤ 1, (vi, vj) ∈ e(w6) where vi ∈ d and vj ∈ v (w6) − d. 93 riyaz ur rehman a and a mohamed ismayil therefore γeqed(w6) = 4. case(iii): if n is odd and n ≥ 5 we have the wheel graph of order w5, w7, w9, w11, . . . if v ∈ v (wn) then |e(v)| = n − 4. there will always be n − 4 vertices which form the eccentric vertex e(v) for every vertex v. and for any wheel graph where ′n′ is odd. the set d ⊆ v (wn) forms an eccentric dominating sets only when |d| = ⌊n 2 ⌋. then for every v ∈ v (wn) − d there exists a vertex u ∈ d such that |deg(u) − deg(v)| ≤ 1 and (u, v) ∈ e(wn). therefore γeqed(wn) = ⌊n2 ⌋. case(iv): the wheel graph wn where n is even and n ≥ 8 has n − 4 eccentric vertices. we have wheel graphs w8, w10, w12, . . . for every vertex v ∈ v (wn), |e(v)| = n − 4. from theorem-3.6,3.5, γeqed(wn) ̸= 1 and the central vertex vi ∈ d then d contains other vertices of wn where cardinality of d is of the form ⌊n+1 3 ⌋ + 1. for every v ∈ v − d there exists a vertex u ∈ d such that e(v) lies in d and |deg(u) − deg(v)| ≤ 1 such that there exists an edge between u and v. therefore γeqed(wn) = ⌊n+13 ⌋ + 1. theorem 3.8. an eqed-set d is a minimal eqed-set if one of the following conditions holds, 1. for every vertex u in v − d there does not exists v in d such that e(u) = {v} ie, u has no eccentric vertex in d. 2. there exists some u ∈ v − d such that n(u) ⋂ d = {v}, e(u) ⋂ d = {v} and |d(u) − d(v)| ≤ 1 where uv ∈ e(g). proof. suppose d is a minimal eqed-set of g. then for every vertex v in d, d − {v} is not an eqed-set. thus there exists some vertex u in v − d ⋃ {v} which is not dominated by any vertex in d −{v} or there exists u ∈ v −d ⋃ {v} such that u does not have an eccentric vertex in d − {v} ie, e(u) ̸= d − {v} or |d(u) − d(v)| ≰ 1 or uv /∈ e(g). ∴ the concept of equitable condition does not hold. case(i): if v = u then u does not have an eccentric vertex in d ie, e(u) ̸= d. case(ii): if v ̸= u, (a) if u ∈ v −d and u is not dominated by d−{v}, but dominated by d then u is adjacent to only v in d ie,n(u) ⋂ d = {v}. (b) if u ∈ v −d and u does not have an eccentric vertex in d−{v} but u has an eccentric vertex in d. thus v is the only eccentric vertex of u in d ie, e(u) ⋂ d = {v}. (c) if u ∈ v − d and |d(u) − d(x)| ≰ 1 or ux /∈ e(g) where x ∈ d − {v} but |d(u) − d(v)| ≤ 1 and uv ∈ e(g). conversely, suppose d is an eqed-set and for each v ∈ d, one of the two conditions holds. now we show that d is a minimal eqed-set. suppose d is not an minimal eqed-set ie, there exists a vertex v ∈ d such that d − {v} is an eqed-set. hence v is adjacent to at least one vertex x in d −{v}, v has an eccentric vertex in d −{v} ie, e(v) ∈ d −{v} and |d(u) − d(x)| ≤ 1 where ux ∈ e(g). ∴ equitable condition holds and eqed-set exists. also if d − {v} is an eqed-set, then every vertex u in v − d is adjacent to at least one vertex x in d − {v}, u has an eccentric vertex in d − {v} ie,e(u) ∈ d−{v} and |d(u)−d(x)| ≤ 1 and ux ∈ e(g). therefore condition-(2) 94 equitable eccentric domination does not hold. hence neither condition-(1) nor (2) holds, which is a contradiction to our assumption. hence for each v ∈ d one of the 2 conditions holds. the equitable eccentric dominating set, γeqed(g), upper equitable eccentric dominating set and γeqed(g) of standard graphs are tabulated. graph figure d minimum eqed set. |d| = γeqed(g) γeqed(g) s upper eqed set. |s| = γeqed(g) γeqed(g) diamond graph v1 v4 v2 v3 {v1, v2}, {v1, v3}, {v2, v3}, {v2, v4}, {v3, v4}. 2 {v1, v2}, {v1, v3}, {v2, v3}, {v2, v4}, {v3, v4}. 2 tetrahedral graph v2 v1 v3 v4 {v1}, {v2}, {v3}, {v4}. 1 {v1}, {v2}, {v3}, {v4}. 1 claw graph v2 v3 v1 v4 does not exist 0 does not exist 0 paw graph v2 v3 v1 v4 {v1, v3}, {v2, v3}, {v3, v4}. 2 {v1, v3}, {v2, v3}, {v3, v4}. 2 bull graph v3 v4 v5 v2v1 {v1, v2, v3}, {v1, v2, v4}, {v1, v2, v5}. 3 {v1, v2, v3}, {v1, v2, v4}, {v1, v2, v5}. 3 butterfly graph v3 v2 v5 v1 v4 {v1, v2, v3}, {v1, v3, v5}, {v2, v3, v4}, {v3, v4, v5}. 3 {v1, v2, v3}, {v1, v3, v5}, {v2, v3, v4}, {v3, v4, v5}. 3 banner graph v3 v4 v1 v2 v5 {v1, v2, v5}, {v1, v3, v5}, {v2, v3, v5}, {v2, v4, v5}, {v3, v4, v5}. 3 {v1, v2, v5}, {v1, v3, v5}, {v2, v3, v5}, {v2, v4, v5}, {v3, v4, v5}. 3 95 riyaz ur rehman a and a mohamed ismayil graph figure d minimum eqed set. |d| = γeqed(g) γeqed(g) s upper eqed set. |s| = γeqed(g) γeqed(g) fork graph v2 v3 v1 v4 v5 {v1, v2, v3, v4}, {v1, v2, v4, v5}, {v1, v3, v4, v5}. 4 {v1, v2, v3, v4}, {v1, v2, v4, v5}, {v1, v3, v4, v5}. 4 (3,2)-tadpole graph v2 v3 v4 v1 v5 {v1, v4}, {v4, v5}. 2 {v1, v2, v3, v5}. 4 kite graph v3 v4 v1 v5 v2 {v1, v2, v4}, {v1, v3, v4}, {v2, v3, v4}, {v2, v4, v5}, {v3, v4, v5}. 3 {v1, v2, v4}, {v1, v3, v4}, {v2, v3, v4}, {v2, v4, v5}, {v3, v4, v5}. 3 (4,1)-lollipop graph v3 v4 v1 v5 v2 {v1, v4}, {v2, v4}, {v3, v4}, {v4, v5}. 2 {v1, v4}, {v2, v4}, {v3, v4}, {v4, v5}. 2 house graph v2 v3 v1 v4 v5 {v2, v4}, {v3, v5}. 2 {v1, v2, v3}, {v1, v4, v5}. 3 house x graph v2 v3 v1 v4 v5 {v1, v2}, {v1, v3}, {v1, v4}, {v1, v5}. 2 {v1, v2}, {v1, v3}, {v1, v4}, {v1, v5}. 2 gem graph v1 v2 v5 v3 v4 {v1, v2}. 2 {v1, v3, v4}, {v2, v3, v4}, {v3, v4, v5}. 3 96 equitable eccentric domination graph figure d minimum eqed set. |d| = γeqed(g) γeqed(g) s upper eqed set. |s| = γeqed(g) γeqed(g) dart graph v3 v4 v1 v5 v2 {v2, v4}. 2 {v1, v2, v3, v5}. 4 cricket graph v4 v5v3 v1 v2 {v1, v3, v4, v5}, {v2, v3, v4, v5}. 4 {v1, v3, v4, v5}, {v2, v3, v4, v5}. 4 pentatope graph v1 v4 v5 v2 v3 {v1}, {v2}, {v3}, {v4}, {v5}. 1 {v1}, {v2}, {v3}, {v4}, {v5}. 1 johnson solid skeleton-12 graph v2 v1 v3 v4 v5 {v1, v2}, {v1, v3}, {v1, v4}, {v1, v5}, {v2, v3}, {v3, v4}, {v3, v5}. 2 {v1, v2}, {v1, v3}, {v1, v4}, {v1, v5}, {v2, v3}, {v3, v4}, {v3, v5}. 2 cross graph v3 v1 v2 v4 v5 v6 {v1, v2, v3, v4, v5}, {v1, v2, v3, v4, v6}. 5 {v1, v2, v3, v4, v5}, {v1, v2, v3, v4, v6}. 5 net graph v5 v6 v3 v4 v1 v2 {v1, v2, v3, v6}, {v1, v2, v4, v6}, {v1, v2, v5, v6}. 4 {v1, v2, v3, v6}, {v1, v2, v4, v6}, {v1, v2, v5, v6}. 4 fish graph v4 v2 v5 v1 v6 v3 {v2, v3, v4}, {v3, v4, v5}. 3 {v1, v2, v4, v5, v6}. 5 97 riyaz ur rehman a and a mohamed ismayil graph figure d minimum eqed set. |d| = γeqed(g) γeqed(g) s upper eqed set. |s| = γeqed(g) γeqed(g) a graph v3 v4 v1 v5 v2 v6 {v1, v2, v5, v6}, {v1, v3, v5, v6}, {v1, v4, v5, v6}, {v2, v3, v5, v6}, {v2, v4, v5, v6}, {v3, v4, v5, v6}. 4 {v1, v2, v5, v6}, {v1, v3, v5, v6}, {v1, v4, v5, v6}, {v2, v3, v5, v6}, {v2, v4, v5, v6}, {v3, v4, v5, v6}. 4 r graph v3 v4 v1 v5 v2 v6 {v2, v3, v5, v6}. 4 {v1, v3, v4, v5, v6}. 5 4-polynomial graph v2 v3v1 v5v4 v6 {v1, v2, v3}, {v1, v3, v4}, {v2, v3, v4}, {v3, v4, v5}, {v3, v4, v6}, {v4, v5, v6}. 3 {v1, v2, v3}, {v1, v3, v4}, {v2, v3, v4}, {v3, v4, v5}, {v3, v4, v6}, {v4, v5, v6}. 3 (2,3)-king graph v2 v3v1 v5v4 v6 {v1, v2, v3}, {v1, v2, v6}, {v1, v3, v5}, {v1, v5, v6}, {v2, v3, v4}, {v2, v4, v6}, {v3, v4, v5}, {v4, v5, v6}. 3 {v1, v2, v3}, {v1, v2, v6}, {v1, v3, v5}, {v1, v5, v6}, {v2, v3, v4}, {v2, v4, v6}, {v3, v4, v5}, {v4, v5, v6}. 3 antenna graph v2 v1 v3 v4 v5 v6 {v1, v2, v5}, {v1, v2, v6}, {v1, v3, v5}, {v1, v3, v6}, {v1, v4, v5}, {v1, v4, v6}. 3 {v1, v2, v3, v4}. 4 3-prism graph v2 v3 v4 v1 v5 v6 {v1, v2}, {v3, v5}, {v4, v6}. 2 {v1, v5, v6}, {v2, v3, v4}. 3 octahedral graph v4 v3v2 v1 v5 v6 {v1, v2, v3}, {v1, v2, v5}, {v1, v3, v6}, {v1, v5, v6}, {v2, v3, v4}, {v2, v4, v5}, {v3, v4, v6}, {v4, v5, v6}. 3 {v1, v2, v3}, {v1, v2, v5}, {v1, v3, v6}, {v1, v5, v6}, {v2, v3, v4}, {v2, v4, v5}, {v3, v4, v6}, {v4, v5, v6}. 3 98 equitable eccentric domination 4 conclusions inspired by eccentric dominating set and equitable dominating set we introduce the equitable eccentric dominating set. we find minimum equitable eccentric dominating set, minimum equitable eccentric domination number γeqed(g), upper equitable eccentric dominating set and upper equitable eccentric domination number γeqed(g) of different standard graphs. we have discussed the properties and proved theorems related to equitable eccentric dominating set of family of graphs. acknowledgements the authors express their gratitude to the managementratio mathematica for their constant support towards the successful completion of this work. we wish to thank the anonymous reviewers for the valuable suggestions and comments. references [1] a. anitha, s. arumugam, and e. sampathkumar. degree equitable sets in a graph. international j. math. combin, 3:32–47, 2009. 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[8] a. m. ismayil and a. r. u. rehman. equal eccentric domination in graphs. malaya journal of matematik (mjm), 8(1, 2020):159–162, 2020. 99 riyaz ur rehman a and a mohamed ismayil [9] t. janakiraman, m. bhanumathi, and s. muthammai. eccentric domination in graphs. international journal of engineering science, computing and bio-technology, 1(2):1–16, 2010. [10] m.-j. jou. upper domination number and domination number in a tree. ars combinatoria, 94:245–250, 2010. [11] o. ore. theory of graphs. providence, american mathematical society, 1962. [12] v. swaminathan and k. m. dharmalingam. degree equitable domination on graphs. kragujevac journal of mathematics, 35(35):191–197, 2011. 100 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica issn (print): 1592-7415 issue n. 30 (2016) 23-33 (online): 2282-8214 23 verification of the mathematically computed impact of the relief gradient to vehicle speed martin bureš, filip dohnal department of military geography and meteorology university of defence in brno, czech republic martin.bures@unob.cz abstract terrain traficability is one of the key activities of military planning, firefighting and emergency interventions. terrain traficability is affected by many factors and terrain slope is one of them. deceleration ratio that represents the influence of slope inclination is dependent on a technical attributes of vehicle. the results of field terrain tests suggest that deceleration ratio established via calculation does not have to correspond with practical experience. keywords: cross-country movement, deceleration ratio, terrain slope doi: 10.23755/rm.v30i1.6 1 introduction the basis for the planning of vehicle movement in terrain is the knowledge of natural conditions, which influence the movement itself. with respect to the driving characteristics, which are characterized by a whole range of technical parameters, there is a modelling process of the impact of natural conditions on the movement in the field [1], [2], [3]. landscape represents very complicated system and therefore, during the modelling of the natural conditions impact on the movement, the landscape elements are evaluated separately. one of these elements is terrain relief, whose slope characteristics have a direct influence on the speed of a moving vehicle [4]. compared to the other terrain characteristics, martin bureš and filip dohnal 24 the relief slope can be successfully analyzed with the gis tools (considering the accuracy and quality of spatial data) [5]. 2 theory of cross country movement the vehicle mobility in the field is based on the mutual effect of the tree basic components, which influence: operation in terrain (maneuver), used technique and geographical conditions. the mutual influence of these components, related to the military operations, shows fig. 1. fig. 1: the influence of geographic conditions to combat action and combat equipment [6]. modern methods of conducting military operations are supported by a range of operational analysis. one of the very important area represent the terrain analysis, which are these days conducted especially with the usage of digital geographic data and gis tools. in this area we classify also terrain trafficability analysis, which results may not be used only for military purposes, but also for the fulfilment of the tasks of irs or emergency management authorities. the terrain trafficability can be defined as a mobile ability of units, which is influenced especially by geographical factors of the territory and technical parameters of vehicles, or (according to [6]) as the level of technical competence of individual vehicles to move in terrain and overcame different geographic features and phenomena. evaluation of geographic factors which influence the terrain trafficability mainly concentrates on the impact of relief gradient, microrelief forms, soil condition, vegetation, waters, climate and weather condition, settlements and communications. these factors are later divided to other components [6]. the evaluation is also influenced by technical data of used vehicles and driver’s capability. but it is very difficult to mathematically evaluate driver’s influence. verification of the mathematically computed impact of the relief gradient to vehicle speed 25 all these factors are closely related and influence each other. their combined influence on vehicle cause deceleration or even stopping. the real speed of the vehicle can be expressed by this formula [2]: (1) where vj means vehicle speed at j-section of vehicle path [km·h -1 ], vmax maximum vehicle speed at communications [km·h -1 ], ci i-coefficient of deceleration due to factor fi computed for j-section with invariable values ci, n number of geographic factors effecting at given section of terrain and k number of sections on vehicle path. the terrain trafficability is very complex and it is not possible to identify effect of all the terrain factors, therefore it is necessary to proceed systematically. first comes identifying basic terrain factors influence, such as relief gradient, then comes their combined influence and last comes less important components. the impact of relief gradient to cross-country movement a relief gradient represents one of the most fundamental factor implicating cross-country movement. the calculation of total resulting coefficient of vehicle deceleration by relief and microrelief impact is given for determinate by relation as follows [7]: (2) where c11 is deceleration coefficient by impact of gradient factor and c12 deceleration coefficient by impact of microrelief factor. 3 calculation of coefficient of vehicle deceleration of impact of relief gradient (c11) it is possible to express a relief gradient by various terrain models such as: raster model, tin and others. the coefficient of deceleration of gradient factor c11 is determinable by three methods as follows [6]: 1) according to dma method (defence mapping agency); 2) on the basic of tractive charts of particular vehicles; 3) by the terrain operation tests. determination of c11 according to dma method: martin bureš and filip dohnal 26 according to the formula listed below, which contains values of relief gradient and parameters of vehicle, is possible to acquire deceleration ratio [8]: (3) where gradtmax [%,°] is maximum climbing capability of a vehicle on terrain; gradkmax [%,°] maximum climbing capability of a vehicle on road and sh [%,°] mean value of slope gradient obtained from the table 1. category slope [%] sh [%] slope [°] sh [°] 1 < 0 0 < 0 0 2 0 – 3 1,5 0,00 – 1,35 0,68 3 3 – 10 6,5 1,35 – 4,50 2,93 4 10 – 20 15 4,50 – 9,00 6,75 5 20 – 30 25 9,00 – 13,50 11,25 6 30 – 45 37,5 13,50 – 20,25 16,88 7 > 45 slope [%] > 20,25 slope [°] table 1: determination of mean value of the slope gradient (sh) from the measured range of slopes [6]. determination of c11 at the basis of tractive charts: the deceleration ratio of impact of gradient factor can by also determinable on the basis of tractive charts of particular vehicles [6]. to calculate running characteristic on the route of vehicle it must be started from the presupposition that this route is described at particular section by longitudinal gradient (α), transversal inclination (β), coefficient of rolling resistance (f) and coefficient of static friction (φ). particularly significant from the point of view of crosscountry movements evaluation are also following data:  attainable driving speed (eventually an acceleration);  conditions whereat coming to a swerving either of longitudinal or transversal direction;  conditions whereat coming to loss of maneuverability and longitudinal or transversal rollover. a tractive chart is the formulation of tractive power dependence on vehicle driving speed. the driving speed is plotted on the horizontal axis on the chart and on the vertical axis are plotted tractive power and forces of resistance. the tractive power ft depends on engine torque and total ratio, whereas both quantities are changeable in running. providing that transmission efficiency is constant, the tractive power at particular speed gear is adequate to engine torque verification of the mathematically computed impact of the relief gradient to vehicle speed 27 at that moment. considering total ratio changeability, we can say that each vehicle has as much tractive power curves as the number of vehicle speed gears. the process of calculating the course of curves of tractive power has following parts [6]:  number of points are selected on external torque characteristics of engine that are characterizing engine torque curve;  from the characteristic we determine corresponding quantity of engine torque mm and proper engine revolutions nm;  the coordinates (mm, nm) are read out of selected points on the engine torque characteristics;  the coordinates (mm, nm) are then transformed to coordinates (ft, v) by formula (4) and points with the coordinates (ft, v) for particular speed gears create the curves of tractive power at the tractive chart. (4) where ft [n] is tractive power, mm [nm] engine torque, ηm [%] mechanical efficiency of transmissions, ic(j) total transmission ratio, rd [m] wheel dynamic radius, v [km·h -1 ] vehicle driving speed and nm [min -1 ] engine revolutions. the curves of rolling resistance by even speed movements are marked by proper terrain gradient and tractive power curve is marked by relating speed gear. for the ideal course of tractive power each tractive power curve tangents a hyperbolic curve. the contact points of both curves at every speed gear corresponds to engine revolutions at maximum power. the chart is completed under horizontal axis and scales of motor revolutions at given speed in particular speed gears. this diagram also presents a survey of driving characteristics of vehicle [6]:  climb capability at particular speed gears (by the interpolation among curves of rolling resistance – slopes);  what speed gear is to be used during uphill driving on particular slope;  what speed is achievable on a particular slope;  maximum speed on a plain field (vmax). determination of c11 at the basis of operational testing: for the basic type of vehicles was relief gradient deceleration ratio determine based on terrain tests. for the particular vehicles was used following procedure [6]: 1. the tractive chart is calculated. martin bureš and filip dohnal 28 2. the readings of maximum available driving speeds and driving positions used were made from the tractive diagram for each partial parts of section given. 3. the passage time was calculated for each mentioned partial parts (at all 19 sections of terrain). 4. the results were compared with operational driving tests and on that basis; the resulting coefficient of deceleration was defined for each section (at all 19 sections of terrain). 5. there were calculated mean values of the multiple coefficients of deceleration for each vehicle for: terrain; cartways and forest ways; roads. to calculate presupposed driving speed on communications, cartways and forest ways can be used following relation: (5) where vest [km·h -1 ] means estimated driving speed, vmax [km·h -1 ] maximum driving speed indicated for a vehicle and c11 multiple coefficient of deceleration according to the table 2. carriageway type passenger off-road vehicle medium off road utility vehicles heavy off road lorries infantry combat vehicles tanks terrain 0.22 0.31 0.28 0.42 0.41 cartways and forest ways 0.43 0.53 0.52 0.58 0.53 roads 0.72 0.86 0.84 0.72 0.72 table 1: the mean multiple coefficients of deceleration of military vehicle movements on free terrain and on communications [6]. 4 field testing and data processing to verify the theoretical values field tests were used. tests were conducted in the military training area libava in 2015, there were tested eight types of vehicles, including tatra 810 6x6 (t810). for analysis of the impact of the relief gradient there were selected rides on the training circuit, which contains a tank track. unpaved surface of the tank track was not covered by vegetation and contained lots of micro-relief forms, especially the waves od soil that have approximately 20 m in length with an amplitude up to 1 m and ruts. the width of the tank tracks ranges from 10 to 30 m. the maximum slope of the test area reaches only to values of 16 °. verification of the mathematically computed impact of the relief gradient to vehicle speed 29 the vehicle routes were recorded by a gps receiver trimble geoexplorer 3000 geoxt equipped with an external antenna external mini. the vehicle speed was calculated from locations and times of the records. all records have been checked and the wrong or unnecessary ones have been removed (an error in position, parking, turning at the end of the route). next step was to add the value of the terrain slope from the most precise digital elevation model of czech republic (dmr5g) [7] in the spot of each record with use of the arcgis 10.2.1 [8]. correction of the estimated speed values of the speed calculated by formula (3) and derived from the traction diagram are acceptable only in case of ideal conditions, where the only factor influencing the drive is terrain slope. the analysed rides took place under invariant but still not ideal conditions, such as after rain with muddy and slippery surface. after the elimination of micro-relief affected records all other influences can be considered constant. maximum speed t810 vehicle in terrain mode (vmax) is 65 km·h -1 , the maximum reached speed in the given conditions was 36 km·h -1 (vmax‘). then the deceleration coefficient cs (influence of surroundings) has the value 0.55. all the following values have been corrected by this coefficient (equation xxx). (6) method of predicting the vehicle speed in general terrain, which neglects the influence of the slope, was not corrected, because all the factors have been already included. verification of theoretical values of speed the results of all three methods of calculating the velocity field were compared with the measured data. unfortunately, the measured data do not represent the whole range of terrain slope which t810 can pass through, for example up to 30 °. frequency distribution of the slope gradient in the records is shown in fig. 2. small counts in the higher slopes reduce their credibility. however, at least in the lower slopes below 7 ° the data can be probably used to verify mathematical apparatus. martin bureš and filip dohnal 30 fig. 2: counts of measured values the table 3 compares the calculated values according to the methodology of dma and speed read from t810 tractive chart with the measured speed. the same comparing is represented on the fig. 3. tab. 3. the comparison of the calculated and measured values. the difference between both estimated value is not significant in small slopes, but with a growing slope the difference increase up to 5 km·h -1 . the measured speed is much lower in slopes 0 ° – 7 ° and the same situation applies to the predicted average speed, which is 28 km·h -1 , but the measured speed was 23 km·h -1 . verification of the mathematically computed impact of the relief gradient to vehicle speed 31 fig. 3. the illustration of the comparison of the calculated and measured values the achieved results do not correspond with the expectations and it is probably not possible to use this data at this point of the research to verify impact of the slope gradient to the vehicle speed. the reason of very slow ride along the entire length of the route and a reason of unusable results seem to be less experienced driver, who drove a given car. significant distortion of measured data due to unexperienced driver was confirmed by comparison with the another lorry, tatra 815 8x8. its technical specifications are slightly different, but the maximum surmountable slope remains the same value. the fig. 4 illustrates both measured rides – t810 and t815. data measurement by other vehicles proves that the main impact on the t810 ride was the driver. the relatively high speed at inclinations of 11 ° 16 ° are caused by a too short climb to slowdown the vehicles marginally. fig. 4: the comparison of the calculated and measured values (t810, t815) martin bureš and filip dohnal 32 5 conclusions unfortunately, the data from the t810 cannot be used to verify the mathematical apparatus used to calculate the impact of the slope gradient on vehicle speed. the first obstacle is the number of data from higher terrain slopes and the other is a distortion caused by inexperience of the driver. even this result has a positive contribution in the form of experience needed for planning field tests and obtaining relevant data. to determine the impact of the slope gradient on the speed of t810 is necessary to get more data from multiple passes through the high slopes near the limits of the vehicle. the next step to successful verification of mathematical calculations is testing several drivers. it will significantly reduce the influence of experience of the driver and also possibly his mental state. 6 acknowledgement the work presented in this paper was supported within the project for “development of the methods of evaluation of environment in relation to defense and protection of the czech republic territory” (project code naturenvir) by the ministry of defence the czech republic. verification of the mathematically computed impact of the relief gradient to vehicle speed 33 bibliography [1] collins, j., m., (1998). military geography for professionals and the public. 1st brassey's ed. washington, d. c.: brassey's, xxiv, 437 p. isbn 15-748-8180-9. [2] rybansky, m., hofmann, a., hubacek, m. et al., (2015). modeling of cross-country transport in raster format. environ. earth sci., 74: 7049. doi:10.1007/s12665-015-4759-y. [3] hofmann, a., hošková-mayerová, š., talhofer, v. et al., (2015). creation of models for calculation of coefficients of terrain passability. quality & quantity, 49: 1679. doi:10.1007/s11135-014-0072-1. [4] vala, m., (1992). evaluation of the passability of the military vehicles (in czech). /habilitation thesis/. va brno, 152 pp. [5] rybansky, m., vala, m., (2010). relief impact to cross-country movement. in: proceedings of the joint 9th asia-pacific istvs conference, sapporo, japan, 16 pp. [6] rybansky, m., (2009). cross-country movement: the impact and evaluation of geographic factors. 1st ed. brno. academical publishing cerm, 113 p. isbn 978-80-7204-661-4. [7] talhofer, v., hofmann, a., kratochvil, v., hubacek m., zerzan, p., (2015). verification of digital analytical models case study of the cross-country movement. in: icmt´15 – international conference on military technologies 2015, brno, (czech republic), 7 pp. isbn 978-14799-7785-7. [8] rybansky, m., hofmann, a., hubacek, m. et al., (2015). modelling of cross-country transport in raster format. environ. earth sci., 74: 7049. doi:10.1007/s12665-015-4759-y. [9] hubacek, m., kovarik, v., kratochvil, v., (2016). analysis of influence of terrain relief roughness on dem accuracy generated from lidar in the czech republic territory. in: international archives of the photogrammetry, remote sensing and spatial information sciences isprs archives, 41 25-30. doi:10.5194/isprsarchives-xli-b4-25-2016. [10] esri. (2013). user documentation. copyright © 1995–2013 esri. approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 41, 2021, pp. 137-145 137 a characterization of strong fuzzy diameter zero in intuitionistic fuzzy metric spaces s. yahya mohamed* e. naargees begum† abstract the idea of intuitionistic fuzzy metric space introduced by park (2004). in this paper, we introduce the notion of strong intuitionistic fuzzy diameter zero for a family of subsets based on the intuitionistic fuzzy diameter for a subset of 𝐴. then we introduce nested sequence of subsets having strong intuitionistic fuzzy diameter zero using their intuitionistic fuzzy diameter. keywords: strong fuzzy diameter; intuitionistic fuzzy metric space; strong completeness. 2010 ams subject classification: 05c72, 54e50, 03f55.‡ *pg and research department of mathematics, affiliated to bharathidasan university, government arts college, trichy, tamilnadu, india. yahya_md@yahoo.com. † department of mathematics, dr. r.k. shanmugam college of arts and science, indili, tamilnadu, india; mathsnb@gmail.com. ‡ received on september 18, 2021. accepted on december 1, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.661. issn: 1592-7415. eissn: 2282-8214. © the authors. this paper is published under the cc-by licence agreement. s. yahya mohamed and e. naargees begum 138 1 introduction the theory of fuzzy sets was introduced by l.a. zadeh [17] in 1965. kramosil and michalek [6] introduced the fuzzy metric spaces (fm-spaces) by generalizing the concept of probabilistic metric spaces to fuzzy situation. george and veeramani [4] modified the concept of fuzzy metric space introduced by kramosil and michalek [6] with a view to obtain a hausdorff topology on fuzzy metric spaces which have very important applications in quantum particle particularly in connection with both string and e-infinity theory. in 2004, park [8] defined the concept of intuitionistic fuzzy metric space with the help of continuous t-norms and continuous t-conorms. several researchers have shown interest in the intuitionistic fuzzy metric space successfully applied in many fields, it can be found in [5, 10, 11, 13, 14, 15, 16]. theory of fuzzy sets have been widely used and developed in different fields of sciences, including mathematical programing, theory of modeling, theory of optimal control, theory of neural network, engineering and medical sciences, coloured image processing, etc. in this paper, the concept of characterization of strong fuzzy diameter zero in intuitionistic fuzzy metric spaces are introduced and also discuss some properties of strong fuzzy diameter zero in intuitionistic fuzzy metric spaces. 2 preliminaries definition 2.1[17] let 𝑋 be a nonempty set. a fuzzy set 𝐴 in 𝑋 is characterized by its membership function 𝜇𝐴 ∶ 𝑋 → [0, 1] and 𝜇𝐴(𝑥) is interpreted as the degree of membership of element 𝑥 in fuzzy set 𝐴 for each 𝑥 ∈ 𝑋. it is clear that 𝐴 is completely determined by the set of tuples 𝐴 = {(𝑥, 𝜇𝐴(𝑥))|𝑥 ∈ 𝑋}. definition 2.2[4] the 3-tuple (𝐴, 𝑀,∗) is said to be a fuzzy metric space if 𝐴 be a non-empty set and ∗ be a continuous t-norm. a fuzzy set 𝐴2 × (0, ∞) is called a fuzzy metric on 𝐴 if 𝑎, 𝑏, 𝑐 ∈ 𝐴 and 𝑠, 𝑡 > 0, the following condition holds: 1. 𝑀 (𝑎, 𝑏, 𝑡) = 0 2. 𝑀 (𝑎, 𝑏, 𝑡) = 1 if and only if 𝑎 = 𝑏 a characterization of strong fuzzy diameter zero in intuitionistic fuzzy metric spaces 139 3. 𝑀 (𝑎, 𝑏, 𝑡) = 𝑀(𝑏, 𝑎, 𝑡 ) 4. 𝑀 (𝑎, 𝑏, 𝑡 + 𝑠) ≥ 𝑀(𝑎, 𝑏, 𝑡) ∗ 𝑀(𝑎, 𝑏, 𝑠) 5. 𝑀 (𝑎, 𝑏, •): (0, ∞) → [0, 1] is left continuous the function 𝑀(𝑎, 𝑏, 𝑡) denote the degree of nearness between 𝑎 and 𝑏 with respect to t respectively. definition 2.3[1, 2] let a set 𝐸 be fixed. an intuitionistic fuzzy set 𝐴 in 𝐸 is an object of the following 𝐴 = {(𝑥, 𝜇𝐴(𝑥), 𝜐𝐴(𝑥)), 𝑥 ∈ 𝐸 } where the functions 𝜇𝐴(𝑥): 𝐸 → [0, 1] and 𝜐𝐴 (𝑥 ): 𝐸 → [0, 1] determine the degree of membership and the degree of non-membership of the element 𝑥 ∈ 𝐸, respectively, and for every 𝑥 ∈ 𝐸: 0 ≤ 𝜇𝐴(𝑥) + 𝜐𝐴(𝑥) ≤ 1, when 𝜐𝐴(𝑥) = 1 − 𝜇𝐴(𝑥) for all 𝑥 ∈ 𝐸 is an ordinary fuzzy set. in addition, for each ifs 𝐴 in 𝐸, if 𝜋𝐴(𝑥) = 1 − 𝜇𝐴(𝑥) − 𝜐𝐴(𝑥). then 𝜇𝐴(𝑥) is called the degree of indeterminacy of 𝑥 to 𝐴 or called the degree of hesitancy of 𝑥 to 𝐴. it is obvious that 0 ≤ πa(x) ≤ 1, for each 𝑥 ∈ 𝐸. definition 2.4 [7] a 5-tuple (𝐴, 𝑀, 𝑁,∗,∘) is said to be an intuitionistic fuzzy metric space if 𝐴 is an arbitrary set, ∗ is a continuous t-norm, ∘ is a continuous tconorm and, 𝑀, 𝑁 are fuzzy sets on 𝐴2 × [0, ∞) satisfying the conditions: 1. 𝑀(𝑎, 𝑏, 𝑡) + 𝑁(𝑎, 𝑏, 𝑡) ≤ 1, for all 𝑎, 𝑏 ∈ 𝐴 and 𝑡 ˃ 0 2. 𝑀(𝑎, 𝑏, 0) = 0, for all 𝑎, 𝑏 ∈ 𝐴 3. 𝑀(𝑎, 𝑏, 𝑡) = 1, for all 𝑎, 𝑏 ∈ 𝐴 and 𝑡 ˃ 0 if and only if 𝑎 = 𝑏 4. 𝑀(𝑎, 𝑏, 𝑡) = 𝑀(𝑏, 𝑎, 𝑡), for all 𝑎, 𝑏 ∈ 𝐴 and 𝑡 > 0 5. 𝑀(𝑎, 𝑏, 𝑡) ∗ 𝑀(𝑏, 𝑐, 𝑠) ≤ 𝑀(𝑎, 𝑐, 𝑡 + 𝑠), for all 𝑎, 𝑏, 𝑐 ∈ 𝐴 and 𝑠, 𝑡 ˃ 0 6. 𝑀(𝑎, 𝑏,•): [0, ∞) → [0, ∞] is left continuous for all 𝑎, 𝑏 ∈ 𝐴 7. 𝑙𝑖𝑚 𝑡→∞ 𝑀(𝑎, 𝑏, 𝑡) = 1, for all 𝑎, 𝑏 ∈ 𝐴 and 𝑡 > 0 8. 𝑁(𝑎, 𝑏, 0) = 1, for all a, 𝑏 ∈ 𝐴 9. 𝑁(𝑎, 𝑏, 𝑡) = 0, for all 𝑎, 𝑏 ∈ 𝐴 and 𝑡 > 0 if and only if 𝑎 = 𝑏 10. 𝑁(𝑎, 𝑏, 𝑡) = 𝑁(𝑏, 𝑎, 𝑡), for all 𝑎, 𝑏 ∈ 𝐴 and 𝑡 > 0 11. 𝑁(𝑎, 𝑏, 𝑡) ∘ 𝑁(𝑏, 𝑐, 𝑠) ≥ 𝑁(𝑎, 𝑐, 𝑡 + 𝑠), for all 𝑎, 𝑏, 𝑐 ∈ 𝐴 and 𝑠, 𝑡 > 0 12. 𝑁(𝑎, 𝑏,•): [0, ∞) → [0,1] is right continuous for all 𝑎, 𝑏 ∈ 𝐴 13. 𝑙𝑖𝑚 𝑡→∞ 𝑁(𝑎, 𝑏, 𝑡) = 0, for all 𝑎, 𝑏 ∈ 𝐴. the functions 𝑀(𝑎, 𝑏, 𝑡) and 𝑁(𝑎, 𝑏, 𝑡) denote the degree of nearness and the degree of non-nearness between 𝑎 and 𝑏 w.r.t 𝑡 respectively. definition 2.5 [9] the fuzzy diameter of a non-empty set 𝐵 of a fuzzy metric space 𝐴, with respect to t, is the function 𝜑𝐵 : (0, +∞) → [0, 1] given by φb(t) = 𝑖𝑛𝑓{𝑀(𝑎, 𝑏, 𝑡): 𝑎, 𝑏 ∈ 𝐵} for each 𝑡 ∈ 𝑅 +. s. yahya mohamed and e. naargees begum 140 definition 2.6 [9] a collection of sets {𝐵𝑖 }𝑖∈𝐼 is said to have fuzzy diameter zero if given 𝑟 ∈ (0, 1) and 𝑡 ∈ 𝑅+ there exists 𝑖 ∈ 𝐼 such that m(a, b, t) > 1 − r for all 𝑎, 𝑏 ∈ 𝐵𝑖. 3 strong fuzzy diameter zero in intuitionistic fuzzy metric spaces definition 3.1 the fuzzy diameter of a non-empty set 𝐵 of a intuitionistic fuzzy metric space (𝐴, 𝑀, 𝑁,∗,∘), with respect to 𝑡, is the function φb: (0, +∞) → [0, 1] given by 𝜑𝐵 (𝑡) = 𝑖𝑛𝑓{𝑀(𝑎, 𝑏, 𝑡): 𝑎, 𝑏 ∈ 𝐵} and 𝜓𝐵 : (0, +∞) → [0, 1] given by 𝜓𝐵 (𝑡) = 𝑠𝑢𝑝{𝑁(𝑎, 𝑏, 𝑡): 𝑎, 𝑏 ∈ 𝐵} for each 𝑡 ∈ 𝑅+ definition 3.2 a collection of sets {𝐵𝑖}𝑖∈𝐼 of a intuitionistic fuzzy metric space (𝐴, 𝑀, 𝑁,∗,∘) is said to have fuzzy diameter zero if given 𝑟 ∈ (0, 1) and 𝑡 ∈ 𝑅+ there exists 𝑖 ∈ 𝐼 such that 𝑀(𝑎, 𝑏, 𝑡) > 1 − 𝑟 𝑁(𝑎, 𝑏, 𝑡) < 𝑟 for all 𝑎, 𝑏 ∈ 𝐵𝑖 . theorem 3.3 let {𝐵𝑛}𝑛∈ℕ be a nested sequence of sets of the intuitionistic fuzzy metric space (𝐴, 𝑀, 𝑁,∗,∘). then the following statements are equivalent: (i) {𝐵𝑛}𝑛∈ℕ has fuzzy diameter zero. (ii) 𝑙𝑖𝑚 𝑛→∞ 𝜑𝐵𝑛 (𝑡) = 1, 𝑙𝑖𝑚𝑛→∞ 𝜓𝐵𝑛 (𝑡) = 0 for all 𝑡 ∈ 𝑅 +. proof: (i)→(ii): let 𝑡 ∈ 𝑅+. given 𝑟 ∈ (0, 1) exists 𝑛𝑟,𝑡 ∈ ℕ such that 𝑀(𝑎, 𝑏, 𝑡) > 1 – 𝑟, 𝑁(𝑎, 𝑏, 𝑡) < 𝑟 for each 𝑎, 𝑏 ∈ 𝐵𝑛 with 𝑛 ≥ 𝑛𝑟,𝑡 . then, 𝜑𝐵𝑛 (𝑡) = 𝑖𝑛𝑓{𝑀(𝑎, 𝑏, 𝑡): 𝑎, 𝑏 ∈ 𝐵𝑛} ≥ 1 – 𝑟 and 𝜓𝐵𝑛 (𝑡) = 𝑠𝑢𝑝{𝑁(𝑎, 𝑏, 𝑡): 𝑎, 𝑏 ∈ 𝐵𝑛} ≤ 1 – 𝑟 for all n ≥ nr,t. hence, 𝑙𝑖𝑚 𝑛→∞ 𝜑𝐵𝑛 (𝑡) = 1 and 𝑙𝑖𝑚𝑛→∞ 𝜓𝐵𝑛 (𝑡) = 0, since r is arbitrary in (0,1). (ii)→(i): suppose 𝑙𝑖𝑚 𝑛→∞ 𝜑𝐵𝑛 (𝑡) = 1 and 𝑙𝑖𝑚𝑛→∞ 𝜓𝐵𝑛 (𝑡) = 0, for all 𝑡 ∈ 𝑅 +. let 𝑡 ∈ 𝑅+and let 𝑟 ∈ (0, 1). we can find 𝑛𝑟,𝑡 ∈ ℕ such that 𝜑𝐵𝑛 (𝑡) > 1 − 𝑟 and 𝜓𝐵𝑛 (𝑡) < 𝑟 for all ≥ 𝑛𝑟,𝑡. thus, 𝑀(𝑎, 𝑏, 𝑡) > 1 − 𝑟 and 𝑁(𝑎, 𝑏, 𝑡) < 𝑟 for each 𝑎, 𝑏 ∈ 𝐵𝑛 with 𝑛 ≥ 𝑛𝑟,𝑡 i.e., {𝐵𝑛}𝑛∈ℕ has fuzzy diameter zero. a characterization of strong fuzzy diameter zero in intuitionistic fuzzy metric spaces 141 definition 3.4 a family of non-empty sets {𝐵𝑖}𝑖∈𝐼 of a intuitionistic fuzzy metric space (𝐴, 𝑀, 𝑁,∗,∘) has strong fuzzy diameter zero if for 𝑟 ∈ (0, 1) there exists 𝑖 ∈ 𝐼 such that 𝑀(𝑎, 𝑏, 𝑡) > 1 – 𝑟 and 𝑁(𝑎, 𝑏, 𝑡) < 𝑟 for each 𝑎, 𝑏 ∈ 𝐵𝑛 and all 𝑡 ∈ 𝑅 +. theorem 3.5 let (𝐴, 𝑀, 𝑁,∗,∘) be an intuitionistic fuzzy metric space and let {𝐵𝑛}𝑛∈ℕ be a nested sequence of sets of 𝐴. then the following statements are equivalent. (i) {𝐵𝑛}𝑛∈ℕ has strong fuzzy diameter zero. (ii) 𝑙𝑖𝑚 𝑛→∞ 𝜑𝐵𝑛 (𝑡𝑛) = 1, 𝑙𝑖𝑚𝑛→∞ 𝜓𝐵𝑛 (𝑡𝑛) = 0 for every decreasing and increasing sequence of positive real numbers {𝑡𝑛}𝑛∈ℕ that converges and diverges respectively. proof: (i) → (ii): let {𝑡𝑛}𝑛∈ℕ be a decreasing increasing sequence of positive real numbers that converges and diverges respectively. given 𝑟 ∈ (0, 1), we can find 𝑛𝑟 ∈ ℕ such that 𝑀(𝑎, 𝑏, 𝑡) > 1 – 𝑟 and 𝑁(𝑎, 𝑏, 𝑡) < 𝑟 for each a, b ∈ bn with 𝑛 ≥ 𝑛𝑟 and all 𝑡 ∈ 𝑅 +. in particular, 𝑀(𝑎, 𝑏, 𝑡𝑛) > 1 – 𝑟 and 𝑁(𝑎, 𝑏, 𝑡𝑛 ) < 𝑟 for all a, b ∈ bn with 𝑛 ≥ 𝑛𝑟 , i.e., 𝜑𝐵𝑛 (𝑡𝑛) ≥ 1 − 𝑟, 𝜓𝐵𝑛 (𝑡𝑛) ≤ 𝑟 for all 𝑛 ≥ 𝑛𝑟 , i.e., 𝑙𝑖𝑚𝑛→∞ 𝜑𝐵𝑛 (𝑡𝑛) = 1,𝑙𝑖𝑚𝑛→∞ 𝜓𝐵𝑛 (𝑡𝑛) = 0. (ii) → (i): suppose that {𝐵𝑛}𝑛∈ℕ has not strong fuzzy diameter zero. let r ∈ (0, 1) such that 𝐼 = { 𝑛 ∈ ℕ: 𝑀(𝑎, 𝑏, 𝑡) ≤ 1 – 𝑟, 𝑁(𝑎, 𝑏, 𝑡) ≥ 𝑟 for some 𝑎, 𝑏 ∈ 𝐵𝑛 and some 𝑡 ∈ 𝑅 +}, is infinite. take 𝑛1 = 𝑚𝑖𝑛 𝐼. then, there exist 𝑎𝑛1 , 𝑏𝑛1 ∈ 𝐵𝑛1 such that 𝑀(𝑎𝑛1 , 𝑏𝑛1 , 𝑡𝑛1 ) ≤ 1 – 𝑟, 𝑁(𝑎𝑛1 , 𝑏𝑛1 , 𝑡𝑛1 ) ≥ 𝑟 with 0 < 𝑡𝑛1 < 1. take 𝑛2 > 𝑛1, with 𝑛2 ∈ ℕ, such that 𝑀(𝑎𝑛1 , 𝑏𝑛1 , 𝑡𝑛1 ) ≤ 1 – 𝑟, 𝑁(𝑎𝑛1 , 𝑏𝑛1 , 𝑡𝑛1 ) ≥ 𝑟 for some 𝑎𝑛2 , 𝑏𝑛2 ∈ 𝐵𝑛2 and 0 < 𝑡𝑛2 < 𝑚𝑖𝑛{𝑡𝑛1 , 1 2 }. in this way, we construct, by induction, a sequence {𝑡𝑛𝑖 }𝑖∈ℕ such that 𝑀(𝑎𝑛𝑖 , 𝑏𝑛𝑖 , 𝑡𝑛𝑖 ) ≤ 1 – 𝑟, 𝑁(𝑎𝑛𝑖 , 𝑏𝑛𝑖 , 𝑡𝑛𝑖 ) ≥ 𝑟 for some 𝑎𝑛𝑖 , 𝑏𝑛𝑖 ∈ 𝐵𝑛𝑖 , 𝑛𝑖 ∈ ℕ with 𝑛𝑖 > 𝑛𝑖−1 and 0 < 𝑡𝑛𝑖 < 𝑚𝑖𝑛{𝑡𝑛𝑖−1 , 1 𝑖 }. then, 𝜑𝐵𝑛𝑖 (𝑡𝑛𝑖 ) = 𝑖𝑛𝑓{𝑀(𝑎, 𝑏, 𝑡𝑛𝑖 ): 𝑎, 𝑏 ∈ 𝐵𝑛𝑖 } ≤ 1 – 𝑟, 𝜓𝐵𝑛𝑖 (𝑡𝑛𝑖 ) = 𝑠𝑢𝑝{𝑁(𝑎, 𝑏, 𝑡𝑛𝑖 ): 𝑎, 𝑏 ∈ 𝐵𝑛𝑖 } ≥ 𝑟 for all 𝑖 ∈ ℕ . hence {𝜑𝐵𝑛𝑖 (𝑡𝑛𝑖 )}𝑖∈ℕ, {𝜓𝐵𝑛𝑖 (𝑡𝑛𝑖 )}𝑖∈ℕ does not converge and diverge s. yahya mohamed and e. naargees begum 142 respectively. now, {𝑡𝑛𝑖 }𝑖∈ℕ is a subsequence of the decreasing and increasing sequence {tn}n∈ℕ that converges and diverges respectively, given by 𝑡𝑛 = { 𝑡𝑛1 𝑛 ≤ 𝑛1 𝑡𝑛 𝑖+1 𝑛𝑖 ≤ 𝑛 ≤ 𝑛𝑖+1 and the sequence {𝜑𝐵𝑛 (𝑡𝑛)}𝑛∈ℕ, {𝜓𝐵𝑛 (𝑡𝑛)}𝑛∈ℕ does not converge and diverge respectively. thus, we get the contradiction. theorem 3.6 let {𝐵𝑛}𝑛∈ℕ be a nested sequence of sets with fuzzy diameter zero in a intuitionistic fuzzy metric space (𝐴, 𝑀, 𝑁,∗,∘). {𝐵𝑛}𝑛∈ℕ has strong fuzzy diameter zero if and only if {𝐵𝑛} is a singleton set after a certain stage. proof: suppose {𝐵𝑛}𝑛∈ℕ is not eventually constant. put 𝑝𝑛 = 𝑠𝑢𝑝{𝑑(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝐵𝑛 }, 𝑞𝑛 = 𝑖𝑛𝑓{𝑑(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝐵𝑛 } and take 𝑡𝑛 = 𝑝𝑛 and 𝑡𝑛 = 𝑞𝑛 for all 𝑛 ∈ ℕ. then, {𝑡𝑛 }𝑛∈ℕ is a decreasing and increasing sequence of positive real numbers converges and diverges respectively. then, 𝑙𝑖𝑚 𝑛→∞ 𝜑𝐵𝑛 (𝑡) = 𝑙𝑖𝑚 𝑖𝑛𝑓{𝑀𝑑(𝑎, 𝑏, 𝑡𝑛): 𝑎, 𝑏 ∈ 𝐵𝑛 } 𝑛→∞ = 𝑙𝑖𝑚 𝑛→∞ 𝑡𝑛 𝑡𝑛+𝑑𝑖𝑎𝑚(𝐵𝑛) = 𝑙𝑖𝑚 𝑛→∞ 𝑝𝑛 𝑝𝑛+𝑝𝑛 = 1 2 and 𝑙𝑖𝑚 𝑛→∞ 𝜓𝐵𝑛 (𝑡) = 𝑙𝑖𝑚 𝑠𝑢𝑝{𝑁𝑑(𝑎, 𝑏, 𝑡𝑛 ): 𝑎, 𝑏 ∈ 𝐵𝑛} 𝑛→∞ = 𝑙𝑖𝑚 𝑛→∞ 𝑡𝑛 𝑡𝑛+𝑑𝑖𝑎𝑚(𝐵𝑛) = 𝑙𝑖𝑚 𝑛→∞ 𝑞𝑛 𝑞𝑛+𝑞𝑛 = 1 2 hence {𝐵𝑛}𝑛∈ℕ has not strong fuzzy diameter zero. theorem 3.7 let (𝐴, 𝑀, 𝑁,∗,∘) be a intuitionistic fuzzy metric space. if {𝐵𝑛}𝑛∈ℕ is a nested sequence of sets of a which has strong fuzzy diameter zero then {𝐵𝑛}𝑛∈ℕ has strong fuzzy diameter zero. proof: first, we prove that 𝜑�̅� (𝑡 ) = 𝜑𝐵 (𝑡 ), 𝜓�̅� (𝑡 ) = 𝜓𝐵 (𝑡 ) for every subset 𝐵 of 𝐴 . indeed, take 𝑎, 𝑏 ∈ 𝐵. then, we can find two sequences {𝑎𝑛}𝑛∈ℕ and a characterization of strong fuzzy diameter zero in intuitionistic fuzzy metric spaces 143 {𝑏𝑛}𝑛∈ℕ in 𝐵 that converge to 𝑎 and 𝑏, respectively. let 𝑡 ∈ 𝑅 + and an arbitrary ε ∈ (0, 1). we have that 𝑀(𝑎, 𝑏, 𝑡 + 2𝜀) ≥ 𝑀(𝑎, 𝑏𝑛, 𝜀) ∗ 𝑀(𝑎𝑛, 𝑏𝑛, 𝑡) ∗ 𝑀(𝑏𝑛, 𝑏, 𝜀) ≥ 𝑀(𝑎, 𝑎𝑛, 𝜀) ∗ 𝜑𝐵 (𝑡 ) ∗ 𝑀(𝑏𝑛, 𝑏, 𝜀), 𝑁(𝑎, 𝑏, 𝑡 + 2𝜀) ≤ 𝑁(𝑎, 𝑏𝑛, 𝜀) ∘ 𝑁(𝑎𝑛, 𝑏𝑛, 𝑡) ∘ 𝑁(𝑏𝑛, 𝑏, 𝜀) ≤ 𝑁(𝑎, 𝑎𝑛, 𝜀) ∘ 𝜓𝐵 (𝑡 ) ∘ 𝑁(𝑏𝑛, 𝑏, 𝜀) and taking limit on the inequality when n tends to ∞, we obtain 𝑀(𝑎, 𝑏, 𝑡 + 2𝜀) ≥ 1 ∗ 𝜑𝐵 (𝑡 ) ∗ 1 = 𝜑𝐵 (𝑡 ) , 𝑁(𝑎, 𝑏, 𝑡 + 2𝜀) ≤ 1 ∘ 𝜑𝐵 (𝑡 ) ∘ 1 = 𝜑𝐵 (𝑡 ) . since ε is arbitrary, due to the continuity of 𝑀(𝑎, 𝑏, 𝑡), 𝑁(𝑎, 𝑏, 𝑡) we obtain 𝑀(𝑎, 𝑏, 𝑡) ≥ 𝜑𝐵 (𝑡 ), 𝑁(𝑎, 𝑏, 𝑡) ≤ 𝜓𝐵 (𝑡 ) and then 𝜑�̅� (𝑡 ) ≥ 𝜑𝐵 (𝑡 ), 𝜓�̅� (𝑡 ) ≥ 𝜓𝐵 (𝑡 ). on the other hand, we have 𝜑�̅� (𝑡 ) ≤ 𝜑𝐵 (𝑡 ), 𝜓�̅� (𝑡 ) ≤ 𝜓𝐵 (𝑡 ) and hence 𝜑�̅� (𝑡 ) = 𝜑𝐵 (𝑡 ), 𝜓�̅� (𝑡 ) = 𝜓𝐵 (𝑡 ). let {𝑡𝑛}𝑛∈ℕ be a decreasing and increasing sequence of positive real numbers convergent and divergent respectively. by theorem 3.5, we have that 𝑙𝑖𝑚 𝑛→∞ 𝜑𝐵𝑛 (𝑡𝑛) = 1, 𝑙𝑖𝑚 𝑛→∞ 𝜓𝐵𝑛 (𝑡𝑛) = 0. then, by our last argument, we have that, 𝑙𝑖𝑚 𝑛→∞ 𝜑𝐵𝑛 (𝑡𝑛) = 𝑙𝑖𝑚𝑛→∞ 𝜑�̅�𝑛 (𝑡𝑛) = 1, 𝑙𝑖𝑚 𝑛→∞ 𝜓𝐵𝑛 (𝑡𝑛) = 𝑙𝑖𝑚𝑛→∞ 𝜓�̅�𝑛 (𝑡𝑛) = 0, and consequently, by theorem 3.5, {𝐵𝑛}𝑛∈ℕ has strong fuzzy diameter zero. 4 conclusion intuitionistic fuzzy set theory plays a vital role in uncertain situations in all aspects. in this paper, the characterizations of strong fuzzy diameter zero in intuitionistic fuzzy metric spaces are discussed and proved that the nested sequences having the strong fuzzy diameter zero in intuitionistic fuzzy metric space. we have also provided that nested sequences of subsets has strong fuzzy diameter zero if and only if singleton set after a certain stage in a intuitionistic fuzzy metric spaces. s. yahya mohamed and e. naargees begum 144 references [1] k.t.atanassov. more on intuitionistic fuzzy sets. fuzzy sets and systems, 33(1), 37-45. 1989. 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[17] l.a. zadeh, fuzzy sets. information and control, 8(3), 338-356. 1965. ratio mathematica volume 44,2022 reach energy of digraphs v. mahalakshmi* b. vijaya praba† k. palani‡ abstract a digraph d consists of two finite sets (𝑉, 𝒜), where 𝑉 denotes the vertex set and 𝒜 denotes the arc set. for vertices 𝑢, 𝑣 ∈ 𝑉, if there exists a directed path from 𝑢 to 𝑣 then 𝑣 is said to be reachable from 𝑢 and vice versa. the reachability matrix of d is the 𝑛 × 𝑛 matrix 𝑅(𝐷) = [𝑟𝑖𝑗], where 𝑟𝑖𝑗 = 1, if 𝑣𝑗 is reachable from 𝑣𝑖 and 𝑟𝑖𝑗 = 0 otherwise. the eigen values corresponding to the reachability matrix are called reach eigen values. the reach energy of a digraph is defined by 𝐸𝑅(𝐷) = ∑ |𝜆𝑖| 𝑛 𝑖=1 where 𝜆𝑖 is the eigen value of the reachability matrix. in this paper we introduce the reach spectrum of a digraph and study its properties and bounds. moreover, we compute reach spectrum for some digraphs. keywords: reachable, reachability matrix, reach eigen values, reach spectrum, reach energy. 2010 ams subject classification: 05c50,05c90, 15a18§ *assistant professor (pg & research department of mathematics, a.p.c. mahalaxmi college for women thoothukudi, india); email mahalakshmi@apcmcollege.ac.in †research scholar reg.no: 21212012092002 (pg & research department of mathematics, a.p.c. mahalaxmi college for women thoothukudi, india); email vijayapraba.b@gmail.com ‡ third author’s affiliation (pg & research department of mathematics, a.p.c. mahalaxmi college for women thoothukudi, india); email palani@apcmcollege.ac.in affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamilnadu, india; §received on june 19th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.926. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 363 v. mahalakshmi, b. vijaya praba, and k. palani 1. introduction in this paper we considered simple and connected graph. a directed graph or digraph d consists of two finite sets (𝑉, 𝒜 ) where 𝑉 denotes the vertex set and 𝒜 denotes the arc set. for two vertices 𝑢 and 𝑣, an arc from 𝑢 to 𝑣 is denoted by 𝑢𝑣. two vertices 𝑢 and 𝑣 is said to be adjacent if either 𝑢𝑣 ∈ 𝒜 or 𝑣𝑢 ∈ 𝒜 . in 1978 gutman [4] defined the energy of a simple graph as the sum of the absolute values of its eigen values and it is denoted by 𝐸(𝐺). i.e., 𝐸(𝐺) = ∑ |𝜆𝑖| 𝑛 𝑖=1 . the concept of graph energy was extended to digraph by pena and rada [8] and adiga et al. [1]. khan et al. [5] defined a new notion of energy of digraph called iota energy. in this paper, we investigate the properties and some bounds on reach energy. definition 1.1. a path is said to be directed path in which all the edges are directed either in clockwise or in anticlockwise direction and it is denoted as 𝑃𝑛⃗⃗⃗⃗ ⃗. let {𝑣1, 𝑣2, … , 𝑣𝑛} be the vertex set of a directed path. then the set {𝑣𝑖𝑣𝑖+1| 𝑖 = 1,2, . . . , 𝑛 − 1} is the arc set of 𝑃𝑛⃗⃗⃗⃗ ⃗. definition 1.2. a path is said to be alternate path in which the edges are given alternate direction and it is denoted as 𝐴𝑃𝑛⃗⃗ ⃗⃗⃗⃗ ⃗⃗ definition 1.3. a star graph 𝐾1,𝑛 in which all the edges are directed towards the root vertex is called an instar and is denoted as 𝑖𝐾1,𝑛⃗⃗⃗⃗⃗⃗⃗⃗ ⃗ definition 1.4. a star graph 𝐾1,𝑛 in which all the edges are directed away from the root vertex is called an outstar and is denoted as 𝑜𝐾1,𝑛⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ 2. reach energy definition 2.1. let 𝐷 = (𝑉, 𝐴) be a directed graph with 𝑛 vertices. the reachability matrix [2] 𝑅(𝐷) = [𝑟𝑖𝑗] is the 𝑛 × 𝑛 matrix with 𝑟𝑖𝑗 = 1, if 𝑣𝑗 is reachable from 𝑣𝑖 and 𝑟𝑖𝑗 = 0 otherwise. we assume that each vertex is reachable from itself. the characteristic polynomial of 𝑅(𝐷) = [𝑟𝑖𝑗] is denoted by 𝑓(𝐷, 𝜆) = 𝑑𝑒𝑡(𝑅(𝐷) − 𝐼𝜆). let {𝜆1, 𝜆2, … , 𝜆𝑛} be the reach eigen values of 𝐷. the reach eigen values of the graph 𝐷 are the eigen values of 𝑅(𝐷) and is called as reach spectrum of 𝐷. the spectrum of d is denoted by 𝑠𝑝𝑒𝑐 𝐷 = { 𝜆1 𝜆2 … 𝜆𝑛 𝑚1 𝑚2 … 𝑚𝑛 } where 𝑚𝑖 is the algebraic multiplicity of the eigen values 𝜆𝑖, for 1 ≤ 𝑖 ≤ 𝑛 then the reach energy of 𝐷 is defined as the sum of absolute values of reach spectrum of 𝐷. i.e., 𝐸𝑅(𝐷) = ∑ |𝜆𝑖| 𝑛 𝑖=1 364 reach energy of digraphs example 2.2. figure 1. reachability matrix is 𝑅(𝐷) = ( 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1) characteristic polynomial of 𝑅(𝐷) is given by 𝑓(𝐷, 𝜆) = 𝜆6 − 6𝜆5 + 15𝜆4 − 20𝜆3 + 15𝜆2 − 6𝜆 + 1. hence, the reach spectrum is { 1 6 }. therefore, the reach energy of d is 𝐸𝑅(𝐷) = 6. 3. reach energy of some graphs theorem 3.1. directed path and alternate path attains same reach energy. proof: let 𝑃𝑛(𝐷) be the directed path with vertex set 𝑉 = {𝑣1, 𝑣2, … , 𝑣𝑛} let 𝐴𝑃𝑛(𝐷) be the alternate path with vertex set 𝑉 = {𝑣1′, 𝑣2′, … , 𝑣𝑛′}. the reachability matrix of 𝑃𝑛(𝐷) is in the upper triangular matrix form with the entries 1. the reachability matrix of 𝐴𝑃𝑛(𝐷) is of the form 𝑅(𝐴𝑃𝑛(𝐷)) = [ 1 1 0 ⋯ 0 0 0 1 0 ⋯ 0 0 0 1 ⋯ 1 0 0 0 0 0 ⋱ 1 0 ⋮ ⋮ ⋮ 0 ⋮ ⋮ 0 0 ⋯ 0 1 1] 𝑣1 𝑣2 𝑣3 𝑣4 𝑣5 𝑣6 365 v. mahalakshmi, b. vijaya praba, and k. palani characteristic polynomial of 𝑃𝑛 is 𝑓(𝑃𝑛(𝐷), 𝜆) = det(𝑅(𝑃𝑛(𝐷)) − 𝜆 𝐼𝑛) 𝑓(𝑃𝑛(𝐷), 𝜆) = | 1 − 𝜆 1 ⋯ 1 0 1 − 𝜆 1 ⋮ ⋮ ⋮ ⋱ 1 0 ⋯ 0 1 − 𝜆 | = (−1)𝑛 (𝜆 − 1)𝑛. characteristic polynomial of 𝐴𝑃𝑛 is 𝑓(𝐴𝑃𝑛(𝐷), 𝜆) = det(𝑅(𝐴𝑃𝑛(𝐷)) − 𝜆 𝐼𝑛) 𝑓(𝐴𝑃𝑛(𝐷), 𝜆) = | | 1 − 𝜆 1 0 ⋯ 0 0 0 1 − 𝜆 0 ⋯ 0 0 0 1 ⋯ 1 0 0 0 0 0 ⋱ 1 0 ⋮ ⋮ ⋮ 0 ⋮ ⋮ 0 0 ⋯ 0 1 1 − 𝜆 | | = (−1)𝑛 (𝜆 − 1)𝑛. clearly, 𝑓(𝑃𝑛(𝐷), 𝜆) = 𝑓(𝐴𝑃𝑛(𝐷), 𝜆). since the characteristic polynomial of 𝑃𝑛(𝐷) and 𝐴𝑃𝑛(𝐷) are same, spectrum of 𝑅(𝑃𝑛) and 𝑅(𝐴𝑃𝑛) are same. hence, the reach spectrum of 𝑅(𝑃𝑛(𝐷)) and 𝑅(𝐴𝑃𝑛(𝐷)) are { 1 𝑛 } and its reach energy is 𝐸𝑅(𝑃𝑛(𝐷)) = 𝐸𝑅(𝐴𝑃𝑛(𝐷)) = ∑ 1 𝑛 1 = 𝑛 therefore, the directed path and alternate path attains same reach energy. theorem 3.2. reach energy of directed star is independent of its orientation. proof: let 𝐾1,𝑛−1(𝐷) be the directed instar with vertex set 𝑣1, 𝑣2, … , 𝑣𝑛−1 let 𝐾′1,𝑛−1(d) be the directed outstar with vertex set 𝑣1′, 𝑣2′, … , 𝑣𝑛−1′ the reachability matrix of 𝐾1,𝑛−1(𝐷) is of the form 𝑅 (𝐾1,𝑛−1(𝐷)) = 𝐼𝑛 + ( 0 01×𝑛−1 𝐽𝑛−1×1 0𝑛−1×𝑛−1 ) the reachability matrix of 𝐾′1,𝑛−1(𝐷) is of the form 𝑅 (𝐾′1,𝑛−1(𝐷)) = 𝐼𝑛 + ( 0 𝐽1×𝑛−1 0𝑛−1×1 0𝑛−1×𝑛−1 ) characteristic polynomial of 𝐾1,𝑛−1(𝐷) is 366 reach energy of digraphs 𝑓(𝐾1,𝑛−1(𝐷), 𝜆) = det(𝑅 (𝐾1,𝑛−1(𝐷)) − 𝜆 𝐼𝑛) 𝑓(𝐾1,𝑛−1(𝐷), 𝜆) = | 1 − 𝜆 0 ⋯ 0 1 1 − 𝜆 0 ⋮ ⋮ 0 ⋱ 0 1 0 0 1 − 𝜆 | = (−1)𝑛 (𝜆 − 1)𝑛 characteristic polynomial of 𝐾′1,𝑛−1(𝐷) is 𝑓(𝐾′1,𝑛−1(𝐷), 𝜆) = det(𝑅 (𝐾′1,𝑛−1(𝐷)) − 𝜆 𝐼𝑛) 𝑓(𝐾′1,𝑛−1(𝐷), 𝜆) = | 1 − 𝜆 1 ⋯ 1 0 1 − 𝜆 0 0 ⋮ 0 ⋱ 0 0 ⋯ 0 1 − 𝜆 | = (−1)𝑛 (𝜆 − 1)𝑛. clearly, 𝑓(𝐾1,𝑛−1(𝐷) , 𝜆) = 𝑓(𝐾′1,𝑛−1(𝐷) , 𝜆). since the characteristic polynomial of 𝐾1,𝑛−1(𝐷) and 𝐾′1,𝑛−1(𝐷) are same, spectrum of 𝑅(𝐾1,𝑛−1(𝐷) ) and 𝑅(𝐾′1,𝑛−1(𝐷) ) are same. hence, the reach spectrum of 𝑅(𝐾1,𝑛−1(𝐷) ) and 𝑅(𝐾′1,𝑛−1(𝐷) ) are { 1 𝑛 and its reach energy 𝐸𝑅 (𝐾1,𝑛−1(𝐷)) = 𝐸𝑅 (𝐾′1,𝑛−1(𝐷)) = ∑ 1 𝑛 1 = 𝑛 therefore, the reach energy of directed star is independent of its orientation. 4. properties of reach eigen values theorem 4.1: let 𝐷 be any digraph. if 𝜆1, 𝜆2, … , 𝜆𝑛 are the reach eigen values of 𝑅(𝐷), then the following condition holds. i. ∑ 𝜆𝑖 = 𝑛 𝑛 𝑖=1 ii. ∑ 𝜆𝑖 2𝑛 𝑖=1 = 𝑛 + 𝛼 + 𝛽; where 𝛼 = ∑ 𝑟𝑖𝑗𝑟𝑗𝑖𝑖>𝑗 and 𝛽 = ∑ 𝑟𝑖𝑗𝑟𝑗𝑖𝑖<𝑗 proof: i. sum of eigen values of 𝑅(𝐷) is same as the trace of 𝑅(𝐷). i.e., ∑ 𝜆𝑖 = ∑ 𝑟𝑖𝑖 𝑛 𝑖=1 𝑛 𝑖=1 ; since each vertex is reachable from itself, all the diagonal entries must be 1. = 1 + 1 + 1 + ⋯ + 1 (𝑛 times) 367 v. mahalakshmi, b. vijaya praba, and k. palani therefore, ∑ 𝜆𝑖 = 𝑛 𝑛 𝑖=1 ii. since, the sum of squares of the eigen values of r is the trace of [𝑅(𝐷)]2 ∑ 𝜆𝑖 2 𝑛 𝑖=1 = ∑ ∑ 𝑟𝑖𝑗 𝑟𝑗𝑖 𝑛 𝑖=1 𝑛 𝑖=1 = ∑ 𝑟𝑖𝑖𝑟𝑖𝑖 𝑛 𝑖=𝑗=1 + ∑ 𝑟𝑖𝑗𝑟𝑗𝑖 𝑛 𝑖≠𝑗=1 = ∑(𝑟𝑖𝑖) 2 𝑖=𝑗 + ∑ 𝑟𝑖𝑗𝑟𝑗𝑖 𝑖>𝑗 + ∑ 𝑟𝑖𝑗𝑟𝑗𝑖 𝑖<𝑗 = 𝑛 + 𝛼 + 𝛽 ; where 𝛼 = ∑ 𝑟𝑖𝑗𝑟𝑗𝑖𝑖>𝑗 and 𝛽 = ∑ 𝑟𝑖𝑗𝑟𝑗𝑖𝑖<𝑗 therefore, ∑ 𝜆𝑖 2 𝑛 𝑖=1 = 𝑛 + 𝛼 + 𝛽 5. bounds for reach energy theorem 5.1: let 𝐷 be a directed graph. let z be the absolute value of determinant of the reachability matrix 𝑅 of 𝐷 i.e., 𝑍 = |det 𝑅(𝐷)| then 𝑛√𝑛 + 𝛼 + 𝛽 ≤ 𝐸𝑅(𝐷) ≤ √(𝑛 + 𝛼 + 𝛽) + 𝑛(𝑛 − 1)𝑍 2/𝑛 proof: we know that cauchy schwarz inequality is (∑ 𝑎𝑖𝑏𝑖 𝑛 𝑖=1 ) 2 ≤ (∑ 𝑎𝑖 𝑛 𝑖=1 ) 2 (∑ 𝑏𝑖 𝑛 𝑖=1 ) 2 put 𝑎𝑖 = 1, 𝑏𝑖 = |𝜆𝑖| (∑|𝜆𝑖| 𝑛 𝑖=1 ) 2 ≤ (∑ 1 𝑛 𝑖=1 ) 2 (∑|𝜆𝑖| 2 𝑛 𝑖=1 ) [𝐸𝑅(𝐷)] 2 ≤ 𝑛2(𝑛 + 𝛼 + 𝛽) 𝐸𝑅(𝐷) ≤ 𝑛√𝑛 + 𝛼 + 𝛽 (1) since arithmetic mean is not smaller than geometric mean, we have 368 reach energy of digraphs 1 𝑛(𝑛 − 1) ∑|𝜆𝑖||𝜆𝑗| 𝑖≠𝑗 ≥ (∏|𝜆𝑖||𝜆𝑗| 𝑖≠𝑗 ) 1 𝑛(𝑛−1) = (∏|𝜆𝑖| 2(𝑛−1) 𝑛 𝑖=1 ) 1 𝑛(𝑛−1) = ∏|𝜆𝑖| 2 𝑛 𝑛 𝑖=1 = |∏ 𝜆𝑖 𝑛 𝑖=1 | 2 𝑛 = |det 𝑅(𝐷)| 2 𝑛 = 𝑍 2 𝑛 therefore, ∑ |𝜆𝑖||𝜆𝑗|𝑖≠𝑗 ≥ 𝑛(𝑛 − 1) 𝑍 2 𝑛 (2) now consider, [𝐸𝑅(𝐷)] 2 = (∑|𝜆𝑖| 𝑛 𝑖=1 ) 2 = ∑|𝜆𝑖| 2 𝑛 𝑖=1 + ∑|𝜆𝑖||𝜆𝑖| 𝑖≠𝑗 ≥ (𝑛 + 𝛼 + 𝛽) + 𝑛(𝑛 − 1) 𝑍 2 𝑛); 𝑏𝑦 (2) hence, 𝐸𝑅(𝐷) ≥ √(𝑛 + 𝛼 + 𝛽 ) + 𝑛(𝑛 − 1) 𝑍 2 𝑛 from (1) and (2), 𝑛√𝑛 + 𝛼 + 𝛽 ≤ 𝐸𝑅(𝐷) ≤ √(𝑛 + 𝛼 + 𝛽) + 𝑛(𝑛 − 1)𝑍 2/𝑛 references [1] adiga.c, balakrishnan, r., & so, w. (2010). the skew energy of a digraph. linear algebra and its applications, 432(7), 1825-1835. [2] arumugam.s, ramachandran.s, invitation to graph theory, scitech. [3] bapat.r.b. b, graphs and matrices hindustan book agency, (2011) [4] gutman. i, the energy of a graph, ber. math – statist. sekt. for schungsz. graz 103 (1978), 1-22. [5] mehtab khan, (2021). a new notion of energy of digraphs. iranian journal of mathematical chemistry, 12(2), 111-125. 369 v. mahalakshmi, b. vijaya praba, and k. palani [6] palani, k., & kumari, m. l. (2022). minimum hop dominating energy of a graph, advan.appl.math.sci. 21(3), 2022, pp. 1169-1179. [7] palani, k., & lalitha kumari, m. (2021, march). total energy of a graph. proceedings of the second international conference on applied mathematics and intellectual property rights (pp. 9-10). [8] pena. i and rada. j, energy of digraphs, linear multilinear algebra 56(5) (2008) 565-579. 370 ratio mathematica weaker forms of nano irresolute and its contra functions a.yuvarani * s. vijaya† p. santhi‡ abstract in this paper the concept of some weaker forms of irresolute and contra irresolute functions in nano topological spaces are studied and its related characteristics are discussed. also we introduced the notion called contra nano alpha irresolute function, contra nano semi irresolute function, contra nano pre irresolute function and its properties are examined. finally, we have revealed some applications related to recent scenario of online teaching and covid-19 which can be expressed as nano irresolute functions and contra irresolute functions respectively. keywords: ns-irresolute function, np-irresolute function, contra nαirresolute function, contra ns-irresolute function, contra np-irresolute function. 2020 ams subject classifications: 54b05 1 *the american college, madurai, india; yuvamaths2003@gmail.com. †thiagarajar college, madurai, india; viviphd.11@gmail.com. ‡the standard fireworks rajaratnam college for women, sivakasi, saayphd.11@gmail.com. 1 india; received on april 18, 2022. accepted on august 25, 2022. published on september 25, 2022. doi: 10.23755/rm.v43i0.764. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. volume 43, 2022 a. yuvarani, s. vijaya, p. santhi 1 introduction in 1982 pawlak [pawlak, 1982] investigated about approximate operations, equality and inclusion on sets. in [crossley and hildebrand, 1972], irresolute functions was introduced and analysed by crossley and hildebrand in topological spaces. weak and strong forms of irresolute functions in topology were discussed by maio and noiri [maio and noiri, 1988]. the conception of nano-topology was initiated by lellis thivagar [thivagar and richard, 2013b],[m. lellis thivagar and richard, 2013] and [m. lellis thivagar and devi, 2017]. also in [thivagar and richard, 2013a], nano continuous functions, nano interior and nano closure was look over by lellis and carmel richard. bhuvaneshwari and ezhilarasi[bhuvaneshwari and ezhilarasi, 2016] introduced irresolute maps and semigeneralized irresolute maps in nano topological spaces. new functions called nsirresolute and np-irresolute functions are originated and look into its behaviour in this article. further the notions called contra nα-irresolute function, contra ns-irresolute function, contra np-irresolute function were introduced and examined their properties. throughout this article we use the notation nts, n-open, nα-open, ns-open, np-open, nα-continuous, ns-continuous, np-continuous for ”nano topological spaces, nano open, nano α-open, nano semi-open, nano preopen sets, nano α-continuous, nano semi-continuous, nano pre-continuous” respectively. similar notation is used for their respective closed sets. 2 nano irresolute functions definition 2.1. let u1 and u2 be nts with respect to τr(x) and τr′ (y). then h : u1 → u2 is called 1. ns-irresolute if h−1(s) is ns-open set in u1 for each ns-open set s in u2, 2. np-irresolute if h−1(s) is np-open set in u1 for each np-open set s in u2. example 2.1. take u1 = {w,x,y,z} with u1/r = {{x,z},{y,w}} and x = {x,z}. then τr(x) = {u1,φ,{x,z}}. let u2 = {q,r,s,t} with u2/r′ = {{q},{r,s},{t}} and y = {q,t}. then τr′ (y) = {u2,φ,{q,t}}. we define h : (u1, τr(x)) → (u2, τr′ (y)) as h(x) = q, h(y) = r, h(z) = t, h(w) = s. then the inverse image of any ns-open in u2 is ns-open in u1 and the inverse image of any np-open in u2 is np-open in u1. therefore h is ns-irresolute and np-irresolute. theorem 2.1. let u1 and u2 be the nts with reference to τr(x) and τr′ (y) and h : u1 → u2 be a mapping. then the statements given below are equivalent. 1. h is nα-irresolute. weaker forms of nano irresolute and its contra functions 2. h−1(s) is nα-closed in u1, for each nα-closed set s in u2. 3. h(nαcl(s)) ⊆ nαcl(h(s)) for each s ⊆ u1. 4. nαcl(h−1(s)) ⊆ h−1(nαcl(s)) for each s ⊆ u2. 5. h−1(nαint(s)) ⊆ (nαint(h−1(s)) for each s ⊆ u2. 6. h is nα-irresolute for each x ∈ u1. proof. (i) =⇒ (ii). it is obvious. (ii) =⇒ (iii). let s ⊆ u1. then, nαcl(h(s)) is a nα-closed set of u2. by (ii), h−1(nαcl(h(s))) is a nα-closed set in u1 and nαcl(s) ⊆ nαcl(h−1h(s)) ⊆ nαcl(h−1(nαcl((h(s)))) = h−1(nαcl(h(s))). so h(nαcl(h(s)) ⊆ nαcl(h(s)). (iii) =⇒ (iv). let s be a subset of u2. by (iii) h(nαcl(h−1(s))) ⊆ nαcl(hh−1(s)) ⊆ nαcl(s). so nαcl(h−1(s)) ⊆ h−1h(nαcl(h−1(s))) ⊆ h−1(nαcl(s)). (iv) =⇒ (v). let s be a subset of u2. by (iv), h−1(nαcl(u2-s)) ⊇ nαcl(h−1(u2− s)) = nαcl(u1−h−1(s)). since u1−nαcl(u1−s) = nαint(s), then h−1(nαint(s)) = h−1(u2−nαcl(u2−s)) = u1−h−1(nαcl(u2−s)) ⊆ u1−nαcl(u1−h−1(s)) = nαint(h−1(s)). (v) =⇒ (vi). let s be a nα-open set of u2, then s = nαint(s). by (v), h−1(s)= h−1(nαint(s)) ⊆ nαint(h−1(s)) ⊆ h−1(s). so, h−1(s) = nαint(h−1(s)). thus, h−1(s) is a nα-open set of u. therefore h is nα-irresolute. (i) =⇒ (vi). let h be nα-irresolute, x ∈ u1 and any nα-open set s of u2, such that h(x) ⊆ s. then x ∈ h−1(s) = nαint(h−1(s)). let b = h−1(s), then b is a nα-open set of u1 and so h(b) = hh−1(s) ⊆ s. thus h is nα-irresolute for each x ∈ u1. (vi) =⇒ (i). let s be a nα-open set of u2, x ∈ h−1(s). then h(x) ∈ s. by hypothesis there exists a nα-open set b of u1 such that x ∈ b and h(b) ⊆ s. hence x ∈ b ⊆h−1(h(b)) ⊆h−1(s) and x ∈ b = nαint(b) ⊆ nαint(h−1(s)). so, h−1(s) ⊆ nαint(h−1((s)). hence h−1(s) = nαint(h−1(s)). thus h is nα-irresolute.2 theorem 2.2. let u1 and u2 be the nts with respect to τr(x) and τr′ (y) and h : u1 → u2 be a 1-1 and onto function. then h is nα-irresolute iff nαint(h(s)) ⊆ h(nαint(s)) for each s ⊆ of u1. proof. let s be any subset of u1. by theorem 2.1 and since h is 1-1 and onto, h−1(nαint(h(s))) ⊆ nαint(h−1(h(s))) = nαint(s). so, hh−1(nαint(h(s))) ⊆h(n αint(s)). thus nαint(h(s)) ⊆ h(nαint(s)). conversely, let s be a nα-open set of u2. then s = nαint(s). by hypothesis, h(nαint(h−1(s))) ⊇ nαint(h(h−1(s))) = nαint(s) = s. thus we get h−1h(nαint (h−1(s))) ⊇h−1(s). since h is 1-1 and onto, nαint(h−1(s))=h−1h(nαint(h−1(s))) ⊇h−1(s). hence h−1(s) = nαint(h−1(s)). so h−1(s) is nα-open set of u. thus h is nα-irresolute.2 a. yuvarani, s. vijaya, p. santhi lemma 2.1. let u1 be a nts with respect to τr(x) then 1. nαcl(s) ⊆ ncl(s) for every subset s of u1, 2. ncl(s) = nαcl(s) for every α-open subset s of u1. theorem 2.3. let u1 and u2 be the nts with respect to τr(x) and τr′ (y) and h : u1 → u2 be a nα-irresolute. then ncl(h−1(s)) ⊆h−1(ncl(s)) for every s ⊆ u2. proof. let s be any n-open subset of u2. since h is nα-irresolute and nαcl(h−1 (s)) is equal to ncl(h−1(a)). by theorem 2.1, nαcl(h−1(s)) ⊆h−1(nαcl(s)) and by lemma 2.1 h−1(nαcl(s) ⊆ h−1(ncl(s)). then nαcl(h−1(s)) ⊆ h−1(ncl(s)). therefore ncl(h−1(s)) ⊆ h−1(ncl(s)).2 theorem 2.4. let u1 and u2 be the nts with respect to τr(x) and τr′ (y).then h : u1 → u2 is a nsirresolute iff for each ns-closed subset h−1(s) is ns-closed in u1. proof. if h is ns-irresolute, then h−1(b) is ns-open in u1 for each ns-open set b ⊆ u2. if s is any ns-closed subset of u2, then u2−s is ns-open. thus h−1(u2−s) is ns-open in u1, but h−1(u2−s) = u1−h−1(s) so that h−1(s) is ns-closed in u1. conversely, if for all ns-closed set s ⊆ u2, h−1(s) is ns-closed in u1 and if b is any ns-open subset of u2, then u2−b is ns-closed. also h−1(u2−b) = u1−h−1(b) which is ns-closed in u1. therefore h−1(b) is ns-open set in u1. hence h is ns-irresolute.2 theorem 2.5. if h : u1 → u2 and g : u2 → u3 is ns-irresolute(np-irresolute) then g◦h : u1 → u3 is ns-irresolute(np-irresolute). proof. (i) if a ⊆ u3 is ns-open(np-open), then g−1(s) is ns-open(np-open) set in u2 because g is ns-irresolute(np-irresolute). consequently since h is nsirresolute(np-irresolute), h−1(g−1(s))= (g◦h)−1(s) is ns-open(np-open) set in u1. hence g◦h is ns-irresolute(np-irresolute). theorem 2.6. if h : u1 → u2 is nα-irresolute(ns-irresolute, np-irresolute) and g : u2 → u3 is nα-continuous(ns-continuous, np-continuous) then g◦h : u1 → u3 is nα-continuous(ns-continuous, np-continuous). proof. let s ⊆ u3 is n-open. since g is nα-continuous(ns-continuous, npcontinuous), g−1(s) is nα-open(ns-open, np-open)set in u2. consequently since h is nα-irresolute(ns-irresolute, np-irresolute), h−1(g−1(s)) = (g◦h)−1(s) is nαopen(ns-open, np-open) set in u1. hence g◦h is nα-continuous(ns-continuous, np-continuous).2 weaker forms of nano irresolute and its contra functions theorem 2.7. let u1 and u2 be the nts with respect to τr(x) and τr′ (y). a function h : u1 → u2 is 1. ns-irresolute and np-irresolute then h is nα-irresolute, 2. nα-continuous iff it is ns-continuous and np-continuous. proof. it is obvious. 3 nano contra irresolute functions here we introduce contra irresolute functions and its characteristics are discussed. the notations used are ncα-open, ncs-open, ncp-open for ”nano contra α-open, nano contra semi-open, nano contra pre-open functions” respectively. definition 3.1. let u1 and u2 be the nts with respect to τr(x) and τr′ (y). then h : u1 → u2 is said to be 1. ncα-open if h(s) is nα-closed in u2 for each n-open s in u1, 2. ncs-open if h(s) is ns-closed in u2 for each n-open s in u1, 3. ncp-open if h(s) is np-closed in u2 for each n-open s in u1. example 3.1. 1. let u1 = {j,k,l} with u1/r = {{l},{j,k}} and x = {k,l}. then τr(x) = {u1,φ,{l},{j,k}}. let u2 = {x,y,z}, u2/r’ = {{y},{x,z}} and y = {y,z}. subsequently τr′ (y) = {vu2,φ,{y},{x,z}}. we label h : (u1, τr(x)) → (u2, τr′ (y)) as h(j) = x, h(k) = z, h(l) = y. subsequently h(s) is nα-closed in u2 for every n-open set s in u1. hence h is ncα-open. 2. let u1 = {j,k,l,m} with u1/r = {{j},{l},{k,m}} and x = {j,k}. subsequently τr(x) = {u1,φ,{j},{k,m},{j,k,m}}. let u2 = {p,q,r,s} with u2/r’ = {{p},{s},{q,r}} and y = {p,r}. subsequently τr′ (y) = {u2,φ,{p},{q,r},{p, q,r}}. we label h : (u1, τr(x)) → (u2, τr′ (y)) as h(j) = s, h(k) = r, h(l) = p, h(m) = q. then h(s) is ns-closed in u2 for every n-open set s in u1. hence h is ncs-open. 3. let u1 = {j,k,l,m} with u1/r = {{l},{m},{j,k}} and x = {j,l}. subsequently τr(x) = {u1,φ,{l},{j,k},{j,k,l}}. let u2 = {p,q,r,s} with u2/r’ = {{q},{r},{p,s}} and y = {p,r}. subsequently τr′ (y) = {u2,φ,{r},{p,s},{p,r, s}}. we define h : (u1, τr(x)) → (u2, τr′ (y)) as h(j) = q, h(k) = s, h(l) = p, h(m) = r. then h(s) is np-closed in u2 for every n-open set s in u1. hence h is ncp-open. a. yuvarani, s. vijaya, p. santhi definition 3.2. let u1 and u2 be the nts with respect to τr(x) and τr′ (y). then h : u1 → u2 is said to be cnα-irresolute(cns-irresolute, cnp-irresolute) if h−1(s) is nα-closed(ns-closed, np-closed)set in u1 for every nα-open set(nsopen, np-open) in u2 . example 3.2. 1. let u1 = {j,k,l,m} with u1/r = {{j},{k},{l},{m}} and x = {j}. then τr(x) = {u1,φ,{j}}. let u2 = {w,x,y,z} with u2/r’={{w},{x},{y },{z}} and y = {x,y,z}. then τr′ (y) = {v,φ,{x,y,z}}. we label h : (u1, τr(x)) → (u2, τr′ (y)) as h(j) = w, h(k) = x, h(l) = y, h(m) = z. then h−1(s) is ns-closed in u1 for every ns-open set s in u2 . therefore h is cnα-irresolute and cns-irresolute. 2. let u1 = {p,q,r} with u1/r = {{p},{q,r}}and x={q,r}. then τr(x) = {u1,φ,{q,r}}. let u2 = {j,k,l} with u2/r’ = {{j},{k,l}} and y = {j}. then τr′ (y) = {u2,φ,{j}}. we define h : (u1,τr(x)) → (u2, τr′ (y)) as h(p) = j, h(q) = k, h(r) = l. then h−1(s) is np-closed in u1 for every np-open set s in u2 . so h is cnp-irresolute. theorem 3.1. consider u1 and u2 be the nts with respect to τr(x) and τr′ (y). then h : u1 → u2 is cnα-irresolute iff for each nα-closed subset s of u2, h−1(s) is nα-open in u1. proof. if h is cnα-irresolute, then for each nα-open subset b in u2, h−1(b) is nα-closed in u1. if s is any nα-closed subset in u2, then u2− s is nα-open. thus h−1(u2− s) is nα-closed but h−1(u2− s) = u1 − h−1(s) so that h−1(s) is nα-open in u1. conversely, if, for all nα-closed subsets s of u2, h−1(s) is nα-open in u1 and if b is any nα-open subset of u2, then u2 − b is nα-closed. also h−1(u2− b) = u1 − h−1(b) is nα-open. thus h−1(b) is nα-closed in u1. hence h is cnα-irresolute.2 corolary 3.1. let u1 and u2 be the nts with respect to τr(x) and τr′ (y). then h : u1 → u2 is cns-irresolute(cnp-irresolute) if and only if for each ns-closed subset(np-closed subset) s of u2, h−1(s) is ns-open(np-open) in u1. theorem 3.2. if the functions h : u1 → u2 and g : u2 → u3 are cnα-irresolute then g◦h : u1 → u3 is nα-irresolute. proof. if s ⊆ u3 is nα-open, then g−1(s) is nα-closed in u2 because g is cnαirresolute. consequently since h is cnα-irresolute, h−1(g−1(s))= (g◦h)−1(s) is nα-open set in u1, by corollary 4.6. hence g◦h is nα-irresolute.2 corolary 3.2. if the functions h : u1 → u2 and g : u2 → u3 are cns-irresolute (cnp-irresolute) then g◦h : u1 → u3 is ns-irresolute(np-irresolute). weaker forms of nano irresolute and its contra functions theorem 3.3. if the function h : u1 → u2 is cnα-irresolute and the function g : u2 → u3 is ncα-continuous then g◦h : u1 → u3 is nα-continuous. proof. let s ⊆ u3 is n-open. since g is ncα-continuous, g−1(s) is nα-closed in u2. consequently since h is cnα-irresolute, h−1(g−1(s))= (g◦h)−1(s) is nα-open set in u1, by theorem 4.5. hence g◦h is nα-continuous.2 corolary 3.3. if the function h : u1 → u2 is cns-irresolute(cnp-irresolute) and the function g : u2 → u3 is ncs-continuous(ncp-continuous) then g◦h : u1 → u3 is ns-continuous(np-continuous). theorem 3.4. let u1 and u2 be the nts with respect to τr(x) and τr′ (y). then h : u1 → u2 is cns-irresolute and cnp-irresolute then h is cnα-irresolute. proof. it is obvious. theorem 3.5. let u1 and u2 be the nts with respect to τr(x) and τr′ (y). then h : u1 → u2 is cnα-irresolute then it is ncα-continuous. proof. consider the n-open set t ⊆ u2. which implies t is a nα-open set in u2. but h is cnα-irresolute so h−1(t) is a nα-closed set in u1. it shows that h is ncα-continuous function.2 4 applications finally, we discuss the application of nano irresolute functions and its contra functions. example 4.1. advances in technology and some pandemic situations allow students to study entirely online. consider the impact of e-learning on students characteristics, as a function of, the innovative strategies used in online teaching. let us consider some of the strategies used in online teaching are powerpoint presentation (p), videos (v), mind map (m), online quiz (q), group discussion (g) and its impact on students characteristics are intellectually curious (i), good time management (t), self-driven (s), enhanced communication skills (c). let u1={p,v,m,q,g} be the universe of the innovative strategies used in online teaching with u1/r ={{p,v},{m,g}, {q}} and x1 ={p,q}. subsequently τr(x1)={u1,φ,{q},{p,v},{p,v,q}}. let u2 ={i,t,s,c} be the universe on students characteristics with u2/r’ = {{i,s},{t,c}} and x2 = {t,c}. then τr′ (x2) = {u2, φ, {t,c}}. we define h : (u1, τr(x1)) → (u2, τr′ (x2)) as h(p) = c, h(v) = c, h(m) = i, h(q) = t and h(g) = s. then for every nα-open set in u2, inverse image is nα-open set in u1 and also for every ns-open set in u2, inverse image is ns-open set in u1. hence h is nα-irresolute a. yuvarani, s. vijaya, p. santhi and ns-irresolute. thus, the impact of e-learning on students characteristics, as a function of the innovative strategies used in online teaching, are nα-irresolute and ns-irresolute function. example 4.2. the main cause of illness is the infectious diseases. however, some initial precautions may help to prevent infections. if not, it leads to serious medical conditions and sometimes to death. consider the precautionary measures to be adopted to prevent affecting from covid-19, as a function of, its symptoms. let the symptoms of covid-19 are dry cough (k), fever (f), shortness of breath (b), loss of taste/smell (l) and the precautionary measures to be adopted are sanitizing (s), social distancing (d), wearing mask (m), boosting immunity power (i). let u1 = {k,f,b,l} be the universe of symptoms of covid-19 with u1/r = {{k},{f},{b},{l}} and x1 = {k}. then τr(x1) = {u1, φ, {k}}. let u2 = {s,d,m,i} be the universe of the precautionary measures to be adopted with u2/r’ = {{s},{d},{m},{i}} and x2 = {d,m,i}. then τr′ (x2) = {u2, φ, {d,m,i}}. we define h : (u1, τr(x1)) → (u2, τr′ (x2)) as h(k) = s, h(f) = d, h(b) = m and h(l) = i. then for every nα-open set in u2, inverse image is nα-closed set in u1 and also for every ns-open set in u2, inverse image is ns-closed set in u1. thus h is contra nα-irresolute and contra ns-irresolute. thus, the precautionary measures to be adopted to prevent affecting from covid19, as a function of its symptoms, are contra nα-irresolute and contra ns-irresolute function. 5 conclusions through the above discussions we have summarized the conceptulation of irresolute functions and contra irresolute functions in nts along with examples. further, we have revealed some applications related to current scenario of online teaching and covid-19 which can be expressed as nano irresolute functions and contra irresolute functions respectively. thus these notions can be applied in many real time situations. references k. bhuvaneshwari and a. ezhilarasi. on nano semi-generalized irresolute maps in nano topological spaces. international journal of mathematical archive, 7(3):68–75, 2016. s. crossley and s. hildebrand. semi topological properties. fundamenta mathematicae, 74:233–254, 1972. weaker forms of nano irresolute and its contra functions s. j. m. lellis thivagar and v. s. devi. on new class of contra continuity in nano topology. italian journal of pure and applied mathematics, 41:1–12, 2017. s. j. m. lellis thivagar and c. richard. remarks on weak forms of nano continuity. iiste, 3(7):1–13, 2013. g. d. maio and t. noiri. weak and strong forms of irresolute functions. rend.circ.mat.palermo, 18:255–273, 1988. z. pawlak. rough sets. international journal of computer and information sciences, 11(5):341–356, 1982. m. l. thivagar and c. richard. on nano continuity. mathematical theory and modelling, 3(7):32–37, 2013a. m. l. thivagar and c. richard. on nano forms of weakly open sets. international journal of mathematics and statistics invention, 1(1):31–37, 2013b. ratio mathematica 27 (2014) 3-25 issn:1592-7415 multi-criteria media mix decision model for advertising multiple product with segment specific and mass media sugandha aggarwala, anshu guptab, p.c. jhaa adepartment of operational research, university of delhi, delhi-110007, india sugandha\_or@yahoo.com anshu@aud.ac.in bsbppse, ambedkar univeristy delhi, delhi-110006, india jhapc@yahoo.com abstract judicious media planning decisions are crucial for successful advertising of products. media planners extensively use mathematical models supplemented with market research and expert opinion to devise the media plans. media planning models discussed in the literature largely focus on single products with limited studies related to the multi-product media planning. in this paper we propose a media planning model to allocate limited advertising budget among multiple products advertised in a segmented market and determine the number of advertisements to be given in different media. the proposed model is formulated considering both segment specific and mass media vehicles to maximize the total advertising reach for each product. the model also incorporates the cross product effect of advertising of one product on the other. the proposed formulation is a multi-objective linear integer programming model and interactive linear integer goal programming is discussed to solve the model. a real life case study is presented to illustrate the application of the proposed model. 3 s. aggarwal, a. gupta and p. c jha key words: multiple products, mass advertising, segment specific advertising, spectrum effect, media planning, multi-objective decision making, interactive approach. 2000 ams: 90b60, 90c10, 90c29. 1 introduction a firm’s market share and profit are driven by consumer demand and spending. advertising carried by the firms to promote their products play a crucial role in fuelling consumer demand. it is through media that consumers receive advertising messages. it acts as a link between the advertisers and the consumers. media such as television, radio, newspapers, magazines, and the internet act as distributors of the advertising messages. media planning is a challenging process and the media choices are made such that the advertising objectives are met. the goal of a media planner is to reach the target audience with the right message through the right media. advertising reach and frequency are the critical elements in setting up a media plan [19]. this study proposes a mathematical programming media allocation model to maximize the advertising reach of a firm that markets multiple products advertised through different media in a segmented market. there are two major aspects of media planning, viz. selection of the media and allocation of the advertising budget. a media planner focuses on reaching its target customers with a right message that can convert them into potential buyers. the target market of a product can be taken as uniform or it can be bifurcated in to various segments based on the customer profile characteristics. when the market is considered as uniform, the advertising is carried at the mass level through the media that could reach all the customers. though, the customers in the target market possesses some common characteristics that identify them as the potential customers still there exist differences in how they respond to the products and the advertising messages. if the product is advertised only at the mass level with a uniform advertising strategy, due to the differential behaviour of the potential market customers it may not be effective in influencing the customers to buy the product. in the recent years firms have tried to reach its customers with advertising that is tailored with respect to their individual characteristics so that the advertising not only reaches them but also convert them into potential buyers. segmentation is an important concept of marketing that helps the advertisers to develop a media plan with respect to the customer’s characteristics. given the importance of segment driven marketing, importance of mass mar4 media mix decision model for multiple product keting can’t be undermined as it creates a wider spectrum of reach. hence the marketers choose to adopt the advertising strategy such that the product is advertised using mass media and also with segment driven advertising media. the reach obtained in segments can thus be obtained both from segment specific advertising and mass advertising. the model proposed in this paper incorporates this idea and develops a media plan that allocates advertising budget for both mass and segment specific media. companies are increasingly extending their products into product lines that are related or fall into distant categories. marketing product lines instead of single product gives a competitive edge to the firms. it helps in meeting the diversified demand of products that are related which customer tend to use together and also provides a variety to the customers. firms have limited resources in terms of value that can be used for advertising. for the case of single product advertised in a segmented market, the segments compete for media budget allocation among themselves and with mass media allocation. if a firm markets several products the competition for advertising budget first exists between the products and then at the segment and mass level. at any instant of time if several products are marketed by a firm advertising reach of an individual product no longer remains independent of other products. due to substitution or complimentary effect that one product may have on other the advertising reach is also affected. very limited research has investigated media planning model for multiple products jointly [16] . in this paper we propose a multi-objective linear integer media planning model to allocate advertising budget between several products marketed by a firm through various media in a segmented market. the model allocates media budget and also determines the number of advertisements for each product, in all chosen media both at segment and mass level. it also incorporates cross product effect of advertising among products and maximizes the total advertising reach taking all products together. interactive goal programming technique is discussed to solve the formulated model. the paper is organized as follows: literature review is carried in section 2. in section 3, the model formulation and solution procedure are discussed. a case study is presented in section 4 illustrating the solution methodology. concluding remarks are made in section 5. 2 literature review the researchers have worked on various aspects of media planning such as the models for media selection, models concerning the ”timing” aspect, market segmentation studies, budget allocation models, media scheduling 5 s. aggarwal, a. gupta and p. c jha models, media effectiveness models. broadbent [3] presented a review of the simulation and optimization procedures for the media planning models. the author discussed a number of media planning models and classified them into two approaches: mathematical model approach and algorithmic approach. a linear programming media allocation model was proposed by bass and lonsdale [1] with an objective to maximize the media exposures for one product for a single time unit. authors explored the influence of several types of constraints on the model solution. little and lodish [14] formulated a media planning model based on a heuristic search algorithm to select and schedule media maximizing the total market response in different segments over the several time periods. zufryden [20] developed media planning models with an objective of maximizing sales and determine the optimal media schedule over time. they considered stochastic response behavior in the objective function and later developed heuristics for solving the model [21]. dwyer and evans [7] proposed an optimization model for to select the best set of mailing lists in the direct mail advertising maximizing the proportion of customers reachable with direct mail pieces. the formulated binary integer model is solved through the branch and bound algorithm. korhonen et al. [12] proposed an evolutionary approach to media selection model. the model constraints and objective have interchangeable role in this approach. the iterations are performed for different set of objectives and constraints, computing the decision maker’s value function in each iteration. then the value function most suited to the decision maker is chosen as the solution. the study was carried out for a software company in finland. doyle and saunders [6] developed a model to determine the spending on the promotion of multiple products for a retail store. the model optimally allocates budget to the promotional campaigns where each campaign is for a specific product. they considered cross product effect of advertising campaigns that lags or leads a particular campaign for up to four periods. a logarithmic linear regression model was proposed by the authors. danaher and rust [5] developed a model with an objective of maximizing the return on investment considering the diminishing return on the advertising and calculated the optimal amount of expenditure on the media campaign. a media planning model based on the analytic hierarchy process was developed by kwak et al. [13]. the model is developed to allocate the budget in the media categories and determine the number of advertisements for different media categories for digital products. three criterion customer, advertising and budget were considered to be fulfilled through the model. the solution methodology based on pre-emptive goal programming technique was used. buratto et al. [4] analyzed the media selection problem to choose an 6 media mix decision model for multiple product advertising channel for the pre-launch campaign for a new product (as cited in [11]). authors considered a segmented market with several advertising channels that have different diffusion spectra and efficiencies. the problem is analyzed in two steps. first, an optimal control problem is solved explicitly in order to determine the optimal advertising policy for each channel. then a maximum profit channel is chosen. they discussed a simulation where the choice of a newspaper among six italian newspapers is presented. grosset and viscolani [8] proposed a dynamic profit maximizing advertising model comparing the model performance under two strategies viz. 1) single medium advertising for a segmented market, that reaches segments with the same message but with varying effectiveness and 2) advertising independently for each segment through a single segment specific medium. the profit is measured in terms of goodwill where the growth of goodwill depends on the advertising effort and the goodwill decays due to forgetting of the advertised brand. viscolani [18] proposed a non-linear programming advertising model for a segmented market to select a set of advertising media with an objective of maximizing profit. using the approach similar to the grosset and viscolani [8] they suggested to use multiple media. hsu et al. [9] gave a fuzzy model using genetic algorithm to determine the optimum advertising mix and the number of insertions in different promotional instruments based on linguistic preferences of the domain experts. bhattacharya [2] proposed an integer programming model to determine the optimal number of insertions in different media with an objective of maximizing the reach to the target population for a single product. jha et al. [10] extended the model for the multiple products and a segmented market. saen [17] proposed a model for the selection of media through the approach of data envelopment analysis in presence of flexible factors and imprecise data. royo et al. [16] proposed an advertising budget allocation model for multiple products considering cross elasticity of products. they optimised the investment on advertising in multiple media for multiple products. this model was further extended by royo et al. [16] under stochastic environment. jha et al. [11] proposed an integer linear programming model of media planning for a single product advertised with multiple media with mass and segment specific advertising strategies. the model is developed with reach maximization objective. as discussed above an extensive literature has been developed on optimization of media planning decisions. most of the researchers have focused on media planning models for single product. in the present age, firms market several products simultaneously to provide product variety to the customer. the advertisement budget is to be divided among the products judiciously. in case of multi-product offering it is also observed that the one product ad7 s. aggarwal, a. gupta and p. c jha vertising affects the advertising of other product[16]. the effect can either be substitutive or complimentary. it is important to measure and take account of this effect in media planning decisions. this necessitates joint media planning for the range of products such that the advertising budget can be shared between the products judiciously at the same time accounting for the crossproduct effect of advertising which is considered in this paper. the study carried also integrates concept of media planning for multiple products with segmentation aspect. another distinguished feature of the study is that we consider two types of advertising strategies viz. mass and segment specific in the model development. this differentiation between advertising strategies has been recently carried in some recent studies [11]. both strategies affect advertising message reach in the potential market in different manner. while the mass advertising spread reach over the entire market widely, segment specific advertising plays crucial role in targeting segments. the model developed in this paper maximizes the total reach of all the products taking in to consideration budgetary restrictions and bounds on the decision variables. the reach function is formulated considering the cross product effect of advertising. the model is tested on a real life case study. 8 media mix decision model for multiple product 3 model development 3.1 notation i index for segments (i = 0, 1, ...,n) j index for advertising media (j = 1, 2, ...,mi) k index for media options (k = 1, 2, ...,kij) l index for slot in a media (l = 1, 2, ...,lij) p index for products (p = 1, 2, ...,p ) q index for customer profile characteristics (q = 1, 2, ...,q) jkl jth media, kth media option, lth slot a p ijkl reach per advertisement for p th product in ith segment, jklth media driver cijkl average number of readers/viewers of jkl th media driver in segment i cijkl cost of inserting one advertisement in jkl th media driver in segment i v p ijkl lower bound on the number of advertisements in jkl th media driver of segment i for pth product u p ijkl upper bound on the number of advertisements in jkl th media driver of segment i for pth product x p ijkl decision variable denoting the number of advertisements to be given in jklth media driver of segment i for pth product e p irjkl percentage of people who follow jkl th media driver in segment i, and are pth product’s potential customers possessing rth profile characteristic. α p ijk spectrum effect of k th media option of jth mass media vehicle on ith segment for pth product; ; 0 < α p ijk < 1 wrp relative importance of r th customer profile characteristic for pth product r minimum proportion of budget allocated for mass advertisement g total advertising budget zp total reach of p th product ap reach component solely due to advertisement of product p θpf constant of proportionality representing cpe of advertising of product p on reach of product f 3.2 model formulation assuming a firm markets p products in a segmented market and the segments index vary from 1 to n and index 0 represents the mass media. 9 s. aggarwal, a. gupta and p. c jha the mathematical model to maximize the total reach of advertising for each product through the mass and segment specific media is formulated as follows vector maximizez = [z1,z2, ....,zp ] t (1) subject to (p1) p∑ p=1 n∑ i=0 mi∑ j=1 kij∑ k=1 lij∑ l=1 cijklx p ijkl ≤ g (2) p∑ p=1 m0∑ j=1 k0j∑ k=1 l0j∑ l=1 c0jklx p 0jkl ≥ rg (3) x p ijkl ≥ v p ijkl ∀p = 1, 2, ...p ;i = 0, 1, 2, . . . n; j = 1, 2, . . . mi; k = 1, 2, . . . kij; l = 1, 2, . . . lij (4) x p ijkl ≤ u p ijkl ∀p = 1, 2, ...p ;i = 0, 1, 2, . . . n; j = 1, 2, . . . mi; k = 1, 2, . . . kij; l = 1, 2, . . . lij (5) x p ijkl ≥ 0 and integers ∀p = 1, 2, ...p ;i = 0, 1, 2, . . . n; j = 1, 2, . . . mi; k = 1, 2, . . . kij; l = 1, 2, . . . lij (6) where zp = ap + p∑ f=1 f 6=p θpfaf (7) ap = n∑ i=1   mi∑ j=1 kij∑ k=1 lij∑ l=1 a p ijklx p ijkl + m0∑ j=1 k0j∑ k=1 l0j∑ l=1 α p ijkl ( a p 0jklx p 0jkl ) (8) a p ijkl = { r∑ r=1 wrp e p irjkl } cijkl (9) equation (1) represented by z is a vector of objective functions with the components denoting the advertising reach of each product p. component of z denoted by zp (expressed mathematically as (7)) represents the combined reach from advertising for the product p and the cross product effect from advertising of other products. where the reach expected to obtain from advertising for the product p is expressed as ap (given by (8)). the individual 10 media mix decision model for multiple product advertising reach of each product as given by equation (8) is the sum of the reach from segment specific advertising and spectrum effect of the mass advertising in the segments. the per unit advertisement reach as given in equation (9) is the product of the readership/viewership of the media driver and the relative proportion of potential customers among them. equation (2) represents the budgetary constraint. knowing the importance of mass advertising it is likely that media planner specify a lower bound on the budget to be spent on mass advertising as otherwise very little budget may be allocated to the mass media. equation (3) represents the lower bound constraint on the mass media budget allocation. constraint (4) and (5) are the lower and upper bounds specified by the media planner on the number of advertisements in different media for different products to ensure the diversity in advertising budget allocations rather than allocating the entire budget to some specific set of media. constraint (6) imposes the decision variable to take integral values. in the literature authors have suggested to formulate evolutionary model [12] wherein constraints and objectives roles can interchange. this allows flexibility to the decision maker, tradeoff the model variables and ensures that an efficient solution is obtained. in this direction in order to obtain an efficient solution and ensure some minimum reach for every product first we solve the model (p1) for each reach objective one by one to obtain the advertising reach aspirations for all products. these aspirations are set as lower bound constraints on reach objective and the resulting model is formulated as follows vector maximizez = [z1,z2, ....,zp ] t subject to constraints (2)-(6) and zp ≥ z∗p ∀p = 1, 2, ...p (p2) weighted sum multi-objective model using scalar weights µp; ∑ µp = 1; (p = 1, 2, ...p) according to the relative importance of the products [15] is formulated using (p2) to obtain the media planning model as given in (p3) maximize p∑ p=1 µpzp subject to constraints (2)-(6) and zp ≥ z∗p ∀p = 1, 2, ...p (p3) the weights in the model (p3) can be given by the decision maker or computed through the interactive approach (discussed in detail in [11]). the 11 s. aggarwal, a. gupta and p. c jha linear integer optimization model (p3) is solved by coding on lingo optimization modelling software. the solution to model (p3) may result in infeasibility due to high aspirations on reach objective for products. further a goal linear integer model is formulated for model (p2) to obtain a compromised solution and trade off the reach aspirations and budget. solution methodology: goal programming in goal programming, the solution is obtained such that the deviations from the goals are minimized. deviations may be either positive or negative. problem (p2) is solved in two stages. in stage 1 the model is solved to minimize the deviations of the rigid constraints and in the second stage goal deviations are minimized incorporating the solution of first stage. the formulations of the two stages of goal programming are given as follows stage 1 minimize ρ1 + η2 + p∑ p=1 n∑ i=0 mi∑ j=1 kij∑ k=1 lij∑ l=1 ηp ijkl + p∑ p=1 n∑ i=0 mi∑ j=1 kij∑ kj=1 lij∑ lj=1 ρ ′p ijkl subject to constraints (p4) p∑ p=1 n∑ i=0 mi∑ j=1 kij∑ k=1 lij∑ l=1 cijklx p ijkl + η1 −ρ1 = a (10) p∑ p=1 m0∑ j=1 k0j∑ k=1 l0j∑ l=1 c0jklx p 0jkl + η2 −ρ2 = ra (11) x p ijkl + η p ijkl −ρ p ijkl = v p ijkl ∀p = 1, 2, ...p ;i = 0, 1, 2, . . . n; j = 1, 2, . . . mi; k = 1, 2, . . . kij; l = 1, 2, . . . lij (12) 12 media mix decision model for multiple product x p ijkl + η p ijkl −ρ p ijkl = u p ijkl ∀p = 1, 2, ...p ;i = 0, 1, 2, . . . n; j = 1, 2, . . . mi; k = 1, 2, . . . kij; l = 1, 2, . . . lij (13) x p ijkl ≥ 0 and integers ∀p = 1, 2, ...p ;i = 0, 1, 2, . . . n; j = 1, 2, . . . mi; k = 1, 2, . . . kij; l = 1, 2, . . . lij (14) ηp ijkl ,ρp ijkl ,η ′p ijkl ,ρ ′p ijkl ≥ 0 ∀p = 1, 2, ...p ;i = 0, 1, 2, . . . n; j = 1, 2, . . . mi; k = 1, 2, . . . kij; l = 1, 2, . . . lij (15) ηi,ρi ≥ 0∀i = 1, 2 (16) stage 2 minimize g (η,ρ,x) = p∑ p=1 λp+2ηp+2 subject to constraints (10)-(15) and zp + ηp+2 −ρp+2 = z∗p ∀p = 1, 2, ...p ηi,ρi ≥ 0 ∀i = 1, 2, ..., (p + 2) (p5) where g(η,ρ,x) is objective function of (p5) and ηp+2, ρp+2, are negative and positive deviational variables of goals for pth product objective function. 4 case study to illustrate the application of the proposed model a case study is presented in this section demonstrating the media planning decision of a firm marketing five products (p1-p5) in the market. the name of the firm has not been disclosed due to the commercial confidentiality. the firm has to devise an advertising plan for its products for a period of one quarter. on the basis of geographic segmentation, the market for all the products is divided into fourteen segments (say s1-s14). the company wants to promote all products at the mass level as well as at the segment level. the firm’s potential market is described on the basis of demographic characteristics: gender and income level, that is the potential market to which these products are targeted to, are females belonging to middle class group. for segment level advertising in each segment, up to four newspapers (rnp1-rnp4), and two television channels (rch1, rch2) are selected. for the mass advertising four newspapers (nnp1-nnp4), and two television channels (nch1, nch2) are selected. each of these media is chosen 13 s. aggarwal, a. gupta and p. c jha based on the potential market preferences, expert opinion and the market research. further in each media there are different slots, such as in case of newspapers we can advertise on front page (fp) and/or other pages (op). similarly in case of television, slots can be classified as prime time (pt) and other time (ot). the total budget given by the firm for the media planning is rs. 800 millions. the minimum proportion of the budget allocated to mass media is set as 30 %. the data given by the firm is confidential and used with appropriate rescaling (given in tables 3-12 in the appendix). the potential customer profile matrix corresponding to each media is computed for all segments by conducting a survey of on a sample. the percentage profile matrix computed for product 1 is given in table 3. similarly profile matrices are computed for all products. the weights defining relative importance of the potential customer profile characteristics gender and income level is given in table 4. the values of the relative importance are inferred from the primary and secondary data with expert opinion. the cross product effect coefficient matrix is shown in table 5. table 6 gives the spectrum effect coefficient of the mass media on the various segments of the potential market. the cost of advertisement in newspapers is measured in per square cm and an advertising space of 4cm x 6cm is considered. in case of television advertisement rates are given per 10 second slot and 30 second advertisement duration is preferred by the media planner. the advertising costs used in the study are given in table 7. the media planner has also provided the lower and upper bounds on number of advertisements to be given for different products in different media as given in table 8-12. these bounds are set to ensure the minimum visibility of ads in every media and distribute the advertising resources judiciously such that all chosen media can be used for advertising. the optimization model (p1) is coded on lingo optimization modelling software. in order to compute the target goals on the reach objectives for each product, first model (p1) is solved for each of the five products as a single objective model taking reach objective of one product at a time. the branch and bound method is used in the software to solve the model. using these aspirations as the lower bounds on reach objectives for all products, the media planning model (p3) is coded. as the scalar weights of relative importance of product are not known, so we use interactive technique (for details of the method reader may refer [11]) to determine these weights. for the first iteration of interactive technique, 125 (=v ) dispersed weighing vectors are generated randomly such that the components of each vector lie in the range [0, 1] and the sum of all the components of each vector is equal to one. taking a suitable value of d (computed using mathematical 14 media mix decision model for multiple product expression given in the algorithm) and through forward filtering approach 10 non-dominated distinct vectors (w) are filtered with l2 metric distances between each set of vector. the problem (p3) is solved for all these 10 filtered weighing vectors. the model shows infeasibility with these filtered weighting vectors. thus we form an interactive weighted sum goal programming model for (p1) to obtain a compromised solution using the reach targets as goals on the reach objectives. the goal programming model is solved in two stages. in stage 1, the deviations corresponding to the rigid constraints are minimized and in stage 2 the deviations from the reach goals are minimized. first, the model (p4) is coded and solved in lingo. in the next stage of goal programming, model (p5) is coded incorporating the solution obtained in stage 1. the weights given to the reach deviations are determined using interactive approach. using the ten non-dominated distinct vectors generated earlier, the problem (p5) is solved 10 times. the solution and the objective function values are tabulated for all the runs and 5 (=p) best criterion vectors are filtered from 10 runs which are presented to the decision maker. on discussion with the decision maker, most preferred solution is selected. using the weighing vector corresponding to the selected solution, the reduction factor is calculated and a new interval is formed between which new generation of weighting vectors is generated and the iteration is repeated. five iterations of the interactive approach is carried based on the termination criteria (t . k) of the algorithm. note that the parameters of the interactive algorithms are defined in jha et al. [11] and same notations are used in this paper. all the calculations are carried out on a computing device with intel core duo 1.40 ghz processor and 4 gb ram. the average time taken to solve each problem is 2-4 seconds. it can be seen from solution in table 1 that as we move from iteration 1 to iteration 5, the total reach obtained from all the five products together improves. but the percentage change in the total objective function value decreases in successive iterations (except one iteration). as per the termination criteria of the interactive algorithm should converge in five iterations and we can see that the solutions of iteration 4 and 5 are very close to each other (% change=.09%), so the algorithm is terminated in five iterations. the budget is fully utilized with 24.27 % of the total budget allocated to newspaper and the rest of 75.73% to tv. with these budget allocations among media it is expected to obtain approximately 20% of the reach from newspaper advertising and rest 80% from tv advertising. the distribution of budget among mass and segment level media is 31% and 69% (approx.) respectively. the product wise percentage allocation of the total budget and expected reach is given in table 2. the optimal number of advertisements for different media for all the products is given in table 13-17 in the appendix. 15 s. aggarwal, a. gupta and p. c jha table 1: iteration parameters and the solution obtained iteration 1 (h = 0) iteration 2 (h = 1) iteration 3 (h = 2) iteration 4 iteration 5 (h = 4) interval width [λh+11 ,λ h+1 1 ] [0, 1] [0, .732] [0, .536] [0.056, .449] [0.0715, .359] [λh+12 ,λ h+1 2 ] [0, 1] [0, .732] [0, .536] [0, .392] [0.101, .389] [λh+13 ,λ h+1 3 ] [0, 1] [0, .732] [0, .536] [0, .392] [0.013, .301] [λh+14 ,λ h+1 4 ] [0, 1] [0, .732] [0.015, .552] [0, .392] [0.0848, .373] [λh+15 ,λ h+1 5 ] [0, 1] [0, .732] [0, .536] [0.075, .468] [0.0104, .298] d 0.066 0.0545 0.0536 0.0445 0.026 reduction factor 0.732 0.536 0.392 0.2877 0.2108 vector 1 vector 5 vector 1 vector 1 vector 39 [0.1039 0.1854 [0.1624 0.1752 [0.2524 0.1577 [0.2154 0.2447 [0.2025 0.1600 vector selected 0.2556 0.1966 0.1178 0.2834 0.1956 0.1230 0.1569 0.2287 0.2225 0.2485 0.2584] 0.2611] 0.2713] 0.1543] 0.1725] reach 1858245000 1882864000 1899541000 1916792000 1918657000 % increase in reach 1.32% 0.88% 0.91% 0.09% table 2: product wise allocations from iteration 5 products reach achieved reach aspired % reach achieved from aspired % budget utilized p1 641926400 720325700 89% 0.33% p2 313761900 469876000 67% 0.16% p3 572235900 607432200 94% 0.27% p4 192188100 266835600 72% 0.13% p5 198544300 310087000 64% 0.11% 5 conclusion a media planning model is proposed in this study to allocate advertising budget jointly among multiple products advertised in a segmented market. media vehicles are chosen with respect to two types of advertising strategies namely, segment driven and mass media advertising. segment specific media targets the segment potential while the mass media reaches the wider market with spectrum effect on the segments. the model determines the number of advertisement to be given in each of the media within the bounds suggested by media planner. when several products are advertised by a firm to serve the diverse need of a market, advertising of one product shows the cross product effect on other products. the study considers this effect in the model. model applicability and solution methodology based on interactive linear integer goal programming is discussed with a case study and lingo is used for computational support. the proposed model incorporates the cross product effect of advertising of a firms own products. effect of competitive products can also be included in the future studies. the scope of the model is limited to media planning for a single period. the model can be further extended for dynamic media planning incorporating the retention and diminishing effect of advertising. 16 media mix decision model for multiple product appendix table 3: customer percentage profile matrix for newspapers and television for product 1 segment rnp1 rnp2 rnp3 rnp4 rch1 rch2 gender income gender income gender income gender income gender income gender income fp op fp op fp op fp op fp op fp op fp op fp op pt ot pt ot pt ot pt ot s1 0.29 0.13 0.12 0.04 0.35 0.08 0.12 0.06 0.2 0.14 0.19 0.06 0.24 0.13 0.15 0.09 0.23 0.19 0.15 0.12 0.27 0.17 0.13 0.08 s2 0.3 0.15 0.14 0.09 0.2 0.1 0.1 0.08 0.15 0.12 0.19 0.1 0.27 0.1 0.09 0.12 0.25 0.12 0.14 0.08 0.22 0.1 0.08 0.03 s3 0.29 0.14 0.16 0.07 0.19 0.07 0.12 0.04 0.17 0.15 0.18 0.09 0.22 0.1 0.1 0.05 0.32 0.14 0.21 0.13 0.27 0.13 0.14 0.09 s4 0.15 0.08 0.15 0.08 0.15 0.06 0.08 0.04 0.25 0.12 0.18 0.07 − − − − 0.4 0.23 0.21 0.11 0.23 0.07 0.15 0.06 s5 0.27 0.17 0.2 0.1 0.1 0.05 0.07 0.03 0.15 0.11 0.18 0.06 0.33 0.09 0.07 0.06 0.37 0.15 0.22 0.08 0.18 0.06 0.12 0.03 s6 0.22 0.11 0.13 0.06 0.14 0.06 0.07 0.03 0.21 0.17 0.14 0.04 0.26 0.12 0.18 0.06 0.39 0.2 0.19 0.08 0.24 0.1 0.12 0.09 s7 0.3 0.18 0.18 0.08 0.24 0.14 0.2 0.12 0.26 0.17 0.13 0.05 − − − − 0.3 0.2 0.13 0.09 0.16 0.07 0.11 0.08 s8 0.31 0.17 0.19 0.08 0.12 0.07 0.12 0.06 0.28 0.14 0.16 0.09 0.27 0.08 0.17 0.11 0.31 0.14 0.15 0.09 0.19 0.11 0.13 0.11 s9 0.26 0.12 0.16 0.06 0.25 0.1 0.16 0.07 0.21 0.13 0.17 0.1 − − − − 0.28 0.2 0.17 0.11 0.25 0.16 0.18 0.1 s10 0.29 0.13 0.17 0.07 0.2 0.13 0.12 0.06 0.22 0.14 0.18 0.07 − − − − 0.33 0.18 0.19 0.11 0.27 0.19 0.16 0.1 s11 0.28 0.13 0.13 0.08 0.2 0.13 0.15 0.07 0.22 0.17 0.15 0.09 − − − − 0.31 0.13 0.13 0.1 0.25 0.2 0.14 0.09 s12 0.23 0.13 0.15 0.08 0.15 0.1 0.1 0.05 0.2 0.16 0.2 0.08 0.22 0.12 0.07 0.07 0.46 0.15 0.19 0.07 0.32 0.12 0.23 0.13 s13 0.28 0.13 0.18 0.08 0.23 0.1 0.13 0.07 − − − − − − − − 0.33 0.17 0.3 0.22 0.27 0.19 0.23 0.11 s14 0.25 0.11 0.17 0.1 0.21 0.08 0.12 0.05 − − − − − − − − 0.25 0.17 0.13 0.08 0.13 0.08 0.08 0.06 mass rnp1 rnp2 rnp3 rnp4 rch1 rch2 gender income gender income gender income gender income gender income gender income media 0.25 0.1 0.12 0.06 0.15 0.07 0.11 0.03 0.16 0.06 0.12 0.06 0.15 0.07 0.11 0.06 0.3 0.2 0.21 0.11 0.22 0.16 0.19 0.09 table 4: weights cr1 cr2 p1 0.65 0.35 p2 0.6 0.4 p3 0.36 0.64 p4 0.3 0.7 p5 0.55 0.45 table 5: cross product effect matrix product p1 p2 p3 p4 p5 p1 0 0.0109 0.034 0.02345 0.0034 p2 0.0234 0 0.0234 0.009 0.0054 p3 0.0195 0.0134 0 0.0156 0.00493 p4 0.0214 0.0093 0.0041 0 0.0067 p5 0.0145 0.011 0.0013 0.0078 0 17 s. aggarwal, a. gupta and p. c jha table 6: spectrum effect coefficient of national newspapers and tv channels on regions segments nnp1 nnp2 nnp3 nnp4 nch1 nch2 s1 0.09 0.12 0.1 0.12 0.1 0.11 s2 0.08 0.09 0.13 0.1 0.09 0.06 s3 0.12 0.09 0.09 0.15 0.12 0.1 s4 0.06 0.04 0.05 0.03 0.1 0.11 s5 0.07 0.07 0.08 0.09 0.03 0.04 s6 0.13 0.09 0.1 0.06 0.1 0.08 s7 0.03 0.06 0.04 0.07 0.04 0.05 s8 0.07 0.07 0.07 0.07 0.07 0.05 s9 0.1 0.14 0.1 0 0.04 0.04 s10 0.04 0.05 0.04 0.06 0.05 0.05 s11 0.06 0.06 0.05 0.02 0.04 0.05 s12 0.08 0.08 0.12 0.15 0.1 0.12 s13 0.04 0.02 0.01 0.05 0.05 0.05 s14 0.03 0.02 0.02 0.03 0.07 0.1 table 7: ad cost in different media segments rnp1 rnp2 rnp3 rnp4 rch1 rch2 fp op fp op fp op fp op pt ot pt ot s1 3750 1944 2423 1385 1400 1000 1719 1665 65480 26968 27548 12988 s2 2751 917 2221 1610 1650 900 1300 650 40400 19800 26400 12000 s3 3940 2225 2138 950 2040 1060 1285 1045 43628 15376 30464 13980 s4 1750 500 900 400 790 380 − − 33800 12220 20908 9964 s5 3310 1572 2500 1200 1767 1010 1375 1100 19384 9408 14000 9100 s6 3800 2000 2331 1665 2200 1340 1400 900 45928 16480 41948 21472 s7 1200 600 1150 670 1100 550 − − 8924 5948 6600 3200 s8 1200 600 1160 600 1000 550 900 500 14400 9000 8924 3964 s9 3700 1800 3960 2100 2500 1450 − − 30980 12500 16700 9700 s10 1700 1000 1650 900 1450 850 − − 17848 8956 14956 6980 s11 2500 1200 1640 1040 1100 870 − − 8948 3980 5948 2980 s12 2920 1530 2100 1400 1100 890 2047 1575 41448 18984 30464 17476 s13 1800 900 1000 500 − − − − 12250 6700 9945 4350 s14 700 527 595 424 − − − − 34080 18900 21000 12340 mass media nnp1 nnp2 nnp3 nnp4 nch1 nch2 fp op fp op fp op fp op pt ot pt ot 9800 5640 8690 4250 6900 3540 5500 2900 104390 61019 86814 46570 18 media mix decision model for multiple product table 8: upper and lower bounds on advertisements in different media for p1 segments rnp1 rnp2 rnp3 rnp4 rch1 rch2 fp op fp op fp op fp op pt ot pt ot s1 [1, 22] [12, 85] [1, 15] [12, 65] [1, 20] [12, 70] [1, 12] [11, 68] [8, 36] [18, 92] [6, 39] [15, 85] s2 [1, 14] [9, 50] [1, 12] [8, 41] [1, 12] [7, 42] [1, 13] [7, 48] [7, 34] [16, 88] [4, 33] [13, 73] s3 [1, 24] [11, 90] [1, 16] [9, 69] [1, 20] [10, 75] [1, 15] [8, 62] [7, 39] [19, 94] [5, 38] [14, 83] s4 [1, 14] [4, 56] [1, 10] [3, 44] [1, 12] [4, 42] − − [6, 33] [14, 78] [6, 37] [16, 81] s5 [1, 20] [7, 81] [1, 15] [5, 72] [1, 18] [6, 76] [1, 13] [4, 70] [4, 31] [8, 65] [3, 25] [7, 57] s6 [1, 18] [8, 76] [1, 13] [7, 61] [1, 15] [6, 65] [1, 14] [7, 60] [7, 38] [17, 85] [5, 36] [13, 79] s7 [1, 12] [6, 65] [1, 11] [5, 75] [1, 14] [6, 72] − − [6, 31] [10, 68] [3, 31] [10, 68] s8 [1, 12] [9, 49] [1, 10] [7, 45] [1, 8] [5, 40] [1, 8] [4, 38] [7, 32] [12, 72] [3, 33] [10, 72] s9 [1, 18] [10, 76] [1, 14] [11, 58] [1, 14] [8, 60] − − [5, 33] [12, 64] [3, 29] [8, 64] s10 [1, 13] [6, 49] [1, 10] [5, 40] [1, 10] [4, 42] − − [4, 33] [11, 62] [3, 26] [9, 59] s11 [1, 19] [9, 82] [1, 14] [7, 70] [1, 14] [6, 75] − − [5, 34] [12, 66] [4, 27] [10, 62] s12 [1, 16] [8, 71] [1, 12] [7, 55] [1, 15] [6, 71] [1, 14] [8, 63] [8, 36] [16, 79] [7, 40] [16, 86] s13 [1, 12] [4, 48] [1, 9] [3, 40] − − − − [3, 31] [11, 64] [3, 25] [12, 57] s14 [1, 14] [5, 64] [1, 11] [4, 40] − − − − [5, 32] [14, 85] [5, 29] [14, 65] mass media nnp1 nnp2 nnp3 nnp4 nch1 nch2 fp op fp op fp op fp op pt ot pt ot [1, 18] [12, 84] [1, 12] [12, 64] [1, 15] [12, 70] [1, 12] [12, 64] [8, 39] [20, 94] [8, 25] [17, 86] table 9: upper and lower bounds on advertisements in different media for p2 segments rnp1 rnp2 rnp3 rnp4 rch1 rch2 fp op fp op fp op fp op pt ot pt ot s1 [1, 11] [6, 42] [1, 10] [7, 36] [1, 8] [6, 30] [1, 10] [7, 32] [7, 36] [18, 77] [5, 34] [14, 65] s2 [1, 8] [4, 30] [1, 6] [3, 26] [1, 7] [4, 26] [1, 6] [4, 26] [7, 38] [16, 80] [4, 32] [13, 64] s3 [1, 12] [6, 49] [1, 12] [3, 31] [1, 12] [5, 35] [1, 12] [4, 32] [8, 39] [16, 82] [6, 34] [12, 66] s4 [1, 8] [4, 32] [0, 7] [2, 27] [0, 7] [2, 25] − − [7, 29] [13, 64] [3, 35] [12, 62] s5 [1, 11] [3, 49] [1, 10] [3, 32] [1, 9] [3, 36] [1, 10] [3, 31] [5, 29] [11, 65] [3, 25] [7, 59] s6 [1, 13] [5, 44] [1, 12] [4, 29] [1, 12] [4, 34] [1, 12] [3, 32] [6, 33] [15, 78] [7, 33] [10, 66] s7 [1, 8] [4, 29] [0, 9] [3, 21] [1, 8] [2, 25] − − [7, 26] [12, 63] [3, 27] [7, 52] s8 [1, 9] [6, 27] [0, 8] [4, 25] [0, 6] [3, 23] [0, 6] [3, 22] [5, 28] [12, 64] [3, 31] [8, 55] s9 [1, 10] [7, 45] [1, 9] [6, 42] [1, 9] [5, 35] − − [3, 32] [11, 72] [3, 25] [5, 61] s10 [1, 8] [4, 28] [0, 6] [3, 24] [0, 7] [2, 26] − − [3, 27] [10, 74] [4, 29] [8, 60] s11 [1, 10] [6, 50] [0, 10] [4, 48] [0, 11] [3, 36] − − [2, 27] [8, 71] [3, 26] [8, 49] s12 [1, 9] [4, 35] [0, 8] [2, 32] [0, 6] [1, 30] [0, 8] [2, 32] [7, 36] [14, 68] [4, 33] [12, 64] s13 [1, 8] [3, 28] [0, 6] [4, 26] − − − − [5, 27] [8, 65] [5, 23] [9, 59] s14 [1, 8] [4, 26] [0, 4] [3, 24] − − − − [7, 33] [14, 64] [6, 33] [16, 64] mass media nnp1 nnp2 nnp3 nnp4 nch1 nch2 fp op fp op fp op fp op pt ot pt ot [1, 12] [10, 49] [0, 10] [9, 32] [1, 12] [8, 48] [1, 10] [9, 40] [8, 39] [18, 82] [5, 33] [16, 72] 19 s. aggarwal, a. gupta and p. c jha table 10: upper and lower bounds on advertisements in different media for p3 segments rnp1 rnp2 rnp3 rnp4 rch1 rch2 fp op fp op fp op fp op pt ot pt ot s1 [1, 9] [8, 44] [1, 7] [6, 40] [1, 8] [7, 38] [1, 7] [6, 43] [8, 39] [16, 75] [7, 33] [13, 65] s2 [1, 6] [5, 40] [0, 5] [4, 30] [1, 5] [3, 35] [1, 4] [3, 28] [7, 36] [14, 68] [5, 32] [10, 57] s3 [1, 10] [4, 45] [0, 9] [2, 40] [1, 9] [4, 32] [1, 8] [3, 27] [8, 38] [17, 78] [7, 34] [12, 62] s4 [1, 6] [8, 40] [1, 7] [5, 30] [1, 6] [6, 32] − − [8, 38] [14, 73] [6, 35] [14, 64] s5 [1, 7] [4, 44] [0, 6] [3, 29] [0, 7] [3, 32] [1, 6] [3, 27] [2, 31] [5, 50] [3, 22] [7, 47] s6 [1, 8] [7, 43] [1, 7] [5, 32] [1, 6] [6, 30] [1, 5] [5, 23] [7, 36] [15, 70] [6, 28] [11, 59] s7 [1, 6] [5, 40] [0, 4] [4, 25] [1, 5] [4, 27] − − [3, 33] [10, 55] [6, 25] [8, 49] s8 [1, 6] [4, 40] [1, 5] [3, 27] [1, 6] [3, 32] [1, 5] [3, 27] [6, 35] [12, 63] [5, 26] [10, 52] s9 [0, 9] [7, 42] [1, 8] [7, 35] [0, 7] [5, 32] − − [3, 31] [7, 52] [3, 23] [7, 50] s10 [1, 7] [4, 40] [1, 9] [5, 25] [1, 7] [3, 23] − − [4, 33] [10, 61] [6, 25] [8, 51] s11 [1, 8] [7, 43] [0, 7] [5, 32] [1, 7] [5, 27] − − [5, 34] [8, 57] [4, 26] [11, 55] s12 [1, 7] [6, 43] [1, 5] [5, 34] [1, 6] [4, 29] [1, 5] [5, 26] [8, 36] [16, 73] [7, 32] [16, 66] s13 [1, 5] [5, 40] [0, 4] [4, 28] − − − − [5, 34] [9, 59] [3, 21] [10, 48] s14 [1, 6] [4, 40] [0, 4] [3, 31] − − − − [7, 37] [13, 65] [6, 29] [12, 61] mass media nnp1 nnp2 nnp3 nnp4 nch1 nch2 fp op fp op fp op fp op pt ot pt ot [1, 10] [8, 47] [1, 8] [6, 38] [1, 10] [6, 45] [1, 10] [6, 40] [8, 39] [17, 78] [7, 35] [13, 66] table 11: upper and lower bounds on advertisements in different media for p4 segments rnp1 rnp2 rnp3 rnp4 rch1 rch2 fp op fp op fp op fp op pt ot pt ot s1 [1, 5] [3, 20] [1, 3] [3, 20] [1, 4] [2, 16] [0, 3] [3, 19] [4, 36] [14, 65] [4, 27] [12, 57] s2 [0, 4] [2, 18] [0, 2] [1, 16] [0, 3] [1, 14] [0, 2] [1, 13] [5, 35] [14, 61] [3, 27] [8, 49] s3 [0, 4] [3, 17] [0, 2] [2, 13] [0, 3] [2, 14] [0, 2] [2, 14] [5, 37] [10, 70] [4, 27] [10, 55] s4 [0, 3] [2, 15] [0, 1] [1, 13] [1, 2] [2, 13] − − [5, 37] [13, 64] [4, 26] [11, 57] s5 [1, 4] [2, 16] [1, 3] [1, 12] [0, 3] [2, 14] [0, 2] [1, 12] [2, 27] [9, 51] [3, 25] [7, 41] s6 [0, 5] [3, 20] [1, 3] [2, 18] [1, 2] [3, 14] [0, 3] [2, 18] [4, 32] [12, 63] [3, 25] [9, 51] s7 [0, 4] [2, 18] [1, 3] [2, 16] [1, 3] [2, 14] − − [3, 31] [11, 55] [3, 26] [7, 43] s8 [1, 3] [2, 16] [0, 2] [3, 16] [0, 2] [2, 15] [1, 1] [3, 13] [4, 31] [13, 60] [3, 27] [7, 46] s9 [1, 5] [3, 20] [0, 3] [2, 18] [0, 3] [3, 16] − − [2, 29] [10, 52] [3, 24] [7, 42] s10 [1, 4] [1, 15] [1, 3] [2, 14] [1, 4] [1, 12] − − [3, 33] [13, 59] [3, 27] [7, 45] s11 [0, 5] [3, 20] [1, 3] [3, 18] [0, 4] [3, 19] − − [3, 34] [11, 56] [3, 25] [8, 49] s12 [1, 5] [3, 19] [1, 4] [3, 18] [1, 3] [2, 19] [0, 3] [3, 18] [5, 33] [14, 68] [4, 24] [12, 59] s13 [0, 4] [2, 17] [0, 2] [2, 15] − − − − [3, 38] [12, 57] [3, 23] [7, 47] s14 [0, 4] [2, 15] [0, 2] [1, 13] − − − − [4, 33] [12, 59] [3, 26] [10, 54] mass media nnp1 nnp2 nnp3 nnp4 nch1 nch2 fp op fp op fp op fp op pt ot pt ot [1, 5] [3, 20] [1, 3] [3, 20] [1, 4] [3, 18] [1, 4] [3, 20] [5, 37] [14, 70] [4, 27] [12, 59] 20 media mix decision model for multiple product table 12: upper and lower bounds on advertisements in different media for p5 segments rnp1 rnp2 rnp3 rnp4 rch1 rch2 fp op fp op fp op fp op pt ot pt ot s1 [0, 5] [3, 18] [0, 4] [2, 13] [0, 5] [2, 17] [0, 4] [2, 14] [8, 38] [12, 70] [7, 29] [11, 57] s2 [0, 4] [2, 15] [0, 3] [2, 13] [0, 4] [2, 9] [0, 3] [2, 6] [7, 38] [10, 68] [6, 28] [10, 51] s3 [0, 5] [2, 17] [0, 3] [2, 12] [0, 4] [2, 13] [0, 2] [2, 12] [8, 37] [13, 70] [7, 29] [10, 55] s4 [0, 4] [1, 13] [0, 2] [1, 10] [0, 2] [1, 10] − − [7, 37] [10, 64] [6, 29] [10, 55] s5 [0, 3] [3, 14] [0, 3] [2, 11] [0, 3] [2, 12] [0, 3] [2, 11] [2, 31] [5, 57] [3, 25] [5, 45] s6 [0, 4] [2, 16] [0, 3] [2, 12] [0, 3] [2, 13] [0, 3] [1, 14] [7, 35] [12, 62] [7, 27] [9, 51] s7 [0, 3] [2, 14] [0, 2] [1, 7] [0, 2] [2, 12] − − [6, 33] [7, 59] [5, 25] [7, 49] s8 [0, 4] [2, 15] [0, 2] [2, 8] [0, 3] [2, 10] [0, 2] [1, 8] [6, 34] [9, 62] [6, 27] [8, 51] s9 [0, 5] [3, 18] [0, 4] [3, 13] [0, 3] [2, 12] − − [3, 31] [6, 58] [3, 24] [7, 47] s10 [0, 4] [2, 15] [0, 3] [2, 12] [0, 2] [2, 11] − − [4, 33] [9, 60] [5, 25] [8, 51] s11 [0, 5] [3, 18] [0, 4] [2, 12] [0, 3] [2, 11] − − [5, 34] [7, 61] [4, 24] [9, 47] s12 [0, 5] [3, 17] [0, 4] [2, 16] [0, 3] [2, 15] [0, 3] [2, 11] [7, 37] [13, 66] [6, 27] [11, 55] s13 [0, 3] [1, 13] [0, 2] [1, 7] − − − − [5, 34] [8, 62] [2, 24] [8, 45] s14 [0, 3] [2, 13] [0, 2] [2, 8] − − − − [7, 36] [10, 62] [7, 29] [11, 52] mass media nnp1 nnp2 nnp3 nnp4 nch1 nch2 fp op fp op fp op fp op pt ot pt ot [0, 5] [3, 18] [0, 4] [3, 13] [0, 5] [3, 15] [0, 5] [3, 18] [8, 37] [13, 70] [7, 23] [9, 57] table 13: optimal number of advertisements in different media for p1 segments rnp1 rnp2 rnp3 rnp4 rch1 rch2 fp op fp op fp op fp op pt ot pt ot s1 1 12 15 65 20 70 1 11 36 92 39 85 s2 14 50 1 8 1 42 13 48 34 88 33 13 s3 24 90 16 9 20 75 15 8 39 94 38 83 s4 14 56 1 3 12 42 − − 33 78 37 81 s5 1 7 1 5 18 76 1 4 31 8 25 7 s6 18 76 1 7 1 6 14 7 38 85 17 13 s7 12 65 11 75 14 72 − − 31 68 31 68 s8 12 49 10 45 8 40 8 38 32 72 33 72 s9 18 76 14 11 1 8 − − 5 64 29 8 s10 1 6 1 5 1 4 − − 33 62 26 59 s11 1 9 1 7 1 6 − − 34 66 27 62 s12 1 71 1 7 1 6 14 8 36 79 40 86 s13 1 4 1 3 − − − − 31 64 25 57 s14 1 5 1 4 − − − − 32 85 29 14 mass media nnp1 nnp2 nnp3 nnp4 nch1 nch2 fp op fp op fp op fp op pt ot pt ot 18 84 12 12 15 70 12 64 39 94 25 86 21 s. aggarwal, a. gupta and p. c jha table 14: optimal number of advertisements in different media for p2 segments rnp1 rnp2 rnp3 rnp4 rch1 rch2 fp op fp op fp op fp op pt ot pt ot s1 1 6 10 7 8 30 1 7 7 18 5 14 s2 1 30 1 3 1 4 6 26 38 16 4 13 s3 1 6 1 3 12 35 12 4 39 82 6 12 s4 1 32 0 2 0 25 − − 29 64 35 12 s5 1 3 1 3 1 36 1 3 5 11 3 7 s6 1 5 1 4 1 4 1 3 33 15 7 10 s7 1 4 0 3 8 25 − − 26 63 27 52 s8 9 27 0 4 6 23 6 3 28 64 31 55 s9 1 7 1 6 1 5 − − 3 11 3 5 s10 1 4 0 3 0 2 − − 27 10 4 8 s11 1 6 0 4 0 3 − − 27 71 26 49 s12 1 4 0 2 0 1 8 2 36 14 33 12 s13 1 3 0 4 − − − − 27 65 23 9 s14 1 4 0 3 − − − − 33 14 33 16 mass media nnp1 nnp2 nnp3 nnp4 nch1 nch2 fp op fp op fp op fp op pt ot pt ot 12 49 0 9 12 48 1 9 39 18 33 16 table 15: optimal number of advertisements in different media for p3 segments rnp1 rnp2 rnp3 rnp4 rch1 rch2 fp op fp op fp op fp op pt ot pt ot s1 1 8 7 40 8 38 1 6 39 75 33 65 s2 6 40 5 4 1 35 4 28 36 68 32 57 s3 10 4 0 40 9 32 6 3 38 78 34 62 s4 6 40 7 30 6 32 − − 38 73 35 64 s5 1 4 0 3 7 32 1 3 31 50 3 7 s6 8 43 7 5 6 6 5 5 36 70 6 11 s7 6 5 4 25 5 27 − − 33 55 25 49 s8 6 40 5 27 6 32 5 27 35 63 26 52 s9 0 7 1 7 0 5 − − 31 52 3 7 s10 1 4 1 5 1 3 − − 33 61 25 51 s11 1 7 0 5 1 5 − − 34 57 26 55 s12 1 43 1 5 1 4 5 5 36 73 32 66 s13 1 5 0 4 − − − − 34 59 21 48 s14 1 4 0 3 − − − − 37 65 29 61 mass media nnp1 nnp2 nnp3 nnp4 nch1 nch2 fp op fp op fp op fp op pt ot pt ot 10 47 8 38 10 45 10 40 39 78 35 66 22 media mix decision model for multiple product table 16: optimal number of advertisements in different media for p4 segments rnp1 rnp2 rnp3 rnp4 rch1 rch2 fp op fp op fp op fp op pt ot pt ot s1 1 3 3 3 4 16 0 3 4 14 4 12 s2 0 18 0 1 3 1 2 13 35 14 3 8 s3 0 3 0 2 3 14 2 2 37 70 4 10 s4 3 15 0 1 2 13 − − 37 64 26 11 s5 1 2 1 1 3 2 0 1 2 9 3 7 s6 5 3 1 2 1 3 3 2 32 63 3 9 s7 0 2 3 16 3 14 − − 31 55 26 43 s8 3 16 2 3 2 15 1 13 31 60 27 46 s9 1 3 0 2 3 3 − − 2 10 3 7 s10 1 1 1 2 1 1 − − 33 13 3 7 s11 0 3 1 3 0 3 − − 34 56 25 49 s12 1 3 1 3 3 2 3 18 33 14 24 12 s13 0 2 0 2 − − − − 38 57 23 47 s14 0 2 0 1 − − − − 33 12 26 10 mass media nnp1 nnp2 nnp3 nnp4 nch1 nch2 fp op fp op fp op fp op pt ot pt ot 5 20 3 20 4 18 4 3 37 14 27 12 table 17: optimal number of advertisements in different media for p5 segments rnp1 rnp2 rnp3 rnp4 rch1 rch2 fp op fp op fp op fp op pt ot pt ot s1 0 3 0 2 5 17 0 2 8 12 7 11 s2 0 2 0 2 0 2 3 6 38 10 6 10 s3 0 2 0 2 4 13 2 2 37 70 7 10 s4 0 1 0 1 2 10 − − 37 64 29 10 s5 0 3 0 2 3 2 0 2 2 5 3 5 s6 0 2 0 2 0 2 0 1 35 12 7 9 s7 0 2 0 1 2 12 − − 33 57 25 49 s8 4 15 0 2 3 10 2 8 34 9 27 51 s9 0 3 0 3 0 2 − − 3 6 3 7 s10 0 2 0 2 0 2 − − 33 9 5 8 s11 0 3 0 2 0 2 − − 34 61 24 47 s12 0 3 0 2 0 2 3 2 37 13 27 11 s13 0 1 0 1 − − − − 34 62 24 8 s14 0 2 0 2 − − − − 7 10 7 11 mass media nnp1 nnp2 nnp3 nnp4 nch1 nch2 fp op fp op fp op fp op pt ot pt ot 0 3 0 3 5 3 0 3 37 13 23 9 23 s. aggarwal, a. gupta and p. c jha acknowledgment the authors are very much thankful to the editors and anonymous referees for their valuable comments 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[21] f. s. zufryden, media scheduling and solution approaches, operational research quarterly (1975), 283-295. 25 ratio mathematica volume 41, 2021, pp.291-307 291 study of feedback retrial queueing system with working vacation, setup time and perfect repair poonam gupta1 abstract this manuscript analyses a retrial queueing system with working vacation, interruption, feedback, and setup time with the perfect repair. in the proposed model, the server takes vacation whenever the system gets empty but it still serves the customers at a relatively lower rate. to save power, the concept of setup time is included in the model. at vacation completion instant, the server is immediately turned off as soon as the system gets empty. the customer, who arrives during the closed-down state, activates the server and waits for his turn till the server is turned on. the unreliable server may sometimes fail to activate during setup. the failed server will resume service on being repaired. in the paper, explicit expressions for system size, sojourn times, and probabilities of various states of the server are obtained and results are analyzed graphically using matlab software. keywords: retrial queue; working vacation; feedback; setup time. 2010 ams subject classification2: 60k25, 60k30 1 hindu girls college, sonipat, india; poonammittal2207@gmail.com. 2 received on september 4, 2021. accepted on december 20, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.655. issn: 1592-7415. eissn: 2282-8214. ©gupta. this paper is published under the cc-by licence agreement. p. gupta 292 1. introduction retrial queueing systems with vacations and feedback of customers have attracted many researchers due to their widespread applications in real-life situations such as cellular networks, call centers, computer systems, inventory systems, and production management. in such systems, the customer, who finds the server busy, joins the orbit (free pool) and after a random period, it retries for the service. the retrials may follow a constant or classical retrial policy. in a constant retrial policy, the customer at the head in retrial orbit can reattempt for the service, but in a classical one, all the customers can retry for their turn independent of all others in the orbit. many researchers analyzed the applications of retrial queueing systems. falin and templeton [6] did pioneering work on retrial queues. a good survey on the retrial queueing system was done by artalejo et al. [2] and yang et al. [20]. queueing systems with vacations play a vital role in many real-life systems. the vacations may be due to many reasons. in classical vacation policy, no service is provided to the customers during vacation. servi and finn [18] introduced a new vacation class i.e., working vacation, in which service is provided to customers but at a comparatively lower rate. readers may refer do [5], arivudainambi et al. [1] and chandrasekaran et al. [3] for reference. furthermore, the concept of vacation interruption has been widely used in queueing systems. in this policy, at service completion instant, the server interrupts the ongoing vacation and returns to a normal working state on finding waiting customers. keeping in view the importance of the concept, many researchers have analyzed the queueing systems with vacation interruption. li and tian [12] analyzed m/m/1 queueing system with working vacation and interruption using the matrix geometric method. a pioneer work on m/g/1 queueing model with vacation interruption was done by zhang and hou [21]. later, gupta and kumar [7, 8, 9] studied retrial queues with different vacation policies, impatient behaviour of customers and obtain closed-form expressions for important performance measures. the retrial queueing system with bernoulli feedback of customers is characterized by the feature that the unsatisfied customers may rejoin the system with some probability until they receive satisfactory service. choi et al. [4] analyzed a retrial queueing model with geometric loss and feedback. kumar et al. [10,11] studied m/g/1 retrial queueing model with feedback and starting failure. mokaddis [15] analyzed feedback queueing systems with vacations and system failure. varalakshmi et al. [19] discussed a single server queue with study of feedback retrial queueing system with w.v., setup time, and perfect repair 293 immediate feedback and server breakdown. the concept of power saving is very important in today's scenario. to save power, the system should be turned off when not in use. realizing the need for power saving, many researchers studied queueing model with setup time. phungduc [16, 17] incorporated the concept of setup time to retrial queueing system. we may refer the reader to [13, 14] for the related works. in this manuscript, a single server retrial queueing system with feedback, setup time, working vacation, and interruption under perfect repair is analyzed. if setup time is taken as zero, the model reduces to m/m/1 feedback retrial queueing system with working vacation and interruption. further, the model changes to a retrial queueing system with feedback and setup time with perfect repair if vacation time tends to zero. thus, our model generalizes some of the retrial queueing models existing in queueing literature. 2. practical application of the model consider a manufacturing system consisting of an iron re-rolling mill, a foreman(server), and a worker(assistant) to operate the mill. the foreman will operate the mill if the raw material is available and produce the products i.e., iron angles, iron rods, etc. if the raw material is not available due to transport issues, an increase in the price of raw material, etc., then the foreman may go on vacation(rest). during the vacation period of the foreman, if raw material becomes available then the worker will operate the mill, but the production will be relatively at a slow speed. when a batch of the product is completed, then the worker will call the foreman to resume the production at a higher speed by interrupting the vacation of the foreman. in another situation, if the foreman’s vacation period completes, he will return to the production to operate the mill. if the raw material is available then he will manage the production at a higher speed otherwise, if the raw material is not available, to save power, he may turn off the mill. again, the availability of new raw material will initiate the setup of the machine (re-rolling mill) and production starts again if setup occurs successfully otherwise the machine will be sent for repair, and during this period there will be no production. p. gupta 294 3. model description and assumptions we considered a markovian retrial queueing system with working vacation, interruption, feedback, and setup time under perfect repair. the following assumptions are taken for the proposed model. 1. the arrival of customers is by the poisson process with rate λ. the customer who finds the server busy; joins the free pool (orbit) and waits for his turn. the customers in the orbit are assumed to follow classical retrial policy. the retrial time is exponentially distributed with parameter ξ. 2. the service time in the normal state of the server is assumed to be exponentially distributed with rate µ. when all the customers are served, the server goes to a working vacation state in which it still provides service to customers with a slow rate θ. the vacation time and service time in vacations again follow an exponential distribution with vacation rate ϕ. 3. the unsatisfied customers may rejoin the orbit as feedback customers with probability ‘f’ or may leave the system with complementary probability 𝑓=̅ (1-f). 4. on completion of vacation, if customers are found still waiting for their turn, the normal service period resumes otherwise, the server is turned off immediately to save power. 5. the customer who arrives in the off-state of the server; waits for his turn in front of the server till it is turned on. the setup time is required to restart the server. the setup time is assumed to follow an exponential distribution with a mean of 1/s. the customers arriving in the setup state, have to join the orbit. 6. the server is assumed to be unreliable i.e., during the set-up state, activation of the server may fail with probability �̅� = (1-p). the failed server is sent for repair and repair time is exponentially distributed with parameter r. 7. the inter-arrival time, service time, vacation time, retrial time, and setup time are all mutually independent. taking n(t) as the number of customers in the orbit at time t and j(t) as the state of the server. where, study of feedback retrial queueing system with w.v., setup time, and perfect repair 295 𝐽(𝑡) = { 0, 𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑖𝑠 𝑏𝑢𝑠𝑦 𝑖𝑛 𝑎 𝑛𝑜𝑟𝑚𝑎𝑙 𝑠𝑡𝑎𝑡𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 1, 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑓𝑟𝑒𝑒 𝑖𝑛 𝑎 𝑛𝑜𝑟𝑚𝑎𝑙 𝑠𝑡𝑎𝑡𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 2, 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑏𝑢𝑠𝑦 𝑖𝑛 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑣𝑎𝑐𝑎𝑡𝑖𝑜𝑛 𝑠𝑡𝑎𝑡𝑒 3, 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑓𝑟𝑒𝑒 𝑖𝑛 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑣𝑎𝑐𝑎𝑡𝑖𝑜𝑛 𝑠𝑡𝑎𝑡𝑒 4, 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑖𝑛 𝑠𝑒𝑡𝑢𝑝 𝑜𝑟 𝑐𝑙𝑜𝑠𝑒 − 𝑑𝑜𝑤𝑛 𝑠𝑡𝑎𝑡𝑒 5, 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑢𝑛𝑑𝑒𝑟 𝑟𝑒𝑝𝑎𝑖𝑟 𝑠𝑡𝑎𝑡𝑒 {n(t), j(t)} represents a markov process with following state-space {(n, j), n ≥ 0, j=0,1,2,4,5} u {(0,3)}. here (n, 0), n ≥ 0 represents that server is busy in a regular service period with n customers waiting in the orbit. the state (0, 1) represents that system is in a close-down period. states (n, 1), n ≥1 shows that system is free in the regular service period. states (n, 2), n ≥0 represents the state that the system is busy in the working vacation period. the state (0, 3) represents that the server is free during the vacation period. states (n, 3), n≥1 do not exist due to the inclusion of the concept of vacation interruption. states (n, 4), n≥0 represent that system is in a setup state with n customers in the orbit and one customer waiting in the service area for successful set up of the system. states (n, 5), n≥0, show that the system is under repair due to sudden breakdowns with n customers waiting in the orbit for their turn. 4. steady-state equations and stationary probabilities denoting by 𝑝𝑛 𝑗, the probability of n customers waiting in the orbit when the system is in state j and using markov process for the quasi-birth death model, the stationary state equations governing the model are (𝜆 + 𝜇)𝑝0 0 = 𝜙𝑝0 2 + 𝜉𝑝1 1 + 𝑠𝑝𝑝0 4 + 𝑟𝑝0 5 (1) (𝜆 + 𝜇)𝑝𝑛 0 = 𝜆𝑝𝑛−1 0 + 𝜙𝑝𝑛 2 + (𝑛 + 1)𝜉𝑝𝑛+1 1 + 𝜆𝑝𝑛 1 + 𝑠𝑝𝑝𝑛 4 + 𝑟𝑝𝑛 5, 𝑛 ≥ 1 (2) 𝜆𝑝0 1 = 𝜙𝑝0 3 (3) (𝜆 + 𝜉)𝑝𝑛 1 = 𝑓�̅�𝑝𝑛 2 + 𝑓𝜃𝑝𝑛−1 2 + 𝑓𝜇𝑝𝑛−1 0 + 𝑓�̅�𝑝𝑛 0 , 𝑛 ≥ 1 (4) (𝜆 + 𝜃 + 𝜙)𝑝0 2 = 𝜆𝑝0 3 (5) (𝜆 + 𝜃 + 𝜙)𝑝𝑛 2 = 𝜆𝑝𝑛−1 2, 𝑛 ≥ 1 (6) p. gupta 296 (𝜆 + 𝜙)𝑝0 3 = 𝜇𝑓�̅�0 0 + 𝑓�̅�𝑝0 2 (7) (𝜆 + 𝑠)𝑝0 4 = 𝜆𝑝0 1 (8) (𝜆 + 𝑠)𝑝𝑛 4 = 𝜆𝑝𝑛−1 4, 𝑛 ≥ 1 (9) (𝜆 + 𝑟)𝑝0 5 = 𝑠�̅�𝑝0 4 (10) (𝜆 + 𝑟)𝑝𝑛 5 = 𝑠�̅�𝑝𝑛 4 + 𝜆𝑝𝑛−1 5 , 𝑛 ≥ 1 (11) defining probability generating functions 𝐺𝑖(𝑧) = ∑𝑝𝑛 𝑖𝑧 𝑛 ∞ 𝑛=0 , 𝑖 = 0,1,2,4,5 (12) multiplying equations (1) and (2) with appropriate power of z and taking summation for all possible values of n and using above-defined generating functions, (𝜆 + 𝜇 − 𝜆𝑧)𝐺0(𝑧) = 𝜙𝐺2(𝑧) + 𝜆𝐺1(𝑧) + 𝜉𝐺1 ′(𝑧) + 𝑠𝑝𝐺4(𝑧) + 𝑟𝐺5(𝑧) − 𝜆𝑝0 1 (13) multiplying equations (3) and (4) with appropriate power of z and taking summation for all possible values of n and using generating functions we get, (𝜆 + 𝜉)𝐺1(𝑧) = (𝑓𝑧 + 𝑓 ̅)𝜇𝐺0(𝑧) + (𝑓̅ + 𝑓𝑧)𝜃𝐺2(𝑧) − 𝐴 𝑝0 1 (14) 𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝑓�̅�𝑝0 2 + 𝜇𝑓�̅�0 0 − 𝜉𝑝0 1 − 𝜙𝑝0 3 𝑝0 1 (15) 𝑝0 2 = 𝜆2 𝜙(𝜆 + 𝜃 + 𝜙) 𝑝0 1 (16) 𝑝0 3 = 𝜆 𝜙 𝑝0 1 (17) 𝑝0 0 = ( 𝜆(𝜆 + 𝜙) 𝜇𝜙𝑓̅ − 𝜆2𝜃 𝜇𝜙(𝜆 + 𝜃 + 𝜙) )𝑝0 1 (18) again using probability generating functions along with equations (5) and (6) (𝜆 + 𝜃 + 𝜙 − 𝜆𝑧)𝐺2(𝑧) = 𝜆𝑝0 3 = 𝜆2 𝜙 𝑝0 1 study of feedback retrial queueing system with w.v., setup time, and perfect repair 297 𝐺2(𝑧) = 𝜆2 𝜙(𝜆 + 𝜃 + 𝜙 − 𝜆𝑧) 𝑝0 1 (19) using equations (8), (9), and (12) together (𝜆 + 𝑠)𝐺4(𝑧) = 𝜆𝑧𝐺4(𝑧) + 𝜆𝑝0 1 𝐺4(𝑧) = 𝜆 (𝜆 + 𝑠 − 𝜆𝑧) 𝑝0 1 (20) similar calculations in equations (10) and (11) yield (𝜆 + 𝑟 − 𝜆𝑧)𝐺5(𝑧) = 𝑠(1 − 𝑝)𝐺4(𝑧) makin use of equation (19) in the above equation, we obtain 𝐺5(𝑧) = 𝜆𝑠(1 − 𝑝) (𝜆 + 𝑟 − 𝜆𝑧)(𝜆 + 𝑠 − 𝜆𝑧) 𝑝0 1 (21 using the value of 𝐺0(𝑧) from equation (14) in equation (13), and rearranging the terms we get the following differential equations 𝐺1 ′(𝑧) + 1 𝜉 (𝜆 − (𝜆 + 𝜉)(𝜆 + 𝜇 − 𝜆𝑧) 𝜇(𝑓̅ + 𝑓𝑧) )𝐺1(𝑧) = 𝐵(𝑧) (22) 𝑤ℎ𝑒𝑟𝑒 𝐵(𝑧) = 1 𝜉 [(𝜆 + 𝐴(𝜆 + 𝜇 − 𝜆𝑧) 𝜇(𝑓̅+ 𝑓𝑧) )𝑝0 1 − (𝜙 + 𝜃(𝜆 + 𝜇 − 𝜆𝑧) 𝜇 )𝐺2(𝑧) − (𝑠𝑝𝐺4(𝑧) + 𝑟𝐺5(𝑧))] (23) to solve the differential equation we first find integrating factor, 𝐼.𝐹 = 𝑒 𝜆𝑧 𝜉 (1+ 𝜆+𝜉 𝜇𝑓 ) ((𝑓̅ + 𝑓𝑧) − (𝜆+𝜉)(𝜆+𝜇𝑓) 𝜇𝜉𝑓2 ) (24) the solution of the differential equation (23) is 𝐺1(𝑧) = 𝑒 −𝜆𝑧 𝜉 (1+ 𝜆+𝜉 𝜇𝑓 ) ((𝑓̅ + 𝑓𝑧) (𝜆+𝜉)(𝜆+𝜇𝑓) 𝜇𝜉𝑓2 ) ∫𝑒 𝜆𝑧 𝜉 (1+ 𝜆+𝜉 𝜇𝑓 ) ((𝑓̅ + 𝑓𝑧) − (𝜆+𝜉)(𝜆+𝜇𝑓) 𝜇𝜉𝑓2 ) 𝑧 0 𝐵(𝑧)𝑑𝑧 (25) 𝐺0(𝑧) is obtained from equation (14) as follows p. gupta 298 𝐺0(𝑧) = (𝜆 + 𝜉)𝐺1(𝑧)− (𝑓̅ + 𝑓𝑧)𝜃𝐺2(𝑧) + 𝐴𝑝0 1 (𝑓𝑧 + 𝑓 ̅)𝜇 (26) taking limit 𝑧 → 1 in equations (19), (20), (21), (25), and (26) we obtain the expressions 𝐺2(1) = 𝜆2 𝜙(𝜃 + 𝜙) 𝑝0 1 (27) 𝐺4(1) = 𝜆 𝑠 𝑝0 1 (28) 𝐺5(1) = 𝜆(1 − 𝑝) 𝑟 𝑝0 1 (29) 𝐺1(1) = 𝑒 −𝜆 𝜉 (1+ 𝜆+𝜉 𝜇𝑓 ) ∫𝑒 𝜆𝑧 𝜉 (1+ 𝜆+𝜉 𝜇𝑓 ) ((𝑓̅ + 𝑓𝑧) − (𝜆+𝜉)(𝜆+𝜇𝑓) 𝜇𝜉𝑓2 ) 1 0 𝐵(𝑧)𝑑𝑧 (30) 𝐺0(1) = (𝜆 + 𝜉)𝐺1(1) − 𝜃𝐺2(1) + 𝐴𝑝0 1 𝜇 (31) differentiating equations (19), (20), (21) and taking limit 𝑧 → 1we get 𝐺2 ′(1) = 𝜆3 𝜙(𝜃 + 𝜙)2 𝑝0 1 (32) 𝐺4 ′(1) = 𝜆2 𝑠2 𝑝0 1 (33) 𝐺5 ′(1) = 𝜆2(1 − 𝑝)(𝑟 + 𝑠) 𝑠𝑟2 𝑝0 1 (34) equation (23) on taking limit 𝑧 → 1 implies 𝐺1 ′(1) = 1 𝜉 [𝜉𝐺1(1) − (𝜃 + 𝜙)𝐺2(1) − 𝑠𝑝𝐺4(1) − 𝑟𝐺5(1) + (𝜆 + 𝐴)𝑝0 1](35) similarly differentiating equation (26) and taking limits we obtain 𝐺0 ′(1) = 1 𝜇 [(𝜆 + 𝜉)𝐺1 ′(1) − 𝜃𝐺2 ′(1) − 𝑓𝜃𝐺2(1) − 𝜇𝑓𝐺0(1)] (36) study of feedback retrial queueing system with w.v., setup time, and perfect repair 299 we observe that all the closed-form expressions for 𝐺𝑖(𝑧) and their derivatives for i=0, 1, 2, 4, 5 are implicitly expressed in terms of 𝑝0 1. 𝑝0 1 may be obtained by using the normalization condition 𝐺0(1) + 𝐺1(1) + 𝐺2(1) + 𝐺4(1) + 𝐺5(1) = 1 (37) 5. performance measures in the present section, we obtain some important system performance measures of our proposed model as follows. expected orbit length 𝐸[𝐿𝑂] = 𝐺0 ′(1) + 𝐺1 ′(1) + 𝐺2 ′(1) +𝐺4 ′(1) + 𝐺5 ′(1) (38) expected sojourn time in orbit 𝐸[𝑊𝑂] = 𝐸[𝐿𝑂]/𝜆 = 𝐺0 ′(1) + 𝐺1 ′(1) + 𝐺2 ′(1) + 𝐺4 ′(1) + 𝐺5 ′(1) 𝜆 (39) expected system length 𝐸[𝐿𝑆] = 𝐸[𝐿𝑂] + 𝐺0(1) + 𝐺2(1) (40) expected sojourn time in system 𝐸[𝑊𝑆] = 𝐸[𝐿𝑆]/𝜆 = 𝐸[𝐿𝑂] + 𝐺0(1) + 𝐺2(1) 𝜆 (41) probability of server being in off state = 𝑝0 1 probability of server being in working vacation state (𝑃𝑟𝑤𝑣) = 𝐺2(1) + 𝜆 𝜙 𝑝0 1 = 𝜆2 𝜙(𝜃 + 𝜙) 𝑝0 1 + 𝜆 𝜙 𝑝0 1 (42) probability of server being in setup state (𝑃𝑟𝑆) = 𝐺4(1) = 𝜆 𝑠 𝑝0 1 (43) probability of server in repair state (𝑃𝑟𝑅) = 𝐺5(1) p. gupta 300 = 𝜆(1 − 𝑝) 𝑟 𝑝0 1 (44) 6. numerical and graphical analysis in the present section, the numerical and graphical interpretation of derived closed-form expressions of various system performance measures, for the proposed mathematical model is performed. for this purpose, some of the system parameters are assumed to be fixed as λ=3, μ=7, ξ=1.8, ϕ=2, θ=3, f=0.7, p=0.7, r=0.8, s=0.6, unless otherwise mentioned. the behaviours of important performance measures, for a different set of values of one or more of the parameters is analyzed in the below-plotted graphs. figure1: off-state probability versus setup rate for different values of p from figure 1 we see that with an increase in setup rate, the probability of the server being in off-state increases. this is due to the reason that with an increase in the setup rate, the setup time decreases which causes early return in a normal state of the server hence increasing the probability of the server being in an off state. this probability of off-state increases with an increase in p, for a fixed value of set up rate; this is again due to an increase in chances of successful set up of server that further increases the off state probability. study of feedback retrial queueing system with w.v., setup time, and perfect repair 301 figure2: off-state probability versus repair rate for different values of p we observe from figure 2 that the off-state probability of server increases with repair rate r, for a fixed value of p. this is due to a reduction in repair time with an increase in repair rate which leads to quick repair hence faster return to normal service thereby increasing the off-state probability. figure3: effect of setup rate on mean orbit length for different repair rates figure 3 reveals that the expected orbit length decreases with an increase in setup rate. this is because with an increase in setup rate, the time required for set up of server decreases which results in a quick return to normal service p. gupta 302 period of server thereby reducing the orbit length. for the same reason, mean orbit length decreases with a decrease in repair time. figure 4: effect of repair rate on mean orbit length for different arrival rates we see from figure 4, the expected orbit length decreases with an increase in repair rate. as expected the mean orbit length increases with an increase in arrival rate, for a fixed repair rate. this is due to a reduction in inter-arrival time which increases mean orbit length. figure 5: effect of repair rate on repair state probability of server for different values of p study of feedback retrial queueing system with w.v., setup time, and perfect repair 303 figure 5 depicts that with an increase in repair rate, the probability of the server being in repair state decreases. this agrees with our expectations. as the repair rate increases, the repair is done in a lesser time that makes a faster return to the normal state from the repair state hence the probability of the server being in the repair state decreases. again with an increase in p, for a fixed repair rate, the chances of successful activation (set up) of server raise hence the probability of server being in repair state decreases. figure 6: probability of server in vacation versus vacation rate for different service rates we observe from figure 6 that the probability of the server being in vacation state decreases with an increase in the rate of working vacation. the reason behind the observation is a decrease in the duration of vacation with an increase in the vacation rate. further, the probability of the server in vacation state increases with service rate μ; this is due to faster service which promotes server vacations. p. gupta 304 figure 7: probability of server in set up versus service rate for different setup rates figure 7 depicts that with an increase in service rate, the probability of the server being in setup state increases, this is as expected intuitively. for fixed service rate, as set up rate s increases, the probability decreases; this is due to faster activation of server with reduced setup time. figure 8: variation in expected system length with setup and repair rate study of feedback retrial queueing system with w.v., setup time, and perfect repair 305 figure 9: variation in off state probability of server with setup and repair rate figures 8 and 9 represent the graphical behaviour of mean system length and off-state probability with setup and repair rate respectively. as expected, the mean system length decreases whereas off-state probability increases with an increase in the setup rate, for a fixed value of repair rate. 7. conclusion and future scope this paper analyses a single server retrial queueing system with working vacation, vacation interruption, bernoulli feedback and setup time under perfect repair. the closed-form expressions for expected system size along with the probability of various system state probabilities, closed-down state have been obtained via the probability generating functions approach. the variation of the derived expressions against some of the system parameters is graphically studied by using matlab software. the observed graphical results are analyzed and are found to agree with the theoretically expected behaviour. the retrial queueing model with imperfect repair and multiple waiting servers can be considered for future investigations. conflict of interests the authors declare that there is no conflict of interest. p. gupta 306 references [1] arivudainambi, d., godhandaraman, p., rajadurai, p. 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(2010). performance analysis of m/g/1 queue with working vacations and vacation interruption. journal of computational and applied mathematics, 234(10), 2977–2985. https://doi.org/10.1016/j.cam.2010.04.010 https://doi.org/10.1007/s11518-006-5030-6 http://dx.doi.org/10.1007/978-3-319-22267-7_9 http://dx.doi.org/10.1007/978-3-319-22267-7_9 https://doi.org/10.1016/s0166-5316(02)00057-3 https://www.researchgate.net/profile/m-varalakshmi https://www.researchgate.net/profile/chandrasekaran-m https://www.researchgate.net/profile/saravanarajan-mc https://www.researchgate.net/journal/international-journal-of-engineering-and-technology-0975-4024 https://www.researchgate.net/journal/international-journal-of-engineering-and-technology-0975-4024 http://dx.doi.org/10.14419/ijet.v7i4.10.21044 https://doi.org/10.1016/j.cam.2010.04.010 ratio mathematica volume 42, 2022 on integer cordial labeling of some families of graphs s sarah surya* alan thomas † lian mathew ‡ abstract an integer cordial labeling of a graph g(p, q) is an injective map f : v → [−p 2 ...p 2 ]∗ or [−⌊p 2 ⌋...⌊p 2 ⌋] as p is even or odd, which induces an edge labeling f∗ : e → {0, 1} as f∗(uv) = { 1, f(u) + f(v) ≥ 0 0, otherwise such that the number of edges labelled with 1 and the number of edges labelled with 0 differ at most by 1. if a graph has integer cordial labeling, then it is called integer cordial graph. in this paper, we have proved that the banana tree, k1,n ∗ k1,m, olive tree, jewel graph, jahangir graph, crown graph admits integer cordial labeling. keywords: banana tree, k1,n ∗ k1,m, olive tree, jewel graph, jahangir graph, crown graph, integer cordial labeling. 2020 ams subject classifications: 05c78 1 *department of mathematics, stella maris college(autonomous), chennai, affiliated to the university of madras, india; e-mail: sara24solomon@gmail.com. †department of mathematics, st. aloysius college, edathua, india; e-mail: alanampalathara@gmail.com. ‡department of mathematics, stella maris college(autonomous), chennai, affiliated to the university of madras, india; e-mail: lianmathew64@gmail.com. 1received on january 28th, 2022. accepted on june 9th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v39i0.709. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 105 s sarah surya, alan thomas, lian mathew 1 introduction let g be a simple, finite, undirected graph. for terms not defined here, we refer to harary [3]. a graph labeling is an assignment of integers to the vertices or edges, or both, under some conditions. it has wide applications in mathematics as well as in other fields such as circuit design, communication network addressing, date base management and so on. in this paper, we have proved that the banana tree, k1,n ∗ k1,m, olive tree, jahangir graph, jewel graph, crown graph are integer cordial. 2 preliminaries definition 2.1 [8] a mapping f : v (g) → {0, 1} is called binary vertex labeling of g and f(v) is called the label of the vertex v of g under f. if for an edge e = uv, the induced edge labeling f∗ : e(g) → {0, 1} is given by f∗(e) = |f(u) − f(v)|. definition 2.2 [1] a binary vertex labeling of a graph g is called a cordial labeling if |vf(0) − vf(1)| ≤ 1 and |ef(0) − ef(1)| ≤ 1, where vf(i) = number of vertices having label i under f and ef(i) = number of edges having label i under f∗. a graph g is cordial if it admits cordial labeling. i. cahit [1] introduced the concept of cordial labeling as a weaker version of graceful and harmonious graphs. definition 2.3 [7] an integer cordial labeling of a graph g(p, q) is an injective map f : v → [−p 2 ...p 2 ]∗ or [−⌊p 2 ⌋...⌊p 2 ⌋] as p is even or odd, which induces an edge labeling f∗ : e → {0, 1} defined by f∗(uv) = { 1, f(u) + f(v) ≥ 0 0, otherwise such that the number of edges labelled with 1 and the number of edges labelled with 0 differ at most by 1. if a graph has integer cordial labeling, then it is called integer cordial graph. definition 2.4 [4] a banana tree bn,k is a graph obtained by connecting one leaf of each of n copies of a k star graph with a single root vertex. it has nk + 1 vertices and nk edges. definition 2.5 [5] k1,n ∗ k1,m is the graph obtained from k1,n by attaching root of a star k1,m at each pendant vertex of k1,n. definition 2.6 [6] olive tree tk is a rooted tree consisting of k branches where the i th branch is a path of length i and it consists of k(k+1) 2 + 1 vertices. definition 2.7 [5] the jewel graph jn is the graph with vertex set v (jn) = {u, v, x, y, wi : 1 ≤ i ≤ n} and edge set e(jn) = {ux, uy, xy, xv, yv, uwi, vwi : 1 ≤ i ≤ n}. definition 2.8 [2] jahangir graph jn,m for m ≥ 3, is a graph on nm + 1 vertices, consisting of a cycle cnm with one additional vertex which is adjacent to m ver106 on integer cordial labeling of some families of graphs tices of cnm at a distance n to each other on cnm. definition 2.9 [5] the crown cn ⊙ k1 is the graph obtained from a cycle by attaching a pendant edge to each vertex of the cycle. in [7], nicholas et al. introduced the concept of integer cordial labeling of graphs and proved that some standard graphs such as path pn, star graph k1,n, cycle cn, helm graph hn, closed helm graph chn are integer cordial. kn is not integer cordial, kn,n is integer cordial iff n is even and kn,n\m is integer cordial for any n, where m is perfect matching of kn,n. in [8], sarah et al. proved that the sierpinski sieve graph, the graph obtained by joining two friendship graphs by a path of arbitrary length, (n, k)− kite graph and prism graph are integer cordial. 3 main results theorem 3.1. banana tree bn,k is integer cordial. proof. case1: when n is even (the total number of vertices is odd). let u denote the root vertex. let v1, v2, . . . , vnk 2 denote the vertices on n 2 leaves and vnk 2 +1, vnk 2 +2, . . . , vnk denote the vertices on the remaining n 2 leaves of bn,k. we define f : v → [−⌊p 2 ⌋...⌊p 2 ⌋] as follows: f(u) = 0 f(vi) = −i; 1 ≤ i ≤ nk 2 f(vi) = i − nk 2 ; nk 2 + 1 ≤ i ≤ nk case 2: when n is odd and number of vertices is odd. let u denote the root vertex. let v1, v2, . . . , v(n−1)k 2 denote the vertices on ⌊n 2 ⌋ leaves and v(n−1)k 2 +1 , v(n−1)k 2 +2 , . . . , v(n−1)k denote the vertices of remaining ⌊n2 ⌋ leaves. let u1, u2, . . . , uk denote the k vertices of the another leaf such that u1 is adjacent to u and uk is adjacent to ui where 1 ≤ i ≤ k. 107 s sarah surya, alan thomas, lian mathew we define f : v → [−⌊p 2 ⌋...⌊p 2 ⌋] as follows: f(u) = 0 f(vi) = −i; 1 ≤ i ≤ (n − 1)k 2 f(vi) = i − (n − 1)k 2 ; (n − 1)k 2 + 1 ≤ i ≤ k(n − 1) f(ui) = −( nk 2 + 1 − i); 1 ≤ i ≤ k 2 f(ui) = k(n + 1) 2 + 1 − i; k 2 + 1 ≤ i ≤ k case 3: when n is odd and the number of vertices is even. let u denote the root vertex. let v1, v2, . . . , v(n−1)k 2 denote the vertices on ⌊n 2 ⌋ leaves and v(n−1)k 2 +1 ,v(n−1)k 2 +2 ,...,vk(n−1) denote the vertices of another ⌊n2 ⌋ leaves where v1 not adjacent to u. let u1, u2, ..., uk denote the k vertices of the remaining leaf such that u1 is adjacent to u and uk is adjacent to ui, 1 ≤ i ≤ k. we define f : v → [−p 2 ...p 2 ]∗ as follows: f(u) = 1 f(vi) = −i; 1 ≤ i ≤ (n − 1)k 2 f(vi) = i + 1 − (n − 1)k 2 ; (n − 1)k 2 + 1 ≤ i ≤ k(n − 1) f(ui) = − ( ⌈ nk 2 ⌉ + 1 − i ) ; 1 ≤ i ≤ ⌈ k 2 ⌉ f(ui) = k(n + 1) 2 + 2 − i; ⌈ k 2 ⌉ + 1 ≤ i ≤ k hence in all the possible cases, we have |ef(1) − ef(0)| ≤ 1. therefore, banana tree bn,k admits integer cordial labeling(see figure 1). theorem 3.2. the graph k1,n ∗ k1,m is integer cordial. proof. case 1: when n is even and m can be either odd or even. let u1, u2, ...un 2 (1+m) be the vertices of n 2 leaves and let w1, w2, ...wn 2 (1+m) be the vertices of the other n 2 leaves and let u0 be the center vertex. we define f : v → [−⌊p 2 ⌋...⌊p 2 ⌋] as follows. f(u0) = 0 f(ui) = i, 1 ≤ i ≤ n 2 (1 + m) f(wi) = −i, 1 ≤ i ≤ n 2 (1 + m) 108 on integer cordial labeling of some families of graphs figure 1: integer cordial labeling of b3,5 case 2: when n is odd. let the center vertex be u0. let u1, u2, ...u⌊ n 2 ⌋(1+m) be the vertices of ⌊n2 ⌋ leaves and let w1, w2, ...w⌊ n 2 ⌋(1+m) be the vertices of the other ⌊n2 ⌋ leaves and let v1, v2, ...v(m+1) be the remaining vertices on the left out leaf, where v1 is adjacent to u. case 2.1: when m is odd. we define f : v → [−⌊p 2 ⌋...⌊p 2 ⌋] as follows: f(u0) = 0 f(ui) = i, 1 ≤ i ≤ ⌊ n 2 ⌋(1 + m) f(wi) = −i, 1 ≤ i ≤ ⌊ n 2 ⌋(1 + m) f(v1) = − ( ⌊ n 2 ⌋(2 + m) ) f(vi) = ⌊ n 2 ⌋(1 + m + i), 2 ≤ i ≤ ⌊ m + 1 2 ⌋ f(v⌊ m+1 2 ⌋+i) = −⌊ n 2 ⌋(1 + m + i), ⌈ m + 1 2 ⌉ ≤ i ≤ m + 1 case 2.2: when m is even. we define f : v → [−p 2 ...p 2 ]∗ as follows: f(u0) = 1 f(ui) = i + 1, 1 ≤ i ≤ ⌊ n 2 ⌋(1 + m) f(wi) = −(i + 1), 1 ≤ i ≤ ⌊ n 2 ⌋(1 + m) f(v1) = −1 f(vi) = ⌊ n 2 ⌋(1 + m + i), 2 ≤ i ≤ m + 1 2 f(v⌊ m+1 2 ⌋+i) = − ( ⌊ n 2 ⌋(1 + m + i) ) , m + 1 2 ≤ i ≤ m + 1 109 s sarah surya, alan thomas, lian mathew here, for all possible cases, we have |ef(1) − ef(0)| ≤ 0. therefore k1,n ∗ k1,m is integer cordial(see figure 2). figure 2: integer cordial labeling of k1,3 ∗ k1,3 theorem 3.3. olive tree tk admits integer cordial labeling. proof. let ui denote the (k + 1 − i)th branch and u denote the root vertex. case 1: when k(k+1) 2 + 1 is an odd number. an integer cordial labeling of tk is obtained by assigning the positive integer from 1 to k(k+1) 4 to the vertices of the branches namely u2, u3, u6, u7, u10, u11, u14,..., in any order and the negative integers from −1 to −k(k+1) 4 to the vertices of the branches namely u1, u4, u5, u8, u9, u12, u13, ..., in any order and let u = 0. case 2: when k(k+1) 2 + 1 is an even number. an integer cordial labeling of tk is obtained by assigning the positive integers from 2 to (k(k+1)+2) 4 to the vertices of the branches namely u2, u3, u6, u7, u10, u11, ..., in any order and the negative integers from −1 to −(k(k+1)+2) 4 to the vertices of the branches namely u1, u4, u5, u8, u9, u12, u13, ..., in any order and let u = 1. hence, we have |ef(1) − ef(0)| ≤ 1. therefore, olive tree admits integer cordial labeling(see figure 3). theorem 3.4. the jewel graph jn admits integer cordial labeling. proof. let v (gn) = {u, v, x, y, wi : 1 ≤ i ≤ n} and e(g) = {ux, uy, xy, xv, yv, uwi, vwi : 1 ≤ i ≤ n}. case 1: when n is even. 110 on integer cordial labeling of some families of graphs figure 3: integer cordial labeling of t4 we define f : v → [−p 2 ...p 2 ]∗ as follows: f(u) = 1 f(v) = −1 f(x) = 2 f(y) = −2 f(wi) = i + 2, 1 ≤ i ≤ ⌊ n 2 ⌋ f(wi) = −(i − ⌊ n 2 ⌋ + 2), ⌈ n 2 ⌉ ≤ i ≤ n − 1 case 2: when n is odd. we define f : v → [−⌊p 2 ⌋...⌊p 2 ⌋] as follows: f(u) = 1 f(v) = −1 f(x) = 2 f(y) = −2 f(wi) = i + 2, 1 ≤ i ≤ ⌊ n 2 ⌋ f(wi) = −(i − ⌊ n 2 ⌋ + 2), ⌈ n 2 ⌉ ≤ i ≤ n − 1 f(wn) = 0 here, for both the cases, we have n + 3 edges with label 1 and n + 2 edges with label 0. hence in all possible cases, we have |ef(1) − ef(0)| = 1. therefore, jn is integer cordial(see figure 4). theorem 3.5. jahangir graph jn,m is integer cordial except when n = 1. 111 s sarah surya, alan thomas, lian mathew figure 4: integer cordial labeling of j3 proof. when n = 1, we have j1,m to be a complete graph with m+1 vertices and hence not integer cordial. let u denote the central vertex adjacent to m vertices of cnm and let v1, v2, ..., vnm denote the vertices in the cycle cnm. case 1: when the number of vertices (nm + 1) is odd. we define f : v → [−⌊p 2 ⌋...⌊p 2 ⌋] as follows: f(u) = 0 f(vi) = i; 1 ≤ i ≤ nm 2 f(v( nm 2 +i) = −i; 1 ≤ i ≤ nm 2 case 2: when the number of vertices (nm + 1) is even. we define f : v → [−p 2 ...p 2 ]∗ as follows. f(u) = 1 f(vi) = i + 1; 1 ≤ i ≤ ⌊ nm 2 ⌋ f(v⌊ nm 2 ⌋+i) = −i; 1 ≤ i ≤ ⌈ nm 2 ⌉ hence in all possible cases, we have |ef(1) − ef(0)| ≤ 1. therefore jn,m admits integer cordial labeling except when n = 1(see figure 5). theorem 3.6. the crown cn ⊙ k1 admits integer cordial labeling. 112 on integer cordial labeling of some families of graphs figure 5: integer cordial labeling of j2,4 proof. let v1, v2, ..., vn be the vertices of the inner cycle and let u1, u2, ..., un be the pendent vertices where ui is adjacent to vi. we define f : v → [−p 2 ...p 2 ]∗ as follows: f(vi) = −i f(ui) = i here, we have n edges with label 1 and n edges with label 0. hence, |ef(1) − ef(0)| = 0. therefore, the crown cn ⊙ k1 is integer cordial(see figure 6). figure 6: integer cordial labeling of c4 ⊙ k1 4 conclusion in this paper, we proved that the banana tree, k1,n ∗ k1,m, olive tree, jewel graph, jahangir graph, crown graph are integer cordial. obtaining the integer cordial labeling of other class of graphs is still open. further investigation can be done for all the networks. 113 s sarah surya, alan thomas, lian mathew references [1] i. cahit, cordial graphs, a weaker version of graceful and harmonious graphs, ars combinatoria, 1987, vol 23, 201 – 207. [2] gao, w., a. s. i. m. a. asghar, and waqas nazeer, computing degree-based topological indices of jahangir graph, engineering and applied science letters, 2018, 16-22. [3] f. harary, graph theory, addison wesley, reading, massachusetts, 1972. [4] lokesha, v., r. shruti, and a. sinan cevik, m-polynomial of subdivision and complementary graphs of banana tree graph, j. int. math, virtual inst, 2020, 157-182. [5] lourdusamy, a., and f. patrick, sum divisor cordial graphs, proyecciones (antofagasta), 2016, 119-136. [6] marykutty p. t., and k. a. germina, open distance pattern edge coloring of a graph, annals of pure and applied mathematics, 2014, 191-198. [7] t. nicholas and p. maya, some results on integer cordial graph, journal of progressive research in mathematics (jprm), 2016, vol 8, issue 1,11831194. [8] s. sarah surya, sharmila mary arul, and lian mathew, integer cordial labeling for certain families of graphs, advances in mathematics: scientific journal, 2020, no.9, 7483–7489. 114 ratio mathematica volume 42, 2022 comparative study between local and global optimization for the heston model mohammed bouasabah* abstract the objective of this study is to estimate the calibration parameters of the heston stochastic volatility model by the two optimization methods: local and global, then to compare their performances and finally to recommend one of the two methods. to predict the prices of eur/usd currency options, we use the heston stochastic volatility model. we will first present the model and the two optimization methods: local and global, then we will estimate the calibration parameters using the two optimization methods with matlab software, then compare the two and recommend the most efficient method. results have shown that the local optimization provides excellent calibration parameters with a reduced computational time compared to the global optimization. therefore, we can clearly recommend it for the heston model. keywords: heston model; local calibration; global calibration. 2010 ams subject classification: 46n10, 91b60, 91b70, 91g20, 91g30 *national school of business and management, ibn tofail university, kenitra, morocco. mohammed.bouasabah@uit.ac.ma doi: 10.23755 / rm.v41i0.773. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 217 received on april 26th, 2022. accepted on june 26th, 2022. published on june 30th, 2022. m. bouasabah 1. introduction calibration is the process of adjusting the parameters of a model by integrating the uncertainty of the parameters and/or the model to obtain a representation of the modeled system that satisfies a predefined criterion. the model used is the one proposed by heston in 1993, which is a stochastic volatility model. the success of heston's model lies in the calibration of its parameters and in the analytical form of its characteristic function. in this study, first, we will give the explicit form of call currency option prices based on the heston model as well as the calibration parameters that must be computed. then we will present the two optimization methods: local and global as well as the criterion according to which we look for its parameters. finally, we calculate the parameters of the model for each method and we do an analysis of the performances. this work will help the trader who seeks to estimate the prices of currency options on the eur/usd parity using the heston model, to choose the optimization method that provides a more accurate calibration of the model. 2. theoretical principles of heston model the present value of the expected payoff from a european currency option is given by: 𝐸[𝑆𝑡 𝑆⁄ ] = 𝑆0𝑒 (𝑟𝑑− 𝑟𝑓)𝑡 (1) under the equivalent martingale measure, heston model is given by: [1]: 𝑑𝑆𝑡 = (𝑟𝑑 − 𝑟𝑓 )𝑆𝑡 𝑑𝑡 + √𝑉𝑡 𝑆𝑡 𝑑�̅�𝑡 (2) 𝑑𝑉𝑡 = 𝜅 ∗(𝜃∗ − 𝑉𝑡 )𝑑𝑡 + 𝜎√𝑉𝑡 𝑑�̅�𝑡 where 𝑑�̅�𝑡 𝑑�̅�𝑡 = 𝜌𝑑𝑡 𝒓𝒅 and 𝒓𝒇 are the domestic and forgein interest rates respectively. 𝜽 is the long-term mean of variance. 𝜿 is the rate of mean reversion. 𝝈 is the coefficient of volatility. 𝑺𝒕 and 𝑽𝑡 are the price and volatility of the process. 𝑩𝒕 and 𝒁𝒕 are correlated wiener process, and the correlation coefficients is 𝝆. according to feller the following condition will be satisfied (feller condition): 𝟐𝜿𝜽 > 𝝈² [3]. we assume that the rates 𝒓𝒅 and 𝒓𝒇 are constant. according to heston and nandi [8] currency call option at time t, given a strike price k, that expires at time t is given by: 𝐶(𝑆, 𝑣, 𝑡) = 𝑆𝑒−𝑟𝑓 𝜏𝑃1 − 𝐾𝑒 −𝑟𝑑 𝜏𝑃2 (3) where 𝜏 = 𝑇 − 𝑡 , 𝑃1 and 𝑃2 can be defined via the fourier inversion transformation [2]: 𝑃𝑗 = 1 2 + 1 𝜋 ∫ 𝑅𝑒[ 𝑒 −𝑖𝜓𝑙𝑛𝐾𝑓𝑗(𝑥,𝑣,𝜏,𝜓) 𝑖𝜓 ] ∞ 0 𝑑𝜓 (4) 218 comparative study between local and global optimization for heston model where j=1,2. and the characteristic function for the logarithm of the exchange rate, 𝑥 = 𝑙𝑛(𝑆𝑡 ) is given by [10]: 𝑓𝑗 (𝑥, 𝑣, 𝜏, 𝜓) = 𝑒 𝐶(𝜏,𝜙)+𝐷(𝜏,𝜙)𝑣𝑡+𝑖𝜙𝑥 where: 𝐶(𝜏, 𝜙) = (𝑟𝑑 − 𝑟𝑓 )𝜙𝑖𝜏 + 𝑎 𝜎² [(𝑏𝑗 − 𝜌𝜎𝜙𝑖 + 𝑑)𝜏 − 2ln ( 1−𝑔𝑒 𝑑𝜏 1−𝑔 )] 𝐷(𝜏, 𝜙) = 𝑏𝑗 − 𝜌𝜎𝜙𝑖 + 𝑑 𝜎2 ( 1 − 𝑒𝑑𝜏 1 − 𝑔𝑒𝑑𝜏 ) 𝑔 = 𝑏𝑗 − 𝜌𝜎𝜙𝑖 + 𝑑 𝑏𝑗 − 𝜌𝜎𝜙𝑖 − 𝑑 𝑑 = √(𝜌𝜎𝜙𝑖 − 𝑏𝑗 ) 2 − 𝜎²(2𝑢𝑗 𝜙𝑖 − 𝜙 2) 𝑢1 = 1 2 ; 𝑢2 = − 1 2 ; 𝑎 = 𝜅𝜃; 𝑏1 = 𝜅𝜆 − 𝜌𝜎; 𝑏2 = 𝜅 + 𝜆 3. materials and methods 3.1 calibration methods the parameters to be estimated are: 𝒗, 𝜿, 𝝈, 𝜽 and 𝝆. in this study, we use inverse problem method, which consists of collecting market data and then using it to search for the model parameters while minimizing the difference between the heston model prices and the real market prices. in other words, we look for the parameters that minimize the squared error between the market prices of european currency options and those of the model. we assume that for a call option, there are several realizations of the parameters of the heston model. then we can write: 𝑀𝑖𝑛 𝑆(ω) = min(ω) ∑ (𝐶𝑖 𝐻 (𝐾𝑖 , 𝑇𝑖 ) − 𝐶𝑖 𝑀 (𝐾𝑖 , 𝑇𝑖 )) 2 𝑁 𝑖=1 (5) ω is a vector of realizations 𝐶𝑖 𝐻 (𝐾𝑖 , 𝑇𝑖 ), 𝑎𝑛𝑑 𝐶𝑖 𝑀 (𝐾𝑖 , 𝑇𝑖 ) are prices estimated by the model and the market prices of the option 𝑖 respectively, with strike 𝐾𝑖 and maturity 𝑇𝑖. 𝑁 is the number of options used for calibration. it is then a nonlinear optimization problem under the constraint: 2𝜅𝜃 − 𝜎2 > 0 .[5] since it is a nonlinear function then there are several local extrema depending on the initially estimated value, and even with a judicious choice of the initial value, convergence to the global optimum is not certain. therefore, we will try both local and global optima. 3.1.1 local optimization if there are multiple local minima of the objective function, we cannot know whether the minimum that is found is the optimal one. in other words, we cannot know if it is a local or global minimum and there is no easy way to measure the distance to the global solution. for this, we will define an acceptance threshold, if the minimum found is acceptable, we 219 m. bouasabah can at least ensure that any solution found is consistent with our criteria and if no solution was found, we start the process again with another initial value [6]. in this work, we will consider the following set of acceptable solutions: ∑|𝐶𝑖 𝐻 (𝐾𝑖 , 𝑇𝑖 ) − 𝐶𝑖 𝑀 (𝐾𝑖 , 𝑇𝑖 )| ≤ 𝑁 𝑖=1 1 2 ∑|𝑏𝑖𝑑𝑖 − 𝑎𝑠𝑘𝑖 | 𝑁 𝑖=1 (6) bid and ask are the market prices for buying and selling respectively. for local optimization, we use the lsqnonlin (least-squares non-linear) function in matlab [12], which uses a reflective trust-region minimization algorithm (see yuan (1999) [9]. in addition, we will also define lower and upper bounds for the optimal parameters. these bounds allow us to reject solutions that are not economically acceptable. we will consider the following bounds [7]: • we consider the [0,1] interval for both for 𝜽 and 𝑽𝟎. • 𝝆 takes values from -1 to 1. • for 𝝈 an interval of 0 to 5 will be considered in the calibration. • to have a mean reversion, the parameter 𝜿 must take positive values. the maximum values of κ will be set dynamically during calibration as a consequence of the non-negativity constraint. • to this we add the feller condition which allows the model to avoid negative or zero values 2𝜅𝜃 − 𝜎² > 0, [3]. 3.1.2 global optimization global optimization allows to search for the global optimum using stochastic methods, even if a local minimum is found the search process does not stop. however, stochastic methods also have some drawbacks. the mathematical properties of these algorithms are less easy to handle than those of local algorithms. moreover, despite their name, their convergence to the global minimum is not guaranteed. indeed, the output sequence being stochastically determined, the algorithm may decide to stop prematurely and, in some cases, the solution obtained may be inferior to that of a local search. in sum, even if global optimization is theoretically more efficient, when working with functions of unknown form, it is not easy to decide which calibration procedure will be the most powerful [6]. in order to compare the results of the global optimization with the local one, we use the simulated annealing framework. this algorithm proceeds by iteration taking into account the previous information but also introducing randomization. initially, the algorithm starts with a high tolerance for random shocks, and different regions are examined during the first phase. as a result, even if a minimum is found, the algorithm continues to look for improved solutions. as time progresses, the algorithm reduces its tolerance until it eventually settles on the best optimum reached. 220 comparative study between local and global optimization for heston model in this work, we use the matlab function asamin which was developed by professor shinichi sakata and involves the adaptive simulated annealing (asa) algorithm, with dynamic adjustment for random shock tolerance [4]. goel and stander (2009) showed that the asa framework provides good results among a range of different global optimizers. for comparison purposes, we will use the same parameter bounds as outlined above. 4. results and discussion for the calibration of the parameters of the heston model, we use the eur/usd parity as an underlying asset. to price the currency option, eur libor (us libor) data is used to determine the domestic (foreign) interest rate and the following market data had to be taken from the forex market. this data includes: eur/usd exchange rate, strike price, bid and ask [11] the currency call option prices have been obtained from the bloomberg database. this data will be used to calibrate the model. the time period is from may 28, 2020 to may 7, 2021 and we use 5 maturities and for each maturity we consider 5 strike prices (25 currency options in all). the domestic and foreign risk-free interest rates are also provided by bloomberg and the available maturities match those of the options. we use in-themoney (itm) currency call options sorted by 𝑆𝑡 𝐾 . 4.1 local calibration results we have developed matlab scripts to obtain the following parameters: 𝝊 0 𝜽 0.0041 𝝈 0.3163 𝝆 0.9925 𝜿 12.4467 using these parameters to estimate the following currency option prices: [ 0.02332; 0.0206; 0.01800; 0.0155; 0.0132; 0.0506; 0.0456; 0.0408; 0.0363; 0.0321; 0.0501; 0.0463; 0.0427; 0.0393; 0.0361; 0.0556; 0.0521; 0.0487; 0.0455; 0.0424; 0.0560; 0.0527; 0.0496; 0.0466; 0.0437] by taking the difference between these estimated prices and the market price (the price considered is the median price i.e., 𝑀𝑖𝑑 𝑝𝑟𝑖𝑐𝑒 = bid+ask 2 ) and comparing this difference with the difference between the market bid and ask prices, we find that all the options considered meet the selection criteria chosen. 221 m. bouasabah the table above shows that the calibrated heston model gives an accurate match for all traded options. all currency options have a predicted value within the observed bid-ask spread. moreover, the average deviation of the model from the average market price is 0.00068153, which is less than the average deviation of the bid-ask spreads (0.003978) when evaluated against our acceptance criterion. the time elapsed for the local calibration is 2.377027 seconds. 4.2 global calibration results we have developed matlab scripts to obtain the following parameters: 𝝊 0 𝜽 0.0051 𝝈 0.2898 𝝆 1 𝜿 12.5445 𝜿 12.4467 using these parameters to estimate the following currency option prices: [0.0234; 0.0208; 0.0183; 0.0161; 0.0139; 0.0509; 0.0463; 0.0419; 0.0378; 0.0340; 0.0524; 0.0488; 0.0453; 0.0421; 0.0390; 0,0587; 0.0553; 0.0521; 0.0490; 0.0460; 0.0597; 0.0565; 0.0534; 0.0505; 0.0477] by calculating the difference between these estimated prices and the market price (the price considered is the median price i.e., 𝑀𝑖𝑑 𝑝𝑟𝑖𝑐𝑒 = bid+ask 2 ) by comparing this difference with the difference between the market bid and ask prices, we find that 21 options out of the 25 options considered meet the selection criterion. as can be noticed, the parameters found with the asa algorithm are somewhat different from those found in the local optimization case. moreover, in the global calibration, 21 out of 25 satisfied the chosen selction criterion and the average deviation from the average market price is 0.00252278. therefore, the quality of the asa solution is less accurate than the results obtained by matlab's lsqnonlin. in addition, the main drawback of asa is its significantly higher computation time (590.013684 seconds in asa versus 6.5 seconds in matlab's lsqnonlin). based on these analyses, we can confirm that matlab's lsqnonlin provides excellent calibration results (the average distance to the average market price is 0.00068153 versus 0.00252278 for asa and all currency call option prices are within the observed bid-ask spreads), and also uses lower computation times. however, these results depend on the complexity of the objective function; it is possible that this function is not complicated enough to take advantage of the asa algorithm. in particular, since we generally do not know whether the objective function can have 222 comparative study between local and global optimization for heston model multiple local minima, a prudent choice would be to use both calibration approaches. the drawback is that a global search does not necessarily improve the results delivered by a local search. however, progress in computing capacity and numerical methods is reducing the time required for global calibration. in our calibration, the execution time of asa was less than 10 minutes, which for many practical applications is worth testing better solutions. 5. conclusion in this study, we used local and global optimization to look for the calibration parameters of the heston stochastic volatility model used to estimate the prices of eur/usd currency options. our results showed that local optimization provides excellent calibration parameters with reduced computational time compared to global optimization. therefore, we can clearly recommend the local optimization for the heston model. references [1] heston, steven l. "a closed-form solution for options with stochastic volatility with applications to bond and currency options." the review of financial studies 6.2: 327-343. 1993. [2] carr, peter & stanley, morgan & madan, dilip. option valuation using the fast fourier transform. j. comput. finance. 2. 10.21314/jcf.1999.043. doi:10.21314/jcf.1999.043. 2001. [3] gikhman, ilya i., a short remark on feller's square root condition .2011. available at: http://dx.doi.org/10.2139/ssrn.1756450. [4] goel, t. and stander, n. adaptive simulated annealing for global optimization in ls-opt. 7th european ls-dyna conference. 2009. [5] jahn, johannes. introduction to the theory of nonlinear optimization. springer nature, 2020. [6] milan mrázek and jan pospíšil, calibration and simulation of heston model , open mathematics, https://doi.org/10.1515/math-2017-0058. 2017. [7] ricardo crisostomo. "an analysis of the heston stochastic volatility model: implementation and calibration using matlab," ,ssrn electronic journal 2015. doi:10.2139/ssrn.2527818. [8] s. heston and s. nandi, “a closed-form garch option valuation model”. the review of financial studies, volume 13, issue 3, july 2000, pages 585–625. 2000. https://doi.org/10.1093/rfs/13.3.585. [9] yuan, ya-xiang. a review of trust region algorithms for optimization. icm99: proceedings of the fourth international congress on industrial and applied mathematics. 1999. 223 http://dx.doi.org/10.21314/jcf.1999.043 https://doi.org/10.1515/math-2017-0058 m. bouasabah [10] wu, l. from characteristic functions and fourier transforms to pdfs/cdfs and option prices. zicklin school of business, baruch college. 2007. [11] chicago board options exchange, s and p 500 index options specications on the cboe website, accessed at http://www.cboe.com [12] the mathworks, help files on the mathworks website-lsqnonlin, accessed at http://www.mathworks.com. 224 microsoft word kapur lavoro finale.doc ratio mathematica 18 (2008), 62 90 62 effect of introduction of fault and imperfect debugging on release time p. k. kapur*, deepali gupta@, anshu gupta*, p. c. jha* abstract one of the most important decisions related to the efficient management of testing phase of software development life cycle is to determine when to stop testing and release the software in the market. most of the testing processes are imperfect once. in this paper first we have discussed an optimal release time problem for an imperfect faultdebugging model due to kapur et al considering effect of perfect and imperfect debugging separately on the total expected software cost. next, we proposed a srgm incorporating the effect of imperfect fault debugging and error generation. the proposed model is validated on a data set cited in literature and a release time problem is formulated minimizing the expected cost subject to a minimum reliability level to be achieved by the release time using the proposed model. solution method is discussed to solve such class of problem. a numerical illustration is given for both type of release problem and finally a sensitivity analysis is performed. keywords: software reliability, non-homogeneous poisson process, imperfect debugging, error generation, release time. 1. introduction last decade of the twentieth century is marked in history for the incredible growth in the information technology. consequently computers and computer-based systems have entered in every walk * department of operational research, university of delhi, delhi –07, india. @ department of mathematics, jaypee institute of information technology, noida p. k. kapur, d. gupta, a. gupta, p. c. jha 63 and talk of our lives. we have become heavily dependent on automated tools and intelligent systems for almost every activity. a mere delay in the operation of these systems can led to big financial loses. our lives depend critically on the correct functioning of these systems. there are already numerous instances where failures of computer-controlled systems have led to colossal loss of human lives and economy. with the increased dependence of human kind on software systems, software systems are also becoming complex and large and a major concern for the software developers is to deliver more reliable software in smaller development time. it is the testing stage of the software development in which attempts are made to remove most of the faults lying dormant in software. a successful test strategy begins by considering the requirement specification and continues by specifying test cases based on this requirement specification, to be executed later to find the corresponding faults, which might have been introduced during the various stages of the sdlc. the growing field of software reliability engineering deals in building mathematical models that describe the failure\removal phenomenon with respect to time\testing efforts and consequent enhancement in reliability of the software due to fault removal known as software reliability growth modeling (srgms). several srgms have been discussed and validated by the various researchers under the varying set of assumptions. most of these models depict either exponential or s-shaped relationship between the testing time\effort and the corresponding number of faults removed [2,9]. most of the earlier software reliability models assume the fault removal process (fault debugging) to be perfect i.e. when an attempt is made to remove a fault, it is removed with certainty and no new faults are introduced. but this assumption is not realistic due to the complexity of the software system and incomplete understanding of the user’s requirements or specifications by the testing team. the software testing team may not be able to fix the cause of the failure properly or they may introduce new faults during removal. therefore effect of introduction of fault and imperfect debugging on release time 64 it is necessary to incorporate the effect of imperfect debugging into the software reliability growth modeling. in recent years, several imperfect debugging srgms have been proposed and studied (pham [10], kapur and younes [5], slud [12], and obha and chou [8], etc.) there are two type of imperfect debugging possibilities-first, on a failure the corresponding fault is identified, but just because of incomplete understanding of the software, the detected fault is not removed completely and hence the fault content of the software remains unchanged on the removal action, proposed by kapur [3] known as imperfect fault debugging, second, when on a failure the corresponding fault is identified and removed with certainty but some new faults are added to the software during the removal process, proposed by obha and chou[8]. this type of imperfect debugging led to an increase in the fault content of the software known as error generation. no software can be tested indefinitely in order to make it bug free since users of the software want faster deliveries and constraint on development cost. as discussed above an important objective of developing srgm is to predict software performance using the measure of software reliability and use the information for decisionmaking. an important decision problem of practical concern is to determine when to stop testing and release the software system to the user known as “release time problem”. this decision depends on the model used for describing the failure phenomenon and the criterion used for determining system readiness. the optimization problem of determining the optimal time of software release can be formulated based on goals set by the management. firstly the management may wish to determine the optimal release time such that total expected cost of testing in the testing and operation phase is minimum. secondly they may set a reliability level to be achieved by the release time. thirdly they may wish to determine the release time such that the total expected cost of the software is minimum and reliability of the software is achieved to a certain desired level. such a problem is known as a bi-criteria release time problem. for bi-criteria release p. k. kapur, d. gupta, a. gupta, p. c. jha 65 time problem release time is determined by carrying a trade off between cost and reliability. many researchers in literature have studied various release time problems for different srgms [2,3,6,7,9]. min xie [13] attempted to determine the optimal release time of software using the srgm proposed by obha and chou [8] incorporating the second type of imperfect debugging i.e. error generation. whereas the author is referring to imperfect fault debugging that is due to the fault not fixed properly, in his cost model, which creates confusion between two types of imperfect debugging. the cost model used by the author is incomplete, as he considered the cost of fixing an error to be same for due to perfect and imperfect fault debugging during testing and operation phase. the mathematical form of srgm by obha and chou [8] is equivalent to the kapur [3] model of imperfect fault debugging but the two models are based on different set of assumptions, obha and chou model incorporate the effect of error generation whereas kapur model incorporate the effect of imperfect fault debugging. in this paper we have determined the optimal time when software is ready to be release for use using the imperfect fault-debugging model due to kapur [3] modifying the cost model of min xie. we incorporated separate cost of fixing an error due to perfect and imperfect fault debugging during testing and operation phase in the cost model and determined the release time in the way as determined by min xie, which gives the optimal values of the release time and the level of perfect debugging p. however it is imperative to estimate the level of perfect fault debugging i.e. p from the srgm used to describe the failure phenomenon using the collected failure data, and not as a decision to be obtained from release time problem. at the optimal release time of software determined minimizing the cost, we may not obtain the desired reliability level. hence if we have a reliability level to be achieved by the optimal time of software release we should incorporate the desired reliability level either as a constraint of a release time problem or as an objective of bi-criteria release time problem. however we may not obtain a minimum cost at the desired reliability level, therefore release time is determined by a trade-off between reliability and cost. in this paper we have proposed a srgm incorporating two types of imperfect effect of introduction of fault and imperfect debugging on release time 66 debugging simultaneously. the proposed model is validated on software failure data sets used in literature. we then determined the release time for the proposed model minimizing the total expected software cost subject to minimum level of reliability to be achieved by the release time incorporating the effect of imperfect debugging and error generation on cost model. this paper is organized as follows: in the section 2.1 we have discussed a release time problem for perfect debugging srgm due to goel okumoto. in section 2.2.1 we have discussed we have reviewed imperfect fault debugging srgm due to kapur et al. then in section 2.2.2 we have discussed the effect of imperfect debugging on total expected software cost and then finally in section 2.2.3 we formulated a release time problem for imperfect fault debugging srgm due to kapur et al and derived the optimal release time of the software minimizing the total expected software cost. in section 3 first we proposed a srgm incorporating the effect of imperfect debugging and fault generation in section 3.1. parameters of the proposed model are estimated in section 3.2. further we discuss the effect of imperfect fault debugging and error generation on total expected software cost in section 3.3 and finally a release time problem is formulated and solved minimizing the total expected software testing cost subject to minimum reliability level constraint. in section 4.1 a numerical illustration is given for both type of release problem and finally a sensitivity analysis is performed to determine the effect of variations in minimum reliability level to be achieved, in cost of fixing an error perfectly and imperfectly in operation phase and in level of perfect debugging.. 2. release time problem for imperfect fault debugging srgm 2.1 determination of release time for perfect debugging srgm among all srgms developed so far a large family of stochastic reliability models based on a non-homogeneous poisson process known as nhpp reliability models, has been widely used. some of p. k. kapur, d. gupta, a. gupta, p. c. jha 67 them depict exponential growth while others show s-shaped growth depending on nature of growth phenomenon during testing. most commonly cost model seen in literature for determination of release time for perfect debugging nhpp models is [2,13] 1 3c c m(t) c (m( ) m(t)) ct     …(2.1) using goel okumoto nhpp [2] model, for which the mean value function is btm(t) a(1 e )  …(2.2) the optimal release time minimizing the total expected software cost defined as (1) is given by * 3 1 ab(c c )1 t ln b c        …(2.3) maximum likelihood estimates (mle) of a and b for the software failure data cited in zhang and pham [10], are obtained as a = 142.32 and b = 0.1246. assuming c1 = $200, c3 = $1500, and c = $5, from (3), the optimal release time is calculated as 67.70556 and the minimum expected software cost is found to be $28,842. however, the model assumes a perfect testing process. it would be of interest to study the effect of imperfect debugging on total expected software testing cost. in the next section we have discussed the effect of imperfect fault debugging on expected software testing cost, briefly discussing the imperfect fault debugging srgm due to kapur [3]. 2.2 release time problem for imperfect fault debugging srgm ( kapur []) 2.2.1 imperfect fault debugging srgm effect of introduction of fault and imperfect debugging on release time 68 a simple imperfect fault debugging model proposed by kapur [3] assume on a failure the corresponding fault is identified and when an attempt is made to remove the fault it is not fixed properly, which does not lead to any change in the initial fault content of the software. the model is formulated as follows model assumptions 1. software system is subject to failures at random times caused by faults remaining in the software. 2. failure rate of the software is equally affected by errors remaining in the software. 3. at any time the failure rate of the software is proportional to the faults remaining in the software. 4. on a instantaneous repair effort starts and the following may occur: (a) fault contents are reduced by one, with probability p (b) fault contents are unchanged with probability 1-p. 5. the error removal phenomenon in the software is modeled by nhpp. notations a : initial error content. b : proportionality constant(fault removal rate per remaining fault). p : probability of perfect debugging. mf(t) : mean number of failures detected in (0,t]. mr(t) : mean number of faults removed in the software till time t. (t) : intensity function or fault detection rate per unit time. the differential equation describing the rate of change of )(tmr with respect to time under the assumptions specified above and following the notations is given by  r rm (t) bp a-m (t)  …(2.4) p. k. kapur, d. gupta, a. gupta, p. c. jha 69 solving equation (2.4) under the initial condition 0)0( rm is given by bpt rm (t) a 1 e     …(2.5) corresponding mean number of failures in (0,t] is given by  dt(t)mab)( 0 r  t f tm …(2.6) bpt f a m (t) 1 e p     …(2.7) the nhpp intensity function is given by (t) ab exp( bpt)   …(2.8) it can be seen that (t) is a decreasing function in t with λ(0) = ab and λ(∞)=0. in the next section we have proposed the cost model incorporating the effect of imperfect debugging. 2.2.2 effect of imperfect debugging on the cost model a major concern in software development is the cost. it is well known that the development of a software system is time-consuming and costly. since most software testing processes are imperfect debugging ones, it is of great importance for the management to know the effect of the imperfect debugging on software cost (ammann et al. [1], shanthikumar [11], and pham [10]). on the other hand, if the release time of the software is determined by the minimum cost criterion, the imperfect debugging will affect the release time as well. effect of introduction of fault and imperfect debugging on release time 70 the parameter p representing the probability of perfect debugging can also represent the testing level, indicating “how perfect” the testing process is. testing level parameter p is usually influenced by a number of factors, such as the experience of the testing personnel, the testing strategy adopted, and the number of reviews in debugging. when the testing level is low, it is possible to increase it to a certain extent, but usually this has to be achieved at a higher testing cost. total expected software cost includes cost of testing and the cost of fixing a fault during testing and operation phase for perfect and imperfect debugging. cost of fixing an error is different for both perfect and imperfect debugging. also the cost of testing is a function of perfect debugging probability p. since the testing cost parameter c depends on the testing team composition and testing strategy used, if the probability of perfect debugging is to be increased, it is expected that extra financial resources will be needed to engage more experienced testing personnel, and this will result in an increase of c. in other words, c should be a function of the testing level, denoted by c(p) and hence this function should possess the following two properties: 1. c(p) is a monotonous increasing function of p. 2. when p1, c(p). the second property implies that perfect debugging is impossible in practice or the cost of achieving it is extremely high. notations: c1(c2): cost incurred on a prefect (imperfect) debugging effort before release of the software system. c3(c4): cost incurred on a prefect (imperfect) debugging effort after release of the software system. (c3 > c1, c4 > c2). c : testing cost per unit time. t : release time of the software. t* : optimal release time. p. k. kapur, d. gupta, a. gupta, p. c. jha 71 r0 : desired level of software reliability at the release time(0 < r0 < 1). although there are many cost functions that can satisfy these conditions, a simple, but reasonable function that meets the two properties above is given by:   c c(p) 1 p   …(2.9) hence, the cost model (2.1) can be modified as     1 2 f 3 4 f f ct min c(t,p) c p c (1 p) m (t) c p c (1 p) m ( ) m (t) (1 p)           ...(2.10) if the release time remains at 67.70556, the software cost under different probabilities of perfect debugging or testing levels is calculated and summarized in table 1. it is clear that the software cost changes significantly as the testing level, p, changes. obviously, if the management has not taken into consideration the effect of imperfect debugging on software cost, the model may give a wrong estimate of the system reliability and\or cost at the release time. table 1. in the next section we will determine optimal release time and optimal testing level such that the total expected software cost is minimized. p cost($103) 0.7 37037 0.75 35485.03 0.8 34345.28 0.85 33652.61 0.9 33693.12 0.95 36123.14 1 ∞ effect of introduction of fault and imperfect debugging on release time 72 2.2.3 optimal release policy the optimization problem minimizing the total expected software cost in order to determine optimal release time t* and optimal testing level p* can be formulated as follows      1 2 f 3 4 f f ct min c(t,p) c p c (1 p) .m (t) c p c (1 p) . m ( ) m (t (1 p)           subject to 0 p 1 and t 0   …(2.11) using the principles of calculus the above optimization problem can be solved as follows: taking partial derivates of c(p,t) with respect to p and t and equate them to zero, we have that      bpt bpt1 2 3 4c cc p c (1 p) .abe c p c (1 p) . abe 0t (1 p)              …(2.12) and             bpt bpt bpt 1 2 1 2 2 2 bpt bpt bpt 3 4 3 4 2 2 c a a a abt c c 1 e c p c (1 p) e e p p pp p a a abt ct (c c ) e c p c (1 p) e e 0 p pp 1 p                                        …(2.13) from (2.12) t can be expressed in terms of p as 2 1ab(d d )(1 p)1t g(p) ln bp c          …(2.14) where )1(,)1( 432211 pcpcdpcpcd  it is clear that, when p takes values between (0,1), the condition t > 0 is always satisfied. p. k. kapur, d. gupta, a. gupta, p. c. jha 73 substituting the value of t from (2.14) into (2.13), we get   2 1 2 2 1 2 2 1 4 2 2 2 2 1 ab d d 1 p (2p 1)c(d d ) ln c ab(d d )(1 p) c(c c )(1 p) cc 0 p bp (1 p) (d d )                    …(2.15) or, equivalently h(p) = 0.   2 1 2 2 1 2 2 1 4 2 ab d d 1 p h(p) (2p 1)c(d d )ln c ab(d d )(1 p) c(c c )(1 p) 0 c                   …(2.16) h(p) is a continuous function of p on(0,1) and p 0 p 1 lim h(p) k lim h(p)        …(2.17) where  4 2 2 4 2 ab(c c ) k c ln abc c c c c             …(2.18) now, taking the derivative of h(p) with respect to p, we have that           ab d d 1 p 12 1h (p) 2c(d d )ln (2p 1)c d d c(c c )2 1 2 1 4 2c 1 p ab d d 1 p2 122c ab(d d )(1 p) c ab(1 p) (2p 1)cln c c c c 02 2 1 2 3 1 4 2c                                           it can be seen that h(p) is a continuous and strictly decreasing function on (0,1) and     ab c c4 2lim h (p) 2k cln c ab c c c c lim h (p)2 4 3 2 1cp 0 p 1                    the following theorem summarizes some analytical results regarding the existence and uniqueness of the optimal solution. effect of introduction of fault and imperfect debugging on release time 74 theorem 1. the optimal values of p and t, denoted by p* and t*, which minimize the expected software cost given by (9) are as follows: case 1. if k  0, then p* = inf{p : h(p)< 0}and t* = g(p*). case 2. if k >0, then define p = inf(p:dh/dp < 0}and 1. if h(p) > 0, then p* = min[c(p1, t1),c(p2, t2)] and t* =g(p*),where p1 and p2 are the solutions to the equation of h(p) = 0 and t1 = g(p1), t2 = g(p2). 2. if h(p) = 0, then p* equals the unique solution to the equation of h(p) = 0 and t* = g(p*). 3. if h(p) < 0, then p* and t* does not exist within 0<p<1 and t>0. using the above procedure to find the optimal release time first we need to determine the value inf{p : h(p)< 0 or p = inf(p:dh/dp < 0} what ever is the case assuming a perfect debugging environment i.e p=1 as both h(p) and h(p) function of p in order to determine the optimal value of p and then using this optimal value of p we estimate the other parameters of the srgm based on the collected failure data and then determine the optimal release time. the procedure if repeated for this optimal value and more dense data we will obtain another set of optimal values and hence it is a iterative approach. hence the solution procedure adopted by min xie does not terminate in one step to give the optimal values. however it is imperative to estimate the level of perfect fault debugging i.e. p from the srgm used to describe the failure phenomenon using the collected failure data over a period of time, and not as a decision to be obtained from release time problem by minimizing cost function. the effect of level of perfect debugging on release time can be obtained by carrying a sensitivity analysis on the release problem. whenever a decision is made to release the software the management evaluate the reliability of the software as quality metric at the release time. in the numerical example given in this paper we found that the reliability level at the optimal release time is 0.9398, where as for the p. k. kapur, d. gupta, a. gupta, p. c. jha 75 problem discussed by min xie it is 0.9117. however if the management desires to obtain a reliability level 0.95 by the release time, the approach followed above to find the optimal solution couldn’t be used. therefore we must consider the level of reliability to be achieved while formulating such class of problem. before we discuss the release time problem by minimizing the cost under reliability constraint we propose and validate a srgm incorporating the effect of both imperfect fault debugging and error generation in the next section. 3. release time problem for an srgm incorporating two types of imperfect debugging 3.1 srgm with two types of imperfect debugging during the testing process when a fault in encountered, corresponding fault is identified and an attempt is made to remove the fault, there are three possibilities, first the fault is removed perfectly, secondly the fault is not removed perfectly due to which the fault content remains unchanged known as imperfect fault debugging, third the fault is removed perfectly, but when the test case that led to the failure is reexecuted some other fault is encountered, known as error generation. in fact while removing the fault the programmer has introduced a new fault leading to an increase in total fault content of the software. newly introduced fault leads to a failure only when the original fault is removed perfectly. model assumptions 1. software system is subject to failures at random times caused by faults remaining in the software. 2. failure rate of the software is equally affected by errors remaining in the software. 3. at any time the failure rate of the software is proportional to the faults remaining in the software. effect of introduction of fault and imperfect debugging on release time 76 4. on a instantaneous repair effort starts and the following may occur: (a) fault contents are reduced by one with probability p (b) fault contents are unchanged with probability 1-p. 5. the error removal phenomenon in the software is modeled by nhpp. 6. during the fault removal process faults are generated with a constant probability . under the assumptions specified above the differential equation for the proposed model is given by  'r rm ( t ) bp a(t)-m (t) …(3.1) where a(t) can be expressed as ra(t) a m (t)   …(3.2) substituting (3.2) in (3.1) we have  'r r rm (t) bp a + m (t) m (t)  …(3.3) solving equation (3.3) under the initial condition ' rm (0) 0 we get bp(1 α)tr a m (t) 1 e 1 α       …(3.4) corresponding mean number of failures in (0,t] is given by  dt(t)ma(t)b)( 0 r  t f tm …(3.5)  α)tbp(1f e1α)p(1 a (t)m    …(3.6) the nhpp intensity function is given by )exp()( bptabt  …(3.7) p. k. kapur, d. gupta, a. gupta, p. c. jha 77 it can be seen that (t) is a decreasing function in t with (0) ab and ( ) 0     . in the next section we validate and compare the model with some existing models. 3.2 parameter estimation method of least squares or maximum likelihood has been suggested and widely used for estimation of parameter of mathematical models. the model proposed in this paper is a non-linear and it is difficult to find solution for nonlinear models using least square method and require numerical algorithms to solve it. statistical software packages such as spss help to overcome this problem. spss is a statistical package for social sciences. it is a comprehensive and flexible package for statistical analysis and data management system. spss can take data from almost any type of file and use them to generate tabulated reports, charts and plots of distributions and trends, descriptive statistics, and conduct complex statistical analysis. spss regression models enables the user to apply more sophisticated models to the data using its wide range of nonlinear regression models. for the estimation of the parameters of the proposed model method of least square has been used. non-linear regression is a method of finding a nonlinear model of the relationship between the dependent variable and a set of independent variables. unlike traditional linear regression, which is restricted to estimating linear models, nonlinear regression can estimate models with arbitrary relationships between independent and dependent variables. 3.2.1 comparison criteria 1. mean square error (mse): effect of introduction of fault and imperfect debugging on release time 78 the model under comparison is used to simulate the fault data, the difference between the expected values, n(t) and the observed data ni is measured by mse as follows. 2k i i i 1 ( n ( t ) n ) m s e k    …(3.8) where k is the number of observations. the lower mse indicates less fitting error, thus better goodness of fit. 2. coefficient of multiple determination (r2): we define this coefficient as the ratio of the sum of squares resulting from the trend model to that from constant model subtracted from 1. 2 residual ss r 1 corrected ss  … (3.9) r2 measures the percentage of the total variation about the mean accounted for the fitted curve. it ranges in value from 0 to 1. small values indicate that the model does not fit the data well. the larger value of r2 explains the better fit of the model. 3.2.2 data analysis and model comparison to validate the proposed model we have carried out the parameter estimation on a data set from a real time command and control system, which represents 136 failures, observed during system testing for 25 hours of cpu time [9]. parameters of the model are estimated by the nonlinear least squares method in spss using cumulative failure data against time. estimated parameter values are given in table2. the mse and r2 values are also given. the fitting of the models is illustrated graphically in figure 1 and figure 2. p. k. kapur, d. gupta, a. gupta, p. c. jha 79 table 2: figure 1: figure 2: cumulative failures curve 0 20 40 60 80 100 120 140 160 1 4 7 10 13 16 19 22 25 time m (t ) actual data estimated values non-cumulative failure curve 0 5 10 15 20 25 30 1 4 7 10 13 16 19 22 25 time m (t ) actual data estimated values in the next section we have proposed the cost model incorporating the effect of imperfect fault debugging and error generation. 3.3 effect of imperfect fault debugging and error generation on cost model like knowing the effect of the imperfect debugging on software cost it is also of great importance for the management to know the effect of fault generation on cost. since due to fault generation amount of fault content of software increases, it has a direct effect on the reliability level of the software achieved by the release time. the parameter  representing the probability of error generation is usually influenced by a number of factors, such as the experience of the testing personnel, the testing strategy adopted, and the number of estimated parameter values for the proposed model parameters goodness of fit criteria a b p  rmspe r2 134 0.140238 0.998417 0.0125628 30.64387 .96641 effect of introduction of fault and imperfect debugging on release time 80 reviews in debugging etc. it is possible to decrease the value of  to a certain extent, but usually this has to be achieved at a higher testing cost. as specified above total expected software cost includes cost of testing and the cost of fixing a fault during testing and operation phase for perfect and imperfect debugging. cost of fixing an error is different for both perfect and imperfect debugging however it remains unchanged due to error generation. but the cost of testing is a function of both perfect debugging probability p and fault generation probability  . since the testing cost parameter c depends on the testing team composition and testing strategy used, if the probability of perfect debugging is to be increased and probability of error generation is to be decreased, it is expected that extra financial resources will be needed to engage more experienced testing personnel, and this will result in an increase of c. in other words, c should be a function of the testing level and error generation, denoted by c(p,) and hence this function should possess the following two properties: 1. c(p,) is a monotonous increasing function of p and (1- ). 2. when p1, and 0  , c(p,). the second property implies that perfect debugging is impossible in practice or the cost of achieving it is extremely high. although there are many cost functions that can satisfy these conditions, a simple, but reasonable function that meets the two properties above is given by:    c c(p) 1 p 1    …(3.10) hence, the cost model (2.9) can be modified as         ct min c(t,p) c p c (1 p) m (t) c p c (1 p) m ( ) m (t)1 2 f 3 4 f f 1 p 1            …(3.11) p. k. kapur, d. gupta, a. gupta, p. c. jha 81 in the next section we will determine optimal release time for software minimizing the total expected cost subject to the desired reliability constraint. 3.4 optimal release policy the optimization problem minimizing the total expected software cost in order to determine optimal release time t* subject to the software reliability not less than a specified reliability objective can be formulated as follows    1 2 f 3 4 f f ct min c(t) c p c (1 p) .m (t) c p c (1 p) . m ( ) m (t) 1 p(1 )              subject to 0))]()((exp[)|( rtmxtmtxr  where 0 < r0 < 1 and x > 0. using the principles of calculus and assuming that the values of all the parameters of the proposed srgm have been estimated including p and  form the past failure data, the above optimization problem can be solved as follows: taking partial derivates of c(t) with respect to t and equating it to zero, we have    bp(1 )t bp(1 )t1 2 3 4c cc p c (1 p) abe c p c (1 p) abe 0t 1 p(1 )                  …(3.12) from (3.12) we observe that 2 1 c (t) (d d )(1 p(1 ))       …(3.13) where 1 1 2 2 3 4d c p c (1 p) , d c p c (1 p )      …(3.14) (t) ab exp( bp(1 )t) (0) ab ( ) 0         …(3.15) from (3.15) it can be seen that (t) is a decreasing function in time. effect of introduction of fault and imperfect debugging on release time 82 result 1: if   2 1 c ab > d 1 (1 )  d p  then c(t) is decreasing for 0t < t and increasing for t > t0 thus, there exist a finite and unique t=t0 (>0) minimizing the total expected cost. and if   2 1 c ab d 1 (1 )    d p  then c '(t) 0 for t 0  and hence c(t) is minimum for t = 0. further reliability of software defined as “given that the testing has continued up to time t, the probability that a software failure does not occur in time interval (t, t x) (x 0)  ”. hence the reliability of software is represented mathematically as  m(t x) m(t)r(x | t) r (t x | t) exp     …(3.16) using (3.16) we obtain ( )( | 0) , ( | ) 1m xr x e r x   …(3.17) result 2: from (3.17) it is observed that ( | ) , 0r x t t is a increasing function of time. thus 0( | 0) r x r there exist t=t1(>0) such that 0( | ) r x t r and if 0( | 0) r x r then 0( | ) 0  r x t r t and t=t1=0. combining the cost and reliability requirements we state the following theorem for optimal release policy for the proposed srgm of imperfect fault debugging and error generation. theorem 2: assuming 3 1 4 2 0c c 0, c c 0, c 0, x 0, and 0 r 1        (a)    0 0 12 1 c if ab > & ( | 0) 1, * max( , ) d 1 (1 ) r x r t t t d p        (b)    0 02 1 c if ab > & ( | 0) 0, * d 1 (1 ) r x r t t d p        p. k. kapur, d. gupta, a. gupta, p. c. jha 83 (c)    0 12 1 c if ab & ( | 0) 1, * d 1 (1 ) r x r t t d p         (d)    02 1 c if ab & 0 ( | 0), * 0 d 1 (1 ) r r x t d p         using the above theorem we can determine the optimal release time minimizing the total expected software cost under a desired reliability level constraint. 4. numerical examples and sensitivity analysis 4.1 numerical example of release time problem for imperfect fault debugging srgm assuming that the parameters a and b of imperfect fault debugging srgm due to kapur et al the srgm have already been estimated using the collected failure data and estimated values of a and b are 142.32 and 0.1246 respectively. further assuming that cost of perfect fault debugging during testing and operation phase i.e. c1 and c2 to be $200 and $110 respectively, cost of imperfect fault debugging during testing and operation phase to be same i.e. c3 = c4 =$1500 and cost of per unit testing c=$10. following the theorem 1 we obtain the optimal release time t* = 56.28, optimal level of perfect debugging p* = 0.8897 and optimal total expected software cost c(t*) = 33365.047 and achieved level of reliability r(t*) = 0.9398. where as for the release time problem discussed by min xie t* = 55.196, optimal level of perfect debugging p* = 0.85 and optimal total expected software cost c(t*) = 37931.44 and achieved level of reliability r(t*) = 0.9117. thus we can see that if we include separate cost of fixing faults perfectly and imperfectly it has significant effect on optimal release time and cost depending upon the values of the various costs associated with the cost model. note that the above release time problem is solved in way as done by min xie which gives optimal values of p and t* however it is imperative to estimate the level of perfect fault debugging i.e. p from the srgm used to describe the failure phenomenon using the collected failure data, and not as a decision to be obtained from release time problem to be obtained from effect of introduction of fault and imperfect debugging on release time 84 release time problem. in the next numerical example we have determined the optimal release time minimizing the cost function subject to reliability constraint assuming that value of perfect debugging and error generation parameters are estimated using collected failure data. 4.2 numerical example of release time problem for an srgm incorporating two types of imperfect debugging assuming that the parameters a, b, p and α of proposed srgm have already been estimated using the collected failure data and estimated values of a, b, p and α are 134, 0.14024, 0.99842 and 0.01256 respectively. further assuming that cost of perfect and imperfect fault debugging during testing i.e. c1 and c2 to be $200 and $110 respectively, cost of perfect and imperfect fault debugging during operation phase to be same i.e. c3 = c4 =$1500 and cost of per unit testing c=$10. if minimum reliability requirement by the release time is 0.85, following result 1 and 2 we obtain t0 = 25.6162 and t1 =38.3983. then finally following theorem 2 we obtain t* = 38.3983. the minimum total expected software cost at t* i.e. c(t*) = $55235.55 and number of faults removed by the release time m(t*) = 135. 4.3 sensitivity analysis we have conducted, a sensitivity analysis of the release time problem formulated for the proposed model to study the effect of variations in minimum reliability requirement by the release time, most sensitive costs involved in cost function and level of perfect debugging, on the optimal release time and total expected software testing cost. although we can analyze the sensitivity of all the parameters of the srgm and cost model but due to the limitation on size of paper we still can evaluate the optimal release time problem for various conditions by examining about the behavior of some parameters and costs that have the most significant influence. p. k. kapur, d. gupta, a. gupta, p. c. jha 85 we define mov oov re lative change (rc) oov   …(4.1) where oov is the original optimal values and mov is the modified optimal values obtained when there is a variation is some attribute of the release time problem. 4.3.1 effect of variations in minimum reliability requirement by the release time the optimal value of the release time obtained for the desired reliability level may be too late as compared to the scheduled delivery time, in such a case the management and/or the user of a project based software may agree to release the software at some lower reliability level with some warranty on the failures, which in turn will change the optimal release time to an earlier time and consequently lower the cost. on the other hand if the scheduled delivery is later than the optimal release time the management may wish to increase the desired reliability level at some addition testing cost. assuming the values of parameters and various costs associated with cost model to be same as in section 4.2. if minimum reliability requirement by the release time increased to 0.95 (about 12% increase) then we obtain t* = 46.73 (about 21.7% increase) and its rc is 0.217229. the minimum total expected software cost at t* i.e. c(t*) = $60542.43 (about 9.6% increase), its rc is 0.096077 and number of faults removed by the release time m(t*) = 136 and if minimum reliability requirement by the release time decreased to 0.75 (about 12% decrease) then we obtain t* = 34.27 (about 10.7% decrease) and its rc is -0.10757. the minimum total expected software testing cost at t* i.e. c(t*) = $52984.85 (about 12.48% decrease), its rc is -0.12483 and number of faults removed by the release time m(t*) = 134. figure 2 plots the relative change in the optimal release time and cost for the case of 12% increase and decrease in reliability objective. effect of introduction of fault and imperfect debugging on release time 86 figure 2: relative change in optimal release time and cost for 12% increase and decrease in reliability -0.2 -0.1 0 0.1 0.2 0.3 1 2 variation r c time cost reliability 4.3.2 effect of variations in costs involved in the cost model here we investigate the sensitivity of variations in various costs involved in the cost model. if any of the cost fixing an error in testing phase or operation phase for perfect and \or imperfect debugging and cost of per unit testing time varies during the testing process, it will have significant changes in optimal testing cost and release time. we have studied the sensitivity of cost of perfectly fixing an error in testing and operation phase. sensitivity for the rest of the costs can be carried in a similar manner. if we assume that the values of parameters of the srgm to be same given in section 4.2 and assuming that cost of perfect and imperfect fault debugging during testing i.e. c1 and c2 to be $200 and $110 respectively, cost of perfect and imperfect fault debugging during operation phase to be same i.e. c3 = c4 =$2000 and cost of per unit testing c=$2. if minimum reliability requirement by the release time is 0.85, following result 1 and 2 we obtain t0 = 39.61 and t1 =38.3983. then finally following theorem 2 we obtain t* = 39.61. the minimum total expected software cost at t* i.e. c(t*) = $55958.87 and number of faults removed by the release time m(t*) = 134. now if cost of fixing a fault perfectly and imperfectly in operation phase i.e. c3 and c4 is increased by 25% i.e. from $2000 to $2500, p. k. kapur, d. gupta, a. gupta, p. c. jha 87 then we obtain t* = 41.38 (about 0.4% increase) and its rc is 0.0447562. the minimum total expected software cost at t* i.e. c(t*) = $57053.21 (about 1.9% increase), its rc is 0.019556 and number of faults removed by the release time m(t*) = 136 and if c3 and c4 is decreased by 25% i.e. from $2000 to $1500, then we obtain t* = 38.398 (about 3% decrease) and its rc is –0.030605. the minimum total expected software cost at t* i.e. c(t*) = $55235.55 (about 1.2% decrease), its rc is –0.01293 and number of faults removed by the release time m(t*) = 135. figure 3 plots the relative change in the optimal release time and cost for the case of 25% increase and decrease in cost of fixing an error in operation phase. figure 3. relative change in optim al release tim e and cost for 25% increase and decrease in cost of fixing errors in operation phase -0.05 0 0.05 1 2 variation r c time cost 4.3.3 effect of variations in level of perfect fault debugging finally we investigate the sensitivity of variations in level of perfect fault debugging parameter p. if the testing personals were skilled personal the level of perfect fault debugging would be more or vice versa. variations in level of perfect debugging have significant effect on the optimal time of software release. if the level of perfect debugging increases for a testing process it is expected that the software can be released earlier as compared to the optimal release time determined otherwise and vice versa effect of introduction of fault and imperfect debugging on release time 88 if we assume that the values of parameters a, b and α of the srgm and the cost involved in cost function to be same given in section 4.2 and a reliability level of 0.85 is desired to be achieved and assuming value of perfect fault debugging parameter p is 0.9. following result 1 and 2 we obtain t0 = 39.369 and t1 = 40.737. then finally following theorem 2 we obtain t* = 40.737. the minimum total expected software cost at t* i.e. c(t*) = $56649.53 and number of faults removed by the release time m(t*) = 135. now if p is increase by 5%, then we obtain t* = 39.369 (about 0.03% decrease) and its rc is –0.03357. the minimum total expected software cost at t* i.e. c(t*) = $55812.96 (about 1.4% decrease), its rc is –0.01477 and number of faults removed by the release time m(t*) = 135 and if p is decrease by 5%, then we obtain t* = 42.93 (about 5.3 % increase) and its rc is 0.053869. the minimum total expected software cost at t* i.e. c(t*) = $58037.50 (about 2.4% increase), its rc is 0.024501 and number of faults removed by the release time m(t*) = 136. figure 4 plots the relative change in the optimal release time and cost for the case of 5% increase and decrease perfect fault debugging parameter p. figure 4. relative change in optim al release tim e and cost for 5% increase and decrease level of perfect fault debugging -0.1 -0.05 0 0.05 0.1 1 2 variation r c time cost reliability a similar conclusion can be obtained for the other costs and parameters of the srgm such as c1, c2, c, a, b and α taking the simultaneous changes in two or more costs and srgm parameters. p. k. kapur, d. gupta, a. gupta, p. c. jha 89 5. conclusion in this paper first we have formulated and derived optimal release time minimizing the expected software cost subject for an imperfect faultdebugging model due to kapur et al considering effect of perfect and imperfect debugging separately on the total expected software cost. next, we proposed a srgm incorporating the effect of imperfect fault debugging and error generation. the proposed model is validated a data set cited in literature. then a release time problem is formulated and solved minimizing the expected software cost subject to a minimum reliability level to be achieved by the release time for the proposed model. a numerical illustration is given for both type of release problem and finally a sensitivity analysis is performed to determine the effect of variations in minimum reliability level to be achieved by release time and various costs involved in cost model on optimal release time and cost. reference: 1. ammann p.e., brilliant s.s., and knight j.c, “the effect of imperfect error detection on reliability,” ieee trans. software eng., vol. 20, pp. 142-148, 1994. 2. kapur p.k., garg r.b., and s. kumar, “contributions to hardware and software reliability”, world scientific, singapore.1999. 3. kapur p.k., garg r.b., “optimal software release policies for software reliability growth models under imperfect debugging”, recherché operationanelle/operations research, vol 24, pp. 295305,1990. 4. kapur p.k., agarwal s., garg r.b., “ bi-criterion release policy for exponential software reliability growth models ”, recherche operationanelle/operations research, vol 28, pp. 165-180,1994. 5. kapur p.k. and younes s., “modeling an imperfect debugging phenomenon in software reliability,” microelectronics and reliability, vol. 36, pp. 645-650, 1996. effect of introduction of fault and imperfect debugging on release time 90 6. kapur pk, bhalla vk. “optimal release policy for a flexible software reliability growth model” reliability engineering and system safety 1992; 35: 49-54. 7. kapur pk, garg rb, bahlla vk. “release policies with random software life cycle and penalty cost” microelectronics reliability 1993; 33 (1): 7-12. 8. ohba m. and chou x.m., “does imperfect debugging affect software reliability growth?” proc. 11th int’l conf. software eng., pp. 237-244, 1989. 9. pham h., “software reliability”, springer-verlang singapore pte. ltd. 2000. 10. pham h, “a software cost model with imperfect debugging, random life cycle and penalty cost,” int’l j. systems science, vol. 27, pp. 455-463, 1996. 11. shanthikumar j.g., “a state and time-dependent occurrence rate software reliability model with imperfect debugging,” proc. nat’l computer conf., pp. 311-315, 1981. 12. slud e., “testing for imperfect debugging in software reliability,” scandinavian j. statistics, vol. 24, pp. 555-572, 1997. 13. xie m., “ a study of the effect of imperfect debugging on software development cost”, ieee transactions on software engineering, vol 29, no 5, may 2003. microsoft word tomasini.doc ratio mathematica 18 (2008), 107 132 107 notizie sulla vita e sulle opere di ‘gioseffo mari’ matematico e idraulico nella mantova del settecento giuliana tomasini1 sunto. si illustrano alcuni documenti inediti, relativi alla vita e alle opere di giuseppe mari, in particolare sulla sua attività di ‘regio matematico’ e di autorevole membro dell’accademia virgiliana di mantova. conosciuto anche, e forse ancora di più, come ‘gioseffo mari’, fu matematico e idraulico di primo piano nella mantova della seconda metà del settecento, ricoprendovi importanti cariche nella gestione delle acque del mantovano. abstract. we have illustrated same unpublished documents, about life and works of giuseppe mari, best known as ‘gioseffo mari’. he was a very famous mathematician and hydraulician in mantua during the second half of the seventh century and he had important charges on the management of the mantuan waters. parole chiave: matematica, idraulica, mantova, settecento 1. introduzione questo lavoro si inserisce in una più vasta ricerca che pone la città di mantova quale punto d’incontro di un ideale crocevia d’europa dell’arte e della scienza. lo scopo principale di questa ricerca è quello di approfondire la conoscenza di quegli accadimenti culturali, 1 politecnico di milano, polo regionale di mantova, dipartimento best; e-mail: giuliana.tomasini@virgilio.it. lavoro eseguito nell’ambito del gruppo di ricerca d’ateneo del politecnico di milano e presentato al congresso nazionale mathesis, tenutosi a trento dal 2 al 4 novembre 2006. ratio mathematica 18 (2008), 107 132 108 ambientali, sociali e politici attraverso i quali il ducato di mantova acquisì una posizione di preminenza nella pianura padana. mantova, infatti, per essere stata la più importante piazzaforte della pianura padana e anche per la sua collocazione geografica e per la conformazione specifica del suo territorio (con i conseguenti problemi idraulici che ne derivano), fu per secoli una fucina di scienziati. questi erano fondamentalmente degli importanti matematici e idraulici che godevano di una posizione privilegiata e spesso di ragguardevole potere politico, sociale e culturale. in un siffatto contesto trova la sua esatta collocazione questo lavoro, che si ricollega agli studi ora in corso sullo scenario mantovano di cui si è detto, studi dei quali una prima traccia si trova nel convegno nazionale mathesis sul tema «contributi di scienziati mantovani allo sviluppo della matematica e della fisica», tenutosi nel maggio 2001, proprio a mantova.1 assai conosciuto dai suoi contemporanei,2 il nome di gioseffo3 mari riemerge periodicamente nella letteratura dell’ottocento4 e del 1 il convegno è stato patrocinato dall’accademia nazionale virgiliana di mantova e dalla prima facoltà di architettura del politecnico di milano – sede di mantova. gli “atti del convegno contributi di scienziati mantovani allo sviluppo della matematica e della fisica, a cura di f. mercanti e l. tallini, mantova 17-19 maggio 2001, c. u. m., mantova 2001”, sono stati pubblicati nel novembre 2001 ad opera del consorzio universitario mantovano. 2 cfr. storia letteraria d’italia, vi, modena, soliani 1754, p. 44; ibid., xiv, modena, remondini 1759, pp. 176, 181, 202; giornale de’ letterati d’italia, xxx, modena, società tipografica 1785, pp. 262-278; ibid. xxxi, pp. 278-282; 3 il vero nome di mari è giuseppe (appendice 1), anche se in molti atti ufficiali e su frontespizi di suoi libri o nella intestazione di suoi manoscritti o, addirittura, nella sua firma si trova spessissimo il nome ‘gioseffo’. nel seguito verranno utilizzati indifferentemente i due nomi. 4 cfr. l. rosso, biografia degli uomini illustri mantovani mancati nel secolo xix, i, mantova, s. e. 1830, pp. 72-81; a. mainardi, storia di mantova, mantova, benvenuti 1865, p. 365; id., dello studio pubblico di mantova, mantova, segna 1871, pp. 21, 27, 32; c. sommervogel, bibliothèque de la compagnie de jésus, v, bruxelles-paris, schepens-picard 1894, coll. 544-546;. ratio mathematica 18 (2008), 107 132 109 novecento,1 fino ai nostri giorni del nuovo millenio.2 in tal senso, uno stimolo per approfondire le scarne conoscenze sulla sua vita e sulle sue opere mi è stato offerto, in particolare, da alcune annotazioni di maria teresa borgato3, ugo baldini4 e luigi pepe5. in esse si evidenziano, in particolare, il valore scientifico e la peculiarità, anche in una visione moderna, del pensiero e dell’attività di questo importante scienziato, che fu anche ‘regio matematico’ e ‘prefetto generale alle acque’ del mantovano, direttore della facoltà di matematica dell’accademia virgiliana di mantova e ‘matematico nazionale’ in seno all’istituto nazionale della repubblica italiana. in tal senso è in corso di studio una monografia sulla sua vita e sulle sue opere, sostanziata da molti documenti inediti, conservati per la maggior parte a mantova.6 nel prosieguo si anticipano alcune tra le più importanti notizie biografiche che lo riguardano (§ 2). si elencano, inoltre, le opere a stampa e le dissertazioni manoscritte giacenti presso l’accademia virgiliana di mantova, nonché alcuni progetti idraulici (§ 3). infine, nell’appendice documentaria si riporta il contenuto di quattro emblematici documenti 1 cfr. l. mazzoldi, la legislazione sulle acque del mantovano nel ’700, in politica ed economia a mantova e nella lombardia durante la dominazione austriaca (1707-1866). atti del convegno storico, a cura di r. giusti, bollettino storico mantovano, ii, mantova 1959, pp. 166-168, 171; p. carpeggiani, un progetto idraulico della fine del settecento, in civiltà mantovana, vi, 35, 1972, pp. 325, 329, 331. 2 cfr. m. t. borgato, agostino masetti e i suoi progetti idraulici nel periodo napoleonico, atti del convegno contributi di scienziati mantovani, cit., pp. 31-32, 35, 38; u. baldini, s. rocco e la scuola scientifica della provincia veneta: il quadro storico (1600-1773), in gesuiti e università in europa (secoli xvi-xviii). atti del convegno di studi. parma, 13-15 dicembre 2001, a cura di g. p. brizzi e r. greci, bologna, clueb 2002, p. 316; l. pepe, istituti nazionali, accademie e società scientifiche nell’europa di napoleone, firenze, olschki 2005, pp. 45, 6263, 148, 166, 186, 440, 468, 475. 3 m. t. borgato, agostino masetti, cit. ivi a p. 31 si legge che giuseppe mari, per la sua «attività di matematico e idraulico meriterebbe uno studio particolare». 4 u. baldini, s. rocco e la scuola scientifica, cit. 5 l. pepe, istituti nazionali, cit. 6 detti documenti si trovano negli archivi dell’accademia nazionale virgiliana di mantova (nel seguito aanv), di stato di mantova (asmn) e in quello di milano (asmi). ratio mathematica 18 (2008), 107 132 110 inediti, relativi, rispettivamente, due alla vita (appendici 1 e 3), e due alle opere idrauliche e scientifiche (appendici 2 e 4). 2. notizie biografiche nato il 9 febbraio 1730 a canneto,1 «figlio del signor carlo mari, e della signora susanna2 sua legittima consorte»,3 gioseffo mari compì i primi studi di carattere umanistico nel collegio dei gesuiti di mantova fino al 1744, quando entrò a far parte della compagnia di gesù.4 nel 1746 fu presente nel noviziato di s. ignazio a bologna e tra il 1747 e il 1750 si dedicò allo studio della filosofia nella scuola gesuitica bolognese di s. lucia, nella quale ebbe come docente delle discipline matematiche e fisiche vincenzo riccati (1707-1775).5 successivamente insegnò lettere e fu maestro di retorica nel collegio di brescia (17511754), nei sei anni seguenti (1755-62) ancora lettere nel collegio di reggio emilia dedicandosi anche allo studio della teologia nalla scuola di s. lucia in bologna. divenuto sacerdote, venne richiamato dai 1 piccolo paese situato sulla riva destra del fiume oglio, denominato oggi canneto sull’oglio, a circa trenta chilometri di distanza da mantova. 2 il cognome della madre è poli, come si desume dall’atto di morte di mari conservato nell’archivio storico diocesano di mantova, parrocchia di s. maria della carità, mortuorum liber, 1775-1811, c. 168/49. 3 appendice 1. 4 come emerge dai cataloghi dell’archivum romanum societatis jesu (nel seguito arsi), in essa rimase fino alla soppressione della stessa avvenuta, come è noto, nel 1773 per disposizione di papa clemente xiv. dai medesimi cataloghi affiorano le notizie riguardanti tutte le sedi in cui si mosse mari come novizio, sacerdote e professo, nel seguito riportate. 5 illustre matematico trevigiano, legò il suo nome, in particolare, agli sviluppi in serie delle funzioni iperboliche. vincenzo riccati era figlio dell’altro grande matematico jacopo (1676-1754), studioso dei metodi di risoluzione di particolari equazioni differenziali (cfr, l. pepe, jacopo riccati, i nuovi calcoli e i ‘principia mathematica’, «i riccati e la cultura della marca nel settecento europeo», a cura di g. piaia e m. l. soppelsa, firenze, olschki 1992, pp. 111-125). ratio mathematica 18 (2008), 107 132 111 superiori a mantova, dove, dal 1762 al 1771, fu lettore di matematica e confessore nel collegio mantovano.1 in questo periodo iniziò la sua fervida attività scientifica, che durerà per tutta la vita, con la pubblicazione di importanti opere scientifiche, la formulazione di numerose dissertazioni lette presso l’accademia reale di scienze, e belle lettere di mantova2 e la progettazione di importanti opere idrauliche. nel 1772, sempre presso il collegio mantovano, ebbe l’incarico dell’insegnamento, oltre che della matematica e della fisica sperimentale, anche dell’idraulica. con la soppressione dell’ordine dei gesuiti, nel 1773 mari, sprovvisto di una abitazione, «ricovrossi in casa di ottimo amico il signor luigi asti imperial regio direttore delle poste».3 il crescente sviluppo degli studi idraulici e la situazione idrica del mantovano produssero la conseguente pressante richiesta di formazione professionale di «giovani capaci agl’impieghi soggetti al dipartimento delle acque, strade, e confini, siccome una tale provincia richiede».4 per la realizzazione di questo programma mari, con «il disposto da sua maestà l’imperatrice regina con suo reale dispaccio 11. maggio 1780», assumeva «la carica del nuovo regio matematico aggregata alla provincia delle acque ».5 tra i compiti del regio matematico6 vi era quello di istituire una scuola di idraulica, dato che per suprema disposizione viene addossato al regio matematico di dare le lezioni fisse alla gioventù nella teoria, e pratica della scienza delle acque; 1 nel frattempo, il 15 agosto 1763, professò a mantova i voti di povertà, obbedienza, castità e completa obbedienza al papa, suggellando in tal modo la sua completa appartenenza all’ordine dei gesuiti. anche questa notizia si trova nei cataloghi dell’arsi. 2 attualmente accademia nazionale virgiliana di mantova. a tale prestigiosa accademia mari fu associato già dal 1769 per poi ricoprirne i vari gradi accademici e leggervi numerose dissertazioni (cfr. § 3. 2). 3 l. rosso, biografia degli uomini illustri, cit, p. 73. 4 appendice 3. 5 ibidem. 6 tale nomina gli comportò anche la carica di prefetto generale alle acque. ratio mathematica 18 (2008), 107 132 112 in correspettività della quale provida prescrizione rimarrà all’effetto suddetto stabilita una scuola, la quale avendo anch’essa rapporto alla pubblica istruzione, sarà sistemata come le altre tutte del regio ginnasio1 dove, peraltro, già mari operava. l’avviso dell’istituzione della scuola teorica e pratica d’idrostatica e d’idraulica porta la data del 9 aprile 1781 e il 26 novembre dello stesso anno «il ridetto regio matematico signor abate mari»2 tenne la sua prima lezione. nel 1782 trovò una nuova e duratura dimora presso la nobile famiglia del conte anselmo zanardi, che «gli affidò la numerosa e scelta sua biblioteca».3 in particolare, tra il 1784 e il 1786 pubblicò il primo tomo de le teorie idrauliche4, dedicato al conte zanardi (figura 1), e i tomi i (figura 2) e ii de l’idraulica pratica ragionata.5 nei due lustri che seguirono predispose anche i tomi successivi,6 si dedicò all’insegnamento, agli studi scientifici, in particolare di idraulica e, soprattutto, alla risoluzione di svariati problemi fluviali. figura 1 1 ibidem. 2 asmn, magistrato camerale antico, b. 370, istruzione ginnasio, aa. 1761-85. 3 l. rosso, biografia, cit, p. 74. 4 g. mari, le teorie idrauliche concordate colle sperienze proposte a’ suoi discepoli, i, guastalla, costa 1784. 5 g. mari, l’idraulica pratica ragionata proposta a’ suoi discepoli, i, guastalla, costa 1784 e ibidem, ii, guastalla, costa 1786. 6 tali tomi furono pubblicati ai primi dell’ottocento. ratio mathematica 18 (2008), 107 132 113 figura 2 con l’ingresso delle armate francesi in mantova nel 1797 la posizione di prestigio di mari non mutò. il 22 marzo 1798 venne costituita la repubblica romana e pubblicato il regolamento dell’istituto nazionale della medesima. esso, come l’institut di parigi, raccoglieva in sè gli uomini che più si distinguevano nel campo del sapere. tra i quarantotto membri associati non residenti in roma venne nominato nella «classe di scienze matematiche e fisiche sezione di matematica»,1 giuseppe mari. successivamente fu proclamata la costituzione della repubblica italiana e l’istituto nazionale, con la legge del 17 agosto 1802, divenne istituto nazionale della repubblica italiana. mari 1 l. pepe, istituti nazionali, cit., p. 45. ratio mathematica 18 (2008), 107 132 114 continuò a farne parte.1 nel 1802 furono finalmente pubblicati anche il terzo e il quarto tomo de l’idraulica pratica ragionata2 e, nel 1804, il secondo tomo de le teorie idrauliche3. sempre nel 1804 ebbe a mantova l’ultimo prestigioso incarico di insegnamento del nascente calcolo decimale, che però dovette abbandonare l’anno successivo per il precario stato di salute, che si aggravò fino a condurlo alla morte, avvenuta in mantova il 9 giugno 1807. 3. le opere a stampa, le dissertazioni e i progetti idraulici in questo paragrafo si elencano le principali opere riguardanti l’attività pubblicistica e quella progettistica di mari. 3. 1. opere a stampa  le teorie idrauliche concordate colle sperienze proposte a’ suoi discepoli, i, guastalla, costa 1784.  l’idraulica pratica ragionata proposta a’ suoi discepoli, i, guastalla, costa 1784.  l’idraulica pratica ragionata proposta a’ suoi discepoli, ii, guastalla, costa 1786.  inutilità, e danno del ritirare gli argini nelle corrosioni,4 in «memorie della reale accademia di scienze, e belle lettere ed arti», mantova, pazzoni 1795, pp. 177-224.  sopra il trasporto del canale di busseto che intendesi di fare nella villa di polesine sopra i di lei5 fondi, ed a danno di detta marchesa, parma, rossi e ubaldi 1798. 1 cfr. ibidem, p. 148. 2 g. mari, l’idraulica pratica ragionata proposta a’ suoi discepoli, iii, guastalla, costa 1802 e ibidem, iv, guastalla, costa 1802. 3 g. mari, le teorie idrauliche concordate colle sperienze proposte a’ suoi discepoli, ii, guastalla, costa 1804. 4 unica dissertazione data alle stampe. 5 trattasi della marchesa dorotea pallavicini vidoni alla quale l’opera è indirizzata sotto forma di lettera (biblioteca di storia delle scienze ‘carlo viganò’ ratio mathematica 18 (2008), 107 132 115  l’idraulica pratica ragionata proposta a’ suoi discepoli, iii, guastalla, costa 1802.  l’idraulica pratica ragionata proposta a’ suoi discepoli, iv, guastalla, costa 1802.  le teorie idrauliche concordate colle sperienze proposte a’ suoi discepoli, ii, guastalla, costa 1804. 3. 2. dissertazioni presso l’accademia reale di scienze, e belle lettere ricoprì tra il 1775 e il 1793 la carica di censore della facoltà di matematica, per divenirne direttore nel 1794. dalle dissertazioni che ivi egli lesse tra il 1769 e il 1804, per quasi tutto l’arco della sua vita, emerge la poliedricità dei suoi interessi scientifici di tipo storico, geografico, matematico, fisico, astronomico e chimico, anche se i suoi studi riguardarono prevalentemente l’idraulica. le dissertazioni che giacciono, manoscritte, presso l’archivio dell’accademia nazionale virgiliana di mantova sono le seguenti.  dissertazione contro la quasi comune opinione, che le inondazioni del nilo abbiano prodotto il rialzamento de’ campi dell’egitto.1  dissertazione in difesa delle due leggi astronomiche newtoniane dedotte dalle kepleriane.2  ripari al po.3  sulla forza centrifuga.4 dell’università cattolica del sacro cuore, brescia, fa 7 b 623). 1 questa dissertazione venne completata in tempi diversi (prima parte, 1769, seconda parte, 1772, aanv, b. 45/8). 2 dissertazione letta nel 1775 (aanv, b. 61/29). 3 dissertazione letta nel 1788 (aanv, b. 45/3). 4 in questa dissertazione del 1790, che è il completamento di un’altra del 1772 andata perduta, mari esprime il meglio del suo sapere scientifico. tra gli esempi emblematici della sua brillante esposizione, per spiegare il concetto di forza centrifuga illustrava «il fenomeno de corpi, che girano sulla superficie della terra nel suo moto diurno. per questa rotazione terrestre, oggi è oramai fuor di dubbio, che qualunque corpo pesa meno all’equatore, che in qualunque parallello, come a ratio mathematica 18 (2008), 107 132 116  memoria fisica sui termometri.1  per l’apparecchio chimico del cittadino astier presentato all’accademia scientifica di mantova.2  delle cagioni diradatrici delle tenebre dell’ecclissi.3 3. 3. progetti idraulici numerosissime sono le relazioni lasciate da mari manoscritte, oltre alle dissertazioni, riguardanti la sua attività di idraulico, tra le quali si possono ricordare almeno le seguenti.4  27 maggio 1770 relazione su un rimedio provvisorio da apportare alla chiusa di governolo in attesa di uno stabile intervento.  1780 circa progetto per irrigare l’orto botanico di mantova (figura 3).5  17 novembre 1780 relazione intorno alla navigazione del po fino a mantova.  12 dicembre 1780 relazione sulla chiusa di governolo.  12 aprile 1781 relazione con osservazioni su un progetto del colonnello lorgna relativo alla chiusa di governolo.  12 luglio 1786 relazione per salvaguardare il froldo di revere. rimedi necessari e modalità d’intervento.  20 novembre 1786 relazione sull’irrigazione del boschetto. cagion d’esempio al parallello di mantova, lontano 45° poco più, e meno anche in mantova, che a pietroburgo, il cui parallello dista dal nostro 15°», cfr. appendice 4. 1 questa dissertazione fu letta in due giorni diversi del 1798, aanv, b. 60/18. 2 dissertazione letta nel 1803 (aanv, b. 59/25). 3 dissertazione letta nel 1804 (aanv, b. 60/13). 4 tali relazioni giacciono presso asmn e asmi. 5 per questo progetto, «l’artificio consisterà in far zampillare l’acqua nella vasca con un getto, che riesca superiore alquanto al piano dell’orto, col mezzo d’una ruota mossa dall’acqua ad un’altezza conveniente di circa sei braccia ½ », cfr. appendice 2. ratio mathematica 18 (2008), 107 132 117 figura 3  2 gennaio 1787 massime per troncare alla radice la moltitudine di abusi in materia d’acque.  30 gennaio 1787 elenco dei canali che devono essere scavati per la conservazione delle acque.  21 giugno 1787 progetto idraulico per collegare la città di mantova alla favorita.1  11 settembre 1787 progetto di una macchina per regolare l’acqua del lago superiore al vaso di porto.  12 maggio 1788 intera riforma degli scoli per le valli di sermide presentata sotto forma di dissertazione suddivisa in dieci proposizioni (nella quinta cavo mari).  21 maggio 1789 relazione sull’intervento all’argine di gazzuolo.  14 marzo 1792 piano di regolamento per togliere i disordini all’argine di luzzara.  28 marzo 1792 relazione del regio matematico sugli scandagli fatti nel po. 1 questo progetto è stato pubblicato in p. carpeggiani, un progetto idraulico cit., pp. 324-349. ratio mathematica 18 (2008), 107 132 118  11 aprile 1793 riflessioni e lavori da eseguire all’argine di luzzara onde evitare dannose inondazioni.  20 dicembre 1795 relazione dettagliata intorno alle inondazioni dei fiumi ostigliesi. appendice documentaria 1. atto di battesimo di giuseppe mari, del 1730. archivio della parrocchia di s. antonio abate in canneto sull’oglio, registro dei battezzati dal 1715 al 1768, vol. 8, c. 90, n. 12. adì 10 febbrajo 1730. giuseppe figlio del signor carlo mari, e della signora susanna sua legittima consorte è stato battezzato da me antonio maria monda rettore di s. michele. padrini l’eccellentissimo signor dottor domenico rebusca, e la molto illustre signora costanza arrivabene. nacque jeri alle ore sei. 2. progetto per irrigare l’orto botanico, del 1780 circa. asmi, a. g., studi p. a., b. 255. avendosi sperimentato a qual altezza poteva levarsi l’acqua del vivo, che teglia l’orto botanico in tempo d’acqua ordinaria del lago, da cui derivasi, si è trovato, che non monta, che all’altezza di due braccia e ¼ sopra il suo pelo ordinario, quando sostentisi con una chiusa, da cui si lasci stramazzare. in tempo d’acqua bassa del lago non sarà molto minore l’altezza: poiché in esso si rattiene l’acqua sempre all’altezza necessaria pei mulini. il pelo presente è oncie 55 ¼ più basso del piano su cui aprirassi la vasca. sostenendo due braccia l’acqua, e anche se si vuole due oncie di più, il pelo dell’acqua così sostenuta verrebbe ad esser più basso del pianterreno della vasca oncie 29 ¼. scavando dunque il terreno 5 braccia per formare la vasca, verrebbonvi sul fondo di esso 30 ¾ oncie di acqua: ma la superficie di essa sarebbe distante dalla superficie del piano oncie 29¼ e se attorno alla vasca si faccia un parapetto, o un orto erboso alto solo oncie 6 ¾, la superficie dell’acqua sarà 36 oncie più bassa del ciglio dell’orto, e riuscirà assai incomodo l’irrigar l’orto. è dunque necessario di servirsi dell’innalzamento dell’acqua per darvi una caduta capace di muovere una ruota, da cui si sollevi l’acqua ad ratio mathematica 18 (2008), 107 132 119 un’altezza, che basti ai bisogni dell’innaffiamento. questo richiede, che possa farsi col minor incomodo possibile, e nel più breve tempo. il minor incomodo possibile potrebbe aversi, quando l’innaffiamento si potesse eseguire da una sola persona, e colla monor fatica, e si farebbe nel minor tempo, quando nell’atto istesso si potessero irrigare tutti i quarti dell’orto. se l’acqua entrasse nella vasca posta nel mezzo dei quarti a modo di getto, prima entro un piccol bacino, dal cui labbro cadesse poi nella vasca, e se il bacino fosse alquanto superiore al piano dell’orto, così che si potesser porre sott’esso piccoli canaletti di legno, che rivolgesser l’acqua ai diversi quarti, chiaro è, che una sola persona potrebbe irrigare i quarti tutti, e in poco tempo, perché contemporaneamente. tutto dunque l’artificio consisterà in far zampillare l’acqua nella vasca con un getto, che riesca superiore alquanto al piano dell’orto, col mezzo d’una ruota mossa dall’acqua ad un’altezza conveniente di circa sei braccia ½ . non suggerisco le trombe perché il rio è infetto di sabbie sul fondo, dalle quali spesso sarebber guastate. per animar questa ruota non si può dar all’acqua, che un braccio e ¼ di caduta. dandone una maggiore, potrebbe avvenire, che in acqua bassa del lago non potesse entrar l’acqua nella corsia con quell’altezza e corpo capace a far girare la ruota. il soprapiù, a che può sollevarsi, servirà colla pressione a dar maggior forza alla caduta. un’usciara sopra la soglia della corsia, che formerà la caduta ne regolerà la quantità necessaria al movimento della ruota. come dunque nelle bassezze del lago, l’acqua sulla soglia della vasara può essere scarsa, e quindi di poco momento, converrà formare una ruota, che possa muoversi anche da poca forza. una ruota, che non porti sempre l’acqua alla circonferenza, ma da questa la passi al centro, e la ruota, che si muoverà colla minor fatica, e tanto più, quanto che in essa il suo raggio, che è il braccio di leva della potenza, dee esser più lungo, e leverà l’acqua pel cammin più corto. non può dunque convenire al bisogno dell’orto, che la ruota inventata dal signor faya. convien però addattarla ad un canale, che scorra sopra un piano inclinato, qual formerassi dalla caduta dell’acqua. ecco come ne ho imaginata la maniera. piantarsi in un albero i crocili necessarj per formare la ruota per le pale. questa avrà 6 braccia e mezzo di raggio netto dall’albero: porterà 24 pale, ciascuna delle quali sarà di 11 oncie di larghezza, e di 8 di altezza. tutto ciò secondo l’arte ordinaria. in distanza di mezzo braccio circa dai primi crocili, altrettanti ne sortiran dall’albero, che richiudano e fermino i condotti, che debbon portar l’acqua all’albeero. questi condotti saranno evolute di un circolo, che abbia la sua circonferenza uguale all’altezza, a cui vuolsi levar l’acqua. l’acqua monterà con una direzione verticale, tangente all’albero, e ratio mathematica 18 (2008), 107 132 120 perpendicolare al canale in qualunque sito essa si trovi. l’azione del suo peso corrispondendo sempre all’estremità di un raggio orizzontale, che sarà il braccio di leva costante; la potenza, che leverà questo peso coll’ajuto della ruota, sarà sempre la stessa. la corsia sarà larga al principio della caduta oncie 14. e, ove porterà l’acqua perpendicolarmente alle pale, oncie 12. i condotti, che prenderan l’acqua saran 4; la lor bocca sarà di tre oncie in quadro. salteran fuori della circonferenza delle pale 6 oncie misurate sul raggio della ruota, pescheranno in acqua oncie 4. il fondo della corsia sarà 2 oncie più alto del pelo dell’acqua, e avrà uno scalino, che si abbasserà le dette due oncie nel sito, ove l’acqua urta perpendicolarmente la pala, per formare una corsia più bassa, che riceva l’acqua, che ha urtato senza che dia rigurgito alla sopravvegnente. essendovi poca caduta convien levare tutti gl’impedimenti possibili. per questo effetto si può tenere il raggio dell’albero più corto di 2 oncie del figato, o l’albero 2 oncie più alto. i condotti alzan l’acqua fino all’albero, dal quale sarà trasmessa in un vaso di legno, o di marmo, nel fondo del quale sarà il tubo, che porterà l’acqua alla vasca. il vaso dourà empiersi; il sovra più cadrà nel rio, o in un cavo vicino per alimentar l’erbe palustri. il tubo farà zampillare l’acqua dal bacino della vasca, e empirà la vasca fino al segno, che si vorrà. sarà bene tenerla alta, perché se ne possa anche cavare a mano, volendo, con facilità. a quel segno si farà la bocca di un piccol canale, che porterà l’avanzo alla piccola peschiera, e da essa al rio, a cui è vicinissima, o ai boschetti vicini. la strada che si fa fare all’acqua è la più breve. ciò si rileverà dal piano dell’orto, ove è marcata. il pelo dell’acqua del rivo è più basso del piano, ove verrà posta la vasca, oncie 55 ¼. facendo il raggio della ruota netto dall’albero di braccia 6 ½, si può far cadere l’acqua in un vaso o conservatorio colla superficie più alta del piano, ove andrà posta la vasca, oncie 22 ¾. tenendo l’orlo della vasca alto 4 oncie dal terreno, e sovresso il bacino alto altre 4; il getto sarà un braccio, meno quello, che porteranno le risistenze, e darà tutto il comodo per le irrigazioni. la vasca sarà profonda tre braccia. oncie 30 sopra il fondo si farà un canaletto con un mattone disposto pel lungo, e chiuso da’ lati da altri due mattoni posti verticalmente, e superiormente da mattoni posti di traverso. questo col fondo di sotto sarà 7 oncie sotto il terreno del viale di mezzo, e colla parte superiore 3 oncie: correrà il canaletto rettamente fino alla peschiera, che avrà simile bochetto laterale per trasmettere l’acqua soprabbondante nel rivo. la peschiera per mantenere il pesce dovrà contenere acqua all’altezza di 4 braccia almeno, e nel suo fondo avrà un bocchetto verso il rio da potersi aprire, quando si volesse cangiarvi acqua. il ratio mathematica 18 (2008), 107 132 121 tubo di condotta dal conservatorio alla vasca, che si può far di cotto, avrà due pollici di diametro non considerata la grossezza delle sue pareti. il foro da cui l’acqua zampillerà fuor del bacino potrà avere il diametro di un pollice e ½ e l’acqua da esso tramandata potrà bastare alla irrigazione. abbate gioseffo mari regio matematico camerale. 3. istruzioni per la sistemazione della carica di regio matematico, del 1780. asmi, a. g., studi p. a., b. 125. ritenuto il disposto da sua maestà l’imperatrice regina con suo reale dispaccio 11. maggio 1780 successivamente abbassato a questo regio detto magistrato camerale con venerata lettera primo giugno prossimo passato di sua eccellenza il signor conte ministro plenipotenziario, e vice-governatore sarà la carica del nuovo regio matematico aggregata alla provincia delle acque, e ne’ seguenti articoli saranno dettagliate le incombenze che la riguardano. primo: la carica di regio matematico essendo instituita principalmente alla migliore economia direzione e distribuzione delle acque del mantovano, l’oggetto delle quali forma una delle provincie di questo regio detto magistrato camerale, s’intenderà quindi soggetto il regio matematico stesso, ed addetto al tribunale per tutti quegl’incombenti, in cui credes’egli di doverlo impiegare. secondo: tre possono essere segnatamente le cause, per cui convenga al magistrato valersi della lui opera. primo: per avere un di lui sentimento verbale, e sarà della di lui puntualità il prestarsi alle premure del dicastero, dove avravvi conveniente luogo, e sessione. secondo: per riferire in iscritto a norma degl’eccitamenti, e sarà lui impegno di soddisfarvi colla maggior esattezza. terzo: per visite locali secondo le occorrenze, e sarà del di lui zelo disimpegnarle in tutta quella estensione, che la materia richiederà, nel qual caso sarà provveduto in conformità della pratica, e colli riguari dovuti al carattere del suo impiego; dipendendo poi dall’autorità superiore la riflessione delle sue diete in evento di visite che non riguardino interesse della regia camera. terzo: qualora dal tribunale economico passi a quello di giustizia alcuna causa agitata tra privati se pure il regio matematico non avrà opinato presso il magistrato, sarà della natura del suo ufficio l’essere prescelto a dare il suo ratio mathematica 18 (2008), 107 132 122 sentimento, particolarmente se si tratti di causa, il cui scioglimento dipenda dall’applicazione delle teorie dottrinali della scienza delle acque. quarto: le convocazioni, che annualmente si tengono nanti il magistrato delle digagne, essendo quelle, che danno un lume generale de’ bisogni delle loro arginature, e de’ loro scoli, sarà conveniente che v’intervenga il ridetto regio matematico per suo lume, e potere alla circostanza prestarvi i suoi suggerimenti. quinto: per suprema disposizione viene addossato al regio matematico di dare le lezioni fisse alla gioventù nella teoria, e pratica della scienza delle acque; in correspettività della quale provida prescrizione rimarrà all’effetto suddetto stabilita una scuola, la quale avendo anch’essa rapporto alla pubblica istruzione, sarà sistemata come le altre tutte del regio ginnasio. sesto: quanto al tempo, ed al luogo in cui farla il regio matematico, e professore né passerà d’intelligenza colla soprintendenza generale ai regi studj: quanto poi al numero e durata delle lezioni, come pure al principio, e fine dell’anno scolastico s’uniformarà al diario, ed orario, che si pubblica in ogni anno per le regie scuole di mantova. settimo: lo scopo di questa istituzione diriggendosi interamente al formar giovani capaci agl’impieghi soggetti al dipartimento delle acque, strade, e confini, siccome una tale provincia richiede cognizioni teoriche, e pratiche sul regolamento de’ fiumi de’ canali d’irrigazione, de’ scoli primarj, e secondarj, degl’edifizj idraulici, delle arginature, vie pubbliche, e linee de’ confini territoriali; verseranno quindi le sue lezioni su de’ principj, che hanno analogia agl’oggetti suddetti. ottavo: conducendo a poter meglio perfezionare la studiosa gioventù in questa carriera, oltre lo studio delle teoriche leggi la cognizione de’ tipi, e delle mappe generali, e particolari, sarà della premura del regio matematico il formarne una metodica, e ben ordinata collezione in sussidio de’ suoi allievi. giovando parimenti alla pratica la cognizione de’ trattati, e delle relazioni antiche de’ passati ingegneri, assieme a quelle più importanti de’ moderni; questi mezzi pure saranno impiegati dal regio matematico per istruzione della gioventù addetta alla sua scuola, ed regio detto magistrato si presterà a somministrargliene le occorrenze. nono: non convenendo poi che tutti indistintamente i giovani siano ammessi a questa scuola, se prima non consti di una sufficiente loro disposizione, e capacità; dovranno perciò preventivamente presentarsi al regio matematico, perché egli mediante un esame conoscere possa la loro idoneità per l’ammissione. ratio mathematica 18 (2008), 107 132 123 decimo: siccome per sovrana disposizione gl’impieghi d’ingegneri camerali ed altri uffizj, che vi hanno rapporto, saranno in occasione di sopravegnente vacanza assegnati a quelli che avranno fatto il corso di questa scuola, e successivamente l’alunnato con aver dato saggio della loro abilità; così il regio matematico terrà esatto conto de’ progressi, qualità personali, e meriti di ciascheduno, affine di potere alla circostanza informarne il dicastero. undicesimo: per dare allla gioventù maggior campo di formare la pratica negl’esercizj del loro istituto sarà in facoltà del regio matematico scegliersi uno de’ suoi scolari all’atto di qualche visita locale, per le cui diarie supplirà la regia camera. 4. dissertazione del signor abate giuseppe mari ‘sulla forza centrifuga’, del 1796. aanv, b. 60/10. altra volta1 io vi ragionai, non ven rimembra forse accademici ornatissimi per la lunghezza del tempo infrappostosi, della vera indole, e natura di quella forza, che dicesi centrifuga, di quella cioè, di cui è animato un corpo, che viaggia per una linea curva, e sempre tende a sottrarsi dal centro del suo moto. conoscer vi feci, non esser questa una forza reale di natura, con che essa operi veracemente, ma una forza in essa immaginata da geometri, che altre pure ideali surrogarono alle vere, con grandi vantaggi nella fisica. trattato allora l’argomento in tutta la sua confaccente ampiezza, rimaneva solo un dubbio, a che profitto tornasse cotesta ideale surrogazione d’una forza immaginaria ad una vera. a compimento di materia io vi promisi allora, ben lo ricordo, di dimostrarvene a mio agio i sommi vantaggi, e la quasi necessità, che di essa si avea. altri suggetti più acconcj alle circostanze mi han fatto divagare dal propostomi, nelle altre volte, che ho avuto l’onor d’intertenervi da questo luogo. sento ora il rimorso di una troppa dilazione, e vengo oggi a liberarmene. l’astrusità dell’argomento non torrà il diletto alla spiegazione de’ leggiadri sperimenti, che verrò producendo. parlerò prima di quelli, che nascono nei fluidi, e negli ammassi di piccoli corpicciuoli nell’atto in che ruotano intorno parallelli all’orizzonte, e sì nei fluidi semplici, e che ne’ fluidi misti di materie, e più leggiere, e più gravi. scenderò indi a parlare di quelli, che nascono nei fluidi, che raggiransi perpendicolari all’orizzonte in compagnia di corpi, o più gravi, o più lievi. di quelli in terzo luogo, che 1 mari allude alla dissertazione del 1772 andata perduta, cfr. nota 37. ratio mathematica 18 (2008), 107 132 124 ruotansi obbliqui all’orizzonte con corpi più o meno specificamente pesanti. infine ancor de’ corpi posti alla circonferenza de’ globi rotanti prima al loro equatore, poi ne’ diversi circoli parallelli. darò un saggio in tutti questi modi di rotazione, a’ quali riduconsi agevolmente gli altri sperimenti prodottisi in questo genere. l’ampiezza della materia non mi farà abusare della vostra sofferenza, e la vaghezza degli oggetti, che vengo a schierarvi avanti, mi concilierà, io mel prometto, tutta la gentil vostra attenzione. non negano i geometri, che i fenomeni de’ corpi viaggianti per curve non ricevano spiegazione in un qualche modo almeno, e per la più parte colle forze reali di natura. credono però, e sentenziano, che assai meglio comprendansi colla scorta della lor forza immaginaria, che tanti è presta a dichiararne, quanti ne può fornir la natura, o inventarne l’artifizio. senza di questa avviluppansi essi di modo, e di buona grazia il confessano, tra mezzo le oscurità che trovano appena traccia di sortirne. l’arte con che opera natura è spesse volte un enigma. non basta loro il penetrarvi, né con gli occhj della mente, né con quelli del corpo. sfugge agli uni, e agli altri il magistero, con che ella agisce, e combina le sue forze, buona parte delle quali per ultima giunta è al tutto sconosciuta. laddove la teoria della forza centrifuga riduce i fenomeni tutti alle sole leggi dell’equilibrio, leggi semplicissime, e affatto note, e sicure. senza il presidio di questa mendicar debbonsi le spiegazioni dalle leggi de’ diversi movimenti di natura, or troppo oscure, ora poco stabilite, onde, e sorgon poi implicatissimi raziocinj, e spiegazioni sì poco eloquenti, alle quali non sa arrendersi, che a puro stento, l’intelletto. non basta ciò solo, o signori miei, a riconoscere vantaggiosa non solo, ma come a dire necessaria la forza, ch’io vi commendo? vedetelo alle prove de’ fenomeni, che con essa verrò rischiarando, poste a fronte delle implesse e antiche spiegazioni. parliam dunque prima di quelli, che appajon nei fluidi, e negli ammassi de’ solidi, che ruotan parallelli all’orizzonte, ora semplici, ora trasmischiati a materie più leggiere, ora più gravi. molte volte vi sarà avvenuto di osservare, che nel rotarsi alcun poco intorno al suo asse una bottiglia, o altro vaso non ripieno di liquore, come costumasi a sollevarne e rimescelar col fluido qualche materia depostasi sul fondo; nell’atto della rotazione vedesi il fluido racchiuso assorgere, e come rampicarsi alle pareti del vaso, e deprimersi vieppiù, quanto più appressasi al centro di sua rotazione, o sia all’asse verticale del vaso. di un sì piano fenomeno già ricorderete l’antica astrusa spiegazione. intendevasi diviso il fluido rotante in molte, e sottili zone, o fascie cilindriche concentriche, che sorgevan perpendicolari dal fondo, e montavan fino alla superficie del fluido, e stendevansi in quel numero, che più vi capiva, dal ratio mathematica 18 (2008), 107 132 125 centro, o asse del vaso alla circonferenza, o alle pareti di esso. fate un cavo conico nel mezzo d’una cipolla, e le varie tonache, di che va vestita, vi daran l’idea delle zone acquee, ch’io vi descrivo. ogniuna di queste zone acquee descrive, dicevasi, il suo circolo, in rotandosi, più piccolo le più aderenti al centro più largo le più distanti. dunque deve avervi una forza atta a ritenerle nell’intrapreso circolo, che chiamanla centripeta. ciò non può porsi in dubbio. ma ove rintracciar questa forza? eccovi accademici ornatissimi le prime ambiguità. niuna non ne traspare al certo in niuna delle zone, che si aggira, che capace sia di tener vincolata al suo circolo l’acqua, che il descrive, colla piccola fascia, di che è composta. e ciò è pur vero per qualunque indagine se ne prenda. dunque cotesta forza ritenente ciascuna zona in circolo sarà estrinseca a ciascuna di essa. la deduzione non può riprendersi. ma se tal forza è forestiera a ciascuna zona; non potrà nascer d’altronde, che da pressione immediata dalla vicina esterna zona, che opponsi allo sforzo della interna, e vietale di scostarsi dal centro, intorno a cui ruota, e di passar oltre alle pareti del vaso, e fa quindi le veci della centripeta. e dal non vedersi altra forza ritenente, che la pressione della prossima esterna zona; passasi di lancio a stabilire, che dunque ciascuna zona acquea ritiensi nel suo circolo dalla pressione della sua vicino, che immediatamente la cinge e la circoscrive. se dunque la forza, che ritiene la prima più angusta zona nel mezzo del vaso nel suo circolo, risiede nella zona seconda esteriore che la serra, e preme, affinché non si scosti dal centro; non potrà questa seconda zona tener compressa la prima all’indentro senza premere d’altrettanto la terza sua vicina al di fuori, dalla quale essa pure è circondata: e ciò per quella infallibil legge ne’ fluidi di premere, e sfiancare ugualmente verso ogni parte. dunque alla forza propria della terza zona si fa aggiunta ancor della pressione nata dalla centrifuga della zona seconda, e della prima, e quindi la forza della terza avanzerà quella detta seconda, e per la ragione stessa la forza della quarta zona la vincerà molto più sopra quella della terza, e così mano mano fino alle pareti del vaso, ove l’ultime zone saran più premute, che le antecedenti. dalla quale pressione poi le più premute debbon rialzarsi più che le meno, e così l’acqua assorgere ai lati, mentre si deprime al centro. se sarò apparso alquanto disuso in questa sposizione, non fu ciò a motivo di dar taccia di troppa prolissità alla vecchia spiegazione, ma per renderla soltanto, quanto potevasi più chiara. qualunque chiarezza, o brevità, che vi abbiate scorta, scomparisce al lampo quasi della spiegazione, che diffonde la forza centrifuga. ammirate. le zone esteriori descrivono maggior circolo, che le interiori. posseggon dunque maggior forza centrifuga, che trovasi in ragione de’ raggi de’ circoli descritti. distrae ratio mathematica 18 (2008), 107 132 126 dunque nelle esterne zone maggior parte di gravità dal premere al basso, che nelle interne. converrà dunque per mettersi in equilibrio colle lor vicine, che quelle tanto crescan d’altezza, quanto sceman di gravità. negate signori miei, se il potete, che non appaja questa la legitima figlia di natura, e l’altra al paragone suppositiva. analogo a questo si è l’altro genere di sperimenti col fluido, o coll’ammasso de’ solidi, che gira orizzontalmente, e con cui tramischiansi corpi più leggieri, come quando cribrasi nel vaglio il grano a purgarlo dalle sue buccie, o dalle paglie, o quando rotasi una bottiglia d’alcun liquore schiumoso. le paglie, e le buccie raccolgonsi nel centro del crivello, e le schiume nel mezzo del vaso. ma la spiegazione non è meno analoga all’altra, quantunque più dotta, e raffinata. le schiume, e le paglie prendono la forza a rotarsi, o dal fluido, o dai grani, che le sostentano, e si ruotano con esse. ma la forza è in ragion della massa stando l’altre cose pari; dunque han più forza a scostarsi dal centro il fluido, e il grano, che le buccie, e le schiume soggette. ma non può scostarsi dal centro il fluido, e il grano senza gittar verso il centro le paglie, e le schiume, dovendo quelli di necessità occupare il posto di queste, nel portarsi alle pareti del cribro, o del vaso, e non avendo le paglie, e le schiume altro agio a muoversi, che alla parte, ove scontrano minor resistenza, che è il centro, ov’è la menoma velocità. in quella guisa, che i corpi più leggieri d’ugual volume d’acqua immersi in un vaso sospingonsi all’insù, non da altra forza, che da quella delle particelle acquee vicine, e più gravi, che discendono con maggior conato, che quelli più leggieri vi oppongano. così come nel discendere le particelle acquee, e nell’occupare il luogo de’ corpi più leggieri, fan questi ascendere; per ugual modo i grani dotati di maggior forza centrifuga, che portansi alle pareti del cribro, nell’occupare i posti delle paglie, e delle buccie le cacciano al centro. la parità è introdotta ad ornamento di dottrina; non per esigenza dell’esposto. la spiegazione pertanto se è semplice, non è men breve. voglio risparmiare a’ vostri ingegni il tormento della spiegazione antica, e intendo, che me ne siate grati sulla mia parola. invece di nojarvi, sonomi prefisso di dilettarvi, e avanzomi nel mio suggetto col fenomeno occorso ad hugenio e da lui registrato nella dottissima dissertazione de caussa gravitatis, in cui coll’acqua rotar vedrete orizzontalmente un corpo d’essa più pesante, siccome vi ho data promessa. in un vaso cilindrico tutto ripieno d’acqua, e ben chiuso gittò molti piccoli pezzi di cera spagna, e fece rotar il vaso verticalmente, e l’acqua orizzontale. osservò, i° che i minuzzoli della cera più gravi dell’acqua eransi adagiati sul fondo levigatissimo, e che seguivan assai meglio il moto del fondo del vaso, di quel che facesse l’acqua circonfusa, e che nel rotarsi del cilindro ratio mathematica 18 (2008), 107 132 127 concorser tutti alle pareti di quello. arrestato in ii° luogo improvvisamente il moto del vaso, o cilindro, ma proseguendo a girarvi entro l’acqua in forza del concepito movimento di rotazione, vide ricorrer tutti i pezzetti di cera dalla circonferenza al centro del vaso, e dove’ rimarcare in terzo luogo, che nel portarsi al centro descriveva ciascuno una spirale. osservate la leggiadria della triplice ristrettissima spiegazione. la cera in i° luogo già caduta al fondo era specificamente più grave dell’acqua. come più grave contraeva maggior forza centrifuga per allontanarsi dal centro, e quindi accorreva a’ lati del vaso. eccovi la ragione del primo fenomeno. arrestato in ii° luogo il moto del vaso, ma non dell’acqua, i pezzetti di cera più gravi, e più scabri più presto dovean perdere la forza centrifuga di quello che potesse far l’acqua. dunque allor l’acqua più robusta cacciar dovea di luogo la cera per appoggiarsi essa alle pareti, e spinger questa nel mezzo; come i grani fan colle buccie, e colle paglie. e questa è pur la ragione del secondo fenomeno. nel portarsi in iii° luogo i pezzetti di cera al centro, traversar doveano zone d’acqua, che rotavansi orizzontali colle particelle loro, e più, o meno, quanto più, o meno distavan dal centro. nell’atto dunque che direttamente spingevansi al centro, raccoglievan dalle zone che traversavano, moti di rotazione. da questi due moti composti nasce, come già sapete, la spirale. ed eccovi la ragione anche del terzo fenomeno. io vi acerto, o signori, che la spiegazione di questi tre fenomeni è così rapida, che non eccede in lunghezza il racconto, che di essi tesse hugenio. questo eccellente uomo trova simile il fenomeno natogli dal rotarsi corpi più gravi entro un fluido, a quello avvenuto a cartesio, che facendo girare pur orizzontalmente in un vaso molte palline di piombo con limatura di legno, questa fu confinata da quelle nel mezzo del vaso. la ragione n’è la medesima, che ne’ grani colle paglie, e nell’acqua prevalente alla forza de’ minuzzoli della cera spagna. ne vuole però avvertiti hugenio, difficilmente aver luogo il fenomeno, quando non percotansi di continuo le pareti del vaso, affine di dividere, e staccare l’una dall’altre le materie, che vi si aggirano. i fenomeni, che in ii° luogo vi ho proposti de’ fluidi, che ruotano verticalmente con corpi di diversa gravità, presentanci da bulfingero nella sua dissertazione de caussa gravitatis physica coronata dalla reale accademia di parigi. fe rotare bulfingero sull’asse orizzontale, come a un di presso si fa girare il globo della macchina elettrica, una sfera di vetro quasi tutta ripiena d’acqua in cui posta avea alquanta limatura di ferro. nel rotarsi orizzontalmente la sfera, il fluido dovea rotarsi verticalmente. varj fenomeni ne risultarone. ratio mathematica 18 (2008), 107 132 128 scelgo que’ soli, che dipendono dalla forza centrifuga. la limatura in i° luogo formò come un equatore, o sia una fascia sull’interna superficie del globo equidistante dai poli, sui quali questo aggiravasi, e perpendicolare all’asse orizzontale della rotazione. l’aria in ii° luogo, che stanziava in grossa bolla alla sommità del globo, quando rimaneva quieto; al raggirarsi di questo, concentrossi attorno l’asse di rotazione divisa in piccole bollicine. queste bollicine formaronsi in iii° luogo in un cilindro, il cui asse era l’asse stesso della rotazione, ma tenevan con seco da principio avviluppata molt’acqua, dalla quale in progresso liberatesi, ritrassersi in forma cubica al centro del globo rotante. l’abate nollet ripetendo lo stesso sperimento pose olio di trementina ad occupare il sito, che teneva l’aria. quello come più leggiero galleggiava al di sopra dell’acqua, come l’aria di bulfingero. nollet ottenne gli stessi fenomeni, e l’olio seguì le traccie stesse dell’aria nell’altro sperimento. la mia spiegazione però si acconcierà all’uno, e all’altro. la limatura in i° luogo come più pesante dell’acqua discesa era, non v’ha dubbio, alla parte infima del globo. al rotarsi di questo sull’asse orizzontale, la limatura, che toccava immediatamente il vetro posandovi sopra, contrar dovea la prima dal vetro girante il moto di rotazione verticale, indi quell’altra limatura, che succedeva alla prima ad appoggiarsi alla parte infima del globo. acquistata l’una, e l’altra la forza centrifuga dal moto del globo, siccome più pesanti dell’acqua, d’una maggior forza erano animate, che l’acqua stessa, e quindi dovean tendere al maggior circolo, escludendone l’acqua di minor forza, perché men grave. il maggior circolo trovavasi appunto nell’equatore del globo. dunque per la prevalente forza dovea la limatura disporsi in fascia, o zona circolare all’equatore del globo, perpendicolare all’asse orizzontale del globo girante, escludendone l’acqua, e in fascia tanto più grande, quanto più abbondasse la limatura. così il primo fenomeno non può ricevere più nitida e facile soluzione. l’aria in ii° luogo, o l’olio di trementina, che galleggiavano alla sommità del globo anzi pure dell’equatore di esso, dovetter prima balzarsi di sito dalla limatura, che agognò a quel posto, e perseguirsi poi dall’acqua ne’ circoli prossimi all’equatore. sconcertata l’aria da tali forze, e tanto superiori lasciavasi da esse scompartire in quante bolle la volevan ridotta i suoi persecutori. assalita però eziandio in ciascuna bolla dalle zone roteanti dell’acqua, ceder dovea il campo di battaglia, e prender la ritirata a poco a poco al centro delle zone stesse, e all’asse del rotamento, ove il fluido loro inimico poteva meno, perché ivi menomi erano i suoi giri e le sue forze. così le bolle sempre più bersagliate nella lor fuga, nel ricoverarsi all’asse della rotazione, ove quasi in sicura trincea trovavansi al coperto dell’assalto ratio mathematica 18 (2008), 107 132 129 nimico; dovevano appunto squadronarsi in cilindro attorno di esso, miste a piccole particelle d’acqua, che tra i lor volumi avviluppatele traeansi nella lor fuga dietro prigioniere. che se l’acqua assalitrice accresciuta di forze per una maggior rotazione data al globo, movea alle bolle dell’aria, o dell’olio un nuovo assalto; ristringevansi queste nel loro trinciamento all’intorno dell’asse di rotazione quel tanto che bastava a sottrarle alla maggior furia dell’attacco; in luogo cioè, ove ancor meno l’acqua si aggirava. se infine astrette erano a sloggiarne dal prepotente moto dell’acqua; allora serrandosi tra loro vie maggiormente, e dando libertà alle imprigionate particelle acquee per ristringersi di volume; componevansi per ultimo scampo in un nocciolo cilindrico, e in mezzo alla sfera appostavansi protette al di sopra dalla ferrea fascia delle limature. non milita soltanto negli eserciti la scienza delle ritirate. la natura insegna l’arte di eseguirle, se sapessero studiarla. aggiunge l’abate nollet, che se inchiudasi nel globo coll’acqua girante una pallottola di cera; questa al nascere della rotazione scende a poco a poco verso l’asse orizzontale del globo, e del moto rotatorio, e a qualunque punto dell’asse pervenga discendendo, ivi gettasi essa pure a girare senza propender nullo né al centro né ai poli del globo. che se poi l’asse di rotazione non tengasi esattamente orizzontale, ma elevato da alcuna parte, e quindi obbliquo all’orizzonte, come porta il terzo genere degli sperimenti, che sonomi proposto a dilucidarvi; la palla allora a qualunque punto appostisi dell’asse in discendendo; arrampicasi poi lungo l’asse medesimo al polo più elevato. ad ottenere, che piuttosto discenda al polo più depresso, forza è render la palla più greve d’ugual volume acqueo con parte di piombo infusavi in seno. se raggunisi similmente nel girar dell’acqua alcuna bolla di quell’aria, che va sparsa per gl’interstizj dell’acqua; accorre questa pure al polo di rotazione più elevato. l’eleganza di questi sperimenti sta attendendo la spiegazione. la palla di cera men pesante dell’acqua dee premersi dalla maggior forza centrifuga dell’acqua al centro di quel qualunque circolo parallello all’equatore, in cui scontrisi, come le bolle aeree, o come l’olio di trementina. vero è che l’eccesso della gravità dell’acqua sopra la cera sospinge la palla all’insù verso la superficie del globo; ma questo impulso vinto è dalla centrifuga dell’acqua troppo superiore di possa. non ostante adunque la maggior leggierezza della palla, dee questa confinarsi all’asse di rotazione. ivi, stando l’asse orizzontale non è spinta la palla che all’insù per la sua leggierezza, e tenuta obbligata al tempo stesso al basso, dalla forza maggiore centrifuga dell’acqua, che agisce direttamente contro la forza, che la solleva. non sentendosi spinta la palla da niun altro lato, arrestasi a quel punto dell’asse, a cui fu depressa, rotando ratio mathematica 18 (2008), 107 132 130 semplicemente intorno ad esso per la forza impressavi dalla rotazione dell’acqua. questa è la soluzione del primo fenomeno, non men di esso a mio parere elegante. che se l’asse di rotazione inclini con angolo sensibile all’orizzonte, dovrà la palla più leggiera dell’acqua, e più ancora qualunque bolla aerea poggiare al polo elevato dell’asse. se la palla al contrario rendasi più grave discendere al polo depresso. imperocchè da una parte la forza centrifuga non oppone niuna resistenza alla palla nella direzione dell’asse, attorno a cui volgesi l’acqua, e il globo, non dispiegando lungo l’asse niuna forza. infatti si mantiene la palla rotando pacificamente in qualunque punto di esso, ove venga abbassata. dall’altra parte la forza dell’acqua, che sospinge la palla allo insù per la prevalente gravità specifica, non è più perpendicolare all’asse, quando questo è inclinato all’orizzonte, come perpendicolare vi è sempre la forza centrifuga. spingendosi allora adunque obbliquamente la palla all’alto dall’eccesso del peso d’un ugual volume di fluido circostante, e non contrastandovi la forza centrifuga dell’acqua nella stessa direzione, può e deve la palla poggiare verso il polo elevato di rotazione, tenendo però la via dell’asse stesso sempre più sgombra di resistenza. così l’eccesso di gravità nella palla renduta più pesante col piombo intrusovi, d’un ugual volume di fluido, deve trarre la palla stessa a calarsi al polo più depresso. non vi stancherò, o signori col racconto delle vecchie spiegazioni. o queste mancano al tutto, o sono così involute, che non è possibile seguirne l’orme ascoltando per qualunque sforzo di attenzione, che vi si adoperi. contentatevi per ultimo, ch’io vi faccia vedere l’intralciamento terribile della vecchia spiegazione sul fenomeno de’ corpi, che posano sulla superficie d’altri corpi rotanti, come in ultimo luogo debbo esporvi. scelgo il fenomeno de corpi, che girano sulla superficie della terra nel suo moto diurno. per questa rotazione terrestre, oggi è oramai fuor di dubbio, che qualunque corpo pesa meno all’equatore, che in qualunque parallello, come a cagion d’esempio al parallello di mantova, lontano 45° poco più, e meno anche in mantova, che a pietroburgo, il cui parallello dista dal nostro 15°. poiché, ecco ciò che una volta dicevasi, armatevi di sofferenza. poiché la direzione della gravità de’ corpi, che tendono al centro della terra è obbliqua ne’ circoli parallelli all’equatore, e tanto più obbliqua, quant’essi dall’equatore si scostano, e si approssimano al polo, e si esercita oltre a ciò in un piano da essi diverso, dirigendosi la gravità per un piano, che passa pel centro dell’equatore, laddove il piano de’ circoli parallelli è perpendicolare al punto dell’asse della terra, che loro corrisponde; per ciò è, dicevasi, che la gravità per esempio di un sasso in qualche circolo parallello, che descrivesi nel moto diurno della terra attorno a se stessa; per ratio mathematica 18 (2008), 107 132 131 ciò è, io dissi, che la gravità del sasso deve risolversi in due parti giusta il costume. e prega a considerar quella di queste due parti, che impiegasi a ritenere il sasso girante colla terra nel suo circolo; questa pure in altre due forze è a risolversi, perché essa pure è obbliqua ai circoli parallelli, e in diversi piani. una di queste due ultime forze già risolute trovasi nel piano de’ parallelli, l’altra fuor d’esso, ma perpendicolare al piano. or la parte sola della gravità, che trovasi nel piano, quella è che impiegasi a ritenere il corpo nel circolo, e che si dice centripeta. essendo dato il circolo di ciascun parallello, è dato il raggio eziandio, dunque ancor la forza necessaria a ritenere il sasso nel suo circolo, dunque ancor la perpendicolare al piano del circolo. dunque la prima delle due parti, nelle quali fu divisa la gravità primitiva del sasso, ne è già conosciuta. or sottraendo questa dalla gravità stessa primitiva, ne risulterà l’altra seconda parte della gravità. non vi stancate, vi supplico, accademici ornatissimi sostenete ancor per poco. il fenomeno non è ancora spiegato col ritrovamento di quella parte di gravità, che ci è comparsa dal calcolo. convien cercare di più la potenza premente. a dedur questa, or è forza comporre la seconda parte di gravità già trovata con quella, che è perpendicolare al piano del circolo parallello. l’equivalente a queste due forze composte sarà la potenza premente, che esprimerassi dalla diagonale di un parallelogrammo, i lati de’ quali siano le due forze, che compongonsi. or eccoci alla risoluzione. ma questa potenza così composta è sempre maggiore ne’ parallelli più distanti dall’equatore, come appare poi di leggieri. dunque è maggior la potenza premente ne’ circoli parallelli, più che si scostano dall’equatore. dunque lo stesso sasso deve pesar più a pietroburgo, che a mantova, e a mantova più che all’equatore. s’io non sono riuscito intelligibile, o signori in questo raziocinio, non ne ho punto rossore, e quasi me ne compiaccio. so bene che non può ascriversi a colpa della mia esposizione, ma bensì dell’inviluppo delle cose, che ho dovuto svolgere. ho dovuto dividere la gravità primitiva del sasso in due parti; una di queste risolvere in altre due, e indi comporre la seconda di queste colla seconda parte della gravità. or voglio compensarvi con altrettanto piacere della noja, di che sento avervi colmati sotto la vecchia spiegazione. presentovi col mezzo della forza centrifuga spiegato il fenomeno in brevi tratti, e all’ultima evidenza. sotto l’equatore descrive il sasso un maggior circolo nel diurno moto della terra, che in mantova. essendo la forza centrifuga in ragion de’ raggi de’ circoli descritti, stando l’altre cose pari, sarà investito il sasso all’equatore di maggior forza centrifuga che a mantova, e molto più che a pietroburgo. dunque una maggior forza di gravità all’ingiù nel sasso viene elisa dalla ratio mathematica 18 (2008), 107 132 132 centrifuga all’equatore, che a mantova, e qui più, che a pietroburgo. ma quella porzion di gravità che si elide non preme. dunque una maggior parte di gravità nel sasso non preme all’equatore, che a mantova, o a pietroburgo. dunque il sasso men pesa all’equatore, che a mantova, e meno qui che a pietroburgo. non ho intralasciata niuna idea intermedia per abbreviarvi il mio discorso. io l’ho difuso in tutta la sua ampiezza. per questa minor gravità all’equatore, la terra deve esser colà più elevata come lo è in fatti che ai poli. non son pago però se non vi fo comprendere tutta la finezza di questo metodo. quantunque sia minore, come vi ho mostrato, la forza centrifuga ne’ circoli parallelli all’equatore, che all’equatore stesso; riflettete di più, che non tutta in essi, come all’equatore, non impiegasi a distrarre la gravità dalla sua tendenza al centro della terra. imperocché ed è colà obbliqua alla tendenza di gravità, ed è in diverso piano, come di sopra si è dichiarato. per iscoprir dunque quella parte di centrifuga, che opponsi alla gravità, risolvesi in due forze. ma tosto che ne’ circoli parallelli ha da risolversi in due la centrifuga; certo è, che minor sarà in essi la parte, che toglie alla gravità, e tanto minore, quanto la linea rappresentante questa parte facciasi più breve. dunque ne’ circoli parallelli la centrifuga minore che all’equatore, agisce anche meno ad elidere la forza di gravità, e tanto meno, quanto più obbliqua alla direzione della gravità. in questo discorso non vi ha che una semplice risoluzione di forze, e nell’antico ben due, ed una composizione. quale è, o signori, più semplice, quale più chiaro, e persuadente? non avrò io dunque ragione di sostenere il metodo della forza centrifuga, sebbene forza immaginaria, nel senso da me stabilito nella prima dissertazione, molto utile alla fisica. se per esso intendiamo ciò, che per altra guisa ci è sparso di molte tenebre; ci si fa chiara la sua utilità. se per esso giugniamo facilmente a dichiarare ciò che fuor d’esso è inesplicabile, o incomprensibile; ci si mostra eziandio la quasi necessità. i maestri dell’arte infatti tutti oggimai, e l’hanno adottato ed esteso alla spiegazione de’ più interessanti problemi della fisica e dell’astronomia. di questi non ho parlato, perché troppo conosciuti. ho stimato di farvela veder utile anche ne’ più piccoli fenomeni, perché riconosciate la sua utilità nelle somme, e nelle infime opere di natura e dell’arte. avete veduti interpretati i fenomeni de’ fluidi e degli ammassi de’ solidi, che ruotan i° parallelli, poi perpendicolari, e in iii° luogo obbliqui all’orizzonte con corpi, or più leggieri, or più gravi intermisti, e infine il gran fenomeno de’ solidi, che posano alla circonferenza de’ globi rotanti, e ciò a fronte delle vecchie interpretazioni. così spero d’aver compiuta la materia propostavi altra volta, e di essermi sciolto dalle promesse, che ho sembrato forse aver dimenticate. approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 41, 2021, pp. 181-196 181 the sequence of trifurcating fibonacci numbers parimalkumar a. patel * devbhadra v. shah † abstract one of the remarkable generalizations of fibonacci sequence is a 𝑘-fibonacci sequence and subsequently generalized into the ‘bifurcating fibonacci sequence’. in this paper, further generalization as the sequence of ‘trifurcating fibonacci numbers’ is studied and binet-like formula for these numbers is obtained. the analogous of cassini’s identity, catalan’s identity, d’ocagne’s identity and some fundamental identities for the terms of this sequence has also been investigated. keywords: fibonacci sequence, bifurcating fibonacci sequence, generalization of fibonacci sequence, binet formula, identities related with the fibonacci sequence. 2010 ams subject classification‡: 11b39, 11b83, 11b99. * veer narmad south gujarat university, surat, india; parimalpatel4149@gmail.com. † veer narmad south gujarat university, surat, india; drdvshah@yahoo.com. ‡ 1received on september 30, 2021. accepted on december 10, 2021. published on december 31, 2021.doi: 10.23755/rm.v41i0.668. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. parimalkumar a. patel, devbhadra v. shah 182 1. introduction the fibonacci sequence {𝐹𝑛}𝑛≥0 is a sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … , where each term is the sum of two preceding terms. the corresponding recurrence relation is 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 ; 𝑛 ≥ 2. various generalizations of this sequence have appeared in recent years [5, 7, 8]. related to this work (i) some work alters the first two terms of the sequence from 0,1 to any arbitrary integers 𝑎, 𝑏 while maintaining the recurrence relation (ii) some more work preserves the first two terms of the sequence but alters the recurrence relation (iii) even the combined approach of altering the initial terms as well as recurrence relation was considered by several authors. for further details about this sequence, one can refer koshy [6], patel, shah [7], singh, sikhwal, bhatnagar [8] and related papers available in the literature. one interesting generalization depending on exactly one real parameter 𝑘 is the sequence of 𝑘-fibonacci numbers {𝐹𝑘,𝑛} which is defined using a linear recurrence relation 𝐹𝑘,𝑛 = 𝑘𝐹𝑘,𝑛−1 + 𝐹𝑘,𝑛−2 ; 𝑛 ≥ 2 where 𝐹𝑘,0 = 0 and 𝐹𝑘,1 = 1. for 𝑘 = 1, we get the standard fibonacci sequence and for 𝑘 = 2, we get the sequence of pell numbers. this sequence was studied by arvadia and shah [1]. edson and yayenie [4] generalized this sequence to a sequence which depends on two real parameters 𝑎, 𝑏. they defined the bifurcating sequence {𝐹𝑛 (𝑎,𝑏) } 𝑛≥0 by the recurrence relation 𝐹𝑛 (𝑎,𝑏) = { 𝑎𝐹𝑛−1 (𝑎,𝑏) + 𝐹𝑛−2 (𝑎,𝑏) ; if 𝑛 is even 𝑏𝐹𝑛−1 (𝑎,𝑏) + 𝐹𝑛−2 (𝑎,𝑏) ; if 𝑛 is odd ; 𝑛 ≥ 2 where 𝐹0 (𝑎,𝑏) = 0, 𝐹1 (𝑎,𝑏) = 1. diwan and shah [2, 3], yayenie [10], verma and bala [9] studied this sequence extensively and obtained significant results. it is easy to observe that (i) by considering 𝑎 = 𝑏 = 1, we get standard fibonacci sequence (ii) by considering 𝑎 = 𝑏 = 2, we get the sequence of pell numbers and (iii) by considering 𝑎 = 𝑏 = 𝑘, we get the sequence of 𝑘-fibonacci numbers. in this paper, we further generalize this sequence to a sequence of trifurcating fibonacci numbers, which depends on the three real parameters 𝑎, 𝑏, 𝑐. definition: for any three nonzero positive integers 𝑎, 𝑏 and 𝑐, the trifurcating fibonacci sequence {𝐹𝑛 (𝑎,𝑏,𝑐) } 𝑛≥0 is defined recursively by 𝐹0 (𝑎,𝑏,𝑐) = 0, 𝐹1 (𝑎,𝑏,𝑐) = 1 and the recurrence relation the sequence of trifurcating fibonacci numbers 183 𝐹𝑛 (𝑎,𝑏,𝑐) = { 𝑎𝐹𝑛−1 (𝑎,𝑏,𝑐) + 𝐹𝑛−2 (𝑎,𝑏,𝑐) ; if 𝑛 ≡ 0(𝑚𝑜𝑑 3) 𝑏𝐹𝑛−1 (𝑎,𝑏,𝑐) + 𝐹𝑛−2 (𝑎,𝑏,𝑐) ; if 𝑛 ≡ 1(𝑚𝑜𝑑 3) 𝑐𝐹𝑛−1 (𝑎,𝑏,𝑐) + 𝐹𝑛−2 (𝑎,𝑏,𝑐) ; if 𝑛 ≡ 2(𝑚𝑜𝑑 3) . to avoid cumbersome notation, we denote 𝐹𝑛 (𝑎,𝑏,𝑐) by 𝑃𝑛. few terms of this trifurcating generalized fibonacci sequence are shown in the table 1. in this paper we obtain various interesting results for this sequence. 𝒏 𝑷𝒏 0 0 1 1 2 𝑐 3 𝑎𝑐 + 1 4 𝑎𝑏𝑐 + 𝑏 + 𝑐 5 𝑎𝑏𝑐2 + 𝑏𝑐 + 𝑐2 + 𝑎𝑐 + 1 6 𝑎2𝑏𝑐2 + 2𝑎𝑏𝑐 + 𝑎𝑐2 + 𝑎2𝑐 + 𝑎 + 𝑏 + 𝑐 7 𝑎2𝑏2𝑐2 + 2𝑎𝑏2𝑐 + 2𝑎𝑏𝑐2 + 𝑎2𝑏𝑐 + 𝑎𝑏 + 2𝑏𝑐 + 𝑏2 + 𝑐2 + 𝑎𝑐 + 1 table 1 2. fundamental identities for the trifurcating sequence {𝑷𝒏}𝒏≥𝟎 in this section, we derive some interesting identities for the terms of the sequence {𝑃𝑛}𝑛≥0. we first show that any two consecutive terms of {𝑃𝑛}𝑛≥0 are always relatively prime. theorem 2.1. gcd(𝑃𝑛, 𝑃𝑛−1) = 1; for all 𝑛 = 1, 2, … . proof. we prove the result by considering three cases when 𝑛 = 3𝑘, 3𝑘 + 1 or 3𝑘 + 2. we present the proof only for the case 𝑛 = 3𝑘 and other cases follows accordingly. now euclidean algorithm leads to the following system of equations: 𝑃3𝑘 = 𝑎𝑃3𝑘−1 + 𝑃3𝑘−2 𝑃3𝑘−1 = 𝑐𝑃3𝑘−2 + 𝑃3𝑘−3 𝑃3𝑘−2 = 𝑏𝑃3𝑘−3 + 𝑃3𝑘−4 ⋮ parimalkumar a. patel, devbhadra v. shah 184 𝑃4 = 𝑏𝑃3 + 𝑃2 𝑃3 = 𝑎𝑃2 + 𝑃1 𝑃2 = 𝑐𝑃1 + 0 it now easily follows from euclidean algorithm that gcd(𝑃𝑛, 𝑃𝑛−1) = 𝑃1 = 1. we now prove certain summation formulae for the terms of {𝑃𝑛}𝑛≥0. lemma 2.2. a) 𝑃3𝑛+2 = (𝑏𝑐 + 1)(𝑃3 + 𝑃6 + ⋯ + 𝑃3𝑛) + (𝑃2 + 𝑃5 + ⋯ + 𝑃3𝑛−2) + 𝑐 b) 𝑃3𝑛+1 = (𝑎𝑏 + 1)(𝑃2 + 𝑃5 + 𝑃8 + ⋯ + 𝑃3𝑛−1) +(𝑏 − 1)(𝑃1 + 𝑃4+. . . +𝑃3𝑛−2) + 1 c) 𝑃3𝑛 = (𝑎𝑐 + 1)(𝑃1 + 𝑃4+. . . +𝑃3𝑛−2) + (𝑎 − 1)(𝑃0 + 𝑃3 + ⋯ + 𝑃3𝑛−3). proof. since, 𝑃3𝑛+2 = 𝑐𝑃3𝑛+1 + 𝑃3𝑛 we get 𝑃2 = 𝑐𝑃1 + 𝑃0 𝑃5 = 𝑐𝑃4 + 𝑃3 𝑃8 = 𝑐𝑃7 + 𝑃6 ⋮ 𝑃3𝑛−1 = 𝑐𝑃3𝑛−2 + 𝑃3𝑛−3 𝑃3𝑛+2 = 𝑐𝑃3𝑛+1 + 𝑃3𝑛. adding all these equations we get 𝑃2 + 𝑃5 + 𝑃8+. . . +𝑃3𝑛+2 = 𝑐(𝑃1 + 𝑃4+. . . +𝑃3𝑛+1) + (𝑃0 + 𝑃3 + ⋯ + 𝑃3𝑛) (2.1) again, 𝑃3𝑛 = 𝑎𝑃3𝑛−1 + 𝑃3𝑛−2 gives 𝑃3 = 𝑎𝑃2 + 𝑃1 𝑃6 = 𝑎𝑃5 + 𝑃4 𝑃9 = 𝑎𝑃8 + 𝑃7 ⋮ 𝑃3𝑛−3 = 𝑎𝑃3𝑛−4 + 𝑃3𝑛−5 𝑃3𝑛 = 𝑎𝑃3𝑛−1 + 𝑃3𝑛−2 adding these equations, we get 𝑃3 + 𝑃6 + 𝑃9 + ⋯ + 𝑃3𝑛 = 𝑎(𝑃2 + 𝑃5 + 𝑃8 + ⋯ + 𝑃3𝑛−1) +(𝑃1 + 𝑃4+. . . +𝑃3𝑛−2) (2.2) also, since 𝑃3𝑛+1 = 𝑏𝑃3𝑛 + 𝑃3𝑛−1, we have 𝑃4 = 𝑏𝑃3 + 𝑃2 𝑃7 = 𝑏𝑃6 + 𝑃5 𝑃10 = 𝑏𝑃9 + 𝑃8 ⋮ the sequence of trifurcating fibonacci numbers 185 𝑃3𝑛−2 = 𝑏𝑃3𝑛−3 + 𝑃3𝑛−4 𝑃3𝑛+1 = 𝑏𝑃3𝑛 + 𝑃3𝑛−1 adding all these equations we get 𝑃4 + 𝑃7 + ⋯ + 𝑃3𝑛−2 + 𝑃3𝑛+1 = 𝑏(𝑃3 + 𝑃6 + ⋯ + 𝑃3𝑛) + (𝑃2 + 𝑃5+. . . +𝑃3𝑛−1) (2.3) using (2.3) in (2.1) we get 𝑃2 + 𝑃5 + 𝑃8 + ⋯ + 𝑃3𝑛+2 = 𝑐(1 + 𝑏(𝑃3 + 𝑃6 + ⋯ + 𝑃3𝑛) + (𝑃2 + 𝑃5 + ⋯ + 𝑃3𝑛−1)) +(𝑃0 + 𝑃3 + ⋯ + 𝑃3𝑛) = 𝑐 + (𝑏𝑐 + 1)(𝑃0 + 𝑃3 + 𝑃6 + ⋯ + 𝑃3𝑛) + 𝑐(𝑃2 + 𝑃5+. . . +𝑃3𝑛−1) this finally gives 𝑃3𝑛+2 = (𝑏𝑐 + 1)(𝑃0 + 𝑃3 + 𝑃6 + ⋯ + 𝑃3𝑛) +(𝑐 − 1)(𝑃2 + 𝑃5+. . . +𝑃3𝑛−1) + 𝑐 (2.4) other results can be proved accordingly by considering the pair of equations (2.1), (2.2) and further (2.1), (2.3) together. lemma 2.3. a) ∑ 𝑃3𝑖 𝑛 𝑖=0 = 𝑎𝑃3𝑛(𝑏−1)(𝑐−1)−(𝑃3𝑛+1−1)(𝑎𝑐+1)(𝑐−1)+(𝑃3𝑛+2−𝑐)(𝑎𝑏+1)(𝑎𝑐+1) (𝑎𝑏+1)(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑏−1)(𝑐−1) . b) ∑ 𝑃3𝑖+1 = 𝑛 𝑖=0 (𝑃3𝑛+1−1)(𝑎−1)(𝑐−1)−(𝑃3𝑛+2−𝑐)(𝑎𝑏+1)(𝑎−1)+𝑎𝑃3𝑛(𝑎𝑏+1)(𝑏𝑐+1) (𝑎𝑏+1)(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑏−1)(𝑐−1) . c) ∑ 𝑃3𝑖+2 = (𝑃3𝑛+2−𝑐)(𝑎−1)(𝑏−1)−𝑎𝑃3𝑛(𝑏𝑐+1)(𝑏−1)+(𝑃3𝑛+1−1)(𝑎𝑐+1)(𝑏𝑐+1) (𝑎𝑏+1)(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑏−1)(𝑐−1) 𝑛 𝑖=0 . proof. we only prove result (a) here and other two results can be proved in a similar way. using (b) and (c) of lemma 2.2, we get 𝑎𝑃3𝑛 = (𝑎𝑐 + 1) (𝑏 − 1) {(𝑃3𝑛+1 − 1) − (𝑎𝑏 + 1)(𝑃2 + 𝑃5 + 𝑃8 + ⋯ + 𝑃3𝑛−1)} +(𝑎 − 1)(𝑃0 + 𝑃3 + ⋯ + 𝑃3𝑛) then 𝑎𝑃3𝑛 − (𝑎𝑐+1) (𝑏−1) (𝑃3𝑛+1 − 1) − (𝑎 − 1)(𝑃0 + 𝑃3 + ⋯ + 𝑃3𝑛) = − (𝑎𝑐+1)(𝑎𝑏+1) (𝑏−1) (𝑃2 + 𝑃5 + 𝑃8 + ⋯ + 𝑃3𝑛−1) using (2.4) we get 𝑎𝑃3𝑛 − (𝑎𝑐+1) (𝑏−1) (𝑃3𝑛+1 − 1) − (𝑎 − 1)(𝑃0 + 𝑃3 + ⋯ + 𝑃3𝑛) = − (𝑎𝑏+1)(𝑎𝑐+1) (𝑏−1)(𝑐−1) ((𝑃3𝑛+2 − 𝑐) − (𝑏𝑐 + 1)(𝑃0 + 𝑃3 + 𝑃6 + ⋯ + 𝑃3𝑛)) then, ((𝑎𝑏 + 1)(𝑎𝑐 + 1)(𝑏𝑐 + 1) + (𝑎 − 1)(𝑏 − 1)(𝑐 − 1))(𝑃0 + 𝑃3 + ⋯ + 𝑃3𝑛) = 𝑎(𝑏 − 1)(𝑐 − 1)𝑃3𝑛 − (𝑎𝑐 + 1)(𝑐 − 1)(𝑃3𝑛+1 − 1) parimalkumar a. patel, devbhadra v. shah 186 +(𝑎𝑏 + 1)(𝑎𝑐 + 1)(𝑃3𝑛+2 − 𝑐). hence, 𝑃0 + 𝑃3 + ⋯ + 𝑃3𝑛 = 𝑎𝑃3𝑛(𝑏−1)(𝑐−1)−(𝑃3𝑛+1−1)(𝑎𝑐+1)(𝑐−1)+(𝑃3𝑛+2−𝑐)(𝑎𝑏+1)(𝑎𝑐+1) (𝑎𝑏+1)(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑏−1)(𝑐−1) . we now obtain the sum of first 𝑘 trifurcating fibonacci numbers. theorem 2.4. ∑ 𝑃𝑛 𝑘 𝑛=1 = {(𝑏−1)(𝑐−1)−⌊1− 𝜒(𝑘) 3 ⌋(𝑏𝑐+1)(𝑏−1)}⌊ 4−𝜒(𝑘) 3 ⌋𝑎𝑃 3⌊ 𝑘 3 ⌋−3 +{⌊ 1+𝜒(𝑘) 3 ⌋(𝑏−1)(𝑐−1)+(𝑎𝑏+1)(𝑏𝑐+1)−⌊ 2+𝜒(𝑘) 3 ⌋(𝑏𝑐+1)(𝑏−1)}𝑎𝑃 3⌊ 𝑘 3 ⌋ +{⌊1− 𝜒(𝑘) 3 ⌋[(𝑎−1)(𝑐−1)+(𝑎𝑐+1)(𝑏𝑐+1)]−(𝑎𝑐+1)(𝑐−1)}⌊ 4−𝜒(𝑘) 3 ⌋(𝑃 3⌊ 𝑘 3 ⌋−2 −1) +⌊ 2+𝜒(𝑘) 3 ⌋{(𝑎−1)(𝑐−1)+(𝑎𝑐+1)(𝑏𝑐+1)−⌊ 1+𝜒(𝑘) 3 ⌋(𝑎𝑐+1)(𝑐−1)} +⌊ 4−𝜒(𝑘) 3 ⌋{⌊1− 𝜒(𝑘) 3 ⌋(𝑎−1)(𝑏−1)+(𝑎𝑏+1)(𝑎𝑐+1)}(𝑃 3⌊ 𝑘 3 ⌋−1 −𝑐) +{⌊ 2+𝜒(𝑘) 3 ⌋(𝑎−1)(𝑏−1)+⌊ 1+𝜒(𝑘) 3 ⌋(𝑎𝑏+1)(𝑎𝑐+1)−(𝑎𝑏+1)(𝑎−1)}(𝑃 3⌊ 𝑘 3 ⌋+2 −𝑐) (𝑎𝑏+1)(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑏−1)(𝑐−1) . proof. we first obtain the value of ∑ 𝑃𝑛 𝑘 𝑛=1 in three cases when 𝑘 is of the form 3𝑚 − 2, 3𝑚 − 1 and 3𝑚 and then combine the results to obtain a single result. for 𝑘 = 3𝑚 − 2, using the above lemma we get ∑ 𝑃𝑛 𝑘 𝑛=1 = (𝑃3 + ⋯ + 𝑃3𝑚−3) + (𝑃1 + 𝑃4 + ⋯ + 𝑃3𝑚−2) +(𝑃2 + 𝑃5+. . . +𝑃3𝑚−4) = { 𝑎𝑃3𝑚−3(𝑏−1)(𝑐−1)−(𝑃3𝑚−2−1)(𝑎𝑐+1)(𝑐−1)+(𝑃3𝑚−1−𝑐)(𝑎𝑏+1)(𝑎𝑐+1)+ (𝑃3𝑚−2−1)(𝑎−1)(𝑐−1)−(𝑃3𝑚+2−𝑐)(𝑎𝑏+1)(𝑎−1)+𝑎𝑃3𝑚(𝑎𝑏+1)(𝑏𝑐+1)+ (𝑃3𝑚−1−𝑐)(𝑎−1)(𝑏−1)−𝑎𝑃3𝑚−3(𝑏𝑐+1)(𝑏−1)+(𝑃3𝑚−2−1)(𝑎𝑐+1)(𝑏𝑐+1) } (𝑎𝑏+1)(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑏−1)(𝑐−1) on simplification, we get, ∑ 𝑃𝑛 𝑘 𝑛=1 = { 𝑎𝑃3𝑚−3{(𝑏−1)(𝑐−1)−(𝑏𝑐+1)(𝑏−1)}+(𝑃3𝑚−2−1){(𝑎−1)(𝑐−1)+ (𝑎𝑐+1)(𝑏𝑐+1)−(𝑎𝑐+1)(𝑐−1)}+(𝑃3𝑚−1−𝑐){(𝑎−1)(𝑏−1)+(𝑎𝑏+1)(𝑎𝑐+1)} −(𝑃3𝑚+2−𝑐)(𝑎𝑏+1)(𝑎−1)+𝑎𝑃3𝑚(𝑎𝑏+1)(𝑏𝑐+1) } (𝑎𝑏+1)(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑏−1)(𝑐−1) next, for the case 𝑘 = 3𝑚 − 1, we get ∑ 𝑃𝑛 𝑘 𝑛=1 = (𝑃3 + ⋯ + 𝑃3𝑚−3) + (𝑃1 + 𝑃4 + ⋯ + 𝑃3𝑚−2) +(𝑃2 + 𝑃5+. . . +𝑃3𝑚−1) the sequence of trifurcating fibonacci numbers 187 = { 𝑎𝑃3𝑚−3(𝑏−1)(𝑐−1)−(𝑃3𝑚−2−1)(𝑎𝑐+1)(𝑐−1)+(𝑃3𝑚−1−𝑐)(𝑎𝑏+1)(𝑎𝑐+1)+ (𝑃3𝑚+1−1)(𝑎−1)(𝑐−1)−(𝑃3𝑚+2−𝑐)(𝑎𝑏+1)(𝑎−1)+𝑎𝑃3𝑚(𝑎𝑏+1)(𝑏𝑐+1)+ (𝑃3𝑚+2−𝑐)(𝑎−1)(𝑏−1)−𝑎𝑃3𝑚(𝑏𝑐+1)(𝑏−1)+(𝑃3𝑚+1−1)(𝑎𝑐+1)(𝑏𝑐+1) } (𝑎𝑏+1)(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑏−1)(𝑐−1) simplifying this we get ∑ 𝑃𝑛 𝑘 𝑛=1 = 𝑎𝑃3𝑚−3(𝑏−1)(𝑐−1)+(𝑃3𝑚−1−𝑐)(𝑎𝑏+1)(𝑎𝑐+1)−(𝑃3𝑚−2−1)(𝑎𝑐+1)(𝑐−1)+ {(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑐−1)}(𝑃3𝑚+1−1)+(𝑃3𝑚+2−𝑐){(𝑎−1)(𝑏−1) −(𝑎𝑏+1)(𝑎−1)}+𝑎𝑃3𝑚{(𝑎𝑏+1)(𝑏𝑐+1)−(𝑏𝑐+1)(𝑏−1)} (𝑎𝑏+1)(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑏−1)(𝑐−1) finally, for 𝑘 = 3𝑚, we get ∑ 𝑃𝑛 𝑘 𝑛=1 = (𝑃3 + ⋯ + 𝑃3𝑚) + (𝑃1 + 𝑃4 + ⋯ + 𝑃3𝑚−2) +(𝑃2 + 𝑃5+. . . +𝑃3𝑚−1) = { 𝑎𝑃3𝑚(𝑏−1)(𝑐−1)−(𝑃3𝑚+1−1)(𝑎𝑐+1)(𝑐−1)+(𝑃3𝑚+2−𝑐)(𝑎𝑏+1)(𝑎𝑐+1)+ (𝑃3𝑚+1−1)(𝑎−1)(𝑐−1)−(𝑃𝑚𝑛+2−𝑐)(𝑎𝑏+1)(𝑎−1)+𝑎𝑃3𝑚(𝑎𝑏+1)(𝑏𝑐+1)+ (𝑃3𝑚+2−𝑐)(𝑎−1)(𝑏−1)−𝑎𝑃3𝑚(𝑏𝑐+1)(𝑏−1)+(𝑃3𝑚+1−1)(𝑎𝑐+1)(𝑏𝑐+1) } (𝑎𝑏+1)(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑏−1)(𝑐−1) this on simplification gives ∑ 𝑃𝑛 𝑘 𝑛=1 = 𝑎𝑃3𝑚{(𝑏−1)(𝑐−1)+(𝑎𝑏+1)(𝑏𝑐+1)−(𝑏𝑐+1)(𝑏−1)}+(𝑃3𝑚+1−1){(𝑎−1)(𝑐−1)+(𝑎𝑐+1) (𝑏𝑐+1)−(𝑎𝑐+1)(𝑐−1)}+(𝑃3𝑚+2−𝑐){(𝑎−1)(𝑏−1)+(𝑎𝑏+1)(𝑎𝑐+1)−(𝑎𝑏+1)(𝑎−1)} (𝑎𝑏+1)(𝑎𝑐+1)(𝑏𝑐+1)+(𝑎−1)(𝑏−1)(𝑐−1) combining all these three results, we finally get the required result. the following are the interesting identities related with the summation of trifurcating fibonacci numbers as well as its squares. theorem 2.5. ∑ 𝑎 ⌊1− 𝜒(𝑘+1) 3 ⌋ 𝑏 ⌊1− 𝜒(𝑘) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑘+2) 3 ⌋𝑛 𝑘=0 𝑃𝑘 = 𝑃𝑛 + 𝑃𝑛+1 − 1. proof. using the definition of trifurcating fibonacci numbers, we have 𝑃3𝑛 = 𝑎𝑃3𝑛−1 + 𝑃3𝑛−2 ; 𝑃3𝑛+1 = 𝑏𝑃3𝑛 + 𝑃3𝑛−1; 𝑃3𝑛+2 = 𝑐𝑃3𝑛+1 + 𝑃3𝑛 this can be written as 𝑎𝑃3𝑛−1 = 𝑃3𝑛 − 𝑃3𝑛−2 ; 𝑏𝑃3𝑛 = 𝑃3𝑛+1 − 𝑃3𝑛−1 ; 𝑐𝑃3𝑛+1 = 𝑃3𝑛+2 − 𝑃3𝑛 thus, we have the following system of equations: 𝑐𝑃1 = 𝑃2 − 𝑃0 𝑎𝑃2 = 𝑃3 − 𝑃1 𝑏𝑃3 = 𝑃4 − 𝑃2 ⋮ 𝑎𝑃𝑛 = 𝑃𝑛+1 − 𝑃𝑛−1; if 𝑛 ≡ 0(𝑚𝑜𝑑 3) 𝑏𝑃𝑛 = 𝑃𝑛+1 − 𝑃𝑛−1; if 𝑛 ≡ 1(𝑚𝑜𝑑 3) 𝑃𝑛 = 𝑃𝑛+1 − 𝑃𝑛−1; if 𝑛 ≡ 2(𝑚𝑜𝑑 3) adding all the above equations and using the fact that 𝑃0 = 0 and 𝑃1 = 1, we get ∑ 𝑎 ⌊1− 𝜒(𝑘+1) 3 ⌋ 𝑏 ⌊1− 𝜒(𝑘) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑘+2) 3 ⌋𝑛 𝑘=0 𝑃𝑘 = 𝑃𝑛 + 𝑃𝑛+1 − 1. parimalkumar a. patel, devbhadra v. shah 188 theorem 2.6. ∑ 𝑎 ⌊1− 𝜒(𝑘+1) 3 ⌋ 𝑏 ⌊1− 𝜒(𝑘) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑘+2) 3 ⌋𝑛 𝑘=0 𝑃𝑘 2 = 𝑃𝑛𝑃𝑛+1. proof. we prove this result only for the case 𝑛 ≡ 0(𝑚𝑜𝑑 3) and remaining cases can be proved accordingly. we let 𝑛 = 3𝑚 and apply induction on 𝑚. for 𝑚 = 1, we have ∑ 𝑎 ⌊1− 𝜒(𝑘+1) 3 ⌋ 𝑏 ⌊1− 𝜒(𝑘) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑘+2) 3 ⌋3 𝑘=0 𝑃𝑘 2 = 𝑏𝑃0 2 + 𝑐𝑃1 2 + 𝑎𝑃2 2 + 𝑏𝑃3 2 since 𝑃0 = 0, 𝑃1 = 1, 𝑃2 = 𝑐 and 𝑃3 = (1 + 𝑎𝑐), we get ∑ 𝑎 ⌊1− 𝜒(𝑘+1) 3 ⌋ 𝑏 ⌊1− 𝜒(𝑘) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑘+2) 3 ⌋3 𝑘=0 = 𝑐(1 + 𝑎𝑐) + 𝑏(1 + 𝑎𝑐) 2 = 𝑃3𝑃4 we next assume that the result holds for some positive integer 𝑚 = 𝑙 > 1. that is let ∑ 𝑎 ⌊1− 𝜒(𝑘+1) 3 ⌋ 𝑏 ⌊1− 𝜒(𝑘) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑘+2) 3 ⌋3𝑙 𝑘=0 𝑃𝑘 2 = 𝑃3𝑙𝑃3𝑙+1 holds. now, ∑ 𝑎 ⌊1− 𝜒(𝑘+1) 3 ⌋ 𝑏 ⌊1− 𝜒(𝑘) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑘+2) 3 ⌋3(𝑙+1) 𝑘=0 𝑃𝑘 2 = ∑ 𝑎 ⌊1− 𝜒(𝑘+1) 3 ⌋ 𝑏 ⌊1− 𝜒(𝑘) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑘+2) 3 ⌋3𝑙 𝑘=0 𝑃𝑘 2 + 𝑐𝑃3𝑙+1 2 + 𝑎𝑃3𝑙+2 2 + 𝑏𝑃3𝑙+3 2 . = 𝑃3𝑙𝑃3𝑙+1 + 𝑐𝑃3𝑙+1 2 + 𝑎𝑃3𝑙+2 2 + 𝑏𝑃3𝑙+3 2 = 𝑃3𝑙+1𝑃3𝑙+2 + 𝑎𝑃3𝑙+2 2 + 𝑏𝑃3𝑙+3 2 = 𝑃3𝑙+2𝑃3𝑙+3 + 𝑏𝑃3𝑙+3 2 = 𝑃3(𝑙+1)𝑃3(𝑙+1)+1 thus, by induction, the result to be proved holds for every positive integer 𝑛. 3. binet-like formula for the trifurcating fibonacci sequence: generating function is used to solve the linear homogeneous recurrence relations. in this section, the generating function for the trifurcating fibonacci sequence is derived and it is used to obtain binet-like formula for these numbers. we first prove a result which will be needed to obtain the generating function of 𝑃𝑛. lemma 3.1. 𝑃𝑛+3 − (𝑎𝑏𝑐 + 𝑎 + 𝑏 + 𝑐)𝑃𝑛 + 𝑃𝑛−3 = 0. proof. we prove the result by considering the three cases when 𝑛 = 3𝑘, 3𝑘 + 1 or 3𝑘 + 2. we present the proof only for the case 𝑛 = 3𝑘 and other cases follows accordingly. using the definition of 𝑃𝑛, we get 𝑃3𝑘+3 = 𝑎𝑃3𝑘+2 + 𝑃3𝑘+1 = 𝑎(𝑐𝑃3𝑘+1 + 𝑃3𝑘) + 𝑏𝑃3𝑘 + 𝑃3𝑘−1 the sequence of trifurcating fibonacci numbers 189 = 𝑎𝑐𝑃3𝑘+1 + (𝑎 + 𝑏)𝑃3𝑘 + 𝑐𝑃3𝑘−2 + 𝑃3𝑘−3. now, by definition we have 𝑃3𝑘+1 = 𝑏𝑃3𝑘 + 𝑃3𝑘−1. multiplying this by 𝑎𝑐 we get 𝑎𝑐𝑃3𝑘+1 = 𝑎𝑏𝑐𝑃3𝑘 + 𝑎𝑐𝑃3𝑘−1 = 𝑎𝑏𝑐𝑃3𝑘 + 𝑐(𝑃3𝑘 − 𝑃3𝑘−2) substituting this value in above equation we get 𝑃3𝑘+3 = (𝑎𝑏𝑐 + 𝑎 + 𝑏 + 𝑐)𝑃3𝑘 − 𝑃3𝑘−3 hence, 𝑃3𝑘+3 − (𝑎𝑏𝑐 + 𝑎 + 𝑏 + 𝑐)𝑃3𝑘 + 𝑃3𝑘−3 = 0. lemma 3.2. the generating function of the subsequence {𝑃𝑚}𝑚≥0 of {𝑃𝑛}𝑛=0 ∞ is (i) 𝑓(𝑥) = 𝑐𝑥2+𝑥5 (1−(𝑎𝑏𝑐+𝑎+𝑏+𝑐)𝑥3−𝑥6) ; when 𝑚 ≡ 2(𝑚𝑜𝑑 3) (ii) 𝑔(𝑥) = 𝑥−𝑎𝑥4 (1−(𝑎𝑏𝑐+𝑎+𝑏+𝑐)𝑥3−𝑥6) ; when 𝑚 ≡ 1(𝑚𝑜𝑑 3) (iii) ℎ(𝑥) = (𝑎𝑐+1)𝑥3 (1−(𝑎𝑏𝑐+𝑎+𝑏+𝑐)𝑥3−𝑥6) ; when 𝑚 ≡ 0(𝑚𝑜𝑑 3). proof. we present the proof only for the case 𝑚 ≡ 2(𝑚𝑜𝑑 3). the other cases can be proved accordingly. we first let 𝑓(𝑥) = ∑ 𝑃3𝑛−1𝑥 3𝑛−1∞ 𝑛=1 = 𝑃2𝑥 2 + 𝑃5𝑥 5 + 𝑃8𝑥 8 + ⋯. then (𝑎𝑏𝑐 + 𝑎 + 𝑏 + 𝑐)𝑥3 𝑓(𝑥) = (𝑎𝑏𝑐 + 𝑎 + 𝑏 + 𝑐)𝑃2𝑥 5 + (𝑎𝑏𝑐 + 𝑎 + 𝑏 + 𝑐)𝑃5𝑥 8 +(𝑎𝑏𝑐 + 𝑎 + 𝑏 + 𝑐)𝑃8𝑥 11 … using lemma 3.1, we get (1 − (𝑎𝑏𝑐 + 𝑎 + 𝑏 + 𝑐)𝑥3 − 𝑥6)𝑓(𝑥) = 𝑐𝑥2 + 𝑥5 − ∑ (𝑝3𝑘+2 − (𝑎𝑏𝑐 + 𝑎 + 𝑏 + 𝑐)𝑝3𝑘−1 − 𝑝3𝑘−4) ∞ 𝑚=2 = 𝑐𝑥 2 + 𝑥5. thus, 𝑓(𝑥) = 𝑐𝑥2+𝑥5 (1−(𝑎𝑏𝑐+𝑎+𝑏+𝑐)𝑥3−𝑥6) , as required. the following result gives the generating function for 𝑃𝑛. theorem 3.3. the generating function for the trifurcating fibonacci sequence {𝑃𝑛} is 𝐹(𝑥) = 𝑥(1+𝑐𝑥+𝑥2+𝑎𝑐𝑥2−𝑎𝑥3+𝑥4) 1−(𝑎𝑏𝑐+𝑎+𝑏+𝑐)𝑥3−𝑥6 . proof. we begin with the formal power series representation of the generating function for {𝑃𝑛}. let 𝐹(𝑥) = 𝑃0 + 𝑃1𝑥 + 𝑃2𝑥 2 + ⋯ + 𝑃𝑘𝑥 𝑘 + ⋯ = ∑ 𝑃𝑚𝑥 𝑚∞ 𝑚=0 then, 𝑐𝑥𝐹(𝑥) = 𝑐𝑃0𝑥 + 𝑐𝑃1𝑥 2 + 𝑐𝑃2𝑥 3 + ⋯ + 𝑐𝑃𝑘𝑥 𝑘+1 + ⋯ = ∑ 𝑐𝑃𝑚𝑥 𝑚+1∞ 𝑚=0 = ∑ 𝑐𝑃𝑚−1𝑥 𝑚∞ 𝑚=1 . also, 𝑥2𝐹(𝑥) = 𝑃0𝑥 2 + 𝑃1𝑥 3 + 𝑃2𝑥 2 + ⋯ + 𝑃𝑘𝑥 𝑘 + ⋯ = ∑ 𝑃𝑚𝑥 𝑚∞ 𝑚=0 since 𝑃3𝑘+2 = 𝑐𝑃3𝑘+1 + 𝑃3𝑘, we get (1 − 𝑐𝑥 − 𝑥2)𝐹(𝑥) = 𝑥 + ∑ (𝑃3𝑛 − 𝑐𝑃3𝑛−1 − 𝑃3𝑛−2)𝑥 3𝑛∞ 𝑛=1 parimalkumar a. patel, devbhadra v. shah 190 + ∑ (𝑃3𝑛+1 − 𝑐𝑃3𝑛 − 𝑃3𝑛−1)𝑥 3𝑛+1∞ 𝑛=1 since 𝑃3𝑘 = 𝑎𝑃3𝑘−1 + 𝑃3𝑘−2 and 𝑃3𝑘+1 = 𝑏𝑃3𝑘 + 𝑃3𝑘−1 we get (1 − 𝑐𝑥 − 𝑥2)𝐹(𝑥) = 𝑥 + (𝑎 − 𝑐) ∑ 𝑃3𝑛−1𝑥 3𝑛∞ 𝑛=1 + (𝑏 − 𝑐) ∑ 𝑃3𝑛𝑥 3𝑛+1∞ 𝑛=1 . for convenience, we let 𝑓(𝑥) = ∑ 𝑃3𝑛−1𝑥 3𝑛∞ 𝑛=1 and 𝑔(𝑥) = ∑ 𝑃3𝑛𝑥 3𝑛∞ 𝑛=1 using lemma 3.2 (a) and 3.2 (b) we get (1 − 𝑐𝑥 − 𝑥2)𝐹(𝑥) = 𝑥 + (𝑎 − 𝑐) (𝑎𝑐+1)𝑥3 (1−(𝑎𝑏𝑐+𝑎+𝑏+𝑐)𝑥3−𝑥6) +(𝑏 − 𝑐) 𝑐𝑥2+𝑥5 (1−(𝑎𝑏𝑐+𝑎+𝑏+𝑐)𝑥3−𝑥6) on simplification, we get the required result. we now obtain the binet-like formula for the sequence of trifurcating fibonacci numbers. theorem 3.4. the terms of the trifurcating fibonacci sequence {𝑃𝑛} are given by 𝑃𝑛 = 𝛾(𝑛)𝛼 ⌊ 𝑛 3 ⌋ −𝛿(𝑛)𝛽 ⌊ 𝑛 3 ⌋ 𝛼−𝛽 where 𝛾(𝑛) = (⌊ 𝜒(𝑛)+2 3 ⌋ 𝑐 ⌊ 𝜒(𝑛)+1 3 ⌋ 𝛼 + (−1)𝜒(𝑛)𝑎 ⌊ 4−𝜒(𝑛) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑛) 3 ⌋ + ⌊1 − 𝜒(𝑛) 3 ⌋) and 𝛿(𝑛) = (⌊ 𝜒(𝑛)+2 3 ⌋ 𝑐 ⌊ 𝜒(𝑛)+1 3 ⌋ 𝛽 + (−1)𝜒(𝑛)𝑎 ⌊ 4−𝜒(𝑛) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑛) 3 ⌋ + ⌊1 − 𝜒(𝑛) 3 ⌋) with 𝛼 = 𝑢+√𝑢2+4 2 , 𝛽 = 𝑢−√𝑢2+4 2 , 𝑢 = 𝑎 + 𝑏 + 𝑐 + 𝑎𝑏𝑐 and 𝜒(𝑛) = { 0 𝑖𝑓 𝑛 ≡ 0(𝑚𝑜𝑑 3) 1 𝑖𝑓 𝑛 ≡ 1(𝑚𝑜𝑑 3) 2 𝑖𝑓 𝑛 ≡ 2(𝑚𝑜𝑑 3) . proof. from the generating function of {𝑃𝑛}, we have 𝐹(𝑥) = − 𝑥(1+𝑐𝑥+(1+𝑎𝑐)𝑥2−𝑎𝑥3+𝑥4) (𝑥3+𝛼)(𝑥3+𝛽) this can be rewritten as 𝐹(𝑥) = − 1 𝛼−𝛽 [ (1+𝑎𝑐)𝛼−(𝑎𝛼+1)𝑥+(𝛼−𝑐)𝑥2 (𝑥3+𝛼) − (1+𝑎𝑐)𝛽−(𝑎𝛽+1)𝑥+(𝛽−𝑐)𝑥2 (𝑥3+𝛽) ] using mclaurin series expansion, we get 𝐹(𝑥) = − 1 𝛼−𝛽 [ ∑ (−1)𝑛(1+𝑎𝑐)𝛼 𝛼𝑛+1 ∞ 𝑛=0 𝑥 3𝑛 − ∑ (−1)𝑛(𝑎𝛼+1) 𝛼𝑛+1 ∞ 𝑛=0 𝑥 3𝑛+1 + ∑ (−1)𝑛(𝛼−𝑐) 𝛼𝑛+1 ∞ 𝑛=0 𝑥 3𝑛+2 − ∑ (−1)𝑛(1+𝑎𝑐)𝛽 𝛽𝑛+1 ∞ 𝑛=0 𝑥 3𝑛 + ∑ (−1)𝑛(𝑎𝛽+1) 𝛽𝑛+1 ∞ 𝑛=0 𝑥 3𝑛+1 − ∑ (−1)𝑛(𝛽−𝑐) 𝛽𝑛+1 ∞ 𝑛=0 𝑥 3𝑛+2 ] the sequence of trifurcating fibonacci numbers 191 = − 1 𝛼−𝛽 [ ∑ (−1) 𝑛(1 + 𝑎𝑐) ( 𝛽𝑛−𝛼𝑛 (𝛼𝛽)𝑛 )∞𝑛=0 𝑥 3𝑛 − ∑ (−1)𝑛 ( (𝑎𝛼+1)𝛽𝑛+1−(𝑎𝛽+1)𝛼𝑛+1 (𝛼𝛽)𝑛+1 )∞𝑛=0 𝑥 3𝑛+1 + ∑ (−1)𝑛 ( (𝛼−𝑐)𝛽𝑛+1−(𝛽−𝑐)𝛼𝑛+1 (𝛼𝛽)𝑛+1 )∞𝑛=0 𝑥 3𝑛+2 ] now, if 𝛼, 𝛽 are the roots of 1 − (𝑎 + 𝑏 + 𝑐 + 𝑎𝑏𝑐)𝑥 − 𝑥2 = 0 then 𝛼 = 𝑢+√𝑢2+4 2 , 𝛽 = 𝑢−√𝑢2+4 2 . if we let 𝑢 = 𝑎 + 𝑏 + 𝑐 + 𝑎𝑏𝑐, then it is easy to observe that 𝛼𝛽 = −1, 𝛼 + 𝛽 = 𝑢 and 𝛼 − 𝛽 = √𝑢2 + 4 . then 𝐹(𝑥) = [ ∑ (1 + 𝑎𝑐) ( 𝛼𝑛−𝛽𝑛 𝛼−𝛽 )∞𝑛=0 𝑥 3𝑛 − ∑ ( (𝑎𝛽+1)𝛼𝑛+1−(𝑎𝛼+1)𝛽𝑛+1 𝛼−𝛽 )∞𝑛=0 𝑥 3𝑛+1 + ∑ ( (𝛽−𝑐)𝛼𝑛+1−(𝛼−𝑐)𝛽𝑛+1 𝛼−𝛽 )∞𝑛=0 𝑥 3𝑛+2 ] thus, 𝐹(𝑥) = ∑ 1 𝛼−𝛽 ( ( ⌊1 − 𝜒(𝑛) 3 ⌋ + (−1)𝜒(𝑛)𝑎 ⌊ 4−𝜒(𝑛) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑛) 3 ⌋ + ⌊ 𝜒(𝑛)+2 3 ⌋ 𝑐 ⌊ 𝜒(𝑛)+1 3 ⌋ 𝛼 ) 𝛼⌊ 𝑛 3 ⌋ − ( ⌊1 − 𝜒(𝑛) 3 ⌋ + (−1)𝜒(𝑛)𝑎 ⌊ 4−𝜒(𝑛) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑛) 3 ⌋ + ⌊ 𝜒(𝑛)+2 3 ⌋ 𝑐 ⌊ 𝜒(𝑛)+1 3 ⌋ 𝛽 ) 𝛽⌊ 𝑛 3 ⌋ ) ∞ 𝑛=0 𝑥 𝑛 for convenience if we write 𝜒(𝑛) = { 0 𝑖𝑓 𝑛 ≡ 0(𝑚𝑜𝑑 3) 1 𝑖𝑓 𝑛 ≡ 1(𝑚𝑜𝑑 3) 2 𝑖𝑓 𝑛 ≡ 2(𝑚𝑜𝑑 3) and 𝛾(𝑛) = (⌊ 𝜒(𝑛)+2 3 ⌋ 𝑐 ⌊ 𝜒(𝑛)+1 3 ⌋ 𝛼 + (−1)𝜒(𝑛)𝑎 ⌊ 4−𝜒(𝑛) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑛) 3 ⌋ + ⌊1 − 𝜒(𝑛) 3 ⌋), 𝛿(𝑛) = (⌊ 𝜒(𝑛)+2 3 ⌋ 𝑐 ⌊ 𝜒(𝑛)+1 3 ⌋ 𝛽 + (−1)𝜒(𝑛)𝑎 ⌊ 4−𝜒(𝑛) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑛) 3 ⌋ + ⌊1 − 𝜒(𝑛) 3 ⌋) then 𝐹(𝑥) can be written as 𝐹(𝑥) = ∑ 𝛾(𝑛)𝛼 ⌊ 𝑛 3 ⌋ −𝛿(𝑛)𝛽 ⌊ 𝑛 3 ⌋ 𝛼−𝛽 ∞ 𝑛=0 𝑥 𝑛 this gives 𝑃𝑛 = 𝛾(𝑛)𝛼 ⌊ 𝑛 3 ⌋ −𝛿(𝑛)𝛽 ⌊ 𝑛 3 ⌋ 𝛼−𝛽 , as desired. the following results are the easy consequence from this theorem. corollary 3.5. (i) 𝑃3𝑛 = (1 + 𝑎𝑐) ( 𝛼𝑛−𝛽𝑛 𝛼−𝛽 ) parimalkumar a. patel, devbhadra v. shah 192 (ii) 𝑃3𝑛+1 = ( (𝑎𝛽+1)𝛼𝑛+1−(𝑎𝛼+1)𝛽𝑛+1 𝛼−𝛽 ) (iii) 𝑃3𝑛+2 = ( (𝛽−𝑐)𝛼𝑛+1−(𝛼−𝑐)𝛽𝑛+1 𝛼−𝛽 ). 4. some more identities relating trifurcating fibonacci numbers: in this section, we use the above binet-like formula to derive some interesting properties for the terms of trifurcating fibonacci sequence. if we let 𝑤 = 𝛾(𝑛)𝛿(𝑛), then we observe that 𝑤 = ⌊1 − 𝜒(𝑛) 3 ⌋ 2 + 2(−1)𝜒(𝑛) ⌊1 − 𝜒(𝑛) 3 ⌋ 𝑎 ⌊ 4−𝜒(𝑛) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑛) 3 ⌋ + ⌊1 − 𝜒(𝑛) 3 ⌋ ⌊ 𝜒(𝑛)+2 3 ⌋ 𝑐 ⌊ 𝜒(𝑛)+1 3 ⌋ (𝛼 + 𝛽) + 𝑎 2⌊ 4−𝜒(𝑛) 3 ⌋ 𝑐 2⌊1− 𝜒(𝑛) 3 ⌋ +(−1)𝜒(𝑛) ⌊ 𝜒(𝑛)+2 3 ⌋ 𝑎 ⌊ 4−𝜒(𝑛) 3 ⌋ 𝑐 ⌊ 𝜒(𝑛)+1 3 ⌋+⌊1− 𝜒(𝑛) 3 ⌋ (𝛼 + 𝛽) + ⌊ 𝜒(𝑛)+2 3 ⌋ 2 𝑐 2⌊ 𝜒(𝑛)+1 3 ⌋ 𝛼𝛽 this on simplification gives the value of 𝑤 = 𝛾(𝑛)𝛿(𝑛) as 𝑤 = { (1 + 𝑎𝑐)2 ; if 𝑛 ≡ 0(𝑚𝑜𝑑 3) (1 + 𝑎𝑐)(1 + 𝑏) ; if 𝑛 ≡ 1(𝑚𝑜𝑑 3) (1 + 𝑎𝑐)(1 + 𝑏𝑐); if 𝑛 ≡ 2(𝑚𝑜𝑑 3) this can be further written as 𝑤 = { 𝑃3 2 ; if 𝑛 ≡ 0(𝑚𝑜𝑑 3) (1 + 𝑏)𝑃3 ; if 𝑛 ≡ 1(𝑚𝑜𝑑 3) (1 + 𝑏𝑐)𝑃3; if 𝑛 ≡ 2(𝑚𝑜𝑑 3) we first obtain an identity for the terms of {𝑃𝑛}, which is analogous to that of catalan’s identity for fibonacci numbers. theorem 4.1. for any two nonnegative integers 𝑘 and 𝑟 (≤ 𝑘 3 ), we have 𝑃𝑘−3𝑟𝑃𝑘+3𝑟 − 𝑃𝑘 2 = (−1)𝑙−𝑟+1𝑤 ( 𝑃3𝑟 𝑃3 ) 2 . proof. if 𝑘 ≡ 0(𝑚𝑜𝑑 3) then taking 𝑘 = 3𝑙 and using corollary 3.5, we get 𝑃3𝑙−3𝑟𝑃3𝑙+3𝑟 − 𝑃3𝑙 2 = (1 + 𝑎𝑐) ( 𝛼𝑙−𝑟−𝛽𝑙−𝑟 𝛼−𝛽 ) (1 + 𝑎𝑐) ( 𝛼𝑙+𝑟−𝛽𝑙+𝑟 𝛼−𝛽 ) − (1 + 𝑎𝑐)2 ( 𝛼𝑙−𝛽𝑙 𝛼−𝛽 ) 2 = (1+𝑎𝑐)2 (𝛼−𝛽)2 {(𝛼𝑙−𝑟 − 𝛽𝑙−𝑟)(𝛼𝑙+𝑟 − 𝛽𝑙+𝑟) − (𝛼𝑙 − 𝛽𝑙)2} = (1+𝑎𝑐)2 (𝛼−𝛽)2 {𝛼2𝑙 − (−1)𝑙 ( 𝛽 𝛼 ) 𝑟 − (−1)𝑙 ( 𝛼 𝛽 ) 𝑟 + 𝛽2𝑙 − (𝛼2𝑙 − 2(−1)𝑙 + 𝛽2𝑙)} the sequence of trifurcating fibonacci numbers 193 = (1+𝑎𝑐)2 (𝛼−𝛽)2 (−1)𝑙−𝑟+1{𝛼2𝑟 − 2(−1)𝑟 + 𝛽2𝑟} = (1+𝑎𝑐)2 (𝛼−𝛽)2 (−1)𝑙−𝑟+1(𝛼𝑟 − 𝛽𝑟)2 = (−1)𝑙−𝑟+1 ( (1+𝑎𝑐)(𝛼𝑟−𝛽𝑟) (𝛼−𝛽) ) 2 thus, 𝑃3𝑙−3𝑟𝑃3𝑙+3𝑟 − 𝑃3𝑙 2 = (−1)𝑙−𝑟+1𝑃3𝑟 2 next, if we let 𝑘 ≡ 1(𝑚𝑜𝑑 3) then by considering 𝑘 = 3𝑙 + 1, we get 𝑃3𝑙−3𝑟+1𝑃3𝑙+3𝑟+1 − 𝑃3𝑙+1 2 = ( (𝑎𝛽+1)𝛼𝑙−𝑟+1−(𝑎𝛼+1)𝛽𝑙−𝑟+1 𝛼−𝛽 ) ( (𝑎𝛽+1)𝛼𝑙+𝑟+1−(𝑎𝛼+1)𝛽𝑙+𝑟+1 𝛼−𝛽 ) − ( (𝑎𝛽+1)𝛼𝑙+1−(𝑎𝛼+1)𝛽𝑙+1 𝛼−𝛽 ) 2 = 1 (𝛼−𝛽)2 [ 𝑎2(𝛼𝑙−𝑟 − 𝛽𝑙−𝑟)(𝛼𝑙+𝑟 − 𝛽𝑙+𝑟) −𝑎(𝛼𝑙−𝑟 − 𝛽𝑙−𝑟)(𝛼𝑙+𝑟+1 − 𝛽𝑙+𝑟+1) − 𝑎(𝛼𝑙−𝑟+1 − 𝛽𝑙−𝑟+1)(𝛼𝑙+𝑟 − 𝛽𝑙+𝑟) +(𝛼𝑙−𝑟+1 − 𝛽𝑙−𝑟+1)(𝛼𝑙+𝑟+1 − 𝛽𝑙+𝑟+1) −{𝑎2(𝛼𝑙 − 𝛽𝑙)2 − 2𝑎(𝛼𝑙 − 𝛽𝑙)(𝛼𝑙+1 − 𝛽𝑙+1) + (𝛼𝑙+1 − 𝛽𝑙+1)2}] = 1 (𝛼−𝛽)2 [ 𝑎2(−1)𝑙−𝑟+1[𝛽2𝑟 + 𝛼2𝑟 + 2(−1)𝑟] −(−1)𝑙−𝑟+1[𝛽2𝑟 + 𝛼2𝑟 + 2(−1)𝑟] −𝑎(−1)𝑙+1 [( 𝛽 𝛼 ) 𝑟 (𝛼 + 𝛽) + ( 𝛼 𝛽 ) 𝑟 (𝛼 + 𝛽) + 2(𝛼 + 𝛽)] ] = (−1)𝑙−𝑟+1 (𝛼−𝛽)2 {𝑎2(𝛼𝑟 − 𝛽𝑟)2 − 𝑎𝑢[𝛽2𝑟 + 𝛼2𝑟 + 2(−1)𝑟] − (𝛼𝑟 − 𝛽𝑟)2} = (−1)𝑙−𝑟+1 (𝛼−𝛽)2 (𝛼𝑟 − 𝛽𝑟)2{𝑎2 − 𝑎𝑢 − 1} = (−1)𝑙−𝑟+1 (1+𝑎𝑐)2 {𝑎2 − 𝑎𝑢 − 1}𝑃3𝑟 2 thus, 𝑃3𝑙−3𝑟+1𝑃3𝑙+3𝑟+1 − 𝑃3𝑙+1 2 = (−1)𝑙−𝑟+1𝑤𝑃3𝑟 2 𝑃3 −2. finally, if 𝑘 ≡ 2(𝑚𝑜𝑑 3) then by considering 𝑘 = 3𝑙 + 2, we get 𝑃3𝑙−3𝑟+2𝑃3𝑙+3𝑟+2 − 𝑃3𝑙+2 2 = ( (𝛽−𝑐)𝛼𝑙−𝑟+1−(𝛼−𝑐)𝛽𝑙−𝑟+1 𝛼−𝛽 ) ( (𝛽−𝑐)𝛼𝑙+𝑟+1−(𝛼−𝑐)𝛽𝑙+𝑟+1 𝛼−𝛽 ) − ( (𝛽−𝑐)𝛼𝑙+1−(𝛼−𝑐)𝛽𝑙+1 𝛼−𝛽 ) 2 . = 1 (𝛼−𝛽)2 [ (𝛼𝑙−𝑟 − 𝛽𝑙−𝑟)(𝛼𝑙+𝑟 − 𝛽𝑙+𝑟) + 𝑐(𝛼𝑙−𝑟 − 𝛽𝑙−𝑟)(𝛼𝑙+𝑟+1 − 𝛽𝑙+𝑟+1) +𝑐(𝛼𝑙−𝑟+1 − 𝛽𝑙−𝑟+1)(𝛼𝑙+𝑟 − 𝛽𝑙+𝑟) +𝑐2(𝛼𝑙−𝑟+1 − 𝛽𝑙−𝑟+1)(𝛼𝑙+𝑟+1 − 𝛽𝑙+𝑟+1) −(𝛼𝑙−𝑟 − 𝛽𝑙−𝑟)2 − (𝛼𝑙 − 𝛽𝑙)(𝛼𝑙+1 − 𝛽𝑙+1) − 𝑐2(𝛼𝑙+1 − 𝛽𝑙+1)2 ] = 1 (𝛼−𝛽)2 [ (−1)𝑙−𝑟+1[𝛽2𝑟 + 𝛼2𝑟 + 2(−1)𝑟] −𝑐2(−1)𝑙−𝑟+1[𝛽2𝑟 + 𝛼2𝑟 + 2(−1)𝑟] −𝑐(−1)𝑙+1 [( 𝛽 𝛼 ) 𝑟 (𝛼 + 𝛽) + ( 𝛼 𝛽 ) 𝑟 (𝛼 + 𝛽) − 2(𝛼 + 𝛽)] ] = (−1)𝑙−𝑟+1 (𝛼−𝛽)2 {(𝛼𝑟 − 𝛽𝑟)2 − 𝑐𝑢[𝛽2𝑟 + 𝛼2𝑟 + 2(−1)𝑟] − 𝑐2(𝛼𝑟 − 𝛽𝑟)2} parimalkumar a. patel, devbhadra v. shah 194 = (−1)𝑙−𝑟+1 (𝛼−𝛽)2 (𝛼𝑟 − 𝛽𝑟)2{1 + 𝑐𝑢 − 𝑐2} = (−1)𝑙−𝑟+1 (1+𝑎𝑐)2 {1 + 𝑐𝑢 − 𝑐2}𝑃3𝑟 2 thus, 𝑃3𝑙−3𝑟+2𝑃3𝑙+3𝑟+2 − 𝑃3𝑙+2 2 = (−1)𝑙−𝑟+1𝑤𝑃3𝑟 2 𝑃3 −2 hence, in general we write 𝑃𝑘−3𝑟𝑃𝑘+3𝑟 − 𝑃𝑘 2 = (−1)𝑙−𝑟+1𝑤𝑃3𝑟 2 𝑃3 −2. the following identity is analogous to the cassini’s identity for fibonacci numbers which follows easily from the above theorem. corollary 4.2. 𝑃𝑘−3𝑃𝑘+3 − 𝑃𝑘 2 = (−1)𝑛𝑤 for any integer 𝑘 ≥ 3. the following identity is similar to d’ocagne’s identity of fibonacci numbers. theorem 4.3. 𝑃𝑚𝑃𝑛+3 − 𝑃𝑚+3𝑃𝑛 = (−1) 𝑛𝑃𝑚−𝑛 ( 𝑤 𝑃3 ) where 𝑚, 𝑛 are nonnegative integers such that 𝑚 ≥ 𝑛 and 𝑚 ≡ 𝑛(𝑚𝑜𝑑 3). proof. since 𝑚 ≡ 𝑛(𝑚𝑜𝑑 3), we first let both 𝑚, 𝑛 to be of the form 3𝑗, 3𝑘 respectively, for positive integers 𝑗 and 𝑘 ≤ 𝑗. then 𝑃3𝑗𝑃3𝑘+3 − 𝑃3𝑗+3𝑃3𝑘 = (1 + 𝑎𝑐) ( 𝛼𝑗−𝛽𝑗 𝛼−𝛽 ) (1 + 𝑎𝑐) ( 𝛼𝑘+1−𝛽𝑘+1 𝛼−𝛽 ) −(1 + 𝑎𝑐) ( 𝛼𝑗+1−𝛽𝑗+1 𝛼−𝛽 ) (1 + 𝑎𝑐) ( 𝛼𝑘−𝛽𝑘 𝛼−𝛽 ) = (1+𝑎𝑐)2 (𝛼−𝛽)2 {𝛼𝑗𝛽𝑘(𝛼 − 𝛽) − 𝛼𝑘𝛽𝑗(𝛼 − 𝛽)} = (1 + 𝑎𝑐)2(−1)𝑘 ( 𝛼𝑗−𝑘−𝛽𝑗−𝑘 𝛼−𝛽 ) = (−1) ⌊ 𝑛−1 3 ⌋ 𝑤𝑃𝑚−𝑛𝑃3 −1 if 𝑚, 𝑛 are of the form 3𝑗 + 1 and 3𝑘 + 1 respectively, then for positive integers 𝑗 and 𝑘 ≤ 𝑗, we have 𝑃3𝑗+1𝑃3𝑘+3+1 − 𝑃3𝑗+3+1𝑃3𝑘+1 = [ ( (𝑎𝛽+1)𝛼𝑗+1−(𝑎𝛼+1)𝛽𝑗+1 𝛼−𝛽 ) ( (𝑎𝛽+1)𝛼𝑘+2−(𝑎𝛼+1)𝛽𝑘+2 𝛼−𝛽 ) − ( (𝑎𝛽+1)𝛼𝑗+2−(𝑎𝛼+1)𝛽𝑗+2 𝛼−𝛽 ) ( (𝑎𝛽+1)𝛼𝑘+1−(𝑎𝛼+1)𝛽𝑘+1 𝛼−𝛽 ) ] = 1 (𝛼−𝛽)2 [ 𝑎 2 (𝛼𝑗𝛽𝑘(𝛼 − 𝛽) − 𝛼𝑘𝛽𝑗(𝛼 − 𝛽)) −𝑎 (𝛼𝑗𝛽𝑘(𝛼2 − 𝛽2) − 𝛼𝑘𝛽𝑗(𝛼2 − 𝛽2)) −(𝛼𝑗𝛽𝑘(𝛼 − 𝛽) − 𝛼𝑘𝛽𝑗(𝛼 − 𝛽) ] = 1 (𝛼−𝛽) [(𝑎2 − 𝑎𝑢 − 1)𝛼𝑘𝛽𝑘(𝛼𝑗−𝑘 − 𝛽𝑗−𝑘)] = (−1)𝑘(𝑎2 − 𝑎𝑢 − 1) ( 𝛼𝑗−𝑘−𝛽𝑗−𝑘 𝛼−𝛽 ) = (−1) ⌊ 𝑛−1 3 ⌋ 𝑤𝑃𝑚−𝑛𝑃3 −1 the sequence of trifurcating fibonacci numbers 195 finally, if 𝑚, 𝑛 are of the form 3𝑗 + 2 and 3𝑘 + 2 respectively, then for positive integers 𝑗 and 𝑘 ≤ 𝑗, we have 𝑃3𝑗+2𝑃3𝑘+3+2 − 𝑃3𝑗+3+2𝑃3𝑘+2 = ( (𝛽−𝑐)𝛼𝑗+1−(𝛼−𝑐)𝛽𝑗+1 𝛼−𝛽 ) ( (𝛽−𝑐)𝛼𝑘+2−(𝛼−𝑐)𝛽𝑘+2 𝛼−𝛽 ) − ( (𝛽−𝑐)𝛼𝑗+2−(𝛼−𝑐)𝛽𝑗+2 𝛼−𝛽 ) ( (𝛽−𝑐)𝛼𝑘+1−(𝛼−𝑐)𝛽𝑘+1 𝛼−𝛽 ) = 1 (𝛼−𝛽)2 [ 𝛼𝑗𝛽𝑘(𝛼 − 𝛽) − 𝛼𝑘𝛽𝑗(𝛼 − 𝛽) − 𝑐2(𝛼𝑗𝛽𝑘(𝛼 − 𝛽) − 𝛼𝑘𝛽𝑗(𝛼 − 𝛽) +𝑐 (𝛼𝑗𝛽𝑘(𝛼2 − 𝛽2) − 𝛼𝑘𝛽𝑗(𝛼2 − 𝛽2)) ] = 1 (𝛼−𝛽) [(1 + 𝑐𝑢 − 𝑐2)𝛼𝑘𝛽𝑘(𝛼𝑗−𝑘 − 𝛽𝑗−𝑘)] = (−1)𝑘(𝑎2 − 𝑎𝑢 − 1) ( 𝛼𝑗−𝑘−𝛽𝑗−𝑘 𝛼−𝛽 ) = (−1) ⌊ 𝑛−1 3 ⌋ 𝑤𝑃𝑚−𝑛𝑃3 −1 combining all the above cases, we finally get 𝑃𝑚𝑃𝑛+3 − 𝑃𝑚+3𝑃𝑛 = (−1) ⌊ 𝑛−1 3 ⌋ 𝑤𝑃𝑚−𝑛𝑃3 −1. we use above binet-like formula to prove the following identity which combines four consecutive 𝑃𝑛’s. theorem 4.4. 𝑎 ⌊1− 𝜒(𝑘+2) 3 ⌋ 𝑏 ⌊1− 𝜒(𝑘+1) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑘) 3 ⌋ 𝑃𝑘+1 2 + 𝑎 ⌊1− 𝜒(𝑘) 3 ⌋ 𝑏 ⌊1− 𝜒(𝑘+2) 3 ⌋ 𝑐 ⌊1− 𝜒(𝑘+1) 3 ⌋ 𝑃𝑘+2 2 = 𝑃𝑘+2𝑃𝑘+3 − 𝑃𝑘𝑃𝑘+1. proof. we prove the result only for the case 𝑘 = 3𝑛 and the remaining cases 𝑘 = 3𝑛 + 1 and 𝑘 = 3𝑛 + 2 can be handled accordingly. now, 𝑐𝑃3𝑛+1 2 + 𝑎𝑃3𝑛+2 2 = 𝑐 {( (𝑎𝛽+1)𝛼𝑛+1−(𝑎𝛼+1)𝛽𝑛+1 𝛼−𝛽 )} 2 + 𝑎 {( (𝛽−𝑐)𝛼𝑛+1−(𝛼−𝑐)𝛽𝑛+1 𝛼−𝛽 )} 2 . = 1 (𝛼−𝛽)2 [ 𝑎2𝑐(𝛼2𝑛 − 2(−1)𝑛 + 𝛽2𝑛) −2𝑎𝑐(𝛼2𝑛+1 − (−1)𝑛𝛼 − (−1)𝑛𝛽 + 𝛽2𝑛+1) +𝑐(𝛼2𝑛+2 − 2(−1)𝑛+1 + 𝛽2𝑛+2) +2𝑎𝑐(𝛼2𝑛+1 − (−1)𝑛𝛽 + (−1)𝑛𝛼 + 𝛽2𝑛+1) +𝑐2(𝛼2𝑛+2 − 2(−1)𝑛+1 + 𝛽2𝑛+2) + 𝑎(𝛼2𝑛 − 2(−1)𝑛 + 𝛽2𝑛)] = (1+𝑎𝑐) (𝛼−𝛽)2 {𝑎(𝛼𝑛 − 𝛽𝑛)2 + 𝑐(𝛼𝑛+1 − 𝛽𝑛+1)2} also, 𝑃3𝑛+2𝑃3𝑛+3 − 𝑃3𝑛𝑃3𝑛+1 = ( (𝛽−𝑐)𝛼𝑛+1−(𝛼−𝑐)𝛽𝑛+1 𝛼−𝛽 ) (1 + 𝑎𝑐) ( 𝛼𝑛+1−𝛽𝑛+1 𝛼−𝛽 ) −(1 + 𝑎𝑐) ( 𝛼𝑛−𝛽𝑛 𝛼−𝛽 ) ( (𝑎𝛽+1)𝛼𝑛+1−(𝑎𝛼+1)𝛽𝑛+1 𝛼−𝛽 ) parimalkumar a. patel, devbhadra v. shah 196 = (1+𝑎𝑐) (𝛼−𝛽)2 [ (𝛼𝑛 − 𝛽𝑛)(𝛼𝑛+1 − 𝛽𝑛+1) + 𝑐(𝛼𝑛+1 − 𝛽𝑛+1)2 +𝑎(𝛼𝑛 − 𝛽𝑛)2 − (𝛼𝑛 − 𝛽𝑛)(𝛼𝑛+1 − 𝛽𝑛+1) ] = (1+𝑎𝑐) (𝛼−𝛽)2 {𝑎(𝛼𝑛 − 𝛽𝑛)2 + 𝑐(𝛼𝑛+1 − 𝛽𝑛+1)2} this proves the required result. 4. conclusions in this paper, we considered the sequence of ‘trifurcating fibonacci numbers’ and obtained its binet-like formula. we also obtained the analogous of cassini’s identity, catalan’s identity, d’ocagne’s identity and some fundamental identities for the terms of this sequence. references [1] arvadia m. p., shah d. v. left k-fibonacci sequence and related identities. journal club for applied sciences. 2 (1), 20 – 26, july 2015. [2] diwan d. m., shah d. v. explicit and recursive formulae for the class of generalized fibonacci sequence. international journal of advanced research in engineering, science and management. 1 (10), 1 – 6, july 2015. [3] diwan d. m., shah d. v. extended binet’s formula for the class of generalized fibonacci sequences. vnsgu journal of science and technology. 4 (1), 205 – 210, 2015. [4] edson m., yayenie o. a new generalization of fibonacci sequence and extended binet’s formula. integers, electron. j. comb. number theor., 9, 639 – 654, 2009. [5] gupta v. k., panwar y. k., o. sikhwal. generalized fibonacci sequences. theoretical mathematics & applications. 2 (2), 115 – 124, 2012. [6] koshy thomas. fibonacci and lucas numbers with applications. john wiley and sons, inc., n. york., 2001. [7] patel vandana r., shah devbhadra v. generalized fibonacci sequence and its properties. int. journal of physics and mathematical sciences. 4 (2), 118 – 124, 2014. [8] singh b., sikhwal o., bhatnagar s. fibonacci-like sequence and its properties. int. j. contemp-math. sciences. 5 (18), 857 – 868, 2010. [9] verma, ankur bala. on properties of generalized bi-variate bi-periodic fibonacci polynomials. international journal of advanced science and technology. 29 (3), 8065 – 8072, 2020. [10] yayenie o. a note on generalized fibonacci sequence. applied mathematics and computation. 217, 5603 – 5611, 2011. ratio mathematica volume 44, 2022 motor imagery classification using rough neural network j. anila maily 1 dr. c. velayutham2 dr. m. mohamed sathik3 abstract brain computer interface is a system which provides a communication channel between the user and a computer without using the normal neuromuscular pathways. with bci a user will be able to communicate with the mind. in a bci system the brain activities are measured using eeg acquisition system. the acquired brain signals are analyzed and classified to identify the user’s intention. motor imagery bci works by making the user imagine their body parts without actually moving it. prominent features are extracted from the acquired brain signals and the extracted features are classified to find the motor imagery performed by the user. this study uses datasets are provided by the dr. cichocki's lab (lab for advanced brain signal processing). we propose the rough neural network (rnn) for motor imagery classification. the experimental results show that rnn classifier gives higher accuracy than backpropagation classifier. keywords: bci, eeg, motor imagery, rnn 2010 ams subject classification: 06b10, 06d994. 1 research scholar, (part-time internal) (register number: 12336), dept of computer science, sadakathullah appa college, rahmath nagar, tirunelveli. affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india associate professor, st. mary‟s college (autonomous), thoothukudi. email: anilamaily@gmail.com 2 head and associate professor, department of computer science, aditanar college of arts and science tiruchendur affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamilnadu, india. email:cvsir22@gmail.com 3 principal, sadakathullah appa college, rahmath nagar, tirunelveli -627011, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india. email: mmadsadiq@gmail.com 4received on june21st, 2022. accepted on aug 10, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.902. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by license agreement 155 mailto:cvsir22@gmail.com j. anila maily, dr. c. velayutham and dr. m. mohamed sathik 1. introduction brain computer interfacing (bci) is to provide a communication channel between man, and machine, bci users brain signals are captured, analyzed into commands for communication and control (birbaumer et. al., 2003, wolpow et. al.,2002) [1, 2]. bci is a boon for people with severe motor disorders like amyotrophic lateral sclerosis (als), brain stem stroke, cerebral palsy and spinal cord injury. a user’s intent, as reflected by brain signals, is translated by the bci system into a desired output: computer-based communication or control of an external device (joseph et. al., 2009) [3]. motor imagery works by making the people imagine the moving their body parts without actually doing it. the brain signals during the imagination are recorded and analyzed to identify the intent of the person. in bci systems the brain signals are recorded from multiple channels to preserve high spatial accuracy. the rough set theory (rst) enables the discovery of data dependencies and the reduction of the number of attributes contained in the data set using the data alone requiring no additional information. given a dataset with discretized attribute values, it is possible to find a subset (of the original attributes using rough set theory that are the most informative. all other attributes can be removed from the dataset with minimal information loss. (velayutham, c., et. al., 2011) [4]. 2. classification by bpn and rnn neural networks are the classifier used mostly in bci. neural network integrated with rough set theory (rst) is known as rough set neural network (rnn). the rough set theory and neural network are the two important methods of intelligent information processing. optimizing the net for correct responses to the training input data set is done by backpropogation. more than one hidden layer may be beneficial for some applications, but one hidden layer is sufficient if enough hidden neurons are used ((r. jensen, and q. shen, 2008, c. velayutham, and k. thangavel, 2011, simon haykin, 2005) [5 7]. rough set and neural networks can solve complex and high-dimensional problems, which are called rough neural networks (rnns) (weidong zhao, and guohua chen. 2002) [8]. this paper investigates the motor imagery classification accuracy using backpropagation network classifier and propose rough neural network classifier. the rest of the paper is organized as follows. section ii briefly describes classification technique used with eeg. section iii presents the methodology adopted, section iv is about results and discussion. section v concludes the paper. 156 motor imagery classification using rough neural network 3. literature review hong k-s et al., (2018), [9] presented a brain-computer interface (bci) framework for hybrid functional near-infrared spectroscopy (fnirs) and electroencephalography (eeg) for locked-in syndrome (lis) patients. for classification, linear discriminant analysis has been most widely used. siavash sakhavi et al., (2018) [10] proposed a classification framework for motor imagery (mi) data by introducing a new temporal representation of the data and utilized a convolutional neural network (cnn) architecture for classification. the framework classified bci competition iv-2a 4-class mi data set by 7% increase in average subject accuracy. vladimir a. maksimenko et al., (2018) [11] applied artificial neural network (ann) for recognition and classification of eeg patterns associated with motor imagery in untrained subjects. ann optimization is proposed by pre-processing the eeg signals with a low pass filter and it is shown filtration of high frequency spectral components enhances the classification performance up to 90±5%. han, c., kim, y., kim, d.y. et al., (2019) [12] investigated the possibility of using an eeg-based endogenous bci paradigm for online binary communication by a patient in complete locked in syndrome (clis). an online classification accuracy of 87.5% was achieved when riemannian geometry-based classification was applied to real-time eeg data recorded while the patient was performing one of two mental-imagery tasks for 5 s. shiu kumar et al., (2019) [13] introduced a novel scheme for classifying motor imagery (mi) tasks using electroencephalography (eeg) signal that can be implemented in real-time having high classification accuracy between different mi tasks. they proposed a new predictor, optical, that uses a combination of common spatial pattern (csp) and long short-term memory (lstm) network for obtaining improved mi eeg signal classification. 4. methodology the methodology followed for motor imagery classification is shown in figure 1. data sets of motor imagery eeg the datasets provided by the dr. cichocki's lab (lab. for advanced brain signal processing), is used for this study. data files and format all data sets are stored in the matlab format (*.mat). the file name consists of subject id, channel number, imagery tasks and session number. for example, 'subc_6chan_3lrf_s1': subject c, 6 channels, 3-class imagery tasks of left hand, right hand and feet and session 1. each file contains one session which consists of several runs separated by short breaks. some subjects have many sessions conducted on different days. the detailed information of the dataset is given in table 1. 157 j. anila maily, dr. c. velayutham and dr. m. mohamed sathik figure 1: methodology feature extraction and feature selection prominent features are extracted from the signals using statistical parameter mean correlation. the extracted features are preprocessed by applying minmax normalization and discretized with k-means algorithm. the preprocessed datasets are further reduced by using supervised feature selection algorithm quick reduct based on rough set theory. classification by bpn and rnn the reduced feature set selected from the feature selection algorithms are assigned to the input neurons. the number of hidden neurons is greater than or equal to the number of input neurons, and there is only one output neuron. initial weights are assigned randomly. the output from each hidden neuron is calculated using the sigmoid function 𝑆1 = 1 1+𝑒−𝜆𝑥 where  = 1 and 𝑥 = ∑ 𝑤𝑖ℎ𝑖 𝑘𝑖 (1) where wihis the weight assigned between input and hidden layer and k is the input value. the output from the output layer is calculated using the sigmoid function. s2 = 1 1+𝑒−𝜆𝑥 , where  = 1 and 𝑥 = ∑ 𝑤ℎ𝑜𝑖 𝑆𝑖 (2) where who is the weight assigned between hidden and output layer and si is the output value from hidden neurons. s2 is subtracted from the desired output. using this error (e) value, the updating of weight is performed as: 𝛿 = 𝑒𝑆2(1 − 𝑆2) (3) the weights assigned between the input and the hidden layer and the hidden and output layer are updated as: classification using rough neural network dr. cichocki's lab’s motor imagery eeg data feature extraction by mean correlation preprocessing by normalization and discretization feature selection by quick reduct algorithm 158 motor imagery classification using rough neural network s l. n o . d a ta se t s u b je c t c la ss c h a n n e l d u ra tio n (se c ) t ria l n u m b e r 1 0 x 1 0 c v (a c c .± std .) s a m p le ra te d e v ic e 1 suba_5chan_3lrf a lh/rh/f 5 4s 270 0.92±0.004 256hz g. tec 2 subb_5chan_3lrf b lh/rh/f 5 4s 174 0.86±0.01 250hz neuroscan 3 subb_6chan_3lrf 6 150 0.80±0.03 4 subc_5chan_3lrf c lh/rh/f 5 4s 180 0.86±0.01 256hz g.tec 5 subc_6chan_3lrf_s1 6 3s 300 0.89±0.01 6 subc_6chan_3lrf_s2 300 0.84±0.01 7 subc_6chan_3lrf_s3 204 0.89±0.01 8 subc_5chan_3lrf _day1 c lh/rh/f 5 4s 210 0.72±0.02 256hz g.tec 9 subc_5chan_3lrf _day2 210 0.81±0.01 10 subc_5chan_3lrf _day3 180 0.81±0.01 11 subc_5chan_3lrf _day4 180 0.83±0.02 12 subc_5chan_3lrf _day5 234 0.87±0.01) 13 subc_5chan_3lrf _day6 150 0.88±0.01 14 subc_5chan_3lrf _day7 180 0.88±0.01 15 subc_14chan_3lrr c lh/rh/r 14 4s 350 0.78±0.01 250hz neuroscan table 1: the detailed information of three class dataset 159 j. anila maily, dr. c. velayutham and dr. m. mohamed sathik 5. results the classification accuracy of three class datasets using backpropagation classifier and rough neural network classifier is tabulated in table 2. figure 2 and figure 3 depicts the classification accuracy of suba_5chan_3lrf. classification accuracy of subb_6chan_3lrf using bpn and rnn classifier is shown in figure 4 and figure 5. figure 6 and figure 7 depicts the regression plot of suba_5chan_3lrf using bpn and rnn. it is observed that the performance of rnn classifier shows higher performance than the bpn classifier for all the datasets. figure 8 shows the accuracy of all three class datasets using bpn and rnn classifier. sl no. dataset accuracy bpn rnn suba_5chan_3lrf 98.80 99.70 subb_6chan_3lrf 97.20 99.00 subb_5chan_3lrf. 96.00 99.10 subc_5chan_3lrf. 98.00 99.50 subc_6chan_3lrf_s1 95.30 98.70 subc_6chan_3lrf_s2 99.40 99.70 subc_6chan_3lrf_s3 98.40 99.60 subc_5chan_3lrf_day1 99.60 100.0 subc_5chan_3lrf_day2 98.30 99.60 subc_5chan_3lrf_day3. 93.70 97.90 subc_5chan_3lrf_day4 97.80 99.50 subc_5chan_3lrf_day5 96.80 99.30 subc_5chan_3lrf_day6 97.50 99.90 subc_5chan_3lrf_day7 98.70 99.90 subc_14chan_3lrr 98.90 99.10 table 2: classification accuracy of three class data sets by bpn and rnn 160 motor imagery classification using rough neural network figure2: classification performance of suba_5chan_3lrf by bpn classifier figure 3 : classification performance of suba_5chan_3lrf by rnn classifier 161 j. anila maily, dr. c. velayutham and dr. m. mohamed sathik figure 4: classification performance of subb_6chan_3lrf bpn classifier figure 5: classification performance of subb_6chan_3lrf rnn classifier 162 motor imagery classification using rough neural network figure 6: regression plot of suba_5chan_3lrf by bpn classifier 163 j. anila maily, dr. c. velayutham and dr. m. mohamed sathik figure 7: regression plot of suba_5chan_3lrf by rnn classifier 164 motor imagery classification using rough neural network figure 8: classification accuracy by bpn and rnn 5. conclusion the main aim of bci is to identify the user’s intention through the brain signals. once the prominent features are extracted and selected, they have to be classified using the classification algorithm. the performance of bci depends on the classification accuracy. rough neural network (rnn) is proposed for motor imagery classification. it is tested on three class datasets with feature extraction using mean correlation and supervised feature selection method using quick reduct algorithm. the rnn classifier gives higher accuracy for the three class datasets than the neural network backpropagation (bpn) classifier. acknowledgements the authors thank dr. cichocki's lab (lab. for advanced brain signal processing, for providing the motor imagery eeg data-set). references [1] n. birbaumer, t. hinterberger, a. kubler, n. neumann. the thought-translation device (ttd): neurobehavioral mechanisms and clinical outcome, ieee trans. neural systems rehab. eng. 11 120–123, 2003 [2] j. wolpaw, n. birbaumer, d. mcfarland, g.pfurtscheller, t.vaughana, brain– computer interfaces for communication and control, clin.neurophysiol.113 767– 79,12002. 165 j. anila maily, dr. c. velayutham and dr. m. mohamed sathik [3] mak, joseph n, and jonathan r wolpaw, “clinical applications of brain-computer interfaces: current state and future prospects.” ieee reviews in biomedical engineering vol. 2: 187-199. doi:10.1109/rbme.2009.2035356,2009. [4] velayutham, c., and k. thangavel, "improved rough set algorithms for optimal attribute reduct. “journal of electronic science and technology 9.2 (2011): 108117,2011. [5] r. jensen, and q. shen, “computational intelligence and feature selection: rough and fuzzy approaches”, ieee press/wiley & sons,2008. [6] c. velayutham, and k. thangavel, “unsupervised feature selection using rough set”, proceedings of the international conference on emerging trends in computing (icetc 2011), coimbatore, pp. 307-315,2011. [7] simon haykin, “neural networks”, second edition by, prentice hall of india, 2005. [8] weidong zhao, and guohua chen, “a survey for the integration of rough set theory with neural networks”, systems engineering and electronics, vol. 24, no. 10, pp. 103107,2002. [9] hong k-s, khan mj and hong mj (2018) feature extraction and classification methods for hybrid fnirs-eeg brain-computer interfaces. front. hum.neurosci. 12:246.doi:10.3389/fnhum.2018.00246,2018. [10] sakhavi, siavash & guan, cuntai & yan, shuicheng, learning temporal information for brain-computer interface using convolutional neural networks. ieee transactions on neural networks and learning systems. pp. 1-11. 10.1109/tnnls.2018.2789927,2018. [11] vladimir a. maksimenko, semen a. kurkin, elena n. pitsik, et al., “artificial neural network classification of motor-related eeg: an increase in classification accuracy by reducing signal complexity,” complexity, vol. 2018, article id 9385947, 10 pages, https://doi.org/10.1155/2018/9385947. [12] han, c., kim, y., kim, d.y. et al., “electroencephalography-based endogenous brain–computer interface for online communication with a completely locked-in patient. j neuroengineering rehabil 16, 18 (2019) doi:10.1186/s12984-019-0493-0 [13] kumar, s., sharma, a. & tsunoda. t, brain wave classification using long shortterm memory network based optical predictor. sci rep 9, 9153 doi:10.1038/s41598019-45605 ,2019. 166 https://doi.org/10.1155/2018/9385947 microsoft word mondi_virtuali9_50.doc 9 mondi virtuali: second life e la sua economia franco eugeni, valentina giunco e laura manuppella introduzione nell’arco di alcuni decenni le comunicazioni umane hanno subito rilevanti cambiamenti. ci si chiede dove siano finiti carta e penna, se qualcuno li usa ancora per potersi scambiare messaggi a distanza. la vera distanza in realtà, è quella che l’uomo, complice delle sempre più sofisticate tecnologie, è riuscito ad abbattere con l’invenzione e l’utilizzo, oramai immancabile, delle tecnologie telematiche. il canadese marshall mcluhan, studioso di letteratura e teorico dei mezzi di comunicazione, descrisse nei primi anni sessanta nel suo libro più noto, understading media (il titolo in italiano è gli strumenti del comunicare) , la condizione dell’uomo moderno come quella dell’abitante di un “villaggio globale” intendendo che, grazie ai mezzi di comunicazione moderni, che consentono di congiungere tra loro “in tempo reale” i punti più lontani del pianeta, il mondo si è rimpicciolito, al punto da poter essere paragonato ad un villaggio. in realtà mcluhan dava alla sua metafora un significato più profondo: il villaggio globale e l’evoluzione degli strumenti di comunicazione sono il frutto di un processo rivoluzionario all’interno della società stessa, che ha portato all’evoluzione dei mezzi di comunicazione elettrici e all’evoluzione dell’uomo attraverso lo sviluppo di nuovi modi di comunicare. 10 il virtuale e le comunità virtuali quando si parla di realtà virtuale il primo pensiero che si ha, è rivolto ad un computer. ma riflettendo bene la realtà virtuale la viviamo ogni giorno nei sogni. quando sogniamo, infatti, noi viviamo una specie di realtà virtuale poiché ci troviamo in un mondo nel quale siamo completamente immersi: possiamo muoverci, parlare, guardare, a volte addirittura volare. ed è proprio attraverso il sogno che possiamo esprimere al meglio le nostre pulsioni più profonde perché non abbiamo vincoli e possiamo fare idealmente qualunque cosa e vivere qualunque esperienza, anche al di fuori delle regole fisiche che governano la realtà. vivendo un’esperienza virtuale di questo tipo si sperimenta un mondo che non esiste nella realtà, ma che esiste nella nostra mente e nel quale ci si può muovere con assoluta libertà. partendo dal significato delle due parole, possiamo notare che, unite, formano un ossimoro, ovvero un termine composto da due vocaboli che sono uno l’esatto opposto dell’altro: la parola virtuale è l’opposto di reale. associano i due termini, che fra loro si annullano, nasce l’idea di costruire idealmente dei mondi che, in realtà, non esistono. inoltre, poiché non è in nessun modo possibile rendere materialmente reale un mondo inventato, la realtà virtuale ha il compito di rendere ogni mondo fittizio il più possibile coerente con le situazioni reali, tanto da poter far credere a chi vive un’esperienza virtuale di far parte veramente di quel mondo, esattamente come se lo stesse vivendo in quel momento, con le stesse emozioni, le stesse sensazioni, gli stessi stimoli: come se stesse vivendo un sogno in piena coscienza. 11 esistono tre modi fondamentali per definire quella che è l’esperienza di vivere in un mondo fittizio: realtà virtuale, realtà artificiale, cyberspazio. le differenze che distinguono questi termini sono sottili, ma fondamentalmente tutti e tre hanno in sé il concetto di realtà virtuale. secondo la maggior parte degli esperti in questo campo, per realtà virtuale s'intende la possibilità di entrare in un mondo verosimile, e quindi esistente, almeno per il fatto di rispondere esattamente a tutte le leggi della fisica. così in un mondo di realtà virtuale dobbiamo stare attenti per esempio a non sporgerci troppo da un balcone, perché potremmo cadere, come dobbiamo prestare attenzione ai muri perché ci sbatteremmo contro. in definitiva, grazie alla realtà virtuale, possiamo esplorare sì il nostro mondo fantastico, ma ancora una volta saremo soggetti a tutte quelle leggi della fisica alle quali siamo abituati da sempre e con le quali conviviamo quotidianamente. la realtà artificiale, invece, non solo permette di vivere in mondo che non esiste, ma permette anche di viverci infrangendo le leggi della fisica, permette cioè di esprimersi in assoluta libertà, senza condizioni, senza vincoli, senza limiti. ed è per questo che gli studiosi hanno esaltato il mondo di realtà artificiale, a vera e propria espressione artistica, poiché è proprio nel poter spaziare senza alcun limite in quelli che sono i percorsi della mente e dell’anima, che si può accedere ad un mondo completamente nuovo. l’ultima denominazione è quella di cyberspazio, ideata nel 1984 da william gibson nel suo romanzo neuromancer. il cyberspazio prevede una sorta di realtà virtuale che può essere sperimentata simultaneamente da più persone in tutto il mondo. 12 questo tipo di definizione, apparentemente fine a se stessa, nasconde invece una realtà molto interessante che, forse, è lo scopo ultimo della realtà virtuale stessa: quello di far coesistere, con una sorta di unione tecnologico-culturale, mondi completamente diversi e lontani. l’obiettivo del cyberspazio è quindi quello di creare una realtà sostitutiva, parallela a quella reale, in grado di far comunicare allo stesso modo e su piani identici, tutti i popoli del mondo. uno dei problemi che ci si può porre di fronte al concetto di realtà virtuale è quello che riguarda la conoscenza o, per dirla con un termine filosofico, il suo aspetto gnoseologico. infatti, se è vero che attraverso i sistemi virtuali è possibile essere coinvolti sensorialmente e vivere esperienze diverse da quelle quotidiane, appartenenti magari a mondi fantastici o comunque distanti dai nostri, è anche vero che grazie a questo tipo di realtà veniamo a conoscenza di nuove classi di realtà. molti filosofi e studiosi considerano la realtà virtuale un mezzo capace di far conoscere nuove realtà, anche se forse, ciò che cambia è il come si conosce. in effetti esistono due grandi categorie di apprendimento: quella che ha un rapporto prevalentemente legato al pensiero e quindi alla logica, e quella che ha un rapporto basato prevalentemente sul pensiero figurativo e quindi legato alla percezione delle immagini. possiamo affermare che la realtà virtuale appartiene alla seconda categoria del come si apprende e che ci si trova davanti ad una forma della conoscenza umana che ci riporta alle origini: l’uomo, infatti, fin dalle sue origini, prima di esprimersi attraverso il linguaggio, lo ha fatto attraverso l’immagine, attraverso un linguaggio che si esprimeva per mezzo dell’uso di simboli. 13 la realtà virtuale è l’evolversi di quello che una volta era rappresentato dal graffito, un evolversi che rappresenta la possibilità di vedere il mondo in modo più ampio. infatti, se attraverso l’evoluzione del linguaggio è possibile dire sempre di più e in modo più preciso, è anche vero che un linguaggio più evoluto, come quello che la realtà virtuale mette a disposizione, allarga molto le zone del dicibile e quindi garantisce una visione nettamente più ampia e completa del mondo e, di conseguenza, della conoscenza. attraverso i servizi blogging, messagging e le tecnologie come il voip 2 (voice over ip), milioni di persone comunicano con amici e utenti in tutto il mondo, spesso senza averli mai incontrati di persona. il punto cardine del social network 3 è proprio quello di favorire la nascita di comunità virtuali che si sviluppano intorno a grandi centri di interazione digitali come ad esempio i blog 4 o programmi di messaggistica istantanea. tutto ciò nasce dalla necessità dei navigatori del web 5 di inserire le proprie idee e/o i propri commenti all’interno di uno spazio virtuale come il blog, un sito proprio che costituisca, nella navigazione, un punto di riferimento fisso, un’identità digitale in cui 1 voip (voice over internet protocol): è una tecnologia che rende possibile effettuare una conversazione telefonica sfruttando una connessione internet o un’altra rete dedicata che utilizza il protocollo ip, anziché passare attraverso la linea telefonica tradizionale (pstn). ciò consente di eliminare le relative centrali di commutazione e di economizzare sulla larghezza di banda occupata. vengono instradati sulla rete pacchetti di dati contenenti le informazioni vocali, codificati in forma digitale, e ciò solo nel momento in cui è necessario, cioè quando uno degli utenti collegati sta parlando. 3 i social network sono comunità virtuali che favoriscono la comunicazione e l’incontro fra utenti legati da interessi comuni. 4 il blog è un diario pubblico in internet, tenuto da uno o più persone dette blogger. e’ uno spazio sul web dove poter raccontare storie, esperienze e pensieri, da condividere con gli altri. non vi sono delle tematiche prefissate e ognuno è libero di scrivere su qualsiasi argomento. esistono blog d’informazione giornalistica, di pettegolezzi o semplicemente di racconti quotidiani circa le proprie esperienze di viaggi, sport poesia, musica, letture,ecc. 5 il termine navigare viene utilizzato in riferimento al contatto tra tanti porti e punti diversi costituiti dai numerosi siti web 14 ritrovarsi. in ambito italiano, questo scenario è dominato da uno dei servizi blogging usato più frequentemente dai navigatori: windows live space. oltre ad essere tra i più frequentati, è anche il “luogo” dove gli utenti amano intrattenersi più a lungo. nonostante il limite fisico del veicolo comunicativo e della ristretta larghezza di banda, gli utenti tendono a soddisfare i propri bisogni di riduzione dell’incertezza e di affinità nei confronti dell’altro adattando le proprie strategie comunicative alle possibilità offerte dal mezzo e facendo in modo di esprimere anche le emozioni. . ciò è possibile grazie all’accurato utilizzo della punteggiatura, delle maiuscole e di simpatici segni ideografici come le emoticon 6. oggi invece, le innovazioni della tecnologia informatica, hanno risolto anche ciò che poteva sembrare un ostacolo alla comunicazione. come comprendersi e destreggiarsi tra le diverse lingue del globo. dialogare con un giapponese, con un tedesco o con chiunque altro non parli la nostra lingua, o non essere “skillato” 7 in inglese, non è più un problema. sono state varcate anche queste barriere: la pratica si chiama conlang (constructed languages), è molto 6 le emoticon (o smilies, in italiano faccine) sono riproduzioni stilizzate di quelle principali espressioni facciali umane che si manifestano in presenza di un’emozione (sorriso, broncio, ghigno, ecc). vengono utilizzate prevalentemente su internet e negli sms (acronimo dell’inglese short message service) per sopperire alla mancanza del linguaggio non verbale nella comunicazione scritta. alcuni esempi di principali emoticon sono:  :-) sta per sorriso o tono scherzoso, amichevole  :-( broncio o tristezza  :-d riso  :-p linguaccia o ‘oops’  :-o stupore (‘a bocca aperta’) ve ne sono tanti altri e frequentemente i software on-line (forum, chat, ecc) o i software per telefoni cellulare sostituiscono automaticamente i segni d’interpunzione con immagini. 7 il termine ‘skillato’ deriva dall’inglese skill che significa ‘abilità’, ossia la capacità di svolgere mansioni complesse in modo ben finalizzato, organizzato, razionale ed usando l’esperienza per adattarsi a circostanze specifiche. e’ divenuto uso comune utilizzare il termine inglese skill per indicare il medesimo concetto, in particolare nel campo dell’informatica, per indicare il livello di preparazione tecnica di cui gode un individuo, o nei videogiochi. 15 diffusa in rete, e si riferisce a qualsiasi nuovo idioma. il primo conlang fu il toki pona, nato nel 2001 per mano di una linguista canadese: sonja elen 8. il mondo di second life uno degli universi paralleli più frequentati al giorno d’oggi nel web è second life: un mondo virtuale tridimensionale multi-utente on-line. questo sistema è stato creato nel 2003 dalla società californiana linden lab con l’obiettivo di creare un modo innovativo di condividere esperienze. tale progetto si è concretizzato in second life che è in grado di fornire ai suoi utenti (definiti "residenti") gli strumenti per raggiungere e creare in questo luogo virtuale nuovi contenuti grafici: oggetti, fondali, fisionomie dei personaggi, contenuti audiovisivi, ecc.. tecnicamente è un gioco di ruolo di massa, in realtà si è trasformato in un fenomeno sociale a livello planetario che oggi conta circa otto milioni di residenti 9. infatti, in sl si combinano assieme sia gli elementi di una chat (che rappresenta la modalità di comunicazione base con la quale si interagisce), sia gli elementi di un gioco di ruolo, ossia la capacità di muoversi in questo determinato cyberspazio tramite un personaggio che ci rappresenta e che in questo mondo ha una vita lavorativa e sociale. in questo modo ognuno può far entrare in gioco la propria immaginazione e costruire il proprio business, la propria abitazione e la propria immagine ed entrare in 8 la traduzione offerta dal sito ufficiale www.tokipona.org viene fatta dal toki pona all’inglese o dal toki pona all’esperanto. la traduzione in italiano invece è stata effettuata attraverso un sistema di traduzione offerto dal sito www.google.it che permette di ottenere, in questo caso, la traduzione di tutte le categorie di parole, in cui è suddiviso l’insieme di sostantivi, aggettivi e verbi del vocabolario del toki pona. 8 il numero degli utenti iscritti varia da fonte a fonte. otto milioni è il numero approssimativo degli iscritti, circa due milioni, invece, sono coloro che “giocano” regolarmente. 16 relazione con gli altri avatar secondo gli schemi di socializzazione tipici del mondo reale. tutto in sl 10 è costruito da chi entra a far parte di questa realtà virtuale ed è un infinito e bellissimo esercizio di creatività. l’opportunità alternativa concessa da questo nuovo strumento capace di fare evadere dalla realtà, parte dall’idea di dare all’individuo la possibilità di potersi creare il proprio aspetto esteriore, quello che può essere più adatto alla propria seconda vita. gli utenti, infatti, possono decidere come apparirà il loro io digitale (avatar 11). con i capelli biondi o scuri, con le fattezze di donna o sotto le sembianze di un uomo, o addirittura farlo apparire come un super eroe, magari con i capelli fucsia, e così via. il sistema, infatti, mette a disposizione dell’utente un’ampia gamma di caratteristiche somatiche e di abiti. l’identità reale è celata dietro un nickname12, che può essere fantasioso oppure essere attinto da una lista fornita dai programmatori, prima di effettuare il primo accesso in sl. la modalità di spostamento utilizzata in questo mondo alternativo, non è esattamente identica alla dinamica con la quale ci muoviamo nel mondo reale. gli avatar, infatti, gesticolano e si muovono come umani, ma in più hanno la possibilità di realizzare uno dei sogni che accompagna da sempre gli uomini: volare per spostarsi da un ambiente ad un altro. l’immersione in questo mondo comporta quindi l’assunzione di un’identità diversa e quindi anche il comportarsi di conseguenza. significa lasciarsi alle spalle la propria 10 second life 11 l’avatar è un alter ego virtuale, il cui nome deriva, secondo la definizione dello zingarelli, dal bramanesimo e dall’induismo dove rappresentava ciascuna delle dieci incarnazioni del dio visnù. 12 il nickname, o semplicemente nick, nella cultura di internet è uno pseudonimo utilizzati dagli utenti della rete telematica per identificarsi in un determinato contesto o in un determinata comunità virtuale. il nickname può essere un soprannome, una sigla o semplicemente un insieme di lettere e numeri. 17 immagine, appartenente alla vita reale, per diventare chi si desidera essere, o chi si detesta o per essere semplicemente se stessi. l’avatar, in questo universo di pixel, non deve dipendere dagli stimoli della fame o della sete, dal freddo, né tantomeno deve temere le malattie o la morte. l’unica condizione necessaria per una sopravvivenza decorosa, è quella di costruirsi una casa, per non ritrovarsi a vagabondare per le vie delle diverse isole. nel momento in cui l’utente/avatar non è in grado di costruirsi da solo un’abitazione, magari perché ha poca dimestichezza con i vari programmi 3d, può sempre rivolgersi ad apposite agenzie immobiliari dalle quali comprare, seppur a prezzi piuttosto elevati, ad esempio, delle bellissime villette a schiera, anche in zone residenziali e con vista mare. ma ciò che distingue second life dai comuni giochi 3d sta nel fatto che questo mondo virtuale, non solo è popolato, ma è anche costruito dagli utenti stessi che lo costruiscono secondo le loro preferenze, progettando case o addirittura, interi quartieri. i programmatori mettono a disposizione la tecnologia di base, ma poi sono gli utenti che aggiungono di volta in volta nuovi contenuti al sito. la sua peculiarità è quella di lasciare agli utenti la libertà di usufruire dei diritti d’autore sugli oggetti che essi creano, che possono essere venduti e scambiati tra i residenti utilizzando una moneta solo apparentemente virtuale: il linden dollar, che può essere convertito in veri dollari americani. secondo il creatore di second life, philip rosedale: “è l’economia del gioco la rivoluzione che sta attirando numerose aziende”. questo trentottenne, uomo d’affari 18 americano, ha cominciato la sua attività vendendo sistemi di database a 17 piccole imprese, ha ottenuto un diploma di laurea in scienze della fisica presso l’università della california, a san diego, e nel 1996 è entrato a far parte della realnetworks in qualità di vice presidente. nel 1999 ha lasciato la realnetworks per fondare la linden lab, una piccola azienda con circa trenta dipendenti che progetta ambienti virtuali in 3d. successivamente ha riunito attorno a sé un gruppo di sviluppatori informatici con i quali si è messo all’opera per creare sistemi in grado di comprimere le immagini virtuali al punto da creare un mondo del tutto simile a quello reale. il risultato finale è stato proprio second life, un metamondo in grado di contenere e veicolare strategie di marketing per le imprese. l’obiettivo di rosedale era quello di dimostrare l’esistenza di un modello per un’economia virtuale o una società virtuale. il suo intento, nella creazione di questo mondo virtuale, era quello di far condividere l’esperienza dei singoli individui e di mettere a disposizione di questi ultimi, i mezzi per poter esprimere ciascuno le proprie possibilità in piena libertà invece di creare antagonismo tra i partecipanti e di sviluppare quel continuo clima di sfida che caratterizza tutti i videogiochi.. investendo nel mondo virtuale, infatti, si possono guadagnare soldi reali: lo hanno capito, oltre a quelli della linden lab, anche gli utenti del gioco. questi spesso, nella prima vita, sono designer, artisti, creativi, programmatori di giochi e altro ancora e utilizzano sl come enorme vetrina e mezzo per tradurre i linden dollar in dollari americani, ad esempio vendendo l’idea proposta. 19 sl è comunemente utilizzato dai suoi utenti per proporre agli altri partecipanti conferenze, file musicali e video, opere d’arte, messaggi politici ecc; si è inoltre assistito alla creazione di numerose sottoculture all’interno dell’universo sl, che è stato studiato in numerose università come modello virtuale di un’interazione umana, falsa, ma in cui si proietta ciò che si vorrebbe essere. di queste sottoculture fanno parte, però, anche politici, che hanno visto in questo mondo, la possibilità di aprire spazi al fine di propagandare le proprie idee politiche: oggi il più tecnologico dei politici italiani, ad esempio, è antonio di pietro, il primo segretario di partito ad aprire un blog, ad inviare video da visualizzare su youtube 13 e, addirittura, ad organizzare proprio su sl una conferenza stampa. un’altra categoria che ha compreso a pieno le potenzialità di questo mondo virtuale è quella dei cantanti: gli u2 ad esempio, hanno tenuto dei concerti direttamente nel virtuale, mentre la cantante irene grandi, ha ambientato, proprio su sl, una delle sue più recenti canzoni. per poter entrare a far parte di questa “seconda vita” il requisito fondamentale è quello di essere maggiorenni. il primo passo per entrare nel vivo di questo spazio virtuale è quello di iscriversi (e ci sono due alternative: free, cioè gratuita e premium che permette con dieci dollari al mese di avere soldi da spendere in second life) e di costruirsi un avatar. il secondo, per quanto riguarda l’ambito italiano, è frequentare i forum di second life italia (www.secondlifeitalia.com), un sito ad hoc per gli abitanti del bel paese dove interagire e chiedere informazioni sul gioco, una 13youtube è’ un sito nel quale è possibile condividere la visione di qualsiasi tipo di video, anche amatoriale, che viene inserito direttamente dagli utenti, sul sito stesso. 20 community fondata da riccardo ranieri e luca lodini. in questo spazio interamente dedicato al popolo made in italy, lo scopo è principalmente quello di mettere a proprio agio tutti gli italiani che, appena entrati nel gioco, si sentono un po’spaesati, e di ricreare loro ambienti quanto più simili ai panorami delle più famose città italiane. 4. immagini in second life queste, ad esempio, sono alcune immagini che ci mostrano il quartiere dei parioli su sl14: 14 e’ possibile visionare le foto direttamente su http://web20.exite.it 21 un’altra riproduzione virtuale di ciò caratterizza lo spaccato popolare italiano nel reale, è stata l’elezione di miss itland 2007, eletta dal club itland, che ha proclamato reginetta dell’italia di sl giuly lowey. la manifestazione virtuale si è svolta come un comune concorso di bellezza che si rispetti, con sfilate e cambi d’abito. di seguito, alcune immagine a testimonianza dell’evento, e la più bella dell’itland: 22 23 5. l’economia e gli investimenti virtuali di second life per rendere più semplice e chiaro l’ingresso in sl a chi si addentra per la prima volta in questo mondo virtuale, la maggioli editore ha edito un mensile on-line di alcune decine di pagine, che si propone di trattare alcuni temi principali che riguardano questa nuova realtà alternativa. nel numero 1 di ottobre 2007 di second life magazine, ad esempio, vengono toccate tematiche riguardanti l’economia che regola sl, le controversie legali che si possono creare e risolvere on-line, gli sport che dal reale sono approdati direttamente nel virtuale, e molti altri argomenti. come una normale (pseudo)realtà che si rispetta, anche second life è organizzata sulla base di un modello economico ed è in grado di provvedere al suo mantenimento e persino alla ricchezza dei singoli individui che la caratterizzano. questo modello economico è stato pensato, dai creatori del gioco, in modo tale da lasciar spazio a coloro che con un po’ di coraggio e con un minimo d’investimento aspirano ad ottenere buoni profitti. il solo pensiero di fare soldi dentro e attraverso un computer potrebbe sembrare un concetto un po’ azzardato, eppure è proprio ciò che accade nel cyberspazio di sl. su second life, infatti, chiunque può mettersi in affari, aprendo ad esempio un negozio o trovandosi un lavoro stipendiato. l’avatar stesso può diventare un affare: il proprio taglio di capelli, la propria casa, gli oggetti che si posseggono possono essere ritenuti interessanti e la popolarità permette di entrare a far parte dei personaggi più 24 importanti di sl, che il sistema stesso provvederà a ripagare aumentando il loro budget mensile. infatti, progettare oggetti o svolgere attività (lo stilista, il parrucchiere, ecc..) consente principalmente di disporre di un capitale iniziale (la cosiddetta proprietà intellettuale) che può essere scambiata in maniera illimitata, visto che un oggetto è riproducibile all’infinito, al fine di soddisfare i bisogni degli utenti che aspirano, in questo ambiente, a massimizzare il proprio status. le modalità d'interazione sono diverse in second life, infatti solo ad un livello più evoluto è possibile acquistare case, automobili, vestiti, che danno agli avatar la possibilità di interagire in modo più completo fra di loro. ciò è possibile acquistando, con dollari veri, i linden dollar che saranno utilizzati nel virtuale e che, alla fine del mese, saranno poi convertiti, al netto di una commissione, attraverso carta di credito, in dollari americani. quello che potrebbe sembrare un semplice ambiente virtuale ispirato ai videogiochi di tipo immersivo, sta diventando un vero e proprio universo parallelo nel quale gli abitanti replicano fedelmente tutto quello che avviene nella loro real life, la vita normale, economia compresa. il fattore vincente di sl è proprio l’immersività. il programma, che funge da finestra aperta visivamente su questo mondo virtuale, disseminato di isole sulle quali è possibile costruire case, strade e altri ambienti popolati da oggetti, consente di vivere ed esplorare questi ambienti in modo realistico, anche grazie ad una grafica ben curata. per alcuni sl è diventato realmente un vero e proprio lavoro, a volte molto più remunerativo di quello reale. secondo un’inchiesta di business week, si sono 25 quadruplicati i residenti che, grazie all’inventiva e ad un irrilevante investimento economico iniziale, sono arrivati a guadagnare più di 5,000 dollari americani al mese, lasciando, nella maggior parte dei casi, il lavoro reale. per quanto riguarda le diverse tipologie di occupazioni c’è da dire che all’inizio, trovare un lavoro effettivamente remunerativo in sl, può sembrare un’operazione piuttosto complessa. quando si comincia, i pochi soldi si guadagnano solamente scommettendo nei casinò, oppure ballando nei locali, o addirittura stando semplicemente stesi a prendere il sole, anche se non è strettamente necessario lavorare in sl, in quanto vi sono oggetti gratis per tutti, servizi liberi e luoghi da esplorare e nei quali risiedere che sono completamente gratuiti. poi però, nel momento in cui si cerca qualcosa di redditizio, con le giuste conoscenze, ci si può imbattere in lavori più remunerativi come ad esempio hostess, commessi, organizzatori di eventi o baristi per circa dieci dollari l’ora. per sfondare però, c’è bisogno di qualcosa di più: la maggior parte dei "paperoni" di sl infatti, è costituita da persone che hanno creato un’attività in proprio come stilisti e creatori di vestiti, costruttori di oggetti, consulenti, venditori di appezzamenti di terra, ecc. ma la cerchia di attività non è limitata a questi settori. c’è perfino chi, come reuben steiger, nome in codice reuben millionsofus, ha addirittura lanciato la propria azienda virtuale, la millionsofus. il suo lavoro è quello di consulente di marketing per grossi clienti. ogni giorno qualche importante azienda o politico, apre un proprio quartier generale in sl, per farsi pubblicità. ebbene, aiutare le aziende ad interfacciarsi con questo nuovo mondo, sembra in assoluto, il lavoro più 26 remunerativo. la sede è nel mondo virtuale, ma i contatti di consulenza marketing sono reali e coinvolgono soggetti come microsoft, toyota, coca-cola e warner bros. un’altra rinomata azienda che ha visto in questo mondo parallelo un’ulteriore possibilità di guadagno è stata la gabetti. non possono essere venduti prodotti veri, ma si ha, in ogni caso, la possibilità di inserire un collegamento al loro negozio sul web. da questi collegamenti sta nascendo uno scambio fiorente. proprio nel marzo scorso la gabetti ha acquistato un’isola di second life. un’isola equivale a 65 metri quadri ed ha un costo di 1,300 euro una tantum, più 250 euro al mese di manutenzione; una tale spesa è richiesta in quanto per costruire un edificio col software, un programmatore può arrivare a chiedere fino a 15mila euro. la compravendita e l’acquisto di case hanno fruttato alla gabetti circa 2,3 milioni di linden dollar, che equivalgono a circa 6mila euro. numerose sono le aziende ed i personaggi famosi che hanno scelto second life come vetrina per il mondo: nelle sedi virtuali infatti c’è la possibilità di essere visti a livello internazionale, essere conosciuti quindi da diversi milioni di abitanti e di stabilire con essi interazioni utili ai propri interessi commerciali. uno dei metodi più usati per far conoscere il brand (la marca) della propria azienda in sl è l’architettura degli edifici e delle sedi, realizzate con linee molto accattivanti, al fine di attirare l’attenzione, al pari di un cartellone pubblicitario che, al contrario, non può essere utilizzato. molte aziende, addirittura, utilizzano questo mondo virtuale per testare i gusti e le esigenze degli avatar e lo fanno attraverso veri e propri sondaggi dai quali ricavano spunti per creare i prodotti che poi andranno a finire nel mercato reale. ciò è quello 27 che ha attuato la casa automobilistica della toyota, che ha creato nel virtuale un prototipo di macchina che, una volta valutato positivamente dagli avatar, avrebbe fatto il proprio ingresso nel mondo reale. nel mondo di sl ha fatto il suo ingresso anche una rinomata azienda di telefonia mobile: la vodafone che sta mettendo in atto una sorta di ridefinizione nel rapporto tra reale e virtuale. ha messo a disposizione dei suoi utenti un nuovo servizio: vodafone insideout. questo servizio consente di chiamare all’interno del metamondo di sl, ma anche e soprattutto di effettuare chiamate da sl verso apparecchi reali. la particolarità sta nel mantenere la separazione tra avatar e utente, infatti, chi riceve la chiamata vede comparire il nome dell’avatar, ma l’identità della persona reale resta celata. l’avatar in questo modo, acquisisce una personalità definita: la telefonata non arriva all’utente del mondo reale, bensì alla rappresentazione che l’individuo dà di sé nel mondo virtuale. per quanto riguarda i costi che l’utente/avatar deve sostenere, fino al 30 novembre sono nulli per poi diventare a pagamento in un prossimo futuro(300 linden dollar per ogni messaggio o minuto di chiamata, circa 1,44 dollari americani). altro aspetto da tenere in considerazione è che i numeri assegnati agli avatar sono presi dall’utenza tedesca e ciò significa che chiamare da un numero reale ad un cellulare virtuale viene a costare come una chiamata in germania. anche la telecom si è cimentata nella comunicazione virtuale e lo ha fatto attraverso il first life communicator. e’ stato più che altro un esperimento, terminato il 18 settembre c.a. ed è consistito nell’indossare un object, reperibile sull’isola telecom italia 1. questa proposta è stata 28 realizzata dalla tim in collaborazione con isn virtual worlds, una società milanese che opera nel campo della tecnologia informatica in veste di fornitore di servizi per il marketing e la comunicazione con i nuovi mezzi tecnologici. il first life communicator è stato pensato per effettuare telefonate tra avatar, spedire 40 sms, email, per fare telefonate della durata massima di tre minuti e per essere usato verso il mondo esterno su numeri nazionali, verso tutti gli operatori fissi e mobili, nell’arco di quattro settimane. su sl sono stati riprodotti una gran parte dei settori commerciali presenti nella vita reale che hanno fruttato agli utenti/avatar giri d’affari che non hanno nulla da invidiare a quelli realizzati nella loro prima vita. per quanto riguarda le personalità che hanno saputo sfruttare al meglio le possibilità di guadagno del mondo di sl vi sono: anshe chung, una delle personalità più note nell’universo 3d, soprannominata la “rockfeller di second life”. i suoi anshe chung studios sono i più ricercati da chi 29 ha bisogno di case o isole virtuali. nella realtà la sua attività non ha nulla a che vedere con quella virtuale in quanto, ailin philip linden: la sua mente geniale è riuscita a fargli guadagnare, attraverso la sua stessa creazione, due volte, in quanto philip rosedale, questo è il suo vero nome, oltre ad essere uno dei creatori di sl, è anche lui, un venditore di appezzamenti. 30 cristiano midnight, il cui vero cognome è diaz. e’ stato un innovatore in sl, che ha creato un’applicazione per la condivisione delle foto nel mondo, snapzilla. barnesworth anubis, nella realtà adam anders nel mondo virtuale di sl gestisce un negozio di vestiti e un’agenzia immobiliare. 31 stroker serpentine, il cui nome reale è kevin alderman, è colui che fino a non molto tempo fa, ha gestito il porno ed i quartieri a luci rosse della virtuale amsterdam. ora la sua attività è stata acquistata all’asta, per cinquanta mila dollari, da uno sconosciuto olandese, ma il suo intento ora è quello di creare una sorta di campo giochi per adulti, sempre legato al tema dell’eros. in sl, infatti, oltre a quartieri ricchi di case, negozi, musei e strutture architettoniche di ogni genere, oltre al lussuoso quartiere dei parioli (creato dall’immobiliarista bruno cerboni), possiamo incappare anche nei quartieri a luci rosse. second life quindi, è da considerare come un vero e proprio mondo parallelo, in cui è possibile lavorare, studiare, intrattenersi e realizzare a pieno il proprio ego superando anche i limiti imposti dal reale e dalla morale. in alcuni siti, infatti, si possono leggere curiosità riguardanti truffe ed inganni a livello civile e commerciale, e a tal proposito, la società californiana linden lab, si sta attivando per realizzare un tribunale virtuale a cui i residenti possono rivolgersi in caso di denunce e comportamenti scorretti tra avatar, al fine di mantenere sempre alto il livello di sicurezza e di lealtà tra i residenti che, cercano spesso di ottimizzare nel virtuale i loro guadagni reali, anche attraverso mestieri che nella real life sono considerati fuori legge, come la prostituzione. credo che ciò che più attiri di questo mondo alternativo, siano proprio le numerose possibilità di guadagno concesse dai più svariati mestieri che è possibile 32 intraprendere, e a sostegno di ciò, la mancanza di limiti legali che nel mondo reale non sarebbero possibili perché porterebbero alla più completa anarchia. l’idea di usufruire di un modello di educazione virtuale, a molti potrebbe sembrare poco idoneo al fine di un raggiungimento culturale vero e proprio, in quanto il mezzo utilizzato per la trasmissione degli insegnamenti è un computer 15. esistono invece strumenti telematici in grado di fornire insegnamenti accurati anche attraverso la rete. l’e-learning, così è definita un’attività formativa on-line, rivolta ad utenti adulti, prevede infatti, l’utilizzo della connessione alla rete per usufruire dei materiali didattici e dei corsi specifici che vengono trasmessi attraverso una “piattaforma tecnologica” (learning management sistem). gli utenti che usufruiscono di questa modalità di istruzione sono studenti universitari, docenti, studenti di scuole superiori, che hanno a loro disposizione la possibilità di tenere sempre sotto controllo il loro grado di apprendimento, sia attraverso un piano di studi da seguire, sia attraverso controlli periodici on-line di apprendimento con schede di valutazione ed autovalutazione. ci si iscrive ad un corso, fornendo nome e password, e si accede di volta in volta ad una piattaforma fornendo questi dati. una volta entrati nella piattaforma di e-learning si inizia ad interagire con gli elementi del corso: al suo interno è possibile trovare dispense, che è possibile scaricare direttamente dalla rete, esercitazioni o addirittura immagini con le quali potersi esercitare a casa. questi corsi sono seguiti da un tutor che, in un apposito forum, si mette a disposizione degli utenti al fine di offrire chiarimenti e delucidazioni sulle 15 si faccia riferimento a http://www.repubblica.it nella sezione tecnologia & scienze, articolo del 4 maggio 2007 di valerio maccari 33 materie trattate. gli utenti naturalmente hanno la possibilità di interagire anche con altri iscritti, e questo rappresenta una possibilità in più di confrontarsi con altre persone su più argomenti. al termine di questi corsi, gli utenti, dopo aver superato una prova finale, ricevono un attestato, a giustificare quanto appreso. questa tipologia d’insegnamento è stata adottata anche da alcuni atenei che permettono appunto di seguire i corsi universitari direttamente dal proprio pc, scaricare dalla rete il materiale didattico senza recarsi necessariamente nella sede universitaria e senza dipendere dai vincoli di tempo degli orari di corso. l’insegnamento nel virtuale potrebbe essere un’altra grande innovazione partorita dal mondo di sl,infatti, la vera novità di questa nuova tipologia di insegnamento è da vedere nel fatto che queste lezioni virtuali rendono possibile annullare le distanze che intercorrono fra gli studenti/avatar che si affacciano da ogni parte del mondo, creando un senso di comunità e di vicinanza anche se il proprio compagno di banco si collega da un altro continente. 6. conclusioni non è possibile stabile con precisione fino a che punto la realtà virtuale possa essere considerata realtà né tantomeno stabilire se una conoscenza ottenuta attraverso il virtuale possa sortire gli stessi effetti di una conoscenza ottenuta attraverso nel mondo reale. 34 in termini filosofici, ciò che è reale può essere considerato tale attraverso due strade completamente diverse fra loro: idealismo ed empirismo. il primo sostiene che può essere considerata realtà tutto ciò di cui abbiamo conoscenza, mentre il secondo presuppone che la realtà sia rappresentata solo da ciò di cui abbiamo un dato sensibile, tangibile, altrimenti il mondo esterno rimane assolutamente inconoscibile. in un mondo virtuale, la coscienza di aver appreso, avviene attraverso l’esperienza sensibile, di conseguenza sono soddisfatte entrambe le teorie, empirica e idealistica; ciò significherebbe che fra realtà virtuale e realtà, non sussiste nessuna differenza e che non sarebbe possibile fare alcuna distinzione fra i due tipi di realtà. una differenza fondamentale invece c’è, ed è quella che caratterizza e distingue nettamente i due tipi di realtà. nel mondo reale l’uomo è costretto a viverci, in quanto ci si trova automaticamente inserito appena nasce, in quello virtuale, invece, decide di inserirsi e, di conseguenza, decide di immergersi in un mondo che non si è trovato, ma che si è costruito in base alle proprie conoscenze. tuttavia, se si andasse verso un uso incondizionato di sistemi virtuali, probabilmente si andrebbe incontro ad una serie di problematiche. una di queste, ad esempio, riguarda la possibilità di un costante estraniarsi dal mondo reale e di conseguenza un disadattamento sempre più marcato alla realtà, con la perdita della propria identità e forse con una pretesa di autosufficienza che può arrivare all’incapacità di socializzare. le libertà che la realtà virtuale può offrire all’individuo, infatti, sono numerose e spesso allettanti ed hanno la capacità di estendere le capacità umane, sia quelle comunicative, sia quelle fisiche, rappresentando senz’altro un evolversi della conoscenza e un ampliamento delle 35 proprie libertà. però è necessario fare degli sforzi affinché la realtà virtuale possa veramente considerarsi un mezzo nuovo e straordinario capace di aprire nuovi orizzonti a chiunque, in grado di eliminare tutte le barriere e di dare all’uomo la possibilità di sentirsi realmente più libero di conoscere. e’ possibile affermare che, indipendentemente da ciò che s’intende per realtà, l’esperienza virtuale rappresenta indubbiamente un mezzo per ampliare le proprie conoscenze e in particolare, se non permette di conoscere realmente mondi ipotetici, permette almeno di conoscere più a fondo la realtà stessa, dando all’uomo l’opportunità vera e concreta di vivere non un’esperienza di realtà, ma un’esperienza del suo rapporto con la realtà e quindi del suo rapporto con se stesso e col mondo. second life è considerato da molti un vero e proprio paese dei balocchi in quanto, a partire dall’aspetto esteriore, è possibile diventare e fare ciò che si è sempre desiderato. in questo mondo infatti, non vigono le leggi che regolano la società del reale, quindi è possibile fare ciò che si desidera, senza vincoli. da un lato questo aspetto dà la possibilità di dare libero sfogo alla propria fantasia ed immaginazione, ma dall’altro annienta quelli che sono i limiti che ciascun individuo possiede: la mia libertà non si esaurisce dove comincia quella dell’altro. questo potrebbe portare ad uno stato di anarchia assoluta che distruggerebbe l’uomo, questa volta però nel reale, dove risiedono emozioni, sentimenti ed una economia assolutamente reale. 36 7. lettura riassuntiva e dati si sente parlare, sempre più spesso, di sistemi immersivi di realtà virtuale e di mondi virtuali. le descrizioni più comuni del virtuale fanno riferimento per lo più alla tecnologia utilizzata (mouse,guanti,occhiali ecc.) mentre le spiegazioni delle esperienze umane nei mondi virtuali utilizzando categorie concettuali provenienti dall’ambito disciplinare della semiotica come: simulazione visiva. proprio grazie a questi motivi negli ultimi anni è nato un fenomeno come il “second life”:è un mondo virtuale tridimensionale multi-utente inventato nel 2003 dalla società americana linden lab. un mondo virtuale insomma che assomiglia tantissimo a internet, che offre migliori prospettive di sviluppo, tanto che sinonimo di second life è l’espressione “internet 3d”. alla bidimensionalità e alla staticità del web tradizionale, sl sostituisce la tridimensionalità (anche se solo grafica), personalizzazione ( la navigazione avviene mediante un avatar gestito direttamente da una persona fisica tramite tastiera e mouse), maggiore dinamicità e inoltre la sua relativa facilità di introdurre contenuti di qualsiasi genere, mediante tecniche di costruzione, di animazione e di linguaggio (liden script). il sistema fornisce ai suoi utenti (residenti) gli strumenti per aggiungere e creare nel mondo virtuale nuovi contenuti grafici: oggetti, fondali, fisionomie dei personaggi, contenuti audiovisivi ecc. la peculiarità del mondo di second life è quella di lasciare agli utenti la libertà di usufruire dei diritti d’autore sugli oggetti che essi creano, potendoli poi scambiare e vendere con i residenti mediante l’uso della moneta virtuale “ linden dollar” che 37 può essere convertita in veri dollari americani mediante il sito www.eldexchange.eu (1000l$ equivalgono a 4 $ statunitensi). per molti versi anche qui come nel vita reale ricopre un ruolo importante l’aspetto economico, ogni giorno vengono effettuati dei movimenti finanziari pari a un 1 milione e 500 mila dollari tra i circa 4 milioni di avatar esistenti sulla rete. quindi è evidente l’interesse suscitato da un tale movimento di denaro, che serve alle transazioni più disparate: dall’acquisto di abbigliamento e oggetti personali creati da avatar ( ovvero le figure virtuali dei residenti in second life, dotati di un proprio nome, sembianza, forma, movimento ecc.), all’impiego nel gioco d’azzardo con tavoli da poker e casinò; dal pagamento di prestazioni sessuali con accompagnatrici e accompagnatori virtuali o dotati della webcam reale, all’investimento bancario; dall’affitto di case e negozi, alla compravendita di porzioni di terreno o di vere e proprie isole (le sim equivalenti ad uno spazio di 65536 metri quadrati) gestite direttamente e finanziariamente dai server di linden lab oppure da server privati collegati in rete. attualmente partecipano alla creazione del mondo di second life oltre 400 mila utenti attivi di tutto il pianeta (circa 9 milioni totali se si considerano gli utenti registrati, il che comprende gli utenti inattivi) , e ciò che distingue "second life" dai normali giochi 3d online è che ogni personaggio che partecipa alla "seconda vita" rappresenta, secondo la fantasia dell'utente, l'utente stesso. gli incontri all'interno del mondo virtuale appaiono dunque come reali scambi tra esseri umani attraverso la mediazione "figurata" degli avatar. l'iscrizione è gratuita, anche se è obbligatorio essere maggiorenni. per costruire e vendere oggetti all'interno di "second life", inoltre, occorre "comprare" 38 aree di terreno nel mondo virtuale. molti personaggi che partecipano alla vita di "second life" sono programmatori in 3d. qualcuno di essi ha guadagnato somme di (vero) denaro vendendo gli script dei propri oggetti creati per essere utilizzati dentro il mondo virtuale. second life viene comunemente utilizzato dai suoi utenti per proporre agli altri partecipanti conferenze, file musicali e video, opere d'arte, messaggi politici, ecc.; si è inoltre assistito alla creazione di numerose sottoculture all'interno dell'universo simulato sl, che è stato studiato in numerose università come modello virtuale di una interazione umana, falsa, ma in cui si proietta ciò che si vorrebbe essere. non sono solo gli investimenti ad attirare molti visitatori e a far aumentare questa nuova moda sempre più crescente negli ultimi anni. del milione e mezzo di persone che si sono collegate nell’ultimo mese di giugno e degli 8 milioni e 200 mila residenti non tutti sono attratti dai soldi da spendere e guadagnare ma anche dai iniziative sociali, culturali e religiose. le università americane insieme a quelle europee hanno svolto iniziative culturali di un certo rilievo, sperimentando forme di condivisione tra ricercatori o iniziative di e-learning in aule virtuali; numerosi gli istituti culturali: ad esempio il nostro ministero degli affari esteri ha acquistato uno spazio per collocarvi un virtuale istituto italiano di cultura che attualmente ospita una mostra sull’atre italiana contemporanea; tante le librerie che offrono e-book scaricabili gratuitamente o a pagamento dalla rete; molti i musei virtuali con interessanti mostre tematiche; tanti i teatri , i cinema, i luoghi di riflessione religiosa (chiese, templi, moschee), gli eventi e le iniziative che si aggiungono quotidianamente a un’offerta ha come unico limite la fantasia umana. l’idea generale 39 del sl è quella di creare una realtà virtuale nella quale sono permesse un’infinità di libere espressioni senza alcun vincolo e restrizione se non legata all’aspetto tecnologico, che per la maggior parte degli avatar è superabile nel tempo. in realtà in materia nascono alcune problematica già messe al vaglio da diverse nazioni e istituzioni sull’espressione di “villaggio globale”:  una globalizzazione non totale, che riguarda una piccola porzione del mondo reale in quanto: a. è rivolta alla parte tecnologicamente avanzata del pianeta, basti pensare che il 61% degli avatar sia europeo, il 19% nordamericano e il solo 13% asiatico b. ai ricchi che possiedono l’uso delle fonti energetiche e culturali c. a causa delle profonde divisioni dal resto di un mondo non globale in cui il web non giunge d. a causa di una forte diffidenza basti guardare la cina che concorre tecnologicamente con l’occidente  l’uso improprio della rete per scopi di pedofilia  la proiezione di una realtà e di un sistema economico, culturale e religioso esistente e che come tale è in assoluto contrasto con l’utopia di una società felice e perfetta che non è e che dovrebbe essere  alcuni psichiatri lo considerano un gioco pericoloso nel caso l'utente sia persuaso, anche a causa della forte influenza psicologica praticata dalla ditta 40 produttrice per ragioni di guadagno, il pericolo più grande è che "second life" diventi veramente una seconda vita, anziché una finzione. tutti questi aspetti hanno indotto diverse nazioni verso politiche di controllo messe in atto dalle forze di polizia per controllare e cercare di gestire nel miglior modo possibile la complessità del problema. second life è caratterizzata da una vera e propria struttura nella quale si possono delineare le differenti caratteristiche che rendono questa realtà virtuale unica nel suo genere: 41 l’elemento politico in quanto tale è largamente diffuso in second life: dalla politica pragmatica come i luoghi di propaganda ( ad esempio i siti di supporto o principali caratteristiche del second life possiede una propria geografia virtuale sim è l’unità fondamentale del territorio le unità sono indipendenti e si ha una trasparenza architettonica e sociale uniformità, eguaglianza sociale,illimitatezza del risorse scarsa presenza di una legislazione, estrema libertà di movimento e visione negli spazi virtuali 42 di opposizione all’amministrazione americana) o come quelli in cui si dibattono temi particolari alla politica ideale, come i luoghi costruiti da gruppi socialisti, comunisti, anarchici che sono abbastanza numerosi. con delle analisi più approfondite però si è potuto constatare una certa superficialità e la prevalenza di elementi commerciali e di svago e quindi di condizioni abbastanza agevolate all’espressione di ideali utopici, le quali invece non si realizzano, almeno finora, compiutamente. il fattore denaro (che da virtuale diventa reale) è uno degli apici del second life in quanto muove gran parte dell’attività dei naviganti e che fa diventare sl un luogo assai appetibile non solo per investitori virtuali, ma per protagonisti dell’economia reale. un’ulteriore riflessione consiglierebbe di indagare sull’età dei residenti. i dati statistici rilevabili dal web indicano un’età media di 32 anni. da una ricerca effettuata dalla linden lab si evince che a sorpresa gli italiani, partiti in sordina nell’apprezzare la possibilità di inventarsi una seconda vita, già lo scorso gennaio balzavano tra i paesi top ten con i loro 1.93% di residenti attivi (al primo posto gli stati uniti con il 31,19%) convalidando la convinzione che fa della nostra una nazione scarsamente propensa al rischio, più portata ad osservare e solo quando tutto è più che consolidato, a partecipare. certo che osservando anche l’età dei residenti (13-17 anni con 1,24%; 18-24 anni con 27,46%; 25-34 anni con 38,78%, 35-44 anni con 21 %, e 45+ anni con 11,52%) e verificando come questo nuovo mondo sia prevalentemente “giovane”, viene da chiedersi se mai arriverà l’era di una spinta innovativa tutta nostrana. come si può constatare consultando le tabelle a pagina 5 come 43 l’aumento della popolazione residente nel second life abbia avuto un’impennata esponenziale dopo un primo tentennamento nei mesi iniziali. nelle tabelle successive si possono controllare: il numero di residenti, numero di registrati, percentuale dei residenti attivi per paese (condizione per essere tale è il collegamento per più di un’ora al mese) e il totale delle ore di utilizzo da parte di tutti i residenti. sl mette in relazione più persone di varie culture contemporaneamente, finisce per sfociare in un uso altrettanto consumistico delle relazioni interpersonali: il disagio sociale che probabilmente è almeno in parte fattore di interesse per esperienze come le comunità virtuali, comporta la ricerca di relazioni numerose ma superficiali e momentanee che non consentono l’interscambio profondo di idee e opinioni intime; al pari di quanto accade d’altronde nei luoghi di ritrovo e di divertimento odierni, dove la compulsività della ricerca di relazioni cela gravi solitudini. 44 numero di registrati percentuali dei residenti attivi per paese 2003 gennaio 144 2003 febbraio 167 2003 marzo 209 2003 aprile 286 2003 maggio 411 2003 giugno 618 2003 luglio 881 2003 agosto 1.095 2003 settembre 1.294 2003 ottobre 1.477 2003 novembre1.745 2003 dicembre 2.098 2004 gennaio 2.740 2004 febbraio 3.419 2006 gennaio 124.175 2006 febbraio 144.830 2006 marzo 165.054 2006 aprile 198.104 2006 maggio 228.445 2006 giugno 323.467 2006 luglio 427.817 2006 agosto 597.269 2006 settembre 805.638 2006 ottobre stati uniti 31,19% francia 12,73% germania 10,46% regno unito 8,09% olanda 6,55% spagna 3,83% brasile3,77% canada 3,30% belgio 2,63% italia 1,93% australia 1,48% svizzera 1,29% giappone 1,29% svezia 0,95% danimarca 0,88% cina 0,61% austria 0,56% repubblica ceca 0,11% sud africa 0,11% hong kong 0,11% taiwan 0,09% croazia 0,08% estonia 0,08% lussemburgo 0,08% europa 0,08% colombia 0,07% malesia 0,07% marocco 0,06% emirati arabi 0,06% filippine 0,06% tailandia 0,06% afghanistan 0,06% 45 2004 marzo 4.026 2004 aprile 5.159 2004 maggio 6.351 2004 giugno 8.133 2004 luglio 9.443 2004 agosto 10.910 2004 settembre 12.485 2004 ottobre 14.038 2004 novembre 15.481 2004 dicembre 17.131 2005 gennaio 18.959 2005 febbraio 20.913 2005 marzo 23.058 2005 aprile 25.361 2005 maggio 1.203.244 2006 novembre 1.727.229 2006 dicembre 2.251.416 2007 gennaio 3.117.287 grecia 0,55% turchia 0,51% messico 0,48% argentina 0,45% irlanda 0,39% portogallo 0,39% polonia 0,36% ungheria 0,34% norvegia 0,33% venezuela 0,29% israele 0,27% singapore 0,26% nuova zelanda 0,21% romania 0,20% finlandia 0,19% india 0,18% cile 0,17% corea del nord 0,12% bulgaria 0,05% peru 0,05% slovenia 0,04% tunisia 0,04% cipro 0,04% porto rico 0,04% algeria 0,04% egitto 0,04% russia 0,03% arabia saudita 0,03% costa rica 0,03% ucraina 0,03% andorra 0,03% slovakia 0,03% vietnam 0,03% latvia 0,03% martinica 0,03% lithuania 0,03% albania 0,02% 46 27.642 2005 giugno 30.820 2005 luglio 36.663 2005 agosto 40.200 2005 settembre 52.635 2005 ottobre 69.524 2005 novembre 84.155 uruguay 0,02% all’interno di second life si vive una vera e propria vita. come prima cosa si deve creare l’identità tanto desiderata ( si può assumere la sembianza che si vuole:alti,biondi,belli ecc.) 47 e poi cercare di cominciare una vita da zero. bisogna ad esempio vestirsi, comprare da mangiare, lavorare ovvero vivere in tutti i sensi. ma per fare qualsiasi cosa bisogna cominciare a guadagnare qualche lined dollar e come fare????? si possono guadagnare dei soldi trovando un lavoro vero e proprio ad esempio come ballerino oppure esistono dei luoghi dove si possono guadagnare qualche ld ballando, prendendo il sole o semplicemente stando seduti. buffo vero? ebbene si anche senza fare nulla, stando seduti in uno dei numerosi locali ed in mille altre maniere apparentemente banali si può “lavorare”. ed in mille altre maniere, apparentemente banali. certo non si guadagna granchè ma per una cifra che varia da 10 a 20 l$ l'ora vale anche la pena, per non girare proprio senza soldi. ecco una lista di luoghi che offrono questa possibilità: money island una intera isola per guadagnare ballando, stando seduti (circa 5 l$ ogni 10 minuti) o compilando sondaggi (più remunerativo, ma vengono richiesti dati personali). carduccis mansion qui sembrano un po’ più taccagni. per una ora seduti di fronte alle slot (senza giocare) pagano 12 l$ . tonline beach .qui puoi prendere il sole guadagnando 2 l$ ogni 10 minuti perchè qualcuno dovrebbe pagare per non fare nulla? e' l'altro quesito che nasce spontaneo. successivamente però, curiosando tra i vari terreni in vendita e in affitto in second life e tramite delle ricerche sulla rete si capisce il perchè. i terreni o comunque i luoghi più frequentati vengono restituiti con più rilevanza nelle ricerche, conseguentemente sono più affollati ed acquistano valore, sia per la vendita che per l'affitto. i luoghi in secondo life sostanzialmente hanno le stesse dinamiche dei siti 48 internet: più visitatori, più valore. poiché si simula una vita vera e propria in sd si diventa anche residenti e quindi si viene assunti per un lavoro vero proprio. in quasi tutti luoghi si trovano oggetti gratuiti da prendere, ma quasi sempre di tratta di merchandising o oggetti sponsorizzati. esistono quasi dappertutto negozi di abbigliamento ma i capi si pagano. alcuni sono economici, altri decisamente più cari, ma sempre a pagamento. a meno che non si conoscano i posti giusti. la bellezza di second life oltre che di crearsi una nuova identità è anche quella di scoprire dei luoghi nuovi pieni di curiosità. help island: è un'isola che ti permette di esercitarti nell'iterazione con il mondo esterno, l'utilizzo di oggetti ed anche di veicoli (aerei, auto, barche) e insegna le tecniche di base della creazione. esistono anche manuali e corsi più avanzati di programmazione per la creazione di oggetti avanzati con iterazioni sul mondo esterno. anche su help island esiste un negozio gratis, che offre addirittura un aereo, una bicicletta, una casa ed altri oggetti decisamente interessanti. secret reflection : una grotta meravigliosa tutta da esplorare. babylon, babylonia : un’ isola dedicata alla civiltà di sumer, in particolare alla dea ishtar / inanna, meglio conosciuta come aphrodite per i greci o venus per i 49 romani. c'è un vero e proprio museo ed alcuni templi dell'antica sumer ricostruiti. akk horse ranch island, aeos : un ranch fantastico circondato dal mare. e' possibile andare a cavallo o godere del meraviglioso tramonto. skydive sl skydiving on st. pom, pamran : un centro di addestramento per lancio con il paracadute e tante altre invenzioni da scoprire e soprattutto da creare liberamente in una realtà virtuale in cui l’unico vincolo e l’immaginazione. ora di seguito sono riportate alcune immagini e siti interessanti per capire meglio il principio di questa nuova realtà virtuale: www.secondlife.com (sito che consente di convertire l$ in $ americani), www.youmark.it (sito che riporta le statistiche di liden lab) 50 alcuni classici paesaggi presenti nel second life complesso residenziale una delle numerose vie museo dell’arte paesaggio microsoft word articoloprob_ratio.doc 31 logica fuzzy e calcolo delle probabilità: due facce della stessa medaglia? danilo pelusi1 gianpiero centorame2 sunto: il seguente articolo illustra le possibili analogie e differenze tra il calcolo delle probabilità e la logica fuzzy. in particolare, sono messi a confronto gli insiemi tradizionali con quelli fuzzy in base alle molteplici definizioni che si possono attribuire alla probabilità di un evento. parole chiave: insieme fuzzy, evento, funzione di appartenenza, sottoinsiemità. 1. logica bivalente e polivalente: confronto fin dai tempi di aristotele la scienza, la matematica, la logica e la cultura in generale si sono basate su una concezione del mondo abbastanza semplice, fatta di cose assolutamente bianche o nere, di affermazioni vere o false totalmente, di oggetti classificabili in insiemi ben definiti e con confini ben precisi. una logica di natura bivalente domina e persuade la mente degli scienziati sempre più convinti di poter spiegare la natura intrinseca delle cose, i misteri più profondi della scienza con due semplici valori, 0 e 1. tutto è riconducibile ad essi, non ci sono vie di mezzo, la verità si fonda su due unici valori, piccoli ma capaci di spiegare e semplificare tutti i mondi in cui le varie scienze affondano le loro radici. la logica binaria, nata e diffusasi con aristotele, si riduce fondamentalmente ad una sola legge: a o non-a. o questo o non questo. il cielo è blu o non blu; non può essere blu e non blu. non può essere a e non-a. ciononostante la fede binaria ha sempre sollevato 1 pelusid@virgilio.it 2 cianpino78@yahoo.it 32 dubbi, ha sempre prodotto reazioni, critiche, insofferenze da parte di coloro che guardavano la natura sotto altri punti di vista tesi a far emergere le innumerevoli contraddizioni nascoste in essa, fatte di cose e non cose, di a e non-a. la filosofia ci racconta che questa forte contrapposizione alla logica binaria ebbe inizio in realtà ancora ben prima che nascesse aristotele, cioè circa due secoli avanti con budda, il quale diffuse in oriente la sua dottrina basata principalmente su un unico obiettivo: quello di squarciare il velo bivalente e vedere il mondo così com’è. in sostanza lo scontro si riduce ad una disputa fra i due più grandi capostipiti della filosofia, occidentale da una parte, e orientale dall’altra; le loro teorie hanno affascinato per secoli la mente di innumerevoli scienziati, ma la scienza stessa in generale sembra aver seguito nel corso del tempo due cammini ben differenti e contrastanti: quello occidentale e quello orientale. il calcolo delle probabilità si basa essenzialmente su una logica di natura bivalente. infatti quando noi parliamo di probabilità non facciamo altro che attribuire ad un certo evento un numero compreso tra zero e uno che rappresenta la possibilità che l’evento stesso si verifichi; ma la sua natura è bivalente, o si o no, o accade o non accade, non esistono scelte intermedie. la probabilità interviene in qualsiasi situazione dove ci troviamo di fronte ad incertezza dovuta a mancanza di dati, di informazioni, ecc.; gli eventi non hanno una natura deterministica bensì sono governati dalla casualità, da forze oscure che ne impediscono la determinazione a priori. nella logica fuzzy il mondo è caratterizzato da completa vaghezza, incertezza; più aumentano le informazioni, più emerge la natura fuzzy delle cose. più dati ci aiutano a fissare la sfumatura mediante la quale gli insiemi si sovrappongono; essa elimina i confini che segnano dove una cosa cessa di essere quella cosa. la probabilità per contro si dissolve quando i dati noti diventano numerosi. gli oggetti sono e non sono, appartengono e, allo stesso tempo, non appartengono ad un insieme, ma tutti in una certa misura. a ciascun elemento attribuiamo un valore, sempre compreso tra zero e uno, che esprime l’appartenenza ad un determinato insieme fuzzy. tale valore non è una probabilità, non fa riferimento al 33 verificarsi o non dell’evento stesso, bensì rappresenta la misura di un fatto deterministico, ma “vago” in una certa misura, che non risponde ad una natura bivalente ma polivalente, con infiniti gradi di appartenenza tra 0 e 1. per esempio, affermare che la probabilità che una determinata mela cada domani dal ramo è del 60%, è un’affermazione probabilistica, riguardante l’evento stesso, ma pur sempre di natura bivalente (infatti la mela o cadrà o resterà attaccata al ramo). dire invece che la stessa mela ha un grado di appartenenza all’insieme delle mele “dolci” del 70% (vedi figura 1), significa che per il 70% la mela appartiene all’insieme fuzzy d = dolce e per il restante 30% all’insieme a = acerbo. abbiamo a che fare con degli insiemi fuzzy in quanto non possiamo definire dei confini ben precisi in cui classificare le mele, inoltre abbiamo una logica a più valori in quanto per altre persone con gusti diversi i gradi di appartenenza potrebbero cambiare. 2. la funzione di appartenenza e la variabile casuale gli insiemi fuzzy in generale sono determinati da coppie ordinate del tipo: a=[(x,a(x))] (1) con x elemento e a(x) funzione di appartenenza, cioè una funzione che a ciascun valore x associa un determinato grado di appartenenza. in fig.1 insieme fuzzy relativo al tipo di mela. 34 generale la funzione di appartenenza ha una rappresentazione triangolare o trapezoidale, ma nulla vieta di utilizzarne altre come per esempio la gaussiana; in essa la x può assumere infiniti gradi di appartenenza compresi tra 0 e 1, leggibili direttamente dal grafico. la sua forma non è scelta a caso ma esemplifica in maniera chiara l’intento di rappresentare insiemi in cui ci sono elementi che vi appartengono al 100% (ad esempio in corrispondenza del vertice di una funzione triangolare o della base minore di una funzione trapezoidale) e altri che ne fanno parte in una certa misura. i valori della funzione di appartenenza crescono e decrescono “dolcemente” in maniera tale che gli insiemi si sovrappongano e le x possano cadere nelle zone d’intersezione degli intervalli di definizione dei vari insiemi. nel calcolo delle probabilità accade qualcosa di molto analogo; supponiamo infatti di avere un insieme di eventi e di farne una partizione; in base ad un determinato criterio logico assegniamo a ciascun evento un valore xi, successivamente definiamo una variabile casuale cioè una funzione f(x) che associa a ciascuna xi un valore pi di probabilità compreso tra 0 e 1, rispettando sempre la condizione di normalizzazione, cioè: pi=1 (2) mentre nel caso di variabili casuali discrete la f(x) ci fornisce la probabilità di un determinato evento, nelle variabili continue la p è definita da un integrale del tipo: f(x)dx (3) infatti non esiste la probabilità in un punto, bensì in un’area. nella logica fuzzy invece è sempre la f(x) a fornirci il grado di appartenenza e non l’area sottostante la funzione. inoltre in questo caso la probabilità è definita in un certo intervallo infinitesimo, mentre nella logica fuzzy, il grado di appartenenza è sempre valutato in un punto ben preciso. 35 3. insiemi fuzzy e insiemi tradizionali: analogie e differenze nella logica fuzzy gli insiemi seguono delle regole apparentemente molto simili a quelle della logica tradizionale su cui si basa appunto il calcolo delle probabilità. infatti volendo andare a definire le principali operazioni che possiamo compiere su di essi, abbiamo che:  unione (a or b, x) = max (a(x), b(x)) (4)  intersezione (a and b, x) =min (a(x), b(x)) (5)  complementazione  a (x)= 1-a(x) (6) le tre espressioni precedentemente elencate sono analoghe a quelle degli insiemi tradizionali, basta andare a sostituire alla funzione di appartenenza o membership a(x), la funzione caratteristica che esprime appunto l’appartenenza o meno di un elemento generico ad un determinato insieme. la differenza fondamentale risulta a livello grafico, dove non esiste corrispondenza tra le due logiche. infatti supponiamo di avere il seguente insieme fuzzy p così rappresentato (figura 2): p 0 0,5 xmin xmax fig.2 insieme fuzzy. 1 36 il complementare di p, applicando la sua definizione, risulta essere come in figura 3: e’ evidente che nella logica tradizionale il risultato sarebbe stato differente, in quanto saremmo andati a prendere l’area esterna all’insieme p (fig.2), rispettando il principio di non contraddizione il quale afferma che un elemento non può appartenere contemporaneamente ad un insieme p ed al suo complementare pc. nel calcolo delle probabilità si può considerare l’area di p come la probabilità del nostro evento e l’area totale come la probabilità del nostro insieme universo; si avrà quindi: p(p)+p( p ) = 1 (7) espressione analoga a p(x)+p(x) = 1 (8) ma il risultato è differente in quanto, nella probabilità graficamente facciamo un discorso di aree, mentre nella logica fuzzy l’espressione viene applicata a ciascun punto x sulla funzione di appartenenza. l’intersezione tra p e pc è così rappresentata (fig.4): xmin xmax 0 1 0,5 fig.3 insieme complementare. pc 37 si noti che tale insieme non coincide con l’insieme vuoto , come si vorrebbe nella logica tradizionale, o nel calcolo delle probabilità, dove la probabilità dell’evento composto pp  è zero: p( pp  ) = p(p)p( p/p ) (9) ma p( p/p ) = 0 (10) in quanto sono due eventi incompatibili, cioè il verificarsi dell’uno esclude il verificarsi dell’altro, sempre secondo il principio di non contraddizione. l’unione tra p e pc sarà quindi data dalla figura 5: anche qui osserviamo che tale insieme non corrisponde all’insieme universo , quindi viene a cadere anche il principio del terzo escluso, il quale afferma che l’unione di un insieme con il suo complemento fornisce l’insieme universo. in termini probabilistici avremo: p and pc xmin xmax 1 0,5 0 fig.4 intersezione di due insiemi fuzzy. xmin xmax 1 0,5 0 fig.5 unione di due insiemi fuzzy. p or pc 38 p( pp  ) = p(p)+p( p )p( pp  ) (11) ma, poiché p e p sono incompatibili, avremo: p( pp  ) = p(p)+p( p ) = p() = 1 (12) con  insieme universo. 4. la frequenza relativa: un concetto fuzzy una delle definizioni principali di probabilità è sicuramente quella frequentista, o statistica a posteriori, in cui viene definita come rapporto tra il numero delle prove favorevoli e il totale delle prove effettuate: n f limp n   con 1p0  (13) condizione fondamentale è che il numero delle prove sia molto elevato, e che le stesse siano identiche ed effettuate nelle medesime condizioni. tale probabilità la ritroviamo ad esempio nel gioco d’azzardo, o lanciando una moneta, oppure giocando ai dadi. se ipotizziamo di giocare a testa o croce possiamo tranquillamente costruirci il nostro insieme x di eventi totali, che rappresenta il totale dei lanci effettuati, e a sua volta suddividerlo in due sottoinsiemi rappresentanti da una parte gli esiti favorevoli e dall’altra gli esiti sfavorevoli (vedi figura 6): esiti favorevoli esiti sfavorevoli fig. 6 partizione dell’insieme universo. esiti favorevoli 39 l’insieme dei tentativi favorevoli non interseca quello dei tentativi sfavorevoli e i due insiemi esauriscono o riempiono l’insieme universo x, cioè quello del totale delle prove. gli eventi al loro interno sono bivalenti, o testa o croce, non esistono situazioni intermedie, non ci sono tentativi che cadono sulla linea di demarcazione, o ci si schiera da una parte o dall’altra. il principio di non contraddizione è salvo, la legge del terzo escluso pure, l’a o nona vale sempre. eppure tale diagramma nasconde un aspetto fuzzy, apparentemente difficile da scovare ma semplicissimo se si pensa al concetto di sottoinsiemità. generalmente nella logica tradizionale siamo abituati a parlare di insiemi che ne contengono altri, o totalmente o non affatto, quindi il concetto di appartenenza continua sempre a mantenere una natura bivalente. ma cosa succede se l’insieme non è contenuto totalmente e se a contenere non è più l’intero, o se si voglia insieme universo, bensì la parte o sottoinsieme? ecco allora il concetto di sottoinsiemità, cioè la maniera in cui un insieme è contenuto in un altro. ma anche in questo caso continuiamo a non vedere nulla di buono; i due insiemi di figura 6 sono disgiunti, non si contengono affatto, mentre entrambi appartengono al 100% all’insieme universo, che cioè li contiene totalmente. ma dov’è allora la sottoinsiemità? semplice, nella maniera in cui la parte contiene l’intero. sembra un concetto tanto ostico, come può infatti contenere l’intero essendo un suo sottoinsieme? l’unica eccezione è il caso in cui la parte è uguale allo stesso intero, ma in generale differisce da esso. tuttavia essa lo contiene sempre parzialmente, lo contiene cioè in una certa misura. consideriamo un altro insieme non fuzzy a, sottoinsieme di x, rappresentante sempre i nostri esiti favorevoli (figura 7): a x fig. 7 insieme degli esiti favorevoli. 40 il nostro insieme è una parte di x e non è fuzzy, non ha niente in comune con il suo opposto o complementare non-a. supponiamo ora di ridurre a ad un punto fino a svanire nel nulla, ossia nell’insieme vuoto. in questo caso la parte non contiene l’intero, come può infatti il nulla contenere un qualcosa? ora facciamo crescere di nuovo il nostro insieme a fino a eguagliare completamente il nostro rettangolo: in questo caso la parte contiene al 100% l’intero. ma il passaggio non è stato immediato, la nostra parte è andata da un’inclusione nulla, 0, a una completa inclusione, 1, passando per vie intermedie in cui ha contenuto il nostro intero in misura parziale direttamente proporzionale alla superficie che mano a mano ha occupato nel nostro diagramma. dunque abbiamo definito l’intero nella parte, ma in realtà cosa rappresenta? è la probabilità della parte. volendo conoscere la probabilità di fare testa o croce, tornando al lancio della moneta, non dobbiamo far altro che osservare la misura in cui il nostro insieme di esiti favorevoli contiene l’insieme di tutti i tentativi. in generale la probabilità di un insieme o di un evento a è uguale alla misura in cui la parte a contiene lo “spazio campionario” x. la sottoinsiemità coincide con quella che abitualmente definiamo frequenza relativa e la dimostrazione è semplicissima. sia s(x,a) la misura in cui x è un sottoinsieme di a. poiché a è totalmente un sottoinsieme di x, s(a,x)=1, ma in generale l’opposta sottoinsiemità s(x,a) è compresa fra gli estremi bivalenti: 0<s(x,a)<1. supponiamo che x contenga n tentativi e a gli na tentativi coronati da successo. allora il teorema della sottoinsiemità implica che: n/n xinelementilidegnumero xainelementilidegnumero )a,x(s a   (14) vale a dire la frequenza relativa degli esiti positivi di tutti i tentativi. 41 5. la soggettività nelle due logiche nella logica fuzzy prevale un certo grado di soggettività soprattutto nella parte iniziale della scelta degli insiemi, in cui bisogna andare a definire quale rappresentazione adottare e gli intervalli in cui gli stessi cadono. tale fase è molto importante, infatti da essa dipendono i tre processi fondamentali di fuzzificazione, inferenza e defuzzificazione. è chiaro che costruire un insieme, per esempio triangolare, non è la stessa cosa di adottare una rappresentazione trapezoidale: cambiano le pendenze, stessi valori di input corrispondono a gradi di appartenenza differenti, il che vuol dire attivazione degli insiemi output in punti completamente diversi. la scelta iniziale tuttavia non è del tutto casuale, dipende dal fenomeno che si sta studiando, spesso si ragiona anche a ritroso, cioè supponiamo di avere una determinata regola e volere per la stessa un’attivazione dell’output in un determinato punto: si costruisce l’insieme input in maniera che a un punto x stabilito del nostro intervallo corrisponda il valore di attivazione desiderato. molto importante è osservare come possono cambiare gli intervalli di definizione degli insiemi a seconda delle persone; supponiamo infatti di voler considerare l’insieme delle donne alte: cosa significa alto? per esempio potrei scegliere di far appartenere a questo insieme le donne che superano un metro e 65cm, ma un’altra persona potrebbe ritenere alta anche una donna di un metro e 63cm, quindi intervalli differenti a seconda delle persone. anche nel calcolo delle probabilità la soggettività è presente e la ritroviamo nella definizione soggettiva. in base ad essa ciascun individuo assegna un valore differente ad un evento che rappresenta la fiducia attribuitagli riguardo al suo verificarsi. e’ tipica delle scommesse, per esempio immaginiamo di avere una corsa clandestina di otto cavalli: in base alla definizione classica ciascun cavallo avrebbe 8 1 di probabilità di vincere; per la definizione frequentista occorrerebbe effettuare un numero di prove molto elevato e in base ai risultati attribuire in seguito le probabilità. la somma che noi scommettiamo sulla vittoria di un cavallo rapportata al totale delle scommesse rappresenta il nostro grado di fiducia. 42 tuttavia mentre nel calcolo delle probabilità vale il criterio di coerenza, cioè ciascun giocatore deve essere disposto a scambiare la propria posizione con l’altro, il quale conferisce un carattere deterministico alla teoria, definendo in maniera univoca la probabilità, ciò non è vero nella logica fuzzy, dove la scelta degli intervalli e relativi insiemi resta un fatto del tutto soggettivo e dipende da colui che effettua lo studio. bibliografia [1] cicchitelli g., (1984) probabilità e statistica, maggioli, perugia [2] cammarata s., (1994) sistemi fuzzy, etaslibri, bologna [3] dall’aglio g., (1987), calcolo delle probabilità, zanichelli, bologna [4] kosko b., (1995), il fuzzy pensiero. teoria e applicazioni della logica fuzzy, baldini e castaldi, milano [5] pesarin f., (1989), introduzione al calcolo delle probabilità, la nuova italia scientifica, roma ratio mathematica volume 46, 2023 some new odd prime graphs s. meena* g. gajalakshmi† abstract for a graph g, a bijection f is called an odd prime labeling , if f from v to {1, 3, 5, ...., 2|v |−1} for each edge uv in g the greatest common divisor of the labels of end vertices (f(u), f(v)) is one. in this paper we investigate the existence of odd prime labeling of some new classes of graphs and we prove that the graphs such as the z−pn graph, fish graph, umbrella graph, cocount tree , f -tree, y -tree and double sunflower graph are odd prime graphs. keywords: odd prime graph, z − pn graph, fish graph, umbrella graph,cocount tree , f -tree, y -tree, double sunflower graph. 2020 ams subject classifications: 05c78 1 *department of mathematics, govt, arts & science college, chidambaram 608 102, india; meenasaravanan14@gmail.com. †department of mathematics, govt, arts & science college, chidambaram; gaja61904@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1056. issn: 1592-7415. eissn: 2282-8214. ©s. meena et al. this paper is published under the cc-by licence agreement. 44 s. meena and g. gajalakshmi 1 introduction in this paper by a graph g = 〈v (g), e(g)〉 we mean a simple graph. for graph theoretical terminologies we refer j .a.bondy and u .s. r.murthy [1976]. a graph labeling is an assignment of integers to the vertices or edges or both subject to some constraints. for entire survey of graph labeling we refer gallian [2009]. the concept of prime labeling was introduced by roger etringer and was studied further by many researchers deretsky et al. [1991], tout et al. [1982].meena and vaithilingam [2013] proved that crown related graphs are prime graphs. meena and naveen [2018] investigated about the prime labeling of graphs related to bicyclic graphs. prime labeling in the concept of graph operation was discussed by meena and kavitha [2015]. the existence of odd prime labeling for some new classes of graph was discussed by meena et al. [2021]. the notion of odd prime labeling was introduced by prajapati and shah [2018] and they have proved many researchers. we give some families of odd prime graphs and some necessary condition for a graph to be odd prime graph. motivated by this study, further studied by in this paper we investigate the existence of odd prime labeling of some new classes of odd prime graphs. definition 1.1. let g = 〈v (g), e(g)〉 be a graph. a bijection f : v (g) → o|v | is called an odd prime labeling if for each edge uv ∈ e, greatest common divisor 〈f(u), f(v)〉is1. a graph is called an odd prime graph if its admits odd prime labeling. here o|v | = {1, 3, 5, ...2|v |−n} definition 1.2. z − pn is a graph obtained from a pair of p1 and p2 of path of length n in which the ith vertex of a path p1 is joined with (i + 1)th vertex of a path p2. definition 1.3. fish graph is a graph obtained by attaching one of the vertex of k3 to any one of the vertex of cn.it is denoted by cn@k3. definition 1.4. for any integers m > 2, n > 1.an umbrella graph u(m, n) is the graph obtained by identifying the end vertex of path pn with the central vertex of a fan graph fm. definition 1.5. coconut tree graph is obtained by identifying the central vertex of k1,m with a pendant vertex of the path pn. definition 1.6. f tree on n + 2 vertices, denoted by fn is obtained from a path pn by attaching exactly two pendant vertices of the n−1 and nth vertex of pn. 45 some new odd prime graphs definition 1.7. y tree on n + 1 vertices, denoted by yn is obtained from a path pn by attaching a pendant vertex of the nth vertex of pn. definition 1.8. a double sunflower graph order n, denoted by dsfn, is a graph obtained from the graph sfn by intserting a new vertex ci on each edges aiai+1 and adding edges for each i. 2 main results theorem 2.1. z − (pn) is an odd prime graph for all integers n ≥ 3. proof. let g = z − (pn) be the graph v (g) = {ui, vi/1 ≤ i ≤ n} e(g) = {(uiui+1), (vivi+1)/1 ≤ i ≤ n−1}∪{(viui+1)/1 ≤ i ≤ n−1} now |v (g)| = 2n and |e(g)| = 3(n-1) define a labeling f : v → o2n as follows f(ui) = 4i-5 for 1 ≤ i ≤ n f(vi) = 4i+1 for 1 ≤ i ≤ n clearly vertex labels are distinct. for each e = uv ∈ e, if gcd(f(u), f(v)) = 1 (i) e = u1u2, gcd(f(u1), f(u2)) = gcd(1, 3) = 1 (ii) e = uiui+1, gcd(f(ui), f(ui+1)) = gcd(4i−5, 4i−1) = 1 for 1 ≤ i ≤ n (iii)e = vivi+1, gcd(f(vi), f(vi+1)) = gcd(4i + 1, 4i + 5) = 1 for 1 ≤ i ≤ n (iv) e = viui+1, gcd(f(vi), f(ui+1)) = gcd(4i + 1, 4i−5) = 1 for 1 ≤ i ≤ n (v) e = vn−1vn, gcd(f(vn−1)), f(vn) = gcd(4n−3, 4n−1) = 1 for 1 ≤ i ≤ n thus f admits odd prime labeling on z − (pn) and hence z − (pn) is an odd prime graph. figure 1: z − (pn) and its odd prime labeling theorem 2.2. fish graph is an odd prime graph for n ≥ 3. proof. let g = cn@k3 be the graph v (g) = {ui/1 ≤ i ≤ n}∪{(v1v2)} e(g) = {(uiui+1)/1 ≤ i ≤ n−1}∪{(u1un)}∪{u1vi/1 ≤ i ≤ 2} ∪ {(v1, v2)} 46 s. meena and g. gajalakshmi now |v (g)|= n + 2 and |e(g)| = n + 3 define a labeling f : v → on+2 as follows. f(u1) = 1 f(v1) = 3 f(v2) = 5 f(ui) = 2i + 3 for 2 ≤ i ≤ n clearly vertex labels are distinct. for each e = uv ∈ e, if gcd(f(u), f(v)) = 1 (i) e = v1v2, gcd(f(v1), f(v2)) = gcd(3, 5) = 1 (ii) e = u1v1, gcd(f(u1), f(v1)) = gcd(1, 3) = 1 for 1 ≤ i ≤ n (iii)e = u1v2, gcd(f(u1), f(v2)) = gcd(1, 5) = 1 for 1 ≤ i ≤ n (iv) e = uiui+1, gcd(f(ui), f(ui+1) = gcd(2i + 3, 2i + 5) = 1 for 2 ≤ i ≤ n−1 (v) e = u1u2, gcd(f(u1), f(u2) = gcd(1, f(u2) = 1 (vi) e = u1un, gcd(f(u1), f(un) = gcd(1, f(un) = 1 thus f admits odd prime labeling on cn@k3 and hence fish graph is an odd prime graph. figure 2: fish graph cn@k3 and its odd prime labeling theorem 2.3. the umbrella graph u(m, n) is an odd prime graph. proof. consider the umbrella graph u(m, n) with vertex set. v (u(m, n)) = {xi, yi/1 ≤ i ≤ m, 1 ≤ i ≤ n} e(u(m, n)) = {xixi+1/1 ≤ i ≤ m−1} ∪{xiy1/1 ≤ i ≤ m}∪{yiyi+1/1 ≤ i ≤ n−1} now |v (u(m, n))| = m + n and |e(u(m, n))| = 2m + n−2 define f : v → om+n as follows. f(xi) = 2i + 1 for 1 ≤ i ≤ m f(y1) = 1 f(yi) = 2(i + m)−1 for 2 ≤ i ≤ n clearly vertex labels are distinct. with this labeling for each e = uv ∈ e, if gcd(f(u), f(v)) = 1. 47 some new odd prime graphs (i) e = xixi+1, gcd(f(xi), f(xi+1)) = gcd(2i+1, 2i+3) = 1 for 1 ≤ i ≤ m−1 (ii) e = xiy1, gcd(f(xi), f(y1)) = gcd(2i + 1, 1) = 1 for 1 ≤ i ≤ m (iii)e = y1y2, gcd(f(y1), f(y2)) = gcd(1, 2m + 3) = 1 (iv) e = yiyi+1, gcd(f(yi), f(yi+1) = gcd(2(i + m) − 1, 2(i + m) + 1) = 1 for 2 ≤ i ≤ n−1 as they are consecutive odd integers. this f is a odd prime labeling on u(m, n) and hence it is an odd prime graph. figure 3: u(m, n) and its odd prime labeling theorem 2.4. cocount tree ct(m, n) is an odd prime graph. proof. v (g) = {ui, vi/1 ≤ i ≤ m, 1 ≤ i ≤ n} and edge set e(g) = {uiv1/1 ≤ i ≤ m}∪{vivi+1/1 ≤ i ≤ n−1} now |v (ct(m, n))| = m + n and |e(ct(m, n))| = m + n−1 define a labeling f : v (g) →{1, 3, 5, ....2m + 2n−1} as follows f(ui) = 2(n + i)−1 for 1 ≤ i ≤ m f(vi) = 2i−1 for 1 ≤ i ≤ n clearly vertex labels are distinct. for each e = uv ∈ e, if gcd(f(u), f(v)) = 1 (i) e = uiv1, gcd(f(ui), f(v1)) = gcd(2(n + 1)−1, 1) = 1 for 1 ≤ i ≤ n; (ii) e = vivi+1, gcd(f(vi), f(vi+1)) = gcd(2i−1, 2i + 1)=1 for 1 ≤ j ≤ m−1. thus f admits odd prime labeling on ct(m, n) and hence ct(m, n) is an odd prime graph. 48 s. meena and g. gajalakshmi figure 4: ct(m, n) and its odd prime labeling theorem 2.5. let g be the graph obtained by identifying a pendant vertex of pm with a leaf of k1,n then g is an odd prime graph for all m and n. proof. v (g) ={u, ui, vj/1 ≤ i ≤ n, 2 ≤ i ≤ m} and the edge set e(g) = {uvi/1 ≤ i ≤ n}∪{vjvj+1/2 ≤ j ≤ m−1}∪{uv2} here u = v1 now |v (g)| =m+n and |e(g)| = m + n−1 define a labeling f : v (g) →{1, 3, 5, ....2m + 2n−1} as follows f(u) = 1 f(ui) = 2i + 1 for 1 ≤ i ≤ n f(vi) = 2(n + i)−1 for 2 ≤ i ≤ n clearly the vertex labels are distinct. for each e = uv ∈ e if gcd(f(u), f(v)) = 1 (i) e = uui, gcd(f(u), f(ui)) = gcd(1, 2i + 1) = 1 for 1 ≤ i ≤ n; (ii) e = uv2, gcd(f(u), f(v2)) = gcd(1, 2n + 3) = 1 for 1 ≤ i ≤ n; (iii)e = vivi+1, gcd(f(vi), f(vi+1))= gcd(2(n + i) − 1, 2(i + n) + 1) =1 for 2 ≤ i ≤ n; thus f admits odd prime labeling on g and hence g is an odd prime graph. 49 some new odd prime graphs figure 5: g and its odd prime labeling theorem 2.6. f tree fpn n ≥ 3 is an odd prime graph. proof. let v (g) = {u, v, vi, /1 ≤ i ≤ n−1} e(g) = {vivi+1/1 ≤ i ≤ n−1}∪{uv2, vv1} be the vertex set and edge set of fpn now |v (fpn)| = n + 2 and |e(fpn)| = n + 1 define a labeling f : v (g) →{1, 3, 5, ....2n + 3} as follows f(u) = 3 f(v) = 1 f(vi) = 2i+3 for 1 ≤ i ≤ n clearly vertex labels are distinct. for each e = uv ∈ e if gcd(f(u), f(v)) = 1 (i) e = vv1, gcd(f(v), f(v1)) = gcd(1, 5) = 1 (ii)e = uv2, gcd(f(u), f(v2)) = gcd(3, 7) = 1 (iii)e = vivi+1, gcd(f(vi), f(vi+1)) = gcd(2i + 3, 2i + 5) = 1 thus f admits odd prime labeling on fpn and hence f tree fpn is an odd prime graph. figure 6: fpn and its odd prime labeling theorem 2.7. y tree is an odd prime graph. proof. let v (g) = {uvi, /1 ≤ i ≤ n} e(g) = {vivi+1, vn−1u /1 ≤ i ≤ n−1} be the vertex set and edge set of y-tree now |v (g)| = n + 1 and |e(g)| = n 50 s. meena and g. gajalakshmi define a labeling f : v (g) →{1, 3, 5, ....2n + 1} as follows f(u) = 2n+1 f(vi) = 2i-1 for 1 ≤ i ≤ n clearly vertex labels are distinct. for each e = uv ∈ e if gcd(f(u), f(v)) = 1 (i) e = uvn−1, gcd(f(u), f(vn−1)) = gcd(2n + 1, 2n−1) = 1 (iii) e = vivi+1, gcd(f(vi), f(vi+1)) = gcd(2i−1, , 2i + 1) = 1 for 1 ≤ i ≤ n−1 thus f admits odd prime labeling on y -tree and hence y tree is an odd prime graph. figure 7: y -tree and its odd prime labeling theorem 2.8. for any natural numbers k ≥ 3 graph dsfn is an odd prime graph. proof. the vertex set and edge set of dsfn of order k respectively are v (dsfn) = {li, mi, ni/1 ≤ i ≤ k} e(dsfn) = {limi, lini, mini, mili+1, lkmk, lknn, l1nk/1 ≤ i ≤ n−1} now |v (dsfn)| = 3k and |e(dsfn)| = 5k define a labeling f : v (dsfn) →{1, 3, 5, ....6n−1} as follows f(li) = 6i−5 for 1 ≤ i ≤ n f(mi) = 6i−3 for 1 ≤ i ≤ n f(ni) = 6i−1 for 1 ≤ i ≤ n clearly all the vertex labels are distinct. with this labeling for each e = uv ∈ e if gcd(f(u), f(v)) = 1 (i) e = limi, gcd(f(li), f(mi)) = gcd(6i−5, 6i−3) = 1 for 1 ≤ i ≤ n (ii) e = lini, gcd(f(li), f(ni)) = gcd(6i−5, 6i−1) = 1 for 1 ≤ i ≤ n (iii) e = mini, gcd(f(mi), f(ni) = gcd(6i−3, 6i−1) = 1 for 1 ≤ i ≤ n (iv) e = mili+1, gcd(f(mi), f(li+1)) = gcd(6i−3, 6i−5) = 1 for 1 ≤ i ≤ n−1 (v) e = mkl1, gcd(f(mk), f(l1)) = gcd(f(mk), f(l1)) = 1 thus f is a odd prime labeling on dsfn and hence dsfn is an odd prime graph. 51 some new odd prime graphs figure 8: dsfn and its odd prime labeling 3 conclusions odd prime labelings of various classes of graphs such as z − pn graph, fish graph, umbrella graph, cocount tree, f -tree, y -tree and double sunflower graph are investigated. to derive similar results for other graph families is an open area of research. references t. deretsky, s. lee, and j. mitchem. on vertex prime labelings of graphs, in graph theory, combinatorics and applications, j. alavi, g. chartrand, o. oellerman, and a. schwenk, eds.,. proceedings 6th international conference theory and applications of graphs (wiley, new york),, 1:359 – 369, 1991. j. gallian. a dynamic survey of graph labeling. ds6, 2009. j .a.bondy and u .s. r.murthy. graph theory and applications. (northholland), new york, 1976. 52 s. meena and g. gajalakshmi s. meena and p. kavitha. prime labeling of duplication of some star related graphs. international journal of mathematics trends and technology, 23:26 – 32, 2015. s. meena and j. naveen. on prime vertex labeling of corona product of bicyclic graphs. journal of computer and mathematical sciences, 9:1512 – 1526, 2018. s. meena and k. vaithilingam. prime labeling for some crown related graphs. international journal of scientific & technology research, 2:92 – 95, 2013. s. meena, p. kavitha, and g. gajalakshmi. prime labeling of h super subdivision of cycle related graph. aip conference proceedings (communicated). s. meena, g. gajalakshmi, and p. kavitha. odd prime labeling for some new classes of graph (communicated). seajm, 2021. u. prajapati and k. shah. on odd prime labeling. international journal of research and analytical reviews, 5:284 – 294, 2018. a. tout, a. dabboucy, and k. howalla. prime labeling of graphs. nat. acad. sci letters, 11:365 – 368, 1982. 53 ratio mathematica volume 44, 2022 odd fibonacci stolarsky-3 mean labeling of some special graphs sree vidya. m 1 , sandhya s. s 2 abstract let g be a graph with p vertices and q edges and an injective function where each is a odd fibonacci number and the induced edge labeling are defined by and all these edge labeling are distinct is called odd fibonacci stolarsky-3 mean labeling. a graph which admits a odd fibonacci stolarsky-3 mean labeling is called a odd fibonacci stolarsky-3 mean graph. keywords: stolarsky-3 mean labeling of graphs, odd fibonacci stolarsky-3 mean labeling of graphs, bull graph, wheel graph, (m, n)-tadpole graph, fire cracker graph, pan graph, gear graph, star graph. ams subject classification: 05c78 3 1 research scholar, sree ayyappa college for women, chunkankadai. 2 assistant professor, department of mathematics, sree ayyappa college for women, chunkankadai. [affiliated to manonmaniam sundaranar university, abishekapatti – tirunelveli 627012, tamilnadu, india] email: witvidya@gmail.com 1 sssandhya2009@gmail.com 2 3 received on june 10th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.916. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 281 mailto:witvidya@gmail.com1 mailto:sssandhya2009@gmail.com2 sree vidya. m, sandhya. s. s 1. introduction the graph considered here will be finite, undirected and simple graph with p vertices and q edges. for all detailed survey of graph labeling, we refer to galian [1]. for all other standard terminology and notations, we follow harary [2]. s. s. sandhya, s. somasundaram and s. kavitha introduced the concept of stolarsky 3 mean labeling of graphs in [3]. in this paper, we introduced a new concept namely odd fibonacci mean labeling of graphs. definition: 1.1 the fibonacci numbers can be defined by linear recurrence . this generates the infinite sequence of integer beginning 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144, 233, …. definition: 1.2 let g be a graph with p vertices q edges. an injective function where each isa odd fibonacci number and the induced edge labeling defined by edge e =uv is labeled with f=(e=uv)= (or) , then all the edge labels are distinct and are from odd fibonacci number where . a graph that admits odd fibonacci stolarsky-3 mean labeling is called odd fibonacci stolarsky-3 mean graph. definition: 1.3 the bull graph is a planar undirected graph with 5 vertices and 5 edges in the form of a triangle with two disjoint pendent edges. definition: 1.4 the wheel graph is join of the graphs , . definition: 1.5 the (m, n)-tadpole graph is a graph is a special consisting of a graph on m (at least 3) vertices and a path graph on n vertices connected with a bridge. definition: 1.6 an firecracker is a graph obtained by the concatenation of stars by linking one leaf. definition: 1.7 gear graph is obtained from the wheel by adding a vertex between every pair of adjacent vertices of the cycle. the gear graph has 2n+1 vertices and 3n edges. definition:1.8 the pan graph is the graph obtained by joining a cycle graph to a singleton graph with a bridge. definition: 1.9a star graph with n vertices is a tree with one vertex having degree n-1 282 odd fibonacci stolarsky-3 mean labeling of some special graphs and other n-1 vertices having degree 1. a star graph with n+1 vertices . 2. main results theorem 2.1: the bull graph is a odd fibonacci stolarsky-3 mean labeling. proof. let be a bull graph. here , define a function where each is a odd fibonacci number. then the induced edge labeling is all distinct. hence, we proved a bull graph admits odd fibonacci stolarsky-3 mean labeling. example 2.2: odd fibonacci stolarsky-3 mean labeling of bull graph is shown below. figure 2.1: bull graph theorem 2.3: the wheel graph admits odd fibonacci stolarsky-3 mean labeling. proof. let be a wheel graph. here , define a function where each is a odd fibonacci number. then the induced edge labeling are all distinct. hence, we proved a wheel graph admits odd fibonacci stolarsky-3 mean labeling. example 2.4: odd fibonacci stolarsky-3 mean labeling of wheel graph is shown below. 283 sree vidya. m, sandhya. s. s figure 2.2 wheel graph theorem 2.5: the (m, n)tadpole graph admits odd fibonacci stolarsky-3 mean labeling. proof. let g be a (m, n)tadpole graph. here , define a function where each is a odd fibonacci number. then the induced edge labeling is all distinct. hence, we proved a (m, n)tadpole graph admits odd fibonacci stolarsky-3 mean labeling. example 2.6: odd fibonacci stolarsky-3 mean labeling of (5,1)tadpole graph is shown below. figure 2.3 (5, 1)tadpole graph theorem 2.7: a fire cracker graph admits odd fibonacci stolarsky-3 mean labeling. proof. let g be a fire cracker graph. here , define a function where each is a odd fibonacci number and assignment of vertex labeling are 284 odd fibonacci stolarsky-3 mean labeling of some special graphs then the induced edge labeling is all distinct. hence, we proved a fire cracker admits odd fibonacci stolarsky-3 mean labeling. example 2.8: odd fibonacci stolarsky-3 mean labeling of fire cracker graph is shown below. figure 2.4: fire cracker graph theorem 2.9: the pan graph admits an odd fibonacci stolarsky-3 mean labeling. proof. let g be a pan graph. here , define a function where each is a odd fibonacci number and assignment of vertex labeling are then the induced edge labeling are all distinct. hence, we proved a pan admits odd fibonacci stolarsky-3 mean labeling. example 2.10: odd fibonacci stolarsky-3 mean labeling of pan graph is shown below. figure 2.5: pan graph theorem 2.11: the gear graph admits odd fibonacci stolarsky-3 mean labeling. proof. let be gear graph. 285 sree vidya. m, sandhya. s. s here , define a function where each is a odd fibonacci number. then the induced edge labeling is all distinct. hence, we proved a gear graph admits odd fibonacci stolarsky-3 mean labeling. example 2.12: odd fibonacci stolarsky-3 mean labeling of gear graph is shown below. figure 2.6 gear graph theorem 2.13: the star graph admits an odd fibonacci stolarsky-3 mean labeling. proof. let g be a star graph. here , define a function where each is a odd fibonacci number and assignment of vertex labeling are then the induced edge labeling is all distinct. hence, we proved a star graph admits odd fibonacci stolarsky-3 mean labeling. example 2.14: odd fibonacci stolarsky-3 mean labeling of star graph is shown below. 286 odd fibonacci stolarsky-3 mean labeling of some special graphs figure 2.7 star graph 3. conclusion we have introduced a new labeling namely odd fibonacci stolarsky-3 mean labeling of graphs. we prove that bull graph, wheel graph, (m,n)-tadpole graph, fire cracker graph, pan graph, gear graph, star graph. extending the study to other families of graphs is an open area of research. references [1] galian, j. a. (2019) a dynamic survey of graph labeling. the electronic journal of combinatories. [2] harary, f. (1988) graph theory. narosa publishing house reading, new delhi. [3] s. somasundaram and r. ponraj, “mean labeling of graphs”, national academy of science letters vol.26, p.210-213. [4] s. s. sandhya, s. somasundaram, s. kavitha, stolarsky 3 mean labeling of graphs global journal of pure and applied mathematics, vol.14, 2018, no.14, pp.39 47. [5] k. thirugnanasambandam, g. chitra, y. vishnupriya, prime odd mean labeling of some special graphs, journal of emerging technologies and innovative research (jetir), volume 5, issue 6, 2018 jetir june 2018. [6] sree vidya. m & sandhya, s.s 2020, degree splitting of stolarsky-3 mean labeling of graphs, international journal of computer science, issn 2348-6600, vol.8, no.2, pp. 2413 – 2420. [7] sree vidya. m & sandhya, s.s 2022, decomposition of stolarsky-3 mean labeling of graphs, international journal for innovative engineering research, volume, issue 1, pp. 08-12. 287 ratio mathematica volume 44,2022 product signed domination in graphs t. m. velammal * a. nagarajan † k. palani ‡ abstract let be a simple graph. the closed neighborhood of , denoted by , is the set . a function is a product signed dominating function, if for every vertex where . the weight of , denoted by , is the sum of the function values of all the vertices in . . the product signed domination number of is the minimum positive weight of a product signed dominating function. in this paper, we establish bounds on the product signed domination number and estimate product signed domination number for some standard graphs. keywords: graphs, product signed dominating function, product signed domination number. 2010 ams subject classification: 05c69 § . * research scholar (reg. no. 21212232092010), pg & research department of mathematics, v.o. chidambaram college, thoothukudi-628008, tamil nadu, india. affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india.avk.0912@gmail.com † associate professor (retd.), v.o. chidambaram college, thoothukudi-628008, tamil nadu, india. affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india.nagarajan.voc@gmail.com ‡ associate professor, a.p.c. mahalaxmi college for women, thoothukudi-628002, tamil nadu, india. affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india. palani@apcmcollege.ac.in § received on june 8th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.923. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement 340 mailto:avk.0912@gmail.com mailto:nagarajan.voc@gmail.com mailto:palani@apcmcollege.ac.in t. m. velammal, a. nagarajan, and k. palani 1. introduction the fundamental thought of graphs was first presented in eighteenth era by swiss mathematician leonhard euler. it has numerous applications in natural sciences, technology, information system research and so on. the quickest developing region in theory of graph is domination. ore introduced the terms “dominating set” and “domination number”. dunbar et al. introduced signed domination number [1],[2],[4],[5]. hosseini gave a lower and upper bound for the signed domination number of any graph [3]. in this paper, we introduce the concept of product signed domination number and find bounds on product signed domination number. 2. preliminaries definition 2.1: a comb graph is a graph obtained by joining a pendant edge to each vertex of a path. definition 2.2: a star graph is a tree on vertices with one vertex having degree and the other vertices having degree . definition 2.3: a tree containing exactly two non-pendant vertices is called a double star. it is denoted by 3. main results definition 3.1: let be a simple graph. the closed neighborhood of , denoted by , is the set . a function is a product signed dominating function, if for every vertex where . the weight of , denoted by , is the sum of the function values of all the vertices in . the product signed domination number of is the minimum positive weight of a product signed dominating function. observation 3.2. (i) in a graph , a pendant vertex and its corresponding support vertex get the same functional values (i.e.) either or since otherwise . (ii) in a product signed dominating function, all the vertices of a graph should not be assigned since product signed domination number is positive. 341 product signed domination in graphs (iii) in a product signed dominating function, for every vertex contains either zero or even number of vertices with functional value , since otherwise . (iv) if and denote the number of vertices with functional values and respectively, then . theorem 3.3: the total number of vertices. proof: let be a complete graph on vertices. let and . since each pair of vertices is connected by an edge, in a product signed dominating function the number of vertices with functional value must be even. case 1: define a function as follows. when , every vertex should be assigned under , since otherwise would not be a product signed dominating function with a positive weight. the total number of vertices. case 2: and is odd subcase 2.1: is even partition the vertex set into two sets and such that , and . define as obviously, for every , and hence is a product signed dominating function. 342 t. m. velammal, a. nagarajan, and k. palani subcase 2.2. is odd partition the vertex set into two sets and such that , and . here is odd and is even. define as clearly, for every , .hence is a product signed dominating function. also, . since is odd. this function gives the minimum value for product signed domination number. case 3: and is even subcase 3.1: is even if we partition the vertex set into two sets and such that , and and assign to all the vertices in and to all the vertices in , then the function would be a product signed dominating function but the weight would be zero. since is even, is odd. partition the vertex set into two sets and such that , and . define as therefore is a product signed dominating function. also . since is even, this function gives the minimum value for product signed domination number as before. subcase 3.2: is odd we have is even. partition the vertex set into two sets and such that , and . define as correspondingly, for every , . hence is a product signed dominating function. also . proceeding as above, this function gives the minimum value for product signed domination number. 343 product signed domination in graphs the total number of vertices. theorem 3.4: for the comb graph, , the product signed domination number , the total number of vertices. proof: let be a comb graph . let be the vertex set with ’s representing the pendant vertices and be the edge set. since is the pendant vertex to , both and must be either or for (by 3.2(i)). case 1: is odd define as follows. if and are both assigned , then and should be assigned since otherwise would be . further if , again . hence , , , and so on. then and is a product signed dominating function. correspondingly, the weight of the graph is a negative integer which is a contradiction to the weight is positive. hence, let us start with . then , since otherwise is not a product signed dominating function. hence is the only product signed dominating function having a positive weight. hence it is the unique product signed dominating function. the total number of vertices of case 2: is even define as follows. if and are both assigned , then and should be assigned since otherwise would be . further if , again . hence 344 t. m. velammal, a. nagarajan, and k. palani , , , and so on. then and hence . therefore, this is not a product signed dominating function. hence, let us start with . then , since otherwise is not a product signed dominating function. hence is the unique product signed dominating function. the total number of vertices of by cases 1 and 2, . observation 3.5: for any graph , total number of vertices of . here the bounds are sharp since and total number of vertices. theorem 3.6: the product signed domination number of a path on vertices is equal to . proof: let be a path on vertices. if then (or) if then (or) by the above observation, if is assigned , then must be assigned so that . then must be assigned so that . so must be assigned so that . proceeding like this, we define a function as follows. for so when is not a product signed dominating function since when is not a product signed dominating function since when is a product signed dominating function having a negative weight. so let us try with assigned to . if is assigned , must be assigned so that . again must be assigned so that . again must be assigned so that and so on. therefore, . and is a minimum positive weight product signed dominating function. 345 product signed domination in graphs the weight of this function , the total number of vertices. therefore, , the total number of vertices. theorem 3.7: the product signed domination number of a cycle on vertices is equal to . proof: let be a path on vertices. if then (or) if then (or) by the above observation, if is assigned , then must be assigned so that . then must be assigned so that . so must be assigned so that . proceeding like this, we define a function as follows. for so, when is a product signed dominating function having negative weight. when is not a product signed dominating function since . when is not a product signed dominating function since . so let us try with assigned to . if is assigned , must be assigned so that . again must be assigned so that . again must be assigned so that and so on. therefore, . and is a minimum positive weight product signed dominating function. the weight of this function , the total number of vertices. therefore, , the total number of vertices. theorem 3.8: the product signed domination number of a star graph on vertices is equal to . proof: let be a star graph on vertices. let and . by 3.2(i), and should be assigned same functional value. 346 t. m. velammal, a. nagarajan, and k. palani if , then the weight of is negative. therefore must be equal to and hence define as . and obviously is the minimum positive weight product signed dominating function. therefore, , the total number of vertices. theorem 3.9: the product signed domination number of a double star graph is equal to . proof: let be a double star graph on vertices. let and . case 1: number of pendant vertices to atleast one of is odd. without loss of generality, assume that number of pendant vertices to is odd. if we assign to , then all the pendant vertices to must be assigned (by 3.2(i)). since number of pendant vertices to is odd, must be assigned . hence again by 3.2(i), all the pendant vertices to get . but here . so this is not a product signed dominating function. hence define as clearly, is the minimum positive weight product signed dominating function. therefore, , the total number of vertices. case 2: number of pendant vertices to both and is even. if we assign to , then all the pendant vertices to must be assigned (by 3.2(i)). since number of pendant vertices to is even, must be assigned . hence again by 3.2(i), the pendant vertices to get . here this is a product signed dominating function having a negative weight. so, the only possible positive weight product signed dominating function is therefore, , the total number of vertices. 347 product signed domination in graphs references [1] j. dunbar, s.t. hedetniemi. henning, and p.j. slater (1995), signed domination in graph theory, in: graph theory, combinatorics and applications, john wiley & sons, new york, 311-322. [2] ernest j. cockayneand christina m. mynhardt (1996), on a generalisation of signed dominating funtions of graphs, ars combinatoria, 43, 235-245. [3] s.m. hosseini moghaddam (2015), new bounds on the signed domination numbers of graphs, australasian journal of combinatorics, 61(3), 273-280. [4] izak broere, johannes h. hattingh, michael a. henning, and alice a. mcrae (1995), majority domiation in graphs, discrete mathematics, 138, 125-135. https://doi.org/10.1016/0012-365x(94)00194-n [5] odile favaron (1996), signed domination in regular graphs, discrete mathematics, 158, 287-293. https://doi.org/10.1016/001-365x(96)00026-x 348 https://doi.org/10.1016/0012-365x(94)00194-n https://doi.org/10.1016/001-365x(96)00026-x ratio mathematica volume 43, 2022 existence and uniqueness of solution of volterra integrodifferential equation of fractional order via s-iteration haribhau l. tidke ∗ gajanan s. patil † rupesh t. more ‡ abstract in this paper, we study the existence, uniqueness and other properties of solutions of volterra integrodifferential equation of fractional order involving the caputo fractional derivative. the tool employed in the analysis is based on application of s− iteration method. since the study of qualitative properties in general required differential and integral inequalities, but here s−iteration method itself has equally important contribution to study various properties such as dependence on initial data, closeness of solutions and dependence on parameters and functions involved therein. an example in support of the allestablished results is given. keywords: existence and uniqueness; normal s−iterative method; fractional derivative; continuous dependence; closeness; parameters. 2020 ams subject classifications: 35a01, 35a02, 34a12, 26a33, 35b30, 45d05, 47h10. 1 ∗(department of mathematics, school of mathematical sciences, kavayitri bahinabai chaudhari north maharashtra university, jalgaon, india); tharibhau@gmail.com. †(department of mathematics, psgvpm’s asc college, shahada, india); gajanan.umesh@rediffmail.com. ‡(department of mathematics, arts, commerce and science college, bodwad, india); rupeshmore82@gmail.com. 1received on may 28th, 2022. accepted on december 25th, 2022. published on december 30th, 2022. doi: 10.23755/rm.v41i0.791. issn: 1592-7415. eissn: 2282-8214. c©the authors. this paper is published under the cc-by licence agreement. haribhau l. tidke, gajanan s. patil and rupesh t. more 1 introduction we consider the following volterra integrodifferential equation of fractional order involving the caputo fractional derivative of the type: ( dα∗a ) y(t) = f ( t,y(t), ∫ t a h ( s,y(s) ) ds ) , (1) for t ∈ i = [a,b], n− 1 < α ≤ n, n ∈ n, with the given initial conditions y(j)(a) = cj, j = 0, 1, 2, · · · ,n− 1, (2) where f : i × x × x → x, h : i × x → x are continuous functions and cj (j = 0, 1, 2, . . . ,n− 1) are given elements in x. several researchers have introduced many iteration methods for certain classes of operators in the sense of their convergence, equivalence of convergence and rate of convergence etc. (see [1, 3, 4, 5, 6, 8, 9, 18, 19, 20, 21, 22, 23, 24, 31, 32]). the most of iterations devoted for both analytical and numerical approaches. the s− iteration method, due to simplicity and fastness, has attracted the attention and hence, it is used in this paper. the problems of existence, uniqueness and other properties of solutions of special forms of ivp (1)-(2) and its variants have been studied by several researchers under variety of hypotheses by using different techniques, [2, 7, 10, 11, 12, 13, 14, 15, 16, 26, 27, 29, 30] and some of references cited therein. in recently, soltuz and grosan [33] have studied the special version of equation (1) for different qualitative properties of solutions. authors are motivated by the work of sahu [31] and influenced by [5,33]. the main objective of this paper is to use normal s−iteration method to establish the existence and uniqueness of solution of the initial value problem (1)-(2) and other qualitative properties of solutions. 2 preliminaries before proceeding to the statement of our main results, we shall setforth some preliminaries and hypotheses that will be used in our subsequent discussion. let x be a banach space with norm ‖ · ‖ and i = [a,b] denotes an interval of the real line r. for the fractional order α, n − 1 < α ≤ n, n ∈ n, we define b = cr(i,x), (where r = n for α ∈ n and r = n− 1 for α /∈ n), as a banach space of all r times continuously differentiable functions from i into x, endowed with the norm existence and uniqueness of solution via s-iteration ‖y‖b = sup{‖y(t)‖ : y ∈ b}, t ∈ i. definition 2.1 (28). the riemann liouville fractional integral (left-sided) of a function h ∈ c1[a,b] of order α ∈ r+ = (0,∞) is defined by iαa h(t) = 1 γ(α) ∫ t a (t−s)α−1h(s) ds, t ∈ i where γ is the euler gamma function. definition 2.2 (28). let n− 1 < α ≤ n, n ∈ n. then the expression dαah(t) = dn dtn [ in−αa h(t) ] , t ∈ [a,b] is called the (left-sided) riemann liouville derivative of h of order α whenever the expression on the right-hand side is defined. definition 2.3 (25). let h ∈ cn[a,b] and n − 1 < α ≤ n, n ∈ n. then the expression ( dα∗a ) h(t) = in−αa h (n)(t), t ∈ [a,b] is called the (left-sided) caputo derivative of h of order α. lemma 2.1 (17). if the function f = (f1, · · · ,fn) ∈ c1[a,b], then the initial value problems( dαi∗a ) y(t) = fi(t,y1, · · · ,yn), y (k) i (0) = c i k, i = 1, 2, · · · ,n, k = 1, 2, · · · ,mi where mi < αi ≤ mi + 1 is equivalent to volterra integral equations: yi(t) = mi∑ k=0 cik tk k! + iαia fi(t,y1, · · · ,yn), 1 ≤ i ≤ n. as a consequence of the lemma 2.1, it is easy to observe that if y ∈ b and f ∈ c1[a,b], then y(t) satisfies the integral equation y(t) = n−1∑ j=0 cj j! (t−a)j + 1 γ(α) ∫ t a (t−s)α−1f ( s,y(s), ∫ s a h ( σ,y(σ) ) dσ ) ds, (3) which is equivalent to (1)-(2). we need the following pair of known results: haribhau l. tidke, gajanan s. patil and rupesh t. more theorem 2.1. ([31], p.194) let c be a nonempty closed convex subset of a banach space x and t : c → c a contraction operator with contractivity factor m ∈ [0, 1) and fixed point x∗. let αk and βk be two real sequences in [0, 1] such that α ≤ αk ≤ 1 and β ≤ βk < 1 for all k ∈ n and for some α,β > 0. for given u1 = v1 = w1 ∈ c, define sequences uk,vk and wk in c as follows: s-iteration process: { uk+1 = (1 −αk)tuk + αktyk, yk = (1 −βk)uk + βktuk,k ∈ n. picard iteration: vk+1 = tvk,k ∈ n. mann iteration process: wk+1 = (1 −βk)wk + βktwk,k ∈ n. then we have the following: (a) ‖uk+1 −x∗‖≤ mk [ 1 − (1 −m)αβ ]k ‖u1 −x∗‖, for all k ∈ n. (b) ‖vk+1 −x∗‖≤ mk‖v1 −x∗‖, for all k ∈ n. (c) ‖wk+1 −x∗‖≤ [ 1 − (1 −m)β ]k ‖w1 −x∗‖, for all k ∈ n. moreover, the s-iteration process is faster than the picard and mann iteration processes. definition 2.4. ([31], p.194) in particular, for αk = 1, k ∈ n ∪ {0} in the s-iteration process, then it reduces to as follows:  u0 ∈ c, uk+1 = tyk, yk = (1 − ξk)uk + ξktuk, k ∈ n∪{0}. (4) this is called normal s−iteration method. note: for our convenience, we replaced βk in the s-iteration process by ξk. lemma 2.2. ([33], p.4) let {βk}∞k=0 be a nonnegative sequence for which one assumes there exists k0 ∈ n, such that for all k ≥ k0 one has satisfied the inequality βk+1 ≤ (1 −µk)βk + µkγk, (5) where µk ∈ (0, 1), for all k ∈ n ∪{0}, ∞∑ k=0 µk = ∞ and γk ≥ 0, ∀k ∈ n ∪{0}. then the following inequality holds 0 ≤ lim sup k→∞ βk ≤ lim sup k→∞ γk. (6) existence and uniqueness of solution via s-iteration 3 existence and uniqueness of solutions via s−iteration now, we are able to state and prove the following main theorem which deals with the existence and uniqueness of solutions of the problem (1)-(2). theorem 3.1. assume that there exist functions p, q ∈ c(i,r+) such that ‖f ( t,u1,u2 ) −f ( t,v1,v2 ) ‖≤ p(t) [ ‖u1 −v1‖ + ‖u2 −v2‖ ] (7) and ‖h(t,u1) −h(t,v1)‖≤ q(t)‖u1 −v1‖, for t ∈ i. if θ = iaαp(t) ( 1 + (b − a)q ) < 1 ( where q = sup a≤t≤b q(t) ) , then the iterative sequence {yk}∞k=0 generated by normal s− iteration method (4) with the real control sequence {ξk}∞k=0 in [0, 1] satisfying ∞∑ k=0 ξk = ∞, converges to a unique point y ∈ b, which is the required solution of the equations (1)-(2) with the following estimate: ‖yk+1 −y‖b ≤ θk+1 e ( 1−θ )∑k i=0 ξi ‖y0 −y‖b. (8) proof. let y(t) ∈ b and define the operator (ty)(t) = n−1∑ j=0 cj j! (t−a)j + 1 γ(α) ∫ t a (t−s)α−1f ( s,y(s), ∫ s a h ( σ,y(σ) ) dσ ) ds, t ∈ i. (9) let {yk}∞k=0 be iterative sequence generated by normal s−iteration method (4) for the operator given in (9). we will show that yk → y as k →∞. from (4), (9) and assumption, we obtain ‖yk+1(t) −y(t)‖ = ‖(tzk)(t) − (ty)(t)‖ = ‖ n−1∑ j=0 cj j! (t−a)j + 1 γ(α) ∫ t a (t−s)α−1f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) ds haribhau l. tidke, gajanan s. patil and rupesh t. more − n−1∑ j=0 cj j! (t−a)j − 1 γ(α) ∫ t a (t−s)α−1f ( s,y(s), ∫ s a h ( σ,y(σ) ) dσ ) ds‖ ≤ 1 γ(α) ∫ t a (t−s)α−1‖f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) −f ( s,y(s), ∫ s a h ( σ,y(σ) ) dσ ) ‖ds ≤ 1 γ(α) ∫ t a (t−s)α−1p(s) [ ‖zk(s) −y(s)‖ + ∫ s a q(σ)‖zk(σ) −y(σ)‖dσ ] ds. (10) now, we estimate ‖zk(t) −y(t)‖ = [ (1 − ξk)‖yk(t) −y(t)‖ + ξk‖(tyk)(t) − (ty)(t)‖ ] ≤ (1 − ξk)‖yk(t) −y(t)‖ + ξk 1 γ(α) ∫ t a (t−s)α−1p(s) × [ ‖yk(s) −y(s)‖ + ∫ s a q(σ)‖yk(σ) −y(σ)‖dσ ] ds. (11) now, by taking supremum in the inequalities (10) and (11), we obtain ‖yk+1 −y‖b ≤ 1 γ(α) ∫ t a (t−s)α−1p(s) [ ‖zk −y‖b + ∫ s a q(σ)‖zk −y‖bdσ ] ds ≤ 1 γ(α) ∫ t a (t−s)α−1p(s) [ ‖zk −y‖b + (b−a)q‖zk −y‖b ] ds ≤ iaαp(t) ( 1 + (b−a)q ) ‖zk −y‖b = θ‖zk −y‖b (12) and ‖zk −y‖b ≤ [ (1 − ξk)‖yk −y‖b + ξkθ‖yk −y‖b ] = [ 1 − ξk ( 1 − θ )] ‖yk −y‖b, (13) respectively. therefore, using (13) in (12), we have ‖yk+1 −y‖b ≤ θ [ 1 − ξk ( 1 − θ )] ‖yk −y‖b. (14) thus, by induction, we get ‖yk+1 −y‖b ≤ θk+1 k∏ j=0 [ 1 − ξk ( 1 − θ )] ‖y0 −y‖b. (15) existence and uniqueness of solution via s-iteration since ξk ∈ [0, 1] for all k ∈ n∪{0}, the definition of θ and ξk ≤ 1 yields, ⇒ ξkθ < ξk ⇒ ξk ( 1 − θ ) < 1, ∀ k ∈ n∪{0}. (16) from the classical analysis, we know that 1 −x ≤ e−x = 1 −x + x2 2! − x3 3! + · · · , x ∈ [0, 1]. hence by utilizing this fact with (16) in (15), we obtain ‖yk+1 −y‖b ≤ θk+1e− ( 1−θ )∑k j=0 ξj‖y0 −y‖b = θk+1 e ( 1−θ )∑k i=0 ξi ‖y0 −y‖b. (17) since ∞∑ k=0 ξk = ∞, e − ( 1−θ )∑k j=0 ξj → 0 as k →∞. (18) hence, using this, the inequality (17) implies lim k→∞ ‖yk+1 −y‖b = 0 and therefore, we have yk → y as k →∞. remark: it is an interesting to note that the inequality (17) gives the bounds in terms of known functions, which majorizes the iterations for solutions of the problem (1)-(2) for t ∈ i. 4 continuous dependence via s−iteration in this section, we shall deal with continuous dependence of solution of the problem (1) on the initial data, functions involved therein and also on parameters. 4.1 dependence on initial data suppose y(t) and y(t) are solutions of (1) with initial data y(j)(a) = cj, j = 0, 1, 2, · · · ,n− 1, (19) haribhau l. tidke, gajanan s. patil and rupesh t. more and y(j)(a) = dj, j = 0, 1, 2, · · · ,n− 1, (20) respectively, where cj,dj are elements of the space x. then looking at the steps as in the proof of theorem 3.1, we define the operator for the equation (1) with the initial conditions (20): (ty)(t) = n−1∑ j=0 dj j! (t−a)j + 1 γ(α) ∫ t a (t−s)α−1f ( s,y(s), ∫ s a h ( σ,y(σ) ) dσ ) ds, t ∈ i. (21) we shall deal with the continuous dependence of solutions of equations (1) on initial data. theorem 4.1. suppose the function f in equation (1) satisfies the condition (7). consider the sequences {yk}∞k=0 and {yk} ∞ k=0 generated normal s− iterative method associated with operators t in (9) and t in (21), respectively with the real sequence {ξk}∞k=0 in [0, 1] satisfying 1 2 ≤ ξk for all k ∈ n ∪{0}. if the sequence {yk} ∞ k=0 converges to y, then we have ‖y −y‖b ≤ 3m( 1 − θ ), (22) where m = n−1∑ j=0 ‖cj −dj‖ j! (b−a)j. proof. from iteration (4) and equations (9); (21) and assumptions, we obtain ‖yk+1(t) −yk+1(t)‖ = ‖(tzk)(t) − (tzk)(t)‖ = ‖ n−1∑ j=0 cj j! (t−a)j + 1 γ(α) ∫ t a (t−s)α−1f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) ds − n−1∑ j=0 dj j! (t−a)j − 1 γ(α) ∫ t a (t−s)α−1f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) ds‖ ≤ n−1∑ j=0 ‖cj −dj‖ j! (b−a)j existence and uniqueness of solution via s-iteration + 1 γ(α) ∫ t a (t−s)α−1‖f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) −f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) ‖ds ≤ m + 1 γ(α) ∫ t a (t−s)α−1p(s) [ ‖zk(s) −zk(s)‖ + ∫ s a q(σ)‖zk(σ) −zk(σ)‖dσ ] ds. (23) recalling the equations (12) and (13), the above inequality becomes ‖yk+1 −yk+1‖b ≤ m + θ‖zk −zk‖b, (24) and similarly, it is seen that ‖zk −zk‖b ≤ ξkm + [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b. (25) therefore, using (25) in (24) and using hypothesis θ < 1, and 1 2 ≤ ξk for all k ∈ n∪{0}, the resulting inequality becomes ‖yk+1 −yk+1‖b ≤ m + ‖zk −zk‖b ≤ m + ξkm + [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b ≤ 2ξkm + ξkm + [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b ≤ [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b + ξk ( 1 − θ ) 3m( 1 − θ ). (26) we denote βk = ‖yk −yk‖b ≥ 0, µk = ξk ( 1 − θ ) ∈ (0, 1), γk = 3m( 1 − θ ) ≥ 0. the assumption 1 2 ≤ ξk for all k ∈ n ∪{0} implies ∞∑ k=0 ξk = ∞. now, it can be easily seen that (26) satisfies all the conditions of lemma 2.2 and hence, we have 0 ≤ lim sup k→∞ βk ≤ lim sup k→∞ γk haribhau l. tidke, gajanan s. patil and rupesh t. more ⇒ 0 ≤ lim sup k→∞ ‖yk −yk‖b ≤ lim sup k→∞ 3m( 1 − θ ) ⇒ 0 ≤ lim sup k→∞ ‖yk −yk‖b ≤ 3m( 1 − θ ). (27) using the assumptions, lim k→∞ yk = y, lim k→∞ yk = y, we get from (27) that ‖y −y‖b ≤ 3m( 1 − θ ), (28) which shows that the dependency of solutions of the equations (1)-(2) and (1) with the initial conditions (20) on given initial data. 4.2 closeness of solution via s−iteration consider the problem (1)-(2) and the corresponding problem ( dα∗a ) y(t) = f ( t,y(t), ∫ t a h ( s,y(s) ) ds ) , (29) for t ∈ i = [a,b], n− 1 < α ≤ n, n ∈ n, with the given initial conditions y(j)(a) = dj, j = 0, 1, 2, · · · ,n− 1, (30) where f is defined as f and dj (j = 0, 1, 2, . . . ,n−1) are given elements in x. then looking at the steps as in the proof of theorem 3.1, we define the operator for the equations (29)(30) (ty)(t) = n−1∑ j=0 dj j! (t−a)j + 1 γ(α) ∫ t a (t−s)α−1f ( s,y(s), ∫ s a h ( σ,y(σ) ) dσ ) ds, t ∈ i. (31) the next theorem deals with the closeness of solutions of the problems (1)-(2) and (29)-(30). theorem 4.2. consider the sequences {yk}∞k=0 and {yk} ∞ k=0 generated normal s− iterative method associated with operators t in (9) and t in (31), respectively with the real sequence {ξk}∞k=0 in [0, 1] satisfying 1 2 ≤ ξk for all k ∈ n ∪{0}. assume that existence and uniqueness of solution via s-iteration (i) all conditions of theorem 3.1 hold, and y(t) and y(t) are solutions of (1)(2) and (29)-(30) respectively, (ii) there exist non negative constant � such that ‖f ( t,u1,u2 ) −f ( t,u1,u2 ) ‖≤ �, ∀ t ∈ i. (32) if the sequence {yk} ∞ k=0 converges to y, then we have ‖y −y‖b ≤ 3 [ m + �(b−a)α γ(α+1) ] ( 1 − θ ) . (33) proof. from iteration (4) and equations (9); (31) and hypotheses, we obtain ‖yk+1(t) −yk+1(t)‖ = ‖(tzk)(t) − (tzk)(t)‖ = ‖ n−1∑ j=0 cj j! (t−a)j + 1 γ(α) ∫ t a (t−s)α−1f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) ds − n−1∑ j=0 dj j! (t−a)j − 1 γ(α) ∫ t a (t−s)α−1f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) ds‖ ≤ n−1∑ j=0 ‖cj −dj‖ j! (b−a)j + 1 γ(α) ∫ t a (t−s)α−1‖f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) −f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) ‖ds ≤ m + 1 γ(α) ∫ t a (t−s)α−1‖f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) −f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) ‖ds + 1 γ(α) ∫ t a (t−s)α−1‖f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) −f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ ) ‖ds ≤ m + 1 γ(α) ∫ t a (t−s)α−1�ds haribhau l. tidke, gajanan s. patil and rupesh t. more + 1 γ(α) ∫ t a (t−s)α−1p(s) [ ‖zk(s) −zk(s)‖ + ∫ s a q(σ)‖zk(σ) −zk(σ)‖dσ ] ds ≤ m + �(b−a)α γ(α + 1) + 1 γ(α) ∫ t a (t−s)α−1p(s) [ ‖zk(s) −zk(s)‖ + ∫ s a q(σ)‖zk(σ) −zk(σ)‖dσ ] ds. (34) recalling the derivations obtained in equations (12) and (13), the above inequality becomes ‖yk+1 −yk+1‖b ≤ m + �(b−a)α γ(α + 1) + θ‖zk −zk‖b, (35) and similarly, it is seen that ‖zk −zk‖b ≤ ξk [ m + �(b−a)α γ(α + 1) ] + [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b. (36) therefore, using (36) in (35) and using hypothesis θ < 1, and 1 2 ≤ ξk for all k ∈ n, the resulting inequality becomes ‖yk+1 −yk+1‖b ≤ [ m + �(b−a)α γ(α + 1) ] + ‖zk −zk‖b ≤ [ m + �(b−a)α γ(α + 1) ] + ξk [ m + �(b−a)α γ(α + 1) ] + [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b ≤ 2ξk [ m + �(b−a)α γ(α + 1) ] + ξk [ m + �(b−a)α γ(α + 1) ] + [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b ≤ [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b + ξk ( 1 − θ )3[m + �(b−a)α γ(α+1) ] ( 1 − θ ) . (37) we denote βk = ‖yk −yk‖b ≥ 0, µk = ξk ( 1 − θ ) ∈ (0, 1), γk = 3 [ m + �(b−a)α γ(α+1) ] ( 1 − θ ) ≥ 0. existence and uniqueness of solution via s-iteration the assumption 1 2 ≤ ξk for all k ∈ n ∪{0} implies ∞∑ k=0 ξk = ∞. now, it can be easily seen that (37) satisfies all the conditions of lemma 2.2 and hence, we have 0 ≤ lim sup k→∞ βk ≤ lim sup k→∞ γk ⇒ 0 ≤ lim sup k→∞ ‖yk −yk‖b ≤ lim sup k→∞ 3 [ m + �(b−a)α γ(α+1) ] ( 1 − θ ) ⇒ 0 ≤ lim sup k→∞ ‖yk −yk‖b ≤ 3 [ m + �(b−a)α γ(α+1) ] ( 1 − θ ) . (38) using the assumptions, lim k→∞ yk = y, lim k→∞ yk = y, we get from (38) that ‖y −y‖b ≤ 3 [ m + �(b−a)α γ(α+1) ] ( 1 − θ ) , (39) which shows that the dependency of solutions of ivp (1)-(2) on the function involved on the right hand side of the given equation. remark: the inequality (39) relates the solutions of the problems (1)-(2) and (29)-(30) in the sense that, if f and f are close as � → 0, then not only the solutions of the problems (1)-(2) and (29)-(30) are close to each other (i.e. ‖y−y‖b → 0), but also depends continuously on the functions involved therein and initial data. 4.3 dependence on parameters we next consider the following problems ( dα∗a ) y(t) = f ( t,y(t), ∫ t a h ( s,y(s) ) ds,µ1 ) , (40) for t ∈ i = [a,b], n− 1 < α ≤ n, n ∈ n, with the given initial conditions y(j)(a) = cj, j = 0, 1, 2, · · · ,n− 1, (41) and ( dα∗a ) y(t) = f ( t,y(t), ∫ t a h ( s,y(s) ) ds,µ2 ) , (42) haribhau l. tidke, gajanan s. patil and rupesh t. more for t ∈ i = [a,b], n− 1 < α ≤ n, n ∈ n, with the given initial conditions y(j)(a) = dj, j = 0, 1, 2, · · · ,n− 1, (43) where f : i × x × x × r → x is continuous function, cj, dj (j = 0, 1, 2, . . . ,n− 1) are given elements in x and constants µ1, µ2 are real parameters. let y(t), y(t) ∈ b and following steps from the proof of theorem 3.1, define the operators for the equations (40) and (42), respectively (ty)(t) = n−1∑ j=0 cj j! (t−a)j + 1 γ(α) ∫ t a (t−s)α−1f ( s,y(s), ∫ s a h ( σ,y(σ) ) dσ,µ1 ) ds, t ∈ i; (44) and (ty)(t) = n−1∑ j=0 dj j! (t−a)j + 1 γ(α) ∫ t a (t−s)α−1f ( s,y(s), ∫ s a h ( σ,y(σ) ) dσ,µ2 ) ds, t ∈ i. (45) the following theorem states the continuous dependency of solutions on parameters. theorem 4.3. consider the sequences {yk}∞k=0 and {yk} ∞ k=0 generated normal s− iterative method associated with operators t in (44) and t in (45), respectively with the real sequence {ξk}∞k=0 in [0, 1] satisfying 1 2 ≤ ξk for all k ∈ n∪{0}. assume that (i) y(t) and y(t) are solutions of (40)-(41) and (42)-(43) respectively, (ii) there exist functions p, r ∈ c(i,r+) such that ‖f ( t,u1,u2,µ1 ) −f ( t,v1,v2,µ1 ) ‖≤ p(t) [ ‖u1 −v1‖ + ‖u2 −v2‖ ] , and ‖f ( t,u1,u2,µ1 ) −f ( t,u1,u2,µ2 ) ‖≤ r(t) ∣∣∣µ1 −µ2∣∣∣. existence and uniqueness of solution via s-iteration if the sequence {yk} ∞ k=0 converges to y, then we have ‖y −y‖b ≤ 3 [ m + |µ1 −µ2|iaαr(t) ] ( 1 − θ ) , (46) where θ = ia αp(t) ( 1 + (b−a)q ) < 1, t ∈ i. proof. from iteration (4) and equations (44); (45) and hypotheses, we obtain ‖yk+1(t) −yk+1(t)‖ = ‖(tzk)(t) − (tzk)(t)‖ = ‖ n−1∑ j=0 cj j! (t−a)j + 1 γ(α) ∫ t a (t−s)α−1f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ,µ1 ) ds − n−1∑ j=0 dj j! (t−a)j − 1 γ(α) ∫ t a (t−s)α−1f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ,µ2 ) ds‖ ≤ n−1∑ j=0 ‖cj −dj‖ j! (b−a)j + 1 γ(α) ∫ t a (t−s)α−1‖f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ,µ1 ) −f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ,µ2 ) ‖ds ≤ m + 1 γ(α) ∫ t a (t−s)α−1‖f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ,µ1 ) −f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ,µ1 ) ‖ds + 1 γ(α) ∫ t a (t−s)α−1‖f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ,µ1 ) −f ( s,zk(s), ∫ s a h ( σ,zk(σ) ) dσ,µ2 ) ‖ds ≤ m + 1 γ(α) ∫ t a (t−s)α−1r(s)|µ1 −µ2|ds + 1 γ(α) ∫ t a (t−s)α−1p(s) [ ‖zk(s) −zk(s)‖ + ∫ s a q(σ)‖zk(σ) −zk(σ)‖dσ ] ds ≤ m + |µ1 −µ2|iaαr(t) haribhau l. tidke, gajanan s. patil and rupesh t. more + 1 γ(α) ∫ t a (t−s)α−1p(s) [ ‖zk(s) −zk(s)‖ + ∫ s a q(σ)‖zk(σ) −zk(σ)‖dσ ] ds. (47) recalling the derivations obtained in equations (12) and (13), the above inequality becomes ‖yk+1 −yk+1‖b ≤ m + |µ1 −µ2|ia αr(t) + θ‖zk −zk‖b, (48) and similarly, it is seen that ‖zk −zk‖b ≤ ξk [ m + |µ1 −µ2|iaαr(t) ] + [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b. (49) therefore, using (49) in (48) and using hypothesis θ < 1, and 1 2 ≤ ξk for all k ∈ n∪{0}, the resulting inequality becomes ‖yk+1 −yk+1‖b ≤ [ m + |µ1 −µ2|iaαr(t) ] + ‖zk −zk‖b ≤ [ m + |µ1 −µ2|iaαr(t) ] + ξk [ m + |µ1 −µ2|iaαr(t) ] + [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b ≤ 2ξk [ m + |µ1 −µ2|iaαr(t) ] + ξk [ m + |µ1 −µ2|iaαr(t) ] + [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b ≤ [ 1 − ξk ( 1 − θ )] ‖yk −yk‖b + ξk ( 1 − θ )3[m + |µ1 −µ2|iaαr(t)]( 1 − θ ) . (50) we denote βk = ‖yk −yk‖b ≥ 0, µk = ξk ( 1 − θ ) ∈ (0, 1), γk = 3 [ m + |µ1 −µ2|iaαr(t) ] ( 1 − θ ) ≥ 0. the assumption 1 2 ≤ ξk for all k ∈ n ∪{0} implies ∞∑ k=0 ξk = ∞. now, it can be easily seen that (50) satisfies all the conditions of lemma 2.2 and hence we have 0 ≤ lim sup k→∞ βk ≤ lim sup k→∞ γk existence and uniqueness of solution via s-iteration ⇒ 0 ≤ lim sup k→∞ ‖yk −yk‖b ≤ lim sup k→∞ 3 [ m + |µ1 −µ2|iaαr(t) ] ( 1 − θ ) ⇒ 0 ≤ lim sup k→∞ ‖yk −yk‖b ≤ 3 [ m + |µ1 −µ2|iaαr(t) ] ( 1 − θ ) . (51) using the assumption lim k→∞ yk = y, lim k→∞ yk = y, we get from (51) that ‖y −y‖b ≤ 3 [ m + |µ1 −µ2|iaαr(t) ] ( 1 − θ ) , (52) which shows the dependence of solutions of the problem (1)-(2) is on parameters µ1 and µ2. remark: the result dealing with the property of a solution called “dependence of solutions on parameters”. here the parameters are scalars. notice that the initial conditions do not involve parameters. the dependence on parameters is an important aspect in various physical problems. 5 example we consider the following problem: ( dα∗ ) y(t) = 3t 5 [t− sin(y(t)) 2 + 1 9 ∫ t 0 e−s (2 + s)2 y(s)ds ] , (53) for t ∈ [0, 1], n− 1 < α ≤ n, n ∈ n, with the given initial conditions y(j)(0) = cj, j = 0, 1, 2, · · · ,n− 1. (54) comparing this equation with the equation (1), we get f ∈ c(i ×r2,r), with f ( t,y(t), ∫ t 0 h(s,y(s))ds ) = 3t 5 [t− sin(y(t)) 2 + 1 9 ∫ t 0 e−s (2 + s)2 y(s)ds ] and h(t,y(t)) = 3t 45 e−t (2 + t)2 y(t). haribhau l. tidke, gajanan s. patil and rupesh t. more now, one can easily show that∣∣∣f(t,y(t),z(t)) −f(t,y(t),z(t))∣∣∣ ≤ 3t 5 [1 2 ∣∣∣ sin(y(t)) − sin(y(t))∣∣∣ + 1 9 ∣∣∣z(t) −z(t)∣∣∣] ≤ 3t 10 [∣∣∣y −y∣∣∣ + ∣∣∣z −z∣∣∣], (55) and ∣∣∣h(t,y(t)) −h(t,z(t))∣∣∣ ≤ 3t 45 e−t (2 + t)2 ∣∣∣y −z∣∣∣, (56) where p(t) = 3t 10 , and q(t) = 3t 45 e−t (2 + t)2 . therefore, we have q = sup t∈[0,1] {q(t)} = 3 180 = 1 60 . thus, we the estimate θ = ia αp(t) ( 1 + (b−a)q ) = ia α 3t 10 ( 1 + 1 60 ) = 3 10 ( 1 + 1 60 ) (ia α)(t) = 61 200 (ia α)(t) = 61 200 tα+1 γ(α + 2) ≤ 1 γ(α + 2) , (t ≤ 1). (57) therefore, the condition θ < 1 is satisfied only if 1 γ(α + 2) < 1. we define the operator t : b → b by (ty)(t) = n−1∑ j=0 cj j! tj + 1 γ(α) ∫ t 0 (t−s)α−1 3s 5 [s− sin(y(s)) 2 + 1 9 ∫ s 0 e−σ (2 + σ)2 y(σ)dσ ] ds, (58) for t ∈ i. since all conditions of theorem 3.1 are satisfied and so by its conclusion, the sequence {yn} associated with the normal s−iterative method (4) for the operator t in (58) converges to a unique solution y ∈ b. existence and uniqueness of solution via s-iteration 6 conclusions firstly, we proved the main result, which address the existence and uniqueness of the solution to the ivp (1)-(2) by the method of normal s−iteration. next, we discussed various properties of solutions like continuous dependence on the initial data, closeness of solutions, and dependence on parameters and functions involve therein. finally, we provided an appropriate example to support all of the findings. acknowledgement: the authors are very grateful to the referees for their comments and remarks. references [1]. r. agarwal, d. o’regan, and d. sahu, iterative construction of fixed points of nearly asymptotically nonexpansive mappings. journal of nonlinear and convex analysis, 8(2007), pp. 61-79. 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[33]. s. soltuz and t. grosan, data dependence for ishikawa iteration when dealing with contractive-like operators, fixed point theory appl, 2008, 242916 (2008). https://doi.org/10.1155/2008/242916. microsoft word argoli_ratio.doc 43 uno scienziato abruzzese: andrea argoli. raffaele mascella, danilo pelusi dipartimento di scienze della comunicazione, università degli studi di teramo nato a tagliacozzo nel 1570, andrea argoli proveniva da famiglia illustre che in quegli anni diede tanti personaggi alla chiesa e alle armi, nonché alle scienze, alle lettere e alla diplomazia. compiuti gli studi di lettere, ancora giovane, si trasferì a napoli dedicandosi agli studi di medicina, matematica e astronomia. qui rimase a lungo prima di trasferirsi a roma ove fu fatto lettore di matematica alla sapienza. a roma godé della protezione del cardinale bixia che mise a sua disposizione la sua ricca biblioteca. dunque, ottenuta la cattedra di matematica nel 1621, la mantenne sino al 1627 quando fu sostituito da b. castelli. si ignorano i motivi della sostituzione, ma forse la causa principale furono i sospetti, suscitati negli animi dei regnanti dal suo studio dell’astrologia, come dice in una nota biografica, certamente ispirata dallo stesso argoli, il segretario dell’accademia degli incogniti di venezia, di cui lo scienziato abruzzese faceva parte. infatti per esse subì delle persecuzioni tanto da non poter godere molta serenità negli studi. dopo la sostituzione, l’argoli rimase ancora a roma, forse addetto alla biblioteca del cardinale bixia, sino al 1632, quando accettò l’invito a trasferirsi nella repubblica veneta dove il senato gli somministrò i mezzi per compiere le sue osservazioni astronomiche. qui assunse l’incarico dell’insegnamento della matematica nell’università di padova, come successore di b. sovero nella cattedra ch’era stata in precedenza anche di g. galilei. del suo insegnamento il senato veneto fu tanto soddisfatto che lo insignì dell’onorificenza di cavaliere di s. marco e, terminati i sei anni di contratto, gli raddoppiò lo stipendio, aumentato ulteriormente più volte in seguito. riavutosi da una grave malattia, si ritirò, negli ultimi tempi, tra i francescani conventuali, vestendovi l’abito. morì nel 1657 a padova e qui fu sepolto nella basilica del santo, ove fin dal 1648 era stato scolpito il monumento con l’epigrafe da lui stesso dettata. l’argoli ebbe grande rinomanza fra i contemporanei per la sua erudizione e le qualità di carattere e d’umanità. i suoi scritti furono numerosi, anche se diversi rimasero manoscritti, e abbracciarono l’enorme mole di conoscenza e di ricerca che lo scienziato di tagliacozzo perseguì in modo concatenato e controverso. fra le sue prime pubblicazioni troviamo i problemi astronomici dei triangoli dimostrati per mezzo dei seni, tangenti e secanti e con la sola moltiplicazione senza divisione (1604), le tavole del primo mobile, colle quali, lasciate le 44 prolissità degli antichi, facilissimamente si compongono le direzioni (1610), nella quale erano annesse le tavole delle posizioni, ed il trattato sull’arte medica, e osservazione riguardanti l’agricoltura e l’arte della navigazione (1621). come si può già intuire i suoi scritti vanno dalla matematica all’astronomia, dall’astrologia dalla medicina, toccando di fatto quasi ogni ramo dello scibile. quando si trasferì a roma cominciò a scrivere le cosiddette effemeridi, diari che contengono osservazioni astronomiche secondo il sistema copernicano, molto note e consultate ai suoi tempi, sebbene non esenti da errori. di queste fu infaticabile compilatore, a cominciare dalle effemeridi dei moti celesti alla longitudine di roma dal 1620 al 1640 (1629), sostenute dalle tavole pruteniche e congruenti con quelle daniche, rodolfine e ticoniane, e dalle effemeridi del moto del sole dal 1621 al 1624 (1623). in esse l’argoli illustra un nuovo sistema (fig. a lato), ideato nella volontà di conciliare il sistema solare copernicano con quello tolemaico. l’introduzione alle stesse effemeridi è un capolavoro di erudizione dove si rivela anche che l’autore conosceva a fondo il greco, l’ebraico e il latino. altre effemeridi tolemaiche, copernicane e ticoniane furono da lui compilate descrivendo i moti celesti fino al 1660. opera di altra natura è invece l’isagoge che contiene i canoni sui precetti dell’astrologia. nelle sue intenzioni quest’opera rappresentava una introduzione alla teoria degli astrologi. la sua opera maggiore è il pandosion sphaericum (fig. a lato), un trattato di astronomia tolemaica pubblicato a padova nel 1644 e, con aggiunte, ripubblicato nel 1653. il pandosion è un trattato completo delle sue dottrine astronomiche, cosmografiche, metereologiche, trigonometriche e metriche, nonché contenente elementi di astrologia. nel pandosion il matematico-astronomo abruzzese affronta il problema della delimitazione e definizione del concetto di astronomia, di universo e procede poi alla trattazione degli «elementi» cosmici, dei meridiani, dei paralleli, dei pianeti, di altri mondi, di giorni critici (l’influsso ippocratico-galenico qui si mescola con l’interpretazione astrologica) e di altri argomenti. altra opera notevole, anche se ormai d’interesse solamente storico, è il ptolomeus parvus, un commentario ai quattro volumi tolemaici col testo greco e latino, ristampato più volte nel corso del secolo. esso è preceduto da una prefazione in cui si esalta l’astrologia e si sostiene che essa non viola le leggi ecclesiastiche, né nega il libero arbitrio, perché il cielo «è come un libro che contiene in sé ogni scritta sul futuro», che tuttavia dio può distruggere o variare a suo piacimento. i due trattati anzidetti sono documenti della diffusione dell’astrologia nel xvii secolo. 45 queste opere furono considerate molto accurate, più di quanto lo erano generalmente opere di quel genere, e si lodò il suo ingegno nell’aver profittato dei libri più scelti della biblioteca del cardinale bixia, nella quale godeva, appunto, libero accesso; così si lodò anche il fatto di aver facilitato le dottrine di ticone. sebbene tolemaico, subì il fascino di g. galilei. nel 1635, sollecitato da roma a scrivergli contro, diede invece una risposta che il micanzio così comunicò a galileo: «degna di un virtuoso, d’un servitore di questo principe e della stima che si deve far di v.s.».1 nello stesso tempo confidò di avere scritto un discorso sul sistema del mondo, nel quale attribuiva alla terra un solo moto, ma temeva «d’incontrar mala ventura» per cui non fu mai pubblicato e non si ha alcuna notizia del suo contenuto e della sua sorte. si interessò della teoria euclidea scrivendo dei commentari come il libro sui problemi di euclide, dimostrati in diversi modi, anche se con lacune ed euclidis, più volte da lui ricordato tra le sue opere inedite, ma mai pubblicato e successivamente disperso. tra le molte sue opere di carattere matematico e astronomico (è il caso di ricordare che con le sue effemeridi aveva stabilito anche una specie di astronomia medica) c’è pure un trattato di medicina, nel quale il contenuto complessivo era mascherato dal titolo due libri sui giorni critici e sul decubito degli infermi, che ebbe due edizioni in padova nel 1639 e nel 1652 e da p. de castro fu giudicata una delle più necessarie a un medico erudito. buona parte del contenuto di quest’opera è oggi superato o, comunque, non preso in considerazione dagli orientamenti della medicina attuale, pure se certa parte, di carattere prognostico (indizi da attingere in base al decubito dei pazienti), ad esempio, mantenga il suo vivo interesse. ciò che colpisce tuttavia è la profonda, ampia erudizione dell’autore, che si muove abilmente e con sicurezza tra la tematica squisitamente ippocratica dell’argomento e i richiami storicoletterari, tra le opinioni degli alessandrini, di galeano, avicenna, alberto magno e tanti altri medici da un lato e le vedute di astrologi e filosofi dall’altro. l’opera è poi corredata di una congerie di schemi medico-astrologici, il cui studio accurato potrebbe tuttora dare qualche spunto di interesse, nei quali si teorizza la soggezione delle interne ed esterne parti del corpo ai pianeti e ai segni zodiacali. sono infine dati gli oroscopi di personalità ragguardevoli tutte morte, papi e re, principi e cardinali. non deve stupire la presenza contemporanea, ancora nel secolo xvii, di interessi astrologici e astronomici con interessi medici. il legame astrologia-medicina, così intenso e operante nel secolo precedente non fu facile a sciogliersi. titolo di grande vanto perviene all’argoli dal fatto di essersi inserito, con la sua speculazione, in quella «filosofia dei circoli», movimento di pensiero che ha esercitato influsso, in certo senso determinante, sulla scoperta di w. harvey sulla circolarità del moto sanguigno ed egli stesso fu uno tra i più autorevoli assertori della nuova concezione dello scienziato inglese. la perfetta aderenza dell’argoli alla 1 g. galilei, le opere, ediz. naz., xvi, firenze 1905, p. 256. 46 tesi harveyana, si incentra in primo luogo in un passo assai significativo del pandosion sphaericum in cui un capitolo è dedicato specificamente al moto circolare sanguigno. in questo intreccio di nuovo e di vecchio, che risulta essere il pandosion, l’illustrazione della circolazione sanguigna si inserisce con naturalezza, pur se concepita pochi anni prima, in quanto alcuni fondamenti di essa costituivano già materia per gli scienziati da più secoli. che la propensione del medico e matematico abruzzese per il «movimento circolare» non desti meraviglia, pur se egli vive a 20 secoli di distanza da aristotele, che aveva riconosciuto tra i movimenti semplici il circolare e il rettilineo2, ciò dipende dal fatto che l’ambiente padovano era, ancora a metà del secolo xvii, impregnato di aristotelismo. tra gli studi e le ricerche più propriamente matematiche è infine da rilevare che l’argoli, come si evidenzia nell’histoire de l’astronomie di ferdinando keefer (parigi 1879), sapeva ridurre tutte le operazioni trigonometriche ad una semplice addizione 10 anni prima che si scoprissero i logaritmi. oggi la sua città natale lo ricorda con una piazza e un asilo infantile a lui intitolati. andrea argoli rappresenta, come asserisce il premuda [4], un simbolo di «quell’umiltà che si cela dietro al lavoro operoso e fecondo delle generazioni abruzzesi» nonché illustre rappresentante di quei «diversi personaggi abruzzesi dei tempi trascorsi, poco noti e viceversa assai fertili nella loro operosità scientifica, più spesso originale». bibliografia [1] r. aurini, dizionario bibliografico della gente d’abruzzo, vol. 1, cooperativa tipografica ars et labora, teramo 1952, pp. 422-429. [2] a. antinori, raccolta di memorie historiche delle tre provincie degli abruzzi, tomo iii, ed. giuseppe campo, napoli 1782, pp. 283-285. [3] a. paoluzi, tagliacozzo, notizie storiche – le chiese, gli edifici – personaggi celebri, escursioni, studio bibliografico a. polla, avezzano 1983, pp. 63-65. [4] l. premuda, medici abruzzesi nello studio di padova, in abruzzo rivista dell’istituto di studi abruzzesi, anno vi, n. 2-3, maggio 1968, pp.505-512. [5] dizionario bibliografico degli italiani, vol. 4, ed. enciclopedia treccani, roma 1962, pp. 132-134. [6] http://es.rice.edu/es/humsoc/galileo/catalog/files/argoli.html, catalogo della comunità scientifica. 2 aristotele, de coelo, ii, 3. l’epigrafe sulla tomba di a. argoli, da lui stesso scritta. ratio mathematica volume 48, 2023 topology via graph ideals p.gnanachandra* k.lalithambigai† abstract the study of ideal topological space has started since 1933 and till date it is being developed by several mathematicians. various classes of open sets, different types of operators and exploration of elementary topological results in ideal topological spaces have been discussed in various research papers. methods of generating topologies using various relations have been explored by many researchers. many researchers explored the methods of inducing topologies via graphs. this paper, introduces the notions of graph ideals, graph local function and characterizes some of their properties. it also describes a method of generating a new graph topology on the vertex set of a graph from the graph adjacency topology using kuratowski closure operator and depicts the nature of open sets with respect to the new topology. further, it explores the condition for compatibility of the graph adjacency topology with graph ideal. keywords: graph ideal, open neighbourhood function, graph local function, graph adjacency topological space 2020 ams subject classifications: 54h99, 57m15, 54a10, 54a05 1 *centre for research and post graduate studies in mathematics, ayya nadar janaki ammal college, sivakasi (affiliated to madurai kamaraj university, madurai), india; pgchandra07@gmail.com. †sri kaliswari college, sivakasi (affiliated to madurai kamaraj university, madurai), india; lalithambigaimsc@gmail.com. 1received on november 14, 2022. accepted on july 9, 2023. published on august 1, 2023. doi: 10.23755/rm.v39i0.956. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. p.gnanachandra, k.lalithambigai 1 introduction topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending. for a very long time, it was believed that abstract topological structures have limited applications in the generalization of real line and complex plane or some connections to algebra and other branches of mathematics. further, it seems that there is a large gap between these structures and real life applications. generating topologies by relations and the representation of topological concepts through binary relations will narrow the gap between topology and its applications. a relation on graph represents a key for bridging graph theory and topological structures. the relation induces new types of topological structures to the graph. in 1967, j.w.evans et al.[5] showed that there is a one-to-one correspondence between the labelled transitive directed graph with n points and the labelled topologies on n points. in 1967, s.s.anderson and g.chartrand[1] investigated the lattice-graph of the topologies of transitive directed graphs presented by j.w.evans et al.[5]. in 2010, c.marijuan[13] studied the relation between directed graphs and finite topologies. in 2013, s.m.amiri et al.[2] induced a topology on vertex set of undirectd graph. in 2018, a.kilicman and k.abdulkalex[8] associated an incidence topology with vertex set of simple graphs without isolated vertices. in 2018, shokry nada et al.[15] generated topologies using the post class relations on the vertex set of graphs and discussed some of its applications in the biomedical field. k.lalithambigai and p.gnanachandra in [12], described a method of generating topology using adjacency, incidence relations on vertex set of graphs and studied the properties of closure and interior of vertex set of subgraphs in the graph adjacency topological space. further, k.lalithambigai and p.gnanachandra in [11], introduced graph grills and explored the properties of topologies induced by graph grills on vertex set of graphs. the topic of ideals in general topological spaces is treated in the classic text by kuratowski[9],[10] and also by r.vaidyanathaswamy in [16],[17]. the properties which a topological space possess locally and the conditions through which those properties become a global one was studied by o.njastad in 1966 [14]. generation of new topologies from the old one via ideals in general topological space was studied by d.jankovic and t.r.hamlett[7]. ideal on a topological space can be used to study the similarity of two structures. till now, scholars have mostly contributed to the study of ideals in general topological space. it can be noticed that most of the real life problems can be modelled as a graph and can be solved using the graph theory concepts. the aim of this paper is to bridge the gap between abstract and concrete concepts in ideal topological spaces, by defining graph ideals and graph local functions in graph adjacency topological space. this paper explores the basic facts of graph local function and describes the method of generating a topology via graph ideal new topology from the older via graph ideals. further, this paper characterizes the nature of the open sets of the new topology in terms of the closure operator and investigates the compatibility of graph adjacency topology with the graph ideal. 2 preliminaries fundamental definitions and preliminaries of graph theory and topology may be found in the sources [3],[4],[6]. a graph g consists of a pair (v(g),x(g)), where v(g) is a nonempty finite set whose elements are called vertices and x(g) is a set of unordered pairs of distinct elements of v(g). the elements of x(g) are called edges of the graph g. an edge joining a vertex to itself is called a loop. edges joining the same vertices are called multiple edges. a graph without loops and multiple edges is called a simple graph. a graph g is called a bipartite graph if v can be partitioned into two disjoint subsets v1 and v2 such that every edge of g joins a vertex of v1 to a vertex of v2. (v1,v2) is called a bipartition of g. if further g contains every edge joining the vertex of v1 to the vertex of v2, then g is called a complete bipartite graph. if v1 contains m vertices and v2 contains n vertices then the complete bipartite graph g is denoted by km,n. it should be noticed that k1,m is called a star for m⩾1. the degree of a vertex v in a graph g is the number of edges incident with v and it is denoted by deg(v). a graph in which degree of every vertex are the same is called a regular graph. a vertex of degree 0 is called an isolated vertex. the concept of ideals in topological spaces has been studied by kuratowski[9] and vaidyanathaswamy[14]. an ideal on a topological space (x, τ) is a non empty collection i of subsets of x satisfying the following two conditions: (i) if a ∈ i and b ⊂ a, then b ∈ i (ii) if a ∈ i and b ∈ i, then a ∪ b ∈ i an ideal topological space is a topological space (x, τ) with an ideal i on x, and is denoted by (x, τ, i) throughout the paper, the graphs under discussion are the simple undirected graphs which are not star graphs. 3 graph ideal and graph local function in this section, graph ideal and graph local function in a graph adjacency topological space are defined with illustrations. definition 3.1. let g = (v(g),x(g)) be a graph. for v ∈ v(g), the neighbourhood set nv of v is defined as nv = {u ∈ v (g) : uv ∈ x(g)}. p.gnanachandra, k.lalithambigai definition 3.2. let g = (v(g),x(g)) be a graph without isolated vertices. define sn as the family of nv for all v ∈ v(g), i.e., sn = {nv : v ∈ v (g)}. then sn forms a subbase for a topology ta on v(g) and the pair (v(g),ta) is called graph adjacency topological space. if w is a vertex induced subgraph of g, then the closure of v(w) is defined by cl(v(w)) = v(w) ∪{v ∈ v (g) : nv ∩ v (w) ̸= ϕ} and the interior of v(w) is defined by int(v(w)) = {v ∈ v (g) : nv ⊆ v (g)}. definition 3.3. let g = (v(g),x(g)) be a graph for which p(v ) and p(x) are the power sets of v(g) and x(g) respectively. the set i = {g′ : g′ = (v ′, x′ ), where v ′ ⊆ v, x′ ⊆ x} is said to be a graph ideal on a graph adjacency topological space (v(g),ta) if it satisfies the following two conditions i) if g′ and g′′ ∈ i, then g′ ∪ g′′ ∈ i ii) if g′ is a subgraph of g′′ and g′′ ∈ i, then g′ ∈ i. example 3.1. consider the following graph 1 2 3 4 5 e 1 e 2 e 4 e 3 e 5 e 6 figure. 3.1 for the graph in figure 3.1, i = {({1, 2}, {e1}), ({1}, ϕ), ({2}, ϕ), ({1, 2}, ϕ)} is a graph ideal. i = {({1, 2, 3}, {e2}), ({4, 5}, {e6}), ({1, 2, 3, 4, 5}, {e2, e6}), ({1, 2}, {e1}) is not a graph ideal, since, ({1}, ϕ) /∈ i, ({2}, ϕ) /∈ i, ({3}, ϕ) /∈ i, ({4}, ϕ) /∈ i, ({5}, ϕ) /∈ i. definition 3.4. let (v (g), ta) be a graph adjacency topological space such that v ∈ v(g). the open neighbourhood system at v denoted by n(v) is defined as n(v) = {u ∈ ta : v ∈ u}. definition 3.5. let (v (g), ta) be a graph adjacency topological space with a graph ideal i. let w be a subgraph of g. then (v (w))∗(i, ta) = {v ∈ v (g) : for every u ∈ n(v), v (w) ∩ u ̸= v (g′ ) for any g′ ∈ i} is called the graph local function of v(w) with respect to i and ta. topology via graph ideal example 3.2. consider the following graph 1 2 3 4 5 6 e 1 e 2 e 4 e3 figure 3.2 graph local function of a subgraph of the graph in figure 3.2 is illustrated below: sn = {{4, 5}, {6}, {5}, {1}, {1, 3}, {2}}. b = {ϕ, {4, 5}, {6}, {5}, {1}, {1, 3}, {2}}. ta = {ϕ, {4, 5}, {6}, {5}, {1}, {1, 3}, {2}, {4, 5, 6}, {4, 5, 1}, {1, 3, 4, 5}, {2, 4, 5}, {5, 6}, {1, 6}, {1, 3, 6}, {2, 6}, {1, 5}, {1, 3, 5}, {2, 5}, {1, 2}, {1, 5, 6}, {1, 4, 5}, {1, 4, 5, 6}, {1, 5, 6}, {1, 3, 5, 6}, {1, 3, 4, 5, 6}, {2, 4, 5, 6}, {2, 4, 5, 1}, {1, 2, 3, 4, 5}, {2, 5, 6}, {1, 2, 6}, {1, 2, 3, 6}, {1, 2, 5}, {1, 2, 3, 5}, {1, 2, 5, 6}, {1, 2, 4, 5}, {1, 2, 4, 5, 6}, {1, 2, 5, 6}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6}}. let i = {({1}, ϕ), ({5}, ϕ), ({1, 5}, ϕ), ({1, 5}, e2)} be a graph ideal. let w = ({1, 2, 3}, ϕ) be a subgraph of the given graph. then (v (w))∗(i, ta) = {2, 3}. note 3.6. 1. let g be a graph with n vertices. for each i = 1,2, ...n, {({vi}, ϕ)} is a graph ideal. in this case, for any subgraph w , (v (w))∗(i, ta) = cl(v (w)). 2. let p(g) be the collection of all subgraphs of a graph g. clearly p(g) is a graph ideal. in this case, for any subgraph w, (v (w))∗(i, ta) = ϕ. 4 facts concerning graph local function this section describes some basic facts concerning the graph location function which are useful in the generation of new topology from the old one. theorem 4.1. let (v (g), ta) be a graph adjacency topological space with graph ideals i and j . let h and w be subgraphs of g. then 1. v(h) ⊆ v(w) ⇒ (v(h))∗ ⊆ (v (w))∗ 2. i ⊆ j ⇒ (v (w))∗(j ) ⊆ (v (w))∗(i) p.gnanachandra, k.lalithambigai 3. (v(w))∗ ⊆ cl(v(w)) 4. ((v(w))∗)∗ ⊆ (v (w))∗ 5. (v(h ∪ w))∗ = (v (h))∗ ∪ (v (w))∗ 6. (v (h))∗ − (v (w))∗ ⊆ (v (h) − v (w))∗ 7. u ∈ ta ⇒ u ∩ (v (w))∗ = u ∩ (u ∩ v (w))∗ ⊆ (u ∩ v (w))∗ 8. (v(w))∗(i) ∪ (v (w))∗(j ) = (v (w))∗(i ∩ j ). 9. (v (h) ∩ v (w))∗ ⊆ (v (h))∗ ∩ (v (w))∗. 10. i = v(g ′ ) for some g ′ ∈ i ⇒ (v (h) ∪ i)∗ = (v (h))∗ = (v (h) − i)∗. 11. (v(h))∗ ((v(h))∗)∗ ⊆ (v (h) − (v (h))∗)∗. proof. 1. let v /∈ (v (w))∗. hence there exists u ∈ n(v) such that v (w) ∩ u = v (g′), for some g′ ∈ i. since v (h) ⊆ v (w), v (h) ∩ u ⊆ v (w) ∩ u. hence there exists u ∈ n(v) such that v (h) ∩ u = v (g′), for some g′ ∈ i. so v /∈ (v (h))∗. hence (v (h))∗ ⊆ (v (w))∗. 2. let v ∈ (v (w))∗(j ). hence, for every u ∈ n(v), v (w) ∩ u ̸= v (g′), for any g′ ∈ j . since i ⊆ j , v (w) ∩ u ̸= v (g′), for any g′ ∈ i. hence for every u ∈ n(v), v (w) ∩ u ̸= v (g′), for any g′ ∈ i. hence (v (w))∗(j ) ⊆ (v (w))∗(i) 3. let v ∈ (v (w))∗. hence for every u ∈ n(v), v (w) ∩ u ̸= v (g′), for any g′ ∈ i. thus v (w) ∩ nv ̸= ϕ and v ∈ cl(v (w)). hence (v (w))∗ ⊆ cl(v (w)). 4. let v ∈ ((v (w))∗)∗. hence for every u ∈ n(v), (v (w))∗ ∩ u ̸= v (g′), for any g′ ∈ i. for every u ∈ n(v), v (w) ∩ u ̸= v (g′), for any g′ ∈ i. so v ∈ (v (w))∗ and ((v (w))∗)∗ ⊆ (v (w))∗. 5. let v /∈ (v (h))∗ ∪ (v (w))∗. then v /∈ (v (h))∗ and v /∈ (v (w))∗. so there exists u1 ∈ n(v) such that v (h) ∩ u1 = v (g ′ ), for some g′ ∈ i and u2 ∈ n(v) such that v (w) ∩ u2 = v (g ′′ ), for some g′′ ∈ i. hence (v (h) ∪ v (w)) ∩ u = v (g′′′), where u = u1 ∪ u2 and v (g ′′′ ) = v (g′) ∪ v (g′′). so there exists u ∈ n(v) such that (v (h) ∪ v (w)) ∩ u = v (g′′′), for some g′′′ ∈ i. topology via graph ideal hence v /∈ (v (h ∪ w))∗ and (v (h ∪ w))∗ ⊆ (v (h))∗ ∪ (v (w))∗ . thus v (h) ⊆ v (h ∪ w) and v (w) ⊆ v (h ∪ w). by (1), (v (h))∗ ⊆ (v (h ∪ w))∗ and (v (w))∗ ⊆ (v (h ∪ w))∗. therefore (v (h))∗ ∪ (v (w))∗ ⊆ (v (h ∪ w))∗. hence (v (h))∗ ∪ (v (w))∗ = (v (h ∪ w))∗. 6. let v /∈ (v (h) − v (w))∗. hence there exists u ∈ n(v) such that (v (h) − v (w)) ∩ u = v (g′), for some g′ ∈ i. so (v (h) ∩ u) − (v (w) ∩ u) = v (g′), for some g′ ∈ i. thus (v (h) ∩ u) = v (g′) and v (w) ∩ u ̸= v (g′), for some g′ ∈ i. so v /∈ (v (h))∗ and v ∈ (v (w))∗. thus v /∈ (v (h))∗ − (v (w))∗. hence (v (h))∗ − (v (w))∗ ⊆ (v (h) − v (w))∗. 7. let u ∈ ta and v ∈ u ∩ (v (w))∗. v ∈ u and v ∈ (v (w))∗. v ∈ u and u ∈ ta ⇒ u ∈ n(v). v ∈ (v (w))∗ ⇒ for every u1 ∈ n(v), v (w) ∩ u1 ̸= v (g ′ ), for any g′ ∈ i. u ∩ (v (w) ∩ u1) ̸= v (g ′ ), for any g′ ∈ i. (u ∩ v (w)) ∩ u1 ̸= v (g ′ ), for any g′ ∈ i. hence v ∈ (u ∩ v (w))∗. so v ∈ u ∩ (u ∩ v (w))∗. u ∩ (v (w))∗ ⊆ u ∩ (u ∩ v (w))∗. by reserving the above steps, it follows that u ∩ (u ∩ v (w))∗ ⊆ u ∩ (v (w))∗. so u ∩ (v (w))∗ = u ∩ (u ∩ v (w))∗. moreover, u ∩ (u ∩ v (w))∗ ⊆ (u ∩ v (w))∗. 8. since i ∩ j ⊆ i and i ∩ j ⊆ j , (v (w))∗(i) ⊆ (v (w))∗(i ∩ j ) and (v (w))∗(j ) ⊆ (v (w))∗(i ∩ j ). hence (v (w))∗(i) ∪ (v (w))∗(j ) ⊆ (v (w))∗(i ∩ j ). let v ∈ (v (w))∗(i ∩ j ). so, for every u ∈ n(v), v (w) ∩ u ̸= v (g′), for any g′ ∈ i ∩ j . for every u ∈ n(v), v (w)∩u ̸= v (g′), for any g′ ∈ i or v (w)∩u ̸= v (g′), for any g′ ∈ j . so v ∈ (v (w))∗(i) or v ∈ (v (w))∗(j ) and so v ∈ (v (w))∗(i)∪(v (w))∗(j ). (v (w))∗(i ∩ j ) ⊆ (v (w))∗(i) ∪ (v (w))∗(j ). so (v (w))∗(i ∩ j ) = (v (w))∗(i) ∪ (v (w))∗(j ). 9. since v (h) ∩ v (w) ⊆ v (h) and v (h) ∩ v (w) ⊆ v (w), by (1), (v (h) ∩ v (w))∗ ⊆ (v (h))∗ and (v (h) ∩ v (w))∗ ⊆ (v (w))∗. hence (v (h) ∩ v (w))∗ ⊆ (v (h))∗ ∩ (v (w))∗. 10. since v (h) ⊆ v (h) ∪ i and v (h) − i ⊆ v (h), by (1), (v (h))∗ ⊆ (v (h) ∪ i)∗ and (v (h) − i)∗ ⊆ (v (h))∗. now, v /∈ (v (h) − i)∗ p.gnanachandra, k.lalithambigai ⇒ there exists u ∈ n(v) such that u ∩ (v (h) − i) = v (g′), for some g′ ∈ i. ⇒ there exists u ∈ n(v) such that u ∩ (v (h) − v (g′′)) = v (g′), for some g′, g′′ ∈ i. so there exists u ∈ n(v) such that (u ∩ v (h)) − (u ∩ v (g′′)) = v (g′), for some g′, g′′ ∈ i. this implies that there exist u ∈ n(v) such that u ∩ v (h) = v (g′), for some g′ ∈ i. ⇒ v /∈ (v (h))∗. so (v (h))∗ ⊆ (v (h) − i)∗. let v ∈ (v (h) ∪ i)∗. for every u ∈ n(v), u ∩ (v (h) ∪ i) ̸= v (g′), for any g′ ∈ i. for every u ∈ n(v), (u ∩ v (h)) ∪ (u ∩ i) ̸= v (g′), for any g′ ∈ i. but i = v (g′′), for some g′′ ∈ i. so u ∩ v (h) ̸= v (g′′′), for any g′′′ ∈ i. so v ∈ (v (h))∗. hence (v (h) ∪ i)∗ ⊆ (v (h))∗. so (v (h) ∪ i)∗ = (v (h))∗ = (v (h) − i)∗. 11. v ∈ (v (h))∗ − ((v (h))∗)∗. so v ∈ (v (h))∗ and v /∈ ((v (h))∗)∗. for every u ∈ n(v), u ∩ v (h) ̸= v (g′), for any g′ ∈ i and there exists u1 ∈ n(v) such that u1 ∩ (v (h))∗ = v (g ′′ ), for some g′′ ∈ i . for every u ∈ n(v), (u ∩ v (h)) − (u ∩ (v (h))∗) ̸= v (g′′′), for any g′′′ ∈ i. for every u ∈ n(v), u ∩ (v (h) − (v (h))∗) ̸= v (g′′′), for any g′′′ ∈ i. v ∈ (v (h) − (v (h))∗)∗. so (v (h)∗ − ((v (h))∗)∗ ⊆ (v (h) − (v (h))∗)∗. 5 topology via graph ideals this section describes a method of generating a new topology from the old one using kuratowski closure operator cl∗. definition 5.1. let (v (g), ta) be a graph adjacency topological space with a graph ideal i. let w be a subgraph of g. define cl∗(v (w)) = v (w)∪(v (w))∗ and t ∗a(i) = {v (h) ⊆ v (g) : cl ∗(v (g) − v (h)) = v (g) − v (h)}. t ∗a(i) is called the graph adjacency topology generated by cl∗. when there is no ambiguity, it is denoted as t ∗a . example 5.1. consider the following graph topology via graph ideal 1 2 34 e 1 e 2 e 4 e3 sn = {{2}, {1, 3, 4}, {2, 4}, {2, 3}} b = {ϕ, {2}, {1, 3, 4}, {2, 4}, {2, 3}, {4}, {3}} ta = {ϕ, {2}, {1, 3, 4}, {2, 4}, {2, 3}, {4}, {3}, {3, 4}, {2, 3, 4}, {1, 2, 3, 4}} i = {({1, 2}, {e1}), ({1}, ϕ), ({2}, ϕ), ({1, 2}, ϕ)} n(1) = {{1, 3, 4}, {1, 2, 3, 4}}, n(2) = {{2}, {2, 4}, {2, 3}, {2, 3, 4}, {1, 2, 3, 4}}, n(3) = {{1, 3, 4}, {2, 3}, {3}, {3, 4}, {2, 3, 4}, {1, 2, 3, 4}}, n(4) = {{1, 3, 4}, {2, 4}, {4}, {3, 4}, {2, 3, 4}, {1, 2, 3, 4}} t ∗a = {ϕ, {3}, {4}, {3, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}. observation 5.2. 1. if i = {({vi}, ϕ)} is a graph ideal, then (v (w))∗(i, ta) = cl(v (w)) for every v (w) ⊆ v (g). hence cl∗(v (w)) = cl(v (w)). t ∗a = {v (h) ⊆ v (g) : cl ∗(v (g) − v (h)) = v (g) − v (h)} = {v (h) ⊆ v (g) : cl(v (g) − v (h)) = v (g) − v (h)} = ta. 2. if p(g) is the collection of all subgraphs of a graph g. then p(g) is a graph ideal. hence for any subgraph w, (v (w))∗(i, ta) = ϕ and cl∗(v (w)) = v (w). t ∗a = {v (h) ⊆ v (g) : cl ∗(v (g) − v (h)) = v (g) − v (h)} = {v (h) ⊆ v (g) : v (g) − v (h) = v (g) − v (h)} = p(g) hence t ∗a is a discrete topology. 3. for every graph ideal i, {({vi}, ϕ)} ⊆ i ⊆ p(g). hence ta ⊆ t ∗a ⊆ discrete topology. proposition 5.1. if i and j are graph ideals such that i ⊆ j , then t ∗a(i) ⊆ t ∗a(j ). proof. i ⊆ j ⇒ (v (h))∗(j ) ⊆ (v (h))∗(i). thus {v (h) ∪ (v (h))∗(j )} ⊆ {v (h) ∪ (v (h))∗(i)}. therefore cl∗(v (h))(j ) ⊆ cl∗(v (h))(i). this means cl∗(v (g) − v (h))(i) ⊆ cl∗(v (g) − v (h))(j ). hence t ∗a(i) ⊆ t ∗ a(j ). p.gnanachandra, k.lalithambigai definition 5.3. the derived set of v(h) in v(g), ta) denoted by (v (h))d is defined by v ∈ (v (h))d if and only if (u − {v}) ∩ v (h) ̸= ϕ for every u ∈ n(v). the derived set of v(h) in v(g), t ∗a) is denoted by (v (h)) d∗. example 5.2. consider the following graph 1 5 2 4 3 e 1 e 2 e 4 e 5 e3 sn = {{1}, {5, 4}, {4}, {3, 2, 5}, {1, 2, 4}} b = {ϕ, {1}, {5, 4}, {4}, {3, 2, 5}, {1, 2, 4}, {2}} ta = {ϕ, {1}, {5, 4}, {4}, {3, 2, 5}, {1, 2, 4}, {2}, {1, 2, 4, 5}, {2, 5}, {2, 3, 4, 5}, {2, 4, 5}, {2, 4}, {1, 2, 3, 4, 5}} let v (h) = {3, 5}. then (v (h))d = {3}. note 5.4. 1. v ∈ (v (h))d∗ ⇔ v ∈ cl∗(v (h) − {v}) ⇔ v ∈ (v (h) − {v}) ∪ (v (h) − {v})∗ ⇔ v ∈ (v (h) − {v})∗ ⇔ for every u ∈ n(v), (v (h) − {v}) ∩ u ̸= v (g′), for any g′ ∈ i hence v ∈ (v (h))d∗ ⇔ for every u ∈ n(v), (v (h)−{v})∩u ̸= v (g′), for any g ′ ∈ i. 2. (v (h))d∗ ⊆ (v (h))∗ definition 5.5. a graph which has either an infinite number of vertices or edges or both is called an infinite graph. definition 5.6. let g be an infinite graph. let if be the graph ideal of finite subgraphs of g and h be a subgraph of g. a vertex v is said to be an ω-accumulation point of v(h) if and only if u ∩ v (h) is infinite for every u ∈ n(v). the set of all ω-accumulation points of v(h) is denoted by (v (h))ω. observation 5.7. (v (h))ω = (v (h))∗(if). definition 5.8. let g be an infinite graph. let ic be the graph ideal of countable subgraphs of g and h be a subgraph of g. a vertex v is said to be a condensation point of v(h) if u ∩ v(h) is uncountable for every u ∈ n(v). the set of all condensation points of v(h) is denoted by (v (h))∗(ic). topology via graph ideal definition 5.9. a graph adjacency topological space (v (g), ta) is said to be a t1 space if for each pair of distinct vertices vi and vj, vi belong to every u ∈ n(vi) for which vi does not belong to any u ∈ n(vj) and vj belongs to every u ∈ n(vj) and vj does not belong to any u ∈ n(vi). remark 5.1. 1. let if be a graph ideal of finite subgraphs of g. since ({vi}, ϕ) ∈ if for each vi ∈ v (g), (v (h))∗ = (v (h))d∗. also (v (h))∗ = (v (h))ω. hence ω-accumulation points of v(h) in (v (g), ta) are precisely the limit points of v(h) in (v (g), t ∗a(if)). since ({vi}, ϕ) ∈ if, ({vi}, ϕ)∗ = ϕ. so cl∗({vi}) = {vi}.hence (v (g), t ∗a) is t1. in t1 spaces, ω-accumulation point and limit point coincides. hence the set of ω-accumulation points of v(h) in (v (g), ta) and (v (g), t ∗a) coincide. (v (h))∗(if) = (v (h))d iff ta = t ∗a(if)) iff (v (g), ta) is t1. 2. let ic be a graph ideal of countable subgraphs of g. since (v (h))∗ is the set of condensation points of v(h) and (v (h))∗ = (v (h))d∗, the condensation points of v(h) in (v(g), ta) are precisely the limit points of v(h) in (v (g), t ∗a(ic)). definition 5.10. let h be a subgraph of a graph g. we say that v(h) is closed and discrete in (v (g), ta) if and only if (v (h))d = ϕ. lemma 5.1. let (v (g), ta) be a graph adjacency topological space with a graph ideal i. if i ∈ i, then i is closed and discrete in (v (g), t ∗a). proof. i ∈ i ⇒ i∗ = ϕ ⇒ id∗ = ϕ. remark 5.2. let (v (g), ta) be a graph adjacency topological space. let icd = {h : h is a subgraph of g and (v (h))d = ϕ}. let us prove icd is a graph ideal. h1, h2 ∈ icd ⇒ (v (h1))d = ϕ and (v (h2))d = ϕ. hence for every v ∈ v (g), (u −{v})∩v (h1) = ϕ and (u −{v})∩v (h2) = ϕ. so for every v ∈ v (g), (u − {v}) ∩ v (h1 ∪ h2) = ϕ. so h1 ∪ h2 ∈ icd. let h1 ∈ icd and h2 be a subgraph of h1. h1 ∈ icd ⇒ (v (h1))d = ϕ. hence for every v ∈ v (g), (u − {v}) ∩ v (h1) = ϕ and so for every v ∈ v (g), (u − {v}) ∩ v (h2) = ϕ. so (v (h2))d = ϕ which implies h2 ∈ icd . icd is a graph ideal. also (v (h))d ⊆ (v (h))∗. hence t ∗a = ta. lemma ?? implies that icd is the largest graph ideal with the property that t ∗a = ta. finally, (v (h))∗ = (v (h))d iff (v (g), ta) is t1. p.gnanachandra, k.lalithambigai example 5.3. let (v (g), ta) be a graph adjacency topological space. let i = {g′ : g′ is a subgraph of g and int(cl(v (g′))) = ϕ}, i.e., i is the collection of nowhere dense subgraphs of (v (g), ta). let g ′ , g ′′ ∈ i. hence int(cl(v (g′))) = ϕ and int(cl(v (g′′))) = ϕ. now, int(cl(v (g ′ ∪ g′′))) = int(cl(v (g′) ∪ cl(v (g′′))) ⊇ int(cl(v (g′)) ∪ int(cl(v (g ′′ )) = ϕ. hence int(cl(v (g ′ ∪ g′′))) ⊇ ϕ. so i is not a graph ideal. 6 the open sets of t ∗a let (v (g), ta) be a graph adjacency topological space with a graph ideal i. let h be a subgraph of g. then v(h) is said to be t ∗a -closed if and only if ((v (h))∗ ⊆ v (h). hence v(h) is t ∗a -open if and only if v(g) v(h) is t ∗ a closed, i.e., if and only if (v (g) − v (h))∗ ⊆ v (g) − v (h), i.e., if and only if v (h) ⊆ v (g) − (v (g) − v (h))∗. hence v(h) is t ∗a -open if and only if v ∈ v (h) ⇒ v /∈ (v (g) − v (h)) ∗. so v(h) is t ∗a -open if and only if v ∈ v (h) ⇒ there exists u ∈ n(v) such that u ∩ (v (g) − v (h)) = v (g′), for some g′ ∈ i. let w = u ∩ (v (g) − v (h)). v(h) is t ∗a -open if and only if, for v ∈ v (h) there exists u ∈ n(v) such that w = v (g′), for some g′ ∈ i. if v ∈ v (h) then v /∈ v (g)−v (h). so v /∈ w . also v ∈ u. hence v ∈ u −w . v(h) is t ∗a -open if and only if, for v ∈ v (h), there exists u ∈ n(v) such that v ∈ u − w and w = v (g′) for some g′ ∈ i. let β(i, t ∗a) = {u − w : u ∈ ta, w = v (g ′ ) for some g ′ ∈ i}. β(i, t ∗a) is a basis for t ∗a . note 6.1. if i and j are graph ideals then i ∨ j = {i ∪ j : i ∈ i, j ∈ j } is also a graph ideal. theorem 6.1. let (v (g), ta) be a graph adjacency topological space with a graph ideals i and j . let h be a subgraph of g. then (v (h))∗(i ∨ j , ta) = (v (h))∗(i, t ∗a(j )) ∩ (v (h)) ∗(j , t ∗a(i)). proof. let v /∈ (v (h))∗(i ∨ j , ta). hence there exists u ∈ n(v) such that u ∩v (h) = v (g′), for some g′ ∈ i ∨j . let g ′′ ∈ i and g′′′ ∈ j such that u ∩ v (h) = v (g′′′′) where g′′′′ = g′′ ∪ g′′′ . assume that v (g ′′ ) ∩ v (g′′′) = ϕ. hence (u ∩ v (h)) − v (g′′) = v (g′′′) and (u ∩ v (h)) − v (g′′′) = v (g′′). so (u −v (g′′))∩v (h) = v (g′′′), where g′′′ ∈ j and (u −v (g′′′))∩v (h) = v (g ′′ ), where g ′′ ∈ i. topology via graph ideal ⇒ v /∈ (v (h))∗(j , t ∗a(i)) or v /∈ (v (h)) ∗(i, t ∗a(j )). ⇒ v /∈ (v (h))∗(j , t ∗a(i)) ∩ (v (h)) ∗(i, t ∗a(j )). hence (v (h))∗(i, t ∗a(j )) ∩ (v (h)) ∗(j , t ∗a(i)) ⊆ (v (h)) ∗(i ∨ j , ta). v /∈ (v (h))∗(i, t ∗a(j )) implies that there exists u ∈ n(v) such that (u − v (g ′′ )) ∩ v (h) = v (g′) for some g′ ∈ i, g′′ ∈ j . assume that v (g ′′ ) ⊆ v (h). hence u ∩ v (h) = v (g′) ∪ v (g′′) = v (g′′′), where g ′′′ ∈ i ∨ j . so v /∈ (v (h))∗(i ∨ j , ta). (v (h))∗(i ∨ j , ta) ⊆ (v (h))∗(i, t ∗a(j )). similarly, (v (h))∗(i ∨ j , ta) ⊆ (v (h))∗(j , t ∗a(i)). (v (h))∗(i ∨ j , ta) ⊆ (v (h))∗(i, t ∗a(j )) ∩ (v (h)) ∗(j , t ∗a(i)). hence (v (h))∗(i ∨ j , ta) = (v (h))∗(i, t ∗a(j )) ∩ (v (h)) ∗(j , t ∗a(i)). remark 6.1. given a graph adjacency topological space (v (g), ta) and a graph ideal i, t ∗∗a = (t ∗ a(i)) ∗(i) is a topology on v (g) and t ∗∗a is finer than t ∗ a . corollary 6.1. let (v (g), ta) be a graph adjacency topological space with a graph ideal i. then (v (h))∗(i, ta) = (v (h))∗(i, t ∗a) and hence t ∗ a = t ∗∗ a . proof. (v (h))∗(i, ta) = (v (h))∗(i ∨ i, ta) = (v (h))∗(i, t ∗a(i)) ∩ (v (h)) ∗(i, t ∗a(i)) = (v (h))∗(i, t ∗a(i)) t ∗∗a = t ∗ a(i)) ∗(i) = t ∗a . 7 compatibility of ta with i this section introduces the definition of compatibility of ta with the graph ideal i and explores its significance. definition 7.1. let (v (g), ta) be a graph adjacency topological space with a graph ideal i. the graph adjacency topology ta is said to be compatible with the graph ideal i, denoted ta ∽ i, if the following holds for every vertex induced subgraph h of g : if for every v ∈ v (h), there exists a u ∈ n(v) such that u ∩ v (h) = v (g′), for some g ′ ∈ i, then v (h) = v (g′′), for some g′′ ∈ i. example 7.1. consider the following graph p.gnanachandra, k.lalithambigai 1 4 3 2 e 1 e 2 e 4 sn = {{4}, {3}, {2, 4}, {1}}. b = {ϕ, {4}, {3}, {2, 4}, {1}}. ta = {ϕ, {4}, {3}, {2, 4}, {1}, {3, 4}, {1, 4}, {2, 3, 4}, {1, 3}, {1, 2, 4}, {1, 3, 4}, {1, 2, 3, 4}}. n(1) = {{1}, {1, 4}, {1, 3}, {1, 2, 4}, {1, 3, 4}, {1, 2, 3, 4}}. n(2) = {{2, 4}, {2, 3, 4}, {1, 2, 4}, {1, 2, 3, 4}}. n(3) = {{3}, {3, 4}, {2, 3, 4}, {1, 3}, {1, 3, 4}, {1, 2, 3, 4}}. n(4) = {{4}, {2, 4}, {3, 4}, {1, 4}, {2, 3, 4}, {1, 2, 4}, {1, 3, 4}, {1, 2, 3, 4}}. ta is compatible with the graph ideal i = {({1}, ϕ), ({2}, ϕ), ({1, 2}, ϕ)} and not compatible with the graph ideal i = {({1}, ϕ), ({4}, ϕ), ({1, 4}, ϕ)}. observation 7.2. let (v (g), ta) be a graph adjacency topological space with a graph ideal i. then the following are implied by ta ∽ i : (a) for every vertex induced subgraph h of g, v (h) ∩ (v (h))∗ = ϕ ⇒ (v (h))∗ = ϕ. (b) for every vertex induced subgraph h of g, (v (h) − (v (h))∗)∗ = ϕ. (c) for every vertex induced subgraph h of g, (v (h)∩(v (h))∗)∗ = (v (h))∗. (d) β(i, t ∗a) = {u − w : u ∈ ta, w = v (g ′ ), for some g ′ ∈ i} = t ∗a . theorem 7.1. let (v (g), ta) be a graph adjacency topological space with a graph ideal i. then the following are equivalent: (1) ta ∽ i (2) for every vertex induced subgraphs h of g, v (h)∩(v (h))∗ = ϕ ⇒ v (h) = v (g ′ ), for some g ′ ∈ i. (3) for every vertex induced subgraphs h of g, v (h)−(v (h))∗ = ϕ or v (h)− (v (h))∗ = v (g ′ ), for some g ′ ∈ i. (4) for every t ∗a closed subgraph h, v (h)−(v (h)) ∗ = v (g ′ ) for some g ′ ∈ i. (5) for every vertex induced subgraphs h of g, if v(h) contains no induced subgraph w with v (w) ⊆ (v (w))∗, then v (h) = v (g′) for some g′ ∈ i. topology via graph ideal proof. (1) ⇒ (2) is obvious. (2) ⇒ (3): for every vertex induced subgraphs h of g, v (h) − (v (h))∗ = ϕ or (v (h)−(v (h))∗)∩(v (h)−(v (h))∗)∗ = ϕ. so by (2), v (h)−(v (h))∗ = ϕ or v (h) − (v (h))∗ = v (g′), for some g′ ∈ i. (3) ⇒ (4) is straightforward. (4) ⇒ (1): let h be a vertex induced subgraph of g and assume that for every v ∈ v(h), there exists u ∈ n(v) such that u ∩ v (h) = v (g′) for some g′ ∈ i. then v (h) ∩ (v (h))∗ = ϕ. since (v (h)∪(v (h))∗)∗ = (v (h))∗∪((v (h))∗)∗ ⊆ (v (h))∗∪v (h), v (h)∪ (v (h))∗ is t ∗a closed. so by (4), (v (h) ∪ (v (h))∗) − (v (h) ∪ (v (h))∗)∗ = v (g′) for some g′ ∈ i. but (v (h)∪(v (h))∗)−(v (h)∪(v (h))∗)∗ = (v (h)∪(v (h))∗)−((v (h))∗∪ (v (h))∗∗) = (v (h) ∪ (v (h))∗) − (v (h))∗ = v (h). hence v (h) = v (g ′ ), for some g ′ ∈ i. so ta ∽ i (3) ⇒ (5): let h be a vertex induced subgraph of g and assume that v(h) contains no induced subgraph w with v (w) ⊆ (v (w))∗. since v (h)−(v (h))∗ = v (g′), for some g′ ∈ i, v (h)∩(v (h))∗ ⊆ (v (h)∩ (v (h))∗)∗, v (h) ∩ (v (h))∗ = ϕ. so v (h) = v (h) − (v (h))∗ and v (h) = v (g′), for some g′ ∈ i. (5) ⇒ (3) : since (v (h)−(v (h))∗)∩(v (h)−(v (h))∗)∗ = ϕ, v (h)−(v (h))∗ contains no vertex induced subgraph w such that v (w) ⊆ (v (w))∗. so by (5), v (h) − (v (h))∗ = v (g′) for some g′ ∈ i. theorem 7.2. let (v (g), ta) be a graph adjacency topological space with a graph ideal i and ta ∽ i. let h be a vertex induced subgraph of g. if v(h) is t ∗a closed then it is the union of v(w), where (v (w)) ∗ ⊆ cl(v (w)), and v (g′) for some g ′ ∈ i. proof. let v(h) be t ∗a closed. then (v (h)) ∗ ⊆ v (h). so v (h) = (v (h) − (v (h))∗) ∪ (v (h))∗. by theorem ??(3), v (h) − (v (h))∗ = v (g′) for some g′ ∈ i. also (v (h))∗ ⊆ cl(v (h)). so v(h)is the union of v(w), where (v (w))∗ ⊆ cl(v (w)), and v (g′) for some g ′ ∈ i. 8 conclusions graph ideal and graph local function in a graph adjacency topological space have been defined and the basic facts concerning graph local function have been p.gnanachandra, k.lalithambigai proved. the method of generating a new topology from the older one using kuratowski closure operator via graph ideal has been discussed. further, the characteristics of the open sets of the new topology in terms of the closure operator have been analyzed. new topologies generated by two different ideals have been compared. the feature of graph location function based on the union of two ideals has been studied. the compatibility of the graph adjacency topology with graph ideal has been defined and the equivalent conditions for compatibility of the graph adjacency topology with graph ideal have been investigated. further, the results in this paper are useful in the study of some new sets and topologies in graph adjacency topological space with graph ideal. these concepts can be studied further by stating graph local function of induced subgraphs of a graph with respect to graph adjacency topology using graph prime ideal and graph principal ideal. the open sets of topologies generated via prime ideals and principal ideals can be compared and investigated in future studies. acknowledgement we are overwhelmed to acknowledge our thankfulness to those who helped us to put these ideas, well above the level of simplicity and into something concrete. we would like to express our special thanks to editor in chief, ratio mathematica for his time and efforts for the successful completion of this paper. his useful advice and suggestions were really helpful to us. the generosity and expertise of the reviewers have improved this study in innumerable ways and saved us from many errors;those that inevitably remain are entirely our own responsibility. we gratefully acknowledge the reviewers who stood by us and encouraged us to work on this paper. references s.s.anderson,g.chartrand,the lattice-graph of the topology of a transitive directed graph,mathematica scandinavica,21(1967),105-109. s.m. amiri,a.jafarzadeh,h.khatibzadeh an alexandroff topology on graphs,bulletin of iranian mathematical society,39(2013),647-662 j.a.bondy,u.s.r.murthy,graph 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c.marijuan,finite topology and digraphs,proyecciones,29,(2010),291-307. o. njastad,remarks on topologies defined by local properties,avh.norske vid.akad oslo i(n.s),8(1966),1-16 shokry nada, abd el fattah el atik and mohammed atef,new types of topological structures via graphs,mathematical methods in the applied sciences,(2018),1-10. r.vaidyanathaswamy,set topology,chelsea publishing company,(1960). r.vaidyanathaswamy,the localization theory in set-topology,proc.indian. acad. sci.20(1945),51-61. ratio mathematica volume 41, 2021, pp. 173-180 tikhonov type regularization for unbounded operators e shine lal* p ramya† abstract in this paper, we introduce a tikhonov type regularization method for an ill-posed operator equation tx = y, where t is a closed densely defined unbounded operator on a hilbert space h. keywords: densely defined operator, closed operator, tikhonov type regularization. 2020 ams subject classifications: 47a10, 47a52 1 *e shine lal, (department of mathematics, university college, thiruvananthapuram, kerala, india -695034); shinelal.e@gmail.com †p ramya, (department of mathematics, n.s.s college, nemmara, kerala, india-678508); ramyagcc@gmail.com 1received on september 20, 2021. accepted on december 10, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.663. ©the authors. this paper is published under the cc-by licence agreement. 173 e shine lal, p ramya 1 introduction most of the problems arise in the field of science and engineering can be modelled as an operator equation tx = y (1) where t : x → y is a bounded linear map from a normed linear space x to a normed linear space y . in most of the cases (1) is ill-posed. certain regularization procedures are known for solving ill-posed operator equation (1). for example tikhonov regularization, mollifier method, ritz method [5, 3]. in this paper we introduce a tikhonov type regularization method for solving an ill-posed operator equation (1), where t is a closed densely defined operator on a hilbert space h and we study the order of convergence. 2 preliminaries let l(h),c(h) and b(h) denote the space of all linear, closed linear and bounded linear operators on a hilbert space h respectively. for t ∈ l(h), the domain, range of t are denoted by d(t),n(t) respectively. an operator t ∈ l(h) is said to be densely defined if d(t) = h. for example let t : l2(n) → l2(n) defined by t(x1,x2,x3, ....,xn, ....) = (x1, 2x2, 3x3, .....,nxn, ....) with domain d(t) = {(x1,x2,x3, ....,xn, ....) ∈ h : σ∞j=1|jxj| 2 < ∞}. then t is closed and unbounded. since c00 ⊆ d(t) and c00 is dense in l2(n), d(t) is dense in l2(n). proposition 2.1. let t ∈ c(h) be a densely defined operator. then there exist a unique operator t∗ ∈ c(h) such that 〈tx,y〉 = 〈x,t∗y〉 ∀x ∈ d(t), ∀y ∈ d(t∗). proof. let d(t∗) = {y ∈ h : 〈tx,y〉 is continuous for every x ∈ d(t) }. for y ∈ d(t), define f : d(t) → c by f(x) = 〈tx,y〉 ∀x ∈ d(t). extend f to f0 : h → c by f0(x) = lim n→∞ 〈txn,y〉 where (xn) is a sequence in d(t) such that xn → x. 174 tikhonov type regularization for unbounded operators next we prove that f0 is well defined. for, let (xn) and (yn) be two sequences in d(t) converges to x. since t is closed, t(xn−yn) → 0. if 〈txn,y〉→ 〈x,y〉, then |〈tyn,y〉−〈x,y〉| = |〈tyn −txn + txn −x,y〉| ≤ ‖t(yn −xn)‖‖y‖ + |〈txn −x,y〉| → 0 as n →∞ hence f0 is well defined. since f0 is a bounded linear functional on the hilbert space h, by riesz representation theorem there exist a unique y∗ ∈ h such that f0(x) = 〈x,y∗〉. thus 〈tx,y〉 = 〈x,y∗〉 ∀x ∈ d(t). define t∗ : d(t∗) → h by t∗y = y∗. then t∗ is well-defined. also 〈tx,y〉 = 〈x,t∗y〉 ∀x ∈ d(t), ∀y ∈ d(t∗). consider an ill-posed operator equation tx = y (2) where t is a closed densely defined operator on h. definition 2.1. [7] let t ∈ c(h) be densely defined. then there exist a unique densely defined operator t† ∈ c(h) with domain d(t†) = r(t)⊕r(t)⊥ satisfies the following properties (i) tt†y = p r(t) y for all y ∈ d(t†), (ii) t†tx = qn(t)⊥x for all x ∈ d(t). (iii) n(t†) = r(t)⊥. where p and q are the orthogonal projection on to r(t) and n(t⊥) respectively. the operator t† is called the moore-penrose inverse of t . for y ∈ d(t†), let sy = {x ∈ d(t) : ‖tx−y‖ ≤ ‖tu−y‖ ∀u ∈ d(t)}. then u ∈ sy is called least square solution of the operator equation (2). note that ‖t†y‖ ≤ ‖x‖ ∀x ∈ sy, is called least square solution of minimal norm and is denoted by x̂ [7]. if r(t) is not closed, then t† is not continuous. now we introduce a tikhonov type regularization procedure for finding an approximate solution for t†y. 3 tikhonov type regularization in this section we introduce a tikhonov type regularization procedure for solving (2). 175 e shine lal, p ramya lemma 3.1. let t ∈ c(h) be densely defined and α > 0. then t∗t + αi and tt∗+αi are bijective closed densely defined operators on h. also (tt∗+αi)−1 and (t∗t + αi)−1 are bounded, self adjoint operators on h. proof. let t ∈ c(h) and α > 0. by proposition 2.1, we have t∗ ∈ c(h). hence, (tt∗ + αi) and (t∗t + αi) are closed densely defined operators on h. since 〈(tt∗ + αi)x,x〉 = 〈t∗x,t∗x〉 + α〈x,x〉 ≥ 0,∀x ∈ d(t∗), we have (tt∗+αi) is a positive operator. similarly (t∗t +αi) is also a positive operator. since t∗t + αi is positive, ‖(t∗t + αi)x‖‖x‖≥ 〈(t∗t + αi)x,x〉 = 〈t∗tx,x〉 + α‖x‖2 ≥ α‖x‖2 ∀x ∈ h thus ‖(t∗t + αi)x‖≥ α‖x‖∀x ∈ h (3) since t∗t + αi is bounded below, it is one-one and its inverse from the range is continuous. also r(t∗t + αi) is closed. since t∗t + αi is also self adjoint, r(t∗t + αi) = n(t∗t + αi)⊥ = h. hence t∗t + αi is onto. therefore (t∗t + αi)−1 ∈ b(h). similary (tt∗ + αi)−1 ∈ b(h). from (3), ‖(t∗t + αi)−1‖≤ 1 α . theorem 3.1. let t ∈ c(h) be densely defined. then t∗(tt∗ + αi)−1 and t(t∗t + αi)−1 are bounded operators on h. also ‖ t∗(tt∗ + αi)−1 ‖≤ 1√ α and ‖ t(t∗t + αi)−1 ‖≤ 1√ α . proof. we have (t∗t + αi)−1t∗t = i −α(t∗t + αi)−1 since 〈(t∗t + αi)−1x,x〉≥ 0 ∀x ∈ h, 〈(t∗t + αi)−1t∗tx,x〉 = 〈i −α(t∗t + αi)−1x,x〉 = 〈x,x〉−α〈(t∗t + αi)−1x,x〉≤ 〈x,x〉. since (t∗t + αi)−1t∗t self adjoint, ‖(t∗t + αi)−1t∗t‖≤ 1. let x ∈ h. ‖t∗(tt∗ + αi)−1x‖2 = 〈t∗(tt∗ + αi)−1x,t∗(tt∗ + αi)−1x〉 = 〈tt∗(tt∗ + αi)−1x, (tt∗ + αi)−1x〉 = 〈(tt∗ + αi)−1tt∗x, (tt∗ + αi)−1x〉 ≤ ‖(tt∗ + αi)−1tt∗x‖‖(tt∗ + αi)−1x‖ ≤ 1 α ‖x‖2 176 tikhonov type regularization for unbounded operators we have ‖t∗(tt∗ + αi)−1x‖2 ≤ 1 α ‖x‖2 ∀x ∈ h. thus ‖t∗(tt∗ + αi)−1‖≤ 1 √ α . hence t∗(tt∗ + αi)−1 is bounded. similarly t(t∗t + αi)−1 is bounded. lemma 3.2. [7] let t ∈ c(h) be densely defined. then (i) (tt∗ + i)−1t ⊆ t(t∗t + i)−1 (ii) (t∗t + i)−1t∗ ⊆ t∗(tt∗ + i)−1 remark 3.1. from theorem 3.1 , we have t∗(tt∗ + αi)−1 and t(t∗t + αi)−1 are bounded. therefore from lemma 3.2, we have (tt∗ + αi)−1t and (t∗t + αi)−1t∗ are bounded. lemma 3.3. let t ∈ c(h) be densely defined. for every x ∈ d(t) ∩ n(t)⊥ ‖α(t∗t + αi)−1x‖−→ 0,as α → 0. proof. let tα = α(t∗t + αi)−1, α > 0. from (3.1) we have ‖(t∗t + αi)−1‖≤ 1 α . hence ‖tα‖≤ 1 for every α > 0. let u ∈ r(t∗t) then there exist v ∈ d(t∗t) such that t∗tv = u. ‖tαu‖ = ‖tαt∗tv‖ = α‖(t∗t + αi)−1t∗tv‖ ≤ α‖(t∗t + αi)−1t∗t‖‖v‖ ≤ α‖v‖ hence ‖tαu‖≤ α‖v‖∀u ∈ r(t∗t). thus for every u ∈ r(t∗t), ‖α(t∗t + αi)−1u‖ −→ 0 as α −→ 0. since r(t∗t) = n(t)⊥, ‖α(t∗t + αi)−1x‖−→ 0, ∀x ∈ d(t) ∩n(t)⊥. theorem 3.2. let t ∈ c(h) be densely defined and rα = (t∗t + αi)−1t∗. then {rα}α>0 is a regularization family for (2). proof. let y ∈ d(t∗). then (t∗t + αi)x̂ = t∗y + αx̂. hence x̂ = (t∗t + αi)−1(t∗y + αx̂). thus t†y −rαy = x̂− (t∗t + αi)−1t∗y = (t∗t + αi)−1(t∗y + αx̂) − (t∗t + αi)−1t∗y = (t∗t + αi)−1αx̂ 177 e shine lal, p ramya hence ‖t†y −rαy‖ = α‖(t∗t + αi)−1x̂‖. since x̂ ∈ d(t) ∩n(t)⊥, by lemma 3.3, ‖t†y −rαy‖−→ 0 as α −→ 0. thus {rα}α>0 is a regularization family for (2). 4 order estimate in this section we find an error estimate for the regularization family rα = (t ∗t + αi)−1t∗, where t is a closed densely defined operator. we use the following lemmas. lemma 4.1. [7] for t ∈ c(h) we have the following (i) if µ ∈ c and λ ∈ σ(t) then λ + µ ∈ σ(t + µi) (ii) if α ∈ c and λ ∈ σ(t) then αλ ∈ σ(αt ) (iii) σ(t2) = {λ2 : λ ∈ σ(t)} lemma 4.2. [7] let t ∈ l(h) be a positive operator. then the following results bold. (i) t† is positive. (ii) σ(t) = σa(t) (iii) 0 /∈ σ(i + t) that is (i + t)−1 ∈ b(h) (iv) if 0 /∈ σ(t) then 0 6= λ ∈ σ(t) if and only if 1 λ ∈ σ(t−1) theorem 4.1. suppose t ∈ c(h) is densely defined positive operator. then for every α > 0 σ ( (t + αi)−2t ) = { λ (λ + α)2 : λ ∈ σ(t) } proof. since t is positive, t + αi is bijective. also (t + αi)−2t = (t + αi)−1 −α(t + αi)−2. from lemmas 4.1, 4.2 for α,λ > 0, we have λ ∈ σ(t) if and only if (λ + α)−1 ∈ σ ( (t + αi)−1 ) . hence σ ( (t + αi)−2t ) = { µ−αµ2 : µ ∈ σ ( (t + αi)−1 )} = { 1 λ + α − α (λ + α)2 : λ ∈ σ(t) } = { λ (λ + α)2 : λ ∈ σ(t) } . 178 tikhonov type regularization for unbounded operators corolary 4.1. let t ∈ c(h) be densely defined and α > 0. then ‖(t∗t + αi)−1t∗‖ = sup { √λ λ + α : λ ∈ σ(t∗t) } 6 1 2 √ α proof. we have rα = (t∗t + αi)−1t∗. hence r∗αrα = t(t ∗t + αi)−2t∗. from lemma 2.2 in [2], we have r∗αrα = (tt ∗ + αi)−2tt∗. since r∗αrα is self adjoint and bounded, ‖rα‖2 = ‖r∗αrα‖ = sup { |k| : k ∈ σ(r∗αrα) } = sup { λ (λ + α)2 : λ ∈ σ(tt∗) } ‖rα‖ = sup { √λ λ + α : λ ∈ σ(tt∗) } . since 2 √ αλ(λ + α)−1 6 1 for λ,α > 0, we have ‖rα‖ 6 1 2 √ α . now we find an order estimate for rα. corolary 4.2. let t ∈ c(h) is densely defined and rα = (t∗t + αi)−1t∗. for every α > 0 and δ > 0, let yδ ∈ h be such that ‖y − yδ‖ 6 δ. then ‖rαy −rαyδ‖ 6 δ 2 √ α . proof. for ‖y −yδ‖ 6 δ, ‖rαy −rαyδ‖ 6 ‖rα‖‖y −yδ‖ 6 1 2 √ α ‖y −yδ‖ 6 δ 2 √ α theorem 4.2. let t ∈ c(h) is densely defined and rα = (t∗t +αi)−1t∗. then ‖x̂−rαyδ‖ 6 ‖x̂−rαy‖ + δ 2 √ α . if α = α(δ) is chosen such that α(δ) −→ 0 and δ √ α(δ) −→ 0 as δ −→ 0, then ‖x̂−rδα(δ)‖−→ 0 as δ −→ 0. proof. ‖x̂−rαyδ‖≤‖x̂−rαy‖ + ‖rαy −rαyδ‖ ≤‖x̂−rαy‖ + δ 2 √ α by theorem 3.6, ‖x̂−rαy‖−→ 0 as α −→ 0. 179 e shine lal, p ramya references [1] p mathe b hofmann and h. r.v weizsacker. regularization in hilbert space under unbounded operators and general source conditions. iop publishing inverse problems, 2009. 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[7] m. t nair s. h kulkarni and g ramesh. some properties of unbounded operators with closed range. proc.indian acadamy of science, 118:613–625, 2008. 180 ratio mathematica volume 47, 2023 υg-operator in grill n-topology antony george a* davamani christober m† abstract in 2017, lellis thivagar et.al [4] introduced a closure operator nτgcl by using the local function φg in grill n-topology. in this article, we introduce a new operator υg in the same topological space. we study the properties of this new operator which helps us to derive a few equivalent expressions and a characterizing condition, in terms of υg. then a suitability condition for a grill in n-topological space x is formulated. also, we discuss the characterizing condition for the discussed suitability condition. in addition, we introduce and study υ̂g–sets and utilize the υg -operator to define a generalized open set and their properties. keywords: grill n-topology, n-topology suitable for grill, relatively g-dense, anti-co dense. 2020 ams subject classifications: 54a05, 54a99, 54c10. 1 *the american college, madurai, india; email: antonygeorge@americancollege.edu.in. †the american college, madurai, india; email: christober.md@gmail.com. 1received on november 29, 2022. accepted on april 28, 2023. published on june 30, 2023. doi: 10.23755/rm.v41i0.967. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 385 antony george a and davamani christober m 1 introduction the grill concept is a powerful supporting tool, like nets and filters, in dealing with many a topological concept quite effectively. the idea of grill on a topological space was first introduced by coquet [1]. later cattopadhyay and thorn [2] proved the grills are always unions of ultra-filters. further roy and mukherjee [6] established typical topology associated with grill on a topological space (x,τ). lellis thivagar et.al [4] initiated the concept of grill in n-topological space and a topology nτg was introduced in terms of an operator φg, constructed rather naturally from a grill g on a n-topological space (x,τ). in this paper, we endeavour for an investigation in grill-associated n-topology with a new orientation. we introduce a new operator υg, defined in terms of the previously introduced operator φg, as a kind of dual of φg. we study the properties of this new operator which helps us to derive equivalent expressions for the operator υg and a characterizing condition, in terms of υg. also, a suitability condition is formed for a grill in n-topological space. also, we discuss the characterizing condition for the discussed suitability condition. finally, we introduce and study υ̂g–sets and utilize the υg -operator to define a generalized open set and their properties. 2 prerequisites in this section we recollect some definitions and results which are beneficial in the sequent. by a space x, we mean a grill n-topological space (x,nτ,g) with n-topology nτ and grill g on x on which no separation axioms are assumed unless explicitly stated. definition 2.1. [4] a non-empty collection g of non empty subsets of a ntopological space (x,nτ) is called a grill on x if, (i) a ∈ g and a ⊆ b ⊆ x =⇒ b ∈ g, (ii) a,b ⊆ x and a∪b ∈ g =⇒ a ∈ g or b ∈ g. then a n-topological space (x,nτ) together with a grill g is called a grill ntopological space and it is denoted by (x,nτ,g). 386 υg-operator in grill n-topology remark 2.2. [4] in (x,nτ), the following are true. (i) the grill g = p(a)−{∅} is maximal grill in any n-topological space (x,nτ) (ii) the grill g = {x} is the minimal grill in any n-topological space (x,nτ). definition 2.3. [4] let (x,nτ,g) be a grill n-topological space and for each a ⊆ x, the operator φg(a,nτ) = {x ∈ x | a ∩ u ∈ g,∀ u ∈ nτ(x)} is called the local function associated with the grill g and the n-topology nτ. it is denoted as φg(a). for any point x of a n-topological space (x,nτ), nτ(x) means the collection of all nτ-open sets containing x. theorem 2.4. [4] let (x,nτ) be a n-topological space. then the following are true. (i) if g is any grill on x, then φg is an increasing function in the sense that a ⊆ b implies φg(a,nτ) ⊆ φg(b,nτ). (ii) if g1 and g2 are two grills on x with g1 ⊆ g2, then φg1 (a,nτ) ⊆ φg2 (a,nτ) for all a ⊆ x. (iii) for any grill g on x and if a /∈ g, then φg(a,nτ) = ∅. theorem 2.5. [4] let (x,nτ,g) be a grill n-toplogical space. then for all a,b ⊆ x. (i) φg(a∪b) ⊇ φg(a) ∪ φg(b) (ii) φg(φg(a)) ⊆ φg(a) = nτ-cl(φg(a)) ⊆ nτ-cl(a). theorem 2.6. [4] if g is a grill on a n-toplogical space (x,nτ) with nτ −{∅} ⊆ g, then for all u ∈ nτ, u ⊆ φg(u). lemma 2.7. [4] for any grill g on a n-topological space (x,nτ) and any a,b ⊆ x, φg(a) − φg(b) = φg(a−b) − φg(b). corolary 2.8. [4] let g is a grill on a n-topological space (x,nτ) and suppose a,b ⊆ x with b /∈ g. then φg(a∪b) = φg(a) = φg(a−b). proposition 2.9. [4] corresponding to a grill g on a n-topological space (x,nτ), the operator nτg-cl : p(x) → p(x) defined by nτg-cl(a) = a ∪ φg(a), for all a ⊆ x, satisfies kuratowski’s closure axioms and also there exists a unique topology nτg = {u ⊆ x | nτg-cl(uc) = uc} which is finer than nτ. 387 antony george a and davamani christober m proposition 2.10. [4] in a grill n-topological space (x,nτ,g), nτ ⊆ β(g,nτ) ⊆ nτg and in particular if g = p(x) −{∅} then nτ = β(g,nτ) = nτg. proposition 2.11. [4] in a grill n-topological space (x,nτ,g) and a ⊆ x such that a ⊆ φg(a) , then nτ-cl(a) = nτg-cl(a) = nτ-cl(φg(a)) = φg(a). definition 2.12. [9] in (x,nτ), s ⊆ x then s is nτ-dense if nτ-cl(s) = x. 3 υg-operator via grills in this section, we suggest a new operator is called υg(a,nτ) (upsilon) in grill n-topological space, and take up some basic associated results. throughout this section (x,nτ,g) denotes a grill n-topological space. definition 3.1. let g be a grill on a n-topological space (x,nτ). we define a map υg : p(x) → p(x), given by υg(a,nτ) = x − φg(x − a) for any a ⊆ x. we shall simply write υg(a), assuming that the grill g under consideration is understood. remark 3.2. it follows from theorem 2.5 (ii) that υg is open in (x,nτ) for any subset a of x. thus υg treated as a mapping from p(x) to nτ. remark 3.3. in view of theorem 2.4(ii) it turns out that for two grills g1 and g2 on x, g1 ⊆ g2 =⇒ υg1 (a) ⊇ υg2 (a). but for a given grill g on x, υg is increasing in the sense that whenever a ⊆ b ⊆ x, then υg(a) ⊆ υg(b). this is again an immediate consequence of theorem 2.4 (i); however it may so happen that υg(a) ⊆ υg(b) even if a * b. the following is an example to justify our contention. example 3.4. let n = 3 and x = {s,t,u} and consider τ1o(x) = {∅,x,{t,u}}, τ2o(x) = {∅,x,{s}} and τ3o(x) = {∅,x,{s},{t,u}}. then 3τo(x)={∅,x,{s}, {t,u}} is a tri topology and consider the grill g = {{s},{u},{s,u},{s,t},x}. thus (x, 3τ,g) is a grill tri topological space on x. now, φg({s}) = {s} and φg({t}) = ∅. then υg({t,u}) = x − φg({s}) = {t,u} and υg({s,u}) = x − φg({t}) = x. thus υg({t,u}) ⊆ υg({s,u}) although {t,u} * {s,u}. theorem 3.5. let (x,nτ,g) be a grill n-topological space. then the following statements are true: 388 υg-operator in grill n-topology (i) if s ∈ nτg then s ⊆ υg(s). (ii) if s,t ⊆ x then υg(s ∩t) = υg(s) ∩ υg(t). (iii) if s ⊆ x and s /∈ g, then υg(s) = x − φg(x). (iv) if s,t ⊆ x with t /∈ g, then υg(s) = υg(s −t) = υg(s ∪t). (v) if s,t ⊆ x with (s −t) ∪ (t −s) /∈ g, then υg(s) = υg(t). proof. (i) in fact, s ∈ nτg =⇒ φg(x −s) ⊆ x −s by result 2.9, s ⊆ x − φg(x −s) = υg(s). (ii) υg(s ∩t) = x − φg(x − (s ∩t)) = x − φg[(x −s) ∪ φg(x −t)] = [x − φg[(x −s)] ∩ φg[(x −t)] = υg(s) ∩ υg(t). (iii) υg(s) = x −φg(x −s) = x − [φg(x −s)−φg(s)] = x − [φg(x)− φg(s)] = x − φg(x). (iv) υg(s − t) = x − φg((x − s) ∪ t) = x − [φg(x − s) ∪ φg(t)] = x − φg(x −s) = x − υg(s). (v) let (s−t)∪(t −s) /∈ g so that s−t,t −s /∈ g. then by using corollary 2.8 we have υg(s) = υg((t−(t−s))∪(s−t) = υg(t−(t−s)) = υg(t). remark 3.6. from (ii) of the above theorem we see that the operator υg is distributive over finite intersection. that is not necessarily true for finite union which is shown below. example 3.7. let n = 2 and x = {1, 2, 3} and consider τ1o(x)={∅,x,{1, 2}}, τ2o(x)={∅,x}. then 2τo(x)={∅,x,{1, 2}}. consider the grill g = {{1},{1, 2}, {2},{1, 3},{2, 3},x}. now φg({1, 3}) = {1, 2, 3} =x= φg{2, 3} and φg({3}) = φ. then υg({1})=x − φg({2, 3}) = ∅, υg({2}) = x − φg({1, 3}) = ∅ and υg({1, 2}) = x − υg({3}) = x. thus υg({1}) ∪ υg({2}) 6= υg({1, 2}). next we derive two equivalent expressions for υg(a) where a ⊆ (x,nτ). theorem 3.8. in (x,nτ,g), let a ⊆ x. then the following statements are true: (i) υg(a) = {x ∈ x : ∃v ∈ nτ(x) such that v −a /∈ g}. (ii) let υg(a) = ∪{v ∈ nτ : v −a /∈ g}. 389 antony george a and davamani christober m proof. (i) x ∈ υg(a) iff x /∈ φg(x−a) ⇐⇒ there exist v ∈ nτ(x) such that v −a = v ∩ (x −a) /∈ g ⇐⇒ x ∈ r.h.s. (ii) let a# = ∪{v ∈ nτ : v −a /∈ g}. now x ∈ a# then there exists v ∈ nτ with x ∈ v such that v − a /∈ g which implies that there exist v ∈ nτ(x) such that v − a /∈ g. thus by (i), x ∈ υg(a). from this it is clear that υg(a) ⊆ a#. remark 3.9. let g = p(x)−{∅}, then by theorem 3.8 (ii) υg(a) = ∪{u ∈ nτ : u −a = ∅} = ∪{u ∈ nτ : u ⊆ a} = nτ–int(a), for any space (x,nτ). corolary 3.10. let (x,nτ,g) be a grill n-topological space and a ⊆ x .then a∩ υg(a) = nτg-inta proof. we have, x ∈ a ∩ υg(a) =⇒ x ∈ a and x ∈ υg(a) =⇒ x ∈ a and there exist v ∈ nτ(x) such that v − a /∈ g (by theorem 3.8 (i)) which implies v − (v − a) is a nτg-open neighbourhood of x such that v − (v − a) ⊆ a =⇒ x ∈ a. again x ∈ nτg-inta implies there exist a nτg-open neighbourhood u −b of x, where u ∈ nτ and b /∈ g, such that x ∈ u −b ⊆ a =⇒ u − a ⊆ b and u − a /∈ g. so by theorem 3.8 (i) x ∈ υg(a). thus x ∈ a∩ υg(a) = nτg-inta. theorem 3.11. let (x,nτ,g) be a grill n-topological space and if a ∈ nτ then υg(a) = ∪{s ∈ nτ : s∆a /∈ g}. proof. let a# = ∪{s ∈ nτ : s∆a /∈ g}. then by theorem 3.8 (ii), a# ⊆ υg(a). now, x ∈ υg(a) which implies there exist s ∈ nτ(x) such that s−a /∈ g (by theorem 3.8 (i)). let v = s ∪ a ∈ nτ. then v ∆a = s − a /∈ g and x ∈ v ∈ nτ. thus x ∈ a#. from the result so far, we arrive at the following simple and alternative description of the topology nτg in terms of our introduced operator. theorem 3.12. if (x,nτ,g) is a grill n-topological space then nτg = {s ⊆ x,s ⊆ υg(s)}. proof. t = {s ⊆ x : s ⊆ υg(s)}. in fact, ∅ ⊆ υg(∅) =⇒ ∅ ∈ t . υg(x) = x − φg(x −x) = x − φg({∅}) = x ∈ t . now a1,a2 ∈ t then a1 ⊆ υg(a1) and a2 ⊆ υg(a2) which implies a1 ∩a2 ⊆ υg(a1)∩υg(a2) = 390 υg-operator in grill n-topology υg(a1 ∩ a2)(by theorem 3.5(ii)). again {ai : i ∈ λ} ∈ t which implies ai ⊆ υg(ai) for each i ∈ λ. this implies ai ⊆ υg(∪i∈λai) for each i ∈ λ (by remark 3.3) hence ∪i∈λai ⊆ υg(∪i∈λai) this implies that ∪i∈λai ∈ t . we will show that t = nτg. indeed, v ∈ nτg, then v ∈ υg(v ) (by theorem 3.5 (i)). this implies that v ∈ t . conversely, a ∈ t =⇒ a ⊆ υg(a) this implies a = a∩υg(a) = nτg-inta (remark 3.9) which implies that a ∈ nτg. 4 suitable for a grill n-topology in this segment, we intend to do some investigations in respect of nτg, along with certain applications, under the assumption of such a suitability conditions imposed on the concerned grills. throughout this section (x,nτ,g) denotes a grill n-topological space. definition 4.1. let (x,nτ,g) be a grill n-topological space and nτ is called suitable for the grill g if a− φg(a) /∈ g, for all a ⊆ x. example 4.2. let x = {1, 2, 3}. for n = 3, consider τ1 = {∅,x,{1}},τ2 = {∅,x,{2}},τ3 = {∅,x,{1, 2}} and 3τ = {∅,x,{1},{2},{1, 2},x}. let g = {{1},{1, 2},{1, 3},x}. for every a ⊆ x, a−φg(a) /∈ g. hence 3τ is suitable for g. theorem 4.3. in a grill n -topological space (x,nτ,g), the following are equivalent: (i) nτ is suitable for the grill g. (ii) for any nτg-closed subset a of x, a− φg(a) /∈ g. (iii) for any a ⊆ x and each x ∈ a, there corresponds some u ∈ nτ(x) with u ∩a /∈ g, it follows that a /∈ g. (iv) a ⊆ x and a∩ φg(a) = φ =⇒ a /∈ g. proof. (i) =⇒ (ii) it is obvious. 391 antony george a and davamani christober m (ii) =⇒ (iii) let a ⊆ x and suppose for every x ∈ a there exist u ∈ nτ(x) such that u∩a /∈ g. then x /∈ φg(a) so that a∩φg(a) = φ. now as a∪φg(a) is nτg-closed, by (ii) we have (a ∪ φg(a)) − φg(a ∪ φg(a)) /∈ g.that is (a∪φg(a))−φg(a)∪φg(φg(a)) /∈ g =⇒ (a∪φg(a))−φg(a) /∈ g (by theorem 2.5 (i)) that is a /∈ g. (iii) =⇒ (iv) if a ⊆ x and a∩φg(a) = ∅ then a ⊆ x −φg(a). let x ∈ a. then x /∈ φg(a) implies there exist u ∈ nτ(x) such that u ∩a /∈ g. then by (iii), a /∈ g. (iv) =⇒ (i) let a ⊆ x. we first claim that (a−φg(a))∩φg(a−φg(a)) = ∅. in fact x ∈ (a− φg(a)) ∩ φg(a− φg(a)) =⇒ x ∈ a− φg(a) =⇒ x ∈ a and x /∈ φg(a) implies there exist u ∈ nτ(x) such that u ∩ a /∈ g. now u ∩ (a − φg(a)) ⊆ u ∩ a /∈ g =⇒ x /∈ φg(a − φg(a)), which is contradiction. hence by (iv), a− φg(a) /∈ g. now we derive, in term of the υg, a characterizing condition for a n-topology nτ to be suitable for a grill g on a n-topological space x. theorem 4.4. let g be a grill on a n-topological space x then nτ is suitable for g iff υg(a) −a /∈ g for any a ⊆ x. proof. let nτ be suitable for g and a ⊆ x, we first observe that x ∈ υg(a)−a iff υg(a) and x /∈ a iff there exists u ∈ nτ(x) such that x ∈ u −a /∈ g. thus to each x ∈ υg(a)−a, there exist u ∈ nτ(x) such that u ∩(υg(a)−a) /∈ g. as nτ is suitable for g, by theorem 4.2 we have υg(a) − a /∈ g. conversely, let a ⊆ x, suppose that to each x ∈ a there corresponds some u ∈ nτ(x) such that u ∩ a /∈ g. we need to show by virtue of theorem 4.2 that a /∈ g. now, by theorem 3.8 (i) we have, υg(x − a) = {x ∈ x : there exists u ∈ nτ(x) such that u − (x − a) /∈ g} = {x ∈ x : there exists u ∈ nτ(x) such that u ∩ a /∈ g}. thus a ⊆ υg(x − a) and hence a = υg(x − a) ∩ a = υg(x −a) − (x −a) /∈ g. corolary 4.5. if the n-topology nτ of a space x is suitable for a grill g on x, then υg is an idempotent operator i.e., υg(υg(a)) = υg(a) for any a ⊆ x. proof. since υg is nτ-open in x, hence υg(a) ∈ nτ for any a ⊆ x and so υg(a) ∈ nτg. hence by theorem 3.5(i) , υg(a) ⊆ υg(υg(a)) for any 392 υg-operator in grill n-topology a ⊆ x. also nτ is suitable for g, so υg(a) ⊆ υg(a ∪ b) for some b /∈ g. thus υg(υg(a)) ⊆ υg(a∪b) = υg(a). theorem 4.6. in (x,nτ,g), nτ is suitable for g . let a ⊆ x and v be a nonnull open set such that v ⊆ φg(a) ∩ υg(a) . then v −a /∈g and v ∩a ∈ g. proof. v ⊆ φg(a)∩υg(a) =⇒ v ⊆ υg(a) =⇒ v −a ⊆ υg(a)−a /∈ g by theorem 4.3 we get v −a /∈ g. again v ⊆ φg(a) and v 6= ∅ =⇒ v ∩a ∈ g. theorem 4.7. in (x,nτ,g), the following assertions are similar: (i) nτ −{∅}⊆ g (ii) υg(∅) = ∅ (iii) if a is nτ-closed then υg(a) −a = ∅ (iv) if a ⊆ x then nτ-int(nτ-cl(a)) = υg(nτ-int(nτ-cl(a))) (v) if a is nτ-regular open in x then a = υg(a) (vi) if v ∈ nτ then υg(v ) ⊆ nτ-int(nτ-cl(v )) ⊆ φg(v ) proof. (i) =⇒ (ii) : υg(∅) = ∪{v ∈ nτ : v −∅ /∈ g} by theorem 3.8 (ii) υg(∅) = ∪{v ∈ nτ : v /∈ g} = ∅. (ii) =⇒ (iii) : let x ∈ υg(a) −a then there exists v ∈ nτ(x) such that x ∈ v −a /∈ g. since a is nτ-closed, we obtain x ∈ v −a ∈{u ∈ nτ : u /∈ g}, a contradiction to υg(∅) = ∅. (iii) =⇒ (iv) : since nτ-int(nτ-cl(a)) is nτ-open, by theorem 3.5 (i) we get nτ-int(nτ-cl(a)) ⊆ υg(nτ-int(nτ-cl(a))). again using (iii), we get υg(nτ-cl(a)) ⊆ nτ-cl(a). by remark 3.2, υg(nτ-cl(a)) = nτ-int(υg(nτcl(a)) ⊆ nτ-int(nτ-cl(a)).since υg(nτ-int(nτ-cl(a))) ⊆ υg(nτ-cl(a)) ⊆ nτ-int(nτ-cl(a)). thus nτ-int(nτ-cl(a)) = υg(nτ-int(nτ-cl(a))). (iv) =⇒ (v): it is trivial. (v) =⇒ (vi): let v ∈ nτ. by assumption, ∅ = υg(∅) = ∪{v ∈ nτ : u /∈ g}. by theorem 3.8(ii) we obtain nτ − {∅} ⊆ g. then by theorem 2.6 we get v ⊆ φg(v ) and hence by proposition 2.11, we have φg(v ) = nτ-cl(v ). now, v ⊆ nτ-int(nτ-cl(v )) ⊆ nτ-cl(v ) = φg(v ). since υg(v ) ⊆ υg(nτint(nτ-cl(v ))) = nτ-int(nτ-cl(v )) ⊆ φg(v ). (vi) =⇒ (i): if v ∈ nτ − g, by theorem 3.5 (i), v ⊆ υg(v ) ⊆ φg(v ) = ∅. that is v = ∅ by theorem 2.4(i). 393 antony george a and davamani christober m 5 υ̂g -sets in this segment, we discuss about a new open set υ̂g in grill n-topological space and investigate some of its properties. definition 5.1. in (x,nτ,g) a subset s of x is called a υ̂g-set if s ⊆ nτcl(υg(s)). the group of all υ̂g-sets in (x,nτ,g) is signify by υ̂g(x,nτ). proposition 5.2. if {sα : α ∈ ∆} is a group of nonempty υ̂g-sets in (x,nτ,g), then ∪α∈∆sα ∈ υ̂g(x,nτ). proof. for each α ∈ ∆, sα ⊆ nτ-cl(υg(sα)) ⊆ nτ-cl(υg(∪α∈∆sα)) this implies that ∪α∈∆sα ⊆ nτ-cl(υg(∪α∈∆sα)). thus ∪α∈∆sα ∈ υ̂g(x,nτ). remark 5.3. the intersection of two υ̂g-sets need not be a υ̂g-set and it is shown in the following example. example 5.4. let x = {1, 2, 3, 4}. let n = 3. consider τ1 = {∅,x,{2, 3}},τ2 = {∅,x,{1, 2, 3}} and τ3 = {∅,x,{1},{1, 2, 3}}. then 3τ = {∅,x,{1},{1, 2, 3}, {2, 3}} and the grill g = {{1},{2},{1, 3},{1, 2},{1, 4},{2, 3},{2, 4},{1, 2, 3}, {2, 3, 4},{1, 2, 4},{1, 3, 4},{2, 3, 4},x}. let a = {1, 4} and b = {2, 3, 4} are υ̂g-sets but a ∩ b is not a υ̂g-set. for let a = {1, 4}, φg(x − a) = {2, 3, 4} and υ̂g(a) = {1}. hence a ⊆ nτ-cl(υ̂g(a)) implies that a is υ̂gset. for b = {2, 3, 4}, φg(x − b) = {1, 4} and υ̂g(b) = {2, 3}. hence b ⊆ nτ-cl(υ̂g(b)) implies that b is υ̂g-set. on the other hand, since a∩b = {4}, φg(x−(a∩b)) = x and υ̂g(b) = ∅. hence a∩b * nτ-cl(υ̂g(a∩b)) implies that a∩b is not a υ̂g-set. remark 5.5. the intersection of an nτα-set and υ̂g-set is a υ̂g-set. corolary 5.6. in (x,nτ,g), if for any s ∈ nτ then s ⊆ υg(s) . theorem 5.7. let a ∈ υ̂g(x,nτ) on (x,nτ,g). if u ∈ nτα then u ∩ a ∈ υ̂g(x,nτ). proof. assume that a is nτ-open for every a ⊆ x, g ∩ nτ-cl(a) ⊆ nτcl(g∩a). let u ∈ nτα and a ∈ υ̂g(x,nτ). by corollary 5.6, we have u∩a ⊆ 394 υg-operator in grill n-topology nτ-int(nτ-cl(nτ-int(u)))∩nτ-cl(υg(a)) ⊆ nτ-int(nτ-cl(υg(u)))∩nτcl(υg(a)) ⊆ nτ-cl(nτ-int(nτ-cl(υg(u))) ∩ υg(a) = nτ-cl(nτ-int(nτcl(υg(u) ∩ υg(a))) = nτ-cl(υg(u) ∩ υg(a)) = nτ-cl(υg(u ∩a)). hence u ∩a ∈ υ̂g(x,nτ). corolary 5.8. let a ∈ υ̂g(x,nτ) on (x,nτ,g). if u ∈ nτ then u ∩ a ∈ υ̂g(x,nτ). definition 5.9. in (x,nτ,g), if for every relatively nonempty open set l∩k,l ∈ nτ and (l∩k) ∩e ∈ g then the set e is relatively g-dense in a set k. next we prove a necessary and sufficient condition for a /∈ υ̂g(x,nτ). theorem 5.10. a set a /∈ υ̂g(x,nτ) if and only if there exits x ∈ a such that there is a neighbourhood vx ∈ nτ(x) for which x −a is relatively nτg-dense in vx. proof. let a /∈ υ̂g(x,nτ). we are to show that there exists x ∈ a and a neighbourhood vx ∈ nτ(x) satisfying that x − a is relatively g-dense in vx. since a * nτ-cl(υg(a)) , there exits x ∈ x such that x ∈ a but x /∈ nτcl(υg(a)). hence there exists a neighbourhood vx ∈ nτ(x) such that vx ∩ υg(a) = ∅. this implies that vx ∩ (x − φg(x − a)) = ∅ and hence vx ⊆ φg(x − a). let u be any non empty open set in vx. since vx ⊆ φg(x − a), therefore u ∩ (x −a) ∈ g which implies that (x −a) is relatively g-dense in vx. converse is obvious. definition 5.11. a space (x,nτ,g) is said to be anti-co-dense grill if nτ-{∅}⊆ g theorem 5.12. let g be a anti-co dense grill on a space (x,nτ,g). then so(x,nτg) = υ̂g(x,nτ). definition 5.13. a set a ⊆ x in (x,nτ,g), a is called a υa-set if a ⊆ nτint(nτ-cl(υg(a))). the collection υa-sets in (x,nτ,g) is denoted by nτa. from definitions of 5.1 and 5.13 it follows that nτa ⊆ υ̂g(x,nτ). the collection nτa forms a topology finer than nτ. theorem 5.14. let g be a anti-co dense grill on (x,nτ,g). then the collection nτa = {a ⊆ x : a ⊆ nτ-int(nτ-cl(υg(a)))} forms a n-topology on x. 395 antony george a and davamani christober m proof. (i) it is observed that ∅ ⊆ nτ-int(nτ-cl(υg(∅)) and x ⊆ nτ-int(nτcl(υg(x), and thus ∅ and x ∈ nτα. (ii) let {aα : α ∈ ∆}⊆ nτa, then υg(aα) ⊆ υg(∪aα) for every α ∈ ∆. thus aα ⊆ nτ-int(nτ-cl(υg(aα))) ⊆ nτ-int(nτ-cl(υg((∪aα))) for every α ∈ ∆, which implies that ∪aα ⊆ nτ-int(nτ-cl(υg(∪aα))). therefore, ∪aα ∈ nτa. (iii) let a,b ∈ nτa. since υg(a) is open in (x,nτ), we obtain a ∩ b ⊆ nτ-int(nτ-cl(υg(a))) ∩ nτ-int(nτ-cl(υg(a))) = nτ-int(nτ-cl(υg(a) ∩ υg(b))). therefore a∩b ⊆ nτ-cl(nτ-int(υg(a∩b))). proposition 5.15. let (x,nτ,g) be a grill n-topological space. then υg(a) 6= ∅ if and only if a contains a nonempty nτg-interior. corolary 5.16. let (x,nτ,g) be a grill n-topological space. then {x} ∈ υ̂g(x,nτ) if and only if {x}∈ nτa . proof. let {x} ∈ υ̂g(x,nτ), therefore by proposition 5.15, {x} is open in (x,nτ,g). since {x} ⊆ υg({x}) and υg({x}) is nτ-open in (x,nτ), therefore {x}⊆ nτ-int(nτ-cl(υg{x}). 6 conclusion in this paper, we introduced a new operator in grill n-topological space, using this operator some important properties and equivalent expressions are derived. we arrived a topology nτg using the introduced operator. in addition, suitability condition of a grill with the n-topological space x is formulated. we discuss the characterizing condition for a n-topology to be a suitable for a grill g on x. also we introduce and study υ̂g–sets and utilize the υg-operator to define a generalized open set and their properties. this concept can be extended to other applicable research areas of topology such as nano topology, fuzzy topology, intuitionistic topology, digital topology and so on. 396 υg-operator in grill n-topology references [1] g. choquet, sur les notions de filter et grill, comptes rendus acad. sci. paries. 224, 171-173, 1947. [2] k.c. chattopadhyay, o. njastad and w. j. thron, mertopic spaces and extensions of closure spaces, can. j. math. vol. xxxv (4), 613-629, 1983. [3] m. lellis thivagar, v. ramesh and m. arockia dasan, on new structure of n-topology, cogent mathematics, 3, 1204104, 1-10, 2016. [4] m. lellis thivagar, i. l. reilly, m. arockia dasan and v. ramesh, generalized open sets in grill n-topology, appl. gen. topol. vol.18 (2), 289-299. 2017. [5] m. lellis thivagar, m.arockia dasan, new topologies via weak ntopological open sets and mappings, journal of new theory, vol.29, 49-57. 2019. [6] b. roy and m. n. mukherjee, on a typical topology induced by a grill, soochow j. math. vol 33(4), 771-786, 2007. [7] n. levine, semi-open sets in and semi continuity in topological space, the american mathematical monthly, vol.70(1), 36-41, 1963. [8] o.njastad, on some classes of nearly open sets, pacific j.math, vol.15(3), 961970. 1965. [9] j. loyala foresith spencer, m. davamani christober, door spaces on ntopology, ratio mathematica, vol 47, 2023. 397 ratio mathematica volume 44, 2022 gaussian tribonacci r-graceful labeling of some tree related graphs dr. sunitha. k 1 sheriba. m 2 abstract let r be any natural number. an injective function kallforgtkikikigv rq }...,,2,2,1,1,,0{)(: 1  , where 1rq gt is the th rq )1(  gaussian tribonacci number in the gaussian tribonacci sequence is said to be gaussian tribonacci r-graceful labeling if the induced edge labeling },...,,{)(: 121 *   rq gtgtgtge such that |)()(|)( * vuuv   is bijective. if a graph g admits gaussian tribonacci r-graceful labeling, then g is called a gaussian tribonacci r-graceful graph. a graph g is said to be gaussian tribonacci arbitrarily graceful if it is gaussian tribonacci r-graceful for all r. in this paper we investigate the path graph np , the comb graph 1kpm , the coconut tree graph ,),( nmct the regular caterpillar graph 1nkpm , the bistar graph nmb , and the subdivision of bistar graph ][ ,nm bs are gaussian tribonacci arbitrarily graceful. keywords: gaussian tribonacci sequence, gaussian tribonacci graceful labeling, path graph, comb graph, coconut tree graph, regular caterpillar graph, bistar graph and subdivision of bistar graph. subject classification:05c78 3 1 assistant professor, department of mathematics, scott christian college (autonomous), nagercoil629003 email: ksunithasam@gmail.com 2 part time research scholar, department of mathematics, scott christian college (autonomous), nagercoil-629003.affiliated to manonmaniam sundaranar university, tirunelveli-627012 email: sheribajerin@gmail.com 3 received on june 19th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.906. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 188 mailto:ksunithasam@gmail.com mailto:sheribajerin@gmail.com dr. sunitha. k and sheriba. m 1. motivation and main results graphs considered throughout this paper are finite, simple, undirected and nontrivial. labeling of graph is an assignment of values to vertices and edges or both subject to certain conditions. in 1967, rosa [6] introduced the concept of graceful labeling. in 1982, slater [4] introduced the concept of k-graceful labeling of graphs and is defined as follows: let g be a simple graph with p vertices and q edges. let k be any natural number. define an injective mapping }1...,,2,1,0{)(:  kqgv that induces bijective mapping }1,,....1,{)(: *  kqkkge where |)()(|)( * vuuv   for all )(geuv and ),(, gvvu  then  is called k-graceful labeling while *  is called an induced edge k-graceful labeling and the graph g is called k-graceful graph. graphs that are k-graceful for all k are sometimes called arbitrarily graceful. yuksel soykan e tal. [11] introduced the concept of gaussian generalised tribonacci numbers. in this sequel, we introduce a new concept gaussian tribonacci r-graceful labeling of graphs. we follow d. b. west [10] and j. a. gallian [2], for standard terminology and notations. definition 1.1 let g be a graph with p vertices and q edges. let r be any natural number. an injective function kallforgtkikikigv rq }...,,,2,1,1,,0{)(: 1  ,where 1rq gt is the th rq )1(  gaussian tribonacci number in the gaussian tribonacci sequence is said to be gaussian tribonacci graceful labeling if the induced edge labeling }...,,,{)(: 121 *   rq gtgtgtge such that |)()(|)( * vuuv   is bijective. if a graph g admits gaussian tribonacci graceful labeling, then g is called a gaussian tribonacci graceful graph. a graph g is said to be gaussian tribonacci arbitrarily graceful if it is gaussian tribonacci r-graceful for all r. remark 1.1 [11] the gaussian tribonacci sequence is obtained as follows: 31,1,0 321210   ngtgtgtgtandigtgtgt nnnn ...},1324,713,47,24,2,1,1,0{, iiiiiiie  is the gaussian tribonacci sequence. definition 1.2 [9] the comb graph 1 kp n  is obtained by joining a single pendent edge to each vertex of the path n p . definition 1.3 [3] a regular caterpillar graph 1 nkp m  is obtained from the path m p by joining 1 nk vertices to each vertices of the path m p . 189 gaussian tribonacci r-graceful labeling of some tree related graphs definition 1.4 [8] the bistar graph nm b , is obtained from 2 k by attaching m pendent edges to one end of 2 k and n pendent edges to the other end of 2 k . definition 1.5 [10] the subdivision of bistar graph )( ,nm bs is obtained by subdividing each edge of a bistar graph nm b , . definition 1.6[9] a coconut tree graph ct(m,n) is obtained from the path n p by appending n new pendent edges at an end vertex of n p . 2. main results theorem 2.1 the path graph np is gaussian tribonacci arbitrarily graceful for all .2n proof. let np be a path graph of length 1n with vertex set }1/{)( niupv in  and edge set }11/{)( 1   niuupe iin such that 1|)(||)(|  nqpeandnppv nn define kallforgtkikikipv rqn }...,,2,2,1,1,,0{)(: 1  by 01 )( tu  1,2,)1()()( 11   rnigtuu irq i ii  thus admits gaussian tribonacci graceful labeling for all r . hence the path graph np is gaussian tribonacci arbitrarily graceful for all 2n . example 2.1 the gaussian tribonacci 2-graceful labeling of path graph 5p is given in figure 2.1 1+i2+i4+2i7+4i 0 7+4i 3+2i 5+3i 4+2i figure 2.1 theorem 2.2 the comb graph 1kpn is gaussian tribonacci arbitrarily graceful for all 2n . proof. let niui 1, be the vertices of the path np and let nivi 1, be the vertices which are joined to each vertices of the path np .the resultant graph 1kpn whose vertex set is }1/,{)( 1 nivukpv iin  and edge set is }}11/{}1/{{)( 11   niuunivukpe iiiin such that npkpv n 2|)(| 1  and 12|)(| 1  nqkpe n case 1 190 dr. sunitha. k and sheriba. m for 2n define kallforgtkikikipv rqn }...,,2,2,1,1,,0{)(: 1  by 1,)(,)( 1201   rgtugtu r  , 1,)()(,1,)( 2211   rgtuvrgtv rrq  case 2 for 3n define kallforgtkikikipv rqn }...,,2,2,1,1,,0{)(: 1  by 01 )( gtu  , 1,)(,1,2,)1()()( 111   rgtvrnigtuu rirq i ii  1,2,)()( 1   rnigtuv irii  thus admits gaussian tribonacci graceful labeling for all r. hence the comb graph 1kpn is gaussian tribonacci arbitrarily graceful for all 2n . example2.2 the gaussian tribonacci 3-graceful labeling of comb graph 12 kp  is given in figure 2.2 4+2i 7+4i 2+i 7+4i 2+i 0 4+2i figure 2.2 the gaussian tribonacci 2-graceful labeling of comb graph 15 kp  is given in figure 2.3 149+81i 81+44i 24+13i 1+i 2+i 4+2i 7+4i 13+7i 1+i 147+80i 64+35i 105+57ii 75+41i 149+81i 68+37i 112+61i0 88+48i 44+24i figure 2.3 theorem 2.3 the coconut tree graph ct (m, n) is gaussian tribonacci arbitrarily graceful for all .2, nm proof. let miui 1, be the vertices of the path mp and let nivi 1, be the new vertices which are attached to the th m vertex of the path mp . the resultant graph is ct (m, n) whose vertex set is }}1/{}1/{{)],([ nivmiunmctv ii   and edge set 191 gaussian tribonacci r-graceful labeling of some tree related graphs is }}1/{}11/{{)],([ 1 nivumiuunmcte imii    such that nmpnmctv |)],([| and 1|)],([|  nmqnmcte define kallforgtkikikipv rqn }...,,2,2,1,1,,0{)(: 1  by ,)( 01 gtu  1,2,)1()()( 11   rmigtuu irq i ii  1,1,)()( 1   rnigtuv irmi  thus  admits gaussian tribonacci graceful labeling for all r. hence the coconut tree graph ct(m,n) is gaussian tribonacci arbitrarily graceful for all 2, nm . example 2.3 the gaussian tribonacci1-graceful labeling of coconut tree graph )5,5(ct is given in figure 2.4 81+44i 44+24i 24+13i 13+7i 7+4i 4+2i 2+i 1+i 1 0 81+44i 37+20i 61+33i 48+26i 41+22i 44+24i 46+25i 47+26i 41+25i ` figure 2.4 theorem 2.4 the regular caterpillar graph 1nkpm is gaussian tribonacci arbitrarily graceful for all 2, nm proof. let mivi 1, be the vertices of the path mp and let njmivij  1,1, be the vertices attached to each vertices of the path .mp the resultant graph is 1nkpm whose vertex set is }}1,1/{}1/{{][ 1 njmivmivnkpv ijim   and edge set is }}1,1/{}11/{{][ 11 njmivvmivvnkpe ijiiim    such that 1|][||][| 11  mnmqnkpeandmnmpnkpv mm define a function kallforgtkikikipv rqn }...,,2,2,1,1,,0{)(: 1  by 1,2,)1()()(,)( 1101   rmigtvvgtv irq i ii  1,1,)( )2(11   rnjgtv mjrqj  1,1,2,)()( )2()1(1   rnjmigtvv mjnirqiij  thus admits gaussian tribonacci r-graceful labeling for all r . 192 dr. sunitha. k and sheriba. m hence the regular caterpillar graph 1nkpm are gaussian tribonacci arbitrarily graceful for all .2, nm example 2.4 the gaussian tribonacci 2-graceful labeling of regular caterpillar graph 14 2kp  is given in figure 2.5 0 504+274i 274+149i 504+274i 230+125i 149+81i 379+206i 2+i 1+i 377+205i 378+205i 4+2i7+4i 13+7i24+13i 44+24i81+44i 81+44i 44+24i 223+121i 226+123i 480+261i 491+267i figure 2.5 theorem 2.5 the bistar graph nm b , is gaussian tribonacci arbitrarily graceful for all .2, nm proof. let vu, be the vertices of 2 k and let miui 1, be the m vertices attached to one end of 2 k and njv j 1, be the n vertices attached to the other end of 2k .the resultant graph is nm b , whose vertex set is }1,1/,,,{)( , njmivuvubv jinm  and edge set is }}{}1/{}1/{{)( , uvnjvvmiuube jinm   such that 2|)(| ,  nmpbv nm and 1|)(| ,  nmqbe nm define a function kallforgtkikikipv rqn }...,,2,2,1,1,,0{)(: 1  by 1,1,)()(,)(,)( 110   rnigtvvgtvgtu irqirq  1,1,)( 1   rmigtu inrqi  thus  admits gaussian tribonacci r-graceful labeling for all r. hence nm b , is gaussian tribonacci arbitrarily graceful for all .2, nm example 2.5 the gaussian tribonacci 1-graceful labeling of bistar graph 4,2 b is given in figure 2.6 193 gaussian tribonacci r-graceful labeling of some tree related graphs 13+7i 7+4i 4+2i 2+i 1+i 1 0 24+13i 24+13i 11+6i 17+9i 20+11i 22+12i 1+i 1 figure 2.6 theorem 2.6 the subdivision of the bistar graph )( ,nm bs is gaussian tribonacci arbitrarily graceful for all 1, nm proof. let vu, be the central vertices of the bistar graph nm b , and let nivandmiu ii  1,1, be the vertices joined with u and v respectively. let nivandmius ii  1,1,, 11 be the new vertices obtained by subdividing the edges nivvandmiuuuv ii  1,1,, respectively. the resulting graph is )( ,nm bs whose vertex set is }}{},{}1/,{}1/,{{)]([ 11 , svunivvmiuubsv iiiinm   and edge set is }}1/,{}{}{}1/,{{)]([ 1111 , nivvvvsvusmiuuuubse iiiiiinm   such that 3)(2|)]([| ,  nmpbsv nm and 2)(2|)]([| ,  nmqbse nm define a function kallforgtkikikipv rqn }...,,2,2,1,1,,0{)(: 1  by 1,1,)()()(,1,)(,0)( 1 1 21   rmigtuugturgtvs irirqrq  1,1,)()(,1,1,)()( 2 1 2 1   rnigtvvrmigtuu irqiirmii  1,1,)()( 12 1   rnigtvv rimii  thus  admits tribonacci r-graceful labeling for all r . hence the subdivision of the bistar graph )( ,nm bs is gaussian tribonacci arbitrarily graceful for all 1, nm . example 2.6 the gaussian tribonacci 1-graceful labeling of subdivision of bistar graph s( 2,2 b ) is given in figure 2.7 194 dr. sunitha. k and sheriba. m 80+44i 4+2i 76+42i 2+1i 78+42i 80+43i 1+1i 1 81+44i 81+44i 0 149+81i 149+81i 44+24i 105+57i 7+4i 112+61i 24+13i 125+68i 13+7i 138+85i figure 2.7 3. conclusion in this paper, we investigate the path graph, the comb graph, the coconut tree graph, the regular caterpillar graph, the bistar graph and the subdivision of bistar graph are gaussian tribonacci arbitrarily graceful. in future, we investigate gaussian tribonacci arbitrarily graceful labeling of cycle related graphs. references [1] david.w.and anthoney. e. barakaukas, “fibonacci graceful graphs”. [2] j. a. gallian, a dynamic survey of graph labeling, the electronic journal of combinatorics, (2013). [3] murugesan. n and uma. r, “super vertex gracefulness of some special graphs”, iqsr journal of mathematics, vol.:11, issue 3 ver (may june 2015), pp. 07-15 [4] p. j. slater, on k-graceful graphs, in: proc. of the 13 th south eastern conference on combinatorics, graph theory and computing (1982),53-57. [5] p. prathan and kamesh kumar, “on k-graceful labeling of some graphs”, journal of applied mathematics, vol.: 34 (2016), no.1 – 2, pp. 09-17 [6] rosa. a “on certain valuation of vertices of graph”, (1967). [7] steven k lee, hunterlehmann, andrew park, prime labeling of families of trees with gaussian integers, akce international journal of graphs and combinatorics 13 (2016), 165-176 [8] thirugnanasambandan, k and chitra g.,” fibonacci mean anti-magic labeling of graphs”, international journal of computer applications (0975-8887), vol.134, no,15, january 2016. [9] uma. r and amuthavalli. d, “fibonacci graceful labeling of some star related graphs”, international journal of computer applications (0975-8887) vol.134, no.15, january 2016. 195 gaussian tribonacci r-graceful labeling of some tree related graphs [10] west.d.b. b, introduction to graph theory, prentice-hall of india, new delhi, (2001). [11] yuksel soykan, ekan tasdemir, inci okumus, melih gocen,” gaussian generalized tribonacci numbers”, journal of progressive research in mathematics (jprm), vol.14, issue:2. 196 ratio mathematica volume 45, 2023 n�̂�*s-continuous functions in nano topological spaces m. anto * j. carolinal † abstract the aim of this paper is to introduces n�̂�*s-continuous function in nano topological spaces and we also study the relation between n�̂�*s-irresolute functions and n�̂�*scontinuous functions in different closed sets. keywords: �̂�*s-closed set, �̂�*s-continuous functions, n�̂�*s-irresolute. 2010 ams subject classification: 54c05‡ * associate professor, pg and research department of mathematics, annai velankanni college, tholayavattam 629157, india. e-mail: antorbjm@gmail.com. † assistant professor, pg and research department of mathematics, annai velankanni college, tholayavattam 629157, india. e-mail carolinalphonse@gmail.com. ‡ received on july 21, 2022. accepted on october 15, 2022. published on january 30, 2023.doi: 10.23755/rm.v45i0.987. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 97 m. anto and j. carolinal 1. introduction topology which is a branch of mathematics, is formally defined as the study of qualitative properties of certain objects that are invariant under a certain kind of transformation, especially those properties that are invariant under a certain kind of equivalence. the term topology was introduced by a german mathematician called johnn benedict listing in 1847. the modern topology largely based on the idea of set theory was developed by george cantor in the later part of 19th century. since its inception the topic has been growing in different level and in various fields. nano topology is one of the latest feathers in topology that applies to real life situations. lellis thivagar was the main brain behind developing the concept of nano topology. it is constructed in terms of lower and upper approximations and boundary region of a subset of a universe. the term “nano” can be ascribed to any unit of measure. the concept of continuity plays a very major role in general topology and they are now the research topics of many topologists worldwide. indeed, a significant theme in general topology concerns the variously modified forms of continuity, separation axioms etc., by utilizing generalized open sets. n. levine [7] introduced the concept of generalized closed sets in 1970. the concept of �̂�*s –closed sets was introduced by m. anto [12]. in 2013, m. lellis thivagar [6] has introduced nano topological space with respect to a subset x of a universe u, which is defined in terms of lower and upper approximation of x. he has also defined nano-closed sets, nano-interior and nanoclosure of a set. he has also introduced, among other, some certain weak form of nano open sets such as nano -open sets, nano semi open sets and nano pre-open sets. the aim of this paper is to introduce a new class of sets on nano topological spaces called n�̂�*s –closed sets. further, we investigate and discuss the relation of this new sets with existing ones. 2.preliminaries definition 2.1 let u be a non-empty finite set of objects called the universe and r be an equivalence relation on u named as the in-discernibility relation. then u is divided into disjoint equivalence class, elements belonging to the same equivalence class are said to be in discernible with one another. the pair (u, r) is said to be the approximation space. let x ⊆ u. • the lower approximation of x with respect to r is the set of all object which can be for certain classifies as x with respect to r and it is denoted by 𝐿𝑅(x). that is 𝐿𝑅(x) = ⋃𝑥∈𝑈{r(x): r(x) x}, where r(x) denotes the equivalence classes determined by x u. • the upper approximation of x with respect to r is the of all objects, which can be for certain classified as x with respect to r and it is denoted by 𝑈𝑅(x). that is, 𝑈𝑅(x) = ⋃𝑥∈𝑈{r(x): r(x) ∩ 𝑋 ≠ ∅}. 98 n�̂�*s-continuous functions in nano topological spaces • the boundary of the region of x with respect to r is the set of all objects, which can be classified neither as x nor as not x with respect to r and it is denoted by 𝐵𝑅(x) = 𝑈𝑅(x) 𝐿𝑅(x). definition 2.2 if (ur) is an approximation space and x, y u, then 1. 𝐿𝑅(x) ⊆ x ⊆ 𝑈 (x) 2. 𝐿𝑅(∅) = 𝑈𝑅(∅) =∅ and 𝐿𝑅(u) 𝑈𝑅(𝑈) = u 3. 𝑈𝑅(𝑋 ∪ 𝑌) = 𝑈𝑅(𝑋) ∪ 𝑈𝑅(𝑌) 4. 𝑈𝑅(𝑋 ∩ 𝑌) ⊆ 𝑈𝑅(𝑋) ∩ 𝑈𝑅(𝑌) 5. 𝑈𝑅(𝑋𝑌) ⊇ 𝑈𝑅(𝑈𝑅(y) 6. 𝑈𝑅(𝑋 ⋂𝑌) = 𝑈𝑅(𝑋 𝑈𝑅(𝑌) 7. 𝐿𝑅( ) 𝐿𝑅(𝑌) and 𝑈𝑅(x) ⊆ 𝑈𝑅(𝑌) whenever x u 8. 𝑈 ((𝑋𝑐)) [𝐿 [(𝑋)]𝑐 and 𝐿 [(𝑋𝑐)] [𝑈 (𝑋)]𝑐 9. 𝑈𝑅(𝑈𝑅((x) = 𝐿 (𝑈𝑅((x) 𝑈𝑅(x) 10. 𝐿𝑅(𝐿𝑅(x) 𝑈𝑅((𝐿𝑅(x) 𝐿𝑅(x) definition2.3 let u be the universe, r be an equivalence relation on u and 𝜏𝑅(x) = {u, ∅, 𝐿𝑅(x), 𝑈𝑅(x), 𝐵𝑅(x)} where x ⊆ u. 𝜏𝑅(x)satisfies the following axioms: 1. u and ∈ 𝜏 (x) 2. the union of elements of any sub collection of 𝜏𝑅(x) is in 𝜏𝑅(x). 3. the intersection of the elements of any finite sub collection of 𝜏𝑅(x) is in 𝜏𝑅(x). that is, 𝜏𝑅(x) forms a topology on u is called the nano topology on u with respect to x. we call (u, 𝜏𝑅(x)) is called the nano topological space. definition2.4 if (u, 𝜏𝑅(x)) is a nano topological space with respect to x where x ⊆ u and if a ⊆ u, then the nano interior of a is defined as the union of all nano open subsets of a and it is de noted by nint(a). that is, nint(a) is the largest nano open subset of a. the nano closure of a is defined as the intersection of all nano closed sets containing a and its denoted by ncl(a). ncl(a) is the smallest nano closed set containing a. remark 2.5 if 𝜏𝑅(x) is the nano topology on u with respect to x, then the set b= {u, 𝐿𝑅(x), 𝑈𝑅(x), 𝐵𝑅(x)} is the basis for 𝜏𝑅(x). definition 2.6 a function 𝑓: (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is called • nano-continuous [2] if 𝑓−1(a) is nano-closed in (x, 𝜏) for every nano-closed set a in (y, 𝜎). • nano -continuous [7] if 𝑓−1(a) is nano 𝛼-closed in (x, 𝜏) for every nano-closed set a in (y, 𝜎). • nano semi-continuous [4] if 𝑓−1(a) is nano semi closed in (x, 𝜏) for every nanoclosed set a in (y, 𝜎). • nano regular-continuous if 𝑓−1(a) is nano regular closed in (x, 𝜏) for every nanoclosed set a in (y, 𝜎). 99 m. anto and j. carolinal definition 2.7 if (u, 𝜏𝑅(x)) is a nano topological space if a u., then a is said to be • n𝑔#𝛼-closed if n cl(a) ⊆ v whenever a v and v is nĝ-open in (u, 𝜏𝑅(x) • n𝑔∗-closed if ncl(a) ⊆ v whenever a v and v is ng-open in (u, 𝜏𝑅(x) • n𝑔∗sclosed if nscl(a) ⊆ v whenever a v and v is ng-open in (u, 𝜏𝑅(x)). • nsĝ-closed if ncl(a) ⊆ u whenever a u and u is nsg-open in (u, 𝜏𝑅(x)). • nĝ -closed if n cl(a) ⊆ u whenever a u and u is nĝ-open in (u, 𝜏𝑅(x)) • n𝛼𝑔∗s-closed if n cl(a) ⊆ u whenever a u and u is n g-open in (u, 𝜏𝑅(x)). • n𝑔𝛼𝑔 closed if ncl(a) ⊆ u whenever a u and u is n g-open in (u, 𝜏𝑅(x)) definition 2.8 if (u, 𝜏𝑅(x)) is a nano topological space if a u., then a is said to be • n𝑔#𝛼-continuous [9] if 𝑓−1(a) is n𝑔#𝛼-closed in (u, 𝜏𝑅(x)) for every nano-closed set a in (v, 𝜎𝑅(y)) • n𝑔∗-continuous [12] if 𝑓−1(a) is n𝑔∗-closed in (u, 𝜏𝑅(x)) for every nano-closed set a in (v, 𝑅(y)) • n𝑔∗s-continuous [12] if 𝑓−1(a) is n𝑔∗s-closed in (u, 𝜏𝑅(x)) for every nano-closed set a in (v, (y)) • nĝ-continuous [6] if 𝑓−1(a) is nĝ-closed in (u, 𝜏𝑅(x)) for every nano-closed set a in (v, 𝜎𝑅(y)) • nsĝ-continuous [11] if 𝑓−1(a) is nsĝ-closed in (u, 𝜏𝑅(x)) for every nano-closed set a in (v, 𝜎𝑅(y)) • nĝ -continuous [10] if 𝑓−1(a) is nĝ -closed in (u, 𝜏𝑅(x)) for every nano-closed set a in (v, 𝜎𝑅(y)) • n𝛼𝑔∗s-continuous [8] if 𝑓−1(a) is n𝛼𝑔∗s-closed in (u, 𝜏𝑅(x)) for every nano-closed set a in (v, 𝜎𝑅(y)) • n𝑔𝛼g-continuous [5] if 𝑓−1(a) is n𝑔𝛼g-closed in (u, 𝜏𝑅(x)) for every nano-closed set a in (v, 𝜎𝑅(y)) definition 2.9[1] a subset a of a nano topological space (u, 𝜏𝑅(x)) is called n𝑔 *s closed set if nscl(a) u whenever a u and u is nĝ-open in (u, 𝜏𝑅(x)). 3. n𝒈 *s-continuous functions in this section we define n𝑔 *s-continuous functions and discuss some of their properties. definition 3.1 let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be two nano topological spaces. then a mapping : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is called n𝑔 *s-continuous if 𝑓−1(s) is n𝑔 *s-closed in (u, 𝜏𝑅(x)) for every nano-closed set s in (v, 𝜎𝑅(y)). definition 3.2 a function 𝑓: (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is n𝑔 *s-irresolute if 𝑓−1(s) is n𝑔 *s closed in (u, 𝜏𝑅(x)) for each n𝑔 *s -closed set s in (v, 𝜎𝑅(y)). example 3.3. let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be two nano topological spaces. let u = {a, b, c, d} with u/r = {{b, c}, {a}, {d}} and x = {a}. then the nano topology 𝜏𝑅(x) 100 n�̂�*s-continuous functions in nano topological spaces ={ ,u,{a}}. then n𝑔 *s-c(u, 𝜏𝑅(x)) = { , u,{b},{c},{a, b},{a, c}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d }, {b, c, d}}. let v = {a, b, c, d} with v/r = {{a}, {c}, {b, d}} and y = {a, b, c}. 𝜎𝑅(y) = { , v, {a, c}, {b, d}}. (𝜎𝑅(y)) = {∅, v, {a, c}, {b, d}}. n𝑔 *s-c (v, 𝜎𝑅(y)) = { , u, {b}, {c}, {a, b}, {a, c}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}. define 𝑓: u v as f (a) = b, (b) = a, (c) = d, (d) = c. we have 𝑓−1 (a, c) = {b, d}, 𝑓−1 (b, d) = {a, c}. thus, the inverse image of every n-closed set in v is n𝑔 *s -closed u. therefore, f is n𝑔 *s –continuous. 4. main results proposition 4.1 let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be two nano topological spaces. a function : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is nano-continuous then is n𝑔 *s-continuous. proof: let a be any nano closed set in (v, 𝜎𝑅(y)). since is nano continuous, 𝑓−1(a) is nano closed in (u, 𝜏𝑅(x)). since every nano closed set is n𝑔 *s-closed. therefore 𝑓−1(a) is n𝑔 *s-closed in (u, 𝜏𝑅(x)). hence is n𝑔 *s-continuous. remark 4.2 the converse of the above theorem need not be true, as proved by the following example. example 4.3 let (u, 𝜏𝑅(x)) be a nano topological space where u = {a, b, c, d} with u/r = {{b}, {c}, {a, d}} and x = {a, b}. 𝜏𝑅(x) = { , u, {b}, {a, d}, {a, b, d}}. (𝜏𝑅(x))𝑐 = { , u, {a, c, d}, {c}, {b, c}}. nĝ*s c (u, 𝜏𝑅(x)) = { , u, {b}, {c},{a, c}, {a, d}, {b, c}, {c, d}, {a, b, c}, {a, c, d},{b, c, d}}. nsc (u, 𝜏𝑅(x)) = { , u, {b}, {c}, {b, c}, {a, d}, {a, c, d}}. n c (u, 𝜏𝑅(x)) = { , u, {b}, {c}, {b, c}, {a, d}, {a, c, d}}. let (v, 𝜎𝑅(y)) be a nano topological space where v = {a, b, c, d} with v/ r = {{b}, {d}, {a, c}} and y = {a, d}. let 𝜎𝑅(y) = { , v, {d}, {a, c}, {a, c, d}}. (𝜎𝑅(y))𝑐 = { , v, {a, b, c}, {b}, {b, d}}. define (a) = d, (b) = a, (c) = b, (d) = c. let 𝑓−1(b) = {c}, 𝑓−1(b, d) = {a, c}, 𝑓−1(a, b, c) = {b, c, d}. here {b, c, d} is nĝ*s-closed but not n-closed. proposition 4.4 let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be two nano topological spaces. a function 𝑓: (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is n -continuous then is n𝑔 *s-continuous. proof let a be any nano closed set in (v, 𝜎𝑅(y)). since is -continuous, 𝑓−1(a) is nano -closed in (u, 𝜏𝑅(x)). since every nano -closed set is n𝑔 *s-closed. therefore 𝑓−1(a) is n𝑔 *s-closed in (u, 𝜏𝑅(x)). hence is n𝑔 *s-continuous. remark 4.5 the converse of the above theorem need not be true, as proved by the following example. from the example 3.6, the sub set a= {b, c, d} is not nano -closed set in (u, 𝜏𝑅(x)). hence f is not nano -continuous. proposition 4.6 let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be two nano topological spaces. a function : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is nsemi-continuous then is n𝑔 *s-continuous. 101 m. anto and j. carolinal proof let a be any nano closed set in (v, 𝜎𝑅(y)). since is semi-continuous, 𝑓−1(a) is nano semi-closed in (u, 𝜏𝑅(x)). since every nano semi-closed set is n𝑔 *s-closed. therefore 𝑓−1(a) is n𝑔 *s-closed in (u, 𝜏𝑅(x)). hence is n𝑔 *s-continuous. remark 4.7 the converse of the above theorem need not be true, as proved by the following example. from the example 3.6, the sub set a= {b, c, d} is not nano semi-closed set in (u, 𝜏𝑅(x)). hence f is not nano semi-continuous. proposition 4.8 let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be two nano topological spaces. a function : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is n𝑔#𝛼-continuous then is n𝑔 *s-continuous. proof let a be any nano closed set in (v, 𝜎𝑅(y)). since is n𝑔#𝛼-continuous, 𝑓−1(a) is n𝑔#𝛼-closed in (u, 𝜏𝑅(x)). since every n𝑔#𝛼-closed set is n𝑔 *s-closed. therefore 𝑓−1(a) is n𝑔 *s-closed in (u, 𝜏𝑅(x)). hence is n𝑔 *s-continuous. remark 4.9 the converse of the above theorem need not be true, as proved by the following example. example 4.10 let (u, 𝜏𝑅(x)) be a nano topological space where u = {a, b, c, d} with u/r = {{a}, {c}, {b, d}} and x = {a}. 𝜏𝑅(x) = { , u, {a}}. nĝ*sc (u, 𝜏𝑅(x)) = { , u, {b},{c}, {d}, {a, b}, {a, d}, {a, c}, {b, c}, {b, d}, {c, d}, {a, b, d}, {a, b, c,},{a, c, d}}. n𝑔#𝛼c (u, 𝜏𝑅(x)) = { , u, {b}, {c}, {d}, {b, c}, {b, d}, {c, d}, {b, c, d}}. let (v, 𝜎𝑅(y)) be a nano topological space where v = {a, b, c, d} with v/ r = {{a}, {c}, {b, d}} and y = {a, b}. 𝜎𝑅(y) = { , v, {a}, {b, d}, {a, b, d}, (𝜎𝑅(y))= { , v, {c}, {a, c}, {b, c, d}}. define (a) = b, (b) = a, (c) = c, (d) = d. let 𝑓−1(c) = {c}. 𝑓−1(a, c) ={b, c}, 𝑓−1(b, c, d) ={a, c, d}. here {a, c, d} is nĝ*s closed but not n𝑔#𝛼-closed. proposition 4.11 let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be two nano topological spaces. a function : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is n𝑔 𝛼-continuous then is n𝑔 *s-continuous. proof let a be any nano closed set in (v, 𝜎𝑅(y)). since is n𝑔 𝛼-continuous, 𝑓−1(a) is n𝑔 𝛼-closed in (u, 𝜏𝑅(x)). since every n𝑔 𝛼-closed set is n𝑔 *s-closed. therefore 𝑓−1(a) is n𝑔 *s-closed in (u, 𝜏𝑅(x)). hence is n𝑔 *s-continuous. remark 4.12 the converse of the above theorem need not be true, as proved by the following example. example 4.13 let (u, 𝜏𝑅(x)) be a nano topological space where u = {a, b, c, d} with u/r = {{a}, {b}, {c, d}} and x = {b, d}. 𝜏𝑅(x) = { , u, {b}, {c, d}, {b, c, d}}. nĝ*sc (u, 𝜏𝑅(x)) = { , u, {a}, {b}, {a, b}, {a, d}, {a, c}, {c, d}, {a, b, d}, {a, b, c,}, {a, c, d}}. n𝑔 𝛼c (u, 𝜏𝑅(x)) = { , u, {a}, {a, b}, {a, c}, {a, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}}. let (v, 𝜎𝑅(y)) be a nano topological space where v = {a, b, c, d} with v/ r = {{a}, {c}, {b, d}} and y = {a, b, d}. 𝜎𝑅(y) = { , v, {a, b, d}}, (𝜎𝑅(y)) = { , v, {c}}. 102 n�̂�*s-continuous functions in nano topological spaces define (a) = b, (b) = c, (c) = d, (d) = a. let 𝑓−1(c) = {b}. here {b} is nĝ*s-closed but not n𝑔 𝛼-closed. proposition 4.14 let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be two nano topological spaces. a function : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is ng*-continuous then is nĝ*s -continuous. proof let a be any nano closed set in (v, 𝜎𝑅(y)). since is ng*-continuous, 𝑓−1(a) is ng*-closed in (u, 𝜏𝑅(x)). since every ng*-closed set is n𝑔 *s-closed. therefore 𝑓−1(a) is n𝑔 *s-closed in (u, 𝜏𝑅(x)). hence is n𝑔 *s-continuous. remark 4.15 the converse of the above theorem need not be true, as proved by the following example. example 4.16 let (u, 𝜏𝑅(x)) be a nano topological space where u = {a, b, c, d} with u/r = {{a}, {c}, {b, d}} and x = {b, c}. 𝜏𝑅(x) = { , u, {c}, {b, d}, {b, c, d}}. (𝜏𝑅(x))𝑐 = { , u, {a, b, d}, {a}, {a, c}}. nĝ*sc (u, 𝜏𝑅(x)) = { , u, {a}, {c}, {a, b}, {a, c}, {a, d}, {b, d}, {a, b, d}, {a, b, c,}, {a, c, d}}. ng*c (u, 𝜏𝑅(x)) = { , u, {a}, {a, b}, {a, c}, {a, d}, {b, d}, {a, b, c}, {a, b, d}, {a, c, d}}. let (v, 𝜎𝑅(y)) be a nano topological space where v = {a, b, c, d} with v/ r = {{b}, {c}, {a, d}} and y = {a, c}. 𝜎𝑅(y)= { , v, {c}, {a, d}, {a, c, d}}. (𝜎𝑅(y))𝑐= { , v, {a, b, d}, {b}, {b, c}}. define (a) = c, (b) = d, (c) = b, (d) = a. let 𝑓−1(a, b, d) = {a, b, c}, 𝑓−1(b, c) = {a, c}, 𝑓−1(b) = {c}. here {c} is nĝ*s-closed but not ng*-closed. proposition 4.17 let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be two nano topological spaces. a function : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is ng*s-continuous then is nĝ*s -continuous. proof let a be any nano closed set in (v, 𝜎𝑅(y)). since is ng*s-continuous, 𝑓−1(a) is ng*s-closed in (u, 𝜏𝑅(x)). since every ng*s-closed set is n𝑔 *s-closed. therefore 𝑓−1(a) is n𝑔 *s-closed in (u, 𝜏𝑅(x)). hence is n𝑔 *s-continuous. remark 4.18 the converse of the above theorem need not be true, as proved by the following example. example 4.19 let (u, 𝜏𝑅(x)) be a nano topological space where u = {a, b, c, d} with u/r = {{a}, {b}, {c, d}} and x = {a, c}. 𝜏𝑅(x) = { , u, {a}, {c, d}, {a, c, d}}. (𝜏𝑅(x))𝑐 = { , u, {b, c, d}, {b}, {a, b}}. nĝ*sc (u, 𝜏𝑅(x)) = { , u, {a}, {b}, {a, b}, {b, c}, {b, d}, {c, d}, {a, b, d}, {a, b, c,}, {b, c, d}}. ng*sc (u, 𝜏𝑅(x)) = { , u, {a}, {a, b}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {b, c, d}}. let (v, 𝜎𝑅(y)) be a nano topological space where v = {a, b, c, d} with v/ r = {{a}, {c}, {b, d}} and y = {a, b}. 𝜎𝑅(y) = { , v, {a}, {b, d}, {a, b, d}}. (𝜎𝑅(y))𝑐= { , v, {b, c, d}, {c}, {a, c}}. define (a) = d, (b) = c, (c) = b, (d) = a. let 𝑓−1(b, c, d) = {a, b, c}, 𝑓−1(a, c) = {b, d}, 𝑓−1(c) = {b}. here {b} is nĝ*s-closed but not ng*s-closed. 103 m. anto and j. carolinal proposition 4.20 let (u, 𝜏𝑅(x)) and (v, σr′(y)) be two nano topological spaces. a function : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is nsĝ-continuous then is nĝ*s -continuous. proof let a be any nano closed set in (v, σr′(y)). since is nsĝ-continuous, 𝑓−1(a) is nsĝ -closed in (u, 𝜏𝑅(x)). since every nsĝ-closed set is n𝑔 *s-closed. therefore 𝑓−1(a) is n𝑔 *s closed in (u, 𝜏𝑅(x)). hence is n𝑔 *s-continuous. remark 4.21 the converse of the above theorem need not be true, as proved by the following example. let (u, 𝜏𝑅(x)) be a nano topological space where u = {a, b, c, d} with u/r = {{b}, {c}, {a, d}} and x = {a, b}. (u) = { , u, {b}, {a, d}, {a, b, d}}. (𝜏𝑅(x))𝑐 = { , u, {a, c, d}, {c}, {b, c}}. nĝ*sc (u, 𝜏𝑅(x)) = { , u, {b}, {c}, {a, c}, {a, d}, {b, c}, {c, d}, {a, c, d}, {a, b, c,}, {b, c, d}}. nsĝc (u, 𝜏𝑅(x)) = { , u, {c}, {b, c}, {a, c, d}}. let (v, 𝜎𝑅(y)) be a nano topological space where v = {a, b, c, d} with v/ r = {{a}, {b}, {c, d}} and y = {b, c}. 𝜎𝑅(y) = { , v, {b}, {c, d}, {b, c, d}}. (𝜎𝑅(y))𝑐= { , v, {a, c, d}, {a}, {a, b}}. define (a) = b, (b) = a, (c) = d, (d) = c. let 𝑓−1(a, c, d) = {b, c, d}, 𝑓−1(a, b) = {a, b}, 𝑓−1(a) = {b}. here {b} is nsĝ-closed but not ng*s-closed. proposition 4.22 let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be two nano topological spaces. a function : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is ng𝛼𝑔-continuous then is nĝ*s -continuous. proof let a be any nano closed set in (v, 𝜎𝑅(y)). since is ng𝛼𝑔-continuous, 𝑓−1(a) is ng𝛼𝑔-closed in (u, 𝜏𝑅(x)). since every ng𝛼𝑔-closed set is n𝑔 *s-closed. therefore 𝑓−1(a) is n𝑔 *s-closed in (u, 𝜏𝑅(x)). hence is n𝑔 *s-continuous. remark 4.23 the converse of the above theorem need not be true, as proved by the following example. example 4.24 let (u, 𝜏𝑅(x)) be a nano topological space where u = {a, b, c, d} with u/r = {{b}, {c}, {a, d}} and x = {a, b}. 𝜏𝑅(x) = { , u, {b}, {a, d}, {a, b, d}}. (𝜏𝑅(x))𝑐 = { , u, {a, c, d}, {c}, {b, c}}. nĝ*sc (u, 𝜏𝑅(x)) = { , u, {b}, {c}, {a, c}, {a, d}, {b, c}, {c, d}, {a, c, d}, {a, b, c,},{b, c, d}}. nsĝc (u, 𝜏𝑅(x)) = { , u, {c}, {b, c}, {a, c, d}}. let (v, 𝜎𝑅(y)) be a nano topological space where v = {a, b, c, d} with v/ r' = {{a}, {b}, {c, d}} and y = {b, c}. 𝜎𝑅(y) = { , v, {b}, {c, d}, {b, c, d}}. (𝜎𝑅(y))𝑐= { , v, {a, c, d}, {a}, {a, b}}. define (a) = b, (b) = a, (c) = d, (d) = c. let 𝑓−1(a, c, d) = {b, c, d}, 𝑓−1(a, b) = {a, b}, 𝑓−1(a) = {b}. here {b} is nsĝ-closed but not ng*s-closed. proposition 4.25 let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be two nano topological spaces. a function : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is n𝛼𝑔∗𝑠-continuous then is nĝ*s -continuous. proof let a be any nano closed set in (v, 𝜎𝑅(y)). since is n𝛼𝑔∗𝑠-continuous, 𝑓−1(a) is n𝛼𝑔∗𝑠-closed in (u, 𝜏𝑅(x)). since every n𝛼𝑔∗𝑠-closed set is n𝑔 *s-closed. therefore 𝑓−1(a) is n𝑔 *s-closed in (u, 𝜏𝑅(x)). hence is n𝑔 *s-continuous. 104 n�̂�*s-continuous functions in nano topological spaces remark 4.26 the converse of the above theorem need not be true, as proved by the following example. example 4.27 let (u, 𝜏𝑅(x)) be a nano topological space where u = {a, b, c, d} with u/r = {{b}, {c}, {a, d}} and x = {b, c}. 𝜏𝑅(x) = { , u, {b, c}}. (𝜏𝑅(x))𝑐 = { , u, {a, d}} nĝ*sc(u, 𝜏𝑅(x)) = { , u, {a},{d},{a, b}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, d}, {a, c, d}, {a, b, c,},{b, c, d}}. n𝛼𝑔∗𝑠c (u, 𝜏𝑅(x)) = { , u, {a}, {d}, {a, d}}. let (v, σr′(y)) be a nano topological space where v = {a, b, c, d} with v/ r' = {{b}, {c}, {a, d}} and y = {a, c}. 𝜎𝑅(y) = { , v, {c}, {a, d}, {a, c, d}}. (𝜎𝑅(y))𝑐= { , v, {a, b, d}, {b}, {b, c}}. define (a) = b, (b) = c, (c) = d, (d) = a. let 𝑓−1(a, b, d) = {a, c, d}, 𝑓−1(b, c) = {a, b}, 𝑓−1(b) = {a}. here {a, b} is n𝑔 *s-closed but not ng*s-closed. proposition 4.28 composition of two nĝ*s-continuous function need not be nĝ*s-continuous. let (u, 𝜏𝑅(x)) be a nano topological space where u = {a, b, c, d} with u/r = {{b}, {c}, {a, d}} and x = {a, b}. 𝜏𝑅(x) = { , u, {b}, {a, b, d}, {a, d}}. (𝜏𝑅(x))𝑐 = { , u, {a, c, d}, {c}, {b, c}} nĝ*sc(u, 𝜏𝑅(x)) = { , u, {b},{c},{a, c}, {a, d}, {b, c}, {c, d}, {a, b, c}, {a, c, d}, {b, c, d}}. define (a) = d, (b) = a, (c) = b, (d) = c. let 𝑓−1(a, b, c) = {b, c, d}, 𝑓−1(b, d) = {a, c}, 𝑓−1(b) = {c}. here {a, b} is n𝑔 *s-but not ng*s-closed. let (v, 𝜎𝑅(y)) be a nano topological space let v = {a, b, c, d} with v/ r = {{b}, {d}, {a, c}} and y = {a, d}. 𝜎𝑅(y) = { , v, {d}, {a, c}, {a, c, d}}. (𝜎𝑅(y))𝑐= { , v, {a, b, c}, {b}, {b, d}}. nĝ*sc(u, 𝜏𝑅(x)) = { , u, {b},{d},{a, b}, {a, c}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {b, c, d}}. let (w, (z)) be a nano topological space let w = {a, b, c, d} with w/ r = {{a}, {c}, {b, d}} and z = {b, c}. (z) = { , w, {c}, {b, d}, {b, c, d}}. ((z) )𝑐 = { , v, {a, b, d}, {a}, {a, c}}. define (a) = b, (b) = a, (c) = d, (d) = c. let 𝑔−1(a, b, d) = {a, b, c}, 𝑔−1(a, c) = {b, d}, 𝑔−1(a) = {b}. now, ∶ (u, 𝜏𝑅(x)) (w, (z)) by ( (a) = c, ( (b) = b, ( c) = a, ( (d) = d, 𝑔−1 (𝑓−1(a)) = a, 𝑔−1(𝑓−1(a, c)) =(a, c), 𝑔−1(𝑓−1(a, b, d)) =(a, b, d) is not n𝑔 *s-closed in u but {a, b, d} is closed in z. therefore, is not n𝑔 *s-continuous. proposition 4.29 let f (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) be n𝑔 *s be a function. then following are equivalent. (i) f is n𝑔 *s -continuous. (ii) 𝑓−1(a) is n𝑔 *s -open for each open set a in y. proof: (i) (ii) suppose that f is a n𝑔 *s -continuous. let a be n-open in u. then ac is n-closed in v. since is n𝑔 *s -continuous, we have 𝑓−1 (ac) is n𝑔 *s -closed in u. but 𝑓−1(ac) = [𝑓−1(a)]c. hence f−1 (a) is nĝ*s-open in u. 105 m. anto and j. carolinal (ii) (i) suppose that 𝑓−1 (a) is n𝑔 *s -open for each n-open set a in v. let s be nclosed in v. then sc is nano open in v. by assumption, 𝑓−1 (sc) is n𝑔 *s -open in u and hence 𝑓−1(s) is n𝑔 *s -closed in u. hence f is nĝ*s-continuous. proposition 4.30 let 𝑓 ∶ (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) be function (i) f is n𝑔 *s-continuous. (ii) for each u in u and for each n-open set b containing f (u), there is a n𝑔 *s -open set a containing u such that (a) b. (iii) (n𝑔 *s cl(a)) ncl( (a)) for each subset a of u. (iv) n𝑔 *scl 𝑓−1(b)) 𝑓−1(ncl(f(b)) for each subset b of v. then (i) (ii) (iii) (i) proof: (i) (ii) let u u and b be an open set containing f (u). then, by (i) f -1(b) is n𝑔 *s -open set of u containing u. if a = 𝑓−1(b), then (a) = (𝑓−1(b)) b. (ii) (iii) let a be a subset of a space u and f(u) ncl(f(a)). then there exists nopen set b of v containing (u) such that b f (a) = . now, by (ii), there is a n𝑔 *s open set g containing u such that (u) f (g) b. hence (a) f (g) = . that is, (a g) = . i.e., a g) = . therefore, u n𝑔 *scl (a) and also (u) n𝑔 *scl 𝑓 (a). therefore (n𝑔 *s cl(a) ncl( (a)) (iii) (iv) let b be a subset of v such that a = 𝑓−1(b). by (iii), (n𝑔 *scl(a) ncl( (a) for each subset a of u. therefore, (n𝑔 *scl 𝑓−1(b)) ncl ( (𝑓−1 (b))). i.e., (n𝑔 *s cl 𝑓−1(b)) ncl(b). i.e., n𝑔 *scl 𝑓−1 (b) ⊆ 𝑓−1(ncl(b)). lemma 4.31 a subset a of a nano topological space (u, 𝜏𝑅(x)) is n𝑔 *s -open iff f nsint (a) whenever f a and f is nĝ-closed. proposition 4.32 let b be a n𝑔 *s open (or n𝑔 *s -closed) subset of (v, 𝜏𝑅′(y)) (satisfying nsint(b) =nint(b). then 𝑓−1 (b) is n𝑔 *s -open (or n𝑔 *s -closed) in (u, 𝜏𝑅(x)). if 𝑓 ∶ (u, 𝜏𝑅(x)) (v, 𝜏𝑅′(y)) is nĝ*s-continuous and if image of a nĝ-closed set in u under is nĝ closed set in v. proof. let b be a n𝑔 *s -open set in v. let f 𝑓−1 (b) where f is a nĝ-closed set in u. then f(f) b holds. by our assumption, f(f) is nĝ-closed set in v and b be a n𝑔 *s open set in v. therefore, by lemma 3.7 f(f) nsint(b) holds. again, by our assumption, (f) nint(b) and hence f ⊆ 𝑓−1 (nint(b)) holds. since f is n𝑔 *s continuous and nint (b) is n open in v, f -1(int (b)) is n𝑔 *s -open in u so, by lemma 3.7, f nsint (𝑓−1 (nint (b))) holds. i.e., f nsint (𝑓−1 (nint (b))) nsint (𝑓−1 (b)) holds. therefore 𝑓−1 (b) is n𝑔 *s -open. by taking complements, we can show that if b is n𝑔 *s -closed in v, then 𝑓−1 (b) is n𝑔 *s -closed in u. proposition 4.33 let 𝑓 ∶ (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) be a function. then the following are equivalent. (i) is nĝ*s-irresolute. (ii) 𝑓−1(b) is n𝑔 *s -open for each n𝑔 *s -open set b in v. 106 n�̂�*s-continuous functions in nano topological spaces proof: (i) (ii) suppose that f is n𝑔 *s -irresolute. let b be n𝑔 *s -open in v. then is n𝑔 *s -closed in v. since is n𝑔 *s -irresolute, we have 𝑓−1(𝐵𝑐) is n𝑔 *s -closed in u. but −1 (𝐵𝑐) = [𝑓−1 (b)]c. therefore 𝑓−1(b) is nĝ*s-open in u. (ii) (i) n𝑔 *s suppose that 𝑓−1 (b) is n𝑔 *s -open for each is n𝑔 *s -open set b in v. let h be is n𝑔 *s closed in v. then 𝐻𝑐 is n𝑔 *s -open in v. therefore 𝑓−1 (𝐻𝑐) is n𝑔 *s -open in u. therefore 𝑓−1 (h) is n𝑔 *s -closed in u. therefore, f is n𝑔 *s -irresolute. proposition 4.34 if a function 𝑓 ∶ (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is n𝑔 *s -irresolute, then is n𝑔 *s -continuous. proof: let b be a n-closed set of v. but every n-closed set is n𝑔 *s -closed. therefore, b is a n𝑔 *s -closed set of v. since is n𝑔 *s -irresolute, 𝑓−1(b) is n𝑔 *s closed in u. hence, by definition 3.2 is n𝑔 *s -continuous. remark 4.35 the converse of proposition 3.10 need not be true as seen from the following example. example 4.36 let (u, 𝜏𝑅(x)) and (v, 𝜎𝑅(y)) be a nano topological spaces where u = v = {a, b, c, d} with u/r = {{b}, {c}, {a, d}} and x = {a, c}. then the nano topology 𝜏𝑅(x) = { , u, {c}, {a, c, d}, {a, d}}. (𝜏𝑅(x)) = { , u, {a, b, d}, {b}, {b, c}}. then n𝑔 *s c (u, 𝜏𝑅(x)) = { , u, {b}, {c}, {a, b}, {a, d}, {b, c}, {b, d},{a, b, d}, {b, c, d}}. let v/ r = {{a}, {b, c}, {d}} and y = {a, c}. then the nano topology 𝜏𝑅(x) = { , v, {a}, {a, b, c}, {b, c}}. (𝜎𝑅(y))𝑐= { , u, {b, c, d}, {d}, {a, d}}. then n𝑔 *sc (u, 𝜏𝑅(x)) = { , u, {a}, {d}, {b, d},{c, d}, {a, d}, {b, c, d}, {a, b, d}, {a, c, d}}. define 𝑓 ∶ u v as (a) = c, (b) = d, (c) = a, (d) = b. we have 𝑓−1(a) = {c}, f -1(d) = {b}, 𝑓−1(b, d) = {a, d}, 𝑓−1(c, d) = {a, b}, 𝑓−1(a, d) = {b, c}, 𝑓−1(a, b, d) = {b, c, d}, 𝑓−1(a, c, d) = {a, b, c}, 𝑓−1(b, c, d) = {a, b, d}. 𝑓−1(b) is nĝ*s-closed for each nano-closed set b in v. hence f: u v is n𝑔 *s continuous. but, 𝑓−1(a, c, d) = {a, b, c} is not n𝑔 *s -closed. therefore 𝑓 ∶ u v is not n𝑔 *s -irresolute. proposition 4.37 if a function 𝑓 ∶ (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is n𝑔 *s -irresolute, then for every subset a of u. then (n𝑔 *scl(a)) nscl( (a)). proof: let a u. we know that every ns-closed set is n𝑔 *s-closed set in v. therefore, we have nscl( (a)) is n𝑔 *s -closed in v. since is n𝑔 *s -irresolute, then 𝑓−1(nscl( (a))) is n𝑔 *s-closed in u. also a 𝑓−1 ( (a)) ⊆ 𝑓−1 (nscl ( (a))). since 𝑓−1(nscl( (a))) is n𝑔 *s -closed, we have n𝑔 *scl(a) 𝑓−1 (nscl(f(a))). therefore 𝑓( n𝑔 *scl(a)) {𝑓−1 (nscl( (a)))} nscl( (a)). proposition 4.38 if a function : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is bijective, n𝑔 *s continuous, nscl (a) = ncl(a) for all subsets b in v and if image of a nĝ-open set is nĝ-open under , then is n𝑔 *s -irresolute. 107 m. anto and j. carolinal proof: let b be n𝑔 *s -closed set of v. let 𝑓−1 ((b)) a where a is nĝ-open in u. then 𝑓−1( (b)) (a). since f is surjective, b (a). since (a) is nĝ-open and since b is n𝑔 *s -closed in v, we have nscl(b) (a). by our assumption, ncl(b) (a). since is injective, 𝑓−1(ncl(b)) a. since is n𝑔 *s -continuous and since ncl(b) is n-closed in v, 𝑓−1(ncl(b)) is nĝ*s-closed in u. therefore 𝑓−1((b)) is n𝑔 *s closed in u and hence is n𝑔 *s irresolute. definition 4.39 a function 𝑓 ∶ (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is called a n𝑔 *s -closed map if (a) is n𝑔 *s -closed in v for every n-closed set a of u definition 4.40 a function 𝑓 ∶ (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is called a nĝ*s-open map if (a) is nĝ*s-open in v for every n-open set a of u proposition 4.41 if 𝑓 ∶ (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is n𝑔 *s -irresolute and a is a n𝑔 *s closed subset of u, then (a) is n𝑔 *s -closed in v. proof: let (a) b and b is nĝ-open in v. then 𝑓−1( (a)) 𝑓−1 (b). i.e., a ⊆ 𝑓−1 (b). since f is nĝ-irresolute, 𝑓−1(b) is nĝ-open in u. since a is n𝑔 *s -closed, ncl(a) 𝑓−1 (b). so, (ncl(a)) (𝑓−1 (b)). i.e., (ncl(a)) u. since is n𝑔 *s -closed and ncl(a) is n closed in u, f(ncl(a)) is n𝑔 *s -closed in v. therefore nscl ( (ncl (a)) b. since (a) ⊆ 𝑓 (ncl (b)), we have nscl ( (a)) nscl ( (ncl (b)) b. therefore (a) is n𝑔 *s -closed in v. proposition 4.42 if 𝑓 ∶ (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is n-closed and: v w is n𝑔 *s closed, then o is n𝑔 *s -closed. proof: let a be a n-closed set of u. since is n-closed, (a) is n-closed in v. since is n𝑔 *s -closed, (a)) is n𝑔 *s -closed in w. hence o : u w is n𝑔 *s -closed. proposition 4.43 let 𝑓 ∶ (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is n-closed and : (v, 𝜎𝑅(y)) (w, (z)) be two maps such that 𝑔o : (u, 𝜏𝑅(x)) (w, 𝜇𝑅(z)) is n𝑔 *s -open map, if is n continuous and surjective. proof: let b be a n-open v. since f is n-continuous, 𝑓−1 (b) is n-open in u. since 𝑓−1(b) is n-open in u, o (𝑓−1 (b))) is n𝑔 *s -open in w. i.e., (b) is n𝑔 *s -open in w. therefore, is a n𝑔 *s -open map. proposition 4.44 for any bijection 𝑓 (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)), the following are equivalent: (i) 𝑓−1 : (v, 𝜎𝑅(y)) u is nĝ*s-continuous. (ii) is n𝑔 *s -open. (iii) is n𝑔 *s -closed. 108 n�̂�*s-continuous functions in nano topological spaces proof: (i) (ii) let a be n-open in u. then u-a is n-closed in u. since 𝑓−1 is n𝑔 *s continuous, (𝑓−1)-1 (u-a) = (u – f) = v – (f) is n𝑔 *s -closed in v. then (f) is n𝑔 *s -open in v. hence is n𝑔 *s -open. (ii) (iii) let a be n-closed in u. then u-a is n-open in u. since 𝑓 is n𝑔 *s open, (u – f) = v – (f) is n𝑔 *s -open in v. then (f) is n𝑔 *s -closed in v. hence is n𝑔 *s closed. (iii) (i) let a be n-closed in u. since : (u, 𝜏𝑅(x)) (v, 𝜎𝑅(y)) is n𝑔 *s closed. (a) is n𝑔 *s closed in v. i.e., (𝑓−1)-1(a) n𝑔 *s -closed in v. therefore 𝑓−1is n𝑔 *s -continuous. 5. conclusion in this paper, we introduced and studied the concepts of nĝ*s-continuous and n𝑔 *s irresolute in nano topological spaces and we compare it with other nano-continuous and irresolute function and proved that composition of two nĝ*s-continuous functions need not be a nĝ*s-continuous functions. we also investigate some of its properties and give suitable examples for the reverse which is not true. in future this work will be extended with some real-life applications. references [1] m. anto and j. carolinal, “n𝑔 *s-continuous functions in nano topological spaces” (communicated) in seajmms. [2] m. lellis thivagar. m and carmel richard, ‘on nano continuity’ mathematical theory of modeling vol.3, no.7, 2013. [3] pious missier, m. anto, ĝ*s-closed sets in topological spaces, international journal of modern engineering research, (ijmer),22496645/vol.4/iss.11/nov.2014/32 [4] p. sathishmohan, v. rajendran, a. devika and r. vani on ‘nano semicontinuity and nano pre-continuity’, international journal of applied research 2017;3(2):76-79. [5] qays hatem imaran, murtadha m abdulkadhim and mustafa. h. hadi. ‘on nano generalised alpha generalized closed sets’ in nano topological spaces, general mathematics notes: vol 34, issue 2. 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[10] v. rajendran, p. sathishmohan and m. malarvizhi ‘on n𝑔 𝛼-continuous function in nano topological spaces, malaya journal of mathematics vol.5, no.1, 355360,2019. [11] v. rajendran, p. sathishmohan and r. mangayarkarasi,’ ns𝑔 -continuous function in nano topological spaces’, turkish journal of computer and mathematics education, vol.12, no.10, 2021. [12] v. rajendran, p. sathishmohan and r. nithya kala, ‘on new class of continuous functions in nano topological spaces’, malaya journal of mathematics vol.6, no.2, 385-389, 2018 110 ratio mathematica 22 (2012) 69-84 issn:1592-7415 hypergroups and geometric spaces maria scafati tallini viale ippocrate, 97, 00161-roma, italy. tallini@mat.uniroma1.it abstract we explain some links between hypergrpoups and geometric spaces. we show that for any given hypergroup it is possible to define a particular geometric space and then a canonical homomorphism between the hypergroup and a group. key words: hypergroup, geometric space 2000 ams subject classifications: 22n22. 1 hypergroups and their properties a hypergroupoid is a pair (g, ◦) where g is a non-empty set and ◦ : g × g → p ′(g) is a mapping of g × g into the set of non-empty subsets of g, denoted as p ′(g). a semihypergroup is a hypergroupoid satisfying the following associative property: ∀x, y, z ∈ g, (x ◦ y) ◦ z = x ◦ (y ◦ z), (1) where the left hand side of (1) is the set (x ◦ y) ◦ z = ∪ u∈x◦y u ◦ z and the right hand side is the set x ◦ (y ◦ z) = ∪ v∈y◦z x ◦ v. the associative property means that the two set theoretical unions coincide. we say that (g, ◦) satisfies the reproducibility property (both left and right), if ∀a, b ∈ g, ∃x ∈ g : b ∈ a ◦ x and ∃y ∈ g : b ∈ y ◦ a. (2) if (2) is satisfied, the family b2 = {a ◦ b : a, b ∈ g} is a covering of g (the index 2 under b, means that we consider the hyper product of two elements of g). 69 m. scafati tallini a hypergroup is an associative hypergroupoid satisfying the reproducibility property. we remark that a hypergroup (g, ◦) having a single valued product (that is, such that ∀x, y ∈ g, |x ◦ y| = 1) is a group. this is because the following result holds: theorem 1.1. an associative groupoid (g, ◦) is a group if, and only if, ∀a, b ∈ g, ∃x ∈ g : ax = b and ∃y ∈ g : ya = b. (3) ((3) is also called right and left quotient axiom). proof. if (g, ◦) is a classical group, (3) obviously holds. assume that the associative groupoid (g, ◦) satisfies (3). let us proove that it is a group. fix a ∈ g, let u be one of the elements z ∈ g such that: az = a (see (3)). for any c ∈ g, there is y ∈ g such that ya = c. then we have au = a =⇒ y(au) = (ya)u = ya =⇒ cu = (ya)u = ya = c =⇒ cu = c. hence, ∀c ∈ g, cu = c. (4) similarly, by (3), we prove that there exists v ∈ g such that: ∀c ∈ g, vc = c. (5) by (4), for c = v and by (5), for c = u, we get vu = v, vu = u, that is v = u and then ∀c ∈ g, uc = cu = c. therefore the unity of (g, ◦) exists and it is unique. for any a ∈ g, there is at least an element a′ ∈ g and a′′ ∈ g such that: aa′ = u = a′′a, (see 3). then a′ = ua′ = (a′′a)a′ = a′′(aa′) = a′′u = a′′, that is in g there is an element a′(= a′′), such that a′ = a′a = u. such an element a′ is obviously unique and it is the inverse, a−1, of a. then (g, ◦) is a classical group and the theorem is proved. a substructure of the hypergroup (g, ◦) is a subset h(̸= ∅) such that ∀x, y ∈ h, x ◦ y ∈ h. the pair (h, ◦) is a semihypergroup, if it satisfies (1). in particular it is a hypergroup if it satisfies also (2). let f be the set of all the substructures of (g, ◦). two cases may occur.∩ t∈f t /∈ ∅ or ∩ t∈f t = ∅. set s = f in the first case and s = f ∪ {∅} in the second. in both cases it is: g ∈ s and ∀i ∈ i, ti ∈ s =⇒ ∩ i∈i ti ∈ s, where i is a non empty set of 70 hypergroups and geometric spaces indices. in this way, s is a closure system of g. hence, for any x ⊆ g, the closure x̄ of x in g is: x = ∩ t ∈ s, t ⊆ x t if x̄ = ∅, x̄ is the least (from the set theoretical perspective) substructure containing x. so the following closure operator is defined as follows. :̄ x ⊆ g −→ x̄ ∈ s. (6) note that this closure operator satisfies the following properties: x ⊆ x̄; x ⊆ s, s ∈ s =⇒ x̄ ⊆ s; x = x̄ ⇐⇒ x ∈ s; ∀ x ⊆ g, x̄ = ¯̄x; ∀ x, y ⊆ g, x ⊆ y =⇒ x̄ = ȳ ; ∀ i ∈ i, xi ⊆ g =⇒ ∪ i∈i x̄i ⊆ ∪ i∈i xi and ∩ i∈i xi ⊆ ∩ i∈i x̄i. for all x ⊆ g, we define: x independent def⇐⇒ ∀ x ∈ x, x /∈ x r {x}, (7) x dependent def⇐⇒ ∈ x ∈ x, x ∈ x r {x}, (8) x generator def⇐⇒ x̄ = g, (9) x base def⇐⇒ x is independent and x̄ = g. (10) the pair (g, ◦) is finitely generated if, and only if, there is a finite subset x of g such that x̄ = g. we can easily prove that theorem 1.2. if (g, ◦) has a finite generator x then there is a finite base contained in x. a hypergroup (g, ◦) is called monic if, and only if, it does not contain any substructure different from (g, ◦). if (g, ◦) is a hypergroup and n ∈ n+ = n − {0} then we let bn def = {x1 ◦ x2 ◦ . . . ◦ xn ∈ g : (x1, x2, . . . , xn) ∈ gn}, (11) b def= {bn : n ∈ n+}. (12) 71 m. scafati tallini we call complete part of the hypergoup (g, ◦) a subset a such that b ∈ b, b ∩ a ̸= ∅ =⇒ b ⊆ a. (13) obviously, ∅ and g are complete parts. furthermore, the union and the intersection of complete parts are complete parts. moreover the complement of a complete part is a complete part. hence, the complete parts of (g, ◦) form a topology, where every open set it is also closed. we remark that (g, ◦) is a group if, and only if, every subset is a complete part (getting the discrete topology). we easily prove that if (g, ◦) is a hypergroup such that ∀ x, y ∈ g, ∈ b ∈ b such that x, y ∈ b then the only complete parts of (g, ◦) are ∅ and g (getting the trivial topology). (14) in section 5, we characterize the complete parts of (g, ◦) and we prove that the intersection of the subhypergroups which are also complete parts is a subhypergroup which is a complete part. we remark that the intersection of two subhypergroups may not be a subhypergroup in general. as a matter of fact, the intersection of two subhypergroups may be even the emptyset; as we will see in an example down below. we define heart of (g, ◦) and we denote it by ω, the intersection of all the subhypergroups which are also complete parts of (g, ◦); that is, the least subhypergroup complete part of (g, ◦). we easily prove that (see (14)): ∀x, y ∈ g, ∃b ∈ b : x, y ∈ b =⇒ ω = g. (15) as a matter of fact, by (14), the only complete parts of (g, ◦) are ∅ and g and the only subhypergroup which is a complete part is g. the following statements hold: if g ∈ b then ω = g. if (g, ◦) does not contain proper subhypergroups complete parts then ω = g. if (g, ◦) is monic then ω = g. we define scalar unity of (g, ◦) an element u ∈ g such that: ∀a ∈ g, a ◦ u = u ◦ a = {a}. obviously, if a scalar unity in (g, ◦) exists then it is unique. moreover, {u} is a subhypergroup of (g, ◦) and generally it is not a complete part. we get: {u} is a complete part ⇐⇒ [∀a, b ∈ g : u ∈ a ◦ b =⇒ a ◦ b = u] ⇐⇒ {u} = ω. in section 5 we prove that u ∈ ω in any case. finally, we remark that the hypergroups having a scalar unity are rather difficult to discover. 72 hypergroups and geometric spaces 2 examples of hypergroups example 2.1. let g be any non-empty set. define the multivalued product ∀x, y ∈ g, x ◦ y = g. then (g, ◦) is a hypergroup which we call trivial. this hypergroup is monic, its complete parts are only ∅ and g and its heart coincides with g. example 2.2. let g be any non-empty set. define the product ∀x, y ∈ g, x ◦ y = {x, y}. the (g, ◦) is called discrete hypergroup. every non-empty set is a subhypergroup and then there are subhypergroups whose intersection is the empty set. by (14) the only complete parts of (g, ◦) are ∅ and g and then ω = g. example 2.3. let (g, ◦) be a group and n a normal sugroup of (g, ◦). set ∀ x, y ∈ g, x ◦ y = xyn. then (g, ◦) is a hypergroup. the condition (2) is obvious (it suffices to set x = a−1b and y = ba−1); as regard the associativity, we have: ∀ a, b ∈ g, (a ◦ b) ◦ c = abnc = abcn = a ◦ (b ◦ c). moreover, bn coincides with the cosets of n in g and then ω = n and (n, ◦) is the trivial hypergroup (that is x ◦ y = n). example 2.4. let (gi, ◦i)i∈i be a family of hypergroups (in particular of groups) such that |i| ≥ 2, gi ∩ gj = ∅, i ̸= j. set g = ∪ i∈i gi and ∀x, y ∈ g, x ◦ y def= { x ◦i y if x, y ∈ (gi, ◦i), g if x ∈ gi and x ∈ gj. it is easy to prove that the pair (g, ◦) is a hypergroup. for any i ∈ i we get that (gi, ◦i) is a subhypergroup of (g, ◦) and such hypergroups are two by two disjoint. the only complete parts of (g, ◦) are ∅ and g and then ω = g. example 2.5. let (g, ◦) be any group with |g| ≥ 3. set: ∀x, y ∈ g, x ◦ y = g r {xy}. (16) let us prove that (g, ◦) is a hypergroup. we have: ∀x, y, z ∈ g, (x ◦ y) ◦ z = ∪ t∈gr{xy} g r {tz}. (17) since |g| ≥ 3, for any c ∈ g there is an element t′ in g such that t′ ∈ g r {xy, cz−1}. we have c ∈ g r {t′z}, with t′ ∈ g r {xy}, whence, by 73 m. scafati tallini (17), c ∈ (x ◦ y) ◦ z. therefore (x ◦ y) ◦ z = g. similarly, we prove that x ◦ (y ◦ z) = g. so, the associative property of (g, ◦) follows. now, let us prove the reproducibility property (2). for any a, b ∈ g there is x ∈ g r {a−1b} and y ∈ g r {ba−1}. so, b ∈ g r {ax} = a ◦ x, b ∈ g r {ya} = y ◦ a. this implies (2). let us prove that (g, ◦) is monic. let s be a subhypergroup of (g, ◦) and a ∈ s. we have a ◦ a = g r {a2} ⊆ s, then |s| ≥ 2 (because |g| ≥ 3). now, let b ∈ s, with b ̸= a. it is a ◦ b = g r {ab} ⊆ s, hence, if s ̸= g then s = g r {ab} = g r {a2} which is impossible because a ̸= b. therefore, s = g and (g, ◦) is monic. we easily prove that the only complete parts of (g, ◦) are ∅ and g, and so, ω = g. example 2.6. let g = rn. for any x, y ∈ rn, with x ̸= y, set x ◦ x = {x}, x ◦ y = the closed interval whose extremal points are x and y. we prove that (rn, ◦) is a hypergroup. in fact, the reproducibility property is obvious and the associativity holds because (x ◦ y) ◦ z coincides with the triangle (eventually, degenerate), t(x, y, z), with vertices x, y and z. the same happens for x ◦ (y ◦ z). and so, (x ◦ y) ◦ z = t(x, y, z) = x ◦ (y ◦ z). in (rn, ◦) every convex set is a substructure and viceversa. the open convexes are subhypergroups. therefore, disjoint subhypergroups exist. moreover, by (14) the only complete parts are ∅ and g = rn, and then ω = g. example 2.7. let g = p(d, k) be the d-dimensional projective space over the field k. for all x, y ∈ p(d, k), with x ̸= y, set x ◦ x = {x}, x ◦ y = the line through x and y. it is easy to prove that (g, ◦) is a hypergroup. in fact, the reproducibility property is obvious and the associativity holds because (x ◦ y) ◦ z coincides with the subspace spanned by x, y and z. the substructures of (g, ◦) are hypergroups and coincide with the subspaces of g = p(d, k). from (14), the only complete parts of (g, ◦) are ∅ and g = p(d, k), and then ω = g. example 2.8. let g = vk be a vector space over the field k. set ∀ x, y ∈ g, x ◦ y = {a(x + y) : a ∈ k}. we can prove that (g, ◦) is a hypergroup. the substructure of (g, ◦) are the subspaces of vk and, hence, they are hypergroups. moreover (14) is satisfied and the only complete parts of (g, ◦) are ∅ and g = vk. hence, ω = g. 74 hypergroups and geometric spaces example 2.9. let (g1, ◦1) and (g2, ◦2) be two hypergroups. set g = g1 × g2. furthermore, if x = (x1, x2) and y = (y1, y2) are two elements of g then x◦y def= (x1 ◦1 y1, x2 ◦2 y2. it is easy to prove that such a (g, ◦) is a hypergroup which we call the cartesian product of (g1, ◦1) and (g2, ◦2). similarly, we define the cartesian product of a family of hypergroups {(gi, ◦i) : i ∈ i}. in particular, we set g = ∏ i∈i gi, x ◦ y = {xi ◦i yi : i ∈ i}, where x = {xi : i ∈ i} and y = {yi : i ∈ i}. we remark that if each (gi, ◦i) has a scalar unity ui then the cartesian product has a scalar unity u = {ui : i ∈ i}. example 2.10. in the examples from 2.1 to 2.8, the hypergroups do not have a scalar unity. now we give an example of a hypergroup with a scalar unity of order 2. and this is also the simplest hypergroup which is not a group. consider the set g = {u, a} and set u ◦ u def= {u}, u ◦ a def= a ◦ u def= {a}, a ◦ a def= {u, a}. it can be easily checked that (g, ◦) is a hypergroup which as u as scalar unity. from (14), the only complete parts of (g, ◦) are ∅ and g because a ◦ a = g. hence, ω = g. the only subhypergroup of (g, ◦) is {u} which is not a complete part. the cartesian product of copies of such hypergroup is a wide class of hypergroups with scalar unity. 3 homomorphisms of hypergroups let (g, ◦) and (g′, ◦′) be two hypergroupoids. we call homomorphism of (g, ◦) and (g′, ◦′) a mapping f : g −→ g′ such that ∀x, y ∈ g, f(x ◦ y) ⊆ f(x) ◦′ f(y). (18) we remark that in general f(g) is a substructure of (g′, ◦′). let us prove: theorem 3.1. if g −→ g′ is a homomorphism of (g, ◦) to (g′, ◦′), for any substructure h′ of (g′, ◦′) such that f−1(h′) ̸= ∅ then h′ = f−1(h′) is a substructure of (g, ◦). moreover, if (g, ◦) satisfies the reproducibility property, we have: ∀a′, b′ ∈ f(g), ∃x′ ∈ f(g) : b′ ∈ a′ ◦′ x′ =⇒ ∃y′ ∈ g′ : b′ ∈ y′ ◦′ a′. (19) finally, if (g, ◦) and (g′, ◦′) are semihypergroups and a’ is a complete part of (g′, ◦′) then a = f−1(a′) is a complete part of (g′, ◦′) and a = f−1(a′) is a complete part of (g, ◦). 75 m. scafati tallini proof. let x, y ∈ h. it is x′ = f(x), y′ = f(y) ∈ h′ and then a′ ◦′ y′ ∈ h′. by (18) we get: f(x ◦ y) ⊆ f(x) ◦′ f(y) = x′ ◦′ y′ ⊆ h′, where x ◦ y ⊆ f−1(h′) = h. therefore h is a substructure of (g, ◦). if (g, ◦) satisfies the reproducibility property we get: ∀a′, b′ ∈ f(g) =⇒ ∃a, b ∈ g : a = f(a), b′ = f(b) =⇒ ∃x ∈ g : b ∈ a ◦ x, ∃y ∈ g : b ∈ y ◦ a =⇒ ∃x′ = f(x) ∈ f(g) : b′ ∈ f(a ◦ x) ⊆ f(a) ◦′ f(x) = a′ ◦′ x′ and ∃y′ = f(y) ∈ f(g) : b′ = f(y ◦ a) ⊆ f(y) ◦′ f(a) = y′ ◦′ a′ =⇒ ∃x′ ∈ f(g), b′ ∈ a′ ◦′ x′, ∃y′ ∈ f(g) : b′ ∈ y′ ◦′ a′. if (g, ◦) and (g′, ◦′) are semihypergroups, that is the associativity property holds, for any complete part a′ of (g′, ◦′), setting a = f−1(a′), we get, since f(a) = a′ and setting x′i = f(xi), xi ∈ gi: (x1 ◦ x2 ◦ . . . ◦ xn) ∩ a ̸= ∅ =⇒ f(x1 ◦ x2 ◦ . . . ◦ xn) ∩ f(a) ̸= ∅ =⇒ ∅ ̸= f(x1 ◦ x2 ◦ . . . ◦ xn) ∩ f(a) ⊆ (x′1 ◦ ′ x′2 ◦ ′ . . . ◦′ x′n) ⊆ a ′ =⇒ (x1 ◦ x2 ◦ . . . ◦ xn) ⊆ f−1(x′1 ◦ ′ x′2 ◦ ′ . . . ◦′ x′n) ⊆ a =⇒ (x1 ◦ x2 ◦ . . . ◦ xn) ⊆ a, therefore, a is a complete part of (g, ◦) and the theorem is proved. a homomorphism is called strong if in (18) the equality holds. obviously, if (g′, ◦′) is a groupoid (that is, if the operation ◦′ is single valued) then every homomorphism between any two hypergroupoids (g, ◦) and (g′, ◦′) is strong. now, let us prove theorem 3.2. let f be a strong homomorphism between the hypergroupoid (g, ◦) and (g′, ◦′). then im(f) = f(g) is a hypergroup; (20) if h is a substructure of (g, ◦) then f(h) is a substructure of f(g); (21) if h is a subhypergroup of (g, ◦) then f(h) is a subhypergroup of f(g); (22) if h′ is a substructure of f(g) then f−1(h′) is a substructure of (g, ◦); (23) if a′ is a complete part of (g′, ◦′) then f−1(a′) is a complete part of (g, ◦). (24) 76 hypergroups and geometric spaces proof. since f is a strong homomorphism, if h is a substructure of (g, ◦) then ∀x′, y′ ∈ f(h) =⇒ ∃x, y ∈ h: x′ = f(x), y′ = f(y) =⇒ x ◦ y ⊆ h, x′ ◦′ y′ = f(x) ◦′ f(y) = f(x ◦ y) ⊆ f(h) =⇒ x′ ◦′ y′ ⊆ f(h) ⊆ f(g), that is if h is a substructure of (g, ◦) then f(h) is a substructure of (g′, ◦′). (25) from (25) with h = g, we have that f(g) is a substructure of (g′, ◦′), that is f(g) is closed with respect to the product. furthermore, from theorem 3.1 (second part), we have that (f(g), ◦′) satisfies the reproducibility property. let us prove that (f(g), ◦′) is associative. in fact, ∀x′, y′, z′ ∈ f(g) ∃x, y, z ∈ g: x′ = f(x), y′ = f(y), z′ = f(z) =⇒ (x′◦′y′)◦′z′ = f((x◦y)◦z) = f(x ◦ (y ◦ z)) = x′ ◦′ (y′ ◦′ z′); and so, (f(g), ◦′) is a hypergroup. hence, the property (20) holds. the property (21) follows from (25) and (20). the property (22) follows from (21) and theorem 3.1 (second part). the property (23) is trivial. the property (24) follows from theorem 3.1 (third part). hence, the theorem is proved. theorem 3.3. let (g, ◦) be a hypergroup, ω be the heart of (g, ◦), (g′, ◦) be a group and u be the unity of (g′, ◦). if f : g −→ g′ is a homomorphism (necessarily, strong) of (g, ◦) in (g′, ◦) then 1. if h′ is a subgroup of (g′, ◦) then f−1(h′) is a subhypergroup of (g, ◦); 2. ∀a′ ⊆ g′, f−1(a′) is a complete part of (g, ◦); 3. ∀x′ ∈ f(g), f−1(x′) is a complete part ( ̸= ∅) of (g, ◦); 4. ∀b ∈ b (see (12), |f(b)| = 1; 5. ω ⊆ f−1(u). proof. if h′ is a subgroup of g′, from (20), we have that h′ ∩ f(g) is a subgroup of f(g) and hence, from (23), f−1(h′ ∩ f(g)) = f−1(h′) is a substructure of (g, ◦). let us prove that (2) holds for f−1(h′). we have that: ∀a, b ∈ f−1(h′), ∃x, y ∈ g: b ∈ a ◦ x, b ∈ y ◦ a =⇒ f(a), f(b) ∈ h′, f(b) = f(a) · f(x) = f(y) · f(a) =⇒ f(x) = (f(a))−1 · f(b) ∈ h′, f(y) = f(b) · (f(a))−1 ∈ h′ =⇒ ∃x, y ∈ f−1(h′): b ∈ a ◦ x, b ∈ y ◦ a. this implies that (2) holds for f−1(h′). hence (3.3,1) holds. 77 m. scafati tallini since every subset a′ of the group g′ is a complete part in g′, from theorem 3.1 (third part), we have that (3.3, 2) and (3.3, 3) hold. for all b = (x1 ◦ x2 ◦ . . . ◦ xn) ∈ b, let x ∈ b. if x′ def = f(x) ∈ f(b) then x ∈ f−1(x′) and hence b ∩ f−1(x′) ̸= ∅. but, f−1(x′) is a complete part of (g, ◦), hence b ⊆ f−1(x′), and so f(b) = {x′} and |f(b)| = 1. this implies (3.3, 4). the set {u} is a subgroup of g′, so from (3.3, 1), f−1(u) is a subhypergroup of (g, ◦) which is also a complete part because of (3.3, 3). this implies ω ⊆ f−1(u) because ω is the intersection of all the subhypergroups complete parts of (g, ◦). this implies (3.3, 5) and hence the theorem. . the composition of two homomorphisms is a homomorphism, likewise the composition of two strong homomorphisms is a strong homomorphism. furthermore the identity is a strong homomorphism. hence, the hypergroups form a category with respect to the homomorphisms and a category with respect to the strong homomorphisms which is a subcategory of the first category. 4 geometric spaces a geometric space is a pair (p, b), where p is a non-empty set, whose elements we call points and b is a family of parts of p, whose elements we call blocks. the set p is called the support of the geometric space and b is called the geometric structure. let (p, b) and (p ′, b′) be two geometric spaces. we call isomorphism between them a bijection f : p → p ′, such that ∀b ∈ b, f(b) ∈ b′, ∀b′ ∈ b′, f−1(b′) ∈ b. the composition of two isomorphisms is an isomorphism and the identity is an isomorphism. it follows that within the geometric spaces the relation of isomorphism is an equivalence relation. so, we study the equivalence classes of such spaces. the isomorphisms of (p, b) onto itself are called automorphisms. the automorphisms of (p, b) form a group under the composition which is called aut(p, b). this gives rise to a geometry of the geometric space (p, b). more precisely, two subsets f and f ′ of (p, b) are called “equal”, if there is an automorphism of (p, b) which changes f onto f ′. such an equality relation is an equivalence relation. the geometry of (p, b) is the study of the properties of the subsets of (p, b) which are invariant under the group aut(p, b). two geometric spaces (p, b) and (p ′, b′) are called equivalent, 78 hypergroups and geometric spaces if, and only if, aut(p, b) = aut(p ′, b′); (26) that is, if the geometry determined by aut(p, b) coincides with that arising from aut(p ′, b′). we remark that, given a geometric space (p, b), if b′ consists of the complements of the elements of b in p, then (p, b) and (p, b′) are ,because of (26), two distinct geometric spaces, which have the same geometry. example 4.1. let (s, a) be a topological space whose open sets are such that ∅ ∈ a, s ∈ a, aii∈i, ai ∈ a =⇒ ∪ i∈i ai ∈ a, and a1, a2 ∈ a =⇒ a1 ∩ a2 ∈ a. the complements of the open sets are the closed sets of the topology. we denote by c the family of the closed sets. the two structures (p, a) and (p, c) are two distinct geometric spaces, admitting the same geometry. example 4.2. let r be the family of the lines of the real plane r2. the pair (r2, r) is a geometric space which is called real affine plane. the group aut(r2, r) is called the group of the affinities of (r2, r) and we prove that every affinity is an invertible linear transformation; that is: x′ = ax + by + c, y′ = a1x + b1y + c1, with ab1 − a1b ̸= 0. example 4.3. let c be the family of the circles of r2. the pair (r2, c) is a geometric space. an automorphism of such a space is a bijection which changes circles to circles and therefore changes also lines to lines (because it changes three collinear points to three collinear points, and conversely). it follows that an automorphism of (r2, c) is an affinity which changes circles to circles and therefore it is a similitude. it follows that aut(r2, c) is the group of the similitudes of the real plane and the geometry of (r2, c) is the similitude geometry. example 4.4. let c1 be the family of the circles with radius 1 in the real plane r2. the automorphisms of the geometric space (r2, c1) is the set of the bijections changing the circles of radius 1 to themselves, we can prove that such bijections change circles to circles and, then, lines to lines; hence, aut(r2, c1) is the group of the movements in the plane and the geometry of (r2, c1) is the euclidean geometry. example 4.5. let k be a field and p(r, k) be the r-dimensional projective space over the field k. its points are the (r + 1)-tuples not all zero in k, defined up to a non-zero multiplicative factor. the lines consist of those 79 m. scafati tallini points x = (x0, x1, . . . , xr) ∈ p(r, k) each of which is the linear combination of two distinct fixed points y = (y0, y1, . . . , yr), z = (z0, z1, . . . , zr) ∈ p(r, k): x = λy + µz ⇐⇒ xi = λyi + µzi, i = 0, 1, . . . , r let l be the family of the lines of p(r, k). the geometric space (p(r, k), l) is the r-dimensional projective space over the field k. we prove that the automorphisms of such a space are of the form x′i = r∑ j=0 aijϑxj, i = 0, 1, . . . , r, where det(ai,j) ̸= 0 and ϑ is a collineation (that is, an automorphism ϑ : k → k. the geometry of (p(r, k), l) is called projective geometry. let (p, b) be any geometric space. an n-tuple of blocks (b1, b2, . . . , bn) is called polygonal of (p, b) if, and only if, bi ∩ bi+1 ̸= ∅, i = 1, 2, . . . , n − 1. assume that b is a covering of p. in p the following relation γ is defined (connectedness by polygonals): ∀ x, y ∈ p, x γ y ⇐⇒ there is a polygonal (b1, b2, . . . , bn) such that x ∈ b1 and y ∈ bn. (27) the relation γ is an equivalence. in fact, γ is reflexive because b is a covering of p , and it is also obviously symmetric and transitive. for any x ∈ p , the equivalence class γ(x) of x is the union of the polygonals through x and it is called connected component of x. if γ(x) = p then the space (p, b) is connected by polygonals. note that, in any geometric space (p, b), for all given x ∈ p: ∀ b ∈ b, b ∩ γ(x) ̸= ∅ =⇒ b ⊆ γ(x). this implies that if bγ(x) indicates the family of blocks contained in γ(x) then the pair (γ(x), bγ(x) is a connected geometric space. this space is called connected component of (p, b). note that (p, b) is the disjoint union of its connected components. example 4.6. let ω be a non-empty open set in rn and let b be the family of the closed segments contained in ω. consider the geometric space (ω, b). actually, every classical polygonal is a polygonal according to the above definition. conversely, every above polygonal contains a classical polygonal. hence, the connected components of ω in the classical sense coincide with the connected components of ω previously defined. 80 hypergroups and geometric spaces example 4.7. let (v, e) be a graph. this is a geometric space (p, b) in which p = v and b = v ∪ e. note that every block has cardinality which is less than or equal to 2. the classical notion of polygonal (or, path) of a graph coincide with the above definition of polygonal. also, the connected components of (v, e) according to the classical definition coincide with the connected components of (p = v, b = v ∪ e). example 4.8. a semilinear space (p, l), where the elements of l are called lines, is a geometric space such that every line has at least two points and through two distinct points there is at most a line. for instance, every graph without loops is a semilinear space. another example is given by any ruled algebraic variety of p(r, k). the notion of polygonal actually coincide with the classical one (a n-tuple of lines (l1, l2, . . . , ln) such that li ∩ li+1 ̸= ∅) and then the notion of connected component of a semilinear space just given, coincide with the classical one. 5 hypergroups and geometric spaces let (g, ◦) be a hypergroupoid and let b be the family of parts of g consisting of all the hyper products of more than one element in g (see (11) and (12). then the geometric space (p = g, b) remains defined. if in (g, ◦) the reproducibility property (2) holds then b is a covering of g. if x, y ∈ g, x is in relation τ with y if, and only if, there is an element b ∈ b containing x and y. equivalently, x τ y def⇐⇒ there exists an hyper product containing both x and y. the relation τ is reflexive because b is a covering of g and obviously symmetric. however, τ may not be transitive in general. we recall that a relation ρ defined in g can be regarded as a subset (called graph of ρ) of the cartesian product g × g. this implies that it is possible to define a partial ordering relation in the set of all the relations defined in g given by the usual set-theoretical inclusion. moreover, if {ρi : i ∈ i} is a family of equivalence relations in g then ρ def = ∩ i∈i ρi is an equivalence relation defined in g. this is because if a, b, c ∈ g, then a ρ b ⇐⇒ a ρi b, ∀ i ∈ i; and so, ρ is reflexive, symmetric, and transitive. this implies that the equivalences in g form a closure system (also because the relation whose graph is g × g is an equivalence relation). now, let τ∗ be the intersection of all the equivalences containing τ. note that τ∗ is the smallest equivalence relation (and hence, transitive) which contains the possibly non-transitive relation τ. for this reason, τ∗ is called transitive closure of τ. as a matter of fact, if g 81 m. scafati tallini is an hypergroup then τ∗ = τ as the following theorem states; and hence, τ is transitive. theorem 5.1. let g be a hypergroup,n∗ = n − 0, z be an element of the heart ω of g and p(z) def = {a ∈ p ′(g) : z ∈ a, ∃m ∈ n∗, ∃(a1, a2, ..., am) ∈ gm : a = m∏ j=1 aj}. set m = ∪ a∈p(z) a, then ω = m, τ ∗ = τ. proof. first we prove that m is a complete part of g. let (z1, z2, ..., zn) ∈ gn such that ∏ i∈i z∩m ̸= ∅. if a ∈ ∏ i∈i zi ∩ m then a product a ∈ p(z) such that a ∈ a exists. from the reproducibility property of g, a pair (w, b) of elements of g exists such that zn ∈ wz and z ∈ ab. hence: z ∈ ab ⊂ n−1∏ i=1 zib = n−1∏ i=1 ziznb ⊂ n−1∏ i=1 ziwzb ⊂ n−1∏ i=1 ziwab; and so ∏n−1 i=1 ziwab ⊂ m. it then follows that: n∏ i=1 zi = n−1∏ i=1 zizn ⊂ n−1∏ i=1 ziwz ⊂ ∏ i = 1n−1ziwab ⊂ n−1∏ i=1 ziwab ⊂ m, and therefore m is a complete part of g. for any non-empty subset a of g,we denote by c(a) the complete closure of a; that is, the intersection of all the complete parts of g containing a. now, since z ∈ ω ∩ m and m is a complete part of g, we have ω = c(z) ⊂ c(m) = m. moreover, for any product a of p(z), it is z ∈ ω ∩ a and, as ω is a complete part of g, then a ⊂ ω and consequently the inclusion m ⊂ ω holds; hence, m = ω. now, let x τ∗ y. we have x ∈ c(y) = yω = ym and so a product a ∈ p(z) exists such that x = ya. by the reproducibility property of ω, there exists b ∈ ω such that z ∈ bz. since b ∈ ω = m, there exist a product b ∈ p(z) such that b ∈ b. by the reproducibility property of g, there is v ∈ g such that y = vz. now, from z ∈ a, {b, z} ⊂ b, y ∈ vz, z ∈ bz and x ∈ ya, we get: x ∈ ya ⊂ vza ⊂ vbza ⊂ vbba, and y ∈ vz ⊂ vbz ⊂ vbbz ⊂ vbba. consequently, it follows {x, y} ⊂ vbba and then x τ y. so τ = τ∗. we note that a somewhat similar argument can be found in [1]. 82 hypergroups and geometric spaces note that, the above theorem does not hold in the more general case of g being a semihypergroup, as the following example shows. example 5.1. let g be a set such that |g| ≥ 4 and let a, b, c, d be four distinct elements of g. let a ◦ a = {b, c} and, for any pair (x, y) ∈ g × g, with (x, y) ̸= (a, a), let x ◦ y = {b, d}. it can be easily shown that the set g equipped with the hyperproduct just defined is a semihypergroup such that, for all n ∈ n, n ≥ 3 and for all (x1, x2, . . . , xn) ∈ gn, ∏n i=1 xi = {b, d}. furthermore, c τ b because a◦a = {b, c}, and b τ d because, say, a◦b = {b, d}. this implies c τ∗ b. however, c τ d does not hold. if ρ is an equivalence relation in g containing τ (that is, such that x τ y =⇒ x ρ y) then ρ contains the connectedness by polygonals relation γ defined in (27); that is, ρ ⊇ τ =⇒ ρ ⊃ γ. in fact, since ρ ⊇ τ (that is, x τ y =⇒ x ρ y), if x γ y =⇒ ∃ (b1, b2, . . . , bm): bi ∈ b, bi ∩ bi−1 ̸= ∅, x ∈ b1 and y ∈ bm =⇒ x τ x1, x1 ∈ b2 ∩ b1, x1 τ x2, x2 ∈ b3 ∩ b2, . . ., xm−1 τ y, x1 ∈ bm−1 ∩ bm ̸= ∅ =⇒ x ρ x1, x1 ρ x2, . . ., xm−1 ρ y =⇒ x ρ y because ρ is an equivalence and, hence, it is transitive. it then follows that γ = τ∗. (28) note that ∀ x, y ∈ g, x◦y ∈ b. hence, all the elements of x◦y belong to the same connected component of (g, b) which we denote by γ(x ◦ y). if x τ x′ and y τ y′ then x, x′ ∈ b1 ∈ b and y, y′ ∈ b2 ∈ b, then x ◦ y ⊆ b1 ◦ b2 and x′ ◦ y′ ⊆ b1 ◦ b2 and then γ(x ◦ y) = γ(x′ ◦ y′). this proves, x τ x′ and y τ y′ =⇒ γ(x ◦ y) = γ(x′ ◦ y′). (29) in the set g/γ = g/τ∗ (see (28)) of the connected components of (g, b) it is then possible to define the following single valued product. ∀ γ(x), γ(y) ∈ g/γ, where x, y ∈ g, γ(x) · γ(y) def= γ(x ◦ y). (30) the pair (g/γ, ·) just defined is a groupoid. furthermore, the mapping φ : (g, ◦) → g/γ x → γ(x), (31) is a surjective homomorphism because of (30). this implies that (g/γ, ·) is a group, because the associativity and (2) hold in g. let u be the unity of (g/γ, ·). it can be proved that every complete part a of (g, ◦) is the counterimage through φ defined in (31) of a subset of (g/γ, ·), and so a is the union of connected components. moreover, the image under φ of every subhypergroup complete part of (g, ◦) is a subgroup of (g/γ, ·), and viceversa. 83 m. scafati tallini this implies that the intersection of a set of some subhypergroups complete parts of (g, ◦) is a subhypergroup complete part of (g, ◦). now, since the heart ω of (g, ◦) is the intersection of all the subhypergroups complete parts of (g, ◦) it follows that ω = φ−1(u). references [1] freni, d.: une note sur le coeur d’un hypergroupe et sur la cloture transitive β∗ de β, riv. mat. pura e applicata univ. udine, n. 8, (1991), 153–156. [2] corsini, p.: prolegomena of hypergroup theory, aviani editore, udine, 1986. [3] scafati tallini, m.: a-ipermoduli e spazi ipervettoriali, riv. mat. pura e applicata univ. udine, n. 3, (1988), 39-48. [4] scafati tallini, m.: hypervector spaces, proc. fourth int: congress on ”algebraic hyperstructures and applications”, xanthi, greece, (1990), 167-174. [5] scafati tallini, m.: sottospazi, spazi quozienti ed omomorfismi tra spazi ipervettoriali, riv. mat. pura e applicata univ. udine,n. 18, (1996), 71-84. 84 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 43, 2022 some results in hausdorff neutrosophic metric spaces on hutchinson-barnsley operator vinchu balan shakila* maduraiveeran jeyaraman† abstract the main purpose of this paper is to prove the neutrosophic contraction properties of the hutchinson-barnsley operator on the neutrosophic hyperspace with respect to the hausdorff neutrosophic metrics. also we discuss about the relationships between the hausdorff neutrosophic metrics on the neutrosophic hyperspaces. our theorems generalize and extend some recent results related with hutchinson-barnsley operator in the metric spaces to the neutrosophic metric spaces. keywords: contraction, hutchinson-barnsley operator, metric space, hausdorff neutrosophic metric spaces, hyperspace. 2010 ams subject classification: 03e72, 54e35, 54a40, 46s40‡ *research scholar, p.g. and research department of mathematics, raja doraisingam government arts college, sivagangai. affiliated to alagappa university, karaikudi, tamilnadu, india. *department of mathematics with ca (sf), sourashtra college, madurai, tamilnadu, india. e-mail: shakilavb.math@gmail.com. † p.g. and research department of mathematics, raja doraisingam government arts college, sivagangai. affiliated to alagappa university, karaikudi, tamilnadu, india. e-mail: jeya.math@gmail.com, orcid: https://orcid.org/0000-0002-0364-1845. ‡received on may 16th, 2022. accepted on june 28th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v41i0.782. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. mailto:jeya.math@gmail.com https://orcid.org/0000-0002-0364-1845 v. b. shakila and m. jeyaraman 1. introduction the fractal analysis was introduced by mandelbrot in 1975 [8] and popularized by various mathematicians [6], [3], [4]. sets with non-integral hausdorff dimension, which exceeds its topological dimension, are called fractals by mandelbrot [8]. hutchinson [6] and barnsley [3] initiated and developed the hutchinson-barnsley theory (hb theory) in order to define and construct the fractal as a compact invariant subset of a complete metric space generated by the iterated function system (ifs) of contractions. that is, hutchinson[6] introduced an operator on hyperspace called as hutchinson-barnsley operator (hb operator) to define a fractal set as a unique fixed point by using the banach contraction theorem in the metric spaces. recently in [5], [15], hb operator properties were analyzed in fuzzy metric spaces. here we introduce the concepts and properties of hb operator in the intuitionistic fuzzy metric spaces. atanassov [2] introduced and studied the notion of intuitionistic fuzzy set by generalizing the notion of fuzzy set. park [10] defined the notion of intuitionistic fuzzy metric space as a generalization of fuzzy metric space. in 1998, smarandache [12,13] characterized the new concept called neutrosophic logic and neutrosophic set and explored many results in it. in the idea of neutrosophic sets, there is t degree of membership, i degree of indeterminacy and f degree of non-membership. basset et al. [1] explored the neutrosophic applications in different fields such as model for sustainable supply chain risk management, resource levelling problem in construction projects, decision making. in 2019, kirisci et al [9] defined nms as a generalization of ifms and brings about fixed point theorems in complete nms. later jeyaraman at el., [7, 11] proved fixed point results in non-archimedean generalized intuitionistic fuzzy metric spaces. in 2020, sowndrarajan jeyaraman and florentin smarandache [14] proved some fixed point results for contraction theorems in neutrosophic metric spaces. in this paper, we prove the neutrosophic contraction properties of the hb operator on the neutrosophic hyperspace with respect to the hausdorff neutrosophic metrics. also we discuss about the relationships between the hausdorff neutrosophic metrics on the neutrosophic hyperspaces. here our theorems generalize and extend some recent results related with hutchinson-barnsley operator in the metric spaces. 2.preliminaries definition 2.1. [3] let (σ, 𝑑) be a metric space and 𝒦0(σ) be the collection of all non-empty compact subsets of σ. http://fs.unm.edu/nss2/index.php/111/article/view/753 some results in hausdorff neutrosophic metric spaces on hutchinson-barnsley operator define 𝑑(𝜁, 𝑄) ≔ 𝑖𝑛𝑓𝑦∈𝑄 𝑑(𝜁, 𝜂) and 𝑑(𝑃, 𝑄) ∶= 𝑠𝑢𝑝𝑥∈𝑃 𝑑(𝜁, 𝑄) for all 𝜁 ∈ σ and 𝑃, 𝑄 ∈ 𝒦0(σ). the hausdorff metric or hausdorff distance (𝐻𝑑) is a function (𝐻𝑑) ∶ 𝒦0(σ)x 𝒦0(σ) → ℝ defined by 𝐻𝑑(𝑃, 𝑄) = 𝑚𝑎𝑥{𝑑(𝑃, 𝑄), 𝑑(𝑄, 𝑃)}. then 𝐻𝑑 is a metric on the hyperspace of compact sets 𝒦0(σ) and hence (𝒦0(σ), 𝐻𝑑) is called a hausdorff metric space. theorem 2.2. [3] if (σ, 𝑑) is a complete metric space, then (𝒦0(σ), 𝐻𝑑 ) is also a complete metric space. definition 2.3. [3] let (σ, 𝑑) be a metric space and 𝑓𝑛 ∶ σ → σ, 𝑛 = 1,2, , … 𝑁0(𝑁0 ∈ ℕ) be 𝑁0 contraction mappings with the corresponding contractivity ratios 𝑘𝑛 , 𝑛 = 1,2, … 𝑁0. the system {σ; 𝑓𝑛, 𝑛 = 1,2, … 𝑁0} is called an iterated function system (ifs) or hyperbolic iterated function system with the ratio 𝑘 = 𝑚𝑎𝑥𝑛=1 𝑁0 𝑘𝑛. then the hutchinson barnsley operator (hbo) of the ifs is a function 𝐹 ∶ 𝒦0(σ) → 𝒦0(σ) defined by 𝐹(𝑄) = ⋃ 𝑓𝑛 (𝑄), 𝑁0 𝑛=1 for all 𝑄 ∈ 𝒦0(σ). theorem 2.4. [3] let (σ, 𝑑) be a metric space. let {σ; 𝑓𝑛 , 𝑛 = 1,2, … 𝑁0; 𝑁0 ∈ ℕ } be an ifs. then, the hbo (f) is a contraction mapping on (𝒦0(σ), 𝐻𝑑 ). theorem 2.5. [3] let (σ, 𝑑) be a complete metric space and {σ; 𝑓𝑛 , 𝑛 = 1,2,3 … 𝑁0; 𝑁0 ∈ ℕ } be an ifs. then, there exists only one compact invariant set 𝑃∞ ∈ 𝒦0(σ) of the hbo (f) or equivalently, f has a unique fixed point namely 𝑃∞ ∈ 𝒦0(σ). definition 2.6. [3] the fixed point 𝑃∞ ∈ 𝒦0(σ) of the hbo f described in the theorem (2.5) is called the attractor (fractal) of the ifs. sometimes 𝑃∞ ∈ 𝒦0(σ) is called as fractal generated by the ifs and so called as ifs fractal. definition 2.7. a binary operation ∗ ∶ [0,1] x [0,1] → [0,1] is a continuous tnorm, if * satisfies the following conditions: (a) * is commutative and associative; (b) * is continuous (c) 𝑎 ∗ 1 = 𝑎 for all 𝑎 ∈ [0,1]; (d) 𝑎 ∗ 𝑏 ≤ 𝑐 ∗ 𝑑 whenever 𝑎 ≤ 𝑐 and 𝑏 ≤ 𝑑, and 𝑎, 𝑏, 𝑐, 𝑑 ∈ [0,1]. v. b. shakila and m. jeyaraman definition 2.8. a binary operation ⋄ ∶ [0,1] x [0,1] → [0,1] is a continuous tnorm, if ⋄ satisfies the following conditions: (a) ⋄ is commutative and associative; (b) ⋄ is continuous (c) 𝑎 ⋄ 0 = 𝑎 for all 𝑎 ∈ [0,1]; (d) 𝑎 ⋄ 𝑏 ≤ 𝑐 ⋄ 𝑑 whenever 𝑎 ≤ 𝑐 and 𝑏 ≤ 𝑑, and 𝑎, 𝑏, 𝑐, 𝑑 ∈ [0,1]. definition 2.9. a 6-tuple (σ, ξ, θ, υ,∗,⋄) is said to be an neutrosophic metric space (shortly nms), if σ is an arbitrary set, ∗ is a neutrosophic ctn, ⋄ is a neutrosophic ctc and ξ, θ 𝑎𝑛𝑑 υ are neutrosophic on σ x σ satisfying the following conditions: for all 𝜁, 𝜂, 𝛿,𝜔 ∈ σ, 𝜆, 𝜇 ∈ 𝑅+. (i) 0 ≤ ξ ( 𝜁, 𝜂, 𝜆) ≤ 1; 0 ≤ θ ( 𝜁, 𝜂, 𝜆) ≤ 1; 0 ≤ υ ( 𝜁, 𝜂, 𝜆) ≤ 1; (ii) ξ ( 𝜁, 𝜂, 𝜆) + θ ( 𝜁, 𝜂, 𝜆) + υ ( 𝜁, 𝜂, 𝜆) ≤ 3; (iii) ξ ( 𝜁, 𝜂, 𝜆) = 1 if and only if 𝜁 = 𝜂; (iv) ξ ( 𝜁, 𝜂, 𝜆) = ξ ( 𝜂, 𝜁, 𝜆) for 𝜆 > 0 (v) ξ ( 𝜁, 𝜂, 𝜆)∗ ξ ( 𝜂, 𝛿, 𝜇) ≤ ξ ( 𝜁, 𝛿, 𝜆 + 𝜇), for all 𝜆 , 𝜇 > 0; (vi) ξ ( 𝜁, 𝜂, .) : [0, ∞) → [0 , 1] is neutrosophic continuous ; (vii) lim 𝜆→∞ ξ ( 𝜁, 𝜂, 𝜆) = 1 for all 𝜆 > 0; (viii) θ ( 𝜁, 𝜂, 𝜆) = 0 if and only if 𝜁 = 𝜂; (ix) θ ( 𝜁, 𝜂, 𝜆) = θ ( 𝜂, 𝜁, 𝜆) for 𝜆 > 0; (x) θ ( 𝜁, 𝜂, 𝜆) ⋄ θ ( 𝜂, 𝛿, 𝜇) ≥ θ ( 𝜁, 𝛿, 𝜆 + 𝜇), for all 𝜆 , 𝜇 > 0; (xi) θ ( 𝜁, 𝜂, .) : [0, ∞) → [0,1] is neutrosophic continuous; (xii) lim 𝜆→∞ θ ( 𝜁, 𝜂, 𝜆) = 0 for all 𝜆 > 0; (xiii) υ ( 𝜁, 𝜂, 𝜆) = 0 if and only if 𝜁 = 𝜂 ; (xiv) υ ( 𝜁, 𝜂, 𝜆) = υ ( 𝜂, 𝜁, 𝜆) for 𝜆 > 0; (xv) υ ( 𝜁, 𝜂, 𝜆) ⋄ υ ( 𝜂, 𝛿, 𝜇) ≥ υ ( 𝜁, 𝛿, 𝜆 + 𝜇), for all 𝜆 , 𝜇> 0; (xvi) υ( 𝜁, 𝜂, .) : [0, ∞) → [0,1] is neutrosophic continuous; (xvii) lim 𝜆→∞ υ ( 𝜁, 𝜂, 𝜆) = 0 for all 𝜆 > 0; then, (𝛯, 𝛩, 𝛶) is called an nms on 𝛴. the functions 𝛯, 𝛩 𝑎𝑛𝑑 𝛶denote degree of closedness, neturalness and non-closedness between 𝜁 𝑎𝑛𝑑 𝜂 with respect to 𝜆 respectively. example 2.10. let (𝛴, 𝑑) be a metric space. let 𝛯𝑑, 𝛩𝑑 and 𝛶𝑑 be the functions defined on 𝛴2 x (0, ∞) by 𝛯𝑑 (𝜁, 𝜂, 𝜆) = 𝜆 𝜆+𝑑(𝜁,𝜂) , 𝛩𝑑(𝜁, 𝜂, 𝜆) = 𝑑(𝜁,𝜂) 𝜆+𝑑(𝜁,𝜂) and 𝛶𝑑 (𝜁, 𝜂, 𝜆) = 𝑑(𝜁,𝜂) 𝜆 , for all 𝜁, 𝜂 ∈ 𝛴 and 𝜆 > 0. then (𝛴, 𝛯𝑑 , 𝛩𝑑, 𝛶𝑑 ,∗,⋄) is a some results in hausdorff neutrosophic metric spaces on hutchinson-barnsley operator nms which is called standard nms, and (𝛯𝑑 , 𝛩𝑑, 𝛶𝑑 ) is called as standard nm induced by the metric d. definition 2.11. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a nms. the open ball 𝐵(𝜁, 𝑟, 𝜆) with 𝜁 ∈ 𝛴 and radius r, 0 < 𝑟 < 1, with respect to 𝜆 > 0 is defined as 𝐵(𝜁, 𝑟, 𝜆) = {𝜂 ∈ 𝛴 ∶ 𝛯 ( 𝜁, 𝜂, 𝜆) > 1 − 𝑟, 𝛩 ( 𝜁, 𝜂, 𝜆) < 𝑟 𝑎𝑛𝑑 𝛶 ( 𝜁, 𝜂, 𝜆) < 𝑟 }. define 𝜏(𝛯,𝛩,𝛶) = { 𝑃 ⊂ 𝛴 ∶ 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝜁 ∈ 𝑃, ∃ 𝜆 > 0 𝑎𝑛𝑑 𝑟 ∈ (0,1) 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐵(𝜁, 𝑟, 𝜆) ⊂ 𝑃 }. then 𝜏(𝛯,𝛩,𝛶) is a topology on 𝛴 induced by a nfm (𝛯, 𝛩, 𝛶). the topologies induced by the metric and the corresponding standard nm are the same. proposition 2.12. the metric space (𝛴, 𝑑) is complete if and only if the standard nms (𝛴, 𝛯𝑑 , 𝛩𝑑, 𝛶𝑑 ,∗,⋄)is complete. definition 2.13. a neutrosophic fuzzy b-contraction (neutrosophic fuzzy sehgal contraction) on an nms (σ, ξ, θ, υ,∗,⋄) is a self –mapping f on σ for which ξ(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≥ ξ(𝜁, 𝜂, λ), θ(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ θ(𝜁, 𝜂, λ) and υ(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ υ(𝜁, 𝜂, λ) for all 𝜁, 𝜂 ∈ σ and λ > 0, where k is a fixed constant in (0,1). 3. hausdorff neutrosophic metric spaces definition 3.1. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a nms and 𝜏(ξ,θ,υ) be the topology induced by the nm (ξ, θ, 𝛶).we shall denote by 𝒦0(𝑋), the set of all non-empty compact subsets of (𝛴, 𝜏(ξ,θ,𝛶)). define ξ(𝜁, 𝑄, λ) ∶= sup 𝜂 ∈ 𝑄 ξ(𝜁, 𝜂, λ) , ξ(𝑃, 𝑄, λ) ∶= inf 𝜁 ∈ 𝑃 ξ(𝜁, 𝑄, λ), 𝛩(𝜁, 𝑄, λ) ∶= inf 𝜂 ∈ 𝑄 𝛩(𝜁, 𝜂, λ) , 𝛩(𝑃, 𝑄, λ) ∶= sup 𝜁 ∈ 𝑃 𝛩 (𝜁, 𝑄, λ) and 𝛶(𝜁, 𝑄, λ) ∶= inf 𝜂 ∈ 𝑄 𝛶(𝜁, 𝜂, λ) , 𝛶(𝑃, 𝑄, λ) ∶= sup 𝜁 ∈ 𝑃 𝛩(𝜁, 𝑄, λ), for all 𝜁 ∈ 𝛴 and 𝑃, 𝑄 ∈ 𝒦0(𝑋). then, we define the hausdorff nm (𝐻ξ, 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) as 𝐻ξ(𝑃, 𝑄, λ) = 𝑚𝑖𝑛{ξ(𝑃, 𝑄, λ), ξ(𝑄, 𝑃, λ)}, v. b. shakila and m. jeyaraman 𝐻𝛩 (𝑃, 𝑄, λ) = 𝑚𝑎𝑥{𝛩(𝑃, 𝑄, λ), 𝛩(𝑄, 𝑃, λ)}and 𝐻𝛶 (𝑃, 𝑄, λ) = 𝑚𝑎𝑥{𝛶(𝑃, 𝑄, λ), 𝛶(𝑄, 𝑃, λ)}. here (𝐻ξ, 𝐻𝛩 , 𝐻𝛶 ) is a nm on the hyperspace of compact sets, 𝒦0(σ) and hence (𝒦0(σ), 𝐻ξ, 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) is called a hausdorff nms. proposition 3.2. let (σ, 𝑑) be a metric space. then the hausdorff nm (𝐻ξ𝑑 , 𝐻𝛩 𝑑, 𝐻𝛶 𝑑) of the standard nm (ξ𝑑, 𝛩𝑑, 𝛶𝑑 ) coincides with the standard nm (𝐻ξ𝑑 , 𝐻𝛩 𝑑, 𝐻𝛶 𝑑) of the hausdorff metric (𝐻𝑑) on 𝒦0(σ). ie., 𝐻ξ𝑑 (𝑃, 𝑄, λ) = ξh 𝑑(𝑃, 𝑄, λ), 𝐻𝛩 𝑑 (𝑃, 𝑄, λ) = 𝛩h𝑑 (𝑃, 𝑄, λ) and 𝐻𝛶 𝑑 (𝑃, 𝑄, λ) = 𝛶h 𝑑(𝑃, 𝑄, λ), for all 𝑃, 𝑄 ∈ 𝒦0(σ) and λ > 0. proof: fix λ > 0 and 𝑃, 𝑄 ∈ 𝒦0(σ).we recall that sup 𝛽 ∈ 𝑄 ξ𝑑 (𝛼, 𝛽, λ) = λ λ+ inf 𝛽∈𝑄 𝑑(𝛼,𝛽) , inf 𝛽 ∈ 𝑄 𝛩𝑑(𝛼, 𝛽, λ) = 1 1+ λ inf 𝛽∈𝑄 𝑑(𝛼,𝛽) and inf 𝛽 ∈ 𝑄 𝛶𝑑 (𝛼, 𝛽, λ) = 1 λ inf 𝛽∈𝑄 𝑑(𝛼,𝛽) , for all 𝛼 ∈ 𝑃.it follows that ξ𝑑 (𝛼, 𝑄, λ) = λ λ+d(𝛼,𝑄) ,𝛩𝑑(𝛼, 𝑄, λ) = 1 1+ λ d(𝛼,𝑄) and 𝛶𝑑 (𝛼, 𝑄, λ) = 1 λ d(𝛼,𝑄) for all 𝛼 ∈ 𝑃. then inf 𝛼 ∈ 𝑃 ξ𝑑 (𝛼, 𝑄, λ) = λ λ+ sup 𝛼∈ 𝑃 d(𝛼,𝑄) , sup 𝛼 ∈ 𝑃 𝛩𝑑 (𝛼, 𝑄, λ) = 1 1+ sup 𝛼∈ 𝑃 λ d(𝛼,𝑄) and sup 𝛼 ∈ 𝑃 𝛶𝑑 (𝛼, 𝑄, λ) = 1 sup 𝛼∈𝑃 λ d(𝛼,𝑄) . it follows that ξ𝑑 (𝑃, 𝑄, λ) = λ λ+d(𝑃,𝑄) , 𝛩𝑑(𝑃, 𝑄, λ) = 1 1+ λ d(𝑃,𝑄) = d(𝑃,𝑄) λ+d(𝑃,𝑄) and 𝛶𝑑 (𝑃, 𝑄, λ) = 1 λ d(𝑃,𝑄) = d(𝑃,𝑄) λ . similarly, we obtain some results in hausdorff neutrosophic metric spaces on hutchinson-barnsley operator ξ𝑑 (𝑄, 𝑃, λ) = λ λ+d(𝑄,𝑃) , 𝛩𝑑(𝑄, 𝑃, λ) = d(𝑄,𝑃) λ+d(𝑄,𝑃) and 𝛶𝑑 (𝑄, 𝑃, λ) = d(𝑄,𝑃) λ . therefore, 𝐻ξ𝑑 (𝑃, 𝑄, λ) = ξh𝑑 (𝑃, 𝑄, λ), 𝐻𝛩 𝑑 (𝑃, 𝑄, λ) = 𝛩h𝑑 (𝑃, 𝑄, λ) and 𝐻𝛶 𝑑(𝑃, 𝑄, λ) = 𝛶h𝑑 (𝑃, 𝑄, λ). the proof is complete. using the proposition 3.2., we can easily compute distances with respect to the hausdorff nm (𝐻ξ𝑑, 𝐻𝛩 𝑑 , 𝐻𝛶 𝑑) of the standard nm (ξ𝑑, 𝛩𝑑, 𝛶𝑑 ) by computing distances with respect to the hausdorff metric (𝐻𝑑) implied by the metric d. here, we illustrate this situation with two examples. example 3.3. let (ℝ, 𝑑) be the euclidean metric space and 𝑃 = [𝛼1, 𝛼2] and 𝑄 = [𝛽1, 𝛽2] be two compact intervals of ℝ. then 𝑑(𝑃, 𝑄) = |𝛼1 − 𝛽1| and 𝑑(𝑄, 𝑃) = |𝛼2 − 𝛽2|and hence 𝐻𝑑 (𝑃, 𝑄) = 𝑚𝑎𝑥{|𝛼1 − 𝛽1|, |𝛼2 − 𝛽2|}; so, by proposition (3.2), we have ξ𝑑 (𝑃, 𝑄, λ) = λ λ+𝑚𝑎𝑥{|𝛼1−𝛽1|,|𝛼2−𝛽2|} , 𝛩𝑑(𝑃, 𝑄, λ) = 𝑚𝑎𝑥{|𝛼1−𝛽1|,|𝛼2−𝛽2|} λ+𝑚𝑎𝑥{|𝛼1−𝛽1|,|𝛼2−𝛽2|} and 𝛶𝑑 (𝑃, 𝑄, λ) = 𝑚𝑎𝑥{|𝛼1−𝛽1|,|𝛼2−𝛽2|} λ , for all λ > 0. example 3.4. let (σ, 𝑑) be the discrete metric space such that |σ| ≥ 2. let 𝑃 and 𝑄 be two non-empty finite subsets of σ, with 𝑃 ≠ 𝑄. then 𝑑(𝑃, 𝑄) = 1 = 𝑑(𝑄, 𝑃) and hence 𝐻𝑑(𝑃, 𝑄) = 1; so by proposition 3.2., we have 𝐻ξ𝑑 (𝑃, 𝑄, λ) = λ λ+1 ,𝐻𝛩 𝑑 = 1 1+λ and 𝐻𝛶 𝑑 = 1 λ , for all λ > 0. definition 3.5. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be an nms and 𝜏(𝛯,𝛩,𝛶) be the topology induced by (ξ, θ, 𝛶). we observe that (𝒦0(𝒦0(σ)), 𝐻𝐻𝛯 , ℋ𝐻𝛩 , ℋ𝐻𝛶 ,∗,⋄) is also an nms, where 𝒦0(𝒦0(σ)) is the hyperspace of all non-empty compact subsets of (𝒦0(σ), 𝐻ξ, 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) and (ℋ𝐻𝛯 , ℋ𝐻𝛩 , ℋ𝐻𝛶 ) is the hausdorff nm on 𝒦0(𝒦0(σ)) implied by the hausdorff nm (ξ𝑑 , 𝛩𝑑, 𝛶𝑑 ) on 𝒦0(σ). that is, for all 𝑃 ∈ 𝒦0(σ) and 𝔓, 𝔔 ∈ 𝒦0(𝒦0(σ)), 𝐻𝐻𝛯 (𝔓, 𝔔) = 𝑚𝑖𝑛{𝐻ξ𝔓, 𝔔, 𝐻ξ(𝔔, 𝔓)},𝐻𝐻𝛩 (𝔓, 𝔔) = 𝑚𝑎𝑥{𝐻𝛩 (𝔓, 𝔔), 𝐻𝛩 (𝔔, 𝔓)} and𝐻𝐻𝛶 (𝔓, 𝔔) = 𝑚𝑎𝑥{𝐻𝛶 (𝔓, 𝔔), 𝐻𝛶 (𝔔, 𝔓)} where v. b. shakila and m. jeyaraman 𝐻ξ(𝔓, 𝔔) ≔ inf 𝑝 ∈ 𝔓 𝐻ξ(𝑃, 𝔔), 𝐻ξ(𝑃, 𝔔) ≔ sup 𝑄 ∈ 𝔔 𝐻ξ(𝑃, 𝑄), 𝐻𝛩 (𝔓, 𝔔) ≔ sup 𝑝 ∈ 𝔓 𝐻𝛩 (𝑃, 𝔔), 𝐻𝛩 (𝑃, 𝔔) ≔ inf 𝑄 ∈ 𝔔 𝐻𝛩 (𝑃, 𝑄) and 𝐻𝛶 (𝔓, 𝔔) ≔ sup 𝑝 ∈ 𝔓 𝐻𝛶 (𝑃, 𝔔), 𝐻𝛶 (𝑃, 𝔔) ≔ inf 𝑄 ∈ 𝔔 𝐻𝛶 (𝑃, 𝑄). proposition 3.6. let (σ, 𝑑) be a metric space and let (𝒦0(σ), 𝐻d) and (𝒦0(𝒦0(σ)), ℋ𝐻𝑑 ) be the corresponding hausdorff metric spaces. then, the hausdorff nm (ℋ𝛯𝐻𝑑 , ℋ𝛩𝐻𝑑 , ℋ𝛶𝐻𝑑 ) of the standard nm(𝛯𝐻𝑑 , 𝛩𝐻𝑑 , 𝛶𝐻𝑑 ) coincides with the standard nm (𝛯ℋ𝐻𝑑 , 𝛩ℋ𝐻𝑑 , 𝛶ℋ𝐻𝑑 ) of the hausdorff metric (ℋ𝐻𝑑 ) on 𝒦0(𝒦0(σ)), ie. ℋ𝛯𝐻𝑑 (𝔓, 𝔔, λ) = 𝛯ℋ𝐻𝑑 (𝔓, 𝔔, λ), ℋ𝛩𝐻𝑑 (𝔓, 𝔔, λ) = 𝛩ℋ𝐻𝑑 (𝔓, 𝔔, λ) and ℋ𝛶𝐻𝑑 (𝔓, 𝔔, λ) = 𝛶ℋ𝐻𝑑 (𝔓, 𝔔, λ) for all 𝔓, 𝔔 ∈ 𝒦0(𝒦0(σ)) and λ > 0. proof: proposition 3.2. completes the proof. 4. neutrosophic hutchinson-barnsley operator in this section, we define the neutrosophic iterated function system (nifs) and neutrosophic hb operator on the nms. definition 4.1. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄) be an nms and 𝑓𝑛 : 𝛴 → 𝛴, 𝑛 = 1,2,3 … 𝑁0(𝑁0 ∈ ℕ) be 𝑁0 neutrosophic b-contractions. then the system {𝛴; 𝑓𝑛 , 𝑛 = 1,2,3 … 𝑁0} is called a nifs of neutrosophic b-contractions in the nms (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄). definition 4.2. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a nms. let {𝛴; 𝑓𝑛 , 𝑛 = 1,2,3 … 𝑁0} be an nifs of neutrosophic b-contractions. then the neutrosophic hutchinson-barnsley operator (nhbo) of the nifs is a function 𝐹 ∶ 𝒦0(σ) → 𝒦0(σ) defined by 𝐹(𝑄) = ⋃ 𝑓𝑛 (𝑄) 𝑁0 𝑛=1 , for all 𝑄 ∈ 𝒦0(σ). definition 4.3. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a complete nms. let 𝑓𝑛 : 𝛴 → 𝛴, 𝑛 = 1,2,3 … 𝑁0(𝑁0 ∈ ℕ) be a nifs of neutrosophic b-contractions and f be the nhbo of the nifs. we say that the set 𝑃∞ ∈ 𝒦0(σ) is neutrosophic attractor (neutrosophic fractal) of the given nifs, if 𝑃∞ is a unique fixed point of the nhbo f. such 𝑃∞ ∈ 𝒦0(σ)is also called as fractal generated by the nifs and so called nifs fractal of neutrosophic b-contractions. some results in hausdorff neutrosophic metric spaces on hutchinson-barnsley operator properties of nhbo now, we prove the interesting results about the neutrosophic b-contraction properties of operators with respect to the hausdorff neutrosophic metric on 𝒦0(σ). theorem 4.4. let (σ, 𝑑) be a metric space. let 𝑓: 𝛴 → 𝛴 be a contraction function on (σ, 𝑑), with a contractivity ratio k. then h𝛯𝒅 (𝑓(𝑃), 𝑓(𝑄), λ) ≥ h𝛯𝒅 (p, q, λ), h𝛩𝒅 (𝑓(𝑃), 𝑓(𝑄), λ) ≤ h𝛩𝒅 (p, q, λ)and h𝛶𝒅 (𝑓(𝑃), 𝑓(𝑄), λ) ≤ h𝛶𝒅 (p, q, λ), for all 𝑃, 𝑄 ∈ 𝒦0(σ) and λ > 0. proof: fix λ > 0 and let 𝑃, 𝑄 ∈ 𝒦0(σ). since f is contraction on (σ, 𝑑) with the contractivity ratio 𝑘 ∈ (0,1) and by theorem 2.4. for the case 𝛩 = 1, we have 𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄)) ≤ 𝑘𝐻𝑑 (𝑃, 𝑄). since λ > 0 and 𝑘 ∈ (0,1), 𝑘λ 𝑘λ+𝐻𝑑(𝑓(𝑃),𝑓(𝑄)) ≥ 𝑘λ 𝑘λ+k𝐻𝑑(𝑃,𝑄) = λ λ+𝐻𝑑(𝑃,𝑄) , 𝐻𝑑(𝑓(𝑃),𝑓(𝑄)) 𝑘λ+ 𝐻𝑑(𝑓(𝑃),𝑓(𝑄)) ≤ 𝑘𝐻𝑑(𝑃,𝑄) 𝑘λ+𝑘𝐻𝑑(𝑃,𝑄) = 𝐻𝑑(𝑃,𝑄) λ+𝐻𝑑(𝑃,𝑄) and 𝐻𝑑(𝑓(𝑃),𝑓(𝑄)) 𝑘λ ≤ 𝑘𝐻𝑑(𝑃,𝑄) 𝑘λ = 𝐻𝑑(𝑃,𝑄) λ . by using the above inequalities and the proposition 3.2., we have h𝛯𝒅 (𝑓(𝑃), 𝑓(𝑄), kλ) = 𝛯𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄), kλ) = kλ kλ + 𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄)) ≥ λ λ + 𝐻𝑑 (𝑃, 𝑄) = 𝛯𝐻𝑑 (𝑃, 𝑄, λ) = h𝛯𝒅 (𝑃, 𝑄, λ), h𝛩𝒅 (𝑓(𝑃), 𝑓(𝑄), kλ) = 𝛩𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄), kλ) = 𝑘𝐻𝑑(𝑓(𝑃), 𝑓(𝑄)) kλ + 𝐻𝑑(𝑓(𝑃), 𝑓(𝑄)) ≤ 𝐻𝑑(𝑃, 𝑄) λ + 𝐻𝑑(𝑃, 𝑄) = 𝛩𝐻𝑑 (𝑃, 𝑄, λ) = h𝛩𝒅 (𝑃, 𝑄, λ) and similarly, h𝛶𝒅 (𝑓(𝑃), 𝑓(𝑄), kλ) = 𝛶𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄), kλ) = 𝑘𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄)) kλ + 𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄)) ≤ 𝐻𝑑 (𝑃, 𝑄) λ + 𝐻𝑑 (𝑃, 𝑄) = 𝛶𝐻𝑑 (𝑃, 𝑄, λ) = h𝛶𝒅 (𝑃, 𝑄, λ). the above theorem 4.4. shows that f is a neutrosophic b-contraction on 𝒦0(σ) with respect to the hausdorff neutrosophic metric (h𝛯𝒅 , h𝛩𝒅 , h𝛶𝒅 ) implied by the standard metric (𝛯𝑑 , 𝛩𝑑, 𝛶𝑑 ), if f is contraction on a metric space (σ, 𝑑). the following theorem is somewhat generalization of the theorem 4.4. v. b. shakila and m. jeyaraman theorem 4.5. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄) be a nms. let (𝒦0(σ), 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) be the corresponding hausdorff nms. suppose 𝑓: 𝛴 → 𝛴 be a neutrosophic b-contraction on (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄). then for 𝑘 ∈ (0,1), 𝐻𝛯 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≥ 𝐻𝛯 (𝑃, 𝑄. λ), 𝐻𝛩 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝐻𝛩 (𝑃, 𝑄. λ) and 𝐻𝛶 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝐻𝛶 (𝑃, 𝑄. λ) for all 𝑃, 𝑄 ∈ 𝒦0(σ) and λ > 0. proof: fix λ > 0. let 𝑃, 𝑄 ∈ 𝒦0(σ). for given 𝑘 ∈ (0,1), we get 𝛯(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≥ ξ(𝜁, 𝜂, λ), for all 𝜁, 𝜂 ∈ 𝛴, 𝛯(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≥ ξ(𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃 𝑎𝑛𝑑 𝜂 ∈ 𝑄, sup 𝜂 ∈ 𝑄 𝛯 (𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≥ sup 𝑦 ∈ 𝑄ξ (𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃, 𝛯(𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≥ ξ(𝜁, 𝑄, λ),for all 𝜁 ∈ 𝑃, inf 𝜁 ∈ 𝑃 𝛯(𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≥ inf 𝜁 ∈ 𝑃 ξ(𝜁, 𝑄, λ), 𝛯(𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≥ ξ(𝑃, 𝑄, λ). similarly 𝛯(𝑓(𝑄), 𝑓(𝜁), 𝑘λ) ≥ ξ(𝑄, 𝑃, λ). hence 𝐻𝛯 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≥ 𝐻𝛯 (𝑃, 𝑄. λ) now, 𝛩(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ 𝛩(𝜁, 𝜂, λ), for all 𝜁, 𝜂 ∈ 𝛴, 𝛩(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ 𝛩(𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃 𝑎𝑛𝑑 𝜂 ∈ 𝑄, inf 𝜂 ∈ 𝑄 𝛩(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ inf 𝑦 ∈ 𝑄 𝛩(𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃, 𝛩(𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≤ 𝛩(𝜁, 𝑄, λ), for all 𝜁 ∈ 𝑃, sup 𝜁 ∈ 𝑃 𝛩 (𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≤ sup 𝜁 ∈ 𝑃 𝛩 (𝜁, 𝑄, λ). 𝛩(𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝛩(𝑃, 𝑄, λ). similarly 𝛩(𝑓(𝑄), 𝑓(𝜁), 𝑘λ) ≤ 𝛩(𝑄, 𝑃, λ). hence 𝐻𝛩 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝐻𝛩 (𝑃, 𝑄. λ) and 𝛶(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ 𝛶(𝜁, 𝜂, λ), for all 𝜁, 𝜂 ∈ 𝛴, 𝛶(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ 𝛶(𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃 𝑎𝑛𝑑 𝜂 ∈ 𝑄 inf 𝜂 ∈ 𝑄 𝛶(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ inf 𝑦 ∈ 𝑄 𝛶(𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃 𝛶(𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≤ 𝛶(𝜁, 𝑄, λ), for all 𝜁 ∈ 𝑃, sup 𝜁 ∈ 𝑃 𝛶(𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≤ sup 𝜁 ∈ 𝑃 𝛶(𝜁, 𝑄, λ). 𝛶(𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝛶(𝑃, 𝑄, λ). similarly 𝛶(𝑓(𝑄), 𝑓(𝜁), 𝑘λ) ≤ 𝛶(𝑄, 𝑃, λ). hence 𝐻𝛶 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝐻𝛶 (𝑃, 𝑄. λ). this completes the proof. some results in hausdorff neutrosophic metric spaces on hutchinson-barnsley operator the above theorem 4.5. shows that f is a neutrosophic b-contraction on 𝒦0(σ) with respect to the hausdorff neutrosophic metric 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶, if f is neutrosophic b-contraction on neutrosophic metric space (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄). lemma 4.6. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a nms. if 𝑄, 𝑅 ⊂ 𝛴 such that 𝑄 ⊂ 𝑅, then 𝛯(𝜁, 𝑄, λ) ≤ 𝛯(𝜁, 𝑅, λ), 𝛩(𝜁, 𝑄, λ) ≥ 𝛩(𝜁, 𝑅, λ) and 𝛶(𝜁, 𝑄, λ) ≥ 𝛶(𝜁, 𝑅, λ) for all 𝜁 ∈ 𝛴 and λ > 0. proof: fix λ > 0. let 𝜁 ∈ 𝛴 and 𝑄, 𝑅 ⊂ 𝛴 such that 𝑄 ⊂ 𝑅. then, ξ(𝜁, 𝑄, λ) = sup 𝑞 ∈ 𝑄 ξ (𝜁, 𝑞, λ) ≤ sup 𝑞 ∈ 𝑅 ξ (𝜁, 𝑞, λ) = ξ(𝜁, 𝑅, λ), 𝛩(𝜁, 𝑄, λ) = inf 𝑞 ∈ 𝑄 𝛩(𝜁, 𝑞, λ) ≥ inf 𝑞 ∈ 𝑅 𝛩(𝜁, 𝑞, λ) = 𝛩(𝜁, 𝑅, λ) and 𝛶(𝜁, 𝑄, λ) = inf 𝑞 ∈ 𝑄 𝛶(𝜁, 𝑞, λ) ≥ inf 𝑞 ∈ 𝑅 𝛶(𝜁, 𝑞, λ) = 𝛶(𝜁, 𝑅, λ). lemma 4.7. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a nms. if 𝑄, 𝑅 ⊂ 𝛴 such that 𝑄 ⊂ 𝑅, then 𝛯(𝑃, 𝑄, λ) ≤ 𝛯(𝑃, 𝑅, λ), 𝛩(𝑃, 𝑄, λ) ≥ 𝛩(𝑃, 𝑅, λ) and 𝛶(𝑃, 𝑄, λ) ≥ 𝛶(𝑃, 𝑅, λ) for all 𝑃 ⊂ 𝛴 and λ > 0. proof: fix λ > 0. let 𝑃, 𝑄, 𝑅 ⊂ 𝛴 such that 𝑄 ⊂ 𝑅. by the lemma 4.6., we have 𝛯(𝑃, 𝑄, λ) = inf 𝑝 ∈ 𝑃 𝛯(𝑝, 𝑄, λ), 𝛯(𝑃, 𝑄, λ) ≤ 𝛯(𝑝, 𝑄, λ), for all 𝑝 ∈ 𝑃 𝛯(𝑃, 𝑄, λ) ≤ 𝛯(𝑝, 𝑅, λ), for all 𝑝 ∈ 𝑃, 𝛯(𝑃, 𝑄, λ) ≤ inf 𝑝 ∈ 𝑃 𝛯(𝑝, 𝑅, λ) , 𝛯(𝑃, 𝑄, λ) ≤ 𝛯(𝑃, 𝑅, λ). similarly, by the lemma 4.6. 𝛩(𝑃, 𝑄, λ) = sup 𝑝 ∈ 𝑃 𝛩 (𝑝, 𝑄, λ), 𝛩(𝑃, 𝑄, λ) ≥ 𝛩(𝑝, 𝑄, λ) for all 𝑝 ∈ 𝑃, 𝛩(𝑃, 𝑄, λ) ≥ 𝛩(𝑝, 𝑅, λ) for all 𝑝 ∈ 𝑃, 𝛩(𝑃, 𝑄, λ) ≥ sup 𝑝 ∈ 𝑃 𝛩 (𝑝, 𝑅, λ), 𝛩(𝑃, 𝑄, λ) ≥ 𝛩(𝑃, 𝑅, λ) and 𝛶(𝑃, 𝑄, λ) = sup 𝑝 ∈ 𝑃 𝛶 (𝑝, 𝑄, λ), 𝛶(𝑃, 𝑄, λ) ≥ 𝛶(𝑝, 𝑄, λ) for all 𝑝 ∈ 𝑃, 𝛶(𝑃, 𝑄, λ) ≥ 𝛶(𝑝, 𝑅, λ) for all 𝑝 ∈ 𝑃, 𝛶(𝑃, 𝑄, λ) ≥ sup 𝑝 ∈ 𝑃 𝛶 (𝑝, 𝑅, λ) 𝛶(𝑃, 𝑄, λ) ≥ 𝛶(𝑃, 𝑅, λ). lemma 4.8. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a nms. if 𝑃, 𝑄, 𝑅 ⊂ 𝛴, then v. b. shakila and m. jeyaraman 𝛯(𝑃 ∪ 𝑄, 𝑅, λ) = 𝑚𝑖𝑛{𝛯(𝑃, 𝑅, λ), 𝛯(𝑄, 𝑅, λ)}, 𝛩(𝑃 ∪ 𝑄, 𝑅, λ) = 𝑚𝑎𝑥{𝛩(𝑃, 𝑅, λ), 𝛩(𝑄, 𝑅, λ)} and 𝛶(𝑃 ∪ 𝑄, 𝑅, λ) = 𝑚𝑎𝑥{𝛶(𝑃, 𝑅, λ), 𝛶(𝑄, 𝑅, λ)},for all λ > 0. proof: fix λ > 0. let 𝑃, 𝑄, 𝑅 ⊂ 𝛴. then 𝛯(𝑃 ∪ 𝑄, 𝑅, λ) = inf 𝜁 ∈ 𝑃 ∪ 𝑄 𝛯(𝜁, 𝑅, λ) = 𝑚𝑖𝑛 { inf 𝑝 ∈ 𝑃 𝛯(𝑝, 𝑅, λ), inf 𝑞 ∈ 𝑄 𝛯(𝑞, 𝑅, λ),} = 𝑚𝑖𝑛{𝛯(𝑃, 𝑅, λ), 𝛯(𝑄, 𝑅, λ)}, 𝛩(𝑃 ∪ 𝑄, 𝑅, λ) = sup 𝜁 ∈ 𝑃 ∪ 𝑄 𝛩 (𝜁, 𝑅, λ) = 𝑚𝑎𝑥 { sup 𝑝 ∈ 𝑃 𝛩 (𝑝, 𝑅, λ), sup 𝑞 ∈ 𝑄 𝛩 (𝑞, 𝑅, λ),} = 𝑚𝑎𝑥{𝛩(𝑃, 𝑅, λ), 𝛩(𝑄, 𝑅, λ)} and 𝛶(𝑃 ∪ 𝑄, 𝑅, λ) = sup 𝜁 ∈ 𝑃 ∪ 𝑄 𝛶(𝜁, 𝑅, λ) = 𝑚𝑎𝑥 { sup 𝑝 ∈ 𝑃 𝛶 (𝑝, 𝑅, λ), sup 𝑞 ∈ 𝑄 𝛶(𝑞, 𝑅, λ),} = 𝑚𝑎𝑥{𝛶(𝑃, 𝑅, λ), 𝛶(𝑄, 𝑅, λ)} lemma 4.9. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄) be a nms. let (𝒦0(σ), 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) be the corresponding hausdorff nms. if 𝑃, 𝑄, 𝑅, 𝑆 ∈ 𝒦0(σ) then 𝐻𝛯 (𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) ≥ 𝑚𝑖𝑛{𝐻𝛯 (𝑃, 𝑅, λ), 𝐻𝛯 (𝑄, 𝑆, λ)}, 𝐻𝛩 (𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) ≤ 𝑚𝑎𝑥{𝐻𝛩 (𝑃, 𝑅, λ), 𝐻𝛩 (𝑄, 𝑆, λ)} and 𝐻𝛶 (𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) ≤ 𝑚𝑎𝑥{𝐻𝛶 (𝑃, 𝑅, λ), 𝐻𝛶 (𝑄, 𝑆, λ)}, for all λ > 0. proof: fix λ > 0. let 𝑃, 𝑄, 𝑅, 𝑆 ∈ 𝒦0(σ). by using lemma 4.7. and lemma 4.8., we get 𝛯(𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) = 𝑚𝑖𝑛{𝛯(𝑃, 𝑅 ∪ 𝑆, λ), 𝛯(𝑄, 𝑅 ∪ 𝑆, λ)} ≥ 𝑚𝑖𝑛{𝛯(𝑃, 𝑅, λ), 𝛯(𝑄, 𝑆, λ)} ≥ 𝑚𝑖𝑛{𝐻𝛯 (𝑃, 𝑅, λ), 𝐻𝛯 (𝑄, 𝑆, λ)}. similarly, 𝛯(𝑅 ∪ 𝑆, 𝑃 ∪ 𝑄, λ) ≥ 𝑚𝑖𝑛{𝐻𝛯 (𝑃, 𝑅, λ), 𝐻𝛯 (𝑄, 𝑆, λ)}. hence, 𝐻𝛯 (𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) ≥ 𝑚𝑖𝑛{𝐻𝛯 (𝑃, 𝑅, λ), 𝐻𝛯 (𝑄, 𝑆, λ)}. 𝛩(𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) = 𝑚𝑎𝑥{𝛩(𝑃, 𝑅 ∪ 𝑆, λ), 𝛩(𝑄, 𝑅 ∪ 𝑆, λ)} ≤ 𝑚𝑎𝑥{𝛩(𝑃, 𝑅, λ), 𝛩(𝑄, 𝑆, λ)} ≤ 𝑚𝑎𝑥{𝐻𝛩 (𝑃, 𝑅, λ), 𝐻𝛩 (𝑄, 𝑆, λ)}. similarly, 𝛩(𝑅 ∪ 𝑆, 𝑃 ∪ 𝑄, λ) ≤ 𝑚𝑎𝑥{𝐻𝛩 (𝑃, 𝑅, λ), 𝐻𝛩 (𝑄, 𝑆, λ)}. hence, 𝐻𝛩 (𝑅 ∪ 𝑆, 𝑃 ∪ 𝑄, λ) ≤ 𝑚𝑎𝑥{𝐻𝛩 (𝑃, 𝑅, λ), 𝐻𝛩 (𝑄, 𝑆, λ)} and 𝛶(𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) = 𝑚𝑎𝑥{𝛶(𝑃, 𝑅 ∪ 𝑆, λ), 𝛶(𝑄, 𝑅 ∪ 𝑆, λ)} ≤ 𝑚𝑎𝑥{𝛶(𝑃, 𝑅, λ), 𝛶(𝑄, 𝑆, λ)} ≤ 𝑚𝑎𝑥{𝐻𝛶 (𝑃, 𝑅, λ), 𝐻𝛶 (𝑄, 𝑆, λ)}. similarly, 𝛶(𝑅 ∪ 𝑆, 𝑃 ∪ 𝑄, λ) ≤ 𝑚𝑎𝑥{𝐻𝛶 (𝑃, 𝑅, λ), 𝐻𝛶 (𝑄, 𝑆, λ)}. hence, 𝐻𝛶 (𝑅 ∪ 𝑆, 𝑃 ∪ 𝑄, λ) ≤ 𝑚𝑎𝑥{𝐻𝛶 (𝑃, 𝑅, λ), 𝐻𝛶 (𝑄, 𝑆, λ)}. this completes the proof. some results in hausdorff neutrosophic metric spaces on hutchinson-barnsley operator the following theorem is a generalized version of the theorem 4.5. theorem 4.10. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a nms. let (𝒦0(σ), 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) be the corresponding hausdorff nms. suppose 𝑓𝑛 : 𝛴 → 𝛴, 𝑛 = 1,2, … 𝑁0; 𝑁0 ∈ ℕ, is a neutrosophic b-contraction on (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄). then the neutrosophic hbo is also a neutrosophic b-contraction on (𝒦0(σ), 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ,∗,⋄). proof: fix λ > 0. let 𝑃, 𝑄 ∈ 𝒦0(σ). by using the lemma 4.9. and theorem 4.5. for a given 𝑘 ∈ (0,1), we get 𝐻𝛯 (𝐹(𝑃), 𝐹(𝑄), 𝑘λ) = 𝐻𝛯 (⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑃), ⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑄), 𝑘λ) ≥ 𝑁0 min 𝑛 = 1 𝐻𝛯 (𝑓𝑛(𝑃), 𝑓𝑛 (𝑄), 𝑘λ) ≥ 𝐻𝛯 (𝑃, 𝑄, λ), 𝐻𝛩 (𝐹(𝑃), 𝐹(𝑄), 𝑘λ) = 𝐻𝛩 (⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑃), ⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑄), 𝑘λ) ≤ 𝑁0 max 𝑛 = 1 𝐻𝛩 (𝑓𝑛 (𝑃), 𝑓𝑛 (𝑄), 𝑘λ) ≤ 𝐻𝛩 (𝑃, 𝑄, λ) and 𝐻𝛶 (𝐹(𝑃), 𝐹(𝑄), 𝑘λ) = 𝐻𝛶 (⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑃), ⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑄), 𝑘λ) ≤ 𝑁0 max 𝑛 = 1 𝐻𝛶 (𝑓𝑛(𝑃), 𝑓𝑛 (𝑄), 𝑘λ) ≤ 𝐻𝛶 (𝑃, 𝑄, λ). this completes the proof. from the above theorem 4.10., we conclude that the operator f is a neutrosophic b-contraction on 𝒦0(𝛴) with respect to the hausdorff neutrosophic metric (𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ), if 𝑓𝑛 is neutrosophic b-contraction on an neutrosophic metric space(𝛴, 𝛯, 𝛩, 𝛶,∗,⋄) for each 𝑛 ∈ {1,2, … 𝑁0}. 5. hausdorff neutrosophic metrics on 𝓚𝟎(σ) and 𝓚𝟎(𝓚𝟎(σ)) now, we investigate the relationships between the hyperspaces 𝒦0(σ) and 𝒦0(𝒦0(σ)) and the hausdorff neutrosophic metrics 𝐻𝛯 and ℋ𝐻𝛯 . v. b. shakila and m. jeyaraman theorem 5.1. let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄) be a nms. let (𝒦0(σ), 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) and (𝒦0(𝒦0(σ)), ℋ𝐻𝛯 , ℋ𝐻𝛩 , ℋ𝐻𝛶 ,∗,⋄) be the corresponding hausdorff neutrosophic hyper spaces. let 𝔓, 𝔔 ∈ 𝒦0(𝒦0(σ)) be such that {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔} ∈ 𝒦0(σ). then 𝐻𝛯 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄 ∶ 𝑄 ∈ 𝔔}, λ) ≥ ℋ𝐻𝛯 (𝔓, 𝔔, λ), 𝐻𝛩 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ : 𝑄 ∈ 𝔔}, λ) ≤ ℋ𝐻𝛩 (𝔓, 𝔔, λ) and 𝐻𝛶 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, λ) ≤ ℋ𝐻𝛶 (𝔓, 𝔔, λ) for all λ > 0. proof: fix λ > 0. firstly, we note that 𝛯(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = inf 𝑞 ∈ 𝑄 𝛯(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = inf 𝑞 ∈ 𝑄 sup {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓} 𝛯(𝑞, 𝑝, λ) = inf 𝑞 ∈ 𝑄 sup 𝑃 ∈ 𝔓 sup 𝑝 ∈ 𝑃 𝛯 (𝑞, 𝑝, λ) ≥ sup 𝑃 ∈ 𝔓 inf 𝑞 ∈ 𝑄 sup 𝑝 ∈ 𝑃 𝛯 (𝑞, 𝑝, λ) = sup 𝑃 ∈ 𝔓 𝛯(𝑄, 𝑃, λ). it follows that 𝛯({𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = inf {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔} 𝛯(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = inf 𝑄 ∈ 𝔔 inf 𝑞 ∈ 𝑄 𝛯(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = inf 𝑄 ∈ 𝔔 𝛯(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) ≥ inf 𝑄 ∈ 𝔔 sup 𝑃 ∈ 𝔓 𝛯(𝑄, 𝑃, λ). similarly, 𝛯({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, λ) ≥ inf 𝑃 ∈ 𝔓 sup 𝑄 ∈ 𝔔 𝛯(𝑃, 𝑄, λ). hence, 𝐻𝛯 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, λ) = 𝑚𝑖𝑛 { 𝛯({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, λ), 𝛯({{𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) } ≥ 𝑚𝑖𝑛 { inf 𝑃 ∈ 𝔓 sup 𝑄 ∈ 𝔔 𝛯(𝑃, 𝑄, λ) , inf 𝑄 ∈ 𝔔 sup 𝑃 ∈ 𝔓 𝛯(𝑄, 𝑃, λ)} ≥ 𝑚𝑖𝑛 { inf 𝑃 ∈ 𝔓 sup 𝑄 ∈ 𝔔 𝐻𝛯 (𝑃, 𝑄, λ) , inf 𝑄 ∈ 𝔔 sup 𝑃 ∈ 𝔓 𝐻𝛯 (𝑄, 𝑃, λ)} = 𝑚𝑖𝑛{𝐻𝛯 (𝔓, 𝔔, λ), 𝐻𝛯 (𝔔, 𝔓, λ)} = ℋ𝐻𝛯 (𝔓, 𝔔, λ). secondly, we note that 𝛩(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑞 ∈ 𝑄 𝛩(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) some results in hausdorff neutrosophic metric spaces on hutchinson-barnsley operator = 𝑠𝑢𝑝 𝑞 ∈ 𝑄 inf {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓} 𝛩(𝑞, 𝑝, λ) = sup 𝑞 ∈ 𝑄 inf 𝑃 ∈ 𝔓 inf 𝑝 ∈ 𝑃 𝛩(𝑞, 𝑝, λ) ≤ inf 𝑃 ∈ 𝔓 sup 𝑞 ∈ 𝑄 inf 𝑝 ∈ 𝑃 𝛩(𝑞, 𝑝, λ) = inf 𝑃 ∈ 𝔓 𝛩(𝑄, 𝑃, λ). it follows that 𝛩({𝑞 ∈: 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔} 𝛩(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑄 ∈ 𝔔 sup 𝑞 ∈ 𝑄 𝛩(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑄 ∈ 𝔔 𝛩(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) ≤ sup 𝑄 ∈ 𝔔 inf 𝑃 ∈ 𝔓 𝛩(𝑄, 𝑃, λ). similarly, 𝛩({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, λ) ≤ sup 𝑃 ∈ 𝔓 inf 𝑄 ∈ 𝔔 𝛩(𝑃, 𝑄, λ). hence, 𝐻𝛩 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, λ) = 𝑚𝑎𝑥 { 𝛩({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, λ), 𝛩({{𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) } ≤ 𝑚𝑎𝑥 { sup 𝑃 ∈ 𝔓 𝑖𝑛𝑓 𝑄 ∈ 𝔔 𝛩(𝑃, 𝑄, λ) , sup 𝑄 ∈ 𝔔 𝑖𝑛𝑓 𝑃 ∈ 𝔓 𝛩(𝑄, 𝑃, λ)} ≤ 𝑚𝑎𝑥 { sup 𝑃 ∈ 𝔓 𝑖𝑛𝑓 𝑄 ∈ 𝔔 𝐻𝛩 (𝑃, 𝑄, λ) , sup 𝑄 ∈ 𝔔 𝑖𝑛𝑓 𝑃 ∈ 𝔓 𝐻𝛩 (𝑄, 𝑃, λ)} = 𝑚𝑎𝑥{𝐻𝛩 (𝔓, 𝔔, λ), 𝐻𝛩 (𝔔, 𝔓, λ)} = ℋ𝐻𝛩 (𝔓, 𝔔, λ) and lastly, we note that 𝛶(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑞 ∈ 𝑄 𝛶 (𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = 𝑠𝑢𝑝 𝑞 ∈ 𝑄 inf {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓} 𝛶(𝑞, 𝑝, λ) = sup 𝑞 ∈ 𝑄 inf 𝑃 ∈ 𝔓 inf 𝑝 ∈ 𝑃 𝛶(𝑞, 𝑝, λ) ≤ inf 𝑃 ∈ 𝔓 sup 𝑞 ∈ 𝑄 inf 𝑝 ∈ 𝑃 𝛶(𝑞, 𝑝, λ) = inf 𝑃 ∈ 𝔓 𝛶(𝑄, 𝑃, λ) it follows that 𝛶({𝑞 ∈: 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔} 𝛶(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑄 ∈ 𝔔 sup 𝑞 ∈ 𝑄 𝛶 (𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑄 ∈ 𝔔 𝛶(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) ≤ sup 𝑄 ∈ 𝔔 inf 𝑃 ∈ 𝔓 𝛶(𝑄, 𝑃, λ). similarly, 𝛶({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈: 𝑄 ∈ 𝔔}, λ) ≤ sup 𝑃 ∈ 𝔓 inf 𝑄 ∈ 𝔔 𝛶(𝑃, 𝑄, λ). hence, 𝐻𝛶 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, λ) v. b. shakila and m. jeyaraman = 𝑚𝑎𝑥 { 𝛶({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, λ), 𝛶({{𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) } ≤ 𝑚𝑎𝑥 { sup 𝑃 ∈ 𝔓 𝑖𝑛𝑓 𝑄 ∈ 𝔔 𝛶(𝑃, 𝑄, λ) , sup 𝑄 ∈ 𝔔 𝑖𝑛𝑓 𝑃 ∈ 𝔓 𝛶(𝑄, 𝑃, λ)} ≤ 𝑚𝑎𝑥 { sup 𝑃 ∈ 𝔓 𝑖𝑛𝑓 𝑄 ∈ 𝔔 𝐻𝛶 (𝑃, 𝑄, λ) , sup 𝑄 ∈ 𝔔 𝑖𝑛𝑓 𝑃 ∈ 𝔓 𝐻𝛶 (𝑄, 𝑃, λ)} = 𝑚𝑎𝑥{𝐻𝛶 (𝔓, 𝔔, λ), 𝐻𝛶 (𝔔, 𝔓, λ)} = ℋ𝐻𝛶 (𝔓, 𝔔, λ). the proof is complete. 6. conclusions in this paper, we proved the neutrosophic contraction properties of the hutchinson-barnsley operator on the neutrosophic hyperspace with respect to the hausdorff neutrosophic metrics. also we discussed about the relationships between the hausdorff neutrosophic metrics on the neutrosophic hyperspaces. this paper will lead our direction to develop the hutchinson-barnsley theory in the sense of neutrosophic b-contractions in order to define a fractal set in the neutrosophic metric spaces as a unique fixed point of the neutrosophic hbo. references [1] m. abdel-basset, m. saleh, abduallah gamal, florentin smarandache, an approach of topsis technique for developing supplier selection with group decision making under type-2 neutrosophic number, applied soft computing, 77, 2019, 438-452. [2] k. atanassov, intuitionistic fuzzy sets, fuzzy sets and systems, 20, 1986, 87-96. [3] m. barnsley, fractals everywhere, 2nd ed., academic press, usa, 1993. [4] m. barnsley, super fractals, cambridge university press, new york, 2006. [5] d. easwaramoorthy and r. uthayakumar, analysis on fractals in fuzzy metric spaces, fractals, 19(3), 2011, 379-386. [6] j. e. hutchinson, fractals and self similarity, indiana university mathematics journal, 30, 1981, 713-747. [7] m. jeyaraman, m. suganthi, s. sowndrarajan, fixed point results in nonarchimedean generalized intuitionistic fuzzy metric spaces, notes on intuitionistic fuzzy sets, 25, 2019, 48-58. some results in hausdorff neutrosophic metric spaces on hutchinson-barnsley operator [8] b. b. mandelbrot, the fractal geometry of nature, w. h. freeman and company, new york, 1983. [9] murat kirisci and necip simsek, neutrosopohic metric spaces, mathematical sciences, islamic azad university, 2020. [10] j. h. park, intuitionistic fuzzy metric spaces, chaos, solitons and fractals, 22, 2004, 1039-1046. [11] m. rajeswari, m. jeyaraman, s. durga, some new fixed point theorems in generalized intuitionistic fuzzy metric spaces, notes on intuitionistic fuzzy sets, 25(3), 2019, 42-52. [12] f. smarandache, a unifying field in logics: neutrosophic logic. neutrosophy, neutrosophic set, neutrosophic probability and statistics. xiquan, phoenix, 3rd edn. 2003. [13] f. smarandache, neutrosophic set a generalization of the intuitionistic fuzzy set, international journal of pure and applied mathematics, 24(3), 2005, 287-297. [14] sowndrarajan, jeyaraman and florentin smarandache, fixed point results for contraction theorems in neutrosophic metric spaces, neutrosophic sets and systems, 36, 2020, 308-318. [15] r. uthayakumar and d. easwarmoorthy, hutchinson-barnsley operator in fuzzy metric spaces, international journal of engineering and natural sciences, 5(4), 2011, 203-207. ratio mathematica volume 46, 2023 properties of nano generalized pre c-interior in a nano topological space. padmavathi p* abstract the aim of this paper is to introduce and study the properties the nano generalized pre cinterior of a set such as nano generalized pre c-border and nano generalized pre c-exterior in a nano topological space. keywords:nano generalized pre c-border, nano generalized pre cexterior. 2020 ams subject classifications: 06f20, 06f15, 20cxx. 1 *department of mathematics, sri g.v.g visalakshi college for women (autonomous), udumalpet, tamilnadu, india. padmasathees74@gmail.com 1received on september 15, 2022. accepted on march, 2023. published on march 20, 2023. doi: 10.23755/rm.v46i0.1075. issn: 1592-7415. eissn: 2282-8214. ©p.padmavathi. this paper is published under the cc-by licence agreement. 194 p.padmavathi 1 introduction the concept of generalized-semi closed sets to characterize the s-normality axiom was introduced by s.p.arya et.al. the semi-generalized mappings and generalized-semi mappings were studied. in 2013 , govindappa navalagi investigated some of the regularity axioms, normality axioms and continuous functions through gs-open sets and sg-open sets. also, govindappa navalagi continued the study of gs-continuous and sg-continuous functions to introduce the new notions like generalized semiclosure and generalized semi-interior operators. lellis thivagar [1] obtained the notion of nano topology and he studied the various forms of nano sets, their closures and interiors and their homeomorphisms lellis thivagar et al introduced nano topological space with respect to a subset of a universe which is defined in terms of approximations and boundary region. in this paper, i have introduced the properties of nano generalized pre c-interior in a nano topological space. 2 preliminaries definition 2.1. [3] let = be a non empty finite set of objects called the universe and < be an equivalence relation on = named as indiscernibility relation. then = is divided into disjoint equivalence classes. elements belonging to the same equivalence class are said to be indiscernible with one another. the pair (=,<) is said to be approximation space. let ℵ⊆=. then (i) the lower approximation of ℵ with respect to < is the set of all objects, which can be for certain classified as ℵ with respect to < and it is denoted by γ<(ℵ). γ<(ℵ) = =x∈=<(x) : <(x) ⊆ℵ by γ<(ℵ). where <(ℵ) denotes the equivalence class determined. (ii) the upper approximation of ℵ with respect to < is the set of all objects which can be possibly classified as ℵ with respect to < and it is denoted by τ<(ℵ). τ<(ℵ) = =x∈=<(x) : <(x) ⋂ ℵ 6= 0. (iii) the boundary region of ℵ with respect to < is the set of all objects which can be classified neither as ℵ nor as not-x with respect to < and it is denoted by b<(ℵ).b<(ℵ) = τ<(ℵ)−γ<(ℵ). proposition 2.1. [3] if (=,<) is an approximation space and ℵ,y ⊆=,then 1. γ<(ℵ) ⊆ℵ⊆ τ<(ℵ) 2. γ<(φ) = τ<(ℵ) = φ 195 properties of nano generalized pre c-interior in a nano topological space 3. γ<(u) = τ<(=) = = 4. τ<(ℵ∪y ) = τ<(ℵ)∪ τ<(y ) 5. τ<(ℵ∩y ) ⊆ τ<(ℵ)∩ τ<(y ) 6. γ<(ℵ∪y ) ⊇ γ<(ℵ)∪γ<(y ) 7. γ<(ℵ∩y ) = γ<(ℵ)∩γ<(y ) 8. γ<(ℵ) ⊆ γ<(y )andτ<(ℵ) ⊆ τ<(y ), whenever ℵ⊆ y . 9. τ<(ℵc) = [γ<(ℵ)]candγ<(ℵc) = [τ<(ℵ)]c 10. τ<[τ<(ℵ)] = γ<[τ<(ℵ)] = τ<(ℵ) 11. γ<[γ<(ℵ)] = τ<[γ<(ℵ)] = γ<(ℵ) definition 2.2. [1] let = be the universe, < be an equivalence relation on = and r<(ℵ) = {=,φ,γ<(ℵ),τ<(ℵ),b<(ℵ)} where ℵ ⊆ =. then r<(ℵ) satisfies the following axioms. 1. = and φ ∈ r<(ℵ). 2. the union of all the elements of any sub-collection of r<(ℵ) is in r<(ℵ). 3. the intersection of the elements of any finite sub collection of r<(ℵ) is in r<(ℵ).then r<(ℵ) is a topology on = called the nano topology on = with respect to ℵ. the elements of r<(ℵ) are called as nano open sets in = and (=,r<(ℵ)) is called as a nano topological space. the complement of the nano open sets are called nano closed sets. definition 2.3. [1] if (=,r<(ℵ)) is a nano topological space with respect to ℵ, where ℵ⊆= and if a ⊆=, then 1. the nano interior of a is defined as the union of all nano open subsets contained in a and is denoted by nint(a). that is nint(a) is the largest nano open subset of a . 2. the nano closure of a is defined as the intersection of all nano closed sets containing a and is denoted by ncl(a). that is ncl(a) is the smallest nano closed set containing a. definition 2.4. [2] a subset a of a nano topological space (=,r<(ℵ)) is called a nano generalized pre c-closed set (briefly ngpc−closed set) if npcl(a) ⊆ g whenever a ⊆ g and c is nano c-set. the complement of a ngpc−closed set is called ngpc−open set. 196 p.padmavathi definition 2.5. [2] the nano generalized pre c-interior of a set a in (=,r<(ℵ)) is defined as the union of all ngpc−open sets of u contained in a and it is denoted by ngpc−int(a). that is ngpc−int(a) is the largest ngpc−open subset of a. definition 2.6. [2] the nano generalized pre c-closure of a set a in (=,r<(ℵ)) is defined as the intersection of all ngpc−closed sets of u containing a and it is denoted by ngpc − cl(a). that is ngpc − cl(a) is the smallest ngpc−closed superset of a in iu. remark 2.1. [2] 1. a subset a of (=,r<(ℵ)) is ngpc−open if and ony if ngpc− int(a) = a. 2. a subset a of (=,r<(ℵ)) is ngpc−closed if and only if ngpc−cl(a) = a. theorem 2.1. [2] let a and b be subsets of (=,r<(ℵ)). then 1. ngpc− int(=) = = and ngpc− int(φ) = φ. 2. ngpc− int(a) ⊂ a. 3. if b is any ngpc−open set contained in a, then b ⊂ ngpc− int(a). 4. if a ⊂ b then ngpc− int(a) ⊆ ngpc− int(b). 5. ngpc− int(ngpc− int(a)) = ngpc− int(a). theorem 2.2. [2] if a and b are subsets of =, then the following statements are true. 1. ngpc− int(a)∪ngpc− int(b) ⊂ ngpc− int(a∪b). 2. ngpc− int(a∩b) = ngpc− int(a)∩ngpc− int(b). theorem 2.3. [2] if a is a subset of (=,r<(ℵ)), then nint(a) ⊂ ngpc−int(a). theorem 2.4. [2] for the subsets a and b of =, the following statements are true. 1. =−ngpc− cl(a) ⊂ ngpc− cl(=−a). 2. if a is ngpc−closed then ngpc−cl(a)−ngpc−cl(b) ⊂ ngpc−cl(a− b). 197 properties of nano generalized pre c-interior in a nano topological space 3 properties of nano generalized pre c-interior in this section the nano generalized pre c-border and nano generalized pre cexterior of a set are defined in terms of nano generalized pre c-interior and some of their properties are derived. definition 3.1. the nano generalized pre c-border of a set a in (=,r<(ℵ)) is defined as a−ngpc− int(a) and it is denoted by ngpc−bd(a). definition 3.2. the nano generalized pre c-exterior of a set a in (=,r<(ℵ)) is defined as ngpc− int(=−a) and it is denoted by ngpc−ext(a). example 3.1. let = = {a,b,c,d} with =/< = {{a} ,{b} ,{c,d}} and ℵ = {b,d}. then r<(ℵ) = {=,φ,{b} ,{c,d} ,{b,c,d}} is a nano topology on u with respect to ℵ. the complement of r<(ℵ) is given by rc(ℵ) = {u,φ,{a} ,{a,b} , {a,c,d}. ngpc−closed sets are {φ,=,{a} ,{c} ,{d} , {a,b} ,{a,c} ,{a,d} ,{a,b,c} ,{a,b,d} ,{a,c,d}. ngpc−open sets are φ,=, {b} ,{c} ,{d} ,{b,c} ,{b,d} , {a,b,c} ,{a,b,d} ,{b,c,d}. here ngpc − int({a}) = φ, ngpc − int({b}) = {b}, ngpc − int({a,c,d}) = {c,d}, ngpc− int({c,d}) = {c,d} and ngpc− int({a,b,d}) = {a,b,d}. then ngpc−bd({a}) = {a} ,ngpc−bd({b}) = φ , ngpc−bd({a,b,d}) = φ and ngpc−bd({a,c,d}) = {a}. ngpc−ext({a}) = {b,c,d}, ngpc−ext({b}) = {c,d}, ngpc−ext({a,b}) = {c,d} and ngpc−ext({a,c,d}) = {b}. theorem 3.1. for a subset a of = the following statements hold. 1. ngpc−bd(φ) = ngpc−bd(=) = φ. 2. ngpc−bd(a) ⊂ nbd(a). 3. a = ngpc− int(a)∪ngpc−bd(a). 4. ngpc− int(a)∩ngpc−bd(a) = φ. 5. ngpc− int(a) = a−ngpc−bd(a). 6. ngpc− int(ngpc−bd(a)) = ngpc−bd(ngpc− int(a)) = φ. 7. a is ngpc−open if and only if ngpc−bd(a) = φ. 8. ngpc−bd(ngpc−bd(a)) = ngpc−bd(a). proof. 1. the proof is an immediate consequence of definition (3.1). 198 p.padmavathi 2. let x ∈ ngpc−bd(a). ⇒ x ∈ a−ngpc− int(a). by theorem (2.3), nint(a) ⊂ ngpc− int(a) ⇒ a−ngpc− int(a) ⊂ a−nint(a). hence x ∈ a−ngpc− int(a) ⇒ x ∈ a−nint(a). ⇒ x ∈ nbd(a). therefore ngpc−bd(a) ⊂ nbd(a). 3. ngpc−int(a)∪ngpc−bd(a) = ngpc−int(a)∪(a−ngpc−int(a)) = a. 4. ngpc−int(a)∩ngpc−bd(a) = ngpc−int(a)∩(a−ngpc−int(a)) = φ. 5. the proof directly follows from definition (3.1). 6. let x ∈ ngpc−int(ngpc−bd(a)). then x ∈ ngpc−bd(a) as ngpc− bd(a) ⊂ a. also x ∈ ngpc− int(ngpc−bd(a)) ⊂ ngpc− int(a)r. therefore x ∈ ngpc − int(a) ∩ ngpc − bd(a) which is a contradiction to (d). hence ngpc− int(ngpc−bd(a)) = φ. 7. by result (2.8), a is ngpc−open ⇔ ngpc− int(a) = a ⇔ a−ngpc− int(a) = φ ⇔ ngpc−bd(a) = φ. (by definition (3.1)) 8. in definition (3.1) let a = ngpc−bd(a). then ngpc−bd(ngpc−bd(a)) = ngpc−bd(a)−ngpc−int(ngpc− bd(a)) = ngpc−bd(a)−φ = ngpc−bd(a). (using (6)). theorem 3.2. for the subsets a and b of = the following statements hold. 1. ngpc−ext(φ) = = and ngpc−ext(=) = φ. 2. next(a)ngpc−ext(a). 3. if a ⊂ b then ngpc−ext(b) ⊂ ngpc−ext(a). 4. ngpc−ext(a) is ngpc−open. 5. ngpc−ext(a) = =−ngpc− cl(a). 6. a is ngpc−closed if and only if ngpc−ext(a) = =−a. 7. ngpc−ext(ngpc−ext(a)) = ngpc− int(ngpc− cl(a)) 199 properties of nano generalized pre c-interior in a nano topological space 8. ngpc − ext(ngpc − ext(a)) = ngpc − ext(ngpc − int(= − a)) = ngpc−ext(=−ngpc− cl(a)). 9. ngpc−ext(a∪b) ⊂ ngpc−ext(a)∪ngpc−ext(b). 10. ngpc−ext(a∪b) = ngpc−ext(a)∩ngpc−ext(b). 11. ngpc−ext(a)∩ngpc−ext(b) ⊂ ngpc−ext(a∩b). proof. 1. the proof is immediate from definition (3.2). 2. next(a) ⊂ ngpc−ext(a) follows from theorem (2.3). 3. if a ⊂ b then =−b ⊂=−a. by (iv) of theorem (2.2), ngpc− int(=− b) ⊂ ngpc− int(=−a). hence ngpc−ext(b) ⊂ ngpc−ext(a). 4. consider ngpc−int(ngpc−ext(a)) = ngpc−int(ngpc−int(=−a)) = ngpc − int(=− a) = ngpc − ext(a). (by (v) of theorem (2.9)) by remark (2.1), ngpc−ext(a) is ngpc−open. 5. ngpc−ext(a) = ngpc− int(=−a) = =−ngpc−cl(a). (from (ii) of theorem (2.4)). 6. by remark (2.1), a is ngpc−closed ⇔ ngpc−cl(a) = a ⇔=−ngpc− cl(a) = =−a−ngpc− int(=−a) = =−a ⇔ ngpc−ext(a) = =−a. 7. in definition let a = ngpc−ext(a).then ngpc−ext(ngpc−ext(a)) = ngpc− int(=−ngpc−ext(a) = ngpc− int(ngpc− cl(a)). (using (5)). 8. it follows from definition (3.2) and (5). 9. we know that a ⊂ a ∪ b and b ⊂ a ∪ b. from (c) ngpc − ext(a ∪ b) ⊂ ngpc−ext(b) and ngpc−ext(a∪b) ⊂ ngpc−ext(b). hence ngpc−ext(a∪b) ⊂ ngpc−ext(a) ⊂ ngpc−ext(b). 10. ngpc−ext(a∪b) = ngpc− int(=− (a∪b)). (by definition (3.2)) = ngpc− int((=−a)∩ (=−b)). = ngpc− int(=−a)∩ngpc− int(=−b). (by (ii) of theorem(2.4)) = ngpc−ext(a)∩ngpc−ext(b). hence ngpc−ext(a∪b) = ngpc−ext(a)∩ngpc−ext(b). 200 p.padmavathi 11. we know that a ∩ b ⊂ a and a ∩ b ⊂ b. from (c) ngpc − ext(a) ⊂ ngpc − ext(a ∩ b) and ngpc − ext(b) ⊂ ngpc − ext(a ∩ b). hence ngpc−ext(a)∩ngpc−ext(b) ⊂ ngpc−ext(a∩b). references [1] m.lellis thivagar and c. richard. note on nano topological spaces (communicated). [2] p.padmavathi and r. nithyakala. a note on nano generalized pre c-closed sets. international journal of advanced science and technology, 29(3s):194 – 2021, 2020. [3] z.pawlak. rough sets, theoretical aspects of reasoning about data. kluwer academic publishers, boston, 1991. 201 ratio mathematica volume 47, 2023 bi-amalgamated algebra with (n, p)-weakly clean like properties aruldoss antonysamy* selvaraj chelliah† abstract let f : a −→ b and g : a −→ c be two ring homomorphisms and let k and k′ be two ideals of b and c, respectively such that f−1(k) = g−1(k′). in this paper, we give a characterization for the bi-amalgamation of a with (b, c) along (k, k′) with respect to (f, g) (denoted by a ▷◁f,g (k, k′)) to be a (n, p)-weakly clean ring. keywords: (n, p)-clean ring ; (n, p)weakly clean ring; bi-amalgamation algebra along ideals. 2020 ams subject classifications:16n40, 16u40, 16s99, 16u60. 1 *periyar university, salem, india; aruldossa529@gmail.com. †periyar university, salem, india; selvavlr@yahoo.com. 1received on september 15, 2022. accepted on december 15, 2022. published online on january 10, 2023. doi: 10.23755/rm.v41i0.937. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 324 a. aruldoss and c. selvaraj 1 introduction throughout this paper all rings are commutative with identity elements. let a and b be two rings with unity, let k be an ideal of b and let f : a → b be a ring homomorphism. in d’anna et al. [2009], the authors introduced and studied the new ring structure the following subring of a × b: a ▷◁f k := {(a, f(a) + k) | a ∈ a, k ∈ k} called the amalgamation of a with b along k with respect to f. this new ring structure construction is a generalization of the amalgamated duplication of a ring along an ideal. the amalgamated duplication of a ring along an ideal was introduced and studied in (d’anna [2006], d’anna and fontana [2007]). in [d’anna et al., 2009, section 4], the authors studied the amalgamation can be in the frame of pullback constructions and also the basic properties of this construction (e.g., characterizations for a ▷◁f k to be a noetherian ring, an integral domain, a reduced ring) and they characterized those distinguished pullbacks that can be expressed as an amalgamation. let α : a −→ c, β : a −→ c and f : a −→ b be ring homomorphisms. in d’anna et al. [2009], the authors studied amalgamated algebras within the frame of pullback α × β such that α = β ◦ f [d’anna et al., 2009, proposition 4.2 and 4.4]. in this motivation, the authors created the new constructions, called bi-amalgamated algebras which arise as pullbacks α × β such that the following diagram of ring homomorphims c d a b f g α β is commutative with α ◦ πb(α × β) = α ◦ f(a), where πb denotes the canonical projection of b×c over b. namely, let f : a −→ b and g : a −→ c be two ring homomorphisms and let k and k′ be two ideals of b and c, respectively, such that f−1(k) = g−1(k′). the bi-amalgamation of a with (b, c) along (k, k′) with respect to (f, g) is the subring of b × c given by a ▷◁f,g (k, k′) := {(f(a) + k, g(a) + k′)|a ∈ a, (k, k′) ∈ k × k′} following kabbaj et al. [2013], the above definition was introduced by and studied by kabbaj, louartiti and tamekkante. following nicholson [1977], an element a in a ring a is called a clean if a is a sum of a unit and an idempotent in a. a ring is clean if all its elements are clean. 325 bi-amalgamated algebra with (n, p)-weakly clean like properties clean rings were initially developed by nicholson [1977], as a natural class of rings which have the exchange property. this paper aims at studying the transfer of the notion of n-clean rings, (n, p)weakly clean rings to bi-amalgamation of algebra along ideals. we denote by u(a), set of all unit elements of a. 2 (n, p)-clean ring we start with definition of (n, p)-clean rings. definition 2.1. (chen and qua [2014]) an element a ∈ a is said to be (n, p)clean if a = u1 + ... + un + x for some unit ui ∈ a(i = 1, ..., n) and xp = x, where x is called p-potent element. the ring a is said to be (n, p)-clean if all of its elements are (n, p)-clean. remark 2.1. every clean rings are (1, 2)-clean rings and n-clean rings are (n, 2)clean rings. following chhiti [2018], the above definition is studied the authors in 2018. now, we start with following example of (2, 2)-clean ring. example 2.1. let a := z4, b := z4 × z4 and c := z2 and let j := 0 × z4 and j′ := z2 are ideals of b and c respectively. consider the map f : a −→ b is defined by f(a) = (a, 0) for all a ∈ a and the map g : a −→ c is defined by g(0) = g(2) = 0 and g(1) = g(3) = 1. hence, a ▷◁f,g (j, j′) ={ ((0, 0), 0), ((0, 1), 0), ((0, 2), 0), ((0, 3), 0), ((2, 0), 0), ((2, 1), 0), ((2, 2), 0), ((2, 3), 0), ((1, 0), 1), ((1, 1), 1), ((1, 2), 1), ((1, 3), 1), ((3, 0), 1), ((3, 1), 1), ((3, 2), 1), ((3, 3), 1) } is a 2-clean ring. therefore, by the remark 2.1 a ▷◁f,g (j, j′) is (2, 2)-clean ring. proposition 2.1. the class of (n, p)-clean ring is closed under homomorphic images. proof. the proof is straightforward.2 proposition 2.2. if a ▷◁f,g (k, k′) is a (n, p)-clean ring then f(a) + k and g(a) + k′ are (n, p)-clean rings. proof. clearly, by proposition 2.1 (n, p)-clean ring is a (n, p)-clean ring. thus, in view of kabbaj et al. [2013] [proposition 4.1], we have the following isomorphism of rings a▷◁ f,g(k,k′) 0×k′ ∼= f(a) + k and a▷◁ f,g(k,k′) k×0 ∼= g(a) + k′. hence, f(a) + k and g(a) + k′ are (n, p)-clean rings.2 326 a. aruldoss and c. selvaraj definition 2.2. a ring is called uniquely (n, p)-clean ring if each element in a can be written as unique way. theorem 2.1. assume that a is (n, p)-clean ring and f(a) + k k and g(a) + k′ k′ are uniquely n-clean rings. then a ▷◁f,g (k, k′) is (n, p)-clean ring if and only if f(a) + k and g(a) + k′ are (n, p)-clean rings. proof. if a ▷◁f,g (k, k′) is a (n, p)-clean ring, then so f(a) + k and g(a)+k′ by proposition 2.1. conversely, assume that f(a)+k and g(a)+k′ are (n, p)-clean rings. since a is a (n, p)-clean ring, we can write a = u1+...+un+x for some unit ui ∈ a(i = 1, ..., n) and xp = x. on the other hand, since f(a)+k is a (n, p)-clean ring, f(a)+k = (f(x1)+k1)+...+(f(xn)+kn)+f(p)+k∗ with f(xi) + ki(i = 1, ..., n) and f(y) + k∗ are respectively units and (f(p) + k∗) p = f(p) + k∗ element of f(a) + k. it is clear that f(x1) = f(x1) + k1 (resp., f(u1)),...,f(xn) = f(xn) + kn (resp., f(un)) and f(p) = f(p) + k∗ (resp., f(x)), are respectively units and an p-potent element of f(a) + k k , and we have f(a) = f(u1)+ ...+f(un)+f(p) = f(x1)+ ...+f(xn)+f(x). thus, f(u1) = f(x1),..., f(un) = f(xn) and f(p) = f(x) since f(a) + k k is an uniquely (n, p)-clean ring. consider k∗1, ..., k ∗ n, k ∗ l ∈ k such that f(x1) = f(u1)+k ∗ 1,..., f(xn) = f(un)+k ∗ n and f(x) = f(p) + k∗l and also since g(a) + k ′ is a (n, p)-clean ring, g(a) + k′ = (g(x′1) + k ′ 1) + ... + (g(x ′ n) + k ′ n) + g(p ′) + k′∗ with g(x′i) + k ′ i(i = 1, ..., n) and g(p′) + k′∗ are respectively units and p-potent element of g(a) + k′. it is clear that g(x′1) = g(x ′ 1) + k ′ 1 (resp., g(u1)),...,g(x′n) = g(x′n) + k′n (resp., g(un)) and g(p′) = g(p′) + k′∗ (resp., g(x)), are respectively units and an p-potent element of g(a) + k′ k′ , and we have g(a) = g(u1) + ... + g(un) + g(p) = g(x′1) + ... + g(x′n) + g(p ′). thus, g(u1) = g(x′1),..., g(un) = g(x′n) and g(x) = g(p′) since g(a) + k′ k′ is an uniquely (n, p)-clean ring. consider k′∗1 , ..., k ′∗ n , k ′∗ l ∈ k such that g(x′1) = g(u1) + k ′∗ 1 ,..., g(x ′ n) = g(un) + k ′∗ n and g(y ′) = g(p) + k′∗l . we have (f(a)+k, g(a)+k′) = {(f(u1)+k∗1 +k1)+...+(f(un)+k∗n +kn)+(f(p)+k∗l + k∗), (g(u1)+k ′∗ 1 +k ′ 1)+...+(g(un)+k ′∗ n +k ′ n)+(g(p ′)+k′∗l +k ′∗)} = (f(u1)+k∗1+ k1, g(u1)+k ′∗ 1 +k ′ 1)+...+(f(un)+k ∗ n+kn, g(un)+k ′∗ n +k ′ n)+(f(p)+k ∗ l +k ∗, g(p′)+ k′∗l +k ′∗). it is clear that (f(p)+k∗l +k ∗, g(p′)+k′∗l +k ′∗) is an p-potent in a ▷◁f,g (k, k′). hence, we have only to prove that (f(u1) + k∗1 + k1, g(u1) + k ′∗ 1 + k ′ 1) is invertible in a ▷◁f,g (k, k′). since f(u1) + k∗1 + k1 is invertible in f(a) + k, there exists an element f(β) + k0 such that (f(u1) + k∗1 + k1)(f(β) + k0) = 1. thus f(u1)f(β) = 1. then f(β) = f(u −1 1 ). so there exists k ∗ 0 ∈ k such that f(β) = f(u−11 ) + k ∗ 0. similarly, g(u1) + k ′∗ 1 + k ′ 1 is invertible in g(a) + k ′. hence, (f(u1)+k∗1 +k1, g(u1)+k ′∗ 1 +k ′ 1)(f(u −1 1 )+k ∗ 0 +k0, g(u −1 1 )+k ′∗ 0 +k ′ 0) = 327 bi-amalgamated algebra with (n, p)-weakly clean like properties (f(u1) + k ∗ 1 + k1, g(u1) + k ′∗ 1 + k ′ 1)(f(β) + k0, g(γ) + k ′ 0) = (1, 1). therefore, (f(u1) + k ∗ 1 + k1, g(u1) + k ′∗ 1 + k ′ 1) is invertible in a ▷◁ f,g (k, k′). similarly, each term are invertible. this completes the proof.2 proposition 2.3. let f : a → b and g : a → c be two surjective ring homomorphisms, let k and k′ be two ideals of b and c respectively such that f−1(k) = g−1(k′) = i0 and let a is a (n, p)-clean ring and a/i0 is an uniquely (n, p)-clean ring. then a ▷◁f,g (k, k′) is a (n, p)-clean ring. proof. it is clear that b and c are (n, p)-clean rings. so, since f(a)+k = b and g(a) + k′ = c, we conclude that a ▷◁f,g (k, k′) is a (n, p)-clean ring by theorem 2.1.2 let a be a commutative ring with identity and let m be a unitary a-module. the idealization of m in a(or trivial extension of a by m) is the commutative ring a ∝ m = {(a, m)|a ∈ r, m ∈ m} under the usual addition and the multiplication defined as (a1m1)(a2m2) = (a1a2, a1m2 +a2m1) for all (a1, m1), (a2, m2) ∈ a ∝ m. theorem 2.2. consider n and p two positive integers (p ≥ 2). let f : a −→ c, g : a −→ c be two ring homomorphisms. assume that a is (n, p)-clean ring. let k be an ideal of b such that f(u) + k is invertible (in b) for each u ∈ u(a) and k ∈ k and k′ be an ideal of c such that g(u) + k′ is invertible (in c) for each u ∈ u(a) and k′ ∈ k′. then a ▷◁f,g (k, k′) is a (n, p)-clean ring if and only if f(a) + k and g(a) + k′ are (n, p)-clean ring. proof. in light of proposition 2.1, homomorphic image of (n, p)-clean ring is (n, p)-clean ring. thus, in view of kabbaj et al. [2013] [proposition 4.1], we have the following isomorphism of rings a▷◁ f,g(k,k′) 0×k′ ∼= f(a) + k and a▷◁ f,g(k,k′) k×0 ∼= g(a) + k′. hence, f(a) + k and g(a) + k′ are (n, p)-rings. conversely, we assume that a is (n, p)-clean and k be an ideal of b such that f(u)+k is invertible (in b) and k′ be an ideal of c such that g(u) + k′ is invertible (in c). then there exist v1 ∈ b such that (f(u1 + k)v1) = 1 and there exist v2 ∈ c such that (g(u1 + k ′)v2) = 1. hence, (f(u1) + k)(f(u −1 1 ) − v1f(u −1 1 )k) = f(u1)f(u −1 1 ) + kf(u−11 ) − (f(u1) + k)v1f(u −1 1 )k) = 1 + kf(u −1 1 ) − f(u −1 1 )k = 1 and (g(u1) + k′)(g(u−11 )−v2g(u −1 1 )k ′) = g(u1)g(u −1 1 )+k ′g(u−11 )−(g(u1)+k′)v2g(u −1 1 )k ′) = 1 + k′g(u−11 ) − g(u −1 1 )k ′ = 1. thus, (f(u1) + k, g(u1) + k′) is invertible in a ▷◁f,g (k, k′). hence, (f(a) + k, g(a) + k′) = (f(u1 + u2 + ... + un + x) + k, g(u1 + u2 + ... + un + x) + k ′) = (f(u1) + k, g(u1) + k ′) + (f(u2), g(u2)) + ... + (f(un), g(un)) + (f(x), g(x)), where (f(u1) + k, g(u1) + k′) ∈ u(a ∝ m), (f(ui), g(ui) ∈ u(a ∝ m)(i = 2, 3, ..., n) and (f(x), g(x)p = (f(x), g(x)). consequently, a ▷◁f,g (k, k′) is (n, p)-clean ring.2 328 a. aruldoss and c. selvaraj 3 (n, p)-weakly clean ring now we introduce the new class of ring. definition 3.1. a ring a is called (n, p)-weakly clean ring if a = u1 +...+un +x or a = u1 + ... + un − x for some unit ui ∈ a(i = 1, ..., n) and xp = x, where x is p-potent element. if the above representation is unique, we say that a is uniquely (n, p)-weakly clean ring. note that (n, p)-clean ring is weakly (n, p)-clean ring. in this section we study the tranfer of (n, p)-weakly clean ring property to the ring a ▷◁f,g (k, k′) is defined above. we establishes necessary and sufficient conditions for a ▷◁f,g (k, k′) to be (n, p)-weakly clean. proposition 3.1. the class of (n, p)-weakly clean is closed under homomorphic images. proof. the proof is straightforward.2 proposition 3.2. a ring a is called uniquely (n, p)-weakly clean ring if the representation of a (n, p)-weakly clean element in a unique way. our first main result gives a necessary and sufficient conditions for a ▷◁f,g (k, k′) to be (n, p)-weakly clean ring. theorem 3.1. let f : a → b and g : a → c be two ring homomorphisms, let k and k′ be two ideals of b and c respectively such that f−1(k) = g−1(k′) = i0. assume that the following conditions hold: a) a is a (n, p)-weakly clean ring and a/i0 is an uniquely (n, p)-weakly clean ring. b)f(a) + k and g(a) + k′ are (n, p)-weakly clean rings and atmost one of them is not a (n, p)-clean ring. then a ▷◁f,g (k, k′) is a (n, p)-weakly clean ring. proof. without loss of generality, we assume that f(a) + k is a n-weakly clean ring and g(a) + k′ is a n-clean ring. let a ∈ a. then a can be written as a = u1 + ... + un + x or a = u1 + ... + un − x for some ui ∈ a(i = 1, ..., n) and p-potent x ∈ a. since a is (n, p)-weakly clean ring and since f(a) + k is a (n, p)-weakly clean ring, f(a)+k = (f(x1)+k1)+...+(f(xn)+kn)+(f(p)+k∗) or f(a)+k = (f(x1)+k1)+ ...+(f(xn)+kn)−(f(p)+k∗) with f(xi)+ki(i = 1, ..., n) and f(p) + k∗ are respectively units and p-potent element of f(a) + k. therefore, f(a) = f(u1) + ... + f(un) + f(x) or f(a) = f(u1) + ... + f(un) − f(x). then, in f(a) + k/k we have: f(a) = f(u1) + ... + f(un) + f(x) 329 bi-amalgamated algebra with (n, p)-weakly clean like properties or f(a) = f(u1) + ... + f(un) − f(x). it is clear that f(x1) = f(x1) + k1 (resp., f(u1)),...,f(xn) = f(xn) + kn (resp., f(un)) and f(p) = f(p) + k∗ (resp., f(x)), are respectively units and p-potent element of f(a) + k k , and we have f(a) = f(u1) + ... + f(un) + f(x) = f(x1) + ... + f(xn) + f(p) or f(a) = f(u1)+...+f(un)−f(x) = f(x1)+...+f(xn)−f(p). thus, f(u1) = f(x1),..., f(un) = f(xn) and f(x) = f(p) since f(a) + k k is an uniquely (n, p)-weakly clean ring. consider k∗1, ..., k ∗ n, k ∗ l ∈ k such that f(x1) = f(u1) + k ∗ 1,..., f(xn) = f(un) + k ∗ n and f(p) = f(x) + k ∗ l . hence, f(a) + j = (f(u1) + k ∗ 1 + k1) + ... + (f(un)+k ∗ n+kn)+(f(x)+k ∗ l +k ∗) or f(a)+j = (f(u1)+k∗1 +k1)+...+(f(un)+ k∗n + kn) − (f(x) + k∗l + k ∗). thus, using the same technique of the previous part g(a) + k′ = (g(u1) + k ′∗ 1 + k ′ 1) + ... + (g(un) + k ′∗ n + k ′ n) + (g(p) + k ′∗ l + k ′∗) or g(a) + k′ = (g(u1) + k ′∗ 1 + k ′ 1) + ... + (g(un) + k ′∗ n + k ′ n) − (g(p) + k′∗l + k ′∗) since g(a)+k′/k′ ∼= a/i0 is an uniquely n-weakly clean ring. this implies that (f(a)+k, g(a)+k′) = {(f(u1)+k∗1 +k1)+...+(f(un)+k∗n +kn)+(f(x)+k∗l + k∗), (g(u1)+k ′∗ 1 +k ′ 1)+...+(g(un)+k ′∗ n +k ′ n)+(g(x)+k ′∗ l +k ′∗)} = (f(u1)+k∗1 + k1, g(u1)+k ′∗ 1 +k ′ 1)+...+(f(un)+k ∗ n+kn, g(un)+k ′∗ n +k ′ n)+(f(x)+k ∗ l +k ∗, g(x)+ k′∗l + k ′∗). now, the same argument follows from theorem 2.1 in the remaining case, f(a) + j = (f(u1) + k∗1 + k1) + ... + (f(un) + k ∗ n + kn) − (f(x) + k∗l + k ∗). let g(a)+k′ = (g(u1)+k′∗1 +k ′ 1)+...+(g(un)+k ′∗ n +k ′ n)−(g(x)+k′∗l +k ′∗). we have (f(a)+k, g(a)+k′) = {(f(u1)+k∗1 +k1)+...+(f(un)+k∗n +kn)+(f(x)+ k∗l + k ∗), (g(u1) + k ′∗ 1 + k ′ 1) + ... + (g(un) + k ′∗ n + k ′ n) + (g(x) + k ′∗ l + k ′∗)} = (f(u1) + k ∗ 1 + k1, g(u1) + k ′∗ 1 + k ′ 1) + ... + (f(un) + k ∗ n + kn, g(un) + k ′∗ n + k′n) − (f(x) + k∗l + k ∗, g(x) + k′∗l + k ′∗). in all cases, (f(a) + k, g(a) + k′) is a (n, p)-weakly clean elements of a ▷◁f,g (j, j′). this completes the proof. 2 4 conclusions through the above we have studied the characterization for the bi-amalgamation of a with (b, c) along (k, k′) with respect to (f, g) to be a (n, pclean ring along with an example. further, we have studied the necessary and sufficient conditions for a ▷◁f,g (k, k′) to be a (n, p)-weakly clean ring. acknowledgements the first author is partially supported by periyar university research fellowship (letter no: pu/ad-3/urf/015723/2020 dated 16th november 2020). the second author is supported by dst fist (letter no: sr/fst/msi-115/2016 dated 10th november 2017). 330 a. aruldoss and c. selvaraj references a. chen and k. qua. some properties of n-weakly clean rings. aip conference proceedings, 1605, 2014. m. chhiti. on (n, p)-clean commutative rings and n-almost clean rings. palestine journal of mathematics, 7:23–27, 2018. m. d’anna. a construction of gorenstein rings. j. algebra, 306:507–519, 2006. m. d’anna and m. fontana. an amalgamated duplication of a ring along an ideal: the basic properties. journal of algebra and its applications, 6:443–459, 2007. m. d’anna, c. finocchiaro, and m. fontana. amalgamated algebras along an ideal. comm algebra and applications, 306:241–252, 2009. s. kabbaj, k. louartiti, and m. tamekkente. bi-amalgameted algebras along ideals. j. commut. algebra, 9:65–87, 2013. w. nicholson. lifting idempotents and exchange rings. trans. amer. math. soc, 229:269–278, 1977. 331 ratio mathematica volume 43, 2022 a common fixed point theorem for three weakly compatible selfmaps of a s-metric space kiran virivinti* niranjan goud javaji† rajani devi katta‡ abstract fixed point theorems were established by using contractive conditions. in this paper we prove a common fixed point theorem for three weakly compatible selfmaps of a s-metric space by utilizing a contractive condition of rational type.further we deduce a common fixed point theorem for two weakly compatible selfmaps of a s-metric space. keywords: s-metric space; fixed point; weakly compatible mappings; associated sequence of a point relative to three selfmaps. 2020 ams subject classifications: 54h25,47h10. 1 *department of mathematics, osmania university, hyderabad, india; kiranmathou@gmail.com. †department of mathematics, m.v.s government college, mahaboobnagar, telangana, india; jngoud1979@gmail.com. ‡department of mathematics, k.v.r(w) government degree college, kurnool, andhra pradesh, india; dr.rajanidevi@gmail.com. 1received on april 21, 2022. accepted on september 1, 2022. published on october 1, 2022. doi: 10.23755/rm.v42i0.767. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. v. kiran, j. niranjan goud, k. rajani devi 1 introduction fixed point theory is an important branch of non-linear analysis due to its application potential. in proving fixed point theorems, we use completeness, continuity, convergence and various other topological aspects. banach’s contraction principle or banach’s fixed point theorem is one of the most important results in nonlinear analysis. this theorem has been generalized in many directions by generalizing the underlying space or by viewing it as a common fixed point theorem along with other selfmaps. in the past few years, a number of generalizations of metric spaces like g -metric spaces, partial metric spaces and cone metric spaces were initiated. these generalizations were used to extend the scope of the study of fixed point theory. recently one more generalization, namely s -metric spaces, was introduced by sedghi s , shobe n, aliouche.a [2012]. among all generalizations, s-metric spaces evinced a lot of interest in many researchers as they unified, extended, generalized and refined several existing results onto these s -metric spaces. commutativity plays an important role in proving common fixed point theorems. as it is a stronger requirement, sessa [1982] introduced the notion of weakly commuting maps as a generalization of commuting maps. afterwards the idea of compatibility was introduced by g. jungck [1986]. later on jungck and rhoades [1998]introduced the notion of weakly compatibility as a generalization of compatibility. they also proved that compatible mappings are weakly compatible but not conversely. in this paper we establish a common fixed point theorem for three weakly compatible selfmaps of a s-metric space using a contractive condition of rational type. our theorem which is established in the framework of s-metric spaces generalizes the theorem of sumit chandok [2018] which is proved in metric space. now we recall some basic definitions required in the sequel in section 2 and establish main results in section 3. 2 preliminaries we now recollect the essential definitions which are useful for our discussion. definition 2.1. let y be a nonempty set. a function s : y 3 → [0,∞) is said to be s − metric if it satisfies the following conditions for each β1,β2,β3,β4 ∈ y (i) s(β1,β2,β3) ≥ 0, (ii) s(β1,β2,β3) = 0 if and only if β1 = β2 = β3, (iii) s(β1,β2,β3) ≤ s(β1,β1,β4) + s(β2,β2,β4) + s(β3,β3,β4). then (y,s) is said to be a s-metric space. a common fixed point theorem for three weakly compatible selfmaps of a s-metric space example 2.1. let y = r and s : r3 → [0,∞) be defined by s(β1,β2,β3) = |β2 + β3 − 2β1| + |β2 − β3| for β1,β2,β3 ∈ r, then (y,s) is a s-metric space. remark 2.1. it is shown in a s-metric space that s(β1,β1,β2) = s(β2,β2,β1) for all β1,β2 ∈ y . definition 2.2. let (y,s) be an s-metric space. a sequence {tn} in y said to convergent, if there is a t ∈ y such that s(tn, tn, t) → 0; that is for each ϵ > 0, there exists an n0 ∈ n such that for all n ≥ n0, we have s(tn, tn, t) < ϵ and we denote this by lim n→∞ tn = t. definition 2.3. suppose (y,s) is an s-metric space. a sequence {tn} in y is called a cauchy sequence if to each ϵ > 0, there exists n0 ∈ n such that s(tn, tn, t) < ϵ for each n,m ≥ n0 . definition 2.4. let (y,s) be an s-metric space, if there exists sequences {tn} and {un} such that lim n→∞ tn = t and lim n→∞ un = u then lim n→∞ s(tn, tn,un) = s(t, t,u), then we say that s(t,u,v) is continuous in t and u . definition 2.5. suppose ϕ and ψ self maps of a s-metric space (y,s) such that for every sequence {tn} in y with lim n→∞ ψtn = lim n→∞ ϕtn = t for some t ∈ x we have lim n→∞ s(ψϕtn,ψϕtn,ϕψtn) = 0, then ψ and ϕ are called compatible mappings. definition 2.6. in a s-metric space (y,s),two selfmaps ϕ and ψ of y are said to be weakly compatible if ϕψt = ψϕt whenever ϕt = ψt for t ∈ y . definition 2.7. if ψ, µ and ϕ are self maps of a non empty set y such that ψ(y ) ⊆ ϕ(y ), and µ(y ) ⊆ ϕ(y ) then for any t0 ∈ y , if {tn} is a sequence in y such that ϕt2n+1 = ψt2n and ϕt2n+2 = µt2n+1 for n ≥ 1 then {tn} is called an associated sequence of t0 relative to three selfmaps ψ, µ and ϕ. 3 main theorem we now state our main theorem of the section. theorem 3.1. let p be a subset of a s-metric space (y,s), ψ,µ and ϕ are three selfmaps of p such that (i) ψ(y ) ∪ µ(y ) ⊆ ϕ(y ) and (ϕ(p),s) is complete. v. kiran, j. niranjan goud, k. rajani devi (ii) s(µy1,µy1,ψy2) ≤ k1{ s(ϕy1,ϕy1,µy1).s(ϕy2,ϕy2,ψy2) s(ϕy1,ϕy1,ϕy2) + s(ϕy1,ϕy1,ψy2) + s(ϕy2,ψy2,µy1) } + k2s(ϕy1,ϕy1,ϕy2) for every y1,y2 ∈ p and k1,k2 ∈ [0,1) with 2k1 + k2 < 1. (iii) the pairs (ϕ,ψ) and (ϕ,µ) are weakly compatible. then ψ,µ and ϕ have a unique common fixed point. proof. let t0 be a point in y. since ψ(y )∪µ(y ) ⊆ ϕ(y ), we obtain an associated sequence {tn} in y such that ϕt2n+1 = µt2n,ϕt2n+2 = ψt2n+1. from the condition (ii) of theorem 3.1 we have, s(ϕt2n+1,ϕt2n+1,ϕt2n+2) = s(µt2n,µt2n,ψt2n+1) ≤ k1 [ s(ϕt2n,ϕt2n,µt2n).s(ϕt2n+1,ϕt2n+1,ψt2n+1) s(ϕt2n,ϕt2n,ϕt2n+1) + s(ϕt2n,ϕt2n,ψt2n+1) + s(ϕt2n+1,ϕt2n+1,µt2n) ] + k2s(ϕt2n,ϕt2n,ϕt2n+1) ≤ k1 [ s(ϕt2n,ϕt2n,ϕt2n+1).s(ϕt2n+1,ϕt2n+1,ϕt2n+2) s(ϕt2n,ϕt2n,ϕt2n+1) + s(ϕt2n,ϕt2n,ϕt2n+2) + s(ϕt2n+1,ϕt2n+1,ϕt2n+1) ] + k2s(ϕt2n,ϕt2n,ϕt2n+1) ≤ 2k1 [ s(ϕt2n,ϕt2n,ϕt2n+1).s(ϕt2n+1,ϕt2n+1,ϕt2n+2) s(ϕt2n+1,ϕt2n+1,ϕt2n+1) ] + k2s(ϕt2n,ϕt2n,ϕt2n+1) ≤ (2k1 + k2) s(ϕt2n,ϕt2n,ϕt2n+1). similarly, we can prove s(ϕt2n,ϕt2n,ϕt2n+1) ≤ (2k1 + k2)s(ϕt2n−1,ϕt2n−1,ϕt2n) therefore s(ϕtn,ϕtn,ϕtn+1) ≤ (2k1 + k2) s(ϕtn−1,ϕtn−1,ϕtn) ≤ (2k1 + k2)2 s(ϕtn−2,ϕtn−2,ϕtn−1) ≤ (2k1 + k2)3 s(ϕtn−3,ϕtn−3,ϕtn−2) · · · · · · · · · · · · · · · · · · ≤ (2k1 + k2)n s(ϕt0,ϕt0,ϕt1) → 0, a common fixed point theorem for three weakly compatible selfmaps of a s-metric space since (2k1 + k2)n → 0 as n → ∞. now we claim that {ϕtn} is a cauchy sequence. for any m,n ∈ n such that m > n we have, s(ϕtn,ϕtn,ϕtm) ≤ 2[s(ϕtn,ϕtn,ϕtn+1) + s(ϕtn+1,ϕtn+1,ϕtn+2) + · · · + s(ϕtm−1,ϕtm−1,ϕtm)] ≤ 2cns(ϕt0,ϕt0,ϕt1) + cn+1s(ϕt0,ϕt0,ϕt1) + · · · + cms(ϕt0,ϕt0,ϕt1) ≤ 2cn(1 + c + c2 + · · · + cm−n)s(ϕt0,ϕt0,ϕt1) ≤ 2cn 1 − cm−n 1 − c s(ϕt0,ϕt0,ϕt1) ≤ 2 cn 1 − c s(ϕt0,ϕt0,ϕt1) → 0, since c < 1 then cn → 0 as n → ∞. therefore {ϕtn} is a cauchy sequence in x. since (ϕ(p),s) is complete, there is a t ∈ p such that ϕtn → ϕt as n → ∞. we now prove that t is a point of coincidence of µ,ψ and ϕ. from the condition (ii) of theorem 3.1 we have, s(ϕt2n+1,ϕt2n+1,ψt) = s(µt2n,µt2n,ψt) ≤ k1 [ s(ϕt2n,ϕt2n,µt2n).s(ϕt,ϕt,ψt) s(ϕt2n,ϕt2n,ϕt) + s(ϕt2n,ϕt2n,ψt) + s(ϕt,ϕt,µt2n) ] + k2s(ϕt2n,ϕt2n,ϕt) ≤ k1 [ s(ϕt2n,ϕt2n,ϕt2n+1).s(ϕt,ϕt,ψt) s(ϕt2n,ϕt2n,ϕt) + s(ϕt2n,ϕt2n,ψt) + s(ϕt,ϕt,ϕt2n+1) ] + k2s(ϕt2n,ϕt2n,ϕt), which gives s(ϕt,ϕt,ψt) = 0 as n → ∞ and hence ϕt = ψt. also we have, s(µt,µt,ϕt) = s(µt,µt,ψt) ≤ k1 [ s(ϕt,ϕt,µt).s(ϕt,ϕt,ψt) s(ϕt,ϕt,ϕt) + s(ϕt,ϕt,ψt) + s(ϕt,ϕt,µt) ] + k2s(ϕt,ϕt,ϕt), which implies s(µt,µt,ϕt) = 0 proving µt = ϕt. therefore we have ϕt = ψt = µt = a(say), v. kiran, j. niranjan goud, k. rajani devi proving that t is a coincident point of µ,ψ and ϕ. since the pairs (ϕ,µ) and (ϕ,ψ) are weakly compatible, we have ϕµt = µϕt and ϕψt = ψϕt which implies ϕa = ψa = µa. now we have, s(ϕa,ϕa,a) = s(µa,µa,ψt) ≤ k1 [ s(ϕa,ϕa,µa).s(ϕt,ϕt,ψt) s(ϕa,ϕa,ϕt) + s(ϕa,ϕa,µt) + s(ϕt,ϕt,µa) ] + k2s(ϕa,ϕa,ϕt) ≤ k1 [ s(ϕa,ϕa,ϕa).s(a,a,a) s(ϕa,ϕa,a) + s(ϕa,ϕa,a) + s(a,a,ϕa) ] + k2s(ϕa,ϕa,a) ≤ k2 s(ϕa,ϕa,a), leading to a contradiction, giving that s(ϕa,ϕa,a) = 0 implies ϕa = a. hence ϕa = ψa = µa = a, showing that a is a common fixed point of µ,ψ and ϕ. we now prove that the common fixed point is unique. suppose a′(̸= a) is another common fixed point of µ,ψ and ϕ. that is a′ = ϕa′ = ψa′ = µa′. we have s(a,a,a′) = s(µa,µa,ψa′) ≤ k1 [ s(ϕa,ϕa,µa).s(ϕa′,ϕa′,ψa′) s(ϕa,ϕa,ϕa′) + s(ϕa,ϕa,ψa′) + s(ϕa′,ϕa′,µa) ] + k2s(ϕa,ϕa,ϕa ′) ≤ k1 [ s(a,a,a).s(a′,a′,a′) s(a,a,a′) + s(a,a,a′) + s(a,a,a′) ] + k2s(a,a,a ′) ≤ k2s(a,a,a′), which is a contradiction since k2 < 1. therefore s(a,a,a′) = 0 implies a = a′, proving the uniqueness. corolary 3.1. let p be a subset of a s-metric space (y,s). suppose that ϕ,µ are two selfmaps of p satisfy (i) µ(y ) ⊆ ϕ(y ) and (ϕ(p),s) is complete. (ii) s(µy1,µy1,µy2) ≤ k1{ s(ϕy1,ϕy1,µy1).s(ϕy2,ϕy2,µy2) s(ϕy1,πy1,ϕy2) + s(ϕy1,ϕy1,µy2) + s(ϕy2,ϕy2,µy1) } + k2s(ϕy1,ϕy1,ϕy2) a common fixed point theorem for three weakly compatible selfmaps of a s-metric space for every y1,y2 ∈ p and k1,k2 ∈ [0,1) with 2k1 + k2 < 1. (iii) the pair (µ,ϕ) is weakly compatible. then µ and ϕ have a unique common fixed point. proof. on taking ψ = µ in theorem 3.1, the corollary follows. 4 conclusion in this paper, a common fixed theorem for three weakly compatible selfmaps of a s-metric space is established with the aid of an associated sequence of three selfmaps.moreover, we deduce a common fixed point theorem for two selfmaps. as s-metric space is a robust generalization of metric space, our theorem generalizes the theorem in literature. references n. aliouche.a, sedghi. a generalization of fixed point theorems in s-metric spaces. mat. vesnik, 3:258–266, 2012. s. chandok. common fixed theorem for generalized contractions mappings. thai journal of mathematics, 16:305–314, 2018. jungck and rhoades. fixed point for set valued functions with out continuity. indian j pure appl.math.sci, 29:227–238, 1998. g. jungck. compatible mappings and common fixed points. int j of math and math.sci, 4:771–779, 1986. sessa. on a weak commutativity condition of mappings in a fixed point considerations,. publ.l’institut math, 62:149–153, 1982. ratio mathematica volume 44, 2022 soft igδs-closed functions y. rosemathy 1 dr. k. alli 2 abstract in this paper, we have introduced a new class of open and closed functions called soft igδs-closed and soft igδs-open functions in ideal topological spaces and also investigated some of its characterizations and properties with the existing sets. key words and phrases. soft sets, soft topological spaces, soft regular open, soft δ-cluster point, soft igδs-closed functions, soft strongly igδs-closed functions. mathematics subject classification. 54a10, 54a20, 54c08 3 . 1 research scholar (reg.no-18111072092002) the m.d.t hindu college, tirunelveli-627010, tamilnadu, india. (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli627012, tamil nadu, india). email: ravimathy18@gmail.com 2 assistant professor, department of mathematics, the m.d.t hindu college, tirunelveli-627010, tamil nadu, india. (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india) 3 received on june 26 th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.912. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by license agreement 246 mailto:ravimathy18@gmail.com y. rosemathy, dr. k. alli 1. introduction the concept of soft sets was first introduced by molodtsov [12] in 1999 as a general mathematical tool for dealing with uncertain objects. in [12, 13], molodtsov successfully applied the soft theory in several directions, such as smoothness of functions, game theory, operations research, riemann integration, perron integration, probability, theory of measurement, and so on. after presentation of the operations of soft sets [11], the properties and applications of soft set theory have been studied increasingly [3, 8, 13]. in [14] o.ravi et all decompositions of ï g-continuity via idealization and in recent years, many interesting applications of soft set theory have been expanded by embedding the ideas of fuzzy sets [1, 2, 4, 9, 10, 11, 13]. to develop soft set theory, the operations of the soft sets are redefined and a uni-int decision making method was constructed by using these new operations [5]. recently, in 2011, shabir and naz [15] initiated the study of soft topological spaces. they defined soft topology on the collection τ of soft sets over x. consequently, they defined basic notions of soft topological spaces such as soft open and soft closed sets, soft subspace, soft interior, soft closure, soft neighborhood of a point, soft separation axioms, soft regular spaces and soft normal spaces and established their several properties. hussain and ahmad [6] investigated the properties of soft open, soft closed, soft interior, soft closure, soft neighborhood of a point. they also defined and discussed the properties of soft interior, soft exterior and soft boundary which are fundamental for further research on soft topology and will strengthen the foundations of the theory of soft topological spaces. in [16] s. tharmar and r. senthilkumar introduced soft locally closed sets in soft ideal topological spaces. in this paper, we have introduced a new class of open and closed functions called soft igδs-closed and soft igδs-open functions in ideal topological spaces and also investigated some of its characterizations and properties with the existing sets. 2. preliminaries in this section, we present some basic definitions and results which are needed in further study of this paper which may found in earlier studies. throughout this paper, x refers to an initial universe, e is a set of parameters, ℘(x) is the power set of x, and a⊂ e definition 2.1. [12] a soft set fa over the universe x is defined by the set of ordered pairs fa={(e, fa(e)) : e ∈ e, fa(e)∈℘(x)} where fa : e→℘(x), such that f a ( e ) = ∅, if e∈a⊂e and fa(e)=∅ if e∈/a . the family of all soft sets over x is denoted by ss(x). 247 https://dergipark.org.tr/en/download/article-file/105177 soft igδs-closed functions a a definition 2.2. [11] the soft set f∅ over a common universe set x is said to be null soft set, denoted by ∅. here f∅(e)=∅, ∀e∈e. definition 2.3. [11] a soft set fa over x is called an absolute soft set, denoted by ã , if e∈a, fa(e)=x. definition 2.4. [11] let fa, gb be soft sets over a common universe set x. then fa is a soft subset of gb, denoted fa⊂gb if fa(e)⊂gb(e), ∀e∈e. definition 2.5. [11] let fa, gb be soft sets over a common universe set x. the union of fa and gb, is a soft set hc defined by hc(e)=fa(e)∪gb(e), ∀e∈e, where c=a∪b. that is, hc=fa∪gb. definition 2.6. [11] let fa, gb be soft sets over a common universe set x. the intersection of fa and gb, is a soft set hc defined by hc(e)=fa(e)∩gb(e), ∀e∈e, where c=a∩b. that is, hc=fa∩gb. definition 2.7. [16] the complement of the soft set fa over x, denoted by f c is defined by a c (e)=x−fa(e), ∀e∈e. definition 2.8. [16] let fa be a soft set over x and x∈x. we say that x∈fa if x∈fa(e), ∀e∈a. for any x∈x, x∈/ f a if x∈/ fa (e) for some e∈a. definition 2.9. [20] the soft set fa∈ss(x) is called a soft point in ss(x) if there exist x∈x and e∈e such that f(e)={x} and f(e c )=∅ for each e c ∈e−{e} and the soft point fa is denoted by xε. definition 2.10. [16] a soft topology τ is a family of soft sets over x satisfying the following properties. (1) ∅, x̃ belong to τ . (2) the union of any number of soft sets in τ belongs to τ . (3) the intersection of any two soft sets in τ belongs to τ . the triplet (x, τ , e) is called a soft topological space. definition 2.11. [15] let (x, τ , e) be a soft topological space over x. then (1) the members of τ are called soft open sets in x. (2) a soft set fa over x is said to be a soft closed set in x if f c ∈τ . (3) a soft set fa is said to be a soft neighborhood of a point x∈x if x∈fa and fa is soft open in (x, τ , e) (4) the soft interior of a soft set fa is the union of all soft open subsets of fa. the 248 y. rosemathy, dr. k. alli soft interior of fa is denoted by int(fa). (5) the soft closure of fa is the intersection of all soft closed super sets of fa. the soft closure of fa is denoted by cl(fa) or fa. definition 2.12. [18] a soft set fa in a soft topological space (x, τ , e) is said to be a soft regular open (resp. soft regular closed) if fa=int(cl(fa)) (resp. fa=cl(int(fa))). definition 2.13. let i be a non-null collection of soft sets over a universe x with the same set of parameters e. then i⊂ss(x) is called a soft ideal on x with the same set e if (1) fa∈i and ga∈i⇒fa∪ga∈i. (2) fa∈i and ga⊂fa⇒ga∈i. definition 2.14. let (x, τ , e) be a soft topological space and i be a soft ideal over x with the same set of parameters e. then f∗a=∪{xe∈x : oxe ∩ f a ∈/ i , for all oxe ∈τ} is called the soft local function of fa with respect to i and τ , where oxe is a τ -open set containing xe. theorem 2.15. let i and j be any two soft ideals with the same set of parameters e on a soft topological space (x, τ , e). let fa, ga∈ss(x). then (1) (∅)∗=∅. (2) fa⊂ga⇒ f∗a⊂g ∗ a. (3) i⊂j ⇒ f∗a(j )⊂fa ∗(i). (4) f∗a⊂cl(fa), where cl is the soft closure w.r.t τ . (5) f∗a is τ -closed soft set. (6) (f∗a) ∗ ⊂ f∗a. (7) (fa∪ga) ∗=f∗a∪ga ∗. definition 2.16. let (x, τ , e) be a soft topological space, i be a soft ideal over x with the same set of parameters e and cl∗ : ss(x) →ss(x) be the soft closure operator. then there exists a unique soft topology over x with the same set of parameters e, finer than τ , called the ⋆-soft topology, defined by τ∗, given by τ∗={ fa∈ss(x) : cl∗(x−fa)=x−fa}. definition 2.17. [7] let fa be a soft subset of soft topological space (x, τ , e). then (1) xε is called a soft δ-cluster point of fa if fa∩int(cl(ua))= ∅ for every soft open set ua containing xε. (2) the family of all soft δ-cluster point of fa is called the soft δ-closure of fa and is denoted by clδ(fa). (3) a soft subset fa is said to be soft δ-closed if clδ(fa)=fa. the complement of a soft δ-closed set of x is said to be soft δ-open. 249 soft igδs-closed functions lemma 2.18. [7] let fa be a soft subset of soft topological space (x, τ , e). then, the following properties hold: (1) int (cl (fa)) is soft regular open, (2) every soft regular open set is soft δ-open, (3) every soft δ-open set is the union of a family of soft regular open sets. (4) every soft δ-open set is soft open. proposition 2.19. [7] intersection of two soft regular open sets is soft regular open. lemma 2.20. [7] let fa and ga be soft subsets of soft topological space (x, τ , e). then, the following properties hold. (1) fa ⊂ clδ(fa), (2) if fa⊂ga, then clδ(fa)⊂clδ(ga), (3) clδ(fa)=∩{ga∈ss(x): fa⊂ga and ga is soft δ-closed}, (4) if (fa)α is a soft δ-closed set of x for each α∈△, then ∩{(fa)α: α∈△} is soft δclosed, (5) clδ(fa) is soft δ-closed. theorem 2.21. [7] let (x, τ , e) be a soft topological space and τδ={fa∈ss(x) : fa is a soft δ-open set}. then τδ is a soft topology weaker than τ . definition 2.22. a soft subset fa of a soft ideal topological space (x, τ , e, i) is said to be (1) soft pre-i-open if fa⊂int (cl (fa)), (2) soft semi-i-open if faa ⊂ cl (int (fa)), (3) soft α-i-open if fa⊂int (cl (int (fa))). the complement of soft pre-i-open (resp. soft semi-i-open, soft α-i-open) set is called a soft pre-i-closed (resp. soft semi-i-closed, soft α-i-closed). definition 2.23. the soft semi-i-closure of fa is defined by the intersection of all soft semiiclosed sets containing fa and is denoted by siscl-(fa) definition 2.24. a soft set fa of soft ideal topological space x is called soft generalized δ semi closed (briefly soft igδs-closed) set if siscl (fa) ⊂ ga whenever fa ⊂ ga and ga are soft δopen over x. a soft set fa of x is called soft generalized δ semi-open (briefly soft sigδs-open) set if fa c is soft sigδs-closed. the family of all soft sigδs-closed subsets of the space x is denoted by sigδs-c(x) and soft sigδs-open subsets of the space x is denoted by sigδs-o(x). 3. soft sigδs-closed and soft sigδs-open functions definition 3.1. a function f: (x, τ, e) → (y, σ, k, i) is said to be soft sigδs-closed (resp. soft 250 y. rosemathy, dr. k. alli sigδs-open) if f (va) is soft sigδs-closed (resp. sost sigδs-open) over y for every soft closed (resp. soft open) set va over x. definition 3.2. (1) a function f: (x, τ, e, i) → (y, σ, k, j) is soft igδs-irresolute if f −1 (va) is soft sigδs-closed over x for every soft sigδs-closed set va over y. (2) a function f: (x, τ, e, i) → (y, k, σ) is soft sigδs-continuous if f −1 (va) is soft sigδs closed over x for every soft closed set va over y. theorem 3.3. a function f: (x, τ, e) → (y, σ, k, i) is soft sigδs-closed if and only if f (va) is soft sigδs-open over y for every soft open set va over x. proof: suppose f: (x, τ, e) → (y, σ, k, i) is soft sigδs-closed function and va is a soft open set over x. then x̃ − va is soft closed over x. by hypothesis f ( x̃ − va) = ỹ − f (va) is a soft sigδs-closed set over y and hence f (va) is soft sigδs-open set over y. on the other hand, if fa is soft closed set over x, then x̃ − fa is a soft open set over x. by hypothesis f ( x̃ −fa) = ỹ −f (fa) is soft sigδs-open set over y, implies f (fa) is soft sigδs-closed set over y. therefore, f is soft sigδs-closed function. definition 3.4. a soft ideal topological space x is said to be soft tigδs-space if every soft sigδs closed set is soft closed over x. definition 3.5. a soft ideal topological space x is said to be soft sigδs-t2 space if every soft sigδs-closed set is soft semi-closed over x. theorem 3.6. if f: (x, τ, e) → (y, σ, k, i) is soft sigδs-closed function and y is soft tigδsspace, then f is a soft closed function. proof: let va be a soft closed set over x. since f is a soft sigδs-closed function, implies f (va) is soft sigδs-closed over y. now y is soft tigδs-space, implies f (va) is a soft closed set over y. therefore, f is a soft closed function. theorem 3.7. if f: (x, τ, e) → (y, σ, k, i) is soft sigδs-closed function and y is soft sigδst2 space, then f is soft semi-closed function. proof: let va be a soft closed set over x. since f is a soft sigδs-closed function, f (va) soft is sigδs-closed set over y. now y is soft sigδs-t2 space, implies f (va) is a soft semi-closed set over y. therefore, f is a soft semi-closed function. theorem 3.8. for the function f: (x, τ, e) → (y, σ, k, i), the following statements are equivalent. (1) f is a soft sigδs-open function. (2) for each soft subset fa of x, f (int (fa)) ⊂ sigδs – int (f (fa)) (3) for each xe ∈ x̃ , the image of every soft nhd of xe is soft sigδs-nhd of f (xe). proof: (1) → (2) suppose (1) holds and fa ⊂ x. then int (fa) is a soft open set over x. by (1), f (int (fa)) is a soft sigδs-open set over y. 251 soft igδs-closed functions therefore sigδs – int (f (int (fa))) = f (int (fa)). since f (int (fa)) ⊂ f (fa), implies sigδs – int (f (int (fa))) ⊂ sigδs – int (f (fa)). that is f (int (fa)) ⊂ sigδs – int (f (fa)). (2) → (3) suppose (2) holds. let xe ∈ x̃ and fa be an arbitrary soft nhd of xe over x. then there exists a soft open set ha in x such that xe ∈ ha ⊂ fa. by (2), f (ha) = f (int (ha)) ⊂ sigδs −int (f (ha)). but sigδs −int (f (ha)) ⊂ f (ha) is always true. therefore, f (ha) = sigδs−int (f (ha)) and hence f (ha) is soft sigδsopen set over y. further f (xe) ∈ f (ha) ⊂ f (fa), this implies, f (fa) is a soft sigδs-nhd of f (xe) over y. hence (3) holds. (3) → (1) suppose (3) holds. let va be any soft open set over x and xe ∈ va. then ye = f (xe) ∈ f (va). by (3), for each ye ∈ f (va), there exists a soft sigδs-nhd (za)ye of ye over y. since (za)ye is a soft sigδs-nhd of ye, there exists a soft sigδs-open set (va)ye in va such that ye ∈ (va)ye ⊂ (za)ye. therefore f (va) = ∪{(va)ye: ye ∈ f (va)}, which is union of soft sigδs-open sets and hence soft sigδs-open set over y. therefore, f is soft sigδs-open function. theorem 3.9. a function f: (x, τ, e) → (y, σ, k, i) is soft sigδs-closed if and only if for each soft subset ha over y and for each soft open set ua over x containing f −1 (ha), there exists a soft sigδs-open set va over y such that ha ⊂ va and f −1 (va) ⊂ ua. proof: assume that f is soft sigδs-closed function. let ha ⊂ y and ua be a soft open set over x containing f −1 (ha). since f is a soft sigδs-closed function and x̃ − ua is soft closed over x, implies f ( x̃ − ua) is soft sigδs-closed set over y. then va = ỹ − f ( x̃ − ua) is soft sigδs-open set over y such that ha ⊂ va and f −1 (va) ⊂ ua. conversely, let fa be a soft closed set over x, then x̃ − fa is a soft open set over x and f −1 (ỹ − f (fa)) ⊂ x̃ − fa. by hypothesis, there is a soft sigδs-open set va over y such that ỹ −f (fa) ⊂ va and f −1 (va) ⊂ x˜ −fa. therefore, ỹ −va ⊂ f (fa) ⊂ f ( x̃ −f −1 (va)) ⊂ ỹ −va, this implies f (fa) = ỹ − va. since va is a soft sigδs-open set over y and so f (fa) is soft sigδsclosed over y. hence f is soft sigδs-closed function. theorem 3.10. if f: (x, τ, e) → (y, σ, k, i) is soft sigδs-closed, then for each soft sigδs-closed set ha over y and each soft open set ga over x containing f −1 (ha), there exists soft sigδs-open set va containing ha such that f −1 (va) ⊂ ua. proof: suppose f: (x, τ, e) → (y, σ, k, i) is soft sigδs-closed function. let ha be any soft sigδs-closed set over y and ua is a soft open set over x containing f −1 (ha), by theorem 3.9, there exists a soft sigδs-open set ga over y such that ha ⊂ ga and f −1 (ga) ⊂ ua. since ha is soft sigδs-closed set and ga is soft sigδs-open set containing ha implies ha ⊂ igδs-int (ga). put va = igδs-int (ga), then ha ⊂ va and va are soft sigδs-open set over y and f −1 (va) ⊂ ua. 252 y. rosemathy, dr. k. alli theorem 3.11. a function f: (x, τ, e) → (y, σ, k, i) is soft sigδs-closed, if and only if sigδscl(f(fa)) ⊂ f (cl (fa)), for every soft subset fa over x. proof: suppose f: (x, τ, e) → (y, σ, k, i) is a soft sigδs-closed and fa ⊂ x. then f (cl (fa)) is soft sigδs-closed over y. since f (fa) ⊂ f (cl (fa)), implies sigδs-cl(f(fa)) ⊂ sigδs-cl (f (cl (fa))) = f (cl (fa)). hence sigδs-cl(f(fa)) ⊂ f (cl (fa)). conversely, let fa be any soft closed set over x. then cl (fa) = fa. therefore, f (fa) = f (cl (fa)). by hypothesis, sigδs-cl(f(fa)) ⊂ f (cl (fa)) = f (fa) implies sigδs-cl(f(fa)) ⊂ f (fa). but f (fa) ⊂ sigδs-cl(f(fa)) is always true. this shows, f (fa) = sigδs-cl(f(fa)). therefore f (fa) is soft sigδs-closed set over y and hence f is soft sigδs-closed. theorem 3.12. let f : (x, τ, e) → (y, σ, k, i) and g : (y, σ, k, i) → (z, µ, l, j ) be any two functions. then (g ◦ f ) : (x, τ, e) → (z, µ, l, j ) is soft sigδs-closed function if f and g satisfy one of the following conditions (1) f , g are soft sigδs-closed functions and y is soft tigδs-space. (2) f is soft closed and g is soft sigδs-closed function. proof: (1) suppose fa is soft closed set over x. since f is soft sigδs-closed function f (fa) is soft sigδs-closed set over y. now y is soft tigδs-space, implies f (fa) is soft closed set over y. also, g is soft sigδs-closed function, implies g (f (fa)) = (g ◦ f) (fa) is soft sigδs-closed set over z. hence (g ◦ f) is soft sigδs-closed function. (2) suppose fa is soft closed set over x. since f is soft closed function f (fa) is soft closed set over y. now g is soft sigδs-closed function, implies g(f (fa)) = (g ◦ f )(fa) is soft sigδs-closed set over z. hence (g ◦ f ) is soft sigδs-closed function. theorem 3.13. let f: (x, τ, e) → (y, σ, k, i) and g : (y, σ, k, i) → (z, µ, l, j ) be any two functions such that (g ◦ f ) : x → z be soft sigδs-closed function. then following results hold (1) if f is soft continuous surjection, then g is soft sigδs-closed function. (2) if g is soft sigδs-irresolute and injective, then f is soft sigδs-closed function. proof: (1) suppose fa is a soft closed set over y. since f is soft continuous and surjective, f −1 (fa) is a soft closed set over x. therefore, (g ◦ f) (f −1 (fa)) = g(fa) is soft sigδs-closed set over z and hence g is soft sigδs-closed function. (2) suppose ha is soft closed set over x. then (g ◦ f) (ha) is soft sigδs-closed set over z. since g is soft sigδs-irresolute, g −1 ((g ◦ f) (ha)) = f (ha) is soft sigδs-closed set over y. hence f is soft sigδs-closed function. theorem 3.14. for any bijection f: (x, τ, e) → (y, σ, k, i), the following statements are equivalent: (1) f −1 is soft sigδs-continuous. (2) f is a soft sigδs-open function. 253 soft igδs-closed functions (3) f is a soft sigδs-closed function. proof: (1) → (2) suppose fa is a soft open set over x, then by (1), (f −1 ) −1 (fa) = f (fa) is soft sigδs-open set over y and hence f is soft sigδs-open function. (2) → (3) suppose fa is a soft closed set over x, then x̃ − fa is a soft open set over x. by (2), f ( x̃ − fa) = ỹ − f (fa) is soft sigδs-open over y, implies f (fa) is soft sigδs-closed over y. therefore, f is soft sigδs-closed function. (3) → (1) let fa be a soft closed set over x. by (3), f (fa) = (f −1 ) −1 (fa) is soft sigδs-closed over y. therefore f −1 is soft sigδs continuous function. 4. soft sipgδs-closed functions definition 4.1. a function f: (x, τ, e) → (y, σ, k, i) is said to be soft sipgδs closed (resp. soft sipgδs open) if f (va) is soft sigδs-closed (resp. soft sigδs-open) over y for every soft semiclosed (resp. soft semi-open) set va over x. definition 4.2. (1) a function f: (x, τ, e) → (y, σ, k, i) is said to be soft semi-closed if f(va) is soft semi-closed over y for every soft semi-closed set va over x. (2) a function f: (x, τ, e) → (y, σ, k) is said to be soft pre-closed if f(va) is soft closed over y for every soft semi-closed set va over x. (3) a function f: (x, τ, e) → (y, σ, k) is said to be soft δ-continuous if f −1 (va) is soft δ-closed over x for every soft δ-closed set va over y. theorem 4.3. a function f: (x, τ, e) → (y, σ, k, i) is soft sipgδs-closed if and only if f (va) is soft sigδs-open over y for every soft semi-open set va over x. proof: similar to the proof of theorem 3.3. remark 4.4. every semi-open function is igδs-open function. theorem 4.5. if f: (x, τ, e) → (y, σ, k, i) is soft sipgδs-closed function and y is soft igδst1/2 space, then f is soft semi closed function. proof: suppose va is a soft semi-closed set over x. since f is a soft sipgδsclosed function f (va) is soft sigδs-closed set over y. now y is soft igδs-t1/2 space f (va) is a soft semi-closed set over y. therefore, f is a soft semi-closed function. theorem 4.6. a function f: (x, τ, e) → (y, σ, k, i) is soft sipgδs-closed if and only if for each soft subset ha over y and for each soft semi-open set ua over x containing f −1 (ha), there exists 254 y. rosemathy, dr. k. alli a soft sigδs-open set va over y such that ha ⊂ va and f −1 (va) ⊂ ua. proof: similar to the proof of theorem 3.9. theorem 4.7. if f: (x, τ, e) → (y, σ, k, i) is soft sipgδs-closed, then for each soft sigδs-closed set ha over y and each soft semi-open set ga over x containing f −1 (ha), there exists soft sigδs-open set va over y containing ha such that f −1 (va) ⊂ ua. proof: similar to the proof of theorem 3.10. theorem 4.8. if f is soft δ-continuous, soft sipgδs-closed, then f (ha) is soft sigδs-closed over y for each soft sigδs-closed ha over x, with x is soft igδs-t1/2 space. proof: suppose ha is any soft sigδs-closed set over x and va is a soft δ-open set over y containing f (ha). this implies ha ⊂ f −1 (va). since f is soft δcontinuous, f −1 (va) is a soft δopen set containing ha, therefore, sigδs-cl (ha) ⊂ f −1 (va) and hence f (igδs-cl (ha)) ⊂ va. since f is soft sipgδs-closed, implies f (igδs-cl (ha)) is soft sigδs-closed set contained over y, implies sigδs-cl (f(sigδs-cl (ha))) ⊂ va. thus, sigδs-cl(f(ha)) ⊂ sigδs-cl (f (sigδs-cl (ha))) ⊂ va. that is, sigδs-cl(f(ha)) ⊂ va. this shows that f (ha) is soft sigδs-closed over y. theorem 4.9. let f: (x, τ, e) → (y, σ, k, i) and g: (y, σ, k, i) → (z, µ, l, j) be any two functions. then (g ◦ f): x → z is soft sipgδs-closed function if f and g satisfy one of the following conditions: (1) f , g are soft sipgδs-closed functions and y is soft igδs-t1/2 space. (2) f is soft pre-closed and g is soft sigδs-closed function. (3) f is soft semi-closed and g is soft sipgδs-closed function. (4) f is soft sipgδs-closed function and g is soft δ-continuous, soft sipgδs-closed function and y is soft igδs-t1/2-space. proof: (1) suppose fa is soft semi-closed set over x. since f is soft sipgδsclosed function f (fa) is soft sipgδs-closed set over y. now y is soft igδs-t1/2-space, therefore f (fa) is soft semi closed set over y. also g is soft sipgδs-closed function, implies g(f (fa)) = (g ◦ f )(fa) is soft sigδs-closed set over z. hence (g ◦ f) is soft sipgδs-closed function. (2) suppose fa is soft semi-closed set over x. since f is soft pre-closed, f (fa) is soft closed set over y. now g is soft sigδs-closed function, implies g(f (fa)) = (g ◦ f )(fa) is soft sigδs-closed set over z. hence (g ◦ f) is soft sipgδs-closed function. (3) suppose fa is soft semi-closed set over x. since f is soft semi-closed function, f (fa) is soft semi-closed set over y. now g is soft sipgδs-closed function, implies g (f (fa)) = (g ◦ f) (fa) is soft sigδs-closed set over z. hence (g ◦ f) is soft sipgδs-closed function. (4) suppose ha is a soft semi-closed set over x. since f is soft sipgδs-closed function f (ha) is soft sigδs-closed set over y. since g is soft δ-continuous, soft sipgδs-closed function by theorem 4.8, g (f (ha)) = (g ◦ f) (ha) is soft sigδs-closed set over z. hence (g ◦ f) is soft sipgδs-closed function. 255 soft igδs-closed functions 5. strongly soft sigδs-closed and soft quasi sigδsclosed functions definition 5.1. a function f: (x, τ, e) → (y, σ, k, i) is said to be strongly soft sigδsclosed (resp. strongly soft sigδs-open), if f (fa) is soft sigδs-closed (resp. soft sigδs-open) set over y for every soft sigδs-closed (resp. soft sigδs-open) set fa over x. remark 5.2. every strongly soft sigδs-closed function is soft sigδs-closed function. theorem 5.3. a surjective function f: (x, τ, e) → (y, σ, k, i) is strongly soft sigδsclosed (resp. strongly soft sigδs-open), if and only if for any soft subset ga over va and each soft sigδsopen (resp. soft sigδs-closed) set ua over x containing f −1 (ga), there exists a soft sigδsopen (resp. soft sigδs-closed) set va over y containing ga and f −1 (va) ⊂ ua. proof: similar to the proof of theorem 3.9. theorem 5.4. if a function f: (x, τ, e) → (y, σ, k, i) is a strongly soft sigδs closed function, then for each soft sigδs-closed set ha over y and each soft sigδs-open set ua over x containing f −1 (ha), there exists soft sigδs-open set va over y containing ha such that f −1 (va) ⊂ ua. proof: similar to the proof of theorem 3.10. theorem 5.5. a function f: (x, τ, e) → (y, σ, k, i) is strongly soft sigδs-closed, if and only if sigδs – cl (f (fa)) ⊂ f (sigδs-cl (fa)) for every soft subset fa over x. proof: let f be strongly soft sigδs-closed function and fa ⊂ x. then f (sigδs-cl (fa)) is soft sigδs-closed over y. since f (fa) ⊂ f (sigδs-cl (fa)), implies sigδscl(f(fa)) ⊂ sigδs-cl (f (sigδscl (fa))) = f (sigδs-cl (fa)). therefore, sigδscl(f(fa)) ⊂ f (sigδs-cl (fa)). conversely, fa is any soft sigδs-closed set over x. then sigδs-cl (fa) = fa, implies, f (fa) = f (sigδs-cl (fa)). by hypothesis, sigδs-cl(f(fa)) ⊂ f (sigδs-cl (fa)) = f (fa). hence sigδscl(f(fa)) ⊂ f (fa). but f (fa) ⊂ sigδs-cl(f(fa)) is always true. this shows, f (fa) = sigδscl(f(fa)). therefore, f (fa) is soft sigδsclosed set over y. hence f is strongly soft sigδs-closedclosed function. theorem 5.6. let f: (x, τ, e) → (y, σ, k, i) and g: (y, σ, k, i) → (z, µ, l, j) be two functions, such that (g ◦ f ) : x → z is strongly soft sigδs-closed function. then (1) f is soft sigδs-irresolute and surjective implies g is strongly soft sigδs-closed. (2) g is soft sigδs-irresolute and injective implies f is strongly soft sigδs-closed. proof: (1) let fa be soft sigδs-closed set over y. since f is soft sigδs irresolute and surjective, f −1 (fa) is soft sigδs-closed set over x. also since (g ◦ f) is strongly soft sigδs-closed function, implies (g ◦ f)(f −1 (fa)) = g(fa) is soft sigδsclosed over z. therefore, g is strongly soft sigδsclosed. (2) let fa be soft sigδs-closed set over x. since (g ◦ f) is strongly soft sigδs256 y. rosemathy, dr. k. alli closed function (g ◦ f) (fa) is soft sigδs-closed over z. also, since g is soft sigδsirresolute and injective, g −1 (g ◦ f) (fa) = f (fa) is soft sigδs-closed set over y. therefore, f is strongly soft sigδs closed. theorem 5.7. for any bijection, f: (x, τ, e) → (y, σ, k, i) the following statements are equivalent. (1) f −1 is soft sigδs-irresolute. (2) f is a strongly soft sigδs-open function. (3) f is a strongly soft sigδs-closed function. proof: similar to the proof of theorem 3.14. definition 5.8. a function f: (x, τ, e) → (y, σ, k, i) is said to be soft quasi sigδs-closed (resp. soft quasi sigδs-open), if for each soft sigδs-closed (resp. soft sigδs-open) set fa over x, f(fa) is soft closed (resp. open) set over y. remark 5.9. every soft quasi sigδs-closed function is soft closed, strongly soft sigδs closed and soft sigδs-closed function. remark 5.10. every soft quasi sigδs-closed function is soft sipgδs-closed. remark 5.11. following diagram is obtained from the definitions. soft sipgδs-closed ↗ ↑ ↘ soft quasi sigδs-closed → strongly soft sigδs-closed → soft sigδs-closed ↘ ↗ sot closed theorem 5.12. a surjective function f: (x, τ, e, i) → (y, σ, k) is soft quasi sigδs-closed (resp. soft quasi sigδs-open), if and only if for any soft subset ga over y and each soft sigδs-open (resp. soft sigδs-closed) set ua over x containing f −1 (ga), there exists a soft open (resp. soft closed) set va over y containing ga and f −1 (va) ⊂ ua. proof: similar to the proof of theorem 3.9. theorem 5.13. a function f: (x, τ, e, i) → (y, σ, k) is soft quasi sigδs-closed if and only if cl (f (fa)) ⊂ f (sigδs-cl (fa)) for every soft subset fa over x. proof: suppose that f is soft quasi sigδs-closed function and fa ⊂ x. then sigδs – cl (fa) is soft sigδs-closed set over x. therefore f (sigδs-cl (fa)) is soft closed over y. since f (fa) ⊂ f (sigδs-cl (fa)), implies cl (f (fa)) ⊂ cl (f (sigδs-cl (fa))) = f (sigδs-cl (fa)). this implies, cl (f (fa)) ⊂ f (sigδs-cl (fa)). conversely, fa is any soft sigδs-closed set over x. then sigδs-cl (fa) = fa. therefore, f (fa) = f (sigδs-cl (fa)). by hypothesis, cl (f (fa)) ⊂ f (sigδs-cl (fa)) = f (fa). hence cl (f (fa)) ⊂ f (fa). but f (fa) ⊂ cl (f (fa)) is always 257 soft igδs-closed functions true. this shows, f (fa) = cl (f (fa)). this implies f (fa) is soft closed set over y. therefore, f is soft quasi sigδs-closed function. theorem 5.14. let f: (x, τ, e, i) → (y, σ, k, j) be a function from a space x to a soft tigδs-space y. then following are equivalent (1) f is strongly soft s-igδs-closed function. (2) f is soft quasi-s-igδs-closed function. proof: (1) ⇒ (2) suppose (1) holds. let fa be a soft sigδs-closed set over x. then f (fa) is soft s-igδs-closed over y. since y is soft tigδs-space, f (fa) is soft closed over y. therefore, f is soft quasi sigδs-closed function. (2) ⇒ (1) suppose (2) holds. let fa be a soft s-igδs-closed set over x. then f (fa) is soft closed and hence soft sigδs-closed over y. therefore, f is strongly soft s-igδs-closed. function. references [1] b. ahmad and a. kharal, on fuzzy soft sets, advances in fuzzy systems, (2009), 1-6. [2] h. aktas and n. cagman, soft sets and soft groups, information sciences, 1(77) (2007), 2726-2735. [3] m. i. ali, f. feng, x. liu, w. k. min and m. shabir, on some new operations in soft set theory, computers and mathematics with applications, 57(2009), 1547-1553. [4] n. cagman, f. citak and s. enginoglu, fuzzy parameterized fuzzy soft set theory and its applications, turkish journal of fuzzy systems, 1(1) (2010), 21-35. 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[20] i. zorlutuna, m. akdag, w.k. min and s. atmaca, remarks on soft topological spaces, annals of fuzzy mathematics and informatics, 3(2012), 171-185. 259 https://dergipark.org.tr/en/download/article-file/105177 https://dergipark.org.tr/en/download/article-file/105177 ratio mathematica volume 42, 2022 a study on the number of edges of some families of graphs and generalized mersenne numbers sreekumar k g∗ ramesh kumar p† manilal k ‡ abstract the relationship between the nandu sequence of the sm family of graphs and the generalized mersenne numbers is demonstrated in this study. the sequences obtained from the peculiar number of edges of sm family of graphs are known as nandu sequences. nandu sequences are related to the two families of sm sum graphs and sm balancing graphs. the sm sum graphs are established from the inherent relationship between powers of 2 and natural numbers, whereas the sm balancing graphs are linked to the balanced ternary number system. in addition, some unusual prime numbers are discovered in this paper. these prime numbers best suit as an alternate for the mersenne primes in the case of the public key cryptosystem. keywords: nthsm balancing graphs, nthsm sum graphs, nandu sequence, generalized mersenne numbers, bipartite kneser type-1 graphs. 2020 ams subject classifications: 05c99, 40a30.1 ∗department of mathematics, university of kerala, thiruvananthapuram, india; sreekumar3121@gmail.com. †department of mathematics, university of kerala, thiruvananthapuram, india; ramesh.ker64@gmail.com. ‡department of mathematics, university college, university of kerala, thiruvananthapuram, india.; manilalvarkala@gmail.com. 1received on january 12th, 2022. accepted on may 12th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v39i0.704. issn: 1592-7415. eissn: 2282-8214. c©the authors. this paper is published under the cc-by licence agreement. 61 sreekumar k g, ramesh kumar p, and manilal k. 1 introduction in computer science, number systems and related ideas are used, particularly in cryptography. the binary number system, which is used in binary computers, and the balanced ternary number system, which is used in ternary computers, are two number systems used in computers. the balanced ternary number system was used in the russian setun computer. graph theory is used to investigate the combinatorial structure of these two number systems. sm sum graphs and sm balancing graphs are the two categories of graphs based on these number systems that we will focus on here. these graphs are groups of graphs that have been structured in a specific order. sm family of graphs consists of sm sum graph, sm balancing graphs and its complement graphs. for large values of n, these graphs are all non-asymmetric graphs [10] with bigger automorphism groups. the properties of these graphs lead to the discovery of some classical sequences. these sequences are called nandu sequences or ne-sequences. these nandu sequences have a relation with mersenne primes as well as generalized mersenne numbers [8]. furthermore, this is related to the low weight polynomial form of integers [4] which was used in elliptical curve cryptography. the residue number system (rns) [2, 3] has an important role in modular multiplications in computer science. this rns modular multiplication is used in the prime field based in elliptic curve cryptosystems too. generalized mersenne numbers are used in rns modular multiplication for more efficiency. the newly defined sequence {ℵn} in this paper are the particular cases of the generalized mersenne numbers. at present, the modular multiplication requires a maximum of modulus 521. eventually, the relationship between the nandu sequence and the generalized mersenne numbers are established in this work. the use of this relationship in elliptic curve cryptography is yet to be worked out. also, a study on mersenne primes in real quadratic fields was done by sushma palimar and b.r. shankar [6]. the primality testing of large numbers in maple was given in the work of s.y.yan [12]. a study on low weight polynomial form of integers for efficient modular multiplication was done by jaewook chung and m.anwar hasan [4]. a study on generalized double fibonomial numbers was done by mansi shah and devbhadra shah [7]. in this paper, we established some properties of nandu sequences. the nandu sequence {nt∑ n } for sm( ∑ n) satisfies the recurrence relation nt ∑ n+1 = 2nt∑ n + 2n + n− 1, nt∑ 2 = 2. let ℵn = ntbn −nt∑n n , where ntbn is the nandu sequence of sm(bn). we derived a closed form of the generating function of the sequence ℵn and is given by g(x) = n−2∑ r=0 2n−2 ( 3 2 )r xr. the convergence of ∑ 1 ℵn is then obtained. some preliminaries are given below. 62 a study on number of edges and generalized mersenne numbers 2 preliminary in this section, we provide the basic definitions and some results from the related work mainly from [9, 10]. we begin with the definition of sm families of graphs. let’s look at how sm balancing graphs are defined [9]. consider the set tn = {3m : m is an integer, 0 ≤ m ≤ n− 1} for a fixed positive integer n ≥ 2. let i = {−1, 0, 1}. let x ≤ 1 2 (3n − 1) be any positive integer which is not a power of 3. then x can be expressed as x = n∑ j=1 αjyj, (1) where αj ∈ i, yj ∈ tn and y′js are distinct. each yj such that αj 6= 0 is called a balancing component of x. consider the simple digraph g = (v,e), where v = {v1,v2, . . . ,v1 2 (3n−1)} and adjacency of vertices is defined by: for any two distinct vertices vx and vyj , (vx,vyj ) ∈ e if (1) holds and αj = −1, and (vyj,vx) ∈ e if (1) holds and αj = 1. this digraph g is called the nthsmd balancing graph, denoted by smd(bn). its underlying undirected graph is called the nth sm balancing graph, denoted by sm(bn). let us now look at the definition of sm sum graphs. if p < 2n, is a positive integer which is not a power of 2, then p = ∑n 1 xi, with xi = 0 or 2 m, for some integer m, 0 ≤ m ≤ n− 1 and xi’s are distinct. here each xi 6= 0 is called an additive component of p. for a fixed integer n ≥ 2, the simple graph sm( ∑ n), called n th sm sum graph [9], is a graph with vertex set {v1,v2, . . . ,v2n−1} and adjacency of vertices defined by, vi and vj are adjacent if either i is an additive component of j or j is an additive component of i. in sm( ∑ n), degree of the vertex v2n−1 is n and ∑ v∈v deg v = 2n(2n−1 − 1). in sm(bn), the number of vertices is 1 2 (3n − 1) and ∑ v∈v deg v = 2n(3n−1 − 1). note: for a fixed integer n ≥ 2, let tn = {3m : m is an integer, 0 ≤ m ≤ n − 1}, n = {1, 2, 3, . . . , t}, where t = 1 2 (3n − 1). also, let pn = {2m : m is an integer, 0 ≤ m ≤ n − 1}, m = {1, 2, 3, . . . , 2n − 1}. then consider pcn = m −pn , tcn = n −tn throughout this paper unless otherwise specified. the hamming weight of a string was defined as the number of 1’s in the strings of 0 and 1. here the number of additive components gives the hamming weight of string (binary) representation of all numbers in pcn. the hamming weight of string (binary) representation of numbers in pn is always 1. 63 sreekumar k g, ramesh kumar p, and manilal k. bipartite kneser type-1 graphs let sn = {1, 2, · · · ,n}, for an integer n > 1. for any two integers k ≥ 1 and n ≥ 2k + 1, the bipartite kneser graph [10] h(n,k) has all the k-element subsets and all the (n − k)-element subsets of sn as vertices, and two vertices are adjacent if and only if one of them is a subset of the other. here we define the bipartite kneser type-1 graph as follows. definition 2.1. [10] let sn = {1, 2, 3, . . . ,n} for a fixed integer n > 1. let φ(sn) be the set of all non-empty subsets of sn. let v1 be the set of 1element subsets of sn and v2 = φ(sn) −v1. define a bipartite graph with adjacency of vertices as: a vertex a ∈ v1 is adjacent to a vertex b ∈ v2 if and only if a ⊂ b. this graph is called a bipartite kneser type-1 graph. this bipartite kneser type-1 graph is isomorphic to the graph sm( ∑ n) for each n. to study the structure of the bipartite kneser type-1 graph, we can make use of sm( ∑ n) graph. the automorphism groups of the bipartite kneser type-1 graphs are isomorphic to the symmetric group sn for each n > 2. 3 nandu sequences of sm( ∑ n) and sm(bn) the nandu sequence or ne-sequence {ntm}n−1m=1 of sm graphs are the sequence of numbers whose terms are the half of the sum of degrees of the vertices of sm( ∑ n) or sm(bn) for all n ≥ 2. here we assume n ≥ 2 for both the sm sum graph and sm balancing graphs unless otherwise specified. kinkar das and i gutman [5] estimated the wiener index by means of number of vertices, number of edges, and diameter. 3.1 nandu sequence for the graph sm( ∑ n) definition 3.1. for the sm sum graph sm( ∑ n), with vertex set v = {vi : 1 ≤ i ≤ 2n − 1}, the nandu sequence {nt∑ n } is defined as a sequence with nth term as nt∑ n = 1 2 ∑ v∈v deg v and the sequence {dnt∑ n } defined by dnt∑ n =∑ v∈v deg v as double nandu sequence. i.e., {nt∑ n } = 2, 9, 28, 75, 186, . . . . theorem 3.1. let {nt∑ n } be the nandu sequence for the sm sum graph sm( ∑ n), n ≥ 2. then nt∑ n+1 = 2nt∑ n + 2n + n− 1, nt∑ 2 = 2. 64 a study on number of edges and generalized mersenne numbers proof. consider the graph sm( ∑ n) with vertex set v = {vi : 1 ≤ i ≤ 2 n − 1}. the nandu sequence is {nt∑ n } with nt∑ n = 1 2 ∑ v∈v deg v. then we have nt∑ n = n(2n−1 − 1). nt∑ n+1 = (n + 1)(2n − 1) = n(2n − 1) + 2n − 1 = 2n(2n−1 − 1) + 2n − 1 + n = 2nt∑ n + 2n + n− 1. hence proved. theorem 3.2. let nt∑ n be a nandu sequence of sm sum graph. then the following holds. 1. nt∑ n is a composite number for all n > 2 and is always divisible by n. 2. if nt∑ n n is a prime, then n− 1 is a prime. 3. 1 2n ∑ v∈v deg v = nt∑ n n is a mersenne number. 4. nt∑ n n gives the number of times each element of pn is used to make numbers in pcn, the complement of pn for a fixed n. proof. the proof is obvious from the definition of the sequence nt∑ n . definition 3.2. let v be the vertex set of g = sm( ∑ n). let ∆ be the maximum degree of g and δ be the minimum degree of g. the vertex degree polynomial of g is given by deg(g,x) = ∆∑ m=δ deg(g,m) ·xm = n∑ k=2 ( n k ) xk + n ·x2n−1−1, where deg(g,m) is the number of vertices of degree m. let g = sm( ∑ n) be an sm sum graph with vertex set v . the derivative of deg(g,x) at x = 1 is dnt∑ n , the (n− 1)th term of the double ne-sequence of g. now let us see the summation of terms of nandu sequence of sm sum graph. let {nt∑ n }, where nt∑ n = 1 2 ∑ v∈v deg v, be the nandu sequence for the sm sum graphs. then its summation is given by n∑ r=1 nt∑ r = n2n+1 − n(n+1) 2 −n. 65 sreekumar k g, ramesh kumar p, and manilal k. lemma 3.1. [9] if g = sm( ∑ n), pn = {2 m : m is an integer, 0 ≤ m ≤ n−1}, then d(vi, vj) =   1 if i is an additive component of j or j is an additive component of i, 2 if i, j ∈ pn or i, j 6∈ pn, i and j have at least one common additive component, 3 if neither i nor j is an additive component but exactly one of them belongs to pn, 4 if i, j 6∈ pn, i and j have no common additive components. proposition 3.1. [9] let g = sm( ∑ n) be an n th sm sum graph. let dr(vi,vj) denote the number of unordered pairs of vertices for which d(vi,vj) = r. then dr(vi,vj) =   n.(2n−1 − 1) if r = 1, n(n− 1) 2 + [ (2n −n− 2)(2n −n− 1) 2 − δn ] if r = 2, (n + 1) · 2n − (n + 2)2n−1 −n2 if r = 3, δn if r = 4, where δn = 1 2 n−2∑ r=2 [( n r )n−2∑ k=2 ( n−r k )] . remark 3.1. for n = 2 or 3, we get that δn = 0. in these cases, the diameter of the graph sm( ∑ n) is 2 or 3. theorem 3.3. suppose g = sm( ∑ n) n ≥ 2 be an n th sm sum graph. the (n− 1)th term of the nandu sequence is equal to d1(vi,vj), where vivj is an edge of g. proof. the proof follows from proposition 3.1 . 3.2 nandu sequence for the graph sm(bn) we introduced two new sequences, called nandu sequence and double nandu sequence for the sm balancing graphs also. here we discuss some of their properties. proposition 3.2 ([9]). for the nth sm balancing graph sm(bn), let dr(vi,vj) be the number of unordered pairs of vertices for which d(vi,vj) = r. let t = 1 2 (3n − 1). then dr(vi,vj) =   n · (3n−1 − 1) if r = 1, n(n− 1) 2 + [ (t−n)(t−n− 1) 2 −σn ] if r = 2, 1 2 (n · 3n−1 + n− 2n2) if r = 3, σn if r = 4, 66 a study on number of edges and generalized mersenne numbers where σn = 1 2 n−2∑ r=2 [( n r )n−2∑ k=2 ( n−r k ) 2r+k−2 ] . definition 3.3. consider the sm balancing graph sm(bn), n ≥ 2 , with vertex set v = {vi : 1 ≤ i ≤ 12 (3 n − 1)}. the sequence {ntbn} with ntbn = 1 2 ∑ v∈v deg v is called the nandu sequence and the sequence {dntbn} defined by dntbn = ∑ v∈v deg v is called the double nandu sequence. i.e.,{ntbn} = 4, 24, 104, 400, 1452, . . . . definition 3.4. let g = sm(bn) with vertex set v . let ∆ be the maximum degree of g and δ be the minimum degree of g. the vertex degree polynomial of g is given as deg(g,x) = ∆∑ m=δ deg(g,m) ·xm = n∑ k=2 2k−1 ( n k ) xk + n ·x3 n−1−1. example 3.1. for n = 5, deg(g,x) = 16x5 + 40x4 + 40x3 + 20x2 + 5x80. theorem 3.4. let g = sm(bn) be a sm balancing graph with vertex set v . the derivative of deg(g,x) at x = 1 is dntbn , the (n− 1)th term of the double ne-sequence of g. proof. let deg(g,x)′ be the derivative of deg(g,x) w.r.to x. deg(g,x)′ = n(2x + 1)n−1 −n + n · (3n−1 − 1) ·x3 n−1−2 hence, deg(g, 1)′ = n · 3n−1 −n + n · (3n−1 − 1) = 2n(3n−1 − 1) = dntbn. here we provide the summation for the nandu sequence of sm balancing graphs. let {ntbn}, where ntbn = 1 2 ∑ v∈v deg v, be the nandu sequence for the sm balancing graphs. then n∑ r=1 ntbr = 3 4 [ 2n · 3n + 3n − 1 ] −n− n(n+1) 2 . theorem 3.5. let sm(bn) be the nth sm balancing graph and ntbn be the nandu sequence. then ntbn+1 = 3ntbn + 3n + 2n−1, ntb2 = 4, for all n ≥ 2. 67 sreekumar k g, ramesh kumar p, and manilal k. proof. consider the graph sm(bn). then we have ntbn = n(3 n−1 − 1), ntb2 = 4. for all n ≥ 2, ntbn+1 = (n + 1)(3 n − 1) = n(3n − 1) + 3n − 1 = 3n(3n−1 − 1) + 3n − 1 + n = 3ntbn + 3 n + 2n− 1. hence proved. 4 relationship between nandu sequence and generalized mersenne numbers mersenne numbers are numbers of the form mn = 2n−1. if a mersenne number is prime, then n is a prime. but the converse is not true. mersenne numbers are a particular case of a larger class of numbers, the generalized mersenne numbers [11], ga,n = an−(a−1)n characterised by their base a and exponent n. the idea of generalized mersenne numbers was introduced by solinas [8] in 1999 for the use in elliptic curve cryptography. the use of generalised mersenne numbers in modular arithmetic to perform fast modular multiplications is well known. theorem 4.1. let ntbn and nt∑n be the terms of the nandu sequence of sm balancing graph and sm sum graph respectively. then ntbn−nt∑n = n(3n−1− 2n−1). proof. the proof follows from the definition of nandu sequences of sm sum graph and sm balancing graphs. definition 4.1. let sm( ∑ n) and sm(bn) be the sm sum graphs and sm balancing graphs respectively, n ≥ 3. then the sequence {ℵn} is defined as ℵn = ntbn −nt∑n n . theorem 4.2. for n ≥ 3 and when n is odd, ℵn ≡ 0( mod 5) proof. we have ℵn = ntbn −nt∑n n = 3n−1 − 2n−1 since n is odd, then n− 1 is even, say n− 1 = 2m, for some integer m. also, 3n−1 − 2n−1 = 32m − 22m, which is a multiple of 5. hence proved. theorem 4.3. ∑ 1 ℵn converges. 68 a study on number of edges and generalized mersenne numbers proof. suppose sm( ∑ n) and sm(bn) be the sm sum graphs and sm balancing graphs respectively. then we have the sequence {ℵn} as ℵn = ntbn −nt∑n n . therefore ℵn = 3n−1 − 2n−1 = (3 − 2)(3n−2 + 3n−3 · 2 + · · · + 2n−2) ≥ 2n−2 + 2n−2 + · · · + 2n−2 = (n− 1) · 2n−2 but 1 3n−1 − 2n−1 ≤ 1 (n− 1).2n−2 = 4 (n− 1) · 2n . now consider the series ∑ 1 (n− 1) · 2n . we have 2n > n− 1 , for n ≥ 3. to check the convergence of ∑ 1 (n− 1) · 2n , it is enough to check the convergence of ∑ 1 (n− 1)2 . but ∑ 1 (n− 1)2 is convergent. therefore, by comparison test, ∑ 1 ℵn converges. theorem 4.4. let sm( ∑ n) and sm(bn) be the sm sum graphs and sm balancing graphs. then ℵn+1 = 2ℵn + 3n−1, for all n ≥ 2, given ℵ2 = 1. proof. consider the graph sm( ∑ n) and sm(bn). then we have ℵn = ntbn −nt∑n n . ℵn+1 = 3n − 2n = 3.3n−1 − 2.2n−1 = 2(3n−1 − 2n−1) + 3n−1 = 2ℵn + 3n−1. hence proved. lemma 4.1. a closed form of the generating function of the sequence ℵn is given by g(x) = n−2∑ r=0 2n−2 ( 3 2 )r xr. proof. we have ℵn+1 = 2ℵn + 3n−1, for all n ≥ 2, given ℵ2 = 1. when n = 2, ℵ3 = 2ℵ2 + 3 ℵ4 = 2ℵ3 + 32=2(2ℵ2 + 3) + 32=22ℵ2 + 2 · 3 + 32 similarly, ℵ5 = 2ℵ4 + 32=23ℵ2 + 22 · 3 + 2 · 32 + 33 ℵ6 = 24ℵ2 + 23 · 3 + 22 · 32 + 2 · 33 + 34. 69 sreekumar k g, ramesh kumar p, and manilal k. continuing in this way we get, ℵn = n−2∑ r=0 2n−2 ( 3 2 )r therefore, the required generating function is g(x) = n−2∑ r=0 2n−2 ( 3 2 )r xr. theorem 4.5. let sm( ∑ n) and sm(bn) be the sm sum graphs and sm balancing graphs. then ℵn+1 = g3,n. theorem 4.6. if ℵn is prime, then n is even. but the converse is not true. proof. the proof follows from theorem 4.2. definition 4.2. let sm( ∑ n) and sm(bn) be the sm sum graphs and sm balancing graphs. then ℵn+1 = g3,n. for some values of n, g3,n is a prime. these prime numbers are called sm prime numbers. these prime numbers can be used to replace the mersenne primes in the new public key cryptosystem introduced by d. aggarwal, et al [1]. in their work, they propose a new public-key cryptosystem whose security is based on the computational intractability of the problem: given a mersenne number p = 2n−1, where n is a prime, a positive integer h and an n-bit integer h, decide whether there exist nbit integers f, g each of hamming weight less than h such that h = f g modulo p. theorem 4.7. the series ∑ 1 nt∑ n and ∑ 1 ntbn converges. proof. we have for n ≥ 3, 2n − 1 > n. so ∑ 1 n(2n − 1) ≤ 1 n2 . as ∑ 1 n2 is convergent, ∑ 1 n(2n − 1) = ∑ 1 nt∑ n is convergent. the same way ∑ 1 ntbn also converges. hence proved. we observed that ℵn is always an odd number and is a prime when n = 3, 4, 6, 18, 30 and 32 so on. there exists g3,n primes. currently, the largest modulus required for modular multiplication is 521. for n = 6, ℵn = 665. consider the function ag3,n (x) as a function which gives the number of sm primes among the g3,n generalized mersenne numbers which are less than or equal to the corresponding ℵn. we get that for n = 2, ag3,n (x) = 1, for n = 3, ag3,n (x) = 2, for n = 4, ag3,n (x) = 3 and for n = 5, ag3,n (x) = 3 etc. in fact the sum of degrees of vertices vx, x ∈ tcn minus the sum of degrees of vertices vx, x ∈ pcn is the same as the sum of degrees of vertices vx, x ∈ tn minus the sum of degrees of vertices vx, x ∈ pn. if these difference on either side is divided by n, then the quotient is equal to the ℵn. 70 a study on number of edges and generalized mersenne numbers theorem 4.8. if p1,p2,p3, ...,pn are some odd prime numbers, then p = ∏n i=1 pi is having 1 as an additive component. proof. since p1,p2,p3, ...,pn are odd prime numbers, each is having 1 as an additive component. then clearly p = ∏n i=1 pi also has 1 as an additive component. this proves the theorem. corolary 4.1. let the number of odd prime numbers less than 2n be denoted by n(β). then n(β) ≤ 2n−1 − 1 for all n ≥ 3. 5 conclusion the nandu sequences of the two sm families of graphs were examined, and a relation between these sequences and the generalized mersenne numbers was established. as a consequence, we have a new sequence of integers called g3,n, which is a type of generalized mersenne number that may be employed in elliptical curve cryptography. these sequences are important in using a graph theory method to investigate the nature and structure of the two number systems binary and balanced ternary. it has been noted that the (n− 1)th term of the nandu sequence of sm( ∑ n) is identical to unordered pairs of vertices which are at distance one. the ℵn functions can be studied further in relation to elliptic curve cryptography. conflict of interest the authors hereby declare that we have no potential conflict of interest. 6 acknowledgements the authors would like to thank the anonymous referee for their helpful comments which greatly improved the paper. this research has been promoted/supported by the university of kerala, india. references [1] divesh aggarwal, antoine joux, anupam prakash, and miklos santha, a new public-key cryptosystem via mersenne numbers, annual international cryptology conference, springer, 2018, pp. 459–482. 71 sreekumar k g, ramesh kumar p, and manilal k. [2] j-c bajard and laurent imbert, a full rns implementation of rsa, ieee transactions on computers 53 (2004), no. 6, 769–774. [3] jean-claude bajard, marcelo kaihara, and thomas plantard, selected rns bases for modular multiplication, 2009 19th ieee symposium on computer arithmetic, ieee, 2009, pp. 25–32. [4] jaewook chung and m anwar hasan, low-weight polynomial form integers for efficient modular multiplication, ieee transactions on computers 56 (2006), no. 1, 44–57. [5] kinkar ch das and ivan gutman, estimating the wiener index by means of number of vertices, number of edges, and diameter, match commun. math. comput. chem 64 (2010), no. 3, 647–660. [6] sushma palimar et al., mersenne primes in real quadratic fields, arxiv preprint arxiv:1205.0371 (2012). [7] mansi shah and shah devbhadra, generalized double fibonomial numbers, ratio mathematica 40 (2021), 163. [8] jerome a solinas et al., generalized mersenne numbers, citeseer, 1999. [9] kg sreekumar and k manilal, hosoya polynomial and harary index of sm family of graphs, journal of information and optimization sciences 39 (2018), no. 2, 581–590. [10] , automorphism groups of some families of bipartite graphs, electronic journal of graph theory and applications 9 (2021), no. 1, 65–75. [11] tao wu and li-tian liu, elliptic curve point multiplication by generalized mersenne numbers, journal of electronic science and technology 10 (2012), no. 3, 199–208. [12] sy yan, primality testing of large numbers in maple, computers & mathematics with applications 29 (1995), no. 12, 1–8. 72 microsoft word janovitz.doc ratio mathematica 18 (2008), 91 106 91 studi liceali di matematici ebrei nella mantova del tardo ottocento alessandro janovitz1 sunto. si illustrano gli studi effettuati, le scuole frequentate, le materie seguite e i professori avuti da sei illustri matematici ebrei mantovani (gino fano, aldo finzi, gino loria, cesare rimini, adolfo viterbi e giulio vivanti), nati in un’area geografica e temporale ristrettissime. si evidenzia il positivo influsso dell’insegnamento ginnasiale e liceale da essi ricevuto nella mantova del tardo ottocento. abstract. gino fano, aldo finzi, gino loria, cesare rimini, adolfo viterbi and giulio vivanti are six remarkable mantuan jewish mathematicians. this paper deals with their good high-school education received in mantua before their university studies, in the second half of the nineteenth century. parole chiave: formazione matematica, liceo mantovano, matematici ebrei 1. introduzione mantova può vantare una plurisecolare tradizione nell’ambito degli studi scientifici, in particolare quelli idraulici, indispensabili in un territorio circondato, come quello mantovano, dalle acque lacustri. basti citare, a titolo esemplificativo, i nomi ben noti di giovanni 1 dipartimento best del politecnico di milano, via bonardi, 3, 20133 milano. email: a.janovitz@alice.it. il presente lavoro, svolto nell’ambito del gruppo di ricerca d’ateneo del politecnico di milano, è stato presentato al congresso nazionale mathesis, tenutosi a trento dal 2 al 4 novembre 2006. ratio mathematica 18 (2008), 91 106 92 benedetto ceva (milano 1647 – mantova 1734) e giuseppe (o gioseffo) mari (canneto sull’oglio 1730 – mantova 1807), entrambi oggetto di recentissime trattazioni storiche.1 proprio in questo filone di studi, volto a rivalutare l’importanza delle scienze, in particolare matematiche, nel mantovano, si colloca il presente lavoro, dedicato alla analisi degli studi preuniversitari compiuti a mantova nella seconda metà dell’ottocento da sei giovani ebrei che sarebbero divenuti illustri matematici: gino fano, aldo finzi, gino loria, cesare rimini, adolfo viterbi e giulio vivanti. dopo aver presentato una breve biografia di ciascuno, ponendo in evidenza le scuole frequentate (§ 2), se ne descrivono gli studi effettuati, illustrando le materie seguite (§ 3) e ricostruendo le personalità dei professori avuti (§ 4). 2. gino fano, aldo finzi, gino loria, cesare rimini, adolfo viterbi, giulio vivanti come già detto, fano, finzi, loria, rimini, viterbi e vivanti nacquero tutti a mantova tra il 1859 e il 1882, compiendo ivi i primi studi, come meglio si vedrà nelle rispettive sintetiche biografie, poste in ordine cronologico, a loro dedicate. 2.1. giulio vivanti giulio benedetto isacco vivanti2 nacque a mantova il 24 maggio 1859 da famiglia di posizione socialmente elevata. i suoi primi studi, di tipo classico, si svolsero privatamente e da privatista affrontò annualmente gli esami al ginnasio-liceo virgilio di mantova,3 con risultati assai positivi. laureatosi, nel 1881, in ingegneria civile al politecnico di torino, conseguì due anni dopo la laurea in matematica all’università di bologna. libero docente in calcolo infinitesimale dal 1 si veda, per esempio, [25], [26] e l’articolo di giuliana tomasini in questa rivista. 2 a proposito di giulio vivanti si veda [15], [23] e [28]. 3 [5], bb. 193, 194, 195. ratio mathematica 18 (2008), 91 106 93 1892, insegnò all’università di pavia e, dal 1895, in quella di messina. ordinario dal 1901, tornò nel 1907 all’università di pavia, ove rimase fino al 1924, anno in cui si trasferì presso la neonata università di milano. emerito dal 1934, pubblicò opere scientifiche fino al 1947. i suoi principali campi di interesse di furono le funzioni analitiche (in tale ambito elaborò un teorema, detto teorema di vivanti) e la storia della matematica, con particolare riguardo all’analisi. fondò e diresse la sezione pavese e poi quella milanese della mathesis. ebbe fra i suoi allievi bruno de finetti (1906-1985). morì a milano, il 19 novembre 1949. 2.2. gino loria gino benedetto loria1 nacque a mantova il 19 maggio 1862 da famiglia abbiente e di notevole levatura culturale. si iscrisse, nel 1875, all’istituto tecnico provinciale e scuola agraria carpi di mantova, ove si diplomò nel 1879,2 anno in cui si iscrisse all’università di torino. ivi laureatosi, nel 1883, in matematica, divenne ordinario di geometria superiore nell’università di genova dal 1891, ove insegnò fino al 1935. abbandonò genova, per le persecuzioni razziali, rifugiandosi nelle valli valdesi del piemonte. al termine della guerra pubblicò svariate opere. studiò prima la geometria delle sfere e della retta, le trasformazioni razionali dello spazio, le funzioni ellittiche; si occupò poi di storia delle matematiche e delle scienze, raggiungendo presto fama internazionale. fondatore e direttore del «bollettino di bibliografia e storia delle scienze matematiche», collaborò ai più importanti periodici scientifici e fece parte di moltissime prestigiose accademie e istituzioni scientifiche. si occupò anche di questioni didattiche, in particolare per conto della commissione internazionale per l’insegnamento della matematica, e fu promotore e direttore della scuola di magistero in genova. si distinse tra i soci più attivi della mathesis, della cui sezione ligure fu a lungo presidente e instancabile animatore. morì a genova, il 30 gennaio 1954. 1 a proposito di gino loria si veda [19], [23], [29] e [33]. 2 [1]. ratio mathematica 18 (2008), 91 106 94 2.3. gino fano gino angelo fano1 nacque a mantova il 5 gennaio 1871. la famiglia d’origine era particolarmente in vista per censo e per le attività politiche e sociali svolte. iscrittosi al ginnasio-liceo virgilio di mantova nel 1880, dopo tre anni si trasferì, per volontà del padre, al collegio militare di milano e poi all’accademia militare di torino. opponendosi ai voleri paterni, tornò a mantova per iscriversi al reale istituto tecnico alberto pitentino e scuola agraria carpi, ove, nel 1888, ottenne la licenza.2 iscrittosi, a torino, a ingegneria, scelse però presto il corso di matematica, in cui si laureò con lode. dopo un anno di assistentato a torino e il perfezionamento a gottingen con felix klein (1849-1925), insegnò nella università di messina e, dal 1901, in quella di torino. ordinario di geometria proiettiva e descrittiva dal 1905, fu sospeso dall’insegnamento nel 1938, per le leggi razziali, e si rifugiò in svizzera, tornando in italia dopo la guerra. fu tra i massimi geometri dell’epoca: tra i suoi lavori più importanti si ricordano quelli sulle geometrie finite e sulle varietà algebriche tridimensionali (rispettivamente piano di fano e varietà di fano). medaglia d’oro dei benemeriti della pubblica istruzione, si occupò anche di didattica, partecipando attivamente alla vita della mathesis. morì a verona l’8 novembre 1952. 2.4. adolfo viterbi adolfo davide graziadio viterbi3 nacque a mantova il 27 settembre 1873 da famiglia agiata, culturalmente elevata e assai attiva nel contesto politico locale. studiò al ginnasio-liceo virgilio di mantova dal 1883 al 1891, anno in cui ottenne la licenza4 e si iscrisse all’università di bologna, prima in ingegneria, poi in matematica. laureatosi a messina nel 1899, specializzatosi a pisa e a gottingen, 1 a proposito di gino fano si veda [16], [18], [23] e [32]. 2 [2]. 3 a proposito di adolfo viterbi si veda [9], [20], [22], [23], [30] e [31]. 4 [5], bb. 162, 163, 164, 165, 166, 167, 168, 169, 203, 204 e 210. ratio mathematica 18 (2008), 91 106 95 conseguita una seconda laurea a padova in ingegneria civile, divenne ivi libero docente in meccanica razionale. insegnò geodesia teoretica all’università di pavia, ordinario dal 1914, distinguendosi per le particolari doti di tipo didattico. volontario nella grande guerra, morì al fronte il 18 novembre 1917. i suoi studi, orientati dapprima all’analisi, si rivolsero per lo più alla meccanica razionale e alla geodesia, in particolare alla risoluzione approssimata del problema di dirichlet. fu collaboratore della mathesis. ebbe particolare generosità occupandosi, con importanti donazioni, dei concittadini in condizioni di bisogno. fu maestro di rocco serini (1886-1964), ordinario di fisica matematica a pavia. 2.5. aldo finzi aldo finzi1 nacque a mantova il 20 dicembre 1878. dopo aver compiuto i primi studi privatamente, frequentò, dal 1893, il ginnasioliceo virgilio di mantova, presso il quale si licenziò nel 1895.2 laureatosi con lode in matematica all’università di padova, sotto la guida di gregorio ricci-curbastro (1853-1925), iniziò la carriera accademica all’università di messina, in qualità di assistente di giulio vivanti. si dedicò anche all’insegnamento negli istituti tecnici di messina, reggio calabria, bari e napoli, per divenire preside, ispettore e infine provveditore della campania, incarico che ricoprì per più di dieci anni.3 libero docente in geometria analitica, ebbe incarichi di insegnamento all’università di napoli. oggetto prediletto dei suoi studi fu il calcolo differenziale assoluto; a lui si deve un teorema sulla curvatura conforme di una varietà. collaborò alla «enciclopedia delle matematiche elementari» e pubblicò svariati testi per studenti delle scuole superiori. morì a roma il 18 novembre 1934. 1 a proposito di aldo finzi si veda [24]. 2 [5], bb. 174 e 175. 3 [7]. ratio mathematica 18 (2008), 91 106 96 2.6. cesare rimini cesare giacomo rimini1 nacque a mantova il 18 febbraio 1882 da famiglia economicamente assai debole. a soli otto anni iniziò gli studi al ginnasio-liceo virgilio di mantova, ottenendo con ottimi voti la licenza nel 1898.2 si laureò con lode all’università di pisa, da enfant prodige, a vent’anni. ivi perfezionatosi in analisi, si laureò, nel 1907, in ingegneria a bologna, esercitandovi, per oltre vent’anni, la professione di ingegnere. libero docente, dal 1927, in elettrotecnica generale, incaricato all’università e nella scuola postuniversitaria di perfezionamento in radiocomunicazioni di bologna, ne fu espulso per le leggi razziali. dopo la guerra, riassunse le cariche accademiche fino al 1955. si occupò di teoria delle superfici, eteromografie e iperomografie, calcolo tensoriale e elettrotecnica, in particolare delle proprietà geometriche delle correnti alternative. fu presidente della sezione di bologna dell’associazione elettrotecnica italiana. autore di apprezzati testi di analisi per studenti universitari, fu premiato dall’accademia d’italia per il suo «elementi di radiotecnica». morì a bologna l’1 aprile 1960. 3. l’istruzione secondaria a mantova nella seconda metà dell’ottocento come si è visto nel § 2, i matematici di cui si sta trattando ebbero una formazione scolastica legata al liceo classico e all’istituto tecnico di mantova.3 in seguito all’annessione, nel 1866, di mantova al regno d’italia, il liceo divenne, nel 1867, «regio ginnasio-liceo virgilio». la scuola tecnica, invece, fu sostituita nel 1868 dall’«istituto tecnico provinciale e scuola agraria carpi», governativo dal 1879, denominato 1 a proposito di cesare rimini si veda [14], [17] e [23]. 2 [5], bb. 169, 170, 172, 174, 175, 176, 177 e 178. 3 durante la dominazione asburgica, l’istruzione secondaria a mantova era garantita dall’esistenza dell’«imperial regio liceo di mantova» e di una scuola tecnica attiva per il primo triennio. ratio mathematica 18 (2008), 91 106 97 nel 1884 «r. istituto tecnico alberto pitentino e scuola agraria carpi di mantova», tuttora funzionante come «istituto tecnico commerciale statale alberto pitentino».1 per comprendere meglio quale fosse l’organizzazione degli studi nelle due scuole nel periodo storico qui considerato, si propongono due tabelle.2 a n n o reli gion e ita lia no te de sc o la ti no gr ec o st. ge og r ma te ma t st. nat. fisi ca fi lo so f gi nn as t. di se gn o fr an ce se i '64 '75 '90 2 0,5 4 7 7 8 9 7 3 3 2 3 1 1 2 1* 1 1 3 2 ii '64 '75 '90 2 0,5 3 7 7 2 7 11 7 3 3 4 3 1 2 2 1 4 3 2,5 iii '64 '75 '90 2 0,5 2 7 7 2 6 8 6 4 3 3 4 3 1 1 2 1 1 4 3 2,5 iv '64 '75 '90 2 0,5 2 5 5 2 6 6 6 4 5 3 3 4 4 3 3 2 2 2 1 4 v '64 '75 '90 2 0,5 2 4 6 3 6 6 5 4 5 3 3 4 4 4 3 2 2 2 1 4 1 '64 '75 '90 2 0,5 2 6 4 3 7 5 5 4 3 3 3 5 4 3 6 3 2 1+1 2 4 2 '64 '75 '90 2 0,5 2 3 4 2 4 4 3 4 3 3 3 3 4 3 3 3 3 2+4, 5 2+3 3 4 2 4 3 '64 '75 '90 2 0,5 2 3 4 2 5 3 3 5 3 3 3 3 3 1 3 3 3 3+4, 5 2+3 3 4 2 4 * scienze naturali tabella n. 1 – orario settimanale delle lezioni presso il 1 si veda a tal proposito [21]. 2 le tabelle n. 1 e n. 2 sono state elaborate sulla scorta dei documenti conservati nell’archivio dell’istituto tecnico commerciale statale ‘a. pitentino’ di mantova e nell’archivio di stato di mantova. ratio mathematica 18 (2008), 91 106 98 «regio ginnasio-liceo virgilio» di mantova. nella prima, relativa al liceo, alla prima colonna corrisponde l’anno di corso, alla seconda l’anno di riferimento (e precisamente il 1864, il 1875 e il 1890), a ciascuna delle successive una materia di insegnamento; a ogni colonna corrisponde, invece, l’anno di corso e per ognuno di essi si presenta il numero di ore settimanali a seconda dell’anno di riferimento. nella tabella n. 2, relativa invece all’istituto tecnico (sezione fisico-matematica), a ciascuna riga corrisponde una disciplina, mentre le colonne (raggruppate per ogni anno di corso) sono a loro volta suddivise nei due anni di riferimento, e precisamente il 1872 e il 1885, ognuno corrispondente all’anno di applicazione di una riforma scolastica. i anno ii anno iii anno iv anno '72 '85 '72 '85 '72 '85 '72 '85 lettere italiane 6 6 6 5 5 4 5 6 geografia 2 3 2 3 2 2 storia 3 31 3 31 3 21 3 lingua francese 3 3 2 3 2 2 2 lingua tedesca o inglese 3 4 3 4 5 4 5 matematica elementare 6 5 matematica 6 5 5 5 5 5 geometria descrittiva e disegno 4 4 storia naturale 3 3 3 3 2 fisica 3 3 3 52 33 nozioni generali di chimica 3 34 44 chimica 3 3 elementi di meccanica 3 disegno ornamentale 6 6 6 6 6 6 disegno architettonico 4 6 totale 35 30 37 31 39 30 37 29 1 con il nome di storia generale 2 con il nome di fisica generale 3 con il nome di fisica complementare 4 con il nome di chimica generale tabella n. 2 orario settimanale delle lezioni presso l’istituto tecnico di mantova, sezione fisico-matematica. ratio mathematica 18 (2008), 91 106 99 la lettura delle due tabelle mostra come l’insegnamento della matematica abbia subito una evidente riduzione al liceo classico (ove già rivestiva un ruolo piuttosto marginale), rimanendo invece di centrale importanza nell’istituto tecnico. 4. i docenti il seguente elenco indica i docenti di matematica dei sei scienziati oggetto del presente lavoro: a fianco del nome di ciascuno compare il periodo di insegnamento e la scuola di servizio («tecnico» rappresenta l’istituto tecnico di mantova, mentre «classico» indica il liceo classico di mantova). pietro caminati 1883-96 tecnico adolfo coen 1895-1905 classico antonio c. dall’acqua 1883-1908 tecnico vespasiano fattorini 1868-83 tecnico ruggero grilli 1894-95 classico alessandro sterza 1870-79 tecnico 1874-1908 classico giuseppe tezza 1863-94 classico dall’esame del materiale documentario originale si evince che caminati ebbe come allievo fano; coen rimini; dall’acqua fano; fattorini loria; grilli finzi e loria; sterza fano, loria, rimini e viterbi; tezza finzi e viterbi. nel seguito si espongono, per ciascun docente, gli elementi più significativi utili per delinearne le personalità.1 4.1. pietro caminati nato a genova il 7 aprile 1837, ottenne all’università di genova il diploma di magistero ne1 1855 e la laurea in ingegneria idraulica nel 1 le notizie, salvo diverso avviso, sono state desunte da [4] e [6]. ratio mathematica 18 (2008), 91 106 100 1859, diplomandosi poi in architettura presso l’accademia di belle arti di genova nel 1862. professore di matematica in svariate scuole (pierdarena, modica, terni, sondrio, palermo, mantova e foggia), ottenne numerosi riconoscimenti per le proprie attività, tra i quali una menzione onorevole nell’esposizione nazionale in torino per «l’opera pubblica sopra i logaritmi di somma e sottrazione» (1884), la medaglia di bronzo nell’esposizione generale italiana in palermo «per lavoro didattico musicale col titolo “nuova curva fonica da darsi agli istrumenti musicali ad arco”» (1892) e la medaglia d’oro nell’esposizione internazionale delle industrie e del lavoro in torino per un goniografo e un compasso goniografico, entrambi brevettati (1911). le sue ventitré pubblicazioni sono per lo più dedicate allo studio della geometria e dei logaritmi. fondò e diresse, inoltre, «il tartaglia. periodico di scienze fisico-matematiche elementari per gli alunni delle scuole secondarie pubblicato per cura del prof. ing. pietro caminati», pubblicato dal 1898 al 1899. 4.2. adolfo coen nato a livorno il 9 gennaio 1848 da famiglia ebraica, studiò dapprima all’università di pisa, poi in quella di bologna, ivi laureandosi con lode in matematica nel 1885. insegnò matematica in diverse città (tempio, cagliari, vicenza, cosenza, salerno e mantova). fu membro onorario della società scientifica degli studi psicologici di francia, socio fondatore dell’unione matematica italiana e ufficiale della corona d’italia. pubblicò, tra il 1887 e il 1895, alcuni libri di testo per le scuole superiori. morì a livorno il 17 dicembre 1926. 4.3. antonio carlo dall’acqua nato a dolo (venezia) il 22 marzo 1838, conseguita la laurea in ingegneria e architettura all’università di padova nel 1861, ottenne il diploma di architetto presso l’accademia di belle arti di venezia nel 1864. fu docente di matematica a forlì, como, pesaro e mantova, collocato a riposo nel 1908. fece parte della commissione conservatrice dei monumenti ed oggetti d’arte e di antichità per la ratio mathematica 18 (2008), 91 106 101 provincia di pesaro e urbino dal 1881 e di quella per la provincia di mantova dal 1889; fu socio corrispondente della r. accademia raffaello di urbino dal 1882 e socio effettivo residente della r. accademia virgiliana di mantova dal 1890, divenendone prefetto nel 1907. fece altresì parte della civica commissione degli studi di mantova. pubblicò una quarantina di opere, composte da monografie, articoli e necrologi, tutte riguardanti la storia dell’arte e dell’architettura. morì a mantova l’11 ottobre 1928. 4.4. vespasiano fattorini nato a mantova il 13 settembre 1836, si laureò in ingegneria civile e architettura all’università di pavia nel 1860. fu docente di matematica a vigevano, ferrara, venezia e mantova, divenendo preside dell’istituto tecnico di cremona dal 1883. morì a napoli il 19 gennaio 1886. eugenio togliatti (1890-1977) lo descrive come «persona modesta, ma didatticamente assai capace e piena di entusiasmo per la sua scienza.»1 4.5. ruggero grilli nato a russi (ravenna) il 17 settembre 1848, iniziò i propri studi in matematica presso l’università di bologna nel 1866. fu docente di matematica nel liceo di mantova nell’anno scolastico 1894-95. pubblicò, nel 1889, un trattato di algebra per i licei. si trasferì a treviso nel 1895. 4.6. alessandro sterza nato a verona il 13 giugno 1851, conseguito il diploma di perito agrimensore, seguì privatamente i corsi di matematica e di meccanica a monaco di baviera e i corsi dell’accademia di belle arti di milano, ottenendo la licenza per l’insegnamento fisico-matematico e di agronomia e agrimensura nonché il diploma della reale accademia di 1 [34], p. 115. ratio mathematica 18 (2008), 91 106 102 milano per l’insegnamento del disegno. fu professore di matematica dapprima a verona, successivamente a mantova, insegnando ivi anche disegno presso la scuola d’arte del comune, la regia scuola normale e la scuola serale d’arte. prestò altresì servizio, sempre a mantova, nella scuola superiore femminile e nella scuola d’arte applicata all’industria. fu, a partire dal 1880, socio effettivo residente della r. accademia virgiliana,1 ove tenne, tra il 1901 e il 1904, alcune conferenze, sintetizzate negli «atti e memorie» dell’accademia stessa, sulla telegrafia senza fili, sulla navigazione aerea e sulla radioattività.2 fu, infine, cavaliere dell’ordine di leopoldo i del belgio e vicepresidente del giurì internazionale per le scienze all’esposizione internazionale di bruxelles del 1897. pubblicò una ventina di opere, fra le quali numerosi testi scolastici e un romanzo. morì a aosta il 15 aprile 1912. 4.7. giuseppe tezza nato a azzago, verona, il 16 luglio 1840, laureatosi in matematica, la insegnò per oltre trent’anni al liceo classico di mantova. pubblicò due opere scientifiche, una della quali dedicata alle scienze applicate. morì a milano il 27 giugno 1897. 5. considerazioni conclusive da quanto precedentemente visto, emergono alcuni dati significativi che caratterizzano positivamente gli insegnamenti della matematica impartiti a mantova nella seconda metà dell’ottocento, come si vedrà nel dettaglio in seguito. anzitutto, la essenziale stabilità nel tempo del corpo docente. i professori prima citati mantennero, infatti, la propria cattedra nella medesima istituzione scolastica mantovana assai a lungo: caminati per trentatré anni, coen per venti, dall’acqua per venticinque, fattorini 1 [12], p. v; [13], p. vi. 2 [9], pp. xxx-xxxii; [10], pp. xix-xxi; [11], pp. xvii-xx. ratio mathematica 18 (2008), 91 106 103 per quindici, sterza per trentaquattro, tezza per trentuno.1 in secondo luogo, il possesso di titoli di studio adeguati: coen e tezza erano laureati in matematica; caminati, fattorini e dall’acqua erano ingegneri; sterza aveva seguito privatamente corsi di matematica e di meccanica a monaco di baviera. le pubblicazioni, poi, rappresentano un indizio della attenzione posta sia alle questioni didattiche sia alla ricerca scientifica: caminati scrisse venticinque pubblicazioni di carattere matematico e didattico; coen quattro manuali di aritmetica e algebra; sterza undici libri di testo scolastici e vari articoli di divulgazione scientifica; tezza un articolo e un saggio sulle scienze; dall’acqua trentasei pubblicazioni sulla storia dell’arte. infine, gli interessi coltivati da molti in ambiti extrascolastici indicano una propensione a allargare le proprie attività e i propri saperi a svariati livelli, consentendo favorevoli ricadute anche nella didattica. bibliografia fondi archivistici2 [1] apmn, registri delle classificazioni, anni scolastici 1875-76, 187677, 1877-78 e 1878-79. [2] apmn, registro inscrizioni e classificazioni. anni 1885-86-86-8787-88-88-89. completo. [3] apmn, stati – esami licenza. dall’anno 1881-82 all’anno 1888-89 inclusi. [4] apmn, stato del personale. [5] asmn, liceo ginnasio «virgilio», parte i, bb. 162, 163, 164, 165, 166, 167, 168, 169, 170, 172, 174, 175, 176, 177, 178, 193, 194, 195, 1 fa eccezione grilli, insegnante per un solo anno a mantova. 2 si sono utilizzati i seguenti acronimi: apmn, archivio dell’istituto tecnico commerciale statale ‘a. pitentino’, mantova; archivio della provincia, mantova; asmn, archivio di stato, mantova; avmn, archivio dell’accademia nazionale virgiliana, mantova. ratio mathematica 18 (2008), 91 106 104 203, 204 e 210. [6] asmn, liceo ginnasio «virgilio», parte ii, b. 35, regg. stato personale 1887-1940, stato personale dal 1887 al 1923. [7] avmn, fascicolo di aldo finzi. opere a stampa [8] (1918) adolfo viterbi, bollettino della “mathesis” società italiana di matematica, 2, 83-86. 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(2002) giorgio todesco e cesare rimini, in: la cattedra negata. dal giuramento di fedeltà al fascismo alle leggi razziali nell’università di bologna, clueb, bologna, 185-194. [18] fano u. (2000) the memories of an atomic physicist for my children and grandchildren, physics essays, 2-3, 176-197. [19] giacardi l. (1999) gino loria, in: la facoltà di scienze matematiche fisiche naturali di torino, 1848-1998, tomo secondo, i docenti, a cura di c. s. roero, deputazione subalpina di storia patria, ratio mathematica 18 (2008), 91 106 105 torino, 520-525. [20] il municipio di mantova (1918) in memoria del prof. comm. adolfo viterbi, mondadori, ostiglia. [21] janovitz a. insegnamenti della matematica a mantova nella seconda metà dell’ottocento (in corso di pubblicazione). [22] janovitz a. 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(1959) adolfo viterbi, bollettino di geodesia e scienze affini, 2, 291-295. [32] terracini a. (1953) commemorazione del socio gino fano, rendiconti accademia nazionale dei lincei, 14, 702-715. [33] terracini a. (1954) commemorazione del socio gino loria, rendiconti accademia nazionale dei lincei, classe scienze matematiche, fisiche e naturali, 17, 402-421. [34] togliatti e. (1954) necrologio. gino loria, bollettino della unione matematica italiana, 9, 115-118. ratio mathematica volume 44, 2022 some graph parameters of clique graph of cyclic subgroup graph on certain nonabelian groups s. ragha* r. rajeswari† abstract the aim of this paper is to examine various graph parameters of clique graph of cyclic subgroup graph on certain non-abelian groups and also we obtain some theorems and results in detail. keywords: cyclic subgroup graph, clique graph, hub number, topological indices 2010 ams subject classification: 05c09,05c25,05c12,05c50‡ *research scholar (reg. no. 20212012092008), pg & research department of mathematics, a.p.c mahalaxmi college for women, thoothukudi-628002, tamil nadu, india. affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamil nadu, india.raghasankar810@gmail.com †assistant professor, pg &research department of mathematics, a.p.c mahalaxmi college for women, thoothukudi-628002, tamil nadu, india. rajimuthuram@gmail.com ‡received on june 17th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.927. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 371 s. ragha and r. rajeswari 1. introduction algebraic graph theory is a branch of mathematics in which graphs are constructed from the algebraic structures such as groups, rings etc. j. john arul singh and s. devi [3] have introduced the notion of cyclic subgroup graph of a finite group. the concept of clique graphs was discussed at least as early as 1968 by hamelink and ronald. c. after that, roberts and spencer have given the concept of a characterization of clique graphs [2] in 1971. gutman introduced the concept of energy in 1978. a topological index of a graph g is a numerical parameter mathematically derived from the graph structure. it is a graph invariant which does not depend on the labeling or pictorial representation of the graph and it is the graph invariant number calculated from a graph representing a molecule. our present work is provoked by the above study. in section 2, we discuss some graph parameters for the clique graph of cyclic subgroup graph on certain non-abelian groups. in section 3, we examine some energies on clique graph of cyclic subgroup graph for dihedral group and generalised quaternion group. in section 4, we give some topological indices for the clique graph of cyclic subgroup graph on certain non-abelian groups. in this paper, 𝑝 represents prime number and 𝑝𝑞 represents distinct primes where 𝑞 > 𝑝. before entering, let us look into some necessary definitions and notations. the cyclic subgroup graphγ𝑧 (𝐺) for a finite group 𝐺 is a simple undirected graph in which the cyclic subgroups are vertices and two distinct subgroups are adjacent if one of them is a subset of the other. the clique graph𝒦(𝐺) of an undirected simple graph 𝐺, is a graph with a node for each maximal cliques in 𝐺. two vertices in 𝒦(𝐺) are adjacent when their corresponding maximal cliques in 𝐺 share at least one vertex in common. for an integer 𝑛 ≥ 3, the dihedral group𝐷2𝑛 of order 2𝑛 is 𝐷2𝑛 =< 𝑟, 𝑓: 𝑟 𝑛 = 𝑓 2 = 1, 𝑓𝑟𝑓 = 𝑟−1 >. the generalized quaternion group 𝑄4𝑛 =< 𝑎, 𝑏: 𝑏 2 = 𝑎𝑛, 𝑎2𝑛 = 𝑒, 𝑏𝑎𝑏−1 = 𝑎−1 >, where 𝑒 is the identity element.a hub set in a graph 𝐺 is a set 𝐻 of vertices in 𝐺, such that any two vertices outside 𝐻 are connected by a path whose all internal vertices lie in 𝐻. the hub number of 𝐺, denoted by ℎ(𝐺), is the minimum cardinality of a hub set in 𝐺. let g be a simple undirected graph, note that 𝑢𝑖 ~𝑢𝑗 denotes that 𝑢𝑖 is adjacent to 𝑢𝑗 for 1 ≤ 𝑖 ≠ 𝑗 ≤ 𝑛. the adjacency matrix of g denoted by 𝐴(𝐺) = (𝑎𝑖𝑗 ) is an 𝑛 × 𝑛 matrix defined as 𝑎𝑖𝑗 = 1 when 𝑢𝑖 ~𝑢𝑗 and 0 otherwise. the sum of the absolute values of the eigen values of its adjacency matrix is called the energy i.e., 𝐸(𝐺) = ∑ |𝜆𝑖 | 𝑛 𝑖=1 . the closed neighborhood matrix𝑁 = [𝑛𝑖,𝑗 ] = 𝐴 + 𝐼𝑛 has 𝑛𝑖,𝑗 = 1 if and only if 𝑢𝑖 ∈ 𝑁[𝑢𝑗 ]. the sum of the absolute values of the eigen values of its closed neighborhood matrix is called the closed neighborhood energy. the laplacian matrix and signless laplacian matrix is defined as 𝐿(𝐺) = 𝐷(𝐺) − 𝐴(𝐺) and 𝑆𝐿(𝐺) = 𝐷(𝐺) + 𝐴(𝐺) where 𝐴(𝐺) is the adjacency matrix and 𝐷(𝐺) is the diagonal matrix of vertex degrees. 372 some graph parameters of clique graph of cyclic subgroup graph on certain non-abelian groups 2. some graph parameters of clique graph of cyclic subgroup graph on certain non-abelian groups theorem 2.1: if 𝒦(γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on a non-abelian group, then 𝒦(γ𝑧 (𝐺)) is biconnected. proof: let 𝒦(γ𝑧(𝐺)) be the clique graph of cyclic subgroup graph on a non-abelian group. now, 𝑊1, 𝑊2, … , 𝑊𝑛 be the maximal cliques of γ𝑧 (𝐺). by the definition of clique graph, 𝑉(𝑈) = 𝑊1, 𝑊2, … , 𝑊𝑛 and (𝑊𝑖 , 𝑊𝑗 ) ∈ 𝐸(𝑈) if only if 𝑖 ≠ 𝑗 and 𝑊𝑖 ∩ 𝑊𝑗 ≠ ∅ and take 𝑈 = 𝒦(γ𝑧(𝐺)). for γ𝑧 (𝐺), there exists a universal vertex which is adjacent to rest of its vertices. now, any two vertices in 𝒦(γ𝑧(𝐺)) are adjacent only when their corresponding maximal cliques in γ𝑧 (𝐺) have atleast one vertex in common. it is clear that, there is a path in between every starting vertex and ending vertex. even after removing any vertex, the graph remains connected. now concluding that 𝐺 is connected and it does not contain any articulation point which results to a biconnected graph. theorem 2.2: for a clique graph of cyclic subgroup graph on any non-abelian group, the hub number is 0. proof: based on the proof of 2.1, for 𝒦(γ𝑧 (𝐺)), every pair of vertices are adjacent. hence, there does not exist an intermediate vertex lies in the hub set. in this case, the minimum hub set is a null set. hence, ℎ (𝒦(γ𝑧 (𝐺))) = 0. theorem 2.3: let 𝒢 = 𝒦(γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on any non-abelian group and |𝑉(𝒢)| = 𝑚. then ℊ(𝒢) = 𝓂(𝒢) = 𝑚. proof: let 𝒢 = 𝒦(γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on any nonabelian group. the geodetic closure of a vertex set 𝑆 ⊂ 𝒱 is the set of all vertices 𝑦 ∈ 𝒱which lies in some geodesic in 𝒢 joining two vertices 𝑢 and 𝑣 of 𝑆. clearly for 𝒦(γ𝑧 (𝐺)), there exists 𝑚 maximal cliques which is connected by at least one vertex in common. now, the resulting graph consists of 𝑚 independent vertices in it. hence ℊ(𝒢) = 𝑚. consider, a set 𝐷 of vertices of 𝒦(γ𝑧(𝐺)) is a monophonic set of 𝒦(γ𝑧(𝐺)), if each vertex 𝑣 ∈ 𝒦(γ𝑧(𝐺)) lies on an 𝑢 − 𝑤 monophonic path in 𝒦(γ𝑧 (𝐺)) for some 𝑢, 𝑤 ∈ 𝐷 and the minimum cardinality of a monophonic set of 𝒦(γ𝑧 (𝐺)), 𝓂(𝒢) = 𝑚. theorem 2.4: for any non-abelian group, 𝒦(γ𝑧(𝐺)) is non-planar. proof directly follows from theorem 2.1. theorem 2.5: let 𝒢 = 𝒦(γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on any non-abelian group and |𝑉(𝒢)| = 𝑚. then 𝜅(𝒢) = 𝑚 − 1. proof: for 𝒦(γ𝑧 (𝐺)), by removing 𝑚 − 1 vertices which makes the graph disconnected. 373 s. ragha and r. rajeswari theorem 2.6: the independence number of 𝒦(γ𝑧 (𝐺)) is 1. proof follows from direct computation. theorem 2.7: for any non-abelian group, ℎ (𝒦(γ𝑧 (𝐺))) ≠ 𝛾 (𝒦(γ𝑧(𝐺))). proof: let 𝒦(γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on any non-abelian group. here, every pair of vertices are adjacent. choose any one vertex 𝑢1 ∈ 𝒦(γ𝑧(𝐺)) as a dominating set, which is adjacent to all other vertices. hence, the domination number is 1. by theorem 2.2, this theorem can be proved. 3.various graph energies on 𝓚(𝚪𝒛(𝑫𝟐𝒏)) and 𝓚(𝚪𝒛(𝑸𝟒𝒏)) theorem 3.1: the adjacency energy on clique graph of cyclic subgroup graph for a dihedral group of order 2𝑛, 𝑛 ∈ ℕ and 𝑛 > 2 is 𝐸 (𝒦(γ𝑧 (𝐷2𝑛))) = { 2𝑛 𝑖𝑓 𝑛 = 𝑝, 𝑝2 2(𝑛 + 1) 𝑖𝑓 𝑛 = 𝑝𝑞 proof: case i: for 𝑛 = 𝑝, 𝑝2 the vertex set of 𝒦(γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+1}. the adjacency matrix can be written as 𝐴 = 𝐽𝑛+1 − 𝐼𝑛+1. the obtained characteristic polynomial is (𝑥 − 𝑛)(𝑥 + 1)𝑛 . the spectrum of 𝒦(γ𝑧 (𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧(𝐷2𝑛))) = { 𝑛 1 −1 𝑛 }. for 𝑛 = 𝑝, 𝑝2, 𝐸 (𝒦(γ𝑧(𝐷2𝑛))) = 2𝑛 case ii: for 𝑛 = 𝑝𝑞 the vertex set of 𝒦(γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. the adjacency matrix can be written as 𝐴 = 𝐽𝑛+2 − 𝐼𝑛+2. the obtained characteristic polynomial is (𝑥 − (𝑛 + 1))(𝑥 + 1)𝑛+1. the spectrum of 𝒦(γ𝑧 (𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧(𝐷2𝑛))) = { 𝑛+1 1 −1 𝑛+1 }. for 𝑛 = 𝑝𝑞, 𝐸 (𝒦(γ𝑧 (𝐷2𝑛))) = 2(𝑛 + 1). theorem 3.2: if 𝑛 = 𝑝, then the eigen values of 𝐴 (𝒦(γ𝑧 (𝑄4𝑛))) are 𝑛 + 1 with multiplicity 1 &−1 with multiplicity 𝑛 + 1 and 𝐸 (𝒦(γ𝑧 (𝑄4𝑛))) = 2(𝑛 + 1). proof: the vertex set of 𝒦(γ𝑧(𝑄4𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. the adjacency matrix can be written as 𝐴 = 𝐽𝑛+2 − 𝐼𝑛+2. the obtained characteristic polynomial is −(𝑥 − (𝑛 + 1))(𝑥 + 1)𝑛+1. the spectrum of 𝒦(γ𝑧 (𝑄4𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧(𝑄4𝑛))) = { 𝑛+1 1 −1 𝑛+1 }. for 𝑛 = 𝑝, 𝐸 (𝒦(γ𝑧 (𝑄4𝑛))) = 2(𝑛 + 1). 374 some graph parameters of clique graph of cyclic subgroup graph on certain non-abelian groups theorem 3.3: the closed neighborhood energy on clique graph of cyclic subgroup graph for a dihedral group of order 2𝑛, 𝑛 ∈ ℕ and 𝑛 > 2 is 𝐸𝑁 (𝒦(γ𝑧 (𝐷2𝑛))) = { 𝑛 + 1 𝑖𝑓 𝑛 = 𝑝, 𝑝2 𝑛 + 2 𝑖𝑓 𝑛 = 𝑝𝑞 proof: case i: for 𝑛 = 𝑝, 𝑝2 the vertex set of 𝒦(γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+1}. the closed neighborhood matrix can be written as 𝑁 = 𝐽𝑛+1. the obtained characteristic polynomial is (𝑥 − (𝑛 + 1))𝑥𝑛. the closed neighborhood spectrum of 𝒦(γ𝑧(𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧 (𝐷2𝑛))) = { 𝑛+1 1 0 𝑛 }. for 𝑛 = 𝑝, 𝑝2, 𝐸𝑁 (𝒦(γ𝑧(𝐷2𝑛))) = 𝑛 + 1. case ii: for 𝑛 = 𝑝𝑞 the vertex set of 𝒦(γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. the closed neighborhood matrix can be written as 𝑁 = 𝐽𝑛+2. the obtained characteristic polynomial is (𝑥 − (𝑛 + 2))𝑥𝑛+1. the closed neighborhood spectrum of 𝒦(γ𝑧(𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧 (𝐷2𝑛))) = { 𝑛+2 1 0 𝑛+1 }. for 𝑛 = 𝑝𝑞, 𝐸𝑁 (𝒦(γ𝑧 (𝐷2𝑛))) = 𝑛 + 2. theorem 3.4: if 𝑛 = 𝑝, then the eigen values of 𝑁 (𝒦(γ𝑧 (𝑄4𝑛))) are 𝑛 + 2 with multiplicity 1 &0 with multiplicity 𝑛 + 1 and 𝐸𝑁 (𝒦(γ𝑧 (𝑄4𝑛))) = 𝑛 + 2. proof: the vertex set of 𝒦(γ𝑧(𝑄4𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. the closed neighbourhood matrix can be written as 𝑁 = 𝐽𝑛+2. the obtained characteristic polynomial is −(𝑥 − (𝑛 + 2))𝑥𝑛+1. the closed neighborhood spectrum of 𝒦(γ𝑧(𝑄4𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧 (𝑄4𝑛))) = { 𝑛+2 1 0 𝑛+1 }. for 𝑛 = 𝑝, 𝐸𝑁 (𝒦(γ𝑧 (𝑄4𝑛))) = 𝑛 + 2. theorem 3.5: the laplacian spectrum on clique graph of cyclic subgroup graph for a dihedral group of order 2𝑛, 𝑛 ∈ ℕ and 𝑛 > 2 is (i) for 𝑛 = 𝑝, 𝑝2, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧 (𝐷2𝑛))) = { 𝑛+1 𝑛 0 1 } (ii) for 𝑛 = 𝑝𝑞, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧(𝐷2𝑛))) = { 𝑛+2 𝑛+1 0 1 }. proof: case i: for 𝑛 = 𝑝, 𝑝2 the vertex set of 𝒦(γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+1}. the laplacian matrix can be written as 𝐿 = (𝑛 + 1)𝐼𝑛+1 − 𝐽𝑛+1. the obtained characteristic polynomial is (𝑥 − (𝑛 + 1))𝑛𝑥. 375 s. ragha and r. rajeswari the laplacian spectrum of 𝒦(γ𝑧 (𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧(𝐷2𝑛))) = { 𝑛+1 𝑛 0 1 }. case ii: for 𝑛 = 𝑝𝑞 the vertex set of 𝒦(γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. the laplacian matrix can be written as 𝐿 = (𝑛 + 2)𝐼𝑛+2 − 𝐽𝑛+2. the obtained characteristic polynomial is(𝑥 − (𝑛 + 2)) 𝑛+1 𝑥. the laplacian spectrum of 𝒦(γ𝑧 (𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧(𝐷2𝑛))) = { 𝑛+2 𝑛+1 0 1 }. theorem 3.6: if 𝑛 = 𝑝, then the eigen values of 𝐿 (𝒦(γ𝑧 (𝑄4𝑛))) are 𝑛 + 2 with multiplicity 𝑛 + 1 &0 with multiplicity 1. proof: the vertex set of 𝒦(γ𝑧(𝑄4𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. the laplacian matrix can be written as 𝐿 = (𝑛 + 2)𝐼𝑛+2 − 𝐽𝑛+2. the obtained characteristic polynomial is −(𝑥 − (𝑛 + 2)) 𝑛+1 𝑥. the laplacian spectrum of 𝒦(γ𝑧 (𝑄4𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧(𝑄4𝑛))) = { 𝑛+2 𝑛+1 0 1 }. theorem 3.7: the signless laplacian spectrum on clique graph of cyclic subgroup graph for a dihedral group of order 2𝑛, 𝑛 ∈ ℕ and 𝑛 > 2 is (𝑖)𝐹𝑜𝑟 𝑛 = 𝑝, 𝑝2, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧 (𝐷2𝑛))) = { 2𝑛 1 𝑛 − 1 𝑛 } (𝑖𝑖)𝐹𝑜𝑟 𝑛 = 𝑝𝑞, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧 (𝐷2𝑛))) = { 2𝑛 + 2 1 𝑛 𝑛 + 1 } proof: case i: for 𝑛 = 𝑝, 𝑝2 the vertex set of 𝒦(γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+1}. the signless laplacian matrix can be written as 𝑆𝐿 = 𝐽𝑛+1 + (𝑛 − 1)𝐼𝑛+1. the obtained characteristic polynomial is (𝑥 − 2𝑛)(𝑥 − (𝑛 − 1))𝑛. the signless laplacian spectrum of 𝒦(γ𝑧(𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧 (𝐷2𝑛))) = { 2𝑛 1 𝑛−1 𝑛 }. case ii: for 𝑛 = 𝑝𝑞 the vertex set of 𝒦(γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. the signless laplacian matrix can be written as 𝑆𝐿 = 𝐽𝑛+2 + 𝑛𝐼𝑛+2. the obtained characteristic polynomial is (𝑥 − (2𝑛 + 2))(𝑥 − 𝑛)𝑛+1. the signless laplacian spectrum of 𝒦(γ𝑧(𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧 (𝐷2𝑛))) = { 2𝑛+2 1 𝑛 𝑛+1 }. 376 some graph parameters of clique graph of cyclic subgroup graph on certain non-abelian groups theorem 3.8: if 𝑛 = 𝑝, then the eigen values of 𝑆𝐿 (𝒦(γ𝑧 (𝑄4𝑛))) are 2𝑛 + 2 with multiplicity 1 &𝑛 with multiplicity 𝑛 + 1. proof: the vertex set of 𝒦(γ𝑧(𝑄4𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. the signless laplacian matrix can be written as 𝑆𝐿 = 𝐽𝑛+2 + 𝑛𝐼𝑛+2. the obtained characteristic polynomial is (𝑥 − (2𝑛 + 2))(𝑥 − 𝑛)𝑛+1. the signless laplacian spectrum of 𝒦(γ𝑧(𝑄4𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(γ𝑧 (𝑄4𝑛))) = { 2𝑛+2 1 𝑛 𝑛+1 }. 4.some topological indices on clique graph of cyclic subgroup graph for certain non-abelian groups theorem 4.1: if 𝒦(γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on a nonabelian group and |𝑉 (𝒦(γ𝑧(𝐺)))| = 𝑚, then the balaban index is 𝐽 (𝒦(γ𝑧 (𝐺))) = 𝑚3−𝑚2 2(𝑚2−3𝑚+4) proof:𝐽 (𝒦(γ𝑧 (𝐺))) = 𝑦 𝑦−𝑥+2 ∑ 1 √𝑤(𝑢).𝑤(𝑣)𝑢𝑣∈𝐸(𝒦(γ𝑧(𝐺))) , where the sum is taken over all edges of a connected graph 𝐺, 𝑥 and 𝑦 are the cardinalities of the vertex and the edge set of 𝐺, 𝑤(𝑢) and 𝑤(𝑣) denoted the sum of distances from u (resp.v) to all other vertices of g. 𝐽 (𝒦(γ𝑧 (𝐺))) = 𝑚2−𝑚 𝑚2−𝑚−2𝑚+4 ( 𝑚(𝑚−1) 2(𝑚−1) ) = 𝑚2−𝑚 𝑚2−3𝑚+4 ( 𝑚 2 ) = 𝑚3−𝑚2 2(𝑚2−3𝑚+4) theorem 4.2: if 𝒦(γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on a nonabelian group and |𝑉 (𝒦(γ𝑧(𝐺)))| = 𝑚, then atom bond connectivity status index is 𝐴𝐵𝐶𝑆 (𝒦(γ𝑧 (𝐺))) = 1 √2 𝑚√𝑚 − 2. proof: 𝐴𝐵𝐶𝑆 (𝒦(γ𝑧 (𝐺))) = ∑ √ 𝜎(𝑢)+𝜎(𝑣)−2 𝜎(𝑢)𝜎(𝑣)𝑢𝑣∈𝐸(𝒦(γ𝑧(𝐺))) = (√ (𝑚 − 1) + (𝑚 − 1) − 2 (𝑚 − 1)(𝑚 − 1) ) 𝑚(𝑚 − 1) 2 = 1 √2 𝑚√𝑚 − 2. theorem 4.3: if 𝒦(γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on a nonabelian group and |𝑉 (𝒦(γ𝑧(𝐺)))| = 𝑚, then the arithmetic-geometric status index is 𝐴𝐺𝑆 (𝒦(γ𝑧(𝐺))) = 𝑚(𝑚−1) 2 . 377 s. ragha and r. rajeswari proof: 𝐴𝐺𝑆 (𝒦(γ𝑧 (𝐺))) = ∑ 𝜎(𝑢)+𝜎(𝑣) 2√𝜎(𝑢)𝜎(𝑣)𝑢𝑣∈𝐸(𝒦(γ𝑧(𝐺))) = 𝑚 − 1 + 𝑚 − 1 2√(𝑚 − 1)(𝑚 − 1) × 𝑚(𝑚 − 1) 2 = 𝑚(𝑚 − 1) 2 theorem 4.4: if 𝒦(γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on a nonabelian group and |𝑉 (𝒦(γ𝑧(𝐺)))| = 𝑚, then (𝑖)the first zagrebdegree eccentricityindex, 𝐷𝐸1 (𝒦(γ𝑧 (𝐺))) = 𝑚 3 (𝑖𝑖)the second zagreb degree eccentricity index, 𝐷𝐸2 (𝒦(γ𝑧 (𝐺))) = 𝑚3(𝑚−1) 2 . proof: (i) 𝐷𝐸1 (𝒦(γ𝑧 (𝐺))) = ∑ (𝑒𝑖 + 𝑑𝑖 ) 2 𝑣𝑖∈𝑉(𝒦(γ𝑧(𝐺))) (where 𝑒𝑖 be the eccentricity and 𝑑𝑖 be the degree) = 𝑚(1 + 𝑚 − 1)2 = 𝑚3 (𝑖𝑖)𝐷𝐸2 (𝒦(γ𝑧(𝐺))) = ∑ (𝑒𝑖 + 𝑑𝑖 )(𝑒𝑗 + 𝑑𝑗 ) 𝑣𝑖𝑣𝑗∈𝐸(𝒦(γ𝑧(𝐺))) = 𝑚(𝑚 − 1) 2 (1 + 𝑚)(1 + 𝑚) = 𝑚3 (𝑚−1) 2 . references [1] s. arumugam, ramachandran, invitation to graph theory, scitech publications pvt. ltd, india, 2006. [2] fred s. roberts and joel h. spencer, a characterizations of clique graphs, journal of combinatorial theory 10,102-108(1971). [3] j. john arul singh, s. devi, cyclic subgroup graph of a finite group, international journal of pure and mathematics, vol. 111 no.3 2016, 403-408. [4] kulli v.r, computation of status neighborhood indices of graphs, international journal of recent scientific research, vol 11, issue 04(b), pp. 38079-38085, april 2020. [5] subarsha banerjee, laplacian spectra of comaximal graph of ℤ𝑛, arxiv:2005.02316v2[math.co] 23 nov 2020. [6] subarsha banerjee, prime coprime graph of a finite group, arxiv:1911.02763v2 [math.co] 3feb 2021. [7] veena mathad and sultan senan mahde, the minimum hub energy of a graph, palestine journal of mathematics, vol.6(1) (2017), 247 – 256. 378 ratio mathematica volume 44, 2022 vlsi implementation of multi-bit error detection and correction codes for space communications poongodi.s1 asoda sunayana rani2 abstract data transmission in advanced space communications are suffering with the different types of noises. further, these noises causeburst errors indata. thus, the error correction codes (ecc) plays the major role to detect and correct the errors. however, the conventional hamming encoders, decoderswere detected and corrected only one bit error. therefore, this work implementation the multi-bit error detection and correctioncodes (mbe-dcc) for multiple bits error detection and correction. initially, mbe-dcc encoding operation is implemented by using generator matrix, which contains both identity bits and parity bits. then, encoded code word is transmitted into the channel of space communication, where encoded data corrupted by different types of noises, errors. therefore, the mbe-dcc decoding operation performed at receiver side of space communications, which corrected all the errors using syndrome detection, error location detection, and error correction modules. the simulations revealed that the proposed mbe-dcc resulted in superior performance than conventional ecc methods. keywords: multi-bit error detection and correction codes, encoding, decoding, syndrome decoding, error analysis, error correction modules. ams classification:05c123 1professor, dept. of ece, cmr engineering college, hyderabad. dr.poongodi@cmrec.ac.in 2pg student, cmr engineering college, hyderabad. sunayana7007@gmail.com 3received on june 9th, 2022. accepted on sep 1st, 2022. published on nov30th, 2022. doi: 10.23755/rm.v44i0.897. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 107 mailto:dr.poongodi@cmrec.ac.in mailto:sunayana7007@gmail.com poongodi.s & asoda sunayana rani 1. introduction the space communicationscontain the channel and to store a value memory is used in channel. in which random access memory (ram) [1] has been the developing and most compatible device in particular for microprocessor and microcontroller. main memory helps to keep the processor or controller busy and to perform any task in a minimum time [2]. to achieve this on-chipmemory termed as cache memory is used. this type of main memory is developed and placed at different levels close to the processor on same die, additionally to reduce the access time of the processor for the data that to be processed from the main memory (ram) [3]. moreover, the data saved in the cache memory need to be the actual that has been stored during the write operation. nevertheless, stored value need not be the same as that was stored. probability of change in stored value is increasing linearly because of rapid change in semiconductor processing technology. to overcome this issue of reliability in cache memory ecc is imperative [4]. various works [5,6] proposed three models on memory with ecc, of which first model is data communication between memory and cache, second model is the same that includes the additional copy. last model is the cross-switch communication between the original and additional memory with ecc. for all the model’s memory designed is a simple single row address memory with ecc as a separate block [7]. ecc block for the memory is an additional hardware design which increases the delay in area, howeveradditional overhead in area and latency enhances the performance degradation of the system. performance degradation of a system that includes cache memory can be improved, provided the decoding time for detection [8], correction and the ability to correct or detect with some failure should be a fraction of total memory design size. in recent studies memory with ecc is inherent for protecting the memory from errors in particular soft errors. multiple cell upset (mcu) [9] denotes arbitrary number of cells which is repeatedly affected by soft error is random and most of the cases it is not adjacent. however due to scaling down of the gate width of transistors, density of transistors increases in this case occurrence of soft error has the tendency of dissemination. to mitigate this new and optimized coding algorithms are proposed and being proposed on adjacent error also called as burst errors. most common adjacent errors are two, three termed as double adjacent [10], triple adjacent [11] respectively. however, the performance degradation of a system cannot be compromised for which optimized syndrome computation is considered. therefore, the major contributions of this work are as follows: • implementation of mbe-dcc for multiple bits error-detection and correction in space communication applications. • mbe-dcc encoder is developed with generator matrix, which generates the encoded code word. • implementationof mbe-dcc decoder with syndrome detection, error location detection, and error correction modules. 108 vlsi implementation of multi-bit error detection and correction codes for space communications • implementation of syndrome detection is introduced for detecting status of error, which results error presented or absented in encoded data. • implementation of error location detection module for identifying the number of error bits with their position. • implementation of error correction module for correcting all the errors in encoded data. rest of the article is organized as follows: section 2 deals with literature survey, section 3 deals with the proposed mbe-dcc implementation, section 4 deals with analysis of results with performance comparison, section 5 concludes the article with possible future directions. 2. literature survey cosmic rays and ic packaging dye made from radioactive elements are few significant sources of soft error in cache memory. furthermore, recent studies on cache memories have shown that reliability issues in cache memory for application that are used to store or during processing of data is significant concern. especially graphical processing units [12] currently developed by nividia, amd and intel are designed with high bandwidth cache memories and frequently subjected to reliability problems. to substantive this, in [13] authors presented a flexible ecc designed for 32-byte cache memory which also consumes low energy for data fetching. in [14] authors compared different error detection techniques and developed a method which has error guard coverage of 97.9 % using tag in cache. the tags are used for index identification. a limitation is that the adjacent location tag in cache memory may be having the same bits that leads to a faulty read or write operation in the cache memory. the change in bits of tag is due to alpha particles. this can be reduced by deploying ecc, sec – ded, in-cache replication (icr), and in combination of sec– ded [15], icr and sec – ded parity. however, the proposed icr method has good fault coverage. in [16] authors proposed counter, shifter, multiplexer and comparator were used and the additional peripherals added to the development of icr contributed to additional overhead in area. even after adding the addition peripherals, the delay and area are found to be less. this technique [17] lags because the disadvantage is if the size of the cache increases in terms of level 1, 2, the method proposed increases the complexity of peripherals in 11 proportion to the chance in the size of cache. also, here only one error is possible to correct. moreover, triple adjacent location is not addressed. in [18] authors proposed a method to reduce energy overhead in dram (cache) achieved through ecc. dram is not modified in design instead, for the usual dram a new ecc that access or decodes only the error word is presented. conventionally only encoded input data is stored, which is decoded to correct or detect the soft error using ecc in a memory. in contrary only error word is corrected using hamming code and error in a word is detected by parity codes. in [19] authors also proposed a dram (cache) with ecc, unlike other methods [20] in particular proposed a method that detects hard errors using a 109 poongodi.s & asoda sunayana rani bist and build in self repair circuits on a chip. ecc that is capable to detect and correct physical fault and soft error is achieved through redundancy circuit or spare circuit and ecc respectively. however, study outlined above of dram with ecc [21] is limited to detect and correct two bits and single bit respectively and correcting soft error beyond one error is not possible. most widely implemented memory circuit is made to store data in sram. there are few applications in which sram are neither used as cache nor main memory. for instance, sram is used as a configurable switch, programmed to connect the logic blocks, technically termed as routing this architecture [22] is implemented in commercial fpga ‘s such as xilinx virtex 4 and altera stratix, moreover sram ‘s is used as configuration frames, which occupies more than 80% area in fpga, such as xilinx virtex 6 and fpga of altera family. however, both reconfigurable and configurable frames designed using sram are most probably sensitive to soft errors. in [23] authors proposed a scheme to correct multiple bits upset in configurable frame of fpga. to detect mbu by combining scrubbing and erasure code additionally to detect error interleaving-n-dimensional (ind) method is also proposed. this is also implemented in virtex-6 xlv240t, which has less overhead in area. since in proposed ind method [24], with reduced repeated parity bits in all dimensions at regular intervals helps for less area occupation in the sram. in [25] authors proposed also presented error correction for switch boxes that are built using sram, also mitigation in soft error is achieved through redundancy method. in which zero optimized sram are used for interconnection and one optimized sram are redundant interconnection termed short and open faults respectively. however major work presented is on optimized routing algorithm. 3. proposed method this section gives the detailed analysis of proposed mbec method. figure 1 shows the flowchart of proposed mbec method.theproposed mbe-dcc is implemented for multiple bits error-detection and correction in space communication applications.initially, mbe-dcc encoder is developed with generator matrix, which generates the encoded code word. here, the matrix multiplication is operation is performed that the generator matrix and data input, which generates the code word. then, the code-word is transmitted in channel of space engineering, where data bits are corrupted by different types of errors and noises. further, mbe-dcc decoder is developed with syndrome detection, error location detection, and error correction modules.here, syndrome detection is implemented for detecting status of error, which results error presented or absented in encoded data.then, error location detection module is introduced for identifying the number of error bits with their position. then, error correction module is developed for correcting all the errors in encoded data. 110 vlsi implementation of multi-bit error detection and correction codes for space communications figure 1.proposed mbe-dcc flowchart 3.1. mbe-dcc encoding the operation of mbe – dccencoding is achieved by performing the mathematicalmatrix multiplication between generator matrix and data input. 𝑉 = 𝐷𝐺 (1) here, 𝑉 is the encoded code word, g is the generator matrix, 𝐷 is the input data. all of them are binary linear block codes. the process used to design these codes is based on some rules for linear block codes construction. in this paper, the proposed codes are also binary linear block codes and obey similar construction rules. normally, the binary codes are described by the number of data-bits, k, redundancy bits, (n − k), and the block size of the encoded-word, n. an (n, k) code is defined by its generator matrix g or parity check matrix h in (2) where 𝐼𝑘×𝑘 is the identity matrix, p is the matrix with size k × (n − k), and 𝑃 𝑇 is the transpose of p. in the encoding process, the generator matrix g is used to encode the data bits through the process. table 1. construction of generator matrix. c 0 c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 c 10 c 11 c 12 c 13 c 14 c 15 c 16 c 17 c 18 c 19 c 20 c 21 c 22 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 111 poongodi.s & asoda sunayana rani 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 table 1 shows the final constructed generator matrix, which contains identity matrix and parity bits. here, parity bits contain size as 16x7, which are ranged from all rows with c0c6 columns. further, the size of identity bits contains size as 16x16 i.e., all rows with c7c22columns. the size of data is 16 bits as follows 𝐷 = [𝐷1, 𝐷2, 𝐷3, 𝐷4, 𝐷5, 𝐷6, 𝐷7, 𝐷8, 𝐷9, 𝐷10, 𝐷11, 𝐷12, 𝐷13, 𝐷14, 𝐷15, 𝐷16] (3) finally, encoding operation is achievedthrough calculation of check bits (c1 to c7)as follows: 𝐶1 = 𝐷1⨁𝐷4⨁𝐷6⨁𝐷8⨁𝐷9⨁𝐷10⨁𝐷14 (4) 𝐶2 = 𝐷2⨁𝐷4⨁𝐷5⨁𝐷7⨁𝐷8⨁𝐷11⨁𝐷15 (5) 𝐶3 = 𝐷3⨁𝐷7⨁𝐷11⨁𝐷13⨁𝐷16⨁𝐷10 (6) 𝐶4 = 𝐷1⨁𝐷4⨁𝐷8⨁𝐷10⨁𝐷12⨁𝐷13 (7) 𝐶5 = 𝐷2⨁𝐷5⨁𝐷6⨁𝐷7⨁𝐷8⨁𝐷13⨁𝐷14 (9) 𝐶6 = 𝐷2⨁𝐷6⨁𝐷7⨁𝐷11⨁𝐷13⨁𝐷16 (10) 𝐶7 = 𝐷3⨁𝐷6⨁𝐷9⨁𝐷11⨁𝐷12⨁𝐷13⨁𝐷15⨁𝐷16 (11) finally, encoding operation is achieved as follows: 𝑉 = [ 𝐶1, 𝐶2, 𝐶3, 𝐶4, 𝐶5, 𝐶6, 𝐶7, 𝐷1, 𝐷2, 𝐷3, 𝐷4, 𝐷5, 𝐷6, 𝐷7, 𝐷8, 𝐷9, 𝐷10, 𝐷11, 𝐷12, 𝐷13, 𝐷14, 𝐷15, 𝐷16 ] (12) 3.2 mbe-dcc decoding the mbe-dcc decoding is consisting of syndrome detection, error location detection and error correction stages. the encoded output 𝑉 is transmitted into space communication channel, where different types of noises were added. 𝑅 = 𝑉 + 𝐸 𝑅𝐶 = [𝐶1, 𝐶2, 𝐶3, 𝐶4, 𝐶5, 𝐶6, 𝐶7] + [𝐸1, 𝐸2, 𝐸3, 𝐸4, 𝐸5, 𝐸6, 𝐸7] 𝑅𝐷 = [𝐷1, 𝐷2, 𝐷3, 𝐷4, 𝐷5, 𝐷6, 𝐷7, 𝐷8, 𝐷9, 𝐷10, 𝐷11, 𝐷12, 𝐷13, 𝐷14, 𝐷15, 𝐷16] + [𝐸8, 𝐸9, 𝐸10, 𝐸11, 𝐸12, 𝐸13, 𝐸14, 𝐸15, 𝐸16, 𝐸17, 𝐸18, 𝐸19, 𝐸20, 𝐷𝐸21, 𝐸22, 𝐸23] 𝑅 = [𝑅𝐶, 𝑅𝐷] 𝑅 = [ 𝑅𝐶1, 𝑅𝐶2, 𝑅𝐶3, 𝑅𝐶4, 𝑅𝐶5, 𝑅𝐶6, 𝑅𝐶7, 𝑅𝐷1, 𝑅𝐷2, 𝑅𝐷3, 𝑅𝐷4, 𝑅𝐷5, 𝑅𝐷6, 𝑅𝐷7, 𝑅𝐷8, 𝑅𝐷9, 𝑅𝐷10, 𝑅𝐷11, 𝑅𝐷12, 𝑅𝐷13, 𝑅𝐷14, 𝑅𝐷15, 𝑅𝐷16 ] here, 𝑉 is the encoded code word, which stores into memory or transmitted into channel. further, 𝐸 represents the error and 𝑅 represents the received vector with error. 112 vlsi implementation of multi-bit error detection and correction codes for space communications 𝑆 = 𝑅. 𝐻𝑇 (6) here, 𝑆 represents the syndrome value, 𝐻 represents the parity check matrix and it is constructed from generator matrix as shown in table 2. the size of identity bits (𝐼) are 7x7 i.e., all rows with h1-h7columns and size of the parity bits (𝑃) are 7x16 i.e., all rows with h8-h23columns. further, if the 𝑆 is zero, which indicates no errors in received data. further, if the 𝑆 is not equal to zero, which indicates errors presented in received data. table 2. construction of parity check matrix. h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 h 1 0 h 1 1 h 1 2 h 1 3 h 1 4 h 1 5 h 1 6 h 1 7 h 1 8 h 1 9 h 2 0 h 2 1 h 2 2 h 2 3 1 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 1 1 0 1 1 r c 1 r c 2 r c 3 r c 4 r c 5 r c 6 r c 7 r d 1 r d 2 r d 3 r d 4 r d 5 r d 6 r d 7 r d 8 r d 9 r d 1 0 r d 1 1 r d 1 2 r d 1 3 r d 1 4 r d 1 5 r d 1 6 finally, the syndrome calculation is simplified as follows: 𝑆1 = 𝑅𝐶1⨁𝑅𝐷1⨁𝑅𝐷4⨁𝑅𝐷6⨁𝑅𝐷8⨁𝑅𝐷9⨁𝑅𝐷10⨁𝑅𝐷14 𝑆2 = 𝑅𝐶2⨁𝑅𝐷2⨁𝑅𝐷4⨁𝑅𝐷5⨁𝑅𝐷7⨁𝑅𝐷8⨁𝑅𝐷11⨁𝑅𝐷15 𝑆3 = 𝑅𝐶3⨁𝑅𝐷3⨁𝑅𝐷7⨁𝑅𝐷11⨁𝑅𝐷13⨁𝑅𝐷16⨁𝑅𝐷10 𝑆4 = 𝑅𝐶4⨁𝑅𝐷1⨁𝑅𝐷4⨁𝑅𝐷8⨁𝑅𝐷10⨁𝑅𝐷12⨁𝑅𝐷13 𝑆5 = 𝑅𝐶5⨁𝑅𝐷2⨁𝑅𝐷5⨁𝑅𝐷6⨁𝑅𝐷7⨁𝑅𝐷8⨁𝑅𝐷13⨁𝑅𝐷14 𝑆6 = 𝑅𝐶6⨁𝑅𝐷2⨁𝑅𝐷6⨁𝑅𝐷7⨁𝑅𝐷11⨁𝑅𝐷13⨁𝑅𝐷16 𝑆7 = 𝑅𝐶7⨁𝑅𝐷3⨁𝑅𝐷6⨁𝑅𝐷9⨁𝑅𝐷11⨁𝑅𝐷12⨁𝑅𝐷13⨁𝑅𝐷15⨁𝑅𝐷16 then, based on syndrome values error locations were identified using bit pattern matching process. it is explained by following example. 𝐷 = [1,1,0,1,1,1,0,0,1,1,0,0,1,1,1,1] then, 𝐶values are calculated using equations (4)-(11) and resulted as follows: 𝐶 = [0,0,1,0,1,0,1] then, encoded codeword v becomes, 𝑉 = [0,0,1,0,1,0,1,1,1,0,1,1,1,0,0,1,1,0,0,1,1,1,1] 113 poongodi.s & asoda sunayana rani consider 4 bits are corrupted and error occurred in [d2, d3, d4, d5] positions of 𝑉. then, r becomes 𝑅 = [0,0,1,0,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0,1,1,1,1] the error locations are identified performing all combinations of xor in h-matrix columns and the xor outcome will be matched with syndrome for anyone of the combination.the process is repeated until the syndrome is matched. table 3 illustrates the error location identification process. table 2. error location identification. h9 h10 h11 h12 xor (h9, h10, h11, h12) s 0 0 1 0 1 1 1 0 1 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 1 1 the syndrome is matched with the xor (h9, h10, h11, h12) combination according to table 3, so error locations become [𝐷2, 𝑅𝐷3, 𝑅𝐷4, 𝑅𝐷5].the error occurred positions in r vector is [𝑅𝐷2, 𝑅𝐷3, 𝑅𝐷4, 𝑅𝐷5], which are equivalent to h vector positions as [h9, h10, h11, h12]. finally, error correction operation is implemented by performing the complement of error corrected bits. the resultant error corrected outcome is obtained as follows 𝑂𝑢𝑡 = [0,0,1,0,1,0,1,1,1,0,1,1,1,0,0,1,1,0,0,1,1,1,1] 4. results and discussions xilinx ise software was used to create all of the mbe – dccdesigns. this software programmed gives two types of outputs: simulation and synthesis. the simulation results provide a thorough examination of the mbe – dccarchitecture in terms of input and output byte level combinations. decoding procedure approximated simply by applying numerous combinations of inputs and monitoring various outputs through simulated study of encoding correctness. the use of area in relation to the transistor count will be accomplished as a result of the synthesis findings. in addition, a time summary will be obtained with regard to various path delays, and a power summary will be prepared utilizing the static and dynamic power consumption. 114 vlsi implementation of multi-bit error detection and correction codes for space communications figure 2. simulation outcome of mbe-dcc. figure 5 represents the simulation outcome of mbe-dcc. here, data denotes the initial input (in), error_in is the manual error input, and enc_out denotes the encoded operand as a whole. the dec_out is the decoded output data that was error-free and is identical to the input data. figure 3. design summary. figure 3 shows the design (area) summary of proposed method. here, the proposed method utilizes the low area in terms of slice luts i.e., 55 out of available 17600. figure 4. time summary figure 4 shows the time summary of proposed method. here, the proposed method consumed total 0.321ns of time delay, which is entirely route delay. 115 poongodi.s & asoda sunayana rani figure 5. power summary. figure 12 shows the power consumption report of propsoed mbe-dcc. here, the proposed method consumed power as 1.065 watts. table 4 compares the performance evaluation of various mbe-dccapproches. here, the propsoed mbe-dccresulted in superior (reduced) performance in terms of luts, time-delay, and power consumption as compared to conventional approaches such as lbc [22], stbc [23], and turbo [24]. table 4. performance evaluation. metric lbc[22] stbc [23] turbo[24] proposed mbe-dcc luts 78 72 64 55 time delay (ns) 3.28 2.284 1.453 0.321 power consumption (w) 3.45 2.34 1.79 1.065 5. conclusion the mbe-dcc for multiple bits error detection and correction is implemented in this work. the initial implementation of mbe-dcc encoding employs a generator matrix that has both identity bits and parity bits. then, encoded data that has been corrupted by various noises and errors is transmitted into the space communication channel. thus, using error location detection, syndrome detection, and error correction modules, the mbe-dcc decoding operation was carried out at the receiver side of space communications. the simulations showed that the proposed mbe-dcc performed better than traditional ecc techniques.this work can be extended with advanced multiple bit error detection and correction codes for real time applications. 116 vlsi implementation of multi-bit error detection and correction codes for space communications references [1] somashekhar, vikas maheshwari, and r. p. singh. 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[30] s.poongodi and dr.b. kalaavathi,” data hiding in watermarking technique with effective key length using spread spectrum technique”, australian journal of basic and applied sciences,sep 2014,100-105. 119 ratio mathematica volume 44, 2022 the upper and forcing fault tolerant geodetic number of a graph t. jeba raj 1 k. bensiger 2 abstract a fault tolerant geodetic is said to be minimal fault tolerant geodetic set of if no proper subset of is a fault tolerant geodetic set of is called the upper fault tolerant geodetic number of is denoted by . some general properties satisfied by this concept are studied. for connected graphs of order with to be is given. it is shown that for every pair of with , there exists a connected graph such that and , where is the fault tolerant geodetic number of and is the upper fault tolerant geodetic number of a graph. let s be a -set of . a subset is called a forcing subset for if is the unique -set containing t. a forcing subset for of minimum cardinality is a minimum forcing subset of . the forcing fault tolerant geodetic number of s, denoted by , is the cardinality of a minimum forcing subset of . the forcing fault tolerant geodetic number of , denoted by is , where the minimum is taken over all -sets in . the forcing fault tolerant geodetic number of some standard graphs are determined. some of its general properties are studied. it is shown that for every pair of positive integers and with and there exists a connected graph such that and keywords: tolerant geodetic, connected graphs, minimum cardinality. 2010 ams subject classification:05c12, 05c69 3 1 assistant professor, department of mathematics, malankara catholic college, mariagiri, kaliyakkavilai 629 153, india, email: jebarajmath@gmail.com 2 register number. 20123082091004, research scholar, department of mathematics, malankara catholic college, mariagiri, kaliyakkavilai 629 153, india. email: bensigerkm83@gmail.com (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamil nadu, india.) 3 received on june23rd, 2022. accepted on aug 10 th , 2022. published on nov30th, 2022.doi: 10.23755/rm.v44i0.903. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by license agreement. 167 t. jeba raj & k. bensiger 1. introduction by a graph we mean a finite, undirected connected graph without loops or multiple edges. the order and size of are denoted by and respectively. for basic graph theoretic terminology, we refer to [9]. two vertices and of said to be adjacent in if the neighborhood of the vertex in is the set of vertices adjacent to . the degree of the vertex is if is an edge of a graph with and then we call an end edge, a leaf and a support vertex. for any connected graph , a vertex is called a cut vertex of if is disconnected. the subgraph induced by set of vertices of a graph is denoted by with and . a vertex is called an extreme vertex of if is complete. a vertex is an internal vertex of an path if is a vertex of and an edge of is an internal edge of an path if is an edge of with both of its ends or in . the distance between two vertices and in a connected graph is the length of a shortest path in . an path of length is called an geodesic. a vertex is said to lie on an geodesic if isa vertex of including the vertices and . for a vertex of the eccentricity is the distance between and a vertex farthest from . the minimum eccentricity among the vertices of is the radius, and the maximum eccentricity is its diameter, . we denote by and by the closed interval consists of and all vertices lying on some geodesic of . for a non-empty set , the set is the closure of a set is called a geodetic set if thus every vertex of is contained in a geodesic joining some pair of vertices in . the minimum cardinality of a geodetic set of is called the geodetic number of and is denoted by . a geodetic set of minimum cardinalities is called -set of . for references on geodetic parameters in graphs see [4, 5, 6, 7, 10]. let be a geodetic set of . be the set of extreme vertices of . then is said to be a fault tolerant geodetic set of , if is also a geodetic set of for every the minimum cardinality of a fault tolerant geodetic set is called fault tolerant geodetic number and is denoted by . the minimum fault tolerant geodetic dominating set of is denoted by -set of the following theorem is used in the sequel. theorem 1.1. [6] each extreme vertex of a connected graph belongs to every geodetic set of g. 168 the upper and forcing fault tolerant geodetic number of a graph 2. the upper fault tolerant geodetic number of a graph definition 2.1. a fault tolerant geodetic is said to be minimal fault tolerant geodetic set of if no proper subset of is a fault tolerant geodetic set of is called the upper fault tolerant geodetic number of is denoted by example 2.2. for the graph given in figure 2.1, is a -setof so that . let then is a minimal faulttolerant geodetic set of and so it is easily verified that there is no faulttolerant geodetic set of with cardinality more than six. therefore . observation 2.3. (i) for a connected graph of order (ii) no cut vertex of belongs to any minimal fault tolerant geodetic set of . (iii) each extreme vertex of belong to any minimal fault tolerant geodetic set of g. theorem 2.4. for the complete graph proof: this follows from observation 2.3(iii). ∎ theorem 2.5. for any non-trivial tree, number of end vertices proof: this follows from observation 2.3(ii) and (iii). ∎ theorem 2.6. for the cycle = proof: let n be even. let be the antipodal vertex of and be the antipodal vertex of , where . then } is a minimal fault tolerant geodetic set of and so . we prove that = . on the contrary, suppose that . 169 t. jeba raj & k. bensiger then there exists a geodetic set such that . then there exist at least two pair of antipodal vertices of g. hence it follows that , which is a contradiction. therefore = 4. let be odd. let and be two adjacent vertices of . let and be antipodal vertices of and be two antipodal vertices of . then is a minimal fault tolerant geodetic set of and so ≥ 5. we prove that = 5. on the contrary, suppose that ≥ 6. then there exists a fault tolerant geodetic set of such that then contains two pair of antipodal vertices. which implies , which is a contradiction. therefore = 5. ∎ theorem 2.7. let g be the complete bipartite graph = proof: let and be the two bipartite sets of . let where is a fault tolerant geodetic set of and so . we prove that = . on the contrary, suppose that ≥ . then there exists a fault tolerant geodetic set of such that . which implies , which is a contradiction. therefore .∎ theorem 2.8. for every pair of with , there exists a connected graph such that and proof: let be a path on five vertices and let be a graph obtained from and by joining each with and . let be the graph obtained from by introducing the vertices and join each with . the graph is shown in figure 2.2. first, we prove that . let be the set of end vertices of . by observation 2.3(iii), is a subset of every fault tolerant geodetic set of . it is easily verified that there is no fault tolerant geodetic set of cardinality less than a and so let is a fault tolerant geodetic set of so that next, we prove that . let then is a minimal fault tolerant geodetic set of and so we prove that on the contrary, suppose that then there existsa fault tolerant geodetic set of such that since and it follows that either or , which is a contradiction. therefore ∎ 170 the upper and forcing fault tolerant geodetic number of a graph 3. the forcing fault tolerant geodetic number of a graph definition 3.1. let be a -set of . a subset is called a forcing subset for if is the unique -set containing . a forcing subset for of minimum cardinality is a minimum forcing subset of . the forcing fault tolerant geodetic number of , denoted by is the cardinality of a minimum forcing subset of . the forcing fault tolerant geodetic number of , denoted by , is where the minimum is taken over all sets in example 3.2. for the graph given in figure 3.1, are the only two -sets of so that so that and the next theorem follows immediately from the definition of the forcing fault tolerant geodetic number of the graph. 171 t. jeba raj & k. bensiger theorem 3.3. for any connected graph in the following we determine the forcing fault tolerant geodetic number of some standard graphs. theorem 3.4. for a non-trivial tree proof: since for a tree , the set of end vertices of is the unique -set of hence it follows that = 0. ∎ theorem 3.5. for the complete graph proof: since is the unique -set of , ∎ theorem 3.6. for the cycle = proof: let case 1. let is even. for , ) is the unique -set of so that = 0. so, let . let let be a -set of , where is the antipodal vertex of and is the antipodal vertex of . since , -set of is not unique and so = 1. since is the unique -set of containing , ( ) = 1 so that = 1. case 2. let is odd. for is the unique -set of so that = 0. so, let . let then it is easily verified that no singleton or two element subsets of a -set is not a forcing subset of . let = then is the -set of containing { . therefore = 3. ∎ theorem 3.7. for the fan graph = 0. proof: let and is the unique -set of so that = 0. ∎ theorem 3.8. for the wheel graph , = 0. proof: let and is the unique -set of so that = 0. ∎ theorem 3.9. for the complete bipartite graph = proof: let } and be the two partite sets of let . then by theorem, = 0. 172 the upper and forcing fault tolerant geodetic number of a graph for and . then is the unique -set of so that = 0. so let . then is the unique -set of so that = 0.for , let be a -set of . then any two element subsets of is a forcing subset of and so ≥ 3. let . then { } is a forcing subset of so that = 3. since this is true for al set of , = 3. let and . let be a -set of . then one or two or three element subsets of is a forcing subset of and so ≥ 4. let = { }. then is a forcing subset of so that = 4. since this is true for all -set of , = 4. ∎ theorem 3.10. for every pair of positive integers and with and there exists a connected graph such that = and proof: let be a path on three vertices. let be a copy of path on three vertices. let be the graph obtained from and by adding new vertices and introducing the edges and . the graph is shown in figure 3.2. first, we prove that . let then is a subset of every -set of . let then every -set of contains at least one vertex from each and so ≥ . let then is a -set of so that = b. next, we prove that . by theorem now since every is a subset of every -set of and every -set contains at least one vertex from each every -set is of the form where let be a forcing subset with then there exists for some such that therefore ∎ 173 t. jeba raj & k. bensiger references [1] h.a. ahangar, s. kosari, s.m. sheikholeslami and l. volkmann, graphs with large geodetic number, filomat, 29:6 (2015), 1361 – 1368. [2] h. abdollahzadehahangar, v. samodivkin, s. m. sheikholeslami and abdollah khodkar, the restrained geodetic number of a graph, bulletin of the malaysian mathematical sciences society, 38(3), (2015), 1143-1155. [3] h. abdollahzadeh ahangar, fujie-okamoto, f. and samodivkin, v., on the forcing connected geodetic number and the connected geodetic number of a graph, ars combinatoria, 126, (2016), 323-335. [4] h. abdollahzadehahangar and maryam najimi, total restrained geodetic number of graphs, iranian journal of science and technology, transactions a: science, 41, (2017), 473–480. [5] f. buckley and f. harary, distance in graphs, additionwesley, redwood city, ca, (1990). [6] g. chartrand, p. zhang, the forcing geodetic number of a graph, discuss. math. graph theory, 19 (1999), 45–58. [7] g. chartrand, f. harary and p. zhang, “on the geodetic number of a graph”, networks, 39(1), (2002), 1 6. [8] h. escaudro, r. gera, a. hansberg, n. jafari rad and l. volkmann,” geodetic domination in graphs”, journal of combinatorial mathematics and combinatorial computing, 77, (2011), 88101. [9] t.w. hayes, p.j. slater and s.t. hedetniemi, fundamentals of domination in graphs, boca raton, ca: crc press, (1998). [10] a. hansberg and l. volkmann, on the geodetic and geodetic domination numbers of a graph, discrete mathematics, 310, (2010), 2140-2146. [11] mitre c. dourado, fabio protti, dieter rautenbach and jayme l. szwarcfiter, some remarks on the geodetic number of a graph, discrete mathematics,310, (2010), 832-837. [12] h.m. nuenay and f.p. jamil,” on minimal geodetic domination in graphs”, discussiones mathematicae graph theory, 35, (3), (2015), 403-418. 174 ratio mathematica volume 46, 2023 linear predictor and autocorrelation for noisy and delayed digital signal g.vinu priya* jothilakshmi r† abstract this paper deals with the association between the linear prediction and digital signal modeling and ends up with the suitable ways to predict the signal by considering a stationary signal yn. the linear prediction of signal modeling based on the finite past and the solutions are arrived in a recursive manner. further we analyzed the wiener filter along with spectral theorem and autocorrelation in terms ofpredictive analysis. this estimates the gap function along with delay and noise. the delayed signal’sproperties are analyzed like causal, stability and applied these into optimum filtering. finally the predicted error is compared with linear predictor and wiener filter. then transfer function is applied to estimate the interval function and gap function along with delay. keywords: linear predictor, weiner filter, gapped function, delay, autocorrelation, orthogonal. 2020 ams subject classifications: 39 a10, 39 a45. 1 *pg and research department of mathematics, d.k.m. college for women (autonomous), india. e-mail:vinupriya14@gmail.com, †pg and research department of mathematics, mazharul uloom college, tamil nadu, india. e-mail:jothilakshmiphd@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1083. issn: 1592-7415. eissn: 2282-8214. ©g.vinu priya et al. this paper is published under the cc-by licence agreement. 271 g.vinu priya and jothilakshmi r 1 introduction due to recent developments in digital signal processing and communication technology, and its subfields of spectrum estimation, real-time adaptive signal processing and prediction algorithms comes into an attention of many researchers chen et al. [2006], dogariu et al. [2021] and gland and oudjane [2003]. the unified extension of this digital signal processing is developed in terms of analysis of geometrical point of view, linear estimator and applied various algorithms like gram-schmidt orthogonalizations, lattice realizations and so on. further these concepts deals with the autoregressive extensions and singular autocorrelation matrices and their sinusoidal representations mao et al. [2017]. this motivates us to proceed further with linear prediction makhoul [1975] and pituk [2004]. this paper is organized as follows. section ii focuses the linear prediction and digital signal modeling welch et al. [2006]. section iii applies the autoregressive models into the prediction coefficients. in section iv linear predictions and levinson’s formula are applied in the random input signal. finally, section v concludes the paper. 2 linear predictor the linear prediction and digital signal modeling and ends up with the suitable ways to predict the signal by considering a stationary signal yn. this rules the signal pattern as follows syy(z) = σ 2 �b(z)b(z −1)�n → (b(z)) → yn (1) here b(z) be any filter as bounded, �n be a sequence of noise term by spectral factoring theorem. let ryy(k) be the autocorrelation of yn: ryy(k) = e[yn + kyn] this is used to predict the present value through the past values by using yn−1 = {yi,−∞ < i ≤ n − 1}. if y1(n) = yn−1, then the linear prediction is identified and compared with the optimum wiener filtering and estimated the signal y1(n). now we identify y1(z) = z−1y (z) with the spectral value b(z). now define the optimum filter h(z) as h(z) = 1 σ2�b(z) [ σ2�b(z)b(z −1) b(z−1) ], (2) 272 linear predictor and autocorrelation for noisy and delayed digital signal here b(z) be a causal and stable filter, and extend the causal and stale filter as zb(z) is then zb(z) = z(b1z −1 + b2z −2 + b3z −3 + · · ·) the optimum filter h(z) is then h(z) = z[1− 1 b(z) ] (3) yn → (z−1) −→y1(n) (h(z)) → ŷn/n−1 this filter output is y1(n) and the consequent output is predicted by yn/n−1. the predicted error is defined as �n. in the figure 1, the indicator line separates the figure 1: error predictor through wiener filter linear predictor part and wiener filter part in the signal error prediction [2, 7]. apply the reduction equation (1) in terms of the predicted error filter a(z) as, syy(z) = σ2� a(z)a(z−1 (4) and a(z)syy(z) = σ2� a(z−1) (5) this follows that s�y(z) = a(z)syy(z) (6) 273 g.vinu priya and jothilakshmi r furthermore r�y(k) = e[�nyn−k] = ∞∑ i=0 airyy(k − i) (7) which is recognized by the interval function [3]. now construct �n from the orthogonal complement of yn−1 = yn−k,k = 1,2, . . ., and hence yn−k is orthogonal to all k = 1,2, . . .. therefore, the equation (7) implies r�y(k) = e[�nyn−k] = ∞∑ i=0 airyy(k − i) = 0 (8) this result follows from the z-domain equation of (6) and interval function. applying the symmetry property in (7) provided k = 0 and we get σ2� = e[� 2 n] = e[�nyn] = ryy(0) + a1ryy(1) + a2ryy(2) + ... (9) combined the equations (8) and (9), ∞∑ i=0 airyy(k − i) = σ2�δ(k),k ≥ 0 (10) this normal equation is extended with the parameters {a1,a2, . . . ,σ2�} based on the output signal? yn and this is computed with ryy(k). 3 autoregressive models in general, the prediction coefficients are infinite since the predictor is predicated on the infinite past. when yn is autoregressive, then the signal model b(z) is defined as b(z) = 1 (1 + a1z−1 + a2z−2 + . . . + apz−p) (11) this shows that the prediction filter is polynomial a(z) = 1 + a1z −1 + a2z −2 + . . . + apz −p (12) the output function yn is defined for uncorrelated sequence �n, we get yn + a1yn−1 + a2yn−2 + ... + apyn−p = �n (13) 274 linear predictor and autocorrelation for noisy and delayed digital signal further optimum prediction of yn is written like ŷn/n−1 = −[a1yn−1 + a2yn−2 + ... + apyn−p] (14) here most effective prediction of yn is calculated based on the past p samples. the infinite set of equations (10) or (11) remains valid and the primary p + 1 samples coefficients {1,a1,a2, . . . ,ap}are nonzero [8, 14]. these primary past samples are a part of the equation (11) and these samples are enough to define the parameters of {a1,a2, . . . ,ap;σ2�}:  ryy(0) ryy(1) · · · ryy(p) ryy(1) ryy(0) · · · ryy(p−1) ryy(2) ryy(1) · · · ryy(p−2) ... ... ... ... ryy(p) ryy(p−1) · · · ryy(0)   =   1 a1 a2 ... ap   =   σ2� 0 0 ... 0   (15) these equations are solved efficiently through levinson’s algorithm and this algorithm needs o(p2) operations and o(p) memory locations. o(p3) and o(p2) which is necessary to calculate the inverse of the autocorrelation matrix ryy. the parameters {a1,a2, . . . ,ap;σ2�} completely determines yn. by considering z = ejω in the equation (5) we determine syy(ω) = σ2� |a(ω)|2 = σ2� |1 + a1e−jω + a2e−2jω + . . . + ape−jωp| (16) the normal equations (16) build is used to approximate and estimates the parameters {a1,a2, . . . ,ap;σ2�}. there are many various ways to extract the estimates and the parameters. here are the few methods 1. yule-walker methodology 2. variance methodology and 3. burg’s methodology. autocorrelations ryy(k) of equation (16) is wrriten based on the yule-walker methodology, is ryy(k) = 1 n n−1−k∑ n=0 yn+kyn (17) the primary p + 1 changes are required in (16) as like p ≤ n − 1 based on the parameters {â1, â2, . . . , âp; σ̂2�}. this represents the block of n samples and filter 275 g.vinu priya and jothilakshmi r parameters (i.e. p + 1). to synthesize the random samples, variance σ̂2� would be generated and pass through the generator filter whose coefficients are calculated like, b̂(z) = 1 â(z) = 1 |1 + â1z−1 + â2z−2 + +âpz−p|2 (18) 4 linear predictions and levinson’s formula in this section, we come accross that if the autoregressive random input signal is of order p, then the optimum linear predictor reduces to a predictor of order p. a geometrical method to perceive this property could be extended in to the projection of yn onto the topological subspace based on the output signal {yn−i,1 ≤ i < ∞} and the same could be reduced based on past samples; i.e. {yn−i,1 ≤ i ≤ p}. this generates the output function yn. consider a stationary series (based on time) yn with the autocorrelation function r(k) = e[yn+kyn]. for any given p, the output function takes the following new form consider a stationary series (based on time) yn with the autocorrelation function r(k) = e[yn+kyn]. for any given p, the output function takes the following new form ŷn = −[a1yn−1 + a2yn−2 + ... + apyn−p] (19) the prediction coefficients are chosen to reduce the mean square error as ε = e[e2n] (20) where en is the predicted error and define en as follows en = yn − ŷn = yn + a1yn−1 + a2yn−2 + ... + apyn−p (21) e[enyn−i] = 0, (22) by substituting (21) in the equation (22), we get p linear equations p∑ j=0 aje[yn−jyn−i] = p∑ j=0 r(i− j)aj = 0 (23) by (22), we found the reduced value as σ2� = e[enyn] (24) 276 linear predictor and autocorrelation for noisy and delayed digital signal equations (23) and (24) may be combined into the matrix equation like (p + 1)× (p + 1),  ryy(0) ryy(1) · · · ryy(p) ryy(1) ryy(0) · · · ryy(p−1) ryy(2) ryy(1) · · · ryy(p−2) ... ... . . . ... ryy(p) ryy(p−1) · · · ryy(0)   =   1 a1 a2 ... ap   =   σ2� 0 0 ... 0   (25) which is identical for equation (16) for the autoregressive case. it was necessary to connect the order of the predictor associate with the previous one. hence the lower order optimum predictors also are calculated. consider the gap function as gp(k) = e[( p∑ i=0 apiyn−i)yn−k] = p∑ i=0 apir(k − i) (26) figure 2: gap conditions for the delay these gap conditions are an equivalent because of the orthogonal equations (22) which is illustrated in figure 2. utilizing gp(k) construct a new function with space gp+1(k) from the past p + 1 hence we get, gp(k) → gp(−k). a delay of (p + 1) time can realigned and illustrated in the following figure. this shows the minimum of p and choosen the parameter γp+1 and gp+1(k) adds an additional delay which deviates the length p + 1 are illustrated in the figure 3. 277 g.vinu priya and jothilakshmi r figure 3: gap (delay) conditions for gp(k) and gp(p + 1−k) 5 conclusions in this paper, linear prediction is predicted the present value through the past values. the linear prediction of signal modeling related to finite past and the solutions are arrived in a recursive manner. further we analyzed the wiener filter along with spectral theorem and autocorrelation in terms of predictive analysis. this estimates the gap function along with delay and noise. there will be an indicator line which separates the linear predictor part and wiener filter part in the signal error prediction. this normal equation is extended with the signal parameters based on the output signal yn and this is computed with ryy(k). then the infinite matrix equation is reduced to a finite form and, moreover, the ryy(k) is obviously measurable. finally the predicted error is compared with linear predictor and wiener filter. then transfer function is applied to estimate the interval function and gap function along with delay. finally the gapped function gp(k) and gp(p + 1−k) possess the same value. references j. chen, j. benesty, y. huang, and s. doclo. new insights into the noise reduction wiener filter. audio, speech, and language processing, volume 14. ieee transactions, 2006. l. dogariu, j. benesty, c. paleologu, and s. ciochina. an insightful overview of the wiener filter for system identification. appl. sci., 11(7774), 2021. f. l. gland and n. oudjane. a robustification approach to stability and to uni278 linear predictor and autocorrelation for noisy and delayed digital signal form particle approximation of nonlinear filters: the example of pseudomixing signals. stochastic processes and their applications, 106(2):279–316, 2003. j. makhoul. linear prediction a tutorial review. proceedings of the ieee, 63(4): 561–580, 1975. j. mao, d. ding, y. song, and fe. alsaadi, event-based recursive filtering for time-delayed stochastic nonlinear systems with missing measurements. signal processing, 134:158–165, 2017. m. pituk. a criterion for the exponential stability of linear difference equations, volume 17. 2004. t. b. welch, g. w. c., h., and g. m. m. real-time digital signal processing from matlab to c with the tms320c6x dsk. taylor and francis, 2006. 279 ratio mathematica volume 46, 2023 solving fuzzy linear programming problems by using the fuzzy exponential barrier method a.nagoor gani* r.yogarani† abstract in order to resolve the fuzzy linear programming problem, the fuzzy exponential barrier approach is the major strategy employed in this article. to overcome the problems with fuzzy linear programming, this method uses an algorithm. in this concept, a fuzzy inequality constraint is produced since the objective functions are convex.numerical examples are provided. keywords:fuzzy exponential barrier function, fuzzy exponential barrier convergence,fuzzy optimality solution. 2020 ams subject classifications: 54e20, 54h25. 1 *research department of mathematics, jamal mohamed college(autonomous),affiliated to bharathidasan university,tiruchirappalli-620020,tamil nadu,india. ganijimc@yahoo.co.in. †research department of mathematics, jamal mohamed college(autonomous), affiliated to bharathidasan university,tiruchirappalli-620020, tamil nadu, india.yogaranimaths@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022.published on march 20, 2023. doi: 10.23755/rm.v46i0.1088. issn: 1592-7415. eissn: 2282-8214. ©a.nagoor gani et al. this paper is published under the cc-by licence agreement. 310 a.nagoor gani and r.yogarani 1 introduction fuzzy set theory has aided mathematical modelling and control theory. unsolved decision-making problems can be applied in real-world situations using a variety of techniques. the fuzzy set theory developed by a. et al. [2012] has presented in 1965. fuzzy linear programming was first established in 1997 by tanaka and asai [1984], and fuzzy numbers were published in dubois [1983]. a function was proposed by hsieh [1999] to cope with fuzzy arithmetical operations. using a technique that divides, adds, and multiplies fuzzy integers in addition to subtracting and adding them. the fuzzy linear programming problem, as formulated by mahadevi et al. [2009], a. and yogarani [2021] and fiacco and v. [1990] zimmermann [1978] likewise has a duality. the process of defuzzifying involves converting fuzzy values into clear crispvalues. since a few years ago, these techniques have been thoroughly researched and applied in fuzzy systems mahadevi et al. [2006]. a representative value from a given set, according to some characters, was the main objective of these processes. the defuzzification method establishes a link between all fuzzy sets and all real numbers. given the high cost ofan infeasible solution, moengin and parwadi [2011] et al. fiacco and anthony [1976] developed the exponential barrier function technique. the fuzzy exponential barrier approach is a different way to solve fuzzy linear programming problems. the fuzzy exponential barrier technique includes a fuzzy objective function that specifies a severe penalty for violating the fuzzy requirements in order to approximate fuzzy constraints. a fuzzy barrier function in fuzziness optimization problems that has a value that grows exponentially with the separation from the viable zone. the fuzzy exponential barrier parameter, a positive decreasing parameter used in this process, establishes how close to the original fuzzy unconstrained problem is. in section 2 of this study, some fundamental ideas of fuzzy set theory and the algebraic operation of triangular fuzzy numbers are provided. a fuzzy exponential barrier technique is used to build an algorithm in section 3 to tackle the problem of fuzzy linear programming problem. section 4 includes an example. 2 preliminaries 2.1 : fuzzy set if m̃ = {(x,µ̃m(x)) : x ∈ m,µm̃(x) ∈ [0,1]} defines a fuzzy set m̃ the pair (x,µ m̃ (x))that makes up the membership function has two elements: x, which is a member of the classical set m̃,and the µ m̃ (x) belongs to the interval [0,1]. 2.2 : triangular fuzzy number the triangular fuzzy number is the fuzzy set m̃ = (m1,m2,m3),m1 ≤ m2 ≤ 311 solving fuzzy linear programming problems by using the fuzzy exponential barrier method m3 ∈ r, if the membership function of m̃ is defined by µ m̃ (y) =   y−m1 m2−m1 , m1 ≤ y ≤ m2 m3−y m3−m2 , m2 ≤ y ≤ m3 0, otherwise 3 : fuzzy exponential barrier method primal-dual of the fuzzy linear programming problems maxz̃ = f̃t(ỹ) subject to mjỹ ≥ ñk,j = 1,2. where m1 ∈ rm×n, f̃, ỹ ∈ rn, ñk ∈ rm f̃t = (f1,f2,f3), ñ = (n1,n2,n3) (1) minz̃ = ñ(ỹ) mtj (x̃) ≤ f̃,m2 = mt1 ∈ rm×n, ñtk , f̃ tt = f̃, we may consider the rank of the matrix m1 without losing generality. the triangular fuzzy number is used to specify the fuzzy components in the preceding problem. consider that every problem has at least one feasible solution (1) the fuzzy exponential barrier function is considered by ẽ(ỹ,γ) : rn → rfor every scalar δ > 0 as follows. ẽ(ỹ,γ) = f̃t(ỹ)−γ m∑ i=1 e(bi(ỹ)) β (2) consider ẽ : rn → (−∞,∞) is a function given by bi(ỹ) =mjỹ − ñk ≤ 0,mjỹ − ñk = 0forallj, mjỹ − ñk > 0,mjỹ − ñk 6= 0forallj. 3.1 : fuzzy exponential barrier lemma this lemma is based on the local and global behaviour of the unconstraintsmaximizer of the fuzzy exponential barrier function. statement: an exponential barrier parameter γk is a fuzzy sequence that increased in size. the fuzzy linear programming problems and fuzzy exponential barrier function then for each k condition are below. (i).ẽ(ỹk+1,γk+1) ≥ ẽ(ỹk,γk) (ii).ẽ(ỹk ≤ ẽ(ỹk+1) (iii).f̃t(ỹk) ≥ f̃t(ỹk+1) (iv).f̃t(ỹ∗) ≤ ẽ(ỹk,γk) ≤ f̃t(ỹk) 312 a.nagoor gani and r.yogarani proof : (i).ẽ(ỹk+1,γk+1) =f̃t(ỹk+1)−γk+1 m∑ i=1 e(β(mjỹ k+1−ñk)) ≥ f̃t(ỹk) −γk m∑ i=1 e(β(mjỹ k−ñk)) = ẽ(x̃k,γk) (ii).f̃t(ỹk)−γk m∑ i=1 e(β(mjỹ k+1−ñk)) ≤ f̃t(ỹk+1)γk+1 m∑ i=1 e(β(mjỹ k+1−ñk)) (3) f̃t(ỹk+1)−γk+1 m∑ i=1 e(β(mjỹ k+1−ñk)) ≤ f̃t(ỹk)−γk m∑ i=1 e(β(mjỹ k−ñk)) (4) the inequalities adding (3) and (4), we get f̃t(ỹk)−γk m∑ i=1 e(β(mjỹ k−ñk))+ f̃t(ỹk+1)− m∑ i=1 e(β k+1(mjỹ k−ñk)) ≤ f̃t(ỹk+1)− m∑ i=1 e(β k+1(mjỹ k+1−ñk)) + f̃t(ỹk)−γk m∑ i=1 e(β(mjỹ k−ñk))...(f̃t(ỹk) + f̃t(ỹk+1))− (f̃t(ỹk+1) + (f̃t(ỹk)) + γk m∑ i=1 e(β k(mjỹ k−ñk)) + γk+1 m∑ i=1 e(β k+1(mjỹ k−ñk))) ≤ γk m∑ i=1 e(β k(mjỹ k−ñk)) + γk+1 m∑ i=1 e(β k+1(mjỹ k−ñk))). however (f̃t(ỹk) + f̃t(ỹk+1))− (f̃t(x̃k+1) + f̃t(x̃k)) = 0. then (γk +γk+1)γk ∑m i=1 e (βk(mjỹ k−ñk)) ≥ (γk +γk+1) ∑m i=1 e (βk+1(mjỹ k−ñk)) 313 solving fuzzy linear programming problems by using the fuzzy exponential barrier method since δk ≤ δk+1, ẽ(ỹk+1) ≥ ẽ(ỹk). (iii). from the proof of (i), it can be obtained that f̃t(ỹk)−γk ∑m i=1 e (β(mjỹ k−ñk)) ≥ f̃t(ỹk+1)−γk+1 ∑m i=1 e (β(mjỹ k+1−ñk)). ẽ(ỹk) ≤ ẽ(ỹk+1). then f̃t(ỹk+1) ≥ f̃t(ỹk). (iv). from the proof of (ii) ∑m i=1 e (β(mjỹ k−ñk)) ≤ ∑m i=1 e (β(mjỹ k−ñk)), f̃t(ỹk) ≥ f̃t(ỹk+1). f̃t(ỹ∗) ≥ f̃t(ỹk)−γk m∑ i=1 e(β(mjỹ k−ñk)) = ẽ(ỹk,γk) ≥ f̃t(ỹk). 3.2 : a fuzzy exponential barrier convergence theorem suppose that the fuzzy linear programming problem and the fuzzy exponential barrier functions are both continuous functions. in a fuzzy linear programming problem, an increasing series of positive fuzzy exponential barrier parameters k is necessary such that {γk} → ∞,k → ∞ to ỹ of z̃there is an optimal solution. if the limit pointxis within the boundaries of the range of ỹof z̃, then it exists. proof : let ỹ be a limit point of {ỹk} ẽ(ỹk, γ̃k) = f̃t(ỹk)−γk m∑ i=1 e(β(mjỹ k−ñk)) ≤ f̃t(ỹ∗) lim k→∞ f̃t(ỹk) = f̃t(ỹ), lim k→∞ e(β(mjỹ k−ñk)) ≤ 0 from the lemma 3.2 (iv) we get, lim k→∞ ẽ(ỹk, γ̃k) = f̃t(y∗) ỹ is feasible. 3.2 fuzzy exponential barrier function algorithm 1. acknowledge the fuzzy objective function and constraints of the problem and repharse the problem standard forms to reflect them. write max z̃ = f̃t(ỹ). subject to mjỹk − ñk ≤ 0. 2.the fuzzy exponential barrier function defined by 314 a.nagoor gani and r.yogarani ẽ(ỹ, γ̃) = f̃t(ỹ)−γ m∑ i=1 e(β(mjỹ k−ñk)). 3.3. maxinequality constraints with the fuzzy exponential barrier functions maxẽ(ỹ, γ̃) = f̃t(ỹ)−γ m∑ i=1 e(β(mjỹ k−ñk)). 4.first-order needed conditions for optimality γ →∞, which are used to find the best solution to the specified fuzzy linear programming problems. 5.find ẽ(ỹk, γ̃k) = maxx≥0 ẽ(ỹ, γ̃), then minimize ỹk and γ = 1,k = 1,2, ......,k = i then stop. alternatively, cancontinue to step 5. the fuzzy exponential barrier approach employs the same procedure as the earlier algorithm for the dual fuzzy linear programming problems. 4. numerical example the primal fuzzy linear programming problem min z̃ = (3.75,4,4.25)ỹ1 + (2.75,3,3.25)ỹ2 2ỹ1 + 3ỹ2 ≥ (5.75,6,6.25), 4ỹ1 + ỹ2 ≥ (3.75,4,4.25). solution : figure 1: finding an infeasible solution for primal fuzzy linear programming problem using the fuzzy exponential barrier function algorithm we get, the fuzzy exponential barrier method is given by ẽ(ỹ,γ) = ((3.75,4,4.25)ỹ1 + (2.75,3,3.25)ỹ2 −γ m∑ i=1 e(bi(y)) β according to this strategy, you can convert fuzzy linear programming problems into a standard form of unconstrained problems. ỹ1 = (0.45,0.8,1.15)− 1.2104 γ , ỹ2 = (0.0727,0.6,1.1273)− 1.3951 γ x̃1 = 0.0060 γ + (0.35,0.60,0.85), x̃2 = (0.9518,1.60,2.25) + 0.0180 γ , 315 solving fuzzy linear programming problems by using the fuzzy exponential barrier method for different values of γj,j → ∞ the optimal values of ỹ1(γ), ỹ2(γ) are calculated as listed in tables (1) and (2) we get, table1: primal fuzzy linear programming problem solution(i) no δk ỹ1 ỹ2 1 10 (0.3290,0.6790,1.0290) (-0.0668,0.4605,0.9878) 2 102 (0.4379,0.7879,1.1379) (0.0587,0.5860,1.1133) 3 103 (0.4488,0.7988,1.1488) (0.0713,0.5986,1.1259) 4 104 (0.4499,0.7999,1.1499) (0.0726,0.5999,1.1272) 5 105 (0.45,0.8,1.15) (0.0727,0.6,1.1273) table 2: dual fuzzy linear programming problem solution(i) no γj ỹ1 ỹ2 1 10 (0.3506,0.6006,0.8506) (0.9518,1.6018,2.2518) 2 102 (0.3501,0.6001,0.8501) (0.9502,1.6002,2.2502) 3 103 (0.3500,0.600,0.8500) (0.9500,1.600,2.2500) 4 104 (0.3500,0.600,0.8500) (0.9500,1.600,2.2500) 5 105 (0.3500,0.600,0.8500) (0.9500,1.600,2.2500) the optimal value of the given problem (1) can be obtained by ỹ1 = (0.45,0.8,1.15), ỹ2 = (0.0727,0.6,1.1273),min z̃ = (2.9908,7.2,11.4092). the corresponding given problem of the optimal values for x̃1, x̃2 are to be obtained by x̃1 = (0.35,0.6,0.85), x̃2 = (0.95,1.6,2.25),max z̃ = (4.25,7.2,10.15). 3 conclusions this article outlines a method for solving the primal-dual fuzzy linear programming problem more effectively by combining the fuzzy exponential barrier function and the fuzzy exponential barrier parameter.the table for the primal-dual problem demonstrates how quickly the best solution can be reached using the fuzzy exponential barrier primal-dual algorithm we developed when the fuzzy exponential barrier parameter is used. 316 a.nagoor gani and r.yogarani figure 1: finding an infeasible solution for primal fuzzy linear programming problem references n. a. and r. yogarani. solving fuzzy linear programming problem with a fuzzy polynomial barrier method. international journal of aquatic science, 12:169– 174, 2021. n. a., assarudeen, and s. mohamed. a new operation on triangular fuzzy number for solving fuzzy linear programming problem. applied mathematical science, 6(11):525–532, 2012. h. p. dubois, didier. the mean value of a fuzzy number, volume 24(3). fuzzy sets and system, 1983. fiacco and v. anthony. sensitivity analysis for nonlinear programming using penalty methods, volume 10(1). mathematical programming, 1976. fiacco and p. g. a. m. v., anthony. nonlinear programming: sequential unconstrained minimization techniques, volume 4. 1990. 317 solving fuzzy linear programming problems by using the fuzzy exponential barrier method s.-h. c. hsieh, chihhsun. similarity of generalized fuzzy numbers with graded mean integration representation. proc. 8th int. fuzzy systems association world congr., (2):551–555, 1999. mahadevi, a. n., and s. nasseri. duality in fuzzy number linear programming by use of a certain linear ranking function. applied mathematics and computation, 180(1):206–216, 2006. mahadevi, s.-n. n., amiri, and a. yazdhani. fuzzy primal simplex algorithms for solving fuzzy linear programming problems. 2009. moengin and parwadi. exponential barrier method in solving linear programming problems. international journal of engineering and technology, 12(3), 2011. h. tanaka and k. asai. fuzzy linear programming problems with fuzzy numbers. fuzzy sets and systems, 13(1):1–10, 1984. h. zimmermann. fuzzy programming and linear programming with several objective functions. fuzzy sets and systems, 1(1):45, 1978. 318 ratio mathematica volume 45, 2023 gaussian twin neighborhood prime labeling on fan digraphs palani k * shunmugapriya a † abstract gaussian integers are complex numbers of the form 𝛾 = 𝑥 + 𝑖𝑦 where 𝑥 and 𝑦 are integers and𝑖2 = −1. the set of gaussian integers is usually denoted by ℤ[𝑖]. a gaussian integer 𝛾 = 𝑎 + 𝑖𝑏 ∈ ℤ[𝑖] is prime if and only if either𝛾 = ±(1 ± 𝑖), 𝑁(𝛾) = 𝑎2 + 𝑏2 is a prime integer congruent to 1 (mod 4), or 𝛾 = 𝑝 + 0𝑖 or = 0 + 𝑝𝑖 where 𝑝 is a prime integer and |𝑝| ≡ 3(mod 4). let 𝐷 = (𝑉,𝐴) be a digraph with |𝑉| = 𝑛. an injective function 𝑓: 𝑉(𝐷) → [𝛾𝑛] is said to be a gaussian twin neighborhood prime labeling of 𝐷, if it is both gaussian in and out neighborhood prime labeling. a digraph which admits a gaussian twin neighborhood prime labeling is called a gaussian twin neighborhood prime digraph. in this paper, we introduce some definitions of fan digraphs. further, we establish the gaussian twin neighborhood prime labeling in fan digraphs using gaussian integers. keywords: gaussian integers, neighborhood prime, labeling, digraphs. 2010 ams subject classification: 05c78‡ * pg & research department of mathematics (a.p.c. mahalaxmi college for women, thoothukudi-628 002, tamil nadu, india); palani@apcmcollege.ac.in. † department of mathematics, sri sarada college for women (autonomous), tirunelveli-627 011. (research scholar-19122012092005, a.p.c. mahalaxmi college for women, thoothukudi-628 002, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamil nadu, india); priyaarichandran@gmail.com. ‡ received on july 24, 2022. accepted on september 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.973. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 9 mailto:palani@apcmcollege.ac.in mailto:priyaarichandran@gmail.com palani k and shunmugapriya a 1. introduction graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. the concept of graph labeling was introduced by rosa in [1]. an useful survey on graph labeling by j.a. gallian can be found in [2]. spiral ordering of the gaussian integer was first introduced by hunter lehmann and andrew park in [4]. t. j. rajesh kumar and t. k. mathew varkey [7] introduced the concept of gaussian neighborhood prime labeling of a graph. k. palani [6] et.al, introduced the concept of gaussian twin neighborhood prime labeling in digraphs. let 𝐷 = (𝑉,𝐴) be a digraph of order𝑛. then 𝑉 is the set of vertices of 𝐷 with|𝑉| = 𝑛, and a is the set of arcs of 𝐷 consisting of ordered pairs of distinct vertices. the in-degree 𝑑−(𝑣) of a vertex 𝑣 in a digraph 𝐷 is the number of arcs having 𝑣 as its terminal vertex. the out-degree 𝑑+(𝑣) of 𝑣 is the number of arcs having 𝑣 as its initial vertex [2]. throughout this article we use only digraphs. in this article, we introduce the definition of fan and double fan digraphs by orienting fan and double fan graphs. also, we investigate the existence of gaussian twin neighborhood prime labeling in fan digraphs. 2. preliminaries the following basic definitions and properties are from [4] definition 2.1. gaussian integers are complex numbers of the form 𝛾 = 𝑥 + 𝑖𝑦 where 𝑥 and 𝑦 are integers and𝑖2 = −1. the set of gaussian integers is usually denoted byℤ[𝑖]. a gaussian integer is even if 1 + 𝑖 divides𝛾. otherwise it is an odd gaussian integer. definition 2.2. a gaussian integer 𝛾 = 𝑎 + 𝑖𝑏 ∈ ℤ[𝑖] is prime if and only if either (i) 𝛾 = ±(1 ± 𝑖) (ii) 𝑁(𝛾) = 𝑎2 + 𝑏2 is a prime integer congruent to 1 (mod 4), or (iii) 𝛾 = 𝑝 + 0𝑖 or = 0 + 𝑝𝑖 where 𝑝 is a prime integer and|𝑝| ≡ 3(mod 4). definition 2.3. the spiral ordering of the gaussian integers is recursively defined ordering of the gaussian integers. we denote the 𝑛th gaussian integer in the spiral ordering by 𝛾𝑛. the ordering is defined beginning with 𝛾1 = 1 and continuing as: 𝛾𝑛+1 = { 𝛾𝑛 + 𝑖 if re(𝛾𝑛) ≡ 1 (mod 2),re(𝛾𝑛) > im(𝛾𝑛) + 1 𝛾𝑛 − 1 if im(𝛾𝑛) ≡ 0 (mod 2),re(𝛾𝑛) ≤ im(𝛾𝑛) + 1,re(𝛾𝑛) > 1 𝛾𝑛 + 1 if im(𝛾𝑛) ≡ 1 (mod 2),re(𝛾𝑛) < im(𝛾𝑛) + 1 𝛾𝑛 + 𝑖 if im(𝛾𝑛) ≡ 0 (mod 2),re(𝛾𝑛) = 1 𝛾𝑛 − 𝑖 if re(𝛾𝑛) ≡ 0 (mod 2),re(𝛾𝑛) ≥ im(𝛾𝑛) + 1,im(𝛾𝑛) > 0 𝛾𝑛 + 1 if re(𝛾𝑛) ≡ 0 (mod 2), im(𝛾𝑛) = 0 and [𝛾𝑛] denotes the set of first 𝑛 gaussian integers in the spiral ordering. properties 2.4. 1. a gaussian integer 𝛾 = 𝑥 + 𝑖𝑦 is called a prime gaussian integer if its only divisors are ±1,±𝑖,±𝛾 or ±𝛾𝑖. 10 gaussian twin neighborhood prime labeling of fan digraphs 2. two gaussian integers 𝑥 and 𝑦 are relatively prime if their only common divisors are the units inℤ[𝑖]. 3. let 𝛾 be a gaussian integer and let 𝑢 be a unit. then 𝛾 and 𝛾 + 𝑢 are relatively prime. 4. in the spiral ordering, consecutive gaussian integers are relatively prime. 5. in the spiral ordering, consecutive odd gaussian integers are relatively prime. 6. let 𝛼 be a prime gaussian integer and 𝛾 be a gaussian integer. then 𝛾 and 𝛾 + 𝛼 are relatively prime if and only if𝛼 ∤ 𝛾. 7. let 𝛾 be an odd gaussian integer, let 𝑡 be a positive integer and 𝑢 be a unit. then 𝛾 and 𝛾 + 𝑢(1 + 𝑖)𝑡 are relatively prime. the following definitions are taken from [6] definition 2.5. let 𝐷 = (𝑉,𝐴) be a digraph with |𝑉| = 𝑛. an injective function 𝑓: 𝑉(𝐷) → [𝛾𝑛] is called gaussian in-neighborhood prime labeling of 𝐷, if for every vertex 𝑣 ∈ 𝑉 (𝐷) with𝑑−(𝑣) > 1, the gaussian integers in the set {𝑓(𝑢):𝑢 ∈ 𝑁−(𝑣)} are relatively prime where 𝑁−(𝑣) = {𝑢 ∈ 𝑉 (𝐷) ∶ 𝑢𝑣 ⃗⃗⃗⃗⃗⃗ ∈ 𝐴(𝐷)}. definition 2.6. let 𝐷 = (𝑉,𝐴) be a digraph with|𝑉| = 𝑛. an injective function 𝑓: 𝑉(𝐷) → [𝛾𝑛] is called gaussian out-neighbourhood prime labeling of𝐷, if for every vertex 𝑣 ∈ 𝑉 (𝐷) with𝑑+(𝑣) > 1, the gaussian integers in the set {𝑓(𝑢):𝑢 ∈ 𝑁+(𝑣)} are relatively prime where𝑁+(𝑣) = {𝑢 ∈ 𝑉 (𝐷) ∶ 𝑣𝑢 ⃗⃗⃗⃗⃗⃗ ∈ 𝐴(𝐷)}. definition 2.7. let 𝐷 = (𝑉,𝐴) be a digraph with|𝑉| = 𝑛. a function 𝑓: 𝑉(𝐷) → [𝛾𝑛] is said to be a gaussian twin neighbourhood prime labeling of𝐷, if it is both gaussian in and out neighborhood prime labeling. a digraph which admits gaussian twin neighborhood prime labeling is called a gaussian twin neighborhood prime digraph. observation 2.8. 1. if 𝐷 is a digraph such that 𝑁+(𝑣) or 𝑁−(𝑣) are either 𝜑 or singleton set, then 𝐷 admits a gaussian twin neighborhood prime labeling. 2. a neighborhood prime digraph 𝐷 in which every vertex is such that either its indegree or out-degree at most 1 is gaussian twin neighborhood prime. the following definitions are referred from [8]. definition 2.9. fan graph is defined as the graph𝑃𝑛 + 𝐾1, 𝑛 ≥ 2 where 𝐾1 is the empty graph on one vertex and 𝑃𝑛, a path graph on 𝑛 vertices. definition 2.10. a double fan 𝐷𝐹𝑛 consists of two fan graphs with a common path. in other words𝐷𝐹𝑛 = 𝑃𝑛 + 𝐾2̅̅ ̅,𝑛 ≥ 2. 3. fan digraphs in this section, some new digraphs are introduced by orienting fan graphs in different possible ways and named accordingly. also we investigate the existence of the gaussian twin neighborhood prime labeling of those digraphs. 11 palani k and shunmugapriya a definition 3.1. in a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the edges of the path 𝑃𝑛 clockwise or anticlockwise and the spoke edges towards the central vertex. call the resulting digraph as in-fan and denote it by𝑖𝐹𝑛⃗⃗ ⃗. definition 3.2. in a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the edges of the path 𝑃𝑛 clockwise or anticlockwise and the spoke edges away from the central vertex. call the resulting digraph as out-fan and denote it by𝑜𝐹𝑛⃗⃗ ⃗. definition 3.3. a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, is said to be an alternating fan (𝐴𝐹𝑛⃗⃗ ⃗) if the edges of the path 𝑃𝑛 are oriented clockwise or anticlockwise and the spoke edges alternately. definition 3.4. in a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the path edges alternately and the spoke edges towards the central vertex. call the resulting digraph as alternating in-fan and denote it by𝐴𝑖𝐹𝑛⃗⃗ ⃗. definition 3.5. in a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the edges of the path 𝑃𝑛 alternately and the spoke edges away from the central vertex. call the resulting digraph as alternating outfan and denote it by𝐴𝑜𝐹𝑛⃗⃗ ⃗. definition 3.6. in a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the edges of the path 𝑃𝑛 alternately and the spoke edges such that either 𝑑+(𝑣) > 0 or𝑑−(𝑣) > 0 ∀ 𝑣 ∈ 𝑉(𝑃𝑛). call the resulting digraph as sole double alternating fan and denote it by𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗. definition 3.7. in a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the edges of the path 𝑃𝑛 alternately and the spoke edges such that neither 𝑑+(𝑣) > 0 nor𝑑−(𝑣) > 0 ∀ 𝑣 ∈ 𝑉(𝑃𝑛). call the resulting digraph as di-double alternating fan and denote it by𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗. theorem 3.8. in-fan (𝑖𝐹𝑛⃗⃗ ⃗) admits gaussian twin neighborhood prime labeling for 𝑛 ≥ 2. proof: let 𝑛 ≥ 2 and let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the directed path 𝑃𝑛⃗⃗ ⃗ and 𝑢 be the apex vertex. then 𝐴(𝑖𝐹𝑛⃗⃗ ⃗) = {𝑣𝑖𝑣𝑖+1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑣𝑖𝑢⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛} is the arc set. this digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. define an injective function 𝑓:𝑉(𝑖𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1 and 𝑓(𝑣𝑖) = 𝛾𝑖+1 for1 ≤ 𝑖 ≤ 𝑛. here,𝑑−(𝑢) > 1. further, the labels of the in-neighborhood vertices of 𝑢 are consecutive gaussian integers in the spiral ordering and so they are relatively prime. 𝑁−(𝑣1) = 𝜙 and𝑁 −(𝑣𝑖) = {𝑣𝑖−1} for 2 ≤ 𝑖 ≤ 𝑛. therefore, 𝑓 is a gaussian in-neighborhood prime labeling. next to prove 𝑓 is also a gaussian out-nighborhood prime labeling. 12 gaussian twin neighborhood prime labeling of fan digraphs now 𝑑+(𝑣𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛 − 1. here, the out-neighborhood vertices of 𝑣𝑖(1 ≤ 𝑖 ≤ 𝑛 − 1) contains the gaussian integer 𝛾1 = 1 and 𝛾1 is relatively prime to all the gaussian integers. further, 𝑁+(𝑣𝑛) = {𝑢} and𝑁 +(𝑢) = 𝜙. therefore, 𝑓 is a gaussian out-neighborhood prime labeling. which implies 𝑓 is a gaussian twin neighborhood prime labeling. hence, in-fan (𝑖𝐹𝑛⃗⃗ ⃗) admits gaussian twin neighborhood prime labeling for𝑛 ≥ 2. theorem 3.9. out-fan (𝑜𝐹𝑛⃗⃗ ⃗) admits gaussian twin neighborhood prime labeling for𝑛 ≥ 2. proof: let 𝑛 ≥ 2 and let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the directed path 𝑃𝑛⃗⃗ ⃗ and 𝑢 be the apex vertex. then 𝐴(𝑜𝐹𝑛⃗⃗ ⃗) = {𝑣𝑖𝑣𝑖+1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑢𝑣𝑖⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛} is the arc set. this digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. define an injective function 𝑓:𝑉(𝑜𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1 and 𝑓(𝑣𝑖) = 𝛾𝑖+1 for1 ≤ 𝑖 ≤ 𝑛. now 𝑑−(𝑣𝑖) > 1 for 2 ≤ 𝑖 ≤ 𝑛. in the above labeling, the in-neighborhood vertices of 𝑣𝑖 contains the gaussian integer 𝛾1 = 1 which is relatively prime to all gaussian integers. further, 𝑁−(𝑣1) = {𝑢} and𝑁 −(𝑢) = 𝜙. therefore, 𝑓 is a gaussian in-neighborhood prime labeling. next to prove 𝑓 is also a gaussian out-nighborhood prime labeling. now 𝑑+(𝑢) > 1 and the labels of the out-neighborhood vertices of 𝑢 are consecutive gaussian integers in the spiral ordering and so they are relatively prime. also, 𝑁+(𝑣𝑖) = {𝑣𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛 − 1 and 𝑁 −(𝑣𝑛) = 𝜙 therefore, 𝑓 is a gaussian out-neighborhood prime labeling. which implies 𝑓 is a gaussian twin neighborhood prime labeling. hence, out-fan (𝑜𝐹𝑛⃗⃗ ⃗) admits gaussian twin neighborhood prime labeling for 𝑛 ≥ 2. theorem 3.10. alternating fan (𝐴𝐹𝑛⃗⃗ ⃗) admits gaussian twin neighborhood prime labeling for 𝑛 ≥ 2. proof: let 𝑛 ≥ 2 and let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the directed path 𝑃𝑛⃗⃗ ⃗ and 𝑢 be the apex vertex. this digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. case (i): 𝒏 is odd 13 palani k and shunmugapriya a 𝐴(𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖𝑣2𝑖+1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } is the corresponding arc set. define 𝑓:𝑉(𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by𝑓(𝑢) = 𝛾1; 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛+1 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1) 2 +𝑖+1 for1 ≤ 𝑖 ≤ 𝑛−1 2 . clearly 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . now 𝑁−(𝑢) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤ 𝑛+1 2 and the labels of the in-neighborhood vertices of 𝑢 are consecutive gaussian integers in the spiral ordering and hence are relatively prime. further, 𝑁−(𝑣2𝑖) = {𝑢,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the label set of the in-neighbors of 𝑣2𝑖 contains𝛾1 = 1. also, 𝑁−(𝑣1) = 𝜙 and 𝑁 −(𝑣2𝑖−1) = {𝑣2𝑖−2} for 2 ≤ 𝑖 ≤ 𝑛+1 2 therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is odd. next to prove 𝑓 is also a gaussian out-neighborhood prime labeling. now 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for1 ≤ 𝑖 ≤ 𝑛+1 2 . 𝑁+(𝑢) = {𝑣2,𝑣4,…,𝑣2𝑖} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the out-neighborhood vertices of 𝑢 are labeled with the consecutive gaussian integers in the spiral ordering and so by the result 1.4(4), the labels are relatively prime. now 𝑁+(𝑣2𝑖−1) = {𝑢,𝑣2𝑖} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and 𝑁+(𝑣𝑛) = {𝑢}. also, the out-neighborhood vertices of 𝑣2𝑖−1(1 ≤ 𝑖 ≤ 𝑛−1 2 ) contains the gaussian integer 𝛾1 = 1. further, 𝑁+(𝑣2𝑖) = {𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−1 2 . therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is odd. 𝑓 is a gaussian twin neighborhood prime labeling when 𝑛 is odd. case (ii): 𝒏 is even 𝐴(𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖𝑣2𝑖+1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−2 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } is the corresponding arc set. define 𝑓:𝑉(𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by𝑓(𝑢) = 𝛾1; 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 2 and 𝑓(𝑣2𝑖) = 𝛾𝑛 2 +𝑖+1 for1 ≤ 𝑖 ≤ 𝑛 2 . clearly 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for1 ≤ 𝑖 ≤ 𝑛 2 . now 𝑁−(𝑢) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤ 𝑛 2 and the labels of the in-neighborhood vertices of 𝑢 are consecutive gaussian integers in the spiral ordering and so are relatively prime. 14 gaussian twin neighborhood prime labeling of fan digraphs further, 𝑁−(𝑣2𝑖) = {𝑢,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤ 𝑛 2 and the label set of in-neighbors of 𝑣2𝑖 contains the gaussian integer𝛾1 = 1. also, 𝑁−(𝑣1) = 𝜙 and 𝑁 −(𝑣2𝑖−1) = {𝑣2𝑖−2} for 2 ≤ 𝑖 ≤ 𝑛 2 therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is even. next to prove 𝑓 is also a gaussian out-nighborhood prime labeling. now 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for1 ≤ 𝑖 ≤ 𝑛 2 . 𝑁+(𝑢) = {𝑣2,𝑣4,…,𝑣2𝑖} for1 ≤ 𝑖 ≤ 𝑛 2 and the out-neighborhood vertices of 𝑢 are labeled with the consecutive gaussian integers in the spiral ordering and so they are relatively prime. further, 𝑁+(𝑣2𝑖−1) = {𝑢,𝑣2𝑖} for 1 ≤ 𝑖 ≤ 𝑛 2 and the out-neighborhood vertices of 𝑣2𝑖−1(1 ≤ 𝑖 ≤ 𝑛 2 ) contains the gaussian integer𝛾1 = 1. also, 𝑁+(𝑣2𝑖) = {𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−2 2 and𝑁+(𝑣𝑛) = 𝜙. therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is even. 𝑓 is a gaussian twin neighborhood prime labeling when 𝑛 is even. cases (i) and (ii) imply 𝑓 is a gaussian twin neighborhood prime labeling. hence, alternating fan (𝐴𝐹𝑛⃗⃗ ⃗) admits gaussian twin neighborhood prime labeling for𝑛 ≥ 2. theorem 3.11. alternating in-fan (𝐴𝑖𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph for𝑛 ≥ 2. proof. let 𝑛 ≥ 2 and let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the directed path 𝑃𝑛⃗⃗ ⃗ and 𝑢 be the apex vertex. case (i): 𝒏 is odd 𝐴(𝐴𝑖𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } is the corresponding arc set. define an injective function 𝑓:𝑉(𝐴𝑖𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛+1 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1) 2 +𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . here 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . clearly, the label set of in-neighborhood vertices of 𝑢 contains consecutive gaussian integers in the spiral ordering and so they are relatively prime. now 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the labels of the in-neighborhood vertices of 𝑣2𝑖 are consecutive gaussian integers in the spiral ordering further, 𝑁−(𝑣2𝑖−1) = 𝜙 for 1 ≤ 𝑖 ≤ 𝑛+1 2 therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is odd. next to prove 𝑓 is also a gaussian out-neighborhood prime labeling. 15 palani k and shunmugapriya a now 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛+1 2 and the labels of vertices in 𝑁+(𝑣2𝑖−1) contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. further, 𝑁+(𝑣2𝑖) = {𝑢} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and 𝑁+(𝑢) = 𝜙. therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is odd. by (1) and (2), 𝑓 is a gaussian twin neighborhood prime labeling when 𝑛 is odd. case (ii): 𝒏 is even 𝐴(𝐴𝑖𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−2 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } is the arc set. define an injective function 𝑓:𝑉(𝐴𝑖𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by𝑓(𝑢) = 𝛾1; 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for1 ≤ 𝑖 ≤ 𝑛 2 and 𝑓(𝑣2𝑖) = 𝛾𝑛 2 +𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 2 . here 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for1 ≤ 𝑖 ≤ 𝑛−2 2 . clearly, the label set of in-neighborhood vertices of 𝑢 contains consecutive gaussian integers in the spiral ordering and so they are relatively prime. now 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−2 2 and the label set of in-neighborhood vertices of 𝑣2𝑖 are consecutive gaussian integers in the spiral ordering. also, 𝑁−(𝑣𝑛) = {𝑣𝑛−1} and 𝑁 −(𝑣2𝑖−1) = 𝜙 for 1 ≤ 𝑖 ≤ 𝑛+1 2 . therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is even. now 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 and the out-neighborhood vertices of 𝑣2𝑖−1 contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. further, 𝑁+(𝑣2𝑖) = {𝑢} for 1 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is even. by (3) and (4), 𝑓 is a gaussian twin neighborhood prime labeling when 𝑛 is even. cases (i) and (ii) imply 𝑓 is a gaussian twin neighborhood prime labeling. thus, an alternating in-fan (𝐴𝑖𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph for 𝑛 ≥ 2. theorem 3.12. alternating out-fan (𝐴𝑜𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph for 𝑛 ≥ 2. proof: let 𝑛 ≥ 2. let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the directed path 𝑃𝑛⃗⃗ ⃗ and 𝑢 be the apex vertex. this digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. case (i): 𝒏 is odd 𝐴(𝐴𝑜𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑢𝑣2𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } is the corresponding arc set. define an injective function 𝑓:𝑉(𝐴𝑜𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 16 gaussian twin neighborhood prime labeling of fan digraphs 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for1 ≤ 𝑖 ≤ 𝑛+1 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1) 2 +𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . now 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the label set of in-neighbors of 𝑣2𝑖 are the consecutive gaussian integers in the spiral ordering. further, 𝑁−(𝑣2𝑖−1) = {𝑢} for 1 ≤ 𝑖 ≤ 𝑛+1 2 and𝑁−(𝑢) = 𝜙. therefore, 𝑓 is a gaussian inneighborhood prime labeling when 𝑛 is odd. next to prove 𝑓 is a gaussian out-neighborhood prime labeling. now 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for2 ≤ 𝑖 ≤ 𝑛−1 2 . clearly, the label set of out-neighborhood vertices of 𝑢 contains all the vertices of the path 𝑃𝑛⃗⃗ ⃗ which are labeled with the consecutive gaussian integers in the spiral ordering. further, 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} for 2 ≤ 𝑖 ≤ 𝑛−1 2 and the labels of the outneighborhood vertices of 𝑣2𝑖−1 are consecutive gaussian integers in the spiral ordering. also, 𝑁+(𝑣1) = {𝑣2} , 𝑁 +(𝑣𝑛) = {𝑣𝑛−1} and 𝑁 +(𝑣2𝑖) = 𝜙 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is odd. (1) and (2) imply 𝑓 is a gaussian twin neighborhood prime labeling if 𝑛 is odd. case (ii): 𝒏 is even 𝐴(𝐴𝑜𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−2 2 } ∪ {𝑢𝑣2𝑖−1⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } is the corresponding arc set. define an injective function 𝑓:𝑉(𝐴𝑜𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 2 and 𝑓(𝑣2𝑖) = 𝛾𝑛 2 +𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 2 . now 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 and the label set of the in-neighborhood vertices of 𝑣2𝑖 are consecutive gaussian integers in the spiral ordering and so they are relatively prime. further, 𝑁−(𝑢) = 𝜙 and 𝑁−(𝑣2𝑖−1) = {𝑢} for 1 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is even. next to prove 𝑓 is also a gaussian out-neighborhood prime labeling. here 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤ 𝑛 2 . clearly, the labels out-neighborhood vertices of 𝑢 contains consecutive gaussian integers in the spiral ordering and so are relatively prime. further, 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} for 2 ≤ 𝑖 ≤ 𝑛 2 and the labels of out-neighborhood vertices of 𝑣2𝑖−1 are consecutive gaussian integers in the spiral ordering and so they are relatively prime. also, 𝑁+(𝑣1) = {𝑣2} and 𝑁 +(𝑣2𝑖) = 𝜙 for 1 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 is a gaussian outneighborhood prime labeling when 𝑛 is even. (3) and (4) imply 𝑓 is a gaussian twin neighborhood prime labeling if 𝑛 is even. from the cases (i) and (ii), 𝑓 is a gaussian twin neighborhood prime labeling. thus, an alternating outfan (𝐴𝑜𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph for 𝑛 ≥ 2. 17 palani k and shunmugapriya a theorem 3.13. sole double alternating fan (𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph for 𝑛 ≥ 2. proof: let 𝑉(𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑢,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} be the vertex set where 𝑣𝑖 represent the ith vertex of the directed path 𝑃𝑛⃗⃗ ⃗ and 𝑢 is the apex vertex. this digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. case (i): 𝒏 is odd 𝐴(𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } is the corresponding arc set. define an injective function 𝑓:𝑉(𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for1 ≤ 𝑖 ≤ 𝑛+1 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1) 2 +𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . now 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for1 ≤ 𝑖 ≤ 𝑛−1 2 . the label set of in-neighborhood vertices of 𝑢 are consecutive gaussian integers in the spiral ordering and so are relatively prime. also, the in-neighborhood vertices of 𝑣2𝑖 contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. further, 𝑁−(𝑣2𝑖−1) = {𝑢} for1 ≤ 𝑖 ≤ 𝑛+1 2 . therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is odd. next to prove 𝑓 is also a gaussian out-nighborhood prime labeling. now 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for1 ≤ 𝑖 ≤ 𝑛+1 2 . the label set of out-neighborhood vertices of 𝑢 are consecutive gaussian integers in the spiral ordering and so those are relatively prime. also, the label set of out-neighborhood vertices of 𝑣2𝑖−1 contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. 𝑁+(𝑣2𝑖) = 𝜙 for1 ≤ 𝑖 ≤ 𝑛−1 2 . therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is odd. case (ii): 𝒏 is even 𝐴(𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−2 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } is the arc set. define an injective function 𝑓:𝑉(𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 2 and 𝑓(𝑣2𝑖) = 𝛾𝑛 2 +𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 2 . now 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 . the label set of of in-neighborhood vertices of 𝑢 are consecutive gaussian integers in the spiral ordering and so those are relatively prime. 18 gaussian twin neighborhood prime labeling of fan digraphs further, the label set of in-neighborhood vertices of 𝑣2𝑖 contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. also, 𝑁−(𝑣2𝑖−1) = 𝜙 for 1 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is even. next to prove 𝑓 is also a gaussian out-nighborhood prime labeling. now 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 . the label set of out-neighborhood vertices of 𝑢 are consecutive gaussian integers and so they are relatively prime. then the out-neighborhood vertices of 𝑣2𝑖−1 contains the gaussian integer𝛾1 = 1. also, 𝑁+(𝑣2𝑖) = 𝜙 for1 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is even. case (i) and (ii) imply 𝑓 is a gaussian twin neighborhood prime labeling. hence, the sole-double alternating fan (𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph for𝑛 ≥ 2. theorem 3.14. di-double alternating fan 𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗ is a gaussian twin neighborhood prime digraph for𝑛 ≥ 2. proof: let 𝑛 ≥ 2 and let 𝑉(𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑢,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} be the vertex set where 𝑣𝑖 represent the ith vertex of the directed path 𝑃𝑛⃗⃗ ⃗ and 𝑢 is the apex vertex. this digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. case (i): 𝒏 is odd 𝐴(𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑢𝑣2𝑖−1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } is the arc set. define an injective function 𝑓:𝑉(𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for1 ≤ 𝑖 ≤ 𝑛+1 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1) 2 +𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . now𝑑−(𝑢) > 1. the label set of in-neighborhood vertices of 𝑢 are consecutive gaussian integers in the in the spiral ordering and so they are relatively prime. also, 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the in-neighborhood vertices of 𝑣2𝑖 are labeled with the consecutive gaussian integers. further, 𝑁−(𝑣2𝑖−1) = {𝑢} for 1 ≤ 𝑖 ≤ 𝑛+1 2 . therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is odd. now 𝑑+(𝑢) > 1 and the out-neighborhood vertices of 𝑢 are consecutive gaussian integers in the labeling and so they are relatively prime. also, 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤ 𝑛−1 2 and the label set of the out-neighborhood vertices of 𝑣2𝑖−1 contains the consecutive gaussian integers in the spiral ordering. 19 palani k and shunmugapriya a further, 𝑁+(𝑣1) = {𝑣2} and 𝑁 +(𝑣𝑛) = {𝑣𝑛−1}. 𝑁+(𝑣2𝑖) = {𝑢} for 1 ≤ 𝑖 ≤ 𝑛−1 2 . therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is odd. case (ii): 𝒏 is even 𝐴(𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−2 2 } ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑢𝑣2𝑖−1⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } is the arc set. define an injective function 𝑓:𝑉(𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 2 and 𝑓(𝑣2𝑖) = 𝛾𝑛 2 +𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 2 . now 𝑑−(𝑢) > 1 and labeling of the in-neighborhood vertices of 𝑢 are consecutive gaussian integers in the spiral ordering and they are relatively prime. also, 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−2 2 and the in-neighborhood vertices of 𝑣2𝑖 are labeled with the consecutive gaussian integers and so are relatively prime. further, 𝑁−(𝑣𝑛) = {𝑣𝑛−1} and 𝑁 −(𝑣2𝑖−1) = {𝑢} for 1 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is even. next to prove 𝑓 is also a gaussian out-nighborhood prime labeling. now 𝑑+(𝑢) > 1 and the labeling of the out-neighborhood vertices of 𝑢 are consecutive gaussian integers. also, 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤ 𝑛 2 and the label set of the out-neighborhood vertices of 𝑣2𝑖−1(2 ≤ 𝑖 ≤ 𝑛 2 ) are consecutive gaussian integers in the spiral ordering and so are relatively prime. also, 𝑁+(𝑣1) = {𝑣2} and 𝑁 +(𝑣2𝑖) = {𝑢} for 1 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is even. cases (i) and (ii) imply 𝑓 is a gaussian twin neighborhood prime labeling. thus the di-double alternating fan (𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph for𝑛 ≥ 2. 4. double fan digraphs in this section, some new digraphs are introduced by orienting double fan graphs in different possible ways and named accordingly. also, the gaussian twin neighborhood prime labeling is proved for those digraphs. definition 4.1. in a double fan𝐷𝐹𝑛 = 𝑃𝑛 + 𝐾2̅̅ ̅, orient the edges of the common path 𝑃𝑛 clockwise or anticlockwise and the spoke edges are towards the central vertex. call the resulting digraph as double in-fan and denote it by𝐷𝑖𝐹𝑛⃗⃗ ⃗. 20 gaussian twin neighborhood prime labeling of fan digraphs definition 4.2. in a double fan𝐷𝐹𝑛 = 𝑃𝑛 + 𝐾2̅̅ ̅, orient the edges of the common path 𝑃𝑛 clockwise or anticlockwise and the spoke edges away from the central vertex. call the resulting digraph as double out-fan and denote it by𝐷𝑜𝐹𝑛⃗⃗ ⃗. definition 4.3. a double fan𝐷𝐹𝑛 = 𝑃𝑛 + 𝐾2̅̅ ̅, is said to be a double alternating fan (𝐷𝐴𝐹𝑛⃗⃗ ⃗) if the edges of the common path 𝑃𝑛 are oriented in clockwise or anticlockwise and the spoke edges alternately. definition 4.4. in a double fan𝐷𝐹𝑛, orient the edges of the common path 𝑃𝑛 alternately and the spoke edges are towards the central vertices. call the resulting digraph as double alternating in-fan and denote it by𝐷𝐴𝑜𝐹𝑛⃗⃗ ⃗. definition 4.5. in a double fan𝐷𝐹𝑛, orient the edges of the common path 𝑃𝑛 alternately and the spoke edges are away the central vertices. call the resulting digraph is called a double alternating out-fan and denote it by𝐷𝐴𝑖𝐹𝑛⃗⃗ ⃗. definition 4.6. in a double fan 𝐷𝐹𝑛, orient the edges of the common path 𝑃𝑛 alternately and the spoke edges such that either 𝑑+(𝑣) = 0 or 𝑑−(𝑣) = 0 ∀ 𝑣 ∈ 𝑉(𝑃𝑛). the resulting digraph is called a double sole-double alternating fan and denoted it as 𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗. definition 4.7. in a double fan 𝐷𝐹𝑛, orient the edges of the common path 𝑃𝑛 alternately and the spoke edges such that neither 𝑑+(𝑣) = 0 nor 𝑑−(𝑣) = 0 ∀ 𝑣 ∈ 𝑉(𝑃𝑛). the resulting digraph is called a double di-double alternating fan and denote it by 𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗. theorem 4.8. double in-fan (𝐷𝑖𝐹𝑛⃗⃗ ⃗ )is a gaussian twin neighborhood prime digraph. proof: let 𝑉(𝐷𝑖𝐹𝑛⃗⃗ ⃗) = {𝑢,𝑤,𝑣𝑖| 1 ≤ 𝑖 ≤ 𝑛} where 𝑢 and 𝑤 are the apex vertices and 𝑣𝑖 represent the 𝑖𝑡ℎ vertex of the directed path 𝑃𝑛⃗⃗ ⃗. then 𝐴(𝐷𝑖𝐹𝑛⃗⃗ ⃗) = {𝑣𝑖𝑣𝑖+1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ | 1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑣𝑖𝑢⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑖𝑤⃗⃗⃗⃗⃗⃗ | 1 ≤ 𝑖 ≤ 𝑛} is the arc set. this digraph graph has 𝑛 + 2 vertices and 3𝑛 − 1 edges. define an injective function 𝑓:𝑉(𝐷𝑖𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by(𝑢) = 𝛾1 , 𝑓(𝑤) = 𝛾2 and 𝑓(𝑣𝑖) = 𝛾𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛. here 𝑑−(𝑢) > 1, 𝑑−(𝑤) > 1. by the definition of 𝑓, the in-neighborhood vertices of 𝑢 and 𝑤 are labeled by consecutive gaussian integers 𝛾1 and 𝛾2 in the spiral ordering and so are relatively prime. now, 𝑁−(𝑣1) = 𝜙 and 𝑁 −(𝑣𝑖) = {𝑣𝑖−1} for 2 ≤ 𝑖 ≤ 𝑛. therefore, 𝑓 is a gaussian in-neighborhood prime labeling. 21 palani k and shunmugapriya a next to prove 𝑓 is also gaussian out-neighborhood prime labeling. now, 𝑑+(𝑣𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛. further, 𝑁+(𝑣𝑖) = {𝑢,𝑣𝑖+1,𝑤} for 1 ≤ 𝑖 ≤ 𝑛 − 1 and the labels of vertices in 𝑁 +(𝑣𝑖) contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. also, 𝑁+(𝑣𝑛) = {𝑢,𝑤} and labels of 𝑢 and 𝑤 are consecutive gaussian integers. further, 𝑁+(𝑢) = 𝑁+(𝑤) = 𝜙. therefore, 𝑓 is a gaussian outneighborhood prime labeling. 𝑓 is a gaussian twin neighborhood prime labeling. thus, double fan(𝐷𝑖𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph. theorem 4.9. double out-fan(𝐷𝑜𝐹𝑛⃗⃗ ⃗) admits gaussian twin neighborhood prime labeling. proof: let 𝑉(𝐷𝑜𝐹𝑛⃗⃗ ⃗) = {𝑢,𝑤,𝑣𝑖| 1 ≤ 𝑖 ≤ 𝑛} where 𝑢 and 𝑤 are the apex vertices and 𝑣𝑖 represent the 𝑖𝑡ℎ vertex of the directed path 𝑃𝑛⃗⃗ ⃗. then 𝐴(𝐷𝑜𝐹𝑛⃗⃗ ⃗) = {𝑣𝑖𝑣𝑖+1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ | 1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑢𝑣𝑖⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑤𝑣𝑖⃗⃗ ⃗⃗ ⃗⃗ | 1 ≤ 𝑖 ≤ 𝑛} is the arc set. this digraph has 𝑛 + 2 vertices and 3𝑛 − 1 edges. define an injective function 𝑓:𝑉(𝐷𝑜𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by 𝑓(𝑢) = 𝛾1 , 𝑓(𝑤) = 𝛾2 and 𝑓(𝑣𝑖) = 𝛾𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛. now, 𝑑−(𝑣𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛. 𝑁−(𝑣1) = {𝑢,𝑤} and 𝑁 −(𝑣𝑖) = {𝑢,𝑣𝑖−1,𝑤} for 2 ≤ 𝑖 ≤ 𝑛. clearly, label set of vertices in 𝑁−(𝑣𝑖) contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. also, 𝑁−(𝑢) = 𝑁−(𝑤) = 𝜙. therefore, 𝑓 is a gaussian in-neighborhood prime labeling. next to prove 𝑓 is also gaussian out-neighborhood prime labeling. here 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1. by the definition of 𝑓, the set of out-neighborhood vertices of 𝑢 and 𝑤 are labeled by the consecutive gaussian integers in the spiral ordering and which are relatively prime. further, 𝑁+(𝑣𝑖) = {𝑣𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛 − 1 and 𝑁 +(𝑣𝑛) = 𝜙. therefore, 𝑓 is a gaussian out-neighborhood prime labeling. 𝑓 is a gaussian twin neighborhood prime labeling. hence, double out-fan (𝐷𝑜𝐹𝑛⃗⃗ ⃗) admits gaussian twin neighborhood prime labeling. theorem 4.10. double alternating fan (𝐷𝐴𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph. proof: let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the path 𝑃𝑛⃗⃗ ⃗ and 𝑢,𝑤 be the apex vertices. let 𝑉(𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑢,𝑤,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} be the vertex set. this digraph has 𝑛 + 2 vertices and 3𝑛 − 1 arcs. case (i): 𝑛 is even 𝐴(𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖𝑣2𝑖+1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−2 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } is the arc set. 22 gaussian twin neighborhood prime labeling of fan digraphs define an injective function 𝑓:𝑉(𝐷𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by 𝑓(𝑢) = 𝛾1;𝑓(𝑤) = 𝛾2 and 𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 and 𝑓(𝑣2𝑖) = 𝛾𝑛 2 +𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛 2 . here, 𝑑−(𝑢) > 1, 𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 . clearly, the label set of in-neighborhood vertices of 𝑢 and 𝑤 are consecutive gaussian integers in the spiral ordering. further, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛 2 . by the definition of 𝑓, the vertex 𝑢 is labeled as 𝛾1 = 1, which is relatively prime to all the gaussian integers. also, 𝑁−(𝑣1) = 𝜙 and 𝑁 −(𝑣2𝑖−1) = {𝑣2𝑖−2} for 2 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 is gaussian in-neighborhood prime labeling when 𝑛 is even. next to prove 𝑓 is also gaussian out-neighborhood prime labeling. now 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 . clearly, the label set of out-neighborhood vertices of 𝑢 and 𝑤 are labeled by consecutive gaussian integers in the spiral ordering. further, 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖,𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛 2 and by the definition of 𝑓, the label of 𝑢 is 𝛾1 = 1, which is relatively prime to all the gaussian integers. also, 𝑁+(𝑣2𝑖) = {𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−2 2 and 𝑁+(𝑣𝑛) = 𝜙. therefore, 𝑓 is a gaussian out neighborhood prime labeling when 𝑛 is even. 𝑓 is a gaussian twin neighborhood prime labeling when 𝑛 is even. case (ii): 𝑛 is odd. then, 𝐴(𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖𝑣2𝑖+1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛−1 2 } is the arc set. define an injective function 𝑓:𝑉(𝐷𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by 𝑓(𝑢) = 𝛾1;𝑓(𝑤) = 𝛾2 and 𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1 2 )+𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . here,𝑑−(𝑢) > 1, 𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . clearly, the label set of in-neighborhood vertices of 𝑢 and 𝑤 are consecutive gaussian integers in the spiral ordering. further, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and by the definition of𝑓, the vertex 𝑢 has the label 𝛾1 = 1, which is relatively prime to all the gaussian integers. also, 𝑁−(𝑣1) = 𝜙 and 𝑁 −(𝑣2𝑖−1) = {𝑣2𝑖−2} for 2 ≤ 𝑖 ≤ 𝑛+1 2 . therefore, 𝑓 is a gaussian inneighborhood prime labeling when 𝑛 is odd. next to prove 𝑓 is also a gaussian out-nighborhood prime labeling. now𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛+1 2 . by the definition of𝑓, the label set of out-neighborhood vertices of 𝑢 and 𝑤 consecutive gaussian integers in the spiral ordering. also, 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖,𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the label of 𝑢 is 𝛾1 = 1 is relatively prime to the gaussian integers. 23 palani k and shunmugapriya a 𝑁+(𝑣2𝑖) = {𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−1 2 . further,𝑁+(𝑣𝑛) = {𝑢,𝑤}. the vertices 𝑢 and 𝑣 are labeled by 𝛾1 and 𝛾2 respectively. since 𝛾1 and 𝛾2 are consecutive gaussian integers in the spiral ordering and so they are relatively prime. therefore, 𝑓 is a gaussian outneighborhood prime labeling when 𝑛 is odd. 𝑓 is a gaussian twin neighborhood prime labeling. from both the cases, 𝑓 is a gaussian twin neighborhood prime labeling. hence, double alternating fan (𝐷𝐴𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph. theorem 4.11. double alternating in-fan (𝐷𝐴𝑖𝐹𝑛⃗⃗ ⃗) admits gaussian twin neighborhood prime labeling. proof. let 𝑉(𝐷𝐴𝑖𝐹𝑛⃗⃗ ⃗) = {𝑢,𝑤,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} where 𝑢,𝑤the apex vertices are and 𝑣𝑖 represent the 𝑖th vertex of the common path𝑃𝑛⃗⃗ ⃗. this digraph has 𝑛 + 2 vertices and 3𝑛 − 1 arcs. case (i): 𝑛 is odd. 𝐴(𝐷𝐴𝑖𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖𝑤⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛−1 2 } is the corresponding arc set. define an injective function 𝑓:𝑉(𝐷𝐴𝑖𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by(𝑢) = 𝛾1, 𝑓(𝑤) = 𝛾2 , 𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛+1 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1 2 )+𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . here, 𝑑−(𝑢) > 1, 𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . 𝑁−(𝑢) = 𝑁−(𝑤) = {𝑣1,𝑣2,…,𝑣𝑛}. the vertices 𝑣1,𝑣2,…,𝑣𝑛 are labeled with the consecutive gaussian integers in the spiral ordering and so they are relatively prime. further, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the labels of 𝑁−(𝑣2𝑖) are consecutive gaussian integers in the spiral ordering. also, 𝑁−(𝑣2𝑖−1) = 𝜙 for1 ≤ 𝑖 ≤ 𝑛+1 2 . therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is odd. next to prove 𝑓 is also gaussian out-neighborhood prime labeling. here 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 and 𝑑+(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 further, 𝑁+(𝑣1) = {𝑣2,𝑢,𝑤} and 𝑁 +(𝑣2𝑖−1) = {𝑢,𝑤,𝑣2𝑖−2,𝑣2𝑖} for2 ≤ 𝑖 ≤ 𝑛−1 2 . and the label set of out-neighborhood vertices 𝑣2𝑖−1(1 ≤ 𝑖 ≤ 𝑛−1 2 ) contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. also, 𝑁+(𝑣2𝑖) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the vertices 𝑢 and 𝑤 are labeled by the gaussian integers 𝛾1 = 1 and 𝛾2 = 1 + 𝑖 respectively. now, 𝑁+(𝑢) = 𝑁+(𝑤) = 𝜙. therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is odd. 𝑓 is a gaussian twin neighborhood prime labeling when 𝑛 is odd case (ii): 𝑛 is even 24 gaussian twin neighborhood prime labeling of fan digraphs 𝐴(𝐷𝐴𝑖𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−2 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖𝑤⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } is the arc set. define an injective function 𝑓:𝑉(𝐷𝐴𝑖𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by 𝑓(𝑢) = 𝛾1;𝑓(𝑤) = 𝛾2 and 𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 and𝑓(𝑣2𝑖) = 𝛾𝑛 2 +𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛 2 . here, 𝑑−(𝑢) > 1, 𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−2 2 . by the definition of𝑓, the label set of the in-neighborhood vertices of 𝑢 and 𝑤 are consecutive gaussian integers in the spiral ordering and so they are relatively prime. further, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−2 2 and the vertices in 𝑁−(𝑣2𝑖) are labeled by consecutive gaussian integers in the spiral ordering. also, 𝑁−(𝑣𝑛) = {𝑣𝑛−1} and 𝑁 −(𝑣2𝑖−1) = 𝜙 for 1 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 admits a gaussian in-neighborhood prime labeling when 𝑛 is even. now 𝑑+(𝑣2𝑖−1) > 1 and 𝑑 +(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 . here, 𝑁+(𝑢) = 𝑁+(𝑤) = 𝜙. also, 𝑁+(𝑣1) = {𝑣2,𝑢,𝑤} and 𝑁 +(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖,𝑢,𝑤} for 2 ≤ 𝑖 ≤ 𝑛 2 and the labels of 𝑁+(𝑣2𝑖−1)(1 ≤ 𝑖 ≤ 𝑛 2 ) contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. 𝑁+(𝑣2𝑖) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛 2 and the vertices 𝑢 and 𝑤 are labelled by the consecutive gaussian integers 𝛾1 = 1 and 𝛾2 = 1 + 𝑖 respectively. so the labels of vertices in 𝑁+(𝑣2𝑖) are relatively prime. therefore, 𝑓 is a gaussian outneighborhood prime labeling when 𝑛 is even. 𝑓 is a gaussian twin neighborhood prime labeling. hence, double alternating in-fan (𝐷𝐴𝑖𝐹𝑛⃗⃗ ⃗) admits gaussian twin neighborhood prime labeling. theorem 4.12. double alternating out-fan (𝐷𝐴𝑜𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph. proof. let 𝑉(𝐷𝐴𝑜𝐹𝑛⃗⃗ ⃗) = {𝑢,𝑤,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} where 𝑢,𝑤 are the apex vertices and 𝑣𝑖 represent the 𝑖th vertex of the common path 𝑃𝑛⃗⃗ ⃗. this digraph has 𝑛 + 2 vertices and 3𝑛 − 1 arcs. case (i): 𝑛 is odd. 𝐴(𝐷𝐴𝑜𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑢𝑣2𝑖−1⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑤𝑣2𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛−1 2 } is the arc set. define an injective function 𝑓:𝑉(𝐷𝐴𝑜𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by (𝑢) = 𝛾1,𝑓(𝑤) = 𝛾2 , 𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛+1 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1 2 )+𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . here, 𝑑−(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛+1 2 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . the vertices 𝑢 and 𝑤 has no in-neighbors. that is, 𝑁−(𝑢) = 𝑁−(𝑤) = 𝜙. 25 palani k and shunmugapriya a now 𝑁−(𝑣2𝑖−1) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛+1 2 and the vertices 𝑢 and 𝑤 are labelled by the consecutive gaussian integers 𝛾1 = 1 and 𝛾2 = 1 + 𝑖 respectively. so the labels of vertices in 𝑁−(𝑣2𝑖) are relatively prime. further, 𝑁−(𝑣2𝑖) = {𝑢,𝑤,𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the label set of inneighborhood vertices 𝑣2𝑖 contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is odd. next to prove 𝑓 is also gaussian out-neighborhood prime labeling. now, 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤ 𝑛−1 2 . the out-neighborhood vertices of 𝑢 and 𝑤 are labeled by consecutive gaussian integers in the spiral ordering and so they are relatively prime. also, 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} for 2 ≤ 𝑖 ≤ 𝑛−1 2 and the vertices 𝑣2𝑖−2 and 𝑣2𝑖 are labeled by 𝛾 ( 𝑛+1 2 )+𝑖+1 and 𝛾 ( 𝑛+1 2 )+𝑖+2 which are consecutive gaussian integers in the spiral ordering. 𝑁+(𝑣1) = {𝑣2} and 𝑁 +(𝑣𝑛) = {𝑣𝑛−1}. then, 𝑁+(𝑣2𝑖) = 𝜙 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is odd. 𝑓 is a gaussian twin neighborhood prime labeling when 𝑛 is odd case (ii): 𝑛 is even 𝐴(𝐷𝐴𝑜𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−2 2 } ∪ {𝑢𝑣2𝑖−1⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑤𝑣2𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } is the corresponding arc set. define an injective function 𝑓:𝑉(𝐷𝐴𝑜𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by (𝑢) = 𝛾1,𝑓(𝑤) = 𝛾2 , 𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛 2 )+𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛 2 . here, 𝑑−(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 . further, 𝑁−(𝑢) = 𝑁−(𝑤) = 𝜙. also, 𝑁−(𝑣2𝑖−1) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛 2 and the vertices 𝑢 and 𝑤 are labeled with the consecutive gaussian integers 𝛾1 = 1 and 𝛾2 = 1 + 𝑖 respectively. here, 𝑁−(𝑣2𝑖) = {𝑢,𝑤,𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−2 2 and 𝑁−(𝑣𝑛) = {𝑢,𝑤,𝑣𝑛−1}. further, the label of the in-neighborhood vertices of 𝑣2𝑖(1 ≤ 𝑖 ≤ 𝑛 2 ) contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is even. next to prove 𝑓 is also a gaussian out-neighborhood prime labeling. now, 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤ 𝑛 2 . clearly, the label of the out-neighborhood vertices of 𝑢 and 𝑤 are labeled by consecutive gaussian integers in the spiral ordering and so are relatively prime. also, 𝑁+(𝑣1) = {𝑣2}. 26 gaussian twin neighborhood prime labeling of fan digraphs 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} for2 ≤ 𝑖 ≤ 𝑛 2 . the vertices 𝑣2𝑖−2 and 𝑣2𝑖 are labeled by 𝛾𝑛 2 +𝑖+1 and 𝛾𝑛 2 +𝑖+2 which are consecutive gaussian integers in the labeling in spiral ordering. also, 𝑁+(𝑣2𝑖) = 𝜙 for1 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is even. therefore, 𝑓 is a gaussian twin neighborhood prime labeling when 𝑛 is even. thus, double alternating out-fan (𝐷𝐴𝑜𝐹𝑛⃗⃗ ⃗) is a gaussian twin neighborhood prime digraph. theorem 4.13. double sole double alternating fan (𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) admits gaussian twin neighborhood prime labeling. proof: let 𝑉(𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑢,𝑤,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} be the vertex set where 𝑣𝑖 represent the ith vertex of the common path 𝑃𝑛⃗⃗ ⃗ and 𝑢,𝑤 be the apex vertices. this digraph has 𝑛 + 2 vertices and 3𝑛 − 1 arcs. case (i) 𝑛 is odd. 𝐴(𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛−1 2 } is the arc set. define an injective function 𝑓:𝑉(𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by (𝑢) = 𝛾1,𝑓(𝑤) = 𝛾2 , 𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛+1 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1 2 )+𝑖+2 for1 ≤ 𝑖 ≤ 𝑛−1 2 . here, 𝑑−(𝑢) > 1,𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for1 ≤ 𝑖 ≤ 𝑛+1 2 . further, 𝑁−(𝑢) = 𝑁−(𝑤) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤ 𝑛+1 2 and the labels of vertices in 𝑁−(𝑢) and 𝑁−(𝑤) are consecutive gaussian integers in the spiral ordering and hence are relatively prime. also 𝑁−(𝑣2𝑖) = {𝑢,𝑤,𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the label of the vertex 𝑢 is 𝛾1 = 1 which is relatively prime to all the gaussian integers. here, 𝑁−(𝑣2𝑖−1) = 𝜙 for1 ≤ 𝑖 ≤ 𝑛+1 2 . therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is odd. next to prove 𝑓 is also gaussian out-neighborhood prime labeling. here, 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛+1 2 . now 𝑁+(𝑢) = 𝑁+(𝑤) = {𝑣2,𝑣4,…,𝑣2𝑖} for1 ≤ 𝑖 ≤ 𝑛−1 2 . clearly, the label set of out-neighborhood vertices of 𝑢 and 𝑤 are labeled by the consecutive gaussian integers in the spiral ordering and so they are relatively prime. also, 𝑁+(𝑣1) = {𝑢,𝑤,𝑣2}, 𝑁 +(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖,𝑢,𝑤} for 2 ≤ 𝑖 ≤ 𝑛−3 2 and 𝑁+(𝑣𝑛) = {𝑢,𝑤,𝑣𝑛−1}. further, the label of vertices in 𝑁+(𝑣2𝑖−1) for 1 ≤ 𝑖 ≤ 𝑛−1 2 contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. also, 𝑁+(𝑣2𝑖) = 𝜙 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . 27 palani k and shunmugapriya a therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is odd. therefore, 𝑓 is a gaussian twin neighborhood prime labeling when 𝑛 is odd. case (ii): 𝑛 is even 𝐴(𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−2 2 } ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } is the arc set. define an injective function 𝑓:𝑉(𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by 𝑓(𝑢) = 𝛾1,𝑓(𝑤) = 𝛾2, 𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛 2 )+𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛 2 . here, 𝑑−(𝑢) > 1,𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 . now 𝑁−(𝑢) = 𝑁−(𝑤) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤ 𝑛 2 and the label set of vertices in 𝑁−(𝑢) and 𝑁−(𝑤) are consecutive gaussian integers in the spiral ordering. further, 𝑁−(𝑣2𝑖) = {𝑢,𝑤,𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−2 2 and 𝑁−(𝑣𝑛) = {𝑢,𝑤,𝑣𝑛−1}. by the definition of 𝑓, the label of the vertex 𝑢 is 𝛾1 which is relatively prime to all the gaussian integers. 𝑁−(𝑣2𝑖−1) = 𝜙 for 1 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is even. next to prove 𝑓 is also a gaussian out-nighborhood prime labeling. here 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 . now 𝑁+(𝑢) = 𝑁+(𝑤) = {𝑣2,𝑣4,…,𝑣2𝑖} for 1 ≤ 𝑖 ≤ 𝑛 2 and the out-neighborhood vertices of 𝑢 and 𝑤 are labeled by the consecutive gaussian integers𝛾 ( 𝑛 2 )+3 ,𝛾 ( 𝑛 2 )+4 , …, 𝛾 ( 𝑛 2 )+𝑖+2 and so they are relatively prime. further, 𝑁+(𝑣1) = {𝑢,𝑤,𝑣2} and 𝑁 +(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖,𝑢,𝑤} for 2 ≤ 𝑖 ≤ 𝑛 2 . and the labels of vertices in 𝑁+(𝑣1) and 𝑁 +(𝑣2𝑖−1) for 2 ≤ 𝑖 ≤ 𝑛 2 contains the gaussian integer 𝛾1 = 1 which is relatively prime to all the gaussian integers. also, 𝑁+(𝑣2𝑖) = 𝜙 for 1 ≤ 𝑖 ≤ 𝑛 2 . therefore, 𝑓 is a gaussian outneighborhood prime labeling when 𝑛 even. from both the cases, 𝑓 is a gaussian twin neighborhood prime labeling. hence double sole double alternating fan (𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗ ⃗) admits a gaussian twin neighborhood prime labeling. theorem 4.14. double di-double alternating fan 𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗ is a gaussian twin neighborhood prime digraph. proof: let 𝑉(𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑢,𝑤,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} be the vertex set where 𝑣𝑖 represent the ith vertex of the common path 𝑃𝑛⃗⃗ ⃗ and 𝑢,𝑤 be the apex vertices. this digraph has 𝑛 + 2 vertices and 3𝑛 − 1 arcs. case (i): 𝑛 is odd 𝐴(𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑢𝑣2𝑖−1⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−1 2 } ∪ {𝑤𝑣2𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛+1 2 } ∪ {𝑣2𝑖𝑤⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛−1 2 } is the arc set. 28 gaussian twin neighborhood prime labeling of fan digraphs define an injective function 𝑓:𝑉(𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by 𝑓(𝑢) = 𝛾1;𝑓(𝑤) = 𝛾2 and 𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛+1 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1 2 )+𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . here, 𝑑−(𝑢) > 1,𝑑−(𝑤) > 1, 𝑑−(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛+1 2 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . now 𝑁−(𝑢) = 𝑁−(𝑤) = {𝑣2,𝑣4,…,𝑣2𝑖} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the label set of inneighborhood vertices of 𝑢 and 𝑤 are consecutive gaussian integers in the spiral ordering and so those are relatively prime. also, 𝑁−(𝑣2𝑖−1) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛+1 2 and the vertices 𝑢 and 𝑤 are labeled with consecutive gaussian integers 𝛾1 = 1 and 𝛾2 = 1 + 𝑖. since the consecutive gaussian integers in the spiral ordering are relatively prime. further, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−1 2 and the labels of the vertices in 𝑁−(𝑣2𝑖) are consecutive gaussian integers and so are relatively prime. therefore, 𝑓 is a gaussian in-neighborhood prime labeling when 𝑛 is odd. here𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1, 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤ 𝑛−1 2 and 𝑑+(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−1 2 . now 𝑁+(𝑢) = 𝑁+(𝑤) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤ 𝑛+1 2 . by the definition of𝑓, the out-neighborhood vertices of 𝑢 and 𝑤 are labeled by the consecutive gaussian integers𝛾3, 𝛾4, …, 𝛾𝑖+2. since the consecutive gaussian integers in the spiral ordering are relatively prime. 𝑁+(𝑣1) = {𝑣2} and 𝑁 +(𝑣𝑛) = {𝑣𝑛−1}. now 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} for 2 ≤ 𝑖 ≤ 𝑛−1 2 . the vertices 𝑣2𝑖−2and 𝑣2𝑖 are labeled by the consecutive gaussian integers and which are relatively prime. also, 𝑁+(𝑣2𝑖) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛−1 2 . since the label of the vertex 𝑢 is 𝛾1 = 1 which is relatively prime to all the gaussian integers. therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is odd. (i) and (ii) imply, 𝑓 is a gaussian twin neighborhood prime labeling when 𝑛 is odd. case (ii): 𝑛 is even 𝐴(𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑛−2 2 } ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑢𝑣2𝑖−1⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑣2𝑖𝑤⃗⃗⃗⃗⃗⃗⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } ∪ {𝑤𝑣2𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗|1 ≤ 𝑖 ≤ 𝑛 2 } is the arc set. define an injective function 𝑓:𝑉(𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗) → [𝛾𝑛+2] by (𝑢) = 𝛾1,𝑓(𝑤) = 𝛾2 , 𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛 2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛 2 )+𝑖+2 for 1 ≤ 𝑖 ≤ 𝑛 2 . here, 𝑑−(𝑢) > 1,𝑑−(𝑤) > 1, 𝑑−(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛−2 2 . now 𝑁−(𝑢) = 𝑁−(𝑤) = {𝑣2,𝑣4,…,𝑣2𝑖} for 1 ≤ 𝑖 ≤ 𝑛 2 . further, the labels of vertices in 𝑁−(𝑢) and 𝑁−(𝑤) are consecutive gaussian integers in the spiral ordering and so those are relatively prime. 29 palani k and shunmugapriya a also, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛−2 2 and the vertices in 𝑁−(𝑣2𝑖) are labeled with the consecutive gaussian integers and so those are relatively prime. 𝑁−(𝑣𝑛) = {𝑣𝑛−1}. 𝑁−(𝑣2𝑖−1) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛 2 . since the vertices 𝑢 and 𝑤 are labeled with the consecutive gaussian integers 𝛾1 = 1 and 𝛾2 = 1 + 𝑖 respectively. then 𝛾1 and 𝛾2 are relatively prime. therefore, 𝑓 is a gaussian inneighborhood prime labeling when 𝑛 is even. next to prove 𝑓 is also gaussian out-neighborhood prime labeling. here 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1, 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤ 𝑛 2 and 𝑑+(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛 2 . now 𝑁+(𝑢) = 𝑁+(𝑤) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤ 𝑛 2 . by the definition of 𝑓, the label set of the out-neighborhood vertices of 𝑢 and 𝑤 are labeled by the consecutive gaussian integers 𝛾3, 𝛾4, …, 𝛾𝑖+2 and so those are relatively prime. also, 𝑁+(𝑣1) = {𝑣2}. now 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} for 2 ≤ 𝑖 ≤ 𝑛 2 and the vertices 𝑣2𝑖−2 and 𝑣2𝑖 are labeled by the consecutive gaussian integers 𝛾𝑛 2 +𝑖+1 and 𝛾𝑛 2 +𝑖+2 . 𝑁+(𝑣2𝑖) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤ 𝑛 2 . since the label of the vertex 𝑢 is 𝛾1 = 1 which is relatively prime to all the consecutive gaussian integers. therefore, 𝑓 is a gaussian out-neighborhood prime labeling when 𝑛 is even. (iii) and (iv) imply 𝑓 is a gaussian twin neighborhood prime labeling when 𝑛 is even. from cases (i) and (ii), 𝑓 is a gaussian twin neighborhood prime labeling. hence, double di-double alternating fan 𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗ ⃗ is a gaussian twin neighborhood prime digraph. 5 conclusions in this article, we established gaussian twin neighborhood prime labeling in fan and double fan digraphs. in this way, we extend our thoughts to different types of labeling in digraphs. references [1] alex rosa, on certain valuation of the vertices of a graph, in theory of graphs, gordon and breach, 1967. [2] gallian j a, a dynamic survey of graph labeling, the electronic journal of combinatorics, #ds6, 2019. [3] harary f, graph theory, addison wesley publishing company, reading, mass, 1972. 30 gaussian twin neighborhood prime labeling of fan digraphs [4] klee s, lehmann h and park a, prime labeling of families of trees with gaussian integers, akce international journal of graphs and combinatorics, 13(2): 165-176, 2016. [5] lehmann h and park a, prime labeling of small trees with gaussian integers, rose hulman undergraduate mathematics journal, 17(1): 72-97, 2016. [6] palani k and shunmugapriya a, gaussian twin neighborhood prime labeling on wheel digraphs, industrial engineering journal, 15(11): 226-243, 2022. [7] rajesh kumar t j & mathew varkey t k, gaussian prime labeling of some classes of graphs and cycles, annals of pure and applied mathematics, 16 (1): 133-140, 2018. [8] rokad a h, product cordial labeling of double wheel and double fan related graphs, kragujevac journal of mathematics, 43(1): 7-13, 2019. 31 on 2-repeated burst codes b. k. dass department of mathematics university of delhi delhi 110 007, india e-mail: dassbk@rediffmail.com poonam garg∗ department of mathematics deen dayal upadhyaya college (university of delhi) shivaji marg, karam pura new delhi 110 015, india e-mail: poonamgarg 68@yahoo.co.in abstract. there are several kinds of burst errors for which error detecting and error correcting codes have been constructed. in this paper, we consider a new kind of burst error which will be termed as ‘2-repeated burst error of length b(fixed)’. linear codes capable of detecting such errors have been studied. further, codes capable of detecting and simultaneously correcting such errors have also been dealt with. the paper obtains lower and upper bounds on the number of parity-check digits required for such codes. an example of such a code has also been provided. ∗corresponding author. ratio mathematica, 19, pp. 11-24 11 1. introduction investigations in coding theory have been made in several directions but one of the most important aspects considered has been the detection and correction of errors. the beginning was made with the detection and correction of random errors [refer hamming (1950)] and thereafter the advent of bch codes for multiple error correction was taken up. though there is a long history concerning the growth of the subject and many of the codes developed have found applications in numerous areas of practical interest, one of the areas of practical importance in which a parallel growth of the subject took place is that of burst error detecting and correcting codes. it has also been observed that in many communication channels the likelihood of the occurrence of errors is more in adjacent digits rather than their occurrence in a random manner. extending the work of hamming (1950), abramson (1959) developed codes which dealt with the correction of single and double adjacent errors. the work due to fire (1959) depicted a more general concept of clustered errors which in the literature are known as ‘burst errors’. a burst of length b may be defined as follows: definition 1. a burst of length b is a vector whose only non-zero components are among some b consecutive components, the first and the last of which is non-zero. fire (1959) considered two kinds of bursts viz. open-loop burst which are popularly refered to simply a burst (as in definition 1) and the other is called as ‘closed-loop burst’ defined as follows: ratio mathematica, 19, pp. 11-24 12 definition 2. let b be an integer and x = (ξ1, . . . ,ξn) be a vector in v n(q) , a vector space of n-tuples over gf(q) . if 2 6 b 6 n + 1 2 , then x is called a ‘closed-loop burst vector of length b’ whenever there is an i such that 1 6 i 6 b− 1 , ξi · ξn−b+i+1 6= 0 , ξi+1 = ξi+2 = · · · = ξn−b+i = 0 . stone (1961), and bridwell and wolf (1970) considered multiple bursts. it was noted by chien and tang (1965) that in several channels errors occur in the form of a burst but not in the end digits of the burst. channels due to alexander, gryb and nast (1960) fall in this category. this prompted chien and tang to propose a modification in the definition of a burst and they defined a burst of length b which shall be called as ct burst of length b as follows: definition 3. a ct burst of length b is a vector whose only non-zero components are confined to some b consecutive positions, the first of which is non-zero. this definition was further modified by dass (1980) as follows: definition 4. a burst of length b(fixed) is an n-tuple whose only non-zero components are confined to b consecutive positions, the first of which is non-zero and the number of its starting positions in an n-tuple is among the first n− b + 1 components. it is clear that the nature of burst errors differ from channel to channel depending upon the behaviour of channels or the kind of errors which occur during the process of transmission. also, in very busy communication channels, errors repeat themselves. so is a situation when ratio mathematica, 19, pp. 11-24 13 errors occur in the form of a burst. in a way, we need to consider repeated bursts. codes that detect and correct repeated open-loop bursts have been studied by berardi, dass and verma (2009). in this paper, a 2-repeated burst (open-loop) of length b has been defined as follows: definition 5. a 2-repeated burst of length b is an n-tuple whose only non-zero components are confined to two distinct sets of b consecutive digits, the first and the last component of each set being non-zero. the development of codes detecting and correcting repeated burst errors will economize in the number of parity-check digits required not only in comparison with codes dealing with detection and correction of the same number of random errors but also in comparison to the usual burst error detecting and correcting codes while considering such repeated bursts as single bursts. in this paper, we introduce yet another kind of a repeated burst and define a ‘2-repeated burst of length b(fixed)’ as follows: definition 6. a 2-repeated burst of length b(fixed) is an n-tuple whose only non-zero components are confined to two distinct sets of b consecutive digits, the first component of each set is non-zero and the number of its starting positions is among the first n− 2b + 1 components. for example, (1000001000) is a 2-repeated burst of length up to 4 (fixed) whereas (0000100100) is a 2-repeated burst of length at most 3 (fixed). ratio mathematica, 19, pp. 11-24 14 these 2-repeated burst patterns of length b(fixed) include several 2repeated bursts of length b or less in an obvious manner. moreover, these are four times in number than the 2-repeated burst patterns of the same length in the binary case, and in the q-nary case these are q2 (q − 1)2 -times the number of 2-repeated bursts. it is clear from the fact that the number of 2-repeated burst vectors of length b is (q−1)4(q)2(b−2) and the number of 2-repeated burst vectors of length b(fixed) is (q−1)2(q)2(b−1) giving the ratio as q2 (q − 1)2 . in section 2, we obtain bounds for codes detecting 2-repeated bursts of length b(fixed). section 3 presents a bound for codes which can detect and simultaneously correct such 2-repeated bursts. in what follows a linear code will be considered as a subspace of the space of all n-tuples over gf(q) . the distance between two vectors shall be considered in the hamming sense. 2. 2-repeated burst error detecting codes in this section, we consider linear codes that are capable of detecting any 2-repeated burst of length b(fixed). clearly, the patterns to be detected should not be code words. in other words we consider codes that have no 2-repeated burst of length b(fixed) as a code word. firstly, we obtain a lower bound over the number of parity-check digits required for such a code. theorem 1. any (n,k) linear code over gf(q) that detects any 2-repeated burst of length b(fixed) must have at least 2b parity-check digits. ratio mathematica, 19, pp. 11-24 15 proof. the result will be proved on the basis that no detectable error vector can be a code word. let v be an (n,k) linear code over gf(q) . consider a set x that has all those vectors which have their non-zero components confined to some two fixed distinct b consecutive components in the first n − b + 1 components. we claim that no two vectors of the set x can belong to the same coset of the standard array, else a code word shall be expressible as a sum or difference of two error vectors. assume on the contrary that there is a pair, say x1,x2 in x belonging to the same coset of the standard array. their difference viz. x1−x2 must be a code vector. but x1−x2 is a vector all of whose non-zero components are confined to the same two fixed b consecutive components and so is a member of x , i.e., x1−x2 is a 2-repeated burst of length b(fixed), which is a contradiction. thus all the vectors in x must belong to distinct cosets of the standard array. the number of such vectors over gf(q) is clearly q2b . the theorem follows since there must be at least this number of cosets. remark 1. incidentally, this result coincides with [theorem 1, berardi, dass and verma (2009)] when bursts considered are open-loop bursts. an upper bound on the number of check digits required for the construction of a linear code is provided in the following theorem. this bound assures the existence of a linear code that can detect all 2repeated bursts of length b(fixed). the bound has been obtained by first constructing a matrix under certain constraints and then by reversing the ratio mathematica, 19, pp. 11-24 16 order of its columns altogether giving rise to a parity-check matrix for the requisite code, a technique given by dass (1980). theorem 2. there exists an (n,k) linear code that has no 2-repeated burst of length b(fixed) as a code word provided that qn−k > qb−1[1 + (n− 2b + 1)(q − 1)qb−1] . (1) proof. the existence of such a code will be shown by constructing an appropriate (n − k) × n parity-check matrix h . firstly, we construct a matrix h′ from which the requisite parity-check matrix h shall be obtained by reversing the order of its columns altogether. any non-zero (n−k) -tuple is chosen as the first column h1 of h′. subsequent columns are added to h′ such that after having selected the first j − 1 columns h1,h2, . . . ,hj−1 , j -th column hj is added provided that hj 6= (αj−b+1hj−b+1 + αj−b+2hj−b+2 + · · · + αj−1hj−1) + (βihi + βi+1hi+1 + · · · + βi+b−1hi+b−1) (2) where either all βi are zero or if βt is the last nonzero coefficient then b 6 t 6 j − b, αj ’s and βi ’s in gf(q) . this condition ensures that no 2repeated burst of length b(fixed) will be a code word. the number of ways in which the coefficients αj can be selected is clearly q b−1 . to enumerate the coefficients βi is equivalent to enumerate the number of bursts of length b(fixed) amongst the first j − b components. this number, including the vector of all zeros, is [theorem 1, dass (1980)] 1 + (j − 2b + 1)(q − 1)qb−1 . ratio mathematica, 19, pp. 11-24 17 thus, the total number of possible combinations that hj can not be equal to, is qb−1[1 + (j − 2b + 1)(q − 1)qb−1] . (3) at worst, all these linear combinations might yield a distinct sum. therefore a column hj can be added to h ′ provided that qn−k > (3) . the required parity-check matrix h = [h1h2 . . .hn] can be obtained from h′ by reversing the order of its columns altogether (hi → hn−i+1 ). for a code of length n, replacing j by n gives the result. remark 2. in view of the fact that the result obtained in theorem 2 is the same as the result for the correction of bursts of length b(fixed), such a code can serve dual purpose viz. it can either be used to correct bursts of length b(fixed) or can be used to detect 2-repeated bursts of length b(fixed). 3. simultaneous detection and correction of repeated burst errors in this section we determine extended reiger’s bound [reiger (1960); also refer theorem 4.15, peterson and weldon (1972)] for simultaneous detection and correction of 2-repeated bursts of length b(fixed). the following theorem gives a bound on the number of parity-check digits for ratio mathematica, 19, pp. 11-24 18 a linear code that simultaneously detects and corrects 2-repeated bursts of length b(fixed). theorem 3. an (n,k) linear code over gf(q) that corrects all 2repeated bursts of length b(fixed) must have at least 4b parity-check digits. further, if the code corrects all 2-repeated bursts of length b(fixed) and simultaneously detects 2-repeated bursts of length d(fixed) (d > b) then the code must have at least 2(b + d) parity-check digits. proof. we first prove the first part. consider a burst of length 4b(fixed) in the first n − b + 1 components. such a vector is expressible as a sum or difference of two vectors, each of which is a 2-repeated burst of length b(fixed). these component vectors must belong to different cosets of the standard array because both such errors are correctable errors. accordingly, such a vector viz. burst of length 4b(fixed) can not be a code vector. in view of theorem 1, such a code must have at least 4b parity-check digits. further, consider a burst of length 2(b+d) (fixed), the burst confining to the first n − b + 1 components. such a vector is expressible as a sum or difference of two vectors, one of which is a 2-repeated burst of length b(fixed) and the other is a 2-repeated burst of length d(fixed). both such component vectors, one being a detectable error and the other being a correctable error, can not belong to the same coset of the standard array. therefore such a vector can not be a code vector, i.e., a burst of length 2(b+d) (fixed) can not be a code vector. hence the code must have at least 2(b + d) parity-check digits. ratio mathematica, 19, pp. 11-24 19 remark 3. incidentally, this result coincides with [theorem 3, berardi, dass and verma (2009)], when bursts considered are open-loop bursts. example. we conclude the paper with an example. consider a (7, 2) binary code with parity check matrix h =   0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0   this matrix has been constructed by the synthesis procedure, outlined in the proof of theorem 2, by taking b = 3 . it can be seen from table 1 that the syndromes of the different 2-repeated bursts of length 3 (fixed) are nonzero, showing thereby that the code that is the null space of this matrix can detect all bursts of length 3 (fixed). table 1 error vectors syndromes 1000000 00001 1001000 01001 1001100 11001 1001010 01110 1001110 11110 1000100 10001 1000110 10110 1000101 11111 1000111 11000 1100000 00010 1101000 01010 1101100 11010 1101010 01101 (contd.) ratio mathematica, 19, pp. 11-24 20 error vectors syndromes 1101110 11101 1100100 10010 1100110 10101 1100101 11100 1100111 11011 1010000 00101 1011000 01101 1011100 11101 1011010 01010 1011110 11010 1010100 10101 1010110 10010 1010101 11011 1010111 11100 1110000 00110 1111000 01110 1111100 11110 1111010 01001 1111110 11001 1110100 10110 1110110 10001 1110101 11000 1110111 11111 0100000 00011 0100100 10011 0100110 10100 0100101 11101 0100111 11010 0110000 00111 0110100 10111 0110110 10000 0110101 11001 0110111 11110 0101000 01011 0101100 11011 (contd.) ratio mathematica, 19, pp. 11-24 21 error vectors syndromes 0101110 11100 0101101 10101 0101111 10010 0111000 01111 0111100 11111 0111110 11000 0111101 10001 0111111 10110 0010000 00100 0011000 01100 0010100 10100 0011100 11100 0001000 01000 0001100 11000 0001010 01111 0001110 11111 0000100 10000 0000110 10111 0000101 11110 0000111 11001 references [1] abramson, n.m. (1959), a class of systematic codes for nonindependent errors, ire trans. on information theory, it-5, no. 4, pp. 150–157. [2] alexander, a.a., gryb, r.m. and nast, d.w. (1960), capabilities of the telephone network for data transmission, bell system tech. j., vol. 39, no. 3, pp. 431–476. [3] berardi, l., dass, b.k. and verma, rashmi (2009), on 2-repeated burst error detecting codes, journal of statistical theory and practice, vol. 3, no. 2, pp. 381–391. [4] bridwell, j.d. and wolf, j.k. (1970), burst distance and multipleburst correction, bell system tech. j., vol. 49, pp. 889–909. ratio mathematica, 19, pp. 11-24 22 [5] campopiano, c.n. (1962), bounds on burst error correcting codes, ire trans., it-8, pp. 257–259. [6] chien, r.t. and tang, d.t. (1965), on definitions of a burst, ibm journal of research and development, vol. 9, no. 4, pp. 292–293. [7] dass, b.k. (1980), on a burst-error correcting code, journal of information and optimization sciences, vol. 1, no. 3, pp. 291–295. [8] fire, p. (1959), a class of multiple-error-correcting binary codes for non-independent errors, sylvania report rsl-e-2, sylvania reconnaissance systems laboratory, mountain view, calif. [9] hamming, r.w. (1950), error-detecting and error-correcting codes, bell system technical journal, vol. 29, pp. 147–160. [10] peterson, w.w. and weldon, e.j., jr. (1972), error-correcting codes, 2nd edition, the mit press, mass. [11] reiger, s.h. (1960), codes for the correction of “clustered errors”, ire trans. inform. theory, it-6, pp. 16–21. [12] stone, j.j. (1961), multiple burst error correction, information and control, vol. 4, pp. 324–331. ratio mathematica, 19, pp. 11-24 23 ratio mathematica volume 47, 2023 door spaces on ntopology loyala foresith spencer j * davamani christober m† abstract in this article, we explore the idea of door space on n-topological space. here, we discuss which door spaces in this space are related with nτ submaximal. the equivalent conditions shows how it connects a n-topological property. also, we derive various door spaces using separation axioms and discuss the characteristics of such door spaces. we take a strong form of open set in n-topological space and introduce a new door space called nτβ door space. in addition, we analyze nτβ-door space and discuss the relationship between a nτβ-locally closed set and nτ-closed set. keywords: n-topology; door space; sub-maximal; nτdβ-open sets. 2020 ams subject classifications: 54a05, 54a10, 54c05 1 *the american college, madurai, india; e-mail: spencerjraja@gmail.com. †the american college, madurai, india; e-mail:christober.md@gmail.com. 1received on september 15, 2022. accepted on march 10, 2023, published on april 4, 2023. doi: 10.23755/rm.v39i0.957. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 367 loyala foresith spencer j, davamani christober m 1 introduction in 1955, j.l. kelly [2], introduced the term ’door’ in classical topology. he investigated the relationship between the different topological spaces and door spaces. mccartan [5] found three types of door spaces and established the concepts of connected door spaces and maximally connected. muckenhoupt and williams [6] concluded that there exists a non-zero borel measure in every connected door space. mathew [4]enquired about hyper-connected door space and proposed its related concepts. in 1983, monsef et al.[1] initiated the study of β-open sets and β-continuity in a topological space. this article explores the idea of door space in n topological space. here, we introduce nτ-door space and discuss which door spaces are related with nτ submaximal. the equivalent conditions shows how it connects with nτ frontier, a n-topological property. also, we derive various door spaces using separation axioms and discuss the characteristics of such door spaces. we take a strong form of open set in n-topological space and introduce a new door space called nτβ door space . finally, we introduce nτβ locally closed sets and its properties which are all the essential tool for the future development of this concept. 2 nτ door spaces in this section, we establish various door spaces in n-topological space and explicit its properties. moreover, discussed about the sub maximal concept and analyze its characterization in this space. definition 2.1. a (p, nτ) space is called hausdorff space of n topological space if for every given pair of different points z, q ∈ p ,there exists r, s ∈ nτ o(p) such that z ∈ r, q ∈ s, r ∩ s = ϕ and is denoted by nτ hausdorff space. definition 2.2. a (p, nτ) space is called semi hausdorff space of n topological space if for every given pair of different points z, q ∈ p ,there exists r, s ∈ nτ so(p) such that z ∈ r, q ∈ s, r∩s = ϕ and is denoted by nτ semi hausdorff space. theorem 2.1. each nτ hausdorff space of (p, nτ) is nτ semi hausdorff space. proof. suppose (p, nτ) is a nτ hausdorff space then there exisits two disjoint points which can be isolated by disjoint nτ open set. since each nτ open set is a nτ semi open then we may conclude that each nτ hausdorff space of (p, nτ) is nτ semi hausdorff space. 368 door spaces on ntopology remark 2.1. the inverse of the theorem 5.2.3 need not be true which is shown below. let p = {l, m, n, o, p}. for n = 3, consider τ1 = {ϕ, {l}, {l, p}, {l, m, o, p}, p}, τ2 = {ϕ, {l, m, o}, {m, o}, p} and τ3 = {ϕ, {m, o, p}, {l, m, o, p}, p}. then 3τ = {ϕ, {l}, {p}, {l, p}, {m, o}, {l, m, o}, {m, o, p}, {l, m, o, p}, p}. here {l, n} is 3τ semi open set but not 3τ open. definition 2.3. if every subset of (p, nτ) is either nτ open or nτ-closed then (p, nτ) is called as nτ door space. example 2.1. let s = {x, y, z}. for n = 2, τ1 = {ϕ, {z}, {x, z}, s}. then 2τ o(s) = {ϕ, {x}, {z}, {x, z}, s}, 2τ c(s) = {ϕ, {y, z}, {x, y}, {y}, s}. therefore s is a 2τ door space. theorem 2.2. every subspace g of a nτ door space is nτ door space. proof. let (p, nτ) be a door space. let g ⊆ p and u ∈ g. since p is a nτ door space then u is either nτ open or nτ closed in p and hence in g. therefore g is also a nτ door space. definition 2.4. a subset s of (p, nτ) is nτ dense if nτ cl(s) = p . definition 2.5. a n-topological space (p, nτ) is sub maximal if every nτ dense subset of p is nτo(p) and is denoted as nτsub maximal. theorem 2.3. every door space (p, nτ) is nτ sub maximal. proof. let (p, nτ) be a door space and v ⊂ p be a nτ dense. if v is not a nτ open then it is nτ closed since p is a nτ door space. now v = nτcl(v ) = p and v is nτ open. hence p is nτ sub maximal. theorem 2.4. every subspace l of a sub maximal space (p, nτ) is again a nτ sub maximal. proof. let a be a nτ dense subset of l. then nτcl(a) ∩ l = l and so l ⊂ nτcl(a). since a∪(p −nτcl(a)) is nτ dense in p then it is an nτ open subset of p . hence s ∩ (a ∪ (p − nτcl(a))) = a is nτ open in l or equivalently l is nτ submaximal. theorem 2.5. in (p, nτ), the following conditions are equivalent 1. p is nτ submaximal. 2. for any a ⊂ p , the subspace nτ fr(a) = nτ-cl(a)− nτ int(a) = nτ-cl(a) ∩ nτ-cl(p − a) is discrete. 369 loyala foresith spencer j, davamani christober m proof. (1) =⇒ (2) let y ∈ nτfr(a). since a is nτ dense in nτcl(a) then so is a ∪ {y}. since nτcl(a) is sub maximal according to theorem 5.2.11 then a ∪ {y} = nτcl(a) ∩ u where u is nτ open in p . in the same way it can seen that ac ∪ {y} = nτcl(ac) ∩ v where v is nτ open in p . thus {y} = (a ∪ {y}) ∩ (ac ∪ {y}) = nτcl(a) ∩ nτcl(p − a) ∩ u ∩ v . hence {y} is nτ open in nτfr(a) and so nτfr(a) is discrete. (2) =⇒ (1) let a be a nτ dense in p . by assumption, nτcl(a)−nτint(a) = p − nτint(a) is discrete and thus a − nτint(a) is its nτ open subset. hence a − nτint(a) = (p − nτint(a)) ∪ u where u is nτ open in p . thus a − nτint(a) ⊂ u and so a−nτint(a) ⊂ u −nτint(a). for the reverse inclusion if y ∈ u − nτint(a) then y ∈ (p − nτint(a)) ∩ u = a − nτint(a). this shows that a − nτint(a) = u − nτint(a) and hence a = u ∪ nτint(a). thus a is nτ open in p . theorem 2.6. in (p, nτ), the following conditions are equivalent 1. p is nτ submaximal. 2. every nτ pre open subset of p is nτ open. proof. (1) =⇒ (2) let a be a nτ pre open in p . then a ⊂ nτint(nτcl(a)). since a is nτ dense in nτcl(a) and nτcl(a) is nτ sub maximal according to theorem 5.2.11 then a is nτ open in nτcl(a). thus a is nτ open in nτint(nτcl(a)). since nτint(nτcl(a)) is nτ open in p then we may conclude that a is nτ open in p . (2) =⇒ (1) let a be a nτ dense in p . since a ⊂ p = nτint(p) = nτint(nτcl(p)) then a is nτ pre open and by assumption, nτ open. this shows that p is nτ sub maximal. theorem 2.7. if (p, nτ) is submaximal and u ⊂ p then u is nτ open iff it is the intersection of a nτ dense and nτ regular open [3]. proof. it is enough to prove that for every nτ open set u, we have u = d ∩ v where d is nτ dense and v is nτ regular open since the reverse inclusion is trivial. clearly u ⊂ nτint(nτcl(u)). thus u = nτcl(u) − (nτcl(u) − u) = nτcl(u) ∩ (p − (nτcl(u) − u)) = nτint(nτcl(u)) ∩ (u ∪ p − nτcl(u)) where nτint(nτcl(u)) = v is nτ regular open and u ∪ (p − nτcl(u)) = d is nτ dense. theorem 2.8. the homeomorphic image of nτ door space is a nσ door space. proof. consider (z, nτ) and (q, nσ) are door spaces and η : z → q be a homeomorphism. let u ⊆ q. consider η−1(u) ⊆ z. since z is nτ door space then η−1 is either nτ open or nτ closed in z. now η(η−1(u)) = u and u is either nσopen or nσ closed in q. 370 door spaces on ntopology definition 2.6. in (p, nτ), if every subset of p is either nτ semi-open or nτ semi-closed then (p, nτ) is semi-door space and is denoted by nτ semi-door space theorem 2.9. the homeomorphic image of nτ semi-door space is a nσ semidoor space. proof. let (z, nτ) and (q, nτ) are n topological spaces and (z, nτ) be nτ semi-door space. a mapping η : z → q be a homeomorphism. let p ⊂ q. consider η−1(p) ⊂ z, since z is nτ semi door space then η−1(p) is either nτ semi -open or nτ semi closed. so η(η−1(p)) = p is either nτ semi open or nτ semi closed. hence q is a nτ semi door space. theorem 2.10. a nτ clopen subspace of a nτ semi-door space is nτ semidoor space. proof. let (z, nτ) be semi-door space. let q be a nτ-clopen subset of z. let a ⊆ q and a ⊆ z. since z is a nτ semi-door space, then a is either nτ semi open or nτ semi closed in z. since q is nτ open and nτ closed, then a is either nτ semi open or nτ semi closed in q. hence q is nτ semi-door space. theorem 2.11. if (p, nτ) be a door space and if z ∈ p , s is a nτ neighbourhood of z, then s − z ∈ nτ and s ∈ nτ. proof. let s be an neighbourhood of a point v and if v is nτint(s) then it is enough to prove that s − v is nτ open. if we assume s − v is not nτ open then p − (s − v) = (p − s) ∪ v should be nτ open. this contradicts that v = s ∩ ((p − s) ∪ v) should be nτ open. definition 2.7. in (p, nτ), if every two disjoint points in p can be isolated by disjoint nτ open sets then (p, nτ) is a hausdorff door space and is denoted by nτ-hausdorff door space. definition 2.8. in (p, nτ), if every two disjoint points in p can be isolated by disjoint nτ semi open sets then (p, nτ) is a semi hausdorff door space and is denoted by nτ semi-hausdorff door space. definition 2.9. a nτ semi door space is said to be hausdorff semi-door space of (p, nτ) if a given pair of different points r, s ∈ p , there exist m, n ∈ nτo(p) such that r ∈ m, s ∈ n, m ∩ n = ϕ and is denoted by nτ-hausdorff semi door space. proposition 2.1. every nτ hausdorff door space is nτ semi-hausdorff door space. 371 loyala foresith spencer j, davamani christober m proposition 2.2. if (p, nτ) be hausdorff door space and z ∈ p then r ∈ nτ so(p) ⇐⇒ r ∈ nτ. proposition 2.3. if (p, nτ) be hausdorff door space and z, q ∈ (p, nτ). if z ∈ nτso(p) and q ∈ nτ then z ∩ q ∈ nτ so(p). theorem 2.12. nτ semi-hausdorff door space has atmost one limit point. proof. consider (p, nτ) be a hausdorff space. let a, b are distinct limit points in p. since p is nτ semi-hausdorff, ∃ g, h ∈ so(p) : a ∈ g, b ∈ h and g ∩ h = ϕ. since p is nτ door space then u = {g − {a}} ∪ {b} is either nτ open or nτ closed. suppose if it is nτ open then by theorem 5.2.19 u ∩ h = {b} is nτ semi open and hence by theorem 5.2.20 u ∩ h is nτ open. otherwise uc is nτ open and uc ∩ g = {a} is nτ semi open and nτ open by propositions 5.2.24 and 5.2.25. hence at least one of the two point will be isolated in p and by contradiction the result is proved. proposition 2.4. a nτ hausdorff semi-door space has atmost one limit point. proof. proof is similar as discussed in the previous result. 3 nτβ-door space and locally nτβ-closed set in this section, we introduce and analyze nτβ door space and nτβ locally closed sets. definition 3.1. a subset u of p is said to be locally nτ-closed set if u = r ∩ s where r is a nτ closed in p . the set of all locally nτ closed sets are denoted by lnτ-cl(p) definition 3.2. a subset u of p is said to be locally nτβ-closed set if u = r∩s where r is a nτβ open and s is nτβ-closed in p . the set of all locally nτβ closed sets are denoted by lnτβ-cl(p) definition 3.3. if every subset of (p, nτ) is either nτβ-open or nτβ closed then (p, nτ) is called nτβ door space and signified by nτβd. example 3.1. let s = {x, y, z}. for n = 2, τ1 = {ϕ, s}, τ2 = {ϕ, {x, y}, s}. then 2τ o(s) = {ϕ, {x, y}, s}, 2τcl(s) = {ϕ, {z}, s}. here s is a 2τβ door space. lnτ-cl(s) = {ϕ, {x, y}, {z}, s} and lnτβ-cl(p) = {ϕ, {x}, {y}, {z}, {x, y}, {y, z}, {z, x}, s}. remark 3.1. from the example 5.3.4, we get 372 door spaces on ntopology 1. every locally nτ-closed set is locally nτβclosed set but its converse need not be true. 2. every nτ-door space is a nτβd but its converse need not be true. remark 3.2. if (p, nτ) is nτβ door space then lnτβ-cl(p) = ℘(p). theorem 3.1. let g ⊆ p . then the following are equivalent: 1. g is locally nτβ closed sets 2. g = f ∩ nτ-cl(g) for some nτβ-open set f . proof. (1) =⇒ (2) let g ∈ lnτβcl(p). now there will be a nτβ open set f and a nτβ closed subset e such that g = f ∩ e since g ⊆ f and g ⊆ nτβcl(p) then nτβcl(p) ⊆ e. hence f ∩ nτβcl(p) ⊆ f ∩ e = g. therefore g = f∩ nτβcl(g). this proves (1) =⇒ (2) . (2) =⇒ (1) by definition, nτβcl(g) is nτβ closed. ∴ g = f∩lnτβcl(g) ∈ nτβlc(p). 4 conclusion in this paper, we introduced the idea of door spaces in n-topological space. their structural properties have been discussed and emphasized. some of the important results arrived through illustrated examples. the importance is to analyse the relationship with other ntopological properties. so, we investigated submaximal concepts and locally closed sets through n-topology. in addition, we introduce nτβ-door space and discuss the relationship between a nτβ-locally closed set and nτ-closed set. with the help of these locally closed sets it can be extend to introduce locally continuous maps in this topological space. these prime ideas can open the future scope of this concept and extended to other research areas of topology such as fuzzy topology, digital topology, and so on. references [1] abd el-monsef, m.e., el-deeb s.n. and mahmoud r.a. β-open sets and βcontinuous mappings, bull. fac. sci., 12:77 – 90,1983. [2] kelly j.l. general topology, princeton, nj. d. van nastrand, 1955. [3] loyala f. spencer j. and davamani christober m. theta open sets in ntopology, ratio mathematica, 43:163 – 175, 2022. 373 loyala foresith spencer j, davamani christober m [4] mathew p.m. on hyper connected spaces, indian j. pure appl. math..1988. [5] mccartan s.d. door spaces are identifiable, proceedingds of roy irish acad. sect. a, 87 (1): 13 – 16, 1987. [6] muckenhoupt b. and williams v. borel measures on connected door spaces, riv. mat. univ. parma., 3(2): 103 – 108, 1973. 374 ratio mathematica issue n. 30 (2016) pp. 45-58 issn (print): 1592-7415 issn (online): 2282-8214 dealing with randomness and vagueness in business and management sciences: the fuzzy-probabilistic approach as a tool for the study of statistical relationships between imprecise variables fabrizio maturo department of management and business administration university g. d’annunzio, chieti pescara f.maturo@unich.it abstract in practical applications relating to business and management sciences, there are many variables that, for their own nature, are better described by a pair of ordered values (i.e. financial data). by summarizing this measurement with a single value, there is a loss of information; thus, in these situations, data are better described by interval values rather than by single values. interval arithmetic studies and analyzes this type of imprecision; however, if the intervals has no sharp boundaries, fuzzy set theory is the most suitable instrument. moreover, fuzzy regression models are able to overcome some typical limitation of classical regression because they do not need the same strong assumptions. in this paper, we present a review of the main methods introduced in the literature on this topic and introduce some recent developments regarding the concept of randomness in fuzzy regression. keywords: fuzzy data; fuzzy regression; fuzzy random variable; tools for business and management sciences 2010 ams subject classifications: 62j05; 62j86; 03b52; 62a86; 97m10 doi: 10.23755/rm.v30i1.8 45 fabrizio maturo 1 introduction regression analysis offers a possible solution to study the dependence between two sets of variables. standard classical statistical linear regressions take the form [27]: yi = b0 + b1xi1 + b2xi2 + ... + bjxij + .... + bpxip + ui (1) where: • i=1,.....,n is the i-th observed unit; • j=1,...,p is the j-th observed variable; • yi is the dependent variable, observed on n units; • xij are the p independent variables observed on n units; • b0 is the crisp intercept and bj are the p crisp coefficients of the p variables; • ui are the random error terms that indicate the deviation of y from the model; • yi, xij, bj, ui are all crisp values. in classical regression model it is assumed that: • e(ui) = 0 • σ2ui = σ 2 • σui,uj = 0 ∀ i,j with i 6= j in matrix form, the classical regression model is expressed as: y = xβ + u (2) where y = (y1, y2, ..., yn)′, b = (b0, b1, b2, ..., bp)′, u = (u1, u2, ..., un)′ are vectors and x is a matrix: x =   1 x11 . . . x1p 1 x21 . . x2p 1 . . . . . 1 . . . . . 1 xn1 . . . xnp   46 dealing with randomness and vagueness in business and management sciences the aim of statistical regression is to find the set of unknown parameters so that the model gives is a good prediction of the dependent variable y. the most widely used regression model is the multiple linear regression model (mlrm), as well as the ordinary least squares (ols) [12] is the most widespread estimation procedure. under the ols assumptions the estimates are blue (best linear unbiased estimator), as stated by the famous gauss-markov theorem. ols is based on the minimization of the sum of squared deviations: min (y − xb)′(y + xb) (3) the optimal solution of the minimization problem is the following vector: b̂ = (x′x)−1x′y (4) the ols model is comfortable but its assumptions are every restrictive. several phenomena violate these assumptions causing biased and inefficient estimators [9]. in particular the assumptions e(u|x) ≈ n(0,σ2i) is very strong and rarely it is respected in real phenomena. moreover in case of ”quasi” multicollinearity (many highly correlated explanatory variables), although this does not violate ols assumption there is a bad impact on the variance of b. in these circumstance the ols estimators are efficient and unbiased but have large variance, making estimation useless from a practical point of view. the effects of the quasi multi-collinearity are more evident when the sample size is small [1]. the generally proposed solution consists in removing correlated exploratory variables. this solution is unsatisfying in many applications fields where the user would keep all variables in the model. in general, we can observe that classical statistical regression has many useful applications but presents troubles in the following situations [26]: • number of observations is inadequate (small data set); • difficulties verifying distribution assumptions; • vagueness in the relationship between input and output variables; • ambiguity of events or degree to which they occur; • inaccuracy and distortion introduced by linearization; furthermore, there are many variables that, for their own nature, are better described by a pair of ordered values, like daily temperatures or financial data. by summarizing this measurement with a single value, there is a loss of information. in these situations data are better described by interval values rather than by single 47 fabrizio maturo values. interval arithmetic studies and analyzes this type of imprecision; but if the intervals has no sharp boundaries, fuzzy set theory is the better tool. in particular fuzzy regression model are able to overcome some typical limitation of classical regression because they don’t need the same strong assumptions. furthermore, some nuanced concepts that exist in economic and social sciences, need to be necessarily treated with linguistic variables, which for their nature, are imprecise concepts. 2 fuzzy linear regression models (flr) there are two general ways, not mutually exclusive, to develop a fuzzy regression model: • models where the relationship of the variables is fuzzy; • models where the variables themselves are fuzzy; therefore fuzzy linear regression (flr) can be classified in: • partially fuzzy linear regression (pflr), that can be further divided into: – pflr with fuzzy parameters and crisp data; – pflr with fuzzy data and crisp parameters; • totally fuzzy linear regression (tflr) where data and parameters are both fuzzy. fuzzy least squares regression is more close to the traditional statistical approach. in fact, following the least squares line of thought [13], the aim is to minimize the distance between the observed and the estimated fuzzy data. this approach is referred as fuzzy least squares regression (flsr). in case of one independent variable, the model take the form: ỹi = b0 + b1x̃i + ũi i=1,2,...,n (5) where: • i=1,.....,n is the i-th observed unit; • yi is the dependent fuzzy variable, observed on n units; • xi is the independent fuzzy variable, observed on n units; 48 dealing with randomness and vagueness in business and management sciences figure 1: relation between output and input variables • b0 and b1 are the crisp intercept and the crisp regression coefficient; • ui are the fuzzy random error terms; from a graphical point of view [26] the relation between output and input variables can be represented as shown in fig.1 in case of several independent variables, the model take the form: ỹi = b0 + b1x̃i1 + b2x̃i2 + ... + bjx̃ij + .... + bp x̃ip + ũi (6) where: • i=1,.....,n is the i-th observed unit; • j=1,...,p is the j-th observed variable; • yi is the dependent fuzzy variable, observed on n units; • xij are the p independent fuzzy variables, observed on n units; • b0 is the crisp intercept and bj are the p crisp regression coefficients measured for the p fuzzy variables; • ui are the fuzzy random error terms; limiting the reasoning to the first model, the error term can be expressed as follows: ũi = ỹi − b0 − b1x̃i i=1,2,...,n (7) 49 fabrizio maturo therefore, from a least square perspective, the problem becomes as follows: min n∑ i=1 [ỹi − b0 − b1x̃i]2 i=1,2,...,n (8) many criteria for measuring this distance have been proposed over the years; however, the most common are two methods: • the diamond’s approach; • the compatibility measures approach. 2.1 flsr using distance measures the diamond’s approach is also known as fuzzy least squares regression using distance measures. this is the most close approach to the traditional statistical one. following the least squares line of thought, the aim is to minimize the distance between the observed and the estimated fuzzy data, by minimizing the output quadratic error of the model. since the model contains fuzzy numbers the minimization problem considers distances between fuzzy numbers [5, 17, 20, 15, 19, 18]. diamond defined an l2-metric between two triangular fuzzy numbers; it measures the distance between two fuzzy numbers based on their modes, left spread and right spread as follows d[(c1, l1,r1), (c2, l2,r2)] 2 = = (c1 − c2)2 + [(c1 − l1) − (c2 − l2)]2 + [(c1 + r1) − (c2 + r2)]2 (9) the methods of diamond are rigorously justified by a projection-type theorem for cones on a banach space containing the cone of triangular fuzzy numbers, where a banach space is a normed vector space that is complete as a metric space under the metric d(x,y) = ||x−y|| induced by the norm [25]. in the case of crisp coefficients and fuzzy variables, the problem is the following: min n∑ i=1 d[ỹi ∗ − ỹi]2 i=1,2,...,n (10) where, ỹi ∗ = b0 + b1x̃i (11) 50 dealing with randomness and vagueness in business and management sciences figure 2: compatibility measure therefore the optimization problem can be written as follows: min n∑ i=1 d[b0 + b1x̃i − ỹi]2 i=1,2,...,n (12) using diamond’s difference in this minimization problem, we can obtain the parameters. if the solutions exist, it is necessary to solve a system of six equations in the same number of unknowns; of course, these equations arise from the derivatives being set equal to zero. 2.2 flsr using compatibility measures the second type of fuzzy least squares regression model is based on celmins’s compatibility measures [3]. a compatibility measure can defined by γ(ã,b̃) = maxmin(µa(x),µb(x)) (13) this index is included in the interval [0,1] as shown in fig. 2. a value of ”0” means that the membership functions of the fuzzy numbers a and b are mutually exclusive as shown in fig. 3. a value of ”1” means that the membership functions a coincides with that one of b as shown in fig.4. the basic idea is to maximize the overall compatibility between data and model. thus, the objective may be reformulated in a minimization problem with the following objective function: 51 fabrizio maturo figure 3: zero compatibility figure 4: max compatibility 52 dealing with randomness and vagueness in business and management sciences min n∑ i=1 [1 −γi]2 i=1,2,...,n (14) 3 fuzzy regression models with fuzzy random variables recent studies have reintroduced the concept of fuzzy random variables (frvs) [24] firstly introduced by puri and ralescu [23]. the need for frvs arises when the data are not only affected by imprecision but also by randomness [11]. several papers deal with this topic that it is called fuzzy-probabilistic approach. it consists in explicitly taking into account randomness for estimating the regression parameters and assessing their statistical properties [22, 7, 8]. the membership function of a fuzzy number can be expressed, in term of spreads as: µ ã (x) =   lam−x al for x ≤ am, al > 0 1 for x ≤ am, al = 0 rx−am ar for x > am, ar > 0 0 for x > am, ar = 0 (15) where the functions l, r : <− > [0, 1] are convex upper semi-continuous functions so that l(0) = r(0) = 1 and l(z) = r(z) = 0, for all z ∈ </[0, 1] [6] and am is the center, al and ar are the left and the right spread. of course these functions must be chosen by the researcher in advance and must be the same for all the data. in particular, for a triangular fuzzy number we obtain: µ ã (x) =   0 for x ≤ am −al 1 − am−x al for am −al ≤ x ≤ am 1 − x−am ar for am ≤ x ≤ am + ar 0 for x ≥ am + ar (16) a distance for these functions [21] could be: d2(ã,b̃) = (am−bm)2+[(am−λal)−(bm−λbl)]2+[(am+ρar)−(bm+ρbr)]2 (17) where, 53 fabrizio maturo λ = ∫ 1 0 l−1(α)dα ρ = ∫ 1 0 r−1(α)dα these functions consider the shape of the membership functions; for example, for triangular fuzzy numbers λ and ρ = 1/2. to avoid the problem of the non-negativity of the spreads of ỹ , it is possible to solve a non negative regression problem [14], or to transform the spreads of ỹ by means of the centers and the spreads of the p regressors x. in this context, we use the latter method introducing two invertible functions [7]: g : (0, +∞) −→< h : (0, +∞) −→< thus the linear regression model take the form   ym = xb ′ m + am + um g(yl) = xb ′ l + al + ul h(yr) = xb ′ r + ar + ur (18) where ul,um,ur are the real valued random variables with e(ul|(x)) = 0, e(um|(x)) = 0, e(ur|(x)) = 0. the row vector of length 3p of all the components of the regressors is: x = (xm1, xl1, xr1, ....., xmp, xlp, xrp) the row vectors of length 3p of the parameters related to x are: bm = (bmm1, bml1, bmr1, ..., bmmp, bmlp, bmrp) bl = (blm1, bll1, blr1, ..., blmp), bllp, blrp) br = (brm1, brl1, brr1, ..., brmp, brlp, brrp) the generic element bijt is the regression coefficient between the component i�[m,l,r] of ỹ , where m,l,r refer to center and the transformed spread of ỹ , and the component j�[m,l,r] of the regressor x̃t with t=1,....,p, where m,l,r refer to 54 dealing with randomness and vagueness in business and management sciences the corresponding center, left spread and right spread. for example bmr2 is the relationship between the right spread of x̃2 and the center of y. of course, am, al, and ar are the intercepts. the covariance matrix of x is denoted by: σ(x) = e[(x − e(x))′(x − e(x))] (19) the covariance matrix of um, ul, ur is indicated with σ and contains the variances σ2um , σ 2 ul and σ2ur . the regression parameters can be expressed as: bm ′ = [σ(x)] −1e[(x − e(x))′(ym − e(ym)] bl ′ = [σ(x)] −1e[(x − e(x))′(g(yl) − e(g(yl))] br ′ = [σ(x)] −1e[(x − e(x))′(h(yr) − e(h(yr))] am = e(ym|x) − [σ(x)]−1e[(x − e(x))′(ym − e(ym)] al = e(g(yl)|x) − [σ(x)]−1e[(x − e(x))′(g(yl) − e(g(yl))] ar = e(h(yr)|x) − [σ(x)]−1e[(x − e(x))′(h(yr) − e(h(yr))] (20) since the total variation of the response can be written in terms of variances and covariances of real random variables, it can be decomposed in the variation not depending on the model and that explained by the model. thus, we can obtain a determination coefficient for the fuzzy model based on the decomposition of the total variance given by: e[d2(yt,e(yt)] = e[d 2(yt,e(yt|x)]+ + e[d2(e(yt|x, e(yt))] (21) therefore, the linear determination coefficient r2 can be defined as: r2 = e[d2(e((yt|x), e(yt))] e[d2(yt,e(yt)] = = 1 − e[d2(yt,e(yt|x)] e[d2(yt,e(yt)] (22) the meaning of this index is the same of the classical regression model. the estimation problem of the regression parameters is faced by means of the ls criterion. as shown in [6], applying the appropriate substitutions and using the concept of distance between two fuzzy numbers, like in the diamond’s approach, it is possible to find the equation of the estimators of all parameters. 55 fabrizio maturo 4 conclusions fuzzy regression models are able to overcome some limitations of classical regression because they do not need the same strong assumptions. in this paper, we have presented a review of the main methods introduced in the literature on this topic and some recent developments regarding the concept of randomness in fuzzy regression. in practical applications relating to business and management sciences, fuzzy regression models with fuzzy random variables are more suitable for the characteristics of the data. however, some of the main issues of zadeh’s operations with these models are the following: the addition and the multiplication between fuzzy numbers lead to a considerable increase of the spreads; the multiplication of two symmetric fuzzy numbers does not provide a symmetric fuzzy number or at least a fuzzy number with equal spreads; spreads of zadeh’s product depend heavily on the modes of the numbers; some important algebraic properties, such as the distributive property, are valid only in particular circumstances; the product of two triangular fuzzy numbers does not provide a triangular fuzzy number. therefore, alternative operations in order to overcome some problems connected to the addition and the product between fuzzy numbers in fuzzy linear regression models are strongly necessary. moreover, our research prospects include considering finite geometric spaces [16, 2], multivalued functions [4] and algebraic hyperoperations [10] in fuzzy regression models. references [1] achen, c., 1982. interpreting and using regression. sage publications, california. [2] ameri, r., nozari, t., 2012. fuzzy hyperalgebras and direct product, ratio mathematica. 24. [3] celmins, a., 1987. least squares model fitting to fuzzy vector data. fuzzy sets and systems 22(3), 245–269. [4] corsini, p., mahjoob, r., 2010. multivalued functions, fuzzy subsets and join spaces. ratio mathematica 20, 1–41. [5] diamond, p., 1988. fuzzy least squares. information sciences 46 (3), 141– 157. [6] ferrraro, m., colubi, a., gonzales-rodriguez, a., coppi, r., 2010. a linear regression model for imprecise response. int j approx reason 21, 759–770. 56 dealing with randomness and vagueness in business and management sciences [7] ferrraro, m., colubi, a., gonzales-rodriguez, a., coppi, r., 2011. a determination coefficient for a linear regression model with imprecise response. environmetrics 22, 516–529. [8] gonzales-rodriguez, a., blanco, a., colubi, a., lubiano, m., 2009. estimation of a simple linear regression model for fuzzy random variables. fuzzy sets syst 160, 357–370. [9] gujarati, d., 2003. basic econometrics. mcgraw-hill, new york. [10] hošková-mayerová, š., maturo, a., 2016. fuzzy sets and algebraic hyperoperations to model interpersonal relations, in: recent trends in social systems: quantitative theories and quantitative models. springer nature, pp. 211–221. url: http://dx.doi.org/10.1007/978-3-319-40585-8 19, doi:10.1007/978-3-319-40585-8 19. [11] klir, g., 2006. uncertainty and information: foundations of generalized information theory. wiley, new york. [12] kratschmer, v., 2006a. strong consistency of least-squares estimation in linear regression models with vague concepts. j multivar anal 97, 633–654. [13] kratschmer, v., 2006b. strong consistency of least-squares estimation in linear regression models with vague concepts. j multivar anal 97, 1044– 1069. [14] lawson, c., r.j. hanson, 1995. solving least squares problems. classics in applied mathematics 15. [15] maturo, a., maturo, f., 2013. research in social sciences: fuzzy regression and causal complexity, in: multicriteria and multiagent decision making with applications to economics and social sciences. springer, pp. 237–249. url: http://dx.doi.org/10.1007/978-3-642-35635-3 18, doi:10.1007/978-3-642-35635-3 18. [16] maturo, a., maturo, f., 2014. finite geometric spaces, steiner systems and cooperative games. analele universitatii ”ovidius” constanta seria matematica 22. url: http://dx.doi.org/10.2478/auom-2014-0015, doi:10.2478/auom-2014-0015. 57 fabrizio maturo [17] maturo, a., maturo, f., 2016. fuzzy events, fuzzy probability and applications in economic and social sciences, in: recent trends in social systems: quantitative theories and quantitative models. springer nature, pp. 223–233. url: http://dx.doi.org/10.1007/978-3-319-40585-8 20, doi:10.1007/978-3-319-40585-8 20. [18] maturo, f., 2016. fuzziness: teorie e applicazioni. aracne editrice, roma, italy. chapter la regressione fuzzy. pp. 99–110. [19] maturo, f., fortuna, f., 2016. bell-shaped fuzzy numbers associated with the normal curve, in: topics on methodological and applied statistical inference. springer. url: http://dx.doi.org/10.1007/978-3-319-44093-4 13, doi:10.1007/978-3-319-44093-4 13. [20] maturo, f., hošková-mayerová, š., 2016. fuzzy regression models and alternative operations for economic and social sciences, in: recent trends in social systems: quantitative theories and quantitative models. springer nature, pp. 235–247. url: http://dx.doi.org/10.1007/978-3-319-40585-8 21, doi:10.1007/978-3-319-40585-8 21. [21] m.s. yang, c.k., 1996. on a class of fuzzy c-numbers clustering procedures for fuzzy data. fuzzy sets and syst 84, 49–60. [22] nather, w., 2006. regression with fuzzy random data. comput stat data anal 51, 235–252. [23] puri, m., ralescu, d., 1986. fuzzy random variables. j math anal appl 114, 409–422. [24] ramos-guajardo, a., colubi, a., gonzales-rodriguez, a., 2010. onesample tests for a generalized frã¨chet variance of a fuzzy random variable. metrika 71, 185–202. [25] shapiro, a., 2004. fuzzy regression and the term structure of interest rates revisited. proceedings of the 14th international afir colloquium vol.1, 29–45. [26] shapiro, a., 2005. fuzzy regression models. arc . [27] stock, j., watson, m., 2009. introduction to econometrics. pearson addison wesley. 58 ratio mathematica volume 45, 2023 strong perfect cobondage number of standard graphs govindalakshmi t.s.1 meena n.2 abstract let g be a simple graph. a subset s  v(g) is called a strong (weak) perfect dominating set of g if |ns(u) ∩ s| = 1(|nw(u) ∩ s| = 1) for every u ∊v(g) s where ns(u) = {v ∊ v(g) / uv ∈ 𝐸(𝐺), deg v ≥ deg u} (nw(u) = {v ∊v(g) / uv ∈ 𝐸(𝐺), deg v ≤ deg u}. the minimum cardinality of a strong (weak) perfect dominating set of g is called the strong (weak) perfect domination number of g and is denoted by γsp(g)(γwp(g)). the strong perfect cobondage number bcsp(g) of a nonempty graph g is defined to be the minimum cardinality among all subsets of edges x ⊆ e(g) for which 𝛾sp (g + x) < 𝛾sp(g). if bcsp(g) does not exist, then bcsp(g) is defined as zero. in this paper study of strong perfect cobondage number of standard graphs and some special graphs are determined. keywords: strong perfect dominating set, strong perfect domination number and strong perfect cobondage number. ams subject classification: 05c693 1research scholar, reg. no. 12566, p.g. & research department of mathematics, the m.d.t. hindu college, tirunelveli –10 & assistant professor of mathematics, manonmaniam sundaranar university college, puliangudi-627855. affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli -627 012, tamilnadu, india. email: laxmigopal2005@gmail.com 2assistant professor, p.g. & research department of mathematics, the m.d.t. hindu college, tirunelveli-10. affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627 012, tamil nadu, india. email: meena@mdthinducollege.org. 3 received on july 21th, 2022. accepted on october 15th, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.983. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement 66 govindalakshmi t s and meena n 1. introduction let g be a simple graph with vertex set v(g) and edge set e(g). a dominating set d of g is a subset of v(g) such that every vertex in v – d is adjacent to at least one vertex in d [6]. a set d  v(g) is a strong(weak) dominating set of g [5] if every vertex in v – d is strongly(weakly) dominated by at least one vertex in d. the strong (weak) domination number γs(g) (γw(g)) is the minimum cardinality of a strong (weak) dominating set of g. a dominating set s is a perfect dominating set of g [1] if |n(v) ∩s| = 1 for each v ∊ v – s. motivated by these definitions, the authors defined strong perfect domination in graphs [2]. in [4], v.r. kulli and b. janakiram introduced the concept of cobondage number of a graph. the cobondage number of graph g is the minimum cardinality among the set of edges x such that 𝛾(g+x) < 𝛾(g). in this paper strong perfect cobondage number is defined and strong perfect cobondage number of paths, cycle, bistar, complete bipartite graph and some special graphs are determined. for all graph theoretic terminologies and notations harary [3] is followed. 2. preliminaries definition 2.1. [2] let g be a simple graph. a subset s  v(g) is called a strong (weak) perfect dominating set of g if |ns(u) ∩ s| = 1(|nw(u) ∩ s| = 1) for every u ∊v(g) s where ns(u) = {v ∊ v(g)/uv ∈ 𝐸(𝐺), deg v ≥ deg u} (nw(u) = {v ∊v(g)/uv ∈ 𝐸(𝐺), deg v ≤ deg u}. remark 2.2. [2] the minimum cardinality of a strong (weak) perfect dominating set of g is called the strong (weak) perfect domination number of g and is denoted by γsp(g)(γwp(g)). definition 2.3. bi star is the graph obtained by joining the apex vertices of two copies of star k1, n. definition 2.4. the wheel wn is defined to be the graph cn 1 + k1, n ≥ 4. definition 2.5. the helm hn is the graph obtained from the wheel wn with n spokes by adding n pendant edges at each vertex on the wheel’s rim. definition 2.6. the corona g1⨀ g2 of two graphs g1 and g2 (where gi has pi points and qi lines) is defined as the graph g obtained by taking one copy of g1 and p1 copies of g2, and then joining the ith point of g1 to every point in the i th copy of g2. definition 2.7. the lily graph ln, n ≥ 2 can be constructed by two-star graphs 2k1, n, n ≥ 2 joining two path graphs 2pn, n ≥ 2 with sharing a common vertex. theorem 2.8. [2] for any path pm, 67 strong perfect cobondage number of standard graphs then 𝛾sp (pm) = { 𝑛 𝑖𝑓 𝑚 = 3𝑛, 𝑛𝜖𝑁 𝑛 + 1 𝑖𝑓 𝑚 = 3𝑛 + 1, 𝑛𝜖𝑁 𝑛 + 2 𝑖𝑓 𝑚 = 3𝑛 + 2, 𝑛 𝜖𝑁 theorem 2.9 [2] for any cycle cm, then 𝛾sp (cm) = { 𝑛 𝑖𝑓 𝑚 = 3𝑛, 𝑛𝜖𝑁 𝑛 + 1 𝑖𝑓 𝑚 = 3𝑛 + 1, 𝑛𝜖𝑁 𝑛 + 2 𝑖𝑓 𝑚 = 3𝑛 + 2, 𝑛 𝜖𝑁 theorem 2.10. [2] let g be a connected graph with |v(g)| = n. then 𝛾sp (g ⨀k1) = n. remark 2.11. [2] (i) 𝛾sp (d r, s) = 2, r, s 𝜖 n (ii) 𝛾sp (km, n) = { 2 𝑖𝑓 𝑚 = 𝑛, 𝑚, 𝑛 ≥ 2 𝑚 + 𝑛 𝑖𝑓 𝑚 ≠ 𝑛 , 𝑚, 𝑛 ≥ 2 (iii) 𝛾sp (hn) = n, 𝑛 ≥ 5 𝑎𝑛𝑑 𝛾sp (h4) = 3. 3. main results definition 3.1. the strong perfect cobondage number bcsp(g) of a graph is defined to be the minimum cardinality among all subsets of edges x ⊆ e(g) for which 𝛾sp (g + x)< 𝛾sp(g). if bcsp(g) does not exist, then bcsp(g) is defined as zero. example 3.2. consider the graphs g and g + e in the following figures 1 and 2 respectively. fig. 1. fig. 2. 68 govindalakshmi t s and meena n {v3, v7, v6}, {v3, v7, v5, v4} and {v3, v8, v9, v4, v5} are some strong perfect dominating sets of g. hence 𝛾sp(g) ≤ 3. since there is no full degree vertex in g, 𝛾sp(g) ≥ 2. there are five pendent vertices. any strong perfect dominating sets contains either pendent vertices or their support vertices. v3, v7, v5 are the support vertices adjacent with pendent vertices. therefore 𝛾sp(g) ≥ 3. hence 𝛾sp(g) = 3. let e = v6v7. {v3, v7}, {v7, v4, v1, v2} are some strong perfect dominating sets of g + e. therefore 𝛾sp(g + e)≤ 2. since there is no full degree vertex in g + e, 𝛾sp(g) ≥ 2. hence 𝛾sp(g + e) = 2< 𝛾sp(g). hence bcsp(g) = 1. theorem 3.3. bcsp (pm) = { 1 𝑖𝑓 𝑚 = 3𝑛 + 1, 𝑛 ≥ 1 1𝑖𝑓 𝑚 = 3𝑛 + 2, 𝑛 ≥ 1 3 𝑖𝑓 𝑚 = 3𝑛, 𝑛 ≥ 2 proof. let g = pm, m ≥ 4. let v(g) = {vi / 1≤i≤m}. case (1). let m = 3n+1, n ≥ 1. 𝛾sp(g) = n+1. let e = v3n – 1v3n+1, deg v3n-1= △ (𝐺 + 𝑒) = 3. t = {v2, v5, v8, …, v3n -1} is a strong perfect dominating set of g + e. |t| = n. therefore 𝛾sp (g + e) ≤ n. let s be any strong perfect dominating set of g + e. since v3n – 1 is the unique maximum degree vertex in g + e, v3n – 1 belongs to s. the subgraph induced by the vertices v1, v2, …, v3n – 2 is p3n – 3. therefore 𝛾sp (p3n 3) = n – 1. |s| = n. hence 𝛾sp(g + e) ≥ n. therefore 𝛾sp (g + e) = n < n+1 = 𝛾sp(g). hence bcsp(g) = 1. case (2). let m = 3n+2, n ≥ 1. 𝛾sp(g) = n+2. let e = v3n v3n+2, deg v3n = △(g + e) = 3. t1 = {v1, v3, v6, …, v3n}, t2 = {v2, v5, …, v3n 4 v3n-3, v3n} are some strong perfect dominating sets of g + e. |t1| = |t2 | = n + 1. therefore 𝛾sp (g + e) ≤ n+1. let s be any strong perfect dominating set of g +e. v3n is strongly dominates v3n + 2. therefore v3n 𝜖 s. the subgraph induced by the vertices v1, v2, …, v3n – 2 is p3n – 2. therefore 𝛾sp(p3n 2) = 𝛾sp(p3(n – 1) + 1) = n. therefore |s| = n + 1. 𝛾sp (g + e) ≥ n+1. therefore 𝛾sp (g + e) = n+1 < n+2 = 𝛾sp(g). hence bcsp(g) = 1. case (3). let g = p3n, n ≥ 2. 𝛾sp(g) = n. when one or two edges added with p3n strong perfect domination number of the resulting graph either increases or remains same. hence bcsp(g) ≥ 3. let e1 = v1v3, e2 = v3v5, e3 = v3v6. let x = {e1, e2, e3}. t = {v3, v8, v11, …, v3n-1}is a strong perfect dominating set of g+x. |t| = n – 1. therefore 𝛾sp (g+x) ≤ n 1. let s be any strong perfect dominating set of g + x. deg v3 = 5. v3 is the unique maximum degree vertex of g + x. v3 belongs to s. the subgraph induced by the vertices v7, v8, …, v3n is a p3n – 6. therefore 𝛾sp(p3n 6) = n 2. therefore |s| = n – 2 + 1 = n 1. 𝛾sp (g + x) ≥ n 1. therefore 𝛾sp (g + x) = n 1 < n = 𝛾sp(g). hence bcsp(g) = 3. remark. let m = 3. 𝛾sp(g) = 1. hence bcsp(g) = 0. theorem 3.4. bcsp (cm) = { 1 𝑖𝑓 𝑚 = 3𝑛 + 1, 𝑛 ≥ 1 1 𝑖𝑓 𝑚 = 3𝑛 + 2, 𝑛 ≥ 1 3 𝑖𝑓 𝑚 = 3𝑛, 𝑛 ≥ 2 proof. let g = cm, m ≥ 4. let v(g) = {vi / 1 ≤ 𝑖 ≤ m}. case (1). let m = 3n+1, n ≥ 1. 𝛾sp(g) = n+1. let e = v1v3. t= {v1, v5, v8, …, v3n-1} is a strong perfect dominating set of g + e. |t| = n. therefore 𝛾sp (g +e) ≤ n. let s be any strong perfect dominating set of g + e. since v1, v3 are maximum degree vertices in 69 strong perfect cobondage number of standard graphs g+ e, either v1 belongs to s or v3 belongs to s. suppose v1 belongs to s (proof is similar if v3 belongs to s). the subgraph induced by the vertices v4, v5, …, v3n is the path p3n – 3. 𝛾sp(p3n 3) = n 1. hence 𝛾sp (g + e) ≥ n. therefore 𝛾sp (g + e) = n < n+1 = 𝛾sp(g). hence bcsp(g) = 1. case (2). let m = 3n+2, n ≥ 2. 𝛾sp(g) = n+2. let e = v1v3n + 1. t = {v1, v4, v7, …, v3n-2, v3n 1} is a strong perfect dominating set of g + e. |t| = n+1. therefore 𝛾sp (g + e) ≤ n+1. let s be any strong perfect dominating set of g + e. since v1, v3n + 1 are maximum degree vertices in g + e, either v1 belongs to s or v3n + 1 belongs to s. suppose v1 belongs to s (proof is similar if v3n + 1 belongs to s). the subgraph induced by the vertices v3, v4, …, v3n is the path p3n – 2. therefore 𝛾sp(p3n 2) = 𝛾sp (p3(n – 1) +1) = n. hence𝛾sp (g + e) ≥ n+1. therefore 𝛾sp (g + e) = n+1 < n+2 = 𝛾sp(g). hence bcsp(g) = 1. subcase (2a). if n = 1, 𝛾sp(c5) = 3. let e = v1v4. {v1, v2} is the unique 𝛾sp set of g + e. 𝛾sp (g + e) = 2 < 𝛾sp(g). hence bcsp(c5) = 1. case (3). let m = 3n, n ≥ 2. 𝛾sp(g) = n. when one or two edges added with c3n strong perfect domination number of the resulting graph either increase or remain same. hence bcsp(g) ≥3. let e1 = v1v3, e2 = v1v4, e3 = v1v5. x = {e1, e2, e3}.t = {v1, v7, v10, …, v3n2} is a strong perfect dominating set of g + x. |t| = n – 1.therefore 𝛾sp (g +x) ≤ n – 1. let s be any strong perfect dominating set of g + x. since v1 is the unique maximum degree vertex in g + x, v1 belongs s. the subgraph induced by the vertices v6, v7, …, v3n 1 is the path p3n – 6. therefore 𝛾sp(p3n 6) = 𝛾sp (p3(n – 2) ) = n 2. hence 𝛾sp (g + x) ≥ n 1. therefore 𝛾sp (g + x) = n 1 < n = 𝛾sp(g). hence bcsp(g) = 3. theorem 3.5. bcsp(dr, s) = s, r ≥ s, r, s ≥ 1. proof. let g = dr,s , r, s ≥ 1. let v(g) = {u, v, ui, vj/ 1 ≤ i ≤ r, 1 ≤ j ≤ s}, e(g) = {uv, uui, vvj/1 ≤ i ≤ r, 1 ≤ j ≤ s }. 𝛾sp(g) = 2. let r ≥ s, r, s ≥ 1, 𝛾sp(g + x) = 1 if and only if g + x must have a full degree vertex. it is possible only if u is adjacent with v1, v2, …, vs or v is adjacent with u1, u2, …, ur. since r ≥ s, x = {uvj/ 1≤j≤s}. u is the unique full degree vertex of g + x. therefore 𝛾sp(g + x) = 1 < 𝛾sp(g). hence bcsp(g) = s. theorem 3.6. bcsp(hn) = 1, n ≥ 4. proof: let g = hn, n ≥ 4. let v(g) = {v, vi, ui / 1≤ i ≤ n 1}, e(g) = {vvi, viui /1≤ 𝑖 ≤ 𝑛 − 1} ∪{ vnv1} ∪ {vivi+1 /1≤ i ≤ n-2}. 𝛾sp(g) = n, n ≥ 5. deg v = n 1, deg vi = 4, deg ui = 1, 1 ≤ 𝑖 ≤ 𝑛 − 1. case (1). let n ≥ 5. let e = u1un 1. t = {v, u1, u2, …, un-2} and {v, u2, u3, …, un 1} are some strong perfect dominating sets of g + e. |t| = n 1. therefore 𝛾sp (g + e) ≤n 1. let s be any strong perfect dominating set of g + e. there are n – 3 pendent vertices, either n – 3 pendent vertices belong to s or their support vertices belong to s. v is the unique maximum degree vertex of g + e. hence v belongs to s. since u1 and un 1 are adjacent either u1 or un 1 belong to s. therefore 𝛾sp(g + e) ≥ n 1. hence 𝛾sp(g + e) = n − 1 < 𝛾sp(g). hence bcsp(g) = 1. case (2). let n = 4. 𝛾sp(g) = 3. let e = v1u3. {v1, u2} is the unique 𝛾sp set of g + e. therefore 𝛾sp (g + e) =2 <3 = 𝛾sp(g). hence bcsp(g) = 1. 70 govindalakshmi t s and meena n theorem 3.8. bcsp (km,n) = { 𝑛 − 1 𝑖𝑓 𝑚 = 𝑛, 𝑚, 𝑛 ≥ 2 1 𝑖𝑓 𝑚 = 2, 𝑛 ≥ 3, 𝑚 ≠ 𝑛, 𝑚 < 𝑛 𝑛 − 𝑚 𝑖𝑓 𝑚 ≠ 𝑛, 𝑚 < 𝑛, 𝑚 ≥ 3, 𝑛 ≥ 4 proof. let g = km, n, m, n ≥ 2. v(g) = {vi, uj / 1≤ i≤ m, 1≤ j≤ n}. case (1). let m = n, m, n ≥ 2. 𝛾sp(g) = 2. 𝛾sp(g + x) = 1 if and only if g + x must have a full degree vertex. it is possible only if any ui is adjacent with all other uj or vi is adjacent with all other vj, i ≠ j,1 ≤ j≤ n. therefore, ui or vi is a full degree vertex of g + x. let x = {uiuj /1 ≤ j≤n and j ≠i} or {vivj /1 ≤ j≤ n and i ≠j}. |x| = n – 1. therefore 𝛾sp (g + x) =1 < 𝛾sp(g). hence bcsp(g) = n 1. case (2). let m ≠ n, m <n, m = 2, n ≥ 2. g = k2, n. 𝛾sp(g) = n +2. let e = v1v2. v1 and v2 are full degree vertices of g + e. therefore 𝛾sp (g + e) =1 < 𝛾sp(g). hence bcsp(g) = 1. case (3). let m ≠ n, m < n, m ≥ 3, n ≥ 4. 𝛾sp(g) = m + n. if any two vertices vi and vj are adjacent then g′ is the new graph 𝛾sp(g') = m + n, otherwise the vertices uj are strongly dominated by two vertices vi and vj, a contradiction. suppose degree of the vertex uj is increased so that deg vi = deg uj. uj strongly dominates the vertices vi, 1 ≤i≤m and also strongly dominates (n – m) ui’s. without loss of generality, let u1 be adjacent with u2, u3, …, un-m+1. let x = {u1u2,u1u3, …, u1un – m + 1}. |x| = n – m. t = {u1, un – m + 2, un – m + 3,…,un} is a strong perfect dominating set of g + x. 𝛾sp(g + x) ≤ m. let s be any strong perfect dominating set of g. vi’s ,1≤i≤m are maximum degree vertices and they are mutually non adjacent if they belong to s then u1 is strongly dominated by more than one vertex in s, a contradiction. if u1 is included along with vi’s then uj, j ≠ i, is dominated by more than one vertex, a contradiction. hence s contains all the vertices of g + x. |s| = n + m. any 𝛾sp – set does not contain v1,v2,…,vm. therefore, t is the minimum strong perfect dominating set of g + x. therefore 𝛾sp(g + x) = m < m + n = 𝛾sp(g). hence bcsp(g) = n m. theorem 3.9. let g ⊙ k 1 = g' be a connected graph. then bcsp(g') = 1. proof. let v(g') = {vi, ui / 1≤i≤n}. 𝛾sp(g') = n. deg vi ≥ 2, deg ui = 1,1 ≤ i≤ n. let t = {vi/1 ≤ i ≤ n} be a strong perfect dominating set of g′. |t| = n. join any vertex vi with uj, without loss of generality, let e = v2u1. v2 strongly dominates v1,v3,u1 and u2. the subgraph induced by the remaining vertices vi, ui, 3 ≤ i ≤ n, is h⊙ k1, where |v(g')| = n – 2. therefore 𝛾sp(h ⊙k1) = n – 2. these n – 2 vertices along with v1 form a strong perfect dominating set of g⊙ k1. 𝛾sp(g') ≥ n – 2 + 1 = n – 1. let t = { vi/ 2 ≤i ≤n } is a strong perfect dominating set of the resulting graph g' + e. therefore 𝛾sp (g' + e) < 𝛾sp(g'). hence bcsp(g') = 1. theorem 3.10. 𝛾sp(ln) = { 2𝑘 + 1 𝑖𝑓 𝑛 = 3𝑘, 3𝑘 + 2, 𝑘 ≥ 1 2𝑘 + 3 𝑖𝑓 𝑛 = 3𝑘 + 1, 𝑘 ≥ 1 proof. let g = ln, n ≥ 3. v(g) = {ui, vj/1≤i≤2n, 1≤j≤2n 1}. |v(g)| = 4n – 1. deg vn = 2n + 2 = △(g), deg ui =1,1≤ i ≤2n, deg v1 = deg v2n – 1 = 1, deg vj = 2, 2 ≤ j ≤ 2n 2. case (1). let n = 3k, k ≥ 1. t = {v2, v3, v6, …, v3k – 3, v3k, v3k + 3, v3k + 6, …, v6k – 6, v6k – 3, v6k – 2} is a strong perfect dominating set of g. |t| = 1+ 3𝑘−3−3 3 + 2 + 6𝑘−6−3𝑘−3 3 + 71 strong perfect cobondage number of standard graphs 3 = 2𝑘 + 1. 𝛾sp (g) ≤ 2k + 1. let s be any strong perfect dominating set of g. since v3k is unique the maximum degree vertex in g. v3k belongs to s. the subgraph induced by the vertices v1, v2, …, v3k – 2 ∪ v3k + 2, v3k + 3, …, v6k – 1 is 2 p3k – 2. 𝛾sp(2p3k 2) = 2𝛾sp (p3(k – 1) +1) = 2k. |s| = 2k+ 1. therefore 𝛾sp (g) ≥ 2k + 1. hence 𝛾sp (g) = 2k + 1. case (2). let n = 3k+1, k ≥ 1. t = {v2, v3, v4, v7, …, v3k – 2, v3k + 1, v3k + 4, v3k + 7, …, v6k – 2, v6k – 1, v6k} is a strong perfect dominating set of g. |t| = 1+ 3𝑘−2−4 3 + 3 + 6𝑘−2−3𝑘−4 3 + 3 = 2𝑘 + 3. 𝛾sp (g) ≤ 2k + 3. let s be any strong perfect dominating set of g. since v3k + 1 is unique the maximum degree vertex in g. v3k + 1 belongs to s. the subgraph induced by the vertices v1, v2, …, v3k – 1∪ v3k + 3, v3k +4, …, v6k +1 is 2 p3k – 1. 𝛾sp(2p3k 1) = 2𝛾sp (p3(k – 1) +2) = 2k + 2. |s| = 2k+ 3. therefore 𝛾sp (g) ≥ 2k + 3. hence 𝛾sp (g) = 2k + 3. case (3). let n = 3k+2, k ≥ 1. t = {v2, v5, …, v3k – 1, v3k + 2, v3k + 5, v3k + 8, …, v6k + 2} is a strong perfect dominating set of g. |t| = 1+ 3𝑘−1−2 3 + 1 + 6𝑘+2−3𝑘−5 3 + 1 = 2𝑘 + 1. 𝛾sp (g) ≤ 2k + 1. let s be any strong perfect dominating set of g. since v3k + 2 is unique the maximum degree vertex in g. v3k + 2 belongs to s. the subgraph induced by the vertices v1, v2, …, v3k ∪ v3k + 4, v3k +5, …, v6k + 3 is 2 p3k. 𝛾sp(2p3k) = 2𝛾sp (p3k) = 2k. |s| = 2k+ 1. therefore 𝛾sp (g) ≥ 2k + 1. hence 𝛾sp (g) = 2k + 1. remark. l2 = k 1,6. therefore 𝛾sp(l2) = 1. theorem 3.11. bcsp(ln) = { 1 𝑖𝑓 𝑛 = 3𝑘, 3𝑘 + 1, 𝑘 ≥ 1 3 𝑖𝑓 𝑛 = 3𝑘 + 2, 𝑘 ≥ 1 proof. let g = ln, n ≥ 3. v(g) = {ui, vj/1 ≤ i ≤ 2n, 1 ≤ j ≤ 2n 1}. case (1). let n = 3k, k ≥ 1. let e = v3k 2v3k. deg v3k = 6k + 3 = △(g + e), deg ui = 1, 1 ≤ i ≤ 6k, deg v1 = deg v6k – 1 = 1, deg vj = 2, 2 ≤ j ≤ 6k – 2, j ≠ 3k – 2, deg v3k – 2 = 3. t = {v2, v5, …, v3k – 4, v3k, v3k + 3, v3k + 6, …, v6k – 3, v6k – 2} is a strong perfect dominating set of g + e. |t| = 3𝑘−4−2 3 + 2 + 6𝑘−3−3𝑘−3 3 + 2 = 2𝑘. 𝛾sp (g + e) ≤ 2k. let s be any strong perfect dominating set of g + e. since v3k is unique the maximum degree vertex in g+ e. v3k belongs to s. the subgraph induced by the vertices v1, v2, …, v3k – 3 ∪ v3k + 2, v3k +3, …, v6k 1 is p3k – 3 ∪ p3k – 2. 𝛾sp(p3(k–1)) + 𝛾sp (p3(k – 1) + 1) = 2k 1. |s| = 2k. therefore 𝛾sp (g + e) ≥ 2k. hence 𝛾sp (g + e) = 2k. therefore 𝛾sp (g + e) < 2k +1 = 𝛾sp(g). hence bcsp(g) = 1. case (2). let n = 3k+1, k ≥ 1. let e = v3k 1v3k + 1. deg v3k + 1 = 6k + 5 = △(g +e), deg ui = 1,1 ≤ i ≤ 6k + 2,deg v1 = deg v6k+ 1 = 1, deg vj = 2, 2 ≤ j ≤ 6k, j≠ 3k – 1, deg v3k – 1 = 3. t = {v2, v3, v6, …, v3k – 3, v3k + 1, v3k + 4, …, v6k – 2, v6k – 1, v6k} is a strong perfect dominating set of g + e. |t| = 1+ 3𝑘−3−3 3 + 2 + 6𝑘−2−3𝑘−4 3 + 3 = 2𝑘 + 2. 𝛾sp (g + e) ≤ 2k + 2. let s be any strong perfect dominating set of g +e. since v3k + 1 is unique the maximum degree vertex in g +e. v3k + 1 belongs to s. the subgraph induced by the vertices v1, v2, …, v3k – 2 ∪ v3k + 3, v3k + 4, …, v6k +1 is p3k – 2∪ p3k – 1. 𝛾sp(p3(k–1) + 1)+ 𝛾sp (p3(k – 1) +2) 72 govindalakshmi t s and meena n = 2k + 1. |s| = 2k+ 2. therefore 𝛾sp (g + e) ≥ 2k + 2. hence 𝛾sp (g + e) = 2k + 2< 2k + 3 = 𝛾sp(g). hence bcsp(g) = 1. case (3). let n = 3k+2, k ≥ 1. 𝛾sp(g) = 2k +1. to reduce the strong perfect domination number, at least one edge must be added. since v3k + 2 is the unique maximum degree vertex, it belongs to any strong perfect dominating set. make v3k+2 adjacent with v3k, v3k – 1, v3k – 2. so that strong perfect domination number decreases by at least one. let x = {v3kv3k + 2, v3k 1v3k + 2, v3k 2v3k + 2}. t = {v2, v5, …, v3k 4, v3k + 2, v3k + 5, v3k + 8, …, v6k + 2} is a strong perfect dominating set of g + x. hence 𝛾sp (g + x) = 2k< 2k +1 = 𝛾sp(g). hence bcsp(g) = 3. 4. conclusion in this paper, strong perfect cobondage number of some standard graphs and some special graphs are studied. further study can be observed for path related graphs, cycle related graphs, subdivision graphs, middle graphs and product graphs. bounds for strong perfect cobondage number of graphs and nordhaus gaddum type results can be determined. references [1] j.v. changela and g. j. vala, perfect domination and packing of a graph, int. j. eng. innov., (ijeit), vol 3, issue 12, pp 61-64, june 2014. [2] t.s. govindalakshmi and n. meena, strong perfect domination in graphs, international journal of mathematics trends and technology, vol 58, issue 3, pp 29-33, june 2018. [3] f. harary, graph theory, addison-wesley, reading, mass,1969 [4] v. r. kulli and b. janakiram, the cobondage number of a graph, discuss. math. graph theory., vol 16., pp 111 – 117., 1996. [5] e. sampathkumar and l. pushpalatha, strong weak domination and domination balance in a graph, discrete. math., vol 161, pp 235-242,1996. [6] teresa w. haynes, stephen t. hedetniemi, peter j. slater, fundamentals of domination in graphs, marcel decker, inc., new york, 1998. 73 ratio mathematica volume 41, 2021, pp. 79-100 on extended quasi-mv algebras mengmeng liu* hongxing liu† abstract in this paper, we introduce a new algebraic structure called extended quasi-mv algebras, which are generalizations of quasi-mv algebras. the notions of ideals, ideal congruences and filters in equasi-mv algebras were introduced and their mutual relationships were investigated. there is a bijection between the set of all ideals and the set of all ideal congruences on an equasi-mv algebra. keywords: equasi-mv algebras; quasi-mv algebras; idempotent elements; ideal congruences; filters 2020 ams subject classifications: 03g27 1 *school of mathematics and statistics, shandong normal university, 250014, jinan, p. r. china; 535236069@qq.com. †school of mathematics and statistics, shandong normal university, 250014, jinan, p. r. china; lhxshanda@163.com. 1received on october 15, 2021. accepted on december 18, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.680. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 79 mengmeng liu, hongxing liu 1 introduction mv-algebras were introduced by chang chang [1958] as an algebraic counterpart of infinite valued logic. there are many papers on mv-algebras. also, many algebraic structures are defined, which extend the notion of mv-algebras. quantum computation logics m. l. dalla chiara and leporini [2005] received more attention in recent years, which are new forms of quantum logics g. cattaneo and leporini [2004]. these logics determine the meaning of a sentence with a mixture of quregisters m. l. dalla chiara and greechie [2013]. corresponding to quantum computational, ledda, konig, paoli and giuntini introduced the notion of quasi-mv algebras in a. ledda and giuntini [2006], which are generalizations of mv-algebras. the element 0 in a quasi-mv algebra is not necessarily a neutral element of the operation ⊕. since then, many authors continued to study quasi-mv algebras. for example, ledda etc. studied some properties of quasi-mv algebras and √ ′ quasi-mv algebras f. bou and freytes [2008], f. paoli and freytes [2009]; chen introduced pseudo-quasi-mv algebras which are non-commutative generalizations of quasi-mv algebras liu and chen [2016]. emv-algebras (extended mv-algebras) dvurečenskij and zahiri [2019] are also generalizations of mv-algebras. an emv-algebra does not necessarily have a top element. dvurečenskij and zahiri gave some properties of emv-algebras. the notions of ideals, congruences and filters in emv-algebras were also introduced and the relationships between them were investigated. one of the main results is that every emv-algebra can be embedded into an emv-algebra with a top element. liu presented ebl-algebras in liu [2020], which extended the notion of bl-algebras. the author gave some properties of ebl-algebras. also, the concepts of ideals, congruences and filters were introduced and the relationships between them were studied. inspired by dvurečenskij and zahiri [2019], we shall give the definition of equasi-mv algebras. in these algebras, 0 is not necessarily the neutral element and the complement element of 0 does not necessarily exist. the structure of this paper is as follows. in sect.2, we give some definitions and results of quasi-mv algebras. in sect.3, we introduce equasi-mv algebras and present some examples of equasi-mv algebras. in sect.4, we define ideals and ideal congruences in equasi-mv algebras. and we study the relationships between them. in sect.5, we introduce the notions of filters and prime ideals. moreover, every equasi-mv algebra has at least one maximal ideal. 80 on extended quasi-mv algebras 2 preliminaries in this section, we will give some notions and results on quasi-mv algebras, which will be used in the following. a quasi-mv algebra a. ledda and giuntini [2006] is an algebra a=〈a,⊕,′,0,1〉 of type 〈2,1,0,0〉 satisfying the following conditions: qmv1) x⊕ (y ⊕z) = (x⊕z)⊕y; qmv2) x′′ = x; qmv3) x⊕1 = 1; qmv4) (x′ ⊕y)′ ⊕y = (y′ ⊕x)′ ⊕x; qmv5) (x⊕0)′ = x′ ⊕0; qmv6) (x⊕y)⊕0 = x⊕y; qmv7) 0′ = 1. in any quasi-mv algebra a, we can define the following operations: x⊗y = (x′ ⊕y′)′; x d y = x⊕ (x′ ⊗y); x e y = x⊗ (x′ ⊕y). it is obvious that x d y = (x d y)⊕0 and x e y = (x e y)⊕0. moreover, we can also define an binary relation 6 on a as follows: x 6 y iff x e y = x ⊕ 0. the relation 6 is a preordering of a, but not a partial ordering. lemma 2.1. [a. ledda and giuntini, 2006, lemma 8] let a be a quasi-mv algebra. for all x,y,z ∈ a, the following statements are equivalent. (i) x 6 y; (ii) x′ ⊕y = 1; (iii) x d y = y ⊕0. in the following, we give some properties of quasi-mv algebras, including a few properties of preordering 6 and the operations e and d. lemma 2.2. [a. ledda and giuntini, 2006, lemma 11] let a be a quasi-mv algebra. for all x,y,z,w ∈ a: (i) x⊕0 6 y⊕0, y⊕0 6 x⊕0 imply x⊕0 = y⊕0; (vi) x 6 x⊕0 and x⊕0 6 x; (ii) x 6 y and z 6 w imply x⊕z 6 y⊕w; (vii) x⊗y 6 z iff x 6 y′⊕z; (iii) x 6 y and z 6 w imply x⊗z 6 y⊗w; (viii) if x 6 y, then y′ 6 x′; (iv) x 6 y and z 6 w imply xez 6y ew; (ix) 0 6 x, x 6 1. (v) x 6 y and z 6 w imply xdz 6 ydw; lemma 2.3. [a. ledda and giuntini, 2006, lemma 12] let a be a quasi-mv 81 mengmeng liu, hongxing liu algebra. for all x,y,z ∈ a: (i) xey = yex; (vii) x⊗(ydz) = (x⊗y)d(x⊗z); (ii) xdy = ydx; (viii) xe(yez) = (xey)ez; (iii) xey 6 x, y and x, y 6 xdy; (ix) xd(ydz) = (xdy)dz; (iv) if x 6 y, z, then x 6y ez; (x) x 6 xex and xex 6 x; (v) if x, y 6 z, then xdy 6 z; (xi) (xey)′ = x′dy′ and (xdy)′ = x′ey′. (vi) x⊕(yez) = (x⊕y)e(x⊕z); the following lemma gives the distributivity between e and d on quasi-mv algebras. lemma 2.4. let a be a quasi-mv algebra. for all x,y,z ∈ a, (i) (x d y) e z = (x e z) d (y e z); (ii) (x e y) d z = (x d z) e (y d z); (iii) x e (y ⊕z) 6 (x e y)⊕ (x e z); (iv) (x d y)⊗ (x d z) 6 x d (y ⊗z). proof. (i) for any x,y ∈ a, we have x,y 6 xdy and so xez, yez 6 (xdy)ez by lemma 2.2 (iv). it follows from lemma 2.3 (v) that (xez)d(yez)6(xdy)ez. conversely, we have (x d y) e z = (x d y)⊗ ((x d y)′ ⊕z) = (x d y)⊗ ((x′ ⊕z) e (y′ ⊕z)) (lemma 2.3 (xi) and (vi)) 6 (x⊗ (x′ ⊕z)) d (y ⊗ (y′ ⊕z)) (lemma 2.3 (vii) and (iii)) = (x e z) d (y e z). then ((x e z) d (y e z)) ⊕ 0 6 ((x d y) e z) ⊕ 0 and ((x d y) e z) ⊕ 0 6 ((x e z) d (y e z))⊕0. note that ((x e z) d (y e z))⊕0 = (x e z) d (y e z) and ((x d y) e z) ⊕ 0 = (x d y) e z. it follows that (x e z) d (y e z) = (x d y) e z by lemma 2.2 (i). similarly, we can prove (ii). (iii) for any x,y,z ∈ a, since x 6 x⊕0 6 x⊕y, we have (x e y)⊕ (x e z) = ((x e y)⊕x) e ((x e y)⊕z) (lemma 2.3 (vi)) = (x⊕x) e (y ⊕x) e (x⊕z) e (y ⊕z) > x e x e x e (y ⊕z) = (x⊕0) e x e (y ⊕z) (lemma 2.3 (x)) = (x⊕0) e (y ⊕z). note that (x⊕0)e(y⊕z) = xe(y⊕z). it follows that xe(y⊕z) 6 (xey)⊕(xez). (iv) for any x,y,z ∈ a, it follows from (x⊗y)′ ⊕y = x′ ⊕y′ ⊕y = 1 that 82 on extended quasi-mv algebras x⊗y 6 y. then we have (x d y)⊗ (x d z) = ((x d y)⊗x) d ((x d y)⊗z) (lemma 2.3 (vii)) = (x⊗x) d (y ⊗x) d (x⊗z) d (y ⊗z) 6 x d x d x d (y ⊗z) = (x⊕0) d x d (y ⊗z) (lemma 2.3 (x)) = (x⊕0) d (y ⊗z). note that (x ⊕ 0) d (y ⊗ z) = x d (y ⊗ z). it follows that (x d y) ⊗ (x d z) 6 x d (y ⊗z).2 let a be a quasi-mv algebra and a ∈ a. if a ⊕ a = a, we call a to be idempotent. we use i(a) to denote the set of all idempotent elements of a. for a ∈ a, we call a regular if a⊕0 = a. we denote the set of all regular elements of a by r(a). lemma 2.5. let a be a quasi-mv algebra. for any x ∈ a, a ∈i(a), we have (i) x⊕a = x d a; (ii) x⊗a = x e a. proof. (i) for any x ∈ a and a ∈ i(a), we have x,a 6 x ⊕ a. then x d a 6 x⊕a by lemma 2.3 (v). conversely, (x⊕a)⊗ (x d a)′ = (x⊕a)⊗ (x′ e a′) (lemma 2.3 (xi)) 6 ((x⊕a)⊗x′) e ((x⊕a)⊗a′) (lemma 2.2(iii) and 2.3(iv)) = (a e x′) e (x e a′) = (a e a′) e (x e x′) = 0 e (x e x′) = 0. this means that (x⊕a)′ ⊕ (x d a) = 1. it follows that x⊕a 6 x d a. (ii) by (i), we have x′ ⊕a′=x′ d a′, that is (x′ ⊕a′)′ = (x′ d a′)′ = x e a. it follows that x e a = x⊗a.2 the application of the above lemma will be reflected in the following proof process. example 2.1. [a. ledda and giuntini, 2006, example 3] the diamond is the 4element quasi-mv algebra, where the operations ⊕ and ′ are defined as following tables: ⊕ 0 a b 1 0 0 b b 1 a b 1 1 1 b b 1 1 1 1 1 1 1 1 ′ 0 1 a a b b 1 0 remark that a⊕a = 1, but aea = (a′⊕(a′⊕a)′)′ = (a⊕(a⊕a)′)′ = b 6= 1. 83 mengmeng liu, hongxing liu 3 equasi-mv algebras in the section, we shall define the notion of extended quasi-mv algebras, which are generalizations of quasi-mv algebras. some basic properties of these algebras are presented. definition 3.1. a extended quasi-mv algebra (abbreviated as equasi-mv algebra) is an algebra a=〈a,⊕,0〉, if the following conditions are satisfied: eqmv1) 〈a,⊕,0〉 is a commutative preordered semigroup and (x⊕y)⊕0 = x⊕y for all x,y ∈ a; eqmv2) for each x ∈ a, there is b ∈i(a) such that x 6 b, and the element λb(x) = min{z ∈ [0,b] : z ⊕x = b} exists in a for all x ∈ [0,b] such that 〈[0,b],⊕,λb,0,b〉 is a quasi-mv algebra. note that for any x,y ∈ a, there exist a,b ∈i(a) such that x 6 a and y 6 b. then there exists c ∈i(a) such that a,b 6 c. in fact, take c = a⊕b. it is obvious that a,b 6 a⊕b and a⊕b ∈i(a). therefore, an equasi-mv algebra has enough idempotent elements. that is, for all x ∈ a, there is a ∈i(a) such that x 6 a. let a be an equasi-mv algebra. for all n ∈ n and x ∈ a, we define 0.x = 0, 1.x = x, · · · , (n + 1).x = n.x⊕x. an equasi-mv algebra 〈a,⊕,0〉 is called a proper equasi-mv algebra if 0 has no complement element. example 3.1. if 〈a,⊕,′ ,0,1〉 is a quasi-mv algebra, then 〈a,⊕,0〉 is an equasimv algebra. also, if 〈a,∨,∧,⊕,0〉 is an emv-algebra, it is obvious that 〈a,⊕,0〉 is an equasi-mv algebra. example 3.2. let 〈a,⊕,′ ,0,1〉 be a quasi-mv algebra and 〈b,∨,∧,⊕,0〉 be an emv-algebra. we define that the operation on the algebra a×b is point by point. that is, for any 〈x1,x2〉,〈y1,y2〉 ∈ a×b, 〈x1,x2〉⊕〈y1,y2〉 = 〈x1 ⊕y1,x2 ⊕y2〉. and the least element of a×b is 0 = 〈0,0〉. for any x ∈ b, there exists b ∈i(b) such that x 6 b. then for any 〈x1,x2〉 ∈ a × b, there exists 〈1,b〉 ∈ i(a) × i(b). it suffices to show that 〈[〈0,0〉,〈1,b〉],⊕,λ〈1,b〉,〈0,0〉,〈1,b〉〉 is a quasi-mv algebra. we define λ〈1,b〉(〈x1,x2〉) = 〈(x1)′,λb(x2)〉, for all 〈x1,x2〉 ∈ a×b. as a result, a×b is an equasi-mv algebra. example 3.3. let e be a smallest idempotent of an equasi-mv algebra a. then an equasi-mv algebra is the algebra s=〈a×a,⊕s,0s〉, where: (i) 0s = 〈0, e 2 〉; (ii) xs ⊕s ys = 〈x1 ⊕y1, e2〉, for all x s = 〈x1,x2〉 and ys = 〈y1,y2〉. for any a ∈ i(a), we define as = 〈a, e 2 〉. then as = as ⊕as ∈ i(s). now we show that 〈[0s,as],⊕s,λas,0s,as〉 is a quasi-mv algebra, where λas(xs) = 84 on extended quasi-mv algebras 〈λa(x1),x2〉 and a ∈ i(a). it is easy to show that λas(xs) is the least element such that xs ⊕zs = as for all xs ∈ [0s,as]. it is clear that λasλas(x s) = λas〈λa(x1),x2〉 = 〈x1,x2〉 = xs. and λas(xs⊕s 0s) = λas〈x1 ⊕ 0, e2〉 = 〈λa(x1) ⊕ 0, e 2 〉, λas(xs) ⊕ 0s = 〈λa(x1),x2〉⊕ 0s = 〈λa(x1)⊕0, e2〉. what’s more, λas(0 s) = 〈λa(0), e2〉 = 〈a, e 2 〉 = as. example 3.4. let 〈a,∨,∧,0〉 be a generalized boolean algebra conrad and darnel [1997]. for any x,y ∈ [0,b], where ⊕ = ∨ and λb(x) is the unique relative complement of x in [0,b]. then 〈a,⊕,0〉 is an emv-algebra by example 3.2 (2) in dvurečenskij and zahiri [2019]. hence, 〈a,⊕,0〉 is an equasi-mv algebra. example 3.5. let 〈a,⊕,′ ,0,1〉 be a quasi-mv algebra and 〈b,∨,∧,0〉 be a generalized boolean algebra. it is easy to show that a×b is an equasi-mv algebra. proof. the operation ⊕ on a × b is defined pointwise. for all 〈x,y〉 ∈ a × b, there exist a ∈ i(a) and b ∈ i(b) such that 〈x,y〉 6 〈a,b〉 and 〈[〈0,0〉,〈a,b〉],⊕,λ〈a,b〉,〈0,0〉,〈a,b〉〉 is a quasi-mv algebra. let’s give a specific description of the above example. let the diamond (example 2.6) be the 4-element quasi-mv algebra a and m = 〈m,∨,∧,0〉 be the generalized boolean algebra conrad and darnel [1997], where m is the set of components of any positive element n+ and the least element 0 := ∅. that is, m = {n : n ⊆ n+}. then every element n in m is idempotent. it is easily shown that a×m with the pointwise operation is an equasi-mv algebra. 2 example 3.6. let s = 〈[0,1] × [0,1],⊕,′ ,0,1〉 be a standard quasi-mv algebra a. ledda and giuntini [2006, example 5]. let a = s⊕s⊕s⊕··· . then a is an equasi-mv algebra. proof. obviously, 〈a,⊕,0〉 is a commutative preordered semigroup and (x⊕ y) ⊕ 0 = x⊕y for all x,y ∈ a. for any x,y ∈ a. suppose x = (xi), y = (yi). if xi 6= 0 or yi 6= 0, there exists ui ∈ i(a) such that xi,yi 6 ui for all i > 1. if xi = yi = 0, take ui = 0. we have an idempotent u = (ui) ∈ a such that x,y 6 u and 〈[0,u],⊕,λu,0,u〉 is a quasi-mv algebra. 2 remark 3.1. let a be an equasi-mv algebra. for all x,y ∈ a, there exists b ∈ i(a) such that x,y ∈ [0,b]. in the quasi-mv algebra 〈[0,b],⊕,λb,0,b〉, we denote x db y = λb(λb(x)⊕y)⊕y, x eb y = λb(λb(x)⊕λb(λb(x)⊕y)). proposition 3.1. let a be an equasi-mv algebra and a,b ∈ i(a) such that a 6 b. for each x ∈ [0,a], we have (i) λb(a) is an idempotent, and λa(a)=0; (ii) λa(x)⊕0 = λb(x) e a; (iii) λb(x)⊕0 = λa(x)⊕λb(a); (iv) λa(x) 6 λb(x). 85 mengmeng liu, hongxing liu proof. since 〈[0,b],⊕,λb,0,b〉 is a quasi-mv algebra and a ∈ i(a), by lemma 2.5 (i) we get that x⊕a = x d a for all x ∈ [0,b]. (i) since 〈[0,b],⊕,λb,0,b〉 is a quasi-mv algebra, λb(a) is also an idempotent element by lemma 26 in a. ledda and giuntini [2006]. it is obvious λa(a) = 0 in the quasi-mv algebra 〈[0,a],⊕,λa,0,a〉. (ii) for all x ∈ [0,a], we have (λb(x) e a)⊕ (x⊕0) = (λb(x)⊕ (x⊕0)) e (a⊕ (x⊕0)) (lemma 2.3 (vi)) = b e a = a. it follows that λa(x) ⊕ 0 = λa(x ⊕ 0) 6 λb(x) e a in the quasi-mv algebra 〈[0,b],⊕,λb,0,b〉. conversely, since b = a ⊕ λb(a) = x ⊕ (λa(x) ⊕ λb(a)), we get λb(x) 6 λa(x) ⊕ λb(a). since λb(a) is an idempotent, by lemma 2.5 (i) we have λa(x)⊕λb(a) = λa(x) d λb(a). hence, λb(x) 6 λa(x) d λb(a). thus λb(x) e a 6 (λa(x) d λb(a)) e a (lemma 2.2 (iv)) = λa(x)⊕0 (lemma 2.4 (i)). summary of the above results, we get that λa(x)⊕0 = λb(x) e a. (iii) by (ii) we have λa(x)⊕λb(a) = (λa(x)⊕0)⊕λb(a) = (λb(x) e a)⊕λb(a) = λb(x) d λb(a) (lemma 2.3 (vi) and lemma 2.5 (i)). it follows from x 6 a that λb(a) 6 λb(x). then λb(x) d λb(a) = λb(x) ⊕ 0. therefore, λb(x)⊕0 = λa(x)⊕λb(a). (iv) it follows from (ii) or (iii).2 the following statement shows that da and ea on [0,a] are coincide with d and e on a, respectively. proposition 3.2. let a be an equasi-mv algebra. for all x,y ∈ a, there exist a,b ∈i(a) such that x,y ∈ [0,a] and x,y ∈ [0,b]. then we have (i) x ea y = x eb y; (ii) x da y = x db y. proof. (i) by definition 3.1, for all a,b ∈ i(a), there exists c ∈ i(a) such that a,b 6 c. then we have x dc y = x⊕λc(x⊕λc(y)⊕0) = x⊕λc(x⊕λa(y)⊕λc(a)) (proposition 3.1 (iii)) = x⊕ (λc(x⊕λa(y))⊗c a) (the definition of ⊗c) = x⊕ (λc(x⊕λa(y)) e a) (lemma 2.5 (ii)) = x⊕ ((λa(x⊕λa(y)) d λc(a)) e a) (proposition 3.1(iii), lemma 2.5(i)) = x⊕ (λa(x⊕λa(y)) e a) (lemma 2.4 (i)) = x⊕ (λa(x⊕λa(y))⊕0) = x da y. 86 on extended quasi-mv algebras similarly, we can show that x dc y = x db y. hence, x da y = x db y. (ii) we also have xecy = λc(λc(x)⊕λc(λc(x)⊕y)) = λc(λc(x)⊕λc(λa(x)⊕y⊕λc(a))) (proposition 3.1 (iii)) = λc(λc(x)⊕(λc(λa(x)⊕y)⊗ca)) (definition of ⊗c) = λc(λc(x)⊕((λa(λa(x)⊕y)⊕λc(a))⊗c a)) (proposition 3.1 (iii)) = λc(λc(x)⊕((λa(λa(x)⊕y)⊕λc(a))ea)) (lemma 2.5 (i)) = λc(λc(x)⊕(λa(λa(x)⊕y)ea)) ( lemma 2.4 (i)) = λc(λc(x)⊕λa(λa(x)⊕y)) = λc(λa(x)⊕λa(λa(x)⊕y)⊕λc(a)) (proposition 3.1 (iii)) = λc(λa(x)⊕λa(λa(x)⊕y))⊗ca (definition of ⊗c) = (λa(λa(x)⊕λa(λa(x)⊕y))⊕λc(a))⊗ca (proposition 3.1 (iii)) = λa(λa(x)⊕λa(λa(x)⊕y))ea = xeay. similarly, we can show that x ec y = x eb y and so x ea y = x eb y.2 definition 3.2. let a be an equasi-mv algebra and x,y ∈ [0,a] where a ∈ i(a). a preordering 6a on the quasi-mv algebra 〈[0,a],⊕,λa,0,a〉 defined as follows: x 6a y ⇐⇒ x ea y = x⊕0. by proposition 3.2, for any x,y 6 a,b, where a,b ∈ i(a), we have x 6a y ⇐⇒ x 6 y ⇐⇒ x 6b y. then we can also define a preordering 6 on a by x 6 y ⇐⇒ x e y = x⊕0, where x e y = x ea y. lemma 3.1. let a be an equasi-mv algebra. for all x,y ∈ a, the operation ⊗: a × a → a defined by x ⊗ y = λa(λa(x) ⊕ λa(y)), where a ∈ i(a) and x,y 6 a. then (i) the well-defined binary operation ⊗ on a is not determined by the choice of a and is also order preserving and associative. (ii) if x,y ∈ a, x 6 y, then y⊗λa(x) = y⊗λb(x) and y⊕0 = x⊕(y⊗λa(x)) for all a,b ∈i(a) and x,y 6 a,b. (iii) if x,y ∈ [0,a] and a ∈ i(a), then x ⊗ λa(y) = x ⊗ λa(x e y) and x⊕0 = (x e y)⊕ (x⊗λa(y)). (iv) an element a ∈ a is idempotent iff a⊗a = a. proof. (i) let x,y ∈ a and a,b ∈ i(a) such that x,y 6 a,b. we claim that λa(λa(x)⊕λa(y)) = λb(λb(x)⊕λb(y)). indeed, there exists an element c ∈i(a) 87 mengmeng liu, hongxing liu such that a,b 6 c. then λc(λc(x)⊕λc(y)) = λc(λa(x)⊕λc(a)⊕λa(y)⊕λc(a)) (proposition 3.1 (iii)) = λc(λa(x)⊕λa(y))⊗c λc(λc(a)) (propsition 3.1 (i)) = λc(λa(x)⊕λa(y)) e a (lemma 2.5 (ii)) = (λa(λa(x)⊕λa(y))⊕λc(a)) e a (lemma 3.1 (iii)) = (λa(λa(x)⊕λa(y)) d λc(a)) e a (lemma 2.5 (i)) = λa(λa(x)⊕λa(y)) e a = λa(λa(x)⊕λa(y)). similarly, we have λc(λc(x)⊕λc(y)) = λb(λb(x)⊕λb(y)). let x,y,z ∈ a. there exists c ∈ i(a) such that x,y,z 6 c. it follows from the definition of ⊗ that x⊗y, y ⊗z ∈ [0,c]. then (x⊗y)⊗z = λc(λc(x⊗y)⊕λc(z)) = λc((λc(x)⊕λc(y))⊕λc(z)) = λc(λc(x)⊕ (λc(y)⊕λc(z))) = λc(λc(x)⊕λc(y ⊗z)) = x⊗ (y ⊗z). this proves that ⊗ is associative. it is easy to prove that ⊗ is order preserving. (ii) let x 6 y and x,y 6 a,b, where a,b ∈i(a). there exists c ∈i(a) such that a,b 6 c. by proposition 3.1, we have y ⊗λa(x) = λc(λc(y)⊕λc(λa(x))) = λc(λc(y)⊕λc(λa(x))⊕0) = λc(λc(y)⊕λc(λa(x)⊕0)) = y ⊗ (λa(x)⊕0). then y ⊗λc(x) = y ⊗ (λc(x)⊕0) = y ⊗ (λa(x)⊕λc(a)) = y ⊗ (λa(x) d λc(a)) (lemma 2.5 (i)) = (y ⊗λa(x)) d (y ⊗λc(a)) (lemma 2.3 (vii)). since λc(a) 6 λc(y), we have y⊗λc(a) 6 y⊗λc(y) = 0, where y 6 a 6 c. this implies y ⊗ λc(x) = y ⊗ λa(x). similarly, we have y ⊗ λc(x) = y ⊗ λb(x). it follows that y ⊗λa(x) = y ⊗λb(x). in the quasi-mv algebra 〈[0,a],⊕,λa,0,a〉, we have x⊕ (y ⊗λa(x)) = x⊕λa(λa(y)⊕x) = x d y = y ⊕0. (iii) let x,y 6 a and a ∈i(a). we have x⊗λa(x e y) = x⊗ (λa(x) d λa(y)) = (x⊗λa(x)) d (x⊗λa(y)) (lemma 2.3 (vii)) = x⊗λa(y). 88 on extended quasi-mv algebras (x e y)⊕ (x⊗λa(y)) = (x e y)⊕ (x⊗λa(x e y)) = (x e y)⊕λa(λa(x)⊕ (x e y)) = x⊕λa(x⊕λa(x e y)) (qmv 4) = x⊕λa(x⊕λa(x)⊕λa(λa(x)⊕y)) = x⊕0. (iv) =⇒: suppose a,b ∈i(a) with a 6 b. we have λb(a)⊕λb(a) = λb(a) by proposition 3.1 (i). in the quasi-mv algebra 〈[0,b],⊕,λb,0,b〉, we have a⊗a = λb(λb(a)⊕λb(a)) = λb(λb(a)) = a. ⇐=: for each a ∈ a, there exists b ∈ i(a) such that a 6 b. suppose a⊗a = a. we have λb(λb(a)⊕λb(a)) = a. then λb(λb(λb(a)⊕λb(a))) = λb(a). it follows from λb(a) ⊕ λb(a) = λb(a) that λb(a) ∈ i(a). by proposition 3.1 (i), we have λb(λb(a)) ⊕ λb(λb(a)) = λb(λb(a)). that is a ⊕ a = a. it implies a ∈i(a). 2 theorem 3.1. let a be an equasi-mv algebra. then 〈r(a),dr,er,⊕r,0r〉 is an emv-subalgebra of a. proof. it is obvious that r(a) is closed under the operations dr,er,⊕r,0r. for all x,y ∈r(a), there exists a ∈i(a) such that x,y 6 a. then [0,a]∩r(a) is an mv-algebra of [0,a] by lemma 15 in a. ledda and giuntini [2006]. this means that r(a) is an emv-subalgebra of a. 2 4 ideals and congruences in this section, we give the notions of ideals and ideal congruences of equasimv algebras. we also give an equivalent definition of ideals. moreover, there is a one-to-one correspondence between the set of all ideals and the set of all ideal congruences. definition 4.1. let a be an equasi-mv algebra. an equivalence relation θ on a is called a congruence, if the following conditions hold: (i) θ is compatible with ⊕; (ii) for all b ∈ i(a),θ ∩ ([0,b] × [0,b]) is a congruence on the quasi-mv algebra 〈[0,b],⊕,λb,0,b〉. the set of all congruences on a represented by con(a). definition 4.2. let a1,a2 be two equasi-mv algebras. we call a map f : a1 −→ a2 to be an equasi-mv homomorphism, if it satisfies the following statements: (i) f(x⊕y) = f(x)⊕f(y) and f(0) = 0, for all x,y ∈ a1; (ii) for all x,y ∈ [0,a] and a ∈i(a1), f(λa(x)) = λf(a)(f(x)). 89 mengmeng liu, hongxing liu example 4.1. let f : a1 → a2 be an equasi-mv homomorphism. we can define θ = {(x,y) ∈ a1 ×a1 : f(x) = f(y)}, then θ is a congruence. let a be an equasi-mv algebra and θ be a congruence on a. we denote a/θ = {x/θ : x ∈ a}, where x/θ = {y ∈ a : 〈x,y〉 ∈ θ}. we define operations e, d, ⊕ on a/θ as follows: for any x,y ∈ a, x/θ e y/θ = (x e y)/θ, x/θ d y/θ = (x d y)/θ, x/θ ⊕y/θ = (x⊕y)/θ. suppose x/θ 6 y/θ. then (x e y)/θ > x/θ. for all z ∈ a, we have x/θ ⊕z/θ = (x⊕z)/θ 6 ((x e y)⊕z)/θ 6 (y ⊕z)/θ = y/θ ⊕z/θ. this proves that 〈a/θ,⊕,0/θ〉 is a commutative preordered semigroup and (x/θ⊕ y/θ)⊕0/θ = x/θ ⊕y/θ. for all x ∈ a, there exists a ∈ i(a) such that x 6 a. it is easily shown that a/θ is an idempotent element and x/θ 6 a/θ. since a is an equasi-mv algebra, we have that 〈[0,a],⊕,λa,0,a〉 is a quasi-mv algebra. and let θa = θ∩ ([0,a]× [0,a]) be an ideal congruence on 〈[0,a],⊕,λa,0,a〉. for any x/θa ∈ [0/θa,a/θa], we define λa/θa(x/θa) = λa(x)/θa. then [0/θa,a/θa] is a quasi-mv algebra. now we show that 〈[0/θ,a/θ],⊕,λa/θ,0/θ,a/θ〉 is a quasi-mv algebra. for all x/θ ∈ [0/θ,a/θ], there exists y/θ ∈ [0/θ,a/θ] such that x/θ ⊕y/θ = a/θ. it follows that 〈x ⊕ y,a〉 ∈ θ. and since x,y 6 a, we have 〈x ⊕ y,a〉 ∈ θa. that is, x/θa ⊕ y/θa = a/θa. thus y/θa > λa(x)/θa and so y/θ > λa(x)/θ. this implies that λa/θ(x/θ) exists and equals to λa(x)/θ. it can be easily shown that 〈[0/θ,a/θ],⊕,λa/θ,0/θ,a/θ〉 is a quasi-mv algebra. thus, 〈a/θ,⊕,0/θ〉 is an equasi-mv algebra. and the map π : 〈a,⊕,0〉 −→ 〈a/θ,⊕,0/θ〉 defined by x 7−→ x/θ is an equasi-mv homomorphism from a onto a/θ. definition 4.3. let a be an equasi-mv algebra and i be a nonempty subset of a. we call i to be an ideal of a if the following conditions hold: (i1) 0 ∈ i; (i2) for all x,y ∈ i, then x⊕y ∈ i; (i3) x ∈ i and y 6 x imply y ∈ i. if i is an ideal of a and x ∈ a, we have x ∈ i iff x⊕0 ∈ i by (i3). definition 4.4. let a be an equasi-mv algebra and i be a nonempty subset of a. if the following statements hold, i is a weak ideal of a: (w1) 0 ∈ i; (w2) for all x,y ∈ i, then x⊕y ∈ i; (w3) x ∈ i and y ∈ a imply x⊗y ∈ i. 90 on extended quasi-mv algebras lemma 4.1. let i be an ideal of an equasi-mv algebra a. then i is a weak ideal. proof. let i be an ideal of a and x ∈ i. if y ∈ a with y 6 x, there exists b ∈i(a) such that x,y 6 b. then we have (x⊗y) e x = λb(λb(x)⊕λb(y)) e x = λb(λb(x)⊕λb(y)⊕λb(λb(x)⊕λb(y)⊕x)) = λb(λb(x)⊕λb(y)⊕λb(b)) = x⊗y. it follows that x⊗y 6 x. thus x⊗y ∈ i and so i is a weak ideal of a.2 the converse of lemma 4.1 is not true. for example, {0} is a weak ideal, but not an ideal. proposition 4.1. let i be a nonempty subset of an equasi-mv algebra a and 0 ∈ i. then i is an ideal iff for all x,y ∈ a, a ∈ i(a) with x,y 6 a, λa(x) ⊗y ∈ i and x ∈ i implies y ∈ i. proof. =⇒: let i be an ideal of a. for all x,y ∈ a and a ∈ i(a) with x,y 6 a, if λa(x)⊗y ∈ i and x ∈ i, we have (λa(x)⊗y)⊕x ∈ i. since λa(y)⊕ ((λa(x)⊗y)⊕x) = λa(y)⊕ (λa(x⊕λa(y))⊕x) = λa(y)⊕ (λa(λa(x)⊕y)⊕y) (qmv4) = λa(y)⊕y ⊕λa(λa(x)⊕y) = a, we have y 6 (λa(x)⊗y)⊕x ∈ i and y ∈ i. ⇐=: for any x,y ∈ i and a ∈ i(a) with x 6 y and x,y 6 a, we have λa(x)⊗y = 0 ∈ i. hence, y ∈ i is obtained from propositional conditions. and then λa(x)⊗ (x⊕y) = λa(x⊕λa(x⊕y)) = λa(x) e y 6 y ∈ i. then λa(x)⊗ (x⊕y) ∈ i. it follows from x ∈ i that x⊕y ∈ i.2 definition 4.5. let a be an equasi-mv algebra. we define a binary relation 4 as follows: for all x,y ∈ a, x 4 y iff x e y = x. the binary relation 4 satisfies antisymmetry and transitivity, but when x is a regular element, it satisfies reflexivity. lemma 4.2. let a be an equasi-mv algebra and x,y ∈ a. then x 4 y iff x 6 y and x ∈r(a). 91 mengmeng liu, hongxing liu proof. if x 4 y, we have x e y = x and x e y = (x e y) ⊕ 0 = x ⊕ 0. it follows that x 6 y and x ⊕ 0 = x. thus x ∈ r(a). conversely, if x 6 y and x ∈r(a), we have x e y = x⊕0 = x and so x 4 y.2 lemma 4.3. let a be an equasi-mv algebra and j ⊆ a. then the following statements are equivalent: (i) j is a weak ideal of a; (ii) (1) if x,y ∈ j, then x⊕y ∈ j; (2) if x ∈ j, y 4 x, then y ∈ j. proof. (i)=⇒(ii): suppose x ∈ j and y 4 x. there exists b ∈i(a) such that x 6 b. then x⊗ (λb(x)⊕y) = x e y ∈ j. since y 4 x, we have x e y = y ∈ j. (ii)⇐=(i): for any x ∈ j, y ∈ a, there exists b ∈ i(a) such that x,y 6 b. since x⊗y 6 x and x⊗y ∈r(a) by lemma 4.2, we have x⊗y 4 x. therefore, x⊗y ∈ j.2 let a be an equasi-mv algebra and h be a subset of a. the ideal generated by h is the smallest ideal of a containing h, denoted by 〈h〉. lemma 4.4. let a be an equasi-mv algebra and h ⊆ a, then (i) 〈h〉={x∈a: there exist h1,· · ·,hn∈h,n ∈ nsuch that x6h1⊕···⊕hn}; (ii) 〈0〉 is the smallest ideal of a; (iii) if i is an ideal of a and x ∈ a, we have 〈i ∪{x}〉 = {z ∈ a : z 6 a⊕n.x for some a ∈ i and n ∈ n}. proof. (i) we write m ={x∈a : there existh1, · · ·,hn∈h,n∈n such that x6 h1⊕···⊕hn}. then m is an ideal of a. now we show that m is the smallest ideal of a containing h. suppose m ′ is an ideal of a containing h. for any x ∈ m, there exist h1, · · · ,hn ∈ h such that x 6 h1 ⊕ ···⊕ hn. as h ⊆ m ′, we get x ∈ m ′ and so m ⊆ m ′. (ii) by (i) we obvious get the result.2 definition 4.6. an ideal i of an equasi-mv algebra a is maximal if for all x ∈ a\ i, 〈i ∪{x}〉 = a. definition 4.7. let a be an equasi-mv algebra and θ be a congruence on a. θ is an ideal congruence if for all x,y ∈ a, (x⊕0)θ(y ⊕0) ⇒ xθy. example 4.2. let a be an equasi-mv algebra and x,y ∈ a. a binary relation χ defined as follows: xχy iff x 6 y and y 6 x. it is easy to show that χ is compatible with ⊕. we now show that for all b ∈ i(a), χ∩([0,b]×[0,b]) is congruence on the quasi-mv algebra 〈[0,b],⊕,λb,0,b〉. suppose 〈x,y〉 ∈ χ ∩ ([0,b] × [0,b]). it follows from 〈x,y〉 ∈ χ that x 6 y and y 6 x. hence, λb(y) 6 λb(x) and λb(x) 6 λb(y). therefore, 〈λb(x),λb(y)〉 ∈ χ ∩ ([0,b] × [0,b]). that is, χ is a congruence on a. as a result, χ is an ideal congruence. 92 on extended quasi-mv algebras definition 4.8. let a be an equasi-mv algebra, i be an ideal of a and θ be an ideal congruence on a. we define two relations f(j) on a×a and g(θ) on a as follows: 〈x,y〉 ∈ f(j) iff there exists b ∈i(a) such that x⊗λb(y),y ⊗λb(x) ∈ j; g(θ) = 0/θ = {x ∈ a : xθ0}. theorem 4.1. let a be an equasi-mv algebra, j be an ideal of a and θ be an ideal congruence on a. (i) f(j) is an ideal congruence on a; (ii) g(θ) is an ideal of a; (iii) j = g(f(j)); (iv) θ = f(g(θ)). proof. (i) obviously, f(j) is a congruence on a. now we show that f(j) is an ideal congruence. let 〈x⊕0,y ⊕0〉 ∈ f(j). there exists b ∈i(a) such that x,y 6 b. then λb(x ⊕ 0) ⊗ (y ⊕ 0), λb(y ⊕ 0) ⊗ (x ⊕ 0) ∈ j. it follows that λb(x)⊗y=λb(x⊕0)⊗(y⊕0) ∈ j. similarly, λb(y)⊗x ∈ j. thus, 〈x,y〉 ∈ f(j). therefore, f(j) is an ideal congruence on a. (ii) suppose 〈x,0〉 ∈ θ and y 6 x. we have 〈λb(x),b〉 ∈ θ. that implies 〈λb(x)⊕y,b〉 ∈ θ and so 〈x⊗(λb(x)⊕y), x⊗b〉 ∈ θ. that is, 〈xey,x⊕0〉 ∈ θ. it follows from y 6 x that x e y = y ⊕0. thus, 〈y ⊕0,x⊕0〉 ∈ θ. since θ is an ideal congruence on a, we have 〈y,x〉 ∈ θ. this together with 〈0,x〉 ∈ θ implies that 〈y,0〉 ∈ θ and so y ∈ g(θ). therefore, g(θ) is an ideal of a. (iii) it is easily seen that g(f(j)) = {x ∈ a : x⊕ 0 ∈ j}. for all x ∈ a, we have x ∈ j iff x⊕0 ∈ j. thus g(f(j)) = {x ∈ a : x ∈ j}. (iv) for any x,y ∈ a, if 〈x,y〉 ∈ f(g(θ)), there exists b ∈ i(a) such that x,y 6 b, 〈λb(x)⊗y,0〉 ∈ θ and 〈λb(y)⊗x,0〉 ∈ θ. then 〈(λb(x)⊗y)⊕x,0⊕x〉 ∈ θ. by (λb(x) ⊗ y) ⊕ x = x d y, we get 〈x d y,0 ⊕ x〉 ∈ θ. similarly, we have 〈x d y,0 ⊕ y〉 ∈ θ. thus, 〈0 ⊕x,0 ⊕ y〉 ∈ θ. since θ is an ideal congruence on a, we have 〈x,y〉 ∈ θ. therefore, f(g(θ)) ⊆ θ. conversely, if 〈x,y〉 ∈ θ, there exists b ∈ i(a) such that x,y 6 b and so 〈y ⊗ λb(x),x ⊗ λb(x)〉 ∈ θ. this together with x ⊗ λb(x) = 0 implies 〈y ⊗ λb(x),0〉 ∈ θ. similarly, 〈x⊗λb(y),0〉 ∈ θ. thus, 〈x,y〉 ∈ f(g(θ)). therefore, θ ⊆ f(g(θ)).2 let i be an ideal of an equasi-mv algebra a. the relation θi is defined as follows: for all x,y ∈ a, (x,y)∈θi ⇐⇒ ∃b ∈i(a) withx,y 6 bsuch that λb(λb(x)⊕y),λb(λb(y)⊕x) ∈ i. proposition 4.2. let a be an equasi-mv algebra. if i is an ideal of a, the relation θi is an ideal congruence on a. proof. let i be an ideal of a. suppose 〈x,y〉,〈y,z〉 ∈ θi. we have λb(λb(x)⊕ y), λb(λb(y)⊕x) ∈ i and λb(λb(z)⊕y), λb(λb(y)⊕z) ∈ i where b ∈i(a) such 93 mengmeng liu, hongxing liu that x,y,z 6 b. since i is an ideal of a, we have λb(λb(x)⊕y)⊕λb(λb(y)⊕z) ∈ i and λb(λb(y) ⊕ x) ⊕ λb(λb(z) ⊕ y) ∈ i. and (λb(x) ⊕ z) ⊕ (λb(λb(x) ⊕ y) ⊕ λb(λb(y)⊕z)) = b. it follows that λb(λb(x)⊕z) ∈ i. similarly, λb(λb(z)⊕x) ∈ i. then 〈x,z〉 ∈ θi. the reflexivity and symmetry is clear. it is easy to prove that θi is compatible with ⊕. for all u ∈ i(a) such that x,y,z 6 u. now, we show that θiu = θi ∩ ([0,u]× [0,u]) is a congruence on the quasi-mv algebra 〈[0,u],⊕,λu,0,u〉. suppose 〈x,y〉 ∈ θiu , we have λu(λu(x)⊕ y), λu(λu(y)⊕x) ∈ i ∩ ([0,u]× [0,u]). then (λu(x⊕z)⊕ (y ⊕z))⊕λu(λu(x)⊕y) =λu(x⊕z)⊕x⊕z ⊕λu(λu(y)⊕x) =λu(λu(x)⊕λu(z))⊕λu(z)⊕z ⊕λu(λu(y)⊕x) =u. it follows that λu(λu(x⊕z)⊕(y⊕z)) 6 λu(λu(x)⊕y) ∈ θi. then λu(λu(x⊕z)⊕ (y⊕z)) ∈ θi. similarly, λu(λu(y⊕z)⊕(x⊕z)) ∈ θi. thus, 〈x⊕z,y⊕z〉 ∈ θiu . and 〈λu(x),λu(z)〉 ∈ θiu is obvious. therefore, θi is a congruence on a. for each 〈x ⊕ 0,y ⊕ 0〉 ∈ θi, we have λb(λb(x ⊕ 0) ⊕ (y ⊕ 0)), λb(λb(y ⊕ 0) ⊕ (x⊕ 0)) ∈ i. that is, λb(λb(x) ⊕y), λb(λb(y) ⊕x) ∈ i. thus 〈x,y〉 ∈ θi. therefore, θi is an ideal congruence.2 theorem 4.2. let a be an equasi-mv algebra. there is a one-to-one correspondence between the set of all ideals and the set of all ideal congruences. proof. let i be an ideal of a and θi be an ideal congruence induced by i. now we show that i = 0/θi. since 0 ∈ i, we have 〈x,0〉 ∈ θi, for all x ∈ i. it follows that x ∈ 0/θi. conversely, suppose x ∈ 0/θi. there exists a ∈ i(a) such that x 6 a. by proposition 4.1, since λa(x) ⊗ 0 ∈ i and 0 ∈ i, we have x ∈ i. hence, i = 0/θi. let θ be an ideal congruence on a. let i = 0/θ. suppose 〈x,y〉 ∈ θi. there exists a ∈ i(a) such that x,y 6 a and λb(λb(x) ⊕ y), λb(λb(y) ⊕ x) ∈ i = 0/θ. that is, 〈λb(λb(x) ⊕ y),0〉 ∈ θ and 〈λb(λb(y) ⊕ x),0〉 ∈ θ. hence, 〈λb(λb(x) ⊕ y) ⊕ y,0 ⊕ y〉 ∈ θ and 〈λb(λb(y) ⊕ x) ⊕ x,0 ⊕ x〉 ∈ θ. since λb(λb(x)⊕y)⊕y = λb(λb(y)⊕x)⊕x, we have 〈x⊕0,y⊕0〉 ∈ θ. and since θ is an ideal congruence on a, we have 〈x,y〉 ∈ θ. conversely, let 〈x,y〉 ∈ θ. there exists a ∈ i(a) such that x,y 6 a. then 〈λa(x),λa(y)〉 ∈ θ and 〈λa(x)⊗y,λa(y)⊗y〉 ∈ θ. since λa(y)⊗y = 0, we have λa(x) ⊗ y ∈ 0/θ. similarly, λa(y) ⊗ x ∈ 0/θ. that is, 〈x,y〉 ∈ θi. therefore, θ = θi.2 theorem 4.3. let a be an equasi-mv algebra. then f(i)◦f(j) = f(j)◦f(i) is vaild, where i and j are ideals of a. 94 on extended quasi-mv algebras proof. suppose f(i),f(j) ∈ coni(a) and 〈x,y〉 ∈ f(i)◦f(j) for x,y ∈ a. so there exists z ∈ a such that 〈x,z〉 ∈ f(i) and 〈z,y〉 ∈ f(j). there exists b ∈i(a) such that x,y,z 6 b. let p be a ternary term defined as follows: pb(x,y,z) = (x⊗ (λb(y)⊕ (y e z))) d (z ⊗ (λb(y)⊕ (y e x))). then (x⊗ (λb(z)⊕ (z e y))) d (y ⊗ (λb(z)⊕ (z e x))) f(i) pb(z,z,y) = y ⊕0 and (x⊗ (λb(z)⊕ (z e y))) d (y ⊗ (λb(z)⊕ (z e x))) f(j) pb(x,y,y) = x⊕0. let (x⊗ (λb(z)⊕ (z e y))) d (y ⊗ (λb(z)⊕ (y e x))) = t, where t 6 b ∈i(a). it follows from 〈t,y⊕0〉 ∈ f(i) and 〈t,x⊕0〉 ∈ f(j) that (y ⊕0)⊗λb(t), λb(y ⊕0)⊗ t ∈ i; (x⊕0)⊗λb(t), λb(x⊕0)⊗ t ∈ j. now, y⊗λb(t) 6 (y⊕0)⊗λb(t) ∈ i, x⊗λb(t) 6 (x⊕0)⊗λb(t) ∈ j. similarly, λb(y)⊗t 6 λb(y⊕0)⊗t ∈ i, λb(x)⊗t 6 λb(x⊕0)⊗t ∈ j. thus, 〈t,y〉 ∈ f(i) and 〈t,x〉 ∈ f(j). that is, 〈x,y〉 ∈ f(j)◦f(i).2 lemma 4.5. if a is an equasi-mv algebra, the lattice coni(a) of ideal congruences on a is a sublattice of con(a). proof. let i, j be two ideals of a. it is easy to prove that f(i ∩ j) = f(i)∩f(j). now we show that f(i ∨j) = f(i)∨f(j). since g(f(i ∨ j)) = i ∨ j and g(f(i)) ∨ g(f(j)) = i ∨ j, we claim that g(f(i) ∨ f(j)) = g(f(i)) ∨ g(f(j)). let x ∈ g(f(i)) ∨ g(f(j)) such that x 6 y ⊕ z where y ∈ g(f(i)) and z ∈ g(f(j)). then we get 〈y,0〉 ∈ f(i), 〈z,0〉 ∈ f(j) and 〈y,z〉 ∈ f(i)◦f(j) = f(i)∨f(j). it follows that 〈z⊕0,0〉 ∈ f(j), 〈y⊕z,z⊕0〉 ∈ f(i) and 〈y⊕z,0〉 ∈ f(i)◦f(j) = f(i)∨f(j). and then x 6 y ⊕z ∈ g(f(i)∨f(j)). therefore, g(f(i))∨g(f(j) ⊆ g(f(i)∨f(j)). conversely, for any x ∈ g(f(i) ∨ f(j)), we have 〈x,0〉 ∈ f(i) ∨ f(j) = f(i) ◦ f(j). then there exist z ∈ a and b ∈ i(a) such that 〈x,z〉 ∈ f(i) and 〈z,0〉 ∈ f(j). and 〈x⊗λb(z),0〉 ∈ f(i), 〈z,0〉 ∈ f(j). then x 6 (x⊗λb(z))⊕ z. since x⊗λb(z) ∈ g(f(i)) and z ∈ g(f(j)), we have x ∈ g(f(i)) ∨g(f(j)). thus, g(f(i)∨f(j)) ⊆ g(f(i))∨g(f(j)).2 theorem 4.4. coni(a) is distributive. proof. by theorem 4.2, we only need to prove that the lattice of ideals on a is distributive. suppose i,j,k are ideals on a and x ∈ i ∩ (j ∨k). then x ∈ i and x 6 y ⊕z, for some y ∈ j, z ∈ k. hence, x 6 (x e y)⊕ (x e z). it follows from x e y ∈ i ∩j, x e z ∈ i ∩k that x ∈ (i ∩j)∨ (i ∩k).2 95 mengmeng liu, hongxing liu 5 filters and prime ideals in this section, we introduce the notions of filters and prime ideals of equasimv algebras. moreover, we study some properties of them. we prove that every equasi-mv algebra has at least one maximal ideal. also, we get prime theorem on equasi-mv algebras. definition 5.1. let 〈a,⊕,0〉 be an equasi-mv algebra and f be a nonempty subset of a. f is called a filter if the following conditions are satisfied: (i) for all x,y ∈ a, if x 6 y and x ∈ f , then y ∈ f ; (ii) for all x,y ∈ f , then x⊗y ∈ f . definition 5.2. we call a filter f is proper if f 6= a. a proper filter f is maximal, if for all x ∈ a\f , 〈f ∪{x}〉 = a. let a be an equasi-mv algebra. for x ∈ a and n ∈ n, we define x1 = x, · · · , xn = xn−1 ⊗x, n > 2. proposition 5.1. let a be an equasi-mv algebra and f be a filter of a. then if is an ideal of a, where if := {λa(x) : x ∈ f,∃a ∈i(a),x 6 a}. proof. for all x ∈ a, we have x ∈ if ⇐⇒ ∃a ∈i(a) s.t. x 6 a,λa(x) ∈ f. it is obvious that 0 ∈ if . suppose x,y ∈ if . there exist a,b ∈ i(a) such that x 6 a and y 6 b. it follows λa(x), λb(y) ∈ f . let c ∈ i(a) such that a,b 6 c. then λc(x), λc(y) ∈ f by proposition 3.1 (iv). that implies λc(x) ⊗λc(y) ∈ f . since λc(x), λc(y) 6 c and λc(x)⊗λc(y) = λc(x⊕y), we have x⊕y ∈ if . suppose x,y ∈ a with x ∈ if and y 6 x. there exists a ∈ i(a) such that x 6 a and λa(x) ∈ f . since x,y ∈ [0,a] and y 6 x, we have λa(x) 6 λa(y). it implies λa(y) ∈ f and y ∈ if . 2 in the following, we give an equivalent condition of maximal filters. proposition 5.2. let a be an equasi-mv algebra and f be a proper filter of a. (i) for all x ∈ a, 〈f ∪{x}〉 = {z ∈ a : z > y ⊗xn,∃n ∈ n,y ∈ f}; (ii) f is a maximal filter iff for all x /∈ f , there exist n ∈ n and b ∈ i(a) with x 6 b such that λb(xn) ∈ f . proof. (i) it is obvious. (ii) let f be a maximal filter and x /∈ f . we have 0 ∈ 〈f ∪{x}〉 by (i) and so there exist n ∈ n and y ∈ f such that 0 = y⊗xn. there exists b ∈i(a) such that x,y 6 b. then b = λb(y ⊗xn) = λb(y) ⊕λb(xn), it follows that y 6 λb(xn) and λb(xn) ∈ f . conversely, for any x ∈ a \ f , there exist n ∈ n, b ∈ i(a) such that λb(xn) ∈ f . then 0 = λb(xn) ⊗xn and 0 ∈ 〈f ∪{x}〉. it follows that 〈f ∪{x}〉 = a and f is a maximal filter.2 96 on extended quasi-mv algebras lemma 5.1. let f be a proper filter of an equasi-mv algebra a. (i) if a ∈ f ∩i(a), we have a /∈ if . (ii) if a ∈ f ∩i(a), then for all b ∈i(a) with a < b, we have λb(a) ∈ if . (iii) if f is a maximal filter of a, then for all a ∈i(a), a /∈ if implies a ∈ f . (iv) if j is a maximal ideal of a, then ∀a ∈i(a)\j =⇒ λb(a) ∈ j, where b ∈i(a) and a < b. (∗) (v) if j is an ideal of a satisfying (∗), then fj is a filter of a, where fj := {λa(x) : x ∈ j,a ∈i(a)\j,x < a}. proof. (i) suppose a ∈ f ∩i(a) and a ∈ if . there exists b ∈ i(a) such that a 6 b and λb(a) ∈ f . it follows from λb(a), a ∈ f that 0 = a⊗λb(a) ∈ f , which is a contradiction. (ii) it is obvious. (iii) let a ∈ i(a) and a /∈ if . for all b ∈ i(a) with a 6 b, we have λb(a) /∈ f by proposition 5.1. suppose a /∈ f . since f is a maximal filter, we have 〈f ∪{a}〉 = a. by proposition 5.2, there exist n ∈ n and x ∈ f such that 0 = x⊗an. we have u ∈ i(a) such that x,a 6 u and 0 = x⊗an = x⊗u an. since a ∈ i(a), we get an = a and so u = λu(x) ⊕ λu(a). it follows that x 6 λu(a) and λu(a) ∈ f , which is a contradiction. (iv) suppose a ∈ i(a) and a /∈ j. for any b ∈ i(a) and a < b, we have λb(a) ∈ 〈j ∪{a}〉 = a. by lemma 4.4, there exist n ∈ n and x ∈ j such that λb(a) 6 x⊕n.a. since a,λb(a) ∈ [0,b], we have λb(a) = λb(a)⊕0 = λb(a) e (x⊕n.a) 6 (λb(a) e x)⊕ (λb(a) e n.a) (lemma 2.4 (iii)) = λb(a) e x. it follows λb(a) 6 x ∈ j and so λb(a) ∈ j. (v) suppose x,y ∈ a with x 6 y and x ∈ fj. there exists a ∈i(a)\j such that x < a and λa(x) ∈ j. let b ∈i(a) and a,y 6 b. we have λb(y) 6 λb(x) 6 λa(x) ⊕λb(a). by (iv), we have λb(a) ∈ j and λa(x) ⊕λb(a) ∈ j. that implies λb(y) ∈ j and y ∈ fj. let x,y ∈ fj. there exist a,b ∈ i(a) \ j such that x 6 a, y 6 b and λa(x),λb(y) ∈ j. let c ∈ i(a) and a,b 6 c. we have λc(a),λc(b) ∈ j by (iv) and λc(x) 6 λc(x)⊕0 = λa(x)⊕λc(a) ∈ j, λc(y) 6 λc(y)⊕0 = λb(y)⊕λc(b) ∈ j by proposition 3.1. it follows that λc(x),λc(y) ∈ j and λc(x) ⊕ λc(y) ∈ j. thus λc(λc(x)⊕λc(y)) ∈ fj. that is, x⊗y = x⊗c y ∈ fj.2 definition 5.3. let a be an equasi-mv algebra and i be an ideal of a. we call i to be prime if for all x,y ∈ a, x e y ∈ i implies that x ∈ i or y ∈ i. proposition 5.3. let i be an ideal of an equasi-mv algebra a. then i is prime iff 97 mengmeng liu, hongxing liu for any x,y ∈ a, there exists a ∈i(a) with x,y 6 a such that λa(λa(x)⊕y) ∈ i or λa(λa(y)⊕x) ∈ i. proof. ⇐=: let π: a −→ a/i be the canonical projection and θ be an ideal congruence. if x e y ∈ i, we have (x e y)/θ = x/θ e y/θ ∈ π(i). let x/θ = [i] or y/θ = [j], where i,j ∈ i. there exists a ∈ i(a) such that x,y,i,j 6 a, λa(x) ⊗ i ∈ i, λa(i) ⊗ x ∈ i or λa(y) ⊗ j ∈ i, λa(j) ⊗ y ∈ i. it follows from proposition 4.1 that x ∈ i or y ∈ i. =⇒: for any x,y ∈ a, there exists a ∈i(a) such that x,y 6 a. we have (λa(x)⊕y)d(λa(y)⊕x) =λa(x)⊕y⊕λa(λa(x)⊕y⊕λa(λa(y)⊕x)) =λa(x)⊕λa(λa(x)⊕λa(λa(y)⊕x))⊕λa(λa(λa(x)⊕λa(λa(y)⊕x))⊕λa(y)) =λa(y)⊕x⊕λa(λa(y)⊕x⊕x)⊕λa(λa(λa(x)⊕λa(λa(y)⊕x))⊕λa(y)) =λa(x)⊕λa(λa(y)⊕x))⊕λa((λa(x)⊕λa(λa(y)⊕x))⊕y)⊕x⊕λa(λa(y)⊕x⊕x) =a. it follows λa((λa(x) ⊕ y) d (λa(y) ⊕ x)) = 0 ∈ i. that is, λa(λa(x) ⊕ y) e λa(λa(y)⊕x) = 0 ∈ i. therefore, λa(λa(x)⊕y) ∈ i or λa(λa(y)⊕x) ∈ i.2 example 5.1. let a × m be an equasi-mv algebra mentioned in example 3.6. it can be easily proved that p = {0,b} is a prime ideal of a quasi-mv algebra a. now we show that p ×m is a prime ideal of an equasi-mv algebra a×m. obviously, 〈0,0〉 ∈ p × m and 〈0,m〉 ⊕ 〈b,m〉 = 〈b,m〉 ∈ p × m. and for any 〈x,m〉 6 〈b,m〉, we have 〈x,m〉 ∈ a × m. then p × m is an ideal of a × m. for any 〈x1,y1〉,〈x2,y2〉 ∈ a × m, suppose 〈x1,y1〉 e 〈x2,y2〉 = 〈x1 ex2,y1∧y2〉 ∈ p×m, we have x1 ∈ p or x2 ∈ p . that is, 〈x1,y1〉 ∈ p×m or 〈x2,y2〉 ∈ p ×m. let a be a proper equasi-mv algebra and a ∈i(a)\{0}. we define ↑ a = {x ∈ a : x > a}. then ↑ a is a filter of a. moreover, ↑ a is a proper filter of a. proposition 5.4. let f be a maximal filter of an equasi-mv algebra a. then if = {λa(x) : x ∈ f,∃a ∈i(a),x 6 a} is a maximal ideal of a. proof. we know that if is an ideal of a by proposition 5.1. as f 6= ∅, we have a ∈i(a)∩f and so a /∈ if by lemma 5.1 (i). let j be an ideal of a and if ⊆ j. suppose a /∈ j and a ∈ i(a), we have a /∈ if and so a ∈ f by lemma 5.1 (iii). then for any b ∈ i(a) with a 6 b, we have λb(a) ∈ if ⊆ j. hence, j satisfies condition (∗) in lemma 5.1 (iv). it follows from lemma 5.1 (iv) that fj is a filter of a. 98 on extended quasi-mv algebras suppose x ∈ f and w ∈i(a)\j. there exists u ∈i(a) such that x,w 6 u. since j is a proper ideal, we have u /∈ j. it follows from the definition of if that λu(x) ∈ if ⊆ j and then x ∈ fj. that implies f ⊆ fj. since f is a maximal filter, we have fj = f or fj = a. if fj = a, then there exist x ∈ j and a ∈ i(a) such that x < a and λa(x) = 0, which is a contradiction. thus fj = f . by lemma 5.1 (v), for all x ∈ j, there exists a ∈ i(a) \ j such that x < a and λa(x) ∈ fj = f . hence, we have x ∈ if . that is, j ⊆ if . thus j = if . this proves that if is a maximal ideal of a.2 theorem 5.1. let a be a proper equasi-mv algebra. then a has at least one maximal ideal. proof. suppose 0 6= a ∈ a. note that ↑ a is a filter and {0} 6=↑ a. by zorn’s lemma, we know that the set of all filters that does not contain 0 has a maximal element, which is a maximal filter of a, denoted by f . it follows from proposition 5.4 that if is a maximal ideal.2 the following statement gives the prime theorem on equasi-mv algebras. theorem 5.2. let i be a proper ideal of an equasi-mv algebra a and a ∈ a\i. then there exists a maximal ideal p which contains i and a ∈ a\p . moreover, p is prime. proof. let m = {j : i ⊆ j,a /∈ j} where i,j are ideals of a. by zorn’s lemma, m has a top element p . it follows from i ∈ m that m 6= ∅. we claim that p is prime. suppose x e y ∈ p and x,y /∈ p . we have a ∈ 〈p ∪{x}〉 and a ∈ 〈p ∪{y}〉. then there exist n ∈ n and u,v ∈ p such that a 6 u⊕n.x and a 6 v ⊕n.y. it follows that a 6 (u⊕n.x) e (v ⊕n.y) 6 (u⊕v ⊕n.x) e (u⊕v ⊕n.y). by lemma 2.4 (iii), we have a 6 (u⊕v⊕n.x)e(u⊕v⊕n.y) = (u⊕v)⊕(n.xen.y) 6 (u⊕v)⊕n2.(xey) ∈ p. it follows that a ∈ p , which is a contradiction. thus, we have x ∈ p or y ∈ p .2 6 conclusion in this paper, we introduce the notion of equasi-mv algebras, which are generalizations of quasi-mv algebras. we study some basic properties of equasi-mv algebras, such as ideals, ideal congruences and filters and investigate their mutual relationships. we show that there is a one-to-one correspondence between the set of all ideals and the set of all ideal congruences on an equasi-mv algebra. and we also studied some results on maximal ideals and prime ideals. there are many topics that deserve further study. for example, (1) can any equasi-mv algebra be embedded into an equasi-mv algebra with a top element? 99 mengmeng liu, hongxing liu (2) does any simple equasi-mv algebra have a top element? (3) the author introduced me-algebras and studied the categorical equivalence between equality algebras and abelian lattice-ordered groups in liu [2019]. we will study the relationships between monadic equasi-mv algebras and monadic equality algebras. references f. paoli a. ledda, m. konig and r. giuntini. mv-algebras and quantum computation. studia logica, 82(2):245–270, 2006. c. c. chang. algebraic analysis of many valued logics. transactions of the american mathematical society, 88(2):467–490, 1958. p. f. conrad and m. r. darnel. generalized boolean algebras in lattice-ordered groups. order, 14(4):295–319, 1997. a. dvurečenskij and o. zahiri. on emv-algebras. fuzzy sets and systems, 373: 116–148, 2019. a. ledda f. bou, f. paoli and h. freytes. on some properties of quasi-mv algebras and √ ′ quasi-mv algebras. part ii. soft computing, 12(4):341–352, 2008. r. giuntini f. paoli, a. ledda and h. freytes. on some properties of quasi-mv algebras and √ ′ quasi-mv algebras. reports math. log., 44:31–63, 2009. r. giuntini g. cattaneo, m. l. dalla chiara and r. leporini. an unsharp logic from quantum computation. international journal of theoretical physics, 43 (7):1803–1817, 2004. h. liu. on categorical equivalences of equality algebras and monadic equality algebras. logic journal of the igpl, 27(3):267–280, 2019. h. liu. ebl-algebras. soft computing, 24(19):14333–14343, 2020. j. liu and w. chen. a non-commutative generalization of quasi-mv algebras. in 2016 ieee international conference on fuzzy systems (fuzz-ieee), pages 122–126. ieee, 2016. r. giuntini m. l. dalla chiara and r. greechie. reasoning in quantum theory: sharp and unsharp quantum logics, volume 22. springer science & business media, 2013. r. giuntini m. l. dalla chiara and r. leporini. logics from quantum computation. international journal of quantum information, 3(02):293–337, 2005. 100 ratio mathematica 22 (2012) 45-59 issn:1592-7415 atanassov’s intuitionistic fuzzy index of hypergroupoids mohsen asghari-larimia, irina cristeab adepartment of mathematics, golestan university, gorgan, iran bcentre for systems and information technologies, university of nova gorica, slovenia asghari2004@yahoo.com, irinacri@yahoo.co.uk abstract in this work we introduce the concept of atanassov’s intuitionistic fuzzy index of a hypergroupoid based on the notion of intuitionistic fuzzy grade of a hypergroupoid. we calculate it for some particular hypergroups, making evident some of its special properties. key words: (atanassov’s intuitionistic) fuzzy set, (atanassov’s intuitionistic) fuzzy grade, hypergroup. 2000 ams subject classifications: 20n20; 03e72. 1 introduction in 1986 atanassov [5, 8] defined the notion of intuitionistic fuzzy set as a generalization of the elder one of fuzzy set introduced by zadeh [30]. since then this new tool of investigation of various uncertain problems bears his name: atanassov’s intuitionistic fuzzy set. more exactly, for any element x in a finite nonempty set x one assigns two values in the interval [0, 1]: the membership degree µ(x) and the non-membership degree λ(x) such that 0 ≤ µ(x) + λ(x) ≤ 1. for any x ∈ x, the value π(x) = 1 − µ(x) − λ(x) is called the atanassov’s intuitionistic fuzzy index or the hesitation degree of x to x. this index is an important characteristic of the intuitionistic fuzzy sets since it provides valuable information on each element x in x (see [6, 7]). recently, bustince et al. [9] have given a construction method to obtain a generalized atanassov’s intuitionistic fuzzy index. for other applications of this index see [10, 11]. 45 m. asghari-larimi and irina cristea the concept of fuzzy grade appeared in hypergroup theory in 2003 and seven years later it was extended to the intuitionistic fuzzy case. based on the two connections between hypergroupoids and fuzzy sets introduced by corsini [13, 14], one may associate with any hypergroupoid h a sequence of join spaces and fuzzy sets, whose length is called the fuzzy grade of h. corsini and cristea determined the fuzzy grade of all i.p.s. hypergroups of order less than 8 (see [15, 16]). the same problem was treated by cristea [21], angheluţă and cristea [1] for the complete hypergroups, by corsini et al. for the hypergraphs and hypergroupoids obtained from multivalued functions [18, 19, 20]. in 2010 cristea and davvaz [22] introduced and studied the atanassov’s intuitionistic fuzzy grade of a hypergroupoid as the length of the sequence of join spaces and intuitionistic fuzzy sets associated with a hypergroupoid. these sequences have been determined for all i.p.s. hypergroups of order less than 8 and for the complete hypergroups of order less than 7 by davvaz et al. [23, 24, 25]. any hypergroupoid h may be endowed with an intuitionistic fuzzy set ā = (µ̄, λ̄) in the sense of cristea-davvaz [22]. based on this construction, we define here the notion of atanassov’s intuitionistic fuzzy index of a hypergroupoid h. throughout the paper we use, for simplicity, the term of intuitionistic fuzzy set instead of atanassov’s intuitionistic fuzzy set. the paper is structured as follows. after some background information regarding hypergroups theory, we recall in section 2 the construction of the sequence of join spaces and intuitionistic fuzzy sets associated with a hypergroupoid h, presenting some technical results for the membership functions µ̃, µ̄, λ̄. in section 3 we introduce the notion of intuitionistic fuzzy index of a hypergroupoid, giving its formula for some particular hypergroups. we conclude with final remarks and some open problems. 2 atanassov’s intuitionistic fuzzy grade of hypergroupoids first we recall some definitions from [12, 17], needed in what follows. let h be a nonempty set and let p∗(h) be the set of all nonempty subsets of h. a hyperoperation on h is a map ◦ : h × h −→ p∗(h) and the couple (h, ◦) is called a hypergroupoid. this hyperoperation can be extended to a binary operation on p∗(h). if 46 atanassov’s intuitionistic fuzzy index of hypergroupoids a and b are nonempty subsets of h, then we denote a ◦ b = ∪ a∈a, b∈b a ◦ b, x ◦ a = {x} ◦ a and a ◦ x = a ◦ {x}. a hypergroupoid (h, ◦) is called a semihypergroup if, for all x, y, z in h, we have the associative law: (x ◦ y) ◦ z = x ◦ (y ◦ z), which means that∪ u∈x◦y u ◦ z = ∪ v∈y◦z x ◦ v. we say that a semihypergroup (h, ◦) is a hypergroup if, for all x ∈ h, we have the reproducibility axiom: x ◦ h = h ◦ x = h. a hypergroup (h, ◦) is called total hypergroup if, for any x, y ∈ h, x ◦ y = h. for each pair of elements a, b ∈ h, we denote: a/b = {x ∈ h | a ∈ x ◦ b} and b\a = {y ∈ h | a ∈ b ◦ y}. a commutative hypergroup (h, ◦) is called a join space if the following condition holds: a/b ∩ c/d ̸= ∅ =⇒ a ◦ d ∩ b ◦ c ̸= ∅. a commutative hypergroup (h, ◦) is canonical if and only if it is a join space with a scalar identity. the notion of join space has been introduced and studied for the first time by prenowitz. later on, together with jantosciak, he reconstructed, from an algebraic point of view, several branches of geometry: the projective, the descriptive and the spherical geometry (see [28]). several connections between hypergroups and (intuitionistic) fuzzy sets have been investigated till now (see for example [2, 3, 4, 26]). here we focus our study on that initiated by corsini [14] in 2003, when he defined a sequence of join spaces associated with a hypergroupoid endowed with a fuzzy set. based on this idea, cristea and davvaz [22] extended later on this connection to the intuitionistic fuzzy grade. we recall now briefly these constructions. for any hypergroup (h, ◦), corsini [14] defined a fuzzy subset µ̃ of h in the following way: for any u ∈ h, one considers: µ̃(u) = ∑ (x,y)∈q(u) 1 |x ◦ y| q(u) , (1) where q(u) = {(a, b) ∈ h2 | u ∈ a ◦ b}, q(u) = |q(u)|. if q(u) = ∅, set µ̃(u) = 0. 47 m. asghari-larimi and irina cristea on the other hand, with any hypergroupoid h endowed with a fuzzy set α, we can associate a join space (h, ◦α) as follows (see [13]): for any (x, y) ∈ h2, x ◦α y = {z ∈ h | α(x) ∧ α(y) ≤ α(z) ≤ α(x) ∨ α(y)}. (2) then, from (1h, ◦1) we obtain, in the same way as in (1), a membership function µ̃1 and then the join space 2h and so on. a sequence of fuzzy sets and of join spaces (rh, µ̃r) is determined. we denote µ̃0 = µ̃, 0h = h. if two consecutive hypergroups of the obtained sequence are isomorphic, then the sequence stops. the length of this sequence has been called by corsini and cristea [15, 16] the (strong) fuzzy grade of the hypergroupoid h. let now (h, ◦) be a finite hypergroupoid of cardinality n, n ∈ n∗. cristea and davvaz [22] defined on h an atanassov’s intuitionistic fuzzy set (µ, λ) in the following way: for any u ∈ h, we set: µ(u) = ∑ (x,y)∈q(u) 1 |x ◦ y| n2 , λ(u) = ∑ (x,y)∈q(u) 1 |x ◦ y| n2 , (3) where q(u) = {(x, y) | (x, y) ∈ h2, u ∈ x ◦ y}, q(u) = {(x, y) | (x, y) ∈ h2, u /∈ x ◦ y}. if q(u) = ∅, then we put µ(u) = 0 and similarly, if q(u) = ∅, then we put λ(u) = 0. it follows immediately the following relation. corolary 2.1. for all u ∈ h, µ̃(u) ≥ µ(u). proof. let |h| = n. then q(u) ≤ n2, for all u ∈ h. thus, by definitions (1) and (3), we have µ̃(u) ≥ µ(u). moreover, for all u ∈ h, we define: a(u) = ∑ (x,y)∈q(u) 1 |x◦y|, a(u) =∑ (x,y)∈q(u) 1 |x◦y|. remark 2.1. there exist hypergroups h such that there exists u ∈ h with (i)|q(u)| = |q(u)| and µ(u) ̸= λ(u). (ii)|q(u)| ̸= |q(u)| and µ(u) = λ(u). 48 atanassov’s intuitionistic fuzzy index of hypergroupoids an example for the case (i) is the hypergroup h1 and the case (ii) is illustrated by the hypergroup h2 given below; both hypergroups are commutative and are represented by the following tables: h1 1 2 3 4 1 1,4 1,2 1,3 1,4 2 2 2,3 2,4 3 3 3,4 4 1,2,3,4 where |q(1)| = |q(1)| = 8, a(1) = 15 4 , a(1) = 20 4 and by consequence λ(1) = 15 64 ̸= 20 64 = µ(1). h2 1 2 3 1 1 1,2 1,3 2 2 2,3 3 3 where |q(1)| = 5 ̸= 4 = |q(1)| and by consequence λ(1) = µ(1) = 1 3 . 3 atanassov’s intuitionistic index of hypergroupoids definition 3.1. [5] let (µ, λ) be an intuitionistic fuzzy set on the nonempty set h. we call the hesitation degree or the intuitionistic index of the element x in the set h the following expression π(x) = 1 − µ(x) − λ(x). evidently 0 ≤ π(x) ≤ 1, for all x ∈ h. remark 3.1. let (µ, λ) be the intuitionistic fuzzy set associated with a hypergroupoid (h, ◦) like in cristea-davvaz [22]. since µ(u)+λ(u) = ∑ (x,y)∈h2 1 |x◦y| n2 , for all u ∈ h, according with (ω′), it follows that π(u) = constant, for all u ∈ h. therefore we introduce the following definition. definition 3.2. let (h, ◦) be a hypergroupoid and let µ, λ be the membership functions defined in (w′). for all u ∈ h, we define π(h) = π(u) = 1 − µ(u) − λ(u), and we call it the intuitionistic index of the hypergroupoid h. 49 m. asghari-larimi and irina cristea in the following we determine the intuitionistic index for some particular hypergroupoids. proposition 3.1. let (h, ◦) be a total hypergroup of cardinality n. then π(u) = n − 1 n , for any u ∈ h. proof. let (h, ◦) be a total hypergroup and |h| = n. then x ◦ y = h, for all x, y ∈ h. thus a(u) = n, a(u) = 0, for all u ∈ h. since q(u) = ∅, it follows that µ(u) = 1 n , λ(u) = 0, which imply that π(u) = n−1 n . proposition 3.2. let h = {x1, x2, x3, ..., xn−1, xn}, be the hypergroupoid defined as it follows: xi ◦ xj = {xi, xj}, 1 ≤ i, j ≤ n. then µ(x) = 1n, π(x) = n−1 2n , for any x ∈ h. proof. the table of the commutative hyperoperation ◦ is the following one: h x1 x2 x3 ... xn−1 xn x1 x1 x1, x2 x1, x3 ... x1, xn−1 x1, xn x2 x2 x2, x3 ... x2, xn−1 x2, xn x3 x3 ... x3, xn−1 x3, xn ... ... ... ... xn−1 xn−1 xn−1, xn xn xn let xi ∈ h. then |q(xi)| = {(xi, xi), (xi, xs), (xp, xi), 1 ≤ s, p ≤ n, s, p ̸= i}. it follows that |q(xi)| = 2n−1. thus |q(xi)| = n2−(2n−1) = (n−1)2. so, we have µ(xi) = 1 n2 [1+ 1 2 (2n−2)] = 1 n . similarly, we obtain λ(xi) = π(xi) = n−1 2n . corolary 3.1. let (h, ◦) be the hypergroupoid defined in proposition 3.2. then, for any x ∈ h, we obtain µ(x) = 1 n > λ(x) = π(x) = n−1 2n , for any n < 3, and µ(x) = 1 n < λ(x) = π(x) = n−1 2n for any n ≥ 3. with any hypergroupoid h one may associate a sequence of join spaces and fuzzy sets denoted by ((ih, ◦i), µi(u))i≥1. then, for any i ≥ 1, we can divide ih in the classes {icj}rj=1, where x, y ∈ icj ⇐⇒ µi−1(x) = µi−1(y). moreover, we define the following ordering relation: j < k if, for elements x ∈ icj and y ∈ ick, we have µi−1(x) < µi−1(y). we need the following notations: for all j, s, set kj = |icj|, sc = ∪ 1≤j≤s icj, sc = ∪ s≤j≤r icj, sk = |sc|, sk = |sc|. 50 atanassov’s intuitionistic fuzzy index of hypergroupoids therefore with any ordered chain (ic1, ic2, ..., icr) we may associate an ordered r-tuple (k1, k2, ..., kr), where kl = |icl|, for all l, 1 ≤ l ≤ r. using these notations we determine the general formula for calculating the values of the membership functions µi, λi. theorem 3.1. [22] for any u ∈ ics, i ≥ 1, s ∈ {1, 2, ..., r}, we find that µi(z) = ks + 2 ∑ l≤s≤m l̸=m klkm∑ l≤t≤m kt n2 and λi(z) = ∑ l ̸=s kl + 2 ∑ s<l<m≤r klkm∑ l≤t≤m kt + 2 ∑ 1≤l<m<s klkm∑ l≤t≤m kt n2 . we immediately obtain also the formula for calculating the intuitionistic fuzzy index. theorem 3.2. for any u ∈ ics, i ≥ 1, s ∈ {1, 2, ..., r}, πi(u) = 1 −  ∑rl=1 kl + 2∑1≤l<m≤r klkm∑l≤t≤m kt n2   . one of the natural problems regarding the intuitionistic fuzzy grade of a hypergroupoid is that of constructing a hypergroupoid h with i.f.g.(h) = p, for any arbitrary natural number p ≥ 2. a such example is given in the next result. lemma 3.1. [22] let h = {x1, x2, x3, ..., xn−1, xn}, where n is an even number, be the hypergroupoid defined as follows: i) xi ◦ xi = xi, 1 ≤ i ≤ n, ii) xi ◦ xj = xj ◦ xi = {xi, xi+1, ..., xj}, 1 ≤ i < j ≤ n. then, for any s ∈ {1, 2, ..., n 2 }, we have µ(xs) = µ(xn−s+1), λ(xs) = λ(xn−s+1) and µ(xs) ≤ µ(xs+1), λ(xs) ≥ λ(xs+1). 51 m. asghari-larimi and irina cristea moreover, we obtain µ(x1) = 1 n2 ( 1 + 2 n∑ m=2 1 m ) , λ(x1) = 1 n2 ( 1 − n + 2n n∑ m=2 1 m ) µ(xs+1) = µ(xs)+ 2 n2 n−s∑ m=s+1 1 m , λ(xs) = λ(xs+1)+ 2 n2 n−s∑ m=s+1 1 m , ∀s, 2 ≤ s ≤ n 2 . notation 3.1. in the following we denote the hypergroup of lemma 3.1 by hn, n ∈ n∗. for hn, we denote the membership functions µ and λ introduced in (ω′) by µn and λ n . example 3.1. let us consider again the hypergroup hn that has the following table h x1 x2 x3 ... xn−1 xn x1 x1 x1, x2 x1 → x3 ... x1 → xn−1 x1 → xn x2 x2 x2, x3 ... x2 → xn−1 x2 → xn x3 x3 ... x3 → xn−1 x3 → xn ... ... ... ... xn−1 xn−1 xn−1, xn xn xn where we use the notation xi → xj = {xi, xi+1, ..., xj}, with i < j. then we obtain µn(x1) µ n(x2) µ n(x3) µ n(x4) λ n (x1) λ n (x2) λ n (x3) λ n (x4) π n(xk) n = 2 0.500 0.250 0.250 n = 3 0.296 0.407 0.333 0.222 0.370 n = 4 0.197 0.302 0.354 0.250 0.447 n = 5 0.142 0.229 0.256 0.353 0.266 0.240 0.504 n = 6 0.108 0.179 0.212 0.342 0.273 0.240 0.547 n = 7 0.085 0.144 0.176 0.186 0.332 0.257 0.241 0.231 0.581 n = 8 0.075 0.125 0.145 0.159 0.226 0.176 0.156 0.141 0.698 where k ∈ {1, ..., n}. the missing values in the previous table are equal to the other written values as in the formulas in lemma 3.1. lemma 3.2. for the hypergroup hn, with n an even number, for any s ∈ {1, 2, ..., n 2 } and n ≥ 6, we obtain µn(xs) < λ n (xs). 52 atanassov’s intuitionistic fuzzy index of hypergroupoids corolary 3.2. let us consider the hypergroup hn. then, for any s ∈ {1, 2, ..., n}, we obtain π(hn) = πn(xs) = 1 − 1 n2 [ 2 − n + 2(n + 1) n∑ m=2 1 m ] . proof. since µn(x1) + λ n (x1) = 1 n2 [(1 + 2 ∑n m=2 1 m ) + (1 − n + 2n ∑n m=2 1 m )] = 1 n2 [2 − n + 2(n + 1) ∑n m=2 1 m ], the required result is proved. theorem 3.3. let us consider the hypergroups (hn, ◦) and (hn+2, ◦), with n an even natural number. then π(hn) < π(hn+2). proof. by corollary 3.2, we will prove that 1 n2 [2 − n + 2(n + 1) n∑ m=2 1 m ] > 1 (n + 2)2 [2 − (n + 2) + 2(n + 3) n+2∑ m=2 1 m ]. denoting ∑n m=2 1 m by a, we will show that: 1 n2 [2 − n + 2(n + 1)a] > 1 (n + 2)2 [−n + 2(n + 3)(a + 2n + 3 (n + 1)(n + 2) )], that is equivalent with (2 − n)(n + 1)(n + 2)3 + 2(n + 1)2(n + 2)3a > > −n3(n + 1)(n + 2) + 2n2(n + 1)(n + 2)(n + 3)a + 2n2(2n + 3)(n + 3). therefore we have 4(n + 1)(n + 2)(n2 + 4n + 2)a > 2(3n4 + 10n3 + n2 − 16n − 8) if and only if 2a > 3n4 + 10n3 + n2 − 16n − 8 n4 + 7n3 + 16n2 + 14n + 4 . 53 m. asghari-larimi and irina cristea we prove the last relation by induction on n. for n = 3 the relation becomes 2(1 2 + 1 3 ) > 466 460 , that is 5 3 > 466 460 , that is true. we suppose that p(n) : 2 n∑ m=2 1 m > 3n4 + 10n3 + n2 − 16n − 8 n4 + 7n3 + 16n2 + 14n + 4 is true and we prove that p(n) : 2 n+1∑ m=2 1 m > 3(n + 1)4 + 10(n + 1)3 + (n + 1)2 − 16(n + 1) − 8 (n + 1)4 + 7(n + 1)3 + 16(n + 1)2 + 14(n + 1) + 4 is fulfilled. since 2 n+1∑ m=2 1 m = 2 n∑ m=2 1 m + 2 n + 1 > 3n4 + 10n3 + n2 − 16n − 8 n4 + 7n3 + 16n2 + 14n + 4 + 2 n + 1 , it remains to prove that 3n5 + 15n4 + 25n3 + 17n2 + 4n n5 + 8n4 + 23n3 + 30n2 + 18n + 4 > 3n4 + 22n3 + 49n2 + 28n − 10 n4 + 11n3 + 43n2 + 71n + 42 . after simple computations that we omit here, we prove that the last relation is true, for any natural number n. now the proof is complete. generalizing this theorem we obtain the following result. corolary 3.3. for the hypergroups hn and hn′, with n, n′ two even natural numbers such that n < n′, we have the following relation: π(hn) < π(hn′). proof. let n′ = n + 2k, k ∈ n∗. we will prove the relation by induction on k. for k = 1, by theorem 3.3 we have π(hn) < π(hn+2). assume the corollary is true for k = m, i.e., π(hn) < π(hn+2m). again by theorem 3.3 we have π(hn+2m) < π(hn+2(m+1)). thus, the thesis of the corollary is true for k = m + 1. this completes the induction and the proof. we conclude this section with a result regarding the membership function µn. theorem 3.4. let us consider the hypergroups hn and hn+2, with n an even natural number. then, for any s ∈ {1, 2, ..., n/2}, we have the relation µn(xs) > µ n+2(xs). 54 atanassov’s intuitionistic fuzzy index of hypergroupoids proof. for any s ∈ {1, 2, ..., n/2}, we will prove that 1 n2 [2s − 1 + 2 n−2s+1∑ m=1 s s + m + 2 s−1∑ m=1 m n − m + 1 ] > > 1 (n + 2)2 [2s−1+2 n−2s+1∑ m=1 s s + m +2s( 1 n − s + 2 + 1 n − s + 3 )+2 s−1∑ m=1 m n − m + 3 ]. denote a = 2s − 1 + 2 n−2s+1∑ m=1 s s + m . since 1 n2 ( 2 s−1∑ m=1 m n − m + 1 ) > 1 (n + 2)2 ( 2 s−1∑ m=1 m n − m + 3 ) , to prove that µn(xs) > µ n+2(xs) it is enough to show that (n + 2)2a > n2 [ a + 2s(2n − 2s + 5) (n − s + 2)(n − s + 3) ] , that is true if and only if a > sn2(2n − 2s + 5) (2n + 2)(n − s + 2)(n − s + 3) . since n−2s+1∑ m=1 s s + m > s(n − 2s + 1) n − s + 1 , it follows that a > (4s − 1)n − 6s2 + 5s − 1 n − s + 1 . it remains to prove that (4s − 1)n − 6s2 + 5s − 1 n − s + 1 > sn2(2n − 2s + 5) (2n + 2)(n2 + (5 − 2s)n + s2 − 5s + 6) = = 2sn3 + n2(−2s2 + 5s) 2n3 + n2(−4s + 12) + n(2s2 − 14s + 22) + 2s2 − 10s + 12 , and this is true if and only if [2n3+n2(−4s+12)+n(2s2−14s+22)+2s2−10s+12][(4s−1)n−6s2+5s−1] > 55 m. asghari-larimi and irina cristea > (n − s + 1)[2sn3 + n2(−2s2 + 5s)], which is equivalent with e(n) = n4(6s − 29 + n3(−24s2 + 55s − 14) + n2(30s3 − 143s2 + 161s − 34)+ +n(−12s4 + 102s3 − 246s2 + 182s − 34) − 12s4 + 70s3 − 124s2 + 70s − 12 > 0, whenever 2 ≤ 2s ≤ n. then we find e(4)(n) = 144s − 48 > 0; it follows that e′′′(n) is a strictly increasing function, so e′′′(n) ≥ e′′′(2s) = 144s2 + 234s − 84 > 0. it follows that e′′(n) is a strictly increasing function, so e′′(n) ≥ e′′(2s) = 60s3 + 278s2 + 154s − 68 > 0. then e′(n) is a strictly increasing function, thus e′(n) ≥ e′(2s) = 12s4 + 126s3 + 230s2 + 46s − 34 > 0. thus, we obtain that e(n) is a strictly increasing function, so e(n) ≥ e(2s) = 28s4 + 110s3 + 104s2 + 2s − 12 > 0, for any s ≥ 1, and the required result is proved. a generalized corollary follows. corolary 3.4. let us consider the hypergroups hn and hn′, with n, n′ even numbers such that n < n′. then µn(xs) > µ n′(xs), ∀s ∈ {1, 2, ..., n/2}. proof. the proof is similar to the proof of corollary 3.3. 4 conclusions given a hypergroupoid h, one may calculate two numerical functions associated with it: the fuzzy grade and the intuitionistic fuzzy grade. in this note we define another one, called the atanassov’s intuitionistic fuzzy index of a hypergroupoid. this function depends on the values of the first membership functions in the sequence of join spaces and intuitionistic fuzzy sets associated with h as in [22]. we have determined it for some particular hypergroups. we will investigate further properties and connections with the intuitionistic fuzzy grade 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[30] l.a. zadeh, fuzzy sets, inform and control, 8 (1965), 338-353. 59 moderate-density close-closed loop burst error detecting codes 15 moderate-density close-closed loop burst error detecting codes bal kishan dass1 sapna jain2 abstract: in this paper, we study cyclic codes detecting a subclass of close-closed loop bursts viz. moderate-density close-closed loop bursts. a subclass of ct close-closed loop bursts called ct moderate-density close-closed loop bursts is also studied. a comparative study of the results obtained in this paper has also been made. keywords: cyclic codes, moderate-density bursts, close-closed loop bursts, error detecting codes. 1. introduction burst errors are the most common type of errors that occur in several communication channels. codes developed to detect and correct such errors have been studied extensively by many authors. the most successful early burst error correcting codes were due to fire (1959). fire in his report gave the idea of open and closed loop bursts defined as follows: definition 1. an open loop burst of length b is a vector all of whose non-zero components are confined to some b consecutive components, the first and the last of which is non-zero. definition 2. a closed loop burst of length b is a vector all of whose non-zero components are confined to some b consecutive components, the first and the last of which is non-zero and the number of positions from where the burst can start is n (i.e. it is possible to come back cyclically at the first position after the last position for enumeration of the length of the burst). definition 2 of closed loop burst can also be formulated mathematically on the lines campopiano (1962) as follows: definition 2a. let )(qv n be the set of all ordered ntuples with components belonging to gf(q). let x = ),...,,( 110 −naaa be a vector in )(qv n . then x is 1 department of mathematics, university of delhi, delhi 110 007, india 2 department of mathematics, university of delhi, delhi 110 007, india 16 called a closed loop burst of length b, ,2 nb ≤≤ if ∃ an i , 10 −≤≤ ni , such that 0. ≠ji aa where j = ( i + b 1) modulo n and     <======== >==== −++− −++ j i if 0...... j i if 0... 121110 121 njji ijj aaaaaa aaa there is yet another definition of a burst due to chien and tang (1965) which runs as follows: definition 3. a ct burst of length b is a vector all of whose non-zero components are confined to some b consecutive components, the first of which is non-zero. based on these definitions, dass & jain (2000) defined close-closed loop bursts, open-closed loop burst, ct close-closed loop burst, and ct open-closed loop burst and proved results for close-closed loop bursts and ct close-closed loop bursts. the definitions and the results proved by dass & jain (2000) are as follows: definition 4. let ),...,,( 110 −= naaax be a vector in )(),( qgfaqv i n ∈ and let nb ≤≤2 . then x is called a close-closed loop burst of length b if ∃ an i , 11 −≤≤ bi such that .0...,0. 1111 ====≠ −+−+−+− ibniiibn aaaaa definition 5. the class of open loop burst as considered in definition 1 may be termed as open-closed loop bursts. definition 6. let x= ),...,,( 110 −naaa be a vector in )(qv n and nb ≤≤2 . then x is called a ct close-closed loop bursts of length b if ∃ an i , 11 −≤≤ bi such that 0≠+− ibna ; at least one of 110 ,...,, −iaaa is non-zero and 0... 11 ==== −+−+ ibnii aaa . definition 7. the class of ct open loop burst as considered in definition 3 may be termed as ct open-closed loop bursts. theorem a. an (n, k) cyclic can not detect any close-closed loop burst of length b where 12 +≤≤ kb . theorem b. the fraction of close-closed loop bursts of length b ( 12 +≤≤ kb ) that goes undetected to the total number of close-closed loop bursts in any (n, k) cyclic code is 17 = 2 132 )1)(1( )1( −− −−+− qb qq bbk . theorem c. an (n, k) cyclic code can not detect any ct close-closed loop burst of length b where 12 +≤≤ kb . theorem d. the fraction of ct close-closed loop burst of length b )12( +≤≤ kb that goes undetected to the total number of ct close-closed loop bursts in any (n, k) cyclic code is = ( ) 1)1( )1( 1 11 +−− − − −+− bqbq qq b bbk . there are of course many situations in which errors occur in the form of bursts but not all digits inside the burst get corrupted. usually, the weight of the burst lies between two numbers 1w and 2w such that 212 ww ≤≤ ≤ length of burst. such bursts are known as moderate-density bursts. moderate-density bursts with respect to close-closed loop burst are known as moderate-density close-closed loop bursts and are defined as follows: definition 8. a close-close loop burst of length b whose weight lies between 1w and 2w , bww ≤≤≤ 212 , is called a moderate-density close-closed loop burst. the development of codes which detect/correct moderate-density close-closed loop bursts can economize in the number of parity check digits required, suitably reducing the redundancy of the code or in the other words, suitably increasing the efficiency of transmission. in the second section of this paper, we obtain results similar to theorem a and b for moderate-density close-closed loop bursts whereas in the third section, we obtain results similar to theorems c and d for ct moderatedensity close-closed loop bursts. the last section viz. section 4 gives a comparison of the results obtained in section 2 and section 3. in what follows, an (n, k) cyclic code over gf(q) is taken as an ideal in the algebra of polynomials modulo the polynomial 1−nx . 18 2. moderate-density close-closed loop burst error detection in this section, we obtain results of theorems a and b for moderatedensity closeclosed loop bursts. theorem 1. an (n, k) cyclic codes can not detect any moderate-density close-closed loop burst of length b with weight lying between 1w and 2w )( 21 bww ≤≤ where 12 +≤≤ kb . proof. there is no deviation in the final conclusion of this theorem from that of theorem a because the proof is based on the length of the burst giving rise to a polynomial which is of the same degree even when the weight consideration over the burst is considered. hence the proof is omitted. q.e.d. theorem 2. the fraction of moderate-density close-closed loop bursts of length b )12( +≤≤ kb with weight lying between 1w and 2w that goes undetected to the total number of moderate-density close-closed loop bursts in any (n, k) cyclic code is = } }∑     ∑    ∑ −      −−      −− − − = − = − 〉−〈= − − + − +− 1 1 1 1 1, 1 1 1 1 1 2 1 12 112 2 2 1 1 )1(1)1(1 )1( b i w r rw rwr r r r r bk qiqib qq where }1,.{max1, 1111 rwrw −=− proof. let r(x) denote a moderate-density close-closed loop burst of length )12( +≤≤ kbb with weight w lying between 1w and 2w )( 21 bww ≤≤ . let g(x) denote the generator polynomial of the code of degree k.n − now r (x) will be of the form r(x)= )...( 111 ib nibnibn ibn xaxaax −−−++−+− +− +++ );...( 11 2 210 − −+++++ i i xaxaxaa ibnabi +−−≤≤ ,11 0, 1 ≠−ia and the number of non-zero coefficients, including 1, −+− iibn aa lies between 1w and 2w . ),()( 21 xrxrx ibn += +− say where ibniibnibn xaxaaxr −− −++−+− +++= 1 11 ...)( and ....)( 11 2 2102 − −++++= i i xaxaxaaxr let 1r be the number of non-zero coefficients in 1r (x) and 2r be the number of non-zero coefficients in 2r (x), 19 where 11 21 −≤≤ wr and 11 22 −≤≤ wr such that 2211 wrrw ≤+≤ . for any fixed value of i, let us give different values of 1r . (i) let 11 =r . then 11 ,1 221 −≤≤〉−〈 wrw ( ,122112211 rwrrwwrrw −≤≤−⇒≤+≤q also 12 ≥r )1 , 12211 rwrrw −≤≤〉−〈∴ where { }.1 ,max1 , 1111 rwrw −=〉−〈 we have then number of polynomials of type ()1()(1 −= qxr ) 0 0 )1(1 −−− qib number of polynomials of type 1 1 1,1 1 2 2 2 12 2 )1(1)1()( − − 〉−〈= − −∑       −−= r w wr r qiqxr ∴number of polynomials of type r(x) = 1 1 1,1 1 2 0 2 2 12 2 )1(1)1(1 − − 〉−〈= − −∑       −−     −− r w wr r qiqib (ii) for 21 =r we get 21 ,2 221 −≤≤〉−〈 wrw number of polynomials of type ()1()(1 −= qxr ) )1(1 1 −−− qib number of polynomials of type 1 2 1,2 1 2 2 2 12 2 )1(1)1()( − − 〉−〈= − −∑       −−= r w wr r qiqxr ∴ number of polynomials of type r(x) = 1 2 1,2 1 3 1 2 2 12 2 )1(1)1(1 − − 〉−〈= − −∑       −−     −− r w wr r qiqib continuing the computation for various values of ,...,4,31 =r we finally, have 11 221 =⇒−= rwr and number of polynomials of type =)(1 xr 2 2 2 2 )1(1)1( − − −      −−− w w qibq number of polynomials of type )(2 xr = = 11 1 1 2 2 2 )1(1)1( − = − −∑       −− r r r qiq ∴ number of polynomials of type r(x) = 2 2 )1(1 2 w w qib −      −− − 11 1 1 2 2 2 )1(1 − = − −∑       − r r r qi so, for a fixed value of i, 20 number of polynomials of type r(x) = }1 1, 1 1 1 1 1 2 12 112 2 1 2 1 1 )1(1)1(1 − − 〉−〈= − + − = − −∑       −−∑          −− r rw rwr r r w r r qiqib summing over i, we get total number of polynomials of type r(x) = { }1 1, 1 1 1 1 1 1 1 2 12 112 2 1 2 1 1 )1(1)1(1 − − 〉−〈= − + − = − − = −∑       −−∑          −−∑ r rw rwr r r w r r b i qiqib again, r(x) will go undetected if g(x) divides r(x) ⇒ r(x) =g(x)q(x) for some polynomials q(x) ⇒ )()( 21 1 xrxrx bn ++− = g(x)q(x) now, number of polynomials of type q(x) = )11( +−− bk qq (refer[3]) ∴ratio of moderate-density close-closed loop bursts that goes undetected to the total number of moderate-density close-closed loop bursts is = } }∑     ∑    ∑ −      −−      −− − − = − = − 〉−〈= − − + − +− 1 1 1 1 1, 1 1 1 1 1 2 1 12 112 2 2 1 1 )1(1)1(1 )1( b i w r rw rwr r r r r bk qiqib qq where }1 ,.{max1 , 1111 rwrw −=− hence the proof. q.e.d. special cases. (i) for ,221 === wwb the ratio obtained in the preceding theorem reduces to the ratio given in theorem b for b=2 and the ratio in each case becomes )1( 1 − − q q k . (ii) for ,21 =w the result obtained in the preceding theorem reduces to the case of low-density close-closed loop bursts considered by dass & jain (2000). (iii) for ,2 bw = the result obtained in the preceding theorem reduces to the case for high-density close-closed loop bursts, considered by dass & jain (2000). 21 3. ct moderate-density close-closed loop burst error detection in this section we extend the studies made in section 2 for ct moderate-density close-closed loop bursts. firstly, we obtain the following result, the proof of which is omitted. theorem 3. an (n, k) cyclic code can not detect any ct moderate-density closeclosed loop burst of length )12( +≤≤ kbb with weight lying between 1w and ).( 212 bwww ≤≤ we now prove the following result. theorem 4. the fraction of ct moderate-density close-closed loop bursts of length b )12( +≤≤ kb with weight lying between 1w and 2w that goes undetected to the total number of ct moderate-density close-closed loop bursts in any (n, k) cyclic code is = } }∑     ∑    ∑ −      −      −− − − = − = − 〉−〈=− +− 1 1 1 1 1,1 1 2 1 12 112 2 2 1 1 )1()1(1 )1( b i w r rw rwr r r r r bk qiqib qq where }1 ,.{max1 , 1111 rwrw −=− proof. let r(x) denote a ct moderate-density close-closed loop burst of length b )12( +≤≤ kb with weight lying between 1w and 2w )( 21 bww ≤≤ . let g(x) denote the generator polynomial of the code of degree kn − . now r(x) will be of the form r(x) = )...( 111 ib nibnibn ibn xaxaax −−−++−+− +− +++ );...( 11 2 210 − −+++++ i i xaxaxaa ibnabi +−−≤≤ ,11 0≠ and the number of non-zero coefficients, including 1, −+− iibn aa lies between 1w and 2w . ),()( 21 xrxrx ibn += +− say where ibniibnibn xaxaaxr −− −++−+− +++= 1 11 ...)( and ....)( 11 2 2102 − −++++= i i xaxaxaaxr let 1r be the number of non-zero coefficients in )(1 xr and 2r be the number of non-zero coefficients in )(2 xr , where 11 21 −≤≤ wr and 11 22 −≤≤ wr 22 such that 2211 wrrw ≤+≤ . for any fixed value of i, let us give different values of 1r . (i) let 11 =r . then 11 ,1 221 −≤≤〉−〈 wrw and number of polynomials of type ()1()(1 −= qxr ) 0 0 )1(1 −−− qib number of polynomials of type 2 2 12 2 )1()( 1 1,1 2 r w wr r qixr −∑       = − 〉−〈= ∴ number of polynomials of type r(x) = 2 2 12 2 )1()1(1 1 1,10 r w wr r qiqib −∑       −     −− − 〉−〈= (ii) let 21 =r we get 21 ,2 221 −≤≤〉−〈 wrw number of polynomials of type ()1()(1 −= qxr ) )1(1 1 −−− qib number of polynomials of type 2 2 12 2 )1()( 2 1,2 2 r w wr r qixr −∑       = − 〉−〈= ∴ number of polynomials of type r(x) = 2 2 12 2 )1()1(1 2 1,2 2 1 r w wr r qiqib −∑       −     −− − 〉−〈= continuing the computation for various values of ,...,4,31 =r we finally, have 11 221 =⇒−= rwr and number of polynomials of type =)(1 xr 2 2 2 2 )1( 1 )1( − − −      −−− w w qibq number of polynomials of type )(2 xr = 2 2 2 )1( 1 1 r r r qi −∑       = number of polynomials of type r(x) = 1 2 2 2 )1(1 − − −      −− w w qib 2 2 2 )1( 1 1 r r r qi −∑       = so, for a fixed value of i, number of polynomials of type r(x) = }212 112 2 1 2 1 1 )1()1(1 1, 1 1 1 r rw rwr r r w r r qiqib −∑       −∑          −− − 〉−〈= − = − summing over i , we get total number of polynomials of type r(x) = { }212 112 2 1 2 1 1 )1()1(1 1, 1 1 1 1 1 r rw rwr r r w r r b i qiqib −∑       −∑          −−∑ − 〉−〈= − = − − = again, r(x) will go undetected if g(x) divides r(x) ⇒ r(x) = g(x)q(x) for some polynomials q(x) 23 ⇒ )()( 21 xrxrx ibn ++− = g(x)q(x) now, number of polynomials of type q(x) = )11( +−− bk qq (refer[3]) ∴ratio of moderate-density close-closed loop bursts that goes undetected to the total number of moderate-density close-closed loop bursts is = }∑     ∑    ∑ −      −      −− − − = − = − 〉−〈=− +− 1 1 1 1 1,1 1 2 1 12 112 2 2 1 1 )1()1(1 )1( b i w r rw rwr r r r r bk qiqib qq where }1,.{max1, 1111 rwrw −=− hence the proof. q.e.d. special cases. (i) for ,221 === wwb the ratio obtained in the preceding theorem reduces to the ratio given in theorem b for b=2 and the ratio in each case becomes )1( 1 − − q q k . (ii) for ,21 =w the result obtained in the preceding theorem reduces to the case of low-density close-closed loop bursts considered by dass & jain (2000). (iii) for ,2 bw = the result obtained in the preceding theorem reduces to the case for high-density close-closed loop bursts, considered by dass & jain (2000). 4. comparative study in this section, we present the comparison of the results obtained in section 2 and section 3 viz. theorem 2 and theorem 4. the comparison has been presented in the form of a table by taking specific values of b, 1w and 2w in the binary case. for 221 === wwb , both definitions viz. of moderate-density close-closed loop burst and of ct moderate-density close-closed loop burst coincide. therefore, we start comparing the results for b=3, and onwards. 24 table [ ]2=q ____________________________________________________________________ moderate-density close-closed ct moderate-density close-closed loop bursts loop bursts (theorem 2) (theorem 4) ________________________________________________________________ [ ]2,2;3 21 === wwb 00.64 00.33 50.12 = = = k k k 00.4 00.2 00.1 [ ]3,2;3 21 === wwb 00.34 50.13 75.02 = = = k k k 40.2 20.1 60.0 [ ]3,3;3 21 === wwb 00.64 00.33 50.12 = = = k k k 00.6 00.3 50.1 ________________________________________________________________ [ ]2,2;4 21 === wwb 33.95 66.44 33.23 = = = k k k 66.4 33.2 16.1 [ ]3,2;4 21 === wwb 11.35 55.14 77.03 = = = k k k 00.2 00.1 50.0 [ ]4,2;4 21 === wwb 33.25 66.14 58.03 = = = k k k 64.1 82.0 41.0 [ ]3,3;4 21 === wwb 66.45 33.24 16.13 = = = k k k 50.3 75.1 87.0 25 [ ]4,3;4 21 === wwb 11.35 55.14 77.03 = = = k k k 54.2 27.1 63.0 [ ]4,4;4 21 === wwb 33.95 66.44 33.23 = = = k k k 33.9 66.4 33.2 ________________________________________________________________ [ ]2,2;5 21 === wwb 00.156 50.75 75.34 = = = k k k 00.6 00.3 50.1 [ ]3,2;5 21 === wwb 75.36 87.15 93.04 = = = k k k 00.2 00.1 50.0 [ ]4,2;5 21 === wwb 14.26 07.15 53.04 = = = k k k 33.1 66.0 33.0 [ ]5,2;5 21 === wwb 87.16 93.05 46.04 = = = k k k 22.1 61.0 30.0 [ ]3,3;5 21 === wwb 00.56 50.25 25.14 = = = k k k 00.3 50.1 75.0 [ ]4,3;5 21 === wwb 50.26 25.15 62.04 = = = k k k 71.1 85.0 42.0 26 [ ]5,3;5 21 === wwb 14.26 07.15 53.04 = = = k k k 53.1 76.0 38.0 [ ]4,4;5 21 === wwb 00.56 50.25 25.14 = = = k k k 00.4 00.2 00.1 [ ]5,4;5 21 === wwb 75.36 87.15 93.04 = = = k k k 15.3 57.1 78.0 [ ]5,5;5 21 === wwb 00.156 50.75 75.34 = = = k k k 00.15 50.7 75.3 note. the fractions have been calculated up to 2 decimal places. acknowledgement. the second author wishes to thank university grants commission for providing grant (vide ref. no. f-13-3/99(sr-i)) under minor research project to carry out this research work. references 1. c.n. campopiano (1962), bounds on burst error correcting codes, ire trans., it-8, pp. 257-259. 2. r.t. chien and d.t. tang (1965), on definitions of a burst ibm j. res. & devlop. , july pp. 292-293. 3. b.k. dass and sapna jain (2000), on a class of closed loop bursts error detecting codes, international journal of nonlinear sciences and numerical simulation 2, pp. 305306, 2001. 4. b.k. dass and sapna jain (2000), low-density close-closed loop burst error detecting codes, accepted for publication in korean journal of computational and applied mathematics. 5. b.k. dass and sapna jain (2000), high-density close-closed loop burst error detecting codes, submitted. 6. p. fire (1959), a class of multiple-error-correcting binary codes for non-independent errors, sylvania report rsl-e-2, sylavania recon. sys. lab., mountain view, california. 7. w.w. peterson (1961), error correcting codes, cambridge, mass: the m.i.t. press. microsoft word bkdas 90 on recognition of cipher bit stream from different sources using majority voting fusion rule shri kant*, veena sharma*, b. k. dass** abstract in the present paper, majority-voting rule has been investigated for its possible application in cryptological sciences. a novel approach is proposed to address the complex identification problem of overlapping classes. the method for representing patterns using different measurements has been discussed and the majority voting rule is used to fuse the results obtained in different measurement spaces. the proposed approach is quite natural and simple to implement in comparison with usual fusion strategies. the scheme has been implemented for three-class problem and results were tabulated and presented graphically. keywords decision fusion, representation space, pattern space, expert classifiers, majority logic, stream ciphers and cryptology. -------------------------------------------------------------------------------------- address for correspondence: scientific analysis group, defence r & d organization, metcalfe house complex, delhi-ll0054 tel no.: (011) 23813862 email: shrikant@scientist.com, shrikant.ojha@gmail.com *e**deptt. of mathematics faculty of mathematical sciences university of delhi delhi-110054 *scientific analysis group defence r & d organization metcalfe house complex delhi-ll0054 91 1. introduction identification of cipher bit streams generated from different sources is the primary step for a cryptanalyst. it requires cipher bit stream to be represented in the form of a pattern vector. in the measurement space, the analyst can take various measurements for patterns. based on specific perception and scale, patterns are represented as points in some multidimensional feature space. the feature space is partitioned using the discriminant functions made on the basis of patterns of known classes, referred as training/learning patterns. the performance of the discriminant function is measured by categorization of independent patterns, known as test patterns, to their own partitions. a higher percentage of correct classification of the patterns in the test set indicates a better discriminator. the fundamental goal of an analyst is to arrive at the highest probable correct classification of a given set of patterns. this objective leads to the design and development of different type of classifiers to solve a particular pattern recognition problem. here, the accuracy in classification attained by different classifiers may be different. also, the set of patterns correctly classified by one classifier may differ with the set of patterns correctly classified by another classifier. thus, instead of searching for the best among the set of classifiers, it is found better to combine the decisions of individual classifiers. by applying a combination strategy to the set of classifiers such that the participating classifiers work complementary to each other, we are likely to get a classification rate better than that of a single best classifier. various combination strategies or decision fusion techniques have been proposed and studied by many researchers. lam and suen ([1]:1997), kittler, et. al. ([2]:1998), alkoot, et. al. ([3]:1999), kuncheva, et. al. ([4]:2001), chen and cheng ([5]:2001) and alexandre, et. al. ([6]:2001) etc. made a detailed study of different aspects of these combination strategies. 92 we first give a brief description of these fusion schemes. let x be a pattern which is to be assigned to one of m possible classes w1, w2, …, wm with the help of anyone of m individual classifiers. each classifier approximates the a posteriori probability p(wi/x), i= 1, 2, … , m, that is the probability that pattern x belongs to class wi, given that x was observed. a classifier assigns x to class wk if )1( )/(max)/( ,...,1 −−−= − xwpxwp i mi k for convenience, let we denote the a posteriori probabilities computed by classifier cj by pj(wi/x), where j = 1, 2, …, m and i = 1, 2, …, m. it is assumed that these estimates of a posteriori probabilities given by individual classifiers are independent and identically distributed according to some pre-assumed distribution function. here, aim is to get improved estimates p(wi/x) by applying some combination rule ‘f’ to the individual estimates pj(wi/x) given by each of the m classifiers i.e. ( ) )2( m...,,2 ,1,)/(...,,)/(..., ,)/()/( 1 −−−== ixwpxwpxwpxwp imijii f pattern x is finally allocated to class wk according to the rule (1). thus the rule for decision fusion becomes ( ){ })/(...,,)/(max )/( 1 1 xwpxwpxwpifwx imi m i kk f = =∈ some prevalent decision fusion rules are the average rule, geometric mean rule, maximum rule, minimum rule, median rule, and majority vote rule. the theoretical and experimental comparative studies about the performance of decision fusion approaches have been carried out by kittler, et. al. ([2]:1998), alkoot and kittler ([3]:1999), chen & cheng ([5]:2001) and kuncheva ([7]:2002) etc., using different data sets. sensitivity to estimation errors of these schemes under different 93 assumptions and different approximations has been analyzed, kittler, et. al. ([2]:1998), alkoot and kittler ([3]:1999). it has been found that relative performance of various combination schemes changes under different conditions. the main emphasis has been given to comparison of the two basic schemes i.e. sum rule and product rule, kittler, et. al. ([2]:1998), alkoot and kittler ([3]:1999), alexandre, et. al. ([6]:2001). the sum rule is found easy to implement and less sensitive to errors than product rule, in most of the scenarios, kittler, et. al. ([2]:1998). the product rule and strategies devised from it perform better when all the experts produce small errors. further, the number of classifiers employed in fusion and number of classes in the problem also has an effect on the relative performance of different experts, alkoot and kittler ([3]:1999). in general as stated earlier, these fusion rules, with an exception of majority voting rule, use the probabilities obtained by different classifiers to take the final fused decision about the class-memberships of the patterns. these probabilities given by different classifiers are called soft decisions. on the other hand, majority-voting rule works on hard decisions. that is, in majority voting rule, different classifiers first give their respective decisions about the class-memberships called the hard decisions, and then the decision taken by maximum number of classifiers is taken as the final decision. instead of handling the probabilities, it simply works on the decisions given by different classifiers and therefore, is easiest to implement, lee and srihari ([8]:1993) and lam and suen ([1]:1997). and yet, experiments show that majority-voting rule is just as effective as other combination schemes, which are more complex in nature. also, majority-voting scheme is found to be one of the schemes, which are relatively stable. keeping all these facts into mind, we have chosen majority-voting scheme for experimentation to support our approach of fusion, which is slightly different from the usual approach. let us first formulate majority-voting scheme mathematically. 94 in majority voting rule, the individual a posteriori probabilities pj(wi/x) are used to produce hard decisions δij where ⎪⎩ ⎪ ⎨ ⎧ = =δ = otherwise 0 )x/w(pmax)x/w(p if 1 kj m 1k ij ij then we assign the pattern x to class wk if ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ δ= ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ δ ∑∑ = = = m 1j ij m 1i m 1j kj max in the literature, mostly, two-class problem have been addressed with the help of decision fusion rules, although the rules can be implemented for any number of classes and any number of features representing a pattern. however, when the number of classes increases, the computational complexity also increases and the final decision may be costly for overlapping classes. we address this difficulty by proposing in section 2, a simple and easy to implement approach, working on the basis of consensus of decisions taken in different representation spaces. section 3 presents the problem definition and a description about various representation spaces. section 4 contains the algorithm and section 5 contains details of experimentation and results. finally, in section 6 we present our observations and conclusions. 2. proposed approach for classification before discussing our approach, let us put the usual fusion approach in a form, which can be compared with proposed one. let, there are m classes and m classifiers. as discussed before, the a posteriori probabilities pij, where i = 1, 2, …, m and j = 1, 2, …, m, are computed for a given pattern x to be classified in one of the pre-specified class. 95 classes pattern x w1 w2 … wm classifier c1 p11 p12 … p1m classifier c2 p21 p22 … p2m : : : : : : … : : classifier cm pm1 pm2 … pmm table 2.1 the pattern x gets its class membership in class wi if a predefined function f as described in section 1, gives optimum value for class wi i.e. i j ),p,,p,(p )p,,p,(p mj2j1jmi2i1i ≠∀−−−>−−− f f now, instead of considering different types of classifiers, we propose to consider different representations of same set of patterns and allow a single classifier to take decision about class memberships. going this way, in spite of having only one classifier, one can have different probable decisions and can apply any of the traditional fusion schemes. further, if one have only two or very few classifiers available, then there will be more chances of having a tie instead of having a decision due to lack of majority of a single decision, specially when we are going to deal a multiclass problem. in that case, our approach presents a way to use fusion to have more authenticated decisions by considering many representations of set of patterns, according to the underlying problem. as stated before, we have chosen majority voting rule for fusion i.e. we accept the decision obtained in majority of the representation/feature spaces using a single classifier. let we have ‘r’ representation spaces to observe a pattern x in ‘r’ different ways. with the help of a classifier c, we wish to classify x in one of the pre-specified m classes. let pij, i = 1, 2, …, r and j = 1, 2, …, m be the probability for x of membership in jth class, while the ith representation is used to present the pattern. first we convert these soft decisions into hard decisions ∆ij, by allocating one class wj to the pattern x in ith representation space i.e. 96 ⎪⎩ ⎪ ⎨ ⎧ = =∆ = otherwise 0 pmaxp if 1 ik m 1k ij ij table 2.2(a) soft decisions: table2.2(a) hard decision: table2.2(b) from the table 2.2(b), it is clear that a pattern will get its class membership in class wk if ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∆= ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∆ ∑∑ = = = r 1i ij m 1j r 1i ik max classes classifier c w1 w2 --wm representation x1 p11 p12 --p1m representation x2 p21 p22 --p2m | | | | | | -- | | representation xr pr1 pr2 --prm classes classifier c w1 w2 --wm representation x1 ∆11 ∆12 --∆1m representation x2 ∆21 ∆22 --∆2m | | | | | | -- | | representation xr ∆r1 ∆r2 --∆rm 97 3. problem definition and feature computation in the present day communication scenario, any type of information viz. visual scenes, voice and text, is stored and communicated digitally. the authorized recipient at the other end recovers the same with precise accuracy and correctness. the adversary may intercept, record and retrieve all the plain transmission with some trial and error, using available means and technology. but, he will not be able to make any sense of it if the information is transmitted after encipherment by applying some cryptographic techniques. to experiment with the said problem, enciphered bit streams of scenes, voice and text have been generated from three independent stream ciphers respectively. the stream ciphers used are clock-controlled shift registers, geffe generator and cascade of linear shift registers with nonlinear combiner. the details are described in geffe ([9]:1973), rueppel ([10]:1986), schneier ([11]:1996), kumar ([12]:1997) and menezes et. al. ([13]:1997). we consider each fixed length sample (now onwards referred as a message) of enciphered bit stream as a pattern. these patterns require their representation in pattern space as multidimensional feature vectors so that these can become suitable for further analysis. the process of feature extraction from each message to form a suitable mathematical pattern is like an art and this is improved by experimentation and practice. next, we will describe the procedure followed by us to extract significant feature vectors from these bit streams. 3.1 mathematical representation let us denote the samples of enciphered binary streams by mlk, where l = 1, 2, 3 and k = 1, 2, …, n. in this representation, l =1 stands for encrypted scene, l =2 stands for encrypted voice and l =3 stands for encrypted text. the number of messages taken from each respective source is ‘n’. all messages are assumed to be of a sufficiently long length of ‘c’ bits, where 1000 ≤ c ≤ 5000 bits usually. from each message mlk, binary pattern word (i.e. small blocks of bits) of a suitable fixed length ‘b’ are read, 98 where b = 5 or 7 bits etc. now, these binary words can be read from a message in two (overlapping and non-overlapping) ways. in overlapped reading, we proceed bit by bit i.e. first pattern word starts from the first bit of the message and second pattern word starts from the second bit of the message and so on. and in non-overlapped reading, we move block by block i.e. we divide the whole message into blocks of given pattern word length and then these blocks are taken as pattern words. one can take a pattern word of any length depending upon the prior knowledge of assignable character for a fixed group of bits. for a binary pattern word of length ‘b’, we have possibility of 2b different words. if we do a certain computation on given message, for each of these 2b possible words, then we will have 2b computed quantities. these 2b quantities or measures together will constitute a 2b-dimensional feature vector. so, by varying pattern word length ‘b’, we will get feature vectors of different dimensions from a particular message. for example, for b=5 and b=7, 32dimensional and 128-dimensional feature vector will be obtained respectively. in each case, we get different feature space with different components and different dimensions. following this method, we can have different representations of a particular raw pattern. in a message mlk, number of total occurrences of pattern words,‘t’ is given by ⎪ ⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎨ ⎧ =⎥⎦ ⎤ ⎢⎣ ⎡ =−− = )case goverlappin-onn ( t b c (3)-- )case verlappingo ( t1)(bc t d c in the subsequent sub-sections, we present further, the two different types of computations done to compute the feature vectors. in these subsections, we refer 'ith pattern word’ for binary equivalent of decimal number 'i', where 0 ≤ i ≤ 2b –1. for example, if b = 5 then dimension of the vector = 99 25 =32 and the indices of the vector will vary from 0 to 31. it can be better understood with the help of table given below. binary word equivalent decimal feature component 00000 0 f[0] 00001 1 f[1] | | | 00111 7 f[7] 01000 8 f[8] | | | 11111 31 f[31] 3.1.1 percentage frequency vector (pfv): first, we compute the frequency vector f. the ith component of the vector f, 'fi' is the frequency of ith pattern word in a particular message, where 0 ≤ i ≤ 2b -1. so, ith component of the percentage frequency vector p, 'pi' is the percentage of the ith component of the frequency vector f. each component 'pi' where 0 ≤ i ≤ 2b –1, can be computed as ⎪ ⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎨ ⎧ = × = × = )case goverlappin-onn ( p t 100f (4)-- )case verlappingo ( p t 100f p d i d i c i c i i 3.1.2 average distance vector (adv): in any message, any pattern word can occur more than once. based on these different occurrences of the same pattern word and distances between them, we compute average distance vector. each occurrence in 100 the message is marked by the first bit of a pattern word. let pi,j be the position of the first bit of ith pattern word, in jth occurrence in the message then the ith component of the average distance vector a is defined as ( ) )5( 2 1,, −−− − = ∑ = − i f j jiji i f pp a i as defined above, fi is the number of times the ith pattern word occur in the message sequence, where 0 ≤ i ≤ 2b -1. now, from each given message bit-stream for a fixed pattern word length, four types of feature vectors can be generated with the help of (4) and (5) depending upon different choices, as shown below. one message (or raw pattern) pattern word length: ‘b’ bits reading ways: overlapping non-overlapping vectors: pfv. adv pfv adv in further discussion, we will use the following notations described in table 3 to refer the four possible cases. the notations have been taken as first letter of each words near the arrow in small letters. 101 notation pattern word length: b= 5 or7 way of reading type of vector bop ‘b’ bits overlapping percentage frequency boa ,, ,, average distance bnp ,, nonoverlapping percentage frequency bna ,, ,, average distance table 3 4. algorithm: suppose there are m classes w1, w2, ---, wm and n is the total number of raw patterns taken from each class. thus, in total we have mn patterns. let we denote the number of patterns to be taken for learning from each class by l. the remaining (n-l) patterns will be used for testing. let we present all the patterns in ‘r’ different representations taking different combinations of choices i.e. varying the pattern word length, way of reading and type of vector. the dimension of each pattern in any representation will depend upon the pattern word length chosen for that representation. let we denote the dimension in the pth representation by np, where p = 1, 2, ---, r. step 1: make a set of raw patterns (or message bit streams), keeping the patterns of all classes together. from this set, further compute ‘r’ sets by converting these patterns into vectors in ‘r’ different representations. step 2: select one of the classification technique such as minimum distance classifier, bayes classifier or perceptron algorithm etc. as discussed in tou and gonzalez ([14]:1974), bow ([15]:1984), kant and sharma ([16]:2000) etc. step 3: pass each representation of patterns to the classifier one by one i.e. for p = 1, 2, ---, r, apply classification algorithm to pth 102 representation which is a set containing np-dimensional vectors. store class allotted to each pattern in each representation. step 4: set j = 1. step 5:for jth pattern, initialize count [ i ] = 0, where i = 1,2,---, m. step 6: set p = 1. step 7: if jth pattern in pth representation goes to class wk, increment the count [ k ] by 1. step 8: repeat step 7 for p = 2, ---, r. step 9: finally, assign jth pattern to class wk if { }]i[countmax]k[count m 1i= = if there are more than one class such that the quantity count [k] of these classes are equal to the maximum value computed in the equation, then there arise uncertainty about the final classmembership of the pattern under consideration. in that case, the pattern is kept into the category of rejection. step 10: repeat step 5 to step 9 for j = 2, ---, mn. 5. experimentation and results: as discussed earlier, we have experimented with the problem of identification among the encrypted bit streams of scenes, speech and the text respectively. to deal with this three-class problem, we have first computed different suitable representations from these bit streams. each representation is a set of vectors computed from the bit streams. various techniques have been applied to classify the patterns for each 103 representation. here, we are showing the classification results by two classifiers namely, maximum likelihood classifier and minimum distance classifier for each individual representation of patterns. and finally we have shown the results obtained by proposed fusion approach. . maximum likelihood classifier: in the tables 5.1(a) to 5.1(d), we have shown the percentage self-classification given by the maximum likelihood classifier for the four different representations of the same set of patterns. notation used for each representation can be understood with the help of table 3. we have taken 150 patterns for learning of the classifier from each of the class. table 5.2 shows the results obtained by fusion of classification results in individual representations. in table 5.2, we have included the percentage of patterns, which cannot be allocated to any class due to uncertainty in deciding the final class membership. representation: ‘5na’ % classification encrypted scene encrypted speech encrypted text encrypted scene 82.67 10 7.33 encrypted speech 8 84.67 7.33 encrypted text 8.67 13.33 78 table 5.1(a) representation: ‘5oa’ % classification encrypted scene encrypted speech encrypted text encrypted scene 82.67 9.33 10 encrypted speech 16.67 80 3.33 encrypted text 13.33 8.67 78 table 5.1(b) representation: ‘7na’ % classification encrypted scene encrypted speech encrypted text encrypted scene 98 0 2 encrypted speech 1 97 2 encrypted text 0.67 0 99.33 table 5.1(c) 104 representation: ‘7oa’ % classification encrypted scene encrypted speech encrypted text encrypted scene 97.33 1.33 1.33 encrypted speech 1 97 2 encrypted text 0 4.67 95.33 table 5.1(d) proposed approach % classification encrypted scene encrypted speech encrypted text rejecte d encrypted scene 96 0 0 4 encrypted speech 0 98.67 0 1.33 encrypted text 0 0 96 4 table 5.2 in tables 5.1(a) to 5.1(d), we observe some wrong classifications i.e. the percentage of patterns, which are misclassified to other classes to whom they do not belong actually. but in table 5.2, we can see that there are no wrong classifications among classes, though we have some rejections here. this means that misclassification occurred in case of individual representations is somewhat corrected by our approach of fusion. and the patterns, which cannot be still correctly classified due to lack of consensus, are shifted to rejection category. knowing that a misclassification is costly than a rejection, we found our classification approach to be advantageous. this phenomenon is illustrated in the graph(1) displayed next. in the graph, three series are plotted to show the percentage of number of misclassified patterns in each of the three classes. in each series, the classification results obtained in individual representations and by proposed fusion approach are compared. it is clear from the graph that using the proposed approach of fusion, we get a decrease to zero in percentage of misclassification in each of the class. 105 g r a p h ( 1 ) : c o m p a r i s i o n o f % m i s c l a s s i f i c a t i o n b y m a x i m u m l i k e l i h o d c l a s s i f i e r 0 5 1 0 1 5 2 0 2 5 5 n a 5 o a 6 n a 6 o a p r o p o s e d s c h e m e r e p r e s e n t a t i o n s % m is cl as si fic at io n e n c r y p t e d s c e n e e n c r y p t e d s p e e c h e n c r y p t e d t e x t minimum distance classifier: the percentage self-classification for different representation of patterns with minimum distance classifier has been summarized in table 5.3(a) to 5.3(f). after fusing the classification results in these six representations, we get improved results as shown in table 5.4. here also, we observe that by using fusion there is a great decrement in number of misclassified patterns, in each of the class. the patterns, which cannot be allocated to any class due to a tie of votes, are kept in rejection category representation: ‘5oa’ % classification encrypted scene encrypted speech encrypted text encrypted scene 34 31.33 34.67 encrypted speech 27.33 48 24.67 encrypted text 25.33 30.67 44 table 5.3(a) 106 representation: ‘5na’ % classification encrypted scene encrypted speech encrypted text encrypted scene 55.33 18 26.67 encrypted speech 28.67 50 21.33 encrypted text 28.66 22 49.33 table 5.3(b) representation: ‘7na’ % classification encrypted scene encrypted speech encrypted text encrypted scene 52.67 24 23.33 encrypted speech 20 60 20 encrypted text 26 22 52 table 5.3(c) representation: ‘7na’ % classification encrypted scene encrypted speech encrypted text encrypted scene 66.67 20.67 12.67 encrypted speech 21.33 55.33 23.33 encrypted text 20.67 20 59.33 table 5.3(d) representation: ‘5np’ % classification encrypted scene encrypted speech encrypted text encrypted scene 51.33 25.33 23.33 encrypted speech 24.67 47.33 28 encrypted text 28 26.67 45.33 table 5.3(e) representation: ‘7np’ % classification encrypted scene encrypted speech encrypted text encrypted scene 66.67 16.67 16.67 encrypted speech 16.67 64.67 18.67 encrypted text 20 16 64 table 5.3(f) 107 proposed approach % classification encrypted scene encrypted speech encrypted text rejected encrypted scene 62.67 12.67 12 12.67 encrypted speech 8 60 11.33 20.67 encrypted text 14.67 10.67 59.33 15.33 table 5.4 with minimum distance classifier, we are able to get more than 55% classification consistently for each class. these results of classification may not be very high for the practical application, but this consistency is extremely useful from a cryptanalysis point of view. the graph(2) plotted to compare the results by minimum distance classifier, in different representations and those obtained by fusion, presents a similar trend as shown by maximum likelihood classifier. as compared to individual representations, the proposed fusion approach gives the least number of wrongly classified patterns. 108 graaph(2): comparison of % misclassification by minimum distance classifier 0 10 20 30 40 50 60 70 5oa 5na 6na 7na 5np 7np proposed approach representations % m is cl as si fic at io n encrypted scene encrypted speech encrypted text using both the classifiers, we have tested several sets of patterns from each class, and the test-classification is also found to be quite encouraging. 6. observations and conclusion: the experimentation done for the present work has given us enough idea about handling the problem of discrimination among various random sources. the proposed idea is quite general in nature and can be applied to other kind of classification problem as well, if it is possible to compute different measurements for the same set of patterns. also, experimentation can be done for any number of classes as described in algorithm, instead of restricting to a three class problem. according to the nature of underlying problem and knowledge of significant features, different measurements may be computed to get different representations of 109 patterns. for our problem, we adhered to the most suitable representations of patterns where the classification is more transparent. dealing with the said problem, the following observations and constraints are found to be important: 1. while assigning class membership to a pattern in each of the representation space, the classifier has three possibilities. either the pattern will be correctly classified, wrongly classified or the classifier will remain uncertain about the class membership of the pattern. the third possibility of uncertainty of decision arises due to tie between values of discriminating function for the possible classes. this situation of neutral position of the classifier leads to no decision or rejection, i.e. classification is neither correct, nor wrong. here, for convenience, we have considered only those representation spaces in which classifier had only two alternatives, of being correct or wrong and no rejections. again, while taking the final decision by fusion as proposed, there may be cases of no consensus. this situation of uncertainty in deciding final class membership of a pattern leads to a rejection. we have kept these patterns in a separate category. 2. after applying proposed fusion, it has been observed that there are no wrong classifications with maximum likelihood classifier, though there are few patterns, which cannot be allocated to any class and have been kept in no decision category. minimum distance classifier shows the similar trend with less wrong classifications by fusion as compared to those obtained in individual representations. here also, the patterns about which the classifier is not certain are kept in rejected category. for both the classifiers, graphs have also been plotted to compare the percentage misclassification of patterns, in individual representations and after fusion by proposed approach. it is clear from the graphs by using the proposed fusion approach that we are getting reduced percentage of misclassification, which is the merit of our approach. 110 3. final results, obtained by fusion by proposed approach, are better than the results obtained by using single representation spaces. 4. as we have discussed earlier, each of the representation space has dimension as np, p = 1, 2, ---, r. the general observation is that we obtain consistently better performance when the size of learning set is more than 5x np. acknowledgement we would like to express our sincere gratitude and deep veneration to dr. p k saxena, director sag and dr. laxmi narain sc. ‘f’ for giving us this opportunity to carry out the present work. we are also thankful to ms. neelam verma, sc. ‘e’ for her constructive suggestion made during the preparation this paper. references: [1]. lam, l., suen, c. y., 1997. application of majority voting to pattern recognition: an analysis of its behaviour and performance. ieee transactions on systems, man, and cybernetics 27(5), 553-568. [2]. kittler, j., hatef, m., duin, r., matas, j., 1998. on combining classifiers. ieee trans. pami 20(3), 226-239. [3]. alkoot, f. m., kittler, j., 1999. experimental evaluation of expert fusion strategies. pattern recognition letters 20, 1361-1369. [4]. kuncheva, l., bazdek, j., duin, r., 2001. decision templates for multiple classifier fusion: an experimental comparison. pattern recognition letters 34(2), 299-314. [5]. chen, d., cheng, x., 2001. an asymptotic analysis of some expert fusion methods. pattern recognition letters 22, 901-904. 111 [6]. alexandre, luis a., campilho, aurelio c., kamel, m., 2001. on combining classifiers using sum and product rules. pattern recognition letters 22, 1283-1289. [7]. kuncheva, ludmilla i., 2002. a theoretical study on six classifier fusion strategies. ieee trans. pami 24(2), 281-286. [8]. lee, d. s., srihari, s. n., 1993. handprinted digit recognition: a comparison of algorithms. in proc. 3rd int. workshop frontiers handwriting recognition. buffalo, ny, pp. 153-162. [9]. geffe, p. r., 1973. how to protect data with ciphers that are really hard to break. electronics, 46(1), 99-101. [10]. rueppel, r. a., 1986. analysis & design of stream ciphers. springerverlag. [11]. schneier, b., 1996. applied cryptography, second edition john wiley & sons, inc. [12]. kumar, i. j., 1997. cryptology: system identification and key clustering. agean park press, ca, usa. [13]. menezes, a. j., van oorschot, p. c., vanstone, s. a., 1997. handbook applied cryptography. crc press, boca raton. [14]. tou, j. t., gonzalez, r. c., 1974. pattern recognition principles. addison-wesley publishing company. [15]. bow, sing-tze, 1984. pattern recognition: application to large dataset problems. marcel dekker, inc., new york & basel. [16]. kant, s., sharma, v., 2000. discrimination among various type of encrypted bitstream. international conference on quality reliability and information technology, 21-23 dec, new delhi. ratio mathematica volume 44, 2022 on the study of edge monophonic vertex covering number k. a. francis jude shini* s. durai raj† x. lenin xaviour‡ a. m. anto§ abstract for a connected graph g of order n ≥ 2, a set s of vertices of g is an edge monophonic vertex cover of g if s is both an edge monophonic set and a vertex covering set of g. the minimum cardinality of an edge monophonic vertex cover of g is called the edge monophonic vertex covering number of g and is denoted by 𝒎𝒆𝜶(𝑮). any edge monophonic vertex cover of cardinality 𝒎𝒆𝜶(𝑮) is a 𝒎𝒆𝜶(𝑮)-set of g. some general properties satisfied by edge monophonic vertex cover are studied. keywords: monophonic set; edge monophonic set; vertex coveringset; edgemonophonic vertex cover; edge monophonic vertex covering number. 2010 ams subject classification: 05c12**. *research scholar, reg no: 20213132092001, department of mathematics, (pioneer kumaraswamy college, nagercoil-629003, tamil nadu, india.); shinishini111@gmail.com †associate professor and principal, department of mathematics, (pioneer kumaraswamy college, nagercoil-629003, tamil nadu, india.); durairajsprinc@gmail.com ‡assistant professor, department of mathematics, (nesamony memorial christian college, marthandam629165, tamil nadu, india.); leninxaviour93@gmail.com §assistant professor, department of mathematics, (st. alberts college(autonomous), ernakulam, kochi, india.); antoalexam@gmail.comaffiliated to manonmaniamsundaranar university, abishekapatti, tirunelveli – 627 012, tamilnadu, india. ** received on june 10 th, 2022. accepted on sep 1st, 2022. published on nov30th, 2022. doi: 10.23755/rm.v44i0.907. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 197 mailto:shinishini111@gmail.com mailto:durairajsprinc@gmail.com mailto:leninxaviour93@gmail.com mailto:antoalexam@gmail.com k. a. francis jude shini, s. durai raj, x. lenin xaviour and a. m. anto 1. introduction by a graph g=(v,e) , we mean a finite undirected connected graph without loops and multiple edges. the order and size of g are denoted by n and m respectively. also 𝛿(𝐺) is the minimum degree in a graph g. for basic graph theoretic terminology, we refer to harary[7]. the distance d(u,v) between two vertices u and v in a connected graph g is the length of a shortest u-v path in g(1) for a vertex v of g, the eccentricity e(v) is the distance between v and a vertex farthest fromv. the minimum eccentricity among the vertices of g is the radius, rad g and the maximum eccentricity is its diameter, diamg. the neighbourhood of a vertex v of g is the set n(v) consisting of all vertices which are adjacent with v. a vertex v is a simplicial vertex or an extreme vertex of g if the subgraph induced by its neighbourhood n(v) is complete. a caterpillar is a tree of order 3 or more, the removal of whose end vertices produces a path called the spine of the caterpillar. a diametralpath of a graph is a shortest path whose length is equal to the diameter of the graph. a tree containing exactly two non-pendent vertices is called a doublestar denoted by 𝑆𝐾1𝑘2 where 𝑘1 and 𝑘2 are the number of pendent vertices on these two non-pendent vertices. a graph g is called triangle free if it does not contain cycles of length 3. a set of vertices no two of which are adjacent is called an independentset. by a matching in a graph g, we mean an independent set of edges of g. a maximalmatching is a matching m of a graph g that is not a subset of any other matching. the independencenumber of is the maximum number of vertices in an independent set of vertices of g. a geodeticset of is a set (g) such that every vertex of is contained in a geodesic joining some pair of vertices in s. the geodeticnumber of is the minimum cardinality of its geodetic sets and any geodetic set of cardinalities is a minimumgeodeticset or a geodeticbasis or a -set of g. the geodetic number of a graph was introduced in [2, 8] and further studied in [3-5]. a chord of a path is an edge joining two non-adjacent vertices of p. a path is called a monophonic path if it is a chordless path. a set of vertices of is a monophonic set of if each vertex of lies on an monophonic path for some . the minimum cardinality of a monophonic set of is the monophonicnumber of and is denoted by . any monophonic set of cardinalities is a minimummonophonicsetoramonophonicbasisora -set of g. the monophonic number of a graph was studied and discussed in [9, 12]. a set of vertices in is called an edgemonophonicset of if every edge of lies on a monophonic path joining some pair of vertices in and the minimum cardinality of an edge monophonic set is the edgemonophonicnumber of g. an edge monophonic set of cardinalities is called an -set of g. the edge monophonic number of a graph was introduced in and further studied in [10]. a subset is said to be a vertexcoveringset of if every edge has at least one end vertex in s. a vertex covering set of with the minimum cardinality is called a 198 on the study of edge monophonic vertex covering number minimumvertexcoveringset of g. the vertexcoveringnumber of is the cardinality of any minimum vertexcovering set of g. it is denoted by vertex covering number was studied in [14]. for a connected graph of order , a set of vertices of is anmonophonicvertexcover of if is both a monophonic set and a vertex covering set of g. the minimum cardinality of ae monophonic vertex cover of is called the monophonicvertexcoveringnumber of and is denoted by . any monophonic vertex cover of cardinality is a -set of g. a subset is a dominatingset if every vertex in v-s is adjacent to at least one vertex in s. the minimum cardinality of a dominating set in a graph is called the dominating umber of and denoted by . the dominating number of a graph was studied in [6]. a set of vertices of is said to be monophonicdominationset if it is both a monophonic set and a dominating set of g. the minimum cardinality of a monophonic domination set of is called a monophonicdominationnumber of and denoted by . the monophonic domination number was studied in [11]. a set of vertices of a graph is an edgemonophonicdominationset if it is both edge monophonic set and a domination set of g. the minimum cardinality of an edge monophonic domination set of is called an edgemonophonicdominationnumber of and denoted by . the edge monophonic domination number was studied in [13]. the following theorems will be used in the sequel. theorem 1.1. [10] every extreme vertex of a connected graph belongs to every edge monophonic set of g. in particular, each end vertex of belongs to every edge monophonic set of g. theorem 1.2. [10] let be a connected graph with cut-vertices and be an edge monophonic set of g. if is a cut-vertex of , then every component of g-v contains an element of s. 2.the edge monophonic vertex coverofa graph definition 2.1. let be a connected graph of order 2. aset of vertices of is an edgemonophonicvertexcover of if is both an edge monophonic set and a vertex cover of g. the minimum cardinality of an edge monophonic vertex cover of is called the edgemonophonicvertexcoveringnumber of and is denoted by . any edge monophonic vertex cover of cardinality is a -set of g. example 2.2. for the graph given in figure 2.1, is a minimum edge monophonic set of so that and is a minimum edge monophonic vertex cover of so that . thus, the edge monophonic number and the edge monophonic vertex covering number of a graph are different. 199 k. a. francis jude shini, s. durai raj, x. lenin xaviour and a. m. anto figure 2.1: remark 2.3. for the graph given in figure 2.2, is a minimum monophonic vertex cover of so that and is a minimum edge monophonic vertex cover of so that . hence the monophonic vertex covering number is different from the edge monophonic vertex covering number of a graph. figure 2.2: remark 2.4. for the graph given in figure 2.3, is a minimum edge monophonic set of so that is a minimum edge monophonic dominating set of so that and is a minimum edge monophonic vertex cover of so that . hence the edge monophonic vertex covering number of a graph is different from the edge monophonic number and the edge monophonic dominating number of a graph. figure 2.3: 200 on the study of edge monophonic vertex covering number theorem 2.5. for any connected graph proof. any edge monophonic set of needs at least 2 vertices and so . from the definition of edge monophonic vertex cover of , we have, . clearly v(g) is an edge monophonic vertex cover of g. hence . thus remark 2.6. the bounds in theorem 2.5 are sharp. for the complete graph . in remark 2.3, we have, the bounds are strict in example 2.2 as here 2 remark 2.7. clearly union of a vertex covering set and an edge monophonic set of is an edge monophonic vertex cover of g. in figure 2.1, is an edge monophonic vertex cover, in figure 2.2, is an edge monophonic vertex cover and in figure 2.3, is an edge monophonic vertex cover. thus figure 2.4: for the graph in figure 2.4, we observe that is a minimum vertex cover of so that is a minimum edge monophonic set of so that and is a -set of and so theorem 2.8. each extreme vertex of belongs to every edge monophonic vertex cover of g. in particular, each end vertex of belongs to every edge monophonic vertex cover of g. proof. from the definition of -set, every -set of is a -set of g. hence the result follows from theorem 1.1. corollary 2.9. for any graph with extreme vertices, max{2, } proof. the result follows from theorem 2.5 and theorem 2.8. corollary 2.10. let be a star. then 201 k. a. francis jude shini, s. durai raj, x. lenin xaviour and a. m. anto proof. let be the centre and be the set of all extreme vertices of . clearly is a minimum edge monophonic vertex cover of by theorem 2.8. hence corollary 2.11. for the complete graph proof. since every vertex of the complete graph is an extreme vertex, by theorem 2.8, the vertex set is the unique edge monophonic vertex cover of . thus remark 2.12. the converse of corollary 2.11 need not be true. for the graph given in figure 2.5, is an -set of so that and is not complete. figure 2.5: theorem 2.13. let be a connected graph with cut-vertices and be an edge monophonic vertex cover of g. if is a cut-vertex of , then every component of g-v contains an element of s. proof. from the definition of -set, every -set of is a -set of g. hence the result follows from theorem 1.2. remark 2.14. the cut-vertex of in theorem 2.13 need not belong to s. for the graph given in figure 2.6, is aa edge monophonic vertex cover of g. here 4 is a cut-vertex which does not belong to and 3 is a cut-vertex which belong to s. figure 2.6: 202 on the study of edge monophonic vertex covering number theorem 2.15. if a and are positive integers such that , then there exists a connected graph of order with proof. we prove this theorem by considering two cases. case (i): . let . then by theorem 2.11, case (ii): . consider , the complete graph on a-l vertices …, . add new vertices ... to by joining the vertices …, to both and and the graph is shown in figure 2.7. let be the set of all extreme vertices of g. then by theorem 1.1, they must belong to every edge monophonic set. also, we observe that is a minimum edge monophonic set. also the edges of and the edges are covered by the vertices of ’. now to cover the edges , we must include at least the vertex to . hence is a minimum edge monophonic vertex cover of g. thus figure 2.7: 3.conclusions in this paper we analysed the edge monophonic vertex covering number of a graph. it is more interesting to continue my research in this area and it is very useful for further research. acknowledgements the authors are grateful to the reviewers for their valuable remarks to improve this paper. 203 k. a. francis jude shini, s. durai raj, x. lenin xaviour and a. m. anto references [1] f. buckley and f. harary. distanceingraphs. addison‐wesley, 1990. [2] f. buckley, f. harary, and l. quintas. extremal results on the geodetic number of a graph.scientiaa, 2, 1988. [3] g. chartrand, f. harary, and p. zhang. on the geodetic number of a graph. networks: aninternationaljournal, , 2002. [4] g. chartrand, g. l. johns, and p. zhang. on the detour number and geodetic number of a graph. arscombinatoria, 72:3-15, 2004. [5] g. chartrand, e. m. palmer, and p. zhang. the geodetic number of a graph: a survey. congressusnumeration, pages 37-58, 2002. [6] a. hansberg and l. volkmann. on the geodetic and geodetic domination numbers of a graph. discretemathematics, , 2010. [7] f. harary. graph theory. addison wesley publishing company. reading, ma, usa., 1969. [8] f. harary, e. loukakis, and c. tsouros. the geodetic number of a graph. mathematicalandcomputer modelling, , 1993. [9] j. john and s. panchali. the upper monophonic number of a graph. internationalj. math. combin, 4:46-52, 2010. [10] j. john and p. a. p. sudhahar. on the edge monophonic number of a graph. filomat, 26(6):1081‐1089, 2012. [11] j. john, p. a. paul sudhahar, and d. stalin. on the number of a graph. proyecciones(antofagasta), , 2019. [12] a. santhakumaran, p. titus, and k. ganesamoorthy. on the monophonic number of a graph. journalofappliedmathematics&informatics, , 2014. [13] p. a. p. sudhahar, m. m. a. khayyoom, and a. sadiquali. edge monophonic domination number of graphs. j. . inmathematics, , 2016. [14] d. thakkar and j. bosamiya. vertex covering number of a graph. mathematicstoday, 27:30 35, 2011. 204 ratio mathematica volume 43, 2022 a new form of continuity in fuzzy soft topological spaces sandhya g venkatachala rao* anil p narappanavar† p. g. patil‡ abstract the current work introduces a new class of fuzzy soft b continuous functions such as slightly b continuous, semi b continuous, pre b continuous functions and their relation with the existing fuzzy soft continuous functions in fuzzy soft topological spaces. further optimal definitions of totally b continuous functions have also been brought out in the paper. a new space such as fuzzy soft b compact space is also initiated. keywords: fuzzy soft (fs) open set, fs semi-open set, fs pre-open set, fs b-open set, fs continuous functions. 2020 ams subject classifications: 54a05, 54a40.1 *global academy of technology, bengaluru, india;sandhya.gv@gat.ac.in †global academy of technology, bengaluru, india; anilpn@gat.ac.in ‡karnatak university, dharwad, karnataka, india ; pgpatil@kud.ac.in 1received on august 23, 2021. accepted on july 15, 2022. published on september 25, 2022. doi: 10.23755/rm.v39i0.647. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. sandhya g. v., anil p. n. and p. g. patil. 1 introduction topology is significant and a significant zone of arithmetic, and it can give numerous connections between logical regions and numerical models. both mathematicians and computer scientists have concentrated on fuzzy set theory, and numerous utilizations of these have emerged throughout the long term. the soft set hypothesis has been applied to various fields with incredible achievement and rich potential for application in every engineering and sciences gambit. the idea of fuzzy soft sets is presented as a comprehensive numerical tool for managing vulnerability. in the past years, issues in the field of engineering, physics, social sciences, and medical sciences etc., in recent times involving uncertainties, cannot be dealt with crisp data. zadeh (30) in 1965 introduced a general mathematical device recognized as a ”fuzzy set” to address uncertainties. the topological structure of fuzzy sets was introduced by chang (4). to overcome the existing difficulties in fuzzy set theory, soft sets were introduced by molodtsov (9) in 1999. this theory of soft sets can be successfully applied in many directions such as game theory, riemann integration, smoothness of functions, probability theory etc. maji et al. (8) introduced the merger of fuzzy set and soft set known as a fuzzy soft set. the notion of the topological structure of fuzzy soft (fs) set was introduced by tanay and kandemir (26) in 2011 and studied further by varol and aygun (29), roy and samanta (16) and pradeep(14). continuity is the core of any topological space. the authors patil et. al. (11) and missier and rodrigo(12) many others contributed significantly to the continuous functions in topology. kharal and ahmad (7) studied mappings of fuzzy soft classes. the concept of a fuzzy soft semi-open set was introduced by a.kandil et al. (6). fuzzy soft pre-open and regular open sets were introduced by sabir hussain in 2016 (19). sabir hussain(20) has also proposed fuzzy soft semi-open and semi-permanent functions in fuzzy soft topological spaces. the concept of fs bopen sets was introduced by anil p. n. (2) in 2016. anil p. n et al. (3) has also introduced fs strongly b-continuous and perfectly b-continuous functions. fuzzy soft pre continuous functions were introduced and studied by ponselvakumari and selvi (13). sabir hussain (21) has also introduced fs locally connected spaces and the concept of fs semi pre-open set. the idea of generalized fs b open set and fs gb continuous functions were initiated by sandhya and anil p. n. (23). fuzzy soft connectedness through fs b open set was formed by rodyna (15). further abbas et al. (1) and ruth and selvam (18) contributed to the concept of fuzzy soft connectedness in 2018. ibedou and abbas (5) defined a fuzzy soft net consisting of fuzzy soft points and their convergence. this powerful tool called net a new form of continuity in fuzzy soft topological spaces is applied to study some important properties of fuzzy soft topological spaces by rui gao and jianrong wu (17) in 2018. sabir hussain (22) introduced compactness and locally compactness in fuzzy soft topological spaces. smitha and sindhu (24) introduced gb-closed and gb-open sets in intuitionistic fuzzy soft topological spaces in 2019. tingshui ping (27) investigated a few mappings on fuzzy soft topological spaces. alkouri (28) introduced a new mathematical tool called complex generalised fuzzy soft set, a combination of generalised fuzzy soft set and complex fuzzy set. parimala and karthika (10) reviewed fuzzy soft topological spaces and neutrosophic soft topological spaces in 2020. smitha and sindhu (25) studied gb-continuous functions in intuitionistic fuzzy soft topological spaces in 2021. zhi kong and lifu wang (31) applied a fuzzy soft set in decision-making problems based on grey theory in 2021. in this work, a new class of fs b continuous functions known as fs slightly b continuous, fs semi b continuous functions, fs pre b continuous, and fs totally b continuous mappings in fs topological spaces are introduced, and some of their properties are studied. further, the concept of fs b compact spaces is initiated. 2 preliminaries definition 2.1. (8) let u be the initial universe and k be the set of parameters. iu be the set of all fuzzy sets on u. let a ⊆ k and a mapping f : a → iu . a pair (f,a) is called fuzzy soft (fs) set over u. it is also denoted by fa. that is for each a ∈ a, f(a) = fa : u → i is a fuzzy set on u. example 2.1. (8) let the set of shirts be u, and the set of parameters be k. a fs set describes the attractiveness of shirts with respect to the given parameters. the set of all fuzzy sets of u is iu x = {x1,x2,x3} and k = {e1,e2,e3,e4,e5}. let e = {e1,e2,e3} be the subset of k. a fs set is denoted by (f,e) or fe . where e1= colourful, e2= bright, e3= reasonable price, e4= good quality, e5=modern (f,e) = {{0.5/x1,0.9/x2,1/x3},{0.3/x1,0.6/x2,0/x3},{0.2/x1,0.9/x2,1/x3}} describes three shirts with respect to parameters e1, e2, and e3 . the shirt x1 w.r.t e1 = colouful has a graded value 0.5 out of 1. similarly,x2 with respect to e1 has a graded value of 0.9, and x3 has 1 out of 1. . . so on. definition 2.2. (26) let τ be a collection of all fs sets over a universe u and k be a fixed parameter set. a triplet (u,τ,k) is called fuzzy soft topological space [fsts] if the following hypotheses are satisfied: i. õk, ĩk ∈ τ sandhya g. v., anil p. n. and p. g. patil. ii. arbitrary union of members of τ is a member of τ. iii. finite intersection of members of τ is a member of τ. each member of τ is called fs open set. if fk ∈ τ then 1 − fk is known as fs closed set. definition 2.3. if (u,τ,k) is fsts, then a fs set fk in u is called a i. fs semi-open (6) if fk ≤ fsclfsint(fk), fs semi-closed if fsintfscl(fk) ≤ fk . ii. fs pre-open(19) if fk ≤ fsintfscl(fk), fs pre-closed if fsclfsint(fk) ≤ fk . iii. fs b-open (2) if fk ≤ fsclfsint(fk) ∨ fsintfscl(fk). the complement of the fs b open set is fs b closed. a fs set that is both b open and b closed is called fs b clopen. and fs b open set is referred to as fsbo. iv. fs semi pre-open (20) if fk ≤ fsclfsintfscl(fk), fs semi pre-closed fsclfsintfscl(fk) ≤ fk . v. fs generalised b open(23) if fsbint(fk) ≥ gk whenever (fk) ≥ gk and gk is fs closed set in u. example 2.2. consider a fsts (u,τ,k) and k = {e1,e2} where u = {a,b}, τ = {õ, 1̃,(f1,k),(f2,k),(f3,k)} (f1,k) = {{0.6/a,0.8/b},{0.7/a,1/b}}, (f2,k) = {{0.4/a,0.4/b},{0.3/a,0.1/b}}, (f3,k) = {{0.3/a,0.3/b},{0.2/a,0.1/b}}. in (u,τ,k), (g1,k) = {{0.5/a,0.6/b},{0.4/a,0.3/b}}, is fs semi open. (g2,k) = {{0.4/a,0.3/b},{0.3/a,0.1/b}}, is fs pre open, fs b open and also fs gb open. (g3,k) = {{0.6/a,0.4/b},{0.3/a,0.2/b}}, is fs semi pre open. definition 2.4. (8) if fk is a fs set, then i. the intersection of all fs closed supersets of fk is fs closure of fk . ii. the union of all fs open subsets of fk is called fs interior of fk . definition 2.5. (2) if fk is a fs set, then i. the intersection of all fs b closed supersets of fk is fs b closure (fsbcl) of fk . ii. the union of all fs b open subsets of fk is called fs b interior (fsbint) of fk . a new form of continuity in fuzzy soft topological spaces definition 2.6. let (u,τ,k) and (v,σ,k) be fsts and f be a function from u to v . then f is said to be a i. fs continuous (7) if the inverse of every fs open set in v is fs open in u. ii. fs semi-continuous(resp.fs pre continuous) (13) if the inverse of every fs open set in v is fs semi-open (respfs pre-open)in u. iii. fs semi pre continuous (21) if the inverse of every fs open set in v is fs semi pre-open in u. iv. fs b-continuous (23) if the inverse of every fs open set in v is fs b open in u. v. fs b-irresolute (23) if the inverse of every fs b open set in v is fs b open in u. vi. fs contra b continuous (3) if the inverse of every fs open set in v is fs b closed in u. vii. fs strongly continuous (3) if the inverse of every fs set in v is fs clopen in u. viii. fs perfectly continuous (3) if the inverse of every fs open set in v is fs clopen in u. ix. fs strongly b-continuous (3) if the inverse of each fs b open set in v is fs open set in u. x. fs perfectly b-continuous (3) if for each fs b-open set in v its inverse is fs clopen in u. xi. fs gb continuous (23) if for each fs open set in v its inverse is fs gb open set in u. definition 2.7. any fsts (u,τ,k) is called i. fs discrete space (23) if every fs set is fs open in τ. ii. fs locally indiscrete space(21) if every fs open set is closed in τ . iii. fs bt1/2 space (3) if every fs b open set is fs open. iv. fs b connected (15) if there are no fs b separations of 1̃k , otherwise (u,τ,k) is said to be fs b disconnected space. definition 2.8. (15) let (u,τ,k) be a fsts. an fs b separation on 1̃k is a pair of non-null proper fs b open sets fk and gk where fk ∩ gk = 0̃k , 1̃k = fk ∪ gk . sandhya g. v., anil p. n. and p. g. patil. 3 fuzzy soft slightly b-continuous functions consider two fsts (u,τ,k), (v,σ,k) and f is a function from u to v and k is the set of parameters throughout this section. definition 3.1. a function f is said to be fs slightly continuous (fssc) if the inverse of each fs clopen set in v is fs open in u. definition 3.2. a function f is said to be fs slightly b continuous (fssbc) if the inverse of each fs clopen set in v is fs b open (fsbo) in u. example 3.1. suppose f is an identity map and u = {a,b} v = {c,d}, k = {e1,e2}, τ = {õ, 1̃,(f1,k),(f2,k)} and σ = {õ, 1̃,(g1,k),(g2,k)} (f1,k) = {{1/a,0.9/b},{0.8/a,0.8/b}}, (f2,k) = {{0/a,0.1/b},{0.2/a,0.2/b}}, (g1,k) = {{0.7/c,0.6/d},{0.5/c,0.6/d}}, (g2,k) = {{0.3/c,0.4/d},{0.5/c,0.4/d}}. the inverse images of (g1,k) and (g2,k) are fs b open sets. therefore f is fs slightly b continuous. theorem 3.1. every fs slightly continuous function is fs slightly b continuous. proof: let f be fs slightly continuous. let (g,k) be fs clopen set in v , then f−1(g,k) is fs open and hence fs b-open in u. hence f is fs slightly bcontinuous. converse need not be confirmed, as seen from the below example. in example 3.1, f−1(g1,k) and f−1(g2,k) are fs b-open sets but not fs open in u. therefore, f is slightly b continuous but not fs slightly continuous. theorem 3.2. every fs contra b continuous function is fs slightly b-continuous. proof: if f is fs contra b continuous map and (g,k) is fs clopen set in v , then f−1(g,k) is fs b open in u. hence the theorem. the reverse implication is not valid. example 3.2. let f be an fs identity map. let u = {a,b}, v = {c,d} and k = {e1,e2} and τ = {õ, 1̃,(f1,k),(f2,k)} and σ = {õ, 1̃,(g1,k),(g2,k)}, where (f1,k) = {{1/a,0.9/b},{0.8/a,0.8/b}}, (f2,k) = {{0/a,0.1/b},{0.2/a,0.2/b}}, (g1,k) = {{0.7/c,0.6/d},{0.5/c,0.6/d}}, (g2,k) = {{0.3/c,0.4/d},{0.5/c,0.4/d}} (g3,k) = {{0.2/c,0.3/d},{0.4/c,0.3/d}}. it is verified that f−1(g1,k) and f−1(g2,k) are fs b open sets but f−1(g3,k) is not fs b closed in u. thus f is fs slightly b-continuous but not fs contra b continuous. theorem 3.3. every fs b continuous function is fs slightly b continuous. proof: let (g,k) be fs clopen set in v and f be fs b continuous. then f−1(g,k) is fs b clopen in u. hence f is fs slightly b continuous. the reverse implication need not be true in general. a new form of continuity in fuzzy soft topological spaces example 3.3. consider fs identity map. let u = {a,b}, v = {c,d}, k = {e1,e2}, τ = {õ, 1̃,(f1,k),(f2,k)} and σ = {õ, 1̃,(g1,k),(g2,k)}, where (f1,k) = {{0.5/a,0.4/b},{0.3/a,0.4/b}}, (f2,k) = {{0.3/a,0.3/b},{0.2/a,0.3/b}}, (g1,k) = {{0.4/c,0.5/d},{0.4/c,0.6/d}}, (g2,k) = {{0.6/c,0.5/d},{0.6/c,0.4/d}}, (g3,k) = {{0.4/c,0.5/d},{0.3/c,0.3/d}}. since the inverse of fs clopen sets (g1,k) and (g2,k) are fs b open sets in u, but f is fs slightly b continuous and f−1(g3,k) is not fs b open in u. hence, f is not fs b continuous. theorem 3.4. composition of fs slightly b continuous functions need not be fs slightly b continuous. example 3.4. let f : (u,τ,k) → (v,τ ′,k) and g : (v,τ ′,k) → (w,σ,k) be fs identity mappings. so, g ◦ f : (u,τ,k) → (w,σ,k) is also fs identity map. let u = {a,b}, v = {c,d}, w = {g,h}, k = {e1,e2}, τ = {õ, 1̃,(f1,k),(f2,k)}, τ ′ = {õ, 1̃,(g1,k),(g2,k)} and σ = {õ, 1̃,(h1,k),(h2,k)} be fuzzy soft topological spaces. here, (f1,k) = {{0.4/a,0.3/b},{0.4/a,0.3/b}}, (f2,k) = {{0.5/a,0.4/b},{0.4/a,0.4/b}}, (g1,k) = {{0.5/c,0.4/d},{0.5/c,0.5/d}}, (g2,k) = {{0.5/c,0.6/d},{0.5/c,0.5/d}}, (h1,k) = {{0.7/g,0.8/h},{0.6/g,0.5/h}} (h2,k) = {{0.3/g,0.2/h},{0.4/g,0.5/h}}. then f−1(g1,k), f−1(g2,k) in u and g−1(h1,k), g−1(h2,k) in v are fs b open sets but (g ◦ f)−1(h2,k) is not fs b open in u. theorem 3.5. let f : (u,τ,k) → (v,τ ′,k) and g : (v,τ ′,k) → (w,σ,k) be two fs mappings, then i. if f is fs b-irresolute and g is fssbc, then g ◦ f is fssbc . ii. if f is fs b-irresolute and g is fs b-continuous, then g ◦ f is fssbc. iii. if f is fs b-irresolute and g is fssc, then g ◦ f is fs b continuous . iv. if f is fs b-continuous and g is fssc, then g ◦ f is fssbc. v. if f is strongly b continuous and g is fssbc, then g ◦ f is fssc. vi. if f is fssbc and g is fs perfectly b continuous, then g ◦ f is fs b-irresolute. sandhya g. v., anil p. n. and p. g. patil. vii. if f is fssbc and g is fs contra continuous, then g ◦ f is fssbc. viii. if f is fs b irresolute and g is fs contra b continuous, then g ◦ f is fssbc. proof: i. let (h,k) be fs clopen set in w , since g is fssbc, g−1(h,k) is fs b open in v and f is fs b irresolute (g ◦ f)−1(h,k) = f−1(g−1(h,k)) is fs b open in u. thus g ◦ f is fssbc. ii. let (h,k) be fs clopen set in w , since g is fs b continuous, g−1(h,k) is fs b open in v and f is fs b irresolute, (g ◦ f)−1(h,k) = f−1(g−1(h,k)) is fs b open in u. therefore g ◦ f is fssbc. iii. let (h,k) be fs clopen set in w , since g is fssc, g−1(h,k) is fs open set in v and also fs b open in v since f is fs b irresolute, (g ◦ f)−1(h,k) = f−1(g−1(h,k)) is fs b-open. therefore, g ◦ f is fssbc. iv. let (h,k) be fs clopen set in w , since g is fs slightly continuous, g−1(h,k) is fsb open set in v , since f is fsb continuous, (g◦f)−1(h,k) = f−1(g−1(h,k)) is fs b-open in u and hence g ◦ f is fssbc. v. let (h,k) be fs clopen set in w . since g is fs slightly continuous, g−1(h,k) is fs b open in v and f is fs strongly b continuous. (g ◦ f)−1(h,k) = f−1(g−1(h,k)) is fs open in u. consequently g ◦ f is fssc. vi. let (h,k) be fs b open set in w , since g is fs perfectly b continuous, g−1(h,k) is fs open. fs closed in v since f is fssbc, (g ◦ f)−1(h,k) = f−1(g−1(h,k)) is fs b open in u. accordingly g ◦ f is fs b irresolute. vii. let (h,k) be fs clopen set in w , since g is fs contra continuous, g−1(h,k) is fs open and fs closed in v since, f is fssbc, (g◦f)−1(h,k) = f−1(g−1(h,k)) is fs b open in u. hence g ◦ f is fssbc. a new form of continuity in fuzzy soft topological spaces viii. let (h,k) be fs b clopen set in w , since g is fs contra b continuous, g−1(h,k) is fs b open and fs b closed in v . since f is fs b irresolute, (g ◦ f)−1(h,k) = f−1(g−1(h,k)) is fs b open, and fs b closed in u. so g ◦ f is fssbc. theorem 3.6. if f is fssbc and u is fs bt1/2 topological space, then f is fssc. proof: let (h,k) be fs clopen set in w . since f is fssbc, f−1(h,k) is fs b open in the space u and u is fs bt1/2 space, so f−1(h,k) is fs open in u. hence f is fssc. theorem 3.7. if f is fssbc and u is fs b connected space, then v is not fs discrete space. proof: let us assume v as fs discrete space. let (h,k) be a proper non-empty fs open subset of v . since, f is fs slightly b continuous, so f−1(h,k) is proper non-empty fs b clopen subset of u, which contradicts that u is fs b connected. therefore v is not fs discrete space. theorem 3.8. if f is fssbc and v is fs locally indiscrete space, then f is fs b continuous. proof: let (h,k) be fs open set in v and v is locally indiscrete space with (h,k) is fs closed in v . and function is fs slightly b continuous, f−1(h,k) is fs b open in u. hence f is fs b continuous. remark 3.1. from the above observations of stronger and weaker forms of fs slightly b continuous functions in fsts we have the following implications. figure 1: 4 fuzzy soft semi b continuous functions throughout this section, (u,τ,k) and (v,σ,k) be any two fsts where k is the set of parameters and f be a mapping from u to v sandhya g. v., anil p. n. and p. g. patil. definition 4.1. a function f is fs semi b continuous if the inverse of every fs b open (fsbo) set is fs semi-open. the family of all fs semi b continuous functions is denoted by fssmbc. theorem 4.1. if f is a member of fssmbc then it is fs semi-continuous. proof: if f ∈ fssmbc and (g,k) is fs open set in v , since every fs open set is fsbo, f−1(g,k) is fs semi-open in u. hence f is fs semi-continuous. the converse of the this theorem need not be true in general. example 4.1. consider the fs identity map f from u to v . let τ = {õ, 1̃,(f1,k),(f2,k)}, σ = {õ, 1̃,(g1,k),(g2,k)} be fsts. let u = {a,b}, v = {c,d}, k = {e1,e2}, where (f1,k) = {{0.5/a,0.3/b},{0.2/a,0.4/b}}, (f2,k) = {{0.3/a,0.1/b},{0.2/a,0.3/b}}, (g1,k) = {{0.5/c,0.7/d},{0.3/c,0.5/d}}, (g2,k) = {{0.4/c,0.5/d},{0.2/c,0.3/d}}. consider (h,k) = {{0.3/c,0.4/d},{0.3/c,0.1/d}} a fsbo set in v . since f−1(h,k) it is not fs semi-open in u, f does not belong to fssmbc. but it is fs semi-continuous. theorem 4.2. if f ∈ fssmbc then f is fs b continuous. proof: if f is in fssmbc, the inverse of every fsbo set is fs semi-open. consider fs open set (g,k) in v , f−1(g,k) is fs semi-open and hence fsbo in u, f is fs b continuous. but the converse is not as seen from the above example 4.1, f−1(g1,k) and f−1(g2,k) are fs b-open sets in u. therefore, f is fs b continuous. but f−1(h,k) is not fs semi-open in u, hence f /∈ fssmbc. theorem 4.3. if f ∈ fssmbc then f is fsgb continuous. proof: for an fs semi b-continuous function, the inverse image of a fsbo set is fs semi-open. each fs semi-open set is fs gb open. hence f is fsgb continuous. with the counter example, we can prove that converse is not valid. example 4.2. let f be fs identity mapping and τ = {õ, 1̃,(f1,k),(f2,k)}, σ = {õ, 1̃,(g1,k),(g2,k)} be fsts. let u = {a,b}, v = {c,d}, k = {e1,e2}, where (f1,k) = {{0.3/a,0.4/b},{0.5/a,0.6/b}}, (f2,k) = {{0.4/a,0.4/b},{0.6/a,0.6/b}}, (g1,k) = {{0.5/c,0.4/d},{0.1/c,0.8/d}}, (g2,k) = {{0.4/c,0.3/d},{0.1/c,0.6/d}}. consider fs b open set, (h,k) = {{0.4/c,0.2/d},{0.5/c,0.4/d}} in v . then f−1(g1,k) and f−1(g2,k) are fs gb-open sets in u but f−1(h,k) is not fs semi available in u. thus f is fs gb continuous but f /∈ fssmbc. theorem 4.4. if f is a member of fssmbc then it is fs semi pre continuous. proof: every fs semi-open set is fs semi pre-open proof is evident. a new form of continuity in fuzzy soft topological spaces theorem 4.5. if θ ∈ fssmbc and u is fs bt1/2 space, then θ is fs continuous. proof: since θ is fs semi b continuous function, for any fs open set (g,k) in v , θ−1(g,k) is fs semi-open in u. and every fs semi-open set is fsbo and hence fs open in fs bt1/2 space, θ is fs continuous. the converse is not true. example 4.3. let θ : (u,τ,k) → (v,σ,k) be a fs mapping defined by θ(a) = d and θ(b) = c. let u = {a,b}, v = {c,d} and k = {e1,e2}. let τ = {õ, 1̃,(f1,k),(f2,k)} and σ = {õ, 1̃,(g1,k),(g2,k)} be fsts, where (f1,k) = {{0.6/a,0.5/b},{0.4/a,0.5/b}}, (f2,k) = {{0.5/a,0.4/b},{0.3/a,0.4/b}}, (g1,k) = {{0.5/c,0.6/d},{0.5/c,0.4/d}}, (g2,k) = {{0.4/c,0.5/d},{0.4/c,0.3/d}}. consider fs b open set (h,k) = {{0.6/c,0.5/d},{0.7/c,0.7/d}} in v and θ−1(g1,k), θ−1(g2,k) are fs open sets but θ−1(h,k) is not fs semi-open in u. thus θ is fs continuous but θ /∈ fssmbc . theorem 4.6. if α : (u,τ,k) → (v,τ ′,k) is fs semi b continuous and β : (v,τ ′,k) → (w,σ,k) is fs b continuous, then β ◦ α : (u,τ,k) → (w,σ,k) is fs gb continuous. proof: let (h,k) be fs open set in w , since β is fs b continuous, β−1(h,k) is fsbo in v and α is fs semi b continuous, so (β ◦ α)−1(h,k) = α−1(β−1(h,k)) is fs semi-open and hence fs gb open in u. thus (β ◦ α) is fs gb continuous. theorem 4.7. if α : (u,τ,k) → (v,τ ′,k) is fs semi b continuous and β : (v,τ ′,k) → (w,σ,k) is fs semi-continuous, then β◦α : (u,τ,k) → (w,σ,k) is fs semi pre continuous. proof: let (h,k) be fs open set in w , since β is fs semi-continuous, β−1(h,k) is fs semi-open and also fsbo in v . since α is fs semi b continuous (β ◦ α)−1(h,k) = α−1(β−1(h,k)), is fs semi-open in u. every fs semi-open set is fs semi pre-open. hence (β ◦ α) is fs semi pre continuous. remark 4.1. the relations of stronger and weaker forms of fs semi-continuous functions in fsts is represented as : figure 2: sandhya g. v., anil p. n. and p. g. patil. 5 fuzzy soft pre b continuous functions in this section η : (u,τ,k) → (v,τ ′,k) is defined as fs mapping, and parameter set be k where u and v are fsts. definition 5.1. a function η is said to be fs pre b continuous if the inverse of each fsbo in v is fs pre-open in u. the family of fs pre b continuous functions is denoted by fspbc. theorem 5.1. every fs pre b continuous function is fs pre continuous. proof: let η be fs pre b continuous mapping, (g,k) be fs open set in v . since every fs open set is fsbo, η−1(g,k) is fs pre-open in u. hence η is fs pre continuous. but converse need not be true in general. example 5.1. let η : (u,τ,k) → (v,τ ′,k) be a function defined by η(x1) = y2 and η(x2) = y1 where u = {x1,x2}, v = {y1,y2}, k = {e1,e2}. let τ = {õ, 1̃,(f1,k),(f2,k)} and τ ′ = {õ, 1̃,(g1,k),(g2,k)} be fsts. (f1,k) = {{0.5/x1,0.6/x2},{0.3/x1,0.4/x2}} (f2,k) = {{0.4/x1,0.3/x2},{0.2/x1,0.4/x2}} (g1,k) = {{0.3/y1,0.3/y2},{0.2/y1,0.2/y2}} (g2,k) = {{0.5/y1,0.3/y2},{0.3/y1,0.3/y2}}. consider (h,k) = {{0.4/y1,0.1/y2},{0.3/y1,0.2/y2}} which is a fsbo in v . η−1(g1,k) and η−1(g2,k) are fs pre-open sets in u, but η−1(h,k) is not fs pre-open in u. therefore, η is fs pre-continuous but not fs pre b continuous. theorem 5.2. if η ∈ fspbc then η is fs b continuous. proof: let η be fs pre b-continuous function. so the inverse of every fsbo set is fs pre-open and each fs pre-open set is fsbo. hence η is fs b continuous. converse of the above theorem need not be accurate. example 5.2. let η : (u,τ,k) → (v,τ ′,k) be fs identity map, where u = {x1,x2}, v = {y1,y2} and k = {e1,e2}. let τ = {õ, 1̃,(f1,k),(f2,k)} and τ ′ = {õ, 1̃,(g1,k),(g2,k)} be fuzzy soft topological spaces. (f1,k) = {{0.3/x1,0.2/x2},{0.2/x1,0.3/x2}} (f2,k) = {{0.3/x1,0.3/x2},{0.8/x1,0.5/x2}} (g1,k) = {{0.5/y1,0.6/y2},{0.2/y1,0.3/y2}} (g2,k) = {{0.4/y1,0.3/y2},{0.2/y1,0.3/y2}}. consider (h,k) = {{0.5/y1,0.6/y2},{0.3/y1,0.3/y2}} which is a fsbo set in v . η−1(h,k) is not fs pre-open in u. hence η /∈ fspbc. but it is fs b continuous. theorem 5.3. if η ∈ fspbc then η is fsgb continuous. proof: let η : (u,τ,k) → (v,τ ′,k) be fs pre bcontinuous function and a new form of continuity in fuzzy soft topological spaces (g,k) be fs open set in v since every fs open set is fsbo and η is fs pre-bcontinuous, η−1(g,k) is fs pre-open, and hence fs gb open in u. converse is not be true. example 5.3. let η : (u,τ,k) → (v,τ ′,k) be fs identity mapping, where u = {x1,x2}, v = {y1,y2} and k = {e1}. let τ = {õ, 1̃,(f1,k),(f2,k)} and τ ′ = {õ, 1̃,(g1,k),(g2,k)} be fuzzy soft topological spaces. (f1,k) = {{0.3/x1,0.2/x2}} (f2,k) = {{0.3/x1,0.3/x2}} (g1,k) = {{0.5/y1,0.6/y2}} (g2,k) = {{0.4/y1,0.3/y2}}. consider fs b open set (h,k) = {{0.8/y1,0.7/y2}} in v . since, η−1(g1,k) and η−1(g2,k) are fs gb-open but η−1(h,k) is not fs pre-open in u, η is fs gb-continuous but not fs pre bcontinuous. theorem 5.4. if η ∈ fspbc then it is fs semi pre continuous. proof: let η : (u,τ,k) → (v,τ ′,k) be fs pre bcontinuous function. let (g,k) be fs open set in v . hence fs b open and η−1(g,k) is fs pre-open in u. every fs pre-open set is fs semi pre-open, η is fs semi pre continuous. converse need not be accurate as seen from the example 6.1, η−1(g1,k) and η−1(g2,k) are fs semi pre-open sets in u. therefore η is fs semi pre continuous. but η−1(h,k) is not fs pre-open in u. hence η is not fs pre bcontinuous. theorem 5.5. if η is fs pre b continuous and (u,τ,k) is fs bt1/2 space, then η is fs continuous. proof: let η : (u,τ,k) → (v,τ ′,k) be fs pre bcontinuous function. so the inverse of each fsbo set is fs pre-open and hence fsbo in u. but u is fs bt1/2 space each fs b open set is fs open, η is fs continuous. example 5.4. let η : (u,τ,k) → (v,τ ′,k) be defined by η(x1) = y2 and η(x2) = y1, where u = {x1,x2}, v = {y1,y2}, k = {e1,e2} τ = {õ, 1̃,(f1,k),(f2,k)}and τ ′ = {õ, 1̃,(g1,k),(g2,k)} be fuzzy soft topological spaces. (f1,k) = {{0.6/x1,0.5/x2},{0.4/x1,0.5/x2}} (f2,k) = {{0.5/x1,0.4/x2},{0.3/x1,0.4/x2}} (g1,k) = {{0.5/y1,0.6/y2},{0.5/y1,0.4/y2}} (g2,k) = {{0.4/y1,0.5/y2},{0.4/y1,0.3/y2}}. consider fs b open set (h,k) = {{0.6/y1,0.5/y2},{0.6/y1,0.7/y2}} in v . since η−1(g1,k) and η−1(g2,k) are fs open sets and η−1(h,k) is not fs preopen in u, η is fs continuous but not fs pre b continuous. theorem 5.6. if η : (u,τ,k) → (v,τ ′,k) is fs pre b continuous and µ : (v,τ ′,k) → (w,σ,k) is fs b continuous, then µ ◦ η : (u,τ,k) → (w,σ,k) is fs gb continuous. sandhya g. v., anil p. n. and p. g. patil. proof: let (h,k) be fs open set in w , since g is fs b continuous, µ−1(h,k) is fsbo in v and η is fs pre b continuous (µ ◦ η)−1(h,k) = η−1(µ−1(h,k)) is fs pre-open in u. every fs pre-open set is fs gb open, (µ ◦ η) is fs gb continuous. remark 5.1. from the above observations we have the following implication: figure 3: 6 fuzzy soft totally b continuous functions definition 6.1. a function ψ : (u,τ,k) → (v,τ ′,k) is fs totally continuous if the inverse of every fs open set is fs clopen. definition 6.2. a function ψ : (u,τ,k) → (v,τ ′,k) is fs totally b continuous if inverse image of every fs open set in v is fs b clopen in u. theorem 6.1. every fs totally continuous function is fs totally b continuous. proof: since every fs open(closed) set is fs b open (b closed), it was evident that every fs totally continuous function is fs totally b-continuous. but converse need not be true. example 6.1. let ψ : (u,τ,k) → (v,τ ′,k) be defined by ψ(x1) = y2 and ψ(x2) = y1. let τ = {õ, 1̃,(f1,k),(f2,k)}, τ ′ = {õ, 1̃,(g1,k),(g2,k)}, be fuzzy soft topological spaces. let u = {x1,x2}, v = {y1,y2} and k = {e1,e2} (f1,k) = {{1/x1,0.9/x2},{0.8/x1,0.8/x2}} (f2,k) = {{0/x1,0.1/x2},{0.2/x1,0.2/x2}} (g1,k) = {{0.7/x1,0.6/x2},{0.5/x1,0.6/x2}} (g2,k) = {{0.3/x1,0.4/x2},{0.5/x1,0.4/x2}}. then ψ−1(g1,k) and ψ−1(g2,k) are neither fs open, nor fs closed sets in u. but they are fs b clopen sets in u. therefore ψ, fs totally b continuous but not fs totally continuous. theorem 6.2. every fs perfectly b-continuous function is fs totally b continuous. proof: let ψ : (u,τ,k) → (v,τ ′,k) be fs perfectly b continuous function. consider an fs open set fk in v . since fk is fs b open and ψ is fs perfectly b continuous. ψ−1(fk) is fs open and fs closed in u. every fs open (closed) set is fsbo (closed), thus ψ is fs totally b continuous. but the converse is not valid. a new form of continuity in fuzzy soft topological spaces in example 6.1, ψ−1(g1,k) and ψ−1(g2,k) are fs b open, and b closed sets in u. therefore ψ is fs totally b-continuous. consider a fsbo set (h,k) = {{0.2/y1,0.3/y2},{0.3/y1,0.4/y2}}. but ψ−1(h,k) it is neither fs open nor fs closed in u. therefore ψ is not perfectly b continuous. remark 6.1. the concepts of fs strongly b-continuous function and fs totally b continuous functions are independent of each other. in example 6.1, ψ−1(g1,k) and ψ−1(g2,k) are fs b open, and b closed sets in u. therefore ψ is fs totally b-continuous. consider a fsbo set (h,k) = {{0.2/y1,0.3/y2},{0.3/y1,0.4/y2}} in v , ψ−1(h,k) is not fs open in u. therefore ψ is not strongly b-continuous. theorem 6.3. every fs totally b-continuous function is fs b continuous. proof: let ψ : (u,τ,k) → (v,τ ′,k) be fs totally b continuous function. then inverse of each fs open set is fsbo, and fs b closed in u. so ψ is fs b continuous. following example shows that, fs b continuous function need not be fs totally b continuous. example 6.2. let ψ : (u,τ,k) → (v,τ ′,k) be a fs identity map. u = {x1,x2}, v = {y1,y2} and k = {e1,e2}, let τ = {õ, 1̃,(f1,k),(f2,k),(f3,k),(f4,k),(f5,k),(f6,k),(f7,k)}, τ ′ = {õ, 1̃,(g1,k),(g2,k)} be fsts. (f1,k) = {{ 1/2 x1 , 1/3 x2 } , { 1/4 x1 , 2/3 x2 }} (f2,k) = {{ 1/3 x1 , 1/4 x2 } , { 0 x1 , 1/6 x2 }} (f3,k) = {{ 1/2 x1 , 1 x2 } , { 2/3 x1 , 1/6 x2 }} (f4,k) = {{ 1/5 x1 , 1/3 x2 } , { 1/4 x1 , 1/6 x2 }} (f5,k) = {{ 1/5 x1 , 1/4 x2 } , { 0 x1 , 1/6 x2 }} (f6,k) = {{ 1/2 x1 , 1 x2 } , { 2/3 x1 , 2/3 x2 }} (f7,k) = {{ 1/3 x1 , 1/3 x2 } , { 1/4 x1 , 1/6 x2 }} (g1,k) = {{ 1/2 y1 , 1/4 y2 } , { 1/5 y1 , 0 y2 }} (g2,k) = {{ 1/4 y1 , 1/5 y2 } , { 1/6 y1 , 0 y2 }} then ψ−1(g1,k) and ψ−1(g2,k) are fsbo sets, but they are not fs b closed sets in u. therefore ψ is fs b-continuous but not fs totally b continuous. theorem 6.4. every fs totally b-continuous function is fsgb continuous. proof: let ψ : (u,τ,k) → (v,τ ′,k) be fs totally b-continuous function. then inverse of fs open set is fs b-open, and fs b-closed in u. since the inverse of every fsbo set is fs gb-open, ψ is fs gb-continuous. example 6.2, give the sandhya g. v., anil p. n. and p. g. patil. converse is not true, ψ−1(g1,k) and ψ−1(g2,k) are fs gb open sets but not fs b closed sets in u. therefore ψ, fsgb continuous but not fs totally b-continuous. theorem 6.5. every fs totally b continuous function is fs semi pre continuous. proof: let ψ : (u,τ,k) → (v,τ ′,k) be fs totally bcontinuous function. let (g,k) be fs open set in v , then ψ−1(g,k) is fs b open and fs b closed in u and every fs b open set is fs semi pre-open, ψ is fs semi pre continuous. in example 6.2, ψ−1(g1,k) and ψ−1(g2,k) are fs semi pre-open sets but not fs b open, and fs b closed sets in u. therefore ψ, fs semi pre continuous but not fs totally b continuous. hence converse of this is not true in general. theorem 6.6. if f : (u,τ,k) → (v,τ ′,k) is fs totally b continuous and λ : (v,τ ′,k) → (w,σ,k) is fs b continuous, then λ ◦ f : (u,τ,k) → (w,σ,k) is fs gb continuous. proof: let (h,k) be fs open set in w , since λ is fs b continuous, λ−1(h,k) is fsbo in v and f is fs totally b continuous, (λ ◦ f)−1(h,k) = f−1(λ−1(h,k)) is fs b open and fs b closed in u. every fsbo is fs gb open, (λ ◦ f) is fsgb continuous. theorem 6.7. if f : (u,τ,k) → (v,τ ′,k) is fs totally b continuous and λ : (v,τ ′,k) → (w,σ,k) is fs b continuous, then λ ◦ f : (u,τ,k) → (w,σ,k) is fs semi pre continuous. proof: let (h,k) be fs open set in w , since λ is fs b continuous, λ−1(h,k) is fsbo in v and f is fs totally b continuous, (λ ◦ f)−1(h,k) = f−1(λ−1(h,k)) is fs b open and fs b closed in u. since every fs b open set is fs semi pre-open, (λ ◦ f) is fs semi pre continuous. remark 6.2. the concept of fs pre b continuous and totally b continuous functions in fsts are independent of each other. example 6.3. suppose ψ : (u,τ,k) → (v,τ ′,k) is defined by λ(x1) = y2 and λ(x2) = y1 and τ = {õ, 1̃,(f1,k),(f2,k)}, τ ′ = {õ, 1̃,(g1,k),(g2,k)}, be any two fsts. let u = {x1,x2}, v = {y1,y2} and k = {e1,e2}, (f1,k) = {{ 1 x1 , 0.9 x2 } , { 0.8 x1 , 0.8 x2 }} (f2,k) = {{ 0 x1 , 0.1 x2 } , { 0.2 x1 , 0.2 x2 }} (g1,k) = {{ 0.7 y1 , 0.6 y2 } , { 0.5 y1 , 0.6 y2 }} (g2,k) = {{ 0.3 y1 , 0.4 y2 } , { 0.5 y1 , 0.4 y2 }} here, λ−1(g1,k) and λ−1(g2,k) are fs b open, and b closed sets in u. thus λ is fs totally b continuous. consider an fsb open set (h,k) = {{0.2/y1,0.3/y2},{0.3/y1,0.4/y2}} . since λ−1(h,k) it is not fs pre-open in u, λ it is not fs pre b continuous. remark 6.3. from the above observations, we have the following : a new form of continuity in fuzzy soft topological spaces figure 4: 7 fuzzy soft b compact spaces definition 7.1. let (u,τ,k) be fsts and fk ∈ fss(u,τ,k) the set of all fs sets. an fs set fk is called b compact if each fs b open cover of fk has a finite subcover. also (u,τ,k) is called fs b compact if each fs open cover of 1̃k has a finite subcover. remark 7.1. a fsts is fs b compact if u is finite. example 7.1. let (u,τ,k) and (v,σ,k) be two fsts and τ ⊂ σ. then fsts (u,τ,k) is fs b compact if (v,σ,k) is fs b compact. definition 7.2. a fsts (u,τ,k) is called a i. strongly compact if and only if every fs pre-open cover of u has a finite subcover. ii. semi-compact if and only if every fs semi-open cover of u has a finite subcover. iii. semi pre compact if and only if every fs semi pre-open cover of u has a finite subcover. iv. s-closed if and only if every fs semi-open cover of u has a finite subcollection whose closures cover u. remark 7.2. each fs semi-open and fs pre-open sets implies fs b open sets. every fs b compact space means each of fs strongly compact and fs semi-compact spaces. also, since fs b-open set implies fs semi pre-open set, it is clear that fs semi pre compact space means fs b compact space. the finite intersection property for fs b compact spaces is provided as follows. definition 7.3. a family ψ of fs b open sets has the finite intersection property if the intersection of members of each finite subfamily of ψ is not the null fs set. sandhya g. v., anil p. n. and p. g. patil. theorem 7.1. a fsts u is fs b compact if and only if each family of fs b closed sets with the finite intersection property has a non-null intersection. proof: let ψ be an arbitrary family of fs b closed sets with the finite intersection property. we assume that ⋂ i∈i {(fi,k) : (fi,k) ∈ ψ} is non-null, that is⋂ i∈i (fi,k) = õk . then ( ⋂ i∈i (fi,k)) c = ⋃ i∈i (fi,k) c = 1̃k . since each (fi,k) is b closed, the family {(fi,k)c : i ∈ i} is fs b open cover of u. but u is fs b compact, therefore ⋃ i∈i (fi,k) = 1̃k . thus we have k = ( ⋂ i∈i (fi,k)) c ⋂ i∈i (fi,k) = õk a contradiction to assumption. suppose u is such that each family of fs b closed sets with the finite intersection property has a non-null intersection. let ψ = {(fi,k) : i ∈ i} be a family of fs b open sets. let ψ has a finite subfamily that also covers u. assume that⋃ i∈j (fi,k) = 1̃k for any finite j < i. then ⋂ i∈j (fi,k) c = ( ⋃ i∈j (fi,k)) c ̸= 0̃k , since j is finite. thus {(fi,k)c : i ∈ i} has finite intersection property. by assumption ⋂ i∈i (fi,k) c ̸= õk , and we have⋃ i∈i (fi,k) ̸= 1̃k . this is a contradiction. thus u is fs b compact. theorem 7.2. let gk be fs closed set in fs b compact space (u,τ,k). then gk is also fs b compact. proof: let ψ = {(hi,k) : i ∈ i} be fs b open cover of gk . then 1̃k ⊆ {( ⋃ i∈i (hi,k)) ∪ (g,k)c}. therefore there exists a finite sub covering (h1,k),(h2,k),(h3,k). . . ..(g,k) c. hence we get 1̃k ⊆ (h1,k) ∪ (h2,k) ∪ (h3,k) ∪ . . . ...... ∪ (hn,k) ∪ (g,k)c. therefore (g,k) ⊆ (h1,k) ∪ (h2,k) ∪ (h3,k)∪. . . .....∪(hn,k)∪(g,k)c, which implies (g,k) ⊆ (h1,k)∪(h2,k)∪ (h3,k)∪. . . .....∪(hn,k) since (g,k)∩(g,k)c = 0̃k . hence (g,k) has a finite subcover thus gk is fs b compact. 8 conclusion the present work zeroed in on introducing slightly b, semi b, pre b and totally b continuous mappings in fuzzy soft topological spaces. the correlation with the existing fs continuous functions are studied, established and compared. it is proved that every fs is slightly continuous, fs contra b continuous, and fs b continuous function is fs slightly b continuous. in contrast, the composition of fs slightly b continuous function need not be fs slightly b continuous. in fs bt1/2 space fs slightly b continuous function becomes fs slightly continuous. counter a new form of continuity in fuzzy soft topological spaces examples have been shown to illustrate and evidence that the reverse implications do not imply either. it is deduced that every fs pre b continuous and fs semi b continuous function is fs b continuous, fs gb continuous, and fs semi pre continuous function. it is also enumerated that the converse is not valid with evidence using a counter example. in addition, it is implicated that fs totally b continuous function is also fs b continuous, fs gb continuous, and fs semi pre continuous with clear inferences that the reverse implication is not true. further, it is also concluded that fs pre b continuous and fs totally b continuous, fs strongly b continuous and fs totally b continuous functions are independent of each other. a new form of topological space 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[31] zhi kong and lifu wang, application of fuzzy soft set in decision-making problems based on grey theory, vol 236 (2011), 1521-1530, (2021). ratio mathematica volume 47, 2023 edge coloring in complement of bipolar fuzzy graphs s.yahya mohamed* n.subashini† abstract graph theory is rapidly moving into the mainstream of mathematics because of its applications in diverse fields which include chemistry, bio-chemistry, electrical engineering (communications networks and coding theory), computer science (algorithms and computations) and operations research (scheduling). graph coloring is one of the most important concepts in graph theory and is used in many real time applications like job scheduling, aircraft scheduling, computer network security, map coloring, automatic channel allocation for small wireless local area networks. two types of coloring namely vertex coloring and edge coloring are usually associated with any graph. in this paper, we analyze edge coloring of the complement bipolar fuzzy graphs using concept of as bipolar fuzzy numbers through the 𝛼 − cuts of bipolar fuzzy graphs. for different values of 𝛼 − cuts which depend on edge and vertex membership value of the graph, we will get different graph and different chromatic number. keywords: complement bipolar fuzzy graph; edge coloring; chromatic number; 𝛼 − cut of bfg. 2010 ams subject classification: 05c72‡ * assistant professor, department of mathematics, rani anna government college for women (affiliated to manonmaniam sundaranar university, tirunelveli-8, india; yahya_md@yahoo.com † assistant professor, department of mathematics, saranathan college of engineering, tiruchirappalli, india; yazhinisubal@gmail.com ‡ received on september 15, 2022. accepted on december 15, 2022. published on march 10, 2023. doi:10.23755/rm.v39i0.969. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 398 mailto:yahya_md@yahoo.com mailto:yazhinisubal@gmail.com s.yahya mohamed and n.subashini 1. introduction in 1994, zhang [13] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets. bipolar fuzzy sets are an extension of fuzzy sets whose membership degree range is [-1,1]. in a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0,1] of an element indicates that the element somewhat satisfies the property, and the membership degree [-1,0) of an element indicates that the element somewhat satisfies the implicit counter-property. although bipolar fuzzy sets and intuitionistic fuzzy sets look similar to each other, they are essentially different sets. in many domains, it is important to be able to deal with bipolar fuzzy information. it is noted that positive information represents what is granted to be possible, while negative information represents what is considered to be impossible. this domain has recently motivated new research in several direction. akram[1] introduced the concept of bipolar fuzzy graphs and defined different operations on it. sunil mathew, m.s.sunitha and a.vijaya kumar[8] introduced the concept of complement bipolar fuzzy graphs in 2014 and discussed about some connectivity concepts in bipolar fuzzy graphs. as an advancement coloring of fuzzy graph was outlined by authors eslahchi and onagh[4] in 2004, and later developed by them a fuzzy vertex coloring in 2006. this fuzzy vertex coloring was extended to fuzzy total coloring in terms of family of fuzzy set by m.ananthanarayanan and s.lavanya [2]. the concept of chromatic number of fuzzy graphs was introduced by munoz.s[6]. anjali and sunitha [3] developed algorithms to the chromatic number of fuzzy graphs. types of paths and strong cycle connectivity in bipolar fuzzy graphs discussed by s. y.mohamed and subashini[11,12] a.tahmasbpour and r.a.borzooei[9] introduced the concept of chromatic number of bipolar fuzzy graphs. in this paper, we determine edge chromatic number of complement bipolar fuzzy graphs using 𝛼 −cut value. 2. preliminaries definition 2.1 by a bipolar fuzzy graph, we mean a pair 𝐺 = (𝐴, 𝐵) where 𝐴 = (𝜇𝐴 𝑃 , 𝜇𝐴 𝑁 ) is a bipolar fuzzy set in v and 𝐵 = (𝜇𝐵 𝑃 , 𝜇𝐵 𝑁 ) is a bipolar fuzzy relation on e such that 𝜇𝐵 𝑃 (𝑥𝑦) ≤ min(𝜇𝐴 𝑃 (𝑥), 𝜇𝐴 𝑃 (𝑦)) and 𝜇𝐵 𝑁 (𝑥𝑦) ≥ max(𝜇𝐴 𝑁 (𝑥), 𝜇𝐴 𝑁 (𝑦)) for all(𝑥, 𝑦) ∈ 𝐸. definition 2.2 we call a the bipolar fuzzy vertex set of v, b the bipolar fuzzy edge set of e respectively. note that b is symmetric bipolar fuzzy relation on a. we use the notation xy for an element of e. thus, g = (a,b) is a bipolar fuzzy graph of 𝐺 ∗ = (𝑉, 𝐸) if 𝜇𝐵 𝑃 (𝑥𝑦) ≤ min (𝜇𝐴 𝑃 (𝑥), 𝜇𝐴 𝑃 (𝑦)) and 𝜇𝐵 𝑁 (𝑥𝑦) ≥ max (𝜇𝐴 𝑁(𝑥), 𝜇𝐴 𝑁(𝑦)) for all 𝑥𝑦 ∈ 𝐸. 𝐻 = (𝛼, 𝛽) (where 𝛼 = (𝜇𝐴 𝑃, 𝜇𝐴 𝑁 ) is a bipolar fuzzy subset of a set a and 𝛽 = (𝜇𝐵 𝑃, 𝜇𝐵 𝑁 ) is a bipolar fuzzy relation on b) is called a partial bipolar fuzzy subgraph of g if 𝛼 ≤ 𝐴 and 𝛽 ≤ 𝐵. we call 𝐻 = (𝛼, 𝛽) a spanning bipolar fuzzy subgraph of 𝐺 = (𝐴, 𝐵) if 𝛼 = 𝐴. 399 edge coloring in complement of bipolar fuzzy graphs definition 2.3 an arc ((𝜇𝐴 𝑝 (𝑢), 𝜇𝐴 𝑁 (𝑢)) , (𝜇𝐴 𝑝 (𝑣), 𝜇𝐴 𝑁 (𝑣))) of g is called m-strong if 𝜇𝐵 𝑃 (𝑢, 𝑣) = min (𝜇𝐴 𝑃(𝑢), 𝜇𝐴 𝑃 (𝑣)) and 𝜇𝐵 𝑁 (𝑢, 𝑣) = max (𝜇𝐴 𝑁(𝑢), 𝜇𝐴 𝑁(𝑣)). suppose g : (a,b) be a bipolar fuzzy graph. the complement of 𝐺 is denoted as 𝐺 ̅: (�̅�, �̅�) where �̅� = 𝐴 and �̅�𝐵 𝑃 (𝑥𝑦) = min(𝜇𝐴 𝑃 (𝑥), 𝜇𝐴 𝑃 (𝑦)) − 𝜇𝐵 𝑃 (𝑥𝑦) and �̅�𝐵 𝑁 (𝑥𝑦) = max(𝜇𝐴 𝑁 (𝑥), 𝜇𝐴 𝑁(𝑦)) − 𝜇𝐵 𝑁 (𝑥𝑦). definition 2.4 ∝ − cut set of bipolar fuzzy set a is denoted as 𝐴∝ is made up of members whose positive membership is not less than ∝ and negative membership is not greater than ∝. 𝐴∝ 𝑃 = {𝑥 ∈ 𝑋 , 𝜇𝐴 𝑃 (𝑥) ≥∝} and 𝐴∝ 𝑁 = {𝑥 ∈ 𝑋 , 𝜇𝐴 𝑁 (𝑥) ≤∝} where ∝ − cut set of fuzzy set is crisp set. the ∝ − cut of bfg defined as 𝐺∝ = ( 𝑉∝, 𝐸∝) where 𝑉∝ = {𝑣 ∈ 𝑉/𝜎 ≥∝} and 𝐸∝ = {𝑒 ∈ 𝐸/𝜇 ≥∝}. 3. complement of bipolar fuzzy graph complement of fg has been defined by moderson[5]. complement of a bfg 𝐺 ∶ (𝜎, 𝜇) as a bfg �̅� ∶ (𝜎, �̅�) where 𝜎 = 𝜎 and 𝜇�̅� 𝑃 (𝑥𝑦) = 0 𝑖𝑓 𝜇(𝑥𝑦) > 0, 𝜇�̅� 𝑁 (𝑥𝑦) = 0 𝑖𝑓 𝜇(𝑥𝑦) < 0 and 𝜇�̅� 𝑃 (𝑥𝑦) = 𝑚𝑖𝑛 [𝜇𝐴 𝑃 (𝑥), 𝜇𝐴 𝑃 (𝑦)], 𝜇�̅� 𝑁 (𝑥𝑦) = 𝑚𝑎𝑥 [𝜇𝐴 𝑁 (𝑥), 𝜇𝐴 𝑁 (𝑦)] otherwise. from the definition �̅� is a bfg even if 𝐺 is not and �̿� = 𝐺 if and only if g is m-strong bfg. also, automorphism graph of 𝐺 and �̅� are not identical. but there is some drawbacks in the definition of complement of a bfg mentioned above. in fig 3.3, �̿� ≠ 𝐺 and note that they are identical provided g is m strong bfg. figure 3.1: bipolar fuzzy graph 𝐺 now the complement of a bfg 𝐺 ∶ (𝜎, 𝜇) is the bfg �̅� ∶ (𝜎, �̅�) where 𝜎 = 𝜎 and 𝜇�̅� 𝑃 (𝑥𝑦) = 𝑚𝑖𝑛 [𝜇𝐴 𝑃 (𝑥), 𝜇𝐴 𝑃 (𝑦)] − 𝜇𝐵 𝑃 (𝑥𝑦) and 𝜇�̅� 𝑁 (𝑥𝑦) = 𝑚𝑎𝑥 [𝜇𝐴 𝑁 (𝑥), 𝜇𝐴 𝑁 (𝑦)] − 𝜇𝐵 𝑁 (𝑥𝑦) 400 s.yahya mohamed and n.subashini figure 3.2: complement of bipolar fuzzy graph �̅� figure 3.3: complement of complement bipolar fuzzy graph �̿� figure 3.4: bipolar fuzzy graph 𝐺 401 edge coloring in complement of bipolar fuzzy graphs figure 3.5: complement of bipolar fuzzy graph �̅� figure 3.6: complement of complement bipolar fuzzy graphs now 𝜎 = 𝜎 and 𝜇�̅� 𝑃 (𝑥𝑦) = 𝑚𝑖𝑛 [𝜇𝐴 𝑃 (𝑥), 𝜇𝐴 𝑃 (𝑦)] − 𝜇𝐵 𝑃 (𝑥𝑦) and 𝜇�̅� 𝑁 (𝑥𝑦) = 𝑚𝑎𝑥 [𝜇𝐴 𝑁 (𝑥), 𝜇𝐴 𝑁(𝑦)] − 𝜇𝐵 𝑁 (𝑥𝑦). hence �̿� = 𝐺. this shows that complement of complement bfg is a bfg. 4. edge coloring in complement of bfg we find all the different membership value of vertices and edges in the complement of a bfg. this membership value will work as a cut of this complement bfg. depend upon the values of ∝ − cut we find different types of bipolar fuzzy sets for the same complement bfg. then we color all the edges of the complement bfg so that no incident edges will not get the same color and find the minimum number of color will need to color the complement bfg is known as chromatic number. for solving this problem we have the calculation into three cases. in first case we take a bfg (𝐺) which have 5 vertices and 5 edges. all the vertices and edges have membership value. in second case we find the complement of this graph (𝐺1). in third case we define the edge coloring function to color the complement bfg. 402 s.yahya mohamed and n.subashini case 1: consider a bfg which have five vertices a, b, c, d, e and corresponding membership values {(– 0.8,0.9), (– 0.8,0.75), (– 0.85,0.95), (– 0.85,0.95), (– 0.8,0.9)}. graph consist of five edges 𝑒1 𝑒2, 𝑒3, 𝑒4, 𝑒5 with their corresponding membership value {(– 0.8,0.75), (– 0.8,0.9), (– 0.6,0.85), (– 0.85,0.95), (– 0.4,0.6)} corresponding bfg is shown in fig 4.1. figure 4.1: bipolar fuzzy graph 𝐺 case 2. we find the complement of a bfg. figure 4.2: complement of bipolar fuzzy graph �̅� case 3. given a bfg 𝐺 = (𝑉, 𝐸) its edge chromatic number is bipolar fuzzy number 𝜒(𝐺) = {(𝑋𝛼 , 𝛼)} where 𝑋𝛼 is the edge chromatic number of 𝐺𝛼 where 𝛼 values are the 403 edge coloring in complement of bipolar fuzzy graphs different membership value of vertex and edge of graph g. in this bfg there are five 𝛼 −cuts. there are {(– 0.2,0.05), (– 0.4,0.15), (– 0.8,0.75), (– 0.8,0.9), (– 0.85,0.95)}. for every value of 𝛼, we find graph 𝐺𝛼 and find its fuzzy edge chromatic number. for = (−0.2,0.05) , bfg 𝐺 = (𝜎, 𝜇) where 𝜎 = {(– 0.8,0.9), (– 0.8,0.75), (– 0.85,0.95), (– 0.85,0.95), (– 0.8,0.9)}. figure 4.3: 𝜒(−0.2,0.05) = 4 for 𝛼 − cut (-0.2,0.05) we find the graph 𝐺(−0.2,0.05) (figure 4.3). then we proper color all the edges of this graph and the chromatic number of this graph is 4. for 𝛼 = (−0.4,0.15) bfg 𝐺 = (𝜎, 𝜇) where 𝜎 = {(– 0.8,0.9), (– 0.8,0.75), (– 0.85,0.95), (– 0.85,0.95), (– 0.8,0.9)} figure 4.4: 𝜒(−0.4,0.15) = 4 for 𝛼 − cut value (-0.4,0.15) we find the graph 𝐺(−0.4,0.15) (figure 4.4). then we proper color all the edges of this graph and the chromatic number of this graph is 404 s.yahya mohamed and n.subashini 4(four). for 𝛼 = (−0.8,0.75) bfg 𝐺 = (𝜎, 𝜇) where 𝜎 = {(– 0.8,0.9), (– 0.8,0.75), (– 0.85,0.95), (– 0.85,0.95), (– 0.8,0.9)}. figure 4.5: 𝜒(−0.8,0.75) = 3 now for 𝛼 − cut value (-0.8,0.75), we find the graph 𝐺(−0.8,0.75) (figure 4.5). then we proper color all the edges of this graph and the chromatic number of this graph is 3. for 𝛼 = (−0.8,0.9) bfg 𝐺 = (𝜎, 𝜇) where 𝜎 = {(– 0.8,0.9), (– 0.85,0.95), (– 0.85,0.95), (– 0.8,0.9)}. figure 4.6: 𝜒(−0.8,0.9) = 2 for 𝛼 − cut value (-0.8,0.9) we find the graph 𝐺(−0.8,0.9) (figure 4.6). then we proper color all the edges of this graph and the chromatic number of this graph is 2. for = (−0.85,0.95) , bfg 𝐺 = (𝜎, 𝜇) where 𝜎 = {(– 0.85,0.95)} and 405 edge coloring in complement of bipolar fuzzy graphs figure 4.7: 𝜒(−0.8,0.9) = 0 for 𝛼 − cut value (-0.85,0.95), we find the graph 𝐺(−0.85,0.95) (figure 4.7). then we proper color all the edges of this graph and the chromatic number of this graph is 0. in the above example five crisp graph 𝐺𝛼 = (𝑉𝛼 , 𝐸𝛼 ) are obtained by considering different values of 𝛼. now for the edge chromatic number 𝜒𝛼 for any 𝛼, it can be shown that the chromatic number of bfg is 𝜒(𝐺) = {(4, −0.2,0.05), (4, −0.4,0.15), (3, −0.8,0.75), (2, −0.8,0.9), (0, −0.85,0.95)}. 4 conclusions in this paper, we have found the complement bipolar fuzzy graph and color all the edges of that complement bfg through the 𝛼 − cuts. also, we observed that edge chromatic number depends on 𝛼 − cut value. references [1] m. akram. bipolar fuzzy graphs. information sciences, doi 10.1016/j.ins 2011.07.037, 2011. [2] m. ananthanarayanan, s.lavanya. fuzzy graph coloring using ∝ −cuts. international journal of engineering and applied science, 4, 2014. [3] anjali kishore, m.s.sunitha. chromatic number of fuzzy graphs. annals of fuzzy mathematics and informatics, 7, 543-551. 2014. [4] c.eslahchi, b.n.onagh. vertex strength of fuzzy graphs. international journal of mathematics and mathematical science, 1-9. 2006. [5] j.n.mordeson, p.s.nair. fuzzy graphs and fuzzy hypergraphs. physica verlag 2000. [6] s.munoz, t.ortuno, j.ramirez, j.yanez. coloring of fuzzy graphs. omega: the international journal of management science. 33, 211-221. 2005. [7] s.samanta, m.pal. bipolar fuzzy hypergraphs. international journal of fuzzy logic systems, 2(1), 17-28. 2012. [8] m.s.sunitha and a.vijayakumar, complement of a fuzzy graph, indian journal of pure and applied mathematics, 9(33), 1451-1464. 2002. 406 s.yahya mohamed and n.subashini [9] a.tahmasbpoor, r.a.borzooe. chromatic number of bipolar fuzzy graphs. j.appli.math. and informatics, 34(1-2), 49-60. 2016. [10] s. yahya mohamed, n. subashini. some types of paths in a bipolar fuzzy graphs. journal of applied science and computations, 6(6), 305-308. 2019. [11] s. yahya mohamed, n. subashini. strength of strong cycle connectivity in bipolar fuzzy graphs. advances in mathematics: scientific journal 9(1), 15-25. 2020. [12] l.a.zadeh. fuzzy sets. information and control 8, 338-353. 1965. [13] w. r. zhang. bipolar fuzzy sets and relations. a computational framework for cognitive modeling and multi-agent decision analysis, proceedings of the first international joint conference of the north american fuzzy information processing society biannual conference, 305–309. 1994. 407 ratio mathematica volume 42, 2022 mixed picture fuzzy graph myithili kothandapani * nandhini chandrasekar † abstract a new form of picture fuzzy graph has been identified and extended here as mixed picture fuzzy graph (mpfg). the picture fuzzy set is formed from the fuzzy set and the intuitionistic fuzzy set. it is helpful when there are multiple options, such as yes, no, rejection and abstain. mpfg, which is dependent on the picture fuzzy relation, is defined in this paper. the properties of various types of degrees, order and size of mpfg are examined. also some types of mpfg such as regular, strong, complete and complement of mpfg are introduced and their properties were analysed. as an application part, the concept of mpfg has been applied in instagram and the result has been discussed here. keywords: picture fuzzy graph, mpfg, degree, size & order of mpfg, regular, strong, complete mpfg and complement of mpfg 2020 ams subject classifications: 05c72 1 *associate professor and head (department of mathematics(ca), vellalar college for women, erode, india); myithili@vcw.ac.in. †research scholar (department of mathematics, vellalar college for women, erode, india); c.nandhini@vcw.ac.in. 1received on april 29th, 2022. accepted on june 25th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v41i0.776. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 225 k. k. myithili, c. nandhini 1 introduction many decision-making problems in unpredictable environments have been modelled using fuzzy graphs. a variety of generalisations of fuzzy graphs have really been implemented to deal with the uncertainty of complex real-life circumstances. zadeh’s(19) fuzzy set theory played a significant role in decision making in unpredictable environments. rosenfeld(14), developed the basic conception of fuzzy graph 10 years after zadeh’s seminal article on fuzzy sets. as compared to the graph, the fuzzy graph seems to be a beneficial tool for modelling those problems because it is more efficient, flexible and compatible with any real-world problem. mordeson & nair(8) introduced the idea of a complement fuzzy graph, in which sunitha & kumar(17) expanded the concept. the principle of atanassov’s(2) intuitionistic fuzzy set (ifs) allocates a membership and non-membership degree individually, with the sum of the two degrees not exceeding the value one. shannon and atanassov proposed a description for intuitionistic fuzzy relations and intuitionistic fuzzy graphs, as well as a list of properties in (16). different operations on intuitionistic fuzzy graphs were defined by parvathi et al.(10; 11). nagoor gani and shajitha begum(9) has characterised about degree, order and size of intuitionistic fuzzy graphs. the picture fuzzy set (pfs) is a new idea that deals with uncertainties and is a direct continuation of the ifs. it can simulate uncertainty in circumstances including multiple types of answers: yes, abstention, rejection, and no. it is shown about one of the most fundamental concepts of degree of neutrality goes absent from the ifs principle. cuong & kreinovich(6) proposed pfs, a direct extension of fuzzy set and ifs that integrates the principle of positive, negative, and neutral membership degree of an element. cuong(4) investigated some pfs properties and proposed distance measures between them. phong and co-authors(13) investigated some picture fuzzy relation compositions. then, cuong and hai(5) extended some fuzzy logic operators for pfss, including such conjunctions, complements, and disjunctions. peng & dai(12) proposed and implemented an algorithmic solution for pfs in a decision-making problem. new concepts of pfg with application was published by cen zuo et al.,(3). l. t. koczy et al.(7) analyzed the study of social networks and wi-fi networks using the concept of picture fuzzy graphs. wei xiao, arindam dey, and le hoang son(18) spoke about their research on regular picture fuzzy graphs and how they can be used in communication networks. sankar das and ganesh ghorai(15) investigated the creation of a road map based on a multigraph using picture fuzzy information. and abdelkadir muzey mohammed(1) explained about mixed graph representation. 226 mixed picture fuzzy graph 2 basic definitions we stepped over some fundamental definitions in this section that are related to our main concept. definition 2.1. (3) let g∗pf = (v, e) be a graph. a pair gpf = (a, b) is called a picture fuzzy graph on g∗ where a = (µa, ηa, νa) is a picture fuzzy set on v and b = (µb, ηb, νb) is a picture fuzzy set on e ⊆ v × v such that for each arc vu ∈ e. µb(v, u) ≤ min(µa(v), µa(u)) ηb(v, u) ≤ min(ηa(v), ηa(u)) νb(v, u) ≥ max(νa(v), νb(u))   (1) denotes the degree of positive membership, neutral membership & membership membership of the edge (v, u) ∈ e. definition 2.2. (1) a mixed graph gm = (v, e, a) is a graph consists from set of vertices v , set of undirected edges e & set of directed edges(or arcs) a. 2.1 notations the following mathematical symbols were used throughout the paper: gpf –picture fuzzy graph gm–mixed graph gmpf –mixed picture fuzzy graph µa(v), ηa(v), νa(v)–positive, neutral & negative membership of a vertex v in gmpf µb(v, u), ηb(v, u), νb(v, u)–positive, neutral & negative membership of an edge vu in gmpf µ→ b (v, u), η→ b (v, u), ν→ b (v, u)–positive, neutral & negative membership of an arc vu in gmpf d(vi)–degree of a vertex vi in gmpf δ(gmpf)–minimum degree of a gmpf ∆(gmpf)–maximum degree of a gmpf (gmpf) c–complement of a gmpf (gcmpf) c–complement of complement gmpf o(gmpf)–order of a gmpf s(gmpf)–size of a gmpf sp –strength of a path p h ′ –subgraph of gmpf cdmpf(vi, vj)–circle-distance between vi and vj of gmpf c(s)–centrality of a squad 227 k. k. myithili, c. nandhini 3 mixed picture fuzzy graph[mpfg] the popularity of social media sites and networks are growing every day. positive, neutral & negative membership of a vertex can be classified as good, neutral and bad activities in pfg. the situation now is why we should have to switch from pfg to mpfg? mpfg is the combination of both directed and undirected edges. many real-life situations take the shape of mpfg. further a real life problem has been identified and resolved using this mpfg. definition 3.1. let g∗mpf = (v, e, → e) be a graph. an ordered triple gmpf = (a, b, → b) is called mixed picture fuzzy graph on g∗mpf , where a = (µa, ηa, νa) is a picture fuzzy set on v, b = (µb, ηb, νb) is a picture fuzzy relation on the undirected edge e ⊆ v × v and → b= (µ→ b , η→ b , ν→ b ) is a picture fuzzy relation on the directed edge → e ⊆ v × v , which satisfies, µb(v, u) ≤ min(µa(v), µa(u)) ηb(v, u) ≤ min(ηa(v), ηa(u)) νb(v, u) ≥ max(νa(v), νa(u))   ∀(v, u) ∈ e & µ→ b (v, u) ≤ min(µa(v), µa(u)) η→ b (v, u) ≤ min(ηa(v), ηa(u)) ν→ b (v, u) ≥ max(νa(v), νa(u))   ∀(v, u) ∈ → e (2) also → b must not have a symmetric relation. figure 1: mixed picture fuzzy graph 228 mixed picture fuzzy graph definition 3.2. consider a graph h ′ = (v ′ , e′, → e ′ ) is mixed picture fuzzy subgraph (mpfsg) of mpfg if v ′ ⊆ v, e′ ⊆ e and → e ′ ⊆ → e if, µ ′ a(v) ≤ µa(v), η ′ a(v) ≤ ηa(v), ν ′ a(v) ≥ νa(v), µ ′ b(v, u) ≤ µb(v, u), η ′ b(v, u) ≤ ηb(v, u), ν ′ b(v, u) ≥ νb(v, u), µ ′ → b (v, u) ≤ µ→ b (v, u), η ′ → b (v, u) ≤ η→ b (v, u), ν ′ → b (v, u) ≥ ν→ b (v, u). theorem 3.1. a mpfg is a expandation of ifg. proof. the statement becomes trivial by assuming the neutral membership/abstain is equal to zero. hence mpfg can reduce to ifg. similarly, the statement “a mpfg is a generalization of pfg” is also true. theorem 3.2. if v = {v1, v2, ..., vn} is vertex set of mpfg, gmpf = (v, e, → e ). then total number of edges denoted by |empf| in mpfg gmpf is given by, |empf| = 1/2 [ ∑ v∈v deg(v) + ∑ v∈v degin(v) ] or |empf| = 1/2 [ ∑ v∈v deg(v) + ∑ v∈v degout(v) ] proof. let gu = (v, e) be undirected subgraph of gmpf and gd = (v, → e) with directed edges which are disjoint mpfsgs of mpfg gmpf = (v, e, → e) such that empf = e ∪ → e handshaking theorem and elementary counting principle, which states that |e| = 1 2 ∑ deg(v) and | → e| = ∑ v∈v degin(v) = ∑ v∈v degout(v) (3) |empf| = |e ∪ → e| = | → e| + |e| − |e ∩ → e| (4) since, gu = (v, e) and gd = (v, → e) are disjoint mpfsgs, |e ∩ → e|=0 then (4) is reduced to |empf| = | → e| + |e| (5) substituting (3) in (5), we get |empf| = 1/2 [ ∑ v∈v deg(v) + ∑ v∈v degin(v) ] or |empf| = 1/2 [ ∑ v∈v deg(v) + ∑ v∈v degout(v) ] 229 k. k. myithili, c. nandhini [∵ ∑ v∈v degin(v)= ∑ v∈v degout(v)] hence proved definition 3.3. the degree of a vertex in a mpfg denoted as, d(vi) = (dµ(vi), dη(vi), dν(vi)) where, dµ(vi) = ∑ v ̸=u µb(v, u) + 1 2 [ ∑ v ̸=u µ→ bin (v, u) + ∑ v ̸=u µ→ bout (v, u) ] dη(vi) = ∑ v ̸=u ηb(v, u) + 1 2 [ ∑ v ̸=u η→ bin (v, u) + ∑ v ̸=u η→ bout (v, u) ] dν(vi) = ∑ v ̸=u νb(v, u) + 1 2 [ ∑ v ̸=u ν→ bin (v, u) + ∑ v ̸=u ν→ bout (v, u) ]   (6) from figure 1, we get, d(v1)=(0.45,0.3,0.3), d(v2)=(0.95,0.95,0.9), d(v3)=(0.4,0.45,0.4), d(v4)=(0.5,0.75,1.0), d(v5)=(0.8,0.9,0.85), d(v6)=(0.55,0.45,0.35), d(v7)=(0.45,0.3,0.3), δ(gmpf)=(0.4,0.3,0.3) and ∆(gmpf)=(0.95,0.95,1.0) definition 3.4. consider gmpf = (v, e, → e) be a mpfg. the neighbourhood of a vertex is represented as, nh(v) = (nhµ(v), nhη(v), nhν(v)) where, nhµ(v) = {u ∈ v/µb(v, u) = min(µa(v), µa(u)), µ→ b (v, u) = min(µa(v), µa(u))} nhη(v) = {u ∈ v/ηb(v, u) = min(ηa(v), ηa(u)), η→ b (v, u) = min(ηa(v), ηa(u))} nhν(v) = {u ∈ v/νb(v, u) = max(νa(v), νa(u)), ν→ b (v, u) = max(νa(v), νa(u))}   (7) and nh[v] = nh(v) ∪ {v} represents closed neighbourhood of a vertex. definition 3.5. the neighbourhood degree of a vertex is represented as, dnh(v) = (dnhµ(v), dnhη(v), dnhν (v)) where, dnhµ(v) = ∑ u∈nh(v) µa(u) dnhη(v) = ∑ u∈nh(v) ηa(u) dnhν (v) = ∑ u∈nh(v) νa(u)   (8) 230 mixed picture fuzzy graph note: if a vertex is an isolated vertex then nh(v) = ∅ definition 3.6. the closed neighbourhood degree of a vertex is denoted as, dnh[v] = (dnhµ[v], dnhη[v], dnhν [v]) where, dnhµ[v] = ∑ u∈nh(v) µa(u) + µa(v), dnhη[v] = ∑ u∈nh(v) ηa(u) + ηa(v) dnhν [v] = ∑ u∈nh(v) νa(u) + νa(v)   (9) definition 3.7. a path in gmpf = (a, b, → b) is a distinct vertices sequence v0, v1, v2, ..., vk one of the succeeding responses are satisfied with both directed & undirected edges, µb(vi−1, vi), ηb(vi−1, vi) > 0 and νb(vi−1, vi) = 0 µb(vi−1, vi), ηb(vi−1, vi) = 0 and νb(vi−1, vi) > 0 µb(vi−1, vi), ηb(vi−1, vi), νb(vi−1, vi) > 0 µ→ b (vi−1, vi), η→ b (vi−1, vi) > 0 and ν→ b (vi−1, vi) = 0 µ→ b (vi−1, vi), η→ b (vi−1, vi) = 0 and ν→ b (vi−1, vi) > 0 µ→ b (vi−1, vi), η→ b (vi−1, vi), ν→ b (vi−1, vi) > 0 i = 1, 2, ..., k. where k denotes the length of the path. definition 3.8. a mpfg gmpf = (a, b, → b) seems to be connected, if each set of vertices possesses atleast 1 mixed picture fuzzy path connecting them, else it is said to be disconnected. definition 3.9. if there is a path p = vn, v1, ..., vn for n ≥ 3 then it’s a cycle. definition 3.10. the complement of a gmpf = (a, b, → b) is a gcmpf = (a c, bc, → bc) iff it follows, µa c = µa, ηac = ηa, νac = νa and µcb(v, u) = min(µa(v), µa(u)) − µb(v, u) ηcb(v, u) = min(ηa(v), ηa(u)) − ηb(v, u) νcb(v, u) = max(νa(v), νa(u)) − νb(v, u) µ→ b c(v, u) = min(µa(v), µa(u)) − µ→ b (v, u) η→ b c(v, u) = min(ηa(v), ηa(u)) − η→ b (v, u) ν→ b c(v, u) = max(νa(v), νa(u)) − ν→ b (v, u)   (10) 231 k. k. myithili, c. nandhini figure 2: complement of mixed picture fuzzy graph theorem 3.3. if gcmpf be a complement of mpfg, then (g c mpf) c = g note: a mpfg is self-complementary if (gcmpf) c = g definition 3.11. the order of a gmpf is represented by, o(gmpf) = (oµ(gmpf), oη(gmpf), oν(gmpf)) where, oµ(gmpf) = ∑ u∈v µa(v) oη(gmpf) = ∑ u∈v ηa(v) oν(gmpf) = ∑ u∈v νa(v)   (11) here oµ(gmpf), oη(gmpf) & oη(gmpf) are the order of positive, neutral & negative membership degree respectively. definition 3.12. let gmpf = (a, b, → b) is mpfg. the size of a gmpf is represented by, s(gmpf) = (sµ(gmpf), sη(gmpf), sν(gmpf)) where, sµ(gmpf) = ∑ v,u∈v µb(v, u) + ∑ v,u∈v µ→ b (v, u) sη(gmpf) = ∑ v,u∈v ηb(v, u) + ∑ v,u∈v η→ b (v, u) sν(gmpf) = ∑ v,u∈v νb(v, u) + ∑ v,u∈v ν→ b (v, u), ∀j ̸= i.   (12) here sµ(gmpf), sη(gmpf) and sν(gmpf) are the size of positive, neutral & negative membership respectively. 232 mixed picture fuzzy graph definition 3.13. for a path p, sµ = min v,u∈v {µb(v, u)} + min v,u∈v {µ→ b (v, u)} sη = min v,u∈v {ηb(v, u)} + min v,u∈v {η→ b (v, u)} sν = max v,u∈v {νb(v, u)} + max v,u∈v {ν→ b (v, u)}   (13) the strength of a path sp = (sµ, sη, sν). definition 3.14. a gmpf = (a, b, → b) is said to be strong mpfg if, µb(v, u) = min(µa(v), µa(u)) ηb(v, u) = min(ηa(v), ηa(u)) νb(v, u) = max(νa(v), νa(u)), ∀(v, u) ∈ e & µ→ b (v, u) = min(µa(v), µa(u)) η→ b (v, u) = min(ηa(v), ηa(u)) ν→ b (v, u) = max(νa(v), νa(u)), ∀(u, v) ∈ → e.   (14) figure 3: strong mixed picture fuzzy graph note:(gcmpf) c = gmpf iff g is strong mpfg definition 3.15. a mpfg gmpf = (a, b, → b) is said to be complete mpfg if, µb(v, u) = min(µa(v), µa(u)) ηb(v, u) = min(ηa(v), ηa(u)) νb(v, u) = max(νa(v), νa(u)) and µ→ b (v, u) = min(µa(v), µa(u)) η→ b (v, u) = min(ηa(v), ηa(u)) ν→ b (v, u) = max(νa(v), νa(u)), ∀v, u ∈ v   (15) 233 k. k. myithili, c. nandhini figure 4: complete mixed picture fuzzy graph note: every complete mpfg becomes a strong mpfg. but the contrary, does not have to be true. theorem 3.4. the order of a complete mpfg is equal to the closed neighbourhood degree of every vertex (i.e), oµ(gmpf) = {dnµ[v]|v ∈ v }, oη(gmpf) = {dnη[v]|v ∈ v }, oν(gmpf) = {dnν [v]|v ∈ v }. proof. consider gmpf = (v, e, → e) be a complete mpfg. the µ, η and ν-order of gmpf , is the sum of the positive, neutral and negative membership value of each vertex respectively. we know that, if gmpf is a complete mpfg, then the closed neighbourhood µ, η and ν-degree of every vertex is the sum of the positive membership, neutral membership & negative membership values of the vertices respectively. therefore, oµ(gmpf) = {dnµ[v]|v ∈ v }, oη(gmpf) = {dnη[v]|v ∈ v }, oν(gmpf) = {dnν [v]|v ∈ v }. hence the result. definition 3.16. a mpfg gmpf = (a, b, → b) is defined as regular mpfg if, µb(v, u) = min(µa(v), µa(u)) and ∑ u̸=v µb(u, v) = constant, ηb(v, u) = min(ηa(v), ηa(u)) and ∑ u̸=v ηb(u, v) = constant, νb(v, u) = max(νa(v), νa(u)) and ∑ u̸=v νb(u, v) = constant, µ→ b (v, u) = min(µa(v), µa(u)) and ∑ u̸=v µ→ b (u, v) = constant, η→ b (v, u) = min(ηa(v), ηa(u)) and ∑ u̸=v η→ b (u, v) = constant, ν→ b (v, u) = max(νa(v), νa(u)) and ∑ u̸=v ν→ b (u, v) = constant.   (16) 234 mixed picture fuzzy graph figure 5: regular mixed picture fuzzy graph theorem 3.5. every complete mpfg is a regular mpfg. proof. consider gmpf = (v, e, → e) be a mpfg. from the definition of complete mpfg we have, µb(v, u) = min(µa(v), µa(u)), ηb(v, u) = min(ηa(v), ηa(u)), νb(v, u) = max(νa(v), νa(u)) and µ→ b (v, u) = min(µa(v), µa(u)), η→ b (v, u) = min(ηa(v), ηa(u)), ν→ b (v, u) = max(νa(v), νa(u)) ∀v, u ∈ v. then, the closed neighbourhood µ, η and ν-degree of every vertex is the sum of the positive membership, neutral membership & negative membership values of the vertices and itself respectively. as a result, the closed neighbourhood µdegree, closed neighbourhood η-degree, & closed neighbourhood ν-degree were the same for all vertices. therefore, min. closed neighbourhood degree is equal to max. closed neighbourhood degree. hence gmpf is a regular mpfg. definition 3.17. let gmpf = (a, b, → b) be a mpfg. if two vertices v & u are linked by a length of a path k in gmpf is p : v0, v1, v2, ..., vn−1, vn then µb(v, u), ηb(v, u), νb(v, u) and µ→ b (v, u), η→ b (v, u), ν→ b (v, u) are described as follows µb k(v, u) = min{µb(v, v1), µb(v1, v2), ..., µb(vk−1, u)} ηb k(v, u) = min{ηb(v, v1), ηb(v1, v2), ..., ηb(vk−1, u)} νb k(v, u) = max{νb(v, v1), νb(v1, v2), ..., νb(vk−1, u)} µ→ b k(v, u) = min{µ→ b (v, v1), µ→ b (v1, v2), ..., µ→ b (vk−1, u)} η→ b k(v, u) = min{η→ b (v, v1), η→ b (v1, v2), ..., η→ b (vk−1, u)} ν→ b k(v, u) = max{ν→ b (v, v1), ν→ b (v1, v2), ..., ν→ b (vk−1, u)} let µ∞(v, u), η∞(v, u), ν∞(v, u) is strength of connectedness between the 235 k. k. myithili, c. nandhini two nodes v & u of mpfg. µ∞b (v, u) = sup{µb k(v, u)/k = 1, 2, ...} η∞b (v, u) = sup{ηb k(v, u)/k = 1, 2, ...} ν∞b (v, u) = inf{νb k(v, u)/k = 1, 2, ...} µ∞→ b (v, u) = sup{µ→ b k(v, u)/k = 1, 2, ...} η∞→ b (v, u) = sup{η→ b k(v, u)/k = 1, 2, ...} ν∞→ b (v, u) = inf{ν→ b k(v, u)/k = 1, 2, ...} here inf has been used to determine the minimum membership value and sup is used to determine the maximum membership value. figure 6: strength of connectedness consider a conneted mpfg as shown in the figure 6 the possible paths between v1 to v4 are p1 : v1 − v4 along with the value of membership (0.4, 0.3, 0.2) p2 : v1 − v2 − v4 along with the value of membership (0.4, 0.3, 0.3) p3 : v1 − v3 − v4 along with the value of membership (0.3, 0.2, 0.3) p4 : v1 − v2 − v3 − v4 along with the value of membership (0.3, 0.2, 0.3) p5 : v1 − v2 − v3 − v4 along with the value of membership (0.3, 0.2, 0.3) p6 : v1 − v3 − v2 − v4 along with the value of membership (0.3, 0.2, 0.3) we’ve arrived to this conclusion through routine calculations, µ∞(v1, v4) = sup{0.4, 0.4, 0.3, 0.3, 0.3, 0.3} = 0.4 η∞(v1, v4) = sup{0.3, 0.3, 0.2, 0.2, 0.2, 0.2} = 0.3 ν∞(v1, v4) = inf{0.2, 0.3, 0.3, 0.3, 0.3, 0.3} = 0.2 the strength of connectedness between 2 vertices v1 & v4 of a mpfg is (0.4, 0.3, 0.2) 236 mixed picture fuzzy graph definition 3.18. consider gmpf = (v, e, → e) be a mpfg & v, u be any two distinct vertices. in gmpf , eliminating an edge or arc (v, u) decreases the strength between some pair of vertices and is described to as a bridge. definition 3.19. let g ′ mpf = (a1, b1, → b1) and g ′′ mpf = (a2, b2, → b2) be two mpfgs. a homomorphism h : g ′ mpf → g ′′ mpf is a mapping function h from v1 to v2 if: • µa1(v1) ≤ µa2(h(v1)) ηa1(v1) ≤ ηa2(h(v1)) νa1(v1) ≥ νa2(h(v1)) • µb1(v1, u1) ≤ µb2(h(v1), h(v2)) ηb1(v1, u1) ≤ ηb2(h(v1), h(v2)) νb1(v1, u1) ≥ νb2(h(v1), h(v2)), ∀v1 ∈ v1 & v1, u1 ∈ e1 • µ→ b1 (v1, u1) ≤ µ→ b2 (h(v1), h(v2)) η→ b1 (v1, u1) ≤ η→ b2 (h(v1), h(v2)) ν→ b1 (v1, u1) ≥ ν→ b2 (h(v1), h(v2)), ∀v1 ∈ v1 & v1, u1 ∈ → e1 definition 3.20. let g ′ mpf = (a1, b1, → b1) and g ′′ mpf = (a2, b2, → b2) be two mpfgs. an isomorphism h : g ′ mpf → g ′′ mpf is a bijective mapping function h from v1 to v2 if: • µa1(v1) = µa2(h(v1)) ηa1(v1) = ηa2(h(v1)) νa1(v1) = νa2(h(v1)) • µb1(v1, u1) = µb2(h(v1), h(v2)) ηb1(v1, u1) = ηb2(h(v1), h(v2)) νb1(v1, u1) = νb2(h(v1), h(v2)), ∀v1 ∈ v1 & v1, u1 ∈ e1 • µ→ b1 (v1, u1) = µ→ b2 (h(v1), h(v2)) η→ b1 (v1, u1) = η→ b2 (h(v1), h(v2)) ν→ b1 (v1, u1) = ν→ b2 (h(v1), h(v2)), ∀v1 ∈ v1 & v1, u1 ∈ → e1 theorem 3.6. isomorphism of mpfg is an equivalence relation. 237 k. k. myithili, c. nandhini proof. for show that mpfg isomorphism is an equivalence relation, we must first prove that it is reflexive, symmetric, and transitive. reflexive: consider θ : gmpf → gmpf is a mapping, therefore θ is an identity function. hence it is reflexive. symmetric: in isomorphic mpfg gmpf & hmpf , there exist a 1-1 correspondence θ : gmpf → hmpf which sustains adjacency. from θ is 1-1 correspondence from gmpf to hmpf , here 1-1 correspondence θ−1 from hmpf to gmpf which sustains adjacency. hence isomporphism of mpfg is symmetric. transitive: if gmpf is isomorpic to hmpf and hmpf is isomorphic to kmpf , then there are 1-1 correspondences between θ & ϕ from gmpf to hmpf & hmpf to kmpf respectively, which sustains adjacency. it follows ϕ ◦ θ is a 1-1 correspondence between from gmpf to kmpf which sustains adjacency. hence it is transitive. therefore, isomorphism of mpfg is an equivalene relation. definition 3.21. let gmpf = (a, b, → b) be a mpfg. a vertex v of gmpf is said to be busy vertex if µa(v) ≤ dµ(v), ηa(v) ≤ dη(v), νa(v) ≥ dν(v). otherwise, it is called free vertex. definition 3.22. let gmpf = (a, b, → b) be a mpfg. then an edge (v, u) is defined as an effective edge iff µb(v, u) = min(µa(v), µa(u)), ηb(v, u) = min(ηa(v), ηa(u)), νb(v, u) = max(νa(v), νa(u)), µ→ b (v, u) = min(µa(v), µa(u)), η→ b (v, u) = min(ηa(v), ηa(u)), ν→ b (v, u) = max(νa(v), νa(u)). note: when all edges in a graph are effective, the graph is complete. 4 application of mpfg in instagram social media has grown gaining popularity in latest years of its user-friendliness. social media services such as whatsapp, facebook, twitter, and instagram allow people to communicate across long distances. to put it another way, social media has made the entire globe available at the touch of a button. social media sites are also valuable resources for public awareness creation, as they rapidly distribute information about natural disasters and terrorist/criminal attacks to a mass audience. social network is a collection of vertices and edges. persons, groups, countries, associations, locations, business and other entities are represented by vertices, while edges define the relationship between vertices. we commonly use a classical graph to describe a social network, with vertices representing persons and edges representing relationships/flows between vertices. several manuscripts 238 mixed picture fuzzy graph have been shared on social media platforms. however, a classical graph cannot accurately model a social network. since all vertices in a classical graph are extremely significant. as a result, in today’s social networks, every social units (personal or organisational) are given equal weight. in fact, however, not all social units are equal in importance. in a classical graph, all edges (relationships) have the same weight. for example, a person may be well-versed in certain practises. on the other hand, they have no experience of certain activities, and he has a very little knowledge of others. we can easily represent these three kinds (positive, neutral and negative) of vertex and edge membership degrees with a picture fuzzy set, which has three membership values for each element. in instagram, we can classify three activities namely good, neutral and bad activities which is represented in pfg as positive, neutral & negative membership values of a vertex. similarly, edge membership value can be used to describe the strength of relationship between two vertices. since social media has such a vast number of clients, it also contains mutual and single-sided relationships; it is not restricted to directed or undirected relationships. as a result, we have introduced a mixed picture fuzzy graph which includes both directed and undirected edges. it provides a more accurate result than previous methods. for example, in instagram an undirected edge exists when two friends have a mutual relationship. similarly, if a friend-1 follows friend-2 but friend-2 doesn’t then there occurs directed edge. the vertex effect on a social media platform is identified via centrality, which is one of the most significant concepts in social networking. the degree of centrality determines how closely a social squad is linked to other social squads. it essentially provides the social squad’s/person’s participation in the social network. a vertex’s centrality seems more central than that of other vertex’s. the centre people are muchis closer to the others and has access to more information. it should be noticed that a person’s information is shared by a friend of a friend. however, friends of friends communicate less information than direct friends. as a result, the importance of the relationship gradually decreases as it passes from one member to the next along a connected path. in mpfg, suppose a friend-1 directly connected to a friend-2, then we say v1 is circle distance-1(cd-1) friend of v2. the set of all cd-1 friends of v represented as cd1(v). i.e., cd1(v) = {vi ∈ v ; vi is a cd-1 friends of v}. correspondingly, suppose there is a shortest path between v1 & v2 with m edges, then v1 is a cd-m friend of v2. that is, cdm(v) = {vi ∈ v ; vi is a cd-m friends of v}. now, consider cd ′ m(v) = cdm(v) − cd ′ m−1(v), where m = 2, 3, ... and cd ′ m(v) = cdm(v). cd-1 friends are obviously more significant than cd-2 friends, and cd-2 friends are more significant than cd-3 friends, and so on. the linguistic term “more significant” could be denoted by weights(wm). let 0 ≤ wm ≤ 1 have been the weights that gradually decreases, when the cd between the friends increases. then w1 ≥ w2 ≥ ... ≥ wm ≥ .... 239 k. k. myithili, c. nandhini let u1(= vi), u2, u3, ...um(= vj) are the vertices upon the path between vi and vj. we have to derive mpfcd cdmpf(vi, vj) between vi and vj with this path as cdmpf(vi, vj) = m−1∑ n=1 µ(un, un+1) + m−1∑ n=1 η(un, un+1) + m−1∑ n=1 ν(un, un+1) there could be several paths connecting two vertices in a networks. let us assume these paths of equal length whose mpfcd (cdmpf ) seems to be the maximum in gmpf . suppose these are n edgeds in this path of maximum mpfcd, we designate cdnmpf i.e, cd n mpf(vi, vj) represents the mpfcd between the vertices vi & vj in mpfg with the particular path having accurately n edges. we stated that social squad s with atmost cd-p friends. the centrality c(s) of a social squad is defined as follows: c(s) = ∑ u1∈cd11(v) w1cd 1 mpf(v, u1) + ∑ u2∈cd ′ 2(v) w2cd 2 mpf(v, u2) + ... + ∑ up∈cd ′ p(v) wpcd p mpf(v, up) (17) close friends are valued more than the next closest friends, while the significance of the furthest friend gradually decreases. the significance is established by including the weight wi, which stands for cd-i friend, i = 1,2,3,... for example, mpfg of 7 people after 7 days is shown in figure 7. also the link membership values are shown in same figure. in the definition of centrality of a social unit, p can be taken as fixed for a social network. here we assumed that p = 3 and measure the centrality of social squad. here, we take w1 = 1 and wi+1 = 1/2wi, i = 1,2,... 4.1 centrality of v1 here cd1(v1) = {v2, v4, v3} = cd ′ 1(v1), cd2(v1) = {v3, v5, v7, v2}, cd ′ 2(v1) = cd2(v1) − cd ′ 1(v1) = {v5, v7}, cd3(v1) = {v6, v7, v3}, cd ′ 3(v1) = cd3(v1) − cd ′ 2(v1) = {v6, v3}. 240 mixed picture fuzzy graph figure 7: mixed picture fuzzy network now, ∑ u1∈(cd1) ′ (v1) (cdmpf) 1(v1, u1) = {membership values of(v1, v2)}+{membership values of(v1, v4)} + {membership values of (v1, v3)} = (µ(v1, v2), η(v1, v2), ν(v1, v2))+((µ(v1, v4), η(v1, v4), ν(v1, v4)) + (µ(v1, v3), η(v1, v3), ν(v1, v3)) = (0.37, 0.23, 0.26)+(0.3, 0.33, 0.27)+(0.4, 0.32, 0.2) = (1.07, 0.88, 0.73). ∑ u2∈(cd2) ′ (v1) (cdmpf) 2(v1, u2) = (cdmpf) 2(v1, v5) + (cdmpf) 2(v1, v7) = {membership values of{(v1, v5) + (v1, v5)}}+ {membership values of{(v1, v3) + (v3, v7)}} = {(0.3, 0.33, 0.27) + (0.2, 0.3, 0.2)}+ {(0.4, 0.32, 0.2) + (0.33, 0.3, 0.252)} = (1.23, 0.25, 0.922). 241 k. k. myithili, c. nandhini ∑ u3∈(cd3) ′ (v1) (cdmpf) 3(v1, u3) = (cdmpf) 3(v1, v6) + (cdmpf) 3(v1, v3) = {membership values of{(v1, v3) + (v3, v7) +(v7, v6)}}+{membership values of{(v1, v4) + (v4, v5) + (v5, v3)}} = {(0.4, 0.32, 0.2) + (0.33, 0.3, 0.252) + (0.4, 0.2, 0.23)} + {(0.3, 0.33, 0.27) + (0.2, 0.3, 0.2) + (0.52, 0.27, 0.25)} = (2.15, 1.72, 1.402). the centrality of v1 is c(v1) = ∑ u1∈cd ′ 1(v1) w1cd 1 mpf(v1, u1) + ∑ u2∈cd ′ 2(v1) 0.5 × cd2mpf(v1, u2) + ∑ u3∈cd ′ 3(v1) 0.25 × cd3mpf(v1, u3) = (2.2225, 1.935, 1.5415) similarly, we can calculate centralities of other vertices. c(v2)=(1.8125,1.6125,1.363), c(v3)=(0.985,0.755,0.66975), c(v4)=(1.715,1.781,1.4775), c(v5)=(2.818,2.317,1.91225), c(v6)=(2.8005,2.1905,1.824), c(v7)=(3.442,2.33,2.042). 4.1.1 disscussion suppose there are more than one paths between two vertices, we have to choose the shortest distance path to calculate the centrality. from the results, we have centrality of v3 is comparatively less than other vertices. because v3 has less number of mutual friends. so degree of centrality depends on mutual friends and friends of circle distance-i. social networks are built on the backs of millions of users and massive amounts of data. we used a simple numerical example of a mpfg to describe a small social network problem in this study. the smaller examples are really useful in understanding the benefits of our suggested model. 242 mixed picture fuzzy graph 5 conclusion the prime goal of this paper is to just introduce the terms and concepts of a mpfg and examined the various types of mpfg. initially, we present a definition of an mpfg built from a picture fuzzy graph in this paper. few types of degrees were discussed with its properties. we discuss about regular, strong, complete and complement of mpfg are some of the different forms of mpfg. the isomorphic property has also been analysed in mpfg. when comparing to picture fuzzy graph models, the mpfg can boost effectiveness, reliability, flexibility and comparability in modelling complex real-world scenarios. a model has been developed to represent a social network problem using mpfg. the concept of a mpfg can be used to a database system, a computer network, a traffic signal system, a social network, a transportation network and image processing among other things. 6 acknowledgements future work is to develop this concept in the field of transversals of mpfg. references [1] abdelkadir muzey mohammed, mixed graph representation and mixed graph isomorphism, gazi university journal of science, 2017, 30(1), 303310. 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[19] l.a.zadeh, fuzzy sets, information and control, 1965, 8(3), 338-356. 244 ratio mathematica volume 46, 2023 transversal core of intuitionistic fuzzy k-partite hypergraphs myithili k.k* keerthika r† abstract in graph theory, a transversal is a set of vertices incident to every edge in a graph but in intuitionistic fuzzy k-partite hypergraph(ifk-phg), the transversal is a hyperedge which cuts every hyperedges. in this article, intuitionistic fuzzy transversal(ift), minimal ift, locally minimal ift, iftc(intuitionistic fuzzy transversal core) of ifkphg has been defined. it has been proved that every ifk-phg has a nonempty ift. also few of the properties relating to the transversal of ifk-phg were discussed. keywords: ift; minimal ift; locally minimal ift; iftc of ifkphg. 2020 ams subject classifications: 34k36, 57q65, 05c65,93b20. 1 *associate professor (department of mathematics(ca),vellalar college for women, erode638012, tamil nadu, india.); mathsmyth@gmail.com. †assistant professor (department of mathematics, vellalar college for women, erode-638012, tamil nadu, india.); keerthibaskar18@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1076. issn: 1592-7415. eissn: 2282-8214. ©myithili k.k et. this paper is published under the cc-by licence agreement. 202 myithili k.k, keerthika r 1 introduction euler was the first author who found graph theory in 1736. the graph theoretical approach is widely used to solve numerous issues in different areas like computer science, optimization, algebra and number theory. as an application part, the concept of graph has been extended to hypergraph, an edge with more than one or two vertices. an idea of graph and hypergraph was popularized by berge [1976] in 1976. fuzzy graph and fuzzy hypergraph concepts are developed by the authors in j.n.mordeson and nair [2000]. in k.t.atanassov [1999], the author wrote ideas of intuitionistic fuzzy sets(ifs). according to k.t.atanassov [2002], k.t.atanassov [2012], the researcher putforth ideas of intuitionistic fuzzy relations and cartesian products are defined. in myithili and keerthika [2020a], the authors proposed the notion of k-partite hyperedges in ifhgs(intuitionistic fuzzy hypergraphs). certain operations like union, intersection, ringsum, cartesian product were discussed in myithili and keerthika [2020b] . it has numerous application problems in decision-making. in myithili and parvathi [2015], myithili and parvathi [2016], myithili et al. [2014] transversals and its properties on intuitionistic fuzzy directed hypergraphs were discussed. the authors in goetschel [1995], goetschel [1998], goetschel et al. [1996] initiated the concepts like fuzzy transversal and fuzzy coloring in fuzzy hypergraph. in this article an attempt has been made to analyze the transversal and its related properties in ifk-phgs. 2 symbolic representation mnmv-membership and non-membership values fsv-finite set of vertices fifs-family of intuitionistic fuzzy subsets ifh-intuitionistic fuzzy hyperedge onv-open neighborhood of the vertex cnv-closed neighborhood of the vertex ℵ = (∨, ξ,ψ) intuitionistic fuzzy(if) k-partite hypergraph with edge set ξ, vertex set ∨ and k-partite hyperedge ψ h(ℵ) height of ifk-phg fk(ℵ) fundamental sequence (fs) of ifk-phg c(ℵ) core set(cs) of ifk-phg ik(ℵ) induced fundamental sequence(ifs) of ifk-phg 203 transversal core of intuitionistic fuzzy k-partite hypergraphs ℵ(ai,bi) (ai,bi)-level of ifk-phg (ai,bi) edge membership(em) and non-membership values(enmv) t r(ℵ) minimal intuitionistic fuzzy transversal(mift) of ifk-phg 3 preliminaries definition 3.1. myithili and keerthika [2020a] the ifk-phg ℵ is an ordered triple ℵ = (∨, ξ,ψ) where, •∨ = {g1,g2,g3, · · · ,gn} is a fsv, • ξ = {ξ1, ξ2, ξ3, · · · , ξm} is a fifs of ∨, • ξj = {(gi,ωj(gi),νj(gi)) : ωj(gi),νj(gi) ≥ 0, ωj(gi) + νj(gi) ≤ 1}, 1 ≤ j ≤ m, • ξj 6= ∅, 1 ≤ j ≤ m, • ⋃ j supp(ξj) = ∨, 1 ≤ j ≤ m. for all gi ∈ ξ ∃ k disjoint sets ψi, i = 1, 2, · · · ,k and no two vertices in the same set are adjacent such that ξk = k⋂ i=1 ψi = ∅ definition 3.2. myithili and keerthika [2020a] let an ifk-phg be ℵ = (∨, ξ,ψ). the height of ifk-phg is defined by h(ℵ) = {max(min(ωkij )),max(max(νkij ))} for all 1 ≤ i ≤ m and 1 ≤ j ≤ n. also ωkij and νkij are mnmv of the k-partite hyperedge ψij. definition 3.3. myithili and keerthika [2020a] let ℵ be an ifk-phg. suppose ψj,ψk ∈ψ and 0 < δ,ε ≤ 1. the (δ,ε)-level is defined by (ψj,ψk)(δ,ε) = {gi ∈ ∨|min(ωδkij (gi)) ≥ δ,max(ν ε kij (gi)) ≤ ε}. definition 3.4. myithili and keerthika [2020a] let ℵ be ifk-phg, ℵai,bi = 〈 ∨ai,bi, ξai,bi 〉 be the (ai,bi)-level of ℵ. the sequence of real numbers {a1,a2, · · · ,ak; b1,b2, · · · ,bk} 3 0 ≤ ai ≤ hω(ℵ) and 0 ≤ bi ≤ hν(ℵ), satisfies: (i) if a1 < δ ≤ 1 & 0 ≤ ε < b1 then ψδ,ε = ∅, (ii) if ai+1 ≤ δ ≤ ai; bi ≤ ε ≤ bi+1 then ψδ,ε = ψai,bi , (iii) ψai,bi @ ψai+1,bi+1 is fundamental sequence of ifk-phg and it is denoted as fk(ℵ). definition 3.5. myithili and keerthika [2020a] let c(ℵ) = {ℵa1,b1,ℵa2,b2, · · · ,ℵak,bk} be core set of ℵ. the analogous set of (ai,bi)-level hypergraphs is ℵa1,b1 ⊂ ℵa2,b2 ⊂ ··· ⊂ ℵak,bk is said to be ℵ-ifs and it is denoted by ik(ℵ). the (ak,bk)level is known as support level of ℵ. ℵak,bk is known as the support of ℵ. definition 3.6. myithili and keerthika [2020a] let ℵ = (∨, ξ,ψ) & ℵ′ = (∨′, ξ′,ψ′ ) are ifk-phgs, ℵ is known as partial ifk-phg of ℵ′ , if 204 myithili k.k, keerthika r ∨ ′ = { min (supp (ωkij )) |ωkij ∈ ψ ′ max (supp (νkij )) |νkij ∈ ψ ′ the partial ifk-phg generated by ψ ′ and is represented as ℵ ⊆ ℵ′ . also, if ℵ⊆ℵ′ and ℵ 6= ℵ′ exists then ℵ⊂ℵ′ . definition 3.7. myithili and keerthika [2020a] let ℵ be the ifk-phg, c(ℵ) = {ℵa1,b1,ℵa2,b2, · · · ,ℵak,bk}. ℵ is called as ordered if c(ℵ) is ordered (i.e) ℵa1,b1 ⊂ ℵa2,b2 ⊂ ··· ⊂ ℵak,bk . the ifk-phg is known as simply ordered if {ℵai,bi|i = 1, 2, · · · ,k} is simply ordered, (i.e) if it is ordered and if ψ ∈ ℵai+1,bi+1\ℵai,bi then ψ * ℵai,bi . 4 main results definition 4.1. consider an ifk-phg ℵ. an ift t of ifk-phg is an if subset of ∨ with t (ψj,ψk) ∩ a (ψj,ψk) 6= ∅ for each a ∈ ψ where ψj = min(ωkij ) and ψk = max(νkij ) ∀ 1 ≤ i ≤ m, 1 ≤ j ≤ n. also ωkij and νkij is the mnmv of kth partition of jth edge in ith vertex. definition 4.2. a minimal ift t for ifk-phg be a transversal of ℵ, which satisfies the condition that if t1 ⊂ t , then t1 is not ift of ℵ. note: the set of minimal ift of ifk-phg is denoted as t r(ℵ). always t r(ℵ) 6= ∅. example 4.1. an ifh (intuitionistic fuzzy hypergraph) with ∨ = {g1,g2,g3,g4,g5,g6,g7,g8}, ξ = {ξ1, ξ2, ξ3, ξ4} has been considered. 205 transversal core of intuitionistic fuzzy k-partite hypergraphs figure 1: intuitionistic fuzzy hypergraph using the above figure we can construct an ifk-phg ℵ, with ψ = {ψ1,ψ2,ψ3} disjoint hyperedges which are represented below as incidence matrix the minimal ift of ifk-phg is attained as follows, 206 myithili k.k, keerthika r the corresponding graph is shown below. figure 2: ℵ and minimal ift of ℵ definition 4.3. if t is an ifs with t (ai,bi) as minimal ift (mift) of ℵ(ai,bi) for each (ai,bi) ∈ (0, 1) then t is called as locally minimal ift (lmift) of ifk-phg. the set containing lmift of ifk-phg is written as t r∗(ℵ) theorem 4.1. if t is an ift of ℵ then h(t ) ≥ h(ψj) for ψj ∈ ψ. also, if t is the minimal ift of ifk-phg, then 207 transversal core of intuitionistic fuzzy k-partite hypergraphs h(t ) = {max(min(ωkij )),max(max(νkij )) | ωkij,νkij ∈ ψ} = h(ℵ). theorem 4.2. every ifk-phg has a nonempty ift. note: every ift of ifk-phg contains a mift. theorem 4.3. if t ′ ∈ t r(ℵ) and for every g ∈ ∨, t ′ (g) ∈ fk(ℵ), then fk(t r(ℵ)) ⊆ fk(ℵ). theorem 4.4. t r(ℵ) is sectionally elementary. proof. let fk(t r(h )) = a1,a2, · · · ,ak; b1,b2, · · · ,bk. assume that t ′ ∈ t r(h ) and some δ,ε ∈ (ai,bi) such that t (ai,bi) ⊂ t (δ,ε). since t r(ℵai,bi ) = t r(ℵδ,ε), ∃ some a ∈ t r(h ) 3 a ai,bi = t δ,ε. then t δ,ε ⊂ a ai,bi implies the ifs ∨(gi) defined by ∨(gi) = { (δ,ε) if x ∈ a ai,bi \ t ai,bi a (gi) otherwise is an ift of ifk-phg. here ∨ < a , implies the contradiction of minimality (cm) of a . theorem 4.5. for every a ∈ t r(ℵ), a a1,b1 is a minimal ift of ℵa1,b1 . proof. for any ifk-phg ℵ = (∨, ξ,ψ), consider a minimal ift t of ℵa1,b1 such that t ⊂ a a1,b1 . define the ifs ∨(gi) where ∨(gi) = { (a2,b2) if x ∈ a a1,b1 \ t a (gi) otherwise by the above theorem, ∨ is an ift of ifk-phg, cm of a . definition 4.4. let ℵ be ifk-phg. the intuitionistic fuzzy transversal core (iftc) of ℵ is ℵ′ = (∨′, ξ′,ψ′ ) with the following condition that (i) min t r(ℵ) = min t r(ℵ′ ), (ii) ℵ′ = ∪ min t r(ℵ), (iii) ψ \ ψ′ is exactly the set containing vertices of ℵ which does not belong to t r(ℵ),where ψ′ is the remaining hyperedge set, after deleting hyperedges that are correctly contained in another hyperedge. the remarks of the statement is, (i) for any ifk-phg without spike hyperedges, ∃ transversal core which are always unique. (ii) the definition also holds good for ifk-phgs with spike (a hyperedge with single vertex) hyperedges. 208 myithili k.k, keerthika r definition 4.5. in ifk-phg, the onv gi is the set containing adjacent vertices of gi except itself in a k-partite hyperedge and is denoted as nk(gi). example 4.2. consider an ifk-phg with ∨ = {g1,g2,g3,g4,g5,g6,g7}, ξ = {ξ1, ξ2, ξ3} where, ξ1 = {g1 〈0.5, 0.2〉 ,g2 〈0.3, 0.4〉 ,g3 〈0.6, 0.3〉}, ξ2 = {g2 〈0.3, 0.4〉 ,g4 〈0.2, 0.5〉 ,g5 〈0.3, 0.4〉}, ξ3 = {g3 〈0.6, 0.3〉 ,g6 〈0.4, 0.3〉 ,g7 〈0.1, 0.7〉} with ψ1 = {g1 〈0.5, 0.2〉 ,g4 〈0.2, 0.5〉 ,g7 〈0.1, 0.7〉}, ψ2 = {g2 〈0.3, 0.4〉 ,g6 〈0.4, 0.3〉}, ψ3 = {g3 〈0.6, 0.3〉 ,g5 〈0.3, 0.4〉} here g1 and g7 are the onv g4 in ψ1. definition 4.6. in ifk-phg, the cnv gi is the set containing adjacent vertices of gi including the vertex in a k-partite hyperedge and is denoted as nk[gi]. example 4.3. from the above example it is clear that the closed neighborhood of the vertex g3 is g3 and g5 in ψ3. theorem 4.6. in ℵ, the following claims are related (i) t is an ift of ifk-phg, (ii) t ai,bi∩a ai,bi 6= ∅, for all ifh a ∈ ψ and every (ai,bi) with 0 < ai ≤ hω(ℵ), 0 < bi ≤ hν(ℵ), (iii) t ai,bi is an ift of ℵai,bi , for each (ai,bi) with 0 < ai ≤ δ, 0 < bi ≤ ε. proof. from the definition, ”a minimal ift t for ifk-phg is a transversal of ℵ, which satisfies the property that if t1 ⊂ t , then t1 is not an ift of ℵ” the result is immediate. theorem 4.7. for a simple ifk-phg, t r(t r(ℵ)) = ℵ. theorem 4.8. for any ifk-phg, t r(t r(ℵ)) ⊆ℵ. proof. from definition 4.4, ∃ a ℵ′ (partial hypergraph) of a simple ifk-phg 3 t r(ℵ′ ) = t r(ℵ). from theorem 4.7, t r(t r(ℵ)) = t r(t r(ℵ′ )) implies ℵ′ ⊆ℵ. theorem 4.9. let ℵ be an ifk-phg and suppose that t ∈ t r(ℵ). if ℵ′ ⊆ supp(t ) ⊆ ℵ, then ∃ a hyperedge of ifk-phg a , (ai,bi) ∈ a represents the mnmv of a 3 (i) (ai,bi) = h(a ) = h(t ai,bi ) > 0, (ii) th(a ) ∩ ah(a ) = ℵ. proof. let 0 < h(t ai,bi ) ≤ 1 and ψ′ be the set of all if k-partite hyperedges where h(τai,bi ) ≥ h(t ai,bi ) for each τ ∈ ψ′ . 209 transversal core of intuitionistic fuzzy k-partite hypergraphs since t ai,bi is an ift of ℵai,bi and ℵ′ ⊆ t ai,bi is nonempty. further, for each τ ∈ ψ′ , h(τ) ≥ h(τai,bi ) ≥ h(t ai,bi ) is true. also, assume that t ai,bi is the mift, then for all τ ∈ ψ′ , h(τ) > h(t ai,bi ) and ∃ℵτ 6= ℵ with ℵτ ∈ τh(τ)∩th(τ). define an ifk-phg ℵ1 3 ℵ1(u) =   t (u) whenever u 6= ℵ′, min (h(a )/h(a ) < h(t ai,bi )),max (h(a )/h(a ) < h(t ai,bi )) whenever u = ℵ′ hence ℵ1 is an ift of ifk-phg and h(ℵ ai,bi 1 ) < h(t ai,bi ), it does not meet the basic requirement of t . for each τ ∈ ψ′ satisfies the first part of the theorem 4.9 and has ℵτ which is not in ℵ with ℵτ ∈ τh(τ) ∩ th(τ). the procedure is repeated, and the argument of (i) provides a contradiction and bringing close to the proof. theorem 4.10. let ℵ be an ifk-phg. then, ∃t ∈ t r(ℵ) with ℵ′ ⊆ supp(t ) ⊆ ℵ, if and only if for a ∈ ψ it meets the following requirements: (i) (ai,bi) = h(a ), (ii) the level cut (aj,bj) of h(a ′ ) is not a subhypergraph of the level cut (ai,bi) of h(a ), for all a ′ ∈ ψ with h(a ′ ) > h(a ), (iii) the level cut (ai,bi) of h(a ) does not contain any other hyperedge of ℵh(a ), where (ai,bi) denotes mnmv of a . proof. necessary part: (i) let t ∈ t r(ℵ) and 0 < h(t ai,bi ) ≤ 1. condition (i) is followed from theorem 4.9. (ii) suppose that for each a satisfying (i) ∃ a ′ ∈ ψ 3 h(a ′ ) > h(a ) and a ′ h(a ′ ) ⊆ ah(a ), then ∃ u 6= ℵ ′ , with u ∈ a ′h(a ′) ∩ th(a ′) ⊆ ah(a ) ∩ th(a ) which differs from the concept of theorem 4.9. (iii) assume for each a satisfying (i) and (ii) then ∃ a ′ ∈ ψ so that ∅ 6= a ′ h(a ) ⊂ ah(a ). since a ′ h(a ) 6= ∅ and by (ii), we have h(a ′ ) = h(a ) = (ai,bi). if (aj,bj) = h(a ′) and a ′′ ∈ ψ such that ∅ 6= a ′′h(a ) ⊂ a ′h(a ) ⊂ ah(a ). the process is continued and the chain ends finitely, without loss of abstraction assume (ai,bi) < h(a ). but, ∃u 6= ℵ ′ 3u ∈ a ′h(a )∩th(a ) ⊆ ah(a )∩th(a ), which contradicts theorem 4.9. sufficient part: let a ∈ ψ satisfy the condition (i), (ii) and (iii). by condition (i), the process is trivial. by condition (ii) and (iii) ∃ u ∈ a ′h(a ′) \ ah(a ) for every a ′ ∈ ψ 3 a ′ 6= a and h(a ′ ) ≥ h(a ). let ∨a be the set of all vertices of ℵ 3 ∨a ∩ ah(a ) = ∅. the initial sequence of transversals are constructed. so τs ⊆ ∨ for all s, 210 myithili k.k, keerthika r 1 ≤ s < i and τi ⊆∨a ∪∨i. hence, ∨i ∈ τi for each i. the process is terminated till it reaches a minimal ift with (ai,bi) = h(a ) = h(t ai,bi ). theorem 4.11. let ℵ be an ifk-phg with fk(ℵ) = {a1,a2, · · · ,ak; b1,b2, · · · ,bk} so that 0 ≤ ai ≤ hω(ℵ), 0 ≤ bi ≤ hν(ℵ). also, ℵai,bi ⊆ a ′ , be the elementary ifk-phg if and only if h(a ′ ) = (ai,bi) and supp(a ′ ) is a hyperedge of ℵai,bi . then t r(t r(ℵ)) is the partial ifk-phg of ℵai,bi . proof. from theorem 4.5 and by the construction of minimal ift, the (ai,bi)level hypergraph of t r(ℵ) is t r(ℵai,bi ) which means that (t r(ℵ))ai,bi = t r(ℵai,bi ). let τ belongs to t r(t r(ℵ)). from theorem 4.9, h(τ(∨i)) > 0, this implies that ∃ t ∈ t r(ℵ) with h(τ(∨i)) = h(t ). from theorem 4.1, h(t ) = (max(min(ωkij )),max(max(νkij ))) = h(ℵ) for all minimal ift t . hence τ is elementary with h(ai,bi). since supp(τ) = τai,bi , theorem 4.5 suggest that supp(τ) is the minimal ift of (t r(ℵ))ai,bi . it is obvious that supp(τ) is a hyperedge of ℵai,bi . hence τ is a hyperedge of ℵai,bi . theorem 4.12. let ℵ be an ifk-phg with ℵai,bi is a simple. then t r(t r(h )) = ℵai,bi . proof. by the above theorem, t r(t r(ℵ)) ⊆ ℵai,bi . let τ be an elementary with h(t ) = (ai,bi) and supp(τ) ∈ ℵai,bi . by theorem 4.11, supp(τ) is a minimal ift of (t r(ℵ))ai,bi . since each minimal ift of t r(ℵ) is elementary by definition of minimal ift the process ends at (ai,bi)-level and τ ∈ t r(t r(ℵ)). hence ℵai,bi ⊆ t r(t r(ℵ)) which implies ℵai,bi = t r(t r(ℵ)). 5 conclusion in this article, some interesting concepts like ift, minimal ift, locally minimal ift and iftc of ifk-phgs were discussed. it is important to note that iftc exists for both spike and non-spike intuitionistic fuzzy k-partite hyperedges. in future, the authors planned to work on robotics with multi-task concept as an application part of ifk-phg. references c. berge. graphs and hypergraphs. north holland, new york, 1976. r. h. goetschel. introduction to fuzzy hypergraphs and hebbian structures. fuzzy sets and systems, 76:113–130, 1995. 211 transversal core of intuitionistic fuzzy k-partite hypergraphs r. h. goetschel. fuzzy colorings of fuzzy hypergraphs. fuzzy sets and systems, 94:185–204, 1998. r. h. goetschel, w. l. craine, and w. voxman. fuzzy transversals of fuzzy hypergraphs. fuzzy sets and systems, 84:235–254, 1996. j.n.mordeson and p. nair. fuzzy graphs and fuzzy hypergraphs. physica verlag, new york, 2000. k.t.atanassov. intuitionistic fuzzy sets theory and applications. physica verlag, new york, 1999. k.t.atanassov. on index matrix representation of the intuitionistic fuzzy graphs. notes on intuitonistic fuzzy set, 4:73–78, 2002. k.t.atanassov. on intuitionistic fuzzy sets theory. springer, new york, 2012. k. myithili and r. keerthika. types of intuitionistic fuzzy k-partite hypergraphs. aip conference proceedings, 2261:030012–1 – 030012–13, 2020a. k. myithili and r. keerthika. properties of strong and complete intuitionistic fuzzy k-partite hypergraphs. turkish journal of computer and mathematics education, 11(2):784–791, 2020b. k. myithili and r. parvathi. transversals of intuitionistic fuzzy directed hypergraphs. notes on intuitionistic fuzzy sets, 21(3):66–79, 2015. k. myithili and r. parvathi. properties of transversals of intuitionistic fuzzy directed hypergraphs. advances in fuzzy sets and systems, 21(1):93–105, 2016. k. myithili, r. parvathi, and m. akram. certain types of intuitionistic fuzzy directed hypergraphs. international journal of machine learning and cybernetics, 2:1–9, 2014. 212 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 48, 2023 inventory model with preservation technology and exponential holding cost in fuzzy scenario hemalatha shunmugam * annadurai karuppuchamy † abstract inventories are ubiquitous in the business sector. since inventory is most frequently incurring expense, stock control is critical for an organization and it must be scrimping and saving in contemplation of function the merchandising fruitfully. in this paper, an inventory model for a deteriorating item under exponential holding cost with collaborative preservation technology investment under carbon policy is considered. also, this study is developed in a fuzzy scenario by employing triangular fuzzy numbers. signed distance method is utilized to enhance decision making and optimization. further the convexity of the total cost function for both the crisp and the fuzzy case is established. the objective is to determine the optimal investment in preservation technology and the optimal cycle length so as to minimize the total cost. moreover, some managerial results are obtained by using sensitivity analysis and graphical representation is also carried out. the applications of the proposed model is used in the fields of constructing machinery or heavy duty construction equipment, specific chemicals and processed food. keywords: carbon emission; preservation technology; signed distance method; stock dependent demand; triangular fuzzy number. 2010 ams subject classification: 94d05, 35q93.‡ *ph. d scholar, mother teresa women’s university, kodaikanal. ssm institute of engineering and technology, dindigul, india; shemalatha1974@gmail.com. † m.v. muthiah government arts college for women, dindigul, india; drkannadurai@gmail.com. ‡ received on november 3, 2022. accepted on july 10, 2023. published on august 1, 2023. doi: 10.23755/rm.v39i0.951. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement inventory model with preservation technology and exponential holding cost in fuzzy scenario 1. introduction the displayed inventory level is a promotional device in today's globalized technology to boost income. according to numerous researches, having a significant number of products on display may attract more consumers. this implies a positive relationship between demand and stock levels. as a result, demand in this model is considered as stock-dependent, which is more practical. bardhan et al. [1] investigated an optimal replenishment policy and preservation technology investment for a noninstantaneous deteriorating item with stock-dependent demand. preservation technology is a critical component in reducing deterioration. giri et al. [7] have explored a supply chain model for time-dependent deteriorating item with preservation technology investment. for a single-vendor multi-buyer model, setiawan and endrayanto [11] implemented a coordination strategy and synchronization in the production flow, including adjustable lead time. khanna et al. [9] adopted an optimizing preservation strategy for deteriorating items with time-varying holding cost and stock dependent demand. global warming poses a significant hazard to our planet. the world's attention is currently focused on reducing carbon emissions. dye and yang [6] anticipated that ordering and storing inventory causes carbon emissions. they looked at sustainability in the context of a collaborative trade credit arrangement, where demand is tied to the credit period. daryanto and wee [4] considered a production lot size decision of a manufacturer incorporating environmental impact of carbon emission. tao and xu [14] developed an inventory model concerning emission-regulation policies with consumer’s low carbon awareness, providing decision support. shen et al. [13] developed a production inventory model for deteriorating items with collaborative preservation technology investment under carbon tax. yu et al. [3] presented an inventory model of a deteriorating product considering carbon emissions. patel et al. [5] decided optimal order quantity for the industries especially chemical industries with trended demand under trade credit with existence of cap and trade structure to reduce carbon emissions. tripathi and mishra [12] investigated an eoq model with linear time dependent demand and different holding cost functions. in the above analysis, it is presumed that all parameters are precisely known. but in real world, parameters are imprecise in nature and one has to deal with approximation of numbers that are close to real numbers. fuzzy number provides a way to model this epistemic uncertainty and its propagation. bjork [2] analysed an eoq model in a fuzzy environment. alrefaei and tuffaha [10] studied an intuitionistic polygonal fuzzy numbers. hemalatha and annadurai [8] proposed an integrated production-distribution inventory system for deteriorating products in fuzzy environment by ensuring extra investment thereby reducing setup cost. we develop the model including some points which highlight the novelty of our model. in our model, the deterioration effect of the product is considered and preservation technology is addressed to regulate the deterioration rate. demand is cogitated as stock-dependent and the holding cost is ruminated as an exponential. under hemalatha shunmugam and annadurai karuppuchamy carbon emission regulations, the goal is to resolute the optimal investment in preservation technology and cycle time (dye & yang, [6]). as the path of developing eoq models with uncertainty expressed as fuzzy numbers are quite profitable, the fuzzy model is discussed in this study. the cost parameters are considered as a triangular fuzzy numbers. the total cost function is defuzzified and proven to be convex using the signed distance method. following an introduction, the remainder of the article is organized as follows: the second section is devoted to the related preliminary definitions. in section 3, notations and assumptions are shown. in section 4, a mathematical model for the crisp model is developed and another mathematical model for a fuzzy model is developed in section 5. numerical example is provided to illustrate the crisp and fuzzy models in section 6. in section 7, sensitivity analysis and managerial insights are provided to validate the concept. comparative study is given in section 8. finally, conclusion and future research direction are given in the last section. 2. preliminaries the following definitions of fuzzy sets are relevant to the method used in the proposed model. definition 2.1 triangular fuzzy numbers: let ( )1 2 3 1 2 3, , ,d p p p p p p=   be a triangular fuzzy number with membership function: 1 1 2 2 1 3 2 3 3 2 , ( ) , 0 , . b x p p x p p p p x x p x p p p otherwise −   −  − =   −     definition 2.2 signed distance method (bjork [2]): the signed distance of  ,l r  measured from 0 is  ( )0 ( ) ( ) d , , 0 . 2 l r d d l r    + = for the triangular fuzzy number ,b r −  the distance from d to 0 is written as ( , 0)d d ( )1 2 3 1 2 . 4 p p p= + + 3. notations and assumptions to develop the proposed model, we adopt the following notations and assumptions. notations: the notations used in our model are listed as follows: e t length of cycle e  investment in preservation technology per unit time ece a fixed carbon emission per order ece c carbon emission per unit per order eoc c cost of ordering (per order) inventory model with preservation technology and exponential holding cost in fuzzy scenario edc c unit cost due to deterioration ehc c holding cost per unit time t ( ( ) e rt ehc ehc c t h= ) epc c purchasing cost per unit ece h carbon emission for inventory per unit time ( ) eh i t level of inventory at time , 0 et t t  q size of order ( ),ep e etc t  total cost of the system 0 y rate of deterioration in the absence of preservation technologies ( )y  investment in preservation technologies reduces the rate of deterioration assumptions: the following assumptions are made in the model: 1) the rate of demand is directly proportional to the stock level. i.e., ( ( )) ( ), 0, 0 1. eh eh d i t a bi t a b= +    2) the time horizon is infinite with negligible lead time. 3) shortages are not allowed. 4) preservation technology investment reduces the rate of deterioration gradually. the reduced deterioration rate ( ), e y  is a function of preservation technology cost e  such that 0 ( ) ,e u e y y e   − = which satisfies the conditions / 0, ep e tc    2 2/ 0 ep e tc    and 0 (0) ,y y= where u is sensitivity parameter of investment 0 1u  . 5) the holding cost is considered to be dependent on time as ( ) e , 0 1. rt ehc ehc c t h r=   6) the total amount of carbon emissions includes emissions from ordering, holding and purchasing inventory. 4. mathematical model in this section, a mathematical model is developed to determine the cycle time and optimal investment in preservation technology. deteriorating items are likewise regarded with low carbon emission cost and preservation technology in our inventory system. the equation governing the inventory level (khanna et al. [9]) can be expressed as follows: ( ) ( ) ( ) ( ( )), 0 t .eh eh eh e di t y i t a bi t t dt + = − +   (1) the solution of equation (1) using the boundary condition (t ) 0, eh e i = is given by ( ) ( ) ( )( ) 2 ( ) ( ) / 2 , eh e e e i t a t t t t y b = − + − +   (2) and the initial inventory level is ( )( )( )2(0) ( ) / 2 .eh e e eq i a t t y b = = + +  (3) the total cost of the system is calculated by adding the following costs. ordering cost: , eoc c hemalatha shunmugam and annadurai karuppuchamy preservation technology investment: , e e pt t= holding cost: 0 ( )= et rt ehc eh ehc h e i t dt ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )2 2 2 2 3 3/ / 1/ ( ) / 2 / 2 / 2 / 2 / ,− = − + − + + − − + +  e ert rt ehc e e e e ah t r e r r y b t r t r e r r deterioration cost: ( )( ) 0 ( ) et edc eh edc c q d i t dt= − ( )( )2 2 3( ) 3 / 6 ,edc e e e ec t y a abt ab t = − −  and purchasing cost: ( )( )( )2 ( ) / 2 .epc epc e e eepc c q ac t t y b = = + +  then total carbon emissions tec in a finite time horizon t (dye & yang, [6]) is ( ) ( )( )( ) ( ) ( )( )( ) 0 2 2 3 , ( ) ( ) / 2 / 2 ( ) / 6 . et ep e e ece ece ece eh ece ece e e e ece e e e tc t a c q h i t dt a ac t t y b ah t t y b    = + + = + + + + + +  ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( )( )( ) 2 2 2 2 3 3 2 2 2 3 2 . ., , / / / / 1 / ( ) / 2 / 2 / 2 / 2 / / ( ) / 2 / ( ) 3 / 6 / 1 ( ) / 2 e e rt ep e e eoc e ehc e e rt e e epc e e e edc e e e e e e e e ece ece e e e ece e e i e tc t c t ah t t r e r r y b t r t r e r r ac t t t y b c t t y a abt ab t t t a ac t t y b ah t t − = + − + − + +    − − + + + + +    + − − +   + + + + +        ( ) ( )( )( ) ( )2 3/ 2 ( ) 6 . 4/e et y b  + +    the objective is to minimize the total cost by jointly optimizing the cycle time e t and the investment in preservation technology e  . to establish optimality, taking the first order partial derivative and equate it into zero, we get / 0 ep e tc t  = and / 0,ep etc   = (5) that is ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( ) 2 3 2 2 2 0 0 3 2 2 0 0 0 2 0 0 / / 1 / / 2 / / 2 / 6 3 2 2 / 2 / / 6 3 2 0, − − − − − − − − + + + − + + + − − + + + − −  + + − + + + =   e e e e e e e e rt u u eoc e ehc e e ehc e u u u ehc ehc e epc edc e e u u ece ece e ece e c t ah e r t r y e b rt ah r t y e b ah r ah r t ac y e b c ay e abt y e ab t ac y e b a t ah t y e b        (6) and ( ) ( )( ( ) ( )) ( ) ( ) ( )( ) ( ) 3 2 3 0 0 0 2 0 0 1 / / 2 / / (3 ) / 6 / 6 / 2 0. − − − − − − − + + − − − − − + = e e e e e e u rt u u ehc e ehc e ehc ehc e dc e e u u ece e epc ece e h auy e r t uy e ah t r ah r ah r t c uy e t a abt ah t uy e a c c t uy e      (7) we now derive the optimal values of e t and e as et  and e   by simultaneously solving equations (6) and (7). we derive the total cost of the system by replacement these values into equation (4). equation (3) is used to determine the optimal order quantity. by examining the second order sufficient conditions, for the total cost equation (4) to be minimum are inventory model with preservation technology and exponential holding cost in fuzzy scenario ( )( )( ) ( ) ( ) ( ) 2 2 2 02 3 3 3 2 3 0 0 03 3 3 2 2 2 2 2 2 2 0 , 3 3 e e e e e rt ep ueoc ehc ehc e e e e e e u eceu uehc edc ece e e tc c ah e ah r y e b rt r t t t r t r t ah y e bah c ab a y e b y e b r t t     − − − − −  = + − + + + −  + + + − + + +  ( ) 2 ( )2 20 02 3 2 3 22 2 00 0 2 (3 ) 0, 6 3 2 e e e e e e u rt ep uehc ehc e ehc ehc e e uu u epc ece eedc e e ece e tc h au y e ah t ah ah u y e r t r r r t a c c t u y ec au y e t bt ah t u y e      − + − −− −    = − + − +     +− + + +  and ( ) ( ) 2 2 ( ) 0 03 3 3 2 00 0 1 1 1 2 (3 2 ) 2 . 6 2 3 e e e e e e u rt ep ep uhc e hc e e e e e e uu u epc eceedc e ece e tc tc h auy e rt ah uy e t t r t r r t a c c uy ec auy e bt ah t uy e       − + − −− −     = = − + − −        +− − − − then, we have ( )( ) ( )( )( )( ( )( ) ( )( )( ))2 2 2 2 2 2, / . , / , / . , / 0.ep e e e ep e e e ep e e e e ep e e e etc t t tc t tc t t tc t t          −        (8) since all the second order derivatives are highly nonlinear, the optimality is determined graphically (figure 1). 5. fuzzy model the fuzzy inventory model, including the fuzzification and defuzzification processes, is described in this section.. the method of fuzzification involves transforming crisp parameters into fuzzy parameters. fuzzy variables can be represented by the membership function given in the preliminary section for triangular fuzzy numbers. here, we consider the ordering cost, holding cost, holding cost component, demand parameters, and purchasing cost as uncertain. they are represented as triangular fuzzy numbers as follows: 1 2 1 2 ( , , ), 0 , 0, eoc eoc eoc eoc eoc c c c c c= − +      3 4 3 4 ( , , ), 0 , 0, edc edc edc edc edc c c c c c= − +      5 6 5 6 ( , , ), 0 , 0 ,a a a a a= − +      7 8 7 8 ( , , ), 0 , 0,b b b b b= − +      9 10 9 10 ( , , ), 0 , 0, ehc ehc ehc ehc ehc h h h h h= − +      (9) 11 12 11 12 (r , , ), 0 , 0 , hc hc hc hc hc r r r r= − +       13 14 13 14 ( , , ), 0 , 0. epc epc epc epc epc c c c c c= − +      then the left and right  cuts of the various parameters eoc c , edc c , ehc h , r , ,a b and epc c are given by ( ) ( )1 1 2 20; 0,l reoc eoc eoc eocc c c c   = − +   = − +   hemalatha shunmugam and annadurai karuppuchamy ( ) ( )3 3 4 40; 0,l redc edc edc edcc c c c   = − +   = − +   ( ) ( )5 5 6 60; 0,l ra a a a   = − +   = − +   ( ) ( )7 7 8 80; 0,l rb b b b   = − +   = − +   (10) ( ) ( )9 9 10 100; 0,l rehc ehc ehc ehch h h h   = − +   = − +   ( ) ( )11 11 12 120; 0,l rr r r r   = − +   = − +   ( ) ( )13 13 14 140; 0.l repc epc epc epcc c c c   = − +   = − +   hence, when the costs eoc c , edc c , ehc h , r , ,a b and epc c in equation (4) are fuzzified with triangular fuzzy numbers eoc c , edc c , ehc h , r , ,a b and epc c as expressed in equation (10). thus, the total cost is obtained in fuzzy sense is given by ( ) ( ) ( ) ( ) 2 2 2 2 3 3 2 2 2 3 ( ) 21 2 2 , 2 ( ) ( ) 3 2 6 1 − +  − −   = + + − + − + +         + − −    + + + +         + + e ert rt e eoc ehc e e e ep e e e e e e e e e eepc edc e e e e e e ece ec e y bc ah t t te e tc t t t r r r r r r r t y b t y a abt ab tac c t t t t t a ac t      ( ) ( )2 32( ) ( ) . 2 2 6     + +     + + +           e e e e e e e ece t y b t y bt t ah   (11) from equation (11), we obtain the left and right  cuts of ( ),ep e etc t  is as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 2 2 2 2 2 3 3 2 ( )1 , 2 ( )2 2 2 2 ( ) 3 −  +−  = + + − +     +−   − + + + +         − + el l l l l l l el l l l l l l r t e leoc l ehc e ep e e e e r t e e ll epce e e e e e l ledc e y bc a h t e tc t t t r r r t y ba ct t e t r tr r r t y a a bc t                     ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) 2 3 2 3 2 6 ( ) ( )1 , 2 2 6  −   +         + +     + + + + +           l e l l e e e e e e l e e l e ece l ece e l ece e t a b t t t t y b t y bt a a c t a h t           inventory model with preservation technology and exponential holding cost in fuzzy scenario ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 2 2 2 2 2 3 3 2 ( )1 , 2 ( )2 2 2 2 ) ( 3 − −  +−  = + + − +      +−   − + + +         − + er r r r r r r er r r r r r r r t e reoc r ehc e ep e e e e r t e e rr epce e e e e e r redc e y bc a h t e tc t t t r r r t y ba ct t e t r tr r r t y a a bc t                     ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) 2 3 2 3 2 6 ( ) ( )1 . 2 2 6  −   +         + +     + + + + +           r e r r e e e e e e r e e r e ece r ece e r ece e t a b t t t t y b t y bt a a c t a h t           the process of defuzzifying involves turning the fuzzified results into quantifiable quantities. hence, the fuzzified total cost equation (11) narrated with triangular fuzzy number is transformed into the crisp function by utilizing signed distance formula. then the defuzzified total cost is calculated and is given by ( ) ( ) ( ) ( ) 6 62 43 51 2 2 2 3 3 6 6 6 6 6 6 6 2 2 2 3 4 3 3 4 3 43 7 2 2 3 ( ) 21 2 2 ( , , 0) 2 ( ) ( ) 2 6 (1 3 − − +  − −  = + + − + − + +         + − − + + + +        + + + e eh t h t ee e e ep e e e e e e e e e e e e e e e e e e ece ece e e y hh h t t th e e d tc t t t h h h h h h h t y h t y h h h t h h th h th t t t t t y a h c t t      ( ) ( )324 4 3 ) ( ) . 2 2 6     + + + +            e ee ece h t y ht h h  (12) where, 2 1 1 0, 4 eoc h c  − = +  2 4 3 1 ( ) 4 edc h c= +  −  , 3 6 5 1 ( ) 4 h a= +  −  , 4 8 7 1 ( ) 4 h b= +  −  , 5 10 9 1 ( ) 4 ehc h h= +  −  , 6 12 11 1 ( ) 4 hc h r= +  −  , and 7 14 13 1 ( ) 4 epc h c= +  −  since all the second order derivatives are highly nonlinear, the optimality is determined graphically (figure 2). 6. numerical example in this section, a suitable example is given to illustrate the model. here, we consider an inventory system with the same data as in khanna et al. [9] and yu et al. [3]. eoc c = 40/order, edc c = 50/year, u = 0.05, 0 y = 0.09, ehc h = 0.7 per unit per year, r = 0.5, a = 25, b = 0.07, epcc = 90 per unit, 0.02ecea = , 0.1ecec = , 0.1eceh = . moreover, we summarize the input parameters as fuzzy triangular values and defuzzified values in table 1. the total cost for the crisp model is ( ),ep e et c t t = 2460, hemalatha shunmugam and annadurai karuppuchamy the cycle time is et = 0.8 year and investment in preservation technology e t = 40.52. the total cost of the fuzzy model is ( ), ep e etc t  = 1631, the cycle time is et = 0.92 year and investment in preservation technology et =37.59. 7. sensitivity analysis we inspect the effects of variations in the system variables eoc c , a , b , ehc h , r , and epc c on the optimal ordering quantity q the cycle time is e t and investment in preservation technology e  with minimum total expected cost. the optimal values of q , e t , e  and ( ),ep e et c t t are derived, when one of the parameters changes (increases or decreases) by 25% and all other parameters remain unchanged. the results of sensitivity analysis are presented for both the cases in table 2 and are graphically shown in figures 3 – 8. on the basis of the results of table 2 and figures 3 – 8, we see that fuzzy model provides best optimal solution as compared to crisp model. 7. 1 managerial insights in this section, we study the effect of changes in the cost components of the system on the optimal length of the cycle e t  , the optimal ordering quantity q  , the optimal investment in preservation technology e   and the minimum total cost for crisp model as  ep t c and for fuzzy model as  ep t c . a sensitivity analysis is carried out by considering the same numerical example and computed results are shown in table 2. based on the computational results, we obtain the following managerial insights. (1) it's interesting to note that increasing the holding costs components hc r has a positive effect. this will lead to a decrease in q , e t , e  and ( ),ep e et c t t . but increase in the values of the holding costs components hc h will lead to an increase in q , e t , e  and ( ),ep e et c t t . (2) the optimal solution for several values of d , increase in demand parameter a results increase in q and e  . this result has implication on the holding cost, ordering cost as well as delivery cost. therefore, an increase in a will lead to an increase of ( ),ep e et c t t and decrease in et . (3) from table 2, the values of q , et , and e decrease with decrease in the values of parameter b but increases of ( ),ep e et c t t (4) it is foreseeable that if the buyer's ordering cost eoc c rises, ( ),ep e et c t t and q will increases. this is because, for high values of ordering cost, departing from the inventory model with preservation technology and exponential holding cost in fuzzy scenario optimal solution has a substantial effect on et and e respectively. as a result, an increase in eoc c will result in an increase in et and e in both circumstances. (5) in table 2, with an increase in purchasing cost epc c , e and ( ),ep e et c t t increases but decrease of et andq . fuzzy input parameters triangular fuzzy numbers defuzzified values eoc c (30, 40, 50) 37.5 edc c (40, 50, 60) 45 a (20, 25, 30) 22.5 b (0.06, 0.07, 0.08) 0.06 ehc h (0.6, 0.7, 0.8) 0.6 r (0.4, 0.5, 0.6) 0.45 epc c (80, 90, 100) 65 table 1. fuzzy input parameters as triangular values 8. comparative study from table 2, it is observed that triangular fuzzy number gives the best optimum solution. the fuzzy model with triangular fuzzy numbers generates a better result than the crisp model with the total cost with 33.70% savings. in this paper, it is shown that the knowledge of the crisp model is gradually improved to a fuzzy model with triangular fuzzy and fine-tuned our model into more specific knowledge with minimum total cost. the main reason for this situation is the low carbon emission cost and exponential holding cost which helps to increase the sales and a positive impact on customer preference. the triangular fuzzy model finds lower values of =e 37.59 and total cost ( ), ep e etc t  = 1631 (better) at each performance criterion than the crisp model e t = 40.52 and total cost ( ),ep e et c t t = 2460 indicating that total cost is higher than the fuzzy model with triangular fuzzy numbers. thus fuzzy model gives a better result than the crisp model. hence, fuzzy model gives the advantages of the application of fuzzy in real-world environment on supply chain management. hemalatha shunmugam and annadurai karuppuchamy figure 1. graphical representation figure 2. graphical representation for crisp model for fuzzy model figure 3. effect of holding cost ehc h figure 4. effect of holding cost component r figure 5. effect of demand parameter a figure 6. effect of demand parameter b figure 7. effect of ordering cost eoc c figure 8. effect of purchasing cost epc c 0 0.5 1 1.5 2 0 50 100 150 200 2.4 2.5 2.6 2.7 2.8 2.9 3 x 10 4 cycle timeinvestment in preservation technology t o ta l c o s t 0 0.5 1 1.5 2 0 50 100 150 200 1600 1650 1700 1750 1800 1850 cycle timeinvestment in preservation technology t o ta l c o s t 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 holding cost t o ta l c o st crisp fuzzy 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 holding cost component r t o ta l c o st crisp fuzzy 10 15 20 25 30 35 40 500 1000 1500 2000 2500 3000 3500 4000 demand parameter a t o ta l c o st crisp fuzzy 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 demand parameter b t o ta l c o st crisp fuzzy 15 20 25 30 35 40 45 50 55 60 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 ordering cost t o ta l c o st crisp fuzzy 20 40 60 80 100 120 140 500 1000 1500 2000 2500 3000 3500 4000 purchasing cost t o ta l c o st crisp fuzzy inventory model with preservation technology and exponential holding cost in fuzzy scenario table 2. effects of parameters on optimal solution p a ra m e te r % crisp parameter value s fuzzy parameter value s crisp optimal values fuzzy optimal values e t  e   q   ep t c e t  e   q   ep t c eoc c -50 20 18.75 0.6 38.63 15.37 2431 0.76 35.72 17.59 1608 -25 30 26.25 0.7 39.59 18.01 2446 0.83 36.48 19.25 1618 0 40 37.50 0.75 40.52 19.33 2460 0.92 37.59 21.40 1631 +25 50 41.25 0.81 41.40 20.92 2472 0.95 37.95 22.12 1635 +50 60 52.50 0.87 42.24 22.52 2484 1.04 38.98 24.29 1646 a -50 12.5 10.9 0.91 28.99 11.85 1271 1.10 25.29 12.55 824.8 4 -25 18.75 16 0.81 35.65 15.71 1867 1.00 31.74 16.63 1181 0 25 22.5 0.75 40.52 19.33 2460 0.92 37.59 21.40 1631 +25 31.25 25 0.71 44.38 22.82 3050 0.90 39.43 23.23 1803 +50 37.5 29.63 0.68 47.59 26.18 3640 0.86 42.42 26.26 2122 b -50 0.04 0.038 0.84 41.85 21.45 2423 0.99 38.45 22.84 1609 -25 0.05 0.045 0.8 41.30 20.49 2435 0.96 38.13 22.21 1616 0 0.07 0.06 0.75 40.52 19.33 2460 0.92 37.59 21.40 1631 +25 0.09 0.08 0.72 39.98 18.66 2483 0.88 37.08 20.62 1650 +50 0.1 0.09 0.70 39.77 18.19 2495 0.87 36.89 20.46 1660 ehc h -50 0.35 0.34 0.69 36.87 17.75 2440 0.86 34.53 19.98 1617 -25 0.525 0.47 0.72 38.86 18.54 2450 0.89 36.19 20.69 1624 0 0.70 0.60 0.75 40.52 19.33 2460 0.92 37.59 21.40 1631 +25 0.875 0.73 0.78 41.93 20.12 2469 0.95 38.82 22.12 1637 +50 1.05 0.83 0.80 43.17 20.64 2478 0.97 39.66 22.59 1642 r -50 0.25 0.23 1.46 58.14 38.50 2595 1.76 54.65 41.89 1728 -25 0.375 0.31 0.94 47.10 24.37 2496 1.25 46.43 29.33 1670 0 0.5 0.45 0.75 40.52 19.33 2460 0.92 37.59 21.40 1631 +25 0.625 0.55 0.67 36.58 17.22 2444 0.82 33.91 19.03 1619 +50 0.75 0.65 0.63 34.19 16.18 2436 0.77 31.53 17.85 1613 epc c -50 45 35 0.97 37.24 25.24 1295 1.13 34.83 26.51 930.1 7 -25 67.5 50 0.84 38.99 21.73 1879 1.01 36.30 23.58 1281 0 90 65 0.75 40.52 19.33 2460 0.92 37.59 21.40 1631 +25 112.5 85 0.69 41.85 17.73 3039 0.83 39.11 19.24 2095 +50 135 93 0.64 43.04 16.41 3616 0.81 39.66 18.76 2281 hemalatha shunmugam and annadurai karuppuchamy 9. conclusions in this study, we examined the impacts of cycle time and investment in preservation technology on an inventory model with exponential holding cost in a fuzzy scenario. our research revealed that various integrated inventory models would be beneficial for both the seller and buyer in cases where the cost parameters take the form of a triangular fuzzy number. we obtained more information about the cost parameters in relation to the decision variables and total fuzzy profit from the managerial insights. the model’s viability is investigated using numerical analysis and sensitivity analysis. the present study can be extended with the trade credit financing policy, seasonal and expiry products, inflation and multi items. references [1] s. bardhan, h. pal and b. c. giri. optimal replenishment policy and preservation technology investment for a non-instantaneous deteriorating item with stock-dependent demand. operational research, 19(2): 347-368, 2019. 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[11] r. setiawan and i. endrayanto. analysis of the single-vendor multi-buyer inventory model for imperfect quality with controllable lead time. iaeng international journal of applied mathematics, 51(3): 2021. [12] r. p. tripathi and s. m. mishra. eoq model with linear time dependent demand and different holding cost functions. international journal of mathematics in operational research, 9(4): 2016. [13] y. shen, k. shen and c. yang. a production inventory model for deteriorating items with collaborative preservation technology investment under carbon tax. sustainability, 11(18): 5027, 2019. [14] z. tao and j. xu. carbon-regulated eoq models with consumer’s low-carbon awareness. sustainability, 11(4): 1004, 2019. ratio mathematica volume 44, 2022 split domination decomposition of path graphs e. ebin rajamerly1 praisy b2 abstract a decomposition (g1, g2, g3, …, gn) of g is said to be a split domination decomposition (sdd), if the following conditions are satisfied:(i) each gi is connected(ii)γs(gi) = i, 1≤ i ≤ n. in this paper, we prove that path, path corona and subdivision of path graph admit sdd. keywords: split domination, decomposition, split domination decomposition. 2010 ams subject classification: 05c12, 05c693 1associate professor, research department of mathematics, nesamony memorial christian college, marthandam. tamilnadu, india. mail id: ebinmerly@gmail.com 2research scholar, research department of mathematics, nesamony memorial christian college, marthandam, tamilnadu, india. mail. id: praisyblessed@gmail.com 3received on june 28th, 2022. accepted on aug 10th, 2022. published on nov30th, 2022.doi: 10.23755/rm.v44i0.904. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by license agreement. 175 e. ebin raja merly and praisy b 1. introduction graph theory in mathematics refers to the study of graphs. the theory of domination is one of the rapidly developing areas in graph theory. the concept of split domination was developed by veerabhadrappa r. kulli and bidarahalli janakiram [2]. another important concept in graph theory is decomposition of graphs. decompositions are imposed by applying several conditions on gi in the decompositions by several authors based on their studies. we introduce a new concept split domination decomposition of a graph which is motivated by the concepts of linear path decomposition [3] and connected domination decomposition [4]. we have considered here simple undirected graphs without loops or multiple edges. the order and size of the graph are indicated by p and q respectively. terms not defined here are used in the sense of frank harary [1]. 2. preliminaries definition2.1: a dominating set d of a graph g = (v, e) is a set of vertices such that each vertex of g is either in d or has at least one neighbor in d. definition 2.2: a dominating set d of a graph g = (v, e) is a split dominating set if the induced sub graph <v-d> is disconnected. the split domination number γs(g ) of g is the minimum cardinality of a split dominating set. definition2.3: if g1, g2, g3, …, gn are edge disjoint sub graphs of g such that e(g) = e (g1) e (g2) … e (gn), then (g1, g2, g3,…, gn) is said to be decomposition of g. definition2.4: the corona pp ⊙ k1 is the graph constructed from a copy of pp, where for each vertexu ∈ v(pp), a new vertex u ′and a pendent edge uu′ are added. it is denoted by pp + and is called comb. definition 2.5: a subdivision of a graph g is a graph obtained by inserting a new vertex in each edge of g and is denoted by s(g). 3. split domination decomposition definition 3.1: a decomposition (g1, g2, g3, …, gn) of g is said to be a split domination decomposition (sdd), if the following conditions are satisfied: (i). each gi is connected (ii). 𝛾𝑠(𝐺𝑖 ) = 𝑖, 1≤ 𝑖 ≤ 𝑛. 176 split domination decomposition of path graphs example 3.2: g g1 g2 figure 1: graph g and its sdd (g1, g2) remark 3.3: a path with 3 vertices have split domination number 1 and path having 3k-2,3k-1 and 3k vertices have the split domination number k, k. 2. theorem 3.4: a pathp𝑝, 3n2−3n+6 2 ≤ p ≤ 3n2+n+2 2 admits split domination decomposition (g1, g2, g3, …, gn) if and only if∑ γs(gi) = n i=1 𝑛(𝑛+1) 2 . proof: let pp=u1u2 … up be a path of order p. assume that pp , 3n2−3n+6 2 ≤ p ≤ 3n2+n+2 2 admits split domination decomposition (g1, g2, g3,…, gn). clearly γs(gi) = i , 1 ≤ i ≤ n. therefore γs(g1) + γs(g2) + ⋯ + γs(gn) = 1 + 2 + ⋯ + n ∑ γs(gi) = n i=1 n(n + 1) 2 conversely, assume that ∑ γs(gi) = n i=1 n(n+1) 2 . clearly γs(gi) = i , 1≤ i ≤ n. therefore pp admits split domination decomposition (g1, g2, g3, …,gn). next, we have to find the bound for p. by remark 3.3, the subgraphs g1, g2, g3, …, gnof pp having minimum possible vertices are g1 = u1u2u3 g2 = u3u4u5u6 g3 = u6u7u8u9u10u11u12 ⁞ gn = umum+1 … up wherem = 3n2−9n+12 2 , p = 3n2−3n+6 2 clearly |v(pp)| = |v(g1)| + |v(g2)| + ⋯ + |v(gn)| u5 u2 𝑢1 u3 u4 u1 u1 u2 u3 u5 u3 u4 u1 177 e. ebin raja merly and praisy b p =(3+4+… +3𝑛 − 2) − (𝑛 − 1)= 3n2−3n+6 2 next, the maximum possible vertices of subgraphsg1, g2, g3,…,gn of p𝑝 are g1 = u1u2u3 g2 = u3u4u5u6u7u8 g3 = u8u9u10u11u12u13u14u15u16 ⁞ gn = umum+1 … up wherem = 3n2−5n+4 2 , p = 3n2+n+2 2 clearly |v(pp)| = |v(g1)| + |v(g2)| + ⋯ + |v(gn)| p = (3+6+… +3n) − (n − 1) = 3n2+n+2 2 therefore 3n2−3n+6 2 ≤ p ≤ 3n2+n+2 2 . corollary 3.5: if 3n2+n+2 2 < 𝑝 < 3n2+3n+6 2 , then ppdoes not admit split domination decomposition. theorem 3.6:pp + admits split domination decomposition (g1, g2, g3, …, gn) if and only if pp has n2+n 2 (n > 1)vertices. proof: let pp=u1u2 … up be a path with p vertices. if we join the verticesu1 ′ , u2 ′ , … , up ′ to u1, u2, … , up respectively, then we get pp +. assume that pp has n2+n 2 (n > 1)vertices. to prove pp + admits split domination decomposition (g1, g2, g3,…,gn). suppose p = n2+n 2 g1 =< {𝑢1, u2, u1 ′ } > g2 =< {u2, u3, u2 ′ , u3 ′ } > g3=< {𝑢3, u4, u5, u6, u4 ′ , u5 ′ , u6 ′ } > ⁞ gn =< {𝑢l, ul+1, … , up, ul+1 ′ , … , up ′ } > notice that the minimum split dominating set of gn has vertices and pp has 1+2+3+…. + = n(n+1) 2 = n2+n 2 vertices. clearly γs(gi) = i , 1 ≤ i ≤ n. therefore (g1, g2, g3, . . . ,gn)is a split domination decomposition of pp +. conversely, suppose pp + admits split domination decomposition. to prove pp has n2+n 2 , (n > 1)vertices. suppose not, case (i):|v(pp)| > n2+n 2 we join m vertices in pp where m=1, 2, 3, …, or n. constructing (g1, g2, g3…, gn) in the above, we have remaining m vertices where m=1, 2, 3, …, or n. we cannot arrange 178 split domination decomposition of path graphs the m vertices in the minimum split dominating set of gi otherwise (g1, g2, g3, …, gn) would not be a split domination decomposition for pp +. if these m vertices alone to give a sub graph gkm , then (g1,g2, …,gn , gkm ) would not be a split domination decomposition for pp +which is a contradiction. case (ii):|𝑉(pp)| < n2+n 2 . we eliminate m vertices in pp where =1, 2, 3, … or n − 1.constructing(g1, g2, g3, …, gn) in the above, we have remaining m vertices where 𝑚 = 𝑛 − 1, n-2, .., or n-(n-1). we cannot arrange the m vertices in the minimum split dominating set of gi otherwise (g1, g2, g3,…, gn) would not be a split domination decomposition for pp +. if these m vertices alone to give a sub graph gkm , then(g1,g2 ,…,gn−1 , gkm ) would not be a split domination decomposition for pp +which is a contradiction. therefore pp has n2+n 2 , (n > 1)vertices. note 3.7: in general, if pp admits split domination decomposition, then s (pp) need not admit split domination decomposition and vice-versa. so we cannot use the range of p as in theorem 3.4 to s (𝑃𝑝). theorem 3.8: let pp be a (p, q) -path. subdivision of the path graph s (pp) admits split domination decomposition (g1, g2, g3,…,gn)if and only if 2n2−6n+14 2 ≤ p ≤ n2+5n−8 2 . proof: let pp=u1u2 … up be a path with p vertices. then s (pp) has 2p − 1vertices. assume that s (pp) admits split domination decomposition. now we can find the range of p if and only if s (pp) admits split domination decomposition. from note 3.7, we can’t apply the range of 𝑝 in ppas in theorem 3.4 tos (pp). hence using the range of p in pp to s (pp), the following table shows the probabilities for s (pp) admits split domination decomposition. table:1 no. of decompositions (𝑛) 2 3 4 5 6 7 no. of vertices in𝑃𝑝(𝑝) 4 7 8 11 12 13 14 17 18 19 20 21 25 26 27 28 29 34 35 36 37 38 39 no. of vertices in 𝑆( 𝑃𝑝) 7 13 15 21 23 25 27 33 35 37 39 41 49 51 53 55 57 67 69 71 73 75 77 179 e. ebin raja merly and praisy b using newton’s forward difference formula, we have to find the upper and lower bound of p for pp such that s (𝑃𝑝) admitssplit domination decomposition. to find the lower bound of p, using table-1. by newton’s forward formula,p = p0+ u 1! ∆p0+ u(u−1) 2! ∆2p0 + ⋯ where u= n−𝑛0 ℎ = n − 3 ( n0 = 3 andh = 1) here p0 = 7, ∆p0 = 4, ∆ 2p0 = 2 , ∆ 3p0 = 0 therefore = 2n2−6n+14 2 next, to find the upper bound of p, using table-1. by newton’s forward formula, = p0+ u 1! ∆p0+ u(u−1) 2! ∆2p0 + ⋯ where u = n−𝑛0 ℎ = n − 3 ( n0 = 3 and h = 1) here p0 = 8, ∆p0 = 6, ∆ 2p0 = 1 , ∆ 3p0 = 0 therefore = n2+5n−8 2 therefore s(pp) admits split domination decomposition, if 2n2−6n+14 2 ≤ p ≤ n2+5n−8 2 . conversely, assume that 2n2−6n+14 2 ≤ p ≤ n2+5n−8 2 . to prove s (pp) admit split domination decomposition. suppose not, consider the lower bound of p, if we eliminate one vertex frompp,then the corresponding s(pp) will not admit split domination decomposition. hence p = 2n2−6n+14 2 − 1 < 2n2−6n+14 2 .which is a contradiction. consider the upper bound of p, if we join one vertex to pp, then the corresponding s (pp) will not admit split domination decomposition. hencep= n2+5n−8 2 +1> n2+5n−8 2 .which is a contradiction. therefore s (pp) admits split domination decomposition. 4. conclusion in this paper, we deal that path, path corona and subdivision of path graph admits split domination decomposition. further investigations could also be done to get the condition at which some graphs admit split domination decomposition. 180 split domination decomposition of path graphs references [1] harary f, “graph theory”, narosa publishing house, new delhi (1998) [2] kulli, v. r and janakiram, b., “the split domination number of a graph”, graph theory notes of new york, xxxii,16-19(1997). [3] e. ebin raja merly and n. gnanadhas, (2011), “linear path decomposition of lobster”, international journal of mathematics research. volume 3, number 5. 447455. [4] jeya jothi d, e. ebin raja merly, “connected domination decomposition of helm graph”, international journal of scientific research and review.vol.7, issue10,2018. [5] sr little femilin jana. d., jaya. r., arokia ranjithkumar, m., krishnakumar. s., “resolving sets and dimension in special graphs”, advances and applications in mathematical sciences 21 (7) (2022), 3709 – 3717. 181 https://scholar.google.com/citations?view_op=view_citation&hl=en&user=xwcp70yaaaaj&sortby=pubdate&authuser=1&citation_for_view=xwcp70yaaaaj:ijcspb-oge4c the transposition in hypercompositional structures ratio mathematica, 21, 2011, pp. 75-90 75 t h e t r a n s p o s i t i o n a x i o m i n h y p e r c o m p o s i t i o n a l s t r u c t u r e s ch. g. massouros a,b and g. g. massouros a,b a technological institute of chalkis, gr34400, evia, greece b 54, klious st., gr15561, cholargos-athens, greece abstract. the hypergroup (as defined by f. marty), being a very general algebraic structure, was subsequently quickly enriched with additional axioms. one of these is the transposition axiom, the utilization of which led to the creation of join spaces (join hypergroups) and of transposition hypergroups. these hypergroups have numerous applications in geometry, formal languages, the theory of automata and graph theory. this paper deals with transposition hypergroups. it also introduces the transposition axiom to weaker structures, which result from the hypergroup by the removal of certain axioms, thus defining the transposition hypergroupoid, the transposition semi-hypergroup and the transposition quasi-hypergroup. finally, it presents hypercompositional structures with internal or external compositions and hypercompositions, in which the transposition axiom is valid. such structures emerged during the study of formal languages and the theory of automata through the use of hypercompositional algebra. ams-classification number: 20n20, 68q70, 51m05 1. the transposition axiom in hypergroups hypercompositional structures are algebraic structures equipped with multivalued compositions, which are called hyperoperations or hypercompositions. a hypercomposition in a non-void set h is a function from the cartesian product h h to the powerset  p h of h . hypercompositional structures came into being through the notion of the hypergroup. the hypergroup was introduced by f. marty in 1934, during the 8 th congress of the scandinavian 76 mathematicians [18]. f. marty used hypergroups in order to study problems in non-commutative algebra, such as cosets determined by non-invariant subgroups. a hypergroup, which is a generalization of the group, satisfies the following axioms: i.     ab c a bc  for all a,b,c h  (associativity), ii. ah ha h   for all a h  (reproduction). note that, if «» is a hypercomposition in a set h and a, b are subsets of h , then a b signifies the union  a ,b a b a b    . in both cases, aa and aa have the same meaning as  a a and  a a respectively. generally, the singleton  a is identified with its member a . in [18], f. marty also defined the two induced hypercompositions (right and left division) that result from the hypercomposition of the hypergroup, i.e.   a x h | a xb b     and   a x h | a bx b     . it is obvious that the two induced hypercompositions coincide, if the hypergroup is commutative. for the sake of notational simplicity, w. prenowitz [48] denoted division in commutative hypergroups by a / b . later on, j. jantosciak used the notation a / b for right division and b\ a for left division [14]. notations :a b and ..a b have also been used correspondingly for the above two types of division [21]. in [14] and then in [15], a principle of duality is established in the theory of hypergroups. more precisely, two statements of the theory of hypergroups are dual statements, if each results from the other by interchanging the order of the hypercomposition, i.e. by interchanging any hypercomposition ab with the hypercomposition ba . one can observe that the associativity axiom is self-dual. the left and right divisions have dual definitions, thus they must be interchanged in a construction of a dual statement. therefore, the following principle of duality holds: given a theorem, the dual statement resulting from interchanging the order of hypercomposition “” (and, necessarily, interchanging of the left and the right divisions), is also a theorem. this principle is used throughout this paper. the following properties are direct consequences of axioms (i) and (ii) and the principle of duality is used in their proofs [see also 20, 21]: 77 property 1.1. ab   is valid for all the elements a,b of a hypergroup h . p r o o f. suppose that ab   for some ,a b h . per reproduction, ah h and bh h . hence,    h ah a bh ab h h       , which is absurd. property 1.2. a / b   and a\ b   for all the elements a,b of a hypergroup h . p r o o f. per reproduction, hb h for all b h . hence, for every a h there exists x h , such that a xb . thus, /x a b and, therefore, /a b   . dually, a\ b   . property 1.3. in a hypergroup h , the non-empty result of the induced hypercompositions is equivalent to the reproduction axiom. p r o o f. suppose that /x a   for all ,x a h . thus, there exists y h , such that x ya . therefore, x ha for all x h , and so h ha . next, since ha h for all a h , it follows that h ha . per duality, h ah . conversely now, per property 1.2, the reproduction axiom implies that a / b   and a\ b   for all a,b in h . property 1.4. in a hypergroup h equalities (i) h h / a a / h  and (ii) h a\ h h \ a  are valid for all a in h . p r o o f. (i) per property 1.1, the result of hypercomposition in h is always a non-empty set. thus, for every x h there exists y h , such that y xa , which implies that /x y a . hence, /h h a . moreover, /h a h . therefore, h h / a . next, let .x h since h xh , there exists y h such that a xy , which implies that /x a y . hence, /h a h . moreover, /a h h . therefore, h a / h . (ii) follows by duality. the hypergroup (as defined by f. marty), being a very general algebraic structure, was enriched with additional axioms, some less and some more powerful. these axioms led to the creation of more specific types of hypergroups. one of these axioms is the transposition axiom. it was introduced by w. prenowitz, who used it in commutative hypergroups. w. prenowitz called the resulting hypergroup join space [48]. thus, join space (or join hypergroup) is defined as a commutative hypergroups h , in which / /a b c d   implies ad bc   for all , , ,a b c d h (transposition axiom) is true. this type of hypergroup has been widely utilized in the study of 78 geometry via the use of hypercompositional algebra tools which function without any need of cartesian or other coordinate-type systems [48, 49]. later, j. jantosciak generalized the transposition axiom in an arbitrary hypergroup as follows: \ /b a c d   implies ad bc   for all , , ,a b c d h . he named this particular hypergroup transposition hypergroup and studied its properties in [14]. the transposition axiom also emerged in the hypercompositional structures which surfaced during the study of formal languages through the use of hypercompositional algebra tools [see, for example, 6, 7, 27, 32, 33, 35, 36, 42, 44; see also 7, 12, 13 for other occurrences of the join space]. the manner in which these structures emerged will be discussed in paragraph 3. in the present paragraph we will only deal with the mathematical description of join space classes which resulted from the theory of formal languages and automata. the basic concept which generated these types of join spaces is the incorporation of a special neutral element e into a transposition hypergroup. this neutral element e possesses the property  ,ex xe e x  for every element x of the hypergroup and was named strong. thus, the fortification of transposition hypergroups by an identity element came into being. therefore a fortified transposition hypergroup is a transposition hypergroup h for which the following axioms are valid: i. ee e , ii. x ex xe  for all x h , iii. for every  x h e  there exists a unique  y h e  , such that e xy and, furthermore, y satisfies e yx . if the commutativity is valid in h , then h is called a fortified join hypergroup. theorem 1.1. in a fortified transposition hypergroup h, the identity is strong. p r o o f. it must be proven that  ex e, x for all x in h . this is true for x e . let x e . suppose that y ex . then, x e\ y . however, 1 x e / x   , since 1 e xx   . thus, 1 e\ y e / x   and transposition yields 1 e ee yx    . hence,  y x,e . theorem 1.2. in a fortified transposition hypergroup h, the strong identity is unique. 79 p r o o f. suppose that u is an identity distinct from e . it then follows that there exists z distinct from u , such that u ez . but,  ez e, z , so  u e, z , which is a contradiction. it is worth noting that a transposition hypergroup h becomes a quasicanonical hypergroup, if it incorporates a scalar identity, i.e. an identity e with the property ex xe x  for all x in h . moreover, a join hypergroup is a canonical hypergroup, if it contains a scalar identity [14, 20, 23]. a hypergroup h with a strong identity e has a natural partition. let      | , and c= |a x h ex xe e x x h e ex xe e        . then, h a c  and a c  . a member of a is an attractive element and a member of c is a canonical element. see [39] for the origin of terminology. fortified join hypergroups and fortified transposition hypergroups have been studied in a series of papers [see, for example, 15, 22, 33, 37, 39, 43], in which several very interesting properties of these types of hypergroups were revealed. the following was proven, among others [15]: structure theorem. a transposition hypergroup h containing a strong identity e is isomorphic to the expansion of a quasicanonical hypergroup  c e by the transposition hypergroup a of all attractive elements through the idempotent e . moreover, from the theory of automata resulted the transposition polysymmetrical hypergroup [24, 42, 45], i.e. a transposition hypergroup h , having an identity (or neutral) element e , such that ee e , x ex xe  for all x h and also, for every { }x h e  there exists at least one element { }x h e  , (called symmetric or two-sided inverse of x ), such that e xx and e x x . the set of the symmetric elements of x is denoted by ( )s x and is called the symmetric set of x . a commutative transposition polysymmetrical hypergroup is called a join polysymmetrical hypergroup. theorem 1.3. if a polysymmetrical transposition hypergroup contains a strong identity e , then this identity is unique. analytical examples of the above hypergoup types are presented in [28]. a thorough study of transposition hypergroups with idempotent identity is presented [30] 80 2. the transposition axiom in hypergroupoids in the previous paragraph it was mentioned that the hypergroup was enriched with further axioms, a fact which led to the creation of specific hypergroup families. however, mathematical research also followed the reverse course. certain axioms were removed from the hypergroup and the resulting weaker structures were studied. thus, the pair  ,h  , where h is a non-empty set and " " a hypercomposition, was named partial hypegroupoid, while it was called hypegroupoid if ab   for all ,a b h . a hypergroupoid in which the associativity is valid, was called semi-hypergroup, while it was called quasihypergroup, if only the reproductivity is valid. the quasi-hypergroups in which the weak associativity is valid, i.e.    ab c a bc   for all , ,a b c h , were named hv-groups [55]. certain properties of these structures, which are analogous to those of hypergroups, are presented herein. property 2.1. if the weak associativity is valid in a hypergroupoid, then this hypergroupoid is not partial. p r o o f. suppose that ab   for some ,a b h . then,  ab c   for any c h . therefore,    ab c a bc   , which is absurd. hence, ab is nonvoid. the following is a direct consequence of the above property: property 2.2. the result of the hypercomposition in an hv-group h is always a non-empty set. property 2.3. a hypergroupoid h is a quasi-hypergroup, if the results of induced hypercompositions in it are non-void. p r o o f. suppose that /x a   is valid for all ,x a h . then, there exists y h , such that x ya . therefore, x ha for all x h and so h ha . but ha h is also valid for all a h . hence, h ha . by duality, ah h . thus, h is a quasi-hypergroup. property 2.4. a / b   and b\ a   is valid for all the elements a,b of a quasi-hypergroup h . p r o o f. per equality h hb , there exists y h , such that a yb for every a h . thus, /y a b and, therefore, /a b   . b\ a   , per the principle of duality. 81 property 2.5. in a quasi-hypergroup h , the equalities h a / h h \ a  are valid for all a in h . p r o o f. let .x h since h xh , there exists y h such that a xy , which implies that /x a y . hence, /h a h . moreover, /a h h . therefore, h a / h . the other equality follows by duality. property 2.6. in any non-partial hypergroupoid h, the equalities h h / a a\ h  are valid for all a in h . p r o o f. since the result of the hypercomposition in a non-partial hypergroupoid is always a non-empty set, there exists y h such that y xa for every x h . this implies that /x y a . hence, /h h a . moreover, /h a h . therefore, h h / a . the other equality follows by duality. the following is a direct consequence of properties 2.5 and 2.6 above: property 2.7. in any hv-group h, the equalities (i) h h / a a / h  and (ii) h a\ h h \ a  are valid for all a in h . extensive work has been done on the construction of hypergroupoids, on their enumeration and on the study of their structure (see, for example, [3, 4, 5, 6, 9, 10, 11, 29, 50, 51, 52, 54]). as mentioned above, this direction pertained to researching hypercompositional structures resulting from the weakening of the structure of the hypergoup. the opposite direction pertained to researching hypercompositional structures resulting from the reinforcement of the structure of the hypergoup. these two directions are combined in [31], via the introduction of the transposition axiom into the hv-group, thus leading to the following definition: definition 2.1. an hv-group (h,) is called transposition hv-group, if it satisfies the transposition axiom: b\ a c / d   implies ad bc   for all a,b,c,d h . a transposition hv-group (h,) is called join hv-group, if h is a commutative hvgroup, while it is called weak join hv-group, if h is an hv-commutative group. the fortified transposition hv-group was also defined in [31], in a manner analogous to the definition of the fortified transposition hypergroup, as follows: definition 2.2. a transposition hv-group (h,) is called fortified, if h contains an element e , which satisfies the axioms: i. ee e , ii. x ex xe  for all x h , 82 iii. for every  x h e  there exists a unique  y h e  , such that e xy and, furthermore, y satisfies e yx . if “” is commutative, then h is called a fortified join hv-group. properties of the structure above, as well as relevant examples are presented in [31]. the elements of the fortified transposition hv-group are partitioned into canonical and attractive, exactly as in hypergroups. proposition 2.1. let h be a fortified transposition hv-group and suppose that x, y are attractive elements with 1 y x   . then, x, y xy and x, y yx . p r o o f. since x is an attractive element,  ex xe e, x  is valid. therefore,  1e / x x\ e e, x  . moreover,  y / y z | y zy  . hence, e y / y . thus, y / y x\ e   which, per the transposition axiom, results into ey yx   or, equivalently,  e, y yx   . since 1y x , it follows that y yx . similarly, x yx and, per duality, x, y xy . corollary 2.1. a fortified transposition hv-group containing exclusively attractive elements is weakly commutative. as can be observed, the transposition axiom is not dependent on the two hypergroup axioms (asssosiativity and reproduction) and their consequences. therefore, the transposition axiom can be introduced even into a partial hypergroupoid. thus, the notions of the transposition hypergroupoid, of the transposition quasi-hypergroup and of the transposition semi-hypergroup emerge. if the commutativity is also valid in the above, the notions of the join hypergroupoid, of the join quasi-hypergroup and of the join semi-hypergroup emerge as well. the following proposition is analogous to the one used in [31] for the construction of transposition hv-groups. the proof of this proposition, as well as of proposition 2.3 below, is quite straightforward, albeit long, since all the possible cases must be verified. proposition 2.2. let h be a hypergroupoid (either partial or nonpartial) or a quasi-hypergroup. also, let an arbitrary subset abi of h be associated to each pair of elements   2,a b h . if ab a ,b h i    , then h endowed with the hypercomposition: * aba b ab i  , ,a b h is a transposition hypergroupoid or a transposition quasi-hypergroup respectively, while it is a join hypergroupoid or 83 a join quasi-hypergroup, if the commutativity is valid in h and ab bai i for all ,a b h . corollary 2.1. if h is a hypergroupoid (either partial or nonpartial) or a quasi-hypergroup and w is an arbitrary element of h, then h endowed with the hypercomposition  x y xy x, y,w   is a transposition hypergroupoid or a transposition quasi-hypergroup respectively, while it is a join hypergroupoid or a join quasi-hypergroup, if the commutatity is valid in h. proposition 2.3. let h be a set with more that two elements and let w be an arbitrary element in h . two hypercompositions are defined in h as follows:  ,la b a w for all ,a b h and  ,ra b b w for all ,a b h . then,  , lh and  , rh are transposition semi-hypergroups. 3. the transposition axiom in hypercompositional structures with internal compositions m. krasner was the first to expand hypercompositional structures via the creation of structures containing composition and hypercompositions. thus, in 1956, he replaced the additive group of a field with a special hypergroup, thereby introducing the hyperfield. he then used the hyperfield as the proper algebraic tool, in order to define a certain approximation of complete valued fields by sequences of such fields [16, 17]. later, he introduced a more general structure, which relates to hyperfields in the same way rings relate to fields. he called this structure hyperring. additional hypercompositional structures, similar to the above, introduced by various researchers, soon followed. examples of those are the superring and the superfield, in which both the addition and the multiplication are hypercompositions [47]. additionally, the study of formal languages introduced structures in which the hypercompositional component is a join hypergroup. indeed, let a be an alphabet, let *a denote the set of the words defined over a and let  be the empty word. then, set *a is a semigroup with regard to the concatenation of the worlds. this semigroup has  as its neutral element, since a a a   for all a in *a . in addition, the expression a b , where a and b are words over a, is used in formal languages theory to denote «either a or 84 b ». based on the fact that a b is in essence a biset, hypercomposition  ,a b a b  appears in the word set *a . it has been proven that *a is a join hypergroup [32, 33] with regard to this hypercomposition. this hypergroup was named b(iset)-hypergroup. however, since *a is a semigroup with regard to world concatenation and since it has been proven that world concatenation is distributive with regard to the hypercomposition, a new hypercompositional structure thus emerged. this structure was named hyperringoid. definition 3.1. a hyperringoid is a non-empty set y equipped with an operation “” and a hyperoperation "+" , such that: i) (y,+) is a hypergroup, ii) (y, ) is a semigroup, iii) the operation “” is distributive on both sides of the hyperoperation “+ ”. if (y,+) is a join hypergroup, (y,+,) is called join hyperringoid. the join hyperringoid that results from a b-hypergroup is called b-hyperringoid and the special b-hyperringoid that appears in the theory of formal languages is the linguistic hyperringoid. join hyperringoids are studied in [38, 40, 41]. another notion in the theory of formal languages is the null word, the introduction of which resulted from the theory of automata. the null word is symbolized with 0 and is bilaterally absorbing with regard to word concatenation. therefore, the extension of the composition and of the hypercomposition onto  * 0a  results into the following: 0 0 0a a  ,  0 0 0,a a a    for all *a a . with these extensions, structure  ( * 0 , , )a    continues to be a hyperringoid, which, however now also has an absorbing element. the additive structure of these hyperringoid comprises a fortified join hypergroup. thus, a new hypercompositional structure appeared: definition 3.2. if the additive part of a hyperringoid is a fortified join hypergroup whose zero element is bilaterally absorbing with respect to the multiplication, then, this hyperringoid is named join hyperring. a join hyperdomain is a join hyperring which has no divisors of zero. a proper join hyperring is a join hyperring which is not a krasner hyperring. a join hyperring k is called join hyperfield if  * 0k k  is a multiplicative group. join hyperrings are studied in [25, 41]. moreover, hypercompositional structures having external operations and hyperoperations on hypergroups appeared [see, for example, 1, 2, 19, 56]. the 85 notions of the set of operators and hyperoperators from a hyperringoid y over an arbitrary non-void set m were introduced in [33, 34], in order to describe the action of the state transition function in the theory of automata. y is a set of operators over m , if there exists an external operation from m y to m , such that ( ) ( )s s   for all s and , y  and, moreover, 1s s for all s , when y is a unitary hyperringoid. if there exists an external hyperoperation from m y to  p m which satisfies the above axiom, with the variation that 1s s when y is a unitary hyperringoid, then y is a set of hyperoperators over m . if m is a hypergroup and y is a hyperringoid of operators over m , such that, for each , y  and ,s t m , the axioms: (i)  s t s t     , (ii) ( )s s s      hold, then m is called right hypermoduloid over y . if y is a set of hyperoperators, then m is called right supermoduloid. if the second of the above axioms holds as an equality, then the hypermoduloid is 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hadronic press (1994), pp.207-214. 90 ch. g. massouros, e mail: masouros@teihal.gr url: http://www.teihal.gr/gen/profesors/massouros/index.htm g. g. massouros, e mail: germasouros@gmail.com url: https://sites.google.com/site/gerasimosgmassouros/ mailto:masouros@teihal.gr mailto:germasouros@gmail.com ratio mathematica volume 47, 2023 common fixed point theorems for (ϕ,f)integral type conractive mapping on c∗-algebra valued b-metrix space jahir hussain rasheed* maheshwaran kanthasamy† abstract the object of this paper, we establish the concept of integral type of common fixed point theorem for new type of generalized c∗-valued contractive mapping. the main theorem is an existence and uniqueness of common fixed-point theorem for self-mappings with (ϕ,f)contractive conditions on complete c∗-algebra valued b-metric space. moreover, some illustrated examples are also provided. keywords: c∗-algebra valued, common fixed point, b-metric spaces. 2020 ams subject classifications: 47h10, 54h25, 54m20.1 *department of mathematics, jamal mohamed college (autonomous) (affiliated to bharathidasan university), tiruchirappalli-620020, tamilnadu, india; hssn jhr@yahoo.com. †department of mathematics, jamal mohamed college (autonomous) (affiliated to bharathidasan university), tiruchirappalli-620020, tamilnadu, india; mahesksamy@gmail.com. 1received on august 22, 2023. accepted on february 23, 2023. published on april 4, 2023. doi: 10.23755/rm.v41i0.834. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 186 r. jahir hussain and k. maheshwaran 1 introduction in 2002, branciari [2002] introduced the concept of integral type contraction on fixed point solution. many writers researched at the presence of fixed points for a variety of integral type contractive mappings, see liu et al. [2018]. especially, liu et al. [2014] several more fixed point theorems for integral type contractive mappings in complete metric spaces. after that ma et al. [2014] and ma and jiang [2015] presented the notion of c∗-algebra-valued metric space, c∗-algebravalued b-metric space and investigated certain fixed point results for self-mapping under certain contractive conditions. alsulami et al. [2016] investigated that fixed point theorem in the classical banach fixed point theorem can be used to produce c∗-algebra-valued bmetric space in fixed point results kamran et al. [2016]. we symbolize a as an unital c∗-algebra, and ah= {a ∈a :a =a∗}. especially, an element a ∈a is a positive factor, if a = a∗. a natural partial order on ah given by a ≤b if fθ ≤ (b − a), where q signifies the zero element in a. then, let a+ and a′ symbolize the set {a ∈a :θ ≤ a} and the set {a ∈a :ab = ba, ∀ b∈ a}, respectively and |a| = (a∗a) 1 2 . 2 preliminaries definition 2.1 (ma and jiang [2015]). let χ be a non-empty set and ω∈ a such that ω ≥ i. suppose that the mapping db : x × x → a is held, the following constraints exist. (i) θ ≤ db(ζ, η) and db(ζ, η) = θ iff ζ = η; (ii) db(ζ, η) = db(η, ζ); (iii) db(ζ, η) ≤ ω(db(ζ, ϑ) + db(ϑ, η)) for all ζ, η, ϑ ∈ χ. then, db is called c∗-algebra-valued b-metric on x and (χ, a, db) is called c∗-algebra-valued b-metric space. definition 2.2 (ma and jiang [2015]). let (χ, a, db) be c∗-algebra valued bmetric space. assume that {ζn} is a sequence in χ and ζ ∈ χ. if for each ϵ > θ, there exists n such that ∀ n > n , ||d(ζn, ζ)|| ≤ ϵ then {ζn} is alleged to be convergent with regard to a, and {ζn} converges to ζ, i.e., we take limn→∞ ζn = ζ. if for each ϵ > θ, there exists n such that ∀ l n,m > n , ||d(ζn, ζm)|| ≤ ϵ, then {ζn} is referred to as a cauchy sequence in χ. (χ, a, db) is referred to as a complete c∗-algebra-valued b-metric space if every cauchy sequence is convergent in χ. 187 common fixed point theorems for (ϕ,f) integral type contractive mapping on c∗-algebra valued b-metrix space definition 2.3 (mustafa et al. [2021]). let the non-decreasing function f : a+ → a+ be positive linear mapping satisfying following constraints: (i) f is continuous; (ii) f (a) = θ iff a = θ; (iii) limn→∞ fn (a) = θ. definition 2.4 (mustafa et al. [2021]). suppose that a and b are c∗-algebra. a mapping f :a→ b is said to be c∗-homomorphism if : (i) f (aζ + bη) = af(ζ) + bf(η) for all a,b ∈c and ζ,η ∈ a; (ii) f(ζη) = f(ζ)f(η) for all ζ,η ∈ a; (iii) f (ζ∗) = f(ζ)∗ for all ζ ∈ a; (iv) f maps the unit in a to the unit in b. lemma 2.1. let (χ,a,db) be a c∗-algebra valued b-metric space such that db(ζ,η) ∈ a, for all ζ,η ∈ χ where ζ ̸= η. let ϕ : a+ → a+ be a function with the following properties: (i) ϕ(a) = θ if and only if a = θ; (ii) ϕ(a) < a, for all a ∈ a; (iii) either ϕ(a) ≤ db(ζ,η) or db(ζ,η) ≤ ϕ(a), where a ∈ a and ζ,η ∈ χ. corolary 2.1. every c∗-homomorphism is bounded. lemma 2.2. every ∗-homomorphism is positive. definition 2.5 (branciari [2002]). the function ξ : χ → χ is called sub-additive integrable function iff ∀ a,b ∈ χ, ∫ a+b 0 ξdt ≤ ∫ a 0 ξdt + ∫ b 0 ξdt. 188 r. jahir hussain and k. maheshwaran 3 main results definition 3.1. let (χ,a,db) is a complete c∗-algebra valued b-metric space. let l,m, :χ → χ be a integral c∗-valued contractive mapping and f (∫ db(lζ,mη) 0 ξdt ) ≤ f (∫ i(ζ,η) 0 ξdt ) − ϕ (∫ db(ζ,η) 0 ξdt ) (1.1) i(ζ,η) ≤ ( α ∫ db(ζ,η) 0 ξdt + γ ∫ [db(ζ,lζ)+db(η,mη)] 0 ξdt +δ ∫ [db(ζ,mη)+db(η,lζ)] 0 ξdt ) for all ζ,η ∈ χ, where ω ∈ a′+, α + γ + δ ≥ 0 with ωα + γ(ω + 1) + δ (ω(ω + 1)) < 1. f ∈ψ and ϕ ∈ φ and ξ : χ → χ is the lebesgus-integral function. theorem 3.1. let (χ,a,db) is a complete c∗-algebra valued b-metric space. 3.1 are ∗−homomorphisms and with the constraint f ( a) ≤ ϕ( a) and ξ : χ → χ is a lebesgus-integral mapping which is summable, non-negative and such that for all ε > 0, ∫ ε 0 ξdt > 0. then l and m have a unique common fixed point in χ. proof. let ζ0 ∈ χ and define ζn= lζn−1, ζn+1= mζnwe have f (∫ db(ζn,ζn+1) 0 ξdt ) = f (∫ db(lζn−1,mζn) 0 ξdt ) ≤ f (∫ i(ζn−1,ζn) 0 ξdt ) − ϕ (∫ db(ζn−1,ζn) 0 ξdt ) = f ( α ∫ db(ζ,η) 0 ξdt + γ ∫ [db(ζ,lζ)+db(η,mη)] 0 ξdt +δ ∫ [db(ζ,mη)+db(η,lζ)] 0 ξdt ) − ϕ (∫ db(ζn−1,ζn) 0 ξdt ) =  f(α)f (∫ db(ζn−1, ζn) 0 ξdt ) +f(γ)f (∫ [db(ζn−1,lζn−1)+db(ζn,mζn)] 0 ξdt ) +f(δ)f (∫ [db(ζn−1,mζn)+db(ζn,lζn−1)] 0 ξdt ) − ϕ (∫ db(ζn−1,ζn) 0 ξdt ) .   therefore, ∥ f (∫ db(ζn,ζn+1) 0 ξdt ) ∥=∥ f (∫ db(lζn−1,mζn) 0 ξdt ) ∥ ≤   ∥ f(α)∥ ∥ f (∫ db(ζn−1,ζn) 0 ξdt ) ∥ +∥ f(γ)∥ ∥ f (∫ [db(ζn−1,lζn−1)+db(ζn,mζn)] 0 ξdt ) ∥ +∥ f(δ)∥ ∥ f (∫ [db(ζn−1,mζn)+db(ζn,lζn−1)] 0 ξdt ) ∥ − ∥ ϕ (∫ db(ζn−1,ζn) 0 ξdt ) ∥ → 0 as n → +∞.   189 common fixed point theorems for (ϕ,f) integral type contractive mapping on c∗-algebra valued b-metrix space give that ϕ (2.1) and f (2.4) are strongly monotone functions. we have∫ db(ζn,ζn+1) 0 ξdt = ∫ db(lζn−1,mζn) 0 ξdt ≤ ( α ∫ db(ζn−1,ζn) 0 ξdt + γ ∫ [db(ζn−1,lζn−1)+db(ζn,mζn)] 0 ξdt +δ ∫ [db(ζn−1,mζn)+db(ζn,lζn−1)] 0 ξdt ) = ( α ∫ db(ζn−1,ζn) 0 ξdt + γ ∫ [db(ζn−1,ζn)+db(ζn,ζn+1)] 0 ξdt +δ ∫ [db(ζn−1,ζn+1)+db(ζn,ζn)] 0 ξdt ) ≤ (α + γ) ∫ db(ζn−1, ζn) 0 ξdt + γ ∫ db(ζn,ζn+1) 0 ξdt + δ ∫ db(ζn−1,ζn+1) 0 ξdt ≤ (α + γ) ∫ db(ζn−1, ζn) 0 ξdt + γ ∫ db(ζn,ζn+1) 0 ξdt + ωδ ∫ (db(ζn−1,ζn)+db(ζn,ζn+1)) 0 ξdt ≤ (α + γ + ωδ ) ∫ db(ζn−1, ζn) 0 ξdt + (γ + ωδ) ∫ db(ζn,ζn+1) 0 ξdt. this implies that ∫ db(ζn,ζn+1) 0 ξdt ≤ α + γ + ωδ γ + ωδ ∫ db(ζn,ζn−1) 0 ξdt ∫ db(ζn,ζn+1) 0 ξdt ≤ h ∫ db(ζn,ζn−1) 0 ξdt where, h = α+γ+ωδ γ+ωδ < 1. thus, we have ∥ ∫ db(ζn−1,ζn) 0 ξdt ∥ ∥ ∫ db(ζn,ζn+1) 0 ξdt ∥≤∥ h ∥ ∥ ∫ db(ζn,ζn−1) 0 ξdt ∥→ 0, asn,m → +∞. if n > m ∫ db(ζn,ζm) 0 ξdt ≤ ( ω ∫ db(ζn,ζn−1) 0 ξdt + ω 2 ∫ db(ζn−1,ζn−2) 0 ξdt + . . . + ωn−m ∫ db(ζm−1,ζm) 0 ξdt ) . applying the constraint of theorem then, f (∫ db(ζn,ζm) 0 ξdt ) ≤  f ( ω ∫ db(ζn,ζn−1) 0 ξdt ) +f ( ω2 ∫ db(ζn−1,ζn−2) 0 ξdt ) + . . . + f ( ωn−m ∫ db(ζm−1,ζm) 0 ξdt )   190 r. jahir hussain and k. maheshwaran ≤  f(ω)f (∫ db(ζn,ζn−1) 0 ξdt ) +f (ω2) f (∫ db(ζn−1,ζn−2) 0 ξdt ) + . . . + f (τn−m) f (∫ db(ζm−1,ζm) 0 ξdt )   ≤   f ( ω ∫ i(ζn,ζn−1) 0 ξdt ) − ϕ ( ω ∫ db(ζn,ζn−1) 0 ξdt ) +f ( ω2 ∫ i(ζn−1,ζn−2) 0 ξdt ) − ϕ ( ω2 ∫ db(ζn−1,ζn−2) 0 ξdt ) + . . . + f ( ωn−m ∫ i(ζm−1,ζm) 0 ξdt ) − ϕ ( ωn−m ∫ db(ζm−1,ζm) 0 ξdt )   =   f   αω ∫ db(ζn,ζn−1) 0 ξdt +γω ∫ [db(ζn−1,ζn)+db(ζn,ζn+1)] 0 ξdt +δω ∫ [db(ζn−1,ζn+1)+db(ζn,ζn)] 0 ξdt   −ϕ ( ω ∫ db(ζn,ζn−1) 0 ξdt ) + . . . +f   αω n−m ∫ db(ζm−1,ζm) 0 ξdt +γωn−m ∫ [db(ζm,ζm−1)+db(ζm−1,ζm−2)] 0 ξdt +δωn−m ∫ [db(ζm,ζm−2)+db(ζm−1,ζm−1)] 0 ξdt   −ϕ ( ωn−m ∫ db(ζm−1,ζm) 0 ξdt )   therefore f (∫ db(ζn,ζm) 0 ξdt ) = f(α)f(ω)f (∫ db(ζn,ζn−1) 0 ξdt ) +f(γ)f(ω)f (∫ [db(ζn−1,ζn)+db(ζn,ζn+1)] 0 ξdt ) +f(δ)f(ω)f (∫ [db(ζn−1,ζn+1)+db(ζn,ζn)] 0 ξdt ) −ϕ ( ω ∫ db(ζn,ζn−1) 0 ξdt ) + . . . +f(α)f (ωn−m) f (∫ db(ζm−1,ζm) 0 ξdt ) +f(γ)f (ωn−m) f (∫ [db(ζm,ζm−1)+db(ζm−1,ζm−2)] 0 ξdt ) +f(δ)f (ωn−m) f (∫ [db(ζm,ζm−2)+db(ζm−1,ζm−1)] 0 ξdt ) −ϕ ( ωn−m ∫ db(ζm−1,ζm) 0 ξdt ) . since the property of ϕ (2.1) and f (2.4) is strongly monotone, we have 191 common fixed point theorems for (ϕ,f) integral type contractive mapping on c∗-algebra valued b-metrix space ∫ db(ζn,ζm) 0 ξdt ≤ αω ∫ db(ζn,ζn−1) 0 ξdt + γω ∫ [db(ζn−1,ζn)+db(ζn,ζn+1)] 0 ξdt +δω ∫ [db(ζn−1,ζn+1)+db(ζn,ζn)] 0 ξdt + . . . +αωn−m ∫ db(ζm−1,ζm) 0 ξdt +γωn−m ∫ [db(ζm,ζm−1)+db(ζm−1,ζm−2)] 0 ξdt +δωn−m ∫ [db(ζm,ζm−2)+db(ζm−1,ζm−1)] ξdt so we get ∫ db(ζn,ζm) 0 ξdt ≤   ∥ α ∥∥ ω ∥∥ ∫ db(ζn,ζn−1) 0 ξdt ∥ + ∥ γ ∥∥ ω ∥∥ ∫ [db(ζn−1,ζn)+db(ζn,ζn+1)] 0 ξdt ∥ + ∥ δ ∥∥ ω ∥∥ ∫ [db(ζn−1,ζn+1)+db(ζn,ζn)] 0 ξdt ∥ + . . . + ∥ α ∥∥ ωn−m ∥∥ ∫ db(ζm−1,ζm) 0 ξdt ∥ + ∥ γ ∥∥ ωn−m ∥∥ ∫ [db(ζm,ζm−1)+db(ζm−1,ζm−2)] 0 ξdt ∥ + ∥ δ ∥∥ ωn−m ∥∥ ∫ [db(ζm,ζm−2)+db(ζm−1,ζm−1)] 0 ξdt ∥   → 0, as n.m → +∞. then {ζn} is cauchy sequence. since (χ,a,db) is a complete c∗-algebra valued b-metric space there exists u ∈ χ such that ζn → u as n → ∞. now since ∫ db(u,mu) 0 ξdt ≤ ω [∫ db(u,ζn+1) 0 ξdt + ∫ db(ζn+1,mu) 0 ξdt ] = ω [∫ db(ζn+1,mu) 0 ξdt + ∫ db(u,ζn+1) 0 ξdt ] = ω [∫ db(lζn,mu) 0 ξdt + ∫ db(u,ζn+1) 0 ξdt ] f (∫ db(u,mu) 0 ξdt ) = ω [ f (∫ db(lζn,mu) 0 ξdt ) +f (∫ db(u,ζn+1) 0 ξdt )] ≤ ω [ f (∫ i(ζn,u) 0 ξdt ) − ϕ (∫ db(ζn,u) 0 ξdt )] + ω [ f (∫ db(u,ζn+1) 0 ξdt )] 192 r. jahir hussain and k. maheshwaran ∥ f (∫ db(lζn,mu) 0 ξdt ) ∥≤   ∥ ω ∥∥ f (∫ db(u,ζn+1) 0 ξdt ) ∥ + ∥ ω ∥∥ fα ∥∥ ∫ db(ζn,u) 0 ξdt ∥ + ∥ ω ∥∥ fγ ∥ ∥ ∫ [db(ζn,ζn+1)+db(u,mu)] 0 ξdt ∥ + ∥ ω ∥∥ fδ ∥∥ ∫ [db(ζn,mu)+db(u,lζn)] 0 ξdt ∥ − ∥ ω ∥∥ ϕ (∫ db(ζn,u) 0 ξdt ) ∥ .   using the property of ϕ (2.1), we get ∥ f (∫ db(lζn,mu) 0 ξdt ) ∥ ≤   ∥ ω ∥∥ f (∫ db(u,ζn+1) 0 ξdt ) ∥ + ∥ ω ∥∥ fα ∥∥ ∫ db(ζn,u) 0 ξdt ∥ + ∥ ω ∥∥ fγ ∥ ∥ ∫ [db(ζn,ζn+1)+db(u,mu)] 0 ξdt ∥ + ∥ ω ∥∥ fδ ∥∥ ∫ [db(ζn,mu)+db(u,lζn)] 0 ξdt ∥   . where f (2.4) is strongly monotone, then ∥ ∫ (db(lζn,mu)) 0 ξdt ∥≤   ∥ ω ∥∥ ∫ (db(u,ζn+1)) 0 ξdt ∥ + ∥ ω ∥∥ α ∥∥ ∫ db(ζn,u) 0 ξdt ∥ + ∥ ω ∥∥ γ ∥ ∥ ∫ [db(ζn,ζn+1)+db(u,mu)] 0 ξdt ∥ + ∥ ω ∥∥ δ ∥∥ ∫ [db(ζn,mu)+db(u,lζn)] 0 ξdt ∥   = ∥ ω ∥∥ ∫ (db(u,ζn+1)) 0 ξdt ∥ + ∥ ω ∥   ∥ α ∥∥ ∫ db(ζn,u) 0 ξdt ∥ + ∥ γ ∥ ∥ ∫ [db(ζn,ζn+1)+db(u,mu)] 0 ξdt ∥ + ∥ δ ∥∥ ∫ [db(ζn,mu)+db(u,ζn+1)] 0 ξdt ∥   as ζn → u and ζn+1 → u as n → ∞, we get ∥ 1−ωγ−ωδ ∥ ∥ ∫ db(u,mu) 0 ξdt ∥≤ [ ∥ ω ∥∥ α ∥∥ ∫ db(ζn,u) 0 ξdt ∥ + ∥ ω ∥∥ 1 + δ ∥∥ ∫ db(u,ζn+1) 0 ξdt ∥ ] → 0 as n → ∞. 193 common fixed point theorems for (ϕ,f) integral type contractive mapping on c∗-algebra valued b-metrix space hence ∥ ∫ db(mu,u) 0 ξdt ∥= 0 since ∥ 1 − ωγ − ωδ ∥> 0. as a result, mu = u that is u is a fixed point of m. similarly we are able to demonstrate that lu = u. hence lu = mu = u. this demonstrates that u is common fixed point of l and m. let v be a different fixed point common to l and m. (i.e) lv = mv = v such that u ̸= v we have ∫ db(u,v) 0 ξdt = ∫ db(lu,mv) 0 ξdt then f (∫ db(u,v) 0 ξdt ) = f (∫ db(lu,mv) 0 ξdt ) ≤ f (∫ i(u,v) 0 ξdt ) − ϕ (∫ db(u,v) 0 ξdt ) ∥ f (∫ db(lu,mv) 0 ξdt ) ∥≤   ∥ fα ∥∥ ∫ db(u,v) 0 ξdt ∥ + ∥ fγ ∥∥ ∫ [db(u,lu)+db(v,mv)] 0 ξdt ∥ + ∥ fδ ∥∥ ∫ [db(u,mv)+db(v,lu)] 0 ξdt ∥ − ∥ ϕ (∫ db(u,v) 0 ξdt ) ∥   using the property of ϕ (2.1), we get ∥ f (∫ db(lu,mv) 0 ξdt ) ∥≤   ∥ fα ∥∥ ∫ db(u,v) 0 ξdt ∥ + ∥ fγ ∥ ∥ ∫ [db(u,lu)+db(v,mv)] 0 ξdt ∥ + ∥ fδ ∥∥ ∫ [db(u,mv)+db(v,lu)] 0 ξdt ∥   where f (2.4) is strongly monotone, then ∥ ∫ (db(lu,mv)) 0 ξdt ∥ ≤   ∥ α ∥∥ ∫ db(u,v) 0 ξdt ∥ + ∥ γ ∥ ∥ ∫ [db(u,lu)+db(v,mv)] 0 ξdt ∥ + ∥ δ ∥∥ ∫ [db(u,mv)+db(v,lu)] 0 ξdt ∥   ≤ ∥ α + 2δ ∥ ∥ ∫ db(u,v) 0 ξdt ∥ ≤ ∥ ωα + (ω + 1)γ + ω(ω + 1)δ ∥ ∥ ∫ db(u,v) 0 ξdt ∥ < ∥ ∫ db(u,v) 0 ξdt ∥ . which is a contradiction. hence ∥ ∫ db(u,v) ξdt ∥= 0 and u = v. thus u is a unique common fixed point of l and m. corolary 3.1. let (χ,a,db) is a complete c∗-algebra valued b-metric space. let l :χ → χ be a contractive mapping and f (∫ db(lnζ,lnη) 0 ξdt ) ≤ f (∫ i(ζ,η) 0 ξdt ) − ϕ (∫ db(ζ,η) 0 ξdt ) 194 r. jahir hussain and k. maheshwaran i(ζ,η) ≤  α ∫ db(ζ,η) 0 ξdt + β ∫ [1+db(ζ,lnζ)]db(η,lnη) 1+db(ζ,η) 0 ξdt +γ ∫ [db(ζ,lnζ)+db(η,lnη)] 0 ξdt +δ ∫ [db(ζ,lnη)+db(η,lnζ)] 0 ξdt   for all ζ,η ∈ χ, where ω ∈ a′+, α + β + γ + δ ≥ 0 with ωα + β + γ(ω + 1) + δ (ω(ω + 1)) < 1. f and ϕ are ∗−homomorphisms and with the constraint f ( a) ≤ ϕ( a) and ξ : χ → χ is a lebesgus-integral mapping which is summable, non-negative and such that for all ε > 0, ∫ ε 0 ξdt > 0. then l have a unique fixed point in χ. example 3.1. let χ = [0,1] and a =r2 with a norm ∥ ζ ∥= |ζ| be a real c∗algebra. we define p = {(ζ,η ) ∈ r2 : ζ ≥ 0, η ≥ 0}. the partial order ≤ with respect to the c∗-algebra r2. ζ1 ≤ ζ2 and η1 ≤ η2 for all (ζ1,η1) ,(ζ2,η2) ∈ r2. let db : χ×χ → r2 suppose that db(ζ,η) = 2 (|ζ − η|, |ζ − η|) for ζ,η ∈ χ. then, (χ,a,db) is a c∗-algebra valued b-metric space where ω = 1 in theorem 3.1. let f,ϕ : p → p be the mappings defined as follows: for t =(ζ,η) ∈ p f (t ) =   (ζ,η), if ζ ≤ 1 and η ≤ 1, (ζ2,η) , if ζ > 1 and η ≤ 1, (ζ,η2) , if ζ ≤ 1 and η > 1, (ζ2,η2) , if ζ > 1 and η > 1. and for s = (s1,s2) ∈ p with v = min {s1,s2} , ϕ(s) = {( v2 2 , v 2 2 ) , if v ≤ 1( 1 2 , 1 2 ) , if v > 2 then, f and ϕ have the properties mentioned in (2.4) and (2.1). let l,m :χ → χ be defined as follows: l(ζ) = { 1 32 , if 0 ≤ ζ ≤ 1 2 0, if 1 2 < ζ ≤ 1 ; m(ζ) = 1 32 , for ζ ∈ χ then, l and m have the required properties mentioned in theorem 3.1. let α = 1 16 , β = 0 , γ = 1 64 and δ = 1 64 . it can be verified that: f (db (lζ,mη)) ≤ f (n(ζ,η)) − ϕ(db(ζ,η)) , ∀ ζ,η ∈ χ with η ≤ ζ . hence, theorem 3.1 is satisfied. then demonstrate that 0 is a unique common fixed point of l and m. 195 common fixed point theorems for (ϕ,f) integral type contractive mapping on c∗-algebra valued b-metrix space 4 conclusions in theorem 3.1 we have formulated a contractive conditions to modify and extend the concept of common fixed point theorem for c∗-algebra valued b-metric space via (ϕ,f)-integral type contractive mapping. the existence and uniqueness of the result is presented in this article. we have also given some example which satisfies the contractive condition of our main result. our result may be the vision for other authors to extend and improve several results in such spaces and applications to other related areas. acknowledgements the authors thanks the management, ratio mathematica for their constant support towards the successful completion of this work. we wish to thank the anonymous reviewers for a careful reading of manuscript and for very useful comments and suggestions. references h. alsulami, r. agarwal, e. karapä±nar, and f. khojasteh. short note on c∗valued contraction mappings. journal of inequalities and application, 2016:50: 1–3, 2016. a. branciari. a fixed point theorem for mappings satisfying a general contractive condition of integral type. international journal of mathematics and mathematical sciences, 29:531–536, 2002. t. kamran, m. postolache, a. ghiura, s. batul, and r. ali. the banach contraction principle in c∗-algebra-valued b-metric spaces with application. fixed point theory and applications, 2016:10:1–7, 2016. z. liu, h. wu, j. ume, and s. kang. some fixed point theorems for mappings satisfying contractive conditions of integral type. fixed point theory and applications, 2014:69:1–14, 2014. z. liu, m. he, x. liu, and l. zhao. common fixed point theorems for four mappings satisfying contractive inequalities of integral type. nonlinear functional analysis and applications, 23:473–501, 2018. z. ma and l. jiang. c∗-algebra-valued b-metric spaces and related fixed point theorems. fixed point theory and applications, 1:1–12, 2015. 196 r. jahir hussain and k. maheshwaran z. ma, l. jiang, and h. sun. c∗-algebra-valued metric spaces and related fixed point theorems. fixed point theory and applications, 2014:206:1–11, 2014. r. mustafa, s. omran, and q. nguyen. fixed point theory using ψ contractive mapping in c∗−algebra valued b-metric space. mathematics, 9,92:1–8, 2021. 197 ratio mathematica volume 44, 2022 steiner domination decomposition number of graphs mahiba m1 ebin raja merly e2 abstract in this paper, we introduce a new concept steiner domination decomposition number of graphs. let 𝐺 be a connected graph with steiner domination number𝛾𝑠(𝐺). a decomposition 𝜋 = {𝐺1, 𝐺2, … , 𝐺𝑛 } of 𝐺 is said to be a steiner domination decomposition (𝑆𝐷𝐷) if 𝛾𝑠 (𝐺𝑖 ) = 𝛾𝑠(𝐺), 1 ≤ 𝑖 ≤ 𝑛. steiner domination decomposition number of 𝐺 is the maximum cardinality obtained for an 𝑆𝐷𝐷 of 𝐺 and is denoted as 𝜋𝑠𝑡𝑑 (𝐺). bounds on 𝜋𝑠𝑡𝑑 (𝐺) are presented. also, few characteristics of the subgraphs belonging to 𝑆𝐷𝐷 of maximum cardinality are discussed. keywords: subgraphs; domination; decomposition number. ams subject classification: 05c12, 05c693 1research scholar (reg.no: 20213112092013), research department of mathematics, nesamony memorial christian college, marthandam-629165. affiliated to manonmaniam sundaranar university, tirunelveli-627012, tamil nadu, india. mahibakala@gmail.com 2associate professor, research department of mathematics, nesamony memorial christian college, marthandam-629165. affiliated to manonmaniam sundaranar university, tirunelveli-627012, tamil nadu, india. ebinmerly@gmail.com 3received on june 18th, 2022. accepted on aug 10th, 2022. published on nov30th, 2022. doi: 10.23755/rm.v44i0.896. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 100 mailto:mahibakala@gmail.com mailto:ebinmerly@gmail.com mahiba. m and ebin raja merly. e 1. introduction let 𝐺 be a simple, connected and undirected graph with vertex set 𝑉(𝐺)and edge set 𝐸(𝐺). the order and size of 𝐺 are 𝑝 and 𝑞 respectively. for standard terminologies and notations, we refer to [1]. steiner domination number of a graph is a concept introduced by john 𝑒𝑡 𝑎𝑙. further studies on this concept is found in [7], [8]. in [5], we introduced the concept of steiner decomposition number of graphs and in [6] we presented the steiner decomposition number of complete 𝑛 − sun graph. in this paper, a new decomposition concept called steiner domination decomposition number of graphs is studied. the following are the basic definitions and results needed for the subsequent section. definition 1.1. [2] let 𝐺 be a connected graph. for a set 𝑊 ⊆ 𝑉(𝐺), a tree 𝑇 contained in 𝐺 is a steiner tree with respect to 𝑊 if 𝑇 is a tree of minimum order with 𝑊 ⊆ 𝑉(𝑇). the set 𝑆(𝑊) consists of all vertices in 𝐺 that lie on some steiner tree with respect to 𝑊. the set 𝑊 is a steiner set for 𝐺 if 𝑆(𝑊) = 𝑉(𝐺). the minimum cardinality among the steiner sets of 𝐺 is the steiner number 𝑠(𝐺). definition 1.2. [3] a set 𝐷 ⊆ 𝑉(𝐺)in a graph 𝐺 is called a dominating set if every vertex 𝑣 ∈ 𝑉(𝐺) is either an element of 𝐷 or is adjacent to an element of 𝐷. the domination number 𝛾(𝐺) is the minimum cardinality of a dominating set of 𝐺. definition 1.3. [4] for a connected graph 𝐺, 𝑊 ⊆ 𝑉(𝐺)is called a steiner dominating set if 𝑊 is both a steiner set and a dominating set. the minimum cardinality of a steiner dominating set of 𝐺 is said to be steiner domination number and is denoted by 𝛾𝑠(𝐺). a steiner dominating set of cardinalities𝛾𝑠(𝐺) is said to be a 𝛾𝑠 − 𝑠𝑒𝑡 of 𝐺. definition 1.4. the decomposition 𝜋 of a graph 𝐺 is a collection of edge disjoint subgraphs 𝐺1, 𝐺2 , … , 𝐺𝑛 such that each 𝐺𝑖 , 1 ≤ 𝑖 ≤ 𝑛is connected and 𝐸(𝐺) = 𝐸(𝐺1) ∪ e(𝐺2) ∪ … ∪ e(𝐺𝑛). definition 1.5. [5] for a connected graph 𝐺 with steiner number 𝑠(𝐺), a decomposition 𝜋 = {𝐺1, 𝐺2 , … , 𝐺𝑛}of 𝐺 is said to be a steiner decomposition(𝑆𝐷) if 𝑠(𝐺𝑖 ) = 𝑠(𝐺) for all 𝑖, (1 ≤ 𝑖 ≤ 𝑛).the maximum cardinality obtained for the steiner decomposition 𝜋 of 𝐺 is called the steiner decomposition number of 𝐺 and is denoted by 𝜋𝑠𝑡 (𝐺). steiner decomposition of cardinality 𝜋𝑠𝑡 (𝐺) is denoted as 𝑆𝐷𝑚𝑎𝑥 . theorem 1.6. [4] for any connected graph 𝐺 of order 𝑝 ≥ 2, 𝛾𝑠(𝐺) = 2 if and only if there exists a steiner dominating set 𝑊 = {𝑢, 𝑣}of 𝐺 such that 𝑑(𝑢, 𝑣) ≤ 3. theorem 1.7. [4] for a connected graph 𝐺 of order 𝑝 ≥ 2, 𝛾𝑠(𝐺) = 𝑝 if and only if 𝐺 = 𝐾𝑝. result 1.8. [7] for the path graph on 𝑝 vertices (𝑝 ≥ 2), 𝛾𝑠(𝑃𝑝) = 101 steiner domination decomposition number of graphs { ⌈ 𝑝−4 3 ⌉ + 2 𝑖𝑓 𝑝 ≥ 5 2 𝑖𝑓 𝑝 = 2,3,4 notation 1.9. ℱ𝑝 denotes the family of trees of order 𝑝 with the property that each vertex is either a pendant vertex or a support vertex. 2. steiner domination decomposition definition 2.1. a decomposition 𝜋 = {𝐺1, 𝐺2 , … , 𝐺𝑛}of a graph 𝐺 is called a steiner domination decomposition (𝑆𝐷𝐷) if 𝛾𝑠(𝐺𝑖)=𝛾𝑠(𝐺), (1 ≤ 𝑖 ≤ 𝑛). the maximum cardinality obtained for 𝜋 is called the steiner domination decomposition number of 𝐺 and is denoted by 𝜋𝑠𝑡𝑑 (𝐺). an 𝑆𝐷𝐷 of cardinality 𝜋𝑠𝑡𝑑 (𝐺)is denoted as 𝑆𝐷𝐷𝑚𝑎𝑥 . a graph 𝐺 with 𝜋𝑠𝑡𝑑 (𝐺) = 1 is said to be non-steiner domination decomposable graph. if 𝜋𝑠𝑡𝑑 (𝐺) ≥ 2 then 𝐺 is said to be steiner domination decomposable graph. example 2.2. consider the graph 𝐺 in figure 1. figure 1. graph 𝐺 and its steiner domination decomposition 𝜋 = {𝐺1, 𝐺2} the set 𝑊 = {𝑣1, 𝑣2 , 𝑣5, 𝑣9}is a 𝛾𝑠 − 𝑠𝑒𝑡 of 𝐺. hence 𝛾𝑠(𝐺) = 4. since 𝛾𝑠(𝐺1) = 𝛾𝑠(𝐺2) = 4 = 𝛾𝑠(𝐺), 𝜋 = {𝐺1, 𝐺2} is a 𝑆𝐷𝐷. it can be easily verified that 𝜋 is a 𝑆𝐷𝐷𝑚𝑎𝑥 . thus 𝜋𝑠𝑡𝑑 (𝐺) = 2. theorem 2.3. if 𝜋𝑠𝑡𝑑 (𝐺) = 𝑞 then 𝑑𝑖𝑎𝑚 𝐺 < 4. proof. steiner domination decomposition number of 𝐺, 𝜋𝑠𝑡𝑑 (𝐺) = 𝑞 ⟺ 𝜋 = {𝐺𝑖 ≅ 𝐾2 /1 ≤ 𝑖 ≤ 𝑞} is a 𝑆𝐷𝐷𝑚𝑎𝑥 . steiner domination number of 𝐾2 is , hence 𝛾𝑠(𝐺) = 2. also, we have 𝛾𝑠(𝐺) = 2 implies 𝑑𝑖𝑎𝑚 𝐺 < 4. therefore if 𝜋𝑠𝑡𝑑 (𝐺) = 𝑞 then 𝑑𝑖𝑎𝑚 𝐺 < 4. hence proved. theorem 2.4. let 𝐺 be a connected graph with 𝛾𝑠(𝐺) ≥ 3. then 1 ≤ 𝜋𝑠𝑡𝑑 (𝐺) ≤ 102 mahiba. m and ebin raja merly. e ⌊ 𝑞 𝛾𝑠(𝐺) ⌋. proof. from definition 2.1, it is obvious that 𝜋𝑠𝑡𝑑 (𝐺) ≥ 1. let 𝜋 = {𝐺𝑖 /1 ≤ 𝑖 ≤ 𝑛} be a 𝑆𝐷𝐷 of 𝐺. first to prove |𝐸(𝐺𝑖 )| ≥ 𝛾𝑠(𝐺) for all 𝑖. assume to the contrary that |𝐸(𝐺𝑖 )| < 𝛾𝑠(𝐺) for some 𝑖. without loss of generality, let |𝐸(𝐺1)| < 𝛾𝑠(𝐺). then |𝑉(𝐺1)| ≤ 𝛾𝑠(𝐺). case (i):|𝑉(𝐺1)| < 𝛾𝑠 (𝐺) if |𝑉(𝐺1)| < 𝛾𝑠(𝐺) then 𝛾𝑠(𝐺1) < 𝛾𝑠(𝐺). therefore 𝐺1 ∉ 𝜋. case (ii): |𝑉(𝐺1)| = 𝛾𝑠(𝐺) in order to satisfy 𝛾𝑠 (𝐺1) = 𝛾𝑠(𝐺), 𝐺1 must be a complete graph on 𝛾𝑠(𝐺)vertices. but we have |𝑉(𝐺1)| > |𝐸(𝐺1)|. hence 𝐺1 is non isomorphic to 𝐾𝛾𝑠(𝐺). therefore 𝐺1 ∉ 𝜋. in both the cases, we arrive at a contradiction to our assumption that 𝐺1 ∈ 𝜋. hence |𝐸(𝐺1)| ≥ 𝛾𝑠(𝐺). since 𝐺1 is chosen arbitrarily, we can conclude |𝐸(𝐺𝑖 )| ≥ 𝛾𝑠(𝐺) for all 𝑖.thus subgraphs of 𝐺 belonging to any steiner domination decomposition should have atleast 𝛾𝑠(𝐺)edges and so 𝜋𝑠𝑡𝑑 (𝐺) ≤ ⌊ 𝑞 𝛾𝑠(𝐺) ⌋. hence 1 ≤ 𝜋𝑠𝑡𝑑 (𝐺) ≤ ⌊ 𝑞 𝛾𝑠(𝐺) ⌋. theorem 2.5. let 𝐺 be a steiner domination decomposable graph with 𝑞 edges. for 𝛾𝑠(𝐺) = 3, 𝜋𝑠𝑡𝑑 (𝐺) = 𝑞 3 if and only if each 𝐺𝑖 ∈ 𝑆𝐷𝐷𝑚𝑎𝑥 is isomorphic to either 𝐾1,3or 𝐾3 and for 𝛾𝑠(𝐺) > 3, 𝜋𝑠𝑡𝑑 (𝐺) = 𝑞 𝛾𝑠(𝐺) if and only if each 𝐺𝑖 ∈ 𝑆𝐷𝐷𝑚𝑎𝑥 is isomorphic to 𝐾1,𝛾𝑠(𝐺). proof. let 𝐺 be a steiner domination decomposable graph. assume 𝛾𝑠(𝐺) = 3 and 𝜋𝑠𝑡𝑑 (𝐺) = 𝑞 3 . then for any 𝐺𝑖 ∈ 𝑆𝐷𝐷𝑚𝑎𝑥 , |𝐸(𝐺𝑖 )| = 3 and hence|𝑉(𝐺𝑖 )| ≤ 4. if |𝑉(𝐺𝑖 )| ≤ 3 for some 𝑖, then the only graph that satisfies 𝛾𝑠(𝐺𝑖 ) = 3 is 𝐾3. if |𝑉(𝐺𝑖 )| = 4 for some 𝑖, then 𝐺𝑖 is a tree. star graph 𝐾1,3 is the unique tree which satisfies the required properties. thus if 𝜋𝑠𝑡𝑑 (𝐺) = 𝑞 3 then 𝐺𝑖 ≅ 𝐾1,3or 𝐾3 for all 𝐺𝑖 ∈ 𝑆𝐷𝐷𝑚𝑎𝑥 . converse part is obvious. now, assume𝛾𝑠(𝐺) > 3and 𝜋𝑠𝑡𝑑 (𝐺) = 𝑞 𝛾𝑠(𝐺) .then|𝐸(𝐺𝑖 )| = 𝛾𝑠(𝐺) for every 𝐺𝑖 ∈ 𝑆𝐷𝐷𝑚𝑎𝑥 and so |𝑉(𝐺𝑖 )| ≤ 𝛾𝑠(𝐺) + 1. there doesn't exist any graph 𝐺𝑖 with the properties |𝑉(𝐺𝑖 )| ≤ 𝛾𝑠(𝐺) and 𝛾𝑠(𝐺𝑖 ) = 𝛾𝑠(𝐺). since 𝐾1,𝛾𝑠(𝐺) is the unique graph on 𝛾𝑠(𝐺) + 1 vertices that has steiner domination number same as 𝐺, we have |𝑉(𝐺𝑖 )| = 𝛾𝑠(𝐺) + 1 implies 𝐺𝑖 ≅ 𝐾1,𝛾𝑠(𝐺). hence if 𝜋𝑠𝑡𝑑 (𝐺) = 𝑞 𝛾𝑠(𝐺) then 𝐺𝑖 ≅ 𝐾1,𝛾𝑠(𝐺) for all 𝐺𝑖 ∈ 𝑆𝐷𝐷𝑚𝑎𝑥 . converse is obvious. theorem 2.6. let 𝐺 be a connected graph with 𝛾𝑠(𝐺) ≥ 3 and ⌊ 𝑞 𝛾𝑠(𝐺) ⌋ = 𝑚, (𝑚 > 1). if 𝜋𝑠𝑡𝑑 (𝐺) = 𝑚 − 𝑛 , (0 ≤ 𝑛 < 𝑚 − 1) then 𝛾𝑠(𝐺) ≤ |𝐸(𝐺𝑖 )| ≤ (𝑛 + 2)𝛾𝑠(𝐺) − 1 for all 𝐺𝑖 ∈ 𝑆𝐷𝐷𝑚𝑎𝑥 . proof. let 𝐺 be a connected graph such that 𝛾𝑠(𝐺) ≥ 3 . let ⌊ 𝑞 𝛾𝑠(𝐺) ⌋ = 𝑚, (𝑚 > 1). assume 𝜋𝑠𝑡𝑑 (𝐺) = 𝑚 − 𝑛 where 0 ≤ 𝑛 < 𝑚 − 1. let 𝜋 = {𝐺1, 𝐺2 , … , 𝐺𝑚−𝑛}be a 𝑆𝐷𝐷𝑚𝑎𝑥 of 𝐺. to prove 𝛾𝑠(𝐺) ≤ |𝐸(𝐺𝑖 )| ≤ (𝑛 + 2)𝛾𝑠(𝐺) − 1 for all 𝐺𝑖 ∈ 𝜋. the requirement of edges in each subgraph belonging to any 𝑆𝐷𝐷 of 𝐺 is atleast 𝛾𝑠(𝐺). hence|𝐸(𝐺𝑖 )| ≥ 𝛾𝑠(𝐺)for every 𝐺𝑖 ∈ 𝜋. without loss of generality, assume 103 steiner domination decomposition number of graphs |𝐸(𝐺𝑚−𝑛)| ≥ |𝐸(𝐺𝑖 )|, 1 ≤ 𝑖 ≤ 𝑚 − (𝑛 + 1). since⌊ 𝑞 𝛾𝑠(𝐺) ⌋ = 𝑚, we get 𝑚𝛾𝑠(𝐺) ≤ 𝑞 ≤ (𝑚 + 1)𝛾𝑠(𝐺) − 1. we know that ∑ |𝐸(𝐺𝑖 )| = 𝑞 𝑚−𝑛 𝑖=1 and |𝐸(𝐺𝑖 )| ≥ 𝛾𝑠(𝐺)for 1 ≤ 𝑖 ≤ 𝑚 − (𝑛 + 1). therefore, ∑ |𝐸(𝐺𝑖 )| ≤ (𝑚 + 1)𝛾𝑠(𝐺) − 1 𝑚−𝑛 𝑖=1 (𝑚 − (𝑛 + 1))𝛾𝑠(𝐺) + |𝐸(𝐺𝑚−𝑛)| ≤ (𝑚 + 1)𝛾𝑠(𝐺) − 1 ⇒ |𝐸(𝐺𝑚−𝑛)| ≤ (𝑛 + 2)𝛾𝑠(𝐺) − 1 thus, the possible number of edges in a subgraph belonging to 𝑆𝐷𝐷𝑚𝑎𝑥 is atmost (𝑛 + 2)𝛾𝑠(𝐺) − 1. hence 𝛾𝑠(𝐺) ≤ |𝐸(𝐺𝑖 )| ≤ (𝑛 + 2)𝛾𝑠(𝐺) − 1 for all 𝐺𝑖 ∈ 𝑆𝐷𝐷𝑚𝑎𝑥 . theorem 2.7. let 𝐺 be a connected graph with 𝛾𝑠(𝐺) ≥ 5 and ⌊ 𝑞 𝛾𝑠(𝐺) ⌋ = 𝑚, (𝑚 > 1). if 𝜋𝑠𝑡𝑑 (𝐺) = 𝑚 − 𝑛 , (0 ≤ 𝑛 < 𝑚 − 1) then the number of path graphs belonging to 𝑆𝐷𝐷𝑚𝑎𝑥 is strictly less than 𝑛 + 1. proof. let 𝐺 be a connected graph with 𝑞 edges. let 𝛾𝑠(𝐺) = 𝑘 + 1 where 𝑘 ≥ 4. assume𝜋𝑠𝑡𝑑 (𝐺) = 𝑚 − 𝑛, (0 ≤ 𝑛 < 𝑚 − 1).let 𝜋 = {𝐺𝑖 /1 ≤ 𝑖 ≤ 𝑚 − 𝑛} be a 𝑆𝐷𝐷𝑚𝑎𝑥 . let 𝑁 denotes the number of path graphs belonging to 𝜋. first we try to prove 𝑁 ≠ 𝑛 + 1. suppose 𝑁 = 𝑛 + 1. consider 𝐺1, 𝐺2 , … , 𝐺𝑛+1 ∈ 𝜋 as path graphs. path graphs with steiner domination number 𝑘 + 1 are 𝑃3𝑘−1, 𝑃3𝑘and 𝑃3𝑘+1. therefore 3𝑘 − 2 ≤ |𝐸(𝐺𝑖 )| ≤ 3𝑘 for 1 ≤ 𝑖 ≤ 𝑛 + 1. now, ∑ |𝐸(𝐺𝑖 )| 𝑚−𝑛 𝑖=1 = ∑ |𝐸(𝐺𝑖 )| 𝑛+1 𝑖=1 + ∑ |𝐸(𝐺𝑖 )| 𝑚−𝑛 𝑖=𝑛+2 ≥ (𝑛 + 1)(3𝑘 − 2) + (𝑚 − 2𝑛 − 1)(𝑘 + 1) ∑ |𝐸(𝐺𝑖 )| 𝑚−𝑛 𝑖=1 ≥ (𝑛 + 2)𝑘 − (4𝑛 + 3) + 𝑚(𝑘 + 1) for 𝑘 ≥ 4, 𝑞 ≤ (𝑚 + 1)(𝑘 + 1) − 1 < (𝑛 + 2)𝑘 − (4𝑛 + 3) + 𝑚(𝑘 + 1). . this is a contradiction since ∑ |𝐸(𝐺𝑖 )| = 𝑞 𝑚−𝑛 𝑖=1 and 𝑞 ≤ (𝑚 + 1)(𝑘 + 1) − 1. hence 𝑁 ≠ 𝑛 + 1. if 𝑁 > 𝑛 + 1 then ∑ |𝐸(𝐺𝑖 )| > (𝑛 + 2)𝑘 − (4𝑛 + 3) + 𝑚(𝑘 + 1). 𝑚−𝑛 𝑖=1 this again results in a contradiction. thus 𝑁 < 𝑛 + 1 and so number of path graphs belonging to 𝜋 is strictly less than 𝑛 + 1. hence the proof. theorem 2.8. if 𝑇 ∈ ℱ𝑝 then 𝜋𝑠𝑡𝑑 (𝑇) = 1. proof. assume 𝑇 ∈ ℱ𝑝 . every vertex of 𝑇 is either a pendant vertex or a support vertex. let 𝑙 and 𝑚 be the number of pendant vertices and support vertices of 𝑇 respectively. clearly the set of all pendant vertices of 𝑇 forms the 𝛾𝑠 − 𝑠𝑒𝑡. hence 𝛾𝑠(𝑇) = 𝑙. number of edges of 𝑇 is 𝑙 + 𝑚 − 1. also, 𝑚 ≤ 𝑙 for any graph. hence by theorem 2.4, 𝜋𝑠𝑡𝑑 (𝑇) = 1. remark 2.9. if 𝑠(𝐺) = 𝛾𝑠(𝐺) then 𝜋𝑠𝑡 (𝐺)need not be equal to 𝜋𝑠𝑡𝑑 (𝐺). 104 mahiba. m and ebin raja merly. e figure 2. graph 𝐺 and its 𝑆𝐷𝐷𝑚𝑎𝑥 , 𝜋 = {𝐺1, 𝐺2} for the graph 𝐺 in figure 2, minimum steiner set= 𝛾𝑠 − 𝑠𝑒𝑡 = {𝑣1, 𝑣6, 𝑣8, 𝑣11}. hence 𝑠(𝐺) = 𝛾𝑠(𝐺) = 4. steiner domination decomposition 𝜋 = {𝐺1, 𝐺2}is a 𝑆𝐷𝐷𝑚𝑎𝑥 of 𝐺 and so 𝜋𝑠𝑡𝑑 (𝐺) = 2. also 𝜋𝑠𝑡 (𝐺) = 1. therefore 𝜋𝑠𝑡 (𝐺) ≠ 𝜋𝑠𝑡𝑑 (𝐺). theorem 2.10. let 𝐺 be a connected graph such that 𝑠(𝐺) = 𝛾𝑠(𝐺) = 𝑘 (𝑠𝑎𝑦). if there exist some 𝑆𝐷𝑚𝑎𝑥 and 𝑆𝐷𝐷𝑚𝑎𝑥 for 𝐺 satisfying the condition that each subgraph in the decompositions is of order 𝑘 + 1 and has a cutvertex of degree 𝑘 then 𝜋𝑠𝑡 (𝐺) = 𝜋𝑠𝑡𝑑 (𝐺). proof. consider a connected graph 𝐺 with 𝑠(𝐺) = 𝛾𝑠(𝐺) = 𝑘. let 𝜋1 and 𝜋2 be the 𝑆𝐷𝑚𝑎𝑥 and 𝑆𝐷𝐷𝑚𝑎𝑥 respectively which satisfies the condition that each subgraph in both the decompositions is of order 𝑘 + 1 and has a cutvertex of degree 𝑘. first to prove, 𝜋1is a 𝑆𝐷𝐷. let 𝜋1 = {𝐺𝑖 /1 ≤ 𝑖 ≤ 𝑛 }. 𝜋1is a 𝑆𝐷 implies 𝑠(𝐺𝑖 ) = 𝑘 for all 𝑖. each 𝐺𝑖 (1 ≤ 𝑖 ≤ 𝑛 )is of order 𝑘 + 1 and has a cutvertex of degree 𝑘.therefore minimum steiner set of 𝐺𝑖 = 𝛾𝑠 − 𝑠𝑒𝑡 of 𝐺𝑖 for all 𝑖 and so 𝛾𝑠(𝐺𝑖 ) = 𝑘. thus 𝜋1 is a 𝑆𝐷𝐷. in the similar way, we can prove 𝜋2 is a 𝑆𝐷. now to prove, 𝜋𝑠𝑡 (𝐺) = 𝜋𝑠𝑡𝑑 (𝐺). suppose 𝜋𝑠𝑡 (𝐺) > 𝜋𝑠𝑡𝑑 (𝐺) then |𝜋1| > |𝜋2|. since 𝜋1 is a 𝑆𝐷𝐷, we get a contradiction to 𝜋2 is a𝑆𝐷𝐷𝑚𝑎𝑥 . suppose 𝜋𝑠𝑡 (𝐺) < 𝜋𝑠𝑡𝑑 (𝐺) then |𝜋1| < |𝜋2|. since 𝜋2 is a 𝑆𝐷, we get a contradiction to 𝜋1 is a 𝑆𝐷𝑚𝑎𝑥 . therefore 𝜋𝑠𝑡 (𝐺) = 𝜋𝑠𝑡𝑑 (𝐺). 3. conclusion in this paper, we initiate a study on steiner domination decomposition number of graphs. it is quite interesting to investigate this new parameter and study the properties of the subgraphs belonging to 𝑆𝐷𝐷. future works can be done on calculating the steiner 105 steiner domination decomposition number of graphs domination decomposition number for families of graphs and finding the bounds in terms of other graph theoretical parameters. references [1] j.a. bondy and u.s.r. murty, graph theory with applications, macmillan press, london, 1976. [2] g. chartrand and p. zhang, the steiner number of a graph, discrete mathematics, 242, pp.41-54, 2002. [3] t.w. haynes, s.t. hedetniemi and p.j. slater, fundamentals of domination in graphs, crc press, 2013. [4] j. john, g. edwin and p.a.p. sudhahar, the steiner domination number of a graph, international journal of mathematics and computer applications research, 3(3), pp.3742, 2013. [5] e. e. r. merly and m. mahiba, steiner decomposition number of graphs, malaya journal of matematik, special issue, pp.560-563, 2021. [6] e. e. r. merly and m. mahiba, steiner decomposition number of complete𝑛 − sun graph, journal of physics: conference series,1947, 2021. [7] k. ramalakshmi and k. palani, on steiner domination number of graphs, international journal of mathematics trends and technology, 41(2), pp.186-190, 2017. [8] s.k. vaidya and r.n. mehta, on steiner domination in graphs, malaya journal of matematik, 6(2), pp.381-384, 2018. [9] sr little femilin jana. d., jaya. r., arokia ranjithkumar, m., krishnakumar. s., “resolving sets and dimension in special graphs”, advances and applications in mathematical sciences 21 (7) (2022), 3709 – 3717. 106 https://scholar.google.com/citations?view_op=view_citation&hl=en&user=xwcp70yaaaaj&sortby=pubdate&authuser=1&citation_for_view=xwcp70yaaaaj:ijcspb-oge4c ratio mathematica volume 43, 2022 group decision making in conditions of uncertainty using fermat’s weak fuzzy graphs and beal’s weak fuzzy graphs t m nishad* b.mohamed harif† a.prasanna‡ abstract decision making is a process of solving problems for choosing the best alternative. the best way to illustrate the alternatives and relation between them is a graph. developing a fuzzy graph is the convenient way of illutration if there is uncertainty in alternatives or in their relation. in group decision making problems, according to a group of experts, the relation between alternatives involves measure of preference and non preference. intuitionistic fuzzy graph has limitations to model such problems. in npythagorean fuzzy graphs the hesitancy degree and other decision tools are restricted to second degree.to overcome the flaws of intuitionistic fuzzy graphs and npythagorean fuzzy graphs, we introduced fermat’s fuzzy graphs in 2022. in this paper the decision tools are generalized for fermat’s fuzzy graphs. a practical example of selection of investement scheme is illustrated. finally, beal’s fuzzy graphs is developed as generalization of fermat’s fuzzy graphs. keywords: fuzzy graph; weak fuzzy graph; fermat’s fuzzy graph; fermat’s weak fuzzy graphs; beal’s fuzzy graph; beal’s weak fuzzy graphs. 2010 ams subject classification: 03e72,05c72. *research scholar, department of mathematics, rajah serfoji government college (autonomous), (affiliated to bharathidasan university),thanjavur-613005,tamilnadu, india.; email :nishadtmphd@gmail.com. †assistant professor and research supervisor, department of mathematics, rajah serfoji government college (autonomous),(affiliated to bharathidasan university),thanjavur-613005, tamilnadu, india.; email:bmharif@rsgc.ac.in. ‡assistant professor, p g and research department of mathematics, jamal mohamed college (autonomous),(affiliated to bharathidasan university), tiruchirappalli,-620020, tamilnadu, india.; email: apj_jmc@yahoo.co.in. ‡received on september 15, 2022. accepted on december15, 2022. published on december30, 2022. doi:10.23755/rm.v39i0.854. issn: 1592-7415. eissn: 2282-8214. © nishad et al. this paper is published under the cc-by licence agreement. 208 mailto:nishadtmphd@gmail.com mailto:bmharif@rsgc.ac.in mailto:apj_jmc@yahoo.co.in t.m.nishad, b.mohamed harif and a prasanna 1. introduction l a zadeh in 1965[1] introduced fuzzy sets to describe the vagueness phenomena in real world problems.in 1975[2], azriel rosenfeld introduced fuzzy graphs. a.prasanna and t m nishad introduced weak fuzzy graphs in 2021 [3]. in 2009, hongmei and lianhua defined interval valued fuzzy graph (ivfg) [4] and in 2013 talebi and rashmanlou studied properties of isomorphism and complement of an ivfg[5]. to overcome the flaws of intuitionistic fuzzy graph in simulation, muhammed akram, amna habib, etc. discussed specific types of pythagorean fuzzy graphs and applications to decision making in 2018[6]. fermat’s weak fuzzy graph and hesitancy degree in general scale are discussed by b.m harif and t.m nishad in 2022[7]. the american banker and amateur mathematician mr.daniel andrew beal formulated the beal’s conjecture in1993 [8] as a generalization of fermat’s conjecture. the contents of this article are as follows. in section 2 some fundamental concepts of fuzzy graphs and fermat’s fuzzy graphs are reviewed.section 3 illustrates the mathematical model of a group decision making problem using fermat’s weak fuzy graph. section 4 describes the generalized decision tools for fermat’s fuzzy graphs. a practical example of selection of investment scheme is illustrated in section 5. in section 6 , the fundamental concepts of beal’s fuzzy graph and some theorems are developed.the whole article is concluded in section 7. 2.some fundamental concepts of fuzzy graphs a mapping : [0,1]m a → from a non empty set a is a fuzzy subset of a. a fuzzy relation r on the fuzzy subset m , is a fuzzy subset of a a . a is assumed as finite non empty set. definition 2.1: suppose a is the underlying set. a fuzzy graph is a pair of functions g : (𝑚, 𝑟) where fuzzy subset : [0,1]m a → , the fuzzy relation r on 𝑚 is denoted by 𝑟 : a a →[0,1], such that for all ,u v a , we have 𝑟(𝑢, 𝑣) ≤ 𝑚(𝑢) ∧ 𝑚(𝑣) where stands minimum. g*: (𝑚∗, 𝑟∗) denotes the underlying crisp graph of a fuzzy graph g : (𝑚, 𝑟) where * { / ( ) 0}m u a m u=   and * {( , ) / ( , ) 0}r u v a a r u v=    . the nodes u and v are known as neighbours if 𝑟(𝑢, 𝑣) > 0. definition 2.2.: a fuzzy graph g:(𝑚, 𝑟) is a strong fuzzy graph if 𝑟(𝑢, 𝑣) = 𝑚(𝑢) ∧ 𝑚(𝑣),∀(𝑢, 𝑣) ∈ 𝑟∗ . definition 2.3: a fuzzy graph g :(𝑚, 𝑟) is a weak fuzzy graph if 𝑟(𝑢, 𝑣) < 𝑚(𝑢) ∧ 𝑚(𝑣)for all (𝑢, 𝑣) ∈ 𝑟∗ . 209 group decision making in conditions of uncertainty using fermat’s weak fuzzy graphs and beal’s weak fuzzy graphs definition 2.4: a fermat’s fuzzy set (ffs) on a universal set a is a set of 3 tuples of the form f={(u, 𝐼𝐹 (𝑢), 𝑂𝐹(𝑢))} where 𝐼𝐹 (𝑢) and 𝑂𝐹(𝑢) represents the membership and non membership degrees of u a and 𝐼𝐹 (𝑢), 𝑂𝐹(𝑢) satisfy 0 ≤ 𝐼𝐹 𝑛 (𝑢) + 𝑂𝐹 𝑛 (𝑢) ≤ 1 for all u a , n n={1,2,3,..} . definition 2.5: a fermat’s fuzzy relation (ffr) r on a a is a set of 3 tuples of the form r = { ( uv, 𝐼𝑅 (𝑢𝑣), 𝑂𝑅 (𝑢𝑣) } where 𝐼𝑅 (𝑢𝑣), and 𝑂𝑅 (𝑢𝑣) represents the membership degree and non membership degree of uv in r and ( ), ( ) r r i uv o uv satisfy 0 ( ) ( ) 1 n n r r i uv o uv +  for all uv a a . ffr need not be symmetric. hence 𝐼𝑅 (𝑢𝑣) need not be equal to ( ) r i vu . definition 2.6: a fermat’s fuzzy graph (ffg(n)) on a non empty set a is a pair g : (𝜎, µ) with 𝜎 as ffs on a and µ as ffr on a such that 𝐼𝜇 (𝑢𝑣) ≤ 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) ≥ 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) and 0 ( ) ( ) 1 n n i uv o uv    +  for all u,v a , n n={1,2,3,..} .where 𝐼𝜇 : a a →[0,1] and 𝑂𝜇 : a a →[0,1] represents the membership and non membership functions of µ respectively. definition 2.7: a fermat’s fuzzy preference relation (ffpr) on the set of nodes n ={x1, x2, … xn} is represented by a matrix m = (mij)nxn, where mij = ( xixj, i(xixj) , o(xixj) ) for all i,j =1,2,3..n. let mij = (iij ,oij) where iij indicates the degree to which the node xi is preferred to node xj and oij denotes the degree to which the node xi is not preferred to the node xj and 𝜋𝑖𝑗 = √1 − 𝐼𝑖𝑗 𝑛 − 𝑂𝑖𝑗 𝑛𝑛 is interpreted as hesitancy degree ,with the conditions, iij ,oij [0,1], 0 ≤ 𝐼𝑖𝑗 𝑛 + 𝑂𝑖𝑗 𝑛 ≤ 1 , iij = oji , iii = oii = 0.5 for all i,j =1,2,3..n. definition 2.8: a fermat’s fuzzy graph g : (𝜎, µ) is said to be fermat’s strong fuzzy graph fsfg(n) with underlying crisp graph g*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) = 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) = 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) for all uv µ ∗ definition 2.9: a fermat’s fuzzy graph g : (𝜎, µ) is said to be fermat’s weak fuzzy graph fwfg(n) with underlying crisp graph g*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) < 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) > 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) for all uv µ ∗ definition 2.10 : a fermat’s fuzzy graph g : (𝜎, µ) is said to be complete ffg with underlying crisp graph g*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) = 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) = 𝑂𝜎 (𝑢) ∨ 210 t.m.nishad, b.mohamed harif and a prasanna 𝑂𝜎 (𝑣) for all u,v 𝜎 ∗. 3.modeling of group decision making problem example 3.1: mr. x from india wish to invest money in any of the following 5 schemes that helps him better financial security in future. 1. public provident fund s1 2. national saving certificate s2 3. atal pension yojana s3 4. national pension scheme s4 5. sovereign gold bonds s5 he consulted with 4 experts and they advised the merits and demerits of each particular scheme comparing with other. the aggregate of information ffpr is prepared as relation matrices.how can he select the best scheme? modeling: suppose the 5 schemes are s1,s2,s3,s4 and s5. consider the discrete set of alternatives a = {s1,s2,s3,s4,s5}.since the alternatives are present , assign the membership degree as 1 and non membership degree 0 to each alternatives. consider the set of experts as {e1,e2,e3,e4}. since each experts gives the acceptance and rejection reasons comparing every pair of alternatives, the aggregate of information ffpr can be represented as relation matrices. this data represents a ffg(n). if in the given ffprs , all the membership values are in (0,1) and non membership values are greater than 0 then the given ffg(n) will be fwfg(n). 4. decision tools for fermat’s fuzzy graph in decision making, the optimal score having maximum rank is considered as best choice. the scores to rank the alternatives can be calculated using score function . here is the collective fermat’s fuzzy element which can be obtained using fermat’s fuzzy weighted averaging operator ffwa. the weight of each expert can be obtained using deviations of each experts and the deviations can be calculated from difference matrices. the entries in difference matrix is calculated using fermats fuzzy hamming distance between fermat’s fuzzy elements. 4.1 fuzzy averaging operator ffa fermat’s fuzzy element (ffe) indicates preference of each expert ek over each pair of alternatives. it is determined using fermat’s fuzzy averaging operator ffa ( )is p ip ( ) ( ) ( )( ) ( ) ( ) 1 1 1 2 1 1 , ,...., 1 1 , , 1, 2, 3,.., . ij ij m mm m k k k n n i i im j j ffa p p p i o i m   = =        = − − =              ( )k i p 211 group decision making in conditions of uncertainty using fermat’s weak fuzzy graphs and beal’s weak fuzzy graphs ffe is used in calculation of ffwa. 4.2 fermat’s fuzzy hamming distance between ffes. from the given ffpr, where are hesitancy degree. 4.3 difference matrix 4.4 average values of difference matrix the equation to determine average values of difference matrix 4.5 deviation of expert er from remaining experts the equation to determine deviation of expert er from remaining experts 4.6 weight of experts wr. the equation to determine weight of experts wr. 4.7 fermat’s fuzzy weighed averaging operator ffwa fermat’s fuzzy weighed averaging operator ffwa to compute collective fermat’s fuzzy element pi over other alternatives is 4.8 score function score function to rank the alternatives , 1, 2,..., i u i m= , ( ) ( )( )( ) ( ) , , 1, 2,..., . i i n n i p p i i i s p i o i u o u u i m   = − − = ( ) ( ) ( ) ( )( )1 2, ,...., , 1, 2, 3,.., . k k k k i i i im p ffa p p p i m= = ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , , 2 l k l k l k ij ij p p p pij ij ij ij l k n n n n n n ij ij p p d p p i i o o         = − + − + −      ( ) ( ) ( ) ( ) ( ) ( ) 1 , 1l k l l k kij ij p p p p ij ij ij ij n n n n n n p p i o i o      = − − = − − ( )( ) ( ) ( )( ),lk l klk ij ij ij mxm mxm d d d p p= = ( ) 2 1 1 1 m m lk lk ij i j d d m = = =  1, s r rk k k r d d =  =  ( ) ( ) 1 1 1 , 1, 2,.., . r r s r r d w r s d − − = = =  ( ) ( ) ( )( ) ( ) ( ) ( )1 2 1 1 , ,...., 1 1 , , k k i ik k p p i i w w s s s n n i i i i p p k k p ffwa p p p i o i o   = =        = = − − =             ( )is p 212 t.m.nishad, b.mohamed harif and a prasanna 5. illustration of a practical example in example 3.1, suppose the aggregate of information ffpr is given as following relation matrices. data 1. the information from e1 in the form of relation matrix. data 2. the information from e2 in the form of relation matrix. data 3. the information from e3 in the form of relation matrix. data 4. the information from e4 in the form of relation matrix. the above data represents a ffg(n). among the relations,0.8+0.8= 0.9+0.7= 0.7+0.9 = 1.6 is the maximum sum among measures of acceptance and corresponding rejection. since there is more than one pair with the same sum, we break the tie by comparing the sum of powers and selecting the pair that brings maximum sum.here 0.82+0.82 =1.28 < ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 0.5, 0.5 0.6, 0.8 0.7, 0.9 0.6, 0.9 0.3, 0.91 0.8, 0.6 0.5, 0.5 0.5, 0.8 0.6, 0.7 0.7, 0.92 0.9, 0.7 0.8, 0.5 0.5, 0.5 0.4, 0.9 0.8, 0.43 0.9, 0.6 0.7, 0.6 0.9, 0.4 0.5, 0.5 0.8, 0.54 5 0.9, 0.3 0.9, 0.7 0.4, 0.8 0.5, 0.8 0.5, 0.5 c c c c c c c c c c                  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 0.5, 0.5 0.6, 0.7 0.7, 0.8 0.8, 0.8 0.3, 0.81 0.7, 0.6 0.5, 0.5 0.5, 0.8 0.6, 0.7 0.7, 0.82 0.8, 0.7 0.8, 0.5 0.5, 0.5 0.4, 0.9 0.8, 0.53 0.8, 0.8 0.7, 0.6 0.9, 0.4 0.5, 0.5 0.8, 0.54 5 0.8, 0.3 0.8, 0.7 0.5, 0.8 0.5, 0.8 0.5, 0.5 c c c c c c c c c c                  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 0.5, 0.5 0.5, 0.8 0.6, 0.9 0.7, 0.9 0.4, 0.91 0.8, 0.5 0.5, 0.5 0.5, 0.8 0.6, 0.7 0.5, 0.92 0.9, 0.6 0.8, 0.5 0.5, 0.5 0.4, 0.9 0.6, 0.43 0.9, 0.7 0.7, 0.6 0.9, 0.4 0.5, 0.5 0.8, 0.54 5 0.9, 0.4 0.9, 0.5 0.4, 0.6 0.5, 0.8 0.5, 0.5 c c c c c c c c c c                  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 0.5, 0.5 0.6, 0.8 0.7, 0.9 0.7, 0.9 0.3, 0.91 0.8, 0.6 0.5, 0.5 0.5, 0.8 0.7, 0.7 0.7, 0.92 0.9, 0.7 0.8, 0.5 0.5, 0.5 0.4, 0.8 0.8, 0.43 0.9, 0.7 0.7, 0.7 0.8, 0.4 0.5, 0.5 0.8, 0.54 5 0.9, 0.3 0.9, 0.7 0.4, 0.8 0.5, 0.8 0.5, 0.5 c c c c c c c c c c                  213 group decision making in conditions of uncertainty using fermat’s weak fuzzy graphs and beal’s weak fuzzy graphs 0.92+0.72 =1.3. so we consider sum of higher powers of 0.9 and 0.7 till we get a sum ≤1. note that 0.93+0.73 =1.072 > 1.but 0.94+0.74 = 0.8962 < 1 .so the ffg(n) is ffg(4).since alternatives are present , the membership value 1 and non membership value 0 have to be assigned to each alternatives. in the given ffrs , all the membership and non membership values are in (0,1). hence the given ffg(4) is fwfg(4). fermat’s fuzzy eelements are p1 (1) = (0.5854,0.7816) , p2 (1) = (0.6624,0.6853) , p3 (1) = (0.7732,0.5753) , p4 (1) = (0.8196,0.5144) , p5 (1) = (0.7786,0.5827) p1 (2) = (0.6544,0.7090) , p2 (2) = (0.6231,0.6694) , p3 (2) = (0.7301,0.6015) , p4 (2) = (0.7896,0.5448) , p5 (2) = (0.6855,0.5827) p1 (3) = (0.5725,0.7816) , p2 (3) = (0.6304,0.6608) , p3 (3) = (0.7435,0.5578) , p4 (3) = (0.8196,0.5305) , p5 (3) = (0.7786,0.5448) p1 (4) = (0.6129,0.7816) , p2 (4) = (0.6803,0.6853) , p3 (4) = (0.7732,0.5619) , p4 (4) = (0.7896,0.5471) , p5 (4) = (0.7786,0.5827) from the difference matrices and the average values of difference matrices we get the deviations d1= 0.196528, d2= 0.341216, d3= 0.261024 and d4= 0.242128. then the weights of experts are w1= 0.31842, w2= 0.18340, w3= 0.23974 and w4= 0.25845. now the collective fermat’s fuzzy elements are p1= ( ip1, op1) = (0.60466,0.76775), p2 = ( ip2, op2) = (0.65365,0.67642), p3 = ( ip3, op3) = (0.75922,0.57224), p4 = ( ip4, op4) = (0.80715,0.53211) and p5 = ( ip5, op5) = (0.76520,0.57338). the corresponding score function gives the following scores s(p1) = 0.21377, s(p2) = 0.02680, s(p3) = 0.22503, s(p4) = 0.34427 and s(p5) = 0.23476 since s(p4) is the maximum score ,the best choice is s4, the national pension scheme. 6. beal’s fuzzy graph bfg(m,n) if the membership value of acceptance (or rejection) is given a limit ( say α ) then the membership value ( say β) of rejection (or acceptance ) is assumed to be governed by the in equation βn ≤ 1αm for some m,n n={1,2,3,..} .therefore the generalization of ffg(n) has importance. 214 t.m.nishad, b.mohamed harif and a prasanna definition 6.1: a beal’s fuzzy set (bfs) on a universal set a is a set of 3 tuples of the form f={(u, 𝐼𝐹 (𝑢), 𝑂𝐹(𝑢))} where 𝐼𝐹 (𝑢) and 𝑂𝐹 (𝑢) represents the membership and non membership degrees of u a and 𝐼𝐹 (𝑢), 𝑂𝐹(𝑢) satisfy 0 ≤ 𝐼𝐹 𝑚(𝑢) + 𝑂𝐹 𝑛 (𝑢) ≤ 1 for all u a , m,n n={1,2,3,..} . definition 6.2: a beal’s fuzzy relation (bfr) r on a a is a set of 3 tuples of the form r = { ( uv, 𝐼𝑅 (𝑢𝑣), 𝑂𝑅 (𝑢𝑣) } where 𝐼𝑅 (𝑢𝑣), and 𝑂𝑅 (𝑢𝑣) represents the membership degree and non membership degree of uv in r and ( ), ( ) r r i uv o uv satisfy 0 ≤ 𝐼𝑅 𝑚(𝑢𝑣) +)𝑂𝑅 𝑛(𝑢𝑣) ≤ 1 for all uv a a . bfr need not be symmetric. hence 𝐼𝑅 (𝑢𝑣) need not be equal to ( )ri vu . definition 6.3:a beal’s fuzzy graph bfg(m,n) on a non empty set a is a pair g : (𝜎, µ) with 𝜎 as bfs on a and µ as bfr on a such that 𝐼𝜇 (𝑢𝑣) ≤ 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) ≥ 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) and 0 ≤ 𝐼µ 𝑚(𝑢𝑣) +)𝑂µ 𝑛(𝑢𝑣) ≤ 1 for all u,v a , m,n n={1,2,3,..} .where 𝐼𝜇 : a a →[0,1] and 𝑂𝜇 : a a →[0,1] represents the membership and non membership functions of µ respectively. definition 6.4: a beal’s fuzzy graph g : (𝜎, µ) is said to be beal’s strong fuzzy graph bsfg(m,n) with underlying crisp graph g*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) = 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) = 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) for all uv µ ∗ . definition 6.5: a beal’s fuzzy graph g : (𝜎, µ) is said to be beal’s weak fuzzy graph bwfg(m,n) with underlying crisp graph g*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) < 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) > 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) for all uv µ ∗ definition 6.6 : a beal’s fuzzy graph g : (𝜎, µ) is said to be complete bfg with underlying crisp graph g*: (𝜎∗, µ∗) 𝑖𝑓 𝐼𝜇 (𝑢𝑣) = 𝐼𝜎 (𝑢) ∧ 𝐼𝜎 (𝑣), 𝑂𝜇 (𝑢𝑣) = 𝑂𝜎 (𝑢) ∨ 𝑂𝜎 (𝑣) for all u,v 𝜎 ∗. theorem 6.1: when m = n, bfg (m,n) ffg(n) and bfg (1,1) ffg(1) which is an intuitionistic fuzzy graph. proof. directly follows from the definitions. i.e, beal’s fuzzy graph is generalization of fermat’s fuzzy graph and fermat’s fuzzy graph is generalization of intuitionistic fuzzy graph. theorem 6.2: bwfg (m-1,n-1)  bwfg(m,n) but the converse is not true. proof. let g : (𝜎, µ) be a bwfg (n-1) with 𝜎 as bfs on a and µ as bfr on a . since 𝐼𝜇 (𝑢𝑣) < 1 , 𝐼µ 𝑚−1(𝑢𝑣) < 1  𝐼µ 𝑚(𝑢𝑣) < 𝐼µ 𝑚−1(𝑢𝑣) < 1, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑚 𝑁 → (1) similarly since 1 1 ( ) 1, ( ) 1 ( ) ( ) n n n o uv o uv o uv o uv     − −       215 group decision making in conditions of uncertainty using fermat’s weak fuzzy graphs and beal’s weak fuzzy graphs therefore 𝐼µ 𝑚−1(𝑢𝑣) +)𝑂µ 𝑛−1(𝑢𝑣) ≤ 1  𝐼µ 𝑚(𝑢𝑣) + 𝑂µ 𝑛(𝑢𝑣) ≤ 1. hence bwfg(m-1,n-1)  bwfg(m,n). it is obvious from equation (1) that the converse is not true. 7 conclusion in this article some fundamental concepts of fuzzy graphs and fermat’s fuzzy graphs are reviewed.the decision tools are generalized for fermat’s fuzzy graphs. application of fermats weak fuzzy graph in modeling group decision making problem is illustrated with a practical example. the fundamental concepts of beal’s fuzzy graph are developed. the applications of ffg(n) and bfg(m,n) in various fields of science, social science and engineering are under research. acknowledgement the authors wish to thank all the reviewers of this article. references [1] l. a. zadeh fuzzy sets,inform.and control 8 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[2] a. rosenfeld, fuzzy graphs, in: l.a. zadeh, k.s. fu, k.tanaka, m. shimura fuzzy sets and their applications to cognitive and decision process, academic press,1975,pp.77-95. [3] a. prasanna and t.m. nishad,some properties of weak fuzzy graphs, math. comput. sci.11(3) (2021), 3594-3601. [4] ju hongmei and wang lianhua, interval-valued fuzzy sub semigroups and subgroups associated by interval-valued fuzzy graphs, global congress on intelligent systems (2009), 484-487. [5] a. a.talebi and h. rashmanlou, isomorphism on interval-valued fuzzy graphs, annals of fuzzy mathematics and informatics 6(1) (2013), 47-58. [6] muhammed akram, et al., specific types of pythagorean fuzzy graphs and applications to decision-making, math. comput. appl. 23(3) (2018),42. [7] b.mohamed harif and t.m nishad ,interval valued weak fuzzy graphs and fermat’s weak fuzzy graphs,advances and applications in mathematical sciences,volume 22,issue 2, dec.2022. [8] r.daniel mauldin (1997). a generalization of fermat's last theorem: the beal conjecture and prize problem (pdf). notices of the ams. 44 (11): 1436–1439. 216 ratio mathematica volume 44, 2022 antagonistic intuitionistic fuzzy sub commutative ideals of subtraction g-algebra b. lena 1 c. ragavan 2 abstract the notions of over antagonistic intuitionistic fuzzy sub commutative ideals of subtraction g-algebras are introduced. the characterization properties of antagonistic intuitionistic fuzzy sub commutative ideals are obtained. we investigate the relations between antagonistic intuitionistic fuzzy sub implicative ideals and antagonistic intuitionistic fuzzy sub commutative ideals of subtraction g-algebra. keywords: subtraction g-algebra, aifsi ideals, aifsc ideals. 2020 ams subject classifications: 05a15, 11b68, 34a05. 3 1 department of mathematics, trinity college for women, namakkal, tamil nadu, india;lenasenthil1977@gmail.com 2 department of mathematics, svm college, uthangarai, krishnagiri, tamil nadu, india. 3 received on april 12th, 2022.accepted on sep 1st, 2022.published on nov 30th, 2022.doi: 10.23755/rm.v44i0.913. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by license agreement. 260 mailto:lenasenthil1977@gmail.com b. lena, c. ragavan, s. jeyanthi 1. introduction the introduction of fuzzy sets by zadeh [17], there have been a number of generalizations of this fundamental concept. bck-algebras and bci-algebras are two classes of logical algebra. extensive applications of fuzzy set theory have been found in various fields, for example, artificial intelligence, computer science, control engineering, expert system, management science, operation research and many others which were initiated by k. iseki [3, 4]. the notion of fuzzy sets, invented by l. a. zadeh [17], has been applied to many fields. since then, fuzzy bci/bck algebras have been extensively investigated by several researchers. for bck-algebras, y. b. jun et al. [7, 10] introduced the notions of fuzzy positive implicative ideals. bandaru and rafi [1] introduced a new notion called g-algebra, since the fuzzy g-algebras have been extensively investigated by several researches. the properties of fuzzy sub commutative ideals and fuzzy sub implicative ideals are obtained. ragavan and solairaj [14] some new results on intuitionistic fuzzy h-ideals in bci algebra. 2. preliminaries definition 2.1. [8] a subtraction g-algebra we mean a nonempty set x with a binary operation and a constant 0 satisfying the following conditions: (f1) xx = 0, (f2) x – (x y) = y, for all x, y ϵ x. definition 2.2. [15] (g-subalgebra) a non-empty subset s of a subtraction g-algebra x is called a subtraction g-subalgebra of x if x-y ∈s [0,1]. definition 2.3. [16] a fuzzy set f of a universe x is a function from x to the unit closed interval [0, 1], that is f: x → [0, 1]. definition 2.4. [2] an intuitionistic fuzzy set a in a finite universe of discourse x = {x1, x2, x3, x4, ··· xn} is given by a = {<x, ψa(x), ωa(x) >: x ∈x}, where ψa: x → [0, 1] and ωa : x → [0, 1] such that 0 ≤ ψa(x) + ωa (x) ≤ 1. the number ψa(x) and ωa (x) denote the degree of membership and non-membership of x ∈x to a, respectively. for each ifs a in x, if πa(x) = 1 − ψa(x)− ωa(x),for all x ∈x. definition 2.5. a nonempty subset i of x is called an ideal of x if (i1): 0 ∈ i, and (i2): x y ∈ i and y ∈ i imply x ∈ i. definition 2.6. a fuzzy subset ψa of x is said to be a fuzzy ideal of x if it satisfies (f3) ψa(0) ≥ ψa (x) for all x ∈ x, (f4) ψa (x) ≥ min {ψa (x y), ψa (y)} for all x, y ∈ x. 261 antagonistic intuitionistic fuzzy sub commutative ideals of subtraction g-algebra definition 2.7. an intuitionistic fuzzy set a = {x, ψa(x), ωa(x): x ∈ x} in x is called an intuitionistic fuzzy ideal of x if it satisfies (f5) ψa (0) ≥ ψa (x), ωa(0) ≤ ωa (x), (f6) ψa (x) ≥ min {ψa (x y), ψa (y)} and (f7) ωa(x) ≤ max {ωa (x y), ωa(y)} for all x, y ∈x. definition 2.8. an intuitionistic fuzzy set a = {x, ψa(x), ωa(x): x ∈x} in x is called an antagonistic -intuitionistic fuzzy ideal of x if it satisfies (f8) ψa (0) ≤ ψa (x), ωa (0) ≥ ωa (x), (f9) ψa(x) ≤ max { ψa (x y), ψa (y)} and (f10) ωa(x) ≥ min {ωa (x y), ωa(y)} for all x,y ∈x. definition 2.9. [9, 11] a nonempty subset i of x is called a positive implicative ideal (i.e., weakly positive implicative ideal) of x if it satisfies (i1) 0 ∈ i and (i3): ((x z) z) (y z) ∈ i and y ∈ i imply x – z ∈ i. definition 2.10. [9, 14] a nonempty subset i of x is called a sub-implicative ideal of x if it satisfies (i1) 0 ∈ i and (i3): ((x (x y)) (y x)) z ∈ i and z ∈ i imply y (y x) ∈ i. definition 2.11. [9, 13] a nonempty subset i of x is called a sub-commutative ideal of x if it satisfies (i1) 0 ∈ i and (i4): (y (y (x (x y)))) z ∈ i and z ∈ i imply x (x y) ∈ i. definition 2.12. an anti-fuzzy subset ψaof x is called an antagonistic -fuzzy subimplicative ideal of x if it satisfies (f11) ψa(0) ≤ ψa (x) for all x ∈ x, and (f12) ψa(y (y x)) ≤ max { ψa(((x (x y)) (y x)) z), ψa(z)} for all x, y, z ∈ x. definition 2.13. an anti-fuzzy subset ψa of x is called an antagonistic -fuzzy subcommutative ideal (briefly, afsc-ideals) of x if it satisfies (f1) and (f13) ψa(x (x y)) ≤ max { ψa((y (y (x (x y)))) z), ψa(z)} for all x, y, z ∈ x. definition 2.14. an anti-fuzzy subset (ψa, ωa) of x is called an antagonistic intuitionistic fuzzy sub-implicative ideal of x if it satisfies (f16) ψa(0) ≤ ψa (x), ωa (0) ≥ ωa (x) for all x ∈ x, and (f17) ψa(y (y x)) ≤ max { ψa (((x (x y)) (y x)) z), ψa(z)} for all x, y, z ∈ x. (f18) ωa (y (y x)) ≥ min {ωa (((x (x y)) (y x)) z), ωa(z)} for all x, y, z ∈x. definition 2.15. an anti-fuzzy subset (ψa, ωa) of x is called an antagonistic intuitionistic fuzzy sub-commutative ideal (briefly, afsc-ideals) of x if it satisfies (f19) ψa (0) ≤ ψa (x), ωa (0) ≥ ωa (x) for all x ∈ x, and (f20) ψa (x (x y)) ≤ max { ψa((y (y (x (x y)))) z), ψa(z)} for all x, y, z ∈ x. 262 b. lena, c. ragavan, s. jeyanthi (f21) ωa (x (x y)) ≥ min {ωa ((y (y (x (x y)))) z), ωa(z))} for all x, y, z ∈ x. 3. over antagonistic-intuitionistic fuzzy sub commutative ideal of subtraction g-algebras theorem 3.1. if gα, β(a) and gα, β(b) are antagonistic -intuitionistic fuzzy subcommutative ideal of subtraction g-algebra then gα, β(a+b) is also antagonistic intuitionistic fuzzy sub-commutative ideal of subtraction g-algebra. proof: given gα, β(a) and gα, β(b) are antagonistic -intuitionistic fuzzy subcommutative ideal of x. (1) ψa (0) ≤ ψa(x), ωa (0) ≥ ωa (x) for all x ∈ x, and (2) ψa (x (x y)) ≤ max {ψa ((y (y (x (x y)))) z), ψa (z)} for all x, y, z ∈ x. (3) ωa (x (x y)) ≥ min { ωa ((y (y (x (x y)))) z), ωa (z))} for all x, y, z ∈ x. (4) ψb (0) ≤ ψb (x), ωb (0) ≥ ωb (x) for all x ∈ x, and (5) ψb (x (x y)) ≤ max {ψb ((y (y (x (x y)))) z), ψb (z)} for all x, y, z ∈ x. (6) ωb (x (x y)) ≥ min {ωb ((y (y (x (x y)))) z), ωb (z))} for all x, y, z ∈ x. case1: ψa (0) + ψb (0) ψa (0). ψb (0) ≤ ψa (x) + ψb (x) ψa (x). ψb (x) gα, β {ψa (0) + ψb (0) ψa (0). ψb (0)} ≤ gα, β {ψa (x) + ψb (x) ψa (x). ψb (x)} {αψa (0) + αψb (0) αψa (0). α ψb (0)} ≤ {α ψa(x) + α ψb (x) α ψa (x). α ψb (x)} {αψa (0) + αψb (0) – α 2 ψa (0). ψb (0)} ≤ {αψa (x)+ αψb (x)α 2 ψa (x). ψb (x)} gα, β (a + b) (0) ≤ gα, β (a + b) (x) ωa (0). ωb (0) ≥ ωa (x). ωb (x) gα, β {ωa (0). ωb (0)} ≥ gα, β {ωa (x). ωb (x)} {β ωa (0). β ωb (0)} ≥ {β ωa(x). β ωb(x)} {β 2 ωa (0). ωb (0)} ≥ {β 2 ωa (x). ωb (x)} gα, β (a + b) (0) ≥ gα, β (a + b) (x) case 2: { ψa (x (x y)) + ψb (x (x y)) ψa (x (x y)). ψb (x (x y))} ≤ max {[ ψa ((y -(y (x (x y))) z), ψa(z)]} + max [ψb ((y (y (x (x y))) z), ψb (z)]max {ψa ((y (y (x (x y))) z), ψa (z)}. max {ψb ((y (y (x (x y))) z), ψb (z)}} gα, β {ψa (x (x y)) + ψb (x (x y)) ψa (x (x y)). ψb (x (x y))} ≤ gα, β max{{μa ((y (y (x (x y))) z) + μb ((y (y (x (x y))), μa(z) + μb (z)}max{ μa ((y (y (x (x y))) z).μb ((y (y (x (x y))), μa (z).μb (z)}} {αψa(x-(x-y)) + αψb(x-(x-y)) αψa(x-(x-y)). αψb(x-(x-y))} ≤ max {{α ψa ((y-(y-(x(x-y)))-z) + α ψb ((y-(y-(x-(x-y)))-z), α ψa (z) + α ψb(z)}max {α ψa ((y-(y-(x-(x-y)))z). α ψb((y-(y-(x-(x-y)))-z)-, α ψa (z). α ψb (z)}} {αψa(x-(x-y)) + αψb(x-(x-y)) – α 2 ψa(x-(x-y)). ψb(x-(x-y))} ≤ max {{ α ψa ((y-(y-(x(x-y)))-z) + αψb ((y-(y-(x-(x-y)))-z) – α 2 ψa ((y-(y-(x-(x-y)))-z). ψb ((y-(y-(x-(x-y)))z)}, {α ψa (z) + αψb (z)α 2 ψa (z). ψb (z)}} gα, β(a+b) (x-(x-y)) ≤ max {gα, β (a+b) ((y-(y-(x-(x-y)))-z), gα, β (a+b) (z)} ωa (x (x y)). ωb (x (x y)) ≥ min {ωa ((y (y (x (x y)))) z), ωa (z)}. min {ωb ((y (y (x (x y)))) z), ωb (z)} 263 antagonistic intuitionistic fuzzy sub commutative ideals of subtraction g-algebra ωa (x (x y)). ωb (x (x y)) ≥ min {ωa ((y (y (x (x y)))) z). ωb ((y (y (x (x y))))-z}, min {ωa (z). ωb(z)} gα,β{ ωa(x (x y)). ωb (x (x y))}≥ gα,β{ min { ωa ((y (y (x (x y)))) z). ωb ((y (y (x (x y))))-z}, min{ ωa (z). ωb(z)}} {β ωa(x (x y)). β ωb (x (x y))} ≥min {β ωa ((y (y (x (x y)))) z). β ωb ((y (y (x (x y))))-z}, min {α ωa(z)).α ωb (z)} { β 2 ωa (x (x y)). ωb (x (x y))} ≥min {{β 2 ωa((y (y (x (x y)))) z). ωb ((y (y (x (x y))))-z},{ β 2 ωa(z)).α ωb (z}} gα,β(a+b)(x(x y))≥min{ gα,β(a+b)((y (y (x (x y)))), gα,β(a+b) (a+b)(z)} therefore, the two numbers of a+b is an antagonisticintuitionistic fuzzy subcommutative ideal in subtraction g-algebra of x. theorem 3.2. if gα,β(a) and gα,β(b) are antagonistic -intuitionistic fuzzy subcommutative ideal of subtraction g-algebra then gα,β(a•b) is also antagonistic intuitionistic fuzzy sub-commutative ideal of subtraction g-algebra. proof: given gα,β(a) and gα,β(b) are antagonistic -intuitionistic fuzzy subcommutative ideal of x. (1) ψa (0)≤ ψa (x), ωa(0)≥ωa(x) for all x ∈ x, and (2) ψa(x (x y)) ≤ max { ψa ((y (y (x (x y)))) z), ψa(z)} for all x, y, z ∈ x. (3) ωa (x (x y)) ≥min { ωa((y (y (x (x y)))) z), ωa (z))} for all x, y, z ∈ x. (4) ψb (0)≤ ψb (x), ωb(0)≥ ωb(x) for all x ∈ x, and (5) ψb (x (x y)) ≤ max { ψb ((y (y (x (x y)))) z), ψb (z)} for all x, y, z ∈ x. (6) ωb (x (x y))≥ min { ωb ((y (y (x (x y)))) z), ωb (z))} for all x, y, z ∈ x. case1: {ψa(0). ψb(0)} ≤ {ψa(x). ψb (x)} gα,β{ψa(0). ψb(0)}≤ gα,β{ψa (x). ψb (x)} {αψa(0). αψb(0)}≤ { αψa (x). αψb (x)} {α 2 ψa(0). ψb(0)}≤ { α 2 ψa (x). ψb (x)} gα,β(a•b)(0)≤ gα,β(a•b)(x) {ωa(0)+ ωb(0) ωa(0). ωb(0)} ≥ {ωa(x)+ ωb (x)ωa (x). ωb (x)} gα,β{ ωa(0)+ ωb(0) ωa(0). ωb(0)} ≥ gα,β{ ωa(x)+ ωb (x)ωa (x). ωb (x)} {β ωa(0)+ β ωb(0) β ωa(0). β ωb(0)} ≥{ β ωa(x)+ β ωb (x)β ωa (x). β ωb (x)} {β ωa(0)+ β ωb(0) – β 2 ωa(0). ωb(0)} ≥ { β ωa(x)+ β ωb (x)β 2 ωa (x). ωb (x)} gα,β(a•b)(0) ≥ gα,β(a•b)(x) case2: {ψa(x(xy)). ψb (x(xy))} ≤ max {ψa ((y-(y-(x-(x-y)))-z), ψa (z)}. max {ψb ((y-(y(x-(x-y)))-z), ψb (z)}} gα,β{ ψa(x-(x-y)). ψb(x-(x-y))} ≤ gα,β max{ ψa ((y-(y-(x-(x-y)))-z). ψb((y-(y-(x-(xy))), ψa(z). ψb(z)}} {αψa(x-(x-y)). αψb(x-(x-y))} ≤ max {α ψa ((y-(y-(x-(x-y)))-z). α ψb((y-(y-(x-(x-y)))z)-, α ψa (z). α ψb (z)}} {α 2 ψa(x-(x-y)). ψb(x-(x-y))} ≤ max {{α 2 ψa ((y-(y-(x-(x-y)))-z). ψb ((y-(y-(x-(x-y)))z)}, {α ψa (z) + αψb (z)α 2 ψa (z). ψb (z)}} 264 b. lena, c. ragavan, s. jeyanthi gα,β(a•b)( x-(x-y))≤ max{ gα,β (a•b) ((y-(y-(x-(x-y)))-z), gα,β (a•b) (z)} { ωa(x(xy)) + ωb (x(xy)) ωa(x(xy)). ωb (x(xy))} ≥min {[ ωa((y-(y-(x-(xy)))-z), ωa(z)]} + max [ωb((y-(y-(x-(x-y)))-z), ωb (z)]max {ωa ((y-(y-(x-(x-y)))-z), ωa (z)}. max { ωb ((y-(y-(x-(x-y)))-z), ωb (z)}} { ωa(x(xy)) + ωb (x(xy)) ωa(x(xy)). ωb (x(xy))} ≥{min { ωa((y-(y-(x-(xy)))-z) +ωb((y-(y-(x-(x-y)))-z) ωa ((y-(y-(x-(x-y)))-z). ωb ((y-(y-(x-(x-y)))-z)}, min{ ωa (z)+ ωb (z) ωa (z). ωb (z)}} ωa(x(xy)) + ωb (x(xy)) ωa(x(xy)). ωb (x(xy))} ≥{min { ωa((y-(y-(x-(xy)))-z) +ωb((y-(y-(x-(x-y)))-z) ωa ((y-(y-(x-(x-y)))-z). ωb ((y-(y-(x-(x-y)))-z)}, {ωa (z)+ ωb (z) ωa (z). ωb (z)}} gα,β{ωa(x-(x-y))+ ωb(x-(x-y)) ωa(x-(x-y)). ωb(x-(x-y))} ≥min gα,β {{ωa ((y-(y-(x(x-y)))-z) + ωb ((y-(y-(x-(x-y))), ωa (z) + ωb (z)}max{ωa((y-(y-(x-(x-y)))-z). ωb((y(y-(x-(x-y))), ωa(z). ωb(z)}} {αωa(x-(x-y))+ αωb(x-(x-y)) αωa(x-(x-y)). αωb(x-(x-y))} ≥min{{αωa ((y-(y-(x-(xy)))-z) + αωb ((y-(y-(x-(x-y)))-z), αωa (z) + αωb(z)}max {αωa ((y-(y-(x-(x-y)))-z). αωb((y-(y-(x-(x-y)))-z), αωa (z). αωb (z)}} {α ωa (x (x y)) + α ωb (x (x y)) – α 2 ωa (x (x y)). ωb (x (x y))} ≥ min {{ α ωa ((y (y (x (x y))) z) + α ωb ((y (y (x (x y))) z) – α 2 ωa ((y (y (x (x y))) z). ωb ((y (y (x (x y))) z)}, {αωa (z) + αωb (z)α 2 ωa (z). ωb (z)}} gα, β(a • b) (x (x y)) ≥ min {gα, β (a • b) ((y (y (x (x y))) z), gα,β (a • b) (z)} therefore, the two numbers of a b is an antagonisticintuitionistic fuzzy subcommutative ideal in subtraction g-algebra of x. example 3.1. let x = {0, a, b, c} be a subtraction g-algebra with the following cayley table. 0 a b c 0 0 a b c a 0 0 b c b 0 a 0 c c 0 a b 0 we define an antifuzzy set a then a = < x, ψa, ωa> by routine calculation a is an antagonistic -intuitionistic fuzzy sub-implicative ideal of x. x 0 a b c ψa .40 .42 .62 .75 ωa .50 .45 .29 .21 we define an antifuzzy set b then b = < x, ψb, ωb > by routine calculation b is an antagonistic -intuitionistic fuzzy sub-implicative ideal of x. x 0 a b c ψb .51 .53 .61 .72 ωb .47 42 .31 .26 265 antagonistic intuitionistic fuzzy sub commutative ideals of subtraction g-algebra if a and b are antagonistic -intuitionistic fuzzy sub-implicative ideal of x. obviously the union a b, since x = b, y = c, z = a, α =.41, β=.46. {α 2 ψa (0). ψb (0)} ≤ {α 2 ψa(x). ψb(x)}. 0343 ≤ .0636 {βωa (0) +βωb (0)-β 2 ωa (0). ωb (0)} ≥ {βωa(x)+β ωb(x)β 2 ωa(x). ωb(x)}. 4605 ≥ .2569 {α 2 ψa (x (x y)). ψb (x (x y))} ≤ max {{α 2 ψa ((y (y (x (x y)))) z). ψb ((y (y (x (x y)))) z)}, {α 2 ψa(z). ψb(z)}}. 0920 .0636 {βωa (b (b c)) + βωb (b (b c))β 2 ωa (b (b c)). ωb (b (b c)) ≥ min {{βωa ((c (c (b (b c)))) a) + β ωb ((c (c (b (b c)))) a)β 2 ωa ((c (c (b (b c)))) a). ωb ((c (c (b (b c)))) a)}, {βωa(a)) + βωb(a))β 2 ωa(a)). ωb(a))}}. 2047 . 3603 therefore, the given two products of antagonistic -intuitionistic fuzzy sub-implicative ideal of x is not an antagonistic -intuitionistic fuzzy sub-implicative ideal of x. references [1] atanassov k. intuitionistic fuzzy sets vii itkr’s session, sofia, june 1983(deposed in central sci, techn. library of bulg. acad. of sci., 1697/84) (in bulg.). 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[14] l. a. zadeh, fuzzy sets, information and control 8 (1965), 338–353. 267 ratio mathematica volume 44, 2022 properties of intuitionistic multi-anti fuzzy normal ring dr. r. muthuraj1 s. yamuna2 abstract in this paper, we discuss the properties of an intuitionistic multi-anti fuzzy normal ring of a ring is defined and discussed its properties. some results based on cartesian product, homomorphism and anti homomorphism of an intuitionistic multi-anti fuzzy normal ring of a ring are also discussed. keywords –r intuitionistic multi-anti fuzzy ring, h intuitionistic multi-anti fuzzy normal subring, r1, r2rings. mathematics subject classification:03e72, 47s40,08a05,08a72,16y30,08a20n253. 1pg and research department of mathematics, h.h. the rajah’s college, pudukkottai 622001, tamilnadu, india. e-mail: rmr1973@yahoo.co.i; rmr1973@gmail.com 2 part time research scholar, pg and research, department mathematics, h.h. the rajah’s college, pudukkottai 622001, tamilnadu, india. e mail: bassyam1@gmail.com (affiliated to bharathidasan university, tiruchirappalli 24) 3received on june 29th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.910. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 222 mailto:rmr1973@gmail.com r. muthuraj and s. yamuna 1. introduction the idea of fuzzy sets introduced by l. a. zadeh 1965 [19] is an approach to mathematical representation of vagueness in everyday curriculum, the idea of fuzzy set is welcome because it handles uncertainty and vagueness which ordinary set could not address. in fuzzy set theory membership function of an element is single value between 0 and 1. therefore, a generalization of fuzzy set was introduced by attanassov [1], 1983 called intuitionistic fuzzy set (ifs) which deals with the degree of non-membership function and the degree of hesitation. after several year, sabu sebastian [13] introduced the theory of multi-fuzzy sets in terms of multi-dimensional membership function. r. muthuraj and s. balamurugan [15] introduced the concept of multi-anti fuzzy subgroup and discussed some of its properties. r. muthuraj and s. balamurugan [17] introduced the concept of multi-anti fuzzy ideal of a ring under homomorphism. in this paper, we discuss the properties of an intuitionistic multi-anti fuzzy normal ring of a ring is defined and discussed its properties. some results based on cartesian product, homomorphism and anti homomorphism of an intuitionistic multi-anti fuzzy normal ring of a ring. 2. preliminaries 2.1 definition [1, 2, 4] a fuzzy subset a of a ring r is called a fuzzy subring of r if for all x, y r i. a (x– y) ≥ min {a(x), a (y)} and ii. a (xy) ≥ min {a (x), a (y)}. 2.2 definition [2, 7] a fuzzy subset a of a ring r is called an anti-fuzzy subring of r if for all x, y r i. a (x– y) ≤ max {a(x), a (y)} and ii. a (xy) ≤ max {a (x), a (y)}. 2.3 proposition [7] let r1 and r2be rings and let f be a homomorphism from r1 onto r2. if a is a anti fuzzy ideal of r2 then f -1(a) is a anti fuzzy ideal of r1. 2.4 definition [2] let r be a ring. let g ={  x, a(x), b(x)  / xr} be an intuitionistic fuzzy set defined on a ring r, where a: r→[0,1], b: r→[0,1] such that 0  a(x) + b(x)  1. an intuitionistic fuzzy subset g of r is called an intuitionistic fuzzy ring on r if the following conditions are satisfied. for all x, y r, i. a ( x – y) ≥ min {a (x), a(y)} , ii. a (xy) ≥ min {a(x), a(y)}, iii. b (x− y)  max {b (x), b (y)}, iv. b (xy)  max {b (x), b (y)}. 223 properties of intuitionistic multianti fuzzy normal ring 2.5 definition [7,17] let r be a ring. let g = {x, a(x), b(x) / xr} be an intuitionistic fuzzy set defined on a ring r, where a: r→[0,1], b: r→[0,1] such that 0  a(x) + b(x)  1. an intuitionistic fuzzy subset g of r is called an intuitionistic antifuzzy ring on r if the following conditions are satisfied. for all x, y r, i. a (x – y)  max {a (x), a (y)}, ii. a (xy)  max {a(x), a (y)}, iii. b (x− y) ≥ min {b (x) , b (y)}, iv. b (xy) ≥ min {b (x) , b (y)}. 2.6 definition [18] an intuitionistic multi-anti fuzzy ring g = {x, a(x), b(x) / xr} on a ring r is said to be an intuitionistic multi-anti fuzzy normal ring on r if for every x, y  r, a (xy) = a (yx) and b(xy) = b(yx). 2.7 example [18] consider the intuitionistic fuzzy sets, g = {x, a(x), b(x) / xr} of dimension 2 on z is defined as, a1(x) = 0.2 if x = 0; a1(x) = 0.7 if x 0 and a2(x) = 0.3 if x = 0; a2(x) = 0.9 if x 0. b1(x) = 0.7 if x = 0; b1(x) = 0.2 if x 0 and b2(x) = 0.6 if x = 0; b2(x) = 0.1 if x 0. then the intuitionistic multi-fuzzy set g = (a, b) of dimension 2 on z is defined as,      = 0 x if(0.7,0.9) 0 x if(0.2,0.3) = (x)) 2 a (x), 1 (a = a(x) .b(x)= (b1(x), b2(x)) = { (0.7,0.6)if x = 0 (0.2,0.1)if x ≠ 0 clearly, g is an intuitionistic multi-anti fuzzy normal ring on z. 3. properties of intuitionistic multi-anti fuzzy normal ring in this section, the properties of an intuitionistic multi-anti fuzzy normal ring is discussed. 3.1 theorem let g = {x, a(x), b(x) / xr} and h = {x, c(x), d(x) / xr} be any two intuitionistic multi-anti fuzzy normal subrings of rings r1 and r2 respectively. then their anti cartesian product g  h is an intuitionistic multi-anti fuzzy normal subring of r1  r2. proof let g and h be any two intuitionistic multi-anti fuzzy normal subrings of rings r1 and r2 respectively. then, by theorem 2.2.5, g  h is an intuitionistic multi-anti fuzzy subring of r1 r2. let (p, q), (r, s)  r1r2.for each i = 1, 2, ..., k, now, (a  c) ((p, q)(r, s)) = (( ai  ci )(pr, qs )) 224 r. muthuraj and s. yamuna = (max {ai(pr), ci(qs)}) = (max {ai(rp), ci(sq)}) = ((ai ci )( rp, sq )) = (a  c) ((r, s) (p, q)) therefore, (a  c)((p, q)(r, s)) = (a  c) ((r, s) (p, q)) and (b d) ((p, q)(r, s)) = (( bi di )(pr, qs )) = (min {bi(pr), di(qs)}) = (min {bi(rp), di(sq)}) = (( bi di )( rp, sq )) = (b  d) ((r, s) (p, q)) (b d) ((p, q)(r, s))=(b  d) ((r, s) (p, q)) hence, (g  h) ((p, q)(r, s) = (g  h) ((r, s) (p, q)). hence, the anti cartesian product gh is an intuitionistic multi-anti fuzzy normal subring of r1  r2. 3.2 theorem let g = {x, a(x), b(x) / xr}and h = {x, c(x), d(x) / xr }be intuitionistic multi-fuzzy subsets of r1 and r2 respectively, such that c(02)  a(x) and d(02)  b(x) for all x in r1, where 02 is the additive identity element of r2. the anti cartesian product g  h is an intuitionistic multi-anti fuzzy normal subring of r1  r2, and then g is an intuitionistic multi-anti fuzzy normal subring of r1. proof let p, r  r1 and 02r2.let g  h be an intuitionistic multi-anti fuzzy normal subring of r1 r2. then, by theorem 2.2.7, g is an intuitionistic multi-anti fuzzy subring of r1. for each i = 1, 2,..., k, a (pr) = (a1(pr ), a2(pr ), ... , ak(pr )) = (max {a1(pr ), c1(0202)} , ... , max{ak(pr ), ck(0202)}) a (pr) = (max (ai(pr ), ci(0202))) and b (pr) = (b1(pr ), b2(pr ), ... , bk(pr )) = (min {b1(pr ), d1(0202)} , ... , min{bk(pr ), dk(0202)}) b(pr) = (min (bi(pr ), di(0202))) that is, a (pr) = (max (ai (pr), ci (0202))) and b(pr) = (min (bi(pr ), di(0202))). hence, (g  h )( pr, 0202 ) = (g  h)(pr, 0202) = (g  h)((p, 02)  (r, 02)) = (g  h)((r, 02)  (p, 02)) (g  h )( pr, 0202) = (g  h)(pr, 0202). that is, a(pr) = a(rp) and b(pr) = b(rp). 225 properties of intuitionistic multianti fuzzy normal ring 3.3 theorem. let g = {x, a(x), b(x) / xr}and h = {x, c(x), d(x) / xr} be intuitionistic multi-fuzzy subsets of r1 and r2 respectively, such that a (01)  c(y) and b (01)  d(y)for all y in r2, where 01 is the additive identity element of r1. the anti cartesian product g  an intuitionistic multi-anti fuzzy normal subring of r1  r2, then his a multi-anti fuzzy normal subring of r2. proof let q, s  r2 and 01r1.let g  h is an intuitionistic multi-anti fuzzy normal subring of r1  r2. then, by theorem 2.2.8, an intuitionistic multi-anti fuzzy subring of r1. for each i = 1, 2, k, c (qs) = (c1(qs), c2(qs), ..., ck(qs)) = (max {c1(qs), a1(0101)}, ..., max {ck (qs), ak (0101)}) c (qs) = (max (ci(qs), ai (0101))) and d (qs) = (d1(qs), d2(qs), ..., dk(qs)) = (min {d1(qs), b1(0101)}, ..., min {dk (qs), bk (0101)}) d(qs) = (min (bi (0101), di (qs))) that is, c (qs) = (max (ai (0101), ci (qs))) and d(qs) = (min (bi(0101), di(qs))). hence, (g  h )( 0101,qs ) = (g  h )(0101, qs) = (g  h )(01 ,q)  (01 , s) ) = (g  h )((01 , s)  ( 01 ,p) ) (g h) (01 01, qs) = (g  h) (01 01, sq). that is, c(qs) = c(sq) and d(qs) = d(sq). hence, h is an intuitionistic multi-anti fuzzy normal subring of r1. 3.4 remark let g = {x, a(x), b(x) / xr}and h = {x, c(x), d(x) / xr}be intuitionistic multi-fuzzy subsets of rings r1 and r2 respectively. the anti cartesian product g  h is an intuitionistic multi-anti fuzzy normal subring of r1  r2, then it is not necessarily that both g and h are intuitionistic multi-anti fuzzy normal subrings of r1 andr2 respectively. 226 r. muthuraj and s. yamuna 4. properties of intuitionistic multi-anti fuzzy normal subring of a ring under homomorphism and anti homomorphism in this section, the properties of intuitionistic multi-anti fuzzy normal subring of a ring under homomorphism and anti homomorphism are discussed. 4.1 theorem let r1 and r2 be any two rings. let f: r1 → r2 be an onto homomorphism. if g = {x, a(x), b(x) / xr1} is an intuitionistic multi-anti fuzzy normal subring of r1, then f(g) is an intuitionistic multi-anti fuzzy normal subring of r2, if g has inf property and g is f-invariant. proof let g be an intuitionistic multi-anti fuzzy normal subring of r1. then, by theorem 2.3.2, f(g) is an intuitionistic multi-anti fuzzy subring of r2. then if x, yr1, then f(x), f(y)r2. now, f(a)(f(x)f(y)) = f(a)(f(xy)) = a(xy) = a(yx) = f(a)(f(yx)) = f(a)(f(y)f(x)) there fore, f(a)(f(x)f(y)) = f(a)(f(y)f(x)) and f(b)(f(x)f(y)) = f(b)(f(xy)) = b(xy) = b(yx) = f(b)(f(yx)) = f(b)(f(y)f(x)) there fore, f(b)(f(x)f(y)) = f(b)(f(y)f(x)). hence, g(f(x)f(y) = g(f(y)f(x)). hence, f(g) is an intuitionistic multi-anti fuzzy normal subring of r2. 4.2 theorem let r1 and r2 be any two rings. let f: r1 → r2 be a homomorphism. if h = { x, c(x), d(x) / xr1} is an intuitionistic multi-anti fuzzy normal subring of r2, then f–1(h) is an intuitionistic multi-anti fuzzy normal subring of r1. proof let h be an intuitionistic multi-anti fuzzy normal subring of r2. then, by theorem 2.3.3, f–1(h) is an intuitionistic multi-anti fuzzy subring of r1. for any x, yr1, now, f–1(c)(xy) = c(f(xy)) = c(f(x)f(y)) = c(f(y)f(x)) 227 properties of intuitionistic multianti fuzzy normal ring = c(f(yx)) = f–1(c)(yx) therefore, f–1(c)(xy) = f–1(c)(yx) and f–1(d)(xy) = d(f(xy)) = d(f(x)f(y)) = d(f(y)f(x)) = d(f(yx)) = f–1(d)(yx) therefore, f–1(d)(xy) = f–1(d)(yx). hence, f–1(h)(xy) = f–1(h)(yx). hence, f–1(h) is an intuitionistic multi-anti fuzzy normal subring of r1. 4.3 theorem let r1 and r2 be any two rings. let f: r1 → r2 be an onto anti homomorphism. if g = { x, a(x), b(x) / xr1} is an intuitionistic multi-anti fuzzy normal subring of r1, then f(g) is an intuitionistic multi-anti fuzzy normal subring of r2, if g has inf property and g is f-invariant. proof let g be an intuitionistic multi-anti fuzzy normal subring of r1. then, by theorem 2.3.4, f(g) is an intuitionistic multi-anti fuzzy subring of r2. then if x, yr1, then f(x), f(y)r2. now, f(a)(f(x)f(y)) = f(a)(f(yx)) = a(yx) = a(xy) = f(a)(f(xy)) = f(a)(f(y)f(x)) there fore, f(a)(f(x)f(y)) = f(a)(f(y)f(x)) and f(b)(f(x)f(y)) = f(b)(f(yx)) = b(yx) = b(xy) = f(b)(f(xy)) = f(b)(f(y)f(x)) there fore, f(b)(f(x)f(y) = f(b)(f(y)f(x)). hence, g(f(x)f(y)) = g(f(y)f(x)). hence, f(g) is an intuitionistic multi-anti fuzzy normal subring of r2. 4.4 theorem let r1 and r2 be any two rings. let f: r1 → r2 be an anti homomorphism. if h = {x, a(x), b(x) / xr1} is an intuitionistic multi-anti fuzzy normal subring of r2, then f –1(h) is an intuitionistic multi-anti fuzzy normal subring of r1. proof let b be an intuitionistic multi-anti fuzzy normal subring of r2. then, by theorem 2.3.5, f–1(h) is an intuitionistic multi-anti fuzzy subring of r1. 228 r. muthuraj and s. yamuna for any x, yr1, now, f–1(c)(xy) = c(f(xy)) = c(f(y)f(x)) = c(f(x)f(y)) = c(f(yx)) = f–1(c)(yx) therefore, f–1(c)(xy) = f–1(c)(yx) and f–1(d)(xy) = d(f(xy)) = d(f(y)f(x)) = d(f(x)f(y)) = d(f(yx)) = f–1(d)(yx) therefore, f–1(d)(xy) = f–1(d)(yx). f–1(h)(xy) = f–1(h)(yx). hence, f–1(h) is an intuitionistic multi-anti fuzzy normal subring of r1. 5. conclusion in this paper, we discuss the properties of an intuitionistic multi-anti fuzzy normal ring of a ring is defined and discussed its properties. homomorphism and anti homomorphism of an intuitionistic multi-anti fuzzy normal ring of a ring. references [1] attanassov, intuitionistic fuzzy sets, fuzzy sets and systems, 20,87-89,1986. 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[19] zadeh. l. a, fuzzy sets, information and control 8, 338-353,1965. 230 ratio mathematica volume 46, 2023 wheat crop yield forecasting using various regression models shakila c v* khadar babu sk † abstract the prediction of crop yield, particularly paddy production is a challenging task and researchers are familiar with forecasting the paddy yield using statistical methods, but they have struggled to do so with greater accuracy for a variety of factors. therefore, machine learning methods such as elastic net, ridge regression, lasso and polynomial regression are demonstrated to predict and forecast the wheat yield accurately for all india-level data. assessment metrics such as coefficient of determination (r2), root mean square error (rmse), mean absolute error (mae), mean squared error (mse) and mean absolute percentage error (mape) are used to evaluate the performance of each developed model. finally, while evaluating the prediction accuracy using evaluation metrics, the performance of the polynomial regression model is shown to be high when compared to other models that are already accessible from various research in the literature. keywords: elastic net; ridge regression; lasso regression; polynomial regression; ordinary least squares; forecast 2020 ams subject classifications: 62j05, 62j07, 62-04, 62-06. 1 *department of mathematics, vellore institute of technology (vit), vellore, tamil nadu, india-632014; ajjimaths@gmail.com. †department of mathematics, vellore institute of technology (vit), vellore, tamil nadu, india-63201; khadar.babu36@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1085. issn: 1592-7415. eissn: 2282-8214. ©shakila c v et al. this paper is published under the cc-by licence agreement. 280 shakila c v and khadar babu sk 1 introduction agriculture has been the economic backbone of many nations. there are more than 118.9 million farmers in india, and as the population expands, there will be a demand for food. as a result, we need new methods to produce more food products in a shorter amount of time. however, since agriculture is not a profitable industry, not many people choose it as a career. bhosale et al. [2018]. agriculture has always been recognized as a vital and great culture that india has traditionally practiced. in the past, people used the land where they lived and made crop choices based on the local weather and conditions. however, due to the greenhouse effect and changes in climatic conditions, farmers today cannot predict when it will rain, snow, or whether there will be water available for their crops, among other things. with this serious issues and demand, it is important to estimate sustainable agricultural production using a system that can accurately measure crop conditions, crop type, and yield. freie et al. [1999]. there are a few approaches to dealing with constructing the suitable improvement in the agricultural industry. there are several methods to approach these issues by utilizing some of the most significant technological advancements. one of the greatest and simplest technologies we can employ is ai-based and machine learning prediction principles. furthermore, recently developed machine learning (ml) algorithms are more capable than statistical techniques to find yield estimations. artificial neural networks, decision trees, regression analysis, clustering, bayesian networks, time series analysis, and markov chain models are just a few of the mathematical and statistical techniques used in machine learning (ml) approaches for crop prediction. due to the availability of multiple data from various sources to expose hidden information, the use of these machine learning techniques in crop production demonstrates even more significant advantages. 2 literature survey machine learning addresses problems when the relationship between information and yield variables is uncertain or difficult to comprehend (shaik et al. [1999] and veenadhari et al. [2014]). unlike traditional measuring techniques, machine learning explicitly displays the data whose trade-mark is beneficial to conduct modeling of complex and non-linear practices, such as a capacity for crop output forecasting (praveen and rama [2019] and kumar et al. [2019]). using supervised learning, the majority of machine learning algorithms are successfully used to predict crop yields (praveen et al. [2017] and ravikumar et al. [2019]). the preparation measures will continue until the model attained the desired level of 281 wheat crop yield forecasting using various regression models accuracy on the preparation data. the majority of research in the past have developed statistical agricultural production prediction models using multiple linear regressions (mlrs) (rai et al. [2013];kumar et al. [2014]; dhekale et al. [2014]). das et al. [2017] has studied about the statistical approaches for feature selection or feature extraction, such as least absolute shrinkage and selection operator (lasso), stepwise multiple linear regression (smlr) or elastic net (enet) method, can be utilized to address these issues. yousefi et al. [2015] discussed to forecast the output energy of rice production in iran, several researchers used the polynomial and radial basis function kernels of support vector regression (svr). paidipati et al. [2021] developed a model using svr approach with various non-linear patterns for forecasting rice cultivation in india. there are few studies comparing the accuracy of feature selection, feature extraction, and both approaches combined for agricultural yield forecasting. with the following objectives: (i) to develop overall crop yield prediction models using various multivariate models; and (ii) to assess the analytical performance of the developed models, our study has found scope to develop and select a statistical forecasting model for rice using various regression techniques for the india level. elastic net regression, ridge regression, lasso regression, and polynomial regression are some of the techniques we employed to construct this work. there are many comparable projects on the market, but what sets our project apart from the competition is how we’ve integrated python with machine learning to cut down on the number of lines of code and production costs while still producing accurate results (pramod et al. [2019]; tutun et al. [2016]). 3 material and methods 3.1 ridge regression in ridge regression µ is a penalty term and that penalty function is equal to the squared root of the coefficient. the square of the coefficients magnitude corresponds to the l2 term. to regulate that penalty term, we additionally incorporate the coefficient µ. in this instance, if µ is zero, the formula is the fundamental ols; however, if µ is more than zero, a constraint will be added to the coefficient. this constraint makes the quantity of the coefficient tend towards zero as we raise the amount of µ. this results in a tradeoff between smaller variance and increased bias. lr = arg.minα̂ ( ‖y −α∗x‖2 + µ∗‖α‖2 ) . where µ is regularization penalty. 282 shakila c v and khadar babu sk because it never reaches a coefficient of zero but only minimises it, ridge regression lowers a model’s complexity without lowering the number of variables. as a result, this model is unable to achieve feature reduction. 3.2 lasso regression least absolute shrinkage and selection operator is short for lasso regression. it extends the cost function’s penalty term. the whole sum of the coefficients is represented by this phrase. when the value of the coefficients increases from 0 to 1, this term penalises, causing the model to lower the value of variables in order to minimise loss. while lasso regression usually makes the value of the coefficient to absolute zero, ridge regression never does. llasso = arg.minα̂ ( ‖y −α∗x‖2 + µ∗‖α‖1 ) . with various data types, lasso occasionally has difficulties.if the number of predictors (p) is more than the number of observations, lasso will choose at most n predictors as non-zero even if all of the predictors are significant (n). the lasso regression method chooses one of the highly collinear variables at random when there are two or more, which is bad for data interpretation. 3.3 elastic net to address the drawbacks of ridge and lasso regression, ? formed an elastic net regression. in general, ridge regression performs best with highly correlated variables, whereas lasso regression performs well with less correlated variables. however, there are many models that represent a significant number of variables but lack information on attributes like correlation. ridge regressions and lasso are not very helpful in these circumstances. to get away from this problem, the function is estimated using enr since it takes into account the consequences of both lasso and ridge regressions. l1 and l2 norms can be used to define the lasso and ridge regression penalties, respectively. for accurate prediction, enr take into account the l1 and l2 penalties by the following equations. lenr = arg.min α̂ ∑ i (yi −αxi)2 + β1 1∑ k=1 |αk|+ β2 1∑ k=1 αk2. where ‖l1‖ = β1 ∑1 k=1 |αk|and‖l2‖ = beta 2 ∑1 k=1 αk2. l1 is sum of the weights and l2 is the sum of the square of the weights. 283 wheat crop yield forecasting using various regression models 3.4 polynomial regression in polynomial regression, a kind of linear regression, the relationship between the random variable x and the dependent variables y is represented as a nth-degree polynomial. polynomial regression is used to fit a nonlinear relationship between the value of x and the corresponding dependent mean of y, denoted by the notation e(y|x). here is a polynomial regression model’s general equation. l = s0 + s1x1 + s2x12 + s2x13 + ... + snx1n some correlations may be curvy, according to a researcher’s hypothesis. such scenarios will undoubtedly have a polynomial term. the assumption in common multiple linear regression analysis is that every independent variable is independent of every other independent variable. in the case of polynomial regression models, this assumption is incorrect. 3.5 model validation 3.5.1 mean absolute percentage error (mape) to determine the mean absolute percentage error (mape), the absolute error for each period is subtracted from predicted values then as follows the procedures. mape = 100 n n∑ t=1 ∣∣∣∣predictedi −actualiactuali ∣∣∣∣ 4 results and discussion 4.1 dataset directorate of economics and statistics department of agricultural and farmers welfare, and government of india provided the time series data of wheat yield at india level (1966 to 2017). the research used characteristics like area under cultivation (thousand / hectares), production (thousand / tons), and yield (kg / hectare) to evaluate data from all of india. elastic net, ridge regression, lasso regression, and polynomial regression were constructed and compared to determine the best-fit model. 4.2 overview of wheat parameter statistics the elastic net, ridge regression, lasso regression, and polynomial regression models were separately applied to the wheat yield data to investigate 284 shakila c v and khadar babu sk table 1: statistical measures measures elastic net regression lasso regression ridge regression polynomial regression r2 0.9252 0.9252 0.9252 0.9688 mse 38964456.45 40910114.34 42898765.78 17083450.77 rmse 5996.1034 6396.1015 6876.3675 4133.2131 mae 4894.87 5194.88 5794.86 3302.65 mape 20.435 55.405 69.203 8.232 their connection, and the effectiveness of each model was evaluated using mse, r2, and mape. figure 1: all india level wheat production vs area (1966-2017) figure 1shows that the all-india level wheat production wise area. the stronger correlation and the error higher form will be regarded as the most effective method for predicting agricultural production (kg/acre). the statistical approach’s results are shown in the first case r2 values were verified among all the particular regression models (table 1). 285 wheat crop yield forecasting using various regression models 4.3 regression models numerous research showed that machine learning techniques could forecast wheat production. for a consistent wheat yield, it is necessary to increase prediction accuracy. to ensure a consistent wheat production, it is necessary to improve prediction accuracy. the accuracy of the suggested elastic net regression, ridge regression, lasso regression, and polynomial regression for wheat yield prediction is assessed using the r2, rmse, mae, mse, and mape metrics, as was previously stated. figure 2: forecasting using elastic net regression analysising the data by using elastic net regression, we have got the values of r2, mse, rmse, mae, and mape of around 0.9252, 38964456.45, 5996.1034, 4894.87, and 20.435, respectively, the forecasting utilizing elastic net regression is shown in fig.2 here 92% of data are used to fit the model. fig. 3 shows the forecasting using the ridge regression, which is evaluation metrics with values of r2, mse, rmse, mae, and mape of about 0.9252, 42898765.78, 6876.3675, 5794.86 and 69.203 respectively. here 92% of data are used to fit the model. and the mape values is 69.2 which large in model fitting. by using lasso regression model the values of r2, mse, rmse, mae, and mape of around 0.9251, 40910114.34, 6396.10, 5194.88 and 55.405, respectively, the forecasting utilizing lasso regression is shown in fig 4. here also 92% of data are used to fit the model. but the mape values are huge as 55% for the model fit. in fig. 5 forecasting the wheat yield data with polynomial regression, estimate the r2, mse, rmse, mae, and mape of around is 0.9687, 17083450.77, 286 shakila c v and khadar babu sk figure 3: forecasting using ridge regression figure 4: forecasting using lasso regression 4133.213, 3302.674 and 8.23 respectively. comparing the all above various measure value the r2 is 96% data using to fit the model. mean-while, the mape values is 8.23 which is highly acceptably accurate level. fig. 6 shows the forecasting of india-wide level wheat crop production using various regression models. the accuracy of the polynomial regression model exhibits superior scale than other chosen machine learning models, as like regression 287 wheat crop yield forecasting using various regression models figure 5: forecasting using polynomial regression figure 6: forecasting using various regression models. models. 5 conclusions statistical and machine learning methods are used to predict agricultural yield. the statistical analysis of various regression approaches and machine learning, specifically elastic net regression, ridge regression, lasso regression, and polynomial regression are among the techniques that are evaluated to acquire 288 shakila c v and khadar babu sk higher accurate crop yield forecast. to evaluate the level of accuracy of the various methods, model performance measures are updated. the main findings are drawn from the results seen: • assessment metrics such as coefficient of determination (r2), root mean square error (rmse), mean absolute error (mae), mean squared error (mse) and mean absolute percentage error (mape) are used to evaluate the performance of each developed model. • the obtained study showed that the polynomial regression method produces superior evaluation metrics with values of r2, mse, rmse, mae, and mape of about 0.9687, 17083450.77, 4133.21, 3302.67, and 8.2326 respectively. • the r2 metrics of the polynomial regression is 4.498 percent better than other existing models from other regression models. • hence, its improved performance metrics, the suggested machine learning algorithm in particular, polynomial regression reduces the risk factor for wheat yield. references s. bhosale, r. thombare, p. dhemey, and a. chaudhari. crop yield prediction using data analytics and hybrid approach. in 2018 fourth international conference on computing communication control and automation (iccubea),ieee, pages 1–5, 2018. b. das, r. sahoo, s. pargal, g. krishna, r. verma, v. chinnusamy, v. sehgal, and v. gupta. comparison of different uniand multi-variate techniques for monitoring leaf water status as an indicator of water-deficit stress in wheat through spectroscopy. biosystems engineering, 160:69–83, 2017. doi: 10.1016/j.biosystemseng.2017.05.007. b. dhekale, 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elastic net regression model for electricity consumption forecasting, 2016. s. veenadhari, b. misra, and c. d.singh. machine learning approach for forecasting crop yield based on climatic parameters. in 2014 international conference on computer communication and in-formatics, 8:1– 5, 2014. m. yousefi, b. khoshnevisan, s. band, s. motamedi, m. md nasir, m. arif, and r. ahmad. support vector regression methodology for prediction of output energy in rice production. stochastic environmental research and risk assessment, 29, 2015. doi: 10.1007/s00477-015-1055-z. 290 ratio mathematica volume 45, 2023 a study on �̂�∗∗𝐬 − 𝑹𝟎 and �̂� ∗∗𝐬 − 𝑹𝟏 spaces in topological spaces m. anto* s. andrin shahila† abstract in this paper, we introduces the concept of 𝑅0-space, 𝑅1-space, door space, submaximal using �̂�∗∗𝑠-closed set and investigate its properties. we have also study their relationship with some other higher separation axioms. we have also, introduced a new definition 𝑆�̂�∗-space by using semi-closed and �̂�∗-closed sets and study its relationship with other closed sets using �̂�∗∗𝑠-closed set. keywords. ĝ∗∗s − r0, ĝ ∗∗s − r1 space, ĝ ∗∗s -door spaces, ĝ∗∗s –submaximal, sĝ∗space. 2010 ams subject classification: 54d15, 54g05‡ *associate professor, p.g and research department of mathematics annai velankanni college, tholayavattam, kanyakumari district ,629157,affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamilnadu, india; e-mail: andrinshahila@gmail.com. †phd scholar (reg.no:19213012092006), p.g and research department of mathematics annai velankanni college, tholayavattam, kanyakumari district, 629157, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamilnadu, india; e-mail: antorbjm@gmail.com. ‡ received on july 22, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.989. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 111 m. anto and s. andrin shahila 1. introduction topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects. it emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. ideas that are now classified as topology were expressed as early as 1736. by the middle of the 20th century, topology had become an important area of study within mathematics. the word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. topology includes many subfields namely point-set topology, algebraic topology and geometric topology. in the literature of general topology, the concept of semi open sets was introduced by levine in 1963 [11] and g-closed sets in 1970 [12]. m.k.r.s. veera kumar defined �̂�-closed sets in 2001 [10] and �̂�∗-closed sets in 1996 [9]. in the year 1995, j. dontchev [2, 3], defined on door spaces and submaximal spaces. the notion of 𝑅0 topological space is introduced by n. a. shanin [13] in 1943. later, a. s. davis [13] rediscovered it and studied some properties of this weak separation axiom. in 2010, s. balasubramanian defined separation axioms on generalized sets [14]. several topologists further investigated properties of 𝑅0topological spaces and many interesting results have been obtained in various contexts. in the same paper, a. s. davis also introduced the notion of 𝑅1 topological space which are independent of both 𝑇0 and 𝑇1 but strictly weaker than𝑇2. m. g. murdeshwar and s. a. naimpally [8] studied some of the fundamental properties of the class of 𝑅1 topological spaces. in 1963, n. levine [11] offered a new notion to the field of general topology by introducing semi-open sets. he defined this notion by utilizing the known notion of closure of an open set, i.e., a subset of a topological space is semiopen if it is contained in the closure of its interior. since the advent of this notion, several new notions are defined in terms of semi-open sets of which two are semi-𝑅0 and semi-𝑅1 introduced by s. n. maheshwari and r. prasad [15] and c. dorsett [1], respectively. these two notions are defined as natural generalizations of the separation axioms 𝑅0 and 𝑅1 by replacing the closure operator with the semi closure operator and openness with semi-openness. since then, this notion received wide usage in general topology. in this paper, we continue the study of the above mentioned classes of topological spaces satisfying these axioms by introducing two more notions in terms of �̂�∗∗𝑠-closed sets [7] called �̂�∗∗𝑠-𝑅0and �̂� ∗∗𝑠-𝑅1. we have also applied the definition in [7] to door space and submaximal space.in this paper we introduced sĝ∗-space using semi-closed and �̂�∗-closed sets. 112 a study on �̂�∗∗𝑠 − 𝑅0 and �̂� ∗∗𝑠 − 𝑅1 spaces in topological spaces 2. preliminaries throughout this paper (𝑋 , 𝜏) represent the non-empty topological spaces on which no separation axioms are assumed unless otherwise mentioned. for a subset a of a space (𝑋 , 𝜏), 𝑐𝑙(𝐴)and 𝑖𝑛𝑡 (𝐴) denote the closure and interior of a respectively. definition. 2.1 i) a space (x, τ) is r0 [13] if for each open set u of x, x ∈ u implies cl({x}) ⊆ u. ii) a topological space (x, τ) is r1 [13] if for x, y ∈ x such that cl({x}) ≠ cl({y}), there are disjoint open sets u and v such that cl({x})⊆ u and cl({y}) ⊆ v. iii) a topological space (x, τ) is a door space [3] if every subset of x is either open or closed in x. iv) a topological space (x, τ) is a submaximal space [4] if every dense subset of x is open in x. 3.�̂�∗∗𝐬 − 𝑹𝟎 and �̂� ∗∗𝐬 − 𝑹𝟏 spaces definition. 3.1. a space (𝑋, 𝜏) is said to be ĝ∗∗s − 𝑅0 if for any open set𝑈, 𝑥 ∈ 𝑈, then ĝ∗∗scl{x} ⊆ u. definition. 3.2. a space (𝑋, 𝜏) is said to be ĝ∗∗s − 𝑅1 if for x and y in x, withĝ∗∗scl({x}) ≠ ĝ∗∗scl({y}), there exist two disjoint ĝ∗∗s-open sets u and v such that ĝ∗∗scl({x}) ⊆ 𝑈 and ĝ∗∗scl({y}) ⊆ 𝑉 theorem. 3.3. every ĝ∗∗s − 𝑅0-space is ĝ ∗∗s − 𝑇0-space. proof. let (𝑋, 𝜏) be a ĝ∗∗s − 𝑅0-space. let x and y be two distinct points of x. by hypothesis, for any open set u and𝑥 ∈ 𝑈 ⇒ ĝ∗∗scl({x}) ⊆ u. also, every open set is ĝ∗∗s-open, which implies u is ĝ∗∗s-open and𝑥 ∈ 𝑈. therefore, for any distinct points x and y, there exist a ĝ∗∗s-open set u containing x and not y. hence, (𝑋, 𝜏) is a ĝ∗∗s − 𝑇0space. theorem. 3.4. every ĝ∗∗s − 𝑅0-space is ĝ ∗∗s − 𝑇2-space. proof. let (𝑋, 𝜏) be a ĝ∗∗s − 𝑅0-space. let x and y be two distinct points of x. by hypothesis, for any open set u and𝑥 ∈ 𝑈 ⇒ ĝ∗∗scl{x} ⊆ u. ifĝ∗∗scl({x}) ⊈ u, there exist another open set v not containing x, so𝑦 ∈ 𝑉. also,𝑈 ∩ 𝑉 = ∅. therefore, for any distinct points x and y, there exist two disjoint ĝ∗∗s-open sets u and v. hence, (𝑋, 𝜏) is a ĝ∗∗s − 𝑇2-space. theorem. 3.5. if (𝑋, 𝜏) isĝ∗∗s − 𝑅0, then for any closed set u and𝑥 ∉ 𝑈, there exist an ĝ∗∗s-open set g such that 𝑈 ⊆ 𝐺 and𝑥 ∉ 𝐺. proof. suppose (𝑋, 𝜏) isĝ∗∗s − 𝑅0. let u be any closed set and 𝑥 ∉ 𝑈 ⇒ 𝑋\𝑈 is open and𝑥 ∈ 𝑋\𝑈. by assumption,ĝ∗∗scl({x}) ⊆ x\u ⇒ u ⊆ x\ĝ∗∗scl({x}). put𝐺 = 113 m. anto and s. andrin shahila 𝑋\ĝ∗∗scl{x}.since ĝ∗∗scl{x} is ĝ∗∗s-open. also, 𝑈 ⊆ 𝐺 and𝑥 ∉ 𝐺. therefore, for any closed set u and 𝑥 ∉ 𝑈, there exist an ĝ∗∗s-open set g such that 𝑈 ⊆ 𝐺 and𝑥 ∉ 𝐺. conversely, suppose for any closed set u and 𝑥 ∉ 𝑈, there exist an ĝ∗∗s-open set g such that 𝑈 ⊆ 𝐺 and 𝑥 ∉ 𝐺 theorem. 3.6. every ĝ∗∗s − 𝑅0-space is ĝ ∗∗s-regular. proof. let (𝑋, 𝜏) be a ĝ∗∗s − 𝑅0-space. let f be a closed set and𝑥 ∈ 𝑋 − 𝐹. by theorem: 3.3, there exist an ĝ∗∗s-open set g such that 𝑈 ⊆ 𝐺 and𝑥 ∉ 𝐺. put𝐻 = {𝑥}. since every singleton set is open, so h is open. also, we know that, every open set is ĝ∗∗s-open. also, 𝐺 ∩ 𝐻 = 𝐺 ∩ {𝑥} = ∅. thus, g and h are disjoint ĝ∗∗s-open sets containing x and h respectively. therefore, (𝑋, 𝜏) is ĝ∗∗s-regular. theorem. 3.7. a topological space (𝑋, 𝜏) is ĝ∗∗s − 𝑅0 iff for any points x and y in x, 𝑥 ≠ 𝑦 impliesĝ∗∗scl({x}) ∩ ĝ∗∗scl({y}) = ∅. proof. let x be ĝ∗∗s − 𝑅0 and 𝑥 ≠ 𝑦 in 𝑋 ⇒ {𝑥} be an open set and𝑦 ∉ {𝑥}. since,𝑥 ∈ {𝑥}, we have ĝ∗∗scl({x}) ⊆ {𝑥}. thusĝ∗∗scl({x}) = {𝑥}. now,ĝ∗∗scl({x}) ∩ ĝ∗∗scl({y}) = {𝑥} ∩ {𝑦} = ∅. conversely, suppose for any points 𝑥 and 𝑦 in𝑋, 𝑥 ≠ 𝑦 ⇒ ĝ∗∗scl({x}) ∩ ĝ∗∗scl(y) = ∅. let v be an open set and 𝑥 ∈ 𝑉 and let 𝑦 ∈ ĝ∗∗scl({x}) (1). suppose𝑦 ∉ 𝑉. by assumption, ĝ∗∗scl({x}) ∩ ĝ∗∗scl({y}) = ∅ ⇒ 𝑦 ∉ ĝ∗∗scl({x})this is a contradiction to (1). therefore, 𝑦 ∈ 𝑉 andĝ∗∗scl({x}) ⊆ 𝑉. thus (𝑋, 𝜏) isĝ∗∗s − 𝑅0. corollary. 3.8. a topological space (𝑋, 𝜏) is ĝ∗∗s − 𝑅0 iff for any points x and y in x, 𝑥 ≠ 𝑦 ⇒ 𝑐𝑙({𝑥}) ∩ 𝑐𝑙({𝑦}) = ∅. proof. directly follows from theorem.3.5 theorem. 3.9. if (𝑋, 𝜏) isĝ∗∗s − 𝑅0, then it isĝ ∗∗s − 𝑇1. proof. let (𝑋, 𝜏) be a ĝ∗∗s − 𝑅0-space and 𝑥 ∈ 𝑋 ⇒ {𝑥} is open. by hypothesis, ĝ∗∗scl({x}) ⊆ {𝑥} ⇒ ĝ∗∗scl({x}) = {𝑥} ⇒ {𝑥} is ĝ∗∗s-closed ⇒every singleton set is ĝ∗∗s-closed⇒ (𝑋, 𝜏) isĝ∗∗s − 𝑇1. theorem. 3.10. every ĝ∗∗s − 𝑅0-space is ĝ ∗∗s − 𝑅1-space. proof. let (𝑋, 𝜏) isĝ∗∗s − 𝑅0. let𝑥, 𝑦 ∈ 𝑋, withĝ ∗∗scl({x}) ≠ ĝ∗∗scl({y}). by above theorem, ĝ∗∗scl({x}) = {𝑥} and ĝ∗∗scl({y}) = {𝑦} ⇒ {𝑥} and {𝑦} are ĝ∗∗s-open sets and {𝑥} ∩ {𝑦} = ∅ ⇒ (𝑋, 𝜏) is ĝ∗∗s − 𝑅1-space. theorem. 3.11. for any ĝ∗∗s-closed set h, ĝ∗∗scl({x}) ∩ 𝐻 = ∅, for every𝑥 ∈ 𝑋\𝐻, then (𝑋, 𝜏) is ĝ∗∗s − 𝑅0-space. proof. assume that for any ĝ∗∗s-closed set h,ĝ∗∗scl({x}) ∩= ∅, for every 𝑥 ∈ 𝑋\𝐻. let g be any open set and 𝑥 ∈ 𝐺. then 𝑥 ∈ 𝐺 ⇒ 𝑥 ∈ 𝑋(𝑋\𝐺) and 𝑋\𝐺 is closed. therefore, by assumption, ĝ∗∗scl({x}) ∩ (𝑋\𝐺) = ∅ ⇒ ĝ∗∗scl({x}) ⊆ 𝐺. hence proved. 114 a study on �̂�∗∗𝑠 − 𝑅0 and �̂� ∗∗𝑠 − 𝑅1 spaces in topological spaces corollary. 3.12. for any ĝ∗∗s-closed set h,𝑐𝑙({𝑥}) ∩ 𝐻 = ∅, for every 𝑥 ∈ 𝑋\𝐻, then (𝑋, 𝜏) is ĝ∗∗s − 𝑅0-space. proof. assume that for any ĝ∗∗s-closed set h,𝑐𝑙({𝑥}) ∩ 𝐻 = ∅, for every 𝑥 ∈ 𝑋\𝐻. let v be any open set in x and 𝑥 ∈ 𝑉. then, 𝑥 ∈ 𝑉 = 𝑋\(𝑋\𝑉) and 𝑋\𝑉 is closed. since every closed set is closed. since, every closed set is ĝ∗∗s-closed which implies that 𝑋\𝑉 is ĝ∗∗s-closed. by assumption, 𝑐𝑙({𝑥}) ∩ (𝑋\𝑉) = ∅ ⇒ 𝑐𝑙({𝑥}) ⊆ 𝑉 ⇒ ĝ∗∗scl({x}) ⊆ 𝑐𝑙({𝑥}) ⊆ 𝑉 ⇒ ĝ∗∗scl({x}) ⊆ v ⇒ (𝑋, 𝜏) is ĝ∗∗s − 𝑅0-space. theorem. 3.13. if a topological space, (𝑋, 𝜏) is ĝ∗∗s − 𝑅1 and ĝ ∗∗s − 𝑇0-space. proof. since every ĝ∗∗s − 𝑇1 is ĝ ∗∗s − 𝑇0. then, the result is obvious. theorem. 3.14. if (𝑋, 𝜏) is ĝ∗∗s − 𝑅0 and ĝ ∗∗s − 𝑇0-space, then it is also a ĝ ∗∗s − 𝑇1space. proof. let 𝑥 ≠ 𝑦 be any two points of x. since (𝑋, 𝜏) isĝ∗∗s − 𝑅0, there exist an open set u such that 𝑥 ∈ 𝑈 andĝ∗∗scl({x}) ⊆ 𝑈 ⇒ 𝑥 ∉ 𝑋\𝑈. since, 𝑦 ∉ 𝑈, there exist another ĝ∗∗s-open set 𝑋\𝑈 = 𝑉(𝑠𝑎𝑦) containing y but not x. therefore, for any two distinct points x and y, there exist two distinct ĝ∗∗s-open sets u and v. hence, (𝑋, 𝜏) is ĝ∗∗s − 𝑇1-space. theorem. 3.15. if a topological space (𝑋, 𝜏) isĝ∗∗s − 𝑅1, then either 𝑐𝑙({𝑥}) = 𝑋, for each 𝑥 ∈ 𝑋 or 𝑐𝑙({𝑥}) ≠ 𝑋, for each𝑥 ∈ 𝑋. proof. assume that (𝑋, 𝜏) isĝ∗∗s − 𝑅1. if 𝑐𝑙({𝑥}) = 𝑋, for each𝑥 ∈ 𝑋, then the theorem is obvious. if not, then there exist 𝑦 ∈ 𝑋 such that𝑐𝑙({𝑦}) ≠ 𝑋. suppose not, there exist 𝑧 ∈ 𝑋 such that𝑐𝑙({𝑧}) = 𝑋. now,𝑐𝑙({𝑦}) ≠ 𝑋 = 𝑐𝑙({𝑧}). since (𝑋, 𝜏) isĝ∗∗s − 𝑅1, there exist disjoint ĝ∗∗s-open sets u and v containing 𝑐𝑙({𝑦}) and 𝑐𝑙({𝑧}) respectively. since 𝑐𝑙({𝑧}) = 𝑋, we have 𝑉 = 𝑋 ⇒ 𝑈 ∩ 𝑉 = 𝑈 ≠ ∅ which a contradiction is to𝑈 ∩ 𝑉 = ∅. hence proved. 4. �̂�∗∗𝐬-door space definition. 4.1. a topological space (x, τ) is called a ĝ∗∗s-door space if every subset is either ĝ∗∗s-open or ĝ∗∗s-closed. definition. 4.2. a topological space (x, τ) is called ĝ∗∗s-submaximal if every dense subset of x is ĝ∗∗s-open. definition. 4.3. a topological space (x, τ) is called ĝ∗∗s-extremally disconnected space in which the ĝ∗∗sclosure of every ĝ∗∗s-open subset is ĝ∗∗s-open. 115 m. anto and s. andrin shahila definition. 4.4. a topological space is called sĝ∗-space if the intersection of a semiclosed set with ĝ∗-closed set is ĝ∗-closed. definition. 4.5. a topological space (x, τ) is called ĝ∗∗s-hyperconnected if every nonempty ĝ∗∗s-open subset of x is ĝ∗∗s-dense. theorem. 4.6. every subspace of ĝ∗∗s-door space is a ĝ∗∗s-door space. proof. let s be a subspace of x and a ⊆ s is a subset of x. since x is ĝ∗∗s-door space, a is either ĝ∗∗s-open or ĝ∗∗s-closed in x. hence a is either ĝ∗∗s-open or ĝ∗∗s-closed in s. therefore, s is ĝ∗∗s-door space. theorem. 4.7. in a ĝ∗∗s-hyperconnected space, every ĝ∗∗s-submaximal space is a ĝ∗∗sdoor space. proof. let a ⊂ x and if a is dense in x, so a is ĝ∗∗s-open. (1) if a is not dense, there exist a non-empty open setb ⊂ ac. since every open set is ĝ∗∗sopen, there exist a non-empty ĝ∗∗s-open setb ⊂ ac. since x is ĝ∗∗s-hyperconnected, b is dense and ac is also dense. again by definition of ĝ∗∗s-submaximal, ac is ĝ∗∗s-open which implies a is ĝ∗∗s-closed. (2) from (1) and (2), a is either ĝ∗∗s-open or ĝ∗∗s-closed. therefore, (x, τ) is a ĝ∗∗s-door space. theorem. 4.8. in(x, τ), every door space is ĝ∗∗s-door space. proof. let a be a door space. every subset of x is either open or closed. since every closed set is ĝ∗∗s-closed and every open set is ĝ∗∗s-open, every subset of x is either ĝ∗∗s-open or ĝ∗∗s-closed. therefore, a is a ĝ∗∗s-door space. remark. 4.9. converse of above theorem is not true. example. 4.10. let x = {a, b, c}, τ = {∅, x, {a}}, τc = {∅, x, {b, c}}, ĝ∗∗so(x, τ) = {∅, x, {a}, {a, b}, {a, c}} and ĝ∗∗sc(x, τ) = {∅, x, {b}, {c}, {b, c}}. hence (x, τ) is a ĝ∗∗s-door space but it is not a door space. theorem. 4.11. for a subset a of a sĝ∗-space (x, τ) the following are equivalent: i) a is ĝ∗∗s-closed ii) ĝ∗cl({x}) ∩ a ≠ ∅, for each x ∈ scl(a) iii) scl(a) − a contains no non-empty ĝ∗-closed set. proof.i) ⇒ ii) letx ∈ scl(a). supposeĝ∗cl({x}) ∩ a = ∅. thena ⊆ x − ĝ∗cl({x}). since a is ĝ∗∗s-closed, scl(a) ⊆ x − ĝ∗cl({x}) which is a contradiction tox ∈ scl(a). therefore, ĝ∗cl({x}) ∩ a ≠ ∅ 116 a study on �̂�∗∗𝑠 − 𝑅0 and �̂� ∗∗𝑠 − 𝑅1 spaces in topological spaces ii) ⇒ iii) let f be a ĝ-closed set such thatf ⊆ scl(a) − a. supposex ∈ f. thenĝ∗cl({x}) ⊆ f. therefore ∅ ≠ ĝ∗cl({x}) ∩ a ⊆ f ∩ a ⊆ [scl(a) − a] ∩ a = ∅ which is a contradiction. iii) ⇒ i) let a ⊆ g and g be ĝ∗-open in x, suppose scl (a) not a subset of g. then scl(a) ∩ (x − g) is non-empty ĝ∗-closed subset of scl(a) − a which is a contradiction. therefore, scl(a) ⊆ g and hence a is ĝ∗∗s-closed. theorem. 4.12. if b is a cl open subset of(x, τ), then b is a ĝ∗∗s-closed set. proof. since b is cl open, b is both open and closed. let u be a ĝ∗-open set in x andb ⊆ u. since b is clopen, we have int(cl(b)) = b ⇒ int(cl(b)) ⊆ b and b ⊆ int(cl(a)), (ie)int(cl(b)) ⊆ b ⇒ b is semi-closed andscl (b) − b. therefore, scl(b) ⊆ u. hence b is ĝ∗∗s-closed in x. theorem. 4.13. in(x, τ), every ĝ∗∗s-dense is dense. proof. let a ⊆ x be ĝ∗∗s-dense in(x, τ) ⇒ ĝ∗∗scl(a) = x. since, x = ĝ∗∗scl(a) ⊆ cl(a) ⇒ x ⊆ cl(a). always, cl(a) ⊆ x ⇒ cl(a) = x ⇒ a ⊆ x be dense in (x, τ). remark. 4.14. converse of above theorem is not true. example. 4.15. letx = {a, b, c}, τ = {∅, x, {a, b}}, τc = {∅, x, {c}}, ĝ∗∗sc(x, τ) = {∅, x, {c}, {b, c}, {a, c}} and ĝ∗∗so(x, τ) = {∅, x, {a}, {b}, {a, b}}. let a = {a} is a dense subset of x but it is not ĝ∗∗s-dense. remark. 4.16. every �̂�∗∗𝑠-door space need not be a �̂�∗∗𝑠-submaximal. example. 4.17. let 𝑋 = {𝑎, 𝑏, 𝑐}, 𝜏 = {∅, 𝑋, {𝑎, 𝑏}}, �̂�∗∗𝑠𝑐(𝑋, 𝜏) = {∅, 𝑋, {𝑐}, {𝑏, 𝑐}, {𝑎, 𝑐}} and�̂�∗∗𝑠𝑜(𝑋, 𝜏) = {∅, 𝑋, {𝑎}, {𝑏}, {𝑎, 𝑏}}. here (𝑋, 𝜏) is a �̂�∗∗𝑠-door space but it is not �̂�∗∗𝑠-submaximal. remark. 4.18. every �̂�∗∗𝑠-submaximal space need not be �̂�∗∗𝑠-door space. example. 4.19. let 𝑋 = {𝑎, 𝑏, 𝑐, 𝑑}, 𝜏 = {∅, 𝑋, {𝑎}, {𝑑}, {𝑎, 𝑑}, {𝑎, 𝑏, 𝑐}}, 𝑠𝑐(𝑋, 𝜏) = {∅, 𝑋, {𝑏}, {𝑐}, {𝑑}, {𝑏, 𝑐}, {𝑐, 𝑑}, {𝑏, 𝑑}, {𝑎, 𝑏, 𝑐}, {𝑏, 𝑐, 𝑑}}, �̂�∗𝑐(𝑋, 𝜏) = {∅, 𝑋, {𝑏}, {𝑐}, {𝑑}, {𝑎, 𝑏}, {𝑏, 𝑐}, {𝑎, 𝑐}, {𝑏, 𝑑}, {𝑐, 𝑑}, {𝑎, 𝑏, 𝑐}, {𝑏, 𝑐, 𝑑}, {𝑎, 𝑐, 𝑑}, {𝑎, 𝑏, 𝑑}}, �̂�∗∗𝑠𝑐{𝑋, 𝜏} = {∅, 𝑋, {𝑏}, {𝑐}, {𝑑}, {𝑏, 𝑐}, {𝑐, 𝑑}, {𝑏, 𝑑}, {𝑎, 𝑏, 𝑐}, {𝑏, 𝑐, 𝑑}}, �̂�∗∗𝑠𝑜(𝑋, 𝜏) = {∅, 𝑋, {𝑎}, {𝑑}, {𝑎, 𝑏}, {𝑎, 𝑐}, {𝑎, 𝑑}, {𝑎, 𝑏, 𝑐}, {𝑎, 𝑐, 𝑑}, {𝑎, 𝑏, 𝑑}} . remark. 4.20. every 𝑆�̂�∗-space need not be a door space. 117 m. anto and s. andrin shahila example. 4.21. let 𝑋 = {𝑎, 𝑏, 𝑐}, 𝜏 = {∅, 𝑋, {𝑎}, {𝑏, 𝑐}}, 𝜏𝑐 = {∅, 𝑋, {𝑎}, {𝑏, 𝑐}}, 𝑠𝑐(𝑋, 𝜏) = {∅, 𝑋, {𝑎}, {𝑏, 𝑐}} 𝑎𝑛𝑑 �̂�∗𝑐(𝑋, 𝜏) = {∅, 𝑋, {𝑎}, {𝑏, 𝑐}}. here 𝑠𝑐 ∩ �̂�∗𝑐 = �̂�∗𝑐 but it is not a door space. theorem. 4.22. in 𝑆�̂�∗-space i) (𝑠𝑔)∗-closed set coincide with �̂�∗∗𝑠-closed set. ii) 𝑔𝑠∗∗-closed set coincide with �̂�∗∗𝑠-closed set. iii) �̂�∗-closed set coincide with �̂�∗∗𝑠-closed set. iv) 𝛼𝑔-closed set coincide with �̂�∗∗𝑠-closed set. v) 𝑝𝑠-closed set coincide with �̂�∗∗𝑠-closed set. vi) 𝑔𝑠𝑝-closed set coincide with �̂�∗∗𝑠-closed set. vii) 𝑠𝑝-closed set coincide with �̂�∗∗𝑠-closed set. viii) 𝑠𝑔-closed set coincide with �̂�∗∗𝑠-closed set. ix) 𝑔𝛼-closed set coincide with �̂�∗∗𝑠-closed set. 𝑔𝑠-closed set coincide with �̂�∗∗𝑠-closed set. theorem. 4.23. in 𝑆�̂�∗-space, semi-closed and �̂�-closed are independent. theorem. 4.24. the property of being �̂�∗∗𝑠-door space is a topological property. proof. let (𝑋, 𝜏) be a �̂�∗∗𝑠-door space and let 𝑓: 𝑋 → 𝑌 be a homeomorphism. let𝐴 ⊆ 𝑌, consider 𝑓 −1(𝐴) ⊆ 𝑋, since x is a �̂�∗∗𝑠-door space, then 𝑓 −1(𝐴) is either �̂�∗∗𝑠-open or �̂�∗∗𝑠-closed in x. now,𝑓(𝑓 −1(𝐴)) = 𝐴. then a is either �̂�∗∗𝑠-open or �̂�∗∗𝑠-closed in y. therefore, y is a �̂�∗∗𝑠-door space. theorem. 4.25. the property of being a �̂�∗∗𝑠-door space is an expensive property. proof. suppose (𝑋, 𝜏) is a �̂�∗∗𝑠-door space. let 𝜏 ⊂ 𝜏1 and𝐴 ⊆ 𝑋. since, (𝑋, 𝜏) is a �̂�∗∗𝑠-door space, then a is either �̂�∗∗𝑠 − 𝜏-open or �̂�∗∗𝑠 − 𝜏-closed. since𝜏1 ⊂ 𝜏, a is either �̂�∗∗𝑠 − 𝜏1-open or �̂� ∗∗𝑠 − 𝜏1-closed. then (𝑋, 𝜏1) is a �̂� ∗∗𝑠-door space. theorem. 4.26. let (𝑋, 𝜏) be a �̂�∗∗𝑠-door space and 𝑌 ⊆ 𝑋 be a clopen subset of y, then (𝑌, 𝜏𝑌) is also a �̂� ∗∗𝑠-door space. proof. let 𝐴 ⊆ 𝑌 be a subset of y. now,𝐴 ⊆ 𝑋. by hypothesis, (𝑋, 𝜏) is a �̂�∗∗𝑠-door space which implies that a is either �̂�∗∗𝑠-open or �̂�∗∗𝑠-closed in x. since y is both open and closed. then a is either �̂�∗∗𝑠-open or �̂�∗∗𝑠-closed in y. therefore, y is also a �̂�∗∗𝑠-door space. proposition. 4.27. let a and b be ĝ∗∗s-closed subsets of x such that 𝑐𝑙(𝐴) = 𝑠𝑐𝑙(𝐴) and𝑐𝑙(𝐵) = 𝑠𝑐𝑙(𝐵). thus 𝐴 ∪ 𝐵 is ĝ∗∗s-closed. 118 a study on �̂�∗∗𝑠 − 𝑅0 and �̂� ∗∗𝑠 − 𝑅1 spaces in topological spaces proof. let u be ĝ∗-open such that𝐴 ∪ 𝐵 ⊆ 𝑈 ⇒ 𝐴 ⊆ 𝑈 𝑎𝑛𝑑 𝐵 ⊆ 𝑈 ⇒ 𝑠𝑐𝑙(𝐴) ⊆ 𝑈 𝑎𝑛𝑑 𝑠𝑐𝑙(𝐵) ⊆ 𝑈 ⇒ 𝑐𝑙(𝐴) ⊆ 𝑈𝑎𝑛𝑑 𝑐𝑙(𝐵) ⊆ 𝑈 ⇒ 𝑐𝑙(𝐴 ∪ 𝐵) = 𝑐𝑙(𝐴) ∪ 𝑐𝑙(𝐵) ⊆ 𝑈 ⇒ 𝑐𝑙(𝐴 ∪ 𝐵) ⊆ 𝑈 ⇒ 𝑠𝑐𝑙(𝐴 ∪ 𝐵) ⊆ 𝑐𝑙(𝐴 ∪ 𝐵) ⊆ 𝑈. therefore, 𝐴 ∪ 𝐵 is ĝ∗∗s-closed. proposition. 4.28. if b is a regular open subset of a topological space(𝑋, 𝜏), then b is ĝ∗∗s-closed. proof. let u be ĝ∗-open set such that𝐵 ⊆ 𝑈. since b is regular open, 𝑖𝑛𝑡(𝑐𝑙(𝐵)) = 𝐵 ⇒ 𝑖𝑛𝑡(𝑐𝑙(𝐵)) ⊆ 𝐵 ⊆ 𝑈 ⇒ 𝐵 is semi − closed ⇒ 𝑠𝑐𝑙(𝐵) = 𝐵 ⊆ 𝑈 ⇒ 𝑠𝑐𝑙(𝐵) ⊆ 𝑈. therefore, b is ĝ∗∗s-closed. theorem. 4.29. if a and b are subsets of x such that 𝐴 ⊆ 𝐵 and a is 𝑔∗-closed, then b is ĝ∗∗s-closed. proof. assume 𝐵 ⊆ 𝑈 and u is ĝ∗-open. therefore, 𝐴 ⊆ 𝐵 ⇒ 𝐴 ⊆ 𝑈. also, every ĝ∗open set is g-open which implies u is g-open in x. therefore, 𝐴 ⊆ 𝑈 and u is g-open ⇒ 𝑐𝑙(𝐴) ⊆ 𝑈 ⇒ 𝑐𝑙(𝐵) ⊆ 𝑈 ⇒ 𝑠𝑐𝑙(𝐵) ⊆ 𝑐𝑙(𝐵) ⊆ 𝑈 ⇒ 𝑠𝑐𝑙(𝐵) ⊆ 𝑈. therefore b is ĝ∗∗sclosed. theorem. 4.30. if a and b are subsets of x such that 𝐴 ⊆ 𝐵 and a is 𝑔∗𝑠-closed, then b is ĝ∗∗s-closed. proof. assume that 𝐵 ⊆ 𝑈 and u is ĝ∗-open. since𝐴 ⊆ 𝐵 ⇒ 𝐴 ⊆ 𝑈. also, every ĝ∗open set is gs-open which implies u is gs-open in x. therefore, 𝐴 ⊆ 𝑈 and u is gsopen⇒ 𝑠𝑐𝑙(𝐴) ⊆ 𝑈 ⇒ 𝑠𝑐𝑙(𝐴) ⊆ 𝑠𝑐𝑙(𝐵) ⊆ 𝑈 ⇒ 𝑠𝑐𝑙(𝐵) ⊆ 𝑈. hence proved. 5. conclusion we have studied the concept of 𝑅0-space, 𝑅1-space, door space, and submaximal space via ĝ∗∗s-closed set and examples are provided to state the converse doesn’t implies. it shows that every ĝ∗∗s − 𝑅0-space is ĝ ∗∗s − 𝑇2-space, ĝ ∗∗s-regular and ĝ∗∗s − 𝑅1-space. in addition we have shown that every �̂� ∗∗𝑠-door space need not be 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[15] s. n. maheshwari and r. prasad, on r0 spaces, portug. math. 34(1975), 213217. 120 ratio mathematica volume 45, 2023 on the edge covering transversal edge domination in graphs e. sherin danie1 s. robinson chellathurai2 abstract let 𝐺 = (𝑉, 𝐸) be any graph with 𝑛 vertices and 𝑚 edges. an edge dominating set which intersects every minimum edge covering set in a graph 𝐺is called an edge covering transversal edge dominating set of 𝐺. the minimum cardinality of an edge covering transversal edge dominating set is called an edge covering transversal edge domination number of 𝐺and is denoted by𝛾𝑒𝑒𝑐𝑡 (𝐺). any edge covering transversal edge dominating set of cardinalities𝛾𝑒𝑒𝑐𝑡 (𝐺) is called a 𝛾𝑒𝑒𝑐𝑡-set of 𝐺. the edge covering transversal edge domination number of some standard graphs are determined. some properties satisfied by this concept are studied. keywords: domination number, edge domination number, edge covering, edge covering number, edge covering transversal edge domination number. mathematics subject classification code: 05c69, 05c703. 1 research scholar, reg. no: 19213162092014, department of mathematics, scott christian college (autonomous), nagercoil 629 003, kanyakumari district, tamilnadu, india. (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627 012, tamil nadu, india.) e-mail: *sherindanie24@gmail.com. 2 associate professor, department of mathematics, scott christian college (autonomous), nagercoil 629 003, kanyakumari district, tamilnadu, india. (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627 012, tamil nadu, india.) e-mail: robinchel@rediffmail.com. 3 received on july 14, 2022. accepted on october 15, 2022. published on january 30, 2023. doi: 10.23755/rm.v45i0.977. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 45 e. sherin danie and s. robinson chellathurai 1. introduction let 𝐺 = (𝑉, 𝐸) be any graph with 𝑛vertices and 𝑚edges. for any graph theoretic terminlogies not defined here, refer to the book of bondy and murthy [3]. one of the fastest growing areas in graph theory is the study of domination and related subset problems such as independence, covering and matching. by a graph 𝐺, we mean a nontrivial, finite, undirected graph with neither loops nor multiple edges. two vertices 𝑢and 𝑣are said to be adjacent if 𝑢𝑣is an edge of 𝐺. the open neighbourhood of a vertex 𝑣in a graph 𝐺is defined as the set 𝑁𝐺 (𝑣) = {𝑣 ∈ 𝑉 (𝐺) ∶ 𝑢𝑣 ∈ 𝐸(𝐺)}, while the closed neighbourhood of 𝑣in 𝐺is defined as 𝑁𝐺 [𝑣] = 𝑁𝐺 (𝑣) ∪ {𝑣}. for any vertex 𝑣in a graph 𝐺, the number of vertices adjacent to 𝑣is called the degree of v in 𝐺, denoted by 𝑑𝑒𝑔𝐺 (𝑣). if the degree of a vertex is 0, it is called an isolated vertex, while if the degree is 1, it is called an end-vertex. the minimum degree of vertices in 𝐺is defined by 𝛿(𝐺) = 𝑚𝑖𝑛{𝑑𝑒𝑔(𝑣)/𝑣 ∈ 𝑉 (𝐺). the maximum degree of vertices in 𝐺is defined by ∆(𝐺) = 𝑚𝑎𝑥{𝑑𝑒𝑔(𝑣)/𝑣 ∈ 𝑉 (𝐺). a vertex 𝑣is called a universal vertex if 𝑑𝑒𝑔𝐺(𝑣) = 𝑛 − 1.two edges are said to adjacent edges if they have a common vertex. for any set 𝑆of vertices of g, the induced subgraph < 𝑆 >is the maximal subgraph of g with vertex set. a subset 𝑆 ⊆ 𝑉 (𝐺) is called a dominating set [3, 6, 7, 8]if every vertex 𝑣 ∈ 𝑉 (𝐺) \ 𝑆is adjacent to a vertex 𝑢 ∈ 𝑆. the domination number, 𝛾(𝐺), of a graph 𝐺denotes the minimum cardinality of such dominating sets of 𝐺. a minimum dominating set of a graph 𝐺is hence often called as a γ-set of 𝐺. a subset 𝑆 ⊆ 𝐸 (𝐺) is called an edge dominating set [1, 2] if every edge𝑓 ∈ 𝐸 (𝐺) \ 𝑆is adjacent to an edgeℎ ∈ 𝑆. the edge domination number, 𝛾𝑒 (𝐺), of a graph 𝐺denotes the minimum cardinality of such edge dominating sets of 𝐺. a minimum edge dominating set of a graph 𝐺is hence often called as a 𝛾𝑒 -set of 𝐺. an edge cover [5] of a graph is a set of edges such that every vertex of the graph is incident to atleast one edge of the set. a minimum edge covering is an edge covering of smallest possible size. the edge covering number𝜌(𝐺) is the size of a minimum edge covering.given a graph 𝐺 and a collection of subsets of its vertices, a subset of 𝑉(𝐺) is called a transversal of 𝐺 if it intersects each subset of the collection [4]. in this paper, we studied the concept of the edge covering transversal edge domination number of𝐺. 2. on the edge covering transversal edge domination number of a graph definition 2.1. an edge dominating set which intersects every minimum edge covering set in a graph 𝐺is called an edge covering transversal edge dominating set of 𝐺.the minimum cardinality of an edge covering transversal edge dominating set is called an edge covering transversal edge domination number of 𝐺and is denoted by 𝛾𝑒𝑒𝑐𝑡 (𝐺). any edge covering transversal edge dominating set of cardinalities𝛾𝑒𝑒𝑐𝑡 (𝐺) is called a 𝛾𝑒𝑒𝑐𝑡-set of 𝐺. 46 on the edge covering transversal edge domination in graphs example 2.2.for the graph 𝐺given in fig. 2.1, 𝐷1 = {𝑣2𝑣3, 𝑣4𝑣5} and 𝐷2 = {𝑣2𝑣5, 𝑣3𝑣4} are the only two minimum edge dominating sets of 𝐺. also,𝑆1 = {𝑣1𝑣5, 𝑣2𝑣3, 𝑣3𝑣4},𝑆2 = {𝑣1𝑣2, 𝑣2𝑣3, 𝑣4𝑣5}, 𝑆3 = {𝑣1𝑣2, 𝑣2𝑣5, 𝑣3𝑣4}, 𝑆4 = {𝑣1𝑣5, 𝑣2𝑣3, 𝑣4𝑣5} are the only 4 minimum edge covering sets of 𝐺. since 𝐷1 ∩ 𝑆3= ϕ and 𝐷2 ∩ 𝑆4 = 𝜙, 𝐷1and 𝐷2are not an edge covering transversal edge dominating sets of g and so 𝛾𝑒𝑒𝑐𝑡 (𝐺) = 3.now,𝐷3 = {𝑣1𝑣2, 𝑣2𝑣3, 𝑣4𝑣5} is a 𝛾𝑒𝑒𝑐𝑡-set of g, so that 𝛾𝑒𝑒𝑐𝑡 (𝐺) = 3. g figure 2.1 theorem 2.3. a set 𝑆of edges of 𝐺 = 𝐾𝑟,𝑟 (𝑟 ≥ 2) is a minimum edge covering transversal edge domination of 𝐺if and only if 𝑆consists of 𝑛-independent edges. proof. let 𝑆 be any set of 𝑟-independent edges of 𝐺 = 𝐾𝑟,𝑠 (2 ≤ 𝑟 ≤ 𝑠). then s is a minimum edge covering set of 𝐺. since 𝑆 ∩ 𝐷 = 𝜙for any edge covering transversal edge dominating set 𝐷of 𝐺, 𝑆is an edge covering transversal edge dominating set of 𝐺. hence, it follows that 𝛾𝑒𝑒𝑐𝑡 (𝐺) ≤ 𝑟. if 𝛾𝑒𝑒𝑐𝑡 (𝐺) ≤ 𝑟, then there exists an edge covering transversal edge dominating set 𝑆’, such that |𝑆′| < 𝑟. therefore, there exists atleast one vertex v of g, such that v is not incident with any edge of 𝑆’ and so 𝑆’ is not an edge covering transversal edge dominating set of 𝐺, which is a contradiction. hence, s is a minimum edge covering transversal edge dominating set of 𝐺. conversely, let 𝑆be a minimum edge covering set of g. let 𝑆’ be any set of nindependent set of edges of 𝐺.then as in first part of this theorem, 𝑆'′ is a minimum edge covering set of 𝐺. therefore, | 𝑆′| = 𝑟. hence, |𝑆| = 𝑟. if 𝑆is not independent, then there exists a vertex 𝑣of 𝐺such that, 𝑣is not independent with any edge of 𝑆. hence 𝑆is not an edge covering transversal edge dominating set of g, which is a contradiction. thus, s consists of r-independent edges. theorem 2.4. a set s of edges of 𝐺 = 𝐾𝑟,𝑠(2 ≤ 𝑟 ≤ 𝑠 ) is a minimum edge covering transversal edge domination of 𝐺, if and only if 𝑆consists of 𝑟 − 1 independent edges of 𝐺and 𝑠 − 𝑟 + 1 adjacent edges of𝐺. proof. let 𝑋 = {𝑢1, 𝑢2, . . . . . , 𝑢𝑟 } and 𝑌 = {𝑣1, 𝑣2, . . . . . , 𝑣𝑠 } be a bipartition of 𝐺. let 𝑆be any set of 𝑟 − 1 independent edges of g and 𝑠 − 𝑟 + 1 adjacent edges of 𝐺.since each vertex of 𝐺is incident with an edge of 𝑆, it follows that 𝛾𝑒𝑒𝑐𝑡 (𝐺) < 𝑠. v 5 v 2 v 4 v 3 v 1 47 e. sherin danie and s. robinson chellathurai if 𝛾𝑒𝑒𝑐𝑡 (𝐺) < 𝑠, then there exists an edge covering transversal edge dominating set s ′ of 𝐺such that |𝑆′| < 𝑠. therefore, there exists atleast one vertex 𝑣of 𝐺such that 𝑣is not incident with any edge of 𝑆′ and so 𝑆′ is not an edge covering transversal edge dominating set of 𝐺, which is a contradiction. hence, 𝑆is a minimum edge covering transversal edge dominating set of 𝐺. conversely, let𝑆be a minimum edge covering transversal edge dominating set of 𝐺. let s′ be any set of 𝑟 − 1 independent edges of 𝐺and 𝑟 − 𝑠 + 1 adjacent edges of 𝐺. then as in the first part of this theorem, 𝑆is a minimum edge covering transversal edge dominating set of 𝐺. therefore |𝑆′| = 𝑠. hence |𝑆| = 𝑠. let us assume that 𝑆 = 𝑆1 ∪ 𝑆2, where 𝑆1consists of independent edges and 𝑆2consists of adjacent edges of 𝐺. if |𝑆1| ≤ 𝑠 − 2, then 𝑆2must contain atmost edges𝑠 − 𝑟. then there exists atleast one vertex 𝑣of 𝑋, such that 𝑣is not incident with any edge of 𝑆and so 𝑆is not an edge covering transversal edge dominating set of 𝐺, which is a contradiction. therefore 𝑆consists of 𝑠 − 1 independent edges of 𝐺and 𝑠 − 𝑟 + 1 adjacent edges of 𝐺. corollary 2.5. for the complete bipartite graph, 𝐾𝑟,𝑠(2 ≤ 𝑟 ≤ 𝑠 ), 𝛾𝑒𝑒𝑐𝑡 (𝐺) = 𝑠. theorem 2.6. for the complete graph 𝐺 = 𝐾𝑛(𝑛 ≥ 4) with 𝑛even, a set 𝑆of edges of 𝐺is a minimum edge covering transversal edge domination of 𝐺if and only if 𝑆consists of 𝑛 2 independent edges. proof. the proof is similar to the proof of the theorem 2.4. theorem 2.7. for the complete graph 𝐺 = 𝐾𝑛(𝑛 ≥ 5)g with𝑛odd, a set 𝑆of edges of 𝐺is a minimum edge covering transversal edge dominating set of 𝐺if and only if s consists of 𝑛−3 2 independent edges and two adjacent edges of 𝐺. proof. the proof is similar to the proof of the theorem 2.4. . corollary2.8. for the complete graph 𝐺 = 𝐾𝑛(𝑛 ≥ 4), 𝛾𝑒𝑒𝑐𝑡 (𝐺) = { 𝑛 2 if𝑛 is even 𝑛 + 1 2 if 𝑛 is odd theorem 2.9. for the cycle 𝐺 = 𝐶𝑛(𝑛 ≥ 4), 𝛾𝑒𝑒𝑐𝑡 (𝐺) = { 𝑛 2 if 𝑛 is even ⌈ 𝑛 2 ⌉ if 𝑛 is odd proof. let 𝐶𝑛: {𝑣1, 𝑣2, 𝑣3, . . . . , 𝑣𝑛 , 𝑣1} be the cycle. we consider the following two cases case (i). 𝑛is even. let 𝑛 = 2𝑘(𝑘 ≥ 2), then 𝑆1 = {𝑣1𝑣2, 𝑣3𝑣4, 𝑣5𝑣6, … , 𝑣2𝑘−1𝑣2𝑘 } and 𝑆2 = {𝑣2𝑣3, 𝑣4𝑣5, 𝑣6𝑣7, … , 𝑣2𝑘−2𝑣2𝑘−1, 𝑣2𝑘 𝑣1}are the only two minimum edge covering sets of 𝐺. let d be an edge dominating sets of 𝐺. then it is clear that 𝐷 ∩ 𝑆1 ≠ 𝜙and𝐷 ∩ 48 on the edge covering transversal edge domination in graphs 𝑆2 ≠ 𝜙. therefore 𝑆1and 𝑆2are the only two minimum edge covering transversal edge dominating sets of 𝐺, so that𝛾𝑒𝑒𝑐𝑡 (𝐺) = 𝑘 = 𝑛 2 case (ii). 𝒏 is odd let. 𝑛 = 2𝑘 + 1(𝑘 ≥ 2), then𝑆1 = {𝑣1𝑣2, 𝑣3𝑣4, 𝑣5𝑣6, … , 𝑣2𝑘−1𝑣2𝑘 , 𝑣2𝑛𝑘 𝑣2𝑘+1} is a minimum edge covering transversal edge dominating set of 𝐺, so that𝛾𝑒𝑒𝑐𝑡 (𝐺) = ⌈ 𝑛 2 ⌉. theorem 2.10. for the path 𝐺 = 𝑃𝑛 (𝑛 ≥ 4), 𝛾𝑒𝑒𝑐𝑡 (𝐺) = { 𝑛 2 if 𝑛 is even 𝑛+1 2 if 𝑛 is odd proof. the proof is similar to the proof of the theorem 2.9. theorem 2.11. for the star graph, 𝐺 = 𝐾1,𝑛−1, 𝛾𝑒𝑒𝑐𝑡 (𝐺) = 𝑛 − 1. theorem 2.12. for the wheel graph 𝐺 = 𝐾1 + 𝐶𝑛−1(𝑛 ≥ 5), 𝛾𝑒𝑒𝑐𝑡 (𝐺) = { 𝑛 − 1 2 if 𝑛 − 1 𝑖s even 𝑛 + 1 2 if 𝑛 − 1 is odd figure 2.2 proof.let𝑉 (𝐾1) = {𝑥} and 𝐶𝑛−1𝑏𝑒 𝑣1, 𝑣2, 𝑣3, . . . . , 𝑣𝑛−1, 𝑣1. we consider the following cases. case (i).𝒏 – 𝟏 is even. let 𝑛 − 1 = 2𝑘(𝑘 ≥ 3). then 𝑆1 = {𝑣1𝑣2, 𝑣3𝑣4, 𝑣5𝑣6, … , 𝑣2𝑘−1𝑣2𝑘 } and 𝑆2 = {𝑣2𝑣3, 𝑣4𝑣5, 𝑣6𝑣7, … , 𝑣2𝑘−2𝑣2𝑘−1, 𝑣2𝑘 𝑣1}are the only two minimum edge covering sets of 𝐺. then it is clear that 𝐷 ∩ 𝑆1 ≠ 𝜙and that 𝐷 ∩ 𝑆2 ≠ 𝜙. therefore 𝑆1and 𝑆2are the only two minimum edge covering transversal edge dominating sets of 𝐺, so that𝛾𝑒𝑒𝑐𝑡 (𝐺) = 𝑘 = 𝑛−1 2 . v 7 v 1 v 6 v 5 v 2 v 4 v 3 49 e. sherin danie and s. robinson chellathurai case (ii). 𝒏 – 𝟏 is odd. let 𝑛 − 1 = 2𝑘 + 1(𝑘 ≥ 2). then𝑆1 = {𝑣1𝑣2, 𝑣3𝑣4, 𝑣5𝑣6, … , 𝑣2𝑘−1𝑣2𝑘 , 𝑣2𝑘 𝑣2𝑘+1} is a minimum edge covering transversal edge dominating set of 𝐺, so that 𝛾𝑒𝑒𝑐𝑡 (𝐺) = ⌈ 𝑛−1 2 ⌉. theorem 2.13. let g be a connected graph withmedges (m≥ 2), then 1 ≤ γeect (g) ≤ m. theorem 2.14. for any graph 𝐺, 𝛾𝑒 (𝐺) ≤ 𝛾𝑒𝑒𝑐𝑡 (𝐺) ≤ 𝛾𝑒 (𝐺) + ∆(𝐺). proof. let 𝐷be a 𝛾𝑒𝑒𝑐𝑡 t-set of 𝐺. then 𝐷itself is an edge dominating set. therefore 𝛾𝑒 (𝐺) ≤ |𝐷| = 𝛾𝑒𝑒𝑐𝑡 (𝐺). now, let s be a 𝛾𝑒 -set in g and let v be a vertex of maximum degree. therefore 𝑑𝑒𝑔(𝑣) = ∆(𝐺). then every minimum edge covering transversal edge dominating set of 𝐺contains an edge of < 𝐸[𝑁(𝑣)] > so 𝑆 ∪< 𝐸[𝑁(𝑣)] >is an edge covering transversal edge dominating set. also, since 𝑆intersects < 𝐸[𝑁(𝑣)] > |𝑆 ∪< 𝐸[𝑁(𝑣)] > | ≤ 𝛾𝑒 (𝐺) + ∆(𝐺). then𝛾𝑒 (𝐺) ≤ 𝛾𝑒𝑒𝑐𝑡 (𝐺) ≤ 𝛾𝑒 (𝐺) + ∆(𝐺). remark 2.15. the bound in the theorem 2.14 is sharp. figure 2.3 for the graph given in figure 2.3,𝑆1 = {𝑣1𝑣2, 𝑣3𝑣4} is a 𝛾𝑒 -set of g and 𝛾𝑒𝑒𝑐𝑡-set of 𝐺so that 𝛾𝑒 (𝐺) = 𝛾𝑒𝑒𝑐𝑡 (𝐺). 3. conclusions in this paper, we obtain some results related to edge covering transversal domination in graphs and this work gives the scope for an extensive study of various domination parameters of these graphs. acknowledgment the authors are highly thankful to the anonymous referees for their kind comments and fruitful suggestions on the first draft of this paper. v 4 v 3 v 2 v 1 g 50 on the edge covering transversal edge domination in graphs references [1] araya chaemchan, the edge domination number of connected graphs, australian journal of combinatorics, 48, (2010), 185–189 [2] s. arumugam and s. velammal, edge domination in graphs, 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[8] s. kavitha, s.r chellathurai, j. john, on the forcing connected domination number of a graph, journal of discrete mathematical sciences and cryptography, 20 (3), (2017), 611–-624. 51 microsoft word articolo1.doc i numeri reali franco eugeni, daniela tondini, annamaria viceconte department of communication science,univeristy of teramo. e-mail : {eugeni, dtondini}@unite.it 1. il grande problema dei pitagorici e la "maledizione" di ippaso. fin dal vi secolo a.c., anche se con modalità logiche differenti e con un simbolismo molto diverso dall’attuale, il campo dei numeri razionali si poteva considerare operativamente conquistato. naturalmente l'uso dei numeri razionali venne rivolto alla geometria allo scopo di costruire una teoria della misura per i segmenti. la scuola pitagorica (vi sec. a.c.) che operava a crotone propose una teoria della misura molto semplice, basata sulla seguente assunzione arbitraria: "ogni segmento contiene un numero infinito di punti" allora se il segmento ab ha m punti e il segmento cd ha n punti la misura di ab rispetto a cd è m/n. l'ingenua soluzione dei pitagorici portava facilmente ad un assurdo. infatti se si considera un quadrato e la sua diagonale, è di immediata verifica, anche sperimentale, che il quadrato costruito sulla diagonale si ripartisce in quattro mezzi quadrati, tali che: (1) ac2 = 2ab2 a d c da questa relazione segue che la teoria dei pitagorici è falsa! infatti, ammessa la teoria della misura dei pitagorici, si avrebbe che ac ab m n = , cioè: (2) m2 = 2n2 che è certamente falsa. infatti è falsa se m è dispari, essendo il secondo membro pari. quindi m deve essere pari ed n dispari altrimenti m n si potrebbe semplificare. posto allora m = 2a, si ha: 4a2 = 2n2 ⇒ n2 = 2a2 ancora assurda per essere n dispari ed il secondo membro pari la conseguenze che ne risulta è che vi sono due tipi di coppie di segmenti. vi sono coppie di segmenti ab e cd che sono confrontabili, nel senso che il segmento, ad esempio, cd è m volte la n-sima parte del segmento ab. cioè: cd m n ab= . in tal caso ab e cd si dicono commensurabili. vi sono però coppie di segmenti per le quali un siffatto confronto non si può fare; un esempio è rappresentato appunto dal lato e dalla diagonale di un quadrato dato per i quali una siffatta relazione non può essere scritta. coppie siffatte si dicono incommensurabili. la scoperta dei pitagorici riguardò dunque l'incommensurabilità. tale scoperta si tenne segreta a lungo, ma ippaso da metaponto (crotone), che sembra fosse un pitagorico fuoriuscito, la rivelò a platone, che da buon pettegolo quale era, la diffuse nel modo scientifico di allora. fu da allora che ippaso venne maledetto. notizia di tale maledizione ci giunge dal x libro di euclide, comunemente attribuito a proclo: "è fama che colui il quale per primo rese di pubblico dominio la teoria degli irrazionali sia perito in un naufragio, ciò perché l'inesprimibile e b a b c d l'inimmaginabile avrebbero dovuto essere sempre celati. perciò il colpevole, che fortuitamente toccò e rivelò questo aspetto delle cose viventi, fu trasportato al suo luogo d'origine e viene in perpetuo flagellato dalle onde". non sappiamo bene a chi si deve la prima trattazione dei numeri irrazionali; molti la fanno risalire a eudosso di cnido (iv sec. a.c.) al quale si attribuisce solitamente la stesura del libro v degli “elementi” di euclide, il più ammirato dell'opera euclidea. tale libro, sotto l'apparenza di una teoria generale delle frazioni, fornisce una teoria generale delle grandezze commensurabili e non. nei libri aritmetici di euclide, dal vii al ix, non si parla degli irrazionali quadratici, menzionati solo nel x. tuttavia la questione degli irrazionali rimase oscura per molti secoli, certamente fino a luca pacioli (1445-1517) e dopo di lui se ne occuparono a fondo michael stiefel (1486-1567), gerolamo cardano (1501-1576) e simon stevin (1548-1620) tanto per citare i primi. 2. i nuovi numeri (maledetti dai pitagorici): i numeri reali. i numeri scoperti dai pitagorici nei due secoli successivi, derivanti dall'uso del teorema di pitagora, furono i soli radicali quadratici. in questo paragrafo vogliamo ampliare il campo dei razionali in una nuova struttura, quella dei numeri reali. denotiamo con ℚ il campo dei numeri razionali. è ben noto che con il metodo delle divisioni successive ogni numero razionale si può esprimere come un’espressione decimale periodica. esempi. ( )1 2 0 5 0= , ( )4 3 1 3= , ( )7 6 11 6= , le espressioni date indicano ordinatamente un decimale finito (0-periodico), un decimale con un solo periodo dopo la virgola, un decimale con antiperiodo e periodo. questo si evince dal fatto che nel denominatore della frazione, ridotta ai minimi termini, ci sono, nel primo caso, solo divisori del 10, nel secondo caso nessun divisore del 10, nel terzo caso infine sono presenti sia divisori del 10 (nel caso è 2) che non divisori del 10 (nel nostro caso 3). ricordiamo che è sempre facile risalire da una rappresentazione decimale periodica alla corrispondente frazione generatrice. esempi. ( )213 553 0 213553 1000 , = ; ( )511 53 51153 511 99 , = − ; ( )51 34 123 5134123 5134 99900 , = − le regole di passaggio dai decimali alle frazioni generatrici, del resto evidenti dagli esempi, non si ritiene di doverle enunciare in questa sede. (una regola che utilizza uno strumento matematico per fare alcune somme infinite: la serie geometrica, utilizzabile per una spiegazione di queste formule delle frazioni generatrici è riportata inell’appendice 2). consideriamo ora, da un punto di vista puramente formale, una scrittura del tipo: 2,3132121 … 15,1231411 … a,a1a2a3 … si tratta di semplici scritture nelle quali i termini dopo la virgola … non presentano alcuna caratteristica di periodicità … una vasta classe di numeri di questo tipo si rappresenta facendo ricorso ai cosiddetti "radicali". sia a un numero reale positivo che chiameremo radicando. si dimostra che esiste sempre uno ed un sol numero reale b, positivo che diremo radicale tale che nb = a. l’intero b così introdotto si chiama radicale n-simo ovvero radice n-ma aritmetica di a, e si denota con: n a ovvero con il simbolo na 1 la teoria dei radicali insegna a moltiplicare tra loro espressioni di questo tipo ed in generale ad operare su espressioni conteneti radicali. nella storia del mondo matematico si sono spesso presentate situazioni di questo tipo specie in relazione a particolari operazioni che escono decisamente dal mondo razionale delle quattro operazioni. vediamo alcuni esempi. esempio 1. in primo luogo presentiamo una successione di cifre chiaramente non periodiche e costruibili secondo una regola precisa : n = 0,12345678910111213141516171819202122232425 …… (nel quale i numeri dopo la virgola sono la successione dei naturali). esempio 2. una seconda successione di cifre, banalmente non periodiche, certamente interessanti, sono quelle del numero: q = 0,101001000100001 … la cui regola di formazione è evidente. esempio 3. un altro esempio esempio è la lunghezza della diagonale del quadrato unitario denotata con 2 . abbiamo osservato come 2 , non essendo una frazione − altrimenti sarebbe m2 = 2n2 − non può essere espressa da una rappresentazione periodica. questo a priori non disse al tempo che 2 , possedesse una rappresentazione aperiodica, la cosa venne scoperta con l’algoritmo di estrazione della radice quadrata. nel i secolo venne scoperto un algoritmo di estrazione di radice quadrata, con un procedimento che non ha termine, e che non da luogo a periodicità. mediante questo algoritmo si hanno le prime 15 cifre decimali aperiodiche del numero irrazionale: 2 = 1,41421 35623 73095 04880 16887 24209 69807 85697 … è interessante la regola di teone di smirne (125 a.c.) asserente che se a b è un'approssimazione di 2 allora 2a b a b + + è un'approssimazione migliore. così una buona sequenza di frazioni che approssimano è la seguente: 1 1 1 = ; 3 1, 5 2 = ; 14 7 1, 4 10 5 = = ; 34 17 1, 41(6) 24 12 = = ; 82 41 1, 41379 58 29 = = ; 99 1, 4142857 70 = …; 239 1, 4142011 169 = ; 577 1, 4142156 408 = … nota storica. le prime tracce dell’algoritmo di estrazione si trovano in un commento (del i secolo) del chiu-chang suanshu (ovvero: nove capitoli sull’arte matematica. il , risale al 250 a.c.) forse la più influente fra le opere matematiche cinesi, assieme al chou pei suan ching (1200 a.c.), che si occupa di corpi celesti e misura del tempo. per tornare al discorso generale, ogni scrittura del tipo a,a1a2a3 … si chiamerà numero reale. se poi la sequenza dopo la virgola è periodica il numero reale si dirà razionale e – seguendo le note regole – potrà essere scritto utilizzando anche la sua frazione generatrice. se invece, come nel caso dei due esempi precedenti e in quelli subito che seguono, la sequenza dopo la virgola è aperiodica, allora il numero reale si dirà irrazionale. questo è il caso del numero q dell’esempio 1, chiaramente irrazionale ma anche del numero simbolicamente indicato con 2 . l'insieme dei numeri reali (ovvero degli allineamenti decimali pensati come oggetti) si denota con ℝ. nota storica. l'espressione "numerus irrationalis" è stata introdotta da gerardo da cremona (1114–1187); altri contributi nel lento progredire della teoria relativa agli irrazionali si hanno da parte di luca pacioli (sec. xv), m. stiefel, cardano e stevin (xvi). le classi contigue sono state introdotte da capelli (1897) e la sistemazione rigorosa è dovuta a weierstrass, dedekind e peano in pubblicazioni più o meno contemporanee e più o meno equivalenti. nel linguaggio comune la parola irrazionale fa pensare a qualcosa oltre la ragione e questo termine allora, imposto dalla storia e dall'uso, sta ad indicare la meraviglia dei greci all'atto della loro scoperta. continuiamo ora l’esame di questi oggetti presentando alcuni numeri irrazionali che presentano un interesse notevole in tutto il corso dell’evoluzione della matematica: il numero π, il numero e (base dei logaritmi naturali), il numero φ legato alla sezione aurea. esempio 4. in questo esempio trattiamo il numero π, del quale indichiamo le prime 30 cifre decimali, π = 3,14159 26535 89793 33846 26433 83279 … questo numero, geometricamente, esprime … il rapporto della lunghezza di una circonferenza al suo diametro … la lettera π è l’iniziale sia della parola περιφερεια (periferia) che della parola περιµετροζ (perimetro) collegate idealmente con la lunghezza della circonferenza. la seguente tabella prova come il miglioramento delle approssimazione di questo oggetto irrazionale vada più o meno di pari passo con la storia del mondo. π = 3 (bibbia, talmud, tradizione antica) π = 49 16 = 3,0625 (bodhaiana, india, 200 a.c.) π = 2 8 9 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = 3,1 604 (egiziani, amhes – in paphirus rind –1650 a.c.) (henry rind è lo scopritore del papiro (30cm. per 5,46m.) nel 1858 e amhes l’esecutore della trascrizione nel 1650 a.c. – london, british museum) π = 10 = 3,162277 folclore π = 92 29 = 3,17 folclore π = 142 45 = 3,155555 folclore π = 157 50 = 314 100 = 3,14 (cinesi, desunto da opere arabe) π = 377 120 = 173 20 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = 3,141666 valore tolemaico (tolomeo, 200 d.c., 250 a.c.) π = 22 7 = 3,142 valore di archimede (250 a.c.) e dei cinesi π tra 103 71 + = 3,140845 e 10 3 70 + = 3,1428571 (limitazioni di archimede) π = 142 45 = 3,1555 (wang fan, cina, 256 d.c.) π = 3,14 da un poligono di 96 lati (cina, liu hui, cina, 300 d.c.) π = 3,14159 da un poligono di 3072 lati (cina, liu hui, cina, 300 d.c.) π = 355 113 = 377 120 − − 22 7 = 3,14159292 (tsu ch’ung-chih 470 d.c.) si sottrae membro a membro il valore tolemaico dall’archimedeo π = 3,1415927 (ancora tsu ch’ung-chih, cina, 472 d.c.) …calcoli π = 3227 1250 = 3,1416 (bhaskara, india, 1150) π = 3,14159265 589793 (al kashi, 1400) … calcoli π con 100 cifre decimali (shanks, 1871) … calcoli π con 707 cifre decimali (shanks, 1873) … calcoli il famoso matematico indiano srinivasa ramanujan (india, 1900), diede varie approssimazioni di π quali ad esempio le seguenti: π = 4 2143 22 = 3,14159265 58 (snivrasta ramanujan, india, 1900) 299 2 2 1103 π = (fornisce π con nove decimali esatti) π con 100265 cifre decimali è stato calcolato al centro ibm di new york nel 1961, utilizzando ibm 70900 (8 ore e un minuto). nota storica. l’uso della lettera π sembra risalire al xvii secolo in william oughtred, the key of mathematics, london 1647 e isaac barrow, matematicae lectiones in scolis publicis, accademiae cantabrigensis, londini, 1684. la usava w.jones nel 1706, mentre bernouilli usava la c, eulero nel 1734 las lettera p e nel 1736 la c, ancora cr. goldbach nel 1732 usava di nuovo π che divenne simbolo universale dopo la pubblicazione dell'analisi di eulero. esempio 5. il numero e = 2,7 1828 1828 45904 52353 60287 47135 26624 77572 47093 699 … è utilizzato come base dei logaritmi di nepero. un'approssimazione per le prime cifre è data da 878 2, 71826 ... 323 = facciamo un esperimento e calcoliamo i numeri del tipo (si possono collegare con la nozione di interesse composto): 1 1 n ne n ⎛ ⎞ = +⎜ ⎟ ⎝ ⎠ con n = 1, 2, 3, 4, 5, 6, … si ottiene: e1 = 2; e2 = 2,25; e3 = 2,369; e4 = 2,441; e5 = 2,48; … continuando appare un numero che cresce verso il numero di prime cifre 2,71 … possono essere trattate varie questioni tra cui, ad esempio, la catenaria. esempio 6. la sezione aurea: il numero φ = 5 + 1 2 = 1,61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 … il numero, del quale sopra sono date le prime 50 cifre decimali, si chiama sezione aurea dell’unità e si indica con φ, in omaggio allo scultore fidia. le proprietà di tale rapporto, come esempio per il lettore, vanno ricercate in una enciclopedia, anche tenedo in conto le relazioni con l’arte e l’estetica. esempi. alcuni radicali che sono numeri irrazionali sono i seguenti: 2 = 1,41421 35623 73095 04880 16887 24209 69807 85697 … 3 = 1,73205 08075 68877 29352 74463 41505 87236 6942 … φ = 5 + 1 2 = 1,61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 … 5 = 2,30258 50929 94045 68401 79914 5468436420 7601 … numeri per i quali si hanno indicazioni particolari nel 1900 david hilbert (1862-1943) al congresso dei matematici di quell’anno pose 23 famosi problemi rimasti irrisolti. tra questi vi era il provare se 22 = 2,665144 … eπ = 23,14069 26327 79269 00572 9086 … risultavano o no trascendenti lo studio della loro trascendenza (cioè soluzioni di equazioni algebriche a coefficienti interi) proposta da david hilbert (1862-1943) al congresso dei matematici del 1900, come settimo dei suoi famosi 23 problemi. per il secondo la trascendenza fu provata nel 1929, per il primo nel 1930). il numero π 2e = 0,20787 95763 50761 90854 6955 … è irrazionale. il numero π e = 22, 45915 77183 61045 47342 7152 … non si sa se è razionale o irrazionale. il primo dei due numeri sopra indicati eguaglia il numero ii , dove i è l’unità immaginaria. la prova si fa facendo ricorso alla famosa : eiπ + 1 = 0. esprimente un bel legame tra e, π, l’unità immaginaria i, l’unità moltiplicativa 1 e lo zero! il simbolo γ denota il numero a cui tende la quantità 1 1 1 1 1 ... 2 3 4 logn n + + + + + − quando n tende a crescere indefinitamente. fu introdotto nel 1781 proprio da eulero – che ne calcolò pure le prime sedici cifre decimali – e studiato anche da mascheroni. talvolta γ viene detta costante di eulero– mascheroni. una sua approssimazione è data da: γ = 0, 57721 56649 01532 86060 65120 90082 402431 … allo stato attuale delle conoscenze non si sa se γ sia un numero razionale oppure irrazionale. si sa solo che se fosse a b γ = , allora sarebbe b > 1010000! un'ultima questione è la seguente: come si fanno i calcoli con i numeri reali? la risposta è semplice: di fatto non si fanno! immaginiamo di avere una espressione del tipo: 2 3,14 2, 71 1, 41 0, 76 2, 411 2 eπ − + − + ≅ = + si pongono varie questioni: a) il valore trovato è approssimato! b) di quanto è approssimato: è maggiore o minore dell'espressione? c) come si valuta l'errore commesso? la teoria dei calcoli approssimati insegna a valutare gli errori che sull’espressione quando si approssimano le parti che la compongono. tuttavia spesso occorre prima delle elaborazioni lavorare sulle espressione nelle quali i numeri che intervengono sono espressi o tramite radicali o tramite lettere che li indicano. questa parte di calcolo si effettua conoscendo due tipi di calcolo e precisamente: − la teoria dei radicali − il calcolo letterale e di queste parti ci impadroniremo nei prossimi capitoli. appendice 1. − alternativa mediante le classi contigue. − denotiamo con ℚ il campo dei numeri razionali. siano a e b due parti o classi di ℚ entrambe non vuote. la coppia (a, b) costituisce una coppia di classi contigue se: (1) proprietà di separazione. comunque presi a∈a e b∈b risulta a < b. (2) proprietà di apertura. a non ha massimo, b non ha minimo. (3) proprietà di avvicinamento. comunque preso un numero razionale ε > 0 esistono a∈a e b∈b tali che b – a < ε. un numero razionale c è elemento separatore della coppia di classe contigue (a, b) se • esso non appartiene ad alcuna delle due classi, cioè c∉a, c∉b; • vale la seguente limitazione: ∀ a ∈ a, ∀ b ∈ b ⇒ a < c < b l’elemento separatore di una coppia di classi contigue è unico. esempio 1. si consideri un qualsiasi numero razionale, ad esempio 4 3 . quindi siano: 4 : , 3 a x x x⎧ ⎫= ∈ <⎨ ⎬ ⎩ ⎭ ¤ 4 : , 3 b y y y⎧ ⎫= ∈ >⎨ ⎬ ⎩ ⎭ ¤ per le classi a e b le proprietà (1) e (2) sono banalmente verificate. per la (3) è sufficiente osservare che, se ε > 3 è un razionale arbitrario, é sempre possibile trovare un elemento di a appartenente all'intervallo 4 3 2 4 3 −⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ε , ed un elemento di b appartenente all'intervallo 4 3 4 3 2 , +⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ε tali che la loro differenza sia minore di ε. quindi (a, b) è una coppia di classi contigue che ammette 4 3 come elemento separatore. esempio 2. siano due classi così definite: a = {x : x ∈ ℚ, x < 2} b = {y : y ∈ ℚ, y > 2} la proprietà (1) è verificata; infatti da x ∈ a ed y ∈ b segue x < 2 < y, cioè x < y. le proprietà (2) e (3) si possono provare ma non sono immediate. vale invece la pena osservare che, supposto che esista un elemento separatore c, non potrà mai risultare c < 2 oppure c > 2; quindi dovrebbe essere c = 2 con c razionale: assurdo! di conseguenza la coppia (a, b) è una coppia di classi contigue che non ammette elemento separatore. dicesi numero reale una qualsiasi coppia di classi contigue. se la coppia di classi contigue (a, b) ammette un elemento separatore c (razionale) allora il numero reale (a, b) si dice razionale e si identifica con l'elemento separatore della classe stessa, ponendo: c = (a, b) se la coppia di classi contigue(a, b) non ammette elemento separatore, allora il numero reale (a, b) si pensa come un nuovo oggetto, che amplia il campo razionale e viene detto numero irrazionale. ora è facile convincersi che tra l’idea di numero razionale/irrazionale introdotto come allineamento decimale e l’idea di numero razionale/irrazionale introdotto come coppia di classi contigue vi è un profondo legame come provato dal ragionamento che segue e dai successivi esempi. sia dato un numero reale rappresentato da: a,a1a2a3 … dove a è un intero e a1, a2, a3, …. sono cifre tra 0 e 9. si suppone inoltre esclusa una rappresentazione eventuale con infiniti 9 sostituita da quella corrispondente con infiniti zeri. se si considera l’insieme a dei razionali del tipo: a; a,a1; a,a1a2; a,a1a2a3; ……………… e l’insieme b dei razionali del tipo: a + 1; a,(a1 + 1); a,a1(a2 + 1); a,a1a2(a3 + 1); ……………… la coppia (a, b) costituisce una coppia di classi contigue che definisce un numero irrazionale. considerato, come esempio, q = 0,101001000100001 …… chiaramente non periodico, esso definisce un numero irrazionale che può ritenersi individuato dalle classi: 0; 0,1; 0,10; 0,101; 0,1010; ………… 0; 1 10 ; 1 10 ; 101 1000 ; 1010 10000 ; ………… 1; 0,2; 0,11; 0,102; 0,1011; ………… 1; 2 10 ; 11 100 ; 102 1000 ; 1011 10000 ; ………… si può dimostrare che le classi (a, b) sono contigue. nota storica. l'espressione “numerus irrationalis” è stata introdotta da gerardo da cremona (1114-1187); altri contributi nel lento progredire della teoria relativa agli irrazionali si hanno da parte di luca pacioli (sec. xv), m. stiefel, cardano e stevin (xvi). le classi contigue sono state introdotte da capelli (1897) e la sistemazione rigorosa è dovuta a weierstrass, dedekind e peano in pubblicazioni più o meno contemporanee e più o meno equivalenti. nel linguaggio comune la parola irrazionale fa pensare a qualcosa oltre la ragione e questo termine allora, imposto dalla storia e dall'uso, sta ad indicare la meraviglia dei greci all'atto della loro scoperta. appendice 2. comunque le regole usate per giungere a tali relazioni sono strettamente connesse alla serie geometrica. se x è un numero razionale (o anche reale) tale che 0 < x < 1, allora vale la seguente relazione, che noi non proviamo, richiedendo la prova delle conoscenze più avanzate di quelle possedute al momento attuale: 2 3 0 1 1 ... ... 1 n k k x x x x x x ∞ = + + + + + + = = −∑ mediante la formula sopra riportata si ha immediatamente: ( ) ( ) ( ) 0 9 0 9 0 09 0 009 9 10 9 10 9 10 9 10 1 1 10 1 1002 3 , , , , ... ...= + + + = + + + = + + +⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ... = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ⋅ − = ⋅ = = ∞ ∑ 9 10 1 10 9 10 1 1 1 10 9 10 10 9 1 0 k k la relazione "0,(9) = 1" porta di fatto ad identificare 0,(9) con 1,(0) e quindi ad abolire quella scrittura che può essere chiamata "scrittura degli infiniti nove …". come conseguenza, ad esempio, è immediato verificare che risulta: a: b: 12,31(9) = 12,32 infatti: ( ) ( ) ( )12, 31 9 12, 31 0, 00 9 12, 31 0, 01 12, 32 0= + = + = essendo appunto: 0,(9) = 1 ( )1 10, 09 0, 9 0,1 10 10 = = = ( )1 10, 009 0, 9 0, 01 100 100 = = = …………. appendice 3. − quanto sono infiniti i numeri irrazionali? − un insieme si dice numerabile quando non lo si può porre in corrispondenza biunivoca con i numeri naturali. proviamo che i numeri reali – anzi solo quelli tra 0 ed 1 − non si possono numerare! supponiamo, per assurdo di averlo fatto, allora avremo l’elenco che segue: 0, 77158964110 0, 3411080076 0, 561108065 … ……….. il numero 0,abcd … dove a diverso da 7, b diverso da 4, c diverso da 1, … non è compreso tra quelli indicati nell’elenco. un assurdo! dunque i reali non sono numerabili! esercizi. 1. − (svolto) − calcolare, con l'uso della serie geometrica, la frazione generatrice di 12,1(2). si ha: ( )12 1 2 12 1 0 02 0 002 121 10 2 100 2 1000 121 10 2 100 1 1 10 1 100 , , , ,= + + + + + + = + + + +⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ... = ... ... = + − ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ = + ⋅ = + = ⋅ + = = −121 10 2 100 1 1 1 10 121 10 2 100 10 9 121 10 2 90 121 9 2 90 1091 90 1212 121 90 2. − osservazione − utilizzando l’appendice 2 si possono desumere le regole seguenti: a) parte intera, antiperiodo(0) = parteinteraantiperiodo/1seguitoda tanti 0quanto periodo; b) parte intera, antiperiodo(periodo) = = parteinteraantiperiodoperiodo parteinteraantiperiodo/tanti 9 quanto periodotanti 0quanto antiperiodo; c) parte intera, (periodo) = parteinteraperiodo − parteintera/tanti 9 quanto periodo 3. − applicare le regole 2 ai tre casi seguenti − 1,34; 12,1(2); 12,31(9) 4. − verificare i calcoli nel seguente ragionamento − dato il numero decimale come ad esempio 1,34 la sua frazione generatrice può essere calcolata in due modi. uno semplice è quello di scrivere 134 100 ! (regola a)). un secondo modo, più complicato, è quello di leggere 1,34 = 1,34(0) come un numero dotato di antiperiodo 34 e periodo zero. occorre allora calcolare: 1340 134 900 − (regola b)). 5. − verificare le seguenti eguaglianze − si noti che se nel numeratore compaiono solo 2 ovvero 5 o entrambi il numero è decimale, se non compare 2 e nemmeno 5 allora è periodico semplice altrimenti periodico misto! 1 0, 5 2 = ; 6 1 1 0, 015625 64 2 = = ; 1 0, 7 3 = ; 1 0, (142857) 7 = ; 1 0, (1) 9 = ; 1 0, (09) 11 = ; 1 0,(0588235294117647) 17 = : si osservi che, in tal caso, il periodo è lungo 16!; 1 0,(012345679) 81 = : si osservi che, in tal caso, tutte le cifre, ad eccezione del numero 8, sono nel periodo!; 1 0,1(6) 6 = ; 1 0,0(714285) 14 = 6. − nota − la lunghezza massima del periodo del numero 1 n è data da n – 1 cifre, come si può vedere per 1 7 che ha 6 cifre di periodo e per 1 17 che presenta 16 cifre (cfr. esercizio 5). si sa ad esempio che : 1 97 ha un periodo di 96 cifre, le cui prime sono: 1 0,(010309278350515...) 97 = 1 0,0005373... 1861 = ha 1860 cifre decimali mentre 1 17389 ne ha 17388. calcolare ora altre cifre di 1 97 e 1 1861 ! (si noti che 97 e 1861 sono primi, per cui le due frazioni hanno periodicità semplice!) 7. − verificare le seguenti eguaglianze scritte nel papiro rhind: (nell’antichità vi era la tendenza ad usare solo frazioni dell’unità con l’unica eccezione di 2 3 , utilizzata dagli egiziani) 2 1 1 7 4 28 = + ; 2 1 1 11 6 66 = + ; 2 1 1 1 97 56 679 776 = + + 7. − verificare le identità seguenti − 2 1 1 7 4 28 = + 2 1 1 1 97 56 679 776 = + + 1 1 3 1 3 5 1 3 5 7 ... 3 5 7 7 9 11 9 11 13 15 + + + + + + = = = = + + + + + + 8. − esercizio facoltativo − sapendo che, per definizione, valgono le seguenti relazioni: 0! = 1! =1, 2! = 2 × 1; 3! = 3 × 2 × 1 = 6; 4! = 4 × 3 × 2 × 1 = 24 … n! = n × (n − 1) × (n − 2) × … 2 × 1 (il simbolo n! fu introdotto nel 1808 dal tedesco cristian kramp in segno di stupore per la rapidità di crescita) eseguire qualcuno dei calcoli seguenti: n! + 1 è un quadrato per n = 4, 5, 7. per n = 22, 23, 24, n! ha n cifre. 10! = 6! × 7!; 145 = 1! + 4! + 5!. 40585 = 4! + 0! + 5! + 8! + 5!. 9. − rispondere seguendo il consiglio − con quali numeri può terminare il quadrato di un numero intero? (si considerino i quadrati delle 10 cifre: 0, 1, 2, …, 9 cioè 0, 1, 4, 9, 16, 25, 36, 49, 64, 81). 10. − un numero che termina con 2, 3, 7, 8 può essere un quadrato perfetto? ed un numero che termina con 21? 11. − tra due interi cade almeno un razionale? (dove cade la media della somma, il terzo della somma, il quarto della somma e così via …?) 12. − tra due numeri razionali cade sempre almeno un irrazionale! leggere almeno l’esempio che segue! tra 0,(1234) … e 0,(123513) cade ad esempio 0,1234567891011… (tutti i numeri a partire da 5) che è certamente minore del secondo! 13. − tra due irrazionali cade almeno un razionale (semplice!). 14. − può accadere che la radice quadrata di un numero sia maggiore del numero dato? se si, fare un esempio. 15. − dovendo estrarre la radice quadrata di 2500 si può estrarre la radice di 25 e poi aggiungere uno zero? 16. − risulta 2 = 1,414213562...= quanto vale il quadrato di 1,414213562? perché è meno di 2? 17. − dire tra quali interi sono comprese le radici quadrate di 21, 42, 73, 109? e le radici cubiche? 18. − portare fuori dal segno di radice uno o più fattori: esempio: 2773493147 2 =⋅=⋅= applicare l’idea alle seguenti radici e ad altre simili: 28 ; 3 192 ; 4 162 19. − calcolare il prodotto: 2 15 2 15 + ⋅ − ! può dirsi che i due numeri sono uno l’inverso dell’altro? curiosità ed osservazioni 1. − il lettore si convinca che le relazioni sono giuste nonostante le semplificazioni errate! 16 1 64 4 = (ma non perché si semplifica il 6!!!); 19 1 95 5 = ; 26 2 65 5 = ; 49 4 98 8 = ; 16666 1 66664 4 = ; 3544 344 7531 731 = ; 143185 1435 17018560 170560 = 2. − il numero irrazionale log22, le cui prime cifre sono log22 = 0,30102 99956 63981 …, moltiplicato per una potenza del 2 ne fornisce, in parte intera per eccesso, il numero di cifre decimali. 2127 ha 39 cifre decimali perché 127 × 0,301 = 38,227! 3. − il 27mo numero di mersenne, scoperto da gillies presso l’università dell’illinois nel 1963, fu celebrato con un famoso annullo postale con la scritta "211213 – 1 is a prime", … 4. − il numero 2 11213 – 1 ha 3376 cifre decimali poiché 11213 × 0,301 = 3375,773! (oggi il primato del mondo dei numeri di mersenne è un numero con circa 4 milioni di cifre!) 5. − la sezione aurea può ottenesi da somme infinite φ = λ.1111 ++++ φ = 1 + λ+ + + + 1 1 1 1 1 1 1 1 le cui somme parziali sono : 3 1 2 1, 5 2 − − = ; 5 1, (6) 3 = . 6. − i numeri π ed e possono ottenesi da : 2 2 4 4 2 2 1 1 1 1 1 2 4 1 ... 1 3 3 5 2 1 2 1 3 5 7 9 11 a a a a π ⎛ ⎞ ⎛ ⎞= × × × × × × × × = × − + − + − +⎜ ⎟ ⎜ ⎟− +⎝ ⎠ ⎝ ⎠ l l l 2 + 1 1 1 1 1 2 1 ... ... 1 1! 2! 3! !1 2 2 3 3 4 4 5 e n = + = + + + + + + + + + +l l 7. − 878 2, 71826 323 = è la migliore approssimazione di e con un rapportodi tre cifre (entrambi palindromici e di differenza 555), mentre 87 2, 7187 32 = 87/32 = 2,7187 è la migliore approssimazione di e con un rapporto di due cifre. 8. − il numero n! = n × (n − 1) × … 3 × 2 × 1 cresce molto rapidamente: 20! = 243290200817664000 e con buona approssimazione è dato da: ! 2n nn n e nπ−≈ 9. − secondo varie teorie gli atomi dell’universo sono tra 1080 e 1087. 10. − le miriadi secondo archimede sono numeri dell’ordine di 104. 11. − tutte le volte che 1 n ha periodo massimo n − 1 le cifre decimali dei numeri dopo la virgola possono essere ordinate in un quadrato (n − 1) × (n − 1)18 che scritto interamente costituisce un quadrato semi−magico, nel senso che la somma delle righe e delle colonne è una costante. l’esempio di 1 7 , 2 7 , …, 6 7 seguente è ben chiaro. la somma delle righe e delle colonne è sempre 27. la somma della diagonale principale è 31 e quella della diagonale secondaria è 23. 1 = 0,(142857) 7 2 = 0,(285714) 7 3 = 0,(428571) 7 4 = 0,(571428) 7 5 = 0,(714285) 7 6 = 0,(857142) 7 se scriviamo i numeri che vanno da 1 19 a 18 19 , ciascuno di essi ha periodo massimo lungo 18. le cifre dopo la virgola possono essere ordinate in un quadrato 18 × 18 (del quale riportiamo le prime 7 righe) che scritto interamente costituisce un quadrato magico proprio. la proprietà è che la somma delle righe, quella delle colonne e delle due diagonali è una costante, che in questo caso vale esattamente 81. 1 = 0, (052631578947368421) 19 18 cifre!!! 2 = 0, (105263157894736842) 19 3 = 0, (157894736842105263) 19 4 = 0, (210526315789473684) 19 5 = 0, (263157894736842105) 19 6 = 0, (315789473684210526) 19 7 = 0, (368421052631578947) 19 …………………………………. …. e così via fino a 18 19 . f.eugeni-d.tondini-a.viceconte www.eiris.it costante 27 ratio mathematica volume 46, 2023 some results on range labeling j. senthamizh selvan* r. jahir hussain† abstract let g=( , ) be a graph with finite and simple . let (g) and (g) be the vertex set and edge set of g respectively. a range labeling of a graph g is an injective function. (g) →{1, 2. . . .} such that the edge labeling α∗ : (g) → {1, 2. . . ..} is defined by α[]= maximum value ( ) minimum value (). a graph which admits such labeling is called a range graph. in this paper the range labeling is introduced and range labeling for a some trees as for star tree, spider tree and banana tree are calculated. keywords: graph; labeling; graceful labeling; range labeling; star trees; spider trees; banana trees. 2020 ams subject classifications: 05c78 1 *department of mathematics (jamal mohamed college (autonomous), bharathidasan university, thiruchirappalli-620020, tamil nadu, india); senthamizh16@gmail.com †department of mathematics (jamal mohamed college (autonomous), bharathidasan university, thiruchirappalli-620020, tamil nadu, india); hssnjhr@yahoo.com 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1080 issn: 1592-7415. eissn: 2282-8214. ©j. senthamizh selvan et al. this paper is published under the cc-by licence agreement. 236 j. senthamizh selvan and r. jahir hussain 1 introduction graph labeling is an assignment of set of integers to the set of vertices, edges or both based on certain conditions, in 1967 rosa was first introduced the graph labeling. a graph labeling are useful family of mathematical models applied in many areas such as radar, missile guidance, radio frequency modulation and many more. a graph labeling is one topic in graph theory. so many kind of graph labeling among others: magic labeling, graceful labeling afsana ahmed munia [2014], o edge magic labeling, 1edge magic labeling, mean labeling, and etc. every year and updated survey comes about various types of labeling by j. a. gallian. from the survey, various types of labeling analyzed and introduced a new type of labeling called range labeling moinin al aziz md [2014]. in this paper are to proof that some trees namely star tree, spider tree, and banana tree admit range labeling. this article assumed for all graphs are finite and simple. the graph g = (ψ,τ) where ψ(g) set of vertices and τ(g) set of edges. a labeling is a most one of the part of the graph theory, r.uma [2012]. a labeling is the assignment of labels, traditionary defined by integers to edges or vertices or both of vertices and edges moinin al aziz md [2014]. the origin of labeling is rosa by 1967. r. jahir hussain [2022] and afsana ahmed munia [2014] was first developed range labeling in 2022. the article range labeling apply for star trees, spider trees, banana tress a. n.mohamed [2013]. 2 preliminary definition 2.1. r.uma [2012] a graceful labeling of a graph g is a vertex labeling α : ψ → [0,m] such that α is injective and induced mapping α(τ) = |α(ψk) −α(ψk+1)|, for every ψkψk+1 ∈ τ(g). assigns different labels to different edges of g. the differences |α(ψk)−α(ψk+1)| is labeled weight of the edges ψkψk+1. a graph g is called graceful. definition 2.2. let g = (ψ,τ) be a graph with n vertices. a bijection on α : ψ → {1, 3, 6, 10, 15, ........n 2+n 2 } is called a range labeling if for each edge τ is distinct and τ is defined by α∗(τ) = maximum value (ψk, ψk+1)− minimum value (ψk, ψk+1). definition 2.3. a tree for 1-internal vertex and k edges is called star s1,k that appear to be complete by bipartite graph k1,k. 237 some results on range labeling definition 2.4. a spider tree with atmost one vertex of degree greater than 2. definition 2.5. an (m,t)-banana tree is a graph attained by attaching 1-edge of all m copies of an t-star graph for a 1-root vertex is different from each stars. 3 main results theorem 3.1. all star tree take a range labeling. proof. let g = (ψ,τ), be a graph for ψ1 is an interval vertex, 10 edges. consider, α : ψ →{1, 3, 6, 10, 15, ........n 2+n 2 } α∗(τ) = maximum value (ψk, ψk+1)− minimum value (ψk, ψk+1) α∗(τ1) = maximum value (ψ1, ψ2)− minimum value (ψ1, ψ2) if ψ1 is a maximum value, ψ2 is a minimum value. α∗(τ1) = (ψ1 − ψ2). = 1 is an integer. suppose ψ2 is a maximum value, ψ1 is a minimum value. α∗(τ1) = (ψ2 − ψ1). = 1 is an integer. α∗(τ2) = maximum value (ψ1, ψ3)− minimum value (ψ1, ψ3) if ψ1 is a maximum value, ψ3 is a minimum value. α∗(τ2) = (ψ1 − ψ3). = 1 is an integer. suppose ψ3 is a maximum value, ψ1 is a minimum value. α∗(τ2) = (ψ3 − ψ1). = 1 is an integer. α∗(τ9) = maximum value (ψ1, ψ10)− minimum value (ψ1, ψ10) if ψ1 is a maximum value, ψ10 is a minimum value. α∗(τ9) = (ψ1 − ψ10). = 1 is an integer. suppose ψ10 is a maximum value, ψ1 is a minimum value. α∗(τ9) = (ψ10 − ψ1). = 1 is an integer. α∗(τ10) = maximum value (ψ1, ψ11)− minimum value (ψ1, ψ11) if ψ1 is a maximum value, ψ11 is a minimum value. α∗(τ10) = (ψ1 − ψ11). = 1 is an integer. suppose ψ11 is a maximum value, ψ1 is a minimum value. α∗(τ10) = (ψ11 − ψ1). 238 j. senthamizh selvan and r. jahir hussain = 1 is an integer. hence, every star tree is a received range labeling therefore, any star tree is a range graph. example 3.1. 1 28 36 4555 66 3 6 10 15 21 27 35 4454 65 2 5 9 14 20 fig.1. range labelling for star tree s10. theorem 3.2. every spider tree take a range labeling. proof. let g = (ψ,τ), be a graph. let α : ψ →{1, 3, 6, 10, 15, ........n 2+n 2 } a spider tree for atmost 1-node of degree greater than 2 and this node is said to be section node and is denoted by ψ0. a stage of a spider graph is a path from the section node to edge of the tree. the edge is denoted by ψ1, ψ2, ψ3. α∗(τ) = maximum value (ψk, ψk+1)− minimum value (ψk, ψk+1) α∗(τ1) = maximum value (ψ0, ψ1)− minimum value (ψ0, ψ1) if ψ0 is a maximum value, ψ1 is a minimum value. α∗(τ1) = (ψ0 − ψ1). = 1 is an integer. suppose ψ1 is a maximum value, ψ0 is a minimum value. α∗(τ1) = (ψ1 − ψ0). = 1 is an integer. 239 some results on range labeling α∗(τ2) = maximum value (ψ0, ψ2)− minimum value (ψ0, ψ2) if ψ0 is a maximum value, ψ2 is a minimum value. α∗(τ2) = (ψ0 − ψ2). = 1 is an integer. suppose ψ2 is a maximum value, ψ0 is a minimum value. α∗(τ2) = (ψ2 − ψ0). = 1 is an integer. α∗(τ3) = maximum value (ψ0, ψ3)− minimum value (ψ0, ψ3) if ψ0 is a maximum value, ψ3 is a minimum value. α∗(τ3) = (ψ0 − ψ3). = 1 is an integer. suppose ψ3 is a maximum value, ψ0 is a minimum value. α∗(τ3) = (ψ3 − ψ0). = 1 is an integer. so, all spider tree accepted range labeling thus, every spider tree is a range graph. example 3.2. 1 10 6 3 9 5 2 fig.2. range labelling for spider tree. 1015 6 1 3 75 2 3 fig.3. range labelling for spider tree. example 3.3. 240 j. senthamizh selvan and r. jahir hussain 6 3 36 1 136 10 15 105 28 91786621 120 5545 2 25 63 77 50 3 9 5 2021 121 34 45 99 15 fig.1. range labelling for banana tree b(3,5). theorem 3.3. all banana tree take a range labeling. proof. let g = (ψ,τ), be a graph. let α : ψ →{1, 3, 6, 10, 15, ........n 2+n 2 } b(3, 5) is a banana tree by attaching 1-edge of every 3 copies (ψ1, ψ2, ψ3) of an 5−star (ψ1., ψ2., ψ3.) graph for 1-root vertex is different from every stars. α∗(τ) = maximum value (ψk, ψk+1)− minimum value (ψk, ψk+1) α∗(τ1) = maximum value (ψ0, ψ1)− minimum value (ψ0, ψ1) if ψ0 is a maximum value, ψ1 is a minimum value. α∗(τ1) = (ψ0 − ψ1). = 1 is an integer. suppose ψ1 is a maximum value, ψ0 is a minimum value. α∗(τ1) = (ψ1 − ψ0). = 1 is an integer. α∗(τ2) = maximum value (ψ0, ψ2)− minimum value (ψ0, ψ2) if ψ2 is a maximum value, ψ0 is a minimum value. α∗(τ2) = (ψ2 − ψ0). = 1 is an integer. suppose ψ0 is a maximum value, ψ2 is a minimum value. α∗(τ2) = (ψ0 − ψ2). = 1 is an integer. α∗(τ3) = maximum value (ψ0, ψ3)− minimum value (ψ0, ψ3) 241 some results on range labeling if ψ0 is a maximum value, ψ3 is a minimum value. α∗(τ3) = (ψ0 − ψ3). = 1 is an integer. suppose ψ3 is a maximum value, ψ0 is a minimum value. α∗(τ3) = (ψ3 − ψ0). = 1 is an integer. α∗(τ4) = maximum value (ψ1, ψ11)− minimum value (ψ1, ψ11) if ψ1 is a maximum value, ψ11 is a minimum value. α∗(τ4) = (ψ1 − ψ11). = 1 is an integer. suppose ψ11 is a maximum value, ψ1 is a minimum value. α∗(τ4) = (ψ11 − ψ1). = 1 is an integer. α∗(τ7) = maximum value (ψ1, ψ14)− minimum value (ψ1, ψ14) if ψ1 is a maximum value, ψ14 is a minimum value. α∗(τ7) = (ψ1 − ψ14). = 1 is an integer. suppose ψ14 is a maximum value, ψ1 is a minimum value. α∗(τ7) = (ψ14 − ψ1). = 1 is an integer. α∗(τ8) = maximum value (ψ2, ψ21)− minimum value (ψ2, ψ21) if ψ2 is a maximum value, ψ21 is a minimum value. α∗(τ8) = (ψ2 − ψ21). = 1 is an integer. suppose ψ21 is a maximum value, ψ2 is a minimum value. α∗(τ8) = (ψ21 − ψ2). = 1 is an integer. α∗(τ11) = maximum value (ψ2, ψ24)− minimum value (ψ2, ψ24) if ψ2 is a maximum value, ψ24 is a minimum value. α∗(τ11) = (ψ2 − ψ24). = 1 is an integer. suppose ψ24 is a maximum value, ψ2 is a minimum value. α∗(τ11) = (ψ24 − ψ2). = 1 is an integer. α∗(τ12) = maximum value (ψ3, ψ31)− minimum value (ψ3, ψ31) if ψ3 is a maximum value, ψ31 is a minimum value. α∗(τ12) = (ψ3 − ψ31). = 1 is an integer. 242 j. senthamizh selvan and r. jahir hussain suppose ψ31 is a maximum value, ψ3 is a minimum value. α∗(τ12) = (ψ31 − ψ3). = 1 is an integer. α∗(τ15) = maximum value (ψ3, ψ34)− minimum value (ψ3, ψ34) if ψ3 is a maximum value, ψ34 is a minimum value. α∗(τ15) = (ψ3 − ψ34). = 1 is an integer. suppose ψ34 is a maximum value, ψ3 is a minimum value. α∗(τ15) = (ψ34 − ψ3). = 1 is an integer. hence, every banana tree received range labeling. therefore, all banana tree is a range labeling. 4 conclusions in this article discussed for some trees received range labeling, so this trees star tree, spider tree, banana tree is also a range graph. further more analysis for this labeling apply for some special graphs. references a. n.mohamed. the combination of spider graphs with star graphs forms graceful. international journal of advanced research in engineering and applied sciences, 2(5), 2013. s. t. m. k. afsana ahmed munia, jannatul marwa. new class of gracefull tree. international journal of scientific and engineering research, 5(2), 2014. m. f. h. moinin al aziz md. graceful labeling of trees, methods and applications. 17th international conference on computer and information technology, 2014. j. s. s. r. jahir hussain. range labeling for some graphs. international journal of advances and application mathematical sciences (accepted), 2022. n. m. r.uma. graceful labeling of some graphs and their subgraphs. asian journal of current engineering and maths, 1, 2012. 243 ratio mathematica connections between ideals of semisimple emv-algebras and set-theoretic filters xiaoxue zhang∗ hongxing liu† abstract in this paper, we mainly study connections between ideals of the semisimple emv-algebra m and filters on some nonempty set ω. we show that there is a bijection between the set of all closed ideals of m and the set of all filters on ω. we get that this correspondence also holds between the set of all closed prime ideals of m and the set of all weak ultrafilters on ω. we prove that the topological space of all closed prime ideals of m and the topological space of all weak ultrafilters on ω are homeomorphic. keywords: semisimple emv-algebra; ideal; filter; closure operation; closed ideal 2020 ams subject classifications: 06d99 1 ∗school of mathematics and statistics, shandong normal university, 250014, jinan, p. r. china; zhangxiaoxuexz@163.com. †school of mathematics and statistics, shandong normal university, 250014, jinan, p. r. china; lhxshanda@163.com. 1 is published under the cc-by licence agreement. received on january 12, 2022. accepted on may 12, 2022. published on september 25, 2022. volume 43, 2022 doi: 10.23755/rm.v43i0.786. issn: 1592-7415. eissn: 2282-8214. © the authors. this paper xiaoxue zhang, hongxing liu 1 introduction an mv-algebra is an algebra (m;⊕,∗, 0) of type (2, 1, 0, 0) which has the top element 1. the study of mv-algebras is very in-depth and comprehensive, which has important applications in other areas of mathematical research. there are close connections between ideals of a semisimple mv-algebra and filters on some associated nonempty set. moreover, there exists a bijection between the set of all closed ideals of a semisimple mv-algebra and the set of all filters on some nonempty set. for more details about it, we recommend the monographs cignoli et al. [2013], lele et al. [2021]. an emv-algebra is an algebra (m;∨,∧,⊕, 0) of type (2, 2, 2, 0), which is a new class of algebraic structures. emv-algebras cannot guarantee the existence of the top element 1, which are the generalizations of mv-algebras. mv-algebras are termwise equivalent to emv-algebras with the top element, dvurečenskij and zahiri [2019]. we shall mainly study connections between ideals of a semisimple emvalgebra m and filters on ω, where m ⊆ [0, 1]ω and [0, 1]ω is an emv-clan of fuzzy functions on some nonempty set ω. this paper is organized as follows. in section 2, we give some basic notions and theorems on emv-algebras, which will be used in the paper. in section 3, we start by introducing the limits of f ∈ m along a filter f on ω. we study the connections between ideals of m and filters on ω. in section 4, we define a closure operation on m. we exhibit a one-to-one correspondence between the set of all closed ideals of m and the set of all filters on ω. we show that there is a homeomorphism between the topological space of all closed prime ideals of m and the topological space of all weak ultrafilters on ω. in addition, there is an example of an ideal that is a non-closed ideal, and some properties of closed ideals are listed. 2 preliminaries in this section, we introduce some basic notions and theorems on an emvalgebra, which will be used in the following sections. a filter f on a nonempty set ω is a collection of subsets of ω satisfying (i) the intersection of two elements in f again belongs to it and (ii) for all s ∈ f , s ⊆ t ⊆ ω implies that t ∈ f . by (ii), we have ω ∈ f for any filter f on ω. a filter f is called proper if ∅ /∈ f . it is obvious that if f1 and f2 are filters on ω, f1 ∩ f2 is also a filter of ω. in fact, for all s1,s2 ∈ f1 ∩ f2, we get s1 ∩ s2 ∈ f1 ∩ f2. moreover, for any s ∈ f1 ∩ f2 and s ⊆ t ⊆ ω, which implies t ∈ f1 and t ∈ f2. so t ∈ f1 ∩f2. we have shown that f1 ∩f2 is a filter on ω. connections between ideals of semisimple emv-algebras and set-theoretic filters definition 2.1. ([cignoli et al., 2013, definition 1.1.1]) an mv-algebra is an algebra (m;⊕,∗, 0, 1) of type (2, 1, 0, 0) such that (m;⊕, 0) is a commutative monoid, and for all x,y ∈ m satisfying the following axioms: (mv1) x∗∗ = x; (mv2) x⊕ 0∗ = 0∗; (mv3) (x∗ ⊕y)∗ ⊕y = (y∗ ⊕x)∗ ⊕x. for all x,y ∈ [0, 1], the real interval [0, 1] with the operations x⊕y = min{x+ y, 1} and x∗ = 1 −x is an mv-algebra. let (m; +, 0) be a monoid. an element a ∈ m is called idempotent if it satisfies the equation a + a = a. we denote the set of all idempotent elements of m by i(m). we recommend cignoli et al. [2013] for mv-algebras. emv-algebras as the generalizations of mv-algebras have many important properties. we recommend dvurečenskij and zahiri [2019] for emv-algebras. definition 2.2. ([dvurečenskij and zahiri, 2019, definition 3.1]) an emv-algebra is an algebra (m;∨,∧,⊕, 0) with type (2, 2, 2, 0) satisfying the followings: (emv1) (m;∨,∧, 0) is a distributive lattice with the least element 0; (emv2) (m;⊕, 0) is a commutative ordered monoid with the neutral element 0; (emv3) for all a,b ∈ i(m) with a ≤ b and for each x ∈ [a,b], the element λa,b(x) = min{y ∈ [a,b] | x⊕y = b} exists in m, and ([a,b];⊕,λa,b,a,b) is an mv-algebra; (emv4) for any x ∈ m, there is a ∈i(m) such that x ≤ a. emv-algebras cannot guarantee the existence of the top element 1. an ideal i of an emv-algebra m is a nonempty subset satisfying (i) for all x,y ∈ i, x⊕y ∈ i and (ii) for each y ∈ i and x ∈ m, x ≤ y can deduce x ∈ i. let ideal(m) to denote the set of all ideals of m. an ideal i of m is proper if i 6= m. a proper ideal i is called prime if for any x,y ∈ m, x ∧ y ∈ i implies that x ∈ i or y ∈ i. we use p(m) to denote the set of all prime ideals of m. an ideal i of m is maximal if for all x ∈ m\i, we have 〈i ∪{x}〉 = m, where 〈i ∪{x}〉 = {z ∈ m | z ≤ a⊕n.x for some a ∈ i and some n ∈ n}. the set of all maximal ideals of m is denoted by maxi(m). it is well known that any maximal ideal of m must be prime ([dvurečenskij and zahiri, 2019]). an emv-algebra m is semisimple if and only if rad(m) = {0}, where rad(m) , ∩{i | i ∈ maxi(m)}. the set rad(m) is called the radical of m. for two emv-algebras (m1;∨,∧,⊕, 0) and (m2;∨,∧,⊕, 0), a mapping φ : m1 −→ m2 is called an emv-homomorphism if φ preserves the operations ∨,∧,⊕ and 0, and for each b ∈i(m1) and for each x ∈ [0,b], we have φ(λb(x)) = λφ(b)(φ(x)). every mv-homomorphism is also an emv-homomorphism, but the converse is not necessarily true ([dvurečenskij and zahiri, 2019]). a mapping s : m −→ [0, 1] is said a state-morphism on m if s is an emv-homomorphism from xiaoxue zhang, hongxing liu the emv-algebra m into the emv-algebra of the real interval ([0, 1];∨,∧,⊕, 0) with top element, such that there exists an element x ∈ m with s(x) = 1. the set ker(s) = {x ∈ m | s(x) = 0} is called the kernel of the state-morphism s ([dvurečenskij and zahiri, 2019]). theorem 2.1. ([dvurečenskij and zahiri, 2019, theorem 4.2 (ii)]) let m be an emv-algebra and s be a state-morphism on m. then ker(s) is a maximal ideal of m. in addition, there is a unique maximal ideal i of m such that s = si, where si : x 7−→ x/i for all x ∈ m. definition 2.3. ([dvurečenskij and zahiri, 2019, definition 4.9]) let ω be a nonempty set. a system t ⊆ [0, 1]ω is called an emv-clan if it satisfies the following conditions: (1) 0 ∈ t such that 0(w) = 0 for all w ∈ ω; (2) if a ∈ t is a 0-1-valued function, then a − f ∈ t for each f ∈ t with f(w) ≤ a(w) for all w ∈ ω, and if f,g ∈ t with f(w),g(w) ≤ a(w) for all w ∈ ω, then f ⊕ g ∈ t , where (f ⊕ g)(w) = min{f(w) + g(w),a(w)} for all w ∈ ω; (3) for each f ∈ t , there exists a 0-1-valued function a ∈ t such that f(w) ≤ a(w) for all w ∈ ω; (4) for given w ∈ ω, there exists f ∈ t such that f(w) = 1. from dvurečenskij and zahiri [2019, proposition 4.10], we see that any emvclan can be organized into an emv-algebra. that is, every emv-clan on some ω 6= ∅ is an emv-algebra, see dvurečenskij and zahiri [2019]. 3 ideals of semisimple emv-algebras and filters on associated nonempty sets let m be a semisimple emv-algebra. by dvurečenskij and zahiri [2019, theorem 4.11], there is an emv-clan [0, 1]ω on some ω 6= ∅ such that m is an emv-subalgebra of [0, 1]ω. in this section, for a semisimple emv-algebra m ⊆ [0, 1]ω, we shall define the notion of limits along a filter. the connections between ideals of m and filters on ω are studied. for each f ∈ m and for all ε > 0, we denote d(f,ε) = {x ∈ ω | f(x) < ε}. definition 3.1. let m be a semisimple emv-algebra and f be a filter on ω such that m ⊆ [0, 1]ω. for any f ∈ m and t ∈ [0, 1], we call that f converges to t along f if for every ε > 0, there is s ∈ f such that | f(s) − t |< ε. proposition 3.1. let m be a semisimple emv-algebra and f be a proper filter on ω such that m ⊆ [0, 1]ω. then for each f ∈ m, there has at most one limit along f. connections between ideals of semisimple emv-algebras and set-theoretic filters proof. the proof is similar to lele et al. [2021, proposition 2.2].2 for any f ∈ m, the limit of f along a proper filter f on ω does not necessarily exist. but it would be unique if it exists by proposition 3.1. we denote it by limff. let i be an ideal of m and f be a filter on ω. we define fi = {s ⊆ ω | d(f,ε) ⊆ s for some f ∈ i and ε > 0} and if = {f ∈ m | f converges to 0 along f}={f ∈ m | d(f,ε) ∈ f for all ε > 0}. proposition 3.2. let m be a semisimple emv-algebra and f be a filter on ω such that m ⊆ [0, 1]ω. for all f,g ∈ m: (1) if limff and limfg exist, then limf (f⊕g) exists and limf (f⊕g) = limff⊕ limfg. (2) if limff exists, then limfλa(f) exists and limfλa(f) = λa(limff), where a is an idempotent element of m such that f ∈ [0,a]. proof. (1) suppose that f,g ∈ m, limff and limfg exist. there exists an idempotent element a ∈ i(m) such that f,g ∈ [0,a]. also, we have limff,limfg ≤ a(x) for all x ∈ ω. in the mv-algebra ([0,a];⊕,λa, 0,a), limff and limfg also exist. by lele et al. [2021, lemma 2.4], we have limf (f ⊕ g) exists and limf (f ⊕g) = limff ⊕ limfg. (2) recall that λa(f) = min{z ∈ [0,a] | z ⊕ f = a}, where a ∈ i(m) with f ∈ [0,a]. since ([0,a];⊕,λa, 0,a) is an mv-algebra, the result follows from lele et al. [2021, lemma 2.4].2 recall that an ultrafilter u on ω is a filter which is maximal, in other words, any filter that contains it is equal to it. an ultrafilter u on ω is equally a collection of subsets of ω satisfying (i) u is proper, (ii) the intersection of two subsets in the collection belongs to it and (iii) for any subset v , v ∈ u if and only if ω\v /∈ u, see garner [2020, definition 2]. from (iii), we see that ω ∈ u for any ultrafilter u on ω. we shall show that the limits along an ultrafilter exist. proposition 3.3. let m be a semisimple emv-algebra and u be an ultrafilter on ω such that m ⊆ [0, 1]ω. then, for any f ∈ m, there has a unique limit along u. proof. suppose that there is no t ∈ [0, 1] such that limuf = t. that is, for any t ∈ [0, 1], there exists ε0 > 0 such that f−1(ot) /∈ u, where ot = (t−ε0, t + ε0). in fact, if for all ε > 0, there exists t0 ∈ [0, 1] such that f−1(ot0 ) ∈ u, where ot0 = (t0 − ε,t0 + ε). it follows that limuf = t0, which is a contradiction. since [0, 1] is compact, for each open covering {ot | t ∈ [0, 1]} of [0, 1], where ot = (t − ε,t + ε), there exists a finite subset {ot1,ot2, ......,otn} such that [0, 1] = ⋃n i=1 oti . since u is an ultrafilter on ω, we have ⋃n i=1 f −1(oti ) = xiaoxue zhang, hongxing liu f−1( ⋃n i=1 oti ) = f −1([0, 1]) = ω ∈ u. by garner [2020, definition 2], there is j ∈ {1, 2, ......,n} such that f−1(otj ) ∈ u, which is a contradiction. hence, f has at least one limit along u. by proposition 3.1, the uniqueness of the limit is clear.2 theorem 3.1. let m be a semisimple emv-algebra and u be an ultrafilter on ω such that m ⊆ [0, 1]ω. consider the mapping φu : m −→ [0, 1] given by φu (f) = limuf, where f ∈ m. then φu is an emv-homomorphism with ker(φu ) = iu . proof. let φu : m −→ [0, 1] be a mapping defined by φu (f) = limuf, where f ∈ m. by proposition 3.3, the limit of f along u is unique. so φu is well-defined. for all f,g ∈ m, there is a ∈ i(m) such that f,g ∈ [0,a] and ([0,a];⊕,λa, 0,a) is an mv-algebra. now we consider the restriction of φu on [0,a]. from lele et al. [2021, proposition 2.6] we see that φu |[0,a] is an mvhomomorphism. clearly, φu (0) = 0. also, we have φu (f⊕g) = φu (f)⊕φu (g), φu (f ∨g) = φu (f)∨φu (g) and φu (f ∧g) = φu (f)∧φu (g). that is, φu is an emv-homomorphism. in addition, ker(φu ) = {f ∈ m | limuf = 0} = iu .2 theorem 3.2. let m be a semisimple emv-algebra such that m ⊆ [0, 1]ω. we have the followings: (1) for each ideal i of m, fi is a filter on ω. moreover, if i is proper, then fi is proper. (2) for each filter f on ω, if is an ideal of m. moreover, if f is proper, then if is proper. proof. (1) let i be an ideal of m. (i) for all ε > 0 and f ∈ i, we have d(f,ε) = {x ∈ ω | f(x) < ε} ⊆ ω. then ω ∈ fi. (ii) let s1 ⊆ s2 ⊆ ω and s1 ∈ fi. there exist f ∈ i and ε > 0 such that d(f,ε) ⊆ s1 ⊆ s2. this implies that s2 ∈ fi. (iii) suppose that s1,s2 ∈ fi. there exist f,g ∈ i and ε,δ > 0 such that d(f,ε) ⊆ s1 and d(g,δ) ⊆ s2. it follows that d(f,ε) ∩d(g,δ) ⊆ s1 ∩s2. in addition, since d(f ⊕g, min(ε,δ)) ⊆ d(f,ε) ∩d(g,δ) and f ⊕g ∈ i, we have d(f,ε)∩d(g,δ) ∈ fi. by (ii), it now follows that s1∩s2 ∈ fi. so fi is a filter on ω. let i be a proper ideal. suppose that fi is not proper. then ∅ ∈ fi. so there exist f ∈ i and ε > 0 such that f(x) ≥ ε for all x ∈ ω. we choose n ≥ 1 such that f(x) ≥ ε ≥ 1 n . then nf ∈ i and nf(x) ≥ 1. it implies that 1 ∈ i and i = m, which is a contradiction. therefore, fi is proper. (2) let f be a filter on ω. (i) since 0 ∈ if , we have if 6= ∅. connections between ideals of semisimple emv-algebras and set-theoretic filters (ii) for all f,g ∈ if , by proposition 3.2, we have limf (f ⊕ g) = limff ⊕ limfg = 0. so f ⊕g ∈ if . (iii) suppose that f ∈ m, g ∈ if and f ≤ g. we have limff ≤ limfg = 0. then f ∈ if . therefore, if is an ideal of m. let f be a proper filter. if if is not proper, then if = m. for all f ∈ if = m, for all ε > 0, we have d(f,ε) ∈ f . there exists a ∈ i(m) such that f ≤ a and a ∈ m = if . so for any x ∈ ω, there is g(x) > 0 such that a(x) ≥ g(x), where g ∈ [0,a]. it follows that ∅ = d(a,g(x)) ∈ f , which is a contradiction. hence, if is proper.2 proposition 3.4. let m be a semisimple emv-algebra such that m ⊆ [0, 1]ω. then we have the followings: (1) for each ideal i of m, i ⊆ ifi . (2) for each filter f on ω, fif ⊆ f . (3) for each filter f on ω, fif = f if {0, 1} ω ⊆ m. proof. the proof is similar to lele et al. [2021, proposition 2.8].2 proposition 3.5. let m be a semisimple emv-algebra such that m ⊆ [0, 1]ω. we have the followings: (1) if {0, 1}ω ⊆ m, then for each maximal ideal k of m, fk is an ultrafilter on ω. (2) iu is a maximal ideal of m if u is an ultrafilter on ω. (3) if {0, 1}ω ⊆ m, the converse of (2) is true. proof. (1) let k be a maximal ideal of m and s ⊆ ω. suppose s /∈ fk. we will show that ω\s ∈ fk. we define f ∈ m by f(x) = { 0 x ∈ s, 1 x /∈ s. then we have d(f, 0.5) = s /∈ fk. it follows that f /∈ k. let b ∈ i(m) such that f ∈ [0,b]. it follows from f /∈ k that f /∈ kb, where kb = k ∩ [0,b]. since k is a maximal ideal of m, by dvurečenskij and zahiri [2019, proposition 3.22], kb is a maximal ideal of the mv-algebra ([0,b];⊕,λb, 0,b). by the maximality of kb, there exists n ≥ 1 such that λb(nf) ∈ kb. then λb(nf) ∈ k. notice that nf = f, which follows that λb(f) = λb(nf) ∈ k. in addition, we also have ω\s = ω\d(f, 0.5) = d(λb(f), 0.5) ∈ fk. hence, by freiwald [2014, chapter ix, theorem 3.5], fk is an ultrafilter on ω. (2) let u be an ultrafilter on ω. from theorem 3.1, there is an emv-homomorphism φu : m −→ [0, 1] defined by φu (f) = limuf. since m ⊆ [0, 1]ω is semisimple, for given w ∈ ω, there is f ∈ m such that f(w) = 1. so for xiaoxue zhang, hongxing liu {w} ⊆ ω ∈ u and all ε > 0, we have f({w}) ⊆ (1 − ε, 1 + ε), which implies that there exists f ∈ m such that φu (f) = limuf = 1. hence, φu is a state-morphism on m. by theorem 2.1, ker(φu ) = iu is a maximal ideal of m. (3) if iu be a maximal ideal of m. then fiu is an ultrafilter on ω by (1). by proposition 3.4 (3), u = fiu is an ultrafilter.2 proposition 3.6. let m be a semisimple emv-algebra and f be a filter on ω such that {0, 1}ω ⊆ m ⊆ [0, 1]ω. then for any f ∈ m, f is an ultrafilter if and only if f has a unique limit along f. proof. ⇒: if f is an ultrafilter. by proposition 3.3 we see that f has a unique limit along f. ⇐: suppose that f has a unique limit along f , where f ∈ m. consider the mapping φf : m −→ [0, 1] defined by φf (f) = limff. we have that φf is well-defined. by the proof of proposition 3.5, φf is a sate-morphism on m. so ker(φf ) = if is a maximal ideal of m by theorem 2.1. therefore, f is an ultrafilter on ω by proposition 3.5 (3).2 4 closed ideals of semisimple emv-algebras in this section, we introduce the notions of closure operations and c-closed ideals on emv-algebras. we get a bijection between the set of all closed ideals of m and the set of all filters on ω. we exhibit a homeomorphism between the topological space of all closed prime ideals of m and the topological space of all weak ultrafilters on ω. definition 4.1. a closure operation on an emv-algebra m is a mapping c : ideal(m) −→ ideal(m) satisfying the following conditions: for all i,j ∈ ideal(m), (c1) i ⊆ ic; (c2) if i ⊆ j, then ic ⊆ jc; (c3) icc = ic; where ic=c(i). proposition 4.1. let m be a semisimple emv-algebra and m ⊆ [0, 1]ω. for each ideal i of m, we denote ic = ifi . then c is a closure operation on m. proof. the proof is similar to lele et al. [2021, proposition 3.1].2 an ideal i of m is called c-closed if ic = i. we frequently prefer to call an ideal is closed instead of c-closed. the set of all closed ideals of m is denoted by c(m). in the subsequent sections, we shall mainly study closed ideals of m, where the closure operation is given by proposition 4.1. now we show that any maximal ideal must be contained in c(m). connections between ideals of semisimple emv-algebras and set-theoretic filters proposition 4.2. let m be a semisimple emv-algebra and m ⊆ [0, 1]ω. every maximal ideal of m is a closed ideal. proof. let i be a maximal ideal of m. ifi is a proper ideal by theorem 3.2. by proposition 3.4 (1), we have i ⊆ ifi . suppose i & ifi . for any f ∈ ifi\i, by the maximality of i, we have m = 〈i ∪{f}〉⊆ ifi , which is a contradiction. so i = ifi . we have shown that i is closed.2 theorem 4.1. let m be a semisimple emv-algebra such that {0, 1}ω ⊆ m ⊆ [0, 1]ω. then there is a bijection between the set of all closed ideals of m and the set of all filters on ω. proof. let f(ω) to denote the set of all filters on ω. define two mappings: θ : c(m) −→ f(ω) by θ(i) = fi and υ : f(ω) −→c(m) by υ(f) = if . by theorem 3.2 and proposition 3.4(3), θ and υ are well-defined. for any i ∈ c(m) and f ∈ f(ω), we get θυ(f) = θ(if ) = fif = f and υθ(i) = υ(fi) = ifi = i. so θυ and υθ are identical mappings. hence, θ is a bijection.2 remark 4.1. from theorem 4.1, we get a one-to-one correspondence between the set of all closed ideals of m and the set of all filters on ω. we shall study the restriction of this correspondence. we define cm (m) = {i ∈ c(m) | i ∈ maxi(m)} and fu (ω) = {f | f is an ultrafilter on ω}. suppose that {0, 1}ω ⊆ m ⊆ [0, 1]ω. it is easy to verify that there is also a bijection between cm (m) and fu (ω). in fact, define two mappings ψ : fu (ω) −→ cm (m) given by ψ(u) = iu and ψ′ : cm (m) −→ fu (ω) given by ψ′(i) = fi. from proposition 3.4 (3) and proposition 3.5 we see that ψ and ψ′ are well-defined. similar to theorem 4.1, we can prove that ψ is a bijection. next, we will study a special class of filters on ω, which corresponds to closed prime ideals of m. a filter f on ω is called a weak ultrafilter if if is a prime ideal of m. we denote the set of all weak ultrafilters on ω by w(ω). proposition 4.3. let m be a semsimple emv-algebra and m ⊆ [0, 1]ω. every ultrafilter on ω is a weak ultrafilter. proof. let f be an ultrafilter on ω. then if is a maximal ideal of m by proposition 3.5 (2). so if is prime ([dvurečenskij and zahiri, 2019]). hence, f is a weak ultrafilter.2 proposition 4.4. let m be a semisimple emv-algebra and m ⊆ [0, 1]ω. if i is a prime ideal of m, fi is a weak ultrafilter on ω. xiaoxue zhang, hongxing liu proof. let i be a prime ideal of m. then fi is proper. it follows that ifi is a proper ideal by theorem 3.2. suppose that f ∧ g ∈ ifi for f,g ∈ m. we get d(f ∧ g,ε) ∈ fi for all ε > 0. since d(f,ε),d(g,ε) ⊆ d(f ∧ g,ε) ∈ fi, we have that at least one of d(f,ε) and d(g,ε) is nonempty. that is, f ∈ ifi or g ∈ ifi . in fact, suppose that d(f,ε) and d(g,ε) are empty sets. it follows that ∅ = d(f ∧g,ε) ∈ fi, which is a contradiction. we have shown that fi is a weak ultrafilter on ω.2 theorem 4.2. let m be a semisimple emv-algebra such that {0, 1}ω ⊆ m ⊆ [0, 1]ω. then there is a bijection between the set of all closed prime ideals of m and the set of all weak ultrafilters on ω. proof. let pc(m) to denote the set of all closed prime ideals of m. define two mappings: φ : pc(m) −→ w(ω) defined by φ(i) = fi and γ : w(ω) −→ pc(m) defined by γ(f) = if . the mappings φ and γ are well-defined by proposition 4.4, proposition 3.4 (3) and the definition of weak ultrafilters. for any i ∈pc(m) and f ∈ w(ω), we have γφ(i) = γ(fi) = ifi = i and φγ(f) = φ(if ) = fif = f . so φγ and γφ are identical mappings. hence, φ is a bijection.2 lemma 4.1. let m be a semisimple emv-algebra such that m ⊆ [0, 1]ω. then there is a topology on the space w(ω) which has bw , {uw(f) | f ∈ m} as a basis, where uw(f) = {f ∈ w(ω) | f /∈ if} for f ∈ m. proof. for any f ∈ w(ω), there is f ∈ m\if such that f ∈ uw(f) ∈ bw since if is prime. furthermore, for all f,g ∈ m, suppose that f ∈ uw(f) ∩ uw(g). then f /∈ if and g /∈ if . we have f ∧ g /∈ if since if is a prime ideal of m, which follows that uw(f) ∩uw(g) ⊆ uw(f ∧ g). for any f ∈ uw(f ∧ g), we have f ∧ g /∈ if . it implies that d(f ∧ g,ε0) /∈ f for some ε0 > 0. it follows from d(f,ε0),d(g,ε0) ⊆ d(f ∧ g,ε0) /∈ f and f ∈ w(ω) that f /∈ if and g /∈ if . then uw(f ∧g) ⊆uw(f) ∩uw(g). so uw(f ∧g) = uw(f) ∩uw(g). that is, for any f ∈ uw(f) ∩uw(g), there is uw(f ∧ g) ∈ bw such that f ∈ uw(f ∧ g) ⊆ uw(f) ∩uw(g). we have shown that the sets uw(f) form a basis of the topology on w(ω).2 from lemma 4.1, we get a space w(ω) whose topology is the topology generated by bw. the open sets on w(ω) are sets ⋃ uw(f)∈bw′ uw(f), where bw′ ⊆ bw and f ∈ m. when we refer to the topological space w(ω), it will be with reference to the topology { ⋃ uw(f)∈bw′ uw(f) | bw′ ⊆bw} ([munkres, 2000]). connections between ideals of semisimple emv-algebras and set-theoretic filters lemma 4.2. let m be a semisimple emv-algebra and m ⊆ [0, 1]ω. the sets uc(f),f ∈ m form a basis of the topology on pc(m), where uc(f) = {i ∈ pc(m) | f /∈ i} for f ∈ m. proof. we denote bc = {uc(f) | f ∈ m}. for any i ∈ pc(m), there is f ∈ m\i such that i ∈ uc(f) ∈ bc since i is proper. it is obvious that uc(f) ∩uc(g) ⊆ uc(f ∧ g). suppose that i ∈ uc(f ∧ g). then f ∧ g /∈ i = ifi , where f,g ∈ m. similar to lemma 4.1, we have f /∈ ifi = i and g /∈ ifi = i. it implies that uc(f ∧ g) ⊆ uc(f) ∩ uc(g). so uc(f ∧ g) = uc(f) ∩ uc(g). that is, for any i ∈ uc(f) ∩ uc(g), there is uc(f ∧g) ∈bc such that i ∈uc(f ∧g) ⊆uc(f) ∩uc(g). hence, we have shown that bc as the basis of the topology on pc(m).2 by lemma 4.2 and munkres [2000], the topology on pc(m) is the topology generated by bc where the open sets are sets ⋃ uc(f)∈bc′ uc(f), where bc′ ⊆ bc and f ∈ m. theorem 4.3. let m be a semisimple emv-algebra such that {0, 1}ω ⊆ m ⊆ [0, 1]ω. then the two topological spaces pc(m) and w(ω) are homeomorphic. proof. consider the two well-defined bijections φ and γ defined by theorem 4.2. (1) φ is continuous. without lost of generality, we shall prove that the preimage of any uw(f) in w(ω) is open in pc(m). we have φ−1(uw(f)) = γ(uw(f)) = {if | f /∈ if}. for any if ∈ γ(uw(f)), where f ∈ w(ω) and f /∈ if , by proposition 3.4 (3), we have if ∈ pc(m). then if ∈ uc(f). so γ(uw(f)) ⊆ uc(f). moreover, for any i ∈ uc(f), then i ∈ pc(m) and f /∈ i. we have fi ∈ w(ω) and f /∈ i = ifi . it implies that i ∈ γ(uw(f)). so uc(f) ⊆ γ(uw(f)). hence, φ−1(uw(f)) = γ(uw(f)) = uc(f) is an open set in pc(m). (2) γ is continuous. we shall prove γ−1(uc(f)) = uw(f). we have γ−1(uc(f)) = φ(uc(f)) = {fi | f /∈ i}. for any f ∈ uw(f), we get f ∈ w(ω) and f /∈ if . by proposition 3.4 (3), we see that if ∈ pc(m) and f = fif ∈ φ(uc(f)). so uw(f) ⊆ φ(uc(f)). for each fi ∈ φ(uc(f)), where i ∈ pc(m) and f /∈ i = ifi . it follows that fi ∈ uw(f). so φ(uc(f)) ⊆ uw(f). thus γ−1(uc(f)) = φ(uc(f)) = uw(f) is an open set in w(ω). we have shown that φ is a homeomorphism between pc(m) and w(ω).2 example 4.1. there exist non-closed ideals. let m be a semisimple emv-algebra such that m ⊆ [0, 1]ω. suppose that i is an ideal of m. it is obvious that ifi = {f ∈ m | ∀ε > 0,∃δ > 0 and g ∈ i such that g−1([0,δ)) ⊆ f−1([0,ε))}. in fact, for each f ∈ ifi , we have d(f,ε) ∈ fi xiaoxue zhang, hongxing liu for all ε > 0. so there exist g ∈ i and δ > 0 such that d(g,δ) ⊆ d(f,ε). it follows that g−1([0,δ)) ⊆ f−1([0,ε)). let m = [0, 1]z + , where all operations given by definition 2.3 and dvurečenskij and zahiri [2019, proposition 4.10]. let i = {f ∈ m | for all but finitely many n ∈ z+ such that f(n) = 0}. it follows from (f⊕g)(n) = min{f(n)+g(n),a(n)} and simple exercises that i is an ideal of m, where f,g ∈ i and a ∈ m is a 0-1valued function such that f(n),g(n) ≤ a(n) for all n ∈ z+. consider f given by f(n) = n+1 n2+1 (n ∈ z+). clearly, f ∈ m\i. it is easy to see that f(n) → 0 when n → ∞. that is, for all ε > 0, there is n ∈ z+ such that f(n) < ε when n > n. now we consider g ∈ m defined by g(n) = { 1 n 1 ≤ n ≤ n, 0 n > n. then g ∈ i and d(g,δ) ⊆ d(f,ε) for δ = min{ 1 n+1 ,ε}. it implies that g−1([0,δ)) ⊆ f−1([0,ε)). so f ∈ ifi . we have shown that i is a non-closed ideal. definition 4.2. let m be an emv-algebra and i be an ideal of m. then i is called radical if i = rad(m), where rad(m) is the radical of m. proposition 4.5. let m be a semisimple emv-algebra such that m ⊆ [0, 1]ω. the following conditions are satisfied: (1) the intersection of closed ideals of m is also a closed ideal. (2) an ideal i of m is closed if i is radical. proof. (1) let {iα | α ∈ λ} be a family of closed ideals of m. for each β ∈ λ, it follows from ⋂ α∈λ iα ⊆ iβ that ( ⋂ α∈λ iα) c ⊆ iβc = iβ. then ( ⋂ α∈λ iα) c ⊆ ⋂ β∈λ iβ = ⋂ α∈λ iα. since ⋂ α∈λ iα ⊆ ( ⋂ α∈λ iα) c, we have ( ⋂ α∈λ iα) c = ⋂ α∈λ iα. so ⋂ α∈λ iα ∈c(m). (2) suppose that i is radical. it implies that i = ∩{k | k ∈ maxi(m)}. so by proposition 4.2 and (1), i is closed.2 5 conclusion for a semisimple emv-algebra m such that m ⊆ [0, 1]ω, we introduce the notion of limits along a filter on ω, which is unique if it exists. for all ultrafilters u on ω and for all f ∈ m, we give an emv-homomorphism φu with kernel equal to iu , which is defined by φu (f) = limuf. we study connections between ideals of m and filters on ω. we define closure operations and closed ideals on emvalgebras. we show that there is a bijection between the set of all closed ideals of m and the set of all filters on ω. we show that there is a homeomorphism connections between ideals of semisimple emv-algebras and set-theoretic filters between the topological space pc(m) and the topological space w(ω). we give an example of a non-closed ideal and some properties of closed ideals. assume that f is a filter of the proper emv-algebra m and i is an ideal of m. we can show that if = {λa(x) | x ∈ f,a ∈i(m),x ≤ a} is an ideal of m. if f is a maximal filter of m, if is a maximal ideal of m can be proved. we can also get that fi = {λa(x) | x ∈ i,a ∈i(m)\i,x < a} is a filter of m under the assumption that ∀a ∈i(m),a /∈ i =⇒ (∀b ∈i(m),a < b)λb(a) ∈ i. references r. l. cignoli, i. m. d’ottaviano, and d. mundici. algebraic foundations of manyvalued reasoning. springer science & business media, dordrecht, 2013. a. dvurečenskij and o. zahiri. on emv-algebras. fuzzy sets and systems, 373: 116–148, 2019. r. c. freiwald. an introduction to set theory and topology. washington university in st. louis, 2014. r. garner. ultrafilters, finite coproducts and locally connected classifying toposes. annals of pure and applied logic, 171(10):102831, 2020. c. lele, j. b. nganou, and c. m. oumarou. ideals of semisimple mv-algebras and convergence along set-theoretic filters. fuzzy sets and systems, 2021. j. munkres. topology (2nd edition). prentice-hall, inc, london, 2000. cardona5-14.pdf 5 on davidson's problem in the collective risk theory elena cardona* abstract. in this paper davidson's classic problem concerning the solution of an integrodifferential equation regarding the collective risk theory with the aim of examining the probability of the failure of an insurance company is further analysed. the validity of a new representation’s formula to the solution of the problem is demonstrated after having discussed the question of the existence of that solution. keyword. ruin probability. integro -differential equation. 1. introduction. as it is well known, the collective risk theory, introduced by lundberg and subsequently developed by various authors during the last hundred years, has been a fundamental contribution to questions concerning the probability of the failure of an insurance company in a finite time. the usual approach to such problems consists in examining the dynamic over time of the risk reserve’s fund, which the company assigns in the starting time to the management of non-life insurance portfolio with homogeneous policies covering repeatable accidents. the topic will now be briefly reviewed in order that the problem in question can then be discussed. 2. recalls on the collective point of view with reference to the perio d (0,t), +∈ rt , we put w(t) = size at time t of the risk reserve’s fund, which an insurance company above specified assigns to whole portfolio or its part; in particular: w(0) = x. iz = random variable (r.v.) “ company's outlay relative to i-th claim”; * department of mathematics and statistics, university “federico ii”, naples (italy). e-mail: cardona@unin a.it work carried out within the framework of the research project cnr00c91c_006 funded by cnr agenzia 2000. 6 c = flow of the company’s receipts without fund’s yield, assumed constant over time (lundberg's hypothesis, 1903); n(t) = r.v. “number of claims in (0,t)”; s(t) = r.v. “company's total outlays relative to n(t) claims”. therefore ∑ == )( 1 )( tn i i zts (2.1) w t x ct s t( ) ( )= + − (2.2) in the sequel we assume the r.v . zi independent and identically distributed (i.i.d.) with c.d.f. p(z) absolutely continuous, and therefore with p.d.f. p(z) continuous, in + 0r . with these hypotheses and positions, the two factors which determine the net risk premiums flow are: i) the average outlay per claim, given by i dzzp z zdp z ze i ∀===µ ∫∫ +∞+∞ ,)()()( 001 (2.3) ii) the average number e[n(1)] of claims in the unit of time, which, in the hypothesis that n(t) follows poisson distribution with parameter νt = e[n(t)], is given by intensity ν. in the same hypothesis, the r.v. s(t) follows compound-poisson distribution with intensity ν for the arrivals process and its expected value is given by t tse νµ= 1)]([ (2.4) owing to loading on the premiums with mean rate λ>0, the parameters must be fixed in such a way that ) +( = c λνµ 11 (2.5) for the sake of exposition’s simplicity, the unit of time (or operational time if the arrivals process were non-homogeneous) will be chosen in such a way that ν=1. moreover, one will be assumed that the average outlay per claim is the unit of amounts, so µ1 = 1. therefore (2.5), giving µ1 ν = 1, becomes c = 1 + λ (2.5’) 7 in the aforementioned paper [4] davidson dealt with fundamental questions concerning the theory of risk and of ruin in the hypothesis that the safety loading is variable in function of the initial level x of the risk reserve’s fund. he introduces the following values: ψ(u, x) = probability that the fund initially at x level falls below x-u (u≥0); χ(u, v, x)dv = probability that the fund, initially at x level, falls below x-u (u≥0) and that, when it does so for the first time, its value is between x-(u+v) e x-(u+v+dv). therefore, 0≥∀ u it results ψ(u, x) = ( ) vd xv,u, ∫ +∞ χ 0 . that given, davidson analyses all the mutually exclusive events whose probability is χ(u, v, x)dv. by means of differential arguments, he obtains the equation ( ) ( ) ( ) ( )ux,v, x,,u x,v,u v x,v,u u −χχ=χ ∂ ∂ −χ ∂ ∂ 00 (2.6) by evaluating the risk reserve through the various possibilities about the number of claims in (0,t), davidson demonstrates that the proces s is regulated by the following differential equation 0 = (v)p x +)x(v, x -)x,( +)x(v, v +)x(v, x )(1 1 )(1 1 0 λ+ ν      λ+ νν ∂ ∂ ν ∂ ∂ (2.7) where ν(v, x) =χ(0, v, x)dv and being λ(x) the safety loading rate, which is supposed a function of the initial level x of the risk reserve. laurin and lundberg (see [11], [14]) had already obtained (2.7) via other methods. from (2.7) the integro -differential equation in the unknown ψ(u, x) [ ] [ ] + )x,u( )x,u()x,u( )x( xu ψ−ψ+ψλ+1 ( ) dz)z(pzx,zu u ∫ −−ψ 0 + 01 = )u(p + − (2.8) follows. the (2.8), with the positions u-x = ξ (2.9) 8 u)+(u, = ),u( ξψξψ (2.10) becomes [ ∫ ⋅ξψξψξλξψ u 0 u ),z( )(u, u)+(+1 1 = ),u( ] )p(u + 1 -dz )zu(p −⋅ (2.11) that davidson, in the aforementioned work, resolved by means of a procedure based on the theory of integral equations, assumi ng the initial condition )(0, = ξψξψ ),0( (2.12) the (2.11) with the condition (2.12) is often cited in the literature as “davidson's problem”. let us remark that, put )f(u, = u ξξψ 1),( (2.13) (2.11) leads to [ ] = uf )+(u+1 u f u ),(),( ξξλξ ∫ ξ= u 0 zd (z)p )z,-u(f (2.14) with the initial condition 0f = f ≡ξψξ ),0(1),0( (2.15) in which f 0 is a constant suitably assigned . moreover, the assignment of f0 gives rise to some difficulties. really, remembering (2.9) and the meaning of ψ(u,x), if one puts u = x or ξ = 0, ψ(u,x) signifies the asymptotic probability of ruin in proper sense, or rather that the risk reserve, initially at x level, will sooner or later fall to zero. in such a case, let us write ψ in the form ψ(x,x)= ψ*(x) and put f(x) = 1 ψ*(x), that is the asymptotic probability of non-ruin when x is the initial fund. that stated, it results that the constant f0 , which appears in (2.11), supposing a variable loading, must be fixed in such a way as to satisfy the condition: f ( )+∞ = 1 (obviously non-ruin is assured if the initial reserve is infinitely large). due to this problem, a resolving procedure has not yet been found in the case of an infinitely large initial fund. algorithms of asymptotic calculus of the constant f0 can, however, be applied with reference to the similar problem f(k) = 1, for a sufficiently large k (see [2], [10]). 9 3. new thoughts on davidson's problem. let us consider the following integro -differential problem [ ] ( , ) 1 + ( + ) f u u f u u f u z p z dz f f f c r u o ξ λ ξ ∂ ξ ∂ ξ ξ ( , ) ( , ) ( ) ( , ) ( ) = − = ∈      ∫ + 0 1 00 (3.1) where: 0 < f0 < 1, c r 1 0( ) + is the class of continuous functions with continuous derivatives in r0 + e p(z) , p(z) are defined as in § 2. besides, it results: [ ] = 11 0 − ∞ ∫ p z dz( ) taking as amount’s unit the mean outlay per claim. it is known that the integro -differential problem (3.1) admits only one solution. with the parameter ξ fixed, the existence of the solution can be demonstrated by using the successive approximation method, which however allows to find a representation’s formula for the solution, performed in the following § 4. to this aim, we observe that the first of (3.1) yields: ∂ ξ ∂ λ ξ ξ λ ξ ) f u u u f u u f u z p z dz u ( , ) ( ) ( , ( ) ( ) ( )= + + − + + −∫ 1 1 1 1 0 (3.2) which, integrated between 0 and u, becomes: [ ] + d f + f = uf u τ ξ+τλ+ ξτ ξ ∫0 )(1 ),( )0(),( ( )[ ] ∫∫ τ τξ−τ ξ+τλ+ − 00 .)(),( 1 1 d dz zp zf u (3.3) putting: ),0(0 ξf = f (3.4) [ ] + d f f = u f u n n τξ+τλ+ ξτ +ξξ ∫ − 0 1 )(1 ),( ),0(),( 10 [ ]∫ ∫ τ − τξ−τξ+τλ+ − u n d dz zp z f 0 0 1 )(),( )(1 1 ; ,...)3,2,1( =n (3.5) ),(),(),( 1 ξξξβ − uf uf = u nnn ; ,...)3,2,1( =n (3.6) 00 f = β . (3.7) one obtains ),(...),(),(),0(),( 21 ξβ++ξβ+ξβ+ξξ uuuf = uf nn ; ( , , ,... )n = 1 2 3 (3.8) and also [ ] [ ]∫∫ ⋅ ξ+τλ+ −τ ξ+τλ+ ξτβ ξβ + uu n n d = u 00 1 )(1 1 )(1 ),( ),( d dz zp z n∫ τ τξ−τβ 0 )(),( ; ( , , ,... )n = 1 2 3 (3.9) we now prove that: ∫∫ ∫ −τ − τ ⋅ ξ+τλ+ ⋅⋅⋅⋅ ξ+τλ ⋅ ξ+τλ+ ξβ 2 0 10 1 0 )(1 1 ))(1 1 ),( n1 n u 0 2 n (+1 1 f = u [ ] 1121 )(1 τ⋅⋅⋅⋅τ⋅ττ−τ−⋅⋅⋅⋅ − ddd p nn (3.10) dim.: proceeding by induction, for n=1 one obtains β ξ λ τ ξ λ τ ξ τ τ 1 0 10 10 0 1 0 1 1 1 1 ( , ) ( ) ( ) ( )u f f p z dz d u u == + + − + +∫ ∫ ∫ = − + +∫ f p d u 0 1 10 1 1 1 ( ) ( ) τ λ τ ξ τ (3.11) being p(0) = 0. let us now verify that (3.10), supposed to be true for n=k , is also valid for n=k+1. in fact : 11 = d dz zpzd = u k u 1 u k k ∫∫∫ τ + τξ−τβξ+τλ+ −τ ξ+τλ+ ξτβ ξβ 1 0 11 0 10 1 1 1 )(),( )(1 1 )(1 ),( ),( ( ) ( ) ( ) ( ) ⋅ ξ+τλ+ ⋅⋅⋅⋅ ξ+τλξ+τλ+ξ+τλ+ = ∫∫∫∫ −τττ +1 1 f k k u o 121 00 30 21 0 1 1 1 1 1 1 ( )[ ] 11321 τ⋅⋅⋅τττ−τ−⋅⋅⋅ + d d d p kk d d dk kτ τ τ+ ⋅ ⋅ ⋅1 1 − d 1 + ( 0 u 2 2 d p z zτ λ τ ξ τ λ τ ξ τ ττ1 1 0 1 0 2 0 1 1 + + +∫ ∫ ∫∫ − ( ) ( ) ) ........ (3.12) }∫ ∏ ∏ τ + = +++ τττ−τ− ξ+τλ+ τ−k dd p p k i k 2=j kjj i k0 1 3 1311 ...)](1[ )(1 1 )](1[ as a consequence of ( ) ( ) ( )[ ]⋅τ−τ−⋅⋅⋅⋅ ξ+τλ+   ξ+τλ ∫∫∫ ∫ τ + ττ −τ p +1 1 zp 1-k1 1 o kk oo z o 2 1 3 1 1 1 )( 2 ( )[ ] ( )[ ] =τ    τ⋅⋅⋅⋅τ⋅ττ−τ−⋅⋅⋅⋅τ−τ−⋅ +− dz d d d d p p kkkk 231321 11 ( ) ( ) ( )[ ]⋅τ−τ−⋅⋅⋅⋅ ξ+τλ   ξ+τλ+= ∫∫ ∫ −τ + τ τ p +1 1 k o kk 3 11 2 1 0 02 1 1 1 ( )[ ] ( )[ ] ∫ τ−τ +− τ    τ⋅⋅⋅⋅ττ−τ−⋅⋅⋅⋅τ−τ−⋅ 21 0 231321 )(11 d dz zp d d p p kkk (3.13) we obtain {b u x f p t l t xk k tktku i i k + = += + + ⋅∫∫∫ ∏1 0 1000 1 1 1 1 1 ( , ) .... [ ( )] ( ) + . ∏ +τ−τ− k 2=j jj p )(1[ 1 +τττ + 11 ....dd d] kk )]∫ ∫ ∫ τ τ τ u 0 0 0 1+k 1 k (p-[1....∏ += ξ+τλ+τ−τ 1 121 )(1 1 )( k i i . p }=ττττ−τ− ++=∏ 1112 ....)](1[ dd d p . kkjj k j ( ) ( ) ( ) ⋅⋅⋅ ξ+τλξ+τλξ+τλ = ∫∫∫ ττ +1 1 +1 1 +1 1 f o 32 u o 1 o 21 0 ( )[ ] d d d ttp kk k 1=j jj 111 1 τ⋅⋅⋅⋅ττ− ++∏ . (3.14) 12 let us now prove ( ) ( ),...3,2,1 ! , 0 =≤ξβ n n u f u n n (3.15) dim: proceeding also here by induction’s process, owing to (3.11) it results ( ) ( ) β ξ λ τ ξ τ1 0 1 1 0 1 1 u f d f u, 0 u ≤ + + ≤∫ (3.16) moreover (3.15), supposed to be true for n=k , is also valid for n=k +1. in fact, we obtain ( ) ( ) d k ! f d u o ku o k k 1 1 01 1 1 1 1 , τ τ ≤τ ξ+τλ+ ξτβ ≤β ∫∫+ ( )! k u f 1+k 1 0 + = because of (3.15), the series β ξn n u = ∞ ∑ 1 ( , ) for each fixed ξ is absolutely and uniformly convergent in every limited interval +⊂ 0r i and therefore the succession { } nnn uf ∈ξ ),( converges uniformly to a function ),( uf ξ . considering now in (3.5) the limit for ∞→n , one obtains ( ) ( ) ( ) ⋅ ξ+τλ+ −τ ξ+τλ+ τ =ξ ∫∫ d f +f uf uu 00 1 1 1 )( )0(, ( ) τξ−τ∫ τ d dz zp zf o )(, (3.17) then the existence of the solution of (3.1) (davidson's problem) is proved. about the uniqueness of the solution, see [5]. 4. on the representation of the solution to davidson's problem. a representation’s formula for the function f(u,ξ) of problem (3.1) and therefore for the asymptotic probability of non-ruin will now be evaluated. substituting (3.10) into (3.8) we obtain: 13 ( ) [ ( ) ( ){∂ ∂ ξ λ τ ξ τ u f u f p u , = + + + − +∫0 10 11 1 1 1 + ⋅∫∑ = 0 1 2 τ k n ( ) [ ] ⋅ + += − ∏∫ i k i k kp2 0 1 1 1 1 . λ τ ξ τ τ ( ) ( )[ ].. j=1k -1 ∏ − − +1 1p j jτ τ } ] d d dkτ τ τ⋅ ⋅ ⋅ ⋅ 2 1 (4.1) setting ( )11 1 τα p = (4.2) and, more generally, [ ] ( ) α τ λ τ ξ ττ k k io k p = 1 1 + i= 2 k ⋅⋅ ⋅ + ⋅∏∫∫ − 1 1 0 1 ( ) ( )[ ]⋅ − − = = − ∏ j k j j kp d d k 1 1 21 2 3 + 1τ τ τ τ...... ( , ,.....) (4.3) we observe that, being ( )α τ τk 1 10 0≥ ∀ ≥ (4.4) the series ( )∑ τα k k 1 results to be regular and, according to the theorem of monotonous convergence, calculating the limit under the sign of integral in (4.1) one obtains the solution ( ) ( ) d f = uf k k u         τγ ξ+τλ+ +ξ ∑∫ ∞ =1 1 0 1 0 1 1 1, (4.5) the foregoing results can be summed up as follows. theorem: let λ(z) be a continuous and positive function in [0,+∞] and p(z) be a continuous and non-negative function in [0,+∞]. given ( )p z p t dt z ( ) = 0 ∫ the solution f of the problem (3.1) is provided by: 14 ( ) ( ) ( ) f u f dk k u ,ξ ξ λ τ ξ α τ = 0 1 1 10 1 1 1 + + +        = ∞ ∑∫ (4.6) where αk is given by (4.2) and (4.3). references 1 arfwedson g., some problems in the collective theory of risk , skand. aktuartidskr, (1950). 2 badolati e., sulla probabilità di rovina in ipotesi di caricamenti variabili, pubbl. n. 26, istituto statistica e matematica, ist. univ. navale, napoli, (1987). 3 badolati e., una nuova equazione integro -differenziale della teoria del rischio; pubbl. n. 28, istituto statistica e matematica, ist. univ. navale, napoli, (1988). 4 davidson a., on the ruin problem in the collective theory of risk; skand. aktuartidskr. , 3-4 supplement, (1969). 5 di lorenzo e., su un'equazione integro -differenziale della teoria collettiva del rischio in ipotesi di caricamenti variabili; g.i.i.a, (1989). 6 di lorenzo e., una formula di rappresentazione per la probabilità asintotica di non rovina; pubbl. ist. statistica, univ. salerno, (1990). 7 favati l., gosio c., lisei g., su una successione di approssimanti per difetto della probabilità di rovina nel caso di investimento della riserva; pubbl. ist. matem. finanz., univ. genova, (1988). 8 feller w., an introduction to probability theory; wiley, (1971). 9 gerber h.u., an introduction to mathematical risk theory, huebner found., irwin; philadelphia, (1979). 10 gosio c., alcune limitazioni per la probabilità di rovina nel caso di caricamento non costante; pubbl. ist. matem. finanz., univ. genova, (1989). 11 laurin i., an introduction into lundberg's theory of risk; skand. aktuartidskr., 1-2, (1930). 12 lisei g., su un confine inferiore della probabilità di rovina nell'ambito della teoria collettiva del rischio; g.i.i.a., (1983). 13 lisei g., una successione di approssimanti per difetto della probabilità di rovina; pubbl. ist. matem. finanz., univ. genova, (1988). 14 lundberg f., forsakrangsteknisk riskutjamming. i. teori; stockholm, (1926). 15 marina borghesani m.e., alcune osservazioni sulla probabilità di rovina nel caso di caricamento non costante; pubbl. ist. matem. finanz., univ. genova, (1988). ratio mathematica volume 44, 2022 the detour domination and connected detour domination values of a graph r. v. revathi 1 m. antony 2 abstract the number of -sets that belongs to in g is defined as the detour domination value of indicated by for each vertex . in this article, we examined at the concept of a graph’s detour domination value. the connected detour domination values of a vertex represented as , are defined as the number of -sets to which a vertex belongs to g. some of the related detour dominating values in graphs’ general characteristics are examined. this concept’s satisfaction of some general properties is investigated. some common graphs are established. keywords: domination number; detour number; detour domination value; connected detour domination value; etc. 2010 ams subject classification: 05c15, 05c69 3 1 register number 19133232092002, research scholar, department of mathematics, st. jude’s college, thoothoor 629 176, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamil nadu, india; revathi87gowri@gmail.com. 2 associate professor, department of mathematics, st. jude’s college, thoothoor 629 176, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627 012, tamil nadu, india; antony.michael@yahoo.com 3 received on june 9 th, 2022. accepted on aug 10 th , 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.908. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 205 mailto:revathi87gowri@gmail.com r.v. revathi, and m. antony 1. introduction graph having the type g = (v, e) is a finite, undirected connected graph without loops or numerous edges. the order and size of the letter g are represented by the characters n and m, respectively. we refer to [3] for the fundamental terms used in graph theory. if uv is an edge of g, then two vertices u and v are said to be adjacent. if two edges of g connect a vertex, they are said to be adjacent. the distance between two vertices and v in a connected graph is thelength of a shortest path in . a u-v geodesic is a u-v path of length . the longest path in g is also referred to as detour distance between two vertices u and v in a linked graph g from u to v. a detour is a path of length . if x is a vertex of p that also contains the vertices u and v, then x is said to lie on a detour. every vertex of g is contained in a detour connecting some pair of vertices in s, which is the definition of a detour set of g. any detour set of order (g) is referred to as a minimum detour set of g or a -set of g. the detour number (g) of g is the minimum order of a detour set. these ideas have been researched in [4, 5, 6]. if for every v v \ d is adjacent to a vertex in d, then the set d⊆ v is a dominant set of g. if no subset of a dominating set d is a dominating set of g’s, then d is said to be minimal. the symbol denotes the domination number of g, which is the least cardinality of a minimal set of g dominating sets. in [4], the graph's domination number was studied. if a set s is both a detour and a dominating set of g’s, then it is referred to as a detour dominating set of g. any detour dominating set of order is referred to as a – set of g. the detour domination number of g is the minimal order of its detour dominating set. in [8], the detour domination number of a graph studied. if a set s is a detour dominating set of g and its induction by s is connected, the set s ⊆ v(g) is referred to as a connected detour dominating set of . any connected detour dominating set with order is referred to as a set of g. the connected detour domination number of of g is the maximum order of its connected detour dominating sets. in [8,9], the connected detour domination number of a graph was investigated. the subsequent theorem is applied thereafter. theorem 1.1[3] every detour set of a connected graph g contains each end vertex. theorem 1.2[3] let be a connected graph then if and only if theorem 1.3[3] let be a connected graph of order then if and only if 206 the detour domination and connected domination values of a graph 2.the detour domination value of a graph definition 2.1. for each vertex we define the detour domination value of denoted by to be the number of -sets to which belongs to example 2.2. in relation to the graph g in figure 2.1, = , = = = = , = are the onlysix minimum -sets of such that , , theorem 2.3. for the complete graph = ( for each proof. since any two sets of g's vertices is the -set of thus since each vertex of belongs to exactly -sets, it follows that for each ∎ theorem 2.4. for a star = ( for each proof. we have . let s represent the set of all of the end vertices in g. then is the unique -set of thus therefore for each ∎ theorem 2.5. for the complete bipartite graph with bipartite sets and and if then if for any in figure 2.1 207 r.v. revathi, and m. antony if with then proof. let and be the two bipartite sets of since any two adjacent vertices of is a -sets of it follows that if if then it has only one a set of such that if then it only one -set of such that . if then any vertex in belongs to a -set of hence also if ,then any vertex in belongs to a -set thus for if then for any in if with , then , ∎ theorem 2.6.for the wheel graph , and proof. let and . let then , , , , , , are -sets of such that , , let then any two adjacent vertices of is a -set of so that for , hence lies in exactly three -set of so that for all since is adjacent to vertices of ∎ theorem 2.7. for the cycle graph , proof. let . let where here a -set comprises and is fixed by the choice of the first there exists exactly one -set containing the vertex and there are two -sets omitting the vertex such as containing the vertex and containing the vertex . thus let where here a -set is constituted in exactly one of the following two ways. i) comprises ’s and one ii) comprises ’s 208 the detour domination and connected domination values of a graph case(i) note that is fixed by the choice of the single choosing a in the same as choosing its initial vertex in the counter clockwise order. hence case(ii) it is clear that each dominates three vertices, exactly there are two vertices, say and each of whom is adjacent to two distinct ’s in . and is fixed by the placements of and there are ways of choosing consider the (a sequence of slots) obtained as a result of cutting from the centered about vertex. may be placed in the first slot of any of the . as the order of selecting the two vertices and is immaterial . summing over the two disjoint cases, we get let where here a -set comprises of only and is fixed by the placement of the only vertex which is adjacent to two distinct in hence ∎ 3. the connected detour domination value of a graph definition 3.1. for each vertex we define the connected detour domination values of denoted by to be the number of -sets to which belongs to example 3.2. for the graph given in figure 3.1, = , = = = = = , = = are the only eight minimum -sets of such that , , and figure 3.1 209 r.v. revathi, and m. antony proposition 3.3. let be a graph with vertices without cut vertices and then and and equality holds if and only if . proof. let be a universal vertex of . let then is a -set of so that since is a universal vertex of belongs to every -set of since contains at most -sets, let . hence it follows that belongs to every -set of therefore the converse is clear. ∎ theorem3.4. for and proof. let then and are the -sets of so that . as is vertex transitive for all . since belongs to -sets of it follows that for all ∎ theorem3.5. for and for each vertex . proof. since is the unique -sets of the results follow theorem. ∎ theorem3.6. for the complete graph for each vertex . proof. since any two set of vertices of is the -set of , it follows that since each vertex of belongs to exatly -sets, it follows that for each vertex . ∎ theorem3.7. for the wheel graph and proof. let and let then are the sets of such that and . let then any two adjacent vertices of is a -sets of so that for lies in excatly three sets of so that for all . since is adjacent to vertices of ∎ theorem3.8. let and and 210 the detour domination and connected domination values of a graph then for is odd and and for is even and proof. let and case (i) is odd. are the only three -sets of such that so that . case (ii) is even. arethe only four -sets of so that , and . ∎ theorem3.9. proof. let be a -sets of of cardinality where if then and any two adjacent vertices form a -set i.e. , are all possible -sets of . if there is a unique -set so let by lemma 2.2 either or (and not both). let . as to maintain connectedness of and to dominate we have in the same way, thus since contains elements, let the other 2 vertices in be to dominate and one of and (say must be either or similarly is either or since there are two choices each for and such that forms a -set, the number of -sets containing is 4. similarlythe number of -sets containing is 4. hence by lemma 2.2, we get for ∎ theorem3.10. let be a rectangular grid with and let or if then for all if then and , if then proof. the proof is clear for and theorem 2.10, so we assume that let be a vertex in case 1: [ let then using the line of proof of theorem 3.10, the -sets containing are precisely those where and is either or i.e, same for the case when or or . 211 r.v. revathi, and m. antony case 2: [ let note that any connected dominating set contains either also total number of minimum connected dominating sets is 8, out of which only two does not contain namely and thus now,as there exist isomorphisms which maps to , respectively, by proposition 2.2, we have . case 3: [ in this case, from the proof of theorem 2.10 we have . references [1] buckley and f.harary, distance in graph, addition-wesly-wood city, ca, (1990). [2] g. chartrand, g. johns and p. zhang, on the detour number and geodetic number of a graph, ars combinatoria, 72(2004), 3-15. [3] g. chartrand, g. johns and p. zhang, the detour number of a graph, utilitas mathematica, 64(2003), 97-113. [4] t.w. haynes, p.j. slater and s.t. hedetniemi, fundamentals of domination in graphs, marcel dekker, new york, (1998). [5] j. john and n. arianayagam, the detour domination number of a graph, discrete mathematics algorithms and applications, 09(01), 2017, 17500069. [6] j. john and n. arianayagam, the total detour number of a graph, journal of discrete mathematics sciences and cryptography, 17(4), 2014, 337-350. [7] j. john and v. r. sunil kumar, the open detour number of a graph, discrete mathematics algorithms and applications, 13(01), 2021, 2050088. [8] a. p. santhakumaran and s. athisayanathan, the connected detour number of graphs, j. combin. math. combin. compute;69, (2009), 205-218. [9] j. vijaya xavier parthian, and c. caroline selvaraj, connected detour domination number of some standard graphs, journal of applied science and computations, 5(11), (2018), 486-489. 212 https://www.researchgate.net/profile/n-arianayagam?_sg%5b0%5d=ai8km30mfubw6vp_hpl9ftasnlnak-i2emhkkkua4zw9oc2x4mkobwn8oj0nliaunjubrji.cbgvuuninztp-y4ajv71iucmrwfnpmzmurx7jtz-dqq4fjb4p4k7gzvko4hmn_bxkxtfm7jaruysm8nqoxbaca&_sg%5b1%5d=dv6mvihbocf9vaih5rrsso-je9wrghyj5y-_-alrb6kxac5qhxggeecwfsveuhtndaueuna.xxdoc98db55iy810uoym2f2ldxhc9xgoazclmuj62k8ksi7zhyajx2fffiqylzpqjtky1x4hbnbexgwtp14clw http://dx.doi.org/10.1142/s1793830917500069 https://www.researchgate.net/profile/n-arianayagam?_sg%5b0%5d=ai8km30mfubw6vp_hpl9ftasnlnak-i2emhkkkua4zw9oc2x4mkobwn8oj0nliaunjubrji.cbgvuuninztp-y4ajv71iucmrwfnpmzmurx7jtz-dqq4fjb4p4k7gzvko4hmn_bxkxtfm7jaruysm8nqoxbaca&_sg%5b1%5d=dv6mvihbocf9vaih5rrsso-je9wrghyj5y-_-alrb6kxac5qhxggeecwfsveuhtndaueuna.xxdoc98db55iy810uoym2f2ldxhc9xgoazclmuj62k8ksi7zhyajx2fffiqylzpqjtky1x4hbnbexgwtp14clw http://dx.doi.org/10.1142/s1793830917500069 ratio mathematica volume 48, 2023 w8 curvature tensor in generalized sasakian-space-forms gyanvendra pratap singh* rajan† anand kumar mishra‡ pawan prajapati§ abstract the generalized sasakian-space-forms and their properties have been examined by various researchers such as alegre and carriazo [2008], prakasha [2012], sarkar and akbar [2014], shanmukha et al. [2018], sarkar and sen [2012], rajan and singh [2020] and sarkar and sen [2012]. motivated by the results of these works, we have proposed the idea of the w8−curvature tensor in generalized sasakian-spaceforms. the main goal of this paper is to investigate the curvature properties of generalized sasakian-space-forms that satisfy the conditions ξ − w8− flatness, ϕ − w8−semi-symmetric, w8 · q = 0, w8 · r = 0 and to prove some interesting results. keywords: sasakian-space-form, generalized sasakian-space-form, ϕ−recurrent, ϕ−symmetric, ϕ−semi-symmetric, w8− curvature tensor, einstein manifold, η−einstein manifold. 2020 ams subject classifications:53c15, 53c25, 53d15 1 *department of mathematics and statistics, deen dayal upadhyaya gorakhpur university, gorakhpur-273009 (up) india; gpsingh.singh700@gmail.com. †department of mathematics and statistics, deen dayal upadhyaya gorakhpur university, gorakhpur-273009 (up) india; rajanvishwakarma497@gmail.com. ‡department of mathematics and statistics, deen dayal upadhyaya gorakhpur university, gorakhpur-273009 (up) india; aanandmishra1796@gmail.com. §department of mathematics and statistics, deen dayal upadhyaya gorakhpur university, gorakhpur-273009 (up) india; pawanpra123@gmail.com. 1received on november 19, 2022 . accepted on july 5, 2023. published on august 1, 2023. doi: 10.23755/rm.v39i0.961. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. gyanvendra pratap singh, rajan, anand kumar mishra, pawan prajapati 1 introduction the nature of a riemannian manifold depends on the curvature tensor r of the manifold. it is well known that the sectional curvatures of a manifold determine its curvature tensor completely. real space-forms are riemannian manifolds with constant sectional curvature c, and their curvature tensor is given by r(x, y )z = c{g(y, z)x − g(x, z)y }. representation for these spaces are hyperbolic spaces (c < 0), spheres (c > 0) and euclidean spaces (c = 0). the sasakian manifold determines the ϕ-sectional curvature of a sasakian-spaceform, and it has a specific form for its curvature tensor. the kenmotsu and cosymplectic space-forms use the same notation. alegre et al. [2004] developed and researched generalized sasakian-space-forms in an effort to generalize such space -forms in a shared frame. a generalized sasakian-space-form is an almost contact metric manifold (m2n+1, ϕ, ξ, η, g), whose curvature tensor is given by r(x, y )z = f1{g(y, z)x − g(x, z)y } + f2{g(x, ϕz)ϕy − g(y, ϕz)ϕx + 2g(x, ϕy )ϕz} + f3{η(x)η(z)y − η(y )η(z)x + g(x, z)η(y )ξ − g(y, z)η(x)ξ}. the riemannian curvature tensor of a generalized sasakian-space-form m2n+1(f1, f2, f3) is simply given r = f1r1 + f2r2 + f3r3, where f1, f2, f3 are differential functions on m2n+1(f1, f2, f3) and r1(x, y )z = g(y, z)x − g(x, z)y, (1) r2(x, y )z = g(x, ϕz)ϕy − g(y, ϕz)ϕx +2g(x, ϕy )ϕz, and (2) r3(x, y )z = η(x)η(z)y − η(y )η(z)x +g(x, z)η(y )ξ − g(y, z)η(x)ξ, (3) where f1 = c+3 4 , f2 = f3 = c−1 4 . here c denotes the constant ϕ-sectional curvature. numerous geometers, including alegre and carriazo [2008], de and majhi [2015], de and sarkar [2010], kim [2006], prakasha [2012], sarkar and akbar [2014], sarkar and sen [2012], shanmukha et al. [2018], singh [2016], w8 curvature tensor in generalized sasakian-space-forms have explored the characteristics of the generalized sasakian-space-form. numerous writers have addressed the idea of local symmetry of a riemannian manifold in various ways and to different extents. takahashi introduced the sasakian manifold’s locally ϕ-symmetry in toshio [1977]. this is extended by de, shaikh, and sudipta to the notation of ϕ-symmetry in de et al. [2003], after which they introduce the notation of ϕ-recurrent sasakian manifold. on the kenmotsu manifold de et al. [2009], lp-sasakian manifold venkatesha [2008], w8curvature tensor in the lorentzian sasakian manifold rajan and singh [2020] and (lcs)nmanifold shaikh et al. [2008], the ϕ-recurrent condition was further studied. in tripathi and gupta [2012] have define the w8-curvature tensor, given by w8(x, y )z = r(x, y )z + 1 (n − 1) [s(x, y )z − s(y, z)x], where r and s are curvature tensor and ricci tensor of the manifold respectively. a new class of almost contact riemann manifold was presented by k. kenmotsu [1972], sometimes referred to as a kenmotsu manifold. kenmotsu studied at the underlying characteristics of these manifolds local structure. kenmotsu manifolds have a one-dimensional basis, a kahler fibre, and are locally isometric to warped product spaces. according to kenmotsu’s research, a kenmotsu manifold has a negative curvature of -1 if r(x,y)z = 0, where r is the riemannian curvature tensor and r(x,y)z is the derivative of the tensor algebra at each point of the tangent space. because odd dimensions hyperbolic spaces cannot admit sasakian structures, unlike odd dimensional spheres, which are well known to do so, odd dimensional hyperbolic kenmotsu structure is permitted in spaces. normal kenmotsu manifolds the almost contact riemannian manifolds. several properties of kenmotsu manifold have been studied by many authors like bagewadi et al. [2007], blair [1976], chaubey and ojha [2010], de [2008], ingalahalli and bagewadi [2012], hui and chakraborty [2017], baishya and chowdhury [2016], nagaraja et al. [2018], özgür [2006], prakasha and balachandra [2018], ali shaikh and kumar hui [2009], sinha and srivastava [1991]. these concepts served as our inspiration as we made an effort to research the characteristics of generalized sasakian-space-form. the structure of the current paper is as follows. in section 2, we review some preliminary results. in section 3, we study ξ−w8−flat generalized sasakian-space-forms. section 4, deals with the ϕ−w8−semi-symmetric condition in generalized sasakian-space-form and found to be einstein manifold. in section 5, we discuss generalized sasakian-space-form satisfying w8 · q = 0 and also found to be einstein manifold. finally in the last gyanvendra pratap singh, rajan, anand kumar mishra, pawan prajapati section, we discuss the generalized sasakian-space-form satisfying w8 ·r = 0 and found to be η−einstein manifolds. 2 generalized sasakian-space-forms the riemannian manifold m2n+1 is called an almost contact metric manifold if the following result holds blair [1976, 2002]: ϕ2x = −x + η(x)ξ, (4) η(ξ) = 1, ϕξ = 0, η(ϕx) = 0, g(x, ξ) = η(x), (5) g(ϕx, ϕy ) = g(x, y ) − η(x)η(y ), (6) g(ϕx, y ) = −g(x, ϕy ), g(ϕx, x) = 0, (7) (∇xη)(y ) = g(∇xξ, y ), ∀x, y ∈ (tpm). (8) a almost contact metric manifold is said to be sasakian if and only if blair [1976], sasaki [1965] (∇xϕ)y = g(x, y )ξ − η(y )x, (9) ∇xξ = −ϕx. (10) again we know that alegre et al. [2004] in (2n+1)-dimensional generalized sasakianspace -form: s(x, y ) = (2nf1 + 3f2 − f3)g(x, y ) − (3f2 + (2n − 1)f3)η(x)η(y ), (11) s(ϕx, ϕy ) = s(x, y ) + 2n(f1 − f3)η(x)η(y ), (12) qx = (2nf1 + 3f2 − f3)x − (3f2 + (2n − 1)f3)η(x)ξ, (13) r = 2n(2n + 1)f1 + 6nf2 − 4nf3, (14) r(x, y )ξ = (f1 − f3){η(y )x − η(x)y }, (15) r(ξ, x)y = (f1 − f3){g(x, y )ξ − η(y )x}, (16) η(r(x, y )z) = (f1 − f3){g(y, z)η(x) − g(x, z)η(y )}, (17) s(x, ξ) = 2n(f1 − f3)η(x), (18) qξ = 2n(f1 − f3)ξ, (19) for any vector fields x,y,z where r,s,q and r are the riemannian curvature tensor, ricci tensor, ricci operator g(qx, y ) = s(x, y ) and scalar curvature tensor w8 curvature tensor in generalized sasakian-space-forms of generalized sasakian-space-forms in that order. 3 ξ − w8−flat generalized sasakian-space-form in this section, we study ξ − w8−flat in generalized sasakian-space-form: definition 3.1. a generalized sasakian-space-form is said to be ξ − w8−flat if w8(x, y )ξ = 0, (20) for any vector fields x, y on m. w8-curvature tensor tripathi and gupta [2012] is defined as w8(x, y )z = r(x, y )z + 1 (n − 1) [s(x, y )z − s(y, z)x], (21) where r and s are curvature tensor and ricci tensor of the manifold respectively. replacing z by ξ in (21), we get w8(x, y )ξ = r(x, y )ξ + 1 (n − 1) [s(x, y )ξ − s(y, ξ)x]. (22) by using (20) in (22), we get r(x, y )ξ + 1 (n − 1) [s(x, y )ξ − s(y, ξ)x] = 0. (23) by virtue of (15), (18) in (23) and on simplification, we obtained (f1−f3){η(y )x−η(x)y }+ 1 (n − 1) [s(x, y )ξ−2n(f1−f3)η(y )x] = 0. (24) by taking inner product with ξ in (24) and on simplification, we have s(x, y ) = 2n(f1 − f3)η(y )η(x). (25) hence above discussion, we state the following theorem: theorem 3.1. if a generalized sasakian-space-forms satisfying ξ − w8−flat condition then the generalized sasakian-space-form is a special type of η−einstein manifolds. gyanvendra pratap singh, rajan, anand kumar mishra, pawan prajapati 4 ϕ − w8−semi-symmetric condition in generalized sasakian-space-form in this section, we study ϕ − w8−semi-symmetric condition in generalized sasakian-space-form: definition 4.1. a generalized sasakian-space-form is said to be ϕ − w8−semisymmetric if w8(x, y ) · ϕ = 0, (26) for any vector fields x,y on m. now, (26) turns into (w8(x, y ) · ϕ)z = w8(x, y )ϕz − ϕw8(x, y )z = 0. (27) from equation (21), we get w8(x, y )z = r(x, y )z + 1 (n − 1) [s(x, y )z − s(y, z)x]. (28) replace z by ϕz in (28), we obtain w8(x, y )ϕz = r(x, y )ϕz + 1 (n − 1) [s(x, y )ϕz − s(y, ϕz)x]. (29) making use of (28) and (29) in (27) and on simplification, we get r(x, y )ϕz − ϕr(x, y )z + 1 (n − 1) [s(y, z)ϕx − s(y, ϕz)x] = 0. (30) putting x = ξ in (30) and by virtue of (16) and on simplification, we obtain (f1 − f3)g(y, ϕz)ξ − 1 (n − 1) s(y, ϕz)ξ = 0. (31) replace ϕz by z in (31) and on simplification, we get s(y, z)ξ = (n − 1)(f1 − f3)g(y, z)ξ. (32) by taking inner product with ξ in (32), we get s(y, z) = (n − 1)(f1 − f3)g(y, z). (33) hence, we state the following theorem: w8 curvature tensor in generalized sasakian-space-forms theorem 4.1. if a generalized sasakian-space-form satisfying ϕ − w8−semisymmetric condition then the generalized sasakian-space-form is an einstein manifolds. 5 generalized sasakian-space-form satisfying w8·q = 0 in this section, we study generalized sasakian-space-form satisfying w8 · q = 0. then we have w8(x, y )qz − q(w8(x, y )z) = 0. (34) putting y = ξ in (34), we obtain w8(x, ξ)qz − q(w8(x, ξ)z) = 0. (35) by virtue of (21) in (35), we get r(x, ξ)qz + 1 (n − 1) [s(x, ξ)qz − s(ξ, qz)x] − q{r(x, ξ)z + 1 (n − 1) [s(x, ξ)z − s(ξ, z)x]} = 0. (36) by using (16), (18) in (36), we obtain −(f1 − f3)[g(x, qz)ξ − η(qz)x] + 1 (n − 1) [2n(f1 − f3)η(x)qz − 2n(f1 − f3)η(qz)x] −q[−(f1 − f3)g(x, z)ξ − η(z)x] + 1 (n − 1) [2n(f1 − f3)η(x)z − 2n(f1 − f3)η(z)x] = 0, (37) −(f1 − f3)s(x, z)ξ + (f1 − f3)qη(z)x + 2n (n − 1) (f1 − f3)η(x)qz − 2n (n − 1) (f1 − f3)qη(z)x +(f1 − f3)g(x, z)qξ − (f1 − f3)η(z)qx − 2n (n − 1) (f1 − f3)η(x)qz + 2n (n − 1) (f1 − f3)qη(z)x = 0. (38) gyanvendra pratap singh, rajan, anand kumar mishra, pawan prajapati using (19) and simplifying (38), we have s(x, z)ξ = 2n(f1 − f3)g(x, z)ξ. (39) taking inner product with ξ in (39) and on simplifaction, we have s(x, z) = 2n(f1 − f3)g(x, z). (40) hence, we state the following theorem: theorem 5.1. a generalized sasakian-space-form satisfying w8 · q = 0 is an einstein manifolds. 6 generalized sasakian-space-form satisfying w8 · r = 0 in this section, we study generalized sasakian-space-form satisfying w8 · r = 0. then we have w8(ξ, u)r(x, y )z − r(w8(ξ, u)x, y )z −r(x, w8(ξ, u)y )z − r(x, y )w8(ξ, u)z = 0. (41) putting z = ξ in (41), we have w8(ξ, u)r(x, y )ξ − r(w8(ξ, u)x, y )ξ −r(x, w8(ξ, u)y )ξ − r(x, y )w8(ξ, u)ξ = 0. (42) by using (15) in (42) and on simplification, we get (f1 − f3)η(w8(ξ, u)x)y − (f1 − f3)η(w8(ξ, u)y )x −r(x, y )w8(ξ, u)ξ = 0. (43) by using (21) in (43), we get (f1 − f3)η[r(ξ, u)x + 1 (n − 1) {s(ξ, u)x − s(u, x)ξ}]y −(f1 − f3)η[r(ξ, u)y + 1 (n − 1) {s(ξ, u)y − s(u, y )ξ}]x −r(x, y )[r(ξ, u)ξ + 1 (n − 1) {s(ξ, u)ξ − s(u, ξ)ξ}] = 0. (44) w8 curvature tensor in generalized sasakian-space-forms by using (16), (17), (18) in (44) and on simplification, we get (f1 − f3){g(u, x)y − g(u, y )x} + 2n (n − 1) (f1 − f3)η(u)η(x)y − 2n (n − 1) (f1 − f3)η(u)η(y )x + 1 (n − 1) {s(u, y )x − s(u, x)y } +r(x, y )u = 0. (45) putting y = ξ in (45), we get (f1 − f3){g(u, x)ξ − g(u, ξ)x} + 2n (n − 1) (f1 − f3)η(u)η(x)ξ − 2n (n − 1) (f1 − f3)η(u)η(ξ)x + 1 (n − 1) {s(u, ξ)x − s(u, x)ξ} +r(x, ξ)u = 0. (46) by using (5), (16), (18) in (46) and on simplification, we get s(x, u)ξ = 2n(f1 − f3)η(u)η(x)ξ. (47) by taking inner product with ξ in (47), we have s(x, u) = 2n(f1 − f3)η(u)η(x). (48) hence, we state the following theorem: theorem 6.1. if a generalized sasakian-space-form satisfies w8 · r = 0, then the manifold is a special type of η− einstein manifolds. 7 conclusions in this paper, we proposed the notion of the w8−curvature tensor in generalized sasakian-space-forms drawing inspiration from the generalized sasakianspace-form and the w8−curvature tensor. several definitions are provided to support this new mechanism. the concept of generalized sasakian-space-forms has been extensively studied by various authors,including alegre et al. [2004], prakasha [2012], sarkar and akbar [2014] and shanmukha et al. [2018]. their research has shed light on the properties and characteristics of these space-forms. gyanvendra pratap singh, rajan, anand kumar mishra, pawan prajapati the main objective of this study is to investigate the curvature properties of the generalized sasakian-space-form. the analysis begins by examining the ξ − w8− flat generalized sasakian-space-forms which are revealed to be a specific type of einstein manifolds. furthermore, the paper explores the ϕ − w8− semi-symmetric condition in the generalized sasakian-space-form establishing it as an einstein manifold. additionally, the generalized sasakian-space-form satisfying w8 · q = 0 is discussed demonstrating its status as an einstein manifold. lastly, the paper presents the discovery of generalized sasakian-spaceforms that satisfy w8 · r = 0 and are identified as η− einstein manifolds. statements and declarations funding: this work is supported by council of scientific and industrial research (csir), india, under senior research fellowship with file.no. 09/057(0226)/2019emr-i. informed consent statement: not applicable. data availability statement: data from the previous studies have been used and they are cited at the relevant places according as the reference list of the paper. conflict of interest: the authors declare no conflict of interest. acknowledgements the authors are very greatful to prof. s.k. srivastava and dr. vivek kumar sharma, department of mathematics and statistics, deen dayal upadhyaya gorakhpur university gorakhpur (u.p) india and prof. ljubica s. velimiroric, department of mathematics, university of nis, serbia for their continuous support and suggestions to improve the quality and presentation of the paper. references p. alegre and a. carriazo. structures on generalized sasakian-space-forms. differential geometry and its applications, 26(6):656–666, 2008. p. alegre, d. e. blair, and a. carriazo. generalized sasakian-space-forms. israel journal of mathematics, 141(1):157–183, 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venkatesha. on concircular ϕ-recurrent lp-sasakian manifolds. differ. geom. dyn. syst, 10:312–319, 2008. ratio mathematica volume 41, 2021, pp. 19-27 encryption using semigroup action anooja i* vinod s† biju g.s‡ abstract an enciphering transformation is a function f that converts any plaintext message into a ciphertext message and deciphering transformation is a function f−1, which reverse the process. such a set-up is called a cryptosystem. in this paper, we extend a generalization of the original diffie-hellman key exchange and elgamal cryptosystem in (z/pz)∗ by constructing a semigroup action on a finite dimensional vector space t over f2. keywords: semigroup action; enciphering; plaintext; ciphertext; cryptosystem 2010 ams subject classifications: 94a60, 08a70, 08a62. 1 *department of mathematics, cms college kottayam (autonomous), kottayam 686 001 kerala, india; anoojai@gmail.com. †department of mathematics, government college for women, thiruvananthapuram, kerala, india; wenod76@gmail.com. ‡department of mathematics, college of engineering, thiruvananthapuram-695016, kerala, india; gsbiju@cet.ac.in. 1received on august 23, 2021. accepted on december 14, 2021. published on december 31, 2021. doi: 10.23755/rm.v41i0.649. issn: 1592-7415. eissn: 2282-8214. ©anooja et al. this paper is published under the cc-by licence agreement. 19 anooja i, vinod s, biju g.s 1 introduction recently there has been a lot of on-going research work to find more secure and efficient public key cryptosystems based on algebraic structures such as nonabelian groups, linear groups, semigroups and power series rings (see anshel et al. [1999], baumslag et al. [2006], maze et al. [2007], shpilrain and zapata [2006]), and where the security is based on hard algorithmic problems from combinatorial group theory. the hard problems from combinatorial group theory include the conjugacy search problem, the decomposition search problem and the subgroup membership search problem. most common public key cryptosystems and public key exchange protocols presently in use, such as the rsa algorithm, diffiehellman, and elliptic curve methods are number theory based and hence depend on the structure of abelian groups. the idea of using semigroups as platforms for public key cryptosystems has appeared in several papers. yamamura [1998] has considered a group action of sl2(z). blackburn and galbraith [1999] have analyzed the system of yamamura and they have shown that it is insecure. maze et al. [2007] showed that the discrete logarithm problem over a group can be considered as a special case of an action by a semigroup on a set. they showed that every semigroup action by an abelian semigroup on a set gives rise to a diffie-hellman key exchange. by taking the action of the semigroup on itself, a semigroup can then be used as a platform for a public key cryptosystem. kropholler et al. [2010] studied the potential of the semigroup 〈a, b ; ap = br, aq = bs〉 as platforms for the diffie-hellman key exchange protocol. special instances of semigroup actions appears in anshel et al. [1999], shpilrain and ushakov [2005], ko et al. [2000] and slavin [2007]. in this paper, we try to extend a generalization of the original diffie-hellman key exchange and elgamal cryptosystem in (z/pz)∗ by constructing a semigroup from a (p, q)-graph g and defining a semigroup action on a finite vector space of dimension q over the field f2. 2 notations and basic results most of the notations, definitions and results we mentioned here are standard and can be found in menezes et al. [1996], koblitz [1998], lyndon and schupp [1977], maze et al. [2007] and diffie and hellman [1976]. most of the public key cryptosystems and public key exchange protocols currently in use, like the diffie and hellman [1976] key exchange protocol, the elgamal [1985] public key cryptosystem, the digital signature algorithm (dsa) and the elgamal’s signature scheme, use the discrete logarithm problem as the basis of their security. 20 encryption using semigroup action the discrete logarithm problem can be defined as follows. problem 2.1. (discrete logarithm problem) let g be a group and a, b ∈ g. find an integer n ∈ n such that an = b. problem (2.1) has a solution if and only if b ∈ 〈a〉, the cyclic group generated by a. if b ∈ 〈a〉 then there is a unique integer n satisfying 1 ≤ n ≤ ord(a) such that an = b. this unique integer is called the discrete logarithm of b with base a and denote it by loga b. discrete logarithm problem plays important role in the diffie-hellman key agreement and the elgamal public key cryptosystem, the digital signature algorithm and elgamal’s signature scheme. currently the multiplicative group (z/pz)∗ of integers modulo n where n is a prime is widely used as the platform group. protocol 2.1. (diffie-hellman key exchange protocol) let g be a group. 1. alice and bob publicly agree on an element g ∈ g. 2. alice chooses n ∈ n and computes gn . alice’s private key is n, her public key is gn. 3. bob chooses m ∈ n and computes gm . bob’s private key is m, his public key is gm. 4. their common secret key is then gmn. the elgamal public key cryptosystem works as follows: alice chooses n ∈ n, a, b ∈ g where b = an. the private key of alice is (a, b, n), the public key is (a, b). bob chooses a random integer r ∈ n and he applies the encryption function ϕ : g → g×g m → (c1, c2) = (ar, mbr) alice computes m from the ciphertext (c1, c2) by m = c2(cn1) −1. 3 construction of a semigroup from a (p, q)graph let g be a finite (p, q)-graph and h be a subgraph of g. let xh denote a vector corresponding to h such that xh = (x1, x2, . . . , xq) where xi = { 1 if ei is in h 0 otherwisef 21 anooja i, vinod s, biju g.s figure 1: graph g let s be a set of such vectors. then s is a semigroup under the operation defined by xh yk = (x1, x2, . . . , xq)(y1, y2, . . . , yq) = (x1y1, x2y2, . . . , xqyq) we shall illustrate this with the following example. example 3.1. consider the graph g with p = 6 and q = 10 given in figure 1. let us consider seven subgraphs of g, which are displayed in figure 2. let xh1 , xh2, . . .,xh7 be the vectors corresponding to the subgraphs h1, h2, . . ., h7 respectively. let s = {xh1, xh2, . . . , xh7}. then xh1 = (1, 0, 1, 0, 0, 1, 0, 0, 1, 1) xh2 = (1, 1, 0, 0, 0, 1, 1, 0, 0, 0) xh3 = (1, 1, 1, 0, 0, 0, 0, 0, 1, 0) xh4 = (1, 0, 0, 0, 0, 1, 0, 0, 0, 0) xh5 = (1, 0, 1, 0, 0, 0, 0, 0, 1, 0) xh6 = (1, 1, 0, 0, 0, 0, 0, 0, 0, 0) xh7 = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0) 22 encryption using semigroup action figure 2: subgraphs of the graph g 23 anooja i, vinod s, biju g.s now, xh1xh2 = xh4, xh1xh3 = xh5, xh1xh4 = xh4, xh1xh5 = xh5, xh1xh6 = xh7, xh1xh7 = xh7, xh2xh3 = xh6, xh2xh4 = xh4, xh2xh5 = xh7, xh2xh6 = xh6, xh2xh7 = xh7, xh3xh4 = xh7, xh3xh5 = xh5, xh3xh6 = xh6, xh3xh7 = xh7, xh4xh5 = xh7, xh4xh6 = xh7, xh4xh7 = xh7, xh5xh6 = xh7, xh5xh7 = xh7, xh6xh7 = xh7 also, xhi(xhjxhk) = (xhixhj)xhk, i, j, k = 1, 2, . . . , 7. hence s is a semigroup. 4 key exchange using s-action let t be a q dimensional vector space over f2. define the left action of s on t , ϕ : s × t → t such that ϕ(x, t) = xt. we call this action as an s-action on the vector space t . the right action is similarly defined. let g be a (p, q)-graph, s an abelian semigroup associated with the graph g, t be a q dimensional vector space over f2, and an s-action on t as defined above. diffie-hellman key exchange using s-action is as follows: 1. alice and bob agree on an element t ∈ t . 2. alice chooses x ∈ s and computes xt. alice’s private key is x, her public key is xt. 3. bob chooses y ∈ s and computes yt. bobss private key is y, his public key is yt. 4. their common secret key is then x(yt) = (xy)t = (yx)t = y(xt). example 4.1. consider the semigroup s in the example 3.1. let t be a 10 dimensional vector space over f2. suppose alice and bob want to agree on a key. suppose they choose t = (0, 1, 1, 0, 1, 0, 1, 1, 0, 0) ∈ t . then alice chooses xh1 = (1, 0, 1, 0, 0, 1, 0, 0, 1, 1) ∈ s and computes xh1t = (0, 0, 1, 0, 0, 0, 0, 0, 0, 0). then send it to bob. similarly, bob chooses xh6 = (1, 1, 0, 0, 0, 1, 0, 0, 1, 1) ∈ s and computes xh6t = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0). then send it to alice. their common key is xh1(xh6t) = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0). 24 encryption using semigroup action s-action problem let g be a (p, q)-graph and s be a semigroup associated with the graph g, acting on a q dimensional vector space t over f2. given elements t ∈ t and y ∈ s, find x ∈ s such that xt = y. 4.1 diffie-hellman problem using s-action let g be a (p, q)-graph, s be a semigroup associated with the graph g, t be a q dimensional vector space over f2 and ϕ be an s-action on t . given r, s, t ∈ t with s = xr and t = yr for some x, y ∈ s, find (xy)r ∈ t . 5 cryptosystem using s-action let g be a (p, q)-graph, s be a semigroup associated with the graph g, t be a q dimensional vector space over f2, t is an additive abelian group and an action on t as defined above. elgamal cryptosystem using s-action is as follows: 1. alice chooses elements t ∈ t and x ∈ s. alice’s public key is (t, xt). 2. bob chooses a random element y ∈ s and encrypts a message m using the encryption function (m, y) 7→ (yt, (y(xt)) + m) = (c1, c2). 3. alice can decrypt the message using m = (y(xt))−1 + (y(xt)) + m = (xc1) −1 + c2 note: message m is also represented as vectors. each letter in the message represents a vector (x1, x2, . . . , xq), q ≥ 26 such that xi = { 1 if the corresponding letter is in ith position of the alphabet 0 otherwise example 5.1. let g be any (p, q)-graph with q = 26. let t be a 26 dimensional vector space over f2, t be an additive abelian group and s be the semigroup associated with the graph g. the action of s on t is as defined earlier. suppose alice wants to receive a message. 25 anooja i, vinod s, biju g.s 1. alice chooses t = (0, 1, 1, 0, 1, 0, 1, 1, 0, . . . , 0) ∈ t . then chooses x = (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, . . . , 0) ∈ s corresponding to one subgraph h1 of g and compute xt = (0, 0, 1, 0, 1, 0, 1, 0, 0, . . . , 0). her public key is (t, tx). 2. bob wishes to send a message m = meet me tomorrow to alice. he send it letter by letter. so, first he wants to send the letter m = m1(m) = (0, 0, . . . , 0, 0, 1, 0, 0, . . . , 0, 0). for, he chooses y = (0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, . . . , 0, 0) ∈ s that is a vector corresponding to one subgraphh2 of g and compute yt = (0, 0, 1, 0, 1, 0, 0, 1, 0, 0, . . . , 0, 0) = c1 y(xt) = (0, 0, 1, 0, 1, 0, 0, . . . , 0, 0) and y(xt) = (0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, . . . , 0, 0) = c2 then he sends (c1, c2) to alice. 3. after receiving this, alice decrypt the message by computing (xc1)−1 + c2. xc1 = (0, 0, 1, 0, 1, 0, 0, . . . , 0, 0) (xc1) −1 = (0, 0, 1, 0, 1, 0, 0, . . . , 0, 0) (xc1) −1 + c2 = (0, 0, 0, . . . , 0, 0, 1, 0, . . . , 0, 0) = m1(m) = m similarly, they transfer each letter in the message. references iris anshel, michael anshel, and dorian goldfeld. an algebraic method for public key cryptography. mathematical research letters, 6(3–4):287–291, 1999. gilbert baumslag, benjamin fine, and xiaowei xu. cryptosystems using linear groups. applicable algebra in engineering, communication and computing, 17(3):205–217, 2006. simon r blackburn and steven galbraith. cryptanalysis of two cryptosystems based on group actions. in international conference on the theory and application of cryptology and information security, volume 3531, pages 52–61. springer, 1999. 26 encryption using semigroup action whitfield diffie and martin hellman. new directions in cryptography. ieee transactions on information theory, 22(6):644–654, 1976. taher elgamal. a public key cryptosystem and a signature scheme based on discrete logarithms. ieee transactions on information theory, 31(4):469–472, 1985. ki hyoung ko, sang jin lee, jung hee cheon, jae woo han, ju-sung kang, and choonsik park. new public-key cryptosystem using braid groups. in annual international cryptology conference, pages 166–183. springer, 2000. neal koblitz. algebraic methods of cryptography. berlin heidelberg new york: springer, 1998. ph kropholler, sj pride, wam othman, kb wong, and pc wong. properties of certain semigroups and their potential as platforms for cryptosystems. in semigroup forum, volume 81, pages 172–186. springer, 2010. roger c lyndon and paul e schupp. combinatorial group theory. berlin heidelberg new york: springer, 1977. gérard maze, chris monico, and joachim rosenthal. public key cryptography based on semigroup actions. advances of mathematics of communications, 1 (4):489–507, 2007. alfred j menezes, paul c van oorschot, and scott a vanstone. handbook of applied cryptography. discrete mathematics and its applications. crc press, new york, 1996. vladimir shpilrain and alexander ushakov. thompson’s group and public key cryptography. in international conference on applied cryptography and network security, pages 151–163. springer, 2005. vladimir shpilrain and gabriel zapata. combinatorial group theory and public key cryptography. applicable algebra in engineering, communication and computing, 17(3):291–302, 2006. keith r slavin. public key cryptography using matrices. 2007. us patent 10260818, http://www.patentstorm.us/patents/7184551-fulltext.html. akihiro yamamura. public-key cryptosystems using the modular group. in international workshop on public key cryptography, pages 203–216. springer, 1998. 27 ratio mathematica volume 47, 2023 moduli of continuity of functions in hölder’s class hα,2ωk [0, 1) by first kind chebyshev wavelets and its applications in the solution of lane-emden differential equations shyam lal * deepak kumar singh † abstract in this paper, two new moduli of continuity w (( f − s2k−1,0f ) , 1 2k ) , w (( f − s2k−1,mf ) , 1 2k ) and two estimators e2k−1,0(f) and,e2k−1,m(f) of a functions f in hölder’s class hα,2ωk [0,1) by first kind chebyshev wavelets have been determined. these moduli of continuity and estimators are new and best possible in wavelet analysis. applying this technique, lane -emden differential equations have been solved by the first of kind chebyshev wavelet method. these solutions obtained by first kind chebyshev wavelet method approximately coincided with their exact solutions. this is a significant achievement of this research paper in wavelet analysis. keywords: chebyshev wavelet, modulus of continuity, wavelet approximation, hölder’s class, orthonormal basis, operational matrix of integration . 2020 ams subject classifications: 42c40, 65t60, 65l10 , 65l60. 1 *department of mathematics, institute of science, banaras hindu university, varanasi 221005, india; shyam lal@rediffmail.com. †department of mathematics, institute of science, banaras hindu university, varanasi 221005, india; 24deepak97@gmail.com. 1received on june 12, 2022. accepted on december 25, 2022. published online on january 10, 2023. doi: 10.23755/rm.v39i0.794. issn: 1592-7415. eissn: 2282-8214. ©shyam lal and deepak kumar singh. this paper is published under the cc-by licence agreement. 52 shyam lal and deepak kumar singh 1 introduction recently, researchers are making attention on wavelets. wavelets have connections between several branches of mathematical sciences and play an important role in signal processing, engineering and technology.the approximation of functions of a certain class by trigonometric series is a common places of analysis.approximation of functions belonging to some class by wavelet method has been discussed by many researchers like devore[1], morlet[4] , meyer[3] and debnath[2]. wavelets are new tools to solve differential equations and to estimate the moduli of continuity & the approximation of functions.wavelets help in the most accurate representation of functions f ∈ hα,2ωk [0,1) class. several wavelets are known like that haar wavelet , legendre wavelet , chebyshev wavelet. haar wavelet is one of simplest in wavelet analysis . due to its simplicity and better applications, it is used in solution of integral as well as differential equations . haar wavelet contains a non-smooth character. this is a difficiency of haar wavelet to estimate the moduli of continuity and the approximation of the smooth function by it. this weak point is almost removed by chebyshev wavelets and more accurate the moduli of continuity and approximations of functions are obtained. sripathy[14] discussed the chebyshev wavelet based approximation for solving linear and non-linear differential equations. moduli of continuity of functions have been studies by alexander babenko[10]. in best of our knowledge, there is no work associated to the modulus of continuity and approximation of a function f in hölder’s class by first kind chebyshev wavelet method. to make an advanced study in this direction, in this paper, the moduli of continuity and approximation of functions have been determined in hölder’s class hα,2ωk [0,1). several linear, as well as non-linear differential equations are solvable by galerkin, collocation, and other known methods. these equations can be solved by chebyshev wavelet technique in very efficient and suitable manners. this motivates us to consider first kind chebyshev wavelet method for the solution of differential equations. also, babolian and fattahzadeh[5] suggested a method to solve the differential equations by using chebyshev wavelet operational matrix of integration. in this paper, the lane-emden differential equations has been solved by chebyshev wavelet technique. the main characteristic of this techique is that it reduces the problem to a system of algebraic equations. the approach is based on converting the given differential equations into integral equations through integration by approximating various signals involved in the equation through truncated orthogonal chebyshev wavelet series and using the operational matrix p of integration , to eliminate the integral operations. this paper is organized as follows: section(1) is introductory in which the 53 moduli of continuity of functions in hölder’s class.... importance of moduli of continuity and chebyshev wavelet of first kind related literature are studied. in section(2), chebyshev wavelet of first kind ,approximation of function and moduli of continuity of functions in class hα,2ωk [0,1) are defined.in section(3), theorem concerning the moduli of continuity of f−s2k−1,m(f) has been established and also its detail proof is discussed in section(4).in section(5) corollaries are deduced from theorem of section (3) . in section(6) first kind chebyshev wavelet operational matrix of integration has been constructed and the product operational matrix has been obtained in section(7). in section(8), lane-emden differential equations of index 0, 1 & 2 are solved by chebyshev wavelet method. finally, the main conclusions are summarized in section (9). 2 definitions and preliminaries 2.1 chebyshev wavelets of first kind wavelets constitute a family of functions constructed from dialation and translation of a single function ψ ∈ l2(r) called mother wavelet .we write ψb,a(t) = |a| −1 2 ψ ( t − b a ) , a ̸= 0. ( daubechies [6]) if we restrict the values of dialation and translation parameter to a = a−k0 ,b = (2n − 1)b0a0−k,a0 > 1,b0 > 0 respectively,the following family of discrete wavelets are constructed: ψk,n(t) = |a0| k 2 ψ ( ak0t − (2n − 1)b0 ) . now, taking a0 = 2,ψ(t) = t̃m(t) & b0 = 1 the chebyshev wavelet ψ(k,n,m,t) of first kind , generally denoted by ψ (c1) n,m(t) over the interval [0,1), is obtained as (babolian [5]) ψ (c1) n,m(t) = { 2 k 2 t̃m(2 kt − 2n + 1), n − 1 2k − 1 ≤ t < n 2k − 1 0 , otherwise (1) where t̃m(t) =   1 √ π , m = 0√ 2 π tm(t) , otherwise. where n = 1,2, ...,2k−1 , m = 0,1,2, ...m and k is the positive integer. in above definition, tm are the first kind chebyshev polynomials of degree m on the interval [-1,1] which are defined by tm(t) = cos(mθ), θ = arccos(t) (2) 54 shyam lal and deepak kumar singh and also satisfy the following recursive formula: t0(t) = 1 t1(t) = t, tm+1(t) = 2ttm(t) − tm−1(t), m = 1,2,3, ...... the set of {tm(t) : m = 0,1,2,3, ...} in the hilbert space l2[−1,1] is a orthogonal set with respect to the weight function ω(t) = 1√ 1−t2 . orthogonality of chebyshev polynomial of first kind on the interval [-1,1] implies that ⟨tm(t),tn(t)⟩ = ∫ 1 −1 tm(t)tn(t)√ 1 − t2 dt =   π, m = n = 0 0, n ̸= m. π 2 , n = m ̸= 0. in dealing with chebyshev wavelets, the weight function ω(t) for orthogonal chebyshev polynomials has to dilated and translated to construct orthonomal wavelets. so the first kind chebyshev wavelets are an orthonormal set with weight function ( s. dhawan[7]) ωk(t) =   ω1,k(t), 0 ≤ t < 12k−1 , ω2,k(t), 1 2k−1 ≤ t < 2 2k−1 , ... ω2k−1,k(t), 2k−1−1 2k−1 ≤ t < 1, (3) where ωn,k(t) = ω(2kt−2n+1). furthermore, the set of wavelets ψn,m(t) makes an orthonormal basis in hilbert space l2ωk[0,1), i.e. ⟨ψ (c1) n,m ,ψ (c1) n′,m′⟩ωk = ∫ 1 0 ψ (c1) n,m(t)ψ (c1) n ′ m ′ (t)ωk(t)dt = δn,n′δm,m′ in which δ denotes kronecker delta function defined by δn,n′ = { 1, n=n’ 0, otherwise 2.2 first kind chebyshev wavelet expansion and approximation of function the function f ∈ l2ωk[0,1) is expressed in the chebyshev wavelet series as f(t) = ∞∑ n=1 ∞∑ m=0 cn,mψ (c1) n,m(t), (4) 55 moduli of continuity of functions in hölder’s class.... where cn,m = ⟨f,ψ (c1) n,m⟩ωk. the (2 k−1,m + 1)th partial sums of above series (4) is given by s2k−1,m(f)(t) = 2k−1∑ n=1 m∑ m=0 cn,mψ (c1) n,m(t) = c t ψ (c1) (t) (5) in which c and ψ (c1)(t) are 2k−1(m + 1) vectors of the form ct = [c1,0,c1,1, ...c1,m,c2,0,c2,1...,c2,m, ......,c2k−1,0, ...,c2k−1,m] and ψ (c1) = [ψ (c1) 1,0 ,ψ (c1) 1,1 , ...,ψ (c1) 1,m,ψ (c1) 2,0 ,ψ (c1) 2,1 , ...,ψ (c1) 2,z , ...,ψ (c1) 2k−1,0, ...,ψ (c1) 2k−1,m−1] t the chebyshev wavelet approximation e2k−1,m(f) of a function f ∈ l2ωk[0,1) by (2k−1,(m + 1))th partial sums s2k−1,m(f) of its chebyshev wavelet series is given by e2k−1,m(f) = min s 2k−1,m (f) ∥f − s2k−1,m(f)∥2 where, ∥f∥2 = (∫ 1 0 |f(t)|2 ωk(t)dt )1 2 if e2k−1,m(f) → 0 as k,m → ∞ then e2k−1,m(f) is called the best approximation of f of order (2k−1,m + 1) ( zygmund[8]). 2.3 modulus of continuity the modulus of continuity of a function f ∈ l2ωk[0,1) is defined as w (f,δ) = sup 0<h≤δ ||f(· + h) − f(·)||2 = sup 0<h≤δ (∫ 1 0 |f(t + h) − f(t)|2ωk(t)dt )1 2 . it is remarkable to note that w (f,δ) is a non-decresing function of δ and w (f,δ) → 0 as δ → 0+ (chui[9]). 2.4 function of hölder’s class a function f is said to be in hölder’s class h α ωk [0,1) of order α, 0 < α ≤ 1, ( g.das[11] ) if f satisfies( f(x + t) − f(x) ) ωk(x) = o (|t|α) , ∀x,t,x + t ∈ [0,1). if (∫ 1 0 |f(x + t) − f(x)|2 ωk(x)dx )1 2 = o (|t|α) , ∀x,t,x + t ∈ [0,1). 56 shyam lal and deepak kumar singh then f is said to be a function of hölder’s class h α,2 ωk [0,1) of order α, 0 < α ≤ 1. the class h α ωk [0,1) is a proper subclass of h α,2 ωk [0,1). let f ∈ hαωk[0,1),α = 1. then( f(x + t) − f(x) ) ωk(x) = o(|t|)(∫ 1 0 |f(x + t) − f(x)|2ωk(x)dx ) = ∫ 1 0 o (|t|)2 ωk(x) dx = o ( |t|2 )2k−1∑ n=1 ∫ n 2k−1 n−1 2k−1 √ 1 − (2kx − 2n + 1)2dx = o ( |t|2 )2k−1∑ n=1 ∫ π 0 sin2 θ dθ 2k = o ( |t|2 ) π 4(∫ 1 0 |f(x + t) − f(x)|2ωk(x)dx )1 2 = ( o (|t|)2 )1 2 √ π 2 = o (|t|) i.e f ∈ h α,2 ωk [0,1), α = 1. let, f(t) = { t2, 0 ≤ t < 1 2 1, 1 2 ≤ t < 1 f ( 1 2 + ϵ 2 ) − f ( 1 2 − ϵ 2 ) = 1 − ( 1 2 − ϵ 2 )2 = 3 4 + ϵ 2 − ϵ2 4 .( f ( 1 2 + ϵ 2 ) − f ( 1 2 − ϵ 2 ) ϵ ) ωk(x) = ( 3 4ϵ + 1 2 − ϵ 4 ) ωk ( 1 2 − ϵ 2 ) → +∞ as ϵ → 0+ therefore, f /∈ h α ωk [0,1),α = 1.∫ 1 0 ∣∣f(x + t) − f(x)∣∣2ωk(x)dx = ∫ 12 0 ∣∣f(x + t) − f(x)∣∣2ωk(x)dx + ∫ 1 1 2 ∣∣f(x + t) − f(x)∣∣2ωk(x)dx = ∫ 1 2 0 |(x + t)2 − x2|2ωk(x)dx + 0 ≤ ∫ 1 2 0 ∣∣2xt + t2∣∣ 2√ 3 dx, 1 ≤ ω(x) ≤ 2 √ 3 = 2 √ 3 ∫ 1 2 0 ( 4x2t2 + t4 + 4xt3 ) dx 57 moduli of continuity of functions in hölder’s class.... ∫ 1 0 ∣∣f(x + t) − f(x)∣∣2ωk(x)dx = 2√ 3 [ 4 3 t2 × 1 8 + t4 × 1 2 + 2t3 × 1 4 ] = 2 √ 3 [ 1 6 × t2 + 1 2 × t2 + 1 2 × t2 ] = 7 3 √ 3 t2 (∫ 1 0 ∣∣f(x + t) − f(x)∣∣2ωk(x)dx) 1 2 ≤ √ 7 3 √ 3 ∣∣t∣∣ (∫ 1 0 |f(x + t) − f(x)|2ωk(x)dx )1 2 = o (|t|) f ∈ h α,2 ωk [0,1),α = 1 thus,h α ωk [0,1) ⊊ h α,2 ωk [0,1), α = 1. 3 theorems in this paper, following theorems have been proved. theorem 3.1. if f ∈ hα,2ωk [0,1) and its first kind chebyshev wavelet expansion be f(t) = ∞∑ n=1 ∞∑ m=0 cn,mψ (c1) n,m(t) having (2k−1,m + 1)th partial sums ( s2k−1,m(f) ) (t) = 2k−1∑ n=1 m∑ m=0 cn,mψ (c1) n,m(t) then the moduli of continuity of f − ( s2k−1,m(f) ) satisfies i.w((f − s2k−1,0f), 1 2k ) = sup 0<h≤ 1 2k ∥(f − s2k−1,0f)(· + h) − (f − s2k−1,0f)(·)∥ = o ( 1 2(k−1)α ) ii.w((f − s2k−1,mf), 1 2k ) = sup 0<h≤ 1 2k ∥(f − s2k−1,mf)(· + h) − (f − s2k−1,mf)(·)∥ = o ( 1 2k(α+1/2) √ m + 1 ) , m ≥ 1. 58 shyam lal and deepak kumar singh 4 proof proof of the theorem (3.1) (i) error between f(x) and its chebyshev wavelet expansion in interval [ n−1 2k−1 , n 2k−1 ) is given by en(t) = cn,0ψ (c1) 1,0 (t) − fχ[ n−1 2k−1 , n 2k−1 ) (t) = (∫ n 2k−1 n−1 2k−1 f(x)ψ (c1) 1,0 (x)ωk(x)dx ) ψ (c1) 1,0 (t) − fχ[ n−1 2k−1 , n 2k−1 ) (t) = 2 k 2 √ π (∫ n 2k−1 n−1 2k−1 f(x)ωk(x)dx ) 2 k 2 √ π − fχ[ n−1 2k−1 , n 2k−1 ) (t) = 2k π (∫ n 2k−1 n−1 2k−1 f(x)ωk(x)dx ) − 2k π fχ[ n−1 2k−1 , n 2k−1 ) (t) ∫ n 2k−1 n−1 2k−1 ωk(x)dx = 2k π (∫ n 2k−1 n−1 2k−1 (f(x) − f(t))ωk(x)dx ) . |en(t)| ≤ 2k π (∫ n 2k−1 n−1 2k−1 |(f(x) − f(t))||ωk(x)|dx ) = 2k π (∫ n 2k−1 n−1 2k−1 |(f(x) − f(t))|| √ ωk(x)|| √ ωk(x)|dx ) ≤ 2k π (∫ n 2k−1 n−1 2k−1 (|(f(x) − f(t))|)2 |ωk(x)dx )1 2 (∫ n 2k−1 n−1 2k−1 |ωk(x)|dx )1 2 ∫ n 2k−1 n−1 2k−1 |ωk(x)|dx = ∫ n 2k−1 n−1 2k−1 |ω(2kx − 2n + 1)|dx = π 2k |en(t)| ≤ 2k π (∫ n 2k−1 n−1 2k−1 (|(f(x) − f(t))|)2 |ωk(x)dx )1 2 ( π 2k )1 2 = 2k π (∫ n 2k−1 n−1 2k−1 (|(f(x) − f(t))|)2 |ω(2kx − 2n + 1)dx )1 2 ( π 2k )1 2 = 2k π (∫ 1 −1 ( |f ( v + 2n − 1 2k ) − f(t)| )2 |ω(v)| dv 2k )1 2 ( π 2k )1 2 59 moduli of continuity of functions in hölder’s class.... |en(t)| ≤ 1 √ π (∫ 1 −1 ( |f ( v + 2n − 1 2k ) − f(t)| )2 |ω(v)|dv )1 2 = 1 √ π (∫ 1 0 ∣∣∣∣f ( v + 2n − 1 2k ) − f(t) ∣∣∣∣2 |ω(v)|dv + ∫ 0 −1 ∣∣∣∣f ( v + 2n − 1 2k ) − f(t) ∣∣∣∣2 |ω(v)|dv )1 2 = 1 √ π (∫ 1 0 ∣∣∣∣f ( v + 2n − 1 2k ) − f(t) ∣∣∣∣2 |ω(v)|dv + ∫ 1 0 ∣∣∣∣f ( −v + 2n − 1 2k ) − f(t) ∣∣∣∣2 |ω(−v)|dv )1 2 = 1 √ π (∫ 1 0 ∣∣∣∣f ( v + 2n − 1 2k ) − f(t) ∣∣∣∣2 |ω(v)|dv )1 2 + 1 √ π (∫ 1 0 ∣∣∣∣f ( −v + 2n − 1 2k ) − f(t) ∣∣∣∣2 |ω(v)|dv )1 2 , v + 2n − 1 2k ∈ [ n − 1 2 2k−1 , n 2k−1 ) and −v + 2n − 1 2k ∈ ( n − 1 2k−1 , n − 1 2 2k−1 ] |en(t)| ≤ 1 √ π o ( 1 2(k−1) )α + 1 √ π o ( 1 2(k−1) )α = 2 √ π o ( 1 2(k−1) )α = o ( 1 2(k−1)α ) (6) ∥en∥22 = ∫ n 2k−1 n−1 2k−1 |en(t)|2|ωk(t)|dt = o ( 1 2(k−1) )2α ∫ n 2k−1 n−1 2k−1 |ωk(t)|dt = o ( 1 22(k−1)α ) π 2k = 1 2k o ( 1 22(k−1)α ) ≤ 1 2k−1 o ( 1 22(k−1)α ) ∥en∥22 = o ( 1 2(k−1)(2α+1) ) (7) 60 shyam lal and deepak kumar singh ∥∥f − (s2k−1,0f)∥∥22 = ∫ 1 0  2k−1∑ n=1 en(t)  2 ωk(t)dt = ∫ 1 0  2k−1∑ n=1 (en(t)) 2ωk(t)  dt (8) + ∑ ∑ 1⩽ n̸=n′≤ 2k−1 ∫ 1 0 en(t)en′(t)ωk(t)dt, due to disjointness of supports of en(t)&en′(t), equation(8) becomes, = ∫ 1 0  2k−1∑ n=1 (en(t)) 2ωk(t)  dt = 2k−1∑ n=1 ∫ 1 0 (en(t)) 2ωk(t) = 2k−1∑ n=1 ∥en(t)∥22 = 2k−1∑ n=1 o ( 1 2(k−1)(2α+1) ) = o ( 2k−1 2(k−1)(2α+1) ) = o ( 1 22(k−1)α ) ∥∥f − (s2k−1,0f)∥∥2 = o ( 1 2(k−1)α ) (9) w (( f − s2k−1,0f ) , 1 2k ) = sup 0<h≤ 1 2k ∥ (f − s2k−1,0f)(· + h) − (f − s2k−1,0f)(·)∥2 ≤ || ( f − s2k−1,0f ) ||2 + || ( f − s2k−1,0f ) ||2 = 2|| ( f − s2k−1,0f ) ||2 = 2.o ( 1 2(k−1)α ) = o ( 1 2(k−1)α ) w (( f − s2k−1,0f ) , 1 2k ) = o ( 1 2(k−1)α ) . (10) 61 moduli of continuity of functions in hölder’s class.... (ii) consider f(x) = ∞∑ n=1 ∞∑ m=0 cn,mψ (c1) m,n(t) cn,m = ⟨f,ψ (c1) m,n(t)⟩ωk = ∫ n 2k−1 n−1 2k−1 f(t)ψ (c1) n,m(t)ωk(t)dt = 2 k+1 2 √ π (∫ n 2k−1 n−1 2k−1 f(t)tm(2 kt − 2n + 1)ωk(t)dt ) = 2 k+1 2 √ π ∫ n 2k−1 n−1 2k−1 ( f(t) − f ( 2n − 1 2k ) + f ( 2n − 1 2k )) g(t)ωk(t)dt = 2 k+1 2 √ π (∫ n 2k−1 n−1 2k−1 ( f(t) − f ( 2n − 1 2k )) g(t)ωk(t)dt ) + 2 k+1 2 √ π f ( 2n − 1 2k )∫ n 2k−1 n−1 2k−1 g(t)ωk(t)dt (11) where g(t) = tm(2 kt − 2n + 1)∫ n 2k−1 n−1 2k−1 g(t)ωk(t)dt = ∫ n 2k−1 n−1 2k−1 tm(2 kt − 2n + 1)ω(2kt − 2n + 1)dt = ∫ 0 π tm(cosθ)ω(cosθ) (− sinθ) 2k dθ = ∫ π 0 cosmθ sinθ (sinθ) 2k dθ = 1 2k [ sinmθ m ]π 0 = 0 (12) therefore equation (11) reduces to, cn,m = 2 k+1 2 √ π (∫ n 2k−1 n−1 2k−1 ( f(t) − f ( 2n − 1 2k )) g(t)ωk(t)dt ) = 2 k+1 2 √ π (∫ n 2k−1 n−1 2k−1 ( f(t) − f ( 2n − 1 2k )) g(t) √ ωk(t)ωk(t)dt ) |cn,m| ≤ 2 k+1 2 √ π (∫ n 2k−1 n−1 2k−1 ∣∣∣∣f(t) − f ( 2n − 1 2k )∣∣∣∣2 |ωk(t)|dt )1 2 × ( ∫ n 2k−1 n−1 2k−1 ∣∣tm(2kt − 2n + 1)∣∣2 |ωk(t)| dt )1 2 62 shyam lal and deepak kumar singh = 2 k+1 2 √ π (∫ n 2k−1 n−1 2k−1 ∣∣∣∣f(t) − f ( 2n − 1 2k )∣∣∣∣2 |ω(2kt − 2n + 1)|dt )1 2 × (∫ n 2k−1 n−1 2k−1 ∣∣tm(2kt − 2n + 1)∣∣ |ω(2kt − 2n + 1)|dt ) = 2 k+1 2 √ π (∫ 1 −1 ∣∣∣∣f ( v + 2n − 1 2k ) − f ( 2n − 1 2k )∣∣∣∣2 |ω(v)|dv2k )1 2 × (∫ π 0 |tm(cosθ)| |ω(cosθ)| sinθ 2k dθ ) = 1 2k √ 2 π (∫ 1 −1 ∣∣∣∣f ( v + 2n − 1 2k ) − f ( 2n − 1 2k )∣∣∣∣2 |ω(v)|dv )1 2 (∫ π 0 |cosmθ| ∣∣∣∣ 1sinθ ∣∣∣∣sinθdθ ) = 1 2k √ 2 π (∫ 1 −1 ∣∣∣∣f ( v + 2n − 1 2k ) − f ( 2n − 1 2k )∣∣∣∣2 |ω(v)|dv )1 2 (∫ π/2 − π 2 ∣∣∣cos(mπ 2 − mt) ∣∣∣ dt ) = √ 2 π 2k (∫ 1 −1 ∣∣∣∣f ( v + 2n − 1 2k ) − f ( 2n − 1 2k )∣∣∣∣2 |ω(v)|dv)12∫ π 2 − π 2 ∣∣∣cos mπ 2 cosmt + sin mπ 2 sinmt ∣∣∣dt ≤ √ 2 π 2k−1 (∫ 1 −1 ∣∣∣∣f ( v + 2n − 1 2k ) − f ( 2n − 1 2k )∣∣∣∣2 |ω(v)|dv )1 2 (∫ π/2 0 (|cos(mt)| + |sin(mt)|) dt ) ≤ 1 2k−1 √ 2 π (∫ 1 −1 ∣∣∣∣f ( v + 2n − 1 2k ) − f ( 2n − 1 2k )∣∣∣∣2 |ω(v)|dv )1 2 × [ |sin(mt)| m + |cos(mt)| m ]π 2 0 = 1 m.2k √ 2 π (( 1 2k )α + ( 1 2k )α) = 1 m.2k−1 √ 2 π ( 1 2kα ) 63 moduli of continuity of functions in hölder’s class.... |cn,m| ≤ 1 m.2k−1 √ 2 π ( 1 2kα ) . (13) let i(t) = f(x) − s2k−1,m(f)(t) i(t) = ∞∑ n=1 ∞∑ m=0 cn,mψ (c1) n,m(t) − 2k−1∑ n=1 m∑ m=0 cn,mψ (c1) n,m(t) = (2k−1∑ n=1 + ∞∑ n=1 2k−1+1 )( m∑ m=0 + ∞∑ m= m+1 ) cn,mψ c1 n,m(t) − 2k−1∑ n=1 m∑ m=0 cn,mψ c1 n,m(t) = 2k−1∑ n=1 ∞∑ m=m+1 cn,mψ (c1) n,m(t), by def n of ψ (c1) n,m (i(t))2 =  2k−1∑ n=1 ∞∑ m=m+1 cn,mψ (c1) n,m(t)  2 = 2k−1∑ n=1 ∞∑ m=m+1 c2n,m ( ψ (c1) n,m(t) )2 + 2k−1∑ n=1 ∑ ∑ m+1≤m̸=m′≤∞ cn,mcn,m′ψ (c1) n,m(t)ψ (c1) n,m′(t) + ∑ ∑ 1⩽ n̸=n′≤ 2k−1 ∞∑ m=m+1 cn,mcn′,mψ (c1) n,m(t)ψ (c1) n′,m(t) + ∑ ∑ 1⩽ n̸=n′≤ 2k−1 ∑ ∑ m+1≤m̸=m′≤∞ cn,mcn′,m′ψ (c1) n,m(t)ψ (c1) n′,m′(t) ||i||22 = ∫ 1 0 |f(t) − s2k−1,m(f)(t)|2ωn(t)dt = 2k−1∑ n=1 ∞∑ m=m+1 |cn,m|2 ∫ 1 0 |ψ (c1) n,m(t)| 2ωn(t)dt + 2k−1∑ n=1 ∑ ∑ m+1≤m̸=m′≤∞ cn,mcn,m′ ∫ 1 0 ( ψ (c1) n,m(t)ψ c n,m′(x) ) ωn(t)dt + ∑ ∑ 1⩽ n̸=n′≤ 2k−1 ∞∑ m=m+1 cn,mcn′,m ∫ 1 0 ( ψ (c1) n,m(t)ψ (c1) n′,m(t) ) ωn(t)dt + ∑ ∑ 1⩽ n̸=n′≤ 2k−1 ∑ ∑ m+1≤m̸=m′≤∞ cn,mcn′,m′ ∫ 1 0 ψ (c1) n,m(t)ψ (c1) n′,m′(t)ωn(t)dt = 2k−1∑ n=1 ∞∑ m=m+1 |cn,m|2, by orthonormality of {ψ (c1) n,m}. 64 shyam lal and deepak kumar singh ||f − s2k−1,m(f)||22 ≤ 2k−1∑ n=1 ∞∑ m=m+1 [ 1 m.2k−1 √ 2 π ( 1 2kα )]2 = 2 π .2k−1. 1 22(k−1) . 1 22kα ∞∑ m=m+1 1 m2 = 22 π 2k 22kα ∞∑ m=m+1 1 m2 ≤ 22 π 2k 22kα [ 1 (m + 1)2 + ∫ ∞ m+1 1 m2 dm ] by cauchy integral test, ≤ 22 π 2k 22kα [ 1 (m + 1) + ( −1 m )∞ m+1 ] = 22 π 2k 22kα [ 1 (m + 1) + 1 m + 1 ] ∥f − s2k−1,m(f)∥2 ≤ 2 √ 2 π 1 2k(α+1/2) √ m + 1 = o ( 1 2k(α+1/2) √ m + 1 ) w (( f − s2k−1,mf ) , 1 2k ) = sup 0<h≤ 1 2k ∥ ( f − s2k−1,mf ) (· + h) − ( f − s2k−1,mf ) (·)∥2 ≤ || ( f − s2k−1,0f ) ||2 + || ( f − s2k−1,mf ) ||2 = 2|| ( f − s2k−1,mf ) ||2 = 2.o ( 1 2k(α+1/2) √ m + 1 ) = o ( 1 2k(α+1/2) √ m + 1 ) . thus, this theorem is completely established. 65 moduli of continuity of functions in hölder’s class.... 5 corollary corolary 5.1. if f ∈ hα,2ωk [0,1) and its first kind chebyshev wavelet expansion be f(t) = ∞∑ n=1 ∞∑ m=0 cn,mψ (c1) n,m(t) having (2k−1,m + 1)th partial sums ( s2k−1,m(f) ) (t) = 2k−1∑ n=1 m∑ m=0 cn,mψ (c1) n,m(t) then the first kind chebyshev wavelet approximation e2k−1,m(f) of f is given by (i)e2k−1,0(f) = min||f − (s2k−1,0f)||2 = min||f − 2k−1∑ n=1 cn,0ψ (c1) n,0 (t)||2 = o ( 1 2(k−1)α ) (ii)e2k−1,m(f) = min||f − (s2k−1,mf)||2 = min||f − 2k−1∑ n=1 m∑ m=0 cn,mψ (c1) n,m(t)||2 = o ( 1 2k(α+1/2) √ m + 1 ) the proof of corollary (5.1) can be developed parallel to the proof of theorem (3.1) independently. remark if f ∈ hα,2ωk [0,1), then the moduli of continuity w (( f − s2k−1,mf ) , 1 2k ) = o ( 1 2k(α+1/2) √ m+1 ) and approximation e2k−1,m(f) = o ( 1 2k(α+1/2) √ m+1 ) tends to 0 as k → ∞ , m→ ∞ . hence w (( f − s2k−1,mf ) , 1 2k ) and e2k−1,m(f) are best possible modulus of continuity and approximation of functions respectively in wavelet analysis.it is also observed that w (( f − s2k−1,mf ) , 1 2k ) ≤ e2k−1,m(f). hence modulus of continuity is sharper than the approximation of function in h α,2 ωk [0,1) by first kind chebyshev wavelet method. 66 shyam lal and deepak kumar singh 6 first kind chebyshev wavelet operational matrix of integration in this section, the operational matix p of integration of order 2k−1(m + 1) × 2k−1(m + 1)has been derived. this plays a great role in dealing with the problem of lane-emden differential equations ( babolian [5]) . first the matrix p8×8 has been obtained for m = 3 and k = 2 . there are eight basis wavelet functions given below: ψ1,0(t) = { 2√ π , 0 ≤ t < 1 2 , 0, 1 2 ≤ t < 1, ψ1,1(t) = { 2 √ 2 π (4t − 1), 0 ≤ t < 1 2 , 0, 1 2 ≤ t < 1, ψ1,2(t) = { 2 √ 2 π (32t2 − 16t + 1), 0 ≤ t < 1 2 , 0, 1 2 ≤ t < 1, ψ1,3(t) = { 2 √ 2 π (256t3 − 192t2 + 36t − 1), 0 ≤ t < 1 2 , 0, 1 2 ≤ t < 1, ψ2,0(t) = { 0, 0 ≤ t < 1 2 , 2√ π , 1 2 ≤ t < 1, ψ2,1(t) = { 0, 0 ≤ t < 1 2 , 2 √ 2 π (4t − 3), 1 2 ≤ t < 1, ψ2,2(t) = { 0, 0 ≤ t < 1 2 , 2 √ 2 π (32t2 − 48t + 17), 1 2 ≤ t < 1, ψ2,3(t) = { 0, 0 ≤ t < 1 2 , 2 √ 2 π (256t3 − 576t2 + 420t − 99), 1 2 ≤ t < 1. let, ψ(t) = [ψ10,ψ11,ψ12,ψ13,ψ20,ψ21,ψ22,ψ23] t (14) by integrating the basis functions from 0 to t and expressing them in term of chebyshev wavelet series , 67 moduli of continuity of functions in hölder’s class.... ∫ t 0 ψ1,0(x) = { 2√ π t, 0 ≤ t < 1 2 , 1√ π , 1 2 ≤ t < 1, = [ 1 4 , 1 4 √ 2 ,0,0, 1 2 ,0,0,0 ] ψ(t) ∫ t 0 ψ1,1(x) = { 2 √ 2 π (4t − 1), 0 ≤ t < 1 2 , 0, 1 2 ≤ t < 1, = [ − 1 8 √ 2 ,0, 1 16 ,0,0,0,0,0 ] ψ(t) ∫ t 0 ψ1,2(x) = { 2 √ 2 π (32t2 − 16t + 1), 0 ≤ t < 1 2 , 0, 1 2 ≤ t < 1, = [ − 1 6 √ 2 ,− 1 8 ,0, 1 24 ,− 1 3 √ 2 ,0,0,0 ] ψ(t) ∫ t 0 ψ1,3(x) = { 2 √ 2 π (256t3 − 192t2 + 36t − 1), 0 ≤ t < 1 2 , 0, 1 2 ≤ t < 1, = [ 1 16 √ 2 ,0,− 1 16 ,0,0,0,0,0 ] ψ(t)∫ t 0 ψ2,0(x) = { 0, 0 ≤ t < 1 2 , 2√ π , 1 2 ≤ t < 1, = [ 0,0,0,0, 1 4 , 1 4 √ 2 ,0,0 ] ψ(t) ∫ t 0 ψ2,1(x) = { 0, 0 ≤ t < 1 2 , 2 √ 2 π (4t − 3), 1 2 ≤ t < 1, = [ 0,0,0,0,− 1 8 √ 2 ,0, 1 16 ,0 ] ψ(t) ∫ t 0 ψ2,2(x) = { 0, 0 ≤ t < 1 2 , 2 √ 2 π (32t2 − 48t + 17), 1 2 ≤ t < 1, = [ 0,0,0,0,− 1 6 √ 2 ,− 1 8 ,0, 1 24 ] ψ(t) ∫ t 0 ψ2,3(x) = { 0, 0 ≤ t < 1 2 , 2 √ 2 π (256t3 − 576t2 + 420t − 99), 1 2 ≤ t < 1. = [ 0,0,0,0, 1 16 √ 2 ,0,− 1 16 ,0 ] ψ(t) 68 shyam lal and deepak kumar singh thus, ∫ t 0 ψ(x)dx = p8×8ψ(t) ( m. a. fariborzi[13]) (15) where p8×8 =   1 4 1 4 √ 2 0 0 1 2 0 0 0 − 1 8 √ 2 0 1 16 0 0 0 0 0 − 1 6 √ 2 −1 8 0 1 24 − 1 3 √ 2 0 0 0 1 16 √ 2 0 − 1 16 0 0 0 0 0 0 0 0 0 1 4 1 4 √ 2 0 0 0 0 0 0 − 1 8 √ 2 0 1 16 0 0 0 0 0 − 1 6 √ 2 −1 8 0 1 24 0 0 0 0 1 16 √ 2 0 − 1 16 0   (16) 7 the product operation matrix (pom) in this section , the product operation matrix has been obtained. this has important role in solving differential equations ( babolian [5]). by equation (14), ψ ψt =  ψ1,0ψ1,0 ψ1,0ψ1,1 ψ1,0ψ1,2 ψ1,0ψ1,3 ψ1,0ψ2,0 ψ1,0ψ2,1 ψ1,0ψ2,2 ψ1,0ψ2,3 ψ1,1ψ1,0 ψ1,1ψ1,1 ψ1,1ψ1,2 ψ1,1ψ1,3 ψ1,1ψ2,0 ψ1,1ψ2,1 ψ1,1ψ2,2 ψ1,1ψ2,3 ψ1,2ψ1,0 ψ1,2ψ1,1 ψ1,2ψ1,2 ψ1,2ψ1,3 ψ1,2ψ2,0 ψ1,2ψ2,1 ψ1,2ψ2,2 ψ1,2ψ2,3 ψ1,3ψ1,0 ψ1,3ψ1,1 ψ1,3ψ1,2 ψ1,3ψ1,3 ψ1,3ψ2,0 ψ2,1ψ2,1 ψ1,3ψ2,2 ψ1,3ψ2,3 ψ2,0ψ1,0 ψ2,0ψ1,1 ψ2,0ψ1,2 ψ2,0ψ1,3 ψ2,0ψ2,0 ψ2,0ψ2,1 ψ2,0ψ2,2 ψ2,0ψ2,3 ψ2,1ψ1,0 ψ2,1ψ1,1 ψ2,1ψ1,2 ψ2,1ψ1,3 ψ2,1ψ2,0 ψ2,1ψ2,1 ψ2,1ψ2,2 ψ2,1ψ2,3 ψ2,2ψ1,0 ψ2,2ψ1,1 ψ2,2ψ1,2 ψ2,2ψ1,3 ψ2,2ψ2,0 ψ2,2ψ2,1 ψ2,2ψ2,2 ψ2,2ψ2,3 ψ2,3ψ1,0 ψ2,3ψ1,1 ψ2,3ψ1,2 ψ2,3ψ1,3 ψ2,3ψ2,0 ψ2,3ψ2,1 ψ2,3ψ2,2 ψ2,3ψ2,3   substituting the values of each entry of above symmetric matrix , and after simplification expressing them in term of chebyshev wavelet series , ψ ψt =   a o4×4 o4×4 b   where o4×4 is a 4 × 4 zero matrix and a =   2√ π ψ1,0 2√ π ψ1,1 2√ π ψ1,2 2√ π ψ1,3 2√ π ψ1,1 2√ π ψ1,0 + √ 2 π ψ1,2 √ 2 π ψ1,1 + √ 2 π ψ1,3 √ 2 π ψ1,2 2√ π ψ1,2 √ 2 π ψ1,1 + √ 2 π ψ1,3 2√ π ψ1,0 √ 2 π ψ1,1 2√ π ψ1,3 √ 2 π ψ1,2 √ 2 π ψ1,1 2√ π ψ1,0   69 moduli of continuity of functions in hölder’s class.... b =   2√ π ψ2,0 2√ π ψ2,1 2√ π ψ2,2 2√ π ψ2,3 2√ π ψ2,1 2√ π ψ1,0 + √ 2 π ψ1,2 √ 2 π ψ2,1 + √ 2 π ψ2,3 √ 2 π ψ2,2 2√ π ψ2,2 √ 2 π ψ2,1 + √ 2 π ψ2,3 2√ π ψ2,0 √ 2 π ψ2,1 2√ π ψ2,3 √ 2 π ψ2,2 √ 2 π ψ2,1 2√ π ψ2,0   8 solution of the lane -emden differential equation by first kind chebyshev wavelet many problems in the literature of mathematical physics can be distinctively formulated as equations of lane-emden type as follows: d2y dt2 + 2 t dy dt + l(y) = 0 ( a-m wazwaz[12]) where l(y) is some given function of y. a difficult element in the analysis of this type of equations is the singularity behaviour that occurs at t = 0. most algorithms currently in use for handling the lane-emden type problems are based on either series solution or perturbation techniques. in recent years, a lot of attention has been devoted to the study of wavelet theory. the first kind chebyshev wavelet method accurately computes the solution of differential equations and it is of great interest to solve this type of problem. the most popular form of l(y) attracted by the scientific community is l(y) = yn where n is constant parameter. the standard lane-emden equation of index n is of the form d2y dt2 + 2 t dy dt + yn = 0, with y(0) = 1, y′(0) = 0 (17) it is a basic equation in the theory of stellar structure. the equation describes the temperature variation of a spherical gas cloud under the mutual attraction of its molecules . it was physically shown that interesting values of n lie in the interval [0,5] . in addition, exact solutions exist only for n = 0,1,5. notice that equation (17) is linear for n = 0,1 and non-linear otherwise. example (1). for n = 0, the lane-emden differential equation (17) reduces to ty′′ + 2y′ + t = 0 with y(0) = 1,y′(0) = 0 (18) 70 shyam lal and deepak kumar singh the exact solution of the equation (18) is y(t) = 1 − t 2 6 . the equation (18) has been solved using first kind chebyshev wavelet. first it is assumed that y′′(t) can be expanded in terms of chebyshev wavelet as y′′(t) = ct ψ(t) (19) where ψ(t) is given by equation (14). by integrationg equation (19) twice with respect to t from 0 to t and using conditions of equation (18) & (15) following equations are obtained , y′(t) = ct p ψ(t) (20) and, y(t) = ct p2 ψ(t) + dt ψ(t) (21) where, dt = [ √ π 2 ,0 , 0 ,0 , √ π 2 ,0 ,0, 0] (22) using equation (14), t is expressed as t = et ψ(x) (23) where, et = [ √ π 8 , √ π 2 8 , 0, 0, 3 √ π 8 , √ π 2 8 , 0, 0] (24) substituting values from equation (19) to (24) in equation (18), et ψ(t)ψt (t)c + 2ctpψ(t) + et ψ(t) = 0 (25) the following property of the product of two first kind chebyshev wavelet vectors will also be used: et ψ(t)ψt (t) ≃ ψt (t)ẽ (26) where ẽ is 8×8 matrix. let us establish equation (26) & find ẽ, et ψψt = 1 4 [ ψ1,0+ 1 √ 2 ψ1,1, 1 √ 2 ψ1,0+ψ1,1+ 1 2 ψ1,2, ψ1,2+ 1 2 ψ1,1+ 1 2 ψ1,3, ψ1,3 + ψ1,2 2 , 3ψ2,0+ ψ2,1√ 2 , 3ψ2,1+ 1 √ 2 ψ2,0+ ψ2,2 2 , ψ2,1 2 +3ψ2,2+ 1 2 ψ2,3, 3ψ2,3+ 1 2 ψ2,2 ] et ψψt =   ψ1,0 ψ1,1 ψ1,2 ψ1,3 ψ2,0 ψ2,1 ψ2,2 ψ2,3   · 1 4   1 1√ 2 0 0 0 0 0 0 1√ 2 1 1 2 0 0 0 0 0 0 1 2 1 1 2 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 0 3 1√ 2 0 0 0 0 0 0 1√ 2 3 1 2 0 0 0 0 0 0 1 2 3 1 2 0 0 0 0 0 0 1 2 3   71 moduli of continuity of functions in hölder’s class.... where ẽ = 1 4   1 1√ 2 0 0 0 0 0 0 1√ 2 1 1 2 0 0 0 0 0 0 1 2 1 1 2 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 0 3 1√ 2 0 0 0 0 0 0 1√ 2 3 1 2 0 0 0 0 0 0 1 2 3 1 2 0 0 0 0 0 0 1 2 3   8×8 (27) using equation (26), equation (25) becomes (ψt (t)ẽ)c + ψt (t)(2ptc) + ψt (t)e = 0 ψt (t) [ ẽc + (2pt c) + e ] = 0 ẽc + (2pt c) + e = 0 (28) equation (28) is a set of algebraic equations which is solved for c in following form c = ( ẽ + 2pt )−1 (−e) c =   −0.295408975150919 0.000000000000000 −0.000000000000000 −0.000000000000000 −0.295408975150919 0.000000000000000 −0.000000000000000 0.000000000000000   (29) by substituting the chebyshev wavelet coefficients c from equation (29) into equation (21), the explicit form of the approximate solution of equation (18) is, y(t) = [ 0.872379629742559,−0.013055355597036,−0.003263838899259,0, 0.798527385954829,−0.039166066791109,−0.003263838899259,0 ] ψ(t) using the simplified value of ψ(t) from equation (14), y(t) becomes y(t)= { 1 − 0.166666666666660t2, 0 ≤ t < 1 2 , 1 − 6.357719098970234.10−15 t − 0.166666666666661t2, 1 2 ≤ t < 1. the exact solution and the solution obtained by chebyshev wavelet method of differential equation (18) at different values of t are given in table (1): 72 shyam lal and deepak kumar singh variable (t) exact solution chebyshev solution absolute error 0 1.000000000000000 1.000000000000000 0 0.1 0.998333333333333 0.998333333333333 0 0.2 0.993333333333333 0.993333333333334 0.000000000000001 0.3 0.985000000000000 0.985000000000001 0.000000000000001 0.4 0.973333333333333 0.973333333333334 0.000000000000001 0.5 0.958333333333333 0.958333333333333 0 0.6 0.940000000000000 0.939999999999999 0.000000000000001 0.7 0.918333333333333 0.918333333333333 0 0.8 0.893333333333333 0.893333333333333 0 0.9 0.865000000000000 0.865000000000000 0 table(1): comparison table for the exact and chebyshev wavelet solution of lane-emden equation of index 0. this table shown that the solution of equation (18) obtained by first kind chebyshev wavelet method nearly coincides with its exact solution.the graphs of exact and first kind chebyshev wavelet solutions of lane -emden differential equation for n = 0 are drawn in figure (1). 73 moduli of continuity of functions in hölder’s class.... example (2). for n = 1, the lane-emden differential equation (17) reduces to ty′′ + 2y′ + ty = 0 with y(0) = 1,y′(0) = 0 (30) the exact solution of the equation (30) is y(t) = { sin(t) t , t ̸= 0 1 t = 0. substituting values from equation (19) − (24) in equation (30) , et ψ(t)ψt (t)c + 2ctpψ(t) + et ψ(t)(ct p2 + dt )ψ(t) = 0 c = [ ẽ + 2pt + ẽ(p2)t ]−1 (−ẽd) c = [−0.2871902685,0.0077317973,0.0019076755,−0.0000134466, −0.2449764764,0.0218383818,0.0015733213,−0.0000401720]t (31) by substituting the chebyshev wavelet coefficients c from equation (31) into equation (21), the explicit form of the approximate solution of equation (30) is, y(t) = [ 0.872505218190184,−0.012913303990151,−0.003192905781766, 0.000020169906045,0.801594934932526,−0.036835535048721, −0.002723022009203, 0.000056975401025 ] ψ(t) or, y(t)=   0.9999960361853 + 2.5435469133605.10−4 t − 0.1692243050960t2+ 0.0082397473925t3, 0 ≤ t < 1 2 0.9979746687982 + 0.0116372814448t − 0.1914197434042t2+ 0.0232754139255t3 1 2 ≤ t < 1. table (2): comparisons between the exact solution and numerical solutions for various values of t variable (t) exact solution chebyshev solution absolute error 0 1 0.999996036185 0.0396381467104×10−4 0.1 0.99833416646 0.998337468350 0.0330188261299×10−4 0.2 0.99334665397 0.993343852898 0.0280107641192×10−4 0.3 0.98506735553 0.985064628313 0.0272722411598×10−4 0.4 0.97354585577 0.973549233079 0.0337730799104×10−4 0.5 0.95885107720 0.958847800410 0.0327679807199×10−4 0.6 0.94107078899 0.941073419447 0.0263045585602×10−4 0.7 0.92031098176 0.920308558518 0.0242325006294×10−4 0.8 0.89669511362 0.896692870105 0.0224351906097×10−4 0.9 0.87036323291 0.870366006692 0.0277377353607×10−4 74 shyam lal and deepak kumar singh this table shown that the solution of equation (30) obtained by first kind chebyshev wavelet method nearly coincides with its exact solution.the graphs of exact and first kind chebyshev wavelet solutions of lane -emden differential equation for n = 1 are drawn in figure (2). example (3). for n = 2 , the lane-emden differential equation is ty′′ + 2y′ + ty2 = 0 with y(0) = 1,y′(0) = 0 (32) the taylor series expansion of function l(y) about a point y = 1 is given as, l(y) = ∞∑ m=0 (y − 1)m m! dml(y) dym ∣∣∣∣ y=1 y2 = 1 + 2 (y − 1) + (y − 1)2 from equation (21) & (22), = dt ψ(t) + 2ct p2 ψ(t) + ct p4ψ(t) substituting the taylor series exapansion of y2 and values from equation (19) to (24) in equation (32) , et ψ(t)ψt (t)c + 2ctpψ(t) + et ψ(t)ψt (t) [ d + 2 (pt )2 c + (pt )4 c ] = 0 75 moduli of continuity of functions in hölder’s class.... c = [ ẽ + 2pt + 2 ẽ(pt )2 + ẽ(pt )4 ]−1 (−ẽd) (33) c = [−0.279001698770205,0.015429503002548,0.003798307476540, − 0.000031441187989,−0.195293820350412,0.043103314900479, 0.003011315345934,−0.000095144707040]t (34) by substituting the chebyshev wavelet coefficients c from equation (34) into equation (21), the explicit form of the approximate solution of equation (32) is, y(t) = [ 0.872630597117721,−0.012771501223632,−0.003122128096052, 0.000040262875496,0.804642516403461,−0.034524067315646, −0.002189080129024,0.000112495988561 ] ψ(t) or, y(t)=   0.9999921077841 + 5.0666971784118.10−4 t − 0.1717663081650t2 +0.016448064886675t3, 0 ≤ t < 1 2 0.996061399263954 + 0.022704430920508t − 0.215186679101439t2 +0.045956511961640t3 1 2 ≤ t < 1. 9 conclusion (i) in theorem (3.1) , the moduli of continuity has been estimated as following: w (( f − s2k−1,0f ) , 1 2k ) = o ( 1 2(k−1)α ) → 0 as k → ∞, w (( f − s2k−1,mf ) , 1 2k ) = o ( 1 2k(α+1/2) √ m + 1 ) → 0 as k → ∞,m → ∞. (ii) in corollary (5.1) , e2k−1,0(f) = o ( 1 2(k−1)α ) → 0 as k → ∞, e2k−1,m(f) = o ( 1 2k(α+1/2) √ m + 1 ) → 0 as k → ∞,m → ∞. thus w (( f − s2k−1,0f ) , 1 2k ) , w (( f − s2k−1,mf ) , 1 2k ) , e2k−1,0(f) & e2k−1,m(f) are best possible estimators in wavelet analysis. 76 shyam lal and deepak kumar singh (iii) from theorem (3.1) and corollary (5.1) , it is observed that w (( f − s2k−1,0f ) , 1 2k ) ≤ 2e2k−1,0(f) w (( f − s2k−1,mf ) , 1 2k ) ≤ 2e2k−1,m(f) hence moduli of continuity w (( f − s2k−1,0f ) , 1 2k ) , w (( f − s2k−1,mf ) , 1 2k ) are better and sharper than approximations e2k−1,0(f), e2k−1,m(f) respectively. (iv) solution of lane-emden differential equations by first kind chebyshev wavelet is approximately same as exact solution of differential equations. this is the significant achievement of this research paper . calculations performed in section(8) demonstrate that the accuracy of chebyshev wavelet method is quite high. in this method , there is no complex integral or methodology. application of this proposed method is very simple & gives the explicite form of approximate solutions to the lane emden differential equations. these are the main advantages of the method. this method is also very convenient for solving the boundary value problems. hence, this proposed method is very reliable, simple, fast & computationally efficients method. acknowledgments shyam lal, one of the authors, is thankful to dst cims for encouragement to this work. deepak kumar singh, one of the authors, is grateful to u.g.c. (india) for providing financial assistance in the form of ugc non-net fellowship for his research work. authors are greatful to the referee for his valuable comments and suggestions, to improve the quality of this research paper. references [1] r.a.devore, nonlinear approximation,1998, acta numerica, cambridge university press ( cambridge), vol.7, pp. 51-150 . [2] l.debnath, wavelet transform and their applications,2002, birkhauser , massachusetts(bostoon) . 77 moduli of continuity of functions in hölder’s class.... [3] y.meyer, progress in wavelet analysis and applications , 1993,frontieres, gif-sur-yvette, pp. 9-18 . [4] j.morlet, g.arens, e.fourgeau and d.giard, wave propagation and sampling theory-part ii: sampling theory and complex waves, geophysics,1982,vol. 47(2),pp. 222-236 . [5] e.babolian, and f. fattahzadeh, numerical solution of differential equations by using chebyshev wavelet operational matrix of integration, appl. math. and comput.,2007, vol. 188(1), pp. 417–426 . 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[14] b.sripathy, p.vijayaraju, g.hariharan, chebyshev wavelet based approximation method to some non-linear differential equations arising in engineering, int. j. of math. anal,2015, vol.9, pp.993–1010 . 78 ratio mathematica volume 44, 2022 relationship between weight function and 1 – norm m. melna frincy* j. robert victor edward† abstract the function on a subset of is the function defined by for 𝑥 , we define . the hamming weight of is the number of non – zero coordinates of , where from this one could see that , where is the 1 – norm of given by where . this gives a relationship between the weight function and the 1 – norm.in this paper we establish certain properties of the weight function using the properties of norms. keywords:mininorm, mininormed space, 1-norm, weight function. 2010 ams subject classification: 90b06‡ *research scholar, department of mathematics, scott christian college(autonomous) nagercoil629003, tamilnadu, india. affiliated to manonmaniumsundaranar university, tirunelveli – 627012, tamilnadu, india.email: melnabensigar84@gmail.com. †department of mathematics, scott christian college(autonomous)nagercoil629003,tamilnadu,indiaaffiliated to manonmaniumsundaranar university, tirunelveli – 627012, tamilnadu, india.email: jrvedward@gmail.com. ‡received on june 24, 2022. accepted on aug 10th , 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.914. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 268 mailto:melnabensigar84@gmail.com mailto:jrvedward@gmail.com m. melna frincy and dr. j. r. v. edward 1. introduction let , where is the set of real numbers. then, is a vector space over of dimension the hamming weight function on is the function given by number of non –zero co-ordinates of for . thus satisfies the conditions: for all and if and only if (1) for all and . (2) for all (3) a norm on is a function satisfying for all and if and only if (4) for all and . (5) and for all (6) we see that satisfies the condition of a norm except the condition (5). instead, it satisfies (2). we may call such a function a mininorm. let us formalize the definition. definition:1.1let be a vector space over or a mininorm on is a function satisfying the following conditions: for all and if and only if (7) for all and (8) for all (9) a vector space with a mininorm defined on it is called a mininormed spaces. it is clear that is a mininorm on 2.the weight function and the 1norm the 1norm or on is defined by , where (10) we cannot connect the weight function with the 1norm using the function. the – function on is defined by (11) the – function can be extended to in the following way: , (12) where this – function satisfies the following 269 relationship between weight function and 1norm for all and if and only if (13) for all and (14) and for all (15) hence the partial order relation on is defined as follows: for and in if and only if (16) now let then, now, hence = number of nonzero components of thus, (17) this gives the connection between the hamming weight function and the 1norm, via the – function. 3.topological properties of the – function proposition:3.1the – function on is bounded. proof:let since for all hence is bounded. proposition:3.2the – function on is not continuous. proof:first we show that is not continuous, let then as that is, in with . 270 m. melna frincy and dr. j. r. v. edward but for all n. so, hence is not continuous. now for all thus, where denotes the composition of functions. is continuous . suppose is continuous. hence is continuous, since the composition of two continuous functions is continuous. this is not possible. hence is not continuous acknowledgements the authors thank the referees for their valuable suggestions and comments. references [1] m. melnafrincy and j.r.v. edward – extension of the – function to . turkish online journal of qualitative inquiry . volume 12, issue 6, june 2021: 714-718. [2]justesan and hoholdt – a course in error correcting codes. hindustan bork agency, new delhi, 2004. [3] e.kreyszig – introductory functional analysis with applications. john wiley & sons, new york, 1978. [4] b.v.limaye – functional analysis, new age international publishers, new delhi, 1996. [5] g. f. simmons – introduction to topology and modern analysis. mc – graw hill, tokyo, 1963. 271 ratio mathematica volume 44, 2022 on intuitionistic semi * continuous functions g. esther rathinakani* m. navaneethakrishnan† abstract in this paper we introduce intuitionistic semi * continuous and intuitionistic contra semi * continuous functions via the concept of intuitionistic semi * open and intuitionistic semi * closed set respectively. also, we investigate their properties and characterization. keywords: intuitionistic semi * open, intuitionistic semi * closed, intuitionistic semi * continuous, intuitionistic contra-semi * closed. 2010 ams subject classification: 54c05‡ *research scholar, reg. no. 19222102092011, pg and research department of mathematics, kamaraj college, thoothukudi, tamilnadu, india. affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli, tamilnadu, india; e-mail estherrathinakani@gmail.com. †associate professor, pg and research department of mathematics, kamaraj college, thoothukudi, tamilnadu, india. affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli, tamilnadu, india; e-mail navaneethan65@yahoo.co.in. ‡ received on june 8th, 2022. accepted on aug 10th, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.891. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement 65 g. esther rathinakani and m. navaneethakrishnan 1. introduction the concept of intuitionistic set introduce by d. coker [2] in 1996 and also he [1] has introduced the concept of intuitionistic topological space. in [5], we introduced the concept of intuitionistic generalized closure operator and defined a new intuitionistic topology τ* and studied their properties. also, in [6] we introduced a new open set namely intuitionistic semi * open and studied their properties. in this paper we define intuitionistic semi * continuous and intuitionistic contra semi * continuous functions via the concept of intuitionistic semi * open and intuitionistic semi * closed set respectively. also, we investigate their properties and characterization. 2.preliminaries definition 2.1 [1] let x be a nonempty fixed set. an intuitionistic set (is in short) ã is an object having the form ã =< 𝑋, 𝐴1, 𝐴2 > where 𝐴1 and 𝐴2 are subsets of 𝑋 such that 𝐴1 ∩ 𝐴2 = ∅. the set 𝐴1 is called the set of member of ã , while 𝐴2 is called the set of non member of ã . definition 2.2 [1] an intuitionistic topology (it in short) by subsets of a nonempty set x is a family 𝜏 of is’s satisfying the following axioms. (a) ∅̃i , xĩ ∈ 𝜏, (b) g̃1 ∩ g̃2 ∈ 𝜏 for every g̃1 , g̃2 ∈ 𝜏, and (c) ∪ g̃ 𝑖 ∈ 𝜏 for any arbitrary family {g̃ : 𝑖 ∈ 𝐽} ⊆ 𝜏. the pair (𝑋, 𝜏) is called an intuitionistic topological space (its in short) and any is ã in 𝜏 is called an intuitionistic open set (ios). the complement of an io set ã in is called an intuitionistic closed set (ics). definition 2.3 [1] let (𝑋, 𝜏) be an its and ã =< 𝑋, 𝐴1, 𝐴2 > be an is in x. then the interior and the closure of 𝐴 are denoted by iint(ã ) and i𝑐𝑙(ã ), and are defined as follows. iint(ã ) = ∪ {g̃ | g̃ is an ios and g̃ ⊆ ã } and icl(ã ) = ∩ {k̃ | k̃ is an ics and ã ⊆ k̃}. definition 2.4 [2] let x be a nonempty set and p ∈ x be a fixed element. then the is p̃ defined by p̃ =< x, {p}, {p} c > is called an intuitionistic point (in short, ip). definition 2.5 [10] let (𝑋, 𝜏) be an its and ã =< 𝑋, 𝐴1, 𝐴2 > be an is in x, ã is said to be intuitionistic generalized closed set (briefly ig – closed set ) icl(ã ) ⊆ ũ whenever ã ⊆ ũ and ũ is ios in x. 66 on intuitionistic semi * continuous functions definition 2.6 [5] if �̃� is an is of an its (x, τ), then the intuitionistic generalized closure of �̃� is defined as the intersection of all ig – closed sets in x containing �̃� and is denoted by icl*(�̃�). definition 2.7 [6] the is �̃� of an its (x, τ) is called intuitionistic semi * open sets if there is an intuitionstic open set �̃� in x such that �̃� ⊆ �̃� ⊆ icl*(�̃�). definition 2.8 [6] the intuitionistic semi * interior of �̃� is defined as the union of all intuitionistic semi * open sets of x contained in �̃�. it is denoted by is*int(�̃�). definition 2.9 an intuitionistic set �̃� of a its (x, τ) is called an intuitionistic semi * closed set if x �̃� is intuitionistic semi * open. definition 2.10 the semi * closure of an is �̃� is defined as the intersection of all intuitionistic semi * closed sets in x that containing �̃�. it is denoted by is*cl(�̃�). theorem 2.11 let (x, τ) be an its and �̃� be an is of x. then (i) is*cl(x �̃�) = x – is*int(�̃�) (ii) i s*int(x �̃�) = x – is*cl(�̃�) definition 2.12[9] a function f : x → y is said to be intuitionistic semi continuous if 𝑓 −1(�̃�) is iso in x for every ios �̃� in y. theorem 2.13[6] let (x, τ) be an its and �̃� be an is of x. then (i) every ios is is*o. (ii) every is*o is iso. (iii) every ics is is*c. (iv) every is*c is isc. theorem 2.14[6] let (x, τ) be an its. then (i) if {�̃�𝛼} is a collection of is*o in x then ⋃ �̃�𝛼 is is*o. (ii) if �̃� is is*o in x and �̃� is an ios in x, then �̃� ∩ �̃� is is*o in x. theorem 2.15[6] let (x, τ) be an its and �̃� be an is of x. then (i) �̃� is is*o if and only if is*int(�̃�) = �̃�. (ii) �̃� is is*c if and only if is*cl(�̃�) = �̃�. 67 g. esther rathinakani and m. navaneethakrishnan theorem 2.16[6] let (x, τ) be an its, �̃� be an is of x and 𝑝 ∈ x. then 𝑝 ∈ is*cl(�̃�) if and only if every is*o in x containing 𝑝 intersects �̃�. definition 2.17[7] let (x, τ) be an its and �̃� be an is of x. then the intuitionistic semi * frontier of �̃� (denoted by is*fr (�̃�)) is defined by is*fr (�̃�) = is*cl(�̃�) is*int (�̃�). theorem 2.18[7] let (x, τ) be an its and �̃� be an is of x. then is*fr (�̃�) = is*cl(�̃�) ⋂ is*cl (�̃�). 3. intuitionistic semi * continuous functions definition 3.1 a function f : x →y is said to be intuitionistic semi * continuous at 𝑝 ∈ x if for each intuitionistic open set �̃� of y containing f(𝑝), there is an intuitionistic semi open set �̃� in x such that 𝑝 ∈ �̃� and f(�̃�) ⊆ �̃�. definition 3.2 a function f : x → y is said to be intuitionistic semi * continuous if 𝑓 −1(�̃�) is is*o in x for every ios �̃� in y. theorem 3.3 every intuitionistic continuous function is intuitionistic semi * continuous. proof: let f : x → y be intuitionistic continuous and �̃� be io in y. then 𝑓 −1(�̃�) is io in x. therefore by theorem 2.13(i), 𝑓 −1(�̃�) is is*o in x. hence f is intuitionistic semi * continuous function. remark 3.4 the converse of the above theorem need not be true as seen from the succeeding example example 3.5 let x = {i, j, k} = y and τ1= {𝑋�̃�, ∅̃𝐼 , < x, {i},{j, k} >, < x, {j}, {i, k} >, < x, {i, j}, {k}>}, τ2= {𝑋�̃�, ∅̃𝐼 , < x, {j},{i, k} >, < x, {i}, {j} >, < x, {i, j}, ∅>}. let f: (x, τ1) → (y, τ2) be defined by f(i) = j, f(j) = i, f(k) = k. then f is intuitionistic semi * continuous. let �̃� = < x, {i, j}, ∅>. then 𝑓 −1(�̃�) = < x, {k, i}, ∅> is not ios in τ1. therefore f is not an intuitionistic continuous. corollary 3.6 every constant function is intuitionistic semi * continuous function. proof: we know that every constant function is intuitionistic continuous function. therefore by theorem 3.3 every constant function is intuitionistic semi * continuous function. 68 on intuitionistic semi * continuous functions theorem 3.7 let ß be the intuitionistic basis of the intuitionistic topological space y. then the function f: x →y is intuitionistic semi * continuous if and only if inverse image of every basic ios in y under the function f is is*o in x. proof: let f: x →y be intuitionistic semi * continuous. then the inverse image of every ios in y is is*o in x. in particular, the inverse image of every basic ios in y is is*o in x. coversely, assume that �̃� be an ios in y. then �̃� = ⋃ �̃�α where �̃�α∈ ß. now 𝑓 −1(�̃�) = 𝑓 −1(⋃ �̃�α) = ⋃ 𝑓 −1(�̃�α). therefore by hypothesis, 𝑓 −1 (�̃�α) is is*o for each α. then by theorem 2.14(i), 𝑓 −1 (�̃�) is is*o. hence the function f is intuitionistic semi * continuous. theorem 3.8 every intuitionistic semi * continuous function is intuitionistic semi continuous. proof: let f: x →y be intuitionistic semi * continuous function and �̃� be an ios in y. then 𝑓 −1(�̃�) is is*o in x. therefore by theorem 2.13(ii), 𝑓 −1(�̃�) is iso in x. hence f is intuitionistic semi continuous. remark 3.9 the following example shows that the converse of the above theorem need not be true. example 3.10 let x = {i, j, k} = y and τ1= {𝑋�̃�, ∅̃𝐼 , < x, {j},{i, k} >, < x, {i}, {j} >, < x, {i, j}, ∅>}, τ2= {𝑋�̃�, ∅̃𝐼 , < x, {i},{j} >, < x, {i, j}, ∅>}. let f: (x, τ1) → (y, τ2) be defined by f(i) = j, f(j) = i, f(k) = k. then f is intuitionistic semi continuous. let �̃� = < x, {j}, {i}>. then 𝑓 −1(�̃�) = < x, {j}, {i}> is not an is*o in τ1. therefore f is not an intuitionistic semi * continuous. lemma 3.11 let (x. τ) be an its and �̃� be an is of x. then (i) �̃� is is*o in x if and only if icl*(iint(�̃�)) = icl*(�̃�). (ii) �̃� is is*c in x if and only if iint*(icl(�̃�)) = iint*(�̃�). proof: (i) let �̃� be an is*o. then by definition of is*o we have �̃� ⊆ icl*(iint(�̃�)). hence icl*(�̃�) ⊆ icl*(iint(�̃�)). also we have iint(�̃�) ⊆ �̃� , icl*(iint(�̃�)) ⊆ icl*(�̃�). thus icl*(iint(�̃�)) = icl*(�̃�). on the other hand, let icl*(iint(�̃�)) = icl*(�̃�). then by definition of is*o, �̃� is is*o. (ii) �̃� is is*c if and only if x �̃� is is*o. then by (i) �̃� is is*c if and only if icl*(iint(x �̃�)) = icl*(x �̃�). hence �̃� is is*c if and only if iint*(icl(�̃�)) = iint*(�̃�). 69 g. esther rathinakani and m. navaneethakrishnan theorem 3.12 let f: x → y be a function. then the following are equivalent. (i) f is intuitionistic semi * continuous. (ii) f is intuitionistic semi * continuous at each ip of x. (iii) 𝑓 −1(�̃�) is is*c in x for every ics �̃� in y. (iv) f(is*cl(�̃�)) ⊆ icl(f(�̃�)) for every is �̃� of x. (v) is*cl(𝑓 −1 (�̃�)) ⊆ 𝑓 −1(icl(�̃�)) for every is �̃� of y. (vi) iint*(icl(𝑓 −1 (�̃�))) = iint*(𝑓 −1 (�̃�)) for every ics �̃� in y. (vii) icl*( iint(𝑓 −1 (�̃�))) = icl*(𝑓 −1 (�̃�)) for every ios �̃� in y. (viii) 𝑓 −1 (iint(�̃�) ⊆ is*int(𝑓 −1 (�̃�)) for every is �̃� in y. proof: (i) ⟹ (ii). let f: x → y be an intuitionistic semi * continuous. let 𝑝 ∈ x and �̃� be an ios in y containing f(�̃�). then 𝑝 ∈ 𝑓 −1(�̃�). since f is intuitionistic semi * continuous, �̃� = 𝑓 −1(�̃�) is an is*o in x containing 𝑝 such that f(�̃�) ⊆ �̃�. hence f is intuitionistic semi * continuous at each ip of x. (ii)⟹ (iii). let �̃� be an ics in y. then �̃� = y �̃� is an ios in y. let 𝑝 ∈ 𝑓 −1(�̃�). then f(𝑝) ∈ �̃�. by hypothesis, there is a is*o set �̃�𝑝 in x containing 𝑝 such that f(𝑝) ∈ f(�̃�𝑝) ⊆ �̃�. therefore �̃�𝑝 ⊆ 𝑓 −1(�̃�). hence 𝑓 −1(�̃�) = ∪ {�̃�𝑝 : 𝑝 ∈ 𝑓 −1(�̃�)}. by theorem 2.14(i), 𝑓 −1(�̃�) is is*o in x. thus 𝑓 −1(�̃�) = 𝑓 −1(y �̃�) = x 𝑓 −1(�̃�) is is*c in x. hence 𝑓 −1(�̃�) is is*c in x for every ics �̃� in y. (iii)⟹ (iv). let �̃� be an is of x and let �̃� be an ics containing f(�̃�). then by (iii), 𝑓 −1(�̃�) is is*c containing �̃�. this implies that is*cl(�̃�) ⊆ 𝑓 −1(𝑈) and hence f(is*cl(�̃�)) ⊆ �̃�. thus f(is*cl(�̃�)) ⊆ icl(f(�̃�)). (iv)⟹ (v).let �̃� be an is of y. let �̃� = 𝑓 −1(�̃�). by assumption, f(is*cl(�̃�)) ⊆ icl(f(�̃�) ⊆ icl(�̃�).this implies (is*cl(�̃�)) ⊆ 𝑓 −1(icl(�̃�)). hence is*cl(𝑓 −1(�̃�)) ⊆ 𝑓 −1(icl(�̃�)). (v)⟹ (vi). let �̃� be an ics in y. then by (v), is*cl(𝑓 −1(�̃�)) ⊆ 𝑓 −1(icl(�̃�)) = 𝑓 −1 (�̃�). also we have 𝑓 −1 (�̃�) ⊆ is*cl(𝑓 −1(�̃�)). hence is*cl(𝑓 −1(�̃�)) = 𝑓 −1 (�̃�). thus by theorem 2.15(ii), 𝑓 −1 (�̃�) is closed. therefore by lemma 3.11 (ii) iint*(icl(𝑓 −1(�̃�))) = iint*(𝑓 −1 (�̃�)). (vi)⟹ (vii). let �̃� be an ios in y. then y �̃� is ics in y. therefore by assumption, iint*( icl(𝑓 −1(y �̃�))) = iint*(𝑓 −1 (y �̃�)).this implies that icl*(iint (𝑓 −1(�̃�))) = icl*(𝑓 −1 (�̃�)). (vii)⟹ (i). let �̃� be an ios in y. then by assumption, icl*(iint (𝑓 −1(�̃�))) = icl*(𝑓 −1 (�̃�)). now by lemma 3.11 (i), 𝑓 −1 (�̃�) is is*o in x. hence f is intuitionistic semi * continuous. (i)⟹ (viii). let �̃� be any is of y. then iint(�̃�) is ios in y. by intuitionistic semi * continuity of f, 𝑓 −1(iint(�̃�)) is is*o in x and it is contained in 𝑓 −1 (�̃�). hence 𝑓 −1(iint(�̃�)) ⊆ is*int(𝑓 −1(�̃�)). 70 on intuitionistic semi * continuous functions (viii)⟹ (i). let �̃� be an ios in y. then iint(�̃�) = �̃�. by (viii) 𝑓 −1(�̃�) ⊆ is*int(𝑓 −1(�̃�)) and hence 𝑓 −1(�̃�) = is*int(𝑓 −1(�̃�)). therefore by theorem 2.15(i), 𝑓 −1(�̃�) is is*o in x. thus f is intuitionistic semi * continuous. theorem 3.13 the function f: x → y is not an intuitionistic semi * continuous at an ip 𝑝 in x if and only if 𝑝 belongs to the intuitionistic semi * frontier of the inverse image of some ios in y containing f (𝑝). proof: let f be not an intuitionistic semi * continuous at an ip 𝑝. then there is an ios �̃� in y containing f(𝑝) such that f(�̃�) is not an is of �̃� for every is*o set �̃� in x containing 𝑝. hence �̃� ∩ (x 𝑓 −1(�̃�)) ≠ ∅̃𝐼 for every is*o set �̃� containing 𝑝. by theorem 2.16, 𝑝 ̃ ∈ is*cl(x 𝑓 −1(�̃�)). also we have 𝑝 ̃ ∈ 𝑓 −1(�̃�) ⊆ is*cl (𝑓 −1(�̃�)). thus 𝑝 ̃ ∈ is*cl (𝑓 −1(�̃�)) ∩ is*cl(x 𝑓 −1(�̃�)). hence by theorem 2.18, 𝑝 ∈ is*fr(𝑓 −1(�̃�)). conversely, let f be an intuitionistic semi * continuous at an ip 𝑝. let �̃� be any ios in y containing f(�̃�). then 𝑓 −1(�̃�) is an is*o set in x containing 𝑝. hence by theorem 2,15(i), 𝑝 ∈ is*int(𝑓 −1(�̃�)). thus �̃� ∉ is*fr(𝑓 −1(�̃�)). this proves the theorem. theorem 3.14 let f: x → ∏ 𝑋𝛼 be an intuitionistic semi * continuous where ∏ 𝑋𝛼 is the intuitionistic product topology and f(𝑝)= (fα(𝑝)). then each coordinate function fα : x → xα is an intuitionistic semi * continuous. proof: let �̃� be an ios in xα. then 𝑓𝛼 −1(�̃�) = (pα ° f) -1(�̃�) = 𝑓 −1(𝑃𝛼 −1 (𝑈)), where pα : ∏ 𝑋𝛼 → x, the projection map. since pα is intuitionistic continuous, 𝑃𝛼 −1 (�̃�) is ios in ∏ 𝑋𝛼. since f is intuitionistic semi * continuous, 𝑓𝛼 −1(�̃�) = 𝑓 −1(𝑃𝛼 −1 (�̃�)) is is*o in x. thus each fα is intuitionistic semi * continuous. remark 3.15 the converse of the above theorem is not true in general. however the converse is true if is*o(x) is ics under finite intersection as seen in the following theorem. theorem 3.16 let f: x → ∏ 𝑋𝛼 be defined by f(𝑝)= (fα(𝑝)) and ∏ 𝑋𝛼 be the intuitionistic product topology. let is*o(x) be ics under finite intersection. if each coordinate function fα : x → xα is intuitionistic semi * continuous, then f is intuitionistic semi * continuous. proof: let �̃� be the basic ios in ∏ 𝑋𝛼. then �̃� = ∩ 𝑃𝛼 −1(�̃�) where each �̃� is ios in xα, the intersection being taken over finitely many α’s and where pα : ∏ 𝑋𝛼 → x is the 71 g. esther rathinakani and m. navaneethakrishnan projection map. now 𝑓 −1(�̃�) = 𝑓 −1(∩(𝑃𝛼 −1(�̃�𝛼))) = ∩ 𝑓 −1 (𝑃𝛼 −1(�̃�𝛼)) = ∩ (pα ° f) 1(�̃�𝛼) = ∩ 𝑓𝛼 −1(�̃�𝛼) is is*o, by hypothesis. thus by theorem 3.7, f is intuitionistic semi * continuous. theorem 3.17 let f: x → y be intuitionistic continuous and g: x → z be intuitionistic semi * continuous. let h: x → y × z be defined by h(𝑝) = (f(𝑝), g(𝑝)) and y × z be the intuitionistic product topology. then h is intuitionistic semi * continuous. proof: using theorem 3.7, it is sufficient to prove that the inverse image under h of every basic ios in y × z is is*o in x. let �̃� × �̃� be the basic ios in y × z. then ℎ−1 (�̃� × �̃�) = 𝑓 −1 (�̃�) ∩ 𝑔−1 (�̃�). now by intuitionistic continuity of f, 𝑓 −1 (�̃�) is io in x and by intuitionistic semi * continuity of g, 𝑔−1 (�̃�) is is*o in x. therefore by theorem 2.14(ii), ℎ−1 (�̃� × �̃�) = 𝑓 −1 (�̃�) ∩ 𝑔−1 (�̃�) is is*o in x. hence h is intuitionistic semi * continuous. theorem 3.18 let f: x → y be intuitionistic semi * continuous and h: y → z be an intuitionistic continuous. then h ° f: x → z is intuitionistic semi * continuous. proof: let �̃� be an ios in z. since h is intuitionistic continuous, ℎ−1 (𝑈) is ios in y. since f is intuitionistic semi * continuous, 𝑓 −1 (ℎ−1 (�̃�)) is ios in x. therefore 𝑓 −1 (ℎ−1 (�̃�)) = (ℎ ° 𝑓)−1) (�̃�) is is*o in x. hence h ° f is intuitionistic semi * continuous. remark 3.19 from the above theorem it can be seen that the composition of two intuitionistic semi * continuous need not be intuitionistic semi * continuous. example 3.20 let x = y = z ={i, j, k} and τ1= {𝑋�̃�, ∅̃𝐼 , < x, {i},{j, k} >, < x, {k}, {i, j} >, < x, {i, k}, {b}>}, τ2= {𝑋�̃�, ∅̃𝐼 , < x, {j},{i, k} >, < x, {i},{j} >, < x, {i, j}, ∅>} τ3= {𝑋�̃�, ∅̃𝐼 , < x, {j},{i} >, < x, {i, j}, ∅>}. let f: (x, τ1) → (y, τ2) be defined by f(i) = i, f(j) = k, f(k) = j and let g: (y, τ2) → (z, τ3) be defined by g(i) = j, g(j) = i, g(k) = k. then f and g are intuitionistic semi * continuous. let g ° f : (x, τ1) → (z, τ3) and �̃� = < x, {j}, {i}>. then (𝑔 ° 𝑓 )−1(�̃�) = g(f(< x, {j}, {i}>)) = g(<x, {k},{i}>) = <x, {k},{i}> is not an is*o in (z, τ3). therefore g ° f is not an intuitionistic semi * continuous. definition 3.21 a function f: x → y is said to be intuitionistic contra semi * continuous if 𝑓 −1 (�̃�) is is*c in x for every ios �̃� in y. remark 3.22 the concept of intuitionistic semi * continuity is free from intuitionistic contra semi * continuity. theorem 3.23 let f: x → y be the function. then the following are equivalent 72 on intuitionistic semi * continuous functions (i) f is intuitionistic contra semi * continuous. (ii) for each 𝑝 ̃ ∈ x and each isc �̃� in y containing f(𝑝), there exists a is*o set �̃� in x containing 𝑝 such that f(�̃�) ⊆ �̃�. (iii) the inverse image of each isc in y is is*o in x. (iv) icl*(iint(𝑓 −1 (�̃�))) = icl*(𝑓 −1 (�̃�)) for every ics �̃� in y. (v) iint*(icl(𝑓 −1 (�̃�))) = iint*(𝑓 −1 (�̃�)) for every ios �̃� in x. proof: (i) ⇒ (ii). let f: x → y be the intuitionistic contra semi * continuous function. let 𝑝 ∈ x and �̃� be an ics in y containing f(𝑝). take �̃� = y �̃�. then �̃� is an ios in y not containing f(𝑝). since f is intuitionistic contra semi * continuous, 𝑓 −1 (�̃�) is a intuitionistic semi * closed set in x not containing 𝑝. therefore 𝑓 −1 (�̃�) = x 𝑓 −1 (�̃�) is a is*c in x not containing 𝑝. thus �̃� = 𝑓 −1 (�̃�) is a is*o in x containing 𝑝 such that f(�̃�) ⊆ �̃�. hence (i). (ii) ⇒ (iii). let �̃� be an ics in y and 𝑝 ∈ 𝑓 −1 (�̃�). then f(𝑝) ∈ �̃�. by assumption, there is an is*o set �̃�𝑝 in x containing 𝑝 such that f(𝑝) ∈ f(�̃�𝑝) ⊆ �̃�. therefore �̃�𝑝 ⊆ 𝑓 −1 (�̃�). thus 𝑓 −1 (�̃�) = ∪ { �̃�𝑝 : 𝑝 ∈ 𝑓 −1 (�̃�)}. by theorem 2.14 (i) 𝑓 −1 (�̃�) is an is*o in x. hence (ii). (iii) ⇒ (iv). let �̃� be an isc in y. then by hypothesis, 𝑓 −1 (�̃�) is is*o in x. hence from lemma 3.11(i), we have icl*(iint(𝑓 −1 (�̃�))) = icl*(𝑓 −1 (�̃�)). (iv) ⇒ (v). let �̃� be an ios in y, then y �̃� is an ics in y. by assumption, icl*(iint(𝑓 −1 (y �̃�))) = icl*(𝑓 −1 (y �̃�)). therefore [icl*(iint(𝑓 −1 (y �̃�)))]c = [icl*(𝑓 −1 (y �̃�))]c. hence iint*(icl(𝑓 −1 (�̃�))) = iint*(𝑓 −1 (�̃�)). (v) ⇒ (i). let �̃� be an ios in y. then by assumption, iint*(icl(𝑓 −1 (�̃�))) = iint*(𝑓 −1 (�̃�)). therefore by lemma 3.11 (ii), 𝑓 −1 (�̃�) is is*c in x. thus f is an intuitionistic contra semi * continuous. theorem 3.24 every intuitionistic contra continuous is intuitionistic contra semi * continuous. proof: let f: x → y be the intuitionistic contra continuous function and 𝑈 be an ios in y. then 𝑓 −1 (�̃�) is an ics in x. hence f is intuitionistic contra semi * continuous. theorem 3.25 every intuitionistic contra semi * continuous is intuitionistic contra semi continuous. proof: let f: x → y be the intuitionistic contra semi * continuous function and �̃� be an ios in y. then 𝑓 −1 (�̃�) is an is*c in x. hence f is intuitionistic contra semi continuous. 73 g. esther rathinakani and m. navaneethakrishnan theorem 3.26 let (x, τ1) and (y, τ2) be an its. (i) if f: x → y is intuitionistic contra semi * continuous and g: y → z is intuitionistic contra continuous, then g ° f: x → z is intuitionistic semi * continuous. (ii) if f: x → y is intuitionistic semi * continuous and g: y → z is intuitionistic contra continuous, then g ° f: x → z is intuitionistic contra semi * continuous. (iii) if f: x → y is intuitionistic contra semi * continuous and g: y → z is intuitionistic continuous, then g ° f: x → z is intuitionistic contra semi * continuous. proof: let �̃� be an ios in z. then 𝑔−1 (�̃�) is an ics in y. since f is intuitionistic contra semi * continuous, (g ∘ f)¯1(�̃�) = 𝑓 −1 (𝑔−1 (�̃�)) is is*c in x. thus g ° f is intuitionistic semi * continuous. (ii) and (iii) can be proved in a similar way. 4. conclusions in this paper, we dealt with intuitionistic semi * continuous and intuitionistic contra semi * continuous. in future we wish to do our research work in intuitionistic semi * separated, intuitionistic semi * connected, intuitionistic semi * compact, intuitionistic semi * irresolute continuous function and so on. references [1] d. coker, an introduction to intuitionistic topological spaces preliminary report, akdeniz university, mathematics department, turkey, 1995. [2] d. coker, a note on intuitionistic sets and intuitionistic points, turk. j. math., 20(3),1996,343-351. [3] d. coker, an introduction to intuitionistic fuzzy topological spaces, fuzzy sets and systems, 88, 1997, 81-89. [4] d. coker, an introduction to intuitionistic topological spaces, busefal81, 2000, 51 -56. [5] rathinakani, g. esther, and m. navaneethakrishnan, "a new closure operator in intuitionistic topological spaces." [6] rathinakani, g. esther, and m. navaneethakrishnan, "a study on intuitionistic semi * open set." design engineering (2021): 5043-5049. [7] g. esther rathinakani and m. navaneethakrishnan, some new operators on intuitionistic semi * open set, (communicated). [8] robert. a and pious missier. s, functions associated with semi * open sets, international journal of modern sciences and engineering technology (ijmset) vol.1, issue 2, 2014, pp. 39-46. [9] s. girija, s. selvanayaki and gnanambal ilango, semi closed and semi continuous mapping in intuitionistic topological space, journal of physics: conf. series 1139 (2018) 012057. [10] younis j. yaseen and asmaa g. raouf (2009) "on generalization closed set and generalized continuity on intuitionistic topological spaces" university of tikritcollege of computer science and mathematics. 74 ratio mathematica volume 45, 2023 modular coloring and switching in some planar graphs sanma. g. r1 p. maya2 abstract for a connected graph g, let c: v (g) →ℤk (k ≥ 2) be a vertex coloring of g. the color sum σ(v) of a vertex v of g is defined as the sum in ℤk of the colors of the vertices in n (v) that is (v) = ∑ c(u)u∈n(v) (mod k). the coloring c is called a modular k-coloring of g if 𝜎(x) ≠ 𝜎(y) in ℤk for all pairs of adjacent vertices x, y ∈ g. the modular chromatic number or simply the mc-number of g is the minimum k for which g has a modular kcoloring. a switching graph is an ordinary graph with switches. for many problems, switching graphs are a remarkable straight forward and natural model, but they have hardly been studied. a vertex switching gv of a graph g is obtained by taking a vertex v of g, removing the entire edges incident with v and adding edges joining v to every vertex which are not adjacent to v in g. in this paper we determine the modular chromatic number of wheel graph, friendship graph and gear graph after switching on certain vertices. here, we first define switching of graphs. next, we investigating several problems on finding the mc(g) after switching of graphs and provide their characterization in terms of complexity. keywords. modular coloring, modular chromatic number, switching, wheel graph, friendship graph, gear graph. msc ams classification 2020: 05c153 1assistant professor, department of mathematics, sree narayana college, sivagiri, varkala, india. phone no:9446069833 email id: sanmagr@gmail.com. 2 assistant professor, department of mathematics, sree devi kumari women’s college, kuzhithurai, kanyakumari dist, tamil nadu, india phone no:9442588393. email id: drmaya009@gmail.com 3 received on july 22, 2022. accepted on october 15, 2022. published on january 30, 2023.doi: 10.23755/rm.v45i0.984. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 74 mailto:sanmagr@gmail.com sanma. g. r, p. maya 1. introduction we are encouraged by the modular colorings and the modular chromatic number of different graphs, where the chromatic number is defined as the color sum of all the neighboring vertices in 𝕫k. at this point of view, to the curiosity for minimizing the modular chromatic number, determined to switching in certain vertices in some graphs. for a vertex v of a graph g, let n (v) denote the neighborhood of v (the set of adjacent vertices to vertex v). for a graph g without isolated vertices, let c: v (g) → 𝕫k (k ≥ 2) be a vertex coloring of g where adjacent vertices may be colored the same. the color sum 𝜎(v) of a vertex v of g is defined as the sum in 𝕫k of the colors of the vertices in n(v) ,that is 𝜎(v) =∑ 𝑐(𝑢)𝑢𝜖𝑁(𝑣) [1, 2, 3]. the coloring c is called a modular sum k-coloring or simply a modular k-coloring of g, if 𝜎(x) ≠ 𝜎(y) in 𝕫k for all pairs x, y of adjacent vertices of g. a coloring c is called modular coloring if c is a modular k-coloring for some integar k ≥ 2.the modular chromatic number mc(g) is the minimum k for which g has a modular kcoloring.this concept was introduced by okamoto, salehi and zhang [4, 5, 6, 8]. in order to distinguish the vertices of a connected graph and to differntiate the adjacent vertices of a graph with the minimum number of colors, the concept of modular coloring was put forward by okamoto, salehi and zhang [6]. a graph h is the switching of a graph g with respect to the vertex v of g if v (g) = v (h) and e(h) = (e(g)\{ uv : u ∈ ng(v) }) ∪ { uv : u ∈n’g(v) }[7,9]. the switching of g with respect to v is denoted gs (v). the operation of creating gs (v) is called switching on v in g. in other words, switching on a vertex v of a graph has the effect of removing all edges incident with the vertex v and joining the vertex v to all vertices to which it was formerly non-adjacent. here, we first define switching of graphs. next, we investigating several problems on finding the mc (g) after switching of graphs and provide their characterization in terms of complexity. in this paper we find the modular chromatic number of wheel graph, friendship graph and gear graph after switching on certain vertices at different levels. 2. modular coloring of wheel graph after switching the switching of a vertex in a wheel having n vertices is denoted by ws (n). in wheel switching is not possible in w (3) since it is a complete graph. switching in a wheel is obtained in two ways. they are 1)the switching at the vertex u ∈ ℓ0. 2)the switching at the vertex vi ∈ ℓ1.be the vertices let u ∈ ℓ0 be the central vertex and v1, v2, …, vn ∈ ℓ1be the vertices which are adjacent to u ∈ ℓ0. the switching at u ∈ ℓ0 makes the graph w(n) is a cycle having n vertices with a central vertex u ∈ ℓ0. 75 modular coloring and switching in some planar graphs theorem 2.1 the modular coloring of a graph obtained after the switching of a vertex vi ∈ ℓ1 is ws (n) = 3 for n = 4k, 4k+1, [4k+2; k > 1]; ws (6) = 4; ws (n) = 4 for n = 4k+3, k ≥ 1. proof: for a wheel w (n), let the vertex u ∈ ℓ0 ,vi ∈ ℓ1 for i = 1, 2, …., n be the vertices. case (i) mc [ws (4)] = 3. let v1, v2, v3, v4 ∈ ℓ1be the vertices at level ℓ=1.switching is taken forv1 ∈ ℓ1. consider the modular coloring c(v):v[ws(4)]→ 𝕫3 defined by c(v)={ 0 for v1 ∈ ℓ1, u ∈ ℓ0 1 otherwise then 𝜎(v)={ 0 𝑓𝑜𝑟 𝑢 ∈ ℓ0 2 𝑓𝑜𝑟 𝑣3 ∈ ℓ1 1 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4). ∴ mc [ws (4)]=3. hence the proof. case (ii) mc [ws (5)] = 3. let v1, v2, v3, v4, v5∈ ℓ1be the vertices at level ℓ=1.switching is taken for v1 ∈ ℓ1. consider the modular coloring c (v):v[ws(5)]→ 𝕫3 defined byc(v)={ 1 for v3 ∈ ℓ1 2 for v4 ∈ ℓ1 0 otherwise then σ(v) ={ 0 for u ∈ ℓ0, v1 ∈ ℓ1 1 for v2,v4 ∈ ℓ1 2 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀x,y of adjacent vertices in ws(5). ∴ mc [ws (5)]=3.hence the proof. case (iii)mc[ws(6)] = 4.let v1,v2,v3,v4,v5,v6∈ ℓ1be the vertices at level ℓ=1.switching is taken for v1 ∈ ℓ1 consider the modular coloring c(v):v[ws(6)]→ 𝕫3 defined by c(v) = { 1 for v3 ∈ ℓ1 2 for v4 ∈ ℓ1 0 otherwise then σ (v) = { 3 for u ∈ ℓ0, v1 ∈ ℓ1 1 for v2,v4 ∈ ℓ1 2 for v3,,v5 ∈ ℓ1 0 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(6). ∴ mc [ws (6)] = 3. hence the proof. case (iv) mc [ws (4k+3)] = 4. let u∈ ℓ0 be the central vertex. let v1,v2,…vi-1,vi,vi+1,….,v4k+3∈ ℓ1be the vertices at level ℓ =1.switching is taken for vi ∈ ℓ1 . after switching vi is adjacent to the vertices vi+2,vi+3,…v4k+3,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. the 4k vertices which are adjacent to vi is renamed as r1, r2, …., r4k. 76 sanma. g. r, p. maya subcase (i) mc [ws (4k+3)] = 4 for k = 1 + 4j, j = 0, 1, 2, …. [eg: ws (7), ws (23), ws (39), …. consider the modular coloring c(v):v[ws(4k+3)]→ 𝕫4 defined by c(v)={ 2 if r4k ∈ ℓ1 1 if r1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 3 for u ∈ ℓ0, vi ∈ ℓ1 2 for vi−1,r4k−1 ∈ ℓ1 1 for vi+1,,r2j ∈ ℓ1, for j = 1,2, … (2k − 1) 0 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k+3). ∴ mc[ws(4k+3)]=4 for k=1+4j ,j=0,1,2,….hence the proof. eg: figure1. switching with modular coloring in ws(7) subcase(ii) mc[ws(4k+3)]=4 for k=2+4j ,j=0,1,2,….[ws(11),ws(27), ws(43),…] consider the modular coloring c(v):v[ws(4k+3)]→ 𝕫4 defined by c(v)={ 2 if r4k ∈ ℓ1 1 if u ∈ ℓ0, r1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 0 for u ∈ ℓ0, vi ∈ ℓ1 3 for r4k−1 ∈ ℓ1 2 for vi+1,,𝑣𝑖−1, 𝑅2𝑗 ∈ ℓ1, for j = 1,2, … (2k − 1) 1 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k+3). ∴ mc[ws(4k+3)]=4 for k=2+4j ,j=0,1,2,…. hence the proof. eg: 77 modular coloring and switching in some planar graphs figure 2. switching with modular coloring in ws(11) subcase(iii) mc[ws(4k+3)]=4 for k=3+4j ,j=0,1,2,….[eg: ws(15), ws(31),ws(47),…] consider the modular coloring c(v):v[ws(4k+3)]→ 𝕫4 defined by c(v)={ 2 if u ∈ ℓ0, r4k ∈ ℓ1 1 if r1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 0 for 𝑣𝑖−1 , r4k−1 ∈ ℓ1 1 for u ∈ ℓ0, 𝑣𝑖 ∈ ℓ1 3 for vi+1,,𝑅2𝑗 ∈ ℓ1, for j = 1,2, … (2k − 1) 2 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k+3). ∴ mc[ws(4k+3)]=4 for k=3+4j ,j=0,1,2,….hence the proof. eg: figure 3. switching with modular coloring in ws(15) subcase(iv) mc[ws(4k+3)]=4 for k=4+4j ,j=0,1,2,….[ws(19),ws(35),ws(51),…] 78 sanma. g. r, p. maya consider the modular coloring c(v):v[ws(4k+3)]→ 𝕫4 defined by c(v)={ 3 if u ∈ ℓ0 1 if r1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 2 if r4k ∈ ℓ1 0 elsewhere then σ(v)={ 1 for vi−1 , r4k−1 ∈ ℓ1 2 for u ∈ ℓ0, vi ∈ ℓ1 0 for vi+1,,r2j ∈ ℓ1, for j = 1,2, … (2k − 1) 3 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k+3). ∴ mc[ws(4k+3)]=4 for k=4+4j ,j=0,1,2,….hence the proof. eg: figure 4. switching with modular coloring in ws(19) case(v)mc[ws(4k)] = mc[ws(4k+1)] = mc[ws(4k+2)]=3. subcase(i) mc[ws(4k)] = 3 for k=2+3j,j=0,1,2….[ws(8),ws(20),ws(32),…] after switching vi is adjacent to the vertices vi+2,vi+3,…v4k,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. let the 4k-3 vertices which are adjacent to vi is renamed as r1,r2,…..,r4k-3 respectively. consider the modular coloring c(v):v[ws(4k)]→ 𝕫3 defined by c(v)={ 1 if r1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 2 for u ∈ ℓ0, vi ∈ ℓ1 1 for vi+1,,vi−1, r2j ∈ ℓ1, for j = 1,2, … . ,2(k − 1) 0 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k). ∴ mc[ws(4k)]=3 for k=2+3j ,j=0,1,2,….hence the proof. eg: 79 modular coloring and switching in some planar graphs figure 5. switching with modular coloring in ws(8) subcase(ii) mc[ws(4k)] = 3 for k=3+3j,j=0,1,2….[ws(12),ws(24),ws(36),..] after switching vi is adjacent to the vertices vi+2,vi+3,…v4k,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. let the 4k-3 vertices which are adjacent to vi is renamed as r1, r2, ….., r4k-3 respectively. consider the modular coloring c(v):v[ws(4k)]→ 𝕫3 defined by c(v)={ 1 if u ∈ ℓ0, r1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 0 for u ∈ ℓ0, vi ∈ ℓ1 1 r1+2j ∈ ℓ1, for j = 0,1,2, … . , (k + 1) 2 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k). ∴ mc[ws(4k)]=3 for k=3+3j ,j=0,1,2,….hence the proof. eg: figure 6. switching with modular coloring in ws(12) subcase(iii) mc[ws(4k)] = 3 for k=4+3j,j=0,1,2….[ws(16),ws(28),ws(40),…] after switching vi is adjacent to the vertices vi+2,vi+3,…v4k,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. let the 4k-3 vertices which are adjacent to vi is renamed as r1, r2, ….., r4k-3 respectively. 80 sanma. g. r, p. maya consider the modular coloring c(v):v[ws(4k)]→ 𝕫3 defined by c(v)={ 2 if u ∈ ℓ0 1 forr1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 1 for u ∈ ℓ0, vi ∈ ℓ1 2 r1+2j ∈ ℓ1, for j = 0,1,2, … . ,2(k − 1) 0 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k). ∴ mc[ws(4k)]=3 for k=4+3j ,j=0,1,2,…. hence the proof. eg: figure7. switching with modular coloring in ws(16) subcase(iv) mc[ws(4k+1)] = 3 for k=2+3j,j=0,1,2….[ws(9),ws(21),ws(33),…] after switching vi is adjacent to the vertices vi+2,vi+3,…v4k+1,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. let the 4k-2 vertices which are adjacent to vi is renamed as r1, r2… r4k-2 respectively. consider the modular coloring c(v): v[ws(4k+1)]→ 𝕫3 defined by c(v) = { 1 forr1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v) = { 2 for u ∈ ℓ0, vi ∈ ℓ1 1 for vi+1 , r2j ∈ ℓ1, for j = 1,2, … . , (2k − 1) 0 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x, y of adjacent vertices in ws (4k+1). ∴ mc [ws (4k+1)] = 3 for k = 2+3j, j = 0, 1, 2, …. hence the proof. 81 modular coloring and switching in some planar graphs eg: figure 8. switching with modular coloring in ws(9) subcase (v) mc[ws (4k+1)] = 3 for k = 3+3j, j = 0, 1, 2…. [ws (13), ws (25), ws (37), …] after switching vi is adjacent to the vertices vi+2, vi+3, …v4k+1, v1, v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. let the 4k-2 vertices which are adjacent to vi is renamed as r1, r2, ….., r4k-2 respectively. consider the modular coloring c(v):v[ws(4k+1)]→ 𝕫3 defined by c(v)={ 1 for u ∈ ℓ0 ; r1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v) = { 0 for u ∈ ℓ0, vi ∈ ℓ1 2 for vi+1 , r2j ∈ ℓ1, for j = 1,2, … . , (2k − 1) 1 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k+1). ∴ mc[ws(4k+1)] = 3 for k = 3+3j, j = 0, 1, 2, …. hence the proof. eg: figure 9. switching with modular coloring in ws(13) subcase (vi) mc [ws(4k+1)] = 3 for k = 4+3j, j = 0, 1, 2…. [ws (17), ws (29), ws (41), …] 82 sanma. g. r, p. maya after switching vi is adjacent to the vertices vi+2,vi+3,…v4k+1,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. let the 4k-2 vertices which are adjacent to vi is renamed as r1, r2, ….., r4k-2 respectively. consider the modular coloring c(v):v[ws(4k+1)]→ 𝕫3 defined by c(v)={ 2 for u ∈ ℓ0 1 for r1+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 1 for u ∈ ℓ0, vi ∈ ℓ1 0 for 𝑣𝑖+1 , r2j ∈ ℓ1, for j = 1,2, … . , (2k − 1) 2 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k+1). ∴ mc[ws(4k+1)]=3 for k=4+3j ,j=0,1,2,….hence the proof. eg: figure 10. switching with modular coloring in ws(17) subcase(vii) mc[ws(4k+2)] = 3 for k=2+3j,j=0,1,2….[ws(10),ws(22),ws(34),…] after switching vi is adjacent to the vertices vi+2, vi+3, …v4k+2, v1, v2, …, vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. let the 4k-2 vertices which are adjacent to vi is renamed as r1, r2, ….., r4k-1 respectively. consider the modular coloring c(v):v[ws(4k+2)]→ 𝕫3 defined by c(v)={ 1 for r2+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 2 for u ∈ ℓ0, vi ∈ ℓ1 1 for r1+2j ∈ ℓ1, for j = 0,1,2, … . , (2k − 1) 0 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k+2). ∴ mc [ws (4k+2)] = 3 for k = 2+3j, j = 0, 1, 2, …. hence the proof. 83 modular coloring and switching in some planar graphs eg: figure 11. switching with modular coloring in ws(10) subcase(viii) mc[ws(4k+2)] = 3 for k=3+3j,j=0,1,2….[ws(14),ws(26),ws(38),…] after switching vi is adjacent to the vertices vi+2,vi+3,…v4k+2,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. let the 4k-2 vertices which are adjacent to vi is renamed as r1,r2,…..,r4k-1 respectively. consider the modular coloring c(v):v[ws(4k+2)]→ 𝕫3 defined by c(v)={ 1 for u ∈ ℓ0 , r2+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 0 for u ∈ ℓ0, vi ∈ ℓ1 2 for r1+2j ∈ ℓ1, for j = 0,1,2, … . , (2k − 1) 1 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k+2). ∴ mc[ws(4k+2)]=3 for k=3+3j ,j=0,1,2,….hence the proof. eg: figure 12. switching with modular coloring in ws(14) subcase(ix)mc [ws(4k+2)] = 3 for k=4+3j,j=0,1,2….[ws(18),ws(30),ws(42),…] 84 sanma. g. r, p. maya after switching vi is adjacent to the vertices vi+2,vi+3,…v4k+2,v1,v2,…,vi-2 respectively and not adjacent to the vertices vi-1 and vi+1. let the 4k-2 vertices which are adjacent to vi is renamed as r1,r2,…..,r4k-1 respectively. consider the modular coloring c(v):v[ws(4k+2)]→ 𝕫3 defined by c(v)={ 2 for u ∈ ℓ0 1 𝑓𝑜𝑟 r2+4j ∈ ℓ1, j = 0,1,2 … (k − 1) 0 elsewhere then σ(v)={ 1 for u ∈ ℓ0, vi ∈ ℓ1 0 for r1+2j ∈ ℓ1, for j = 0,1,2, … . , (2k − 1) 2 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in ws(4k+2). ∴ mc[ws(4k+2)]=3 for k=4+3j ,j=0,1,2,….hence the proof. eg: figure 13. switching with modular coloring in ws(18) 3. modular colorings after switching on friendship graph let u ∈ ℓ0 be the center v1, v2, v3, …. v2n be the vertices in ℓ1 where each of 2 consecutive vertices forms an edge for the respective cycles since a friendship graph is constructed by joining n copies of the cycle c3 with a common vertex. the vertices in ℓ1is taken in the clockwise direction. modular coloring after switching of a friendship graph with n vertices is denoted by mc[fss(n)].here switching can be taken only for vi∈ ℓ1 for any i=1,2,… 2n.we cannot form a switching with u ∈ ℓ0 since it is adjacent to all vertices of ℓ1. theorem 3.1. the modular coloring of a friendship graph after switching a vertex in ℓ1 then (i)mc[fss(2)]=3. (ii) mc[fss(n)]=4;n≥ 3. proof: case (i) mc[fss(2)] = 3. 85 modular coloring and switching in some planar graphs switching is taken for v1∈ ℓ1.therefore v1 is adjacent to v3 and v4 after switching in fss(2). consider a modular coloring c(v):v[fss(2)] → 𝕫3 defined by c(v)= { 2 for u ∈ ℓ0 1 for v4 ∈ ℓ1 0 elsewhere then σ(v)={ 1 for u ∈ ℓ0, v1 ∈ ℓ1 2 for v4, v2 ∈ ℓ1 0 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀x,y of adjacent vertices in fss(2) ∴mc[fss(2)]=3. hence the proof. case(ii) mc[fss(n)]=4;n≥3. subcase (i) mc[fss(n)]=4;n≥3 for n is odd. switching is taken for v1∈ ℓ1.therefore v1 is adjacent to v3 ,v4 ,….v2n after switching in fss(n). consider a modular coloring c(v):v[fss(n)] → 𝕫4 defined by c(v)= { 3 for u ∈ ℓ0 2 for v2j ∈ ℓ1 ; j = 1,2, … , n 0 elsewhere then σ(v)={ 2 for u ∈ ℓ0 3 v2j ∈ ℓ1; 𝑗 = 1,2, … . , 𝑛 0 for v1 ∈ ℓ1 1 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in fss(n) ∴ mc[fss(n]=4 for n≥3 for n is odd. hence the proof. eg: figure 14. switching with modular coloring in fss(5) 86 sanma. g. r, p. maya subcase (ii).mc[fss(n)]=4;n≥3 for n is even. switching is taken for v1∈ ℓ1.therefore v1 is adjacent to v3 ,v4 ,….v2n after switching in fss(n). consider a modular coloring c(v):v[fss(n)] → 𝕫4 defined by c(v)= { 3 for u ∈ ℓ0 2 for v2j ∈ ℓ1 ; j = 1,2, … , n 0 elsewhere then σ(v)={ 0 for u ∈ ℓ0 3 v2j ∈ ℓ1; 𝑗 = 1,2, … . , 𝑛 2 for v1 ∈ ℓ1 1 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in fss(n) ∴ mc[fss(n]=4 for n≥3 for n is even. hence the proof. eg: figure 15. switching with modular coloring in fss(4) 4.modular colorings after switching on gear graph. let u ∈ ℓ0 be the center of a gear graph.let v1,v3,v5,….v2n-1 be the vertices in ℓ1 are adjacent to u ∈ ℓ0 and v2,v4,v6,….v2n be the vertices in ℓ1 .the switching in g(n)is denoted by gs(n).switching in gear graph is obtained in two ways. that is (i)switching of the vertex u ∈ ℓ0 and (ii)switching of a vertex vi ∈ ℓ1; 𝑖 = 1,2, … . ,2𝑛. (i)by switching of the vertex u ∈ ℓ0 in a gear graph g(n) result in another gear graph g’(n) in which vertices in ℓ1which are not adjacent with u ∈ ℓ0 in g(n) become adjacent with g’(n).therefore gs(n)=g(n)=g’(n).hence mc[gs(n)]=mc[g(n)]=mc[g’(n)]. (ii) switching of a vertex vi ∈ ℓ1; 𝑖 = 1,2, … . ,2𝑛 .here specifying the vertex vi ∈ ℓ1 which are adjacent with u ∈ ℓ0 is taken for switching.in general take vi as v1. 87 modular coloring and switching in some planar graphs theorem 4.1 the modular coloring of the graph obtained after the switching of a vertex in gear graph in ℓ1 (which are adjacent with u ∈ ℓ0).ie (i) mc[gs(2)]=2.(ii) mc[gs(n)]=3;n>2. proof. case (i) mc[gs(2)]=2. switching is taken for v1∈ ℓ1in g (2).therefore v1 is adjacent to v3 after switching in gs (n). consider a modular coloring c (v):v[gs(2)] → 𝕫2 defined by c(v)= { 1 for v3 ∈ ℓ1 0 elsewhere then σ(v)={ 0 v3 ∈ ℓ1 1 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in gs(2) ∴mc [gs(2]=2. hence the proof. case (ii) mc[gs(n)]=3 for n>2. switching is taken for v1∈ ℓ1 in g(n).therefore v1 is adjacent to v3 ,v4,….v2n-1 after switching in gs(n) and not adjacent to the remaining vertices in gs(n). consider a modular coloring c(v):v[gs(n)] → 𝕫3 defined by c(v)= { 1 for u ∈ ℓ0; v1 ∈ ℓ1 0 elsewhere then σ(v)={ 2 for v3+2j ∈ ℓ1 ; 𝑗 = 0,1,2, … . (𝑛 − 2). 1 for v4+2j ∈ ℓ1 ; 𝑗 = 0,1,2, … . (𝑛 − 3) 0 otherwise here 𝜎(𝑥) ≠ 𝜎(𝑦) ∀ x,y of adjacent vertices in gs(n) ∴ mc[gs(n)]=3 for n>2. hence the proof. eg: figure 16. switching with modular coloring in gs(6). 88 sanma. g. r, p. maya 5. conclusions in a wheel graph the modular coloring of a graph obtained after the switching of a vertex vi ∈ ℓ1 is ws(n)=3 for n=4k,4k+1,[4k+2;k>1]; ws(6)=4; ws(n)=4 for n=4k+3,k≥1.the labeling is quite similar to one other and differs according to the change in number of vertices. also in a friendship graph the modular coloring after switching a vertex in ℓ1 then (i)mc[fss(2)]=3. (ii) mc[fss(n)]=4;n ≥ 3. similarly in gear graph the modular coloring obtained after the switching of a vertex in ℓ1 (which are adjacent with u ∈ ℓ0).ie (i) mc[gs(2)]=2.(ii) mc[gs(n)]=3;n>2. altogether it is explicitly clear that after switching in different levels of the graphs, the modular chromatic number varies in between two to four. we cannot expect a higher level of modular chromatic number after switching in vertices at different levels. studying this problem and related problems in the context of switching graphs may help in answering the long open question whether all of these problems have a polynomial algorithm. we conclude this paper by listing a number of switching graph problems of which we do not know the complexity references [1] g. chartrand and p. zhang. chromatic graph theory chapman and hall/crc press, boca raton (2009). [2] t. nicholas, sanma. g. r. modular colorings of circular halin graphs of level two, asian journal of mathematics and computer research 17(1): 48-55, 2017 [3] n. paramaguru, r. sampathkumar. modular colorings of join of two special graphs. electronic journal of graph theory and applications 2(2) (2014), 139149. [4] f. okamoto, e. salehi, p. zhang. a checkerboard problem and modular colorings of graph bull. inst. combin. appl58 (2010), 2947. [5] f. okamoto, e. salehi, p. zhang. a solution to checkerboard problem j. comput. appl. math5(2010)447-458. [6] f. okamoto, e. salehi, p. zhang. on modular colorings of graphs pre-print. [7] patrick neisink. the vertex switching reconstruction problem. university of ottawa, canada. [8] ryan jones, thesismodular and graceful edge coloring of graphs. western michigan university. [9] j f groote, bas ploeger. switching graphs international journal of foundations of computer science vol. 20, no 05, pp 869-886 (2009). 89 ratio mathematica volume 46, 2023 on fzdomination number of fuzzy graphs lekha a* parvathy k.s† abstract given a fuzzy graph g = (v,µ,σ), the fzdomination number, γfz(g), is the least scalar cardinality of an fzdominating set of g. in this article, we examine several features of fz-domination number of fuzzy graphs as a result of various fuzzy graph operations. we find bounds for the fz-domination number of a few graph products and look at the requirements for the sharpness of these bounds. keywords: fuzzy graph; fz-dominating sets; fz-domination number; graph operations 2020 ams subject classifications:05c69, 05c72. 1 *maharaja’s technological institute, thrissur, kerala, india. alekharemesh@gmail.com, lekha.a.res@smctsr.ac.in. †st. mary’s college, thrissur, kerala, india. parvathy.math@gmail.com (corresponding author). 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1078. issn: 1592-7415. eissn: 2282-8214. ©lekh a et al. this paper is published under the cc-by licence agreement. 213 lekh a and parvathy k.s 1 introduction since the initial introduction of fuzzy graphs by rosenfeld [1975], a large number of researchers have studied the subject. the notion of domination in fuzzy graphs was first proposed by somasundaram and somasundaram [1998]. somasundaram [2005], gani and chandrasekaran [2006], manjusha and sunitha [2015], bhutani and sathikala [2016] also studied domination in fuzzy graphs. mordeson and chang-shyh [1994] developed operations of fuzzy graphs that are comparable to those in crisp graphs. different variations of domination in fuzzy graphs found in literature do not consider the situations where we need to take all the non-zero edges incident at a vertex into consideration. these definitions use either the effective or the strong edges of the fuzzy graph.but our model of fuzzy domination in fuzzy graphs [2022] takes into account all the non-zeroedges incident at a vertex, even if they are small in strength. further most variations of domination in fuzzy graphs found in literature do not considerthe fuzzy subsets of the vertex set, instead considered the crisp subsets of the fuzzy vertex set.but while considering fuzzy graphs and their subset problems it is more apt toconsider fuzzy subsets of the vertex set than their crisp subsets. by taking all these into consideration we defined fzdomination in fuzz graphs[2022]. we, lekha and parvathy [2022] developed fz-domination in fuzzy graphs, which coincides with fractional domination in crisp graphs presented by hedetniemi and wimer [1987] and explored by hedetniemi and mynhardt [1990]. in this article, we examine the effects of several graph operations on fz-domination. for basic definitions, terminology and notation in fuzzy graphs we refer to mordeson and nair [2000]. definition 1.1. (lekha and parvathy [2022]). given a fuzzy graph g = (v,µ,σ), a fuzzy subset µ′ of µ is defined as an fz-dominating set of g, if for every v ∈ v , µ′(v) + ∑ x∈v ( σ(x,v)∧µ′(x) ) ≥ µ(v). a fuzzy subset µ′ is a minimal fz-dominating set, if µ′′ ⊂ µ′ is not an fzdominating set. definition 1.2. (lekha and parvathy [2022]) fuzzy domination number or fzdomination number of g, denoted by γfz(g), is defined as γfz(g) = min {|µ′| : µ′ is a minimal fz-dominating set of g} example 1.1. for f = (µ,σ) shown in fig. 1, µ1 = {(x,0.1),(y,0.5),(z,0.2)} 214 on fzdomination number of fuzzy graphs µ2 = {(x,0.6),(y,0),(z,0.6)} µ3 = {(x,0.4),(y,0.2),(z,0.4)} µ4 = {(x,0.5),(y,0.1),(z,0.5)} are all minimal fz-dominating sets of f. µ1 is a minimum fz-dominating set and γfz(f) = 0.8. figure 1: fuzzy graph, f 2 fzdomination in union of fuzzy graphs let g = (v1,µ1,σ1) and h = (v2,µ2,σ2). g∪h = (v,µ,σ) where v = v1 ∪v2 µ(u) = µ1(u) if u ∈ v1 \v2 = µ2(u) if u ∈ v2 \v1 = µ1(u)∨µ2(u) if u ∈ v1 ∩v2 and σ(u,v) = σ1(u,v) if u ∈ v1 \v2,v ∈ v1 = σ2(u,v) if u ∈ v2 \v1,v ∈ v2 = σ1(u,v)∨σ2(u,v) if u,v ∈ v1 ∩v2 = 0 otherwise the following theorem gives a general upper bound for the fz-domination number of union of two fuzzy graphs. theorem 2.1. for any two nontrivial fuzzy graphs g and h, γfz(g∪h) ≤ γfz(g) + γfz(h) 215 lekh a and parvathy k.s proof. consider the fuzzy graphs g = (v1,µ1,σ1) and h = (v2,µ2,σ2). let µ′1 and µ′2 be the minimum fz-dominating sets of g and h respectively. let the fuzzy subset µ′ of v be defined by µ′(u) = µ′1(u) if u ∈ v1 \v2 = µ′2(u) if u ∈ v2 \v1 = µ′1(u)∨µ ′ 2(u) if u ∈ v1 ∩v2 now let v ∈ v . case (i) if v ∈ v1 \v2, then µ(v) = µ1(v) ≤ ( µ′1(v) + ∑ x∈v1 σ1(x,v)∧µ′1(x) ) = µ′(v) + ∑ x∈v σ(x,v)∧µ′(x). case (ii) if v ∈ v2 \v1, then µ(v) = µ2(v) ≤ ( µ′2(v) + ∑ x∈v2 σ2(x,v)∧µ′2(x) ) = µ′(v) + ∑ x∈v σ(x,v)∧µ′(x). case (iii) if v ∈ v1 ∩v2 µ(v) = µ1(v)∨µ2(v) ≤ ( µ′1(v) + ∑ x∈v1 σ1(x,v)∧µ′1(x) ) ∨ ( µ′2(v) + ∑ x∈v2 σ2(x,v)∧µ′2(x) ) ≤ (µ′1(v)∨µ ′ 2(v)) + ( ∑ x∈v1\v2 (σ1(x,v)∧µ′1(x)) + ∑ x∈v2\v1 (σ2(x,v)∧µ′2(x)) + ∑ x∈v1∩v2 (σ1(x,v)∨σ2(x,v))∧ (µ′1(x)∨µ ′ 2(x)) ) ≤ µ′(v) + ∑ x∈v σ(x,v)∧µ′(x). thus µ′ is an fz-dominating set of g∪h and µ′(v) ≤ µ′1(v) + µ′2(v). hence |µ′| ≤ |µ′1|+ |µ′2|. thus, γfz(g∪h) ≤ γfz(g) + γfz(h). 216 on fzdomination number of fuzzy graphs remark 2.1. obviously equality holds in the above theorem if the vertex sets of g and h are disjoint. the following example shows that equality may hold even if they are not disjoint. for the graphs in fig. 2, γfz(g) = 0.5, γfz(h) = 0.6 and γfz(g∪h) = 1.1 so that γfz(g∪h) = γfz(g) + γfz(h). figure 2: fuzzy graphs g, h and g∪h 3 fzdomination in join of fuzzy graphs let g = (v1,µ1,σ1) and h = (v2,µ2,σ2) whose vertex sets are disjoint. the join g +h is defined by g +h = (v,µ,σ) where v = v1 ∪v2, µ(u) = µ1(u) if u ∈ v1 = µ2(u) if u ∈ v2 and σ(u,v) = σ1(u,v) if u,v ∈ v1 = σ2(u,v) if u,v ∈ v2 = µ1(u)∧µ2(v) if u ∈ v1 and v ∈ v2 theorem 3.1. for any two nontrivial fuzzy graphs g and h whose vertex sets are disjoint, γfz(g +h) ≤ max{γfz(g),γfz(h)} 217 lekh a and parvathy k.s proof. let g = (v1,µ1,σ1) and h = (v2,µ2,σ2) be two fuzzy graphs such that v1 ∩v2 = φ. let γfz(g) ≥ γfz(h) and let µ′1 be a minimum fz-dominating set of g. define µ′ ⊂ µ by µ′(u) = µ′1(u) if u ∈g = 0 if u ∈h let m be such that m = max{µ2(u);u ∈h}. now m ≤ γfz(h) ≤ γfz(g) implies that µ′ is an fz-dominating set of g +h. hence, γfz(g +h) ≤ max{γfz(g),γfz(h)} in the following discussion m, m1 and m2 denote the maximum membership value of a vertex in g +h, g and h respectively. observation 3.1. it is possible that γfz(g +h) ≤ min{γfz(g),γfz(h)} for example, if m ≤ γfz(h) ≤ γfz(g), then µ′ ⊂ µ defined by µ′(u) = µ′2(u) if u ∈h = 0 if u ∈g is an fz-dominating set of g +h and hence γfz(g +h) ≤ γfz(h). here equality occurs if m = γfz(h). the following example shows that strict inequality can also occur in this relation. example 3.1. consider the fuzzy graphs g1, g1 and g1 +g2 given in fig.3. here γfz(g1) = 1.6, γfz(g2) = 1, m = 0.8, γfz(g1 +g2) = 0.9 so that γfz(g1 +g2) < γfz(g2) observation 3.2. if γfz(h) ≤ m ≤ |µ2|, then γfz(g +h) = m. claim: define µ′′2 ⊃ µ′2 in h such that |µ′′2| = m. then, µ′′2 is an fz-dominating set of g+h. hence γfz(g+h) ≤ m. also, since there is a vertex of membership value m in g +h, we get γfz(g +h) = m. observation 3.3. if m1 ≤ |µ2| and m2 ≤ |µ1|, then γfz(g +h) ≤ m1 + m2. 218 on fzdomination number of fuzzy graphs figure 3: fuzzy graphs g1, g2 and g1 +g2 claim: define µ′1 ⊂ µ1 in g such that |µ′1| = m2 and µ′2 ⊂ µ2 in h such that |µ′2| = m1. now µ′ defined by µ′(u) = µ′1(u) if u ∈g = µ′2(u) if u ∈h is an fzdominating set in g +h. hence γfz(g +h) ≤ m1 + m2. observation 3.4. if |µ2| ≤ m, then µ′ ⊂ µ in g +h defined by µ′(u) = µ2(u) if u ∈h = max{0,µ′1(u)−|µ2|} if u ∈g is an fzdominating set in g + h. then, γfz(g + h) ≤ γfz(g) −n|µ2| where n is the number of vertices u ∈g having µ1(u) ≥ |µ2|. observation 3.5. if γfz(h) ≤ m ≤ γfz(g), then µ′ ⊂ µ in g +h defined by µ′(u) = µ′2(u) if u ∈h = max{0,µ′1(u)−γfz(h)} if u ∈g is an fzdominating set in g +h. then, γfz(g +h) ≤ γfz(g)−nγfz(h) where n is the number of vertices u ∈g having µ1(u) ≥ γfz(h). 219 lekh a and parvathy k.s 4 fzdomination in corona of fuzzy graphs the corona of g = (v1,µ1,σ1) and k1 = (u,µ2(u)) is the fuzzy graph g◦k1 obtained by attaching a copy of k1 to each vertex vi ∈ v1 such that σ(vi,ui) = µ1(vi)∧µ2(ui) where ui represents the vertex in the copy of k1 corresponding to vi ∈ v1. observation 4.1. the following two results are obvious. 1. γfz(g ◦k1) ≥ γfz(g) 2. γfz(g ◦k1) ≥ nµ2(u) figure 4: fuzzy graph g ◦k1 remark 4.1. the example below shows that equality may occur in observation 4.1(a). consider g ◦k1 in figure 4. µ′ = {(a, 13),(b, 1 3 ),(c, 1 3 ),(d, 1 3 )} is a minimum fzdominating set of g and γfz(g) = 43 . µ ′ is an fz-dominating set of g ◦k1 also. therefore, γfz(g ◦k1) ≤ 43 = γfz(g). on the other hand from observation 4.1(a), γfz(g ◦k1) ≥ γfz(g). thus we get γfz(g ◦k1) = γfz(g). theorem 4.1. γfz(g ◦k1) ≤ γfz(g) + nµ2(u) where n = |v1|. proof. let µ be the fuzzy subset of g ◦k1 and µ′1 be a minimum fz-dominating set of g. let µ′ ⊂ µ be such that µ′(v) = µ′1(v) if v ∈ v1 = µ2(v) otherwise 220 on fzdomination number of fuzzy graphs then µ′ is an fz-dominating set of g ◦k1 and |µ′| = |µ′1|+ n|µ2| = γfz(g) + nµ2(u) therefore γfz(g ◦k1) ≤ γfz(g) + nµ2(u) theorem 4.2. if µ2(u) ≥ µ1(v) for all v ∈ v1, then γfz(g ◦k1) = nµ2(u) proof. it is clear that γfz(g ◦k1) ≥ nµ2(u) consider µ′ where µ′(v) = 0 if v ∈ v1 = µ2(v) if v = u then, µ′ is fz-dominating set of g ◦k1. therefore, γfz(g ◦k1) ≤ |µ′| = nµ2(u). hence γfz(g ◦k1) = nµ2(u) figure 5: g′ ◦k′1 remark 4.2. the condition µ2(u) ≥ µ1(v) for all v ∈ v1 is not necessary to get γfz(g ◦k1) = nµ2(u). for example, consider g′ ◦k′1 given in figure 5. here, µ2(u) < µ1(v) for all v ∈ v1. now γfz(g ◦k1) ≥ nµ2(u) implies that γfz(g′ ◦k′1) ≥ 2. also µ′ = {(a, 1 2 ),(b, 1 2 ),(c, 1 2 ),(d, 1 2 )} is fz-dominating set of g′ ◦k′1. hence γfz(g′ ◦k′1) = 2 = nµ2(u) 221 lekh a and parvathy k.s 5 fzdomination in cartesian product let g1 = (v1,µ1,σ1) and g2 = (v2,µ2,σ2). the cartesian product is the fuzzy graph g12g2 = (v,µ1 ×µ2,σ1 ×σ2) where v = v1 ×v2, (µ1 ×µ2)(a,b) = µ1(a)∧µ2(b) and (σ1 ×σ2) ( (a1,b2),(a2,b2) ) = µ1(a1)∧σ2(b1,b2) if a1 = a2 = σ1(a1,a2)∧µ2(b1) if b1 = b2 = 0 otherwise theorem 5.1. for any two nontrivial fuzzy graphs g and h, γfz(g2h) ≤ min{nγfz(g),mγfz(h)}, m, n are the number of vertices with nonzero membership values in g and h respectively. ‘ proof. let g = (v1,µ1,σ1) and h = (v2,µ2,σ2) where v1 = {(u1,µ1(u1)),(u2,µ1(u2)), ...,(um,µ1(um))} and v2 = {(v1,µ2(v1)),(v2,µ2(v2)), ...,(vn,µ2(vn))} g2h = (v,µ,σ) where v = v1 ×v2, µ(u,v) = µ1(u)∧µ2(v) and σ ( (ui,vj),(u ′ i,v ′ j) ) = σ1(ui,u ′ i) if vj = v ′ j = σ2(vj,v ′ j) if ui = u ′ i = 0 otherwise let gj denotes the fuzzy sub-graph of g2h induced by v1 ×vj ⊂ v1 ×v2. then, v (gj) = {(u1,vj),(u2,vj), ...,(um,vj)} µ(ui,vj) = µ1(ui)∧µ2(vj) ≤ µ1(ui) and σ ( (ui,vj),(u ′ i,vj) ) = min{σ1(ui,u′i),µ2(vj)}≤ σ(ui,u ′ i) claim: γfz(gj) ≤ γfz(g). define µ′j on gj as µ′j(ui,vj) = µ′1(ui)∧µ2(vj) consider (ui,vj) ∈gj. µ′1 is an fz-dominatng set of g implies that 222 on fzdomination number of fuzzy graphs µ1(ui) ≤ µ′1(ui) + ∑ uk∈g σ1(uk,ui)∧µ ′ 1(uk). hence, µ1(ui)∧µ2(vj) ≤ µ′1(ui)∧µ2(vj) + ∑ uk∈g σ1(uk,ui)∧µ′1(uk)∧µ2(vj) ≤ µ′j(ui,vj) + ∑ uk∈g σ((uk,vj),(ui,vj))∧µ′j(uk,vj) that is, µ(ui,vj) ≤ µ′j(ui,vj) + ∑ uk∈g (σ((uk,vj),(ui,vj))∧µ′j(uk,vj) thus we get µ′j is an fzdominating set of gj for j = 1,2, ...,n also |µ′j| ≤ |µ′1| shows that γfz(gj) ≤ γfz(g) for j = 1.2....,n hence γfz(g2h) ≤ nγfz(g) similarly, γfz(g2h) ≤ mγfz(h) thus we get, γfz(g2h) ≤ min{nγfz(g),mγfz(h)} in the previous theorem, equality might hold. for g,h and g2h given in fig. 6, γfz(g) = 0.2, γfz(h) = 0.2 and γfz(g2h) = 0.4 so that γfz(g2h) = min{nγfz(g),mγfz(h)} figure 6: fuzzy graph g,h and g2h v. g. vizing presented the following conjecture regarding the cartesian product of crisp graphs in 1968. γ(g2h) ≥ γ(g)γ(h), for every pair of finite crisp graphs g and h. 223 lekh a and parvathy k.s possibly the most significant unsolved issue in the field of domination theory is vizing’s conjecture. here, we investigate the applicability of vizing’s like inequality to fz-dominantion in fuzzy graphs. vizing’s conjecture is said to be satisfied by a fuzzy graph g, if γfz(g2h) ≥ γfz(g)γfz(h) for every fuzzy graph h. definition 5.1. a fuzzy graph h = (v1,µ1,σ1) is known as a partial fuzzy subgraph of g = (v,µ,σ) induced by v1 if v1 ⊂ v , µ1(u) = µ(u) if u ∈ v1, 0 otherwise and σ1(u,v) = σ(u,v)∧µ(u)∧µ(v) for all u,v ∈ v . definition 5.2. the spanning fuzzy subgraph of g = (v,µ,σ) is the partial fuzzy subgraph g′ = (v1,µ′,σ′) where v = v1 and µ = µ′ if g′ is a spanning fuzzy subgraph of the fuzzy graph g, then γfz(g′) ≥ γfz(g)). theorem 5.2. if g satisfies vizing’s conjecture and g′ is a spanning fuzzy subgraph of g such that γfz(g′) = γfz(g), then g′ also satisfies vizing’s conjecture. proof. g′2h is a spanning fuzzy subgraph of g2h for every fuzzy graph h. hence γfz(g′2h) ≥ γfz(g2h) ≥ γfz(g)γfz(h) = γfz(g′)γfz(h) the example below illustrates that in general this inequality does not hold for fz-domination in fuzzy graphs. example 5.1. consider the fuzzy graphs g = (v1,µ1,σ1) and h = (v2,µ2,σ2) given in figure 7. for g, v1 = {(a,1),(b,1),(c,1)}, σ1(a,b) = σ1(b,c) = 1,σ1(a,c) = 0. for h, v2 = {(u,1),(v,1),(w,1)}, σ2(u,v) = σ2(v,w) = 1,σ2(u,w) = 0. µ′1 = {(a,0.8),(b,0.6),(c,0.8)} is a minimum fz-dominating set of g. hence γfz(g) = 2.2. similarly γfz(h) = 2.2 now µ′ = {((a,u),0.6),((a,v),0.4),((a,w),0.6)},((b,u),0.4),((b,v),0.2), ((b,w),0.4),((c,u),0.6),((c,v),0.4),((c,w),0.6) is a minimum fz-dominating set of g2h. hence γfz(g2h) = 4.2 here γfz(g2h) < γfz(g)γfz(h) there are fuzzy graphs for which 1. γfz(g2h) < γfz(g)γfz(h) 2. γfz(g2h) = γfz(g)γfz(h) 3. γfz(g2h) > γfz(g)γfz(h) 224 on fzdomination number of fuzzy graphs figure 7: fuzzy graphs g, h and g2h 6 conclusions graph operations are techniques for creating new graphs from ones that already exist, and they are crucial in the design and analysis of large networks. in this article, we investigate various characteristics of the fz-domination number of fuzzy graphs under the influence of some graph operations. it is possible to derive bounds for the fz-domination number of the union, join, corona, and cartesian product of fuzzy graphs. through examples, the sharpness of these bounds are demonstrated and the factors that contribute to the sharpness are examined. references s. a. k. bhutani and l. sathikala. on (r,s)-fuzzy domination in fuzzy graphs. new mathematics and natural computation 12(01):1-10, 2016. n. gani and v. chandrasekaran. domination in fuzzy graph. advances in fuzzy sets and systems, 1, 01 2006. e. c. g. s. hedetniemi and c. mynhardt. properties of minimal dominating functions of graphs. technical report, dms-547-ir, 1990. s. h. s. hedetniemi and t. wimer. linear time resourse allocation algorithms for trees. technical report url-014, department of mathematics, clemson university, 1987. 225 lekh a and parvathy k.s a. lekha and k. s. parvathy. fuzzy domination in fuzzy graphs. journal of intelligent and fuzzy systems, 2022. doi: 10.3233/jifs-220987. o. t. manjusha and m. s. sunitha. strong domination in fuzzy graphs. fuzzy information and engineering, 7(3):369-377, 2015. j. n. mordeson and p. chang-shyh. operations on fuzzy graphs. information sciences, 79(3):159–170, 1994. issn 0020-0255. j. n. mordeson and p. s. nair. fuzzy graphs and fuzzy hypergraphs. physicaverlag, 2000. a. rosenfeld. fuzzy graphs. in fuzzy sets and their applications to cognitive and decision processes, pages 77–95. academic press, 1975. a. somasundaram. domination in products of fuzzy graphs. international journal of uncertainty, fuzziness and knowledge-based systems vol. 13, no. 2 (2005) 195-204, world scientific publishing company, 2005. a. somasundaram and s. somasundaram. domination in fuzzy graphs – i. pattern recognition letters, 19(9):787–791, 1998. issn 0167-8655. 226 ratio mathematica volume 44, 2022 a new class of nano generalized closed sets in nano topological spaces anbarasi rodrigo p* subithra p† abstract in this paper, we introduce a new class of nano generalized closed sets in nano topological spaces namely nano generalized 𝛼∗-closed sets. then we discuss some of its properties and investigate their relation with many other nano closed sets. also, we define nano generalized 𝛼∗-open set and discuss its relation with other open sets. finally, we define the properties of nano generalized 𝛼∗-interior and nano generalized 𝛼∗-closure. keywords: ℕ𝑔𝛼∗closed sets, ℕ𝑔𝛼∗open sets, ℕ𝑔𝛼∗int, ℕ𝑔𝛼 ∗cl. 2010 ams subject classification: 54a05‡ *assistant professor, department of mathematics, st. mary’s college (autonomous), (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli) thoothukudi-1, tamilnadu, india; anbu.n.u@gmail.com. † research scholar, reg.no. 21212212092003, department of mathematics, st. mary’s college (autonomous), (affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli) thoothukudi-1, tamilnadu, india;p.subithra18@gmail.com. ‡ received on june 4, 2022. accepted on september 1, 2022. published on nov 30, 2022. doi: 10.23755/rm.v44i0.900. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 132 mailto:anbu.n.u@gmail.com mailto:p.subithra18@gmail.com p. anbarasi rodrigo and p. subithra 1. introduction the theory of nano topology proposed by lellis thivagar [4] and carmel richard is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. the elements of a nano topological space are called the nano open set. the author has defined nano topological space in terms of lower and upper approximations and boundary region. he has defined nano closed sets, nano-interior and nano-closure of a set. he also introduced certain weak forms of nano open sets such as nano 𝛼-open set, nano semi-open sets and nano pre-open sets. levine [5] introduced the class of 𝑔-closed sets in 1970. k. bhuvaneswari introduced nano 𝑔closed [1], nano 𝑔𝑠-closed [3], nano 𝛼𝑔-closed [6], nano 𝑔𝑝-closed [2], nano 𝑔𝑟closed [12] sets and studied their properties. nano 𝑔∗𝑝-closed sets was introduced by rajendran [10] and investigated. the aim of this paper is to introduce and study the properties of nano 𝑔𝛼∗-closed sets and nano 𝑔𝛼∗-open sets in nano topological spaces. finally, we define the properties of nano generalized 𝛼∗-interior and nano generalized 𝛼∗-closure. 2. preliminaries throughout this paper (𝕌, 𝜏ℝ(𝕏)) represent nano topological spaces on which no separation axioms are assumed unless and otherwise mentioned. for a subset 𝕊 of (𝕌, 𝜏ℝ(𝕏)), 𝑁𝑐𝑙(𝕊) and 𝑁𝑖𝑛𝑡(𝕊) denote the nano closure of 𝕊 and nano interior of 𝕊 respectively. we recall the following definitions which are useful in the sequel. definition 2.1. [4] let 𝕌 be a non-empty finite set of objects called the universe and ℝ be an equivalence relation on 𝕌 named as the indiscernibility relation. elements belonging to the same equivalence class are said to be indiscernible with one another. the pair (𝕌,ℝ) is said to be the approximation space. let 𝕏 ⊆ℕ 𝕌. then 1) the lower approximation of 𝕏 with respect to ℝ is the set of all objects, which can be for certain classified as 𝕏 with respect to ℝ and it is denoted by 𝕃ℝ(𝕏). 𝕃ℝ(𝕏) = ⋃ {ℝ(𝑥): ℝ(𝑥) ⊆ℕ 𝕏} 𝑥∈𝕌 2) the upper approximation of 𝕏 with respect to ℝ is the set of all objects, which can be possibly classified as 𝕏 with respect to ℝ and it is denoted by 𝕌ℝ(𝕏). 𝕌ℝ(𝕏) = ⋃ {ℝ(𝑥): ℝ(𝑥) ∩ 𝕏 ≠ ϕ} 𝑥∈𝕌 3) the boundary region of 𝕏 with respect to ℝ is the set of all objects, which can be classified neither as 𝕏 nor as not –𝕏 with respect to ℝ and it is denoted by 𝔹ℝ(𝕏). 𝔹ℝ(𝕏). = 𝕌ℝ(𝕏) − 𝕃ℝ(𝕏). definition 2.2. [4] let 𝕌 be the universe, ℝ be an equivalence relation on 𝕌 and 𝜏ℝ(𝕏) = {𝕌, 𝜙, 𝕌ℝ(𝕏), 𝕃ℝ(𝕏), 𝔹ℝ(𝕏)} where 𝕏 ⊆ℕ𝕌. then ℝ(𝕏) satisfies the following axioms: 1) 𝕌 and ϕ ∈ 𝜏ℝ(𝕏), 2) the union of the elements of any sub collection of 𝜏ℝ(𝕏) is in 𝜏ℝ(𝕏), 133 a new class of nano generalized closed sets in nano topological spaces 3) the intersection of the elements of any finite sub collection of 𝜏ℝ(𝕏) is in 𝜏ℝ(𝕏). that is, 𝜏ℝ(𝕏)is a topology on 𝕌 called the nano topology on 𝕌 with respect to 𝕏. we call (𝕌, 𝜏ℝ(𝕏)) as the nano topological space(𝑁𝑇𝑆). the elements of 𝜏ℝ(𝕏) are called as nano open sets. the complement of nano-open sets is called nano closed sets. definition 2.3. [4] if (𝕌, 𝜏ℝ(𝕏)) is a 𝑁𝑇𝑆 with respect to 𝕏 and if 𝕊 ⊆ℕ𝕌, then • the nano interior of 𝕊 is defined as the union of all nano open subsets of 𝕊 and it is denoted by ℕ𝑖𝑛𝑡(𝕊). that is, ℕ𝑖𝑛𝑡(𝕊) is the largest open subset of 𝕊. • the nano closure of 𝕊 is defines as the intersection of all nano closed sets containing 𝕊 and it is denoted by ℕ𝑐𝑙(𝕊). that is, ℕ𝑐𝑙(𝕊) is the smallest nano closed set containing 𝕊. definition 2.4. a subset 𝕊 of a 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)) is called; 1) nano pre-open [4] if 𝕊 ⊆ℕ ℕ𝑖𝑛𝑡(ℕ𝑐𝑙(𝕊)) 2) nano semi-open [4] if 𝕊 ⊆ℕ ℕ𝑐𝑙(ℕ𝑖𝑛𝑡(𝕊)) 3) nano 𝛼-open [4] if 𝕊 ⊆ℕ ℕ𝑖𝑛𝑡(ℕ𝑐𝑙(ℕ𝑖𝑛𝑡(𝕊))) 4) nano 𝛽-open [11] if 𝕊 ⊆ℕ ℕ𝑐𝑙(ℕ𝑖𝑛𝑡(ℕ𝑐𝑙(𝕊))) 5) nano regular-open [4] if 𝕊 = ℕ𝑖𝑛𝑡(ℕ𝑐𝑙(𝕊)) the complements of the above-mentioned sets are called their respective closed sets. definition 2.5. a subset 𝕊 of a 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)) is called; 1) ℕ𝑔-closed [1] if ℕ𝑐𝑙(𝕊) ⊆ℕ 𝔽, whenever 𝕊 ⊆ℕ𝔽 and 𝔽 is nano open in 𝕌. 2) ℕ𝑔𝑠-closed [3] if ℕ𝑠𝑐𝑙(𝕊) ⊆ℕ 𝔽, whenever 𝕊 ⊆ℕ𝔽 and 𝔽 is nano open in 𝕌. 3) ℕ𝛼𝑔-closed [6] if ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ 𝔽, whenever 𝕊 ⊆ℕ𝔽 and 𝔽 is nano open in 𝕌. 4) ℕ𝑔𝑝-closed [2] if ℕ𝑝𝑐𝑙(𝕊) ⊆ℕ 𝔽, whenever 𝕊 ⊆ℕ𝔽 and 𝔽 is nano open in 𝕌. 5) ℕ𝑔𝛽-closed [7] if ℕ𝛽𝑐𝑙(𝕊) ⊆ℕ 𝔽, whenever 𝕊 ⊆ℕ𝔽 and 𝔽 is nano open in 𝕌. 6) ℕ𝑔𝑟-closed [12] if ℕ𝑟𝑐𝑙(𝕊) ⊆ℕ 𝔽, whenever 𝕊 ⊆ℕ𝔽 and 𝔽 is nano open in 𝕌. 7) ℕ𝑔∗-closed [8] if ℕ𝑐𝑙(𝕊) ⊆ℕ 𝔽, whenever 𝕊 ⊆ℕ𝔽 and 𝔽 is ℕ𝑔-open in 𝕌. 8) ℕ𝑔∗𝑠-closed [9] if ℕ𝑠𝑐𝑙(𝕊) ⊆ℕ 𝔽, whenever 𝕊 ⊆ℕ𝔽 and 𝔽 is ℕ𝑔-open in 𝕌. 9) ℕ𝑔∗𝑝-closed [10] if ℕ𝑝𝑐𝑙(𝕊) ⊆ℕ 𝔽, whenever 𝕊 ⊆ℕ𝔽 and 𝔽 is ℕ𝑔-open in 𝕌. 10) ℕ𝑔∗𝑟-closed [13] if ℕ𝑟𝑐𝑙(𝕊) ⊆ℕ 𝔽, whenever 𝕊 ⊆ℕ𝔽 and 𝔽 is ℕ𝑔-open in 𝕌. theorem 2.6. [1] every nano open set is ℕ𝑔-open. 3. nano generalized 𝜶∗-closed sets definition 3.1. a nano generalized 𝛼∗(in short, ℕ𝑔𝛼∗) closed set is a subset 𝕊 of a 𝑁𝑇𝑆(𝕌, 𝜏ℝ(𝕏)) ifℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽)whenever 𝕊 ⊆ℕ 𝔽and𝔽 𝑖𝑠 ℕ𝑔-open in 𝕌. instance 3.2. let 𝕌 = {p, q, r, s} with 𝕌/ℝ ={{p}, {r}, {q, s}} and 𝕏 = {p, q} ⊆ℕ 𝕌. then 𝜏𝑅 (𝕏) = {𝕌, ϕ, {p}, {q, s}, {p, q, s}}. here ŋ𝑔𝛼 ∗-closed = {𝕌, ϕ, {r}, {p, r}, {q, r}, {r, s}, {p, q, r}, {p, r, s}, {q, r, s}}. 134 p. anbarasi rodrigo and p. subithra instance 3.3. let 𝕌 = {p, q, r} with 𝕌/ ℝ = {{p}, {p, q}} and 𝕏 = {p} ⊆ℕ 𝕌. then 𝜏ℝ(𝕏) = {𝕌, ϕ, {p}, {q}, {p, q}}. here ℕ𝑔𝛼 ∗closed = {ϕ, 𝕌, {r}, {p, r}, {q, r}}. theorem 3.4. every nano closed set is ℕ𝑔𝛼∗-closed. proof: let 𝕊 be a nano closed set in 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)). then we have, ℕ𝑐𝑙(𝕊) = 𝕊. let 𝔽 be a nano open set in 𝕌 such that 𝕊 ⊆ℕ 𝔽 and𝔽 = ℕ𝑖𝑛𝑡(𝔽). by theorem 2.6, we have 𝔽 is ℕ𝑔-open in 𝕌. then ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑐𝑙(𝕊) = 𝕊 ⊆ℕ 𝔽 = ℕ𝑖𝑛𝑡(𝔽) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). thus ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is ℕ𝑔-open in 𝕌. therefore 𝕊 is ℕ𝑔𝛼∗-closed. remark 3.5 the invert of the preceding theorem does not hold as witnessed in the succeeding instance. instance 3.6. let 𝕌 = {p, q, r, s} with 𝕌/ℝ = {{r}, {p, q, s}} and 𝕏 = {q, s}. then 𝜏ℝ(𝕏) = {ϕ, 𝕌, {p, q, s}}. here {𝕌, ϕ, {r}, {p, r}, {q, r}, {r, s}, {p, q, r}, {p, r, s}, {q, r, s}} is ℕ𝑔𝛼∗closed set but the set is not nano closed. theorem 3.7. every ℕ𝑔𝛼∗closed set is ℕ𝑔 closed. proof. let 𝕊 be a ℕ𝑔𝛼∗closed set in 𝑁𝑇𝑆(𝕌, 𝜏ℝ(𝕏)) and let 𝔽 be a nano open set in 𝕌 such that 𝕊 ⊆ℕ 𝔽 and 𝔽 = ℕ𝑖𝑛𝑡(𝔽). by theorem 2.6, we have 𝔽 is ℕ𝑔-open in 𝕌. also, 𝕊 is ℕ𝑔𝛼∗-closed, then ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). then, ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) ⊆ℕ 𝔽. thus ℕ𝑐𝑙(𝕊) ⊆ℕ 𝔽 whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is nano open in 𝕌. hence 𝕊 is ℕ𝑔-closed. remark 3.8. the invert of the preceding theorem does not hold as witnessed in the succeeding instance. instance 3.9. let 𝕌 = {p, q, r, s} with 𝕌/ℝ = {{p, q}, {r, s}} and 𝕏 = {p, q}. then 𝜏ℝ(𝕏) = {𝕌, ϕ, {p, q}}. here {𝕌, ϕ, {r}, {s}, {p, r}, {p, s}, {q, r}, {q, s}, {r, s}, {p, q, r}, {p, q, s}, {p, r, s}, {q, r, s}} is ℕ𝑔 – closed set but the set is not ℕ𝑔𝛼∗closed. theorem 3.10. every ℕ𝑔𝛼∗closed set is ℕ𝛼𝑔closed. proof. let 𝕊 be a ℕ𝑔𝛼∗-closed set in 𝑁𝑇𝑆(𝕌, 𝜏ℝ(𝕏)) and let 𝔽 be a nano open set in 𝕌 such that 𝕊 ⊆ℕ 𝔽 and𝔽 = ℕ𝑖𝑛𝑡(𝔽). by theorem 2.6, we have 𝔽 is ℕ𝑔-open in 𝕌. also, 𝕊 is ℕ𝑔𝛼∗-closed, then ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). then ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) = 𝔽. thus ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ 𝔽 whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is nano open in 𝕌. hence 𝕊 is ℕ𝛼𝑔closed. remark 3.11. the invert of the preceding theorem does not hold as witnessed in the succeeding instance. instance 3.12. let 𝕌 = {p, q, r} with 𝕌/ℝ = {{{r}, {p, q}} and 𝕏 = {r}. then 𝜏ℝ(𝕏) = {ϕ, 𝕌, {r}}. here {ϕ, {p}, {q}, {p, q}, {p, r}, {q, r}} is ℕ𝛼𝑔 closed but the set is not ℕ𝑔𝛼∗closed. 135 a new class of nano generalized closed sets in nano topological spaces theorem 3.13. every ℕ𝑔𝛼∗-closed set is ℕ𝑔𝑠-closed. proof. let 𝕊 be a ℕ𝑔𝛼∗-closed set in 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)). let 𝔽 be a nano open set in 𝕌 such that 𝕊 ⊆ℕ 𝔽 and𝔽 = ℕ𝑖𝑛𝑡(𝔽). by theorem 2.6, 𝔽 is ℕ𝑔-open in 𝕌. since 𝕊 is ℕ𝑔𝛼∗-closed, ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). then ℕ𝑠𝑐𝑙(𝕊) ⊆ℕ ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) = 𝔽. thus ℕ𝑠𝑐𝑙(𝕊) ⊆ℕ 𝔽 whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is nano open in 𝕌. hence 𝕊 is ℕ𝑔𝑠closed. remark 3.14. the invert of the former theorem does not holds as witnessed in the succeeding instance. instance 3.15. let 𝕌 = {p, q, r, s} with 𝕌/ℝ = {{p}, {q}, {r, s}} and 𝕏 = {q, s}. then 𝜏ℝ(𝕏) = {ϕ, 𝕌, {q}, {r, s}, {q, r, s}. here {ϕ, 𝕌, {p}, {q}, {r}, {s}, {p, q}, {p, r}, {p, s}, {r, s}, {p, q, r}, {p, q, s}, {p, r, s}} is ℕ𝑔𝑠 closed but it is not ℕ𝑔𝛼∗closed. theorem 3.16. every ℕ𝑔𝛼∗-closed set is ℕ𝑔𝑝 closed. proof: let 𝕊 be a ℕ𝑔𝛼∗-closed set in 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)). let 𝔽 be a nano open set in 𝕌 such that 𝕊 ⊆ℕ 𝔽 and𝔽 = ℕ𝑖𝑛𝑡(𝔽). by theorem 2.6, we have 𝔽 is ℕ𝑔-open in 𝕌. since 𝕊 is ℕ𝑔𝛼∗-closed, ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). then ℕ𝑝𝑐𝑙(𝕊) ⊆ℕ ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) = 𝔽. thus ℕ𝑝𝑐𝑙(𝕊) ⊆ℕ 𝔽 whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is nano open in 𝕌. hence 𝕊 is ℕ𝑔𝑝closed. remark 3.17. the invert of the former theorem does not holds as witnessed in the succeeding instance. instance 3.18. let 𝕌 = {p, q, r, s} with 𝕌/ℝ = {(p}, {r}, {q, s}} and 𝕏 = {r, s}. then 𝜏ℝ(𝕏) = {ϕ, 𝕌, {r}, {q, s}, {q, r, s}}. here {ϕ, 𝕌, {p}, {q}, {s}, {p, q}, {p, r}, {p, s}, {p, q, r}, {p, q, s} {p, r, s}} is ℕ𝑔𝑝-closed but it is not ℕ𝑔𝛼∗closed. theorem 3.19. every ℕ𝑔𝛼∗-closed set is ℕ𝑔𝛽-closed. proof. let 𝕊 be a ℕ𝑔𝛼∗-closed set in 𝑁𝑇𝑆(𝕌, 𝜏ℝ(𝕏)) and let 𝔽 be a nano open set in 𝕌 such that 𝕊 ⊆ℕ 𝔽 and𝔽 = ℕ𝑖𝑛𝑡(𝔽). by theorem 2.6, we have 𝔽 is ℕ𝑔-open in 𝕌. since 𝕊 is ℕ𝑔𝛼∗-closed, ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). then ℕ𝛽𝑐𝑙(𝕊) ⊆ℕ ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) = 𝔽. thus ℕ𝛽𝑐𝑙(𝕊) ⊆ℕ 𝔽 whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is nano open in 𝕌. hence 𝕊 is ℕ𝑔𝛽closed. remark 3.20. the invert of the former theorem does not hold as witnessed in the succeeding instance. instance 3.21. let 𝕌 = {p, q, r} with 𝕌/ℝ = {{p}, {q, r}} and 𝕏 = {p, q}. then 𝜏ℝ(𝕏) = {ϕ, 𝕌, {p}, {q, r}}. here {ϕ, 𝕌, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}} is ℕ𝑔𝛽-closed but it is not ℕ𝑔𝛼∗-closed. theorem 3.22. everyℕ𝑔𝛼∗closed set is ℕ𝑔𝑟 closed. 136 p. anbarasi rodrigo and p. subithra proof. let 𝕊 be a ℕ𝑔𝛼∗-closed set in 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)). let 𝔽 be a nano open set in 𝕌 such that 𝕊 ⊆ℕ 𝔽 and𝔽 = ℕ𝑖𝑛𝑡(𝔽). by theorem 2.6, we have 𝔽 is ℕ𝑔-open in 𝕌. since 𝕊 is ℕ𝑔𝛼∗-closed, ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). then ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑟𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) = 𝔽. thus ℕ𝑟𝑐𝑙(𝐴) ⊆ℕ 𝔽 whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is nano open in 𝕌. hence 𝕊 is ℕ𝑔𝑟closed. remark 3.23. the transpose of the preceding theorem does not hold as witnessed in the succeeding instance. instance 3.24. let 𝕌 = {p, q, r} with 𝕌/ℝ = {{r}, {p, q}} and 𝕏={r}. then 𝜏ℝ(𝕏) = {ϕ, 𝕌, {r}}. here {ϕ, {p}, {q}, {p, q}, {p, r}, {q, r}} is ℕ𝑔𝑟-closed which is not ℕ𝑔𝛼∗closed. theorem 3.25. every ℕ𝑔∗-closed set is ℕ𝑔𝛼∗-closed. proof: let 𝕊 be a ℕ𝑔∗-closed set in 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)). let 𝔽 be a nano open set in 𝕌 such that 𝕊 ⊆ℕ 𝔽 and𝔽 = ℕ𝑖𝑛𝑡(𝔽). through theorem 2.6, we have 𝔽 is ℕ𝑔-open in 𝕌. since 𝕊 is ℕ𝑔∗-closed, ℕ𝑐𝑙(𝕊) ⊆ℕ 𝔽. then ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑐𝑙(𝕊) ⊆ℕ 𝔽 = ℕ𝑖𝑛𝑡(𝔽) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). thus ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽)whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is ℕ𝑔open in 𝕌. hence 𝕊 is ℕ𝑔𝛼∗-closed. remark 3.26. the transpose of the preceding theorem does not hold as witnessed in the succeeding instance. instance 3.27. let 𝕌 = {p, q, r} with 𝕌/ℝ = {{p}, {q, r}} and 𝕏 = {p}. then 𝜏ℝ(𝕏) = {ϕ, 𝕌, {p}}. here {ϕ, {q}, {r}, {q, r}} is ℕ𝑔𝛼∗-closed which is not ℕ𝑔∗-closed. theorem 3.28. every ℕ𝑔𝛼∗-closed set is ℕ𝑔∗𝑠-closed. proof. let 𝕊 be a ℕ𝑔𝛼∗-closed set in 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)). then ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is ℕ𝑔-open in 𝕌.thus ℕ𝑠𝑐𝑙(𝕊) ⊆ℕ ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) ⊆ℕ 𝔽, we get ℕ𝑠𝑐𝑙(𝕊) ⊆ℕ 𝔽 whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is ℕ𝑔-open in 𝕌. hence 𝕊 is ℕ𝑔 ∗𝑠-closed. remark 3.29. the transpose of the preceding theorem does not hold as witnessed in the succeeding instance. instance 3.30. let 𝕌 = {p, q, r, s} with 𝕌/ℝ = {{p}, {r}, {q, s}} and 𝕏 = {q, r}. then 𝜏ℝ(𝕏) = {ϕ, 𝕌, {r}, {q, s}, {q, r, s}}. here {ϕ, 𝕌, {p}, {r}, {p, q}, {p, r}, {p, s}, {q, s}, {p, q, r}, {p, q, s}, {p, r, s}} is ℕ𝑔∗𝑠-closed which is not ℕ𝑔𝛼∗-closed. theorem 3.31. every ℕ𝑔𝛼∗-closed set is ℕ𝑔∗𝑝-closed. proof: let 𝕊 be a ℕ𝑔𝛼∗-closed set in 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)). then ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is ℕ𝑔-open in 𝕌.thus ℕ𝑝𝑐𝑙(𝕊) ⊆ℕ ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) ⊆ℕ 𝔽, we get ℕ𝑝𝑐𝑙(𝕊) ⊆ℕ 𝔽 whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is ℕ𝑔-open in 𝕌. hence 𝕊 is ℕ𝑔 ∗𝑝-closed. 137 a new class of nano generalized closed sets in nano topological spaces remark 3.32. the polar statement of the preceding theorem does not hold as witnessed in the succeeding instance. instance 3.33. let 𝕌 = {p, q, r, s} with 𝕌/ℝ = {{p, q}, {r, s}} and 𝕏 = {q, r, s}. then 𝜏ℝ(𝕏) = {ϕ, 𝕌, {p, q}, {r, s}}. here {ϕ, 𝕌, {p}, {q}, {r}, {s}, {p, q}, {p, r}, {p, s}, {q, r}, {q, s}, {r, s}, {p, q, r}, {p, q, s}, {p, r, s}, {q, r, s}} is ℕ𝑔∗𝑝-closed but it is not ℕ𝑔𝛼∗-closed. theorem 3.34. every ℕ𝑔∗𝑟-closed set is ℕ𝑔𝛼∗-closed. proof: let 𝕊 be a ℕ𝑔∗𝑟-closed set in 𝑁𝑇𝑆(𝕌, 𝜏ℝ(𝕏)) and let 𝔽 be a nano open set in 𝕌 such that 𝕊 ⊆ℕ 𝔽 and𝔽 = ℕ𝑖𝑛𝑡(𝔽). through theorem 2.6, we have 𝔽 is ℕ𝑔-open in 𝕌. since 𝕊 is ℕ𝑔∗𝑟-closed, ℕ𝑟𝑐𝑙(𝐴) ⊆ℕ 𝔽. then ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑟𝑐𝑙(𝕊) ⊆ℕ 𝔽 = ℕ𝑖𝑛𝑡(𝔽) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). thus ℕ𝛼𝑐𝑙(𝕊) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽)whenever 𝕊 ⊆ℕ 𝔽 and 𝔽 is ℕ𝑔open in 𝕌. hence 𝕊 is ℕ𝑔𝛼∗-closed. remark 3.35. the polar statement of the preceding theorem does not hold as witnessed in the succeeding instance. instance 3.36. let 𝕌 = {p, q, r, s} with 𝕌/ℝ = {{r}, {q, s}} and 𝕏 = {p, r}. then 𝜏ℝ(𝕏) = {ϕ, 𝕌, {r}}. here {ϕ, 𝕌, {p}, {q}, {s}, {p, q}, {p, s}, {q, s}, {p, q, s}} is ℕ𝑔𝛼∗closed but it is not ℕ𝑔∗𝑟-closed. remark 3.37. the concepts of nano semi closed and ℕ𝑔𝛼∗-closed are independent as witnessed in the succeeding instance. instance 3.38. let 𝕌 = {p, q, r, s} with 𝕌/ℝ = {{p}, {r}, {q, s}} and 𝕏 = {p, q} ⊆ 𝕌. then 𝜏ℝ(𝕏) = {𝕌, ϕ, {p}, {q, s}, {p, q, s}}. the set {ϕ, 𝕌, {p}, {r}, {p, r}, {q, s}, {q, r, s}} is nano semi closed yet not ℕ𝑔𝛼∗-closed. the set {ϕ, 𝕌, {r}, {p, r}, {q, r}, {r, s}, {p, q, r}, {p, r, s}, {q, r, s}} is ℕ𝑔𝛼∗-closed but the set is not nano semi closed. remark 3.39. ℕ𝑔𝛼∗-closed set lies between ℕ𝑔∗-closed set and ℕ𝑔-closed set. that is, ℕ𝑔∗-closed⊆ℕ ℕ𝑔𝛼 ∗-closed⊆ℕ ℕ𝑔-closed. remark 3.40. the diagram that follows exhibit the relation between ℕ𝑔𝛼∗-closed sets and other closed sets. ℕ𝑔𝛼∗-closed ℕ𝑔-closed ℕ𝛼𝑔-closed ℕ𝑔𝑠-closed ℕ𝑔𝑝-closed ℕ𝑔𝛽-closed ℕ𝑔𝑟-closed ℕ𝑔∗𝑠-closed ℕ𝑔∗𝑝-closed ℕ𝑔∗-closed ℕ𝑔∗𝑟-closed 138 p. anbarasi rodrigo and p. subithra theorem 3.41. if 𝔾 and ℍ are ℕ𝑔𝛼∗-closed sets in 𝑁𝑇𝑆(𝕌, 𝜏ℝ(𝕏)), then 𝔾 ∪ ℍ is a ℕ𝑔𝛼∗-closed set. proof: let 𝔾and ℍ be ℕ𝑔𝛼∗-closed sets in a 𝑁𝑇𝑆(𝕌, 𝜏ℝ(𝕏)) and let 𝔽 be any ℕ𝑔-open set in 𝕌 containing 𝔾 and ℍ. then 𝔾 ∪ ℍ ⊆ℕ 𝔽. then 𝔾 ⊆ℕ 𝔽 and ℍ ⊆ℕ 𝔽. since 𝔾 and ℍ are ℕ𝑔𝛼∗-closed sets, ℕ𝛼𝑐𝑙(𝔾) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽)and ℕ𝛼𝑐𝑙(ℍ) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). now, ℕ𝛼𝑐𝑙(𝔾 ∪ ℍ) = ℕ𝛼𝑐𝑙(𝔾) ∪ ℕ𝛼𝑐𝑙(𝐻) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). thus, ℕ𝛼𝑐𝑙(𝔾 ∪ ℍ) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) whenever 𝔾 ∪ ℍ ⊆ℕ 𝔽 and 𝔽 is ℕ𝑔-open in 𝕌. hence 𝔾 ∪ ℍ is a ℕ𝑔𝛼∗-closed. theorem 3.42. if 𝔾 and ℍ are ℕ𝑔𝛼∗-closed sets in 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)), then 𝔾 ∩ ℍ is a ℕ𝑔𝛼∗-closed set. proof: let 𝔾 and ℍ be ℕ𝑔𝛼∗-closed sets in a 𝑁𝑇𝑆(𝕌, 𝜏ℝ(𝕏)) and let 𝔽 be a ℕ𝑔-open set in 𝕌 such that 𝔾 ⊆ℕ 𝔽 and ℍ ⊆ℕ 𝔽. then 𝔾 ∩ ℍ ⊆ℕ 𝔽. since 𝔾 and ℍ are ℕ𝑔𝛼 ∗closed sets, ℕ𝛼𝑐𝑙(𝔾) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽)and ℕ𝛼𝑐𝑙(ℍ) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). now,ℕ𝛼𝑐𝑙(𝔾 ∩ ℍ) ⊆ℕ ℕ𝛼𝑐𝑙(𝔾) ∩ ℕ𝛼𝑐𝑙(ℍ) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽). thus ℕ𝛼𝑐𝑙(𝔾 ∩ ℍ) ⊆ℕ ℕ𝑖𝑛𝑡 ∗(𝔽) whenever 𝔾 ∩ ℍ ⊆ℕ 𝔽 and 𝔽 is ℕ𝑔-open in 𝕌. hence 𝔾 ∩ ℍ is a ℕ𝑔𝛼 ∗-closed. corollary 3.43. if 𝔾 is ℕ𝑔𝛼∗-closed and ℍ is nano closed in 𝕌, then 𝔾 ∩ ℍ is ℕ𝑔𝛼∗closed. proof: let ℍ be nano closed in 𝕌. then by theorem 3.4, ℍ is ℕ𝑔𝛼∗-closed. 𝔾 is also ℕ𝑔𝛼∗-closed. by theorem 3.42, 𝔾 ∩ ℍ is ℕ𝑔𝛼∗-closed. corollary 3.44. if 𝔾 is ℕ𝑔𝛼∗-closed and ℍ is nano open in 𝕌, then 𝔾\ℍ is ℕ𝑔𝛼∗closed. proof: let 𝔾\ℍ = 𝔾 ∩ (𝕌\ℍ). since ℍ is nano open in 𝕌, 𝕌\ℍ is nano closed in 𝕌. since 𝔾 is ℕ𝑔𝛼∗-closed and 𝕌\ℍ is nano closed in 𝕌, by corollary 3.43, 𝔾 ∩ (𝕌\ℍ)is ℕ𝑔𝛼∗-closed. hence 𝔾\ℍ is ℕ𝑔𝛼∗-closed. 4. nano generalized 𝜶∗-open sets definition 4.1. a subset 𝕊 of a 𝑁𝑇𝑆(𝕌, 𝜏ℝ(𝕏)) is called nano generalized 𝛼 ∗(in short, ℕ𝑔𝛼∗) open set if its complement is ℕ𝑔𝛼∗-closed. instance 4.2. let 𝕌 = {p, q, r, s} with 𝕌/ℝ = {{q}, {r}, {p, s}} and 𝕏 = {p, r}. then 𝜏ℝ(𝕏) = {𝕌, ϕ, {r}, {p, s}, {p, r, s}}. here {ϕ, 𝕌, {p}, {r}, {s}, {p, r}, {p, s}, {r, s}, {p, r, s}} is ℕ𝑔𝛼∗-open sets. theorem 4.3. every nano open set is ℕ𝑔𝛼∗-open but the invert may not be true. theorem 4.4. every ℕ𝑔𝛼∗-open set is ℕ𝑔-open but the invert may not be true. theorem 4.5. every ℕ𝑔𝛼∗-open set is ℕ𝛼𝑔-open but the invert may not be true. theorem 4.6. every ℕ𝑔𝛼∗-open set is ℕ𝑔𝑠-open but the invert may not be true. 139 a new class of nano generalized closed sets in nano topological spaces theorem 4.7. every ℕ𝑔𝛼∗-open set is ℕ𝑔𝑝-open but the invert may not be true. theorem 4.8. every ℕ𝑔𝛼∗-open set is ℕ𝑔𝛽-open but the invert may not be true. theorem 4.9. every ℕ𝑔𝛼∗-open set is ℕ𝑔𝑟-open but the invert may not be true. theorem 4.10. every ℕ𝑔∗-open set is ℕ𝑔𝛼∗-open but the invert may not be true. theorem 4.11. every ℕ𝑔𝛼∗-open set is ℕ𝑔∗𝑠-open but the invert may not be true. theorem 4.12. every ℕ𝑔𝛼∗-open set is ℕ𝑔∗𝑝-open but the invert may not be true. theorem 4.13. every ℕ𝑔∗𝑟-open set is ℕ𝑔𝛼∗-open but the invert may not be true. theorem 4.14. if 𝔾 and ℍ are ℕ𝑔𝛼∗-open sets in 𝑁𝑇𝑆, then 𝔾 ∪ ℍ is a ℕ𝑔𝛼∗-open set. theorem 4.15. if 𝔾 and ℍ are ℕ𝑔𝛼∗-open sets in 𝑁𝑇𝑆, then 𝔾 ∩ ℍ is a ℕ𝑔𝛼∗-open set. corollary 4.16. if 𝔾 is ℕ𝑔𝛼∗-open and ℍ is nano open in 𝕌, then 𝔾 ∩ ℍ is ℕ𝑔𝛼∗-open. corollary 4.17. if 𝔾 is ℕ𝑔𝛼∗-open and ℍ is nano closed in 𝕌, then 𝔾\ℍ is ℕ𝑔𝛼∗-open. 5. ℕ𝒈𝜶∗-interior and ℕ𝒈𝜶∗-closure definition 5.1. let 𝕌 be a 𝑁𝑇𝑆 and let any point 𝑎 ∈ 𝕌. a subset 𝕊 of 𝕌 is called the ℕ𝑔𝛼∗-nbhd of 𝑎 if there exists a ℕ𝑔𝛼∗-open set 𝕂 such that 𝑎 ∈ 𝕂 ⊆ℕ 𝕊. definition 5.2. let 𝕊 be a subset of the 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)). a point 𝑎 ∈ 𝕊is called ℕ𝑔𝛼 ∗interior point of 𝕊 if 𝕊 is aℕ𝑔𝛼∗-nbhd of 𝑎. the set which contains all ℕ𝑔𝛼∗-interior points of 𝕊 is called ℕ𝑔𝛼∗-interior of 𝕊 and symbolized as ℕ𝑔𝛼∗-int(𝕊). definition 5.3. let 𝕊 be a subset of the 𝑁𝑇𝑆 (𝕌, 𝜏ℝ(𝕏)). then the intersection of all ℕ𝑔𝛼∗-closed sets containing 𝕊 is called ℕ𝑔𝛼∗-closure of 𝕊. that is, ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) =∩ {ℝ: ℝ 𝑖𝑠 ℕ𝑔𝛼∗ − 𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑒𝑡𝑠 𝑎𝑛𝑑 𝕊 ⊆ℕ ℝ}. theorem 5.4. if 𝕊 be a subset of 𝕌, then ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) =∪ {ℝ: ℝ 𝑖𝑠 ℕ𝑔𝛼∗ − 𝑜𝑝𝑒𝑛 𝑠𝑒𝑡 𝑎𝑛𝑑 ℝ ⊆ℕ 𝕊}. proof: let 𝕊 be a subset of 𝕌. 𝑥 ∈ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) ⇔ 𝑥 is a ℕ𝑔𝛼∗-interior point of 𝕊 ⇔𝕊 is a ℕ𝑔𝛼∗-nbhd of the point 𝑥 ⇔there exists ℕ𝑔𝛼∗-open set ℝ such that 𝑥 ∈ ℝ ⊆ℕ 𝕊 ⇔𝑥 ∈∪ {ℝ: ℝ 𝑖𝑠 ℕ𝑔𝛼∗ − 𝑜𝑝𝑒𝑛 𝑠𝑒𝑡 𝑎𝑛𝑑 ℝ ⊆ℕ 𝕊} hence 𝑁𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) =∪ {ℝ: ℝ 𝑖𝑠 ℕ𝑔𝛼∗ − 𝑜𝑝𝑒𝑛 𝑠𝑒𝑡 𝑎𝑛𝑑 ℝ ⊆ℕ 𝕊}. 140 p. anbarasi rodrigo and p. subithra theorem 5.5. let ℝ and 𝕊 be subsets of 𝕌. then a) ℕg𝛼∗ − 𝑖𝑛𝑡(𝕌) = 𝕌 and ℕg𝛼∗ − 𝑖𝑛𝑡(𝜙) = 𝜙 b) ℕg𝛼∗ − 𝑖𝑛𝑡(𝕊) ⊆ℕ 𝕊 c) if ℝ contains any ℕg𝛼∗-open set 𝕊, then 𝕊 ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(ℝ) d) if ℝ⊆ℕ𝕊, then ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(ℝ) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(𝕊) proof: a) since 𝕌 and ϕ are ℕ𝑔𝛼∗-open sets, by theorem 5.4, ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕌) =∪ {ℝ: ℝ 𝑖𝑠 ℕ𝑔𝛼∗ − 𝑜𝑝𝑒𝑛 𝑠𝑒𝑡 𝑎𝑛𝑑 ℝ ⊆ℕ 𝕌}. ⇒ 𝑁𝑔𝛼∗ − 𝑖𝑛𝑡(𝕌) =∪ {𝕊: 𝕊 𝑖𝑠 𝑎 ℕ𝑔𝛼∗ − 𝑜𝑝𝑒𝑛 𝑠𝑒𝑡} ⇒ 𝑁𝑔𝛼∗ − 𝑖𝑛𝑡(𝕌) = 𝕌 since ϕ is the only ℕ𝑔𝛼∗-open set contained in ϕ, ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝜙) = 𝜙. b) let 𝑥 ∈ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) ⇒𝑥 is a ℕ𝑔𝛼∗-interior point of 𝕊. ⇒𝕊 is a ℕ𝑔𝛼∗-nbhd of 𝑥. ⇒𝑥 ∈ 𝕊 thus ℕg𝛼∗ − 𝑖𝑛𝑡(𝕊) ⊆ℕ 𝕊. c) let 𝕊 be any ℕ𝑔𝛼∗-open set such that 𝕊 ⊆ℕ ℝ and let 𝑥 ∈ 𝕊. since 𝕊 is a ℕ𝑔𝛼∗-open set contained in ℝ, 𝑥 is a ℕ𝑔𝛼∗-interior point of ℝ. that is, 𝑥 ∈ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ) hence 𝕊 ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(ℝ). d) let ℝ and 𝕊 be subsets of 𝕌 such that ℝ⊆ℕ𝕊. let 𝑥 ∈ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ). then 𝑥 is aℕ𝑔𝛼∗-interior point of ℝ and so ℝ is aℕ𝑔𝛼∗-nbhd of 𝑥 contained in 𝕊. therefore 𝑥 is aℕ𝑔𝛼∗-interior point of 𝕊. thus 𝑥 ∈ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊). hence ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(𝕊). theorem 5.6. if 𝕊 is ℕ𝑔𝛼∗-open then ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) = 𝕊. proof: let 𝕊 be a ℕ𝑔𝛼∗-open in 𝕌. we know that ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) ⊆ℕ 𝕊. also, 𝕊 is ℕ𝑔𝛼∗-open set contained in 𝕊. by theorem 5.5 c), 𝕊 ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(𝕊). hence ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) = 𝕊. theorem 5.7. if ℝ and 𝕊 are subsets of 𝕌, then ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ) ∪ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(ℝ ∪ 𝕊). proof: we know that ℝ ⊆ℕ ℝ ∪ 𝕊 and 𝕊 ⊆ℕ ℝ ∪ 𝕊. then by theorem 5.5 d), ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(ℝ ∪ 𝕊) and ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(ℝ ∪ 𝕊). thus ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ) ∪ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(ℝ ∪ 𝕊). theorem 5.8. if ℝ and 𝕊 subsets of 𝕌, then ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ ∩ 𝕊) = ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ) ∩ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊). proof: we know that ℝ ∩ 𝕊 ⊆ℕ ℝ and ℝ ∩ 𝕊 ⊆ℕ 𝕊. by theorem 5.5 d), ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ ∩ 𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(ℝ) and ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ ∩ 𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(𝕊). 141 a new class of nano generalized closed sets in nano topological spaces then ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ ∩ 𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(ℝ) ∩ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊)(1) next, let 𝑥 ∈ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ) ∩ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) then 𝑥 ∈ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ) and 𝑥 ∈ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) hence 𝑥 is a ℕ𝑔𝛼∗-interior point of both sets ℝ and 𝕊. it follows that ℝ and 𝕊 is a ℕ𝑔𝛼∗-nbhd of 𝑥. thus 𝑥 ∈ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ ∩ 𝕊) hence ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ) ∩ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑖𝑛𝑡(ℝ ∩ 𝕊)(2) from (1) and (2), we get ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ ∩ 𝕊) = ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(ℝ) ∩ ℕ𝑔𝛼∗ − 𝑖𝑛𝑡(𝕊). theorem 5.9. let 𝕊 be a subset of 𝕌, then a) ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) is ℕ𝑔𝛼∗-closed in 𝕌 and ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) is the smallest ℕ𝑔𝛼∗closed set in 𝕌 containing 𝕊. b) 𝕊 is ℕ𝑔𝛼∗-closed if and only if ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) = 𝕊. proof: a) since the intersection of all ℕ𝑔𝛼∗-closed subsets of 𝕌 containing 𝕊 is ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊), ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) is ℕ𝑔𝛼∗-closed. ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) is contained in every ℕ𝑔𝛼∗-closed set containing 𝕊. hence, the smallest ŋ𝑔𝛼∗-closed set in 𝕌 containing 𝕊 is ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊). b) suppose 𝕊 is ℕ𝑔𝛼∗-closed. by the definition of ℕ𝑔𝛼∗-closure, ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) = 𝕊 conversely, suppose ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) = 𝕊. by theorem 3.42, ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊)is the ℕ𝑔𝛼∗closed set. therefore, 𝕊 is ℕ𝑔𝛼∗-closed. theorem 5.10 let ℝ and 𝕊 be subsets of 𝕌, then a) ℕ𝑔𝛼∗ − 𝑐𝑙(𝜙) = 𝜙 b) ℕ𝑔𝛼∗ − 𝑐𝑙(𝕌) = 𝕌 c) 𝕊 ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑐𝑙(𝕊) d) if ℝ ⊆ℕ 𝕊 then ℕ𝑔𝛼 ∗ − 𝑐𝑙(ℝ) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑐𝑙(𝕊) e) ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ ∪ 𝕊) = ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ) ∪ ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) f) ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ ∩ 𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑐𝑙(ℝ) ∩ ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) proof: a), b), c), d) follows from the definition of ℕ𝑔𝛼∗-closure. e) we know that ℝ ⊆ℕ ℝ ∪ 𝕊 and 𝕊 ⊆ℕ ℝ ∪ 𝕊. by d), ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑐𝑙(ℝ ∪ 𝕊),ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑐𝑙(ℝ ∪ 𝕊). then ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ) ∪ ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑐𝑙(ℝ ∪ 𝕊)(1) next, we prove ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ ∪ 𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑐𝑙(ℝ) ∪ ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) let 𝑥 ∉ ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ) ∪ ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) ⇒𝑥 ∉ ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ) and 𝑥 ∉ ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) by definition of ℕ𝑔𝛼∗ − 𝑐𝑙, ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ) =∩ {𝐹𝑖 : ℝ ⊆ℕ 𝐹𝑖 , 𝐹𝑖 𝑖𝑠 ℕ𝑔𝛼 ∗ − 𝑐𝑙𝑜𝑠𝑒𝑑} and ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊) =∩ {𝐹𝑖 : 𝕊 ⊆ℕ 𝐹𝑖 , 𝐹𝑖 𝑖𝑠 ℕ𝑔𝛼 ∗ − 𝑐𝑙𝑜𝑠𝑒𝑑}. then 𝑥 ∉ 𝐹𝑖 for some i. since ℝ ⊆ℕ 𝐹𝑖 and 𝕊 ⊆ℕ 𝐹𝑖, ℝ ∪ 𝕊 ⊆ 𝐹𝑖. therefore 𝑥 ∉ ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ ∪ 𝕊) hence ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ ∪ 𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑐𝑙(ℝ) ∪ ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊)(2) from (1) and (2) we have, ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ ∪ 𝕊) = ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ) ∪ ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊). f) we know that ℝ ∩ 𝕊 ⊆ℕ ℝ and ℝ ∩ 𝕊 ⊆ℕ 𝕊. 142 p. anbarasi rodrigo and p. subithra by d), ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ ∩ 𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑐𝑙(ℝ) and ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ ∩ 𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑐𝑙(𝕊) then ℕ𝑔𝛼∗ − 𝑐𝑙(ℝ ∩ 𝕊) ⊆ℕ ℕ𝑔𝛼 ∗ − 𝑐𝑙(ℝ) ∩ ℕ𝑔𝛼∗ − 𝑐𝑙(𝕊). 6. conclusions in this paper, we have introduced nano generalized 𝛼∗-closed sets and discussed some of its properties. then we investigated its relation with many other nano closed sets. further nano generalized 𝛼∗-open sets are defined and its properties and relations with other nano open sets are studied. consequently, nano generalized 𝛼∗-interior and nano generalized 𝛼∗-closure are introduced and discussed. references [1] k. bhuvaneshwari, k. mythili gnana priya. on nano generalized closed sets. international journal of scientific and research publications, 4(5), 1-3. 2014. [2] k. bhuvaneshwari, k. mythili gnana priya. on nano generalized pre-sets and nano pre generalized closed sets in nano topological spaces. international journal of innovative research in science, engineering and technology, 3(10), 16825-29. 2014. [3] k. bhuvaneshwari, k. exhilaration nano semi-generalized and nano generalized semi-closed sets. international journal of mathematics and computer applications research, 4(3), 117-124. 2014. [4] m. lellis thivagar, carmel richard. on nano forms of weakly open sets. international journal of mathematical and statistics invention, 1(1), 31-37. 2012. [5] n. levine. on generalized closed sets in topology. rendiconti del circolo matematico dipalermo,19(2), 8996. 1963. [6] r.t nachiyar, k. bhuvaneswari. on nano generalized α –closed sets and nano αgeneralized closed sets in nano topological spaces. international journal of engineering trends and technology, 13(6), 257-260. 2014. [7] i. rajasekaran, m. mehrain and o. nethaji. on nano gβ-closed sets. international journal of mathematics and its applications, 5(4-c), 377-882. 2017. [8] v. rajendran, p. sathish mohan, k. indirani. on nano generalized star closed sets in nano topological space. international journal of applied research,1(9), 04-07. 2015. [9] v. rajendran, b. anand, s. sharmila banu. on nano generalized star semi closed sets in nano topological spaces. international journal of applied research,1(9), 142144. 2015. 143 a new class of nano generalized closed sets in nano topological spaces [10] v. rajendran, p. sathish mohan, n. suresh. on nano generalized star pre-closed sets in nano topological spaces. international journal of recent scientific research. 7(1), 8066-8070. 2016. [11] a. revathy, g. ilango. on nano β-open sets. international journal of engineering, contemporary mathematics and sciences, 1(2), 16. 2015. [12] p. sulochana devi, k. bhuvaneshwari. on nano regular generalized and nano generalized regular closed sets, international journal of engineering trends and technology. 13(8), 386-390. 2014. [13] n. suresh, v. rajendran, p. sathish mohan. on nano generalized star regular closed sets in nano topological spaces. international journal of advanced research, 2(5), 35-38. 2016. 144 ratio mathematica volume 47, 2023 fixed point theorems in uniformly convex banach spaces jahir hussain rasheed* manoj karuppasamy† abstract in this article, we establish a concept of fixed point result in uniformly convex banach space. our main finding uses the ishikawa iteration technique in uniformly convex banach space to demonstrate strong convergence. additionally, we use our primary result to demonstrate some corollaries. keywords: fixed point; mann iteration; ishikawa iteration. 2020 ams subject classifications: 55m20, 46b80, 52a21.1 *jamal mohamed college (autonomous), affiliated to bharathidasan university, tiruchirapplli-620020, tamilnadu india ; hssn jhr@yahoo.com. †jamal mohamed college(autonomous), affiliated to bharathidasan university, tiruchirapplli-620020, tamilnadu india ; manojguru542@gmail.com. 1received on august 25, 2023. accepted on february 27, 2023. published on april 4, 2023. doi: 10.23755/rm.v39i0.835. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 198 r. jahir hussain, k. manoj 1 introduction mann [mann [1953]] defined mean value methods in an iterative scheme in 1953, and ishikawa [ishikawa [1974]] established fixed points using a new iteration method technique in 1954. takahashi [takahashi [1970] ] introduced the idea of convexity in metric spaces and non-expansive mappings in 1970. then machado [machado [1973]] went on to discuss about a classification of convex subsets of normed spaces. after that, luis bernal-gonzalez [bernal-gonzalez [1996]] discussed convex domain in uniformly banach spaces. berinde [berinde [2004]] investigates iterative scheme to finding fixed points using quasi contractive mappings in uniformly convex banach spaces(stands for cbs), extended to uniformly convex banach spaces(cbs). throughout this paper, we use strong convergence of ishikawa iterations to prove such fixed point results in uniformly cbs using different type of contractions. 2 preliminaries definition 2.1. let (x, d) be a metric space and i = [0, 1]. a mapping w : x × x ×i → x is said to be a convex structure on x if for each (x, y, λ) ∈ x ×x ×i and u ∈ x d (u, w(x, y, λ)) ≤ λd(u, x) + (1 − λ)d(u, y). a metric space (x, d) together with a convex structure w is called a convex metric space,which is denoted by (x, d, w). definition 2.2. let (x, d, w) be a convex metric space. a nonempty subset c of x is said to be convex if w(x, y, λ) ∈ c whenever (x, y, λ) ∈ c × c × i. definition 2.3. let f : x → x. a point x ∈ x is called a fixed point of f if f(x) = x. definition 2.4. let e be a uniformly banach space and t : e → e a map for which there is a real constant k1 ∈ (0, 1/5) such that each pair u, v ∈ x, ∥tu − tv∥ ≤ k1{∥u − v∥ + ∥u − tu∥ + ∥v − tv∥ + ∥u − tv∥ + ∥v − tu∥}. then, t has a fixed point by the approximation of picard. definition 2.5. let e be a uniformly banach space and t : e → e a map for which there is a real constant k2 ∈ (0, 1/3) such that for each pair u, v ∈ x, ∥tu − tv∥ ≤ k2{∥u − v∥ + ∥u − tu∥ + ∥v − tv∥ 2 + ∥u − tv∥ + ∥v − tu∥ 2 }. then, t has a fixed point by the approximation of picard. 199 fixed point theorems in uniformly convex banach spaces definition 2.6. let e be a uniformly banach space and t : e → e a given operator. let u0 ∈ e be arbitrary and {αn} ⊂ [0, 1] a sequence of real numbers. the sequence { un} ⊂ e defined by un+1 = (1 − αn)un + αntun, n = 0, 1, 2, · · · (1) is called the mann iteration. definition 2.7. let e be a uniformly banach space and t : e → e a given operator. let u0 ∈ e be arbitrary, {αn} and {βn} ⊂ [0, 1] a sequence of real numbers. the sequence { un} ⊂ e defined by un+1 = (1 − αn)un + αntvn, n = 0, 1, 2, · · · (2) vn = (1 − βn)un + βntun, n = 0, 1, 2, · · · . (3) then {un} is called ishikawa iteration. result 2.1. berinde [2004] the condition of mann and ishikawa iteration for strong convergence are given below (a) let k be a closed convex subset of a uniformly banach space e and t : k → k as an operator satisfying contraction. let {un} be defined by definition 2.6 and x0 ∈ k, with {αn} ∈ [0, 1] satisfying ∞∑ n=0 αn = ∞. (4) then, {un} converges strongly to a fixed point. (b) let k be a closed convex subset of a uniformly banach space e and t : k → k as an operator satisfying the contraction. let {un} be defined by definition 2.7 and u0 ∈ k, with {αn}, {βn} ∈ [0, 1] satisfying ∞∑ n=0 αn(1 − αn) = ∞. (5) then, {un} strongly converges to a fixed point.2 3 main results theorem 3.1. let k be a closed convex subset of a uniformly banach space e and t : k → k an operator satisfying equation ∥tu−tv∥ ≤ k1{∥u−v∥+∥u−tu∥+∥v −tv∥+∥u−tv∥+∥v −tu∥}. (6) 200 r. jahir hussain, k. manoj let {un} be the ishikawa iteration and u0 ∈ k, where {αn}, {βn} ⊂ [0, 1] with {αn} satisfying equation ∑∞ n=0 αn = ∞. then {un} converges strongly to a fixed point of t . proof. suppose t has a fixed point p in k. consider u, v ∈ k and t is an operator satisfies above equation, ∥tu − tv∥ ≤ k1{∥u − v∥ + ∥u − tu∥ + ∥v − tv∥ + ∥u − tv∥ + ∥v − tu∥} = k1{2∥u − v∥ + 2∥u − tv∥ + ∥u − tu∥ + ∥v − tu∥} ≤ k1{2∥u − v∥ + 2[∥u − tu∥ + ∥tu − tv∥] + ∥u − tu∥ + ∥v − tu∥} (1 − 2k1)∥tu − tv∥ ≤ 2k1∥u − v∥ + 3k1∥u − tu∥ + k1∥v − tu∥ ∥tu − tv∥ ≤ 2k1 1 − 2k1 ∥u − v∥ + 3k1 1 − 2k1 ∥u − tu∥ + k1 1 − 2k1 ∥v − tu∥. take δ = k1 1−2k1 , then we have δ ∈ [0, 1), it result that the inequality ∥tu − tv∥ ≤ 2δ∥u − v∥ + 3δ∥u − tu∥ + δ∥v − tu∥∀u, v ∈ k. (7) now let {un}∞n=0 be ishikawa iteration defined on definition 2.7 and u0 ∈ k arbitrary then ∥un+1 − p∥ = ∥(1 − αn)un + αntvn − (1 − αn + αn)p∥ = ∥(1 − αn)(un − p) + αn(tvn − p)∥ ∥un+1 − p∥ ≤ (1 − αn)∥un − p∥ + αn∥tvn − p∥. (8) in equation (7), put u = p and v = vn, we have ∥tvn − p∥ ≤ 2δ∥p − vn∥ + 3δ∥p − tp∥ + δ∥vn − tp∥ = 2δ∥p − vn∥ + δ∥vn − p∥ ∥tvn − p∥ ≤ 3δ∥vn − p∥. (9) furthermore, ∥vn − p∥ = ∥(1 − βn)un + βntun − (1 − βn + βn)p∥ = ∥(1 − βn)(un − p) + βn(tun − p)∥ ∥vn − p∥ ≤ (1 − βn)∥un − p∥ + βn∥tun − p∥. (10) again in equation (7), put u = p and v = un, we get ∥tun − p∥ ≤ 2δ∥p − un∥ + 3δ∥p − tp∥ + δ∥un − p∥ = (2δ + δ)∥un − p∥ 201 fixed point theorems in uniformly convex banach spaces ∥tun − p∥ ≤ 3δ∥un − p∥. (11) using equation (9),(10),(11) in equation (8), we get ∥un+1 − p∥ ≤ (1 − αn)∥un − p∥ + αn∥tvn − p∥ ≤ (1 − αn)∥un − p∥ + 3αnδ∥vn − p∥ ≤ (1 − αn)∥un − p∥ + 3αnδ[(1 − βn)∥un − p∥ + βn∥tun − p∥] = (1 − αn)∥un − p∥ + 3αnδ(1 − βn)∥un − p∥ + 3αnδβn∥tun − p∥ ≤ (1 − αn)∥un − p∥ + 3αnδ(1 − βn)∥un − p∥ + 9αnδ2βn∥un − p∥ = [1 − αn + 3αnδ − 3αnβnδ + 9αnδ2βn]∥un − p∥ = 1 − αn(1 − 3δ) − 3αnβnδ(1 − 3δ)]∥un − p∥ = [1 − (1 − 3δ)αn(1 + 3δβn)]∥un − p∥ which by the inequality, 1 − (1 − 3δ)αn(1 + 3δβn) ≤ 1 − (1 − 3δ)2αn ⇒ ∥un+1 − p∥ ≤ [1 − (1 − 3δ)2αn]∥un − p∥, n = 0, 1, 2, ... (12) by equation (8), we obtain ∥un+1 − p∥ ≤ n∏ k=0 [1 − (1 − 3δ)2αk]∥u0 − p∥. (13) where, δ ∈ (0, 1), αk, βn ∈ [0, 1] and ∞∑ n=0 αn = ∞ by result (a), we get limn→∞ n∏ k=0 [1 − (1 − 3δ)2αk] = 0. by equation (13) which implies limn→∞ ∥un+1 − p∥ = 0. therefore, {un}∞n=0 converges strongly to p. 2 theorem 3.2. let k be a closed convex subset of a uniformly banach space e and t : k → k an operator satisfying equation ∥tu − tv∥ ≤ k2{∥u − v∥ + ∥u − tu∥ + ∥v − tv∥ 2 + ∥u − tv∥ + ∥v − tu∥ 2 }. let {un} be the ishikawa iteration and u0 ∈ k, where {αn} and {βn} are sequences in [0, 1] with {αn} satisfying equation ∑∞ n=0 αn = ∞. then {un} converges strongly to a fixed point of t . proof. consider t has a fixed point p in k. consider u, v ∈ k and t is an operator satisfies equation ∥tu − tv∥ ≤ k2{∥u − v∥ + ∥u−tu∥+∥v−tv∥ 2 + ∥u−tv∥+∥v−tu∥ 2 } ≤ k2{∥u − v∥ + ∥u−v∥+∥v−tu∥+∥v−tv∥ 2 + ∥u−tv∥+∥v−tu∥ 2 } = k2{32∥u − v∥ + ∥v − tu∥ + 1 2 ∥v − tv∥ + ∥u−tv∥ 2 } (1 − k2)∥tu − tv∥ ≤ 32k2{∥u − v∥ + ∥v − tv∥} + k2 2 ∥u − tv∥ ∥tu − tv∥ ≤ 3k2 2(1−k2) {∥u − v∥ + ∥v − tv∥} + k2 2(1−k2) ∥u − tv∥ 202 r. jahir hussain, k. manoj take δ = k2 1 − k2 ∈ [0, 1).∥tu−tv∥ ≤ 3 2 δ{∥u−v∥+∥v −tv∥}+ δ 2 ∥u−tv∥. (14) now, let {un} be ishikawa iteration defined on definition 2.7 and u0 ∈ k then, ∥un+1 − p∥ = ∥(1 − αn)un + αntvn − (1 − αn + αn)p∥ ∥un+1 − p∥ ≤ (1 − αn)∥un − p∥ + αn∥tvn − p∥ (15) in equation (14), put v = p and u = un, ∥tun − p∥ ≤ 3δ2 ∥un − p∥ + δ 2 ∥un − p∥ ∥tun − p∥ ≤ 2δ∥un − p∥ (16) in equation (14), put u = vn and v = p, ∥tvn − p∥ ≤ 3δ2 ∥vn − p∥ + δ 2 ∥vn − p∥ ∥tvn − p∥ ≤ 2δ∥vn − p∥ (17) ∥vn − p∥ = ∥(1 − βn)un + βntun − (1 − βn + βn)p∥ ∥vn − p∥ ≤ (1 − βn)∥un − p∥ + βn∥tun − p∥ (18) using equation (16), (17), (18) in equation (15), we get ∥un+1 − p∥ ≤ (1 − αn) + 2αnδ∥vn − p∥ ≤ (1 − αn) + 2δαn[(1 − βn)∥un − p∥ + βn∥tun − p∥] ≤ (1 − αn) + 2δαn(1 − βn)∥un − p∥ + 4δ2αnβn∥un − p∥ = [1 − αn(1 + 2δβn)(1 − 2δ)]∥un − p∥ which by inequality, 1 − αn(1 + 2δβn)(1 − 2δ) ≤ 1 − (1 − 2δ)2αn ⇒ ∥un+1 − p∥ ≤ [1 − (1 − 2δ)2αn]∥un − p∥ (19) by equation (19), we obtain ∥un+1 − p∥ ≤ n∏ k=0 [1 − (1 − 2δ)2αk]∥u0 − p∥ (20) where δ ∈ (0, 1), αk, βn ∈ [0, 1] and ∞∑ n=0 αn = ∞ by result (a), we get, limn→∞ n∏ k=0 [1 − (1 − 2δ)2αk] = 0. by equation (20), ⇒ limn→∞ ∥un+1 − p∥ = 0. therefore {un}∞n=0 converges strongly to p.2 corolary 3.1. let k be a closed convex subset of a uniformly banach space e and t : k → k an operator satisfying equation ∥tu − tv∥ ≤ k 4 {∥u − tu∥ + ∥v − tv∥ + ∥u − tv∥ + ∥v − tu∥}. let {un} be the ishikawa iteration and u0 ∈ k, where {αn} and {βn} are sequences of positive numbers in [0, 1] with {αn} satisfying equation ∑∞ n=0 αn = ∞. then {un} converges strongly to a fixed point of t . 203 fixed point theorems in uniformly convex banach spaces corolary 3.2. let e be an uniformly banach space, k is a closed convex subset of e and t : k → k an operator satisfying equation ∥tu − tv∥ ≤ k∥u − v∥ (21) let {un}∞n=0 be the ishikawa iteration and u0 ∈ k, where {αn}∞n=0 and {βn}∞n=0 are sequences of positive numbers in [0, 1] with {αn}∞n=0 satisfying ∑∞ n=0 αn = ∞. then {un}∞n=0 converges strongly to a fixed point of t . corolary 3.3. let e be an uniformly banach space, k is a closed convex subset of e and t : k → k an operator satisfying equation ∥tu − tv∥ ≤ k 2 {∥u − tu∥ + ∥v − tv∥} (22) let {un}∞n=0 be the ishikawa iteration and u0 ∈ k, where {αn}∞n=0 and {βn}∞n=0 are sequences of positive numbers in [0, 1] with {αn}∞n=0 satisfying ∑∞ n=0 αn = ∞. then {un}∞n=0 converges strongly to a fixed point of t . 4 conclusions in this work, we presented the result on strong convergence of fixed point of t. we developed different kinds of contractive conditions to prove strong convergence using ishikawa iterative method in uniforly convex banach space. our main result may be vision for other authors using different contraction to prove several converging fixed point result. acknowledgements the editors and referees are greatly appreciated by the authors for their insightful input, which helped the paper’s presentation. references v. berinde. on the convergence of the ishikawa iteration in the class of quasi contractive operators. acta math. univ. comenianae, lxxiii:1–11, 2004. l. bernal-gonzalez. a schwarz lemma for convex domains in arbitrary banach spaces. journal of mathematical analysis and applications, 200:511–517, 1996. 204 r. jahir hussain, k. manoj s. ishikawa. fixed points by a new iteration method. proc. amer. math. soc., 44: 147–150, 1974. h.v. machado. a chracterization of convex subsets of normed spaces. kodai math-sem.rep., 25:307–320, 1973. w. r. mann. mean value methods in iteration. proc. amer. math. soc., 44:506– 510, 1953. w. a. takahashi. convexity in metric space and non-expansive mappings. kodai math.sem rep., 22:142–149, 1970. 205 we remember here some definitions, notations and results which will be the basis of what follows 1 multivalued functions, fuzzy subsets and join spaces piergiulio corsini,* razieh mahjoob** * dept. of biology and agro-industrial economy, via delle scienze 208, 33100 udine (italy) e-mail: corsini2002@yahoo.com web site: http://ijpam.uniud.it/journal/curriculum_corsini.htm ** dept. of mathematics faculty of basic science, university of semnan, semnan, (iran) ra_mahjoob@yahoo.com abstract one has considered the hypergroupoid ηγ = < h;ογ > associated with a multivalued function γ from h to a set d, defined as follows: ∀ x ∈ h, x ογ x = ⎨y⏐ γ(y) ∩ γ(x) ≠ ∅⎬ , ∀ (y,z) ∈ h2 , y ογ z = y ογ y ∪ z ογ z , and one has calculated the fuzzy grade ∂(ηγ) for several functions γ defined on sets h, such that ⎮h⎮ ∈ ⎨3, 4, 5, 6, 8, 9, 16⎬. introduction the analysis of the connections between hyperstructures and fuzzy sets dates since 1993 when corsini defined and studied the join spaces hμ obtained from the fuzzy set < η, μ > , and a little later zahedi and ameri considered fuzzy hypergroups. these subjects were studied in the following years by several scientists in romania, iran, greece, italy, canada. in 1993 corsini associated a hypergroupoid with every fuzzy subset, and he proved that this hypergroupoid is a join space [8]. in 2003 corsini [14] associated a fuzzy set μη with every hypergroupoid < η, o > and considered the sequence of the fuzzy subsets μη and of the join spaces hμ constructed from a hypergroup. this sequence has been studied in depth for several classes of hypergroups by corsini [14], corsini–cristea [16], [17], [18], corsini–leoreanu-fotea [22], corsini– leoreanu–iranmanesh [23], cristea [25], [26], stefanescu–cristea [70], leoreanu-fotea v. – leoreanu l. [ 53] . in this paper one has considered the hypergroupoid < η, oγ > associated with a multivalued function γ from a set h to a set d, defined as follows ∀ x ∈ h, x oγ x = {y⏐γ(y) ∩ γ(x) ≠ ∅ }, ratio mathematica, 20, 2010 2 ∀ (y,z) ∈ h2 , y oγ z = y oγ y ∪ z oγ z and one has calculated the fuzzy grade ∂(hγ) , for several functions γ defined on sets h such that ⏐η⏐ ∈ {3, 4, 5, 6, 8, 9, 16}. we can remark that we have ∂(η) = s+1, for all the examinated cases with the exception of (136 ), (236), (336), (129), if n = 2sq , where m.c.d. (q,2) = 1. we remember here some definitions, notations and results which will be the basis of what follows. with every fuzzy subset (h; μa ) of a set η , it is possible to associate a hypergroupoid <η; ομ>, where the hyperoperation < ομ> is defined by: ∀ (x,y) ∈ h2 , (i) x ομ y = ⎨ z ⏐ min ⎨ μa(x), μa(y)⎬ ≤ μa(z) ≤ max ⎨ μa(x), μa(y)⎬ ⎬ one proved [8] that <η; ομ> is a join space. with every hypergroupoid < η; ο >, it is possible to associate a fuzzy subset, as follows: set ∀ (x,y) ∈ h2 , ∀ u ∈ h , μx,y (u) = 0 ⇔ u ∉ x ο y if u ∈ x ο y , μx,y (u) = 1/⏐x ο y⏐ , set ∀ u ∈ h , a(u) = ∑(x,y)∈h2 μx,y (u) , q(u) = ⎨ (x,y) ⏐u ∈ x ο y ⎬ , q(u) = ⏐ q(u) ⏐ , (ii) μh(u) = a(u) / q(u) , see [14]. so it is clear that, given a hypergroupoid < η; ο >, a sequence of fuzzy subsets and of join spaces is determined μh = μ1 , μ2 ,…. μm+1 ,…, < η; ο > = 0h , 1h, … mh…, such that ∀j ≥ 1, jμ = hj 1−μ , and jh is the join space associated, after (i), with jμ . we call “fuzzy grade of h”, if it exists, the number ∂(h) (or f.g.(h)) = min ⎨s⏐ mh ≈ m+1h ⎬ and “strong fuzzy grade of h” , if it exists, the number s.f.g.(h) = min ⎨s⏐ mh = m+1h ⎬ , see [17]. in this paper one has determined 6 hypergroupoids of 3 elements such that ∂(h) = 0, 4 hypergroupoids of 3 elements such that ∂(h) = 1, 5 hypergroupoids of 4 elements such that ∂(h) = 0, 8 hypergroupoids of 4 elements such that ∂(h) = 1, 12 hypergroupoids of 4 elements such that ∂(h) = 2, 5 hypergroupoids of 4 elements such that ∂(h) = 3, 2 hypergroupoids of 5 elements such that ∂(h) = 1, 2 hypergroupoids of 6 elements such that ∂(h)= 1, 8 hypergroupoids of 6 elements such that ∂(h) = 2, 3 hypergroupoids of 6 elements such that ∂(h) = 3, 1 hypergroupoid of 8 elements such that ∂(h)= 4, 1 hypergroupoid of 9 elements such that ∂(h)= 2, 1 hypergroupoid of 16 elements such that ∂(h)= 5. ratio mathematica, 20, 2010 3 $ 1. let γ be a multivalued function from a set h = {u1, u2,..., un} to a set d, i.e. γ : h → p*(d). then we have the following theorem 1 if there exists d ∈ d, such that ∀i, γ(ui) ∋ d, then ∂(hγ) = 0. indeed, we have ∀i, xi oγ xi = {uj⎮γ(uj) ∩ γ(uj) ≠ ø} = h, therefore ∀(i, j), ui oγ uj = h. whence oh = t, from which ∀s, sh = oh, so ∂(hγ) = 0. theorem 2 let γ be a multivalued function from a set h to a set d, that is γ : h → p*(d), and let < oγ > be the hyperoperation defined in h : ∀x ∈ h, xoγ x = {z⎮γ(z) ∩ γ( x) ≠ ø}, ∀(y, z), y oγ z = y oγ y ∪ z oγ z. then the hypergroupoid < h; oγ > is a commutative quasi-join space, that is ∀(a, b, c, d) ∈ h4, (j) a / b ∩ c / d ≠ ø ⇒ a oγ d ∩ b oγ c ≠ ø. let’s suppose a / b ∩ c / d ∋ v, that is a ∈ b oγ v, c ∈ d oγ v. then, since b o v = b oγ b ∪ v oγ v, d oγ v = d oγ d ∪ v oγ v, and ∀(x, y) ∈ h2, y ∈ x oγ x ⇒ x ∈ y oγ y, at least one of the following cases is verified (i) a ∈ b oγ b, c ∈ d oγ d, (ii) a ∈ b oγ b, c ∈ v oγ v (iii) a ∈ v oγ v, c ∈ d oγ d, (iv) a ∈ v oγ v, c ∈ v oγ v (i) implies b ∈ a oγ a, whence b ∈ a oγ d, and we have also b ∈ b oγ b ⊂ b oγ c (ii) we find b ∈ a oγ d ∩ b oγ c as in (i). (iii) we obtain c ∈ d oγ d ⊂ a oγ d, and also c ∈ c oγ c ⊂ b oγ c. (iv) implies v ∈ a oγ a ⊂ a oγ d and also v ∈ c oγ c ⊂ b oγ c. therefore the implication (j) is always satisfied whence < h; oγ > is a quasi-join space. ratio mathematica, 20, 2010 4 $ 2. set h = {u1, u2, u3}. then there are functions γ : h → p*(d) such that the fuzzy grade of the associated sequence is respectively 0, 1. (103) set γ(u1) = {d1}, γ(u2) = γ(u3) = {d2, d3}. we have clearly so μ1(u1) = 0.467, μ1(u2) = μ1(u3) = 0.417. it follows 1h = 0h. by consequence ∂(103) = 0. (203) set γ(u1) = {d1, d2}, γ(u2) = γ(u3) = {d3}. we have one obtains μ1(u1) = 0.467, μ1(u2) = μ1(u3) = 0.417. so 1h = 0h, then ∂(20 3) = 0. (303) set γ(u1) = {d1, d2}, γ(u2) = {d2, d3}, γ(u3) = {d3, d1}. we have 1h = 0h = t, ∂(303) = 0. (403) set γ(u1) = {d1, d2}, γ(u2) = {d2 }, γ(u3) = {d3}. we have we obtain μ(1) = 0.417 = μ(2) , μ(3) = 0.467 , so ∂(403) = 0. (503) set γ(u1) = γ(u2) = {d1}, γ(u3) = {d3}. we have 0h u1 u2 u3 u1 u1 h h u2 u2 u3 u2 u3 u3 u2 u3 0h u1 u2 u3 u1 u1 h h u2 u2 u3 u2 u3 u3 u2 u3 0h u1 u2 u3 u1 h h h u2 h h u3 h 0h u1 u2 u3 u1 u1 u2 u1 u2 h u2 u1 u2 h u3 u3 ratio mathematica, 20, 2010 5 as in (403), we obtain ∂(503) = 0. (113) let⎮h⎮= 3 = ⎮d⎮. set γ(u1) = {d1}, γ(u2) = {d2}, γ(u3) = {d3}. so we have we have clearly μ1(u1) = μ1(u2) = μ1(u3) = 0.6. therefore we obtain 1h = t, whence ∂(11 3) = 1. (213) set γ(u1) = {d1, d2, d3}, γ(u2) = {d2}, γ(u3) = {d3}. we have we obtain : μ1(u1) = 0.37, μ1(u2) = μ1(u3) = 0.354. by consequence, so we have : μ2(u1) = 0.467, μ2(u2) = μ2(u3) = 0.417. from this, we obtain 2h = 1h, whence ∂(21 3) = 1. (313) set γ(u1) = {d1, d2, d3}, γ(u2) = { d1, d2}, γ(u3) = {d3}. we have 0h u1 u2 u3 u1 u1 u2 u1 u2 h u2 u1 u2 h u3 u3 0h u1 u2 u3 u1 u1 u1 u2 u1 u3 u2 u2 u2 u3 u3 u3 0h u1 u2 u3 u1 h h h u2 u1 u2 h u3 u1 u3 1h u1 u2 u3 u1 u1 h h u2 u2 u3 u2 u3 u3 u2 u3 ratio mathematica, 20, 2010 6 see (213). so we obtain again ∂(313) = 1. (413) set h = {u1, u2, u3}, γ(u1) = {d1}, γ(u2) = {d2, d3}, γ(u3) = {d3, d1}. so we have by consequence μ1(u1) = 0.354 = μ1(u2), μ1(u3) = 0.370. therefore we obtain hence μ2(u1) = μ2(u2) = 0.4167, μ2(u3) = 0.467. it follows 2h = 1h. therefore ∂(413) = 1. $ 3. set h = {u1, u2, u3, u4}. then there are functions γ : h → p*(d) such that the fuzzy grade of the associated sequence is respectively 0, 1, 2, 3. (104) set γ(u1) = {d1, d2}, γ(u2) = γ(u3) = γ(u4) = {d3, d4}. then we have we obtain μ1(u1) = 0.357, μ1(u2) = μ1(u3) = μ1(u4) = 0.300. by consequence 1h = 0h and therefore ∂(104 )= 0. 0h u1 u2 u3 u1 h h h u2 u1 u2 h u3 u1 u3 0h u1 u2 u3 u1 u1 u3 h h u2 u2 u3 h u3 h 1h u1 u2 u3 u1 u1 u2 u1 u2 h u2 u1 u2 h u3 u3 0h u1 u2 u3 u4 u1 u1 h h h u2 u2 u3 u4 u2 u3 u4 u2 u3 u4 u3 u2 u3 u4 u2 u3 u4 u4 u2 u3 u4 ratio mathematica, 20, 2010 7 (204 ) set γ(u1) = {d1, d2, d3}, γ(u2) = γ(u3) = γ(u4) = {d4} . then we have as in (1) 0h = 1h so ∂(20 4) = 0. (304) set γ(u1) = {d1, d2}, γ(u2) = {d3, d4}, γ(u3) = γ(u4) = {d4}. also in this case by consequence 0h = 1h from which ∂(304) = 0. (404 ) set γ(u1) = {d1, d2, d3, d4}, γ(u2) = {d2, d3, d4}, γ(u3) = γ(u4) = {d4}. we have clearly, 1h = 0h = t. so ∂(404) = 0. (114) set γ(u1) = {d1, d2}, γ(u2) = {d2, d3}, γ(u3) = {d3}, γ(u4) = {d4}. we have whence μ1(u1) = μ1(u3) = 0.333 μ1(u2) = 0.344, μ1(u4) = 0.405. from which we obtain 1h : 0h u1 u2 u3 u4 u1 u1 h h h u2 u2 u3 u4 u2 u3 u4 u2 u3 u4 u3 u2 u3 u4 u2 u3 u4 u4 u2 u3 u4 0h u1 u2 u3 u4 u1 u1 h h h u2 u2 u3 u4 u2 u3 u4 u2 u3 u4 u3 u2 u3 u4 u2 u3 u4 u4 u2 u3 u4 0h u1 u2 u3 u4 u1 h h h h u2 h h h u3 h h u4 h 0h u1 u2 u3 u4 u1 u1 u2 u1 u2 u3 u1 u2 u3 u1 u2 u4 u2 u1 u2 u3 u1 u2 u3 h u3 u2 u3 u2 u3 u4 u4 u4 ratio mathematica, 20, 2010 8 hence μ2(u1) = μ2(u3) = 0.36, μ2(u2) = 0.394, μ2(u4) = 0.429. by consequence 2h = 1h , then ∂(11 4 )= 1. (214) set γ(u1) = {d1, d2}, γ(u2) = {d3, d4}, γ(u3) = {d3}, γ(u4) = {d4}. so we have whence μ1(u1) = 0.405 μ1(u2) = 0.344, μ(u3) = μ(u4) = 0.3. we obtain 1h : so we have : μ2(u1) = 0.429, μ2(u2) = 0.394, μ2(u3) = μ2(u4) = 0.361 whence one finds that 2h = 1h. it follows ∂(21 4) = 1. (314) set γ(u1) = {d1}, γ(u2) = {d2}, γ(u3) = {d3}, γ(u4) = {d4}. so then ∀ i, μ1(ui) = 0.571. by consequence 1h = t and therefore ∂(314) = 1. 1h u1 u2 u3 u4 u1 u1 u3 u1 u3 u1 u3 u2 h u2 u1 u3 u1 u3 u2 h u3 u2 u2 u4 u4 u4 0h u1 u2 u3 u4 u1 u1 h u1 u2 u3 u1 u2 u4 u2 u2 u3 u4 u2 u3 u4 u2 u3 u4 u3 u2 u3 u2 u3 u4 u4 u2 u4 1h u1 u2 u3 u4 u1 u1 u1 u2 h h u2 u2 u2 u3 u4 u2 u3 u4 u3 u3 u4 u3 u4 u4 u3 u4 0h u1 u2 u3 u4 u1 u1 u1 u2 u1 u3 u1 u4 u2 u2 u2 u3 u2 u4 u3 u3 u3 u4 u4 u4 ratio mathematica, 20, 2010 9 (414) set γ(u1) = {d1, d2, d3}, γ(u2) = {d1}, γ(u3) = {d2}, γ(u4) = {d3}. we have so μ1(u1) = 0.328, μ1(u2) = μ1(u3) = μ1(u4) = 0.299. hence so μ2(u1) = 0.357 μ2(u2) = μ2(u3) = μ2(u4) = 0.3 from which 2h = 1h. therefore ∂(41 4 )= 1. (514) set γ(u1) = {d1, d2}, γ(u2) = {d2, d3}, γ(u3) = γ(u4) = {d4}. we have μ1(u1) = μ1(u2) = μ1(u3) = μ1(u4) = 0.333. so 1h = t, whence ∂(51 4) = 1. (614) set γ(u1) = {d1, d2}, γ(u2) = {d2}, γ(u3) = {d3, d4}, γ(u4) = {d4}. we have clearly ∀ i, μ1(u2) = μ1(u1). therefore 1h = t and by consequence ∂(61 4 )= 1. (714) set γ(u1) = {d1, d2, d3}, γ(u2) = {d2, d3}, γ(u3) = {d4}, γ(u4) = {d4}. 0h u1 u2 u3 u4 u1 h h h h u2 u1 u2 u1 u2 u3 u1 u2 u4 u3 u1 u3 u1 u3 u4 u4 u1 u4 1h u1 u2 u3 u4 u1 u1 h h h u2 u2 u3 u4 u2 u3 u4 u2 u3 u4 u3 u2 u3 u4 u2 u3 u4 u4 u2 u3 u4 0h u1 u2 u3 u4 u1 u1 u2 u1 u2 h h u2 u1 u2 h h u3 u3 u4 u3 u4 u4 u3 u4 0h u1 u2 u3 u4 u1 u1 u2 u1 u2 h h u2 u1 u2 h h u3 u3 u4 u3 u4 u4 u3 u4 ratio mathematica, 20, 2010 10 see (514) and (614). we have 1h = t, whence ∂(714) = 1. (604) set γ(u1) = {d1, d2}, γ(u2) = {d3, d4}, γ(u3) = γ(u4) = {d4}. we have so ∂(604 )= 0. (504) set γ(u1) = {d1, d2}, γ(u2) = γ(u3) = γ(u4) = {d4}. we have so μ(1) = 0.357 , μ(2) = μ(3) = μ(4)=0.3 . it follows ∂(504 )=0. (124 ) set γ(u1) = {d1, d2}, γ(u2) = γ(u3) = {d2, d3}, γ(u4) = {d3, d4}. one obtains μ1(u1) = 0.256 = μ1(u4), μ1(u2) = μ1(u3) = 0.260, whence we obtain 1h: 0h u1 u2 u3 u4 u1 u1 u2 u1 u2 h h u2 u1 u2 h h u3 u3 u4 u3 u4 u4 u3 u4 0h u1 u2 u3 u4 u1 u1 h h h u2 u2 u3 u4 u2 u3 u4 u2 u3 u4 u3 u2 u3 u4 u2 u3 u4 u4 u2 u3 u4 0h u1 u2 u3 u4 u1 u1 h h h u2 u2 u3 u4 u2 u3 u4 u2 u3 u4 u3 u2 u3 u4 u2 u3 u4 u4 u2 u3 u4 0h u1 u2 u3 u4 u1 u1 u2 u3 h h h u2 h h h u3 h h u4 u4 u2 u3 ratio mathematica, 20, 2010 11 therefore 2h = t (the total hypergroup). then ∂(12 4 )= 2. (224) set γ(u1) = {d1, d2}, γ(u2) = {d2, d3}, γ(u3) = {d3, d4}, γ(u4) = {d4}. then whence μ1(u1) = 0.292 = μ1(u4), μ1(u2) = μ1(u3) = 0.3. so, we have then μ1(u1) = μ1(u4) = μ1(u2) = μ1(u3) , whence 2h = t, and by consequence ∂(224 )= 2. (324) set γ(u1) = {d1, d2, d3}, γ(u2) = {d2, d3, d4}, γ(u3) = { d2, d4}, γ(u4) = {d3}. we have μ1(u1) = μ1(u2) = 0.260, μ1(u3) = μ1(u4) = 0.256, whence we obtain 1h u1 u4 u2 u3 u1 u1 u4 u1 u4 h h u4 u1 u4 h h u2 u2 u3 u2 u3 u3 u2 u3 0h u1 u2 u3 u4 u1 u1 u2 u1 u2 u3 h h u2 u1 u2 u3 h h u3 u2 u3 u4 u2 u3 u4 u4 u3 u4 1h u1 u4 u2 u3 u1 u1 u4 u1 u4 h h u4 u1 u4 h h u2 u2 u3 u2 u3 u3 u2 u3 0h u1 u2 u3 u4 u1 h h h h u2 h h h u3 u1 u2 u3 h u4 u1 u2 u4 ratio mathematica, 20, 2010 12 then 2h = t, from which ∂(32 4)= 2. (424 ) set γ(u1) = {d1, d2, d3}, γ(u2) = {d2, d4} , γ(u3) = { d3}, γ(u4) = {d4}. we have: then μ1(u1) = 0.3 = μ1(u2), μ1(u3) = μ1(u4) = 0.292. it follows therefore 2h = t, whence ∂(42 4 )= 2. (524 ) set γ(u1) = {d1, d2, d3}, γ(u2) = γ(u3) = {d3, d4}, γ(u4) = {d4}. we have so μ1(u1) = μ1(u4) = 0.23, μ1(u2) = μ1(u3) = 0.260. by consequence, we obtain therefore we have 2h = t, whence ∂(52 4 )= 2. 1h u1 u2 u3 u4 u1 u1 u2 u1 u2 h h u2 u1 u2 h h u3 u3 u4 u3 u4 u4 u3 u4 0h u1 u2 u3 u4 u1 u1 u2 u3 h u1 u2 u3 h u2 u1 u2 u4 h u1 u2 u4 u3 u1 u3 h u4 u2 u4 1h u1 u2 u3 u4 u1 u1 u2 u1 u2 h h u2 u1 u2 h h u3 u3 u4 u3 u4 u4 u3 u4 0h u1 u2 u3 u4 u1 u1 u2 u3 h h h u2 h h h u3 h h u4 u2 u3 u4 1h u1 u4 u2 u3 u1 u1 u4 u1 u4 h h u4 u1 u4 h h u2 u2 u3 u2 u3 u3 u2 u3 ratio mathematica, 20, 2010 13 (624) set γ(u1) = {d1}, γ(u2) = {d1, d2}, γ(u3) = {d2, d3, d4}, γ(u4) = {d4}. we have hence μ1(u1) = 0.292 = μ1(u4), μ1(u2) = μ1(u3) = 0.3. we obtain whence ∀ i, μ1(ui) = μ1(u1). then 2h = t. therefore ∂(62 4 )= 2. (724) set γ(u1) = {d1, d2, d3}, γ(u2) = {d2, d4}, γ(u3) = {d3, d4}, γ(u4) = {d4}. we have μ1(u1) = μ1(u4) = 0.256, μ1(u2) = μ1(u3) = 0.260. so, we obtain 1h: then ∀ i, μ2(ui) = 0.389, so 2h = t, and ∂(72 4 )= 2. 0h u1 u2 u3 u4 u1 u1 u2 u1 u2 u3 h h u2 u1 u2 u3 h h u3 u2 u3 u4 u2 u3 u4 u4 u3 u4 1h u1 u4 u2 u3 u1 u1 u4 u1 u4 h h u4 u1 u4 h h u2 u2 u3 u2 u3 u3 u2 u3 0h u1 u2 u3 u4 u1 u1 u2 u3 h h h u2 h h h u3 h h u4 u2 u3 u4 1h u1 u4 u2 u3 u1 u1 u4 u1 u4 h h u4 u1 u4 h h u2 u2 u3 u2 u3 u3 u2 u3 ratio mathematica, 20, 2010 14 (824) set γ(u1) = {d1, d2}, γ(u2) = {d2}, γ(u3) = {d3}, γ(u4) = {d4}. we have we obtain μ1(u1) = 0.389 = μ1(u2), μ1(u3) = 0.476 = μ1(u4). by consequence, therefore 2h = t and ∂(82 4 )= 2. (924) set γ(u1) = γ(u2) = {d1, d2}, γ(u3) = {d3}, γ(u4) = {d4}. we have see (824). therefore ∂(924 )= 2. (1024 ) set γ(u1) = {d1, d2}, γ(u2) = {d3}, γ(u3) = γ(u4) = {d4}. we obtain whence μ1(u1) = μ1(u2) = 0.476, μ(u3) = μ(u4) = 0.389. so, we obtain 0h u1 u2 u3 u4 u1 u1 u2 u1 u2 u1 u2 u3 u1 u2 u4 u2 u1 u2 u1 u2 u3 u1 u2 u4 u3 u3 u3 u4 u4 u4 1h u1 u2 u3 u4 u1 u1 u2 u1 u2 h h u2 u1 u2 h h u3 u3 u4 u3 u4 u4 u3 u4 0h u1 u2 u3 u4 u1 u1 u2 u1 u2 u1 u2 u3 u1 u2 u4 u2 u1 u2 u1 u2 u3 u1 u2 u4 u3 u3 u3 u4 u4 u4 0h u1 u2 u3 u4 u1 u1 u1 u2 u1 u3 u4 u1 u3 u4 u2 u2 u2 u3 u4 u2 u3 u4 u3 u3 u4 u3 u4 u4 u3 u4 ratio mathematica, 20, 2010 15 hence 2h = t, from which ∂(102 4 )= 2. (1124 ) set γ(u1) = {d1}, γ(u2) = {d2, d3}, γ(u3) = {d3, d4}, γ(u4) = {d4, d1}. we have hence μ1(u1) = μ1(u3) = 0.291, μ1(u3) = μ1(u4) = 0.3. one obtains 1h as follows: from 1h, one finds μ2(ui) = μ2(uj), ∀(i, j). therefore 2h = t, so ∂(112 4 )= 2. (1224) let γ(u1) = {d1, d2}, γ(u2) = {d2, d3}, γ(u3) = {d3}, γ(u4) = {d1}. we have then μ1(u1) = 0.3, μ1(u2) = 0.3, μ1(u3) = 0.2917, μ1(u4) = 0.2917. by consequence, 1h u1 u2 u3 u4 u1 u1 u2 u1 u2 h h u2 u1 u2 h h u3 u3 u4 u3 u4 u4 u3 u4 0h u1 u2 u3 u4 u1 u1 u4 h h u1 u3 u4 u2 u2 u3 u2 u3 u4 h u3 u2 u3 u4 h u4 u1 u3 u4 1h u1 u2 u3 u4 u1 u1 u2 u1 u2 h h u2 u1 u2 h h u3 u3 u4 u3 u4 u4 u3 u4 0h u1 u2 u3 u4 u1 u1 u2 u4 h h u1 u2 u4 u2 u1 u2 u3 u1 u2 u3 h u3 u2 u3 h u4 u1 u4 ratio mathematica, 20, 2010 16 therefore we have 2h = t (the total hypergroup) whence ∂(1224 )= 2. (134) set γ(u1) = {d1, d2}, γ(u2) = {d2, d3}, γ(u3) = {d2}, γ(u4) = {d3}. we have μ1(u1) = 0.272 = μ1(u3), μ1(u2) = 0.286, μ1(u4) = 0.271, whence so we have μ2(u2) = 0.405 = μ2(u4), μ2(u1) = μ2(u3) = 0.34. we obtain we have clearly μ3(u1) = μ3(u3) = μ3(u2) = μ3(u4). therefore 3h is the total hypergroup of order 4 and ∂(134 )= 3. 1h u1 u2 u3 u4 u1 u1 u2 u1 u2 h h u2 u1 u2 h h u3 u3 u4 u3 u4 u4 u3 u4 0h u1 u2 u3 u4 u1 u1 u2 u3 h u1 u2 u3 h u2 h h h u3 u1 u2 u3 h u4 u2 u4 1h u2 u1 u3 u4 u2 u2 u2 u1 u3 u2 u1 u3 h u1 u1 u3 u1 u3 u1 u3 u4 u3 u1 u3 u1 u3 u4 u4 u4 2h u1 u3 u2 u4 u1 u1 u3 u1 u3 h h u3 u1 u3 h h u2 u2 u4 u2 u4 u4 u2 u4 ratio mathematica, 20, 2010 17 (234) set γ(u1) = {d1, d2}, γ(u2) = {d2, d3}, γ(u3) = γ(u4) = {d3, d4}. we obtain the following we have μ1(u1) = 0.27083, μ1(u2) = 0.286, μ1(u3) = μ1(u4) = 0.272. therefore the second hypergroupoid is hence μ2(u2) = μ2(u1) = 0.405, μ2(u3) = μ2(u4) = 0.369. by consequence we have again from 2h we obtain 3h = t. then ∂(23 4 )= 3. (334) if γ(u1) = {d1, d2, d3}, γ(u2) = {d3, d4}, γ(u3) = γ(u4) = {d4}. one finds the same sequence as in (234). therefore ∂(334 )= 3 0h u1 u2 u3 u4 u1 u1 u2 h h h u2 h h h u3 u2 u3 u4 u2 u3 u4 u4 u2 u3 u4 1h u2 u3 u4 u1 u2 u2 u2 u3 u4 u2 u3 u4 h u3 u3 u4 u3 u4 u3 u4 u1 u4 u3 u4 u3 u4 u1 u1 u1 2h u2 u1 u3 u4 u2 u2 u1 h h h u1 u2 u1 h h u3 u3 u4 h u4 u3 u4 0h u1 u2 u3 u4 u1 u1 u2 h h h u2 h h h u3 u2 u3 u4 u2 u3 u4 u4 u2 u3 u4 ratio mathematica, 20, 2010 18 (434) set γ(u1) = {d1, d2, d3}, γ(u2) = {d2, d3}, γ(u3) = {d3, d4}, γ(u4) = {d4}. we have so μ1(u1) = 0.272 = μ1(u2), μ1(u3) = 0.286, μ1(u4) = 0.271. hence then μ2(u3) = μ2(u4) = 0.405, μ2(u1) = μ2(u2) = 0.369, from which we obtain hence 3h = t and ∂(43 4 )= 3. $ 4. set h = {u1, u2, u3, u4, u5}. then there are functions γ : h → p*(d) such that the fuzzy grade of the associated sequence is respectively 1, 2. (115) let ⎮h⎮= 5 = ⎮d⎮, γ(u1) = {d1, d2}, γ(u2) = {d2, d3}, γ(u3) = {d3, d4}, γ(u4) = {d4}, γ(u5) = {d5}. we have so μ1(u1) = 0.29167= μ1(u4), μ1(u2) = μ1(u3) = 0.2936, μ1(u5) = 0.370. we obtain 0h u1 u2 u3 u4 u1 u1 u2 u3 u1 u2 u3 h h u2 u1 u2 u3 h h u3 h h u4 u3 u4 1h u3 u1 u2 u4 u3 u3 u3 u1 u2 u3 u1 u2 h u1 u1 u2 u1 u2 u1 u2 u4 u2 u1 u2 u1 u2 u4 u4 u4 2h u1 u2 u3 u4 u1 u1 u2 u1 u2 h h u2 u1 u2 h h u3 u3 u4 u3 u4 u4 u3 u4 0h u1 u2 u3 u4 u5 u1 u1 u2 u1 u2 u3 u1 u2 u3 u4 u1 u2 u3 u4 u1 u2 u5 u2 u1 u2 u3 u1 u2 u3 u4 u1 u2 u3 u4 u1 u2 u3 u5 u3 u2 u3 u4 u2 u3 u4 u2 u3 u4 u5 u4 u3 u4 u3 u4 u5 u5 u5 ratio mathematica, 20, 2010 19 from this we have μ2(u5) = 0.348, μ2(u2) = μ2(u3) = 0.3067, μ2(u1) = μ2(u4) = 0.3. so μ2(u5) > μ2(u2) = μ(u3) > μ(u1) = μ(u4). it follows that 2h = 1h. therefore ∂(11 5) = 1. (225) set⎮h⎮= 5 = ⎮d⎮, γ(u1) = {d1, d2, d3}, γ(u2) = {d2, d3, d4}, γ(u3) = {d3, d4, d5}, γ(u4) = {d4}, γ(u5) = {d5}. so we have whence we obtain μ1(u1) = 0.228, μ1(u2) = 0.234, μ1(u3) = 0.2447, μ1(u4) = μ1(u1) μ1(u5) = 0.231. we have μ2(u3) = 0.3852, μ2(u2) = 0.3644, μ2(u5) = 0.3412, μ2(u1) = μ2(u4) = 0.3208. so 2h = 1h and by consequence ∂(22 5) = 1. $ 5. set h = {u1, u2, u3, u4, u5, u6}. then there are functions γ : h → p*(d) such that the fuzzy grade of the associated sequence is respectively 1, 2, 3. 1h u5 u2 u3 u1 u4 u5 u5 u5 u2 u3 u5 u2 u3 h h u2 u2 u3 u2 u3 u2 u3 u1 u4 u2 u3 u1 u4 u3 u2 u3 u2 u3 u1 u4 u2 u3 u1 u4 u1 u1 u4 u1 u4 u4 u1 u4 0h u1 u2 u3 u4 u5 u1 u1 u2 u3 u1 u2 u3 u4 h u1 u2 u3 u4 u1 u2 u3 u5 u2 u1 u2 u3 u4 h u1 u2 u3 u4 h u3 h h h u4 u2 u3 u4 u2 u3 u4 u5 u5 u3 u5 1h u3 u2 u5 u1 u4 u3 u3 u3 u2 u3 u2 u5 h h u2 u2 u2 u5 u2 u5 u1 u4 u2 u5 u1 u4 u5 u5 u5 u1 u4 u5 u1 u4 u1 u1 u4 u1 u4 u4 u1 u4 ratio mathematica, 20, 2010 20 (116) set ⎮h⎮= 6 = ⎮d⎮, γ(u1) = {d1, d2}, γ(u2) = {d2, d3}, γ(u3) = {d3, d4}, γ(u4) = {d4, d5}, γ(u5) = γ(u6) = { d5, d6}. we have so μ1(u1) = 0.231667, μ1(u2) = 0.2284, μ1(u3) = 0.22654, μ1(u4) = 0.22656, μ1(u5) = μ1(u6) = 0.219. hence we obtain therefore μ2(u1) = 0.348, μ2(u2) = 0.3315, μ2(u4) = 0.317, μ2(u3) = 0.303, μ2(u5) = μ2(u6) = 0.29. by consequence 2h = 1h, hence ∂(11 6) = 1. (216) set γ(u1) = {d1}, γ(u2) = {d2, d3, d4}, γ(u3) = {d3, d4, d5}, γ(u4) = {d4}, γ(u5) = {d5}, γ(u6) = {d5, d6}. we have 0h u1 u2 u3 u4 u5 u6 u1 u1 u2 u1 u2 u3 u1 u2 u3 u4 h u1 u2 u4 u5 u6 u1 u2 u4 u5 u6 u2 u1 u2 u3 u1 u2 u3 u4 h h h u3 u2 u3 u4 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u4 u3 u4 u5 u6 u3 u4 u5 u6 u3 u4 u5 u6 u5 u4 u5 u6 u4 u5 u6 u6 u4 u5 u6 1h u1 u2 u4 u3 u5 u6 u1 u1 u1 u2 u1 u2 u4 u1 u2 u3 u4 h h u2 u2 u2 u4 u2 u4 u3 u2 u4 u3 u5 u6 u2 u4 u3 u5 u6 u4 u4 u4 u3 u4 u3 u5 u6 u4 u3 u5 u6 u3 u3 u3 u5 u6 u3 u5 u6 u5 u5 u6 u5 u6 u6 u5 u6 ratio mathematica, 20, 2010 21 we obtain μ1(u1) = 0.303 μ1(u2) = 0.22469 μ1(u3) = 0.24 μ1(u4) = μ1(u2) = μ1(u5) = μ1(u6). setting {u2, u4, u5, u6} = p, we have one finds 2h = 1h. so ∂(21 6) = 1. (126) set ⎮h⎮= 6 = ⎮d⎮, γ(u1) = {d1, d2}, γ(u2) = {d2, d3}, γ(u3) = {d3, d4}, γ(u4) = {d4, d5}, γ(u5) = {d5}, γ(u6) = {d6}. so we have μ1(u1) = 0.268, μ1(u2) = 0.267, μ1(u3) = 0.260, μ1(u4) = μ1(u2), μ1(u5) = μ1(u1), μ1(u6) = 0.348. 0h u1 u2 u3 u4 u5 u6 u1 u1 u1 u2 u3 u4 h u1 u2 u3 u4 u1 u3 u5 u6 u1 u3 u5 u6 u2 u2 u3 u4 u2 u3 u4 u5 u6 u2 u3 u4 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u3 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u4 u2 u3 u4 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u5 u3 u5 u6 u3 u5 u6 u6 u3 u5 u6 1h u1 u3 u2 u4 u5 u6 u1 u1 u1 u3 h h h h u3 u3 h h h h u2 p p p p u4 p p p u5 p p u6 p 0h u1 u2 u3 u4 u5 u6 u1 u1 u2 u1 u2 u3 u1 u2 u3 u4 u1 u2 u3 u4 u5 u1 u2 u4 u5 u1 u2 u6 u2 u1 u2 u3 u1 u2 u3 u4 u1 u2 u3 u4 u5 u1 u2 u3 u4 u5 u1 u2 u3 u6 u3 u2 u3 u4 u2 u3 u4 u5 u2 u3 u4 u5 u2 u3 u4 u6 u4 u3 u4 u5 u3 u4 u5 u3 u4 u5 u6 u5 u4 u5 u4 u5 u6 u6 u6 ratio mathematica, 20, 2010 22 therefore we obtain so μ2(u6) = μ2(u3) = 0.315 μ2(u1) = μ2(u5) = μ2(u2) = μ2(u4) = 0.279. setting γ = {u1, u5, u2, u4}, q = {u6, u3} we have so we have μ1(u1) = μ1(u5) = μ1(u2) = μ1(u4) = 0.208 μ1(u6) = μ1(u3) = 0.233 it follows 3h = 2h, by consequence ∂(12 6) = 2. (226) set⎮d⎮= 6 = ⎮h⎮, γ(u1) = {d1, d2}, γ(u2) = {d2, d3}, γ(u3) = {d3, d4}, γ(u4) = {d4, d5}, γ(u5) = {d5, d6}, γ(u6) = {d6}. we have so we obtain μ1(u1) = 0.2467 = μ1(u6), μ1(u2) = 0.243 = μ1(u5), μ1(u3) = μ1(u4) = 0.2407 whence 1h u6 u1 u5 u2 u4 u3 u6 u6 u1 u5 u6 u1 u5 u6 u1 u5 u6 u2 u4 u1 u5 u6 u2 u4 h u1 u1 u5 u1 u5 u1 u5 u2 u4 u1 u5 u2 u4 u1 u5 u2 u4 u3 u5 u1 u5 u1 u5 u2 u4 u1 u5 u2 u4 u1 u5 u2 u4 u3 u2 u2 u4 u2 u4 u2 u4 u3 u4 u2 u4 u2 u4 u3 u3 u3 2h u1 u5 u2 u4 u6 u3 u1 p p p p h h u5 p p p h h u2 p p h h u4 p h h u6 q q u3 q 0h u1 u2 u3 u4 u5 u6 u1 u1 u2 u1 u2 u3 u1 u2 u3 u4 u1 u2 u3 u4 u5 u1 u2 u4 u5 u6 u1 u2 u5 u6 u2 u1 u2 u3 u1 u2 u3 u4 u1 u2 u3 u4 u5 h u1 u2 u3 u5 u6 u3 u2 u3 u4 u2 u3 u4 u5 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u4 u3 u4 u5 u3 u4 u5 u6 u3 u4 u5 u6 u5 u4 u5 u6 u4 u5 u6 u6 u5 u6 ratio mathematica, 20, 2010 23 hence μ2(u1) = μ2(u6) = μ2(u3) = μ2(u4) = 0.2667, μ2(u2) = μ2(u5) = 0.2619. therefore we set p = {u1, u6, u3, u4}. we obtain we have clearly 3h = 2h, whence ∂(22 6) = 2. (326) set γ(u1) = {d1, d2, d3}, γ(u2) = { d3, d4}, γ(u3) = { d4, d5}, γ(u4) = {d5}, γ(u5) = {d5, d6}, γ(u6) = {d6}. so we have μ1(u1) = 0.233, μ1(u2) = 0.230, μ1(u3) = 0.228, μ1(u4) = 0.218, μ1(u5) = μ1(u3) = 0.228, μ1(u6) = 0.231. we obtain 1h u1 u6 u2 u5 u3 u4 u1 u1 u6 u1 u6 u1 u6 u2 u5 u1 u6 u2 u5 h h u6 u1 u6 u1 u6 u2 u5 u1 u6 u2 u5 h h u2 u2 u5 u2 u5 u2 u5 u3 u4 u2 u5 u3 u4 u5 u2 u5 u2 u5 u3 u4 u2 u5 u3 u4 u3 u3 u4 u3 u4 u4 u3 u4 2h u1 u6 u3 u4 u2 u5 u1 p p p p h h u6 p p p h h u3 p p h h u4 p h h u2 u2 u5 u2 u5 u5 u2 u5 0h u1 u2 u3 u4 u5 u6 u1 u1 u2 u1 u2 u3 u1 u2 u3 u4 u5 u1 u2 u3 u4 u5 h u1 u2 u5 u6 u2 u1 u2 u3 u1 u2 u3 u4 u5 u1 u2 u3 u4 u5 h u1 u2 u3 u5 u6 u3 u2 u3 u4 u5 u2 u3 u4 u5 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u4 u3 u4 u5 u3 u4 u5 u6 u3 u4 u5 u6 u5 u3 u4 u5 u6 u3 u4 u5 u6 u6 u5 u6 ratio mathematica, 20, 2010 24 we have μ2(u1) = 0.345 > μ2(u6) = 0.326 > μ2(u4) = 0.324> μ2(u2) = 0.306 > μ2(u3) = μ2(u5) = 0.296. therefore we have we obtain now μ3(u1) = 0.348 > μ3(u6) = 0.33158> μ3(u4) = 0.317 > μ3(u2) = 0.303 > μ3(u3) = μ3(u5) = 0.29. therefore 3h = 2h and it follows that ∂(32 6) = 2. (426) set γ(u1) = {d1, d2}, γ(u2) = { d2, d3}, γ(u3) = { d3, d4}, γ(u4) = {d2}, γ(u5) = {d3}, γ(u6) = {d4}. we have 1h u1 u6 u2 u3 u5 u4 u1 u1 u1 u6 u1 u6 u2 u1 u6 u2 u3 u5 u1 u6 u2 u3 u5 h u6 u6 u6 u2 u6 u2 u3 u5 u6 u2 u3 u5 u6 u2 u3 u5 u4 u2 u2 u2 u3 u5 u2 u3 u5 u2 u3 u5 u4 u3 u3 u5 u3 u5 u3 u5 u4 u5 u3 u5 u3 u5 u4 u4 u4 2h u1 u6 u4 u2 u3 u5 u1 u1 u1 u6 u1 u6 u4 u1 u6 u4 u2 h h u6 u6 u6 u4 u6 u4 u2 u6 u4 u2 u3 u5 u6 u4 u2 u3 u5 u4 u4 u4 u2 u2 u3 u5 u2 u3 u5 u2 u2 u2 u3 u5 u2 u3 u5 u3 u3 u5 u3 u5 u5 u3 u5 ratio mathematica, 20, 2010 25 we find : μ1(u1) = 0.210 = μ1(u4) μ1(u2) = 0.221 μ1(u3) = 0.216 μ1(u5) = 0.208 μ1(u6) = 0.219. hence we obtain we have : μ2(u5) = 0.324 μ2(u2) = 0.345 μ2(u6) = 0.326 μ2(u3) = 0.3058 μ2(u1) = μ2(u4) = 0.296. therefore μ2(u2) > μ2(u6) > μ2(u5) > μ2(u3) > μ2(u1) = μ2(u4). therefore we have we can see that 3h = 2h and it follows that ∂(426) = 2. 0h u1 u2 u3 u4 u5 u6 u1 u1 u2 u4 u1 u2 u3 u4 u5 h u1 u2 u4 u1 u2 u3 u4 u5 u1 u2 u3 u4 u6 u2 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 u1 u2 u3 u4 u5 h u3 u2 u3 u5 u6 h u2 u3 u5 u6 u2 u3 u5 u6 u4 u1 u2 u4 u1 u2 u3 u4 u5 u1 u2 u3 u4 u6 u5 u2 u3 u5 u2 u3 u5 u6 u6 u3 u6 1h u5 u4 u1 u3 u6 u2 u5 u5 u5 u4 u1 u5 u4 u1 u5 u4 u1 u3 u5 u4 u1 u3 u6 h u4 u4 u1 u4 u1 u3 u1 u4 u4 u1 u3 u6 u4 u1 u3 u6 u2 u1 u4 u1 u3 u1 u4 u4 u1 u3 u6 u4 u1 u3 u6 u2 u3 u3 u3 u6 u3 u6 u2 u6 u6 u6 u2 u2 u2 2h u2 u6 u5 u3 u1 u4 u2 u2 u2 u6 u2 u6 u5 u2 u6 u5 u3 h h u6 u6 u6 u5 u6 u5 u3 u6 u5 u3 u1 u4 u6 u5 u3 u1 u4 u5 u5 u5 u3 u5 u3 u1 u4 u5 u3 u1 u4 u3 u3 u3 u1 u4 u3 u1 u4 u1 u1 u4 u1 u4 u4 u1 u4 ratio mathematica, 20, 2010 26 (526) set γ(u1) = {d1}, γ(u2) = {d2, d3}, γ(u3) = {d3, d4}, γ(u4) = {d4, d5}, γ(u5) = {d5, d6} γ(u6) = {d6}. we have we obtain μ1(u1) = 0.348, μ1(u2) = 0.268=μ1(u6), μ1(u3) = 0.2667=μ1(u5), μ1(u4) = 0.260. so, we have now we obtain μ2(u1) = 0.324 μ2(u4)= 0.315, μ2(u5) = μ2(u3) =μ2(u2) = μ2(u6) =0.279. setting p = { u2, u6 ,u3, u5 }, we find 2h we have clearly 3h = 2h whence ∂(52 6) = 2. 0h u1 u2 u3 u4 u5 u6 u1 u1 u1 u2 u3 u1 u2 u3 u4 u1 u3 u4 u5 u1 u4 u5 u6 u1 u5 u6 u2 u2 u3 u2 u3 u4 u2 u3 u4 u5 u2 u3 u4 u5 u6 u2 u3 u5 u6 u3 u2 u3 u4 u2 u3 u4 u5 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u4 u3 u4 u5 u3 u4 u5 u6 u3 u4 u5 u6 u5 u4 u5 u6 u4 u5 u6 u6 u5 u6 1h u1 u2 u6 u3 u5 u4 u1 u1 u1 u2 u6 u1 u2 u6 u1 u2 u6 u3 u5 u1 u2 u6 u3 u5 h u2 u2 u6 u2 u6 u2 u6 u3 u5 u2 u6 u3 u5 u2 u6 u5 u4 u3 u6 u2 u6 u2 u6 u3 u5 u2 u6 u3 u5 u2 u6 u5 u4 u3 u3 u3 u5 u3 u5 u5 u4 u3 u5 u3 u5 u5 u4 u3 u4 u4 2h u1 u4 u2 u6 u3 u5 u1 u1 u4 u1 u4 h h h h u4 u1 u4 h h h h u2 p p p p u6 p p p u3 p p u5 p ratio mathematica, 20, 2010 27 (626) set γ(u1) = {d1, d2, d3}, γ(u2) = {d2, d3, d4}, γ(u3) = {d4, d5}, γ(u4) = {d5, d6}, γ(u5) = {d5}, γ(u6) = {d6}. so, denoting {ui, ui+1,…, uj-1, uj} by uij, we have we have μ1(u1) = 0.233, μ1(u2) = 0.2302, μ1(u3) = μ1(u4) =0.228, μ1(u5) = 0.218, μ1(u6) = 0.2308. hence we have μ2(u1) = 0.345454, μ2(u6) = 0.326316, μ2(u2) = 0.305797, μ2(u3) = μ2(u4) = 0.29615, μ2(u5) = 0.32424. from this, we have 2h as follows one can see that 3h = 2h, therefore ∂(626) = 2. 0h u1 u2 u3 u4 u5 u6 u1 u1 u2 u13 u15 h u15 u1 u2 u4 u6 u2 u13 u15 h u15 u14 u6 u3 u25 u26 u25 u26 u4 u36 u36 u36 u5 u35 u36 u6 u4 u6 1h u1 u6 u2 u3 u4 u5 u1 u1 u1 u6 u1 u6 u2 u1 u6 u2 u3 u4 u1 u6 u2 u3 u4 h u6 u6 u6 u2 u6 u2 u3 u4 u6 u2 u3 u4 u6 u2 u3 u4 u5 u2 u2 u2 u3 u4 u2 u3 u4 u2 u5 u3 u4 u3 u3 u4 u3 u4 u3 u4 u5 u4 u3 u4 u3 u4 u5 u5 u5 2h u1 u6 u5 u2 u3 u4 u1 u1 u1 u6 u1 u6 u5 u1 u6 u5 u2 h h u6 u6 u6 u5 u6 u5 u2 u6 u5 u2 u3 u4 u6 u5 u2 u3 u4 u5 u5 u5 u2 u5 u2 u3 u4 u5 u2 u3 u4 u2 u2 u2 u3 u4 u2 u3 u4 u3 u3 u4 u3 u4 u4 u3 u4 ratio mathematica, 20, 2010 28 (726) set γ(u1) = {d1, d2, d3}, γ(u2) = {d4}, γ(u3) = { d3, d4, d5}, γ(u4) = {d4, d5, d6}, γ(u5) = {d5}, γ(u6) = {d6}. we have μ1(u1) = 0.22, μ1(u2) = 0.20864, μ1(u3) = 0.22762, μ1(u4) = μ1(u3) μ1(u5) = μ1(u2) = 0.20864, μ1(u6) = μ1(u1)= 0.22. hence, μ1(u3) = μ1(u4) = 0.22762 > μ1(u1) = μ1(u6) = 0.22 > μ1(u2) = μ1(u5) = 0.20864. we obtain we have μ2(u3) = μ2(u4) = 0.2667, μ2(u1) = 0.2619 = μ2(u6), μ2(u2) = μ2(u5) = μ2(u3) = μ2(u4) = 0.2667. set p = {u3, u4, u2, u5}, q= {u1, u6}. we obtain it follows that μ3(u3) = μ3(u4) = μ3(u2) = μ3(u5) =0.208, μ3(u1) = μ3(u6) = 0.233. we have clearly 3h = 2h, so ∂(72 6) = 2. 0h u1 u2 u3 u4 u5 u6 u1 u1 u3 u1 u2 u3 u4 u1 u2 u3 u4 u5 h u1 u3 u4 u5 u1 u3 u4 u6 u2 u2 u3 u4 u1 u2 u3 u4 u5 u2 u3 u4 u5 u6 u2 u3 u4 u5 u2 u3 u4 u6 u3 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 h u4 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u5 u3 u4 u5 u3 u4 u5 u6 u6 u4 u6 1h u3 u4 u1 u6 u2 u5 u3 u3 u4 u3 u4 u3 u4 u1 u6 u3 u4 u1 u6 h h u4 u3 u4 u3 u4 u1 u6 u3 u4 u1 u6 h h u1 u1 u6 u1 u6 u1 u6 u2 u5 u1 u6 u2 u5 u6 u1 u6 u1 u6 u2 u5 u1 u6 u2 u5 u2 u2 u5 u2 u5 u5 u2 u5 2h u3 u4 u2 u5 u1 u6 u3 p p p p h h u4 p p p h h u2 p p h h u5 p h h u1 q q u6 q ratio mathematica, 20, 2010 29 (826) set γ(u1) = {d1, d2}, γ(u2) = {d2, d3, d4}, γ(u3) = {d3, d4, d5} γ(u4) = {d5, d6}, γ(u5) = {d5}, γ(u6) = {d6}. we obtain μ1(u1) = 0.233, μ1(u2) = 0.230247, μ1(u3) = 0.228125, μ1(u4) = μ1(u3) μ1(u5) = 0.218518, μ1(u6) = 0.230833. from this, we have 1h. from 1h we obtain : μ2(u1) = 0.34545, μ2(u6) = 0.3263, μ2(u5) = 0.324242, μ2(u2) = 0.305797, μ2(u3) = μ2(u4) = 0.296. therefore we find 2h as follows from 2h we obtain : μ3(u1) = 0.34848, μ3(u6) = 0.331579, μ3(u5) = 0.31739 μ3(u2) = 0.302898, μ3(u3) = μ3(u4) = 0.29 we have clearly 3h = 2h, by consequence ∂(82 6) = 2. 0h u1 u2 u3 u4 u5 u6 u1 u1 u2 u1 u2 u3 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 u1 u2 u4 u6 u2 u1 u2 u3 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 u1 u2 u3 u4 u6 u3 u2 u3 u4 u5 u2 u3 u4 u5 u6 u2 u3 u4 u5 u2 u3 u4 u5 u6 u4 u3 u4 u5 u6 u3 u4 u5 u6 u3 u4 u5 u6 u5 u3 u4 u5 u3 u4 u5 u6 u6 u4 u6 1h u1 u6 u2 u3 u4 u5 u1 u1 u1 u6 u1 u2 u6 u1 u6 u2 u3 u4 u1 u6 u2 u3 u4 h u6 u6 u2 u6 u2 u6 u3 u4 u2 u6 u3 u4 u2 u6 u3 u4 u5 u2 u2 u2 u3 u4 u2 u3 u4 u2 u3 u4 u5 u3 u3 u4 u3 u4 u3 u4 u5 u4 u3 u4 u3 u4 u5 u5 u5 2h u1 u6 u5 u2 u3 u4 u1 u1 u1 u6 u1 u6 u5 u1 u6 u5 u2 h h u6 u6 u6 u5 u6 u5 u2 u6 u5 u2 u3 u4 u6 u5 u2 u3 u4 u5 u5 u5 u2 u5 u2 u3 u4 u5 u2 u3 u4 u2 u2 u2 u3 u4 u2 u3 u4 u3 u3 u4 u3 u4 u4 u3 u4 ratio mathematica, 20, 2010 30 (136) set γ(u1) = {d1, d2, d3}, γ(u2) = {d2, d3, d4}, γ(u3) = {d3, d4, d5}, γ(u4) = {d4, d5, d6} γ(u5) = {d5}, γ(u6) = {d6}. we obtain μ1(u1) = 0.2006173, μ1(u2) = 0.2005208, μ1(u3) = 0.20714, μ1(u4) = 0.211905, μ1(u5) = 0.198765, μ1(u6) = 0.206667. by consequence we have 1h. hence we have μ2(u4) = 0.354545 = μ2(u5), μ2(u3) = 0.34035 = μ2(u2) μ2(u6) = 0.33188 = μ2(u1) from which we obtain 2h. from 2h it follows μ3 (u4) = μ3(u5) = μ3(u6) =μ3(u1) =0.26667 μ3 (u2) = μ3(u3) = 0.26190. 0h u1 u2 u3 u4 u5 u6 u1 u1 u2 u3 u1 u2 u3 u4 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 u1 u2 u3 u4 u6 u2 u1 u2 u3 u4 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 u1 u2 u3 u4 u6 u3 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 h u4 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u5 u3 u4 u5 u3 u4 u5 u6 u6 u4 u6 1h u4 u3 u6 u1 u2 u5 u4 u4 u4 u3 u4 u3 u6 u4 u3 u6 u1 u4 u3 u6 u1 u2 h u3 u3 u3 u6 u3 u6 u1 u3 u6 u1 u2 u3 u6 u1 u2 u5 u6 u6 u6 u1 u6 u1 u2 u6 u1 u2 u5 u1 u1 u1 u2 u1 u2 u5 u2 u2 u2 u5 u5 u5 2h u4 u5 u3 u2 u6 u1 u4 u4 u5 u4 u5 u4 u5 u3 u2 u4 u5 u3 u2 h h u5 u4 u5 u4 u5 u3 u2 u4 u5 u3 u2 h h u3 u3 u2 u3 u2 u3 u2 u6 u1 u3 u2 u6 u1 u2 u3 u2 u3 u2 u6 u1 u3 u2 u6 u1 u6 u6 u1 u6 u1 u1 u6 u1 ratio mathematica, 20, 2010 31 set p = {u4, u5, u6, u1}, q = {u3, u2}. then we obtain 3h as follows from 3h, it follows that 4h = 3h and we have finally ∂(136) = 3. (236) set γ(u1) = {d1, d2, d3}, γ(u2) = {d3, d4}, γ(u3) = {d3, d4, d5}, γ(u4) = {d4, d5, d6}, γ(u5) = {d5}, γ(u6) = {d6}. whence ∂(13 6)= ∂(236)=3. (336) set γ(u1) = {d1, d2, d3}, γ(u2) = {d2, d4}, γ(u3) = {d3, d4, d5}, γ(u4) = {d4, d5, d6} γ(u5) = {d5}, γ(u6) = {d6}. we have see (13 6). we have ∂(33 6)= ∂(136)=3. 3h u4 u5 u6 u1 u3 u2 u4 p p p p h h u5 p p p h h u6 p p h h u1 p h h u3 q q u2 q 0h u1 u2 u3 u4 u5 u6 u1 u1 u2 u3 u1 u2 u3 u4 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 u1 u2 u3 u4 u6 u2 u1 u2 u3 u4 u1 u2 u3 u4 u5 h u1 u2 u3,u4,u5 u1,u2 u3 u4 u6 u3 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 h u4 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u2 u3 u4,u5 u6 u5 u3 u4 u5 u3 u4 u5 u6 u6 u4 u6 0h u1 u2 u3 u4 u5 u6 u1 u1 u2 u3 u1 u2 u3 u4 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 u1 u2 u3 u4 u6 u2 u1 u2 u3 u4 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 u1 u2 u3 u4 u6 u3 u1 u2 u3 u4 u5 h u1 u2 u3 u4 u5 h u4 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u5 u3 u4 u5 u3 u4 u5 u6 u6 u4 u6 ratio mathematica, 20, 2010 32 $ 6. (148) set h = {u1, u2, u3, u4, u5, u6, u7, u8} and γ(u1) = {d1, d2, d3}, γ(u2) = {d2, d3, d4}, γ(u3) = {d3, d4, d5}, γ(u4) = { d4, d5, d6}, γ(u5) = {d5, d6, d7}, γ(u6) = {d7, d8}, γ(u7) = {d7}, γ(u8) = {d8}. so, denoting {ui, ui+1,…, uj-1, uj} by uij, we have we have μ1(u1) = 0.1756, μ1(u2) = 0.17470, μ1(u3) = 0.1754978, μ1(u4) = 0.1729, μ1(u5) = 0.1803, μ1(u6) = 0.1813, μ1(u7) = 0.175641 = μ1(u1), μ1(u8) = 0.19073. so μ1(u8) > μ1(u6) > μ1(u5) > μ(u7) = μ(u1) > μ1(u3) > μ1(u2) > μ(u4). one obtains 1h as follows from which μ2(u4) = μ2(u8) = 0.2890, μ2(u2) = μ2(u6) = 0.2724, μ2(u1) = μ2(u7) = 0.247567, μ2(u3) = μ2(u5) = 0.2549. so we obtain 2h. 0h u1 u2 u3 u4 u5 u6 u7 u8 u1 u13 u14 u15 u15 u17 u13 u58 u13 u57 u13 u6 u8 u2 u14 u15 u15 u17 h u17 u14 u6 u8 u3 u15 u15 u17 h u17 u16 u8 u4 u25 u27 u28 u27 u26 u8 u5 u37 u38 u37 u38 u6 u58 u58 u58 u7 u57 u58 u8 u6 u8 1h u4 u2 u3 u1 u7 u5 u6 u8 u4 u4 u4 u2 u4 u2 u3 u4 u2 u3 u1 u7 u4 u2 u3 u1 u7 u4 u2 u3 u1 u7 u5 u4 u2 u3 u1 u7 u5 u6 h u2 u2 u2 u3 u2 u3 u1 u7 u2 u3 u1 u7 u2 u3 u1 u7 u5 u2 u3 u1 u7 u5 u6 u2 u3 u1 u7 u5 u6 u8 u3 u3 u3 u1 u7 u3 u1 u7 u3 u1 u5 u7 u3 u1 u7 u5 u6 u3 u1 u7 u5 u6 u8 u1 u1 u7 u1 u7 u1 u7 u5 u1 u7 u5 u6 u1 u7 u5 u6 u8 u7 u1 u7 u1 u7 u5 u1 u7 u5 u6 u1 u7 u5 u6 u8 u5 u5 u5 u6 u5 u6 u8 u6 u6 u6 u8 u8 u8 ratio mathematica, 20, 2010 33 hence we have : μ3(u4) = μ3(u8) = μ3(u1) = μ3(u7) = 0.22619, μ3(u2) = μ3(u6) = μ3(u3) = μ3(u5) = 0.2197. setting p= {u4, u8, u1, u7}, q = {u3, u5, u2, u6}, we find 3h from this, we obtain : ∀i, μ4(ui) = 0.166. it follows 4h = t, whence ∂(14 8) = 4. $7. (12 9) let h={ ui | 1 ≤ i ≤ 9} and for i ≤ 7 , set γ(ui) = { di, di+1, di+2}, γ(u8) = {d8}, γ(u9) = {d9}. we obtain 2h u4 u8 u2 u6 u3 u5 u1 u7 u4 u4 u8 u4 u8 u4 u8 u2 u6 u4 u8 u2 u6 u4 u8 u2 u6 u3 u5 u4 u8 u2 u6 u3 u5 h h u8 u4 u8 u4 u8 u2 u6 u4 u8 u2 u6 u4 u8 u2 u6 u3 u5 u4 u8 u2 u6 u3 u5 h h u2 u2 u6 u2 u6 u2 u6 u3 u5 u2 u6 u3 u5 u2 u6 u3 u5 u1 u7 u2 u6 u3 u5 u1 u7 u6 u2 u6 u2 u6 u3 u5 u2 u6 u3 u5 u2 u6 u3 u5 u1 u7 u2 u6 u3 u5 u1 u7 u3 u3 u5 u3 u5 u3 u5 u1 u7 u3 u5 u1 u7 u5 u3 u5 u3 u5 u1 u7 u3 u5 u1 u7 u1 u1 u7 u1 u7 u7 u1 u7 3h u4 u8 u1 u7 u2 u6 u3 u5 u4 p p p p h h t h u8 p p p h h h h u1 p p h h h h u7 p h h h h u2 q q q q u6 q q q u3 q q u5 q ratio mathematica, 20, 2010 34 from 0h we have μ1(u9) = 0.1729 > μ1(u7) = 0.1648 > μ1(u1) = 0.1616 > μ1(u3) = 0.160 > μ1(u6) = 0.1599 > μ1(u8) = 0.1597 > μ1(u2) = 0.159169 > μ1(u5) = 0.157387 > μ1(u4) = 0.159218. from these data, we obtain 1h as follows 0h u1 u2 u3 u4 u5 u6 u7 u8 u9 u1 u1u2 u3 u1 u2 u3 u4 u1 u2 u3 u4 u5 u1 u2 u3 u4 u5 u6 u1 u2 u3 u4 u5 u6 u7 u1 u2 u3 u4 u5 u6 u7 u8 u1 u2 u3 u5 u6 u7 u8 u9 u1 u2 u3 u6 u7 u8 u1 u2 u3 u7 u9 u2 u1 u2 u3 u4 u1 u2 u3 u4 u5 u1 u2 u3 u4 u5 u6 u1 u2 u3 u4 u5 u6 u7 u1 u2 u3 u4 u5 u6 u7 u8 h u1 u2 u3 u4 u6 u7 u8 u1 u2 u3 u4 u7 u9 u3 u1 u2 u3 u4 u5 u1 u2 u3 u4 u5 u6 u1u2 u3 u4 u5 u6 u7 u1 u2 u3 u4 u5 u6 u7 u8 h u1 u2 u3 u4 u5 u6 u7 u8 u1 u2 u3 u4 u5 u7 u9 u4 u2 u3 u4 u5 u6 u2 u3 u4 u5 u6 u7 u2 u3 u4 u5 u6 u7 u8 u2 u3 u4 u5 u6 u7 u8 u9 u2 u3 u4 u5 u6 u7 u8 u2 u3 u4 u5 u6 u7 u9 u5 u3 u4 u5 u6 u7 u3 u4 u5 u6 u7 u8 u3 u4 u5 u6 u7 u8 u9 u3 u4 u5 u6 u7 u8 u3 u4 u5 u6 u7 u9 u6 u4 u5 u6 u7 u8 u4 u5 u6 u7 u8 u9 u4 u5 u6 u7 u8 u4 u5 u6 u7 u8 u9 u7 u5 u6 u7 u8 u9 u5 u6 u7 u8 u9 u5 u6 u7 u8 u9 u8 u6 u7 u8 u6 u7 u8 u9 u9 u7 u9 1h u9 u7 u1 u3 u6 u8 u4 u2 u5 u9 u9 u9 u7 u9 u7 u1 u9 u7 u1 u3 u9 u7 u1 u3 u6 u9 u7 u1 u3 u6 u8 u9 u7 u1 u3 u6 u8 u4 u9 u7 u1 u3 u6 u8 u2 u4 h u7 u7 u7 u1 u7 u1 u3 u7 u1 u3 u6 u7 u1 u3 u6 u8 u7 u1 u3 u6 u8 u4 u7 u1 u3 u6 u8 u2 u4 u7 u1 u3 u6 u8 u2 u5 u4 u1 u1 u1 u3 u1 u3 u6 u1 u3 u6 u8 u1 u3 u6 u8 u4 u1 u3 u6 u8 u2 u4 u1 u3 u6 u8 u2 u5 u4 u3 u3 u3 u6 u3 u6 u8 u3 u6 u8 u4 u3 u6 u8 u2 u4 u3 u6 u8 u2 u5 u4 u6 u6 u6 u8 u6 u8 u4 u6 u8 u2 u4 u6 u8 u2 u5 u4 u8 u8 u8 u4 u8 u2 u4 u8 u2 u5 u4 u4 u4 u2 u4 u2 u5 u4 u2 u2 u5 u2 u5 u5 ratio mathematica, 20, 2010 35 from 1h we obtain as follows, μ2 and then 2h. μ2(u9) = μ2(u5) = 0.2740 > μ2(u7) = μ2(u2) = 0.261085 > μ2(u1) = μ2(u4) = = 0.250716 > μ2(u3) = μ2(u8) = 0.24495 > μ2(u6) = 0.243116. from 2h we obtain μ3(u9) = μ3(u5) = 0.211805 , μ3(u7) = μ3(u2) = 0.205433, μ3(u1) = μ3 (u4) = 0.20504, μ3 (u3) = μ3(u8) = 0.21155, μ3(u6) = 0.24407. then we have 3h as follows 2h u9 u5 u7 u2 u1 u4 u3 u8 u6 u9 u5 u9 u5 u9 u5 u9 u7 u2 u5 u9 u7 u2 u5 u9 u7 u2 u1 u4 u5 u9 u7 u2 u1 u4 u5 u9 u7 u2 u3 u8 u1 u4 u5 u9 u7 u3 u8 u2 u1 u4 h u5 u5 u9 u5 u9 u7 u2 u5 u9 u7 u2 u5 u9 u7 u2 u1 u4 u5 u9 u7 u2 u1 u4 u5 u9 u7u2 u1 u4 u3 u8 u5 u9 u7 u2 u1 u4 u3u8 h u7 u7 u2 u7 u2 u7 u2 u1 u4 u7 u2 u1 u4 u7 u2 u1 u4 u3 u8 u7 u2 u1 u4 u3 u8 u7 u2 u6 u1 u4 u3 u8 u2 u7 u2 u7 u2 u1 u4 u7 u2 u1 u4 u7 u2 u1 u4 u3 u8 u7 u2 u1 u4 u3 u8 u7 u2 u6 u1 u4 u3 u8 u1 u1 u4 u1 u4 u1 u4 u3 u8 u1 u4 u3 u8 u1 u4 u6 u3 u8 u4 u1 u4 u1 u4 u3 u8 u1 u4 u3 u8 u1 u4 u6 u3 u8 u3 u3 u8 u3 u8 u3 u8 u6 u8 u3 u8 u3 u8 u6 u6 u6 3h u6 u9 u5 u3 u8 u2 u7 u1 u4 u6 u6 u6 u9 u5 u6 u9 u5 u6 u9 u5 u3 u8 u6 u9 u5 u3 u8 u6 u9 u5 u3 u8 u2u7 u6 u9 u5 u3 u8 u2 u7 h h u9 u9 u5 u9 u5 u9 u5u3 u8 u9 u5 u3 u8 u9 u5 u3 u8 u2 u7 u9 u5 u3 u8 u2 u7 u9 u5 u3u8 u2 u7 u1 u4 u9 u5 u3 u8 u2 u7 u1 u4 u5 u9 u5 u9 u5 u3 u8 u9 u5 u3 u8 u9 u5 u3 u8 u2 u7 u9 u5 u3 u8 u2 u7 u9 u5 u3 u8 u2 u7 u1 u4 u9 u5 u3u8 u2 u7 u1 u4 u3 u3 u8 u3 u8 u3 u8 u2 u7 u3 u8 u2 u7 u3 u8 u2 u7 u1 u4 u3 u8 u2 u7 u1 u4 u8 u3 u8 u3 u8 u2 u7 u3 u8 u2 u7 u3 u8 u2 u7 u1 u4 u3 u8 u2 u7 u1 u4 u2 u2 u7 u2 u7 u2 u7 u1 u4 u2 u7 u1 u4 u7 u2 u7 u2 u7 u1 u4 u2 u7 u1 u4 u1 u1 u4 u1 u4 u4 u1 u4 ratio mathematica, 20, 2010 36 it is possible to verify that the function φ : 2h → 3h defined as follows φ(u3) = u9, φ(u8) = u5, φ(u1) = u3, φ(u4) = u8, φ(u9) = u1, φ( u5) = u4, φ(u7) = u7, φ(u2) = u2, φ(u6) = u6, is a hypergroup isomorphism. it follows that the fuzzy grade of (12 9) is 2. $ 8. (1516) set h = {ui⎮1 ≤ i ≤ 16}, d = {di⎮1 ≤ i ≤ 16}, γ(u1) = {d1, d2, d3}, γ(u2) = {d2, d3, d4}, and ∀ i : i ≤ 13, γ(ui) = {di, di+1, di+2}, γ(u14) = {d15, d16}, γ(u15) = {d15}, γ(u16) = {d16}. since ∀ i, we have ui ο ui = {uj⎮ γ(uj) ∩ γ(ui) ≠ ∅}, it follows that we have u1 ο u1 = { u1, u2, u3}, u2 ο u2 = { u1, u2, u3, u4}, u3 ο u3 = { u1, u2, u3, u4, u5}, ∀ i : 4 ≤ i ≤ 13, ui ο ui = { ui-2, u i-1, ui, ui+1, ui+2}, u14 ο u14 = { u13, u14, u15, u16}, u15 ο u15 = { u13, u14, u15}, u16 ο u16 = { u14, u16}. for 0h we have the following table : ratio mathematica, 20, 2010 37 0h u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15 u16 u1 u13 u14 u15 u16 u17 u18 u13 u59 u13 u610 u13 u711 u13 u812 u13 u913 u13 u1014 u13 u1115 u13 u1316 u13 u1315 u13 u14 u16 u2 u14 u15 u16 u17 u18 u19 u14 u610 u14 u711 u14 u812 u14u913 u14u1014 u14 u1115 u14u1316 u14u1315 u14u14 u16 u3 u15 u16 u17 u18 u19 u110 u15 u711 u15u812 u15u913 u15u1014 u15u1115 u15u1316 u15u1315 u15u14 u16 u4 u26 u27 u28 u29 u210 u211 u26u812 u26u913 u26u1014 u26u1115 u26u1316 u26u1315 u26u14 u16 u5 u37 u38 u39 u310 u311 u312 u37u913 u37u1014 u37u1115 u37u1316 u37u1315 u37u14 u16 u6 u48 u49 u410 u411 u412 u413 u48u1014 u48u1115 u48u1316 u48u1315 u48u14 u16 u7 u59 u510 u511 u512 u513 u514 u59u1115 u59u1316 u59u1315 u59u14 u16 u8 u610 u611 u612 u613 u614 u615 u610u1316 u610u1315 u610u14 u16 u9 u711 u712 u713 u714 u715 u711u1316 u715 u711u14 u16 u10 u812 u813 u814 u815 u816 u815 u812u14 u16 u11 u913 u914 u915 u916 u915 u914u16 u12 u1014 u1015 u1016 u1015 u1014u16 u13 u1115 u1116 u1115 u1116 u14 u1316 u1316 u1316 u15 u1315 u1316 u16 u14 u16 ratio mathematica, 20, 2010 38 from 0h we obtain μ1(u16) = 0.15673, μ1(u15) = 0.13992, μ1(u14) = 0.141293, μ1(u1) = 0.13867, μ1(u2) = 0.134942, μ1(u13) = 0.134215, μ1(u3) = 0.132574, μ1(u4) = 0.129700, μ1(u5) = 0.128076, μ1(u6) = 0.127554, μ1(u7) = 0.126581, μ1(u12) = 0.1283654, μ1(u11) = 0.126441, μ1(u10) = 0.126878, μ1(u8) = 0.12671, μ1(u9) = 0.126608. for 1h, set v1 = u16, v2 = u14, v3 = u15, v4 = u1, v5 = u2, v6 = u13, v7 = u3, v8 = u4, v9 = u12, v10 = u5, v11 = u6, v12 = u10, v13 = u8, v14 = u9, v15 = u7, v16 = u11. ∀(i, j), such that i ≤ j set vij = {vi, vi+1,...vj}. so we have vi ο1 vj = vij. for 2h we have v1 ο2 v1 = v1 ο2 v16 = v16 ο2 v16 = {v1, v16}, v2 ο2 v2 = v2 ο2 v15 = v15 ο2 v15 = {v2, v15}. generally, vi ο2 vi = vi ο2 v16-(i-1) = v16-(i-1) ο2 v16-(i-1) = {vi, v16-(i-1) }. for i < j, vi ο2 vj = u jsi ≤≤ vs ο2 vs. set p1 = {v1, v16, v8, v9}, p2 = {v2, v15, v7, v10}, p3 = {v3, v14, v6, v11}, p4 = {v4, v13, v5, v12}. then for 3h, ∀k: 1 ≤ k ≤ 14, we have ∀( vi, vj) ∈ pk x pk, vi ο3 vj = pk. if s < t, ∀( vi, vj) ∈ ps x pt, we have vi ο3 vj = u tus ≤≤ pu. for 4h, setting q1 = p1 u p4, q2 = p2 u p3, we have ∀( vi, vj) ∈ qi x qj, vi ο4 vj = qi u qj. by consequence, if i ≠ j, vi ο4 vj = h and vi ο4 vi = qi. since ⎮q1⎮=⎮q2⎮, we have ∀ vi ∈ q1, ∀ vj∈ q2, μ4(vi) = μ4(vj). it follows that 5h = t (total hypergroup) and by consequence ∂(15 16) = 5. references [1] ameri r. and zahedi m.m., hypergroup and join space induced by a fuzzy subset, pu.m.a. vol. 8, (1997) [2] ameri r. and shafiiyan, fuzzy prime and primary hyperideals of hyperrings, advances in fuzzy math., n. 1-2, research india publications (2007) [3] ameri r., hedayati h, molaee a., on fuzzy hyperideals of γ-hyperrings, iranian j. of fuzzy systems, vol. 6, n. 2, (2009) [4] bakhshi m., mashinki m., borzooei r.a., representation theorem for some algebraic hyperstructures, international review of fuzzy mathematics (irfm), vol. 1, n.1, (2006) [5] borzooei r.a., jun young bae, intuitionistic fuzzy hyper bckideals of bckalgebras, iranian j. of fuzzy systems, vol. 1, n.1, (2004) [6] borzooei r.a, zahedi m.m., fuzzy structures on hyper k-algebras, international j. of uncertainty fuzzyness and knowledge-based systems 112, (2), (2000) [7] corsini p., prolegomena of hypergroup theory, aviani editore (1993) [8] corsini p., join spaces, power sets, fuzzy sets, proc. fifth international congress on a.h.a. 1993, iasi. romania, hadronic press, (1994) ratio mathematica, 20, 2010 39 [9] corsini p., rough sets, fuzzy sets and join spaces, honorary volume dedicated to prof. emeritus ioannis mittas, aristotle univ. of thessaloniki, 1999-2000, editors m. konstantinidou, k. serafimidis, g. tsagas [10] corsini p., on chinese hyperstructures, proc. of the seventh congress a.h.a., taormina, 1999, journal of discrete mathematical sciences & criptography, vol. 6 (2003) [11] corsini p., fuzzy sets, join spaces and factor spaces, pu.m.a. vol. 11, n. 3, (2000) [12] corsini p., properties of hyperoperations associated with fuzzy sets and with factor spaces, international journal of sciences and technology, kashan university, vol. 1, n. 1, (2000) [13] corsini p., binary relations, interval structures and join spaces, korean j.math comput. appl. math., 9(1) (2002) [14] corsini p., a new connection between hypergroups and fuzzy sets, southeast asian bulletin of math., 27. 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[21]. the important note of this artice is to extend the notion of intimate mappings in fuzzy metric space using recent concepts like the different forms of e.a properties.in this process we prove three unique common fixed point theorems using these concepts. cocequently these results stand as generalizations of some of the existing results like [16] [19]. furthermore, some illustrations are provided to support our findings. 2. definitions and preliminaries definition 2.1 (b.schweizer and a.sklar [7]):a binary operation ∗:[0,1] × [0, 1]→[0,1] is said to be continuous triangular norm (i. e continuous 𝓉 − norm) if the following assertions hold: (ct-i) * is continuous;(ct-ii)𝒶 ∗ 𝒷 ≤ c ∗ 𝒹 where 𝒶 ≤ 𝒷, 𝒸 ≤ 𝒹 and 𝒶, 𝒷, 𝒸, 𝒹 ∈ [0,1];(ct-iii)𝒶 ∗ 1 = 𝒶 for 𝒶 ∈ [0,1]; (ct-iv) ∗ is associative and commutative. definition 2.2 (kramosil and mechalek [2]): a triplet (𝕏,𝑀𝐾𝑀,*) is fuzzy metric space (i.e., fms) if 𝕏 is a arbitrary set, * is continuous 𝓉 − norm and 𝑀𝐾𝑀 is fuzzy set on 𝕏2× (0, ∞) satisfying the following conditions for all 𝓍, 𝓎, 𝓏𝕏 such that 𝓉, 𝓈(0, ∞): (kmfm-i) mkm(𝓍, 𝓎, 0) = 0 (kmfm-ii) mkm(𝓍, 𝓎, 𝓉) = 1 ∀𝓉 > 0 ⟺ 𝓍 = 𝓎 (kmfm-iii) mkm(𝓎, 𝓍, 𝓉) = mkm(𝓍, 𝓎, 𝓉) (kmfm-iv) mkm(𝓍, 𝓏, 𝓉 + 𝓈) ≥ mkm (𝓍, 𝓎, 𝓉) ∗ mkm (𝓎, 𝓏, 𝓈) (kmfm-v) mkm(𝓍, 𝓎, . ): [0.1]→[0,1] left continuous. example 2.3 (george &veeramani [4]): consider(𝕏, 𝒹𝓊) is a metric space and define 𝑀𝐾𝑀(𝓍, 𝓎, 𝓉) = 𝓉 𝓉 + 𝒹𝓊(𝓍, 𝓎) then ( 𝕏, mkm,∗) is fms where ∀𝓍, 𝓎𝕏, 𝓉 > 0 and ∗ is continuous 𝓉 − norm with 𝒶 ∗ 𝒷 = min {𝒶, 𝒷}. 174 some fixed point results in fuzzy metric space using intimate mappings in the entire paper, (𝕏, 𝑀𝐾𝑀,∗) is to be assumed fms with the condition (kffm-6) : lim 𝓉→∞ mkm(𝓍, 𝓎, 𝓉) = 1 for all 𝓍, 𝓎, 𝕏. definition 2.4 (grabiec [3]): let 〈𝓍𝓃 〉 be sequence in fms (𝕏, mkm,∗), 〈𝓍𝓃 〉 then converges to a point ℓ ∈ 𝕏 if lim 𝓃⟶∞ mkm(𝓍𝓃, ℓ, 𝓉) = 1, ∀𝓉 > 0. definition 2.5 (garbaic [3]): let 〈𝓍𝓃 〉 be a sequence in fms (𝕏, mkm,∗), this sequence < 𝓍𝓃 > in 𝕏 is said to be cauchy sequence in fms if lim 𝓃→∞ mkm(𝓍𝓃+𝓅, 𝓍𝓃 , 𝓉) = 1, ∀𝓉 > 0 and 𝓅 > 0. definition 2.6 (garbiec [3]): if every cauchy sequence is convergent in (𝕏, mkm,∗) then we say that it is complete. lemma 2.7 (s.n. mishra et al [5]): let(𝕏, mkm,∗) be a fms if there exists 𝓀 ∈ (0,1) such that mkm(𝓍, 𝓎, 𝓀𝓉) ≥ mkm(𝓍, 𝓎, 𝓉) then 𝓍 = 𝓎. definition 2.8 ([5],[10]): let 𝔖 and 𝔗 be two self mappings of a fms(𝕏, mkm,∗). then 𝔖 and 𝔗 are (1) compatible if lim 𝓃→∞ m𝐾𝑀(𝔖𝔗𝓍𝓃 , 𝔗𝔖𝓍𝓃 , 𝓉) = 1 whenever a sequence 〈𝓍𝑛 〉 in 𝕏 provided lim 𝓃→∞ 𝔖𝓍𝓃 = lim 𝓃→∞ 𝔗𝓍𝓃 = 𝓉 for some 𝓉𝕏 (2) compatible of type (𝒜) if lim 𝓃→∞ m𝐾𝑀(𝔖𝔗𝓍𝓃 , 𝔗𝔗𝓍𝓃 , 𝓉) = 1 lim 𝓃→∞ m𝐾𝑀 (𝔗𝔖𝓍𝓃 , 𝔖𝔖𝓍𝓃 , 𝓉) = 1 whenever 〈𝑥𝓃 〉 in 𝕏 such that lim n→∞ 𝔖𝓍𝓃 = lim n→∞ 𝔗𝓍𝓃 = 𝓉 for some 𝓉𝕏. now we discuss some definitions related to intimate mappings in fms. definition 2.9: let 𝔄 𝑎𝑛𝑑 𝔖 be two mappings of a fms (𝕏, mkm,∗) into itself. then 𝔄 and 𝔖 are said to be (1). 𝒜-intimate mappings if α mkm(𝔄𝔖𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) ≥ α mkm(𝔖𝔖𝓍𝓃 , 𝔖𝓍𝓃 , 𝓉) where α = lim 𝓃⟶∞ sup or lim 𝓃⟶∞ inf and 〈𝓍𝓃 〉 is a sequence in 𝕏 ∋ lim 𝓃→∞ 𝔄𝓍𝓃 = lim 𝓃→∞ 𝔖s𝓍𝓃 = 𝓉 for some 𝓉𝕏. (2). 𝒮-intimate mapping if α m𝐾𝑀(𝔖𝔄𝓍𝓃, 𝔖𝓍𝓃 , 𝓉) ≥ α m𝐾𝑀(𝔄𝔄𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) where α = lim 𝓃⟶∞ sup or lim 𝓃⟶∞ inf and a sequence 〈𝓍𝑛〉 in 𝕏 ∋ lim 𝓃→∞ 𝔄𝓍𝓃 = lim 𝓃→∞ 𝔖𝓍𝓃 = 𝓉 for some 𝓉𝕏. proposition 2.10: let 𝔄 𝑎𝑛𝑑 𝔖 be two self mappings of a fms (𝕏, 𝑀𝐾𝑀 ,∗). suppose 𝔄 and 𝔖 are compatible mappings of type (𝒜) then the pair of mappings 𝔄 𝑎𝑛𝑑 𝔖 are 𝒜 − intimate mappings and 𝒮-intimate mappings. proof:since 𝔄 𝑎𝑛𝑑 𝔖are compatible of type (𝒜), we have lim 𝓃→∞ m𝐾𝑀 (𝔄𝔖𝓍𝓃, 𝔖𝔖𝑥𝓃, 𝓉) = 1 and lim n→∞ m𝐾𝑀(𝔖𝔄𝓍𝓃, 𝔄𝔄𝑥𝓃, 𝓉) = 1 whenever 〈𝓍𝓃〉 in 𝕏 ∋ lim n→∞ 𝔄 𝓍𝓃 = lim 𝓃→∞ 𝔖𝓍𝓃 = 𝓉 for some 𝓉𝕏. now m𝐾𝑀 (𝔄𝔖𝓃𝓃, 𝔄𝓍𝓃, (2 − β)𝓉) = m𝐾𝑀 (𝔄𝔖𝓍𝓃, 𝔄𝓍𝓃, (1 + 𝑘1)𝓉) ≥ m𝐾𝑀 (𝔄𝔖𝓍𝓃, 𝔖𝔖𝓍𝓃, 𝑘1𝓉) ∗ m𝐾𝑀 (𝔖𝔖𝓍𝓃, 𝔄𝓍𝓃, 𝓉). 175 vijayabaskerreddy bonuga and srinivas veladi by taking 𝑘1 = 1 − β and 0 < 𝑘1 < 1 and letting 𝓃 → ∞ and 𝛽 → 1 we obtain m𝐾𝑀(𝔄𝔖𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) ≥ m𝐾𝑀(𝔄𝔖𝓍𝓃 , 𝔖𝔖𝓍𝓃 , 𝑘1𝓉) ∗ m𝐾𝑀(𝔖𝔖𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) = m𝐾𝑀(𝔖𝔖𝓍𝓃, a𝓍𝓃 , 𝓉). by applying limit supremum on both sides, α m𝐾𝑀(𝔄𝔖𝓍𝓃 , 𝔄𝓍𝓃 , 𝓉) ≥ α m𝐾𝑀(𝔖𝔖𝓍𝓃 , 𝔄𝓍𝓃 , 𝓉) this implies 𝔄 𝑎𝑛𝑑 𝔖 are 𝒜-intimate mappings whenever {𝓍𝓃 } is a sequence in 𝕏 such that lim n→∞ 𝔄𝓍𝓃 = lim n→∞ 𝔖𝓍𝓃 = 𝓉 for some 𝓉𝕏. likewise, we can prove that the pair of these mappings is 𝒮-intimate. proposition 2.11: let 𝔄 and 𝔖 be two self mappings on fms.𝔄 and 𝔖 are 𝒜-intimate mappings and 𝔄t1=𝔖t1=𝑝,𝑝𝕏 then m𝐾𝑀(𝔄p, p, 𝓉) ≥ m𝐾𝑀(𝔖p, p, 𝓉). proof: suppose that {𝑥𝓃 } ∈ 𝕏 is a sequence such that 𝔄xn = 𝔖xn → 𝔄t1 = 𝔖t1 = 𝑝 for some 𝑝, 𝓉𝕏. since the pair of mappings 𝔄 and 𝔖 are 𝒜 − intimate, then we obtain 𝑀𝐾𝑀(𝔄𝑝, 𝑝, 𝓉) = lim 𝓃→∞ m𝐾𝑀(𝔄𝔖𝓍𝓃 , 𝔄𝓍𝓃 , 𝓉) ≥ lim 𝓃→∞ m𝐾𝑀(𝔖𝔖𝓍𝓃 , 𝔖𝓍𝓃 , 𝓉) = 𝑀𝐾𝑀 (𝔖𝑝, 𝑝, 𝓉). thus m𝐾𝑀 (𝔄p, p, 𝓉) ≥ m𝐾𝑀 (𝔖p, p, 𝓉). remark 2.12: a pair of mappings 𝔄 and 𝔖 is 𝒜-intimate or 𝒮-intimate but not compatible mapping of type (𝒜). the following example revels the relation between intimate mappings and compatible mappings of type (𝒜). example 2.13: suppose 𝕏 = [0,1]. define two self-mappings 𝔄 and 𝔖 as follows 𝔄(𝓍) = 5 𝓍+5 𝔖(𝓍) = 1 𝓍+1 for every 𝓍 in [0,1]. consider a sequence 〈𝑥𝑛 〉 = 1 𝓃 𝓃 ∈ ℕ.then lim 𝓃→∞ 𝔄𝑥𝑛 = lim𝔖𝑥𝑛 𝓃→∞ = 1. consequently, lim 𝓃→∞ 𝑀(𝔄𝔖𝑥𝑛, 𝔄𝑥𝑛, 𝓉) = 6𝓉 6𝓉+1 and lim 𝓃→∞ 𝑀(𝔖𝔖𝑥𝑛, 𝔖𝑥𝑛 , 𝓉) = 2𝓉 2𝓉+1 . hence lim 𝑛→∞ 𝑀𝐾𝑀 (𝔄𝔖𝑥𝑛, 𝔄𝑥𝑛 , 𝓉) lim 𝑛→∞ 𝑀𝐾𝑀 (𝔖𝔖𝑥𝑛, 𝔖𝑥𝑛 , 𝓉), for all 𝓉 > 0. thus, the pair (𝔄, 𝔖) is 𝒜-intimate. on the other hand, the (𝔄, 𝔖) are not compatible of type (𝒜),since lim 𝓃→∞ m𝐾𝑀(𝔄𝔖𝓍𝓃, 𝔖𝔖𝑥𝓃 , 𝓉) = 3𝓉 3𝓉+1 ≠ 1 and lim n→∞ m𝐾𝑀(𝔖𝔄𝓍𝓃, 𝔄𝔄𝑥𝓃 , 𝓉) = 3𝓉 3𝓉+1 ≠ 1. definition 2.14[20]: define 𝔄 and 𝔖 as two self maps of fms (𝕏, 𝑀𝐾𝑀 ,∗) then we say that 𝔄 and 𝔖 satisfy the property e.a if there exists a sequence 〈𝑥𝓃 〉 ∈ 𝕏 such that lim 𝓃→∞ 𝔄𝓍𝓃 = lim 𝓃→∞ 𝔖𝓍𝓃 = 𝓉 for some 𝓉 ∈ 𝕏. definition 2.15[21]: suppose 𝔄 , 𝔓 , 𝔅 and 𝔗 are four self maps on fms (𝕏, mkm,∗) then we say that (𝔄, 𝔓)and (𝔅, 𝔗) satisfy common property e.a whenever two sequences 〈x𝓃 〉 and 〈γ𝓃 〉 in 𝕏 satisfying lim 𝓃→∞ 𝔄 𝓍𝓃 = lim 𝓃→∞ 𝔖𝓍𝓃 = lim 𝓃→∞ 𝔅 γ𝓃 = lim 𝓃→∞ 𝔗γ𝓃 = 𝓉 for some 𝓉 ∈ 𝕏. 176 some fixed point results in fuzzy metric space using intimate mappings 3. main results 3.1 theorem: let (𝕏, mkm,∗) be a complete fuzzy metric space. suppose 𝔓,𝔔, 𝔖 and 𝔄 are self maps on 𝕏 satisfying the conditions (𝒞 − 1) 𝔓(𝕏)  𝔖(𝕏) and 𝔔(𝕏)  𝔄(𝕏) (𝒞 − 2) m𝐾𝑀(𝔓𝓍, 𝔔γ, k𝓉) ≥ m𝐾𝑀(𝔄𝓍, 𝔖𝛾, 𝓉) ∗ m𝐾𝑀(𝔓𝓍, 𝔄𝓍, 𝓉) ∗ m𝐾𝑀(𝔔γ, 𝔖𝛾, 𝓉) ∗ m𝐾𝑀(𝔓𝓍, 𝔖𝛾, 𝓉) where 𝑘 ∈ (0,1) and for all 𝓍, γ ∈ 𝕏 (𝒞 − 3) 𝔄( 𝕏 ) is complete (𝒞 − 4) the pair of mappings 𝔄 and 𝔓 𝑖𝑠 𝒜 − intimate and the other pair of mappings also 𝔖 and 𝔔 is 𝒮 − intimate. then 𝔓, 𝔔,𝔖 and 𝔄 have a unique common fixed point in 𝕏. proof: let𝓍0 be any arbitrary point of 𝕏. since from the condition 𝔓(𝕏)  𝔖(𝕏) of (𝒞 − 1) , there exists a point 𝓍1∈𝕏 such that 𝔓𝓍0=𝔖𝓍1=𝛾0. now for this 𝓍1 and applying the (𝒞 − 1)[i.e 𝔔(𝕏)  𝔄(𝕏)] ∃𝓍2𝕏 such that 𝔔𝓍1=𝔄𝓍2=𝛾1. inductively, we establish two real sequences < 𝓍𝓃 > and < γ𝓃 > in 𝕏 ∋ 𝛾2𝑛=𝔓𝓍2𝑛= 𝔖𝓍2𝓃+1 and 𝛾2𝑛+1 = 𝔔𝓍2𝓃+1 = 𝔄𝑥2𝓃+2 for 𝓃 0. by taking 𝓍 = 𝓍2𝓃 , 𝛾 = 𝓍2𝓃+1in the inequality (𝒞 − 2), m𝐾𝑀(𝔓𝓍2n, 𝔔𝓍2𝓃+1, k𝓉) ≥ m𝐾𝑀(𝔄𝓍2n, 𝔖𝓍2𝓃+1, 𝓉) ∗ m𝐾𝑀(𝔓𝓍2n, 𝔄𝓍2𝓃 , 𝓉) ∗ m𝐾𝑀(𝔔𝓍2𝓃+1, 𝔖𝓍2𝓃+1, 𝓉) ∗ m𝐾𝑀(𝔓𝓍2n, 𝔖𝓍2𝓃+1, 𝓉) which implies that an 𝓃→∞ m𝐾𝑀(𝛾2𝑛, 𝛾2𝑛+1, k𝓉) ≥ m𝐾𝑀 (𝛾2𝑛−1, 𝛾2𝑛, 𝓉) ∗ m𝐾𝑀(𝛾2𝑛, 𝛾2𝑛−1, 𝓉) ∗ m𝐾𝑀(𝛾2𝑛+1, 𝛾2𝑛, 𝓉) ∗ m𝐾𝑀 (𝛾2𝑛, 𝛾2𝑛, 𝓉). this yield m𝐾𝑀(𝛾2𝑛, 𝛾2𝑛+1, k𝓉) ≥ m𝐾𝑀(𝛾2𝑛−1, 𝛾2𝑛, 𝓉) ∗ m𝐾𝑀(𝛾2𝑛+1, 𝛾2𝑛, 𝓉) ∗ m𝐾𝑀(𝛾2𝑛, 𝛾2𝑛−1, 𝓉) ∗ 1. again, by the condition kmfm-3, we get m𝐾𝑀(𝛾2𝑛, 𝛾2𝑛+1, k𝓉) ≥ m𝐾𝑀(𝛾2𝑛−1, 𝛾2𝑛, 𝓉) ∗ m𝐾𝑀(𝛾2𝑛, 𝛾2𝑛+1, 𝓉) which implies (since 𝔞 ∗ 𝔟 = min{𝔞, 𝔟}.) m𝐾𝑀(𝛾2𝑛, 𝛾2𝑛+1, k𝓉) ≥ m𝐾𝑀(𝛾2𝑛−1, 𝛾2𝑛, 𝓉). in general mkm(𝛾𝓃+1, 𝛾𝓃+2, 𝑘𝓉) ≥ 𝑀𝐾𝑀 (𝛾𝓃 , 𝛾𝓃+1, 𝓉) … . . (𝜎 − 1) for all 𝓃 = 1,2,3. . , and 𝓉 > 0. from (𝜎 − 1), 177 vijayabaskerreddy bonuga and srinivas veladi [m𝐾𝑀(γ𝓃, γ𝓃+1, 𝓉)] ≥ m𝐾𝑀 (γ𝓃−1, γ𝓃 , 𝓉 k ) ≥ m𝐾𝑀 (γ𝓃−2, γ𝓃−1, 𝓉 k2 ) ≥ ⋯ … ≥ m𝐾𝑀 (γ0, γ1, 𝓉 k 𝓃 ) → 1 as 𝓃 → ∞. . . . . . . (𝜎 − 2) for any 𝓉 > 0 and 𝜆𝑀𝐾 ∈ (0,1) we consider ∀ 𝓃 > 𝓃0 ∈ ℕ such that m𝐾𝑀(γ𝓃, γ𝓃+1, 𝓉) > (1 − mk) … (𝜎 − 3). for 𝓂, 𝓃 ∈ ℕ . suppose 𝓂 ≥ 𝓃, then we have that [mmk(γ𝓃 , γ𝓂 , 𝓉)] ≥ min {mmk (γ𝓃 , γ𝓃+1, 𝓉 𝓂 − 𝓃 ) ∗ mmk (γ𝓃+1, γ𝓃+2, 𝓉 𝓂 − 𝓃 ) ∗. . .. mmk (γ𝓂−1, γ𝓂 , 𝓉 𝓂 − 𝓃 ) ≥ (1 − mk) ∗ (1 − mk) ∗. . . (1 − mk). . (𝓂 − 𝓃) times. this implies mmk(γ𝓂−1, γ𝓂 , 𝓉) ≥ (1 − mk) therefore < γ𝓃 > is cauchy sequence in fms. since (𝕏, 𝑀𝐾𝑀,∗) is complete fms, so sequence {γ𝓃 } converges to p*x. further fuzzy cauchy sequence {γ𝓃 } has convergent subsequence {γ2𝓃+1} and {γ2𝓃 }. from the above argument, γ2𝓃+1 = 𝔔𝓍2𝓃+1 = 𝔄𝓍2𝓃+2→p ∗ and γ2𝓃 = 𝔓𝓍2𝓃 = 𝔖𝓍2𝓃+1→p ∗ as 𝓃→∞ … (𝜎 − 4) now suppose that the range set 𝔄(x) is complete then  a point u𝕏 ∋ 𝔄u=p*..(𝜎 − 5). now we claim that 𝔓u=p* from the inequality, put 𝓍 = 𝓊 and γ = 𝓍2𝓃+1 we have m𝐾𝑀(𝔓u, 𝔔𝓃2𝓃+1, k𝓉) ≥ m𝐾𝑀(𝔄u, 𝔖𝓍2𝓃+1, 𝓉) ∗ m𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ m𝐾𝑀(𝔔𝓍2𝓃+1, 𝔖𝓍2𝓃+1, 𝓉) ∗ m𝐾𝑀(𝔓u, 𝔖s𝓍2𝓃+1, 𝓉). taking limit as 𝓃→∞ m𝐾𝑀(𝔓u, p ∗, k𝓉) ≥ m𝐾𝑀(p ∗, p ∗, 𝓉) ∗ m𝐾𝑀(𝔓u, p ∗, 𝓉) ∗ m𝐾𝑀(p ∗, p ∗, 𝓉) ∗ m𝐾𝑀(𝔓u, p ∗, 𝓉). this gives 𝔓u=p*. that is 𝔓u=𝔄u=p*…... (𝜎 − 6) let us prove that qv=p*. using the equation ((𝜎 − 6) with contained inequality 𝔓 (𝕏) ⊆ 𝔖 (𝕏), p*=𝔓u  𝔓(𝕏)  𝔖(𝕏) then ∃ a point v𝕏 ∋ 𝔖v=𝔓u=p*…. (𝜎 − 7). put 𝓍=u and 𝛾 = 𝑣 in (𝒞 − 2) then we obtain m𝐾𝑀(𝔓u, 𝔔v, k𝓉) ≥ m𝐾𝑀 (au, 𝔖v, 𝓉) ∗ m𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ m𝐾𝑀(𝔔v, 𝔖v, 𝓉) ∗ m𝐾𝑀 (𝔓u, 𝔖v, 𝓉). by using(𝜎 − 7) we get m𝐾𝑀(p ∗, 𝔔v, k𝓉) ≥ m𝐾𝑀(p ∗, 𝔖v, 𝓉) ∗ m𝐾𝑀(p ∗, p ∗, 𝓉) ∗ m𝐾𝑀(𝔔v, p ∗, 𝓉) ∗ m𝐾𝑀(p ∗, p ∗, 𝓉) this gives m𝐾𝑀(p ∗, 𝔔v, 𝑘𝓉) ≥ m𝐾𝑀(𝔔v, p ∗, k𝓉). consequently m𝐾𝑀(p ∗, 𝔔v, k𝓉) ≥ m𝐾𝑀(p ∗, 𝔔v, k𝓉) 178 some fixed point results in fuzzy metric space using intimate mappings this implies 𝔔v=p*. this shows that 𝔔v=𝔖v=p*…... (𝜎 − 8) since 𝔓u=𝔄u=p* and (𝔄,𝔓) is 𝒜-intimate we have m𝐾𝑀(𝔄p*, p*,𝓉) ≥ m𝐾𝑀 (𝔓p*, p*, 𝓉)…. (𝜎 − 9). suppose that 𝔓p*≠ p*. put 𝓍 = p ∗, γ = v in (𝒞 − 2) then we get, m𝐾𝑀(𝔓p ∗, 𝔔v, k𝓉) ≥ m𝐾𝑀(𝔄p ∗, 𝔖v, 𝓉) ∗ m𝐾𝑀 (𝔓p ∗, 𝔄p ∗, 𝓉) ∗ m𝐾𝑀(𝔔v, 𝔖v, 𝓉) ∗ m𝐾𝑀(𝔓p ∗, 𝔖v, 𝓉). using (𝜎 − 8) we get, mkm(𝔓p ∗, p ∗, k𝓉) ≥ mkm(𝔄p ∗, p ∗, 𝓉) ∗ mkm(𝔓p ∗, 𝔄p ∗, 𝓉) ∗ mkm(p ∗, p ∗, 𝓉) ∗ mkm(𝔓p ∗, p ∗, 𝓉). by applying (kmfm-iv) we get m𝐾𝑀(𝔓p ∗, p ∗, k𝓉) ≥ m𝐾𝑀 (𝔓p ∗, p ∗, 𝓉) ∗ m𝐾𝑀(𝔓p ∗, p ∗, 𝓉/2) ∗ m𝐾𝑀(p ∗, 𝔄p ∗, 𝓉/2) ∗ m𝐾𝑀(p ∗, p ∗, 𝓉) ∗ m𝐾𝑀 (𝔓p ∗, p ∗, 𝓉). by using (𝜎 − 9) we get m𝐾𝑀(𝔓p ∗, p ∗, k𝓉) ≥ m𝐾𝑀(𝔓p ∗, p ∗, 𝓉/2). this gives 𝔓p*=p* …...(𝜎 − 10). from (𝜎 − 9) and (𝜎 − 10) we write m𝐾𝑀(𝔄p*, p*,𝓉) ≥ 1 this gives 𝔄p*=p*……(𝜎 − 11) using (𝜎 − 10) and (𝜎 − 11) we get 𝔄p*=𝔓p*=p*…... (𝜎 − 12) also, 𝔔v=𝔖v=p* and using the pair (𝔖, 𝔔) as 𝒮-intimate then we have m𝐾𝑀(𝔖p ∗, p ∗, 𝓉) ≥ m𝐾𝑀(𝔔p ∗, p ∗, k𝓉)…. (𝜎 − 13) suppose that 𝔔p*≠p*. put 𝑥 = u and γ = 𝑝 ∗ in the inequality m𝐾𝑀(𝔓u, 𝔔p ∗, k𝓉) ≥ m𝐾𝑀 (𝔄u, 𝔖p ∗, 𝓉) ∗ m𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ m𝐾𝑀(𝔔p ∗, 𝔖p ∗, 𝓉) ∗ m𝐾𝑀 (𝔓u, 𝔖p ∗, 𝓉) using (𝜎 − 6) and (kmfm-iv) we get, m𝐾𝑀(p ∗, 𝔔p ∗, k𝓉) ≥ m𝐾𝑀(p ∗, 𝔖p ∗, 𝓉) ∗ m𝐾𝑀 (p ∗, p ∗, 𝓉) ∗ m𝐾𝑀 (𝔓p ∗, p ∗, 𝓉 2 ) ∗ m𝐾𝑀 (p ∗, 𝔖p ∗, 𝓉 2 ) ∗ m𝐾𝑀(p ∗, 𝔖p ∗, 𝓉) on using (𝜎 − 13) we get m𝐾𝑀(p ∗, 𝔔p ∗, k𝓉) ≥ m𝐾𝑀(p ∗, 𝔔p ∗, 𝓉) ∗ m𝐾𝑀 (𝔔p ∗, p ∗, 𝓉 2 ) ∗ m𝐾𝑀(𝔔p ∗, p ∗, 𝓉/2) ∗ m𝐾𝑀(p ∗, 𝔔p ∗, 𝓉). this implies m𝐾𝑀(p ∗, 𝔔p ∗, k𝓉) ≥ m𝐾𝑀(p ∗, 𝔔p ∗, 𝓉/2). 179 vijayabaskerreddy bonuga and srinivas veladi this gives 𝔔p*=p*…(𝜎 − 14). from (𝜎 − 13) and (𝜎 − 14) we get m𝐾𝑀(𝔖p*, p*,𝓉)≥1 𝔖p*=p*…...(𝜎 − 15) . using (𝜎 − 14) and (𝜎 − 15) we get 𝔔p*=𝔖p*=p*. …. (𝜎 − 16). using (𝜎 − 12) and (𝜎 − 16) we conclude that 𝔄p*=𝔓p*=𝔔p*=𝔖p*=p*. hence the result. we can prove the uniqueness of the fixed point easily. example 3.1.1: suppose (𝕏, mkm, *) is a standard fms with 𝒶 ∗ 𝒶𝒶 ∀𝒶 ∈ [0,1], where 𝔄, 𝔖, 𝔓 and 𝔔:𝕏→𝕏 as 𝔓(𝑥) = 𝔔(x) = { 𝑥 + 0.125 if 0 ≤ 𝑥 < 0.125 0.25 if 0.125 ≤ 𝑥 ≤ 1 𝔄(𝑥) = 𝔖(x) = { 2𝑥 if 0 ≤ 𝑥 < 0.125 0.25 if 0.125 ≤ 𝑥 ≤ 1 𝔓(𝕏) = 𝔔(𝕏) = [0.125,0.25] and 𝔄(𝕏) = 𝔖(𝕏) = [0,0.25] these sets satisfy the condition (𝒞 − 1). now assume 〈𝓍𝓃 〉 = {0.125 + 1 𝓃 } then lim n→∞ 𝔄𝑥𝓃 = lim 𝓃→∞ 𝔓𝑥𝓃 = 0.25. also we have, lim 𝓃→∞ 𝔄𝔓𝓍𝓃 = lim 𝓃→∞ 𝔄𝔓(0.125 + 1 𝓃 ) = lim 𝓃→∞ 𝔄(0.25) = 0.125. lim 𝓃→∞ mkm(𝔄𝔓𝓍𝓃 , 𝔄𝓍𝓃 , 𝓉) lim n→∞ mkm(𝔓𝔓𝓍𝓃 , 𝔓𝓍𝓃 , 𝓉), for 𝓉 > 0. thus, the pair (𝔄,𝔓) is 𝒜-intimate. further lim 𝓃→∞ mkm(𝔖𝔔𝓍𝓃, 𝔖𝓍𝓃 , 𝓉) lim 𝓃→∞ mkm(𝔔𝔔𝓍𝓃 , 𝔔𝓍𝓃 , 𝓉). thus, the pair (𝔖,𝔔) is 𝒮-intimate. moreover, it satisfies the contraction condition of the theorem. clearly 0.25 is the unique common fixed point for these four mappings. theorem.3.2: let (𝕏, mkm,∗)be a fuzzy metric space. suppose 𝔓,𝔔, 𝔖 and 𝔄 are self maps on 𝕏 satisfies the conditions (𝒞 − 1), (𝒞 − 2), (𝒞 − 3) and (𝒞 − 4) with (𝒞 − 5):(𝔓, 𝔄) or (𝔔, 𝔖) satisfy e.a property then 𝔓, 𝔔,𝔖 and 𝔄 have a unique common fixed point in 𝕏. proof: suppose the pair (𝔔, 𝔖) satisfies e.a property then ∃ sequence 〈𝑥𝑛 〉 in 𝕏 such that lim 𝓃⟶∞ 𝔔𝑥𝑛 = lim 𝓃⟶∞ 𝔖𝑥𝑛 = 𝑝 ∗ for some 𝑝 ∗∈ 𝕏. since 𝔔(𝕏) ⊆ 𝔄(𝕏) then ∃ 〈𝑥𝑛〉 in 𝕏 such that 𝔔𝑥𝓃 = 𝔄𝑦𝓃 . hence lim 𝓃⟶∞ 𝔄𝛾𝓃 = p ∗. ….(𝜑 − 1). now we show that lim 𝓃⟶∞ 𝔓𝛾𝓃 = 𝑝 ∗. put 𝑥 = 𝛾𝑛 and 𝛾 = 𝑥𝑛 we obtain, 180 some fixed point results in fuzzy metric space using intimate mappings m𝐾𝑀(𝔓𝛾𝓃, 𝔔𝑥𝑛 , k𝓉) ≥ m𝐾𝑀 (𝔄𝛾𝓃 , 𝔖𝑥𝑛 , 𝓉) ∗ m𝐾𝑀(𝔓𝛾𝓃 , 𝔄𝛾𝓃 , 𝓉) ∗ m𝐾𝑀(𝔔𝑥𝑛, 𝔖𝑥𝑛 , 𝓉) ∗ m𝐾𝑀 (𝔓𝛾𝓃 , 𝔖𝑥𝑛 , 𝓉). letting 𝓃 ⟶ ∞ and using 𝑝𝛾𝓃 ⟶ 𝑝 ∗ we get lim 𝑛→∞ 𝔔𝑥𝑛 = lim 𝑛→∞ 𝔖𝑥𝑛 = lim 𝑛→∞ 𝔄𝛾𝑛 = lim 𝑛→∞ 𝔓𝛾𝑛 = 𝑝 ∗. suppose that 𝔄(𝕏) is closed subspace of 𝕏, ∃ 𝑢 ∈ 𝕏 such that p*=𝔄𝑢. . . (𝜑 − 2). we show that 𝔄𝑢 = 𝔓𝑢. put 𝑥 = 𝑢 and 𝛾 = 𝑥𝑛 in (𝒞 − 2) then we get m𝐾𝑀(𝔓u, 𝔔𝑥𝑛 , k𝓉) ≥ m𝐾𝑀(𝔄u, 𝔖𝑥𝑛 , 𝓉) ∗ m𝐾𝑀 (𝔓u, 𝔄u, 𝓉) ∗ m𝐾𝑀(𝔔𝑥𝑛, 𝔖𝑥𝑛 , 𝓉) ∗ m𝐾𝑀(𝔓u, 𝔖𝑥𝑛, 𝓉). this implies 𝔓𝑢 = 𝑝 ∗…(𝜑 − 3). from (𝜑 − 2) and (𝜑 − 2) we get 𝔄𝑢 = 𝔓𝑢 = 𝑝 ∗ ….(𝜑 − 4). and since (𝔄, 𝔓) is 𝒜 − 𝑖ntimate then we get 𝔄𝑝 ∗= 𝔓𝑝 ∗= 𝑝 ∗. . . . . . . (𝜑 − 5). since 𝔓(𝕏) ⊆ 𝔖(𝕏) then there exists a point 𝑣 ∈ 𝕏 such that 𝔓𝑢 = 𝔖𝑣 = 𝑝 ∗. . . . . . (𝜑 − 6). now put 𝑥 = 𝑢 and 𝛾 = 𝑣 in (𝒞 − 2) then this gives m𝐾𝑀(𝔓u, 𝔔𝑣, k𝓉) ≥ m𝐾𝑀(𝔄u, 𝔖𝑣, 𝓉) ∗ m𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ m𝐾𝑀(𝔔𝑣, 𝔖𝑣, 𝓉) ∗ m𝐾𝑀 (𝔓u, 𝔖𝑣, 𝓉) implies m𝐾𝑀(𝑝 ∗, 𝔔𝑣, k𝓉) ≥ m𝐾𝑀 (p ∗, 𝑝 ∗, 𝓉) ∗ m𝐾𝑀(𝑝 ∗, p ∗, 𝓉) ∗ m𝐾𝑀(𝔔𝑣, p ∗, 𝓉) ∗ m𝐾𝑀 (𝑝 ∗, 𝑝 ∗, 𝓉). this implies 𝔔𝑣 = 𝑝 ∗ therefore 𝔖𝑣 = 𝔔𝑣 = 𝑝 ∗. . . . . . . (𝜑 − 7), and since (𝔖, 𝔔) is 𝒮 − intimate then we get 𝔖𝑝 ∗= 𝔔𝑝 ∗= 𝑝 ∗ …..(𝜑 − 8). using (𝜑 − 7) (𝜑 − 8) and we conclude that 𝔄𝑝 ∗= 𝔓𝑝 ∗= 𝔔𝑝 ∗= 𝔖𝑝 ∗= 𝑝 ∗. we can prove the uniqueness of the common fixed point easily. example 3.2.1: suppose (𝕏, mkm, *) is a standard fms with 𝒶 ∗ 𝒶𝒶 ∀𝒶 ∈ [1,11), where 𝔄, 𝔖, 𝔓 and 𝔔:𝕏→𝕏 as 𝔓(𝓍) = 𝔔(𝓍) = { 1 if x ∈ {1} ∪ (3,11) 1 + 𝓍 if 1 < 𝓍 ≤ 3 𝔖(𝓍) = { 1 if 𝓍 = 1 6 if 1 < 𝓍 ≤ 3 𝓍 − 2 if 3 < 𝓍 < 11 𝔄(x) = { 1 if 𝓍 = 1 4 if 1 < 𝓍 ≤ 3 3𝓍−1 8 if 3 < 𝓍 < 11 181 vijayabaskerreddy bonuga and srinivas veladi 𝔓(𝕏) = 𝔔(𝕏) = {1} ∪ (2,4] and 𝔖(𝕏) = {1} ∪ {6} ∪ (1,9) 𝔄(𝕏) = {1} ∪ {4} ∪ (1,4) = [1,4] these sets satisfy the conditions (𝒞 − 2)and (𝒞 − 3). now assume 〈𝓍𝓃 〉 = {3 + 1 𝓃 } then lim n→∞ 𝔄𝓍𝓃 = lim 𝓃→∞ 𝔓𝓍𝓃 = 1 and this implies (𝔓, 𝔄) satisfies e.a property and also we have, lim n→∞ 𝔄𝔓𝓍𝓃 = lim 𝓃→∞ 𝔓𝔓𝓍𝓃 = 1. this gives lim 𝓃→∞ mkm(𝔄𝔓𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) lim 𝓃→∞ mkm(𝔓𝔓𝓍𝓃, 𝔓𝓍𝓃 , 𝓉) for 𝓉 > 0. thus, the pair (𝔄,𝔓) is 𝒜-intimate. since lim n→∞ 𝔖𝓍𝓃 = lim 𝓃→∞ 𝔔𝓍𝓃 = 1 and lim n→∞ 𝔖𝔔𝓍𝓃 = lim 𝓃→∞ 𝔔𝔔𝓍𝓃 = 1 this gives lim 𝓃→∞ mkm(𝔖𝔔𝓍𝓃 , 𝔖𝓍n, 𝓉) lim 𝓃→∞ mkm(𝔔𝔔𝓍𝓃 𝔔𝓍𝓃 , 𝓉). thus, the pair (𝔖,𝔔) is 𝒮-intimate. moreover, it satisfies the contraction condition of the theorem. clearly 1 is the unique common fixed point for these four mappings. finally, we discuss another theorem. 3.3 theorem: let(𝕏, mkm,∗) be a fms. suppose 𝔓,𝔔, 𝔖 and 𝔄 are self maps on 𝕏 satisfying the conditions (𝒞 − 2) and (𝒞 − 4) in addition to (𝒞 − 6) 𝔄(𝕏) and 𝔖(𝕏) are closed subsets of 𝕏 (𝒞 − 7)the pairs (𝔓, 𝔄) and (𝔔, 𝔖) share the common property e. a. then 𝔓, 𝔔,𝔖 and 𝔄 have a unique common fixed point in 𝕏. proof: in view of the condition (𝒞 − 7)there exists two sequences 〈𝑥𝓃 〉 and 〈γn〉 in 𝕏 such that lim 𝓃⟶∞ 𝔓𝑥𝓃 = lim 𝓃⟶∞ 𝔄𝑥𝓃 = lim 𝓃⟶∞ 𝔔𝛾𝓃 = lim 𝓃⟶∞ 𝔖𝛾𝓃 = 𝑝 ∗ for some 𝑝 ∗∈ 𝕏. from the (𝒞 − 6) we have𝔄(𝕏) is closed subset of 𝕏,consequently lim 𝓃⟶∞ 𝔓𝑥𝓃 = 𝑝 ∗∈ 𝔄(𝕏). this means there exists appoint 𝑢 ∈ 𝕏 such that 𝔄𝑢 = 𝑝 ∗. now we assert that 𝔓𝑢 = 𝔄𝑢. put 𝑥 = 𝑢 and 𝛾 = 𝛾𝑛, we get m𝐾𝑀(𝔓u, 𝔔𝑦𝑛 , k𝓉) ≥ m𝐾𝑀 (𝔄u, 𝔖𝑦𝑛, 𝓉) ∗ m𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ m𝐾𝑀(𝔔𝑦𝑛, 𝔖𝑦𝑛, 𝓉) ∗ m𝐾𝑀 (𝔓u, 𝔖𝑦𝑛, 𝓉) which on making 𝓃 → ∞, with 𝔄𝑢 = 𝑝 ∗ reduces to 𝔓𝑢 = 𝑝 ∗. this implies 𝔓𝑢 = 𝔄𝑢 = 𝑝 ∗ which signifies that 𝑢 is coincident point of the pair (𝔓, 𝔄). on the other hand, 𝔖(𝕏) is closed subset of 𝕏 therefore lim 𝓃⟶∞ 𝔖𝛾𝓃 = 𝑝 ∗ ∈ 𝔖(𝕏) and hence we can find a point 𝑤 ∈ 𝕏 ∋ 𝔖𝑤 = 𝑝 ∗. now we show that 𝔖𝑤 = 𝔔𝑤. on using condition (𝒞 − 2) with 𝑥 = u and γ = w then we get mkm(𝔓u, 𝔔w, k𝓉) ≥ mkm(𝔄u, 𝔖w, 𝓉) ∗ mkm(𝔓u, 𝔄u, 𝓉) ∗ mkm(𝔔w, 𝔖w, 𝓉) ∗ mkm(𝔓u, 𝔖w, 𝓉). this implies 𝔔w = p ∗.this gives 𝔖w = 𝔔w = p ∗. since the pair (𝔔, 𝔖) is 𝒮 − intimate this gives mkm(𝔖p ∗, p ∗, 𝓉) ≥ mkm(𝔔p ∗, p ∗, 𝓉). 182 some fixed point results in fuzzy metric space using intimate mappings suppose that 𝔖𝑝 ∗≠ 𝑝 ∗. put 𝑥 = u and γ = p ∗ in contraction condition (𝒞 − 2) m𝐾𝑀(𝔓u, 𝔔𝑝 ∗, k𝓉) ≥ m𝐾𝑀 (𝔄u, 𝔖𝑝 ∗, 𝓉) ∗ m𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ m𝐾𝑀(𝔔𝑝 ∗, 𝔖𝑝 ∗, 𝓉) ∗ m𝐾𝑀 (𝔓u, 𝔖𝑝 ∗, 𝓉) implies 𝔔p ∗= p ∗. using mkm(𝔖p ∗, p ∗, t) ≥ mkm(p ∗, p ∗, 𝓉) we get 𝔖p ∗= p ∗. therefore 𝔔p ∗= 𝔖p ∗= p ∗…. . . . . . (ψ − 1). since 𝔓u = 𝔄u = p ∗ and using (𝔓, 𝔄) is 𝒜 −intimate then we get 𝔄p ∗= p ∗. by putting 𝑥 = γ = p ∗ we get mkm(𝔓p ∗, 𝔔p ∗, k𝓉) ≥ mkm(𝔄p ∗, 𝔖p ∗, 𝓉) ∗ mkm(𝔓p ∗, 𝔄p ∗, 𝓉) ∗ mkm(𝔔p ∗, 𝔖p ∗, 𝓉) ∗ mkm(𝔓p ∗, 𝔖p ∗, 𝓉). this implies 𝔓p ∗= p ∗ and this gives 𝔄p ∗= 𝔓p ∗= p ∗. . . . . . (ψ − 2). from (ψ − 1) and (ψ − 2) we conclude that 𝔄p ∗= 𝔓p ∗= 𝔔p ∗= 𝔖p ∗= p ∗. we can prove the uniqueness of the fixed point easily. example 3.3.1: suppose (𝕏, mkm, *) is a standard fms with 𝒶 ∗ 𝒶  𝒶 ∀𝒶 ∈ [1,20], where 𝔄, 𝔖, 𝔓 and 𝔔:𝕏→𝕏 as 𝔓(𝑥) = 𝔔(𝑥) = { 1 if 𝑥 = 1, 2 ≤ 𝑥 < 20 𝑥 if 1 ≤ 𝑥 < 2 𝔖(𝑥) = { 1 if 𝑥 = 1 12 if 1 < 𝑥 < 2 𝑥 + 1 3 if 2 ≤ 𝑥 ≤ 20 𝔄(x) = { 1 if 𝑥 = 1 7 if 1 < 𝑥 < 2 2𝑥 + 5 9 if 2 ≤ 𝑥 ≤ 20 𝔓(𝕏) = 𝔔(𝕏) = {1} ∪ (1,2) , 𝔖(𝕏) = {1} ∪ {12} ∪ [1,5] and 𝔄(𝕏) = {1} ∪ {7} ∪ [1,9] these sets satisfy the conditions (𝒞 − 1) and (𝒞 − 3). now assume 〈𝓍𝓃 〉 = {2 + 1 𝓃 } and 〈γ𝓃 〉 = {1} then lim 𝓃→∞ 𝔄𝓍𝓃 = lim 𝓃→∞ 𝔓𝓍𝓃 = lim 𝓃→∞ 𝔖γ𝓃 = lim 𝓃→∞ 𝔔γ𝓃 = 1. this implies the pairs (𝔓, 𝔄) and (𝔖,𝔔) share the common e. a property and also we have, lim 𝓃→∞ 𝔄𝔓𝓍𝓃 = lim 𝓃→∞ 𝔓𝔓𝓍𝓃 = 1 this gives lim 𝓃→∞ mkm(𝔄𝔓𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) lim 𝓃→∞ mkm(𝔓𝔓𝓍𝓃, 𝔓𝓍𝓃 , 𝓉), for 𝓉 > 0. thus, the pair (𝔄,𝔓) is 𝒜-intimate. since lim 𝓃→∞ 𝔖𝓍𝓃 = lim 𝓃→∞ 𝔔𝓍𝓃 = 1 and lim 𝓃→∞ 𝔖𝔔𝓍𝓃 = lim 𝓃→∞ 𝔔𝔔𝓍𝓃 = 1 this gives lim 𝓃→∞ mkm(𝔖𝔔𝓍𝓃 , 𝔖𝓍𝓃 , 𝓉) lim 𝓃→∞ mkm(𝔔𝔔𝓍𝓃 , 𝔔𝓍𝓃 , 𝓉). thus, the pair (𝔖,𝔔) is 𝒮-intimate. moreover, it satisfies the contraction condition of the theorem. clearly 1 is the unique common fixed point for these four mappings. 183 vijayabaskerreddy bonuga and srinivas veladi 4 conclusion this paper aimed to prove three common fixed point theorems to generalize the class of compatible mappings by using the calss of non compatible mappings like different forms of e.a properties along with intimate mappings in fuzzy metric space. in theorem 3.1, one of the range of mappings is assumed to be complete.further, in theorem 3.2, one of the pairs is assumed to satisfy e.a property along with one of the range of mappings is complete without being complete fuzzy metric space. finally in theorem 3.3, improved version of ea property namely common ea property is assumed along with completeness of fuzzy metric space. moreover, all these results are justified with suitable examples. references [1] l. a. zadeh. fuzzy sets. information and control, 8(3), 338-353.1965. 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[21] yicheng liu, jun wu . zhixiang li. common fixed points of single -valued and multivalued maps. international journal of mathematics and mathematical sciences,19, 3045-3055.2005. 185 ratio mathematica volume 47, 2023 relatively prime inverse domination on line graph c. jayasekaran* l. roshini† abstract let g be non-trivial graph. a subset d of the vertex set v (g) of a graph g is called a dominating set of g if every vertex in v − d is adjacent to a vertex in d. the minimum cardinality of a dominating set is called the domination number and is denoted by γ(g). if v −d contains a dominating set s of g, then s is called an inverse dominating set with respect to d. in an inverse dominating set s, every pair of vertices u and v in s such that (deg(u), deg(v)) = 1, then s is called relatively prime inverse dominating set. the minimum cardinality of a relatively prime inverse dominating set is called relatively prime inverse dominating number and is denoted by γ−1rp (g). in this paper we find relatively prime inverse dominating number of some jump graphs. keywords: domination number, inverse domination number, relatively prime domination number. 2020 ams subject classifications: 05c69,05c76 1 *associate professor, department of mathematics, pioneer kumaraswamy college, nagercoil 629003, tamil nadu, india; jayacpkc@gmail.com. †research scholar, department of mathematics, pioneer kumaraswamy college, nagercoil 629003, tamil nadu, india. affliated to manonmaniam sundaranar university, abishekapatti, tirunelveli 627012,tamil nadu, india.; jerryroshini92@gmail.com. 1received on november 10, 2022. accepted on march 24, 2023. published on april 4, 2023. doi: 10.23755/rm.v41i0.954. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 332 c. jayasekaran, l. roshini 1 introduction by a graph, we mean a finite undirected graph with neither loops nor multiple edges. for graph theoretic terminology, we refer to the book by chartrand and lesniak [1]. all graphs in this paper are assumed to be non-trivial. in a graph g = (v, e), the degree of a vertex v is defined to be the number of edges incident with v and is denoted by deg(v). a set d of vertices of graph g is said to be a dominating set if every vertex in v −d is adjacent to a vertex in d. a dominating set d is said to be a minimal dominating set if no proper subset of d is a dominating set. the minimum cardinality of a dominating set of a graph g is called the domination number of g and is denoted by γ(g). kulli v. r. et al. introduced the concept of inverse domination in graphs [8]. let d be a minimum dominating set of g. if v − d contains a dominating set s, then s is called a inverse domination set of g with respect to d. the inverse domination number γ−1(s) is the minimum cardinality taken over all the minimal inverse dominating set of g. the jewel graph jn is a graph with vertex set v (jn) = {u, x, v, y, vi : 1 ≤ i ≤ n} and edge set e(jn) = {ux, vx, uy, vy, xy, uvi, vvi : 1 ≤ i ≤ n}[7]. bistar bm,n is the graph obtained by joining the center vertices of star graphs k1,m and k1,n by an edge. the vertex set of bm,n is {u, v, ui, vj : 1 ≤ i ≤ m, 1 ≤ j ≤ n} where u, v are apex vertices and ui, vi are pendent vertices. the edge set of bm.n is {uv, uui, vvj : 1 ≤ i ≤ n, 1 ≤ j ≤ m} and |v (bm,n)| = m+n+2, |e(bm,n)| = m + n + 1[2]. a spider graph is a tree with at most one vertex of degree greater than 2[2]. let pn be a path graph with n vertices. the comb graph is defined as pn ⊙ k1. it has 2n vertices and 2n − 1 edges[3]. a wounded spider graph is a graph obtained by subdividing at most n − 1 edges of a star k1,n. the wounded spider includes k1, the star k1,n−1[9]. a set s ⊆ v is said to be relatively prime dominating set if it is a dominating set with at least two elements and for every pair of vertices u and v in s such that (deg(u), deg(v)) = 1. the minimum cardinality of a relatively prime dominating set of a graph g is called the relatively prime domination number of g and is denoted by γrpd(g) [5]. the purpose of this paper is to study about the concept of relatively prime inverse domination on line graphs. definition 1.1. [6]let d be a minimum dominating set of a graph g. if v − d contains a dominating set s of g, then s is called an inverse dominating set with respect to d. if every pair of vertices u and v in s such that (deg(u), deg(v)) = 1, then s is called relatively prime inverse dominating set. the minimum cardinality of a relatively prime inverse dominating set is called a relatively prime inverse domination number and is denoted by γ−1rp (g). if the relatively prime inverse dominating set is absent, then γ−1rp (g) = 0. definition 1.2. [4]a line graph l(g) of a simple graph g is obtained by associ333 relatively prime inverse domination on line graph ating a vertex with each edge of the graph and connecting two vertices with an edge if only if the corresponding edges of g have a vertex in common. example 1.1. consider the graphs g and l(g) which are given figure 1. clearly {e1, e4} is a minimum dominating set of l(g) and {e2, e5} is a corresponding minimum inverse dominating set of l(g) and (deg(e1), deg(e4)) = (4, 3) = 1 and so γ−1rp l(g) = 2. g l(g) figure 1: g, l(g) u2 u6 u5 e4 u1 e5 e1u3 e2 u4 e7 e6 e3 e8 e1 e2 e3 e4 e5 e6 e7 e8 we use the following theorem: theorem 1.1. [8] for a path pn, γ−1rp (pn) =   2 if 3 ≤ n ≤ 5 3 if n = 6, 7 0 otherwise 2 relatively prime inverse domination on line graph theorem 2.1. for the spider graph k1,n,n, γ−1rp (l(k1,n,n)) = n. proof. let v be the centre vertex and the end vertices of k1,n be v1, v2, ..., vn. let u1, u2, ..., un represent the vertices connected with v1, v2, ..., vn, respectively. the resulting graph is the spider graph k1,n,n with vertex set v (k1,n,n) = {v, vi, v ′ i : 1 ≤ i ≤ n} and e(k1,n,n) = {vvi, viv ′ i : 1 ≤ i ≤ n}. clearly, deg(v) = n, deg(vi) = 2, and deg(v ′ i) = 1, 1 ≤ i ≤ n. let the line graph of the graph k1,n,n be l(k1,n,n). denote the edges vvi by ei and viv ′ i by e ′ i . clearly v (l(k1,n,n)) = {ei, e ′ i : 1 ≤ i ≤ n} and e(l(k1,n,n)) = {eiej, eie ′ i : 1 ≤ i ̸= j ≤ n}. let d be a minimum dominating set of l(k1,n,n) and s be a corresponding minimum inverse dominating set of l(k1,n,n). although l(k1,n,n) contains n end 334 c. jayasekaran, l. roshini vertices, any minimum dominating set of l(k1,n,n) must include at least n vertices of l(k1,n,n). clearly, d = {ei : 1 ≤ i ≤ n} is a minimum dominating set and s = {e′i : 1 ≤ i ≤ n} is the corresponding minimum inverse dominating set of l(k1,n,n). since deg(e ′ i) = deg(e ′ j) = 1 for 1 ≤ i ̸= j ≤ n, s is a minimum relatively prime inverse dominating set of l((k1,n,n)). as a result, γ−1rp (l(k1,n,n)) = n. k1,4,4 l(k1,4,4) figure 2: k1,4,4, l(k1,4,4) v e1 e2 e3 e4 e ′ 1 e ′ 2 e ′ 3 e ′ 4 v1 v2 v3 v4 v ′ 1 v ′ 2 v ′ 3 v ′ 4 e1 e2 e3e4 e ′ 1 e ′ 4 e ′ 3 e ′ 2 theorem 2.2. for the wounded spider graph k1,n,s, γ−1rp (l(g)) = s + 1 where s < n. proof. let v be the centre vertex and v1, v2, ..., vn be the end vertices of k1,n. attach u1, u2, ..., us to v1, v2, ..., vn as appropriate where s < n. the resulting graph is the wounded spider graph k1,n,s with vertex set v (k1,n,s) = {v, vi, uj : 1 ≤ i ≤ n, 1 ≤ j ≤ s} and e(k1,n,s) = {vvi, vjuj : 1 ≤ i ≤ n, 1 ≤ j ≤ s}. clearly in k1,n,s, deg(v) = n, deg(vi) = 2, 1 ≤ i ≤ s, deg(vk) = 1, s+1 ≤ i ≤ n and deg(ui) = 1, 1 ≤ i ≤ s. let the line graph of the graph k1,n,s be l(k1,n,s) where we denote the edge vvi by ei and vjuj by e ′ j, 1 ≤ i ≤ n, 1 ≤ j ≤ s. clearly, v (l(k1,n,s)) = {ei, e ′ j : 1 ≤ i ≤ n, 1 ≤ j ≤ s} and e(l(k1,n,s)) = {eiek, eje ′ j : 1 ≤ i ̸= k ≤ n, 1 ≤ j ≤ s}. also in l(k1,n,s), deg(ej) = n, deg(e ′ j) = 1 and deg(ei) = n − 1, 1 ≤ j ≤ s and s + 1 ≤ i ≤ n. let d be a minimum dominating set of l(k1,n,s) and s be a minimum inverse dominating set of l(k1,n,s) with respect to d. since l(k1,n,s) contains s end vertices, any minimum dominating set of l(k1,n,s) must include at least s vertices of l(k1,n,s). clearly, d = {ej : 1 ≤ j ≤ s} and s = {en, e ′ j : 1 ≤ j ≤ s} is a corresponding minimum inverse dominating set of l(k1,n,s). since the degree sequence of vertices in s is (n, 1, 1, ..., 1), s is a minimum relatively prime inverse dominating set of l((k1,n,s)) and hence γ−1rp (l(k1,n,s)) = s + 1. 335 relatively prime inverse domination on line graph k1,5,3 l(k1,5,3) figure 3: k1,5,3, l(k1,5,3) v v1 e1 v3 e3 v5 e5 v2 e2 v4 e4 u1 e ′ 1 u2 e ′ 2 u3 e ′ 3 e4 e5 e1 e2 e3 e ′ 2 e ′ 1 e ′ 3 theorem 2.3. for the jewel graph jn, γ−1rp (l(jn)) = 2 if n ≥ 1. proof. consider a 4-cycle xwyux. join x and y. now adding n new vertices vi, 1 ≤ i ≤ n. join vi with u and w, 1 ≤ i ≤ n. the resulting grph is the jewel graph jn with vertex set v (jn) = {x, y, u, w, vi : 1 ≤ i ≤ n} and edge set e(jn) = {ei, e ′ j, e ′′ j : 1 ≤ i ≤ 5, 1 ≤ j ≤ n}, where e1 = xw, e2 = wy, e3 = yu, e4 = ux, e5 = xy, e ′ j = uvj, e ′′ j = wvj. let the line graph of jn be l(jn) where v (l(jn)) = e(jn) = {ei, e ′ j, e ′′ j : 1 ≤ i ≤ 5, 1 ≤ j ≤ n} and e(l(jn)) = {eiei+1, e1e4, e5ej, e ′ jei, e ′ je ′ k, e ′′ j em, e ′′ j e ′′ p : 1 ≤ i ≤ 3, 1 ≤ j ≤ k, p ≤ n, 3 ≤ l ≤ 4, 1 ≤ m ≤ 2}, i ̸= k. let d be a minimum dominating set of l(jn) and s be a corresponding minimum inverse dominating set. in l(jn), the number of vertices is 2n + 5 and the maximum degree is 2n − 1 and so any minimum dominating set contains at least two vertices. now e1 is adjacent to all vertices except e3 and e ′ i, 1 ≤ i ≤ n; e ′ 1 is adjacent to e3, e4, e ′′ 1 and e ′ i, 2 ≤ i ≤ n. hence d = {e1, e ′ 1} is a minimum dominating set of l(jn). clearly, s = {e3, e ′′ 1} ⊆ v − d is also a minimum dominating set of l(jn). hence, s is a minimum inverse dominating set of l(jn). in l(jn), deg(e3) = n + 3, deg(e ′′ 1) = n + 2 and therefore (deg(e3), deg(e ′′ 1)) = (n + 3, n + 2) = 1. this implies that s is a minimum relatively prime inverse dominating set of l(jn) and so γ−1rp (l(jn)) = 2. . 336 c. jayasekaran, l. roshini j1 l(j1) figure 4: j1, l(j1) e4 e1 exy e2e3 e ′ 1 e ′′ 1 u x w y v1 e ′ 1 e ′′ 1 e1 e2 e3 e4 exy theorem 2.4. for the bistar graph bm,n, γ−1rp (l(bm,n)) = { 2 if (m, n) = 1 0 otherwise . proof. a bistar graph bm,n consists of two star graphs k1,m and k1,n having center vertices u0 and v0 respectively. join u0 and v0 with an edge. the resulting graph is a bistar graph bm,n with the vertex set v (bm,n) = {ui, vj : 0 ≤ i ≤ m, 0 ≤ j ≤ n} and edge set e(bm,n) = {u0ui, v0vj, u0v0 : 1 ≤ i ≤ m, 1 ≤ j ≤ n}. let the line graph of bm,n be l(bm,n) with the vertex set v (l(bm,n)) = {e0, ei, e ′ j : 1 ≤ i ≤ m, 1 ≤ j ≤ n} where e0 = u0v0, ei = u0ui, e ′ j = v0vj and edge set e(l(bm,n)) = {e0ei, e0e ′ j, eiek, e ′ je ′ l : 1 ≤ i ̸= k ≤ m, 1 ≤ j ̸= l ≤ n}. clearly in l(bm,n), deg(e0) = m + n, deg(e ′ i) = m and deg(e ′ j) = n, 1 ≤ i ≤ m, 1 ≤ j ≤ n. let d be a minimum dominating set of l(bm,n) and s be a corresponding minimum inverse dominating set of l(bm,n). in l(bm,n), the vertex e0 dominates all other vertices and so the unique minimum dominating set of is d = {e0}. in v − d, each ex dominates e0 and all other ei, 1 ≤ i ≤ m and i ̸= x and also each e′y dominates all other e ′ j, j ̸= y and 1 ≤ j ≤ n. hence a minimum inverse dominating set s = {ex, e ′ y} for some x, y where 1 ≤ x ≤ m, 1 ≤ y ≤ n. now in l(bm,n), (deg(ex), deg(e ′ y)) = (m, n). this implies that s is a minimum relatively prime inverse dominating set if and only if (m, n) = 1. hence the proof. 337 relatively prime inverse domination on line graph b3,4 l(b3,4) figure 5: b3,4, l(b3,4) u1 u0e2 u3 e3 u2 e1 v0 e0 v1 e ′ 1 v2 e ′ 2 v4 e ′ 4 v3e ′ 3 e ′ 3 e ′ 4 e0 e1 e2 e3 e ′ 1 e ′ 2 theorem 2.5. for the comb graph pn ⊙ k1, γ−1rp (l(pn ⊙ k1)) =   2 if n = 2, 3 3 if n = 4, 5 0 otherwise . proof. consider the path pn = v1v2....vn. for 1 ≤ i ≤ n, add vertex ui which is adjacent to vi. the resulting graph g = pn ⊙ k1 is a comb graph with vertex set v (g) = {vi, ui : 1 ≤ i ≤ n} and edge set e(g) = {ei, e ′ j : 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n} where ei = vivi+1, e ′ j = vjuj, 1 ≤ i ≤ n−1, 1 ≤ j ≤ n. let the line graph of comb graph g be l(g) where the vertex set v (l(g)) = e(g) = {ei, e ′ j : 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n} and edge set e(l(g)) = {eiei+1, eje ′ j, eje ′ j+1 : 1 ≤ i ≤ n − 2, 1 ≤ j ≤ n − 1}. clearly in l(g), deg(ei) = 4, 2 ≤ i ≤ n − 2, deg(e1) = deg(en−1) = 3, deg(e ′ j) = 2, 2 ≤ j ≤ n−1 and deg(ei) = deg(e ′ n) = 1. let d be a minimum dominating set of l(g) and s be a corresponding minimum inverse dominating set of l(g). now we cosider the following five cases. case 1. n = 2 then l(g) is p3. by theorem 1.1, γ−1rp (l(g)) = 2. 338 c. jayasekaran, l. roshini p2 ⊙ k1 l(p2 ⊙ k1) figure 6: p2 ⊙ k1, l(p2 ⊙ k1) v1 v2e1 u2 e ′ 2 u1 e ′ 1 e1 e ′ 1 e ′ 2 case 2. n = 3 in l(g), e1 is adjacent to all vertices except e ′ 3, e ′ 3 is adjacent to e2 only. hence, d = {e1, e ′ 3} is a minimum dominating set of l(g) and a corresponding minimum inverse dominating set s = {e2, e ′ 1}. in l(g), (deg(e2), deg(e ′ 1)) = (3, 1) = 1. this implies that s is a minimum relatively prime inverse dominating set of l(g) and so γ−1rp (l(g)) = 2. p3 ⊙ k1 l(p3 ⊙ k1) figure 7: p3 ⊙ k1, l(p3 ⊙ k1) u1 v1 e ′ 1 v2 e1 u2 e ′ 2 v3 e2 u3 e ′ 3 e1 e2 e ′ 1 e ′ 2 e ′ 3 case 3. n = 4 in l(g), e1 is adjacent to all vertices except e3 and e ′ i, i = 3, 4; e ′ 3 is adjacent to all vertices except e1, e ′ i, i = 1, 2. hence, d = {e1, e3} is a minimum dominating set of l(g) and a corresponding minimum inverse dominating set s = {e2, e ′ 1, e ′ 4}. the degree sequence vertices in s is (4,1,1). this implies that s is a minimum relatively prime inverse dominating set of l(g) and so γ−1rp (l(g)) = 3. 339 relatively prime inverse domination on line graph p4 ⊙ k1 l(p4 ⊙ k1) figure 8: p4 ⊙ k1, l(p4 ⊙ k1) u1 v1 e ′ 1 v2e1 u2 e ′ 2 v3e2 u3 e ′ 3 e ′ 4 e1 e2 e3 e ′ 1e ′ 2 e ′ 3 v4e3 u4 e ′ 4 case 4. n = 5 in l(g), e1 is adjacent to all vertices except ei, i = 3, 4, e ′ 1, e ′ 4, e ′ 5; e3 is adjacent to all vertices except e1, e ′ 1, e ′ 2, e ′ 5; e ′ 5 is adjacent to all vertices except e4. hence, d = {e1, e3, e ′ 5} is a minimum dominating set and a corresponding minimum inverse dominating s = {e2, e4, e ′ 1}. in l(g), the degree sequence of vertices in s is (4, 3, 1). this implies that s is a minimum relatively prime inverse dominating set of l(g) and so γ−1rp (l(g)) = 3. case 5. n ≥ 6 the degree sequence of l(g) is {4, 4, · · · , 4(n−3)times, 3, 3, 2, 2, · · · , 2(n− 2)times, 1, 1}. any minimum dominating set must contain at least four vertices and so any minimum inverse dominating set s as at least four vertices of different degrees and thereby, there exists a pair of vertices (x, y) in s such that (deg(x), deg(y)) = 2 or 4. hence, γ−1rp (l(g)) = 0. thus the theorem following five cases. 3 conclusion inspired by inverse dominating set and relatively prime dominating set, we introduce the relatively prime inverse domination number on line graph. we have determined the relatively prime inverse domination on line graph of some standard graphs like spider graph, wounded spider graph, jewel graph, bistar graph, and comb graph. furthermore our results are also justified with suitable examples. 340 c. jayasekaran, l. roshini the relatively prime inverse domination number can be obtained for many more graphs. acknowledgements the authors express their gratitude to the anonymous reviewers for the valuable suggestions and comments to complete the paper. references [1] g. chartrand, lesniak. graphs and digraphs. crc press, boca raton, fourth ed., 2005. 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[9] selvam avadayappan, m. bhuvaeshwari and r. iswariya. γsplitting graphs. international journal of reasearch in applied science and engineering technology(ijraset), 4(3): 670-680, 2016. 341 ratio mathematica volume 46, 2023 bibo stability and decomposition analysis of signals and system with convolution techniques c. b. sumathi * r. jothilakshmi † abstract in this paper control system’s stability is arrived based on bounded input bounded output (bibo) when bounded input is given in the form of discrete values. the control system allows the state estimation constraints to reach the convergence even when fluctuations in the parameters of the input system occur. to overcome this dtft (discrete time fourier transform) is used when the signal is completely absolutely summable. stability of the lti (linear time invariant) system is showed and is depending on the absolute summable of their impulse response. simultaneously for continuous signal the stability occurs if it is absolutely integrable. in addition to that the linearity and time-invariance properties are discussed. this provide a new way to decompose the periodic signals into fourier series by convolving the fundamental signals. continuous and discrete time signals are focused in this paper to get linear time invariant system (lti) through complex exponentials. finally filtering techniques were used to eliminate the noisy frequency component in a signal. keywords: stability, dtft, ctft, dirichlet conditions. 2020 ams subject classifications: 39a12, 39a30, 39a60. 1 *1pg and research department of mathematics, marudhar kesari jain college for women,tamil nadu, india. c.bsumathi@yahoo.in. †p g and research department of mathematics, mazharul uloom college, tamil nadu, india. jothilakshmiphd@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1086. issn: 1592-7415. eissn: 2282-8214. ©c. b. sumathi et al. this paper is published under the cc-by licence agreement. 291 c. b. sumathi and r. jothilakshmi 1 introduction difference equations are described the evolution of applied mathematics phenomenon of latest technology from artificial intelligence to iot. higher order difference equations and time invariant systems are focusing more real world problems in the diverse fields like communication technology srinivasan et al. [2022]. when we classify these difference equations, each category of classification are solving specific engineering and technological problems r.p.agarwal [2000], thamvichai and t.bose [2002]. various methodologies are used to resolve error reduction and quality enhancement techniques. in such cases analytical solutions are possible to address particular specified problems kayar and kaymakçalanbistritz [2022]. to analyze these difference equations finite difference methods can be utilized. in particular when the errors are magnified, then the difference equations will be unstable and for smaller components these equation guarantees the stability camouzis and ladas [2008], liu1 et al. [2022]. when the larger components are involved, tri-diagonal system (based on crank-nicolson method) will be compared with other explicit and implicit methods with higher derivatives camouziss and g.ladas [2010], liu [2008]. time delay is one of the reasons for instability which appears in dynamical systems such as biological systems, chemical systems, communication systems, nuclear systems, electrical systems, etc., and it is one of the key performances in these systems bose [1995], kaczorek [2011], oppenheim et al. [2009]. as the signal is impulsive, it goes to infinity at any time and hence, the system is unstable even when an input is bounded but an output is infinite. bounded signal is a signal which is having a finite value at all instants of time. in general a signal is bounded if it has finite value m > 0, and the signal does not exceed m, i.e. |y(n)| ≤ m,∀n ∈ z for discrete-time signals. this paper is structured as follows: section ii provides the fundamental concepts of lti systems in time as well as frequency domain. the systems represents through linear time invariant difference equations are discussed in section iii. section iv dealt with the development of fourier series in difference equations with time dependent variable. section v depicts the filtering techniques related to lti systems that change the shape of the input signal. finally section vi concludes the paper. 292 bibo stability analysis of signals and system 2 lti systems 2.1 time domain conditions theorem 2.1.1 (sufficient condition) : the bibo stability of discrete time lti system and its impulse response is absolutely summable i.e. ∑∞ n=−∞ |h[n]| < ∞. proof :consider the following by convolution, y[n] = ∞∑ n=−∞ h[k]x[n−k] (1) then |y[n]| = | ∞∑ n=−∞ h[n−k]x[k]| (2) applying triangular inequality |y[n]| = ||x||∞ ∞∑ n=−∞ |h[k]| (3) therefore h[n] is absolutely summable. theorem 2.1.2 (sufficient condition of continuous time) : the bibo stability of continuous time lti is ∫ ∞ −∞ |h(t)|dt < ∞. proof : consider the output function |y(t)| = | ∫ ∞ −∞ x(t−t)h(t)dt|, (4) ≤ ∫ ∞ −∞ m|h(t)|dt = m ∫ ∞ −∞ |h(t)|dt (5) hence the proof. 2.2 lti systems frequency domain conditions discrete time signal in general, for bibo stability a unit circle in the z-plane must contain all the 293 c. b. sumathi and r. jothilakshmi poles of a system. the condition for stability can be obtained by the above time domain condition ∞∑ n=−∞ |h[n]| = ∞∑ n=−∞ |h[n]||e−jwn| continuous time signal in the continuous case, laplace transform must include the imaginary axis. for bibo stability s-plane must contain all the poles of a system. the condition for stability can be obtained by the above condition∫ ∞ −∞ |h(t)|dt = ∫ ∞ −∞ |h(t)||e−jwt|dt (6) = ∫ ∞ −∞ |h(t)|(e−st|dt (7) where s = σ + jw and re(s) = σ = 0 in addition to above condition, the continuous time signals is convergence if it encounter the following dirichlet conditions • if the interval is finite then x is of bounded variation , • for all finite number of points, x is continuous, if the interval is finite. these conditions are satisfied by the periodic interval. assume that ∞∑ m=0 am = 1 (1−a) (8) where |a| < 1. on multiplying 1−a both side we get ∞∑ m=0 am −a ∑ m = 0∞am = 1 a0 = 1 , since |a| < 1, the sums converge. as an example, if h(t) = atu(t) for every t ∈ r, where a > 0. since the integral is infinite if a ≥ 1, it is unstable and it is finite if 0 < a < 1, and thus∫ ∞ 0 atdt = −1 ln a therefore, if 0 < a < 1, the system becomes stable. 294 bibo stability analysis of signals and system 3 representation of systems via lti difference equations the output form of difference equation of discrete time system is defined like haung and p. m.knopf [2012] y[n] = y[n−1] + x[n] if the system h[n] is lti system then y[n] = x[n]∗h[n] (9) in causal system, the impulse response is zero for all t < 0,n < 0 in both discrete and continuous system respectively alzabut et al. [2021], oppenheim et al. [2009], thamvichai and t.bose [2002]. for example consider the following system, y[n]− 1 2 y[n−1] = x[n] if x[n] = δ [n], then we have y[0] = 1 y[1] = 1 2 y[2] = 1 4 . . . y[n] = ( 1 2 )n then h[n] = ( 1 2 )n u[n] 4 development of fourier series in different equations in this section we calculate the transform signals through fourier series hu [2011], bistritz [2004]. we consider some simple transformations with time variable. the ouput function y(t) is y(t) = est ∫ ∞ −∞ h(τ)e−sτdτ 295 c. b. sumathi and r. jothilakshmi the input signal x(t) = cosw0t, where w0 > 0, and then x(t) = 1 2 ejw0t − 1 2 e−jw0t filtering is used to modify or eliminate some frequency components in the discrete signals. consider h(ejw) = 1 (1−ae(−jw)) now define y(t) as y(t) = e(−at) ∫ 1 0 ea τdτ y(t) = 1 a [1−e(−at)] here x(τ) and h(t− τ) does not overlap, and hence y(t) = 0. 5 conclusions this paper analyzed the stability of the bibo system in time as well as frequency domain. based on the state of time domain, a linear time invariant system is stable only if its impulse response is absolutely stable. in addition, it is focused the decomposition of signals into lti system through suitable examples. the development of these techniques has been used in implementation of time-varying convolution filters. the signals are convolved to produce the linear time invariant system and non overlapping systems. references j. alzabut, m.bohner, and s. grace. oscillation of nonlinear third-order difference equations with mixed neutral terms. adv. difference equ, 2021:1–18, 2021. y. bistritz. testing stability of 2-d discrete systems by a set of real 1-d stability tests,. ieee transactions on circuits and systems i,, 51:1312 – 1320, 2004. t. bose. stability of 2-d state-space system with overflow and quantization. ieee transactions on circuits and systems ii,, 42:432–434, 1995. e. camouzis and g. ladas. dynamics of third order rational difference equations with open problems and conjectures,. chapman and hall crc, boca raton, fl,, 2008. 296 bibo stability analysis of signals and system e. camouziss and g.ladas. global results on rational systems in the plane. part i, journal of difference and applications,, 16(8):975–1013, 2010. y. haung and p. m.knopf. global convergence properties of first order homogeneous system of rational difference equations,. journal of difference equations and applications,, 18:1683 – 1707, 2012. d. hu. new stability tests of positive standard and fractional linear systems. circuits and systems, 2(4):261–268, 2011. t. kaczorek. new stability tests of positive standard and fractional linear systems. circuits and systems, 2(4):261–268, 2011. z. kayar and b. kaymakçalanbistritz. applications of the novel diamond alpha hardy–copson type dynamic inequalities to half linear difference equations,. journal of difference equations and applications,, 28:457 – 484, 2022. t. liu. stability analysis of linear 2-d systems. signal processing,, 3(4):2078 – 2084, 2008. z. liu1, l.jiang, and r.qu. a machine-learning based fault diagnosis method with adaptive secondary sampling for multiphase drive systems. ieee transactions on power electronics,, 2022. a. oppenheim, r. w.schaffer, and phi. oscillation of nonlinear third-order difference equations with mixed neutral terms. discrete time signal processing, 2009. r.p.agarwal. difference equations and inequalities. marcel dekker,, new york,ny, usa, 2nd edition,, 2000. r. srinivasan, r.graef, and e. thandapani. asymptotic behaviour of semicanonical third-order functional difference equations. journal of difference equations and applications,, 28(4):547 –560, 2022. r. thamvichai and t.bose. stability of 2-d periodically shift variant filters,. ieee transactions on circuits and systems ii,, 49:61 – 64, 2002. 297 ratio mathematica volume 45, 2023 ` the connected vertex strong geodetic number of a graph c. saritha* t. muthu nesa beula† abstract in this paper we introduce the concept of connected vertex strong geodetic number 𝑐𝑔𝑠𝑥 (𝐺) of a graph 𝐺 at a vertex 𝑥 and investigate its properties. we determinebounds for it and find the same for some special classes of graphs. we prove that𝑠𝑔𝑥 (𝐺) ≤ 𝑐𝑠𝑔𝑥 (𝐺) for any vertex 𝑥 in 𝐺is connected graphs of order 𝑛 ≥ 2with one are characterized for some vertex 𝑥 in 𝐺.necessary conditions for𝑠𝑔𝑥 (𝐺) to be 𝑛 or 𝑛 − 1 are given for some vertex 𝑥 in 𝐺. it is shown for every pair of integers𝑎 and 𝑏 with 2 ≤ 𝑎 ≤ 𝑏, there exists a connected graph 𝐺 such that 𝑠𝑔𝑥 (𝐺) = 𝑎 and 𝑐𝑠𝑔𝑥 (𝐺) = 𝑏 for some vertex 𝑥 in 𝐺. keywords: strong geodetic number;vertex strong geodetic number; connected strong geodetic number. 2010 ams subject classification: 05c15‡. *register number 20123182092003, research scholar, department of mathematics, women’s christian college, nagercoil629 001, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627 012, tamil nadu, india. saritha.c2012@gmail.com. †assistant professor, department of mathematics, women’s christian college, nagercoil 629 001, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627 012, tamil nadu, india.tmnbeula@gmail.com. ‡received on july 28, 2022. accepted on october 15, 2022. published on january 25, 2023. doi: 10.23755/rm.v45i0.978. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 52 mailto:saritha.c2012@gmail.com c. saritha and t. muthu nesa beula 1. introduction by a graph 𝐺 = (𝑉, 𝐸), we mean a finite, undirected connected graph without loops or multiple edges. the order and size of 𝐺 are denoted by 𝑛 and 𝑚 respectively. for basic graph theoretic terminology, we refer to [1]. two vertices 𝑢 and 𝑣 are said to be adjacent if 𝑢𝑣 is an edge of 𝐺. two edges of 𝐺 are said to be adjacent if they have a common vertex. the distance𝑑(𝑢, 𝑣) between two vertices 𝑢 and v in a connected graph 𝐺 is the length of a shortest 𝑢-𝑣 path in 𝐺. an 𝑢−𝑣 path of length 𝑑(𝑢, 𝑣) is called an 𝑢−𝑣geodesic.an 𝑥 − 𝑦 path of length 𝑑(𝑥, 𝑦) is called geodesic. a vertex 𝑣 is said to lie on a geodesic 𝑃 if 𝑣 is an internal vertex of 𝑃. the closed interval 𝐼[𝑥, 𝑦] consists of 𝑥, 𝑦 and all vertices lying on some 𝑥 − 𝑦 geodesic of 𝐺 and for a non-empty set 𝑆 ⊆ 𝑉 (𝐺), 𝐼[𝑆] = ∪𝑥,𝑦∈𝑆 𝐼[𝑥, 𝑦]. a set 𝑆 ⊆ 𝑉 (𝐺) in a connected graph 𝐺 is a geodetic set of 𝐺 if 𝐼[𝑆] = 𝑉 (𝐺). the geodetic number of 𝐺, denoted by 𝑔(𝐺), is the minimum cardinality of a geodetic set of 𝐺.the geodetic concept were studied in [1, 3, 4]. let 𝑥 be a vertex of 𝐺 and 𝑆 ⊆ 𝑉 − {𝑥}. then for each vertex𝑦 ∈ 𝑆, 𝑥 ≠ 𝑦. let �̃�𝑥 [𝑦] be a selected fixed shortest 𝑥-𝑦 path. then we set 𝐼𝑥 [𝑆] = {�̃�𝑥 (𝑦): 𝑦 ∈ 𝑆} and let 𝑉(𝐼𝑥 [𝑆]) = ⋃ 𝑉(𝑃) 𝑝∈𝐼𝑥[𝑆] . if 𝑉(𝐼𝑥 [𝑆]) = 𝑉 for some 𝐼𝑥 [𝑆]then the set 𝑆 is called a vertex strong geodetic set of 𝐺. the minimum cardinality of a vertex strong geodetic set of 𝐺 is called the vertex strong geodetic number of 𝐺 and is denoted by 𝑠𝑔𝑥 (𝐺).the following theorem is used in sequel. theorem 1.1[4] each extreme vertex of a connected graph belong to every geodetic set of 𝐺. 2. the connected vertex strong geodetic number of a graph definition 2.1. let 𝑥 be a vertex of 𝐺 and 𝑆 ⊆ 𝑉 − {𝑥}. then for each vertex𝑦 ∈ 𝑆, 𝑥 ≠ 𝑦. let �̃�𝑥 [𝑦] be a selected fixed shortest 𝑥-𝑦 path. then we set 𝐼𝑥 [𝑆] = {�̃�𝑥 (𝑦): 𝑦 ∈ 𝑆} and let 𝑉(𝐼𝑥 [𝑆]) = ⋃ 𝑉(𝑃) 𝑝∈𝐼𝑥[𝑆] . if 𝑉(𝐼𝑥 [𝑆]) = 𝑉 for some 𝐼𝑥 [𝑆]then the set 𝑆 is called a vertex strong geodetic set of 𝐺. a vertex strong geodetic set s of x of g is called a connected vertex strong geodetic set of g if g[s] is connected. the minimum cardinality of a connected vertex strong geodetic set of 𝐺 is called the connected vertex strong geodetic number of 𝐺 and is denoted by 𝑐𝑠𝑔𝑥 (𝐺). example 2.2.for the graph 𝐺 given in figure 2.1,𝑐𝑠𝑔𝑥-sets and 𝑐𝑠𝑔𝑥 (𝐺) for each vertex 𝑥 is given in the following table 2.1. 53 the connected vertex strong geodetic number of a graph table 2.1 observation 2.3. let 𝑥 be any vertex of a connected graph𝐺. (i) if 𝑦 ≠ 𝑥 be a simplicial vertex of 𝐺, then 𝑦 belongs to every connected 𝑥vertex strong geodetic set of 𝐺. (ii) the eccentric vertices of 𝑥 belong to every connected 𝑥-vertex strong geodetic set of 𝐺. in the following we determine the connected vertex strong geodetic number of some standard graphs 𝐺 for each vertex in 𝐺. theorem 2.4.for the path𝐺 = 𝑃𝑛 (𝑛 ≥ 3), 𝑐𝑠𝑔𝑥 (𝐺) = { 1 𝑖𝑓 𝑥 𝑖𝑠 𝑎𝑛 𝑒𝑛𝑑 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 𝐺 𝑛 𝑖𝑓 𝑥 𝑖𝑠 𝑎 𝑐𝑢𝑡 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 𝐺 proof. let 𝑃𝑛 be 𝑣1, 𝑣2, … , 𝑣𝑛. if 𝑥 = 𝑣1, then 𝑆 = {𝑣𝑛 } is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 1. similarly if 𝑥 = 𝑣𝑛, then 𝑐𝑠𝑔𝑥 (𝐺) = 1. let 𝑥 be a cut vertex of 𝐺. then by observation 2.3 (i) vertex 𝑐𝑠𝑔𝑥-sets 𝑐𝑠𝑔𝑥 (𝐺) 𝑣1 {𝑣3, 𝑣4}, {𝑣4, 𝑣5} 2 𝑣2 {𝑣4, 𝑣5, 𝑣6} 3 𝑣3 {𝑣1, 𝑣6}, {𝑣5, 𝑣6} 2 𝑣4 {𝑣1, 𝑣6}, {𝑣1, 𝑣2} 2 𝑣5 {𝑣1, 𝑣2, 𝑣3} 3 𝑣6 {𝑣3, 𝑣4}, {𝑣2, 𝑣3} 2 𝑣1 𝑣4 𝑣5 𝐺 figure 2.1 𝑣2 𝑣3 𝑣6 54 c. saritha and t. muthu nesa beula {𝑣1, 𝑣𝑛 } is a subset of every 𝑐𝑠𝑔𝑥-set of 𝐺. let 𝑆 be a 𝑐𝑠𝑔𝑥-set of 𝐺. since 𝐺[𝑆] is connected, it follows that 𝑆 = 𝑉(𝐺) is the unique 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛.∎ theorem 2.5.for the cycle 𝐺 = 𝐶𝑛(𝑛 ≥ 4), 𝑐𝑠𝑔𝑥 (𝐺) = 2, for every 𝑥 ∈ 𝐺. proof. let 𝑉(𝐶𝑛) = {𝑣1, 𝑣2, … , 𝑣𝑛}. without loss of generality let us assume that 𝑥 = 𝑣1. case (i) let 𝑛 be even. let 𝑛 = 2𝑘 (𝑘 ≥ 2). then 𝑣𝑘+1 is the eccentric vertex of 𝐺. by observation 2.3(ii) since {𝑣𝑘+1} is not a 𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) ≥ 2. let 𝑆 = {𝑣𝑘+1, 𝑣𝑘+2}. then 𝑆 is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 2. case (ii) let 𝑛 be odd. let 𝑛 = 2𝑘 + 1 (𝑘 ≥ 2). then 𝑆 = {𝑣𝑘+1,𝑣𝑘+2} is the eccentric vertices of 𝐺. by observation 2.3 (ii) 𝑆 is a subset of every 𝑐𝑠𝑔𝑥-set of 𝐺 and so 𝑐𝑠𝑔𝑥 (𝐺) ≥ 2. since 𝑆 is a 𝑠𝑔𝑥-set of 𝐺 and 𝐺[𝑆] is connected, 𝑆 is a 𝑐𝑠𝑔𝑥-set of 𝐺 sothat 𝑐𝑠𝑔𝑥 (𝐺) = 2.∎ theorem 2.6.for the complete graph 𝐺 = 𝐾𝑛(𝑛 ≥ 4), 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 1, for every 𝑥 ∈ 𝐺. proof. let 𝑥 be a vertex of 𝐺. let 𝑆 = 𝑉(𝐺) − {𝑥}. since every vertex of 𝐺 is an extreme vertex of 𝐺, it follows from observation 2.3(i), 𝑆 is the unique 𝑐𝑠𝑔𝑥-set of 𝐺 so that𝑐𝑠𝑔𝑥 (𝐺) ≥ 𝑛 − 1 for every vertex 𝑥 in 𝐺.∎ theorem 2.7.for the fan graph 𝐺 = 𝐾1 + 𝑃𝑛−1(𝑛 ≥ 5). 𝑐𝑠𝑔𝑥 (𝐺) = { 𝑛 − 1 𝑖𝑓 𝑥 ∈ 𝑉(𝐾1) 𝑛 − 3 𝑖𝑓 𝑥 𝑖𝑠 𝑒𝑥𝑡𝑟𝑒𝑚𝑒 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 𝑃𝑛−1 𝑛 − 2 𝑖𝑓 𝑥 𝑖𝑠 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 𝑃𝑛−1 proof. let 𝑉(𝐾1) = 𝑦 and 𝑉(𝑃𝑛−1) = {𝑣1, 𝑣2, … , 𝑣𝑛−1}. case (i) let𝑥 = 𝑦, then 𝑆 = {𝑣1, 𝑣2, … , 𝑣𝑛 } is a set of all eccentric vertices for 𝑥. by observation 2.3 (ii) 𝑆 is a subset of every 𝑐𝑠𝑔𝑥-set of 𝐺 and so 𝑐𝑠𝑔𝑥 (𝐺) ≥ 𝑛 − 1. since 𝐺[𝑆] is connected, 𝑆 is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 1. let 𝑥 ∈ 𝑉(𝑃𝑛−1). let 𝑥 = 𝑣1. then 𝑆 = {𝑣3, 𝑣4, … , 𝑣𝑛−1} are eccentric vertices of 𝐺. by observation 2.3 (ii) 𝑆 is a subset of every 𝑐𝑠𝑔𝑥 -set of 𝐺 and so 𝑐𝑠𝑔𝑥 (𝐺) ≥ 𝑛 − 3. now 𝑆 is a 𝑠𝑔𝑥-set of 𝐺 and 𝐺[𝑆] is connected. therefore 𝑆 is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 3. if 𝑥 = 𝑣𝑛−1, by the similar way we can prove that 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 3. let 𝑥 ∈ {𝑣2, 𝑣3, … , 𝑣𝑛−2}. without loss of generality let us assume that 𝑥 = 𝑣2. then {𝑣1, 𝑣𝑛−1} is set of extreme vertices of 𝐺. by observation 2.3 (i) {𝑣1, 𝑣𝑛−1} is a subset of every 𝑐𝑠𝑔𝑥-set of 𝐺. {𝑣4, 𝑣5, … , 𝑣𝑛−2} is the set of eccentric vertices of 𝑣2. then {𝑣4, 𝑣5, … , 𝑣𝑛−2}is a subset of every 𝑐𝑠𝑔𝑥-set of 𝐺. let 𝑆 ′ = {𝑣1, 𝑣4, 𝑣5, … , 𝑣𝑛−2,𝑣𝑛−1,}. then 𝑆′ is a 𝑠𝑔𝑥-set of 𝐺 but 𝐺[𝑆 ′] is not connected. therefore 𝑆′ ∪ {𝑦} is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 2.∎ theorem 2.8.for the wheel graph 𝐺 = 𝐾1 + 𝐶𝑛−1(𝑛 ≥ 5). 𝑐𝑠𝑔𝑥 (𝐺) = { 𝑛 − 1 𝑖𝑓 𝑥 ∈ 𝑣1 𝑛 − 3 𝑖𝑓 𝑥 ∈ 𝑉(𝐶𝑛−1) proof. let 𝑉(𝐾1) = 𝑦 and 𝑉(𝐶𝑛−1) = {𝑣1, 𝑣2, … , 𝑣𝑛−1}. 55 the connected vertex strong geodetic number of a graph case(i) let𝑥 = 𝑦, then 𝑆 = {𝑣1, 𝑣2, … , 𝑣𝑛−1} is a set of all eccentric vertices for 𝑥. by observation 2.3 (ii) 𝑆 is a subset of every 𝑐𝑠𝑔𝑥-set of 𝐺 and so 𝑐𝑠𝑔𝑥 (𝐺) ≥ 𝑛 − 1. since 𝐺[𝑆] is connected, 𝑆 is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 1. case (ii) let 𝑥 ∈ 𝑉(𝐶𝑛−1). without loss of generality, let us assume that 𝑥 = 𝑣1. then 𝑆 = {𝑣3, 𝑣4, … , 𝑣𝑛−1} are eccentric vertices of 𝐺. by observation 2.3 (ii) 𝑆 is a subset of every 𝑐𝑠𝑔𝑥 -set of 𝐺 and so 𝑐𝑠𝑔𝑥 (𝐺) ≥ 𝑛 − 3. now 𝑆 is a 𝑠𝑔𝑥-set of 𝐺 and 𝐺[𝑆] is connected. therefore 𝑆 is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 3.∎ theorem 2.9.for the star graph 𝐺 = 𝐾1,𝑛−1(𝑛 ≥ 3), 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 1 for every 𝑥 ∈ 𝐺. proof. let𝑦 be the cut vertex of 𝐺 and {𝑣1, 𝑣2, … , 𝑣𝑛−1} is a set of all eccentric vertices of 𝐺.let𝑥 = 𝑦, then 𝑆 = {𝑣1, 𝑣2, … , 𝑣𝑛−1} is a set of all eccentric vertices for 𝑥. by observation 2.3 (ii) 𝑆 is a subset of every 𝑐𝑠𝑔𝑥-set of 𝐺 and so 𝑐𝑠𝑔𝑥 (𝐺) ≥ 𝑛 − 1. since 𝐺[𝑆] is connected, 𝑆 is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 1. let 𝑥 ∈ {𝑣1, 𝑣2, … , 𝑣𝑛−1} without loss of generality, let us assume that 𝑥 = 𝑣1. then 𝑆 = {𝑣2, 𝑣3, … , 𝑣𝑛−1} are set of eccentric vertices of 𝑣1. by observation 2.3 (ii) 𝑆 is a subset of every 𝑠𝑔𝑥-set of 𝐺 and so 𝑐𝑠𝑔𝑥 (𝐺) ≥ 𝑛 − 2. now 𝑆 is a 𝑐𝑠𝑔𝑥-set of 𝐺 but 𝐺[𝑆] is not a 𝑐𝑠𝑔𝑥-set of 𝐺 and so 𝑐𝑠𝑔𝑥 (𝐺) ≥ 𝑛 − 1. let 𝑆 ′ = 𝑆 ∪ {𝑥}. then 𝑆′ is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 1.∎ theorem 2.10.for the peterson graph 𝐺, 𝑐𝑠𝑔𝑥 (𝐺) = 6 for every 𝑥 ∈ 𝐺. proof. case (i) let𝑥 ∈ {𝑣1, 𝑣2, 𝑣3, 𝑣4, 𝑣5}. without loss of generality let us assume that 𝑥 = 𝑣1. then 𝑆 = {𝑣2, 𝑣5, 𝑣7, 𝑣8, 𝑣9, 𝑣10} is the set of all eccentric vertices for 𝑥. by observation 2.3 (ii) 𝑆 is a subset of every 𝑐𝑠𝑔𝑥 -set of 𝐺 and so 𝑐𝑠𝑔𝑥 (𝐺) ≥ 6. since 𝑆 is a 𝑠𝑔𝑥-set of 𝐺 and 𝐺[𝑆] is connected, 𝑆 is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 6. case (ii) let𝑥 ∈ {𝑣6, 𝑣7, 𝑣8, 𝑣9, 𝑣10}. without loss of generality let us assume that 𝑥 = 𝑣6. then 𝑆 = {𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣8, 𝑣9} is the set of all eccentric vertices for 𝑥. by observation 2.3 (ii) 𝑆 is a subset of every 𝑐𝑠𝑔𝑥 -set of 𝐺 and so 𝑐𝑠𝑔𝑥 (𝐺) ≥ 6. since 𝑆 is a 𝑠𝑔𝑥-set of 𝐺 and 𝐺[𝑆] is connected, 𝑆 is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 6.∎ 56 c. saritha and t. muthu nesa beula theorem 2.11.let 𝐺 be a connected graph. then 1 ≤ 𝑠𝑔𝑥 (𝐺) ≤ 𝑐𝑠𝑔𝑥 (𝐺) ≤ 𝑛 for every vertex 𝑥 in 𝐺. proof. let 𝑥 be a vertex of 𝐺. since every 𝑠𝑔𝑥-set of 𝐺 needs at least one vertex 𝑠𝑔𝑥 (𝐺) ≥ 1. since every connected strong vertex geodetic set of 𝐺 is a strong vertex geodetic set of 𝐺, 𝑠𝑔𝑥 (𝐺) ≤ 𝑐𝑠𝑔𝑥 (𝐺). since 𝑉(𝐺) is a connected strong vertex geodetic set of 𝐺, 𝑐𝑠𝑔𝑥 (𝐺) ≤ 𝑛. therefore 1 ≤ 𝑠𝑔𝑥 (𝐺) ≤ 𝑐𝑠𝑔𝑥 (𝐺) ≤ 𝑛.∎ theorem 2.12.let 𝐺 be a connected graph. then 𝑐𝑠𝑔𝑥 (𝐺) = 1 if and only if 𝑥 is an end vertex of 𝑃𝑛(𝑛 ≥ 2). proof. let 𝑥 be an end vertex of 𝑃𝑛. then by theorem 2.4, 𝑐𝑠𝑔𝑥 (𝐺) = 1. conversely let 𝑐𝑠𝑔𝑥 (𝐺) = 1. let 𝑆 = {𝑦} be the 𝑐𝑠𝑔𝑥-set of 𝑥. we prove that 𝑥 is an end vertex of 𝑃𝑛. on the contrary suppose that 𝑥 is not an end vertex of 𝑃𝑛. then there are at least two 𝑥 − 𝑦 geodesics, which is a contradiction to 𝑆 a 𝑐𝑠𝑔𝑥-set of 𝐺. therefore 𝑥 is an end vertex of 𝑃𝑛.∎ theorem 2.13.let 𝐺 be a connected graph and 𝑥 ∈ 𝐺. if 𝑥 is a universal vertex of 𝐺. then 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 1. proof. let 𝑥 be a universal vertex of 𝐺. then 𝑉(𝐺) − {𝑥} is set of all eccentric vertices for 𝑥. by observation 2.3 (ii), 𝑆 is a subset of every 𝑐𝑠𝑔𝑥-set of 𝐺 and so 𝑐𝑠𝑔𝑥 (𝐺) ≥ 𝑛 − 1. since 𝐺[𝑆] is connected, 𝑆 is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛 − 1.∎ theorem 2.14.let 𝐺 be a connected graph and 𝑥 ∈ 𝐺. if 𝑥 is a cut vertex and universal vertex of 𝐺. then𝑐𝑠𝑔𝑥 (𝐺) = 𝑛. proof. since 𝑥 is a universal vertex of 𝐺, then 𝑉(𝐺) − {𝑥} is set of all eccentric vertices for 𝑥. by observation 2.3 (ii), 𝑆 is a subset of every 𝑐𝑠𝑔𝑥-set of 𝐺 and so 𝑣9 𝑣10 g figure 2.2 𝑣4 𝑣2 𝑣6 𝑣8 𝑣3 𝑣5 𝑣1 𝑣7 57 the connected vertex strong geodetic number of a graph 𝑐𝑠𝑔𝑥 (𝐺) ≥ 𝑛 − 1. since 𝐺[𝑆] is not connected, 𝑆 is not a 𝑐𝑠𝑔𝑥-set of 𝐺. therefore 𝑆 = 𝑉(𝐺) is the unique 𝑐𝑠𝑔𝑥-set of 𝐺. hence 𝑐𝑠𝑔𝑥 (𝐺) = 𝑛.∎ theorem 2.15. for every pair of integers𝑎 and 𝑏 with 2 ≤ 𝑎 ≤ 𝑏, there exists a connected graph 𝐺 such that 𝑠𝑔𝑥 (𝐺) = 𝑎 and 𝑐𝑠𝑔𝑥 (𝐺) = 𝑏 for some vertex 𝑥 in 𝐺. proof.for 𝑎 = 𝑏, let 𝐺 = 𝐾𝑎+1. then by theorem 2.11𝑠𝑔𝑥 (𝐺) = 𝑐𝑠𝑔𝑥 (𝐺) = 𝑎 for every vertex 𝑥 in 𝐺. for 𝑏 = 𝑎 + 1,let 𝐺 = 𝐾1,𝑎. let 𝑥 be a universal vertex of 𝐺. then by theorem 2.14,𝑠𝑔𝑥 (𝐺) = 𝑎 and 𝑐𝑠𝑔𝑥 (𝐺) = 𝑎 + 1. so, let 𝑏 ≥ 𝑎 + 2. let 𝑃0: 𝑢0, 𝑢1, 𝑢2, … , 𝑢𝑏−𝑎, 𝑢𝑏−𝑎+1be a path of order 𝑏 − 𝑎 + 2. let 𝐺 be the graph obtained from 𝑃 by adding the new vertices 𝑧1, 𝑧2, … , 𝑧𝑎−1 and introducing the edges 𝑧𝑖 𝑢 (1 ≤ 𝑖 ≤ 𝑏 − 𝑎 + 1). the graph 𝐺is shown in figure 2.3. let 𝑥 = 𝑢𝑏−𝑎+1. first we prove that 𝑠𝑔𝑥(𝐺) = 𝑎.let𝑆 = {𝑢0, 𝑧1, 𝑧2, … , 𝑧𝑎−1, 𝑢𝑏−𝑎+1}be the end vertices of 𝐺. by observation 2.3(i), 𝑆1 = 𝑆 − {𝑢𝑏−𝑎+1} is a subset of every 𝑠𝑔𝑥-set of 𝐺 and so 𝑠𝑔𝑥 (𝐺) ≥ 𝑎. since 𝑆1 is a 𝑠𝑔𝑥-set of 𝐺, 𝑠𝑔𝑥 (𝐺) = 𝑎. next we prove that 𝑐𝑠𝑔𝑥 (𝐺)= 𝑏.by observation, 𝑆1 is a subset of every𝑐𝑠𝑔𝑥-set of 𝐺. since 𝐺[𝑆1] is not connected 𝑆1 is not a 𝑐𝑠𝑔𝑥-set of 𝐺. let 𝑆2 = 𝑆1 ∪ {𝑢1, 𝑢2, … , 𝑢𝑏−𝑎}. then 𝑆2 is a 𝑐𝑠𝑔𝑥-set of 𝐺 and𝐺[𝑆2] is connected. therefore 𝑆2 is a 𝑐𝑠𝑔𝑥-set of 𝐺 so that , 𝑐𝑠𝑔𝑥 (𝐺) = 𝑏.∎ 3. conclusions in this article we explore the concept of the forcing strong geodetic number of a graph. we extend this concept to some other distance related parameters in graphs. 𝑧1 𝑧2 𝑧𝑎−1 𝑢2 𝑢𝑏−𝑎 𝐺 figure 2.3 𝑢0 𝑢1 𝑢𝑏−𝑎+1 58 c. saritha and t. muthu nesa beula references [1] f. buckley and f. harary, distance in graphs, addison-wesley, redwood city, ca, 1990. [2] l. g. bino infanta and d. antony xavier, strong upper geodetic number of graphs, communications in mathematics and applications 12(3), (2021)737–748. [3]g. chartrand and p. zhang, the forcing geodetic number of a graph, discuss. math. graph theory, 19 (1999), 45-58. [4]g. chartrand, f. harary and p. zhang, on the geodetic number of a graph, networks, 39(2002), 1-6. [5] v. gledel, v. irsic, and s. klavzar, strong geodetic cores and cartesian product graphs, arxiv: 1803. 11423 [math.co] (30 mar 2018). [6]huifen ge, zao wang-and jinyu zou strong geodetic number in some networks, journal of mathematical resarch-11(2), (2019), 20-29. [7] v. irsic, strong geodetic number of complete bipartite graphs and of graphs with specified diameter, graphs and combin. 34 (2018) 443–456. [8] v. irsic, and s. klavzar, strong geodetic problem on cartesian products of graphs, rairo oper. res. 52 (2018) 205–216. [9] p. manuel, s. klavzar, a. xavier, a. arokiaraj, and e. thomas, strong edge geodetic problem in networks, open math. 15 (2017) 1225–1235. [10] c. saritha and t. muthu nesa beula, the forcing strong geodetic number of a graph, proceedings of the international conference on advances and applications in mathematical sciences, 2022, 76-80. [11] c. saritha and t. muthu nesa beula, the vertex strong geodetic number of a graph, (communicated). 59 ratio mathematica volume 44, 2022 on forgotten index of stolarsky-3 mean graphs sree vidya.m 1 sandhya. s. s 2 abstract the forgotten index of a graph g is defined as f(g) = over all edges of ,where , are the degrees of the vertices u and v in , respectively. in this paper, we introduced forgotten index of some standard stolarsky-3 mean graphs. keywords: forgotten index, stolarsky-3 mean graphs. ams subject classification: 05c12 3 1 research scholar, sree ayyappa college for women, chunkankadai 2 research supervisor, department of mathematics, sree ayyappa college for women, chunkankadai. [affiliated to manonmaniam sundaranar university, abishekapatti – tirunelveli 627012, tamilnadu, india] email: witvidya@gmail.com & sssandhya2009@gmail.com 3 received on june 9 th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.915. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by licence agreement. 272 mailto:witvidya@gmail.com%20& sree vidya.m & sandhya. s. s 1. introduction let g be a simple graph corresponding to a drug structure with vertex (atom) set v(g) and edge (bond) set e(g). the edge joining the vertices u and v is denoted by uv. thus, if then u and v are adjacent in g. the degree of a vertex u, denoted by d(u), is the number of edge incident to u. several topological indices such as estrada index, zagreb index, pi index, eccentric index, and wiener index have been introduced in the literature to study the chemical and pharmacological properties of molecules. the forgotten topological index of a graph g is defined as the sum of weights over all edges of ,where and are the degrees of the vertices u and v in , respectively. in this paper, we characterize the external properties of f-index (forgotten topological index). we first introduce some graph transformation which increase or decrease this index. recently in 2015 furtula and gutman was introduced another topological index called index or f-index as f(g) = . on the basis of this work, we introduce a new concept forgotten index ofstolarsky3meangraphs.inthispaperweinvestigate forgotten index of some standard graphs which admit forgotten mean graphs. we will provide a brief summary of definitions and other information which are necessary for our present investigation. definition:1.1 a graph with vertices and edges is called a stolarsky-3 mean graph, if each vertex with distinct labels from and eachedge is assigned the distinct labels then the resulting edge labels are distinct. in this case is called stolarsky-3 mean labeling of . definition:1.2 let g be a stolarsky-3 mean graph. the forgotten index of a graph f(g) is defined by f(g) = , where d(u) is the degree of vertex u in g. theorem 1.3: any path is a stolarsky-3 mean graph. theorem 1.4: any cycle c is a stolarsky-3 mean graph. theorem 1.5: any comb ⨀ 1is a stolarsky 3 mean graph. theorem 1.6: the ladder graph is a stolarsky-3 mean graph. theorem 1.7: atriangular snake graph is a stolarsky-3 mean graph. theorem 1.8: aquadrilateral snake graph is a stolarsky-3 mean graph. 273 on forgotten index of stolarsky-3 mean graphs remark 1.9: if is a stolarsky 3 mean graph, then ‘1’ must be a label of one of the vertices of , since, an edge should get label ‘1’. remark1.10: if u gets label ‘1’, then any edge incident with must get label 1 (or) 2 (or) 3. hence this vertex must have a degree ≤ 3. 2. main results theorem 2.1: let g = pn be a stolarsky-3 mean graph. then the forgotten index of a path pn is f(pn) =5n+4. proof. let g = pn be a stolarsky-3 mean graph. figure: 1 path pn we have and . therefore, by the definition of forgotten topological index, we obtain f(g) = = = = = f(g) = example 2.2. forgotten index of p6 is given below figure: 2 path p6 f(p6) = = = = = theorem 2.3. the forgotten index of cycle is . proof. let be a stolarsky-3 mean graph u1 u2 u3 un-1 un 274 sree vidya.m & sandhya. s. s figure: 3 we have and . therefore, by the definition of forgotten topological index, we obtain f(g) = = = = f(g) = example 2.4. forgotten index of is given below figure: 4 f( ) = = = = 48 theorem 2.5. the forgotten index of comb graph ⨀ . proof. let g = pn ʘ k1 be a stolarsky – 3 mean graph. 275 on forgotten index of stolarsky-3 mean graphs figure: 5 comb pnʘk1 we have and . therefore, by the definition of forgotten topological index, we obtain f(g) = = = = f(g) = example 2.6: forgotten index of p6ʘk1 is given below. figure: 6 comb p6ʘk1 f(p6ʘk1) = [ = = theorem 2.7: forgotten index of ladder graph is proof. let g = pn be a stolarsky3 mean graph figure: 7 g = pn case (i) if n = 2 f ( ) = 276 sree vidya.m & sandhya. s. s = = = case (ii) if n 2 f( ) = = = = = f ( ) = example 2.8. forgotten index of l4 is given below. figure: 8 l4 f( ) = = = = = 140 theorem 2.9. forgotten index of triangular snake graph is 58n-6. proof. let us consider a stolarsky-3 mean graph be a stolarsky-3 mean graph. figure: 9 f(g) = 277 on forgotten index of stolarsky-3 mean graphs = = = = = f(g) = example 2.10. forgotten index of t3 is given below. figure: 10 f(t3) = = = = 120+16+32 = 168 theorem 2.11: forgotten index of quadrilateral snake graph is 80n-48. proof. consider be a stolarsky-3 mean graph. figure:11 f ( = = = = 278 sree vidya.m & sandhya. s. s = f ( = example 2.12. forgotten index of is given below. figure: 12 f( ) = = = = = theorem 2.13: forgotten index of crown graph ⨀ is 28n. proof. consider ⨀ be a stolarsky-3 mean graph. f ( = = = = = f ( = example 2.14. forgotten index of ⨀ is given below. figure: 13 ⨀ 279 on forgotten index of stolarsky-3 mean graphs f ( ⨀ ) = = = = references [1] f. harary. graph theory, narosa publishing house: new delhi; 2001. [2] b. furtula and i. gutman, “a forgotten topological index “, journal of mathematical chemistry, vol. 53, no. 4, pp. 1184-1190,2015. [3] toufik mansour, mohammad ali rostami, on the bounds of the forgotten topological index turkish journal of mathematics, (2017) 41:1687-1702. [4] s. s. sandhya, s. somasundaram, and s. kavitha “stolarsky 3 mean labeling of graphs” journal of applied science and computaions, vol.5, issue 9, pp. 59 – 66. [5] sree vidya. m and sandhya. s. s. “degree splitting of stolarsky 3 mean labeling of graphs” international journal of computer science, issn 2348-6600, volume 8, issue 1, no 2, 2020, page no: 2413 – 2420. [6] sree vidya. m and sandhya. s. s. “decomposition of stolarsky 3 mean labeling of graphs” international journal for innovative engineering research, volume 1, issue 1, march, 2022, page no: 08-12. 280 ratio mathematica some fixed point results using (ψ,ϕ)-generalized almost weakly contractive maps in s-metric spaces d.venkatesh* v.naga raju † abstract fixed point theorems have been proved for various contractive conditions by several authors in the existing literature. in this article, we define an (ψ,ϕ)generalized almost weakly contractive map in s-metric spaces and prove an existence and uniqueness of fixed point of such maps. and also we deduce some existing results as special cases of our result. moreover, we give an example in support of the results. keywords: fixed point; generalized almost weakly contractive map; s-metric space; 2020 ams subject classifications: 47h10,54h25 1 *(department of mathematics, osmania university, hyderabad, telangana-500007, india) ; venkat409151@gmail.com. †(department of mathematics, osmania university, hyderabad, telangana-500007, india); viswanag2007@gmail.com. 1 received on september 7, 2022. accepted on march 1, 2021. published on june 30, 2023. doi: 10.23755/rm.v4li0.855. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. volume 47, 2023 244 d.venkatesh, v.naga raju 1 introduction fixed point technique is considered as one of the powerful tools to solve several problems occur in several fields like computer science, economics, mathematics and its allied subjects. in the year 1906, m.frechet [7] introduced metric spaces. later, in the year 1922, stefan banach [4] proved a very famous theorem called ”banach fixed point theorem”. this theorem has been generalized in many directions by generalizing the underlying space or by viewing it as a common fixed point theorem along with other self maps. in the past few years, a number of generalizations of metric spaces like g -metric spaces, partial metric spaces and cone metric spaces were initiated. these generalizations are used to extend the scope of the study of fixed point theory. in 2012, sedghi, shobe and aliouche [13] introduced s-metric spaces and studied some properties of these spaces. we observe that, every g-metric space need not be a s-metric space and vice-versa. for details, see examples 2.1 and 2.2 in [5]. generally, in proving fixed point results for a single self map, we utilize completeness and a contractive condition. nowadays, the study of fixed point theorems for self maps satisfying different contraction conditions is the center of rigorous research activities. in this direction, dutta et al. [6] introduced (ψ,ϕ)-weakly contractive maps in 2008 and obtained some fixed point results for such contractions. later, g.v.r. babu et al. [1] introduced (ψ,ϕ)-almost weakly contractive maps in g-metric spaces in 2014. fixed points of contractive maps on s-metric spaces were studied by several authors [2], [3] and [11]. since then, several contractions have been considered for proving fixed point theorems. the main purpose of this paper is to define an (ψ,ϕ)generalized almost weakly contractive map in s-metric spaces and prove an existence and uniqueness of fixed point of such maps. furthermore we deduce some results as corollaries to our result and provide an example to validate our result. 2 preliminaries definition 2.1. [8] a function ψ : [0,∞) → [0,∞) is said to be an altering distance function if it satisfies (i) ψ is continuous and non decreasing and (ii) ψ(t) = 0 if and only if t = 0. we denote the class of all altering distance functions by ψ. we denote φ = {ϕ : [0,∞) → [0,∞) : (i) ϕ is continuous and (ii) ϕ(t)=0 if and only if t=0}. in the following, dutta and choudhury [6] established the fixed points of (ψ,ϕ)245 some fixed point results in s-metric spaces weakly contractive maps in complete metric spaces. theorem 2.1. [6] let (x,d) be a complete metric space and let h:x→x be a selfmaps of x. if there exist ψ,ϕ ∈ ψ such that ψ(d(hξ,hϑ)) ≤ ψ(d(ξ,ϑ)) − ϕ(d(ξ,ϑ)) for all ξ,ϑ ∈ x. then h has a unique fixed point. definition 2.2. [10] let x be a non-empty set, g:x3 → [0,∞) be a function satisfying the following properties: (i) g(ξ,ϑ,w) = 0 if ξ = ϑ = w, (ii) g(ξ,ξ,ϑ) > 0 for all ξ,ϑ ∈x with ξ ̸= ϑ, (iii) g(ξ,ξ,ϑ) ≤ g(ξ,ϑ,w) for all ξ,ϑ,w ∈x, (iv) g(ξ,ϑ,w) = g(ξ,w,ϑ) = g(w,ξ,ϑ) =...(symmetriy in all three variables), (v) g(ξ,ϑ,w) ≤ g(ξ,a,a) + g(a,ϑ,w) for all ξ,ϑ,w,a ∈x. then the function g is called a generalized metric(g-metric) and the pair (x,g) is called a g-metric space. definition 2.3. [14] let (x,g) be a g-metric space. a self mapping h of x is said to be weakly contractive if for all ξ,ϑ,w ∈ x g(hξ,hϑ,hw) ≤ g(ξ,ϑ,w) − ψ(g(ξ,ϑ,w)) where ψ is an altering distance function. in 2012, khandaqji, al-sharif and al-khaleel [9] proved the following for weakly contractive maps in g-metric spaces. theorem 2.2. [9] let (x,g) be a complete g-metric space and h:x→x be a self map. if there exist ψ ∈ ψ and ϕ ∈ φ such that ψ(g(hξ,hϑ,hw)) ≤ ψ(max{g(ξ,ϑ,w),g(ξ,hξ,hξ),g(ϑ,hϑ,hϑ),g(w,hw,hw), αg(hξ,hξ,ϑ) + (1 − α)g(hϑ,hϑ,w),βg(ξ,hξ,hξ) + (1 − β)g(ϑ,hϑ,hϑ)}) − ϕ(max{g(ξ,ϑ,w),g(ξ,hξ,hξ), g(ϑ,hϑ,hϑ),g(w,hw,hw),αg(hξ,hξ,ϑ) + (1 − α)g(hϑ,hϑ,w),βg(ξ,hξ,hξ) + (1 − β)g(ϑ,hϑ,hϑ)}) (1) for all ξ,ϑ,w ∈x,where α,β ∈ (0,1). then h has a unique fixed point u(say) and h is g-continuous at u. definition 2.4. [13] let a nonempty set x, then we say that a function s:x3 → [0,∞) is s-metric on x if: (s1) s(ξ,ϑ,w) > 0 for all ξ,ϑ,w ∈x with ξ ̸= ϑ ̸= w, (s2)s(ξ,ϑ,w) = 0 if ξ = ϑ = w, (s3) s(ξ,ϑ,w) ≤ [s(ξ,ξ,a) + s(ϑ,ϑ,a) + s(w,w,a)]. for all ξ,ϑ,w,a ∈x. then (x,s) is called an s-metric space. 246 d.venkatesh, v.naga raju example 2.1. [13] let (x,d) be a metric space. define s:x3 → [0,∞) by s(ξ,ϑ,w) = d(ξ,ϑ) + d(ξ,w) + d(ϑ,w) for all ξ,ϑ,w ∈x. then s is an s-metric on x and s is called the s-metric induced by the metric d. example 2.2. [5] let x=r, the set of all real numbers and let s(ξ,ϑ,w) = |ϑ + w − 2ξ| + |ϑ − w| for all ξ,ϑ,w ∈x. then (x,s) is an s-metric space. example 2.3. [12] let x=r, the set of all real numbers and let s(ξ,ϑ,w) = |ξ − w| + |ϑ − w| for all ξ,ϑ,w ∈x. then (x,s) is an s-metric space. example 2.4. let x=[0,1] and we define s:x3 → [0,∞) by s(ξ,ϑ,w) = { 0 if ξ = ϑ = w max{ξ,ϑ,w} otherwise . then s is an s-metric on x. the following lemmas are useful in our main results. lemma 2.1. [13] in an s-metric space, we have s(ξ,ξ,ϑ) = s(ϑ,ϑ,ξ). lemma 2.2. [5] in an s-metric space, we have (i) s(ξ,ξ,ϑ) ≤ 2s(ξ,ξ,w) + s(ϑ,ϑ,w) and (ii) s(ξ,ξ,ϑ) ≤ 2s(ξ,ξ,w) + s(w,w,ϑ). definition 2.5. [13] let (x,s) be an s-metric space. we define the following: (i) a sequence {ξn} ∈x converges to a point ξ ∈x if s(ξn,ξn,ξ) → 0 as n → ∞. that is, for each ϵ > 0, there exists n0 ∈ n such that for all n≥ n0, s(ξn,ξn,ξ) < ϵ and we denote it by limn→∞ ξn = ξ. (ii) a sequence {ξn} ∈x is called a cauchy sequence if for each ϵ > 0, there exists n0 ∈ n such that s(ξn,ξn,ξm) < ϵ for all n,m≥ n0. (iii) (x,s) is said to be complete if each cauchy sequence in x is convergent. definition 2.6. let (x,s) and (y,s’) be two s-metric spaces. then a function h:x→y is s-continuous at a point ξ ∈x if it is s-sequentially continuous at ξ, that is, whenever {ξn} is s-convergent to ξ, we have h(ξn) is s’-convergent to h(ξ). lemma 2.3. [13] let (x,s) be an s-metric space. if the sequences {ξn} in x converges to ξ, then ξ is unique. lemma 2.4. [13] let (x,s) be an s-metric space. if there exist sequences {ξn} and {ϑn} in x such that limn→∞ ξn = ξ and limn→∞ ϑn = ϑ, then limn→∞ s(ξn,ξn,ϑn) = s(ξ,ξ,ϑ). 247 some fixed point results in s-metric spaces definition 2.7. [13] let (x,s) be an s-metric space. a map h:x→x is said to be an s-contraction if there exists a constant 0 ≤ λ < 1 such that s(h(ξ),h(ξ),h(ϑ)) ≤ λs(ξ,ξ,ϑ) for all ξ,ϑ ∈ x. we now introduce the following definition and support it with a subsequent example. definition 2.8. let (x,s) be an s-metric space. a map h:x → x is called (ψ,ϕ) -generalized almost weakly contractive if it satisfies the inequality ψ(s(hξ,hϑ,hw)) ≤ ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)) + l.θ(ξ,ϑ,w) (2) for all ξ,ϑ,w ∈ x, ψ ∈ ψ, ϕ ∈ φ and l ≥ 0, where m(ξ,ϑ,w) = max{s(ξ,ϑ,w),s(ξ,ξ,hξ),s(ϑ,ϑ,hϑ), 1 2 [s(ξ,ξ,hϑ)+s(ϑ,ϑ,hξ)]}, θ(ξ,ϑ,w) = min{s(ξ,ξ,hξ),s(ϑ,ϑ,hϑ),s(w,w,hξ),s(ξ,ξ,hw)}. example 2.5. let x = [0, 8 7 ] and we define h : x → x by hξ = { ξ 10 if ξ ∈ [0,1] ξ − 4 5 if ξ ∈ (1, 8 7 ] . we define s: x3 → [0,∞) by s(ξ,ϑ,w) = |ξ − w| + |ϑ − w| for all ξ,ϑ,w ∈ x. then (x,s) is a complete s-metric space. we now define functions ψ,ϕ : [0,∞) → [0,∞) by ψ(t) = t, for all t≥ 0 and ϕ(t) = { t 2 if t ∈ [0,1] t t+1 if t ≥ 1. . we now show that h satisfies the inequality (2). case(i): let ξ,ϑ,w ∈ [0,1]. without loss of generality, we assume that ξ > ϑ > w. s(hξ,hϑ,hw) = s( ξ 10 , ϑ 10 , w 10 ) = 1 10 (|ξ − w| + |ϑ − w|) and s(ξ,ϑ,w) = |ξ − w| + |ϑ − w|. sub case (i): if |ξ − w| + |ϑ − w| ∈ [0,1]. in this case, s(hξ,hϑ,hw) = 1 10 (|ξ − w| + |ϑ − w|) ≤ 1 2 (|ξ − w| + |ϑ − w|) = 1 2 s(ξ,ϑ,w) ≤ 1 2 m(ξ,ϑ,w) = m(ξ,ϑ,w) − 1 2 m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). 248 d.venkatesh, v.naga raju sub case(ii): if |ξ − ϑ| + |ϑ − w| ≥1. in this case, s(hξ,hϑ,hw) = 1 10 (|ξ − ϑ| + |ϑ − w|) ≤ |ξ − ϑ| + |ϑ − w| − |ξ − ϑ| + |ϑ − w| 1 + |ξ − ϑ| + |ϑ − w| = s(ξ,ϑ,w) − s(ξ,ϑ,w) 1 + s(ξ,ϑ,w) = (s(ξ,ϑ,w))2 1 + s(ξ,ϑ,w) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). case(ii): let ξ,ϑ,w ∈ (1, 8 7 ]. without loss of generality, we assume that ξ > ϑ > w. s(hξ,hϑ,hw) = s(ξ − 4 5 ,ϑ − 4 5 ,w − 4 5 ) = |ξ − w| + |ϑ − w| ≤ 2 7 ≤ 64 65 = 8 5 − 8 13 = s(ξ,ξ,hξ) − s(ξ,ξ,hξ) 1 + s(ξ,ξ,hξ) = (s(ξ,ξ,hξ))2 1 + s(ξ,ξ,hξ) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). case(iii): let ϑ,w ∈ [0,1] and ξ ∈ (1, 8 7 ]. without loss of generality, we assume that ϑ >w. s(hξ,hϑ,hw) = s(ξ − 4 5 , ϑ 10 , w 10 ) = |ξ − 4 5 − w 10 | + | ϑ 10 − w 10 | = ξ − w 10 − 4 5 + ϑ − w 10 = ξ + ϑ 10 − w 5 − 4 5 = 31 70 ≤ 64 65 = 8 5 − 8 13 = s(ξ,ξ,hξ) − s(ξ,ξ,hξ) 1 + s(ξ,ξ,fξ) = (s(ξ,ξ,hξ))2 1 + s(ξ,ξ,hξ) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). 249 some fixed point results in s-metric spaces case(iv): let w∈ [0,1] and ξ,ϑ ∈ (1, 8 7 ]. without loss of generality, we assume that ξ > ϑ. s(hξ,hϑ,hw) = s(ξ − 4 5 ,ϑ − 4 5 , w 10 ) = |ξ − 4 5 − w 10 | + |ϑ − 4 5 − w 10 | = ξ + ϑ − w 5 − 8 5 = 12 35 ≤ 64 65 = 8 5 − 8 13 = s(ϑ,ϑ,hϑ) − s(ϑ,ϑ,hϑ) 1 + s(ϑ,ϑ,hϑ) = (s(ϑ,ϑ,hϑ))2 1 + s(ϑ,ϑ,hϑ) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). case (v): let ξ,ϑ ∈ [0,1] and w ∈ (1, 8 7 ]. without loss of generality, we assume that ξ > ϑ. s(hξ,hϑ,hw) = s( ξ 10 , ϑ 10 ,w − 4 5 ) = | ξ 10 − w + 4 5 | + | ϑ 10 − w + 4 5 | = | 4 5 − (w − ξ 10 )| + | 4 5 − (w − ϑ 10 )| = w − ξ 10 − 4 5 + w − ϑ 10 − 4 5 = 2w − ξ + ϑ 10 − 8 5 = 41 70 ≤ 64 65 = 8 5 − 8 13 = s(ξ,ξ,hξ) − s(ξ,ξ,hξ) 1 + s(ξ,ξ,hξ) = (s(ξ,ξ,hξ))2 1 + s(ξ,ξ,hξ) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). case (vi): let ξ ∈ [0,1] and w,ϑ ∈ (1, 8 7 ]. without loss of generality, we assume that w > ϑ. s(hξ,hϑ,hw) = s( ξ 10 ,ϑ − 4 5 ,w − 4 5 ) = | ξ 10 − w + 4 5 | + |ϑ − w| = w − ξ 10 − 4 5 + w − ϑ = 2w − ξ 10 − 4 5 − ϑ 250 d.venkatesh, v.naga raju ≤ 27 70 ≤ 64 65 = 8 5 − 8 13 = s(ϑ,ϑ,hϑ) − s(ϑ,ϑ,hϑ) 1 + s(ϑ,ϑ,hϑ) = (s(ϑ,ϑ,hϑ))2 1 + s(ϑ,ϑ,hϑ) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). from all the above cases, we conclude that h is an (ψ,ϕ)-generalized almost weakly contraction map on x. lemma 2.5. [5] let (x,s) be an s-metric space and {ξn} be a sequence in x such that limn→∞ sb(ξn,ξn,ξn+1) = 0. if {ξn} is not a cauchy sequence, then there exist an ϵ > 0 and two sequences {mk} and {nk} of natural numbers with nk > mk > k such that s(ξmk,ξmk,ξnk) ≥ ϵ, s(ξmk−1,ξmk−1,ξnk) < ϵ and (i)limk→∞ sb(ξmk,ξmk,ξnk) = ϵ. (ii) limk→∞ sb(ξmk−1,ξmk−1,ξnk) = ϵ. (iii) limk→∞ sb(ξmk,ξmk,ξnk−1) = ϵ. (iv) limk→∞ sb(ξmk−1,ξmk−1,ξnk−1) = ϵ. 3 main results theorem 3.1. let (x,s) be a complete s-metric space and h: x → x be a (ψ,ϕ)generalized almost weakly contractive mapping. then h has a unique fixed point in x. proof. let ξ0 ∈ x be arbitrary. we define a sequence {ξn} by hξn = ξn+1, for n = 0,1,2,.... if ξn = ξn+1, for some n∈ n, then ξn is a fixed point of h. suppose ξn ̸= ξn+1, for all n∈ n. consider, ψ(s(ξn+1,ξn+1,ξn)) = ψ(s(hξn,hξn,hξn−1)) ≤ ψ(max{s(ξn,ξn,ξn−1),s(ξn,ξn,hξn),s(ξn,ξn,hξn), 1 2 [s(ξn,ξn,hξn) + s(ξn,ξn,hξn)]}) − ϕ(max{s(ξn,ξn,ξn−1),s(ξn,ξn,hξn),s(ξn,ξn,hξn), 1 2 [s(ξn,ξn,hξn) + s(ξn,ξn,hξn)]}) + l.min{s(ξn,ξn,hξn),s(ξn,ξn,hξn),s(ξn−1,ξn−1,hξn),s(ξn,ξn,hξn−1)} 251 some fixed point results in s-metric spaces = ψ(max{s(ξn,ξn,ξn−1),s(ξn,ξn,ξn+1),s(ξn,ξn,ξn+1), 1 2 [s(ξn,ξn,ξn+1) + s(ξn,ξn,ξn+1)]}) − ϕ(max{s(ξn,ξn,ξn−1),s(ξn,ξn,ξn+1),s(ξn,ξn,ξn+1), 1 2 [s(ξn,ξn,ξn+1) + s(ξn,ξn,ξn+1)]}) + l.min{s(ξn,ξn,ξn+1),s(ξn,ξn,ξn+1),s(ξn−1,ξn−1,ξn+1),s(ξn,ξn,ξn)} = ψ(max{s(ξn,ξn,ξn−1),s(ξn,ξn,ξn+1)}) − ϕ(max{s(ξn,ξn,ξn−1), s(ξn,ξn,ξn+1)}) + l.0 if max{s(ξn,ξn,ξn−1),s(ξn,ξn,ξn+1)} = s(ξn,ξn,ξn+1), then we get ψ(s(ξn+1,ξn+1,ξn)) ≤ ψ(s(ξn+1,ξn+1,ξn)) − ϕ(s(ξn+1,ξn+1,ξn)) that is, ϕ(s(ξn+1,ξn+1,ξn)) ≤ 0, which implies that s(ξn+1,ξn+1,ξn) = 0. then we get ξn+1 = ξn, which is a contradiction to our assumption that ξn ̸= ξn+1, for each n. therefore, max{s(ξn,ξn,ξn−1),s(ξn,ξn,ξn+1)} = s(ξn,ξn,ξn−1), then we get ψ(s(ξn+1,ξn+1,ξn)) ≤ ψ(s(ξn,ξn,ξn−1)) − ϕ(s(ξn,ξn,ξn−1)) (3) that is ψ(s(ξn+1,ξn+1,ξn)) ≤ ψ(s(ξn,ξn,ξn−1)) therefore we get, s(ξn+1,ξn+1,ξn) ≤ s(ξn,ξn,ξn−1), for all n and the sequence {s(ξn+1,ξn+1,ξn)} is decreasing and bounded. so, there exists r ≥ 0 such that lim n→∞ s(ξn+1,ξn+1,ξn) = r. letting n → ∞ in equation (3), we get ψ(r) ≤ ψ(r) − ϕ(r), which is a contradiction unless r = 0. hence, lim n→∞ s(ξn+1,ξn+1,ξn) = 0. (4) now we prove that {ξn} is a cauchy sequence. if not, then there exists an ϵ > 0 for which we can find subsequences {ξm(k)} and {ξn(k)} of {ξn} and increasing sequence of integers {m(k)} and {n(k)} such that n(k) is the smallest index for which n(k) > m(k) > k, s(ξm(k),ξm(k),ξn(k)) ≥ ϵ (5) 252 d.venkatesh, v.naga raju then, we have s(ξm(k),ξm(k),ξn(k)−1) < ϵ (6) now, ϵ ≤ s(ξm(k),ξm(k),ξn(k)) = s(ξn(k),ξn(k),ξm(k)) ≤ 2s(ξn(k),ξn(k),ξn(k)−1) + s(ξm(k),ξm(k),ξn(k)−1) ≤ ϵ + 2s(ξn(k),ξn(k),ξn(k)−1) (using equation 6) letting k→ ∞, we get lim k→∞ s(ξm(k),ξm(k),ξn(k)) = ϵ. (7) also, s(ξm(k),ξm(k),ξn(k)) ≤ 2s(ξm(k),ξm(k),ξm(k)−1) + s(ξn(k),ξn(k),ξm(k)−1) ≤ 2s(ξm(k),ξm(k),ξm(k)−1) + 2s(ξn(k),ξn(k),ξn(k)−1) + s(ξm(k)−1,ξm(k)−1,ξn(k)−1) (8) and s(ξm(k)−1,ξm(k)−1,ξn(k)−1) ≤ 2s(ξm(k)−1,ξm(k)−1,ξm(k))+s(ξn(k)−1,ξn(k)−1,ξm(k)) = 2s(ξm(k),ξm(k),ξm(k)−1) + s(ξm(k),ξm(k),ξn(k)−1) (9) letting k → ∞ in equation (9) and using equations (4), (6), (7) and (8) we get lim k→∞ s(ξm(k)−1,ξm(k)−1,ξn(k)−1) = ϵ (10) setting ξ = ξm(k)−1,y = ξm(k)−1 and z = ξn(k)−1 in equation (2), we obtain ψ(ϵ) ≤ ψ(s(ξm(k),ξm(k),ξn(k))) = ψ(s(hξm(k)−1,hξm(k)−1,hξn(k)−1)) ≤ ψ(max{s(ξm(k)−1,ξm(k)−1,ξn(k)−1),s(ξm(k)−1,ξm(k)−1,hξm(k)−1), s(ξm(k)−1,ξm(k)−1,hξm(k)−1), 1 2 [s(ξm(k)−1,ξm(k)−1,hξm(k)−1) + s(ξm(k)−1,ξm(k)−1,hξm(k)−1)]}) − ϕ(max{s(ξm(k)−1,ξm(k)−1,ξn(k)−1),s(ξm(k)−1,ξm(k)−1,hξm(k)−1), s(ξm(k)−1,ξm(k)−1,hξm(k)−1), 1 2 [s(ξm(k)−1,ξm(k)−1,hξm(k)−1) + s(ξm(k)−1,ξm(k)−1,hξm(k)−1)]}) 253 some fixed point results in s-metric spaces + l.min{s(ξm(k)−1,ξm(k)−1,hξm(k)−1),s(ξm(k)−1,ξm(k)−1,hξm(k)−1), s(ξn(k)−1,ξn(k)−1,hξm(k)−1),s(ξm(k)−1,ξm(k)−1,hξn(k)−1)} ≤ ψ(max{s(ξm(k)−1,ξm(k)−1,ξn(k)−1),s(ξm(k)−1,ξm(k)−1,ξm(k)), s(ξm(k)−1,ξm(k)−1,ξm(k)), 1 2 [s(ξm(k)−1,ξm(k)−1,ξm(k)) + s(ξm(k)−1,ξm(k)−1,ξm(k))]}) − ϕ(max{s(ξm(k)−1,ξm(k)−1,ξn(k)−1),s(ξm(k)−1,ξm(k)−1,ξm(k)), s(ξm(k)−1,ξm(k)−1,ξm(k)), 1 2 [s(ξm(k)−1,ξm(k)−1,ξm(k)) + s(ξm(k)−1,ξm(k)−1,ξm(k))]}) + l.min{s(ξm(k)−1,ξm(k)−1,ξm(k)),s(ξm(k)−1,ξm(k)−1,ξm(k)), s(ξn(k)−1,ξn(k)−1,ξm(k)),s(ξm(k)−1,ξm(k)−1,ξn(k))} letting k → ∞ and using equation (10) we get ψ(ϵ) ≤ ψ(max{ϵ,0,0,0}) − ϕ(max{ϵ,0,0,0}) + l.min{0,0,0, ϵ} ψ(ϵ) ≤ ψ(ϵ) − ϕ(ϵ) + l.0 this is a contradiction, since ϵ > 0. this shows that {ξn} is a cauchy sequence in the complete s-metric space (x,s). there exists κ ∈x such that {ξn} → κ as n → ∞. now we prove that hκ = κ. put ξ = ξn,ϑ = ξn and w = κ in equation (2), then we get ψ(s(ξn+1,ξn+1,fκ)) = ψ(s(hξn,hξn,hκ)) ≤ ψ(max{s(ξn,ξn,κ),s(ξn,ξn,hξn),s(ξn,ξn,hξn), 1 2 [s(ξn,ξn,hξn) + s(ξn,ξn,hξn)]}) − ϕ(max{s(ξn,ξn,κ),s(ξn,ξn,hξn),s(ξn,ξn,hξn), 1 2 [s(ξn,ξn,hξn) + s(ξn,ξn,hξn)]}) + l.min{s(ξn,ξn,hξn),s(ξn,ξn,hξn),s(κ,κ,hξn),s(ξn,ξn,hκ)} = ψ(max{s(ξn,ξn,κ),s(ξn,ξn,ξn+1),s(ξn,ξn,ξn+1), 1 2 [s(ξn,ξn,ξn+1) + s(ξn,ξn,ξn+1)]}) − ϕ(max{s(ξn,ξn,κ),s(ξn,ξn,ξn+1), s(ξn,ξn,ξn+1), 1 2 [s(ξn,ξn,ξn+1) + s(ξn,ξn,ξn+1)]}) + l.min{s(ξn,ξn,ξn+1),s(ξn,ξn,ξn+1),s(κ,κ,ξn+1),s(ξn,ξn,hκ)} letting n → ∞, we get ψ(s(κ,κ,hκ)) ≤ ψ(s(κ,κ,κ)) − ϕ(s(κ,κ,κ)+l.0 ψ(s(κ,κ,hκ)) ≤ 0. so, we get s(κ,κ,hκ)=0. 254 d.venkatesh, v.naga raju hence hκ = κ. that is κ is a fixed point of h. to prove the uniqueness of κ, let j be a fixed point of h with κ ̸=j. using equation (2),we consider ψ(s(κ,κ,j)) = ψ(s(hκ,hκ,hj)) ≤ ψ(max{s(κ,κ,j),s(κ,κ,hκ),s(κ,κ,hκ), 1 2 [s(κ,κ,hκ) + s(κ,κ,hκ)]}) − ϕ(max{s(κ,κ,j),s(κ,κ,hκ),s(κ,κ,hκ), 1 2 [s(κ,κ,hκ) + s(κ,κ,hκ)]}) + l.min{s(κ,κ,hκ),s(κ,κ,hκ),s(j,j,hκ),s(κ,κ,hj)} that is, ψ(s(κ,κ,j)) ≤ ψ(s(κ,κ,j)) − ϕ(s(κ,κ,j)) is a contradiction, unless s(κ,κ,j) = 0. hence we get κ = j. this shows that the fixed point of h is unique. 2 if l=0 in the theorem 3.1, then we get the following. corollary 3.1. let (x,s) be a complete s-metric space and h:x→x be a mapping. suppose there exist ψ ∈ ψ and ϕ ∈ φ such that s(hξ,hϑ,hw) ≤ ψ(max{s(ξ,ϑ,w),s(ξ,ξ,hξ),s(ϑ,ϑ,hϑ), 1 2 [s(ξ,ξ,hϑ) + s(ϑ,ϑ,hξ)]}) − ϕ(max{s(ξ,ϑ,w),s(ξ,ξ,hξ),s(ϑ,ϑ,hϑ), 1 2 [s(ξ,ξ,hϑ) + s(ϑ,ϑ,hξ)]}), for all ξ,ϑ,w ∈ x. then h has a unique fixed point κ in x. if ψ is the identity map in the above corollary (3.1), then we get the following. corollary 3.2. let (x,s) be a complete s-metric space and h:x→x be a mapping. suppose there exist ϕ ∈ φ such that s(hξ,hϑ,hw) ≤ max{s(ξ,ϑ,w),s(ξ,ξ,hξ),s(ϑ,ϑ,hϑ), 1 2 [s(ξ,ξ,hϑ) + s(ϑ,ϑ,hξ)]} − ϕ(max{s(ξ,ϑ,w),s(ξ,ξ,hξ),s(ϑ,ϑ,hϑ), 1 2 [s(ξ,ξ,hϑ) + s(ϑ,ϑ,hξ)]}) for all ξ,ϑ,w ∈ x. then h has a unique fixed point κ in x. the following example is in support of theorem 3.1. example 3.1. let x = [0, 7 6 ]. we define s:x3 → [0,∞) by s(ξ,ϑ,w) = max{|ξ− w|, |ϑ − w|}, for all ξ,ϑ,w ∈ x. then s is an s-metric on x. we define h:x → x by 255 some fixed point results in s-metric spaces hξ = { 1 2 if ξ ∈ [0,1] 4 3 − ξ if ξ ∈ (1, 7 6 ] . we define ψ,ϕ : [0,∞) → [0,∞) by ψ(t) = t, for all t≥ 0 and ϕ(t) = t 1+t for all t≥0. we now show that h satisfies inequality (2). case(i) let ξ,ϑ,w ∈[0,1]. without loss of generality, we assume that ξ > ϑ > w. s(hξ,hϑ,hw) = s(1 2 , 1 2 , 1 2 ) = 0. then trivially the inequality (2) holds. case(ii) let ξ,ϑ,w ∈ (1, 7 6 ]. without loss of generality, we assume that ξ > ϑ > w. s(hξ,hϑ,hw) = s( 4 3 − ξ, 4 3 − ϑ, 4 3 − w) = max{| 4 3 − ξ − ( 4 3 − w)|, | 4 3 − ϑ − ( 4 3 − w)|} = max{|w − ξ|, |w − ϑ|} = ξ − w ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 ≤ s(ξ,ξ,hξ) − s(ξ,ξ,hξ) 1 + s(ξ,ξ,hξ) = (s(ξ,ξ,hξ))2 1 + s(ξ,ξ,hξ) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). case(iii) let ϑ,w ∈[0,1] and ξ ∈ (1, 7 6 ]. without loss of generality, we assume that ϑ > w. s(hξ,hϑ,hw) = s( 4 3 − ξ, 1 2 , 1 2 ) = max{| 4 3 − ξ − 1 2 |, | 1 2 − 1 2 |} = ξ − 5 6 ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 ≤ s(ξ,ξ,hξ) − s(ξ,ξ,hξ) 1 + s(ξ,ξ,hξ) = (s(ξ,ξ,hξ))2 1 + s(ξ,ξ,hξ) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). case(iv) let w ∈[0,1] and ξ,ϑ ∈ (1, 7 6 ]. without loss of generality, we assume that ϑ > ξ. s(hξ,hϑ,hw) = s( 4 3 − ξ, 4 3 − ϑ, 1 2 ) = max{| 4 3 − ξ − 1 2 |, | 4 3 − ϑ − 1 2 |} 256 d.venkatesh, v.naga raju = max{| 5 6 − ξ|, | 5 6 − ϑ|} = ξ − 5 6 ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 ≤ s(ϑ,,ϑ,hϑ) − s(ϑ,,ϑ,hϑ) 1 + s(ϑ,,ϑ,hϑ) = (s(ϑ,,ϑ,hϑ))2 1 + s(ϑ,,ϑ,hϑ) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). case(v) let ξ,ϑ ∈[0,1] and w ∈ (1, 7 6 ]. without loss of generality, we assume that ξ > ϑ. s(hξ,hϑ,hw) = ( 1 2 , 1 2 , 4 3 − w) = max{| 1 2 − ( 4 3 − w)|, | 1 2 − ( 4 3 − w)|} = w − 5 6 ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 ≤ s(ξ,ξ,hξ) − s(ξ,ξ,hξ) 1 + s(ξ,ξ,hξ) = (s(ξ,ξ,hξ))2 1 + s(ξ,ξ,hξ) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). case(vi) let ξ ∈[0,1] and ϑ,w ∈ (1, 7 6 ]. without loss of generality, we assume that w > ϑ. s(hξ,hϑ,hw) = s( 1 2 , 4 3 − ϑ, 4 3 − w) = max{| 1 2 − ( 4 3 − w)|, | 4 3 − ϑ − ( 4 3 − w)|} = max{w − 5 6 , |w − ϑ|} = w − 5 6 ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 = s(ϑ,ϑ,hϑ) − s(ϑ,ϑ,hϑ) 1 + s(ϑ,ϑ,hϑ) = (s(ϑ,ϑ,hϑ))2 1 + s(ϑ,ϑ,hϑ) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). case(vii) let ϑ ∈[0,1] and ξ,w ∈ (1, 7 6 ]. without loss of generality, we assume that w > ξ. s(hξ,hϑ,hw) = s( 4 3 − ξ, 1 2 , 4 3 − w) = max{| 4 3 − ξ − ( 4 3 − w)|, | 1 2 − ( 4 3 − w)|} 257 some fixed point results in s-metric spaces = max{|w − ξ|,w − 5 6 } = w − 5 6 ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 = s(ϑ,ϑ,hhϑ) − s(ϑ,ϑ,hϑ) 1 + s(ϑ,ϑ,hϑ) = (s(ϑ,ϑ,hϑ))2 1 + s(ϑ,ϑ,hϑ) ≤ (m(ξ,ϑ,w))2 1 + m(ξ,ϑ,w) = m(ξ,ϑ,w) − m(ξ,ϑ,w) 1 + m(ξ,ϑ,w) = ψ(m(ξ,ϑ,w)) − ϕ(m(ξ,ϑ,w)). from all the above cases, we conclude that h is an (ψ,ϕ)-generalized almost weakly contraction map on x and 1 2 is the unique fixed point of h. 4 conclusion in this paper, we establish an existence and uniqueness of a fixed point theorem for (ψ,ϕ)-generalized almost weakly contraction maps in s-metric spaces. as s-metric space is a generalization of metric space, our result in this article extends and improves the result of khandaqji, al-sharif and al-khaleel [9] and also generalize several well-known comparable results in the literature. further, the result in this paper can be extended to several spaces like sb-metric space, partial sb-metric spaces and other spaces. references [1] g.v.r.babu, d.r. babu, kanuri nageswara rao, bendi venkata siva kumar, fixed points of (ψ,ϕ)-almost weakly contractive maps in g-metric spaces, applied mathematics e-notes, 4 (2014) 69–85. 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[14] n.surender and b.k.reddy, common fixed point theorems for weakly compatible mappings satisfying generalized contraction principle in complete g-metric spaces, annals of pure and applied mathematics,10(2)(2015),179–190. 259 introduction preliminaries main results conclusion ratio mathematica volume 45, 2023 translations of bipolar valued multi fuzzy subnearring of a nearring s. muthukumaran1 b. anandh2 abstract in this paper, some translations of bipolar valued multi fuzzy subnearring of a nearing are introduced and using these translations, some theorems are stated and proved. key words. bipolar valued fuzzy subset, bipolar valued multi fuzzy subset, bipolar valued multi fuzzy subnearring, translations, intersection. subject classification. 97h40, 03b52, 03e723. 1research scholar, p. g. and research department of mathematics, h. h. the rajah’s college, pudukkottai, affiliated to bharathidasan university, tiruchirappalli, tamilnadu, india. email: muthumaths28@gmail.com. 2assistant professor, p.g. and research department of mathematics, h. h. the rajah’s college, pudukkottai, affiliated to bharathidasan university, tiruchirappalli, tamilnadu, india. email: drbaalaanandh@gmail.com. 3received on july 10, 2022. accepted on october 15, 2022. published on january 30, 2023.doi: 10.23755/rm. v45i0.999. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by license agreement. 127 s. muthukumaran and b. anandh 1. introduction in 1965, zadeh [9] introduced the notion of a fuzzy subset of a set, fuzzy sets are a kind of useful mathematical structure to represent a collection of objects whose boundary is vague. since then, it has become a vigorous area of research in different domains, there have been a number of generalizations of this fundamental concept such as intuitionistic fuzzy sets, interval-valued fuzzy sets, vague sets, soft sets etc. w. r. zhang [10, 11] introduced an extension of fuzzy sets named bipolar valued fuzzy sets in 1994 and bipolar valued fuzzy set was developed by lee [2, 3]. bipolar valued fuzzy sets are an extension of fuzzy sets whose membership degree range is enlarged from the interval [0, 1] to [−1, 1]. in a bipolar valued fuzzy set, the membership degree 0 means that elements are irrelevant to the corresponding property, the membership degree (0, 1] indicates that elements somewhat satisfy the property and the membership degree [−1, 0) indicates that elements somewhat satisfy the implicit counter property. bipolar valued fuzzy sets and intuitionistic fuzzy sets look similar each other. however, they are different each other [3]. vasantha kandasamy. w. b [7] introduced the basic idea about the fuzzy group and fuzzy bigroup. m.s. anithat et.al [1] introduced the bipolar valued fuzzy subgroup. sheena. k. p and k. uma devi [6] have introduced the bipolar valued fuzzy subbigroup of a bigroup. shanthi. v.k and g. shyamala [5] have introduced the bipolar valued multi fuzzy subgroups of a group. yasodara. s, ke. sathappan [8] defined the bipolar valued multi fuzzy subsemirings of a semiring. bipolar valued multi fuzzy subnearring of a nearing has been introduced by s. muthukumaran and b. anandh [4]. in this paper, the concept of translations of bipolar valued multi fuzzy subnearring of a nearing is introduced and established some results. definition 1.1. ([11])a bipolar valued fuzzy set (bvfs) b in x is defined as an object of the form b = {< x, b+ (u), b −(u) >/ xx}, where b+: x→ [0, 1] and b −: x→ [−1, 0]. the positive membership degree b+(u) denotes the satisfaction degree of an element x to the property corresponding to a bipolar valued fuzzy set b and the negative membership degree b−(u) denotes the satisfaction degree of an element x to some implicit counter-property corresponding to a bipolar valued fuzzy set b. definition 1.2. ([8]) a bipolar valued multi fuzzy set (bvmfs) a in x is defined as an object of the form b = {  x, b1 +(u), b2 +(u), …, bn +(u), b1 −(u), b2 −(u), …, bn −(u)  / xx}, where bi + : x→ [0, 1] and bi −: x→ [−1, 0], for all i. the positive membership degrees bi +(u) denote the satisfaction degree of an element x to the property corresponding to a bipolar valued multi fuzzy set b and the negative membership degrees bi −(u) denote the satisfaction degree of an element x to some implicit counterproperty corresponding to a bipolar valued multi fuzzy set b. 128 translations of bipolar valued multi fuzzy subnearring of a nearring definition 1.3. ([4])let (n, +, ) be a nearring. a bvmfs b of n is said to be a bipolar valued multi fuzzy subnearring of n (bvmfsnr) if the following conditions are satisfied, for all i, (i) bi + (u−v)  min {bi+ (u), bi + (v)} (ii) bi + (uv)  min {bi + (u), bi + (v)} (iii) bi −(u−v)  max {bi −(u), bi −(v)} (iv) bi −(uv)  max{bi −(u), bi −(v)},  u, vn. definition 1.4. ([8])let a =  a1 +, a2 +, …, an +, a1 −, a2 −, …, an − and b =  b1 +, b2 +, …, bn +, b1 −, b2 −, …, bn − be two bipolar valued multi fuzzy subsets with degree n of a set x. we define the following relations and operations: (i) a  b if and only if for all i, ai +(u) ≤ bi +(u) and ai −(u) ≥ bi −(u),  ux. (ii) ab = {  u, min(a1 +(u), b1 +(u)), min(a2 +(u), b2 +(u)), …, min(an +(u), bn +(u)), max (a1 −(u), b1 −(u) ), max (a2 −(u), b2 −(u) ), …, max (an −(u), bn −(u) )  / ux }. definition 1.5. let c =  c1 +, c2 +, …, cn +, c1 −, c2 −, …, cn − be a bipolar valued multi fuzzy subnearring of a nearring r and sr. then the pseudo bipolar valued multi fuzzy coset (sc)p =  (sc1 +)p1 +, (sc2 +)p2 +, …, (scn +)pn +, (sc1 −)p1 −, (sc2 −)p2 −, …, (scn −)pn − is defined by (sci +)pi +(a) = pi +(s) ci +(a) and (sci −)pi −(a) = − pi −(s) ci −(a), for all i and every ar and pp, where p is a collection of bipolar valued multi fuzzy subsets of r. definition 1.6. [8] let a =  a1 +, a2 +, …, an +, a1 −, a2 −, …, an − be a bipolar valued multi fuzzy subset of x. then the height h(a) =  h(a1 +), h(a2 +), …, h(an +), h(a1 −), h(a2 −), …, h(an −)  is defined for all i as h(ai +) = sup ai +(x) for all xx and h(ai −) = inf ai −(x) for all xx. definition 1.7. [6]let a =  a1 +, a2 +, …, ai +, a1 −, a2 −, …, ai − be a bipolar valued multi fuzzy subset of x. then 0a = 0a1 +, 0a2 +,…, 0an +, 0a1 −, 0a2 −,…, 0an −is defined for all i as 0ai +(x) = ai +(x) h (ai +) for all xx and 0ai −(x) = −ai −(x)h(ai −) for all xx. definition 1.8. [6] let a =  a1 +, a2 +, …, an +, a1 −, a2 −, …, an − be a bipolar valued multi fuzzy subset of x. then a = a1 +, a2 +, …, an +, a1 −, a2 −, …, an −is defined for all i as ai +(x) = ai +(x) / h(ai +) for all xx and ai −(x) = −ai −(x) / h(ai −) for all xx. definition 1.9. [6] let a =  a1 +, a2 +, …, an +, a1 −, a2 −, …, an − be a bipolar valued multi fuzzy subset of x. then a = a1 +, a2 +, …, an +, a1 −, a2 −, …, an −is defined for all i as ai +(x) = ai +(x) + 1− h(ai +) for all xx and ai −(x) = ai −(x) −1−h(ai −) for all xx. 129 s. muthukumaran and b. anandh definition 1.10. [6] let a =  a1 +, a2 +, …, an +, a1 −, a2 −, …, an − be a bipolar valued multi fuzzy subset of x. then a is called bipolar valued normal multi fuzzy subset of x if h(ai +) = 1 and h(ai −) = −1 for all i. 2. properties theorem 2.1.([4]) if b =  b1 +, b2 +, …, bn +, b1 −, b2 −, …, bn −and c =  c1 +, c2 +, …, cn +, c1 −, c2 −, …, cn − are two bipolar valued multi fuzzy subnearrings with degree n of a nearring r, then their intersection bc is a bipolar valued multi fuzzy subnearring of r. theorem 2.2.let k =  k1 +, k2 +… kn +, k1 −, k2 −… kn − be a bipolar valued multi fuzzy subnearring with degree n of a nearring r. then the pseudo bipolar valued multi fuzzy coset (a1k) m is a bipolar valued multi fuzzy subnearring of the nearring r, for every a1 in r and m in m, where m is a collection of bipolar valued multi fuzzy subset of r. proof. let b1, c1 in r and a1r. for each i, then ( ) )()()( 111111 cbkamcbka ii m i i −=− +++ + ≥ mi +(a1) min{ki +(b1), ki +(c1)} = min{mi +(a1) ki +(b1), mi +(a1)ki +(c1)}= min{ ( ) )( 11 bka im i + + , ( ) )( 11 cka im i + + }. therefore ( ) )( 111 cbka im i − + + ≥ min { ( ) )( 11 bka im i + + , ( ) )( 11 cka im i + + }, for b1, c1r. and for each i, then ( ) )()()( 111111 cbkamcbka ii m i i +++ = + ≥ mi +(a1) min{ki +(b1), ki +(c1)}= min{mi +(a1)ki +(b1), mi +(a1)ki +(c1)}= min{ ( ) )( 11 bka im i + + , ( ) )( 11 cka im i + + }. therefore ( ) )( 111 cbka im i + + ≥ min { ( ) )( 11 bka im i + + , ( ) )( 11 cka im i + + }, for all b1, c1r. for each i, ( ) )()()( 111111 cbkamcbka ii m i i −=− −−− −  mi −(a1) max {ki −(b1), ki −(c1)} = max{mi −(a1)ki −(b1), mi −(a1)ki −(c1)}= max{ ( ) )( 11 bka im i − − , ( ) )( 11 cka im i − − }. therefore ( ) )( 111 cbka im i − − −  max { ( ) )( 11 bka im i − − , ( ) )( 11 cka im i − − }, for b1, c1r. also for each i, then ( ) )()()( 111111 cbkamcbka ii m i i −−− = −  mi −(a1) max{ki −(b1), ki −(c1)} = max{mi −(a1)ki −(b1), mi −(a1)ki −(c1)}= max{ ( ) )( 11 bka im i − − , ( ) )( 11 cka im i − − }. therefore ( ) )( 111 cbka im i − − max { ( ) )( 11 bka im i − − , ( ) )( 11 cka im i − − }, for all b1, c1r. hence (a1k) m is a bipolar valued multi fuzzy subnearring of the nearring r. theorem 2.3. if k =  k1 +, k2 +, …, kn +, k1 −, k2 −, …, kn − is a bipolar valued multi fuzzy subnearring with degree n of a nearring r, thenk = k1 +, k2 +, …, kn +, k1 −, k2 −, …, kn −is a bipolar valued multi fuzzy subnearring of r. 130 translations of bipolar valued multi fuzzy subnearring of a nearring proof. let a1, b1 in r. for each i, ki +(a1− b1) = ki +(a1− b1)+1–h(ki +) min{ki +(a1), ki +(b1)}+1–h(ki +)= min{ki +(a1)+1–h(ki +), ki +(b1)+1–h(ki +)}= min{ki +(a1), ki +(b1)}implies ki +(a1− b1)  min{ ki +(a1), ki +(b1) } for all a1, b1r. and for all i, ki +( a1b1) = ki +(a1b1)+1–h(ki +) min{ki +(a1), ki +(b1) }+1–h(ki +) = min{ki +(a1)+1– h(ki +), ki +(b1)+1–h(ki +)}= min{ki +(a1), ki +(b1)}which implies ki +(a1b1)  min{ki +(a1), ki +(b1) } for all a1, b1r. also for all i, ki −(a1− b1) = ki −(a1− b1)−1– h(ki −) max{ki −(a1), ki −(b1) }−1–h(ki −) = max{ki −(a1)−1–h(ki −), ki −(b1)−1–h(ki −)}= max{ki −(a1), ki −(b1)}implies ki −(a1− b1)  max{ ki −(a1), ki −(b1) } for all a1, b1r. and for all i, ki −(a1b1) = ki −(a1b1)−1–h(ki −) max{ki −(a1), ki −(b1)}−1–h(ki −)= max{ki −(a1)−1–h(ki −), ki −(b1)−1–h(ki −)}= max{ki −(a1), ki −(b1)}implies ki −(a1b1)  max{ki −(a1), ki −(b1) } for all a1, b1r. hence k is abipolar valued multi fuzzy subnearring of r. corollary 2.4. let k =  k1 +, k2 +, …, kn +, k1 −, k2 −, …, kn − is a bipolar valued multi fuzzy subnearring with degree n of a nearring r. (i) if er, then for each i, ki +(e) = 1 and ki −(e) = −1, where e is an identity element of r; (ii)for each i, there exists er such that ki +(e) = 1 and ki −(e) = −1 if and only if ki +(a1) = ki +(a1) and ki −(a1) = ki −(a1) for all a1r; (iii)for each i, there exists a1r such that ki +(a1) = ki +(e) and ki −(a1) = ki −(e) if and only if ki +(a1) = 1 and ki −(a1) = −1, for somea1r; (iv)for each i, if there exists a1r such that ki +(a1) = 1 and ki −(a1) = −1, then ki +(a1) = 1 and ki −(a1) = −1; (v)for each i, if ki +(e) = 1, ki −(e) = −1, ki +(a1) = 0 and ki −(a1) = 0, then ki +(a1) = 0, ki −(a1) = 0; (vi) (k)= k, (vii)k is a bipolar valued normal multi fuzzy subnearring of r containing k; (viii) k is a bipolar valued normal multi fuzzy subnearring of r if and only if k = k; (ix)if there exists a bipolar valued multi fuzzy subnearring p of r satisfying p  k; then k is a bipolar valued normal fuzzy subnearring of r; (x)if there exists a bipolar valued multi fuzzy subnearring p of r satisfying p  k, then k = k. proof. (i), (ii), (iii), (iv), (v) and (x) are trivial.(vi) let a1, b1r. for each i, then (ki +) +(a1) = ki +(a1)+1– ki +(e)= {ki+(e)+1–ki +(e)}+1–{ki +(e)+1–ki +(e)}= ki +(a1)+1–ki +(e) = ki +(a1). also for each i, (ki −)−(a1) = ki −(a1) –1– ki −(e)= {ki −(a1)–1–ki −(e)}–1– {ki −(e)–1–ki −(e)} = ki −(a1)–1–ki −(e) = ki −(a1).hence  (k)= k.(vii) let er. clearly ki +(e) = 1 and ki −(e) = −1. thus k is a bipolar valued normal multi fuzzy subnearring of r and k k.(viii) if k = k, then it is obvious that k is a bipolar valued normal multi fuzzy subnearring of r. assume that k is a bipolar valued normal multi fuzzy subnearring of r. let a1r. then ki +(a1) = ki +(a1)+1–ki +(e) = ki +(a1) and 131 s. muthukumaran and b. anandh ki −(a1) = ki −(a1)−1–ki −(e) = ki −(a1). hence k = k. (ix) suppose there exists a bipolar valued multi fuzzy subnearring p of h such that p  k. then 1 = pi + (e) ≤ ki + (e) and −1 = pi −(e) ≥ ki −(e). hence ki +(e) =1 and ki −(e) = −1. theorem 2.5. if k =  k1 +, k2 +, …, kn +, k1 −, k2 −, …, kn − is a bipolar valued multi fuzzy subnearring with degree n of a nearring r, then0k = 0k1 +, 0k2 +, …, 0kn +, 0k1 −, 0k2 −, …, 0kn −is a bipolar valued multi fuzzy subnearring of r. proof. let a1, b1 in r. for each i, 0ki +(a1− b1) = ki +(a1−b1)h(ki +) min{ki +(a1), ki +(b1)}h(ki +) = min{ki +(a1)h(ki +), ki +(b1)h(ki +)}= min{0ki +(a1), 0ki +(b1)}implies 0ki +(a1−b1)  min{ 0ki +(a1), 0ki +(b1)} for all a1, b1r. and for all i, 0ki +(a1b1) = ki +(a1b1)h(ki +)  min{ki +(a1), ki +(b1)} h(ki +)= min{ki +(a1)h(ki +), ki +(b1)h(ki +)}= min{0ki +(a1), 0ki +(b1)}. thus 0ki +(a1b1) min{ 0ki +(a1), 0ki +(b1)} for all a1, b1r. also for all i, 0ki −(a1−b1) = −k −(a1− b1)h(ki −) −max{ki −(a1), ki −(b1)}h(ki −)= max{−ki −(a1)h(ki −), −ki −(b1)h(ki −)} = max{0ki −(a1), 0ki −(b1)}implies 0ki −(a1− b1)  max{0ki −(a1), 0ki −(b1)} for all a1, b1r. and for all i, 0ki −(a1b1) = −ki −(a1b1)h(ki −)−max{ki −(a1), ki −(b1)} h(ki −) = max{−ki −(a1)h(ki −), −ki −(b1)h(ki −)} = max{0ki −(a1), 0ki −(b1)}.therefore 0ki −(a1b1) max{ 0ki −(a1), 0ki −(b1)} for all a1, b1r. hence 0kis a bipolar valued multi fuzzy subnearring of r. theorem 2.6. if k =  k1 +, k2 +, …, kn +, k1 −, k2 −, …, kn − is a bipolar valued multi fuzzy subnearring with degree n of a nearring r, thenk = k1 +, k2 +, …, kn +, k1 −, k2 −, …, kn −is a bipolar valued multi fuzzy subnearring of r. proof. let a1, b1 in r. for each i, then ki +(a1− b1) = ki +(a1− b1) / h(ki +) min{ki +(a1), ki +(b1)} / h(ki +) = min{ki +(a1) / h(ki +), ki +(b1) / h(ki +)}= min{ki +(a1), ki +(b1)}implies ki +(a1− b1)  min{ ki +(a1), ki +(b1)} for all a1, b1r. and for all i, ki +(a1b1) = ki +(a1b1) / h(ki +)  min{ki +(a1), ki +(b1)} / h(ki +)= min{ki +(a1) / h(ki +), ki +(b1) / h(ki +)}= min{ki +(a1), ki +(b1)}. therefore ki +(a1b1) min{ ki +(a1), ki +(b1)} for all a1, b1r. also for all i, ki −(a1− b1) = −ki −(a1− b1) / h(ki −) − max{ki −(a1), ki −(b1)} / h(ki −)= max{−ki −(a1) / h(ki −), −ki −(a1) / h(ki −)} = max{ki −(a1), ki −(b1)}implies ki −(a1− b1)  max{ ki −(a1), ki −(b1)} for all a1, b1r. and for all i, ki −(a1b1) = −ki −(a1b1) / h(ki −)−max{ki −(a1), ki −(b1)} / h(ki −) = max{−ki −(a1) / h(ki −), −ki −(b1) / h(ki −)}= max{ki −(a1), ki −(b1)}. therefore ki −(a1b1) max{ ki −(a1), ki −(b1)}, for all a1, b1r. hence k is a bipolar valued multi fuzzy subnearring of r. corollary 2.7. let k =  k1 +, k2 +, …, kn +, k1 −, k2 −, …, kn − be a bipolar valued multi fuzzy subnearring with degree n of a nearring r. (i) if for each i, h(ki +) < 1, then 0ki +< ki +; (ii) if for each i, h(ki −) −1, then 0ki − ki −, (iii) if for each i, h(ki +) < 1 and h(ki −) −1, then 0k< k; 132 translations of bipolar valued multi fuzzy subnearring of a nearring (iv) if for each i, h(ki +) < 1, then ki + ki +; (v) if for each i, h(ki −) −1, then ki − ki −; (vi) if for each i, h(ki +) < 1 and h(ki −) −1, thenk k; (vii) if for each i, h(ki +) < 1 and h(ki −) −1, thenk is a bipolar valued normal multi fuzzy subnearring of r. proof. (i), (ii), (iii), (iv), (v), (vi) and (vii) are trivial. corollary 2.8. if k is a bipolar valued normal multi fuzzy subnearring of a nearring r, then (i) 0k = k, (ii) k = k. proof. the proof follows from definitions 1.8, 1.9 and 1.11. theorem 2.9.let k =  k1 +, k2 +… kn +, k1 −, k2 −… kn − be a bipolar valued multi fuzzy subnearring with degree n of a nearring r. if (a1k) m and (b1k) m are two pseudo bipolar valued multi fuzzy coset of k, then their intersection (a1k) m ∩(b1k) m is also a bipolar valued multi fuzzy subnearring of the nearring r, for every a1, b1r and m in m, where m is a collection of bipolar valued multi fuzzy subset of r. proof. the proof follows from the theorem 2.1 and 2.2. theorem 2.10.let k =  k1 +, k2 +… kn +, k1 −, k2 −… kn − be a bipolar valued multi fuzzy subnearring with degree n of a nearring r. if (a1k) m and (b1k) m are two pseudos bipolar valued multi fuzzy coset of k and m (a1) m(b1) or m(a1)  m(b1), then their union (a1k) m(b1k) m is also a bipolar valued multi fuzzy subnearring of the nearring r, for every a1, b1r and m in m, where m is a collection of bipolar valued multi fuzzy subset of r. proof. the proof follows from the theorem 2.2. theorem 2.11. let k =  k1 +, k2 +, …, kn +, k1 −, k2 −, …, kn − be a bipolar valued multi fuzzy subnearring with degree n of a nearring r. then k is a bipolar valued multi fuzzy subnearring of r if and only if each (ki +, ki −) is a bipolar valued fuzzy subnearring of r. proof. let a1, b1 in r. suppose k is a bipolar valued multi fuzzy subnearring of r, for each i, ki +(a1−b1) min {ki +(a1), ki +(b1)}, ki +(a1b1) min {ki +(a1), ki +(b1)},ki −(a1− b1) ≤ max{ki −(a1), ki −(b1)} and ki −(a1b1) ≤ max{ki −(a1), ki −(b1)}. hence each (ki +, ki −) is bipolar valued fuzzy subnearring of r. conversely, assume that each (ki +, ki −) is bipolar valued fuzzy subnearring of r. as per the definition of bipolar valued multi fuzzy subnearring of r, k is a bipolar valued multi fuzzy subnearring of r. 133 s. muthukumaran and b. anandh references [1] anitha. m.s., muruganantha prasad &arjunan. k, “notes on bipolar valued fuzzy subgroups of a group”, bulletin of society for mathematical services and standards, vol. 2 no. 3 (2013), 52 −59. 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[11] w.r. zhang, bipolar fuzzy sets, proceedings of fuzzy ieee conferences, (1998), 835− 840. 134 ratio mathematica volume 46, 2023 stability of domination in graphs reeja kuriakose* parvathy k s† abstract the stability of dominating sets in graphs is introduced and studied, in this paper. here d is a dominating set of graph g. in this paper the vertices of d and vertices of v − d are called donors and acceptors respectively. for a vertex u in d, let ψd(u) denote the number |n(u) ∩ (v − d)|. the donor instability or simply dinstability ddinst(e) of an edge e connecting two donor vertices v and u is |ψd(u) − ψd(v)|. the d-instability of d, ψd(d) is the sum of d-instabilities of all edges connecting vertices in d. for a vertex u not in d, let φd(u) denote the number |n(u)∩d|. the acceptor instability or simply a-instability adinst(e) of an edge e connecting two acceptor vertices u and v is |φd(u)−φd(v)|. the a-instability of d, φa(d) is the sum of a-instabilities of all edges connecting vertices in v −d. the dominating set d is d-stable if ψd(d) = 0 and a-stable if φa(d) = 0. d is stable, if ψd(d) = 0 and φa(d) = 0. given a non negative integer α, d is α-d-stable, if ddinst(e) ≤ α for any edge e connecting two donor vertices and d is α-a-stable, if adinst(e) ≤ α for any edge e connecting two acceptor vertices. here we study αstability number of graphs for non negative integer α. keywords: domination number; stable domination 2020 mathematics subject classification: 05c69: 1 *department of mathematics, govt. polytechnic college, koratty, thrissur, kerala, india; reejaiykulambil@gmail.com, reeja.kuriakose@smctsr.ac.in †department of mathematics, st. mary’s college, thrissur, kerala, india; parvathy.ks@smctsr.ac.in (corresponding author). 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1081. issn: 1592-7415. eissn: 2282-8214. ©reeja kuriakose et al. this paper is published under the cc-by licence agreement. 244 reeja kuriakose and parvathy k s 1 introduction graphs considered here are the simple undirected graphs. the study of domination in graphs started in 1850 with a problem of finding the minimum number of queens that are needed to place on a chess board such that each field not occupied by queen can be attacked by atleast one. in 1962, ore was the first in publishing about domination in graphs. many researchers including ernie cockayne, michael henning have contributed much to this field. a beautiful survey of results in domination is given in fundamentals of domination in graphs by w.haynes et al. [1998]. the theory of domination has applications in the study of facility location problems, social networking and so on. several types of domination number such as perfect domination number, connected domination number, have been defined and studied. in this paper the elements of a dominating set are called donors and the other vertices are called acceptors. a subset d of vertices in a social network graph with the condition that each member in d dominates almost equally many members in v − d, or that each member in v − d is dominated by almost equally many members in d, or both, plays a key role. this concept of equitable domination in graphs was defined and studied by a. anitha, s. arumugam and mustapha chellali. it is referred in the paper anitha et al. [2011]. in social network problems related to marketing, banking and others the instability affects the system when adjacent acceptors are dominated by unequal number of donors or adjacent donors dominates unequal number of acceptors. this situation become worse when the instability is large. motivated from this idea the concept of α-stability of dominating sets is being introduced. 2 α-d stable domination definition 2.1. let d be a dominating set. for a vertex u in d let ψd(u) = |n(u) ∩ (v −d)|. the donor instability or d-instability of an edge e connecting two donor vertices u and v, ddinst(e)= |ψd(u) − ψd(v)|. let d ⊂ v , the dinstability of d, is the sum of d-instabilities of all edges connecting vertices in d, ψd(d) = ∑ e∈g[d] d d inst(e) . definition 2.2. let d be a dominating set. given a non negative integer α, d is an α-d-stable dominating set, if ddinst(e) ≤ α for any edge e connecting two donor vertices. cardinality of a minimum α-d-stable dominating set is α-d-stable domination number and it is denoted by γαd (g). definition 2.3. a dominating set d is d-stable if ψd(d) = 0. cardinality of a minimum d-stable dominating set is d-stable domination number and it is denoted 245 stability of domination in graphs by γ0d(g). observation 2.1. if α ≥ β, then γ(g) ≤ γαd (g) ≤ γ β d (g). u1 u2 u3 u4 u5 u6 u7 u8 figure 1: graph with γ(g) ≤ γαd (g) ≤ γ β d (g) example 2.1. in figure 1, d = {u1,u2} is the minimum dominating set. ψd(u1) = 4 and ψd(u2) = 2. and ddinst(u1u2) = 2. hence d is a minimum 2-d-stable dominating set. and for α ≥ 2, γαd (g) = 2. if s = {u1,u7,u8}, ψd(u1) = 3 , ψd(u7) = 0 and ψd(u8) = 0. none of them are adjacent. hence γ1d(g) = γ 0 d(g) = 3. observation 2.2. property of being α-d-stable dominating set is neither superhereditary nor hereditary. theorem 2.1. an α-dstable dominating set d is a minimal α-dstable dominating set if and only if for each vertex v in d one of the following conditions holds 1. v is an isolate of d. 2. v has a private neighbour u in v −d. 3. there exist two adjacent vertices u1 and u2 different from v in d, u1 adjacent to v, u2 not adjacent to v and ψd(u1) = ψd(u2) + α. proof. if an α-dstable dominating set d is minimal, then d is an α-dstable dominating set and for each vertex v in d, d −{v} is not an α-dstable dominating set. this means that some vertex u in (v −d) ∪{v} is not dominated by d−{v} or there exist two adjacent vertices u1 and u2 different from v in d with |ψd(u1)−ψd(u2)| ≤ α but |ψd−{v}(u1)−ψd−{v}(u2)| > α. now if some vertex u in (v − d) ∪{v} is not dominated by any vertex in d −{v}, either u = v, means v is an isolate of d or u ∈ v − d. if u is not 246 reeja kuriakose and parvathy k s dominated by d−{v}, then u is adjacent only to vertex v in d. ie, v has a private neighbour u in v −d. if |ψd(u1) − ψd(u2)| ≤ α and |ψd−{v}(u1) − ψd−{v}(u2)| > α, let α = 0, then ψd(u1) = ψd(u2) and |ψd−{v}(u1) − ψd−{v}(u2)| = α + 1. assume ψd−{v}(u1) > ψd−{v}(u2). then u1 is adjacent to v but u2 is not adjacent to v and ψd(u1) = ψd(u2) + α. if α > 0, then assume ψd(u1) > ψd(u2). then ψd−{v}(u1)−ψd−{v}(u2) = α+ 1. then u1 is adjacent to v but u2 is not adjacent to v and ψd(u1) = ψd(u2) + α. conversely, suppose that d is an α-d-stable dominating set and for each vertex v ∈ d, one of the three statements holds. we show that d is a minimal α-d-stable dominating set. if d is not a minimal α-d-stable dominating set, then there exists a vertex v ∈ d such that d−{v} is an α-d-stable dominating set. then each vertex u in (v −d)∪{v} is adjacent with atleast one vertex in d−{v}. then v is not an isolate of d and condition 1 does not hold. and v has no private neighbour in v − d and condition 2 does not hold. d−{v} is an α-d-stable dominating set implies for any two adjacent vertices u1 and u2 in d−{v}, ψd−{v}(u1)−ψd−{v}(u2) ≤ α. hence condition 3 does not hold. hence d is a minimal α-d-stable dominating set. observation 2.3. for non negative integer α, γαd (g) = 1 ⇐⇒ γ(g) = 1. v1 v2 v3 v4 v5 v6 figure 2: graph with γαd (g) = 1 . theorem 2.2. for a graph g and non negative integer α, βo(g) ≥ γαd (g). proof. let s be a maximum independent set. then, every vertex in v −s is adjacent with atleast one vertex in s. thus s is a dominating set. s is an independent set. it follows that s is a d-stable dominating set. hence, βo(g) ≥ γαd (g).for the graph in figure 2, γαd (g) = 2 = βo(g). hence the bound is sharp. theorem 2.3. if d is an α-d-stable dominating set of a graph g and u and v are adjacent vertices in d with d(v) = d(u)+k+α, k ∈ z+, then d contains atleast k elements from (n[v]−n[u]). 247 stability of domination in graphs proof. if d is an α-d-stable dominating set of a graph g and u and v are adjacent vertices in d with d(v) = d(u) + k + α, k ∈ z+, then |ψd(v)−ψd(u)| ≤ α. hence |n[v] ∩ (v − d)| ≤ |n[u] ∩ (v − d)| + α. thus, d(v) − d(u) ≤ |(n[v]−n[u])∩d|+ α. hence d contains atleast k elements from (n[v]−n[u]). corolary 2.1. if d is a d-stable dominating set of a graph g and u and v are adjacent vertices in d with d(v) > d(u), then d contains atleast d(v) − d(u) elements from (n[v]−n[u]). corolary 2.2. if u is a pendant vertex adjacent to v, d is a d-stable dominating set and u,v ∈ d, then n[v] ⊂ d. theorem 2.4. for any non negative integer β, there exist graph g with γ0d(g) > γ1d(g) > γ 2 d(g)...... > γ β d (g) proof. take k = β + 1 construct g as follows, step 1:let h be the complete graph with vertex set {a1,a2, ....,ak}. step 2:let ai = {ai,1,ai,2, ...,ai,i+1} for i = 1,2, ...,k. form g by joining each vertices in ai with ai in h for i = 1,2...,k. let d be a d-stable dominating set, bi = {ai,ai,1,ai,2, ...,ai,i+1} and ci = bi∩d. if ar,as ∈ d with r < s then by corollary 2.1, d contains s−r elements from as and hence by corollary 2.2, n[as] ⊂ d. thus bs ⊂ d. thus ci = bi for all i = 1,2, ...,k. hence d = v (g) or d = a1∪a2∪a3∪......∪as−1∪as+1∪.....∪ak−1∪ak∪ {as}. hence we take d = a1∪a2∪a3∪......∪as−1∪as+1∪.....∪ak−1∪ak∪{as}. thus |ci| = i + 1 for i 6= s and |cs| = 1. then, |d| = 2 + 3 + ..... + (s) + 1 + (s + 2) + .... + (k + 1) = (k+1)(k+2) 2 − (s + 1). hence |d| is minimum when (k+2)(k+1) 2 −(s+1) is minimum. that is when s = k. and d = a1 ∪ a2 ∪ a3 ∪ ...... ∪ ak−1 ∪{ak} will form a dstable dominating set with |d| = (k+2)(k+1) 2 − (k + 1). hence γ0d(g) = (k+2)(k+1) 2 − (k + 1) = (k)(k+1) 2 . similarly, γ0d(g) = (k)(k+1) 2 γ1d(g) = k(k−1) 2 + 1 ... γ β d (g) = (k−β+1)(k−β) 2 + β. figure 3 illustrates the graph with γ0d(g) > γ 1 d(g) > γ 2 d(g). 248 reeja kuriakose and parvathy k s a1 a2 a3 a1,1 a1,2 a2,2 a2,3 a2,2 a3,1 a3,2 a3,4 a3,3 figure 3: graph with γ0d(g) > γ 1 d(g) > γ 2 d(g) 3 α-a-stable domination definition 3.1. let d be a dominating set. for a vertex u not in d, let φd(u) = |n(u) ∩ d|. the acceptor instability or a-instability of an edge e connecting two acceptor vertices u and v is, adinst(e) = |φd(u)−φd(v)|. the a-instability of d, φa(d) is the sum of a-instabilities of all edges connecting vertices in v −d, φa(d) = ∑ e∈g[v−d] a d inst(e). definition 3.2. let d be a dominating set. given a non negative integer α, d is an α-a-stable dominating set, if adinst(e) ≤ α for any edge e connecting two acceptor vertices. cardinality of a minimum α-a-stable dominating set is α-astable domination number and denoted by γαa (g). definition 3.3. the dominating set d is a-stable if φa(d) = 0 . minimum cardinality of an a-stable dominating set is a-stable domination number and denoted by γ0a(g). observation 3.1. if α ≥ β, then γ(g) ≤ γαa (g) ≤ γβa (g) example 3.1. in figure 1, d = {u1,u2} is the minimum dominating set. φd(u3) = φd(u4) = φd(u5) = φd(u6) = φd(u7) = φd(u8) = 1. hence d is a minimum a-stable dominating set and γαa (g) = 2 for all non negative integer α. 249 stability of domination in graphs observation 3.2. property of being α-a-stable dominating set is neither superhereditary nor hereditary. theorem 3.1. an α-astable dominating set d is a minimal α-astable dominating set if and only if for each vertex v in d one of the following conditions holds 1. v is an isolate of d. 2. v has a private neighbour u in v −d. 3. there exist two adjacent vertices u1 and u2 in v-d, u1 adjacent to v, u2 not adjacent to v and φd(u2) = φd(u1) + α. proof. if an α-astable dominating set d is minimal then d is an α-astable dominating set and for each vertex v in d, d − {v} is not an α-astable dominating set. this means that some vertex u in (v − d) ∪{v} is not dominated by d −{v} or there exist two adjacent vertices u1 and u2 in v − d with |φd(u1)−φd(u2)| ≤ α but |φd−{v}(u1)−φd−{v}(u2)| > α. now if some vertex u in (v −d)∪{v} is not dominated by any vertex in d−{v}, either u = v, means v is an isolate of d or u ∈ v −d. if u is not dominated by d −{v}, then u is adjacent only to vertex v in d. ie, v has a private neighbour u in v −d. if |φd(u1) − φd(u2)| ≤ α and |φd−{v}(u1) − φd−{v}(u2)| > α, let α = 0, then φd(u1) = φd(u2) and |φd−{v}(u1) − φd−{v}(u2)| = α + 1. assume φd−{v}(u2) > φd−{v}(u1). then u1 is adjacent to v but u2 is not adjacent to v and φd(u2) = φd(u1) + α. if α > 0, then assume φd(u2) > φd(u1). then φd(u2)−φd(u1) = α and φd−{v}(u2)−φd−{v}(u1) = α+1. then u1 is adjacent to v but u2 is not adjacent to v and φd(u2) = φd(u1) + α. conversely, suppose that d is an α-a-stable dominating set and for each vertex v ∈ d, one of the three statements holds. we show that d is a minimal α-astable dominating set. if d is not a minimal α-a-stable dominating set,then there exists a vertex v ∈ d such that d −{v} is an α-a-stable dominating set. then each vertex u in (v − d) ∪{v} is adjacent with atleast one vertex in d −{v}. then v is not an isolate of d and condition 1 does not hold. and v has no private neighbour in v − d and condition 2 does not hold. if d −{v} is an α-astable dominating set then for any adjacent vertices u1 and u2 in (v −d) ∪{v}, φd−{v}(u2) − φd−{v}(u1) ≤ α. hence condition 3 does not hold. hence d is a minimal α-a-stable dominating set. observation 3.3. for non negative integer α, γαa (g) = 1 ⇐⇒ γ(g) = 1 theorem 3.2. for α ≥ 1, γαa (g) = 2 ⇐⇒ γ(g) = 2 250 reeja kuriakose and parvathy k s proof. if γ(g) = 2, then for a minimum dominating set d, |d| = 2 |d| = 2 ⇒ φd(v) = 1 or φd(v) = 2 ∀v ∈ v −d ⇒ |φd(v1)−φd(v2)| ≤ 1, ∀v1,v2 ∈ v −d ⇒ γαa (g) = 2 conversely, if γαa (g) = 2, then γ(g) 6= 1. if d is a minimum α-astable dominating set , then |d| = 2, and d is a dominating set. thus, γ(g) = 2 theorem 3.3. for any graph g and non negative integer α, γαa (g) ≤ γp(g). and this bound is sharp. proof. if d is a γp set, then |n[v]∩d| = 1, ∀v ∈ (v −d). hence φd(v) = 1, for all v ∈ (v − d). and so d is an a-stable dominating set . thus every perfect dominating set is an a-stable dominating set . hence, γαa (g) ≤ γp(g). for g = p3n, γp(g) = n = γαa (g). so we can see that the bound is sharp. a1 a2 a3 b′1,1 b1,1 b1,1” b2,2” b2,2 b2,1” b2,1 b′2,2 b′2,1 b′3,1 b′3,3 b′3,2 b3,1” b3,1 b3,2 b3,2” b3,3 b3,3” figure 4: graph with γ0a(g) > γ 1 a(g) > γ 2 a(g) 251 stability of domination in graphs theorem 3.4. for any positive integer β, there exist graph g with γ0a(g) > γ1a(g) > γ 2 a(g)...... > γ β a (g). proof. let k = β + 1. construct g as follows step 1:let h be the complete graph with vertex set {a1,a2, ...,ak} step 2: for each i take i copies of p3 with vertex set {bi,j,b′i,j,bi,j”} for j = 1,2, ..., i and join b′i,j with ai for each j = 1,2, ...., i. let aji = {bij,b ′ ij,bij”} for j = 1,2, ...i and ai = {ai}∪∪ij=1{bij,b′ij,bij”} for all i ∈{1,2, ...,k}. let d be an a-stable dominating set. if ai ∈ d then |ai ∩d| ≥ i + 1. let r be the smallest integer such that ar /∈ d. then |ar ∩d| ≥ r. if s > r and as /∈ d, since γainst(ar,as) = 0, b′sj ∈ d for atmost r values of j. and if there exist j for which b′sj /∈ d then bsj,bsj” ∈ d. =⇒ |as ∩d| ≥ r + 2(s− r) = 2s− r ≥ s + 1 hence, |d| ≥ |a1 ∩d|+ |a2 ∩d|+ |a3 ∩d|+ ... + |ar−1 ∩d|+ |ar ∩d|+ |ar+1 ∩d|+ ... + |ak ∩d| ≥ 2 + 3 + ... + (r −1 + 1) + r + (r + 2) + .... + (k + 1) = 1 + 2 + ... + k + k −1 = k(k+1) 2 + (k −1) . thus, γ0d(g) ≥ k(k+1) 2 + (k −1). and d′ = {a1, ....,ak−1}∪∪ki=1{b′i1,b′i2, ...b′ii} is an a-stable dominating set with |d′| = k(k+1) 2 + (k −1). hence, γ0a(g) ≤ k(k+1) 2 + (k −1) thus, γ0a(g) = k(k+1) 2 + (k −1) similarly, γ1a(g) = k(k+1) 2 + (k −2) γ2a(g) = k(k+1) 2 + (k −3) γ3a(g) = k(k+1) 2 + (k −4) ... γβ−1a (g) = k(k+1) 2 + 1 γβa (g) = k(k+1) 2 figure 4 illustrates the graph with γ0a(g) > γ 1 a(g) > γ 2 a(g). 4 α-stable domination definition 4.1. a dominating set d is stable, if ψd(d) = 0 and φa(d) = 0. minimum cardinality of a stable dominating set is called stable domination number and denoted by γ0(g). 252 reeja kuriakose and parvathy k s definition 4.2. if a dominating set d is an α-d-stable dominating set and α-a stable dominating set, then d is called an α-stable dominating set and cardinality of a minimum α-stable dominating set is defined as α-stable domination number and denoted by γα(g) observation 4.1. if a minimum α -a-stable dominating set is an α-d-stable dominating set, then γα(g) = γαa (g). and if a minimum α-dstable dominating set is an α-astable dominating set, then γα(g) = γαd (g). observation 4.2. if α ≥ β, then γ(g) ≤ γα(g) ≤ γβ(g). definition 4.3. minimum α so that γα(g) = γ(g) is called stable dominating index and denoted by isd(g). example 4.1. in figure 1, the minimum d-stable dominating set {u1,u7,u8} is an a-stable dominating set. hence {u1,u7,u8} is a minimum stable dominating set and γ0(g) = 3. a minimum 1-d-stable dominating set {u1,u7,u8} form a 1-astable dominating set and hence {u1,u7,u8} is a minimum 1-stable dominating set and γ1(g) = 3. and minimum dominating set {u1,u2} is a 2-astable dominating set and a 2-dstable dominating set. {u1,u2} form a minimum 2-stable dominating set. hence, γ2(g) = 2. and ∀α ≥ 2, γα(g) = 2 = γ(g). hence, isd(g) = 2. compliment of a minimum α-stable dominating set need not be an α-stable dominating set. in graph figure 1, {u1,u2,u3} is a minimum 1-stable dominating set but its compliment is not a 1-stable dominating set. observation 4.3. property of being α-stable dominating set is neither superhereditary nor hereditary. theorem 4.1. for any graph g and for any non-negative integer α, γ(g) = 1 ⇐⇒ γα(g) = 1. proof. if γ(g) = 1, then the single vertex set {v} which dominates all vertices of g, is an α-d-stable dominating set and an α-a-stable dominating set. then γα(g) = 1. also any α-stable dominating set is a dominating set. so, if γα(g) = 1 then γ(g) = 1. lemma 4.1. for any graph g and for any non negative integer α, γα(g) = n ⇐⇒ g = kn. proof. if g 6= kn, there is atleast one vertex v with d(v) ≥ 1. then v −{v} is an α-stable dominating set. this means that γα(g) ≤ n − 1. hence if γα(g) = n, then g = kn. if g = kn, then γα(g) = n trivially. 253 stability of domination in graphs theorem 4.2. for a graph g with δ(g) ≥ 1, γα(g) ≤ n−1. proof. from lemma 3.9 it is clear that γα(g) ≤ n−1. theorem 4.3. for any non negative integer α, γα(g) = γ(g) for the following graphs • complete graph kn • path pn • cycle cn • wheel graph wn • helm graph hn proof. in these graphs minimum dominating set d, form an α-stable dominating set. hence α-stable domination number is same as its domination number. theorem 4.4. let g and h be two graphs of order n1 and n2 , then for any non-negative integer α , • γαa (g2h) ≤ min{n1γαa (h),n2γαa (g)} • γαd (g�h) ≤ min{n1γ α d (h),n2γ α d (g)} • γα(g�h) ≤ min{n1γα(h),n2γα(g)}. proof. let sh be a minimum α-a-stable dominating set of h. let us see that s = v (g) ×sh is an α-a-stable dominating set of g�h . if (u,v) ∈ (v (g) × v (h)) −s. then (u,v) is adjacent to atleast one vertex in s. and if (u1,v1) ∈ (v (g)×v (h))−s and (u2,v2) ∈ (v (g)×v (h))−s and (u1,v1) adjacent to (u2,v2). then, φs(u1,v1) = |{(u,v) ∈ s : (u1,v1) adjacent to (u,v)}| = |{(u,v) ∈ s : u1 = u and v1 adjacent to v}∪ {(u,v) ∈ s : u1 adjacent to u and v1 = v}| = |{(u,v) ∈ s : u1 = uand v1 adjacent tov}| = φsh (v1). φs(u2,v2) = φsh (v2). and |φs(u1,v1)−φs(u2,v2)| = |φsh (v1)−φsh (v2) ≤ α. hence s is an αa-stable dominating set of g�h. 254 reeja kuriakose and parvathy k s similarly, if sg is a minimum α-a-stable dominating set of g, then sg × v (h) is an α-a-stable dominating set of g�h. thus, γαa (g2h) ≤ min{n1γαa (h),n2γαa (g)}. let sh be a minimum α-d-stable dominating set of h. let s = v (g) ×sh. if (u,v) ∈ (v (g)×v (h))−s, then (u,v) is adjacent to atleast one vertex in s. and if (u1,v1) ∈ s and (u2,v2) ∈ s and (u1,v1) adjacent to (u2,v2). then, ψs(u1,v1) = |{(u,v) ∈ (v (g)×v (h))−s : (u1,v1) adjacent to (u,v)}| = |{(u,v) ∈ (v (g)×v (h))−s : u1 = u and v1 adjacent to v}∪ {(u,v) ∈ (v (g)×v (h))−s : u1 adjacent to u and v1 = v}| = |{(u,v) ∈ (v (g)×v (h))−s : u1 = u and v1 adjacent to v}| = ψsh (v1). similarly ψs(u2,v2) = ψsh (v2). thus |ψs(u1,v1)−ψs(u2,v2)| = |ψsh (v1)−ψsh (v2) ≤ α thus s is an α-d-stable dominating set of g2h. similarly, if sg is a minimum α-d-stable dominating set of g, sg ×v (h) is an α-d-stable dominating set of g2h. thus, γαd (g2h) ≤ min{n1γ α d (h),n2γ α d (g)}. hence, γα(g2h) ≤ min{n1γα(h),n2γα(g)}. remark 4.1. the bound in theorem 3.12 is attained if g = kn and h = k2; because γα(kn2k2) = 2 = min{2γα(kn),nγα(k2)}. theorem 4.5. for any two graphs g and k and non negative integer α, α-stable domination number of its corona, γα(gok) = |v (g)|. proof. let {v1,v2, .....,vn} be the vertices of g and {u1,u2, ....um} be the vertices of k. ki be the ith copy k in gok. to make sure that each vertex of ki is dominated, we need atleast one vertex of ki or vi. thus the dominating set contains atleast n vertices. let d = {v1,v2, .......,vn}. then each vertex of v − d is adjacent with exactly one vertex of d and each vertex of d dominates exactly m vertices in v − d. and so ψd(v) = m, ∀v ∈ d and φd(v) = 1, ∀v ∈ v − d. therefore, |φd(v1) − φd(v2)| = 0,∀v1,v2 ∈ v − d and |ψd(v1) − ψd(v2)| = 0,∀v1,v2 ∈ d. and so d is a minimum α-stable dominating set. thus γα(gok) = n = |v (g)|. 255 stability of domination in graphs 5 conclusions motivated from this idea the concept of α-stability of dominating sets is being introduced. the stability of dominating sets in graphs is introduced and studied, in this paper. in social network problems related to marketing, banking and others the instability affects the system when adjacent acceptors are dominated by unequal number of donors or adjacent donors dominates unequal number of acceptors. several types of domination number have been defined and studied. this situation become worse when the instability is large. here we study αstability number of graphs for non negative integer α. instability of domination in graphs the concept of stability of domination in graphs are defined. the open problems are characterize the graphs with isd(g) = 0. characterize the graphs for which the compliment of minimum α-stable dominating set is an α-stable dominating set. develop an algorithm to find the α-stable domination number of a graph. in future we extended this topic. references a. anitha, s. arumugam, and m. chellali. equitable domination in graphs. discrete mathematics, algorithms and applications, 3:311–321, 2011. t. w.haynes, s. t.hedetniemi, and p. j.slater. fundamentals of domination in graphs. marcel dekker, new york, 1998. 256 approach of the value of a rent when non-central moments of the capitalization factor are known: an r application with interest rates following normal and beta distributions ratio mathematica volume 43, 2022 fair fuzzy matching in middle fuzzy graph s.yahya mohamed* s.suganthi† abstract a fuzzy matching is a set of edges in which an edge does not incident on a vertex with same membership value. if every vertex of fuzzy graph is m-plunged then the fuzzy matching is called as fair fuzzy matching. in this paper, we introduce the new concept of fair fuzzy matching in middle fuzzy graph. we discussed some properties based on these concepts in an absolute fuzzy labeling graph. keywords: fuzzy middle graph; fuzzy fair matching; condensation. 2010 ams subject classification: 03e72, 05c72, 05c78‡ * assistant professor, department of mathematics, government arts college, trichirappalli, india; yahya_md@yahoo.com. † assistant professor, department of science and humanities, dhanalakshimi srinivasan engineering college, perambalur, india; sivasujithsuganthi@gmail.com. ‡ received on september 15, 2022. accepted on december 18, 2022. published on december 30, 2022. doi:10.23755/rm.v39i0.820. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. s.yahya mohamed and s.suganthi 1. introduction the concept of fuzzy set was introduced by lotfi a. zadeh in 1965[13]. he developed a mathematical theory to deal with uncertainty. the advantage of replacing the classical sets by zadeh’s fuzzy sets is that it gives greater accuracy. rosenfeld [8] introduced the notion of fuzzy graphs in the year 1975. fuzzy matching concept in fuzzy graph is introduced by s.yahya mohamed and s. suganthi [11] and complete matching domination in fuzzy labelling graph[12]. middle graph of a fuzzy graph is an unary operation. it is used to extend the graph together with its fair fuzzy matching. here we consider an absolute fuzzy graph with even number of vertices because fair fuzzy matching exists only for even number of vertices. 2. preliminaries definition 2.1 let 𝛾 be a non-empty set. a fuzzy graph 𝐺 = (𝛼, 𝛽) is a combination of two functions, 𝛼: 𝛾 → [0,1] and 𝛽: 𝛾 × 𝛾 → [0,1] where for all 𝑢, 𝑣 belongs to 𝛾 we have 𝛽(𝑢, 𝑣) ≤ min{𝛼(𝑢), 𝛼(𝑣)}. definition 2.2 two edges of a fuzzy labeling graph 𝐺 = (𝛼, 𝛽) is said to be neighbors to each other if they incident on a vertex with same membership value. definition 2.3 a fuzzy subset 𝑀 of 𝛽(𝑣𝑖 , 𝑣𝑖+1), 1 ≤ 𝑖 ≤ 𝑛 is called a fuzzy matching 𝑴 in fuzzy labeling graph 𝐺 = (𝛼, 𝛽) if its elements are links (neither self loop nor parallel edges) and no two are neighboring in 𝐺. the two ends of an edge in 𝑀 are said to be fuzzy tied under 𝑀. example 2.4 figure 2.1 in figure 2.1, 𝛽 = {𝑒1(0.1), 𝑒2(0.3), 𝑒3(0.5)}, and 𝑓𝑀 = {𝑒1(0.1), 𝑒3(0.5)}. definition 2.5 the vertex v is said to be 𝑴 -fuzzy plunged or fuzzy plunged by 𝑴 if it belongs to the circumstance of the elements of matching 𝑀. fair fuzzy matching in middle fuzzy graph example 2.6 consider the following fuzzy graph given in figure 2.2 figure 2.2 in fig 2.2 𝑀 = {𝑒3(0.5), 𝑒5(0.14)} is one of the matching in 𝐺. then the vertices {𝑣1(0.6), 𝑣3(0.52), 𝑣4(0.16) and 𝑣5(0.26)} are fuzzy plunged by 𝑴. definition 2.7 every vertex of the fuzzy labeling graph is fuzzy 𝑀plunged then the fuzzy matching m is said to be fuzzy fair matching. it is denoted by𝐹𝑓𝑚. example 2.8 consider the following fuzzy graph given in figure 2.3 figure 2.3 in fig 2.3, 𝑀 = {𝑒2(0.5), 𝑒4(0.05)} is one of the fuzzy matching in fuzzy labeling graph 𝐺 in which all vertices are fuzzy plunged by 𝑀. hence, 𝑀 is a fuzzy fair matching. definition 2.9 let 𝑀 be a fuzzy matching in a fuzzy labeling graph 𝐺 = (𝛼, 𝛽). then 𝑀 −fuzzy interchanging path (mfip) is a path whose edges appear alternatively in (𝛽 − 𝑀) and 𝑀. s.yahya mohamed and s.suganthi definition 2.10 an absolute fuzzy labeling graph 𝐺 = (𝛼, 𝛽) is a fuzzy graph 𝐺 in which 𝛽 (𝑣𝑖 , 𝑣𝑗 ) > 0 for all 𝑣𝑖 , 𝑣𝑗  𝑉 and 𝛽(𝑣𝑖 , 𝑣𝑗 ) = min{𝛼(𝑣𝑖 ), 𝛼(𝑣𝑖 )} for all 𝑣𝑖 , 𝑣𝑗 ∈ 𝑉. it is denoted by afg. example 2.11 consider the following fuzzy graph given in figure 2.4 figure 2.4 3. main results definition 3.1 let 𝐺 = (𝛼, 𝛽) be a fuzzy graph and 𝑴 be a fair fuzzy matching in 𝐺 = (𝛼, 𝛽).then the condensation of 𝑴 is defined as each fair fuzzy matching convert into a single vertex. definition 3.2 the minimum number of fair matching to cover all the edges of a fuzzy graph is called the condensation number. definition 3.3 let 𝐺 = (𝛼, 𝛽) be a fuzzy graph and m be a fair fuzzy matching in 𝐺 = (𝛼, 𝛽).the middle fuzzy graph 𝑴(𝑮) contains (i) the vertex set as the union of all vertices in 𝐺 and the condensation of fair matching. (ie.,) 𝑉(𝐺) ∪ 𝐶(𝑀). (ii) the edge set as there exists an edge between 𝑣𝑖 and 𝑣𝑗 with 𝛽(𝑣𝑖,𝑣𝑗 ) > 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 , 𝑣𝑗 ∈ 𝐺 and also each 𝐶(𝑀) has an edge 𝛽(𝑣𝑖,𝑣𝑗 ) > 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 , 𝑣𝑗 ∈ 𝐺, 𝛽(𝑣𝑖 , 𝑣𝑗 ) = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 , 𝑣𝑗 ∈ 𝐶(𝑀). fair fuzzy matching in middle fuzzy graph theorem 3.4 let 𝐺 = (𝛼, 𝛽) be a fuzzy absolute labelling graph with fuzzy fair matching 𝑀. then the number of edges in 𝑀(𝐺) is 3𝑛(𝑛 − 1)/2 where 𝑛 is even. proof: consider 𝐺 = (𝛼, 𝛽) be a fuzzy absolute labelling graph with fuzzy fair matching 𝑀. to find the number of edges in middle fuzzy graph of 𝐺. the number of edges in 𝑀(𝐺) = 𝛽(𝑣𝑖 , 𝑣𝑗 ) > 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 , 𝑣𝑗 ∈ 𝐺 + number lines joining 𝐶(𝑀) to all vertices in 𝐺 = 𝑛(𝑛 − 1)/2 + |𝐶(𝑀)| |v(g)| = 𝑛(𝑛−1) 2 + (𝑛 − 1)𝑛 = 𝑛(𝑛 − 1)[1/2 + 1] hence the number of edges in 𝑀(𝐺) is 3𝑛(𝑛 − 1)/2. theorem 3.5 every middle graph of an absolute fuzzy graph with 𝑛 vertices has (𝑛 − 1) condensation number. here 𝑛 is even. proof: let 𝐺 = (𝛼, 𝛽) be an absolute fuzzy graph and 𝑀 be the fuzzy fair matching in 𝐺. now, we can construct the fair matching to cover all edges of 𝐺. the 𝐸 − 𝑐𝑜𝑢𝑛𝑡 of each fair matching is 𝑛/2 and the number of edges in an absolute fuzzy graph is 𝑛(𝑛 − 1)/2. for 𝑛 = 4, three distinct fair matching need to cover all edges of 𝐺 and 𝐸 − 𝑐𝑜𝑢𝑛𝑡 of each fair matching is 2. hence 𝐶(𝑀) contains 3 points. for 𝑛 = 6, 𝐸 − 𝑐𝑜𝑢𝑛𝑡 of each fuzzy fair matching is 3 and five distinct fair matching cover all edges of 𝐺. then these fair matching are converted into five points. hence the condensation number is 5. similarly for 𝑛 = 8, seven fair matching need to cover all edges of 𝐺.then these fair matching are converted into seven points .hence the condensation number is 7. in general, an absolute fuzzy labelling graph with 𝑛 vertices has (𝑛 − 1) condensation number. theorem 3.6 let 𝐺 = (𝛼, 𝛽) be an absolute fuzzy graph with even number of vertices and 𝑀 be a fair matching in 𝐺. then middle graph of 𝐺 does not contain fair fuzzy matching. proof: let 𝐺 = (𝛼, 𝛽) be an absolute fuzzy graph with even number of vertices and 𝑀 be a fair matching in 𝐺. an absolute fuzzy graph with 𝑛 vertices has (𝑛 − 1) condensation point. now we construct the middle graph of 𝐺. also by the definition of middle graph these (𝑛 − 1) points adjacent to all vertices in 𝐺. hence 𝑀(𝐺) contains the vertex set as union of 𝑉(𝐺) 𝑎𝑛𝑑 𝐶(𝑀). s.yahya mohamed and s.suganthi therefore, 𝑀(𝐺) contains (even + odd) number of vertices. always 𝑀(𝐺) contains odd number of vertices. also 𝑀(𝐺) is not an absolute graph. then we can find a fuzzy matching which does not cover all vertices of 𝑀(𝐺) because 𝐶(𝑀) adjacent to all vertices in 𝐺. also there exist β(vi, vj) > 0 ∀ vi ∈ c(m)and vj ∈ g. so that the required matching is not fair. hence 𝑀(𝐺) does not have fair fuzzy matching. definition 3.7 the number of edges incident to 𝒗 is called 𝒈𝒓𝒂𝒅𝒆 𝒐𝒇 𝒗. theorem 3.8 let 𝐺 = (𝛼, 𝛽) be an absolute fuzzy graph with even number of vertices and 𝑀 be a fair matching in 𝐺. then maximum grade is 2(𝑛 − 1) and minimum grade is 𝑛. proof: let 𝐺 = (𝛼, 𝛽) be an absolute fuzzy graph with even number of vertices and 𝑀 be a fair matching in 𝐺. by the definition, the middle fuzzy graph 𝑀(𝐺) contains (i) the vertex set as the union of all vertices in 𝐺 and the condensation of fair matching (ie.,) 𝑉(𝐺) ∪ 𝐶(𝑀). (ii) the edge set as there exists an edge between 𝑣𝑖 and 𝑣𝑗 with 𝛽(𝑣𝑖,𝑣𝑗 ) > 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 , 𝑣𝑗 ∈ 𝐺 and also each 𝐶(𝑀) has an edge 𝛽(𝑣𝑖 , 𝑣𝑗 ) > 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖, 𝑣𝑗 ∈ 𝐺 , 𝛽(𝑣𝑖 , 𝑣𝑗 ) = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖, 𝑣𝑗 ∈ 𝐶(𝑀). but β(vi, vj) = 0 ∀ vi, vj ∈ c(m) . hence the maximum grade of 𝑀(𝐺) 𝑖𝑠 2(𝑛 − 1) also the minimum grade of 𝑀(𝐺) 𝑖𝑠 𝑛. 4 conclusions in this paper, we introduced the new concept of fair fuzzy matching in an middle fuzzy graph. also we defined condensation of fair fuzzy matching and condensation point. in future, we will find total fuzzy graph, central fuzzy graph using fuzzy matching and fair fuzzy matching. references [1] anjaly kishore, sunitha. m.s, chromatic number of fuzzy graph. annals of fuzzy mathematics and informatics, 7(4), 543-551. 2013. [2] arindam dey, dhrubajyoti ghos, anita pal. edge coloring of complement fuzzy graph. international journal of modern engineering research. 2. 2012. [3] frank harary. graph theory. indian student edition. narosa/addison wesley. 1988. [4] logapriya. b , pandian.k. domination parameters of middle graph of sunlet graph. journal of innovative technology and exploring engineering. 8. 2019. [5] muddebihal.m.h, naila anjum. independent middle domination number in graph. international journal of recent scientific research. 6(1), 2434-2437. 2015. fair fuzzy matching in middle fuzzy graph [6] v. nivethana, a. parvathi. fuzzy total coloring and chromatic number of a complete fuzzy graph, international journal of emerging trends in engineering and development, 6(3), 377-384. 2013. [7] k. ranganathan, r.balakrishnan. a text book of graph theory. springer. [8] a. rosenfeld. fuzzy graphs. fuzzy sets and their applications to cognitive and decision process. l. a. zadeh, k. s. fu, k. tanaka and m. shimura, eds., academic press, new york. 75-95. 1975. [9] s.samanta, t.pramanik, m.pal. fuzzy coloring of fuzzy graphs. afrika matematika, 27(1-2), 37-50. 2016. [10] r.seethalakshmi and r.b. gnanajothi. a note on perfect fuzzy matching. international journal on pure and applied mathematics, 94(2), 155-161. 2014. [11] s.yahya mohamad, s.suganthi. properties of fuzzy matching in set theory. journal of emerging technologies and innovative research, 5(2). 2018. [12] s.yahya mohamad, s.suganthi. matching and complete matching domination in fuzzy labelling graph. journal of applied science and computations. 5(10). 2018. [13] l. a. zadeh. fuzzy sets. information and control. 8, 338-353. 1965. ratio mathematica volume 44, 2022 various product on multi fuzzy graphs r. muthuraj 1 k. krithika 2 s. revathi 3 abstract in this paper, the definition of complement of multi fuzzy graph, direct sum of two multi fuzzy graphs are given and derived some theorems related to them. also, we examine the different product on multi fuzzy graphs such as direct product, cartesian product, strong product, composition, corona product and some properties are analyzed. key words: multi fuzzy graph, complement of multi fuzzy graph, direct sum, direct product, cartesian product, strong product, composition, corona product, ams subject classification: 03e72, 05c72, 05c07 4 1 pg and research department of mathematics, h.h. the rajah’s college, pudukkottai -622001, affiliated to bharathidasan university, tiruchirappalli, tamilnadu, india. e-mail: rmr1973@yahoo.co.in;rmr1973@gmail.com 2 part time research scholar, pg and research department of mathematics, h.h. the rajah’s college, pudukkottai, 622001, (affiliated to bharathidasan university, tiruchirappalli), tamilnadu, india. department of mathematics, dhaanish ahmed college of engineering, chennai – 601301, tamilnadu, india e-mail: krithika.cv1982@gmail.com 3 department of mathematics, saranathan college of engineering, trichy – 620012, tamilnadu, india. e-mail: revathi.soundar@gmail.com 4 received on june 26 th, 2022. accepted on sep 1st, 2022. published on nov 30th, 2022. doi: 10.23755/rm.v44i0.911. issn: 1592-7415. eissn: 2282-8214. ©the authors.this paper is published under the cc-by license agreement. 231 mailto:rmr1973@gmail.com mailto:krithika.cv1982@gmail.com mailto:revathi.soundar@gmail.com r. muthuraj, k. krithika and s. revathi 1. introduction the notion of fuzzy set and fuzzy relations were proposed by l.a zadeh [18] in 1965 for representing uncertainty. the concept of fuzzy graph was first introduced by kauffman [2] from the concept fuzzy relation introduced by l.a zadeh in 1973. in 1975, rosenfeld [14] developed the theory of fuzzy graph and several fuzzy analogs of graph theoretic concepts such as paths, cycles and connectedness. thereafter in 1987, bhattacharya [1] defined some remarks on fuzzy graphs. the operations of union, join, cartesian product and composition of two fuzzy graphs were defined by mordeson. j.n, and prem chand s. nair, [3] in 2000. after that m.s. sunitha and a. vijayakumar [17] extended the concept of operations on fuzzy graph in 2002.sebu sebastian, t.v. ramakrishnan [15] defined multi fuzzy set in 2010. radha. k and arumugam. s [11, 12] defined the direct sum of two fuzzy graphs in 2013 and strong product of two fuzzy graphs in 2014.ozgecolakogluhavare and hamza menken [10] defined the corona product of two fuzzy graphs in 2016.in 2020r.muthuraj and s. revathi [5] introduced the concept of multi fuzzy graph which is the extension of a fuzzy graph with single phenomenon into a multi-phenomenon which suits to describe the real-life problems in a better manner than fuzzy graph. later on, and multi anti fuzzy graph defined by muthuraj. ret.al [6]. in this paper complement of multi fuzzy graph, direct sum of two multi fuzzy graphs and various product on multi fuzzy graphs are defined and proved some theorems related to them. 2. preliminaries definition 2.1 [2] a fuzzy graph ),( g defined on the underlying crisp graph ),( evg   where vve  is a pair of functions ]1,0[: v and ]1,0[: vv ,  is a symmetric fuzzy relation on  such that  )(),(min)( vuuv   for vvu , definition 2.2 [15] let x be a non-empty set. a multi fuzzy set a in x is defined as a set of ordered sequences:  xxxxxxa i  :)...)(),......(),(,( 21  where ]1,0[: x i  for all i. definition 2.3 [5] a multi fuzzy graph (mfg) of dimension m defined on the underlying crisp graph ),( evg  where vve  , is denoted as  ),...,(),,...,( 2121 mm g  and ]1,0[: v i  and ]1,0[: vv i  , i is a symmetric fuzzy relation on i  such that  )(),(min)( vuuv iii   for all mi ....3,2,1 where vvu , and euv 232 various product on multi fuzzy graphs 3. complement of multi fuzzy graph definition 3.1 the complement of a multi-fuzzy graph )),....,(),,....,(( 2121 mmg  of dimension m is a multi-fuzzy graph )),...,(),,...,(( 2121 mm g  of dimension m where ii   and ),())()((),( vuvuvu iiii   for all vvu , and for all mi ....3,2,1 example 3.2 theorem 3.3 if g is a strong multi fuzzy graph then g is also strong multi fuzzy graph. proof: let evu , . then ),())()((),( vuvuvu iiii   ))()(())()(( vuvu iiii   0 since g is strong. let evu , . then ),())()((),( vuvuvu iiii   0))()((  vu ii  )()(( vu ii   theorem 3.4 the complement of complete multi fuzzy graph is a null graph. proof: let ),( evg  be a multi-fuzzy graph with the underlying crisp graph ),( evg  is complete. ie., euvvvuvuvu iii  &,))()((),(  let evu , ),())()((),( vuvuvu iiii   ))()(())()(( vuvu iiii   0 since g is complete. so, we have the edge set of g is empty when g is a complete multi fuzzy graph. hence the complement of complete multi fuzzy graph is a null graph. 4. various product on multi fuzzy graphs in this section  ),...,(),,...,( 21211 mm g  denotes the multi fuzzy graph with dimension m with the underlying crisp graph ),( 111 evg   and  ),...,(),,...,( 21212 nn g  denotes the multi fuzzy graph with dimension n with the underlying crisp graph ),( 222 evg   233 r. muthuraj, k. krithika and s. revathi definition 4.1 the operation direct sum between two mfg 1 g and 2 g is defined as follows, )),......,(),,......,(( 2211221121 kkkk gg   with the underlying crisp graph ),( 212121 eevvgg   ,          21 12 21 )(),(max )( )( ))(( vvuifuu vvuifu vvuifu u ii i i ii     for all ki ....3,2,1 and       2 1 ),(),( ),(),( ),)(( evuifvu evuifvu vu i i ii    for all ki ....3,2,1 if , let k = max (m, n). suppose , then let us introduce n – m membership values of multi fuzzy graph g1 into 0 so as to convert the multi fuzzy graphs g1 and g2 have the same dimension as k. theorem 4.2 the direct sum of two multi fuzzy graph is also a multi-fuzzy graph, proof: let  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nn g  be the multi fuzzy graph with dimension m and n respectively to prove: 21 ggg  is also multi fuzzy graph with dimension k where k=max (m, n)          21 12 21 )(),(max )( )( ))(( vvuifuu vvuifu vvuifu u ii i i ii     case (i): let 1 ),( evu  ),(),)(( vuvu iii   ))(),(min( vu ii  )))((),)(min(( vu iiii   )))((),)(min((),)(( vuvu iiiiii   case (ii): let 2 ),( evu  ),(),)(( vuvu iii   ))(),(min( vu ii  )))((),)(min(( vu iiii   )))((),)(min((),)(( vuvu iiiiii   definition4.3 the operation direct product between two mfg 1 g and 2 g is defined as follows, )),......,(),,......,(( 2211221121 kkkk gg   with the underlying crisp graph ),( 21 evgg   where 21 vvv  and   2211212211 ),(&),/(),(),,( evveuuvuvue  with   211121111111 ),(&,)(),(min),)(( vvvuvvvuallforvuvu iiii     22112121212211 ),(&),(),(),,(min)),(),,)((( evveuuallforvvuuvuvu iiii   for all i = 1, 2, 3, … k. if nm  ,let k = min (m, n). suppose nm  , then we take first m dimensions for 2 g so as to convert the mfg 1 g and 2 g have the same dimension k. 234 various product on multi fuzzy graphs example 4.4 figure1 figure2 theorem 4.5 direct product of two multi fuzzy graph is also a multi-fuzzy graph. proof: let  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nn g  be the multi fuzzy graph with dimension m and n respectively to prove: 21 ggg  is a multi-fuzzy graph of dimension k where k=min (m, n)  )(),(min),)(( 1111 vuvu iiii    ),(),,(min)),(),,)((( 21212211 vvuuvuvu iiii    )(),(min()),(),(min(min 2121 vvuu iiii  ))()((())()(( 2121 vvuu iiii   ))()((())()(( 2211 vuvu iiii   )),)((()),)((( 2211 vuvu iiii   )),)((,),)(min(()),(),,)((( 22112211 vuvuvuvu iiiiii   . theorem 4.6 if 1 g and 2 g are strong multi fuzzy graphs then 21 gg  is also a strong multi fuzzy graph. proof:  )(),(min),)(( 1111 vuvu iiii    ),(),,(min)),(),,)((( 21212211 vvuuvuvu iiii    )(),(min()),(),(min(min 2121 vvuu iiii  ))()((())()(( 2121 vvuu iiii   ))()((())()(( 2211 vuvu iiii   )),)((()),)((( 2211 vuvu iiii   235 r. muthuraj, k. krithika and s. revathi )),)((,),)(min(()),(),,)((( 22112211 vuvuvuvu iiiiii   remark: if 1 g and 2 g are complete multi fuzzy graphs then 21 gg  is not a complete multi fuzzy graph. definition 4.7 the operation cartesian product between two mfg 1 g and 2 g as follows, )),...,(),,...,(( 2211221121 kkkk gg   with the underlying crisp graph ),( 21 evgg   where 21 vvv  and   12121221212211 ),(,),(,/),)(,( euuvvorevvuuvuvue  with   211121111111 ),(&)(),(min),)(( vvvuvvandvuallforvuvu iiii            12122121 22112121 2211 ),(&,)(),,(min ),(&,),(),(min )),(),,)((( euuvvallforvvvvuu evvvuallforuuuvvu vuvu ii ii ii    for all i= 1, 2, 3, ... k if nm  let k = min (m, n). suppose nm  then we take first m dimensions for 2g so as to convert the mfg 1 g and 2 g have the same dimension k. example 4.8 figure 3 figure 4 theorem 4.9 cartesian product of two multi fuzzy graph is also a multi-fuzzy graph. proof: let  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nn g  be the multi fuzzy graph with dimension m and n respectively to prove: 21 ggg  is a multi-fuzzy graph of dimension k where k=min (m, n)  )(),(min),)(( 1111 vuvu iiii    ),(),(min)),(),,)((( 2121 vvuvuvu iiii   236 various product on multi fuzzy graphs   )(),(min),(min 21 vvu iii      )(),(min,)(),(minmin 21 vuvu iiii   ),)((),,)((min 21 vuvu iiii    )(),,(min)),(),,)((( 2121 vuuvuvu iiii     )(,)(),(minmin 21 vuu iii      )(),(min,)(),(minmin 21 vuvu iiii   ),)((),,)((min 21 vuvu iiii   theorem 4.10 cartesian product of two strong multi fuzzy graph is also a strong multi fuzzy graph. proof: let  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nn g  be the multi fuzzy graph with dimension m and n respectively to prove: 21 ggg  is a multi-fuzzy graph of dimension k where k=min (m,n)  )(),(min),)(( 1111 vuvu iiii    ),(),(min)),(),,)((( 2121 vvuvuvu iiii     )(),(min),(min 21 vvu iii      )(),(min,)(),(minmin 21 vuvu iiii   ),)((),,)((min 21 vuvu iiii    )(),,(min)),(),,)((( 2121 vuuvuvu iiii     )(,)(),(minmin 21 vuu iii      )(),(min,)(),(minmin 21 vuvu iiii   ),)((),,)((min 21 vuvu iiii   remark: if 1 g and 2 g are complete multi fuzzy graphs then 21 gg  is not a complete multi fuzzy graph. theorem 4.11 if 21 gg  is a strong multi fuzzy graph then at least one 1g or 2g is a strong multi fuzzy graph. proof: suppose assume that the contrary that 1 g and 2 g are not strong fuzzy graphs. )()(),( 1111 vuvu iii   and )()(),( 2222 vuvu iii   (1) without loss of generality, we assume that )()()(),(),( 1111122 uvuvuvu iiiii   let   12121221212211 ),(,),(,/),)(,( euuvvorevvuuvuvue  consider evuvu ),)(,( 2211 , by definition of 21 gg  & inequality (1) )()()(),()()),(),,)((( 2112112111 vvuvvuvuvu iiiiiii   (2) )()(),)(( 1111 vuvu iiii   & )()(),)(( 2121 vuvu iiii   )()()()(),)((),)(( 21112111 vuvuvuvu iiiiiiii   )()()( 211 vvu iii   (3) 237 r. muthuraj, k. krithika and s. revathi from (2) and (3), ),)((),)(()()()()),(),,)((( 21112112111 vuvuvvuvuvu iiiiiiiii   ),)((),)(()),(),,)((( 21112111 vuvuvuvu iiiiii   this implies that 21 gg  is not a strong multi fuzzy graph. this gives a contradiction. so, if 21 gg  is a strong multi fuzzy graph then atleast one 1 g or 2 g is a strong multi fuzzy graph. definition 4.12 the operation strong product between two mfg 1 g and 2 g is defined as follows, )),...,(),,...,(( 2211221121 kkkk gg   with the underlying crisp graph ),( 21 evgg   where 21 vvv  and   22112112121221212211 ),(&),(),(,),(,/),)(,( evveuuoreuuvvorevvuuvuvue  with   211121111111 ),(&)(),(min),)(( vvvuvvandvuallforvuvu iiii                2211212121 12122121 22112121 2211 ),(&),(),(),,(min ),(&,)(),,(min ),(&,),(),(min )),(),,)((( evveuuallforvvuu euuvvallforvvvvuu evvvuallforuuuvvu vuvu ii ii ii ii     for all i= 1, 2, 3, ... k if nm  , let k = min (m, n). suppose nm  , then we take first m dimensions for 2 g so as to convert the mfg 1 g and 2 g have the same dimension k. example 4.13 figure 5 figure 6 theorem 4.14 strong product of two multi fuzzy graph is also a multi-fuzzy graph. proof: let  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nn g  be the multi fuzzy graph with dimension m and n respectively 238 various product on multi fuzzy graphs to prove: 21 ggg  is a multi-fuzzy graph of dimension k where k=min (m, n)   211121111111 ),(&)(),(min),)(( vvvuvvandvuallforvuvu iiii    ),(),(min)),(),,)((( 2121 vvuvuvu iiii     )(),(min),(min 21 vvu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    )(),,(min)),(),,)((( 2121 vuuvuvu iiii     )(,)(),(minmin 21 vuu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    ),(),,(min)),(),,)((( 21212211 vvuuvuvu iiii       )(),(min,)(),(minmin 2121 vvuu iiii  ))()(())()(( 2121 vvuu iiii   ))()(())()(( 2211 vuvu iiii    ),)((),,)((min 2211 vuvu iiii   theorem 4.15 if 1 g and 2 g are strong multi fuzzy graphs then 21 gg  is also a strong multi fuzzy graph. proof: let  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nng  be the multi fuzzy graph with dimension m and n respectively to prove: 21 gg  is a strong multi fuzzy graph of dimension k where k=min (m, n)   211121111111 ),(&)(),(min),)(( vvvuvvandvuallforvuvu iiii    ),(),(min)),(),,)((( 2121 vvuvuvu iiii     )(),(min),(min 21 vvu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    )(),,(min)),(),,)((( 2121 vuuvuvu iiii     )(,)(),(minmin 21 vuu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    ),(),,(min)),(),,)((( 21212211 vvuuvuvu iiii       )(),(min,)(),(minmin 2121 vvuu iiii  ))()(())()(( 2121 vvuu iiii   ))()(())()(( 2211 vuvu iiii    ),)((),,)((min 2211 vuvu iiii   theorem 4.16 if 1 g and 2 g are complete multi fuzzy graphs then 21 gg  is also a complete multi fuzzy graph. proof: let  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nng  be the two complete multi fuzzy graphs with dimension m and n respectively. then 1 g and 2 g 239 r. muthuraj, k. krithika and s. revathi are strong multi fuzzy graphs where  21 gandg are complete graphs. therefore, 21 gg  is a strong multi fuzzy graph by the theorem (4.15) with  21 gandg are complete graphs. hence 21 gg  is a complete multi fuzzy graph. theorem 4.17 the strong product of two multi fuzzy graphs 1 g and 2 g is the direct sum of the cartesian product of 1 g and 2 g and the direct product of 1 g and 2 g . proof: let  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nn g  be the multi fuzzy graph with dimension m and n respectively. let 21 gg  and 21 gg  be the cartesian product and direct product of 1 g and 2 g with dimension k where k = min(m,n) to prove: )()( 212121 gggggg    2111111111 ),()(),(min),)((),)(( vvvuvuvuvu iiiiii   so,   2111111111 ),()(),(min),)((),)(( vvvuvuvuvu iiiiii    )(),(min),)(( 1111 vuvu iiii   ),)((),)((),)(( 111111 vuvuvu iiiiii         12121211 22121211 2211 ),(),()( ),(),()( )),(),,)((( euuandvvifuuv evvanduufvvu vuvu ii ii ii      22112121212211 ),(&),(,),(),,(min)),(),,)(( evveuuifvvuuvuvu iiii            2211212121 12121211 22121211 2211 ),(&),(),(),,(min ),(),()( ),(),()( )),(),,))(()(( evveuuifvvuu euuandvvifuuv evvanduuifvvu vuvu ii ii ii iiii     )),(),,)((( 2211 vuvu ii   result: let  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nn g  be two strong multi fuzzy graph with dimension m and n respectively and 21 gg  & 21 gg  be the cartesian product and direct product of 1 g and 2 g with dimension k where k = min(m,n) and 21 gg  and 21 gg  be the complement of two multi fuzzy graphs then 21212121 gggggggg  . definition 4.18 the operation composition between two mfg 1 g and 2 g as follows )),...,(),,...,((][ 2211221121 kkkk gg   with the underlying crisp graph ),(][ 21 evgg   where 21 vvv  and   1212112121221212211 ),(,),(,),(,/),)(,( euuvvoreuuvvorevvuuvuvue  with   211121111111 ),(&)(),(min),)(( vvvuvvandvuallforvuvu iiii                 1212121 12122121 22112121 2211 ),(),(),(),(min ),(&,)(),,(min ),(&,),(),(min )),(),,)((( euuallforuuvv euuvvallforvvvvuu evvvuallforuuuvvu vuvu iii ii ii ii      for all i= 1, 2, 3, ... k. 240 various product on multi fuzzy graphs if nm  , let k = min (m, n). suppose nm  , then we take first m dimensions for 2 g so as to convert the mfg 1 g and 2 g have the same dimension k example 4.19 figure 7 figure 8 theorem 4.20 composition of two multi fuzzy graph is also a multi-fuzzy graph. proof: let  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nn g  be the multi fuzzy graph with dimension m and n respectively to prove: 21 ggg  is a multi-fuzzy graph of dimension k where k = min (m, n)   211121111111 ),(&)(),(min),)(( vvvuvvandvuallforvuvu iiii     ),(),(min)),(),,)((( 2121 vvuvuvu iiii     )(),(min),(min 21 vvu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    )(),,(min)),(),,)((( 2121 vuuvuvu iiii     )(,)(),(minmin 21 vuu iii      )(),(min,)(),(minmin 21 vuvu iiii  ))()(())()(( 21 vuvu iiii    ),)((),,)((min 21 vuvu iiii    ),(),(),(min)),(),,)((( 21212211 uuvvvuvu iiiii     )(),(min),(),(min 2121 uuvv iiii   ))()((),(),(min 2121 uuvv iiii   ))()(())()(( 2211 vuvu iiii   241 r. muthuraj, k. krithika and s. revathi  ),)((),,)((min 21 vuvu iiii   theorem 4.21 if  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nn g  are two strong multi fuzzy graphs with dimension m and n respectively and 21 gg  is a strong multi fuzzy graph of dimension k where k = min (m, n). prove that 2121 gggg   proof: let )),...,(),,...,(( 2211221121 kkkk ggg    )),...,(),,...,(( 2211221121 kkkk gg     ),...,(),,...,( 21211 mm g   ),...,(),,...,( 21212 nn g  )),...,(),,...,(( 2211221121 kkkk gg    to prove 2121 gggg   it is enough to prove iiii    for all i= 1, 2, 3, ... k. to prove the above result, there are different cases may arise depending upon the edges joining the vertices case(i): consider the edge 22121 ),()),,(),,(( evvvuvue  then ee and g is a strong multi fuzzy graph, so 0)( e ii   also 0)()( 11 e  since 221 ),( evv  if )),(),,(( 21 vuvue  , 22121 ),( evvandvv  then ee 0)),(),,)((( 21 vuvu ii   now )),)((()),)((()( 21 vuvue iiiiii    ))()((())()(( 21 vuvu iiii   )()()( 21 vvu iii   ),()()()( 2111 vvue ii   )()()( 21 vvu iii   iiii    for all i= 1, 2, 3, ... k. case(ii): consider the edge 12121 ),()),,(),,(( euuvuvue  then ee and g is a strong multi fuzzy graph, so 0)( e ii   also 0)()( 11 e  since 121 ),( euu  if )),,(),,(( 21 vuvue  121 ),( euu  then ee 0)),(),,)((( 21 vuvu ii   now )),)((()),)((()( 21 vuvue iiiiii    ))()((())()(( 21 vuvu iiii   )()()( 21 vuu iii   since 121 ),( euu  )(),()()( 2111 vuue ii   )()()( 21 vuu iii   iiii    for all i= 1, 2, 3, ... k. case(iii): consider the edge 211212211 &),()),,(),,(( vveuuvuvue  then ee and g is a strong multi fuzzy graph, so 0)( e ii   since 121 ),( euu  , 0)()( 11 e  if 211212211 &),()),,(),,(( vveuuvuvue  then ee 0))(( e ii   )),)((()),)((()( 2211 vuvue iiiiii    ))()((())()(( 2211 vuvu iiii   242 various product on multi fuzzy graphs since 121 ),( euu  we have )()(),()()( 212111 vvuue iii   )()()()( 2121 vvuu iiii   )(e ii   iiii    for all i= 1, 2, 3, ... k. case(iv): consider the edge 2211212211 ),(&),()),,(),,(( evveuuvuvue  since ee , 0))(( eii   )),)((()),)((()( 2211 vuvue iiiiii    ))()((())()(( 2211 vuvu iiii   if 121 ),( euu  and if 21 vv  then we have case (ii) if 121 ),( euu  and if 21 vv  then we have case (iii) in all the cases we have, iiii    for all i= 1, 2, 3, ... k. definition 4.22 the operation corona product between two mfg 1 g and 2 g is defined as follows, )),...,(),,...,(( 2211221121 kkkk gg   with the underlying crisp graph ),(),( 212121 eevvevgg         2 1 ),( ),( ))(( vuu vuu u i i ii    and          ',)(),(min ),(),,( ),(),,( ),)(( 2 1 euvvu evuvu evuvu vu ii i i ii     where e' is the set of all edges joining by an edge the i th vertex of 1 g to every vertex in the i th copy of 2 g if nm  , let k = max (m, n). suppose nm  then let us introduce n – m membership values of multi fuzzy graph 1 g into 0 so as to convert the multi fuzzy graphs 1g and 2g have the same dimension as k. example 4.23 figure 9 figure 10 243 r. muthuraj, k. krithika and s. revathi theorem 4.24 corona product of two multi fuzzy graph is also a multi-fuzzy graph. proof: let  ),...,(),,...,( 21211 mm g  and  ),...,(),,...,( 21212 nn g  be the multi fuzzy graph with dimension m and n respectively to prove: 21 ggg  is a multi-fuzzy graph of dimension k where k= max (m, n)       2 1 ),( ),( ))(( vuu vuu u i i ii    case(i): if 1 ),( evu  ),(),)(( vuvu iii    )(),(min vu ii   ))((),)((min vu iiii   case(ii): if 2 ),( evu  ),(),)(( vuvu iii    )(),(min vu ii   ))((),)((min vu iiii   case(iii): if '),( evu   )(),(min),)(( vuvu iiii    ))((),)((min vu iiii   5. conclusion in this paper, the complement of multi fuzzy graph and direct sum of two multi fuzzy graphs are defined and proved some results connected to them. also defined various product on multi fuzzy graphs such as direct product, strong product, cartesian product, composition, corona product and proved some properties related to them. references [1] bhattacharya, “some remarks on fuzzy graphs”, pattern recognition letters, 9 (1987)159 -162. 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[14] rosenfeld. a, “fuzzy graphs”, in: zadeh. l. a, fu k. s, m. shimura (eds), “fuzzy sets and their applications”, academic press, new york (1975)77-95 [15] sebu sebastian and ramakrishnan. t.v, "multi fuzzy sets", international mathematical forum,50, pp.2471-2476, 2010. [16] shovan dogra, “different types of product of fuzzy graphs”, progress in nonlinear dynamics and chaos, vol. 3, no. 1, 2015, 41-56 issn: 2321 – 9238 (online) [17] sunitha m.s, vijayakumar. a, complement of a fuzzy graph, indian j. pure appl. math, 33(9) (2002), 14511464. [18] zadeh l.a., fuzzy sets, information and control, 8 (1965), pp.338-353. 245 ratio mathematica volume 44, 2022 remarks on interiors and closures of weak open sets in bigeneralized topological spaces m. anees fathima* r. jamuna rani† abstract we establish the relationships between the interior and closure operators among the µij-semiopen, µij-preopen, αµij-open, βµij-open sets in bigeneralized topological spaces. keywords: generalized topology, bigeneralized topology, µi-open set, µi-closed set. 12020 ams subject classifications: 54a05, 54a10. *research scholar(reg.no: 19121172092012), pg and research department of mathematics, rani anna government college for women, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamilnadu, india. email: afiseyan09@gmail.com. †assistant professor, pg and research department of mathematics, rani anna government college for women, affiliated to manonmaniam sundaranar university, abishekapatti, tirunelveli-627012, tamilnadu, india. email:jamunarani1977@gmail.com. 1 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 75 received on june, 2022. accepted on september, 2022. doi: 10.23755/rm.v39i0.892. issn: m. anees fathima, r. jamuna rani 1 introduction the study of generalized topological spaces (briefly gts) was first initiated by a.csaszar on 2002 (császár, 2002). in that, he established the interior and closure operators in gts. later in 2010 (boonpok, 2010)this study is extended to bigeneralized topological spaces (briefly bgts) by c. boonpak. in his paper, he found the conception of (m, n)-closed and (m, n)-open sets in bgts. also, he introduced (m, n)g-regular open, (m, n)g-semi open, (m, n)g-pre open, (m, n)g − α-open . with these definitions, the properties of the weak open sets were studied by a. jamuna rani and m. anees fathima in (fathima and rani, 2019; rani and fathima, 2020) and some of its characterizations are also analysed. the purpose of this paper is to prove the relationships between the interior and closure operators among µij-semi open, µij-preopen, αµij-open and βµijopen sets in bgts. also we establish some of its characterizations. 2 preliminares in this section, we provide some basic definitions and notations which are most essential to understand the subsequent section. for any non empty set x and ξ ∈ p(x), ξ is said to be gt (császár, 2002) if ϕ ∈ ξ and ξ is closed under arbitrary union. also, the function γ ∈ γ (where γ denotes the collection of all mappings γ : p(x) → p(x) possessing the property of monotony. ie.,if a ⊂ b ⇒ γ(a) ⊂ γ(b)) is said to be µ-friendly (császár, 2006)if γ(a) ∩ l ⊂ γ(a ∩ l) for every a ⊂ x and l ∈ µ. if γ ∈ γ and µ = {a ⊂ x/a ⊂ γ(a)}is the family of all γopen sets, then µ is a gt (császár, 2002) the pair (x, µ) is a called gts. in (sivagami, 2008) the family of all µfriendly functions is denoted by γ4 and (x, γ) is called the γ-space. it is further proved that every γ-space is a quasi-topological space in (császár, 2008) and all the results established in for γ-spaces are valid for quasi-topological spaces. also for γ ∈ γ, define γ∗ : p(x) → p(x) by γ∗(a) = x − γ(x − a) (császár, 1997) for every subset a of x. let x be a non empty set and µ1, µ2 be generalized topologies on x. a triple (x, µ1, µ2) is said to be a bigeneralized topological space. let a be subset of a bigeneralized topological space x. then the closure of a and the interior of a with respect to µm are denoted by cµm(a) and iµm(a) respectively, for m = 1, 2. (boonpok, 2010) a subset a of a bigeneralized topological space(x, µ1, µ2) is said to be µij-semiopen (fathima and rani, 2019)(resp.µij-preopen (rani and fathima, 2020) αµij-openβµij-open (jamuna rani and anees fathima, 2021)) if a ⊂ cµiiµj(a) where i, j = 1, 2 and i ̸= j (resp. a ⊂ iµicµj(a), a ⊂ 76 remarks on interiors and closures of weak open sets in bigeneralized topological spaces iµicµjiµi(a),a ⊂ cµiiµjcµi(a)). proposition 1.1. (min, 2009)let (x, µ) be a generalized topological space. for subsets a and b of x, the following properties holds. (a) cµ(x − a) = x − iµ(a) and iµ(x − a) = x − cµ(a). (b) if (x − a) ∈ µ, then cµ(a) = a and if a ∈ µ, then iµ(a) = a. (c) if a ⊆ b, then cµ(a) ⊆ cµ(b) and iµ(a) ⊆ iµ(b). (d) if a ⊆ cµ(a) and iµ(a) ⊆ a. (e) cµ (cµ (a)) = cµ (a) and iµ (iµ (a)) = iµ (a). proposition 1.2. (jamunarani et al., 2010) let (x, γ) be a γspace . then g ∩ cγ(a) ⊂ cγ(g ∩ a), for every a ⊂ x and γ-open set g of x. proposition 1.3. let (x, µ) be a quasi topological space and a, b ⊆ x, the following holds. (a) if a and b are µ-open sets, then a ∩ b is µ-open (sivagami, 2008) (b) iµ(a∩b) = iµ(a)∩iµ(b), for every subsets a and b of x (császár, 2008) (c) cµ(a ∪ b) = cµ(a) ∪ cµ(b), for every subsets a and b of x(sivagami, 2008) proposition 1.4. (fathima and rani, 2019) let (x, µ) be a generalized topological space. let a be a subset of x. then the following hold. (a) cσij(a) is the smallest µij-semi closed set containing a. (b) a is µij-semi closed if and only if a = cσij(a). (c) x ∈ cσij(a) if and only if for every µij-semi open g containing x, g ∩ a ̸= ϕ. (d) cσij ∈ γ012+. proposition 1.5. (fathima and rani, 2019)let (x, µ1, µ2) be a bigeneralized topological space. let a be a subset of x. then the following hold. (a) ( iσij )∗ = cσij . (b) ( cσij )∗ = iσij . 77 m. anees fathima, r. jamuna rani (c) iσij(x − a) = x − cσij(a) for every subset a of x. (d) cσij(x − a) = x − iσij(a) for every subset a of x. proposition 1.6. (fathima and rani, 2019)let (x, µ1, µ2) be a bigeneralized topological space. let a be a subset of x. then the following hold. (a) a is µij-semi open if and only if a is cµiiµj open if and only if a = icµiiµj (a). (b) iσij(a) = icµiiµj and cσij = ccµiiµj . (c) iσij(a) = a ∩ cµiiµj(a). (d) cσij(a) = a ∩ iµicµj(a). similar results from (jamuna rani and anees fathima, 2021; rani and fathima, 2020; jamuna rani and anees fathima, 2020) are also used in the next section. 3 relationship between the operators: the following theorem gives some of the relationships between iσij , cσij , iαij , cαij , iµi and cµi . theorem 1.1. let (x, µ1, µ2) be a bigeneralized topological space. let a be a subset of x and µi ∈ γ4. then the following hold. (a) iσij(a) = a ∩ cµiiµj(a) (b) cσij(a) = a ∪ iµicµj(a) (c) iαij(a) = a ∩ iµicµjiµi(a) (d) cαij(a) = a ∪ cµiiµjcµi(a) (e) iµi(cσij(a)) = iµicµj(a) (f) cµi(iσij(a)) = cµiiµj(a) (g) cσij(iµi)(a) = iµicµjiµi(a) (h) iσij(cµi)(a) = cµiiµjcµi(a) (i) cσjiiσij(a) = (a ∩ cµiiµj(a)) ∪ iµjcµiiµj(a) for every subset a of x. 78 remarks on interiors and closures of weak open sets in bigeneralized topological spaces (j) iσijcσji(a) = (a ∩ cµiiµjcµi(a)) ∪ iµjcµi(a) for every subset a of x. proof. (a) by theorem 1.6(b), iσij = icµiiµj implies iσij(a) ⊂ a ∩ cµiiµj(a). for the reverse part, a ∩ cµiiµj ⊂ cµiiµj(a) ⊂ cµiiµj ( a ∩ cµiiµj(a) ) . thus a ∩ cµiiµj = icµiiµj (a ∩ cµiiµj) ⊂ iσij(a). (b) the result follows from (a). (c) the proof is similar to the proof of (a). (d) the result follows from (c). (e) iµi ( cσij(a) ) = iµi ( a ∪ iµicµj(a) ) ⊂ iµi ( a ∪ cµj(a) ) = iµicµj(a). also, by (b), iµicσij(a) ⊃ iµiiµicµj(a) = iµicµj(a). (f) the result follows from (e). (g) by (b), cσij(iµi(a)) = iµi(a) ∪ iµicµjiµi(a) = iµicµjiµi(a). (h) by(a), iσij(cµi(a)) = cµi(a) ∩ cµiiµjcµi(a) = cµiiµjcµi(a). (i) by(a),cσjiiσij(a) = (a ∩ cµiiµj(a)) ∪ iµjcµi(a ∩ cµiiµj(a)). here, cµi(a∩cµiiµj(a)) ⊂ cµiiµj(a) and cµi(a∩cµiiµj(a)) ⊃ cµi ( iµj(a) ∩ cµiiµj(a) ) = cµiiµj(a). (j) now, iσijcσji(a) = (a ∪ iµjcµi(a)) ∩ cµiiµj(a ∪ iµjcµi(a)) = (a ∪ iµjcµi(a)) ∩ cµjiµjcσji(a) = ( a ∪ iµjcµi(a) ) ∩ cµiiµjcµi(a) by (e). hence iσijcσji(a) = ( a ∩ cµiiµjcµi(a) ) ∪ iµjcµi(a). 2 the following theorem shows some results for the operators iπij , cπij , iβij , cβij . theorem 1.2. let (x, µ1, µ2) be a bigeneralized topological space. let a be a subset of x and µi ∈ γ4. then the following hold. (a) iπij(a) = a ∩ iµicµj(a) (b) cπij(a) = a ∪ cµiiµj(a) (c) iβij(a) = a ∩ cµiiµjcµi(a) (d) cβij(a) = a ∪ iµicµjiµi(a) (e) iµi(iσji(a)) = iµi(a) (f) cµi(cσji(a)) = cµi(a). proof. (a) in (rani and fathima, 2020), by theorem 3.2(e), we have iπij = iiµicµj and so iπij(a) ⊂ a ∩ iµicµj(a). let x ∈ iµicµj(a). let g be any µi open set containing x such that g ∩ iµicµj(a) is a µi -open set containing x. since x ∈ cµj(a) and g ∩ iµicµj(a) ∩ a ̸= ϕ and so x ∈ cµj ( iµicµj(a) ∩ a ) . therefore, iµicµj(a) ⊂ cµjiµicµj(a) ∩ a and so iµicµj(a) ⊂ iµicµj(a ∩ iµicµj(a)), a ∩ iµicµj(a) ⊂ iµicµj(a) ⊂ iµicµj ( a ∩ iµicµj(a) ) , a ∩ iµicµj(a) 79 m. anees fathima, r. jamuna rani = iiµicµj ( a ∩ iµicµj(a) ) ⊂ iπij(a) ∩ iµicµj(a) ⊂ iπij(a). hence iπij(a) = a ∩ iµicµj(a). (b) the result follows from (a). (c) the proof is similar to the proof of (a). (d) by theorem 3.10(i) in (jamuna rani and anees fathima, 2021), the proof is similar to the proof of (b). (e) iµi(iσji(a)) = iµi(a ∩ cµjiµi(a)) by theorem 1.1(a) and so iµi(iσji(a)) = iµi(a) ∩ iµicµjiµi(a) = iµi(a) by proposition 1.3(b). (f) the result follows from (e). 2 the following theorem gives characterizations of µij-semi open, µij-preopen, αµij-open, βµij-open sets using the interior and closure operators. theorem 1.3. let (x, µ1, µ2) be a bigeneralized topological space. let a be a subset of x and µi ∈ γ4. then the following hold. (a) a ∈ πij(µ) if and only if cσij(µ) = iµicµj(a) (b) a is µij-preclosed if and only if iσij(a) = cµiiµj(a). (c) a ∈ σij(µ) if and only if cπij(a) = cµiiµj(a). (d) a is µij-semiclosed if and only if iπij(a) = iµicµj(a). (e) a ∈ αij(µ)if and only if cβij(a) = iµicµjiµi(a). (f) a is αµij -closed if and only if iβij(a) = cµiiµjcµi(a). (g) a ∈ βij(µ) if and only if cαij(a) = cµiiµjcµi(a). (h) a is βµij -closed if and only if iαij(a) = iµicµjiµi(a). proof. the proof follows from theorem 1.1(b),1.2(b),1.2(d), 1.1(d).2 theorem 1.4. let (x, µ1, µ2) be a bigeneralized topological space. let a be a subset of x and µi ∈ γ4. then the following hold. (a) cµi(cπij(a)) = cµi(a). (b) iµi(iπij(a)) = iµi(a). 80 remarks on interiors and closures of weak open sets in bigeneralized topological spaces (c) cµi(cβji(a)) = cµi(a). (d) iµi(iβji(a)) = iµi(a). (e) iµi(cπji(a)) ⊂ cµjiµi(a) and cµjiµi(cπji(a)) = cµjiµi(a). (f) iµi(cπji(a)) = iµicµjiµi(a). (g) cβij(iµi(a)) = iµicµjiµi(a) = iµi(cβij(a)). (h) cβij(iβji(a)) = iβji(cβij(a)) = ( a ∪ iµicµjiµi(a) ) ∩ cµjiµicµj(a). (i) iβij (cµi(a)) = cµiiµjcµi(a) = cµiiβij(a). (j) cσij ( iσji(a) ) ⊂ iβji ( cβij(a) ) ⊂ iσji ( cσij(a) ) . proof. (a) clearly, cµi(a) ⊂ cµi ( cπij(a) ) . again, cµi ( cπij(a) ) ⊂ cµi (cµi(a)) = cµi(a). (b) the proof follows from (a). the proof of (c) and (d) are similar to (a) and (b). (e) similar proof, so omitted. (f) by (e), iµi ( cπji(a) ) ⊂ iµicµjiµi(a). again, iµi ( cπji(a) ) = iµi ( a ∪ cµjiµi(a) ) ⊃ iµicµjiµi(a). therefore, iµi ( cπji(a) ) = iµicµjiµi(a). (g) by theorem 1.2(d), cβij (iµi(a)) = iµi(a) ∪ iµicµjiµi (iµi(a)) = iµicµjiµi(a). again, iµi ( cβij(a) ) = iµi ( a ∪ iµicµjiµi(a) ) ⊃ iµicµjiµi(a). for converse part, iµi ( cβij(a) ) ⊂ iµi ( cπij(a) ) ⊂ cµjiµi(a) ⊂ iµicµjiµi(a). (h) similar proof, so omitted. (i) the proof follows from (g). (j) the proof follows from 1.1(i) and 1.1(j).2 the following theorem gives a characterization of βij(µ)-open sets. theorem 1.5. let (x, µ1, µ2) be a bigeneralized topological space. let a be a subset of x and µi ∈ γ4. then the following are equivalent. (a) a ∈ βji(µ). (b) a ⊂ iβji ( cβij(a) ) . (c) a ⊂ iσji ( cσij(a) ) . 81 m. anees fathima, r. jamuna rani proof. (a) ⇒ (b) if a ∈ βji(µ), then a =iβji(a) ⊂ iβji ( cβij(a) ) . (b) ⇒ (c) by theorem 1.4(j), the proof follows. (c) ⇒ (a) a ⊂ iσji ( cσij(a) ) ⇒ a ⊂ ( cσij(a) ∩ cµjiµi ( cσij(a) )) = ( cσij(a) ∩ cµjiµicµj(a) ) by theorem 1.1(e) and so a ⊂ cµjiµicµj(a). 2 in the following theorem, we prove some relationships between iαij ,cαij with iµi , cµi . theorem 1.6. let (x, µ1, µ2) be a bigenralized topological space. let a be a subset of x and µi ∈ γ4. then the following hold. (a) cµicαij(a) = cαijcµi(a)= cµi(a). (b) iµiiαij(a) = iαijiµi(a)= iµi(a). (c) cαij ( iµj(a) ) = cµiiµj(a). d) cµj ( iαij(a) ) = cµjiµi(a). (e) iαij ( cµj(a) ) = iµicµj(a). (f) iµj ( cαij(a) ) = iµjcµi(a). proof. (a) the proof follows from the theorem 1.1(d). (b) the proof of (b) follows from (a). (c) the proof follows from the theorem 1.1(d). (d) cµj ( iαij(a) ) ⊂ cµj ( cµj(a) ∩ iµicµjiµi(a) ) = cµjiµi(a). again, cµj ( iαij(a) ) ⊃ cµj (a ∩ iµi(a)) = cµjiµi(a). (e) the proof of (e) follows from (c). (f) the proof of (f) follows from (d). 2 the following theorem shows some relationships between iαij , cαij with iσij , cσij , iπij ,cπij , iβij and cβij . theorem 1.7. let (x, µ1, µ2) be a bigenralized topological space. let a be a subset of x and µi ∈ γ4. then the following hold. (a) iαij ( cσij(a) ) = iµicµj(a). (b) iαij ( cπji(a) ) = iµicµjiµi(a). (c) iαij ( cβji(a) ) = iµicµjiµi(a). 82 remarks on interiors and closures of weak open sets in bigeneralized topological spaces (d) cαij ( iσij(a) ) =cµiiµj(a). (e) cαij ( iπji(a) ) = cµiiµjcµi(a). (f) cαij ( iβji(a) ) = cµiiµjcµi(a). proof. (a) the proof follows from the theorem 1.1(c), 1.1(f). (b) the proof follows from the theorem 1.1(c), 1.4(g) and 1.1(a). (c) the proof follows from the theorem 1.1(c), 1.4(h). (d) the proof of (d) follows from (a). (e) the proof of (e) follows from (b). (f) the proof of (f) follows from (c). 2 theorem 1.8. let (x, µ1, µ2) be a bigeneralized topological space. let a be a subset of x and µi ∈ γ4. then the following hold. (a) iσijcαij(a) = cαij(a) ∩ cµiiµjcµi(a) (b) cσijiαij(a) = iαij(a) ∩ iµicµjiµi(a) (c) cσijcαji(a) =cαji(a) ∩ cσij(a). (d) iσijiαji(a) =iαji(a) ∩ iσij(a). (e) iσijiβij(a) = iσij(a). (f) cσijcβij(a) = cσij(a). proof. (a) the result follows from the theorem 1.1(a) and 1.6(f). (b) the proof of (b) follows from (a). (c) the result follows from the theorem 1.1(b) and 1.6(a). (d) the proof of (d) follows from (c). (e) the result follows from the theorem 1.1(a), 1.4(d) and 1.2(c). (f) the proof of (f) follows from (e). theorem 1.9. let (x, µ1, µ2) be a bigeneralized topological space. let a be a subset of x and µi ∈ γ4. then the following hold. (a) iπijcαji(a) = cαji(a) ∩ iµicµj(a). (b) iπijiβji(a) = iπij(a). 83 m. anees fathima, r. jamuna rani (c) cπijiαji(a) = iαji ∩ cµiiµj(a). (d) cπijcβji(a) = cπij(a). proof. (a) the result follows from the theorem 1.2(a) and 1.6(a). (b) the result follows from the theorem 1.2(a) and 1.4(i). (c) the proof of (c) follows from (a). (d) the proof of (d) follows from (b).2 theorem 1.10. let (x, µ1, µ2) be a bigeneralized topological space. let a be a subset of x and µi ∈ γ4. then the following hold. (a) iβijcαij(a) = cαij(a) ∩ cµiiµjcµi(a)= cµiiµjcµi(a). (b) cβijiαij(a) = iµicµjiµi(a) = iαij(a) ∩ iµicµjiµi(a). proof. (a) the result follows from the theorem 1.2(c) and 1.6(a). (b) the proof of (b) follows from (a). 2 references c. boonpok. weakly open functions on bigeneralized topological spaces. int. journal of math. analysis, 4(18):891–897, 2010. á. császár. generalized open sets. acta mathematica hungarica, 75(1-2):65–87, 1997. a. császár. generalized topology, generized continuity. acta mathematica hungarica, 96(4):351–357, 2002. á. császár. further remarks on the formula for γ-interior. acta mathematica hungarica, 113(4):325–332, 2006. á. császár. remarks on quasi-topologies. acta mathematica hungarica, 119, 2008. m. a. fathima and r. j. rani. µij-semi open sets in bigeneralized topological space. malaya journal of matematik, s (1), pages 12–16, 2019. r. jamuna rani and m. anees fathima. αµij-open sets in bigeneralized topological spaces. proceedings in the national conference on recent trends in algebra and topology, 2020. 84 remarks on interiors and closures of weak open sets in bigeneralized topological spaces r. jamuna rani and m. anees fathima. βµij-open sets in bigeneralized topological spaces. design engineering, pages 5077–5083, 2021. r. jamunarani, p. jeyanthi, and d. sivaraj. more on γ-interior. bull. allahabad math. soc, 25:1–12, 2010. w. min. almost continuity on generalized topological spaces. acta mathematica hungarica, 125(1):121–125, 2009. r. j. rani and m. a. fathima. µij-preopen sets in bigeneralized topological spaces. advances in mathematics:scientific journal, 2020. p. sivagami. remarks on γ-interior. acta mathematica hungarica, 119(1):81–94, 2008. 85 ratio mathematica volume 46, 2023 super fuzzy matrix of inverse in kth order r.deepa* p. sundararajan† abstract in normal matrix and fuzzy matrix working well but sometimes fail to work for classical model problems. because, it is not satisfied the consistency conditions and other parameter. our proposed method to satisfy the all condition including consistency and perform well compared to the other existing models. unexpected event modelling is a affluent area of study in fuzzy matrix (fm) modelling. every fuzzy matrix may be shown as a multi-dimensional concept, but standard matrices cannot achieve this without the proper scale. to solve this issue, a certain kind of classical fuzzy matrix is required. in this study, the idea of an inverse of a k-regular fuzzy matrix is introduced, and some of its key characteristics are listed. as a result, the same regularity indicator is used to describe a matrix. investigation is also done on the relationship between the regular, k-regular, and consistency of fuzzy matrices powers. keywords: inverse; k-regular; super fuzzy matrix; fuzzy matrix; regular 2020 ams subject classifications: 03e72, 15a09, 35d10, 39b42, 58j52. 1 *excel engineering college(autonomous), komarapalayam, tamilnadu, india. deepssengo @gmail.com. †department of mathematics, arignar anna govt. arts college, namakkal. tamilnadu, india. ponsundar03@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1079. issn: 1592-7415. eissn: 2282-8214. ©r.deepa et al. this paper is published under the cc-by licence agreement. 227 r.deepa and p.sundararajan 1 introduction each component of a matrix known as a boolean matrix has a value of 0 or 1. basic values for a fuzzy matrix fall between [0, 1]. by kim and roush, the idea of fuzzy matrix sections was introduced. in all auothers contributed significantly to fuzzy matrix augmentation. later, utilising modified regular inverses of the coefficient matrix, zheng and wang developed the widely used m x n fuzzy linear machine and the inconsistent fuzzy linear system. using matrix-modified common inverse theory, abbasbandy .s and . m [2005] investigated the minimal response of the overall twin fuzzy linear device. we provide a unique way for finding the inverse of a fantastic fuzzy matrix by changing a well-known idea in this research d and h [1978],p [1963], ravi.j and et. al. [2022], kandasamy w.b.v. and llanthenral k. [2007], m.z. and e.g [1994], m.g [1977], k and b [2006], b and k [2006]. the super fuzzy matrix’s initial and core notions are covered in section 2 along with an illustration. the method of our suggested notion is presented in section 3. the main findings, theorems, and numerical examples of super fuzzy inverse matrices of order k are presented in section 4, and the conclusion is derived in section 5. 2 preliminaries noindent the core concepts and notations of a super fuzzy matrix will be described in this section. this article concentrates on the same old number ordering, underlying maxmin (min max) operations, and fuzzy matrices in super (sfm) with [0,1] support. by (sf)mxn and (sf)n, respectively, all fuzzy matrices in super of order m x n and n x n are represented. space created by the row (or) column is indicated by the letters r(a) or c(a) (a) , kaufmann and gupta [1985], j.b [1983], . definition 2.1. deepa.r and sundararajan.p [2020] a matrix containing entries that fall between [0, 1] is referred to as a fuzzy matrix. in the case of matrices, we have a fuzzy related different matrix. definition 2.2. deepa.r and sundararajan.p [2020] let us deem c fuzzy matrix c = [ c11 c12 c c21 c22 c23 ] the fuzzy submatrices c11 and c21 have the same number of columns because they are fuzzy submatrices, along with c12, c13, c21, c22, and c23. the fuzzy matrices c13 and c23, as well as the fuzzy submatrices c12 and c22, all have 228 super fuzzy matrix of inverse in kth order equal columns. the second index of the fuzzy sub-matrices reveals this. the number of rows in fuzzy submatrices c11, c12, and c13 is the same. there are exactly the same number of rows in the c21, c22, and c23 fuzzy submatrices. as a result, we currently possess c generic super fuzzy matrix. c =   c11 c12 . . . c1n c21 c22... c 2n . . . cm1 cm2... cmn   where cij’s i = 1, 2, . . . , m and j =1, 2, . . . , n. definition 2.3. the modified regular inverse of matrix solution of a positive set of equations can be found by altering the regular inverse of a non-singular matrix. the modified regular inverse of any (potentially square) matrix with complex components may be found using any method. it is used here and in other programmes to immediately solve linear matrix issues as well as to get an expression for the principal idempotent components of the matrix. 3 methodology the kth regular super fuzzy matrix is examined in this section. definition 3.1. let c =   c11 c12 . . . c1n c21 c22... c 2n . . . cm1 cm2... cmn   it is a matrix of fuzzy that is ck each aij is 0≤ aij ≤1. the exact strength of each ingredient is examined here. it is the g-name. inverse’s. ck =   1−kc11 1−kc12 . . . 1−kc1n 1−kc21 1−kc22... 1−kc 2n . . . 1−kcm1 1−kcm2... 1−kcmn   229 r.deepa and p.sundararajan here, k is the smallest positive integer (based on probability). it is possible to multiply two fuzzy matrices without requiring that they both be fuzzy. every entry that is positive is changed to a 1, and every entry that is negative is changed to a 0 or a tendency toward zero. definition 3.2. the properties of super fuzzy matrix is,( ak )k = a axak = ak, (a+b)k= ak + bk, (λa)k = λak, (ba)k= akbk, aak=0 implies a=0. definition 3.3. the condition of the supper fuzzy matrix are given below, axa=a xax=x (ax)k = ax (xa)k = xa xxkak = x xaak=ak bakaak=ak x=xxkak=xxkakay=xay=xaakyky=akyky=y. 4 result & application the ck of super fuzzy matrix met all of the requirements for convergence. theorem 4.1. let a be the inverse of the k by krsfm (k regular super fuzzy matrix), with non-zero rows representing the norm. if a meets the maximin conditions for the matrix equation asa=a for some super fuzzy matrix s, then c is k regular. proof. the non-zero rows of the inverse of the krsfm of c serve as the standard basis in this case. if sb=q, then q’s rows are permutations of c’s rows. then, x is an idempotent of krsfm, with the same row space as c and non-zero z rows functioning as a standard basis.. since the standard foundation is different for each s, b=gq. then cst c=sqst sq=sqq=sq=ck 230 super fuzzy matrix of inverse in kth order ⇒csc=ck. so, c is k regular. theorem 4.2. let c,d (sf)nxm and k=p be two inverse of p regular super fuzzy matrix. if c is p regular, then we prove that hence di* = ∑ xija ∗. ⇒d=xcp ⇒d=xcc’c (since cc’c=cp) ⇒d=dc’cp ⇒d=cc’c ⇒d=cc’cy. ⇒d=cpc’d. theorem 4.3. for c∈(sf)n and for any g−∈(sf )n, if ckx=ckg−, where x is a {1k ,3k} inverse of c then, g− is a {1k ,3k} inverse of c. proof. since x is a {1k ,3k} ⇒ckxc=ck and (ckx)t =ckx. the post is multiplied by a on both sides of ckx=ckg− , ckg−c=ckxc=ck. (ckg−)t =(ckx)t =ckx=ckg− . hence g− is a {1k ,3k}inverse of c. theorem 4.4. for c∈(sf )n, x is a {1p ,3p} inverse of c and h− is a {1p ,3p} inverse of c then, cpx=cph−. proof. since x is a {1p ,3p} inverse of c. ⇒cpxc=cp and (cpx)t =cpx. h− is a {1p ,3p} inverse of c, ⇒ch−cp=cp and (ch− )t =ch− . cph−=(cpxc)h−= (cpx)t (ch− )=(cpx)t (ch− )t =xt (ct )p(h− )t ct = xt (ch− cp)t =xt (cp)t =(cpx)t =cpx. theorem 4.5. for d∈(sf)n, if dt d is d right krsfm and r(dp)⊆r(dt d)p then d has a {1p ,3p} inverse. in particular for p=1, u=(dt d)−dt is a {1, 3} inverse of d. proof. let (dt d)p(dt d)−(dt d) = (dt d)p since r(dp)⊆r((dt d)p), dp=x(dt d)p for some x∈(sf)n dnd tape u=(dt d)−dt dpud=(dp)(ud)=(x(dt d)p)((dt d)−dt d) =x((dt d)p(dt d)−(dt d)) = x(dt d)p=(dp). tdpe v=(dt d)−(dp)t . 231 r.deepa and p.sundararajan dpv=(dp)v =(x(dt d)p)((dt d)−(dp)t ) = x(dt d)p(dt d)−(dt d)pdt =x(dt d)p(dt d)−(dt d)(dt d)p−1xt = x(dt d)p(dt d)p−1xt =x(dt d)2p−1xt =(x(dt d)2p−1xt )t = (dpv)t . hence d has d {1p ,3p}inverse. in particular for p=1,y= (dt d)−dt is d {1, 3} inverse of d. theorem 4.6. let c∈(sf)n be a right krsfm and r(ct c)k⊆r(ck) then ct c has a {3k}inverse. proof. since, ckxc=ck. since r((ct c)k)⊆r(ck), (ct c)k=zck for some z∈(sf )n and take y=xc. (ct c)ky=(zck)(xc)=z(ckxc)=zck=(ct c)k=((ct c)k)t =((ct c)ky)t hence ct c has a {3k} inverse. the findings for the other associated fuzzy matrices improve, and the theorem is fulfilled. while some techniques (regular and inverse) meet some conditions, our proposed krsfm meets all of them (regular & inverse). example 4.1. let us consider iksfm a=   0.5 0.7 0.60.3 0.2 0.2 0.4 0.3 0.1   the rows of a are separate and make up the foundation. iksfpm is a form that p =   0 1 01 0 0 0 0 1   we know that apa=a. now for iksfm r=   0.1 0.4 0.20.3 0.4 0.2 0.3 0.1 0.1   there for ra=a. so g inverse of a sis pr=   0.3 0.4 0.20.1 0.4 0.2 0.3 0.1 0.1   232 super fuzzy matrix of inverse in kth order which is satisfy the relation axa=a. example 4.2. let a = [ < 0.4,0.2 > < 0.4,0.5 > < 0.8, 0.1 > < 0.3,0.4 > ] and b = [ < 0.6,0.2 > < 0.5,0.4 > < 0.,7 0.1 > < 0.4,0.4 > ] for which b=bat a holds. example 4.3. let iksfm is, c =   < 0.8,0.2 > < 0.4,0.2 > < 0.3,0.2 >< 0.4,0.2 > < 0.4,0.2 > < 0.3,0.2 > < 0.6,0.2 > < 0.6,0.2 > < 0.8,0.2 >   d =   < 0.8,0.2 > < 0.4,0.3 > < 0.2,0.2 >< 0.4,0.3 > < 0.4,0.2 > < 0.2,0.2 > < 0.4,0.2 > < 0.4,0.2 > < 0.6,0.2 >   e =   < 0.7,0.2 > < 0.4,0.3 > < 0.3,0.3 >< 0.4,0.3 > < 0.4,0.2 > < 0.4,0.2 > < 0.4,0.2 > < 0.6,0.2 > < 0.8,0.2 >   be two of its g inverse of a. then x = cde =   < 0.8,0.2 > < 0.4,0.2 > < 0.3,0.2 >< 0.4,0.2 > < 0.4,0.2 > < 0.3,0.2 > < 0.6,0.2 > < 0.6,0.2 > < 0.8,0.2 >   forth above x, cxc=c and xcx=x holds. so x is a semi inverse of the iksfm. example 4.4. let us consider the symmetric iksfm a = [ < 0.4,0.2 > < 0.4,0.5 > < 0.8, 0.1 > < 0.3,0.4 > ] now, a2 is, a2 = [ < 0.4,0.2 > < 0.4,0.5 > < 0.8, 0.1 > < 0.3,0.4 > ] [ < 0.4,0.2 > < 0.4,0.5 > < 0.8, 0.1 > < 0.3,0.4 > ] = [ < 0.4,0.2 > < 0.4,0.5 > < 0.8, 0.1 > < 0.3,0.4 > ] = a the above matrix is symmetric and idempotent. 233 r.deepa and p.sundararajan example 4.5. let us consider the symmetric iksfm a = [ < 0.4,0.2 > < 0.4,0.5 > < 0.8, 0.1 > < 0.3,0.4 > ] and b = [ < 0.4,0.2 > < 0.4,0.5 > < 0.8, 0 > < 0,0.4 > ] now, a2 is, b2 = [ < 0.4,0.2 > < 0.4,0.5 > < 0.8, 0 > < 0,0.4 > ] [ < 0.4,0.2 > < 0.4,0.5 > < 0.8, 0 > < 0,0.4 > ] = [ < 0.4,0.2 > < 0.4,0.5 > < 0.8, 0.1 > < 0.3,0.4 > ] 6= b the above matrix is not idempotent. 5 conclusions the main conclusions of the study may be presented in a short conclusions section, which may stand alone or form a subsection of a discussion or results. in this paper, the inverse of the krsfm is used to provide several innovative proposals and theorems. the original machine is replaced with matrix coefficient a by two distinct n m matrix equation systems. as a result, fm must resolve the issue. the k-regular, the regularity of the fuzzy matrix powers, and the relationship between each regular are all examples of regularities. our approach addresses any issues that need the inverse of k regular super fuzzy matrices. a appropriate theorem is also shown. in the next essay, we will look at a variety of fundamental properties as well as computer vision applications. in our proposed classical model to proved, it is a consistent. also in future, we are trying this concept in neural networking system and iot. references n. j. abbasbandy .s and a. . m. turning of reachable set in one dimensional fuzzy differential inclusions. chaos solitons fract., 26:1337–1341, 2005. z. b and w. k. general fuzzy linear systems. appl. math. comput., 181: 1276–1286, 2006. d. d and p. h. operations on fuzzy numbers. journal of syst. sci., 9:613–626, 1978. 234 super fuzzy matrix of inverse in kth order deepa.r and sundararajan.p. inverse of k regular super fuzzy matrix. international journal of innovative technology and exploring engineering, 9(5): 566–570, 2020. k. j.b. idempotents and inverses in fuzzy matrices. journal of malaysian math., 6(2):57–61, 1983. w. k and z. b. inconsistent fuzzy linear systems. appl. math. comput., 181: 973–981, 2006. s. f. kandasamy w.b.v. and llanthenral k. elementary fuzzy matrix theory and fuzzy models for social scientists. los angeles, new york, 2007. kaufmann and m. gupta. introduction fuzzy arithmetic. van nostrand reinhold, new york, 1985. t. m.g. convergence of power of fuzzy matrix. j. of math. anal. appl., 57: 476–486, 1977. r. m.z. and e. e.g. the determinant and adjoint of a square fuzzy matrix. fuzzy sets and systems, 61:297–307, 1994. h. p. matrix algebra for social scientists. holt, rinehart and winston, new york, 1963. ravi.j and et. al. fuzzy graph and their applications: a review. international journal for science and advance research in technology, 8(1):107–111, 2022. 235 ratio mathematica volume 47, 2023 elongation of sets in soft lattice topological spaces g. hari siva annam* t. abinaya† abstract the aim of this paper, we investigate some lattice sets such as soft lattice exterior, soft lattice interior, soft lattice boundary and soft lattice border sets in soft lattice topological spaces which are defined over a soft lattice l with a fixed set of parameter a and it is also a generalization of soft topological spaces. further, we develop and continue the initial views of some soft lattice sets, which are deep-seated for further research on soft lattice topology and will consolidate the origin of the theory of soft topological spaces. 2020 ams subject classifications: 54a05, 54a10. 1 *assistant professor, pg and research department of mathematics, kamaraj college, thoothukudi-628003, tamil nadu, india. hsannam84@gmail.com. affiliated to manonmaniam sundaranar university, tirunelveli-627012, tamil nadu, india. †research scholar [21212102092011], pg and research department of mathematics, kamaraj college, thoothukudi, manonmaniam sundaranar university, tirunelveli-627012, tamil nadu, india. rvtpraba77@gmail.com. 1received on august 12, 2022. accepted on january 2, 2023. published on january 10, 2023. doi: 10.23755/rm.v41i0.838. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 206 g. hari siva annam and t. abinaya 1 introduction the concept of soft theory was first originated by molodstov in 1999, which is deal with unpredictable problems meanwhile modeling results in engineering cases such as medical sciences, economics, etc., in 2003, maji. et. al.[8] studied and discussed the fundamental ideas of soft theory. following stage of soft set linked with netrosophic sets are introduced by parimala mani et. al.[9] in 2018 and also, we introduced the new notion of neutrosophic complex αψ-connectedness in neutrosophic complex topological spaces and investigate some of its properties in 2022[5]. in 2019[13], several new generalizations of nano open sets be introduced and investigated by nethaji, ochanan. the study of soft topological spaces (on short s.t.s) is instated by shabir and naz[14] in 2011. they discussed s.t on the collection θ on soft set (on short s.s) over u. accordingly, they discussed fundamental notions of s.t.s such as soft open (on short s.o), soft closed (on short s.c), s closure, s neighborhood of a point, s ti spaces, for (i =1, 2, 3, 4), s regular spaces, s normal spaces, and their specific features are also established. therefore, in 2011[1], naim cagman, serkan karatas, and serdar enginoglu investigated a topology with s.s called s.t and its corresponding features. then they present the foundation of the theory s.t.s. the s.t.s may be the initial stage for the concepts of the soft mathematical opinion of structures which are the foundation of s.s. theoretic operation. from the concept of s.s, the idea of soft lattices (on short s.l) has arisen. in 2010[7], f. li studied and defined this conviction of s.l and primary operations of results on s.l. additional, an application of s.s to lattices has executed by e. kuppusamy in 2011. a different approach towards s.l can be seen in e. kuppusamy apart from what f. li has done. further, the operation and the properties of s.l were studied by v. d. jobish. et. al.[4] in 2013. many theorems related to various types of unions, intersections, and complements including de margon’s laws are obtained. in 2020[12], m. parimala et. al explained the niαg closed sets in nano ideal toplogical spaces with various prevailing closed sets. currently, topology depends toughly on the thoughts of the soft theory. recently, s.l.t.s was first investigated by sandhya. et. al.[11] in 2021 that are discussed throughout an s.l ’l’ with a fixed set of parameters ’a’ and it is also a generalization of s.t.s. they detailed discussed the concept of soft l open (on short s.l o), soft l closed (on short s.l c), s.l closure, s.l – interior point, and s.l neighborhood. in this paper, we continue investigating a soft l – interior (on short s.l i), soft l – exterior (on short s.l e), soft l boundary (on short s.l b), and soft l border (on short s.l bor) which are basics for stimulating research on s.t.s and will build up the fountain of the theory of s.t.s. 207 elongation of sets in soft lattice topological spaces 2 preliminaries definition 2.1 (5,7). let’s take u be a whole set and a be a set parameters. a pair (f, a), where f is a map from a to ℘(u) is called a s.s over u. here, the s.s is simply represented by fa. example 2.1. let say that there are 6 cars in the whole world u = {w1, w2, w3, w4, w5, w6} is the set of cars under regard and that a = {ρ1, ρ2, ρ3, ρ4, ρ5} is a set of parameters denoted as colors. the ra, (a = 1, 2, 3, 4, 5) it means the parameters ‘red’, ‘blue’, ‘black’, ‘white’, and ’ash,’ respectively. consider the mapping fa given by ‘cars’ (.), where (.) is to be complete in by one of the parameters ra ∈ e. for instance, fa(ρ1) means ‘ cars (colors)’. suppose that b = {ρ1, ρ2, ρ5} ⊆ a and fa(ρ1) = {w1, w4}, fa(ρ2) = u, and fa(ρ5) = {w2, w4, w5} then, we can view the s.s fa as consisting of the following collection of approximations: fa = {(ρ1, {w1, w4}), (ρ2, u), (ρ5, {w2, w4, w5})}. definition 2.2 (2,7). in two s.s fa, ga over u, we say that (i) fa is a soft subset of ga if (a) a ⊆ b , and (b) ∀ρ ∈ a , λ (ρ)= µ (ρ) are equal to estimations. (ii) fa is soft equal set to ga denoted by fa = ga if fa ⊆ ga and ga ⊆ fa definition 2.3 (7). let a = {ρ1, .... ρn} be a parameters. the ‘not set of a’, denoted by γa is defined as γa = { γρ1, ..., γρn} , γρi means not ρi ∀ i = 1, 2, 3. . .n. definition 2.4 (7,9). complement of a s.s fa over u, represented by f ′a is defined as f ′a = (f ′, γa), f ′ : γa −→ ℘ (u) such that(on short s.t) f ′(γρ) = u – f(ρ), ∀ γρ ∈ γa. definition 2.5 (9). the relative complement of a s.s fa over u, stand for fca is defined as (fa)c = (f c, a), f c : a −→ ℘ (u) s.t f c(ρ) = u–f(ρ), ∀ ρ ∈ a. definition 2.6 (7,9). let fa be a s.s over u , then fa is null s.s if ∀ ρ ∈ a, f(ρ) = ϕ and is denoted by ϕa. let fa be a s.s over u ,then fa is absolute s.s represented by ua, if ∀ ρ ∈ a, f(ρ) = u. also, uca = ϕa and ϕ c a = ua. 208 g. hari siva annam and t. abinaya definition 2.7 (2,7). union of two s.s fa, gb over u is the s.s hc , c = a ⋃ b and ∀ ρ ∈ c, κ(ρ) =   λ(ρ), if ρ ∈ a − b µ(ρ), if ρ ∈ b − a λ(ρ) ⋃ µ(ρ), if ρ ∈ a ⋂ b we write fa ⋃ gb = hc . definition 2.8 (2,7). the intersection of two s.s fa, gb over a whole set u is the s.s hc , here c = a ⋂ b and ∀ e ∈ c, κ(ρ) = λ(ρ) or µ(ρ). we mark done fa ⋂ gb = hc. definition 2.9 (1). consider fa, ga ∈ s.s (u, a). the soft symmetric difference of these sets is the s.s. ha ∈ to s.s.(u, a), here the map h : a−→℘(u) defined as follows: h(ρ) = ((f(ρ) \ g(ρ)) ⋃ ((g(ρ) \ (f(ρ)) for each ρ ∈ a. we mark down ha = fa ∆ ga. definition 2.10 (3,6,10). a sublatice of a lattice l is a non-void subset of l that is a lattice with the same meet and join operation as l, ie., α, β ∈ l implies α ∧ β, α ∨ β ∀ α, β ∈ l. definition 2.11 (3,6,10). a complete lattice l and a is the parameters of the s.l over l. the triplet m = (f, a, l), f : a−→℘(l) is s.l if f(ρ) is the sublattice of l for each ρ ∈ a. then the s.l is represented by fla. definition 2.12 (10). two s.l. fla and g l a over l its difference is denoted by hla = f l a \ g l a, is stated as h(ρ) = ((f(ρ) \ g(ρ)) ∀ρ ∈ a. definition 2.13 (10). let us consider l be any complete lattice and a be the non void set of parameters. let θ contains complete members, uniquely complemented s.l. over l, then θ is s.l.t, then the condition hold: (i) ϕa, la ∈ θ. (ii) ⋃ a ∈ n ηa ∈ θ, ∀ { ηa : a ∈ n} ⊆ θ (iii) η1 ⋂ η2 ∈ θ, ∀ η1, η2 ∈ θ. then the triplet (l, θ, a) is called a s.l.t.s. (soft l – space or soft l – topological space) over l. the members of θ are called soft lattice open sets in l. also, a soft lattice (fla) is called soft lattice closed if the relative complement (fla) c belongs to θ. 209 elongation of sets in soft lattice topological spaces 3 extension of s.l sets definition 3.1. in s.l.t.s, the s.l i of (fla) is the union of all s.l o sets contained in fla denoted by (f l a) ◦. i.e., (fla) ◦ = ⋃ {gla : g l a ∈ θ and g l a ⊆ f l a}. theorem 3.1. let ( l, θ, a) be a s.l.t.s over l and fla , g l a are s.l. over l. then, (i) ϕ◦a = ϕa and la = l ◦ a (ii) (fla) ◦ ⊆ (fla) (iii) fla is a s.l − o set ⇐⇒ (f l a) ◦ = fla (iv) ((fla) ◦)◦ = (fla) ◦ (v) fla ⊆ g l a ⇒ (f l a) ◦ ⊆ (gla) ◦ (vi) (fla) ◦ ⋂ (gla)◦ = (fla ⋂ gla)◦ (vii) (fla) ◦ ⋃ (gla)◦ ⊆ (fla ⋃ gla)◦ proof results (i), (ii) are trival. (iii) if (fla) is s.l−o set, (f l a) is itself a s.l−o set contained in (f l a). since, (fla) ◦ is the largest s.l − o set contained in (fla) , (f l a) = (f l a) ◦. conversely, suppose that (fla) = (f l a) ◦. since (fla) ◦ is a s.l − o set, so (fla) is s.l − o set over l. (iv) since (fla) ◦ is s.l − o set, by (iii) ((fla) ◦)◦ = (fla) ◦. (v) suppose that (fla) ⊆ (g l a). since,(f l a) ◦ ⊆ (fla) ⊆ (g l a). (f l a) ◦ is a s.l − o subset of (gla), so by the definition of (g l a) ◦, (fla) ◦ ⊆ (gla) ◦. (vi) we have (fla ⋂ gla) ⊆ f l a and (f l a ⋂ gla) ⊆ g l a. this implies (by v) (fla ⋂ gla) ◦ ⊆ (fla) ◦and (fla ⋂ gla) ◦ ⊆ (gla) ◦ so that, (fla ⋂ gla) ◦ ⊆ (fla) ◦ ⋂ (gla)◦. also, since (fla) ◦ ⊆ fla and (g l a) ◦ ⊆ gla implies (fla) ◦ ⋂ (gla)◦ ⊆ (fla ⋂ gla) so that, (fla ⋂ gla)◦ is the largest s.l − o subsets of (fla ⋂ gla). hence, (f l a) ◦ ⋂ (gla)◦ ⊆ (fla ⋂ gla)◦. thus, (fla) ◦ ⋂ (gla)◦ = (fla ⋂ gla)◦. 210 g. hari siva annam and t. abinaya (vii) since, fla ⊆ (f l a ⋃ gla) and, g l a ⊆ (f l a ⋃ gla). so, by (v) (fla) ◦ ⊆ (fla ⋃ gla) ◦ and (gla) ◦ ⊆ (fla ⋃ gla) ◦. so that (fla) ◦ ⋃ (gla)◦ ⊆ (fla ⋃ gla)◦. example 3.1. now the given example to show that the statement of theorem 1(v) may be strict or equal, let l = {sl1, sl2, sl3, sl4, sl5, sl6, sl7, sl8} ;a = {ρ1, ρ2}; θ = {fl1a, f l 2a, f l 3a, f l 4a, f l 5a, la, ϕa} figure 1: complete lattice fl1a = {(ρ1, {sl4, sl7, sl8}), ( ρ2, {sl3, sl6})}, fl2a = {(ρ1, {sl6, sl8}), (ρ2, {sl1, sl4})} fl3a = {(ρ1, {sl4, sl6, sl7, sl8}), (ρ2, {sl1, sl3, sl4, sl6})}, fl4a = {(ρ1, {sl8}), (ρ2, ϕ)} and fl5a = {(ρ1, {sl4, sl7}), (ρ2, {sl3, sl6})} for equal condition, we choose any two s.l from figure:1, flc = {(ρ1, {sl6, sl8}), (ρ2, {sl1, sl4})} and glc = {(ρ1, {sl1, sl6, sl7, sl8}), (ρ2, {sl1, sl3, sl4, sl6})} (flc ) ◦ = fl2a and (g l c) ◦ = fl2a. hence, fla ⊂ g l a implies(f l a) ◦ = (gla) ◦. 211 elongation of sets in soft lattice topological spaces for inclusion condition, we choose any two s.l from figure:1, fld = {(ρ1, {sl8}), (ρ2, {sl3, sl6})} and gld = {(ρ1, {sl4, sl7, sl8}), (ρ2, {sl3, sl6})} (fld) ◦ = fl4a and (g l d) ◦ = fl1a. hence, fla ⊂ g l a implies(f l a) ◦ ⊂ (gla) ◦. example 3.2. now the given example to show that the statement of theorem 1(vii) may be strict or equal, let us consider the lattice and s.l.t given in example: 3.1 for inclusion condition, we choose any two s.l from figure:1, flc = {(ρ1, {sl6, sl8}), (ρ2, {sl1, sl3, sl6})} and glc = {(ρ1, {sl4, sl7, sl8}), (ρ2, {sl1, sl3, sl4, sl6})} (flc ) ◦ = fl4a and (g l c) ◦ = fl1a, which implies (f l c ) ◦ ⋃ (glc)◦ = fl1a. (flc ⋃ glc) ◦ is fl3a. hence, (fla) ◦ ⋃ (gla) ◦ ⊂ (fla ⋃ gla) ◦. for equal condition, we choose any two s.l from figure:1, flc = {(ρ1, {sl6, sl8}), (ρ2, {sl1, sl4})} and glc = {(ρ1, {sl1, sl6, sl7, sl8}), (ρ2, {sl1, sl3, sl4, sl6})} (flc ) ◦ = fl2a and (g l c) ◦ = fl2a, which implies (f l c ) ◦ ⋃ (glc)◦ = fl2a. hence, (fla) ◦ ⋃ (gla) ◦ = (fla ⋃ gla) ◦. definition 3.2. let (l, θ, a) be a s.l.t.s over l, then the s.l e. of s.l fla is denoted by (fla)◦ and is defined as (f l a)◦ = ((f l a) c)◦. 212 g. hari siva annam and t. abinaya theorem 3.2. let fla and g l a be s.l of a s.l.t.s (l, θ, a). then, (i) (fla ⋃ gla)◦ = (f l a)◦ ⋂ (gla)◦. (ii) (fla)◦ ⋃ (gla)◦ ⊆ (f l a ⋂ gla)◦. (iii) fla ⊆ g l a implies (f l a)◦ ⊇ (g l a)◦. proof (i) (fla ⋃ gla)◦ = ((f l a ⋃ gla) c)◦ = ((fla) c ⋂ (gla) c)◦ = ((fla) c)◦ ⋂ ((gla) c)◦ = (fla)◦ ⋂ (gla)◦ (ii) (fla)◦ ⋃ (gla)◦ = ((f l a) c)◦ ⋃ ((gla) c)◦ ⊆ ((fla) c ⋃ (gla) c)◦ = ((fla ⋂ gla) c)◦ = (fla ⋂ gla)◦ (iii) (gla)◦ = ((g l a) c)◦ ⊆ ((fla) c)◦ = (fla)◦ example 3.3. now the given example to show that the statement of theorem 2(ii) may be strict or equal, let la = {sl1, sl2, sl3, sl4, sl5, sl6, sl7}; a = {ρ1, ρ2}; θ = {fl1a, f l 2a, f l 3a, f l 4a, la, ϕa} fl1a = {(ρ1, {sl3, sl6}), (ρ2, {sl4, sl5})}, f l 2a = {(ρ1, {sl6}), (ρ2, ϕ)} fl3a = {(ρ1, {sl2, sl3, sl5, sl6}), (ρ2, {sl4, sl5, sl6})} and fl4a = {(ρ1, {sl2, sl5, sl6}), (ρ2, {sl6})} figure 2: complete lattice 213 elongation of sets in soft lattice topological spaces for inclusion condition, now we take any two s.l from the figure:2, flc = {(ρ1, {sl2, sl6}), (ρ2, {sl6})} and glc = {(ρ1, {sl2, sl3, sl5}), (ρ2, {sl4, sl5})} then, (flc ⋂ glc) = {(ρ1, {sl2}), (ρ2, ϕ)}. (f l c )◦ = ϕa and (g l c)◦ = f l 2a, which implies (flc )◦ ⋃ (glc)◦ = f l 2a. (f l c ⋂ glc)◦ is f l 1a. hence, (fla)◦ ⋃ (gla)◦ ⊂ (f l a ⋂ gla)◦. for equal condition, now we take any two s.l from the figure:2, flc = {(ρ1, {sl5, sl7}), (ρ2, {sl4, sl7})} and glc = {(ρ1, {sl3, sl5, sl7}), (ρ2, {sl4, sl5, sl7})} then, (flc ⋂ glc) = {(ρ1, {sl5, sl7}), ((ρ2, {sl4, sl7})}. (flc )◦ = f l 2a and (g l c)◦ = f l 2a, which implies (f l c )◦ ⋃ (glc)◦ = f l 2a. (flc ⋂ glc)◦ is f l 2a. hence, (fla)◦ ⋃ (gla)◦ = (f l a ⋂ gla)◦. example 3.4. now the given example to show that the statement of theorem 2(iii) may be strict or equal, let us consider the lattice and s.l.t given in example: 3.3 for equal condition, now we take any two s.l from the figure:2, fld = {(ρ1, {sl3, sl6}), (ρ2, {sl4, sl5, sl6})} and gld = {(ρ1, {sl1, sl3, sl5, sl6}), (ρ2, {sl4, sl5, sl6})} (fld)◦ = ϕa and (g l d)◦ = ϕa. hence, fla ⊆ g l a implies (f l a)◦ = (g l a)◦. 214 g. hari siva annam and t. abinaya for inclusion condition, now we take any two s.l from the figure:2, flb = {(ρ1, {sl4, sl5}), (ρ2, {sl2, sl3, sl6, sl7})} and glb = {(ρ1, {sl2, s l4, sl5, sl7}), (ρ2, {sl1, sl2, sl3, sl5, sl6, sl7})} (flb)◦ = f l 1a and (g l b)◦ = f l 2a. hence,fla ⊆ g l a implies (f l a)◦ ⊃ (g l a)◦. definition 3.3. in s.l.t.s, then the s.l b of s.l fla is denoted by (f l a) b and is defined as (fla) b = fla ⋂ (fla) c. theorem 3.3. let (l, θ, a) be a s.l.t.s: (i) (fla) b ⋂ (fla) ◦ = flϕ (ii) (fla) b ⋂ (fla)◦ = f l ϕ proof (i) (fla) b ⋂ (fla) ◦ = ( fla ⋂ (fla) c ) ⋂ (fla) ◦ = fla ⋂ (fla) c ⋂ (fla) ◦ = flϕ (ii) (fla) b ⋂ (fla)◦ = f l a ⋂ (fla) c ⋂ (fla) c)◦ = fla ⋂ (fla) c ⋂ ((fla)) c = flϕ example 3.5. now the given example for find the boundary, let us consider the lattice and s.l.t given in example: 3.1 now we take any s.l from the figure:1, flc = {(ρ1, {sl2, sl8}), (ρ2, {sl1, sl3})} then, (f l c ) b = (fl4a) c definition 3.4. let (l, θ, a) be a s.l.t.s over l, then the s.l bor of s.l fla is denoted by (f l a) • and is defined as (fla) • = fla − (f l a) ◦. theorem 3.4. let (l, θ, a) be a s.l.t.s. then the following hold: 215 elongation of sets in soft lattice topological spaces (i) (fla) • = a ⋂ (la − fla) (ii) (ϕla) • = ϕla (iii) (fla) • ⊆ ((fla) ◦)c (iv) (fla) • ⊆ fla ⊆ fla proof (i) fla ⋂ ((fla) ◦)c = fla ⋂ (fla) c = fla ⋂ (la − fla) (ii) ϕla ⋂ ((ϕla) ◦)c = (ϕla) ⋂ (ϕla) c = ϕla (iii) fla − (f l a) ◦ = fla ⋂ ((fla) ◦)c ⊆ ((fla) ◦)c (iv) by definition of (fla) •, (fla) • ⊆ fla . we know that,fla ⊂ fla therefore,(f l a) • ⊆ fla ⊆ fla example 3.6. now the given example to show that the statement of theorem 4(iv) may be strict or equal, let us consider the lattice and s.l.t given in example: 3.3 we choose any two s.l from the figure:2, glb = {(ρ1, {sl2, sl3, sl5, sl6}), (ρ2, {sl4, sl5, sl6})} now, the closure of glb is la , then border of g l b, is ϕa hence,(glb) • ⊂ glb ⊂ glb. 4 conclusions in the present work, we defined and discussed some s.l – sets of s.l.t.s. we extended some basic results relating to s.l i, s.l e, s.l b, and s.l bor of s.l.t.s. in the interior section, idempotent and monotonicity results are held. formerly the intersection of the boundary and interior soft lattice gives the null set and the intersection of the boundary and exterior soft lattice should not give the non-empty soft sets. in end, this paper is the inception of a novel 216 g. hari siva annam and t. abinaya structure. further, we learned a few viewpoints, it will be needed to carry out a new seeking work to build future applications. acknowledgement i would like to intently acknowledge the beneficial proposals, efforts, and precious time given by g. hari siva annam. their valued supervision and feedback helped me to complete this article. references [1] c¸ a�gman, n., karata¸s, s. and enginoglu, s., 2011. soft topology. computers and mathematics with applications, 62(1), pp.351-358. 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[14] shabir, m. and naz, m., 2011. on soft topological spaces. computers and mathematics with applications, 61(7), pp.1786-1799. 218 ratio mathematica volume 46, 2023 intuitionistic robust fuzzy matrix for the diagnosis of stress, anxiety and hypertension k.revathi* p. sundararajan† abstract the mathematical model given here attempts to improve precision in the diagnosis of stress, anxiety, and hypertension using intuitionistic robust fuzzy matrix (irfm). in practice, the imprecise nature of medical documentation and the uncertainty of patient information frequently do not provide the appropriate level of confidence in the diagnosis. to that purpose, a novel method based on distinct fuzzy matrices and fuzzy relations is devised, which makes use of the capabilities of fuzzy logic in describing, understanding, and exploiting facts and information that are unclear and lack clarity. with the assistance of 30 doctors, a medical knowledge base is created during the procedure. the model obtained 95.55%t accuracy in the diagnosis, demonstrating its utility. keywords: fuzzy logic; fuzzy matrices; max-min principles; robust; intuitionistics. 2020 ams subject classifications: 03b52, 62f35, 70h30, 20n25 1 *district institute of education and training, perambalur, tamil nadu, india. revathiramesha@gmail.com. †department of mathematics, arignar anna govt. arts college, namakkal. tamilnadu, india. ponsundar03@gmail.com. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1087. issn: 1592-7415. eissn: 2282-8214. ©k.revathi et al. this paper is published under the cc-by licence agreement. 298 k.revathi and p.sundararajan 1 introduction the computer scientist zadeh was the first to utilize fuzzy logic as a scientific notion. medicine is one subject where the use of fuzzy logic was recognized as early as the mid-1970s. the ambiguity encountered during the illness diagnostic process has repeatedly been the subject of application of fuzzy set theory in this sector. the greatest and most helpful explanations of clinical symptoms frequently include language concepts that are inevitably ambiguousadlassing [1986]. fuzzy mathematics seeks to derive exact meaning from erroneous information by capturing ambiguity in real life. it is a valuable tool for decision-making systems. using the fuzzy set paradigm, several ways have been created to imitate medical diagnostic procedures. the basic principle behind a medical diagnosis is to associate a patient’s symptoms or signals with potential illnesses based on the expert’s medical knowledge. according to sanchez [1979], meenakshi and m [2011], elizabeth and sujatha.l [2013], ravi.j [2022] and raich and dalal [2009] is approach demonstrates the doctor’s medical skill as a blurry association between symptoms and illnesses. the method to medical diagnostics by employing an interval-valued fuzzy matrix as a representation. the another approach of medical diagnosis by utilising a triangular fuzzy membership matrix representation. saravanan and prasanna [2016] presented a fuzzy matrices application for use in medicine that utilised the idea network and concept matrices. raich and dalal [2009] employed fuzzy matrices for the first time in diabetes research. in this study, we employed irfm to improve medical diagnostic accuracy klir and yuan [1995]. this mathematical model produces diagnoses based on the expertise and experience of the 30 doctors. we created a differentiating strategy for studying indication relationships for diagnosis, which can be described as nonsymptom indication non-occurrence indication conformability indication gupta [1976]. our method is reasonably accurate, as evidenced by a 95% confidence between the genuine diagnoses supplied by physicians and the diagnostic conclusion produced by our algorithm. 2 research method this section shows the consists of the following elements: 2.1 medical terms: the expertise of physicians is depicted as a hazy relationship between symptoms and illnesses. there are two kinds of hazy relationships between symptoms and diseases: 299 intuitionistic robust fuzzy matrix for the diagnosis of stress, anxiety and hypertension (i). an incidence relation ri it describes the possibility or proclivity of a symptom manifesting when a specific condition is present, i.e., how frequently does the symptom i occur with disease p. (ii). a comfortability relation rj it denotes how well the symptoms differentiate between diseases, or how strongly symptom i verifies disease j. the distinction between occurrence and conformability is significant because, whereas a symptom is present in many diseases, its occurrence and conformability differ depending on the condition.the above-mentioned correlations were discovered using medical records from an expert. 2.2 intuitionistic robust fuzzy matrices (irfm) then the membership function of a and b is defined as, a�b = { max [ min ( σij (r1) + σjk (r1) 2 )] ,max [ min ( sij (r2) + sjk (r2) 2 )]} if sij (r2) = sjk (r2) = 0 for all i,j,k then, a�b = { max [ min ( σij (r1) + σjk (r1) 2 )] , 0 } a�b = { max [ min ( a1 + b1 2 ) , ( a2+b2 2 ) , ( a3 + b3 2 ) , . . . , ( an+bn 2 )] , 0 } let d represent a grouping of specific illnesses, s represent a grouping of symptoms, and p represent a grouping of persons in need of a diagnosis. according to the fuzzy relation rs(p, s) (where pm, sn) membership grades, the set ps is supposed to indicate the degree to which the symptom s is present in the patient p. on the set s, d, the fuzzy relation ri (s, d) = ri (m, n) is defined, where ss, dd denotes the recurrence frequency of symptoms s with sickness d. the same set defines the fuzzy relation rj, which shows how strongly a symptom (s) predicts the presence of an illness (d). a fuzzy relation rj is also formed on the same set, where rj (p, d)= r2(n,p) denotes the strength with which a symptom (s) supports the occurrence of an illness (d). using relations rs, ri, and rj, the following four alternative indicator relations supplied on set p×d are computed: (i).relationship between the two r1 = rs � ri (ii).conformability indication relation r2 = rs � rj 300 k.revathi and p.sundararajan (iii). non-occurrence occurrence indication relation indication relation r3 = rs � (1 − ri) (iv). non-symptom indication relation r4 = ri � (1 − rj) the max-min product of fuzzy matrices is used to compose fuzzy relations. example 1 let d= {d1, d2}, s= {s1, s2, s3}, p= {p1, p2} let r1 = rs � ri so for every i=1, 2 and j= 1, 2, r1(pi, dj) = max{min{ rs(pi, s), r1(s, dj)}}s�s example 2 let r1(p1,d1) = max{min{0.2,0.1}/2, min{0.4,0.7}/2, min{0.1,0.2}/2} = max{0.05, 0.2, 0.15} = 0.2. similarly, r1(pi, dj) for every i and j. hence, we get other indication relations are calculated in the same way. 2.3 hypertension overview a blood pressure reading of 120/80 mmhg or less is considered normal. you may aim to keep your blood pressure in a healthy range every day, regardless of your age. one of the most dangerous elements of high blood pressure is that you may be unaware of it. in reality, one-third of persons with high blood pressure are unaware of their condition. this is because indications of high blood pressure are uncommon until blood pressure is quite high. regular blood pressure checks are the best way to discover if your blood pressure is too high. you may also monitor your blood pressure at home. this is especially important if someone in your family has high blood pressure. i regularly suffer from headaches. headaches and nosebleeds are not always signs of high blood pressure. this is feasible when blood pressure increases above 180/120, as it does during a hypertensive crisis. 2.3.1 there are two kinds of hypertension. 301 intuitionistic robust fuzzy matrix for the diagnosis of stress, anxiety and hypertension (i). the primary (essential) hypertension the most patient instances of high blood pressure have no known cause. primary (essential) hypertension is a form of high blood pressure that appears gradually over time. (ii). secondary hypertension some people develop high blood pressure as a result of a more serious condition. secondary hypertension, a more severe variant of primary hypertension, is a kind of high blood pressure that develops suddenly. secondary hypertension can be caused by a variety of medical conditions and medications, including obstructive sleep apnea, kidney disease, adrenal cancer, and thyroid problems. since birth, your body has had certain blood vessel anomalies. drugs include birth control pills, allergy and cold treatments, decongestants, over-the-counter pain relievers, and some prescription prescriptions. illicit drugs include amphetamines and cocaine. disease 1: heart attack or stroke or heart failure. atherosclerosis, or artery hardening and thickening, can lead to heart attacks, strokes, and other complications. to pump blood against the increased pressure in your veins, your heart must work harder. as a result, the walls of the heart’s pumping chamber thicken (left ventricular hypertrophy) (left ventricular hypertrophy). heart failure may occur if the developing muscle is unable to pump enough blood to meet your body’s demands. disease 2: aneurysm is a weakening and narrowing of the blood arteries in your kidneys. high blood pressure can weaken and bulge your blood vessels, resulting in an aneurysm. a ruptured aneurysm can endanger one’s life. disease 3: eye blood vessel enlargement, constriction, or rupture; metabolic syndrome; memory or cognitive problems; dementia as a result, several organs may be unable to operate effectively. this might result in vision loss. this syndrome is characterised by a larger waist, greater triglyceride levels, lower hdl cholesterol (the ”good” cholesterol), higher blood pressure, and higher insulin levels. all of these disorders raise your chances of acquiring diabetes, heart disease, or stroke. uncontrolled high blood pressure may impair your ability to think, remember, and learn. patients with high blood pressure have difficulty grasping and recalling their ideas. one type of dementia is caused by clogged or restricted arteries, which limit the quantity of blood that can reach the brain. a stroke can cause vascular dementia by cutting off blood supply to the brain. 2.3.2 symptoms of severe high blood pressure if your blood pressure is really high, you should be aware of the following symptoms: symptom s1: severe headaches symptom s2: nosebleed symptom s3: fatigue or confusion 302 k.revathi and p.sundararajan symptom s4: vision problems symptom s5: chest pain symptom s6: difficulty breathing symptom s7: irregular heartbeat symptom s8: blood in the urine symptom s9: pounding in your chest, neck, or ears. people sometimes feel that other symptoms may be related to high blood pressure, however it is possible that they are not: symptom s10: dizziness symptom s11: nervousness symptom s12: sweating symptom s13: trouble sleeping symptom s14: facial flushing s represents the collection of all symptoms. s=(s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,s11,s12,s13,s14} let d represent the whole collection of illnesses, d=(d1,d2,d3}. let p represent the precise universal set of all patients, d=(p1,p2,p3}. 3 experimental study as a case study, we picked a serious condition with three types of symptoms, such as hypertension. these ailments were chosen because, according to who (world health organization) figures, they are the most frequent in india. according to who data from 2016, the adult prevalence of hypertension in india is 24%. following the selection of the three illnesses, we chose 14 symptoms that are commonly found in people with those ailments.then, with the help of medical specialists, we constructed a medical knowledge base. we looked at the prevalence of a symptom in a linked sickness and the confirmation of a disease from a connected symptom as we built the database.the phrase ”symptom emergence in the setting of the associated sickness” is related to the question ”how frequently does a symptom appear in the presence of a disease?” when a certain sickness is present, the average or frequency of symptom presentation is high. it alludes to the question, ”to what degree do symptoms strongly confirm disease d?” a symptom’s discriminating power in confirming the presence of an illness is defined by its ability to confirm the presence of a disease based on its own observations. we distributed blank charts (appendix) to 30 randomly selected specialist doctors in namakkal district, tamilnadu, india, in order to build a knowledge base that includes the presence of a symptom in the linked illness and confirmation of a disease based on a specific symptom. we asked the clinicians to keep track of how many patients with the relevant disease exhibited each symptom out of 100. 303 intuitionistic robust fuzzy matrix for the diagnosis of stress, anxiety and hypertension for example, if a doctor checks 100 diabetic patients and finds ”severe thirst” in 80 of them, he or she must document 90 cases of excessive thirst in hypertension. to determine the presence of a disease, we asked the doctors to specify the percentage of the time that they detect the relevant symptom in the patient. these statistics should be multiplied by 100 such that they fall between [0, 1] and indicate an 80% chance of anaemia. as shown in table 1, we calculated the mean of the information obtained to estimate the data’s central tendency. we awarded membership ratings to each of the 14 fuzzification symptoms based on the severity or frequency of the patient’s condition. the following are the grades: table 1: a patient’s symptoms according to their frequency or intensity sr. no. symptom cont.severecont.mild occa. rare no 1. severe headaches 1.0 0.75 0.50 0.25 0.00 2. nosebleed 1.0 0.75 0.50 0.25 0.00 3. fatigue or confusion 1.0 0.75 0.50 0.25 0.00 4. vision problems 1.0 0.75 0.50 0.25 0.00 5. chest pain 1.0 0.75 0.50 0.25 0.00 6. difficulty breathing 1.0 0.75 0.50 0.25 0.00 7. irregular heartbeat 1.0 0.75 0.50 0.25 0.00 8. blood in the urine 1.0 0.75 0.50 0.25 0.00 9. pounding in your chest, neck, or ears. 1.0 0.75 0.50 0.25 0.00 10. dizziness 1.0 0.75 0.50 0.25 0.00 11. nervousness 1.0 0.75 0.50 0.25 0.00 12. sweating 1.0 0.75 0.50 0.25 0.00 13. trouble sleeping 1.0 0.75 0.50 0.25 0.00 14. facial flushing 1.0 0.75 0.50 0.25 0.00 d denotes ”disease 1, disease 2, disease 3,” and s denotes ”set of 14 symptoms, s1, s2,..., s14.” our diagnostic technique will be demonstrated using three hypothetical instances. as a result, p = p1, p2, and p3, and we’ve built a fuzzy relation rs on the set ps in which membership grades rs(p, s) (where pp , ss reflect the severity or frequency of symptoms identified in these three people) suggest: table 2: expert knowledge-base obtained in a robust manner 304 k.revathi and p.sundararajan ssr. no. symptom disease 1 disease 2 disease 3 i j i j i j 1. severe headaches 0.16 0.07 0.50 0.37 0.51 0.41 2. nosebleed 0.20 0.10 0.89 0.80 0.09 0.05 3. fatigue or confusion 0.86 0.74 0.08 0.05 0.05 0.07 4. vision problems 0.47 0.36 0.17 0.11 0.07 0.03 5. chest pain 0.39 0.38 0.16 0.11 0.08 0.09 6. difficulty breathing 0.02 0.02 0.49 0.37 0.80 0.70 7. irregular heartbeat 0.52 0.45 0.25 0.19 0.37 0.27 8. blood in the urine 0.16 0.07 0.50 0.37 0.51 0.41 9. pounding in your chest, neck or ears. 0.71 0.64 0.85 0.75 0.33 0.22 10. dizziness 0.17 0.19 0.52 0.43 0.03 0. 00 11. nervousness 0.79 0.66 0.60 0.54 0.13 0.79 12. sweating 0.19 0.15 0.39 0.27 0.04 0.03 13. trouble sleeping 0.34 0.24 0.61 0.64 0.10 0.07 14. facial flushing 0.15 0.05 0.55 0.54 0.04 0.02 note: i = occurrence of a symptom & j =confirmation of a disease table 3: the presence of a disease is confirmed by the symptoms sr. no. symptoms p1 p2 p3 1. severe headaches 0.51 0.15 0.00 2. nosebleed 1.00 0.00 0.51 3. fatigue or confusion 0.15 1.00 1.00 4. vision problems 0.00 0.15 0.51 5. chest pain 1.00 0.51 0.00 6. difficulty breathing 0.00 1.00 0.26 7. irregular heartbeat 0.00 0.00 0.00 8. blood in the urine 0.00 0.15 0.51 9. pounding in your chest, neck, or ears. 0.51 0.26 1.00 10. dizziness 0.15 0.00 0.00 11. nervousness 1.00 0.00 0.51 12. sweating 0.26 0.00 0.00 13. trouble sleeping 0.51 0.00 1.00 14. facial flushing 1.00 0.51 0.15 305 intuitionistic robust fuzzy matrix for the diagnosis of stress, anxiety and hypertension table 4: the existence of illness is confirmed by the symptoms ‘d’ ssr. no. symptom ri rj disease 1 disease 2 disease 3 disease 1 disease 2 disease 3 1. severe headaches0.16 0.50 0.51 0.07 0.37 0.41 2. nosebleed 0.20 0.89 0.09 0.10 0.80 0.05 3. fatigue or confusion 0.86 0.08 0.07 0.74 0.05 0.05 4. vision problems 0.47 0.17 0.07 0.36 0.11 0.03 5. chest pain 0.39 0.16 0.09 0.38 0.11 0.08 6. difficulty breathing 0.02 0.49 0.80 0.02 0.37 0.70 7. irregular heartbeat 0.52 0.25 0.37 0.45 0.19 0.27 8. blood in the urine 0.16 0.50 0.51 0.07 0.37 0.41 9. pounding in your chest, neck, or ears. 0.71 0.85 0.33 0.64 0.75 0.22 10. dizziness 0.19 0.52 0.03 0.17 0.43 0. 00 11. nervousness 0.79 0.60 0.79 0.66 0.54 0.13 12. sweating 0.19 0.39 0.04 0.15 0.27 0.03 13. trouble sleeping 0.34 0.64 0.10 0.24 0.61 0.07 14. facial flushing 0.15 0.55 0.04 0.05 0.54 0.02 using the relations rs, ri, and rj, we developed four alternative indicator relations, namely r1, r2, r3, and r4. 306 k.revathi and p.sundararajan we may get a number of diagnostic conclusions from these four indicator associations. for example, if (p, d) = 1, we can validate patient p’s diagnosis of disease d. despite the fact that none of our four patients had this condition, it appears to suggest that disease d1 is strongly confirmed for patient p1, sickness d2 is highly confirmed for patient p3, and disease d3 has a 90% probability of occurring for patient p2. using the standards described above, we may conclude that patient p1 has high blood pressure, patient p3 has hypertension, and patient p2 has anaemia and mild hypertension. we utilised our programme to diagnose 50 cases using this method, and we compared our findings to physician diagnoses. in 43 of 50 instances of diabetes, anaemia, and hypertension, our diagnosis and the doctors’ diagnosis were identical. the degree of accuracy of the model may be calculated as follows: finding of accuracy = [(the number of accurate data)/ (the number of total data)]*100 = (86/90)*100 = 95.55% as a consequence, the chi-square test, often known as the ”goodness of fit test,” was used to check that our mathematical model was statistically sound. before utilising the chi-square test, we assumed that the medical professionals’ diagnoses were completely correct. h0: the medical professionals’ and our diagnoses are identical. h1: our diagnosis differs from that of the physicians. aceptance level (α) = 0.05 & df= 2 at critical value= 5.991 table 5: contingency table for chisquare test observed expected stress 50 50 anxiety 47 50 hypertension 46 50 the chi-square test calculation formula is as follows: χ²= σ [(o – e)² /e] where o is observed frequency, e is expected frequency. 1. where e is the expected frequency and o is the actual frequency 2. using the contingency table, we calculated the chi-square (??) as follows: (χ²) as , χ² = ((50-50)²/50) + ((47-50)²/50) + (46-50)²/50) = 0.50. 307 intuitionistic robust fuzzy matrix for the diagnosis of stress, anxiety and hypertension because 0.50 is greater than 5.991, or the ”2” critical value, the null hypothesis h0 is accepted. we got to the conclusion that the physicians’ and our diagnosis are similar. 4 conclusions fuzzy logic can be used to improve the precision and verification of medical diagnosis. the diagnosis offered by the ”max-min” composition of fuzzy relations built using the ”mean” of the physicians’ data matches to the physicians’ diagnosis. according to the chi-square test, which measures dependability, the proposed mathematical model gets a reliability score of 95.55%. this study’s innovative technique tries to quantify the core principle behind the diagnostic procedure. this technique distinguishes itself by thoroughly investigating the connections between the indicators, particularly the ”non-symptom indication-non-occurrence indication-conformability indication.” this method of preserving specialised information allows patients and general practitioners to access it. while it does not replace a doctor’s diagnosis, it does help to reinforce it. the diagnostic implications of this mathematical model might be considered as a second opinion to the doctor’s diagnosis. this mathematical approach is mostly employed to improve diagnostic accuracy. the main flaw of the proposed mathematical model is that it will be off-target if a patient supplies incorrect inputs, resulting in incorrect diagnostic findings. as more doctor data is collected, software and an android app will be created in the future. the development of new algorithms to improve accuracy is one of the future goals. in recent trends they are given for 95 percent of accurate results. but our proposed concept given more than 95 percent accuracy compared to other existing model. also our proposed method performed all direction (360o). in future work we are trying to write the coding with the help of python and analyze the data for the betterment of the concept. references k. adlassing. fuzzy set theory in medical diagnosis. ieee transaction systems man cybernetics, 16:260–265, 1986. 308 k.revathi and p.sundararajan s. elizabeth and sujatha.l. application of fuzzy membership matrix in medical diagnosis and decision making. applied mathematical sciences, 7:6297–6307, 2013. s. gupta. statistical methods. s. chand and sons, new delhi, 1976. g. klir and b. yuan. fuzzy sets and fuzzy logic: theory and applications. prenticehall, address = new jersey, 1995. a. r. meenakshi and k. m. an application of interval-valued fuzzy matrices in medical diagnosis. international journal of mathematical analysis, 5:1791– 1802, 2011. t. r. a. v. raich, v. and s. dalal. application of fuzzy matrix in the study of diabetes. world academy of science, 56:22–28, 2009. ravi.j. fuzzy graph and their applications: a review. international journal for science and advance research in technology, 8:107–111, 2022. e. sanchez. inverse of fuzzy relations, application to possibility distribution and medical diagnosis. fuzzy sets and systems, 2:75–86, 1979. k. saravanan and j. prasanna. applications of fuzzy matrices in medicine. global journal of pure and applied mathematics, 2:80–84, 2016. 309 ratio mathematica volume 42, 2022 the ahpsort ii to evaluate the high-level instruction performances gerarda fattoruso* paola mancini† gabriella marcarelli‡ abstract this paper proposes a model for ranking italian high schools based on several performance outputs. to analyze the performance of italian public high schools we consider the students’ school performance and their academic achievements; also the school characteristics may influence the performance evaluation of high schools, although the importance of these aspects is certainly less than the results achieved by students. data are from eduscopio and scuolainchiaro portals and refers to the 2019/20 school year. we analyze a sample of 263 high schools (hs) in all italian regions. for each school we consider nine outputs related to students’ school and academic performance, and school characteristics. we assess the performance of high schools using a multi-criteria approach. our analysis involves a high number of schools, so we apply the ahpsort ii method which in addition to defining the ranking of schools also defines their classification. our results show that scientific lyceums are all in the first class regardless of the geographic area. keywords: school ranking; academic performance; students’ achievements; ahpsort ii. 2020 ams subject classifications: 90 operations research, mathematical programming. 1 *corresponding author: university of sannio, benevento, italy and neoma bs, rouen, france; fattoruso@unisannio.it. †university of sannio, benevento, italy; paola.mancini@unisannio.it ‡university of sannio, benevento, italy; gabriella.marcarelli@unisannio.it 1received on may 3rd, 2022. accepted on june 12nd, 2022. published on june 30th, 2022. doi: 10.23755/rm.v41i0.809. issn: 1592-7415. eissn: 2282-8214. ©fattoruso et al. this paper is published under the cc-by licence agreement. 283 g. fattoruso, p. mancini, g. marcarelli 1 introduction this paper focuses on the evaluation of italian high schools’ performance. in italy, eduscopio (giovanni agnelli foundation) and scuolainchiaro (ministry of education) represent important sources of information for such an evaluation; they provide annually, for each school, data on students’ school careers, their academic achievements and school characteristics. in particular, eduscopio provides students’ and their families with a ranking of high school in the area of residence based on university performance of school leavers; scuolainchiaro makes available to the community all the information relating to italian schools of all levels, in an organic and structured form. this paper aims to provide the ranking of italian high schools, taking into account the school and academic careers of students as well as the characteristics of the school. the multi-criteria approach may be a useful tool to assess the performance of high schools. by considering the above datasets, [mancini and marcarelli, 2019] derived the ranking among the typologies of schools; more recently, [mancini and marcarelli, 2022] provide a ranking among the school types both at a national level and within each geographic area. furthermore, applying the ahp method and comparing the results with those obtained by a further mcdm method, promethee, they found significant differences between hs according to criteria related to school and academic performance both within and between geographic areas. many studies have dealt with the application of multi-criteria methods in the field of education [giannoulis and ishizaka, 2010, goztepe, 2020, mancini and marcarelli, 2022, stamenkovic et al., 2016]. by taking into account some performance indicators used by eduscopio and scuolainchiaro and according to the approach proposed by [mancini and marcarelli, 2022, 2019], this study analyzes nine performance outputs for a sample of 263 italian high schools. however, unlike [mancini and marcarelli, 2022, 2019] in which the schools were grouped into 6 different types of schools and 3 geographic areas, we provide the ranking among all the schools. due to the characteristics of the problem (e.g., independence among the elements, the high number of alternatives) and the output required, among multi-criteria methods proposed in the literature, this paper focuses on the analytic hierarchy process sort ii (ahpsort ii) method. the available data allows us to avoid some disadvantages of the ahp. there is no inconsistency in the judgment matrices because entries of matrices are ratios between performance indices. furthermore, the ahpsort ii allows to analyze a sorting decision problem through a feasible interaction with the decision makers precisely because it foresees a limited number of interactions. the goal of this paper is to obtain a ranking of high schools in italy (we consider 251 schools in our study) by sorting them into ordered classes. finally, in order to verify the impact (role) of the geographical and/or the school 284 the ahpsort ii to evaluate the high-level instruction performances typology factors on the performance of a school, we compare our results with those obtained by [mancini and marcarelli, 2022, 2019]. the paper is organized as follows: section 2 reports a literature overview on the topic; section 3 defines the methodology; section 4 reports the case study and the main results; sections 5 and 6 provide a discussion and some concluding remarks, respectively. 2 literature overview the literature on school ranking is vast. past studies have mainly focused on the school quality and its student achievements [eide and showalter, 1998], some other on school’s contribution to student academic performance [jamelske, 2009, kelly and downey, 2010] or on the question of “school accountability” affecting the school choice [burgess et al., 2013, hart and figlio, 2015, nunes et al., 2018]. many factors may influence students’ achievements, such as their socioeconomic status, family background, geographical area of residence and the type of school attended [agasisti and murtinu, 2012, lauer, 2003] as far as school and class size, students’ features and school management and resources [giambona and porcu, 2018, masci et al., 2018]. as regards the impact of secondary school on academic performance, recently, aina et al. [2011] and aina et al. [2021] have demonstrated that differences in university students’ achievements across high schools cannot be limited to the first-year and have to consider the geographic differences. in recent study, several authors used multi-criteria methods for analysed school ranking and their performance. bana e costa and oliveira [2012] use the measuring attractiveness by a categorical based evaluation technique (macbeth) which allows a multi-criteria evaluation of the different scientific areas within high schools in order to offer an accurate evaluation for each range of activities proposed by the school. blasco-blasco et al. [2021] analyze the performance of high schools with a particular focus on student achievement indicators. the authors analyze the student support policies of the schools with the use of technique for others preference by similarity to ideal solution (topsis). yendra et al. [2018] also use topsis integrated with ahp for the analysis of the quality of the training offer of the high school and higher level. recently, mancini and marcarelli [2019] analyzed the performance of italian high schools, and derived a ranking among different typologies of schools based on the students’ academic achievements, their school performance and the school characteristics. then, mancini and marcarelli [2022] made an in-deep analysis taking into account the geographic areas. using ahp and promethee methods they derive a ranking among the school typologies both at a national level and within each geographic area. 285 g. fattoruso, p. mancini, g. marcarelli 3 methodology ahpsort ii is a multi-criteria decision analysis (mcda) method for solving sorting problems. this method allows you to use several criteria and a very large number of alternatives [ishizaka et al., 2012]. furthermore, it is a method that provides for a limited interaction with the decision maker [fattoruso et al., 2022]. the procedure can be repeated or easily automated. the ahpsort ii method is described below. we consider a set of alternatives a=(a1, . . . , ai) evaluated respect a set of criteria g=(g1, . . . , gj); therefore, gj(ai) represents the evaluation of alternative ai on criterion gj. moreover, we consider a set of classes c=(c1, . . . , co); the construction of the classes requires the definition by the decision maker (dm) of the limiting profiles lpij or of the central profiles cpij for each considered gj criterion [ishizaka et al., 2020]. the lpij are defined by the dm when he is able to clearly separate the classes from each other, therefore they represent thresholds that separate the classes from each other; alternatively, when this is not possible, the dm opts to define the cpij which represent the centroids of each class for all the considered criteria. the ahpsort ii allows you to analyze a large number of alternatives by using representative profiles rpsj = (rp1j, . . . , rpsj). the rp are points homogeneously distributed in the observed data for each gj criterion. once all the elements that constitute the decision problem have been defined, ahpsort ii foresees the evaluation of local priorities: wj for gj; psj for rpsj; and, poj alternatively for lpij or cpij. the local priorities are determined using pairwise comparison matrices (pcms) with the eigenvalue method [ishizaka et al., 2020]. the determination of the local priority of the alternatives pij is instead defined through the following linear interpolation formula: pij = psj + ps+1j − psj rps+1j − rpsj · (gj(ai) − rpsj). (1) the global priority of the alternative ai is defined as follow: pi = j∑ j=1 pij · wj. (2) while the global priority pk of lpij or cpij is defined as: pk = j∑ j=1 poj · wj (3) the sorting of ai to a c class takes place considering the final global priority; therefore, considering the proximity of pi to pk. 286 the ahpsort ii to evaluate the high-level instruction performances 4 case study 4.1 data our case study concerns the evaluation of the performance of schools in italy. in particular, our reference sample is composed of 56 scientific lyceums (sl), 38 classical high schools (cl), 39 linguistic high schools (ll), 26 high schools of human sciences (hsl), 43 commercial technical high schools (cths), 49 high school technological technician (tths). figure 1 shows the percentage of schools considered in this study divided by typology. figure 1: typologies of high schools in italy each school is evaluated against criteria that determine the performance of the school among these are: maturity score (g1) defined as the weighted average score between the high school graduation score of enrolled students e non-enrolled students; invalsi test score (g2) defined as the average of each student’s math, reading and foreign language test scores; percentage of graduates in good standing (without failures) (g3); students enrolled in the academic year (g4). in addition to the school performance criteria, schools are evaluated for the academic performance of their students; this type of criteria include: the percentage of students who pass the first year (g5); percentage of academic credits achieved at the end of the first year (g6); average exam score (g7). finally, the evaluation of schools also takes into account criteria that consider the characteristics of each school, including the average number of students per class (g8) and the percentage of teachers employed part-time (g9). the data was collected by the eduscopio and scuolainchiaro portals for the 2019-2020 academic year. 287 g. fattoruso, p. mancini, g. marcarelli 4.2 results in our paper we consider a set of alternatives a = (a1, . . . , a251) evaluated respect the criteria set g = (g1, . . . , g9); the evaluation table of gj(ai) is reported in appendix a. the decision-makers involved in the construction of our study are school managers of the different typology of schools considered which from here on we will generically call dms. we report in table 1 the weights wj of criteria defined with the eigenvalue method. g1 g2 g3 g4 g5 g6 g7 g8 g9 wj 0,10 0,15 0,10 0,10 0,15 0,15 0,15 0,05 0,05 table 1: criteria weights wj for each gj criterion considered, we have defined three priority classes c1, c2 and c3 to which we have associated for simplicity the labels of low (c1), medium (c2), and high (c3). classes define the performance of high schools. therefore in the high (c3) the schools with the best performances will be sorted, in the low class (c1) those with the worst performances. in table 2, we report the central profiles cpij defined by the dms, for each class c and each criterion gj. g1 g2 g3 g4 g5 g6 g7 g8 g9 low (c1) 65,79 1 17,4 0,33 0,2 21,26 20 10 1 medium (c2) 75,71 4 53,8 0,64 0,55 54,4 24,44 19 26,14 high (c3) 80 6 80 0,8 0,8 80 28 26 50 table 2: central profiles cpij for each criterion gj as suggested by abastante et al. [2019], we have built the reference profiles rpsj = (rp1j, . . . , rp6j). we report in table 3 the rpsj for each criterion gj. g1 g2 g3 g4 g5 g6 g7 g8 g9 rp1j 0 0 0 0 0 0 0 0 0 rp2j 17,12 1,4 18,04 0,192 0,18 17,50 4 5,6 10,25 rp3j 34,25 2,8 36,08 0,384 0,36 35,01 8 11,2 20,51 rp4j 51,38 4,2 54,12 0,576 0,54 52,52 12 16,8 30,77 rp5j 68,50 5,6 72,16 0,768 0,72 70,03 16 22,4 41,03 rp6j 85,63 7 90,2 0,96 0,9 87,54 20 28 51,29 table 3: reference profiles rpsj for each criterion gj 288 the ahpsort ii to evaluate the high-level instruction performances moreover, we have defined in table 4 the local priorities psj and in table 5 the local priorities poj. ps1 ps2 ps3 ps4 ps5 ps6 ps7 ps8 ps9 rp1j 0,020 0,032 0,030 0,025 0,023 0,022 0,027 0,023 0,030 rp2j 0,031 0,045 0,043 0,038 0,034 0,034 0,038 0,034 0,042 rp3j 0,041 0,058 0,054 0,067 0,063 0,059 0,051 0,062 0,055 rp4j 0,069 0,090 0,092 0,088 0,089 0,087 0,065 0,085 0,090 rp5j 0,104 0,197 0,187 0,155 0,147 0,156 0,086 0,150 0,154 rp6j 0,280 0,234 0,259 0,267 0,297 0,298 0,128 0,300 0,261 table 4: local priorities psj po1 po2 po3 po4 po5 po6 po7 po8 po9 c1 0,084 0,036 0,033 0,046 0,045 0,042 0,128 0,042 0,032 c2 0,174 0,091 0,083 0,108 0,104 0,102 0,193 0,103 0,084 c3 0,196 0,217 0,218 0,205 0,198 0,198 0,282 0,200 0,252 table 5: local priorities poj after, define the local priority pij with the use of (1), we calculate the global priority pi. in figure 2, in appendix, we report the ranking of the high school. finally, we obtained the classification of the high schools in terms of performance; the results are shown in table 6. north center south italy high 27 26 23 76 medium 57 52 60 169 low 2 3 1 6 table 6: schools performance sorted in geographical areas in figure 3, in appendix, it’s shown how the different typologies of high schools have been sorted into the different performance classes. 5 discussion as shown by the results obtained, it emerges that the schools that obtain the best performance throughout italy are the sls and the cls with the best positions for those in the south follow. a portion of cl is positioned in medium-sized performances mainly in northern and central italy. in the medium classes the ll, 289 g. fattoruso, p. mancini, g. marcarelli hsl, cths and tths converge in order. a few cl from the north, south and center are included in the performance of the lower class. sl have the best performance regardless of the geographical area. moving from north to south the first class is almost exclusively composed by sl, in the north, by a great number of sl and a few cl, in the center, and fifty-fifty by sl and cl, in the south. comparing the results with those obtained by mancini and marcarelli [2019] and mancini and marcarelli [2022], an inversion of ordering emerges in the first positions between sl and cl. the other positions are confirmed. it should be noted that in the works of mancini and marcarelli [2019] and mancini and marcarelli [2022] the initial data are represented by the average performance values by type of school for each criterion considered. in this paper, however, the performances are evaluated for individual schools. in this sense, the difference in the overall results may be due to the influence of anomalous values in the calculation of the average values by type of school. for a more detailed comparison, we also checked the ranking of schools for each criterion. figure 4, in appendix, shows in particular the ranking of the sl and cl for each criterion, geographical area and class considered. as can be seen in the figure 4, sls obtain higher performances for all the criteria except for the g3 criterion for the cls of northern italy. 6 conclusions this paper investigates the performance of italian high schools in order to derive a ranking considering the typology and the geographic area. using ahpsort ii, we obtain a classification of italian high schools into different categories. the results show that the ranking among the types of schools does not vary moving from north to south: scientific lyceums are all in the first class regardless the geographic area. however, the limit of this work is the lack of a model validation. the model may assist students in selecting the type of school to attend; the information makes it possible to make an appropriate choice according to their academic perspectives. our future works will address a comparative analysis to test the model proposed: electre tri [corrente et al., 2016] may provide a classification of high schools into different categories such as ‘over-performing schools’, ‘averageperforming schools’ and ‘weak-performing schools’; then we may compare results with those obtained by our model. furthermore, if the decision makers are not sure about the correct level of reference profiles, it could be interesting to perform a sensitivity analysis with several limiting profiles to test the robustness of the process. finally, when applied to small territorial districts, our model may be 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determination with technique methods or others preference by similarity to ideal solution (topsis). international journal of mechanical engineering and technology (ijmet), 9(13):650–657, 2018. 293 g. fattoruso, p. mancini, g. marcarelli t s area g1 g2 g3 g4 g5 g6 g7 g8 g9 sl a1 north 80,20 6 61,9 0,94 0,88 80,2 27,33 27 9,7 sl a2 north 78,61 3 67,9 0,90 0,80 74,6 26,62 21 18,7 sl a3 north 76,47 4 59,2 0,90 0,81 69,1 25,89 23 16,3 sl a4 north 74,87 4 59,2 0,92 0,74 48,2 24,91 24 27,2 sl a5 north 72,85 4 38,0 0,83 0,68 52,4 24,07 24 25,2 sl a6 north 77,94 7 65,6 0,94 0,89 87,2 28,62 26 16,3 sl a7 north 80,68 7 59,3 0,93 0,87 85,6 28,62 23 7,8 sl a8 north 77,29 6 62,8 0,95 0,89 82,0 27,57 26 11,7 sl a9 north 75,02 4 59,1 0,93 0,84 74,2 26,20 22 14,5 sl a10 north 75,70 4 40,7 0,88 0,76 67,1 25,02 21 17,3 sl a11 north 72,22 4 44,0 0,91 0,80 66,5 24,73 22 23,7 sl a12 north 70,66 4 46,9 0,78 0,61 46,3 25,53 21 27,7 sl a13 north 78,54 3 68,2 0,92 0,86 75,8 27,09 22 24,6 sl a14 north 76,10 5 67,6 0,87 0,79 67,6 26,55 21 5,8 sl a15 north 78,91 7 69,1 0,94 0,90 87,5 28,81 22 12,2 sl a16 north 76,66 4 52,1 0,92 0,84 75,8 26,71 24 11,7 sl a17 north 68,76 4 37,7 0,86 0,72 55,7 23,56 22 39,0 sl a18 north 79,52 5 74,6 0,93 0,90 87,2 28,89 24 22,3 sl a19 north 75,18 5 57,4 0,91 0,86 76,1 27,51 19 38,5 sl a20 north 75,54 4 46,3 0,92 0,84 62,6 25,04 23 11,8 sl a21 center 81,37 5 70,3 0,84 0,80 77,2 27,61 22 14,3 sl a22 center 78,38 6 62,9 0,93 0,86 72,1 26,81 23 19,8 sl a23 center 78,03 4 60,0 0,90 0,83 70,7 26,59 21 20,6 sl a24 center 80,23 6 76,6 0,93 0,87 76,7 27,92 25 6,2 sl a25 center 81,64 5 72,6 0,92 0,85 71,5 27,57 20 17,9 sl a26 center 81,01 6 56,3 0,90 0,84 82,7 28,47 25 7,1 sl a27 center 77,34 5 56,6 0,93 0,86 78,1 27,59 22 7,5 sl a28 center 77,44 3 65,4 0,90 0,81 76,6 27,51 24 10,6 sl a29 center 77,35 5 60,2 0,93 0,86 77,4 27,10 24 16,7 sl a30 center 77,78 2 62,4 0,91 0,82 71,3 26,56 23 15,6 sl a31 center 78,45 6 65,8 0,91 0,83 73,1 26,27 23 23,0 sl a32 center 76,15 6 66,1 0,90 0,81 73,5 26,21 22 12,9 sl a33 center 77,36 5 57,1 0,91 0,85 73,0 26,01 23 7,6 sl a34 center 77,11 2 49,9 0,90 0,76 71,1 26,15 24 20,5 sl a35 center 73,94 5 53,1 0,84 0,71 56,1 24,99 24 20,5 sl a36 center 77,02 3 59,9 0,89 0,76 64,3 25,18 22 9,7 sl a37 center 75,24 3 53,2 0,80 0,68 55,0 25,33 22 18,2 sl a38 center 75,8 4 65,1 0,88 0,74 59,9 25,23 23 18,1 sl a39 south 81,32 4 76,9 0,90 0,81 66,7 25,87 22 9,6 sl a40 south 82,01 6 74,6 0,93 0,85 68,4 26,09 22 4,9 appendix a 294 the ahpsort ii to evaluate the high-level instruction performances t s area g1 g2 g3 g4 g5 g6 g7 g8 g9 sl a41 south 77,97 5 66,1 0,94 0,84 74,9 28,12 18 16,7 sl a42 south 81,79 5 57,1 0,93 0,87 80,0 27,29 24 6,8 sl a43 south 79,05 2 62,4 0,94 0,87 76,3 26,67 23 16,5 sl a44 south 78,65 6 61,1 0,90 0,75 61,6 25,44 21 3,4 sl a45 south 81,41 5 55,7 0,78 0,68 65,8 25,42 18 5,9 sl a46 south 79,16 6 59,2 0,89 0,79 68,3 25,95 22 10,0 sl a47 south 79,86 2 52,6 0,87 0,77 67,9 25,94 21 5,2 sl a48 south 74,06 3 20,6 0,80 0,60 43,3 22,65 14 21,2 sl a49 south 81,07 4 84,6 0,93 0,85 74,9 26,15 27 2,7 sl a50 south 79,60 2 61,7 0,96 0,87 68,9 25,57 23 10,0 sl a51 south 82,32 5 77,9 0,93 0,85 74,1 25,89 21 4,3 sl a52 south 78,27 5 62,3 0,91 0,85 76,5 26,57 24 4,3 sl a53 south 77,21 4 50,5 0,83 0,79 69,4 25,18 21 6,4 sl a54 south 72,66 1 39,8 0,80 0,66 53,3 23,32 21 0,0 sl a55 south 78,94 4 51,3 0,91 0,84 74,5 26,88 20 9,9 sl a56 south 74,99 4 50,3 0,84 0,72 58,5 25,38 21 11,9 cl a57 north 78,96 6 66,9 0,95 0,86 75,0 28,31 22 16,9 cl a58 north 79,47 5 67,3 0,86 0,77 67,7 26,48 24 30,8 cl a59 north 79,26 6 61,9 0,93 0,88 80,2 27,89 24 14,0 cl a60 north 79,90 6 68,2 0,94 0,88 78,5 28,41 22 10,1 cl a61 north 74,10 5 55,5 0,91 0,81 64,8 26,41 23 34,4 cl a62 north 65,79 2 73,3 0,86 0,73 55,1 26,08 19 25,0 cl a63 north 75,87 3 74,6 0,88 0,80 66,8 26,88 22 24,6 cl a64 north 80,86 4 65,3 0,87 0,79 68,9 27,76 21 14,7 cl a65 north 80,55 5 72,2 0,95 0,88 72,7 27,73 21 29,6 cl a66 north 78,82 5 68,8 0,92 0,84 64,6 26,63 22 21,2 cl a67 north 79,53 6 72,7 0,93 0,87 71,3 28,29 23 7,7 cl a68 center 81,30 6 60,1 0,92 0,84 71,4 28,27 21 24,6 cl a69 center 79,34 5 69,4 0,91 0,82 60,3 26,98 22 14,3 cl a70 center 83,12 7 75,9 0,92 0,86 67,7 26,80 24 7,0 cl a71 center 84,57 6 88,5 0,95 0,86 65,8 27,62 23 31,9 cl a72 center 81,25 6 59,4 0,92 0,85 76,0 28,86 24 13,3 cl a73 center 78,85 6 62,6 0,90 0,85 73,1 28,23 24 7,0 cl a74 center 82,45 6 73,9 0,93 0,86 78,5 27,54 23 8,6 cl a75 center 82,77 3 62,6 0,95 0,82 69,3 27,28 23 8,6 cl a76 center 82,4 4 64,4 0,87 0,77 68,5 27,19 23 13,1 cl a77 center 81,29 2 72,4 0,93 0,83 69,5 27,07 23 15,0 cl a78 center 81,41 2 78,9 0,79 0,74 56,7 25,80 23 15,5 cl a79 center 80,06 2 66,5 0,90 0,77 55,2 25,09 23 20,8 cl a80 center 75,70 3 67,5 0,86 0,71 50,7 24,62 23 15,8 295 g. fattoruso, p. mancini, g. marcarelli t s area g1 g2 g3 g4 g5 g6 g7 g8 g9 cl a81 center 81,39 4 83,9 0,95 0,90 66,2 27,06 23 5,4 cl a82 south 83,77 5 90,2 0,95 0,88 68,0 26,30 22 8,5 cl a83 south 83,78 5 72,2 0,94 0,87 80,0 27,77 25 9,9 cl a84 south 81,21 6 78,3 0,95 0,89 77,3 27,41 22 6,8 cl a85 south 79,67 6 57,1 0,88 0,74 68,7 26,36 21 3,4 cl a86 south 74,90 3 73,0 0,83 0,68 55,9 25,83 19 12,4 cl a87 south 79,90 3 80,5 0,76 0,62 42,3 25,69 23 26,4 cl a88 south 82,55 6 83,4 0,91 0,83 67,4 26,12 23 0,0 cl a89 south 85,2 4 80,0 0,96 0,82 60,5 26,23 14 3,7 cl a90 south 84,13 5 86,7 0,93 0,84 63,0 24,70 20 3,2 cl a91 south 80,54 5 71,3 0,88 0,80 69,4 26,63 22 3,4 cl a92 south 79,32 5 63,30 0,75 0,62 53,0 24,78 17 23,0 cl a93 south 81,20 3 67,5 0,88 0,79 64,4 26,71 19 2,7 cl a94 south 77,03 3 45,6 0,84 0,73 46,8 23,38 19 14,4 ll a95 north 80,33 5 50,8 0,86 0,77 66,3 26,31 23 0,0 ll a96 north 77,98 4 55,4 0,78 0,72 58,6 25,08 19 11,1 ll a97 north 73,42 6 63,7 0,7 0,56 41,0 24,09 22 37,2 ll a98 north 75,23 4 59,5 0,83 0,76 69,9 26,06 23 10,8 ll a99 north 76,91 7 75,4 0,78 0,74 64,0 25,46 23 11,6 ll a100 north 73,4 4 40,6 0,75 0,65 40,6 24,12 21 27,7 ll a101 north 78,8 3 53,5 0,78 0,63 48,2 24,04 21 33,8 ll a102 north 78,78 4 67,6 0,76 0,70 65,0 24,15 24 20,8 ll a103 north 79,1 4 64,8 0,8 0,71 61,4 26,39 21 14,7 ll a104 north 80,05 4 70,3 0,73 0,61 51,3 24,05 24 7,4 ll a105 north 82,81 4 77,9 0,78 0,74 73,6 26,19 24 21,1 ll a106 north 78,27 3 54,4 0,75 0,64 50,8 24,96 23 17,2 ll a107 center 81,49 5 66,4 0,79 0,71 64,9 26,87 22 14,3 ll a108 center 78,25 5 44,3 0,74 0,66 54,9 25,73 22 12,5 ll a109 center 75,69 3 35,3 0,63 0,50 43,5 25,54 20 25,8 ll a110 center 80,3 4 63,5 0,72 0,64 59,0 25,22 23 4,5 ll a111 center 78,86 5 60,0 0,74 0,65 55,2 24,93 20 17,9 ll a112 center 80,25 3 56,0 0,79 0,71 62,2 26,00 23 15,8 ll a113 center 76,92 6 52,5 0,72 0,66 64,6 25,49 23 14,66 ll a114 center 76,42 4 56,1 0,73 0,65 67,9 24,78 21 13,01 ll a115 center 82,52 4 66,7 0,8 0,62 53,3 25,33 24 20,95 ll a116 center 76,91 4 70,8 0,76 0,63 52,7 24,63 23 16,13 ll a117 center 78,09 6 63,0 0,67 0,56 54,2 25,27 24 26,35 ll a118 center 75,51 4 47,3 0,71 0,60 44,9 23,91 22 28,24 ll a119 center 73,76 3 55,2 0,7 0,55 37,1 24,55 21 23,36 ll a120 center 74,28 3 41,8 0,64 0,51 41,4 23,01 20 26,32 296 the ahpsort ii to evaluate the high-level instruction performances t s area g1 g2 g3 g4 g5 g6 g7 g8 g9 ll a121 center 79,18 4 69,4 0,72 0,64 53,9 24,88 23 5,41 ll a122 south 80,26 5 67,8 0,77 0,75 74,8 25,95 22 9,17 ll a123 south 79,24 4 74,5 0,86 0,78 64,3 26,10 21 13,73 ll a124 south 80,25 3 39,6 0,72 0,67 67,6 25,53 22 7,95 ll a125 south 78,5 4 54,6 0,8 0,66 51,3 25,82 28 3,97 ll a126 south 79,12 3 60,0 0,7 0,58 56,8 24,71 22 13,43 ll a127 south 72,71 4 30,0 0,52 0,41 37,8 24,12 20 2,70 ll a128 south 81,31 5 40,0 0,56 0,44 40,5 23,43 18 5,94 ll a129 south 76,84 4 54,7 0,49 0,35 43,0 25,21 16 4,23 ll a130 south 78,93 5 62,9 0,83 0,72 53,4 24,27 20 6,67 ll a131 south 85,63 6 71,3 0,76 0,72 61,1 24,97 20 7,29 ll a132 south 78,32 3 67,3 0,52 0,49 69,8 24,12 21 19,72 ll a133 south 74,38 4 41,3 0,56 0,47 40,9 23,39 18 29,89 hsl a134 north 78,25 5 63,9 0,76 0,65 55,8 24,11 19 18,62 hsl a135 north 76,15 4 65,0 0,76 0,59 39,8 23,62 23 27,2 hsl a136 north 75,85 4 46,7 0,75 0,65 62,2 25,74 23 24,81 hsl a137 north 75,31 4 68,0 0,84 0,64 37,4 23,04 21 28,57 hsl a138 north 72,69 4 59,5 0,61 0,52 53,3 24,46 24 20,8 hsl a139 north 73,27 3 46,2 0,71 0,56 46,7 24,60 19 26,94 hsl a140 north 76,43 4 51,7 0,73 0,69 59,6 24,48 23 11,8 hsl a141 north 73,01 2 47,0 0,76 0,65 46,5 24,43 22 36,80 hsl a142 center 74,40 5 58,8 0,77 0,71 54,7 24,65 22 14,29 hsl a143 center 77,24 4 59,5 0,77 0,67 50,9 24,37 23 4,5 hsl a144 center 79,55 6 65,7 0,77 0,65 45,1 24,67 23 31,9 hsl a145 center 75,58 4 64,2 0,8 0,68 52,2 24,70 23 16,13 hsl a146 center 78,21 4 42,8 0,67 0,58 50,3 23,55 23 13,1 hsl a147 center 73,24 4 56,6 0,69 0,59 52,1 23,40 22 28,24 hsl a148 center 75,46 3 61,8 0,55 0,43 45,2 23,12 23 21,43 hsl a149 south 76,32 6 81,6 0,64 0,57 43,0 23,88 19 16,43 hsl a150 south 75,13 4 70,3 0,83 0,66 34,3 24,25 23 5,41 hsl a151 south 73,61 3 56,3 0,55 0,47 54,4 25,53 22 13,43 hsl a152 south 77,02 4 61,8 0,80 0,65 44,9 23,58 21 13,73 hsl a153 south 78,46 5 55,1 0,49 0,35 51,6 22,87 18 5,94 hsl a154 south 73,78 3 34,3 0,44 0,33 33,7 23,59 18 13,6 hsl a155 south 77,18 4 66,9 0,52 0,44 53,9 24,67 20 4,82 hsl a156 south 78,76 3 65,9 0,80 0,70 56,1 24,42 22 0,00 hsl a157 south 80,11 6 70,0 0,69 0,61 54,7 23,84 20 7,29 hsl a158 south 74,91 4 52,9 0,67 0,54 47,7 24,02 20 16,45 hsl a159 south 78,07 4 32,4 0,57 0,43 22,1 21,22 18 29,89 cths a160 north 72,22 5 42,4 0,51 0,41 43,6 24,14 19 18,62 297 g. fattoruso, p. mancini, g. marcarelli t s area g1 g2 g3 g4 g5 g6 g7 g8 g9 cths a161 north 73,57 5 38,9 0,56 0,44 42,1 24,18 18 27,41 cths a162 north 73,75 3 32,1 0,50 0,42 44,1 22,47 19 23,97 cths a163 north 75,78 6 54,4 0,48 0,37 32,0 23,52 21 13,50 cths a164 north 74,27 3 27,2 0,42 0,36 61,0 24,47 21 22,92 cths a165 north 74,13 4 34,7 0,47 0,42 57,0 24,33 21 28,57 cths a166 north 74,98 4 37,0 0,45 0,34 52,0 23,41 21 27,71 cths a167 north 73,82 4 53,5 0,51 0,37 41,4 24,28 19 26,39 cths a168 north 71,55 2 35,0 0,52 0,40 35,2 22,30 21 22,22 cths a169 north 70,75 4 36,6 0,38 0,27 23,7 22,98 25 20,69 cths a170 north 76,60 2 60,1 0,39 0,33 47,9 24,13 20 12,96 cths a171 north 74,22 3 34,5 0,47 0,38 47,5 24,33 18 0,00 cths a172 north 74,99 4 58,3 0,61 0,50 54,5 23,94 24 7,41 cths a173 north 72,10 3 35,1 0,47 0,27 24,1 23,29 17 50,29 cths a174 north 74,51 5 55,1 0,55 0,45 58,5 24,02 21 38,5 cths a175 north 74,55 4 49,6 0,64 0,54 46,8 23,47 22 26,00 cths a176 center 75,49 3 42,9 0,63 0,53 51,3 25,73 22 37,14 cths a177 center 74,04 4 44,8 0,56 0,42 34,5 23,27 21 25,81 cths a178 center 74,71 5 49,6 0,54 0,49 60,7 24,57 20 24,24 cths a179 center 78,75 5 52,6 0,64 0,61 56,4 25,03 20 17,86 cths a180 center 73,78 3 47,0 0,36 0,30 57,3 24,67 21 24,66 cths a181 center 75,37 4 32,6 0,45 0,36 52,5 24,72 21 20,39 cths a182 center 75,27 2 49,4 0,42 0,28 47,2 24,57 19 9,93 cths a183 center 72,25 4 57,4 0,50 0,30 34,3 24,37 15 25,30 cths a184 center 73,75 2 57,7 0,43 0,38 53,0 21,91 22 13,16 cths a185 center 75,93 5 37,9 0,35 0,22 32,0 22,22 20 27,97 cths a186 center 71,22 3 58,3 0,36 0,23 30,2 21,72 21 34,52 cths a187 center 71,99 5 53,8 0,53 0,40 32,6 20,95 26 20,55 cths a188 center 80,18 4 75,8 0,67 0,50 35,1 24,75 19 17,47 cths a189 south 73,81 3 61,5 0,47 0,42 53,8 24,17 20 11,82 cths a190 south 76,48 2 43,2 0,43 0,34 50,7 24,35 20 15,38 cths a191 south 74,61 4 27,2 0,43 0,33 44,5 23,95 20 16,67 cths a192 south 73,44 3 36,1 0,40 0,31 42,0 22,98 20 21,58 cths a193 south 75,22 3 50,4 0,38 0,29 36,8 23,24 22 10,58 cths a194 south 76,45 4 38,4 0,36 0,26 25,7 23,32 17 9,70 cths a195 south 69,12 2 33,8 0,38 0,26 31,0 20,00 15 0,00 cths a196 south 73,76 5 59,5 0,48 0,39 52,8 24,27 22 6,40 cths a197 south 76,97 4 49,9 0,53 0,43 52,7 23,90 20 9,40 cths a198 south 73,45 5 49,9 0,58 0,48 49,6 22,36 16 6,67 cths a199 south 76,37 5 62,0 0,49 0,42 47,4 22,60 18 17,20 cths a200 south 74,49 5 36,0 0,56 0,50 60,5 22,89 16 0,00 298 the ahpsort ii to evaluate the high-level instruction performances t s area g1 g2 g3 g4 g5 g6 g7 g8 g9 cths a201 south 73,75 4 21,8 0,35 0,29 44,4 22,77 17 5,00 cths a202 south 74,72 4 31,3 0,49 0,37 36,1 23,14 17 4,39 tths a203 south 71,06 3 31,6 0,52 0,37 50,6 25,04 19 23,97 tths a204 north 71,6 4 32,0 0,45 0,36 55,0 24,15 23 27,23 tths a205 north 73,32 4 39,9 0,70 0,58 49,4 24,52 20 11,11 tths a206 north 73,78 4 37,1 0,36 0,27 35,7 22,47 22 29,81 tths a207 north 74,79 6 39,4 0,37 0,26 43,7 22,99 21 22,40 tths a208 north 74,99 4 36,5 0,77 0,67 68,8 25,97 21 27,69 tths a209 north 69,93 5 38,8 0,37 0,27 46,2 26,39 21 3,64 tths a210 north 67,97 4 30,3 0,41 0,35 58,4 24,64 19 28,28 tths a211 north 72,2 4 43,9 0,50 0,37 41,9 24,59 20 35,00 tths a212 north 70,51 5 26,8 0,48 0,35 51,3 23,42 21 17,65 tths a213 north 69,62 4 34,7 0,35 0,24 39,5 23,62 20 24,81 tths a214 north 70,55 4 52,4 0,36 0,20 30,4 25,26 19 26,39 tths a215 north 73,92 4 28,3 0,40 0,28 39,6 23,93 21 15,61 tths a216 north 76,79 6 48,4 0,65 0,57 64,6 26,16 23 23,26 tths a217 north 74,27 3 48,6 0,33 0,25 46,7 24,91 18 36,46 tths a218 north 79,65 6 73,2 0,37 0,24 42,4 24,86 10 40,38 tths a219 north 72,53 4 32,1 0,61 0,49 53,0 25,41 23 32,82 tths a220 north 73,00 3 55,6 0,45 0,32 39,3 25,17 18 23,40 tths a221 north 73,55 4 43,0 0,51 0,43 56,6 26,87 22 26,00 tths a222 north 74,49 4 39,5 0,44 0,37 60,1 25,41 22 29,72 tths a223 center 71,38 4 30,2 0,43 0,37 62,1 25,65 21 21,71 tths a224 center 73,81 3 33,8 0,49 0,38 47,4 24,36 21 25,84 tths a225 center 73,48 6 39,4 0,44 0,30 39,9 23,52 20 10,29 tths a226 center 77,89 3 41,9 0,68 0,59 53,5 24,47 19 20,00 tths a227 center 75,76 4 54,8 0,58 0,47 52,7 25,36 18 28,42 tths a228 center 71,09 3 46,0 0,33 0,23 48,9 26,39 19 8,85 tths a229 center 72,35 4 45,8 0,50 0,38 52,4 25,59 16 19,53 tths a230 center 73,03 3 28,3 0,45 0,33 54,7 24,67 22 24,66 tths a231 center 70,88 3 47,9 0,53 0,36 33,7 24,48 20 36,36 tths a232 center 72,64 3 30,7 0,41 0,29 38,4 23,84 20 20,35 tths a233 center 71,8 5 46,1 0,50 0,32 40,4 23,50 20 11,67 tths a234 center 68,81 5 31,8 0,33 0,21 36,9 23,11 21 18,60 tths a235 center 73,21 5 50,5 0,42 0,24 26,1 22,66 19 25,81 tths a236 center 70,51 3 23,8 0,37 0,31 49,5 23,32 18 8,48 tths a237 south 71,06 3 70,8 0,35 0,32 65,0 25,87 20 11,82 tths a238 south 79,55 3 46,7 0,51 0,44 70,8 24,43 20 21,58 tths a239 south 77,18 3 42,3 0,40 0,24 39,9 25,48 17 21,70 tths a240 south 77,84 5 48,4 0,36 0,23 43,3 24,63 19 14,86 299 g. fattoruso, p. mancini, g. marcarelli t s area g1 g2 g3 g4 g5 g6 g7 g8 g9 tths a241 south 76,68 4 26,0 0,34 0,26 48,7 24,03 22 16,17 tths a242 south 74,62 3 31,7 0,63 0,47 48,5 23,42 16 6,06 tths a243 south 74,07 4 35,9 0,39 0,26 37,3 23,39 20 6,09 tths a244 south 80,27 3 24,5 0,47 0,27 41,5 21,07 18 21,24 tths a245 south 74,97 3 35,2 0,38 0,29 50,5 24,48 19 9,65 tths a246 south 72,50 4 40,8 0,48 0,39 54,3 24,08 17 0,73 tths a247 south 76,93 4 48,2 0,53 0,40 48,2 23,44 18 8,33 tths a248 south 71,41 3 25,8 0,34 0,29 49,8 26,01 19 9,33 tths a249 south 72,75 4 17,4 0,50 0,42 43,0 22,34 17 5,00 tths a250 south 75,08 5 24,7 0,36 0,30 58,6 23,12 16 27,91 tths a251 south 76,04 4 31,7 0,52 0,26 21,3 22,60 17 29,89 table 7: evaluation table of gj(ai). legend: t=type; s=school; area=geo-position. 300 the ahpsort ii to evaluate the high-level instruction performances figure 2: ranking of school defined by pi 301 g. fattoruso, p. mancini, g. marcarelli figure 3: performance of school sorted for typology and geographical areas 302 the ahpsort ii to evaluate the high-level instruction performances figure 4: performance of sls and cls sorted for criteria and geographical areas 303 repeated burst error detecting linear codes b. k. dass1 and rashmi verma2 ∗ department of mathematics university of delhi delhi 110 007, india 1 e-mail: dassbk@rediffmail.com 2 e-mail: riva 7@rediffmail.com abstract. this paper presents lower bounds on the number of paritycheck digits required for a linear code that is capable of detecting errors which are ‘m-repeated burst errors’. further, codes capable of detecting and simultaneously correcting such errors have also been studied. ∗corresponding author. ratio mathematica, 19, pp. 25-30 25 1 introduction many kinds of errors in coding theory have been dealt with for which codes have been constructed to combat such errors. apart from random errors, one of the widely studied error is a burst error. it has been observed that in several communication systems, errors occur predominantly in bursts. a burst of length b may be defined as follows: definition 1. a burst of length b is a vector whose only non-zero components are among some b consecutive components, the first and the last of which is non-zero. stone (1961), and bridwell and wolf (1970) considered multiple burst errors. chien and tang (1965) observed that in many channels errors occur in the form of a burst but errors do not occur in the end digits of the burst, e.g., channels due to alexander, gryb and nast (1960) fall in this category. the nature of burst errors differ from channel to channel depending upon the behaviour of channels or the kind of errors which occur during the process of transmission. codes that detect and correct 2-repeated open-loop bursts have been studied by berardi, dass and verma (2007). a 2-repeated burst (open-loop) of length b has been defined as follows: definition 2. a 2-repeated burst of length b is a vector of length n whose only non-zero components are confined to two distinct sets of b consecutive components, the first and the last component of each set being non-zero. in very busy communication channels, errors repeat themselves more frequently. in view of this, it is desirable to consider more than two repeated bursts. ratio mathematica, 19, pp. 25-30 26 in this paper, we define ‘m-repeated burst of length b’ as follows: definition 3. an m-repeated burst of length b is a vector of length n whose only non-zero components are confined to m distinct sets of b consecutive components, the first and the last component of each set being non-zero. for example, (001020024100314030100) is a 4-repeated burst of length 3 over gf(5). bounds for the detection and correction of such bursts have been derived in this paper. in what follows, a linear code will be considered as a subspace of the space of all n-tuples over gf(q). the distance between two vectors shall be considered in the hamming sense. 2 m-repeated burst error detecting code in this section, we consider linear codes that are capable to detect m-repeated burst of length b or less. clearly, the patterns to be detected should not be code words. firstly, we obtain a lower bound over the number of parity-check digits for such a code. theorem 1. any (n, k) linear code over gf(q) that detects any mrepeated burst of length b or less must have atleast mb parity-check digits. proof. the result will be proved by showing that no detectable error vector can be a code word. let v be an (n, k) linear code over gf(q). consider a set x that has all those vectors which have their non-zero components confined to some m fixed distinct b consecutive components. ratio mathematica, 19, pp. 25-30 27 we claim that no two vectors of x can belong to the same coset of the standard array, else a code word would be expressible as a sum or difference of two error vectors. assume, on the contrary, that there is a pair say x1, x2 in x belonging to the same coset of the standard array. then their difference viz., x1 − x2 must be a code word. but x1 − x2 is a vector all of whose non-zero components are confined to the same m fixed b consecutive components and so is a member of x , i.e., x1 − x2 is m-repeated burst of length b or less, which is a contradiction. thus all the vectors in x must belong to distinct cosets of the standard array. the number of such vectors overgf(q) is clearly qmb . also, total number of cosets in an (n, k) linear code equals qn−k , so we must have qn−k > qmb , i.e., n − k > mb, which proves the result. remarks. for m = 1, this result reduces to the case of single non-repeated bursts (refer theorem 4.13, peterson and weldon (1972)). for m = 2, this result coincides with theorem 1 due to berardi, dass and verma (2007) when bursts considered are 2-repeated bursts of length b or less. 3 simultaneous detection and correction of m-repeated burst errors in the following, we consider linear codes which are capable to detect and correct simultaneously m-repeated bursts and obtain a necessary condition over the number of parity-checks required for such a code. ratio mathematica, 19, pp. 25-30 28 theorem 2. any (n, k) linear code over gf(q) that corrects m-repeated bursts of length b or less must have at least 2mb parity-check digits. further, if the code corrects m-repeated bursts of length b or less and simultaneously detects m-repeated bursts of length d or less (d > b), then the code must have at least m(b + d) parity-check digits. proof. consider a burst of length 2mb. such a vector is expressible as a sum or difference of two vectors each of which is m-repeated burst of length b or less. these component vectors must belong to different cosets of the standard array because both such errors are correctable errors. accordingly, such a vector viz., a burst of length 2mb or less cannot be a code word. in view of theorem 1, such a code must have atleast 2mb parity-check digits. further, consider a burst of length m(b + d). such a vector cannot be a code word because it is always expressible as a sum or difference of two vectors, one of which is m-repeated burst of length b or less and the other is m-repeated burst of length d or less. as earlier, any pair of such component vectors cannot belong to the same coset of the standard array and so a burst of length m(b + d) cannot be a code word. therefore, the code must have atleast m(b + d) parity-check digits. remarks. for m = 1, this result coincides with reiger’s bound (reiger (1960); also refer theorem 4.15, peterson and weldon (1972)). for m = 2, this result reduces to a result due to berardi, dass and verma (theorem 3, (2007)) when bursts considered are 2-repeated bursts of length b or less. ratio mathematica, 19, pp. 25-30 29 references [1] alexander a.a., gryb r.m. and nast d. w. (1960), capabilities of the telephone network for data transmission, bell system tech. j., vol. 39, no. 3, pp. 431–476. [2] berardi l., dass b.k. and verma rashmi (2009), on 2-repeated burst error detecting codes, journal of statistical theory and practice, vol. 3, no. 2, pp. 381–391. [3] bridwell j.d. and wolf j.k. (1970), burst distance and multiple-burst correction, bell system tech. j., vol. 99, pp. 889–909. [4] chien r.t. and tang d.t. (1965), on definitions of a burst, ibm journal of research and development, vol. 9, no. 4 (july), pp. 292– 293. [5] peterson w.w. and weldon e.j., jr. (1972), error-correcting codes, 2nd edition, the mit press, mass. [6] reiger s.h. (1960), codes for the correction of “clustered errors”, ire trans. inform. theory, it-6 march, pp. 16–21. [7] stone j.j. (1961), multiple burst error correction, information and control, vol. 4, pp. 324–331. ratio mathematica, 19, pp. 25-30 30 microsoft word decomposition method comparison burgers_normeratio172.doc ratio mathematica 18 (2008), 51 61 51 decomposition method in comparison with numerical solutions of burgers equation christos mamaloukas* and stefanos spartalis** abstract – this paper presents a solution of the one-dimension burgers equation using decomposition method and compares this solution to the analytic solution [cole] and solutions obtained with other numerical methods. even though decomposition method is a non-numerical method, it can be adapted for solving nonlinear differential equations. the advantage of this methodology is that it leads to an analytical continuous approximated solution that is very rapidly convergent [2,7,8]. this method does not take any help of linearization or any other simplifications for handling the non-linear terms. since the decomposition parameter, in general, is not a perturbation parameter, it follows that the non-linearities in the operator equation can be handled easily, and accurate solution may be obtained for any physical problem. 1. introduction many problems in fluid mechanics and in physics are governed generally by the navier-stokes equations. these equations can show the behaviour of a certain attribute (e.g. momentum, heat) in space and time. the one-dimension non-linear differential equation which is used as a model for these problems is burgers equation. this equation is applied to laminar and turbulence flows as * athens university of economics and business, dept. of informatics, 76 patision str, 10434 athens, greece, email: mamkris@aueb.gr ** democritus university of thrace, dept. of production engineering and management, school of engineering, university library building, kimeria 67100 xanthi, greece, email: sspart@pme.duth.gr 52 num.18 -2008 ratio mathematica c. mamaloukas, s. spartalis well. the burgers equation which is the one-dimension nonlinear diffusion equation is similar to the one dimension navier-stokes equation without the stress term. many researchers tried to find analytic and numerical solutions of this equation using the appropriate initial and boundary conditions. characteristically in benton and platzman [10] are mentioned almost 35 distinct solutions of burger equation but only the half of them are having physical interest. agas [9] tried to get approximate solution of burger equation using a new numerical solution which is called “group of explicit” method. he also tried the method of finite differences and the method of lines in finite elements. the problem he faced was that these methods could not give solutions for big values of the reynolds number. he also found some problems in convergence. in this paper, a solution obtained by the adomian's decomposition method (adm), which is described briefly in this paper and was used by mamaloukas [12, 13] for the numerical solution of the one-dimensional kortweg-de vries equation and the pulsatile flow of an incompressible viscous fluid through a circular rigid tube provided with constriction, is compared numerically and graphically to the analytic and to some others numerical methods. as it is shown in the diagrams at the end of this paper this method gives a computable and accurate solution of the problem using only a small number of terms. 2. formulation of the problem consider the burgers equation with the following form 2 2 u u u u t x x          (1) with boundary conditions: (0, ) (1, ) 0 0u t u t for t   (2) and initial condition: ( , 0) 4 (1 ) sinu x x x or x  (3) 3. brief description of adm let 2 2andt xxl lt x       . then the equation (1) takes the form: t xxl u nu l u  (4) 53 num.18 – 2008 ratio mathematica c. mamaloukas, et al where the first term is the linear, the third is the highest order term and the second is the non-linear term given by u nu u x    (5) now, solving (4) for and t xxl u l u correspondingly we have t xxl u l u nu  (6)  1xx tl u l u nu   (7) by defining the one and twofold right-inverse operators 1 1andt xxl l   , given by the form    1 1 and t xxl dt l dx dx      , we can formally obtain from (6) and (7)  1 1t t t xxl l u l l u nu   (8)  1 1 1xx xx xx tl l u l l u nu    (9) from relations (8) and (9) we obtain    1 1 10 1 2 xx t t xx u u l l u nu l l u nu          (10) where the term 0u is to be determined from the initial conditions, so, is 0 4 (1 ) sinu x x or x  (11) 4. solution of burger's equation with adm now, we introduce a formal counting parameter λ to write equation (10) in the following form    1 1 10 1 2 xx t t xx u u l l u nu l l u nu           (12) the equation (12) is called parameterized equation and the parameter λ inserted here is not a perturbation parameter; it is used only for grouping the terms. the u and the nonlinear term nu are decomposed into the following parameterised forms 54 num.18 -2008 ratio mathematica c. mamaloukas, s. spartalis 0 n n n u u     (13) 0 n n n n u a     (14) where na are the adomian’s special polynomials [1,2] for the specific nonlinearity to be determined by expanding nu in the ascending power of λ and equating the terms of like powers of λ from both sides of (12). these special polynomials depend only on the 0u to nu components. substituting the expressions (13) and (14) into (12) and then equating the like power terms from both sides of the resulting expression we have             1 1 1 1 0 0 0 0 1 1 1 2 1 1 1 1 1 1 1 1 1 2 1 2 ............................................................................... 1 , 0, 2 xx t t xx xx t t xx n xx t n n t xx n n u l l u a l l u a u l l u a l l u a u l l u a l l u a n                                    1, 2,..., n (15) all components are determinable since 0a depends only on 0u , 1a depends only on 0, 1u u ,…, na depends only on 0 1, , ..., nu u u . so, in order to determine adomian’s special polynomials, from (5) and (14) we write 0 n n n u a u x       (16) substituting (13) into (16) and then comparing like-power terms of λ on both sides of resulting expression we obtain the following polynomials. 0 0 0 01 1 0 1 02 1 2 0 1 2 .................................................. .................................................. u a u x uu a u u x x uu u a u u u x x x                 (17) 55 num.18 – 2008 ratio mathematica c. mamaloukas, et al using the initial condition (11) 0a can be calculated from expression (17). substituting the result in the expression of 1u (15) and then performing all necessary calculations and integrations with respect to t and x respectively, we have 1u which is: 3 5 2 3 4 1 4 4 4 8 24 16 2 3 5 v x v x u v t t x t x t x v x        (18) and 2u which is 3 2 2 2 2 2 2 2 2 2 2 2 5 3 2 3 2 3 4 4 2 4 2 5 2 6 2 7 2 8 5 6 2 16 4 16 32 16 96 3 32 148 2 16 160 80 3 3 5 8 248 10 248 4 15 5 9 15 5 5 v x u v t v t x t x v t x v x v t x t x v x v t x v t x t x v x v t x t x v x v x v x v x v t x v t x                        (19) if we suggest as a solution of u an approximation of only two or three terms then from (11), (18) and (19) we have the solution of (1): 0 1u u u  or 0 1 2u u u u   as an example, if we give the values 0.25, 0.05, 0.001x t v   we get 0.712314u  with two terms and 0.712791u  with three terms. 5. tables of results and diagrams for the solution of this equation the initial conditions ( , 0) 4 (1 )u x x x  and ( , 0) sinu x x were used without restricting generality. the boundary conditions were (0, ) (1, ) 0 0u t u t for t   . the compared methods are analytic solution, the implicit method, the explicit method, the methods of lines with gauss-legendre and hermite, the group of explicit method and the decomposition method. for comparison reasons with the results of other published papers, 0.25x  and time amplitude 0.01 0.25t  were used. 56 num.18 -2008 ratio mathematica c. mamaloukas, s. spartalis in the above diagrams numerical results of burger equation are registered for different values of v. for comparison reasons the viscosity values 1, 0.1, 0.01, 0.001v v v v    were used. we give below the tables of results and some diagrams only for the first initial condition ( ) 4 (1 )u x x x  for values of x = 0.25, 0.5, 0.75 for t = 0.01, 0.05 (0.05) 0.25 and for v = 1, 0.1, 0.01, 0.001. as for the second initial condition we get similar results. for comparison reasons we also use as a solution of u, two and three terms of decomposition method. 57 num.18 – 2008 ratio mathematica c. mamaloukas, et al table 1: comparison results for burger equation for initial condition 4 (1 )x x with 1, 0.1, 0.01, 0.001, 0.25, 0.05v x t     x=.25 v=0.1 4x(1-x) x=.25 v=1 t analytic implicit explicit l-gauss l-hermite group decomp t analytic implicit explicit l-gauss l-hermite group decomp 0,01 0,7422 0,7276 0,7236 0,7274 0,7273 0,7272 0,7399 0,01 0,6724 0,6592 0,6489 0,6584 0,6583 0,6574 0,7163 0,05 0,6621 0,6745 0,67 0,6453 0,6452 0,6471 0,6939 0,05 0,4356 0,4224 0,4213 0,3503 0,3254 0,4206 0,5263 0,1 0,584 0,5675 0,5548 0,5608 0,5592 0,5671 0,6364 0,1 0,2751 0,2619 0,2603 0,4527 -0,053 0,2601 0,2888 0,15 0,5189 0,5043 0,5042 0,4931 0,4853 0,5039 0,5789 0,15 0,1794 0,1662 0,1642 -0,934 -0,408 0,1644 0,0513 0,2 0,4681 0,4535 0,4536 0,4362 0,4213 0,4531 0,5214 0,2 0,1191 0,1059 0,1052 -0,7593 0,1041 -0,1862 0,25 0,4265 0,4119 0,4118 0,4263 0,3652 0,4115 0,4639 0,25 0,0807 0,0675 0,0639 -1,1234 0,0657 -0,4237 x=.50/t analytic implicit explicit l-gauss l-hermite group decomp x=.50/t analytic implicit explicit l-gauss l-hermite group decomp 0,01 0,9917 0,9911 0,9901 0,9916 0,9923 0,9914 1,0027 0,01 0,9184 0,9194 0,9188 0,9197 0,9204 0,918 1,0267 0,05 0,9533 0,9527 0,9423 0,9516 0,9524 0,953 0,9867 0,05 0,639 0,64 0,6438 0,5978 0,5983 0,6386 0,8667 0,1 0,8993 0,8987 0,8815 0,893 0,8933 0,899 0,9667 0,1 0,4019 0,4029 0,4023 0,1275 0,2 0,4015 0,6667 0,15 0,8434 0,8428 0,8326 0,8317 0,8313 0,8431 0,9467 0,15 0,2524 0,2534 0,2514 -0,1408 -0,2023 0,252 0,4667 0,2 0,7889 0,7883 0,7935 0,7352 0,7714 0,7886 0,9267 0,2 0,1585 0,1595 0,1583 -0,6192 0,1581 0,2667 0,25 0,7375 0,7369 0,7328 0,5198 0,7143 0,7372 0,9067 0,25 0,0914 0,1065 0,0916 -1,0642 0,0991 0,0667 x=.75/t analytic implicit explicit l-gauss l-hermite group decomp x=.75/t analytic implicit explicit l-gauss l-hermite group decomp 0,01 0,7417 0,7571 0,7517 0,7567 0,7572 0,757 0,7654 0,01 0,7677 0,6928 0,6913 0,6818 0,6824 0,6927 0,837 0,05 0,7663 0,7818 0,7823 0,778 0,7793 0,7816 0,7795 0,05 0,5065 0,4917 0,4924 0,3355 0,3772 0,4915 0,707 0,1 0,7882 0,8038 0,7906 0,7892 0,7934 0,8035 0,7969 0,1 0,3239 0,309 0,3106 -0,8926 -0,046 0,3089 0,5445 0,15 0,7999 0,8154 0,8013 0,778 0,7923 0,8152 0,8145 0,15 0,2069 0,1921 0,1926 -0,602 -0,5021 0,1919 0,382 0,2 0,802 0,8093 0,8179 0,1649 0,7782 0,8173 0,832 0,2 0,1342 0,1188 0,1171 -0,992 0,1192 0,2195 0,25 0,7955 0,8067 0,8116 0,206 0,7553 0,8108 0,8495 0,25 0,0892 0,0735 0,0793 -1,5243 0,0742 0,057 x=.25 v=0.01 x=.25 v=0.001 t analytic implicit explicit l-gauss l-hermite group decomp t analytic implicit explicit l-gauss l-hermite group decomp 0,01 0,7492 0,7346 0,7342 0,7344 0,734 0,7342 0,7422 0,01 0,7349 0,701 0,6945 0,7351 0,7352 0,735 0,7425 0,05 0,746 0,6766 0,6748 0,675 0,6745 0,6755 0,7106 0,05 0,6746 0,6432 0,5932 0,6778 0,6765 0,6782 0,7123 0,1 0,742 0,6122 0,6087 0,608 0,6075 0,6104 0,6711 0,1 0,6075 0,6015 0,6014 0,6126 0,6126 0,6144 0,6746 0,15 0,738 0,5562 0,5512 0,5494 0,5514 0,5537 0,6316 0,15 0,5538 0,5843 0,5885 0,5555 0,5571 0,5582 0,6369 0,2 0,734 0,5016 0,4998 0,5043 0,5046 0,5921 0,2 0,4979 0,5061 0,5112 0,5094 0,5992 0,25 0,73 0,4534 0,4642 0,4652 0,5526 0,25 0,4543 0,463 0,4743 0,4673 0,5615 x=.50/t analytic implicit explicit l-gauss l-hermite group decomp x=.50/t analytic implicit explicit l-gauss l-hermite group decomp 0,01 0,9992 0,9986 0,9992 0,9984 0,999 0,9989 1,0003 0,01 0,9982 0,9325 0,9632 0,9991 0,9991 0,9996 1 0,05 0,996 0,9873 0,9901 0,9854 0,9875 0,9887 0,9987 0,05 0,9864 0,9432 0,9736 0,9888 0,9901 0,9921 0,9998 0,1 0,992 0,9611 0,9662 0,9556 0,9573 0,9636 0,9967 0,1 0,9558 0,9228 0,9323 0,9617 0,9643 0,9699 0,9997 0,15 0,988 0,9233 0,9299 0,9144 0,9183 0,9263 0,9947 0,15 0,9243 0,8647 0,8842 0,9223 0,9283 0,9343 0,9995 0,2 0,984 0,8835 0,8667 0,8734 0,8796 0,9927 0,2 0,8663 0,8754 0,8872 0,8871 0,9993 0,25 0,98 0,7908 0,8283 0,8256 0,9907 0,25 0,7906 0,8209 0,8453 0,8306 0,9991 x=.75/t analytic implicit explicit l-gauss l-hermite group decomp x=.75/t analytic implicit explicit l-gauss l-hermite group decomp 0,01 0,7556 0,7644 0,7517 0,7643 0,7651 0,7645 0,7583 0,01 0,7652 0,6946 0,7432 0,765 0,7652 0,7651 0,7576 0,05 0,7722 0,8241 0,7823 0,8206 0,8243 0,824 0,7867 0,05 0,8204 0,8417 0,8436 0,8249 0,8284 0,8285 0,7874 0,1 0,8469 0,9027 0,7906 0,886 0,8954 0,902 0,8222 0,1 0,8859 0,9632 0,9348 0,895 0,9062 0,9138 0,8247 0,15 0,925 0,9843 0,8013 0,9381 0,9544 0,9832 0,8577 0,15 0,9379 1,023 1,018 0,9507 0,971 1,0049 0,862 0,2 0,9945 0,8179 0,9631 0,9931 1,0671 0,8932 0,2 0,9628 0,9817 1,013 1,1005 0,8993 0,25 0,9996 0,8116 0,8109 1,005 1,1513 0,9287 0,25 0,8115 0,9514 1,024 1,1988 0,9366 58 num.18 -2008 ratio mathematica c. mamaloukas, s. spartalis diagram 1: comparison results with 1, 0.25, 0.01 0.25v x t    using 2 and 3 terms diagram 2: comparison results with 1, 0.5, 0.01 0.25v x t    using 2 and 3 terms 59 num.18 – 2008 ratio mathematica c. mamaloukas, et al diagram 3: comparison results with 0.1, 0.75, 0.01 0.25v x t    using 2 and 3 terms diagram 4: comparison results with 0.01, 0.5, 0.01 0.25v x t    and 2 terms 60 num.18 -2008 ratio mathematica c. mamaloukas, s. spartalis diagram 5: comparison results with 0.001, 0.75, 0.01 0.25v x t    and 2 terms 6. discussions the analytic solution as it is described by cole [13] is liable to restrictions concerning the values of the coefficient 1 e v r  . for example, if the value of the reynolds number is greater than 1000 then we can not find any solution because fourier series do not converge. for this reason we try numerical approaches, like finite differences and finite elements. concerning finite differences the explicit method give us adequate results if and only if 1 / 2  . otherwise results did not converge. with the implicit method we do not need the covenant 1 / 2  , but we need a large number of calculations. finally, the group of explicit methods gives us adequate results with few calculations and the method is more stable. concerning finite elements, the method of lines with gauss and hermite was used with initial and boundary conditions from madsen and sincovec [14]. these methods, using great values of reynolds number, gave us adequate results without the limitations for x and t , with small number of repetitions and without stability limitations. however, the errors depend first on the choice of the polynomial and second on the choice of x and t 61 num.18 – 2008 ratio mathematica c. mamaloukas, et al concerning the decomposition method from the above diagrams it is obvious how powerful this method is. using only two terms we can obtain similar results with the other numerical methods and the analytic solution. of course, in some cases the present solutions deviate from the solutions given in the table. the decomposition solution can be further improved if more-term approximations of the solution are obtained. as far as accurate results are concerned, computational experience has shown that they can be obtained easily by taking half a dozen terms. in case we do not have a sufficiently high precision by using a few of the na , then accordingly to rach r. [15] there are two alternatives. one is to compute additional terms by any of the available procedures. the second approach is to use the adomianmalakian ''convergence acceleration'' procedure [16]. this unique approach conveniently yields the error-damping effect of calculating many more terms of the an to determine whether further calculation is required. 7. conclusions the great advantage of the decomposition method is that of avoiding simplifications and restrictions which change the non-linear problem into a mathematically tractable one, whose solution is not consistent to physical solution. further study on the stability and the convergence of the solutions will prove the accuracy of the above method. bibliography 1. g. adomian, nonlinear stochastic operator equations, academic press, 1986. 2. g. adomian, nonlinear stochastic systems theory and applications to physics, kluwer academic publishers, 1989. 3. g. adomian, j. math. anal. appl., 119, (1986), 340-360. 4. g. adomian, appld. math. lett., 6, no5, (1993), 35-36. 5. g. adomian, rach r., on the solution of nonlinear differential equations with convolution product non-linearities, j. math. anal. appl., 114, (1986), 171-175. 6. g. adomian, solving frontier problems of physics: the decomposition method, kluwer academic publishers, 1994. 7. y. cherruault, kybernetes, 18, no2, (1989), 31-39. 8. y. cherruault, math. comp. modeling, 16, no2, (1992), 85-93. 9. c. agas, the effect of kinematic viscosity in the numerical solution of burger equation, thessaloniki, 1998. 62 num.18 -2008 ratio mathematica c. mamaloukas, s. spartalis 10. e.r. benton & g.w. platzman, a table of solutions of the onedimensional burgers equation, quart. appl. math., 1972. 11. j. m. burgers, the nonlinear diffusion equation, d. reidel publishing company, univ. of maryland, usa, 1974. 12. c. mamaloukas, numerical solution of one dimensional kortweg-de vries equation, bsg proceedings 6, global analysis, differential geometry and lie algebras, 6, (2001), 130-140. 13. c. mamaloukas, haldar k., mazumdar h. p., application of double decomposition to pulsatile flow, journal of computational & applied mathematics, 10 , issue 1-2, (2002), 193–207. 14. j.d. cole, on a quasilinear parabolic equation occurring in aerodynamics, a.appl. maths, 9, (1951), 225-236. 15. n. k. madsen and r. f. sincovec, general software for partial differential equations in numerical methods for differential system, ed. lapidus l., and schiesser w. e., academic press, inc., 1976. 16. r. rach, a convenient computational form of the adomian polynomials, j. math. anal. appl., 102, (1984), 415-419. 17. g. adomian and malakian, self-correcting approximate solutions by the iterative method for nonlinear stochastic differential equations, j. math. anal. appl., 76, (1980), 309-327. 18. g. adomian and malakian, inversion of stochastic partial differential operators-the linear case, j. math. anal. appl., 77, (1980), 505-512. 19. g. adomian and malakian, existence of the inverse of a linear stochastic operator, j. math. anal. appl., 114, (1986), 55-56. 20. g. adomian and r. rach, inversion of nonlinear stochastic operators, j. math. anal. appl., 91, (1983), 39-46. ratio mathematica volume 46, 2023 superior eccentric domination polynomial tejaskumar r* a mohamed ismayil† abstract superior distance involves the path which travels through the closed neighbourhood of both the vertices and the shortest path between them. this unique distance led to the advent of superior dominating sets and superior eccentric dominating sets. the former has a superior neighbourin its compliment for every vertex in itself and the latter has a superior eccentric vertex in itself for every vertex in its compliment. the domination polynomials disuss the idea of total number of dominating sets and dominating sets of specific cardinality. this inspired us to conceptualise the idea of superior eccentric domination polynomial.in this paper, we introduce the superior eccentric domination polynomial sed(g,φ) = ∑β l=γsed(g) |sed(g,l)|φ l where |sed(g,l)| is the number of all distinct superior eccentric dominating sets with cardinality l and γsed(g) is superior eccentric domination number. we find sed(g,) for family of wheel graphs and different standard graphs. the correlation between the coefficients of different sed polynomials are stated and proved. the motivation for this paper is to find a domination polynomial using distance concept in graphs. eccentricity is a distance and eccentric dominating set was already existing. keywords: superior distance, superior eccentricity, superior eccentric domination polynomial. 2020 ams subject classifications: 05c69, 11b83, 05c12. 1 *jamal mohamed college (affiliated to bharathidasan university), tiruchirappalli, india. tejaskumaarr@gmail.com. †jamal mohamed college (affiliated to bharathidasan university), tiruchirappalli, india. amismayil1973@yahoo.co.in. 1received on september 15, 2022. accepted on december 15, 2022. published on march 20, 2023. doi: 10.23755/rm.v46i0.1082. issn: 1592-7415. eissn: 2282-8214. ©tejaskumar r et al. this paper is published under the cc-by licence agreement. 257 tejaskumar r and a mohamed ismayil 1 introduction the shortest path between any two vertices is known as geodesic. the concept of distance in graphs always yields to cater the needs of applications in technology. there are many variants of distances in graphs. the shortest, longest, path involving degree of vertices and chords. kathiresan et al. [2007] introduced the superior distance in graphs. let the path dpq = n[p] ∪ n[q]. the shortest superior distance between p to q is dd(p,q). superior eccentricity of ed(p) = max{dd(q,p) : p,q ∈ v} superior neighbour of p is dd(p) = min{dd(p,q) : q ∈ v −{p}}. a vertex p(q) is a superior neighbour of p if dd(p,q) = dd(p). the superior distance involves the shortest path between two vertices which travels through all their closed neighbourhoods. using the superior distance the same authors kathiresan and marimuthu [2008] introduced the superior domination(sd-set).a set ⊆ v is called a sd-set if every vertex in s has superior neighbour in s-d. the sd-number is the cardinality of the minimum sd-set, denoted by γsed(g). the superior eccentric vertex p is given by dd (p,q) = ed(p). here the adjacency between the vertices in s and its compliment is not mandatory. bhanumathi and abhirami [2017] introduced the superior eccentric domination(sed-set) in graphs. a set ⊆ v is an sed-set if every vertex in s-dhas a superior eccentric vertex in s. the sed-number is the cardinality of the minimum sed-set, denoted by γsed(g). along with being a superior dominating set if the same set has a superior eccentric vertex in itself for every vertex in the compliment of s then it becomes a superior eccentric dominating set. these two conditions play a vital role in the formation of a sed-set. alikhani and peng [2009] conceptualized the idea of domination polynomial, a domination polynomial consists of a coefficients which gives the number of dominating sets and the power of the variable denotes the cardinality of the dominating set which varies between one and the vertex cardinality of graph. they discussed and proved certain properties which speaks of the corelation between the dominating sets. the motivation for this paper is to find a domination polynomial using distance concept in graphs. eccentricity is a distance and eccentric dominating set was already existing. inspired by this work ismayil and tejaskumar [2020] introduced the eccentric domination polynomial. the eccentric dominating polynomial gives the idea about the number of eccentric dominating sets with different cardinality and the symmetry in the coefficients of their polynomials werediscussed and proved. superior distance, superior eccentric domination existed but there was a gap in the literature we did not have a formula which could find the total number of sed in a graph or a sed of specific cardinality, henceforth the same authors ismayil and tejaskumar extended the idea of domination polynomial to superior eccentric domination polynomial. in this paper, we discuss the concept of superior eccentric domination polynomial with an apt example, this concept was mainly in258 superior eccentric domination polynomial troduced to find the total number of sed-sets of any graph. we found the formula which yields a sed polynomial for the family of wheel graphswhich helps us to easily find the total number of sed-sets of any cardinality at a given point of time for a wheel graph.we obtain the formulas and discuss the corelation between the coefficients of different sed polynomials. we tabulate the sed polynomials and their roots of different standard graphs. for all the undefined terminologies refer the book graph theory by harary [2001]. 2 superior eccentric domination polynomial definition 2.1. the superior eccentric domination polynomial sed(g,φ) =∑β l=γsed(g) |sed(g,l)|φ l where |sed(g,l)| is the number of distinct superior eccentric dominating sets (sed-sets) with cardinality l, β ∈ n and γsed(g) is superior eccentric domination number. example 2.1. . ℘4 ℘5℘3 ℘1 ℘2 figure 1: cricket graph vertex superior eccentricity superior eccentric vertex ed(℘) ℘1 2 ℘2 ℘2 2 ℘1 ℘3 2 ℘5 ℘4 6 ℘1,℘2,℘3,℘5 ℘5 2 ℘3 here we see the cricket graph has a sed-set {℘4} of cardinality 1, {℘1,℘4}, {℘2,℘4}, {℘3,℘4}, {℘4,℘5} sed sets of cardinality 2, {℘1,℘2,℘4},{℘1,℘3,℘4}, {℘1,℘4,℘5}, {℘2,℘3,℘4}, {℘2,℘4,℘5}, {℘3,℘4,℘5} sed sets of cardinality 3, {℘1,℘2,℘3,℘4}, {℘1,℘2,℘4,℘5},{℘1,℘3,℘4,℘5},{℘2,℘3,℘4,℘5} sed sets of cardinality 4 and {℘1,℘2,℘3,℘4,℘5} sed sets of cardinality 5. therefore sed(g,φ) = φ 5 + 4φ 4 + 6φ 3 + 4φ 2 + φ. 259 tejaskumar r and a mohamed ismayil 3 superior eccentric domination polynomial of wheel graph definition 3.1. superior eccentric domination polynomial of a wheel graph wβ is given by sed(wβ,φ) = ∑β l=γsed(wβ) |sed(wβ, l)|φ,l where |sed(wβ, l)| is the number of distinct sed-sets with cardinality l and γsed(wβ) is sed-number of wheel. observation 3.1. . 1. sed(wβ,φ) = (1 + φ)β −1, for β = 4,5. 2. sed(wβ,φ) = (1 + φ)β, for β = 6. theorem 3.1. for a wheel graph wβ of order β, 1. |sed(wβ, l)| = |sed(wβ−1, l−1)|+|sed(wβ−1, l)| where l ≤ β and β ≥ 7. 2. sed(wβ,φ) = φ sed(wβ−1,φ) + sed(wβ−1,φ). 3. sed(sβ,φ) = φ(φ + 1)β−1, for all β ≥ 7. proof: let v (wβ) = {℘1,℘2, . . .℘β}. 1. since |sed(wβ, l)| =β−1 cl−1, |sed(wβ, l−1)| =β−2 cl−2 and |sed(wβ−1, l)| =β−2 cl−1. but β−1cl−1 =β−2 cl−2 +β−2 cl−1. therefore |sed(wβ, l)| = |sed(wβ−1, l−1)|+ |sed(wβ−1, l)|. 2. by theorem-wheelthm01 − (1) we have |sed(wβ, l)| = |sed(wβ−1, l − 1)|+ |sed(wβ−1, l)|. when l = 1, |sed(wβ,1)| = |sed(wβ−1,0)|+ |sed(wβ−1,1)| =⇒ φ, |sed(wβ,1)| = φ, |sed(wβ−1,0)|+ φ, |sed(wβ−1,1)|. when l = 2, |sed(wβ,2)| = |sed(wβ−1,1)|+ |sed(wβ−1,2)| =⇒ φ,2 |sed(wβ,2)| = φ,2 |sed(wβ−1,1)|+ φ,2 |sed(wβ−1,2)|. when l = 3, |sed(wβ,3)| = |sed(wβ−1,2)|+ |sed(wβ−1,3)| =⇒ φ 3|sed(wβ,3)| = φ 3|sed(wβ−2,1)|+ φ 3|sed(wβ−1,3)|. 260 superior eccentric domination polynomial when l = 4, |sed(wβ,4)| = |sed(wβ−1,3)|+ |sed(wβ−1,4)| =⇒ φ 4|sed(wβ,4)| = φ 4|sed(wβ−1,3)|+ φ 4|sed(wβ−1,4)|. ... when l = β −1, |sed(wβ,β −1)| = |sed(wβ−1,β −2)| +|sed(wβ−1,β −1)| =⇒ φ β −1|sed(wβ,β −1)| = φ β −1|sed(wβ−1,β −2) +φ β −1|sed(wβ−1,β −1)|. when l = β, |sed(wβ,β)| = |sed(wβ−1,β −1)|+ |sed(wβ−1,β)| =⇒ φ β|sed(wβ,β)| = φ β|sed(wβ−1,β −1)|+ φ β|sed(wβ−1,β)|. therefore φ |sed(wβ,1)|+φ 2|sed(wβ,2)|+φ 3|sed(wβ,3)|+φ 4|sed(wβ,4)|+ · · ·+ φ β−1|sed(wβ,β −1)|+ φ β|sed(wβ,β)| = φ |sed(wβ−1,0)|+φ |sed(wβ−1,1)|+φ 2|sed(wβ−1,1)|+φ 2|sed(wβ−1,2)|+ φ 3|sed(wβ−2,1)|+ φ 3|sed(wβ−1,3)|+ φ 4|sed(wβ−1,3)|+ φ 4|sed(wβ−1,4)| + · · ·+ φ β−1|sed(wβ−1,β −2)| + φ β−1|sed(wβ−1,β −1)|+ φ β|sed(wβ−1,β −1)|+ φ β|sed(wβ−1,β)|. = φ |sed(wβ−1,0)|+ φ 2|sed(wβ−1,1)|+ φ 3|sed(wβ−2,1)|+ φ 4|sed(wβ−1,3)| + · · ·+ φ β−1|sed(wβ−1,β −2)|+ φ β|sed(wβ−1,β −1)|+ φ |sed(wβ−1,1)| + φ 2|sed(wβ−1,2)|+ φ 3|sed(wβ−1,3)|+ φ 4|sed(wβ−1,4)|+ . . . + φ β−1|sed(wβ−1,β −1)|+ φ β|sed(wβ−1,β)| = φ [φ |sed(wβ−1,1)|+ φ 2|sed(wβ−1,2)|+ φ 3|sed(wβ−1,3)| + φ 4|sed(wβ−1,4)|+ · · ·+ φ β−1|sed(wβ−1,β −1)|] + φ |sed(wβ−1,1)| + φ 2|sed(wβ−1,2)|+ φ 3|sed(wβ−1,3)|+ φ 4|sed(wβ−1,4)| + · · ·+ φ β−1|sed(wβ−1,β −1)| since |sed(wβ−1,0)| = |sed(wβ−1,β)| = 0. = φ ∑β−1 l=1 |sed(wβ−1, l)|φ l + ∑β−1 l=1 |sed(wβ−1, l)|φ l. sed(wβ,φ) = φ sed(wβ−1,φ) + sed(wβ−1,φ). 1. by mathematical induction (mi). 261 tejaskumar r and a mohamed ismayil it is true for β = 7. sed(wβ−1,φ) = φ(φ + 1) 7−1 = φ(φ + 1)6 = φ(φ + 1)3(φ + 1)3 = φ 7 + 6φ 6 + 15φ 5 + 20φ 4 + 15φ 3 + 6φ 2 + 1. assume it is true ∀ n less than ′β′. sed(wβ,φ) = φ(1 + φ) (β−1)−1 = φ(1 + φ)β−2 for′β′, sed(wβ,φ) = φ sed(wβ−1,φ) + sed(wβ−1,φ) usingtheorem−3.1− (2) = φ [φ(φ + 1)β−2] + φ(φ + 1)β−2 = φ(φ + 1)β−1 ∴ proved ∀ ′β′. table: |sed(wβ, l)| is the number of superior eccentric dominating sets of wβ with cardinality l where 1 ≤ l ≤ 15. h h h h hhβ l 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 0 2 0 0 3 0 0 0 4 4 6 4 1 5 5 10 10 5 1 6 1 5 10 10 5 1 7 1 6 15 20 15 6 1 8 1 7 21 35 35 21 7 1 9 1 8 28 56 70 56 28 8 1 10 1 9 36 84 126 126 84 36 9 1 11 1 10 45 120 210 252 210 120 45 10 1 12 1 11 55 165 330 462 462 330 165 55 11 1 13 1 12 66 220 495 792 924 792 495 220 66 12 1 14 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 15 1 14 91 364 1001 2002 3003 3423 3003 2002 1001 364 91 14 1 theorem 3.2. the following properties for the co-efficients of sed(wβ,φ) holds. 1. |sed(wβ,1)| = 1 for all β ≥ 6. 2. |sed(wβ,β)| = 1, ∀ β ≥ 4. 3. |sed(wβ,β −1)| = β −1, ∀ β ≥ 6. 262 superior eccentric domination polynomial 4. |sed(wβ,β −2)| = (β−1)(β−2) 2 , ∀ β ≥ 6. 5. |sed(wβ,β −3)| = (β−1)(β−2)(β−3) 6 , ∀ β ≥ 6. 6. |sed(wβ,β −4)| = (β−1)(β−2)(β−3)(β−4) 24 , ∀ β ≥ 6. 7. |sed(wβ, l)| = |sed(wβ,β − l + 1)|, ∀ β ≥ 6. 8. if sedβ = ∑β l=1 |sed(wβ, l)|, ∀ β ≥ 6, then sedβ = 2(sedβ−1), ∀ β ≥ 7. 9. sedβ =total number of sed-sets in wβ = 2β−1, ∀ β ≥ 6. proof: 1. let v (wβ) = {℘1,℘2,℘3, . . .℘β}. in a wheel graph wβ all the vertices form a superior neighbour of central vertex ℘1 except itself. therefore the only set with single cardinality d = {℘1} forms the superior eccentric dominating set of wheel graph wβ where β ≥ 6. 2. the vertex set v (wβ) forms the superior eccentric dominating set |sed(wβ,β)| = 1 for all β ≥ 4. 3. by mi on ′β′. for β = 6, |sed(w6,6−1)| = |sed(w6,5)| = 5. assume it is true ∀ n less than ′β′. for ′β′, by theorem-3.1-(2) and 3.2-(2) |sed(wβ,β −1)| = |sed(wβ−1,β −2)|+ |sed(wβ−1,β −1)| = (β −2) + 1 = β −1 ∴ proved ∀ ′β′. 4. by mi on ′β′. for β = 6, |sed(w6,6−2)| = |sed(w6,4)| = 10. for β = 7, |sed(w7,7−2)| = |sed(w7,5)| = 15. assume it is true ∀ n less than ′β′. 263 tejaskumar r and a mohamed ismayil for ′β′, by theorem-3.1 and 3.2-(3) |sed(wβ,β −2)| = |sed(wβ−1,β −3)|+ |sed(wβ−1,β −2)| = (β −2)(β −3) 2 + (β −2) = (β −2)(β −3) + 2(β −2) 2 = (β −2)(β −3 + 2) 2 = (β −2)(β −1) 2 ∴ proved ∀ ′β′. 5. by mi on ′β′. for β = 6, |sed(w6,6−3)| = |sed(w6,3)| = 10. for β = 7, |sed(w7,7−3)| = |sed(w7,4)| = 20. assume it is true ∀ n less than ′β′. for ′β′, by theorem-3.1 and 3.2-(4) |sed(wβ,β −3)| = |sed(wβ−1,β −4)|+ |sed(wβ−1,β −3)| = (β −2)(β −3)(β −4) 6 + (β −2)(β −3) 2 = (β −2)(β −3)(β −4 + 3) 6 = (β −1)(β −2)(β −3) 6 ∴ proved ∀ ′β′. 6. by mi on ′β′. the result is true for β = 6, |sed(w6,6−4)| = |sed(w6,2)| = 5. for β = 7, |sed(w7,7−4)| = |sed(w7,3)| = 15. assume it is true ∀, n < β. 264 superior eccentric domination polynomial for ′β′, by theorem-3.1 and 3.2-(5) |sed(wβ,n−4)| = |sed(wβ−1,β −5)|+ |sed(wβ−1,β −4)| = (β −2)(β −3)(β −4)(β −5) 24 + (β −2)(β −3)(β −4) 6 = (β −2)(β −3)(β −4)(β −5 + 4) 24 = (β −1)(β −2)(β −3)(β −4) 24 ∴ proved ∀ ′β′. 7. by mi on ′β′. the result is true for β = 6. |sed(w6,2)| = |sed(w6,6−2 + 1)| = |sed(w6,5)| = 5 |sed(w7,3)| = |sed(w7,7−3 + 1)| = |sed(w7,4)| = 20. assume it is true ∀ n less than ′β′. for ′β′, by theorem-3.1 we have |sed(wβ, l)| = |sed(wβ−l, l−1)|+ |sed(wβ−1, l)| = |sed(wβ−1,(β −1− (l−1) + 1))| + |sed(wβ−1,(β −1− (l) + 1))| = |sed(wβ−1,(β −1− l + 1 + 1))| + |sed(wβ−1,(β −1− l + 1))| = |sed(wβ−1,(β − l + 1))| + |sed(wβ−1,(β − l))| = |sed(wβ,(β − l + 1))| ∴ proved ∀ ′β′. 8. sedβ = ∑β l=1 |sed(wβ, l)| by theorem-3.1 we have sedβ = β∑ l=1 [|sed(wβ−1, l−1)|+ |sed(wβ−1, l)|] = β−1∑ l=1 |sed(wβ−1, l)|+ β−1∑ l=1 |sed(wβ−1, l)| = sedβ−1 + sedβ−1 sedβ = 2(sedβ−1) 265 tejaskumar r and a mohamed ismayil 9. by mi on ′β′. when β = 6, sed6 = 2 6−1 = 25 = 32. sed7 = 2 7−1 = 26 = 64. assume it is true ∀ n less than ′β′. sedβ−1 = 2 β−1−1 = 2β−2 for ′β′, sedβ = 2[sedβ−1] from theorem −3.2− (8) = 2[2β−2] = 2β−1 ∴ proved ∀ ′β′. hence the theorem. remark 3.1. . 1. for any graph g 0 is one of the root of every sed(g,φ). 2. a graph with more than 3 pendant vertices has at least 2 real roots. sed(g,φ) of different standard graphs and their roots are tabulated below: graph figure superior eccentric domination polynomial sed(g,φ) roots diamond graph ℘1 ℘4 ℘2 ℘3 φ 4 + 4φ 3 + 5φ 2 φ 1 = 0, φ 2 = −0.4563, φ 3 = −1.7718 + 1.1151i, φ 4 = −1.7718−1.1151i. tetrahedral graph ℘2 ℘1 ℘3 ℘4 φ 4 + 4φ 3 + 6φ 2 + 4φ φ 1 = 0, φ 2 = −2, φ 3 = −1 + i, φ 4 = −1− i. 266 superior eccentric domination polynomial graph figure superior eccentric domination polynomial sed(g,φ) roots paw graph ℘2 ℘3 ℘1 ℘4 φ 4 + 3φ 3 + 3φ 2 + φ φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −1. banner graph ℘3 ℘4 ℘1 ℘2 ℘5 φ 5 + 4φ 4 + 5φ 3 + 3φ 2 φ 1 = 0, φ 2 = −2.4656, φ 3 = −0.7672 + 0.7926i, φ 4 = −0.7672−0.7926i. graph figure superior eccentric domination polynomial sed(g,φ) roots fork graph ℘2 ℘3 ℘1 ℘4 ℘5 φ 5 + 4φ 4 + 5φ 3 + 2φ 2 φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −2. (3,2)-tadpole graph ℘2 ℘3 ℘4 ℘1 ℘5 φ 5 + 4φ 4 + 3φ 3 + φ 2 φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −2. kite graph ℘3 ℘4 ℘1 ℘5 ℘2 φ 5 + 5φ 4 + 6φ 3 + 2φ 2 φ 1 = 0, φ 2 = −0.378 +0.1877i, φ 3 = −0.378 −0.1877i, φ 4 = −2.122 +1.0538i, φ 5 = −2.122 −1.053i. 267 tejaskumar r and a mohamed ismayil graph figure superior eccentric domination polynomial sed(g,φ) roots house graph ℘2 ℘3 ℘1 ℘4 ℘5 φ 5 + 5φ 4 + 6φ 3 φ 1 = 0, φ 2 = −0.3076 +0.3182i, φ 3 = −0.3076 −0.3182i, φ 4 = −2.1924 +0.5479i, φ 5 = −2.1924 −0.5479i. house x graph ℘2 ℘3 ℘1 ℘4 ℘5 φ 5 + 5φ 4 + 6φ 3 φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −0.382, φ 5 = −2.618. dart graph ℘3 ℘4 ℘1 ℘5 ℘2 φ 5 + 4φ 4 + 6φ 3 + 3φ 2 φ 1 = 0, φ 2 = −1, φ 3 = −32 + √ 3 2 i, φ 4 = −32 − √ 3 2 i. johnson solid skeleton 12 graph ℘2 ℘1 ℘3 ℘4 ℘5 φ 5 + 5φ 4 + 10φ 3 + 10φ 2 + 5φ φ 1 = 0, φ 2 = −0.691 +0.9511i, φ 3 = −0.691 −0.9511i, φ 4 = −1.809 +0.5878i, φ 5 = −1.809 −0.5878i. net graph ℘5 ℘6 ℘3 ℘4 ℘1 ℘1 φ 6 + 3φ 5 + 3φ 4 + φ 3 φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −1. a graph ℘3 ℘4 ℘1 ℘5 ℘2 ℘6 φ 6 + 4φ 5 + 6φ 4 + 4φ 3 + φ 2 φ 1 = 0, φ 2 = −1, φ 3 = −1, φ 4 = −1, φ 5 = −1. 268 superior eccentric domination polynomial graph figure superior eccentric domination polynomial sed(g,φ) roots 4-polynomial graph 1 ℘2 ℘3℘1 ℘5℘4 ℘6 φ 6 + 4φ 5 + 4φ 4 φ 1 = 0, φ 2 = −2. antenna graph ℘2 ℘1 ℘3 ℘4 ℘5 ℘6 φ 6 + 4φ 5 + 3φ 4 + φ 3 φ 1 = 0, φ 2 = −1, φ 3 = −2, φ 4 = −1. octahedral graph ℘4 ℘3℘2 ℘1 ℘5 ℘6 φ 6 + 6φ 5 + 15φ 4 + 20φ 3 +15φ 2 + 6φ φ 1 = 0, φ 2 = −2, φ 3 = −0.5 +0.866i, φ 4 = −0.5 −0.866i, φ 5 = −1.5 +0.866i, φ 6 = −1.5 −0.866i. cubical graph ℘3 ℘4 ℘5 ℘6 ℘1 ℘2 ℘8℘7 φ 8 + 8φ 7 + 28φ 6 + 56φ 5 +68φ 4 + 48φ 3 + 16φ 2 φ 1 = 0, φ 2 = −0.6714 +0.5756i, φ 3 = −0.6714 −0.5756i, φ 4 = −0.8352 +1.4854i, φ 5 = −0.8352 −1.4854i, φ 6 = −2.4934 +0.9097i, φ 7 = −2.4934 −0.9097i. wagner graph ℘1 ℘8 ℘4 ℘5 ℘2 ℘7 ℘3 ℘6 φ 8 + 8φ 7 + 24φ 6 +32φ 5 + 16φ 4 φ 1 = 0, φ 2 = −2, φ 3 = −2, φ 4 = −2, φ 5 = −2. 269 tejaskumar r and a mohamed ismayil 4 conclusions in this paper sed polynomial for a graph was defined. formula to find the sed polynomials of family of wheel graphs were stated and proved. corelation between the coefficients of sed polynomials were discussed. the sed polynomial and its roots for different standard graphs are tabulated. in the course of future work the graphs can be classified based on the roots of sed polynomials. the similarproperties of coefficients based on similar roots for different standard graphs can be discussed. the same concept can be extended to other domination parameter. references s. alikhani and y.-h. peng. introduction to domination polynomial of a graph. arxiv preprint arxiv:0905.2251, 2009. m. bhanumathi and r. m. abirami. superior eccentric domination in graphs. international journal of pure and applied mathematics, 117(14):175–182, 2017. f. harary. graph theory. narosa publishing house, new delhi, 2001. a. m. ismayil and r. tejaskumar. eccentric domination polynomial of graphs. advances in mathematics: scientific journal, 9(4):1729–1739, 2020. k. kathiresan and g. marimuthu. superior domination in graphs. utilitas mathematica, 76:173, 2008. k. kathiresan, g. marimuthu, and w. sivakasi. superior distance in graphs. journal of combinatorial mathematics and combinatorial computing, 61:73, 2007. 270 microsoft word corr. i_modelli_matematici_10.doc ratio mathematica, 19, pp. 31-88 31 i modelli matematici costruiti per l’insegnamento delle matematiche superiori pure e applicate nicla palladino1 – franco palladino1 sunto. nell’articolo che si presenta si vuole documentare, alla luce delle ricerche fino ad oggi condotte, anche per vari siti distribuiti in europa, la vicenda dei modelli matematici costruiti per l’insegnamento delle “matematiche superiori” –pure e applicate– e realizzati nel periodo di maggior impegno creativo che va, all’incirca, dalla seconda metà dell’ottocento agli anni trenta del novecento. nel corso dell’esposizione si verranno, inoltre, ad evidenziare, con brevi descrizioni e servendosi di opportuni esempi, una varietà di legami, tutti molto rilevanti, che mettono in corrispondenza l’ideazione e la costruzione dei modelli con studiosi, istituzioni culturali, specifiche visioni della ricerca e della didattica delle scienze matematiche e, ancora, con il mondo delle arti figurative. abstract. in this paper, we want to document the history of the models of mathematical surfaces used for the didactics of pure and applied “high mathematics”, in italy and in europe. these models were built between the second half of nineteenth century and the 1930s. we want here also to underline several important links that put in correspondence conception and construction of models with scholars, cultural institutes, specific views of research and didactical studies in mathematical sciences and with the world of the figurative arts furthermore, by using short descriptions and opportune examples. parole chiave: matematica, modelli, storia, arte, esposizioni. 1 dipartimento di matematica e informatica. università degli studi di salerno. ratio mathematica, 19, pp. 31-88 32 1. i modelli matematici realizzati (essenzialmente in europa) in un intervallo di tempo che è delimitabile, con buona approssimazione, tra gli inizi della seconda metà dell’ottocento e gli anni trenta del novecento, rappresentarono i prodotti di un’impresa culturale che coinvolse alcuni dei più attivi istituti matematici presenti presso le università e i politecnici europei. essa vide impegnati personaggi di prim’ordine applicati alle scienze matematiche e fu feconda di interazioni con la ricerca e la didattica, di “ordine superiore” (con favorevoli ricadute per l’insegnamento preuniversitario), praticate in queste scienze. l’impresa coinvolse anche importanti centri museali d’europa e, in aggiunta, i modelli matematici, nel loro diffondersi, intrecciarono interessanti legami perfino col settore delle arti figurative e il mondo del cinema allorché caddero sotto lo sguardo sensibile di scultori, pittori e scenografi. le ricerche, condotte dagli autori di questo lavoro, sui modelli matematici e sui molteplici aspetti, prima indicati, connessi alla loro realizzazione e diffusione, sono venute, nel corso degli anni e partendo dall’italia, progressivamente ad estendersi a varie sedi d’europa e naturalmente ad approfondirsi. nell’insieme delle indagini fatte, uno sguardo si è rivolto anche verso gli stati uniti d’america. con questo articolo, pur nei limiti di spazio che ne derivano, si vuole dare un quadro più ricco e commentato, ricorrendo a brevi descrizioni e servendosi di alcuni modelli significativi (per i quali si offrono maggiori informazioni), di un’impresa scientifica che oggi ha ritrovato nuovo vigore, grazie alla computer graphics, e recuperato (potenziandola poi) integralmente la sua utilità per la didattica della matematica intesa in senso molto ampio. tutto lo scritto che si presenta è scrupolosamente documentato mediante dettagliate e numerose note la cui consultazione si può eventualmente rimandare a una seconda, più posata, fase di lettura. i modelli in questione furono costruiti impiegando materiali diversi: ottone, gesso, cartone, filo metallico o di fibra naturale, legno e lamelle di legno, celluloide (un materiale, ottenuto per sintesi chimica, fondamentale, tra l’altro, per la nascente cinematografia e ratio mathematica, 19, pp. 31-88 33 tempestivamente usato anche per realizzare modelli), lamine metalliche ricoperte per via elettrochimica (con processo di galvanostegia, pur esso, allora, di recente concezione). essi servivano a far vedere proprietà notevoli riguardanti il tema di ricerca su cui si investigava e a mostrare alcuni risultati che progressivamente si conseguivano in diversi settori delle matematiche “pure” e “applicate”: geometria descrittiva e proiettiva, geometria analitica, geometria algebrica, topologia, teoria delle funzioni (anche a variabile complessa), meccanica razionale, fisicamatematica, scienze delle costruzioni e finanche, per esempio, ottica applicata alla fisiologia del corpo umano (tra i modelli, vi è, infatti, un’elegante realizzazione di oroptera o horopter geometrico –curva cubica sghemba– in filo metallico2) con i suoi collegamenti alla geometria proiettiva e algebrica. 2 esso è compreso nella serie xxviii (sechs modelle zur theorie der cubischen raumcurve und ihrer anwendung in der physiologischen optik), n° 6, del catalog mathematischer modelle für den höheren mathematischen unterricht, edito da martin schilling in leipzig nel 1911 (è questa la settima edizione del catalog, pubblicazione di cui si parlerà più estesamente). sull’ horopter vi è un’interessante informazione comunicata da luigi cremona (1830-1903) a thomas archer hirst (1830-1892), lettera datata bologna 20 dicembre 1864, in cui cremona scrive: “a proposito delle cube gobiche [sic], se voi consulterete i recenti scritti di helmholtz e di hering negli annali di poggerdoff [annalen der physik und chemie, n.d.r.], troverete che l’horopter, cioè il luogo dei punti dello spazio che (per una data posizione degli occhi) projettano imagini identiche sulle due retine, è una cubica gobba: infatti si hanno a considerare due fasci di raggi, i cui centri sono i punti-nodi degli occhi: assumendo come omologhi due raggi che incontrano le retine in punti corrispondenti (identici dicono i fisiologi) i due fasci sono omografici, epperò il luogo delle intersezioni de’ raggi omologhi è una cubica gobba [un’ellisse cubica, n.d.r.]”. hirst, nella lettera di risposta (londra, 27 dicembre 1864), fa presente a questo proposito: “your remarks about the horopter interested me greatly. some time ago i heard prof. helmholtz at the royal society read a paper on the subject and my curiosity was raised regarding the nature of the curve.” (si veda la corrispondenza di luigi cremona (1830-1903), vol. iv –a cura di l. ratio mathematica, 19, pp. 31-88 34 ideati per le “matematiche superiori” –pure e applicate– e per le scienze delle costruzioni, i modelli, di cui si tratta, furono pure accompagnati da altri esemplari pensati per migliorare la didattica di quelle discipline che si insegnavano nei primi anni dei corsi universitari –per matematici, fisici e ingegneri–, al fine di potenziare negli studenti la componente intuitiva-visiva compresente nell’apprendimento delle stesse discipline e, specialmente, della geometria: scopo, questo, che è ben riflesso, per segnalare un caso notevole, nel volume di geometria evidente, intuitiva, vale a dire nell’anschauliche geometrie, composto da d. hilbert e s. cohnvossen.3 non dovrebbe suscitare meraviglia che il grande matematico germanico david hilbert (1862-1943), ricordato a giusta ragione come l’ideatore del moderno metodo assiomatico –dove, si sa, le proposizioni della matematica si concatenano soltanto per mezzo di regole di deduzione accettate come legittime e dove si fa astrazione di tutte le “evidenze” intuitive che i termini, occorrenti nelle proposizioni stesse, possano suggerire alla mente dello studioso–, si sia impegnato anche in questo settore delle scienze matematiche; tra l’altro, egli non sarà l’unico: si potrebbe ricordare ancora, per esempio, hermann wiener (1857-1939). va tenuto presente, a tal riguardo, che tra la fine dell’ottocento e i primi anni del novecento l’enorme sviluppo, qualitativo e quantitativo, delle scienze (comprese le scienze matematiche), della tecnica e della produzione industriale, vissuto dalla germania, condusse, tra l’altro, a un’accentuata compartimentazione delle scienze medesime e alla specializzazione spinta della figura dello scienziato, e del matematico. questo stato di cose –nel settore industriale porterà al taylorismo– fu avvertito tempestivamente da ernesto pascal –1865-1940–, professore di calcolo infinitesimale all’università di pavia dal 1890, che aveva soggiornato, dopo la laurea conseguita in napoli, per un nunzia–, quaderni p.ri.st.em, università “l. bocconi” – milano, palermo, 1999, pp. 67 e 68). 3 berlin, j. springer, 1932; traduzione italiana geometria intuitiva, torino, boringhieri, 1960. ratio mathematica, 19, pp. 31-88 35 anno a gottinga, grazie a una borsa di studio di perfezionamento per l’estero, dove si era fatto apprezzare da felix klein4 –sotto la cui guida si attuò in germania il prodigioso sviluppo della progettazione e realizzazione dei modelli– allorquando notò che la sua stessa opera, il repertorio di matematiche superiori –uscita in due volumi tra il 1898, analisi, e il 1900, geometria–, che tendeva a offrire un quadro completo dei risultati raggiunti nelle “matematiche superiori”, era accolta con particolare favore in germania:5 lì vige –scrive pascal, nel 4 si possono consultare le lettere i-13 e i-14 di pascal a federico amodeo, contenute in f. palladino, n. palladino, dalla “moderna geometria” alla “nuova geometria italiana”. viaggiando per napoli, torino e dintorni, firenze, olschki, 2006. 5 il repertorio, pubblicato per l’editore hoepli di milano, raccoglieva i principali risultati “superiori” maturati, fino a quella data, in analisi e in geometria. alle edizioni accresciute, succedutesi alla prima, vi collaborarono anche federigo enriques (1871-1946) e francesco severi (1879-1961). l’uscita del repertorio rappresenterà uno dei molteplici segnali secondo i quali viene riconosciuto che la matematica italiana, in tutti i settori, aveva raggiunto, a quel tempo, uno dei primi posti in europa. posizione che occuperà sicuramente nel periodo compreso tra gli ultimi decenni dell’ottocento e la prima guerra mondiale e che sarà suggellata dall’assegnazione, all’italia, del quarto congresso internazionale dei matematici (i primi tre si erano tenuti, rispettivamente, a zurigo, parigi e nella rinomata università tedesca di heidelberg nel land del baden) celebrato a roma nel 1908. a proposito del repertorio, ecco quanto scrive lo stesso pascal in una lettera (milano, 5 giugno 1906) diretta al suo conterraneo ernesto cesàro (1859-1906): “[…] riapro la lettera per annunziarle che del mio repertorio in tedesco [repertorium der höheren mathematik, leipzig, druck und verlag von b.g. teubner, i theil: die analysis, 1900, ii theil: die geometrie, 1902, n.d.r.] si farà la seconda edizione istituendo in germania un comitato di redazione, del quale si occuperà il d.r epstein di strasburgo, che procederà d’accordo con me. così si spera di farne una sorta di opera periodica, ed il mio repertorio diventerà una specie di istituzione matematica tedesca! (dico per celia!). sono contento che esso abbia avuto tanto successo in germania, ed è curioso che i tedeschi abbiano dovuto ricorrere ad un italiano per un’opera che ora corre per le mani di tutti i loro studenti di matematica. essa forse risponde al modo con cui gli studii sono organizzati in germania, per cui lo ratio mathematica, 19, pp. 31-88 36 provare a spiegare le ragioni del successo del repertorium– un’organizzazione degli studi, e della ricerca, fin troppo specialistica. e, in tale contesto, sia la geometria intuitiva sia la costruzione di modelli plastici (pascal ne incrementò l’acquisto da parte dell’università di pavia e li menzionò più volte nel repertorio) tendono a diventare dei settori che assumono una loro autonomia, collegati alla ricerca e all’insegnamento nel campo delle scienze matematiche. ciò precisato, la presenza di hilbert in “spazi” –come direbbero i matematici– distinti non è da vedersi quindi come un’anomalia (o una “contraddizione”) ma è interpretabile piuttosto come segno della vastità del suo ingegno capace di dedicarsi, mantenendo la profondità dello specialista (e con l’aggiunta di una grande laboriosità), a una pluralità di settori: un’attitudine che frequentemente esalta la possibilità di ottenere risultati di valore assoluto, utili per l’intero arco delle scienze matematiche. in particolare, a proposito del tema che in questo articolo si vuole trattare, fu proprio hilbert a stimolare la progettazione di un modello in gesso, realizzato da werner boy (1879-1914), nel 1901, in due versioni, riguardanti la topologia. (erano modelli, ideati da boy –perciò conosciuti poi come boysche fläche, erste und zweite version–, destinati a “dimostrare”, grazie esclusivamente all’evidenza della realizzazione plastica, che era possibile immergere il piano proiettivo reale in uno spazio a tre dimensioni. entrambe le versioni sono presenti, volendo indicare almeno una sede italiana, all’università di napoli).6 studente tedesco finisce alle volte per essere troppo specialista [il corsivo è aggiunto, n.d.r.], e coll’ignorare molte altre parti della matematica che dovrebbe conoscere.” per questo brano si rimanda a f. palladino, n. palladino, dalla “moderna geometria” alla “nuova geometria italiana”. viaggiando per napoli, torino e dintorni, cit., pp. xiii-xiv. 6 i due modelli (modelle für die abbildung der projectiven ebene auf eine im endlichen geschlossene singularitätenfreie fläche), realizzati, all’università di gottinga, da boy –che lì era diventato philosophical doctor avendo avuto hilbert come relatore della tesi– in collaborazione col professore, della stessa università, friedrich georg schilling (1868-1950) –un nome notevole nel campo della costruzione di strumenti e modelli matematici–, furono ratio mathematica, 19, pp. 31-88 37 l’anschauliche geometrie, un testo che ancora oggi conserva un elevato interesse, esce, curiosamente, al tempo in cui l’esperienza della progettazione e costruzione dei modelli vive la sua ultima fase e si prepara l’affermazione del bourbakismo che, con la sua radicale proposizione del punto di vista assiomatico in chiave strutturalista,7 contribuirà alla progressiva estraneazione dagli “istituti di matematica” di strumenti meccanici e modelli plastici i quali vengono chiusi negli armadi o, in alcuni casi, relegati a prendere polvere in soffitte e sottoscala, quando non andranno dispersi a causa di traslochi come accadde, tra la fine degli anni settanta e gli inizi degli ottanta del secolo scorso, nell’ambito dell’università di pisa. una tendenza all’estraneazione che verrà, tra l’altro, a privare l’insegnamento di sussidi didattici che avrebbero potuto dispiegare ancora (si potrebbe aggiungere “come sempre”, qualora si venga ad usarli opportunamente) la loro efficacia: bisognerà attendere gli ultimi inseriti ai primi due posti della serie xxx: gips-modelle verschiedener art (la serie comprendeva 8 modelli in gesso di vario tipo) del catalog di martin schilling (sul quale in seguito si daranno altre informazioni). lo stesso catalog, nella sua parte seconda (dove viene svolto il commento scientifico delle serie di modelli presentati nella prima parte), include le due superfici sotto l’argomento analysis situs, espressione usata da g.w. leibniz, e conservata successivamente per lungo tempo, in riferimento a quelle proprietà delle figure che saranno, due secoli dopo all’incirca, studiate nell’ambito della moderna topologia. una parametrizzazione della superficie di boy fu trovata, mediante l’utilizzazione del computer, solo nel 1978 da bernard morin mentre françois apéry, nel 1984, ne ottenne un’equazione algebrica. nella pubblicazione, mathematical models/mathematische modelle, (edizione bilingue, braunschweig/wiesbaden, friedr. vieweg & sohn, 1986, due voll.) curata da gerd fischer, sotto il cap. 6, models of the real projective plane, redatto da u. pinkall, del vol. i (commentary), p. 64 è scritto: “the first immersion (i.e. non-singular mapping) of rp2 into r2 was constructed in 1901 by w. boy in göttingen”. 7 sull’argomento si può consultare il saggio di g. israel, un aspetto ideologico della matematica contemporanea: il bourbakismo, incluso in matematica e fisica: struttura e ideologia, a cura di e. donini, a. rossi, t. tonietti, bari, de donato, 1977, pp. 35-70. ratio mathematica, 19, pp. 31-88 38 decenni del novecento per vedere reintrodotta la “visione” –che avrà la proprietà, in più, di essere diventata dinamica– nella ricerca e nella didattica delle scienze matematiche, grazie alle possibilità offerte dall’informatica. tendenza all’esclusione che non è un fenomeno tipicamente italiano poiché esso parte dai principali paesi europei e sarà comune al resto del mondo: almeno in questo caso non appare attribuibile, per l’italia, alla opposizione esercitata dall’idealismo di croce e gentile nei confronti della filosofia positivista entro la quale maturarono la progettazione e l’uso dei modelli plastici per le scienze matematiche. le considerazioni fin qui esposte portano a riflettere che in un intervallo di tempo di circa mezzo secolo trascorso tra la seconda metà dell’ottocento e gli anni trenta del novecento, si passa attraverso due “visioni del mondo”, scientifico-matematico, diverse e contrapposte. argomento che si vuole brevemente toccare in questo paragrafo introduttivo per comprendere un po’ meglio la parabola di vita, non molto ampia, dei modelli plastici nati nell’ottocento. la prima delle due “visioni” è caratterizzata dall’alto grado di considerazione goduto dal punto di vista “intuitivo”, emblematicamente espresso dall’affermazione di william thomson (1824-1907), ben noto col titolo di lord kelvin, pronunciata nel pieno del clima positivista e meccanicista: io non sono soddisfatto finché non ho potuto costruire un modello meccanico dell’oggetto che studio. se posso costruire un tale modello, comprendo; altrimenti, non comprendo affatto.8 e, sotto questo punto di vista, volendo anche citare un caso concreto –scelto tra i tanti disponibili– molto significativo e più strettamente attinente alla matematica, si può ricordare che 8 “i am never content until i have constructed a mechanical model of the subject i am studying. if i succeed in making one, i understand; otherwise, i do not”, in w. thomson, molecular dynamics and the wave theory of light: notes of lectures delivered at the johns hopkins university, baltimore, by sir william thomson, … stenographically reported by a.s. hathaway, baltimore (maryland – u.s.a.), johns hopkins university, circa 1884. ratio mathematica, 19, pp. 31-88 39 l’intenzione di dare risalto al contresempio, esibito da peano (la cui fama è, come noto, particolarmente legata a questa forma di indagine scientifica), col quale il matematico torinese smentiva la validità generale del criterio di accertamento (“condizione sufficiente”) fornito da serret (nel suo cours de calcul différentiel et intégral9) per la ricerca dei punti di massimo e minimo, condusse gli studiosi a 9 la prima edizione del cours di joseph-alfred serret, in due tomi, è del 1868 (paris, gauthier-villars). qui si cita dalla terza ediz. del 1886 (stessi luoghi editoriali), p. 219 del t. i. serret, al pf. intitolato des maxima et des minima des fonctions de plusieurs variables indépendantes, parte col dire: “soit f(x,y,z, …) une fonction de plusieurs variables indépendantes x,y,z, … on dit que cette fonction a une valeur maxima pour x=x0 , y=y0, z=z0, …, lorsque la différence f(x0 + h, y0 +k, z0 + l, …) – f(x0, y0, z0, …) est négative pour toutes les valeurs des accroissements h, k, l, … comprises entre –ε et +ε, la quantité positive ε étant d’ailleurs aussi petite que l’on le voudra. si, au contraire, la précédente différence est constamment positive pour les même valeurs de h, k, l, …, la fonction f(x,y,z, …) prend une valeur minima pour x=x0 , y=y0, z=z0, …”; e arriva alla conclusione che “les valeurs de x, y, z, … qui réspondent à un maximum ou à un minimum de la fonction f(x,y,z, …) sont comprise parmi celles qui annulent la differéntielle totale df de cette fonction ou qui la rendent discontinue.” a questo punto serret scrive: “on arrive au même résultat par l’emploi de la formule de taylor. […]”. e così, nel proseguire, con l’impiego della formula di taylor, le sue considerazioni, serret giunge ad affermare che “le maximum ou le minimum a lieu si, pour les valeurs de h, k, l, … qui annulent d2 f et d3 f, d4 f a constamment le signe – ou constamment le signe +”. è su questa affermazione, secondo cui i punti di massimo e minimo (relativi) di una funzione (reale) di più variabili (reali) –che abbia, per un punto di rn, per cui è definita, i differenziali di ordine superiore– sono dati da quei valori di h, k, …, che annullano contemporaneamente d2f e d3f e nei quali d4f è, rispettivamente, minore o maggiore di zero, che si appunta la critica di peano, espressa nell’annotazione n. 133-136 (da lui redatta, così come tutte le altre raccolte ad inizio volume) posta nel testo scritto a nome di angelo genocchi, suo maestro e titolare della cattedra di calcolo infinitesimale, dal titolo calcolo differenziale e principii di calcolo integrale. pubblicato con aggiunte dal d.r giuseppe peano (torino, fratelli bocca, 1884): un trattato che rispecchiava il corso universitario di genocchi e le aggiunte, notevoli, di peano. ratio mathematica, 19, pp. 31-88 40 realizzare, addirittura, un modello plastico (verrà etichettato l’esempio di peano o la superficie di peano o ancora, in germania dove verrà realizzato, fläche von peano) del contresempio stesso: vale a dire della funzione f(x,y)=(y2 – 2px)(y2 – 2qx), ove p>q>0.10 10 peano, infatti, nell’annotazione citata osserva: “non è esatto il criterio enunciato dal serret, calcul, p. 219: «le maximum ou le minimum a lieu si, pour les valeurs de h,k, … qui annulent d2 f et d3 f, d4 f a constamment le signe – ou constamment le signe +». per vedere l’inesattezza di questa proposizione, si consideri p.e. la funzione intera f(x,y)=(y2 – 2px)(y2 – 2qx), ove p>q>0, e fatto x0 = 0, y0 = 0, si avrà f(h,k)=4pqh2 – 2(p+q)hk2 + k4. il sistema dei termini a secondo grado è positivo per tutti i valori di h e k, tolto il valore di h=0, per cui si annullano i termini a terzo grado, e il sistema dei termini a quarto grado è positivo. quindi, secondo il criterio di serret, f(x,y) è minima per x=0. ma è facile assicurarci che questo non è. pongasi invero y2 = 2lx; facendo tendere x a zero anche y tende a zero, e si avrà ( ) ( )( ) .42, 2xqlpllxxf ⋅−−⋅= questa quantità è a nostro arbitrio positiva o negativa, secondoché l è fuori, o dentro all’intervallo (p,q); quindi la funzione f assume in ogni intorno dei valori (0,0) di x e di y valori positivi e valori negativi, ossia valori maggiori e minori di f(0,0)=0, e f non è né massima né minima. lo stesso errore è commesso dal bertrand, calcul, ecc., p. 504; todhunter, calcolo, n. 229, ecc.”. il modello della superficie di peano (o fläche von peano, compresa nella serie xlix, al n° 1, dei modelli editi da martin schilling) è fatto considerando una funzione diversa ma dello stesso tipo, vale a dire la z=f(x,y)=(2x2 – y)(y – x2), anzi, più esattamente, com’è scritto sull’etichetta incollata al modello, è considerata la funzione 200z=(x2 – 10y)(2y – x2) (con (0,0) punto di sella) per rendere meglio evidenti le caratteristiche della superficie stessa. si fa notare che, contrariamente a quanto è affermato in gerd fischer, alla p. 71, vol. i (commentary), dell’opera, citata, essa non è compresa nell’edizione del catalog di m. schilling, anno 1911 (che è composto di 40 serie più una serie iniziale non numerata), né è stato possibile trovare qualche edizione successiva del catalog (che forse non è stata mai stampata) per controllare se la singola superficie o, ragionevolmente, tutta la serie xlix, fosse stata poi inserita. ratio mathematica, 19, pp. 31-88 41 la seconda, successiva, “visione”, molto diversa dalla precedente, è caratterizzata invece dalla sfiducia verso l’uso della comune intuizione sensibile nell’ “accertamento” (o “scoperta”) e, all’occorrenza, nell’ “invenzione” delle strutture matematiche. “accertamento” e “invenzione” fatti allo scopo, rispettivamente, di “unificare” teorie matematiche diverse o usare i “teoremi di struttura” (delle teorie unificate) per saggiare la strutturazione di ulteriori, nuove teorie in fase di ideazione, oppure di precedenti teorie. questo modo di fare matematica –sia detto per inciso– fu impiegato, nell’attuazione del programma di radicale riforma (o, si potrebbe dire, per usare termini consoni a questo discorso, di “ristrutturazione”) dell’insegnamento della matematica, basato sulle cosiddette “mathématiques modernes”, avvenuto, per limitarsi all’europa, specialmente in francia e in belgio, negli anni sessanta del secolo trascorso. programma di riforma attuato con molto rigore, anche a livello di scuola primaria, per cui proprio ciò che, del programma bourbakista, più poteva giovare a un raffinato ricercatore (linguaggio, modo di definire, forma di esposizione, argomenti di studio) si poteva rivelare invece svantaggioso per studenti in tenera o giovane età, dotati di varie propensioni attitudinali e miranti a varie prospettive professionali; studenti interessati, in linea generale, a stabilire un accettabile rapporto con la matematica11 anche attraverso le intorno ai due volumi curati da fischer, corre l’obbligo di avvertire che il vol. ii, costituente il catalogo fotografico, contiene 132 foto di modelli (in alcuni casi è ripreso, con inquadratura diversa, lo stesso oggetto). i modelli fotografati rappresentano campioni che provengono quasi tutti dai dipartimenti di matematica delle università tedesche, in maggioranza da göttingen ma anche da erlangen, heidelberg, münchen, e dal palais de la découverte di parigi. naturalmente, le foto ivi raccolte non riprendono tutti i modelli presenti nel catalog di m. schilling del 1911; mancano, per esempio, i modelli delle superfici dei centri di curvatura, quelli di un’intera serie di superfici sviluppabili, i modelli della fisica-matematica (diffusione del calore, superfici d’onda, ecc.). 11 in questo senso, il più vecchio dei due redattori del presente scritto ricorda un fenomeno curioso che si verificò per alcuni anni accademici –primi anni sessanta del novecento– all’università di napoli. bisogna sapere che scomparso renato caccioppoli (1904-1959) gli succeddette, alla cattedra di ratio mathematica, 19, pp. 31-88 42 multiformi strade dell’ “intuizione” così come, al limite estremo, tante donne (e anche uomini), specialmente del sud d’italia, cui, fino agli anni cinquanta del secolo passato, non veniva data la possibilità neanche di iniziare a frequentare la scuola primaria, si trasmettevano un modo elementare di fare i calcoli oralmente (erano chiamati “i conti alla femminile”), fondato su sottrazioni e addizioni ripetute, continuate divisioni o moltiplicazioni per due, una vaga idea della teoria dei rapporti e proporzioni – questo per i più “raffinati”–, bilanciamenti, e così via. visto dalla parte dei giovanissimi studenti, con l’introduzione spinta del programma bourbakista quasi tutti persero pure la possibilità di ricevere il tradizionale aiuto da parte dei propri genitori, ormai disorientati. il passaggio tra le due “visioni”, rapidamente ricordate, non fu, in italia, così improvviso e radicale. anzi, quando si vadano a scorrere – per considerare un caso, curioso e notevole, che testimonia la compresenza, almeno nella penisola italiana, di vari punti di vista– gli atti della società italiana di matematiche “mathesis”. relazione del congresso di napoli, 13-16 ottobre 1921,12 dove vennero verbalizzati gli interventi dei relatori, saltano agli occhi due “discorsi” molto diversi. vi è quello d’apertura, tenuto nella mattinata della giornata inaugurale, da federigo enriques. e in base all’ampio resoconto che ne dà l’estensore del verbale si viene a sapere: analisi matematica, federico cafiero (1914-1980) mentre l’altra cattedra di analisi era tenuta, già dagli anni precedenti, da carlo miranda (1912-1982). analisi i e ii erano, allora, insegnamenti comuni agli studenti dei corsi di laurea in matematica e in fisica e, anche, a quelli del biennio in ingegneria. ad anni alterni, ciascuno dei due professori partiva col primo anno, svolgendo esclusivamente il corso di analisi i, per tutti gli studenti dei tre corsi di laurea, e completava poi il ciclo, al secondo anno, insegnando, ancora per tutti, analisi ii. il fenomeno curioso consisteva nel fatto che alcuni studenti preferivano rimandare di un anno l’iscrizione ad ingegneria pur di non iniziare il ciclo col professore cafiero il cui stile, molto aderente all’impostazione insiemistica e bourbakista, procurava loro, per ciò stesso, difficoltà nell’apprendimento. 12 «periodico di matematiche», (iv), ii, 1922, p.90 e segg. ratio mathematica, 19, pp. 31-88 43 l’oratore comincia a tratteggiare la natura del matematico in cui il pubblico vede di solito uno sviluppatore di formule e un calcolatore di macchine, e che piuttosto deve essere ravvicinato al poeta e al filosofo; e si ferma a mettere in luce il rapporto storico di parentela e di interdipendenza che appare fra le matematiche e la filosofia nello sviluppo del pensiero europeo. dagli stessi atti emerge pure che nel medesimo giorno –di pomeriggio però– roberto marcolongo (1862-1943), che era riuscito ad allestire, all’università di napoli, un istituto di meccanica razionale tra i primi in europa per la dotazione di libri, modelli e strumenti matematici,13 teneva una relazione, sul materiale didattico d’insegnamento,14 d’ispirazione molto diversa da quella di enriques. nel resoconto è infatti riportato: il prof. marcolongo comincia anzitutto collo sfatare la leggenda che, pel loro insegnamento, i matematici non abbiano bisogno che della lavagna e del gesso. insiste invece sulla necessità che in ogni scuola, di qualsiasi grado, accanto al gabinetto di fisica, di chimica, di scienze naturali, ecc., vi debba essere quello di matematica. anche il professore di matematica deve fare, in varia misura e in varie maniere, delle vere e proprie esperienze; deve avere a propria disposizione e valersene costantemente nell’insegnamento, libri, disegni, modelli, macchine, tavole matematiche, in modo da agevolare lo studio della matematica e renderlo più attraente e più utile. […] perciò il marcolongo fa una lunga e minuta descrizione di tutto quanto l’industria più raffinata, la meccanica di alta precisione e l’ingegno di abili costruttori, ha saputo produrre pei modelli matematici e per gli strumenti. e, nell’insieme dei modelli consigliati, marcolongo poneva pure gli anaglifi geometrici messi a punto da henri vuibert, la cui tecnica di realizzazione fu da questi descritta nell’opuscolo avente per titolo les anaglyphes géométriques, pubblicato nel 1912.15 gli anaglifi –che permettevano di formare una sorta di modelli molto meno “materiali”– facevano acquistare alle figure geometriche, disegnate sul 13 si veda r. marcolongo, quaranta anni di insegnamento, napoli, stabilimento industrie editoriali meridionali, 1935, p. 27 e segg. 14 come annunciato negli atti, il testo completo della relazione di marcolongo, dal titolo modificato in materiale didattico ed esperienze nell’insegnamento, fu riportato nel «giornale di matematiche», lx (13° della 3a serie), 1922. 15 paris, librairie vuibert, boulevard saint germain, 63, 32 pp. ratio mathematica, 19, pp. 31-88 44 piano di un foglio da disegno, un loro rilievo, vale a dire che le figure si trasformavano in oggetti (virtuali) i quali sembravano staccarsi dal foglio ed elevarsi in tre dimensioni (effetto rilievo). una tecnica che permette di vedere anche gli elementi interni di una figura solida e le linee di costruzione della stessa e che può essere considerata come una delle prime che abbiano contribuito al sorgere, al tempo moderno, della realtà virtuale (virtual reality, vr), oggi fiorente settore di applicazione dell’informatica.16 in questi anni, poi, grazie ai calcolatori, a software matematici, alla computer grafica, è indiscussa l’importanza che, nelle scienze matematiche e nel settore della geometria, assumono immagini e realtà virtuale quali strumenti per la didattica e per la ricerca (oggi vi è un rapporto più equilibrato tra “formule” e “figure”); in particolare, si assiste ad un ritrovato interesse, di estensione internazionale, verso quei tipi di modelli e strumenti a cui aveva pure accennato marcolongo, compreso il semplice blocco di fogli di carta quadrettata (3d drawing pad), a linee rosse e verdi, dove si può tracciare, magari con una penna con l’inchiostro di colore nero, una figura su di un foglio e poi osservarla in rilievo mediante gli annessi occhialini (anaglittoscopi) aventi i due “vetri” (che in realtà possono essere fatti anche di carta trasparente) colorati, rispettivamente, in rosso e verde. 16 la visione stereografica e gli anaglifi sono molto utilizzati oggi in un’ampia varietà di applicazioni concernenti geometria, strutturistica chimica, architettura, cinema, realtà virtuale, conservazione dei beni culturali (in quest’ultimo campo, recente è, per esempio, la realizzazione, presso il dipartimento di matematica e informatica dell’università degli studi di salerno, di un filmato in cui, attraverso la tecnica di visualizzazione stereoscopica, lo spettatore diventa visitatore condotto a girare per il sito virtuale –la sua origine reale risale all’età romana, al tempo dell’eruzione del vesuvio del 79 d.c.– di moregine presso pompei). al riguardo si può vedere n. palladino, gli anaglifi di vuibert. origine storica e applicazioni in didattica basata sui modelli di superfici matematiche (preprint n. 19 – 2008. dipartimento di matematica e informatica – università degli studi di salerno), in corso di pubblicazione per il rendiconto dell’accademia delle scienze fisiche e matematiche di napoli; l’ultima parte dell’articolo, § 5, è dedicato alla generazione degli anaglifi nella realtà virtuale. ratio mathematica, 19, pp. 31-88 45 lo scopo di questo articolo, si è detto, è quello di fornire aggiornata documentazione della vicenda storica (e, inoltre, offrire delle riflessioni interpretative) dei modelli matematici plastici, costruiti in europa, venutasi a sviluppare intensamente per poco più di mezzo secolo, tra il 1870, circa, e il 1930, anno, quest’ultimo, in cui la produzione subì un blocco pressoché totale anche per gli sconvolgimenti dovuti alla prima guerra mondiale e per la mancanza di commesse conseguente alla depressione economica scoppiata nel 1929.17 prima di pervenire alla seconda parte dell’articolo, dove l’attenzione verrà orientata sui “fondi” di modelli matematici che si trovano presso le “antiche” sedi universitarie italiane –delle quali si darà un elenco–, informazioni e approfondimenti saranno forniti sulle iniziative più importanti che si ebbero in europa: francia (essenzialmente parigi), regno unito (principalmente londra e manchester), germania (gli istituti di matematica dei politecnici di monaco di baviera, darmstadt, karlsruhe e poi l’università di gottinga). luoghi dove ebbero origine, con slanci di varia potenza e con uno sviluppo di maggiore o minore estensione, l’ideazione e la realizzazione dei modelli matematici e dai quali proviene pressappoco la totalità degli esemplari oggi conservati nei fondi museali delle “antiche” università italiane. (quasi tutti i posti d’europa ora menzionati sono stati raggiunti dagli autori del presente articolo, recentemente essi hanno visitato anche il nucleo di modelli oggi 17 vi è un brano di lettera, del 1932, indirizzata da martin schilling all’istituto di matematica dell’università di gottinga (riportato in gerd fischer, mathematical models/mathematische modelle, cit., ii –catalogo fotografico–, pp. ix-x), in cui s’informa che nell’ultimo anno non era stato prodotto alcun modello, che il primo modello costruito dopo la fine della guerra era stato la superficie di peano, che vi erano nuovi modelli progettati, ma che in conseguenza delle cattive condizioni del mercato la produzione dei modelli era stata sospesa: “[…] dass in den letzten jahren keine neuen modelle erschienen sind. das erste nach dem kriege ist das modell der peano-fläche. es sind verschiedene neue modelle in vorbereitung, die wir aber infolge der schlechten und unübersichtlichen geschäftslage immer wieder zurückgestellt haben”. ratio mathematica, 19, pp. 31-88 46 presente al mathematisches institut della ruprecht-karl-universität di heidelberg). va subito precisato che, purtroppo, in italia non si riuscì a destinare uno spazio istituzionale a favore di questa attività. ciò accadde nonostante la richiesta effettuata, nel 1883, da giuseppe veronese (1854-1917), professore di geometria analitica all’università di padova, di allestire un laboratorio nazionale (“un atelier come quello di monaco”, egli scrive). veronese ebbe, allo scopo, contatti diretti col ministro pro tempore della pubblica istruzione, il medico guido baccelli (1832-1916), e fu sostenuto da illustri professori quali erano brioschi, d’ovidio, de paolis, dini e bertini. sull’episodio si ritornerà più avanti nel corso dell’articolo. vi furono invece delle iniziative locali che produssero anche qualche modello significativo e di buona fattura, tuttavia si trattava di realizzazioni isolate, eseguite da singoli studiosi (un esempio è la cuffia di beltrami18 per le geometrie non euclidee) o da studenti su indicazione dei loro professori, e destinate all’ “uso interno”. forse, in ambito italiano, si può ritenere un’eccezione, sia perché dotata di una sua propria organicità e sia per la finissima fattura, la piccola raccolta sviluppata da alfonso del re (1859-1921) all’università di napoli nell’ambito del gabinetto di geometria descrittiva annesso alla corrispondente cattedra di cui egli era titolare.19 (nella raccolta del re i telai –chiamati, in generale, anche 18 cfr. a.c. capélo, m. ferrari, la «cuffia» di beltrami: storia e descrizione, «bollettino di storia delle scienze matematiche», ii, 1982, pp. 233-247. nell’articolo gli autori fanno notare che “[…] per quanto riguarda la superficie pseudo sferica di tipo ellittico non è possibile ripiegare il modello in questo modo senza effettuare un taglio”. 19 ne dà notizia lo stesso del re nell’opuscolo programma del corso e programma di esame per l’anno scolastico 1906-1907 dove è presentato in appendice l’elenco dei modelli geometrici eseguiti dagli allievi della scuola di geometria descrittiva dell’università di napoli dal 1901 al 1906. sono 36 modelli, dei quali 31 in legno e filo, 3 in legno e ottone, 2 in legno, ottone e crine di cavallo. l’opuscolo riporta ancora altri due elenchi; il primo riguarda 13 modelli acquistati [nel 1901-1902] dal prof. a. del re e donati alla scuola di geometria descrittiva, il secondo i modelli acquistati [nel 1905] dal prof. a. del re sui fondi assegnati alla scuola di geometria ratio mathematica, 19, pp. 31-88 47 “castelli”– che tenevano i fili, in fibra naturale, delle superficie rigate rappresentate, erano in legno lavorato artisticamente con la tecnica del traforo20). solo molto più tardi, con gli anni cinquanta del secolo scorso, luigi campedelli (1903-1978) –professore all’università di firenze–, in seguito ad un deliberato dell’assemblea generale dei soci dell’umi – unione matematica italiana (svoltasi a taormina, nel 1951) produsse una serie fatta di un numero molto limitato (cinquanta) di modelli,21 dei quali quarantatre in gesso, ottenuti “ricalcandoli” (in ciò descrittiva: sono tre intere serie del catalog di m. schilling, e cioè la serie xi (fatta di 8 modelli in filo metallico raffiguranti la relazione tra le singolarità di una curva dello spazio e le singolarità delle proiezioni della medesima curva su tre piani ortogonali tra loro –acht draht-modelle über die rückkehrelemente der projectionen einer unebenen curve von denen der curve selbst–), la xiii (formata da 10 modelli in fili di fibra naturale di superfici rigate del quarto ordine –zehn faden-modelle der regelflächen 4. ordnung–) e la xxviii, qui già citata in nt. 2 (sono sei modelli in fil di ferro di curve cubiche dello spazio considerate per il loro impiego in ottica fisiologica, l’ultimo dei quali è l’horopter di cui si è detto). nell’annuario degli istituti scientifici italiani, diretto da s. pivano, bologna/roma, n. zanichelli/athenaeum, vol. ii, 1920, p. 378, a proposito del “gabinetto di geometria descrittiva e scuola di disegno” diretto da a. del re, è annotato: “fondato nel 1901 dall’attuale direttore [a. del re]. è ricco di modelli geometrici, eseguiti dagli allievi, od acquistati, e di apparecchi per fotogrammetria, per proiezioni, per disegno, ecc.”. 20 uno di questi modelli (l’unico sopravvissuto), rappresentante una superficie rigata, del quarto ordine, limitata da due coniche (rispettivamente un’ellisse e un’iperbole con i suoi due rami) è riprodotto fotograficamente e riportato alla tav. 13 inserita nell’articolo di f. palladino, antichi strumenti e modelli matematici conservati a napoli e a pisa, «physis», vol. xxix (1992) nuova serie, pp. 833-847. 21 essi furono classificati da campedelli in base ai gruppi (tra parentesi tonda è indicato il numero di pezzi per ciascun gruppo): a) quadriche (5); b) curve gobbe del terzo ordine tracciate su cilindri quadrici (4 + 1); c) superfici cubiche non rigate (19); d) superfici rigate gobbe del terzo ordine (4); e) superfici del quarto ordine (6); f) superficie dell’ottavo ordine (1); g) superfici pseudo sferiche (3); superfici rigate: paraboloide iperbolico (con il doppio sistema delle generatrici), iperboloide a una falda (con il ratio mathematica, 19, pp. 31-88 48 avvalendosi dell’opera di artigiani fiorentini) dagli originali, realizzati in germania nell’ottocento, che l’università di pavia aveva acquistati, per la maggior parte, dall’editore ludwig brill, e che aveva avuto la fortuna e la capacità di conservare (a tutt’oggi essi sono ancora a pavia).22 a questi modelli in gesso egli ne aggiunse altri sette in filo di fibra sintetica (filo di nylon di diversi colori fornito gratuitamente dalla rhodiatoce di milano) con il “castello” in ottone. prima di chiudere con il suo impegno, campedelli riprodusse, molto probabilmente, ancora qualche ulteriore modello di superficie rigata, per esempio la “superficie di uguale pendenza” (si è trovata all’università di padova23), di cui si verrà più oltre a parlare in occasione delle collections muret. per quanto nei bollettini-umi non sia data notizia delle fonti ispiratrici utilizzate da campedelli, si è potuto sapere che i modelli originali, da cui egli partì per riprodurre le nuove copie in gesso, furono, come si è accennato, quelli provenienti dall’università di pavia grazie alla dichiarazione fatta, alcuni anni fa –199224–, a franco palladino dalla dott. cesarina dolfi (preside in pensione, negli anni cinquanta collaboratrice di campedelli); mentre si è potuto poi accertare che per i modelli realizzati in filo di nylon (compresa a quanto sembra la “superficie di uguale pendenza”) egli si ispirò al catalogo –una copia del quale si è ritrovata presso il dipartimento di doppio sistema delle generatrici e il cono asintotico); cinque tipi di elicoidi (si veda «bollettino dell’unione matematica italiana – bumi», (iii), vii (1952), pp. 221-222, 362, 465-467; e bumi, (iii), viii (1953), p. 229; bumi, (iii), x (1955), p. 300; bumi, (iii), xi (1956), p. 302). 22 cfr. uno specimen dei giacimenti italiani di modelli e strumenti matematici: il nachlass dell’università di pavia, volumetto di 60 pagine pubblicato per le <<memorie dell’istituto lombardo di scienze e lettere>>, milano, 1997, pp. 315-374 23 cfr. f. palladino, il fondo di strumenti e modelli matematici antichi dell’università di padova e l’iniziativa di giuseppe veronese per un laboratorio nazionale italiano, università di padova – dipartimento di matematica pura ed applicata, padova, 1999, pp. 55-56. 24 al tempo dell’uscita del dossier didattica: dagli strumenti ai modelli, inserito in lettera pristem, milano, università commerciale “l. bocconi”, 6, novembre 1992. ratio mathematica, 19, pp. 31-88 49 matematica “u. dini” dell’università di firenze– dal titolo lehrmodelle für mathematik curato dalla ditta rudolf stoll kg (indirizzo: berlin no 18, oderbuchstrasse 8-14, deutsche demokratische republik –ovvero repubblica democratica tedesca, o germania est come allora pure si diceva–, stato socialista esistito, si ricorderà, fino alla riunificazione germanica avvenuta nel 1990) che, a partire dagli anni cinquanta del xx secolo, cominciò a raccogliere e vendere i modelli matematici prodotti nel “ii mathematischen institut der humboldt-universität” di berlino (est) sotto la guida del direttore dell’institut, kurt schröder (1909-1978). gli oggetti del catalogo stoll sono descritti in tre lingue, tedesco, inglese e francese; per esempio la “superficie di uguale pendenza” (böschungsfläche, in tedesco, sloping surface, in inglese, surface d’égale pente, in francese) avente per curva direttrice un’ellisse è catalogato al posto 401/93. la serie di modelli campedelli, replicata per un certo numero di copie, fu venduta a quelle sedi universitarie (messina, bari, catania, milano-università statale, per citarne alcune) che ne fecero richiesta; ovviamente una copia completa si trova pure all’università di firenze che era sprovvista di modelli ottocenteschi essendo una “giovane” sede universitaria (diversamente da pisa –“antica” università della toscana–, dove vi era una copiosa collezione, ora perduta). per poter gestire questa iniziativa fu attivata una “sezione modelli” dell’umi. 2. alcuni modelli, progenitori di quelli menzionati, al modo di esempi, al precedente pf. 1, cominciarono ad essere costruiti in europa (a quel tempo il continente europeo, per quanto formato da nazioni frequentemente “l’una contro l’altra armata” aveva, considerato nel suo insieme, l’assoluto primato mondiale sotto i profili politico, economico, scientifico, industriale, militare), in rapporto alle esigenze della geometria descrittiva e, successivamente, proiettiva. l’inizio è databile addirittura alla prima metà dell’ottocento, quando in francia (principalmente a parigi) vennero alla luce le prime collezioni di modelli. ratio mathematica, 19, pp. 31-88 50 superata la prima metà del secolo xix, si assiste poi, accanto all’accrescersi della produzione francese che coinvolgeva vari campi delle scienze matematiche (con predilezione verso una tipologia che si potrebbe dire da école polytechnique), alla forte attenzione che, presso vari centri culturali del regno unito di gran bretagna e irlanda, veniva prestata ai modelli matematici (per quanto, in questo regno, la produzione era piuttosto orientata verso la realizzazione di strumenti, tecnologicamente raffinati, ad uso delle scienze matematiche). pochi anni dopo l’unificazione della germania (avvenuta nel 1871 con la formazione del deutsches reich) l’iniziativa consistente nella realizzazione dei modelli matematici assume un grande rilievo scientifico, istituzionale e didattico raggiungendo una propria ben delineata autonomia. viene organizzata e ulteriormente incentivata la produzione degli istituti matematici, fisici, tecnico-meccanici e geodetici esistenti presso le università e i politecnici germanici e, quindi, una ditta (una casa editrice: quella di ludwig brill, appositamente fondata a darmstadt nel 1877 su stimolo di felix klein –1849-1925– e alexander brill –1842-1935–,25 poi continuata, dal 1899, ad opera di martin schilling nella città di halle an der saale, in un primo momento, e a lipsia successivamente) viene a fungere da centro di raccolta (dove però si provvede a dare, in certi casi, un’ “ultima mano” alla preparazione dei modelli costruiti in gesso) con un catalogo unico suddiviso per serie formate quasi sempre da modelli riconducibili al medesimo tema scientifico, ideati e, spesso, costruiti presso una data sede sotto la guida di un dato professore. catalogo che non si presentava come semplice elencazione di pezzi ma, arricchito com’era da puntuali esposizioni dell’argomento coinvolto e dai rimandi ai saggi scientifici ispiratori, rappresentava, nelle sue varie edizioni, la summa descrittiva del “sistema dei modelli plastici” che fiorì in germania per circa quarant’anni.26 la distribuzione dei 25 alexander e ludwig brill erano fratelli. 26 il nome dato al catalogo, che si è già avuto modo di menzionare, era: catalog mathematischer modelle für den höheren mathematischen unterricht. esso era diviso in due parti. nella prima, i modelli erano ordinati per serie (nell’edizione, notevolmente accresciuta, del 1911 le serie erano 40 ratio mathematica, 19, pp. 31-88 51 più una iniziale non numerata); nella seconda parte, i modelli erano omogeneamente raggruppati tenendo conto del loro legame scientifico. così, per esempio, nella sezione xi, il cui titolo è functionentheorie, della seconda parte del catalog, vengono, tra gli altri, compresi tre modelle riemann’sche flächen (modelli di superfici di riemann, realizzati in gesso), appartenenti alla serie xvii, presentata nella prima parte del catalogo; assieme a 16 modelle zur darstellung von functionen einer complexen veränderlichen (pure realizzati in gesso; tra questi sedici vi è, per esempio, il modello rappresentante, insieme, la parte reale e la parte immaginaria – real und imaginärteil– di 12 −= zw , dove w=u+iv e z=x+iy, e, ancora, analogamente, quello riguardante la funzione: w= 4 42 log π π π − + z z appartenenti alla serie xiv, della parte prima del catalogo, e così via). sulle edizioni del catalog si rimanda a f. palladino, uno specimen dei giacimenti italiani di modelli e strumenti matematici: il nachlass dell’università di pavia, cit., e a f. palladino, il fondo di strumenti e modelli matematici antichi dell’università di padova, cit., ivi note 5 e 6. al riguardo, si vuole comunque ricordare ora qualche dato di particolare importanza. delle edizioni del catalog le notevoli e più diffuse sono quelle del 1903 (è la sesta edizione, conta 29 serie di modelli più una serie iniziale, o serie zero, non numerata, di modelli costruiti in cartone) e quella del 1911 (è la settima edizione: conta quaranta serie più quella iniziale che è uguale a quella del 1903), tutte e due allestite da martin schilling. del catalog, curato da l. brill, non si è trovata alcuna copia a stampa in tutti i luoghi d’italia e d’europa che si è avuto modo finora di visitare né, a quanto sembra, altri studiosi hanno comunicato di averlo visto fino ad oggi. vi è però un’esplicita menzione della sua esistenza nell’antico inventario dei modelli di pavia (in f. palladino, uno specimen, cit., p. 320, nt.10, dove è riportata la seguente specificazione: “il catalogo cui si riferiscono i numeri segnati dopo il nome di ciascun modello è il catalog math. modelle von brill, 1885, 2a parte, da pag. 27 a 48”). una generica menzione concernente un’edizione brill del 1882 è fatta pure in gerd fischer, mathematical models/mathematische modelle, cit., ii, p. v, dove si accenna a un catalog der modellsammlung des mathematischen instituts der kgl. technischen ratio mathematica, 19, pp. 31-88 52 modelli (specialmente di quelli in gesso, i quali erano sicuramente superiori, non solo per le teorie matematiche di cui erano interpreti, ma pure per la qualità del materiale e della fattura, ai consimili delle altre ditte straniere, in prima linea francesi), si irradia per la germania, successivamente per l’europa (con varia intensità dalla spagna e dal portogallo alla russia, dalla svezia all’italia) spingendo nell’ombra, tra l’altro, le precedenti, diverse, realizzazioni. verso la fine dell’ottocento, i modelli editi da martin schilling massicciamente si diffondono anche negli stati uniti d’america dove hochschule münchen, aufgestellt im januar 1882 unter leitung von prof. a. brill. a questo punto, bisogna sottolineare, che le serie di modelli, via via che venivano prodotte, erano accompagnate dall’edizione separata di veri e propri saggi illustrativi riguardanti il back ground costituito dalle conoscenze matematiche da cui si era tratto ispirazione. è da ritenersi un piccolo colpo di fortuna l’aver comunque ritrovato, nel 1997, al politecnico di monaco di baviera, dagli autori del presente articolo, la raccolta completa, assemblata nel volume dal titolo abhandlungen zu den durch die verlagshandlung von l. brill in darmstadt veröffentlichten modellen für den höheren mathematischen unterricht, dei saggi relativi alle prime ventuno serie edite da l. brill. il volume non reca la data di edizione ma sulla scheda bibliografica è scritto che fu approntato nel 1892 circa, a darrmstadt. sulla stessa scheda sono riportati i nomi (peraltro noti a ragione della loro attività con la quale ebbe inizio la realizzazione di modelli tra münchen e darmstadt) di quelli che furono probabilmente i curatori della raccolta di saggi: a. brill, j. bacharach, l. schleiermacher, w. dyck, k. rohn; inoltre, l’ex libris è costituito dal segno lasciato da un timbro e dal quale si legge: k.[öniglisch] b.[ayerische] techn. hochschule in münchen – mathem. institut. a questi saggi intende riferirsi gerd fischer quando scrive (si veda mathematical models/mathematische modelle, cit., ii, p. ix): “responding to «instigation from munich» the publishing company l. brill in darmstadt took over the sales of models. they were put together in series, each being accompanied by mathematical explanation. unfortunately it seems that the latter have been lost; at least they cannot be found at the institutes where the collections presently abide.” un ulteriore, curioso, dato si vuole fornire. accanto ai nomi di a. brill, j. bacharach, ecc., elencati, bisogna aggiungervi, tra i primi realizzatori di modelli, quello di rudolf diesel (1858-1913) inventore del motore a combustione interna che ancora oggi è conosciuto col suo nome. ratio mathematica, 19, pp. 31-88 53 il consolidarsi delle prime università della costa atlantica era accompagnato dal sorgere di numerose altre nella direzione degli stati che si incontravano andando verso l’interno e, successivamente, verso il lontano ovest. come si può immaginare, il fenomeno della ideazione e costruzione dei modelli matematici è una metafora della contestuale storia civile e politica, sia europea che, sostanzialmente, del “mondo occidentale”, con l’affacciarsi o l’alternarsi, in posizione di primo piano, di diversi potenti stati nazionali. alcuni segnali che giungono dagli stati uniti d’america, tra la vigilia dello scoppio della prima guerra mondiale e immediatamente dopo la conclusione della stessa, permettono di scandire significativamente la cronologia dell’ultimo segmento del percorso ora delineato e di accorgersi della discesa in campo, anche in questo settore, dell’ “america”. il primo segnale è rappresentato da una recensione, redatta da r.c. archibald (1875-1955), professore di matematica alla brown university di providence –rhode island– il quale, dopo gli studi universitari compiuti negli u.s.a., aveva pure frequentato, come spesso accadeva nell’ottocento ai giovani americani, le università tedesche. archibald era stato a berlino, nel 1898-’99, e a strasburgo, nel 1900, dove aveva conseguito il grado di ph.d. la recensione apparve nel bulletin of the american mathematical society del 1914,27 sotto il titolo di mathematical models, e in essa archibald elencava, commentandoli, i quattro maggiori cataloghi del momento; e cioè: il catalog mathematischer modelle für den höheren mathematischen unterricht di martin schilling, edizione 1911, che si è citato nelle note precedenti. l’abhandlugen zur sammlung mathematischer modelle, in zwanglosen heften herausgegeben von hermann wiener, leipzig, verlag von b.g. teubner. 1. heft von h. wiener, 1907; 2. heft von p. treutlein, 1911. il verzeichnis von h. wieners und p. treutleins sammlung mathematischer modelle für hochschulen, höhere lehranstalten und 27 vol. 20 (1914), pp. 244-247. ratio mathematica, 19, pp. 31-88 54 technische fachschulen. zweite ausgabe mit 6 tafeln, leipzig und berlin, verlag von b.g. teubner, 1912. l’illustrierter spezialkatalog mathematischer modelle und apparate, entworfen von g. koepp und anderen bewährten fachmännern, new york city, eimer and amend.28 si tratta del meglio della produzione specialistica internazionale – comprendente pure il settore della didattica elementare– che appartiene però tutta quanta alla germania (sulle realizzazioni di hermann wiener e josef peter treutlein –1845-1912– si ritornerà) dove si registra l’impegno, in questo settore, anche del grande e rinomato editore g.b. teubner di lipsia. tuttavia è sintomatico che l’ultimo dei titoli elencati da archibald pur essendo ancora in tedesco sia presentato da un editore americano: la “eimer & amend” è un’importante ditta produttrice di apparecchi, chimici e fisici, per i laboratori di ricerca e per l’industria, distribuiti e venduti con il supporto dei relativi cataloghi. il secondo, successivo, segnale è rappresentato dall’autonoma progettazione, all’università dell’illinois (nel laboratorio di matematica installato nell’ambito del dipartimento di matematica), avente sede nella città di urbana, di modelli fatta su iniziativa di arnold emch (1871-1959), professore associato di matematica. emch ne dà dettagliata esposizione in tre numeri dell’university of illinois bulletin (che usciva con cadenza settimanale stampato dall’university of illinois press): il no. 12, vol. xviii, del 22 novembre 1920 (mathematical models. i series); il no. 42, vol. xx, del 18 giugno 1923 (mathematical models. ii series); il no. 35, vol. xxii, del 27 28 su questo catalogo non si è riuscito ad avere altre notizie. g köpp compare invece numerose volte (pp. 243, 244, 245, 246, 257, 258) nel famoso katalog, del 1892, curato da walther dyck in occasione dell’esposizione di monaco di baviera, per quanto riguarda l’ideazione di modelli per l’insegnamento elementare della planimetria, stereometria (compresi solidi stellati), trigonometria, geometria descrittiva e sezioni di quadriche, modelli eseguiti dall’istituto per il materiale didattico “j. ehrhardt & co.” di bensheim (nel land dell’hessen –assia–, in germania). ratio mathematica, 19, pp. 31-88 55 aprile 1925 (mathematical models. iii series).29 nell’introduzione alla prima serie emch precisa che sono stati o verranno realizzati 29 la prima serie presenta 18 “articoli” prodotti e messi in vendita. le novità di maggiore interesse appaiono essere le cauchy surfaces, così definite: “if we consider a function of a complex variable w=f(ξ)=u+iv, and at every point (x,y) erect a perpendicular to the complex ξ-plane equal to the value of u2+v2 at each point, et denote this value by z, we obtain an equation f(x,y,z)=0 which defines a so called cauchy surface. this surface is the locus of the endpoints of those perpendiculars.” vengono proposte due superficie di cauchy, rispettivamente di equazioni 1 1 + − = ξ ξ w , che è una quintica, e 13 −= ξw che è una sestica. il diciottesimo “articolo” è un cinematographic film of poncelet polygon accompagnato dalla corrispondente didascalia: “continuosly appearing movement of a triangle remaining inscribed and circuscribed to two fixed circles respectively.” (un “poligono di poncelet” è un poligono di qualsiasi numero di lati che può essere contemporaneamente inscritto in una curva del second’ordine -una conicae circoscritto ad un’altra dello stesso ordine. al riguardo si può consultare la seguente pubblicazione, a carattere storico: i poligono di poncelet. discorso pronunciato nell’università di genova da g. loria in occasione del solenne accoglimento a dottore aggregato della facoltà di scienze, torino, stamperia reale della ditta g.b. paravia e c., 1889). la seconda serie è fatta di cinque “articoli” che vanno dalla peano surface (espressa dalla quartica z=λf(x,y)=(y2 – 2px)(y2 – 2qx): il modello è realizzato però ponendo λ=1/10, p=2,5, q=0,5) a una curva, una sestics of genus 4, tracciata su di una sfera di vetro trasparente, ecc. della terza serie, composta di sei “articoli”, il più curioso è sembrato essere proprio l’ultimo di essi: models of the symmetric substitution group of order 24 (g24). dopo la breve presentazione dell’argomento: “the symmetric substitution group of n elements of order n! may be represented geometrically in a projective space of n-1 dimensions and this lends to important properties of certain geometric forms. the author has recently considered some of these applications, principally with reference to the g6 and g24. the four elements, numbers, x1, x2, x3, x4 of the g24 are chosen as projective coordinates of a pointing 3-space”, viene enunciato lo specifico teorema interpretato dal modello, che è realizzato in filo. ratio mathematica, 19, pp. 31-88 56 soltanto modelli che non sono disponibili mediante altri cataloghi e che “among the projects considered for future work are designs for stereoscopic and ordinary lantern slides, cinematographic films and charts of mathematical figures”: disegni stereoscopici, proiettori di diapositive, filmati cinematografici, fanno pensare all’agilità, tipicamente americana, nell’utilizzare ampiamente tutte le forme di rappresentazione e comunicazione, comprese naturalmente quelli di recentissima concezione. 3. dopo l’excursus fatto al paragrafo precedente, si vuole ora guardare più da vicino al fenomeno della ideazione e costruzione di modelli matematici nel “vecchio continente”. per quanto riguarda la francia, sono da segnalare la rilevante iniziativa, intrapresa negli anni venti del xix secolo, da irmond bardin –1794-1867– (socialista sansimoniano, egli aggiunse al proprio nome quello di libre e pare che fosse conosciuto anche come bardin de la moselle), allievo, in parigi, dell’école polytechnique poi professore di disegno e fortificazioni presso la stessa scuola oltreché “libero docente” di un corso di geometria descrittiva e di scienze applicata all’industria (sotto questo aspetto collaborò con jean-victor poncelet –1788-1867–) per la costruzione di modelli destinati allo studio della geometria descrittiva (rappresentanti “solidi compenetrati”) e, all’incirca nello stesso tempo, l’iniziativa di théodore olivier (1793-1853), anch’egli allievo dell’école polytechnique e poi professore di geometria descrittiva al conservatoire national des artes et métiers della capitale francese, che progettò una serie di modelli in filo di superfici rigate. la novità espressa dai modelli di olivier consiste principalmente nel fatto che molti di essi erano a conformazione variabile, non fissa come quelle, dei modelli simili, creati in precedenza da gaspard monge (17461818) allorquando fu aperta in francia l’école polytechnique. per esempio, nel caso dell’iperboloide a una falda si potevano ottenere, “storcendo” la configurazione, le posizioni limiti raffiguranti, rispettivamente, un cono o un cilindro, e nel caso di una curva dello spazio, ottenuta dall’intersezione di due superfici realizzate in fili, si ratio mathematica, 19, pp. 31-88 57 potevano, mediante anellini colorati entro cui si facevano passare i fili nei “punti d’intersezione”, evidenziare la curva stessa e i successivi profili che essa veniva ad assumere con lo slittamento degli anellini per effetto dello “storcimento” delle superfici generatrici. gli accorgimenti tecnici ora descritti furono fatti propri dagli autori che vennero dopo e che realizzarono pezzi analoghi conferiti al catalog brill/schilling come attestano alcuni modelli presenti, per esempio, a padova e pavia. una copia della serie olivier (una quarantina di oggetti) fu donata da olivier allo stesso conservatoire e sono oggi esposti nella sezione musée des arts et métiers.30 essi furono materialmente costruiti dalla ditta “pixii père et fils, constructeurs d’instruments de physique, france paris” (che veniva ad essere l’ “auteur materiel”), come tenne a specificare la vedova di olivier. la signora volle, infatti, al momento in cui vendeva, nel 1855, i circa 50 modelli (costituenti una seconda serie posseduta personalmente dal marito) a william m. gillespie, che li portò negli stati uniti d’america, che a ciascuno pezzo fosse applicata, mediante una targhetta, questa annotazione completata dalla complementare sull’ “auteur intellectuel: olivier théodore, professeur, france paris”. l’attività di olivier fu proseguita da fabre de lagrange del quale si dirà nel dare uno sguardo alla realtà del regno unito. di ampia portata e molto conosciuta in europa fu l’impresa, innestatasi su quella di irmond bardin, sviluppata da charles muret, ingegnere della città di parigi (vincitore della medaglia d’oro all’esposizione mondiale di anversa del 1885 per un modello plastico del canale interoceanico di panama), autore di alcune collections delle quali fu edito un catalogue diffuso dall’editore parigino charles delagrave: in una lettera, inviata dalla capitale francese in data 18 agosto 1871, dal muret al matematico eugène catalan (1814-1894), 30 cfr. w.c. stone, the olivier models, schenectady (n.y. u.s.a.), friends of the union college library, 1969 e web-site: www.union.edu/olivier; inoltre l’articolo, siglato by w.m.g., dal titolo: catalogue of the recent additions to the engineering models of union college, schenectady; e ancora il sito: www.math.usma.edu/people/rickey/dms/oliviermodels-france.htm ratio mathematica, 19, pp. 31-88 58 allora professore a liegi (catalan è oggi ricordato, in questo settore, per l’ideazione di una superficie minimale, di cui fu costruito un modello in gesso, conosciuta comunemente col nome del suo ideatore31), si ha modo di constatare che il catalogue (di cui fino ad oggi non è stato possibile trovare un’eventuale edizione a stampa) comprendente le sue realizzazioni contava almeno 323 pezzi.32 le 31 descritta da catalan in mémoire sur les surfaces dont les rayons de courbures en chaque point sont égaux et les signes contraires, «comptes rendus des séances de l’académie des sciences de paris», xli (1855), pp. 1019-1023. l’equazione parametrica di questa superficie, espressa utilizzando le coordinate polari del piano (r,φ), è: x= asin2φ – 2aφ + 1/2av2 sin2φ; y= –acos2φ – 1/2av2 cos2φ; z= 2avsinφ; dove v= 1/r – r. 32 la lettera, che si trova nella corrispondenza epistolare di catalan custodita alla bibliothèque générale de l’université de liège, è citata pure in f. palladino, il fondo di modelli e strumenti matematici antichi dell’università di padova, cit., pp. 5-6. nello scritto di muret si parla non solo dell’esecuzione, che lui va facendo, del modello della “surface cyclotomique”, progettata da catalan, ma vi è un post scriptum che dice: “je joins à ma lettre un catalogue avec supplément. depuis que ce dernier a paru, j’ai construit: 1. l’hélicoïde gauche, en fils de soie montés sur bronze, avec son paraboloïde de raccordement. […] 2. la surface d’égale pente de mr de st venant dont il est question au n° 323 de mon catalogue […].” la surface d’égale pente è generata da una retta (generatrice) che si muove appoggiandosi ai punti di una curva piana (curva direttrice, per esempio un’ellisse) mantenendosi nel piano verticale condotto per la normale alla curva direttrice (per esempio all’ellisse) in ciascun punto di contatto e, inoltre, incontrando il piano della stessa curva direttrice sotto un angolo costante. un modello materiale, semplice e naturale, di “superficie di uguale pendenza” si ottiene scaricando, su di un ampio pavimento granaglia dalla sacca di un silos oppure sabbia da un imbuto: il mucchio (la pila) che viene a generarsi ha la forma di questa superficie (con buona approssimazione è un cono circolare retto). sull’attività di muret nella costruzione di modelli è fatta citazione in un articolo contenuto nella raccolta di lavori matematici, mélanges mathématiques, tome troisième, 1888 (raccolta apparsa in «mémoires de la société royale des sciences», bruxelles, f. hayez, (2), tome xv, 1888); nell’articolo, il ccxxi, che ha per titolo sur une classe de surfaces gauches ratio mathematica, 19, pp. 31-88 59 fotografie dei modelli editi da charles delagrave furono portate all’esposizione tedesca (già ricordata ma della quale si parlerà con maggiore ampiezza), di modelli e strumenti, allestita nel 1893 a monaco di baviera: ne fa menzione il curatore del corrispondente katalog (si veda pag. 24633), walther dyck (1856-1934). non vi sono molte testimonianze, rilevabili presso le sedi universitarie italiane più antiche, degli esemplari appartenuti a queste interessanti collections, ad eccezione di un cospicuo numero di modelli conservati, ancora oggi, al dipartimento di matematica della facoltà di scienze dell’università di genova. è da segnalare, e il dato è molto significativo, che pezzi (probabilmente residui di intere serie) delle collections muret sono presenti nella stessa germania; per esempio, tre sono al mathematisches institut della georg-august universität di gottinga: si tratta, rispettivamente, delle superfici rappresentative delle equazioni z = sin2x + 2siny + sin(x+y) + 10 e cosx + cosy + cosz = 0 e, assieme a queste, del modello relativo alle “querschwingungen eines elastischen stabes” (l’etichetta è scritta in tedesco), cioè delle oscillazioni trasversali di una trave elastica; mentre numerosi altri sono all’institut für mathematik della technische universität (l’erede del mathematisches institut della technische hochschule, vale a dire il politecnico) di monaco di baviera, in barerstrasse, e riguardano, in senso più generale, le scienze delle costruzioni, quale è, ancora ad esempio, il modello descritto dall’etichetta (che non manca di indicare «collections muret. ch. delagrave et c.ie éditeur. rue des écoles 58. paris»): “torsion d’un cylindre à base d’ellipse. surface gauche –novembre 1886–, è scritto in nota a p. 246: “le modèle en a été construit par bardin et par m. muret.” 33 photographische darstellungen der sammlung von gipsmodellen für den mathematischen unterricht von l.j. bardin und ch. muret, ausgestellt von der verlagshandlung von ch. delagrave, paris. proprio nel nachtrag uscito nel 1893, in aggiunta al katalog, vi è un interessante paragrafo (pp. 51-52), historische notiz über die modellsammlung von muret, redatto dal medesimo muret. ratio mathematica, 19, pp. 31-88 60 uv cb cb z 22 22 + − = ϑ affectée par ses sections primitives”. superficie gauche (cioè rigata non sviluppabile34) che reca disegnate, su di una sezione trasversale, le linee “des points dangereux”. oltre a quest’ultimo, altri sei modelli residui, dello stesso tipo, vale a dire riguardanti la teoria dell’elasticità dei corpi solidi, che illustrano sollecitazioni semplici – tipo torsione e flessione– di solidi geometrici (per esempio, prisma a sezione rettangolare o triangolare, cilindro circolare, ecc.), con espliciti riferimenti –che si colgono leggendo le etichette applicate– alle memorie di saint-venant,35 sono pure presenti a monaco di 34 una netta definizione di surface gauche si può leggere in lefebure de fourcy, traité de géométrie descriptive, paris, libraire de l’école polytechnique, 1842 (quatrième édition), p. 92: “surfaces gauches. on nomme ainsi les surfaces qui sont engendrées par une ligne droite, et qui ne sont pas développables.” 35 adhémar jean claude barré de saint-venant (1797-1886), ingegnere e matematico francese, si occupò di resistenza dei materiali e di teoria dell’elasticità, ottenendo risultati che sono alla base della moderna scienza delle costruzioni. di lui si ricordano, in particolare, il problema di saint venant (che consiste nella determinazione dello stato di tensione in un solido elastico-lineare omogeneo ed isotropo di forma cilindrica, libero nello spazio, non soggetto a forze di massa ed in equilibrio sotto l’azione di forze di superficie agenti esclusivamente alle due basi del cilindro) e il principio di saint venant (secondo il quale lo stato di tensione nel cilindro a sufficiente distanza dalle basi non cambia se si sostituiscono le distribuzioni con nuove distribuzioni e purché queste siano staticamente equivalenti alle prime), principio che può anche essere enunciato nella seguente forma equivalente: lo stato di tensione nel cilindro a sufficiente distanza dalle basi dipende unicamente dalla risultante e dal momento risultante dei carichi applicati alle basi stesse, e non dalla loro effettiva distribuzione. al riguardo, un’ampia bibliografia è contenuta nell’articolo di f. foce, saint-venant prima del problema di saint-venant. studi sulla resistenza dei materiale nel periodo 1837-1853, in s. d’agostino (a cura di), atti del ii convegno nazionale di storia dell'ingegneria (napoli, 7-9 aprile 2008), napoli, cuzzolin editore, 2008, vol. 1, pp. 551-562. ratio mathematica, 19, pp. 31-88 61 baviera: conservati assieme ad altri che riguardano corde vibranti, ecc., rappresentano, come si è ipotizzato, i resti di quelle che probabilmente saranno state intere collezioni prodotte dal muret e a suo tempo acquistate. per il regno unito, due appuntamenti espositivi di particolare rilevanza permettono di valutare gli interessi degli studiosi di quel regno per i modelli matematici, delle realizzazioni che essi furono capaci di fare e di quanto –e ciò avvenne in misura maggiore– fossero recepiti i contributi provenienti da matematici e istituzioni matematiche francesi e, ancora, dalle istituzioni, numerose e fiorenti, presenti per tutta quell’area europea che, con a capo la prussia, si andava unificando nell’esteso e progredito stato germanico. il primo dei due eventi fu promosso dal dipartimento di scienze ed arte, costituito nel 1853 e posto, nel 1856, sotto la direzione del comitato del consiglio per l’educazione (dipartimento divenuto perciò: science and art department of the committee of council on education, insediato presso il south kensington museum –oggi science museum– di londra). furono proprio i lords del committee a deliberare, il 22 gennaio del 1875, la formazione di una loan collection of scientific apparatus che doveva […] to include not only apparatus for teaching and for investigation, but also such as possessed historic interest on account of the persons by whom, or the researches in which, it had been employed. fu creato a tal fine un apposito comitato per la formazione della collezione, la quale avrebbe dovuto assumere, in forma stabile, la portata e i compiti istituzionali che in francia aveva il conservatoire des art et métieres. l’esposizione, aperta a maggio del 1876 e che si prevedeva di allestire nelle sale del south kensington museum, fu, a causa del gran numero di oggetti provenienti dal regno unito e dal continente europeo, allestita invece, in questa circostanza, nelle più vaste “galleries on the western side of the horticultural gardens” messe a disposizione dai commissari che avevano curato l’exhibition ratio mathematica, 19, pp. 31-88 62 of the works of industry of all nations del 1851. un handbook e un catalogue36 furono pubblicati in rapporto all’esposizione organizzata. nel consultare quest’ultima pubblicazione si ha modo di riscontrare la presenza della collection of models of ruled surfaces, constructed by fabre de lagrange, in 1872, for the south kensington museum. essa è compresa nella seconda (dal titolo “geometry”) delle sezioni espositive (tutte quante ivi descritte e ora qui elencate in nota). si tratta di 45 superfici, rigate, realizzate in filo, costruite dal francese de lagrange e per le quali era stato pure stampato, nel 1872, un’apposita pubblicazione a cura del già visto science and art department of the committee of council on education, dal titolo a catalogue of a collection of models of ruled surfaces.37 (modelli di superfici rigate di fabre de lagrange furono acquistati anche dalla escola politécnica de lisboa per la cattedra di geometria descritiva. attualmente essi sono al museu de ciência da universidade de lisboa, alcuni conservano ancora l’etichetta originale38). accanto a queste superfici rigate vi erano esposte altre 14 superfici raccolte sotto il nome di plücker model’s: erano delle 36 nella tavola dei contents del catalogue of the special loan collection of scientific apparatus at the south kensington museum (questo è il suo titolo completo), london, printed by george e. eyre and william spottiswoode, 1877 (terza edizione) sono elencate le seguenti 20 sezioni espositive: 1)arithmetic, 2)geometry, 3)measurement, 4)kinematics, statics, and dynamics, 5)molecular physics, 6)sound, 7)light, 8)heat, 9)magnetism, 10)electricity, 11)astronomy, 12)applied mechanics, 13)chemistry, 14)meteorology, 15)geography, 16)geology and mining, 17)mineralogy, crystallography, etc., 18)biology, 19)educational appliances, 20)miscellaneous. 37 constructed by m. fabre de lagrange; with an appendix, containing an account of the application of analysis to their investigation and classification, by c.w. merrifield, london, printed by george e. eyre and william spottswoode. 38 si veda a.r. lozano, s.m. de nápoles, superfícies regradas: manipulação e visulização, projecto matemática em acção, centro de matemática e aplicações fundamentais da universidade de lisboa, reperibile in formato elettronico sul corrispondente web-site. ratio mathematica, 19, pp. 31-88 63 quartiche, in legno (fatte dal costruttore epkens di bonn), possedute dalla london mathematical society. altre singole, interessanti realizzazioni ebbero il loro posto alla mostra, come, per esempio, la famosa steiner’s surface (o superficie romana di steiner, così detta perché, stando alla testimonianza di eugenio beltrami –1835-1900– essa fu ideata da jacob steiner –17961863– durante un soggiorno a roma avvenuto nel 184439): è una superficie razionale del quarto ordine e di terza classe, a simmetria tetraedrale, di equazione cartesiana: x2y2 + y2z2 + z2x2 + xyz = 0; e, ancora per fare un esempio, l’altrettanto famosa superficie diagonale (diagonalfläche) di clebsch, che va considerata come la superficie principale tra quelle regolari (vale a dire senza punti singolari) del terzo ordine: esse, che hanno un’equazione generale della forma 03 =∑ ii i xa , per i=1,…,5 (detta equazione pentaedrale) –dove xi = 0 sono le equazioni di cinque piani formanti il cosiddetto pentaedro di sylvester, mentre tra le cinque x sussiste la relazione identica ,0=∑i ix per i=1,…,5–, contengono 27 rette, ma solo per la diagonalfläche (che si ottiene dall’equazione pentaedrale ponendo ai =1, per i=1,…,5) tutte quante queste rette sono distinte e reali.40 la superficie diagonale affascinò molti artisti, tra i quali man ray e max ernst, come si dirà nella parte finale di questo articolo. sotto vari aspetti –scientifico, didattico, storico– è interessante la testimonianza secondo cui il noto matematico inglese, james joseph sylvester (1814-1897) che molto contribuì anche all’avanzamento della matematica negli stati uniti d’america, aveva in progetto “di far costruire in filo metallico il sistema delle 27 rette esistenti in una superficie di 3° ordine e di farne trarre delle copie stereoscopiche”, 39 la testimonianza di beltrami fu raccolta a voce da francesco gerbaldi (1858-1934) che ne riferisce nel suo articolo la superficie di steiner studiata sulla sua rappresentazione analitica mediante le forme ternarie quadratiche, stamperia reale di torino di i. vigliardi, 1881, p. 5. 40 o, se si vuole, il gruppo delle proiettività che conservano la superficie è il più grande possibile: s5. il prototipo di questo modello, in gesso, fu costruito da christian wiener (1826-1896) su indicazione di alfred clebsch (18331872). ratio mathematica, 19, pp. 31-88 64 intenzione che aveva annunciato in un suo scritto, dal titolo note sur les 27 droites d’une surfaces du 3e dégree, apparso nei comptes rendus des séances de l’académie des sciences di parigi del 1861.41 così si legge in una lettera di luigi cremona (1830-1903) a thomas archer hirst (1830-1892) del 18 gennaio 1865.42 pur essendosi effettuata qualche indagine bibliografica, non è stato possibile accertare se sylvester riuscì a realizzare il suo intento; si è appurato invece che christian wiener pubblicò, nel 1869, un opuscolo, stereoscopische photographien des modelles einer fläche dritter ordnung mit 27 reellen geraden. mit erläuterndem texte,43 in cui vi erano presenti due fotografie stereoscopiche della superficie diagonale. le ultime informazioni fornite danno l’occasione per segnalare che sull’effetto rilievo (visione tridimensionale), realizzato mediante la tecnica degli anaglifi (la quale è strettamente connessa a quella della visione stereoscopica), applicata, modernamente, al tipo di superfici geometriche di cui si sta trattando, è stato compiuto recentemente uno studio da parte di nicla palladino i cui risultati sono raccolti nell’articolo, citato, gli anaglifi di vuibert. origine storica e applicazioni in didattica basata sui modelli di superfici matematiche. l’altro importante evento organizzato nel regno unito fu l’esposizione di edimburgo del 1914 (la napier tercentenary exhibition avutasi in occasione del tricentenario della pubblicazione dell’opera di nepero mirifici logarithmorum canonis descriptio), dove oltre a libri, strumenti e apparecchi per il calcolo numerico e grafico furono esposti modelli di superfici matematiche; e anche in questo caso fu stampato un handbook seguito, nel 1915, da un memorial volume. (l’unico italiano presente con una propria 41 vol. lii, pp. 977-980. 42 la testimonianza si trova in l. nurzia, la corrispondenza tra luigi cremona e thomas archer hirst (1864-1892), in l. nurzia (curatrice), per l’archivio della corrispondenza dei matematici italiani. la corrispondenza di luigi cremona (1830-1903), vol. iv, quaderni p.ri.st.em. – università “l. bocconi”, milano –, n. 11, palermo, 1999, p. 72. 43 leipzig, b.g. teubner. ratio mathematica, 19, pp. 31-88 65 realizzazione all’exhibition fu ernesto pascal che espose uno strumento, un integrafo44). la lettura, ora effettuata, della realtà del regno unito mediante i principali avvenimenti espositivi (che è sembrata essere la chiave interpretativa migliore) non libera però dall’obbligo di avvertire che anche lì furono verosimilmente non pochi i costruttori, di notevole importanza, che produssero anche modelli matematici intesi in senso più o meno stretto. la presenza, a napoli, presso il dipartimento di matematica e applicazioni “r. caccioppoli” di un residuo di 6 model of penetration (“solidi compenetrati”) in legno duro lucidato (hard wood polished) porta a segnalare la ditta costruttrice fondata da george cussons senior, nel 1876, a lower broughton, presso manchester, per la “manifacture of educational and scientific apparatus”. la g. cussons & co. fornì apparecchi e strumenti ai laboratori di numerose sedi universitarie del regno unito, dell’impero britannico e di altri luoghi d’europa e del mondo; una collezione di tali apparecchi scientifici è oggi conservata al museum of science and industry di manchester.45 una copia –da ritenersi abbastanza rara– del catalogo illustrato, edito dalla g. cussons ltd, the technical works, dal titolo apparatus for practical plane and solid geometry, che comprende anche i “solidi compenetrati” (dove è specificato che per ogni singola realizzazione the two solids of interpenetration are made one of dark and one of light wood) accennati è stata ritrovata e fotocopiata dagli autori del presente articolo. la copia non reca l’anno di edizione ma dalla messa in evidenza, sulla copertina, dei premi internazionali ricevuti dalla ditta si deduce che il termine post quem è l’anno 1911. la più importante esposizione internazionale riguardante modelli, apparecchi e strumenti della matematica e della fisica-matematica fu quella che si svolse a monaco di baviera, nel 1893. e così si viene ora a parlare più estesamente della germania. l’esposizione si tenne in 44 si può vedere il citato f. palladino, uno specimen dei giacimenti italiani di modelli e strumenti matematici, pp. 358-360. 45 cfr. j. wetton, scientific instrument making in manchester, 1870-1940. ii: thomas armstrong & brother, and c. cussons & co, «scientific instrument society bulletin», 52, march 1997, pp. 5-8. ratio mathematica, 19, pp. 31-88 66 occasione del convegno dell’associazione dei matematici tedeschi, fondata nel 1890, la deutsche mathematiker–vereinigung. (in verità l’evento, combinato, doveva tenersi nel 1892 a norimberga ma per la “crisi sanitaria” che in quell’anno investì la germania –la città di amburgo fu assalita da un’epidemia di colera– fu rimandato all’anno successivo). per la circostanza fu edito, a cura del citato walther dyck col quale collaborarono “numerosi esperti colleghi”, un katalog mathematischer und mathematisch-physikalischer modelle, apparate und instrumente (seguito da un supplemento, edito nel 1893, ovvero un nachtrag) da considerarsi come una sorta di “sintesi universale”, a carattere teorico-documentale, in cui furono elencati e illustrati tutti i pezzi provenienti dai cataloghi specialistici (compreso ovviamente l’importantissimo catalog di ludwig brill) fino alle singole, non meno interessanti realizzazioni eseguite da isolati autori. il katalog del dyck è diviso in tre sezioni: i) aritmetica, algebra, teoria delle funzioni, calcolo integrale; ii) geometria; iii) matematica applicata. nell’introduzione è scritto che la gran parte degli istituti matematici, fisici, tecnico-meccanici e geodetici delle scuole superiori (università e politecnici) tedesche e straniere avevano messo a disposizione i modelli prodotti negli istituti stessi così come pure gli apparecchi, alcuni dei quali di valore storico, delle loro raccolte. è scritto, inoltre, che da musei, collezioni private, da singoli studiosi, giunsero adesioni, e che oltre alla germania presero parte all’esposizione gli stati uniti d’america, la francia, l’italia,46 i paesi 46 scorrendo il katalog del dyck si trova che l’unico nome di un italiano che avrebbe dovuto esporre già a norimberga è quello di pietro fiorini, nato a sora (regno delle due sicilie, provincia di terra di lavoro –corrispondente con una certa approssimazione all’attuale provincia di caserta; ma sora è dal 1927 in provincia di frosinone–) nel 1852 e morto in torino, città in cui prevalentemente visse ed esercitò la libera professione di ingegnere (si era laureato però in ingegneria a bologna), nel 1912. fiorini presentò un compasso “a quattro punte” (zirkel mit vier spitzen nach fiorini-vergnano, ausgestellt von ingenieur p. fiorini, turin, p. 226 del katalog) e un perspettografo, cioè uno strumento riduttore a proiezione centrale (p. 243 del katalog). mentre se si scorre il nachtrag si ritrova, unico italiano, nicodemo jadanza, nato a campolattaro (regno delle due sicilie, provincia del molise; ma dal 1861 campolattaro appartiene alla provincia di ratio mathematica, 19, pp. 31-88 67 bassi, la norvegia, la svizzera, l’austria-ungheria, la russia e che, in particolare, in inghilterra si formò un comitato, con alla testa lord kelvin, greenhill e henrici, per essere presenti all’esposizione con i migliori pezzi provenienti da raccolte pubbliche e private. è da segnalare che il katalog reca all’inizio ben otto saggi teorici (che si sviluppano per le prime 136 pagine), illustranti numerosi settori d’impiego del materiale esposto, scritti da f. klein, a. voss, a. brill, g. hauck, a. von braunmühl, l. boltzmann, a. amsler, o. henrici. benevento, in campania) nel 1847 e morto a torino nel 1920. jadanza, che si era laureato in matematica (nel 1869) all’università di napoli, era diventato, per concorso, professore di geodesia all’università di torino dove aveva fondato il corrispondente, importante istituto che andò distrutto per i bombardamenti verificatisi nel corso della seconda guerra mondiale. gli strumenti esposti da jadanza a monaco furono cinque: prisma universale jadanza; cannocchiale terrestre accorciato; plesiotelescopio; un nuovo apparato per misurare basi topografiche; microscopio ad ingrandimento costante. essi furono presentati con l’accompagnamento di un ampio commento illustrativo e bibliografico (pp. 110-111 del nachtrag) redatto da sebastian finsterwalder (1862-1951), professore al politecnico di monaco di baviera e già discepolo di l. brill sotto la cui guida aveva eseguito numerosi modelli in gesso appartenenti alle prime dieci serie del catalog brillschilling. il nome di ulisse dini è invece presente nel katalog (p. 292), e precisamente nella didascalia (dove vengono date le indicazioni bibliografiche dei lavori teorici che hanno ispirato la realizzazione del modello) della superficie elicoidale a curvatura costante negativa, la cui curva meridiana è una trattrice (schraubenfläche von constantem negativen krümmungsmass). analogamente, luigi bianchi compare al modello successivo fläche von constantem negativen krümmungsmass mit ebenen krümmungslinien nach enneper (che possiede un sistema di linee di curvatura piane: è del tipo delle superfici di enneper ma, diversamente da queste, che sono esplicitamente formulabili mediante funzioni ellittiche, essa è rappresentabile mediante funzioni elementari). fu studiata, ispirandosi a bianchi, e realizzata da thomas kuen. è la famosa superficie di kuen che tanto interessò i surrealisti: essa sembra raffigurare quel componente di una banda musicale –di quelle che si vedono sfilare, per esempio, nelle feste patronali– che, impettito come un pinguino o come in una posa del famoso attore comico totò, suoni i piatti di ottone. ratio mathematica, 19, pp. 31-88 68 nello stesso 1893, in occasione del congress on mathematics and astronomy, svoltosi a chicago (dal 21 al 26 agosto) nell’ambito della world’s columbian exposition, oltre a una mostra di modelli vi fu una serie di conferenze di felix klein, date anche presso sedi universitarie, in cui questi illustrò le proprietà di alcuni modelli matematici, specialmente di quelli, numerosi, ideati da lui e da alexander brill e costruiti presso l’istituto di matematica del politecnico (allora technische hochschule, oggi technische universität, come si è detto) di monaco di baviera.47 dopo gli appuntamenti espositivi del 1893, il terzo congresso internazionale dei matematici, del 1904, organizzato in germania, ad heidelberg, offrì un’altra consistente esposizione di modelli matematici (fatti in gesso, carta, filo e metallo) realizzati, si badi, nei dieci anni compresi tra il 1893 e il 1904. (anche ad heidelberg furono esposti, accanto ai modelli, strumenti come integrafi, analizzatori armonici, macchine calcolatrici, ecc.). nel complesso, tra modelli e strumenti, furono esposti circa 300 pezzi da parte di quasi 25 espositori e le collezioni (organiche) più importanti presenti furono quelle di hermann wiener (realizzata presso il mathematiches institut der technischen hochschule di darmstadt), di martin shilling (la cui casa editrice era allora in halle an der saale, come si è detto), dei politecnici di darmstadt e karlsruhe, dell’università di gottinga, della zeiss di jena per gli strumenti ottici (che presentò, tra l’altro, una nuova forma di “lavagna luminosa”), della ditta chateau frères di parigi (produttrice di macchine calcolatrici) e di gottlieb coradi (1848-1929) famoso costruttore di strumenti, come integrafi (utili per la risoluzione di equazioni numeriche, per l’integrazione di equazioni 47 “two afternoons were devoted mainly to the explanation and discussion of mathematical models and other appliances, of which an extensive collection had been arranged by prof. klein and dyck. many of the models were unfamiliar to those present, and the opportunity for their examination was highly appreciated.” così è scritto nel resoconto, a cura di h.w. tyler (professore al massachusetts institute of technology), pubblicato nel «bulletin of new york mathematical society» dell’ottobre del 1893. resoconto riportato pure nel «giornale di matematiche» (napoli), xxxi, 1893, p. 377. ratio mathematica, 19, pp. 31-88 69 differenziali, per il calcolo di sforzi e momenti flettenti richiesto nei campi più diversi, da quello dei trasporti terrestri a quello delle costruzioni navali), planimetri (utilizzati per calcolare aree, specialmente in ambito catastale), analizzatori, ecc., la cui ditta aveva sede in zurigo.48 nonostante gino loria (1862-1954) spronasse, con le seguenti parole, i matematici italiani, dalle pagine del bollettino di bibliografia e storia delle scienze matematiche, a partecipare all’esposizione di modelli che si sarebbe allestita ad heidelberg: sarebbe desiderabile che anche l’italia partecipasse a siffatta mostra; ad es. non sarebbe possibile rintracciare ed inviare quel modello di superficie pseudo sferica costruito dal beltrami nell’epoca in cui elaborava il suo celebre saggio d’interpretazione della geometria non euclidea?49 la sua chiamata sortiva lo stesso effetto di una vox clamantis in deserto: pur essendosi portata, la ricerca matematica italiana, su posizioni d’avanguardia (si è accennato, in nota, ai contributi teorici di dini e bianchi), appena però il discorso cominciava a riguardare 48 al riguardo si vedano: verhandlungen des iii.internationalen mathematiker-kongresses in heidelberg vom 8.bis 13.august 1904, hrsg. schriftführer des kongresses dr. a. krazer professor an der technische hochschule karlsruhe, leipzig, b.g. teubner, 1905 (gli atti sono divisi in tre parti, la terza –sotto il nome di die literatur und modellausstellung– dà conto dell’esposizione, fatta di libri, modelli e strumenti, e delle relative conferenze tra le quali vi è una di h. wiener e un’altra del menzionato friedrich schilling); h.w. tyler, the international congress of mathematicians at heidelberg, «bulletin of the american mathematical society», vol. 11, n° 4 (1905), pp. 191-205; j. barrow-green, international congresses of mathematicians from zurich 1897 to cambridge 1912, «the mathematical intelligencer», vol. 16 (1994), n° 2, p. 40, dove è scritto: “another feature of the congress which provoked considerable interest was an extensive exhibition of mathematical literature, apparatus, and models. about 300 models were exhibited, including not only the usual plaster, paper and thread types but also integrators, harmonic analysers, calculating machines, instruments for drawing curves and solving equations, and kinematical models.” 49 anno vii (1904), p.64. ratio mathematica, 19, pp. 31-88 70 applicazioni e investimenti, per quanto di piccolo importo, affioravano le difficoltà e i ritardi. nel settore dei modelli, la novità, la cui maturazione era stata accelerata proprio dalla celebrazione del congresso di heidelberg, era rappresentata dal sorgere di un’altra vasta (e organica) raccolta, quale era quella di hermann wiener (la sammlung mathematischer modelle) che si perfezionerà con l’edizione, per g.b. teubner in leipzig, del corrispondente catalogo (consistente in otto reihen –sarà usato infatti il termine reihe e non più serie, come invece veniva fatto nel catalog di schilling, per indicare una serie di modelli–) pubblicato per la prima volta nel 1905, vale a dire del citato verzeichnis mathematischer modelle.50 l’editore teubner pubblicò poi, nel 1911, il verzeichnis von h. wieners und p. treutlins sammlungen mathematischer modelle, pure questo già menzionato (copie della seconda edizione, del 1912, sono state trovate a napoli e a firenze, ai dipartimenti già indicati) in cui vennero a giustapporsi l’elenco della precedente raccolta (ingrandita però da otto a dodici serie) di h. wiener e quello della raccolta (18 serie per più di 200 modelli) di p. treutlein (direttore del goethe-gymnasium di karlsruhe). si vuol far notare, di passaggio, che peter treutlein, con le sue tesi e le sue esperienze sull’insegnamento intuitivo della geometria, destinato agli scolari fino a circa dodici anni di età, molto influenzò, all’inizio del novecento, l’insegnamento, in giappone, della geometria che veniva ad impartirsi in quelle fasce scolastiche che oggi, in italia, vanno sotto i nomi di scuola primaria e scuola secondaria di primo grado.51 egli godette naturalmente di un forte prestigio anche in germania, infatti ne fu il rappresentante per l’insegnamento di ordine medio (gli altri due delegati erano f. klein, per l’insegnamento universitario, e paul stäckel per la matematica nei 50 due copie di questo catalogo sono state individuate presso il dipartimento di matematiche e applicazioni dell’università degli studi di napoli “federico ii”, una di esse era appartenuta ad ernesto cesàro. 51 si veda t. fujita, k. jones, s. yamamoto, the role of intuition in geometry education: learning from the teaching practice in the early 20th century, «proceedings of 10th international congress on mathematical education, copenhagen 2004; –icme-10 tsg29–». ratio mathematica, 19, pp. 31-88 71 politecnici) nella commissione internazionale dell’insegnamento matematico (presidente lo stesso klein) costituita, nel 1908, per decisione presa nell’ambito dei lavori del iv congresso internazionale dei matematici, svoltosi in quell’anno a roma. in un convegno di questa commissione, inserito nel programma dell’exposition universelle de bruxelles, del 1910, treutlein tenne una conferenza sull’uso dei suoi modelli geometrici nell’insegnamento (mentre klein illustrò quelli del catalog brill/schilling utili all’insegnamento “superiore”).52 per quanto riguarda poi hermann wiener, è da sottolineare che egli, a somiglianza di quanto si è fatto notare per hilbert, mentre sosteneva uno studio della pura geometria fatto senza l’ausilio di immagini ma piuttosto con metodo assiomatico, era impegnato, alternativamente, nella progettazione e, nel suo caso, realizzazione di modelli plastici per gli specifici vantaggi che essi potevano offrire innanzitutto nell’insegnamento, ma anche per mettere in evidenza alcuni aspetti riguardanti la ricerca. 4. lo svolgimento della vicenda connessa all’ideazione e alla costruzione di modelli matematici non vede l’italia collocata in un ruolo di primo piano. l’italia dipenderà dalla produzione degli altri paesi europei come francia, germania, inghilterra. e per tale ragione si è provveduto a dare in precedenza un quadro abbastanza dettagliato di quanto accadeva nei paesi più progrediti in questo settore: nel visitare i cospicui “fondi” museali presenti in italia si vede che i canali di approvvigionamento (anche se non tutti strettamente accompagnati da documenti d’archivio53) che arrivano, per fare degli esempi, a luigi 52 cfr. l. giacardi, the first century of international commissionon mathematical instruction (1908-1910), in web-site: http://www.icmihistory.unito.it; e h. fehr, circulaire n. 3. compte rendu des séances de la commission et des conférences sur l’einseignement scientifique et sur l’enseignement technique moyen, «l’einseignement mathématique», 12, 1910, pp. 353-415. 53 una relazione a stampa, dal titolo la regia università di napoli a s.e. l’on. ministro della pubblica istruzione. voti e deliberazioni della facoltà ratio mathematica, 19, pp. 31-88 72 cremona e poi a guido castelnuovo (1865-1952) a roma, a ulisse di medicina, di scienze naturali e di matematiche sulle dotazioni agli istituti scientifici ed altri bisogni dell’insegnamento (edita a napoli nel 1884 presso la tipografia dell’accademia reale delle scienze, appartenuta probabilmente alla biblioteca dell’ex seminario matematico e custodita oggi tra gli opuscoli della biblioteca del dipartimento di matematica e applicazioni “r. caccioppoli”), contiene –al punto xi. gabinetto per i modelli di geometria– il seguente passo: “a pisa, roma, torino e pavia, ecc. queste collezioni si sono già avute da molti anni; ma qui non è stato ancora possibile averle non ostante le ripetute domande della facoltà di matematica. si ha fiducia che si vorrà una buona volta provvedere”. accompagnato a ciò, vi è una lettera (conservata presso lo stesso dipartimento, fondo documentario francesco siacci) di ettore caporali, del 1884 –anno in cui questi, già professore straordinario di geometria superiore a napoli, dal 1878, passa ad ordinario presso la stessa università– nella quale, rivolgendosi a un collega della medesima sede, egli scriveva: “stimatissimo sig.r professore. sono del parere che convenga acquistare per intero la serie prima (pagina 9 del catalogo) e la serie settima (pag. 20) le quali importano insieme 360 marchi, pari a 450 lire. le rimanenti 50 lire saranno assorbite dalle spese d’imballaggio e trasporto […]”. in mancanza fino ad oggi di una copia del catalog mathematischer modelle di brill, si è effettuato un controllo sul catalog di schilling (che ne rappresenta la naturale continuazione). la prima serie era costituita da cinque modelli in gesso (gipsmodelle nach originalen des mathematischen instituts der kgl. technischen hochschule, münchen) riguardante argomenti diversi (superficie di rotazione della trattrice, superficie focale di un sistema di raggi, superficie dei centri di curvatura [o superficie centro, n.d.r.] di un iperboloide a una falda, ecc.), mentre la settima (gipsmodelle von flächen 3 ordnung nach rodenberg) riguardava superfici del terzo ordine, le hessiane di alcune di esse, ecc. nel catalog dello schilling, edizione 1903, il costo totale delle due intere serie risulta ancora essere di 360 marchi. quasi tutti questi modelli sono a tutt’oggi presenti presso il dipartimento “r. caccioppoli”. per quanto riguarda poi l’università di torino i primi acquisti di modelli risalgono all’anno scolastico (così era chiamato allora l’anno accademico) 1880-’81, al tempo in cui fu rettore enrico d’ovidio (si veda l. giacardi, la collezione di modelli geometrici della biblioteca speciale di matematica “g. peano”, in le memorie della scienza. musei e collezioni dell’università di torino, a cura di g. giacobini, università degli studi di torino e fondazione crt, 2003, pp. 251-266). ratio mathematica, 19, pp. 31-88 73 dini (1845-1918) e poi luigi bianchi (1856-1928) e riccardo de paolis (1854-1892) a pisa, a enrico d’ovidio (1843-1933) e poi corrado segre (1863-1924) a torino, a eugenio beltrami (1835-1900) e poi ernesto pascal a pavia,54 a ettore caporali (1855-1886), pasquale del pezzo (1859-1936) e poi ernesto cesàro a napoli, a giuseppe veronese a padova, a gino loria (1862-1954) a genova, provengono proprio dalle tre nazioni indicate, e, per la maggior parte dei modelli, il grande flusso si avvia, in ordine di tempo, poco dopo il 1877, che è l’anno in cui ha inizio l’attività della casa editrice aperta da ludwig brill in darmstadt. con la formazione del nuovo regno d’italia i problemi istituzionali, finanziari, legislativi, militari, scolastici, e così via, sono molteplici e impegnativi e, si potrebbe aggiungere, non sempre affrontati con “visione unitaria”. in campo scientifico, l’area della ricerca ad indirizzo matematico è, allora, quella che più velocemente progredisce (portandosi, verso la fine dell’ottocento, ai primi posti in europa55), anche perché è la meno condizionata, ai fini della fioritura di validi cultori, dalla necessità di forti investimenti finanziari. 54 nel secondo volume del menzionato repertorio, dedicato alla geometria, pascal fa espliciti riferimenti ai modelli in gesso presenti a pavia. per esempio, a p. 469 si legge: “di tutte queste ciclidi sono stati costruiti modelli in gesso (v. il catalog math. modelle von brill in darmstadt); esemplari di questi modelli sono anche posseduti dall’istituto matematico dell’università di pavia”; e a p. 480: “nel più volte citato catalogo di l. brill, esistono modelli di superficie di steiner; essi fanno anche parte della raccolta posseduta dall’istituto matematico dell’università di pavia”. 55 tanto da spingere rudolf lipschitz (1832-1903) a felicitarsi con beltrami (in una lettera dei primi anni settanta dell’ottocento) poiché in italia e in germania i matematici respiravano ormai la stessa aria: “in italien und deutschland ist es doch dieselbe luft, die wir athmen.” dove athmen è scritto con la h secondo l’ortografia ottocentesca, oggi è atmen. il passo è riportato in f. palladino, n. palladino, dalla “moderna geometria” alla “nuova geometria italiana”. viaggiando per napoli, torino e dintorni, firenze, olschki, 2006, p. xv. questa constatazione si accoppia con l’altra fatta da g. darboux nel corrispondere con j. houël: “je crois que si cela continue les italiens nous dépasseront avant peu. aussi tâchons avec notre bulletin de réveiller ce feu sacré et de faire comprendre aux français qu’il y ratio mathematica, 19, pp. 31-88 74 dai valenti matematici che contribuirono al “risorgimento” politico italiano, come giuseppe battaglini (1826-1894),56 enrico betti (1823-1892), francesco brioschi (1824-1897), luigi cremona,57 vennero molteplici iniziative, scientifiche, didattiche, editoriali, legislative, volte a rinvigorire l’istruzione, in particolare di quella superiore. con senso di responsabilità, i “matematici risorgimentali” non guardarono soltanto allo stretto del loro settore di studio. così, per esempio, brioschi, professore all’università di pavia –in questa città aveva luogo l’unica sede universitaria della lombardia–, prossimo a fondare l’istituto tecnico superiore di milano (legalmente istituito nel 1862 e aperto nel novembre del 1863 –dal 1937 esso prenderà il nome attuale di politecnico–), di cui ne assunse subito la direzione che mantenne fino alla morte, si occupa, da segretario generale del ministero della pubblica istruzione (carica che egli coprì dal luglio del 1861 a dicembre del 1862) di scrivere, da torino (il nuovo regno ha provvisoriamente la città piemontese per capitale), a betti che è professore a pisa:58 a un tas de choses dans le monde dont ils ne se doutent pas, et que si nous sommes toujours la grande nation, on ne s’en aperçoit guère à l’étranger” (si veda la correspondance de g. darboux avec j. houël. chronique d’un rédacteur (déc. 1869-nov. 1871) a cura di h. gispert, «cahiers du séminaire d’histoiredes mathématiques», paris, institut henri poincaré, n° 8 (1987), pp. 67-202, lettera 46, non datata, p. 160). 56 si può vedere giuseppe battaglini. raccolta di lettere (1854-1891) di un matematico al tempo del risorgimento d’italia, a cura di m. castellana e f. palladino, bari, levante, 1996. 57 intorno alle figure di betti, brioschi, cremona, si può consultare anche il volume, recente, per la costruzione dell’unità d’italia. le corrispondenze epistolari brioschi-cremona e betti-genocchi, a cura di n. palladino, a.m. mercurio, f. palladino, firenze, olschki, 2009; e, per cremona e genocchi, si può vedere anche l’epistolario cremona-genocchi (1860-1886). la costituzione di una nuova figura di matematico nell’italia unificata, a cura di l. carbone, r. gatto, f. palladino, firenze, olschki, 2001. 58 passo riportato nel volume, citato, per la costruzione dell’unità d’italia. le corrispondenze epistolari brioschi-cremona e betti-genocchi, p. ix. ratio mathematica, 19, pp. 31-88 75 il ministero ha determinato di mandare all’estero otto o dieci giovani laureati in matematica per dedicarsi allo studio della chimica, della fisica, della mineralogia e della meccanica applicata alle macchine. si darà loro 1500 franchi all’anno e se riescono bene al ritorno saranno destinati all’insegnamento. desidero che tu d’accordo coi tuoi amici e colleghi mi proponesti due o tre giovani toscani laureati recentemente (o da poco tempo) i quali abbiano dati sicure prove di ingegno e di amore allo studio, ed accettassero colle condizioni suddette di dedicarsi a quegli studj. nel clima postrisorgimentale l’allora giovane professore dell’università di padova, giuseppe veronese, studioso di geometria proiettiva degli iperspazi (era stato per un anno –1880-‘81– a perfezionarsi in germania dove aveva soggiornato pure presso felix klein, allora che questi era all’università di lipsia) che di lì a poco verrà ad ideare una famosa “superficie” che porta il suo nome,59 si fa promotore, nel novembre del 1883, di una relazione,60 sostenuta dalle lettere di francesco brioschi, enrico d’ovidio, riccardo de paolis, ulisse dini ed eugenio bertini (1846-1933), al ministro della pubblica istruzione, guido baccelli, in cui chiede che all’università dove insegna si possa aprire un “atelier”, ovvero un laboratorio, per costruire modelli matematici, simile a quelli che funzionano in altri paesi europei, in modo da permettere alle istituzioni universitarie italiane di poter ottenere con minor spesa, evitando di ricorrere ad acquisti all’estero, quel poco di strumentazione utile per “fare matematica”. nel suo scritto egli ricorda preliminarmente: al politecnico di questa città [monaco di baviera] è annesso un istituto matematico, il quale si compone sin dal 1875 d’una collezione, di una biblioteca 59 si tratta di una superficie a due dimensioni di uno spazio a cinque dimensioni che, nella sua più semplice espressione, può essere rappresentata dall’equazione parametrica x1 = u2 , x2 = uv, x3 = v2 , x4 = u, x5 = v, dove x1,…,x5 , sono coordinate non omogenee dello spazio e u e v sono due parametri indipendenti. lo studio di questa superficie è equivalente, dal punto di vista proiettivo, allo studio di tutte le coniche del piano, e una sua proiezione nello spazio ordinario è la superficie romana di steiner. 60 essa è riprodotta integralmente in f. palladino, il fondo di modelli e strumenti matematici antichi dell’università di padova e l’iniziativa di giuseppe veronese per un laboratorio nazionale italiano, cit., pp. 59-62. ratio mathematica, 19, pp. 31-88 76 per i membri (professori e studenti), del seminario matematico, ed un atelier. la collezione conta oltre 500 modelli di geometria superiore, fisica-matematica, e meccanica. l’atelier è fornito di ordigni, apparati, torni per la costruzione di modelli in legno e gesso. questi sono talvolta il risultato di ricerche speciali degli stessi professori dei corsi superiori della facoltà matematica e del politecnico. quando se ne è raccolto un certo numero, i migliori vengono consegnati a una libreria, la quale s’incarica di farli pittare e porli in vendita. […] ma vi sono altri atelier a göttingen , a carlsruhe [oggi karlsruhe], a parigi, etc., i quali avranno forse ordinamenti diversi. nel corso della sua relazione veronese osserva pure: io credo che pel progresso sempre maggiore degli studj matematici in italia, un atelier come quello di monaco sarebbe da noi di grande giovamento, perché le nostre scuole si renderebbero indipendenti anche per questo dall’estero, e potrebbero procurarsi le loro collezioni con minor spese; aggiungendo che le università di roma, torino, pavia, pisa spesero complessivamente 8 mila lire circa in modelli, però sono lontane dall’avere le raccolte complete, mentre tutte le altre università e gli istituti superiori son del tutto sprovvisti. i finanziamenti non si trovarono e l’ “atelier” non s’impiantò, pur trattandosi di un investimento economicamente di piccola entità. le università italiane continuarono a “procurarsi” le loro collezioni, o singoli pezzi, all’estero. limitando le informazioni all’essenziale, si vuole ora dare un elenco delle sedi italiane visitate, rimandando alla bibliografia citata nelle note per maggiori informazioni intorno al lavoro svolto. l’insieme delle sedi italiane, in effetti tutte quelle che erano da visitarsi, raggiunte da franco palladino, tra il 1992 e il 1997 circa, elencate andando da sud verso nord italia, è il seguente: catania (dipartimento di matematica, facoltà di scienze mm. ff. nn.), messina (dipartimento di matematica, facoltà di scienze mm. ff. nn.), bari (dipartimento di matematica, facoltà di scienze mm. ff. nn.), napoli (dipartimento di matematica e applicazioni “r. caccioppoli” (facoltà di scienze mm. ff. nn.) e facoltà di agraria, roma, (dipartimento di matematica “g. castelnuovo”, ratio mathematica, 19, pp. 31-88 77 facoltà di scienze mm. ff. nn.), firenze (dipartimento di matematica “u. dini”, facoltà di scienze mm. ff. nn.), bologna (dipartimento di matematica, facoltà di scienze mm. ff. nn.), modena (dipartimento di matematica pura e applicata “g. vitali”, facoltà di scienze mm. ff. nn.), padova (dipartimento di matematica, facoltà di scienze mm. ff. nn.), parma (dipartimento di matematica, facoltà di scienze mm. ff. nn.), pavia (dipartimento di matematica “f. casorati”, facoltà di scienze mm. ff. nn.), milano (dipartimento di matematica “f. enriques”, facoltà di scienze mm. ff. nn.), torino (dipartimenti di matematica della facoltà di scienze mm. ff. nn. e del politecnico), genova (dipartimento di matematica, facoltà di scienze mm. ff. nn.).61 il lavoro di rilievo fotografico (riproduzione di pezzo per pezzo del fondo di ciascuna sede) e di studio è confluito, nel caso dei fondi più significativi, in un’apposita, corrispondente pubblicazione a stampa.62 61 all’università di pisa, nonostante l’attivo aiuto prestato dal prof. luciano modica e da altri professori, si è ritrovato soltanto l’inventario –a firma del direttore di quello che era, allora, l’istituto di matematica, leonida tonelli (1885-1946)– riguardante i beni dell’istituto, tra cui i modelli. 62 oltre ai lavori già citati nel corso di questo articolo, si segnalano: f. palladino, sui modelli matematici in gesso del dipartimento di matematica dell’università di messina, «rendiconti del seminario di matematica dell’università di messina», serie ii, vol. i (1991), pp. 151-158. f. palladino e g. ferrarese, sulle collezioni di modelli matematici dei dipartimenti di matematica dell’università e del politecnico di torino, «nuncius» (istituto e museo di storia della scienza di firenze), xiii (1998), pp. 169-185; l. carbone, g. cardone e f. palladino, le collezioni di strumenti e modelli matematici del dipartimento di matematica e applicazioni “r.caccioppoli” dell’università “federico ii” di napoli. cataloghi ragionati, «rendiconto dell’accademia delle scienze fisiche e matematiche di napoli», (iv), vol. lxv, anno cxxxvii (1998), pp. 93-257, con 115 foto di modelli e strumenti. f. palladino, le collezioni museali del dipartimento di matematica e applicazioni “r. caccioppoli” dell’università di napoli“federico ii”, in atti del convegno in onore di carlo ciliberto, a cura di t. bruno, p. ratio mathematica, 19, pp. 31-88 78 con la compilazione della tesi di laurea (discussa a dicembre 2000), nicla palladino ha composto il catalogo generale dei pezzi presenti (sono circa 350 distinti esemplari, alcuni dei quali hanno una presenza ripetuta in più sedi, quindi il numero complessivo è molto più alto) in tutti i “fondi” prima indicati, strutturandolo in forma ragionata con numerosi rimandi interni e quasi sempre con la fotografia del singolo oggetto. di ciascun pezzo, o di gruppi degli stessi, è stata messo in rilievo il significato matematico. il lavoro fatto è stato inserito in un web-site corredato dalle riproduzioni tridimensionali (3d) di una ventina di modelli realizzate mediante mathematica®.63 con la compilazione, poi, della tesi di dottorato (discussa nell’autunno del 2004, al termine del ciclo quadriennale di studi, dal titolo e-learning: superfici matematiche in 3d) di nicla è stato mostrato il possibile uso dei modelli nell’insegnamento della geometria mediante e-learning e si è proceduto, con applicazione dell’informatica, alla rappresentazione approssimata di alcuni di essi mediante nurbs. 5. i modelli matematici plastici, in particolar modo quelli costruiti in gesso, oltre ad essere utili alle scienze matematiche, fornirono stimoli e ispirazioni a pittori, scultori, architetti e scenografi. il professore alfredo franchetta, già ordinario di geometria buonocore, l. carbone e v. esposito, napoli, la città del sole, 1997, pp. 119-138, dove, sulla copertina del volume, vi è una bella foto della superficie diagonale di clebsch che è presso il dipartimento napoletano. f. palladino, n. palladino, sulle raccolte museali italiane di modelli per le matematiche superiori. catalogo ragionato e sito web, «nuncius» (istituto e museo di storia della scienza di firenze), xvi (2001), pp. 781-790. 63 consultabile agli indirizzi: www.dmi.unisa.it/people/palladino/modelli; e www.dma.unina.it/~nicla.palladino/catalogo. la ricerca dei modelli può essere effettuata in base ai seguenti criteri: nome del modello, nome della serie, numero della serie, numero del modello, catalogo, etichetta originale, materiale, anno di edizione, progettista, realizzatore, editore, luogo di costruzione, note, sede. ratio mathematica, 19, pp. 31-88 79 all’università di napoli, laureatosi a roma, dove ebbe come professore federigo enriques, interpellato, all’inizio dei passati anni novanta, per informazioni su questi oggetti, ricordava che l’enriques più volte gli aveva sottolineato che scenografi di cinecittà (la “città del cinema” inaugurata a roma nella primavera del 1937) si recavano spesso all’istituto di matematica dell’università (oggi “la sapienza”) per osservare, onde trarre ispirazione, i modelli ivi custoditi. per il versante artistico alcuni elementi di documentazione si trovano nel volume di luigi campedelli dal titolo fantasia e logica della matematica.64 il libro, a carattere divulgativo, contiene tavole illustrative di sculture in gesso di alberto viani (1906-1989) – ricordato, sotto l’angolazione che qui interessa, per la plasticità dei torsi femminili e maschili, per la scultura la cariatide, ecc.– e di un quadro di atanasio soldati (1896-1953) ispirati ai motivi geometrici espressi dai modelli in gesso. anche nel testo, di l. campedelli, esercitazioni complementari di geometria, curato da a. barlotti (con la revisione della succitata c. dolfi),65 ad uso degli studenti universitari, vi sono alcune tavole illustrative (dalla vi alla xii) che raffigurano modelli: sono tutti in gesso e appartengono all’insieme di quelli riprodotti da campedelli, di cui si è parlato, salvo il modello della tav. vi che rappresenta una falda della superficie razionale del quarto ordine, di equazione: (x2 + y2)2 – x2(z2 – 1)=0, “dono della sig.na dott. anna maria sbrana” (recita l’etichetta ad esso incollata), probabilmente allieva di campedelli; il modello è esposto ancora oggi a firenze nelle vetrine che si trovano al dipartimento di matematica “u. dini”. nella recensione al testo di gerd fischer, redatta da jeremy gray,66 sono riportate le foto di alcune sculture del “futurista” umberto boccioni (1882-1916) –si coglie l’occasione per segnalare che nel quadro da lui dipinto, visioni simultanee, del 1911, sono 64 milano, feltrinelli, 1966. 65 padova, cedam, 1975. 66 si veda mathematische modelle/mathematical models (vol. 1-2) edited by gerd fischer, reviewed by jeremy gray, «the mathematical intelligencer», vol. 10, n. 3, 1988, pp. 64-69. ratio mathematica, 19, pp. 31-88 80 evocati tratti, dal colore viola sfumato, che sembrano ispirati, in fondo, dal modello di superficie di riemann con due falde semplicemente connesse67–, del “costruttivista” naum gabo (18901977) –noto anche per le sue sculture dalle forme geometriche costruite con materiali semitrasparenti–, e di costantin brancusi (1876-1957) che richiamano i modelli brill/schilling. un articolo, interessante, di isabelle fortuné68 su man ray (18901976), noto anche per la sua dimensione artistica di “fotografo surrealista”, permette di toccare altri punti riguardanti questo aspetto, poiché l’autrice parla della scoperta da questi fatta, in compagnia di max ernst (1891-1976), intorno al 1934, dei modelli in gesso (appartenenti al catalog brill/schilling) esposti in vetrina all’institut poincaré, fondato a parigi nel 1928.69 nell’articolo l’autrice evidenzia le artistiche fotografie fatte da man ray, ricorda ancora un collage, realizzato da max ernst per illustrare la cartolina-invito per una mostra di man ray (dal titolo man ray, peintures et objets, del 1935), che riproduceva, estratte dal katalog curato dal walther dyck (e dal brill/schilling, va aggiunto), la superficie minimale di enneper,70 una superficie di riemann, la superficie diagonale di clebsch con le sue ventisette rette reali, ecc., e ricorda che ray dipinse, tra il 1948 e il 1954, a partire dalle fotografie fatte all’instiut poincaré, una serie di tavole raggruppate sotto il titolo di équations shakespeariane. l’autrice cita, tra le forme che attrassero l’attenzione di man ray, anche il modello rappresentante la funzione amplitudine di jacobi 67 zweiblättrige einfach zusammenhägende riemann’sche fläche, catalog m. schilling 1911, serie xvii, n. 10. 68 si veda man ray et les objets mathématiques, «études photographiques», 1999, http://etudesphotographiques.revues.org 69 amministrativamente esso è oggi un’ “école interne” dell’université paris vi “pierre et marie curie”. 70 è una superficie minimale del nono ordine, costruita, in base ad un articolo di alfred enneper (1830-1885), del 1871, sotto la direzione di a. brill. essa è rappresentabile parametricamente da un’equazione algebrica e una copia del modello in gesso è, per esempio, a napoli. “un poco di questa superficie”, se potesse passare l’espressione, sembra ritrovarsi nell’opera di antoine pevsner (1886-1962), fratello di naum gabo, dal titolo surface développable, del 1938-1939. ratio mathematica, 19, pp. 31-88 81 (una funzione ellittica),71 esemplari della quale si trovano, per esempio, alle università di pavia, roma (“la sapienza”) e napoli: un modello in gesso che evoca un suolo, o una spiaggia, fatto di dune, profonde, armonicamente ondulate –come un pezzo della costa africana del mediterraneo o come, sull’altra sponda, il litorale campano, da pozzuoli a baia domizia, qual era fino agli anni sessanta del novecento–, da cui si erge una parte, scolpita, di quella che potrebbe essere, nella sua interezza, una sfinge, per esempio. isabelle fortuné interpreta, inoltre, l’attrazione verso questi modelli matematici plastici essere stata generata dal fatto che essi apparivano quali punti d’incontro delle due principali linee di tendenza che connotavano il surrealismo negli anni trenta: le riflessioni sulla nozione di oggetto e, ancora, sulle scoperte della scienza moderna. non si è invece completamente d’accordo con l’autrice, per le ragioni esposte nella parte iniziale di questo articolo, quando individua semplicemente nella “fragilité de ces constructions et leur caractère toujours trop approximatif pour la rigueur mathématiques” le cause della caduta dell’interesse scientifico, nei confronti dei modelli plastici, con la loro conseguente trasformazione, verso gli anni trenta, in “objets de curiosité” o in oggetti destinati a dare al pubblico “une vision moins austère des recherches scientifiques”. intenzione, quest’ultima, che avrebbe poi mosso gli organizzatori dell’exposition internationale des arts et techniques dans la vie moderne, allestita a parigi, nel 1937, al palais de la découverte (appositamente creato, nell’occasione, utilizzando, allo scopo, anche una parte del grand palais costruito nel 1897 in funzione dell’exposition universelle del 1900) a mettere in mostra questi oggetti sotto il tema: l’art et la science.72 71 sul calco in gesso costituente il modello che sta a pavia si legge a malapena l’incisione: φ = am (u,k), mentre su quello che sta a napoli si trova inciso ∫ − ϕ ϕ ϕ 0 22 sin1 k d , dove u è la funzione inversa della precedente ed esprime un integrale ellittico di prima specie. 72 su questa exposition si vedano: il catalogo dal titolo exposition internationale, paris, palais de la découverte, 1937; e la pubblicazione paris ratio mathematica, 19, pp. 31-88 82 interessante e cospicuo è pure il lavoro svolto da angela vierlingclaassen sui models of mathematical surfaces73 dove in un file si tratta dell’argomento mathematical models and art in the early 20th century. ancora uno scultore, attivo negli anni correnti, cayetano ramírez lópez, ha restaurato, nel 2005, quei modelli in gesso appartenenti al “fondo museale” del dipartimento di matematica dell’università di groningen, in olanda; mentre un professore, marius van der put, s’incaricava di restaurare i modelli in filo.74 è quanto emerge dalla tesi di dottorato di irene polo-blanco (il cui titolo è theory and history of geometric models),75 che è rivolta, con molta cura anche per gli aspetti matematici che ne stanno alla base, alla presentazione dei più di cento modelli custoditi a groningen. sono modelli in gesso e filo risalenti, principalmente, alle serie vii, xiii, xvii, xxv del catalog di m. schilling, con in più i “four-dimensional polytopes”, costruiti per “visualizzare” la quarta dimensione, ideati da alicia boole stott (1860-1940).76 a questo proposito, un pezzo rappresentante un 1937. cinquantenaire de l’exposition internationale des arts et des techniques dans la vie moderne, paris, institut française d’architecture/parismusées, 1987. 73 si veda il web site: http://math.harvard.edu, dove è citato anche il lavoro svolto da n. palladino. 74 si coglie qui l’occasione per ricordare che al dipartimento di matematica dell’università di padova, nei trascorsi anni ottanta, il professore tomaso millevoi s’incaricava, aiutato da alcuni studenti, di effettuare un’analoga operazione di restauro per i modelli in filo ivi esistenti. la constatazione fu fatta in loco da f. palladino, osservando le nuove etichette poste sugli oggetti restaurati, e in seguito confermata da francesco bottacin, studente a padova al tempo in cui millevoi si prendeva cura dei modelli, poi professore all’università di salerno. 75 relatori i professori: m. van der put, j. top, j.a. van maanen. la tesi, discussa nel 2007, è stata pubblicata a groningen, per l’academic press europe. 76 sui politopi si può vedere il contributo di h. martini, reguläre polytope und verallgemeinerungen, in o. giering und j. hoschek, geometrie und ihre anwendungen, münchen wien, c. hanser verlag, 1994, pp. 247-281, che è corredato, tra l’altro, da una vasta bibliografia. ratio mathematica, 19, pp. 31-88 83 vierdimensionaler würfel (un cubo a quattro dimensioni), costruito in sbarrette e archi metallici, si è ritrovato anche ad heidelberg, nella recente, menzionata visita che si è effettuata. superficie pseudosferica di tipo parabolico. superficie di kuen. ratio mathematica, 19, pp. 31-88 84 paraboloide iperbolico con il sistema delle rette generatrici. paraboloide iperbolico. ratio mathematica, 19, pp. 31-88 85 superficie di riemann a 2 fogli. superficie di boy. ratio mathematica, 19, pp. 31-88 86 un modello originale di horopter (il profilo della curva, realizzata in filo di ottone, risalta perché più brillante) e, a destra, una riproduzione digitale della stessa. superficie di peano. ratio mathematica, 19, pp. 31-88 87 una vetrina di modelli posseduti dal mathematisches institut della ruprecht-karl-universität di heidelberg. si noti nella penultima posizione (in basso a destra) il cubo a 4 dimensioni. cubo a 4 dimensioni. ratio mathematica volume 42, 2022 closed isometric linear transformations of complex spacetime endowed with euclidean, or lorentz, or generally isotropic metric spyridon vossos* elias vossos† christos g. massouros‡ abstract this paper is the first in a series of documents showing that newtonian physics and einsteinian relativity theory can be unified, by using a generalized real boost (grb), which expresses both the galilean transformation (gt) and the lorentz boost. here, it is proved that the closed linear transformations (clts) in spacetime (st) correlating frames having parallel spatial axes, are expressed via a 4x4 matrix λi, which contains complex cartesian coordinates (ccs) of the velocity of one observer / frame (o/f) wrt another. in the case of generalized special relativity (sr), the inertial os/fs are related via isotropic st endowed with constant real metric, which yields the constant characteristic parameter ωi that is contained in the clt and grb of the specific sr. if ωi is imaginary number, the st can only be described by using complex ccs and there exists real universal speed (ci). the specific value ωi=±i gives the lorentzian-einsteinian versions of clt and grb in st endowed with metric: -gi00η and ci=c, where i; c; gi00; η are the imaginary unit; speed of light in vacuum; time-coefficient of metric; lorentz metric, respectively. if ωi is real number, the corresponding st can be described by using real ccs, but does not exist ci. the specific value ωi=0 gives gt with infinite ci. gt is also the reduction of the clt and grb, if one o/f has small velocity wrt another. the results may be applied to any st * core department, national and kapodistrian university of athens, euripus campus, gr 34400, psahna, euboia, greece; svossos@uoa.gr. † core department, national and kapodistrian university of athens, euripus campus, gr 34400, psahna, euboia, greece; evossos@uoa.gr. ‡ core department, national and kapodistrian university of athens, euripus campus, gr 34400, psahna, euboia, greece; chrmas@uoa.gr. 305 s. vossos, e. vossos, ch. g. massouros endowed with isotropic metric, whose elements (four-vectors) have spatial part (vector) that is element of the ordinary euclidean space. keywords: 5th euclidean postulate; complex space; electromagnetic tensor; euclidean metric; euclidean space; galilean transformation, general relativity; isometry; linear transformation; lorentz boost, lorentz metric, lorentz transformation, minkowski spacetime, newtonian physics, spacetime; special relativity; universal speed. 2010 ams subject classification: 15a04; 83a05.§ abbreviations-annotations ccs: cartesian coordinates ciltocst: closed isometric linear transformation of complex spacetime clt: closed linear transformation ci: universal speed e3: three-dimensional euclidean space e4: euclidean spacetime ert: einsteinian relativity theory esr: einsteinian special relativity gr: general relativity grb: generalized real boost gsr: generalized special relativity gt: galilean transformation io: inertial observer lt: linear transformation lb: lorentz boost m4: minkowski spacetime nps: newtonian physics o/f: observer / frame qms: quantum mechanics rt: relativity theory sr: special relativity st: spacetime (four-dimensional space) tps: theory of physics u: invariant speed wrt: with respect to § received on june 7th, 2022. accepted on june 29th, 2022. published on june 30th, 2022. doi: 10.23755/rm.v41i0.810. issn: 1592-7415. eissn: 2282-8214. © spyridon vossos et al. this paper is published under the cc-by licence agreement. 306 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric 1 introduction linear transformations (lts) are very important in relativity theory (rt) and quantum mechanics (qms) [1]. moreover, there exist many different approaches of rt, which emerge the corresponding qms. for instance, galilean transformation (gt) endowed with the corresponding metric of spacetime (st) produces newtonian physics (nps), which gives the classic qms (schrödinger equation). thus, many low-velocity phenomena, like the atomic spectra (without fine structure) were explained. on the other hand, lorentz transformation (endowed with the lorentz metric of st) produces einsteinian special relativity (esr), which gives relativistic qms (kleingordon equation). thus, many high-velocity phenomena and the fine structure of atomic spectra were explained [2]. in this paper, we prove that there exist two types of closed isometric linear transformation of complex spacetime (ciltocst) with common solution the gt. these can apply not only to special relativity (sr), but also to general relativity (gr), because they are reached without adopting one specific metric of spacetime. in addition, any complex cartesian coordinates (ccs) of the theory may be turned to the corresponding real ccs, in order to be perceived by human senses [3] (pp. 5-6). sr relates the frames of inertial observers (ios), via lts of linear spacetime. esr uses real spacetime (minkowski spacetime) (m4) endowed with lorentz metric (η) and the frames of two ios with parallel spatial axes are always related via lorentz boost (lb). but is known that lb is not closed linear transformation (clt). in contrast, lorentz transformation (combination of spatial euclidean rotation with lb) is clt (e.g. see [4], p. 41, eq. 1.104). thus, if three observers / frames (os/fs): oxyz, o΄x΄y΄z΄ and o΄΄x΄΄y΄΄z΄΄ are related, where the axes of o΄x΄y΄z΄ are parallel not only to the corresponding axes of oxyz, but also to the corresponding axes of ο΄΄x΄΄y΄΄z΄΄, then the axes of oxyz and ο΄΄x΄΄y΄΄z΄΄ are not parallel (figure 1). thus, the transitive attribute in parallelism (which is equivalent to the 5th euclidean postulate) is cancelled, when more than two os/fs are related. this consideration leads to successful results, such as thomas precession, which explains the fine structure of atomic spectra. but this happens only if we take successive observers o, o΄ and o΄΄ with thomas’ order [5]. the reversed order of this sequence yields a result with 200% relative error. 307 s. vossos, e. vossos, ch. g. massouros figure 1: correlation of three successive observers (frames), by using lorentz boost. the frame o΄x΄y΄z΄ has parallel axes to the corresponding of frame oxyz, moving with velocity (β1 c, 0, 0) wrt oxyz. the frame o΄΄x΄΄y΄΄z΄΄ has parallel axes to the corresponding of frame o΄x΄y΄z΄, moving with velocity (0, β2 c, 0) wrt o΄x΄y΄z΄. the correlation of the observers, by using lorentz boost, cancels the absolute character of parallelism. thus, the axes of frame o΄΄x΄΄y΄΄z΄΄ are not parallel to the corresponding of frame oxyz (thomas rotation). in this paper, we prove that there exists clt, which relates os/fs with parallel spatial axes (in case of ios, or observers that have the same acceleration). thus, the transitive attribute in parallelism is valid in complex three-dimensional euclidean space (e3) and the axes rotation that happens in real space, when more than two observers are related, is the equivalent phenomenon of the corresponding generalized real boost (grb) [3] (pp. 56). the clt is divided into two cases: one, where time depends on the position where the event happens, which can have real invariant speed (u) and another, where time is independent from the position and has u=∞. moreover, the demand that the clt is isometric, gives the ciltocst. if the metric of st is independent from the position of the event in st, we have the case of sr and the ciltocst may be applied globally, relating ios. thus, infinite number of sr-theories is produced (each one of which with the corresponding metric of st), keeping the esr-formalism. in the case that the metric of st depends on the position of the event in st, we have the case of gr and the ciltocst may be applied locally, relating os/fs with the same acceleration. thus, infinite number of gr-theories is produced (each one of which with the corresponding metric of st of ios), all of them keeping einsteinian grformalism. of course, zero acceleration leads to the corresponding sr. finally, we present the improper isometric lt in st endowed with euclidean, or lorentz, or generally any isotropic metric. 308 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric 2 the matrix of closed linear transformation of complex spacetime initially, we determine the matrix λ of active interpretation of the clt of complex st endowed with any metric. figure 2: two frames oxyz and o΄x΄y΄z΄ initially coincide. the second is moving with velocity (βc, 0, 0) wrt to oxyz. 2.1 motion in the x-direction we consider one unmoved o/f oxyz, measuring real spacetime and another o/f o΄x΄y΄z΄ with parallel spatial axes, moving to the right, along x-axis with velocity cc  ==  wrt o/f oxyz (figure 2), where c=299,792,458 m s−1 is the speed of light in vacuum and the frames initially coincide. supposing the next linear transformation cdt΄ = bcdt + adx + kdy + νdz (1) dx΄ = gcdt + fdx + δdy + θdz (2) dy΄ = g1cdt + f1dx + hdy + λdz (3) dz΄ = g2cdt + f2dx + ξdy + μdz, (4) we determine the coefficients with the following condition: the space has isotropy. rotating the coordinates system about the x-axis, by one negative right angle (figure 1), we correspond the new axes to the initial axes: t→t, t΄→t΄, x→x, x΄→x΄, y→-z, y΄→-z΄, z→y and z΄→y΄. thus, from (1), we have cdt΄ = bcdt + adx kdz + νdy. (5) (1) compared to (5), gives k=ν=0. besides, from (2) we have dx΄= gcdt + fdx δdz + θdy. (6) (2) compared to (6), gives δ=θ=0. besides, from (3) we obtain -dz΄= g1cdt + f1dx hdz + λdy. (7) (4) compared to (7), gives g2=-g1, f2 =-f1, ξ=-λ and μ=h. besides, from (4), we have dy΄= g2cdt + f2dx ξdz + μdy. (8) (3) compared to (8), gives g2=g1, f2=f1, ξ=-λ and μ=h. so, k=ν=δ=θ=g1=g2=f1=f2=0, ξ=-λ, μ=h and the transformation becomes 309 s. vossos, e. vossos, ch. g. massouros cdt΄ = bcdt +adx (9) dx΄ = gcdt + fdx (10) dy΄ = hdy + λdz (11) dz΄ = -λdy + hdz. (12) using matrices we have the active interpretation of the lt [4] (p. 6):                          − =                 z y x t h h fg ab z y x t d d d cd 00 00 00 00 d d d cd   , (13) or equivalently, dχ΄ = λ1(x) dχ , (14) where the base and the coordinates are     3210 eeeee  = ;             =               = z y x t x x x x x d d d cd d d d d d 3 2 1 0 (15) respectively. besides, the velocities are related in the following way: c c c x x x ab fg    + + = ; c c y x zy ab h    + + = ; c c z x zy ab h    + +− = . (16) 2.2 general linear transformation (motion in a random direction) we then consider one unmoved o/f oxyz and another o/f o΄x΄y΄z΄ with parallel spatial axes, moving with velocity (υx, υy, υz) wrt oxyz, where they initially coincide (figure 3). we rotate oxyz, in order to parallelize the unitary vector x̂ to the velocity vector   of the moving o΄x΄y΄z΄. this is sequentially achieved as following (figure 4). we firstly rotate the coordinate system oxyz about z-axis, through an angle θ: )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( kjizyx → .we then rotate the coordinate system )ˆ,ˆ,ˆ( kji about ĵ , by an angle ω: )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( kjikji → . thus, we have the transformation                      −− −=           z y x z y x    cossinsincossin 0cossin sinsincoscoscos r r r , (17) where 22 sin yx y    + = ; 22 cos yx x    + = ;     z=sin ;     22 cos yx + = . (18) as a result, the 3x3 matrix of (17) becomes 310 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric                         + + − + − ++ −=                   22 2222 2222 0 yx yx zy yx zx yx x yx y zyx r (19) and we define       = r r 0 01~ . (20) the unit means that time is not affected by the spatial rotation. moreover, the transformation )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → is analyzed to the following sequence of successive transformations: )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( kjizyx → ; )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( kjikji → ; )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxkji → . the above simple transformations have active interpretations: xrx ~ r = ; ( ) r1r xx x= ; r t~ xrx = , (21) respectively, where t~ r is the transpose matrix of r ~ . thus, the transformation )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → is actively interpreted: ( ) ( ) xxrrx x t dd ~~ d 1 == . (22) so, we calculate                               +−−−+− +−+−−− −−+−+− = hhfhfhf g hfhhfhf g hfhfhhf g aaa b zxzyyzx z xzyyzyx y yzxzyxx x zyx 2 2 22 22 2 2 222 2 )( )()()( )()()( )()()(                                                . (23) 311 s. vossos, e. vossos, ch. g. massouros figure 3: two frames oxyz and o΄x΄y΄z΄, which initially coincide. the second is moving with random velocity (υx, υy, υz) wrt to oxyz. figure 4: rotation of the initial frame oxyz, in order to achieve parallelization of vector x̂ to the velocity vector   of the moving observer o΄x΄y΄z΄ [ )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( kjiokjiozyxo →→ ]. 2.3 solution of the proper closed linear transformation of complex spacetime (correlation of two perpendicular moving observers / frames) we consider one unmoved o/f oxyz, another o/f o΄x΄y΄z΄ with parallel spatial axes, moving to the right, along x-axis with velocity (βc, 0, 0) wrt oxyz and also a third o/f ο΄΄x΄΄y΄΄z΄΄ with parallel spatial axes, moving upward, along y-axis with velocity (0, βc, 0) wrt οxyz (figure 5). all of them initially coincide and also β > 0, because   = . (24) 312 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric figure 5: two frames o΄x΄y΄z΄ and o΄΄x΄΄y΄΄z΄΄ moving with corresponding velocities (βc, 0, 0) and (0, βc, 0) wrt oxyz. the transformation )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → is analyzed to the following sequence: )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → ; )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → . the above simple transformations have active interpretations, respectively: xx x = −1 )(1 ; xx y )(2 = . thus, the transformation )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → is actively interpreted: xxx xy == −1 )(1)(2 . according to equation (23), it is             − = h h fg ab x   00 00 00 00 )(1 (25) and             − = h fg h ab y 00 00 00 00 )(2   . (26) thus, we have 313 s. vossos, e. vossos, ch. g. massouros                       ++ + − + −− − − − − == − 2222 2222 )(2 1 )(1)(2 00 00 00 00      h h h hh h agbf b agbf g agbf a agbf f π yxy . (27) or equivalently,                       ++−− − + − +− − − + − + − −− − + − +− − − = 22 2 22 2222 2222 2 2222             h h h h agbf b agbf g h f h fh agbf ag agbf gf h h hagbf bh agbf gh h a h ah agbf ab agbf bf π . (28) now, we calculate the velocity factor 4    of observer ο΄΄x΄΄y΄΄z΄΄ wrt ο΄x΄y΄z΄. equation (16) can be applied, because o/f ο΄x΄y΄z΄ is moving in the xdirection and observer ο΄΄ can be considered as the observed body. so, it is x4 = b g , b h y   = 4 , b z   −= 4 (29) and we obtain 22 2 22222 4 )(     ++= ++ = h g bb hg    . (30) replacing the above to (23), yields 4)( 4 λ=  . the condition that the transformation is closed, gives π = λ4. (31) comparing the matrices, element by element, we shall calculate the parameters α, f and g. the transformation must be reduced to gt, if one io has small velocity wrt another io. so, it must be b, g, f, h ≠ 0. we have two cases: (i) λ=0 and (ii) λ≠0. 2.3.1 the case of proper closed linear transformations of complex spacetime with λ=0 (time independent from the position, i.e. a=0). when λ=0, we compare matrices π and λ4 element by element and we also take into account (29). thus, we have: h4=1 (from element π33) and λ4=0 (from element π13). we then obtain f4=1 (from element π12). so, 314 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric h4=f4=1 ; λ4=0. (32) from elements π10 and π20, we get agbf gh h g gg − −= + 2 2 2 4     ; agbf gf h g hg − = + 2 2 2 4   (33) respectively. thus, f h g 2   −= . (34) from elements π01 and π02, we have agbf ab h g ga − −= + 2 2 2 4     ; h a h g ha = + 2 2 2 4   (35) respectively. so, agbf bh h g − −=   . (36) replacing (34) to the above, implies α=0, (37) for the clt, or g=0 for the non-closed lt (because (34) gives h=0 and matrix (25) cannot be identical). thus, element π11 gives f=h (38) and (34) becomes hg   −= . (39) finally, (23) yields the general matrix of clt:       − =             − − − = 3 t )( i o 00 00 00 000 hh b hh hh hh b z y x      ;           =           = 3 2 1        z y x ;           = 0 0 0 . (40) and the typical matrix clt (along x-axis):             − = h h hh b x 000 000 00 000 )(  , (41) where b=b(β) and h=h(β). 315 s. vossos, e. vossos, ch. g. massouros next, we calculate the corresponding ciltocst. the representation of the non-degenerate inner product in basis   e  =  3210 eeee  = ]ˆ,ˆ,ˆ,[ ^ zyxct is the real matrix of metric             = 33 22 11 00 000 000 000 000 g g g g g . (42) in this paper, we consider g00<0 [signature of spacetime: (-+++), or (----)]. the fundamental equation of isometry killing’s equation in a linear space (see e.g. [4], p.10, eq.1.15) is g΄= λτ g λ. (43) the element by element comparison of the above matrices gives 00 2 00332211 ,0 gbgggggg iiii ====== . (44) the isometry of spacetime [see e.g. [4], (p. 240)] is ds΄2= ds2, (45) or equivalently, j iji j iji xgxtgxgxtg dddcdddc 22 00 22 00 +=+ , (46) which combined with (44) and (40) gives b=1 for the clt, (47) or b = -1, ±i for the non-closed lt (because matrix (25) cannot be identical). so, since b=1, clt keeps time invariant. the einstein’s summation convention [4] (p. 3) was used in (46) and will be used in the equations that follow. besides, (44ii) becomes 0000 gg = . (48) thus, for any o/f the metric of the st in accordance with the complex lt is             =  0000 0000 0000 000 00 g g . (49) we observe that detgγ=0. so, this spacetime is degenerate [6] (p. 174). in order to calculate function h, we consider the unmoved o/f oxyz, another o/f o΄x΄y΄z΄ moving to the right, along the x-axis with velocity ( )0,0c, wrt oxyz and a third o/f o΄΄x΄΄y΄΄z΄΄ moving to the left, along the x-axis with velocity ( )0,0c,− wrt oxyz. thus, x΄=λ(x)(β)x and x΄΄=λ(x)(-β)x give xx xx = − − 1 ))(())((  . (50) also, the typical transformation of velocities (16) becomes 316 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric ( ) xx h  +−= c , yy h = , zz h = . (51) thus, the calculation of the velocity factor of observer o΄΄ wrt o/f o΄x΄y΄z΄ gives  h x 2 3 −= , 0 3 = y  , 0 3 = z  . (52) as the transformation is closed, we have ))(( 1 ))(())(( 3 xxx = − − , (53) or equivalently,             =                                3 3 33 000 000 002 0001 1 000 0 1 00 00 1 0001 000 000 00 0001 h h hhh h h h h h hh    , (54) from which it derives that 1 )(3 3 == hh for any value of β. as h depends only on the norm of velocity factor β, the only solution is h=1. hence, there derives the gt, which is expressed by the general matrix ( )       − =             − − − =  3 t i o1 100 010 001 0001      z y x ;           =           = 3 2 1        z y x , (55) and typical matrix along x-axis             − =  1000 0100 001 0001 )(  x , (56) which produces nps with invariant time and infinite universal speed. as unmoved o/f oxyz measures real velocity, the transformation matrix (λγ) contains only real numbers. so, the spacetime is limited to the real domain 4 r . moreover, this st (galilean spacetime) endowed with the galilean metric (49), is degenerate. 2.3.2 the case of proper closed linear transformations of complex spacetime with λ≠0 (time dependent on the position). when λ≠0, we compare matrices π and λ4 element by element and we also take into account (29). we start with (λ4)21+(λ4)12=π21+π12 and we obtain 317 s. vossos, e. vossos, ch. g. massouros ( ) 22 2 22 2 2 44 2      + − − −=           ++ − hagbf ag h g hfgh   . (57) we then use (λ4)32+(λ4)23=π32+π23, which gives ( ) 22 22 2 2 44 2    + − = ++ − h hf h g hfh  . (58) the combination of (57) with the above equation implies ( ) ( ) 22 2 22    + − − −= + − hagbf ag h hfg  . (59) also, (λ4)31+(λ4)13=π31+π13 gives ( ) 22 22 2 2 44 2     + − − =           ++ −− h h agbf b h g hfg   . (60) the combination of the above equation with (58) also gives ( ) ( ) 22 2 22  + − − = + −− h h agbf bh h hfg  . (61) we then add (59) and (61) and get f=h ; f4=h4. (62) moreover, from (λ4)11=π11 and (λ4)00=π00, we have 44 b agbf bf agbf bh h = − = − = , (63) which combined with (62) gives f = h = b. (64) we then use (λ4)22=π2, and we obtain 224 + = h fh h . (65) the combination of the above equation with (63) gives 22 + = − h f agbf b . (66) furthermore, (λ4)01=π01 and (λ4)02=π02 give respectively: 318 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric agbf ab h g ga − −= ++ 22 2 2 4      ; (67) 22 22 2 2 4    + = ++ h a h g a  . (68) the substitution of (68) to (67) gives ( ) agbf b h g − −= + 22   . (69) moreover, the combination of the above equation with (66) gives g = h  − . (70) finally, (66) combined with (64) and (70) yields     h 2 = . (71) the replacement of (64), (70), (71) and b  = , (72) makes the general matrix (23) equivalent to               −− −− −− =                             −− −− −− = 1 1 1 1 222 2 2 2 2 2 2 xyz xzy yzx zyx xy z xz y yz x zyx b bb bb bb bbb b λ                                 . (73) we also define           =           = 3 2 1        z y x ;           =           = 3 2 1        z y x ; ( )           − − − =           − − − = 0 0 0 0 0 0 12 13 23        xy xz yz . (74) it is noted that the antisymmetric matrix a(β) is related to the cross product (external product) [7] (p. 1048), because α(β) δ =      =− . (75) 319 s. vossos, e. vossos, ch. g. massouros thus, the four-vectors of two observers are related, by using the general matrix:         +− =               −− −− −− = )(3 2 222 ),( ai 1 1 1 1 1 λ         t xyz xzy yzx zyx bb . (76) besides, the typical matrix along x-axis is               − − = 100 100 001 001 2 ),)((      bx . (77) so, the proper closed linear transformation of complex spacetime (22) is                            −− −− −− =                 z y x t b z y x t xyz xzy yzx zyx d d d cd 1 1 1 1 d d d cd 222     . (78) the pure mathematical approach is simply obtained by replacing ct→x0. thus,                              −− −− −− =                   3 2 1 0 123 132 231 322212 3 2 1 0 d d d d 1 1 1 1 d d d d x x x x b x x x x     . (79) below, we calculate the corresponding ciltocst. for simplicity reasons, when we write i (the imaginary unit), we mean ±i: ii → ; ii →− . (80) the combination of the fundamental equation of isometry (43) (the killing’s equation in a linear space) with the above, gives ;)1()1(;; 2 2 22 00 222 00200332211    iiii ii g bgbg g ggggg  +=+===== (81) iiii gbg )1( 2 22   += . (82) so, for any o/f, the metric of spacetime in accordance with the ciltocst is isotropic:             = ii ii ii g g g g g 000 000 000 000 00 (83) 320 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric and also 00 2 g g ii= . (84) thus, ω2 is a real number. so, ω is a real or an imaginary number, which only depends on the metric of spacetime. besides, the metrics of the spacetime of two observers (frames) oxyz and o΄x΄y΄z΄, are related using the formulas gbg )1( 2 22   += ; (85) iiii gbg )1( 2 22   += . (86) so, 2 2 2 1 1   +  = ii ii g g b . (87) using the well-known lorentz γ-factor function )(2t )( 1 1 1 1 1 1      = − = − = − = , (88) equation (87) may be written as 2 )(i 2    ii ii g g b  = . (89) besides, the isometry of spacetime (45) combined with (89) and (78) gives iiii gg = . (90) thus, (89) gives 2 )( 2    i b = and we obtain 0 )(i = b . (91) moreover, (81iii) gives 0000 gg = . (92) this means that the metric of st must be affected in the same way for any o/f, in the case of ciltocst. equivalently, the observers that are related must be ios or must have the same acceleration. thus, the metric of spacetime in accordance with the ciltocst is             − − − − −=               =             = 2 2 2 00 2 00 000 000 000 0001 1000 0100 0010 000 1 000 000 000 000     gg g g g g g ii ii ii ii . (93) in case of sr (the frames are moved with constant velocity / the observes are ios), equation (84) becomes 321 s. vossos, e. vossos, ch. g. massouros 00i i2 i g g ii= . (94) so, the time and space metric’s coefficients are independent from the position and they are combined to produce ωι, which is the characteristic parameter of sr. the continuity of the metric of spacetime at the point λ=ω=0 and the matrices (49) and (93) gives 0limlim 0 i 0i == →→ iiii gg  . (95) we also observe that if the metric’s coefficients have the same sign [signature of spacetime: (----)], then the characteristic parameter ωι is a real number, in contrast with the case that the coefficients of metric have different signs [signature of spacetime: (-+++)], where the characteristic parameter ωι is an imaginary number. the representation of the non-degenerate inner product in basis     ]ˆ,ˆ,ˆ,[ ^ 3210 zyxcteeeee ==   for ios is the matrix             − − − − −=                 =               = 2 i 2 i 2 i 00i 2 i i i i i 00i i 000 000 000 0001 1000 0100 0010 000 1 000 000 000 000     gg g g g g g ii ii ii ii . (96) generally, equation (78) gives the active transformation of o/f oxyz to o/f o΄x΄y΄z΄ (if they are accelerated with the same acceleration):                            −− −− −− =                 z y x t z y x t xyz xzy yzx zyx d d d cd 1 1 1 1 d d d cd 222 )(i       . (97) the replacement ct→x0 gives the pure mathematical approach. thus,                              −− −− −− =                   3 2 1 0 123 132 231 322212 )(i 3 2 1 0 d d d d 1 1 1 1 d d d d x x x x x x x x       . (98) using vectors, the above transformation becomes ( )xtt    dcdcd 2 )(i +=   ; ( ) xtxx    dcddd )(i −−=   . (99) moreover, the general and typical matrices of ciltocst are, respectively: 322 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric         +− =               −− −− −− = )(3 2 )(i 222 )(i),( ai 1 1 1 1 1 λ           t xyz xzy yzx zyx ; (100)           = z y x     ; ( )           − − − = 0 0 0 xy xz yz     ;               − − = 100 100 001 001 2 )(i),)((      x . (101) the above matrices λ have the following properties: 4),( i=  ;           = 0 0 0 ; (102) ),( 1 ),(  − − = ; (103) det ),(  = 1. (104) in case of sr, the matrices form a new group (which corresponds to lorentz group) with elements d = ( ),( i   , b) ;             =               = z y x b b b t b b b b b 0 3 2 1 0 c , (105) and operation d1*d2 = ( ),( 2i  ),( 1i  , ),( 2i  b1+ b2), (106) where:  1 b is the μ-coordinate which is measured in o΄x΄y΄z΄, when all the coordinates  x , for ν=0, 1, 2, 3, in oxyz are equal to zero and  2 b is the μ-coordinate which is measured in o΄΄x΄΄y΄΄z΄΄, when all the coordinates  x , for ν=0, 1, 2, 3, in o΄xyz are equal to zero. the above operation expresses the successive transformations: )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → ; )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → . (107) these have active interpretations: 1),( 1i bxx +=  ; 2),( 2i bxx + =  . (108) respectively. thus, the transformation )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx → is actively interpreted: 21),(),(),( 2i1i2i bbxx ++=  . (109) 323 s. vossos, e. vossos, ch. g. massouros as ω2 is a real number, we observe that we always have real time. besides, the norm of the position four-vector for os/fs with the same acceleration / the same metric of st, is the corresponding invariant quantity ( )2222 00 222 2 222 00 2 ddcddc 1 ddcddd xtgxtgxgtgxgxs iiii t    −−−=      +=+== . (110) in the case of sr, the above equation becomes ( )22 i 22 00 222 2 i i 2 i 22 00i 2 ddcddc 1 ddcddd xtgxtgxgtgxgxs iiii t    −−−=         +=+== . (111) if ω is a real number [the coefficients of metric of time and space have the same sign: signature of spacetime: (----)], then s g g ii == 00  (112) with rs . thus, the transformation matrix (λ) contains only real numbers and the st is limited to the real domain r4. finally, the four-vectors of two os/fs have the same metric             − − − − −=               =             = 2 2 2 00 2 00 000 000 000 0001 1000 0100 0010 000 1 000 000 000 000 s s s g s g g g g g g ii ii ii ii (113) and their ccs are related via the matrix: ( )         +− =               −− −− −− = )(3 2 )(i 222 )(i, ai 1 1 1 1 1           s s ss ss ss sss t s xyz xzy yzx zyx s . (114) the typical matrix along x-axis is               − − = 100 100 001 001 2 )(i),)((       s s s sx . (115) if ω is an imaginary number [the coefficients of metric of time and space have different sign: signature of spacetime: (-+++)], then ω = 00 i g g ii − = ξ i ; 00 g g ii − = (116) with +  r . thus, the transformation matrix (λ) contains complex numbers and the spacetime is represented by the complex domain 3 cr . finally, the four-vectors of two os/fs have the same metric 324 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric            − −=               − =             = 2 2 2 00 200 000 000 000 0001 1000 0100 0010 000 1 000 000 000 000     gg g g g g g ii ii ii ii (117) and their ccs are related via the matrix: ( )         +− − =               −− −− −− −−− = )(3 2 )( 222 )(i, aii 1 1ii i1i ii1 1           t xyz xzy yzx zyx . (118) besides, the typical matrix along x-axis is               − − − = 1i00 i100 001 001 2 )()i,)((      x . (119) the substitution of (64), (70), (71) and (72) to (16) gives the velocities typical transformation of ciltocst: c c c 2 x x x    + − = ; c c 2 x zy y    + + = ; c c 2 x yz z    + − = . (120) for the purpose of finding a possible invariant speed (u) for os/fs with the same ω (or equivalently the same acceleration), we assume that a particle is moving to the right with velocity 1 eu  = ; u > 0. so, we have c c c 2 u u x    + − = ; 0= y  ; 0= z  . (121) according to the euclidean metric in the ordinary space e3, the norm of u is 2 22242 222 2 2 2 c c2c cc2 00c c c uu uu u u u     ++ +− =++      + − = , (122) which may be written as ( ) ( ) 0cc2c 3222444 =++−  uuu . (123) so, we obtain 2 2 2 c  −=u , (124) or equivalently, 2 2 2 c u −= . (125) 325 s. vossos, e. vossos, ch. g. massouros since norm u > 0, ω is an imaginary number ( + = ri, ) independent from the velocity (i.e. depends on the acceleration or equivalently the gravitation). thus, we have c 1  =u . (126) so, it is       =       − == − = + = + = u u u u       2)(2 2 22 )(i 1 1 1 1 1 1 1 1 (127) and (110) can also be written as                 +−−=+−== 2 2 22 00 2222 d c dcddddd x u tgxtugxgxs ii t  . (128) in the case of spacetime endowed with constant metric (or equivalently ios), equation (126) becomes c 1 i i  =c . (129) and we obtain the universal speed (ci) of the specific sr. besides, we have          =         − == − = + = + = i ii 2 i )(2 2 i 2 2 i 2 i )(i 1 1 1 1 1 1 1 1 c u c u       (130) and                   +−−=+−== 2 2 i 22 00 222 ii 2 d c dcddddd x c tgxtcgxgxs ii t  . (131) now, let us find the corresponding euclidean ciltocst. we initially define               =               =               = 3 2 1 0 3 2 1 0 d d d d 1 d d d cd 1 d d d d d x x x x z y x t x x x x x    ; 00 d 1 d xx   = , (132) where x0 ; 0 x are the zeroth-coordinates, by using the bases   e  =  3210 eeee  ;  e  =   3210 eeee  (133) of st endowed with metric (93) and euclidean metric, respectively. thus, 0000 gee =  ; 1 00 = ee  , (134) where dot “∙” is euclidean inner product [4] (p. 7). so, we understand that 326 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric 0 00 0 i e g e  − = . (135) then, the ciltocst (97) can be written as                              −− −− −− =                   z y x t z y x t xyz xzy yzx zyx d d d cd 1 1 1 1 1 d d d cd 1 )(i         , (136) or equivalently, dχ΄ω= )( ~ r dχω, (137) where               −− −− −− = 1 1 1 1 ~ )(i)( xyz xzy yzx zyx r       ;           =           = 3 2 1        z y x , (138) or equivalently, } a 0 {i ai 1~ )( 4)(i )(3 )(i)(         − +=         +− =           tt r . (139) the above matrix r ~ is a rotation matrix with the following properties: 4)( i ~ = r ;           = 0 0 0 , (140) )()( 1 )( ~~~  − − == rrr t , (141) det )( ~ r = 1. (142) the corresponding typical matrix along the x-axis in euclidean spacetime (e4) is             − − = 100 100 001 001 ~ )(i))((      xr . (143) now, using vectors the ciltocst of equation (136) becomes )d cd ( cd )(i x tt   +=       ;       −      −= x t xx   d cd dd )(i     , (144) or equivalently, 327 s. vossos, e. vossos, ch. g. massouros )d(dd 0 )(i 0 xxx   +=   ; ( ) xxxx   dddd 0 )(i −−=   . (145) thus, we have the euclidean metric of the position four-vector xω in e 4, and the corresponding invariant quantity is ( ) 2220222 2e 2 d 1 ddddc 1 ddd s g xxxtxgxs ii t =+=+==    , (146) according to (110ii). also, we observe that β-factor can be written as    iii i b x x x x === 00 d d d d ; 0 d d x x b i i = , (147) by using (132ii). the quantity bi is called β-factor and it can substitute the βfactor, in e4. then, equations (136-139) are rewritten:                              −− −− −− =                   3 2 1 0 123 132 231 321 )(i 3 2 1 0 d d d d 1 1 1 1 d d d d x x x x bbb bbb bbb bbb x x x x b   , (148) dχ΄ω= )( ~ b r dχω, (149)               −− −− −− = 1 1 1 1 ~ 123 132 231 321 )(i)( bbb bbb bbb bbb r bb  ;           = 3 2 1 b b b b , (150) } a 0 { ai 1~ )( 4)(i )(3 )(i)(         − +=         +− =   b b i b b r t b b t bb . (151) the above matrix r ~ is a rotation matrix having the following properties: 4)( i ~ =  r ; (152) )()( 1 )( ~~~ b t bb rrr − − == ; (153) det )( ~ b r = 1. (154) besides, the corresponding typical matrix along the x-axis in e4 is             − − = 100 100 001 001 ~ )(i))(( b b b b r bbx  . (155) note that the above transformation can be limited in the real spacetime (r4), because the corresponding lorentz γ-factor is positive for any real β-factor. 328 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric we observe that the above results could be obtained from the initial equations of e4: (136-146), when ω→1 and ct→xω. we also observe that r ~ reminds us of the contravariant electromagnetic tensor [3] (p. 14), [4] (p. 414):               −− −− −− = 0cc c0c cc0 0 12 m 3 1 m 3 m 2 2 m 3 m 1 321 ),( m m bbe bbe bbe eee f be , (156) where e and bm are the intensity of electric field and induction of magnetic field, respectively [4] (p. 396). actually, they are correlated via the formula 4)(i),()( i ~ m bbeb fr += ; j b j be )(i = ; c )(i m j bj b b  = . (157) thus, it is jj be m c= ; ( ) j j j j bbee mm 2 c= (158) where (158ii) is the same as the electromagnetic waves in vacuum, while (158i) means that the vectors of the induction of magnetic field and intensity of electric field are parallel. this reveals a hidden correlation between the spacetime and electromagnetism (maxwell equations). moreover, for any constant value of ωi (or more precisely for any constant metric, i.e. constant values of gi00 and giii), we have a specific ciltocst which correlates ios and the corresponding sr-theory. furthermore, the limit s → si → 0 in the equations (113-115) and their combination with (95) gives gt of complex spacetime with infinite universal speed. in the same way, the limit ξ → ξi → 0 in the equations (117-119) and their combination with (95), gives again gt. thus, the result when λ=0 (gt) is embedded to the case when λ≠0, if we take the corresponding limit to zero (λ→0, or equivalently, ω→0). besides, if one o/f has small velocity wrt another, the ciltocst (even been complex) is reduced to gt. the replacement ξ → ξi=1 to the equations (118) and (119), produces the lorentzian-einsteinian version of ciltocst (λβ) [7] (pp.1047-1048), which is expressed via the general matrix               −− −− −− −−− =  1ii i1i ii1 1 )( xyz xzy yzx zyx      (159) and the typical matrix along the x-axis 329 s. vossos, e. vossos, ch. g. massouros             − − − =  1i00 i100 001 001 ))((      x . (160) from (96), we take the corresponding metric of complex spacetime  iiii ggg ii 1000 0100 0010 0001 =            − =  , (161) which for giii=1 becomes the lorentz metric. thus, we have the sr-theory with universal speed being the speed of light in vacuum (ci=c) [7,8]. this theory gives results that are exactly the same as εsr, when only two os/fs are related. but the results are different, when more than two os/fs are related. besides, it calculates the fine structure peeks of atomic hydrogen’s spectrum [8] (p. 4) more accurately than esr. the explicit form of forward lorentzianeinsteinian ciltocst is ct΄ = γ(ct – βxx – βyy – βzz) (162) x΄ = γ(– βx ct + x + iβzy iβyz) (163) y΄ = γ(– βyct iβz x + y +iβxz) (164) z΄ = γ(– βzct + iβyx iβxy + z) (165) the explicit form of reverse lorentzian-einsteinian ciltocst is ct = γ(ct΄ + βxx΄ + βyy΄ + βzz΄) (166) x = γ(βx ct΄ + x΄ iβzy΄ + iβyz΄) (167) y = γ(βyct΄ + iβz x΄ + y΄ iβxz΄) (168) z = γ(βzct΄ iβyx΄ + iβxy΄ + z΄) (169) when the metric of st depends on the position of the event in spacetime (gr), the transformation is applied locally, not globally (correlating os/fs with the same acceleration / gravitation). a metric is in accordance with the ciltocst, if only the limit of vanishing acceleration leads to the corresponding sr. thus, the usage of (84) and (94) leads to iiii a gg i 0 lim = →  ; (170) 00i2 i i 2 0 00 0 limlim g gg g iiii aa === →→   . (171) 3 proper time – special and general relativity let p be a particle moving with velocity p   wrt observer o ( p  wrt observer o΄) in spacetime. the generalized definition of proper time (τ) is 330 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric 2 00 2 2 c d d g s = . (172) using (84) and (110), we have       +=      += 222 22 2 222 22 00 2 ddc 1 c ddc 1 c d xtxt g g ii      , (173) or equivalently,         +=+= 2 2 2 22 2 2 22 c 1dd c dd p txt     . (174) thus, the relation between the time and the proper time is 2 )(i2 2 d d p t   = . (175) for ω=s with rs , there does not exists real invariant speed (u) and the γ-factor is always positive. so, )(i d d ps t   = . (176) when ω=ξi with r , there exists a real u. if the speed of particle is less than the invariant speed ( p  <u), then the γ-factor is positive again. thus, )( d d p t   = . (177) for gt with s→ξ→0 (the limit of degenerate spacetime), 1 )()( == ppis   . so, dτ = dt΄ = dt (time is invariant) as we know in nps. in the case of st with constant metric (ios), equation (175) becomes 2 )(i2 2 id d p t   = . (178) thus, the lorentzian-einsteinian version of ciltocst with ωi=i (ξi=1), gives u=ci=c. if the speed of a particle is less than the speed of light in vacuum ( c p   ), then γ-factor is positive again. thus, )( d d p t   = . (179) and we have the same result as the esr. in any case, using proper time, we can define four-velocity, fourmomentum etc, building the whole structure of generalized sr and gr. 331 s. vossos, e. vossos, ch. g. massouros 4 the results of closed linear transformation of complex spacetime discussion in this section, we present the typical matrix λ(x)(β), the general matrix λ(β), the covariant matrix of spacetime metric g, the invariant speed u and the domain of the coordinates c4 that corresponds to the transformation of a contravariant infinitesimal four-vector in spacetime: dx΄=λdx . (λ(x)(β), λ(β), g, u, c 4) | | b  = = ? λ≠0 | λ=0 _______________|_______________ | | ( )               − − = 100 100 001 001 2 ))((      bx , ( ) ( ) ( ) ( ) ( )              − =       h h hh b x 000 000 00 000 ))(( , ( ) ( )         +− = )(3 2 ai 1     t b ,  ir,u . ( ) ( ) ( )         − = 3 t )( i o     hh b , u =+∞. | | | isometry | isometry | |               − − = 100 100 001 001 2 )(i),)((      x ,             − =  1000 0100 001 0001 ))((  x , ( )         +− = )(3 2 )(i ai 1      t ,       − =  3 t )( i o1   ,             − − − − −= 2 2 2 00 000 000 000 0001    gg ,             =  0000 0000 0000 000 00 g g , u =+∞, r 4. | | ω=? | 332 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric ii =  | r= s ____________|___________________ | |               − − − = 1i00 i100 001 001 2 )(),)(i(      x ,               − − = 100 100 001 001 2 )(i),)((       s s s sx , ( )         +− − = )(3 2 )(,i aii 1      t , ( )         +− = )(3 2 )(i, ai 1      s s t s ,            − −= 2 2 2 00 000 000 000 0001    gg ,             − − − − −= 2 2 2 00 000 000 000 0001 s s s gg , + = rc 1  u , 3 rcx . u does not exist, 4 rx . the results may be applied to any complex or real isotropic space of dimension four (spacetime), endowed with the corresponding metric, whose elements (four-vectors) have spatial part (vector) with euclidean metric. we simply put ct→x0 ; x→x1 ; y→x2 ; z→x3. (180) so, the β-factor is written as 0 d d x x i i = . (181) new spaces are produced from the initial space: (i) with derivation of the initial four-vector wrt an invariant quantity, such as ‘proper time’, or (ii) with multiplication of the initial four-vector with invariant quantity, such as ‘mass’ (see e.g. [4] p. 109). moreover, there exist applications beyond physics as biometry, econometrics etc, producing suitable vectors and four-vectors. the specific value ωi=i (ξi=1) gives the lorentzian-einsteinian version of ciltocst endowed with lorentz metric (for giii=1), which produces the lorentzian complex relativity theory, which (i) is using the lorentzian-einsteinian version of ciltocst instead of lorentz transformation that is used by esr, (ii) is using complex cartesian coordinates creating a generalized euclidean geometry, (iii) creates the new group of ciltocst with elements complex matrices, instead of the lorentz group of esr, (iv) can produce the lorenz boost of esr [3] (p. 6), 333 s. vossos, e. vossos, ch. g. massouros (v) is successfully applied to mechanics and electromagnetism, (vi) maintains the classical physical laws and the formalism of ert, (vii) is in accordance with quantum mechanics, (viii) gives results that are exactly the same as ert, if only two observers are related, (ix) gives different results than ert, when more than two observers are related, and (x) calculates with better accuracy the fine structure peeks of spectrum of atomic hydrogen than esr [8] (p. 4). finally, we can consider that the value of ωi depends on the cosmic time (tc). thus, lorentz metric is valid, only for ‘events’ near to (nowadays, earth). this could be an explanation for the problem of dark matter and dark energy. 5 improper isometric linear transformations in spacetime endowed with euclidean, or lorentz, or generally isotropic metric in the derivation of proper closed isometric lt (icltocst) (↓↑), we have chosen positive b [the lower sign (↓) in (91)] and via (64) f is positive (↑), too. so, there have remained the following three (3) improper non-closed isometric lts (which do not contain the identity transformation) [see also the lorentzian-einsteinian version [7] (pp. 1049-1050)]: (i) space inversion non-closed isometric linear transformation (↓↓) in isotropic st and e4 with corresponding matrices ( )1~detdetλ −== r : ( )         −− = )(3 t2 i)( ai 1 λ      ; (182) ( ) ( ) )( )(3 t i )(3 t i)( ~ ai 1 ai 1~ b b b r b b r =         −− =         −− =       . (183) the respective typical transformations along the x-axis, have ( )               − −− − = 100 100 001 001 λ 2 i))((      x ; (184) ( ) ( ) ))((ii))(( ~ 100 100 001 001 100 100 001 001 ~ bxbx r b b b b r =             − −− − =             − −− − =        . (185) 334 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric (ii) time inversion non-closed isometric linear transformation (↑↑) in isotropic st and e4 with corresponding matrices ( )1~detdetλ −== r : ( )         +− −− =  )(3 t2 i)( ai 1 λ      ; (186) ( ) ( ) )( )(3 t i )(3 t i)( ~ ai 1 ai 1~ b b b r b b r =         +− −− =         +− −− =       . (187) the respective typical transformations along the x-axis, have ( )               − − −− = 100 100 001 001 λ 2 i))((      x ; (188) ( ) ( ) ))((ii))(( ~ 100 100 001 001 100 100 001 001 ~ bxbx r b b b b r =             − − −− =             − − −− =        . (189) (iii) spacetime inversion non-closed isometric linear transformation (↑↓) in isotropic st and e4 with corresponding matrices ( )1~detdetλ == r : ( )         −− −− = )(3 t2 i)( ai 1 λ      ; (190) ( ) ( ) )( )(3 t i )(3 t i)( ~ ai 1 ai 1~ b b b r b b r =         −− −− =         −− −− =       . (191) the respective typical transformations along the x-axis, have ( )               − −− − −− = 100 100 001 001 λ 2 i))((      x ; (192) ( ) ( ) ))((ii))(( ~ 100 100 001 001 100 100 001 001 ~ bxbx r b b b b r =             − −− − −− =             − −− − −− =        . (193) these matrices are exactly the opposite of the corresponding proper icltocst. the above can be compared to the case of lorentz boost [4] (pp. 30-31), [7] (pp. 1050-1052), where we have: 335 s. vossos, e. vossos, ch. g. massouros (a) space inversion lorentz boost in m4 and e4 with corresponding matrices ( )1~detdetλ ll −== r : ( ) ( ) ( ) ( )           − −−− = t t3 t )l( 1 i          ; (194) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )(li 3i ii 3 )(l ~ 1 i 1 ii i ~ bt t b b t bb t t t r bb bb b b r =           − −−− − =           − −−− − =             . (195) the respective typical transformations along the x-axis, have ( ) ( ) ( ) ( ) ( )             − − −− = 1000 0100 00 00 )l(      x ; (196) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )bx bb bb x r b b r )(l ii ii )(l ~ 1000 0100 00 00 1000 0100 00i 00i ~ =             − − −− − =             − − −− − =        . (197) (b) time inversion lorentz boost in m4 and e4 with corresponding matrices ( )1~detdetλ ll == r : ( ) ( ) ( ) ( )           + − − = t t t          1 i 3 )l( ; (198) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )(lt t i 3i t ii t t3 t )(l ~ 1 i 1 ii i ~ bb b bb r bb bb b b r =           + − −− =           + − −− =             . (199) the respective typical transformations along the x-axis, have ( ) ( ) ( ) ( ) ( )             − − = 1000 0100 00 00 )l(      x ; (200) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )bx bb bb x r b b r )(l ii ii )(l ~ 1000 0100 00 00 1000 0100 00i 00i ~ =             − −− =             − −− =        . (201) (c) spacetime inversion lorentz boost in m4 and e4 with corresponding matrices ( )1~detdetλ ll −== r : 336 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric ( ) ( ) ( ) ( )           + +− −− = t t3 t )l( 1 i          ; (202) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )(lt t i 3i ii t t3 )(l ~ 1 ii i 1 ii i ~ bb b t bb t r bb bb b b r =           + +− − =           + +− − =             . (203) the respective typical transformations along the x-axis, have ( ) ( ) ( ) ( ) ( )             − − −− = 1000 0100 00 00 )l(      x ; (204) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )bx bb bb x r b b r )(l ii ii )(l ~ 1000 0100 00 00 1000 0100 00i 00i ~ =             − − − =             − − − =        . (205) 5 conclusions in a 3d complex ‘space’ endowed with euclidean metric, we consider one frame oxyz, where a ‘position’ vector has real cartesian coordinates. another real independent variable (‘time’) and the aforementioned coordinates produce a real four-vector. there exist two cases of closed linear ‘spacetime’ transformation of this real four-vector: one with the ‘time’ depending on the position, where the ‘event’ happens and another with the ‘time’ being independent from the position. the first case can have real invariant ‘speed’ (u), in contrast to the second case, which has only infinite u. moreover, for transformation having isometry, the first case transformation matrix is totally calculated and contains a parameter ω, with ω2= gii/g00 (the ratio of coefficients of ‘spacetime’ metric) in addition to the ‘velocity’ of the frame o΄x΄y΄z΄ wrt oxyz. the second case is turned to galilean transformation (gt). the assumption that ω→0 in the first case yields gt. so, in isometry, the second case is embedded to the first case transformation. besides the first case is divided to two types: one type, where ‘time’ and ‘space’ have ‘spacetime’ metric coefficients with different signs [signature of spacetime: (-+++)], which leads to complex 3d ‘space’ with real u. the second type, where ‘time’ and ‘space’ have metric coefficients with the same sign [signature of spacetime: (----)], leads to real 3d ‘space’ without u. time remains real, in both cases. if the metric is independent from the ‘position’ of the ‘event’ in ‘spacetime’, it is ω=ωi=constant and we have the case of ‘special relativity’ 337 s. vossos, e. vossos, ch. g. massouros (‘sr’) and the transformation can be applied globally, relating ‘inertial observers / frames’ (‘ios/fs’). thus, infinite number of ‘srs’ are produced (each one with the corresponding metric), all of them keeping einsteinian srformalism. in the case that metric depends on the ‘position’ of the ‘event’ in ‘spacetime’, we have the ‘general relativity’ (‘gr’) and the transformation may be applied locally, relating ‘accelerated observers / frames’. thus, infinite number of ‘grs’ are produced (each one with the corresponding metric of ios’ spacetime), all of them keeping einsteinian gr-formalism. of course, vanishing ‘acceleration’ leads to the corresponding ‘sr’. this new modeling of study allows studying einsteinian relativity theory, newtonian physics (nps), or any other theory of physics that is in accordance with closed linear spacetime transformations simultaneously. this is achieved, because the coefficients of spacetime metric are contained in the transformation matrix. besides, nps is obtained, not only by the low velocity limit, but also by the zero limit of the space coefficient of spacetime metric (gii→0), or equivalently ω→0. finally, we can consider that the value of ωi depends on the cosmic time (tc) and lorentz metric is valid only for ‘events’ near to the (nowadays, earth). this could be used for the explanation of dark matter and dark energy. conflict of interests the authors declare that there is no conflict of interests regarding the publication of this paper. references [1] d.t. finkbeiner. introduction to matrices and linear transformations. san fancisco: freeman and company. 1960. [2] d. kleppner. a short history of atomic physics in the twentieth century. rev. mod. phys. 71 (2), s78-s84. 1999. http://www.cstam.org.cn/upfiles/200732678933.pdf. [3] s. vossos, e. vossos. unification of newtonian physics with einstein relativity theory, by using generalized metrics of complex spacetime and application to the motions of planets and stars, eliminating dark matter. j. phys.: conf. ser. 1141, 012128. 2018. doi: 10.1088/1742-6596/1141/1/012128. [4] m. tsamparlis. special relativity: an introduction with 200 problems and solutions. berlin heidelberg: springer-verlag. 2010. isbn: 978-3-642-03836-5, e-isbn: 978-3-642-03837-2. [5] l.h. thomas. the motion of the spinning electron. nature (london), 117, 514. 1926. doi: 10.1038/117514a0. 338 closed isometric linear transformations of complex spacetime endowed with euclidean or lorentz or generally isotropic metric [6] w. rindler. relativity: special, general and cosmological. new york, usa: oxford university press. 2006. isbn: 978-0-19-856732-5. [7] e. vossos, s. vossos, c. g. massouros. closed linear transformations of complex space-time endowed with euclidean or lorentz metric. ijpam, 442020, 1033-1053. 2020. [8] s. vossos, e. vossos. euclidean complex relativistic mechanics: a new special relativity theory. j. phys.: conf. ser., 633, 012027. 2015. doi: 10.1088/1742-6596/633/1/012027. 339 ratio mathematica volume 47, 2023 families of mappings satisfying a mixed implicit relation rajesh kumar* sanjay kumar† abstract a fixed point for a suitable map or operator is identical to the presence of a solution to a theoretical or real-world problem. as a result, fixed points are crucial in many fields of mathematics, science and engineering. the purpose of this paper is to prove unique common fixed point theorems for families of weakly compatible mappings. given mappings satisfy common limit range property and a mixed implicit relation. our results generalize, extend and improve the results of imdad (2013) and popa (2018). we provide an application for integral type contraction condition. an example is also mentioned to check the authenticity of our results. keywords: common fixed point, weakly compatible mappings, mixed implicit relation, almost altering distance, common limit range property. 2020 ams subject classifications: 47h10, 54h25. 1 *department of mathematics, institute of higher learning, bps mahila vishwavidyalaya, khanpur kalan, sonipat-131305, haryana (india); rkdubaldhania@gmail.com. †department of mathematics, deenbandhu chhotu ram university of science and technology, murthal, sonepat-131039, haryana (india); sanjaymudgal2004@yahoo.com 1received on august 26, 2022. accepted on february 10, 2023. published on march 5, 2023. doi:10.23755/rm.v41i0.839. issn: 1592-7415. eissn: 2282-8214. 219 r. kumar and s. kumar 1 introduction and preliminaries fixed point theory is an important tool of modern mathematics as it helps to find a unique fixed point of multi-valued and single-valued mappings by restricting the condition of the domain of the function. it also helps to find the results of many differential as well as integral equations which can further be used to solve many industrial based problems. the most popular tool in fixed point theory is banach contraction [6] principle which states that every contraction mapping on a complete metric space has a unique fixed point. various authors have extended and generalized this principle in various directions. in 1976, jungck [9] used the concept of commuting maps to prove a common fixed point theorem. several authors have investigated various concepts of minimal commuting maps. a pair of self-mappings (p,q) on a metric space (x ,d) is said to be compatible [10] if limn→∞ d(pqun,qpun) = 0, whenever {un} is a sequence in x such that limn→∞ pun = limn→∞ qun = u for some u ∈ x . a pair of self-mappings (p,q) on a metric space (x ,d) is called weakly compatible [11] if p and q commute at their points of coincidence. pant ([13], [14], [15]) initiated the study of common fixed points for non-compatible mappings. further, aamri and el-moutawakil [1] introduced (e.a) property as a generalization of non-compatible mappings. a pair of self-mappings (p,q) on a metric space (x ,d) is said to satisfy (e.a) property [1] if there exists a sequence {un} in x such that limn→∞ pun = limn→∞ qun = u , for some u ∈ x . in 2011, sintunavarat and kumam [22] introduced the concept of common limit range property. definition 1.1. [22] a pair of self-mappings (p,q) on a metric space (x ,d) is said to satisfy the common limit range property with respect to q, denoted by clrq, if there exists a sequence {un} in x such that lim n→∞ pun = lim n→∞ qun = u, for some u ∈ q(x). thus one can note that the mappings p and q satisfying property (e.a) along with the closedness of the subspace q(x) always have clrq property with respect to q. in 2013, imdad et al. [8] extended the notion of common limit range property for pairs of self mappings. definition 1.2. [8] two pairs of self-mappings (p,q) and (s,t ) on a metric space (x ,d) are said to satisfy common limit range property with respect to q and t , denoted by (clr)(q,t ), if there exists two sequences {un} and {vn} in x such that lim n→∞ pun = lim n→∞ qun = lim n→∞ svn = lim n→∞ t vn = t, 220 families of mappings satisfying a mixed implicit relation where t ∈ q(x) ∩ t (x). in 2018, popa et al. [17] introduced a new type of limit range property as follows. definition 1.3. [17] let p,q, t be self mappings of a metric space (x ,d). the pair (p,q) is said to satisfy common limit range property with respect to t if there exists a sequence {un} in x such that limn→∞ pun = limn→∞ qun = u, for some u ∈ q(x) ∩ t (x). now we extend the definition 1.3 for families of mappings. definition 1.4. let q1,q2,...,q2n and p be self mappings of a metric space (x ,d). the pair (p,q1q3...q2n−1) is said to satisfy common limit range property with respect to q2q4...q2n if there exists a sequence {un} in x such that limn→∞ pun = limn→∞ q1q3...q2n−1un = u, for some u ∈ q1q3...q2n−1(x)∩ q2q4...q2n(x). remark 1.1. [17] let p,q, s and t be self mappings of a metric space (x ,d). if the pairs (p,q) and (s,t ) satisfy the common limit range property with respect to q and t , then (p,q) satisfy the limit range property with respect to t , but the converse does not hold. boyd and wong [5] introduced ϕ contraction condition and generalized the banach contraction principle using this contraction. a self mapping p on a complete metric space (x ,d) is said to satisfy ϕ contraction if d(pα,pβ) ≤ ϕ(d(α,β)), for all α,β ∈ x , where ϕ : [0,∞) → [0,∞) is an upper semi continuous function from right such that 0 ≤ ϕ(t) < t for all t > 0. the theorems of existence of fixed points for self mappings in hilbert spaces satisfying ϕ-weak contraction were studied by alber and guerre-delabriere [3]. further rhoades [21] extended this concept in complete metric space. some fixed point results are proved in [7], [8] and in other papers for mappings with common limit range property satisfying (ϕ,ψ)-weak contractive conditions. the following theorem is proved in [8]. theorem 1.1. [8] let p , q, s and t be self mappings of a metric space (x ,d) satisfying ψ(d(px,qy)) ≤ ψ(m(x,y)) − ϕ(m(x,y)), for all x,y ∈ x and for some ϕ, ψ, where m(x,y) = max{d(sx,ty),d(sx,px),d(ty,qy),d(sx,qy),d(ty,px)} and ϕ, ψ : [0,∞) → [0,∞) such that ϕ is a lower semi-continuous function and ϕ−1(0) = 0 and ψ is a non-decreasing continuous function with ψ−1(0) = 0. if the pairs (p,q) and (s,t) satisfy the (clr)(s,t) property and are weakly compatible, then p , q, s and t have a unique common fixed point. 221 r. kumar and s. kumar definition 1.5. [12] an altering distance is a function ψ : [0,∞) → [0,∞) satisfying: (ψ1): ψ is increasing and continuous, (ψ2): ψ(t) = 0 if and only if t = 0. definition 1.6. [18] a function ψ : [0,∞) → [0,∞) is an almost altering distance if it satisfies: (ψ′1): ψ is continuous, (ψ′2): ψ(t) = 0 if and only if t = 0. example 1.1. define a function ψ : [0,∞) → [0,∞) by ψ(t) = { 2t, t ∈ [0,1] 1 1+t , t ∈ (1,∞). here we note that every altering distance is an almost altering distance, but converse is not true. various authors have unified several common fixed point theorems by using implicit functions. in 2008, ali and imdad [2] introduced a new class of implicit functions. definition 1.7. [2] let f be the family of lower semi-continuous functions f : r6+ → r which are satisfying: (f1) for all u > 0, f(u,0,u,0,0,u) > 0; (f2) for all u > 0, f(u,0,0,u,u,0) > 0; (f3) for all u > 0, f(u,u,0,0,u,u) > 0; definition 1.8. [17] let fd be the set of all lower semi-continuous functions f : r6+ → r which are satisfying: (f1d) for all u > 0, f(u,0,u,0,0,u) ≥ 0; (f2d) for all u > 0, f(u,0,0,u,u,0) ≥ 0; (f3d) for all u > 0, f(u,u,0,0,u,u) ≥ 0; now we provide some examples in support of definition 1.8. 1. let f(u1, ...,u6) = u1 − tmax{u2,u3,u4,u5,u6}, where t ∈ [0,1]. 2. let f(u1, ...,u6) = u1 − tmax{u2,u3,u4, u5+u63 }, where t ∈ [0,1]. 3. let f(u1, ...,u6) = u1 − α max{u2,u3,u4} − β(u5 + u6), where α, β ≥ 0 and α + 2β < 1. 4. let f(u1, ...,u6) = u1 − α max{u2,u3,u4, 12(u5 + u6), u3u4 1+u2 , u5u6 1+u1 }, where α ∈ [0,1). 222 families of mappings satisfying a mixed implicit relation 5. let f(u1, ...,u6) = u1 − max{cu2,cu3,cu4,au5 + bu6)}, where c > 0, a,b ≥ 0 and a + b + c ≤ 1. definition 1.9. [17] let gd be the set of all lower semi-continuous functions g : r5+ → r such that g(t1, ..., t5) > 0 if one of t1, ..., t5 > 0. the following functions belong to the set gd. 1. g(t1, ..., t5) = max{t1, ..., t5}. 2. g(t1, ..., t5) = max{t1, t2+t32 , t4+t5 2 }. 3. g(t1, ..., t5) = t21 + t 2 2 + t 2 3 + t 2 4 + t 2 5. 4. g(t1, ..., t5) = 1 t1+t2+t3+t4+t5 . definition 1.10. [17] a function ϕ(u1, ...,u6, t1, ..., t5) = f(u1, ...,u6)+g(t1, ..., t5) is called a mixed implicit relation. the aim of this paper is to prove general fixed point theorems for families of weakly compatible mappings with common limit range property satisfying a mixed implicit relation. our results generalize, extend and improve the results of popa [17] and imdad [8]. 2 main results in 2018, popa et al. [17] proved the following theorem. theorem 2.1. [17] let (x ,d) be a metric space and p,q,s and t be four self mappings on x satisfying f(ψ(d(px,qy)),ψ(d(sx,ty)),ψ(d(sx,px)),ψ(d(ty,qy)),ψ(d(sx,qy)), ψ(d(ty,px))) + g(ψ(d(sx,ty)),ψ(d(sx,px)),ψ(d(ty,qy)),ψ(d(sx,qy)), ψ(d(ty,px))) ≤ 0, for all x,y ∈ x , for some f ∈ fd, g ∈ gd and ψ is an almost altering distance. if the pairs (p,s) and (q,t) are weakly compatible and (p,s) and t satisfy (clr)(p,s)t property, then p,q,s and t have a unique common fixed point. now we extend the theorem 2.1 for any even number of weakly compatible mappings. 223 r. kumar and s. kumar theorem 2.2. let q1,q2, ...,q2n,p0 and p1 be self mappings on a metric space (x ,d), satisfying the following conditions: (c1) q2(q4...q2n) = (q4...q2n)q2, q2q4(q6...q2n) = (q6...q2n)q2q4, ... q2...q2n−2(q2n) = (q2n)q2...q2n−2, p1(q4...q2n) = (q4...q2n)p1, p1(q6...q2n) = (q6...q2n)p1, ... p1q2n = q2np1, q1(q3...q2n−1) = (q3...q2n−1)q1, q1q3(q5...q2n−1) = (q5...q2n−1)q1q3, ... q1...q2n−3(q2n−1) = (q2n−1)q1...q2n−3, p0(q3...q2n−1) = (q3...q2n−1)p0, p0(q5...q2n−1) = (q5...q2n−1)p0, ... p0q2n−1 = q2n−1p0, (c2) the pairs (p0,q1...q2n−1) and (p1,q2...q2n) are weakly compatible and (p0,q1....q2n−1) and q2...q2n satisfy (clr)(p0,q1...q2n−1)q2...q2n property, (c3) f(ψ(d(p0x,p1y)),ψ(d(q1q3...q2n−1x,q2q4...q2ny)),ψ(d(q1q3...q2n−1x,p0x)), ψ(d(q2q4...q2ny,p1y)),ψ(d(q1q3...q2n−1x,p1y)),ψ(d(q2q4...q2ny,p0x))) + g(ψ(d(q1q3...q2n−1x,q2q4...q2ny)),ψ(d(q1q3...q2n−1x,p0x)), ψ(d(q2q4...q2ny,p1y)),ψ(d(q1q3...q2n−1x,p1y),ψ(d(q2q4...q2ny,p0x))) ≤ 0, for all x,y ∈ x , some f ∈ fd, g ∈ gd and ψ is an almost altering distance. then q1, q2,...,q2n, p0 and p1 have a unique common fixed point in x . proof. let q′1 = q1q3...q2n−1 and q′2 = q2q4...q2n. since (p0,q′1) and q′2 satisfy (clr)(p0,q′1)q′2 property, there exists a sequence {un} in x such that lim n→∞ p0un = lim n→∞ q′1un = lim n→∞ q1q3...q2n−1un = z, where z ∈ q′1(x) ∩ q′2(x) = q1...q2n−1(x) ∩ q2...q2n(x). since z ∈ q2q4...q2n(x), there exists u ∈ x such that z = q2q4...q2nu. using 224 families of mappings satisfying a mixed implicit relation (c3) with x = un and y = u, we get f(ψ(d(p0un,p1u)),ψ(d(q1q3...q2n−1un,q2q4...q2nu)),ψ(d(q1q3...q2n−1un,p0un)), ψ(d(q2q4...q2nu,p1u)),ψ(d(q1q3...q2n−1un,p1u)),ψ(d(q2q4...q2nu,p0un))) + g(ψ(d(q1q3...q2n−1un,q2q4...q2nu)),ψ(d(q1q3...q2n−1un,p0un)), ψ(d(q2q4...q2nu,p1u)),ψ(d(q1q3...q2n−1un,p1u),ψ(d(q2q4...q2nu,p0un))) ≤ 0. taking limits as n → ∞, we have f(ψ(d(z,p1u)),0,0,ψ(d(z,p1u)),ψ(d(z,p1u)),0) + g(0,0,ψ(d(z,p1u)), ψ(d(z,p1u)),0) ≤ 0. if d(z,p1u) > 0, then g(0,0,ψ(d(z,p1u)),ψ(z,p1u),0) > 0, which implies that f(ψ(d(z,p1u)),0,0,ψ(d(z,p1u)),ψ(d(z,p1u),0) < 0, a contradiction of (f2d). hence d(z,p1u) = 0 i.e., z = p1u = q2q4...q2nu. since (p1,q2q4...q2n) is weakly compatible, we have p1z = p1q2q4...q2nu = q2q4...q2np1u = q2q4...q2nz. since z ∈ q1q3...q2n−1(x), which implies z = q1q3...q2n−1v for some v ∈ x . on putting x = v and y = u in (c3), we have f(ψ(d(p0v,p1u)),ψ(d(q1q3...q2n−1v,q2q4...q2nu)),ψ(d(q1q3...q2n−1v,p0v)), ψ(d(q2q4...q2nu,p1u)),ψ(d(q1q3...q2n−1v,p1u)),ψ(d(q2q4...q2nu,p0v))) + g(ψ(d(q1q3...q2n−1v,q2q4...q2nu)),ψ(d(q1q3...q2n−1v,p0v)), ψ(d(q2q4...q2nu,p1u)),ψ(d(q1q3...q2n−1v,p1u),ψ(d(q2q4...q2nu,p0v))) ≤ 0. on simplification, we get f(ψ(d(p0v,z)),0,ψ(d(p0v,z)),0,0,ψ(d(p0v,z))) + g(0,ψ(d(p0v,z)),0,0, ψ(d(p0v,z))) ≤ 0. if d(p0v,z) > 0, then g(0,ψ(d(p0v,z)),0,0,ψ(d(p0v,z))) > 0. 225 r. kumar and s. kumar therefore, we obtain f(ψ(d(p0v,z)),0,ψ(d(p0v,z))0,0,ψ(d(p0v,z)) < 0, a contradiction of (f1d). hence d(p0v,z) = 0, which implies that z = p0v = q1q3...q2n−1v. since (p0,q1q3...q2n−1) is weakly compatible, we get p0z = p0q1q3...q2n−1v = q1q3...q2n−1p0v = q1q3...q2n−1z. now, we prove that z = p1z. on putting x = v and y = z in (c3), we get f(ψ(d(p0v,p1z)),ψ(d(q1q3...q2n−1v,q2q4...q2nz)),ψ(d(q1q3...q2n−1v,p0v)), ψ(d(q2q4...q2nz,p1z)),ψ(d(q1q3...q2n−1v,p1z)),ψ(d(q2q4...q2nz,p0v))) + g(ψ(d(q1q3...q2n−1v,q2q4...q2nz)),ψ(d(q1q3...q2n−1v,p0v)), ψ(d(q2q4...q2nz,p1z)),ψ(d(q1q3...q2n−1v,p1z),ψ(d(q2q4...q2nz,p0v))) ≤ 0, which implies that f(ψ(d(z,p1z)),ψ(d(z,p1z)),0,0,ψ(d(z,p1z)),ψ(d(z,p1z))) + g(ψ(d(z,p1z)),0,0, ψ(d(p1z,z)),ψ(d(z,p1z))) ≤ 0. if d(z,p1z) > 0, then g(ψ(d(z,p1z)),0,0,ψ(d(p1z,z)),ψ(d(z,p1z))) > 0. thus from above, we get f(ψ(d(z,p1z)),ψ(d(z,p1z)),0,0,ψ(d(z,p1z)),ψ(d(z,p1z))) < 0, a contradiction of (f3d). hence d(z,p1v) = 0 i.e., p1z = z and hence p1z = q2q4...q2nz = z. further on putting x = y = z in (c3), we get f(ψ(d(p0z,p1z)),ψ(d(q1q3...q2n−1z,q2q4...q2nz)),ψ(d(q1q3...q2n−1z,p0z)), ψ(d(q2q4...q2nz,p1z)),ψ(d(q1q3...q2n−1z,p1z)),ψ(d(q2q4...q2nz,p0z))) + g(ψ(d(q1q3...q2n−1z,q2q4...q2nz)),ψ(d(q1q3...q2n−1z,p0z)), ψ(d(q2q4...q2nz,p1z)),ψ(d(q1q3...q2n−1z,p1z),ψ(d(q2q4...q2nz,p0z))) ≤ 0. on simplification, we have f(ψ(d(p0z,z)),ψ(d(p0z,z)),0,0,ψ(d(p0z,z)),ψ(d(p0z,z))) + g(ψ(d(p0z,z)),0, 0,ψ(d(p0z,z)),ψ(d(p0z,z))) ≤ 0. 226 families of mappings satisfying a mixed implicit relation if d(p0,z) > 0, then g(ψ(d(p0z,z)),0, ,0,ψ(d(p0z,z)),ψ(d(p0z,z))) > 0, which implies that f(ψ(d(p0z,z)),ψ(d(p0z,z)),0,0,ψ(d(p0z,z)),ψ(d(p0z,z))) < 0, a contradiction of (f3d). hence d(p0z,z) = 0 i.e., p0z = z and hence p0z = q1q3...q2n−1z = z. on putting x = z and y = q4...q2nz in (c3) and using (c1), q′1 = q1q3...q2n−1 and q′2 = q2q4...q2n, we get f(ψ(d(p0z,p1q4...q2nz)),ψ(d(q′1z,q ′ 2q4...q2nz)),ψ(d(q ′ 1z,p0z)), ψ(d(q′2q4...q2nz,p1q4...q2nz)),ψ(d(q ′ 1z,p1q4...q2nz)),ψ(d(q ′ 2q4...q2nz,p0z))) + g(ψ(d(q′1z,q ′ 2q4...q2nz)),ψ(d(q ′ 1z,p0z)),ψ(d(q ′ 2q4...q2nz,p1q4...q2nz)), ψ(d(q′1z,p1q4...q2nz)),ψ(d(q ′ 2q4...q2nz,p0z))) ≤ 0. from this we get f(ψ(d(z,q4...q2nz)),ψ(d(z,q4...q2nz)),0,0,ψ(d(z,q4...q2nz)),ψ(d(q4...q2nz,z))) + g(ψ(d(z,q4...q2nz)),0,0,ψ(d(z,q4...q2nz)),ψ(d(q4...q2nz,z))) ≤ 0. if (d(z,q4...q2nz)) > 0 then g(ψ(d(z,q4...q2nz)),0,0,ψ(d(z,q4...q2nz)),ψ(d(q4...q2nz,z))) > 0. therefore, we have f(ψ(d(z,q4...q2nz)),ψ(d(z,q4...q2nz)),0,0,ψ(d(z,q4...q2nz)), ψ(d(q4...q2nz,z))) < 0, a contradiction to (f3d). hence d(z,q4...q2nz)) = 0 i.e., q4...q2nz = z. hence q2q4...q2nz = q2z = z. continuing like this, we have p1z = q2z = q4z = ... = q2n = z. (1) on putting x = q3...q2n−1z and y = z in (c3) and using (c1), q′1 = q1q3...q2n−1 and q′2 = q2q4...q2n, we get f(ψ(d(p0q3...q2n−1z,p1z)),ψ(d(q′1q3...q2n−1z,q ′ 2z)), ψ(d(q′1q3...q2n−1z,p0q3...q2n−1z)),ψ(d(q ′ 2z,p1z)), ψ(d(q′1q3...q2n−1z,p1z)),ψ(d(q ′ 2z,p0q3...q2n−1z))) + g(ψ(d(q′1q3...q2n−1z,q ′ 2z)),ψ(d(q ′ 1q3...q2n−1z,p0q3...q2n−1z)), ψ(d(q′2z,p1z)),ψ(d(q ′ 1q3...q2n−1z,p1z)),ψ(d(q ′ 2z,p0q3...q2n−1z))) ≤ 0, 227 r. kumar and s. kumar which implies that f(ψ(d(q3...q2n−1z,z)),ψ(d(q3...q2n−1z,z)),0,0,ψ(d(q3...q2n−1z,z)), ψ(d(z,q3...q2n−1z))) + g(d(ψ(d(q3...q2n−1z,z)),0,0,ψ(d(q3...q2n−1z,z)), ψ(d(z,q3...q2n−1z))) ≤ 0. if d(z,q3...q2n−1z) > 0 then g(d(ψ(d(q3...q2n−1z,z)),0,0,ψ(d(q3...q2n−1z,z)),ψ(d(z,q3...q2n−1z))) > 0. thus from above, we obtain f(ψ(d(q3...q2n−1z,z)),ψ(d(q3...q2n−1z,z)),0,0,ψ(d(q3...q2n−1z,z)), ψ(d(z,q3...q2n−1z))) < 0, a contradiction to (f3d). hence d(z,q3...q2n−1z) = 0 i.e., q3...q2n−1z = z. hence q1q3...q2n−1z = q1z = z. continuing like this, we have p0z = q1z = q3z = ... = q2n−1 = z. (2) hence from (1) and (2), we have p0z = p1 = q1z = q2z = q3z = ... = q2n−1 = q2nz = z. therefore, z is a common fixed point of the given self mappings. uniqueness. let w be another fixed point of the given mappings. then p0w = p1w = q1w = q2w = q3w = ... = q2nw = w. suppose that z ̸= w. putting x = z and y = w in condition (c3), we have f(ψ(d(z,w)),ψ(d(z,w)),0,0,ψ(d(z,w)),ψ(d(w,z)) + g(ψ(d(z,w)),0,0,ψ(d(z,w)),ψ(d(w,z))) ≤ 0. if d(z,w) > 0, then g(ψ(d(z,w)),0,0,ψ(d(z,w)),ψ(d(w,z))) > 0. therefore, we obtain f(ψ(d(z,w)),ψ(d(z,w)),0,0,ψ(d(z,w)),ψ(d(w,z)) < 0, a contradiction of (f3d). hence z = w. therefore, z is a unique common fixed point of the given mappings. now we prove a theorem for families of mappings. 228 families of mappings satisfying a mixed implicit relation theorem 2.3. let {sα}α∈j and {qi} 2p i=1 be two families of self-mappings on a metric space (x ,d). suppose that there exists a fixed β ∈ j such that: (c4) q2(q4...q2n) = (q4...q2n)q2, q2q4(q6...q2n) = (q6...q2n)q2q4, ... q2...q2n−2(q2n) = (q2n)q2...q2n−2, sβ(q4...s2n) = (s4...s2n)sβ, sβ(q6...q2n) = (q6...q2n)sβ, ... sβq2n = q2nsβ, q1(q3...q2n−1) = (q3...q2n−1)q1, q1q3(q5...q2n−1) = (q5...q2n−1)q1q3, ... q1...q2n−3(q2n−1) = (q2n−1)q1...q2n−3, sα(q3...q2n−1) = (q3...q2n−1)sα, sα(q5...q2n−1) = (q5...q2n−1)sα, ... sαs2n−1 = s2n−1sα, (c5) the pairs (sα,q1...q2n−1) and (sβ,q2...q2n) are weakly compatible and (sα,q1...q2n−1) and q2...q2n satisfy (clr)(sα,q1...q2n−1)q2...q2n property, (c6) f(ψ(d(sαx,sβy)),ψ(d(q1q3...q2n−1x,q2q4...q2ny)),ψ(d(q1q3...q2n−1x,sαx)), ψ(d(q2q4...q2ny,sβy)),ψ(d(q1q3...q2n−1x,sβy)),ψ(d(q2q4...q2ny,sαx))) + g(ψ(d(q1q3...q2n−1x,q2q4...q2ny)),ψ(d(q1q3...q2n−1x,sαx)), ψ(d(q2q4...q2ny,sβy)),ψ(d(q1q3...q2n−1x,sβy)),ψ(d(q2q4...q2ny,sαx))) ≤ 0, for all x,y ∈ x and some f ∈ fd, g ∈ gd and ψ is an almost altering distance. then all sα and qi have a unique common fixed point in x . proof. let sα0 be a fixed element in {sα}α∈j . by theorem 2.2 with p0 = sα and p1 = sα0 it follows that there exists some u ∈ x such that sαu = sα0u = q1q3...q2n−1u = q2q4...q2nu = u. let β ∈ j be arbitrary. then from (c6), we get f(ψ(d(sαu,sβu)),ψ(d(q1q3...q2n−1u,q2q4...q2nu)),ψ(d(q1q3...q2n−1u,sαu)), ψ(d(q2q4...q2nu,sβu)),ψ(d(q1q3...q2n−1u,sβu)),ψ(d(q2q4...q2nu,sαu))) + g(ψ(d(q1q3...q2n−1u,q2q4...q2nu)),ψ(d(q1q3...q2n−1u,sαu)), ψ(d(q2q4...q2nu,sβu)),ψ(d(q1q3...q2n−1u,sβu)),ψ(d(q2q4...q2nu,sαu))) ≤ 0. 229 r. kumar and s. kumar hence f(ψ(d(u,sβu)),ψ(d(u,u)),ψ(d(u,u)),ψ(d(u,sβu)),ψ(d(u,sβu)),ψ(d(u,u))) +g(ψ(d(u,u)),ψ(d(u,u)),ψ(d(u,sβu)),ψ(d(u,sβu)),ψ(d(u,u))) ≤ 0, i.e., f(ψ(d(u,sβu)),0,0,ψ(d(u,sβu)),ψ(d(u,sβu)),0) +g(0,0,ψ(d(u,sβu)),ψ(d(u,sβu)),0) ≤ 0. if d(u,sβu) > 0, we get g(0,0,ψ(d(u,sβu)),ψ(d(u,sβu)),0) > 0, which implies that f(ψ(d(u,sβu)),0,0,ψ(d(u,sβu)),ψ(d(u,sβu)),0) < 0, a contradiction by (f2d) and hence ψ(d(u,sβu)) = 0 i.e., sβu = u for each β ∈ j. uniqueness follows easily. if we take ψ(t) = t in theorem 2.2, we get theorem 2.4. let q1,q2, ...,q2n,p0 and p1 be self mappings on a metric space (x ,d), satisfying conditions (c1), (c2) and the following condition: (c7) f((d(p0x,p1y)),(d(q1q3...q2n−1x,q2q4...q2ny)),(d(q1q3...q2n−1x,p0x)), (d(q2q4...q2ny,p1y)),(d(q1q3...q2n−1x,p1y)),(d(q2q4...q2ny,p0x))) + g(d(q1q3...q2n−1x,q2q4...q2ny),d(q1q3...q2n−1x,p0x),d(q2q4...q2ny,p1y), d(q1q3...q2n−1x,p1y),d(q2q4...q2ny,p0x)) ≤ 0, for all x,y ∈ x , some f ∈ fd, g ∈ gd and ψ is an almost altering distance. then q1, q2,...,q2n, p0 and p1 have a unique common fixed point in x . if we take ψ(t) = t in theorem 2.3, we get theorem 2.5. let {sα}α∈j and {qi} 2p i=1 be two families of self-mappings on a metric space (x ,d). suppose that there exists a fixed β ∈ j such that conditions (c4) and (c5) are satisfied. moreover, (c8) f(d(sαx,sβy),d(q1q3...q2n−1x,q2q4...q2ny),d(q1q3...q2n−1x,sαx), (d(q2q4...q2ny,sβy),d(q1q3...q2n−1x,sβy),d(q2q4...q2ny,sαx)) + g(d(q1q3...q2n−1x,q2q4...q2ny),d(q1q3...q2n−1x,sαx), d(q2q4...q2ny,sβy),d(q1q3...q2n−1x,sβy),d(q2q4...q2ny,sαx)) ≤ 0, for all x,y ∈ x and some f ∈ fd, g ∈ gd and ψ is an almost altering distance. then all sα and qi have a unique common fixed point in x . 230 families of mappings satisfying a mixed implicit relation remark 2.1. (i). let ψ and ϕ be as in theorem 1.1. then f(u1, ...,u6) = ψ(u1) − ψ(m(x,y)) and g(v1, ...,v5) = ϕ(m(x,y)). then f(u,0,u,0,0,u) = f(u,0,0,u,u,0) = f(u,u,0,0,u,u) = 0 and g(v1, ...,v5) = ϕ(max{v1, ...,v5}) > 0, if one of v1, ...,v5 > 0. hence f ∈ fd and g ∈ gd. then by theorem 2.4, we get a generalization and extension of theorem 1.1 for any even number of weakly compatible mappings. similarly, theorem 2.5 is a generalization and extension of theorem 1.1 for families of weakly compatible mappings. (ii). theorems 2.2 and 2.3 are extension of theorem 2.1 for any even number of weakly compatible mappings and families of weakly compatible mappings respectively. now we give an example in support of our theorems. example 2.1. let x = [0,1] and d be usual metric on x . define sα(x) = x4 1 + x4 for each α ∈ j and all x ∈ x , qi(x) = x n √ 4 for each i ∈ {1,2, ...,2n} and all x ∈ x . then q2q4...q2nx = x4, q1q3...q2n−1x = x4. the pairs (sα,q1...q2n−1) and (sβ,q2...s2n) are weakly compatible.. define implicit function f such that let f(u1, ...,u6) = u1 − 9 10 max{u2,u3,u4,u5,u6}. and g(t1, ..., t5) = 1 100(t1 + t2 + t3 + t4 + t5) . then f ∈ fd and g ∈ gd. thus all the conditions of theorems 2.2 (for α = 0,1) and 2.3 are satisfied for ψ(t) = t and 0 is the unique common fixed point of the mappings. 231 r. kumar and s. kumar 3 application in 2002, branciari [4] obtained banach contraction principle for mappings satisfying an integral type contraction condition. in the same way, we analyze theorem 2.3 for mappings satisfying integral type contraction condition. lemma 3.1. [19] let r : [0,∞) → [0,∞) is a lebesgue measurable mapping which is summable on each compact subset of [0,∞) such that ∫ ∞ 0 r(t)dt > 0, for ϵ > 0. then ψ(t) = ∫ t 0 r(x)dx is an almost altering distance. theorem 3.1. let {sα}α∈j and {qi} 2p i=1 be two families of self-mappings on a metric space (x ,d). suppose that there exists a fixed β ∈ j such that conditions (c4) and (c5) are satisfied. moreover, (c12) f (∫ d(sαx,sβy) 0 r(t)dt, ∫ d(q1q3...q2n−1x,q2q4...q2ny) 0 r(t)dt, ∫ d(q1q3...q2n−1x,sαx) 0 r(t)dt, ∫ d(q2q4...q2ny,sβy) 0 r(t)dt, ∫ d(q1q3...q2n−1x,sβy) 0 r(t)dt, ∫ d(q2q4...q2ny,sαx) 0 r(t)dt ) + g (∫ d(q1q3...q2n−1x,q2q4...q2ny) 0 r(t)dt, ∫ d(q1q3...q2n−1x,sαx) 0 r(t)dt, ∫ d(q2q4...q2ny,sβy) 0 r(t)dt, ∫ d(q1q3...q2n−1x,sβy) 0 r(t)dt, ∫ d(q2q4...q2ny,sαx) 0 r(t)dt ) ≤ 0, for all x,y ∈ x and some f ∈ fd and g ∈ gd. then all sα and qi have a unique common fixed point in x . proof. let ψ(t) be as in lemma 3.1. then ψ(d(sαx,sβy)) = ∫ d(sαx,sβy) 0 r(t)dt,ψ(d(q1q3...q2n−1x,sαx) = ∫ d(q1q3...q2n−1x,sαx) 0 ψ(d(q1q3...q2n−1x,q2q4...q2ny)) = ∫ d(q1q3...q2n−1x,q2q4...q2ny) 0 r(t)dt, ψ(d(q2q4...q2ny,sβy)) = ∫ d(q2q4...q2ny,sβy) 0 r(t)dt, ψ(d(q1q3...q2n−1x,sβy)) = ∫ d(q1q3...q2n−1x,sβy) 0 r(t)dt, ψ(d(q2q4...q2ny,sαx)) = ∫ d(q2q4...q2ny,sαx) 0 r(t)dt. hence the proof of theorem 3.1 follows by theorem 2.3. 232 families of mappings satisfying a mixed implicit relation 4 conclusions in this paper, we have established unique common fixed point theorems for families of weakly compatible mappings satisfying common limit range property and a mixed implicit relation. our results generalize, extend and improve the results of imdad [8] and popa [17]. we provide an application for integral type contraction condition. in the end, we conclude that theory of fixed points can be extended in metric space for some applications as well and that the analogue of many known results can also be obtained in this literature. references [1] a. aamri, d. el-moutawakil, some new common fixed point theorems under strict contractive conditions, math. anal. appl. 270 (2002), 181-188. [2] j. ali, m. imdad, an implicit function implies several contractive conditions, sarajevo j. math., 17 (2008),269–285. [3] ya.i. alber, s. guerre-delabrierre, principle of weakly contractive maps in hilbert spaces. in: new results in operator theory and its applications, adv. appl. math. 98 (y. gahbery, yu. librich, eds.), birkhauser verlag basel, 1997, pp. 7–22. [4] a. branciari, a fixed point theorem for mappings satisfying a general contractive condition of integral type, int. j. math. math. sci. 29(9) (2002), 531-536. [5] d. w. boyd and t. s. w. wong, on nonlinear contractions, proceedings of the american mathematical society, vol. 20, no. 2, pp. 458–464, 1969. [6] s. banach, surles operations dans les ensembles abstraites et leursapplications, fundam. math. 3 (1922), 133-181. [7] m. imdad, s. chauhan, employing common limit range property to prove unified metrical common fixed point theorems, intern. j. anal. 2013, article id 763261. [8] m. imdad, s. chauhan, z. kadelburg, fixed point theorems for mappings with common limit range property satisfying generalized (ψ,ϕ)-weak contractive conditions, math. sci. 7 (2013), doi: 10.1186/2251-7456-7-16. [9] g. jungck, commuting mappings and fixed points, amer. math. monthly 83 (1976), 261-263. 233 r. kumar and s. kumar [10] g. jungck, compatible mappings and common fixed points, int. j. math. math. sci. 9(4)(1986), 771-779. [11] g. jungck, common fixed points for noncommuting nonself mappings on nonnumeric spaces, far east j.math. sci. 4 (1996), 195–215. [12] m.s. khan, m. swaleh, s. sessa, fixed point theorems by altering distance between two points, bull. austral. math. soc. 30 (1984), 1-9. [13] r.p. pant, common fixed points for contractive maps, j. math. anal. appl. 226 (1998), 251–258. [14] r.p. pant, r-weak commutativity and common fixed points of noncompatible maps, ganita 99 (1999), 19–26. 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[22] w. sintunavarat and p. kumam, common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, j. appl. math. (2011), article id 637958. 234 ratio mathematica volume to be assigned, 2023 common fixed point theorem for weakly compatible mappings in sm metric space mallaiah katta * srinivas veladi† abstract in the present paper, at first, we study the structure of the newly smmetric space, which is a combination of s-metric space and multiplicative metric space. we have proved a common fixed point theorem for four self-maps in sm metric space with a new contraction condition by applying the concepts of weakly compatible mappings, semi-compatible mappings, and reciprocally continuous mappings. further, we also provide some examples to support our results. keywords: multiplicative metric space, s-metric space,sm-metric space, weakly compatible mappings, reciprocally continuous mappings, and semi-compatible mappings. 2020 ams subject classifications: 54h25 1 *jn government polytechnic, hyderabad, india; kamanilayam95l@ gmail.com. †university college of science ou, hyderabad,india; srinivasmaths4141@gmail.com. 1received on august 8, 2022. accepted on december 1, 2022. published on january 2, 2023. doi: 10.23755/rm.v39i0.808. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. mallaiah k and srinivas v 1 introduction the notion of multiplicative metric space (mms) was first developed by bashirove [1]. following that, several theorems came to light in this area of mms [2] ,[3] and [4]. on the other side, sedghi.s et al.[5] presented a new structure to s-metric space which modified d-metric and g-metric spaces, and then several fixed point theorems [6] and [7] were obtained. pant et al. [8] generalized the notion of reciprocally continuous mapping which is weaker than continuous and compatible mappings. recently, mukesh kumar jain [9] introduced a more general form of semi-compatible mappings and proved many fixed point theorems in metric space. in this article, we use a new generalized metric space referred to as sm -metric space, which is a combination of both mms and s -metric space. using this concept, we establish a common fixed point theorem by applying weakly compatible mappings(wcm), reciprocally continuous mappings, and semi-compatible mappings. furthermore, some examples are also discussed to support our conclusions. 2 preliminaries: now we give some definitions and examples which are used in this theorem. definition 2.1. [1] “let χ be a non-empty set and δ : χ2 → r+ be a multiplicative metric space (mms) satisfying the properties : (i) δ(ψ,ϕ) ≥ 1 and δ(ψ,ϕ) = 1 ⇐⇒ ψ = ϕ (ii) δ(ψ,ϕ) = δ(ϕ,ψ) (iii) δ(ψ,ϕ) ≤ δ(ψ,σ)δ(σ,ϕ),∀ψ,ϕ,σ ∈ χ.” definition 2.2. [5] “ let χ be a non-empty set defined s : χ3 → [0,∞) satisfying: (i) s(ψ,ϕ,σ) ≥ 0 (ii) s(ψ,ϕ,σ) = 0 ⇐⇒ ψ = ϕ = σ (iii) s(ψ,ϕ,σ) ≤ s(ψ,ψ,ρ) + s(ϕ,ϕ,ρ) + s(σ,σ,ρ),∀ψ,ϕ,σ,ρ ∈ χ. a mapping s together with χ,(χ,s) is called a s-metric space.” definition 2.3. [10] “ let χ be a non-empty set .a function sm : χ3 → r+ satisfying the conditions : sm metric space (i) sm(ψ,ϕ,σ) ≥ 1 (ii) sm(ψ,ϕ,σ) = 1 ⇐⇒ ψ = ϕ = σ (iii) sm(ψ,ϕ,σ) ≤ sm(ψ,ψ,ρ)sm(ϕ,ϕ,ρ)sm(σ,σ,ρ),∀ψ,ϕ,σ,ρ ∈ χ. the pair (χ,sm) is called as sm-metric space”. definition 2.4. [10] “ let (χ,sm) be a sm-metric space, a sequence {ψθ} ∈ χ is said to be (i) cauchy sequence ⇐⇒ sm(ψθ,ψθ,ψl) → 1, for all θ, l → ∞; (ii) convergent ⇐⇒ ∃ψ ∈ χ such that sm(ψθ,ψθ,ψ) → 1 as θ → ∞; (iii) is complete if every cauchy sequence is convergent.” definition 2.5. [11] ” two self-maps m and k of a sm metric space are said to be (i) compatible: if lim θ→∞ sm(mkψθ,mkψθ,kmψθ) = 1, whenever there exist a sequence {ψθ} ∈ χ such that lim θ→∞ sm(mψθ,kψθ,ω) = 1 for some ω ∈ χ. (ii) weaklycompatible mappings: if they commute at their coincidence points, i.e.ω ∈ χ,sm(mω,mω,kω) = 1, =⇒ sm(mkω.mkω,kmω) = 1.” definition 2.6. [9] “two self maps m and k of sm-metric space are said to be semicompatible: if lim θ→∞ sm(mkψθ,mkψθ,kω) = 1 whenever there exists a sequence {ψθ} ∈ x such that lim θ→∞ sm(mψθ,kψθ,ω) = 1 for all ω ∈ χ.” mallaiah k and srinivas v now we present an example in which semi-compatible is weaker than compatible. example 2.6.1 consider χ = [0,∞) with sm(ψ,ϕ,σ) = e|ψ−ϕ|+|ϕ−σ|+|σ−ψ|, for every ψ,ϕ,σ ∈ χ. define two self maps m and k as m(ψ) = { cos2(πψ)+1 2 if 0 < ψ ≤ 1 2 ; sin(πψ) if 1 2 < ψ ≤ 3. and k(ψ) = { 2sin(πψ)−1 2 if 0 < ψ ≤ 1 2 ; 1 − sin(πψ) if 1 2 < ψ ≤ 3. consider a sequence {ψθ} as ψθ = {π2 − 1 θ } for θ ≥ 0. then lim θ→∞ m(ψθ) = lim θ→∞ m( 1 2 − 1 θ ) = lim θ→∞ cos2π(1 2 − 1 θ ) + 1 2 = lim θ→∞ sin2(π θ ) + 1 2 = 1 2 and lim θ→∞ k(ψθ) = lim θ→∞ k( 1 2 − 1 θ ) = lim θ→∞ 2sinπ(1 2 − 1 θ ) − 1 2 = lim θ→∞ 2cos(π θ ) − 1 2 = 1 2 . therefore lim θ→∞ mψθ = lim θ→∞ kψθ = 1 2 = ω (say). now lim θ→∞ mk(ψθ) = lim θ→∞ m( 2cosπ θ − 1 2 ) = lim θ→∞ cos2π( 2cosπ θ −1 2 ) + 1 2 = cos2 π 2 + 1 2 = 1 2 and lim θ→∞ km(ψθ) = lim θ→∞ k( sin2 π θ + 1 2 ) = lim θ→∞ [1 − sinπ( sin2 π θ + 1 2 )] = 0. ∴ lim θ→∞ sm(mkψθ,mkψθ,kmψθ) ̸= 0. this implies these two self-maps m and k are not compatible. but k(ω) = k(1 2 ) = 1 2 . therefore lim θ→∞ sm(mkψθ,mkψθ,kω) = lim θ→∞ sm( 1 2 , 1 2 , 1 2 ) = 1. hence these two self maps m and k are semi-compatible but not compatible. sm metric space definition 2.7. [8] “two self-maps m, k of sm-metric space are said to be reciprocally continuous if lim θ→∞ sm(mkψθ,mkψθ,mω) = 1 and lim θ→∞ sm(kmψθ,kmψθ,kω) = 1, whenever there exist a sequence {ψθ} ∈ χ such that lim θ→∞ sm(mψθ,kψθ,ω) = 1 some ω ∈ χ.” now we present an example in which satisfies reciprocally continuous is weaker but not compatible. example 2.7.1 consider χ = (0,∞) with sm(ψ,ϕ,σ) = e|ψ−ϕ|+|ϕ−σ|+|σ−ψ|, for every ψ,ϕ,σ ∈ χ. define two self maps m and k as m(ψ) = { ψ2 + 2 if 0 < ψ ≤ 1; 4 − ψ if 1 < ψ ≤ 3. and k(ψ) = { 1 − 2ψ if 0 < ψ ≤ 1; ψ − 2 if 1 < ψ ≤ 3. consider a sequence {ψθ} as ψθ = {3 − 1θ}, for θ ≥ 0. now lim θ→∞ m(ψθ) = lim θ→∞ [4 − (3 − 1 θ )] = 1 and lim θ→∞ k(ψθ) = lim θ→∞ [(3 + 1 θ ) − 2] = 1 ∴ lim θ→∞ mψθ = lim θ→∞ kψθ = 1 = ω1 ̸= ϕ. also lim θ→∞ mk(ψθ) = lim θ→∞ m[(3 − 1 θ ) − 2] = lim θ→∞ m(1 − 1 θ ) = 3 and lim θ→∞ km(ψθ) = lim θ→∞ k(4 − (3 − 1 θ ) = lim θ→∞ k(1 + 1 θ ) = −1. ∴ lim θ→∞ sm(mkψθ,mkψθ,kmψθ) = sm(3,3,−1) ̸= 1. this gives the self maps m and k are not compatible in smmetric space. moreover,m(ω1) = 3 and k(ω1) = −1. which gives lim θ→∞ sm(mkψθ,mkψθ,mω1) = sm(3,3,3) = 1, mallaiah k and srinivas v and lim θ→∞ sm(kmψθ,kmψθ,kω1) = sm(−1,−1,−1) = 1. this implies the self-maps m and k are reciprocally continuous but not compatible in sm metric space. now we proceed to the main theorem. 3 main theorem theorem 3.1. let m, h, k, and j be self-mapping of a complete sm-metric space satisfying the following (3.1.1) m(χ) ⊆ j(χ) and h(χ) ⊆ k(χ) (3.1.2) sm(mψ,mψ,hϕ) ≤ { max[sm(mψ,mψ,kψ)sm(hϕ,hϕ,jϕ), sm(mψ,mψ,jϕ)sm(kψ,kψ,hϕ), sm(mψ,mψ,jϕ)sm(hϕ,hϕ,jϕ), sm(mψ,mψ,kψ)sm(hϕ,hϕ,kψ)] }λ where λ ∈ (0, 1 2 ) (3.1.3) the pair m and k are reciprocally continuous and semi-compatible, (3.1.4) the pair h and j are weakly compatible. then the self-maps m, h, k, and j have a unique common fixed point in χ. proof: let there is a point ψ0 ∈ χ, and the sequence {ψθ} be defined as mψ0 = jψ1 = ϕ0. for this point ψ1 then there exists ψ2 ∈ χ such that hψ1 = kψ2 = ϕ1. in general, by induction choose ψθ+1 , construct a sequence {ϕθ} ∈ χ such that ϕ2θ = mψ2θ = jψ2θ+1 and ϕ2θ+1 = hψ2θ+1 = kψ2θ+2, for θ ≥ 0. sm metric space on putting ψ = ψ2θ and ϕ = ϕ2θ+1 in ( 3.1.2) we get. sm(ϕ2θ,ϕ2θ,ϕ2θ+1) = sm(mψ2θ,mψ2θ,hψ2θ+1) ≤ max { sm(mψ2θ,mψ2θ,θψ2θ)sm(hψ2θ+1,hψ2θ+1,jψ2θ+1), sm(mψ2θ,mψ2θ,jψ2θ+1)sm(hψ2θ+1,hψ2θ+1,θψ2θ), sm(mψ2θ,mψ2θ,jψ2θ+1)sm(hψ2θ+1,hψ2θ+1,jψ2θ+1), sm(mψ2θ,mψ2θ,kψ2θ)sm(hψ2θ+1,hψ2θ+1,kψ2θ) }λ sm(ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ max { sm(ϕ2θ,ϕ2θ,ϕ2θ−1)sm(ϕ2θ+1,ϕ2θ+1,ϕ2θ), sm(ϕ2θ,ϕ2θ,ϕ2θ)sm(ϕ2θ+1,ϕ2θ+1,ϕ2θ−1), sm(ϕ2θ,ϕ2θ,ϕ2θ)sm(ϕ2θ+1,ϕ2θ+1,ϕ2θ), sm(ϕ2θ,ϕ2θ,ϕ2θ−1)sm(ϕ2θ+1,ϕ2θ+1,ϕ2θ−1) }λ this implies that sm(ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ sm(ϕ2θ−1,ϕ2θ−1,ϕ2θ+1)λ. sm(ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ {sm(ϕ2θ−1,ϕ2θ−1,ϕ2θ)sm(ϕ2θ,ϕ2θ,ϕ2θ+1)}λ. s1−λm (ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ s λ m(ϕ2θ−1,ϕ2θ−1,ϕ2θ). sm(ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ s λ 1−λ m (ϕ2θ−1,ϕ2θ−1,ϕ2θ). sm(ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ spm(ϕ2θ−1,ϕ2θ−1,ϕ2θ). where p = λ 1 − λ . now this gives sm(ϕθ,ϕθ,ϕθ+1) ≤ spm(ϕθ−1,ϕθ−1,ϕθ) ≤ s p2 m (ϕθ−2,ϕθ−2,ϕθ−1) ≤ · · ·s pn m (ϕ0,ϕ0,ϕn). by using triangular inequality sm(ϕθ,ϕθ,ϕn) ≤ sp θ m (ϕ0,ϕ0,ϕl) ≤ s pθ+1 m (ϕ0,ϕ0,ϕn) ≤ · · ·s pn−1 m (ϕ0,ϕ0,ϕn) mallaiah k and srinivas v sm(ϕθ,ϕθ,ϕn) ≤ s pθ 1−p m (ϕ0,ϕ0,ϕl) for all θ ≥ 1. hence {ϕθ} is a cauchy sequence in sm-metric space. since the self-maps, m and k are weakly reciprocally continuous. lim θ→∞ sm(mkψθ,mkψθ,mω) = 1 or lim θ→∞ sm(kmψθ,kmψθ,θω) = 1. (1) also, the pair ( m, k) is semi compatible, we have lim θ→∞ sm(mkψθ,mkψθ,kω) = 1. (2) from (1) and (2) we get sm(mω,mω,kω) = 1. (3) since m(χ) ⊆ j(χ) which gives then there exists ν ∈ χ such that jν = mψθ, since mψθ → ω as θ → ∞. which implies sm(jν,jν,ω) = 1. (4) now, we have to prove sm(jν,hν,ω) = 1. substitute ψ = ψθ and ϕ = ν in (3.1.2) we have sm(mψθ,mψθ,hν) ≤ { max[sm(mψθ,mψθ,kψθ)sm(hν,hν,jν), sm(mψθ,mψθ,jν)sm(kψ1,kψ1,hν), sm(mψθ,mψθ,jν)sm(hν,hν,jν), sm(mψθ,mψθ,kψθ)sm(hν,hν,kψθ)] }λ sm(ω,ω,hν) ≤{ max[sm(ω,ω,ω)sm(hν,hν,ω),sm(ω,ω,ω)sm(ω,ω,hν), sm(ω,ω,ω)sm(hν,hν,ω),sm(ω,ω,ω)sm(hν,hν,ω)] }λ sm(ω,ω,hν) ≤ {(sm(ω,ω,hν)}λ s(1−λ)m (ω,ω,hν) ≤ 1 =⇒ sm(hν,hν,ω) = 1. ∴ sm(jν,hν,ω) = 1. sm metric space since the pair (h.j) is wcm and ν is a coincidence point then hjν = jhν sm(hω,hω,jω) = 1. (5) substitute ψ = ψθ and ϕ = ω in (3.1.2) we have sm(mψθ,mψθ,hω) ≤{ max[sm(mψθ,mψθ,kψθ)sm(hω,hω,jω), sm(mψθ,mψθ,jω)sm(kψ1,kψ1,hω), sm(mψθ,mψθ,jω)sm(hω,hω,jω), sm(mψθ,mψθ,kψθ)sm(hω,hω,kψθ)] }λ also sm(hω,ω,ω) ≤{ max[sm(ω,ω,ω)sm(hω,hω,ω),sm(ω,ω,ω)sm(ω,ω,hω), sm(ω,ω,ω)sm(hω,hω,ω),sm(ω,ω,ω)sm(hω,hω,ω)] }λ and this gives sm(hω,ω,ω) ≤ sm(hω,ω,ω)λ s(1−λ)m (hω,ω,ω) ≤ 1 =⇒ hω = ω ∴ sm(hω,jω,ω) = 1. (6) replace ψ = ω and ϕ = ν in (3.1.2) then we have sm(mω,mω,hν) ≤ { max[sm(mω,mω,kω)sm(jν,hν,hν), sm(mω,mω,jν)sm(kω,kω,hν), sm(mω,mω,jν)sm(jν,jν,hν), sm(mω,mω,kω)sm(hν,hν,kω)] }λ mallaiah k and srinivas v sm(mω,mω,ω) ≤ { max[sm(mω,mω,mω)sm(ω,ω,ω), sm(mω,mω,ω)sm(mω,mω,ω), sm(mω,mω,ω)sm(ω,ω,ω), sm(mω,mω,mω)sm(ω,ω,mω)] }λ sm(mω,mω,ω) ≤ {sm(mω,mω,ω)}λ s(1−λ)m (mω,mω,ω) ≤ 1 =⇒ mω = ω ∴ sm(mω,jω,ω) = 1. (7) from (6) and (7) we get mω = jω = hω = kω = ω. (8) therefore “ω” is a common fixed point of m, h, k, and j. uniqueness let ρ be one more fixed point, we assume that ρ ̸= ω then we have mρ = kρ = hρ = jρ = ρ. in the condition (3.1.2) put ψ = ω and ϕ = ρ we get sm(mω,mω,hρ) ≤ { max[sm(mω,mω,kω)sm(hρ,hρ,jρ), sm(mω,mω,jρ)sm(kω,kω,hρ), sm(mω,mω,jρ)sm(hρ,hρ,jρ), sm(mω,mω,kω)sm(hρ,hρ,kω)] }λ sm(ω,ω,ρ) ≤{ max[sm(ω,ω,ω)sm(ρ,ρ,ρ),sm(ω,ω,ρ)sm(ω,ω,ρ), sm(ω,ω,ρ)sm(ρ,ρ,ρ),sm(ω,ω,kω)sm(ρ,ρ,ω)] }λ sm(ω,ω,ρ) ≤ { sm(ω,ω,ρ) }λ this implies that sm(ω,ω,ρ) = 1 =⇒ ω = ρ. this shows that “ω” is the unique common fixed point of m.h.j and k. sm metric space now, the following example substantiates our theorem. example 3.2 suppose χ = (0,1),smmetric space by sm(ψ,ϕ,σ) = e|ψ−ϕ|+|ϕ−σ|+|σ−ψ|, when ψ,ϕ,σ ∈ χ. define m ,k ,h j:χxχ → χ as follows m(ψ) = { 2−ψ 5 if 0 < ψ ≤ 1 3 ; ψ if 1 3 < ψ < 1. k(ψ) = { 1 − 2ψ if 0 < ψ ≤ 1 3 ; 1+ψ 2 if 1 3 < ψ < 1. h(ψ) = { 3ψ2 − 3ψ + 1 if 0 < ψ ≤ 1 3 ; 2+ψ 7 if 1 3 < ψ < 1. j(ψ) = { 1 − 6ψ2 if 0 < ψ ≤ 1 3 ; 1 − ψ if 1 3 < ψ < 1. then m(χ) = (1 3 ,1] ⊆ j(χ) = (0,1] and h(χ) = (1 3 ,1] ⊆ k(χ) = (1 3 ,1]. therefore the condition (3.1.1 ) holds. consider a sequence {ψθ} as ψθ = {13 − 1 θ } as θ ≥ 0. then lim θ→∞ m(ψθ) = lim θ→∞ m( 1 3 − 1 θ ) = lim θ→∞ 2 − (1 3 − 1 θ ) 5 = 1 3 and lim θ→∞ k(ψθ) = lim θ→∞ k( 1 3 − 1 θ ) = lim θ→∞ [1 − 2( 1 3 − 1 θ )] = 1 3 . therefore lim θ→∞ m(ψθ) = lim θ→∞ k(ψθ) = 1 3 = ω1. further lim θ→∞ h(ψθ) = lim θ→∞ h( 1 3 − 1 θ ) = lim θ→∞ [3( 1 3 − 1 θ )2 − 3( 1 3 − 1 θ ) + 1] = 1 3 and lim θ→∞ j(ψθ) = lim θ→∞ j( 1 3 − 1 θ ) = lim θ→∞ [1 − 6( 1 3 − 1 θ )2] = 1 3 . therefore lim θ→∞ h(ψθ) = lim θ→∞ j(ψθ) = 1 3 = ω1. mallaiah k and srinivas v moreover lim θ→∞ mk(ψθ) = lim θ→∞ m[1 − ( 2 3 − 2 θ )] = lim θ→∞ m( 1 3 + 2 θ ) = 1 3 and lim θ→∞ km(ψθ) = lim θ→∞ k( 1 3 + 1 5θ ) = lim θ→∞ 1 + 2(1 3 + 1 5θ 2 ) = 2 3 . ∴ lim θ→∞ sm(mkψθ,mkψθ,kmψθ) = sm( 1 3 , 1 3 , 2 3 ) ̸= 1 which implies that the pair ( m, k) is not compatible. furthermore lim θ→∞ hj(ψθ) = lim θ→∞ h( 1 3 + 4 θ − 1 θ2 ) = lim θ→∞ ( 2 + (1 3 + 4 θ − 1 θ2 7 ) = 1 3 and lim θ→∞ jh(ψθ) = lim θ→∞ j( 1 3 + 4 θ − 1 θ2 ) = lim θ→∞ [1 − ( 1 3 + 4 θ − 1 θ2 )] = 2 3 therefore lim θ→∞ sm(hjψθ,hjψθ,jhψθ) = sm( 1 3 , 1 3 , 2 3 ) ̸= 1. which shows that the pair( h,j) is not compatible . also m(1 3 ) = 1 3 ,k(1 3 ) = 1 3 . this implies lim θ→∞ sm(mkψθ,mkψθ,mω1) = sm( 1 3 , 1 3 , 1 3 ) = 1 and lim θ→∞ sm(kmψθ,kmψθ,kω1) = sm( 1 3 , 1 3 , 1 3 ) = 1. this shows that the pair (m, k ) is reciprocally continuous in sm metric space. also lim θ→∞ sm(mkψθ,mkψθ,kω1) = sm( 1 3 , 1 3 , 1 3 ) = 1. this shows that the pair (m, k ) is semi-compatible in sm metric space. hence the inequality (3.1.3)holds. further sm(h( 1 3 ),j( 1 3 ), 1 3 ) = 1 and sm(hj( 1 3 ),jh( 1 3 ), 1 3 ) = 1. sm metric space this implies that sm(hj( 1 3 ),hj(1 3 ),jh(1 3 )) = sm( 1 3 , 1 3 , 1 3 ) = 1. which indicates that the pair ( h, j) is weakly compatible. now, we prove the condition (3.1.2 ) in various cases case-i let ψ,ϕ ∈ [0, 1 2 ],while we have sm(ψ,ϕ,σ) = e|ψ−σ|+|ϕ−σ|. take ψ = 1 4 and ϕ = 1 5 then m(1 4 ) = 7 20 ,k(1 4 ) = 1 2 ,h(1 5 ) = 13 25 and j(1 5 ) = 19 25 substitute the above values in (3.1.2) sm( 7 20 , 7 20 , 13 25 ) ≤{ max[sm( 7 20 , 7 20 , 1 2 )sm( 13 25 , 13 25 , 19 25 ),sm( 7 20 , 7 20 , 19 25 )sm( 13 25 , 13 25 , 1 2 ), sm( 7 20 , 7 20 , 19 25 )sm( 13 25 , 13 25 , 19 25 ),sm( 7 20 , 7 20 , 1 2 )sm( 13 25 , 13 25 , 1 2 )] }λ wehavee0.34 ≤ { max[e0.3e0.48,e0.82e0.34,e0.3e0.04,e0.82e0.48] }λ e0.34 ≤ { max[e0.78,e1.16,e0.0.34,e1.3]}λ =⇒ e0.34 ≤ e1.16λ which gives λ = 0.2 where λ ∈ (0, 1 3 ). case-ii let ψ,ϕ ∈ (1 2 ,1], then sm(ψ,ϕ,σ) = e|ψ−σ|+|ϕ−σ|. take ψ = 1 2 and ϕ = 1 2 then m(1 2 ) = 1 2 ,k(1 2 ) = 3 4 ,h(1 2 ) = 5 14 and j(1 2 ) = 1 2 substitute the above values in (3.1.2) sm( 1 2 , 1 2 , 5 14 ) ≤{ max[sm( 1 2 , 1 2 , 3 4 )sm( 5 14 , 5 14 , 1 2 ),sm( 1 2 , 1 2 , 1 2 )sm( 5 14 ., 5 14 , 3 4 ), sm( 1 2 , 1 2 , 1 2 )sm( 5 14 , 5 14 , 1 2 ),sm( 1 2 , 1 2 , 3 4 )sm( 5 14 , 5 14 , 3 4 )] }λ which implies that e0.285 ≤ { max[e0.5e0.285,e0.0e0.786,e0.0e0.28,e0.5e0.786] }λ e0.285 ≤ { max[e0.785,e0.786,e0.28,e1.286]}λ =⇒ e0.285 ≤ e1.286λ mallaiah k and srinivas v which gives λ = 0.22 where λ ∈ (0, 1 2 ). case-iii let ψ,ϕ ∈ (1 2 ,1], then sm(ψ,ϕ,σ) = e|ψ−σ|+|ϕ−σ| take ψ = 1 4 and ϕ = 1 2 then m(1 4 ) = 7 20 ,k(1 4 ) = 1 2 ,h(1 5 ) = 5 14 and j(1 5 ) = 1 2 substitute the above values in (3.1.2) sm( 7 20 ,0 7 20 , 5 14 ) ≤{ max[sm( 7 20 , 7 20 , 1 2 )sm( 5 14 , 5 14 , 1 2 ),sm( 7 20 , 7 20 , 1 2 )sm( 5 14 , 5 14 , 1 2 ), sm( 7 20 , 7 20 , 1 2 )sm( 5 14 , 5 14 , 1 2 ),sm( 7 20 , 7 20 , 1 2 )sm( 5 14 , 5 14 , 1 2 )] }λ which implies that e0.014 ≤ { max[e0.3e0.28,e0.3e0.28,e0.3e0.28,e0.3e0.28] }λ e0.014 ≤ { max[e0.58,e0.58,e0.58,e0.58]}λ =⇒ e0.014 ≤ e0.5.8λ this gives that λ = 0.14 where λ ∈ (0, 1 2 ). hence the inequality (3.3.2) holds. it can be seen that “1 2 ” is a unique common fixed point for four self mappings m, k h, and j. 4 conclusions in this article, we established a common fixed point theorem in sm-metric space by using weakly-compatible mappings, semi-compatible mappings, and reciprocally continuous mappings for four self-maps. furthermore, our results are also justified with suitable examples. references [1] agamirza e bashirov, emine mısırlı kurpınar, and ali özyapıcı. multiplicative calculus and its applications. journal of mathematical analysis and applications, 337(1):36–48, 2008. sm metric space [2] afrah an abdou. common fixed point results for compatible-type mappings in multiplicative metric spaces. j. nonlinear sci. appl, 9:2244–2257, 2016. 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[11] ma al-thagafi and naseer shahzad. generalized i-nonexpansive selfmaps and invariant approximations. acta mathematica sinica, english series, 24(5):867–876, 2008. ratio mathematica volume 43, 2022 an adaptive neural network approach to predict the capital adequacy ratio giacomo di tollo* gerarda fattoruso† bartolomeo toffano‡ abstract financial institutions, policy makers and regulatory authorities need to implement stress tests in order to test both resilience and the consequences of adverse shocks. the european central bank and the european banking authority regularly conduct these tests, whose importance is more and more evident after the financial crisis of 2007-2008. the stress tests’ nonlinear features of variables and scenarios triggered the need of general and robust strategies to perform this task. in this paper we want to introduce an adaptive neural network approach to predict the capital adequacy ratio (car), which is one of the main ratios monitored to retrieve useful information along many stress test procedures. the neural network approach is based on a comparison between feed-forward and recurrent networks, and is run after a meaningful pre-processing operations definition. results show that our approach is able to successfully predict car by using both neural networks and recurrent networks. keywords: capital adequacy ratio; stress tests, neural network approach.1 *department of law, economics, management and quantitative methods (demm), university of sannio, benevento, italy; giditollo@unisannio.it †corresponding author. department of law, economics, management and quantitative methods (demm), university of sannio, benevento and neoma bs, rouen, france; italy; fattoruso@unisannio.it ‡department of economics, ca’ foscari university, venice, italy; bartolomeo.toffano@unive.it 1received on july 20, 2022. accepted on september 20, 2022. published on september 25, 2022. doi: 10.23755/rm.v43i0.841. issn: 1592-7415. eissn: 2282-8214. ©di tollo et al. this paper is published under the cc-by licence agreement. g. di tollo, g. fattoruso, b. toffano 1 introduction banks’ bankruptcy may have catastrophic effects over the overall economy, since the contagion effect it may trigger could lead to a generalised overall crisis [47]. to this extent the activity of banking supervision, and the role and the authority of bank regulation, play a big role in preventing (or reducing the effect of) the banks’ bankruptcy [11, 44]. although aimed to different targets, many of these supervision exercises are designed to maintain a sufficient banks’ level of capital adequacy to allocate specific reserves aimed to face expected losses and to protect themselves again excessive credit expansion. authorities regulations impose constraints over these reserves, even though banks often imposes themselves reserves higher than the ones imposed by the regulations. these constraints are defined by a minimum capital adequacy ratio that measures the level of capital as a function of the risk bore the bank [27]. in this framework, systemic risk represents the risk of breakdown of the entire financial system: this can be triggered by the misbehavior of a single component of the overall financial system, that triggers negative impacts on the overall system. this scenario can be observed on the timeline of years 2007-2008, starting from some cracks in the subprime mortgage markets leading to a worldwide financial crisis: this confirmed the idea that the more complex and non-linear a system, the higher the probability of the system to fail [23, 28]. in order to prevent these failures (and/or to quantify financial (un)stability), financial institutions resort to stress tests, which are non linear tools used to assess the magnitude of an exogenous shock and to determine a collapse threshold. these tools operate by investigating both the local stress level (i.e. a single bank) and how the shock is globally spread. back in the 1990s, stress tests were intended for testing the resilience and the stability of a financial institution to reach a certain credibility. later, stress tests, have been used to check the stability and vulnerability of single financial institutions and the overall banking systems [16]. based on the evaluation aim and on the implications of the findings, stress tests can be classified into two major groups: microprudential stress tests, which are forward-looking supervisory instruments for determining the liquidity adequacy of individual banks in relation to their portfolio risks [4]; macroprudential stress tests, which consists of two different types of approaches: the bottom-up and the top-down approach [16]. in the bottom-up approach, the effect is measured using data on individual portfolios. on the other hand, in the top-down approach, the impact is estimated by using aggregated data. many computational methods have been introduced to perform stress test and to predict bank failures: discriminant analysis [37], logit and probit analysis [8], neural networks [53], just to name a few. there are also contributions that compared different methods: for instance, [2] investigates different methods such as logistic regression (lr), linear discriminant an adaptive neural network approach to predict the capital adequacy ratio analysis (lda), random forests (rf), support vector machines (svm), neural networks (nn) and random forests of conditional inference trees (crf); [18] compared generalized linear models and generalized additive models, and concluded that generalized linear mixed models have a better ability to predict troubled businesses. during a stress-test procedure many variables are taken into account to monitor the financial institutions, and there exist several studies aimed to assess the relative importance of these variables, in order to select what variable to monitor to get the most accurate information as possible. many contributions focuses in determining the relationships between capital ratios and bank failures [1], and to understand whether these ratios are useful to assess the regulatory capital adequacy [42]. in this context, a careful investigation of capital ratios is crucial for both the regulatory authority and the bank itself, since it has been shown that the familiar banking characteristics for identifying a distress-prone bank identified fragile banks effectively during the global crisis without new information and are likely to continue to work well in the future [40]. amongst many variables (e.g., profitability, liquidity, solvency, productivity, asset quality, see [54]), capital adequacy ratio (car) has a prominent place, and has been used in the predictor set by many works [39, 31, 14, 38, 33, 5, 48]. recently, some contributions proposed to predict it as an indicator of financial health [49]. in our contribution we want to expand this framework by implementing an adaptive neural network approach to predict car from a well established set of indicators, and to provide banks and regulators with useful information about their stress-test activity. our contribution is organised as follows: section 2 reports the main literature on the topic; sections 3 and 4 outline the set of data and the pre-processing operation performed on it; section 5 introduces the methods used in this paper; section 6 comments the main results and section 7 concludes the paper. 2 literature review car represents a particularly relevant topic for assessing the risks to which banks are exposed [6]. in fact, for the construction of the car index, credit risk, market risk, interest rate risk and exchange rate risk are considered. in this sense, the regulatory authorities define the car as a significant indicator of safety and stability as it considers capital as a useful element to absorb losses [34]. currently, the capital adequacy ratios (cars) defined by the minimum ratio of capital to risk weighted assets are 8% under basel ii and 10.5% under basel iii [3, 12]. based on this, car represents a factor of analysis by regulators to determine capital adequacy for banks and to perform stress tests [29]. in order to aggregate the information coming from the literature, we performed an analysis on bibliographic g. di tollo, g. fattoruso, b. toffano data using the software vosviewer (figure 1) to create a keyword co-occurrence map in order to analyze the main car literature in our field of analysis. figure 1: vosviewer: capital adequacy ratio from the analysis of the data, it emerges that several authors analyze the required minimum levels of the car by evaluating macroeconomic indicators [46], [7], financial indicators [50], multi credit rating indicators [45]. furthermore, many authors carry out stress tests on car to verify the effects of economic crises [58], stability [25] and resilience [15] of banks, along with macro stress test for resilience assessment [20]. recent studies are moving towards identifying the most important variables for future projections of the car. in particular, [50] carry out a study on south korean national banks using random forest boruta algorithms, random forest recursive feature elimination, and bayesian regularization neural networks. other contributions use car to benchmark the performances of banks in stress tests [3, 12, 24, 27, 29, 32, 59]. the goal of our contribution is to assess whether we can use stress-testing to effectively benchmark the performance of a bank in a precise scenario, and to this extent we need to choose a metrics that can precisely fill that role. car is apt to measures the financial soundness of banks in absorbing a reasonable amount of loss, and on the basis of the central role that the car assumes in the assessments of banks and on the basis of the guidelines of the literature on the analysis of the minimum levels of the car, our work aims to accurately predict the car by using quantitative methods. in this framework the quantitative research about stress-testing has been twofold: on one side, to predict the banks’ bankruptcy; on the other side, to assess the different variables features and capability to explain the default. according to [54], neural networks are widely used in contributions related to the first side, while its application about the other side are still limited. we can start our discussion by pointing out that along with stress testing, a key topic is the prediction of various risks, that was based on traditional probability an adaptive neural network approach to predict the capital adequacy ratio and statistical theories [9], but that could lead to non-linear formulations or to taking into account just a few variables, hence triggering the we need of complex and non linear models, also due to the needs of a more interconnected world, not only in financial terms. for this reason researchers and risk managers avail themselves of the usage of artificial neural networks and deep learning to stress testing activities and predict high volatility periods. since more hidden layers are in a neural network means a more complex modelling interaction effect, in finance forecasts, large collections of data often require dynamic data relationships that are difficult or impossible to specify under a complete model [9]. on the other hand, deep learning models can identify and manipulate dynamic non-linear data connections that are invisible to any current financial economic theory and may deliver more reliable predictive outcomes than traditional approaches [30]. although artificial neural networks and deep learning have several applications in the financial field such as credit scoring, predictions and forecasts in financial crisis and bankruptcy, we want to focus on how artificial neural networks and deep learning methods are related to stress testing. financial stability is essential to the economic growth of countries and individuals. regulatory agencies and foreign organisations carried out stress testing activities to determine the stability of the financial system even earlier than 2007, but failed to anticipate the unprecedented economic implications of the crisis. for this reason ever more stress testing exercises were created and used from the authorities with a glance to the consequences of an interconnected financial system in the macroeconomic environment. for example, the european banking authority (namely, eba) approach uses simplified assumptions that cover only particular risks to individual bank balance sheets depending on the macro-economic scenario. one of the major drawbacks of the european banking authority approach is the static financial statement expectation, which allows assets and liabilities to stay stable over the horizon considered without any appreciation of management decisions or new loans. macroeconomic feedback impacts, such as the influence of large insolvent firms on the global economy, are not generally welcomed assumptions in these systems. this kind of test aims the planning binaries of an after crisis recovery behaviour. however, [52] show that the main problem with respect to the european banking authority approach is that this mechanism does not provide an early alarm to avoid being completely disarmed in front of a shock. in [52] it is also provided a solution to this weakness of the model. they propose a neural intelligence for which financial or macroeconomic disturbances extend to the bank’s balance sheets while simultaneously building a large neural network with macro and financial factors. the model is capable of gathering more knowledge concealed in a large data set and allows for complex non-linear interactions that materialize under adverse macroeconomic conditions and financial strain. this methodology examines the financial system independently, without relying on the g. di tollo, g. fattoruso, b. toffano forecasts of the single banks. as a result of the cited paper, comparing the static stress test models with dynamic ones, prove that the deep learning framework can become a useful tool and can improve the early warning mechanism’s signaling ability to anticipate future financial issues and failures of individual banks. the authors, finally, compare the performance of the deep learning technique with the classic stress test models, such as the constant balance sheet approach and the dynamic balance sheet approach to satellite modelling. they reveal that the prediction error of the car dropped significantly under the deep learning method due to its improved performance in simulating the one-year gains and losses of financial institutions. for this reason, deep learning architecture may become a useful tool for macro prudential stress testing and can improve the early warning mechanism’s ability to anticipate future financial crises and failures of individual banks. now that we have a measure by which we can benchmark banks, we need to find a way by which we can predict the car of banks based on certain factors which is what we need in order to stress test banks. the more factors we can incorporate in our predictions the better since it will reflect better a real-world situation and make our stress testing much more realistic. 3 data set our data set consists of worldwide banks’ financial indicators; along with stress financial indicators, we have considered also macro-economic indicators, in order to identify the propagation of systemic shocks that propagate into the financial institutions. we have retrieved quarterly observations that covers a period of 12 years (2007 to 2019). data was collected from different sources: stress financial indicators have been collected from the federal deposit insurance corporation2 website3; macroeconomics indicators were collected from the federal reserve economic data4 website5. the sample period covers twelve years: we have collected quarterly data referring to 672 banks and financial institutions between 2007 and 2019, hence we dispose of 34944 observations. the sample includes missing and noisy 2fdic is an independent agency created by the congress to maintain stability and public confidence in the nation’s (usa) financial system. the fdic insures deposits; examines and supervises financial institutions for safety, soundness, and consumer protection; makes large and complex financial institutions resolvable; and manages receiverships 3referred to as https://www.fdic.gov/fdic in what follows 4referred to as https://fred.stlouisfed.org/fred in what follows 5researchers at the st. louis fed contribute to monetary policy discussions by advising on a range of topics, especially in preparation for federal open market committee meetings (from the https://www.fdic.gov/fdic website, accessed on 2021, january 29th). an adaptive neural network approach to predict the capital adequacy ratio table 1: variables used in the experimental phase and the category they belong to: the label fin denotes financial indicators and mac denotes macro-economic indicators. name description fin / mac net loan net loans and leases exposure fin loss allow loss allowance to loans fin dep total deposits fin yield ea yield on earning assets fin fundc ea cost of funding earning assets fin inc aa noninterest income to average assets fin car total risk-based capital ratio fin tot asst average total assets fin tot eq average total equity fin tot loan average total loans fin risk dens risk weight density fin gdp growth gross domestic product growth mac export growth us real exports of goods and services growth mac debt gdp us public debt to gdp mac govex gdp us government expenditure to gdp mac inflat implicit price deflator as a measure of us inflation mac hpi growth house price index growth mac unemp unemployment rate (age 15-64) mac yield 10y 10-year us sovereign bonds yields mac sp500 ret sp 500 quarterly returns mac values. please notice that we have chosen a sample period that does not contain sub-periods denoting the emergence of a crisis, since we want to develop a methodology for ordinary periods, in which systemic shocks are more difficult to detect. collected data show a number of correct entries which is smaller than its theoretical value: this is due to missing and noisy data, and could lead to misbehavior of the neural network approach, hence we had to devise pre-processing operations, that are outlined in what follows. 4 data pre-processing data analysis is a key point in all experimental settings, and it is always performed in order to understand its features, to detect anomalies (if any), and to represent data without loosing useful information. based on the observations oulined by [13, 21], we apply the following data pre-processing operations. g. di tollo, g. fattoruso, b. toffano table 2: variables used in the experimental phase: overall main statistics before pre-processing operations. name mean std kurt. skewn. min max net loan 684062.10 2845178 128.86 10.17 0 75190000 loss allow 10811.40 92866.23 1170.43 29.11 0 5752000 dep 848100.51 4672760 498.48 18.76 68 2180000 yield ea 4.61 1.21 20.01 2.38 0.07 26.96 fundc ea 1.24 0.88 2.67 1.32 0 16.59 inc aa 1.47 18.11 722.73 26.40 -15.95 601.27 car 23.68 15.42 113.18 6.08 0.75 725.80 tot asst 1114221 5587407 302.58 14.46 2816 2110361 tot eq 125720.10 566772.30 129.17 10.33 539.75 14389800 tot loan 683913.22 2852402 129.57 10.19 0 71201027 risk dens 60.09 14.28 0.63 0.06 8.43 192.24 gdp growth 1.73 2.35 4.05 -1.53 -8.45 5.51 exp. growth 3.63 8.17 5.14 -1.19 -28.65 25.84 debt gdp 93.95 11.27 1.38 -1.49 61.65 105.18 govex gdp 0.34 0.01 -1.09 0.37 0.31 0.37 inflat 100.43 4.51 -0.94 0.12 91.70 111.25 hpi growth 201.26 20.39 0.22 0.92 176.86 264.31 unemp 7.22 1.89 -1.40 -0.07 3.78 10.05 yield 10y 2.64 0.72 -0.02 0.70 1.56 4.84 sp500 ret 0.02 0.06 6.76 -1.47 -0.27 0.17 4.1 removal and replacement when collecting data, one may incur in missing and incorrect values. previous contributions related to neural network approaches [21] suggested to remove indicators containing more than 30% of missing and wrong values. our set of data does not contain such indicators, so we are using the whole set of variables in our experimental phase. anyhow, many indicators show missing and wrong values, so we replace missing values (due to computational errors) with the upper limit of the normalization (see what follows), and wrong values with the indicator’s average over time. 4.2 normalization normalization is a general procedure performed in order to feed the neural network with data belonging to the same range: many contributions stress the importance of performing meaningful normalization, and many formulas are sugan adaptive neural network approach to predict the capital adequacy ratio gested [35]. in our case we used the logarithmic transformation that has been already introduced by [21], defined as follows: xi = logu (|min(0, xmin)|+ xi + 1) , (1) where xi represents the value before normalisation of input x for firm i, and xi represents its normalised value. please notice that we have defined u such that u = xmax + 1 this has been imposed in order to have xi ∈ [0, 1]. table 3: main statistics of overall financial and macro-economic indicators after pre-processing operations. name mean std kurtosis skewness min max net loan 0.67 0.08 1.41 0.42 0 1 loss allow 0.48 0.10 1.29 0.25 0 1 dep 0.64 0.07 1.38 0.58 0.22 1 yield ea 0.53 0.06 5.19 -0.07 0.02 1 fundc ea 0.30 0.14 -0.28 0.53 0 1 inc aa 0.41 0.06 6.68 0.05 0 1 car 0.50 0.07 1.23 0.86 0.08 1 tot asst 0.65 0.07 1.38 0.66 0.42 1 tot eq 0.63 0.08 1.36 0.71 0.38 1 tot loan 0.67 0.08 1.43 0.43 0 1 risk dens 0.79 0.05 1.48 -0.74 0.44 1 gdp growth 1.06 0.31 2.58 -1.52 0 1 export growth 0.99 0.2 10.45 -3.09 0 1 debt gdp 0.97 0.02 2.11 -1.71 0.89 1 govex gdp 0.93 0.03 -1.09 0.36 0.86 1 inflat 0.98 ¡ 0.01 -1.17 0.03 0.96 1 hpi growth 0.96 0.01 -0.83 0.01 0.92 1 unemp 0.86 0.10 -1.27 -0.28 0.65 1 yield 10y 0.79 0.11 -0.93 0.10 0.55 1 sp500 ret 1.12 0.57 -1.08 -0.09 0 1 g. di tollo, g. fattoruso, b. toffano 4.3 correlation analysis we have performed a correlation analysis in order to understand whether some kind of correlation arise amongst variables defined in section 3 and to avoid feeding the network with highly-correlated indicators. we have tested pearson’s, kendall’s, and spearman’s correlation, leading to similar trends in the obtained correlations. in what follows we will refer to spearman’s ranked based correlation. we have decided to remove from the predictor set the indicators showing a correlation with a given portion (i.e., j) of other indicators greater than a given threshold (i.e., h). in order to determine the value of j and h we have defined parameter-tuning procedure via revac (see [43]). the values found have been j = 1 3 and h = 0.70. on the basis of these values, we have decided to remove indicators showing a correlation with 30% of the other indicators greater than 0.7. these indicators are: netloan, lossallow, dep, totasst, toteq, totloan. they will not be considered in what follows: 6 indicators have been removed from the predictors set, corresponding to 24 quarterly indicators that will not be used to feed the neural networks’ nodes. as for the neural network experiments, in what follows we are outlining results of the experiments run by using data before the pre-processing operations (referred to as full model) and data after the pre-processing operations (referred to as reduced model). 5 experimental analysis in this section we are introducing the methods used to perform our experimental analysis: the neural network approach will be detailed in section 5.1, along with the main components needed to define its use, i.e., the network topologies (section 5.1.1) and the partitioning of the set of data to enforce generalisation (section 5.1.2). then, we are introducing the methods we are comparing our approach with (linear regression in section 5.2, and generalised linear models in section 5.3), along with the metrics used for our comparisons in section 5.4. 5.1 neural networks in this section we are introducing our neural network approach: artificial neural networks [22] can be referred to as algorithms that mimic the behavior of the human brain to perform complex tasks, and they are used to grasp nonfunctional relationships over the data. they are composed of elementary units (neurons) which are connected to each other via weighted and oriented links (synapses). neurons may have different functions: the input neurons receive data an adaptive neural network approach to predict the capital adequacy ratio from external sources; the output neurons show the computed output values; the hidden neurons are used to perform computations. during the learning phase, the weights associated to the synapses iteratively change over time, accordingly to a specific algorithm: several algorithms have been proposed for this learning phase: back-propagation [56], quasi-newton methods [55], levenberg-marquardt algorithm [26], just to name a few. the learning phase may be organised following three different paradigms: supervised learning [36]; unsupervised learning [10] and reinforcement leaning [57]. in what follows, we are training networks introduced in section 5.1.1 by using back-propagation algorithm, in order to minimize the network test set’s root mean square error (rmse) defined as √√√√1 n n∑ i=1 (ei −ai)2, (2) where n is the test set size, ei is expected output value corresponding to pattern i, and ai is the actual network output corresponding to input pattern i. 5.1.1 network topologies for our experiments we are using two different neural topologies: a feedforward6 architecture with 80 inputs nodes, referred to as standard network (see figure 2), and a variant in which inputs neurons corresponding to the same indicator are grouped by 4 before feeding the first feed-forward layer7 (see figure 3). please notice that, as for the cardinality of hidden neurons, many rules of thumb exist, suggesting different formulas to compute the number of hidden layers and the number of hidden neurons [19]. we have decided not to use any of these rules, resorting to an adaptive method to determine the optimal hidden neurons structure: this procedure has been proposed by [17], and it aims to minimize the network’s error (in our scenario, the eq. 2) calculated for each of the data set at hand. this adaptive procedure starts with a single hidden neuron and iteratively add one neuron until no improvement on the eq. 2 is found over the last userdefined k iterations, and is outlined in algorithm 1. 6a feed-forward network features neurons grouped into layers (1,2, . . . , lmax) : each neuron belonging to layer i (i < lmax) is associated to synapses that connect itself to all neurons belonging to layer i + 1. 7these four values correspond to past observations spreading over one year, since for each indicator i corresponding to time t the input pattern contains the value of the indicators collected at time t, together with the 3 previously quarterly collected values. g. di tollo, g. fattoruso, b. toffano figure 2: standard network. an adaptive neural network approach to predict the capital adequacy ratio figure 3: ad-hoc network: input neurons are grouped by four, indicating the observations over a year of the same variable. g. di tollo, g. fattoruso, b. toffano algorithm 1: adaptive hidden neurons computation for neural networks initialization: observational data set the optimal network topology w.r.t. the error defined in eq. 2. b in 1, . . . , # sub-sampling runs xb ← dataset at bth sub-sample xtrainb ← training set at bth sub-sample run xtestb ← test set at bth sub-sample run i in 1, . . . , # of hidden layer k ← 0 j ← 1 k ≤ k train net netij on xtrainb compute rmseij error with eq. 2 on the xtestb overallij ≤ all rsmeij bestnet ← netij neurons ← (i, j) overallij > rsmeneurons k++ k ← 0 j++ bestnetb ← bestnet neuronsb ← neurons return bestnet, rsme=eq. 2, neurons 5.1.2 training and test set in our experiments we are exploiting the supervised learning, meaning that during the learning phase, for each input pattern, we are also providing the desired output value, that in our scenario corresponds to car: all other indicators considered in table 3 will define the input pattern for each financial institution. in order to grasp the time dynamics, for each input indicator i we are providing to the network the value of the indicators collected at time t, together with the values collected at time (t − 1), (t − 2), and (t − 3). during the neural network learning we have to identify two disjoint sets of observations out of the overall 34944 observations: the training set, that will be used to determine the synapses’ weights, and the test set, that will be used to determine the network performance and to stop the learning. according to [19], we have decided to split the overall data by randomly allocating the 70% of its observations to the training set, and the remaining 30% to the test set. this random allocation has been repeated 50 times, each time determining a different train-test partition. we have then run our neural network approaches on all obtained partitions, and in what follows we are reporting, for each neural approach, the average and standard deviation statistics over the 50 partitions. 5.2 linear regression linear regression is used to model the relationship between two or multiple parameters by fitting a linear equation on the observed data. usually, this is done using the least-square regression that minimizes the sum of squares of the vertical deviation from each data point on the line. the algorithm aims to reduce this sum by selecting the most appropriate constants in the equation representative of the regression line. a linear regression line has an equation of the form y = a + bx where x is the explanatory variable and y is the dependent variable. the slope of the line is b, and a is the intercept (the value of y when x = 0) [60]. an adaptive neural network approach to predict the capital adequacy ratio 5.3 generalised linear models (glms) the basic linear regression predicts a certain value as a linear combination of a specific set of observed values, meaning that a change in one or multiple predictors affects the response variable. however, for complex data, linear regression is not very effective, and in these cases one may resort to generalized linear models [41], that allow response variables to have arbitrary distributions (rather than normal distributions), and define an arbitrary function of the response variable (the link function) to vary linearly with the predictors, rather than assuming that the response itself must vary linearly. a generalized linear model (glm) consists of three elements: linear predictor; link function; probability distribution or exponential family. the linear predictor is the linear combination of parameter b and explanatory variable x. the link function is what links the linear predicted and the probability distribution: there are many link functions and usually, they are used depending on the features of data we are trying to predict and in which range are we expecting it to be. 5.4 evaluation of the model once we build a model through linear regressions (or glms), we need to measure the correctness of this model. this can be done via different statistical measures, that are used to benchmark the performance of predictive models. 5.4.1 root mean squared error the root mean squared errors represents the standard deviation of the prediction errors. by that, we mean that it tells us how concentrated the data is around the line that we predict. to calculate it we can take the square root of the mean squared errors. rmse = √√√√1 n n∑ i=1 (yi − ŷi)2. where n is the number of predictions, yi the observed values, and ŷi being the actual prediction of that variable. 5.4.2 r-squared the r-squared represents how well the data fits on the regression line. more g. di tollo, g. fattoruso, b. toffano generally, it is used to analyze how the difference in one variable can be explained by other variables. in the case of regression, we can reason in percentages and say that the closer the measure is to 1 the closer the points are to the regression line up to 1 where 100% of the points are on the line. generally, this would mean that the higher r-squared is the better the results we have but this can be false in some edge cases. it is calculated by squaring the correlation coefficient calculated with this formula r2 = 1− ssres sstot where ssres is the sum of squares of residuals, and sstot is the residual sum of squares. 5.4.3 f-statistic the f-statistic in a regression is a value that represents how well you improved the regression line compared to a regression line with all the coefficients = 0. if your model significantly improved the model fit then you will get a better fstatistic. but before taking into account the f-value one must first look at the p-value that is calculated at the same time as the f-statistic. with the f-statistic calculation comes the p-value. usually the p-value is looked at before taking the f-statistic into account. if the p-value is lower than the alpha level, then we can reject the null hypothesis and we can consider the f-value, otherwise the f-values is worthless. 6 results and discussion in this section we report the principal results to build a model that is performs well on our benchmarks. all neural approaches have been implemented in python, exploiting the library tensor-flow. experiments have been run on a on a cluster with amd opteron 2216 dual core cpus running at 2.4 ghz with 2x1 mb l2 cache and 4 gb of ram under cluster rocks distribution built on top of centos 5.3 linux. table 4 reports the rmse of the experiments run with both the standard and ad-hoc networks. we have performed 50 runs of the adaptive procedure devised in algorithm 2 and reported the minimum, maximum, mean, median, and standard deviation of the rmse distribution, for each possible instantiation of the pair [network, model]. an adaptive neural network approach to predict the capital adequacy ratio standard standard ad-hoc ad-hoc reduced model full model reduced model full model min 0.070 0.071 0.040 0.040 max 0.074 0.076 0.045 0.051 mean 0.072 0.073 0.042 0.045 std 0.001 0.001 0.001 0.003 median 0.072 0.073 0.042 0.046 table 4: rmse of the experiments run with the standard and ad-hoc networks. for each columns. for each column, statistics over 50 runs of the proposed adaptive procedure are reported. as a first remark, we can see that the pre-processing operation have a valid role in improving the networks’ performances, in both standard and ad-hoc networks. then, we can see that the ad-hoc networks’ error is lower than the standard one. this confirms the results found by [17] and [21], in which authors exploit the fact that the ad-hoc network is able to grasp the temporal dependence of inputs. then, we are presenting the results obtained with linear regression and with glms (both developped using the sci-kit learn library on python[51]), and then, we are comparing them with the neural network approach devised in section 5.1. as a first experiment, we have implemented a regression approach over the whole dataset, and the results are shown in table 5: in this table we report the r2 relative to experiments performed with linear regression and glms, along with two variants: the adjusted r-squared (that takes into account the number of predictors) and the predicted r-squared (that takes into account overfitting). please notice that for glm we have used the pseudo r-squared (for sake of definition) and that we also report the p-value of the f-test (i.e., the probability of obtaining an f-statistic value that is greater than the model’s f-value, under the null hypothesis that the regression model is not significant: low positive values identifies good fit). we have performed experiments by using as predictors both the total set of variables identified after the pre-processing phase (identified by the entry whole set of data in the table), and a limited set of predictors composed of all predictors that are significant for the regression according to their p-value (identified by the entry all observations, limited set of predictors). we see that the reduction of predictors does not improve the goodness of the fit according to the different metrics used, so in what follows we are using the whole set of predictors. in this direction, we remark that generalised linear models lead to a better r−squared, but this comes at the cost of a higher overfitting, as witnessed by the lower value of the predictedr − squared. linear regression instead, leads to a worse (but still acceptable) r−squared, but its difference with the predictedr−squared g. di tollo, g. fattoruso, b. toffano is lower, showing a better robustness of the approach. please notice that all regressions are significant according to the p-value of the f-test. we recall that these experiments have been performed on the whole set of data. in what follows we will describe experiments performed on different partitions of training/test sets, in order to compare these approaches with our neural network approach. table 5: linear regression and glms over the whole set of data: measures of the goodness of the fit. data used model statistic values all observations, linear regression predicted r-squared 0.77 limited set of predictors r-squared 0.78 adj r-squared 0.78 whole set of data linear regression predicted r-squared 0.77 r-squared 0.78 adj r-squared 0.78 p-value of the f-test 0.0 whole set of data glm predicted r-squared 0.71 pseudo r-squared 0.91 p-value of the f-test 0.0 please notice that the previous results have been obtained on the whole set of data. in order to test the generalization capability of our approach, we have split the whole set of data in 50 different training/testing partitions (i.e., the same partitioning generated in section 5.1.2), built our model to fit the training set, and assessed the goodness of the fit on the test set. this has been done for all regression detailed in table 5, along with the neural approaches devised in table 4, and the goodness of fit (assessed by the r − squared) has been reported in table 6, as computed on the error distributions displayed in table 4. by looking the results, we see that the best results are offered by the ad-hoc neural networks, but also the standard network performs fairly well. this is due to the generalisation skill of the network, able to prevent the overfitting we have found on the aforementioned regression approaches. this confirms the goodness of our adaptive procedure, that has been proposed by [17] in a different context, but that can be tailored to the different application scenarios. an adaptive neural network approach to predict the capital adequacy ratio table 6: experiments with linear regression, generalised linear models, and neural networks (standard and ad-hoc). statistics of the r-squared computed on the test sets of 50 different training/testing partitions. model statistic values linear regression min 0.76 max 0.80 mean 0.77 stdd ¡ 0.01 glm min 0.69 max 0.72 mean 0.70 stdd ¡ 0.01 standard nn min 0.71 max 0.90 mean 0.81 stdd 0.15 ad-hoc nn min 0.64 max 0.92 mean 0.84 stdd 0.13 7 concluding remarks the bankruptcy of banks may lead to a huge catastrophic effect over the overall economy, since the contagion effect it may trigger could lead to a generalised overall crisis. to this extent, the activity of banking supervision, and the role and the authority of bank regulation, play a big role, since they may prevent (or reduce the effect of) the banks’ bankruptcy. although aimed to different targets, many of these supervision exercise are designed to maintain a sufficient level of capital adequacy: in a way, the banks have to allocate specific reserves to face expected losses and to protect themselves from excessive credit expansion. bank regulations impose constraints over these reserves, even though banks operate themselves preventively against unexpected crises, and their reserves are often higher than the ones imposed by the regulations. the car is one of the main indicators monitored by the banks themselves and by the supervising authorities in order to assess the bank health, and in our contribution we have devised an adaptive neural network approach to predict the car, and compared the obtained results with standard approaches such as linear regression and generalised linear models. g. di tollo, g. fattoruso, b. toffano results show that neural networks may be successfully used to predict the car, and that their outcomes compare favourably with standard methods when used jointly with meaningful pre-processing operations. in future research, modern recurrent neural networks (long short-term memory or gate recurrent units) and 1d-convolutional neural networks could be used to exploit the time dependency of the data. references [1] s. park s. peristiani a., estrella. capital ratios as predictors of bank failure. economic policy review, 6:33–52, 02 2000. 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[60] x. yan and x. g. su. linear regression analysis: theory and computing. world scientific publishing co., inc., usa, 2009. ratio mathematica volume 47, 2023 superior domination polynomial of cycles tejaskumar r* a mohamed ismayil† abstract superior domination polynomial sd(g, x) = ∑n t=γsd(g) |sd(g, t)|xt is a polynomial in which the power of the variable denotes the cardinality of a superior dominating set and the total number of sets of same cardinality forms the coefficient of the variable. in this paper we find the sd(g, sn) of stars and sd(g, cn) of cycles and properties of the coefficients are discussed. the sd(g, x) different standard graphs are obtained and the roots of the polynomial are tabulated. keywords: superior distance, superior domination, neighbourhood vertex, superior domination polynomial. 2020 ams subject classifications: 05c12, 05c69, 05c31.1 *research scholar, pg & research department of mathematics, jamal mohamed college(affiliated to bharathidasan university), trichy, india; mail id: tejaskumaarr@gmail.com. †associate professor, pg & research department of mathematics, jamal mohamed college(affiliated to bharathidasan university), trichy, india; mail id: amismayil1973@yahoo.co.in. 1received on september 15, 2022. accepted on december 15, 2022. published on march 1, 2023. doi: 10.23755/rm.v41i0.819. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 151 tejaskumar r and a mohamed ismayil 1 introduction the graph g = (v, e) is a finite, undirected, simple, ordred pair where v (g) is a set of vertices and e(g) is the set of edges. in 2009 saeid alikhani and yee-hock peng[1] conceptualized the concept of domination polynomial. domination is a vast arena in graph theory, ore[8] coined the term domination in graphs. a vast literature about domination can be found in domination in graphs[3]. there are different types of distances in graph theory one being superior distance, kathiresan and marimuthu[7] were the pioneers of superior distance in graphs. the same authors[6] put forth the concept of superior domination in 2008. a mohamed ismayil and tejaskumar r[4] introduced eccentric domination polynomial which was the hybrid idea of combining eccentric domination[5] and domination polynomial. in this paper, a distance based domination polynomial called superior domination polynomial is introduced by coalescence of superior domination and domination polynomial. standard formulas to find the coeffcients or the superior dominating sets of stars sn and cycles cn for any value of n. theorems realted to properties of these coefficients are stated and proved. superior domination polynomial sd(g, x) of different standard graphs are calculated, their roots are tabulated. for all the undefined terminologies and basic concepts of graphs refer the book graph theory by frank harary[2]. 2 preliminaries definition 2.1. [1]. let d(g, i) be the family of dominating sets of a graph g with cardinality i and let d(g, i) = |d(g, i)|. then the domination polynomial d(g, x) of g is defined as d(g, x) = ∑|v (g)| i=γ(g) d(g, i)xi, where γ(g) is the domination number of g. definition 2.2. [7]. let duv = n[u] ∪ n[v]. a duv-walk is defined as a u − v walk in g that contains every vertex of duv. the superior distance dd(u, v) from u to v is the length of a shortest du,v walk. definition 2.3. [7]. the superior neighbour of a vertex u is given by dd(u) = min{dd(u, v) : v ∈ v (g)−{u}}. a vertex v(̸= u) is called a superior neighbour of u if dd(u, v) = dd(u). definition 2.4. [6]. a vertex u is said to be a superior dominate a vertex v if v is a superior neighbour of u. 152 superior domination polynomial definition 2.5. [6]. a set s of vertices of g is called a superior dominating set of g if every vertex v (g)−s is superior dominated by some vertex in s. a superior dominating set g of minimum cardinality is a minimum superior dominating set and its cardinality is called superior domination number of g and denoted by γsd(g). theorem 2.1. [6]. for a cycle cn the superior domination number is given by γsd(cn) =   n 3 , if n ≡ 0(mod 3) n+2 3 , if n ≡ 1(mod 3) n+1 3 , if n ≡ 2(mod 3) 3 superior domination polynomial of graphs in this section, we defined superior domination polynomial, properties and results related to superior domination polynomial are observed, stated and proved. definition 3.1. superior domination polynomial is given by sd(g, x) = ∑n t=γsd(g) |sd(g, t)|xt where |sd(g, t)| is the number of distinct superior dominating set with cardinality t and γsd(g) is the superior domination number. example 3.1. . v5 v6 v3 v4 v1 v2 figure 1: net graph vertex minimum superior distance dd superior neighbour v1 3 v2, v6 v2 3 v1, v6 v3 4 v1 v4 4 v2 v5 4 v6 v6 3 v1, v2 153 tejaskumar r and a mohamed ismayil from figure-1 we get {v3, v4, v5} is a superior dominating set with cardinality 3, {v1, v3, v4, v5}, {v2, v3, v4, v5}, {v3, v4, v5, v6} are superior dominating sets of cardinality 4, {v1, v2, v3, v4, v5}, {v1, v3, v4, v5, v6}, {v2, v3, v4, v5, v6} are superior dominating sets of cardinality 5 and {v1, v2, v3, v4, v5, v6} is superior dominating set with cardinality 6. therefore superior domination polynomial is given by sd(g, x) = x6 + 3x5 + 3x4 + x3. theorem 3.1. for a complete graph kn the superior domination polynomial is given by sd(kn, x) = (1 + x)n − 1. proof. the degree of every vertex v ∈ kn is n − 1. for any two vertices u and v the number of vertices on their duv-walk is given by |v (kn)|. since |n[u]| = n and |n[v]| = n both the vertices have common neighbours and both u and v are incident to each other. therefore a du,v-walk between u and v contains all vertices of kn and all the vertices of kn forms the superior neighbour of any v ∈ v (kn) other than itself. by the definition of superior distance, the distance between any two vertices is n − 1. now by the definition of superior domination, for every vertex of v (kn)−s is superior dominated by some vertex in s which is a superior dominating set and every vertex of v (kn)−s has a superior neighbour in s. therefore sd(kn, x) = (1 + x)n − 1. theorem 3.2. if two graphs are isomorphic then sd(g1, x) = sd(g2, x). proof. let g1 and g2 be any two isomorphic graphs. then there exist a oneone and onto function between the vertex sets such that f : v (g1) → v (g2) such that vm and vn are superior neighbours in g1 if and only if f(vm) and f(vn) are superior neighbour of some vertex in g2. therefore |sd(g1, n)| = |sd(g2, n)| ∀ n. therefore sd(g1, x) = sd(g2, x). example 3.2. in the figure 2 and 3 both the tetrahedral graph and complete graph k4 are isomorphic to each other. v3 v4 v1 v2 fig:2-tetrahedral graph tg v3 v4 v1 v2 fig:3-complete graph k4 sd(tg, x) = x 4 + 4x3 + 6x2 + 4x. sd(k4, x) = x 4 + 4x3 + 6x2 + 4x. hence tg ∼= k4 implies sd(tg, x) = sd(k4, x). 154 superior domination polynomial definition 3.2. superior domination polynomial of a star graph sn is given by sd(sn, x) = ∑n t=γsd(sn) |sd(sn, t)|xt where |sd(sn, t)| is the number of distinct superior dominating sets with cardinality t and γsd(sn) is the superior domination number of a star graph. theorem 3.3. for a star graph sn of order n where n ≥ 3, the following are true. 1. |sd(sn, t)| = |sd(sn−1, t − 1)| + |sd(sn−1, t)|, t ∈ z+, t ≤ n. 2. sd(sn, x) = xsd(sn−1, x) + sd(sn−1, x). 3. sd(sn, x) = x(x + 1)n−1. proof. 1. let v (sn) = {v1, v2, . . . vn}. all the pendant vertices form the superior neighbours of central vertex v1 since deg(v1) = ∆(sn) = n − 1. here we have (n−1)ct−1 superior dominating sets of cardinality t. therefore |sd(sn, t)| =(n−1) ct−1, |sd(sn−1, t−1)| =(n−2) ct−2 and |sd(sn−1, t)| =(n−2) ct−1. but (n−1)ct−1 =(n−2) ct−2 +(n−2) ct−1. therefore |sd(sn, t)| = |sd(sn−1, t − 1)| + |sd(sn−1, t)|. 2. by theorem-3.3-(1) we have |sd(sn, t)| = |sd(sn−1, t − 1)| + |sd(sn−1, t)|. when t = 1, |sd(sn, 1)| = |sd(sn−1, 0)| + |sd(sn−1, 1)|. =⇒ x|sd(sn, 1)| = x|sd(sn−1, 0)| + x|sd(sn−1, 1)|. when t = 2, |sd(sn, 2)| = |sd(sn−1, 1)| + |sd(sn−1, 2)|. =⇒ x2|sd(sn, 2)| = x2|sd(sn−1, 1)| + x2|sd(sn−1, 2)|. when t = 3, |sd(sn, 3)| = |sd(sn−1, 2)| + |sd(sn−1, 3)|. =⇒ x3|sd(sn, 3)| = x3|sd(sn−1, 2)| + x3|sd(sn−1, 3)|. when t = 4, |sd(sn, 4)| = |sd(sn−1, 3)| + |sd(sn−1, 4)|. =⇒ x4|sd(sn, 4)| = x4|sd(sn−1, 3)| + x4|sd(sn−1, 4)|. ... when t = n − 1, |sd(sn, n − 1)| = |sd(sn−1, n − 2)| + |sd(sn−1, n − 1)|. =⇒ xn−1|sd(sn, n−1)| = xn−1|sd(sn−1, n−2)|+xn−1|sd(sn−1, n− 1)|. when t = n, |sd(sn, n)| = |sd(sn−1, n − 1)| + |sd(sn−1, n)|. =⇒ xn|sd(sn, n)| = xn|sd(sn−1, n − 1)| + xn|sd(sn−1, n)|. hence x|sd(sn, 1)| + x2|sd(sn, 2)| + x3|sd(sn, 3)| + x4|sd(sn, 4)| + · · · + xn−1|sd(sn, n − 1)| + xn|sd(sn, n)| = x|sd(sn−1, 0)| + x|sd(sn−1, 1)| + x2|sd(sn−1, 1)| + x2|sd(sn−1, 2)| + x3|sd(sn−1, 2)| + x3|sd(sn−1, 3)| + x4|sd(sn−1, 3)|+x4|sd(sn−1, 4)|+· · ·+xn−1|sd(sn−1, n−2)|+xn−1|sd(sn−1, n− 155 tejaskumar r and a mohamed ismayil 1)| + xn|sd(sn−1, n − 1)| + xn|sd(sn−1, n)|. = x|sd(sn−1, 0)| + x2|sd(sn−1, 1)| + x3|sd(sn−1, 2)| + x4|sd(sn−1, 3)| + · · ·+xn−1|sd(sn−1, n−2)|+xn|sd(sn−1, n−1)|+x|sd(sn−1, 1)|+x2|sd(sn−1, 2)|+ x3|sd(sn−1, 3)|+x4|sd(sn−1, 4)|+· · ·+xn−1|sd(sn−1, n−1)|+xn|sd(sn−1, n)|. = x[x|sd(sn−1, 1)|+x2|sd(sn−1, 2)|+x3|sd(sn−1, 3)|+x4|sd(sn−1, 4)|+ · · ·+xn−1|sd(sn−1, n−1)|]+x|sd(sn−1, 1)|+x2|sd(sn−1, 2)|+x3|sd(sn−1, 3)|+ x4|sd(sn−1, 4)| + · · · + xn−1|sd(sn−1, n − 1)| + xn|sd(sn−1, n)|. since |sd(sn−1, 0)| = |sd(sn−1, n)| = 0. = x ∑n−1 t=1 |sd(sn−1, t)|x t + ∑n−1 t=1 |sd(sn−1, t)|x t. sd(sn, x) = x sd(sn−1, x) + sd(sn−1, x). 3. we prove this by mathematical induction. when n = 3, sd(sn, x) = x(x + 1) n−1 = x(x + 1)3−1 = x(x + 1)2 the result is true for n = 3. when n = 4, sd(sn, x) = x(x + 1) 3 the result is true for n = 4. assume the result is true for all natural numbers less than n. sd(sn−1, x) = x(x + 1) (n−1)−1 = x(x + 1)n−2 now we prove the result for n. sd(sn, x) = x sd(sn−1, x) + sd(sn−1, x) using theorem3.3-(2) = x[x(x + 1)n−2] + x(x + 1)n−2 = x(x + 1)n−2[x + 1] = x(x + 1)n−2+1 = x(x + 1)n−1 ∴ the result is true for all n. table: |sd(sn, t)| is the number of superior dominating sets of sn with cardinality t where 1 ≤ t ≤ 15. 156 superior domination polynomial n t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 0 2 2 1 3 1 2 1 4 1 3 3 1 5 1 4 6 4 1 6 1 5 10 10 5 1 7 1 6 15 20 15 6 1 8 1 7 21 35 35 21 7 1 9 1 8 28 56 70 56 28 8 1 10 1 9 36 84 126 126 84 36 9 1 11 1 10 45 120 210 252 210 120 45 10 1 12 1 11 55 165 330 462 462 330 165 55 11 1 13 1 12 66 220 495 792 924 792 495 220 66 12 1 14 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 15 1 14 91 364 1001 2002 3003 3423 3003 2002 1001 364 91 14 1 theorem 3.4. the following properties for the co-efficients of sd(sn, x) holds. 1. |sd(sn, 1)| = 1 for all n > 2. 2. |sd(sn, n)| = 1 for all n ≥ 2. 3. |sd(sn, n − 1)| = n − 1 for all n > 2. 4. |sd(sn, n − 2)| = (n − 1)(n − 2) 2 for all n ≥ 3. 5. |sd(sn, n − 3)| = (n − 1)(n − 2)(n − 3) 6 for all n ≥ 4. 6. |sd(sn, n − 4)| = (n − 1)(n − 2)(n − 3)(n − 4) 24 for all n ≥ 5. 7. |sd(sn, t)| = |sd(sn, n − t + 1)| for all n ≥ 3. 8. if sdn = ∑n t=1 |sd(sn, t)| for all n ≥ 3 then sdn = 2(sdn−1) with initial condition sd3 = 4. 9. sdn =total number of superior dominating sets in sn = 2n−1 for all n ≥ 3. proof. 1. let v (sn) = {v1, v2, . . . vn}. in a star graph sn all the vertices form a superior neighbour of central vertex v1 except itself. therefore the only set with single cardinality d = {v1} forms the superior dominating set of every star graph sn where n > 2. therefore |sd(sn, 1)| = 1 for all n > 2. 157 tejaskumar r and a mohamed ismayil 2. the whole set of vertices v (sn) forms the superior dominating set |sd(sn, n)| = 1 for all n ≥ 2. 3. by mathematical induction on n. the result is true for n = 3, since |sd(s3, 3 − 1)| = |sd(s3, 2)| = 2. assume the result is true for all natural numbers less than n. now we prove it for n ie, |sd(sn−1, n − 2)| = n − 2. by theorem-3.3-(1) and 3.4-(2) |sd(sn, n − 1)| = |sd(sn−1, n − 2)| + |sd(sn−1, n − 1)| = (n − 2) + 1 = n − 1 ∴ the result is true for all n. 4. by mathematical induction on n. for n = 3, |sd(s3, 1)| = 1. for n = 4, |sd(s4, 2)| = 3. assume the result is true for all natural numbers less than n, ie, for n = n − 1, |sd(sn−1, n − 3)| = (n−2)(n−3) 2 now we prove it for n. by theorem-3.3-(1) and 3.4-(3), |sd(sn, n − 2)| = |sd(sn−1, n − 3)| + |sd(sn−1, n − 2)| = (n − 2)(n − 3) 2 + (n − 2) = (n − 2)(n − 3) + 2(n − 2) 2 = (n − 2)(n − 3 + 2) 2 = (n − 2)(n − 1) 2 = (n − 1)(n − 2) 2 ∴ the result is true for all n. 5. by mathematical induction on n. for n = 4, |sd(s4, 1)| = 1. for n = 5, |sd(s5, 2)| = 4. assume the result is true for all natural numbers less than n. for n = n − 1, |sd(sn−1, n − 4)| = (n−2)(n−3)(n−4) 6 158 superior domination polynomial now we prove it for n. by theorem-3.3-(1) and 3.4-(4), |sd(sn, n − 3)| = |sd(sn−1, n − 4)| + |sd(sn−1, n − 3)| = (n − 2)(n − 3)(n − 4) 6 + (n − 2)(n − 3) 2 = (n − 2)(n − 3)(n − 4 + 3) 6 = (n − 1)(n − 2)(n − 3) 6 ∴ the result is true for all n. 6. by mathematical induction on n. the result is true for n = 5 since |sd(s5, 1)| = 1 for n = 6, |sd(s6, 2)| = 5. assume the result is true for all natural numbers less than n. for n = n − 1, |sd(sn−1, n − 5)| = (n−2)(n−3)(n−4)(n−5) 24 now we prove it for n. by theorem-3.3-(1) and 3.4-(5), |sd(sn, n − 4)| = |sd(sn−1, n − 5)| + |sd(sn−1, n − 4)| = (n − 2)(n − 3)(n − 4)(n − 5) 24 + (n − 2)(n − 3)(n − 4) 6 = (n − 2)(n − 3)(n − 4)(n − 5 + 4) 24 = (n − 1)(n − 2)(n − 3)(n − 4) 24 ∴ the result is true for all n. 7. by mathematical induction on n. the result is true for n = 3 since |sd(s3, 1)| = |sd(s3, 3 − 1 + 1)| = |sd(s3, 3)| = 1. for n = 4, t = 2, |sd(s4, 2)| = |sd(s4, 4 − 2 + 1)| = |sd(s4, 3)| = 3. assume the result is true for all natural numbers less than n. for n = n − 1, |sd(sn−1, t − 1)| = |sd(sn−1, (n − t + 1))| now we prove it for n. by theorem-3.3 we have, |sd(sn, t)| = |sd(sn−1, t − 1)| + |sd(sn−1, t)| = |sd(sn−1, (n − 1 − (t − 1) + 1))| + |sd(sn−1, (n − 1 − (t) + 1))| = |sd(sn−1, (n − 1 − t + 1 + 1))| + |sd(sn−1, (n − 1 − t + 1))| = |sd(sn−1, (n − t + 1))| + |sd(sn−1, (n − t))| = |sd(sn, (n − t + 1))| ∴ the result is true for all n. 159 tejaskumar r and a mohamed ismayil 8. sdn = ∑n i=1 |sd(sn, t)| by theorem-3.3 we have sdn = n∑ i=1 [|sd(sn−1, t − 1)| + |sd(sn−1, t)|] = n−1∑ i=1 |sd(sn−1, t)| + n−1∑ i=1 |sd(sn−1, t)| = sdn−1 + sdn−1 sdn = 2[sdn−1] 9. by mathematical induction on n. when n = 3, sd3 = 2 3−1 = 22 = 4. sd4 = 2 4−1 = 23 = 8. assume the result is true for all natural numbers less than n. now we prove it for n. therefore sdn−1 = 2n−1−1 = 2n−2 now sdn = 2[sdn−1] from theorem-3.4-(8) = 2[2n−2] = 2n−2+1 = 2n−1 ∴ the result is true for all n. hence the theorem. 4 superior domination polynomial of cycle let sd(cn, m) be the superior dominating set of cycle cn with cardinality m. k.m. kathiresan and g. marimuttu[6] proved theorem-2.1, for our convenience we reframe theorem-2.1 as γsd(cn) = ⌈n3 ⌉ ∀ n. hereafter we denote the vertex set v (g) = {v1, v2, . . . vn} = [n]. lemma 4.1. for a cycle cn, sd(cn, m) = ∅ if m > n or m < ⌈n3 ⌉ proof. let cn be a cycle, a superior dominating set d has the minimum cardinality among the minimum superior dominating set with cardinality ⌈n 3 ⌉ by theorem-2.1. therefore there is no proper subset of d which forms a superior dominating set. hence sd(cn, m) = ∅ where m < ⌈ n 3 ⌉ = |d| (1) 160 superior domination polynomial there can not exists a superior dominating set greater than the order of the graph. therefore sd(cn, m) = ∅ if m > n (2) from equation-(1) and (2) we obtain the result. observation 4.1. if a cycle cn contains a maximal simple path of length 3k − 1, 3k or 3k + 1 then every dominating set of cn must contain at least k, k + 1 or k + 1 vertices respectively. lemma 4.2. let l be a subset of the vertex set, l ⊆ [n]. if l is in sd(cn−4, m−1) or sd(cn−5, m−1) ∋ l∪{v} ∈ sd(cn, m) for v ∈ [n] then l ∈ sd(cn−3, m−1). proof. let l ∈ sd(cn−4, m − 1) and l ∪ {v} ∈ sd(cn, m) for v ∈ [n] then by lemma-4.3 we consider {1, n − 4}, {2, n − 4} and {1, n − 5} as a subset of l. then l ∈ sd(cn−3, m − 1) suppose l ∈ sd(cn−5, m − 1) and l ∪ {v} ∈ sd(cn, m) for v ∈ [n]. then by lemma-4.3 {1, n − 5} must be a subset of l. hence l ∈ sd(cn−3, m − 1). lemma 4.3. . 1. if sd(cn−1, m − 1) = sd(cn−3, m − 1) = ∅ then sd(cn−2, m − 1) = ∅. 2. if sd(cn−1, m−1) ̸= ∅ and sd(cn−3, m−1) ̸= ∅ then sd(cn−2, m−1) ̸= ∅. 3. if sd(cn−1, m − 1) = sd(cn−2, m − 1) = sd(cn−3, m − 1) = ∅ then sd(cn, m) = ∅. proof. 1. since sd(cn−1, m−1) = sd(cn−3, m−1) = ∅, by lemma-4.1, m−1 > n−1 or m − 1 < ⌈n−3 3 ⌉. in both cases we have sd(cn−2, m − 1) = ∅. 2. suppose that sd(cn−2, m−1) = ∅ by lemma-4.1, m−1 > n−2 or m−1 < ⌈n−2 3 ⌉. if m−1 > n−2 then m−1 > n−3. hence sd(cn−3, m−1) = ∅, a contradiction. hence m−1 < ⌈n−2 3 ⌉. so m−1 < ⌈n−1 3 ⌉, sd(cn−1, m−1) = ∅, also a contradiction. 3. suppose sd(cn, m) ̸= ∅. let l ∈ sd(cn, m) such that at least one vertex labelled as vn or vn−1 is in l. if vn ∈ l, then by observation-4.1 at least one vertex labelled as vn−1, vn−2 or vn−3 is in l. if vn−1 ∈ l or vn−2 ∈ l, then l−{vn} ∈ sd(cn−1, m−1), a contradiction. if vn−3 ∈ l, then l−{vn} ∈ sd(cn−2, m − 1) a contradiction. now suppose that vn−1 ∈ l. then by observation-4.1 at least one vertex labelled vn−2, vn−3 or vn−4 is in l. if vn−2 ∈ l or vn−3 ∈ l, then l−{vn−1} ∈ sd(cn−2, m−1), a contradiction. if vn−4 ∈ l then l−{vn−1} ∈ sd(cn−3, m−1), a contradiction. therefore sd(cn, m) = ∅. 161 tejaskumar r and a mohamed ismayil lemma 4.4. suppose that sd(cn, m) ̸= ∅ then we have 1. sd(cn−1, m − 1) = sd(cn−2, m − 1) = ∅ and sd(cn−3, m − 1) ̸= ∅ if and only if n = 3k and m = k for some k ∈ n. 2. sd(cn−2, m − 1) = sd(cn−3, m − 1) = ∅ and sd(cn−1, m − 1) ̸= ∅ if and only if m = n. 3. sd(cn−1, m − 1) = ∅, sd(cn−2, m − 1) ̸= ∅ and sd(cn−3, m − 1) ̸= ∅ if and only if n = 3k + 2 and m = ⌈3k+2 3 ⌉ for some k ∈ n. 4. sd(cn−1, m − 1) ̸= ∅, sd(cn−2, m − 1) ̸= ∅ and sd(cn−3, m − 1) = ∅ if and only if m = n − 1. 5. sd(cn−1, m − 1) ̸= ∅, sd(cn−2, m − 1) ̸= ∅ and sd(cn−3, m − 1) ̸= ∅ if and only if ⌈n−1 3 ⌉ + 1 ≤ m ≤ n − 2. proof. 1. since sd(cn−1, m−1) = sd(cn−2, m−1) = ∅. by lemma-4.1 m−1 > n−1 or m−1 < ⌈n−2 3 ⌉. if m−1 > n−1 then m > n by lemma-4.1 sd(cn, m) = ∅ a contradiction. therefore m < ⌈n−2 3 ⌉ + 1 since sd(cn, m) ̸= ∅ together we have ⌈n 3 ⌉ ≤ m ≤ ⌈n−2 3 ⌉ + 1. hence n = 3k and m = k for k ∈ n. conversely suppose if n = 3k, m = k for k ∈ n then by lemma-4.1 sd(cn−1, m − 1) = sd(cn−2, m − 1) = ∅ and sd(cn−3, m − 1) ̸= ∅. 2. since sd(cn−2, m−1) = sd(cn−3, m−1) = ∅ by lemma-4.1 m−1 > n−2 or m − 1 < ⌈n−3 3 ⌉. if m − 1 < ⌈n−3 3 ⌉ then m − 1 < ⌈n−1 3 ⌉. hence sd(cn−1, m − 1) = ∅ a contradiction. so we have m > n − 1 also since sd(cn−1, m − 1) ̸= ∅ we have m − 1 ≤ n − 1. hence m = n. conversely suppose if m = n then by lemma-4.1 we have sd(cn−2, m − 1) = sd(cn−3, m − 1) = ∅ and sd(cn−1, m − 1) ̸= ∅. 3. since sd(cn−1, m−1) = ∅ by lemma-4.1 m−1 > n−1 or m−1 < ⌈n−13 ⌉. if m−1 > n−1 then m−1 > n−2 and by lemma-4.1 sd(cn−2, m−1) = sd(cn−3, m − 1) = ∅ a contradiction so we must have m < ⌈n−13 ⌉ + 1. but also we have m−1 ≥ ⌈n−2 3 ⌉ because sd(cn−2, m−1) ̸= ∅. hence we have ⌈n−2 3 ⌉+1 ≤ m < ⌈n−1 3 ⌉+1. therefore n = 3k+2 and m = k+1 = ⌈3k+2 3 ⌉ for some k ∈ n. conversely suppose if n = 3k + 2, m = ⌈3k+2 3 ⌉ for some k ∈ n then by lemma-4.1 sd(cn−1, m−1) = sd(c3k+1, k) = ∅, sd(cn−2, m−1) ̸= ∅ and sd(cn−3, m − 1) ̸= ∅. 162 superior domination polynomial 4. since sd(cn−3, m − 1) = ∅ by lemma-4.1 we have m − 1 > n − 3 or m − 1 < ⌈n−3 3 ⌉ since sd(cn−2, m − 1) ̸= ∅ by lemma-4.1 we have ⌈n−23 ⌉ + 1 ≤ m ≤ n − 1. therefore m − 1 < ⌈n−3 3 ⌉ is not possible. hence we must have m−1 > n−3. then m = n−1 or n but m ̸= n as sd(cn−2, m−1) ̸= ∅. therefore m = n − 1. conversely suppose if m = n − 1 then by lemma-4.1 sd(cn−1, m − 1) ̸= ∅ sd(cn−2, m − 1) ̸= ∅ and sd(cn−3, m − 1) = ∅. 5. since sd(cn−1, m−1) ̸= ∅, sd(cn−2, m−1) ̸= ∅ and sd(cn−3, m−1) ̸= ∅ then by applying lemma-4.1 we have ⌈n−1 3 ⌉ ≤ m − 1 ≤ n − 1, ⌈n−2 3 ⌉ ≤ m − 1 ≤ n − 2 and ⌈n−3 3 ⌉ ≤ m − 1 ≤ n − 3 so ⌈n−1 3 ⌉ ≤ m − 1 ≤ n − 3 and hence ⌈n−1 3 ⌉ + 1 ≤ m ≤ n − 2. conversely suppose if ⌈n−1 3 ⌉ + 1 ≤ m ≤ n − 2 then by lemma-4.1 we have the result. theorem 4.1. for every n ≥ 4 and m ≥ ⌈n 3 ⌉, 1. if sd(cn−1, m − 1) = sd(cn−2, m − 1) = ∅ and sd(cn−3, m − 1) ̸= ∅ then sd(cn, m) = sd(cn, n 3 ) = {{1, 4, . . . n−2}, {2, 5, . . . n−1}, {3, 6, . . . n}}. 2. if sd(cn−2, m − 1) = sd(cn−3, m − 1) = ∅ and sd(cn−1, m − 1) ̸= ∅ then sd(cn, m) = sd(cn, n) = {[n]}. 3. if sd(cn−1, m − 1) = ∅, sd(cn−2, m − 1) ̸= ∅ and sd(cn−3, m − 1) ̸= ∅ then sd(cn, m) = {{1, 4, . . . n − 4, n − 1}, {2, 5, . . . n − 3, n}, {3, 6, . . . n − 2, n}} ⋃ {s ⋃  {n − 2}, if 1 ∈ s {n − 1}, if 1 /∈ s, 2 ∈ s | s ∈ sd(cn−3, m − 1) {n}, otherwise where s ⊆ v (cn). 4. if sd(cn−3, m − 1) = ∅, sd(cn−2, m − 1) ̸= ∅ and sd(cn−1, m − 1) ̸= ∅ then sd(cn, m) = {[n] − {p}|p ∈ [n]} 5. if sd(cn−1, m − 1) ̸= ∅, sd(cn−2, m − 1) ̸= ∅ and sd(cn−3, m − 1) ̸= ∅ then sd(cn, m) = {{n} ⋃ s | s ∈ sd(cn−1, m − 1) ⋃ {s1   {n}, if n − 2 or n − 3 ∈ s1 for s1 ∈ sd(cn−2, m − 1) or sd(cn−1, m − 1) {n − 1}, if n − 2 /∈ s1, n − 3 /∈ s1 or s1 ∈ sd(cn−1, m − 1) ∩ sd(cn−2, m − 1) } ⋃ 163 tejaskumar r and a mohamed ismayil {s2   {n − 2}, if 1 ∈ s2 for s2 ∈ sd(cn−3, m − 1) or s2 ∈ sd(cn−3, m − 1) ∩ sd(cn−2, m − 1) {n − 1}, if n − 3 ∈ s2 or n − 4 ∈ s2 for s2 ∈ sd(cn−3, m − 1) or sd(cn−2, m − 1) } where s, s1 and s2 are subsets of v (cn). proof. 1. sd(cn−1, m − 1) = sd(cn−2, m − 1) = ∅ by lemma-4.4-(1) n = 3k, m = k for some k ∈ n. hence sd(cn, m) = sd(cn, n3 ) = {{1, 4, 7, . . . n− 2}, {2, 5, 8, . . . n − 1}, {3, 6, 9, . . . n}}. 2. sd(cn−2, m − 1) = sd(cn−3, m − 1) = ∅ and sd(cn−1, m − 1) ̸= ∅ by lemma-4.4-(2) m = n. therefore sd(cn, m) = (cn, n) = {[n]}. 3. sd(cn−1, m − 1) = ∅, sd(cn−2, m − 1) ̸= ∅ and sd(cn−3, m − 1) ̸= ∅ by lemma-4.4-(3) n = 3k + 2, m = k + 1 for some k ∈ n we denote the families {{1, 4, . . . 3k − 2, 3k + 1}, {2, 5, . . . 3k − 1, 3k + 2}, {3, 6, . . . 3k, 3k + 3}} and {s ⋃  {3k}, if 1 ∈ s {3k + 1}, if 1 /∈ s, 2 ∈ s | s ∈ sd(c3k−1, k) {3k + 2}, otherwise by l1 and l2 respectively. we shall prove that sd(c3k+2,k+1) = l1 ∪ l2. since sd(ck, 3k) = {{1, 4, 7, . . . 3k−2}, {2, 5, 8, . . . 3k−1}, {3, 6, 9, . . . 3k}} then l1 ⊆ sd(c3k+2, k + 1). also it is obvious that l2 ⊆ sd(c3k+2, k + 1). hence l1 ∪ l2 ⊆ sd(c3k+2, k + 1). now let l ∈ sd(c3k+2, k + 1) then by observation-4.1 at least one of the vertices labelled 3k + 2, 3k + 1 or 3k is in l. suppose that 3k + 2 ∈ l then by observation-4.1 at least one of vertices say 1, 2, 3, 3k + 1, 3k or 3k − 1 are in l. if 3k + 1 and at least one of {1, 2, 3} and also 3k and at least one of {1, 2} are in l. then l − {3k + 2} ∈ sd(c3k+1, k) a contradiction. if {3, 3k} or 2, 3k − 1 is a subset of l, then l = s ∪ {3k + 2} for some s ∈ sd(c3k, k) therefore l ∈ l1 if {1, 3k − 1} is a subset of l then l − {3k + 2} ∈ sd(c3k+1, k) a contradiction. if {3, 3k − 1} is a subset of l and {3k, 3k + 1} is not a subset of l then l − {3k + 2} ∈ sd(c3k−1, k) hence l ∈ l2. if 3k + 1 or 3k is in l we have the result. 4. by lemma-4.4-(4) m = n − 1 therefore sd(cn, m) = sd(cn, n − 1) = {[n] − {x}|x ∈ [n]}. 5. sd(cn−1, m − 1) ̸= ∅, sd(cn−2, m − 1) ̸= ∅ and sd(cn−3, m − 1) ̸= ∅. first suppose that s ∈ sd(cn−1, m − 1) then s ∪ {n} ∈ sd(cn, m). so 164 superior domination polynomial l1 = {{n} ∪ s|s ∈ sd(cn−1, m − 1)} ⊆ sd(cn, m). now suppose that sd(cn−2, m − 1) ̸= ∅. let s1 ∈ sd(cn−2, m − 1) we denote {s1 ⋃ {{n}, if n − 2 or n − 3 ∈ s1 for s1 ∈ sd(cn−2, m − 1) or sd(cn−1, m − 1) {n − 1}, if n − 2 /∈ s1, n − 3 /∈ s1 or s1 ∈ sd(cn−1, m − 1) ∩ sd(cn−2, m − 1) simply by l2 by observation-4.1 at least one vertices labeled n−3, n−2 or 1 is in s1 if n − 2 or n − 3 is in s1. then s1 ∪ {n} ∈ sd(cn, m) otherwise s1 ∪ {n − 1} ∈ sd(cn, m). hence l2 ⊆ sd(cn, m) there we shall consider sd(cn−3, m − 1) ̸= ∅. let s2 ∈ sd(cn−3, m − 1) we denote {s2 ⋃ {{n − 2}, if 1 ∈ s2 for s2 ∈ sd(cn−3, m − 1) or s2 ∈ sd(cn−3, m − 1) ∩ sd(cn−2, m − 1) {{n − 1}, if n − 3 ∈ s2 or n − 4 ∈ s2 for s2 ∈ sd(cn−3, m − 1) or sd(cn−2, m − 1) simply by l3. if n − 3 or n − 4 is in s then s ∪ {n − 1} ∈ sd(cn, m), otherwise s2 ∪ {n − 2} ∈ sd(cn, m). hence l3 ⊆ l. therefore we proved that l1 ∪ l2 ∪ l3 ⊆ sd(cn, m). now suppose that l ∈ sd(cn, m) so by observation-4.1 l contains at least one of the vertices say n, n − 1 or n − 2. if n ∈ l so by observation-4.1 at least one of the vertices labelled n − 1, n − 2 or n − 3 and 1, 2, or 3 in l. if n − 2 ∈ l or n − 3 ∈ l then l = s ∪ {n} for some s ∈ sd(cn−2, m − 1). hence l ∈ l2 otherwise l = s ∪ {n − 1} sor some s ∈ sd(cn−2, m − 1). hence l ∈ l2. if n − 1 or n − 2 is in l, we have the result. theorem 4.2. if sd(cn, m) is the family of superior dominating sets of cn with cardinality m then |sd(cn, m)| = |sd(cn−1, m − 1)| + |sd(cn−2, m − 1)| + |sd(cn−3, m − 1)| proof. we consider the five cases in theorem-4.1 we write theorem-4.1 in the following form. 1. if sd(cn−1, m − 1) = sd(cn−2, m − 1) = ∅ and sd(cn−3, m − 1) ̸= ∅ then sd(cn, m) = {{n − 2} ⋃ s1, {n − 1} ⋃ s2, {n} ⋃ s3|1 ∈ s1, 2 ∈ s2, 3 ∈ s3, s1, s2, s3 ∈ sd(cn−3, m − 1)}. 2. if sd(cn−2, m − 1) = sd(cn−3, m − 1) = ∅ and sd(cn−1, m − 1) ̸= ∅ then sd(cn, m) = {{n} ⋃ s|s ∈ sd(cn−1, m − 1)}. 3. if sd(cn−1, m − 1) = ∅, sd(cn−2, m − 1) ̸= ∅ and sd(cn−3, m − 1) ̸= ∅ then sd(cn, m) = {{n} ⋃ s1, {n − 1} ⋃ s2 or s1, s2 ∈ sd(cn−2, m − 1), 1 ∈ s2} ⋃ (s ⋃   {n − 2}, if 1 ∈ s {n − 1}, if 1 ∈ s, 2 ∈ s ) where s ∈ sd(cn−3, m − 1) {n}, otherwise 4. if sd(cn−3, m − 1) = ∅ and sd(cn−2, m − 1) ̸= ∅, sd(cn−1, m − 1) ̸= ∅ then sd(cn, m) = {{n} ⋃ s1, {n − 1} ⋃ s2 or s1 ∈ sd(cn−1, m − 1), s2 ∈ sd(cn−2, m − 1)}. 165 tejaskumar r and a mohamed ismayil 5. if sd(cn−1, m − 1) ̸= ∅, sd(cn−2, m − 1) ̸= ∅ and sd(cn−3, m − 1) ̸= ∅ then sd(cn, m) = {{n} ⋃ s|s ∈ sd(cn−1, m − 1)} ⋃ {s1 ⋃ {{n}, if n − 2 or n − 3 ∈ s1 for s1 ∈ sd(cn−2, m − 1) or sd(cn−2,m−1) {n − 1}, if n − 2 /∈ s1, n − 3 /∈ s1 or s1 ∈ sd(cn−1, m − 1) ⋂ sd(cn−2, m − 1) } ⋃ {s2 ⋃ {{n − 2}, if 1 ∈ s2 for s2 ∈ sd(cn−3, m − 1) or s2 ∈ sd(cn−3, m − 1) ⋂ sd(cn−2, m − 1) {n − 2}, if n − 3 ∈ s2 or n − 4 ∈ s2 for s2 ∈ sd(cn−3, m − 1) or sd(cn−2, m − 1) } where s1 ∈ sd(cn−2, m−1) or sd(cn−1, m−1) and s2 ∈ {sd(cn−3, m−1) or sd(cn−2, m − 1)} ⋂ sd(cn−1, m − 1). hence we have |sd(cn, m)| = |sd(cn−1, m − 1)| + |sd(cn−2, m − 1)| + |sd(cn−3, m − 1)|. definition 4.1. let sd(cn, m) be the family of superior dominating sets of a cycle cn with cardinality n. then the superior domination polynomial sd(cn, x) of cn is defined as sd(cn, x) = ∑n m=⌈ n 3 ⌉ |sd(cn, m)|x m where sd(cn, m) is the number of distinct superior dominating sets of same cardinality. theorem 4.3. for every n ≥ 4 sd(cn, x) = x[sd(cn−1, x) + sd(cn−2, x) + sd(cn−3, x)] with initial values sd(c1, x) = x, sd(c2, x) = x2+2x, sd(c3, x) = x3 + 3x2 + x. table: the co-efficients of sd(cn, x) for 1 ≤ n ≤ 16 n t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 0 2 0 0 3 3 3 1 4 0 6 4 1 5 0 5 10 5 1 6 0 3 14 15 6 1 7 0 0 14 28 21 7 1 8 0 0 8 38 48 28 8 1 9 0 0 3 36 81 75 36 9 1 10 0 0 0 25 102 150 110 45 10 1 11 0 0 0 11 99 231 253 154 55 11 1 12 0 0 0 3 72 282 456 399 208 66 12 1 13 0 0 0 0 39 273 663 819 598 273 78 13 1 14 0 0 0 0 14 210 786 1372 1372 861 350 91 14 1 15 0 0 0 0 3 125 765 1905 2590 2178 1200 440 105 15 1 16 0 0 0 0 0 56 608 2214 4096 4560 3312 1628 544 120 16 1 theorem 4.4. the following properties hold for co-efficients of sd(cn, x). 1. |sd(c3n, n)| = 3, ∀ n ∈ n. 2. |sd(cn, m)| = |sd(cn−1, m−1)|+ |sd(cn−2, m−1)|+ |sd(cn−3, m−1)|, ∀ n ≥ 4, m ≥ ⌈n 3 ⌉, 166 superior domination polynomial 3. |sd(c3n+2, n + 1)| = 3n + 2, ∀ n ∈ n. 4. |sd(c3n+1, n + 1)| = n(3n+7)+2 2 , ∀ n ∈ n. 5. |sd(cn, m)| = 1, ∀ n ≥ 3. 6. |sd(cn, n − 1)| = n, ∀ n ≥ 3. 7. |sd(cn, n − 2)| = (n−1)n 2 , ∀ n ≥ 3. 8. |sd(cn, n − 3)| = (n−4)(n)(n+1) 6 , ∀ n ≥ 4. 9. ∑3m n=m |sd(cn, m)| = 3 ∑3m−3 n=m−1 |sd(cn, m − 1)|, ∀ m ≥ 4. 10. 1 = |sd(cn, n)| < |sd(cn+1, n)| < |sd(cn+2, n)| < · · · < |sd(c2n−1, n)| < |sd(c2n, n)| > |sd(c2n+1, n)| > · · · > |sd(c3n−1, n)| > |sd(c3n, n)| = 3, ∀ n ≥ 3. 11. if an = ∑n m=⌈ n 3 ⌉ |sd(cn, m)| then for every n ≥ 4, an = an−1 + an−2 + an−3 with initial values a1 = 1, a2 = 3 and a3 = 7. proof. 1. since |sd(cn, 3n)| = {{1, 4, 7, . . . 3n−2}, {2, 5, 8, . . . 3n−1}, {3, 6, 9, . . . 3n}}, so |sd(c3n, n)| = 3. 2. it follows from theorem-4.2. 3. by induction on n, the result is true for n = 1, because |sd(c2, 5)| = {{1, 3}, {1, 4}, {2, 4}, {2, 5}, {3, 5}}. suppose result is true for all n − 1 then we prove for n. by (1),(2) and induction we have |sd(c3n+2, n + 1)| = |sd(c3n+1, n)| + |sd(c3n, n)| + |sd(c3n−1, n)| = 3n + 2 4. by mathematical induction |sd(c2, 4)| = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. so |sd(c4, 2)| = 6, the result is true for n = 1. now suppose that the result is true for all natural numbers less than n and we prove it for n. by (1),(2),(3) and induction we have |sd(c3n+1, n + 1)| = |sd(c3n, n)| + |sd(c3n−1, n)| + |sd(c3n−2, n)| = 3 + 3(n − 1) + 2 + (n − 1) 3(n − 1) + 7) + 2 2 = n(3n + 7) + 2 10 167 tejaskumar r and a mohamed ismayil 5. it is obvious that for a graph with n vertices |sd(g, n)| = 1. 6. it is obvious that for a graph g with n vertices |sd(g, n − 1)| = n. 7. by induction on n. the result is true for n = 3. since |sd(c3, 1)| = 3. assume it is true for all n − 1. we prove for n. by parts (1), (2), (4), (5) and induction |sd(cn, n − 2)| = |sd(cn−1, n − 3)| + |sd(cn−2, n − 3)| + |sd(cn−3, n − 3)| = (n − 2)(n − 1) 2 + n − 2 + 1 = (n − 1)n 2 8. by induction on n. the result is true for n = 4. since |sd(c4, 1)| = 0. assume the result is true for all n − 1. we prove for n, by parts (2), (6), (7) and induction we have |sd(cn, n − 3)| = |sd(cn−1, n − 4)| + |sd(cn−2, n − 4)| + |sd(cn−3, n − 4)| = (n − 5)(n − 1)n 6 + (n − 2)(n − 3) 2 + n − 3 = (n − 4)n(n + 1) 6 9. proof by induction on m. suppose m = 3 then ∑9 n=3 sd(cn, 3) = 54 = 3 ∑6 n=2 |sd(cn, 2)|. now suppose the result is true for every m < t and we prove for m = t 3t∑ n=t |sd(cn, t)| = 3t∑ n=t |sd(cn−1, t − 1)| + 3t∑ n=t |sd(cn−2, t − 1)| + 3t∑ n=t |sd(cn−3, t − 1)| = 3 3(t−1)∑ n=t−1 |sd(cn−1, t − 2)| + 3 3(t−1)∑ n=t−1 |sd(cn−2, t − 2)| + 3 3(t−1)∑ n=t−1 |sd(cn−3, t − 2)| = 3 3t−3∑ n=t−1 |sd(cn, t − 1)| 10. we plan for every m, |sd(cn, m)| < |sd(cn+1, m)| for m ≤ n ≤ 2m − 1 and |sd(cn, m)| > |sd(cn+1, m)| for 2m ≤ n ≤ 3m − 1. we prove first inequality by induction on m. the result hold for m = 3. suppose that result is true for all m ≤ t. now we prove it for m = t + 1. that is |sd(cn, t + 1)| < |sd(cn+1, t + 1)| for t + 1 ≤ n ≤ 2t + 1. |sd(cn, t + 1)| = |sd(cn−1, t)| + |sd(cn−2, t)| + |sd(cn−3, t)| < |sd(cn, t)| + |sd(cn−1, t)| + |sd(cn−2, t)| = |sd(cn+1, t + 1)| the other inequality follows in same way. 168 superior domination polynomial 11. by theorem-4.2, we have an = n∑ m=⌈ n 3 ⌉ |sd(cn, m)| = n∑ m=⌈ n 3 ⌉ |sd(cn−1, m − 1)| + |sd(cn−2, m − 1)| + |sd(cn−3, m − 1)| = n−1∑ m=⌈ n 3 ⌉−1 |sd(cn−1, m)| + n−2∑ m=⌈ n 3 ⌉−1 |sd(cn−2, m)| + n−3∑ m=⌈ n 3 ⌉−1 |sd(cn−3, m − 1)| = an−1 + an−2 + an−3 table: sd(g, x) of different standard graphs and their roots are tabulated in the table below graph figure superior domination polynomial sd(g, x) roots diamond graph v1 v4 v2 v3 x4 + 4x3 + 6x2 + 2x x1 = 0, x2 = −0.4563, x3 = −1.7718 + 1.1151i, x4 = −1.7718 − 1.1151i. claw graph v2 v3 v1 v4 x4 + 3x3 + 3x2 + x x1 = 0, x2 = −1, x3 = −1, x4 = −1. bull graph v3 v4 v5 v2v1 x5 + 3x4 + 3x3 + x2 x1 = 0, x2 = −1, x3 = −1, x4 = −1. butterfly graph v3 v2 v5 v1 v4 x5 + 4x4 + 6x3 + 4x2 + x x1 = 0, x2 = −1, x3 = −1, x4 = −1, x5 = −1. (3,2)-tadpole graph v2 v3 v4 v1 v5 x5 + 4x4 + 5x3 + 2x2 x1 = 0, x2 = −1, x3 = −2, x4 = −1. 169 tejaskumar r and a mohamed ismayil graph figure superior domination polynomial sd(g, x) roots kite graph v3 v4 v1 v5 v2 x5 + 5x4 + 9x3 + 5x2 + x x1 = 0, x2 = −0.378 + 0.1877i, x3 = −0.378 − 0.1877i, x4 = −2.122 + 1.0538i, x5 = −2.122 − 1.0538i. (4,1)-lollipop graph v3 v4 v1 v5 v2 x5 + 4x4 + 6x3 + 4x2 + x x1 = 0, x2 = −1, x3 = −1, x4 = −1, x5 = −1. house graph v2 v3 v1 v4 v5 x5 + 5x4 + 8x3 + 4x2 + x x1 = 0, x2 = −0.3076 + 0.3182i, x3 = −0.3076 − 0.3182i, x4 = −2.1924 + 0.5479i, x5 = −2.1924 − 0.5479i. house x graph v2 v3 v1 v4 v5 x5 + 5x4 + 8x3 + 5x2 + x x1 = 0, x2 = −1, x3 = −1, x4 = −0.382, x5 = −2.618. gem graph v1 v2 v5 v3 v4 x5 + 4x4 + 4x3 x1 = 0, x2 = −2. cricket graph v4 v5v3 v1 v2 x5 + 4x4 + 6x3 + 4x2 + x x1 = 0, x2 = −1, x3 = −1, x4 = −1, x5 = −1. pentatope graph v1 v4 v5 v2 v3 x5 + 5x4 + 10x3 + 10x2 + 5x x1 = 0, x2 = −0.691 = 0.9511i, x3 = −0.691 − 0.9511, x4 = −1.809 + 0.5878i, x5 = −1.809 − 0.5878i. cross graph v3 v1 v2 v4 v5 v6 x6 + 5x5 + 9x4 + 7x3 + 2x2 x1 = 0, x2 = −1, x3 = −2, x4 = −1, x5 = −1. 170 superior domination polynomial graph figure superior domination polynomial sd(g, x) roots fish graph v4 v2 v5 v1 v6 v3 x6 + 5x5 + 10x4 + 9x3 + 3x2 x1 = 0, x2 = −1, x3 = −1, x4 = −32 + √ 3 2 i, x5 = −32 − √ 3 2 i. r graph v3 v4 v1 v5 v2 v6 x6 + 5x5 + 10x4 + 9x3 + 3x2 x1 = 0, x2 = −1, x3 = −1, x4 = −32 + √ 3 2 i, x5 = −32 − √ 3 2 i. (2,3)-king graph v2 v3v1 v5v4 v6 x6 + 4x5 + 6x4 + 4x3 + x2 x1 = 0, x2 = −1, x3 = −1, x4 = −1, x5 = −1. antenna graph v2 v1 v3 v4 v5 v6 x6 + 4x5 + 5x4 + 2x3 x1 = 0, x2 = −1, x3 = −1, x4 = −1. 3-prism graph v2 v3 v4 v1 v5 v6 x6 + 6x5 + 15x4 + 20x3 +15x2 + 6x x1 = 0, x2 = −2, x3 = −0.5 + 0.866i, x4 = −0.5 − 0.866i, x5 = −1.5 + 0.866i, x6 = −1.5 − 0.866i. moser spindle graph v1 v4 v5 v3v2 v6 v7 x7 + 4x6 + 6x5 +4x4 + x3 x1 = 0, x2 = −1, x3 = −1, x4 = −1, x5 = −1. cubical graph v3 v4 v5 v6 v1 v2 v8v7 x8 + 8x7 + 28x6 + 56x5 +70x4 + 48x3 + 16x2 x1 = 0, x2 = −0.6714 + 0.5756i, x3 = −0.6714 − 0.5756i, x4 = −0.8352 + 1.4854i, x5 = −0.8352 − 1.4854i, x6 = −2.4934 + 0.9097i, x7 = −2.4934 − 0.9097i. wagner graph v1 v8 v4 v5 v2 v7 v3 v6 x8 + 8x7 + 24x6 +32x5 + 16x4 x1 = 0, x2 = −2, x3 = −2, x4 = −2, x5 = −2. 171 tejaskumar r and a mohamed ismayil 5 conclusions in this paper we introduced superior domination polynomial, this is a distance based domination polynomial. emphasis was given to the family of stars and cycles. formulas to find the coeffcients of the superior domination polynomials of cycles and stars were stated and proved. these formulas helps us to calculate the number of superior dominating sets of a specific desired cardinality for any given value of n. the superior domination polynomial of different standard graphs and their roots are calculated. 6 acknowledgements the authors express their gratitude to the managemanetratio mathematica for their constant support towards the successful completion of this work. we wish to thank the anonymous reviewers for the valuable suggestions and comments. references [1] s. alikhani and y. hock peng. introduction to domination polynomial of a graph. arxiv preprint arxiv:0905.2251, 2009. [2] f. harary. graph theory, narosa publ. house, new delhi, 2001. [3] t. haynes. domination in graphs: volume 2: advanced topics. routledge, 2017. [4] a. m. ismayil and r. tejaskumar. eccentric domination polynomial of graphs. advances in mathematics: scientific journal, 9:1729–1739, 2020. [5] t. janakiraman, m. bhanumathi, and s. muthammai. eccentric domination in graphs. international journal of engineering science, advanced computing and bio-technology, 1(2):1–16, 2010. [6] k. kathiresan and g. marimuthu. superior domination in graphs. utilitas mathematica, 76:173, 2008. [7] k. kathiresan, g. marimuthu, and s. west. superior distance in graphs. journal of combinatorial mathematics and combinatorial computing, 61:73, 2007. [8] o. ore. path problems. theory of graphs, 1962. 172 ratio mathematica volume 42, 2022 some characteristics of picture fuzzy subgroups via cut set of picture fuzzy set taiwo olubunmi sangodapo* babatunde oluwaseun onasanya‡ abstract given any picture fuzzy set q of a universe y , the set cr,s,t(q) called the (r, s, t)-cut set of q was studied. in this paper, some characteristics of picture fuzzy subgroup of a group are obtained via the cut sets of picture fuzzy set. keywords: fuzzy set, picture fuzzy set, picture fuzzy subgroup, cut set 2020 ams subject classifications: 20n25, 08a72, 03e72.1 *department of mathematics, faculty of science (university of ibadan, oyo state, nigeria); to.ewuola@ui.edu.ng, toewuola77@gmail.com. †corresponding author; department of mathematics, faculty of science (university of ibadan, oyo state, nigeria); babtu2001@yahoo.com, bo.onasanya@ui.edu.ng. 1 10.23755/rm.v39i0.849. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 341 † received on may 24th, 2022. accepted on june 29th, 2022. published on june 30th, 2022. doi: 1 introduction zadeh [18] introduced the notion of fuzzy sets (fss). this theory has been generalised by many researchers. atanassov [1] initiated the concept of intuitionistic fuzzy sets (ifss). for more on intuitionistic fuzzy set, also see [2, 3, 4]. the application of these fuzzy sets have taken different directions such as optimization and decision-making [15] control theory [13] and many others. one particular application is the work of cuong and kreinovich [5] who put forward the notion of picture fuzzy sets (pfss) as a generalisation of fuzzy sets and intuitionistic fuzzy sets. picture fuzzy set has been extensively studied and applied (see [6, 7, 8, 9, 10, 11, 12, 14, 17] for details). rosenfeld [16] generalised fuzzy sets to fuzzy groups (fgs). the idea of cut set of picture fuzzy sets was initiated by dutta and ganju [12] and they obtained some of its properties. dogra and pal [11] corrected the definition of cut set that was given by dutta and ganju [12] and they initiated the notion of picture fuzzy subgroups (pfsgs) of a group. in this paper, motivated by the work of dogra and pal [11], investigation of the characteristics of picture fuzzy subgroups of a group via the cut sets of picture fuzzy sets was done. the organisation of the paper is as follows: section 2 gives the basic definitions and preliminary ideas of pfss; section 3 investigates some properties of picture fuzzy subgroups of a group via cut sets of picture fuzzy sets. 2 preliminaries definition 2.1. [18] let y be a nonempty set. a fs q of y is an object of the form q = {⟨y, σq(y)⟩|y ∈ y } with a membership function σq : y −→ [0, 1] where the function σq(y) denotes the degree of membership of y ∈ q. definition 2.2. [1] let a nonempty set y be fixed. an ifs q of y is an object of the form q = {⟨y, σq(y), τq(y)⟩|y ∈ y } where the functions σq : y → [0, 1] and τq : y → [0, 1] are called the membership and non-membership degrees of y ∈ q, respectively, and for every y ∈ y , 0 ≤ σq(y) + τq(y) ≤ 1. 342 picture fuzzy subgroups via cut set of picture fuzzy set definition 2.3. [5] a picture fuzzy set q of y is defined as q = {(y, σq(y), τq(y), γq(y))|y ∈ y }, where the functions σq : y → [0, 1], τq : y → [0, 1] and γq : y → [0, 1] are called the positive, neutral and negative membership degrees of y ∈ q, respectively, and σq, τq, γq satisfy 0 ≤ σq(y) + τq(y) + γq(y) ≤ 1, ∀y ∈ y. for each y ∈ y , sq(y) = 1 − (σq(y) + τq(y) + γq(y)) is called the refusal membership degree of y ∈ q. definition 2.4. [5] let q and r be two pfss. then, the inclusion, equality, union, intersection and complement are defined as follow: (i) q ⊆ r if and only if for all y ∈ y , σq(y) ≤ σr(y), τq(y) ≤ τr(y) and γq(y) ≥ γr(y). (ii) q = r if and only if q ⊆ r and r ⊆ q. (iii) q ∪ r = {(y, σq(y) ∨ σr(y), τq(y) ∨ τr(y)), γq(y) ∧ γr(y))|y ∈ y }. (iv) q ∩ r = {(y, σq(y) ∧ σr(y), τq(y) ∧ τr(y)), γq(y) ∨ γr(y))|y ∈ y }. definition 2.5. [11] let (g, ∗) be a crisp group and q = {(y, σq(y), τq(y), ηq(y)) | y ∈ g} be a pfs in g. then, q is called picture fuzzy subgroup of g (pfsg) if (i) σq(a∗b) ≥ σq(a)∧σq(b), τq(a∗b) ≥ τq(a)∧τq(b), ηq(a∗b) ≤ ηq(a)∨ηq(b) (ii) σq(a−1) ≥ σq(a), τq(a−1) ≥ τq(a), ηq(a−1) ≤ ηq(a) for all a, b ∈ g. notice that a−1 is the inverse of a ∈ g, or equivalently, q is a pfsg of g if and only if σq(a ∗ b−1) ≥ σq(a) ∧ σq(b), τq(a ∗ b−1) ≥ τq(a) ∧ τq(b), ηq(a ∗ b−1) ≤ ηq(a) ∨ ηq(b). definition 2.6. [11] let (g, ∗) be a crisp group and q = (σq, τq, ηq) be a pfsg of g. then, for a ∈ g the picture fuzzy left coset of q ∈ g is the pfs aq = (σaq, τaq, ηaq) defined by σaq(u) = σq(a −1 ∗ u), τaq(u) = τq(a−1 ∗ u) and ηaq(u) = ηq(a−1 ∗ u) for all u ∈ g. 343 definition 2.7. [11] let (g, ∗) be a crisp group and q = (σq, τq, ηq) be a pfsg of g. then, for a ∈ g the picture fuzzy right coset of q ∈ g is the pfs qa = (σqa, τqa, ηqa) defined by σqa(u) = σq(u ∗ a−1), τqa(y) = τq(u ∗ a−1) and ηqa(u) = ηq(u ∗ a−1) for all u ∈ g. definition 2.8. [11] let (g, ∗) be a crisp group and q = (σq, τq, ηq) be a pfsg of g. then, q is called a picture fuzzy normal subgroup (pfnsg) of g if σqa(y) = σaq(y), τqa(y) = τaq(y), ηqa(y) = ηaq(y) for all a, y ∈ g. remark 2.1. dogra and pal established that pfsg of g is normal if and only if (i) σq(u −1 ∗ a ∗ u) = σq(a) (ii) τq(u −1 ∗ a ∗ u) = τq(a) (ii) ηq(u −1 ∗ a ∗ u) = ηq(a). for all a ∈ q and u ∈ g. cut set of picture fuzzy sets was introduced by dutta and ganju ? but the definition did not capture the cut set very well. thus, dogra and pal ? corrected it in their paper. definition 2.9. [11] let q = {(x, σq, τq, γq)|a ∈ y } be pfs over the universe y . then, (r, s, t)-cut set of q is the crisp set in q, denoted by cr,s,t(q) and is defined by cr,s,t(q) = {a ∈ y |σq(a) ≥ r, τq(a) ≥ s, γq(a) ≤ t} r, s, t ∈ [0, 1] with the condition 0 ≤ r + s + t ≤ 1. theorem 2.1. [11] let (g, ∗) be a crisp group and q = (σq, τq, ηq) be a pfsg of g. then, q is a pfsg if and only if cr,s,t(q) is a crisp subgroup of g. theorem 2.2. [12] if q and r are two pfss of a universe y , then the following results hold (i) cr,s,t(q) ⊆ cu,v,w(q) if r ≥ u, s ≤ v, t ≤ w. (ii) c1−s−t,s,t(q) ⊆ cr,s,t(q) ⊆ cr,1−r−t,t(q). (iii) q ⊆ r implies cr,s,t(q) ⊆ cr,s,t(r). 344 picture fuzzy subgroups via cut set of picture fuzzy set (iv) cr,s,t(q ∩ r) = cr,s,t(q) ∩ cr,s,t(r). (v) cr,s,t(q ∪ r) ⊇ cr,s,t(q) ∪ cr,s,t(r). (vi) cr,s,t(∩qi) = ∩cr,s,t(qi). (vii) c1,0,0(q) = y. 3 main results remark 3.1. it is important to note that [12] misquoted the result of [5]. hence, the results built on this foundation cannot be or at all correct. the counter example in example (3.1) shows that the claims in theorem 2.2 (i), (ii) and (vii) are wrong. the correct version of theorem (2.2) is given in theorem (3.1). example 3.1. let y = {y1, y2, y3, y4}, q = {(y1, 0, 0.1, 0.8), (y2, 0.2, 0.5, 0.3), (y3, 0.4, 0.2, 0.1), (y4, 0.5, 0.3, 0.2)}, and r = {(y1, 0.1, 0.4, 0.5), (y2, 0.3, 0.6, 0.1), (y3, 0.5, 0.3, 0), (y4, 0.5, 0.4, 0.1)} be pfss, taking r = 0.1, s = 0.3, t = 0.5. thus, the (0.1, 0.3, 0.5)-cut set of q are c0.1,0.3,0.5(q) = {y2, y4} and c0.1,0.3,0.5(r) = {y1, y2, y3, y4}. hence, c1−s−t,s,t(q) = c0.2,0.3,0.5(q) = {y2, y4}, cr,s,t(q) = c0.1,0.3,0.5(q) = {y2, y4}, c0.1,0.2,0.3(q) = {y3} and cr,1−r−t,t(q) = c0.1,0.4,0.5(q) = {y2}. thus, c1−s−t,s,t(r) ⊆ cr,s,t(r) ⊊ cr,1−r−t,t(r), which means (i) and (ii) fail. 345 note that the left side of the inclusion which holds obey the condition (i) of our theorem 3.1 furthermore, c1−s−t,s,t(q) ⊆ cr,s,t(q) ⊊ cr,1−r−t,t(q), which means (i) and (ii) fail. note that the left side of the inclusion which holds obey the condition (i) of our theorem 3.1 in addition, c0.1,0.2,0.3(q) = {y3} ⊊= {y2, y4} = c0.1,0.3,0.5(q) which means (i) and (ii) fail. also, c1,0,0(q) = ∅ ≠ y , which means (vii) fails. theorem 3.1. let q and r be two pfss of a universe y . then, the following assertions hold: (i) cr,s,t(q) ⊆ cu,v,w(q) if r ≥ u, s ≥ v, t ≤ w. (ii) c1−s−t,s,t(q) ⊆ cr,s,t(q) ⊆ cr,s,1−r−s(q), (iii) c1−s−t,1−r−t,t(q) ⊆ cr,s,t(q) ⊆ cr,s,1−r−s(q), (iv) cr,1−r−t,t(q) ⊆ cr,s,t(q) ⊆ cr,s,1−r−s(q), (v) c0,0,1(q) = y. proof. (i) let x ∈ cr,s,t(q). using definition 2.9, σq(x) ≥ r ≥ u, τq(x) ≥ s ≥ v, γq(x) ≤ t ≤ w. thus, x ∈ cu,v,w(q) and, as such, cr,s,t(q) ⊆ cu,v,w(q). (ii) since r + s + t ≤ 1 implies that 1 − s − t ≥ r, s ≥ s and t ≤ t, and r ≥ r, s ≥ s and t ≤ 1 − r − s, by theorem 3.1 (i), the result holds. (iii) since r + s + t ≤ 1 implies that 1 − s − t ≥ r, 1 − r − t ≥ s and t ≤ t, and r ≥ r, s ≥ s and t ≤ 1 − s − r, by theorem 3.1 (i), the result holds. (iv) since r + s + t ≤ 1 implies that r ≥ r, 1 − r − t ≥ s and t ≤ t, and r ≥ r, s ≥ s and t ≤ 1 − s − r, by theorem 3.1 (i), the result holds. (v) since ∀ y ∈ y, σ(y) ≥ 0, τ(y) ≥ 0, γ(y) ≤ 1, c0,0,1(q) = y , 346 picture fuzzy subgroups via cut set of picture fuzzy set proposition 3.1. if q is pfsg of g, then cr,s,t(q) is a subgroup of g, where σq(e) ≥ r, τq(e) ≥ s, and ηq(e) ≤ t and e is the identity element of g. proof. clearly cr,s,t(q) ̸= ∅ as e ∈ cr,s,t(q). let a, b ∈ cr,s,t(q) be any two elements. then, σq(a) ≥ r, τq(a) ≥ s, ηq(a) ≤ t and σq(b) ≥ r, τq(b) ≥ s, ηq(b) ≤ t σq(a) ∧ σq(b) ≥ r, τq(a) ∧ τq(b) ≥ s and ηq(a) ∨ ηq(b) ≤ t since q is a pfsg of g, σq(a ∗ b−1) ≥ σq(a) ∧ σq(b) ≥ r, τq(a ∗ b−1) ≥ τq(a) ∧ τq(b) ≥ s and ηq(a ∗ b−1) ≤ ηq(a) ∨ ηq(b) ≤ t. hence, a ∗ b−1 ∈ cr,s,t(q) and cr,s,t(q) is a subgroup of g. proposition 3.2. if q is pfnsg of g, then cr,s,t(q) is a normal subgroup of g, where σq(e) ≥ r, τq(e) ≥ s, and ηq(e) ≤ t and e is the identity element of g. proof. let a ∈ cr,s,t(q) and u ∈ g be any element. then, σq(a) ≥ r, τq(a) ≥ s, ηq(a) ≤ t. also, since q is a pfnsg of g, σq(u −1∗a∗u) = σq(a) ≥ r, τq(u−1∗a∗u) = τq(a) ≥ s, and ηq(u−1∗a∗u) = ηq(a) ≤ t ∀ a ∈ q and u ∈ g. therefore, u−1 ∗ a ∗ u ∈ cr,s,t(q), so a ∗ u ∈ u ∗ cr,s,t(q) which implies cr,s,t(q)∗u ⊆ u∗cr,s,t(q). also, u∗a∗u−1 ∈ cr,s,t(q), so u∗a ∈ cr,s,t(q) ∗ u which implies u ∗ cr,s,t(q) ⊆ cr,s,t(q) ∗ u. hence, u ∗ cr,s,t(q) = cr,s,t(q) ∗ u. thus, cr,s,t(q) is a normal subgroup of g. proposition 3.3. given two pfsgs q and r of a group (g, ∗). then, q ∩ r is a pfsg of g. this proposition has been proved by dogra and pal [11] by using pfsg definition. but, in this paper, a rather simpler approach of cut set of pfs is used to prove it. proof. by theorem 2.1, q ∩ r is a pfsg of g if and only if cr,s,t(q ∩ r) is a crisp subgroup of g. since cr,s,t(q ∩ r) = cr,s,t(q) ∩ cr,s,t(r) (theorem 2.2, iv) and both cr,s,t(q) and cr,s,t(r) are subgroups of g and intersection of two subgroups of a group is its subgroup, cr,s,t(q∩r) is a subgroup of g. therefore, q ∩ r is a pfsg of g. 347 notice that the union of two pfsgs of g need not be a pfsg of g [11]. proposition 3.4. let q be a pfsg of g and a be any fixed element of g. then, (i.) a ∗ cr,s,t(q) = cr,s,t(a ∗ q). (ii.) cr,s,t(q) ∗ a = cr,s,t(q ∗ a), ∀ r, s, t [0, 1] with r + s + t ≤ 1. proof. (i.) cr,s,t(a ∗ q) = {u ∈ g|σa∗q(u) ≥ r, τa∗q(u) ≥ s, ηa∗q(u) ≤ t} with the condition 0 ≤ r + s + t ≤ 1. also a ∗ cr,s,t(q) = a ∗ {b ∈ g|σq(b) ≥ r, τq(b) ≥ s, ηq(b) ≤ t} = {a ∗ b ∈ g|σq(b) ≥ r, τq(b) ≥ s, ηq(b) ≤ t} set a ∗ b = u, so that b = a−1 ∗ u. therefore, a ∗ cr,s,t(q) = { u ∈ g|σq(a−1 ∗ u) ≥ r, τq(a−1 ∗ u) ≥ s, ηq(a−1 ∗ u) ≤ t } = {u ∈ g|σa∗q(u) ≥ r, τa∗q(u) ≥ s, ηa∗q(u) ≤ t} = cr,s,t(a ∗ q) for all r, s, t [0, 1] with r + s + t ≤ 1. (ii.) cr,s,t(q ∗ a) = {u ∈ g|σq∗a(u) ≥ r, τq∗a(u) ≥ s, ηq∗a(u) ≤ t} with the condition 0 ≤ r + s + t ≤ 1. also cr,s,t(q ∗ a) = {b ∈ g|σq(b) ≥ r, τq(b) ≥ s, ηq(b) ≤ t} ∗ a = {b ∗ a ∈ g|σq(b) ≥ r, τq(b) ≥ s, ηq(b) ≤ t} 348 picture fuzzy subgroups via cut set of picture fuzzy set set b ∗ a = u, so that b = u ∗ a−1. therefore, cr,s,t(q) ∗ a = { u ∈ g|σq(u ∗ a−1) ≥ r, τq(u ∗ a−1) ≥ s, ηq(u ∗ a−1) ≤ t } = {u ∈ g|σq∗a(u) ≥ r, τq∗a(u) ≥ s, ηq∗a(u) ≤ t} = cr,s,t(q ∗ a) for all r, s, t [0, 1] with r + s + t ≤ 1. proposition 3.5. let q be a pfsg of g. let a, b ∈ g such that σq(a) ∧ σq(b) = r, τq(a) ∧ τq(b) = s, ηq(a) ∨ ηq(b) = t. then, (i.) a ∗ q = b ∗ q ⇔ a−1 ∗ b ∈ cr,s,t(q) (ii.) q ∗ a = q ∗ b ⇔ a ∗ b−1 ∈ cr,s,t(q) proof. (i.) a ∗ q = b ∗ q ⇔ cr,s,t(a ∗ q) = cr,s,t(b ∗ q) ⇔ a ∗ cr,s,t(q) = b ∗ cr,s,t(q) [by proposition 3.4] ⇔ a−1∗b ∈ cr,s,t(q) [since cr,s,t(q) is a subgroup of g and a, b ∈ cr,s,t(q)]. (ii.) q ∗ a = q ∗ b ⇔ cr,s,t(q ∗ a) = cr,s,t(q ∗ b) ⇔ cr,s,t(q) ∗ a = cr,s,t(q) ∗ b [by proposition 3.4] ⇔ a∗b−1 ∈ cr,s,t(q) [since cr,s,t(q) is a subgroup of g and a, b ∈ cr,s,t(q)]. 349 corolary 3.1. if cr,s,t(q) is pfnsg of g, then proposition 3.4 (i) and (ii) coincide. proof. cr,s,t(a ∗ q) = a ∗ cr,s,t(q) = cr,s,t(q) ∗ a = cr,s,t(q ∗ a). 4 conclusion in this paper, some of the existing results in fuzzy group and fuzzy cut groups of a fuzzy group have been extended to picture fuzzy group and cut subgroup of a picture fuzzy subgroup. in further research, it will be of interest to see the relationship between a picture fuzzy set and bipolar fuzzy set and establish which one is a generalisation of the other. references k.t. atanassov. intuitionistic fuzzy sets. fuzzy sets and systems, 20, 1986, 87-96. k.t. atanassov. more on intuitionistic fuzzy sets. fuzzy sets and systems, 33, 1989, 37-45. k.t. atanassov. new operations defined over the intuitionistic fuzzy sets. fuzzy sets and systems, 61(2), 1994, 137-142. k. t. atanassov. intuitionistic fuzzy sets: theory and applications. springerverlag, new york, 1999. b.c. couong and v. kreinovich. picture fuzzy sets-a new concept for computational intelligence problems. proceeding of the third world congress on information and communication technologies, 2013, 1-6. b.c. couong. picture fuzzy sets. journal of computer science and cybernetics, 30(4), 2014, 409-420. b.c. couong. pythagorean picture fuzzy sets, part 1-basic notions. journal of computer science and cybernetics, 35(4), 2019, 293-304. 350 picture fuzzy subgroups via cut set of picture fuzzy set b.c. couong and p.v. hai. some fuzzy logic operators for picture fuzzy sets. seventh international conference on knowledge and systems engineering, 2015, 132-137. b.c. couong, r.t. ngan and b.d. hai. picture fuzzy sets. seventh international conference on knowledge and systems engineering, 2015, 126-131. b.c. couong, v. kreinovich and r.t. ngan. a classification of representable t-norm operators for picture fuzzy sets. eighth international conference on knowledge and systems engineering, 2016. s. dogra and m. pal. picture fuzzy subgroup. kragujevac journal of mathematics, 47(6), 2023, 911-933. p. dutta and s. ganju. some aspect of picture fuzzy sets. amse journalamse iieta, 54(1), 2017, 127-141. b.o. onasanya, s. wen, y. feng, w. zhang and j. xiong. fuzzy coefficient of impulsive intensity in a non-linear control system. neural processing letter, 53(6), 2021, 4639-4657. h. garg. some picture fuzzy aggregation operators and their applications to multicriteria decision-making. arabian journal for science and engineering, 42(12), 2017, 5275-5290. m.k. eze and b.o. onasanya. case studies on application of fuzzy linear programming in decision-making. ratio mathematica, 35, 2018, 29-45. a. rosenfeld. fuzzy group. journal of mathematical annalysis and applications, 35(3), 1971, 512-517. t.o. sangodapo. some notions on convexity of picture fuzzy sets. journal of mathematical annalysis and applications, 2022. l.a. zadeh. the concept of a linguistic variable and its application to approximate reasoning. information science, i, 1975, 338-353. 351 ratio mathematica volume 47, 2023 on neutrosophic filter of bl-algebras a. ibrahim* s. karunya helen gunaseeli† abstract in this paper, we introduce the notion of neutrosophic filter in bl-algebras and investigate several key characteristics with illustrations. additionally, we obtain few conditions, order properties of neutrosophic filter of bl-algebras. moreover, we prove that intersection of two neutrosophic filters of a blalgebra is also a neutrosophic filter. keywords: bl-algebra; filter; neutrosophic set; neutrosophic filter. 2020 ams subject classification: 03g25,03e72,03f55,06f35 ‡ . 1. introduction in 1965, l. a. zadeh [2, 3, 14] was the first to introduce the notion of fuzzy sets to describe vagueness mathematically. he rectified those problems by designating every feasible individual in the universe a number resembling its grade of membership in the fuzzy set. in lattice implication algebras, xu and qin[12] first proferred the concept of filters. an important aspect of researching various logical algebras is filter theory. in the argument of the completeness of various logical algebra, filters are crucial. several academicians have investigated the filter theory of different logical algebras. in 1998, the ideology of neutrosophy was introduced by smarandache [1,9]. this became a new idea of philosophy to characterize the neutralites. the main idea of neutrosophy is *assistant professor, p.g. and research department of mathematics, h.h. the rajah’s college, pudukkottai, affiliated to bharathidasan university, trichirappalli, india; dribra@hhrc.ac.in, dribrahimaadhil@gmail.com. †research scholar, p.g. and research department of mathematics, h.h. the rajah’s college, pudukkottai, affiliated to bharathidasan university, trichirappalli, india; karjes821@gmail.com. ‡received on august 11, 2022. accepted on may 17, 2023. published on june 30, 2023. doi:10.23755/rm.v39i0.816. issn: 1592-7415. eissn: 2282-8214. ©the authors. this paper is published under the cc-by licence agreement. 141 a. ibrahim and s. karunya helen gunaseeli that behind every concept there also exists an indeterminant degree in addition to truth and falsity. neutrosophy, a discipline of philosophy that has just recently been recognised as a science, examines the genesis, character, and range of neutralities as well as how they interact with various ideational spectra. hajek[6] introduced the concept of bl-algebras (basic logic), is a type of logical algebra. in this paper, the concept of neutrosophic filters of bl-algebras is discussed. in section 2, few fundamental definitions and results are explained. in section 3, we introduce the notion of neutrosophic filters of bl-algebras along with some of its related features. 2. preliminaries in this section, we recall few fundamental definitions and their characteristics that are useful for developing the primary findings. definition 2.1[6,11] a bl-algebra is an algebra (ℬ, ∨, ∧,∘, →, 0,1) of type (2, 2, 2, 2, 0, 0) such that the following are satisfied for all 𝑗1, 𝑘1, 𝑙1 ∈ ℬ, (i) (ℬ, ∨, ∧, 0,1) is a bounded lattice, (ii) (ℬ, ∘, 1) is a commutative monoid, (iii) ′ ∘ ′ and ′ → ′form an adjoint pair, that is, 𝑙1 ≤ 𝑗1→ 𝑘1if and only if 𝑗1 ∘ 𝑙1 ≤ 𝑘1for all 𝑗1, 𝑘1, 𝑙1 ∈ ℬ, (iv) 𝑗1∧ 𝑘1= 𝑗1 ∘ (𝑗1 → 𝑘1), (v) (𝑗1 → 𝑘1) ∨ (𝑘1 → 𝑗1) = 1. proposition 2.2[7,13] the following axioms are satisfied in a blalgebra ℬ for all 𝑗1, 𝑘1, 𝑙1∈ ℬ, (i) 𝑘1 → (𝑗1 → 𝑙1) = 𝑗1→ (𝑘1→ 𝑙1) = (𝑗1 ∘ 𝑘1) → 𝑙1, (ii) 1 → 𝑗1 = 𝑗1, (iii) 𝑗1 ≤ 𝑘1 if and only if 𝑗1 → 𝑘1 = 1, (iv) 𝑗1∨𝑘1 = ((𝑗1→ 𝑘1) → 𝑘1) ∧ ((𝑘1 → 𝑗1) → 𝑗1), (v) 𝑗1 ≤ 𝑘1 implies 𝑘1 → 𝑙1 ≤ 𝑗1 → 𝑙1, (vi) 𝑗1 ≤ 𝑘1 implies 𝑙1 → 𝑗1≤ 𝑙1 → 𝑘1, (vii) 𝑗1 → 𝑘1 ≤ (𝑙1→ 𝑗1) → (𝑙1→ 𝑘1), (viii) 𝑗1 → 𝑘1≤ (𝑘1 → 𝑙1) → (𝑘1→ 𝑙1), (ix) 𝑗1 ≤ (𝑗1→ 𝑘1) → 𝑘1, (x) 𝑗1 ∘ (𝑗1 → 𝑘1) = 𝑗1∧𝑘1, (xi) 𝑗1 ∘ 𝑘1 ≤ 𝑗1∧𝑘1 (xii) 𝑗1→ 𝑘1 ≤ (𝑗1 ∘ 𝑙1) → (𝑘1 ∘ 𝑙1), (xiii) 𝑗1 ∘ (𝑘1 → 𝑙1) ≤ 𝑘1→ (𝑗1 ∘ 𝑙1), (xiv) (𝑗1 → 𝑘1) ∘ (𝑘1 → 𝑙1) ≤ 𝑗1 →𝑙1, (xv) (𝑗1 ∘ 𝑗1 ∗ ) = 0. 142 on neutrosophic filter of bl-algebras note. in the above sequence, ℬ is used to intend the blalgebras and the operations ′ ∨ ′, ′ ∧′, ′ ∘′ have preference on the way to the operations ′ → ′. note. in a blalgebra ℬ,′ ∗ ′is a complement defined as 𝑗1 ∗ = 𝑗1 → 0 for all 𝑗1∈ ℬ. definition 2.3[15] a non-empty subset 𝐹 of a blalgebra ℬ is a filter of ℬ if the following axioms hold for all 𝑗1, 𝑘1∈ ℬ, (i) if 𝑗1, 𝑘1∈𝐹, then 𝑗1 ∘ 𝑘1∈𝐹, (ii) if 𝑗1∈𝐹 and 𝑗1≤ 𝑘1, then 𝑘1∈𝐹. proposition 2.4[15] a nonempty subset 𝐹 of a blalgebra ℬ is a filter of ℬ if and only if the following are satisfied for all 𝑗1, 𝑘1∈ ℬ, (i) 1∈𝐹, (ii) 𝑗1, 𝑗1 → 𝑘1∈𝐹 implies 𝑘1∈𝐹. a filter 𝐹 of a bl-algebra ℬ is proper if 𝐹≠ ℬ. definition 2.5[8,9] let𝑋 be a set.a neutrosophic subset 𝑅 of 𝑋 is a triple (𝑇𝑅 , 𝐼𝑅 , 𝐹𝑅 ) where 𝑇𝑅 : 𝑋→[0,1] is truth membership function, 𝐼𝑅 : 𝑋→[0,1] is indeterminacy function and 𝐹𝑅 : 𝑋 → [0,1] is false membership function and 0 ≤ 𝑇𝑅 (𝑗1) + 𝐼𝑅 (𝑗1) + 𝐹𝑅 (𝑗1) ≤ 3 for all 𝑗1 ∈ 𝑋. hence, for each 𝑗1 ∈ 𝑋, 𝑇𝑅 (𝑗1), 𝐼𝑅 (𝑗1) and 𝐹𝑅 (𝑗1) are all standard real numbers in [0,1] . note. the values of 𝑇𝑅 (𝑗1), 𝐼𝑅 (𝑗1) and 𝐹𝑅 (𝑗1) have no limitations and we have the obvious condition 0 ≤ 𝑇𝑅 (𝑗1) + 𝐼𝑅 (𝑗1) + 𝐹𝑅 (𝑗1) ≤ 3. definition 2.6[4, 9] let 𝑅 and 𝑆 be two neutrosophic sets on 𝑋. define 𝑅 ≤ 𝑆 if and only if 𝑇𝑅 (𝑗1) ≤ 𝑇𝑆 (𝑗1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑆 (𝑗1), 𝐹𝑅 (𝑗1) ≥ 𝐹𝑆 (𝑗1) for all 𝑗1 ∈ 𝑋. definition 2.7[5,9] let 𝑅 and s be two neutrosophic sets on x. define 𝑅 ∧ 𝑆 = (𝑇𝑅 ∧ 𝑇𝑆,𝐼𝑅 ∨ 𝐼𝑆 , 𝐹𝑅 ∨ 𝐹𝑆); 𝑅 ∨ 𝑆 = (𝑇𝑅 ∨ 𝑇𝑆,𝐼𝑅 ∧ 𝐼 𝑆 , 𝐹𝑅 ∧ 𝐹𝑆) where ′ ∧ ′ is the minimum and ′ ∨ ′ is the maximum between real numbers. definition 2.8[10] let 𝑅 be a neutrosophic set in 𝑋 and 𝛼, 𝛽, 𝛾 ∈ [0,1] with 0 ≤ 𝛼 + 𝛽 + 𝛾 ≤ 3 and (𝛼, 𝛽, 𝛾 ) – level set of 𝑅 denoted by 𝑅(𝛼,𝛽,𝛾 ) is defined as 𝑅(𝛼,𝛽,𝛾 ) = {𝑗1 ∈ 𝑋/ 𝑇𝑅 (𝑗1) ≥ 𝛼, 𝐼𝑅 (𝑗1) ≤ β and 𝐹𝑅 (𝑗1) ≤ γ}. 143 a. ibrahim and s. karunya helen gunaseeli 3. properties of neutrosophic filter in this section, we introduce the definition of neutrosophic filter of bl algebra and obtain some relevant properties with illustrations. definition 3.1 a neutrosophic set 𝑅 of algebra ℬ is called a neutrosophic filter if it satisfies the subsequent conditions: (i) 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅 (1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (1) and 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (1), (ii) min{𝑇𝑅 (𝑗1 → 𝑘1), 𝑇𝑅 (𝑗1)}≤ 𝑇𝑅 (𝑘1),min{𝐼𝑅 (𝑗1 → 𝑘1), 𝐼𝑅 (𝑗1)} ≥ 𝐼𝑅 (𝑘1) and min{𝐹𝑅 ( 𝑗1 → 𝑘1), 𝐹𝑅 (𝑗1)} ≥ 𝐹𝑅 (𝑘1)}for all 𝑗1, 𝑘1 ∈ ℬ. example 3.2 let ℬ = {0, 𝑢, 𝑣, 1}. the binary operations ′ ∘ ′ and ′ → ′are given by the subsequent tables (3.1) and (3.2). table3.1: ′ ° ′ operation table 3.2: ′ → ′operation then, (ℬ, ∨, ∧, ∘, →, 0, 1) is a blalgebra. define a neutrosophic set 𝑅 of ℬ as follows: 𝑅 = {(1, [0.9,0.2,0.1]), (𝑢, [0.5, 0.3, 0]), (𝑣, [0.5,0.3,0]), (0, [0.9,0.2,0.1])}. it is evident that 𝑅 is a neutrosophic filter of ℬ and assure the conditions (i) and (ii) of the definition 3.1. example 3.3 let ℬ = {0, 𝑎, 𝑏, 1}. the binary operations are given by the tables (3.3) and (3.4). let 𝑆 of ℬ be a neutrosophic set as follows: 𝑆 = {(1, [0.5,0.3,0.2]), (𝑎, [0.9,0.2,0.1]), (𝑏, [0.9,0.2,0.1]), (0, [0.9,0.2,0.1])} here, 𝑆 is not a neutrosophic filter of ℬ. since 𝑇𝑆 (1) = 0.5 ≱ 0.9 = min{𝑇𝑆 (𝑎 ∘ 1), 𝑇𝑆 (𝑎)}, 𝑇𝑆 (1) = 0.5 ≱ 0.9 =min{𝑇𝑆 (𝑏 ∘ 1), 𝑇𝑆 (𝑏)}. ∘ 0 𝑢 𝑣 1 0 0 0 0 0 𝑢 0 0 𝑢 𝑣 𝑣 0 𝑢 𝑣 𝑣 1 0 𝑢 𝑣 1 → 0 𝑢 𝑣 1 0 1 1 1 1 𝑢 𝑢 1 1 1 𝑣 0 𝑢 1 1 1 0 𝑢 𝑣 1 144 on neutrosophic filter of bl-algebras table 3.3∶ ′ ° ′ operation table 3.4:′ ⟶′operation proposition 3.4 let 𝑅 be a neutrosophic filter in a blalgebra ℬ.if 𝑗1 ≤ 𝑘1then 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅 (𝑘1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (𝑘1), 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1)for all 𝑗1, 𝑘1 ∈ ℬ. proof: let 𝑅 be a neutrosophic filter of a bl-algebraℬ. if 𝑗1 ≤ 𝑘1, then 𝑗1 ⟶ 𝑘1 = 1for all 𝑗1, 𝑘1 ∈ ℬ [from(iii) of the proposition 2.2] then, from (i) and(ii) of the definition 3.1, 𝑇𝑅 (𝑗1) = min{𝑇𝑅( 1 ) , 𝑇𝑅 (𝑗1)} = min{𝑇𝑅(𝑗1 → 𝑘1) , 𝑇𝑅 (𝑗1)} ≤ 𝑇𝑅 (𝑘1), 𝐼𝑅 (𝑗1) = min{𝐼𝑅 ( 1 ), 𝐼𝑅 (𝑗1) } = min{𝐼𝑅 ( 𝑗1 → 𝑘1 ), 𝐼𝑅 (𝑗1) } ≥ 𝐼𝑅 (𝑘1), 𝐹𝑅 (𝑗1) = min{ 𝐹𝑅 ( 1 ), 𝐹𝑅 (𝑗1)} = min{𝐹𝑅 (𝑗1 → 𝑘1), 𝐹𝑅 (𝑗1)} ≥ 𝐹𝑅 (𝑘1) for all 𝑗1, 𝑘1 ∈ ℬ. ∎ proposition 3.5 let 𝑅 be a neutrosophic filter of a bl-algebra ℬ. if 𝑗1 ≤ 𝑘1then 𝑇𝑅 (𝑗1)is order preserving and 𝐼𝑅 (𝑗1), 𝐹𝑅 (𝑗1)are order reversing. proof: let 𝑅 be a neutrosophic filter of a bl-algebra ℬ. to prove: if 𝑗1 ≤ 𝑘1then 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅(𝑘1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (𝑘1), 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1) for all𝑗1, 𝑘1 ∈ ℬ. then, from the proposition 3.4, the proof is straight forward. thus 𝑇𝑅 (𝑗1)is order preserving and 𝐼𝑅 (𝑗1), 𝐹𝑅 (𝑗1)are order reversing. ∎ → 0 𝑎 𝑏 1 0 1 1 1 1 𝑎 𝑎 1 1 1 𝑏 0 𝑎 1 1 1 0 𝑎 𝑏 1 ∘ 0 𝑎 𝑏 1 0 0 0 0 𝑏 𝑎 0 0 𝑎 𝑏 𝑏 0 𝑎 𝑏 𝑏 1 0 𝑎 𝑏 1 145 a. ibrahim and s. karunya helen gunaseeli proposition 3.6 let 𝑅 be a neutrosophic set of a bl–algebra ℬ. 𝑅 is a neutrosophic filter of ℬ if and only if 𝑗1 ⟶ (𝑘1 ⟶ 𝑙1) = 1 implies 𝑇𝑅 (𝑙1) ≥ min{ 𝑇𝑅 (𝑗1) , 𝑇𝑅 (𝑘1) }, 𝐼𝑅 (𝑙1)≤ min{ 𝐼𝑅 ( 𝑗1 ), 𝐼𝑅 (𝑘1)} and 𝐹𝑅 (𝑙1) ≤ min{ 𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1)} for all 𝑗1, 𝑘1, 𝑙1 ∈ ℬ. proof: let 𝑅 be a neutrosophic filter of a blalgebra ℬ. then, from (ii) of the definition 3.1, we have 𝑇𝑅 (𝑙1) ≥ min{𝑇𝑅 ( 𝑙1 ⟶ 𝑘1) , 𝑇𝑅 (𝑘1) }for all 𝑗1,𝑘1,𝑙1 ∈ ℬ. now,𝑇𝑅 ( 𝑙1 ⟶ 𝑘1) ≥ min{ 𝑇𝑅 ( 𝑗1 →(𝑘1 → 𝑙1)) , 𝑇r(𝑗1) }. if 𝑗1 →(𝑘1 → 𝑙1) = 1, then we have 𝑇𝑅 (𝑙1 ⟶ 𝑘1) ≥min{𝑇𝑅 (1) , 𝑇𝑅 (𝑗1) } = 𝑇𝑅 (𝑗1). so, 𝑇𝑅 ( 𝑙1) ≥ min{𝑇𝑅 (𝑗1) , 𝑇𝑅 (𝑘1) }. similarly, 𝐼𝑅 (𝑙1) ≤ min{ 𝐼𝑅 (𝑗1), 𝐼𝑅 (𝑘1)} and 𝐹𝑅 (𝑙1) ≤ min{𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1)}. conversely, let 𝑗1 ⟶ (𝑗1 ⟶ 1) = 1 for all 𝑗1 ∈ ℬ. then,𝑇𝑅 ( 1 ) ≥ min{𝑇𝑅 (𝑗1) , 𝑇𝑅 (𝑗1) } = 𝑇𝑅 (𝑗1). on the other hand, from (𝑗1 → 𝑘1) → (𝑗1 → 𝑘1) = 1 implies 𝑇𝑅 (𝑘1) ≥ min{𝑇𝑅 ( 𝑗1 → 𝑘1),𝑇𝑅 (𝑗1)}, 𝐼𝑅 (𝑘1)≤ min{ 𝐼𝑅 ( 𝑗1 → 𝑘1 ), 𝐼𝑅 (𝑗1)} and 𝐹𝑅 (𝑘1) ≤ min{ 𝐹𝑅 (𝑗1 → 𝑘1), 𝐹𝑅 (𝑗1)}. then, from the definition 3.1, 𝑅 is a neutrosophic filter of ℬ. ∎ corollary 3.7 let 𝑅 be a neutrosophic set of blalgebra ℬ. 𝑅 is a neutrosophic filter of ℬ if and only if 𝑗1 ∘ 𝑘1 ≤ 𝑙1 or 𝑘1 ∘ 𝑙1 ≤ 𝑙1 implies 𝑇𝑅 (𝑙1) ≥min{ 𝑇𝑅 ( 𝑗1) , 𝑇𝑅 (𝑘1 ) }, 𝐼𝑅 (𝑙1)≤ min{𝐼𝑅 (𝑗1), 𝐼𝑅 (𝑘1 )} and 𝐹𝑅 (𝑙1) ≤min{ 𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1 )} for all 𝑗1, 𝑘1 , 𝑙1 ∈ ℬ. proof: from (i) of the proposition 2.2 and the proposition 3.6 the proof is obvious. ∎ proposition 3.8 let 𝑅 be a neutrosophic set of a bl-algebra ℬ. 𝑅is a neutrosophic filter of ℬ if and only if (i) if 𝑗1 ≤ 𝑘1then 𝑇𝑅(𝑗1) ≤ 𝑇𝑅 (𝑘1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (𝑘1) and 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1) , (ii) 𝑇𝑅 (𝑗1 ∘ 𝑘1) ≥ min{𝑇𝑅 ( 𝑗1) , 𝑇𝑅 (𝑘1) },𝐼𝑅 (𝑗1 ∘ 𝑘1)≤ min{𝐼𝑅 ( 𝑗1), 𝐼𝑅 (𝑘1)} and 𝐹𝑅 (𝑗1 ∘ 𝑘1) ≤ min{ 𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1)}for all 𝑗1, 𝑘1 ∈ ℬ. proof: let 𝑅 be a neutrosophic filter of a bl-algebra ℬ. then, from the proposition 3.4, we have 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅 (𝑘1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (𝑘1) and 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1) when 𝑗1 ≤ 𝑘1. 146 on neutrosophic filter of bl-algebras since 𝑗1 ∘ 𝑘1 ≤ 𝑗1 ∘ 𝑘1 and from the corollary 3.7, we have 𝑇𝑅 (𝑗1 ∘ 𝑘1) ≥ min{ 𝑇𝑅 ( 𝑗1), 𝑇𝑅 (𝑘1) }, 𝐼𝑅 (𝑗1 ∘ 𝑘1)≤ min{ 𝐼𝑅 ( 𝑗1), 𝐼𝑅 (𝑘1)} and 𝐹𝑅 (𝑗1 ∘ 𝑘1) ≤ min{ 𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1)}. conversely, let 𝑅 be a neutrosophic set and satisfies (i) and (ii). if 𝑗1 ∘ 𝑘1 ≤ 𝑙1 then from (i) and (ii) we get, 𝑇𝑅 (𝑙1) ≥ min{𝑇𝑅 ( 𝑗1) , 𝑇𝑅 (𝑘1) }, 𝐼𝑅 (𝑙1)≤ min{ 𝐼𝑅 ( 𝑗1), 𝐼𝑅 (𝑘1)} and 𝐹𝑅 (𝑙1) ≤ min{ 𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1)}for all 𝑗1, 𝑘1, 𝑙1 ∈ ℬ. then, from the corollary 3.7, we have 𝑅 is a neutrosophic filter. ∎ proposition 3.9 let 𝑅 be a neutrosophic set of a bl-algebra ℬ. if 𝑅 is a neutrosophic filter of ℬ, then it satisfies the following for all 𝑗1, 𝑘1, 𝑙1 ∈ ℬ. (i) if 𝑇𝑅 ( 𝑗1 ⟶ 𝑘1)= 𝑇𝑅 (1), then 𝑇𝑅 (𝑗1) ≤ 𝑇(𝑘1),𝐼𝑅 ( 𝑗1 ⟶ 𝑘1)= 𝐼 𝑅(1), then 𝐼𝑅 (𝑗1) ≥ 𝐼 𝑅(𝑘1),𝐹𝑅 ( 𝑗1 ⟶ 𝑘1)= 𝐹𝑅(1), then 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1). (ii) 𝑇𝑅 (𝑗1 ⟶ 𝑘1)≤ 𝑇𝑅 (𝑗1 ∘ 𝑙1 ⟶ 𝑘1 ∘ 𝑙1), 𝐼𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐼𝑅 (𝑗1 ∘ 𝑙1 ⟶ 𝑘1 ∘ 𝑙1) and𝐹𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐹𝑅 (𝑗1 ∘ 𝑙1 ⟶ 𝑘1 ∘ 𝑙1). (iii) 𝑇𝑅 ( 𝑗1 ⟶ 𝑘1) ≤ 𝑇𝑅 ((𝑘1 → 𝑙1) → (𝑗1 → 𝑙1)),𝐼𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐼𝑅 (( 𝑘1→ 𝑙1) →(𝑗1 → 𝑙1))and 𝐹𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐹𝑅 ((𝑘1→ 𝑙1) → ( 𝑗1→ 𝑙1)). (iv) 𝑇𝑅 ( 𝑗1 ⟶ 𝑘1) ≤ 𝑇𝑅 ((𝑙1 → 𝑗1) → (𝑙1 → 𝑘1)),𝐼𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐼𝑅 ((𝑙1→ 𝑗1) → (𝑙1→ 𝑘1)) and𝐹𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐹𝑅 ((𝑙1 → 𝑗1) → (𝑙1 → 𝑘1)). proof: (i) let 𝑅 be a neutrosophic filter of a bl-algebra ℬ. then from the definition 3.1, and since 𝑇𝑅 ( 𝑗1 ⟶ 𝑘1)= 𝑇𝑅 (1), we have 𝑇𝑅 (𝑘1) ≥ min{ 𝑇𝑅 (𝑗1) , 𝑇𝑅 (𝑗1 ⟶ 𝑘1) } = min{𝑇𝑅 (𝑗1) , 𝑇𝑅 (1) }= 𝑇𝑅 (𝑗1). thus, 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅 (𝑘1). similarly we get, 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (𝑘1), 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1). from the proposition 2.2 and (i) of the proposition 3.8, we can prove (ii), (iii) and (iv) easily. ∎ proposition 3.10 let 𝑅 be a neutrosophic set of a blalgebra ℬ. 𝑅is a neutrosophic filter of ℬ if and only if (i) 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅 (1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (1) and 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (1), (ii) 𝑇𝑅 ( (𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≥ min{ 𝑇𝑅 ( 𝑗1) , 𝑇𝑅 (𝑘1) }, 𝐼𝑅 ((𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≤ min{𝐼𝑅 ( 𝑗1) , 𝐼𝑅 (𝑘1) } and 𝐹𝑅 ( (𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≤ min{ 𝐹𝑅 ( 𝑗1) , 𝐹𝑅 (𝑘1) }for all 𝑗1,𝑘1,𝑙1 ∈ ℬ. 147 a. ibrahim and s. karunya helen gunaseeli proof: let 𝑅 be a neutrosophic filter of bl-algebra ℬ. from the definition 3.1 (i) is straight forward. since,𝑇𝑅 ((𝑗1 → (𝑘1 → 𝑙1)) → 𝑙1)≥ min{𝑇𝑅 ((𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1), 𝑇𝑅 (𝑘1)} (3.1) now, we have, (𝑗1 →(𝑘1 → 𝑙1)) →(𝑘1 → 𝑙1) = 𝑗1∨(𝑘1 → 𝑙1) ≥ 𝑗1. 𝑇𝑅 ((𝑗1 →(𝑘1 → 𝑙1)) →(𝑘1 → 𝑙1))≥ 𝑇𝑅 (𝑗1) (3.2) [from the proposition 3.5] using (3.2) in (3.1), we have 𝑇𝑅 ( (𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≥ min{𝑇𝑅 (𝑗1) , 𝑇𝑅 (𝑘1) } similarly, we get 𝐼𝑅 ( (𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≤ min{ 𝐼𝑅 ( 𝑗1) , 𝐼𝑅 (𝑘1) } and 𝐹𝑅 ( (𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≤ min{ 𝐹𝑅 ( 𝑗1) , 𝐹𝑅 (𝑘1) }. conversely, assume (i) and (ii) hold. since 𝑇𝑅 (𝑘1) = 𝑇𝑅 (1 → 𝑘1) = 𝑇𝑅 ( (𝑗1 → 𝑘1) → (𝑗1 → 𝑘1) → 𝑘1) ≥ min{𝑇𝑅 ( 𝑗1 → 𝑘1) , 𝑇𝑅 (𝑘1) }. similarly, 𝐼𝑅 (𝑘1)≤ min{𝐼𝑅 (𝑗1 → 𝑘1) , 𝐼𝑅 (𝑘1) }and 𝐹𝑅 (𝑘1) ≤ min{ 𝐹𝑅 ( 𝑗1 → 𝑘1) , 𝐹𝑅 (𝑘1) }. from (i), 𝑅 is a neutrosophic filter of ℬ. ∎ proposition 3.11 the intersection of two neutrosophic filters of ℬ is also a neutrosophic filter of ℬ. proof: let 𝑅 and 𝑆 be two neutrosophic filters of ℬ, to prove: 𝑅 ∩ 𝑆 is a neutrosophic filter of ℬ. we have 𝑇𝑅 (𝑘1) ≥ min{ 𝑇𝑅 (𝑙1) , 𝑇𝑅 (𝑗1) } and 𝑇𝑆(𝑘1 ) ≥ min{ 𝑇𝑆(𝑙1 ) , 𝑇𝑆(𝑗1) } for all 𝑗1, 𝑘1 , 𝑙1 ∈ ℬ. since 𝑇𝑅∩𝑆(𝑘1 ) = min{𝑇𝑅 (𝑘1 ) , 𝑇𝑆(𝑘1 ) } ≥ min{min{ 𝑇𝑅 (𝑙1) , 𝑇𝑅 (𝑗1) }, min{ 𝑇𝑆(𝑙1 ) , 𝑇𝑆(𝑗1) }} = min{min{ 𝑇𝑅 (𝑙1) , 𝑇𝑆(𝑙1 ) }, min{ 𝑇𝑅 (𝑗1) , 𝑇𝑆(𝑗1) }} = min{ min{ 𝑇𝑅∩𝑆(𝑙1) ,𝑇𝑅∩𝑆(𝑗1) }}. similarly,𝐼𝑅∩𝑆(𝑘1 ) = min{ min{ 𝐼𝑅∩𝑆(𝑙1) ,𝐼𝑅∩𝑆(𝑗1) }} and 𝐹𝑅∩𝑆(𝑘1 ) = min{ min{ 𝐹𝑅∩𝑆(𝑙1) ,𝐹𝑅∩𝑆(𝑗1) }}. hence, 𝑇𝑅∩𝑆(𝑘1 ) ≥ min{𝑇𝑅∩𝑆(𝑙1) ,𝑇𝑅∩𝑆(𝑗1) }, 148 on neutrosophic filter of bl-algebras 𝐼𝑅∩𝑆(𝑘1 ) ≤min{𝐼𝑅∩𝑆(𝑙1) ,𝐼𝑅∩𝑆(𝑗1) } and 𝐹𝑅∩𝑆(𝑘1 ) ≤ min{ 𝐹𝑅∩𝑆(𝑙1) ,𝐹𝑅∩𝑆(𝑗1) }. thus 𝑅 ∩ 𝑆 is a neutrosophic filter of ℬ. ∎ corollary 3.12 let 𝑃𝑖 be a family of neutrosophic filters of ℬ, where 𝑖 ∈ 𝐼, 𝐼 is a index set, then ∩𝑖∈𝐼 𝑃𝑖 is a neutrosophic filter of ℬ. ∎ 4. discussion and conclusions in this paper, we have introduced the notion of a neutrosophic filter in bl algebras with illustrations. moreover, several features of the neutrosophic filters are conferred. also, we have derived a few equivalent conditions for a neutrosophic set of a bl-algebra to be a neutrosophic filter. in bl-algebras, fuzzy, vague, and many additional filters have already been defined. the main focus of this paper is to establish the neutrosophical nature of bl-algebras. further, research on bl-algebras structure and the creation of the associated many-valued logical system will benefit from the aforementioned study. in the future, these neutrosophic filters can be extended to include fantastic filters, implicative filters, normal filters, and so on. references [1] h. h. abbass,q. m. luhaib. on smarandache filter of a smarandache bh-algebra. journal of physics, 12-34, 2019. [2] k. t. atanassov. intuitionistic fuzzy sets. itkr’s session: sofia, 1983. [3] k. t. atanassov. intuitionistic fuzzy sets. fuzzy sets and systems, 20, 87-86, 1986. [4] biao long meng. on filters in bealgebras. scientiae mathematicaejaponicae,71(2), 201-207, 2010. [5] r. a. borzooei. neutrosophic deductive filters on bl algebras. intelligent and fuzzy systems, 26, 29933004,2014. [6] p. hajek. metamathematics of fuzzy logic. klower academic publishers: dordrecht, 1999. [7] l.z. liu, k.t. li. fuzzy filters of bl-algebras. information sciences, 173, 141-154,2005. [8] a. a. salama, h. alagamy. neutrosophic filters. journal of computer science engineering, 307-312, 2013. 149 a. ibrahim and s. karunya helen gunaseeli [9] f. smarandache.a unifying field in logics: neutrosophic logic. neutrosophy, neutrosophic set, neutrosophic probability. american research press, 1999. [10] c. a .c. sweety, i. arockiarani. annals of fuzzy mathematics and informatics. rough sets in neutrosophic approximation space,13(4), 449-463,2017. [11] e. turunen. bl-algebras of basic fuzzy logic. mathware and soft computing, 6, 49-61, 1999. [12] y. xu and k.y.qin. on filters of lattice implication algebra. journal of fuzzy maths, 2, 251-260,1993. [13] s. yahya mohamed, p.umamaheshwari. vague filter of blalgebras. computer and mathematicalsciences,9(8), 914-920,2018. [14] l. a. zadeh.fuzzy sets. information and control. 8, 338-353,1965. [15] x. zhang, x. mao, y. wu, x. zhai.neutrosophic filters in pseudo-bci algebras. international journal for uncertainty quantification, 8(6),511526, 2018. 150 ratio mathematica, 21, 2011, pp. 91-105 91 an optimization framework for “build-or-buy” strategy for component selection in a fault tolerant modular software system under recovery block scheme p.c.jha* ritu arora** u.dinesh kumar*** *department of operational research , university of delhi,india **maharaja agrasen institute of technology, ggsip university, delhi,india. ***indian institute of management, bangalore, india *jhapc@yahoo.com ** arora_ritu21@yahoo.co.in ***dineshk@iimb.ernet.in abstract this paper discusses a framework that helps developers to decide whether to buy or build components of software architecture. two optimization models have been proposed. first model is bi-criteria optimization model based on decision variables in order to maximize the software reliability with simultaneous minimization of the overall cost of the system. the second optimization model deals with the issue of compatibility. keywords : modular software, software reliability, software cost, fault tolerance, software components, recovery block scheme 1. introduction science and technology demand high quality software for making improvement and breakthroughs. today, computer hardware and software permeates our modern society. the newest cameras, vcrs, and automobiles cannot be controlled and operated without computers. when the requirement for and dependencies on computer increases, the possibility of crises from computer failures also increases. software systems are developed as per the requirements given by the users. while developing the software, quality and reliability of the software are two key factors. reliability of a software system is defined as the probability that software operates without failure in a specified environment, during a specified exposure period. introduction of redundancy in the parts of the hardware and/or software components is one of the most followed ways to improve the reliability of the system under development. a careful use of redundancy may allow the system to tolerate faults. despite that we still cannot guarantee error free software. a way of handling unknown and unpredictable mailto:*jhapc@yahoo.com mailto:arora_ritu21@yahoo.co.in mailto:***dineshk@iimb.ernet.in 92 software failures is through fault tolerance. one way to reduce the risks of software design faults and thus enhance software dependability is to use software fault tolerance techniques. software fault tolerance techniques are employed during the procurement, or development, of the software. they enable a system to tolerate software faults remaining in the system after its development. when a fault occurs, these techniques provide mechanisms to the software system to prevent system failure from occurring. there are two structural methodologies for fault tolerant system i.e. recovery block scheme and n-version scheme. in this paper, we will discuss optimization model for recovery block. non functional aspects play a significant role in determining software quality. given the fact that lack of proper handling of non functional aspects (cysneiros et al, [5]) of a software application has led to a series of software failures, nonfunctional attributes such as reliability security and performance should be considered during the component selection phase of software development. this paper discusses a framework that helps developers to decide whether buying or building components of software architecture on the base of cost and non functional factors. while developing software, components can be both bought as cots (commercial off-the shelf) products, and probably adapted to work in the software system, or they can be developed in-house. this decision is known as “build-or-buy decision”. this decision affects the overall cost and reliability of the system. most of today’s software systems include one or more cots products. cots are pieces of software that can be reused by software projects to build new systems. benefits of cots based development include significant reduction in the development cost, time and improvement in the dependability requirement. no changes are normally made to their source codes. cots components are used without any code modification and inspection. the components, which are not available in the market or cannot be purchased economically, can be developed within the organization and are known as inhouse built components. kapur et al [8] discussed issues related to reliability of systems through weighted maximization of system quality subject to budgetary constraint. this paper discusses the issues related with reliability of the software systems and cost produced by integrating cots or in-house build components. large software system has modular structure to perform a set of functions. each function is performed by different modules having different alternatives for each module. in case a cots component is selected then different versions are available for each 93 alternative and only one version will be selected for each alternative of a module. if a component is in-house build component, then the alternative of a module is selected. a schematic representation of the software system is given in figure 1. we are selecting the components for modules to maximize the system reliability by simultaneously minimizing the cost. the frequency with which the functions are used is not same for all of them and not all the modules are called during the execution of a function, the software has in its menu. software whose failure can have bad effects afterwards can be made fault tolerant through redundancy at module level (belli and jadrzejowicz, [1]). we assume that functionally equivalent and independently developed alternatives (i.e in-house or cots) for each module are available with an estimated reliability and cost. the first optimization model (optimization model-i) of this paper maximizes the system reliability with simultaneously minimizing the cost. the model contains four problems (p1), (p2), (p3) and (p4). problem (p1) is not in normalized form, therefore, it has been normalized and transformed into problem (p3) and (p4). the second optimization model (optimization model-ii) considers the issue of compatibility between different alternatives of modules as it is observed that some cots components cannot integrate with all the alternatives of another module. the models discussed are illustrated with numerical example. 2. notations r : system quality measure lf : frequency of use, of function l ls : set of modules required for function l ir : reliability of module i l : number of functions, the software is required to perform n : number of modules in the software im : number of alternatives available for module i ijv : number of versions available for alternative j of module i total number of tests performed on the inhouse developed instance (i.e. alternative of module ) number of successful (i .e failure free) test performed on the in-house developed instance (i.e. alternative of module ) 1 t : probability that next alternative is not invoked upon failure of the current alternative : tot ij n j i : suc ij n j i 94 2t : probability that the correct result is judged wrong. 3 t : probability that an incorrect result is accepted as correct. ij x : event that output of alternative j of module i is rejected. yij : event that correct result of alternative j of module i is accepted. sij : reliability of alternative j of module i r : reliability of version of alternative for module ijk k j i ijk c : cost of version k of alternative j for module i ijk r : reliability of version k of alternative j for module i ijk d : delivery time of version k of alternative j for module i ij c : unitary development cost for alternative j of module i ij t : estimated development time for alternative j of module i ij  : average time required to perform a test case for alternative j of module i ij  : probability that a single execution of software fails on a test case chosen from a certain input distribution t y : 0, if constraint is active 1, if constraint is inactive th th t t    : ijk x : 1, if the of cots alternative of the module is chosen 0, otherwise th th th k version j i   ijz : binary variable taking value 0 or 1 1 , if alternative is present in module 0, otherwise j i   3. optimization models the first optimization model is developed for the following situations which also hold good for the second model, but with additional assumptions related to compatibility among alternatives of a module. ijy 1 if the th alternative of th module is in-house developed. 0 otherwise j i   95 the following assumptions are common for the optimization models are: 1. software system consists of a finite number of modules. 2. software system is required to perform a known number of functions. the program written for a function can call a series of modules  n . a failure occurs if a module fails to carry out an intended operation. 3. codes written for integration of modules don’t contain any bug. 4. several alternatives are available for each module. fault tolerant architecture is desired in the modules (it has to be within the specified budget). independently developed alternatives (primarily cots/ in-house components) are attached in the modules and work similar to the recovery block scheme discussed in (berman et al., [2] and kumar, [9]). 5. the cost of an alternative is the development cost, if developed in house; otherwise it is the buying price for the cots product. 6. different inhouse alternatives with respect to unitary development cost, estimated development time, average time and testability of a module are available. 7. cost, reliability and development time of an in-house component can be specified by using basic parameters of the development process, e.g., a component cost may depend on a measure of developer skills, or the component reliability depends on the amount of testing. 8. different versions with respect to cost, reliability and delivery time of a module are available. 9. other than available cost-reliability versions of an alternative, we assume the existence of virtual versions, which has a negligible reliability of 0.001, zero cost and zero delivery time. these components are denoted by index one in the third subscript of , c and . ijk ijk ijk x r for example 1ij r denotes the reliability of first version of alternatives j for module i . 3.1 model formulation let s be a software architecture made of n modules having im alternatives available for each module and each cots alternatives has different versions. 3.1.1 build versus buy decision for each module i , if an alternative is bought (i.e. some 1 ijk x  ) then there is no in-house development (i.e. 0 ij y  ) and vice versa. 96 1 =1; 1, 2,...., and 1, 2,...., ijv ij ijk i k y x i n j m     3.1.2 redundancy constraint the equation stated below guarantees that redundancy is allowed for the components. 2 ; 1, 2,...., and 1, 2,...., ijv ij ijk ij i k y x z i n j m      1 1; 1, 2,...., and 1, 2,...., ij ij i x z i n j m    1 1; 1, 2,.... im ij j z i n    3.1.3 probability of failure free in-house developed components the possibility of reducing the probability that the alternative of module th th j i fails by means of a certain amount of test cases (represented by the variable tot ij n ). cortellessa et al [4] define the probability of failure on demand of an in-house developed alternative of module th th j i , under the assumption that the on-field users’ operational profile is the same as the one adopted for testing (bertolino and strigini, [3]). basing on the testability definition, we can assume that the number suc ij n of successful (i.e. failure free) tests performed on th j alternative of same module.  1 ; 1, 2,...., and 1, 2,....,suc totij ij ij in n i n j m    let a be the event “ suc ij n failure – free test cases have been performed ” and b be the event “ the alternative is failure free during a single run ”.if ij  is the probability that the inhouse developed alternative is failure free during a single run given that suc ij n test cases have been successfully performed, from the bayes theorem we get ( / ) ( ) ( / ) ( / ) ( ) ( / ) ( ) ij p a b p b p b a p a b p b p a b p b     the following equalities come straightforwardly: ( / ) 1; ( ) 1 ; ( / ) (1 ) ; ( ) suc ijn ij ij ijp a b p b p a b p b        97 therefore, we have     1 ; 1, 2,...., and 1, 2,...., 1 1 suc ij ij ij in ij ij ij i n j m             3.1.4 reliability equation of both in-house and cots components the reliability ( ij s ) of th j alternative of th i module of the software. ; 1, 2,...., and 1, 2,...., ij ij ij ij i s y r i n j m    where 1 ; 1, 2,...., and 1, 2,...., ijv ij ijk ijk i k r r x i n j m     3.1.5 delivery time constraint the maximum threshold t has been given on the delivery time of the whole system. in case of a cots components the delivery time is simply given by ijk d , whereas for an inhouse developed alternative the delivery time shall be expressed as ( ) tot ij ij ij t n .   1 1 1 iji vmn tot ij ij ij ij ijk ijk i j k y t n d x t                3.2 objective function 3.2.1 reliability objective function reliability objective function maximizes the system quality (in terms of reliability) through a weighted function of module reliabilities. reliability of modules that are invoked more frequently during use is given higher weights. analytic hierarchy process (ahp) can be effectively used to calculate these weights. 1 maximize ( ) l l l i l i s r x f r     where i r is the reliability of module i of the system under recovery block stated as follows.     ,.........2,1 ; 1 1 1 niypxpzr ij i ij z ij m j j k z ikiji                   1 3 21 1 1ij ij ijp x t s t s t       98    2 1ij ijp y s t  3.2.2 cost objective function cost objective function minimizes the overall cost of the system. the sum of the cost of all the modules is selected from “build – or buy” strategy. the in-house development cost of the alternative j of module i can be expressed as  totij ij ij ijc t n   1 1 1 minimize c(x)= iji vmn tot ij ij ij ij ij ijk ijk i j k c t n y c x              3.3 optimization model i in the optimization model it is assumed that the alternatives of a module are in a recovery block. in recovery block more than one alternative of a program exist. for cots based software multiple alternatives of a module can be purchased from different vendors. each module works under a recovery block. upon invocation of a module the first alternative is executed and the result is submitted for acceptance test. if it is rejected, the second alternative is executed with the original inputs. the same process continues through attached alternative until a result is accepted or the whole recovery block (module) fails. fault tolerance in a recovery block is achieved by increasing the number of redundancies. problem (p1) 1 maximize ( ) l l l i l i s r x f r      (1)   1 1 1 minimize c(x)= iji vmn tot ij ij ij ij ij ijk ijk i j k c t n y c x              (2) subject to  and y are binary variable/ijk ijx s x      ,.........2,1 ; 1 1 1 niypxpzr ij i ij z ij m j j k z ikiji             (3)  1 , 1, 2,...., and 1, 2,....,suc totij ij ij in n i n j m    (4)       1 3 2 1 1 1 ij ij ijp x t s t s t          2 1ij ijp y s t  99     1 ; 1, 2,...., and 1, 2,...., 1 1 suc ij ij ij in ij ij ij i n j m             (5) ; 1, 2,...., and 1, 2,...., ij ij ij ij i s y r i n j m    (6) 1 =1; 1, 2,...., and 1, 2,...., ijv ij ijk i k y x i n j m     (7) 2 ijv ij ijk ij k y x z    ; 1, 2,...., and 1, 2,...., ii n j m  (8) 1 1; 1, 2,...., and 1, 2,...., ij ij i x z i n j m    (9) 1 1 ; 1, 2,...., im ij j z i n    (10)   1 1 1 iji vmn tot ij ij ij ij ijk ijk i j k y t n d x t               (11)  where x is a vector of elements : and ; 1, 2,..... ; 1, 2,...., ; k=1,2,....v ijk ij i ij x y i n j m  3.3.1 normalization the problem (p1) is bicriteria optimization problem in which on one hand system reliability is maximized and other hand cost of selected components to form / assemble the system is minimized. the reliability which is unit free is measured between zero and one whereas cost has its unit. two objectives can be converted to single objective programming problem either if both objectives are of same unit or if both objectives can be made unit free. to make cost function unit free, the following transformation is used. 1 1 1 iji vmn ijk i j k c c      ,   1 1 imn tot ij ij ij ij i j c c t n     now   , and 1 tot ij ij ij ijijk ijk ijijk ij c t nc c c c c c c c c        the resulting problem then can be rewritten as follows. problem (p2) maximize   f 1 1      l l si il l rfx 100 minimize 2 1 1 1 ( ) iji vmn ij ij ijk ijk i j k f x c y c x               subject to sx  the problem (p2) can further be written as vector optimization problem as. problem (p3) vector max  xf subject to sx  where       txfxfxf 21 , 3.3.2 finding properly efficient solution definition 1 (steuer, [10]): a feasible solution sx  * is said to be an efficient solution for the below problem if there exists no sx  such that    *xfxf  and    *xfxf  definition 2 (steuer, [10]): an efficient solution sx  * is said to be an properly efficient solution for the problem (p2) if there exist 0 such that for each r            xfxfxfxf jjrr ** / for some j with    *xfxf jj  and    *xfxf rr  for sx  . using geoffrion’s scalarization the problem (p2) reduces to problem (p4) maxize z= 1 1 2 2f f  subject to sx  0, 1 2121   lemma(geoffrion,[6]):the optimal solution of the problem (p4) for fixed 21 and  is a properly efficient solution for the problem (p3) as well as (p1). 3.4 optimization model ii optimization model ii is an extension of optimization model i. as explained in the introduction, it is observed that some alternatives of a module may not be compatible with alternatives of another module (jung and choi, [7]). the next optimization model ii addresses this problem. it is done, incorporating additional constraints in the optimization models. this constraint can be represented as chugsq t xx  , which means that if alternative s for module g is chosen, then 101 alternative ztu t ,........1 ,  have to be chosen for module h . we also assume that if two alternatives are compatible, then their versions are also compatible. , tgsq hu c t x x my  ghugs msvvq t ,.....,1 , ,......,2 c , ,.......,2  (12)    2 thut vzy (13) constraint (12) and (13) make use of binary variable t y to choose one pair of alternatives from among different alternative pairs of modules. problem (p3) can be transformed to another optimization problem using compatibility constraints and if more than one alternative compatible component is to be chosen for redundancy, constraint (13) can be relaxed as follows.    2 thut vzy (14) 4. illustrative examples consider a software system having two modules with more than one alternative for each module. the data sets for cots and in-house developed components are given in table-1 and table ii, respectively. let      1 2 3 1 2 33, 1, 2,3 , 1,3 , s 2 , 0.5, 0.3 and 0.2l s s f f f       . it is also assumed that 01. and 05. ,01. 321  ttt functions version module module module figure 1 structure of the software 1f 2f lf 1m 2m nm a l t e r n a t i v e s 102 table 1: data set for cots components table 2data set for in-house conponents 4.1 optimization model – i table 3 presents the solution for optimization model i. the problem is solved using software package lingo (thiriez, [11]). the solution to the model gives the optimal component selection for the software system along with the corresponding cost and reliability of the overall system. the sensitivity analysis to the delivery time constraint has been performed. it is clearly seen from the table alternati ves versions 1 2 3 cost reliability delivery time cost reliability delivery time cost reliability delivery time 1 1 0 0.001 0 14 0.90 3 11 0.88 4 2 0 0.001 0 12.5 0.86 4 18 0.92 2 3 0 0.001 0 17 0.90 2 15 0.88 3 2 1 0 0.001 0 13 0.87 4 17. 5 0.86 2 2 0 0.001 0 11 0.91 5 12 0.89 4 3 0 0.001 0 18 0.89 2 15 0.86 3 4 0 0.001 0 13 0.86 4 14 0.88 3 3 1 0 0.001 0 16 0.85 3 18 0.90 2 2 0 0.001 0 16 0.89 3 17 0.87 2 module i alternatives ijt ij  ij c ij  1 1 8 0.005 5 0.002 2 6 0.005 4 0.002 3 7 0.005 4 0.002 2 1 9 0.005 5 0.002 2 5 0.005 2 0.002 3 6 0.005 4 0.002 4 5 0.005 3 0.002 3 1 6 0.005 4 0.002 2 5 0.005 3 0.002 103 that when the delivery time was 10 units, then only cots components were selected. when the delivery time increases along with the cots components, in house build components were also selected. when the delivery time was 12 units, only one in-house component was developed with the minimum cost 79 units attained at reliability level 0.85.our system cost decreases while the corresponding reliability increases because the components developed in-house decreases the cost initially but later if the level of reliability has to be kept at 0.90 then by increasing delivery time by 5 and 9 units respectively, more in-house build components were selected which in turn increases the cost and reliability of the overall system. redundancy is also there in all the four cases. table 3: solution of optimization model i 4.2 optimization model-ii to illustrate optimization model for compatibility, we use previous results. case 1. delivery time is assumed to be 10 units. we assume third alternative of second module is compatible with second and third alternatives of first module. 111 123 133 1x x x   case no. delivery time cots in-house system reliability overall system cost joint objective value 1 10 111 123 132 211 221 232 242 311 322 1 1 1 x x x x x x x x x          nil 0.84 82 0.66 2 12 111 123 132 211 232 241 311 322 1 1 1 x x x x x x x x         22 1y  0.85 79 0.68 3 17 111 123 132 211 221 232 311 1 1 1 x x x x x x x        24 32 1y y  0.93 86 0.74 4 21 111 132 211 221 232 311 1 1 1 x x x x x x       12 24 32 1y y y   0.94 92 0.75 104 211 221 232 242 1x x x x    311 322 1x x  it is observed that due to the compatibility condition, third alternative of first module is chosen as it is compatible with third alternative of second module. the system reliability for the above solution is 0.84 and cost is 81 units. case 2. delivery time is assumed to be 12 units. we assume second alternative of third module is compatible with second and third alternatives of first module. 22 1y  ; 111 123 133 211 232 241 311 322 1 1 1 x x x x x x x x         it is observed that due to the compatibility condition, third alternative of first module is chosen as it is compatible with second alternative of third module. the system reliability for the above solution is 0.85 and cost is 77 units. case 3. delivery time is assumed to be 10 units. we assume third alternative of second module is compatible with second and third alternatives of first module. 24 32 1y y  111 123 133 211 221 232 311 1 1 1 x x x x x x x        it is observed that due to the compatibility condition, third alternative of first module is chosen as it is compatible with third alternative of second module. the system reliability for the above solution is 0.94 and cost is 84 units. 5. conclusions we have presented optimization models that supports the decision whether to buy 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